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--- abstract: | In this paper we study the actions of tori (standard compact tori, as well as their quaternionic analogues) on products of spheres. It is proved that the orbit space of a specific action of a torus on a product of spheres is homeomorphic to a sphere. A similar statement for a real torus $\mathbb{Z}_2^n$ was proved by the second author in 2019. We also provide a statement about arbitrary compact topological groups, generalizing the mentioned results, as well as the results of the first author about the actions of a compact torus of complexity one. address: - Neapolis University, Pafos, Cyprus - Steklov Mathematical Institute, Moscow, Russia author: - Anton Ayzenberg - Dmitry Gugnin title: On actions of tori and quaternionic tori on products of spheres --- [^1] # Introduction {#secIntro} Let $S^m = \{(x_1,\ldots, x_m,x_{m+1})\in \mathbb{R}^{m+1} \mid x_1^2+\ldots+ x_m^2 + x_{m+1}^2 = 1\}$ be a standard sphere. Let $\tau\colon S^m \to S^m$ denote one of standard involutions, namely $\tau(x_1,\ldots,x_m,x_{m+1}) = (x_1,\ldots,x_m, - x_{m+1})$. Consider the product of $k$ spheres of arbitrary positive dimensions: $$S^{m_1}\times S^{m_2}\times \ldots \times S^{m_k}, \quad k\geqslant 2.$$ Commuting involutions $\tau_1, \ldots, \tau_k$ act on this manifold (the action is component-wise, each of the involutions is a permutation of the north and south poles of the corresponding sphere). Since involutions commute, we have a $C^{\omega}$-action of the group $\mathbb{Z}_2^{k}$ on the product of spheres under consideration. The group $\mathbb{Z}_2^{k}$ contains a subgroup $G_k$ of index 2, consisting of orientation-preserving elements. The following result was obtained by the second author [@Gug19]. **Theorem 1** ([@Gug19]). *The quotient space $S^{m_1}\times S^{m_2}\times \ldots \times S^{m_k}/G_k$ is homeomorphic to the sphere $S^m$, $m = m_1+\cdots + m_k$. In this case, the canonical projection onto the space of orbits is given by the following formula: $$\begin{gathered} (x_{1,1}, \ldots, x_{m_1,1}, x_{m_{1} +1,1}, \ldots, x_{1,k}, \ldots, x_{m_k,k}, x_{m_k +1,k}) \mapsto\\ \mapsto \frac{(x_{1,1}, \ldots, x_{m_1,1}, \ldots, x_{1,k}, \ldots, x_{m_k,k} ; x_{m_{1} +1,1} \cdots x_{m_k +1,k})}{\sqrt{ x_{1,1}^2 + \ldots + x_{m_1,1}^2 + \ldots + x_{1,k}^2 + \ldots + x_{m_k,k}^2 + x_{m_{1} +1,1}^2 \cdots x_{m_k +1,k}^2 }}\end{gathered}$$ Moreover, the resulting branched covering $S^{m_1}\times S^{m_2}\times \ldots \times S^{m_k} \to S^m$ is globally $C^{\omega}$-smooth and nondegenerate at the points of the local homeomorphism.* For the sake of convenience, we will call the group $\mathbb{Z}_2^n=O(1)^n$ a real torus. Similarly, we have a complex[^2] torus $T^n=U(1)^n$, and a quaternionic torus $\mathop{\mathrm{Sp}}(1)^n$. In this paper, we formulate and prove a natural generalization of the above theorem to the actions of complex and quaternionic tori; this is done in Section [2](#secMainResults){reference-type="ref" reference="secMainResults"}. More generally, the result can be formulated for an arbitrary compact topological group; this is the subject of Section [3](#secGeneralJoins){reference-type="ref" reference="secGeneralJoins"}. From the general result we deduce that the quotient of some specific action of the quaternionic torus $\mathop{\mathrm{Sp}}(1)^{k-1}$ on the space $\mathbb{H}^k\cong \mathbb{R}^{4k}$ is homeomorphic to the space $\mathbb{R}^{k+3}$, see Proposition [Proposition 1](#propLocal){reference-type="ref" reference="propLocal"}. This assertion is a quaternionic generalization of the previous result of the first author [@AyzCompl] about toric actions of complexity one in general position. # Main results {#secMainResults} Consider $k\geqslant 2$ many spheres of dimensions at least $2$: $$\begin{gathered} S^{m_1} = \{ (\mathbf{x}_1, z_1)\mid \mathbf{x}_1 = (x_{1,1}, x_{2,1}, \ldots, x_{m_1-1,1}) \in \mathbb{R}^{m_1-1}, z_1\in \mathbb{C}, |\mathbf{x}_1|^2 + |z_1|^2 =1 \},\\ \vdots\\ S^{m_k} = \{ (\mathbf{x}_k, z_k)\mid \mathbf{x}_k = (x_{1,k}, x_{2,k}, \ldots, x_{m_k-1,k}) \in \mathbb{R}^{m_k-1}, z_k\in \mathbb{C}, |\mathbf{x}_k|^2 + |z_k|^2 =1 \}.\end{gathered}$$ Consider the direct product $S^{m_1}\times S^{m_2}\times \ldots \times S^{m_k}$. It carries a (left) smooth action of the complex torus $T^{k-1}$. Namely, the element $(r_1,r_2,\ldots, r_{k-1})\in T^{k-1}$ translates a point $$((\mathbf{x}_1, z_1), (\mathbf{x}_2, z_2), \ldots, (\mathbf{x}_k, z_k)) \in S^{m_1}\times S^{m_2}\times \ldots \times S^{m_k}$$ to the point $$((\mathbf{x}_1, z_1r_1^{-1}), (\mathbf{x}_2, r_1z_2r_2^{-1}), (\mathbf{x}_3, r_2z_3r_3^{-1}), \ldots, (\mathbf{x}_k, r_{k-1}z_k)).$$ **Theorem 1**. *The quotient space $S^{m_1}\times S^{m_2}\times \ldots \times S^{m_k}/T^{k-1}$ is homeomorphic to the sphere $S^m, m = m_1+\ldots + m_k - (k-1)$. The canonical projection to the orbit space is given by the formula: $$\label{eqStar} ((\mathbf{x}_1, z_1), (\mathbf{x}_2, z_2), \ldots, (\mathbf{x}_k, z_k)) \mapsto \frac{(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_k, z_1z_2\ldots z_k)}{\sqrt{ |\mathbf{x}_1|^2 + |\mathbf{x}_2|^2 + \ldots + |\mathbf{x}_k|^2 + |z_1z_2\ldots z_k|^2 }}$$* The proof of this theorem is completely analogous to the proof of its quaternionic version, Theorem [Theorem 1](#thmQuaternionic){reference-type="ref" reference="thmQuaternionic"} below. Let $\mathbb{H}$ denote the algebra of quaternions and $\mathop{\mathrm{Sp}}(1)$ --- the Lie group of unit quaternions (i.e. the quaternions of unit length). Consider $k\geqslant 2$ many spheres of dimensions at least 4: $$\begin{gathered} S^{m_1} = \{ (\mathbf{x}_1, q_1)\mid \mathbf{x}_1 = (x_{1,1}, x_{2,1}, \ldots, x_{m_1-3,1}) \in \mathbb{R}^{m_1-3}, q_1\in \mathbb{H}, |\mathbf{x}_1|^2 + |q_1|^2 =1 \}, \\ \vdots \\ S^{m_k} = \{ (\mathbf{x}_k, q_k)\mid \mathbf{x}_k = (x_{1,k}, x_{2,k}, \ldots, x_{m_k-3,k}) \in \mathbb{R}^{m_k-3}, q_k\in \mathbb{H}, |\mathbf{x}_k|^2 + |q_k|^2 =1 \}.\end{gathered}$$ Consider the direct product $S^{m_1}\times S^{m_2}\times \ldots \times S^{m_k}$. It carries a (left) smooth action of the quaternionic torus of rank $k-1$ $$\mathop{\mathrm{Sp}}(1)^{k-1} = \underbrace{\mathop{\mathrm{Sp}}(1)\times \mathop{\mathrm{Sp}}(1)\times \ldots \times \mathop{\mathrm{Sp}}(1)}_{k-1 \text{ times}}.$$ Namely, the element $(r_1,r_2,\ldots, r_{k-1})\in \mathop{\mathrm{Sp}}(1)^{k-1}$ translates a point $$((\mathbf{x}_1, q_1), (\mathbf{x}_2, q_2), \ldots, (\mathbf{x}_k, q_k)) \in S^{m_1}\times S^{m_2}\times \ldots \times S^{m_k}$$ to the point $$((\mathbf{x}_1, q_1r_1^{-1}), (\mathbf{x}_2, r_1q_2r_2^{-1}), (\mathbf{x}_3, r_2q_3r_3^{-1}), \ldots, (\mathbf{x}_k, r_{k-1}q_k)).$$ **Theorem 1**. *The quotient space $S^{m_1}\times S^{m_2}\times \ldots \times S^{m_k}/\mathop{\mathrm{Sp}}(1)^{k-1}$ is homeomorphic to the sphere $S^m, m = m_1+\ldots + m_k - 3(k-1)$. The canonical projection to the orbit space is given by the formula: $$\label{eqTwoStars} ((\mathbf{x}_1, q_1), (\mathbf{x}_2, q_2), \ldots, (\mathbf{x}_k, q_k)) \mapsto \frac{(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_k, q_1q_2\ldots q_k)}{\sqrt{ |\mathbf{x}_1|^2 + |\mathbf{x}_2|^2 + \ldots + |\mathbf{x}_k|^2 + |q_1q_2\ldots q_k|^2 }}.$$* *Proof.* First, let us prove that the canonical projection onto the space of orbits is realized by a simpler formula: $$\label{eq1} ((\mathbf{x}_1, q_1), (\mathbf{x}_2, q_2), \ldots, (\mathbf{x}_k, q_k)) \mapsto (\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_k, q_1q_2\ldots q_k).$$ From the definition of the action of the quaternionic torus on the product of spheres, it is clear that any orbit gets mapped to a single point under the map [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"}. Let us prove that two different orbits cannot map to the same point under [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"}. Indeed, assume $$\label{eq2} (\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_k, q_1q_2\ldots q_k) = (\mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_k, p_1p_2\ldots p_k).$$ It follows that $\mathbf{x}_i = \mathbf{y}_i, 1\leqslant i\leqslant k$. Therefore, $|q_i|=|p_i|$, $1\leqslant i\leqslant k$. It is easily seen that, remaining within a single orbit, we can assume that $0\leqslant q_i=p_i\leqslant 1$, $1\leqslant i\leqslant k-1$. At first, let us assume that $q_1q_2\ldots q_k = p_1p_2\ldots p_k \neq 0$. Then, obviously, $q_k=p_k$, which was to be proved. Now consider the case $q_1q_2\ldots q_k = p_1p_2\ldots p_k = 0$. If $q_k = p_k = 0$, then everything is proved. Let $|q_k| = |p_k| > 0$. Denote the unit quaternion $p_kq_k^{-1}$ by $a$. Let $q_{k-1}=p_{k-1}=0$. Then one can translate the tuple $(q_1,q_2, \ldots, 0, q_k)$ to the tuple $(p_1,p_2,\ldots, 0, p_k)$ using the element $(1,1,\ldots, 1, r_{k-1}= a)\in \mathop{\mathrm{Sp}}(1)^{k-1}$. If $q_{k-1} = p_{k-1} > 0$, $q_{k-2} = p_{k-2} = 0$, then $(1,1,\ldots,1,a,a)$ is the required element of the group $\mathop{\mathrm{Sp}}(1)^{k-1}$. If $q_{k-1} = p_{k-1} > 0$, $q_{k-2} = p_{k-2} > 0$, $q_{k-3} = p_{k-3} = 0$, then the required element is $(1,1,\ldots,1,a,a,a)$. Similar arguments work for other cases. In the most extreme case, we have $q_{k-1} = p_{k-1} > 0$, $q_{k-2} = p_{k-2} > 0$, $\ldots$, $q_2 = p_2 > 0$, $q_1 = p_1 = 0$. In this case, the required element is $(a,a,\ldots,a)$. Thus, we have proved that formula [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} determines a well-defined canonical projection onto the orbit space. It can be seen that the vector on the right hand side of formula [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} is nonzero. Since the formula [\[eqTwoStars\]](#eqTwoStars){reference-type="eqref" reference="eqTwoStars"} on the left hand side involves a compact Hausdorff space (the quotient of a Hausdorff space by a continuous action of a compact Lie group is Hausdorff [@Bredon-ru]), and the sphere appears on the right hand side, it suffices to check *bijectivity* of the map [\[eqTwoStars\]](#eqTwoStars){reference-type="eqref" reference="eqTwoStars"}. **Injectivity**. We need to check that the equality $$\label{eq3} (\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_k, q_1q_2\ldots q_k) = \mu (\mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_k, p_1p_2\ldots p_k),\quad \mu>1$$ never occurs for distinct points. We have $q_1q_2\ldots q_k = \mu p_1p_2\ldots p_k$. Two cases are possible: (A) $q_1q_2\ldots q_k = p_1p_2\ldots p_k = 0$, and (B) $q_i\neq 0, p_i\neq 0, 1\leqslant i \leqslant k$. *Case (A).* We have $p_{i_0} = 0$ for some $1\leqslant i_0\leqslant k$. Then $|\mathbf{y}_{i_0}| = 1$ and $|\mathbf{x}_{i_0}| = \mu |\mathbf{y}_{i_0}| = \mu > 1$, which is impossible. *Case (B).* We have $|\mathbf{x}_i| = \mu |\mathbf{y}_i| \geqslant |\mathbf{y}_i|$, $1\leqslant i\leqslant k$. This implies that $0 < |q_i|\leqslant |p_i|$, $1\leqslant i\leqslant k$. Hence $0 < |q_1q_2\ldots q_k| \leqslant |p_1p_2\ldots p_k|$. On the other hand, we have $|q_1q_2\ldots q_k| = \mu |p_1p_2\ldots p_k| > |p_1p_2\ldots p_k|$, --- a contradiction. **Surjectivity**. Since the image of a compact space under a continuous mapping is always a compact space, then either the surjectivity is proved or the image of the map [\[eqTwoStars\]](#eqTwoStars){reference-type="eqref" reference="eqTwoStars"} is a proper subcompact in the sphere $S^m$. Again, we argue from the contrary. Assume there exists a vector $(\mathbf{t}_1, \mathbf{t}_2, \ldots, \mathbf{t}_k, t)$ such that $\mathbf{t}_i \neq 0$, $1\leqslant i\leqslant k$, $t\neq 0$, and for any $\mu > 0$ the vector $\mu(\mathbf{t}_1, \mathbf{t}_2, \ldots, \mathbf{t}_k, t)$ does not have the form $(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_k, q_1q_2\ldots q_k)$. Denote $\min_{1\leqslant i\leqslant k}\{1 /|\mathbf{t}_i|\}$ by $\mu_0$. As the parameter $\mu$ runs over the interval $(0,\mu_0)$, the lengths of the vectors $\mu \mathbf{t}_i = \mathbf{x}_i$, $1\leqslant i\leqslant k$ increase strictly and continuously and run over the intervals $(0, \mu_0 |\mathbf{t}_i|) \subset (0,1)$, $1\leqslant i\leqslant k$. Moreover, there exists $1\leqslant i_0\leqslant k$ such that $(0, \mu_0| \mathbf{t}_{i_0} |) = (0,1)$. It follows from the length expressions $|q_i| = \sqrt{1- |\mathbf{x}_i|^2}$, that the length of $|q_1(\mu)q_2(\mu)\ldots q_k(\mu)|$ decreases strictly and continuously from $1$ to $0$ (not taking extreme values). In this case, it is possible to achieve collinearity of the nonzero quaternions $t$ and $q_1(\mu)q_2(\mu)\ldots q_k(\mu)$, $0< \mu < \mu_0$. Since the length of $|\mu t|$ increases strictly and continuously from $0$ to $|\mu_0||t|>0$, the Cauchy intermediate value theorem asserts that there exists a parameter $\mu_1\in (0, \mu_0)$, for which the equality $\mu_1(\mathbf{t}_1, \mathbf{t}_2, \ldots, \mathbf{t}_k, t) = (\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_k, q_1(\mu_1)q_2(\mu_1)\ldots q_k(\mu_1))$ holds. The theorem is completely proven. ◻ In the work [@Gug19] of the second author, the proof of Theorem [Theorem 1](#thmReal){reference-type="ref" reference="thmReal"} was omitted due to its simplicity. However, we still give here the proof of the most nontrivial part, namely, the non-degeneracy (local diffeomorphism) of the corresponding branched covering at the points of the local homeomorphism (away from the branching locus). *Proof.* It is easily shown (similar to the above reasoning) that the canonical projection onto the orbit space is given by a simpler formula: $$\begin{gathered} (x_{1,1}, \ldots, x_{m_1,1}, q_1, x_{1,2},\ldots, x_{m_2,2}, q_2, \ldots, x_{1,k},\ldots, x_{m_k,k}, q_k) \mapsto \\ \mapsto (x_{1,1}, \ldots, x_{m_1,1}, x_{1,2},\ldots, x_{m_2,2}, \ldots, x_{1,k},\ldots, x_{m_k,k}, q_1q_2\cdots q_k).\end{gathered}$$ Here, we denote $x_{m_i+1,i} = q_i\in \mathbb{R}$, $1\leqslant i\leqslant k$ for the sake of simplicity. Moreover, for the initial map onto the unit sphere, the points of the local homeomorphism are either $(A)$ points with $q_1q_2\cdots q_k \neq 0$, or $(B)$ there is a unique $1\leqslant j \leqslant k$ with $q_j=0$. It is understood that the points of a local diffeomorphism for the original map of smooth $m$-dimensional manifolds $S^{m_1}\times \ldots \times S^{m_k} \to S^m$ correspond (in both directions) to the points of a local diffeomorphism for the following map of smooth $(m+1)$-dimensional manifolds $F\colon S^{m_1}\times \ldots \times S^{m_k}\times (0,+\infty) \to \mathbb{R}^m\setminus\{0\}$: $$\begin{gathered} F(x_{1,1}, \ldots, x_{m_1,1}, q_1, x_{1,2},\ldots, x_{m_2,2}, q_2, \ldots, x_{1,k},\ldots, x_{m_k,k}, q_k;\mu) =\\ \mu(x_{1,1}, \ldots, x_{m_1,1}, x_{1,2},\ldots, x_{m_2,2}, \ldots, x_{1,k},\ldots, x_{m_k,k}, q_1q_2\cdots q_k),\end{gathered}$$ where the parameter $\mu$ can be taken arbitrarily. Let us verify that in both cases $(A)$ and $(B)$ the Jacobian of the map $F$ is nonzero. **Case $(A)$.** In this case, the string $(x_{1,1}, \ldots, x_{m_1,1}, x_{1,2},\ldots, x_{m_2,2}, \ldots, x_{1,k},\ldots, x_{m_k,k};\mu)$ can be taken as the local coordinates in the preimage. We need to calculate the determinant of order $m+1$. The last column of the desired determinant ($\partial F(\ldots)/\partial\mu$) is equal to (the transposed string) $$(x_{1,1}, \ldots, x_{m_1,1}, \ldots, x_{1,k},\ldots, x_{m_k,k}, q_1q_2\cdots q_k)^\intercal.$$ Further, the column $\partial F(\ldots)/\partial x_{1,1}$ divided by $\mu$ equals $$(1,0,\ldots,0, (\partial q_1/\partial x_{1,1}) q_2q_3\cdots q_k)^\intercal = \left(1,0,\ldots,0, -\frac{x_{1,1}}{q_1} q_2q_3\cdots q_k \right)^\intercal.$$ Similarly, the column $\partial F(\ldots)/\partial x_{2,1}$ divided by $\mu$ equals $$(0,1,0, \ldots,0, (\partial q_1/\partial x_{2,1}) q_2q_3\cdots q_k)^\intercal = \left(0,1,0,\ldots,0, -\frac{x_{2,1}}{q_1} q_2q_3\cdots q_k \right)^\intercal.$$ Making further calculations, we get that the penultimate column $\partial F(\ldots)/\partial x_{m_k,k}$ divided by $\mu$ equals $$(0,0, \ldots,0,1, (\partial q_k/\partial x_{m_k,k}) q_1q_2\cdots q_{k-1})^\intercal = \left(0,0,\ldots,0,1,-\frac{x_{m_k,k}}{q_k} q_1q_2\cdots q_{k-1}\right)^\intercal.$$ Subtracting from the last column $(x_{1,1}, \ldots, x_{m_1,1}, \ldots, x_{1,k},\ldots, x_{m_k,k}, q_1q_2\cdots q_k)$ the first column multiplied by $x_{1,1}$, the second column multiplied by $x_{2,1}$, etc, we obtain, as a result, the lower triangular matrix with the diagonal $$\left(1,1,\ldots, 1, q + \frac{x_{1,1}^2 + x_{2,1}^2 + \ldots + x_{m_1,1}^2}{q_1^2}q + \ldots + \frac{x_{1,k}^2 + x_{2,k}^2 + \ldots + x_{m_k,k}^2}{q_k^2}q\right),$$ where $q=q_1q_2\cdots q_k$. Since $q\neq 0$, the determinant of this matrix is nonzero. The required local diffeomorphism in the case $(A)$ is proved. **Case $(B)$.** Due to certain symmetry of the function $F$ in its arguments, we can assume without loss of generality that $q_1=0$, $q_2q_3\cdots q_k \neq 0$, $x_{1,1} \neq 0$. In this situation, one can choose the string $$q_1, x_{2,1}, x_{3,1}, \ldots, x_{m_1,1}, x_{1,2}, \ldots, x_{m_2,2}, \ldots, x_{1,k}, \ldots, x_{m_k,k}, \mu$$ as local coordinates in the preimage. Recall the definition of $F$: $$F(\ldots) = \mu(x_{i,j}; q).$$ Let us write down the calculations of all first partial derivatives of the function $F$: $$\begin{gathered} \frac{1}{\mu}\frac{\partial F}{\partial q_1} = (0, 0, \ldots, 0; q_2q_3\cdots q_k), \\ \frac{\partial F}{\partial\mu} = (x_{i,j};0), \\ \frac{1}{\mu}\frac{\partial F}{\partial{x_{2,1}}} = \left(-\frac{x_{2,1}}{x_{1,1}}, 1, 0,\ldots, 0; 0\right), \ \frac{1}{\mu}\frac{\partial F}{\partial{x_{3,1}}} = \left(-\frac{x_{3,1}}{x_{1,1}}, 0, 1,0, \ldots, 0; 0\right), \ldots , \\ \frac{1}{\mu}\frac{\partial F}{\partial{x_{m_1,1}}} = \left(-\frac{x_{m_1,1}}{x_{1,1}}, 0, 0,\ldots, 0, 1, 0, \ldots, 0; 0\right), \\ \frac{1}{\mu}\frac{\partial F}{\partial x_{1,2}} = (0, \ldots, 0,1,0, \ldots, 0; 0), \ \ldots \ , \frac{1}{\mu}\frac{\partial F}{\partial x_{m_k,k}} = (0, \ldots,0, 1;0).\end{gathered}$$ By carefully computing the Jacobian of the map $F$ at the given point, we can see that it is nonzero if and only if the following determinant of order $m_1$ is nonzero: $$\begin{vmatrix} x_{1,1}& -x_{2,1} & -x_{3,1} & \ldots & -x_{m_1-1,1} & -x_{m_1,1}\\ x_{2,1}& x_{1,1} & 0 & \ldots & 0 & 0\\ x_{3,1}& 0 & x_{1,1} & \ldots & 0 & 0 \\ \vdots& \vdots & \vdots &\ddots & \vdots & \vdots\\ x_{m_1-1,1} & 0 & 0 & \ldots & x_{1,1} & 0\\ x_{m_1,1}& 0 & 0 &\ldots & 0 & x_{1,1} \end{vmatrix}$$ Denote by $A$ the matrix under this determinant. If this determinant is zero, then the skew-Hermitian matrix $A-x_{1,1}E$ has a nonzero real eigenvalue $\lambda = -x_{1,1}$. However, it is well known, that all eigenvalues of a skew-Hermitian matrix are purely imaginary complex numbers, and it was assumed earlier that $x_1\neq 0$. This contradiction shows that the Jacobian under consideration is nonzero. The desired local diffeomorphism in the case of $(B)$ is completely proved. ◻ *Remark 1*. Similar to the proof above one can show that the maps [\[eqStar\]](#eqStar){reference-type="eqref" reference="eqStar"} and [\[eqTwoStars\]](#eqTwoStars){reference-type="eqref" reference="eqTwoStars"} to the orbit space from Theorem [Theorem 1](#thmComplex){reference-type="ref" reference="thmComplex"} (complex tori) and Theorem [Theorem 1](#thmQuaternionic){reference-type="ref" reference="thmQuaternionic"} (quaternionic tori) are smooth and they are submersions outside the degeneration locus. This means that in the open set of free orbits, the differentials of these maps have maximal possible rank equal to the dimension of the orbit space. In the work [@Gug23b] of the second author, it was shown that the number $k-1$ of commuting involutions on the product of spheres $S^{m_1}\times \ldots \times S^{m_k}$ is the minimal possible if one wants to obtain a rational homological sphere as the orbit space. We pose the following problem related to this fact. **Problem 1**. *Is it true that, for an arbitrary smooth action of the complex torus $T^{k-2}$ on the product $S^{m_1}\times \ldots \times S^{m_k}$ of spheres of dimensions $\geqslant2$, the corresponding orbit space is not a (rational homological) sphere? Is it true that for an arbitrary smooth action of the quaternionic torus $\mathop{\mathrm{Sp}}(1)^{k-2}$ on the product $S^{m_1}\times \cdots \times S^{m_k}$ of spheres of dimensions $\geqslant 4$ the corresponding orbit space is not a (rational homology) sphere?* This conjecture seems hard. It is nontrivial even in the simplest case $k=3$. # General groups {#secGeneralJoins} Consider nonempty compact Hausdorff spaces $X_1,X_2,\ldots, X_k$, $k\geqslant 2$, and an arbitrary compact Hausdorff group $G$. Consider the following (left) continuous action of the group $G^{k-1}$ on the product of joins $\prod_{i=1}^{k}(X_i\ast G)$. Namely, the element $(r_1,r_2,\ldots, r_{k-1})\in G^{k-1}$ translates a point $$(x_1,1-t_1, q_1,t_1; x_2,1-t_2, q_2,t_2; \ldots; x_k,1-t_k, q_k,t_k)$$ to the point $$\label{eqCodiagonalAcn} (x_1,1-t_1, q_1r_1^{-1},t_1; x_2,1-t_2, r_1q_2r_2^{-1},t_2; \ldots; x_k,1-t_k, r_{k-1}q_k,t_k).$$ **Theorem 1**. *The orbit space $\prod_{i=1}^{k}(X_i\ast G)/G^{k-1}$ is homeomorphic to the join $X_1\ast X_2\ast \ldots\ast X_k\ast G$. The canonical projection onto the space of orbits is given by the formula: $$\begin{gathered} \label{eqJoinMap} (x_1,1-t_1, q_1,t_1; x_2,1-t_2, q_2,t_2; \ldots; x_k,1-t_k, q_k,t_k) \mapsto \\ \mapsto (x_1,1-t_1;x_2,1-t_2;\ldots; x_k,1-t_k; q_1q_2\cdots q_k, t_1t_2\cdots t_k)/A,\end{gathered}$$ where $A=t_1\cdots t_k+\sum\nolimits_{i=1}^{k}(1-t_i)$ is the normalizing factor.* *Proof.* Recall that the join of spaces $Y_1,\ldots,Y_m$ is the identification space $$Y_1\times\cdots\times Y_s\times \Delta^{m-1}/\!\!\sim,$$ where $\Delta^{m-1}$ is the standard simplex with barycentric coordinates $(s_1,\ldots,s_m)$, $s_i\geqslant 0$, $\sum s_i=1$, and the equivalence relation $\sim$ is generated by the conditions $$(y_1,\ldots,y_l,\ldots,y_m,(s_1,\ldots,s_m))\sim (y_1,\ldots,y_l',\ldots,y_m,(s_1,\ldots,s_m)), \text{ if }s_l=0$$ for any $l\in[m]$. Notice that $t_i\in[0;1]$ in [\[eqJoinMap\]](#eqJoinMap){reference-type="eqref" reference="eqJoinMap"}, therefore $A>0$. Therefore, all coefficients $\frac{1-t_i}{A}$, $i\in[k]$ and $\frac{t_1\cdots t_k}{A}$ are nonnegative and sum to $1$ due to the choice of the normalizing factor $A$. Hence formula [\[eqJoinMap\]](#eqJoinMap){reference-type="eqref" reference="eqJoinMap"} provides a well-defined continuous map of the form $$h\colon\prod\nolimits_{i=1}^{k}(X_i\times \Delta^1\times G)\to X_1\times\cdots\times X_k\times G\times \Delta^{k-1}.$$ It is easily seen that the map $h$ is surjective. Let us check that $h$ descends to a well-defined map of the quotient spaces, the joins from the statement of the theorem. If $t_i=1$ for some $i\in[k]$, then the corresponding factor $X_i$ collapses to one point both in the space $X_i\ast G$ and in the space $X_i\ast\cdots \ast X_k\ast G$, so equivalent points of this kind are mapped to equivalent points. If $t_i=0$, then $t_1\cdots t_k=0$, and, similarly, equivalent points are mapped to equivalent ones. Therefore, the formula [\[eqJoinMap\]](#eqJoinMap){reference-type="eqref" reference="eqJoinMap"} descends to a well-defined continuous map $$\tilde{h}\colon (X_1\ast G)\times (X_2\ast G)\times \ldots \times (X_k\ast G)/G^{k-1} \to X_1\ast X_2\ast \ldots\ast X_k\ast G.$$ Since all the spaces appearing in this formula are Hausdorff compact, it suffices to show that the map $\tilde{h}$ is bijective. The surjectivity of $\tilde{h}$ follows from the surjectivity of $h$. Let us prove that $\tilde{h}$ is injective. Generally, the proof is similar to the proof of Theorem [Theorem 1](#thmQuaternionic){reference-type="ref" reference="thmQuaternionic"}. Assume that $\tilde{h}(x_1,t_1,q_1,\ldots,x_k,t_k,q_k) = \tilde{h}(x_1',t_1', q_1',\ldots, x_k',t_k',q_k')$. Looking at the real parameters $t_1,\ldots,t_k$ we see that formula [\[eqJoinMap\]](#eqJoinMap){reference-type="eqref" reference="eqJoinMap"} defines a homeomorphism of the $k$-dimensional cube onto the $k$-dimensional simplex. Therefore the equalities $t_i=t_i'$ hold for all $i\in[k]$. If $0<t_i<1$ for all $i$, then the assertion of injectivity reduces to the homeomorphism $G^k/G^{k-1}\cong G$ given by the formula $[(q_1,\ldots,q_k)]\mapsto q_1\cdots q_k$. If $t_i=1$ for some $i$, then the fiber $X_i$ collapses on both sides of the map. If $t_1\cdots t_k=0$, then $t_i=0$ for some $i\in[k]$. This means that the $i$-th component $G_i$ of the product $G^k$ collapses in the target of $\tilde{h}$. Taking the quotient of $G^k$ simultaneously by $G_i$ and the action of the subgroup $G^{k-1}$ described by the formula [\[eqCodiagonalAcn\]](#eqCodiagonalAcn){reference-type="eqref" reference="eqCodiagonalAcn"}, we see that the entire group $G^k$ collapses into a point. Therefore, the identifications on the face of the simplex $\{t_1\cdots t_k=0\}$ are the same in the space $X_1\ast X_2\ast \ldots\ast X_k\ast G$ and in the space $(X_1\ast G)\times (X_2\ast G)\times \ldots \times (X_k\ast G)/G^{k-1}$. Injectivity is proved. ◻ Note that the extreme trivial cases of Theorem [Theorem 1](#thmJoins){reference-type="ref" reference="thmJoins"} appear to be informative. *Example 1*. Let us apply Theorem [Theorem 1](#thmJoins){reference-type="ref" reference="thmJoins"} to the trivial group $G=\{1\}$. We get the standard topological fact: $$\mathop{\mathrm{Cone}}X_1\times \mathop{\mathrm{Cone}}X_2\times \cdots \times \mathop{\mathrm{Cone}}X_k\cong \mathop{\mathrm{Cone}}(X_1\ast X_2\ast \ldots\ast X_k).$$ *Example 1*. Let us apply Theorem [Theorem 1](#thmJoins){reference-type="ref" reference="thmJoins"} to the one-point topological spaces $X_i=\ast$, $i\in[k]$. We get a homeomorphism: $$(\mathop{\mathrm{Cone}}G)^{k}/G^{k-1}\cong\underbrace{\mathop{\mathrm{Cone}}\cdots\mathop{\mathrm{Cone}}}_k G\cong \Delta^{k-1}\ast G.$$ *Remark 1*. The topological part of Theorems [Theorem 1](#thmReal){reference-type="ref" reference="thmReal"}, [Theorem 1](#thmComplex){reference-type="ref" reference="thmComplex"} and [Theorem 1](#thmQuaternionic){reference-type="ref" reference="thmQuaternionic"} is a special case of Theorem [Theorem 1](#thmJoins){reference-type="ref" reference="thmJoins"} if we let $G$ be a torus: real $\mathbb{Z}_2=O(1)$, complex $T^1\cong U(1)$, or quaternionic $\mathop{\mathrm{Sp}}(1)$, respectively; and $X_i$ --- spheres of arbitrary dimensions. Note, however, that the above theorems contain stronger assertions. Since both the spaces $X_i$ and the group $G$ are spheres, their joins are also spheres, and therefore have a natural smooth structure. The question of whether the natural projection onto the orbit space is smooth seems important. For this reason we spent some time proving smoothness in Section [2](#secMainResults){reference-type="ref" reference="secMainResults"}. Example [Example 1](#exTrivialSpace){reference-type="ref" reference="exTrivialSpace"} has a useful treatment in the real, complex, and quaternionic cases. Let $\mathbb{K}$ denote $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, the number $d(\mathbb{K})$ be equal to $1$, $2$, or $4$, respectively, and $S(\mathbb{K})$ denote the compact Lie group of numbers having norm $1$ in the corresponding division algebra. Thus $S(\mathbb{K})$ is $\mathbb{Z}_2$, $U(1)$, or $\mathop{\mathrm{Sp}}(1)$. Topologically, $S(\mathbb{K})$ is a sphere of dimension $d(\mathbb{K})-1$. The group $S(\mathbb{K})^{k-1}$ acts linearly on $\mathbb{K}^k\cong \mathbb{R}^{d(\mathbb{K})k}$ by the formula $$\label{eqCodiagOnEuclid} (g_1,\ldots,g_{k-1})(q_1,\ldots,q_k)=(q_1g_1^{-1},g_1q_2g_2^{-1},\ldots,g_{k-2}q_{k-1}g_{k-1}^{-1},g_{k}q_{k}).$$ **Proposition 1**. *The orbit space $\mathbb{K}^k/S(\mathbb{K})^{k-1}$ is homeomorphic to the space $\mathbb{R}^{d(\mathbb{K})+k-1}$.* *Proof.* **First version of the proof.** The space $\mathbb{K}$ is equivariantly diffeomorphic to the open unit ball in $\mathbb{K}$ that is the interior of the cone $\mathop{\mathrm{Cone}}S(\mathbb{K})$. Apply the homeomorphism from Example [Example 1](#exTrivialSpace){reference-type="ref" reference="exTrivialSpace"} and pass to the interiors of the spaces. **Second version of the proof.** Consider the coordinate-wise action of $S(\mathbb{K})^k$ on $\mathbb{K}^k$. We have $\mathbb{K}^k/S(\mathbb{K})^k\cong \mathbb{R}_{\geqslant 0}^k$. The projection to the quotient space has a natural section, so the space $\mathbb{K}^k$ can be represented as the identification space $$\label{eqQuotConstr} \mathbb{K}^k\cong \mathbb{R}_{\geqslant 0}^k\times S(\mathbb{K})^k/\!\!\sim,$$ similarly to how quasitoric manifolds are defined in the toric topology [@BPnew]. Moding out the second factor in the construction [\[eqQuotConstr\]](#eqQuotConstr){reference-type="eqref" reference="eqQuotConstr"} by the torus action [\[eqCodiagOnEuclid\]](#eqCodiagOnEuclid){reference-type="eqref" reference="eqCodiagOnEuclid"}, we obtain $$\label{eqQuotReduced} \mathbb{K}^k/S(\mathbb{K})^{k-1}\cong \mathbb{R}_{\geqslant 0}^k\times S(\mathbb{K})/\!\!\sim.$$ Note that $\mathbb{R}_{\geqslant 0}^k$ is homeomorphic to the half-space $\mathbb{R}^{k-1}\times \mathbb{R}_{\geqslant 0}$. A careful analysis of the stabilizers shows that the relation $\sim$ in the formula [\[eqQuotReduced\]](#eqQuotReduced){reference-type="eqref" reference="eqQuotReduced"} collapses the component $S(\mathbb{K})$ into a point if and only if the corresponding point from $\mathbb{R}^{k-1}\times \mathbb{R}_{\geqslant 0}$ belongs to the boundary $\mathbb{R}^{k-1}\times\{0\}$, see [@AyzCompl] for details. Hence we get $$\mathbb{K}^k/S(\mathbb{K})^{k-1}\cong \mathbb{R}^{k-1}\times (\mathbb{R}_{\geqslant 0}\times S(\mathbb{K})/\!\!\sim)\cong \mathbb{R}^{k-1}\times\mathbb{K}$$ which completes the proof. ◻ *Remark 1*. The second version of the proof in the general case is completely analogous to the complex case considered in [@AyzCompl Lem.2.11], see also [@Styrt Thm.3.6]. The real case was studied in detail by Mikhailova [@Mikh Thm.2.2]. Proposition [Proposition 1](#propLocal){reference-type="ref" reference="propLocal"} can be understood as a local result. The global consequence follows. **Corollary 1**. *Consider a smooth action of a torus $G$ (real, complex, or quaternionic) on a closed smooth manifold $X$. Assume that the linearized action on the normal space to each orbit is equivalent to a representation of the form [\[eqCodiagOnEuclid\]](#eqCodiagOnEuclid){reference-type="eqref" reference="eqCodiagOnEuclid"} multiplied by a trivial representation. Then the orbit space $X/G$ is a topological manifold.* In the case of a complex torus, many examples of such actions with isolated fixed points were studied in the works of the first author [@AyzCompl; @AyzHP; @AyzMasEquiv]. Actions of real tori whose orbit spaces are manifolds were studied by Gorchakov [@Gorch]. There is also a series of results worth mentioning in this context: $$\label{eqAtiyahArnold} \mathbb{C}P^2/\mathop{\mathrm{conj}}\cong S^4,\qquad \mathbb{H}P^2/U(1)\cong S^7,\qquad \mathbb{O}P^2/\mathop{\mathrm{Sp}}(1)\cong S^{13}.$$ Here the first homeomorphism is the classical Kuiper--Massey theorem (Arnold [@Arn] attributes this result to Pontryagin), the second homeomorphism $\mathbb{H}P^2/U(1)\cong S^7$ is the result of Arnold himself [@Arn Ex.4], and the third one is due to Atiyah--Berndt [@AtBer]. In these examples, the set of fixed points is not discrete, but the linearization of the group action on a normal space to the fixed points' submanifold is equivalent to the linear representations of $\mathbb{Z}_2$ on $\mathbb{R}^2$, $U(1)$ on $\mathbb{C}^2=\mathbb{R}^4$, and $\mathop{\mathrm{Sp}}(1)$ on $\mathbb{H}^2 =\mathbb{R}^8$ respectively. Corollary [Corollary 1](#corOrbitGlobal){reference-type="ref" reference="corOrbitGlobal"} explains why the orbit spaces in all cases [\[eqAtiyahArnold\]](#eqAtiyahArnold){reference-type="eqref" reference="eqAtiyahArnold"} are manifolds; although, by no means, it explains why the orbit spaces are homeomorphic to spheres. In view of Theorems [Theorem 1](#thmReal){reference-type="ref" reference="thmReal"}, [Theorem 1](#thmComplex){reference-type="ref" reference="thmComplex"}, [Theorem 1](#thmQuaternionic){reference-type="ref" reference="thmQuaternionic"} and homeomorphisms [\[eqAtiyahArnold\]](#eqAtiyahArnold){reference-type="ref" reference="eqAtiyahArnold"}, arises a natural question. **Problem 1**. *Describe a class of actions of the groups $\mathbb{Z}_2^k$, $T^k$, and $\mathop{\mathrm{Sp}}(1)^k$ on smooth manifolds with orbit spaces homeomorphic to spheres which is general enough to include both the products of spheres and the manifolds $\mathbb{C}P^2$, $\mathbb{H}P^2$, and $\mathbb{O}P^2$.* 99 V. I. Arnold, Relatives of the quotient of the complex projective plane by complex conjugation, Proc. Steklov Inst. Math., 224, 46--56 (1999). M. Atiyah, J. Berndt, *Projective planes, Severi varieties and spheres*, Surveys in Differential Geometry VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (2003), 1--27. A. Ayzenberg, *Torus actions of complexity 1 and their local properties*, Proc. Steklov Inst. Math., 302 (2018), 16--32. A. Ayzenberg, *Torus action on quaternionic projective plane and related spaces*, Arnold Math. J. 7 (2021), 243--266. A. Ayzenberg, M. Masuda, *Orbit spaces of equivariantly formal torus actions of complexity one*, to appear in Transformation Groups, preprint: [arXiv:1912.11696](https://arxiv.org/abs/1912.11696). G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics V.46 (1972). V. Buchstaber, T. Panov, Toric Topology, Math. Surveys Monogr., 204, AMS, Providence, RI, 2015. V. Gorchakov, *Equivariantly formal 2-torus actions of complexity one*, preprint [arXiv:2304.00936](https://arxiv.org/abs/2304.00936). D. V. Gugnin, *Branched Coverings of Manifolds and nH-Spaces*, Funktsional. Anal. i Prilozhen., 53:2 (2019), 68--71. D. V. Gugnin, *On Nonfree Actions of Commuting Involutions on Manifolds*, Math. Notes, 113:6 (2023), 770--775. M. A. Mikhailova, *On the quotient space modulo the action of a finite group generated by pseudoreflections*, Mathematics of the USSR-Izvestiya 24(1):99 (1985). O. G. Styrt, *On the orbit space of a compact linear Lie group with commutative connected component*, Transactions of the Moscow Mathematical Society, 70 (2009), 171--206. [^1]: This work was supported by the Russian Science Foundation under grant no. 23-11-00143 <https://rscf.ru/en/project/23-11-00143/> [^2]: The adjective "complex" in this context refers to the fact that $T^1=U(1)$ is the set of complex numbers of length $1$. We do not assume any complex structures on complex tori. In this naming convention we are coherent with Arnold's mathematical ideology, where the "real--complex--quaternionic" trinity plays a crucial role.

arxiv_math

--- abstract: | We proved a Bernstein theorem for ancient solution to symplectic mean curvature flow via the complex phase map . author: - "Xiangzhi Cao[^1]  [^2]" bibliography: - "E:/myonlybib/myonlymathscinetbibfrom2023.bib" - "E:/myonlybib/low-quality-bib-to-publish.bib" - "C:/Users/Administrator/Desktop/old/mybib2023.bib" title: A Bernstein theorem for ancient solution to symplectic mean curvature flow --- : Bernstein problem, ancient solution, symplectic mean curvature flow. : 53C24, 53E10 # Introduction Let $X_0: M^n \rightarrow N^{n+m}$ be an isometric immersion from an $n$-dimensional oriented Riemannian submanifold $M^n$ to The Riemannian manifold $N^{n+m}, n \geq 2, m \geq 1$. The mean curvature flow (MCF) is a one-parameter family of smooth immersions $X$ : $M^n \times[-T, 0] \rightarrow \mathbb{R}^{n+m}, T>0$, which satisfies the following evolution equation: $$\left\{\begin{array}{l} \frac{\partial}{\partial t} X(x, t)=H(x, t), \quad x \in M^n, t \in[-T, 0], \\ X(\cdot, 0)=X_0, \end{array}\right.$$ where $H(p, t)$ is the mean curvature vector of $X\left(M^n, t\right) \subset \mathbb{R}^{n+m}$. It is well known that self shrinkers is type I singularity and translating soliton is type II singularity of the mean curvature flow . One can refer to these classical papers([@MR0772132][@MR0799217; @MR0837523; @MR0840280; @MR0892052; @MR0921165] ) for codiemsion one mean curvature flow. One can also refer to Huisken's four lectures [@MR1482034; @MR1482035; @MR1482036; @MR1482037]. One can refer to [@MR1655508; @MR2497990; @MR2483374; @MR2941847; @MR3289845; @MR3078951] for higher codimension mean curvatrue flow. It is intersting to study special mean curvature flow, such as Lagrangian mean curvature flow, symplectic mean curvature flow, hyper-Lagrangian mean curvature flow in Hyper-Kähler manifold. Qiu [@MR4478479] proved a rigidity result via complex phase map. We state it here. **Theorem 1** (cf. Qiu [@MR4478479], Theorem 2). *Let $X: \Sigma^2 \rightarrow \mathbb{R}^4$ be a 2-dimensional complete translating soliton with nonpositive normal curvature. Assume that the image of the complex phase map is contained in a regular ball in $\mathbb{S}^2$, i.e., a geodesic ball $B_R(q)$ disjoint from the cut locus of $q$ and $R<\frac{\pi}{2}$, then $\Sigma$ has to be an affine plane.* As an corollary, in the case of omplete Lagrangian translating soliton with non- positive normal curvature, Corollary 1 in [@MR4478479] is equivalent to [@MR2670053 Theorem 2] . Han and Sun [@MR2670053] proved the nonexistence of translation soliton with nonnegative sectional curvature to the almost calibrated Lagrangian mean curvature flow with the lower bound of the function $\theta$. This papragraph is copied from [@MR4504624 theroem 1.6]. Without loss of generality, we assume that the origin $o \in \mathbb{R}^{n+m}$ lies in $\Sigma^n$. Let $\bar{B}_R^{n+m}$ be an Euclidean closed ball of radius $R$ with the center at $o$ and $B_{R, T}(o)=$ $\bar{B}_R^{n+m} \times[-T, 0] \subset \mathbb{R}^{n+m} \times(-\infty,+\infty)$ be a cylindrical domain in the space-time. Consider $\Sigma_T$ as the space-time domain $$\{(X(p, t), t) \mid p \in M, t \in[-T, 0]\} \subset \mathbb{R}^{n+m} \times(-\infty,+\infty) .$$ Finally, we define the space-time domain $D_{R, T}(o)=\Sigma_T \cap B_{R, T}(o)$. $D_{R, T}(o)$ is compact since $\Sigma_t$ can be written as a complete graph for each $t$. Inspired by the papers([@MR4478479][@MR4145736] [@MR2670053]) , we state our results in the case when $n=2,m=2.$ **Theorem 2**. *Let $X: \Sigma^2 \times[-T, 0] \rightarrow \mathbb{R}^{4}$ be a solution to mean curvature flow with nonpositive normal curvature. Assume that the image of the complex phase map is contained in a regular ball in $\mathbb{S}^2$, i.e., a geodesic ball $B_R(q)$ disjoint from the cut locus of $q$ and $R < \frac{\pi}{2}.$ Assume that there exist a positive constant $C_J$ and a nonnegative constant $C_H$ such that $|dJ| \leq C_{J}|H|$ and $|\vec{H}(p, t)| \leq C_H$ for any point in $\Sigma_T$. Then there exists a constant $C$ which is independent of $R$ and $T$ such that $$\sup _{D_{R / 2, T / 2}(o)} \frac{|H|}{b-\psi\circ J} \leq C\left(\frac{1}{R}+\frac{1}{\sqrt{R}}+\frac{1}{\sqrt{T}}\right),$$ where $b$ is a constant such that $\sup _{\mathcal{M}_T}\psi\circ J \leq 1-c<b<1$.* **Remark 1**. This conditon $|dJ| \leq C_{J}|H|$ can be satisfied by the Lagrangian mean curvature flow in the two dimensional case. **Questions 1**. *It seems that the proof can be carried over to the case that $X: \Sigma^{2n} \times[-T, 0] \rightarrow \mathbb{R}^{4n}$ is a solution to mean curvature flow with nonpositive normal curvature. I guess in this case, there are some gaps needed to fill in.* **Corollary 1**. *Let $X: \Sigma^2 \times (-\infty,0] \rightarrow \mathbb{R}^4$ be an ancient solution to 2-dimensional mean curvature flow with nonpositive normal curvature. Assume that the image of the complex phase map is contained in a regular ball in $\mathbb{S}^2$. Assume that there exist a positive constant $C_J$ and a nonnegative constant $C_H$ such that $|dJ| \leq C_{J}|H|$ and $|\vec{H}(p, t)| \leq C_H$ for any point in $\Sigma_\infty$, then $\Sigma_t$ has to be an affine plane for any $t\in (-\infty,0]$.* **Remark 2**. As we know, self shrinker and tanslating soliton are examples of ancient solution to mean curvatrue flow, so our corollary [Corollary 1](#cor1){reference-type="ref" reference="cor1"} will have wide application. Let $X: \Sigma^2 \rightarrow \mathbb{R}^4$ be an self-shrinkers, which is defined as $$\begin{aligned} H=V^{\perp} \end{aligned}$$ where $V$ is a fixed vection in $\mathbb{R}^4.$ **Corollary 2**. *Let $X: \Sigma^2 \rightarrow \mathbb{R}^4$ be an tanslating solition with nonpositive normal curvature. Assume that the image of the complex phase map is contained in a regular ball in $\mathbb{S}^2$. Assume that there exist a positive constant $C_J$ and a nonnegative constant $C_H$ such that $|dJ| \leq C_{J}|V^{\perp}|$ for any point in $\Sigma$, then $\Sigma$ has to be an affine plane.* Let $X: \Sigma^2 \rightarrow \mathbb{R}^4$ be an self-shrinkers, which is defined as $$\begin{aligned} H=X^{T} \end{aligned}$$ Where $X^{T}$ is the projection of $X$ to the tangent bundle of $\Sigma$. **Corollary 3**. *Let $X: \Sigma^2 \rightarrow \mathbb{R}^4$ be an self-shrinkers with nonpositive normal curvature. Assume that the image of the complex phase map is contained in a regular ball in $S^2$. Assume that there exist a positive constant $C_J$ and a nonnegative constant $C_H$ such that $|dJ| \leq C_{J}|X^{T}|$ and $|X| \leq C_H$ for any point in $\Sigma$, then $\Sigma$ has to be an affine plane.* In the Lagrangian case, we know that(cf. [@MR4478479]) $|dJ |^2=|H |^2$. we can get from Corollary [Corollary 1](#cor1){reference-type="ref" reference="cor1"}. **Theorem 3**. *Let $X: \Sigma^2 \times [-T,0] \rightarrow \mathbb{R}^4$ be a solution to 2-dimensional Lagrangian mean curvature flow with nonpositive normal curvature. Assume that there exist a positive constant $C_J$ and a nonnegative constant $C_H$ such that $|dJ| \leq C_{J}|H|$ and $|\vec{H}(p, t)| \leq C_H$ for any point in $\mathcal{M}_T$. If the cosine of the Lagrangian angle of the initial surface has a positive lower bound, then $\Sigma$ has to be an affine plane.* **Corollary 4**. *Let $X: \Sigma^2 \times (-\infty,0] \rightarrow \mathbb{R}^4$ be an ancient solution to 2-dimensional Lagrangian mean curvature flow with nonpositive normal curvature. Assume that there exist a nonnegative constant $C_H$ such that $|\vec{H}(p, t)| \leq C_H$ for any point in $\mathcal{M}_\infty$. If the cosine of the Lagrangian angle of the initial surface has a positive lower bound, then $\Sigma_t$ has to be an affine plane for any $t\in (-\infty,0]$.* **Remark 3**. As said in in [@MR4478479], the fact that the cosine of the Lagrangian angle of the initial surface has a positive lower bound is preserved along the Lagrangian mean curvatrue flow. Moreover, it implies that the image of the complex phase map is contained in a regular ball in $S^2$, i.e., a geodesic ball $B_R(q)$ disjoint from the cut locus of $q$ and $R < \frac{\pi}{2}$. Since for translationg soliton, the condtion $|\vec{H}(p, t)| \leq C_H$ is automatically satisfied. So, we can derive Corollary 1 in [@MR4478479] and [@MR2670053 Theorem 2] from Corollary [Corollary 4](#cor2){reference-type="ref" reference="cor2"}, **Corollary 5**. *Let $X : \Sigma^2 \to R^4$ be a complete Lagrangian translating soliton with non- positive normal curvature. If the cosine of the Lagrangian angle has a positive lower bound, then $\Sigma$ has to be an affine plane.* # Preliminary We first fix some notations and recall some basical facts about two dimensional Lagrangian mean curvature flow, two dimensional symplectic mean curvature flow and complex phase map. In the end, we give some lemmas used in the proof of this paper. Let $X_0: \Sigma^2 \rightarrow M^4$ be an isometric immersion from an $n$-dimensional oriented Riemannian submanifold $\Sigma$ to the Riemannian manifold $M$. The mean curvature flow (MCF) is a one-parameter family of smooth immersions $X$ : $\Sigma^2 \times[-T, 0] \rightarrow M, T>0$, which satisfies the following evolution equation: $$\left\{\begin{array}{l} \frac{\partial}{\partial t} X(x, t)=H(x, t), \quad x \in \Sigma, t \in[-T, 0], \\ X(\cdot, 0)=X_0, \end{array}\right.$$ In the case where $n=2,m=2$, the Kähler angle of $\Sigma$ in the Kahler-Einstein surface $M$ is denoted by $\alpha$. The surface is called simplectic surface if $\cos \alpha>0$, a Lagrangian surface if $\cos \alpha=0$, holomorphic curve if $\cos \alpha=1.$ If the intial surface is symplectic, then the flow is also symplectic as along as the flow exists(cf. [@MR1864836][@MR1813449][@MR1879229] or [@MR2670053]). The fact is proved via maximum principle of the Kähler angle angle function $\alpha$. If the initial surface is Lagrangian, then the flow pereserve the Lagrangian property(cf.[@smoczyk2000lagrangesche][@MR1922733]). The fact is proved via maximum principle of the Lagrangian angle function $\theta$, whose definiton can be found in [@MR2670053]. Next, we recall the definition of complex phase map of hyper-Lagrangian submanifold $L^{2n}$ of hyperkähler manifold $M^{4n}$. Let $J_1,J_2,J_3$ be three almost complex structure of $M$, where $J_3=J_1J_2,J_1J_2=-J_2J_1$. Let $L$ be a hyper-Lagrangian submanifold of $M$ if there is an almost struction $\hat{J}=\sum\limits_{\alpha=1}^{3}\lambda_{\alpha}J_{\alpha}$ such that the associated symplectic 2-forms $\Omega_{\hat{J}}$ is zero on $L$. Then the complex phase map is defined as $$\begin{aligned} J:L\to \mathbb{S}^2,\quad x \mapsto J(x):=(\lambda_1,\lambda_2,\lambda_3). \end{aligned}$$ Let $L^{2n}$ be hyper-Lagrangian submanifold of hyperkähler manifold $M^{4n}$. We denote the second fundamental form and mean curvature vector by $B$ and $H$ , respectively. **Lemma 1** (cf. [@MR3447682] or [@MR0834612]). *There exists a smooth function $\eta(r, t): \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ supported on $[-R, R] \times$ $-T, 0]$ which has the following properties:* *(1) $\eta(r, t) \equiv 1$ on $[-R / 2, R / 2] \times[-T / 2,0]$ and $0 \leq \psi \leq 1$.* *(2) $\eta(r, t)$ is decreasing if $r \geq 0$, i.e., $\partial_r \eta \leq 0$.* *(3) $\left|\partial_r \eta\right| / \eta^a \leq C_a / R,\left|\partial_r^2 \eta\right| / \eta^a \leq C_a / R^2$ when $0<a<1$.* *(4) $\left|\partial_t \eta\right| / \eta^a \leq C_a / T$ when $0<a<1$.* Next, we consider the mean curvature flow from a closed surface $\Sigma$ in a hyperkähler 4-manifold M, i.e, $X$ : $\Sigma^2 \times[-T, 0] \rightarrow M, T>0$, which satisfies the following evolution equation: $$\label{e5} \begin{aligned} \left\{\begin{array}{l} \frac{\partial}{\partial t} X(x, t)=H(x, t), \quad x \in \Sigma, t \in[-T, 0], \\ X(\cdot, 0)=X_0, \end{array}\right. \end{aligned}$$ For this kind of flow, it is proved in [@MR2320167] that if the image of $J$ lies in a hemisphere of $\mathbb{S}^2$, then this remains so under the mean curvature flow. It is proved in [@MR2320167] that the mean curvature flow of a hyper-Lagrangian submanifold $L$ in $M$ preserves the hyper-Lagrangian condition. Its complex phase map solves harmonic map heat equation. **Theorem 4** (cf. [@MR2320167] ). *The complex phase maps of the mean curvature flow [\[e5\]](#e5){reference-type="eqref" reference="e5"} , $J: \Sigma_t \longrightarrow \mathbb{S}^2$ form an evolving harmonic map heat flow, i.e., $$\frac{\partial J}{\partial t}=\tau(J),$$ where $\tau(J)$ is the tension field of $J$ with respect to the induced metric $g_t$ on $\Sigma_t$.* **Lemma 2** (cf. [@MR2392328] or [@MR3739208]). *For mean curvature flow in Euclidean space, we have $$\label{er} \begin{aligned} \Delta |H|^2-\partial_t |H|^2 \geq 2|\nabla H|^2-2|H |^2|B |^2 \end{aligned}$$ where $H$ and $B$ are mean curvature vector and the second fundmental form of the mean curvature flow, respectively.* Obviously, this inequality [\[er\]](#er){reference-type="eqref" reference="er"} holds for MCF [\[e5\]](#e5){reference-type="eqref" reference="e5"} if the ambient manifold is hyper-Kähler maniold $\mathbb{R}^4$. # The proof *Proof.* We can view $\Sigma$ as a hyper-Lagrangian submanifold in hyper-kähler maniofold $\mathbb{R}^4$ with respect to some almost complex structure. We denote the second fundamental form and mean curvature vector by $B$ and $H$. We use the notation in [@MR4478479] . Let J be the complex phase map of $\Sigma$. Let $\rho$ be the distance function on $S^2$, and $h$ the Riemannian metric of $S^2$. Define $\psi = 1 - \cos \rho$, then $Hess(\psi) = (\cos \rho)h$. It is an standard trick in geometric analysis, we need to compute $$\begin{aligned} \left( \frac{\partial}{\partial t}-\Delta\right) \frac{|H|^2}{(b-\psi\circ J)^2} \end{aligned}$$ Let $\phi= \frac{|H|^2}{(b-\psi\circ J)^2}$. A direct calculation shows that $$\nabla \phi=\frac{\nabla|H|^2}{(b-\psi\circ J)^2}+\frac{2|H|^2 \nabla\psi\circ J}{(b-\psi\circ J)^3} .$$ Similarly we can compute $$\Delta \phi=\frac{\Delta|H|^2}{(b-\psi\circ J)^2}+\frac{4\left\langle\nabla\psi\circ J, \nabla|H|^2\right\rangle}{(b-\psi\circ J)^3}+\frac{2|H|^2 \Delta \psi\circ J}{(b-\psi\circ J)^3}+\frac{6|\nabla\psi\circ J|^2|H|^2}{(b-\psi\circ J)^4} .$$ By Qiu [@MR4478479], we know that $$\begin{aligned} \Delta \psi\circ J= 2\cos \rho |dJ |^2 |H|^2-2|H|^2 \partial_t \psi\circ J \end{aligned}$$ By the above computations, we obtain $$\begin{aligned} \Delta \phi= & \frac{2|\nabla H|^2+\partial_t|H|^2-2|B |^2|H|^2}{(b-\psi\circ J)^2}+\frac{4\left\langle\nabla\psi\circ J, \nabla|H|^2\right\rangle}{(b-\psi\circ J)^3} \\ & +\frac{2\cos \rho |dJ |^2 |H|^2-2|H|^2 \partial_t \psi\circ J}{(b-\psi\circ J)^3}+\frac{6|\nabla\psi\circ J|^2|H|^2}{(b-\psi\circ J)^4} . \end{aligned}$$ On the other hand, the time derivative of $\phi$ is given by $$\partial_t \phi=\frac{\partial_t|H|^2}{(b-\psi\circ J)^2}-\frac{2|H|^2 \partial_t \psi\circ J}{(b-\psi\circ J)^3} .$$ We continue the calculation as $$\begin{aligned} \Delta \phi= & \frac{2|\nabla B|^2-2|H |^2|B |^2}{(b-\psi\circ J)^2}+\frac{4\left\langle\nabla\psi\circ J, \nabla|H|^2\right\rangle}{(b-\psi\circ J)^3} \\ & +\frac{2\cos \rho |dJ |^2 |H|^2}{(b-\psi\circ J)^3}+\frac{6|\nabla\psi\circ J|^2|H|^2}{(b-\psi\circ J)^4}+\partial_t \phi . \end{aligned}$$ Note that the following relations hold: $$\begin{gathered} \frac{2|\nabla H|^2}{(b-\psi\circ J)^2}+\frac{2|\nabla\psi\circ J|^2|H|^2}{(b-\psi\circ J)^4} \geq \frac{4|\nabla B||\nabla\psi\circ J||H|}{(b-\psi\circ J)^3}, \\ \frac{2\left\langle\nabla|H|^2, \nabla\psi\circ J\right\rangle}{(b-\psi\circ J)^3}+\frac{4|H|^2|\nabla\psi\circ J|^2}{(b-\psi\circ J)^4}=\frac{2\langle\nabla\psi\circ J, \nabla \phi\rangle}{(b-\psi\circ J)} . \end{gathered}$$ Hence, we get $$\begin{aligned} \Delta \phi-\partial_t \phi\geq 2(1-b)\frac{|dJ|^2|H |^2}{((b-\psi\circ J))^3}+\frac{2\langle\nabla\psi\circ J,\nabla \phi\rangle}{(b-\psi\circ J)} \end{aligned}$$ Fix a point $(p_0,0)\in \Sigma^2 \times [-T,0]$ such that $X\left(p_0, 0\right)$ is the origin $o$ of $\mathbb{R}^{n+1}$. Let $\eta$ be the function constructed in Lemma [Lemma 1](#lem1){reference-type="ref" reference="lem1"}. We use a cut-off function supported on $D_{R, T}(o)$ given by $\psi(F(p, t)):=\eta(r(X), t)$, where $r(X):=|X|$ is the distance function on $\mathbb{R}^{4}$. Let $L:=-2 \nabla\psi\circ J /(b-\psi\circ J)$. We can calculate $$\begin{aligned} \Delta(\psi \phi) & +\langle L, \nabla(\psi \phi)\rangle-2\left\langle\frac{\nabla \psi}{\psi}, \nabla(\psi \phi)\right\rangle-\partial_t(\psi \phi) \\ = & \psi\left(\Delta \phi-\partial_t \phi\right)+\phi\left(\Delta \psi-\partial_t \psi\right)+\langle\psi L, \nabla \phi\rangle+\langle\phi L, \nabla \psi\rangle-2 \frac{|\nabla \psi|^2}{\psi} \phi \\ \geq & 2(1-b) \psi \frac{|dJ|^2|H |^2}{(b-\psi\circ J)^3}+\phi\left(\Delta \psi-\partial_t \psi\right)+2 \frac{\langle\nabla\psi\circ J, \nabla \psi\rangle}{b-\psi\circ J} \phi-2 \frac{|\nabla \psi|^2}{\psi} \phi . \end{aligned}$$ Note that $D_{R, T}(o)$ is compact, since any time slice $M_t$ can be written as an entire graph. Hence $\psi \phi$ attains its maximum at some point $F\left(p_1, t_1\right)$ in $D_{R, T}(o)$. At this point, we have $$\nabla(\psi \phi)=0, \quad \Delta(\psi \phi) \leq 0, \quad \partial_t(\psi \phi) \geq 0 .$$ Hence , we obtain $$\begin{aligned} 2 \psi((1-b) \frac{|dJ|^2|H |^2}{(b-\psi\circ J)^3} & \leq 2 \phi \frac{\langle\nabla\psi\circ J , \nabla \psi\rangle}{b-\psi\circ J}+2 \phi \frac{|\nabla \psi|^2}{\psi}+\phi\left(\partial_t \psi-\Delta \psi\right) \\ & =I+I I+I I I . \end{aligned}$$ Note that the following holds: $$|\nabla \psi|^2=\left|\partial_r \eta\right|^2|\nabla r|^2 \leq n\left|\partial_r \eta\right|^2 .$$ By [@MR3447682], we know that $$\begin{aligned} |\nabla\psi\circ J|\leq |dJ| \end{aligned}$$ Where $C_1=(\frac{v_1}{2-v_1})^{\frac{5}{2}}$. By using (Young's inequality and the property of $\eta$, we can estimate $I$ as follows: $$\begin{aligned} I & \leq 2 \phi \frac{|\nabla\psi\circ J|}{b-\psi\circ J}|\nabla \psi| \\ &\leq 2 \phi \frac{|dJ|}{b-\psi\circ J}|\nabla \psi| \\ & \leq \frac{\varepsilon}{4} \psi \frac{|H|^{\frac{8}{3}}|dJ|^{\frac{4}{3}}}{(b-\psi\circ J)^4}+\frac{C(\varepsilon)|\nabla \psi|^4}{\psi^3}\\ & \leq \frac{\varepsilon}{4} \psi \frac{|H|^{\frac{8}{3}}|dJ|^{\frac{4}{3}}}{(b-\psi\circ J)^4}+\frac{n^2 C(\varepsilon)\left|\partial_r \eta\right|^4}{\psi^3} \\ & \leq \frac{\varepsilon}{4} \psi \frac{|H|^{\frac{8}{3}}|dJ|^{\frac{4}{3}}}{(b-\psi\circ J)^4}+\frac{C(\varepsilon, n)}{R^4}, \end{aligned}$$ where $\varepsilon>0$ is an arbitrary constant, $C(\varepsilon)$ and $C(\varepsilon, n)$ are constants depending only on $\varepsilon$ and $n$. Similarly, as in [@MR3447682], we can calculate by using Young's inequality and the property of $\eta$, $$I I=2 \phi \frac{|\nabla \psi|^2}{\psi} \leq \frac{\varepsilon}{4} \psi \phi^2+\frac{C(\varepsilon, n)}{R^4} .$$ Now we assume $|\vec{H}(p, t)| \leq C_H$. Since $\partial_r \eta \leq 0$, we have $$\Delta \psi=(\Delta r)\left(\partial_r \eta\right)+|\nabla r|^2\left(\partial_r^2 \eta\right) \geq\left(C_H+\frac{n}{r}\right)\left(\partial_r \eta\right)-n\left|\partial_r^2 \eta\right| .$$ Hence we obtain for the second term of $I I I$ in the same way as [@MR3447682], $$\begin{aligned} -\phi \Delta \psi & \leq \frac{\varepsilon}{4} \psi \phi^2+C(\varepsilon, n)\left(\frac{1}{R^4}+\frac{1}{R^2}\right) . \end{aligned}$$ (Note that we may assume $R / 2 \leq r$ for the second inequality, since $\partial_r \eta \equiv 0$ for $r \leq R / 2$.) As for the first term of $I I I$, as in [@MR3447682] we have $$\begin{aligned} \phi\left(\partial_t \psi\right) & \leq \frac{\varepsilon}{4} \psi \phi^2+C\left(\varepsilon, C_H\right)\left(\frac{1}{R^2}+\frac{1}{T^2}\right) . \end{aligned}$$ Since $$\begin{aligned} |dJ |^2\geq |B |^2\geq \frac{|H |^2}{2}. \end{aligned}$$. Combing the above estimates , we finally obtain $$\frac{1}{2}(1-b)(b-\psi\circ J) \psi \phi^2 \leq \frac{\varepsilon}{4} \psi \frac{|H|^{\frac{8}{3}}|dJ|^{\frac{4}{3}}}{(b-\psi\circ J)^4}+ \frac{3\epsilon}{4} \psi \phi^2+C\left(\varepsilon, n, C_H\right)\left(\frac{1}{R^4}+\frac{1}{R^2}+\frac{1}{T^2}\right) .$$ Noticing our assumption $$\begin{aligned} |dJ|\leq C|H| \end{aligned}$$ Since $$\begin{aligned} \psi <b \end{aligned}$$ So we can take a sufficiently small $\varepsilon$ such that $$\frac{1}{2}(1-b)(b-\psi\circ J)-\varepsilon>0 .$$ Then we have $$(\psi \phi)^2 \leq \psi \phi^2 \leq C\left(\frac{1}{R^4}+\frac{1}{R^2}+\frac{1}{T^2}\right) .$$ Since $\psi \equiv 1$ on $D_{R / 2, T / 2}(o)$, $$\sup _{D_{R / 2, T / 2}(o)} \frac{|H|}{b-\psi\circ J} \leq C\left(\frac{1}{R}+\frac{1}{\sqrt{R}}+\frac{1}{\sqrt{T}}\right) .$$ This completes the proof of Theorem [Theorem 2](#thm1){reference-type="ref" reference="thm1"}. ◻ [^1]: School of Information Engineering, Nanjing Xiaozhuang University, Nanjing 211171, China [^2]: caoxiangzhi\@njxz.edu.cn

arxiv_math

--- abstract: | There are many group-based cryptosystems in which the security relies on the difficulty of solving Conjugacy Search Problem (CSP) and Simultaneous Conjugacy Search Problem (SCSP) in their underlying platform groups. In this paper we give a cryptanalysis of these systems which use certain semidirect product of abelian groups. author: - Delaram Kahrobaei$^{1,2,3,5}$, Carmine Monetta$^4$, Ludovic Perret$^6$, Maria Tota$^4$, Martina Vigorito$^4$ title: Cryptanalysis of protocols using (Simultaneous) Conjugacy Search Problem in certain Metabelian Platform Groups --- $^1$ Department of Computer Science, University of York, UK\ $^2$ Departments of Computer Science and Mathematics, Queens College, City University of New York, USA\ $^3$ Department of Computer Science and Engineering, Tandon School of Engineering, New York University, USA\ $^4$ Department of Mathematics, University of Salerno, IT\ $^5$ Initiative for the Theoretical Sciences, Graduate Center, City University of New York, USA\ $^6$ Sorbonne University, CNRS, LIP6, PolSys, Paris, France # Introduction The field of group-based cryptography began with the seminal work of Anshel, Anshel and Goldfeld in 1999 when they proposed a commutator key-exchange protocol based on the difficulty of simultaneus conjugacy search problem in certain classes of groups, namely braid groups [@AAG]. The search for the platform group for this protocol has been an active area including several cryptanalysis. For a survey on group-based cryptography in the quantum era see [@AMS] and book [@AMSB]. Polycyclic group-based cryptography was introduced by Eick and Kahrobaei in [@EK]. More precisely, they proposed such groups as platform for the Commutator Key-Exchange Protocol, also known as Anshel-Anshel-Goldfeld (a.k.a. AAG) [@AAG], as well as for the non-commutative Diffie-Hellman Key-Exchange Protocol (a.k.a. Ko-Lee) [@KoLee]. The security of these protocols relies on the difficulty to solve the Simultaneous Conjugacy Search Problem (SCSP) and the Conjgacy Search Problem (CSP) in some classes of groups. Their argument is based on experimental results for the CSP for certain metabelian polycyclic groups arising from field extensions. These groups are not virtually nilpotent, hence the CSP cannot be solved using the analysis provided in [@MT]. Nevertheless, some of these groups can be avoided as platform since, in [@KU], Kotov and Ushakov did a cryptanalysis for some groups of this type. A connected work is due to Gryak, Kahrobaei, and Martinez Perez who investigated another class of metabelian groups. Indeed, in [@GKM] they obtain a complexity result concerning the CSP which is proved to be at most exponential for the analyzed class of groups. The methods used to test conjugacy decision problem are different and include experiments conducted with machine learning algorithms, as done by Gryak, Kahrobaei and Haralick, in [@GHK], but also Length-based attack. Garber, Kahrobaei, and Lam, in [@GKL], showed that the Length-based attack is inefficient for certain classes of metabelian polycyclic groups. There are other proposed cryptosystems based on the difficulty of CSP in certain classes of groups, (see the survey by Gryak and Kahrobaei [@GK]), for example Kahrobaei-Koupparis Digital Signature Scheme [@KK], and Khan-Kahrobaei Non-commutative El Gamal Key-exchange [@KahKhan]. In this paper we go further to the results Field-Based-Attack (FBA) in [@KU] and show how to cryptanalyze the CSP and SCSP for some other classes of metabelian groups.\ The authors in [@KU] investigated security properties of the Commutator Key-Exchange Protocol used with certain polycyclic groups. They showed that despite low success of the length based attack the protocol can be broken by a deterministic polynomial-time algorithm. They call this approach FBA and they implemented it in GAP to compare LBA and FBA.\ In this paper we show that FBA could be generalized for protocols based on the difficulty of CSP and SCSP in certain classes of metabelian groups. In particular we prove the followings theorems:\ Theorem [\[theoaag\]](#theoaag){reference-type="ref" reference="theoaag"}: Let $G=M\ltimes N$, where $M\cong\mathbb{Z}^n$ and $N=\mathbb{Z}[m_1^{\pm1},\ldots,m_n^{\pm1}]$ (as additive groups), with $m_1,\ldots,m_n$ positive integers, then there exists a polynomial-time algorithm to break Commutator Key-Exchange protocol for such a group $G$.\ Theorem [\[theodh\]](#theodh){reference-type="ref" reference="theodh"}: Let $G=M\ltimes N$, where $M\cong\mathbb{Z}^n$ and $N=\mathbb{Z}[m_1^{\pm1},\ldots,m_n^{\pm1}]$ (as additive groups), with $m_1,\ldots,m_n$ positive integers, then there exists a polynomial-time algorithm to break Diffie-Hellmann Key-Exchange protocol for such a group $G$.\ This paper is structured as follow: in Section [2](#sec2){reference-type="ref" reference="sec2"}, we recall the definitions of Conjugacy Search Problem and Simultaneous Conjugacy Search Problem and we describe some Key-Exchange Protocols such as Non-commutative Diffie-Hellman and nshel-Anshel-Goldfeld Commutator. Section [3](#sec3){reference-type="ref" reference="sec3"} presents the family of metabelian groups we are interested in with some examples. In Section [4](#nondifproblems){reference-type="ref" reference="nondifproblems"} we prove the main result i.e. how to cryptanalyze the CSP and SCSP in such platform groups and we provide the proofs of Theorem [\[theoaag\]](#theoaag){reference-type="ref" reference="theoaag"} and Theorem [\[theodh\]](#theodh){reference-type="ref" reference="theodh"}. The conclusions of our work are in Section [5](#conclusion){reference-type="ref" reference="conclusion"}. # Background {#sec2} ## (Simultaneous) Conjugacy Search Problem We start out by giving a brief description of two group-theoretic algorithmic problems on which the security of a number of protocols is based. Here and in the following, if $x$ and $g$ are group-elements, the conjugate of $g$ by $x$, which is denoted by $g^x$, is the element $x^{-1}gx$.\ **The Conjugacy Search Problem** (CSP): Let $G$ be a finitely presented group such that the conjugacy decision problem is solvable. Given $g\in G$ and $h=g^x$ for some $x\in G$, the *Conjugacy Search Problem* asks to search such an element $x\in G$.\ **The Simultaneous Conjugacy Search Problem** (SCSP): Given a finitely presented group $G$ and $g_1,\dots ,g_n$, $h_1,\dots ,h_n$ elements of $G$ such that $h_i=g_i^x$, for all $i\in \{1,\dots , n\}$ and some $x\in G$, the *Simultaneous Conjugacy Search Problem* asks to recover such an element $x\in G$.\ Please note that CSP and SCSP are always solvable since we assume that the decision conjugacy problem is solvable in the definitions of these problems. Also, a solution of $g^x=h$ is not unique. In fact, given a solution $x$, the set of solutions is $\{ax\ :\ a\in C_G(g)\}$.\ Examples of well known protocols whose security is based on the difficulty of solving the CSP or the SCSP are the non-commutative Diffie-Hellman (a.k.a Ko-Lee) Key-Exchange Protocol and the Anshel-Anshel-Goldfeld Commutator Key-Exchange Protocol. We recall these protocols below. ## Non-commutative Diffie-Hellman (a.k.a. Ko-Lee) Key Exchange Protocol {#DH} Originally proposed by Ko, Lee, et al. [@KoLee] using braid groups, their non-commutative analogue of Diffie-Hellman key exchange can be generalized to work over other platform groups. Let $G$ be a finitely presented group, with $A,B \leq G$ such that all elements of $A$ and $B$ commute.   An element $g\in G$ is chosen, and $g, G, A, B$ are made public. A shared secret can then be constructed as follows: - Alice chooses a random element $a\in A$ and sends $g^{a}$ to Bob. - Bob chooses a random element $b\in B$ and sends $g^{b}$ to Alice. - The shared key is then $g^{ab}$, as Alice computes $(g^b)^a$, which is equal to Bob's computation of $(g^a)^b$ as $a$ and $b$ commute. The security of such a protocol is based on the difficulty to get $a$ and $b$, which are private, from public information $g,g^a$ and $g^b$. That is to solve the conjugacy equations $$g^x=h\ \ \ \ \ \text{and}\ \ \ \ \ g^y=h'$$ where $h=g^a$ and $h'=g^b$. In other words, the security of Ko-Lee rests upon solving the conjugacy search problem within the subgroups $A, B$. ## Anshel-Anshel-Goldfeld Commutator (a.k.a. AAG) Key-Exchange Protocol {#arit} The Anshel-Anshel-Goldfeld Commutator Key-Exchange Protocol [@AAG] is a two-party protocol performed as follows: - Fix a finitely presented group $G$, called the platform group, a set of generators $g_1,\dots , g_k$ for $G$ and some positive integers $n_1, n_2, l, m$. All this information are made public. - Alice prepares a tuple of elements $\bar{a} = (a_1,\dots, a_{n_1})$ called Alice's public tuple. Each $a_i$ is generated randomly as a product of $g_i$'s and their inverses. - Bob prepares a tuple of elements $\bar{b} = (b_1,\dots, b_{n_2})$ called Bob's public tuple. Each $b_i$ is generated randomly as a product of $g_i$'s and their inverses. - Alice generates a random element $A$ as a product $a_{s_1}^{\epsilon_1}\cdots a_{s_l}^{\epsilon_l}$ of $a_i$'s and their inverses. The element $A$ (or more precisely its factorization) is called the Alice's private element. - Bob generates a random element $B$ as a product $b_{t_1}^{\delta_1}\cdots b_{t_m}^{\delta_m}$ of $b_i$'s and their inverses. The element $B$ (or more precisely its factorization) is called the Bob's private element. - Alice publishes the tuple of conjugates $\bar{b}^A=(A^{-1}b_1A,\dots,A^{-1}b_{n_2}A)$. - Bob publishes the tuple of conjugates $\bar{a}^B=(B^{-1}a_1B,\dots,B^{-1}a_{n_1}B)$. - Finally, Alice computes the element $K_A$ as a product: $$A^{-1}(B^{-1}a_{s_1}^{\epsilon_1}B\cdots B^{-1}a_{s_l}^{\epsilon_l}B)=A^{-1}B^{-1}AB=[A,B]$$ using the elements of Bob's conjugate tuple $\bar{a}^B$. - Similarly, Bob computes the element $K_B$ as a product: $$(A^{-1}b_{t_1}^{\delta_1}A\cdots A^{-1}b_{t_m}^{\delta_m}A)^{-1}B=A^{-1}B^{-1}AB=[A,B]$$ using the elements of Alice's conjugate tuple $\bar{b}^A$. - The shared key is then $K=K_A=K_B=[A,B]$. The security of such a protocol is based on the fact that it is difficult to recover $A$ and $B$ from $\bar{a},\bar{b},\bar{b}^A$ and $\bar{a}^B$, which are public. In practice, if $\bar{b}^A=(b_1',\ ,b_{n_2}')$ and $\bar{a}^B=(a_1',\dots ,a_{n_1}')$, it is achieved by solving a system of conjugacy equations for $A$ and $B$: $$\label{sistema1} \Bigg \{ \begin{array}{ll} X^{-1} b_1 X=b_1' \\ \dots \\ X^{-1} b_{n_2} X=b_{n_2}' \\ \end{array}$$ $$\label{sistema2} \Bigg \{ \begin{array}{ll} Y^{-1} a_1 Y=a_1' \\ \dots \\ Y^{-1} a_{n_1} Y=a_{n_1}' \\ \end{array}$$ This means that the security of AAG rests upon solving the simultaneous conjugacy search problem in $G$. # Examples of Metabelian Groups {#sec3} Here we describe some families of metabelian groups whose CSP and SCSP will be discussed in the next section. To be more precise, we are interested in groups $G$ of the form $G=M\ltimes N$, with both groups $M$ and $N$ abelian. We use multiplicative notation for the whole group $G$ but additive notation for $N$. So if $s\in M$ and $c\in N$, the action of the element $s$ maps $c$ to $$c \cdot s \text{ with additive notation or,}$$ $$c^s=s^{-1}cs\text{ with multiplicative notation.}$$ This kind of groups are metabelian and arise quite naturally in linear algebra and ring theory, as we will show in more details in the following examples. **Example 1**. In [@KU], Kotov and Ushakov studied the security of AAG protocol for some polycyclic platform groups. More precisely they considered the group $M$ as the multiplicative group of a specific field $F$ and the group $N$ as the additive group of the same field $F$; hence $G=F^*\ltimes F$. To construct $F$ they considered an irreducible monic polynomial $f(x)\in \mathbb{Z}[x]$ and put: $$\label{specific field} F=\mathbb{Q}[x]/(f).$$ If $a\in F^*$ and $b\in F$, the action of $a$ maps $b$ to $b\cdot a$. They showed that in such a group it is possible to reduce the systems ([\[sistema1\]](#sistema1){reference-type="ref" reference="sistema1"}) and ([\[sistema2\]](#sistema2){reference-type="ref" reference="sistema2"}) to two systems of linear equations over the field $F$. Then there exist conditions under which each system has a unique solution. **Example 2**. Let $V(+,\cdot)$ be a vector space over a field $F$. Take the group $M$ as the multiplicative group $F^*$ of $F$ and the group $N$ as the additive group of $V$. If $\lambda\in F^*$ and $v\in V$, the action of $\lambda$ maps $v$ to $v\cdot \lambda$. Hence $G=F^*\ltimes V$ has the same structure of the general group we considered before. Notice that, for $V=F$, if $F$ is of the form described in ([\[specific field\]](#specific field){reference-type="ref" reference="specific field"}) we obtain the same example we found in [@KU]. Similarly we could start with a module over a commutative unitary ring. Such examples are interesting from a mathematical point of view but more practical examples, as they have been described in [@GKM], follow. **Example 3**. Split metabelian groups of finite Prüfer rank. We will focus in the case when the group $G$ is given by a presentation of the form $$G=\langle q_1,\ldots,q_n,b_1,\ldots,b_s\mid [q_i,q_j]=1,[b_l,b_t]=1,b_i^{q_l}=q_lb_iq_l^{-1}=b_1^{m_{1i}}b_2^{m_{2i}}\ldots b_s^{m_{si}}\rangle.$$ Observe that $q_1,\ldots,q_n$ generate a free abelian group which we denote by $M$ and $b_1,\ldots,b_s$ generate the abelian group $N$ as normal subgroup of $G$. Then $G=M\ltimes N$. Under these conditions one can show that there is an embedding $N\to\mathbb{Q}^s$ mapping the family $b_1,\ldots,b_s$ to a free basis of $\mathbb{Q}^s$. This means that our group is torsion free metabelian of finite Prüfer rank (meaning that the number of generators needed to generate any finitely generated subgroup is bounded). Observe that the action of $M$ on $N$ can be described using integer matrices: the action of $q_l$ is encoded by the $(s\times s)$-matrix $M_l$ with columns $m_{1i},\ldots,m_{s_i}$. Moreover $G$ is polycyclic if and only if the matrices $M_i$ can be taken to be integer matrices with integral inverses [@AuslanderHall]. One of the main advantages of these groups is that they admit the following fairly simple set of normal forms: $$q_1^{\alpha_1}\ldots q_n^{\alpha_n}b_1^{\beta_1}\ldots b_s^{\beta_s}q_1^{\gamma_1}\ldots q_n^{\gamma_n}.$$ with $\gamma_1,\ldots,\gamma_n>0$. Moreover there is an efficient algorithm (collection) to transform any word in the generators to the corresponding normal form: given an arbitrary word in the generating system, move all of the instances of $q_i$ with negative exponent to the left and all the instances of $q_i$ with positive exponents to the right. [\[genBS\]]{#genBS label="genBS"} **Example 4**. Generalized metabelian Baumslag-Solitar groups. Let $m_1,\ldots,m_n$ be positive integers. We call the group given by the following presentation a *generalized metabelian Baumslag-Solitar group* $$G=\langle q_1,\ldots,q_n,b\mid [q_i,q_j]=1, b^{q_i}=b^{m_i},i,j=1,\ldots,n\rangle.$$ It is a constructible metabelian group of finite Prüfer rank and $G\cong M\ltimes N$ with $M=\langle q_1,\ldots,q_n\rangle\cong\mathbb{Z}^n$ and $N=\mathbb{Z}[m_1^{\pm1},\ldots,m_n^{\pm1}]$ (as additive groups). In [@GKM], the authors showed the CSP in such groups reduce to the Discrete Logarithm Problem. **Example 5**. Let $L:\mathbb{Q}$ be a Galois extension of degree $n$ and fix an integer basis $\{u_1,\ldots,u_k\}$ of $L$ over $\mathbb{Q}$. Then $\{u_1,\ldots,u_k\}$ freely generates the maximal order $\mathcal{O}_L$ as a $\mathbb{Z}$-module. Now, we choose integer elements, $q_1,\ldots,q_n$, generating a free abelian multiplicative subgroup of $L-\{0\}$. Each $q_i$ acts on $L$ by left multiplication and using the basis $\{u_1,\ldots,u_k\}$, we may represent this action by means of an integer matrix $M_i$. Let $N$ be the smallest sub $\mathbb{Z}$-module of $L$ closed under multiplication with the elements $q_i$ and $q_i^{-1}$ and such that $\mathcal{O}_L\subseteq N$, i.e., $$N=\mathcal{O}_L[q_1^{\pm 1},\ldots,q_n^{\pm1}].$$ We then may define $G=M\ltimes N$ where the action of $M$ on $N$ is given by multiplication by the elements $q_i$. The generalized Baumslag-Solitar groups of the previous example are a particular case of this situation for $L=\mathbb{Q}$. If the elements $q_i$ lie in $\mathcal{U}_L$ which is the group of units of $\mathcal{O}_L$, then the group $G$ is polycyclic. # Cryptanalysis of the Commutator and the Non-Commutative Diffie-Hellman key exchange Protocols {#nondifproblems} In this section, we show that the AAG and the Ko-Lee Key Exchange Protocols are not suitable in the case of the generalised metabelian Baumslag-Solitar groups (Example [Example 4](#genBS){reference-type="ref" reference="genBS"}). Similar arguments can be used with minor modifications for the other examples in Section [3](#sec3){reference-type="ref" reference="sec3"}.\ We begin studying the CSP and SCSP in a metabelian group of the form $G=M \ltimes N$, as described in Section [3](#sec3){reference-type="ref" reference="sec3"}. Assume that we have conjugated elements $g,h\in G$ and we want to solve the CSP for $g$, $h$, i.e., we want to find $x\in G$ such that $$g^x=h.$$ We put $g=sc$, $h=s'c'$ and $x=td$, where $s,s',t\in M$ and $c,c',d\in N$. Then $$g^x=x^{-1}gx=d^{-1}t^{-1}sctd=d^{-1}st^{-1}ctd=s(d^{-1})^sc^td.$$ Now $g^x=h$ implies $s'=s$ and $c'=(d^{-1})^sc^td$. Since the element $(d^{-1})^sc^td$ belongs to $N$ we can write it additively as $$-d\cdot s +c\cdot t + d = d \cdot (1-s) + c \cdot t.$$ This means that the CSP above is equivalent to the problem of finding $t\in M$ and $d\in N$ such that $$\label{equation}d \cdot (1-s)+ c\cdot t=c',$$ where $s \in M$ and $c,c' \in N$ are given.\ In particular, if we need to face the SCSP, which means to solve system [\[sistema1\]](#sistema1){reference-type="eqref" reference="sistema1"}, we can apply the reduction process described above. Then, if we put $b_i=s_ic_i$, $b_i'=s_i'c_i'$ with $s_i,s'_i\in M$ and $c_i,c_i'\in N$, for all $i\in\{1,\dots ,n_2\}$, and $X=td$ with $t\in M,\ d\in N$ we will get the following system of equations $$\label{sistema1ridotto} \Bigg \{ \begin{array}{ll} d \cdot (1-s_1)+ c_1 \cdot t = c_1' \\ \dots \\ d \cdot (1-s_{n_2} ) +c_{n_2} \cdot t=c_{n_2}' \\ \end{array}$$ where $s_i \in M$ and $c_i, c_i' \in N$ are given and we need to find $t \in M$ and $d \in N$.\ Then the next results follow.\ We start analyzing the cryptanalysis of AAG protocol in a generalized metabelian Baumslag-Solitar groups, as described in Example [Example 4](#genBS){reference-type="ref" reference="genBS"}. **Theorem 1**. *Let $G=M\ltimes N$, where $M\cong\mathbb{Z}^n$ and $N=\mathbb{Z}[m_1^{\pm1},\ldots,m_n^{\pm1}]$ (as additive groups), with $m_1,\ldots,m_n$ positive integers, then there exists a polynomial-time algorithm to break Commutator Key-Exchange protocol for such a group $G$. [\[theoaag\]]{#theoaag label="theoaag"}* *Proof.* In AAG protocol the attacker knows $b_1^X,b_2^X,\ldots,b_{n_2}^X$ for some $b_1,\ldots,b_{n_2}$ (which are public) and $n_2>1$. To find $X=td$, with $t \in M$ and $d \in N$, the attacker has to solve several equations as ([\[sistema1ridotto\]](#sistema1ridotto){reference-type="ref" reference="sistema1ridotto"}). Let us consider two of them $$d \cdot (1-s)+ c\cdot t=c'$$ $$d \cdot (1-\tilde s)+\tilde c\cdot t=\tilde c'.$$ Here $s,\tilde s, c, \tilde c,c',\tilde c'$ are known and the attacker has to find $t$ and $d$. Recall that $c',\tilde c', c, \tilde c,d$ lie in $N$ which is a subring of $\mathbb{Q}$. If we identify $s$ and $t$ with the integer they act by, then they also lie in $N$. So the above can be seen as a system of two equations in $N$, moreover we know a priori that the system has a solution. This means that unless the second equation is a multiple of the first one, this solution is unique and the standard procedure to solve the system yields then the suitable value of $t$ and $d$ in polynomial time (see [@GKM Section 2]). ◻ The argument in the previous proof applies also when $G$ is as described in Example [Example 2](#vectorspace){reference-type="ref" reference="vectorspace"}, choosing $V=F^n$ with $n\in\mathbb N$.\ Next, let us move to the non-commutative Diffie-Hellmann key exchange protocol (Section [2.2](#DH){reference-type="ref" reference="DH"}). **Theorem 2**. *Let $G=M\ltimes N$, where $M\cong\mathbb{Z}^n$ and $N=\mathbb{Z}[m_1^{\pm1},\ldots,m_n^{\pm1}]$ (as additive groups), with $m_1,\ldots,m_n$ positive integers, then there exists a polynomial-time algorithm to break Diffie-Hellmann Key-Exchange protocol for such a group $G$. [\[theodh\]]{#theodh label="theodh"}* *Proof.* In Ko-Lee protocol the main problem is that Alice and Bob must agree on a set $\Omega$ of pairwise commuting elements and then choose their conjugators $a$ and $b$ from that set. Recall that we are denoting $G=M\ltimes N$. As $M$ is abelian a possible choice would be $\Omega=M$, and if $a$ lies in $M$ then the attacker can find $a$ from $g^{a}$ in polynomial time. Another possibility would be to choose $a,b\in N$. But then $a=d$ and equation ([\[equation\]](#equation){reference-type="ref" reference="equation"}) for $g^{a}$ is $$d \cdot (1-s)+c=c',$$ and the only unknown is $d$ which can be found easily in polynomial time (see [@GKM Section 3]). In the case when $a$ is an arbitrary element not in $M$ or $N$, $\Omega$ must be a subset of the centralizer $C_G(a)$ of $a$ in $G$. As $a\in G\setminus N$, the centralizer of $a$ in $N$ is trivial, which means that $C_G(a)\cap N=1$. The most obvious choice for $\Omega$ is some other complement $M_1$ of $N$ in $G$ and the standard way to specify $M_1$ is via a derivation $\delta:M\in N$ so that $$M_1=\{\delta(x)x\mid x\in M\}.$$ It is a standard fact that $M_1=M^r$ for some $r\in N$ if and only if $\delta(x)=(1-x)\cdot d$, i.e., if and only if $\delta$ is an inner derivation. In our case this can be seen very easily: Things are particularly easy in the case when the element $a$ belongs to $M^r$ for some $r\in N$, which happens if and only if $$a=td=t_1^r=r^{-1}t_1r=t_1 t_1^{-1}r^{-1}t_1r=t_1 r^{-t_1} r,$$ for some $t,t_1 \in M$ and $d \in N$. Additively this is equivalent to $$d=r- r\cdot t=r \cdot (1-t).$$ It is a standard fact that $M^r=\{x\delta(x)\mid x\in M\}$ where $\delta$ is the inner derivation given by $\delta(x)=r \cdot (1-x)$. In this case it is easy to check that $$\Omega\subseteq C_G(a)=M^r.$$ If the attacker has the extra information that $a$ belongs to $M^r$ for some $r$, then the equation that he has to solve is $$r \cdot (1-t)(1-s)+c\cdot t=c'$$ equivalently $$(c-r+ r\cdot s)\cdot t=c'-r+s\cdot r.$$ This can be seen as an equation in $\mathbb{Q}$ and only requires to perform the quotient of $c'-r+ r\cdot s$ by $c-r+ r\cdot s$ thus can be solved in polynomial time (see [@GKM Section 3]). Moreover, we are going to see now that by embedding our group $G$ in a bigger group we may always assume that $a$ lies in some conjugated subgroup of $M$. Let $\tilde G= M\ltimes\tilde N$ where $\tilde N=N\otimes_\mathbb{Z} \mathbb{Q}=\mathbb{Q}$. Then $a=td$ lies in $M^r$ for some $r\in\tilde N$ if and only if $d=r\cdot (1-t).$ This can always be solved in $\mathbb{Q}$, in other words, we can always find a suitable $r\in\mathbb{Q}$. Then one proceeds as we did before with this $r$. The fact that $r$ might not belong to $N$ does not create any troubles: recall that we are dealing not with the conjugacy problem but with the conjugacy *search* problem, meaning that we know a priory that our equations have a solution so the procedure above yields the right values of $t,d$ even if $r$ does not belong to $N$.\ Observe that behind what we said above is the fact that for the group $\tilde G$, the first cohomology group $\rm{H}^1(M,\tilde N)$ is zero, thus all the complements of $\tilde N$ in $\tilde G$ are conjugated. ◻ Notice that exactly the same argument in the previous proof happens for any group $G=M\ltimes N$ with $N\subseteq\mathbb{Q}^n$ for some $n$, so can be extended to the more general version of our groups (Example [Example 3](#split){reference-type="ref" reference="split"}). # Conclusion In this paper we do cryptanalysis for the CSP and SCSP in certain metabelian groups. In particular we show the following. 1. The generalized metabelian Baumslag-Solitar groups can not be used as platform groups in commutator key-exchange protocol. 2. The generalized metabelian Baumslag-Solitar groups can not be used as platform groups in non-commutative Diffie-Hellman protocol. Finally we want to point out that this cryptanalysis could be extended to the other examples in Section [3](#sec3){reference-type="ref" reference="sec3"} and to all cryptosystems based on the difficulty of CSP and SCSP. # Acknowledgement {#acknowledgement .unnumbered} DK thanks the University of Salerno (Italy), where most of this paper was discussed and written. We thank Professor Conchita Martinez-Perez for fruitful discussions. MV thanks Initiative for the Theoretical Sciences at CUNY GC which hosted her Fall 2022. 99 I. Anshel, M. Anshel, and D. Goldfeld, *An algebraic method for public-key cryptography*, Math. Res. Let. 6, 287--291, 1999. I. Anshel, D. Atkins, D. Goldfeld, and P. E. 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--- author: - Jiwei Zheng - "Wei Zhou[^1]" - D. E. Taylor title: Groups with at most 13 nonpower subgroups --- **Abstract.** For a group $G$ and $m\ge 1$, $G^m$ denotes the subgroup generated by the elements $g^m$ where $g$ runs through $G$. The subgroups not of the form $G^m$ are called nonpower subgroups. We extend the classification of groups with few nonpower subgroups from groups with at most 9 nonpower subgroups to groups with at most 13 nonpower subgroups.\ **Keywords**: counting subgroups, power subgroups, nonpower subgroups.\ **2020 Mathematics Subject Classification**: 20D25, 20D60. # Introduction For a group $G$ and $m\ge 1$ the *power* subgroup $G^m$ is the subgroup generated by the elements $g^m$ where $g$ runs through $G$. A subgroup that is not a power subgroup is a *nonpower* subgroup. Let $\mathop{\mathrm{ps}}(G)$ and $\mathop{\mathrm{nps}}(G)$ denote the number of power and nonpower subgroups of $G$. It is immediate that every power subgroup is a characteristic subgroup of $G$. In 1956 Szász [@szasz:1956] proved that $G$ is cyclic if and only if $\mathop{\mathrm{nps}}(G) = 0$. In 2006, Zhou, Shi and Duan [@zhou-shi-duan:2006] showed that a noncyclic group is finite if and only if it has a finite number of nonpower subgroups and it was observed that a finite noncyclic group must have at least three nonpower subgroups. Then Anabanti *et al.* [@anabanti-etal:2022; @anabanti-hart:2022] classified the groups with three or four nonpower subgroups. We summarise these results in the following theorem. (See section [2](#sec:notation){reference-type="ref" reference="sec:notation"} for definitions of the groups.) **Theorem 1**. *For $0\le k\le 4$ a group has exactly $k$ nonpower subgroups if and only if up to isomorphism it is one of the following.* $k=\arabic{Lcount1}$ A cyclic group. No examples. No examples. $C_2\times C_2$, $Q_8$ or $G_{n,3}$ for $n\ge 1$. $C_3\times C_3$. This theorem, combined with the following theorem from our previous paper [@zheng-etal:2023], classifies groups with at most nine nonpower subgroups. **Theorem 2**. *For $5\le k\le 9$ a group has exactly $k$ nonpower subgroups if and only if up to isomorphism it is one of the following.* $k=\arabic{Lcount2}$ $C_2\times C_4$ or $G_{n,5}$ for $n\ge 1$. $C_5\times C_5$, $C_2\times C_2\times C_p$, $Q_8\times C_p$, where $p > 2$ is a prime or $G_{n,3}\times C_q$ for $n\ge 1$, where $q > 3$ is a prime. $D_8$, $\mathop{\mathrm{Alt}}(4)$, $C_2\times C_8$, $Q_{16}$, $M_{4,2}$, $C_3\times C_9$, $M_{3,3}$, $G_{n,7}$ or $F_{n,7}$ for $n\ge 1$. $C_7\times C_7$ or $C_3\times C_3\times C_p$, where $p\ne 3$ is a prime. $C_2\times C_{16}$, $M_{5,2}$, $C_2\times C_2\times C_{p^2}$, $Q_8\times C_{p^2}$, where $p > 2$ is a prime or $G_{n,3}\times C_{q^2}$, where $q > 3$ is a prime. In the present paper we extend the classification to groups with at most 13 nonpower subgroups. **Theorem 3**. *For $10\le k\le 13$ a group has exactly $k$ nonpower subgroups if and only if up to isomorphism it is one of the following.* $k=\arabic{Lcount}$ $C_3\times C_{27}$, $C_2\times C_4\times C_p$, $M_{4,3}$, $\mathop{\mathrm{Sym}}(3)\times C_3$, $A_2 = (C_2\times C_2)\rtimes C_9$, $G^{(2)}_{2,n;\,5,1}$ for $n\ge 2$ or $G_{n,5}\times C_q$ for $n\ge 1$, where $p\ne 2$ and $q\ne 2,5$ are primes. $C_2\times C_{32}$, $C_5\times C_{25}$, $S_{16}$, $M_{6,2}$, $M_{3,5}$, $\mathop{\mathrm{SL}}(2,3)$, $G_{n,11}$ or $G^{(3)}_{5,n;\,11,1}$ for $n\ge 1$. $C_4\times C_4$, $C_{11}\times C_{11}$, $Q_8\times C_{qr}$, $C_2\times C_2\times C_{qr}$, $C_3\times C_3\times C_{s^2}$, $C_5\times C_5\times C_p$, $B^2_{2,2}$, $G_{n,9}$ or $G_{n,3}\times C_{qr}$ for $n\ge 1$, where $p,q,r$ and $s$ are primes such that $p\ne 5$, $3 < q < r$ and $s\ne 3$. $C_2\times C_{64}$, $C_3\times C_{81}$, $M_{7,2}$, $M_{5,3}$, $\mathop{\mathrm{Sym}}(3)\times C_2 = D_{12}$, $C_3\rtimes Q_8$, $A_3 = (C_2\times C_2)\rtimes C_{27}$, $G_{n,13}$ or $F_{n,13}$ for $n\ge 1$. # Notation and definitions {#sec:notation} All groups considered in this paper are finite. Let $\Phi(G)$ denote the Frattini subgroup of $G$. For subgroups $H$ and $K$ of $G$, let $[H,K]$ be the subgroup generated by the commutators $[x,y] = x^{-1}y^{-1}xy$ with $x\in H$ and $y\in K$. Let $G$ be a $p$-group. For all $i\ge 1$, $\Omega_{i}(G)$ denotes the subgroup $\langle\mskip 2mu\relax x\in G\mid x^{p^{i}}=1 \mskip 2mu \rangle$. (In general the exponent of $\Omega_{i}(G)$ may be greater than $p^i$.) We use Gorenstein [@gorenstein:1968 Ch. 5] as a reference for standard results about $p$-groups. Statements of most of the key lemmas from [@gorenstein:1968] used in the proof of the main theorem can be found in our previous paper [@zheng-etal:2023]. Alternatively, see Huppert [@huppert:1967 Ch. III]. Let $C_n$ denote the cyclic group of order $n$ and let $\mathop{\mathrm{Alt}}(n)$ and $\mathop{\mathrm{Sym}}(n)$ denote the alternating and symmetric groups of a set of size $n$. Let $\mathop{\mathrm{SL}}(2,3)$ denote the group of $2\times 2$ matrices of determinant 1 over the field of 3 elements. We use the notation $G = N\rtimes K$ to mean that $G$ is a semidirect product of $N$ by $K$. The action of $K$ on $N$ will be determined by the context. A group $G$ is *metacyclic* if it has a normal cyclic subgroup $N$ such that $G/N$ is cyclic. Almost all the groups that occur in Theorems [Theorem 1](#thm:upto4){reference-type="ref" reference="thm:upto4"}, [Theorem 2](#thm:upto9){reference-type="ref" reference="thm:upto9"} and [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} are metacyclic or the direct product of a metacyclic group and a cyclic group. Many of these groups can be described as a semidirect product $C_{q^m}\rtimes C_{p^n}$ of cyclic groups of prime power orders. Metacyclic groups have been classified by Hempel [@hempel:2000]. The special case that we need is given by the following definition. **Definition 4**. For primes $p$ and $q$, positive integers $m$, $n$ and an integer $r$ such that $r^{p^n}\equiv 1 \pmod{q^m}$, let $G^{(r)}_{p,n;\,q,m}$ denote the group with presentation $$\langle\mskip 2mu\relax a, b \mid a^{p^n} = 1,\ b^{q^m} = 1,\ a^{-1}ba = b^r \mskip 2mu \rangle,$$ If $p^k$ is the order of $r\pmod{q^m}$, the automorphism of $\langle\mskip 2mu\relax b\mskip 2mu \rangle$ induced by conjugation by $a$ has order $p^k$; therefore, if $p\ne q$, then $p^k$ divides $q - 1$ and if the group is nonabelian, then $p$ divides $q-1$. **Remark 5**. For certain values of the parameters the groups $G^{(r)}_{p,n;q,m}$ have standard names (see [@gorenstein:1968]). Where possible we prefer to use these names. 1. $G^{(1)}_{p,n;\,q,m}$ is the direct product $C_{p^n}\times C_{q^m}$. 2. $G^{(-1)}_{2,1;\,q,m}$ is the *dihedral group* $D_{2q^m}$ of order $2q^m$. 3. For $n\ge 4$, $G^{(-1+2^{n-2})}_{2,1;\,2,n-1}$ is the *semidihedral* group $S_{2^n}$ of order $2^n$. 4. For $n\ge 4$ when $p = 2$ and $n\ge 3$ when $p$ is an odd prime, $G^{(1+p^{n-2})}_{p,1;\,p,n-1}$ is the *quasidihedral* group $M_{n,p}$ of order $p^n$. ($M_{3,p}$ is the *extraspecial* group of order $p^3$ and exponent $p^2$.) **Remark 6**. Because of the frequency with which the groups occur in the theorems and for compatibility with [@zheng-etal:2023] we use an abbreviated notation for the following special cases. For $n\ge 1$ and a prime $p$, 1. $G_{n,p^m}$ denotes $G^{(-1)}_{2,n;\,p,m}$ ; 2. $F_{n,p}$ denotes $G^{(r)}_{3,n;\,p,1}$ where the order of $r\pmod p$ is 3 and hence $p\equiv 1\pmod 3$; 3. $B^2_{n,p}$ denotes $G^{(p+1)}_{p,n;\,p,2}$ and we have $B^2_{1,2} = D_8$, $B^2_{2,2}$ is the semidirect product $C_4\rtimes C_4$ and for $p$ odd, $B^2_{1,p}= M_{3,p}$. For completeness we include definitions of two other families of groups with standard names and two families of groups (without standard names), which occur in the proof of the main theorem. **Definition 7**. 1. For $n\ge 3$, $\langle\mskip 2mu\relax a,b\mid a^{2^{n-1}} = b^2 = z,\ z^2 = 1,\ b^{-1}ab = a^{-1}\mskip 2mu \rangle$ is a presentation for the *generalized quaternion* group $Q_{2^n}$ of order $2^n$. 2. For an odd prime $p$, $\langle\mskip 2mu\relax x,y,z \mid x^p = y^p = z^p = 1,\ [x,z] = [y,z] = 1,\ [x,y] = z\mskip 2mu \rangle$ is a presentation for the *extraspecial* group $M(p)$ of order $p^3$ and exponent $p$. 3. For $n\ge 1$, $\langle\mskip 2mu\relax a,b,c \mid a^{3^n} = b^2 = 1,\ bc = cb,\ b^a = c,\ c^a = bc \mskip 2mu \rangle$ is a presentation for the group $A_n = (C_2\times C_2)\rtimes C_{3^n}$ of order $2^2 3^n$. For example, $A_1\simeq\mathop{\mathrm{Alt}}(4)$. 4. For $n\ge 1$ and a prime $p$, $$\langle\mskip 2mu\relax a,b,c\mid [a,b] = c,\ a^p = b^{p^n} = c^p = 1,\ [a,c] = [b,c] = 1 \mskip 2mu \rangle$$ is a presentation for the group $B^1_{n,p}$ of order $p^{n+2}$. Except for $B^1_{1,2} = D_8$, it is non-metacyclic (see [@blackburn:1958 Lemma 2.5]). The quotient mod $\langle\mskip 2mu\relax c\mskip 2mu \rangle$ is $C_p\times C_{p^n}$ and for $p$ odd, $B^1_{1,p}= M(p)$. # Preliminaries {#sec:prelim} First we prove a general result about cyclic subgroups of prime power order. **Theorem 8**. *For all primes $p$ and finite groups $G$, if $m$ divides the exponent $e$ of $G$ and $G^m$ is cyclic of order $p^k$, then $m = e/p^k$. In particular, for all $k$, $G$ has at most one cyclic power subgroup of order $p^k$.* *Proof.* Suppose that for a divisor $m$ of $e$, the power subgroup $G^m$ is cyclic of order $p^k$. Let $q\ne p$ be a prime divisor of $|G|$ and let $y$ be an element of order $q^\ell$, where $q^\ell$ is the exponent of a Sylow $q$-subgroup. If $q^h$ is the highest power of $q$ that divides $m$, there exist integers $s$ and $t$ such that $q^h = sq^\ell + tm$. Therefore $y^{q^h} = y^{tm}\in G^m$, thus $y^{q^h} = 1$ and hence $h = \ell$. This proves that $m = p^jd$ for some $j$ where $d$ is the product of the exponents of the Sylow $q$-subgroups with $q\ne p$. Therefore, for all $g\in G$, $g^d$ is a $p$-element and it follows that $p^{k+j}$ is the exponent of the Sylow $p$-subgroups of $G$. Thus $m = e/p^k$. ◻ **Corollary 9**. *Let $p^f$ be the largest order of a $p$-element of $G$, where $p$ is a prime divisor of $|G|$. Let $p^k$ be the largest order of a cyclic power $p$-subgroup of $G$. If $p$ is odd and a Sylow $p$-subgroup is not cyclic, then $G$ contains at least $pf - k + 1$ cyclic nonpower $p$-subgroups.* *Proof.* Let $P$ be a Sylow $p$-subgroup of $G$ and suppose that $P$ is not cyclic. By a Theorem of Burnside [@burnside:1955 §105], for all $m$ where $1\le m\le f$, $P$ contains more than one subgroup of order $p^m$. Miller[@miller:1929] has proved that for $m > 1$ the number of cyclic subgroups of order $p^m$ is divisible by $p$ and the number of subgroups of order $p$ is congruent to $1+p$ modulo $p^2$ (see also Berkovich [@berkovich:2008 Th. 1.10]). Therefore $P$ has at least $p$ subgroups of order $p^i$ for $2\le i\le f$ and at least $p+1$ subgroups of order $p$. Furthermore, from Theorem [Theorem 8](#thm:power){reference-type="ref" reference="thm:power"} $P$ has exactly one cyclic power subgroup of order $p^i$ for $1\le i\le k$. Therefore $G$ has at least $p + (k-1)(p-1) + p(f-k) = pf-k+1$ cyclic nonpower $p$-subgroups. ◻ The following two lemmas play an essential rôle in the proof of the main theorem. **Lemma 10** ([@anabanti-etal:2022 Lemma 3]). *If $A$ and $B$ are finite groups such that $|A|$ and $|B|$ are coprime, then $\mathop{\mathrm{ps}}(A\times B) = \mathop{\mathrm{ps}}(A)\mathop{\mathrm{ps}}(B)$ and $\mathop{\mathrm{nps}}(A\times B) = \mathop{\mathrm{nps}}(A)s(B) + \mathop{\mathrm{ps}}(A)\mathop{\mathrm{nps}}(B)$, where $s(B)$ is the number of subgroups of $B$.* **Lemma 11** ([@zhou-shi-duan:2006 Lemma 2]). *Suppose $N$ and $H$ are subgroups of $G$ such that $N\trianglelefteq G$. If $HN/N$ is a nonpower subgroup of $G/N$, then $H$ is a nonpower subgroup of $G$. Therefore $\mathop{\mathrm{nps}}(G)\ge \mathop{\mathrm{nps}}(G/N)$.* **Remark 12**. This lemma was stated in [@zheng-etal:2023] with the additional condition $N\subseteq H$. However, the same proof shows that it remains true without this restriction. It often can be used to show that $\mathop{\mathrm{nps}}(G)\ge 2\mathop{\mathrm{nps}}(G/N)$. For example, suppose that $|N|$ is coprime to $|G/N|$. From the Schur--Zassenhaus Theorem [@gorenstein:1968 Th. 6.2.1], $G$ is a semidirect product $N\rtimes K$. In this case, if $H\subseteq K$ and if $HN/N$ is a nonpower subgroup of $G/N$, then both $H$ and $HN$ are nonpower subgroups of $G$. Since $H\ne HN$ it follows that $\mathop{\mathrm{nps}}(G)\ge 2\mathop{\mathrm{nps}}(G/N)$. **Lemma 13** ([@zheng-etal:2023], Theorem 2.12). *There is no finite $p$-group $G$ such that $G/N \simeq M_{n,p}$, where $N$ is a central subgroup of $G$ of order $p$ contained in $G'$.* The following lemma from [@zheng-etal:2023] is a consequence of Lemma 2.2 and Theorem 2.3 of Blackburn [@blackburn:1958]. **Lemma 14**. *For a non-abelian $p$-group $G$ generated by two elements, let $R=\Phi(G')G_{3}$ where $G_{3}=[[G,G],G]$. Then* 1. *$R$ is the only maximal subgroup of $G'$ that is normal in $G$,* 2. *$G$ is metacyclic if and only if $G/R$ is metacyclic,* 3. *If the type of $G/G'$ is $(p,p^{n})$ and $G/R$ has no cyclic maximal subgroup, then $G/R$ is isomorphic to $B_{n,p}^{1}$ or $B_{n,p}^{2}$.* # Nonpower values {#sec:values} Section 3 of [@zheng-etal:2023] provides formulas for the number of nonpower subgroups for many families of groups. For convenience, we summarise this information in the following lemma, then prove formulas for the additional values of $\mathop{\mathrm{nps}}(G)$ needed in the proof of the main result. The formulas for $\mathop{\mathrm{nps}}(M_{n,p})$, $\mathop{\mathrm{nps}}(G_{n,p^k})$ and $\mathop{\mathrm{nps}}(F_{n,p})$ are special cases of Lemmas [Lemma 16](#lemma:Gpp){reference-type="ref" reference="lemma:Gpp"} and [Lemma 17](#lemma:Gnp){reference-type="ref" reference="lemma:Gnp"} below. **Lemma 15**. *For an integer $n$ and a prime $p$ we have:* 1. *for $n\ge 3$, $\mathop{\mathrm{nps}}(D_{2^n}) = 2^n - 1$;* 2. *for $n\ge 3$, $\mathop{\mathrm{nps}}(Q_{2^n}) = 2^{n-1} - 1$;* 3. *for $n\ge 4$, $\mathop{\mathrm{nps}}(S_{2^n}) = 3\cdot 2^{n-2} - 1$;* 4. *for $n\ge 3$, $\mathop{\mathrm{nps}}(M_{n,p}) = p(n-1) + 1$ (when $p = 2$, assume $n\ge 4$); [\[i:QD\]]{#i:QD label="i:QD"}* 5. *$\mathop{\mathrm{nps}}(M(p)) = p^2 + 2p + 2$;[\[i:esp\]]{#i:esp label="i:esp"}* 6. *if $p > 2$, then $\mathop{\mathrm{nps}}(G_{n,p^k}) = p(p^k-1)/(p-1)$;* 7. *if $p\equiv 1\pmod 3$, then $\mathop{\mathrm{nps}}(F_{n,p}) = p$;* 8. *for $n\ge 1$, $\mathop{\mathrm{nps}}(A_n) = 3n + 4$;[\[i:extA4\]]{#i:extA4 label="i:extA4"}* 9. *[\[i:Cp\]]{#i:Cp label="i:Cp"} [@Marius:2010 Th. 3.3] for $n_{2}\ge n_{1}\ge 1$ and a prime $p$ the value of $\mathop{\mathrm{nps}}(C_{p^{n_{1}}}\times C_{p^{n_{2}}})$ is $$\hspace{-4pt}\frac{(n_{2}-n_{1}+1)p^{n_{1}+2}-(n_{2}-n_{1}-1)p^{n_{1}+1}-(n_2+1)p^2+(n_{2}-n_{1}+1)p+n_{2}}{(p-1)^{2}}.$$* **Lemma 16**. *For prime $p>2$, $\mathop{\mathrm{nps}}(G^{(r)}_{p,n;p,m}) = \mathop{\mathrm{nps}}(C_{p^n}\times C_{p^m})$.* *Proof.* Let $G = G^{(r)}_{p,n;\,p,m}$. Since $G'$ is cyclic, we know from [@huppert:1967 III §10] that $G$ is a regular $p$-group. Then from [@hall:1934 Th. 4.21], $G^{p^s}$ is the set $\{\,g^{p^s}\mid g\in G\,\}$. Thus there are exactly $\max(n,m)+1$ power subgroups in $G$. This means that $\mathop{\mathrm{ps}}(G)=\mathop{\mathrm{ps}}(C_{p^n}\times C_{p^m})$. It follows from [@mann:2010 Prop. 1] that $G$ is lattice-isomorphic to $C_{p^n}\times C_{p^m}$. Thus $s(G) = s(C_{p^n}\times C_{p^m})$ and therefore $\mathop{\mathrm{nps}}(G) = \mathop{\mathrm{nps}}(C_{p^n}\times C_{p^m})$. ◻ From (i) and (ix) of Lemma [Lemma 15](#lemma:list){reference-type="ref" reference="lemma:list"} we know that for $p=2$ the conclusion of the Lemma [Lemma 16](#lemma:Gpp){reference-type="ref" reference="lemma:Gpp"} is not valid. **Lemma 17**. *Suppose that $p\ne q$ are primes and that $m$, $n$ and $r$ are positive integers such that $r\ne 1$ and $r^{p^n}\equiv 1\pmod{q^m}$. Let $G = G^{(r)}_{p,n;q,m}$ and let $p^k$ be the order of $r\pmod{q^m}$. Then $p\mid q - 1$ and $\mathop{\mathrm{nps}}(G) = kq(q^m-1)/(q-1)$.* *Proof.* For $0\le i < k$ and $0\le j < m$ let $P_i = \langle\mskip 2mu\relax a^{p^i}\mskip 2mu \rangle$, $Q_j = \langle\mskip 2mu\relax b^{q^{m-j}}\mskip 2mu \rangle$ and $H_{ij} = P_iQ_j$. From the presentation of $G$ we have $Q_j\trianglelefteq G$ and $Z(G)=\langle\mskip 2mu\relax a^{p^k}\mskip 2mu \rangle$. Furthermore, $P_0\in\mathop{\mathrm{Syl}}_p(G)$ and $G$ is the semidirect product $Q\rtimes P_0$, where $Q\in\mathop{\mathrm{Syl}}_q(G)$. It is clear that $P_0\subseteq N_G(P_i)$ and we claim that for $0\le i< k$, $N_G(P_i) = P_0$. If $x\in Q\cap N_G(P_i)$, then $[P_i,\langle\mskip 2mu\relax x\mskip 2mu \rangle]\subseteq P_i\cap Q = 1$; that is, $x\in C_G(P_i)$. It follows from [@gorenstein:1968 Th. 5.2.4] that $x\in C_G(P_0)$ and hence $x\in Z(G)\cap Q = 1$. This establishes the claim. We also have $P_0\subseteq N_G(H_{ij})$ and it follows from the Frattini argument that $N_G(H_{ij}) = Q_jN_G(P_i) = Q_j\rtimes P_0$. Therefore $H_{ij}$ has $q^{m-j}$ conjugates in $G$ and since for $0\le i < k$ and $0\le j < m$ every subgroup of order $p^{n-i}q^j$ is conjugate to $H_{ij}$ we have found $k\sum_{j=0}^{m-1} q^{m-j} = kq(q^m-1)/(q-1)$ subgroups, all of which are nonpower subgroups. For $1\le i\le n$ the power subgroup $G^{p^{n-i}}$ is the unique subgroup of order $p^iq^m$. For $k\le s\le n$ and $0\le j < m$, the power subgroup $G^{p^sq^{m-j}}$ is the unique subgroup of order $p^{n-s}q^j$. This accounts for all subgroups of $G$. Therefore $\mathop{\mathrm{nps}}(G) = kq(q^m-1)/(q-1)$. ◻ **Lemma 18**. *For $n\ge 2$, we have* 1. *$\mathop{\mathrm{nps}}(B^1_{n,p}) = p^2(2n-1) + p(n+1) + 2$, and* 2. *$\mathop{\mathrm{nps}}(B^2_{n,p}) = p^2(n-1) + p(n+1) + 2$.* *Proof.* There are $n+1$ power subgroups of $B_{n,p}^{1}$ and $B_{n,p}^{2}$ . We count all the subgroups of them by considering their exponents. Notice that $\Omega_{n-1}(B^1_{n,p}) = \langle\mskip 2mu\relax a\mskip 2mu \rangle\times\langle\mskip 2mu\relax c\mskip 2mu \rangle\times\langle\mskip 2mu\relax b^p\mskip 2mu \rangle\simeq C_p\times C_p\times C_{p^{n-1}}$. By Corollary 2.2 in [@tarnauceanu-toth:2017], the number of subgroups of exponent at most $p^{n-1}$ is $s(C_p\times C_p\times C_{p^{n-1}}) = 2(n-1)p^2+np+n+2$. Furthermore, there are $p^2+p+1$ subgroups of exponent $p^n$. Thus $\mathop{\mathrm{nps}}(B^1_{n,p}) = p^2(2n-1) + p(n+1) + 2$. Similar to $B^1_{n,p}$, $\Omega_{n-1}(B^2_{n,p}) = \langle\mskip 2mu\relax a\mskip 2mu \rangle\times\langle\mskip 2mu\relax b^p\mskip 2mu \rangle\simeq C_{p^{2}}\times C_{p^{n-1}}$. From Corollary 2.2 in [@tarnauceanu-toth:2017], the number of subgroups of exponent at most $p^{n-1}$ is $s(C_{p^2}\times C_{p^{n-1}}) = (n-2)p^2+pn+n+2$. And there are also $p^2+p+1$ subgroups of exponent $p^n$. Thus $\mathop{\mathrm{nps}}(B^2_{n,p}) = p^2(n-1) + p(n+1) + 2$. ◻ **Lemma 19**. *Let $G = G_{m,3}\times \underbrace{C_3\times\dots\times C_3}_n$. For $n \ge 1$ we have $\mathop{\mathrm{nps}}(G)\ge 4m+6$ and equality holds when $n = 1$.* *Proof.* Similar to the calculation of $\mathop{\mathrm{nps}}(G_{n,3})$ in [@zheng-etal:2023], we have $\mathop{\mathrm{nps}}(G_{m,3}\times C_3) = 4m+6$. And as $n$ increases, the number of subgroups will increase but the number of power subgroups does not increase. Thus $\mathop{\mathrm{nps}}(G)\ge 4m+6$. This completes the proof. ◻ **Lemma 20**. *Let $G = Q_8\times \underbrace{C_2\times\dots\times C_2}_n$. For $n \ge 1$ we have $\mathop{\mathrm{nps}}(G)\ge 16$ and equality holds when $n = 1$.* *Proof.* For all $n\ge 1$ the exponent of $G$ is $4$ and the only proper non-trivial power subgroup is $G^2$ of order $2$. The group $Q_8\times C_2$ has $19$ subgroups, consequently $\mathop{\mathrm{nps}}(Q_8\times C_2) = 16$ and $\mathop{\mathrm{nps}}(G)\ge 16$ for $n\ge 1$. ◻ **Lemma 21**. *By direct calculation we have $\mathop{\mathrm{nps}}(\mathop{\mathrm{Sym}}(4)) = 26$, $\mathop{\mathrm{nps}}(\mathop{\mathrm{SL}}(2,3)) = 11$ and $\mathop{\mathrm{nps}}(C_3\rtimes Q_8) = 13$.* **Lemma 22** ([@zheng-etal:2023 Lemma 3.3]). *Let $G = D_{2p}\times \underbrace{C_2\times\dots\times C_2}_n$. For $n \ge 1$ and a prime $p > 2$, we have $\mathop{\mathrm{nps}}(G)\ge 3p+4$ and equality holds when $n = 1$.* **Lemma 23** ([@zheng-etal:2023 Lemma 3.4]). *Let $X_{n,p} = D_{2p}\times \underbrace{C_3\times\dots\times C_3}_n$. For $n \ge 1$ and a prime $p > 3$, we have $\mathop{\mathrm{nps}}(X_{n,p})=(p+3)s(C_{3}^{n})-6\geq10$ where $C_3^n = \underbrace{C_3\times\dots\times C_3}_n$. For $n > 2$, $\mathop{\mathrm{nps}}(X_{n,3}) > \mathop{\mathrm{nps}}(X_{2,3}) = 48$ and $\mathop{\mathrm{nps}}(X_{1,3}) = 10$.* # Proof of Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} {#proof-of-theorem-thmmain} **Lemma 24**. *Let $P$ be a Sylow $p$-subgroup of a group $G$ such that $P\ne N_G(P)\ne G$. Then $10\le \mathop{\mathrm{nps}}(G)\le 13$ if and only if for some $n\ge 1$ one of the following holds* 1. *$\mathop{\mathrm{nps}}(G) = 10$, $p = 2$ and $G$ is isomorphic to $\mathop{\mathrm{Sym}}(3)\times C_3$ or to $G_{n,5}\times C_r$ where $r\ne 2,5$ is a prime;* 2. *$\mathop{\mathrm{nps}}(G) = 11$, $p = 3$ and $G$ is isomorphic to $\mathop{\mathrm{SL}}(2,3)$;* 3. *$\mathop{\mathrm{nps}}(G) = 12$, $p = 2$ and $G$ is isomorphic to $G_{n,3}\times C_{qr}$ where $q$ and $r$ are primes such that $3 < q < r$.* *In all cases, if $Q\in\mathop{\mathrm{Syl}}_q(G)$ and $q\ne p$, then $Q\trianglelefteq G$.* *Proof.* Since $P\ne N_G(P)\ne G$, both $P$ and $N_G(P)$ have at least $p+1$ conjugates and so $2(p+1)\le\mathop{\mathrm{nps}}(G)\le 13$, hence $p$ is 2, 3 or 5. First we show that $p\ne 5$. Suppose to the contrary that $p = 5$. Then the 6 conjugates of $P$ and the 6 conjugates of $N_G(P)$ are nonpower subgroups, hence every other subgroup is normal. In particular, if $Q\in\mathop{\mathrm{Syl}}_3(G)$, then $Q\trianglelefteq G$. The permutation action of $G$ on $\mathop{\mathrm{Syl}}_5(G)$ defines a homomorphism $G \to \mathop{\mathrm{Sym}}(6)$ with kernel $K$. Then $|KP/K| = 5$ and $KQ/K$ is a normal 3-subgroup of $G/K$. But $\mathop{\mathrm{Sym}}(6)$ does not contain a 3-subgroup normalised by a group of order 5, therefore $p \ne 5$. **Claim.***If $Q\in\mathop{\mathrm{Syl}}_q(G)$ and $q\ne p$, then $Q\trianglelefteq G$.* If $N_G(Q)\ne G$, then $Q = N_G(Q)$ otherwise $G$ would have least $2(p+1)+2(q+1) \ge 14$ nonpower subgroups. There are at least $p+1$ conjugates of $P$ and $N_G(P)$ and at least $q+1$ conjugates of $Q$, therefore $2(p+1) + (q+1)\le 13$ and so $q\le 5$. If $q = 5$, then $p = 2$ and $|G:Q| = 6$. But then $|P| = 2$ and $G$ would have a normal subgroup $H$ of index 2, which must contain the 6 conjugates of $Q$. But $|H:Q| = 3$, which is a contradiction and therefore $q\ne 5$. It follows that $q\le 3$, $G = PQ$ and $|P| = |G:Q|$. Suppose that $p = 2$. Then $q = 3$, $|G : Q| = 4 = |P|$ and the permutation action of $G$ on $\mathop{\mathrm{Syl}}_3(G)$ defines a homomorphism $G\to\mathop{\mathrm{Sym}}(4)$ with kernel $K\subseteq Q$. It follows that $G/K\simeq \mathop{\mathrm{Alt}}(4)$ and therefore $P\simeq C_2\times C_2$. But $N_G(P)\ne G$, therefore $K\ne 1$ and so for each element $x\in P$ of order 2, it follows from Lemma [Lemma 11](#lemma:quo){reference-type="ref" reference="lemma:quo"} that $\langle\mskip 2mu\relax x\mskip 2mu \rangle$ and $\langle\mskip 2mu\relax x\mskip 2mu \rangle K$ are nonpower subgroups. Taking into account the 3 conjugates of $P$, the 3 conjugates of $N_G(P)$ and the 4 conjugates of $Q$, $G$ would have at least 16 nonpower subgroups, contrary to our assumption. This leaves the possibility that $p = 3$, $q = 4$ and hence $|G:Q| = 3 = |P|$. Again the permutation action of $G$ on $\mathop{\mathrm{Syl}}_3(G)$ defines a homomorphism $G\to\mathop{\mathrm{Sym}}(4)$ this time with kernel $K\subseteq N_G(P)$. But $P\nsubseteq K$, therefore $K\subseteq Q$ and Lemma [Lemma 21](#lemma:S4){reference-type="ref" reference="lemma:S4"} implies $G/K\not\simeq\mathop{\mathrm{Sym}}(4)$. Consequently $G/K\simeq \mathop{\mathrm{Alt}}(4)$ and hence $Q\trianglelefteq G$.0◻ If $p = 2$, then $| G : N_G(P)| = 2k + 1$ for some $k$ and so $2(2k+1)\le 13$. Therefore $q = |G : N_G(P)|$ is either 3 or 5 and we choose $Q\in\mathop{\mathrm{Syl}}_q(G)$. Similarly, if $p = 3$, then $|G : N_G(P)| = 4$. In this case we let $q = 2$ and choose $Q\in\mathop{\mathrm{Syl}}_2(G)$. In all cases $G = N_G(P)Q$. **Claim.***$G = PQ\times A$, where $A$ is a cyclic group of order 1, $r$, $r^2$ or $rs$, where $r$ and $s$ are primes different from $p$ and $q$.* Let $R$ be a Sylow $r$-subgroups of $G$, where $r\ne p,q$ is a prime. Then $R$ acts by conjugation on $\mathop{\mathrm{Syl}}_p(G)$ and in all cases $R$ fixes a conjugate of $P$. Thus $R\subseteq N_G(P)$ and since $R\trianglelefteq G$ we have $[P,R]\subseteq P\cap R = 1$. We also have $Q\trianglelefteq G$ and so $[Q,R]\subseteq Q\cap R = 1$. Let $A$ be the product of the Sylow $r$-subgroups for $r\ne p,q$. Then $G = PQ\times A$ and from Lemma [Lemma 10](#lemma:prod){reference-type="ref" reference="lemma:prod"} $$\mathop{\mathrm{nps}}(G) = \mathop{\mathrm{nps}}(PQ)s(A) + \mathop{\mathrm{ps}}(PQ)\mathop{\mathrm{nps}}(A).$$ We have $\mathop{\mathrm{nps}}(PQ)\ge 3$ and $\mathop{\mathrm{ps}}(PQ)\ge 3$. Thus $A$ is cyclic, otherwise $s(A)\ge 3$ and from Theorem [Theorem 1](#thm:upto4){reference-type="ref" reference="thm:upto4"} $\mathop{\mathrm{nps}}(A)\ge 3$ hence $\mathop{\mathrm{nps}}(G) > 13$. Since $A$ is cyclic we have $\mathop{\mathrm{nps}}(A) = 0$ and therefore $\mathop{\mathrm{nps}}(G) = \mathop{\mathrm{nps}}(PQ)s(A)\le 13$. Thus $s(A)\le 4$ and $|A|$ is 1, $r$ or $rs$, where $r\le s$ are primes.0◻ **Claim.***If $p = 2$, then $G$ is isomorphic to $\mathop{\mathrm{Sym}}(3)\times C_3$, $G_{n,5}\times C_r$ where $r\ne 2,5$ is a prime or $G_{n,3}\times C_{rs}$ where $r$ and $s$ are primes such that $3 < r < s$.* We have $G = PQ\times A$ where $A$ is cyclic and $\mathop{\mathrm{nps}}(G) = \mathop{\mathrm{nps}}(PQ)s(A)$. If $A\ne 1$, then $s(A)\ge 2$ and therefore $\mathop{\mathrm{nps}}(G)\le 6$. It follows from Theorems [Theorem 1](#thm:upto4){reference-type="ref" reference="thm:upto4"} and [Theorem 2](#thm:upto9){reference-type="ref" reference="thm:upto9"} that for some $n$, $G\simeq G_{n,3}$ or $G_{n,5}$. If $s(A) = 2$, then $A\simeq C_r$; if $s(A) = 3$, then $A\simeq C_{r^2}$; if $s(A) = 4$, then $A\simeq C_{rs}$, for some primes $r$ and $s$. This leaves the possibility that $A = 1$ and $G = PQ$. If $|G:N_G(P)| = 5$, then $P$ and $N_G(P)$ each have 5 conjugates and if $Q$ is not cyclic it follows from Theorem [Theorem 8](#thm:power){reference-type="ref" reference="thm:power"} that $Q$ contains at least 5 nonpower subgroups of order 5. This contradicts the assumption that $\mathop{\mathrm{nps}}(G)\le 13$ and therefore $|G:N_G(P)| = 3$. We have $[N_Q(P),P] \subseteq P\cap Q = 1$ and therefore $N_Q(P) = C_Q(P)$. Consequently $N_G(P) = P\times C_Q(P)$. Let $K$ be the kernel of the permutation action on $\mathop{\mathrm{Syl}}_2(G)$. Then $G/K\simeq\mathop{\mathrm{Sym}}(3)$ and $K\subseteq N_G(P)$, hence $K\cap Q = C_Q(P)$ and $|Q:C_Q(P))| = 3$. If $P$ is not cyclic, then $\mathop{\mathrm{nps}}(P)\ge 3$ and for each nonpower subgroup $H$ of $P$ it follows from Lemma [Lemma 11](#lemma:quo){reference-type="ref" reference="lemma:quo"} that $H$, $HQ$ and $HC_Q(P)$ are nonpower subgroups of $G$ hence $\mathop{\mathrm{nps}}(G)\ge 15$, which contradicts the assumption $\mathop{\mathrm{nps}}(G)\le 13$. Therefore $P$ is cyclic and so $N_G(P) = C_G(P)$. Let $P = \langle\mskip 2mu\relax a\mskip 2mu \rangle$. Then $P\cap K = \langle\mskip 2mu\relax a^2\mskip 2mu \rangle$ and $K = \langle\mskip 2mu\relax a^2\mskip 2mu \rangle\times C_Q(P)$. Thus $\langle\mskip 2mu\relax a^2\mskip 2mu \rangle$ is a characteristic subgroup of $K$ and therefore $\langle\mskip 2mu\relax a^2\mskip 2mu \rangle\trianglelefteq G$. It follows that $z = a^2\in Z(G)$. Let $b$ be a conjugate of $a$ such that $b\ne a$ and let $x = ab^{-1}$. Then $b^2 = z$, $b^{-1} = bz^{-2}$ and $a^{-1}xa = x^{-1}$. Since $x\in \langle\mskip 2mu\relax z\mskip 2mu \rangle Q$ and $P$ is cyclic, $y = x^{2^k}\ne 1$ belongs to $Q$ for some $k$. Furthermore $a$ has only 3 conjugates therefore $y^3 = 1$. The group $\langle\mskip 2mu\relax y\mskip 2mu \rangle P$ is generated by the conjugates of $P$ and so $\langle\mskip 2mu\relax y\mskip 2mu \rangle P\trianglelefteq G$. Moreover, for some $n$ we have $\langle\mskip 2mu\relax y\mskip 2mu \rangle P\simeq G_{n,3}$. We now have $G = \langle\mskip 2mu\relax y\mskip 2mu \rangle P\times C_Q(P)$. If $C_Q(P)$ is not cyclic, it contains an elementary abelian subgroup $E$ of order 9. Then $\langle\mskip 2mu\relax y\mskip 2mu \rangle\times H$ is elementary abelian of order 27; it contains 13 subgroups of order 3, at most one of which can be a nonpower subgroup (Theorem [Theorem 8](#thm:power){reference-type="ref" reference="thm:power"}). It follows that $C_Q(P)$ is cyclic. If the exponent of $C_Q(P)$ is $3^e$, then $Q^3$ is a power subgroup of order $3^{e-1}$ and from Corollary [Corollary 9](#cor:power){reference-type="ref" reference="cor:power"}, $Q$ contains at least $2e + 2$ nonpower subgroups of $G$. If $e\ge 2$, then $P$, $P\times C_Q(P)$ and $P\times Q^3$ each have 3 conjugates and thus $\mathop{\mathrm{nps}}(G)\ge 15$, which contradicts our assumption. Therefore $G\simeq G_{n,3}\times C_3$ and from Lemma [Lemma 19](#lemma:Gn3C3){reference-type="ref" reference="lemma:Gn3C3"} we have $n = 1$; that is, $G\simeq\mathop{\mathrm{Sym}}(3)\times C_3$. **Claim.***If $p = 3$, then $\mathop{\mathrm{nps}}(G) = 11$ and $G\simeq\mathop{\mathrm{SL}}(2,3)$.* In this case $|G : N_G(P)| = 4$ and the permutation action on $\mathop{\mathrm{Syl}}_3(G)$ defines a homomorphism $G\to\mathop{\mathrm{Sym}}(4)$ whose kernel $K$ is a proper subgroup of $N_G(P)$. Since $Q\trianglelefteq G$ we must have $G/K\simeq \mathop{\mathrm{Alt}}(4)$ and $QK/K\simeq C_2\times C_2$. The assumption $10\le\mathop{\mathrm{nps}}(G)$ implies $K \ne 1$. Let $x_1, x_2, x_3\in Q$ be elements such that $x_iK$ is an involution in $QK/K$. From Lemma [Lemma 11](#lemma:quo){reference-type="ref" reference="lemma:quo"}, $\langle\mskip 2mu\relax x_i\mskip 2mu \rangle$ and $\langle\mskip 2mu\relax x_i\mskip 2mu \rangle K$ are nonpower subgroups. The 4 conjugates of $P$, the 4 conjugates of $N_G(P)$ are nonpower subgroups and since there are at most 13 nonpower subgroups in $G$, it follows that $\langle\mskip 2mu\relax x_i\mskip 2mu \rangle = \langle\mskip 2mu\relax x_i\mskip 2mu \rangle K$. That is, $K\subseteq \langle\mskip 2mu\relax x_i\mskip 2mu \rangle$ and so $K = \langle\mskip 2mu\relax x_i^2\mskip 2mu \rangle$ for $1\le i\le 3$. We now see that $|P| = 3$, $K\subseteq C_G(P)$ and $P$ permutes the $x_i$, hence $x_1^2 = x_2^2 = x_3^2$. It follows that $Q\simeq Q_8$ and thus $G\simeq \mathop{\mathrm{SL}}(2,3)$. ◻ **Lemma 25**. *Let $P$ be a Sylow $p$-subgroup of $G$ such that $P = N_G(P)\ne G$. Then $10\le\mathop{\mathrm{nps}}(G)\le 13$ if and only if for some $n\ge 1$ one of the following holds* 1. *$\mathop{\mathrm{nps}}(G) = 10$, $p = 2$ and $G$ is isomorphic to $G^{(2)}_{n,5}$ for $n\ge 2$.* 2. *$\mathop{\mathrm{nps}}(G) = 10$, $p = 3$ and $G$ is isomorphic to $A_2 = (C_2\times C_2)\rtimes C_9$.* 3. *$\mathop{\mathrm{nps}}(G) = 11$, $p = 2$ and $G$ is isomorphic to $G_{n,11}$.* 4. *$\mathop{\mathrm{nps}}(G) = 11$, $p = 5$ and $G$ is isomorphic to $G^{(3)}_{5,n;\,11,1}$.* 5. *$\mathop{\mathrm{nps}}(G) = 12$, $p = 2$ and $G$ is isomorphic to $G_{n,9}$.* 6. *$\mathop{\mathrm{nps}}(G) = 13$, $p = 2$ and $G$ is isomorphic to $G_{n,13}$, $\mathop{\mathrm{Sym}}(3)\times C_2 = D_{12}$ or $C_3\rtimes Q_8$.* 7. *$\mathop{\mathrm{nps}}(G) = 13$, $p = 3$ and $G$ is isomorphic to $F_{n,13}$ or $A_3 = (C_2\times C_2)\rtimes C_{27}$.* *Proof.* The Sylow subgroup $P$ has $m = |G:N_G(P)|$ conjugates. Since $m\equiv 1\pmod p$ and $N_G(P)\ne G$, we have $m \ge p+1$ and the conjugates of $P$ are nonpower subgroups. The assumption $\mathop{\mathrm{nps}}(G)\le 13$ implies $p\in\{2,3,5,7,11\}$. If $p = 2$, then $m\in\{3,5,7,9,11,13\}$; if $p = 3$, then $m\in\{4,7,10,13\}$; if $p = 5$, then $m\in \{6,11\}$; if $p = 7$, then $m = 8$; if $p = 11$, then $m = 12$. Suppose that $p = 3$ and $m = 10$ or $p = 5$ and $m = 6$. Then $G$ is a group of twice odd order and therefore has a subgroup $H$ of index 2, which contains $P$. Then $|H:P| = m/2$, which is impossible. Suppose that $p = 11$ and $m = 12$. The permutation action on $\mathop{\mathrm{Syl}}_{11}(G)$ defines a homomorphism $G\to\mathop{\mathrm{Sym}}(12)$ whose kernel $K = \bigcap_{g\in G}P^g$ is properly contained in $P$. Let $\overline G = G/K$. Then $N_{\overline G}(P/K) = C_{\overline G}(P/K)$ and therefore by Burnside's normal $p$-complement theorem (see [@gorenstein:1968 Th. 7.4.3]) $P/K$ has a normal 11-complement in $\overline G$, which is a contradiction. Thus in all cases $m$ is a power of a prime $q$ and for $Q\in\mathop{\mathrm{Syl}}_q(G)$ we have $G = PQ$ and $|Q| = m$. We shall show that $Q\trianglelefteq G$. If $N_G(Q)\ne G$, it follows from Lemma [Lemma 24](#lemma:A){reference-type="ref" reference="lemma:A"} that $Q = N_G(Q)$, otherwise $P\trianglelefteq G$. If the order of $Q$ is $q$ or $q^2$, then $Q$ is abelian and it follows from Burnside's normal $p$-complement theorem that $P\trianglelefteq G$. Thus the only possibility is $p = 7$, $m = 8$ and $q = 2$. But then $|G : Q|$ must be a power of 7 and so $\mathop{\mathrm{nps}}(G)\ge 15$, contrary to our assumption. This proves that $Q\trianglelefteq G$. The permutation action on $\mathop{\mathrm{Syl}}_p(G)$ defines a homomorphism $G\to\mathop{\mathrm{Sym}}(m)$ whose kernel $K = \bigcap_{g\in G}P^g$ is properly contained in $P$. Since $Q\trianglelefteq G$ we have $[K,Q]\subseteq K\cap Q = 1$. Moreover $Q$ acts transitively on the $m$ conjugates of $P$ and therefore $K = C_P(Q)$. **Case 1.***$p = 2$ and $m\in\{3,5,7,9,11,13\}$.* We have $G = Q\rtimes P$, where $P$ is a 2-group and $Q$ is an abelian group of order $m$. We treat each value of $m$ separately. **Case 1a.***$p = 2$ and $|Q| = 3$.* by In this case $G/K\simeq\mathop{\mathrm{Sym}}(3)$ and $|P/K| = 2$. Therefore $\Phi(P)\subseteq K = C_P(Q)$ and so $\Phi(P)\trianglelefteq G$. Thus $$G/\Phi(P)\simeq\mathop{\mathrm{Sym}}(3)\times\underbrace{C_2\times\dots\times C_2}_n.$$ From Lemma [Lemma 22](#lemma:dc2){reference-type="ref" reference="lemma:dc2"} $n \le 1$. If $n = 0$, then $P$ is cyclic, $G\simeq G_{n,3}$ and $\mathop{\mathrm{nps}}(G) = 3$, which contradicts the assumption $10\le \mathop{\mathrm{nps}}(G)$. Thus $n = 1$, $G/\Phi(P)\simeq\mathop{\mathrm{Sym}}(3)\times C_2\simeq D_{12}$ and from Lemma [Lemma 22](#lemma:dc2){reference-type="ref" reference="lemma:dc2"} $\mathop{\mathrm{nps}}(G/\Phi(P)) = 13$. The only non-trivial proper power subgroup of $\mathop{\mathrm{Sym}}(3)\times C_2$ is the subgroup of order 3. Therefore, for all $H\subseteq P$ such that $|H\Phi(P)/\Phi(P)| = 2$ it follows from Lemma [Lemma 11](#lemma:quo){reference-type="ref" reference="lemma:quo"} that both $H$ and $H\Phi(P)$ are nonpower subgroups. Therefore $H = H\Phi(P)$ and hence $\Phi(P)\subseteq H$. If $P\setminus \Phi(P)$ contains and element of order 2, then $\Phi(P) = 1$ and thus $G\simeq\mathop{\mathrm{Sym}}(3)\times C_2$. Otherwise $\Phi(P) = \langle\mskip 2mu\relax x^2\mskip 2mu \rangle$ for all $x\in P\setminus\Phi(P$. Thus $P$ is nonabelian and it follows from Lemma [Lemma 11](#lemma:quo){reference-type="ref" reference="lemma:quo"} that $\mathop{\mathrm{nps}}(G)\ge 2\mathop{\mathrm{nps}}(G/Q)$. Then Theorems [Theorem 1](#thm:upto4){reference-type="ref" reference="thm:upto4"} and [Theorem 2](#thm:upto9){reference-type="ref" reference="thm:upto9"} show that the only possibility for $P\simeq G/Q$ is the quaternion group $Q_8$. Thus $G$ is the semidirect product $$C_3\rtimes Q_8 = \langle\mskip 2mu\relax x,y,b\mid x^4 = y^4 = b^3 = [y,b] = 1,\, x^2 = y^2,\,[x,y] = x^2,\,b^x = b^{-1}\mskip 2mu \rangle.$$ **Case 1b.***$p = 2$ and $|Q| = 5$.* by In this case $QK/K$ is a normal subgroup of $G/K\subseteq \mathop{\mathrm{Sym}}(5)$. Therefore $|G/K|$ is either 10 or 20. Suppose $|G/K| = 10$. If $P$ is not cyclic, then $\Phi(P)\subseteq K$ and therefore $\Phi(P)\trianglelefteq G$. We have $G/\Phi(P)\simeq D_{10}\times C_2\times\dots\times C_2$ and Lemmas [Lemma 11](#lemma:quo){reference-type="ref" reference="lemma:quo"} and [Lemma 22](#lemma:dc2){reference-type="ref" reference="lemma:dc2"} imply $\mathop{\mathrm{nps}}(G)\ge 19$ whereas we assume that $\mathop{\mathrm{nps}}(G)\le 13$, hence $P$ is cyclic. If $a$ generates $P$ and $b$ generates $Q$, then $b^a = b^{-1}$ and $G\simeq G_{n,5}$. But then $\mathop{\mathrm{nps}}(G) = 5$ and so $G$ doesn't satisfy the assumption $10\le\mathop{\mathrm{nps}}(G)$. Thus $|G/K| = 20$, $G/K\simeq G^{(2)}_{2,n;\,5,1}$ and $\mathop{\mathrm{nps}}(G/K) = 10$. Choose $a\in P$ such that $aK$ generates $P/K$. The five conjugates of $\langle\mskip 2mu\relax a\mskip 2mu \rangle$ and the five conjugates of $\langle\mskip 2mu\relax a^2\mskip 2mu \rangle$ are nonpower subgroups of $G$. Similarly, the five conjugates of $\langle\mskip 2mu\relax a\mskip 2mu \rangle K$ and the five conjugates of $\langle\mskip 2mu\relax a^2\mskip 2mu \rangle K$ are nonpower subgroups of $G$. Since $\mathop{\mathrm{nps}}(G)\le 13$ we must have $K\subseteq \langle\mskip 2mu\relax a\mskip 2mu \rangle$ and therefore $P$ is cyclic. Therefore $G\simeq G^{(2)}_{2,n;\,5,1}$. **Case 1c.***$p = 2$ and $|Q| = 7$.* by In this case $G/K\simeq D_{14}$ and $P$ has 7 conjugates. Suppose that $a\in P\setminus K$. Then $P$, $\langle\mskip 2mu\relax a\mskip 2mu \rangle$ and their conjugates are nonpower subgroups. But $\mathop{\mathrm{nps}}(G)\le 13$ and so $P = \langle\mskip 2mu\relax a\mskip 2mu \rangle$. Therefore $G\simeq G_{n,7}$ whence $\mathop{\mathrm{nps}}(G) = 7$ and there are no examples with $10\le\mathop{\mathrm{nps}}(G)$. **Case 1d.***$p = 2$ and $|Q| = 9$.* by Suppose that there is a subgroup $R$ of order 3 in $Q$ such that $R\trianglelefteq G$. Then $RP$ is not normal in $G$, otherwise the Frattini argument [@gorenstein:1968 Th. 1.3.7] implies $G = RN_G(P) = RP$, which is a contradiction. The 9 conjugates of $P$ and the 3 conjugates of $RP$ are nonpower subgroups. Therefore all other subgroups are normal. Thus $R$ is the unique subgroup of $Q$ of order 3 and hence $Q$ is cyclic. If $a\in P\setminus K$ and $b$ generates $Q$, then $b^a = b^{-1}$. If $P\ne \langle\mskip 2mu\relax a\mskip 2mu \rangle$, then $\langle\mskip 2mu\relax a\mskip 2mu \rangle\trianglelefteq G$ and so $ab = ba$, which is a contradiction. Therefore $P$ is cyclic, $\mathop{\mathrm{nps}}(G) = 12$ and $G\simeq G_{n,9}$. This leaves us with the possibility that no subgroup of order 3 is normal in $G$. This implies $Q$ is not cyclic, otherwise the subgroup of order three in $Q$ would be a normal subgroup of $G$. Consequently $Q\simeq C_3\times C_3$ and its four subgroups of order 3 are not normal in $G$. Since $\mathop{\mathrm{nps}}(G)\le 13$ and the 9 conjugates of $P$ are nonpower subgroups, it follows that the proper subgroups of $P$ are normal in $G$. As above, this implies that $P$ is cyclic. Furthermore, if $P = \langle\mskip 2mu\relax a\mskip 2mu \rangle$, then $\langle\mskip 2mu\relax a^2\mskip 2mu \rangle\trianglelefteq G$ and so $a^2\in K$. Thus if $x$ is an element of order 3 in $Q$ and $y = x^a\ne x$, then $(xy)^a = xy$ and $\langle\mskip 2mu\relax xy\mskip 2mu \rangle\trianglelefteq G$, which is a contradiction. **Case 1e.***$p = 2$ and $|Q| = 11$.* by In this case $G/K\simeq D_{22}$ and $P$ has 11 conjugates. Thus $P$ is cyclic and $G\simeq G_{n,11}$. **Case 1f.***$p = 2$ and $|Q| = 13$.* by In this case $G/K\simeq D_{26}$ and $P$ has 13 conjugates. Thus $P$ is cyclic and $G\simeq G_{n,13}$. **Case 2.***$p = 3$ and $m\in\{4,7,13\}$.* In this case $G = Q\rtimes P$, where $P$ is a 3-group and $Q$ is a $q$-group with $q\in \{2,7,13\}$. **Case 2a.***$p = 3$ and $|Q| = 4$.* by Considering the homomorphism $G\to \mathop{\mathrm{Sym}}(4)$ (with kernel $K\subset P$), we have $G/K \simeq \mathop{\mathrm{Alt}}(4)$. Thus $Q\simeq C_{2}\times C_{2}$. From $\mathop{\mathrm{nps}}(\mathop{\mathrm{Alt}}(4)) = 7$ and $10\le\mathop{\mathrm{nps}}(G)$ it follows that $K\ne 1$. Then $G$ has 7 nonpower subgroups that properly contain $K$ and 3 nonpower subgroups that are non-trivial proper subgroups of $Q$. By $\mathop{\mathrm{nps}}(G)\leq 13$, we have $\mathop{\mathrm{nps}}(G/Q)\leq 3$. This implies $P\simeq C_{3^n}$ and hence $G\simeq A_{n}$. From Lemma [Lemma 15](#lemma:list){reference-type="ref" reference="lemma:list"} we have $G\simeq A_2$ or $A_3$. **Case 2b.***$p = 3$ and $|Q| = 7$.* by From Theorems [Theorem 1](#thm:upto4){reference-type="ref" reference="thm:upto4"} and [Theorem 2](#thm:upto9){reference-type="ref" reference="thm:upto9"}, Lemma [Lemma 11](#lemma:quo){reference-type="ref" reference="lemma:quo"} and the assumption $\mathop{\mathrm{nps}}(G)\le 13$, $P$ is either cyclic or isomorphic to $C_3\times C_3$. If $P\simeq C_{3} \times C_{3}$ and $H$ is a non-central subgroup of order 3, then $H$ has 7 conjugates. This contradicts $\mathop{\mathrm{nps}}(G)\le 13$ because $P$ has three subgroups of order 3 not in the centre of $G$. Thus $P$ is cyclic. Let $a$ be a generator of $P$. Then there exists a generator $b$ of $Q$ such that $a^{-1}ba = b^2$. Therefore, for some $n\ge 1$, $G\simeq F_{n,7}$. But then $\mathop{\mathrm{nps}}(G) = 7$ whereas we assume that $10\le \mathop{\mathrm{nps}}(G)$. **Case 2c.***$p = 3$ and $|Q| = 13$.* by As in previous cases $P$ is cyclic and thus $G\simeq F_{n,13}$. **Case 3.***$p = 5$ and $|Q| = 11$.* As in previous cases $P$ is cyclic and thus $G\simeq G_{5,n;\,11,1}^{(3)}$. **Case 4.***$p = 7$ and $|Q| = 8$.* The subgroup $P$ acts transitively on the 7 subgroups of order 2 of $Q$ and on the 7 subgroups of order 4 otherwise $Q\subseteq C_G(P)$. Thus $\mathop{\mathrm{nps}}(G)\ge 14$ and so there are no examples with $p = 7$. ◻ *Proof of Theorem [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"}.* From above discussion we may suppose that $G$ is nilpotent so that $G = P\times H$ where $P$ is a non-cyclic $p$-subgroup and $H\ne 1$ is a group whose order is not divisible by $p$. If $H$ is also non-cyclic, then $H$ has at least 3 subgroups and from Lemma [Lemma 10](#lemma:prod){reference-type="ref" reference="lemma:prod"} we have $\mathop{\mathrm{nps}}(G)\ge 15$. Thus $H$ is cyclic and $\mathop{\mathrm{nps}}(P\times H) = \mathop{\mathrm{nps}}(P)s(H)$. If $H\ne 1$, then $s(H)\ge 2$ and therefore the assumption $\mathop{\mathrm{nps}}(G)\le13$ implies $\mathop{\mathrm{nps}}(P)\le 6$. Referring to Theorems [Theorem 1](#thm:upto4){reference-type="ref" reference="thm:upto4"} and [Theorem 2](#thm:upto9){reference-type="ref" reference="thm:upto9"}, $P$ is isomorphic to $C_2\times C_2$, $C_3\times C_3$, $C_2\times C_4$ or $C_5\times C_5$. If $\mathop{\mathrm{nps}}(G) = 10$, then $G\simeq C_2\times C_4\times C_p$, where $p > 3$ is a prime; if $\mathop{\mathrm{nps}}(G) = 12$, then $G\simeq Q_8\times C_{qr}$, $C_2\times C_2\times C_{qr}$, $C_3\times C_3\times C_{s^2}$ or $C_5\times C_5\times C_p$, where $p,q,r$ and $s$ are primes such that $p\ne 5$, $3 < q < r$ and $s\ne 3$. From now on we may suppose that $G$ is a non-cyclic $p$-group. Then $G/\Phi(G)$ is an elementary abelian group of order $p^d$ and every proper non-trivial subgroup of $G/\Phi(G)$ is a nonpower subgroup. Since $\mathop{\mathrm{nps}}(C_p\times C_p\times C_p)\ge \mathop{\mathrm{nps}}(C_2\times C_2\times C_2) = 14$ it follows that $d = 2$ and $G$ is generated by two elements. Thus from Lemma [Lemma 15](#lemma:list){reference-type="ref" reference="lemma:list"} [\[i:Cp\]](#i:Cp){reference-type="eqref" reference="i:Cp"} we have $G/G'\simeq C_2\times C_{2^n}$ for $1\le n\le 6$, $C_3\times C_{3^n}$ for $1\le n\le 4$, $C_5\times C_{5^n}$ for $n\in\{1,2\}$, $C_7\times C_7$, $C_{11}\times C_{11}$ or $C_4\times C_4$. If $G$ is abelian, the proof is complete. From now on we assume that $G'\ne 1$. If $G/G'= C_2\times C_2$ it follows from [@gorenstein:1968 Th. 5.4.5] that $G$ is is isomorphic to $D_{2^m}$, $S_{2^m}$ or $Q_{2^m}$ and then from Lemma [Lemma 15](#lemma:list){reference-type="ref" reference="lemma:list"} the only possibility is $S_{16}$. Since $G'\ne 1$ there exists $R\trianglelefteq G$ such that $|G'/R| = p$. We shall determine the structure of $G/R$ for for each choice of $G/G'$. If $G/G' = C_p\times C_p$ and $p$ is odd, it follows from [@gorenstein:1968 Th. 5.5.1] that $G/R$ is an extraspecial group of order $p^3$ and then from Lemma [Lemma 15](#lemma:list){reference-type="ref" reference="lemma:list"} the only possibility for $G/R$ is $M_{3,5}$. We argue as in [@zheng-etal:2023]. The group $M_{3,5}$ has a cyclic subgroup of order 25 and therefore it is metacyclic. It follows from Lemma [Lemma 14](#lemma:R){reference-type="ref" reference="lemma:R"} that $G$ is metacyclic and so $G$ has a cyclic normal subgroup that properly contains $G'$; that is, $G$ has a cyclic subgroup of index 5 and therefore, by [@gorenstein:1968 Th. 5.4.4] we have $G\simeq M_{n,5}$. From Lemma [Lemma 15](#lemma:list){reference-type="ref" reference="lemma:list"} it follows that $n = 3$ and $G\simeq M_{3,5}$. If $G/G'\simeq C_2\times C_{2^6}$, $C_3\times C_{3^4}$, $C_{11}\times C_{11}$ or $C_4\times C_4$, then $\mathop{\mathrm{nps}}(G/G')\in\{12,13\}$ and all subgroups of $G$ that do not contain $G'$ are normal. But then every subgroup of $G$ is normal because all subgroups that contain $G'$ are normal. Thus $G$ is a Hamiltonian group whence $p = 2$ and $G\simeq Q_8\times C_2\times\dots\times C_2$ (see [@huppert:1967 III, 7.12]). It follows from Lemmas [Lemma 15](#lemma:list){reference-type="ref" reference="lemma:list"} and [Lemma 20](#lemma:hamilton){reference-type="ref" reference="lemma:hamilton"} that no groups of this type satisfy $10\le\mathop{\mathrm{nps}}(G)\le 13$. Similarly, because $\mathop{\mathrm{nps}}(C_5\times C_{5^2}) = 11$, the same argument shows that there are no examples with $G/G' \simeq C_5\times C_{5^2}$. Finally we consider the cases $G/G'\simeq C_2\times C_{2^n}$ for $2\le n\le 5$ and $C_3\times C_{3^{n}}$ for $n = 2,3$. If $G/R$ has a cyclic maximal subgroup, it follows from [@gorenstein:1968 Th. 5.4.4] that $G\simeq M_{n+2,2}$ or $M_{n+2,3}$. From Lemmas [Lemma 13](#lemma:qd){reference-type="ref" reference="lemma:qd"} and [Lemma 15](#lemma:list){reference-type="ref" reference="lemma:list"}, $G$ is isomorphic to $M_{6,2}$, $M_{7,2}$, $M_{4,3}$ or $M_{5,3}$. If the exponent of $G/R$ is $2^n$ or $3^n$, then from Lemmas [Lemma 14](#lemma:R){reference-type="ref" reference="lemma:R"} and [Lemma 25](#lemma:B){reference-type="ref" reference="lemma:B"}, $G/R\simeq B^2_{2,2}$ and $\mathop{\mathrm{nps}}(G/R) = 12$. We may choose generators $a$ and $b$ for $G$ such that their images $\overline a$ and $\overline b$ in $G/R$ satisfy the relations $\overline a^4 = 1$, $\overline b^4 = 1$ and ${\overline a}^{-1}\overline b\overline a = {\overline b}^{-1}$. The nonpower subgroup $\langle\mskip 2mu\relax\overline a\mskip 2mu \rangle$ is the image in $G/R$ of both $\langle\mskip 2mu\relax a\mskip 2mu \rangle$ and $\langle\mskip 2mu\relax a\mskip 2mu \rangle R$. Since $\langle\mskip 2mu\relax a\mskip 2mu \rangle$ is not normal in $G$ we must have $R\subseteq \langle\mskip 2mu\relax a\mskip 2mu \rangle$ and therefore $R = \langle\mskip 2mu\relax a^4\mskip 2mu \rangle$ is cyclic. The group $G$ has at most one power subgroup of order 2 and since $G$ is neither cyclic nor a generalised quaternion group it has at least three subgroups of order 2. Therefore there are elements $x,y\in G\setminus R$ of order 2 such that $\langle\mskip 2mu\relax x\mskip 2mu \rangle$ and $\langle\mskip 2mu\relax y\mskip 2mu \rangle$ are nonpower subgroups. But then $\langle\mskip 2mu\relax x\mskip 2mu \rangle R$ and $\langle\mskip 2mu\relax y\mskip 2mu \rangle R$ are also nonpower subgroups and this is a contradiction. Therefore, $R = 1$, $G\simeq B^2_{2,2}$ and this completes the proof. ◻ **Remark 26**. All groups that occur in Theorems [Theorem 1](#thm:upto4){reference-type="ref" reference="thm:upto4"}, [Theorem 2](#thm:upto9){reference-type="ref" reference="thm:upto9"} and [Theorem 3](#thm:main){reference-type="ref" reference="thm:main"} have the property that for at most one prime $p$ the group has more than one Sylow $p$-subgroup. The smallest example of a group that does not have this property is the semidirect product $C_7\rtimes C_6$ (with trivial centre); it has 21 nonpower subgroups. **Acknowlegments.** The authors are grateful to the referee for her/his valuable comments and for the careful reading of this paper. 99 C. S. Anabanti, A. B. Aroh, S. B. Hart, A. R. Oodo, A question of Zhou, Shi and Duan on nonpower subgroups of finite groups, *Quaest. Math* **45**(6)(2022), 901-910. C. S. Anabanti, S. B. Hart, Groups with a given number of nonpower subgroups, *Bull. Aust. Math. Soc* **106**(2)(2022), 315-319. Y. Berkovich, *Groups of prime power order, Vol. 1*( Walter de Gruyter GmbH & Co. KG, Berlin, 2008). N. Blackburn, On prime-power groups with two generators, *Proc. Cambridge Philos. Soc.* **54**(1958), 327-337. W. Burnside, *Theory of groups of finite order*, 2nd edn.( Dover Publications, Inc., New York, 1955). D. Gorenstein, *Finite Groups*( Harper & Row, New York, 1968). P. Hall, A Contribution to the Theory of Groups of Prime-Power Order. *Proc. London Math. Soc.* **36**(2)(1934), 29-95. C. E. Hempel, Metacyclic groups, *Comm. Algebra* **28**(8)(2000), 3865-3897. B. Huppert, *Endliche Gruppen. I*( Springer-Verlag, Berlin, New York). A. Mann, The number of subgroups of metacyclic groups, *Character theory of finite groups, Contemporary Mathematics* **524** (2010), 93-95. G. A. Miller, Number of Abelian Subgroups in Every Prime Power Group, *Amer. J. Math.* **51**(1)(1929), 31-34. F. Szász, On cyclic groups, *Fund. Math.* **43**(1956), 238-240. M. Tărnăuceanu, An arithmetic method of counting the subgroups of a finite abelian group, *Bull. Math. Soc. Sci. Math. Roumanie (N.S.)*(2010), 373-386. M. Tărnăuceanu, and L. Tŏth, On the number of subgroups of a given exponent in a finite abelian group, preprint(2015), https://arxiv.org/abs/1507.00532. J. W. Zheng, W. Zhou, D.E.Taylor, Groups with few nonpower subgroups. *Bull. Aust. Math. Soc.* (2023), 1-12. W. Zhou, W. J. Shi, Z. Y. Duan, A new criterion for finite noncyclic groups. *Comm. Algebra* **34**(12)(2006), 4453-4457. [^1]: Contact zh_great\@swu.edu.cn

arxiv_math

--- abstract: | The problem of graph reconstruction has been studied in its various forms over the years. In particular, the *Reconstruction Conjecture*, proposed by Ulam and Kelly in 1942, has attracted much research attention and yet remains one of the foremost unsolved problems in graph theory. Recently, Bastide, Cook, Erickson, Groenland, Kreveld, Mannens, and Vermeulen proposed a new model of partial information, where we are given the set of connected triples $T_3$, which is the set of 3-subsets of the vertex set that induce connected subgraphs. They proved that reconstruction is unique within the class of triangle-free graphs, 2-connected outerplanar graphs, and maximal planar graphs. They also showed that almost every graph can be uniquely reconstructed from their connected triples. However, little is known about other classes of non-triangle-free graphs within which reconstruction can occur uniquely, nor do we understand what kind of graphs can be uniquely reconstructed from their connected triples without assuming anything about the classes of graphs to which they belong. The main result of this paper is a complete characterization of all graphs that can be uniquely reconstructed from their connected triples $T_3$. We also show that reconstruction from $T_3$ is unique within the class of regular planar graphs, 5-connected planar graphs, certain strongly regular graphs, and complete multi-partite graphs, whereas it is not unique for the class of $k$-connected planar graphs with $k \leq 4$, Eulerian graphs, or Hamiltonian graphs. author: - Yaxin Qi date: July 2023 title: Graph Reconstruction from Connected Triples --- # Introduction The problem of graph reconstruction that asks how much of an unknown underlying graph of interest can be uniquely determined by specific types of information has been studied in various forms over the years [@N17] [@Cor09]. In particular, the *Reconstruction Conjecture* [@SM60] [@K42], proposed by Ulam and Kelly in 1942, has attracted much research attention and yet remains one of the foremost unsolved problems in graph theory [@Sca16], [@W79] [@Hem77], [@F74]. Recently, Bastide, Cook, Erickson, Groenland, Kreveld, Mannens, and Vermeulen [@BCEGKMV23] proposed a new model of partial information, where we are given the set of connected triples $T_3$, which is the set of 3-subsets of the vertex set that induce connected subgraphs. More generally, Bastide et al. gave the following definition. **Definition 1**. [@BCEGKMV23] For $k \geq 2$ and a finite, simple, connected (labeled) graph $G$, the *set of connected $k$-sets* of $G$, which we will denote as $T_k(G)$, is defined to be $$T_k(G) \coloneqq \{X \subseteq V(G) \mid |X| = k \text{ and the subgraph of $G$ induced by $X$ is connected}\}.$$ In particular, when $k = 3$, we will call $T_3(G)$ *the set of connected triples* of $G$. **Notation 1**. We will use $N(v)$ to denote the set of neighbors of vertex $v \in V(G)$ whereas $N[v]$ denotes $N(v) \cup \{v\}$. We refer to the subgraph of $G$ induced by $X \subseteq V(G)$ as $G[X]$. In particular, we refer to $G[V(G) \setminus \{v\}]$ as $G-v$. Sometimes we drop $G$ if the graph in question is clear or if we don't want to emphasize the graph that gave rise to the set of connected triples. **Remark 1**. When $k =2$, we get back our edge set and thus obtain the entire graph. As always, we are interested in when reconstruction of the underlying (labeled) graph is unique. The observation below establishes a connection among the connected $k$-sets for different values of $k$ and shows that the set of connected triples gives the most information about a graph among all the other non-trivial (i.e. $k \neq 2$) connected $k$-sets. **Observation 1**. [@BCEGKMV23] For $k' \geq k \geq 2$, the connected $k-$sets of a graph are determined by the connected $k'$-sets. In particular, a $(k+1)$-set $X = \{x_1, ...,x_{k+1}\} \subseteq V(G)$ induces a connected subgraph of $G$ if and only if both $G[X \setminus \{y\}]$ and $G[X \setminus \{z\}]$ are connected for some $y,z \in X$. For this reason, we will focus on reconstruction of graphs from $T_3$ and make the following definition. **Definition 2**. A class $\mathcal{C}$ of graphs is $T_3$-reconstructible if all $G_1 \neq G_2 \in \mathcal{C}$ satisfies $T_3(G_1) \neq T_3(G_2)$. In other words, given $T_3(G)$ and the knowledge that the underlying graph $G$ is in $\mathcal{C}$, we are able to uniquely reconstruct $G$. **Remark 2**. Since we are dealing with labeled graphs, we consider a graph unique if no other labeled graph is identical to it. For instance, the path $v_1v_2v_3$ would not be identical to the path $v_1v_3v_2$. Oftentimes the recognition problem of whether a graph $G$ is in a class $\mathcal{C}$ of graphs cannot be solved even from $T_3$. For example, we cannot distinguish a complete graph from an "almost\" complete graph---say a complete graph with an arbitrary edge deleted---because they would both have all $3$-subsets of the vertex set as their connected triples. In the case where the order of the graph is $4$, we cannot even distinguish a complete graph from a cycle. Bastide et al. [@BCEGKMV23] proved that $T_3$-reconstructible classes of graphs include triangle-free graphs on $n \geq 5$ vertices, outerplanar 2-connected graphs on $n \geq 6$ vertices, and maximal planar graphs on $n \geq 7$ vertices. They also showed that almost every graph can be uniquely reconstructed from their connected triples without assuming additional information. However, little is known about other classes of non-triangle-free graphs that are $T_3$-reconstructible, nor do we understand what kind of graphs can be uniquely reconstructed from their connected triples without assuming anything about the classes of graphs to which they belong. The main result of this paper is a complete characterization of all graphs that can be uniquely reconstructed from their connected triples. We also show that regular planar graphs, 5-connected planar graphs, certain strongly regular graphs, and complete multi-partite graphs are reconstructible, whereas $k$-connected planar graphs for $k \leq 4$, Eulerian graphs, and Hamiltonian graphs are not. This paper is organized as follows. In Section 2 we present a few preliminary results not necessarily restricted to reconstruction from connected triples. From Section 3 and onward, we focus exclusively on answering the question of when reconstruction from connected triples is unique. In particular, in Section 3 we prove or disprove the $T_3$-reconstructibility of several families of non-triangle free graphs. In Section 4 we study graphs that can be uniquely reconstructed from their set of connected triples without assuming any additional information, establishing a complete characterization from scratch. Finally in Section 5 we give some future directions. # Preliminary Results Generally, two distinct labeled graphs can have the same connected triples. So if we are only given $T_3$ and the order of a graph $|V(G)| = n$, we cannot always expect to uniquely reconstruct the underlying graph even from its connected triples. However, if we are given the order of $G$, its vertex connectivity $\kappa(G)$ can be uniquely reconstructed from $T_k(G)$ for most values of $k$ that are "not too large.\" Although it is easier to prove this using Observation 1.4, we prove it in a way that will help set up the proof of the next result regarding complete multipartite graphs. **Remark 3**. When we say that "$T_k(G)$ uniquely determines $\kappa(G)$\" in the context of the following theorem, what we really mean, of course, is that if $|V(G)| = n$ and $2 \leq k \leq n - \kappa(G)$ are fixed, then for all graphs $G'$ satisfying $|V(G')| = n$, we have $T_k(G) = T_k(G')$ implies $\kappa(G') = \kappa(G)$. **Theorem 1**. *If the order of a graph $|V(G)| = n$ is known, then for all $2 \leq k \leq n - \kappa(G)$, where $\kappa(G)$ denotes the vertex connectivity of $G$, we have that $T_k(G)$ uniquely determines $\kappa(G)$.* *Proof.* We define a subset $S \subseteq T_k(G)$ to be a *gluing set of $G$* if for all $s_1 \in S$, there exists $s_2 \neq s_1$ such that $s_1 \cap s_2 \neq \emptyset$. We call $g(S) \coloneqq \bigcup_{s \in S}s$ a *glued set of $G$*. It is clear that a graph $H$ with order $m$ is connected if and only if there exists a glued set $g(S)$ of $H$ with cardinality $m$ for some gluing set $S \subseteq T_k(H)$. Indeed, if $H$ were disconnected, then each glued set would be contained in one of the components of $H$. On the other hand, if $H$ were connected, take a maximal glued set, say $g(S)$, and a vertex $v \in g(S)$. For all vertices $u \neq v \in V(G)$, there exists a path from $u$ to $v$. If the path has length at least $k$, then it can be chopped up into overlapping paths on $k$ vertices whose vertex sets all belong to $S$. Otherwise, $u$ and $v$ would exist in a connected subgraph of $G$ with $k$ vertices. Since $S$ gives rise to a maximal glued set, $u$ would be in some set in $S$ and thus in $g(S)$. To uniquely determine $\kappa(G)$, we first drop one vertex at a time and check if all of $v_i \in V(G)$ satisfies that $G-v_i$ has a glued set with order $m-1$. If the answer is yes, we move on to removing two vertices at a time and checking whether all of the corresponding graphs have a glued set with order $m-2$. It is worth noting that when we drop a set of vertices and check for glued sets of the appropriate size, we temporarily delete elements of $T_k(G)$ that contain any of those vertices. We can uniquely determine $\kappa(G)$ by continuing this process and finding the smallest $k$ such that removing $k$ vertices at a time gives a corresponding graph with no glued set of cardinality $n-k$. ◻ **Theorem 2**. *For all complete $n$-partite graphs $K_{r_1,...,r_n}$, where $n \geq 3$ and $r_i \geq 3$ for all $i$, $K_{r_1,...,r_n}$ can be uniquely reconstructed from its connected $k$-sets $T_k$ for all $k \leq \min \{r_1,...,r_n\}$ if we know that it is a complete multi-partite graph.* *Proof.* Observe that a $k$-subset of $V(K_{r_1,...,r_n})$ does not induce a connected subgraph if and only if it is contained within one partite of $K_{r_1,...,r_n}$. Take the complement $\overline{T_k}$ of $T_k$, where $$\overline{T_k(G)} \coloneqq \{X \subseteq V(G) \mid |X| = k \text{ and } G[X] \text{ is not connected}\},$$ and look at the glued sets of $\overline{T_k}$. They form $n$ chains. Taking the upper bounds of the chains give the $n$ partites. ◻ From now on, we will focus on reconstruction of graphs from $T_3$ exclusively. We prove one more preliminary result in this section. **Definition 3**. A strongly regular graph with parameters $G = (v,k, \lambda, u)$ is a $k$-regular graph on $v$ vertices, where every two adjacent vertices share exactly $\lambda$ common neighbors and any two distinct non-adjacent vertices share exactly $u$ common neighbors. **Theorem 3**. *For all strongly regular graphs $G = (v,k, \lambda, u)$ that satisfy $2k-\lambda \neq u +2$ and $v \neq 2k+1$, if we know the value of $k$ and that $G$ is strongly regular, then $G$ can be uniquely reconstructed from $T_3(G)$.* *Proof.* For all $v_1 \neq v_2 \in V(G)$, if $v_1v_2 \in E(G)$, then the number of connected triples in $T_3(G)$ that contain both $v_1$ and $v_2$ is $2(k-1)-\lambda$, which is different from $u$, the number of connected triples in $T_3(G)$ that contain two nonadjacent vertices. So pairs of vertices that appear in the same number of connected triples are either all adjacent or all non-adjacent, resulting in only two possible edge assignments. The only difficulty is that we don't know which of the two numbers is $u$ and which is $2(k-1)-\lambda$. Nevertheless we can temporarily assign $uv$ to be an edge for all $u \neq v \in V(G)$ that are contained in one number of connected triples and all the other pairs that are contained in the other number of connected triples as non-edges. Then pick a vertex $v \in V(G)$, and check if it is contained in exactly $k$ of our assigned edges. If so, our edge assignment was correct. Otherwise, the only other possible edge assignment obtained by flipping all our current edges and non-edges is correct. ◻ # More $T_3$-Reconstructible Classes of Graphs In this section, we prove two results regarding the $T_3$-reconstructibility of the class of $k$-connected planar graphs and the class of regular planar graphs. We will do so by finding information from $T_3$ that almost gives the set of neighbors and then identifying the fake neighbors. We also present a construction that shows that the class of Hamiltonian graphs and the class of Eulerian graphs are not $T_3$-reconstructible. **Definition 4**. The *set of roughly neighbor sets $\mathcal{N}(v)$* of a vertex $v \in V(G)$ is a set whose elements, which we will refer to as $T_3$-neighborhoods, are exactly the maximal subsets of $V(G)$ that satisfy the property that for all $v_1 \neq v_2$ in a $T_3$-neighborhood ${N_v}$, we have $\{v_1, v_2, v\} \in T_3(G)$. Clearly, for every vertex $v \in V(G)$, we can always uniquely determine $\mathcal{N}(v)$ from $T_3(G)$ by, in the worst case scenario, checking each subset of $V(G)$ to see if every pair of its elements form a connected triple with $v$ and then collecting all such distinct sets with maximal cardinality to get $\mathcal{N}(v)$. **Observation 2**. If $\mathcal{N}(v)$ has exactly one element, then either it is exactly the set of neighbors of $v$, which we denote as $N(v)$, or it is $N(v) \cup \{w\}$, where $w \notin N[v]$ is the unique vertex that is adjacent to everything in $N(v)$. If $\mathcal{N}(v)$ has multiple elements, then either they are all of the form of $N(v) \cup \{w_i\}$, for some $w_i \notin N[v]$ that is adjacent to everything in $N(v)$, or there is one element that is exactly $N(v)$ and the rest of the elements are of the form $\left(N(v)\setminus \{v_i\} \right) \cup \{w_j\}$ for some $v_i \in N(v)$ and $w_j$ that is adjacent to everything in $N(v)\setminus \{v_i\}$. Now we prove that the class of 5-connected planar graphs is $T_3$-reconstructible, whereas the class of $k$-connected planar graphs is not for $k \leq 4$. We start by proving two lemmas. **Lemma 1**. *No two adjacent vertices in a 5-connected planar graph $G$ can have more than three vertices in common.* *Proof.* Suppose, for contradiction, that there exist two adjacent vertices $v \neq v_i \in V(G)$ with distinct common neighbors $v_1, v_2, v_3,$ and $v_4$. Since $G$ is 5-connected, there is a path from $v_1$ to $v_2$ that does not contain $v_3, v_4, v$, or $v_i$, which we denote as $P_{v_1v_2}$. Similarly there exist $P_{v_2v_4}$ and $P_{v_1v_4}$ that does not contain any vertex from $\{v_1, v_3, v_i, v\}$ and $\{v_2, v_3, v, v_i\}$, respectively. If any one of the three paths, say $P_{v_1v_2}$, is internally vertex-disjoint with the other two, then we can contract all the edges except the one incident to $v_2$ in $P_{v_2v_4}$ in a way that effectively glues all of its internal vertices to $v_4$, and similarly we glue all the internal vertices of $P_{v_1v_4}$ to $v_4$. After contracting $P_{v_1v_2}$ to an edge between $v_1$ and $v_2$, we notice that $G[\{v, v_i, v_2, v_4, v_1\}]$ contains a $K_5$ minor, which contradicts the premise that $G$ is planar. Otherwise, without loss of generality, assume both $P_{v_1v_2}$ and $P_{v_2v_4}$ intersect $P_{v_1v_4}$ at some internal vertex, respectively. Let $a$ be the vertex in $P_{v_1v_2} \cap P_{v_1v_4}$ that is closet to $v_2$ in $P_{v_1v_2}$. Contract all edges in $P_{v_1v_4}[v_1-a]$, which is the part of $P_{v_1v_4}$ that starts at $v_1$ and ends at $a$, so that $P_{v_1v_4}[v_1-a]$ becomes just the edge $v_1a$. Then, contract $P_{v_1v_4}[a-v_4]$ to $av_4$ and $P_{v_1v_2}[a-v_2]$ to $av_2$. Call the resulting graph $\Tilde{G}$. Now notice that $\Tilde{G}[\{v,a,v_i, v_1, v_2, v_4\}]$ contains a $K_{3,3}$ as its subgraph. This shows that the original graph $G$ contains a $K_{3,3}$ minor and thus can not be planar. This is a contradiction. ◻ **Lemma 2**. *For a 5-connected planar graph $G$ and a vertex $v \in V(G)$, a $T_3$-neighborhood $N_v$ of $v$ is exactly the set of neighbors $N(v)$ if and only if $N_v$ does not contain a vertex $w$ with a $T_3$-neighborhood $N_w$ that contains $N_v \setminus \{w\}$. In particular, no $T_3$-neighborhood of a neighbor $v_i$ of $v$ can contain $N(v) \setminus \{v_i\}$.* *Proof.* It follows from Observation 3.2 that if an element ${N_v}$ in $\mathcal{N}(v)$ is not $N(v)$, then it must contain an element $w$ that is adjacent to everything in ${N_v} \setminus \{w\}$. Since $N(w)$ will always be contained in some element, say ${N_w}$, of $\mathcal{N}(w)$, we have that ${N_w} \in \mathcal{N}(w)$ contains ${N_v} \setminus \{w\}$. For the other direction, suppose, for contradiction, that ${N_v} = N(v)$ contains an element $v_i$ such that there exists ${N_{v_i}} \in \mathcal{N}(v_i)$ that contains $N(v) \setminus \{v_i\}$. Then either $v_i$ is adjacent to everything in $N(v) \setminus \{v_i\}$ or it is adjacent to everything except say $v_h$, in which case $v_h$ would need to be adjacent to at least $\operatorname{deg}(v_i)-1$ many vertices in $N(v_i)$ by Observation 3.2. Since $G$ is 5-connected, every vertex including $v$ would have degree at least 5. Hence the first scenario is impossible because we know by Lemma 3.3 that $v$ and $v_i$ cannot be adjacent and share more than three common neighbors in a planar graph. So $v_i$ is adjacent to $N(v) \setminus \{v_i, v_l\}$ for some $v_l \neq v_i \in N(v)$. If $|N(v)| \geq 6$, then $v_i$ and $v_l$ would share at least four common neighbors within $N(v)$, which, together with $v, v_i$, and $v_l$ would induce a subgraph that contains a $K_{3,3}$, a contradiction. Finally supoose $|N(v)| = 5$ and say $N(v) = \{v_i, v_1, v_2, v_3, v_l\}$. We know $v_l$ cannot be adjacent to all of $v_1, v_2,$ and $v_3$ because otherwise $G[\{v_i, v_l, v, v_1, v_2, v_3\}]$ would contain a $K_{3,3}$, which is a contradiction. Yet the fact that $v_l$ is adjacent to at least $\operatorname{deg}(v_i)-1$ many vertices in $N(v_i)$ implies that $v_l$ is adjacent to two of $v_1, v_2, v_3$, say they are $v_2$ and $v_3$, and shares at least one common neighbor $x$ with $v_i$ outside of $N(v)$. But since $G$ is 5-connected, we know there is a path between $v_l$ and $v_1$ that does not go through any of the vertices in $\{v, v_i, v_2, v_3\}$. Therefore, after we contract this path to a single edge between $v_l$ and $v_1$, the vertex set $\{v,v_i, v_l, v_1, v_2 ,v_3\}$ will induce a subgraph in the resulting graph that contains a $K_{3,3}$. This means that $G$ has a $K_{3,3}$ minor, which is a contradiction. ◻ **Theorem 4**. *For all $k \leq 4$, the class of $k$-connected planar graphs is not $T_3$-reconstructible, whereas the class of $5$-connected planar graphs is $T_3$-reconstructible and the class of $k$-connected planar graphs does not exist for $k \geq 6$.* *Proof.* First, it is clear that if a graph is $k$-connected, then every vertex must have degree at least $k$, otherwise removing the neighbors of a vertex that has degree less than $k$ will disconnect the graph. So a $k$-connected planar graph, where $k \geq 6$, requires every vertex to have degree at least 6, which would give at least $3n$ edges, which is strictly greater than $3n-6$, the maximum number of edges a planar graph on $n$ vertices can have. This is a contradiction. Thus $k$-connected planar graphs do not exist for $k \geq 6$. As for why the class of $k$-connected planar graphs is not $T_3$-reconstructible for all $k \leq 4$, we present an arbitrarily large construction of two 4-connected planar graphs, obtained by switching the labels of $n-1$ and $n-2$, that are not identical as labeled graphs but share the same set of connected triples in Figure [1](#fig:construction){reference-type="ref" reference="fig:construction"}. We next show that the class of 5-connected planar graphs is $T_3$-reconstructible by showing that we can uniquely determine the neighbors of every vertex. Fix a vertex $v \in V(G)$ and look at $\mathcal{N}(v)$. Regardless of which of the four cases mentioned in Observation 3.2 we have, we would be able to identify $N(v)$ if we can identify the existence of $w$ or $w_i$ and precisely which vertex it is. With Lemma 3.4, we can pick out $N(v)$ if it is contained in $\mathcal{N}(v)$. Otherwise, every element of $\mathcal{N}(v)$ must be of the form $N(v) \cup \{w_i\}$, where $N(v) \subseteq N(w_i)$. We can pick a random $N_v = N(v) \cup \{w_i\}$ from $\mathcal{N}(v)$. We know from Lemma 3.4 that $w_i$ is the only vertex in $N_v$ with a $T_3$-neighborhood that contains $N(v)$, since for any $s \in N(v)$, a $T_3$-neighborhood $N_s$ cannot even contain $N(v) \setminus \{s\}$. Thus we can identify $w_i$ as the vertex with a $T_3$-neighborhood that contains all but one element of $N_v$, while all $T_3$-neighborhoods of neighbors of $v$ are necessarily missing at least two elements of $N_v$. Throwing out the $w_i$ from $N_v = N(v) \cup \{w_i\}$ gives us $N(v)$. ◻ {#fig:construction} **Remark 4**. Since the two arbitrarily large graphs in Figure [1](#fig:construction){reference-type="ref" reference="fig:construction"} obtained by switching the labels of $n-1$ and $n-2$ are both Hamiltonian and Eulerian, this construction also shows that the class of Hamiltonian graphs and the class of Eulerian graphs are not $T_3$-reconstructible. Next, we show that the class of regular planar graphs on $n \geq 7$ vertices is $T_3$-reconstructible. To do this, we show that we can first recognize the degree $d$ and then show that we can uniquely reconstruct the underlying graph $G$ knowing that it is a $d$-regular planar graph. Note that $d$ can only range from two to five since we are dealing with connected planar graphs. We break down the proof of the theorem into a series of lemmas and their proofs below. **Lemma 3**. *If $G$ is a 3-regular planar graph, then there exists a vertex $v \in V(G)$ such that every element in $\mathcal{N}(v)$ has order 3.* *Proof.* Suppose, for contradiction, that this is not the case. Then for every $v \in V(G)$, there exists $w \neq v \in V(G)$ such that $N(v) = \{v_1, v_2, v_3\} = N(w)$. Fix $v,w$ that satisfy the above. If $\{v_1, v_2, v_3\}$ forms an independent set, then $v_1$ would have a neighbor, say $x_1$, outside of $N(v) \cup \{v,w\}$. Then only $v_2$ or $v_3$ can have the same exact set of neighbors as $v_1$. Without loss of generality suppose it's $v_3$. If $v_2$ is also a neighbor of $x_1$, then $G[\{v,w, x_1, v_1, v_2, v_3\}]$ would contain a $K_{3,3}$, a contradiction. Thus $v_2$ has a neighbor $x_2 \neq x_1$ outside of $N(v) \cup \{v,w\}$. But neither $v_1$ nor $v_3$ can be adjacent to $x_2$ because they both have degree three. So there does not exist $u \neq v_2 \in V(G)$ with $N(v_2) = N(u)$, which is a contradiction. If $\{v_1, v_2, v_3\}$ does not form an independent set, however, then there would exist at least one edge, say $v_1v_2$, among $\{v_1, v_2, v_3\}$. In this case, there won't exist any $u \neq v_3 \in V(G)$ that satisfies $N(u) = N(v_3)$ because the only possible candidates are $v_1$ and $v_2$, both of which already have three neighbors. This is a contradiction. ◻ **Lemma 4**. *The class of 3-regular planar graphs on greater or equal to 5 vertices is $T_3$-reconstructible.* *Proof.* If $\mathcal{N}(v)$ has only one element $N_v$, then $N(v) = N_v$ if $|N_v| = 3$, otherwise $N_v = N(v) \cup \{w\}$. We will show that we can recognize $w$ in the latter case by showing it is the only vertex $s$ in $N_v$ such that there exists ${N_s}^j \in \mathcal{N}(s)$, where ${N_s}^j = (N_v \cup \{v\}) \setminus \{s\}$. Let $N(v) = \{v_1, v_2, v_3\}$. Suppose, for contradiction, that say $v_1 \in N(v)$ is also such that there exists ${N_{v_1}}^j \in \mathcal{N}(v_1)$, where ${N_{v_1}}^j = (N_v \cup \{v\}) \setminus \{v_1\}$. Then $v_1$ would need to be adjacent to one of $v_2$ and $v_3$ and share all its neighbors with the other, which would result in one of $v_2$ and $v_3$ having degree four, a contradiction. On the other hand, if $\mathcal{N}(v)$ has multiple elements, it cannot be the case that they are each of the form $N(v) \cup\{w_i\}$, where $N(w_i) = N(v)$, otherwise $G[N(v) \cup \{v, w_i, w_j\}]$ would contain a $K_{3,3}$, which is a contradiction. By Observation 3.2, we know that $N(v)$ is an element in $\mathcal{N}(v)$ and all the other elements would be of the form $S_j \coloneqq (N(v) \setminus \{v_i\}) \cup \{w_j\}$, where $w_j \notin N(v)$ and $N(v) \setminus \{v_i\} \subset N(w_j)$. If there is exactly one element $N_v \in \mathcal{N}(v)$ such that $|N_v \cap S_j| = 2$ for all $S_j \neq N_v \in \mathcal{N}(v)$, then we have $N(v) = N_v$. Otherwise, we have $\bigcap_{S_j \in \mathcal{N}(v)} S_j = \{v_1, v_2\} \subset N(v)$ and we just need to distinguish $v_3 \in N(v) \setminus \{v_1, v_2\}$ from all the $w_j$. Note that there can be at most two such $w_j$, say $w_1$ and $w_2$, because of the 3-regularity condition, and that $v_3$ is only contained in the element $N(v) \in \mathcal{N}(v)$. Suppose there are three elements in $\mathcal{N}(v)$ and thus both such $w_1$ and $w_2$ exist. Then $v_3$ cannot be adjacent to $v_1$ or $v_2$, and thus $\{v_1, v_2\}$ is a subset of some element in $\mathcal{N}(w_1)$ and $\mathcal{N}(w_2)$ but not of $\mathcal{N}(v_3)$. Now, suppose $\mathcal{N}(v)$ contains two elements and there exists only one of such $w_1$ and $w_2$ mentioned above. Without loss of generality, let it be $w_1$. If there exists $x \in V(G)$ with $N(x) = N(w_1)$, which happens if and only if elements in $\mathcal{N}(w_1)$ would all have cardinality four, then $v$ would not be contained in any element of $\mathcal{N}(w_1)$. Thus in this case we can distinguish $w_1$ from the neighbors of $v$. If no such $x$ exists, then the cardinality of elements in $\mathcal{N}(w_1)$ is three and at least two of them have intersection $\{v_1, v_2\}$. We can easily check that if the same were true for $v_3$, then $v_3$ would have to be adjacent to both $v_1$ and $v_2$. Then note that $\mathcal{N}(w_1) = \{ N(w_1), \{v_1, v_2, v\}, \{v_1, v_2, v_3\} \}$ would have three elements whereas $\mathcal{N}(v_3) = \{ N(v_3) = \{v,v_1,v_2\}, \{v_1, v_2, w_1\} \}$ would only have two. Thus, we can differentiate between $v_3$ and $w_1$. ◻ **Lemma 5**. *If $G$ is a 4-regular planar graph on $n \geq 7$ vertices, then there exists vertex $v \in V(G)$ such that every element in $\mathcal{N}(v)$ has order 4.* *Proof.* Suppose, for contradiction, that this is not the case. Then for every $v \in V(G)$, there exists $w \neq v \in V(G)$ such that $N(v) = \{v_1, v_2, v_3, v_4\} = N(w)$. Fix a $v \in V(G)$ and let $w \neq v \in V(G)$ be such that $N(w) = N(v) = \{v_1, v_2, v_3, v_4\}$. If there exists $v_1 \in N(v)$ that is adjacent to two other vertices in $N(v)$, say $v_2$ and $v_3$, then $v_4$ is the only vertex that can play the role of satisfying $N(v_4) = N(v_1)$. Hence $G[\{v,w, v_1, v_2, v_3, v_4\}]$ would give the unique (up to isomorphism) 4-regular planar graph on 6 vertices and thus forms a component of $G$. But $G$ has $n \geq 7$ vertices, which means that it will have more than one component, a contradiction. Otherwise, suppose there exists $v_1 \in N(v)$ that is adjacent to one other vertex in $N(v)$, say $v_2$. One of $v_3$ and $v_4$ has to have the same set of neighbors as $v_1$. Without loss of generality, assume it's $v_3$. But now all vertices in $\{v, w, v_1, v_2, v_3\}$ have four neighbors already. Then clearly, no vertex other than $v_4$ can have the same set of neighbors as does $v_4$, which is a contradiction. If not, then no two vertices in $N(v)$ can be adjacent to each other. Then $N(v)$ can be partitioned into pairs that have the same set of neighbors. Suppose $N(v_1) = \{v, w, x_1, x_2\} = N(v_4)$ and $N(v_2) = \{v, w, y_1, y_2\} = N(v_3)$. Observe that we would necessarily have $N(x_1) = N(x_2)$ and $N(y_1) = N(y_2)$. If $y_1$ and $y_2$ are adjacent to $x_1$ and $x_2$, then, after contracting the path $v_4vv_3$ to $v_4v_3$, $G$ contains a $K_{3,3}$ minor with the vertex set being $\{y_1, y_2, v_3, v_4, x_1, x_2\}$, a contradiction. So we need distinct new vertices $y_3$ and $y_4$ to be the other two neighbors of both $y_1$ and $y_2$. If $x_1$ and $x_2$ are adjacent to $y_3$ and $y_4$, then contracting the path $v_4vv_2y_2$ to $v_4y_2$ gives that $G$ contains a $K_{3,3}$ minor with the vertex set being $\{y_3, y_4, v_4, y_2, x_1, x_2\}$, which is also a contradiction. Hence we also need new vertices, say $x_3$ and $x_4$, to be the last two neighbors of both $x_1$ and $x_2$. Applying similar arguments repeatedly shows that we will always need more new vertices to be the last two neighbors of $x_i, x_{i+1}$ and $y_j, y_{j+1}$, respectively. This gives a contradiction because our graph is finite. ◻ **Lemma 6**. *The class of 4-regular planar graphs on greater or equal to 7 vertices is $T_3$-reconstructible.* *Proof.* Fix a vertex $v \in V(G)$. Suppose $\mathcal{N}(v)$ has only one element $N_v$. Then $N(v) = N_v$ if $|N_v| = 4$. Otherwise, $N_v = N(v) \cup \{w\}$ for the unique $w \notin N(v)$ with $N(v) = N(w)$. Note that $w$ satisfies that $(N_v \cup \{v\}) \setminus \{w\} \in \mathcal{N}(w)$. If there exists $v_1 \in N(v)$ that also satisfies this, then $v_1$ would be adjacent to two other vertices in $N(v)$. As we have shown in the proof of Lemma 3.9, this leads to $\{N(v) \cup \{v, w\} \}$ forming a connected component of size 6, which implies that $G$ with $n \geq 7$ vertices would not be connected, a contradiction. Hence we can recognize $w$ and thus determine $N(v)$ in this case. On the other hand, suppose $\mathcal{N}(v)$ has multiple elements. Then $N(v)$ is an element in $\mathcal{N}(v)$ and all the other elements would be of the form $S_j \coloneqq (N(v) \setminus \{v_i\}) \cup \{w_j\}$, where $w_j \notin N(v)$ and $N(v) \setminus \{v_i\} \subset N(w_j)$. The other case for multiple elements recognized in Observation 3.2 would give a $K_{3,3}$ minor, a contradiction. If $\mathcal{N}(v)$ has $\geq 3$ elements, then we can recognize $N(v)$ as the unique element that has an intersection of size three with all other elements, respectively. Now suppose $\mathcal{N}(v)$ has exactly two elements: $N(v) = \{v_1, v_2, v_3, v_4\}$ and $S \coloneqq (N(v) \setminus \{v_4\}) \cup \{w\}$. To recognize $N(v)$ is equivalent to recognizing the existence of $w$. We observe that $\mathcal{N}(w)$ has multiple elements of size four, contains $(S \cup \{v\}) \setminus \{w\}$ as an element, where $S \in \mathcal{N}(v)$, and $v$ is contained in exactly one element of $\mathcal{N}(w)$. Suppose, for contradiction, that the same holds for some $v_1 \in N(v)$. Then either $v_1$ is adjacent to all other vertices in $N(v)$ or all other vertices except for some $v_4 \in N(v)$. The former cannot be possible because for $v$ to exist in exactly one element of $\mathcal{N}(v_1)$ that has multiple elements, there has to exist a vertex $x \neq v_1, v$ that is adjacent to everything in $N(v) \setminus \{v_1\}$, which would result in a $K_{3,3}$ minor, a contradiction. As for the latter, $v_4$ needs to be adjacent to at least three vertices in $N(v_1)$ to be included in some element of $\mathcal{N}(v_1)$. If $v_4$ is adjacent to all four vertices in $N(v_1)$, then elements of $\mathcal{N}(v_1)$ would have size five and not four, which is a contradiction. So $v_4$ is adjacent to three vertices in $N(v_1)$. But in this case $v$ would be contained in both $(N(v_1) \cap N(v_4)) \cup \{v_4\}$ and $N(v_1)$ in $\mathcal{N}(v_1)$, which is also a contradiction. ◻ **Lemma 7**. *The class of 5-regular planar graphs on greater or equal to 5 vertices is $T_3$-reconstructible.* *Proof.* Observe that if $\mathcal{N}(v)$ has multiple elements, then it has to have exactly two elements: $S_1 \coloneqq N(v)$ and $S_2 \coloneqq (N(v) \setminus \{v_1\}) \cup \{w\}$ for some $v_1 \in N(v)$ and $w \notin N(v)$ with $(N(v) \setminus \{v_1\}) \subset N(w)$, otherwise our planar graph would contain a $K_{3,3}$ minor, a contradiction. It's easy to check that $\mathcal{N}(w)$ has multiple elements, including $(S \cup \{v\}) \setminus \{w\}$ for some $S \in \mathcal{N}(v)$ that contains $w$, and that $v$ would belong to exactly one element in $\mathcal{N}(w)$. However, the same cannot be all true for any $v_i \in N(v)$. Thus we will be able to determine the existence of $w$ in elements of $\mathcal{N}(v)$ and pick out $N(v)$. Now suppose $\mathcal{N}(v)$ has only one element $N_v$. If $N_v$ has size 5, then we have $N(v) = N_v$. Else, $N_v = N(v) \cup \{w\}$, where $N(w) = N(v)$. We want to be able to recognize $w$ in $N_v$. Clearly, we have that there exists $N_w \in \mathcal{N}(w)$ such that $N_w = (N_v \cup \{v\}) \setminus \{w\}$. Suppose, for contradiction, that there exists $v_i \in N(v)$ with $(N_v \cup \{v\}) \setminus \{v_i\} \in \mathcal{N}(v_i)$. Then $v_i$ needs to be adjacent to three vertices within $N(v) \setminus \{v_i\}$. But this would mean that $G$ contains a $K_{3,3}$ minor, which is a contradiction. ◻ Piecing together the lemmas above, we obtain a proof of Theorem 3.12 below. **Theorem 5**. *The class of regular planar graphs on greater or equal to 7 vertices is $T_3$-reconstructible.* *Proof.* If $T_3(G) = \{ \{v_{i}, v_{i+1}, v_{i+2} \} \mid i \in \mathbb{Z}/n\mathbb{Z} \}$, then $d =2$ and in particular $G$ is the cycle $v_0v_1v_2...v_{n-2}v_{n-1}v_0$. This is proved in Observation 4.3. Otherwise if there exists vertex $v \in V(G)$ such that every element in $\mathcal{N}(v)$ has order three, then we know $d = 3$. And by Lemma 3.7, we will always be able to recognize when $d = 3$ this way. Otherwise if there exists vertex $v \in V(G)$ such that every element in $\mathcal{N}(v)$ has order four, then we know $d = 4$. Lemma 3.9 shows that we will always be able to recognize when $d = 4$ this way. Otherwise, $d = 5$. Lemma 3.11, together with Lemma 3.10 and Lemma 3.8, concludes the proof of Theorem 3.12. ◻ # Strongly $T_3$-Reconstructible Graphs In this section, we seek to directly answer the question of when reconstruction from connected triples is unique without allowing the additional information of to which classes of graphs our underlying graph of interest belongs. While Bastide et al. have shown that almost every graph can be uniquely reconstructed in this sense, essentially nothing is known about what kind of graphs these are and what it would take to satisfy this unique reconstruction condition. In this section, we explicitly define what we mean by being able to be uniquely reconstructed from $T_3$, give some examples of such graphs, provide a framework for checking if some graph can be uniquely reconstructed from $T_3$, and prove a series of lemmas---including a characterization of triangle-free graphs that can be uniquely reconstructed---that build up to a complete characterization of graphs that can be uniquely reconstructed from $T_3$. **Definition 5**. A finite, simple, connected labeled graph $G$ is *strongly $T_3$-recon\ structible* if for any finite, simple, connected labeled graph $H$ that has the same set of connected triples as $G$, we have that $H$ is identical to $G$. **Question 1**. What kind of graphs are strongly $T_3$-reconstructible? The simplest examples are $l$-cycles and $l$-wheels for all $l \geq 5$, the proofs of which are very straightforward and can be found in [@BCEGKMV23 Observation 4,5]. We will include them here for completeness. **Observation 3**. All cycles on $n \geq 5$ vertices are strongly $T_3$-reconstructible. *Proof.* [@BCEGKMV23 Observation 4] For any cycle $C_n$, we have $T_3(C_n) = \{ \{v_{i}, v_{i+1}, v_{i+2} \} \mid i \in \mathbb{Z}/n\mathbb{Z} \}$. Let $G$ be a graph with $T_3(G) = T_3(C_n)$. If there exists $i \in \mathbb{Z}/n\mathbb{Z}$ such that $v_{i}v_{i+1}$ is not an edge of $G$, then the fact that $\{v_{i-1}, v_{i}, v_{i+1}\}$ and $\{ v_{i}, v_{i+1}, v_{i+2} \}$ are in $T_3(C_n)$ implies that $v_{i-1}v_{i}, v_{i-1}v_{i+1}, v_{i}v_{i+2},$ and $v_{i+1}v_{i+2}$ are edges of $G$. Since $n \geq 5$, there exists $v_{i+3} \notin \{ v_{i-1}, v_{i}, v_{i+1}, v_{i+2}\}$ such that $v_{i+3}v_{i+2} \in E(G)$. This means that $\{v_{i}, v_{i+2}, v_{i+3}\} \in T_3(G) = T_3(C_n)$, which is a contradiction. On the other hand, suppose there exist $i,j \in \mathbb{Z}/n\mathbb{Z}$, where $j \neq 1$ or $n-1$, such that $v_{i}v_{i+j} \in E(G)$. Then both $\{v_{i}, v_{i+1}, v_{i+j}\}$ and $\{v_{i-1}, v_{i}, v_{i+j}\}$ are in $T_3(G) = T_3(C_n)$. But we would need $2 = j = n-2$ for this to happen, which is impossible for all $n \geq 5$. ◻ **Observation 4**. All $l$-wheels are strongly $T_3$-reconstructible for $l \geq 5$. *Proof.* [@BCEGKMV23 Observation 5] A graph $G$ is a $l$-wheel if and only if there exists a vertex $v \in V(G)$ such that $G \setminus v$ is a $l$-cycle and $v$ appears in $\binom{l}{2}$ triples. ◻ **Observation 5**. All paths on $n \geq 5$ vertices are strongly $T_3$-reconstructible. *Proof.* This is a corollary of Theorem 3.8, which we prove later. ◻ We provide some examples of small graphs that are strongly $T_3$-reconstructible in Figure [2](#fig:small){reference-type="ref" reference="fig:small"}. Note that if a graph $G$ contains a strongly $T_3$-reconstructible graph $H$ as an induced subgraph, then we can uniquely reconstruct the embedding of $H$ in $G$ and go from there. {#fig:small} **Definition 6**. An edge $uv \in E(G)$ is *necessary* if there does not exist any graph $H$ with $uv \notin E(H)$ and $T_3(H) = T_3(G)$. We state the following observation in the form of a lemma: **Lemma 8**. *$G$ is strongly $T_3$-reconstructible if and only if it satisfies the following two requirements: 1) Every edge in $G$ is necessary and 2) No additional edge can be added without augmenting $T_3(G)$, which holds if and only if we have $N(v_1) \neq N(v_2)$ for any two non-adjacent vertices $v_1 \neq v_2\in V(G)$.* To understand what graphs are strongly $T_3$-reconstructible, we need to understand the first requirement above. We will find some sufficient conditions for it by looking at several induced subgraphs which our edge could be in that will guarantee the necessity of said edge. Here is a lemma listing simple scenarios, where we know the edge in question would be necessary, that will be important for our full characterization later on. {#fig:lemma} **Lemma 9**. *An edge $uv \in E(G)$ is necessary if it is contained in an induced subgraph of $G$ that belongs to one of the families in Figure [3](#fig:lemma){reference-type="ref" reference="fig:lemma"}:* *Proof.* We will show that in each of these 6 cases, deleting $uv$ would inevitably change the set of connected triples of $G$. $\mathcal{F}_1$: If we were to delete the edge $uv$, then we would need to add $v_1v$ as an edge for $\{u,v,v_1\}$ to remain a connected triple. But this would create a new connected triple $\{v_1, v, v_2\}$. So we need to delete $vv_2$. However, without the edges $uv$ and $vv_2$, $\{u,v,v_2\}$ will not be a connected triple. $\mathcal{F}_2$: If we were to delete the edge $uv$, then we would need to add both $uv_1$ and $uv_2$ as edges to preserve the connected triples $\{u,v,v_1\}$ and $\{u,v,v_2\}$. But this would inevitably create a new connected triple $\{v_1, u, v_2\}$. $\mathcal{F}_3$: If we were to delete the edge $uv$, then we would need to add $vv_2$ as an edge to preserve the connected triple $\{u, v, v_2\}$. To prevent creating the connected triple $\{v, v_1, v_2\}$, we would need to delete edge $v_1v_2$. But we know from $\mathcal{F}_2$ that the edge $v_1v_2$ is a necessary edge here in $\mathcal{F}_3$. $\mathcal{F}_4$: If we were to delete the edge $uv$, then we would need to add $uv_1$ to keep $\{u,v,v_1\}$ as a connected triple. However, since neither $\{u,v_1, v_2\}$ nor $\{u, v_1, v_3\}$ belongs to $T_3(G)$, we would need to delete both $v_1v_2$ and $v_1v_3$, which would destroy $\{v_1, v_2, v_3\}$. $\mathcal{F}_5$: If we were to delete the edge $uv$, we would need to add $vv_3$ as an edge for $\{v, u, v_3\} \in T_3(G)$. But since $\{u, v_3, v_2\} \notin T_3(G)$, we would need to delete $v_2v_3$. Yet $\{v_1, v_2, v_3\} \in T_3(G)$ implies that we also need to add $v_1v_3$ as an edge. Now, we have created a connected triple $\{v_1, v_3, v\} \notin T_3(G)$. $\mathcal{F}_6$: If we were to delete the edge $uv$, then we would need to add $vv_1$ and $uv_3$ as edges for $\{u,v,v_1\}, \{v, u, v_3\} \in T_3(G)$. But since $\{v,v_1,v_2\}, \{u,v_3,v_2\} \notin T_3(G)$, we would need to delete both $v_1v_2$ and $v_2v_3$. But then $\{v_1, v_2, v_3\} \notin T_3(G)$. ◻ With Lemma 4.8, we can give a complete characterization for triangle-free graphs. We first prove a very useful lemma. **Lemma 10**. *Any edge that is not contained in any triangle in a graph $G$ on $n \geq 5$ vertices with the property that $N(v_1) \neq N(v_2)$ for any two non-adjacent vertices $v_1$ and $v_2$ is necessary.* *Proof.* Fix an edge $v_1v_2 \in E(G)$. We will show that $v_1v_2$ is necessary by showing that it is contained in an induced subgraph that belongs to one of the six families of graphs listed in Lemma 4.8. Since $G$ is connected, $v_1v_2$ exists in a connected triple, say $\{v_1, v_2, v_3\}$, with edges $v_1v_2$ and $v_2v_3$. Because $v_1$ and $v_3$ are non-adjacent, we know $N(v_1) \neq N(v_3)$. If there exists a vertex $v \in N(v_1) \setminus N(v_3)$, then the edge $v_1v_2$ exists in an induced subgraph $G[\{v,v_1, v_2, v_3\}]$ that belongs to $\mathcal{F}_1$. Otherwise, there exists a vertex $v \in N(v_3) \setminus N(v_1)$ and $N(v_1) \subsetneq N(v_3)$. If $v_2v \in E(G)$, then $\{v_1, v_2, v_3, v\}$ would induce a graph that belongs to $\mathcal{F}_2$. If this is not the case, then $\{v_1, v_2, v_3, v\}$ would induce a $P_4$. Since $G$ is connected and $n \geq 5$, we know there exists $z \notin \{v_1, v_2, v_3, v\}$ that is adjacent to at least one of the vertices in $\{v_1, v_2, v_3, v\}$. If $v_1z \in E(G)$, then $N(v_1) \subsetneq N(v_3)$ would imply that $v_3z \in E(G)$ and that $\{v_1, v_2, v_3, v, z\}$ induces a graph in $\mathcal{F}_3$. Else if $v_2z \in E(G)$, then $\{v_1, z, v_2, v_3\}$ induces a graph in $\mathcal{F}_2$. Else if $v_3z \in E(G)$, then $G[\{v_1, z, v_2, v_3\}]$ belongs to $\mathcal{F}_4$. Else, we know $vz \in E(G)$ and in which case $G[\{v_1, v_2, v_3, v, z\}]$ belongs to $\mathcal{F}_5$. ◻ **Theorem 6**. *A triangle-free graph $G$ on $n \geq 5$ vertices is strongly $T_3$-reconstructible if and only if it has the the property that $N(v_1) \neq N(v_2)$ for any two non-adjacent vertices $v_1$ and $v_2$.* *Proof.* Note the property in question forms a necessary and sufficient condition for meeting the second requirement in Lemma 4.7. So the "only if\" direction is immediate. We just need to show the "if\" direction (i.e. that satisfying this property also guarantees that every edge in G is necessary, which is the first requirement in Lemma 4.7). Applying Lemma 4.9 to $G$ gives us that every edge in $G$ is necessary. Thus $G$ is strongly $T_3$-reconstructible by Lemma 4.7. ◻ **Remark 5**. Observe that Theorem 4.10, when applied to trees, is equivalent to saying that a tree on $n \geq 5$ vertices is strongly $T_3$-reconstructible if and only if no two leaves share the same parent. If we were to derive a complete characterization of all strongly $T_3$-reconstructible graphs, we would need a necessary and sufficient condition for edges that are contained in a triangle to be necessary. It turns out that the six scenarios, which force their highlighted edges to be necessary, respectively, shown in Lemma 4.8 are exactly what we need. Here we state and prove our full characterization of strongly $T_3$-reconstructible graphs. {#fig:main} {#fig:venn} **Theorem 7**. *A graph $G$ on $n \geq 5$ vertices is strongly $T_3$-reconstructible if and only if $N(v_1) \setminus \{v_2 \} \neq N(v_2) \setminus \{v_1 \}$ for all $v_1 \neq v_2 \in V(G)$, and every edge contained in a triangle is also contained in an induced subgraph belonging to one of the families of graphs in Figure [4](#fig:main){reference-type="ref" reference="fig:main"}:* *Proof of main theorem.* The "if\" direction follows directly from Lemma 4.7, Lemma 4.8, and Lemma 4.9. As for the "only if\" direction, it is clear that if there exists $v_1 \neq v_2 \in V(G)$ such that $N(v_1) \setminus \{v_2\} = N(v_2) \setminus \{v_1\}$, then whether $v_1$ and $v_2$ are adjacent would not make a difference to the set of connected triples of $G$. So if $G$ is strongly $T_3$-reconstructible, then it is definitely the case that $N(v_1) \setminus \{v_2 \} \neq N(v_2) \setminus \{v_1 \}$ for all $v_1 \neq v_2 \in V(G)$. What we need to show is that if an edge is contained in a triangle in a graph that satisfies $N(v_1) \setminus \{v_2 \} \neq N(v_2) \setminus \{v_1 \}$ for all $v_1 \neq v_2 \in V(G)$ but is not contained in an induced subraph that belongs to one of the families in Figure [4](#fig:main){reference-type="ref" reference="fig:main"}, then it is not necessary. To do that, we would take a triangle which our edge is contained in, say triangle $uvw$, and look at how it is connected to the rest of the graph. In particular, we will be interested in how the vertices outside of the triangle but in the neighborhood of $\{u,v,w\}$, which we denote as $N[\{u,v,w\}]$, are connected to the vertices $u, v, w$ respectively. A few words on notation and diction: we will call a vertex outside of the triangle an *exclusive common neighbor* of two vertices of the triangle if said vertex is adjacent to the two vertices but not to the third vertex of the triangle. Since we will only be considering if two vertices have the same set of neighbors outside of themselves, when we write $N(x) \setminus N(y)$ for some adjacent vertices $x$ and $y$, we do not mean to include $y$. Similarly, we will use $N(x) = N(y)$ as short for $N(x)\setminus \{y\} = N(y) \setminus \{x\}$ when $x$ and $y$ are adjacent. We can express all such possibilities by all 2-colorings of the Venn diagram, where the regions colored yellow are exactly the ones that are non-empty, shown in Figure [5](#fig:venn){reference-type="ref" reference="fig:venn"}, up to symmetry. Applying Burnside's lemma gives that there are $40$ scenarios. However, we first note that if there exist at least two vertices in $\{u, v, w\}$ that have private neighbors, say $x$ and $y$, where we say a vertex $u$ in the triangle has a *private* neighbor if there is a vertex outside of the triangle that is adjacent to $u$ but not to $v$ or $w$, then we will find our edge and our triangle $uvw$ in an induced subgraph $G[\{v,u,w,x,y\}]$ that is the bull graph, or the bull graph plus an edge (triangle embedded in a corner of a pentagon), which we know are strongly $T_3$-reconstructible. Furthermore, in either case, our edge would exist in an induced subgraph that belongs to $\mathcal{F}_1$, which would be a contradiction. On the other hand, the condition that $N(v_1) \setminus \{v_2 \} \neq N(v_2) \setminus \{v_1 \}$ for all $v_1 \neq v_2 \in V(G)$ we imposed on our graph would eliminate the cases where there exist two vertices of the triangle---say $u$ and $v$---that share all the same neighbors outside of the triangle $\left( \text{i.e. } N[\{u,v,w\}] \cap N(u) = N[\{u,v,w\}] \cap N(v) \right)$. This leaves us with 12 cases to consider, which we will break into two groups: the group of cases where none of the three vertices have a private neighbor (Cases 1 - 4), and the group of cases where there exists one of $\{u,v,w\}$, say $u$, with a private neighbor (Cases 5 - 12). What we want is to show that for any of the edges in the triangle in each case, if we assume that it is not contained in an induced subgraph that belongs to one of the families in Figure [4](#fig:main){reference-type="ref" reference="fig:main"}, then it would not be a necessary edge. l3.5cm  **Case 1** \[the only non-empty sets are Y: set of exclusive neighbors of $u$ and $v$; Z: set of exclusive neighbors of $v$ and $w$\] *Proof.* By symmetry, we need to show $uv$ and $uw$ are not necessary if they are not contained in an induced subgraph that belongs to one of the families of [4](#fig:main){reference-type="ref" reference="fig:main"}, respectively. Note that $Z = \{z\}$ otherwise $G[\{u,v,z, z_2\}], G[\{u,w,z,z_2\}] \in \mathcal{F}_2$ for some $z_2 \neq z \in Z$, a contradiction. So the only edge we need to add to preserve $T_3(G)$ if we were to delete edge $uv$ is $uz$. For all $p \in N(z) \setminus N(u)$, if it is an exclusive common neighbor of $v$ and $w$, then $G[\{u,v,z,p\}], G[\{u,w,z,p\}] \in \mathcal{F}_2$, a contradiction. Else, $p \notin N[\{u,v,w\}]$. But then we would have $G[\{p,z,v,u,w\}] \in \mathcal{F}_3$, another contradiction. Hence such $p$ does not exist. On the other hand, $z$ has to be adjacent to all $y_i \in Y$, otherwise $G[\{u,y_i, v, z\}] \in \mathcal{F}_1$ for some $y_i \in Y$, which would be a contradiction. Therefore $N(u) \subset N(z)$ and thus $N(u) = N(z)$. This shows that $uv$ is not necessary for it can be replaced by $uz$. As for $uw$, we would need to replace it with $uz$ and $yw$: in this case there does not exist another $y_2 \neq y \in Y$ as $G[\{y, y_2, u, w\}] \notin \mathcal{F}_2$. We have shown that adding $uz$ will not cause trouble when our edge in question is $uv$ or $uw$. The argument for $yw$ is entirely symmetric. ◻ l3.5cm  **Case 2**: \[Case 1 + C: the set of common neighbors of $u$, $v$, and $w$\] *Proof.* We can apply our argument in Case 1. We only need to check that the addition of $uz$ won't create the connected triple $\{u,c_l, z\}$ if $c_lz \notin E(G)$ for some $c_l \in C$. However, if there does exist $c_l \in C$ such that $c_lz \notin E(G)$, then $G[\{c_l, u, v, z\}], G[\{c_l, u, w, z\}] \in \mathcal{F}_1$, a contradiction. This shows that $uv$ and, by symmetry, $vw$ is not necessary. For $uw$, we also need to check the same problem can't happen with the addition of $wy$. But the argument again is entirely symmetric. ◻ l3.5cm  **Case 3**: \[the only non-empty sets are Y: exclusive common neighbors of $u$ and $v$; Z: exclusive common neighbors of $v$ and $w$; M: exclusive common neighbors of $u$ and $w$\] *Proof.* We only need to show $uv$ is not necessary due to symmetry. Again, it's easy to check that both $Z$ and $M$ have only one element, say $z$ and $m$, and that $zy_i, my_i\in E(G)$ for all $y_i \in Y$. Also note $mz \in E(G)$ since $G[\{m,u,v,z\}] \notin \mathcal{F}_1$. Deleting $uv$ would require the addition of $uz$ and $vm$. If there exists $p \in N(z) \setminus N(u)$. Then $p$ is either an exclusive common neighbor of $v$ and $w$ or $p \notin N[\{u,v,w\}]$. The former can't happen since $z$ is the unique exclusive common neighbor of $v$ and $w$ and if it were the latter then $G[\{u,w,z,p,v\}] \in \mathcal{F}_3$, a contradiction. Hence $N(z) \subset N(u)$. On the other hand, having $zy_i, zm, zv, zw \in E(G)$ for all $y_i \in Y$ implies that $N(u) \subset N(z)$ and thus $N(u) = N(z)$. This shows that adding $uz$ as an edge does not create any new connected triples. Similarly, we can show that the same is true for $vm$. ◻ l3.5cm  **Case 4**: \[Case 3 + C: the set of common neighbors of $u$, $v$, and $w$\] *Proof.* Similar to Case 3, we will only need to show it for $uv$ (i.e. $uv$ is not necessary if it is not contained in an induced subgraph belonging to one of the families in [4](#fig:main){reference-type="ref" reference="fig:main"}). As we have shown in Case 3, $u$ and $w$ have an unique exclusive common neighbor, say $m$, and $v$ and $w$ have an unique exclusive common neighbor, say $z$ and $mz \in E(G)$. It is sufficient to add $uz$ and $vm$ to make up for the connected triples destroyed from the deletion of $uv$. We will show that adding $uz$ does not create any new connected triples that were not in $T_3(G)$ by showing that $u$ and $z$ have the same set of neighbors. The argument for $vm$ is symmetric. If there exists $p \in N(z) \setminus N(u)$, then it has to be the case that $p \notin N[\{u,v,w\}]$, which would imply that $G[\{u,v,z,p,w\}] \in \mathcal{F}_3$, a contradiction. On the other hand, recall $mz \in E(G)$ and that $zy_i, zc_l \in E(G)$ for all $y_i \in Y$ and $c_l \in C$, otherwise $G[\{u,v,y_i,z\}], G[\{u,v,c_l,x\}] \in \mathcal{F}_1$ for some $c_l \in C$ or $y_i \in Y$, which would be a contradiction. Therefore $N(u) \subset N(z)$ and thus $N(u) = N(z)$. ◻ For all the cases where $u$ has a private neighbor, say $x_k$, with regards to the triangle $uvw$, both $uv$ and $uw$ are contained in the induced subgraph $G[\{x_k, u, w, v\}]$ in such a way that belongs to $\mathcal{F}_1$, respectively. So the only edge in this triangle that might not be contained in an induced subgraph belonging to any of the families of [4](#fig:main){reference-type="ref" reference="fig:main"} is edge $vw$. Hence we only check when our edge in question is $vw$ in Cases 5 to 12. So when we say some induced subgraph that $vw$ is in belongs to some family in [4](#fig:main){reference-type="ref" reference="fig:main"}, it will always be implicitly assumed that it will be with regards to $vw$, where the $vw$ will correspond to the highlighted edge in the said family in [4](#fig:main){reference-type="ref" reference="fig:main"}. Recall we are assuming that $vw$ is not contained in any induced subgraph that belongs to some family in [4](#fig:main){reference-type="ref" reference="fig:main"} and we want to show that $vw$ is not necessary. l3.5cm   **Case 5**: \[the only non-empty sets are X: set of private neighbors of $u$; Y: set of exclusive common neighbors of $u$ and $v$;\] **Case 6**: \[Case 5 + C: the set of common neighbors of $u$, $v$, and $w$\] *Proof.* If there exists $y_i \neq y_j \in X$, then $G[\{y_i, v, y_j, w\}] \in \mathcal{F}_2$, a contradiction. So $u$ and $v$ have an unique exclusive common neighbor $y$. Also, there exists an unique $x \in X$ such that $xy \in E(G)$. If we were to delete edge $vw$, the only connected triple destroyed would be $\{w,v,y\}$. So we would need to add $wy$ as an edge. If there exists $c_l \in C$ such that $c_ly \notin E(G)$, then $G[\{c_l,w,v,y\}] \in \mathcal{F}_1$, a contradiction. Hence $N(w) \subset N(y)$. For any $p \in N(y) \setminus N(x)$, $p$ is either a private neighbor of $u$ or not in $N[\{u,v,w\}]$ at all. However, the former gives $G[\{u,w,v,y,p\}] \in \mathcal{F}_3$, a contradiction. So $p$ would have to be the unique private neighbor of $u$ that is adjacent to $y$. This means that $\{w,y,p\}$ is the unique new connected triple created by the addition of $wy$ as an edge. Thus we delete the edge $yp$. We will show that $\{v,y,p\}$ is the only connected triple in $T_3(G)$ destroyed by deleting $yp$ and thus we would only need to add $vp$ as an edge to preserve $T_3(G)$. Suppose there exists $p_2 \neq v \in N(y) \setminus N(p)$. If $p_2 \notin N[\{u,v,w\}]$, then we have $G[\{p_2, y, u, w, v\}] \in \mathcal{F}_3$, a contradiction. Else if $p_2$ is a private neighbor of $u$, then $G[\{p_2, p, y, v, w\}] \in \mathcal{F}_2$, a contradiction. Else $p_2$ has to be a common neighbor of all $u$, $v$, and $w$, which would give $G[\{p, y, v, w, p_2\}] \in \mathcal{F}_3$, another contradiction. Thus no such $p_2$ exists and $N(y) \setminus N(p) = \{v\}$. Now suppose there exists $q \neq y \in N(p) \setminus N(y)$. If $q \notin N[\{u,v,w\}]$ or $q$ is a private neighbor of $u$, then $G[\{q, p, y, v, w\}] \in \mathcal{F}_5$, a contradiction. Otherwise, $q$ is a common neighbor of all three vertices, which gives $G[\{v, w, q, y\}] \in \mathcal{F}_1$, which is a contradiction. Lastly, we will show that adding $vp$ does not create any more connected triples. Since $y$ is the unique exclusive common neighbor of $u$ and $v$, any $q \in N(v) \setminus N(p)$ would have to be a common neighbor of $u$, $v$, and $w$. But this would give $G[\{p,w,v,q\}] \in \mathcal{F}_1$, a contradiction. On the other hand, suppose there exists $q \in N(p) \setminus N(v)$. We showed earlier that there does not exist any $q \neq y$ satisfying $qp \in E(G)$ but $qy \notin E(G)$. Thus this $q \in N(p) \setminus N(v)$ is also adjacent to $y$. If $q \notin N[\{u,v,w\}]$, then $G[\{q, y, u, w, v\}] \in \mathcal{F}_3$, a contradiction. Otherwise, $q$ is a private neighbor of $u$, in which case we have $G[\{w,v,y,p,q\}] \in \mathcal{F}_4$, which is a contradiction. Therefore, we have shown that $vw$ is not necessary in this case because deleting $vw$ and $yp$ while adding $yw$ and $vp$ preserves the set of connected triples. ◻ l3.5cm   **Case 7** \[the only non-empty sets are X: the set of private neighbors of $u$; Y: the set of exclusive common neighbors of $v$ and $u$; Z: the set of exclusive common neighbors of $v$ and $w$.\] **Case 8** \[Case 7 + C: the set of common neighbors of $u$, $v$, and $w$\]. *Proof.* First, note that $Y$ has only one element $y$, otherwise $G[\{y, y_i, v, w\}]$ belongs to $\mathcal{F}_2$ (in Figure [4](#fig:main){reference-type="ref" reference="fig:main"}) for some $y_i \neq y \in Y$. If there exists $z_j \in Z$ such that $yz_j \notin E$, then $G[\{y, v, z_j, w\}]$ would be in $\mathcal{F}_2$, a contradiction. So we assume that $yz_j \in E(G)$ for all $z_j$. Similarly, we have $yc_l \in E(G)$ for all $c_l \in C$. If we were to delete edge $vw$, we would need to add $wy$ as an edge to keep $\{v, w, y\}$ as a connected triple. Note that $N(w) \setminus N(v) = \emptyset$ and $N(v) \setminus N(w) = \{y\}$. This shows that deleting $vw$ does not destroy any more connected triples. Now we turn our attention to the addition of $wy$. Since all $c_l$ and $z_j$, as well as $u$ and $v$, are adjacent to $y$, we have that $N(w) \setminus N(y) = \emptyset$. Suppose there exists $p \in N(y) \setminus N(w)$. If $p$ is not in $N[\{u,v,w\}]$, then $G[\{p, y, u, v, w\}]$ would be in $\mathcal{F}_3$, a contradiction. The fact that $Y$ has only one element $y$ and that $p \notin N(w)$ implies that $p$ has to be a private neighbor of $u$. If there exists $p_2 \neq p$ such that $p_2$ is also a private neighbor of $u$ and is adjacent to $y$, then $\{p, p_2, y, v, w\}$ would induce a subgraph that belongs to $\mathcal{F}_4$, a contradiction. Therefore such $p$ is unique (i.e. $N(y) \setminus N(w) = \{p\}$). So we need to delete $py$. Observe that there does not exist $\Tilde{p} \in N(p) \setminus N(y)$ because $\{\Tilde{p}, p, y_i, v, w\}$ cannot induce a subgraph in $\mathcal{F}_5$. Suppose there exists $\Tilde{p} \neq v \in N(y) \setminus N(p)$. If there exists $z_j \in Z$ or $c_l \in C$ such that $pz_j \notin E(G)$ or $pc_l \notin E(G)$, then said $z_j$ or $c_l$, together with $\{p, y, v, w\}$, would induce $\mathcal{F}_3$, which is a contradiction. So $pz_j, pc_l \in E$ for all $z_j \in Z$ and all $c_l \in C$. This means that all common neighbours of $v$ and $w$ are contained in $N(p)$ and thus $\Tilde{p}$ is not a common neighbor of $v$ and $w$. Furthermore, $y$ is the unique exclusive common neighbor of $v$ and $u$. This implies that $\Tilde{p}$ is a private neighbor of $u$. However, then $\{ p, \Tilde{p}, y, v, w\}$ would induce a subgraph that belongs in $\mathcal{F}_4,$ a contradiction. Hence the only vertex in $N(y)\setminus N(p)$ is $v$. To preserve the original connected triple $\{p, y, v\}$, we need to add $pv$ as an edge. Up until the addition of $pv$ as an edge, we have done everything needed to preserve the set of connected triples. Now we want to show that adding $pv$ doesn't create any new connected triples and thus no more changes are needed. We have shown previously that $N(p) \subset N(y)$. So if there exists any $p\prime \in N(p)$ that is not adjacent to $v$, which implies that it is also not adjacent to $w$, then $G[\{p, p\prime, y, v, w\}]$ would belong to $\mathcal{F}_4$, a contradiction. Therefore everything in $N(p)$ must be a neighbor of $v$ and thus $N(p) \setminus N(v) = \emptyset$. On the other hand, $N(v) \setminus N(p)$ is empty as well since every neighbor of $v$ is either $y$ or a common neighbor of $v$ and $w$, all of which we've shown earlier are adjacent to $p$. Since $p$ and $v$ share all the same neighbors, adding $pv$ won't add any new connected triples. ◻ l3.5cm   **Case 9** \[the only non-empty sets are X: the set of private neighbors of $u$; Y: the set of exclusive neighbors of $u$ and $v$; Z: the set of exclusive neighbors of $u$ and $w$\] **Case 10** \[Case 9 + C: the set of common neighbors of $u$, $v$, and $w$ not being empty\] *Proof.* Note that if there exists $z_i \neq z_j \in Z$, then $\{v, w, z_j, z_i\}$ would induce a graph that belongs to $\mathcal{F}_3$, a contradiction. Therefore, we have $Z = \{z\}$ and, very similarly, $Y = \{y\}$. Assume $z$ and $y$ are adjacent, otherwise $vw$ will be contained in an induced subgraph $G[\{z, w, v, y\}]$ that belongs to $\mathcal{F}_1$. Furthermore, assume $yc_l, zc_l \in E(G)$ for all $c_l \in C$, otherwise $\{y, v, w, c_{l_1}\}$ or $\{z, w, c_{l_2}, v\}$ would each induce a subgraph that belongs to $\mathcal{F}_1$ for some $c_{l_1}, c_{l_2} \in C$. If we were to delete $vw$, the only edges we need to add are $vz$ and $wy$. We will show that none of these additions will create new connected triples by showing that the two vertices of any of the two edges have the same set of neighbors outside of themselves. By symmetry, it would be enough to show this for $vz$. Since $N(v) = Y \cup \{w\} = \{w,y\}$ and $zy, zw \in E(G)$, we have that $N(v) \setminus N(z) = \emptyset$. On the other hand, suppose there exists $p \in N(z) \setminus N(v)$. If $p$ is a private neighbor of $u$ with regards to the triangle $uvw$ or if $p \notin N[\{u,v,w\}]$, then $G[\{w,v,y,z,p\}]$ belongs to $\mathcal{F}_6$, a contradiction. Then $p$ must be a common neighbor of $u$ and $w$ that is not adjacent to $v$. (i.e. $p \in Z$). But this would mean that $G[\{z, p, w, v\}]$ belongs to $\mathcal{F}_2$, a contradiction. Thus $N(z) \setminus N(v) = \emptyset$. Therefore $N(z) = N(v)$. This shows that $vw$ is not necessary as replacing it with $wy$ and $vz$ does not change the set of connected triples. ◻ L3.5cm   **Case 11** \[the only non-empty sets are X: the set of private neighbors of $u$; Y: the set of exclusive neighbors of $U$ and $v$; Z: the set of exclusive neighbors of $v$ and $w$; M: the set of exclusive neighbors of $u$ and $w$\] **Case 12** \[Case 11 + C: the set of common neighbors of $u$, $v$, and $w$ not being empty\] *Proof.* If there exists $y_i \neq y_j \in Y$, then $\{y_i, y_j, v, w\}$ will induce a subgraph that belongs to $\mathcal{F}_2$, a contradiction. So $Y = \{y\}$ and similarly $M = \{m\}$. Also assume $ym \in E(G)$ otherwise $G[\{m,w, v, y\}]$ belongs to $\mathcal{F}_1$, which is a contradiction. Since $N(v) \setminus N(w) = \{y\}$ and $N(w) \setminus N(v) = \{m\}$, the only connected triples destroyed by deleting edge $vw$ are $\{w,v,y\}$ and $\{v,w,m\}$. So we need to add $wy$ and $mv$ as edges. We would like to show such additions do not create new connected triples by showing that the two vertices of either one of the two edges have the same set of neighbors outside of themselves. Again, it would be sufficient to check this for one of them, say $wy$, due to symmetry. Recall $ym \in E(G)$ and note that $G[\{y,v, w, z_j\}], G[\{y,v,w,c_l\}] \notin \mathcal{F}_1$ implies that $yz_j, yc_l \in E(G)$ for all $z_j \in Z$ and $c_l \in C$. This shows that $N(w) \setminus N(y) = \emptyset$. On the other hand, suppose there exists $p \in N(y) \setminus N(w)$. Then either $p \notin N[\{u,v,w\}]$ or it is a private neighbor of $u$ (i.e. $p \in X$). However, the former would imply that $G[\{p,y,u,v,w\}] \in \mathcal{F}_3$, a contradiction, whereas the latter $G[\{p,m,w,v,y\}] \in \mathcal{F}_6$, which is also a contradiction. Therefore $N(w) = N(y)$. ◻ This finishes the proof of the main theorem. $\blacksquare$ # Future Directions Given our complete characterization of strongly $T_3$-reconstructible graphs in Theorem 4.12, one might wonder whether the collection of graphs contained in the families listed in the statement is minimal; or what other equivalent characterizations there might exist. Naturally, one could also ask about characterizations of strongly $T_k$-reconstructible graphs for $k \geq 4$, which we can analogously define to be graphs that could be uniquely reconstructed from $T_k$ without any additional information. Furthermore, one could also study the necessary and sufficent conditions on a class of graphs for it to be $T_3$-reconstructible and try to extend it to an arbitrary $k$. We leave these questions for future work. # Acknowledgements {#acknowledgements .unnumbered} This research was conducted at the 2023 University of Minnesota Duluth REU, funded by Jane Street Capital and the National Security Agency. The author would like to thank Noah Kravitz for suggesting the problem, Evan Chen for helping make the figures in the paper, and Yelena Mandelshtam and Andrew Kwon for providing invaluable feedback on the presentation of the paper. Last but certainly not least, the author is deeply grateful to Joe Gallian and Colin Defant for organizing the Duluth REU. 9 Bastide, P., Cook, L., Erickson, J., Groenland, C., Kreveld, M.V., Mannens, I., and Vermeulen, J.l. Reconstructing Graphs from Connected Triples arXiv:2303.06609. Bondy, J.A., Hemminger, R.L. Graph reconstruction -- a survey. Journal of Graph Theory 1(3), 227--268 (1977). Brandes, U., Cornelsen, S. Phylogenetic graph models beyond trees. Discrete Applied Mathematics 157(10), 2361--2369 (2009). Harary, F. A survey of the reconstruction conjecture. Graphs and Combinatorics. pp. 18--28 (1974). Kelly, P.J. On Isometric Transformations. Ph.D. thesis, University of Wisconsin (1942). Lauri, J., Scapellato, R. Topics in Graph Automorphisms and Reconstruction. Cambridge University Press (2016). Mossel, E., Ross, N. Shotgun assembly of labeled graphs. IEEE Transactions on Network Science and Engineering 6(2), 145--157 (2017). Tutte, W. All the king's horses. a guide to reconstruction. Graph Theory and Related Topics pp. 15--33 (1979). Ulam, S.M. A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, vol. 8. Interscience Publishers (1960).

arxiv_math

--- abstract: | Given an $n$-vertex digraph $D$ and a labeling $\sigma:V(D)\to [n]$, we say that an arc $u\to v$ of $D$ is a descent of $\sigma$ if $\sigma(u)>\sigma(v)$. Foata and Zeilberger introduced a generating function $A_D(t)$ for labelings of $D$ weighted by descents, which simultaneously generalizes both Euleraian polynomials and Mahonian polynomials. Motivated by work of Kalai, we look at problems related to $-1$ evaluations of $A_D(t)$. In particular, we give a combinatorial interpretation of $|A_D(-1)|$ in terms of "generalized alternating permutations" whenever the underlying graph of $D$ is bipartite. author: - "Kyle Celano[^1]" - "Nicholas Sieger[^2]" - "Sam Spiro[^3]" bibliography: - refs.bib title: Eulerian Polynomials for Digraphs --- # Introduction Descents and inversions are two of the oldest and most well-studied *permutation statistics* dating back to work of MacMahon [@macmahon2001combinatory; @macmahon1913indices]. A *descent* of a permutation $\sigma$ on the set $[n]:=\{1,2,\dots,n\}$ is an index $i\in [n-1]$ such that $\sigma(i)>\sigma(i+1)$, and an inversion is a pair of integers $(i,j)$ with $1\leq i<j\leq n$ such that $\sigma(i)>\sigma(j)$. The number of descents and inversions of $\sigma$ are denoted by $\mathop{\mathrm{des}}(\sigma)$ and $\mathop{\mathrm{inv}}(\sigma)$, respectively. For example, if $\sigma=23154$ (written in *one-line notation*, meaning $\sigma(1)=2,\ \sigma(2)=3\dots$), then $2$ and $4$ are descents whereas $(1,3),(2,3),(4,5)$ are inversions, so $\mathop{\mathrm{des}}(\sigma)=2$ and $\mathop{\mathrm{inv}}(\sigma)=3$. If $\mathfrak{S}_n$ is the set of permutations of $[n]$, then the generating functions $$A_n(t)=\sum_{\sigma\in \mathfrak{S}_n}t^{\mathop{\mathrm{des}}(\sigma)}\quad M_n(t)=\sum_{\sigma\in \mathfrak{S}_n}t^{\mathop{\mathrm{inv}}(\sigma)}$$ are called the *Eulerian* and *Mahonian* polynomials respectively. Both of these polynomials are important objects of study in many branches of combinatorics and have been generalized in many different ways. In this paper, we consider a polynomial due to Foata and Zeilberger [@FOATA199679] which generalizes both the Eulerian and Mahonian polynomials via directed graphs. A *directed graph* or *digraph* is a pair $D=(V,E)$ consisting of a finite set $V$ of *vertices* and a subset $E\subset V\times V$ of *(directed) edges* or *arcs*. We will sometimes denote arcs $(u,v)\in E$ by $u\to v$ or even $uv$ where convenient. We further assume that $D$ has no loops i.e. no arcs of the form $(v,v)$. We do not allow multiple directed edges from one vertex to another, though it is easy to adapt our forthcoming definitions to accommodate this. A *permutation* of an $n$-vertex digraph $D=(V,E)$ is a bijection $\sigma:V\to [n]$. We will use the notation $\mathfrak{S}_D,\ \mathfrak{S}_V$, or $\mathfrak{S}_n$ to denote the set of permutations of $D$. For a given directed graph $D=(V,E)$ and a permutation $\sigma$ of $D$, a *$D$-descent* (or just *descent* when $D$ is understood) is an arc $u\to v$ such that $\sigma(u)>\sigma(v)$. The total number of $D$-descents of a permutation $\sigma$ is denoted by $\mathop{\mathrm{des}}_D(\sigma)$; see for an example.   [\[fig:basic example with digon\]]{#fig:basic example with digon label="fig:basic example with digon"} **Remark 1**. We claim that the statistic $\mathop{\mathrm{des}}_D$ generalizes both $\mathop{\mathrm{des}}$ and $\mathop{\mathrm{inv}}$. To see this, let $\overrightarrow{P}_n$ be the digraph with vertex set $[n]$ and with arcs $(i,i+1)$ for $i\in [n-1]$, and let $\overrightarrow{K}_n$ be the digraph with vertex set $[n]$ and with arcs $(i,j)$ for integers $1\leq i<j\leq n$. The reader can check that a *$\overrightarrow{P}_n$-descent* is a *descent* (in the classical meaning) and a *$\overrightarrow{K}_n$-descent* is an inversion, and hence $$\mathop{\mathrm{des}}_{\overrightarrow{P}_n}(\sigma)=\mathop{\mathrm{des}}(\sigma)\quad \mathop{\mathrm{des}}_{\overrightarrow{K}_n}(\sigma)=\mathop{\mathrm{inv}}(\sigma).$$ See for an example. With all this in mind, we can now define the central object of study for this paper: the *Eulerian polynomial* of a digraph $D=(V,E)$ is the generating function $$A_D(t)=\sum_{\sigma\in \mathfrak{S}_D}t^{\mathop{\mathrm{des}}_D(\sigma)}.$$ In particular, the previous remark implies $A_{\overrightarrow{P_n}}(t)=A_n(t)$ and $A_{\overrightarrow{K_n}}(t)=M_n(t)$. ## Main results The primary objective of this paper is to study evaluations of $A_D(t)$ at $-1$. This is a problem in the area of *combinatorial reciprocity*, which studies combinatorial polynomials evaluated at negative integers. For example, the classical Eulerian and Mahonian polynomials both have good combinatorial interpretations for their evaluation at $-1$: the former being the number of *alternating permutations* [@foata2005theorie] and the later being the number of *correct proofs of the Riemann hypothesis*[^4]. Many more results on combinatorial reciprocity can be found in the book by Beck and Sanyal [@beck2018combinatorial]. In order to understand $A_D(-1)$ , we utilize the following key observation made by Kalai [@KALAI2002412 Section 8.1]. **Proposition 2**. *If $D,D'$ are orientations of the same graph $G$, then $|A_D(-1)|=|A_{D'}(-1)|$.* This result follows from the fact that if any arc of $D$ is reversed, then the number of descents for any permutation $\sigma\in \mathfrak{S}_V$ changes by exactly 1. With in mind, for any graph $G$ we can define $$\nu(G):=|A_D(-1)|,$$ where $D$ is any orientation of $G$. The problem of studying $\nu(G)$ was first introduced by Kalai [@KALAI2002412] due to its relation with the Condorcet paradox in social choice theory, and a few basic properties of $\nu(G)$ were established by Even-Zohar [@Even-Zohar2017]. Outside of this, nothing seems to be known about $\nu(G)$ despite Kalai raising the problem over 20 years ago. In this paper, we prove three types of results related to $\nu(G)$: we give combinatorial interpretations for $\nu(G)$ for a large class of graphs $G$, we determine the maximum and minimum values achieved by $\nu(G)$ amongst $n$ vertex trees, and we consider the refined problem of determining the multiplicity of $-1$ as a root of $A_D(t)$. ### Combinatorial Interpretations for $\nu(G)$ A key observation towards understanding $\nu(G)$ is a result of Foata and Schützenberger [@foata2005theorie] (see also [@stanley_2011 Exercise 135]) which states that the Eulerian polynomial $A_n(t)$ evaluated at $t=-1$ is equal (up to sign) to the number of alternating permutations of size $n$, i.e. the number of permutations $\sigma$ with $n$ odd and $\sigma(1)<\sigma(2)>\sigma(3)<\cdots >\sigma(n)$. Because $A_n(t)=A_{\overrightarrow{P}_n}(t)$ for $\overrightarrow{P}_n$ the directed path, this result implies $\nu(P_n)$ is equal to the number of alternating permutations of size $n$. Given this observation, it is natural to expect $\nu(G)$ to count "alternating permutations for graphs" for some generalized notion of alternating permutations. There are many such generalizations one could consider, for example, one could force every maximal path of $G$ to be an alternating permutation. However, it turns out that the definition we will want to consider is the following (non-obvious) generalization. **Definition 3**. Given an $n$-vertex graph $G$, we say that an ordering $\pi=(\pi_1,\ldots,\pi_n)$ of the vertex set $V(G)$ is an *even sequence* if each of the subgraphs $G[\pi_1,\ldots,\pi_i]$ induced by the first $i$ vertices of $\pi$ have an even number of edges for all $1\le i\le n$. We let $\eta(G)$ denote the number of even sequences of $G$. For example, depicts an even sequence for the path graph on 5 vertices with vertex set $[5]$. While not immediate, once can verify that even sequences for the path graph $P_n$ with vertex set $[n]$ are exactly inverses of alternating permutations of size $n$ (see ); so $\nu(P_n)=\eta(P_n)$ in this case. Our main result shows that this equality holds for a substantially larger class of graphs. To state this result, we remind the reader that a graph is *complete multipartite* if one can partition its vertices into sets $V_1,\ldots,V_r$ such that $u$ and $v$ are adjacent if and only if $u\in V_i,v\in V_j$ for some $i\ne j$. We say that a graph is a *blowup of a cycle* if one can partition its vertices into sets $V_1,\ldots,V_r$ such that $u$ and $v$ are adjacent if and only if $u\in V_i$ and $v\in V_{i+1}$ for some $i$ (with the indices written mod $r$). With these definitions in mind, we have the following. **Theorem 4**. *If $G$ is a graph which is either bipartite, complete multipartite, or a blowup of a cycle, then $\nu(G)=\eta(G)$.* While the above shows $\nu(G)=\eta(G)$ for a large class of graphs $G$, equality does not hold in general. In fact, the following result shows that the family of graphs from is essentially the largest hereditary family of graphs for which equality holds. **Theorem 5**. *If $G$ is a connected graph such that $\nu(G')=\eta(G')$ for all induced subgraphs $G'\subseteq G$, then $G$ is either bipartite, complete multipartite, or a blowup of a cycle.* Our proof of relies on a structural graph theory result which may be of independent interest; see and for a precise statement. While we do not have a full understanding of $\nu(G)$ for arbitrary graphs, we are able to prove several other results regarding $\nu(G)$, such as the general bound $\nu(G)\le \eta(G)$. We refer the reader to and for a full list of these results. ### Extremal Values of $\nu(G)$ and $\eta(G)$ We next turn to the extremal problem of studying the largest and smallest possible values of $\nu(G)$ and $\eta(G)$. For arbitrary $n$-vertex graphs this is an uninteresting problem, since $\nu(\overline{K_n})=\eta(\overline{K_n})=n!$ and $\nu(K_n)=\eta(K_n)=0$ for $n\ge 2$ are easily seen to achieve the maximum and minimum possible values. However, this problem becomes non-trivial when one looks at smaller classes of graphs. To this end, we consider these extremal problems for trees. To state our result, we recall that a tree is a *star* $K_{1,n}$ if there is a single-non leaf vertex. We say that a tree is a *hairbrush* if it consists of a path $v_0v_1\cdots v_n$ such that each vertex $v_i$ with $i\ge 1$ is adjacent to a leaf $u_i$. That is, hairbrushes are "comb graphs" with an extra vertex $v_0$ attached at the end; see . [\[fig:extremal trees\]]{#fig:extremal trees label="fig:extremal trees"} **Theorem 6**. *If $T$ is a tree on $2n+1$ vertices, then $$n! 2^n\le \nu(T)=\eta(T)\le (2n)!$$ Moreover, equality holds in the lower bound if and only if $T$ is a hairbrush, and equality holds in the upper bound if and only if $T$ is a star.* Note that the equality $\nu(T)=\eta(T)$ follows from , and that $\nu(T)=\eta(T)=0$ if $T$ has an even number of vertices (since $e(T)$ is odd), which is why only considers trees with an odd number of vertices. ### Multiplicity of Roots Lastly, we consider the problem of determining the multiplicity of $-1$ as a root of $A_D(t)$, and we denote this quantity by $\mathop{\mathrm{mult}}(A_D(t),-1)$. Note that studying $\mathop{\mathrm{mult}}(A_D(t),-1)$ is related to studying $\nu(G)=|A_D(-1)|$ in the sense that a graph $G$ has $\nu(G)=0$ if and only if $\mathop{\mathrm{mult}}(A_D(t),-1)>0$ for every orientation $D$ of $G$ One of the first questions one might ask in this setting is to determine how large $\mathop{\mathrm{mult}}(A_D(t),-1)$ can be amongst $n$-vertex digraphs. Trivially, $\mathop{\mathrm{mult}}(A_D(t),-1)\le e(D)$ (since the degree of $A_D(t)$ is at most $e(D)$), which implies $\mathop{\mathrm{mult}}(A_D(t),-1)\le {n\choose 2}$ if $D$ has $n$ vertices. We prove a substantially stronger upper bound which turns out to be sharp. **Theorem 7**. *If $D$ is an $n$-vertex digraph, then $$\mathop{\mathrm{mult}}(A_D(t),-1)\le n-s_2(n),$$ where $s_2(n)$ denotes the number of 1's in the binary expansion of $n$. Moreover, for all $n$, there exist $n$-vertex digraphs $D$ with $A_D(t)=\frac{n!}{2^{n-s_2(n)}}(1+t)^{n-s_2(n)}$.* We also obtain a general lower bound on $\mathop{\mathrm{mult}}(A_D(t),-1)$. **Proposition 8**. *Let $D$ be an orientation of an $n$-vertex graph $G$. If every matching in the complement of $G$ has size at most $m$, then $\mathop{\mathrm{mult}}(A_D(t),-1) \geq \lfloor \frac{n}{2} \rfloor - m$.* Roughly speaking, says that if $G$ is "dense" (i.e. if the complement of $G$ contains small only matchings), then $\mathop{\mathrm{mult}}(A_D(t),-1)$ will be large. While is not tight in general, it turns out to be tight if $D$ is an orientation of the complete graph. **Theorem 9**. *If $D$ is a tournament on $n$ vertices, then $\mathop{\mathrm{mult}}(A_D(t),-1)=\lfloor\frac{n}{2}\rfloor$.* More generally, we suspect that is tight for orientations of complete multipartite graphs, see for more. Given and the fact that $|A_D(-1)|=|A_{D'}(-1)|$ whenever $D,D'$ are orientations of the same graph, it is perhaps natural to guess that $\mathop{\mathrm{mult}}(A_D(t),-1)$ depends only on the underlying graph of $D$. This turns out to be false; see for a counterexample. ## Related Works Before presenting our proofs, we first comment on a variety of similar looking polynomials that appear in the literature. This summary is by no means exhaustive, as there are countless objects adopting the monikers of *Eulerian numbers*, *Eulerian polynomials*, *descents* or *inversions*; most of which have little to no relation to the problems studied here. As previously mentioned, Foata and Zeilberger [@FOATA199679] were the first to define the above polynomial $A_D(t)$. They were primarily interested in finding digraphs $D$ for which the statistic $\mathop{\mathrm{des}}_D$ has the same distribution on $\mathfrak{S}_D$ as another statistic $\mathrm{maj}_D$ which is a generalization of the classical "major index\" permutation statistic. Their problem has a positive answer when $D$ is "bipartitional", which in our terminology is a join of bidirected complete graphs and digraphs with no edges. In our (d) we consider more general joins of digraphs which recovers some of the results from [@FOATA199679]. When $D$ is acyclic, $\mathop{\mathrm{des}}_D$ is a *weighted-inversion statistic* as in Kadell [@KADELL198522] and Degengardt--Milne [@DEGENHARDT200049] which is any function $w:\mathfrak{S}_n\to \mathbb{Z}_{\geq 0}$ of the form $$w(\sigma)=\sum_{\substack{1\leq i<j\leq n\\ \sigma(i)>\sigma(j)}} w_{i,j}$$ for some upper triangular matrix $(w_{i,j})$. In fact, it is easy to see that $\mathop{\mathrm{des}}_D$ encapsulates all weighted inversion statistics $w$ for which $w_{i,j}\in \{0,1\}$ for all $i,j$. Archer et al [@ARCHER2020112041] define a *Eulerian polynomial for a family $\mathfrak{B}_n$ of digraphs* as $$b_n(u)=\sum_{D\in \mathfrak{B}_n}u^{\mathop{\mathrm{des}}(D)}$$ where $\mathop{\mathrm{des}}(D)$ is defined by *fixing* a labeling $\omega_D:V(D)\to[n]$ of the vertices of $D$ for each $D$ and then counting the number of edges that go from larger label to smaller label. The polynomial $A_D(t)$ can be recovered from $b_n(u)$ by selecting $\mathfrak{B}_n$ to be *all* labelings of a fixed digraph, although this particular choice of family of digraphs is not considered in [@ARCHER2020112041]. The polynomial $A_D(t)$ also appears as specializations of the Ellzey *chromatic quasisymmetric function for digraphs* [@Ellzey_2017] as well as the Awan and Bernardi *B-polynomial* [@awan_bernardi_2020]. Lastly, we note that Kalai [@KALAI2002412] explicitly mentioned the determination of $A_D(-1)$ as an interesting problem in the context of the Condocet paradox in social choice theory, and Even-Zohar[@Even-Zohar2017] gave some basic properties of $\nu(G)$ in their study of the random framed knots through permutation statistics. ## Organization of the Paper The rest of this paper is organized as follows. In , we lay out the necessary definitions, notation and elementary properties of $A_D(t)$ for the rest of the paper. In we consider evaluations of $A_D(t)$ at $t = -1$. In particular, we give basic properties of $\nu(G)$ in , we prove in , and we prove in . In we study the bounds for $\nu(T)$ for trees $T$ and prove . In we study the multiplicity of $-1$ as a root of $A_D(t)$ and prove and Theorems [Theorem 7](#thm root multiplicity upper bound){reference-type="ref" reference="thm root multiplicity upper bound"} and [Theorem 9](#thm: -1 multiplicity for tournaments){reference-type="ref" reference="thm: -1 multiplicity for tournaments"}. We conclude the paper in with a few remarks and open problems regarding $A_D(t)$. # Preliminaries {#sec not and basic facts} ## Notation {#sec preliminaries} Graphs in this paper will always be finite and simple. An *oriented graph* is a digraph $D$ obtained by taking a graph $G$ and giving an orientation to each of its edges. In this case we say $G$ is the *underlying graph* of $D$ and that $D$ is an *orientation* of $G$. We will often denote the vertex set of a graph or digraph by $V(G)$ or $V(D)$ respectively, or simply $V$ whenever $G$ or $D$ is understood; and we similarly use the notation $E(G)$ and $E(D)$. For a subset $S\subseteq V(D)$, we write $D[S]$ for the induced subgraph of $D$ on $S$, and we write $D - S$ for the induced subgraph $D[V\setminus S]$. We write $\overline{S}$ for the complement $V(D)\setminus S$. For two sets $S,T\subseteq V(D)$, we write $e_D(S,T)$ for the number of arcs $uv$ whose tail $u$ is in $S$ and whose head $v$ is in $T$. An *integer composition* $\alpha$ of $n\in \mathbb{N}$ of length $\ell$ is a sequence $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_\ell)$ of positive integers such that $\alpha_1+\cdots+\alpha_\ell=n$, and for such a sequence we write $\alpha\vdash n$. The elements $\alpha_1,\dots,\alpha_\ell$ of $\alpha$ are sometimes called the *parts* of $\alpha$. If $\alpha_1 \geq \alpha_2 \geq \dots \geq \alpha_\ell$, we say that $\alpha$ is an *integer partition*. The *type* of an integer composition $\alpha$, denoted by $\lambda(\alpha)$, is the integer partition given by sorting the parts of $\alpha$ in weakly decreasing order. We will use the notation $(n)^m$ to denote the integer partition $({n,\dots,n})$ that has $m$ copies of $n$. An *ordered set partition* of a set $S$ is a sequence $P=(B_1,\dots,B_\ell)$ of mutually disjoint subsets $B_i$ of $S$ called *blocks* such that $\bigcup_{i=1}^\ell B_i= S$. The *type* of a set partition $P$ is the type of the integer composition $(|B_1|,\dots,|B_\ell)|$. An unordered set partition is an ordered set partition with the order forgotten. For positive integers $n_1,\dots,n_r$ and $n=n_1+\cdots+n_r$, the *$t$-multinomial coefficient* is defined to be $$\begin{bmatrix} n\\ n_1,\ldots,n_r \end{bmatrix}_t=\frac{[n]_t!}{[n_1]_t!\cdots [n_r]_t!}$$ where $[n]_t!=(1+t)(1+t+t^2)\cdots (1+t+t^2+\cdots+t^{n-1})$. ## Basic Properties {#sec basic facts} In this subsection, we prove a number of basic facts regarding $A_D(t)$ which will be used throughout the paper. We begin by establishing a list of elementary properties for $A_D(t)$. Some of these properties can be found in Even-Zohar [@Even-Zohar2017]; we have included their proofs for completeness. For this, we recall that a polynomial $f(t)=\sum_{k} a(k) t^k$ is *palindromic with center $d/2$* if $a(k)=a(d-k)$ for all $k$. **Proposition 10**. *Let $D=(V,E)$ be a directed graph with $n$ vertices and $m$ arcs.* 1. *The polynomial $A_D(t)$ is palindromic with center $m/2$.* 2. *If $(u,v),(v,u)\in E$, then $A_D(t)=t\cdot A_{D-(u,v)-(v,u)}(t)$.* 3. *If $D=\bigsqcup_{i=1}^r D_i$ is a disjoint union of digraphs of orders $n_1,\ldots,n_r$, then $$A_D(t)=\binom{n}{n_1,\ldots,n_r} \cdot \prod_{i=1}^r A_{D_i}(t).$$* 4. *If $D$ is formed by taking the disjoint union of digraphs $D_1,\ldots,D_r$ of orders $n_1,\ldots,n_r$ and then adding all arcs of the form $u\to v$ with $u\in V_i,\ v\in V_j$ and with $i<j$, then $$A_D(t)=\begin{bmatrix} n\\ n_1,\ldots,n_r\end{bmatrix}_t \cdot \prod_{i=1}^r A_{D_i}(t).$$* We note that (b) and (c) allows us to reduce our problems to studying digraphs $D$ which are orientations of connected graphs. *Proof.* For (a), consider the map $\varphi:\mathfrak{S}_D\to \mathfrak{S}_D$ which sends $\sigma\in \mathfrak{S}_D$ to $\tau\in \mathfrak{S}_D$ with $\tau(u)=n-\sigma(u)+1$ for all $u\in V$. It is not difficult to see that this is an involution and that $\tau$ has $k$ descents if and only if $\sigma$ has $m-k$ descents. From this it follows that $A_D(t)$ is palindromic with center $m/2$. For (b), we observe that for every permutation of $D$, exactly one of the arcs $(u,v)$ and $(v,u)$ will be a $D$-descent, giving the result. For (c), let $D$, $D_i$ and $n_i$ be as in the statement of the proposition. A permutation $\sigma\in \mathfrak{S}_D$ can be made by choosing an ordered set partition $\pi=(B_1,\dots,B_r)$ of $[n]$ of type $(n_1,\dots,n_r)$ and then choosing a bijection $\sigma_i:V(D_i)\to B_i$ for each $i\in [r]$. For each $i$, we can view $\sigma_i$ as an element of $\mathfrak{S}_{D_i}$. Since $D$ is a disjoint union of digraphs, we have $$\mathop{\mathrm{des}}_D(\sigma)=\mathop{\mathrm{des}}_{D_1}(\sigma_1)+\cdots +\mathop{\mathrm{des}}_{D_r}(\sigma_r).$$ Since there are $\binom{n}{n_1,\dots,n_r}$ ordered set partitions of type $(n_1,\dots,n_r)$, we have $$\begin{aligned} A_D(t)=\sum_{\sigma\in \mathfrak{S}_D}t^{\mathop{\mathrm{des}}_D(\sigma)} =\sum_{(B_1,\dots,B_r)} \prod_{i=1}^r\left(\sum_{\sigma_i\in \mathfrak{S}_{D_i}}t^{\mathop{\mathrm{des}}_{D_i}(\sigma_i)}\right) =\binom{n}{n_1,\dots,n_r}\prod_{i=1}^r A_{D_i} \end{aligned}$$ For (d), we can again view $\sigma\in \mathfrak{S}_D$ as a tuple $(\pi,\sigma_1,\dots,\sigma_r)$ of ordered set partition $\pi$ of type $(n_1,\dots,n_r)$ and permutations $\sigma_i\in \mathfrak{S}_{D_i}$. Let $\mathfrak{M}_{n_1,\dots,n_r}$ be the set of words $w$ with $n_1$ 1's, $n_2$ 2's, etc. Ordered set partitions $\pi$ of type $(n_1,\dots,n_r)$ are in natural bijection with words $w\in \mathfrak{M}_{n_1,\dots,n_r}$ by the map that sends $\pi$ to the word $w$ whose $i$-th letter is $j$ if $i\in B_j$. Then, a $D$-descent $(u,v)$ either has $u,v\in V_i$ or $u\in V_i, v\in V_j$ with $i<j$: the former are counted by $\mathop{\mathrm{des}}_{D_j}(\sigma_j)$, and the latter are counted by the pairs $(w_{\sigma(u)},w_{\sigma(v)})$ with $w$ the word in bijection with $\pi$. Hence, we have $$\mathop{\mathrm{des}}_D(\sigma)=\mathop{\mathrm{inv}}(w)+\mathop{\mathrm{des}}_{D_1}(\sigma_1)+\cdots +\mathop{\mathrm{des}}_{D_r}(\sigma_r).$$ Then we have $$\begin{aligned} A_D(t)=\sum_{\sigma\in \mathfrak{S}_D}t^{\mathop{\mathrm{des}}_D(\sigma)} =\sum_{w\in \mathfrak{M}_{n_1,\dots,n_r}}t^{\mathop{\mathrm{inv}}(w)} \prod_{i=1}^r\left(\sum_{\sigma_i\in \mathfrak{S}_{D_i}}t^{\mathop{\mathrm{des}}_{D_i}(\sigma_i)}\right) =\sum_{w\in \mathfrak{M}_{n_1,\dots,n_r}}t^{\mathop{\mathrm{inv}}(w)} \prod_{i=1}^r A_{D_i} \end{aligned}$$ The result then follows from the well-known result (see [@stanley_2011 Proposition 1.7.1]) that $$\sum_{w\in \mathfrak{M}_{n_1,\dots,n_r}}t^{\mathop{\mathrm{inv}}(w)}=\begin{bmatrix} n\\ n_1,\dots,n_r \end{bmatrix}_t.\qedhere$$ ◻ Next, we have a lemma which allows us to express the Eulerian polynomial of a digraph $D$ in terms of Eulerian polynomials of induced subgraphs of $D$. **Lemma 11**. *If $D=(V,E)$ is an $n$-vertex digraph and $k\in [n]$, then $$A_D(t) = \sum_{S\in \binom{V}{k}} \frac{t^{e_D(S,\overline{S})} + t^{e_D(\overline{S},S)}}{2}A_{D[S]}(t)A_{D - S}(t).$$* *Proof.* For ease of notation we assume $V=[n]$. Fix $\sigma\in \mathfrak{S}_{D}$ and let $S\subseteq V(D)$ be the set such that $\sigma(S)$ is the interval $[k]=\{1,\dots,k\}$. We observe that if $u\to v$ is a descent for $\sigma$ in $D$, then one of the following must hold: - Both $u$ and $v$ are in $S$ - Both $u$ and $v$ are in $\overline{S}$ - $v$ is in $S$ and $u$ is in $\overline{S}$ (since $\sigma(S)=[k]$) Therefore, if we set $\tau=\sigma|_{S}$ and $\tau'=\sigma|_{\overline{S}}$, we have $$\mathop{\mathrm{des}}_D(\sigma) = e_D(\overline{S},S)+\mathop{\mathrm{des}}_{D[S]}(\tau) + \mathop{\mathrm{des}}_{D - S}(\sigma')$$ Hence we have $$\begin{aligned} A_D(t) &=\sum_{\sigma\in \mathfrak{S}_n}t^{\mathop{\mathrm{des}}_D(\sigma)}\\ &= \sum_{S\in \binom{[n]}{k} }\sum_{\substack{\sigma\in \mathfrak{S}_n\\ \sigma(S)=[k]}}t^{\mathop{\mathrm{des}}_D(\sigma)}\\ &= \sum_{S\in \binom{[n]}{k}} \sum_{\substack{\tau\in \mathfrak{S}_S\\ \tau'\in \mathfrak{S}_{[n] - S} }}t^{e_D(\overline{S},S)+\mathop{\mathrm{des}}_{D[S]}(\tau) +\mathop{\mathrm{des}}_{D - S}(\tau') }\\ &= \sum_{S\in \binom{[n]}{k}} t^{e_D(\overline{S},S)}A_{D[S]}(t) A_{D - S}(t) \end{aligned}$$ If we repeat this same argument but consider $\sigma$ and $S$ with $\sigma(S)=\{n - k+1,\dots,n\}$, then we get $$A_D(t) = \sum_{S\in \binom{[n]}{k}} t^{e_D(S,\overline{S})}A_{D[S]}(t) A_{D - S}(t)$$ By adding these two expressions for $A_D(t)$ and dividing by 2, we find that $$A_D(t) = \sum_{S\in \binom{[n]}{k}} \frac{t^{e_D(S,\overline{S})} + t^{e_D(\overline{S},S)}}{2}A_{D[S]}(t)A_{D - S}(t).\qedhere$$ ◻ Finally, we consider a construction which will be useful for and . Given digraphs $D_1,D_2$ and a root vertex $v\in D_2$, the *rooted product digraph*, denoted $D_1\circ_v D_2$, is obtained by gluing a copy of $D_2$ at $v$ to each vertex of $D_1$, see for an example. This product was first defined by Godsil and McKay [@godsil1978new], and it turns out that this operation plays very nicely with the Eulerian polynomial. **Proposition 12**. *Let $D_1$ and $D_2$ be two digraphs on $m$ and $n$ vertices respectively. If $v\in D_2$, then $$A_{D_1\circ_v D_2}(t)=\frac{1}{m!}\binom{mn}{n,\ldots,n}\cdot A_{D_1}(t)A_{D_2}(t)^m.$$ In particular, the polynomial is the same for any choice of root $v\in D_2$.* *Proof.* To create a permutation $\sigma\in \mathfrak{S}_D$, we can do the following 1. Select a vector of permutations $(\sigma_1,\dots,\sigma_m)\in \prod_{i=1}^m\mathfrak{S}_{D_2}$. 2. Select an ordered set partition $\pi=(B_1,\dots,B_m)$ of $[mn]$ of type $(n)^m$. For each $i$, if $B_i=\{b_1<b_2,\dots<b_m\}$, we think of $\sigma_i$ as a map $\sigma:V(D_2)\to B_i$ via $\sigma_i(v_k)=b_j$ if $\sigma(v_k)=j$. 3. Select a permutation $\tau\in \mathfrak{S}_{D_1}$. 4. For each $u\in D_1$, if $\tau(u)=i$, assign the permutation $\sigma_j$ with the $i$-th largest value at the root $v$ to the copy of $D_2$ at $u$. Let $T$ be the set of tuples $(\sigma_1,\dots,\sigma_{m};\pi;\tau)$ of permutations $\sigma_i$ of $D_2$, ordered partitions $\pi$ of type $(n)^m$, and permutations $\tau$ of $D_1$. Then, the above defines a map $$\varphi:T\to \mathfrak{S}_{D}.$$ For a given $\rho\in \mathfrak{S}_m$, we observe that two elements $$(\sigma_1,\dots,\sigma_{m};B_1,\dots,B_m;\tau)\quad\text{\quad and \quad}(\sigma_{\rho(1)},\dots,\sigma_{\rho(m)};B_{\rho(1)},\dots,B_{\rho(m)};\tau)$$ of $T$ produce exactly the same element of $\mathfrak{S}_D$ under $\varphi$. Hence, $\varphi$ is an $m!$-to-1 map. For $\psi\in \mathfrak{S}_D$, let $(\sigma_1\dots,\sigma_{m};\pi;\tau)$ be an element of the preimage $\varphi^{-1}(\psi)$. The $D$-descents of $\psi$ come from either an edge in one of the copies of $D_2$ or an edge in $D_1$. The former are exactly the $D_2$-descents of the $\sigma_i$ (thinking of them as elements of $\mathfrak{S}_{D_2})$. For the latter, $(u,u')$ is a $D$-descent of $\psi$ between two vertices of $D_1$ if and only if the edge is a $D_1$ descent of $\tau$ because $\psi(u)=\sigma_j(v)$ if and only if $\sigma_j(v)$ is the $\tau(u)$-th largest element of $\{\sigma_1(v),\dots,\sigma_j(v)\}$ (thinking of $\sigma_i$ as a map $\sigma_i:V(D_2)\to B_i$). Hence, we have $$\mathop{\mathrm{des}}_D(\psi)=\mathop{\mathrm{des}}_{D_2}(\sigma_1)+\cdots+\mathop{\mathrm{des}}_{D_2}(\sigma_m)+\mathop{\mathrm{des}}_{D_1}(\tau).$$ Since there are $\binom{mn}{n,\ldots,n}$ ordered set partitions of $[mn]$ of type $(n)^m$ and $\varphi$ is $m!$-to-1, we have $$\begin{aligned} A_{D}(t)&=\sum_{\psi\in \mathfrak{S}_D}t^{\mathop{\mathrm{des}}_D(\psi)}\\ &=\frac{1}{m!}\sum_{(\sigma_1,\dots,\sigma_m;\pi;\tau)\in T}t^{\mathop{\mathrm{des}}_{D_2}(\sigma_1)+\cdots+\mathop{\mathrm{des}}_{D_2}(\sigma_m)+\mathop{\mathrm{des}}_{D_1}(\tau)} \\ &=\frac{1}{m!}\binom{nm}{n,\ldots,n}\left(\sum_{\sigma\in \mathfrak{S}_{D_2}}t^{\mathop{\mathrm{des}}_{D_2}(\sigma)}\right)^m\sum_{\tau\in \mathfrak{S}_{D_1}}t^{\mathop{\mathrm{des}}_{D_1}(\tau)}\\ &=\frac{1}{m!}\binom{nm}{n,\ldots,n}A_{D_2}(t)^m A_{D_1}(t).\qedhere \end{aligned}$$ ◻ # Combinatorial Interpretations of $\nu(G)$ {#sec -1 evaluation} In this section, we prove our results regarding $\nu(G):=|A_D(-1)|$ where $D$ is any orientation of the graph $G$. As noted in the introduction, $\nu(G)$ is intimately related to the quantity $\eta(G)$, which is the number of even sequences of $G$, i.e. the number of orderings $\pi$ of $V(G)$ such that the induced subgraphs $G[\pi_1,\ldots,\pi_i]$ have an even number of edges for all $i$. As a warmup, we formally establish the connection between alternating permutations and even sequences of paths mentioned in the introduction. **Proposition 13**. *Let $P_n$ denote the path graph with vertex set $[n]$. A permutation $\pi$ of $[n]$ is an even sequence of $P_n$ if and only if $\pi^{-1}$ is an alternating permutation.* *Proof.* First assume $\pi^{-1}$ is an alternating permutation (which in particular means $n$ is odd), and define $G_j:=P_n[\pi_1,\ldots,\pi_j]$. **Claim 14**. *Fix $1\le i\le n$ and let $j=\pi^{-1}_i$. If $i$ is even then $e(G_j)-e(G_{j-1})=2$, and if $i$ is odd then $e(G_j)-e(G_{j-1})=0$.* *Proof.* First assume $i$ is even (which in particular means $1<i<n$). Since $\pi^{-1}$ is an alternating permutation, $j:=\pi^{-1}_i>\pi^{-1}_{i-1},\pi^{-1}_{i+1}$. This means both of $i$'s neighbors in $P_n$ (namely $i-1$ and $i+1$) lie in $\{\pi_1,\ldots,\pi_{j-1}\}$, so $e(G_j)-e(G_{j-1})=2$ as claimed. Now assume $i$ is odd. Because $\pi^{-1}$ is an alternating permutation, $\pi^{-1}_i$ is less than all of the neighbors of $i$ in $P_n$, so $e(G_j)-e(G_{j-1})=0$ as desired. ◻ Because $e(G_j)-e(G_{j-1})$ is even for all $j$, and since $e(G_1)=0$ is even, we conclude that $e(G_j)$ is even for all $j$, proving that $\pi$ is an even sequence. Now assume $\pi$ is an even sequence, i.e. that the induced subgraphs $G_j:=P_n[\pi_1,\ldots,\pi_j]$ have an even number of edges for all $j$. In particular, $n$ must be odd in order for $G_n=P_n$ to have an even number of edges. **Claim 15**. *For all $1\le i<n$, if $i$ is odd then $\pi_i^{-1}<\pi_{i+1}^{-1}$, and otherwise $\pi_i^{-1}>\pi_{i+1}^{-1}$.* *Proof.* We prove the result by induction on $i$ starting with the base case $i=1$. Assume for contradiction that $\pi_2^{-1}<\pi_1^{-1}:=j$. This implies $e(G_j)-e(G_{j-1})=1$ (since $\pi_j=1$ has exactly one neighbor amongst the set $\{\pi_1,\ldots,\pi_{j-1}\}$, namely $2$), contradicting $e(G_j),e(G_{j-1})$ both being even. Thus we must have $\pi_1^{-1}<\pi_2^{-1}$. Inductively assume we have proven the result up to some value $i>1$. If $i$ is odd and $\pi_{i+1}^{-1}<\pi_i^{-1}:=j$, then $e(G_j)-e(G_{j-1})=1$ (since $\pi_j=i$ has a unique neighbor in $\{\pi_1,\ldots,\pi_{j-1}\}$, namely $\pi_{i+1}^{-1}$ due to the inductive hypothesis $\pi_{i-1}^{-1}>\pi_i^{-1}$), a contradiction. If $i$ is even and $\pi_{i+1}^{-1}>\pi_i^{-1}:=j$, then again $e(G_j)-e(G_{j-1})=1$ (since $\pi_j=i$ has a unique neighbor in $\{\pi_1,\ldots,\pi_{j-1}\}$, namely $\pi_{i-1}^{-1}$ due to the inductive hypothesis $\pi_{i-1}^{-1}<\pi_i^{-1}$). With this we conclude the claim. ◻ This claim, together with the observation that $n$ must be odd, shows that $\pi^{-1}$ is an alternating permutation, completing the proof. ◻ ## Basic Properties of $\nu(G)$ {#subsec:basic} In this subsection we prove several basic properties of $\nu(G)$, some of which will be needed for the proof of . We begin with a basic but important observation. **Lemma 16**. *If $G$ has an odd number of edges, then $\nu(G)=0$.* *Proof.* Let $m=e(G)$ and $D$ be any orientation of $G$. By Proposition [Proposition 10](#prop basic facts){reference-type="ref" reference="prop basic facts"}(a), $A_D(t)$ is palindromic with center $(m-1)/2$. Since $(-1)^k = -(-1)^{m - k}$ for $m$ odd, it follows that $A_D(-1)=0$. ◻ The remainder of our proofs for this section rely heavily on the following special case of . **Corollary 17**. *If $D$ is an $n$-vertex digraph, then $$A_{D}(t) = \sum_{v\in V} \frac{t^{\deg_D^+(v)} + t^{\deg_D^-(v)}}{2}A_{D - v}(t)$$* *Proof.* Applying for $k = 1$ gives $$A_D(t) = \sum_{S\in \binom{V}{1}} \frac{t^{e_D(S,\overline{S})} + t^{e_D(\overline{S},S)}}{2}A_{D[S]}(t)A_{D - S}(t).$$ Note that for $S=\{v\}$, we have $e_D(S,\overline{S}) = \deg_D^+(v)$, $e_D(\overline{S},S) = \deg_D^-(v)$, and $A_{D[S]}(t)=1$, giving the result. ◻ For ease of notation, we will often write the summation symbol $\sum_{v\in V}$ in simply as $\sum_v$ or even $\sum$. This result immediately gives the following. **Corollary 18**. *We have $\nu(G)\le \sum_v \nu(G-v)$.* *Proof.* When $t=-1$, each term in the summation of has coefficients in $\{-1,0,1\}$. Taking absolute values on both sides and using the triangle inequality gives for any orientation $D$ of $G$ that $$\nu(G)=|A_D(-1)|\le \sum_v |A_{D-v}(-1)|=\sum_v \nu(G-v).$$ ◻ An analog of the result above holds for even sequences. **Lemma 19**. *If $G$ is a graph with an odd number of edges, then $\eta(G)=0$. Otherwise $\eta(G)=\sum_v \eta(G-v)$.* *Proof.* If $e(G)$ is odd then there exist no even sequences (since $e(G[\pi_1,\ldots,\pi_n])=e(G)$ is always odd), so $\eta(G)=0$. Assume now that $e(G)$ is even and let $\eta_v(G)$ denote the number of even sequences of $G$ with $v_n=v$. Then $\eta(G)=\sum_v \eta_v(G)$, and it is not difficult to see $\eta_v(G)=\eta(G-v)$ (since $e(G[\pi_1,\ldots,\pi_{n-1},v])=e(G)$ is even for any permutation $\pi$ of $V(G)-v$). This gives the result. ◻ Finally, we introduce two graph operations that play nicely with $\nu(G)$. Given a set of graphs $G_1,\ldots,G_r$ on disjoint vertex sets, the *join* $\bigvee_{i=1}^r G_i$ is the graph obtained by taking the disjoint union of the $G_i$ graphs and then adding all possible edges between each of the $G_i$ graphs. Given graphs $G_1,G_2$ and a root vertex $v\in G_2$, the *rooted product graph* $G_1\circ_v G_2$ is obtained by gluing a copy of $G_2$ at $v$ to each vertex of $G_1$. With all this established, we can state the following results involving $\nu(G)$. **Proposition 20**. *Let $G$ be an $n$-vertex graph.* 1. *We have $\nu(G)\le \eta(G)$.* 2. *We have $\nu(G)\le \sum_{v} \nu(G-v)$.* 3. *If $G=\bigsqcup_{i=1}^r G_i$ is a disjoint union of graphs of orders $n_1,\ldots,n_r$, then $$\nu(G)=\binom{n}{n_1,\ldots,n_r} \cdot \prod_{i=1}^r \nu(G_i).$$* 4. *If $G=\bigvee_{i=1}^r G_i$ is the join of graphs of orders $n_1,\ldots,n_r$, then $$\nu(G)=\left|\begin{bmatrix} n\\ n_1,\ldots,n_r\end{bmatrix}_{-1} \right|\cdot \prod_{i=1}^r \nu(G_i).$$* 5. *If $G=G_1\circ_v G_2$ with $|V(G_i)|=n_i$, then $$\nu(G)=\frac{1}{n_1!}{n_1n_2\choose n_2,\ldots,n_2} \nu(G_1)\nu(G_2)^{n_1}.$$* *Proof.* Note that (b) is , (c) and (d) follow from , and (e) follows from . It thus remains to prove (a), which we do by induction on $V(G)$. The base case is trivial, so assume we have proven the result up to some order $n$ and let $G$ be a graph of order $n$. If $G$ has an odd number of edges then $\nu(G)=\eta(G)=0$ by Lemmas [Lemma 16](#prop odd edges to evaluation at -1){reference-type="ref" reference="prop odd edges to evaluation at -1"} and [Lemma 19](#lem:etaRecurrence){reference-type="ref" reference="lem:etaRecurrence"}. Otherwise by , $$\nu(G)\le \sum_{v} \nu(G-v)\le \sum_{v} \eta(G-v)=\eta(G),$$ where the second inequality used the inductive hypothesis and the equality used . ◻ has a number of nice consequences. For example, (c) and (d) imply that to determine $\nu(G)$ for all graphs $G$, it suffices to do this for graphs $G$ such that $G$ and its complement are both connected. We also have the following immediate consequences. **Corollary 21**. *Let $G$ be an $n$-vertex graph.* 1. *If $G$ has a component with an odd number of edges, then $\nu(G)=0$.* 2. *If every vertex of $G$ has odd degree, then $\nu(G)=0$.* 3. *If $G$ has a vertex $v$ of degree $n-1$, then $\nu(G)=0$ if $n$ is even, and otherwise $\nu(G)=\nu(G-v)$.* *Proof.* Part (a) follows from (c) and the fact that $\nu(G)=0$ whenever $G$ has an odd number of edges by (or alternatively by (a)). For (b), we observe that for any ordering $\pi$ of $V(G)$, either the graph $G=G[\pi_1,\ldots,\pi_n]$ or the graph $G[\pi_1,\ldots,\pi_{n-1}]$ has an odd number of edges. Thus $\eta(G)=0$, and hence $\nu(G)=0$ by (a). For (c), we observe that $G$ is the join of $K_1$ together with $G-v$, so by (d) we have $\nu(G)=|[n]_{-1} |\cdot 1\cdot \nu(G-v)$, and this equals 0 if $n$ is even and otherwise equals $\nu(G-v)$ as desired. ◻ ## Proof of {#subsec:interpretation} In this section, we prove , which we recall says that if $G$ is a graph that is either bipartite, complete multipartite, or a blowup of a cycle, then $\nu(G)=\eta(G)$. The proofs for each of these cases follows the same basic strategy: We first show that for some "natural" orientation $D$ of $G$, we can easily predict the sign of $A_D(-1)$. From this we deduce $\nu(G)=\sum \nu(G-v)$, and hence that $\nu(G)=\eta(G)$ since the statistics $\nu,\eta$ satisfy the same recurrence relation. We begin with the following "natural" orientations for bipartite graphs. **Lemma 22**. *Let $D$ be a digraph such that one can partition its vertex set into $U\cup V$ such that every arc $u\to v$ of $D$ has $u\in U$ and $v\in V$. Then $$A_D(-1)\ge 0,$$ and if $D$ has an even number of arcs, then $$A_D(-1)=\sum_{v\in V(D)}A_{D-v}(-1).$$* *Proof.* We first establish the equality for $D$ with an even number of arcs. By Corollary [Corollary 17](#lemma remove a vertex){reference-type="ref" reference="lemma remove a vertex"}, we have $$A_D(-1)=\sum_{u\in U}\frac{(-1)^{\deg^+(u)}+1}{2}A_{D-u}(-1)+\sum_{v\in V}\frac{1+(-1)^{\deg^-(v)}}{2}A_{D-v}(-1).$$ We claim that for $u\in U$, we have $\frac{(-1)^{\deg^+(u)}+1}{2}A_{D-u}(-1)=A_{D-u}(-1)$. Indeed, this is immediate if $\deg^+(u)$ is even. If $\deg^+(u)$ is odd, then since $D$ has an even number of arcs, $D-u$ has an odd number of arcs. By , $A_{D-u}(-1)=0$, so again the claim trivially holds. An analogous result holds for the $v$ terms, and applying these claims to the inlined equation above gives the result. We next prove $A_D(-1)\ge 0$ by using induction on $|V(D)|$, the base case being trivial. If $D$ has an odd number of arcs then this quantity is 0 by , so we may assume $D$ has an even number of arcs. Thus, by the result proven above, we have $$A_D(-1)=\sum_{v\in V(D)}A_{D-v}(-1)\ge 0,$$ with the last step using the inductive hypothesis on each of the digraphs $D-v$ (each of which continues to satisfy the hypothesis of the lemma). This completes the proof. ◻ **Corollary 23**. *If $G$ is a bipartite graph with an odd number of edges, then $\nu(G)=0$, and otherwise $\nu(G)=\sum_v \nu(G-v)$.* *Proof.* The result when $G$ has an odd number of edges follows from , so assume $G$ has an even number of edges, and let $D$ be an orientation of $G$ as in the previous lemma. Having $A_D(-1)\ge 0$ implies $$\nu(G)=A_D(-1)=\sum_v A_{D-v}(-1)=\sum_v\nu(G-v),$$ where the second equality used the second part of and the last equality used $\nu(G-v)=A_{D-v}(-1)$ since this latter quantity is non-negative by . ◻ We next turn to orientations of complete multipartite graphs. We begin by establishing the following simple criteria for determining if $\nu(G)=0$. **Lemma 24**. *Let $G$ be a complete multipartite graph on $V_1\cup \cdots \cup V_r$. If $|V_i|$ is odd for at least two values of $i$, then $\nu(G)=\eta(G)=0$.* *Proof.* We will show in this case that $\eta(G)=0$, i.e. that there exist no even sequences for $G$. From this it will follow from $\nu(G)\le \eta(G)$ of (a) that $\nu(G)=0$ as well. Assume for contradiction that $\pi$ is an even sequence of $G$. Let $j$ be the smallest integer such that $|V_i\cap \{\pi_1,\ldots,\pi_j\}|$ is odd for at least two values of $i$, noting that such a (smallest) integer exists since this holds for $j=n$ by hypothesis. Since $j$ is the smallest integer with this property, there must be exactly two integers $i$ such that $|V_i \cap \{\pi_1,\ldots,\pi_j\}|$ is odd, say this holds for $i=a,b$. Since $G$ is complete multipartite, the number of edges of $G[\pi_1,\ldots,\pi_j]$ is exactly $$\sum_{i<i'} |V_i\cap \{\pi_1,\ldots,\pi_j\}|\cdot |V_{i'}\cap \{\pi_1,\ldots,\pi_j\}|.$$ Exactly one term in this sum is odd, namely the one with $\{i,i'\}=\{a,b\}$. This implies $G[\pi_1,\ldots,\pi_j]$ has an odd number of edges, contradicting $\pi$ being an even sequence. ◻ We next turn to the "natural" orientation of complete multipartite graphs. **Lemma 25**. *Let $D$ be a digraph with vertex set $V_1\cup \cdots \cup V_r$ and arcs $u\to v$ if and only if $u\in V_i,\ v\in V_j$ and $i<j$. Then $$A_D(-1)\ge 0,$$ and if $|V_i|$ is odd for at most one value of $i$, then $$A_D(-1)=\sum_v A_{D-v}(-1).$$* *Proof.* As in the bipartite case, we begin by establishing the equality. Suppose at most one of the parts of $D$ has odd size. By we have $$A_D(-1)=\sum_{v\in V(D)} \frac{(-1)^{\deg^+(v)}+(-1)^{\deg^-(v)}}{2} A_{D-v}(-1),$$ so it suffices to show that for each $v\in V(D)$, either $\deg^+(v),\deg^-(v)$ are both even or $A_{D-v}(-1)=0$. Suppose $v\in V_i$. Then $\deg^+(v)=|\bigcup_{j>i} V_j|$ and $\deg^-(v)=|\bigcup_{j<i} V_j|$. If $|V_{i'}|$ is even for all $i'\ne i$, then $\deg^+(v)$ and $\deg^-(v)$ will be even. If $|V_{i'}|$ is odd for some $i'\ne i$, then $|V_i|$ must be even by hypothesis, so $|V_i-v|$ is odd. This means $D-v$ is the orientation of a complete multipartite graph with two parts of odd size, namely $V_i-v$ and $V_{i'}$. By the previous lemma this implies $A_{D-v}(-1)=0$, completing the proof of this part. The proof that $A_D(-1)\ge 0$ follows essentially the same inductive proof as in . We omit the details. ◻ From these lemmas, the proof of carries over to give the following. **Corollary 26**. *If $G$ is a complete multipartite graph with at least two parts of odd size, then $\nu(G)=0$, and otherwise $\nu(G)=\sum_v \nu(G-v)$.* Finally, we prove our lemmas for graphs $G$ which are blowups of cycles, which we recall means that one can partition the vertex set of $G$ into sets $V_1,\ldots,V_r$ (which we will call the *parts* of $G$) such that $uv$ is an edge of $G$ if and only if $u\in V_i$ and $v\in V_{i+1}$ for some $i$, where the indices are taken modulo $r$. Again we begin with a simple criteria for having $\nu(G)=0$. **Lemma 27**. *Let $G$ be a blowup of a cycle with parts $V_1,\ldots,V_r$. If $|V_i| |V_{i+1}|$ is odd for an odd number of integers $1\le i\le r$, then $\nu(G)=\eta(G)=0$.* *Proof.* By the definition of $G$ being a blowup of a cycle, we have $e(G)=\sum_{i=1}^r |V_i||V_{i+1}|$. Thus if $|V_i|V_{i+1}|$ is odd for an odd number of integers, then $e(G)$ is odd. This implies $\eta(G)=0$ and hence $\nu(G)=0$ by [Proposition 20](#prop:nuGeneral){reference-type="ref" reference="prop:nuGeneral"}(a). ◻ Our analog of Lemmas [Lemma 22](#lem:bipartiteOrientation){reference-type="ref" reference="lem:bipartiteOrientation"} and [Lemma 25](#lem:multipartiteOrientation){reference-type="ref" reference="lem:multipartiteOrientation"} will be slightly more complex in the setting of blowups of cycles. For this, we define our "natural" directed analog of blowups of cycles as follows: we say a digraph $D$ is a *blowup of a directed $r$-cycle* if it has vertex set $V_1\cup \cdots \cup V_r$ and arcs of the form $u\to v$ if and only if $u\in V_i$ and $v\in V_{i+1}$ for some $i$. For such a digraph, we define $m(D)$ to be the number of integers $1\le i\le r$ such that $|V_i|$ and $|V_{i+1}|$ are both odd, i.e. such that $|V_i||V_{i+1}|$ is odd. **Lemma 28**. *Let $D$ be a blowup of an $r$-cycle.* 1. *If $m(D)$ is odd, then $A_D(-1)=0$.* 2. *If $m(D)$ is even, then $$(-1)^{m(D)/2} A_D(-1)\ge 0$$ and $$(-1)^{m(D)/2}A_D(-1)=\sum_{v} (-1)^{m(D-v)/2} A_{D-v}(-1)$$* Note that the first statement implies that when $m(D-v)$ is odd, $A_{D-v}(-1)=0$. Hence, the sum in the second statement is a well-defined real number. *Proof.* For (a), note that if $m(D)$ is odd, then $D$ is an orientation of a graph $G$ as in , so $A_D(-1)=0$ as desired. It thus remains to prove (b), and we begin by establishing the sum. By , we have $$A_D(-1)=\sum_{v\in V(D)} \frac{(-1)^{\deg^+(v)}+(-1)^{\deg^-(v)}}{2} A_{D-v}(-1),$$ so to prove the desired sum, it suffices to show that for each $v\in V(D)$, either $A_{D-v}(-1)=0$, or $\deg^+(v),\deg^-(v)$ both have the same parity as $(m(D-v)-m(D))/2$. Note that by (a) we have $A_{D-v}(-1)=0$ if $m(D-v)$ is odd, so from now on we may assume $m(D-v)$ is even (and hence it makes sense to talk about the parity of $(m(D-v)-m(D))/2$ since $m(D)$ is assumed to be even). Suppose $v\in V_i$, which means $\deg^+(v)=|V_{i+1}|$ and $\deg^-(v)=|V_{i-1}|$. If $|V_{i-1}|\not\equiv_2 |V_{i+1}|$, then $m(D-v)=m(D)+1$ if $|V_i|$ is even and $m(D-v)=m(D)-1$ if $|V_i|$ is odd. Since $m(D)$ is even, $m(D-v)$ is odd in either case, which we assumed not to be the case. Thus we must have $|V_{i-1}|\equiv_2 |V_{i+1}|$. If both $|V_{i-1}|$ and $|V_{i+1}|$ are even, then $m(D-v)=m(D)$, and hence $\deg^+(v)=|V_{i+1}|,\deg^-(v)=|V_{i-1}|$ have the same parity as $(m(D-v)-m(D))/2$. If instead both these quantities are odd, then $m(D-v)=m(D)+2$ if $|V_i|$ is even and $m(D-v)=m(D)-2$ if $|V_i|$ is odd. Hence, we have $m(D-v)-m(D)=\pm 2$ and so $$(m(D-v)-m(D))/2=\pm 1\equiv_2 |V_{i\pm 1}|=\deg^{\pm}(v),$$ so again in this case the desired result follows. This completes the proof of the equality. The proof that $(-1)^{m(D)/2}A_D(-1)\ge 0$ again follows from essentially the same inductive proof as in . More precisely, by the equality we just proved we find $$(-1)^{m(D)/2} A_D(-1)=\sum_v (-1)^{(m(D-v)} A_{D-v}(-1)\ge0,$$ with this last inequality using the inductive hypothesis. ◻ Again these lemmas give the following corollary. **Corollary 29**. *If $G$ is a blowup of a cycle such that $|V_i| |V_{i+1}|$ is odd for an even number of integers $i$, then $\nu(G)=0$, and otherwise $\nu(G)=\sum_v \nu(G-v)$.* *Proof.* implies the first half of the result. Otherwise, if $D$ is the directed blowup of an $r$-cycle whose underlying graph is $G$, then $m(D)$ is even and inductively gives $$\nu(G)=(-1)^{m(D)/2} A_D(-1)=\sum_v (-1)^{m(D-v)/2}A_{D-v}(-1)=\sum_v \nu(G-v),$$ completing the proof. ◻ We are now ready to prove our main result for this subsection. *Proof of .* We aim to show that $\nu(G)=\eta(G)$ whenever $G$ is bipartite, complete multipartite, or a blowup of a cycle. We first consider the case that $G$ is bipartite. We prove this result by induction on $|V(G)|$, the base case $\nu(K_1)=\eta(K_1)=1$ being trivial. By and , if $G$ has an odd number of edges then $\nu(G)=\eta(G)=0$, and otherwise $$\nu(G)=\sum_v \nu(G-v)=\sum_v \eta(G-v)=\eta(G),$$ where the middle equality used the inductive hypothesis (and that $G-v$ is bipartite whenever $G$ is). Nearly identical arguments work for the cases when $G$ is either complete multipartite or a blowup of a cycle, completing the proof. ◻ It is tempting to try to generalize the approach of this subsection by finding "natural" orientations of other graphs in order to show $\nu(G)=\sum \nu(G-v)$; see for example . However, we emphasize that shows that the inductive proof of can not be extended beyond the class of graphs which are bipartite, complete multipartite, or blowups of cycles. Before moving on, we note the following cute consequence of our results for $\nu(G)$ which gives a combinatorial interpretation for $t$-multinomial coefficients evaluated at $-1$. **Corollary 30**. *If $n_1,\ldots,n_r$ are positive integers and $n=n_1+\cdots+n_r$, then $$\left|\begin{bmatrix} n\\ n_1,\ldots,n_r \end{bmatrix}_{-1}\right|=\begin{cases} 0&\text{at least two parts of odd size}\\ \binom{\lfloor n/2\rfloor}{\lfloor n_1/2\rfloor,\dots,\lfloor n_r/2\rfloor}&\text{otherwise.} \end{cases}$$* It is likely that is already well known in the literature, though the only concrete source we are aware of is [@ajose_2007 Section 5.2] which solves the case $r=2$ (from which the general result can be derived). *Sketch of Proof.* Let $G$ be the complete multipartite graph with parts of sizes $n_1,\ldots,n_r$. Since $G$ is the join of independent sets of size $n_i$, (d) implies $\nu(G)=\left|\begin{bmatrix} n_1+\cdots +n_r\\ n_1,\ldots,n_r \end{bmatrix}_{-1}\right| \prod n_i!$. On the other hand, by using ideas similar to those in , one can work out that the number of even sequences $\eta(G)$ equals $\prod n_i!$ times the number of words $w$ consisting of $n_1$ $1$'s, $n_2$ $2$'s, and so on, with the additional property that each prefix $w_1\cdots w_i$ has all but at most one letter appearing an even number of times. This is equivalent to saying that $w_i=w_{i+1}$ for all odd $i<\sum n_j$, so the number of these words is 0 if $n_i$ is odd for at least two values of $i$, and otherwise equals $\binom{\lfloor n/2\rfloor}{\lfloor n_1/2\rfloor,\dots,\lfloor n_r/2\rfloor}$. By we have $\nu(G)=\eta(G)$, giving the desired result. ◻ ## Proof of {#subsec:induced} In this subsection we characterize which graphs have $\nu(G')=\eta(G')$ for all induced subgraphs $G'\subseteq G$. For this the following will be crucial. **Definition 31**. We define the *odd pan graph* $C_{2k+1}^*$ to be the graph obtained by taking the odd cycle $C_{2k+1}$ and then adding a new vertex $u$ adjacent to exactly one vertex of $C_{2k+1}$; see . We say that a graph $G$ is *odd pan-free* if it contains no induced subgraph which is isomorphic to $C_{2k+1}^*$ for any $k\ge 1$. We note that some authors use the term "odd pan" only to refer to $C_{2k+1}^*$ when $k\ge 2$, but we emphasize that we include the paw graph $C_3^*$ in our definition of odd pans. Our motivation for this definition is the following lemma. **Lemma 32**. *We have $\nu(C_{2k+1}^*)\ne \eta(C_{2k+1}^*)$ for all $k\ge 1$.* *Proof.* We prove this by showing $\nu(C_{2k+1}^*)=0$ and $\eta(C_{2k+1}^*)>0$. Let $v_1,\ldots,v_{2k+1}$ denote the vertices of the odd cycle of $C_{2k+1}^*$ and $u$ the pendant vertex, say with $u$ adjacent to $v_{k+1}$. Define the sequence $(x_1,\ldots,x_{2k+2})$ by having $x_1=u$ and $x_i=v_{k+2i-2}$ for all $i\ge 1$, with these indices for $v$ written modulo $2k+1$; see . The first $k+1$ elements $$\{x_1,\ldots,x_{k+1}\}=\{u,v_{k+2},v_{k+4},\ldots,v_{k-1}\}$$ form an independent set, and hence $C^*_{2k+1}[x_1,\ldots,x_i]$ has no edges for all $1\le i\le k+1$. For $i>k+1$, we have that $$k+2i-2=k+2(i-k-1)+2(k+1)-2\equiv_{2k+1}k+2(i-k-1)-1.$$ Therefore, $x_i=v_{k+2i-2}=v_{k+2(i-k-1)-1}$ is adjacent to $x_{i-k-1}=v_{k+2(i-k-1)-2}$ and $x_{i-k}=v_{k+2(i-k)-2}$. Thus $C_{2k+1}^*[x_1,\ldots,x_i]$ is even for all $i$, so $(x_1,\ldots,x_{2k+2})$ is an even sequence and $\eta(C_{2k+1}^*)>0$. To show $\nu(C_{2k+1}^*)=0$, let $D$ be an orientation of $C_{2k+1}^*$ such that $u\to v_{k+1},$ $v_1\to v_{2k+1}$, and for $0\le i\le k-1$, $v_{k+1+i}\to v_{k+1+(i+1)}$ and $v_{k+1-i}\to v_{k+1-(i+1)}$; see . Define a map $\iota:\mathfrak{S}_V\to \mathfrak{S}_V$ sending $\sigma$ to $\tau:=\iota(\sigma)$ defined by setting $\tau(u)=\sigma(u)$ and $$\tau(v_{k+1+i})=\begin{cases} \sigma(v_{k+1-i})&1\leq |i|\leq k\\ \sigma(v_{k+1})&\text{otherwise} \end{cases}$$ Then, $\iota$ is clearly an involution with no fixed points. By the orientation $D$ of the graph, we have that $(v_{k+1\pm i},v_{k+1\pm (i+1)})$ is a descent of $\tau$ if and only $(v_{k+1\mp i},v_{k+1\mp(i+1)})$ is a descent of $\sigma$ and that $(u,v_{k+1})$ is a descent of $\tau$ if and only if $(u,v_{k+1})$ is a descent of $\sigma$. Finally, we have that $(v_1,v_{2k+1})$ is a descent of $\tau$ if and only if $(v_1,v_{2k+1})$ is not a descent of $\sigma$. Thus, $\iota$ changes the number of descents by exactly 1 and hence it is a sign-reversing involution, proving $A_D(-1)=0$. ◻ Recall that says that if $G$ is a connected graph, then $\nu(G')=\eta(G')$ for all induced subgraphs $G'\subseteq G$ if and only if $G$ is bipartite, complete multipartite, or a blowup of a cycle. In view of the lemma above, it will suffice to prove the following structural graph theory lemma. **Proposition 33**. *If $G$ is a connected graph, then $G$ is odd pan-free if and only if it is either bipartite, complete multipartite, or a blowup of a cycle.* We will prove this proposition through the following two lemmas. **Lemma 34**. *If $G$ be a connected graph which is odd pan-free and which contains a triangle, then $G$ is a complete multipartite graph.* *Proof.* Let $r$ be the maximum size of a clique of $G$, and note that $r\geq 3$ by hypothesis. Let $H$ be a maximal induced subgraph of $G$ which is isomorphic to a complete $r$-partite graph with non-empty parts, say with parts $V_1,\ldots,V_r$. Note that any $r$-clique of $G$ is trivially a complete $r$-partite induced subgraph, so such a maximal induced subgraph exists. We claim that $H=G$. Suppose not, and let $u\in G\setminus H$. Since $G$ is connected, there is a path from $u$ to $H$, so we shall assume that $u$ is adjacent to $H$. Now, $u$ is not adjacent to some $v_i\in V_i$ for all $i$, as this would imply that $u$ together with the $v_i$ form an $(r+1)$-clique in $G$. Hence without loss of generality, we may assume $u$ is not adjacent to any vertex in $V_1$ and that it is adjacent to some $v_2\in V_2$. If $u$ is not adjacent to some $v_k\in V_k$ for $k\geq 3$, then $u,v_1,v_2,v_k$ forms a copy of $C_3^*$ in $G$, which is a contradiction to $G$ being odd pan-free. Thus, $u$ is adjacent to every element of $V_3,\dots,V_r$, and critically we observe that $V_3\ne \emptyset$ since $r\ge 3$. We claim that $u$ is adjacent to every element of $V_2$. Suppose there is some $v_2'\in V_2$ which is not adjacent to $u$. Since $V_3$ is nonempty, we can take any $v_3\in V_3$ (which is adjacent to $u$) and form a $C_3^*$ out of $u,v_1,v_2',v_3$, which is a contradiction. Thus $u$ is adjacent to every element of $V_2$, as well as every element of $V_3,\ldots,V_r$, and is not adjacent to any element of $V_1$. This means $\{u\}\cup V_1,V_2,\dots,V_r$ forms an induced complete $r$-partite subgraph of $G$ that contains $H$, a contradiction to the maximality of $H$. We conclude that $H=G$, completing the proof. ◻ The next lemma deals with the case when $G$ is triangle-free. Here we recall that a graph is a blowup of a cycle if it has vertex set $V_1\cup \cdots \cup V_r$ and edges $uv$ if and only if $u\in V_i$ and $v\in V_{i+1}$ for some $i$. **Lemma 35**. *If $G$ be a connected graph which is odd pan-free and which is triangle-free but not bipartite, then $G$ is a blowup of a cycle.* *Proof.* Assume that the shortest odd cycle of $G$ has length $2k+1$, noting that such a cycle exists with $2k+1\ge 5$ by hypothesis of $G$ being non-bipartite and triangle-free. Let $H$ be a maximal induced subgraph of $G$ which is isomorphic to a blowup of a cycle of length $2k+1$, and let $V_1,\ldots,V_{2k+1}$ be its parts. We claim that $H=G$. Suppose not, and let $u\in G\setminus H$. Since $G$ is connected, there is a path from $u$ to $H$, so we shall assume that $u$ is adjacent to $H$, say that it is adjacent to $v_1\in V_1$. We claim that $u$ is not adjacent to any vertex in $V_2\cup V_{2k+1}$. Indeed, if $u$ was adjacent to some $v_2\in V_2$, then $u,v_1,v_2$ would form a triangle in $G$, a contradiction. A symmetric argument shows $u$ can not be adjacent to any vertex in $V_{2k+1}$. We claim that $u$ is not adjacent to any vertex in $V_i$ for $4\le i\le 2k-1$. Indeed, assume for contradiction that $u$ is adjacent to some $v_i\in V_i$, and for each $j\ne 1,i$ let $v_j$ be some vertex in $V_j$. Observe that if $i$ is odd, then the vertices $v_1,u,v_i,v_{i+1},\ldots,v_{2k+1}$ form an odd cycle of length $2k+1-(i-3)$ (since it excludes vertices from $V_2,\ldots,V_{i-1}$ but includes $u$), contradicting $G$ having no odd cycles of length shorter than $2k+1$. Similarly if $i$ is even then $v_1,u,v_i,v_{i-1},\ldots,v_2$ gives a contradiction. We claim that $u$ can not be adjacent to vertices in both $V_3$ and $V_{2k}$. Indeed, say it were adjacent to some $v_3\in V_3$ and $v_{2k}\in V_{2k}$ and let $v_i\in V_i$ for all other $i$. Then $u,v_3,v_4,\ldots,v_{2k}$ is a cycle of length $2k-1$ in $G$, a contradiction. We claim that there exists some $i\ne 1$ such that $u$ is adjacent to every vertex of $V_i$. Indeed, if for all $i\ne 1$ there existed a $v_i\in V_i$ which $u$ was not adjacent to, then $u,v_1,\ldots,v_{2k+1}$ would induce a $C_{2k+1}^*$ in $G$, a contradiction. With all of the claims above, we can assume $u$ is adjacent to some $v_1\in V_1$, every vertex of $v_3\in V_3$, and that it is adjacent to no vertices in $\bigcup_{i\ne 1,3} V_i$. A symmetric argument to the previous claim shows that $u$ must be adjacent to every vertex of $V_1$. Hence $V_1,V_2\cup \{u\},V_3,\ldots,V_{2k+1}$ induce a larger blowup of a cycle of length $2k+1$ in $G$, a contradiction. We conclude that $H=G$ as desired. ◻ With these two lemmas we can easily prove , and again we recall that this immediately implies when combined with . *Proof of .* It is straightforward to verify that complete multipartite graphs, blowups of cycles, and bipartite graphs are all odd pan-free (with this result also implicitly following from and ). If $G$ is a connected odd pan-free graph which contains a triangle, then implies that $G$ is complete multipartite. Otherwise $G$ is either bipartite or implies that $G$ is a blowup of a cycle, completing the proof. ◻ # Optimal bounds on $\nu(T)$ for trees {#sec bounds for nu on tree} Here we prove , which we recall says that if $T$ is a tree on $2n+1$ vertices, then $$n! 2^n\le \nu(T)=\eta(T)\le (2n)!,$$ with equality holding in the lower bound if and only if $T$ is a hairbrush, and equality holding in the upper bound if and only if $T$ is a star. To aid with our proofs, given a tree $T$, we define $$\widetilde{X}(T)=\{x\in V(T):\textrm{each component of }T-x\textrm{ has an even number of edges}\},$$ and we will denote this simply by $\widetilde{X}$ whenever $T$ is understood. Our motivation for this definition is the following. **Lemma 36**. *If $T$ is a tree with an even number of edges, then $$\nu(T)=\sum_{x\in \widetilde{X}} \nu(T-x).$$* *Proof.* By , we have $$\nu(T)=\sum_{x\in V(T)} \nu(T-x)=\sum_{x\in \widetilde{X}} \nu(T-x)+\sum_{x\in V(T)\setminus \widetilde{X}} \nu(T-x)=\sum_{x\in \widetilde{X}} \nu(T-x),\label{eq:X}$$ where the last equality follows from (a). ◻ With this lemma in mind, the idea for the proofs of the upper and lower bounds is as follows: we first apply and then use induction to bound each of the terms $\nu(T-x)$ in the sum. Finally, we bound our total sum in terms of $|\widetilde{X}|$ and show that equality can only occur when $|\widetilde{X}|=1$. Throughout our proofs, we make heavy use of the fact that if $T'$ is a graph on $2n$ vertices with connected components $T_1,\ldots,T_r$ and $n_i=|V(T_i)|$, then $$\nu(T')={2n\choose n_1,\ldots,n_r} \prod_{i=1}^r \nu(T_i),\label{eq:treeDisjoint}$$ which follows from (c). ## Lower bound for $\nu(T)$ Here we prove that $\nu(T)\geq n!2^n$ for all trees $T$ with $2n+1$ vertices with equality when $T$ is the *hairbrush $H_n$*. Recall that this graph $H_n$ is obtained by starting with a path $v_0-v_1-\cdots-v_n$ and then adding a leaf $u_i$ to each $v_i$ for $i\in [n]$; see . We begin by observing the following. **Lemma 37**. *For $n\geq 0$, the hairbrush $H_n$ has $\nu(H_n)=n!2^n$.* *Proof.* Note that $\nu(H_0)=1=0! 2^0$, so from now on we assume $n>0$. Since $\{v_0,u_1,\dots,u_n\}$ are leaves and $e(H_n)$ is even, none of these vertices are in $\widetilde{X}$. For $i\in [n-1]$, one of the components of $H_n-v_i$ is the subgraph on $\{v_{i+1},u_{i+1},\dots,v_n,u_{n}$}, which has an odd number of edges, so $v_i\notin \widetilde{X}$. On the other hand, $H_n-v_n=H_{n-1}\sqcup \{u_n\}$, so $v_n\in \widetilde{X}$. Hence $\widetilde{X}=\{v_n\}$, so by and [\[eq:treeDisjoint\]](#eq:treeDisjoint){reference-type="eqref" reference="eq:treeDisjoint"} we have $$\nu(H_n)=\nu(H_n-v_n)=\nu(H_{n-1}\sqcup \{u_n\}) =2n\cdot \nu(H_{n-1}).$$ This provides a recurrence relation for $\nu(H_n)$ for $n\ge 1$, which combined with the initial condition $\nu(H_0)=1$ gives the desired formula. ◻ In view of , to show that $\nu(T)>0$ (let alone that $\nu(T)\ge n! 2^n$), it is necessary to show the following. **Lemma 38**. *If $T$ is a tree with an even number of edges, then $\widetilde{X}\ne \emptyset$.* *Proof.* We prove the result by induction on $n=|V(T)|$, the case $n=1$ being trivial. Suppose $n>1$ and let $x_1\cdots x_k$ be a longest path in $T$. Note that every neighbor of $x_2$ other than $x_3$ is a leaf (as otherwise we could extend the path). If $\deg(x_2)$ is even, then $T-x_2$ is the disjoint union of $\deg(x_2)-1$ copies of $K_1$ and a tree $T'$ with $e(T)-\deg(x_2)\equiv_2 0$ edges, so $x_2\in \widetilde{X}$. Thus we may assume $\deg(x_2)$ is odd. Let $T^*$ be $T$ after deleting all of the neighbors of $x_2$ other than $x_3$. Observe that $T^*$ is a tree with an even number of edges and with strictly fewer vertices than $T$ (since we have deleted $x_1$, in particular). By the inductive hypothesis, there exists some vertex $y\in \widetilde{X}(T^*)$, i.e. $y$ is such that each connected component of $T^*-y$ has an even number of edges. Each component of $T^*-y$ is either a component of $T-y$ or it contains $x_2$. In the later case, the component of $T-y$ containing $x_2$ has $\deg(x_2)-1$ more edges than that in $T^*-y$. Since $\deg(x_2)-1$ is even by assumption, we have $y\in \widetilde{X}$, completing the proof. ◻ We will also need the following simple arithmetic inequality. **Lemma 39**. *Let $n$ be a non-negative integer and $(k_1,\ldots,k_r)$ a sequence of non-negative integers such that $r$ is even, and such that $n=r/2+\sum k_i$. Then $$\prod_{i=1}^r \frac{k_i!}{2(2k_i+1)!}\ge \frac{(n-1)!}{2(2n-1)!},$$ with equality if and only if $r=2$ and $\{k_1,k_2\}=\{n-1,0\}$.* *Proof.* Let $\vec{k}=(k_1,\ldots,k_r)$ be a sequence as in the hypothesis of the lemma such that $\prod \frac{k_i!}{2(2k_i+1)!}$ is as small as possible. Without loss of generality, we may assume that $\vec{k}$ is weakly decreasing. We will prove the result by first showing that $\vec{k}$ is of the form $(k_1,0,0,\dots,0)$, and then that $r=2$. First assume for contradiction that $k_1\ge k_2>0$, and define the sequence $(k'_1,\ldots,k'_r)$ by $k'_i=k_i$ if $i>2$ and $k'_1=k_1+1,\ k'_2=k_2-1$. Note that this sequence continues to satisfy the hypothesis of the lemma. We claim that $$\prod_{i=1}^r \frac{k_i!}{2(2k_i+1)!}> \prod_{i=1}^r\frac{k_i'!}{2(2k_i'+1)!}.$$ Since $k_i=k_i'$ for $i>2$, this is equivalent to saying $$\frac{k_1!k_2!}{4(2k_1+1)!(2k_2+1)!}>\frac{(k_1+1)!(k_2-1)!}{4(2k_1+3)!(2k_2-1)!},$$ which further simplifies to $$\frac{k_2}{(2k_2+1)(2k_2)}> \frac{k_1+1}{(2k_1+3)(2k_1+2)} \iff\frac{1}{2k_2+1}> \frac{1}{2k_1+3}$$ and this last bound holds since $k_1\ge k_2$. This contradicts $\vec{k}$ being a minimizer, so we conclude that $k_2=0$. Hence, we must have $\vec{k}=(k_1,0,\ldots,0)$, where necessarily $k_1=n-r/2$ by the hypothesis of the lemma. In this case, $$\label{eq: function of n and r to make increasing} \prod_{i=1}^n \frac{k_i!}{2(2k_i+1)!}= \frac{k_1!}{2(2k_1+1)!}\cdot \frac{1}{2^{r-1}}=\frac{(n-r/2)!}{2^r(2n-r+1)!}.$$ Thus, to conclude the result it suffices to show that this function is strictly increasing for even $r\le n/2$, i.e. that for $r<n/2$ $$\frac{(n-r/2)!}{2^r(2n-r+1)!}<\frac{(n-r/2-1)!}{2^{r+2}(2n-r-1)!}.$$ This is equivalent to saying $$\frac{n-r/2}{(2n-r+1)(2n-r)}<\frac{1}{4},$$ and indeed this quickly follows since $2n-r+1>2n-r\ge 2$. ◻ We now prove our lower bound for trees, which we restate below. **Proposition 40**. *If $T$ is a tree on $2n+1$ vertices, then $\nu(T)\geq 2^nn!$ with equality if and only if $T$ is the hairbrush $H_n$.* *Proof.* We prove the result by induction on $n$, the $n=0$ case being trivial. Assume we have proven the result up to some value $n$ and let $T$ be a tree on $2n+1$ vertices. **Claim 41**. *For each $x\in \widetilde{X}$, we have $\nu(T-x)\ge n!2^n$ with equality only if $T-x$ is the disjoint union of $K_1$ and a hairbrush $H_{n-1}$.* *Proof.* let $T_1,\dots,T_r$ be the connected components of $T-x$, say with $n_i=|V(T_i)|$. Since each $T_i$ has an even number of edges, $n_i=2k_i+1$ for some non-negative integer $k_i$. By [\[eq:treeDisjoint\]](#eq:treeDisjoint){reference-type="eqref" reference="eq:treeDisjoint"} and induction, we have $$\nu(T-x)={2n\choose n_1,\ldots,n_r} \prod_{i=1}^r \nu(T_i)\ge {2n\choose n_1,\ldots,n_r} \prod_{i=1}^r k_i! 2^{k_i}.\label{eq:treeInductive}$$ Using $\sum_{i=1}^r k_i=\sum_{i=1}^r (n_i-1)/2=n-r/2$, we see that the quantity above can be rewritten as $$\begin{aligned} \binom{2n}{n_1,\dots,n_r}\prod_{i=1}^r k_i!2^{k_i}&=\frac{(2n)!}{n_1!\cdots n_r!}2^{k_1+\cdots+k_r}\prod_{i=1}^r k_i! \nonumber\\ \nonumber &=\frac{(2n)!}{(2k_1+1)!\cdots (2k_r+1)!}2^{n-r/2}\prod_{i=1}^r k_i!\\ \nonumber &=(2n)!2^{n-r/2}\prod_{i=1}^r \frac{k_i!}{(2k_i+1)!}\\ \nonumber &=(2n)!2^{n+r/2}\prod_{i=1}^r \frac{k_i!}{2(2k_i+1)!}\\ &\geq (2n)!2^{n+r/2}\frac{(n-1)!}{2(2n-1)!} \label{eq:hairbrush}\\ &= n! 2^{n+r/2-1}\geq n!2^{n},\nonumber \end{aligned}$$ where [\[eq:hairbrush\]](#eq:hairbrush){reference-type="eqref" reference="eq:hairbrush"} used . This proves the desired inequality of the claim. Moreover, implies that equality in [\[eq:hairbrush\]](#eq:hairbrush){reference-type="eqref" reference="eq:hairbrush"} can only occur if $r=2$ and if, say, $T_1$ has one vertex and $T_2$ has $2n-1$ vertices. Moreover, by induction we know [\[eq:treeInductive\]](#eq:treeInductive){reference-type="eqref" reference="eq:treeInductive"} can only hold with equality if $T_2$ is a hairbrush, proving the claim. ◻ By and the claim above, we have $$\nu(T)=\sum_{x\in \widetilde{X}} \nu(T-x)\ge |\widetilde{X}|\cdot n! 2^n\ge n! 2^n,\label{eq:oneX}$$ with this last inequality using . This gives the desired lower bound on $\nu(T)$. Now suppose $\nu(T)=n! 2^n$. This implies both inequalities of [\[eq:oneX\]](#eq:oneX){reference-type="eqref" reference="eq:oneX"} are equalities, which can only hold if $|\widetilde{X}|=1$, say with $\widetilde{X}=\{x\}$; and if $T-x$ consists of an isolated vertex $y$ together with a hairbrush $H_{n-1}$. It remains to show that this implies $T$ must be the hairbrush $H_n$, for which it suffices to show that $x$ is adjacent to $v_{n-1}$ in $H_{n-1}$. If $x$ is adjacent to some $v_i$ or $u_i$ with $i\in [n-2]$, then $T-v_{n-1}$ has 2 components each with an even number of edges ($2n$ and 0 respectively), so $v_{n-1}\in \widetilde{X}$, a contradiction to $\widetilde{X}=\{x\}$. If $x$ is adjacent to $u_{n}$, then $T-v_{n-1}$ has 2 components each with an even number of edges ($2n-2$ and 2 respectively), so $v_{n-1}\in \tilde{X}$. Thus, in all cases, unless $x$ is adjacent to $v_{n-1}$, we have $\widetilde{X}\ne \{x\}$, which is a contradiction. Hence, we conclude that $T$ is $H_{n}$. Finally, provides the other direction. ◻ ## Upper bound for $\nu(T)$ We will prove that $\nu(T)\leq (2n)!$ for all trees $T$ with $2n+1$ vertices with equality only when $T$ is the *star graph* $K_{1,2n}$, which consists of a center vertex $v_0$ adjacent to $2n$ leaves $v_1,\ldots,v_{2n}$. **Lemma 42**. *For all $n\ge 0$, $\nu(K_{1,2n})=(2n)!$.* *Proof.* We prove $K_{1,2n}$ has exactly $(2n)!$ even sequences, from which the result follows by since $K_{1,2n}$ is bipartite. Let $\pi$ be an ordering of the vertices of $T$. If $\pi_{2n+1}\ne v_0$, then the induced subgraph $K_{1,2n}[\pi_1,\ldots,\pi_{2n}]$ is isomorphic to $K_{1,2n-1}$ which has an odd number of edges, so $\pi$ is not an even sequence. Hence, all even sequences $\pi$ have $\pi_{2n+1}=v_0$, and it is easy to see that every $\pi$ with this property is in fact an even sequence. ◻ We next prove some structural results regarding the set $\widetilde{X}$. **Lemma 43**. *Let $T$ be a tree with an even number of edges.* 1. *No vertex $x\in \widetilde{X}$ is a leaf, and no two vertices of $\widetilde{X}$ are adjacent.* 2. *For $x\in \widetilde{X}$, let $T_1,\dots,T_r$, be the connected components of $T-x$ with $n_i=|V(T_i)|$. If $T_i$ contains $\ell_i$ vertices of $\widetilde{X}$, then $n_i\ge 2\ell_i+1$ for each $i\in [r]$.* *Proof.* For (a), if $x$ is a leaf, then $T-x$ is a connected graph with an odd number of edges, so $x\notin \widetilde{X}$. Assume for contradiction that some $x,y\in \widetilde{X}$ are adjacent. Let $T_x$ be the connected component of $T-y$ containing $x$ and similarly let $T_y$ be the component of $T-x$ containing $y$; see . Since $x,y\in \widetilde{X}$, we have $e(T_x),e(T_y)$ even. However, it is not difficult to see that every edge of $T-xy$ appears exactly once in either $T_x$ or $T_y$, which implies $e(T)=e(T_x)+e(T_y)+1$ is odd, a contradiction. For (b), fix $i\in [r]$, set $\ell:=\ell_i$, and write $\{y_1,\ldots,y_\ell\}=\widetilde{X}\cap V(T_i)$. For each $j\in[\ell]$, let $u_j$ be a neighbor of $y_j$ with $\mathrm{dist}_T(u_j,x)>\mathrm{dist}_T(y_j,x)$, which always exists because $y_j$ is not a leaf of $T$ by (a). Let $u_0$ be the unique neighbor of $y_0:=x$ in $T_i$. Note that $u_j\ne y_{j'}$ for any $j,j'$ since $y_j$ and $y_{j'}$ are not adjacent by (a). Also, observe that $u_j\ne u_{j'}$ for $j\ne j'$, as otherwise we would have $\mathrm{dist}_T(y_j,x)=\mathrm{dist}_T(y_{j'},x)$ and that $y_j,y_{j'}$ have a common neighbor $u_j=u_{j'}$ not equal to $x$, which would imply that $T$ has a cycle if $y_j\ne y_{j'}$ (via considering $u_j,y_j,y_{j'}$ and the paths from $y_j,y_{j'}$ to $x$). In total, we conclude that $y_1,\ldots,y_\ell,u_0,u_1,\ldots,u_\ell$ are distinct vertices in $T_i$, proving the second part. ◻ We are now in position to prove our desired bound on $\nu(T)$. **Proposition 44**. *If $T$ is a tree on $2n+1$ vertices, then $\nu(T)\leq (2n)!$ with equality if and only if $T$ is the star $K_{1,2n}.$* *Proof.* We prove the result by induction on $n$, the $n=0$ case being trivial. Fix $x\in \widetilde{X}$ and let $T_1,\ldots,T_r$ be the connected components of $T-x$. By induction, we have $$\nu(T-x)={2n\choose n_1,\ldots,n_r} \prod_{i=1}^r \nu(T_i)\le {2n\choose n_1,\ldots,n_r} \prod_{i=1}^r (n_i-1)!=\frac{(2n)!}{\prod_{i=1}^r n_i}.\label{eq:treeMax}$$ With this and , we find $$\nu(T-x)\le \frac{(2n)!}{\prod_{i=1}^r(2\ell_i+1)}\le \frac{(2n)!}{1+\sum_{i=1}^r 2\ell_i}=\frac{(2n)!}{1+2(|\widetilde{X}|-1)},$$ where the second inequality used repeated applications of the inequality $(\alpha+1)(\beta+1)\ge \alpha+\beta+1$ valid for any $\alpha,\beta\ge 0$, and the equality used that each vertex of $\widetilde{X}-\{x\}$ appears in exactly one $T_i$ subtree. Using this together with gives $$\nu(T)=\sum_{x\in \widetilde{X}} \nu(T-x)\le \frac{|\widetilde{X}|}{2|\widetilde{X}|-1} (2n)!\le (2n)!,$$ proving the desired upper bound. If $\nu(T)=(2n)!$, then the inequalities above must be equalities, which can only happen if $|\widetilde{X}|=1$, say with $\widetilde{X}=\{x\}$, and if $\nu(T-x)=(2n)!$. By [\[eq:treeMax\]](#eq:treeMax){reference-type="eqref" reference="eq:treeMax"}, this can only happen if $n_i=1$ for all $i$, which means $T-x$ consists of $2n$ isolated vertices. This implies $T$ is a star on $2n+1$, completing the proof. ◻ In total, this proposition together with completes the proof of . **Remark 45**. Our proofs yield slightly stronger bounds on $\nu(T)$ whenever $\widetilde{X}$ is large. For example, [\[eq:oneX\]](#eq:oneX){reference-type="eqref" reference="eq:oneX"} gives the lower bound $\nu(T)\ge |\widetilde{X}| n! 2^n$. Bounds of this form are known as *stability results* in extremal graph theory, which roughly are results saying that bounds for a graph $T$ can be substantially improved if $T$ is "far" from a unique extremal construction (in our case, $T$ being "far" from $H_n,K_{1,2n}$ is measured by having $|\widetilde{X}|$ large). # Multiplicity of $-1$ {#section: multiplicity} In this section, we prove Proposition [Proposition 8](#prop:matching){reference-type="ref" reference="prop:matching"} and Theorems [Theorem 9](#thm: -1 multiplicity for tournaments){reference-type="ref" reference="thm: -1 multiplicity for tournaments"} and [Theorem 7](#thm root multiplicity upper bound){reference-type="ref" reference="thm root multiplicity upper bound"} regarding the multiplicity of $-1$ as a root of $A_{D}(t)$. We first prove Theorem [Theorem 7](#thm root multiplicity upper bound){reference-type="ref" reference="thm root multiplicity upper bound"} which we restate for convenience. **Theorem 46**. *If $D$ is an $n$-vertex digraph, then $$\mathop{\mathrm{mult}}(A_D(t),-1)\le n-s_2(n),$$ where $s_2(n)$ denotes the number of 1's in the binary expansion of $n$. Moreover, for all $n$, there exist $n$-vertex digraphs $D$ with $A_D(t)=\frac{n!}{2^{n-s_2(n)}}(1+t)^{n-s_2(n)}$.* *Proof.* We first show the upper bound. Let $m$ be the multiplicity of $-1$ as a root of $A_D(t)$. Observe that $A_D(1) = n!$. Since there is a polynomial $p(t)$ such that $A_D(t) = (1 + t)^mp(t)$ and $p(-1) \neq 0$, we know that $A_D(1) = 2^mp(1) = n!$. Since $A_D(t)$ has integer coefficients and $(1 + t)^m$ also has integer coefficients, it follows that $p(t)$ has rational coefficients. By Gauss's lemma, it follows that $p(t)$ has integer coefficients. Hence $p(1)$ is an integer. It follows that $2^m$ must divide $n!$, which by Legendre's formula implies $m\le n - s_2(n)$. For the lower bound, we first consider the case when $n$ is a power of two. Let $P_2$ be the graph on vertices $v_1,v_2$ with a single arc $v_1\to v_2$. Define the sequence of digraphs $\{L_m\}_{m\in \mathbb{N}}$ by $$L_1 = P_2~~~~\text{and}~~~ L_{m + 1} = L_m\circ_{v_1} P_2,$$ where we recall that this expression for $L_{m+1}$ is the rooted producted mentioned just before . We observe that $L_{m}$ has $2^m$ vertices and $2^m - 1$ arcs. By and induction, we find $$A_{L_m}(t) = (2^m)!\left(\frac{1 + t}{2}\right)^{2^m - 1}.$$ Since $s_2(2^m) = 1$, this gives the desired construction when $n$ is a power of two. For an arbitrary $n$, let $a_1,\dots,a_\ell$ be the indices of the nonzero powers of $2$ in the binary expansion of $n$. Consider the digraph $D$ given via the disjoint union of the digraphs $L_{a_1},\dots,L_{a_\ell}$ defined previously. By Proposition [Proposition 10](#prop basic facts){reference-type="ref" reference="prop basic facts"}, $$A_{D}(t) = \binom{n}{2^{a_1}, \dots, 2^{a_\ell}}\prod_{i = 1}^{\ell} A_{L_{a_i}}(t)=\binom{n}{2^{a_1}, \dots, 2^{a_\ell}}\prod_{i = 1}^{l} (2^{a_i})!\left(\frac{1 + t}{2}\right)^{2^{a_i} - 1}=\frac{n!}{2^{n-s_2(n)}}(1+t)^{n-s_2(n)},$$ giving the desired result. ◻ We next establish our general lower bound, which we restate for convenience. **Proposition 47**. *Let $D$ be an orientation of an $n$-vertex graph $G$. If every matching in the complement of $G$ has size at most $m$, then $\mathop{\mathrm{mult}}(A_D(t),-1) \geq \lfloor \frac{n}{2} \rfloor - m$.* *Proof.* We prove the result by induction on $n$, the base cases $n=0,1$ being trivial. Assume that we have proven the result for $n-2$, and let $D$ be an orientation of an $n$-vertex graph $G$ whose complement has a maximum matching of size $m$. By applying with $k=2$ we find $$A_D(t) = \sum_{S\in \binom{V}{2}} \frac{t^{e_D(S,\overline{S})} + t^{e_D(\overline{S},S)}}{2}A_{D[S]}(t)A_{D - S}(t).$$ We claim that the polynomials $A_{D[S]}(t)A_{D - S}(t)$ in the sum above all have $-1$ as a root with multiplicity at least $\lfloor \frac{n}{2} \rfloor - m$, from which the result will follow. First consider the case that $S$ is not an edge of $G$, which means it is an edge in the complement $\overline{G}$. This implies that every maximal matching of $\overline{G}$ must use at least one vertex of $S$ (as otherwise we could include the edge $S$ into the matching), hence $\overline{G}-S$ is an $n-2$ vertex graph with maximum matching of size at most $m-1$. Inductively this implies $A_{D - S}(t)$ has $-1$ as a root with multiplicity at least $\lfloor \frac{n-2}{2} \rfloor - (m-1)=\lfloor \frac{n}{2} \rfloor - m$, giving the desired result. Next consider the case that $S$ is an edge in $G$. Observe that $\overline{G}-S$ is an $n-2$ vertex graph which continues to have no matching of size larger than $m$, so inductively $A_{D-S}(t)$ has $-1$ as a root with multiplicity at least $\lfloor \frac{n-2}{2} \rfloor - m=\lfloor \frac{n}{2} \rfloor - m-1$. Also note that since $S$ is an edge, $A_{D[S]}(t)=1+t$, so combining these two terms gives the desired multiplicity. This completes the proof of the claim, proving the result. ◻ In particular this result implies $\mathop{\mathrm{mult}}(A_D(t),-1)\ge \lfloor \frac{n}{2} \rfloor$ for tournaments $D$, but proving this holds with equality requires a refinement of which requires some additional notation. Let $OP(\alpha)$ denote the set of all ordered set patitions of type $\alpha$, and let $SP(\lambda)$ denote the set of all unordered set partitions with type $\lambda$. For a digraph $D$ and an ordered set partition $P$ of the vertices of $D$ of length $k$ and $i \in [k]$, define the *$i$-th forward sequence number* of $P$ to be $$FS_{D,P}(i) = \sum_{j = i + 1}^{k} e_D(P_i,P_j)$$ and the *$i$-th reverse sequence number* of $P$ to be $$RS_{D,P}(i) = \sum_{j = i + 1}^{k} e_D(P_j,P_i)$$ where we set $FS_{D,P}(k) = 0$ and $RS_{D,P}(1) = 0$. Note that $FS_{D,P}(i)=e_D(P_i,P_{i+1}\cup \cdots P_k)$ and $RS_{D,P}(i)=e_D(P_{i+1}\cup \cdots P_k,P_i)$. With this notation in hand, we have the following corollary of Lemma [Lemma 11](#lemma split into subgraphs){reference-type="ref" reference="lemma split into subgraphs"}. **Lemma 48**. *If $D$ is a digraph on the vertex set $[n]$ and $\alpha$ is an integer composition of $n$ of length $k$, then $$A_D(t) = \frac{1}{2^{k}}\sum_{P\in OP(\alpha)} \prod_{i = 1}^{k} \left(A_{D[P_i]}(t) \left(t^{FS_{D,P}(i)} + t^{RS_{D,P}(i)}\right)\right)$$* *Proof.* We induct on the number of parts in $\alpha$. If $\alpha$ has one part, then the only partition $P$ is the entire vertex set. Hence, $FS_{D,P}(i) = 0 = RS_{D,P}(i)$ for every $i\in [k]$ and the result follows. Assume the claim holds for any integer composition with $k - 1$ parts, and consider an integer partition $\alpha$ with $k$ parts. Let $\alpha_1$ be the first part of $\alpha$, and let $\alpha'$ be the integer composition given by removing the first part of $\alpha$. By Lemma [Lemma 11](#lemma split into subgraphs){reference-type="ref" reference="lemma split into subgraphs"}, $$A_{D}(t) = \sum_{S\in\binom{[n]}{\alpha_1}} \frac{t^{e_D(S,\overline{S})} + t^{e_D(\overline{S},S)}}{2}A_{D[S]}(t)A_{D - S}(t).\label{eq:partition1}$$ By our inductive hypothesis, for each $S\in \binom{[n]}{\alpha_1}$ $$\label{eq:partition2} A_{D - S}(t) = \frac{1}{2^{k - 1}}\sum_{P\in OP(\alpha')} \prod_{i = 2}^{k} \left(A_{(D - S)[P_i]}(t) \left(t^{FS_{D - S,P}(i)} + t^{RS_{D - S,P}(i)}\right)\right)$$ Observe that for $i\in \{2,\ldots,k\}$ we have $$FS_{D,P}(i) = FS_{D - S,P}(i) ~~\text{and}~~RS_{D,P}(i) = RS_{D - S,P}(i)$$ and that $$FS_{D,P}(1) = e_D(S,\overline{S})~~\text{and}~~RS_{D,P}(1) = e_D(\overline{S},S).$$ It follows from [\[eq:partition1\]](#eq:partition1){reference-type="eqref" reference="eq:partition1"} and [\[eq:partition2\]](#eq:partition2){reference-type="eqref" reference="eq:partition2"} that $$A_D(t) = \frac{1}{2^{k}}\sum_{P\in OP(\alpha)} \prod_{i = 1}^{k} \left(A_{D[P_i]}(t) \left(t^{FS_{D,P}(i)} + t^{RS_{D,P}(i)}\right)\right)$$ as desired. ◻ With the lemma, we can now prove Theorem [Theorem 9](#thm: -1 multiplicity for tournaments){reference-type="ref" reference="thm: -1 multiplicity for tournaments"}. We restate the theorem for convenience. **Theorem 49**. *If $D$ is a tournament on $n$ vertices, then $\mathop{\mathrm{mult}}(A_D(t),-1) = \lfloor \frac{n}{2}\rfloor$.* *Proof.* We first consider the case when $n = 2k$ for some $k\in \mathbb{N}$. By Proposition [Proposition 8](#prop:matching){reference-type="ref" reference="prop:matching"}, $\mathop{\mathrm{mult}}(A_D(t),-1) \geq n/2$. By Lemma [Lemma 48](#lemma ordered set partition expansion){reference-type="ref" reference="lemma ordered set partition expansion"} applied to the partition $(2)^k$, we have $$A_D(t) = \left(1 + t\right)^k\frac{1}{2^{k}}\sum_{P\in OP((2)^k)} t^{FS_{T,P}(i)} + t^{RS_{T,P}(i)},$$ where here we used that $A_{D[P_i]}(t)=1+t$ for all sets $P_i$ of size 2 since $D$ is a tournament. Let $$p(t) = \frac{1}{2^{k}}\sum_{P\in OP((2)^k)}t^{FS_{T,P}(i)} + t^{RS_{T,P}(i)}.$$ We claim that $p(-1) \neq 0$, from which the bound $\mathop{\mathrm{mult}}(A_D(t),-1)\le n/2$ will follow from the inequality above. We first observe that $FS_{T,P}(i)+ RS_{T,P}(i)$ is always even, as both vertices in $P_i$ are adjacent to every vertex in $P_{i + 1}\cup \dots \cup P_{k}$. Thus for every $P\in OP((2)^k)$ and every $i\in [k]$, $$(-1)^{FS_{T,P}(i)}+ (-1)^{ RS_{T,P}(i)} = 2(-1)^{FS_{T,P}(i)}.$$ It follows that $$p(-1) = \frac{1}{2^{k}}\sum_{P\in OP((2)^k)} \prod_{i = 1}^{k}(2(-1)^{FS_{T,P}(i)}) = \sum_{P\in OP((2)^k)} (-1)^{FS_{T,P}} = \sum_{P\in SP((2)^k)} \sum_{\sigma\in \mathfrak{S}_k} (-1)^{FS_{T,\sigma P}}.$$ where $\sigma P$ denotes the ordered set partition $(P_{\sigma(1)},\dots,P_{\sigma(k)})$. We first establish the following claim: **Claim 50**. *For all $\sigma\in \mathfrak{S}_k$, $$FS_{T,\sigma P}\equiv_2 FS_{T,P}~~\text{and}~~ RS_{T,\sigma P}\equiv_2 RS_{T,P}.$$* *Proof.* It suffices to consider the cases where $$P=(P_1,\dots,P_{a-1},P_a,P_{a+1},P_{a+2},\dots,P_k)\qquad P'=(P_1,\dots,P_{a-1},P_{a+1},P_{a},P_{a+2},\dots,P_k)$$ for some $a\in [k-1]$. We then have $$\begin{aligned} {FS}_{D,P} - {FS}_{D,P'} &= \sum_{i = 1}^{k} {FS}_{D,P}(i) - {FS}_{D,P'}(i)\\ &= \sum_{i = 1}^{k} \sum_{j = i + 1}^{k} ({e}_D(P_i,P_j) - {e}_D(P_i',P_j'))\\ &= ({e}_D(P_a,P_{a + 1}) - {e}_D(P_a',P_{a + 1}'))\\ &= {e}_D(P_a,P_{a + 1}) - {e}_D(P_{a + 1},P_{a})\\ &\equiv_2 0 \end{aligned}$$ since ${e}_D(P_a,P_{a + 1}) + {e}_D(P_{a + 1},P_{a})$ is the number of (undirected) edges from $P_a$ to $P_{a + 1}\cup \dots \cup P_k$, which is always even. The result for $RS_{T,\sigma P}$ follows by an identical argument. ◻ With claim [\[eq: fs mod 2 is same for all rearrangements of P\]](#eq: fs mod 2 is same for all rearrangements of P){reference-type="eqref" reference="eq: fs mod 2 is same for all rearrangements of P"} and the fact that $|OP(2^k)|=k!|SP(2^k)|$, we can conclude that $$p(-1)= k!\sum_{P\in SP(2^k)}(-1)^{FS_{T,P}}.$$ Each $P\in SP((2)^k)$ is a perfect matching on $[2k]$, and there are $(2k - 1)!!$ such perfect matchings. Since $(2k - 1)!!$ is odd, it follows that $p(-1) \neq 0$ as desired. Now when $n = 2k + 1$, we can apply the same reasoning as in the above proof to the integer composition $(2^k,1)$. The conclusion follows from the fact that there are $(2k - 1)!!\cdot (2k + 1)$ maximum matchings in $K_{2k + 1}$. ◻ # Concluding Remarks and Open Problems {#sec conclusion} In this paper we studied a notion of Eulerian polynomials $A_D(t)$ for digraphs $D$ and proved a number of results related to evaluations at $t=-1$. We conclude the paper by listing a number of remaining open problems themed around interpreting $\nu(G)$ and multiplicities of $-1$ as a root of $A_D(t)$. **Interpretations for $\nu(G)$**. Recall that for any graph $G$ we define $\nu(G)=|A_D(-1)|$ where $D$ is any orientation of $G$. While provides a combinatorial interpretation for $\nu(G)$ when $G$ is bipartite, complete bipartite, or a blowup of a cycle, we are still far from understanding this quantity for general graphs, which we leave as the main open problem for this paper. **Question 51**. *Can one give a combinatorial interpretation for $\nu(G)$ for arbitrary graphs $G$?* In view of and the bound $\nu(G)\le \eta(G)$ from (a), we suspect that in general $\nu(G)$ should count even sequences of $G$ with some special properties, but what these properties should be remains a mystery. To answer , it might be useful to establish which graphs $G$ satisfy $\nu(G)=\sum_v \nu(G-v)$, as recurrences of this form were a key step in proving . In particular, computational evidence suggests that the following could hold, where here we recall that a graph is *Eulerian* if all of its degrees are even. **Conjecture 52**. *If $G$ is an Eulerian graph, then $\nu(G)=\sum_v \nu(G-v)$.* We note that Eulerian graphs have a "natural" orientation via orienting each edge according to an Eulerian tour. Given that e.g. our proof of relied on "natural" orientations of bipartite graphs, it is plausible that this natural orientation for Eulerian graphs could be used to prove . Our proof of is non-combinatorial, and it would be interesting to have a more direct combinatorial proof of this fact, say for bipartite graphs. **Problem 53**. *For any bipartite graph $G=([n],E)$ and orientation $D$ of $G$, construct an explicit involution $\varphi:\mathfrak{S}_n\to \mathfrak{S}_n$ such that* 1. *The set of fixed points $\mathcal{F}_\varphi$ of $\varphi$ is the set of (inverses of) even sequences of $G$, and* 2. *$(-1)^{\mathop{\mathrm{des}}_D(\sigma)}=-(-1)^{\mathop{\mathrm{des}}_D(\varphi(\sigma))}$ for all $\sigma\notin \mathcal{F}_\varphi$.* Such an involution is known to exist when $G=P_n$ (i.e. when inverses of even sequences are exactly alternating permutations), but this involution is somewhat complex; see [@stanley_2011 Exercise 135] for more. **Multiplicity of Roots**. In we showed every $n$ vertex tournament $D$ has $-1$ as a root of $A_D(t)$ with multiplicity exactly $\lfloor \frac{n}{2} \rfloor$. A natural generalization of this result would be the following. **Conjecture 54**. *If $D$ is the orientation of a complete multipartite graph which has $r$ parts of odd size, then $\mathop{\mathrm{mult}}(A_D(t),-1)=\lfloor \frac{r}{2} \rfloor$.* Observe that the bound $\mathop{\mathrm{mult}}(A_D(t),-1)\ge \lfloor \frac{r}{2} \rfloor$ follows from , so the difficulty lies in proving the upper bound. Another direction is to look at the more general quantity $\mathop{\mathrm{mult}}(A_D(t),\alpha)$, which is defined to be the multiplicity of $\alpha$ as a root of $A_D(t)$. For example, it is not difficult to see that $\mathop{\mathrm{mult}}(A_D(t),0)$ is equal to the minimum number of arcs that one must remove in $D$ to obtain an acyclic digraph. Such a set of arcs is known as a *minimum feedback arc set*, and determining the size of such a set is well known to an NP-Complete problem [@karp2010reducibility]. This connection to feedback arc sets, together with the result of this paper, establishes a number of results for $\mathop{\mathrm{mult}}(A_D(t),\alpha)$ when $\alpha\in \{0,-1\}$, and it is natural to ask what can be said about other integral values of $\alpha$. An immediate obstacle to this is the following. **Question 55**. *Does there exist a digraph $D$ such that $A_D(t)$ has an integral root which is not equal to either $0$ or $-1$?* We have verified that no such digraph exists on at most 5 vertices. We also note that there exist digraphs with real roots of magnitude larger than $2$, so the obstruction to finding these integral roots is not that their magnitudes are too large. ## Acknowledgement {#acknowledgement .unnumbered} This work began as part of the Graduate Student Combinatorics Conference 2022, which was funded through NSF Grant DMS-1933360, UC San Diego, and the Combinatorics Foundation. [^1]: Dept. of Mathematics, Wake Forest University [celanok\@wfu.edu]([email protected]) [^2]: Dept. of Mathematics, University of California San Diego `[email protected]` [^3]: Dept. of Mathematics, Rutgers University `[email protected]`. This material is based upon work supported by the National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship under Grant No. DMS-2202730. [^4]: As of the time of writing.

arxiv_math

--- abstract: | We investigate the number of maximal cliques, i.e., cliques that are not contained in any larger clique, in three network models: Erdős--Rényi random graphs, inhomogeneous random graphs (also called Chung--Lu graphs), and geometric inhomogeneous random graphs. For sparse and not-too-dense Erdős--Rényi graphs, we give linear and polynomial upper bounds on the number of maximal cliques. For the dense regime, we give super-polynomial and even exponential lower bounds. Although (geometric) inhomogeneous random graphs are sparse, we give super-polynomial lower bounds for these models. This comes form the fact that these graphs have a power-law degree distribution, which leads to a dense subgraph in which we find many maximal cliques. These lower bounds seem to contradict previous empirical evidence that (geometric) inhomogeneous random graphs have only few maximal cliques. We resolve this contradiction by providing experiments indicating that, even for large networks, the linear lower-order terms dominate, before the super-polynomial asymptotic behavior kicks in only for networks of extreme size. author: - Thomas Bläsius, Maximillian Katzmann and Clara Stegehuis bibliography: - references.bib title: Maximal Cliques in Scale-Free Random Graphs --- # Introduction While networks appear in many different applications, many real-world networks were found to share some important characteristics. First of all, often their degree distribution is heavy-tailed, which is sometimes denoted as the network being scale-free. Secondly, they often have a high clustering coefficient, implying that it is likely that two neighbors of a vertex are connected themselves as well. For this reason, random graph models that can achieve both scale-freeness and a high clustering coefficient have been at the center of attention over the last years. One example of such a model is the popular hyperbolic random graph (HRG) [@krioukov2010], which has for example been used to model the network of world wide trade [@garcia-perez2016] or the Internet on the Autonomous Systems level [@boguna2010; @kleinberg2007]. This random graph model embeds the vertices in an underlying hyperbolic space and connects them with probabilities depending on their distances, where nearby vertices are more likely to connect. The triangle inequality then ensures the presence of many triangles, while the hyperbolic space ensures the presence of a scale-free degree distribution. Recently, the geometric inhomogeneous random graph (GIRG) was proposed as a generalization of HRG. It combines power-law distributed weights with Euclidean space, making the model simpler to analyze [@bringmann2015]. While the hyperbolic random graph and the GIRG have been designed to exhibit high clustering and a scale-free degree distribution, the question remains whether other properties of this model match real-world data. For this reason, many properties of the GIRG or hyperbolic random graph have been analyzed mathematically, such as the maximum clique size [@Blasius2017], number of $k$-cliques [@michielan2021], spectral gap [@kiwi2018] and separator size [@Hyper_Rando_Graph_Separ_ESA2016; @lengler2017]. In this paper, we focus on another network property: the number of maximal cliques, i.e., cliques that are not part of any larger clique. Cliques in general are an important indicator for structural properties of a network. Indeed, the number of large cliques is a measure of the tendency of a network to cluster into groups. Small cliques of size 3 (triangles) on the other hand, can form an indication of the transitivity of a network or its clustering coefficient. To study these structural clique-based properties, however, all cliques of a given size need to be listed, which can be a computationally expensive process. To list all network cliques, it suffices to list only all maximal cliques, as all smaller cliques can be generated from at least one maximal clique. For this reason, enumerating all maximal cliques of a graph is at the heart of our understanding of cliques in general. For enumerating all maximal cliques, an output-polynomial algorithm [@tsukiyama1977] exists, which can enumerate all maximal cliques efficiently if the graph contains only few of them. There also exist highly efficient implementations to enumerate all maximal cliques [@Listi_Maxim_Cliqu_Spars_ISAAC2010; @Listi_Maxim_Cliqu_Large_jour2013; @Listi_Maxim_Cliqu_Large_SEA2011]. However, for a given graph, it is usually not known a priori how many maximal cliques it has. If this number is large, enumerating all maximal cliques can still take exponential time. However, in practice, enumerating the number of maximal cliques often takes a short amount of time for many real-world instances as well as in realistic network models [@Exter_Valid_Avera_Analy_ESA2022]. In this paper, we therefore focus on the number of maximal cliques in the GIRG random graph. As the GIRG possesses the two main characteristics that are essential to many real-world networks, scale-freeness and an underlying geometry, we believe that investigating the number of maximal cliques in the GIRG can provide insights into in why enumerating the number of maximal cliques can often be done efficiently for many real-world networks. To investigate the influence of the different properties of scale-freeness and clustering, we investigate the number of maximal cliques in three steps. First, we investigate a model without heavy-tailed degrees and with a small clustering coefficient, the Erdős--Rényi model $G(n, p)$; see Section [2](#sec:ER){reference-type="ref" reference="sec:ER"}. We then investigate the GIRG model (Section [3](#sec:GIRG){reference-type="ref" reference="sec:GIRG"}), which has both clustering and scale-free degrees. Finally, in Section [4](#sec:IRG){reference-type="ref" reference="sec:IRG"}, we investigate the Inhomogeneous Random Graph (IRG), a model that is scale-free but has a small clustering coefficient. We complement our theoretical bounds with experiments in Section [5](#sec:experiments){reference-type="ref" reference="sec:experiments"}. Our main findings can be summarized as follows; also see Table [1](#tab:result_summary_detailed){reference-type="ref" reference="tab:result_summary_detailed"} for an overview of our results. - There is a strong dependence on the density of the network. For the Erdős--Rényi model ($G(n, p)$) we obtain a linear upper bound for sparse graphs ($O(n)$ edges) and a polynomial upper bound for non-dense graphs ($O(n^{2 - \varepsilon})$ edges for any $\varepsilon > 0$). For dense graphs on the other hand ($\Omega(n^2)$ edges), we obtain a super-polynomial lower bound. If the density is high enough, our lower bound is even exponential. - This insight carries over to the IRG and GIRG models. Though they are overall sparse, they contain sufficiently large dense subgraphs that allow us to obtain super-polynomial lower bounds. - In the IRG model with power-law exponent $\tau \in (2, 3)$ the small maximal cliques localize: asymptotically maximal cliques of constant size $k>2$ are formed by $k - 2$ hubs of high degree proportional to $n^{1 / (\tau - 1)}$ and two vertices of lower degree proportional to $n^{(\tau-2)/(\tau-1)}$. - We complement our theoretical lower bounds with experiments showing that the super-polynomial growth becomes only relevant for very large networks. #### Discussion and Related Work. Although cliques themselves have been studied extensively in the literature, there is, to the best of our knowledge, only little previous work on the number of *maximal* cliques in network models. In fact, the only theoretical analysis we are aware of is the recent preprint by Yamaji [@yamaji2023], giving bounds for hyperbolic random graphs (HRG) and random geometric graph (RGG), which are also shown in Table [1](#tab:result_summary_detailed){reference-type="ref" reference="tab:result_summary_detailed"}. Interestingly, this includes the upper bound of $\exp(O(n^{\frac{3 - \tau}{6} + \varepsilon}))$ for the HRG model. In contrast to that, we give the asymptotically larger lower bound $\exp(\Omega(n^{\frac{3 - \tau}{4} - \varepsilon}))$ for the corresponding GIRG variant. Thus, there is an asymptotic difference between the HRG and the GIRG model. This is surprising as the GIRG model is typically perceived as a generalization of the HRG model. More precisely, there is a mapping between the two models such that for every HRG with average degree $d_{\mathrm{HRG}}$ there exist GIRGs with average degree $d_{\mathrm{GIRG}}$ and $D_{\mathrm{GIRG}}$ with $d_{\mathrm{GIRG}} \le d_{\mathrm{HRG}} \le D_{\mathrm{GIRG}}$ that are sub- and supergraphs of the HRG, respectively. Moreover, $d_{\mathrm{GIRG}}$ and $D_{\mathrm{GIRG}}$ are only a constant factor apart and experiments indicate that $d_{\mathrm{HRG}} = d_{\mathrm{GIRG}}\cdot(1 + o(1))$, i.e., every HRG has a corresponding GIRG that is missing only a sublinear number of edges [@Effic_gener_geome_inhom_jour2022]. In the case of maximal cliques, however, this minor difference between the models leads to an asymptotic difference. Besides this theoretical analysis, it has been observed empirically that the number of maximal cliques in most real-world networks as well as in the GIRG and the IRG model is smaller than the number of edges of the graph [@Exter_Valid_Avera_Analy_ESA2022]. This indicates linear scaling in the graph size with low constant factors and small lower-order terms, which seems to be a stark contradiction to the super-polynomial lower bounds we prove here. We resolve this contradiction with our experiments in Section [5](#sec:experiments){reference-type="ref" reference="sec:experiments"}, where we observe that the graph size has to be quite large before the asymptotic behavior kicks in, i.e., we observe the super-polynomial scaling as predicted by our theorems but on such a low level that it is overshadowed by the linear lower-order terms. Model Maximal cliques Reference ------- ------------------------------- ----------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- $p = 1 - \Theta(\frac{1}{n})$ $2^{\Omega(n)}$ Theorem [Theorem 3](#thm:erdense){reference-type="ref" reference="thm:erdense"} $p = \Theta(1)$ $n^{\Omega(\log n)}$ Theorem [Theorem 1](#thm:er-lower-bound){reference-type="ref" reference="thm:er-lower-bound"} $p = O(\frac{1}{n^a})$ $n^{O(1)}$ Theorem [Theorem 4](#thm:sparseer){reference-type="ref" reference="thm:sparseer"} $p = O(\frac{1}{n})$ $O(n)$ Theorem [Theorem 4](#thm:sparseer){reference-type="ref" reference="thm:sparseer"} IRG $\exp(\Omega(n^{\frac{3 - \tau}{4} - \varepsilon} \log n))$ Theorem [Theorem 15](#thm:max_cliques_irg){reference-type="ref" reference="thm:max_cliques_irg"} $d$-dim torus, $T = 0$ $\exp(\Omega(n^{\frac{3 - \tau}{4} - \varepsilon}))$ Corollary [Corollary 10](#cor:maxcliquesgirg){reference-type="ref" reference="cor:maxcliquesgirg"} $d$-dim torus, $T > 0$ $\exp(\Omega( {n^{\frac{(3 - \tau)}{5}} (\varepsilon \log n)^{-(1/2)}} ))$ Corollary [Corollary 13](#cor:maxcliquesgirg-non-threshold){reference-type="ref" reference="cor:maxcliquesgirg-non-threshold"} $2$-dim square, $T = 0$ $\exp(\Omega(n^{\frac{3 - \tau}{10} - \varepsilon}))$ Theorem [Theorem 11](#thm:girg_non_torus){reference-type="ref" reference="thm:girg_non_torus"} $2$-dim square, $T > 0$ $\exp(\Omega( n^{\frac{3 - \tau}{10} - \varepsilon}))$ Theorem [Theorem 14](#thmnonzero2dim){reference-type="ref" reference="thmnonzero2dim"} 2-dim, dense $\exp(\Omega(n^{\frac{1}{3}}))$ [@yamaji2023] 2-dim, dense $\exp(O(n^{\frac{1}{3} + \varepsilon}))$ [@yamaji2023] $\exp(\Omega(n^{\frac{3 - \tau}{6}}))$ [@yamaji2023] $\exp(O(n^{\frac{3 - \tau}{6} + \varepsilon}))$ [@yamaji2023] : Summary of our and other results on the number of maximal cliques in different random graph models and their scaling in the number of vertices. # Erdős--Rényi Random Graph {#sec:ER} An Erdős--Rényi random graph $G(n, p)$ has $n$ vertices and each pair of vertices is connected independently with probability $p$. We give bounds on the number of maximal cliques in a $G(n, p)$ depending on $p$. Roughly speaking, we give super-polynomial lower bounds for the dense regime and polynomial upper bounds for a sparser regime. Specifically, we first give a general lower bound that is super-polynomial if $p$ is non-vanishing for growing $n$, i.e., if $p \in \Omega(1)$. Note that $p \in \Omega(1)$ yields a dense graph with a quadratic number of edges in expectation. For super-dense graph with $p = 1 - c/n$ for a constant $c$, we strengthen this lower bound to exponential. In contrast to this, we give a polynomial upper if $p \in O(n^{-a})$ for any constant $a > 0$. For sparse graphs with $p \in O(n^{-1})$, yielding graphs with $\Theta(n)$ edges in expectation, our upper bound on the number of maximal cliques is linear. We start with the general lower bound. **Theorem 1**. *Let $N$ be the number of maximal cliques in a $G(n, p)$. Then $$\mathbb{E}\left[N\right] \ge n^{\frac{\log(n) / 2 - \log\log n + \log\log(1/p)}{\log(1/p)}}\cdot \frac{1 - o(1)}{e}.$$* *Proof.* Let $N_k$ be the number of maximal cliques of size $k$. To estimate $\mathbb{E}\left[N_k\right]$, note that the probability that a fixed subset $C \subseteq V$of $|C| = k$ vertices forms a clique is $p^{k(k-1)/2}$. Moreover, it is maximal if none of the other $n - k$ vertices is connected to all $k$ vertices of $C$, which happens with probability $(1 - p^k)^{n - k}$. As the two events are independent and there are ${n \choose k}$ vertex sets of size $k$, we obtain $$\mathbb{E}\left[N_k\right] = {n\choose k}p^{k(k-1)/2}(1-p^k)^{n-k}.$$ Using that ${n\choose k} \ge (n / k)^k$ and increasing the exponents of the probabilities, we obtain $$\mathbb{E}\left[N_k\right] \ge \left( \frac{n}{k} \right)^kp^{k^2/2}(1-p^k)^{n}.$$ We now set $k=\log(n)/\log(1/p)$, which yields $p^k = n^{-1}$. Thus, in the above bound, the term $n^k p^{k^2/2}$ simplifies to $n^k n^{-k/2} = n^{k/2}$. Moreover, the term $(1 - p^k)^{n}$ simplifies to $(1 - 1/n)^n$, which converges to $1/e$ for $n \to \infty$. Thus, we obtain $$\begin{aligned} \mathbb{E}\left[N_k\right] &\ge n^{k/2} \frac{1}{e k^k} (1 - o(1))\\ &= n^{\frac{\log(n) / 2}{\log(1/p)}} \cdot \left( \frac{\log(n)}{\log(1/p)} \right)^{-\frac{\log(n)}{\log(1/p)}} \cdot \frac{1 - o(1)}{e}.\\ \intertext{Changing the base of the second factor yields} &= n^{\frac{\log(n) / 2}{\log(1/p)}} \cdot e^{-\frac{\log(n)}{\log(1/p)} \cdot \log\left( \frac{\log(n)}{\log(1/p)} \right)} \cdot \frac{1 - o(1)}{e}\\ &= n^{\frac{\log(n) / 2}{\log(1/p)}} \cdot n^{-\frac{\log\log n - \log\log(1/p)}{\log(1/p)}} \cdot \frac{1 - o(1)}{e}\\ & = n^{\frac{\log(n) / 2 - \log\log n + \log\log(1/p)}{\log(1/p)}}\cdot \frac{1 - o(1)}{e}. \end{aligned}$$ As there are clearly at least as many maximal cliques as maximal cliques of size $k$, claimed bound for $\mathbb{E}\left[N\right]$ follows. ◻ This means that in a dense Erdős--Rényi random graph (constant $p$), the expected number of maximal cliques is super-polynomial in $n$. In the following, we show that, when the graph gets even denser, the number of maximal cliques even grows exponentially. For this, we prove the existence of an induced subgraph that has many maximal cliques. Specifically, we aim to find a large *co-matching*, i.e., the complement graph of a matching (or equivalently, a clique minus a matching). **Lemma 2**. *Let $G$ be a co-matching on $2k$ vertices. Then $G$ has $2^k$ maximal cliques.* *Proof.* The complement $\overline G$ of $G$ is a matching with $k$ edges. The maximal independent sets of $\overline G$ are the vertex sets that contain for each edge exactly one of its vertices. Thus, $\overline G$ has $2^k$ maximal independent sets, which implies that $G$ has $2^k$ maximal cliques. ◻ With this, we can show an exponential lower bound for super-dense Erdős--Rényi graphs. **Theorem 3**. *For every $c>0$, there exists a $\zeta>0$ such that $G(n,1-c/n)$ contains at least $2^{\zeta n}$ cliques with high probability.* *Proof.* A co-matching in $G(n,1-c/n)$ corresponds to an induced matching in $G(n,c/n)$. Now fix $M>1$. Then, by [@hofstad2009 Theorem 5.12], with high probability the Erdős--Rényi random graph contains a linear number of vertices of degree at most $M$ and at least $1$. Denote the reduced graph with only vertices of degree at most $M$ by $G_{\leq M}$, which has a linear number of edges. Now we construct an induced matching of linear size in $G_{\leq M}$ as follows. Start with any edge $\{u,v\}$ in $G_{\leq M}$, and add it to the matching. Then, remove $u$, $v$ and all neighbors of $u$ and $v$ from $G_{\leq M}$. This removes at most $2M^2$ edges from $G_{\leq M}$, as all degrees are bounded by $M$. Then, pick another edge and continue this process until $G_{\leq M}$ contains no more edges. As this process removes only a constant number of edges after picking a new edge, at least a linear number of edges will be added before the process finishes. Thus, there is an induced matching of at least $\zeta n$ with high probability, which yields the claim due to Lemma [Lemma 2](#lem:co-matching-nr-cliques){reference-type="ref" reference="lem:co-matching-nr-cliques"}. ◻ Next we consider less dense Erdős--Rényi graphs with $p \in O(n^{-a})$ for a constant $a \in (0, 1]$ and prove a polynomial upper bound on the number of maximal cliques. The degree of the polynomial depends on $a$. For sparse graphs with $p \in O(n^{-1})$, our bound is linear. **Theorem 4**. *Let $p = (c / n)^a$ for constants $c > 0$ and $a \in (0, 1]$ and let $N$ be the number of maximal cliques in a $G(n, p)$. Then $\mathbb{E}\left[N\right] \in O(n^x)$ with $$x = \left\lceil \frac{1}{a}\right\rceil - a \cdot {\left\lceil \frac{1}{a}\right\rceil \choose 2}.$$* *Proof.* As in Theorem [Theorem 1](#thm:er-lower-bound){reference-type="ref" reference="thm:er-lower-bound"}, let $N_k$ be the number of maximal cliques of size $k$. Note that the number of maximal cliques is upper bounded by the number of (potentially non-maximal) cliques. Thus, we obtain $$\mathbb{E}\left[N_k\right] \le {n\choose k}p^{\frac{k(k-1)}{2}}.$$ Using that ${n \choose k} \le (en / k)^k$, inserting $p = (c / n)^a$, and rearranging yields $$\begin{aligned} \mathbb{E}\left[N_k\right] &\le \left( \frac{en}{k} \right)^k \left( \frac{c}{n} \right)^{a\frac{k(k-1)}{2}}\notag\\ &= \left( \frac{ce}{k} \right)^k \left( \frac{c}{n} \right)^{a\frac{k(k-1)}{2} - k}. \label{eq:er-upper-bound-expectation} \end{aligned}$$ We first argue that we can focus on the case where $k$ is constant as the above term vanishes sufficiently quickly for growing $k$. For this, note that $a {k(k-1)} / {2} - k \ge k$ if $k \ge 4/a + 1$. Thus, as $c / n < 1$ for sufficiently large $n$, the second factor of Equation [\[eq:er-upper-bound-expectation\]](#eq:er-upper-bound-expectation){reference-type="eqref" reference="eq:er-upper-bound-expectation"} is upper bounded by $(c / n)^k$. For $k \ge 4/a + 1$, it then follows that $\mathbb{E}\left[N_k\right] \le ( {c^2e}/ (kn) )^k$. For sufficiently large $n$, the fraction is smaller than $1$ and thus the sum over all $N_k$ for larger values of $k$ is upper bounded by a constant due to the convergence of the geometric series. Focusing on $k \in \Theta(1)$ and ignoring constant factors, we obtain $$\mathbb{E}\left[N\right] = O\left( \max_{k \in \mathbb N^+} \left\{n^{x(k)}\right\} \right) \text{ with } x(k) = {k - a\frac{k(k-1)}{2}}.$$ To evaluate the maximum, note that $x(k)$ describes a parabola with its maximum at $k_0 = 1/a + 1/2$. However, $k_0$ may not be integral. To determine the integer $k$ that maximizes $x(k)$, note that for $a \in [\frac{1}{i}, \frac{1}{i - 1}]$ with $i \in \mathbb N^+$, we get $k_0 \in [i - \frac{1}{2}, i + \frac{1}{2}]$. Thus, $i$ is the closest integer to $k_0$. As the parabola is symmetric at its maximum $k_0$, the exponent $x(k)$ is maximized for the integer $k = i = \lceil \frac{1}{a} \rceil$. Substituting $k(k-1)/2 = {k \choose 2}$ yields the claim. ◻ # Geometric Inhomogeneous Random Graphs (GIRG) {#sec:GIRG} While the Erdős--Rényi random graph is homogeneous, and does not contain geometry, we now investigate the number of maximal cliques in a model that contains both these properties, the Geometric Inhomogeneous Random Graph (GIRG) [@bringmann2015]. In this model, each vertex $v$ has a weight, $w_v$ and a position $x_v$. The weights are independent copies of a power-law random variable $W$ with exponent $\tau$, i.e., $$\label{eq:pl} 1-F(w):=\mathbb{P}(W > w) = w^{1-\tau},$$ for all $w\geq 1$. We impose the condition $\tau \in (2,3)$, to ensure that the weights have finite mean but unbounded variance. The vertex positions $x_1,...,x_n$ are independent copies of an uniform random variable on the $d$-dimensional torus $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$. An edge between any two vertices $u,v \in V$ of the GIRG appears independently with a probability $p_{uv}$ determined by the weights and the positions of the vertices $$\label{eq:edgeprob} p_{uv} = \min \left\{\left(\frac{w_u w_v}{n\mu\|x_u-x_v\|^{d}}\right)^{1/T}, 1 \right\},$$ where $\|\cdot\|$ denotes the maximum norm on the torus, $\mu$ is a parameter controlling the average degree, and $0 < T < 1$ is the *temperature* and controls the influence of the geometry. We say that $T=0$ is the *threshold* case of the GIRG. That is, when $T=0$, $$\label{eq:edgeprobthreshold} p_{uv} =\begin{cases} 1 & \frac{w_u w_v}{n\mu\|x_u-x_v\|^{d}}\geq 1 \\ 0 &\text{else}. \end{cases}$$ In the following, we first give a lower bound for the threshold case (Section [3.1](#sec:girg-threshold-case){reference-type="ref" reference="sec:girg-threshold-case"}). The proof makes use of the toroidal structure of the ground space. To prove that this is not essential to obtain a super-polynomial number of maximal cliques, we additionally give a lower bound for a variant of the model where the ground space is a 2-dimensional unit square with Euclidean norm (Section [3.2](#sec:girg-with-2d-square){reference-type="ref" reference="sec:girg-with-2d-square"}). Finally, in Section [3.3](#sec:girg-non-threshold-case){reference-type="ref" reference="sec:girg-non-threshold-case"}, we show how to extend these results to the general case with non-zero temperatures. ## Threshold Case {#sec:girg-threshold-case} Here we show that a $d$-dimensional threshold GIRG $G = (V, E)$ has, with high probability, a super-polynomial number of maximal cliques. To achieve this, we proceed as follows to show that $G$ has a large co-matching as induced subgraph (also see Lemma [Lemma 2](#lem:co-matching-nr-cliques){reference-type="ref" reference="lem:co-matching-nr-cliques"}). We consider the vertex set $S \subseteq V$ containing all vertices whose weight lies between a lower bound $w_\ell$ and an upper bound $w_u$. As a co-matching is quite dense, it makes sense to think of these as rather large weights. We then define disjoint regions $B_1, \dots, B_{2k}$. For $i \in [k]$, we call $B_i$ and $B_{i + k}$ a pair of *opposite* regions. These regions will satisfy the following three properties. First, every $B_i$ contains a vertex from $S$ with high probability. Secondly, pairs of vertices from $S$ in opposite regions are not connected. And thirdly, vertices from $S$ that do not lie in opposite regions are connected. Note that these properties imply the existence of a co-matching on $2k$ vertices, as choosing an arbitrary vertex of $S$ for each region $B_i$ makes it so that each chosen vertex has exactly one partner from the opposite region to which it is not connected, while it is connected to the vertices from all other regions. {#fig:girg-torus-1d} {#fig:girg-torus-2d} In the following we first give a parameterized definition of the regions $B_i$ and then show how to choose the parameters for the above strategy to work; also see Figure [\[fig:girg-torus\]](#fig:girg-torus){reference-type="ref" reference="fig:girg-torus"}. Each $B_i$ is an axis-aligned box, i.e., the cross product of intervals. Let $g, h > 0$ such that $1 / (g + h)$ is an even number. Think of $h$ of as the *height* of each box and of $g$ as the *gap* between the boxes, yielding $2k = 1 / (g + h)$ boxes. Now we define $B_i = [(i - 1) \cdot (g + h), (i - 1) \cdot (g + h) + h] \times [0, \frac{1}{2} - g]^{d - 1}$ for $i \in [2k]$. We call the resulting regions $B_1, \dots, B_{2k}$ the *evenly spaced boxes of height $h$ and gap $g$*. As before, $B_i$ and $B_{i + k}$ for $i \in [k]$ are *opposite* boxes. With this, note that the distance between any pair of points in opposite boxes is at least $u = \frac{1}{2} - h$ (recall that we assume the infinity norm). Moreover, the distance between any pair of points in non-opposite regions is at most $\ell = \frac{1}{2} - g$. This yields the following lemma. **Lemma 5**. *Let $B_1, \dots, B_{2k}$ be evenly spaced boxes of height $h$ and gap $g$ in $\mathbb T^d$. Let $w_\ell = (\frac{1}{2} - g)^{d / 2} \sqrt{\mu n}$ and $w_u = (\frac{1}{2} - h)^{d / 2} \sqrt{\mu n}$. If we place one vertex of weight in $[w_\ell, w_u)$ in each box $B_i$, then these vertices form a co-matching.* *Proof.* As observed above, the vertices in opposite boxes have distance at least $u = \frac{1}{2} - h$. Moreover, the vertices considered here have weight less than $w_u = u^{d / 2} \sqrt{\mu n}$. As $w_u^2 / (\mu n u^d) = 1$, these vertices are not connected (see Equation [\[eq:edgeprobthreshold\]](#eq:edgeprobthreshold){reference-type="eqref" reference="eq:edgeprobthreshold"}). Similarly, vertices in non-opposite boxes have distance at most $\ell = \frac{1}{2} - g$ and weight at least $w_\ell = \ell^{d / 2} \sqrt{\mu n}$. As $w_\ell^2 / (\mu n \ell^d) = 1$, such vertices are connected. Hence, we get a co-matching. ◻ It now remains to choose $g$ and $h$ appropriately. First observe that, for the weight range in Lemma [Lemma 5](#lem:evenly-spaced-boxes-co-matching){reference-type="ref" reference="lem:evenly-spaced-boxes-co-matching"} to be non-empty, we need $w_\ell < w_u$ and thus $g > h$. Beyond that, we want to achieve the following three goals. First, the weight range needs to be sufficiently large such that we actually have a sufficient number of vertices in this range. For this, we want to choose $g$ substantially larger than $h$. Secondly, we want to make each box $B_i$ sufficiently large for it to contain a vertex with high probability. For this, we mainly want $h$ to be large. Thirdly, we want the number of boxes $2k = 1 / (g + h)$ to be large to obtain a large co-matching. For this, we want $g$ and $h$ to be small. Note that these desires of choosing $h$ large, $g$ larger than $h$, and $g + h$ small are obviously conflicting. In the following, we show how to balance these goals out to obtain a co-matching of polynomial size. We start by estimating the number of vertices in the given weight range in the following lemma, which is slightly more general then we need. **Lemma 6**. *Let $a, b > 0$ be constants and let $g, h$ be functions of $n$ such that $g, h \in o(1)$. Let $S$ be the set of vertices with weight in $[(a - g)^b \sqrt{\mu n}, (a - h)^b \sqrt{\mu n}]$. Then $$\mathbb{E}\left[|S|\right] = n^{\frac{3 - \tau}{2}} \cdot \mu^{\frac{1 - \tau}{2}} ba^{b(1 - \tau) - 1} \cdot (g - h \pm O(g^2 + h^2)).$$* *Proof.* Recall from [\[eq:pl\]](#eq:pl){reference-type="eqref" reference="eq:pl"} that the cumulative distribution function for the weights is $F(x) = 1 - x^{1 - \tau}$. Thus, we get $$\begin{aligned} \mathbb{E}\left[|S|\right] \notag &= n \cdot \left( F\left((a - h)^b \sqrt{\mu n}\right) - F\left((a - g)^b \sqrt{\mu n}\right)\right)\\ \notag &= n \cdot \left( ((a - g)^b \sqrt{\mu n})^{1 -\tau} - ((a - h)^b \sqrt{\mu n})^{1 - \tau} \right) \\ \label{eq:nr-vertices-in-range-1} &= \mu^{\frac{1 - \tau}{2}} n^{\frac{3 - \tau}{2}} \left( (a - g)^{b(1 - \tau)} - (a - h)^{b(1 - \tau)} \right). \end{aligned}$$ We can now use the Taylor expansion of $f(x) = (a - x)^c$ at $0$ to obtain the bound $f(x) = a^c - c a^{c - 1} x \pm O(x^2)$, which is valid for $x \in o(1)$. Since $g, h \in o(1)$ we can thus bound the above term in parentheses for $c = b(1 - \tau)$ as $$\begin{aligned} (a - g)^c - (a - h)^c \notag &= - c a^{c - 1} g + c a^{c - 1} h \pm O(g^2 + h^2)\\ \notag &= - c a^{c - 1}(g - h \pm O(g^2 + h^2))\\ \label{eq:nr-vertices-in-range-2} &= b(\tau - 1)a^{b(1 - \tau) - 1} (g - h \pm O(g^2 + h^2)). \end{aligned}$$ Equations [\[eq:nr-vertices-in-range-1\]](#eq:nr-vertices-in-range-1){reference-type="eqref" reference="eq:nr-vertices-in-range-1"} and [\[eq:nr-vertices-in-range-2\]](#eq:nr-vertices-in-range-2){reference-type="eqref" reference="eq:nr-vertices-in-range-2"} together yield the claim. ◻ Note that, if additionally $h \in o(g)$, we can write the last factor as $g(1 - h/g \pm O(g + h^2/g)) = g(1 - o(1) \pm o(1))$ and obtain the following corollary. **Corollary 7**. *Let $a, b > 0$ be constants and let $g, h$ be functions of $n$ such that $g \in o(1)$ and $h \in o(g)$. Let $S$ be the set of vertices with weight in $[(a - g)^b \sqrt{\mu n}, (a - h)^b \sqrt{\mu n}]$. Then $$\label{eq:expS} \mathbb{E}\left[|S|\right] = g n^{\frac{3 - \tau}{2}} \cdot \mu^{\frac{1 - \tau}{2}} ba^{b(1 - \tau) - 1} \cdot (1 \pm o(1)).$$* Consider again the weights $w_\ell$ and $w_u$ as given in Lemma [Lemma 5](#lem:evenly-spaced-boxes-co-matching){reference-type="ref" reference="lem:evenly-spaced-boxes-co-matching"} and let $S$ be the set of vertices in $[w_\ell, w_u)$. Then Corollary [Corollary 7](#col:nr-vertices-in-weight-range-dominant){reference-type="ref" reference="col:nr-vertices-in-weight-range-dominant"} in particular implies that $S$ contains $\Theta(g \cdot n^{\frac{3 - \tau}{2}})$ vertices in expectation. With this, we turn to our second goal mentioned above, namely that each box $B_i$ should be sufficiently large. **Lemma 8**. *Let $B_1, \dots, B_{2k}$ be evenly spaced boxes of height $h$ and gap $g$ in $\mathbb T^d$. If $g \in o(1)$ then each box $B_i$ has volume $h / 2^{d - 1} \cdot (1 - o(1))$.* *Proof.* Recall that the height of $B_i$ is $h$ while its extent in all other dimensions is $\ell = \frac{1}{2} - g$. Thus its volume is $h \cdot (\frac{1}{2} - g)^{d - 1} = h / 2^{d - 1} \cdot (1 - 2g)^{d - 1}$. The claim follows from the fact that $(1 - 2g)^{d - 1}$ approaches $1$ from below for $n \to \infty$ as $g \in o(1)$ and $d$ constant. ◻ Corollary [Corollary 7](#col:nr-vertices-in-weight-range-dominant){reference-type="ref" reference="col:nr-vertices-in-weight-range-dominant"} and Lemma [Lemma 8](#lem:volume-of-boxes){reference-type="ref" reference="lem:volume-of-boxes"} together tell us that the expected number of vertices in each box that have a weight in the desired range is in $\Theta(h \cdot g \cdot n^{\frac{3 - \tau}{2}})$. Recall we want to choose $h$ and $g$ as small as possible such that each box still contains a vertex with high probability. We set $h = c\cdot n^{-\frac{3 - \tau}{4}}$ and $g = c\cdot n^{-\frac{3 - \tau}{4} + \varepsilon}$ for arbitrary constants $c > 0$ and $\varepsilon > 0$. Note that this satisfies the condition $h \in o(g)$ of Corollary [Corollary 7](#col:nr-vertices-in-weight-range-dominant){reference-type="ref" reference="col:nr-vertices-in-weight-range-dominant"} and yields an expected number of $\Theta(n^\varepsilon)$ vertices with the desired weight in each box. Since the number of vertices in a given box follows a binomial distribution and since $\Theta(n^{\varepsilon}) = \omega(\log(n))$, we can apply a Chernoff bound to conclude that actual number of vertices matches the expected value (up to constant factors) with probability $1 - O(n^{-c'})$ for any $c' > 0$ [@bff-espsf-22 Corollaries 2.3 and 2.4]. Together with a union bound, it follows that every box contains $\Theta(n^{\varepsilon})$ vertices (and thus at least one vertex) with probability $1 - O(2k \cdot n^{-c'})$. By choosing $g, h$, and $k$ appropriately, we obtain the following theorem. **Theorem 9**. *Let $G$ be a $d$-dimensional GIRG with $T=0$, and let $s > 0$ and $\varepsilon > 0$ be arbitrary constants. Then, with high probability, $G$ contains a co-matching of size $s \cdot n^{\frac{3 - \tau}{4} - \varepsilon}$ as induced subgraph.* *Proof.* Let $B_1, \dots, B_{2k}$ be evenly spaced boxes of height $h = c\cdot n^{-\frac{3 - \tau}{4}}$ and gap $g = c\cdot n^{-\frac{3 - \tau}{4} + \varepsilon}$ (for appropriately chosen $c > 0$, which will be determined later). Let $w_\ell$ and $w_u$ be defined as in Lemma [Lemma 5](#lem:evenly-spaced-boxes-co-matching){reference-type="ref" reference="lem:evenly-spaced-boxes-co-matching"}. As argued above, Corollary [Corollary 7](#col:nr-vertices-in-weight-range-dominant){reference-type="ref" reference="col:nr-vertices-in-weight-range-dominant"} and Lemma [Lemma 8](#lem:volume-of-boxes){reference-type="ref" reference="lem:volume-of-boxes"} imply that, with high probability, each box $B_i$ includes at least one vertex with weight in $[w_\ell, w_u)$. Thus, by Lemma [Lemma 5](#lem:evenly-spaced-boxes-co-matching){reference-type="ref" reference="lem:evenly-spaced-boxes-co-matching"} these $2k$ vertices form a co-matching. Recall that $2k = 1 / (g + h)$. Thus, we can choose $c$ such that $2k = s \cdot n^{\frac{3 - \tau}{4} - \varepsilon}$. Again, by the above argumentation, it follows that every box contains at least one vertex with probability $1 - O(2k n^{-c'}) = 1 - O(n^{\frac{3 - \tau}{4} - \varepsilon - c'})$ for any constant $c' > 0$. Choosing $c'$ sufficiently large then yields the claim. ◻ This theorem together with Lemma [Lemma 2](#lem:co-matching-nr-cliques){reference-type="ref" reference="lem:co-matching-nr-cliques"} directly imply the following corollary. **Corollary 10**. *Let $G$ be a $d$-dimensional GIRG with $T = 0$, and let $b > 0$ and $\varepsilon > 0$ be arbitrary constants. Then, with high probability, the number of maximal cliques in $G$ is at least $b^{n^{(3 - \tau)/4 - \varepsilon}}$.* Figure [3](#fig:girglb){reference-type="ref" reference="fig:girglb"} shows this lower bound for $b=2$ against $n$. Interestingly, while Corollary [Corollary 10](#cor:maxcliquesgirg){reference-type="ref" reference="cor:maxcliquesgirg"} shows that the number of maximal cliques grows super-polynomially in $n$, for $\tau\gg 2$, this growth may still be slower than the linear slope $n$ for large geometric networks. This is of particular importance as the smaller order terms of the number of maximal cliques contain terms of at least $\Theta(n)$. Indeed, the number of maximal 2-cliques is lower bounded by the number of vertices of degree 1, which scales linearly by Equation [\[eq:pl\]](#eq:pl){reference-type="eqref" reference="eq:pl"}. Thus, for practical purposes, the dominant term could be the linear term instead of the super-polynomial term, especially when the degree exponent is close to 3. {#fig:girglb width="\\linewidth"} {#fig:girglbnotorus width="\\linewidth"} {#fig:IRG lower bound width="\\linewidth"} ## GIRG with 2-Dimensional Square {#sec:girg-with-2d-square} Our previous lower bound for the number of maximal cliques relies on the toroidal structure of the underlying space. We now show that even when the vertex positions are constrained to be positioned in the square $[0,1]^2$ instead, the GIRG still contains a super-polynomial number of maximal cliques. **Theorem 11**. *For any $\varepsilon>0$ and $b>0$, a 2-dimensional GIRG with vertex positions uniformly distributed over $[0,1]^2$ equipped with the 2-norm and $T=0$ contains with high probability at least $$Cb^{n^{\frac{3-\tau}{10}-\varepsilon}}$$ maximal cliques for some $C>0$.* {#fig:circle width="\\linewidth"} {#fig:circleclique width="\\linewidth"} *Proof.* Let $S$ be the set of vertices with weights within $[a\sqrt{\mu n(1-c\cdot n^{-\beta})},a\sqrt{\mu n}]$ for some $0<a<1/4$ (and appropriately chosen $c > 0$, which will be determined later). By Corollary [Corollary 7](#col:nr-vertices-in-weight-range-dominant){reference-type="ref" reference="col:nr-vertices-in-weight-range-dominant"}, $$\mathbb{E}\left[|S|\right] %n\Big(\big(a\sqrt{\mu n(1-n^{-\beta}})\big)^{1-\tau}-(a\sqrt{\mu n})^{1-\tau}\Big)\sim =n^{(3-\tau)/2-\beta}(1+o(1))$$ Let $\mathcal{C}$ be a circle on $[0,1]^2$ of constant radius $R<1/4$. We now create an even number of areas $B_1,\dots,B_{2k}$ of height $h$, evenly distributed over $\mathcal{C}$ as illustrated in Figure [6](#fig:circle){reference-type="ref" reference="fig:circle"}. We ensure that any pair of vertices in two opposite areas $B_i$ and $B_{i+k}$ are disconnected. That is, the distance $t$ between the two ends of these areas should equal $$t=a.$$ We also ensure that any pair of vertices in non-opposite areas connect. This means that the distance $l$ between the rightmost part of $B_i$ and the leftmost part of any non-opposite $B_j$ is at most $$\label{eq:l2dimnotorus} l=a\sqrt{1-c \cdot n^{-\beta}}.$$ The width $q$ of one of the $B_i$'s is given by $$\frac{q}{2}=\sqrt{R^2-(R-h)^2}=\sqrt{h(2R-h)}=\sqrt{h(h+t)}=\sqrt{h(h+a)}.$$ Thus, the area of $B_i$ is given by $$c_1h^{3/2}\sqrt{h+a},$$ for some constant $c_1>0$. The probability that a given area contains no vertices from $S$ is given by $$\label{eq:areaempty} \Big(1-c_1h^{3/2}\sqrt{h+a}\Big)^{|S|}\sim \exp(-c_2n^{(3-\tau)/2-\beta}h^{3/2}),$$ for some $c_2>0$. We now calculate the maximal number of areas that we can pack on $\mathcal{C}$. The circumference of $\mathcal{C}$ is $2\pi R$. The arc length of a single area is at most $q\pi/2$. Furthermore, the arc length of the section with a chord of length $l$, $a(l)$, is given by $$\begin{aligned} \label{eq:arc_length_l} a(l)= 2R\sin^{-1}(l/(2R)) & = 2R\sin^{-1}\Big(\frac{a\sqrt{1-c \cdot n^{-\beta}}}{a+2h}\Big) \nonumber\\ & = 2R\sin^{-1}\Big(1-\frac{a+2h-a\sqrt{1-c\cdot n^{-\beta}}}{a+2h}\Big)\nonumber\\ & = \pi R- 2^{3/2}R\sqrt{\frac{a+2h-a\sqrt{1-c\cdot n^{-\beta}}}{a+2h}} (1+o(1)), \end{aligned}$$ using the Taylor series of $\sin^{-1}(1-x)$ around $x=0$. Now take $h=c\cdot n^{-\gamma}$. Then, by [\[eq:arc_length_l\]](#eq:arc_length_l){reference-type="eqref" reference="eq:arc_length_l"}, $a(l)$ scales as $$\begin{aligned} a(l)= \pi R- R\Theta\Big(\sqrt{(1-\sqrt{1-c\cdot n^{-\beta}})+c\cdot n^{-\gamma}}\Big)= \pi R- R\Theta(\sqrt{c\cdot n^{-\beta}+c\cdot n^{-\gamma}}). \end{aligned}$$ Now the arc length between two adjacent sections $B_i$ and $B_{i+1}$ is equal to $\pi R-a(l)$. This means that the arc length between $B_i$ and $B_{i+1}$ scales as $\pi R-(\pi R-R\Theta\sqrt{c\cdot n^{-\beta}+c\cdot n^{-\gamma})})=R\Theta(\sqrt{c\cdot n^{-\beta}+c\cdot n^{-\gamma}})$. The maximal value of the number of possible areas $2k$, is the total circumference of $\mathcal{C}$ divided by the arc length of an interval $B_i$ and the arc length between $B_i$ and $B_{i+1}$, which yields $$\label{eq:mbound} 2k= \frac{2\pi R}{R\Theta(\sqrt{c\cdot n^{-\beta}+c\cdot n^{-\gamma}})+ \sqrt{h(h+a)}}= \Theta\Big(\min(n^{\beta/2},n^{\gamma/2})\Big) .$$ Thus, by choosing $c$ correctly, we can let $2k= s\cdot \min(n^{\beta/2},n^{\gamma/2})$ for any $s>0$. When all $i\in[2k]$ contain at least one vertex in $S$, any set of $2k$ vertices with exactly one vertex in each $B_i$ forms a clique minus a matching, as illustrated in Figure [7](#fig:circleclique){reference-type="ref" reference="fig:circleclique"}. Furthermore [\[eq:areaempty\]](#eq:areaempty){reference-type="eqref" reference="eq:areaempty"} shows that with high probability, all $B_i$ are non-empty, as long as $n^{(3-\tau)/2-\beta}h^{3/2}\to\infty$ as $n\to\infty$. We therefore choose $\beta=(3-\tau)/5-\varepsilon$ and $\gamma=(3-\tau)/5$. Then, with high probability there is a clique minus a matching of size $2k =s\cdot n^{(3-\tau)/10-\varepsilon}$. Thus, by Lemma [Lemma 2](#lem:co-matching-nr-cliques){reference-type="ref" reference="lem:co-matching-nr-cliques"} and choosing $s$ sufficiently large yields that for fixed $b>0$ the number of maximal cliques can be bounded from below by $$b^{ n^{(3-\tau)/10-\varepsilon}}$$ ◻ Figure [4](#fig:girglbnotorus){reference-type="ref" reference="fig:girglbnotorus"} shows the lower bound of Theorem [Theorem 11](#thm:girg_non_torus){reference-type="ref" reference="thm:girg_non_torus"} against $n$. As for the toroidal case, the super-polynomial growth may be dominated by lower-order linear terms. ## Non-Threshold Case {#sec:girg-non-threshold-case} We now show how our constructions extend to the non-threshold GIRG, where the connection probability is given by Equation [\[eq:edgeprob\]](#eq:edgeprob){reference-type="eqref" reference="eq:edgeprob"} instead of Equation [\[eq:edgeprobthreshold\]](#eq:edgeprobthreshold){reference-type="eqref" reference="eq:edgeprobthreshold"}. **Theorem 12**. *Let $G$ be a $d$-dimensional GIRG with $T > 0$, and let $s > 0$ be an arbitrary constant. Then, there exists an $\varepsilon > 0$ such that, with high probability, $G$ contains a co-matching of size $s \cdot n^{(3 - \tau)/5} \cdot (\varepsilon \log n)^{-(1/2)}$ as induced subgraph.* *Proof.* As before, we consider $2k$ boxes $B_1, \ldots, B_{2k}$ with height $h$ and gap $g$, though now we choose $$\begin{aligned} g = h \cdot (\varepsilon \log n)^{1/2} \qquad \text{and} \qquad h = \frac{1}{2s} \cdot n^{-\frac{3 - \tau}{5}}, \end{aligned}$$ for a constant $\varepsilon > 0$, which we determine below. We again focus on the vertex set $S$ containing all vertices with weights in $[w_\ell, w_u]$, though our choice for $w_u$ is slightly different. In particular, we choose $$\begin{aligned} w_\ell = (1/2 - g)^{d/2} \sqrt{\mu n} \qquad \text{and} \qquad w_u = (1/2 - (T/d + 1)h)^{d/2}\sqrt{\mu n}. \end{aligned}$$ Our goal now is to show that, with high probability, there exists at least one co-matching that contains one vertex from each box. That is, if $M$ denotes the number of such co-matchings, we want to show that $M > 0$ with high probability. We start by bounding the number the number of vertices from $S$ that lie in a given box $B_i$, denoted by $S(B_i)$. Since $T$ and $d$ are constants and $(\varepsilon \log n)^{1/2} \in \omega(1)$, we have $(T/d + 1)h \in o(g)$, allowing us to bound $\mathbb{E}[|S|]$ using Corollary [Corollary 7](#col:nr-vertices-in-weight-range-dominant){reference-type="ref" reference="col:nr-vertices-in-weight-range-dominant"}, which yields $$\begin{aligned} \mathbb{E}[|S|] = \Theta\left(gn^{\frac{3 - \tau}{2}}\right). \end{aligned}$$ Moreover, since the vertices are distributed uniformly at random in the ground space, the fraction of vertices from $S$ that lie in the box $B_i$ is proportional to its volume, which is $\Theta(h)$ according to Lemma [Lemma 8](#lem:volume-of-boxes){reference-type="ref" reference="lem:volume-of-boxes"}. It follows that $$\begin{aligned} \mathbb{E}[|S(B_i)|] = \Theta\left(g h n^{\frac{3 - \tau}{2}}\right) = \Theta\left(h^2 n^{\frac{3 - \tau}{2}} (\varepsilon \log n)^{1/2}\right) = \Theta\left( n^{(3 - \tau)(1/2 - 2/5)} (\varepsilon \log n)^{1/2}\right). \end{aligned}$$ Analogous to the proof of Lemma [Lemma 8](#lem:volume-of-boxes){reference-type="ref" reference="lem:volume-of-boxes"} we can apply a Chernoff bound to conclude that the number of vertices in $S(B_i)$ matches the expected value (up to constant factors) with probability $1 - O(n^{-c})$ for any $c > 0$. Note that the number of boxes is given by $$\begin{aligned} \label{eq:temperature-co-match-size} 2k = \frac{1}{g + h} = \frac{1}{h((\varepsilon \log n)^{1/2} + 1)} = \frac{2s \cdot n^{\frac{3 - \tau}{5}}}{(\varepsilon \log n)^{1/2} + 1}, \end{aligned}$$ which is at most $n$. Thus, applying the union bound yields that with high probability *every* box contains $n' = \Theta(n^{(3 - \tau)(1/2 - 2/5)} (\varepsilon \log n)^{1/2})$ vertices. In the following, we implicitly condition on this event to happen. Now recall that a co-matching consisting of one vertex from each box forms if each vertex is adjacent to the vertices in all other boxes, *except* the vertex from the opposite box. Despite the temperature, vertices in non-opposite boxes are still adjacent with probability $1$, since the weight of two such vertices $i$ and $j$ is at least $w_\ell$ and their distance is at most $\frac{1}{2} - g$ and, thus, according to Equation [\[eq:edgeprob\]](#eq:edgeprob){reference-type="ref" reference="eq:edgeprob"} $$\begin{aligned} p_{ij} = \min \left\{ \left( \frac{w_i w_j}{n \mu ||x_i - x_j||^d}\right)^{1/T}, 1 \right\} \ge \min \left\{ \left(\frac{w_\ell^2}{n \mu (1/2 - g)^d}\right)^{1/T}, 1\right\} = 1. \end{aligned}$$ In contrast to the threshold case, however, the probability for vertices in opposite boxes to be adjacent is no longer $0$. Since two such vertices $i$ and $j$ have distance at least $\frac{1}{2} - h$ and weight at most $w_u = (\frac{1}{2} - (T/d + 1)h)^{d/2}\sqrt{\mu n}$, we can bound the probability for them to be adjacent using Equation [\[eq:edgeprob\]](#eq:edgeprob){reference-type="ref" reference="eq:edgeprob"}, which yields $$\begin{aligned} p_{ij} &= \min \left\{\left(\frac{w_i w_j}{n \mu ||x_i - x_j||^d}\right)^{1/T}, 1\right\} \\ & \le \min \left\{ \left(\frac{w_u^2}{n \mu (\frac{1}{2} - h)^d}\right)^{1/T} , 1\right\} \\ &= \min \left\{ \left(\frac{1 - (T/d + 1)2h}{1 - 2h}\right)^{d/T} , 1\right\} \\ &= \min \left\{ \left(1 - \frac{T}{d} \cdot \frac{2h}{1 - 2h}\right)^{d/T} , 1\right\} \\ &\le \min \left\{ \left(1 - \frac{T}{d} \cdot 2h \right)^{d/T} , 1\right\}. \end{aligned}$$ Since $1 - x \le e^{-x}$, we obtain $p_{ij} \le e^{-2h}$. With this we are now ready to bound the probability $\mathbb{P}[M > 0]$, that at least one co-matching forms that contains one vertex from each box. To this end, we need to find one non-edge in each pair of opposite boxes, i.e., each such pair needs to contain two vertices (one from each box) that are not adjacent. Conversely, the only way to *not* find a co-matching is if there exists one pair of opposite boxes such that all vertices in one box are adjacent to all vertices in the other. This happens with probability at most $(p_{ij})^{(n')^2}$. Since there are $k$ pairs of opposite boxes, applying the union bound yields $$\begin{aligned} \label{eq:pm0} \mathbb{P}[M = 0] \le k(p_{ij})^{(n')^2} \le k \exp(-2h (n')^2). \end{aligned}$$ Now recall that $n' = \Theta(n^{(3 - \tau)(1/2 - 2/5)} (\varepsilon \log n)^{1/2})$ and that $h = 1/(2s) \cdot n^{-(3 - \tau)/5}$. Consequently, we obtain $$\begin{aligned} \mathbb{P}[M = 0] &\le k \exp\left(- \Theta\left(n^{-(3-\tau)/5} \cdot n^{(3 - \tau)(1 - 4/5)} \cdot \varepsilon \log n \right) \right) \\ &= k \exp\left(-\Theta(\varepsilon \log n)\right) \\ &= k n^{-\Theta(\varepsilon)}. \\ \end{aligned}$$ Moreover, since $k = O(n^{(3 - \tau)/5})$ (see Equation [\[eq:temperature-co-match-size\]](#eq:temperature-co-match-size){reference-type="ref" reference="eq:temperature-co-match-size"}), we have $$\begin{aligned} \mathbb{P}[M = 0] = O \left( n^{(3 - \tau)/5 - \Theta(\varepsilon)} \right), \end{aligned}$$ meaning, for sufficiently large $n$, we can choose $\varepsilon$ such that $\mathbb{P}[M = 0] = O(n^{-1})$ and, conversely, $\mathbb{P}[M > 0] = 1 - O(n^{-1})$. So with high probability there exists at least one co-matching of size $$\begin{aligned} 2k = \frac{2s \cdot n^{\frac{3 - \tau}{5}}}{(\varepsilon \log n)^{1/2} + 1} \ge \frac{2s \cdot n^{\frac{3 - \tau}{5}}}{2 (\varepsilon \log n)^{1/2}} = \frac{s \cdot n^{\frac{3 - \tau}{5}}}{(\varepsilon \log n)^{1/2}}, \end{aligned}$$ where the inequality holds for sufficiently large $n$. ◻ Together with Lemma [Lemma 2](#lem:co-matching-nr-cliques){reference-type="ref" reference="lem:co-matching-nr-cliques"} we obtain the following corollary. **Corollary 13**. *Let $G$ be a $d$-dimensional GIRG with $T > 0$, and let $b > 0$ be an arbitrary constant. Then there exists an $\varepsilon > 0$ such that, with high probability, the number of maximal cliques in $G$ is at least $b^{n^{(3 - \tau)/5} \cdot (\varepsilon \log n)^{-(1/2)}}$.* We can extend Theorem [Theorem 11](#thm:girg_non_torus){reference-type="ref" reference="thm:girg_non_torus"} to non-zero temperature in a very similar fashion (proof is in Appendix [7](#app:prooftemperature){reference-type="ref" reference="app:prooftemperature"}) **Theorem 14**. *For any $\varepsilon>0$ and $b>1$, a 2-dimensional GIRG with $T>0$ and vertex positions uniformly distributed over $[0,1]^2$ contains with high probability at least $$b^{n^{\frac{3-\tau}{12}-\varepsilon}\log(n)^{1-\varepsilon}}.$$ maximal cliques.* # Inhomogeneous Random Graphs (IRG) {#sec:IRG} We now turn to a random graph model that is scale-free, but does not contain a source of geometry, the inhomogeneous random graph (IRG). We show that also in this model, the number of maximal cliques scales super-polynomially in the network size $n$. In the IRG, the probability that two vertices of weights $w_u$ and $w_v$ connect is independently given by $$\label{eq:conprob} p(w_u, w_v) = \min\Big(\frac{w_u w_v}{\mu n},1\Big),$$ where $\mu$ controls the expected average degree. Furthermore, as done above, we assume that the weights are independently drawn from the power-law distribution; see Equation [\[eq:pl\]](#eq:pl){reference-type="eqref" reference="eq:pl"}. To show a lower bound on the number of maximal cliques, we make use of the fact that an IRG contains a not too small rather dense subgraph with high probability. The following theorem is obtained by looking just at the subgraph induced by vertices with weights in a certain range. We chose the specific range to satisfy three criteria. First, the range is sufficiently large, such that the subgraph contains many vertices. Second, the range is sufficiently small such that all vertex pairs in the subgraph are connected with a similar probability. And third, the weights are large enough such that a densely connected subgraph forms, but not so large that the vertices merge into a single clique. **Theorem 15**. *Let $G$ be an IRG with $\tau \in (2, 3)$ and let $b > 1$ and $\varepsilon \in (0, \frac{3 - \tau}{4})$ be arbitrary constants. Then, the expected number of maximal cliques in $G$ is in $\Omega\big(b^{n^{(3 - \tau) / 4 - \varepsilon} \log n}\big)$.* *Proof.* We show that already the subgraph $G'$ induced by the vertices in a certain weigh range has the claimed expected number of cliques. To define $G'$, we consider weights in $[w_\ell, w_u]$ with $w_\ell = \sqrt{(1 - g) \mu n}$ and $w_u = \sqrt{(1 - h) \mu n}$. To abbreviate notation, let $$\gamma = n^{\frac{3 - \tau}{4}}.$$ For constants $a$ and $c$ we determine later, we choose $g$ and $h$ as $$g = a h \quad \text{and} \quad h = c n^\varepsilon \log(n) \gamma^{-1}.$$ Note that $w_\ell < w_u$ if and only if $a > 1$. Let $n'$ be the number of vertices in $G'$. From Lemma [Lemma 6](#lem:nr-vertices-in-weight-range){reference-type="ref" reference="lem:nr-vertices-in-weight-range"} it follows $$\mathbb{E}\left[n'\right] \in \Theta(\gamma^2 \cdot n^\varepsilon \log(n) \gamma^{-1}) = \Theta(n^\varepsilon \gamma \log n).$$ As every vertex has independently the same probability to be in $G'$, a Chernoff bound implies that $n' \in \Theta(n^\varepsilon\gamma \log n)$ holds with high probability. Thus, in the following, we implicitly condition on this event to happen. To give a lower bound on the number of maximal cliques in $G'$, we only count the number $N_k$ of maximal cliques of size $k$ with $$k = \frac{3 \varepsilon}{c} n^{-\varepsilon}\gamma.$$ We note that this is the same constant $c$ as in the definition of $h$ above. To compute the expectation for the number $N_k$ of maximal cliques, we consider all vertex sets of size $k$ of the vertices in $G'$, i.e., for a fixed subset $C$ of size $k$, we obtain $$\begin{aligned} \mathbb{E}\left[N_k\right] \ge {n' \choose k} \mathbb{P}\left(C \text{ is a clique}\right) \mathbb{P}\left(C \text{ maximal} \mid C \text{ is a clique.}\right). \end{aligned}$$ In the following, we give estimates for the three terms individually. We start with the event that $C$ is a clique. Due to the lower and upper bound on the weights in $G'$, it follows that any pair of vertices in $G'$ is connected with probability at least $p_\ell = 1 - g$ and at most $p_u = 1 - h$. Thus, for a fixed subset $C$ of vertices of size $|C| = k$, the probability that all $k$ vertices are pairwise connected is at least $p_\ell^{k (k - 1) / 2} = (1 - g)^{k (k - 1) / 2}$. As $g$ goes to $0$ for growing $n$ and $1 - x \in \Omega(\exp(-x))$ in this case, we get $$\label{eq:irg-subset-is-clique} \mathbb{P}\left(C \text{ is a clique}\right) \ge (1 - g)^{k (k - 1) / 2} \in \Omega\left(\exp\left(- \frac{g k (k - 1)}{2}\right)\right).$$ For $C$ to be a maximal clique (conditioning on it being a clique), additionally no other vertex can be connected to all vertices from $C$. This probability is at least $(1 - p_u^k)^{n' - k} = (1 - (1 - h)^k)^{n' - k}$. As $1 - x \le \exp(-x)$, it follows that $(1 - h)^k \le \exp(-hk) = n^{-3\varepsilon}$, where the last equality follows from plugging in the values we chose for $h$ and $k$. Again using $1 - x \in \Omega(\exp(-x))$ for sufficiently small $x$, we can conclude that $$\begin{aligned} \notag \mathbb{P}\left(C \text{ maximal} \mid C \text{ is a clique}\right) &\ge (1 - (1 - h)^k)^{n' - k}\\ \label{eq:irg-clique-is-maximal} &\ge (1 - n^{-3\varepsilon})^{n' - k}\\ \notag &\in \Omega\Big( \exp\big( -n^{-3\varepsilon} (n' - k) \big) \Big). \end{aligned}$$ Finally, for the binomial coefficient, we get $$\label{eq:irg-binomial-coefficient} {n' \choose k} \ge \left( \frac{n'}{k} \right)^k = \exp\left(\log\left(\frac{n'}{k}\right) k\right).$$ Combining Equations [\[eq:irg-subset-is-clique\]](#eq:irg-subset-is-clique){reference-type="eqref" reference="eq:irg-subset-is-clique"}, [\[eq:irg-clique-is-maximal\]](#eq:irg-clique-is-maximal){reference-type="eqref" reference="eq:irg-clique-is-maximal"}, and [\[eq:irg-binomial-coefficient\]](#eq:irg-binomial-coefficient){reference-type="eqref" reference="eq:irg-binomial-coefficient"}, it remains to show that for every constant $b > 1$, we can choose the constants $a > 1$ and $c$ in the definitions of $g$ and $h$ such that $$\log\left(\frac{n'}{k}\right) k - \frac{g k (k - 1)}{2} - n^{-3\varepsilon} (n' - k) \stackrel{\text{\tiny(to be shown)}}{>} n^{-\varepsilon} \gamma \log n \log b = n^{\frac{3 - \tau}{4} - \varepsilon} \log n \log b.$$ This can be achieved by simply plugging in the values for $n'$, $k$, and $g$. For the first (and only positive) term, we obtain $$\begin{aligned} \log\left(\frac{n'}{k}\right) k &= \log\left(\frac{\Theta(n^\varepsilon \log(n) \gamma)}{3\varepsilon / c n^{-\varepsilon}\gamma}\right) \frac{3 \varepsilon}{c} n^{-\varepsilon}\gamma\\ &= \log\left(n^{2\varepsilon}\Theta(\log n)\right) \frac{3 \varepsilon}{c} n^{-\varepsilon}\gamma\\ \intertext{and thus for sufficiently large $n$} &\ge \frac{6 \varepsilon^2}{c} n^{-\varepsilon} \gamma \log n. \end{aligned}$$ For the negative terms, we start with the latter and obtain $$n^{-3\varepsilon} (n' - k) \in \Theta(n^{-3\varepsilon} n^\varepsilon \gamma \log n) = \Theta(n^{-2\varepsilon} \gamma \log n).$$ This is asymptotically smaller than the positive term and can thus be ignored. For the other negative term, first note that $gk = 3 a \varepsilon \log n$. Thus, we obtain $$\frac{g k (k - 1)}{2} \le \frac{ 3 a \varepsilon }{2} \log(n) k = \frac{ 3 a \varepsilon}{2} \log(n) \frac{3 \varepsilon}{c} n^{-\varepsilon}\gamma = \frac{ 9 a \varepsilon^2}{2 c} n^{-\varepsilon} \gamma \log n.$$ Together with the positive term, we obtain that for sufficiently large $n$, it holds $$\begin{aligned} \log\left(\frac{n'}{k}\right) k - \frac{g k (k - 1)}{2} &\ge \frac{6 \varepsilon^2}{c} n^{-\varepsilon} \gamma \log n - \frac{ 9 a \varepsilon^2}{2 c} n^{-\varepsilon} \gamma \log n\\ &= \left(6 - \frac{9 a}{2}\right) \frac{\varepsilon^2}{c} n^{-\varepsilon} \gamma \log n. \end{aligned}$$ With this, we can choose $a > 1$ such that the first factor is positive and we can choose $c$ such that $\varepsilon^2 / c = \log b$, which proves the claim. ◻ Figure [5](#fig:IRG lower bound){reference-type="ref" reference="fig:IRG lower bound"} shows that the lower bound provided by Theorem [Theorem 15](#thm:max_cliques_irg){reference-type="ref" reference="thm:max_cliques_irg"} may still be smaller than linear for networks that are quite large, especially when $\tau\approx 3$. ## Small Maximal Cliques are Rare We now focus on the maximal cliques of a fixed size in the IRG. How many maximal cliques of size $k$ are present in an IRG? Let $N(K_k)$ denote the number of maximal cliques of size $k$. Furthermore, let $M_n(\varepsilon)$ denote $$\label{eq:Mn} M_n(\varepsilon)=\{ ({v_1,\ldots, v_k})\colon w_i\in[\varepsilon,1/\varepsilon] (\mu n) ^{\frac{\tau-2}{\tau-1}}\ \text{ for } i=1,2 \text{ and } w_i\in[\varepsilon,1/\varepsilon] (\mu n) ^{\frac{1}{\tau-1}} \ \forall i\in\{3,\dots,k\} \}.$$ Thus, $M_n(\varepsilon)$ is the set of sets of $k$ vertices such that two vertices have weight approximately $n^{(\tau-2)/(\tau-1)}$, and all other vertices have weights approximately $n^{1/(\tau-1)}$. Denote the number of maximal $k$-cliques with sets of vertices in $M_n(\varepsilon)$ by $N(K_k,M_n(\varepsilon))$. Then, the following theorem shows that these 'typical' maximal cliques are asymptotically all maximal cliques. Furthermore, it shows that all maximal cliques of size $k>2$ occur equally frequently in scaling, and they also appear on the same types of vertices. Here we use $\ensuremath{\stackrel{\mathbb{P}}{\longrightarrow}}$ to denote convergence in probability. **Theorem 16** (Maximal clique localization). *For any fixed $k\geq 3$,* (i) *For any $\varepsilon_n$ such that $\lim_{n\to\infty}\varepsilon_n=0$, $$\frac{N\big(K_k,M_n\left(\varepsilon_n\right)\big) }{N(K_k)}\ensuremath{\stackrel{\mathbb{P}}{\longrightarrow}}1.$$* (ii) *Furthermore, for any fixed $0<\varepsilon<1$, $$\label{eq:Nsubmag} \mathbb{E}\left[N(K_k,M_n(\varepsilon))\right]=\Theta{n^{(3-\tau)(2\tau-3)/(\tau-1)}}.$$* Theorem [Theorem 16](#thm:maxcliqeslocalized){reference-type="ref" reference="thm:maxcliqeslocalized"}(i) states that asymptotically all maximal $k$-cliques are formed between two vertices of weights proportional to $n^{(\tau-2)/(\tau-1)}$ and all other vertices of weights proportional to $n^{1/(\tau-1)}$. Theorem [Theorem 16](#thm:maxcliqeslocalized){reference-type="ref" reference="thm:maxcliqeslocalized"}(ii) then shows that there are proportional to $n^{(3-\tau)(2\tau-3)/(\tau-1)}$ such maximal $k$-cliques. Note that this scaling is significantly smaller than the scaling of the total number of $k$-cliques, which scales as $n^{k/2(3-\tau)}$ [@hofstad2017d]. Interestingly, the scaling of the number of max-cliques is $k$-independent, contrary to the total number of cliques. In particular, the number of $k$ maximal cliques is always $o(n)$, contrary to the number of $k$-cliques which scales larger than $n$ when $\tau<3-2/k$. This shows once more that the large number of maximal cliques in the IRG is caused by extremely large maximal cliques, as fixed-size maximal cliques are only linearly many. To prove this theorem, we need the following technical lemma, which is proven in Appendix [8](#app:proofintfinite){reference-type="ref" reference="app:proofintfinite"}: **Lemma 17**. *$$\label{eq:intmaxclique} \int_0^1\dots\int_0^1 \int_0^\infty\int_{x_1}^\infty x_1^{k-1-\tau}x_2^{1-\tau}x_3^{1-\tau}\cdots x_k^{1-\tau}\prod_{i=3}^k\min\Big(x_2x_i,1\Big)e^{-\mu^{1-\tau}x_1x_2^{\tau-2}}dx_2 dx_1\dots dx_k<\infty$$* Furthermore, we need a lemma that bounds the probability that a given clique on vertices of weights $x_1\leq x_2\dots\leq x_k$ is maximal: **Lemma 18**. *The probability that a given clique between $k$ vertices of weights $x_1\leq x_2\dots\leq x_k$ is maximal is bounded by $$\begin{aligned} & \exp\Big(-C_1n^{2-\tau}\mu^{1-\tau}x_{1}x_{2}^{\tau-2}\Big)(1+o(1)) \leq \mathbb{P}\left(\text{clique on weights }x_1,\dots,x_k\text{ maximal}\right)\nonumber\\ & \leq \exp\Big(-C_2n^{2-\tau}\mu^{1-\tau}x_{1}x_{2}^{\tau-2}\Big), \end{aligned}$$ for some $0<C_1\leq C_2<\infty$.* *Proof.* When $x_{1}\leq x_2\leq \dots \leq x_k$, we can compute the probability that this $k$ clique is part of a larger clique with a randomly chosen vertex as $$\begin{aligned} & \int_1^{\infty}w^{-\tau}\prod_{i\in[k]}\min\Big(\frac{w x_{i}}{\mu n},1\Big)dw \nonumber\\ & = \frac{x_{1}\dots x_k}{(\mu n)^k}\int_1^{\mu n/ x_k}w^{k-\tau}dw + \frac{x_1\dots x_{{k-1}}}{(\mu n)^{k-1}}\int_{\mu n/ x_k}^{\mu n/ x_{{k-1}}}w^{k-1-\tau}dw\nonumber\\ & \quad + \dots + \frac{x_1x_{{2}}}{(\mu n)^{2}}\int_{\mu n/ x_{3}}^{\mu n/ x_{{2}}}w^{2-\tau}dw + \frac{x_1}{\mu n}\int_{\mu n/ x_2}^{\mu n/ x_{{1}}}w^{1-\tau}dw+ \int_{\mu n/ x_1}^{\infty}w^{-\tau}dw\nonumber\\ & = c_k\frac{x_1\dots x_k}{(\mu n)^k}\Big(\frac{\mu n}{x_k}\Big)^{k+1-\tau}+ \dots %+ \frac{x_1x_2}{(\mu n)^2}\Big(\frac{\mu n}{x_2}\Big)^{3-\tau} + c_2\frac{x_1}{\mu n}\Big(\frac{\mu n}{x_2}\Big)^{2-\tau}+c_1\Big(\frac{\mu n}{x_1}\Big)^{1-\tau}, \end{aligned}$$ for some $c_1,\dots, c_k>0.$ When $x_1\leq x_2\leq \dots\leq x_k$, this term becomes $$(\mu n)^{1-\tau}\sum_{l=1}^kc_lx_{l}^{-l+\tau}\prod_{i<l}x_{i}. %\leq k(\mu n)^{1-\tau}\Big(\frac{x_1}{\mu n}\Big(\frac{\mu n}{x_2}\Big)^{2-\tau}+\Big(\frac{\mu n}{x_1}\Big)^{1-\tau}\Big)$$ The ratio between two consecutive terms of this summation equals $$\frac{x_{l}^{\tau-l}x_1\dots x_{{l-1}}}{x_{{l+1}}^{\tau-l-1}x_1\dots x_{{l}}}=\Big(\frac{x_{l}}{x_{{l+1}}}\Big)^{\tau-l-1}.$$ Now as $x_{l}\leq x_{{l+1}}$ and $\tau\in(2,3)$, this ratio is larger than 1 for $l\geq 2$, and smaller than one for $l=1$. This means that the summation can be dominated by $$\label{eq:ubsum} (\mu n)^{1-\tau}\sum_{l=1}^kc_lx_{l}^{-l+\tau}\prod_{i<l}x_{i}\leq C (\mu n)^{1-\tau}x_1x_2^{\tau-2},$$ for some $C>0$. Thus, the probability that a clique on vertices with weights $x_1,\dots, x_k$ is maximal can be upper bounded by $$\begin{aligned} \mathbb{P}\left((x_1,\dots,x_k) \text{ clique maximal }\right) & \leq \Big(1-C (\mu n)^{1-\tau}x_1x_2^{\tau-2}\Big)^{n}\nonumber\\ & \leq \exp\Big(-Cn^{2-\tau}\mu^{1-\tau}x_1x_2^{\tau-2}\Big). \end{aligned}$$ We lower bound the probability that the clique is maximal by using that $$(\mu n)^{1-\tau}\sum_{l=1}^kc_lx_{l}^{-l+\tau}\prod_{i<l}x_{i}\geq c_2 (\mu n)^{1-\tau}x_1x_2^{\tau-2}.$$ Thus, $$\begin{aligned} \mathbb{P}\left((x_1,\dots,x_k) \text{ clique maximal }\right) & \geq \Big(1-c_2 (\mu n)^{1-\tau}x_1x_2^{\tau-2}\Big)^{n}\nonumber\\ & \geq \exp\Big(-c_2n^{2-\tau}\mu^{1-\tau}x_1x_2^{\tau-2}/(1+c_2n^{1-\tau}\mu^{1-\tau}x_1x_2^{\tau-2})\Big)\nonumber\\ & = \exp\Big(-c_2n^{2-\tau}\mu^{1-\tau}x_1x_2^{\tau-2}\Big)(1+o(1)).\end{aligned}$$ ◻ Now we are ready to prove Theorem [Theorem 16](#thm:maxcliqeslocalized){reference-type="ref" reference="thm:maxcliqeslocalized"}: *Proof of Theorem [Theorem 16](#thm:maxcliqeslocalized){reference-type="ref" reference="thm:maxcliqeslocalized"}.* Fix $\ell_i\leq u_i$ for $i\in[k]$. We now compute the expected number of maximal $k$-cliques in which the vertices have weights $n^{(\tau-2)/(\tau-1)}[\ell_i,u_i]$ for $i=1,2$, and $n^{1/(\tau-1)}[\ell_i,u_i]$ for $i\geq 3$. We bound the expected number of such maximal copies of $K_k$ by $$\label{eq:Exp1small} \begin{aligned}[b] &\sum_{\boldsymbol{v}}\mathbb{E}\left[I(K_k, \boldsymbol{v})\mathbbm{1}_{\left\{w_{v_i}\in [\ell_i,u_i]n^{(\tau-2)/(\tau-1)}, \ i =1,2, \ w_{v_i}\in [\ell_i,u_i] n^{1/(\tau-1)},\ i \geq 3 \right\}}\right]\\ & = n^k\int_{\ell_1 n^{(\tau-2)/(\tau-1)}}^{u_1 n^{(\tau-2)/(\tau-1)}}\int_{\ell_2 n^{(\tau-2)/(\tau-1)}}^{u_2 n^{(\tau-2)/(\tau-1)}}\cdots \int_{\ell_k n^{1/(\tau-1)}}^{u_k n^{1/(\tau-1)}}(x_1\cdots x_k)^{-\tau} \prod_{\mathclap{1\leq i<j\leq k}}\min\left(\frac{x_ix_j}{n},1\right) \nonumber\\ & \quad \cdot \mathbb{P}\left((x_1,\dots,x_k) \text{ clique maximal}\right){\rm d}x_k\cdots{\rm d}x_1, \end{aligned}$$ where $I(K_k, \boldsymbol{v})$ is the indicator that a maximal $k$-clique is present on vertices $\boldsymbol{v}$, and the sum over $\boldsymbol{v}$ is over all possible sets of $k$ vertices. Now the probability that a clique is maximal can be upper bounded as in Lemma [Lemma 18](#lem:pmaximal){reference-type="ref" reference="lem:pmaximal"}. We bound the minimum in [\[eq:Exp1small\]](#eq:Exp1small){reference-type="eqref" reference="eq:Exp1small"} by - $x_ix_j/n$ for $\{i,j\}=\{1,2\}$ or $i=1,j\geq 3$; - 1 for $i,j\geq 3$ . Making the change of variables $x_i=y_in^{1/(\tau-1)}$ for $i=3,\dots,k$ and $x_i=y_i/n^{(\tau-2)/(\tau-1)}$ otherwise, we obtain the bound $$\begin{aligned} \label{eq:expnhsmall} &\sum_{\boldsymbol{v}}\mathbb{E}\left[I(K_k, \boldsymbol{v})\mathbbm{1}_{\left\{w_{v_i}\in [\ell_i,u_i]n^{(\tau-2)/(\tau-1)}, \ i =1,2, \ w_{v_i}\in [\ell_i,u_i] n^{1/(\tau-1)}, \ i \geq 3 \right\}}\right]\nonumber\\ & \leq \tilde{K} n^{k}n^{2(\tau-2)/(\tau-1)-k+1}\nonumber\\ & \times \int_{\ell_1}^{u_1}\int_{y_1}^{u_2}\int^{u_3}_{\ell_3}\cdots \int^{u_k}_{\ell_k}y_1^{2-\tau}y_2^{1-\tau}y_3^{1-\tau}\dots y_k^{1-\tau} \prod_{j\geq 3}\min(y_2y_j,1)\exp(-\mu^{1-\tau}y_1y_2^{\tau-2}){\rm d}y_{k}\cdots {\rm d}y_{1} , \end{aligned}$$ for some $\tilde{K}>0$. Because the weights are sampled i.i.d. from a power-law distribution, the maximal weight $w_{\max}$ satisfies that for any $\eta_n\to 0$, $w_{\max}\leq n^{1/(\tau-1)}/\eta_n$ with high probability. Thus, we may assume that $u_i\leq 1/\eta_n$ when $i\geq 3$. Now suppose that at least one vertex has weight smaller than $\varepsilon_n n^{(\tau-2)/(\tau-1)}$ for $i=1,2$ or smaller than $\varepsilon_n n^{1/(\tau-1)}$ for $i\geq 3$. This corresponds to taking $u_i=\varepsilon_n$ and $\ell_i=0$ for at least one $i$, or at least one integral in [\[eq:expnhsmall\]](#eq:expnhsmall){reference-type="eqref" reference="eq:expnhsmall"} with interval $[0,\varepsilon_n]$. Similarly, when vertex 1 or 2 has weight higher than $1/\varepsilon_n n^{(\tau-2)/(\tau-1)}$, this corresponds to taking $\ell_i=1/\varepsilon_n$ and $u_i=\infty$ for $i=1$ or 2, or at least one integral in [\[eq:expnhsmall\]](#eq:expnhsmall){reference-type="eqref" reference="eq:expnhsmall"} with interval $[1/\varepsilon_n,\infty]$. Lemma [Lemma 17](#lem:intfinite){reference-type="ref" reference="lem:intfinite"} then shows that these integrals tends to zero when choosing $u_i=\eta_n$ fixed for $i\geq 3$ and $\varepsilon_n\to 0$. Thus, choosing $\eta_n\to 0$ sufficiently slowly compared to $\varepsilon_n$ yields that $$\begin{aligned} \label{eq:expcontrsmall} \sum_{\boldsymbol{v}}\mathbb{E}\left[I(K_k, \boldsymbol{v})\mathbbm{1}_{\left\{\boldsymbol{v}\notin \Gamma_n(\varepsilon_n,\eta_n)\right\}}\right]= o((n^{(3-\tau)(2\tau-3)/(\tau-1)}), \end{aligned}$$ where $$\Gamma_n(\varepsilon_n,\eta_n) = \{(v_1,\dots,v_k)\colon w_{v_i}\in n^{(\tau-2)/(\tau-1)}[\varepsilon_n,1/\varepsilon_n], i=1,2\ n^{1/(\tau-1)}[\varepsilon_n,1/\eta_n]\}.$$ Let $\bar{\Gamma}_n(\varepsilon_n,\eta_n)$ be the complement of $\Gamma_n(\varepsilon_n,\eta_n)$. Denote the number of maximal cliques with vertices in $\bar{\Gamma}_n(\varepsilon_n,\eta_n)$ by $N(K_k,\bar{\Gamma}_n(\varepsilon_n,\eta_n))$. Since $w_{\max}\leq n^{1/(\tau-1)}/\eta_n$ with high probability, $\Gamma_n(\varepsilon_n,\eta_n)=M_n(\varepsilon_n)$ with high probability. Therefore, with high probability, $$N\Big(K_k,\bar{M}_n\left(\varepsilon_n\right)\Big) = N\Big(K_k,\bar{\Gamma}_n(\varepsilon_n,\eta_n)\Big),$$ where $N\Big(K_k,\bar{M}_n\left(\varepsilon_n\right)\big)$ denotes the number of maximal $k$-cliques on vertices not in $M_n\left(\varepsilon_n\right)$. By [\[eq:expcontrsmall\]](#eq:expcontrsmall){reference-type="eqref" reference="eq:expcontrsmall"} and the Markov inequality, we have for all $\epsilon > 0$ $$\lim_{n \to \infty} \mathbb{P}\left(\left|\frac{N\Big(K_k,\bar{\Gamma}_n(\varepsilon_n,\eta_n)\Big)}{n^{(3-\tau)(2\tau-3)/(\tau-1)}}\right| > \epsilon\right) = 0.$$ Furthermore, Lemma [Lemma 17](#lem:intfinite){reference-type="ref" reference="lem:intfinite"} combined with the lower bound in [\[eq:expnhsmall\]](#eq:expnhsmall){reference-type="eqref" reference="eq:expnhsmall"} shows that when choosing $u_i=1/\varepsilon$ and $\ell_i=\varepsilon$ for some fixed $\varepsilon>0$ for all $i$, $$\begin{aligned} \mathbb{E}\left[N(K_k,M_n(\varepsilon))\right]= \Theta(n^{(3-\tau)(2\tau-3)/(\tau-1)}). \end{aligned}$$ Thus, for fixed $\varepsilon>0$, $$\begin{aligned} N(K_k)&= N(K_k,M_n(\varepsilon))+N(K_k,\bar{M}_n(\varepsilon))=\Theta_p(n^{(3-\tau)(2\tau-3)/(\tau-1)}), \end{aligned}$$ which shows that $$\frac{N\Big(K_k,M_n\left(\varepsilon_n\right)\big)}{N(K_k)}\ensuremath{\stackrel{\mathbb{P}}{\longrightarrow}}1,$$ as required. This completes the proof of Theorem [Theorem 16](#thm:maxcliqeslocalized){reference-type="ref" reference="thm:maxcliqeslocalized"}. ◻ # Experiments {#sec:experiments} As mentioned in the introduction, empirical evidence suggests that the number of maximal cliques in IRGs and GIRGs is small [@Exter_Valid_Avera_Analy_ESA2022]. In fact, all generated networks with $n = \SI{50}{k}$ nodes and expected average degree $10$ have fewer maximal cliques than edges. This stands in stark contrast to our super-polynomial lower bounds. This discrepancy probably comes from the fact that $n = \SI{50}{k}$ is low enough that a linear lower-order term dominates the super-polynomial terms. In this section, we complement our theoretical lower bounds with experiments with an $n$ that is sufficiently large to make the super-polynomial terms dominant. Additionally, we ran some experiments on dense and super-dense Erdős--Rényi graphs. ## Cliques in the Dense Subgraph of GIRGs and IRGs {#sec:girgs-irgs} Our theoretical lower bounds are based on the existence of a dense subgraph among the vertices with weights $\Theta(\sqrt{n})$. To experimentally observe the super-polynomial scaling, we generate IRGs and GIRGs restricted to vertices of high weight. This restriction lets us consider much larger values of $n$. In the following, we first describe the exact experiment setup, before describing and discussing the results. #### Experiment Setup. We generate IRGs and GIRGs with varying number of vertices $n$ and deterministic power-law weights where the $v$th vertex has weight $$w_v = \left( \frac{n}{v} \right)^{\frac{1}{\tau - 1}}.$$ Note that the minimum weight is $w_n = 1$. We use the power-law exponents $\tau \in \{2.2, 2.5, 2.8\}$ and for GIRGs we consider the temperatures $T \in \{0, 0.4, 0.8\}$ and dimension $d = 1$. For each parameter setting, we consider two subgraphs: The subgraph induced vertices with $0.5 \sqrt{n} \le w_i \le \sqrt{n}$ and with just $0.5 \sqrt{n} \le w_i \le n$. In preliminary experiments, we also tried constant factors other than $0.5$, yielding comparable results. As connection probability for the IRGs between the $u$th and $v$th vertex, we use $\min\{1, w_u w_v / n\}$, i.e., vertices of weight $1$ have connection probability $1 / n$ and vertices of weight at least $\sqrt{n}$ are deterministically connected. For GIRGs, we choose the constant factor $\mu$ in Equation [\[eq:edgeprob\]](#eq:edgeprob){reference-type="eqref" reference="eq:edgeprob"} such that we obtain the same expected[^1] average degree as for the corresponding IRG in the considered subgraph. For each of these configurations, we generate 10 graphs. Figure [8](#fig:girg_irg_core_plot){reference-type="ref" reference="fig:girg_irg_core_plot"} shows the average. ![The number of maximal cliques of the dense subgraph of GIRGs and IRGs. The considered subgraphs contain all vertices with weights in $[0.5 \sqrt{n}, \sqrt{n}]$ (left column) and $[0.5 \sqrt{n}, n]$ (right column). The top and bottom plots show the number of cliques with respect to the size of the full graph, and with respect to the size of the considered subgraph, respectively. All axes are logarithmic. Each point is the average of 10 sampled graphs.](irg_girg_core.pdf){#fig:girg_irg_core_plot} #### General Observations. One can clearly see in Figure [8](#fig:girg_irg_core_plot){reference-type="ref" reference="fig:girg_irg_core_plot"} (top row) that the scaling of the number of cliques depending on the graph size is super-polynomial (upward curves in a plot with logarithmic axes). Thus, on the one hand, this agrees with our theoretical analysis. On the other hand, the plots also explain why previous experiments [@Exter_Valid_Avera_Analy_ESA2022] showed a small number of cliques: While the scaling is super-polynomial, the constant factors are quite low. In the top-left plot for $\tau = 2.5$, more than 200 M nodes are necessary to get just barely above 1 M maximal cliques in the dense subgraph. For $\tau = 0.8$ this is even more extreme with $n = \SI{200}{T}$ yielding only 10 k maximal cliques. Thus, unless we deal with huge graphs, the maximal cliques in the dense part of the graph are dominated by the number of cliques in the sparser parts, despite the super-polynomial growth of the former. #### Effect of the Power-Law Exponent $\tau$. The top plots of Figure [8](#fig:girg_irg_core_plot){reference-type="ref" reference="fig:girg_irg_core_plot"} show that a smaller power-law exponent $\tau$ leads to more maximal cliques. The bottom plots show the number of cliques with respect to the size of the dense subgraph and not with respect to the size of the full graph. One can see that the difference for the different power-law exponents solely comes from the fact that the dense subgraph is larger for smaller $\tau$. For the same size of the dense subgraph, the scaling is almost independent of the power-law exponent. #### Effect of the Geometry. In the left plots of Figure [8](#fig:girg_irg_core_plot){reference-type="ref" reference="fig:girg_irg_core_plot"}, we can see that geometry leads to fewer maximal cliques. For $T = 0$, the super-polynomial scaling is only barely noticeable. Higher temperatures lead to a larger number of cliques and we get even more cliques for IRGs. Interestingly, the scaling is slower for IRGs when additionally considering the core of vertices with weight more than $\sqrt{n}$ (see next paragraph). #### Effect of the Core. When not capping the weight at $\sqrt{n}$ but also considering vertices of even higher weight (right plots), we can observe the following. The overall picture remains similar, with a slightly increased number of cliques. However, this increase is higher for GIRGs than it is for IRGs. A potential explanation for this is the following. For IRGs, the core forms a clique and adding a large clique to the graph does not change the overall number of maximal cliques by too much. For GIRGs, however, it depends on the constant $\mu$ controlling the average degree whether this subgraph forms a clique or not. Thus, for the same average degree, the maximum clique is probably somewhat smaller for GIRGs and thus adding the vertices of weight at least $\sqrt{n}$ leads to more additional cliques than in IRGs. ## Cliques in the Dense and Super-Dense Erdős--Rényi Graphs {#sec:cliq-dense-gnp} Here we count the cliques for dense Erdős--Rényi graphs with constant connection probabilities $p \in \{0.6, 0.7, 0.8, 0.9\}$ and super-dense Erdős--Rényi graphs with connection probability $p = 1 - c / n$ for $c \in \{1, 2, 4, 8\}$. Note that the complement of a super-dense Erdős--Rényi graph has constant expected average degree. The scaling of the number of cliques with respect to the number of vertices is shown in Figure [9](#fig:dense_gnp_plot){reference-type="ref" reference="fig:dense_gnp_plot"}, where each point represents 20 samples. {#fig:dense_gnp_plot} Note that for constant $p$, the left plot with logarithmic $y$-axis is curved downward, indicating sub-exponential scaling, while the middle plot with logarithmic $x$- and $y$-axis is bent upwards, indicating super-polynomial scaling. This is in line with our lower bound in Theorem [Theorem 1](#thm:er-lower-bound){reference-type="ref" reference="thm:er-lower-bound"}. For the super-dense case, the right plot indicates exponential scaling, in line with Theorem [Theorem 3](#thm:erdense){reference-type="ref" reference="thm:erdense"}. # Conclusion and Discussion In this paper, we have investigated the number of maximal cliques in three random graph models: the Erdős--Rényi random graph, the inhomogeneous random graph and the geometric inhomogeneous random graph. We have shown that sparse Erdős--Rényi random graphs only contain a polynomial amount of maximal cliques, but in the other two sparse models, the number of maximal cliques scales at least super-polynomially in the network size. This is caused by the degree-heterogeneity in these models, as many large maximal cliques are present close to the core of these random graphs. We prove that there only exist a linear amount of small maximal cliques. Interestingly, these small maximal cliques are almost always formed by two low-degree vertices, whereas all other vertices are hubs of high degree. We have then shown that this dominant super-polynomial behavior of the number of maximal cliques often only kicks for extreme network sizes, and that experimentally, lower-order linear terms instead drive the scaling of the number of maximal cliques until large values of the network size. This explains the dichotomy between the theoretical super-polynomial lower bounds for these models, and the observation that in real-world networks, the amount of maximal cliques is often quite small. Several of our results only constitute lower bounds for the number of maximal cliques. We believe that relatively close upper bounds can be constructed in a similar fashion, but leave this open for further research. While Theorem [Theorem 11](#thm:girg_non_torus){reference-type="ref" reference="thm:girg_non_torus"} only holds for 2-norms, we believe that the theorem can be extended to any $L^p$-norm for $p\neq1,\infty$, by looking at the $L^p$ norm-cycle instead of the regular cycle. For $p=1,\infty$ this approach fails, shortest distance paths to non-opposing segments pass through the center of the cycle. Therefore, opposing segments are just as close as many non-opposing ones. Whether Theorem [Theorem 11](#thm:girg_non_torus){reference-type="ref" reference="thm:girg_non_torus"} also holds with 1 or $\infty$ norms is therefore a question for further research. We also believe that this approach also extends to the underlying space $[0,1]^d$ for general $d$, where instead of looking at a cycle inside $[0,1]^2$, one studies a $d$-ball inscribed in $[0,1]^d$ instead. # Proof of Theorem [Theorem 14](#thmnonzero2dim){reference-type="ref" reference="thmnonzero2dim"} {#app:prooftemperature} **Lemma 19**. *Let $(A_i)_{i\in[k]}$ be a set of areas of size $A$, and let $S$ be a set of vertices, such that $A|S|>n^\varepsilon$ for some $\varepsilon>0$. Then, for any $0<\lambda<1$ and $k<\exp(\lambda A|S|)$, with high probability all areas contain at least $(1-\lambda)A|S|$ vertices.* *Proof.* The Chernoff bound gives for the number of vertices from $S$ within area $A$, $N_{S,A}$: $$\mathbb{P}\left(N_{S,A} < (1-\lambda)A|S| \right)\leq \exp\Big(-\lambda A|S|\Big).$$ This implies that when $A|S|>n^{\varepsilon}$ for some $\varepsilon>0$, then, with high probability, all areas contain at least $(1-\lambda)A|S|$ vertices. ◻ We follow the same construction of areas and sets as in the proof of Theorem [Theorem 11](#thm:girg_non_torus){reference-type="ref" reference="thm:girg_non_torus"}. By [\[eq:mbound\]](#eq:mbound){reference-type="eqref" reference="eq:mbound"} this creates $2k = s \cdot n^{\min(\beta/2,\gamma/2)}$ areas of size $A=n^{-3/2\gamma}$, with on average $\mathbb{E}\left[|S|\right]=n^{(3-\tau)/2-\beta}$ vertices. Thus, Lemma [Lemma 19](#lem:chernoff-areas){reference-type="ref" reference="lem:chernoff-areas"} shows that as long as $\beta+3/2\gamma <(3-\tau)/2$, then all areas contain with high probability at least $$n'= c_1n^{(3-\tau)/2-\beta-3/2\gamma}$$ vertices for some $c_1>0$. From [\[eq:edgeprob\]](#eq:edgeprob){reference-type="eqref" reference="eq:edgeprob"}, it follows that any set of vertices that contains one in each given area still satisfies the requirement that all vertices in non-opposite boxes connect, as in non-opposite boxes, the connection probability equals 1 by [\[eq:l2dimnotorus\]](#eq:l2dimnotorus){reference-type="eqref" reference="eq:l2dimnotorus"}. Now to form a clique minus a matching, vertices in opposite boxes should not connect. With high probability, a positive proportion of vertices in two opposing areas have distance at least $t+h=a+n^{-\gamma}$, by the uniform distribution within areas, and the fact that a positive proportion of the two areas have distance $t+h$. By [\[eq:edgeprob\]](#eq:edgeprob){reference-type="eqref" reference="eq:edgeprob"}, the probability that vertices $i,j\in S$ at distance at least $a+n^{-\gamma}$ are connected is bounded by $$\begin{aligned} \label{eq:opposite_disconnect_prob} p_{ij}& \leq \min\Bigg(\Big(\frac{a^2(1-n^{-\beta})}{(a+n^{-\gamma})^2}\Big)^{1/T},1\Bigg)\nonumber\\ & =(1-n^{-\beta})(1-n^{-\gamma})(1+o(1))\nonumber\\ & =(1-\max(n^{-\beta},n^{-\gamma}))(1+o(1)). \end{aligned}$$ Similarly as in [\[eq:pm0\]](#eq:pm0){reference-type="eqref" reference="eq:pm0"}, $$\mathbb{P}\left(M=0\right)\leq k (1-\max(n^{-\beta},n^{-\gamma})^{(n')^2}\leq k\exp(-\max(n^{-\beta},n^{-\gamma})(n')^2)$$ Using that $n'= c_1n^{(3-\tau)/2-\beta-3/2\gamma}$ therefore yields $$\mathbb{P}\left(M=0\right)\leq k\exp(-c_1^2n^{(3-\tau)-\beta-3\gamma}\max(n^{-\beta},n^{-\gamma})).$$ Thus, choosing $\beta=\gamma=(3-\tau)/5-\varepsilon$ ensures that there is a co-matching of size $k=s \cdot n^{(3-\tau)/10-\varepsilon}$ 0◻ # Proof of Lemma [Lemma 17](#lem:intfinite){reference-type="ref" reference="lem:intfinite"} {#app:proofintfinite} *Proof.* This integral equals $$\begin{aligned} &\int_0^1\dots\int_0^1 \int_0^1\int_0^{x_2} x_1^{k-1-\tau}x_2^{k-1-\tau}x_3^{2-\tau}\cdots x_k^{2-\tau}\exp\Big(-\mu^{1-\tau}x_1x_2^{\tau-2}\Big)dx_1 dx_2\dots dx_k\nonumber\\ & + \int_0^1\dots\int_0^1 \int_1^{\infty}\int_0^{x_2} x_1^{k-1-\tau}x_2^{1-\tau}x_3^{1-\tau}\cdots x_k^{1-\tau}\prod_{i=3}^k\min\Big(x_2x_i,1\Big)\exp\Big(-\mu^{1-\tau}x_1x_2^{\tau-2}\Big)dx_1 dx_2\dots dx_k \end{aligned}$$ Now $$\int_0^1\dots\int_0^1 \int_0^1\int_0^{x_2} x_1^{k-1-\tau}x_2^{k-1-\tau}x_3^{2-\tau}\cdots x_k^{2-\tau}dx_2dx_1\dots dx_k<\infty,$$ as $2-\tau>-1$, and $k-1-\tau>-1$ for $k\geq 3$ as well. We now turn to the second integral. The second integral is finite if $$\begin{aligned} &\int_0^1\dots\int_0^1 \int_1^{\infty} \int_0^{x_2}x_1^{k-1-\tau}x_2^{1-\tau}x_3^{1-\tau}\cdots x_k^{1-\tau}\prod_{i=3}^k\min\Big(x_2x_i,1\Big)\mathbbm{1}_{\left\{x_2^{\tau-2}x_1<1\right\}}dx_1 dx_2\dots dx_k<\infty. \end{aligned}$$ This results in $$\begin{aligned} \label{eq:int2ind} & \int_0^1\dots\int_0^1 \int_1^{\infty} \int_0^{x_2^{2-\tau}}x_1^{k-1-\tau}x_2^{1-\tau}x_3^{1-\tau}\cdots x_k^{1-\tau}\prod_{i=3}^k\min\Big(x_2x_i,1\Big) dx_1 \dots dx_k\nonumber\\ & = \int_0^1\dots\int_0^1 \int_1^{\infty}x_2^{(2-\tau)(k+1-\tau)-1}x_3^{1-\tau}\cdots x_k^{1-\tau}\prod_{i=3}^k\min\Big(x_2x_i,1\Big) dx_2\dots dx_k \end{aligned}$$ W.l.o.g. we assume that $x_3>x_4>\dots>x_k$. Then, the inner integral evaluates to $$\begin{aligned} & \int_1^{\infty}x_2^{(2-\tau)(k+1-\tau)-1}\prod_{i=3}^k\min\Big(x_2x_i,1\Big) dx_2\nonumber\\ & = \int_1^{1/x_3}x_2^{(2-\tau)(k+1-\tau)+k-3}x_3\cdots x_k dx_2+ \dots + \int_{1/x_k}^\infty x_2^{(2-\tau)(k+1-\tau)-1} dx_2\nonumber\\ & = C_3x_3^{(\tau-2)(k+1-\tau)+3-k}x_4\cdots x_k + C_4x_4^{(\tau-2)(k+1-\tau)+4-k}x_5\cdots x_k+ \dots + C_{k} x_k^{(\tau-2)(k+1-\tau)} \end{aligned}$$ We now show that all these terms evaluate to a finite integral when plugged into [\[eq:int2ind\]](#eq:int2ind){reference-type="eqref" reference="eq:int2ind"}. Indeed, $$\begin{aligned} & \int_0^1\int_0^{x_3}\dots\int_0^{x_{k-1}}x_l^{(\tau-2)(k+1-\tau)+l-k}x_{l+1}\cdots x_k x_3^{1-\tau}\cdots x_k^{1-\tau} dx_k dx_{k-1}\dots dx_3\nonumber\\ & = \int_0^1\int_0^{x_3}\dots\int_0^{x_{l-1}}x_l^{(\tau-2)(l-\tau)-1} x_3^{1-\tau}\cdots x_{l-1}^{1-\tau} dx_l dx_{l-1}\dots dx_3\nonumber\\ & = \int_0^1\int_0^{x_3}\dots\int_0^{x_{l-2}}x_{l-1}^{(\tau-2)(l-1-\tau)-1} x_3^{1-\tau}\cdots x_{l-2}^{1-\tau} dx_l dx_{l-2}\dots dx_3<\infty \end{aligned}$$ as the index $l-k$ remains at least 3. Therefore, [\[eq:intmaxclique\]](#eq:intmaxclique){reference-type="eqref" reference="eq:intmaxclique"} is finite as well. ◻ [^1]: We do not sample the positions before computing the expected average degree but we compute the expectation with respect to random positions.

arxiv_math

--- abstract: | Let $p,q,l$ be three distinct prime numbers and let $N$ be a positive integer coprime to $pql$. For an integer $n\ge 0$, we define the directed graph $X_l^q(p^nN)$ whose vertices are given by isomorphism classes of elliptic curves over a finite field of characteristic $q$ equipped with a level $p^nN$ structure. The edges of $X_l^q(p^nN)$ are given by $l$-isogenies. We are interested in when the connected components of $X_l^q(p^nN)$ give rise to a tower of Galois covers as $n$ varies. We show that only in the supersingular case we do get a tower of Galois covers. We also study similar towers of isogeny graphs given by oriented supersingular curves, as introduced by Colò-Kohel, enhanced with a level structure. address: - | Department of Mathematics and Statistics\ University of Ottawa\ 150 Louis-Pasteur Pvt\ Ottawa, ON\ Canada K1N 6N5 - | Département de Mathématiques et de Statistique\ Université Laval, Pavillion Alexandre-Vachon\ 1045 Avenue de la Médecine\ Québec, QC\ Canada G1V 0A6 author: - Antonio Lei - Katharina Müller bibliography: - references.bib title: On towers of isogeny graphs with full level structures --- # Introduction Let $p$ and $l$ be distinct prime numbers. Let $N$ be a positive integer that is coprime to $pl$. Let $G_N^n$ be the directed graph whose vertices are isomorphism classes $(E,P)$, where $E$ is an ordinary elliptic curve defined over a fixed finite field of characteristic $p$ and $P$ is a point of order $Np^n$ on $E$, and the edges of $G_N^n$ are given by $l$-isogenies. In [@LM1], we studied the tower of graph coverings $$G_N^0\leftarrow G_N^1\leftarrow G_N^2\leftarrow \cdots\leftarrow G_N^n\leftarrow\cdots.$$ Note that the degree of the covering $G_N^{n+1}/G_N^n$ is $p$ thanks to the ordinarity condition. We studied the undirected graphs $\tilde G_N^n$ obtained from $G_N^n$ by forgetting the directions of the edges of $G_N^n$. We showed that if $E$ is an elliptic curve representing a non-isolated vertex of $G_1^0$ and $\tilde{{\mathcal{G}}}_N^m$ is a connected component of ${\mathcal{G}}_N^m$ containing a vertex arising from $E$, then there exists an integer $m_0$ such that $$\tilde{\mathcal{G}}_N^{m_0}\leftarrow \tilde{\mathcal{G}}_N^{m_0+1}\leftarrow \tilde{\mathcal{G}}_N^{m_0+2}\leftarrow \cdots\leftarrow \tilde{\mathcal{G}}_N^{m_0+n}\leftarrow\cdots$$ is an abelian $p$-tower in the sense of Vallières and McGown-Vallières [@vallieres; @vallieres2; @vallieres3], i.e. the covering $\tilde{\mathcal{G}}_N^{m_0+r}/\tilde{\mathcal{G}}_N^{m_0}$ is Galois, whose Galois group is isomorphic to the cyclic group $\mathbb{Z}/p^r\mathbb{Z}$ for all $r\ge0$. Such towers exhibit properties that resemble $\mathbb{Z}_p$-towers of number fields studied in classical Iwasawa theory [@iwasawa69; @iwasawa73]. For example, similar to the class numbers of number fields inside a $\mathbb{Z}_p$-tower, the number of spanning trees of $\tilde{\mathcal{G}}_N^{m_0+n}$ can be described explicitly for $n$ sufficiently large (see [@vallieres3; @leivallieres]). In this article, we study the following tower of isogeny graphs as $n$ varies, resulting in non-commutative Galois covers under appropriate hypotheses. Throughout, we fix a positive integer $k$ and a set of representatives of the equivalence classes of elliptic curves over $\mathbb{F}_{q^k}$, which we denote by $S$. **Definition 1**. *Let $p,q,l$ be three distinct prime numbers and let $N$ be a positive integer coprime to $pql$. For an integer $n\ge 0$, we define the directed graph $X_l^q(Np^n)$ whose vertices are given by triples $(E,Q_1,Q_2)$, where $E\in S$ and $\{Q_1,Q_2\}$ is a basis of $E[p^nN]\cong (\mathbb{Z}/p^nN\mathbb{Z})^2$. The edges of $X_l^q(p^nN)$ are given by $l$-isogenies.* The basis $\{Q_1,Q_2\}$ is called a $\Gamma(p^nN)$-level structure on the elliptic curve $E$. As $pN$ is coprime to $q$, we can write $Q_i=R_i+S_i$, where $i\in\{1,2\}$, $R_i\in E[N]$ and $S_i\in E[p^n]$. We shall denote a vertex of $X_l^q(p^nN)$ by $(E,R_1,R_2,P,Q)$ with $R_1,R_2\in E[N]$ and $P,Q\in E[p^n]$ such that $\{R_1\oplus P, R_2\oplus Q\}$ is a $\Gamma(p^nN)$-level structure on $E$. When $n=0$, we may discard $P$ and $Q$, and simply write $(E,R_1,R_2)$ for a vertex in $X_l^q(N)$. The map $(E,R_1,R_2,P,Q)\mapsto(E,R_1,R_2,pP,pQ)$ induces a graph covering $X_l^q(p^{n+1}N)/ X_l^q(p^{n}N)$ (see Corollary [Corollary 10](#cor:cover){reference-type="ref" reference="cor:cover"}). We are interested in when the connected components of $X_l^q(p^nN)$ give rise to a tower of Galois covers as $n$ varies. Let $X$ be a connected component of $X_l^q(N)$. For $n\ge1$, let $X_n$ be the pre-image of $X$ under the covering $X_l^q(p^nN)/ X_l^q(N)$. Note that if $v_1,v_2\in V(X_n)$, the elliptic curves giving rise to $v_1$ and $v_2$ are isogeneous. In particular, they have the same reduction type. Therefore, it makes sense to refer to $X$ as ***ordinary*** or ***supersingular***. It turns out that the coverings $X_n/X$ exhibit different properties depending on the reduction type. In particular, we prove: **Theorem 1** (Corollary [Corollary 16](#cor:ord-not-Galois){reference-type="ref" reference="cor:ord-not-Galois"}). *If the connected component $X$ is ordinary, then $X_{n}/X$ is not a Galois covering when $n$ is sufficiently large.* **Theorem 2** (Corollaries [Corollary 18](#cor:density){reference-type="ref" reference="cor:density"} and [Corollary 20](#cor:connected){reference-type="ref" reference="cor:connected"}). *Suppose that the connected component $X$ is supersingular, $p>2$ and that $N\le 2$. For a positive density set of primes $p$, the covering $X_{n}/ X$ is Galois, with Galois group isomorphic to $\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})$.* In particular, in the setting of Theorem [Theorem 2](#thmB){reference-type="ref" reference="thmB"}, $$X_1\leftarrow X_2\leftarrow\cdots\leftarrow X_n\leftarrow\cdots$$ gives an explicit example that fits in the framework of non-commutative Iwasawa of graph coverings developed in [@KM]. In non-commutative Iwasawa theory of elliptic curves studied in [@CFKSV], the authors conjectured that the Pontryagin dual of the $p$-primary Selmer group of a $p$-ordinary elliptic curve over a $p$-adic Lie extension $\mathcal{K}_\infty$ of a number field $\mathcal{K}_0$ that contains the cyclotomic $\mathbb{Z}_p$-extension should satisfy the so-called $\mathfrak{M}_H(G)$-property. Here, $G=\mathop{\mathrm{Gal}}(\mathcal{K}_\infty/\mathcal{K}_0)$ and $H$ is a subgroup of $G$ such that $\mathcal{K}_\infty^H$ is the cyclotomic $\mathbb{Z}_p$-extension of $\mathcal{K}_0$. The analogue of the $\mathfrak{M}_H(G)$-property in the context of graphs has recently been studied in [@KM]. It is therefore natural to seek an appropriate quotient inside the tower given by Theorem [Theorem 2](#thmB){reference-type="ref" reference="thmB"} that would play the role of the cyclotomic $\mathbb{Z}_p$-extension of a number field. Indeed, the graph $X_l^q(p^nN)$ admits a natural quotient $Y_l^q(p^nN)$ obtained by the map $$(E,R_2,R_2,P,Q)\mapsto (E,R_1,R_2,\langle P,Q\rangle),$$ where $\langle-,-\rangle$ is the Weil pairing (see Definition [Definition 22](#def-zp-graph){reference-type="ref" reference="def-zp-graph"}). We show that these graphs give rise to an abelian $\mathbb{Z}_p$-tower, similar to the ordinary isogeny graphs studied in [@LM1]. In particular, we prove: **Theorem 3** (Corollary [Corollary 27](#cor:Zp-tower){reference-type="ref" reference="cor:Zp-tower"}). *For $N$ sufficiently large, there exists an integer $m_0$ such that if $Y_{m_0}$ is a connected component of $Y_l^q(p^{m_0}N)$ and $Y_{n+m_0}$ is the pre-image of $Y_{m_0}$ in $Y_l^q(p^{n+m_0}N)$, then $Y_{n+m_0}/Y_{m_0}$ is Galois, with Galois group isomorphic to $\mathbb{Z}/p^n\mathbb{Z}$ for all $n\ge0$.* Note that unlike Theorems [Theorem 1](#thmA){reference-type="ref" reference="thmA"} and [Theorem 2](#thmB){reference-type="ref" reference="thmB"}, where a divergence between the ordinary and supersingular cases emerges, Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"} is independent of the reduction type of the chosen connected component. Finally, we study isogeny graphs arising from *oriented supersingular elliptic curves*, which were first introduced in [@colokohel]. An orientation of a supersingular elliptic curve $E$ is given by an embedding $K\hookrightarrow\mathop{\mathrm{End}}(E)\otimes\mathbb{Q}$, where $K$ is an imaginary quadratic field. The authors of [@arpin-et-all] proved that these graphs can be described as volcano graphs, similar to the ordinary isogeny graphs studied in [@kohel]. Let $M\ge1$ be an integer that is coprime to $ql$. We define the isogeny graphs of oriented supersingular elliptic curves equipped with a $\Gamma(M)$-level structure, which we denote by $\mathcal{X}_l^q(M)$ (see Definition [Definition 37](#def:oriented-graph-level){reference-type="ref" reference="def:oriented-graph-level"}). We describe the connected components of these graphs explicitly for $M$ sufficiently large. In particular, we prove in Theorems [Theorem 48](#thm:volcano){reference-type="ref" reference="thm:volcano"}, [Theorem 49](#thm:volcano2){reference-type="ref" reference="thm:volcano2"}, [Theorem 50](#thm:volcano3){reference-type="ref" reference="thm:volcano3"} that depending on the splitting behaviour of $l$ in $K$, a connected component of $\mathcal{X}_l^q(M)$ is a \"double intertwinement\" of either a \"volcano graph\" or a \"tectonic volcano\" (see Definitions [Definition 33](#def:volcano){reference-type="ref" reference="def:volcano"}, [Definition 43](#def:intertwine){reference-type="ref" reference="def:intertwine"} and [Definition 47](#def:crater){reference-type="ref" reference="def:crater"} where these concepts are introduced). The final result of the present article is an analogue of Theorem [Theorem 2](#thmB){reference-type="ref" reference="thmB"} for isogeny graphs of oriented supersingular elliptic curve with level structures: **Theorem 4** (Theorem [Theorem 54](#thm:oriented-tower){reference-type="ref" reference="thm:oriented-tower"}). *Assume that $q\equiv1\pmod{12}$, $N=1$ or $2$ is an integer coprime to $pql$ and that the subgraph $X_l^q(p^nN)^{\mathrm{ss}}$ of $X_l^q(p^nN)$ generated by the vertices arising from supersingular elliptic curves is connected for all $n$. Let $\mathcal{X}_0$ be a connected component of $\mathcal{X}_l^q(N)$ and denote the pre-image of $\mathcal{X}_0$ in $\mathcal{X}_l^q(p^nN)$ by $\mathcal{X}_n$ for $n\ge1$. The covering $\mathcal{X}_n/\mathcal{X}_0$ is a Galois covering with Galois group isomorphic to $\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$.* Theorem [Theorem 4](#thmD){reference-type="ref" reference="thmD"} illustrates an interesting relation between the two classes of isogeny graphs studied in the present paper. We plan to study such relations further in the future. Finally, we remark that a simple necessary and sufficient condition of the connectivity of $X_l^q(p^nN)^{\mathrm{ss}}$ is been proved by Roda [@thesis-roda]. We shall review this result in the main body of the article (see Theorem [Theorem 17](#thm:number-connected-comp){reference-type="ref" reference="thm:number-connected-comp"}). This result is also utilized to prove the non-triviality of the density of primes satisfying Theorem [Theorem 2](#thmB){reference-type="ref" reference="thmB"}. We conclude this introduction by highlighting the growing interest in isogeny graphs of elliptic curves with level structures in recent years. One such example is Goren--Kassaei's work [@gorenkassaei] that delved into the dynamics of Hecke operators on modular curves utilizing such graphs with $\Gamma_1(N)$-level structure. On a different vein, Arpin [@arpin] investigated the implications of isogeny graphs for equivalence classes of supersingular elliptic curves with $\Gamma_0(N)$-level structures in the realm of isogeny-based cryptography. During the preparation of the present article, we learnt about the recent work of Codogni--Lido [@codogni-lido], where they considered more general level structures. Analogous to Theorem [Theorem 17](#thm:number-connected-comp){reference-type="ref" reference="thm:number-connected-comp"}, they studied the number of connected components in these graphs. Moreover, they analyzed the eigenvalues of the associated adjacency matrices, establishing connections between these graphs and modular forms (the special case of $\Gamma_0(N)$-level structure was also studied in [@sugiyama; @LM1]). ## Acknowledgement {#acknowledgement .unnumbered} We thank Pete Clark and Daniel Vallières for helpful exchanges regarding the content of the present article. We thank Sören Kleine for various helpful remarks on the present article. The authors' research is supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096. # Voltage assignment and graph covering In this section, we show that the isogeny graphs with level structures introduced in Definition [Definition 1](#def:intro){reference-type="ref" reference="def:intro"} can be naturally realized as voltage graphs. Our discussion can be regarded as a natural extension of [@LM1 Appendix A] to the non-commutative setting. ## Generalities of voltage graphs Let us first recall the definition of a voltage assignment. **Definition 2**. *Let $X$ be a directed graph. [\[def:voltage\]]{#def:voltage label="def:voltage"}* *The set of vertices and the set of edges of $X$ are denoted by $V(X)$ and $\mathbb{E}(X)$, respectively.* *Let $(G,\cdot)$ a group. A function $\alpha:\mathbb{E}(X)\rightarrow G$ is called a **$G$-valued voltage assignment** on $X$.* *To each voltage assignment $\alpha$ given as in ii), we define the **derived graph** $X(G,\alpha)$ to be the graph whose vertices and edges are given by $V(X)\times G$ and $\mathbb{E}(X)\times G$, respectively. If $(e,\sigma)\in \mathbb{E}(X)\times G$, it links $(s,\sigma)$ to $(t,\sigma\cdot\alpha(e))$, where $e$ is an edge in $X$ from $s$ to $t$.* *A graph arises from a voltage assignment is called a **voltage graph**.* Let $G$, $X$ and $\alpha$ be given as in Definition [\[def:voltage\]](#def:voltage){reference-type="ref" reference="def:voltage"}. There is a natural graph covering $X(G,\alpha)/ X$ given by $$V(X(G,\alpha))\ni(v,\sigma)\mapsto v,\quad \mathbb{E}(X(G,\alpha))\ni(e,\sigma)\mapsto e.$$ If $G$ is finite, then the degree of this covering is $|G|$ (to which we may also refer as a \"$|G|$-sheeted covering\"). Furthermore, there is a natural left action by $G$ on $X(G,\alpha)$ given by $g\cdot (v,\sigma)=(v,g\cdot \sigma)$ and $g\cdot(e,\sigma)=(e,g\cdot \sigma)$ for $g,\sigma\in G$, $v\in V(X)$ and $e\in \mathbb{E}(X)$. **Definition 3**. *Let $X$ be a directed graph.* *We write $X'$ for the undirected graph obtained from $X$ by ignoring the directions of the edges of $X$. The natural map from $X$ to $X'$ is called the **forgetful map**.* *We say that $X$ is **connected** if $X'$ is connected.* *A **connected component** of $X$ is the pre-image of a connected component of $X'$ in $X$ under the forgetful map.* *We call a covering $Y/X$ of directed graphs **Galois** if the corresponding covering of undirected graphs $Y'/X'$ is Galois. When $Y/X$ is Galois, we call $\mathop{\mathrm{Gal}}(Y'/X')$ the **Galois group** of $Y/X$ and write $\mathop{\mathrm{Gal}}(Y/X)=\mathop{\mathrm{Gal}}(Y'/X')$.* *If $Y/X$ is a covering of directed graph, the group of deck transformations of $Y/X$ is denoted by $\mathrm{Deck}(Y/X)$. Similarly, we denote the group of deck transformations of $Y'/X'$ by $\mathrm{Deck}(Y'/X')$.* The notion of connectedness here is sometimes referred to as \"weakly connected\", as opposed to \"strongly connected\", where one requires the existence of a path between any two given vertices. We shall see in Remark [Remark 12](#rk:connected){reference-type="ref" reference="rk:connected"} that these two notions coincide for the graphs of interest in the present paper. **Lemma 4**. *Let $Y/X$ be a $d$-sheeted covering of connected directed graphs. Then $Y'/X'$ is also a $d$-sheeted covering. Furthermore, there is a natural injective group homomorphism $$\Xi:\textup{Deck}(Y/X)\hookrightarrow \textup{Deck}(Y'/X').$$* *Furthermore, if $Y$ is the derived graph of a voltage assignment $\alpha$ on $X$ and $Y'/X'$ is Galois, then $\Xi$ is a group isomorphism.* *Proof.* As $V(X)=V(X')$ and $V(Y)=V(Y')$, it is clear that $Y'/X'$ is a $d$-sheeted covering. It is also clear that every deck transformation of $Y/X$ induces a deck transformation of $Y'/X'$ since the edges of $Y'$ and $X'$ arise from those in $Y$ and $X$, respectively, resulting in the injection $\Xi$. Assume now that $Y$ is the derived graph of a voltage assignment $\alpha$ on $X$ and that $Y'/X'$ (and so $Y/X$) is Galois. As $Y/X$ is a $d$-sheeted cover, $\alpha$ takes values in a finite group $G$ of order $d$. Thus, to show that $\Xi$ is surjective, it suffices to show that $\textup{Deck}(Y/X)$ admits at least $d$ distinct elements. Since $Y=X(G,\alpha)$, the elements of $V(Y)$ are of the form $(v,\sigma)$, where $v\in V(X)$ and $\sigma\in G$. For any $g\in G$, the map $(v,\sigma)\mapsto (v,g\sigma)$ induces a deck transformation of $Y/X$. This results in $d$ distinct elements in $\textup{Deck}(Y/X)$, which proves the surjectivity of $\Xi$. ◻ The following result allows us to determine if a graph covering arising from a voltage assignment is Galois. **Proposition 5**. *Let $X$ be a directed graph, $G$ a finite group equipped with a $G$-valued voltage assignment $\alpha$ on $X$. Let $Y$ denote the derived graph $X(G,\alpha)$.* *If $X$ is connected, then the natural action of $G$ on $Y$ permutes transitively the connected components of $Y$.* *If $Y$ is connected, then $Y/X$ is a Galois cover. Furthermore, its Galois group is isomorphic to $G$.* *Proof.* This follows from [@gonet22 Theorem 2.10] and [@gonet-thesis §2.3], combined with the identification of deck transformations of coverings of directed graphs and undirected graphs given by Lemma [Lemma 4](#lem:decks){reference-type="ref" reference="lem:decks"}. ◻ **Corollary 6**. *Let $X$, $Y$ and $G$ be as in Proposition [Proposition 5](#prop:voltage){reference-type="ref" reference="prop:voltage"}i). Let $d$ be the number of connected components of $Y$. If $d!>|G|$, then $Y/X$ is not Galois.* *Proof.* Let $Y_1$ and $Y_2$ be two connected components of $Y$. Proposition [Proposition 5](#prop:voltage){reference-type="ref" reference="prop:voltage"}i) tells us that there exists $g\in G$ such that $$V(Y_1)\ni(v,\sigma)\mapsto (v,g\cdot \sigma),\quad \mathbb{E}(Y_1)\ni(e,\sigma)\mapsto (e,g\cdot \sigma)$$ induces an isomorphism of graphs from $Y_1$ to $Y_2$. Thus, any permutation of the connected components of $Y$ gives rise to an element of $\mathrm{Deck}(Y/X)$. Since there are $d!$ such permutations and the degree of the covering is $|G|$, the covering cannot be Galois if $d!>|G|$. ◻ *Remark 7*. Assume that $G$ is finite. In the course of the paper we will frequently study the number of connected components of $X(G,\alpha)$. To do so, it suffices to fix a vertex $v\in V(X)$ and study the set $$\left\{ (v,g)\in V(X(G,\alpha))\mid (v,g) \textup{ lies in the same connected component as $(v,1)$}\right\}.$$ Let $d_v$ be the cardinality of this set. As explained in the proof of Corollary [Corollary 6](#cor:not-Galois){reference-type="ref" reference="cor:not-Galois"}, all connected components of $X(G,\alpha)$ are isomorphic to each other. Therefore, the number of connected components of $X(G,\alpha)$ is given by $\frac{\vert G\vert }{d_v}$. ## Realizing isogeny graphs as voltage graphs {#sec:realize} The goal of this section is to realize the graph $X_l^q(p^nN)$ as voltage graph arising from a $\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})$-valued voltage assignment on $X$, similar to the discussion in [@LM1 Appendix A]. For each elliptic curve $E$ in $S$ defined over $\mathbb{F}_{q^k}$, we fix a $\mathbb{Z}_p$-basis $\{s_E,t_E\}$ of the Tate module $T_p(E)$. If $e\in \mathbb{E}(X_l^q(N))$ with source $(E,R_1,R_2)$ and target $(E',R_1',R_2')$, it arises from an $l$-isogeny $\phi:E\rightarrow E'$. Since $p$ and $l$ are coprime, $\phi$ induces a $\mathbb{Z}_p$-isomorphism $$\phi^*:T_p(E)\rightarrow T_p(E').$$ This allows us to define the following voltage assignment: **Definition 8**. *For any given $e\in \mathbb{E}(X_l^q(N))$ arising from an $l$-isogeny $\phi:E\rightarrow E'$, we define $g_e\in \mathop{\mathrm{GL}}_2(\mathbb{Z}_p)$ to be the transpose of the matrix of $\phi^*$ with respect to our chosen bases $\{s_E,t_E\}$ and $\{s_{E'},t_{E'}\}$ so that the following equation holds $$\begin{pmatrix} \phi^*(s_E)\\\phi^*(t_E) \end{pmatrix}=g_e\cdot \begin{pmatrix} s_{E'}\\t_{E'} \end{pmatrix}.$$* *For an integer $n\ge 0$, we define $\alpha_n:\mathbb{E}(X_l^q(N))\rightarrow \mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})$ to be the voltage assignment sending $e$ to the image of $g_e$ under the natural projection $\mathop{\mathrm{GL}}_2(\mathbb{Z}_p)\rightarrow\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})$.* **Theorem 9**. *For all $n\ge0$, the derived graph $X(\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z}),\alpha_n)$ is isomorphic to $X_l^q(p^nN)$.* *Proof.* To simplify notation, we write $Y_n=X(\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z}),\alpha_n)$, $Z_n=X_l^q(p^nN)$ and $G_n=\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})$ in this proof. We first identify the vertices of $Y_n$ and $Z_n$. Recall that $V(Y_n)=Z_0\times G_n$. Let $(E,R_1,R_2,P,Q)\in V(Z_n)$. Since $\{P,Q\}$ is a basis of $E[p^n]$, there exists a unique $\sigma\in G_n$ such that $$\begin{pmatrix} P\\Q \end{pmatrix}=\sigma\cdot\begin{pmatrix} \overline{s}_E\\\overline{t}_E \end{pmatrix},\label{eq:condition-basis}$$ where $\overline{s}_E$ and $\overline{t}_E$ denote the images of $s_E$ and $t_E$ in $E[p^n]$, respectively. Conversely, given any $\left((E,R_1,R_2),\sigma\right)\in V(Y_n)$, we may define a basis of $E[p^n]$ via [\[eq:condition-basis\]](#eq:condition-basis){reference-type="eqref" reference="eq:condition-basis"}. Hence, there is a natural bijection $\Phi$ from $V(Z_n)$ to $V(Y_n)$. It remains to show that the bijection $\Phi$ respects the edges of the two graphs. That is, there is an edge in $Z_n$ from $v$ to $w$ induced by some isogeny if and only if there is an edge in $Y_n$ from $\Phi(v)$ to $\Phi(w)$ induced by the same isogeny. Suppose that $e\in \mathbb{E}(Z_0)$ with $(E,R_1,R_2)$ and $(E',R'_1,R'_2)$ as the source and target, respectively. Let $\phi$ be the corresponding $l$-isogeny. The same isogeny gives rise to an edge from $(E,R_1,R_2,P,Q)$ to $(E',R'_1,R'_2,P',Q')$ in $Z_n$ if and only if $$\phi(P)=P'\quad \text{and} \quad\phi(Q)=Q'.\label{eq:edge-condition}$$ Let us write $$\begin{aligned} \Phi((E,R_1,R_2,P,Q))&=((E,R_1,R_2),\sigma),\\ \Phi((E',R'_1,R'_2,P',Q'))&=((E',R'_1,R'_2),\sigma'). \end{aligned}$$ A direct calculation shows that [\[eq:edge-condition\]](#eq:edge-condition){reference-type="eqref" reference="eq:edge-condition"} is equivalent to the equation $$\sigma\alpha_n(e)=\sigma',$$ which is precisely the condition for there to be an edge going from $((E,R_1,R_2),\sigma)$ to $((E',R'_1,R'_2),\sigma')$ in $Y_n$. This shows that $\Phi$ respects the edges of $Y_n$ and $Z_n$, as desired. ◻ **Corollary 10**. *Let $n\ge1$ be an integer. The map $$\begin{aligned} V(X_l^q(p^nN))&\rightarrow V(X_l^q(p^{n-1}N)),\\ (E,R_1,R_2,P,Q)&\mapsto (E,R_1,R_2,pP,pQ) \end{aligned}$$ induces a graph covering $X_n\rightarrow X_{n-1}$.* *Proof.* Let $X$, $G$ and $\alpha$ be as in Definition [\[def:voltage\]](#def:voltage){reference-type="ref" reference="def:voltage"}. Let $N$ be a normal subgroup of $G$. We write $\tilde\alpha$ for the composition of $\alpha$ with the projection map $G\mapsto G/N$. Then $X(G/N,\tilde\alpha)$ is an intermediate covering of $X(G,\alpha)/X$; the covering map $X(G,\alpha)\rightarrow X(G/N,\tilde\alpha)$ is given by $$V(X(G,\alpha))\ni(v,\sigma)\mapsto (v,\sigma N),\quad \mathbb{E}(X(G,\alpha))\ni(e,\sigma)\mapsto (e,\sigma N).$$ Let $G_n=\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})$ and let $\alpha_n$ be the voltage assignment on $X=X_l^q(N)$ given in Definition [Definition 8](#def:alpha_n){reference-type="ref" reference="def:alpha_n"}. Let $\pi_n:G_n\rightarrow G_{n-1}$ be the natural projection map and $N_n=\ker(\pi_n)$. Then we have a natural graph covering $X(G_n,\alpha_n)/ X(G_n/N_n,\tilde\alpha_n)$ as discussed in the previous paragraph. Let $\Theta_n:V(X(G_n,\alpha_n))\rightarrow V(X_l^q(p^nN))$ be the inverse of the bijection $\Phi$ given in the proof of Theorem [Theorem 9](#thm:iso){reference-type="ref" reference="thm:iso"}. Since $G_{n-1}\cong G_n/N_n$ and $\tilde\alpha_n=\alpha_{n-1}$, we may identify $X(G_n/N_n,\tilde\alpha_{n})$ with $X(G_{n-1},\alpha_{n-1})$, and hence with $X_l^q(p^{n-1}N)$. Finally, it follows from the definition of the Tate module that $$\Theta_{n-1}\circ \pi_n=[p]\circ \Theta_{n},$$ which concludes the proof. ◻ **Corollary 11**. *Let $X$ be a connected component of $X_l^q(N)$ (in the sense of Definition [Definition 3](#def:connected){reference-type="ref" reference="def:connected"}). For an integer $n\ge1$, let $X_n$ denote the pre-image of $X$ in $X_l^q(p^nN)$ under the natural projection map. If $X_n$ is connected, the covering $X_n/X$ is Galois with Galois group isomorphic to $\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$.* *Proof.* This follows directly from Proposition [Proposition 5](#prop:voltage){reference-type="ref" reference="prop:voltage"}ii). ◻ We conclude this section with a couple of remarks on the connected components of $X_l^q(p^nN)$, which will be useful for subsequent sections. *Remark 12*. Let $X_n$ be a connected component of $X_l^q(p^nN)$. Let $e\in \mathbb{E}(X_n)$ with $v=(E,R_1,R_2,P,Q)$ and $w=(E',R'_1,R'_2,P',Q')$ as the source and target, respectively. Then $e$ arises from an $l$-isogeny $\phi:E\rightarrow E'$ such that $\phi(U)=U'$ for $U\in\{R_1,R_2,P,Q\}$. The dual isogeny $\hat\phi$ gives rise to an edge in $X_n$ from $w$ to $(E,lR_1,lR_2,lP,lQ)$. Let $d$ be the order of $[l]$ as an automorphism on $E[p^nN]$. On repeatedly applying $\phi$ and $\hat\phi$ alternatively $d$ times, we obtain a path from $(E,lR_1,lR_2,lP,lQ)$ to $v$. Therefore, there exists a path starting at $w$ and terminating at $v$. In other words, each directed edge in $X_n$ can be \"reversed\" via a path in the opposite direction. Hence, $X_n$ is a strongly connected directed graph. *Remark 13*. Let $v=(E,R_1,R_2,P,Q)$ and $w=(E',R'_1,R'_2,P',Q')$ be two vertices of $X_l^q(Np^n)$. There is a path from $v$ to $w$ if and only if there exists an isogeny $\phi\colon E\to E'$ of $l$-power degree such that $\phi(R_1)=R'_1$, $\phi(R_2)=R'_2$, $\phi(P)=P'$ and $\phi(Q)=Q'$. # The ordinary case The goal of this section is to prove Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"} stated in the introduction. We fix a connected component $X$ of $X_l^q(N)$ and define $X_n$ to be the pre-image of $X$ in $X_l^q(p^nN)$ under the natural projection map. As explained in the introduction, all elliptic curves giving rise to a vertex in $X_n$ have the same reduction type. We assume that these elliptic curves are ordinary in this section. Furthermore, since these curves are isogeneous to each other, their endomorphism rings are orders in the same imaginary quadratic field, which we denote by $K$. We shall refer to $K$ as the CM field of $X$. We begin with the following lemma. **Lemma 14**. *Let $n\ge1$ be an integer. The order of the group $\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ is $p^{4(n-1)}(p^2-1)(p^2-p)$.* *Proof.* Let $G_n=\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})$. Let $\pi_n:G_n\rightarrow G_{1}$ be the group homomorphism induced by the projection $\mathbb{Z}/p^n\mathbb{Z}\rightarrow \mathbb{Z}/p\mathbb{Z}$. Then $\pi_n$ is surjective with $$\ker\pi_n=I_4+p M_{2\times 2}(\mathbb{Z}/p^n\mathbb{Z}).$$ Thus, $|\ker(\pi_n)|=p^{4(n-1)}$, and $|G_n|=p^{4(n-1)}|G_{1}|$. It is a standard fact that $|G_1|=(p^2-1)(p^2-p)$. Hence the lemma follows. ◻ **Theorem 15**. *Assume that $X$ is an ordinary connected component of $X_l^q(N)$ and let $K$ be the CM field of $X$. If $l$ splits in $K$, then there exists a constant $c\ge (p+1)p$ such that the number of connected components of $X_n$ is equal to $cp^{2(n-1)}$ for $n$ sufficiently large. If $l$ is non-split, then the same holds on replacing $cp^{2(n-1)}$ by $cp^{3(n-1)}$.* *Proof.* The degree of the covering $X_l^q(p^nN)/X_l^q(N)$ equals $|\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})|$. As explained in Remark [Remark 7](#rem:connected-comp){reference-type="ref" reference="rem:connected-comp"}, the number of connected components of $X_n$ is given by $|\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})|$ divided the number of pre-images of a fixed $(E,R_1,R_2)\in V(X)$ lying in the same connected component of $X_n$. As all curves with complex multiplication by an order in $K$ are isogenous we can assume that $\textup{End}(E)=\mathcal{O}$ is an order such that $[\mathcal{O}_K:\mathcal{O}]$ is not divisible by $l$. In particular, the number of prime ideals in $\mathcal{O}$ that lie above $l$ is the same as the number of prime ideals in $\mathcal{O}_K$ that lie above $l$. Let $(E,R_1,R_2,P,Q)$ and $(E,R_1,R_2,P',Q')$ be two pre-images of $(E,R_1,R_2)$ in $V(X_n)$. By Remark [Remark 13](#rem:path){reference-type="ref" reference="rem:path"}, they lie in the same connected component of $X_n$ if and only if there is an isogeny $\phi\colon E\to E$ of $l$-power degree such that $\phi(R_1)=R_1$, $\phi(R_2)=R_2$, $\phi(P)=P'$ and $\phi(Q)=Q'$. Recall that $E[p^n]\cong \mathcal {O}/p^n$ and that $\mathcal{O}$ has a natural action on $E[p^n]$. Isogenies of $l$-power degree are given by elements of $l$-power norm in $\mathcal{O}$. Let $U$ be the (multiplicative) set of elements of $l$-power norm that fix $E[N]$. If $l$ splits in $K$, there exists a constant $c'$ such that the image $U_n$ of $U$ in $(\mathcal{O}/p^n)^\times$ has $c'p^{2(n-1)}$ elements for $n$ large enough, with $c'\le (p-1)^2$. In all other cases, $U_n$ has $c'p^{n-1}$ elements for $n$ large enough with $c'\le (p-1)$. Hence, the theorem follows from Lemma [Lemma 14](#lem:order){reference-type="ref" reference="lem:order"} on setting $c=(p^2-1)(p^2-p)/c'\ge (p+1)p$. ◻ We are now ready to prove Theorem [Theorem 1](#thmA){reference-type="ref" reference="thmA"}. **Corollary 16**. *The covering $X_{n}/X$ is not Galois for $n$ sufficiently large.* *Proof.* We prove that $X_{n}/X_{n-1}$ is not Galois, which in turn implies that $X_{n}/X$ is not Galois. It follows from Theorem [Theorem 15](#thm:ordinary-components){reference-type="ref" reference="thm:ordinary-components"} that the number of connected components of $X_{n}$ lying above $X_{n-1}$ is at least $p^2$ when $n$ is sufficiently large. Since the degree of the covering $X_{n}/X_{n-1}$ is $p^4$ by Lemma [Lemma 14](#lem:order){reference-type="ref" reference="lem:order"}, it suffices to show that $(p^2)!>p^4$ in light of Corollary [Corollary 6](#cor:not-Galois){reference-type="ref" reference="cor:not-Galois"}. Indeed, $$(p^2-2)(p^2-3)>\frac{4}{3}\ge\frac{1}{1-\frac{1}{p^2}}.$$ This in turn gives $$(p^2)!\ge p^2(p^2-1)(p^2-2)(p^2-3)>p^4,$$ as required. ◻ # The supersingular case {#sec:ss} We now turn our attention to the supersingular case and prove Theorem [Theorem 2](#thmB){reference-type="ref" reference="thmB"}. We fix once again a connected component $X$ of $X_l^q(N)$ and define $X_n$ accordingly. In contrast to the previous section, we assume that all elliptic curves $E$ giving rise to a vertex in $X$ have supersingular reduction. Let $E$ be any supersingular elliptic curve giving rise to a vertex in $X$. Let $B=\textup{End}(E)\otimes \mathbb{Q}$. Then $B$ is a quaternion algebra, which is only ramified at $q$ and $\infty$. The endomorphism ring $\textup{End}(E)$ is a maximal order in $B$. **Theorem 17** (Roda). *The number of connected components of $X_n$ is given by the index $(\mathbb{Z}/p^nN\mathbb{Z})^\times/\langle l\rangle$. In particular, the number of connected components is uniformly bounded as $n$ varies.* This theorem is the main result of [@thesis-roda Section 3.3]. The proof follows from the arguments given in [@gorenkassaei]. We outline the proof for the convenience of the reader. *Proof.* Let $(E,R_1,R_2,P,Q)$ and $(E,R'_1,R'_2,P',Q')$ be two vertices in $X_n$. Then there is a matrix $C$ in $\textup{GL}_2(\mathbb{Z}/Np^n\mathbb{Z})$ such that $$C\begin{pmatrix} R_1 +P\\R_2+Q \end{pmatrix}=\begin{pmatrix}R'_1+P'\\R'_2+Q'\end{pmatrix}.$$ In light of Remark [Remark 13](#rem:path){reference-type="ref" reference="rem:path"}, we determine when $C$ is induced by an endomorphism of $l$-power degree. Let us first assume that $C\in \textup{SL}_2(\mathbb{Z}/p^nN\mathbb{Z})$. Since $\textup{SL}_2(\mathbb{Z}/p^nN\mathbb{Z})$ has strong approximation, there exists $x\in B$ of norm one that is integral at all places except possibly at $l$ and such that $x$ acts as $C$ on $E[p^nN]$. Let $r$ be a positive integer such that $l^r$ acts trivially on $E[p^nN]$ and such that $l^rx$ is integral at all places. Then $y=l^rx$ is an endomorphism of $E$ of $l$-power degree such that $y(P+R_1)=P'+R'_1$ and $y(Q+R_2)=Q'+R'_2$ (see [@gorenkassaei proof of Theorem 5.3.3] for further details). Assume now that $\det(C)$ is a power of $l$. By [@gorenkassaei proof of Theorem 5.33], $E$ admits an endomorphism $f$ of degree $l^r$ with $r$ odd. As $[l]$ has degree $l^2$, it follows that there is a matrix $D\in \textup{GL}_2(\mathbb{Z}/p^nN\mathbb{Z})$ induced by an $l$-power degree endomorphism such that $\det(C)=\det(D)$. Thus, to determine whether $C$ is induced by an $l$-power degree endomorphism, it suffices to consider $C'=CD^{-1}$. As $C'$ has determinant one this is precisely the first case. Clearly, $C$ can only be induced by an $l$-power degree endomorphism if $\det(C)=l^r$ for some $r$. Thus, we have shown that $C$ is induced by an $l$-power degree endomorphism if and only if $\det(C)$ is an $l$-power, which concludes the proof. ◻ **Corollary 18**. *Assume $N=1$ or $2$ and $p>2$. Let $T$ be the set of primes $l$ such that $X_n$ is connected. Then $T$ has density $\frac{\varphi(p-1)}{p}$, where $\varphi$ is the Euler totient function.* *Proof.* Theorem [Theorem 17](#thm:number-connected-comp){reference-type="ref" reference="thm:number-connected-comp"} says that the graph $X_n$ is connected if and only if $l$ generates $(\mathbb{Z}/p^nN\mathbb{Z})^\times$ for all $n$. This is the case if and only if $l$ generates the cyclic group $(\mathbb{Z}/Np^2\mathbb{Z})^\times$, which is of order $p(p-1)$. The density of primes generating $(\mathbb{Z}/Np^2\mathbb{Z})^\times$ is therefore $\frac{\varphi(p(p-1))}{p(p-1)}=\frac{(p-1)\varphi(p-1)}{p(p-1)}$. ◻ *Remark 19*. If $N>2$ or $p>2$, the group $(\mathbb{Z}/p^nN\mathbb{Z})^\times$ is not cyclic. In particular, $X_n$ is not connected for any choices of $l$. **Corollary 20**. *Assume that $X_n$ is connected for all $n$. Then $X_n/X$ is a Galois covering with Galois group isomorphic to $\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$.* *Proof.* This follows immediately from Proposition [Proposition 5](#prop:voltage){reference-type="ref" reference="prop:voltage"}ii). ◻ Finally, we show that even if the graphs $X_n$ are not connected, we may still obtain a tower of Galois covers. **Corollary 21**. *There exists a minimal integer $m_0$ such that the number of connected components of $X_n$ stabilizes and such that there are no multiple edges. Let $n> m\ge m_0$ and let $G_{n,m}\subset \textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ be the subgroup of matrices congruent to $\begin{pmatrix} 1&0\\0&1 \end{pmatrix} \pmod {p^m}$. Let $Z_n$ and $Z_m$ be connected components of $X_n$ and $X_m$, respectively such that $Z_n/Z_m$ is a graph covering. Then $Z_n/Z_m$ is Galois with $\mathop{\mathrm{Gal}}(Z_n/Z_m)\cong G_{n,m}$.* *Proof.* We have already seen in Theorem [Theorem 17](#thm:number-connected-comp){reference-type="ref" reference="thm:number-connected-comp"} that the number of connected components in $X_n$ stabilizes when $n$ is sufficiently large. The graph $X_n$ contains multiple edges if and only if there are two distinct $l$-isogenies $\phi,\psi\colon E\to E'$, which coincide on $E[Np^n]$, which does not happen when $n$ is large enough. This shows the existence of the integer $m_0$. Assume now that $n>m\ge m_0$. Clearly, $Z_n/Z_m$ is a $\vert G_{n,m}\vert$-sheeted covering and there is a natural embedding $$\psi_m\colon G_{n,m}\hookrightarrow \textup{Deck}(Z_n/Z_m).$$ It remains to show that this embedding is indeed an isomorphism. Let $\sigma\in\textup{Deck}(Z_n/Z_m)$ and let $v$ be a vertex of $Z_n$. Let $\pi \colon Z_n\to Z_m$ be the projection map. By definition $$\pi(\sigma(v))=\pi(v).$$ The graph $Z_n$ is a subgraph of $X_n$. Therefore, there exists a vertex $v_0$ of $X_0$ and elements $h,h'\in\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ such that $v=(v_0,h)$ and $\sigma(v)=(v_0,h')$. As $\pi(\sigma(v))=\pi(v)$, the elements $h$ and $h'$ have the same image in $\textup{GL}_2(\mathbb{Z}/p^m\mathbb{Z})$. In particular, there exists an element $g\in G_{n,m}$ such that $h'=gh$, which is equivalent to $g(v)=\sigma(v)$. We shall show that the action of $g$ on $Z_n$ coincides with that of $\sigma$. Let $w$ be a vertex of $Z_n$ admitting an edge whose source and target are $v$ and $w$, respectively. We need to show that $\sigma(w)=g(w)$. As $Z_m$ does not have any multiple edges by assumption, it admits a unique edge from $\pi(v)$ to $\pi(w)$. It follows that there is a unique edge in $Z_n$ from $\sigma(v)$ to some vertex $w'$ such that $\pi(w)=\pi(w')$. As $\sigma$ and $g$ are both deck transformations of $Z_n/Z_m$, we have $\sigma(w)=g(w)=w'$. Since $Z_n$ is strongly connected by Remark [Remark 12](#rk:connected){reference-type="ref" reference="rk:connected"}, this property propagates to all vertices of $Z_n$. Thus, the actions of $\sigma$ and $g$ agree as desired, proving that $\psi_m$ is indeed an isomorphism. ◻ # An abelian $p$-covering arising from quotients of $X_l^q(p^nN)$ Let $\alpha_n$ be the voltage assignment given in Definition [Definition 8](#def:alpha_n){reference-type="ref" reference="def:alpha_n"}. Let $\beta_n=\det\circ\alpha_n$. Then $\beta_n$ is a voltage assignment with values in $\mathbb{Z}_p^\times$. As explained in the first paragraph of the proof of Corollary [Corollary 10](#cor:cover){reference-type="ref" reference="cor:cover"}, the derived graph $X(X_l^q(N),\beta_n)$ is a subcovering of $X(X_l^q(N),\alpha_n)/X_l^q(N)$. We study this subcovering in detail in this section. Recall that we fixed for any elliptic curve $E$ in $S$ defined over $\mathbb{F}_{q^k}$ a basis $\{s_E,t_E\}$ of the $p$-adic Tate-module $T_p(E)$. Let $$\langle \cdot, \cdot\rangle \colon T_p(E)\times T_p(E)\longrightarrow \mathbb{Z}_p(1)$$ be the Weil pairing. In this section, we make the additional assumption that for all $E$ the pairing $\langle s_E,t_E \rangle$ is a topological generator $\xi$ of $\mathbb{Z}_p(1)$. Let $\xi_{n}$ denote the primitive $p^n$-th root of unity given by the image of $\xi$ under the natural map $\mathbb{Z}_p(1)\rightarrow \mu_{p^n}$. **Definition 22**. *We define a directed graph $Y_l^q(p^nN)$ as follows. The set of vertices of $Y_l^q(p^nN)$ is given by tuples $(E,R_1,R_2,\zeta)$, where $E,R_1,R_2$ are as before and $\zeta$ is a primitive $p^n$-th root of unity. Let $\Psi_n$ be the following surjection $$\Psi_n\colon V(X_l^q(N))\to V(Y_l^q(p^nN)), \quad (E,R_1,R_2,P,Q)\mapsto (E,R_1,R_2,\langle P,Q\rangle).$$ For every vertex $v\in V(Y_l^q(Np^n))$, we fix a pre-image $v'\in V(X_l^q(p^nN))$ under $\Psi_n$. The edges of $Y_l^q(Np^n)$ are given as follows: there is an edge from $v$ to $w$ if and only if there is an edge from $v'$ to some element $w''\in \Psi_n^{-1}(w)$ in $X_l^q(Np^n)$.* Given $(E,R_1,R_2,\zeta)\in V(Y_l^q(p^nN))$, we may write $\zeta=\xi_n^a$ for a unique $a\in(\mathbb{Z}/p^n\mathbb{Z})^\times$. We will frequently denote $(E,R_1,R_2,\zeta)$ by $(E,R_1,R_2,a)$. *Remark 23*. For every elliptic curve $E$ defined over $\mathbb{F}_{q^k}$, the order of the group $\textup{Aut}(E)$ is bounded by a constant $C_q$ that depends only on $q$. Thus, if $N>C_q$, the graph $Y_l^q(Np^n)$ does not admit any multiple edges for all $n\ge0$. **Proposition 24**. *Let $N>C_q$. Then the graph $Y_l^q(Np^n)$ is isomorphic to $X(X_l^q(N),\beta_n)$.* *Proof.* By definition, there is a natural bijective map $$\Theta_n\colon V(Y_l^q(Np^n))\longrightarrow V(X(X_l^q(N),\beta_n)).$$ Thus, it remains to show that $\Theta_n$ respects the edges of the two graphs, as in the proof of Theorem [Theorem 9](#thm:iso){reference-type="ref" reference="thm:iso"}. Let $v=(E,R_1,R_2,a)$ and $w=(E',R'_1,R'_2,a')$ be two vertices of $Y_l^q(Np^n)$. Let $v'$ be the fixed pre-image in $X_l^q(Np^n)$ under $\Psi_n$ as in Definition [Definition 22](#def-zp-graph){reference-type="ref" reference="def-zp-graph"}. There is an edge from $v$ to $w$ if and only if there is an edge from $v'$ to $w''$ for some $w''\in \Psi_n^{-1}(w)$. Theorem [Theorem 9](#thm:iso){reference-type="ref" reference="thm:iso"} allows us to identify $V(X_l^q(p^nN))$ with $V(X_l^q(N))\times\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})$. Two bases of $E[p^n]$ have the same image in $\mu_{p^n}$ under the Weil pairing if and only if the corresponding matrices in $\mathop{\mathrm{GL}}_2(\mathbb{Z}/p^n\mathbb{Z})$ have the same determinant. This implies the existence of $\sigma,\sigma'\in \textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ such that $a=\det(\sigma)$, $a'=\det(\sigma')$ and that $$\begin{aligned} \Psi_n^{-1}(v)&=\left\{(E,R_1,R_2,\sigma\tau)\mid \tau\in \textup{SL}_2(\mathbb{Z}/p^n\mathbb{Z})\right\},\\ \Psi_n^{-1}(w)&=\left\{(E',R'_1,R'_2,\sigma'\tau)\mid \tau\in \textup{SL}_2(\mathbb{Z}/p^n\mathbb{Z})\right\}. \end{aligned}$$ Without loss of generality, we may assume that the chosen pre-image of $v$ is given by $v'=(E,R_1,R_2,\sigma)$. By the definition of $X(X_l^q(N),\alpha_n)$, there exists an edge from $v'$ to an element $w''\in \Psi_n^{-1}(w)$ if and only if there exists an edge $e$ from $(E,R_1,R_2)$ to $(E',R'_1,R'_2)$ in $X_l^q(N)$ and an element $\tau\in \textup{SL}_2(\mathbb{Z}/p^n\mathbb{Z})$ such that $$\sigma'\tau=\sigma \alpha_n(e).$$ The existence of $\tau$ is equivalent to $$a'=\det(\sigma')=\det(\sigma)\det(\alpha_n(e))=a\beta_n(e).$$ This condition is equivalent to the existence of an edge from $(E,R_1,R_2,a)$ to $(E',R'_1,R'_2,a')$ in $X(X_l^q(N),\beta_n)$. Therefore, $\Theta_n$ indeed respects the edges of $Y_l^q(Np^n)$ and $X(X_l^q(N),\beta_n)$, as desired. ◻ *Remark 25*. Proposition [Proposition 24](#prop:iso){reference-type="ref" reference="prop:iso"} tells us that the graph $Y_l^q(p^nN)$ is independent of the choice of pre-image $v'\in V(X_l^q(p^nN))$ of $v\in V(Y_l^q(p^nN))$ when $N$ is sufficiently large. **Theorem 26**. *The number of connected components of $Y_l^q(Np^n)$ is uniformly bounded in $n$.* *Proof.* Let $u$ be the minimal non-negative integer such that $l^u\equiv 1\pmod N$. Since the multiplication by $l^u$ is an endomorphism of $l$-power degree, it follows from Remark [Remark 13](#rem:path){reference-type="ref" reference="rem:path"} that there is a path from $(E,R_1,R_2,a)$ to $(E,R_1,R_2,l^{2u}a)$ in $Y_l^q(Np^n)$ for all choices of $(E,R_1,R_2,a)$. Furthermore, Remark [Remark 7](#rem:connected-comp){reference-type="ref" reference="rem:connected-comp"} tells us that the number of connected components of $Y_l^q(p^nN)$ is bounded by $\vert (\mathbb{Z}/p^n\mathbb{Z})^\times /\langle l^{2u}\rangle\vert$. This index is uniformly bounded in $n$, which concludes the proof of the theorem. ◻ We are now ready to prove Theorem [Theorem 3](#thmC){reference-type="ref" reference="thmC"}. **Corollary 27**. *Assume that $N>C_q$. Let $m_0$ be the minimal integer such that the number of connected components of $Y_l^q(Np^n)$ stabilizes. Let $n>m\ge m_0$. Let $Y_m$ be a connected component of $Y_l^q(p^mN)$ and let $Y_n$ be a connected component of $Y_l^q(p^nN)$ above it. Then $Y_n/Y_m$ is Galois and $\mathop{\mathrm{Gal}}(Y_n/Y_m)$ is the subgroup ${\mathcal{G}}_{n,m}$ of $(\mathbb{Z}/p^n\mathbb{Z})^\times$ of elements $x\equiv 1\pmod {p^m}$.* *Proof.* Clearly, $Y_n/Y_m$ is a $\vert {\mathcal{G}}_{n,m}\vert$-sheeted cover. Via the voltage assignment $\beta_n$, we see that there is an injective group homomorphism $${\mathcal{G}}_{n,m}\hookrightarrow \textup{Deck}(Y_n/Y_m).$$ It suffices to show that this map is surjective. Let $\omega\in\mathrm{Deck}(Y_n/Y_m)$ and $v=(E,R_1,R_2,a)\in V(Y_n)$. Let $v'=(E,R_1,R_2,\sigma)\in X_l^q(Np^n)$ be the fixed pre-image under the map $\Psi_n$ given in Definition [Definition 22](#def-zp-graph){reference-type="ref" reference="def-zp-graph"}. Then there exists $g\in {\mathcal{G}}_{n,m}$ such that $gv=\omega (v)$. Let $g'$ be a lift to $\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$, i.e. $g'$ is a matrix with $\det(g')=g$. In particular, $$\Psi_n(E,R_1,R_2,g'\sigma)=gv.$$ Let $\phi$ be a $l$-isogeny on $E$ and let $w=\phi(v)$. Let $$w'=(\phi(E),\phi(R_1),\phi(R_2),\sigma \alpha_n(e_\phi)),$$ where $e_\phi$ is the edge induced by $\phi$. By definition, $$w=\Psi_n(w')=(\phi(E),\phi(R_1),\phi(R_2), a\beta_n(e_\phi)).$$ As $Y_l^q(N)$ does not have multiple edges we see that $\omega(w)=\phi(\omega(v))$. Furthermore, $$\begin{aligned} \omega(w)&=\phi(\omega(v))=\phi(g v)=\Psi_n(\phi (g' v'))=\Psi_n(\phi(E,R_1,R_2,g'\sigma))\\&=\Psi_n(\phi(E),\phi(R_1),\phi(R_2), g'\sigma \alpha_n(e))=(\phi(E),\phi(R_1),\phi(R_2), g a \beta_n(e_\phi))\\&=gw. \end{aligned}$$ Thus, $\omega$ is uniquely determined by $\omega(v)$. Hence, ${\mathcal{G}}_{n,m}\cong \textup{Deck}(Y_n/Y_m)$, which concludes the proof. ◻ In particular, we see that $$Y_{m_0}\leftarrow Y_{m_0+1}\leftarrow Y_{m_0+2}\leftarrow \cdots \leftarrow Y_{m_0+n}\leftarrow \cdots$$ form an abelian $p$-tower in the sense of [@vallieres Definition 4.1]. # Isogeny graphs of oriented supersingular elliptic curves and Volcano graphs Let $M$ be a positive integer coprime to $ql$. While the structure of an $l$-isogeny graph of ordinary elliptic curves equipped with a $\Gamma_1(M)$-level structure is related to the so-called (tectonic) volcano graphs (see [@LM1 §4]), analogous results for the supersingular counterpart are not available. In [@colokohel], Colò--Kohel introduced the concept of orientations of supersingular elliptic curves. The authors in [@arpin-et-all] showed that the isogeny graph of supersingular elliptic curves equipped with an orientation (without any level structures) is a volcano graph, analogous to the ordinary counterpart studied in [@kohel]. In this section, we review the notion of orientations and describe the structure of oriented supersingular isogeny graphs enhanced with a $\Gamma(M)$-level structure for $M>2$. Furthermore, we will show that the voltage assignment $\alpha_n$ defined in §[2.2](#sec:realize){reference-type="ref" reference="sec:realize"} gives rise to a \"$\textup{GL}_2(\mathbb{Z}_p)$-tower\" of $l$-isogeny graphs of oriented elliptic curves with $\Gamma(Np^n)$-level structure as $n$ varies. Throughout this section, we assume that $q\equiv 1\pmod{12}$ and that $B$ is the quaternion algebra only ramified at $q$ and at $\infty$. Furthermore, we assume that our fixed integer $k$ is even so that $\mathbb{F}_{q^2}\subset \mathbb{F}_{q^k}$. All elliptic curves $E/\mathbb{F}_{q^k}$ are assumed to be supersingular. As $q\equiv 1 \pmod {12}$, we have $\textup{Aut}(E)=\{\pm 1\}$. ## The $l$-isogeny graph of oriented supersingular elliptic curves We recall the notion of oriented supersingular elliptic curves and their isogeny graphs, as given in [@colokohel]. Furthermore, we review a result of Arpin--Chen--Lauter--Scheidler--Stange--Tran [@arpin-et-all], which describes these graphs in terms of \"volcanoes\". We fix once and for all an imaginary quadratic field $K$ in which $q$ does not split. Then there is a natural embedding $\iota \colon K\to B$, where $B$ is a quaternion algebra only ramified at $q$ and $\infty$ as in §[4](#sec:ss){reference-type="ref" reference="sec:ss"}. Let $E$ be a supersingular elliptic curve over ${\mathbb{F}_{q^k}}$. Then $\textup{End}(E)\otimes \mathbb{Q}\cong B$. Thus, we can regard $\iota$ as an embedding of $K$ into $\textup{End}(E)\otimes \mathbb{Q}$. **Definition 28**. *Let $E/\mathbb{F}_{q^k}$ be a supersingular elliptic curve and $\iota$ be an embedding of $K$ into $\mathop{\mathrm{End}}(E)\otimes B$.* *We call the pair $(E,\iota)$ an **oriented elliptic curve**.* *We call an order $\mathcal{O}$ of $K$ **primitive** for $(E,\iota)$, if $\iota(\mathcal{O})=\iota(K)\bigcap \textup{End}(E)$.* *We say that $\mathcal{O}$ is $l$-**primitive** if $l$ does not divide $[\iota(K) \bigcap \textup{End}(E):\iota(\mathcal{O})]$.* *If $\varphi\colon E\to E'$ is an isogeny of elliptic curves, we define the orientation $\varphi_*\iota$ on $E'$ by $$\begin{aligned} \varphi_*\iota:K&\rightarrow\mathop{\mathrm{End}}(E')\otimes \mathbb{Q},\\ \alpha&\mapsto\frac{1}{\deg(\varphi)}\widehat{\varphi}\iota(\alpha)\varphi.\end{aligned}$$* *An **isogeny of oriented elliptic curves** $\varphi\colon (E,\iota)\to (E',\iota')$ is an isogeny of elliptic curves $\varphi\colon E'\to E$ such that $\varphi_*\iota=\iota'$.* *We say that $l$-isogenies $\phi,\psi\colon E\to E'$ are **equivalent isogenies** if they have the same kernel.* *We say that $(E,\iota)$ and $(E,\iota')$ are **equivalent oriented elliptic curves** if there exists an isomorphism of elliptic curves $\phi\colon E\to E'$ such that $\iota'=\phi_*\iota$.* *Remark 29*. Let $\phi,\psi\colon E\to E'$ be equivalent isogenies. Then $(\phi(E),\phi_*\iota)$ and $(\psi(E),\psi_*\iota)$ are equivalent. We are now ready to define the $l$-isogeny graph of oriented elliptic curves. **Definition 30**. *We define $\mathcal{X}_l^q$ to be the directed graph whose vertices are equivalence classes of oriented supersingular elliptic curves $(E,\iota)$ and whose edges are given by $l$-isogenies up to equivalence.* The following notions will allow us to describe the structure of $\mathcal{X}_l^q$. **Definition 31**. *Let $\varphi:(E,\iota)\rightarrow(E',\iota')$ be an isogeny of oriented supersingular elliptic curves. Assume that $\mathcal{O}$ and $\mathcal{O}'$ are orders of $K$ that are primitive for $(E,\iota)$ and $(E',\iota')$, respectively. We say that $\varphi$ is* - ***horizontal** if $\mathcal{O}=\mathcal{O}'$,* - ***descending** if $\mathcal{O}'\subset \mathcal{O}$,* - ***ascending** if $\mathcal{O}'\supset \mathcal{O}$.* The notion of horizontal, ascending and descending isogenies allows us to define the depth of an oriented elliptic curve. **Definition 32**. *Let $(E,\iota)$ be an oriented elliptic curve. Assume that $\mathcal{O}$ is primitive with respect to $(E,\iota)$. We say that $(E,\iota)$ is of depth zero if the conductor of $\mathcal{O}$ is not divisible by $l$. We say that $(E,\iota)$ is of depth $k$ if there is a path of $k$ descending isogenies of degree $l$ starting at a depth zero oriented elliptic curve $(E',\iota')$ and ending at $(E,\iota)$ .* Finally, we introduce the following notions of graphs, which are the last ingredients for our description of $\mathcal{X}_l^q$. **Definition 33**. *An **abstract crater** is either a connected and directed cycle graph or a totally disconnected finite graph.* *A **volcano graph** (or simply a volcano) $G$ is a directed graph whose vertices may be decomposed as $V(G)=\bigcup_{i=0}^\infty V_i$ such that* - *The out-degree of every vertex is $l+1$;* - *The subgraph generated by $V_0$ is an abstract crater;* - *If there is an edge between $v_i$ and $v_j$ then $i-j\in \{\pm 1\}$ or $i=j=0$;* - *Each $v\in V_i$ with $i\ge 1$ has exactly one edge starting at $v$ and ending at a vertex in $V_{i-1}$ and $l$ edges starting at $v$ and ending at vertices in $V_{i+1}$.* *An **undirected volcano** $G$ is an undirected $(l+1)$-regular graph whose vertices may be decomposed as $V(G)=\bigcup_{i=0}^\infty V_i$ such that* - *The subgraph generated by $V_0$, called the **crater** of $G$, is an undirected cycle graph;* - *If there is an edge between $v_i$ and $v_j$ then $i-j\in \{\pm 1\}$ or $i=j=0$;* - *Each $v\in V_i$ with $i\ge 1$ has exactly one neighbour in $V_{i-1}$ and $l$ neighbours in $V_{i+1}$.* *Remark 34*. Note that (undirected) volcanoes are always infinite graphs. **Theorem 35**. *[@arpin-et-all Section 3.4] [\[thm:arpin\]]{#thm:arpin label="thm:arpin"} The graph $\mathcal{X}_l^q$ is an infinite graph. Its connected components are volcano graphs whose craters are generated by the depth zero vertices.* ## The $l$-isogeny graph of oriented supersingular elliptic curve with level structure We would like to consider oriented elliptic curves enhanced with a $\Gamma(M)$-level structure. In other words, instead of pairs $(E,\iota)$ we consider tuples $(E,\iota, R_1,R_2)$, where $\{R_1,R_2\}$ is a basis for $E[M]$. **Definition 36**. *An isogeny of oriented elliptic curves with a $\Gamma(M)$-level structure $\varphi\colon (E,\iota,R_1,R_2)\to (E',\iota',R'_1,R'_2)$ is an isogeny of oriented elliptic curves $(E,\iota)\to (E',\iota')$ such that $\varphi(R_1)=R'_1$ and $\varphi(R_2)=R'_2$.* **Definition 37**. *We fix a set of representatives $S'$ for the equivalence classes of oriented elliptic curves over $\mathbb{F}_{q^k}$.* *Let $\mathcal{X}_l^q(M)$ be the directed graph whose vertices are quadruples $(E,\iota,R_1,R_2)$, where $(E,\iota)\in S'$ and $\{R_1,R_2\}$ is a basis for $E[M]$, and such that the edges are given by $l$-isogenies defined as in Definition [Definition 36](#defn:isogeny-oriented-level){reference-type="ref" reference="defn:isogeny-oriented-level"}.* *Let $e\in\mathbb{E}(\mathcal{X}_l^q(M))$. We say that $e$ is horizontal/descending/ascending if the isogeny giving rise to $e$ is horizontal/descending/ascending.* *Remark 38*. If $\phi:E\rightarrow E'$ is an $l$-isogeny of elliptic curves, the composition $\phi\circ [-1]=[-1]\circ \phi$ is also an $l$-isogeny. Note that $[-1]$ acts non-trivially on a basis of $E[M]$ for $M>2$. Thus, equivalent isogenies induce edges with different targets if $M>2$. For this reason, we have decided to distinguish edges arising from $\phi$ and $\phi\circ[-1]$ in our definition of $\mathcal{X}_l^q(M)$ for all positive integers $M$, even though these two isogenies are equivalent. In particular, the graphs $\mathcal{X}_l^q(1)$ and $\mathcal{X}_l^q$ are not isomorphic to each other. They share the same vertex set, but each edge of $\mathcal{X}_l^q$ corresponds to two distinct edges in $\mathcal{X}_l^q(1)$. In order to describe the structure of $\mathcal{X}_l^q(M)$ in more detail, we recall the following notions from [@arpin-et-all]. **Definition 39**. *Let $E/_{\mathbb{F}_{q^k}}$ be a supersingular elliptic curve and $\theta\in \textup{End}(E)$. Let $f_\theta$ be the minimal polynomial of $\theta$, i.e. the unique polynomial $X^2+aX+b\in \mathbb{Z}[x]$ such that $f_\theta(\theta)=0$. Let $\alpha$ be the (unique up to conjugation) element generating a quadratic extension of $\mathbb{Q}$ with minimal polynomial $f_\theta$. Let $K=\mathbb{Q}(\alpha)$ and let $$\omega_K=\begin{cases} \frac{1+\sqrt{\Delta}}{2} \quad &\Delta\equiv 1 \pmod 4\\ \frac{\sqrt{\Delta}}{2} \quad &\Delta\equiv 0\pmod 4, \end{cases}$$ where $\Delta$ is the fundamental discriminant of $K$.* *We say that $\theta$ is $l$-**primitive** if $\mathbb{Z}[\alpha]$ is $l$-primitive with respect to the orientation $K\to \textup{End}(E)$ given by $\alpha\mapsto \theta$.* *We say that $\theta$ is $l$-**suitable** if $\alpha$ is of the form $f\omega_K+ml$, where $m$ is some integer and $f$ is the conductor of $\mathbb{Z}[\alpha]$.* *Remark 40*. Given a fixed orientation $\iota\colon K\to B$ and a supersingular elliptic curve $E$, there always exists $\theta \in \textup{End}(E)$ that is $l$-primitive and $l$-suitable. Indeed, let $\mathcal{O}$ be an order of $K$, which is primitive with respect to $(E,\iota)$. Let $f$ be the conductor of $\mathcal{O}$ and let $\alpha=f\omega_K$. Then $\mathcal{O}=\mathbb{Z}[f\omega_K]=\mathbb{Z}[\alpha]$. If we set $\theta=\iota(\alpha)$, the endomorphism $\theta$ is $l$-primitive and $l$-suitable. **Proposition 41**. *[@arpin-et-all Proposition 4.8] Let $\theta$ be an element that is $l$-primitive and $l$-suitable for $(E,\iota)$. Assume that $(E,\iota)$ is of depth zero and let $\lambda_1,\lambda_2$ be the eigenvalues of the action of $\theta$ on $E[l]$.* - *If $l$ splits in $K$, then $\lambda_1\neq \lambda_2\in \mathbb{F}_l$. Let $P_{\lambda_i}$ be generators of the $\lambda_i$-eigenspaces, then $E\to E/\langle P_{\lambda_i}\rangle$ are horizontal isogenies for $i=1,2$.* - *If $l$ ramifies in $K$, then $0\neq\lambda_1=\lambda_2\in \mathbb{F}_l$. The $\lambda_i$-eigenspace is one-dimensional, where $i=1, 2$. Let $P$ be a generator of this eigenspace. Then $E\to E/\langle P\rangle$ is a horizontal isogeny.* - *If $l$ is inert in $K$, then $\lambda_1,\lambda_2\in \mathbb{F}_{l^2}\setminus \mathbb{F}_l$ and there are no horizontal isogenies on $E$.* *All other $l$-isogenies are descending in all three cases.* **Lemma 42**. *Assume that $l$ splits in $K$. Let $(E,\iota)$ be a depth zero vertex. Let $\theta\in\mathop{\mathrm{End}}(E)$ be $l$-primitive and $l$-suitable for $(E,\iota)$ and let $\alpha$ be the element attached to $\theta$ given in Definition [Definition 39](#def:suitable){reference-type="ref" reference="def:suitable"}. Let $\lambda_1$ and $\lambda_2$ be given as in Proposition [Proposition 41](#prop:eigenvalues){reference-type="ref" reference="prop:eigenvalues"}. For $i\in\{1,2\}$, let $P^{E}_i$ be a generator of the $\lambda_i$-eigenspace and $\phi^E_i\colon E\to E/\langle P_i^E\rangle$ be the natural projection.* - *Let $\phi\colon (E,\iota)\to (E',\iota')$ be a horizontal isogeny. Then $\theta':=\iota'(\alpha)$ is $l$-primitive and $l$-suitable for $(E',\iota')$. In particular, $\lambda_1$ and $\lambda_2$ are the eigenvalues of the action of $\theta'$ on $E'[l]$.* - *Assume that $\lambda_1\neq \lambda_2$ and let $P_i^{E'}$ be a generator of the $\lambda_i$-eigenspace of $\phi^{E}_i(E)$. Let $i\neq j$. Then $\phi_i^E(P_j)=cP_j^{E'}$, for some $c$ coprime to $l$.* - *Fix $i\in\{1, 2\}$. Let $v_0=(E_0,\iota_0,R_{0,1},R_{0,2})=(E,\iota,R_1,R_2)\in V(\mathcal{X}_l^q(M))$ and define recursively $$v_j=(E_j,\iota_j,R_{j,1},R_{j,2})=\phi_i^{E_{j-1}}(v_{j-1}),\ j\ge1.$$ Then there exists a non-negative integer $r$ such that $v_r=v_0$.* *Proof.* i) and ii) follow directly from definitions. It remains to prove iii). By Theorem [\[thm:arpin\]](#thm:arpin){reference-type="ref" reference="thm:arpin"}, there are only finitely many depth zero vertices in $\mathcal{X}_l^q$ that lie in the same connected component as $v$. Thus, there is a non-negative integer $r_0$ such that $E_{r_0}=E$ and $\iota_{r_0}=\iota$. Hence, there is an isogeny $\gamma:(E,\iota)\rightarrow (E,\iota)$ whose degree is an $l$-power such that $\gamma(R_i)=R_{r_0,i}$ for $i=1, 2$. Let $s$ be the minimal non-negative integer such that $\gamma^s(R_i)=R_i$ for both $i$. Then part iii) follows on setting $r=r_0s$. ◻ We now introduce several notions of graphs that will be utilized in our description of $\mathcal{X}_l^q(M)$. **Definition 43**. *Let $Z$ be a directed graph. We define the **double intertwinement** $Z^{+-}$ of $Z$ to be the graph such that $$V(Z^{+-})=\{+v,-v: v\in V(Z)\}$$ (each vertex of $V(Z)$ gives rise to two vertices in $Z^{+-}$) and that $$\mathbb{E}(Z^{+-})=\{e^{++},e^{+-},e^{-+},e^{--}:e\in\mathbb{E}(Z)\},$$ where $e^{\bullet\circ}$ denotes an edge whose source and target are $\bullet v$ and $\circ w$, respectively if $e$ is an edge from $v$ to $w$ in the original graph $Z$.* *Example 44*. Let us illustrate the double intertwinement of the following graph: Then $Z^{+-}$ is given by The double intertwinement of the directed cycle graph with $4$ edges and $4$ vertices is given by: The following definition was introduced in [@LM1 §5]. **Definition 45**. *Let $\mathfrak r,\mathfrak s,\mathfrak t,\mathfrak c$ be nonnegative integers. We say that a directed graph is an **abstract tectonic crater** of parameters $(\mathfrak r,\mathfrak s,\mathfrak t,\mathfrak c)$ if it satisfies* - *There are $\mathfrak r\mathfrak s\mathfrak t$ vertices;* - *Each edge is assigned a color -- blue or green;* - *At each vertex $v$, there is exactly one blue edge with $v$ as the source, and exactly one blue edge with $v$ as the target, and similarly for green edges;* - *Starting at each vertex, there is exactly one closed blue (resp. green) path without backtracks of length $\mathfrak r\mathfrak s$ (resp. $\mathfrak r\mathfrak t$);* - *After every $\mathfrak s$ (resp. $\mathfrak c\mathfrak t$) steps in the closed blue (resp. green) paths given in d), the two paths meet at a common vertex.* *Example 46*. An abstract tectonic crater with $\mathfrak t=\mathfrak s=1$, $\mathfrak c=2$ and $\mathfrak r=5$. For further examples, the reader is referred to [@LM1 §4.2]. We may enhance the definition of an abstract tectonic crater, giving the \"tectonic\" version of a volcano graph: **Definition 47**. *A **tectonic volcano** $G$ is a directed graph whose vertices may be decomposed into $V(G)=\bigcup_{i=0}^\infty V_i$ such that* - *The out-degree of every vertex is $l+1$;* - *The subgraph generated by $V_0$ is an abstract tectonic crater (we shall refer to $V_0$ as the tectonic crater of $G$);* - *If there is an edge between $v_i$ and $v_j$ then $i-j\in \{\pm 1\}$ or $i=j=0$;* - *Every $v\in V_i$ with $i\ge 1$ has exactly one edge starting at $v$ and ending at a vertex in $V_{i-1}$ and $l$ edges starting at $v$ and ending at a vertex in $V_{i+1}$.* We are now ready to prove our first theorem on the structure of $\mathcal{X}_l^q(M)$. **Theorem 48**. *Assume that $l$ splits in $K$ and that $M>2$. Let $\mathcal{Y}_M$ be a connected component of $\mathcal{X}_l^q(M)$. Then $\mathcal{Y}_M$ is the double intertwinement of a tectonic volcano graph.* *Proof.* There is a natural map $\pi\colon \mathcal{Y}_M\to \mathcal{X}_l^q$ given by $$(E,\iota, R_1,R_2)\mapsto (E,\iota).$$ Note that this is not a covering of graphs, as the vertices of $\mathcal{Y}_M$ have out-degree $2(l+1)$, while those in $\mathcal{X}_l^q$ have out-degree $(l+1)$. It follows from Theorem [\[thm:arpin\]](#thm:arpin){reference-type="ref" reference="thm:arpin"} that $\mathcal{X}_l^q$ is the union of volcanoes. Let $(E,\iota)$ be a depth zero vertex of $\mathcal{X}_l^q$ and let $v$ be a lift of $(E,\iota)$ in $\mathcal{Y}_M$. For each pair $(E_1,E_2)$ of supersingular elliptic curves, we fix a set of representatives $S(E_1,E_2)$ of $\{\textup{$l$-isogenies from $E_1$ to $E_2$}\}/\{\pm 1\}$. Let $\theta\in \textup{End}(E)$ be $l$-primitive and $l$-suitable for $(E,\iota)$ (such $\theta$ always exists as explained in Remark [Remark 40](#rk:primitive-suitable){reference-type="ref" reference="rk:primitive-suitable"}). Let $\alpha\in K$ be the element given as in Definition [Definition 39](#def:suitable){reference-type="ref" reference="def:suitable"}. It follows from Proposition [Proposition 41](#prop:eigenvalues){reference-type="ref" reference="prop:eigenvalues"}i) that the eigenvalues $\lambda_1$ and $\lambda_2$ of the action of $\theta$ on $E[l]$ are distinct elements of $\mathbb{F}_l$. Let $S_0(E_1,E_2)\subset S(E_1,E_2)$ be the subset of horizontal isogenies. Starting at $v$ and only propagating along the edges given by $S_0(E_1,E_2)$, we obtain a finite connected subgraph $Z_0$ of $\mathcal{Y}_M$. We will show that $Z_0$ is an abstract tectonic crater. Let $(E',\iota',R'_1,R'_2)$ be an arbitrary vertex in $Z_0$. We define $\theta'=\iota'(\alpha)$. By Lemma [Lemma 42](#lemma:horizonal-isog.){reference-type="ref" reference="lemma:horizonal-isog."}i) the eigenvalues of the action of $\theta'$ on $E'[l]$ are $\lambda_1$ and $\lambda_2$. For $1\le i \le 2$, let $P^{E'}_i$ be a $\lambda_i$-eigenvector of the action of $\theta'$ on $E[l]$. Let $$\phi^{E'}_i\colon E'\to E'/\langle P^{E'}_i\rangle$$ be the natural projection. We say that an edge in $Z_0$ is blue (resp. green) if it is induced by $\phi^{E'}_1$ or by $\phi^{E'}_1\circ[-1]$ (resp. by $\phi^{E'}_2$ or $\phi^{E'}_2\circ [-1]$). By the definition of $S_0(E_1,E_2)$, exactly one of the two elements in $\{\phi^{E'}_i,\phi^{E'}_i\circ [-1]\}$ induces an edge in $Z_0$. It follows from Lemma [Lemma 42](#lemma:horizonal-isog.){reference-type="ref" reference="lemma:horizonal-isog."}ii) that blue and green edges commute with each other, i.e. starting at a vertex $v'$ any path consisting of $k_1$ blue and $k_2$ green edges will end at the same vertex $w$ -- no matter in which order one follows green and blue edges. For every vertex $v'$ in $Z_0$ we denote by $\phi_{i,j}(v')$ the vertex at the end of a path consisting of $i$ blue and $j$ green edges that starts at $v'$. By a similar proof as the one given in [@LM1 Lemma 4.4], we have: $$\label{analogon-4-4} \textup{If $\phi_{i,j}(v)=\phi_{i',j'}(v)$, then $\phi_{i,j}(v')=\phi_{i',j'}(v')$ for all $v'\in V(Z_0)$.}$$ By Lemma [Lemma 42](#lemma:horizonal-isog.){reference-type="ref" reference="lemma:horizonal-isog."}iii) there exists a non negative integer $h_1$ (resp. $h_2$) such that a blue (resp. green) path of length $h_1$ (resp. $h_2$) sends $v$ to itself. Note that $\phi_{h_1,0}(v)=\phi_{0,h_2}(v)=v$. Choose now $0<s<h_1$ minimal with the property that there exists an integer $j$ such that $\phi_{s,0}(v)=\phi_{0,j}(v)$. Let $t=\gcd(h_2,j)$ and let $c t=j$. Let now $(s',t',c')$ be a triple of integers with the following properties - $s'\mid h_1$. - $t'\mid h_2$. - $c'$ is coprime to $h_2/t'$. - $\phi_{s',0}(v)=\phi_{0,c't'}(v)$. Using [\[analogon-4-4\]](#analogon-4-4){reference-type="eqref" reference="analogon-4-4"} instead of [@LM1 Proposition 4.4] in the proof of [@LM1 proposition 4.6] we can deduce the following claim: $$\label{analogon-4-6} \textup{There exists an integer $d$ such that $s'=ds$ and $t'=dt$.}$$ Substituting Proposition 4.4 in loc. cit by [\[analogon-4-4\]](#analogon-4-4){reference-type="eqref" reference="analogon-4-4"} and Proposition 4.6 by [\[analogon-4-6\]](#analogon-4-6){reference-type="eqref" reference="analogon-4-6"}, it now follows from the same line of argument as given in [@LM1 Section 4.2] that $Z_0$ is indeed a tectonic volcano. Let $Z$ be the subgraph obtained by starting at $v$ and only propagating along the edges in $S(E_1,E_2)$. Then $Z_0$ is a subgraph of $Z$ and all horizontal edges of $Z$ lie in $Z_0$. By Proposition [Proposition 41](#prop:eigenvalues){reference-type="ref" reference="prop:eigenvalues"}, every vertex in $Z_0$ is the source of exactly $l-1$ descending edges in $Z$. By [@arpin-et-all Proposition 2.15], every vertex in $V(Z)\setminus V(Z_0)$ has one ascending edge and $l-1$ descending ones. Therefore, $Z$ is indeed a tectonic volcano with $Z_0$ as its tectonic crater. The last step of our proof constitutes of showing that $\mathcal{Y}_M$ is the double intertwinement of $Z$. For every vertex $v=(E,\iota, R_1,R_2)$ in $Z$, the definition of $S(E_1,E_2)$ implies that the \"opposite\" vertex $v':=(E,\iota, -R_1,-R_2)$ does not lie in $Z$. We may find a path from $v$ to $v'$ by composing an appropriate power of $[l]$ with $[-1]$. Thus, $v'\in V(\mathcal{Y}_M)\setminus V(Z)$. In particular, we may identify $v$ and $v'$ with $+v$ and $-v$ in $V(Z^{+-})$, respectively. If there is an edge $e$ from $v$ to $w$ in $Z$, there is one from $v$ to $-w$ in $\mathcal{Y}_M$ simply by composing the path from $v$ to $w$ with $[-1]$. Similarly, there are edges from $-v$ to $-w$ and from $-v$ to $w$. Therefore, $e\in \mathbb{E}(Z)$ gives rise to four edges in $\mathcal{Y}_M$, which we may identify with the edges $e^{\bullet\circ}$, $\bullet,\circ\in\{+,-\}$ in $Z^{+-}$. As all edges in $\mathcal{Y}_M$ are given by compositions of the edges in the set $S(E_1,E_2)$ and $[-1]$, we conclude that $\mathcal{Y}_M=Z^{+-}$. ◻ In a similar manner, one can prove: **Theorem 49**. *Assume that $l$ ramifies in $K$ and $M>2$. Then $\mathcal{Y}_M$ is the double intertwinement of a volcano with a connected crater.* **Theorem 50**. *Assume that $l$ is inert in $K$ and $M>2$. Then $\mathcal{Y}_M$ is the double intertwinement of a volcano whose crater is disconnected.* *Remark 51*. If $M=2$, the vertices $(E,\iota,R_1,R_2)$ and $(E,\iota,-R_1,-R_2)$ coincide. Furthermore $\phi$ and $\phi\circ [-1]$ define the same edge. So instead of the double intertwinement of a (tectonic) volcano, the graph $\mathcal{Y}_2$ is derived from a (tectonic) volcano by drawing every edge with multiplicity $2$. **Definition 52**. *We define $X_l^q(M)^{\mathrm{ss}}$ to be the subgraph of $X_l^q(M)$ generated by the vertices arising from supersingular elliptic curves.* **Corollary 53**. *Suppose that $M>2$. Then each connected component of $X_l^q(M)^{\mathrm{ss}}$ is the quotient of the double intertwinement of either a volcano or a tectonic volcano.* *Proof.* This follows from the fact that each connected component of $X_l^q(M)^{\mathrm{ss}}$ is the quotient of certain $\mathcal{Y}_M$ given in Theorems [Theorem 48](#thm:volcano){reference-type="ref" reference="thm:volcano"}, [Theorem 49](#thm:volcano2){reference-type="ref" reference="thm:volcano2"} and [Theorem 50](#thm:volcano3){reference-type="ref" reference="thm:volcano3"}, where the covering map is given by sending the vertex $(E,\iota, R_1,R_2)$ to $(E,R_1,R_2)$. ◻ ## Towers of isogeny graphs of oriented supersingular elliptic curves Let $N$ be a positive integer that is coprime to $pql$. We now turn our attention to \"$\textup{GL}_2(\mathbb{Z}_p)$-towers\" that arise from $\mathcal{X}_l^q(p^nN)$ as $n$ varies, proving Theorem [Theorem 4](#thmD){reference-type="ref" reference="thmD"}. **Theorem 54**. *Assume that $q\equiv1\pmod{12}$, $N=1$ or $2$ and that $X_l^q(p^nN)^{\mathrm{ss}}$ is connected for all $n$. Let $\mathcal{X}_0$ be a connected component of $\mathcal{X}_l^q(N)$ and denote the pre-image of $\mathcal{X}_0$ in $\mathcal{X}_l^q(p^nN)$ by $\mathcal{X}_n$ for $n\ge1$. The covering $\mathcal{X}_n/\mathcal{X}_0$ is Galois with Galois group isomorphic to $\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$.* *Remark 55*. The assumption that $X_l^q(p^nN)^{\mathrm{ss}}$ is connected for all $n$ forces $N\in\{1,2\}$ (see Remark [Remark 19](#cyclic-for-large-N){reference-type="ref" reference="cyclic-for-large-N"}). *Proof.* By an abuse of notation, we denote the restriction of the voltage assignment of $\alpha_n$ to the supersingular vertices by the same notation. We have as before $X(\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z}),\alpha_n)=X_l^q(p^nN)^\mathrm{ss}$. By Corollary [Corollary 53](#cor:quotient-volcano){reference-type="ref" reference="cor:quotient-volcano"} there is a natural projection $$\pi_n\colon \mathcal{X}_n\to X_l^q(p^nN)^\mathrm{ss}.$$ Let $\alpha'_n=\alpha_n\circ \pi_0$. By an argument similar to the one employed in the proof of Theorem [Theorem 9](#thm:iso){reference-type="ref" reference="thm:iso"}, we can identify the graph $\mathcal{X}_n$ with $X(\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z}),\alpha'_n)$. Thus, there is an embedding $$\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})\hookrightarrow \textup{Deck}(\mathcal{X}_n/\mathcal{X}_0).$$ Hence, the theorem would follow from the surjectivity of this embedding, which we show below. The projection $\pi$ induces a natural projection $$\pi'_n\colon \textup{Deck}(\mathcal{X}_n/\mathcal{X}_0)\to \textup{Deck}\left(X_l^q(p^nN)^\mathrm{ss}/X_l^q(N)^\mathrm{ss}\right)\cong \textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z}),$$ where the last isomorphism follows from Corollary [Corollary 20](#cor:connected){reference-type="ref" reference="cor:connected"}. Therefore, it suffices to show that $\pi'_n$ is injective. Let $\sigma\in\ker(\pi'_n)$. Let $v=(E,\iota,R_1,R_2,P,Q)\in V(\mathcal{X}_n)$. As $\sigma$ is a deck transformation, $\sigma(v)=(E,\iota,R_1,R_2,P,'Q')$ for some basis $\{P',Q'\}$ of $E[p^n]$. Furthermore, $\sigma\in\ker(\pi'_n)$ means that $$(E,R_1,R_2,P',Q')=\pi_n(\sigma(v))=\pi'_n(\sigma)\pi_n(v)=(E,R_1,R_2,P,Q).$$ It follows that $P=P'$ and $Q=Q'$. Thus, $\sigma(v)=v$. Since this holds for any choice of $v$, the deck transformation $\sigma$ is trivial. Thus, $\pi'_n$ is injective and $\textup{Deck}(\mathcal{X}_n/\mathcal{X}_0)\cong\textup{GL}_2(\mathbb{Z}/p^n\mathbb{Z})$ as desired. ◻

arxiv_math

--- abstract: | Suppose that $S \subseteq [n]^2$ contains no three points of the form $(x,y), (x,y+\delta), (x+\delta,y')$, where $\delta \neq 0$. How big can $S$ be? Trivially, $n \le |S| \le n^2$. Slight improvements on these bounds are obtained from Shkredov's upper bound for the corners problem [@shkredov2006generalization], which shows that $|S| \le O(n^2/(\log \log n)^c)$ for some small $c > 0$, and a construction due to Petrov [@petrovanswer], which shows that $|S| \ge \Omega(n \log n/\sqrt{\log \log n})$. Could it be that for all $\varepsilon > 0$, $|S| \le O(n^{1+\varepsilon})$? We show that if so, this would rule out obtaining $\omega = 2$ using a large family of abelian groups in the group--theoretic framework of [@cu2003; @cohn2005group] (which is known to capture the best bounds on $\omega$ to date), for which no barriers are currently known. Furthermore, an upper bound of $O(n^{4/3 - \varepsilon})$ for any fixed $\varepsilon > 0$ would rule out a conjectured approach to obtain $\omega = 2$ of [@cohn2005group]. Along the way, we encounter several problems that have much stronger constraints and that would already have these implications. author: - Kevin Pratt bibliography: - refs.bib title: On generalized corners and matrix multiplication --- # Introduction The exponent of matrix multiplication $\omega$ is the smallest number such that for any $\varepsilon > 0$, there exists an algorithm for multiplying $n \times n$ matrices using $O(n^{\omega + \varepsilon})$ arithmetic operations. Since Strassen's initial discovery that $\omega < 3$ [@strassen1969gaussian], there has been much work on understanding this fundamental constant, with the end goal being the determination of whether or not $\omega = 2$. It is currently known that $2 \le \omega <2.3716$ [@williams2023new]. The best upper bounds on $\omega$ obtained since 1987 [@strassen1987relative] can be understood as solutions to the following hypergraph packing problem. Let $M_n$ be the *matrix multiplication hypergraph*, the tripartite 3-uniform hypergraph with parts $X_1 = X_2 = X_3 = [n]^2$, and where $((i,j), (k,l),(m,n)) \in X_1 \times X_2 \times X_3$ is a hyperedge if and only if $j=k, l=m, n=i$. Given an abelian group $G$, let $X_G$ be its "addition hypergraph\" with vertex sets $G \sqcup G \sqcup G$, and where $(a_1,a_2,a_3) \in G \times G \times G$ is a hyperedge exactly when $a_1+a_2+a_3=0$. Suppose that $X_G$ contains $k$ disjoint induced copies of $M_n$. Then $$\label{eqn:stppbd} \omega < \log_n(|G|/k).$$ Phrased in terms of the group--theoretic approach proposed by Cohn and Umans [@cu2003] and further developed by Cohn, Kleinberg, Szegedy, and Umans [@cohn2005group], this is equivalent to proving upper bounds on $\omega$ via *simultaneous triple product property* (STPP) constructions in abelian groups. The above inequality was established in [@cohn2005group Theorem 5.5]. It can be also be deduced via the *asymptotic sum inequality* of [@schonhage1981partial]. From this perspective, the best bounds on $\omega$ to date are obtained by taking $G$ to be a large power of a cyclic group --- specifically, ${\mathbb{Z}}_7^\ell$ with $\ell \to \infty$. However, in [@blasiak2017cap] ideas related to the resolution of the cap-set problem in additive combinatorics [@ellenberg2017large] were used to show that one cannot obtain $\omega = 2$ using groups of *bounded exponent* --- such as ${\mathbb{Z}}_7^\ell$ --- via this approach. This obstruction is due to the fact that when $G$ has bounded exponent, there is power-savings over the trivial upper bound on the size of the largest induced matching in $X_G$ (also called a *3-matching* [@sawin2018bounds], or a *tricolored sum-free set* [@blasiak2017cap]). For example, when $G = \mathbb{Z}_7^\ell$ the largest induced matching has size at most $O(6.16^\ell)$. On the other hand, $M_n$ contains an induced matching of size $n^{2-o(1)}$: if we identify vertices in $M_n$ with edges in the complete tripartite graph $K_{n,n,n}$, an induced matching in $M_n$ corresponds to a tripartite graph on at most $3n$ vertices where every edge is contained in a unique triangle, and the number of vertices in the induced matching equals the number of edges in this graph. A well-known construction in extremal combinatorics yields such a graph with $n^{2-o(1)}$ edges (see [@zhao2023graph Corollary 2.5.2])[^1] and hence $M_n$ contains an induced matching of size $n^{2-o(1)}$. Modulo minor details, the claimed barrier then follows, as an efficient packing of copies of $M_n$ into $X_G$ would imply the existence of a large induced matching in $X_G$, a contradiction.[^2] This is the only obstruction to obtaining $\omega = 2$ via the use of that we are aware of. Unfortunately,[^3] this barrier says nothing about the viability of general abelian groups, as their addition hypergraphs may contain large induced matchings. For example, if $A$ is a 3-term arithmetic progression free (hereon abbreviated to 3AP-free) subset of $G$, then the subsets $A,A,-2A$ of the vertex sets of $X_G$ induce a matching of size $|A|$. Hence this barrier cannot apply to any group containing a 3AP-free subset of size $|G|^{1-o(1)}$, such as $\mathbb{Z}_n$ [@behrend1946sets]. Could one achieve $\omega = 2$ using cyclic groups, or perhaps products of cyclic groups of growing orders? In this paper we identify problems in additive combinatorics whose answer we conjecture would rule out obtaining $\omega = 2$ using a large family of abelian groups for which the induced matching barrier is irrelevant. This family includes abelian groups with a bounded number of direct factors --- the "opposite\" condition of that of having bounded exponent. These problems have not been studied before as far as we are aware. Aside from their connections to fast matrix multiplication, we find them intrinsically interesting. We now discuss the simplest-to-state such problem. ## A skew corners problem The *corners problem* in additive combinatorics asks for the size of the largest subset of $[n]^2$ containing no three points of the form $$(x, y), (x,y+\delta),(x+\delta,y)$$ where $\delta \neq 0$. Ajtai and Szemerédi [@ajtai1974sets] settled this problem up to factors of $n^{o(1)}$ by proving an upper bound of $o(n^2)$ and a lower bound of $n^{2-o(1)}$. This problem is significant as it was the first multidimensional case of Szemerédi's theorem to be established, and for its application to the number-on-forehead model in communication complexity [@chandra1983multi]. Here is a subtle strengthening of the condition of the corners problem for which we know essentially nothing: **Question 1**. What is the size of the largest $S \subseteq [n]^2$ which does not contain three points of the form $$(x,y), (x, y+\delta), (x + \delta, y')$$ with $\delta \neq 0$? That is, not only must $S$ avoid all corners, but given any two points in $S$ lying on the same vertical line, the *entire vertical line* passing through the third point that would form a corner with these two points must be absent from $S$! Naturally, we call such a set of points *skew corner-free*. See for an example of such a set. ![The orange points form a skew corner-free subset of $[10] \times [10]$ of size $24$. This is largest possible.](grid.png){#fig:skew10} Note that there is a trivial lower bound of $n$, obtained by taking $S$ to be all points lying on a vertical or horizontal line. We conjecture that this is almost optimal: **Conjecture 2**. *Fix any $\varepsilon > 0$. If $S$ is skew corner-free, then $|S| \le O(n^{1+\varepsilon})$.* A construction due to Petrov [@petrovanswer] () shows that one can have $|S| \ge \Omega(n \log n / \sqrt{\log \log n})$. On the other hand, the best upper bound we know is $O(n^2/(\log \log n)^{ 0.0137 \cdots})$, which follows immediately from Shkredov's upper bound on the corners problem [@shkredov2006generalization]. Two of the main results of this paper are the following. theoremmain1 [\[thm:main1\]]{#thm:main1 label="thm:main1"} If is true, then one cannot obtain $\omega = 2$ via STPP constructions in the family of groups $\mathbb{Z}_q^\ell$, where $q$ is a prime power. Furthermore, a weakening of would rule out obtaining $\omega = 2$ using a specific type of STPP construction in arbitrary abelian groups. In [@cohn2005group], it was conjectured that this type of construction can be used to obtain $\omega = 2$. theoremmain2 [\[thm:main2\]]{#thm:main2 label="thm:main2"} If the largest skew corner-free subset of $[n]^2$ has size $O(n^{4/3 - \varepsilon})$ for some $\varepsilon > 0$, then [@cohn2005group Conjecture 4.7] is false. In fact, seemingly much weaker conjectures than would already have these implications. The weakest conjecture we make is the following. Let $\Delta_n$ be a triangular array of $n(n+1)/2$ points. Suppose that we delete from $\Delta_n$ sets of points lying on lines parallel to the sides of this array, such that the remaining set of points does not contain any equilateral trapezoid with sides parallel to the sides of the array (see ). For example, we might delete all lines in one direction but one. Then, what is the maximum number of points that can remain? By our example, one can achieve at least $n$. We conjecture that this is essentially optimal (). Another condition we introduce, which is intermediate between this and being skew-corner free, is that of a skew corner-free subset of a triangular grid (see ). ## Paper overview In we review the group--theoretic approach of [@cu2003; @cohn2005group]. In we record a very weak lower bound for this approach, which follows easily from the removal lemma in groups of [@serra2009combinatorial]. This lower bound becomes much stronger in $\mathbb{Z}_q^\ell$ (), thanks to the improved bounds on the removal lemma of [@fox2017tight], and we make later use of this fact. In we note that the matrix multiplication hypergraph $M_n$ is an extremal solution to a certain forbidden hypergraph problem. This was our motivating observation. We define the "value\" of a group, $\mathrm{val}(G)$, which captures this forbidden hypergraph problem in a group--theoretic context. This quantity equals the maximum number of triangles in an induced subhypergraph of $X_G$ that does not contain the triforce hypergraph or a cycle of 4 triangles (see ). This can also be expressed in terms of the group operation slightly awkwardly (). The trivial bounds are that $|G| \le \mathrm{val}(G) \le |G|^{3/2}$; using the removal lemma of [@serra2009combinatorial], the upper bound can be improved to $o(|G|^{3/2})$ (). STPP constructions yield lower bounds on the quantity $\mathrm{val}(G)$ (), so ultimately it is upper bounds on $\mathrm{val}(G)$ that we are interested in as a means towards barriers. The quantity $\mathrm{val}(G)$ is super-multiplicative under direct product (), which is one reason why power-improvements over the trivial bound seem to be easier to obtain in direct products of groups. We then focus on the case of abelian groups in . We show that a bound of $\omega = 2$ using the family of groups ${\mathbb{Z}}_q^\ell$ would imply that $\mathrm{val}({\mathbb{Z}}_n) \ge \Omega(n^{1+c})$ for some $c>0$ . We also show that a proof of $\omega = 2$ via *simultaneous double product property* constructions [@cohn2005group] in any family of abelian groups would imply that $\mathrm{val}({\mathbb{Z}}_n) \ge \Omega(n^{4/3 - \varepsilon})$ for any given $\varepsilon>0$ (). We thank Chris Umans for mentioning a related fact to us, which motivated this result. We then relate $\mathrm{val}({\mathbb{Z}}_n)$ to various questions about sets of points in the plane, including (). This gives . We also give an example which shows that one cannot hope to prove strong upper bounds on $\mathrm{val}({\mathbb{Z}}_n)$ via a certain "asymmetric\" averaging argument (). The take-away of this paper is that STPP constructions yield subsets of $G \times G$ which satisfy dramatically stronger properties than that of being corner-free. While subsets satisfying these stronger properties do not imply STPP constructions in any obvious way, we believe that understanding them will be a stepping stone to understanding the power of the group--theoretic approach, and possibly towards improved upper bounds on $\omega$. # Background {#sec:bg} Bounds on $\omega$ from the group--theoretic approach are obtained by designing subsets of groups satisfying the following condition. **Definition 3**. A collection of triples of subsets $S_i, T_i, U_i$ of a group $G$ satisfy the simultaneous triple product property (or STPP for short) if 1. For each $i$, the sets $S_i, T_i, U_i$ satisfy the *triple product property*: if $ss'^{-1}tt'^{-1}uu'^{-1}=I$ with $s, s' \in S_i, t,t' \in T_i, u,u' \in U_i$, then $s=s', t=t', u=u'$. 2. Setting $S_i = A_iB_i^{-1}, T_j = B_jC_j^{-1}, U_k = C_kA_k^{-1}$, $$s_it_ju_k = I \iff i=j=k$$ for all $s_i \in S_i, t_j \in T_j, u_k \in U_k$. The crucial fact is the following: **Theorem 4**. *[@cohn2005group Theorem 5.5] If $S_i, T_i, U_i \subseteq G$ satisfy the STPP, then $$\sum_i (|S_i||T_i||U_i|)^{\omega/3} \le \sum d_i^\omega$$ where $d_i$'s are the dimensions of the irreducible representations of $G$.* The conditions of the STPP imply that the sets involved satisfy a simple "packing bound\" (see the discussion preceding [@blasiak2017cap Definition 2.3]). **Proposition 5**. *If $S_i, T_i, U_i$ satisfy the STPP in a group $G$, then $\sum_i |S_i||T_i| \le |G|$, $\sum_i |T_i||U_i| \le |G|$, and $\sum_i |U_i||S_i| \le |G|$.* A particular type of STPP construction can be obtained from pairs of sets satisfying a condition termed the *simultaneous double product property* in [@cohn2005group]. **Definition 6**. We say that sets $(A_i, B_i)_{i=1}^n$ satisfy the simultaneous double product property (or SDPP for short) if 1. For all $i$, $aa'^{-1} = bb'^{-1}$ only has the solution $a=a', b=b'$ for $a,a' \in A_i, b,b' \in B_i$, 2. $a_i (a_j')^{-1}b_j(b_k')^{-1} = 1$ implies $i=k$, where $a_i \in A_i, a_j' \in A_j, b_j \in B_j, b_k' \in B_k$. In [@cohn2005group] it was conjectured that one can achieve $\omega = 2$ using SDPP constructions in abelian groups. This amounts to the following. **Conjecture 7**. *[@cohn2005group Conjecture 4.7][\[conj:twofamilies\]]{#conj:twofamilies label="conj:twofamilies"} For arbitrarily large $n$, there exists an abelian group $G$ of order $n^{2-o(1)}$ and $n$ pairs of sets $A_i, B_i$ where $|A_i||B_i| > n^{2-o(1)}$ satisfy the SDPP.* If $G$ is a finite group, we let $X_G$ denote the tripartite 3-uniform hypergraph with vertex parts $X_1 = X_2 = X_3 = G$, and where $(g_1, g_2, g_3)$ is a hyperedge (a *triangle*) whenever $g_1g_2g_3=I$. In the event that $G$ is nonabelian, it is important that we fix some ordering on the parts of $X_G$ here. Recall that a 3-uniform hypergraph is said to be *linear* if any two vertices are contained in at most one hyperedge. For example, $X_G$ is linear. The matrix multiplication hypergraph $M_{p,q,r}$ is defined to be the supporting hypergraph of the matrix multiplication tensor; i.e. it is the hypergraph with parts $[p]\times [q], [q] \times [r], [r] \times [p]$, and where $((i,j), (k,l), (m,n))$ is a hyperedge if and only if $j=k, l=m, n=i$. If $X$ is a hypergraph, we sometimes write $E(X)$ for the set of hyperedges of $X$. It is convenient to view STPP constructions from a hypergraph perspective. **Proposition 8**. *There exist sets $S_i, T_i, U_i \subseteq G$, satisfying the STPP if and only if $X_G$ contains as an induced subhypergraph the disjoint union of $M_{|S_i|, |T_i|, |U_i|}$.* *Proof.* It follows from the first condition of the STPP that for all $i$, the subhypergraph induced by $A_{i} := S_iT_i^{-1}, B_{i} := T_iU_i^{-1} , C_{i} := U_iS_i^{-1}$ equals $M_{|S_i|, |T_i|, |U_i|}$. The second condition implies that $A_{i}$ and $A_{j}$ are disjoint when $i \neq j$, and similarly for the subsets of the other parts. The second condition also implies that the only hyperedges in the subhypergraph induced by $\sqcup_i A_{i}, \sqcup_i B_{i},\sqcup_i C_{i}$ are between sets of the form $A_{i}, B_{i}, C_{i}$, so the claim follows. Conversely, suppose that $\sqcup_i A_{i}, \sqcup_i B_{i}, \sqcup_i C_{i}$ induce disjoint hypergraphs $M_{p_i, q_i, r_i}$. Fix some $i$, and for shorthand write $A:=A_{i}, B:=B_{i}, C:=C_{i}$ and let $p :=p_i, q:=q_i, r:=r_i$. Since $A,B,C$ induce $M_{p,q,r}$, we can by definition write $A =\{a_{ij}\}_{i \in [p],j \in [q]}, B = \{b_{ij}\}_{i \in [q],j \in [r]}, C = \{c_{ij}\}_{i \in [r],j \in [p]} \in G$ where $$\label{eqn:tppgen} a_{ij}b_{kl}c_{mn} = I \iff j=k, l=m, n=i.$$ We claim that there exist $X = \{x_i\}_{i \in [p]}, Y = \{y_j\}_{j \in [q]}, Z = \{z_k\}_{k \in [r]}$ such that $a_{ij} = x_iy_j^{-1}$, $b_{jk} = y_jz_k^{-1}$, $c_{ki} = z_kx_i^{-1}$ for all $i \in [p], j \in [q], k \in [r]$. This can be accomplished by taking $x_0 = 1, x_i = a_{i0}a_{00}^{-1}$ for $i>0$, $y_i = a_{0i}^{-1}, z_i = c_{i0}$. Furthermore, implies that $X,Y,Z$ will satisfy the TPP. This shows that for each $i$ there are $X_i, Y_i, Z_i$ such that $A_{1,i} = X_iY_i^{-1}, A_{2,i} = Y_iZ_i^{-1}, A_{3,i} = Z_iX_i^{-1}$, and $X_i, Y_i, Z_i$ satisfy the TPP. The fact that they induce a disjoint union of hypergraphs implies that if $a \in A_{i}, b \in B_j, c \in C_k$, then $abc=I$ implies $i=j=k$, which implies the second condition in the definition of the STPP. ◻ **Remark 9**. The second direction of this proposition is essentially the fact that a complete 2-dimensional simplicial complex has trivial 1-cohomology with coefficients in any group. ## Triangle Removal and the Group-Theoretic approach {#sec:trirem} In [@serra2009combinatorial], a nonabelian generalization of Green's arithmetic removal lemma [@green2005szemeredi] was shown to follow from the directed graph removal lemma of Alon and Shapira [@alon2003testing]. Specifically, they showed the following: **Theorem 10**. *Let $G$ be a finite group of order $N$. Let $A_1, \ldots, A_m, m \ge 2$, be sets of elements of $G$ and let $g$ be an arbitrary element of $G$. If the equation $x_1x_2 \cdots x_m = g$ has $o(N^{m-1})$ solutions with $x_i \in A_i$, then there are subsets $A'_i \subseteq A_i$ with $|A_i \setminus A'_i| = o(N)$ such that there is no solution of the equation $x_1x_2 \cdots x_m = g$ with $x_i \in A_i'$.* The best quantitative bounds for this theorem are due to Fox [@fox2011new], and imply that if there are at most $\delta N^{m-1}$ solutions to $x_1 \cdots x_m = g$, one can remove subsets of $A_i$ of size $\varepsilon N$ and eliminate all solutions, when $\delta^{-1}$ is a tower of twos of height $O(\log \varepsilon^{-1})$. implies the following. **Corollary 11**. *If $X_i,Y_i,Z_i$ satisfy the STPP in a group $G$ of order $n$, then at least one of $\sum |X_i||Y_i|, \sum |X_i||Z_i|, \sum |Y_i||Z_i|$ is at most $o(n)$.* *Proof.* Let $A_1 = \sqcup_i X_iY_i^{-1}, A_2 = \sqcup_i Y_iZ_i^{-1}, A_3 = \sqcup_i Z_iX_i^{-1}$. By definition of the STPP, the equation $x_1x_2x_3=I$ with $x_i \in A_i$ has $\sum_i |X_i||Y_i||Z_i|$ solutions. By the packing bound , $\sum_i |X_i||Y_i| \le n, \sum_i |Y_i||Z_i| \le n, \sum_i |Z_i||X_i| \le n$, so by Cauchy--Schwarz there are at most $n^{3/2} = o(n^2)$ solutions to $a_1a_2a_3=I$. Now suppose that $B_j \subseteq A_j$ satisfy $|B_j|/|A_j| > 0.9999$; we will show that there is a solution to $b_1b_2b_3=I$. For more than a $0.99$ fraction of the values of $i$ we must have $|B_1 \cap X_iY_i^{-1} |/|X_iY_i^{-1}| > 0.99$ (because $0.99 \cdot 1 + 0.01 \cdot 0.99 = 0.9999$) and similarly for the other sets. Hence by the pigeonhole principle there is some $i$ for which $|B_1 \cap X_iY_i^{-1}|/|X_iY_i^{-1}| > 0.99, |B_2 \cap Y_iZ_i^{-1}|/| Y_iZ_i^{-1}| > 0.99, |B_3 \cap Z_iX_i^{-1}|/| Y_iZ_i^{-1}| > 0.99.$ Now consider the tripartite graph with parts $X_i, Y_i, Z_i$, where $(x,y)$ is an edge between $X_i$ and $Y_i$ if $xy^{-1} \in B_1 \cap X_iY_i^{-1}$, $(y,z)$ is an edge between $Y_I, Z_i$ when $yz^{-1} \in B_2 \cap Y_iZ_i^{-1}$, and $(z,x)$ is an edge when $zx^{-1} \in B_3 \cap Z_iX_i^{-1}$. Note that the existence of a triangle in this graph implies that there is a solution to $b_1b_2b_3=I$. First, note that at least $0.9|X_i|$ vertices in $X_i$ have at least $0.9|Y_i|$ neighbors in $Y_i$. (If this were not the case, there would be at most $0.9|X_i||Y_i| + 0.1 \cdot 0.9 \cdot |X_i||Y_i| \le 0.99|X_i||Y_i|$ edges between $X_i$ and $Y_i$, and hence $|B_1 \cap X_iY_i^{-1} |/|X_iY_i^{-1}| \le 0.99$, a contradiction.) Similarly, at least $0.9|X_i|$ vertices in $X_i$ have at least $0.9|Z_i|$ neighbors in $Z_i$. Hence at least $0.8|X_i|$ vertices in $X_i$ have $0.9|Y_i|$ neighbors in $Y_i$ and $0.9|Z_i|$ neighbors in $Z_i$. Pick any such vertex $x_0 \in X_i$. There must be an edge between a neighbor of $x_0$ in $Y_i$ and a neighbor of $x_0$ in $Z_i$, since if not, there would be at most $|Y_i||Z_i| - 0.9^2|Y_i||Z_i| = 0.19|Y_i||Z_i|$ edges between $Y_i$ and $Z_i$. Thus we have found our triangle. By , we can delete subsets of $A_i$ of size $o(n)$ to eliminate all solutions to $x_1x_2x_3=I$. On the other hand, any three subsets of the $A_i$'s of density $0.9999$ contain some such solution. Hence we must have $|A_i| = o(n)$ for some $i$. ◻ As a corollary of this proof, we have the following. **Corollary 12**. *There exists an absolute constant $C > 1$ such that if $X_i,Y_i,Z_i$ satisfy the STPP in ${\mathbb{Z}}_q^\ell$, then at least one of $\sum |X_i||Y_i|, \sum |X_i||Z_i|, \sum |Y_i||Z_i|$ is at most $(q/C)^\ell$.* *Proof.* The proof of shows that $A_1 = \sqcup_i X_iY_i^{-1}, A_2 = \sqcup_i Y_iZ_i^{-1}, A_3 = \sqcup_i Z_iX_i^{-1}$ have the following properties: there are at most $q^{3n/2}$ solutions to $a_1+a_2+a_3=0$, and any subsets of $A_1, A_2, A_3$ of density $0.9999$ each contain some such solution. At the same time, by [@fox2017tight Theorem 1], if $A_1, A_2, A_3 \subseteq {\mathbb{Z}}_q^\ell$ and there are less than $\delta q^{2n}$ solutions to $a_1+a_2+a_3=0$, then we may remove $\varepsilon q^\ell$ elements from $A_1 \cup A_2 \cup A_3$ and eliminate all solutions, when $\delta = (\varepsilon/3)^{\Theta(\log q)}$.[^4] In our setting, $\delta = q^{-n/2}$ and so $\varepsilon = 3 q^{\Theta(-n/\log q)} \le 3 C'^{-n}$ for some universal $C'$. Hence it must have been the case that one of $A_1, A_2, A_3$ had size at most $(q/C)^\ell$ to begin with, for some universal $C$. ◻ One can interpret as saying that the best upper bound on the rank of a direct sum of matrix multiplication tensors provable via the group--theoretic approach is superlinear. We remark the only important property of the matrix multiplication hypergraph for this result was that it satisfies a very weak "regularity\" condition. Specifically, considerations similar to those of the proof of show the following: **Theorem 13**. *Let $\varepsilon > 0$. Let $G$ be a group of order $n$. Let $X = \sqcup_{i=1}^3 A_i$ be a tripartite hypergraph with $o(n^2)$ triangles such that for any $Y_i \subseteq A_i$ with $|Y_i|/n \ge 1-\varepsilon$, there exists $y_i \in Y_i$ such that $(y_1, y_2, y_3) \in E(X)$. Then if $X$ is an induced subhypergraph of $X_G$, $|A_i| \le o(n)$ for $i=1,2,3$.* # Equilateral trapezoid-freeness in hypergraphs and groups {#sec:hyper} We begin with the observation that the matrix multiplication hypergraph is an extremal solution to a certain forbidden hypergraph problem. **Proposition 14**. *Let $X$ be a linear tripartite hypergraph with parts of size $N$ such that any two vertices from different parts are incident to at most one common vertex in the third part. Then the number of triangles in $X$ is at most $N^{3/2}$. Furthermore, when $N$ is a square, an extremal example is the matrix multiplication hypergraph $M_{N^{1/2}}$.* The hypergraphs satisfying the condition of can be equivalently characterized as the linear hypergraphs that do not contain copies of the hypergraphs in . We remark that the proof of the upper bound in is closely related to the upper bound on the Turán density of the 4-cycle. {#fig:forbid} *Proof.* Restricting our attention to one of the parts $X_1$ of $X$, let $d_v$ be the number of triangles that vertex $v \in X_1$ is contained in. Each $v \in X_1$ is contained in $d_v$ triangles, where the vertices of these triangles belonging to $X_2$ and $X_3$ are distinct (as $X$ is linear). Additionally, no pair of such vertices in $X_2$ and $X_3$ can be contained in a triangle incident to another vertex $u \in X_1$, so there are $2 \binom{d_v}{2}$ pairs of vertices in $X_2$ and $X_3$ that are contained in no common triangle. Let $(x_2, x_3)$ be some such pair of vertices. Observe that furthermore, for all $u \neq v \in X_1$, the set of vertices in $X_2$ and $X_3$ incident to the set of triangles containing $u$ cannot also contain both $x_2$ and $x_3$. For if this happened, there would be triangles $(v,x_2,x_3'), (v, x_2', x_3), (u, x_2, x_3''), (u, x_2'', x_3)$, and then $x_2$ and $x_3$ violate the constraint. The total number of triangles equals $m := \sum_{v \in X_1} d_v$, and by the prior observations it follows that $\sum 2\binom{d_v}{2} +m \le N^2$. So $\sum d_v(d_v-1) + m = \sum d_v^2 \le N^2$. The conclusion follows from Cauchy--Schwarz. To see that $M_{N^{1/2}}$ is extremal, note that it contains $N^{3/2}$ triangles, has parts of size $N$, and is linear. To see that it satisfies the second condition, let $(i,j)$ be a vertex in the first part, and let $(k,l)$ be a vertex in the second part. Then $(i,j)$ is contained in a common triangle with exactly the vertices in the third part of the form $(*,i)$, and $(k,l)$ is incident to exactly the vertices in the third part of the form $(l,*)$. Hence $(l,i)$ is the unique neighbor of both. The same argument shows the claim for vertices in any two parts. ◻ The key definition in this paper is that of an "equilateral trapezoid-free\" triple of subsets of a group. The reason for this name will eventually be explained in . **Definition 15**. Let $A, B, C \subseteq G$. We call $(A,B,C)$ equilateral trapezoid-free if the subhypergraph of $X_G$ induced by $A \subseteq X_1,B\subseteq X_2,C \subseteq X_3$ satisfies the conditions of . Equivalently, $(A,B,C)$ is equilateral trapezoid-free if for any fixed $a' \in A , b' \in B , c'\in C$, the following systems of equations in the variables $a \in A, b \in B, c \in C$ each have at most one solution: $$\begin{aligned} I&=a'bc=ab'c,\\ I&=a'bc=abc',\\ I&=ab'c=abc'. \end{aligned}$$ Let $\mathrm{val}(G)$ be the maximum number of solutions to $abc=I$ over all equilateral trapezoid-free triples $(A,B,C)$. The relevance of $\mathrm{val}(G)$ to $\omega$ is due to the following. **Proposition 16**. *Suppose that $X_G$ contains disjoint induced subhypergraphs $M_{n_i,m_i,p_i}$. Then, $\mathrm{val}(G) \ge \sum_{i} n_im_ip_i$.* *Proof.* By the same reasoning as in the second part of the proof of , $M_{n_i, m_i, p_i}$ satisfies the constraints of and contains $n_im_ip_i$ hyperedges. As the disjoint union of these hypergraphs satisfies these constraints as well, the claim follows. ◻ In fact, STPP constructions are essentially the only approach we know of for proving lower bounds on $\mathrm{val}(G)$. To start, we have the following trivial bounds. **Proposition 17**. *For any group $G$, $|G| \le \mathrm{val}(G) \le |G|^{3/2}$.* *Proof.* The lower bound is obtained by the triple $(\{I\}, G, G)$. The upper bound follows from . ◻ The following super-multiplicative behavior of $\mathrm{val}$ is easily checked. **Proposition 18**. *If $(A,B,C)$ is equilateral trapezoid-free in $G$, and $(A',B',C')$ is equilateral trapezoid-free in $H$, then $(A \times A', B \times B', C \times C')$ is a equilateral trapezoid-free in $G \times H$.* It is also easily seen that being equilateral trapezoid-free is preserved by cyclic permutations of the three sets. **Proposition 19**. *If $(A,B,C)$ is equilateral trapezoid-free, then so is $(B,C,A)$.* By an application of combined with the observation that near-extremal solutions to are highly "regular\", we have the following weak improvement to the trivial upper bound of $|G|^{3/2}$. **Proposition 20**. *For any group $G$, $\mathrm{val}(G) \le o(|G|^{3/2})$.* *Proof.* Suppose for contradiction that there exists $\varepsilon_0 > 0$ such that $\mathrm{val}(G) > \varepsilon_0 |G|^{3/2}$, and let $A_0,B_0,C_0 \subseteq G$ witness $\mathrm{val}(G) = \varepsilon_0 |G|^{3/2}$. Next consider the triple $(A,B,C) := (A_0 \times B_0 \times C_0, B_0 \times C_0 \times A_0, C_0 \times A_0 \times B_0)$, which is equilateral-trapezoid free inside of $H:=G^3$ by and , and witnesses $\mathrm{val}(H) \ge \varepsilon |H|^{3/2}$ where $\varepsilon := \varepsilon_0^3$. Let $|H| = N$. Let $X$ be the tripartite hypergraph with parts $A,B,C$ and where there is a triangle between all triples $(a,b,c)$ where $abc=I$. Let $n:=|A| = |B| = |C|$. By we must have $n \ge \varepsilon^{2/3} N$. Note that the number of triangles in $X$ equals $\varepsilon N^{3/2} \ge \varepsilon n^{3/2}$. In what follows, the degree of a vertex in $X$ refers to the number of triangles containing it. Let $Y$ be the random variable that is uniformly distributed over the multiset of vertex degrees from one part of $X$, say $A$. Then ${\mathbb{E}}[Y] \ge \varepsilon n^{1/2}$ and ${\mathbb{E}}[Y^2] \le n$ (this second inequality follows from the use of Cauchy--Schwarz in the proof of ). By the Payley-Zygmund inequality, for any $\theta>0$, $\Pr(Y > \theta \cdot \varepsilon n^{1/2} ) \ge (1-\theta^2)\varepsilon^2$. Taking $\theta = 1/2$, we conclude that at least $p \cdot n := 3n\varepsilon^2/4$ vertices in $A$ have degree at least $\varepsilon n^{1/2} /2$. This holds for $B$ and $C$ as well. Now let $S, T,$ and $U$ be any subsets of $A,B,C$ of size at least $n(1-p/\lambda)$; we'll pick $\lambda \in {\mathbb{N}}$ later. Then the number of triangles incident to any one of these sets, say $S$, is at least $$np(1-\lambda^{-1}) \cdot \varepsilon n^{1/2} /2 = (3/8) n^{3/2} \varepsilon^3 (1-\lambda^{-1}),$$ and the number of triangles incident to $[n] \setminus T$ or $[n] \setminus U$, sets of size at most $np/\lambda$, is at most $$(n^2 \cdot np/\lambda)^{1/2} = (3^{1/2}/2) n^{3/2} \varepsilon \lambda^{-1/2}$$ by Cauchy--Schwarz. It follows that the number of triangles with one vertex in each of $S,T,U$ is at least $$(3/8) n^{3/2} \varepsilon^3 (1-\lambda^{-1}) - 2 \cdot (3^{1/2}/2) n^{3/2} \varepsilon \lambda^{-1/2}$$ which is greater than 1 for $\lambda \gg \varepsilon^{-4}$. In summary, between any three subsets of $A,B,C$ size roughly $n(1-\varepsilon^6)$, there is a triangle. Recall that $n \ge \varepsilon^{2/3} N$. Since $X$ has at most $N^{3/2} \le o(N^2)$ triangles, by we can remove $o(N) = o(n)$ vertices to remove all triangles. But by what we have just shown, after deleting this few vertices some triangle will remain, a contradiction. ◻ **Remark 21**. By combining this proof with [@fox2017tight], it follows that for fixed $n$ and some $\varepsilon > 0$, $\mathrm{val}({\mathbb{Z}}_n^\ell) \le O(n^{3/2(1-\varepsilon)\ell})$. # $\mathrm{val}({\mathbb{Z}}_n)$ and its applications {#sec:zn} Our weakest conjecture is the following. **Conjecture 22**. *For all $\varepsilon > 0$, $\mathrm{val}({\mathbb{Z}}_n) \le O(n^{1+\varepsilon})$.* In this section we give our potential applications of this conjecture. We then introduce several related quantities and make preliminary progress on understanding them. While the quantity $\mathrm{val}({\mathbb{Z}}_n)$ may seem opaque from , it can easily be visualized. This is done by first considering the natural notion of an equilateral trapezoid-free subset of the plane, which is convenient to introduce sooner rather than later. Throughout this section, we let $\Delta_{n+1} = \{(a,b,c) \in {\mathbb{Z}}_{\ge 0}^3: a+b+c=n\}$. A subset of $\Delta_{n+1}$ is said to be corner-free if it contains no configuration $(x+\delta, y, z), (x,y+\delta,z), (x,y,z+\delta)$. **Definition 23**. Let $A, B, C \subseteq \{0, \ldots, n\}$. We call $(A,B,C)$ an equilateral trapezoid-free triple if for any fixed $a', b', c'$, the following systems of equations in the variables $a \in A, b \in B, c \in C$ each have at most one solution: $$\begin{aligned} n&=a'+b+c=a+b'+c\\ n&=a'+b+c=a+b+c'\\ n&=a+b'+c=a+b+c'. \end{aligned}$$ Let $\mathrm{val}(n)$ be the maximum number of solutions to $a+b+c=n$ over all equilateral trapezoid-free triples $(A,B,C)$. We may visualize equilateral trapezoid-free sets as follows. Draw $\Delta_{n+1}$ in the plane as a triangular grid of points. Sets $A,B,C$ correspond to collections of lines parallel to the sides of $\Delta_{n+1}$, and a solution $a+b+c=n$ corresponds a point in $\Delta_{n+1}$ contained in one line in each of these three directions. Let $S \subseteq \Delta_{n+1}$ be the collection of all such points. A violation of a constraint of corresponds to either a subset of 3 points in $S$ forming an equilateral triangle with sides parallel to the sides of $\Delta_{n+1}$, or a subset of 4 points with sides parallel to the sides of $\Delta_{n+1}$ forming an equilateral trapezoid. Equivalently, we are deleting lines parallel to the sides of $\Delta_{n+1}$ to eliminate all of such configurations, while leaving as many points as possible. The maximum possible number of points left equals $\mathrm{val}(n)$. See . {#fig:traps} The following shows that $\mathrm{val}(n)$ and $\mathrm{val}({\mathbb{Z}}_n)$ are essentially the same. **Proposition 24**. 1. *$\mathrm{val}(n) \ge \mathrm{val}(n-1)$.* 2. *$1+2 \cdot \mathrm{val}(2n) \ge \mathrm{val}({\mathbb{Z}}_n) \ge \mathrm{val}(\lfloor n/3 \rfloor)$.* 3. *For $n \ge 6n'$, $\mathrm{val}({\mathbb{Z}}_n) \ge \mathrm{val}({\mathbb{Z}}_n')/2-1$* *Proof.* Suppose that $\mathrm{val}(n)$ is witnessed by sets $A,B,C$. For $N > n$, $A+(N-n), B, C$ then witness $\mathrm{val}(N) \ge \mathrm{val}(n)$, which shows (1). If we take $N = 3n$, we have that $A+2n \subseteq \{0,\ldots, N\}$ and $B, C \subseteq \{0,\ldots, N/3\}$, so $a+b+c \le 5N/3 < 2N$. Since $a+2n + b+c = 0 \bmod N \iff a+2n+b+c =N$, this implies that the sets $A+2n, B, C$ are equilateral trapezoid-free when viewed as subsets of ${\mathbb{Z}}_N$. This shows one direction of (2). In the other direction, suppose $\mathrm{val}({\mathbb{Z}}_n)$ is witnessed by $A,B,C \bmod n$. There are at least $(\mathrm{val}({\mathbb{Z}}_n)-1)/2$ solutions to one of $a+b+c=n, a+b+c=2n$; let $N$ be the right-hand side of the most frequently satisfied equation. Since every solution to $a+b+c=N$ is a solution to $a+b+c = 0 \bmod n$, $A,B,C$ must be equilateral trapezoid-free when viewed as subsets of $\{0, \ldots, N\}$. This shows the other direction of (2). Finally, (3) follows from (1) and (2). ◻ **Theorem 25**. *Suppose that one can achieve $\omega = 2$ via STPP constructions in the family of groups ${\mathbb{Z}}_q^\ell$, $q$ a prime power. Then there exists a constant $c>0$ such that $\mathrm{val}({\mathbb{Z}}_n) \ge \Omega(n^{1+c}).$* *Proof.* By and , any STPP construction with sets $X_i, Y_i, Z_i$ satisfies $\sum |X_i||Y_i| \le (q/C)^\ell$ (we choose the $X$ and $Y$ sets without loss of generality) where $C$ is an absolute constant. By Hölder's inequality, $\sum (|X_i||Y_i||Z_i|)^{2/3} \le q^{2\ell/3}(q/C)^{\ell/3} = (q/C^{1/3})^\ell$. If we can obtain $\omega < 3 - \alpha$ via , then $$\begin{aligned} q^\ell < \sum (|X_i||Y_i||Z_i|)^{2/3 \cdot \alpha + (1-\alpha)} &= \sum (|X_i||Y_i||Z_i|)^{2/3 \cdot \alpha} (|X_i||Y_i||Z_i|)^{1-\alpha}\\ &\le (\sum (|X_i||Y_i||Z_i|)^{2/3})^\alpha (\sum |X_i||Y_i||Z_i|)^{1-\alpha}\\ &\le (q/C^{1/3})^{\alpha \ell} \mathrm{val}(G)^{1-\alpha}\end{aligned}$$ so $\mathrm{val}(G) > q^\ell (C^{\alpha/3(1-\alpha)})^\ell$. By choosing $\alpha$ sufficiently close to 1, $\mathrm{val}(G) > q^\ell 4^\ell$. By taking $k$-fold products of the sets defining the STPP constructions (using that products of STPPs are STPPs [@cohn2005group Lemma 5.4]), we find that $\mathrm{val}({\mathbb{Z}}_q^{k\ell}) > (4q)^{k\ell}$ for all $k$. Let $N = k\ell$. Consider the embedding $\varphi : {\mathbb{Z}}_q^N \to {\mathbb{Z}}_{(3q)^N}$ defined by $\varphi(x_1, \ldots, x_N) = x_1 + x_2 3q + \cdots + x_n (3q)^{N-1}$. Since $\sum y_i (3q)^{i-1}$ has a unique such expression in ${\mathbb{Z}}_{(3q)^N}$ when $y_i < 3q$, it follows that that $$a_1+a_2+a_3 \neq a_4+a_5+a_6 \implies \varphi(a_1) + \varphi(a_2) + \varphi(a_3) \neq \varphi(a_4) + \varphi(a_5) + \varphi(a_6).$$ Hence the image of an STPP under $\varphi$ is an STPP inside of ${\mathbb{Z}}_{(3q)^N}$, so $\mathrm{val}( {\mathbb{Z}}_{(3q)^N}) > (4q)^N$. Because this holds for some particular $q$ and all $N = k\ell$, by part (3) of the theorem follows. ◻ **Corollary 26**. *Suppose that there is a family of STPP constructions obtaining $\omega = 2$ in a family of abelian groups with a bounded number of direct factors. Then there exists a constant $c > 0$ such that $\mathrm{val}({\mathbb{Z}}_n) \ge \Omega(n^{1+c})$.* *Proof.* Suppose we have a family of STPP construction in groups of the form $G = {\mathbb{Z}}_{m_1}\times \cdots \times {\mathbb{Z}}_{m_\ell}$, with $\ell$ fixed. We can then obtain an STPP construction in ${\mathbb{Z}}_p^\ell$, where $p$ is the smallest prime greater than $\max_{i \in k} 3m_i$, by taking the image of this STPP under the map sending $(x_1, \ldots, x_k) \to x_1+x_2p+ \cdots + x_kp^{k-1}$. As $k$ is fixed, it follows from Bertrand's postulate that $p^k \le O(|G|)$. The inequality then implies that one can also obtain $\omega = 2$ in the family of groups ${\mathbb{Z}}_p^\ell$, so we conclude by . ◻ **Remark 27**. Although we expect that is true when the hypothesis is extended to arbitrary abelian groups, we do not know how to generalize to e.g. ${\mathbb{Z}}_n^\ell$ for arbitrary $n$. This is due to the fact that better bounds on the size of 3-matchings in cyclic groups with prime power modului are known than for general moduli (compare Theorems A and A' in [@blasiak2017cap]). To the best of our knowledge, it is an open problem whether the known bounds for non-prime power moduli are tight. For prime power moduli, the known bounds are tight by [@kleinberg2016growth]. Next we show that sufficiently strong simultaneous double product property constructions, which are known to prove $\omega < 2.48$ [@cohn2005group Proposition 4.5], imply strong lower bounds on $\mathrm{val}({\mathbb{Z}}_n)$. We thank Chris Umans for informing us of the fact that if is true, then it is true in cyclic groups, which motivated the following theorem. **Theorem 28**. *If is true, then for any $\varepsilon > 0$, $\mathrm{val}({\mathbb{Z}}_n) \ge O(n^{4/3-\varepsilon})$.* *Proof.* We begin by recalling how to turn an SDPP construction into an STPP construction [@cohn2005group Section 6.2]. Let $S \subset \Delta_n$ be corner-free and of size $n^{2-o(1)}$. For all $v = (v_1,v_2,v_3) \in S$, define the following subsets of $G^3$: $$\begin{aligned} A_v &= A_{v_1} \times \{1\} \times B_{v_3},\\ B_v &= B_{b_1} \times A_{v_2} \times \{1\},\\ C_v &= \{1\} \times B_{v_2} \times A_{v_3}.\end{aligned}$$ It can be verified that the sets $(A_v, B_v, C_v)_{v \in S}$ satisfy the STPP. Hence yields an STPP with $n^{2-o(1)}$ triples of sets of size $n^{2-o(1)}$, inside a group of size $n^{6 - o(1)}$. Now consider the map from $G^3 = {\mathbb{Z}}_{m_1} \times \cdots \times {\mathbb{Z}}_{m_k}$, where $m_1 \le m_2 \le \cdots \le m_k$, to $G' := {\mathbb{Z}}_{\prod_i 3m_i}$ sending $(x_1, \ldots, x_k)$ to $x_1+(3m_1)x_2 + (3m_1)(3m_2)x_3 + \cdots$. First, the image of sets satisfying the STPP under this map still satisfy the STPP. This shows that $\mathrm{val}(G') > n^{2-o(1)} \cdot n^{3(2-o(1))} = n^{8-o(1)}$. Second, for all fixed $c>0$ and $\ell \in {\mathbb{N}}$, $G^3$ cannot contain a subgroup of size $|G^3|^c$ generated by elements of order at most $\ell$ by [@blasiak2017cap Proposition 4.2]. Hence the number of $m_i$'s which are at most $\ell$ is at most $\log_2(|G^3|^c)$. The number of $m_i$'s which are greater than $\ell$ is trivially less than $\log_\ell |G^3|$. So, $$|G'| = \prod_{m_i \le \ell} 3m_i \prod_{m_i> \ell} 3m_i \le 3^{\log_2(|G^3|^c) + \log_\ell |G^3|} \cdot |G^3|.$$ By taking $c$ sufficiently small and $\ell$ sufficiently large, this is at most $n^{6+\delta}$ for any desired $\delta> 0$. The claimed bound follows. ◻ Note that here there is no restriction on the family of abelian groups in consideration, unlike there was in the previous theorem. ## Relaxations of $\mathrm{val}({\mathbb{Z}}_n)$ In this section we explore some strengthenings of which may be easier to understand. We start by discussing an over-strengthening of which *cannot* give any barriers. We then discuss a few strengthenings for which our knowledge is embarrassingly bad, including the notions of skew-corner free sets from the introduction. Considerations of the proof of the $n^{3/2}$ upper bound of reveal that it actually held for a (possibly) much weaker problem, where one only requires that the expected number of solutions of one of the three systems of two equations in is at most 1. We begin by noting that this upper bound is essentially best-possible for this weakened problem. In other words, one cannot hope to prove via an "asymmetric\" averaging argument. **Proposition 29**. *There exist $A, B, C \subseteq {\mathbb{Z}}_n$ such that $${\mathbb{E}}_{a' \in A, b' \in B}\left [\#\{(a,b,c): 0=a'+b+c=a+b'+c\}\right ] \le 1$$ and there are $n^{3/2 - o(1)}$ solutions to the equation $a+b+c=0$ with $a \in A, b \in B, c \in C$.* *Proof.* Let $r(A,B,c)$ denote the number of representations of $c$ as $a+b$. First note that the proposition is equivalent to the statement that $\sum_{c \in C} r(A,B,-c)^2 \le |A| |B|$ and $\sum_{c \in C} r(A,B,-c) = n^{3/2 - o(1)}$. Let $S \subset [n]$ be 3AP-free and of size $n^{1-o(1)}$. Consider the sets $$A = B= [3n^2,4n^2] \cup \bigcup_{x \in S} [x n, xn + n/2],C= -\{2xn+y : x \in S, y \in [n] \}$$ regarded as subsets of ${\mathbb{Z}}_{100n^2}$. By definition, for any $x \in S$ and $y \in [n]$, $-(2xn+y) =c \in C$. If we have any representation $-c = a + b$, then $a, b < 3n^2$. So we have $a = x_1n+y_1, b = x_2n+y_1$ with $x_1, x_2 \in S$ and $1 \le y_1, y_2 \le n$. So $(x_1+x_2)n + (y_1+y_2) = 2xn + y$, and then we are forced to have $x_1 + x_2 = 2x$ and $y_1+y_2 = y$. But because $S$ is 3AP-free, we must have $x_1 = x_2 = x$. Hence $r(A,B,-c)$ is exactly the number of solutions to $y = y_1 + y_2$ with $y_1, y_2 \in [n]$, which is $\Omega(n)$ for $\Omega(n)$ choices of $y \in [n]$. Hence $\sum_{c \in C} r(A,B,-c) = \Theta(|S| n^2) = n^{3-o(1)}$. Also, we have that $\sum_{c \in C} r(A,B,-c)^2 =n^{4-o(1)} < |A| |B| = \Theta(n^4)$, and we are done. ◻ Can one find a construction achieving $n^{3/2 - o(1)}$ for the averaging version of that involves all three systems of equations? That is: **Question 30**. What is the maximum over all $A, B, C \subseteq {\mathbb{Z}}_n$ satisfying $$\begin{aligned} {\mathbb{E}}_{a' \in A, b' \in B}\left [\#\{(a,b,c): 0=a'+b+c=a+b'+c\}\right ] &\le 1,\\ {\mathbb{E}}_{a' \in A, c' \in C}\left [\#\{(a,b,c): 0=a'+b+c=a+b+c'\}\right ] &\le 1,\\ {\mathbb{E}}_{b' \in B, c' \in C}\left [\#\{(a,b,c): 0=a+b'+c=a+b+c'\}\right ] &\le 1,\end{aligned}$$ of the number of solutions to $a+b+c=0$? There are a number of relaxations of the quantity $\mathrm{val}(n)$ for which we know basically nothing. A first relaxation that still seems very stringent is that of a *triforce-free* triple, defined as follows. **Definition 31**. Let $A, B, C \subseteq \{0, \ldots, n\}$. We say that $(A,B,C)$ is triforce-free if there is no solution to $$a+b+c'=a+b'+c=a'+b+c=n$$ with $a \neq a', b \neq b', c \neq c'$. We write $\mathrm{val}(% \begin{tikzpicture}% \newdimen\triforcewidth% \newdimen\triforceheight% \triforcewidth=0.4cm% \pgfmathparse{sqrt(3)}% \pgfmathsetlength{\triforceheight}{\pgfmathresult / 2 * \triforcewidth}% % \foreach \x / \y in {0 / 0, 0.5\triforcewidth / 0, 0.25\triforcewidth / 0.5\triforceheight}% {% \shade[triforcefillshade, xshift=\x, yshift=\y]% (0, 0) -- +(.5\triforcewidth, 0) -- +(60:.5\triforcewidth) -- cycle;% \shade[triforceoutlineshade, xshift=\x, yshift=\y]% (0, 0) -- +(.5\triforcewidth, 0) -- +(60:.5\triforcewidth) -- cycle% (30:.0175\triforcewidth) -- ($(60:.5\triforcewidth) + (-90:.0175\triforcewidth)$) -- ($(0.5\triforcewidth, 0) + (150:.0175\triforcewidth)$) -- cycle;% }% \end{tikzpicture}% ,n)$ for the maximum over all such $A,B,C$ of the number of solutions to $a+b+c=n$. This condition just says that $\{(a,b,c) \in A \times B \times C: a+b+c=n\} \subseteq \Delta_{n+1}$ is corner-free. Equivalently, $(A,B,C)$ is triforce-free if the hypergraph with parts $A,B,C$ and triangles between any triples summing to $n$ does not contain the triforce hypergraph (the second hypergraph in ). As every equilateral trapezoid-free triple of sets also has this property, we have the following. **Proposition 32**. *$\mathrm{val}(% \begin{tikzpicture}% \newdimen\triforcewidth% \newdimen\triforceheight% \triforcewidth=0.4cm% \pgfmathparse{sqrt(3)}% \pgfmathsetlength{\triforceheight}{\pgfmathresult / 2 * \triforcewidth}% % \foreach \x / \y in {0 / 0, 0.5\triforcewidth / 0, 0.25\triforcewidth / 0.5\triforceheight}% {% \shade[triforcefillshade, xshift=\x, yshift=\y]% (0, 0) -- +(.5\triforcewidth, 0) -- +(60:.5\triforcewidth) -- cycle;% \shade[triforceoutlineshade, xshift=\x, yshift=\y]% (0, 0) -- +(.5\triforcewidth, 0) -- +(60:.5\triforcewidth) -- cycle% (30:.0175\triforcewidth) -- ($(60:.5\triforcewidth) + (-90:.0175\triforcewidth)$) -- ($(0.5\triforcewidth, 0) + (150:.0175\triforcewidth)$) -- cycle;% }% \end{tikzpicture}% ,n) \ge \mathrm{val}(n)$.* Here is an even weaker notion than that of being triforce-free. We thank Ryan O'Donnell for suggesting this definition. **Definition 33**. We call $S \subseteq \Delta_n$ skew-corner free if for $(a,b,c), (a,b',c') \in S$, it holds that $(a+b - b', b'', c'') \notin S$ for all $b'', c''$, and this remains true after any permutation of the coordinates of $S$. Pictorially, this says that for any two points lying on an axis-aligned line in $\Delta_n$, the parallel line passing through a third point that would form a corner with these two points must contain no points. As yields corner-free subsets of $\Delta_n$ obtained by deleting axis-aligned lines, it follows that this is a relaxation of being triforce-free. More formally, we have the following. **Proposition 34**. *The largest skew-corner free subset of $\Delta_{n+1}$ is at least $\mathrm{val}(% \begin{tikzpicture}% \newdimen\triforcewidth% \newdimen\triforceheight% \triforcewidth=0.4cm% \pgfmathparse{sqrt(3)}% \pgfmathsetlength{\triforceheight}{\pgfmathresult / 2 * \triforcewidth}% % \foreach \x / \y in {0 / 0, 0.5\triforcewidth / 0, 0.25\triforcewidth / 0.5\triforceheight}% {% \shade[triforcefillshade, xshift=\x, yshift=\y]% (0, 0) -- +(.5\triforcewidth, 0) -- +(60:.5\triforcewidth) -- cycle;% \shade[triforceoutlineshade, xshift=\x, yshift=\y]% (0, 0) -- +(.5\triforcewidth, 0) -- +(60:.5\triforcewidth) -- cycle% (30:.0175\triforcewidth) -- ($(60:.5\triforcewidth) + (-90:.0175\triforcewidth)$) -- ($(0.5\triforcewidth, 0) + (150:.0175\triforcewidth)$) -- cycle;% }% \end{tikzpicture}% ,n)$.* *Proof.* Suppose that $A,B,C \subset \{0, \ldots, n\}$ satisfy the conditions of , and let $S = \{(a,b,c) \subset A \times B \times C: a+b+c=n\} \subseteq \Delta_{n+1}$. Suppose for contradiction that $(a,b,c), (a,b',c') \in S$ and $(a-b+b',b'',c'') \in S$. Since $a-b+b' \in A, b \in B, c' \in C$ and $(a-b+b')+b+c' = a+b'+c' = n$, it follows that $(a-b+b', b,c') \in S$. But this is impossible: the three solutions $a+b'+c' = n, a+b+c=n,(a-b+b') + b +c'$ violate . One reasons similarly about other permutations of coordinates. ◻ The best lower bound that we know on the size of the largest skew-corner free subset of $\Delta_n$ is $\Omega(n)$; $n$ is obtained trivially by taking one line on the side of $\Delta_n$, and it is not hard to improve this to $3n/2$. We have found examples exceeding these bounds with computer search (see ). If we weaken by dropping the requirement that the condition holds for all permutations of coordinates, we are led to the following notion. **Definition 35**. We say $S \subset [n]^2$ is skew corner-free if it contains no configuration $(x,y), (x,y+d), (x+d, y')$ with $d \neq 0$. **Proposition 36**. *The largest skew corner-free subset of $[n]^2$ is at least as big as the largest skew corner-free subset of $\Delta_n$.* *Proof.* Given a skew corner-free set $S \subseteq \Delta_n$, let $S'$ be its projection onto the first two coordinates. This is a subset of $\{0,\ldots, n-1\}^2$ of size $|S|$. By definition, it contains no points $(a,b), (a,b'), (a+b-b',b'')$. By shifting each point by $(1,1)$ we obtain a subset of $[n]^2$ with this property. ◻ As a consequence, we have . *Proof of and .* By , if $\omega = 2$ via STPP constructions in ${\mathbb{Z}}_q^\ell$, then $\mathrm{val}({\mathbb{Z}}_n) \ge \Omega(n^{1+c})$. By , $\mathrm{val}({\mathbb{Z}}_n) = \Theta(\mathrm{val}(n))$, and by , $\mathrm{val}(n)$ is at most the size of the largest skew corner-free subset of $[n]^2$. This proves . One similarly concludes by using . ◻ We have the following nontrivial lower bound for this relaxed problem, due to a MathOverflow answer of Fedor Petrov [@petrovanswer]. **Proposition 37**. *There is a skew corner-free subset of $[n]^2$ of size $\Omega(n \log n/\sqrt{\log \log n})$.* *Proof.* $A \subseteq [n]$ is called *primitive* if for all $a \neq a' \in A, a \nmid a'$. It is easily seen that if $A$ is primitive then the set of points $(a,ka) \subseteq [n]^2$ for all $k\le n/a$ avoids the forbidden configurations. This gives a subset of size $n \sum_{a \in A} 1/a$. At the same time, there exists a $c>0$ and a primitive set $A$ where $\sum_{a \in A} 1/a > c \log n/(\log \log n)^{1/2}$ [@erdos1967theorem]. We note that this is best-possible, matching (up to the constant) an upper bound on $\sum_{a \in A} 1/a$ for primitive $A$ due to Behrend [@behrend1935sequences]. ◻ This construction breaks when we strengthen the definition of skew corner-freeness in $[n]^2$ to forbid skew corners with two points parallel to the $x$ axis. This corresponds to the following notion. **Definition 38**. We say $S \subset [n]^2$ is bi-skew corner-free if it contains no configurations $(x,y), (x,y+d), (x+d, y')$ or $(x,y), (x+d, y), (x',y+d)$, with $d \neq 0$. As far as we know, it is possible that the largest bi-skew corner-free subset has size $O(n)$. # Acknowledgments I thank Ryan O'Donnell for many useful discussions about these problems, and for suggesting . I also thank Chris Umans for making comments which motivated this paper early on, and in particular, which motivated . [^1]: This is the Rusza-Szemerédi problem. The equivalence between induced matchings in $M_n$ and this problem was independently noted in [@alman2023matrix]. [^2]: The techniques involved in the resolution of the cap--set problem (in particular, slice rank) actually give stronger "tensor analogues\" of this barrier; see [@christandl2018barriers; @alman2021limits]. [^3]: Or fortunately, for the optimist. [^4]: While [@fox2017tight Theorem] is only stated for ${\mathbb{Z}}_p^\ell$, it extends to ${\mathbb{Z}}_q^\ell$ by the same argument via the use of [@blasiak2017cap Theorem A'].

arxiv_math

--- abstract: | We show how the relatively initial or relatively terminal fixed points of [@adamek2012relatively] for a well-behaved functor $F$ form a pair of adjoint functors between $F$-coalgebras and $F$-algebras. We use the language of locally presentable categories to find sufficient conditions for existence of this adjunction. We show that relative fixed points may be characterized as (co)equalizers of the free (co)monad on $F$. In particular, when $F$ is a polynomial functor on $\mathsf{Set}$ the relative fixed points are a quotient or subset of the free term algebra or the cofree term coalgebra. We give examples of the relative fixed points for polynomial functors and an example which is the Sierpinski carpet. Lastly, we prove a general preservation result for relative fixed points. author: - | Ezra Schoen\ Strathclyde University, United Kingdom\ ezra.schoen\@strath.ac.uk - | Jade Master[^1]\ Strathclyde University, United Kingdom\ jade.master\@strath.ac.uk - | Clemens Kupke$^*$\ Strathclyde University, United Kingdom\ clemens.kupke\@strath.ac.uk bibliography: - cs.bib title: Relative fixed points of functors --- coalgebra, algebra, fixed points, coalgebra-to-algebra morphisms # Introduction Fixed points of functors are particularly relevant to the study of coalgebras. As in [@jacobs2017introduction; @adamek2018fixed] these fixed points capture the ideas of induction and coinduction on coalgebras. The main focus of this work has thus far been on either the least fixed point of a functor or the greatest fixed point of a functor. However, in general a functor has more fixed points than just these two. We call these additional fixed points "relative fixed points\", after the "relatively terminal coalgebras\" of [@adamek2012relatively]. Other constructions yielding relative fixed points are the rational fixed points of Adámek, Milius, and Velebil [@adamek2006iterative] and the locally finite fixed points of Milius, Pattinson, and Wißmann [@milius2016new]. The main contribution of this paper is a presentation of relative fixed points via a pair of adjoint functors $$\hfill \label{diag:adj} \begin{tikzcd} F\text{-}\mathsf{Coalg}\ar[r,bend left,"\mu"] \ar[r,phantom,"\bot"]& F\text{-}\mathsf{Alg}\ar[l,bend left,"\nu"] \end{tikzcd} \hfill$$ This adjunction reveals the deep connection between relative fixed points and coalgebra-to-algebra homomorphisms (abreviated as ca-morphism). Algebras and coalgebras which have unique ca-morphisms going into or out of them have been studied extensively in [@adámek_lücke_milius_2007; @capretta2009corecursive; @capretta2006recursive; @levy2015final; @adamek2020wellfounded] due to their connection to inductive principles. However, the fixed points studied in this paper are universal with respect to ca-morphisms which may not be unique. In the context of functional programming [@meijer1991functional], not-necessarily unique ca-morphisms are used as data structures for recursion schemes. In [@hauhs2015scientific], the authors argue for the use of non-unique ca-morphisms as a framework for scientific modelling. Whatever the reason for the their relevance, this paper studies relative fixed points in the context of the novel adjunction through examples and results. The paper is organised as follows: In Section [2](#sec:universal){reference-type="ref" reference="sec:universal"}, we will introduce relative fixed points for $F$-(co)algebras, and the adjunction [\[diag:adj\]](#diag:adj){reference-type="eqref" reference="diag:adj"} in the full categorical case. After that we will provide sufficient conditions for the existence of the adjunction. In Section [3](#sec:constructions){reference-type="ref" reference="sec:constructions"}, we will provide examples of these relative fixed points as well as an explicit characterization for polynomial functors. In Section [4](#sec:preservation){reference-type="ref" reference="sec:preservation"}, we will discuss when the adjunction is preserved by a functor and give some important examples of this phenomenon. Finally, in Section [5](#sec:conclusions){reference-type="ref" reference="sec:conclusions"} we will draw conclusions and point to ideas for future work. We now finish the introduction with a brief discussion of relative fixed points for monotone functions to equip the reader with some intuitions before moving to the more general categorical setting that follows. ### Warm-up: Relative fixed points of monotone functions {#warm-up-relative-fixed-points-of-monotone-functions .unnumbered} Consider a monotone function $f: L \to L$ on a complete lattice $L$. The Kleene fixed point theorem provides a construction of least and greatest fixed points for $f$. For example, let $f$ be the following monotone function on $([0,1],\leq)$ the interval of real numbers with the usual ordering:  The function $f$ is overlayed with the function $y=x$. The intersection of the two curves indicate fixed points of $f$. The least fixed point of $f$ is $0$ and the greatest fixed point is $1$ but there are $3$ other fixed points in-between. These relative fixed points have a similar construction to the least and greatest ones. Given a "post-fixed point\" i.e. a point $x \in [0,1]$ such that $x \leq f(x)$ we may find the first fixed point above $x$ as $$\hfill\mu(x) = \sup \{x,f(x),f^2(x),f^3(x),\ldots\} \hfill$$ where the $\ldots$ indicate iteration to a sufficiently large ordinal. Similarly, given a "pre-fixed point\" $f(y) \leq y$, we may find the closest fixed point below $y$ as $$\hfill\nu(y) = \inf \{ y ,f(y),f^2(y),f^3(y),\ldots\}\hfill$$ For a complete lattice $L$, let $Pre(f)$ be the suborder of $L$ consisting of only the post-fixed points $x \leq f(x)$. Similarly, let $Post(f)$ be the suborder of pre-fixed points $f(y) \leq y$. Then there is a Galois connection $$\hfill \begin{tikzcd} Post(f) \ar[r,bend left,"\mu_f"] \ar[r,phantom,"\bot"] & Pre(f) \ar[l,bend left,"\nu_f"] \end{tikzcd} \hfill$$ Being a Galois connection means that $$\hfill\mu(x) \leq y \iff x \leq \nu(y) \hfill$$ In this paper we will generalize this Galois connection to fixed points of functors rather than monotone functions. When generalizing from posets to categories we make the replacements shown in Table [\[table\]](#table){reference-type="ref" reference="table"}. Poset Category ---------------------------------------- ------------------------------------------------------------------------ Monotone Function $f$ Functor $F$ Post-fixed point of $f$ $F$-coalgebra Pre-fixed point of $f$ $F$-algebra $\sup \{f(x),f^2(x),f^3(x), \ldots \}$ $\mathrm{colim}(X \to F(X) \to F^2(X) \to F^3(X) \ldots)$ $\inf\{f(x),f^2(x),f^3(x), \ldots \}$ $\lim (X \leftarrow F(X) \leftarrow F^2(X) \leftarrow F^3(X) \ldots )$ Galois connection Adjunction #### Acknowledgements. For insightful comments and questions, thank you to every member of the Mathematically Structured Programming group, Corina Cirstea, Toby Wilkinson, and Alexandre Goy. # Relative Fixed Points are Adjoint {#sec:universal} In this section, we recall the definition of 'relatively terminal coalgebra' from [@adamek2012relatively], and define the dual notion of 'relatively initial algebra'. As is usual for definitions via universal properties, there may or may not be an object enjoying the property; however, if there is one, it is unique up to unique isomorphism. For an algebra $a$ and a coalgebra $b$, a coalgebra-to-algebra morphism from $a$ to $b$ (abbreviated as ca-morphism) is a morphism $f \colon B \to A$ making the following diagram commute: $$\hfill \begin{tikzcd} FB\arrow[r, "Ff"] & FA\arrow[d, "a"]\\ B\arrow[u, "b"] \arrow[r, "f"] & A \end{tikzcd} \hfill$$ Let $\mathsf{Hylo}(b,a)$ denote the set of coalgebra-to-algebra morphisms from $b$ to $a$. The notation $\mathsf{Hylo}(b,a)$ comes from the name 'hylomorphism' for ca-morphism, which is part of the 'cata-ana' naming scheme in the theory of recursive algorithms [@meijer1991functional]. We have chosen to use the term 'ca-morphism', as in our view this term is more readily understood; but we will still use $\mathsf{Hylo}(b,a)$ for the set of ca-morphisms $b\to a$. [\[universalprop\]]{#universalprop label="universalprop"} Suppose we have an algebra $a : F A \to A$. A coalgebra $a':FA'\to A'$ is called *terminal relative to $a$* if there is a bijection (natural in $b$) $$\hfill\phi\colon F\text{-}\mathsf{Coalg}(b,a')\cong \mathsf{Hylo}(b, a)\hfill$$ Similarly, for a coalgebra $b: B\to FB$, an algebra $b':FB'\to B'$ is called *initial relative to $b$* if there is a bijection (natural in $a$) $$\hfill\psi\colon F\text{-}\mathsf{Alg}(b',a)\cong \mathsf{Hylo}(b, a)\hfill$$ By the Yoneda lemma, if an algebra $a$ admits a relatively terminal coalgebra, it must be unique up to unique isomorphism; hence, we may use the functional notation $\nu(a)$ or $\nu (a)$ to denote *the* coalgebra which is terminal relative to $a$. Similarly, we will write $\mu(b)$ or $\mu (b)$ for *the* algebra which is initial relative to $b$. However, note that so far we have no guarantee as to the existence of $\nu(a)$ and $\mu(b)$; we will adopt the convention that the use of an expression $\nu (a)$ or $\mu (b)$ carries with it the implicit assumption that such an object exists. So for example, Proposition [\[prop:fixed\]](#prop:fixed){reference-type="ref" reference="prop:fixed"} should be read as "For any algebra $a$, *if a relatively terminal coalgebra exists*, it is a fixed point of $F$\". In Theorem [\[existencethm\]](#existencethm){reference-type="ref" reference="existencethm"}, we will show that under appropriate conditions, $\mu$ and $\nu$ define total functors. Let $$\hfill \mathsf{Hylo}(-,=) \colon F\text{-}\mathsf{Coalg}^{\mathrm{op}}\times F\text{-}\mathsf{Alg}\to \mathsf{Set}\hfill$$ be the functor which sends a coalgebra $b$ and an algebra $a$ to the set of coalgebra-to-algebra morphisms from $b$ into $a$. The above definition may be rephrased as follows: $\nu(a)$ is a representing object for $\mathsf{Hylo}(-,a)$ and $\mu(b)$ is a representing object for $\mathsf{Hylo}(b,-)$. As in the Yoneda lemma, of central importance are the maps $\eta = \psi(\text{id}_{\mu (b)})$ and $\epsilon = \phi(\text{id}_{\nu (a)})$. It can easily be verified that for $f:a\to \mu (b)$ and $g:\nu (a) \to b$, we have the equalities $$\begin{aligned} \psi(f) = f\circ\eta\label{eq:psieta}\\ \phi(g) = \epsilon\circ g\label{eq:phieps}\end{aligned}$$ Perhaps surprisingly, the universal properties of $\mu (b)$ and $\nu (a)$ imply that they are always fixed points for $F$. [\[prop:fixed\]]{#prop:fixed label="prop:fixed"} For any algebra $a$, the coalgebra $\nu (a):\nu A\to F\nu A$ is a fixed point of $F$. Similarly, for any coalgebra $b$, the algebra $\mu (b): F\mu (b)\to \mu B$ is a fixed point of $F$. *Proof.* This proposition resembles Lambek's lemma; indeed, it is possible to exhibit $\mu (b)$ as an initial algebra for a well-chosen functor $F_b:\mathsf{C}/B\to \mathsf{C}/B$. This is (up to duality) the approach taken in [@adamek2012relatively]. For concreteness, We have chosen to give an explicit proof. We prove that $\mu (b)$ is a fixed point; the case for $\nu (a)$ follows by duality. We wish to find an inverse $\beta$ to $\mu (b):F(\mu B)\to \mu B$. Since $F(\mu B)$ carries the algebra structure $F(\mu (b)):FF(\mu B)\to F(\mu B)$, it suffices to find a ca-morphism $b\to F(\mu (b))$. This is given by the following diagram: $$\begin{tikzcd} FB\arrow[r, "Fb"] & FFB\arrow[r, "FF\eta"] & FF\mu B\arrow[d, "F\mu (b)"]\\ B\arrow[u, "b"]\arrow[r, "b"] & FB\arrow[r, "F\eta"]\arrow[u, "Fb"] & F\mu B \end{tikzcd}$$ This yields an algebra morphism $\beta:\mu B\to F\mu B$ such that $$\beta\eta = \psi(\beta) = (F\eta) b \label{eq:beta}$$ It remains to show that $\beta$ is a two-sided inverse to $\mu (b)$. Consider the composite $\mu (b)\circ \beta:\mu B\to \mu B$. We claim that under the correspondence $\psi$, this composite corresponds to $\eta$; since $\psi$ is bijective, and $\text{id}_{\mu B}$ corresponds to $\eta$ by definition, this yields $\mu (b)\circ \beta = \text{id}_{\mu B}$. To verify, we use equality [\[eq:psieta\]](#eq:psieta){reference-type="ref" reference="eq:psieta"}: $$\begin{tikzcd} \mu B\arrow[drr, "\beta", bend left]\\ &FB\arrow[r, "F\eta"]&F\mu B\arrow[d, "\mu (b)"]\\ B\arrow[uu, "\eta"]\arrow[ur, "b"]\arrow[rr, "\eta"]&&\mu B \end{tikzcd}$$ The top square is equation [\[eq:beta\]](#eq:beta){reference-type="ref" reference="eq:beta"}, and the bottom square commutes since $\eta$ is a ca-morphism. We may now conclude that $\mu (b)\circ\beta = \text{id}_{\mu B}$. To show that $\beta\circ\mu (b) = \text{id}_{F\mu B}$, we simply note that $\beta$ is an algebra morphism, and hence $$\begin{tikzcd} F\mu B\arrow[r, "F\beta"]\arrow[d, "\mu (b)"]& FF\mu B\arrow[d, "F\mu (b)"]\\ \mu B\arrow[r, "\beta"]&F\mu B \end{tikzcd}$$ commutes. The composite $F\mu (b)\circ F\beta = F(\mu (b)\circ \beta)$ is equal to $F\text{id}_{\mu B} = \text{id}_{F\mu B}$ as already shown, so we also have $\beta\circ\mu (b) = \text{id}_{F\mu B}$. ◻ We now give an adjunction characterizing relative fixed points. [\[thm:adj\]]{#thm:adj label="thm:adj"} Let $\mathsf{C}$ be a category, and $F:\mathsf{C}\to \mathsf{C}$ an endofunctor. Assume that every $F$-algebra has a relatively terminal coalgebra, and every $F$-coalgebra has a relatively initial algebra. Then $\nu:F\text{-}\mathsf{Alg}\to F\text{-}\mathsf{Coalg}$ and $\mu:F\text{-}\mathsf{Coalg}\to F\text{-}\mathsf{Alg}$ are the object parts of two adjoint functors $$\hfill \begin{tikzcd} F\text{-}\mathsf{Coalg}\ar[r,bend left,"\mu"] \ar[r,phantom,"\bot"]& F\text{-}\mathsf{Alg}\ar[l,bend left,"\nu"] \end{tikzcd} \hfill$$ *Proof.* For the action of $\nu$ on morphisms, consider an algebra morphism $$\hfill\begin{tikzcd}FA\arrow[r, "Ff"] \arrow[d, "a"]&FA'\arrow[d, "a'"]\\ A\arrow[r, "f"] & A'\end{tikzcd}\hfill$$ Then we obtain a ca-morphism $$\hfill\begin{tikzcd}F\nu A\arrow[r, "F\epsilon"]& FA\arrow[d, "a"]\arrow[r, "Ff"] & FA'\arrow[d, "Fa'"]\\\nu A\arrow[u, "\nu (a)"] \arrow[r, "\epsilon"] & A\arrow[r, "f"] & A'\end{tikzcd}\hfill$$ So, we can set $\nu(f)$ to be $\phi^{-1}(f\circ \epsilon)$. It is easy to check that this preserves composition. The action of $\mu$ on morphisms is similar. To see that $\mu \dashv \nu$, simply consider the composite isomorphism $$F\text{-}\mathsf{Alg}(\mu (b), a) \cong \mathsf{Hylo}(b,a)\cong F\text{-}\mathsf{Coalg}(b, \nu (a))$$ ◻ We end this section with the following remark: [\[rem:flip\]]{#rem:flip label="rem:flip"} It is easy to see that if $b:B\to FB$ is an isomorphism, then $b^{-1}:FB\to B$ is initial relative to $b$; hence, $\mu (b) = b^{-1}$. Similarly, $\nu \alpha = \alpha^{-1}$ whenever $\alpha:FA\to A$ is an isomorphism. From this, it follows that the monad $\mu\nu : F\text{-}\mathsf{Coalg}\to F\text{-}\mathsf{Coalg}$ maps $b:B\to FB$ to $$(\mu (b))^{-1}:\mu B\to F\mu B$$ ## Existence of the Adjunction {#existence-of-the-adjunction .unnumbered} In this section, we offer sufficient conditions on the endofunctor $F: \mathsf{C}\to \mathsf{C}$ for $\mu$ and $\nu$ to define total functors. We will require that $\mathsf{C}$ is a locally presentable category and that $F$ is an accessible endofunctor. We quickly recall the relevant definitions; for a full explanation of locally presentable categories, see [@adamek1994locally]. Let $\lambda$ be a regular infinite cardinal. $\mathsf{C}$ is a $\lambda$-filtered category if every class of morphisms with size less than $\lambda$ has a cocone in $\mathsf{C}$. The colimit of a functor $D: \mathsf{C}\to \mathsf{D}$ is a $\lambda$-filtered colimit if the category $\mathsf{C}$ is $\lambda$-filtered. An object $A$ of a category $\mathsf{C}$ is $\lambda$-presentable if the representable functor $\mathrm{Hom}(A,-)$ preserves $\lambda$-filtered colimits. [\[lfp\]]{#lfp label="lfp"} A category $\mathsf{C}$ is locally $\lambda$-presentable if - $\mathsf{C}$ is cocomplete, - There are up to isomorphism only a set of $\lambda$-presentable objects, and - every object in $\mathsf{C}$ is colimit of $\lambda$-presentable objects. A category $\mathsf{C}$ is locally presentable if it is $\lambda$-locally presentable for some infinite cardinal $\lambda$. $\mathsf{C}$ is locally finitely presentable if it is $\omega$-presentable for the first infinite cardinal $\omega$. A functor $F \colon\mathsf{C}\to \mathsf{C}$ is $\lambda$-accessible if it preserves $\lambda$-filtered colimits. $F$ is accessible if it preserves $\lambda$-filtered colimits for some $\lambda$. We now state the main theorem of the section: [\[existencethm\]]{#existencethm label="existencethm"} Let $\mathsf{C}$ be a locally presentable category and $F: \mathsf{C}\to \mathsf{C}$ an accessible endofunctor. Then the relatively initial fixed point $\mu(b)$ exists for any coalgebra $b$ and the relatively terminal fixed point $\nu(a)$ exists for any coalgebra $a$. *Proof.* Fix a regular cardinal $\lambda$ such that $\mathsf{C}$ is $\lambda$-presentable, and $F$ is $\lambda$-accessible. Because $\mathsf{C}$ is a locally accessible category, it has colimits of all chains. Hence, given a coalgebra $b:B\to FB$, we can build up the chain $$\hfill \begin{tikzcd} B \ar[r,"b"] & F(B) \ar[r,"Fb"] & F^2(B) \ar[r,"F^2(b)"] & \cdots \end{tikzcd}\label{eq:cochain} \hfill$$ and continue it until the $\lambda$'th iterate $F^\lambda(B) = \mathrm{colim}_{i < \lambda}F^i(B)$. Since the chain of length $\lambda$ is $\lambda$-filtered (recall that $\lambda$ is regular), we know that $F$ preserves this colimit, and hence the chain converges to a universal cocone $m:M\cong FM$, with inclusion maps $j_i:F^iB\to M$. It remains to show that $m^{-1}$ is relatively initial for $b$. Given a ca-morphism $$\hfill \begin{tikzcd} B \ar[d,"f"] \ar[r,"b"] & F(B)\ar[d,"F(f)"] \\ A & \ar[l,"a"] F(A) \end{tikzcd} \hfill$$ we present $A$ as a cocone $(c_i:F^iB\to A)_{i < \lambda}$ over chain [\[eq:cochain\]](#eq:cochain){reference-type="ref" reference="eq:cochain"}. We proceed by ordinal recursion: for $i = 0$, we have $c_0 = f : B\to A$. If $i+1$ is a successor, we set $$\hfill \begin{tikzcd} F^iB\arrow[r, "F^i b"]\arrow[d, "c_i"] & F^{i+1}B\arrow[d, "Fc_i"]\arrow[dl, "c_{i+1}"]\\ A& FA\arrow[l, "a"] \end{tikzcd} \hfill$$ Finally, at successor stages $\alpha$, we have already built up a cocone $(c_i:F^iB\to A)_{i < \alpha}$, hence since $F^\alpha B$ is defined as the colimit over the stages $i < \alpha$, we get a unique mediating arrow $c_\alpha:F^\alpha B\to A$, with $$\hfill \begin{tikzcd} F^iB\arrow[rr]\arrow[dr]\arrow[ddr, bend right, "c_i"] && F^{i'}B\arrow[dl]\arrow[ddl, bend left, "c_{i'}"]\\ &F^\alpha B\arrow[d, dotted, "c_\alpha"]\\ &A \end{tikzcd} \hfill$$ Then too, we get a colimit map $\hat f:M\to A$ such that $\hat f\circ j_i = c_i$. We will verify that $\hat f$ is an algebra morphism - i.e., $\hat f\circ m^{-1} = a\circ Ff$. This is of course equivalent to $\hat f = a\circ Ff\circ m$, and by the colimit property of $M$, it suffices to show that $\hat f\circ j_i = a\circ Ff \circ m^{-1}\circ j_i$ for all $i < \lambda$. This follows by an easy ordinal induction: if $\alpha$ is a limit ordinal, and it holds for all $i < \alpha$, then it also holds for $\alpha$, as $F^\alpha B$ is a colimit over the earlier stages. At successor stages $i + 1$, where it holds for $i$, we get $$\hfill\begin{tikzcd} F^{i+1}B\arrow[dd, bend right, "c_{i+1}", swap]\arrow[dr, "Fj_{i}"]\arrow[d, "j_{i+1}"]\arrow[r, "j_{i+1}"]&M\arrow[d, "m"] \\ M\arrow[d, "\hat f"]& FM\arrow[d, "F\hat f"]\\ A&FA\arrow[l, "a"] \end{tikzcd}\hfill$$ and here the inner square commutes, as $$\hfill c_{i+1} = a\circ F(c_{i}) = a\circ F\hat f\circ Fj_{i}\hfill$$ Next, assume that we have an algebra morphism $g:B'\to A$. Then we obtain a ca-morphism $g\circ j_0:B\to A$ via $$\hfill \begin{tikzcd} B\arrow[r, "b"]\arrow[d, "j_0"] & FB\arrow[d, "Fj_0"]\\ M\arrow[r, "m"]\arrow[d, "g"] & FM\arrow[d, "Fg"]\\ A & FA\arrow[l, "a"] \end{tikzcd}\hfill$$ We now have the two operations $\hat{(-)}:\mathsf{Hylo}(b, a)\to \mathsf{Alg}(m^{-1}, a)$ and $(-)\circ j_0:\mathsf{Alg}(m^{-1}, a)\to \mathsf{Hylo}(b,a)$. We quickly verify that these two operations are inverse. In one direction, it is clear that $j_0\circ \hat f = f$ by construction of $\hat f$. In the other direction, we need to show that $g$ is a mediating arrow for the cone induced by $g\circ j_0:B\to A$. We prove by induction that $g\circ j_i = c_i$ for all $i$. For $i = 0$, this is the definition. If it holds for $i$, then for $i+1$, we get $$\hfill \begin{tikzcd} F^iB\arrow[d, "j_{i+1}"]\arrow[dr, "Fj_i", bend left]\\ M\arrow[r, "m"]\arrow[d, "g"] & FM\arrow[d, "Fg"]\\ A&FA\arrow[l, "a"] \end{tikzcd}\hfill$$ By induction, the right-hand composition is equal to $a\circ F(c_i)$, which is by definition equal to $c_{i+1}$. For limit stages $\alpha$, it follows by the colimit condition. So $g$ is the unique mediating arrow, and hence we get $\widehat{(j_0\circ g)} = g$. This shows that $\mathsf{Hylo}(b,a)\cong F\text{-}\mathsf{Alg}(m^{-1}, a)$, and hence $m$ is initial relative to $b$. Since $b$ was arbitrary, we now know that $\mu (b)$ exists for all $b$. To show the existence of $\nu$ we use the (dual of) the special adjoint functor theorem (e.g. [@riehl2017category Thm. 4.58]). By [@adamek1994locally Exercise 2j], $F\text{-}\mathsf{Coalg}$ is locally presentable and by [@adamek1994locally Corr. 2.75] so is the category $F\text{-}\mathsf{Alg}$. By [@adamek1994locally Thm. 1.58], both these categories are co-wellpowered. The functor $\mu$ preserves colimits, because it is constructed as a colimit and colimits distribute over themselves. Therefore, by the special adjoint functor theorem, $\mu$ has right adjoint $\nu$; to see that that $\nu (a)$ is terminal relative to $a$, consider the natural equivalences $$F\text{-}\mathsf{Coalg}(b, \nu (a)) \cong F\text{-}\mathsf{Alg}(\mu (b) , a)\cong \mathsf{Hylo}(b,a).$$ ◻ As a special case, we may consider functors that preserve both limits and colimits of shape $\omega$. Suppose $C$ is a category with limits of $\omega^\mathrm{op}$-chains and colimits of $\omega$-chains and $F$ is a functor that preserves them. Then $\mu$ and $\nu$ may be calculated as $$\label{eq:omegachain}\hfill\mu(b)\cong\mathrm{colim}(\begin{tikzcd} B \ar[r,"b"] & F(B) \ar[r,"Fb"] & F^2(B) \ar[r,"F^2(b)"] & \cdots \end{tikzcd}) \hfill$$ $$\hfill\nu(a)\cong\lim(\begin{tikzcd} A & F(A) \ar[l,"a",swap] & F^2(A) \ar[l,"Fa",swap] & \ar[l,"F^2(a)",swap] \cdots \end{tikzcd} )\hfill$$ In particular, $\mu(b)$ and $\nu(a)$ exist for all coalgebras $b$ and all algebras $a$. Let $1$ be the terminal object of $\mathsf{C}$ and let $0$ be the initial object. Then there is a unique algebra $1:F1 \to 1$ and $\nu(1)$ is the terminal coalgebra. Similarly, the initial algebra is given by $\mu(0)$ for the unique coalgebra $0: 0 \to F0$. Before moving on to the next section we state a corollary about recursive coalgebras and corecursive algebras. An $F$-algebra $a$ is corecursive if for every $F$-coalgebra $b$ there is a unique ca-morphism from $b \to a$. Dually, an $F$-coalgebra $b$ is recursive if for any $F$-algebra $a$, there is a unique ca-morphism $b \to a$. Recursivity of a coalgebra relates to the termination of that coalgebra when thought of as a program (c.f. [@adámek_lücke_milius_2007]). The following corollary connects (co)recursivity to the $\mu-\nu$ adjunction: A coalgebra $b:B\to FB$ for which $\mu (b)$ exists is recursive if and only if $\mu (b)$ is initial; similarly, an algebra $a:FA\to A$ for which $\nu (a)$ exists is corecursive if and only if $\nu (a)$ is terminal. *Proof.* This can be easily read off: $b$ is recursive if and only if $\mathsf{Hylo}(b, a)$ always has a unique element, and $\mu (b)$ is initial if and only if $F\text{-}\mathsf{Alg}(\mu (b), a)$ always has a unique element. Since $$\mathsf{Hylo}(b,a) \cong F\text{-}\mathsf{Alg}(\mu (b), a)$$ by definition, the equivalence follows. The second statement follows analogously. ◻ # Concrete Constructions of Relative Fixed Points {#sec:constructions} In this section we provide several concrete constructions of relative fixed points, using a presentation of $\mu$ and $\nu$ based on (co)free (co)algebras. In Examples [\[ex1\]](#ex1){reference-type="ref" reference="ex1"}, [\[ex2\]](#ex2){reference-type="ref" reference="ex2"} we explore relative fixed points of polynomial functors and discuss their interpretations. Next, in Proposition [\[polytopos\]](#polytopos){reference-type="ref" reference="polytopos"}, we construct a downward fixed point which classifies cartesian subcoalgebras in the sense of [@adamek2020wellfounded]. In Proposition [1](#sierp){reference-type="ref" reference="sierp"} we illustrate how the Sierpinski carpet may be constructed as a relatively terminal coalgebra. Lastly, we show in Example [\[ex0\]](#ex0){reference-type="ref" reference="ex0"}, how the depleted version of the adjunction, that is the Galois connection between post-fixed points and pre-fixed points mentioned in the introduction, may be useful for something called the safety problem. [\[polyexist\]]{#polyexist label="polyexist"} For a polynomial functor $F: \mathsf{Set}\to \mathsf{Set}$, each coalgebra admits a relatively initial algebra, and every algebra admits a relatively terminal coalgebra. *Proof.* Theorem [\[existencethm\]](#existencethm){reference-type="ref" reference="existencethm"} guarantees their existence if $F$ is accessible. Let $F = \sum_{i\in I}y^{X_i}$, and let $\lambda$ be a regular cardinal, such that $\lambda \geq \sup\{|X_i|\mid i\in I\}$. Then each $y^{X_i}$ is $\lambda$-accessible, and hence so is their coproduct $F$. ◻ In order to give explicit descriptions for $\mu$ and $\nu$ on $\mathsf{Set}$, we exploit the fact that free algebras and cofree coalgebras for polynomial functors on $\mathsf{Set}$ have elegant characterizations: Let $F:\mathsf{Set}\to\mathsf{Set}$ be the polynomial functor given by $FX = \sum_{\sigma\in\Sigma}X^{\mathsf{ar}(\sigma)}$. Then, 1. the free $F$-algebra on $X$, denoted $T^\Sigma(X)$, is given by the set of finite $\Sigma$-branching trees with leaves labeled by elements of $X$. Equivalently, $T^\Sigma(X)$ is the algebra of $\Sigma$-terms over $X$, known from universal algebra (see [@burris1981univalg] for further description of free algebras, as well as quotients of $F$-algebras). 2. The cofree $F$-coalgebra on $X$, denoted $C^\Sigma(X)$, is given by the set of finite and infinite $\Sigma$-branching trees with internal nodes labeled by elements of $X$. In order to make use of (co)free (co)algebras in describing $\mu$ and $\nu$, we employ the following construction: Let $\mathsf{C}$ be a category, and $F:\mathsf{C}\to \mathsf{C}$ an endofunctor. - Assume that every object $X$ in $\mathsf{C}$ admits a free algebra $T^FX$, with unit $\eta:X\to T^FX$ and free algebra structure $\alpha : FT^FX\to T^FX$. Then $\mu(b)$ is given by the coequalizer of the diagram $$\begin{tikzcd} T^{F}(B) \ar[r,shift left=0.5ex,"id"] \ar[r,shift right=0.5ex,"\mathsf{unfold}",swap] & T^{F}(B) \end{tikzcd}$$ in the category of $F$-algebras and where $\mathsf{unfold}$ is the free extension of the following map to $T^F(B)$ $$B \xrightarrow{b} FB \xrightarrow{F \eta} F T^F (B) \xrightarrow{\alpha_B} T^F (B)$$ - Assume that every object $X$ in $\mathsf{C}$ admits a cofree coalgebra $C^FX$, with counit $\eta:C^FX\to X$ and cofree coalgebra structure $\gamma : C^FX\to FC^FX$. Then $\nu(a)$ is given by the equalizer of the diagram $$\begin{tikzcd} C^{F}(A) \ar[r,shift left=0.5ex,"id"] \ar[r,shift right=0.5ex,"\mathsf{pred}",swap] & C^{F}(A) \end{tikzcd}$$ in the category of $F$-coalgebras where $\mathsf{pred}$ is the coextension of the following map to $C^F(A)$ $$C^{F}(A) \xrightarrow{\gamma_A} F C^F (A) \xrightarrow{F \epsilon} FA \xrightarrow{a} A$$ *Proof.* We only prove the statement for $\mu$, since the statement for $\nu$ follows by duality. Let $m:FM\to M$ be the coequalizer of $\text{id}$ and $\mathsf{unfold}$, with quotient map $q:T^F(B)\to M$. Let $a:FA\to A$ be an algebra, and assume $$\hfill\begin{tikzcd} FB \arrow[r, "Ff"] & FA\arrow[d, "a"]\\ B\arrow[u, "b"]\arrow[r, "f"] & A \end{tikzcd}\hfill$$ is a coalgebra-to-algebra morphism. We wish to find an algebra morphism $\tilde f:M\to A$ such that $\tilde f\circ (q\eta) = f$. Since $T^F(B)$ is the free $F$-algebra on $B$, there is a unique algebra morphism $\check f:T^F(B)\to A$ with $\check f\circ \eta = f$; it suffices to show that $\check f$ factors through $q$, or equivalently, that $\check f$ coequalizes $\text{id}$ and $\mathsf{unfold}$. Since $\check f\circ \text{id}= \check f$ and $\check f \circ \mathsf{unfold}$ are both algebra morphisms $T^FB\to A$, we only have to show that they agree on the generators; i.e., $$\label{eq:hatf}\hfill \hat f\circ \eta = \hat f\circ \mathsf{unfold}\circ \eta\hfill$$ To this end, consider the following diagram. $$\begin{tikzcd}[ampersand replacement=\&] \& B \&\&\& FB \\ \\ {}\&{T^F(B)} \& {T^F(B)} \&\& {FT^F(B)} \\ \\ \& A \&\&\& FA \arrow["{\check{f}}", from=3-3, to=5-2] \arrow["{\check{f}}"{description}, from=3-2, to=5-2] \arrow["\eta"{description}, from=1-2, to=3-2] \arrow["\mathsf{unfold}", from=1-2, to=3-3] \arrow["f", swap, from=1-2, to=5-2, bend right = 50] \arrow["a", from=5-5, to=5-2] \arrow["{F \check{f}}"{description}, from=3-5, to=5-5] \arrow["{F \eta}"{description}, from=1-5, to=3-5] \arrow["b", from=1-2, to=1-5] \arrow["{\alpha_B}", from=3-5, to=3-3] \arrow["Ff", from=1-5, to=5-5, bend left = 50] \arrow[phantom, from=3-2, to=3-3, "(*)", description] \end{tikzcd}$$ Facet $(*)$ is equation [\[eq:hatf\]](#eq:hatf){reference-type="ref" reference="eq:hatf"}, which is to be established. The outer square commutes, since by assumption $f$ is a coalgebra-to-algebra morphism. The top right square is the definition of $\mathsf{unfold}$, and the bottom right square commutes as $\hat f$ is an algebra morphism. ◻ Unpacking the above equalizers and coequalizers in the case of polynomial functors on $\mathsf{Set}$ gives the following corollary. The proof of this corollary is left to the reader. [\[prop:polychar\]]{#prop:polychar label="prop:polychar"} Let $F:\mathsf{Set}\to\mathsf{Set}$ be a polynomial functor, say $FX = \sum_{\sigma\in\Sigma}X^{\mathsf{ar}(\sigma)}$. 1. If $b:B\to FB$ is an $F$-coalgebra, then $\mu(b)$ is given by $$\hfill T^\Sigma(B)/\{x\sim b(x)\}\hfill$$ 2. If $a:FA\to A$ is an $F$-algebra, then $\nu(a)$ is given by $$\hfill\{t\in C^\Sigma(A) \mid \text{ if }\begin{tikzpicture}[shorten <=10pt, shorten >=10pt, baseline={([yshift=20pt]current bounding box.south)}] \node at (0,0) {$x$};\node at (-1,1.2) {$y_1$};\node at (1,1.2) {$y_k$};\node at (0,1.2) {$\dots$};\draw (0,0) -- (-1,1.2);\draw (0,0) -- (1,1.2);\node at (0,0.7) {$\sigma$};\end{tikzpicture}\text{ is a height one subtree of }t\text{, then }x = a(\sigma(\vec y))\}\hfill$$ This proposition also shows a connection between the $\nu$-construction, and coequations. To illustrate this, consider what happens in the $\mu$-construction: A coalgebra $b:B\to FB$ is treated as a '(flatly) recursive set of equations' $x \sim b(x)$. Then this set of equations can be used construct a quotient $\mu(b)$ of the free $F$-algebra. Comparing this to the coalgebra-to-algebra picture, it has been noted before that giving a coalgebra-to-algebra morphism $b\to a$ is akin to solving the system of equations presented by $b$ in the algebra $a$ . We propose that there is a dual perspective: rather than solving the system of equations $b$ in $a$, one could also see a coalgebra-to-algebra morphism as *solving the coequation $a$ in $b$*. To our knowledge, the 'coequations-as-algebras' perspective is new. We can leverage $\nu$ to fit it into the wider spectrum of coequational logic. As demonstrated in [@dahlqvist2021coeq], the most general definition of a coequation is 'a subcoalgebra of a cofree coalgebra'. Point (ii) of proposition [\[prop:polychar\]](#prop:polychar){reference-type="ref" reference="prop:polychar"} then shows how each algebra gives rise to a canonical coequation. As a final note, we should highlight an important difference between our current approach to (co)equations, and the one common in universal (co)algebras: in the latter, the main notion is that of *satisfaction* of (co)equations, whereas we focus on *solving* (co)equations. A coequation $E\subseteq C^\Sigma(X)$ is satisfied by $b:B\to FB$ if every coalgebra-to-algebra morphism $B\to C^\Sigma(X)$ factors through $E$. It can quickly be seen that for coequations of the form $\nu(a)$, a coalgebra $b:B\to FB$ satisfies $\nu(a)$ if and only if *every* map $B\to A$ is a coalgebra-to-algebra morphism. Such a situation is exceedingly rare. This also shows that only particular coequations can be described as $\nu(a)$. There are ubiquitous examples of the above theorems. [\[ex1\]]{#ex1 label="ex1"} Let $F: \mathsf{Set}\to \mathsf{Set}$ be the functor given by $FX = \{\times,\checkmark\} \times X$. Let $a$ be the algebra $a \colon F(X) \to X$ with carrier $X=\{0,1\}$ given by $$\hfill (\checkmark, s) \mapsto s \text{ and }(\times, s) \mapsto 1-s \hfill$$ where $s$ is either $0$ or $1$. The algebra may be depicted as $$\hfill\begin{tikzcd} 0 \ar[loop left]{l}{\times} \ar[r,"\checkmark"] & \ar[l] 1 \ar[loop right]{r}{\times} \end{tikzcd} \hfill$$ Then $\nu(a)$ has a carrier given by $$\hfill \{ \begin{pmatrix} u_1 \\ s_1\end{pmatrix}\begin{pmatrix} u_2 \\ s_2\end{pmatrix} \begin{pmatrix} u_3 \\ s_3\end{pmatrix} \cdots \in (\{\times,\checkmark\}\times X)^{\omega} \quad | \quad u_i = \times \implies s_i=s_{i+1} \text{ and } u_i= \checkmark \implies s_i=1-s_{i+1}\} \hfill$$ i.e. the subset of streams in $(\{\times,\checkmark\})^{\omega}$ which follow the action of $a$ *when read from right to left*. Given a coalgebra $b : B \to \{\times,\checkmark\} \times B$, a coalgebra-to-algebra morphism may represent a solution to the constraint represented by $a$. That is, we divide the states of $B$ into two classes, such that the division 'respects the algebra structure on $A$'. If $m$ is such a marking, we obtain $$\hfill \begin{tikzcd} \ar[d,"\hat m ",swap] B \ar[r] & \{\times,\checkmark\} \times B \ar[d,"F{\hat m}"] \\ \nu(a) \ar[r] & \{\times, \checkmark\} \times \nu(a) \end{tikzcd} \hfill$$ via the universal property of $\nu$. Intuitively, $\hat m$ maps a state $x$ to the stream of 'tags and classes' that are observed when running $b$ forwards. The constraint on $m$ then states that whenever a $\checkmark$ is observed, the class must change, whereas whenever a $\times$ is observed, the class must stay the same. A marking satisfying this constraint exists, if and only if on each cycle in $B$, the number of $\checkmark$'s is even. [\[ex2\]]{#ex2 label="ex2"} Consider the coalgebra $b$ for the functor $FX = \{ \times, \checkmark\} \times X^{\{a,b\}}$ as depicted in figure [\[fig:automaton\]](#fig:automaton){reference-type="ref" reference="fig:automaton"} with carrier given by $X=\{q_0,q_1,q_2\}$. Then the carrier of $\mu(b)$ is given by $$\frac{\text{ finite }\{a,b\}\text{ branching trees with $\{\times,\checkmark\}$ labeling internal nodes and $X$ labeling leaves}}{q_0 \cong q_1 \xleftarrow{a} \times \xrightarrow{b} q_2, q_1 \cong q_0 \xleftarrow{a} \times \xrightarrow{b} q_1, q_2 \cong q_2 \xleftarrow{a} \checkmark \xrightarrow{b} q_2}$$ where the quotient denotes a quotient in $\mathsf{Set}$, i.e. the set in the numerator modulo the smallest congruence relation satisfying the tree equations in the denominator. Intuitively, $\mu(b)$ has all finite trees but leaves may be replaced with the equations in the numerator in a recursive and transitive way. One may also see it as terms over the 2 binary operations $\times$ and $\checkmark$ in the three unknowns $\{q_0,q_1,q_2\}$, where $q_0,q_1,q_2$ satisfy a mutual recursive relationship. [\[ex:polytopos\]]{#ex:polytopos label="ex:polytopos"} Let $F:\mathsf{Set}\to\mathsf{Set}$ be the polynomial functor given by $FX = \sum_{\sigma\in \Sigma}X^{\mathsf{ar}(\sigma)}$. Since polynomial functors preserve pullbacks, it follows from Corollary 3.2 in [@johnstone2001topos] that $F\text{-}\mathsf{Coalg}$ is an (elementary) topos. Its subobject classifier $\Omega$ is the coalgebra of 'non-decreasing $\Sigma$-trees'; that is, the points of $\Omega$ are $\mathbf{2}$-labeled $\Sigma$-trees, where the label of a child may not be smaller than the label of its parent. $\Omega$ is not a fixed point unless $F$ is trivial; however, there is a subcoalgebra $\Omega_{\mathsf{cart}}$ which is a fixed point, and arises as $\nu$ of a well-chosen algebra. Consider the algebra $\bigwedge:F\mathbf{2}\to\mathbf{2}$, explicitly $$\hfill \bigwedge:\sigma(x_1,\dots, x_n) \mapsto \begin{cases}1 & x_i = 1\text{ for all }i = 1,\dots, n\\0&\text{ otherwise}\end{cases} \hfill$$ Then $\nu(\bigwedge)$ is a subcoalgebra of $\Omega$; it consists of those non-decreasing $\Sigma$-trees where zeroes 'cannot disappear', i.e. if a node is labeled with $0$, at least one of its children is labeled with $0$. $\Omega_{\mathsf{cart}}$ satisfies a universal property similar to the subobject classifier in $F\text{-}\mathsf{Coalg}$; but instead of classifying *all* subcoalgebras, it classifies only the *cartesian* subcoalgebras i.e., those subcoalgebras $s:S\leq X$ such that the square $$\hfill \begin{tikzcd} S\arrow[r, "s", tail]\arrow[d] & X\arrow[d]\\ FS\arrow[r, "Fs", tail] & FX \end{tikzcd}\hfill$$ is a pullback square. Explicitly, that means that there is a map $\top:Z\to \Omega_\mathsf{cart}$ (with $Z$ the terminal coalgebra), such that for each coalgebra $X$ and each cartesian subobject $S\leq X$, there is a unique map $s:X\to \Omega_\mathsf{cart}$ such that $$\hfill \begin{tikzcd} S\arrow[r]\arrow[d, tail] \arrow[dr, phantom, very near start, " "{pullback}]& Z\arrow[d, "\top"]\\ X\arrow[r, "s"] & \Omega_\mathsf{cart} \end{tikzcd} \hfill$$ is a pullback square. Ordinary subobjects are understood as 'forward stable subsets': they are subsets $S$ such that if $s\in S$, then so are all the successors of $s$. Cartesian subcoalgebras are those subsets which also satisfy the converse implication: if all successors of $s$ are in $S$, then so is $s$. More formally, let $\xi:X\to FX$ be a coalgebra, and consider the 'next-time modality' $\fullmoon:P(X)\to P(X)$ from [@jacobs2017introduction], defined on a subobject $U\leq X$ via the pullback $$\hfill \begin{tikzcd} \fullmoon(U)\arrow[r, tail]\arrow[d]\arrow[dr, phantom, " "{pullback}, very near start]& X\arrow[d, "\xi"]\\ FU\arrow[r, tail] & FX \end{tikzcd} \hfill$$ Then subcoalgebras are subsets $P\subseteq X$ such that $P\subseteq \fullmoon P$; these are classified by $\Omega$. In [@adamek2020wellfounded], they show that Cartesian subcoalgebras are fixed points for $\fullmoon$, i.e. they satisfy $P = \fullmoon P$. [\[polytopos\]]{#polytopos label="polytopos"} $\Omega_\mathsf{cart}$ classifies cartesian subcoalgebras. *Proof.* We wish to show that $\Omega_{\mathsf{cart}} = \nu(\bigwedge)$ classifies cartesian subobjects. We first prove that cartesian subobjects are closed under pullbacks. Assume $P\leq X$ is a cartesian subcoalgebra. Let $y:Y\to X$ be a coalgebra morphism. Then consider the following cube: $$\hfill\begin{tikzcd} Fy^*P && FY \\ & FP && FX \\ y^*P && Y \\ & P && X \arrow[from=1-1, to=1-3] \arrow[from=3-1, to=1-1] \arrow[from=3-3, to=1-3] \arrow[from=3-1, to=3-3] \arrow[from=1-1, to=2-2] \arrow[from=1-3, to=2-4] \arrow[from=2-2, to=2-4, crossing over] \arrow[from=4-4, to=2-4] \arrow[from=4-2, to=2-2, crossing over] \arrow[from=4-2, to=4-4] \arrow[from=3-3, to=4-4] \arrow[from=3-1, to=4-2] \end{tikzcd}\hfill$$ Note that since $F$ preserves pullbacks, we obtain a unique arrow $y^*P\to Fy^*P$. In the above cube, the front square is a pullback since $P$ is strong, and the bottom square is a pullback by definition of $y^*$. Hence, we see that taking the top and back square together, as in $$\hfill \begin{tikzcd} y^*P\arrow[r]\arrow[d] & Y\arrow[d]\\ Fy^*P\arrow[r]\arrow[d] & FY\arrow[d]\\ FP\arrow[r] & FX \end{tikzcd}\hfill$$ the outer square is a pullback. The bottom square is also a pullback, since $F$ preserves pullbacks; hence the top square is a pullback, which shows that $y^*P$ is cartesian. Now consider the terminal object $Z$ in $F\text{-}\mathsf{Coalg}$; this is the coalgebra of finite and infinite $\Sigma$-branching trees. We note that $\top:Z\to \Omega$, which maps a tree $t$ to $t$ constantly labeled with $1$, factors through $\Omega_\mathsf{cart}$; and moreover $\top$ is a *cartesian* subcoalgebra of $\Omega_\mathsf{cart}$. So whenever $P\leq X$ is a pullback of $\top:Z\to\Omega_\mathsf{cart}$, $P$ is a cartesian subcoalgebra. Uniqueness of classifiers $X\to \Omega_\mathsf{cart}$ follows from uniqueness of classifiers $X\to\Omega$, so it suffices to show that if $P\leq X$ is cartesian, there exists a classifier $\ulcorner P\urcorner:X\to \Omega_\mathsf{cart}$. We know that $F\text{-}\mathsf{Coalg}(X, \Omega_\mathsf{cart})\cong \mathsf{Hylo}(X, \bigwedge)$, so we may equivalently provide a coalgebra-to-algebra map $X\to 2$. We claim that the characteristic function $$\hfill \chi_P:x\mapsto \begin{cases}1 & x\in P\\0 & \text{ otherwise}\end{cases} \hfill$$ is a coalgebra-to-algebra morphism. For, consider an arbitrary $x\in X$. Let $\xi(x) = \sigma(x_1,\dots, x_n)$. We consider two cases. 1. If $x\in P$, then since $P$ is a subcoalgebra, we know $x_i\in P$ for all $i$; hence, $$\hfill\bigwedge(\sigma(\chi_P(x_1),\dots, \chi_P(x_n))) = \bigwedge(\sigma(1,\dots, 1)) = 1 = \chi_P(x).\hfill$$ 2. If $x\notin P$, then it suffices to show that at least one of the $x_i$ is also not in $P$. Assume towards a contradiction that $x_i\in P$ for all $i$. Then the following square commutes: $$\hfill \begin{tikzcd} \{*\}\arrow[r, "{*\mapsto x}"]\arrow[d, "{*\mapsto \sigma(x_1,\dots, x_n)}", swap]&X\arrow[d, "\xi"]\\ FP\arrow[r] & FX \end{tikzcd} \hfill$$ hence since $P$ was cartesian, we conclude that the map $*\mapsto x$ factors through the inclusion $P\rightarrowtail X$. But this amounts to saying $x\in P$, which is not the case. We conclude that there is an $x_i$ with $\chi_P(x_i) = 0$, and hence $$\hfill \bigwedge(\sigma(\chi_P(x_1),\dots, \chi_P(x_n))) = 0 = \chi_P(x).\hfill$$ So in both cases, we have $\bigwedge(F\chi_P(\xi(x))) = \chi_P(x)$, which shows that $\chi_P$ is a coalgebra-to-algebra morphism. We conclude that there is a unique coalgebra morphism $\ulcorner P\urcorner:X\to \Omega_\mathsf{cart}$ such that $\chi_P = h\circ \ulcorner P\urcorner$, where $h:\Omega_\mathsf{cart}\to \mathbf{2}$ is the universal coalgebra-to-algebra morphism, mapping a labeled $\Sigma$-tree to the label of its node. We still need to show that $P$ is the pullback of $\top$ along $\ulcorner P\urcorner$. Note, however, that $$\hfill \begin{tikzcd} Z\arrow[r]\arrow[d, "\top"] & 1\arrow[d, "{*\mapsto 1}"]\\ \Omega_\mathsf{cart}\arrow[r, "h"] & \mathbf{2} \end{tikzcd} \hfill$$ is a pullback square, since if the root node of a non-decreasing $\Sigma$-tree $t$ is labeled by 1, then so are all the other nodes in $t$, and hence $t$ is in the image of $\top$. Hence, we can fill in the following diagram: $$\hfill \begin{tikzcd} P\arrow[d, tail]\arrow[r] & Z\arrow[d, "\top"] \arrow[r] & 1\arrow[d, "{*\mapsto 1}"]\\ X\arrow[r, "\ulcorner P\urcorner"]\arrow[rr, "\chi_P", bend right] & \Omega_\mathsf{cart}\arrow[r, "h"] & \mathbf{2} \end{tikzcd} \hfill$$ Here, the outer square is a pullback, since $\chi_P$ classifies $P$ in $\mathsf{Set}$, and we have just shown that the right-hand side is a pullback as well. Therefore, the left-hand square is a pullback, which finishes the proof. ◻ In *Sierpinski Carpet as a Final Coalgebra* ([@noquez2021sierpinski]) Moss and Noquez provide a construction of the Sierpinski carpet as a final coalgebra in a category of 'square metric spaces'. In this section we recall this work and then show how the downward fixed point construction $\nu$ gives a more direct way of constructing the Sierpinski carpet as a final coalgebra. Let $\blacksquare$ denote the set $[0,1]^2$ where $[0,1]$ is the real unit interval. Let $\square$ denote the boundary of $\blacksquare$ or explicitly $$\hfill \square = \{ (i,r) : i \in \{0,1\}, r \in [0,1] \} \cup \{ (r,i) : r \in [0,1], i \in \{0,1\} \} \hfill$$ Let $\mathsf{MS}$ be the category whose objects are metric spaces with diameter less than $2$ and whose morphisms are short maps $f : (X,d) \to (X',d')$ i.e. a function $f : X \to X'$ such that $d(x,y) \leq d'(f(x),f(y))$. We are interested in two different metrics on $\square$: - The path metric $d_p : \square \times \square \to \mathbb{R}$ with $d_p(x,y)$ given by the length of the shortest path in $\square$ between $x$ and $y$. - The taxicab metric $d_t : \square \times \square \to \mathbb{R}$ given by $d_t((s,r),(s',r'))=|s'-s| + |r'-r|$. A square metric space is a metric space $(X,d)$ equipped with an injective function $S : \square \hookrightarrow X$ such that for all $x,y \in \square$, $$\hfill d_t(x,y) \leq d(S(x),S(y)) \leq d_p(x,y) \hfill$$ A morphism of square metric spaces $f : (X,S) \to (X',S')$ is a short map $f : X \to Y$ such that $S = S' \circ f$. This defines a category $\mathsf{SqMS}$ of square metric spaces and their morphisms. $\square$ with the identity function is the initial algebra in square metric spaces. We now define an endofunctor on square metric spaces for which the Sierpinski carpet is a fixed point. We present the following definitions informally. The full definitions may be found in [@noquez2021sierpinski]. Let $M$ be the set $\{0,1,2\}^2 / (1,1)$. For a square metric space $S : \square \to X$, $M \otimes X$ is eight copies of $X$ in a three-by-three grid with the center removed. Mathematically, $M\otimes X$ is the cartesian product $M \times X$ modulo the smallest equivalence relation which identifies the boundaries of the subsquares with each other. We write $m \otimes x$ to denote the equivalence class of $(m,x)$ in $M \otimes X$. We equip $M \otimes X$ with the structure of a square metric space. $M \otimes X$ is equipped with a metric given by scaling down the metric of $X$ by $\frac{1}{3}$ in each copy of $X$. If $x$ and $y$ live in adjacent copies, their distance is set such that the sum of distances to the shared boundary is minimized. For all other points, the distance is set to $2$. There is a map $\square \to M \otimes X$ which maps $\square$ injectively to the outer boundary. For a short map $f : X \to Y$, there is a short map $M \otimes f : M \otimes X \to M \otimes Y$ given by $m \otimes x \mapsto m \otimes f(x)$. This defines a functor $$\hfill M \otimes - \colon\mathsf{SqMS}\to \mathsf{SqMS}\hfill$$ As shown in [@noquez2021sierpinski], $M \otimes -$ has an initial algebra. $\square$ is an initial object in $\mathsf{SqMS}$ so the initial algebra may be found by taking the colimit of the usual chain $$\hfill \square \to M \otimes \square \to M \otimes M \otimes \square \to \ldots \hfill$$ As $\mathsf{SqMS}$ does not have a final object, we cannot construct a final coalgebra by taking the limit of the dual of this chain. However, our construction $\nu$ does not require a final object in the base category. {#sierp} \ The square metric space $M \otimes \blacksquare$ is the same as $\blacksquare$ except with the middle removed. There is an algebra $a \colon M \otimes \blacksquare \to \blacksquare$ given by the natural inclusion. As illustrated in Figure [1](#sierp){reference-type="ref" reference="sierp"}, the Sierpinski carpet is given by the downward fixed point $\nu$ applied to this algebra. The downward fixed point $\nu(\blacksquare \leftarrow M \otimes \blacksquare)$ is the Sierpinski carpet. *Proof.* Because every morphism in the chain $$\hfill \blacksquare \leftarrow M \otimes \blacksquare \leftarrow M \otimes M \otimes \blacksquare \ldots \hfill$$ is an injection, its limit is the intersection $$\hfill\bigcap_{n=0}^{\infty} M^n \otimes \blacksquare \hfill$$ This infinite intersection is the usual definition of the Sierpinski carpet. ◻ We have seen that the Sierpinski carpet may be obtained in a more straightforward way than in [@noquez2021sierpinski] using a relatively initial or terminal fixed point. Other fractals may be generated as downward fixed points in a similar way; for example one can imagine that the Sierpinski triangle may constructed as a downward fixed point in a category of 'triangular metric spaces'. [\[ex0\]]{#ex0 label="ex0"} In [@kori2023exploiting], the authors state the safety problem. This problem may be rephrased in terms of the Galois connection $$\hfill \begin{tikzcd} Post(F) \ar[r,bend left,"\mu_F"] \ar[r,phantom,"\bot"] & Pre(F) \ar[l,bend left,"\nu_F"] \end{tikzcd} \hfill$$ for a particular choice of $F$ and assuming that the set of initial states forms a post-fixed point. A transition system is a triple $(S,I,\delta)$ where $S$ is a set of states, $I \subseteq S$, is a set of initial states, and $\delta \colon S \to \mathcal{P}(S)$ is a transitition relation. Here $\mathcal{P}(S)$ is the power-set of $S$ which is a complete lattice ordered by $\subseteq$. Let $F: \mathcal{P}(S) \to \mathcal{P}(S)$ be the monotone function defined by $F(X)=\bigcup_{x \in X} \delta(x)$ and suppose that $I$ is a post fixed point, i.e., $I \subseteq F(I)$. For a set $P \in \mathcal{P}(S)$, the **the safety problem** for $(I,P,S,F)$ asks if $\mu_F(I) \subseteq P$. The idea here is that $\mu_F(I)$ is the set of reachable states from $I$ and if $\mu_F(I) \subseteq P$, then we say that $I$ is $P$-safe. Now suppose that $P$ is a pre-fixed point $F(P) \subseteq P$. Then the adjunction of this paper says that $$\mu_F(I) \subseteq P \iff I \subseteq \nu_F(P)$$ While $\mu_F(I)$ represents the states reachable from $I$, $\nu_F(P)$ are the states which never go above $P$. In this case the adjunction suggests a strategy for verifying the safety problem. One may answer the safety problem by simultaneously unfolding $I$ and $P$ using $F$. In other words on the first step we check if $I \subseteq P$ if it is then we check $F(I) \subseteq P$ and $I \subseteq F(P)$. If either of those are false, then we know $I$ is not $P$-safe. If both are true then we continue to check $F^2(I) \subseteq P$ and $I \subseteq F^2(P)$. We continue this process indefinitely, checking to see if any of $F^n(I) \subseteq P$ and $I \subseteq F^n(P)$ are false. If we can't falsify any of these inclusions and we arrive at a fixed point (either $\mu_F(I)$ or $\nu_F(P)$), then we know that $I$ is $P$-safe. A major limitation of this approach is that we require $I$ to be increasing and $P$ to be decreasing. In other cases, a different analysis will be necessary to verify safety. Regardless, we believe these ideas may be used to develop an effective algorithm for the safety problem. # Preservation results {#sec:preservation} In this section, we explore when functors preserve $\mu$ and $\nu$. To this end, we take inspiration from [@capretta2006recursive], and focus on an adjoint situation equipped with a 'step' $\theta$. This requires the ingredients depicted in equation [\[eq:step\]](#eq:step){reference-type="ref" reference="eq:step"}. $$\label{eq:step}\hfill \begin{tikzcd}[cells={nodes={minimum size=0.8cm}}] \mathsf{C}\arrow[loop left, "F"] \arrow[r, bend left, "L"{name=L}] & \mathsf{D}\arrow[loop right, "G"]\arrow[l, bend left, "R"{name=R}] \arrow[phantom, "\vdash" marking, from=R, to=L] \end{tikzcd} \qquad \theta:LF\Rightarrow GL \hfill$$ We note that such a $\theta$ comes equipped with its *mate* $\theta^\flat:FR\to RG$ (and indeed this mate correspondence is a bijection, as shown in [@kelly1974el2cat]). This situation covers a wide range of examples. Of particular interest are those cases where $\mathsf{D}$ is an Eilenberg-Moore category $\mathsf{C}^T$ or Kleisli category $\mathsf{Kl}(T)$ for a monad $T$ on $\mathsf{C}$. In these cases, the existence of a lifting $\bar F$ of an endofunctor $F:\mathsf{C}\to\mathsf{C}$ is equivalent to the existence of a step. Consider the data of Scenario [\[eq:step\]](#eq:step){reference-type="ref" reference="eq:step"}. $L$ extends to a functor $\bar L:F\text{-}\mathsf{Coalg}\to G\text{-}\mathsf{Coalg}$ given by $$\hfill \begin{tikzcd} FB\\ B\arrow[u,"b"] \end{tikzcd} \qquad\mapsto \qquad \begin{tikzcd} GLB\\ LFB\arrow[u, "\theta"]\\ LB\arrow[u, "Lb"] \end{tikzcd} \hfill$$ Similarly, $R$ extends to a functor $\bar R:G\text{-}\mathsf{Alg}\to F\text{-}\mathsf{Alg}$ given by $$\hfill \begin{tikzcd} GA\arrow[d, "a"]\\ A \end{tikzcd} \qquad\mapsto\qquad \begin{tikzcd} FRA\arrow[d, "\theta^\flat"]\\ RGA\arrow[d, "Ra"]\\ RA \end{tikzcd} \hfill$$ These functors $\bar L$ and $\bar R$ satisfy something akin to an adjoint relationship. Before stating this relationship, we recall the following ('useful') lemma: [\[lemma:useful\]]{#lemma:useful label="lemma:useful"} If $\theta:LF\to GL$ is a step, with mate $\theta^\flat$, the following two squares commute: $$\hfill \begin{tikzcd} F\arrow[d, "\eta_F"] \arrow[r, "F\eta"] & FRL\arrow[d, "\theta^\flat_L"] & LFR\arrow[d, "\theta_R"] \arrow[r, "L\theta^\flat"] & LRG\arrow[d, "\epsilon_G"]\\ RLF\arrow[r, "R\theta"] & RGL & GLR\arrow[r, "G\epsilon"]& G \end{tikzcd} \hfill$$ See e.g. [@rot2021steps] for a proof. [\[lemma:baradj\]]{#lemma:baradj label="lemma:baradj"} Let $b:B\to FB$ be an $F$-coalgebra, and $a:GA\to A$ a $G$-algebra. The natural isomorphism $\mathrm{Hom}_\mathsf{D}(LB, A)\cong \mathrm{Hom}_\mathsf{C}(B, RA)$ restricts to a natural isomorphism $$\hfill \mathsf{Hylo}(\bar Lb, a)\cong \mathsf{Hylo}(b, \bar Ra) \hfill$$ *Proof.* Fix a coalgebra-to-algebra morphism $\phi:\bar Lb\to a$. Consider $\phi$'s transpose $\tilde\phi = R\phi\circ\eta$ along the adjunction. We claim that $\tilde \phi$ is a coalgebra-to-algebra morphism $b\to \bar Ra$. This can be seen in the following diagram: $$\hfill \begin{tikzcd} FRLB\arrow[rr, "FR\phi"]\arrow[dr, "\theta^\flat"] && FRA\arrow[d, "\theta^\flat"]\\ FB\arrow[u, "F\eta"]\arrow[dr, "\eta"]& RGLB\arrow[r, "RG\phi"] & RGA\arrow[dd, "Ra"]\\ & RLFB\arrow[u, "R\theta"]\\ B\arrow[uu, "b"] \arrow[r, "\eta"] & RLB\arrow[u, "RLb"] \arrow[r, "R\phi"] &RA \end{tikzcd} \hfill$$ Here, the bottom right square is the ca-morphism square for $\phi$; the top right is a naturality square for $\theta^\flat$; the top left is given by lemma [\[lemma:useful\]](#lemma:useful){reference-type="ref" reference="lemma:useful"}; and the bottom left is naturality for $\eta$. The outside of the square is a ca-morphism square for $\tilde\phi$. On the other hand, let $\psi:b\to \bar Ra$ be a ca-morphism. We claim that its transpose is again a ca-morphism $\bar Lb\to a$. This is completely dual to the previous case; but for completeness, it can be seen in the following diagram: $$\hfill \begin{tikzcd} GL(B)\arrow[rr, "GL\psi"] &&GLRA\arrow[d, "G\epsilon"]\\ LF(B)\arrow[u, "\theta"]\arrow[r, "LF\psi"] & LFRA\arrow[ur, "\theta"]\arrow[d, "L\theta^\flat"]&GA\arrow[dd, "a"]\\ &LRGA\arrow[d, "LRa"]\arrow[ur, "\epsilon"]\\ L(B)\arrow[uu, "b"] \arrow[r, "L\psi"] & LRA\arrow[r, "\epsilon"] & A \end{tikzcd} \hfill$$ ◻ In [@capretta2006recursive], it was shown that $\bar L$ preserves recursive coalgebras, and (dually) $\bar R$ preserves corecursive algebras. This now follows directly from the above lemma; however, we can obtain the stronger result that $\bar L$ commutes with the induced comonad $\nu\mu$, and $\bar R$ commutes with the induced monad $\mu\nu$. [\[thm:steppreserve\]]{#thm:steppreserve label="thm:steppreserve"} Consider an adjoint situation as in [\[eq:step\]](#eq:step){reference-type="ref" reference="eq:step"}. Let $b:B\to FB$ be an $F$-coalgebra, and $a:GA\to A$ a $G$-algebra. 1. $\nu\mu(\bar Lb) = \bar L(\nu\mu(b))$ 2. $\mu\nu(\bar Ra) = \bar R(\mu\nu(a))$ *Proof.* We only prove (i), since (ii) follows by duality. Let $b$ be a (fixed) $F$-coalgebra, and $a$ a $G$-algebra. By remark [\[rem:flip\]](#rem:flip){reference-type="ref" reference="rem:flip"}, we know that $\nu\mu(b) = \mu(b)^{-1}$, and $\nu\mu(\bar Lb) = \mu(\bar Lb)^{-1}$; hence, it suffices to show $$\mu(\bar Lb) = (\bar L(\mu (b)^{-1}))^{-1}$$ Using lemma [\[lemma:baradj\]](#lemma:baradj){reference-type="ref" reference="lemma:baradj"}, we have the following chain of equivalences, natural in $a$: $$\begin{aligned} \mathsf{Hylo}(\bar Lb,a)&\cong \mathsf{Hylo}(b, \bar Ra)\\ &\cong \mathrm{Hom}_{\mathsf{Alg}F}(\mu(b), \bar Ra)\\ &\cong \mathsf{Hylo}(\mu(b)^{-1}, \bar Ra)\\ &\cong \mathsf{Hylo}(\bar L(\mu(b)^{-1}),a)\\ &\cong \mathrm{Hom}_{\mathsf{Alg}G}(\bar L(\mu(b)^{-1})^{-1}, a)\end{aligned}$$ ◻ This general theorem can be used to prove preservation in various specific circumstances. [\[coro:monad\]]{#coro:monad label="coro:monad"} Let $T:\mathsf{C}\to\mathsf{C}$ be a monad, let $F:\mathsf{C}\to \mathsf{C}$ be an endofunctor. Write $j\dashv U$ for the Kleisli adjunction of the monad, and $T\dashv |-|$ for the Eilenberg-Moore adjunction. 1. Assume that $F$ extends to a functor $\bar F:Kl(T)\to Kl(T)$ with $\bar Fj = jF$. Then $j$ commutes with $\mu$ and $U$ commutes with $\nu$. 2. Assume that $F$ extends to a functor $\bar F:EM(T)\to EM(T)$ with $\bar FT = TF$. Then $T$ commutes with $\mu$ and $|-|$ commutes with $\nu$. *Proof.* Both of these are instances of adjoint situation [\[eq:step\]](#eq:step){reference-type="ref" reference="eq:step"} with the step given by identities, and hence the statement follows immediately from [\[thm:steppreserve\]](#thm:steppreserve){reference-type="ref" reference="thm:steppreserve"}. ◻ ## $\mu$ and $\nu$ Coincide in a Dagger Category {#mu-and-nu-coincide-in-a-dagger-category .unnumbered} When coalgebras for a polynomial functor $F:\mathsf{Set}\to \mathsf{Set}$ are interpreted as $F$-shaped automata, the initial $F$-algebra serves as finite trace semantics and the terminal $F$-coalgebra gives an infinite trace semantics. When $F$ is no longer a $\mathsf{Set}$-functor this interpretation breaks down. For example if $F: \mathsf{Rel}\to \mathsf{Rel}$, where $\mathsf{Rel}$ is the category of sets and relations, then the initial algebra and terminal coalgebra coincide [@smyth1982category]. In [@karvonen2019way], it is shown that this holds more generally in any dagger category. With this coincidence, the initial algebra/final coalgebra gives a finite trace semantics instead of an infinite trace semantics. To obtain a semantics for infinite traces, Urabe and Hasuo construct an object which is weakly terminal among coalgebras and define the infinite trace semantics as the maximal map into this object [@hasuo2018coalgebraic]. Note that the limit colimit coincidence causes no issues when $\mu(c)$ is interpreted as a semantic object for $c$. However, a generalized limit colimit coincidence also holds for the fixed points generated by $\mu$ and $\nu$. A dagger category $(\mathsf{C},\dag)$ is a category equipped with an identity on objects functor $\dag: \mathsf{C}\to \mathsf{C}^{\mathrm{op}}$ such that $\dag^2=id$. Suppose that $(\mathsf{C},\dag)$ is a dagger category with limits and colimits of countable chains and $F: \mathsf{C}\to \mathsf{C}$ is a dagger functor preserving such limits and colimits. Then there is an isomorphism $$\hfill \mu(c)^\dag \cong \nu(c^\dag)\hfill$$ for each coalgebra $c$. Dually, for each algebra $a$, there is an isomorphism $\nu(a)^\dag \cong\mu(a^\dag)$. This theorem may be viewed as a special case of Theorem [\[thm:steppreserve\]](#thm:steppreserve){reference-type="ref" reference="thm:steppreserve"} but it is simpler to use the construction as a (co)limit. *Proof.* For a coalgebra $X \xrightarrow{c} FX$ we have $$\begin{aligned} \nu(c^{\dag}) & \cong \lim(X \xleftarrow{c^{\dag}} FX \xleftarrow{Fc^{\dag}} F^2X \leftarrow \ldots ) \\ &\cong \mathrm{colim}_{C^{\mathrm{op}}} (X \xleftarrow{c^{\dag}} FX \xleftarrow{Fc^{\dag}} F^2X \leftarrow \ldots ) \\ &\cong \mathrm{colim}(X \xrightarrow{c} FX \xrightarrow{Fc} F^2X \to \ldots )^{\dag} \\ &\cong \mu(c)^{\dag}\end{aligned}$$ The second isomorphism is because limits in $\mathsf{C}$ are colimits in $\mathsf{C}^{op}$ and the third isomorphism is because $\dag$ preserves colimits because it is an equivalence. A similar proof holds for the dual statement. ◻ # Conclusion {#sec:conclusions} In this paper, we have studied the relative fixed points of functors in a variety of contexts. In some of these, the fixed points from these functors have previously been presented as initial algebras or final coalgebras. In other cases, the fixed points are novel, as is the case with polynomial functors. Relative fixpoints provide a fresh perspective on ca-morphisms. Previous work has mostly focused on cases where there is a unique ca-morphisms, via the notions of recursive algebras and corecursive coalgebras [@capretta2009corecursive]. However, in [@hauhs2015scientific], the authors argue that ca-morphisms also hold interest when they are not unique. Using examples in probability, dynamical systems, and game theory, the authors show how non-unique ca-morphisms often represent solutions to problems in these disciplines. This gives us hope that relative fixed points and the results we have proven about them may be useful in these applications as well. In particular, in future work we will develop the algorithm suggested in [\[ex0\]](#ex0){reference-type="ref" reference="ex0"} and expand its capabilities to solve a wider range of problems. Another direction of future work is to understand the connection between relatively terminal coalgebras and coequations. As discussed in section [3](#sec:constructions){reference-type="ref" reference="sec:constructions"}, $\nu (a)$ may be thought of as a 'cofree solution of the coequation $a$'. As such, studying $\nu$ may yield new insights into this class of '(flatly) corecursive coequations', and the kind of properties that may be defined by such. [^1]: Leverhulme Trust Research Project Grant RPG-2020-232

arxiv_math

--- abstract: | For any integer $n>0$, the $n$-th canonical stability index $r_n$ is defined to be the smallest positive integer so that the $r_n$-canonical map $\Phi_{r_n}$ is stably birational onto its image for all smooth projective $n$-folds of general type. We prove the lifting principle for $\{r_n\}$ as follows: $r_n$ equals to the maximum of the set of those canonical stability indices of smooth projective $(n+1)$-folds with sufficiently large canonical volumes. Equivalently, there exists a constant $\mathfrak{V}(n)>0$ such that, for any smooth projective $n$-fold $X$ with the canonical volume $\text{\rm vol}(X)>{\mathfrak V}(n)$, the pluricanonical map $\varphi_{m,X}$ is birational onto the image for all $m\geq r_{n-1}$. address: - Meng Chen, School of Mathematical Sciences, Fudan University, Shanghai 200433, China - Hexu Liu, Shanghai Center for Mathematical Sciences, Fudan University, Jiangwan Campus, Shanghai, 200438, China author: - Meng Chen, Hexu Liu title: A lifting principle for canonical stability indices of varieties of general type --- # Introduction Throughout we work over the complex number field ${\mathbb C}$. Within birational geometrty, an important post-MMP mission might be to work out the exact boundedness of given classes of varieties. The boundedness is usually determined by some key birational invariants such as the canonical (or anti-canonical) volume, the canonical (or anti-canonical) stability index, the Iitaka-fibration index, etc. These are crucial to explicit classifications since they directly disclose the existence of relevant moduli spaces of varieties in question. We start with considering a smooth projective variety $X$ of general type. The *canonical stability index* of $X$ is a birational invariant, which is defined as: $$r_s(X):=\min\{l \in \mathbb{Z}_{>0}|\ \text{$\varphi_{m,X}=\Phi_{|mK_X|}$ is birational for all $m \geq l$}\}.$$ By the work of Hacon-McKernan ([@HM06]), Takayama ([@Tak06]) and Tsuji ([@Tsu06]), for any integer $n > 0$, there is a number $r_n \in \mathbb{Z}_{>0}$ such that the pluricanonical map $\phi_{m}$ is birational onto its image for all $m \geq r_n$ and for all smooth projective $n$-folds of general type. The optimal such number $r_n$ is usually referred to as the $n$-th *canonical stability index*. If one only considers those varieties $X$ with $p_g(X)>0$, the number $r_n^+ \in \mathbb{Z}_{>0}$ is similarly defined as $$r_n^+:=\text{max}\{r_s(X)|\ \text{$X$\ is a smooth proj. $n$-fold of g. t. with $p_g>0$}\}.$$ The sequence $\{r_n\}$ (resp., $\{r_n^+\}$) of canonical stability indices is clearly increasing. Among known values, one has $r_1 = r_1^+ = 3$ and $r_2 = r_2^+ = 5$ by Bombieri ([@Bom73]). In dimension $3$, one has $r_3 \leq 57$ by [@CC15; @C21] and $14 \leq r_3^+ \leq 17$ by [@CHP21]. In dimensions $\geq 4$, no effective upper bound for $r_n$ is known and, however, one has $r_n>2^{\sqrt{2^n}}$ by Esser-Totaro-Wang ([@ETW23]). The motivation of this article is to solve the interesting open problem, of McKernan who first put forward in Mathematics Review (see MR2339333), which is also known as [@CJ17 Conjecture 6.1, Conjecture 6.2]. Our first main result is the following: **Theorem 1**. *For any integer $n \geq 2$, there exists a constant ${\mathfrak{V}}(n)>0$ such that, for any smooth projective $n$-fold $V$ with $\text{\rm vol}(V) > \mathfrak{V}(n)$ (resp., $p_g(V)> \mathfrak{V}(n)$), the inequality holds $$r_s(V)\leq r_{n-1}\ (\text{resp.}\ r_s(V)\leq r_{n-1}^+).$$* The surface case of Theorem [Theorem 1](#C-L-1){reference-type="ref" reference="C-L-1"} was proved by Bombieri ([@Bom73]). The $3$-fold case was proved by Todorov ([@Tod07]) and the first author ([@C03]). The first author and Jiang ([@CJ17]) proved the $4$-fold case and, partially, the $5$-fold case. We noticed that Lacini ([@Lac23]) recently proved that, for a smooth projective $n$-fold $V$ ($n\geq 4$) with sufficiently large canonical volume, $r_s(V)\leq \max\{r_{n-1}, (n-1)r_{n-2} + 2\}$. The equality in the statement "$r_s(V)\leq r_{n-1}$ (resp. $r_{n-1}^+$)" of Theorem [Theorem 1](#C-L-1){reference-type="ref" reference="C-L-1"} can attain for lots of examples. **Example 2**. *Let $X$ be a smooth projective $(n-1)$-fold of general type with $r_s(X) = r_{n-1}$ (resp. $r_s(X) = r_{n-1}^+$ and $p_g(X)>0$). Let $C$ be a smooth curve of genus $g\geq 2$. Take $V:=X \times C$. One sees that, as $g$ is sufficiently large, $\text{\rm vol}(V)$ (resp. $p_g(V)$) can be arbitrarily large and that $r_s(V) = r_{n-1}$ (resp. $r_s(V) = r_{n-1}^+$).* By virtue of Example [Example 2](#exmp1){reference-type="ref" reference="exmp1"}, Theorem [Theorem 1](#C-L-1){reference-type="ref" reference="C-L-1"} has the following equivalent form where the "lifting principle" gets the name: **Theorem 3**. *For any integer $n \geq 1$, there exists a constant ${\mathfrak{V}}(n)>0$ such that, for any number $L>{\mathfrak{V}}(n)$, one has $$r_n=\text{max}\{r_s(X)|\ X\ \text{is a smooth proj. (n+1)-fold with}\ \text{\rm vol}(X)>L\}$$ and, respectively, $$r^+_n=\text{max}\{r_s(X)|\ X\ \text{is a smooth proj. (n+1)-fold with}\ p_g(X)>L\}.$$* Theorem [Theorem 1](#C-L-1){reference-type="ref" reference="C-L-1"} implies the following byproduct. **Corollary 4**. *For any integer $n \geq 2$, there exists a constant ${\mathfrak{V}}(n)>0$ such that, for any smooth projective $n$-fold $V$ satisfying one of the following conditions:* - *$h^j(\mathcal{O}_V)>\mathfrak{V}(n)$ for some integer $j$ with $0<j<n$;* - *$|\chi(\mathcal{O}_V)|>{\mathfrak{V}}(n)$,* *the inequality $r_s(V)\leq r_{n-1}$ holds.* The key step in proving Theorem [Theorem 1](#C-L-1){reference-type="ref" reference="C-L-1"} is to develop the following type of extension theorem: **Theorem 5**. *Let $n$, $d$ be two integers with $n > d >0$ and $\mathcal{P}$ a biraionally bounded set of smooth projective varieties of dimension $d$. Then there exists a positive number $t$, depending only on $d$ and $\mathcal{P}$, such that the following property holds:* > *Let $f : V \rightarrow T$ be a surjective fibration where $V$ and $T$ are smooth projective varieties, $\dim V = n$, $\dim T = n-d$ and $\text{\rm vol}(V)>0$. Assume that the following conditions are satisfied:* > > - *the general fiber $X$ of $f$ is birationally equivalent to an element of $\mathcal{P}$;* > > - *there exists a positive rational number $\delta < t$ and an effective $\mathbb{Q}$-divisor $\Delta \sim_\mathbb{Q}\delta K_V$ such that $X$ is an irreducible non-klt center of $(V,\Delta)$.* > > *Then the restriction map $$H^0(V,pK_V) \to H^0(X_1, pK_{X_1}) \oplus H^0(X_2, pK_{X_2})$$ is surjective for any integer $p \geq 2$ and for any two different general fibers $X_1$, $X_2$ of $f$.* We briefly explain the main idea of this paper. In fact, very little information is known about the canonical stability index sequence $\{r_n\}$. It is believed to be strictly increasing, which is, however, an open question so far (see [@Lac23 Conjecture 1.4]). If one can show that $r_n\geq nr_{n-1}+2$, then the main statement of this paper holds according to a result of Lacini [@Lac23]. Our starting point is [@CJ17 Question 6.6], proposed by the first author and Jiang, which directly induces the solution to the main problem. Hence the key part of this paper is Section 3 where Theorem [Theorem 5](#key){reference-type="ref" reference="key"} (a high co-dimensional version of the extension theorem) is proved. More precisely, if $f:V\to T$ is a fibration with $\dim (T)>1$, then one has the difficulty in directly using the vanishing theorem to study the restriction map $H^0(mK_V)\to H^0(mK_X)$, where $X$ is the general fiber. Roughly speaking, our solution is to blow up $X$ and to use the exceptional divisor over $X$ as a bridge connecting the total space upstairs and X. Nakayama's Zariski decomposition, the adjunction for klt-trivial fibrations, Birkar's boundedness theorem (Theorem [Theorem 18](#BAB2){reference-type="ref" reference="BAB2"}) and Hacon-McKernan's extension theorem (Theorem [Theorem 19](#extend){reference-type="ref" reference="extend"}) play central roles in our argument. Details can be found in Step 1 through Step 8 in the proof of Theorem [Theorem 20](#main1){reference-type="ref" reference="main1"}. # Preliminary ## Divisors, linear systems and contractions. Let $D$ be an $\mathbb{R}$-divisor on a variety $X$ and $P$ a prime divisor. Then denote by $\mu_P({D})$ the coefficient of $P$ in $D$. Write $D = \sum d_i D_i$, where each $D_i$ is a prime divisor and $d_i \in \mathbb{R}$. Define the *roundup* of $D$ to be $\ulcorner{D}\urcorner:=\sum{\ulcorner{d_i}\urcorner D_i}$, the *rounddown* of $D$ to be $\llcorner{D}\lrcorner:= \sum{\llcorner{d_i}\lrcorner D_i}$ and the *fractional part* of $D$ to be $\langle{D}\rangle := \sum{\langle{d_i}\rangle D_i}$ where $\langle{d_i}\rangle = d_i - \llcorner{d_i}\lrcorner$. We say that $D$ is *effective* if $d_i \geq 0$ for all $i$. Given a positive real number $a$, we say that $D < a$ (resp. $D \leq a$) if $d_i < a$ (resp. $d_i \leq a$) for all $i$. Let $\{\hat{D}_j | j \in J\}$ be a set of effective $\mathbb{R}$-divisors where $J$ is a set of index and write $\hat{D}_j = \sum_{i} d_{j,i} D_{j,i}$, where each $D_{j,i}$ is a prime divisor and $d_{j,i} \geq 0$ for all $j$ and $i$. Define the *infimum* of $\{\hat{D}_j| j \in J\}$ to be $\inf\{ \hat{D}_j |j \in J\}:= \sum_{i} (\inf_{j \in J}d_{j,i})D_{j,i}$. Assume that $J$ is an infinite subset of $\mathbb{N}$, that $\liminf_{j \to \infty}d_{j,i} < +\infty$ for all $i$ and that there is only finitely many $i$ with $\liminf_{j \to \infty}d_{j,i} \ne 0$, then define $\liminf_{j \to \infty} \hat{D}_{j} := \sum_{i}(\liminf_{j \to \infty}d_{j,i}) D_{j,i}$, which is also an $\mathbb{R}$-divisor on $X$. If, furthermore, $\lim_{j \to \infty} d_{j,i}$ exists for all $i$, we define $\lim_{j \to \infty} \hat{D}_{j} := \sum_{i}(\lim_{j \to \infty}d_{j,i}) D_{j,i}$. Let $D$ be an integral effective divisor on a normal projective variety $X$. The *linear system* $|D|$ is defined as $|D|:=\{D' \geq 0| D' \sim D\}$ and the *$\mathbb{R}$-linear system* $|D|_{\mathbb{R}}$ is defined as $|D|_{\mathbb{R}}:=\{D' \geq 0| D' \sim_{\mathbb{R}} D\}$. The *fixed part* of $|D|$ is defined as ${\rm Fix}|D|:=\text{inf}\{ D' |D' \in |D|\}$. The *base locus* of $|D|$ is defined to be ${\rm Bs}|D| := \bigcap \{ D' | D' \in |D|\}$. For a prime divisor $\Gamma$ on $X$, define $\sigma_{\Gamma}(D)_{\mathbb{Z}} := \mu_{\Gamma}({\rm Fix}|D|)$. Let $f : X \to Z$ be a projective morphism between varieties. We say that $f$ is a *contraction* if $f_*\mathcal{O}_X = \mathcal{O}_Z$. In particular, a contraction is surjective and has connected fibers. We give a generalization of [@C01 Theorem 2.7]. **Lemma 6**. *Let $Y$ be a smooth projective variety of dimension at least 2 and $X$ be a smooth irreducible divisor on $Y$. Let $D$ and $E$ be two integral effective divisors on $Y$ such that* - *$X$ is not a component of $D + E$, and* - *the image of the restriction map $$H^0(Y,D+E) \to H^0(X, D|_X + E|_X)$$ contains the image of the inclusion $$i:H^0(X,D|_X) \to H^0(X, D|_X + E|_X),$$ which is induced by the effective divisor $E|_X$.* *Denote by $F$ the fixed part of $|D+E|$ on Y, and by $Z_X$ the fixed part of $|D|_X|$ on $X$. Then we have the following inequality $$F|_X \leq Z_X + E|_X.$$* *Proof.* By the second condition, for any effective divisor $D'_X \sim D|_X$ on $X$, there exists an effective divisor $G \sim D+E$ on $Y$ such that $G|_X = D'_X + E|_X$. By the definition of the fixed part, we know $F \leq G$ and that $F$ does not depend on the choice of $D'_X$. Thus $$F|_X \leq \inf\{D'_X \geq 0|D'_X \sim D|_X\} + E|_X = Z_X + E|_X,$$ so we finish the proof. ◻ ## Pairs and singularities. A *sub-pair* $(X,\Delta)$ consists of a normal quasi-projective variety $X$ and an $\mathbb{R}$-divisor $\Delta$ such that $K_X + \Delta$ is $\mathbb{R}$-Cartier. A sub-pair $(X, \Delta)$ with $\Delta \geq 0$ is called a *pair*. If, additionally, $\Delta$ is a $\mathbb{Q}$-divisor, we call $(X,\Delta)$ a $\mathbb{Q}$-pair. If a pair $(X,\Delta)$ satisfies that $X$ is smooth and that the support of $\Delta$ is of simple normal crossing, we say that $(X,\Delta)$ is a *log smooth pair*. Let $(X,\Delta)$ be a sub-pair. Let $f:Y \to X$ be a log resolution of $(X,\Delta)$ and write $K_Y + \Gamma = f^*(K_X + \Delta)$. Then define the *log discrepancy* of a prime divisor $D$ on $Y$ with respect to the sub-pair $(X,\Delta)$ to be $$a(D,X,\Delta) := 1-\mu_D{(\Gamma)}.$$ Fix a number $\epsilon > 0$. We say $(X,\Delta)$ is *sub-lc* (resp., *sub-klt*, *sub-$\epsilon$-lc*) if $a(D,X,\Delta) \geq 0$ (resp., $>0$, $\geq \epsilon$) for every prime divisor $D$ over $X$. If, in addition, $(X,\Delta)$ is a pair, we say $(X,\Delta)$ is *lc* (resp., *klt*, *$\epsilon$-lc*). We say $(X, \Delta)$ is *terminal* if $a(D,X,\Delta) > 1$ for any prime divisor $D$ which is exceptional over $X$. Let $(X,\Delta)$ be a sub-pair and take a log resolution $f:Y \to X$. A *non-klt place* is a prime divisor $D$ over $X$ with $a(D,X,\Delta) \leq 0$ and a *non-klt center* is the image of a non-klt place on $X$. The *non-klt locus* of $(X,\Delta)$ is the union of all non-klt centers of $(X,\Delta)$ which is denoted as ${\rm Nklt}(X,\Delta)$. An irreducible component of ${\rm Nklt}(X,\Delta)$ is called an *irreducible non-klt center*. An *lc place* is a prime divisor $D$ over $X$ with $a(D,X,\Delta) = 0$ and an *lc center* is the image of an lc place on $X$. We say that an lc center $G$ of $(X, \Delta)$ is a *pure lc center* if $(X,\Delta)$ is lc at the generic point of $G$. We say that an lc center $G$ is an *exceptional lc center* if there is a unique lc place over $X$ whose image is $G$ where the unique lc place is called *the exceptional lc place*. Let $(X,\Delta)$ be an lc pair and $L \geq 0$ be an $\mathbb{R}$-Cartier $\mathbb{R}$-divisor. Define the *lc threshold* of $L$ with respect to $(X,\Delta)$ to be $${\rm lct}(X,\Delta,L) := \sup\{t\in \mathbb{R}|(X,\Delta + tL) {\text{ is lc}}\}.$$ Let $A$ be another $\mathbb{R}$-Cartier $\mathbb{R}$-divisor. Define the *lc threshold* of the $\mathbb{R}$-linear system $|A|_{\mathbb{R}}$ with respect to $(X,\Delta)$ to be $${\rm lct}(X,\Delta,|A|_{\mathbb{R}}) := \inf\{{\rm lct}(X,\Delta,L)|0 \leq L \sim_\mathbb{R}A\}.$$ **Lemma 7**. *Let $(X,\Delta)$ be a klt pair. Assume that* - *$D\geq 0$ is a big $\mathbb{R}$-Cartier $\mathbb{R}$-divisor and write $D \sim_{\mathbb{R}} A + E$, where $A$ is ample and $E \geq 0$;* - *$(X,\Delta + D)$ is not klt with an irreducible non-klt center $G$ such that $G$ is not contained in any component of $E$.* *Then, for any $\epsilon >0$, there exist two real numbers $s$, $t$ with $0 \leq t < s \leq 1$, $s + t < 1 + \epsilon$ and an effective divisor $D' \sim_\mathbb{R}(s +t) D$ such that $G$ is a pure and exceptional lc center of $(X,\Delta + D')$.* *If, in addition, $(X,\Delta)$ is a $\mathbb{Q}$-pair and $D$ is a $\mathbb{Q}$-divisor, we can take $s,\ t \in \mathbb{Q}$ and $D'$ a $\mathbb{Q}$-divisor.* *Proof.* Take a log resolution $f:Y \to X$ of $(X, \Delta + D)$ and write $$\begin{aligned} f^*(K_X + \Delta)&=& K_Y + \tilde{\Delta} + \sum{a_i E_i}, \\ f^*(D) &=& \tilde{D} + \sum{b_i E_i}, \end{aligned}$$ where $\tilde{\Delta},\tilde{D}$ are the biratianal transform of $\Delta, D$, respectively, on $Y$ and all $E_i$ are $f$-exceptional divisors on $Y$. Then we take $F:= \sum{c_i E_i}$ to be an $\mathbb{R}$-divisor on $Y$ supported on the exceptional locus with sufficiently small positive coefficients such that $H:=f^*A - F$ is ample on $Y$. Take a general $H_1 \sim_\mathbb{R}H$ and define $A_1 := f_* H_1$ and $D_1 = A_1 + E$. Then $D_1 \sim_\mathbb{R}D$. By the negativity lemma [@Bir12 Lemma 3.3] and the direct computation, one has $$f^*(D_1) = H_1 + \sum{c_i E_i} + E',$$ where $E':=f^*E$. We need to mention that $H_1$ does not contain any exceptional divisors since it is general and that $E'$ does not contain any exceptional divisors mapping onto $G$ since $E$ does not contain $G$. Next, we pick a sufficiently small real number $t \in [0,\epsilon)$, and let $s$ be the largest number such that $(X, \Delta + tD_1 + sD)$ is lc at the generic point of $G$. It follows that $s\leq 1$. We have $$f^*(K_X + \Delta + tD_1 + sD) = K_Y + \tilde{\Delta} + tH_1 + tE' + s \tilde{D} + \sum{(a_i + t c_i + s b_i) E_i}.$$ After appropriately perturbing the coefficients $c_i$, we may and do assume that there exists a unique $E_j$ mapping onto $G$ such that the coefficient $a_j + t c_j + s b_j$ is equal to $1$. In other words, $E_i$ is the unique lc place of $(X, \Delta + tD_1 + sD)$ whose image on $X$ is $G$. Finally, let $D':=tD_1 + sD$, which satisfies all the requirements. For the $\mathbb{Q}$-divisor case, we can assume that $A$, $E$, $F$, $H_1$ are $\mathbb{Q}$-divisors and take $t$ to be a rational number. It follows that $s$ is also a rational number. Thus $D'$ is a $\mathbb{Q}$-divisor. ◻ We need to use the following useful lemma in this paper. **Lemma 8**. *(cf. [@Kol98 Proposition 2.36(1)]) Let $(X,\Delta)$ be a sub-klt sub-pair. Then there is a log resolution $\nu : Y \to X$ of $(X, \Delta)$ such that, if we write $$K_Y + \Gamma - F = \nu^*(K_X + \Delta)$$ where $\Gamma$ and $F$ are effective with no common components, any two components of $\Gamma$ do not intersect. In particular, $(Y,\Gamma - F)$ is terminal.* ## The canonical volume {#vol} Let $X$ be a normal projective $n$-fold with $\mathbb{Q}$-factorial and at worst canonical singularities. Then we define the *canonical volume* of $X$ to be: $$\text{\rm vol}(X) = \varlimsup\limits_{m\to\infty}\frac{h^0(X,mK_X)}{m^n/n!}.$$ The *geometric genus* is defined as $p_g(X):=h^0(X,K_X)$. Note that the canonical volume and the geometric genus are both birational invariants. Here we recall a useful inequality between the canonical volume and the geometric genus: **Lemma 9**. *([@CJ17 Theorem 5.1]) Let $n>0$ be any integer. There exist positive numbers $a_n$ and $b_n$ depending only on the dimension $n$ such that, for all smooth projective n-folds $X$ of general type, the following inequality holds: $$\text{\rm vol}(X) \geq a_n p_g(X) - b_n.$$* Particularly, whenever $p_g(X)$ is sufficiently large, so is $\text{\rm vol}(X)$. ## Nakayama-Zariski decomposition. We mainly refer to [@N04 Chapter III] for the following definitions. Let $X$ be a complex smooth projective variety. For a big $\mathbb{R}$-divisor $D$ on $X$ and any prime divisor $\Gamma$, define $$\sigma_{\Gamma}(D) := \inf\{\mu_{\Gamma}(D')| 0 \leq D' \sim_\mathbb{R}D\}.$$ Then $\sigma_{\Gamma}$ only depends on the numerical class of $D$ and it is a continuous function on the big cone. If $D$ is pseudo-effective, define $\sigma_{\Gamma}(D) := \lim_{\epsilon \to 0}\sigma_{\Gamma}(D + \epsilon A)$, where $A$ is any ample divisor. Note that this definition does not depend on the choice of $A$. Then define $N_\sigma(D) := \sum_{\Gamma}\sigma_{\Gamma}(D)\Gamma$ and $P_\sigma(D):= D -N_\sigma(D).$ We call the decomposition $D = P_\sigma(D) + N_\sigma(D)$ the *$\sigma$-decomposition*. If $P_\sigma(D)$ is nef, we say that $D$ *admits a Zariski-decomposition*. Let $f: X \to Y$ be a projective surjective morphism between smooth varieties and $D$ be a $f$-pseudo-effective $\mathbb{R}$-divisor on $X$. Then one may similarly define the relative version of $\sigma$-decomposition. In fact, on $X$, assume that $D$ is $f$-big and $\Gamma$ is a prime divisor. Then define $$\sigma_{\Gamma}(D;X/Y) := \inf\{\mu_{\Gamma}(D')| 0 \leq D' \sim_\mathbb{R}D \text{ over $Y$}\}.$$ If $D$ is $f$-pseudo-effective, define $\sigma_{\Gamma}(D;X/Y):= \lim_{\epsilon \to 0}\sigma_{\Gamma}(D + \epsilon A;X/Y)$ for a relatively ample divisor $A$. Again it is seen that the definition does not depend on the choice of $A$. If we have $\sigma_{\Gamma}(D;X/Y) < + \infty$ for any prime divisor $\Gamma$ on $X$, then we define that $$N_\sigma(D;X/Y) := \sum_{\Gamma}\sigma_{\Gamma}(D;X/Y)\Gamma,$$ and that $P_\sigma(D;X/Y):= D -N_\sigma(D;X/Y)$. We list some basic properties about the $\sigma$-decomposition. **Lemma 10**. *Let $X$ be a smooth projective variety, $D$ be a pseudo-effective $\mathbb{R}$-divisor on $X$ and $f:X \to Z$ be a projective surjective morphism. Then* 1. *$N_\sigma(D) \geq N_\sigma(D;X/Z)$ always holds.* 2. *If $D$ is integral and effective, then $$H^0(X,D) = H^0(X, D - \ulcorner{N_\sigma(D)}\urcorner).$$* 3. *If $D$ is big, $N_\sigma(D) = \lim_{m \to +\infty}{\frac{1}{m}{\rm Fix}|\llcorner{mD}\lrcorner|}$.* 4. *There is an ample divisor $A_0$, such that for any ample divisor $A$ with $A-A_0$ ample, we have $$N_\sigma(D) = \lim_{m \to +\infty}{\frac{1}{m}{\rm Fix}|\llcorner{mD}\lrcorner + A|}.$$* 5. *If $D$ is big and the ring $R(X,D):= \oplus_{m=0}^\infty H^0(X,\llcorner{mD}\lrcorner)$ is a finitely generated $\mathbb{C}$-algebra, then there exists a birational morphism $g : Y \to X$ from a smooth projective variety such that $P_\sigma(g^*D)$ is a semi-ample $\mathbb{Q}$-divisor. In particular, $g^*D$ admits a Zariski-decomposition.* *Proof.* Statements (1) and (2) are obtained directly from definitions. Statements (3) and (5) are due to [@N04 III, Remark 1.17]. Now we prove Statement (4), for which we mainly refer to [@N04 V, Corollary 1.7(1)]. Let $W:= {\rm NBs}(P_\sigma(D))$ which is defined as [@N04 III, Definition 2.6]. Then, by [@N04 V, Page 168], $W$ is a countable union of subvarieties of codimension at least $2$. In fact, by [@N04 V, Theorem 1.3], there is an ample divisor $A_0$ such that for any ample divisor $A$ with $A-A_0$ ample and, for any $x \in X \setminus W$ and any $m \geq 0$, we have $x \not\in {\rm Bs}|\llcorner{mP_\sigma(D)}\lrcorner + A|$. We can further assume that $A$ has no common components with $D$ and $N_{\sigma}(D)$. Then we only need to prove that, for any prime divisor $\Gamma$, $$\lim_{m \to + \infty}{\frac{1}{m}\sigma_{\Gamma}(\llcorner{mD}\lrcorner + A)_{\mathbb{Z}}} = \sigma_{\Gamma}(D).$$ Since for any prime divisor $\Gamma$, $\Gamma \setminus W$ is non-empty, we can take a closed point $y \in \Gamma \setminus W$. Since $y \not\in {\rm Bs}|\llcorner{mP_\sigma(D)}\lrcorner + A|$, we see that $$\sigma_{\Gamma}(\llcorner{mP_\sigma(D)}\lrcorner + A)_{\mathbb{Z}} = 0.$$ Thus one has $$\begin{aligned} \sigma_{\Gamma}(\llcorner{mD}\lrcorner + A)_{\mathbb{Z}} &\leq \mu_{\Gamma}(\llcorner{mD}\lrcorner - \llcorner{mP_\sigma(D)}\lrcorner) \\ &\leq \mu_{\Gamma}(\ulcorner{mN_{\sigma}(D)}\urcorner) \leq m\sigma_{\Gamma}(D) + 1.\end{aligned}$$ Therefore $$\varlimsup_{m \to \infty} \frac{1}{m}\sigma_{\Gamma}(\llcorner{mD}\lrcorner + A)_{\mathbb{Z}} \leq \sigma_{\Gamma}(D).$$ On the other hand, for any $k > 0$, any sufficienly large real number $m$, and any $\Gamma$ with $\sigma_{\Gamma}(D) > 0$, $$\frac{1}{mk}\sigma_{\Gamma}(\llcorner{mkD}\lrcorner + A)_{\mathbb{Z}} \geq \frac{1}{mk} \sigma_{\Gamma}(\llcorner{m(kD + A)}\lrcorner)_\mathbb{Z}.$$ Since $kD + A$ is big, Statement (3) implies that, for any $k > 0$, $$\begin{aligned} \varliminf\limits_{m \to \infty}\frac{1}{m}\sigma_{\Gamma}(\llcorner{mD}\lrcorner + A)_{\mathbb{Z}} &= \varliminf\limits_{m \to \infty}\frac{1}{mk}\sigma_{\Gamma}(\llcorner{mkD}\lrcorner + A)_{\mathbb{Z}} \\ &\geq \frac{1}{k} \varliminf\limits_{m \to \infty}{\frac{1}{m}\sigma_{\Gamma}(\llcorner{m(kD + A)}\lrcorner)_{\mathbb{Z}}} \\ &= \frac{1}{k}\sigma_{\Gamma}(kD + A) = \sigma_{\Gamma}(D + (1/k)A).\end{aligned}$$ Then, taking the limit as $k\to \infty$, we get $$\varliminf\limits_{m \to \infty}\frac{1}{m}\sigma_{\Gamma}(\llcorner{mD}\lrcorner + A)_{\mathbb{Z}} \geq \sigma_{\Gamma}(D).$$ The lemma is proved. ◻ ## Vertical divisors Let $f:X \to Y$ be a contraction between normal projective varieties and $D$ be an $\mathbb{R}$-divisor on $X$. We say that $D$ is *horizontal* over $Y$ or *$f$-horizontal* if the support of $f(D)$ dominates $Y$, otherwise we say $D$ is *vertical* over $Y$ or *$f$-vertical*. We say that $D$ is *very exceptional* over $Y$ if $D$ satisfies the following conditions: 1. $D$ is vertical over $Y$; 2. For any prime component $P$ of $D$, either $f(P)$ has codimension at least $2$ in $Y$, or there exists another prime divisor $Q$ on $X$ such that $f(Q) = f(P)$, but $Q$ is not contained in the support of $D$. By definition, whenever ${\rm Codim}f(D) \geq 2$, $D$ is automatically very exceptional. **Lemma 11**. *(cf. [@N04 III, Proposition 5.7]) Let $f:X \to Y$ be a contraction between smooth projective varieties and $D$ be an effective $\mathbb{R}$-divisor on $X$. Assume that $D$ is very exceptional over $Y$. Then $D = N_\sigma(D;X/Y)$.* **Lemma 12**. *Let $f:X \to Z$ be a contraction between smooth projective varieties. Let $Q$ be an effective $\mathbb{R}$-divisor on $X$ and be vertical over $Z$. Assume that $A$ is a very ample divisor on $Z$ such that $f^*(aA) - Q$ is pseudo-effective for a positive number $a$. Then there exists an effective divisor $\Delta \sim_{\mathbb{R}} 2aA$ such that $f^*(\Delta) \geq Q$.* *Proof.* By the flattening theorem (see [@Ray72 Chapter 3],[@H75],[@Vil06 Theorem 3.3]), we can take a birational morphism $\sigma: Z' \to Z$ that flattens $f:X \to Z$ such that $Z'$ is smooth. Denote by $U$ the normalization of the main component of $X \times_{Z} Z'$. Then the induced map $\pi: U \to X$ is birational and $f': U \to Z$ is flat. Let $\nu: X' \to U$ be a resolution of $U$, $\rho:= \pi \circ \nu : X' \to X$ and $g:= f' \circ \nu: X' \to Z'$. $$\xymatrix{ X' \ar[r]^\nu \ar[dr]_{g} \ar@/^1.3pc/[rr]^{\rho}& U \ar[d]_{f'} \ar[r]^{\pi} & X \ar[d]^{f}\\ &Z' \ar[r]^{\sigma} & Z }$$ Since $f'$ is flat, we see that for any $f'$-vertical divisor $D$ on $U$, $f'(D)$ is a divisor on $Z'$. Let $P:=\inf\{S \geq 0 \text{ on $Z'$} | f'^*S \geq \pi^*Q\}$. Then $P$ is well-defined. Set $E:= f'^*P - \pi^*Q$. Then $E \geq 0$ and $E$ is very exceptional over $Z'$. Indeed, for any component $D$ of $E$, if we set $S:=f'(D)$ on $Z'$ and there is a prime divisor $D'$ on $U$ such that $f'(D') = S$ and the coefficient of $D'$ in $f'^*P - \pi^*Q$ is zero by the definition of $P$. Let $E' := \nu^*E$ on $X'$. Then $E'$ is also very exceptional over $Z'$. Hence $N_\sigma(E';X'/Z') = E'$ by Lemma [Lemma 11](#m3){reference-type="ref" reference="m3"}. Since $g^*(\sigma^*(aA) - P) + E' = \rho^*(f^*(aA) - Q )$ is pseudo-effective, $$\begin{aligned} N_\sigma(g^*(\sigma^*(aA) - P) + E') &\geq N_\sigma(g^*(\sigma^*(aA) - P) + E';X'/Z') \\ &= N_\sigma(E';X'/Z') = E'.\end{aligned}$$ Then $g^*(\sigma^*(aA) - P) \geq P_\sigma(g^*(\sigma^*(aA) - P) + E')$, which is pseudo-effective. Hence $\sigma^*(aA) - P$ is pseudo-effective by [@N04 Chapter II, Lemma 5.6(2)]. Since $\sigma^*(A)$ is nef and big, there exists an effective divisor $\Theta \sim_\mathbb{R}\sigma^*(2aA) - P$ and then take $\Delta = \sigma_*(\Theta + P)$, which satisfies our requirement. Indeed, since $\Theta + P$ is relatively trivial over $Z$, $\sigma^*(\Delta) = \Theta + P$ by the negativity lemma [@Bir12 Theorem 3.3]. So it follows that $\rho^*(f^*\Delta - Q) = g^*\Theta + E' \geq 0$. Since $\rho$ is birational, we have $f^*(\Delta) - Q \geq 0$. ◻ ## B-divisors We recall the definition of b-divisors introduced by Shokurov. For details, one may refer to [@Amb04] and [@Fuj12] for instance. Let $\mathbb{K}$ be $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$. Let $X$ be a normal projecitve variety. A *b-$\mathbb{K}$-divisor* $\mathbf{D}$ of $X$ consists of a family of $\mathbb{K}$-divisors $\{D_{X'}\}$ indexed by all the birational models $X'$ of $X$ which satisfies that, for any birational morphism $\nu : X'' \to X'$ between two birational models $X'$,$X''$ of $X$, $\nu_*D_{X''} = D_{X'}$ holds. Let $\mathbf{K}$ consist of $\{K_{X'}\}$ for all birational models $X'$, where $K_{X'}$ is the canonical divisor defined by a top rational differential form $(\omega)$ of $X$ such that $\nu_*K_{X''} = K_{X'}$ for any birational morphism $\nu: X'' \to X'$. We call $\mathbf{K}$ the *canonical b-divisor* of $X$. Let $(X,B)$ be a sub-pair. We recall the definition of the *discrepancy b-divisor* $\mathbf{A}(X,B):=\{A_{X'}\}$ as in [@FG14 2.3] and [@Fuj12 Definition 3.6]. For any birational morphism $\nu : X' \to X$ from a normal variety $X'$, we define $$A_{X'}:=K_{X'} - \nu^*(K_X + B).$$ Then we define $\mathcal{O}_{X}(\ulcorner{\mathbf{A}(X,B)}\urcorner) :=\tilde{\nu}_*\mathcal{O}_{Y}(\ulcorner{A_{Y}}\urcorner)$ where $\tilde{\nu}: Y \to X$ is any log resolution of $(X,B)$. ## Adjunction. [\[adjunction\]]{#adjunction label="adjunction"} We recall the adjunction for fiber spaces, for which we mainly refer to [@Amb99],[@Amb04] and [@FG14]. **Definition 13**. **A *klt-trivial fibration* $f: (X,B) \to Z$ consists of a contraction $f:X \to Z$ between normal projective varieties and a sub-pair $(X,B)$, which satisfies the following conditions:** 1. *$(X,B)$ is sub-klt over the generic point of $Z$;* 2. *${\rm rank}\ f_*\mathcal{O}_{X}({\ulcorner{\mathbf{A}(X,B)}\urcorner}) =1$;* 3. *$K_X + B \sim_{\mathbb{R},f} 0$ over $Z$.* We will do adjunction for a klt-trivial fibration $f: (X,B) \to Z$. For any prime divisor $P_i$ on $Z$, define $$t_i := \sup\{a\in \mathbb{R}|(X,B + af^*P_i)\text{ is lc over the generic point of }P_i\}.$$ Then define $B_Z := \sum(1-t_i)P_i$, where the sum is taken over all prime divisors on $Z$. Then it is seen that $B_Z$ is a divisor and we call $B_Z$ the *discriminant part* of $(X,B)$ on $Z$ (see [@FG14 3.4]). Then there exists an $\mathbb{R}$-divisor $M_Z$ such that $K_X + B \sim_\mathbb{R}f^*(K_Z + B_Z + M_Z).$ We call $M_Z$ the *moduli part* of $(X,B)$ on $Z$. Note that $M_Z$ is only determined up to $\mathbb{R}$-equivalence. Here we list several properties about adjunction for fiber spaces. 1. Let $\nu:X' \to X$ be a birational morphism and $K_{X'} + B' = \nu^*(K_X + B)$ be the crepant pullback. One may do adjunction for $f\circ\nu : (X', B') \to Z$. Then the discriminant part of $(X', B')$ on $Z$ is the same as $B_Z$. As being observed in [@Amb99 Remark 3.1], while doing adjunction, we are free to take a crepant model. 2. Let $\sigma:Z' \to Z$ be a birational morphism from a normal projective variety $Z'$. We may take a crepant model $(X', B')$ of $(X, B)$ and assume that $f$ induces a contraction $f': X' \to Z'$. If we do adjunction for $f': (X',B') \to Z'$, we can define the discriminant part $B_{Z'}$ and the moduli part $M_{Z'}$ of $(X',B')$ on $Z'$. Then we have $\sigma_*B_{Z'} = B_Z$. Although the moduli part is only defined up to $\mathbb{R}$-linear equivalence, we can take a compatible moduli part $M_{Z'}$ so that $\sigma_*M_{Z'} = M_Z$. Then we have $$K_{Z'} + B_{Z'} + M_{Z'} = \sigma^*(K_Z + B_Z + M_Z).$$ Moreover, let $A$ be an $\mathbb{R}$-divisor on $Z$ and set $A':= \sigma^*(A)$ on $Z'$. If we do adjunction for $f: (X, B + f^*A) \to Z$, by direct computations, we will get the discriminant part $B_Z + A$ and the moduli part $M_{Z}$. Similarly, if we do adjunction for $f' : (X', B' + {f'}^*(A')) \to Z'$, the discriminant part and the moduli part will be $B_{Z'} + A'$ and $M_{Z'}$, respectively. **Definition 14**. **Let $f : (X, B) \to Z$ be a klt-trivial fibration. Let $\sigma : Z' \to Z$ be a birational morphism from a normal projective variety. We say that $Z'\to Z$ is an *excellent base extension with regard to $f : (X, B) \to Z$* if the following property holds:** > *Let $(X', B')$ be a crepant model of $(X,B)$ and assume that $f$ induces a contraction $f' : X' \to Z'$. If we do adjunction for $f': (X',B') \to Z'$, the moduli part $M_{Z'}$ is nef and, furthermore, for any birational morphism $Z'' \to Z'$ from a normal projective variety $Z''$, the respective moduli part $M_{Z''}$ is the pullback of $M_{Z'}$ (and, hence, is nef as well).* **Lemma 15**. *(see [@Amb04 Theorem 0.2]) Let $f : (X, B) \to Z$ be a klt-trivial fibration. If we assume that $(X,B)$ is a $\mathbb{Q}$-sub-pair, then there exists an excellent base extension with regard to $f : (X, B) \to Z$.* The next lemma with regard to singularities of adjunction is useful to our argument. **Lemma 16**. *Let $f:(X,B) \to Z$ be a klt-trivial fibration. Assume that there exists an excellent base extension $\sigma: Z' \to Z$ with regard to $f: (X,B) \to Z$. Let $(X', B')$ be a crepant model of $(X,B)$ and assume that $f$ induces a contraction $f' : X' \to Z'$ and denote by $B_{Z'}$ and $M_{Z'}$ the discriminant part and the moduli part on $Z'$ of $f' : (X',B') \to Z'$. Then $(X,B)$ is sub-klt provided that $(Z',B_{Z'})$ is sub-klt.* *Proof.* Let $(X',B')$ be a crepant model of $(X,B)$ such that the induced map $f' : X' \dashrightarrow Z'$ from $f$ is a morphism. Denote by $\nu: X' \to X$ the birational morphism. Suppose that $(X,B)$ is not sub-klt. Then there is a prime divisor $D$ over $X$ such that $a(D,X,B) \leq 0$. We will construct a higher model $f'':X''\to Z''$ over $f':X'\to Z'$ as follows. First, take a birational model $X''$ of $X'$ such that $D$ is a divisor on $X''$. Denote by $\pi : X'' \to X$ the birational morphism. Since $(X,B)$ is sub-klt over the generic point of $Z$, we see that $D$ is vertical over $Z$. Next, take a birational morphism $\rho : Z'' \to Z$ that flattens $f \circ \pi : X'' \to Z$. Finally, after possibly blowing-up $Z''$ while replacing $X''$ with a higher birational model and replacing $D$ with its birational transform, we may and do assume that 1. There are birational morphisms $\tau : Z'' \to Z'$ and $\xi : X'' \to X'$ where $X''$ and $Z''$ are both smooth projective; 2. the induced map $f'': X'' \dashrightarrow Z''$ is a morphism; 3. $f''(D)$ is a prime divisor $S$ on $Z''$. $$\xymatrix{ X'' \ar[r]^{\xi} \ar[d]^{f''}& X' \ar[r]^{\nu} \ar[d]^{f'} & X \ar[d]^{f} \\ Z'' \ar[r]^{\tau} & Z'\ar[r]^{\sigma} & Z }$$ Let $K_{X''} + B'' = \xi^*\circ\nu^*(K_X + B)$ be the crepant pullback and let $B_{Z''}$ and $M_{Z''}$ be the respective discriminant part and the moduli part on $Z''$ for $(X'',B'')$. We have $\mu_D(B'')\geq 1$ since $a(D,X,B) \leq 0$. Thus, by the definition of discriminant part, one has $\mu_{S}(B_{Z''}) \geq 1$, which implies that $(Z'',B_{Z''})$ is not sub-klt. On the other hand, since $Z'\to Z$ is an excellent base extension, one has $K_{Z''} + B_{Z''} = \tau^*(K_{Z'} + B_{Z'})$. By assumption that $(Z',B_{Z'})$ is sub-klt, we see that $(Z'',B_{Z''})$ is also sub-klt, a contradiction. ◻ The last part of this subsection is devoted to exploring adjunction for lc centers. Let $(X,B)$ be a pair with a pure and exceptional lc center $G \subset X$ and denote by $\nu:F \to G$ the normalization. Let $f: Y \to X$ be a log resolution of $(X,B)$ and denote by $S$ the unique lc place on $Y$ whose image on $X$ is $G$. Write $K_Y + B_Y = f^*(K_X + B)$, and define $B_S := (B_Y - S)|_S$. Let $g: S \to F$ be the unique morphism induced by the restriction of $f$ to $S$. By [@Amb99 Lemma 4.1], one has 1. $g$ is a contraction; 2. $(K_S + B_S) \sim_{\mathbb{R},g} 0$ over $F$; 3. $(S,B_S)$ is sub-klt over the generic point of $F$; 4. $g_*\mathcal{O}_{S}{(\ulcorner{-B_S}\urcorner)_{\eta_F}} = \mathcal{O}_{F,\eta_F}$, where $\eta_{F}$ is the generic point of $F$. Thus $g: (S,B_S) \to F$ is a klt trivial fibration and we can do adjunction to get the discriminant part $B_{F}$ and the moduli part $M_{F}$ of $(S,B_S)$ on $F$. Thus we can write $$K_S+B_S\sim_\mathbb{R}f^*(K_F+B_F+M_F)$$ which directly implies $$(K_X + B)|_F:=j^*(K_X+B) \sim_\mathbb{R}K_F + B_F + M_F$$ where $j: F\to G\hookrightarrow X$ is the composition. Hence we have: \(viii\) Since $B \geq 0$, by [@Amb99 Lemma 4.2 (3)], one has $B_F \geq 0$. ## The Chow variety. Given a polarized projective variety, one would like to parametrize all its sub-varieties with a fixed dimension and a fixed degree. So we need to recall the Chow variety which was constructed in [@K96 Chapter I]. Let $X$ be a polarized projective variety. Define the following moduli functor $\mathcal{C}how_{n,d}(X)$ by $$\mathcal{C}how_{n,d}(X)(Z):=\left\{ \begin{aligned} &\text{Families of nonnegative algebraic cycles of } \\ &\text{dimension $n$ and degree $d$ of } X \times Z/Z \end{aligned} \right\}.$$ Then, by [@K96 Chapter I, Theorem 3.21], the functor $\mathcal{C}how_{n,d}(X)$ is represented by a universal family $$p: {\rm Univ}_{n,d}(X) \to {\rm Chow}_{n,d}(X),$$ where ${\rm Chow}_{n,d}(X)$ is projective. ## Bounded families. We say that a set $\mathscr{X}$ of varieties is *bounded* (resp., *birationally bounded*) if there is a projective morphism $Z \to S$ such that $S$ is of finite type and that, for any $X \in \mathscr{X}$, there exists a closed point $s \in S$ and an isomorphism $X \to Z_s$ (resp., a birational map $X \dashrightarrow Z_s$), where $Z_s$ is the fiber of $s$. We need the following lemma to slightly relax the condition of boundedness to birational boundedness. **Lemma 17**. *Let $d$ be a positive integer and $M$ be a positive real number. Then there exists a bounded family $\mathcal{P}$ consisting of smooth projective varieties and depending only on $d,M$ such that the following property holds:* > *Assume that* > > 1. *$f : V \to T$ is a contraction between smooth projective varieties with $\dim V - \dim T = d$, and* > > 2. *any general fiber $X$ of $f : V \to T$ is of general type with $\text{\rm vol}(X) \leq M$.* > > *Then there exists the following commutative diagram $$\xymatrix{ > V' \ar[dr]^{g} \ar@{-->}[rr]^{\psi}& &V \ar[dl]_{f} \\ > & T > }$$ satisfying:* > > 1. *$\psi : V' \dashrightarrow V$ is a birational map from a smooth projective variety $V'$;* > > 2. *any general fiber $X'$ of $g$ belongs to the bounded family $\mathcal{P}$.* > > *If we assume further that* > > 1. *$V$ is of general type and for any general fiber $X$ of $f: V \to T$, there is an effective $\mathbb{Q}$-divisor $\Delta \sim_\mathbb{Q}\delta K_V$ on $V$ such that $X$ is an irreducible non-klt center of $(V,\Delta)$,* > > *then we can further require that* > > 1. *for a general fiber $X'$ of $g : V' \to T$, there is an effective $\mathbb{Q}$-divisor $\Delta' \sim_\mathbb{Q}\delta K_{V'}$ on $V'$ such that $X'$ is an irreducible non-klt center of $(V',\Delta')$.* *Proof.* The proof is organized through several steps.\ *Step 1.* The construction of $\mathcal{P}$ using the Chow variety. By the boundedness theorem in [@HM06],[@Tak06] and [@Tsu06], there exists an integer $r_d$ depending only on $d$ such that $|r_dK_X|$ induces a birational map onto its image for any general fiber $X$. Set $S:= {\rm Chow}_{d,\leq {r_d}^d M}(\mathbb{P}^{N})$, the Chow variety of $d$-dimensional varieties in $\mathbb{P}^{N}$ with degree $\leq {r_d}^d M$, where $N:= \llcorner{{r_d}^{d}M}\lrcorner + d$. Then $S$ is projective and of finite type. Let $p:\mathscr{X} \to S$ be the universal family corresponding to the Chow variety. First, take a stratification of $p : \mathscr{X} \to S$ as $$\coprod_{\rm finite} S_i \to S$$ such that each $S_i$ is smooth and that each $p_i$ is flat, where $p_i : \mathscr{X}_i \to S_i$ is the corresponding family. Then, for each $\mathscr{X}_i$, take a resolution $\mathscr{Y}_i \to \mathscr{X}_i$ by finitely many times of blowups of singularities. After possibly replacing $S_i$ with a further stratification, we can assume that each exceptional locus dominates certain $S_i$. Therefore we get a smooth bounded family $q=\coprod q_i : \coprod \mathscr{Y}_i \to \coprod S_i$, and denote by $\mathcal{P}$ this family.\ *Step 2.* Proof that $\mathcal{P}$ satisfies all the requirements. We need to construct $g : V' \to T$ and $\psi : V' \dashrightarrow V$. Let $T_0 \subseteq T$ be an affine open dense subset such that $V_t$ is smooth and $\text{\rm vol}(V_t) \leq M$ for any $t \in T_0$. Let $V_0$ be the preimage of $T_0$ and denote by $f_0 : V_0 \to T_0$ the restriction map. After possibly shrinking $T_0$, we may and do assume $f_0 : V_0 \to T_0$ is flat. Let $\phi : V \dashrightarrow Z$ be the birational map induced by the relative $r_d$-canonical map over $T$ and let $\phi_0 : V_0 \dashrightarrow Z_0$ be the restriction of $\phi$ to $V_0$ where $Z_0$ is the image. After shrinking $T_0$, we may and do assume that, for any $t \in T_0$, $\phi_0|_{V_t}$ is birational onto its image. Then we can embed $Z_0$ into $\mathbb{P}_{T_0}^m = T_0 \times \mathbb{P}^m$, where $m := h^0(r_dK_X) - 1$ for a general fiber $X$ of $f_0 : V_0 \to T_0$ and we denote by $h_0 : Z_0 \to T_0$ the projection morphism. Let $A := \mathcal{O}_{\mathbb{P}^m_{T_0}}(1) |_{Z_0}$ which is a relatively very ample divisor on $Z_0$ over $T_0$. $$\xymatrix{ V_0 \ar@{-->}[r]^{\phi_0} \ar[dr]_{f_0}& Z_0 \ar[d]^{h_0} \ar@{^(->}[r]^i& \mathbb{P}^m_{T_0} \ar[dl]^\pi \\ & T_0 & }$$ Then, for a fiber $Y$ of $h_0$, we have $$\deg{Y} = (A|_Y)^d \leq \text{\rm vol}(r_dK_X) \leq {r_d}^d M.$$ Since $\deg Y \geq h^0(r_dK_X) - d$, we have $m \leq N = \llcorner{{r_d}^{d}M}\lrcorner + d$. Thus we have an embedding $Z_0 \to {\mathbb{P}^N_{T_0}}$ such that the degree of any fiber of $h_0 : Z_0 \to T_0$ is no greater than ${r_d}^d M$. By the representability of the Chow functor, there is a morphism $\rho :T_0 \to S$ such that $Z_0$ is the pull-back of the universal family $\mathscr{X}$. After possibly shrinking $T_0$ again, we may assume that $\rho(T_0)$ is contained in some $S_i$. Since every blowup of $\mathscr{X}_i$ induces a blow up of $Z_0$, we get a birational model $V'_0$ by blowing up $Z_0$ finitely many times such that the fiber of $f'_0 : V'_0 \to T_0$ is smooth and belongs to $\mathcal{P}$. $$\xymatrix{ V'_0 \ar[r] \ar[d] \ar@/_2pc/[dd]_{f'_0}& \mathscr{Y}_i \ar[d] \\ Z_0 \ar[r] \ar[d]^{f_0} & \mathscr{X}_i \ar[d]^{p} \\ T_0 \ar[r]^\rho & S_i }$$ Since $Z_0$ is dense in $Z$, blowing up a center in $Z_0$ also induces a blowup of $Z$ by taking the closure in $Z$. Thus we get a morphism $f' : V' \to T$ such that a general fiber $X'$ of $f'$ is smooth and belongs to $\mathcal{P}$. Finally, after resolving the special fiber of $f'$ and replacing $V'$ with the resulting model, we may and do assume $V'$ is smooth. Then the morphism $g=f':V' \to T$ is what we want. And $\psi$ is taken to be the inverse of the composition of the two birational maps $V \dashrightarrow Z \dashrightarrow V'$.\ *Step 3.* Proof of Statement (iii). Let $\alpha : W \to V$ and $\beta : W \to V'$ be a common resolution of $V$ and $V'$. Write $K_W = \beta^*K_V + E$ and $K_W = \alpha^*K_{V'} + F$, where $E,F$ are effective. For a general fiber $X'$ of $g: V' \to T$, $\psi|_{X'}$ is birational and let $X$ be the birational transform of $X'$ on $V$. By assumption, there is a $\mathbb{Q}$-divisor $\Delta \sim_\mathbb{Q}\delta K_V$ on $V$ such that $X$ is an irreducible non-klt center of $(V,\Delta)$. Let $\Delta':=\beta_*(\alpha^*\Delta + \delta E)$, so $\Delta' \sim_\mathbb{Q}\delta K_{V'}$. Since $X,X'$ are general, they are not contained in the exceptional locus of $p,q$ and $\beta_*{E}$ does not contain $X'$. Thus $\psi$ induces an isomorphism between neighborhoods of the generic points of $X$ and $X'$, then it follows that the log discrepancies over these neighborhoods with respect to $(V,\Delta)$ and $(V', \Delta')$ are the same. Therefore $X'$ is an irreducible non-klt center of $(V',\Delta')$. ◻ We will apply the following boundedness theorem of Birkar in our argument. **Theorem 18**. *([@Bir21 Theorem 1.8]) Let $d$ and $r$ be positive integers and $\epsilon$ be a positive real number. Then there exists a positive number $t$, depending only on $d$, $r$ and $\epsilon$ such that the following property holds:* > *Assume that* > > - *$(X,B)$ is a projective $\epsilon$-lc pair of dimension $d$,* > > - *$A$ is a very ample divisor on $X$ with $A^d \leq r$,* > > - *$A - B$ is pseudo-effective,* > > - *$M \geq 0$ is an $\mathbb{R}$-Cartier $\mathbb{R}$-divisor with $A-M$ being pseudo-effective.* > > *Then one has $${\rm lct}(X,B,|M|_{\mathbb{R}}) \geq {\rm lct}(X,B,|A|_\mathbb{R}) \geq t.$$* ## An extension theorem We need to apply the following useful extension theorem due to Hacon and McKernan. **Theorem 19**. *([@HM06 Corollary 3.17]) Let $(V,\Delta),S,C,H$ satisfy the following conditions:* - *$(V,\Delta)$ is a log smooth lc $\mathbb{Q}$-pair;* - *$S$ is a component of $\Delta$ with coefficient $1$ such that $(K_V + \Delta)|_S$ is pseudo-effective;* - *$C \geq 0$ is an effective $\mathbb{Q}$-divisor whose support does not contain $S$;* - *there is a $\mathbb{Q}$-divisor $G \sim_\mathbb{Q}K_V + \Delta + C$ such that $G$ does not contain any lc centers of $(V,\ulcorner{\Delta}\urcorner)$;* - *$H$ is a sufficiently ample integral divisor which does not contain $S$ and only depends on $(V,\Delta),S,C$ and $A:=(\dim V+1)H$.* *We give some notations of divisors on $S$ as follows: $\Delta_S := (\Delta - S) |_S$, $C_S:= C|_S$, $H_S:=H|_S$, and $A_S :=A|_S$. Then, for any positive integer $m$ such that $m\Delta$ and $mC$ are integral, the image of the restriction map $$H^0(V,m(K_V + \Delta + C) + H + A) \to H^0(S, m(K_S + \Delta_S + C_S) + H_S + A_S)$$ contains the image of the inclusion $$i: H^0(S,m(K_S + \Delta_S) + H_S) \to H^0(S, m(K_S + \Delta_S + C_S) + H_S + A_S)$$ induced by the effective divisor $mC_S + A_S$.* Note that, in Theorem [Theorem 19](#extend){reference-type="ref" reference="extend"}, $H$ and $A$ do not depend on $m$. # An extension theorem for fiber spaces--The proof of Theorem [Theorem 5](#key){reference-type="ref" reference="key"} {#an-extension-theorem-for-fiber-spacesthe-proof-of-theorem-key} We start with considering a special case of Theorem [Theorem 5](#key){reference-type="ref" reference="key"} where only one general fiber is taken into account. **Theorem 20**. *Let $n$ and $d$ be two integers with $n > d >0$ and $\mathcal{P}$ be a birationally bounded set of smooth projective varieties of dimension $d$. Then there exists a positive number $t$, depending only on $d$ and $\mathcal{P}$, such that the following property holds:* > *Let $f : V \to T$ be a contraction between smooth projective varieties and denote by $X$ a general fiber of $f$, which satisfies the following conditions:* > > - *$V$ is of general type;* > > - *$\dim V = n$, $\dim X = d$ and $X$ is birationally equivalent to an element in the bounded set $\mathcal{P}$;* > > - *there exists a positive rational number $\delta < t$ with $\Delta \sim_\mathbb{Q}\delta K_V$ such that $X$ is an irreducible non-klt center of $(V,\Delta)$.* > > *Then, for any integer $p$ with $p \geq 2$, the restriction map $$H^0(V,pK_V) \to H^0(X, pK_X)$$ is surjective.* *Proof.* First of all, by virtue of Lemma [Lemma 17](#m7){reference-type="ref" reference="m7"}, after possibly replacing $\mathcal{P}$ with a bounded set, we may and do assume that $\mathcal{P}$ is a bounded set and that the general fiber $X$ of $f$ belongs to $\mathcal{P}$.\ *Step 1.* Existence of $t$. By the proof of [@CJ17 Theorem 3.5], there exists a finite set $\mathcal{S}$ of positive integers depending only on $d$ and $\mathcal{P}$ such that for any $X \in \mathcal{P}$ the canonical ring $\oplus_{i\geq 0}{H^0(iK_X)}$ is generated by finitely many elements whose degree numbers belong to $\mathcal{S}$. Take $l$ be a common multiple of all numbers in $\mathcal{S}$. Then $l$ only depends on $d$ and $\mathcal{P}$. On the other hand, since $X$ is bounded, there is a very ample Cartier divisor $A$ on $X$ such that $A^d = \tilde{v}$ and that $A - K_X$ is ample and effective, where $\tilde{v}$ is a positive number which depends only on $d$ and $\mathcal{P}$. Since $(X,0)$ is canonical, by Theorem [Theorem 18](#BAB2){reference-type="ref" reference="BAB2"}, there exists a positive number $t$ depending on $d$, $l$ and $\tilde{v}$ (hence only depending on $d$ and $\mathcal{P}$) such that, for any effective divisor $D$ on $X$ satisfying $A-D$ being pseudo-effective, we have such property that $(X,2tlD)$ is klt. In next steps, we prove that this number $t$ satisfies requirements of the theorem.\ *Step 2.* The setup for blowing-up $X$. By Lemma [Lemma 7](#perturb){reference-type="ref" reference="perturb"}, take a small rational number $\epsilon > 0$ with $\delta(1+\epsilon) < t$ and replace $\delta$ with $\delta(1+\epsilon)$, $\Delta$ with another $\Delta'\sim_\mathbb{Q}\delta K_V$. Then we may and do assume that $X$ is a pure and exceptional lc center of $(V,\Delta)$. Let $\nu: W \to V$ be a log resolution of $(V, \Delta)$ and $E$ be the unique lc place on $W$ with $\nu(E)=X$. By the existence of the minimal model [@BCHM10] and Lemma [Lemma 10](#m2){reference-type="ref" reference="m2"}(5), after possibly replacing $W$ with a further resolution, we may and do assume that $P_\sigma(\nu^*(K_V))$ is nef and that the components of $N_\sigma(\nu^*(K_V))$, exceptional divisors of $\nu$ and the birational transform of $\Delta$ have simple normal crossings. Denote by $g:E \to X$ the restriction map of $\nu$ to $E$. We see $g$ is a contraction by the connectedness lemma [@F92 Theorem 17.4]. Then write $$\nu^*(K_V + \Delta) = K_W + E + \Gamma_1 + \Gamma_2 - F,$$ such that $\Gamma_1,\Gamma_2,F$ are effective with no common components, that $\Gamma_{1,E}$ is $g$-horizontal and that $\Gamma_{2,E}$ is $g$-vertical, where $\Gamma_{1,E},\Gamma_{2,E},F_E$ are the restriction $\Gamma_1|_E, \Gamma_2|_E,F|_E$, respectively. Since $E$ is the unique lc place of $(V,\Delta)$ with $\nu(E)=X$, we see that $\Gamma_{1,E} < 1$. In particular, there is no $g$-horizontal non-klt centers of $(E, \Gamma_{1,E} + \Gamma_{2,E} - F_E)$.\ *Step 3.* The pair $(E, \Gamma_{1,E} + \Gamma_{2,E} - F_E)$ is sub-klt. In fact, we would like to prove that, for any effective $\mathbb{R}$-divisor $G \sim_\mathbb{R}A$ on $X$, $(E, \Gamma_{1,E} + \Gamma_{2,E} - F_E + 2\delta(l-1)g^*G)$ is sub-klt. Since $g:(E,\Gamma_{1,E} + \Gamma_{2,E} - F_E) \to X$ is a klt-trivial fibration (see Subsection [\[adjunction\]](#adjunction){reference-type="ref" reference="adjunction"}(iv)$\sim$(vii)), we can do adjunction for $g:(E,\Gamma_{1,E} + \Gamma_{2,E} - F_E) \to X$. Write $$K_E + \Gamma_{1,E} + \Gamma_{2,E} - F_E \sim_\mathbb{R}g^*(K_X + B_X + M_X),$$ where $B_X$ is the discriminant part and $M_X$ is the moduli part, and we have $B_X + M_X \sim_\mathbb{Q}\Delta|_X \sim_\mathbb{Q}\delta K_X$ by direct computations. We have $B_X \geq 0$ by [@Amb99 Lemma 4.2(3)] (see also Subsection [\[adjunction\]](#adjunction){reference-type="ref" reference="adjunction"}(viii)). We are prepared to use Lemma [Lemma 16](#m5){reference-type="ref" reference="m5"} for the contraction $g:E\to X$. By Lemma [Lemma 15](#m4.5){reference-type="ref" reference="m4.5"}, we can take an excellent base extension for $g:(E,\Gamma_{1,E} + \Gamma_{2,E} - F_E) \to X$. Let $\pi : E' \to E$ be a birational morphism such that $E'$ is smooth and that $g' :E' \dashrightarrow X'$ is an induced contraction from $g$. $$\xymatrix{ E' \ar[r]^{\pi} \ar[d]_{g'} & E \ar[d]^g \\ X' \ar[r]^{\sigma} & X }$$ Let $K_{E'} + \Gamma_{1,E'} + \Gamma_{2,E'} - F_{E'} = \pi^*(K_E + \Gamma_{1,E} + \Gamma_{2,E} - F_E)$ be the crepant pullback, where $\Gamma_{1,E'},\Gamma_{2,E'},F_{E'}$ are effective with no common components, $\Gamma_{1,E'}$ is $g'$-horizontal and $\Gamma_{2,E'}$ is $g'$-vertical. Let $B_{X'}$, $M_{X'}$ be the respective discriminant part, the moduli part due to adjunction for $g' : (E', \Gamma_{1,E'} + \Gamma_{2,E'} - F_{E'}) \to X'$. Here we require that $\sigma_*M_{X'} = M_X$ as explained in Subsection [\[adjunction\]](#adjunction){reference-type="ref" reference="adjunction"}(ii). If we do adjunction for $g': (E', \Gamma_{1,E'} + \Gamma_{2,E'} - F_{E'} + 2\delta(l-1)\pi^*g^*G) \to X'$, we will get the discriminant part $B_{X'} + 2\delta(l-1)\sigma^*(G)$ and the moduli part $M_{X'}$ (see [\[adjunction\]](#adjunction){reference-type="ref" reference="adjunction"}(ii)). Thus $X'$ is still an excellent base extension as well for the klt-trivial fibration $g: (E, \Gamma_{1,E} + \Gamma_{2,E} - F_E + 2\delta(l-1)g^*G) \to X$. By Lemma [Lemma 16](#m5){reference-type="ref" reference="m5"}, it suffices to prove that the sub-pair $(X', B_{X'} + 2\delta(l-1)\sigma^*G)$ is sub-klt. Since $M_{X'}$ is nef and $\sigma^*(\delta A)$ is nef and big, there exists an effective $\mathbb{R}$-divisor $D_{X'} \sim_\mathbb{R}\sigma^*(\delta A) + M_{X'}$. It is sufficient to prove that the pair $$(X', B_{X'} + 2\delta(l-1)\sigma^*G + D_{X'})$$ is sub-klt. Since $$K_{X'} + B_{X'} + 2\delta(l-1)\sigma^*G+ D_{X'} \overset{\sigma}\sim_\mathbb{R}0\ \ (\text{over}\ X),$$ let $D_X := \sigma_*(D_{X'}) \sim_{\mathbb{R}} \delta A + M_X$, then we have $$K_{X'} + B_{X'} + 2\delta(l-1)\sigma^*G + D_{X'} = \sigma^*(K_X + B_X + 2\delta(l-1)G + D_X).$$ So it suffices to prove that $(X,B_X + 2\delta(l-1)G + D_X)$ is klt. Since $$\begin{aligned} 0 \leq B_X + 2\delta(l-1)G + D_X &\sim_\mathbb{R}B_X + 2\delta(l-1)A + \delta A + M_X \\ &\sim_\mathbb{R}2\delta(l-1)A + \delta A + \delta K_X \\ &\leq 2\delta l A\end{aligned}$$ and $\delta < t$, we see that $(X,B_X + 2\delta(l-1)G + D_X)$ is klt by the construction of $t$ in Step 1.\ *Step 4.* The pseudo-effectiveness of $g^*(\delta K_X) - \Gamma_{2,E}$. Since $(E, \Gamma_{1,E} + \Gamma_{2,E} - F_E)$ is sub-klt, by Lemma [Lemma 8](#m1.5){reference-type="ref" reference="m1.5"}, there is a log resolution $\rho : \hat{E} \to E$ from a smooth model $\hat{E}$ such that if we write $$K_{\hat{E}} + \Gamma_{1,\hat{E}} + \Gamma_{2,\hat{E}} - F_{\hat{E}} = \rho^*(K_E + \Gamma_{1,E} + \Gamma_{2,E} - F_E)$$ for the crepant pull-back, where $\Gamma_{1,\hat{E}}$, $\Gamma_{2,\hat{E}}$ , $F_{\hat{E}}$ are effective with no common components, $\Gamma_{1,\hat{E}}$ is horizontal over $X$ and $\Gamma_{2,\hat{E}}$ is vertical over $X$, then $(\hat{E}, \Gamma_{1,\hat{E}} + \Gamma_{2,\hat{E}} - F_{\hat{E}})$ is terminal. Let $\hat{g} =g \circ \rho : \hat{E} \to X$ be the composite map. Then take $\tau : X'' \to X$ to be a birational morphism from a smooth model $X''$ that flattens $\hat{g} : \hat{E} \to X$. Let $\xi : E'' \to \hat{E}$ be a birational morphism from a smooth model $E''$ (which factors through the flat model $U\to X''$ of $\hat{g}$) such that the induced birational map $g'' :E'' \dashrightarrow X''$ is a morphism. Let $\eta = \rho \circ \xi : E'' \to E$ be the composition. $$\xymatrix{ E'' \ar@/^1.5pc/[rr]^\eta \ar[r]^{\xi} \ar[d]_{g''} & \hat{E} \ar[r]^\rho \ar[d]^{\hat{g}} & E \ar[dl]^g \\ X'' \ar[r]^{\tau} & X }$$ Let $$K_{E''} + \Gamma_{1,E''} + \Gamma_{2,E''} - F_{E''} = \xi^*(K_{\hat{E}} + \Gamma_{1,\hat{E}} + \Gamma_{2,\hat{E}} - F_{\hat{E}})$$ be the crepant pull-back on $E''$, where $\Gamma_{1,{E''}}$, $\Gamma_{2,{E''}}$ and $F_{{E''}}$ are effective with no common components and $\Gamma_{1,{E''}}$ is horizontal over $X$, and $\Gamma_{2,{E''}}$ is vertical over $X$. Since $(\hat{E}, \Gamma_{1,\hat{E}} + \Gamma_{2,\hat{E}} - F_{\hat{E}})$ is terminal, $\Gamma_{2,E''}$ is just the birational transform of $\Gamma_{2,{\hat{E}}}$. In particular, any component in $\Gamma_{1,\hat{E}}$ and $\Gamma_{2,\hat{E}}$ can not be $\xi$-exceptional. In other words, the image of any prime component of $\Gamma_{2,E''}$ under $g''$ is still a prime divisor on $X''$, since $\tau$ flattens $\hat{g} : \hat{E} \to X$. We may also require that $X''\to X$ is an excellent base extension with respect to the pair $(\hat{E}, \Gamma_{1,\hat{E}} + \Gamma_{2,\hat{E}} - F_{\hat{E}})$. Let $B_X$ and $M_X$ be as in Step 3. Let $B_{X''}$ and $M_{X''}$ be the discriminant part and the moduli part of the adjunction for $g'' : (E'', \Gamma_{1,E''} + \Gamma_{2,E''} - F_{E''}) \to X''$. We write $B_{X''} = B_{X''}^+ - B_{X''}^-$ in the unique way with $B_{X''}^+$ and $B_{X''}^-$ being effective with no common components. Note that $M_{X''}$ is nef in our situation. Since $\eta_*(\Gamma_{2,E''}) = \Gamma_{2,E}$, it is sufficient to prove that $${g''}^*\tau^*(\delta K_X) - \Gamma_{2,E''}$$ is pseudo-effective. Since $${g''}^*\tau^*(\delta K_X) - \Gamma_{2,E''} = {g''}^*(B_{X''}^+) - \Gamma_{2,E''} + {g''}^*(\tau^{*}(\delta K_X) - B_{X''}^+),$$ it is enough to prove that both ${g''}^*(B_{X''}^+) - \Gamma_{2,E''}$ and $\tau^{*}(\delta K_X) - B_{X''}^+$ are pseudo-effective. We prove that ${g''}^*(B_{X''}^+) - \Gamma_{2,E''} \geq 0$. In fact, for any component $D''$ of $\Gamma_{2,E''}$ on $E''$, let $P'' := g''(D'')$. By the above assumption, $P''$ is a prime divisor on $X''$. By the definition of the discriminant part, we have $\mu_{P''}({B_{X''}}) = 1-t$, where $$t = \sup\{a|\Gamma_{1,E''} + \Gamma_{2,E''} - F_{E''} + a g''^*(P'') \text{ is lc over the generic point of $P''$}\}.$$ Let $c:= \mu_{D''}(g''^*(P''))$ and we see $c$ is a positive integer. Then we have $\mu_{D''}(\Gamma_{2,E''} + t g''^*(P'')) = \mu_{D''}(\Gamma_{2,E''}) + tc \leq 1$. Thus, $$\begin{aligned} \mu_{D''}g''^*(B_{X''}) &= c \cdot \mu_{P''}(B_{X''}) \\ &= c(1-t) \\ &\geq 1-ct \geq \mu_{D''}(\Gamma_{2,E''}).\end{aligned}$$ Thus $g''^*(B_{X''}^+) - \Gamma_{2,E''} \geq_{\mathbb{R}} 0$ holds. Then we are left to prove that $\tau^{*}(\delta K_X) - B_{X''}^+$ is pseudo-effective. We rewrite $$K_{X''} + B_{X''} + M_{X''} = \tau^*(K_X + B_X + M_X),$$ as $$\tau^*(B_X + M_X) - M_{X''} = K_{X''/X} + B_{X''}^+ - B_{X''}^-. \label{eq3.1}$$ Note that $M_{X''}$ is actually determined up to $\mathbb{R}$-linear equivalence, we may choose from the very beginning that $\tau_*M_{X''}=M_X$. Since $M_{X''}$ is nef, $\tau^*(M_X) \geq M_{X''}$ by the negativity lemma [@Bir12 Lemma 3.3]. Since $B_X \geq 0$, the left hand side of [\[eq3.1\]](#eq3.1){reference-type="eqref" reference="eq3.1"} is effective. Since $K_{X''/X} \geq 0$ by the smoothness of $X$ and that $B_{X''}^+,B_{X''}^-$ have no common components, we get $$\tau^*(B_X + M_X) - M_{X''} \geq B_{X''}^+.$$ Thus $\tau^*(\delta K_X) - B_{X''}^+ \sim_\mathbb{R}\tau^*(B_X + M_X) - B_{X''}^+ \geq M_{X''}$ is pseudo-effective.\ *Step 5.* A key application of Theorem [Theorem 19](#extend){reference-type="ref" reference="extend"} (yielding Inequality [\[step5\]](#step5){reference-type="eqref" reference="step5"}). By Step 4, $g^*(\delta K_X) - \Gamma_{2,E}$ is pseudo-effective. Noting that $$g^*(\delta K_X) - \Gamma_{2,E} \sim_\mathbb{Q}K_E - g^*K_X + \Gamma_{1,E} - F_E,$$ there exists an ample divisor $H_E$ such that for any sufficiently large and divisible integer $m$, we have $h^0(E,m(K_E - g^*K_X + \Gamma_{1,E} - F_E) + H_E) > 0$ by [@Laz2 Corollary 11.2.13]. After taking a sufficiently ample divisor $H$ on $W$ such that $H|_E - H_E \geq 0$ and replacing $H_E$ with $H|_E$, we can assume $H|_E = H_E$. Note that $H$ does not depend on $m$. For any $D_m \in |m(K_E - g^*K_X + \Gamma_{1,E} - F_E) + H_E|$, there is an inclusion of linear system $$|g^*(mK_X)| \hookrightarrow |m(K_E + \Gamma_{1,E}) + H_E|,$$ induced by the divisor $D_m + mF_E$. To use Theorem [Theorem 19](#extend){reference-type="ref" reference="extend"}, we need to check all the conditions. Actually the notation $(V,\Delta),S,C,H$ in Theorem [Theorem 19](#extend){reference-type="ref" reference="extend"}, respectively, corresponds to $(W,E + \Gamma_1),E,\Gamma_2,H$ here. Besides, the divisor $A$ in Theorem [Theorem 19](#extend){reference-type="ref" reference="extend"} should be $(\dim W+1)H=(n+1)H$. Note that $$K_W + E + \Gamma_1 + \Gamma_2 \sim_\mathbb{Q}\nu^*(K_V + \Delta) + F.$$ Since $X$ is a general fiber of $f : V\to T$, that $F$ has no common components with $E$ and $\Gamma_{1}$, and that ${\rm Supp }(F) \cup {\rm Supp }(E) \cup {\rm Supp } (\Gamma_{1})$ has simple normal crossings, we see there exists an effective $\mathbb{Q}$-divisor $\Lambda \sim_\mathbb{Q}K_W + E + \Gamma_{1} + \Gamma_{2}$ such that $\Lambda$ does not contain any lc center of $(W,E + \ulcorner{\Gamma_1}\urcorner)$. In a word, all conditions of Theorem [Theorem 19](#extend){reference-type="ref" reference="extend"} in our setting here are satisfied. Then, by Theorem [Theorem 19](#extend){reference-type="ref" reference="extend"}, we see that any element of the image of $$i: |m(K_E + \Gamma_{1,E}) + H_E| \to |m(K_E + \Gamma_{1,E} + \Gamma_{2,E}) + (n+2)H_E|$$ can be lifted to an element in $$|m(K_W + E + \Gamma_1 + \Gamma_2) + (n+2)H|=|m(\nu^*((1+\delta) K_V)+F) + (n+2)H|,$$ where $i$ is the injection induced by the divisor $m\Gamma_{2,E} + (n+1)H_E$ and $m$ is any positive integer making all above divisors integral. By the argument above, for any effective member $N_m \in |g^*(mK_X)|$, there exists an effective $\Theta_m \in |m(\nu^*((1+\delta) K_V)+F) + (n+2)H|$, such that $$\Theta_m |_E = N_m + (D_m + mF_E) + (m\Gamma_{2,E} + (n+1)H_E).$$ Denote by $F_m$ the fixed part of $|m(\nu^*((1+\delta) K_V)+F) + (n+2)H|$ and by $Z_m$ the fixed part of $|g^*(mK_X)|$. Then, by Lemma [Lemma 6](#m1){reference-type="ref" reference="m1"}, we get $$F_m|_E \leq Z_m + D_m + mF_E + m\Gamma_{2,E} + (n+1)H_E$$ for any effective member $D_m \in | m (g^*(\delta K_X) - \Gamma_{2,E}) + H_E|$. Let $Y_m := {\rm Fix}|m (g^*(\delta K_X) - \Gamma_{2,E}) + H_E|$. Since $F_m$ does not depend on the choice of $D_m$, we have $$F_m|_E \leq Z_m + Y_m + mF_E + m\Gamma_{2,E} + (n+1)H_E.\label{fix-ineq}$$ Divide Inequality [\[fix-ineq\]](#fix-ineq){reference-type="eqref" reference="fix-ineq"} by $m$ on both sides and let $m$ tends to the infinity. Since $H$ and $H_E$ are sufficiently ample, while applying Lemma [Lemma 10](#m2){reference-type="ref" reference="m2"}, we have $$\begin{aligned} \lim_{m \to \infty}F_m/m &=& N_\sigma{(\nu^*((1+\delta)K_V) +F)} = (1+\delta)N_\sigma{(\nu^*(K_V)) +F},\\ \lim_{m \to \infty}{Z_m/m}& = &N_\sigma{(g^*(K_X))},\\ \lim_{m \to \infty}{Y_m/m}& = &N_\sigma{(g^*(\delta K_X) - \Gamma_{2,E})}. \end{aligned}$$ Hence Inequality [\[fix-ineq\]](#fix-ineq){reference-type="eqref" reference="fix-ineq"} reads $$(1+\delta)N_{\sigma}(\nu^*(K_V))|_E \leq N_\sigma(g^*(K_X))+ Q, \label{step5}$$ where $Q := N_\sigma{(g^*(\delta K_X) - \Gamma_{2,E})} + \Gamma_{2,E}$. We see that $Q \geq 0$. By the argument in Step 4, we have seen that $$g^*(\delta A) - Q=(g^*(\delta A)-\Gamma_{2,E})-N_\sigma{(g^*(\delta K_X) - \Gamma_{2,E})}$$ is pseudo-effective, where $A$ is the very ample divisor constructed in Step 1.\ *Step 6.* The surjective map [\[step6\]](#step6){reference-type="eqref" reference="step6"} by Kawamata-Viehweg vanishing theorem. Fix an integer $p$ with $2\leq p \leq l$ and define $$\begin{aligned} L &:= \llcorner{\nu^*(K_V + \Delta) - K_W - E + (p-1- \delta)N_\sigma(\nu^*K_V)}\lrcorner \\ &= \llcorner{\Gamma_1 + \Gamma_2 - F + (p-1- \delta)N_\sigma(\nu^*K_V)}\lrcorner. \end{aligned}$$ Note that, since $N_\sigma(\nu^*K_V)$ does not contain $E$, the support of $L$ does not contain $E$. Since $$\begin{aligned} && \nu^*(pK_V) - E - K_W - L \\ &\sim_\mathbb{Q}& (p-1-\delta)\nu^*K_V + \Gamma_1 + \Gamma_2 - F - L \\ &\sim_\mathbb{Q}& \langle\Gamma_1 + \Gamma_2 - F + (p-1-\delta)N_\sigma(\nu^*K_V)\rangle + (p-1-\delta)P_\sigma(\nu^*K_V),\end{aligned}$$ where $(W,\langle{\Gamma_1 + \Gamma_2 - F + (p-1-\delta)N_\sigma(\nu^*K_V)}\rangle)$ is a klt pair and $(p-1-\delta)P_\sigma(\nu^*K_V)$ is nef and big, the restriction map $$H^0(W, \nu^*(pK_V) - L) \longrightarrow H^0(E, g^*(pK_X) - L|_E) \label{step6}$$ is surjective, by the Kawamata-Viehweg vanishing theorem.\ *Step 7.* The injective map [\[step7\]](#step7){reference-type="eqref" reference="step7"} between cohomological groups. We explore the divisor $L|_E$. By [\[step5\]](#step5){reference-type="eqref" reference="step5"}, we have $$\begin{aligned} L|_E &=& \llcorner{\Gamma_{1,E} + \Gamma_{2,E} - F_E + (p-1-\delta)N_\sigma(\nu^*(K_V)|_E}\lrcorner \\ &\leq& \llcorner{\Gamma_{1,E} + \Gamma_{2,E} - F_E + \frac{p-1-\delta}{1+\delta}(N_\sigma(g^*(K_X)) + Q)}\lrcorner \\ &\leq& \ulcorner{L_{1,E}}\urcorner + \llcorner{L_{2,E}}\lrcorner,\end{aligned}$$ where $L_{1,E} := \frac{p-1-\delta}{1+\delta} N_\sigma(g^*(K_X))$ and $L_{2,E} := \Gamma_{1,E} + \Gamma_{2,E} - F_E + \frac{p-1-\delta}{1+\delta}Q$. Since $g^*(\delta A) - Q$ is pseudo-effective, by Lemma [Lemma 12](#m4){reference-type="ref" reference="m4"}, there exists an effective $G \sim_\mathbb{R}A$ on $X$ such that $g^*(2\delta G) - Q \geq 0$. Thus $L_{2,E} \leq \Gamma_{1,E} + \Gamma_{2,E} - F_E + (p-1)g^*(2\delta G)$. In Step 3, we have shown that $(E, \Gamma_{1,E} + \Gamma_{2,E} - F_E + (p-1)g^*(2\delta G))$ is sub-klt. Hence $(E, L_{2,E})$ is also sub-klt. It follows that $\llcorner{L_{2,E}}\lrcorner \leq 0$. Also it is clear that one has $L_{1,E} \leq N_\sigma(g^*(pK_X))$. Thus $$\begin{aligned} H^0(E, g^*(pK_X) - L|_E) &\supseteq& H^0(E,g^*(pK_X) - \ulcorner{L_{1,E}}\urcorner - \llcorner{L_{2,E}}\lrcorner) \\ &\supseteq& H^0(E,g^*(pK_X) - \ulcorner{N_\sigma(g^*(pK_X))}\urcorner) \\ &=& H^0(E,g^*(pK_X)).\end{aligned}$$ As the conclusion, there is the following injective map $$j : H^0(E,g^*(pK_X)) \hookrightarrow H^0(E, g^*(pK_X) - L|_E).\label{step7}$$ *Step 8.* Concluding the statement of this theorem. In the following commutative diagram, the first row is surjective by [\[step6\]](#step6){reference-type="eqref" reference="step6"} and the injectivity of $j: H^0(g^*(pK_X)) \to H^0(g^*(pK_X) - L|_E)$ is due to [\[step7\]](#step7){reference-type="eqref" reference="step7"}. Denote by $r$ the restriction map $$H^0(\nu^*(pK_V)) \to H^0(E, g^*(pK_X)).$$ We also see $H^0(\nu^*(pK_V)) = H^0(\nu^*(pK_V) + \ulcorner{F}\urcorner)$ since the support of $F$ is $\nu$-exceptional. Besides, the two vertical injections are due to the fact $-L \leq \ulcorner{F}\urcorner$ and $-L|_E \leq \ulcorner{F|_E}\urcorner$. $$\xymatrix{ H^0(\nu^*(pK_V) - L ) \ar@{^(->}[d] \ar@{->>}[r] & H^0(g^*(pK_X) - L|_E) \ar@{^(->}[d]\\ H^0(\nu^*(pK_V) + \ulcorner{F}\urcorner) \ar@{=}[d] \ar[r] & H^0(g^*(pK_X) + \ulcorner{F|_E}\urcorner) \\ H^0(\nu^*(pK_V)) \ar@{=}[d] \ar[r] ^r & H^0(g^*(pK_X)) \ar@{=}[d] \ar@{^(->}[u] \ar@/_7.5pc/[uu]^{j} \\ H^0(pK_V) \ar[r] & H^0(pK_X) }$$ Now it can be seen that $r$ is surjective by chasing the diagram. In fact, for any element in $H^0(g^*(pK_X))$, it is embedded into $H^0(g^*(pK_X) - L|_E)$ by $j$ and then is lifted to an element in the group $H^0(\nu^*(pK_V) - L)$. Thus we naturally get an element in $H^0(\nu^*(pK_V))$, which is the pre-image we need to find. Finally, we have already proved that, for any integer $p \in [2,l]$, the restriction map $H^0(pK_V) \to H^0(pK_X)$ is surjective. Since for $i \geq l+1$, any element in $H^0(X,iK_X)$ can be written as a sum of products of elements in $H^0(X,pK_X)$ with $2 \leq p \leq l$ by the definition of $l$, we see that $H^0(V,iK_V) \to H^0(X,iK_X)$ is surjective as well for any integer $i \geq 2$. We have proved the theorem. ◻ **Corollary 21**. *(=Theorem [Theorem 5](#key){reference-type="ref" reference="key"}) Under the same condition as that of Theorem [Theorem 20](#main1){reference-type="ref" reference="main1"}, pick two general fibers $X_1$ and $X_2$ of $f:V\to T$. Then, for any integer $p \geq 2$, the restriction map $$H^0(V,pK_V) \to H^0(X_1, pK_{X_1}) \oplus H^0(X_2,pK_{X_2})$$ is surjective. In particular, $r_s(V)\leq \text{max}\{r_s(X),2\}$ where $X$ is the general fiber of $f$.* *Proof.* The spirit of the proof is similar to that of Theorem [Theorem 20](#main1){reference-type="ref" reference="main1"}. So there is no need to repeat the same argument except for the following points. Replacing $t$ with $\frac{1}{2}t$, we may assume that there exists a $\mathbb{Q}$-divisor $\Delta \sim_\mathbb{Q}\delta K_V$ for a positive rational $\delta < 2t$ such that both $X_1$ and $X_2$ are irreducible pure and exceptional lc centers of $(X, \Delta)$. Let $\nu: W \to V$ be a log resolution and $E_1$, $E_2$ be lc places of $X_1$, $X_2$, respectively and denote by $g_i : E_i \to X_i$ for $i=1,2$. Let $$L:= \llcorner{\nu^*(K_V + \Delta) - K_W - E_1 - E_2 + (p-1- \delta)N_\sigma(\nu^*K_V)}\lrcorner.$$ Going on the similar argument as in the proof of Theorem [Theorem 20](#main1){reference-type="ref" reference="main1"}, we will get the surjective map $$H^0(W, \nu^*(pK_V) - L) \to H^0(E_1, g_1^*(pK_X) - L|_{E_1}) \oplus H^0(E_2, g_2^*(pK_X) - L|_{E_2}).$$ Then we explore $L_{E_1}$ and $L_{E_2}$, respectively, in the similar way. All other argument follows similarly to that in Theorem [Theorem 20](#main1){reference-type="ref" reference="main1"} without any problems. So we leave the details to interested readers. ◻ # The proof of Theorem [Theorem 1](#C-L-1){reference-type="ref" reference="C-L-1"} and Corollary [Corollary 4](#C-L-2){reference-type="ref" reference="C-L-2"} {#the-proof-of-theorem-c-l-1-and-corollary-c-l-2} Let us start with recalling the following definition. **Definition 22**. *([@CJ17 Definition 6.5]) Given a birationally bounded set $\mathscr{X}$ of smooth projective varieties and given a positive number $c$, we say that a fibration $f : X \to T$ between smooth projective vareities satisfies condition $(B)_{\mathscr{X},c}$ if* 1. *a general fiber $F$ of $f$ is birationally equivalent to an element of $\mathscr{X}$;* 2. *for a general point $t \in T$, there exists an effective $\mathbb{Q}$-divisor $D_t$ with $D_t \sim_\mathbb{Q}\epsilon K_X$ for a positive rational number $\epsilon < c$, such that the fiber $F_t = f^{-1}(t)$ is an irreducible non-klt center of $(X,D_t)$.* The next theorem reduces our problem to prove the extension theorem in the style of Theorem [Theorem 5](#key){reference-type="ref" reference="key"}. **Theorem 23**. *([@CJ17 Theorem 6.8])[\[fibration\]]{#fibration label="fibration"} Let $n>1$ be an integer. Fix a function $\lambda: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{R}_{>0}$. There exists a constant $\mathcal{K} > 0$ and $n-1$ integers $M_{n-1} > M_{n-2} > \cdots > M_1 > 0$ such that, for any smooth projective $n$-fold $X$ with $\text{\rm vol}(X) \geq \mathcal{K}$, the pluricanonical map $\varphi_{a,X}$ is birational for all integers $a \geq 2$, unless that, after birational modifications, $X$ admits a fibration $f:X \to Z$ which satisfies Condition $(B)_{\mathscr{X}_{k,M_k^k},\lambda(k,M_k^k)}$, where $\mathscr{X}_{k,M_k^k}$ is the set of smooth projective $k$-folds $X$ of general type with $\text{\rm vol}(K_X) \leq M_k^k$.* Now we can prove our main theorem as follows. *Proof of Theorem [Theorem 1](#C-L-1){reference-type="ref" reference="C-L-1"}.* By Theorem [Theorem 5](#key){reference-type="ref" reference="key"}, for any integer $d$ and a birationally bounded set $\mathcal{P}$ consisting of smooth projective varieties of dimension $d$, there is a positive number $t = t(d,\mathcal{P})$ satisfying the conditions in Theorem [Theorem 5](#key){reference-type="ref" reference="key"} (see Step 1 in the proof of Theorem [Theorem 20](#main1){reference-type="ref" reference="main1"}). Then we define the following function $$\begin{aligned} \lambda : \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} &\to \mathbb{R}_{>0} \\ (d,M) &\mapsto t(d,\mathscr{X}_{d,M}), \end{aligned}$$ where $\mathscr{X}_{d,M}$ is the set of smooth projective $d$-folds of general type with the canonical volume $\leq M$. By Theorem [\[fibration\]](#fibration){reference-type="ref" reference="fibration"}, there exists a constant $\mathcal{K}>0$ and $n-1$ positive integers $M_{n-1}, \cdots M_1$ such that, for any smooth projective $n$-fold $V$ with $\text{\rm vol}(V) \geq \mathcal{K}$, the $m$-pluricanonical map $\varphi_{m,V}$ is birational for all integers $m \geq 2$, unless that, after replacing $V$ with a birational modification, there is a fibration $f : V \to T$ which satisfies Property $(B)_{\mathscr{X}_{k,M_k^k},\lambda(k,M_k^k)}$ for some positive integer $k\leq n-1$. Then, by Theorem [Theorem 5](#key){reference-type="ref" reference="key"} while recalling the definition of $\lambda$, the pluricanonical map $\varphi_{m,V}$ is birational for any $m \geq r_s(X_t)$ where $X_t$ is the general fiber of $f$. Noting that $r_s(X_t)\leq r_k$ and that the sequence $\{r_k\}$ is increasing, we have proved that the $m$-canonical map $\varphi_{m,V}$ is birational for all $m \geq r_{n-1}$. This completes the proof of the first statement. We consider a variety with positive geometric genus. On one hand, the canonical volume is linearly increasing with regard to the geometric genus due to the existence of Noether inequality (see Lemma [Lemma 9](#Noether){reference-type="ref" reference="Noether"}). On the other hand, given a fibration $f:V\to T$ with $p_g(V)>0$, each general fiber $X_t$ of $f$ still has $p_g(X_t)>0$. Hence one just simply replace $r_k$ with $r_k^+$ and copy the same proof in the first part. So the second statement follows. ◻ Before proving Corollary [Corollary 4](#C-L-2){reference-type="ref" reference="C-L-2"}, we need to apply the following effective inequality. **Theorem 24**. *([@CJ23 Theorem 1.1]) Fix two integers $n$ and $k$ with $n>0$ and $1\leq k< n$. There exist positive numbers $a_{n,k}$ and $b_{n,k}$ such that the following inequality $$\text{\rm vol}(X)\geq a_{n,k}h^0(X,\Omega_X^k)-b_{n,k}$$ holds for every smooth projective $n$-fold $X$ of general type.* **Theorem 25**. *(=Corollary [Corollary 4](#C-L-2){reference-type="ref" reference="C-L-2"}) For any integer $n \geq 2$, there exists a number $M(n) > 0$. For any smooth projective $n$-fold $V$ with either $|\chi(\mathcal{O}_V)| > M(n)$ or $h^i(\mathcal{O}_V)>M(n)$ for some positive integer $i\leq n$, the pluricanoncial map $\varphi_{m,X}$ is birational for all integers $m \geq r_{n-1}$.* *Proof.* Note that $\chi(\mathcal{O}_V) = \sum_{i=0}^n{(-1)^i h^i(\mathcal{O}_V)}$. If $|\chi(\mathcal{O}_V)|$ is sufficiently large, then so is $h^i(\mathcal{O}_V)$ for some positive number $i \leq n$. By Theorem [Theorem 24](#NN){reference-type="ref" reference="NN"}, $\text{\rm vol}(V)$ is also sufficiently large. Thus there exists a number $M(n) > 0$ such that, whenever $|\chi(\mathcal{O}_V)|>M(n)$ or $h^i(\mathcal{O}_V)>M(n)$, $\text{\rm vol}(V) > \mathfrak{V}(n)$ where $\mathfrak{V}(n)$ is the same constant described in Theorem [Theorem 1](#C-L-1){reference-type="ref" reference="C-L-1"}. Hence the statement follows directly from Theorem [Theorem 1](#C-L-1){reference-type="ref" reference="C-L-1"}. ◻ # **Acknowledgments** {#acknowledgments .unnumbered} The authors appreciate fruitful discussions with Zhi Jiang and Chen Jiang during the preparation of this paper. Especially Chen Jiang pointed out to us an imprecise application of the adjunction in Subsection [\[adjunction\]](#adjunction){reference-type="ref" reference="adjunction"} in an earlier version of this paper. The second author would like to thank Jianshi Yan, Yu Zou, Minzhe Zhu, Mengchu Li, Wentao Chang for useful discussions and help in study. OOOO99 Florin Ambro. *The adjunction conjecture and its applications*, ProQuest LLC, Ann Arbor, MI, 1999, 54 pp. Thesis (Ph.D.)-The Johns Hopkins University. Florin Ambro. *Shokurov's boundary property*, J. Differential Geom. **67**(2004), no.2, 229--255. Caucher Birkar. *Existence of log canonical flips and a special LMMP*, Publ. Math. Inst. Hautes Études Sci. **115** (2012), 325--368. Caucher Birkar. *Singularities of linear systems and boundedness of Fano varieties*, Ann. of Math. (2) **193** (2021), no. 2, 347--405. Caucher Birkar; Paolo Cascini; Christopher D. Hacon; James McKernan. *Existence of minimal models for varieties of log general type*, J. Amer. Math. Soc. **23** (2010), no. 2, 405--468. E. Bombieri. *Canonical models of surfaces of general type*. Inst. Hautes études Sci. Publ. Math. **42** (1973), 171--219. Jungkai A. Chen; Meng Chen. *Explicit biratioanl geometry for 3-folds and 4-folds of general type, III*. Compos Math. **151** (2015), 1041-1082. Meng Chen. *Canonical stability in terms of singularity index for algebraic threefolds*, Math. Proc. Cambridge Philos. Soc. **131** (2001), no. 2, 241--264. Meng Chen. *Canonical stability of 3-folds of general type with $p_g\geq 3$*, Internat. J. Math. **14** (2003), no. 5, 515--528. Meng Chen. *On minimal 3-folds of general type with maximal pluricanonical section index*. Asian J. Math. **22** (2018), no. 2, 257--268. Meng Chen; Yong Hu; Matteo Penegini. *On projective threefolds of general type with small positive geometric genus*, Electron. Res. Arch. **29** (2021), no. 3, 2293--2323. Meng Chen; Zhi Jiang. *A reduction of canonical stability index of 4 and 5 dimensional projective varieties with large volume*, Ann. Inst. Fourier (Grenoble) **67** (2017), no. 5, 2043--2082. Meng Chen; Zhi Jiang. *On projective varieties of general type with many global k-forms*, 2023. arXiv: 2211.0096 Louis Esser; Burt Totaro; Chengxi Wang. *Varieties of general type with doubly exponential asymptotics*, Trans. Amer. Math. Soc. Ser. B **10** (2023), 288--309. Osamu Fujino. *Basepoint-free theorems: saturation, b-divisors, and canonical bundle formula*, Algebra Number Theory **6** (2012), no.4, 797--823. Osamu Fujino; Yoshinori Gongyo.*On the moduli b-divisors of lc-trivial fibrations*, Ann. Inst. Fourier (Grenoble) **64** (2014), no.4, 1721--1735. Christopher D. Hacon; James McKernan. *Boundedness of pluricanonical maps of varieties of general type*. Invent. Math. **166** (2006), no. 1, 1--25. Heisuke Hironaka. *Flattening theorem in complex-analytic geometry*, Amer. J. Math. *97* (1975), 503--547. Yujiro Kawamata. *Subadjunction of log canonical divisors. II*, Amer. J. Math. **120** (1998), no. 5, 893--899. János Kollár. *Flips and abundance for algebraic threefolds*, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992), Société Mathématique de France, Paris, 1992, 1--258. János Kollár. *Rational curves on algebraic varieties*, Ergeb. Math. Grenzgeb. (3), 32. Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, 1996, viii+320 pp. János Kollár; Shigefumi Mori. *Birational geometry of algebraic varieties*, Cambridge Tracts in Math., 134. Cambridge University Press, Cambridge, 1998, viii+254 pp. Justin Lacini. *Boundedness of fibers for pluricanonical maps of varieties of general type*, Math. 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Algebra, **295** (2006), no. 1, 119--140.

arxiv_math

--- abstract: | A long-standing conjecture of Sacks states that it is provable in $\mathsf{ZFC}$ that every locally countable partial order of size continuum embeds into the Turing degrees. We show that this holds for partial orders of height two, but provide evidence that it is hard to extend this result even to partial orders of height three. In particular, we show that the result for height two partial orders holds both in certain extensions of $\mathsf{ZF}$ with only limited forms of choice and in the Borel setting (where the partial orders and embeddings are required to be Borel measurable), but that the analogous result for height three partial orders fails in both of these settings. We also formulate a general obstacle to embedding partial orders into the Turing degrees, which explains why our particular proof for height two partial orders cannot be extended to height three partial orders, even in $\mathsf{ZFC}$. We finish by discussing how our results connect to the theory of countable Borel equivalence relations. address: - College of Engineering, Nihon University, 1 Nakagawara, Tokusada, Tamuramachi, Koriyama, 963-8642, Japan - Department of Mathematics, University of California, Los Angeles author: - Kojiro Higuchi - Patrick Lutz bibliography: - bibliography.bib title: "A Note on a Conjecture of Sacks: It is Harder to Embed Height Three Partial Orders than Height Two Partial Orders" --- # Introduction An enduring goal of computability theory is to determine which structures can be embedded into the Turing degrees. When the structures under consideration are partial orders, there are two obvious restrictions: since the set of Turing degrees has size continuum and every Turing degree has at most countably many predecessors, any partial order which embeds into the Turing degrees must also have these properties. A famous conjecture of Sacks states that these are the only restrictions. More precisely, say that a partial order $(P, \leq_P)$ is **locally countable** if every element $x \in P$ has at most countably many predecessors (i.e. the set $\{y \in P \mid y <_P x\}$ is countable). In 1963, Sacks conjectured that every locally countable partial order of size continuum can be embedded into the Turing degrees [@sacks1963degrees]. Sacks himself proved that this holds in $\mathsf{ZFC}+ \mathsf{CH}$ (by showing that it holds in $\mathsf{ZFC}$ for all locally countable partial orders of size $\omega_1$), but whether it is provable in $\mathsf{ZFC}$ alone is still unknown. We will not resolve Sacks's conjecture in this paper. Instead, we will present a curious phenomenon related to it. Namely, we will demonstrate that it is easy to embed partial orders of height two into the Turing degrees, but hard to embed partial orders of height three. This statement deserves some explanation. First, we will prove (in $\mathsf{ZFC}$) that every locally countable partial order of size continuum and height two embeds into the Turing degrees. We will also show that this result is robust, in the sense that it holds even in settings where only weak forms of choice are available. Second, we will show that there is a general obstacle to embedding partial orders into the Turing degrees which implies that our method of embedding height two partial orders cannot be extended to partial orders of height three. Essentially, our method for height two partial orders embeds the first level of the partial order as a perfect set, but we show that whenever the image of an embedding contains a perfect set, the embedding cannot be extended very much. Moreover, in some settings this obstacle actually yields an outright proof that not all locally countable partial orders of size continuum and height three can be embedded into the Turing degrees, including some of the settings in which our proof for height two partial orders works. Thus, there are some settings in which all locally countable partial orders of size continuum and height two embed into the Turing degrees, but the same does not hold for height three. We will show that this is the case in two particular settings: the $\mathsf{ZF}$ setting, where we work in certain extensions of $\mathsf{ZF}$ which contradict the full Axiom of Choice (but still satisfy weak forms of it), and the Borel setting, where the partial orders and embeddings are required to be Borel measurable.[^1] At the end of the paper, we will mention some connections between our results and the theory of countable Borel equivalence relations. We will now give more precise statements of these results (in particular, we will specify which extensions of $\mathsf{ZF}$ we consider and what we mean by the "Borel setting") and discuss the context for our obstacle to embedding partial orders into the Turing degrees. ## Partial orders of finite height For any natural number $n > 0$, a partial order has **height $n$** if its longest chain has length exactly $n$. We can think of such a partial order as consisting of $n$ "levels": the first level consists of those elements of the partial order with no predecessors, the second level consists of those elements whose predecessors are all in the first level (and which have at least one predecessor in the first level), the third level consists of those elements whose predecessors are all in the first two levels (and which have at least one predecessor in the second level), and so on. The fact that there are no chains of length greater than $n$ implies that every element will end up in one of these $n$ levels (and the fact that there is some chain of length $n$ implies that no level is empty). A typical partial order of height three, stratified into levels in this manner, looks something like this: As stated above, we will prove that Sacks's conjecture holds for partial orders of height two. theoremheighttwozfc [\[thm:height2zfc\]]{#thm:height2zfc label="thm:height2zfc"} Every locally countable partial order of size continuum and height two embeds into the Turing degrees. An equivalent statement was also proved by Kumar and Raghavan in [@kumar2021separating], by a somewhat different technique. However, for reasons that we will discuss later, their proof does not generalize to other settings as well as ours. ## Obstacles to embedding partial orders in the Turing degrees Suppose you have a partial order $(P, \leq_P)$ of size continuum and you want to embed $P$ into the Turing degrees. A reasonable approach is to pick a well-ordering of $P$ of length continuum and define an embedding by transfinite recursion. In other words, pick up elements of $P$ one at a time and show that as long as you have embedded fewer than continuum many elements so far, there is always a place to map the next element to. This is essentially the approach taken by Sacks to embed locally countable partial orders of size $\omega_1$ in [@sacks1963degrees]. A fundamental obstacle to using this approach to solve Sacks's Conjecture was discovered by Groszek and Slaman [@groszek1983independence]. Say that a set $A$ of Turing degrees is **Turing independent** if there is no finite subset of $A$ whose join computes some element of $A$ not in the subset. Groszek and Slaman proved that it is consistent with $\mathsf{ZFC}$ that there is a maximal Turing independent set of size less than continuum. Thus if you want to construct an embedding by transfinite recursion, you have to be careful not to end up with this particular Turing independent set in the image of your embedding at any step in the recursion. For suppose that you do. If you later encounter another element of $P$ which is sufficiently independent of all the elements you have seen so far, there will be nowhere to map it to. Kumar has used Groszek and Slaman's technique to show that a similar problem may occur even when embedding a partial order of height three whose first level has size $\omega_1$ [@kumar2019suborders]. One potential solution to this problem is to make the transfinite recursion satisfy some stronger inductive assumption that prevents it from accidentally building any set like the one constructed by Groszek and Slaman, but no such condition has been identified so far. A different potential solution is to take a more structural approach. To illustrate what we mean, suppose that the partial order $P$ which we want to embed has finite height. Instead of embedding the elements of $P$ one at a time, we can first find an especially nice subset of the Turing degrees to map the first level of $P$ to, then use the niceness of this subset to find another nice subset to map the second level to, and so on. In fact, this is exactly the approach we use to prove Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"}. In particular, we embed the elements of the first level of the partial order as the Turing degrees of a Turing independent perfect set of reals. Since we don't rely on transfinite recursion, this approach is easier to generalize to settings with only limited forms of choice. It might seem reasonable to hope that our proof can be generalized to deal with partial orders of any finite height. For example, we might try to show that the second level of $P$ can also be embedded as a Turing independent perfect set, and more generally that if the $n^\text{th}$ level can be embedded as a Turing independent perfect set then so can the $(n + 1)^\text{st}$ level. However, our next theorem shows that this is not possible. theoremobstacleone [\[thm:obstacle1\]]{#thm:obstacle1 label="thm:obstacle1"} There is a locally countable partial order $(P, \leq_P)$ of size continuum and height three which has the following properties. 1. The first level of $P$ has size continuum. 2. If $f$ is any function from $P$ into the Turing degrees such that the image of $f$ on the first level of $P$ contains a perfect set then $f$ is not an embedding. The fist condition on $P$ may appear somewhat arbitrary, but it is necessary to make the second condition nontrivial: if the first level of $P$ has size less than continuum then its image under $f$ cannot contain a perfect set for cardinality reasons and so the second condition is vacuously true. It is common to phrase obstacles to embedding partial orders into the Turing degrees in terms of obstacles to extending Turing independent sets. We can also do that here. Note that when we say that a set of reals $A$ is Turing independent, we mean that no finite subset of $A$ computes any other element of $A$. theoremobstacletwo [\[thm:obstacle2\]]{#thm:obstacle2 label="thm:obstacle2"} Suppose $A$ is a Turing independent set of reals which contains a perfect set, $A'$, $B$ is a countable, dense subset of $A'$ and $x$ computes every element of $B$. Then $(A\setminus B)\cup \{x\}$ is not Turing independent. To prove Theorems [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} and [\[thm:obstacle2\]](#thm:obstacle2){reference-type="ref" reference="thm:obstacle2"}, we rely on a technical theorem on perfect sets, previously used by the second author and Benjamin Siskind in work on Martin's Conjecture [@lutz2023part]. ## The $\mathsf{ZF}$ setting By examining the proof of Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"} above, we will see that it does not require the full Axiom of Choice. In particular, let us consider two weak choice principles: **Uniformization for Reals**, denoted $\mathsf{Uniformization}_\mathbb{R}$, and **Lusin-Novikov Choice**, denoted $\mathsf{LN}$. - $\mathsf{Uniformization}_\mathbb{R}$ states that every real-indexed family of nonempty sets of reals has a choice function---i.e. if $R$ is a binary relation on $2^\omega$ such that for each $x$, $\{y \mid R(x, y)\}$ is nonempty, then there is some function $f\colon 2^\omega\to 2^\omega$ such that for each $x$, $R(x, f(x))$ holds. - $\mathsf{LN}$ states that for every binary relation $R$ on $2^\omega$, if every section of $R$ is countable (i.e. for each $x\in 2^\omega$, $\{y \in 2^\omega\mid R(x, y)\}$ is countable) then there is a function $f\colon 2^\omega\to (2^\omega)^{\leq \omega}$ enumerating the elements of each section. It is not hard to check that $\mathsf{Uniformization}_\mathbb{R}$ implies $\mathsf{LN}$ (the point is just that the set of enumerations of a countable set of reals can itself be thought of as a set of reals). We will see that it is basically trivial to modify the proof of Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"} to work in $\mathsf{ZF}+ \mathsf{Uniformization}_\mathbb{R}$ and, with only slightly more care, it is also possible to modify the proof to work in $\mathsf{ZF}+ \mathsf{LN}$. Thus we have the following theorem. theoremheighttwozf [\[thm:height2zf\]]{#thm:height2zf label="thm:height2zf"} Every locally countable partial order of size continuum and height two embeds into the Turing degrees. On the other hand, there is also an extension of $\mathsf{ZF}$ in which we can use Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} to show that not every locally countable partial order of size continuum and height three can be embedded into the Turing degrees. Let $\mathsf{PSP}$ denote the **Perfect Set Principle**, which states that every subset of $2^\omega$ is either countable or contains a perfect subset. The key point is that in $\mathsf{ZF}+ \mathsf{PSP}$, the hypothesis of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} is automatically satisfied. Using this observation, we will prove the following theorem. theoremheightthreezf [\[thm:height3zf\]]{#thm:height3zf label="thm:height3zf"} There is a locally countable partial order of size continuum and height three which does not embed into the Turing degrees. Note that the theory $\mathsf{ZF}+ \mathsf{LN}+ \mathsf{PSP}$, in which Theorems [\[thm:height2zf\]](#thm:height2zf){reference-type="ref" reference="thm:height2zf"} and [\[thm:height3zf\]](#thm:height3zf){reference-type="ref" reference="thm:height3zf"} are both provable, is known to be consistent and thus there is a single consistent extension of $\mathsf{ZF}$ in which Sacks's conjecture holds for partial orders of height two but not for partial orders of height three. Here's one way to see that $\mathsf{ZF}+ \mathsf{LN}+ \mathsf{PSP}$ is consistent. Let $\mathsf{AD}_\mathbb{R}$ denote the **Axiom of Real Determinacy**, an axiom which has been extensively studied in inner model theory [@solovay1978independence]. $\mathsf{ZF}+ \mathsf{AD}_\mathbb{R}$ implies the Axiom of Determinacy, and hence $\mathsf{PSP}$ (see [@jech2003set], Theorem 33.3) as well as $\mathsf{Uniformization}_\mathbb{R}$ [@solovay1978independence], and hence $\mathsf{LN}$. Furthermore, $\mathsf{ZF}+ \mathsf{AD}_\mathbb{R}$ is known to be consistent, assuming large cardinals. As a side note, Theorem [\[thm:height3zf\]](#thm:height3zf){reference-type="ref" reference="thm:height3zf"} shows that Sacks's conjecture is independent of $\mathsf{ZF}$. As far as we are aware, this fact has not been published before.[^2] ## The Borel setting Statements analogous to Theorems [\[thm:height2zf\]](#thm:height2zf){reference-type="ref" reference="thm:height2zf"} and [\[thm:height3zf\]](#thm:height3zf){reference-type="ref" reference="thm:height3zf"} also hold in the Borel setting, where we consider only Borel partial orders and Borel embeddings. A **Borel partial order** is simply a partial order $(P, \leq_P)$ such that $P$ and $\leq_P$ are Borel measurable subsets of $2^\omega$ and $2^\omega\times 2^\omega$, respectively.[^3] If $(P, \leq_P)$ is a Borel partial order then a **Borel embedding** of $P$ into Turing reducibility is a Borel measurable function $f\colon P \to 2^\omega$ such that for all $x, y \in P$, $$x \leq_P y \iff f(x) \leq_T f(y).$$ One oddity here is that Turing reducibility itself is not a Borel partial order according to our definition (because it is not a partial order on $2^\omega$, but only a quasi-order). We will return to this point in section [4](#sec:cber){reference-type="ref" reference="sec:cber"}. It is relatively straightforward to modify the proofs of Theorems [\[thm:height2zf\]](#thm:height2zf){reference-type="ref" reference="thm:height2zf"} and [\[thm:height3zf\]](#thm:height3zf){reference-type="ref" reference="thm:height3zf"} to yield the following (essentially we just replace $\mathsf{LN}$ and $\mathsf{PSP}$ with analogous theorems provable in the Borel setting). theoremheighttwoborel [\[thm:height2borel\]]{#thm:height2borel label="thm:height2borel"} Every locally countable Borel partial order of height two has a Borel embedding into Turing reducibility. theoremheightthreeborel [\[thm:height3borel\]]{#thm:height3borel label="thm:height3borel"} There is a locally countable Borel partial order of height three with no Borel embedding into Turing reducibility. Note that in this context, we do not need to explicitly assume that $P$ has size continuum---this follows automatically from the definition of "Borel partial order." ## Acknowledgements {#acknowledgements .unnumbered} Thanks to Steffen Lempp for encouraging us to write this paper and to Benny Siskind and Ted Slaman for several helpful conversations. Also thanks to Ashutosh Kumar for pointing us to the papers [@groszek1983independence; @kumar2019suborders; @kumar2021separating] and for an interesting email exchange and to Elliot Glazer for answering several questions about choiceless set theory. # Embedding height two partial orders is easy {#sec:height2} In this section we will explain how to embed any height two, locally countable partial order of size continuum into the Turing degrees. As discussed in the introduction, we will first prove the theorem in $\mathsf{ZFC}$ and then explain how to modify the proof to work in the theory $\mathsf{ZF}+ \mathsf{LN}$ and in the Borel setting. Here's the basic strategy of the proof. Given a partial order $(P, \leq_P)$ of height two, we will construct a function $f\colon P \to 2^\omega$ such that $$x \leq_P y \iff f(x) \leq_T f(y).$$ It is clear that such a function induces an embedding of $P$ into the Turing degrees, so this is sufficient. In order to construct $f$, we will first pick a perfect set of mutually generic reals. We will then map each element of the first level of $P$ to an element of this perfect set and each element of the second level of $P$ to a sufficiently generic upper bound of the images of its predecessors. There is one wrinkle in this proof: we need to ensure that even when two elements of the second level have exactly the same predecessors, they get mapped to incomparable Turing degrees.[^4] One way to handle this is to insert a unique point below each element of the second level, which is not below any other elements of the partial order. This ensures that the elements of the second level all have distinct sets of predecessors and an embedding of this new partial order yields an embedding of the original partial order by forgetting about the new elements that we added. When we give the proof in detail, we will not quite explain things in this way, but it is essentially what will happen. ## $\mathsf{ZFC}$ case We will break the proof into two lemmas, the first of which tells us that we can find a perfect set of mutually generic reals to map the elements of the first level to and the second of which tells us that we can always find sufficiently generic upper bounds to map the elements of the second level to. Both lemmas are essentially folklore, though we are not aware of anywhere that they are written up in the precise form we would like to use. To state these two lemmas, we need to recall the definition of a perfect tree. **Definition 1**. A tree $T \subseteq 2^{< \omega}$ is called a **perfect tree** if every node in $T$ has incomparable descendants in $T$. The name "perfect tree" is used because the set of infinite paths through $T$ is a perfect subset of $2^\omega$. A perfect tree $T$ can be pictured as a kind of warped version of $2^{< \omega}$: there are no dead ends and if you follow any path for long enough, you will eventually come to a place where you can choose to go left or right and remain in the tree either way. In $2^{< \omega}$ you can make this decision on every step while in an arbitrary perfect tree you may have to take many steps in between each decision. By making this picture more precise, it is possible to prove that the set of paths through a perfect tree is always homeomorphic to $2^\omega$. **Fact 2**. *If $T \subseteq 2^{< \omega}$ is a perfect tree then $[T]$ is homeomorphic to $2^\omega$ and therefore has size continuum.* Instead of stating our first lemma in terms of mutually generic reals we will state it using the notion of a Turing independent set, which was mentioned in the introduction (and which is all we really need for our application of the lemma). Actually, we will not quite use the definition of "Turing independent set" from the introduction, but an essentially equivalent notion which applies to sets of reals rather than sets of Turing degrees. **Definition 3**. A set $A \subseteq 2^\omega$ of reals is called a **Turing independent set** if no finite subset of $A$ computes any other element of $A$---i.e. if $a_0, \ldots, a_n \in A$ and $b$ is any element of $A$ not equal to any $a_i$ then $a_0\oplus \ldots \oplus a_n$ does not compute $b$. We can now actually state our first lemma. The argument is due to Sacks [@sacks1961suborderings Theorem 3], though he did not state it in terms of perfect trees. **Lemma 4** (Sacks). *There is a perfect tree $T$ such that $[T]$ is a Turing independent set.* *Proof.* First consider how one might construct a perfect tree with no computable branches. To do this, we need to ensure that every branch of the tree disagrees with each total computable function in at least one place. We can accomplish this by "growing" the tree from the root node up in a series of stages. At each stage we have built a finite tree and we continue growing it by extending the leaf nodes (i.e. by adding children to the leaf nodes, children to those children, and so on). On alternate stages we can add incomparable children below every leaf node (to make sure the tree is perfect) and extend each leaf node to make sure any branch which extends it disagrees with the next total computable function (to make sure no branch is computable). Note that we do not need the tree itself to be computable so these steps are easy to carry out. To make sure that no finite set of branches computes any other branch, we can do something similar but now instead of extending leaf nodes one at a time to make them disagree with the next computable function, we need to extend finite sets of leaf nodes at the same time to make sure the next computable function which uses those branches as an oracle disagrees with the branches of the tree which extend the other leaf nodes. We will now describe this a bit more formally. We will form a sequence of finite subtrees of $2^{< \omega}$, $T_0 \subseteq T_1 \subseteq T_2 \subseteq \ldots$ such that $T_{n + 1}$ is an end extension of $T_n$ (every node in $T_{n + 1} \setminus T_n$ extends a leaf node of $T_n$). The final tree will be obtained as $T = \bigcup_{n \in \mathbb{N}} T_n$. We can start with $T_0$ as the tree just consisting of a single root node and nothing else (i.e. just the empty sequence). Now we will explain how to extend $T_n$ to $T_{n + 1}$. The idea, again, is to first split every leaf node in $T_n$ and then extend all of them without splitting in order to make sure that no finite subset of them is correctly computing any of the others using the the $n^\text{th}$ Turing functional. To this end, first let $T_n^0$ be the tree formed by adding incomparable children below each leaf node of $T_n$. In other words, $$T_n^0 = T_n \cup \{\sigma^\frown 0 \mid \sigma \text{ is a leaf node of $T_n$}\} \cup \{\sigma^\frown 1 \mid \sigma \text{ is a leaf node of $T_n$}\}.$$ Next, let $\Phi$ be the $n^\text{th}$ Turing functional in some standard enumeration. We will form a finite sequence of end extensions $T_n^0 \subseteq T_n^1 \subseteq \ldots \subseteq T_n^k$ (where $k$ is the number of nonempty subsets of the set of leaves of $T_n^0$) and take $T_{n + 1} = T_n^k$. In each of these extensions, we will not split any nodes. In other words, each leaf of $T_n^0$ will have at most one descendant at each level of $T_n^i$. Suppose we have already formed $T_n^i$ and let $S$ be the $i^\text{th}$ nonempty subset of the set of leaves of $T_n^0$. We will now explain how to form $T_n^{i + 1}$. Our goal is to ensure that no set of branches extending the nodes in $S$ can compute any branch extending any other leaf node of $T_n^0$. Since we never split any nodes in any of the previous extensions of $T_n^0$, each element of $S$ corresponds to a unique leaf of $T_n^i$. Let $\sigma_1, \ldots, \sigma_l$ denote these leaves of $T_n^i$. Let $N$ be a number larger than the height of $T_n^i$. Now either we can find extensions $\tau_1,\ldots, \tau_l$ of $\sigma_1,\ldots,\sigma_l$ such that $\Phi^{\tau_1\oplus\ldots\oplus\tau_l}(N)$ converges or we can't find such extensions. In the former case, define $T_n^{i + 1}$ from $T_n^i$ by extending each $\sigma_j$ to $\tau_j$ and extending all other leaf nodes of $T_n^i$ to strings whose $N^\text{th}$ bit disagrees with $\Phi^{\tau_1\oplus\ldots\oplus\tau_l}(N)$. In the latter case, set $T_n^{i + 1} = T_n^i$. Now let's check that $[T]$ is really a Turing independent set. Let $x_1,\ldots,x_l$ and $y$ be distinct elements of $[T]$ and let $\Phi$ be any Turing functional. Let $n$ be some number large enough that all of $x_1,\ldots, x_l$ and $y$ correspond to distinct leaf nodes in $T_n$ and chosen so that the $n^\text{th}$ Turing functional is equivalent to $\Phi$ (we are assuming that every computable function shows up infinitely often in whatever enumeration we are using). Suppose that $x_1,\ldots, x_l$ correspond to the $i^\text{th}$ set of leaves of $T_n^0$. Then our definition of $T_n^{i + 1}$ ensures that either $\Phi^{x_1\oplus \ldots\oplus x_l}$ disagrees with $y$ in at least one place (this corresponds to the first case in our construction above) or $\Phi^{x_1\oplus\ldots\oplus x_l}$ is not a total function (this corresponds to the second case). ◻ Our second lemma guarantees we can find sufficiently generic upper bounds to map the elements of the second level of our partial order to. The idea of the proof is originally due to Spector, who used it to show that every increasing sequence of Turing degrees has an exact pair of upper bounds (see Theorem 6.5.3 of [@soare2016turing]). **Lemma 5**. *Suppose $T$ is a perfect tree such that $[T]$ is Turing independent. Then every countable subset of $[T]$ has an upper bound in the Turing degrees which does not compute any other element of $[T]$.* *Proof.* Suppose $A$ is a countable subset of $[T]$ and $x_0, x_1, x_2, \ldots$ is an enumeration of the elements of $A$. In order to uniformly handle both the case where $A$ is finite and the case where $A$ is infinite (which will be helpful later) we allow the enumeration to contain repetitions and therefore can assume that it is always infinite. Here's the idea of the proof. We will construct an element $y$ of $2^{\omega \times \omega}$ such that column $n$ of $y$ consists of some finite string followed by $x_n$ and this $y$ will be the upper bound we are after. It is easy to see that any such $y$ computes each element of $A$ and so the bulk of the proof consists of showing that if we choose the finite strings in a sufficiently generic way then $y$ does not compute any other element of $[T]$. The proof crucially depends on the fact that $[T]$ is Turing independent. Formally, we will construct $y$ in a series of stages. At the end of stage $n$ we will have constructed a finite list of finite strings, $\sigma_1, \ldots, \sigma_k$ (where $k$ may not be equal to $n$) and on stage $n + 1$ we will add some more strings onto the end of this list. At the end, we will define column $i$ of $y$ to be $\sigma_i^\frown x_i$. The idea is that on stage $n + 1$ we will ensure that if we run the $n^\text{th}$ program with oracle $y$ it either does not compute any element $[T]$ or it computes one of $x_1,\ldots, x_k$. We will now explain how to complete one step of this construction. Suppose we have just completed stage $n$ and our list of finite strings is $\sigma_0,\sigma_1,\ldots,\sigma_k$. Let us say that a finite string $\tau$ in $2^{<\omega \times < \omega}$ (i.e. a finite initial segment of an element of $2^{\omega\times\omega}$) **agrees with $y$ so far** if for each $i \leq k$, column $i$ of $\tau$ agrees with $\sigma_i^\frown x_i$. In other words, $\tau$ is a possible initial segment of $y$ given what we have built by the current stage. Let $\Phi$ denote the $n^\text{th}$ Turing functional. There are four cases to consider. **Case 1:** There is some finite string $\tau \in 2^{< \omega \times < \omega}$ which agrees with $y$ so far such that $\Phi^\tau$ cannot be extended to a path through $T$. In this case, extend the list $\sigma_0,\sigma_1,\ldots,\sigma_k$ to ensure that $y$ is an extension of $\tau$. This guarantees that $\Phi^y$ is not in $[T]$. **Case 2:** There is some finite string $\tau$ which agrees with $y$ so far and some $m \in \mathbb{N}$ such that for every extension $\tau'$ of $\tau$ which agrees with $y$ so far, $\Phi^{\tau'}(m)$ does not converge. In this case, extend the list $\sigma_0,\sigma_1,\ldots,\sigma_k$ to ensure that $y$ is an extension of $\tau$. This guarantees that $\Phi^y$ is not total. **Case 3:** For every finite string $\tau$ which agrees with $y$ so far, $\Phi^\tau$ is compatible with one of $x_0,x_1,\ldots,x_k$. In this case, do nothing; we are already guaranteed that $\Phi^y$ is either not total or is equal to one of $x_0, x_1, \ldots, x_k$. **Case 4:** None of the first three cases holds. We claim this case actually cannot happen. In particular, in this case we can use $x_0, x_1, \ldots, x_k$ to compute another element of $[T]$, thus violating our assumption that $[T]$ is Turing independent. To do so, first note that because Case 3 does not hold, we can find some finite string $\tau$ which agrees with $y$ so far and such that $\Phi^\tau$ is not compatible with any of $x_0, x_1,\ldots,x_k$. Next, inductively form a sequence $\tau = \tau_0 \prec \tau_1 \prec \tau_2 \prec \ldots$ of finite strings (which all agree with $y$ so far) as follows. Given $\tau_m$, look for some extension $\tau_{m + 1}$ of $\tau_m$ which agrees with $y$ so far and such that $\Phi^{\tau_{m + 1}}(m)$ converges. Because we are not in Case 2, we will always be able to find such a string. Let $z$ be the infinite sequence formed by the $\tau_m$. By construction, $\Phi^z$ is total. Since $z$ extends $\tau$, $\Phi^z$ is not equal to any of $x_0,\ldots,x_k$. Since Case 1 does not hold, $\Phi^{\tau_m}$ is compatible with an element of $[T]$ for each $m$. Since $[T]$ is closed, this implies that $\Phi^z$ itself is in $[T]$. To summarize, $\Phi^z$ is an element of $[T]$ which is not equal to any of $x_0,\ldots, x_k$. The final point is that to carry out the process of choosing the $\tau_m$ described above, we only need to be able to check which finite strings agree with $y$ so far. If we know $x_0,x_1,\ldots,x_k$ then this is easy to do, so $z$ (and hence $\Phi^z$) is computable from $x_0\oplus x_1\oplus\ldots\oplus x_k$. ◻ We will now explain how to put these two lemmas together to prove Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"}. *Proof.* Let $(P, \le_P)$ be a height two, locally countable partial order of size continuum. Let $P_1$ denote the first level of $P$ and let $P_2$ denote the second level. As we mentioned above, instead of directly defining a map of $P$ into the Turing degrees, we will define a map $f\colon P \to 2^\omega$ such that for all $x, y \in P$, $$x \leq_P y \iff f(x) \leq_T f(y).$$ **Definition of $\bm{f}$.** First, let $T$ be a perfect tree such that $[T]$ is Turing independent, as in Lemma [Lemma 4](#lemma:tree){reference-type="ref" reference="lemma:tree"}. Since $P$ and $[T]$ both have size continuum, we can find an injective map $g \colon P \to [T]$. We will define $f(x)$ by cases depending on whether $x$ is in $P_1$ or $P_2$. If $x$ is in $P_1$ then we simply set $f(x) = g(x)$. If $x$ is in $P_2$ then define $f(x)$ as follows. Let $P_{\le x} = \{y \in P \mid y \le_P x\}$ be the set of (non-strict) predecessors of $x$ in $P$; note that $P_{\le x}$ includes $x$ itself. By Lemma [Lemma 5](#lemma:upperbound){reference-type="ref" reference="lemma:upperbound"}, we can find a real which computes every element of $g(P_{\le x})$ but which computes no other elements of $[T]$. Set $f(x)$ equal to some such real. **$\bm{f}$ is an embedding.** Now we need to check that $f$ is an embedding. Let $x$ and $y$ be any two distinct elements of $P$. We need to show that $x \le_P y$ if and only if $f(x) \le_T f(y)$. First suppose $x \leq_P y$. If $x = y$ then we are done, and if not then $x$ must be in the first level of $P$ and $y$ must be in the second level. Therefore $f(x) = g(x)$ and $f(y)$ is an upper bound in the Turing degrees for a set which includes $g(x)$, so $f(y)$ computes $f(x)$. Now suppose that $x \nleq_P y$. We know that no matter which level $x$ is in, $f(x)$ computes $g(x)$. So to show that $f(y)$ doesn't compute $f(x)$, it is enough to show that $f(y)$ doesn't compute $g(x)$. If $y$ is in the first level of $P$ then this is guaranteed by the fact that $g(x)$ and $g(y)$ are distinct elements of the Turing independent set $[T]$ and $f(y) = g(y)$. And if $y$ is in the second level of $P$ then since $x$ is not a predecessor of $y$, our choice of $f(y)$ ensures that it cannot compute $g(x)$. ◻ The theorem we have just proved is very similar to Theorem 2.2 of the paper "Separating Families and Order Dimension of Turing Degrees" by Kumar and Raghavan [@kumar2021separating]. That theorem states that a specific height two, locally countable partial order of size continuum---which the authors refer to as $\mathbb{H}_\mathfrak{c}$---embeds into the Turing degrees.[^5] Obviously this theorem is implied by our Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"}. On the other hand, it is not too hard to show that every height two, locally countable partial order of size continuum embeds into $\mathbb{H}_\mathfrak{c}$ and so the two theorems are actually equivalent. However, there are some differences between our proof and that of Kumar and Raghavan. While we embedded the first level of the partial order as a single Turing independent perfect set and then embedded the elements of the second level essentially independently of each other, Kumar and Raghavan construct their embedding by a transfinite recursion of length continuum. To find places to embed elements of the first level of the partial order, they use the existence of a Turing independent set of size continuum and to find places for elements of the second level, they use the fact that for any countable ideal of Turing degrees there is a set of reals of size continuum, any two of which form an exact pair for the ideal. Kumar and Raghavan's approach has some advantages and disadvantages compared to our approach. On the one hand, since they do not rely on any specific properties of perfect sets their approach does not obviously fall prey to the obstacle presented by our Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} (and there are known constructions of large Turing independent sets which do not produce perfect sets, see [@kumar2023large]). On the other hand, since their approach relies on picking a well-order of size continuum, it cannot be easily adapted to work in the $\mathsf{ZF}$ or Borel settings and also seems more susceptible to the obstacle discovered by Groszek and Slaman (see the note [@kumar2019suborders] by Kumar for more about how this problem applies to their approach). In the end, for all their differences, both methods run into the same problem when embedding partial orders of height three: it's not clear how to make sure the second level of the partial order is embedded as a Turing independent set. ## $\mathsf{ZF}+ \mathsf{LN}$ case We will now show how to modify our proof of Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"} to work in the theory $\mathsf{ZF}+ \mathsf{LN}$. In other words, we will explain how to prove the following theorem. The key observation is that there is actually only one part of the proof that cannot be carried out in $\mathsf{ZF}$: the application of Lemma [Lemma 5](#lemma:upperbound){reference-type="ref" reference="lemma:upperbound"}. Recall that in the proof of Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"}, we had a height two, locally countable partial order $(P, \leq_P)$ of size continuum with first level $P_1$ and second level $P_2$. We first picked a perfect tree $T$ such that $[T]$ is Turing independent and an injection $g\colon P \to [T]$. Up to this point in the proof, everything works in $\mathsf{ZF}$. Then, for each $x \in P_2$, we used Lemma [Lemma 5](#lemma:upperbound){reference-type="ref" reference="lemma:upperbound"} to find an upper bound for $g(P_{\leq x})$ which does not compute any other element of $[T]$. The problem is that while Lemma [Lemma 5](#lemma:upperbound){reference-type="ref" reference="lemma:upperbound"} itself is provable in $\mathsf{ZF}$, it only implies the existence of an appropriate upper bound for $g(P_{\leq x})$, but does not tell us an explicit way to choose such an upper bound. Since we need to choose an upper bound for each $x \in P_2$, mere existence is not enough and so this part of the proof seems to require the use of some form of choice. It is fairly clear that $\mathsf{Uniformization}_\mathbb{R}$ is enough: we can identify the elements of $P_2$ with reals and the set of appropriate upper bounds for each $g(P_{\leq x})$ with a set of reals and thus $\mathsf{Uniformization}_\mathbb{R}$ lets us pick an appropriate upper bound for each $g(P_{\leq x})$. However, by examining the proof of Lemma [Lemma 5](#lemma:upperbound){reference-type="ref" reference="lemma:upperbound"}, we can do slightly better. The only reason the proof of that lemma does not give us an explicit way of constructing an upper bound is because it requires choosing an enumeration of the given countable set. Thus if we could choose an enumeration of each set $g(P_{\leq x})$ then we could use the proof of Lemma [Lemma 5](#lemma:upperbound){reference-type="ref" reference="lemma:upperbound"} to pick upper bounds without any further use of choice. Fortunately, choosing enumerations of countable sets of reals is exactly what $\mathsf{LN}$ lets us do. ## Borel case We will now explain how to modify the proof of Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"} to work in the Borel setting. In other words, how to prove the following theorem. Since the changes are slightly more substantial than for the $\mathsf{ZF}+ \mathsf{LN}$ case, this time we will give a formal proof. But first, we will explain what changes we need to make. There are essentially two problems to overcome in this setting. The first problem is that, just as in the $\mathsf{ZF}+ \mathsf{LN}$ case, we need to deal with the one part of the proof that uses choice. Fortunately, there is an analogue of Lusin-Novikov Choice in the Borel setting: the Lusin-Novikov Theorem.[^6] See Theorem 18.10 and Exercise 18.15 of Kechris's book [@kechris1995classical] for a proof. **Theorem 6** (Lusin-Novikov Uniformization Theorem). *Suppose $R$ is a Borel subset of $2^\omega \times 2^\omega$ with countable sections (i.e. for each $x \in 2^\omega$, the set $\{y \mid R(x, y)\}$ is countable). Then both of the following hold:* 1. *The domain of $R$ (i.e. the set $\mathop{\mathrm{dom}}(R) = \{x \mid \exists y\, R(x, y)\}$) is Borel.* 2. *There is a sequence $\langle f_n \rangle_{n \in \mathbb{N}}$ of Borel functions $f_n \colon \mathop{\mathrm{dom}}(R) \to 2^\omega$ enumerating the sections of $R$. (i.e. for each $x \in \mathop{\mathrm{dom}}(R)$, $\{f_n(x) \mid n \in \mathbb{N}\} = \{y \mid R(x, y)\}$).* The second problem is that in our proof of Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"}, we defined the embedding by cases depending on whether the input was in the first level or second level of the partial order. In $\mathsf{ZF}$, this is not a problem, but in the Borel setting, this will only yield a Borel function if each of the two cases corresponds to a Borel subset of the partial order. However, it turns out that we can use the Lusin-Novikov Theorem to solve this problem as well. When giving the formal proof, it will be helpful to have an explicitly stated, refined version of Lemma [Lemma 5](#lemma:upperbound){reference-type="ref" reference="lemma:upperbound"} (which guarantees the existence of sufficiently generic upper bounds). The proof of the refined version of the lemma just consists of noting that other than choosing an enumeration of the countable set, every part of the construction of the upper bound in the original proof was arithmetically definable and thus yields a Borel map from enumerations to upper bounds. **Lemma 7**. *Suppose $T$ is a perfect tree such that $[T]$ is Turing independent. Then there is a Borel function $h_T \colon (2^\omega)^\omega \to 2^\omega$ such that for any sequence $\overline{x} = \langle x_n \rangle_{n \in \mathbb{N}}$ of elements of $[T]$, $h_T(\overline{x})$ computes all of the $x_n$'s but does not compute any other element of $[T]$.* *Proof of Theorem [\[thm:height2borel\]](#thm:height2borel){reference-type="ref" reference="thm:height2borel"}.* Let $(P, \le_P)$ be a height two, locally countable Borel partial order and let $P_1$ and $P_2$ denote the first and second levels of $P$, respectively. Recall that in the proof of Theorem [\[thm:height2zfc\]](#thm:height2zfc){reference-type="ref" reference="thm:height2zfc"}, we constructed a map $f \colon P \to 2^\omega$ such that $x \le_P y$ if and only if $f(x) \le_T f(y)$. In this proof, we will simply give a definition of the function $f$ which makes it clear why it is Borel; the proof that it is an embedding is unchanged. Let $T$ be a perfect tree such that $[T]$ is Turing independent and let $h_T$ be the function from Lemma [Lemma 7](#lemma:upperboundborel){reference-type="ref" reference="lemma:upperboundborel"}. Let $g \colon 2^\omega \to [T]$ be a homeomorphism (though we really only need $g$ to be a Borel bijection). Now consider the binary relation $R$ on $2^\omega$ defined by $$R(x, y) \iff y <_P x.$$ Note that since $\leq_P$ is Borel, so is $R$. Furthermore, note that the domain of $R$ is exactly $P_2$ and for each $x \in P_2$, the section $\{y \mid R(x, y)\}$ is exactly the set of (strict) predecessors of $x$. Since $P$ is locally countable, the sections of $R$ are all countable. Thus we can apply the Lusin-Novikov Theorem to conclude two things: 1. $P_2$ is Borel. Since $P$ itself is Borel, this implies that $P_1 = P\setminus P_2$ is Borel as well. 2. There are Borel functions $k_n \colon P_2 \to 2^\omega$ such that for each $x \in P_2$, $\{k_n(x) \mid n \in \mathbb{N}\}$ is equal to the set of (strict) predecessors of $x$. It is then straightforward to check that the following definition: $$f(x) = \begin{cases} g(x) &\text{ if } x \in P_1\\ h_T(\langle g(x), g(k_0(x)), g(k_1(x)), \ldots \rangle) &\text{ if } x \in P_2. \end{cases}$$ yields a Borel function. ◻ # Embedding height three partial orders is hard Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} from the introduction gives a general obstacle to embedding height three partial orders in the Turing degrees. In this section, we will give a proof of that theorem and then explain how it yields unconditional non-embeddability results in $\mathsf{ZF}+ \mathsf{PSP}$ and in the Borel setting. First, though, let's recall the statement of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"}. There is one thing we should explain about the statement of this theorem. The function $f$ in the statement is a function from the partial order $P$ to the set of Turing degrees. Thus the image of $f$ on the first level of $P$ is a set of Turing degrees, not a set of reals, so it cannot literally contain a perfect set. What we really mean is that there is a perfect set of reals, $A$, such that the Turing degree of each element of $A$ is in the image of $f$ on the first level of $P$. ## Proof of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} The main tool we will use in the proof of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} is the following technical theorem on perfect sets, which was proved in [@lutz2023part]. **Theorem 8** ($\mathsf{ZF}$). *Suppose that $A$ is a perfect subset of $2^\omega$, $B$ is a countable dense subset of $A$ and $b$ is a real which computes each element of $B$. Then for every $c \in 2^\omega$, there are reals $d_1, d_2, d_3, d_4 \in A$ such that $$b\oplus d_1 \oplus d_2 \oplus d_3 \oplus d_4 \geq_T c.$$* To prove Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"}, we will assume that we have an arbitrary partial order $(P, \leq_P)$ with the properties listed in the theorem statement, along with an embedding $f$ of $P$ into the Turing degrees. We will then assume that the image of $f$ on the first level of $P$ contains a perfect set and try to use Theorem [Theorem 8](#thm:perfect){reference-type="ref" reference="thm:perfect"} to derive a contradiction; the main idea is that Theorem [Theorem 8](#thm:perfect){reference-type="ref" reference="thm:perfect"} puts certain constraints on what configurations of points can be realized in the Turing degrees, but an arbitrary partial order does not necessarily have these constraints. In order to make the proof work, we will make some assumptions about the structure of $P$. We will keep track of these assumptions and, at the end of the proof, check that there is a partial order which satisfies all of them. To begin, let $(P, \leq_P)$ be a partial order of height three and let $P_1, P_2$ and $P_3$ denote the first, second and third levels of $P$, respectively. Implicit in the statement of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} we already have some assumptions about $P$. 1. $P$ is locally countable, size continuum and height three. 2. $P_1$ has size continuum. Next, let $f$ be an embedding of $P$ into the Turing degrees and assume there is a perfect set $A \subseteq 2^\omega$ such that every element of $A$ has Turing degree in $f(P_1)$. We will now attempt to derive a contradiction by finding a configuration of points in $P$ which $f$ cannot take to an isomorphic configuration in the Turing degrees. We will break our construction into a sequence of steps. To follow the construction, it may help to refer to Figure [\[fig:config\]](#fig:config){reference-type="ref" reference="fig:config"}, which depicts the final configuration. **Step 1: Pick a countable dense set.** Let $B$ be a countable, dense subset of $A$. Let $f^{-1}(B)$ denote the subset of $P$ consisting of points which are in the preimage of the set of Turing degrees of reals in $B$. Note that $f^{-1}(B)$ is a countable subset of $P_1$. **Step 2: Pick an upper bound for $B$.** Let $x$ be an element of $P_2$ which is an upper bound for $f^{-1}(B)$. To ensure that such an element exists, we will make another assumption about $P$. 1. Every countable subset of $P_1$ has an upper bound in $P_2$. Now let $b$ be any real whose Turing degree is $f(x)$. Note that since $f$ is an embedding of $P$ into the Turing degrees, $b$ computes each element of $B$. **Step 3: Pick an independent element of $P$.** Let $y$ be an element of $P_2$ which is not equal to $x$. Note that the existence of such an element follows from the assumptions we have made so far: by (A3), each countable subset of $P_1$ has an upper bound in $P_2$, and since $P$ is locally countable and $P_1$ is uncountable, no single element can be an upper bound for all of these countable subsets. Let $c$ be a real whose Turing degree is $f(y)$. **Step 4: Apply Theorem [Theorem 8](#thm:perfect){reference-type="ref" reference="thm:perfect"}.** We are now in position to apply Theorem [Theorem 8](#thm:perfect){reference-type="ref" reference="thm:perfect"}. By that theorem, there are reals $d_1, d_2, d_3, d_4$ in $A$ such that $b\oplus d_1\oplus d_2\oplus d_3\oplus d_4 \geq_T c$. Since $d_1,\ldots,d_4$ are in $A$, their Turing degrees are in $f(P_1)$. Thus there are elements $z_1,z_2,z_3,z_4 \in P_1$ such that for each $i$, $f(z_i)$ is the Turing degree of $d_i$. **Step 5: Reach a contradiction.** Let $w$ be an element of $P_3$ such that $w$ is above $x$ and $z_1,z_2,z_3,z_4$ but *not* above $y$. To ensure that such an element exists, we will make our final assumption about $P$. 1. For every finite subset $Q$ of $P_1\cup P_2$ and every element $q \in P_2$ which is not equal to any element of $Q$, there is an element of $P_3$ which is above every element of $Q$, but not above $q$. Let $e$ be a real whose Turing degree is $f(w)$. Since $f$ is an embedding, $e$ computes $b$ and each of $d_1,d_2,d_3,d_4$. Hence we have $$e \geq_T b \oplus d_1\oplus d_2\oplus d_3\oplus d_4 \geq_T c.$$ However, this contradicts the fact that $w \ngeq_P y$, which, since $f$ is an embedding, should imply that $e \ngeq_T c$. To complete the proof, we just have to show that there is a partial order $P$ which satisfies the assumptions listed above. **Lemma 9**. *There is a partial order $(P, \leq_P)$ such that* 1. *$P$ is locally countable, size continuum and height three.* 2. *$P_1$ has size continuum.* 3. *Every countable subset of $P_1$ has an upper bound in $P_2$.* 4. *For every finite subset $Q$ of $P_1\cup P_2$ and every element $q \in P_2$ which is not equal to any element of $Q$, there is an element of $P_3$ which is above every element of $Q$, but not above $q$.* *Where $P_1, P_2$, and $P_3$ denote the first, second and third levels of $P$, respectively.* *Proof.* Essentially, we can take the "free" locally countable order of height three and size continuum. The construction is not hard, and can be explained very succinctly, but in order to make it clear that the resulting partial order is Borel (which is necessary to prove Theorem [\[thm:height3borel\]](#thm:height3borel){reference-type="ref" reference="thm:height3borel"} below), we will give a more involved definition. It will help to first fix some notation. We will construct $P$ as a subset of $3^\omega$. Given $x \in 3^\omega$, let $\mathop{\mathrm{head}}(x)$ denote the first digit of $x$ and let $\mathop{\mathrm{tail}}(x)$ denote the element of $3^\omega$ given by deleting the first digit of $x$. It will be useful to view some elements of $P$ as coding a countable sequence of elements of $3^\omega$. Given $x \in 3^\omega$, we will use $x_n$ to denote the $n^\text{th}$ element of the sequence coded by $\mathop{\mathrm{tail}}(x)$. We will now define the partial order $(P, \leq_P)$. First, we define the three levels of $P$ as follows. $$\begin{aligned} P_1 &= \{x \in 3^\omega \mid \mathop{\mathrm{head}}(x) = 0\}\\ P_2 &= \{x \in 3^\omega \mid \mathop{\mathrm{head}}(x) = 1 \text{ and for some $n$, } x_n \in P_1\}\\ P_3 &= \{x \in 3^\omega \mid \mathop{\mathrm{head}}(x) = 2 \text{ and for some $n$, } x_n \in P_2\}\end{aligned}$$ and we let $P = P_1\cup P_2\cup P_3$. Next, we define the order $\leq_P$ on $P$. For $x, y \in P$, set $x \leq_P y$ if and only if one of the following conditions holds. 1. $x = y$. 2. $x$ is in a lower level than $y$ and for some $n$, $x = y_n$. 3. $x$ is in $P_1$, $y$ is in $P_3$ and there are some $n, m$ such that $y_n \in P_2$ and $x = (y_n)_m$. Note that the last condition is needed to make $\leq_P$ transitive. It is straightforward to check that $P$ is size continuum, locally countable and has height three and that $P_1$ also has size continuum. It is also straightforward to check that $P_1, P_2,$ and $P_3$ are indeed the first, second and third levels of $P$, respectively. To verify the other two properties required of $P$, we will check the following more general property, which implies both of them: > If $Q$ is a countable, downwards-closed subset of $P$ such that the maximum level of any element of $Q$ is $i \leq 2$ then there is an element of $P_{i + 1}$ whose set of strict predecessors is exactly $Q$. To prove this, simply note that we can take the element $x \in P$ such that $\mathop{\mathrm{head}}(x) = i + 1$ and $\mathop{\mathrm{tail}}(x)$ codes a countable sequence enumerating the elements of $Q$. ◻ ## $\mathsf{ZF}+ \mathsf{PSP}$ case We will now see how to use Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} to prove Theorem [\[thm:height3zf\]](#thm:height3zf){reference-type="ref" reference="thm:height3zf"}. The first key point is that in $\mathsf{ZF}+ \mathsf{PSP}$ we can use the perfect set property to show that the second condition in Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} is always satisfied. The second key point is that the proof of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} still works in $\mathsf{ZF}$. To see this, it suffices to note the following. 1. Theorem [Theorem 8](#thm:perfect){reference-type="ref" reference="thm:perfect"} is provable in $\mathsf{ZF}$. 2. The construction of the partial order $P$ in Lemma [Lemma 9](#lemma:height3exists){reference-type="ref" reference="lemma:height3exists"} still works in $\mathsf{ZF}$ and $P$ still has the claimed properties. 3. It is provable in $\mathsf{ZF}$ that every perfect subset of $2^\omega$ has a countable, dense subset and all other choices made during the proof of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} only involve instantiating a finite number of existential quantifiers. We can now prove Theorem [\[thm:height3zf\]](#thm:height3zf){reference-type="ref" reference="thm:height3zf"}. Let $(P, \leq_P)$ be the partial order in the statement of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"}. Suppose for contradiction that $f$ is an embedding of $P$ into the Turing degrees. Let $P_1$ denote the first level of $P$ and let $A$ denote the set of reals whose Turing degree is in $f(P_1)$. Note that since $P_1$ is uncountable, so is $A$. Now recall that the axiom $\mathsf{PSP}$ states that every uncountable set of reals contains a perfect set. Since we are working in $\mathsf{ZF}+ \mathsf{PSP}$, the set $A$ must contain a perfect set. By our choice of $P$ (from the statement of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"}), this implies that $f$ is not an embedding, which is a contradiction. ## Borel case We will now use Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} to prove Theorem [\[thm:height3borel\]](#thm:height3borel){reference-type="ref" reference="thm:height3borel"}. The proof is similar to the $\mathsf{ZF}+ \mathsf{PSP}$ case; the key point is that we can replace the axiom $\mathsf{PSP}$ with the perfect set theorem for analytic sets. **Theorem 10** (Perfect set theorem for analytic sets; [@kechris1995classical] Exercise 14.13). *Every $\mathbf{\Sigma^1_1}$ subset of $2^\omega$ is either countable or contains a perfect set.* There are two other key points. First, the partial order $P$ we constructed in the proof of Lemma [Lemma 9](#lemma:height3exists){reference-type="ref" reference="lemma:height3exists"} is Borel and second, $P_1$, the first level of $P$, is also Borel. These can both be seen by inspecting the proof. Granting this, we can prove Theorem [\[thm:height3borel\]](#thm:height3borel){reference-type="ref" reference="thm:height3borel"} as follows. Let $f\colon P \to 2^\omega$ be a Borel function and assume for contradiction that for all $x, y \in P$, $$x \leq_P y \iff f(x) \leq_T f(y).$$ In other words, $f$ induces an embedding of $P$ into the Turing degrees, which we will denote $\widetilde{f}$ (i.e. $\widetilde{f}(x)$ is the Turing degree of $f(x)$). Since $P_1$ and $f$ are both Borel, $f(P_1)$ is analytic. And since $P_1$ is uncountable and $f$ is injective, $f(P_1)$ is uncountable. Hence by the perfect set theorem for analytic sets, $f(P_1)$ contains a perfect set. Thus $\widetilde{f}$ satisfies the hypothesis of Theorem [\[thm:obstacle1\]](#thm:obstacle1){reference-type="ref" reference="thm:obstacle1"} and so it is not an embedding, contradicting our assumption. ## Other nonembedding results Using the same techniques we used to prove the main theorems of this section, we can prove a few other related results. Since the proofs do not contain any new ideas, we will keep them brief. We will begin with Theorem [\[thm:obstacle2\]](#thm:obstacle2){reference-type="ref" reference="thm:obstacle2"} from the introduction. *Proof.* We can divide $A'$ into two disjoint perfect sets, $A_0'$ and $A_1'$, such that $B\cap A_0'$ is a countable, dense subset of $A_0'$ (for example, we can take $A_0'$ to be the intersection of $A'$ with some basic open neighborhood of $2^\omega$). Let $y$ be any element of $A_1'\setminus B$. We will show that $x$, together with a finite number of elements of $A_0'\setminus B$, computes $y$, which is enough to show that $(A\setminus B)\cup \{x\}$ is not Turing independent. By Theorem [Theorem 8](#thm:perfect){reference-type="ref" reference="thm:perfect"}, we can find reals $z_1,z_2,z_3, z_4$ in $A_0'$ such that $x\oplus z_1\oplus z_2\oplus z_3\oplus z_4$ computes $y$. This is almost enough, except that some of the $z_i$'s could be in $B$. However, if they are then $x$ computes them by assumption, so we can leave them out. ◻ We will also give an example of another nonembedding result that can be proved using our techniques. Recall that a subset $Q$ of a partial order $(P, \leq_P)$ is **countably directed** if every countable subset of $Q$ has an upper bound in $Q$. **Theorem 11** ($\mathsf{ZF}+ \mathsf{PSP}$). *Suppose $P$ is a locally countable partial order, $Q$ is an uncountable, countably directed subset of $P$ and $x$ is an element of $P$ which is not below any element of $Q$. Then $P$ cannot be embedded into the Turing degrees.* *Proof.* Suppose for contradiction that $f$ is an embedding of $P$ into the Turing degrees. Since $Q$ is uncountable, so is $f(Q)$. Thus by $\mathsf{PSP}$, the set of reals whose Turing degree is in $f(Q)$ must contain a perfect set, $A$. Let $B$ be a countable, dense subset of $A$. Let $y$ be an element of $Q$ such that $f(y)$ is above the Turing degree of each element of $B$ (such a $y$ exists because $Q$ is countably directed). Let $b$ be a real whose Turing degree is $f(y)$ and note that $b$ computes every element of $B$. Let $c$ be a real whose Turing degree is $f(x)$. By Theorem [Theorem 8](#thm:perfect){reference-type="ref" reference="thm:perfect"}, we can find reals $d_1,d_2,d_3,d_4$ in $A$ such that $b\oplus d_1\oplus d_2\oplus d_3\oplus d_4 \geq_T c$. Since the $d_i$ are all in $A$, we can find $z_1, z_2, z_3, z_4 \in Q$ such that $f(z_i)$ is the Turing degree of $d_i$. Again using the fact that $Q$ is countably directed, we can find an upper bound $w$ in $Q$ for $y, z_1,\ldots,z_4$. Thus $f(w) \geq_T f(x)$, but since $f$ is an embedding, this contradicts the fact that $x$ is not below any element of $Q$. ◻ Essentially the same proof can be used to prove the Borel version of this result (note the extra assumption that $Q$ is Borel, which is needed to apply the perfect set theorem). **Theorem 12**. *Suppose $P$ is a locally countable Borel partial order of size continuum, $Q$ is an uncountable, countably directed Borel subset of $P$ and $x$ is an element of $P$ which is not below any element of $Q$. Then $P$ has no Borel embedding into Turing reducibility.* # Countable Borel equivalence relations {#sec:cber} There is something a little odd about our results in the Borel setting. Namely, Theorems [\[thm:height2borel\]](#thm:height2borel){reference-type="ref" reference="thm:height2borel"} and [\[thm:height3borel\]](#thm:height3borel){reference-type="ref" reference="thm:height3borel"} are about whether or not certain Borel partial orders on the reals have Borel embeddings into Turing reducibility. But according to our definitions, Turing reducibility itself is not a Borel partial order. Instead, Turing reducibility (as a relation on $2^\omega$) is a Borel *quasi-order*. This suggests that in the Borel setting, we should consider which Borel quasi-orders embed into the Turing degrees. And since the theory of locally countable Borel quasi-orders in some ways parallels the more well-studied theory of countable Borel equivalence relations, if we phrase our results in this way then it seems natural to compare them to what is known about countable Borel equivalence relations. ## Countable Borel equivalence relations. We will begin by reviewing a few definitions; for a more thorough introduction, see the recent survey by Kechris [@kechris2021theory]. A **countable equivalence relation** is simply an equivalence relation whose equivalence classes are all countable. A **countable Borel equivalence relation** is a countable equivalence relation $(X, \sim_X)$ such that $X$ is a Borel subset of $2^\omega$ and $\sim_X$ is a Borel subset of $2^\omega\times2^\omega$. A **Borel reduction** from $(X, \sim_X)$ to $(Y, \sim_Y)$ is a Borel function $f\colon X \to Y$ such that $x \sim_X y \iff f(x) \sim_Y f(y)$. One of the main focuses of the theory of countable Borel equivalence relations is to determine which countable Borel equivalence relations are Borel reducible to each other. An important role in the theory is played by the **universal countable Borel equivalence relations**. Briefly, a countable Borel equivalence relation is universal if every other countable Borel equivalence relation is Borel reducible to it. Several countable Borel equivalence relations are known to be universal---for example, the orbit equivalence relation of the shift action of the free group on two generators [@dougherty1994structure] and arithmetic equivalence [@marks2016martins]. Kechris has conjectured that Turing equivalence is also universal [@dougherty2000how]. **Conjecture 13** (Kechris). *Turing equivalence is a universal countable Borel equivalence relation.* This conjecture is interesting both for its own sake and because it contradicts Martin's conjecture, a major open question in computability theory; see [@marks2016martins] for more about the connection between the two conjectures. ## Locally countable Borel quasi-orders. The theory of countable Borel equivalence relations has a natural analogue in the theory of locally countable Borel quasi-orders. A **quasi-order** is simply a transitive, reflexive binary relation (essentially a partial order where some elements are allowed to be equivalent to each other) and a quasi-order $(P, \leq_P)$ is **locally countable** if for every $x \in P$, the set $\{y \mid y \leq_P x\}$ is countable. In analogy with the definitions above, a **locally countable Borel quasi-order** is a locally countable quasi-order $(P, \leq_P)$ such that $P$ is a Borel subset of $2^\omega$ and $\leq_P$ is a Borel subset of $2^\omega\times2^\omega$. A **Borel reduction**[^7] from $(P, \leq_P)$ to $(Q, \leq_Q)$ is a Borel function $f\colon P \to Q$ such that $x \leq_P y \iff f(x) \leq_Q f(y)$ and a locally countable Borel quasi-order is **universal** if every other locally countable Borel quasi-order is Borel reducible to it. There is a close connection between countable Borel equivalence relations and locally countable Borel quasi-orders. First, any equivalence relation $(X, \sim_X)$ literally is a quasi-order and if $\sim_X$ is countable as an equivalence relation then it is locally countable as a quasi-order. Second, given a quasi-order $(P, \leq_P)$, there is an associated equivalence relation $\sim_P$ on $P$, defined by $$x \sim_P y \iff x\leq_P y \text{ and } y \leq_P x.$$ If $\leq_P$ is locally countable then $\sim_P$ is countable and if $\leq_P$ is Borel then so is $\sim_P$. Moreover, universality of $\leq_P$ and $\sim_P$ usually go together: it is typically the case that if $\leq_P$ is universal among locally countable Borel quasi-orders then $\sim_P$ is universal among countable Borel equivalence relations, and vice-versa. For example, arithmetic reducibility is a univeral locally countable Borel quasi-order and arithmetic equivalence, its associated equivalence relation, is a universal countable Borel equivalence relation. ## Kechris's conjecture and quasi-orders. As we mentioned above, Turing reducibility, considered as a relation on $2^\omega$, is a locally countable Borel quasi-order. Also, its associated equivalence relation is just Turing equivalence. In light of this, and of the discussion above, there is a natural analogue of Kechris's conjecture for locally countable Borel quasi-orders---namely, the statement that Turing reducibility is a universal locally countable Borel quasi-order. Since every locally countable Borel partial order is also a locally countable Borel quasi-order (and since a Borel reduction of a Borel partial order into a Borel quasi-order is automatically a Borel embedding), Theorem [\[thm:height3borel\]](#thm:height3borel){reference-type="ref" reference="thm:height3borel"} shows that this statement is false (this was also proved by related means in [@lutz2023part]). Since, as we have already mentioned, universality among locally countable Borel quasi-orders and among countable Borel equivalence relations seem to be strongly correlated, this theorem is evidence against Kechris's conjecture. It is also possible to formulate a question that is partway between Kechris's conjecture for countable Borel equivalence relations and for locally countable Borel quasi-orders. Define the **height** of a quasi-order $(P, \leq_P)$ as the length of the longest strictly decreasing chain in $P$ (i.e. the height of the partial order formed by quotienting $P$ by $\sim_P$). Earlier we said that a countable Borel equivalence relation literally is a locally countable Borel quasi-order. Note that this quasi-order always has height one. Also note that if $(X, \sim_X)$ is a countable Borel equivalence relation then a Borel reduction of $\sim_X$ (as an equivalence relation) to Turing equivalence is not quite the same as a Borel reduction of $\sim_X$ (as a quasi-order) to Turing reducibility: the former just needs to send distinct $\sim_X$-equivalence classes to distinct Turing degrees whereas the latter needs to send them to *incomparable* Turing degrees. Thus, one can view the statement that every locally countable Borel quasi-order of height one is Borel reducible to Turing reducibility as a mild strengthening of Kechris' conjecture. In this paper, we have not just shown that Kechris's conjecture is false when countable Borel equivalence relations are replaced by locally countable Borel quasi-orders, we have also shown that the above statement is false when "height one" is replaced by "height three." At this point it may seem that, when viewed from this perspective, the results of this paper actually support Kechris's conjecture. After all, Theorem [\[thm:height2borel\]](#thm:height2borel){reference-type="ref" reference="thm:height2borel"} shows that every height two, locally countable Borel partial order is Borel reducible to Turing reducibility. Doesn't this suggest that the same may be true for quasi-orders? However, upon further consideration, this argument is not very convincing. In a quasi-order of height one, it is possible to have an infinitely long sequence of distinct elements which are all related to each other, but this is impossible in a partial order of finite height. The existence of such sequences seems to cause major problems for the construction we used in the proof of Theorem [\[thm:height2borel\]](#thm:height2borel){reference-type="ref" reference="thm:height2borel"}. Thus we make the following conjecture. **Conjecture 14**. *There is a locally countable Borel quasi-order of height one which is not Borel reducible to Turing reducibility.* [^1]: It has been observed before that there are many similarities between the $\mathsf{ZF}$ setting and the Borel setting. One example can be found in the work of Shani on Borel equivalence relations [@shani2021borel]. [^2]: Also note that unlike the theory $\mathsf{ZF}+ \mathsf{LN}+ \mathsf{PSP}$, which is known to have greater consistency strength than $\mathsf{ZF}$, $\mathsf{ZF}+ \mathsf{PSP}$ is equiconsistent with $\mathsf{ZF}$ by a result of Truss (see Theorem 3.2 of [@truss1974models], though note that Glazer has identified a mistake in that paper which will be fixed in his upcoming thesis [@glazer2023personal]). [^3]: One may instead assume that $P$ is a standard Borel space and $\leq_P$ is a Borel measurable subset of $P \times P$; this does not make a difference for any of the results of this paper. [^4]: A similar problem arises if an element of the second level has exactly one predecessor: we need to make sure it gets sent to a different Turing degree than its predecessor. [^5]: Kumar and Raghavan [@kumar2021separating] also pointed out that both $\mathbb{H}_\mathfrak{c}$ and the Turing degrees have the largest order dimension among all locally countable partial orders of size continuum. On the order dimension of the Turing degrees, see also [@higuchi2020order]. [^6]: Which is why we chose the name "Lusin-Novikov Choice" in the first place. [^7]: Note that earlier we spoke of *Borel embeddings* whereas here we say *Borel reductions*. In descriptive set theory (and in particular, in the theory of Borel equivalence relations and Borel quasi-orders) these have distinct meanings. However, they have the same meaning when the domain is a Borel partial order, which justifies our use of "Borel embedding" earlier.

arxiv_math

--- abstract: | We construct random walks taking place on the $k$-cells of free $G$-CW complexes of finite type. These random walks define operators acting on the cellular $k$-chains that relate nicely to the (upper) cellular $k$-Laplacian. As an application, we use this relation to show that the Novikov-Shubin invariants of a free $G$-CW complex $X$ of finite type can be recovered from quantities related to return probabilities of the random walks on the cells of $X$. author: - "Tim Höpfner[^1]" bibliography: - bibliography.bib title: Laplacians and Random Walks on CW Complexes --- # Introduction In this paper we generalise a classical result that relates the cellular Laplace operator acting on the $0$-chains of a locally finite CW complex $X$ to a random walk taking place on the $0$-cells of the complex. This classical result has proven to be a useful and powerful tool, allowing us to study the Laplace operator and its spectrum by stochastic means. Indeed, if the $1$-skeleton $X^{(1)}$ is a $d$-regular graph, and we denote the propagation operator of the uniform nearest neighbour random walk on $X^{(1)}$ by $P$, then we can interpret $P$ as an operator acting on $L^{2}C^\mathrm{cell}_0(X)$ and it satisfies $P = \mathop{\mathrm{Id}}- \frac{1}{d} \Delta_0$. Thus, we get a direct correspondence between the spectrum of $P$ and the spectrum of $\Delta_0$. While the assumption of pure degree makes the relation between $P$ and $\Delta_0$ particularly nice, the arguments extend without too much trouble to the case of graphs that are not of pure degree. Recently, work of O. Parzanchevski and R. Rosenthal in [@PaRo13] generalises this kind of results to finite simplicial complexes. They constructed a random walk taking place on the $k$-simplices such that the corresponding propagation operator again relates nicely to the upper $k$-Laplacian $\Delta^{\textrm{up}}_k = d_{k+1}^*d_{k+1}$ (see also R. Rosenthal's paper [@Ro14] and S. Mukherjee and J. Steenbergen's paper [@MuSt13] for further details). Using this, they deduce information about the homology of $X$ and the spectral gap of $\Delta_k$ from the random walk. We will generalise their results further, constructing random walks that take place on free $G$-CW complexes of finite type for some group $G$ and relate the resulting random walk to $\Delta^{\textrm{up}}_k$. This builds heavily on the work of O. Parzanchevski and R. Rosenthal but some additional problems arise. In fact, we will define a family of random walks on the *oriented* $k$-cells of $X$ depending on a parameter $q\in [0,1]$, such that the corresponding propagation operators $P_q$ of this random walk induces, in a natural way, an operator $B_q$ acting on the cellular $k$-chains of $X$. We will show that these operators satisfy the equation $$B_q \circ M_{1,q} = \mathop{\mathrm{Id}}- \Delta^{\textrm{up}}_k \circ M_{2,q},$$ where $M_{1,q}$ and $M_{2,q}$ are multiplication operators given explicitly in terms of the local glueing information of the $k$- and $(k+1)$-cells of $X$. Using this, we can recover spectral information of $\Delta^{\textrm{up}}_k$ from this random walk. For finite CW complexes, the classical Betti numbers measure the size of the kernels of the Laplace operators. Passing to infinite CW complexes, these Betti numbers tend to be infinite and provide little information. In these cases, we can use some extra structure, for example coming in the form of a cocompact free proper group action, to define a normalised version called the $L^2$-Betti numbers as well as the so-called Novikov-Shubin invariants that measure the content of subspaces close to the kernel of the Laplace operators. As an application of the constructed random walk, we prove that the $k$th $L^2$-Betti number $b^{(2)}(d_{k+1}^*)$ and the $k$th Novikov-Shubin invariant $\alpha_k(X)$ (measuring the spectrum of $\Delta^{\textrm{up}}_k$ at zero and close to zero respectively) of a free $G$-CW complex of finite type can be described by a quantity $p_q(n)$ of the random walk that is closely related to return probabilities. Concretely, we show that if $X$ is regular enough so that $M_{i,q}\equiv C_{i,q}$ are constant, then for $q<1$ large enough, $$C_{1,q}^{-n} \left( b^{(2)}(d_{k+1}^*) + C^{-1}n^{-\alpha_k(X)/2}\right) \ \leq \ p_q(n) \ \leq\ C_{1,q}^{-n}\left(b^{(2)}(d_{k+1}^*) + Cn^{-\alpha_k(X)/2}\right).$$ For $k=0$, this recovers a classical result of N. Th. Varopoulos [@Varop], stating that the return probability $p$ of the uniform nearest neighbour random walk on $X^{(1)}$ satisfies $p(2n)\sim n^{-\alpha_0(X)/2}$ as $n\to \infty$. This is a key result, as it allows us to compute the zeroth Novikov-Shubin invariant easily by using that these random walks have the same asymptotic behaviour in terms of the growth rate $N(G)$ of the group $G$, meaning that $\alpha_0(X)=N(G)$. In contrast, there are currently only few results for Novikov-Shubin invariants in higher degrees. Recent work in the same direction by L. Bérnard, Y. Chaubet, N. V. Dang and T. Schick [@BCDS23] uses such random walks on higher dimensional skeleta to compute $L^2$-Betti numbers and linking numbers via random walks. # Random Walks on $k$-Cells ## Degree $k$-Upper Random Walks Before we define the random walk, we introduce some quantities that will be useful later on. These quantities capture the local structure of the CW complex described by the incidence numbers of the CW complex. For this, let $X$ be a free $G$-CW complex of finite type[^2]. We denote the set of $k$-cells of $X$ by $I_k$. On each $k$-cell $\alpha\in I_k$ we fix (arbitrarily) one of the two possible orientations. We denote by $\alpha=\alpha_+$ the cell $\alpha$ equipped with this preferred orientation and by $-\alpha=\alpha_-$ the cell $\alpha$ with the other, reversed orientation. Given a pair of $(k+1)$-cell $\beta\in I_{k+1}$ and $k$-cell $\alpha\in I_k$, the incidence number $[\beta:\alpha]\in \mathbb{Z}$ is given by the mapping degree of the map $\chi_{\beta,\alpha}$, $$\begin{tikzcd} \partial\beta_+ \ar[d, phantom, "\simeq" description]\ar[r, "\chi_\beta"] & X^{(k)} \ar[r, two heads] & \frac{X^{(k)}}{X^{(k)}\setminus \{\alpha\}} \ar[r, "\simeq"] & \frac{\alpha_+}{\partial\alpha_+} \ar[d, phantom, "\simeq" description]\\ S^k \ar[rrr, "\chi_{\beta,\alpha}"] & & & S^k, \end{tikzcd}$$ where $\chi_\beta$ denotes the attaching map of $\beta$ to $X^{(k)}$. Note that the sign of $[\beta:\alpha]$ depends on the orientations, where $[\beta_\nu:\alpha_{\nu'}] = \nu\nu'[\beta:\alpha]$ for signs $\nu,\nu'\in\{\pm\}$. **Definition 1**. [\[Def_RW_ds\]]{#Def_RW_ds label="Def_RW_ds"} Let $X$ be a free $G$-CW complex of finite type. For $\alpha\in I_k$ and $\beta\in I_{k+1}$ we define the quantities $$\begin{aligned} \begin{split} d_{+,2}(\alpha) &= \sum_{\beta'\in I_{k+1}} [\beta':\alpha]^2, \\ d_+(\alpha) &= \sum_{\beta'\in I_{k+1}} |[\beta':\alpha]|, \\ d_-(\beta; \alpha) &= \begin{cases} \sum\limits_{\alpha\neq \alpha'\in I_k} |[\beta:\alpha']| &\text{ if } [\beta:\alpha]\neq 0, \\ 0 & \text{ if } [\beta:\alpha]=0, \end{cases} \\ d_-(\alpha) &= \max_{\beta'\in I_{k+1}} d_-(\beta;\alpha), \end{split}\end{aligned}$$ where the maximum in the definition of $d_-(\alpha)$ exists since $d_-(\beta; \alpha)$ is invariant under the $G$-action,[^3] so that it can only assume finitely many different values. Note also that these quantities are independent of the orientations chosen on $\alpha$ and $\beta$.[^4] **Definition 2** (). [\[Def_RW_ds\]]{#Def_RW_ds label="Def_RW_ds"} Let $X$ be a free $G$-CW complex of finite type. For $\alpha\in I_k$ and $\beta\in I_{k+1}$ we define the quantities $$\begin{aligned} \begin{split} d_{+,2}(\alpha) &= \sum_{\beta'\in I_{k+1}} [\beta':\alpha]^2, \\ d_+(\alpha) &= \sum_{\beta'\in I_{k+1}} |[\beta':\alpha]|, \\ d_-(\beta; \alpha) &= \begin{cases} \sum\limits_{\alpha\neq \alpha'\in I_k} |[\beta:\alpha']| &\text{ if } [\beta:\alpha]\neq 0, \\ 0 & \text{ if } [\beta:\alpha]=0, \end{cases} \\ d_-(\alpha) &= \max_{\beta'\in I_{k+1}} d_-(\beta;\alpha), \end{split}\end{aligned}$$ where the maximum in the definition of $d_-(\alpha)$ exists since $d_-(\beta; \alpha)$ is invariant under the $G$-action,[^5] so that it can only assume finitely many different values. Note also that these quantities are independent of the orientations chosen on $\alpha$ and $\beta$.[^6] These quantities generalise the idea of the degree of a vertex to $k$-cells, with $d_+$ being the incoming degree and $d_-$ the (maximal) outgoing degree. They capture the local structure of the CW complex around the $k$-cell $\alpha\in I_k$. We also introduce the notation $$d(\alpha, \alpha', \beta) = -[\beta:\alpha][\beta:\alpha'],$$ measuring how well connected $\alpha$ is to $\alpha'$ along $\beta$.[^7] Note that $d(\alpha,\alpha',\beta)=d(\alpha',\alpha,\beta)$. **Remark 3**. Recall that on the cellular $k$-chains, the upper $k$-Laplacian $\Delta^{\textrm{up}}_k$ is given by the formula $$\begin{aligned} \begin{split} \label{E_Dup_Formula} \Delta^{\textrm{up}}_k&\left( \sum_{\alpha\in I_k} \lambda_\alpha \cdot \alpha \right) \\ &=\sum_{\alpha\in I_k} \left[ \sum_{\beta\in I_{k+1}} [\beta:\alpha]^2 \lambda_\alpha - \sum_{\alpha \neq \alpha'\in I_k} \sum_{\beta\in I_{k+1}} -[\beta:\alpha][\beta:\alpha'] \lambda_{\alpha'} \right] \cdot \alpha\\ &=\sum_{\alpha\in I_k} \left[ d_{+,2}(\alpha) \lambda_\alpha - \sum_{\alpha \neq \alpha'\in I_k} \sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \lambda_{\alpha'} \right] \cdot \alpha, \end{split}\end{aligned}$$ where, for $k=0$, $d_{2,+}(\alpha)=\deg(\alpha)$ is the degree of the vertex $\alpha$ (ignoring loops) and $d(\alpha,\alpha',\beta)=1$ if $\beta$ is an edge from $\alpha$ to $\alpha'$ (again, ignoring loops) and zero otherwise. Note that $\Delta^\mathrm{up}_k$ sees the incidence numbers between $k$- and $(k+1)$-cells in $X$. Hence, they have to appear in the definition of our random walk. Furthermore, both the sign and the size of the incidence numbers have to play a role. **Remark 4** (). Recall that on the cellular $k$-chains, the upper $k$-Laplacian $\Delta^{\textrm{up}}_k$ is given by the formula $$\begin{aligned} \begin{split} \label{E_Dup_Formula} \Delta^{\textrm{up}}_k&\left( \sum_{\alpha\in I_k} \lambda_\alpha \cdot \alpha \right) \\ &=\sum_{\alpha\in I_k} \left[ \sum_{\beta\in I_{k+1}} [\beta:\alpha]^2 \lambda_\alpha - \sum_{\alpha \neq \alpha'\in I_k} \sum_{\beta\in I_{k+1}} -[\beta:\alpha][\beta:\alpha'] \lambda_{\alpha'} \right] \cdot \alpha\\ &=\sum_{\alpha\in I_k} \left[ d_{+,2}(\alpha) \lambda_\alpha - \sum_{\alpha \neq \alpha'\in I_k} \sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \lambda_{\alpha'} \right] \cdot \alpha, \end{split}\end{aligned}$$ where, for $k=0$, $d_{2,+}(\alpha)=\deg(\alpha)$ is the degree of the vertex $\alpha$ (ignoring loops) and $d(\alpha,\alpha',\beta)=1$ if $\beta$ is an edge from $\alpha$ to $\alpha'$ (again, ignoring loops) and zero otherwise. Note that $\Delta^\mathrm{up}_k$ sees the incidence numbers between $k$- and $(k+1)$-cells in $X$. Hence, they have to appear in the definition of our random walk. Furthermore, both the sign and the size of the incidence numbers have to play a role. **Definition 5**. Consider the set of oriented $k$-cells $I_k^\pm = I_k \cup \left\lbrace \alpha_- \:\middle|\: \alpha \in I_k \right\rbrace.$ We call two oriented $k$-cells $\alpha_\nu, \alpha'_{\nu'}\in I_k^\pm$, where $\nu,\nu'\in\{\pm\}$ denote orientations, (upper) neighbours along $\beta\in I_{k+1}$, and write $\alpha_\nu \overset{\beta}{\sim} \alpha'_{\nu'}$ if $$\alpha_\nu\neq -\alpha'_{\nu'} \quad \text{ and } \quad d(\alpha_\nu,\alpha'_{\nu'},\beta) = \nu\nu' d(\alpha,\alpha',\beta) > 0.$$ Note that this condition is independent of the orientation chosen on $\beta\in I_{k+1}$. **Definition 6** (). Consider the set of oriented $k$-cells $I_k^\pm = I_k \cup \left\lbrace \alpha_- \:\middle|\: \alpha \in I_k \right\rbrace.$ We call two oriented $k$-cells $\alpha_\nu, \alpha'_{\nu'}\in I_k^\pm$, where $\nu,\nu'\in\{\pm\}$ denote orientations, (upper) neighbours along $\beta\in I_{k+1}$, and write $\alpha_\nu \overset{\beta}{\sim} \alpha'_{\nu'}$ if $$\alpha_\nu\neq -\alpha'_{\nu'} \quad \text{ and } \quad d(\alpha_\nu,\alpha'_{\nu'},\beta) = \nu\nu' d(\alpha,\alpha',\beta) > 0.$$ Note that this condition is independent of the orientation chosen on $\beta\in I_{k+1}$. **Definition 7**. The random walk $\mathfrak{R}^k = \mathfrak{R}^k(X)$ is given by the state space $I_k^* = I_k^\pm \cup \{\Theta\}$, where $\Theta$ is an auxiliary, absorbing state together with the following moving probabilities: - The moving probabilities starting from the absorbing state $\Theta$ are given by $$\begin{aligned} \mathbb{P}(\Theta\to \Theta) &= 1 \quad \text{and} \quad \mathbb{P}(\Theta\to \alpha_\pm) = 0 \quad \text{for all } \alpha_\pm\in I_k^\pm.\end{aligned}$$ - To define the moving probabilities starting from $\alpha_\nu\in I_k^\pm$, we define first for $\alpha'_{\nu'}\in I^\pm_k$ and $\beta\in I_{k+1}$ the quantities $$\begin{aligned} \mathbb{P}(\alpha_\nu\nearrow \beta) &= \frac{|[\beta:\alpha_\nu]|}{d_+(\alpha)} \qquad \text{and} \qquad \mathbb{P}_\alpha(\beta\searrow \alpha'_{\nu'})= \frac{|[\beta:\alpha'_{\nu'}]|}{d_-(\alpha)}\end{aligned}$$ These probabilities can be seen as an intermediate step of moving first from $\alpha$ to $\beta$ and then from $\beta$ to $\alpha'$ (keeping in mind that we started at $\alpha$). In this sense, we define $$\begin{aligned} &\mathbb{P}(\alpha_\nu\xrightarrow{\beta}\alpha'_{\nu'}) \\ &= \begin{cases} \mathbb{P}(\alpha_\nu \nearrow \beta) \mathbb{P}_\alpha(\beta\searrow\alpha'_{\nu'}) = \frac{-[\beta:\alpha_\nu][\beta:\alpha'_{\nu'}]}{d_+(\alpha)d_-(\alpha)} = \frac{d(\alpha_\nu,\alpha'_{\nu'},\beta)}{d_+(\alpha)d_-(\alpha)} > 0 & \text{if } \alpha_\nu \overset{\beta}{\sim} \alpha'_{\nu'}, \\ 0 & \text{else,} \end{cases} \end{aligned}$$ the probability of moving from $\alpha$ along $\beta$ to $\alpha'$. Recall here that $\alpha\not\sim \pm \alpha$ by definition, so that $\mathbb{P}(\alpha\xrightarrow{\beta} \pm \alpha) = 0$. Finally, we set $$\begin{aligned} \mathbb{P}(\alpha_\nu\to\alpha'_{\nu'}) &= \sum_{\beta\in I_{k+1}} \mathbb{P}(\alpha_\nu\xrightarrow{\beta}\alpha'_{\nu'}) = \sum_{\stackrel{\beta\in I_{k+1}}{\alpha \overset{\beta}{\sim} \alpha'}} \frac{d(\alpha_\nu,\alpha'_{\nu'},\beta)}{d_+(\alpha)d_-(\alpha)} \\ &= \frac{1}{{d_+(\alpha)d_-(\alpha)}}\sum_{\stackrel{\beta\in I_{k+1}}{\alpha_\nu \overset{\beta}{\sim} \alpha'_{\nu'}}} {d(\alpha_\nu,\alpha'_{\nu'},\beta)}\end{aligned}$$ The moving probabilities would add to one if we use $d_-(\beta;\alpha)$ in place of $d_-(\alpha)$, however it will be important later on that we can pull the factor $d_-(\alpha)^{-1}$ out of the sum as it depends only on the starting cell $\alpha$. Consequently, their sum needs not to be one but can be smaller since we might make the denumerator bigger for some of the summands. - The complementary probability will be the probability of moving to $\Theta$, that is $$\mathbb{P}(\alpha_\nu \to \Theta) = 1 - \sum_{\alpha'_{\nu'}\in I_k^\pm} \mathbb{P}(\alpha_\nu \to \alpha'_{\nu'}).$$ This defines a random walk on $I_k^\ast$. The propagation operator $P$ with entries $P_{s,s'} = \mathbb{P}( s'\to s)$ for $s,s'\in I_k^*$ acts on $\ell^2(I_k^*) = \left\lbrace \sum_{s\in I_k^*}\lambda_s \cdot s \:\middle|\: \sum_{s\in I_k^*}|\lambda_s|^2<\infty \right\rbrace$ by $$P\left(\sum_{s\in I_k^*}\lambda_s \cdot s\right) = \sum_{s\in I_k^*}\left[\sum_{s'\in I_k^*} P_{s,s'} \lambda_{s'}\right] \cdot s = \sum_{s\in I_k^*}\left[\sum_{s'\in I_k^*} \mathbb{P}(s'\to s) \lambda_{s'}\right] \cdot s.$$ **Definition 8** (). The random walk $\mathfrak{R}^k = \mathfrak{R}^k(X)$ is given by the state space $I_k^* = I_k^\pm \cup \{\Theta\}$, where $\Theta$ is an auxiliary, absorbing state together with the following moving probabilities: - The moving probabilities starting from the absorbing state $\Theta$ are given by $$\begin{aligned} \mathbb{P}(\Theta\to \Theta) &= 1 \quad \text{and} \quad \mathbb{P}(\Theta\to \alpha_\pm) = 0 \quad \text{for all } \alpha_\pm\in I_k^\pm.\end{aligned}$$ - To define the moving probabilities starting from $\alpha_\nu\in I_k^\pm$, we define first for $\alpha'_{\nu'}\in I^\pm_k$ and $\beta\in I_{k+1}$ the quantities $$\begin{aligned} \mathbb{P}(\alpha_\nu\nearrow \beta) &= \frac{|[\beta:\alpha_\nu]|}{d_+(\alpha)} \qquad \text{and} \qquad \mathbb{P}_\alpha(\beta\searrow \alpha'_{\nu'})= \frac{|[\beta:\alpha'_{\nu'}]|}{d_-(\alpha)}\end{aligned}$$ These probabilities can be seen as an intermediate step of moving first from $\alpha$ to $\beta$ and then from $\beta$ to $\alpha'$ (keeping in mind that we started at $\alpha$). In this sense, we define $$\begin{aligned} &\mathbb{P}(\alpha_\nu\xrightarrow{\beta}\alpha'_{\nu'}) \\ &= \begin{cases} \mathbb{P}(\alpha_\nu \nearrow \beta) \mathbb{P}_\alpha(\beta\searrow\alpha'_{\nu'}) = \frac{-[\beta:\alpha_\nu][\beta:\alpha'_{\nu'}]}{d_+(\alpha)d_-(\alpha)} = \frac{d(\alpha_\nu,\alpha'_{\nu'},\beta)}{d_+(\alpha)d_-(\alpha)} > 0 & \text{if } \alpha_\nu \overset{\beta}{\sim} \alpha'_{\nu'}, \\ 0 & \text{else,} \end{cases} \end{aligned}$$ the probability of moving from $\alpha$ along $\beta$ to $\alpha'$. Recall here that $\alpha\not\sim \pm \alpha$ by definition, so that $\mathbb{P}(\alpha\xrightarrow{\beta} \pm \alpha) = 0$. Finally, we set $$\begin{aligned} \mathbb{P}(\alpha_\nu\to\alpha'_{\nu'}) &= \sum_{\beta\in I_{k+1}} \mathbb{P}(\alpha_\nu\xrightarrow{\beta}\alpha'_{\nu'}) = \sum_{\stackrel{\beta\in I_{k+1}}{\alpha \overset{\beta}{\sim} \alpha'}} \frac{d(\alpha_\nu,\alpha'_{\nu'},\beta)}{d_+(\alpha)d_-(\alpha)} \\ &= \frac{1}{{d_+(\alpha)d_-(\alpha)}}\sum_{\stackrel{\beta\in I_{k+1}}{\alpha_\nu \overset{\beta}{\sim} \alpha'_{\nu'}}} {d(\alpha_\nu,\alpha'_{\nu'},\beta)}\end{aligned}$$ The moving probabilities would add to one if we use $d_-(\beta;\alpha)$ in place of $d_-(\alpha)$, however it will be important later on that we can pull the factor $d_-(\alpha)^{-1}$ out of the sum as it depends only on the starting cell $\alpha$. Consequently, their sum needs not to be one but can be smaller since we might make the denumerator bigger for some of the summands. - The complementary probability will be the probability of moving to $\Theta$, that is $$\mathbb{P}(\alpha_\nu \to \Theta) = 1 - \sum_{\alpha'_{\nu'}\in I_k^\pm} \mathbb{P}(\alpha_\nu \to \alpha'_{\nu'}).$$ This defines a random walk on $I_k^\ast$. The propagation operator $P$ with entries $P_{s,s'} = \mathbb{P}( s'\to s)$ for $s,s'\in I_k^*$ acts on $\ell^2(I_k^*) = \left\lbrace \sum_{s\in I_k^*}\lambda_s \cdot s \:\middle|\: \sum_{s\in I_k^*}|\lambda_s|^2<\infty \right\rbrace$ by $$P\left(\sum_{s\in I_k^*}\lambda_s \cdot s\right) = \sum_{s\in I_k^*}\left[\sum_{s'\in I_k^*} P_{s,s'} \lambda_{s'}\right] \cdot s = \sum_{s\in I_k^*}\left[\sum_{s'\in I_k^*} \mathbb{P}(s'\to s) \lambda_{s'}\right] \cdot s.$$ **Example 9**. Let us consider a small example to see how we can find the moving probabilities. For this, let $X$ be a CW complex and $\alpha\in I_k$ some $k$-cell. In order to find the $k$-cells we can move to from $\alpha=+\alpha$, we proceed as follows: 1. First we consider all $(k+1)$-cells $\beta\in I_{k+1}$ and find those, that have non-zero incidence number $[\beta:\alpha]\neq 0$. For this example, let us say there are two such $(k+1)$-cells $\beta_1$ and $\beta_2$ with $[\beta_1: \alpha] = 1$ and $[\beta_2:\alpha] = -2$. 2. Then we consider all other $k$-cells $\alpha\neq \alpha'\in I_k$ that have non-zero incidence numbers with at least one of the $(k+1)$-cells above, that is $[\beta_1:\alpha']\neq 0$ or $[\beta_2:\alpha']\neq 0$. For this example, let us say there are three such $k$-cells, $\alpha_1$ with $[\beta_1:\alpha_1] = 1$, $\alpha_2$ with $[\beta_1:\alpha_2] = 2$ and $[\beta_2:\alpha_2] = 4$ and $\alpha_3$ with $[\beta_2:\alpha_3] = -2$. We visualise this with the following diagram: $$\begin{tikzcd}[ampersand replacement = \&] \& \beta_1 \ar[dl, "1" description] \ar[d, "2" description] \& \& \& \& \beta_2 \ar[d, "4" description] \ar[dr, "-2" description] \& \& \in I_{k+1} \\ \alpha_1 \& \alpha_2 \& \& \alpha \ar[ull, "1" description] \ar[urr, "-2" description] \& \& \alpha_2 \& \alpha_3 \& \in I_{k} \end{tikzcd}$$ Next, we first change the orientations on the $(k+1)$-cells such that the incidence numbers with $\alpha$ are negative, so here we change the orientation on $\beta_1$.[^8] Then we change the orientations on the $\alpha_i$, $i\in\{1,2,3\}$, for each of the $\beta_j$ independently such that the incidence numbers with the $(k+1)$-cells become positive. This changes our diagram as follows: $$\begin{tikzcd}[ampersand replacement = \&] \& -\beta_1 \ar[dl, "1" description] \ar[d, "2" description] \& \& \& \& \beta_2 \ar[d, "4" description] \ar[dr, "2" description] \& \& \in I_{k+1}^\pm \\ -\alpha_1 \& -\alpha_2 \& \& \alpha \ar[ull, "-1" description] \ar[urr, "-2" description] \& \& \alpha_2 \& -\alpha_3 \& \in I_{k}^\pm \end{tikzcd}$$ We now introduce the auxiliary state $\Theta$. For each of the $(k+1)$-cells, we sum the outgoing incidence numbers. Here, we get $d_-(\beta_1;\alpha) = 1+2 = 3$ for $\beta_1$ and $d_-(\beta_2;\alpha) = 2+4 = 6$ for $\beta_2$. The maximum is therefore $6$, and we add connections from each of the $(k+1)$-cells to the new state $\Theta$ until the sum of outgoing edges is equal to this maximum, i.e., $-\beta_1\xrightarrow{3}\Theta$ in this case: $$\begin{tikzcd}[ampersand replacement = \&] \& \& -\beta_1 \ar[dll, "3" description] \ar[dl, "1" description] \ar[d, "2" description] \& \& \& \& \beta_2 \ar[d, "4" description] \ar[dr, "2" description] \& \& \in I_{k+1}^\pm\\ \Theta \& -\alpha_1 \& -\alpha_2 \& \& \alpha \ar[ull, "-1" description] \ar[urr, "-2" description] \& \& \alpha_2 \& -\alpha_3 \& \in I_{k}^* \end{tikzcd}$$ The moving probabilities can now be read to be proportional to the annotations of the arrows. For the first intermediate step, $\mathbb{P}(\alpha\nearrow -\beta_1) = \nicefrac{1}{3}$ and $\mathbb{P}(\alpha\nearrow \beta_2) = \nicefrac{2}{3}$. For the second step, $$\mathbb{P}(-\beta_1\searrow \Theta) = \nicefrac{3}{6}, \qquad \mathbb{P}(-\beta_1\searrow -\alpha_1) = \nicefrac{1}{6} \quad \text{and} \quad \mathbb{P}(-\beta_1 \searrow -\alpha_2) = \nicefrac{2}{6}$$ and for $\beta_2$ we have $$\mathbb{P}(\beta_2\searrow \alpha_2) = \nicefrac{4}{6} \quad \text{and} \quad \mathbb{P}(\beta_2\searrow -\alpha_3) = \nicefrac{2}{6}.$$ The introduction of $\Theta$ guaranties that the denominator of these (unreduced) fractions is the same everywhere and the moving probabilities are proportional to the incidence numbers even if the first intermediate step leads to different $\beta_i$s. Multiplying these accordingly, we find $$\begin{aligned} {3} &\mathbb{P}(\alpha\xrightarrow{-\beta_1} -\alpha_1) = \nicefrac{1}{3}\cdot\nicefrac{1}{6}, \quad && \mathbb{P}(\alpha\xrightarrow{-\beta_1} -\alpha_2) = \nicefrac{1}{3}\cdot\nicefrac{2}{6}, \quad && \mathbb{P}(\alpha\xrightarrow{-\beta_1} \Theta) = \nicefrac{1}{3}\cdot \nicefrac{3}{6}, \\ &\mathbb{P}(\alpha\xrightarrow{\beta_2} \alpha_2) = \nicefrac{2}{3}\cdot\nicefrac{4}{6}, && \mathbb{P}(\alpha\xrightarrow{-\beta_1} -\alpha_3) = \nicefrac{2}{3}\cdot\nicefrac{2}{6}. &&\end{aligned}$$ Here, every oriented $k$-cell can be reached only via one $(k+1)$-cell, otherwise we would have to sum over all $(k+1)$-cells, that is $\mathbb{P}(\alpha\to s) = \sum_{\pm\beta\in I_{k+1}^\pm} \mathbb{P}(\alpha\xrightarrow{\pm\beta} s)$ for $s\in I_k^*$. Therefore, in this example, a random walker starting at $\alpha$ has the following possible moves, with annotations denoting the probabilities: $$\begin{tikzcd}[ampersand replacement = \&] \& \& \alpha \ar[dll, "\nicefrac{1}{18}" description] \ar[dl, "\nicefrac{1}{9}" description] \ar[d, "\nicefrac{4}{9}" description] \ar[dr, "\nicefrac{2}{9}" description] \ar[drr, "\nicefrac{1}{6}" description] \& \& \\ -\alpha_1 \& -\alpha_2 \& \alpha_2 \& -\alpha_3 \& \Theta \end{tikzcd}$$ Note that the cell $\alpha_2$ can be reached with both possible orientations. If we start at the cell $-\alpha$ with reversed orientation, we obtain the same moving probabilities as for $\alpha$, but now leading to the same cells but with flipped orientations instead. **Example 10** (). Let us consider a small example to see how we can find the moving probabilities. For this, let $X$ be a CW complex and $\alpha\in I_k$ some $k$-cell. In order to find the $k$-cells we can move to from $\alpha=+\alpha$, we proceed as follows: 1. First we consider all $(k+1)$-cells $\beta\in I_{k+1}$ and find those, that have non-zero incidence number $[\beta:\alpha]\neq 0$. For this example, let us say there are two such $(k+1)$-cells $\beta_1$ and $\beta_2$ with $[\beta_1: \alpha] = 1$ and $[\beta_2:\alpha] = -2$. 2. Then we consider all other $k$-cells $\alpha\neq \alpha'\in I_k$ that have non-zero incidence numbers with at least one of the $(k+1)$-cells above, that is $[\beta_1:\alpha']\neq 0$ or $[\beta_2:\alpha']\neq 0$. For this example, let us say there are three such $k$-cells, $\alpha_1$ with $[\beta_1:\alpha_1] = 1$, $\alpha_2$ with $[\beta_1:\alpha_2] = 2$ and $[\beta_2:\alpha_2] = 4$ and $\alpha_3$ with $[\beta_2:\alpha_3] = -2$. We visualise this with the following diagram: $$\begin{tikzcd}[ampersand replacement = \&] \& \beta_1 \ar[dl, "1" description] \ar[d, "2" description] \& \& \& \& \beta_2 \ar[d, "4" description] \ar[dr, "-2" description] \& \& \in I_{k+1} \\ \alpha_1 \& \alpha_2 \& \& \alpha \ar[ull, "1" description] \ar[urr, "-2" description] \& \& \alpha_2 \& \alpha_3 \& \in I_{k} \end{tikzcd}$$ Next, we first change the orientations on the $(k+1)$-cells such that the incidence numbers with $\alpha$ are negative, so here we change the orientation on $\beta_1$.[^9] Then we change the orientations on the $\alpha_i$, $i\in\{1,2,3\}$, for each of the $\beta_j$ independently such that the incidence numbers with the $(k+1)$-cells become positive. This changes our diagram as follows: $$\begin{tikzcd}[ampersand replacement = \&] \& -\beta_1 \ar[dl, "1" description] \ar[d, "2" description] \& \& \& \& \beta_2 \ar[d, "4" description] \ar[dr, "2" description] \& \& \in I_{k+1}^\pm \\ -\alpha_1 \& -\alpha_2 \& \& \alpha \ar[ull, "-1" description] \ar[urr, "-2" description] \& \& \alpha_2 \& -\alpha_3 \& \in I_{k}^\pm \end{tikzcd}$$ We now introduce the auxiliary state $\Theta$. For each of the $(k+1)$-cells, we sum the outgoing incidence numbers. Here, we get $d_-(\beta_1;\alpha) = 1+2 = 3$ for $\beta_1$ and $d_-(\beta_2;\alpha) = 2+4 = 6$ for $\beta_2$. The maximum is therefore $6$, and we add connections from each of the $(k+1)$-cells to the new state $\Theta$ until the sum of outgoing edges is equal to this maximum, i.e., $-\beta_1\xrightarrow{3}\Theta$ in this case: $$\begin{tikzcd}[ampersand replacement = \&] \& \& -\beta_1 \ar[dll, "3" description] \ar[dl, "1" description] \ar[d, "2" description] \& \& \& \& \beta_2 \ar[d, "4" description] \ar[dr, "2" description] \& \& \in I_{k+1}^\pm\\ \Theta \& -\alpha_1 \& -\alpha_2 \& \& \alpha \ar[ull, "-1" description] \ar[urr, "-2" description] \& \& \alpha_2 \& -\alpha_3 \& \in I_{k}^* \end{tikzcd}$$ The moving probabilities can now be read to be proportional to the annotations of the arrows. For the first intermediate step, $\mathbb{P}(\alpha\nearrow -\beta_1) = \nicefrac{1}{3}$ and $\mathbb{P}(\alpha\nearrow \beta_2) = \nicefrac{2}{3}$. For the second step, $$\mathbb{P}(-\beta_1\searrow \Theta) = \nicefrac{3}{6}, \qquad \mathbb{P}(-\beta_1\searrow -\alpha_1) = \nicefrac{1}{6} \quad \text{and} \quad \mathbb{P}(-\beta_1 \searrow -\alpha_2) = \nicefrac{2}{6}$$ and for $\beta_2$ we have $$\mathbb{P}(\beta_2\searrow \alpha_2) = \nicefrac{4}{6} \quad \text{and} \quad \mathbb{P}(\beta_2\searrow -\alpha_3) = \nicefrac{2}{6}.$$ The introduction of $\Theta$ guaranties that the denominator of these (unreduced) fractions is the same everywhere and the moving probabilities are proportional to the incidence numbers even if the first intermediate step leads to different $\beta_i$s. Multiplying these accordingly, we find $$\begin{aligned} {3} &\mathbb{P}(\alpha\xrightarrow{-\beta_1} -\alpha_1) = \nicefrac{1}{3}\cdot\nicefrac{1}{6}, \quad && \mathbb{P}(\alpha\xrightarrow{-\beta_1} -\alpha_2) = \nicefrac{1}{3}\cdot\nicefrac{2}{6}, \quad && \mathbb{P}(\alpha\xrightarrow{-\beta_1} \Theta) = \nicefrac{1}{3}\cdot \nicefrac{3}{6}, \\ &\mathbb{P}(\alpha\xrightarrow{\beta_2} \alpha_2) = \nicefrac{2}{3}\cdot\nicefrac{4}{6}, && \mathbb{P}(\alpha\xrightarrow{-\beta_1} -\alpha_3) = \nicefrac{2}{3}\cdot\nicefrac{2}{6}. &&\end{aligned}$$ Here, every oriented $k$-cell can be reached only via one $(k+1)$-cell, otherwise we would have to sum over all $(k+1)$-cells, that is $\mathbb{P}(\alpha\to s) = \sum_{\pm\beta\in I_{k+1}^\pm} \mathbb{P}(\alpha\xrightarrow{\pm\beta} s)$ for $s\in I_k^*$. Therefore, in this example, a random walker starting at $\alpha$ has the following possible moves, with annotations denoting the probabilities: $$\begin{tikzcd}[ampersand replacement = \&] \& \& \alpha \ar[dll, "\nicefrac{1}{18}" description] \ar[dl, "\nicefrac{1}{9}" description] \ar[d, "\nicefrac{4}{9}" description] \ar[dr, "\nicefrac{2}{9}" description] \ar[drr, "\nicefrac{1}{6}" description] \& \& \\ -\alpha_1 \& -\alpha_2 \& \alpha_2 \& -\alpha_3 \& \Theta \end{tikzcd}$$ Note that the cell $\alpha_2$ can be reached with both possible orientations. If we start at the cell $-\alpha$ with reversed orientation, we obtain the same moving probabilities as for $\alpha$, but now leading to the same cells but with flipped orientations instead. We now define an operator $B$ acting directly on the unoriented $k$-skeleton $I_k$, which is closely related to this random walk. **Definition 11**. We define the projection operator $T\colon \ell^2(I_k^*) \to \ell^2(I_k)$ by $$T\left(\sum_{s\in I_k^*}\lambda_s \cdot s\right) = \sum_{\alpha\in I_k} (\lambda_{\alpha_+} - \lambda_{\alpha_-}) \cdot \alpha$$ and the inclusion operator $I\colon \ell^2(I_k)\to \ell^2(I_k^*)$, using that $I_k=I_k^+\subset I_k^*$, by $$I\left( \sum_{\alpha\in I_k} \lambda_\alpha \cdot \alpha \right) = \sum_{\alpha_+\in I_k^+} \lambda_{\alpha_+}\cdot \alpha_+.$$ Lastly, we define the operator $B\colon \ell^2(I_k)\to \ell^2(I_k)$ by $$B\left(\sum_{\alpha\in I_k} \lambda_\alpha \cdot \alpha \right) = \sum_{\alpha\in I_k} \left[\sum_{\alpha\neq \alpha'\in I_k} \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \lambda_{\alpha'}\right]\cdot \alpha.$$ With $B_{\alpha,\alpha'} = \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\beta\in I_{k+1}} d(\alpha_\nu,\alpha'_{\nu'},\beta)$ for $\alpha\neq\alpha'$ and $B_{\alpha,\alpha}=0$, $$B\left(\sum_{\alpha\in I_k} \lambda_\alpha \cdot \alpha \right) = \sum_{\alpha\in I_k} \left[\sum_{\alpha'\in I_k} B_{\alpha,\alpha'} \lambda_{\alpha'}\right]\cdot \alpha.$$ **Definition 12** (). We define the projection operator $T\colon \ell^2(I_k^*) \to \ell^2(I_k)$ by $$T\left(\sum_{s\in I_k^*}\lambda_s \cdot s\right) = \sum_{\alpha\in I_k} (\lambda_{\alpha_+} - \lambda_{\alpha_-}) \cdot \alpha$$ and the inclusion operator $I\colon \ell^2(I_k)\to \ell^2(I_k^*)$, using that $I_k=I_k^+\subset I_k^*$, by $$I\left( \sum_{\alpha\in I_k} \lambda_\alpha \cdot \alpha \right) = \sum_{\alpha_+\in I_k^+} \lambda_{\alpha_+}\cdot \alpha_+.$$ Lastly, we define the operator $B\colon \ell^2(I_k)\to \ell^2(I_k)$ by $$B\left(\sum_{\alpha\in I_k} \lambda_\alpha \cdot \alpha \right) = \sum_{\alpha\in I_k} \left[\sum_{\alpha\neq \alpha'\in I_k} \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \lambda_{\alpha'}\right]\cdot \alpha.$$ With $B_{\alpha,\alpha'} = \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\beta\in I_{k+1}} d(\alpha_\nu,\alpha'_{\nu'},\beta)$ for $\alpha\neq\alpha'$ and $B_{\alpha,\alpha}=0$, $$B\left(\sum_{\alpha\in I_k} \lambda_\alpha \cdot \alpha \right) = \sum_{\alpha\in I_k} \left[\sum_{\alpha'\in I_k} B_{\alpha,\alpha'} \lambda_{\alpha'}\right]\cdot \alpha.$$ This is captured by the diagram $$\begin{tikzcd}[row sep = 30, column sep = 30] \ell^2(I_k^*) \ar[r, "P"]\ar[d, "T"] & \ell^2(I_k^*)\ar[d, "T"] \\ \ell^2(I_k) \ar[r, "B"] \ar[u, bend left, dashed, "I"] & \ell^2(I_k). \ar[u, bend left, dashed, "I"] \end{tikzcd}$$ This operator $B$ does not describe a random walk since $B_{\alpha,\alpha'}$, the "probability of moving from $\alpha'$ to $\alpha$", may even be negative. However, this operator is closely related to the random walk described by $P$. Indeed, using the operators $T$ and $I$ we can see that $B$ describes the process that arises from the random walk if we consider a random walker arriving at a cell $\alpha_-$ (equipped with the reversed orientation) as the inverse of a random walker at $\alpha_+$ --- that is we allow random walkers at $\alpha_+$ and $\alpha_-$ to cancel each other out.[^10] **Lemma 13**. * The operators $T$, $I$ and $B$ defined above satisfy the equations $$BT = TP, \qquad B=TPI \quad \text{and} \quad B^n = TP^n I.$$ * **Lemma 14** (). * The operators $T$, $I$ and $B$ defined above satisfy the equations $$BT = TP, \qquad B=TPI \quad \text{and} \quad B^n = TP^n I.$$ * *Proof.* We check these equalities by direct computation. For $BT$ we obtain $$\begin{aligned} BT\left( \sum_{s\in I_k^*} \lambda_{s}\cdot s \right) &= B\left(\sum_{\alpha\in I_k} (\lambda_{\alpha_+} - \lambda_{\alpha_-}) \cdot \alpha\right) \\ &= \sum_{\alpha\in I_k} \left[ \sum_{\alpha'\in I_k} \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \cdot (\lambda_{\alpha'_+}-\lambda_{\alpha'_-})\right] \cdot \alpha\end{aligned}$$ and for $TP$ we compute (omitting the coefficient of $\Theta$ as it disappears in the next step) that $$\begin{aligned} TP&\left(\sum_{s\in I_k^*} \lambda_{s} s \right) \\ &= T \left( \sum_{\alpha_\nu \in I_k^\pm} \left[ \sum_{\alpha'_{\nu'}\in I_k^\pm} \frac{1}{d_+(\alpha')d_-(\alpha')} \!\! \sum_{\alpha_\nu \overset{\beta}{\sim} \alpha'_{\nu'}}\!\! -[\beta:\alpha_\nu][\beta:\alpha'_{\nu'}] \lambda_{\alpha'_{\nu'}}\right] {\alpha_\nu} + \cdots \Theta\right) \\ &= \sum_{\alpha\in I_k} \left[ \sum_{\alpha'_{\nu'}\in I_k^\pm} \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\alpha_+ \overset{\beta}{\sim} \alpha'_{\nu'}} -[\beta:\alpha_+][\beta:\alpha'_{\nu'}] \lambda_{\alpha'_{\nu'}} \right] \cdot \alpha \\ &\quad- \sum_{\alpha\in I_k} \left[ \sum_{\alpha'_{\nu'}\in I_k^\pm} \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\alpha_- \overset{\beta}{\sim} \alpha'_{\nu'}} -[\beta:\alpha_-][\beta:\alpha'_{\nu'}] \lambda_{\alpha'_{\nu'}} \right] \cdot \alpha.\end{aligned}$$ Now we use that $[\beta:\alpha] = [\beta:\alpha_+] = -[\beta:\alpha_-]$ and $[\beta:\alpha][\beta:\alpha']\neq 0$ only if either $\alpha_\pm\overset{\beta}{\sim} \alpha'_\pm$ or $\alpha_\pm \overset{\beta}{\sim} \alpha'_\mp$ together with $-[\beta:\alpha_\nu][\beta:\alpha'_{\nu'}] = \nu\nu'd(\alpha,\alpha',\beta)$ to find that $$\begin{aligned} TP\left(\sum_{s\in I_k^*} \lambda_{s}\cdot s \right) &= \sum_{\alpha\in I_k} \left[ \sum_{\alpha'_{\nu'}\in I_k^\pm} \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\alpha_+ \overset{\beta}{\sim} \alpha'_{\nu'}} d(\alpha,\alpha',\beta) \cdot \nu'\lambda_{\alpha'_{\nu'}} \right] \cdot \alpha \\ &\quad + \sum_{\alpha\in I_k} \left[ \sum_{\alpha'_{\nu'}\in I_k^\pm} \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\alpha_- \overset{\beta}{\sim} \alpha'_{\nu'}} d(\alpha,\alpha',\beta) \cdot \nu'\lambda_{\alpha'_{\nu'}} \right] \cdot \alpha \\ &= \sum_{\alpha\in I_k} \left[ \sum_{\alpha\neq \alpha'\in I_k} \frac{1}{{d_+(\alpha')d_-(\alpha')}} \sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \cdot (\lambda_{\alpha'_{+}}-\lambda_{\alpha'_{-}}) \right] \cdot \alpha\end{aligned}$$ showing the first equality. For the second equality, we have $TI = \mathop{\mathrm{Id}}$, hence $TPI = BTI = B$ and lastly $B^n = B^n TI = TP^n I$. ◻ **Corollary 15**. * For the operators $B$ and $P$ defined above, $n\in \mathbb{N}$ and all $\alpha,\alpha' \in I_k$, $$\langle B^n(\alpha),\alpha'\rangle = \langle P^n(\alpha_+), \alpha'_+\rangle - \langle P^n(\alpha_+), \alpha'_-\rangle.$$ * **Corollary 16** (). * For the operators $B$ and $P$ defined above, $n\in \mathbb{N}$ and all $\alpha,\alpha' \in I_k$, $$\langle B^n(\alpha),\alpha'\rangle = \langle P^n(\alpha_+), \alpha'_+\rangle - \langle P^n(\alpha_+), \alpha'_-\rangle.$$ * *Proof.* Using $B^n = TP^n I$, we compute the coefficient $\langle B^n(\alpha),\alpha'\rangle$ of $\alpha'$ in $B(\alpha)$ by $$\begin{aligned} \langle B^n(\alpha),\alpha'\rangle &= \langle TP^n I (\alpha),\alpha'\rangle = \langle TP^n (\alpha_+),\alpha'\rangle = \langle P^n(\alpha_+) ,\alpha'_+ \rangle - \langle P^n(\alpha_+) ,\alpha'_- \rangle. \qedhere\end{aligned}$$ ◻ In particular, we can define the following quantities generalising the idea of return probabilities. **Definition 17**. For the random walk described by $P$ and $\alpha\in I_k$, we define the return probabilities $p_{\alpha,+}$ and the probabilities of returning with reversed orientation $p_{\alpha,-}$ respectively by $$p_{\alpha,+}(n) = \langle P^n(\alpha_+), \alpha_+\rangle \quad \text{and} \quad p_{\alpha,-}(n) = \langle P^n(\alpha_+), \alpha_-\rangle.$$ For the process described by $B$ we define $$p_\alpha(n) = \langle B^n(\alpha), \alpha\rangle.$$ **Definition 18** (). For the random walk described by $P$ and $\alpha\in I_k$, we define the return probabilities $p_{\alpha,+}$ and the probabilities of returning with reversed orientation $p_{\alpha,-}$ respectively by $$p_{\alpha,+}(n) = \langle P^n(\alpha_+), \alpha_+\rangle \quad \text{and} \quad p_{\alpha,-}(n) = \langle P^n(\alpha_+), \alpha_-\rangle.$$ For the process described by $B$ we define $$p_\alpha(n) = \langle B^n(\alpha), \alpha\rangle.$$ Note that all three quantities are independent of the choice of preferred orientations. Notice also that $p_\alpha(n) = p_{\alpha,+}(n) - p_{\alpha,-}(n)$, hence, after summing over all $G$-types of $k$-cells we obtain the following expression for the von Neumann trace of $B^n$. **Corollary 19**. * For $n\in \mathbb{N}$, the von Neumann trace of $B^n$ is given by $$\mathop{\mathrm{tr}}_{\mathcal{N}G} (B^n) = \sum_{\alpha \in I_k(G\setminus X)} p_{\alpha}(n) = \sum_{\alpha\in I_k(G\setminus X)} p_{\alpha,+}(n) - p_{\alpha,-}(n).$$ * **Corollary 20** (). * For $n\in \mathbb{N}$, the von Neumann trace of $B^n$ is given by $$\mathop{\mathrm{tr}}_{\mathcal{N}G} (B^n) = \sum_{\alpha \in I_k(G\setminus X)} p_{\alpha}(n) = \sum_{\alpha\in I_k(G\setminus X)} p_{\alpha,+}(n) - p_{\alpha,-}(n).$$ * Before moving on and relating this random walk to the upper $k$-Laplacian, we will introduce one extra parameter that will prove useful later on when studying the spectra of the operators. ## Lazy Degree $k$-Upper Random Walks Given a random walk $\mathfrak{R} = (V, P)$ with state space $V$ and propagation operator $P$ and $q\in [0,1]$, the $q$-lazy version $\mathfrak{R}_q$ of the random walk $\mathfrak{R}$ is given by $\mathfrak{R}_q = (V, q\mathop{\mathrm{Id}}+ (1+q)P)$, that is the random walk on $V$ that stays put with probability $q$ and moves according to the random walk $\mathfrak{R}$ with probability $1-q$. In particular, the moving probabilities for the $q$-lazy version $\mathfrak{R}^k_q(X)$ of the previously defined random walk on $I_k^*$ are given as follows: - For the absorbing state $\Theta$, $\qquad \mathbb{P}_q(\Theta\to \Theta) = 1, \qquad \mathbb{P}_q(\Theta\to \alpha_\nu) = 0.$ - For $\alpha_\nu \in I_k^\pm$ (that is $\alpha\in I_k$ and $\nu\in \{\pm\}$), $$\begin{aligned} &\mathbb{P}_q(\alpha_\nu \to \alpha_\nu) = q, \qquad \mathbb{P}_q(\alpha_\nu \to -\alpha_\nu) = 0. \end{aligned}$$ - For $\alpha_\nu,\alpha'_{\nu'}\in I_k^\pm$ with $\alpha_\nu \neq \pm \alpha'_{\nu'}$, $$\mathbb{P}_q(\alpha_\nu \to \alpha'_{\nu'}) = \frac{1-q}{{d_+(\alpha)d_-(\alpha)}}\sum_{\stackrel{\beta\in I_{k+1}}{\alpha_\nu \overset{\beta}{\sim} \alpha'_{\nu'}}} d(\alpha_\nu,\alpha_{\nu'},\beta) = (1-q)\mathbb{P}(\alpha_\nu \to \alpha'_{\nu'}).$$ - Lastly, $$\mathbb{P}_q(\alpha_\nu \to \Theta) = 1-\sum_{\alpha'_{\nu'}\in I_k^\pm} \mathbb{P}_q(\alpha_\nu \to \alpha'_{\nu'}) = (1-q)\mathbb{P}(\alpha_\nu \to \Theta).$$ In the same spirit, we define $B_q\curvearrowright\ell^2 I_k$ by $B_q = q\mathop{\mathrm{Id}}+ (1-q)B$. This operator is given by $$\begin{aligned} B_q\left( \sum_{\alpha\in I_k} \lambda_\alpha\cdot \alpha\right) &= \sum_{\alpha\in I_k} \left[ q\lambda_\alpha + \sum_{\alpha\neq \alpha'\in I_k} \frac{1-q}{d_+(\alpha')d_-(\alpha')} \sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta)\lambda_{\alpha'} \right] \cdot \alpha.\end{aligned}$$ **Corollary 21**. * These operators satisfy $B_q T = TP_q$, $B_q = TP_q I$ and $B_q^n = TP_q^nI$. * **Corollary 22** (). * These operators satisfy $B_q T = TP_q$, $B_q = TP_q I$ and $B_q^n = TP_q^nI$. * *Proof.* This follows from the previous equalities since $B_qT = qT+(1-q)BT = qT+(1-q)TP = TP_q$ and $TP_q I = qTI + (1-q)TPI = q + (1-q)B = B_q$. ◻ As before, we consider the probabilities of returning to the same $k$-cell with the same orientation or the reversed orientation respectively. **Definition 23**. For $\alpha\in I_k$, we define the quantities $$\begin{aligned} p_{q,\alpha,+}(n) &= \langle P_q^n(\alpha_+), \alpha_+\rangle, \\ p_{q,\alpha,-}(n) &= \langle P_q^n(\alpha_+), \alpha_-\rangle, \\ p_{q,\alpha}(n) &= \langle B_q^n(\alpha), \alpha\rangle \end{aligned}$$ **Definition 24** (). For $\alpha\in I_k$, we define the quantities $$\begin{aligned} p_{q,\alpha,+}(n) &= \langle P_q^n(\alpha_+), \alpha_+\rangle, \\ p_{q,\alpha,-}(n) &= \langle P_q^n(\alpha_+), \alpha_-\rangle, \\ p_{q,\alpha}(n) &= \langle B_q^n(\alpha), \alpha\rangle \end{aligned}$$ Again, $p_{q,\alpha}(n) = p_{q,\alpha,+}(n) - p_{q,\alpha,-}(n)$, hence we can compute the von Neumann trace of $B_q^n$ using the probabilities of the random walk. We define $$p_q(n) = \mathop{\mathrm{tr}}_{\mathcal{N}G}(B_q^n) = \sum_{\alpha\in I_k(G\setminus X)} p_{q,\alpha,+}(n) - p_{q,\alpha,-}(n)$$ # Relationship to Laplace Operators We now compare the operator $B_q$ to the upper Laplacian $\Delta^\mathrm{up}_k$. Recall from Equation ([\[E_Dup_Formula\]](#E_Dup_Formula){reference-type="ref" reference="E_Dup_Formula"}) that $\Delta^\mathrm{up}_k$ acts on $\ell^2 I_k$ by $$\begin{aligned} \Delta^\mathrm{up}_k\left( \sum_{\alpha\in I_k} \lambda_\alpha\cdot \alpha\right) &= \sum_{\alpha\in I_k}\left[ \sum_{\beta\in I_{k+1}} [\beta:\alpha]^2 \lambda_\alpha - \sum_{\alpha'\neq \alpha\in I_k}\sum_{\beta\in I_{k+1}} -[\beta:\alpha][\beta:\alpha'] \lambda_{\alpha'} \right] \cdot \alpha \\ &= \sum_{\alpha\in I_k}\left[ d_{+,2}(\alpha) \lambda_\alpha - \sum_{\alpha'\neq \alpha\in I_k}\sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \lambda_{\alpha'} \right] \cdot \alpha\end{aligned}$$ **Theorem 25**. * [\[Thm_Bq_Id_Delta\]]{#Thm_Bq_Id_Delta label="Thm_Bq_Id_Delta"} Let $B_q\curvearrowright\ell^2 I_k = \ell^2C^\mathrm{cell}_k(X)$ be the operator $B_q = TP_q I$ defined as above. Then $$\begin{aligned} B_q \circ M_{1,q} = \mathop{\mathrm{Id}}- \Delta^\mathrm{up}_k \circ M_{2,q}, \end{aligned}$$ where $M_{1,q}, M_{2,q}\curvearrowright\ell^2 I_k$ are the non-negative multiplication operators given by $$\begin{aligned} M_{1,q} = &\frac{d_+d_-}{qd_+d_-+(1-q)d_{+,2}}, \\ &\sum_{\alpha\in I_k} \lambda_\alpha \alpha \mapsto \sum_{\alpha\in I_k} \frac{d_+(\alpha)d_-(\alpha)}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)}\cdot\lambda_\alpha \alpha,\\ M_{2,q} = &\frac{1-q}{qd_+d_-+(1-q)d_{+,2}},\\ &\sum_{\alpha\in I_k} \lambda_\alpha \alpha \mapsto \sum_{\alpha\in I_k} \frac{1-q}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)}\cdot\lambda_\alpha \alpha.\end{aligned}$$ * **Theorem 26** (). * [\[Thm_Bq_Id_Delta\]]{#Thm_Bq_Id_Delta label="Thm_Bq_Id_Delta"} Let $B_q\curvearrowright\ell^2 I_k = \ell^2C^\mathrm{cell}_k(X)$ be the operator $B_q = TP_q I$ defined as above. Then $$\begin{aligned} B_q \circ M_{1,q} = \mathop{\mathrm{Id}}- \Delta^\mathrm{up}_k \circ M_{2,q}, \end{aligned}$$ where $M_{1,q}, M_{2,q}\curvearrowright\ell^2 I_k$ are the non-negative multiplication operators given by $$\begin{aligned} M_{1,q} = &\frac{d_+d_-}{qd_+d_-+(1-q)d_{+,2}}, \\ &\sum_{\alpha\in I_k} \lambda_\alpha \alpha \mapsto \sum_{\alpha\in I_k} \frac{d_+(\alpha)d_-(\alpha)}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)}\cdot\lambda_\alpha \alpha,\\ M_{2,q} = &\frac{1-q}{qd_+d_-+(1-q)d_{+,2}},\\ &\sum_{\alpha\in I_k} \lambda_\alpha \alpha \mapsto \sum_{\alpha\in I_k} \frac{1-q}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)}\cdot\lambda_\alpha \alpha.\end{aligned}$$ * *Proof.* For $\alpha, \alpha'\in I_k$ we compare the contributions $(B_q\circ M_{1,q})_{\alpha',\alpha}$ and $(\mathop{\mathrm{Id}}-\Delta^\mathrm{up}_k\circ M_{2,q})_{\alpha',\alpha}$ coming from the coefficient of $\alpha$ in the argument to the coefficient of $\alpha'$ in the image.[^11] For $\alpha\neq \alpha'$ these contributions are given by $$\begin{aligned} (B_q\circ M_{1,q})_{\alpha',\alpha} &= \frac{d_+(\alpha)d_-(\alpha)}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)} \cdot (B_q)_{\alpha',\alpha} \\ &= \frac{d_+(\alpha)d_-(\alpha)}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)} \cdot \frac{1-q}{d_+(\alpha)d_-(\alpha)}\sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \\ &= \frac{1-q}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)} \sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \\ &= 0 - \frac{1-q}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)} \cdot \left(- \sum_{\beta\in I_{k+1}} d(\alpha,\alpha',\beta) \right) \\ &= (\mathop{\mathrm{Id}}-\Delta^\mathrm{up}_k\circ M_{2,q})_{\alpha',\alpha}\end{aligned}$$ and for $\alpha=\alpha'$ by $$\begin{aligned} (B_q\circ M_{1,q})_{\alpha,\alpha} &= q\cdot \frac{d_+(\alpha)d_-(\alpha)}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)} \\ &= 1 - \frac{(1-q)d_{+,2}(\alpha)}{qd_+(\alpha)d_-(\alpha) + (1-q)d_{+,2}(\alpha)} = (\mathop{\mathrm{Id}}- \Delta^\mathrm{up}_k\circ M_{2,q})_{\alpha,\alpha}. \end{aligned}$$ Since all these coefficients agree, the claim follows. ◻ **Remark 27**. [\[R_simplicial_upper_k\_regular\]]{#R_simplicial_upper_k_regular label="R_simplicial_upper_k_regular"} The construction here generalises the one given by O. Parzanchevski and R. Rosenthal on simplicial complexes in [@PaRo13] and the previous theorem generalises Proposition 2.8 (1) of their paper. Considering the random walk of O. Parzanchevski and R. Rosenthal in degree $k=(d-1)$, the incidence numbers of a simplicial complex (viewed as a CW complex) are given as $[\beta:\alpha]\in\{0,\pm 1\}$, where $\pm 1$ occurs if the $(d-1)$-simplex $\alpha$ is in the boundary of the $d$-simplex $\beta$, with sign depending on orientations. Therefore, $$d_+(\alpha) = d_{+,2}(\alpha) = \deg(\alpha), \quad d_-(\alpha) = d,$$ where $\deg(\alpha)$ denoted the number of $d$-simplices $\beta\in I_d$ containing $\alpha$. Hence, $$\frac{d}{q(d-1)+1}B_q = \mathop{\mathrm{Id}}- \frac{1-q}{q(d-1)+1} \cdot \frac{\Delta^\mathrm{up}_{d-1}}{\deg(\alpha)},$$ where $\Delta^\mathrm{up}_w = \frac{\Delta^\mathrm{up}_{d-1}}{\deg(\alpha)}$ is the weighted upper Laplacian used by O. Parzanchevski and R. Rosenthal, defined by using a weighted scalar product on $\ell^2 I_k$. Note that in this case, the diagonal operator $M_{1,q}$ is given by multiplication by a constant (depending on $q$ and $d$ but not on $\alpha\in I_{d-1}$). **Remark 28** (). [\[R_simplicial_upper_k\_regular\]]{#R_simplicial_upper_k_regular label="R_simplicial_upper_k_regular"} The construction here generalises the one given by O. Parzanchevski and R. Rosenthal on simplicial complexes in [@PaRo13] and the previous theorem generalises Proposition 2.8 (1) of their paper. Considering the random walk of O. Parzanchevski and R. Rosenthal in degree $k=(d-1)$, the incidence numbers of a simplicial complex (viewed as a CW complex) are given as $[\beta:\alpha]\in\{0,\pm 1\}$, where $\pm 1$ occurs if the $(d-1)$-simplex $\alpha$ is in the boundary of the $d$-simplex $\beta$, with sign depending on orientations. Therefore, $$d_+(\alpha) = d_{+,2}(\alpha) = \deg(\alpha), \quad d_-(\alpha) = d,$$ where $\deg(\alpha)$ denoted the number of $d$-simplices $\beta\in I_d$ containing $\alpha$. Hence, $$\frac{d}{q(d-1)+1}B_q = \mathop{\mathrm{Id}}- \frac{1-q}{q(d-1)+1} \cdot \frac{\Delta^\mathrm{up}_{d-1}}{\deg(\alpha)},$$ where $\Delta^\mathrm{up}_w = \frac{\Delta^\mathrm{up}_{d-1}}{\deg(\alpha)}$ is the weighted upper Laplacian used by O. Parzanchevski and R. Rosenthal, defined by using a weighted scalar product on $\ell^2 I_k$. Note that in this case, the diagonal operator $M_{1,q}$ is given by multiplication by a constant (depending on $q$ and $d$ but not on $\alpha\in I_{d-1}$). # Application to $L^2$-Invariants As a short reminder on $L^2$-invariants, we briefly recall some important definitions. For a proper introduction we refer to [@KamBook] and [@LckL2]. Let $M$ be a manifold with cocompact free and proper action by a (discrete) group $G$. We denote by $\mathcal{N}G$ the von Neumann algebra of $G$. On $\ell^2 G$, we define the von Neumann trace by $\mathop{\mathrm{tr}}_{\mathcal{N}G}(f) = \langle f(e), e\rangle_{\ell^2 G}$. This notion can be extended to Hilbert $\mathcal{N}G$-modules, that is, Hilbert spaces $V$ with isometric left $G$-action that have an isometric $G$-embedding $V\hookrightarrow H\otimes \ell^2 G$ for some Hilbert space $H$. **Definition 29**. Given two Hilbert $\mathcal{N}G$-modules $U, V$ and a closed densely defined $G$-equivariant operator $f\colon U\to V$, we define the spectral density function $F(f)$ of $f$ using the family of $G$-equivariant spectral projectors of the self-adjoint operator $f^*f$ $\left\{E^{f^*f}_\lambda\right\}_{\lambda\geq 0}$ by $$F(f)\colon \mathbb{R}_{\geq 0} \to [0,\infty], \qquad F(f)(\lambda) = \mathop{\mathrm{tr}}_{\mathcal{N}G}\left( E^{f^*f}_{\lambda^2} \right).$$ We call $f$ Fredholm if there is $\lambda>0$ with $F(f)(\lambda) <\infty$. **Definition 30** (). Given two Hilbert $\mathcal{N}G$-modules $U, V$ and a closed densely defined $G$-equivariant operator $f\colon U\to V$, we define the spectral density function $F(f)$ of $f$ using the family of $G$-equivariant spectral projectors of the self-adjoint operator $f^*f$ $\left\{E^{f^*f}_\lambda\right\}_{\lambda\geq 0}$ by $$F(f)\colon \mathbb{R}_{\geq 0} \to [0,\infty], \qquad F(f)(\lambda) = \mathop{\mathrm{tr}}_{\mathcal{N}G}\left( E^{f^*f}_{\lambda^2} \right).$$ We call $f$ Fredholm if there is $\lambda>0$ with $F(f)(\lambda) <\infty$. This function measures the content of the spectrum of the operator $f$ close to zero in terms of the parameter $\lambda$. Notice that $F$ is monotonously increasing and right-continuous. The value at zero, $F(f)(0)$, measures the size of the kernel of $f$ and is called the $L^2$-Betti number. Additionally, we define the Novikov-Shubin invariants to measure the spectral content close to zero. **Definition 31**. Let $F$ be monotonously increasing and right-continuous. We define the $L^2$-Betti number $b^{(2)}(F) = F(0)$ and the Novikov-Shubin invariant $$\alpha(F) = \liminf_{\lambda\searrow 0} \frac{\log(F(\lambda) - F(0))}{\log(\lambda)}.$$ If $F(f)$ is the spectral density function of $f$, we also write $b^{(2)}(f)=b^{(2)}(F(f))$ and $\alpha(f) = \alpha(F(f))$. **Definition 32** (). Let $F$ be monotonously increasing and right-continuous. We define the $L^2$-Betti number $b^{(2)}(F) = F(0)$ and the Novikov-Shubin invariant $$\alpha(F) = \liminf_{\lambda\searrow 0} \frac{\log(F(\lambda) - F(0))}{\log(\lambda)}.$$ If $F(f)$ is the spectral density function of $f$, we also write $b^{(2)}(f)=b^{(2)}(F(f))$ and $\alpha(f) = \alpha(F(f))$. Here, $\alpha(F)$ measures the asymptotic behaviour of $F$ close to zero. Indeed, if asymptotically $F(\lambda) - F(0) \sim \lambda^{\alpha}$ as $\lambda\searrow 0$, then $\alpha=\alpha(F)$. In particular, we only care for the asymptotic behaviour of $F$ close to zero. This motivates the following equivalence relation defined on such functions. **Definition 33**. Let $F, F'\colon \mathbb{R}_{\geq 0}\to [0,\infty]$ be two monotonously increasing right-continuous functions. We call $F$ and $F'$ dilatationally equivalent, and write $F\sim F'$, if there are constants $C > 0$ and $\lambda_0 > 0$ such that $$F'(C^{-1}\lambda) \leq F(\lambda) \leq F'(C\lambda) \qquad \text{ for all } \lambda\in [0,\lambda_0].$$ **Definition 34** (). Let $F, F'\colon \mathbb{R}_{\geq 0}\to [0,\infty]$ be two monotonously increasing right-continuous functions. We call $F$ and $F'$ dilatationally equivalent, and write $F\sim F'$, if there are constants $C > 0$ and $\lambda_0 > 0$ such that $$F'(C^{-1}\lambda) \leq F(\lambda) \leq F'(C\lambda) \qquad \text{ for all } \lambda\in [0,\lambda_0].$$ Novikov-Shubin invariants are invariant under dilatational equivalence. Given a free $G$-CW complex of finite type, we use the differentials of the cellular $L^2$-chain complex, $d_k\colon \ell^2C^{\mathrm{cell}}_k(X;\mathcal{N}G)\to \ell^2 C^{\mathrm{cell}}_{k-1}(X;\mathcal{N}G)$, and the cellular Laplace operators, $\Delta_k = d_kd_k^* + d_k^*d_k \curvearrowright\ell^2C^{\mathrm{cell}}_k(X;\mathcal{N}G)$, to associate $L^2$-Betti numbers and Novikov-Shubin invariants to $X$. **Definition 35**. Let $X$ be a free $G$-CW complex of finite type. We define the $L^2$-Betti numbers of $X$ by $b^{(2)}_k(X) = b^{(2)}(\Delta_k)$ and the Novikov-Shubin invariants by $\alpha_k(X) = \alpha(d_k).$ **Definition 36** (). Let $X$ be a free $G$-CW complex of finite type. We define the $L^2$-Betti numbers of $X$ by $b^{(2)}_k(X) = b^{(2)}(\Delta_k)$ and the Novikov-Shubin invariants by $\alpha_k(X) = \alpha(d_k).$ Here, we use $d_k$ instead of $\Delta_k$ in the definition of the Novikov-Shubin invariants, as they provide a more detailed picture. Indeed, $\alpha(\Delta_k) = \nicefrac{1}{2}\cdot \min\{\alpha_k(X), \alpha_{k+1}(X)\}$. ## Computing Novikov-Shubin Invariants We now study the connection between the random walk $\mathfrak{R}^k_q=\mathfrak{R}^k_q(X)$ and the Novikov-Shubin invariant $\alpha_k(X)$ for a free $G$-CW complex $X$ of finite type. In degree zero it is reasonable to consider only connected spaces (since the $\ell^2$-spaces and the Laplace operator split as a direct sum with one summand for each connected component). By the same reasoning, we can assume without loss of generality the following analogue in degree $k$. **Definition 37**. Let $X$ be a CW complex. We call $X$ upper $k$-connected if $|I_k|\geq 2$ and for all $\alpha,\alpha'\in I_k$ there are $$\alpha=\alpha_0,\alpha_1,\dots, \alpha_{n-1},\alpha_n=\alpha'\in I_k \quad\text{ and }\quad \beta_1,\dots, \beta_{n}\in I_{k+1}$$ such that $[\beta_i,\alpha_{i-1}]\neq 0$ and $[\beta_i:\alpha_i]\neq 0$ for all $1\leq i\leq n$, that is $\beta_i$ is attached non-trivially to $\alpha_{i-1}$ and $\alpha_i$. **Definition 38** (). Let $X$ be a CW complex. We call $X$ upper $k$-connected if $|I_k|\geq 2$ and for all $\alpha,\alpha'\in I_k$ there are $$\alpha=\alpha_0,\alpha_1,\dots, \alpha_{n-1},\alpha_n=\alpha'\in I_k \quad\text{ and }\quad \beta_1,\dots, \beta_{n}\in I_{k+1}$$ such that $[\beta_i,\alpha_{i-1}]\neq 0$ and $[\beta_i:\alpha_i]\neq 0$ for all $1\leq i\leq n$, that is $\beta_i$ is attached non-trivially to $\alpha_{i-1}$ and $\alpha_i$. This condition implies for $\mathfrak{R}^k_q$ that a random walker can move from any $k$-cell $\alpha_\pm\in I^\pm_k$ to any other (unoriented) $k$-cell $\alpha'$ (that is, to one of the oriented $k$-cells $\alpha'_+$ or $\alpha'_-$). Furthermore, we get bounds on the quantities from Definition [\[Def_RW_ds\]](#Def_RW_ds){reference-type="ref" reference="Def_RW_ds"}. **Lemma 39**. * Let $X$ be an upper $k$-connected free $G$-CW complex of finite type. Then there exists $D\geq 1$ such that $$D\geq d_{+,2}, d_+, d_-\geq 1.$$ In particular, if $q\in [0,1)$ then the operators $M_{1,q}$ and $M_{2,q}$ are positive multiplication operators bounded from below by $$M_{1,q} \geq D^{-2}>0 \quad \text{and} \quad M_{2,q} = (1-q)D^{-2}>0.$$ * **Lemma 40** (). * Let $X$ be an upper $k$-connected free $G$-CW complex of finite type. Then there exists $D\geq 1$ such that $$D\geq d_{+,2}, d_+, d_-\geq 1.$$ In particular, if $q\in [0,1)$ then the operators $M_{1,q}$ and $M_{2,q}$ are positive multiplication operators bounded from below by $$M_{1,q} \geq D^{-2}>0 \quad \text{and} \quad M_{2,q} = (1-q)D^{-2}>0.$$ * *Proof.* Let $\alpha\in I_k$ be arbitrary and let $\alpha\neq \alpha'\in I_k$ be any other $k$-cell. Since $X$ is upper $k$-connected, by definition there is a sequence of $k$-cells $\alpha_i\in I_k$ and $(k+1)$-cells $\beta_i\in I_{k+1}$ connecting $\alpha$ to $\alpha'$. In particular, there exists a $(k+1)$-cell $\beta_1\in I_{k+1}$ such that $[\beta_1:\alpha]\neq 0$ and a $k$-cell $\alpha\neq \alpha_1\in I_k$ such that $[\beta_1:\alpha_1]\neq 0$. Hence $$\begin{aligned} d_{+,2}(\alpha) &\geq [\beta_1:\alpha]^2 \geq 1, \\ d_+(\alpha) &\geq |[\beta_1:\alpha]| \geq 1, \\ d_-(\alpha) &\geq d_-(\beta_1; \alpha) \geq |[\beta_1:\alpha_1]|\geq 1. \end{aligned}$$ Since $X$ is of finite type and these quantities depend only on the $G$-type of $\alpha$, there exists $$D = \sup_{\alpha\in I_k} \{d_{+,2}(\alpha), d_{+}(\alpha), d_{-}(\alpha) \} = \max_{\alpha\in I_k(G\setminus X)} \{d_{+,2}(\alpha), d_{+}(\alpha), d_{-}(\alpha) \} \geq 1.$$ It follows, therefore, that $$\begin{aligned} M_{1,q} &= \frac{d_+ d_-}{qd_+ d_- + (1-q) d_{+,2}} \geq \frac{1}{qD^2 + (1-q)D} \geq \frac{1}{D^2}>0, \\ M_{2,q} &= \frac{1-q}{qd_+ d_- + (1-q) d_{+,2}} \geq \frac{1-q}{qD^2 + (1-q)D} \geq \frac{1-q}{D^2}>0,\end{aligned}$$ and the claim follows. ◻ Generalising the notion of regular graphs, we introduce the following notion of upper $k$-regular free $G$-CW complexes. **Definition 41**. Let $X$ be a free $G$-CW complex of finite type. We call $X$ upper $k$-regular if $X$ is upper $k$-connected and $d_+d_- = d_+(\alpha)d_-(\alpha)$ and $d_{+,2} = d_{+,2}(\alpha)$ are independent of the cell $\alpha\in I_k$. **Definition 42** (). Let $X$ be a free $G$-CW complex of finite type. We call $X$ upper $k$-regular if $X$ is upper $k$-connected and $d_+d_- = d_+(\alpha)d_-(\alpha)$ and $d_{+,2} = d_{+,2}(\alpha)$ are independent of the cell $\alpha\in I_k$. In this case, also the multiplication operators $M_{1,q}$ and $M_{2,q}$ are just multiplication with a constant. Hence, the formula connecting $B_q$ and $\Delta^\mathrm{up}_k$ simplifies further. **Example 43**. This happens, for example, in the case where $X$ is a regular graph for $k=0$ or in the case studied in [@BCDS23] for $k=\dim(X)-1$. While we can compute the Novikov-Shubin invariants in the latter case by Poincaré duality, we can use their values to obtain informations about the random walk considered. **Example 44** (). This happens, for example, in the case where $X$ is a regular graph for $k=0$ or in the case studied in [@BCDS23] for $k=\dim(X)-1$. While we can compute the Novikov-Shubin invariants in the latter case by Poincaré duality, we can use their values to obtain informations about the random walk considered. **Corollary 45**. * Let $X$ be an upper $k$-regular $G$-CW complex of finite type and $q\in [0,1]$. Then $$C_{1,q} B_q = \mathop{\mathrm{Id}}- C_{2,q} \Delta^\mathrm{up}_k,$$ where $C_{1,q}$ and $C_{2,q}$ are positive constants given as $$C_{1,q} = \frac{d_+d_-}{qd_+d_- + (1-q)d_{+,2}} > 0 \qquad \text{and} \qquad C_{2,q} = \frac{1-q}{qd_+d_- + (1-q)d_{+,2}} > 0.$$ * **Corollary 46** (). * Let $X$ be an upper $k$-regular $G$-CW complex of finite type and $q\in [0,1]$. Then $$C_{1,q} B_q = \mathop{\mathrm{Id}}- C_{2,q} \Delta^\mathrm{up}_k,$$ where $C_{1,q}$ and $C_{2,q}$ are positive constants given as $$C_{1,q} = \frac{d_+d_-}{qd_+d_- + (1-q)d_{+,2}} > 0 \qquad \text{and} \qquad C_{2,q} = \frac{1-q}{qd_+d_- + (1-q)d_{+,2}} > 0.$$ * **Definition 47**. Let $X$ be an upper $k$-regular free $G$-CW complex of finite type. We define $$\begin{aligned} \widetilde{B_q} &= C_{1,q} B_q \quad \text{ and } \quad \widetilde{\Delta^\mathrm{up}_{q,k}} = C_{2,q} \Delta^\mathrm{up}_k\end{aligned}$$ so that we have the equality $\widetilde{B_q} = \mathop{\mathrm{Id}}-\widetilde{\Delta^\mathrm{up}_{q,k}}.$ **Definition 48** (). Let $X$ be an upper $k$-regular free $G$-CW complex of finite type. We define $$\begin{aligned} \widetilde{B_q} &= C_{1,q} B_q \quad \text{ and } \quad \widetilde{\Delta^\mathrm{up}_{q,k}} = C_{2,q} \Delta^\mathrm{up}_k\end{aligned}$$ so that we have the equality $\widetilde{B_q} = \mathop{\mathrm{Id}}-\widetilde{\Delta^\mathrm{up}_{q,k}}.$ We now can derive bounds on the spectrum $\sigma\left(\widetilde{\Delta^\mathrm{up}_{q,k}}\right)$ of the operator $\widetilde{\Delta^\mathrm{up}_{q,k}}$. It is well-known that $\Delta^{\textrm{up}}_k$ is bounded, **Lemma 49**. *[\[L:DeltaBounded\]]{#L:DeltaBounded label="L:DeltaBounded"} Let $X$ be a free $G$-CW complex of finite type, then $\Delta^{\textrm{up}}_k\curvearrowright\ell^2C^\mathrm{cell}_k(X)$ is bounded and $\sigma(\Delta^{\textrm{up}}_k) \subset [0, S_{k}]$ for some $S_k<\infty$. * **Lemma 50** (). *[\[L:DeltaBounded\]]{#L:DeltaBounded label="L:DeltaBounded"} Let $X$ be a free $G$-CW complex of finite type, then $\Delta^{\textrm{up}}_k\curvearrowright\ell^2C^\mathrm{cell}_k(X)$ is bounded and $\sigma(\Delta^{\textrm{up}}_k) \subset [0, S_{k}]$ for some $S_k<\infty$. * While the precise bounds won't matter for this paper, we can find explicit bounds by straight-forward computations, for example, $$S_{k} = \max_{\alpha\in I_k(G\backslash X)} \left\{ \sum_{\beta\in I_{k+1}} \sum_{\alpha'\in I_k} |d(\alpha,\alpha',\beta)| \right\} < \infty.$$ Using this, we can prove the following lemma. **Lemma 51**. * [\[L_DeltaTildeSpectrum01\]]{#L_DeltaTildeSpectrum01 label="L_DeltaTildeSpectrum01"} Let $X$ be an upper $k$-regular free $G$-CW complex of finite type. Then there exists $q_0\in (0,1)$ such that for all $q_0\leq q\leq 1$ the spectrum of $\widetilde{\Delta^\mathrm{up}_{q,k}}$ is contained in the unit interval, $\sigma(\widetilde{\Delta^\mathrm{up}_{q,k}}) \subset [0,1].$ * **Lemma 52** (). * [\[L_DeltaTildeSpectrum01\]]{#L_DeltaTildeSpectrum01 label="L_DeltaTildeSpectrum01"} Let $X$ be an upper $k$-regular free $G$-CW complex of finite type. Then there exists $q_0\in (0,1)$ such that for all $q_0\leq q\leq 1$ the spectrum of $\widetilde{\Delta^\mathrm{up}_{q,k}}$ is contained in the unit interval, $\sigma(\widetilde{\Delta^\mathrm{up}_{q,k}}) \subset [0,1].$ * *Proof.* By Lemma [\[L:DeltaBounded\]](#L:DeltaBounded){reference-type="ref" reference="L:DeltaBounded"}, $\sigma(\Delta^\mathrm{up}_k) \subset [0,S]$ for some $S>0$, hence $\sigma(\widetilde{\Delta^\mathrm{up}_{q,k}}) \subset [0, C_{2,q}S]$. Note that $d_+d_-\ge 1$ and $d_{+,2}\geq 1$ so $C_{2,q} = \frac{1-q}{qd_+d_- + (1-q)d_{+,2}}$ is continuous in $q\in (0,1)$ and converges to $0$ as $q\nearrow 1$. In particular, there is $q_0\in (0,1)$ such that $0< C_{2,q} \leq S^{-1}$ for all $q_0\leq q\leq 1$. ◻ **Corollary 53**. * Let $X$ be an upper $k$-regular free $G$-CW complex of finite type and $q\in [q_0, 1)$. Let $\widetilde d_{k+1} = \sqrt{C_{2,q}} d_{k+1}$. Then $\widetilde d_{k+1}^* = \sqrt{C_{2,q}} d_{k+1}^*$ and $$\widetilde{\Delta^\mathrm{up}_{q,k}} = \widetilde d_{k+1}\widetilde d_{k+1}^*$$ is a self-adjoint positive operator with $\sigma(\widetilde{\Delta^\mathrm{up}_{q,k}}) \subset [0,1]$. * **Corollary 54** (). * Let $X$ be an upper $k$-regular free $G$-CW complex of finite type and $q\in [q_0, 1)$. Let $\widetilde d_{k+1} = \sqrt{C_{2,q}} d_{k+1}$. Then $\widetilde d_{k+1}^* = \sqrt{C_{2,q}} d_{k+1}^*$ and $$\widetilde{\Delta^\mathrm{up}_{q,k}} = \widetilde d_{k+1}\widetilde d_{k+1}^*$$ is a self-adjoint positive operator with $\sigma(\widetilde{\Delta^\mathrm{up}_{q,k}}) \subset [0,1]$. * **Remark 55**. Recall that for $q\in [q_0,1)$, since $d$ and $\widetilde{d}$ differ only by a constant factor $\sqrt{C_{2,q}}$, their spectral density functions are dilatationally equivalent and hence their Novikov-Shubin invariants agree, that is, $\alpha_k(X) = \alpha(d_{k+1}) = \alpha(\widetilde d_{k+1}) = \alpha({\widetilde d_{k+1}}^*)$. **Remark 56** (). Recall that for $q\in [q_0,1)$, since $d$ and $\widetilde{d}$ differ only by a constant factor $\sqrt{C_{2,q}}$, their spectral density functions are dilatationally equivalent and hence their Novikov-Shubin invariants agree, that is, $\alpha_k(X) = \alpha(d_{k+1}) = \alpha(\widetilde d_{k+1}) = \alpha({\widetilde d_{k+1}}^*)$. **Lemma 57**. * [\[L_trBq_F\]]{#L_trBq_F label="L_trBq_F"} Let $\chi_I$ denote the indicator function of the interval $I$, then $$\mathop{\mathrm{tr}}_{\mathcal{N}G} (\chi_{[1-\lambda,1]}(\widetilde{B_q})) = F(\widetilde d_{k+1}^*)(\sqrt \lambda).$$ * **Lemma 58** (). * [\[L_trBq_F\]]{#L_trBq_F label="L_trBq_F"} Let $\chi_I$ denote the indicator function of the interval $I$, then $$\mathop{\mathrm{tr}}_{\mathcal{N}G} (\chi_{[1-\lambda,1]}(\widetilde{B_q})) = F(\widetilde d_{k+1}^*)(\sqrt \lambda).$$ * *Proof.* Recall that $\widetilde{B_q} = \mathop{\mathrm{Id}}-\widetilde d_{k+1} \widetilde{d}_{k+1}^*$, hence $$\begin{aligned} \mathop{\mathrm{tr}}_{\mathcal{N}G} \left(\chi_{[1-\lambda,1]}\left(\widetilde{B_q}\right)\right) &= \mathop{\mathrm{tr}}_{\mathcal{N}G}\left(\chi_{[0,\lambda]}\left(\widetilde d_{k+1} \widetilde{d}_{k+1}^*\right)\right) \\ &= \mathop{\mathrm{tr}}_{\mathcal{N}G}\left( E_\lambda^{\widetilde d_{k+1} \widetilde{d}_{k+1}^*}\right) = F\left(\widetilde{d}_{k+1}^*\,\right)\left(\sqrt{\lambda}\right). \qedhere\end{aligned}$$ ◻ We can now proceed in the same way as in degree zero, compare Lück's book [@LckL2 §2.1.4]. **Theorem 59**. * [\[Thm_RW_ak\]]{#Thm_RW_ak label="Thm_RW_ak"} Let $X$ be an upper $k$-regular free $G$-CW complex of finite type and $q\in [q_0, 1)$, with $q_0$ given by Lemma [\[L_DeltaTildeSpectrum01\]](#L_DeltaTildeSpectrum01){reference-type="ref" reference="L_DeltaTildeSpectrum01"}. Then $\alpha_k(X) = 2a$ if and only if there is a constant $C>0$ such that for all $n\in \mathbb{N}$, $$C_{1,q}^{-n} \left( b^{(2)}(d_{k+1}^*) + C^{-1}n^{-a}\right) \quad \leq \quad p_q(n) \quad\leq\quad C_{1,q}^{-n}\left(b^{(2)}(d_{k+1}^*) + Cn^{-a}\right).$$ * **Theorem 60** (). * [\[Thm_RW_ak\]]{#Thm_RW_ak label="Thm_RW_ak"} Let $X$ be an upper $k$-regular free $G$-CW complex of finite type and $q\in [q_0, 1)$, with $q_0$ given by Lemma [\[L_DeltaTildeSpectrum01\]](#L_DeltaTildeSpectrum01){reference-type="ref" reference="L_DeltaTildeSpectrum01"}. Then $\alpha_k(X) = 2a$ if and only if there is a constant $C>0$ such that for all $n\in \mathbb{N}$, $$C_{1,q}^{-n} \left( b^{(2)}(d_{k+1}^*) + C^{-1}n^{-a}\right) \quad \leq \quad p_q(n) \quad\leq\quad C_{1,q}^{-n}\left(b^{(2)}(d_{k+1}^*) + Cn^{-a}\right).$$ * *Proof.* Since by Lemma [\[L_DeltaTildeSpectrum01\]](#L_DeltaTildeSpectrum01){reference-type="ref" reference="L_DeltaTildeSpectrum01"}, $\sigma(\widetilde{\Delta^\mathrm{up}_{q,k}}) \subset [0,1]$ and by construction $\widetilde{\Delta^\mathrm{up}_{q,k}} = \mathop{\mathrm{Id}}- \widetilde{B_q}$, it follows that also $\sigma(\widetilde{B_q})\subset [0,1]$. Therefore, $$(1-\lambda)^n\chi_{[1-\lambda,1]}(\widetilde{B_q}) \quad \leq \quad \widetilde{B_q}^n \quad \leq \quad (1-\lambda)^n\chi_{[0,1-\lambda]}(\widetilde{B_q}) + \chi_{[1-\lambda,1]}(\widetilde{B_q}).$$ Taking traces using Lemma [\[L_trBq_F\]](#L_trBq_F){reference-type="ref" reference="L_trBq_F"} and denoting $\widetilde{p}_q(n) = \mathop{\mathrm{tr}}_{\mathcal{N}G}(\widetilde{B_q}^n)$ yields $$(1-\lambda)^n F(\widetilde d_{k+1}^*)(\sqrt \lambda) \quad \leq \quad \widetilde{p}_q(n) \quad \leq \quad (1-\lambda)^n + F(\widetilde d_{k+1}^*)(\sqrt \lambda).$$ By rearranging these terms and taking logarithms, we obtain the inequalities $$\begin{aligned} \label{Eq_LckRWEqs_1} \frac{\log\left( F(\widetilde d_{k+1}^*)(\sqrt \lambda)- b^{(2)}(d_{k+1}^*)\right) }{\log \lambda} &\leq \frac{\log(\widetilde{p}_q(n)- (1-\lambda)^n b^{(2)}(d_{k+1}^*))}{\log \lambda} - n\cdot \frac{\log(1-\lambda)}{\log \lambda}, \\ \label{Eq_LckRWEqs_2} \frac{\log\left( F(\widetilde d_{k+1}^*)(\sqrt \lambda)- b^{(2)}(d_{k+1}^*)\right)}{\log \lambda} &\geq \frac{\log(\widetilde{p}_q(n)- b^{(2)}(d_{k+1}^*) - (1-\lambda)^n)}{\log \lambda}. \end{aligned}$$ Using $b^{(2)}(d_{k+1}^*) = b^{(2)}(\widetilde d_{k+1}^*)$ and taking the limit inferior for $\lambda\searrow 0$ on the left-hand-sides yields $$\begin{aligned} \liminf_{\lambda\searrow 0} \frac{\log\left( F(\widetilde d)(\sqrt \lambda)- b^{(2)}(d_{k+1}^*)\right)}{\log \lambda} &= \frac{\alpha(\widetilde d_{k+1}^*)}{2} = \frac{\alpha(\widetilde d_{k+1})}{2} = \frac{\alpha_k(X)}{2}.\end{aligned}$$ After substituting $p(n) = \widetilde{p}_q(n)- b^{(2)}(d_{k+1}^*)$, the term on the right-hand-side of Equation ([\[Eq_LckRWEqs_2\]](#Eq_LckRWEqs_2){reference-type="ref" reference="Eq_LckRWEqs_2"}) agrees with the term in [@LckL2 Thm. 2.48], so by the same argument $$\alpha_k(X) \leq 2a \qquad \text{if } \widetilde{p}_q(n)\geq b^{(2)} (d_{k+1}^*) + Dn^{-a} \text{ for } n\geq 1,$$ for some constant $D>0$. For the right-hand-side of Equation ([\[Eq_LckRWEqs_1\]](#Eq_LckRWEqs_1){reference-type="ref" reference="Eq_LckRWEqs_1"}), let $\varepsilon>0$ be arbitrarily small and $n=n(\lambda)$ the largest integer such that $n\leq \lambda^{-\varepsilon}$, that is $n=\lfloor \lambda^{-\varepsilon}\rfloor$. If $\widetilde{p}(n)\geq Cn^{-a} + b^{(2)}(d)$ for some constant $C>0$ and $n\geq 1$, we obtain $$\begin{aligned} &\frac{\log(\widetilde{p}_q(n)- (1-\lambda)^n b^{(2)}(d_{k+1}^*))}{\log \lambda} - n\cdot \frac{\log(1-\lambda)}{\log \lambda} \\ &\qquad\geq \frac{\log(Cn^{-a} + \left[1- (1-\lambda)^n\right] b^{(2)}(d_{k+1}^*))}{\log \lambda} - \frac{\log(1-\lambda)}{\lambda^\varepsilon \log \lambda} \\ &\qquad\geq \frac{\log(Cn^{-a})}{\log \lambda} - \frac{\log(1-\lambda)}{\lambda^\varepsilon \log \lambda},\end{aligned}$$ where we use $\left[1- (1-\lambda)^n\right] b^{(2)}(d_{k+1}^*) \geq 0$ (indeed, even $\left[1- (1-\lambda)^n\right] b^{(2)}(d_{k+1}^*) \xrightarrow{\lambda\searrow 0} 1-e^{-\varepsilon}$). From here, we proceed precisely as in [@LckL2 Thm. 2.48] and find $$\alpha_k(X) \geq 2a \qquad \text{if } \widetilde{p}_q(n)\leq b^{2}(d_{k+1}^*) + Cn^{-a}\text{ for } n\geq 1,$$ concluding the proof of the theorem. ◻ **Remark 61**. This generalises the theorem in degree zero, since in degree zero we have $d_+=d_{+,2}=|S|$, where $|S|$ is the size of a finite generating set of $G$ chosen in the construction of $\mathop{\mathrm{Cayley}}(G)$, and $d_-=1$. Thus $C_{1,q} = 1$ and the exponential decay factor $C_{1,q}^{-n} = 1$ disappears. **Remark 62** (). This generalises the theorem in degree zero, since in degree zero we have $d_+=d_{+,2}=|S|$, where $|S|$ is the size of a finite generating set of $G$ chosen in the construction of $\mathop{\mathrm{Cayley}}(G)$, and $d_-=1$. Thus $C_{1,q} = 1$ and the exponential decay factor $C_{1,q}^{-n} = 1$ disappears. **Example 63**. Let $k\geq 2$ and let $G$ be a finitely generated group with Cayley graph $\mathop{\mathrm{Cayley}}(G)$. Construct a $G$-CW complex $X$ in the following way. - Start with $X^{(1)} = \mathop{\mathrm{Cayley}}(G)$.[^12] - For every $g\in G$ glue a $k$-cell $\alpha_g$ to $X^{(1)}$ by collapsing the boundary of $\alpha_g$ to the vertex $v_g$ corresponding to $g\in G$ in $\mathop{\mathrm{Cayley}}(G)$. This defines $X^{(k)}$. - For every edge $(g,gs)$ in the Cayley graph, glue one $(k+1)$-cell $\beta_{g,gs}$ to $X^{(k)}$ by sending the boundary of $\beta_{g,gs}$ to $\alpha_g\cup (g,gs) \cup \alpha_{gs}$ such that $[\beta_{g,gs}:\alpha_g] = -[\beta_{g,gs}:\alpha_{gs}] \in \{\pm 1\}$. This defines $X^{(k+1)} = X$. On $X$, the degree $k$-upper random walk $\mathfrak{R}^k$ agrees with the random walk $\mathfrak{R}$ on $\mathop{\mathrm{Cayley}}(G)$ when identifying the state corresponding to $\alpha_g = (\alpha_g)_+$ in $\mathfrak{R}^k$ with the state corresponding to $g$ in $\mathfrak{R}$. In particular, for $\mathfrak{R}^k$ we have $p_-(n) \equiv 0$ so that $p(n) = p_+(n)$ is the usual return probability. Further, the values $d_+=d_{2,+}=|S|$ and $d_-=1$ agree with the values on $\mathop{\mathrm{Cayley}}(G)$, so that $C_{1,q}=1$. Therefore, the previous theorem tells us that for $X$ we obtain $\alpha_k(X) = \alpha_0(X)$. Indeed, we can also see this in a different way because $d_{k+1}d_{k+1}^*$ and $d_1d_1^*$ are unitarely equivalent by identifying $\alpha_g$ with $g$ and $\beta_{g,gs}$ with $(g,gs)$. **Example 64** (). Let $k\geq 2$ and let $G$ be a finitely generated group with Cayley graph $\mathop{\mathrm{Cayley}}(G)$. Construct a $G$-CW complex $X$ in the following way. - Start with $X^{(1)} = \mathop{\mathrm{Cayley}}(G)$.[^13] - For every $g\in G$ glue a $k$-cell $\alpha_g$ to $X^{(1)}$ by collapsing the boundary of $\alpha_g$ to the vertex $v_g$ corresponding to $g\in G$ in $\mathop{\mathrm{Cayley}}(G)$. This defines $X^{(k)}$. - For every edge $(g,gs)$ in the Cayley graph, glue one $(k+1)$-cell $\beta_{g,gs}$ to $X^{(k)}$ by sending the boundary of $\beta_{g,gs}$ to $\alpha_g\cup (g,gs) \cup \alpha_{gs}$ such that $[\beta_{g,gs}:\alpha_g] = -[\beta_{g,gs}:\alpha_{gs}] \in \{\pm 1\}$. This defines $X^{(k+1)} = X$. On $X$, the degree $k$-upper random walk $\mathfrak{R}^k$ agrees with the random walk $\mathfrak{R}$ on $\mathop{\mathrm{Cayley}}(G)$ when identifying the state corresponding to $\alpha_g = (\alpha_g)_+$ in $\mathfrak{R}^k$ with the state corresponding to $g$ in $\mathfrak{R}$. In particular, for $\mathfrak{R}^k$ we have $p_-(n) \equiv 0$ so that $p(n) = p_+(n)$ is the usual return probability. Further, the values $d_+=d_{2,+}=|S|$ and $d_-=1$ agree with the values on $\mathop{\mathrm{Cayley}}(G)$, so that $C_{1,q}=1$. Therefore, the previous theorem tells us that for $X$ we obtain $\alpha_k(X) = \alpha_0(X)$. Indeed, we can also see this in a different way because $d_{k+1}d_{k+1}^*$ and $d_1d_1^*$ are unitarely equivalent by identifying $\alpha_g$ with $g$ and $\beta_{g,gs}$ with $(g,gs)$. ## Example: Degree 1-Upper Random Walk on $\mathbb{R}^2$ Consider $\mathbb{R}^2$ as a $\mathbb{Z}^2$-CW complex of finite type as shown on the right, with arrows indicating the chosen preferred orientation. For notation's sake, we will write $\mathbb{Z}^2$ as a multiplicative group with unit element $1\in \mathbb{Z}^2$. Let $x$ and $y$ be two generators of $\mathbb{Z}^2=\langle x,y \:|\: [x,y]=1\rangle$ and the $\mathbb{Z}^2$-action on this CW complex be generated by $x$ shifting to the right by one and $y$ shifting up by one. The red cells indicate $\mathbb{Z}^2$-bases. We will denote the $0$-basis $\mathcal{B}_0 = \{\gamma_\bullet\}$, the $1$-basis $\mathcal{B}_1 = \{\alpha_\uparrow, \alpha_\rightarrow\}$ and the $2$-basis $\mathcal{B}_2 = \{\beta_{\circlearrowright}\}$ in the way suggested by the indices. Given a cell $c$ and $g=x^ay^b\in \mathbb{Z}^2$, we denote by $gc$ the cell obtained by translating $c$ by $g$, that is $a$ units to the right and $b$ units up. The incidence numbers between a $2$-cell $\beta$ and a $1$-cell $\alpha$ are given by $[\beta:\alpha] = 0$ if $\beta$ and $\alpha$ do not touch and $[\beta:\alpha] = \pm 1$ if the cells touch; with sign $+1$ if the orientation $\beta$ induces on $\alpha$ agrees with the orientation on $\alpha$ and $-1$ otherwise. This is an upper $2$-regular CW complex with $$\begin{aligned} d_{+} = 2, \ d_{+,2} = 2, \ d_- = 3, \quad C_{1,q} = \frac{3}{2q+1}, \quad C_{2,q} = \frac{1}{2}\frac{1-q}{2q+1}, \quad C_{1,q}^{-1}C_{2,q} = \frac{1-q}{6}.\end{aligned}$$ The upper Laplacian $\Delta = \Delta^\mathrm{up}_1\curvearrowright\ell^2 \left((\mathbb{R}^2)^{(1)}\right)$ in degree one can be written, with respect to the basis $\mathcal{B}_1$, as the $\mathbb{C}[\mathbb{Z}^2]$-valued matrix $$\Delta = 2- \begin{pmatrix} x+x^{-1} & 1-x-y^{-1}+xy^{-1} \\ 1-x^{-1}-y+x^{-1}y & y+y^{-1} \end{pmatrix}.$$ For the non-lazy random walk on $1$-cells, on $\mathcal{B}_1$ the propagation operator is given as described in Figure [\[F:R2Z2CW_Moves\]](#F:R2Z2CW_Moves){reference-type="ref" reference="F:R2Z2CW_Moves"}. Accounting for changing orientations with signs, this means we can write the corresponding operator $B=TPI$ with respect to $\mathcal{B}_1$ as the $\mathbb{C}[\mathbb{Z}^2]$-valued matrix $$B = \frac{1}{6}\begin{pmatrix} x+x^{-1} & 1-x-y^{-1}+xy^{-1} \\ 1-x^{-1}-y+x^{-1}y & y+y^{-1} \end{pmatrix}.$$ We can readily verify that for $q\in [0,1]$ indeed $$\begin{aligned} B_q &= q\mathop{\mathrm{Id}}+ (1-q)B = q \mathop{\mathrm{Id}}+ \frac{1-q}{6} \left(2\mathop{\mathrm{Id}}-\Delta\right) = C_{1,q}^{-1} \mathop{\mathrm{Id}}- C_{1,q}^{-1}C_{2,q} \Delta,\end{aligned}$$ and thus $C_{1,q} B_q = \mathop{\mathrm{Id}}- C_{2,q} \Delta$. Looking at the boundary of $\beta_\circlearrowright$ given by $$S = (1-x)\alpha_\uparrow + (y-1)\alpha_\rightarrow,$$ it is an eigenstate of $B$ with eigenvalue $\frac{1}{6}\left(x+x^{-1}+y+y^{-1}-2\right)$, compare Figure [\[F:square_ev\]](#F:square_ev){reference-type="ref" reference="F:square_ev"}. Here, $x+x^{-1}+y+y^{-1} = 4P^{\mathbb{Z}^2}$, where $P^{\mathbb{Z}^2}$ can formally also be interpreted as the propagation operator of the uniform nearest neighbour random walk on the grid $\mathop{\mathrm{Cayley}}(\mathbb{Z}^2)$ (or, in this case, rather the $2$-cells of $\mathbb{R}^2$ with this chosen CW structure). We denote $\lambda = \frac{1}{6}(4P^{\mathbb{Z}^2}-2)$. A straight-forward computation shows that $S$ is an eigenstate to $B_q$ with eigenvalue $$\lambda_q = C_{1,q}^{-1}C_{2,q}\left(4P^{\mathbb{Z}^2}+[C_{2,q}^{-1}-4]\right).$$ Since $C_{2,q}(4+[C_{2,q}^{-1}-4]) = 1$, we can set $q' = 1-4C_{2,q} = \frac{4q-1}{2q+1}$ and can formally interpret $$4C_{2,q}P^{\mathbb{Z}^2}+[1-4C_{2,q}] = P^{\mathbb{Z}^2}_{q'}$$ as the propagation operator of the corresponding $q'$-lazy random walk on $\mathop{\mathrm{Cayley}}(\mathbb{Z}^2)$. Note that for $q\in [\nicefrac{1}{4},1]$ we have $4C_{2,q}\in [0,1]$ and $q'\in [0,1]$ so this makes sense. In particular, $$\lambda_q = C_{1,q}^{-1} P^{\mathbb{Z}^2}_{q'} \quad \text{ and } \quad B_q S = C_{1,q}^{-1} P^{\mathbb{Z}^2}_{q'}S.$$ The return quantity $p_q(n)$ that we are interested in is given by $p_q(n) = p_{q,\alpha_\uparrow}(n)+p_{q,\alpha_\rightarrow}(n)$, where $p_{q,\alpha_\uparrow}(n)=\langle B_q^n\alpha_\uparrow, \alpha_\uparrow\rangle$ is the coefficient of $1\alpha_{\uparrow}$ in $B_q^n\alpha_\uparrow$ and similarly for $\alpha_\rightarrow$. By symmetry, $p_{q,\alpha_\uparrow}(n)=p_{q,\alpha_\rightarrow}(n)$ so that $$p_q(n) = 2p_{q,\alpha_\uparrow}(n) = 2\langle B_q^n \alpha_\uparrow, \alpha_\uparrow\rangle.$$ We note from Figure [\[F:R2Z2CW_Moves\]](#F:R2Z2CW_Moves){reference-type="ref" reference="F:R2Z2CW_Moves"} that $$\begin{aligned} \label{Eq_BqWithC1q} B_q(\alpha_\uparrow) &= q\alpha_\uparrow + \frac{1-q}{6}\left(x^{-1}S - S +2\alpha_\uparrow \right) = C_{1,q}^{-1} \alpha_\uparrow + C_{1,q}^{-1}C_{2,q} (x^{-1}-1) S.\end{aligned}$$ Since the random walk is $\mathbb{Z}^2$-invariant, this yields $$\begin{aligned} B_q^n(\alpha_\uparrow) &= C_{1,q}^{-1}B_q^{n-1}(\alpha_\uparrow) + C_{1,q}^{-1}C_{2,q} (x^{-1}-1)B_q^{n-1} S \end{aligned}$$ Resolving this recursive formula we obtain $$\begin{aligned} B_q^n(\alpha_\uparrow) &= C_{1,q}^{-n}\alpha_\uparrow + \sum_{k=0}^{n-1} C_{1,q}^{-n+k} C_{2,q} (x^{-1}-1)B_q^{k} S\\ &= C_{1,q}^{-n} \left(\alpha_\uparrow + \sum_{k=0}^{n-1} C_{2,q} (x^{-1}-1)(P^{\mathbb{Z}^2}_{q'})^{k} S\right)\end{aligned}$$ In order to find the coefficient of $1\alpha_\uparrow$, we notice that $$\begin{aligned} \langle S, 1\alpha_\uparrow \rangle &= 1, \qquad \langle x^{-1}S, 1\alpha_\uparrow \rangle = -1 \quad \text{ and } \quad \langle gS, 1\alpha_\uparrow \rangle = 0\quad \text{ for } g\notin \{1,x^{-1}\}\end{aligned}$$ and therefore it follows that $$\begin{aligned} \langle (P^{\mathbb{Z}^2}_{q'})^{k} S, 1\alpha_\uparrow\rangle &= \langle (P^{\mathbb{Z}^2}_{q'})^{k}, 1-x^{-1}\rangle, \\ \langle x^{-1}(P^{\mathbb{Z}^2}_{q'})^{k} S, 1\alpha_\uparrow\rangle &= \langle (P^{\mathbb{Z}^2}_{q'})^{k}, x-1\rangle.\end{aligned}$$ Using this we obtain $$\begin{aligned} \frac{1}{2}p_q(n) &= \langle B_q^n \alpha_\uparrow, \alpha_\uparrow\rangle \\ &= C_{1,q}^{-n} \left(1 + \sum_{k=0}^{n-1} C_{2,q} \left\langle(P^{\mathbb{Z}^2}_{q'})^{k}, (x-1)-(1-x^{-1})\right\rangle \right) \\ &= C_{1,q}^{-n} \left(1 + \sum_{k=0}^{n-1} C_{2,q} \left\langle(P^{\mathbb{Z}^2}_{q'})^{k}, x+x^{-1}-2\right\rangle \right).\end{aligned}$$ By symmetry, the coefficients of $(P^{\mathbb{Z}^2}_{q'})^{k}$ for $x$ and $x^{-1}$ agree, hence $$\begin{aligned} \frac{1}{2}C_{1,q}^{n}p_q(n) &= 1 - 2C_{2,q} \sum_{k=0}^{n-1} \left\langle(P^{\mathbb{Z}^2}_{q'})^{k}, 1-x\right\rangle \\ &= 1 - 2C_{2,q} \sum_{k=0}^{n-1} \left( p^{\mathbb{Z}^2}_{q'}(k) - p^{\mathbb{Z}^2}_{q'}(e\xrightarrow{k} x) \right)\end{aligned}$$ where $p^{\mathbb{Z}^2}_{q'}(k)$ is the return probability of the $q'$-lazy nearest neighbour random walk on $\mathbb{Z}^2$ after $k$ steps and $p^{\mathbb{Z}^2}_{q'}(e\xrightarrow{k} x)$ the probability of the random walk to be at the vertex $x$ after $k$ steps. If we write $\mathbb E^g_q(n)$ for the expected number of visits of the vertex $g$ in the first $n$ steps for the $q$-lazy nearest neighbour random walk on $\mathbb{Z}^2$ (counting the starting position for $\mathbb E^e_q(n)$, if $q=0$ we suppress it in notation), we can write this as $$\frac{1}{2}C_{1,q}^n p_q(n) = 1- 2C_{2,q}(\mathbb E^e_{q'}(n-1)-\mathbb E^x_{q'}(n-1)).$$ Notice that $q' = 1-4C_{2,q}$ implies that $2C_{2,q} = \frac{1-q'}{2}$. Hence, $$\frac{1}{2}C_{1,q}^n p_q(n) = 1-\frac{1-q'}{2}(\mathbb E^e_{q'}(n-1)-\mathbb E^x_{q'}(n-1)).$$ For $q<1$ large enough, we expect that $$\mathbb E^e_{q'}(n-1)-\mathbb E^x_{q'}(n-1) \sim 1 - \Theta(n^{-1}) \qquad \text{for } n\to\infty.$$ Plugging this back into the equation above, this would imply that $$p_q(n) \sim C^{-n}_{1,q}\left(1 + \Theta(n^{-1})\right) \qquad \text{for } n\to\infty.$$ Here, we can read off $b^{(2)}(d_2^*) = 1$, corresponding to the kernel of $d_2^*$ of $\mathcal{N}G$-dimension one, and the Novikov-Shubin invariant $\alpha_1(\mathbb{R}^2) = \alpha(d_2^*) = 2$. [^1]: This work is part of the author's doctorial thesis at the University of Göttingen. [^2]: Let $G$ be a group and $X$ a CW complex with cellular left action $G\curvearrowright X$. Then $X$ is called a free $G$-CW complex of finite type if the projection $X\twoheadrightarrow {\left.\raisebox{-.15em}{$G$}\middle\backslash\raisebox{.05em}{$X$}\right.}$ is a regular covering and ${\left.\raisebox{-.15em}{$G$}\middle\backslash\raisebox{.05em}{$X$}\right.}$ is a finite CW complex. In particular, the finite CW complex ${\left.\raisebox{-.15em}{$G$}\middle\backslash\raisebox{.05em}{$X$}\right.}$ comes with a CW structure such that there are only finitely many $k$-cells in ${\left.\raisebox{-.15em}{$G$}\middle\backslash\raisebox{.05em}{$X$}\right.}$, exactly one for $G$-type of $k$-cells in $X$. [^3]: $d_-(\beta; \alpha)=d_-(g.\beta; g.\alpha)$ for all $g\in G$. This, and the application in mind, are the reason we restrict to these complexes. The construction makes sense as long as this maximum exists. [^4]: Since in the following, orientations on $(k+1)$-cells will never play a role, we will only take care of orientations on the $k$-cells and work with the preferred orientation on $(k+1)$-cells throughout. [^5]: $d_-(\beta; \alpha)=d_-(g.\beta; g.\alpha)$ for all $g\in G$. This, and the application in mind, are the reason we restrict to these complexes. The construction makes sense as long as this maximum exists. [^6]: Since in the following, orientations on $(k+1)$-cells will never play a role, we will only take care of orientations on the $k$-cells and work with the preferred orientation on $(k+1)$-cells throughout. [^7]: Here, we introduce an extra minus sign to mirror what happens in the case of graphs. There, random walkers can walk from a vertex $v_1$ along an (oriented) edge $e=(v_1,v_2)$ to the vertex $v_2$, where $e$ is an (outgoing) edge for $v_1$ with $[e:v_1]=-1$ and an (incoming) edge for $v_2$ with $[e:v_2]=1$, so that $[e:v_1][e:v_2]=-1$ if $e=\{v_1,v_2\}$. With this extra minus sign, the quantity $d(v_1,v_2,e) = 1$ is positive in this case. [^8]: This orientation of $\beta_1$ has no impact on the formulas in the end and is thus suppressed in the formal definition. [^9]: This orientation of $\beta_1$ has no impact on the formulas in the end and is thus suppressed in the formal definition. [^10]: While it is not clear which, if any, physical process this operator $B$ describes, cancellation between different objects does happen in physics. For example, studying fermions via the Dirac equation suggests that for every particle there is a corresponding anti-particle, such as electrons and positrons. If they meet, they will annihilate each other. [^11]: In the sense that both are operators $\Xi\curvearrowright\ell^2C_k^\mathrm{cell}(X)$ that can be written as $$\Xi\colon \sum_{\alpha\in I_k} \lambda_\alpha \alpha \mapsto \sum_{\alpha'\in I_k}\left[\sum_{\alpha\in I_k} \Xi_{\alpha',\alpha} \lambda_{\alpha}\right]\cdot {\alpha'}.$$ [^12]: If $k\geq 3$ and $G$ is finitely presented, we can further glue in $2$-cells according to the relations in $G$, so that $X^{(2)}$ is the Cayley complex of $G$. In that case the constructed CW complex $X$ satisfies $\pi_1(X)=G$, see for example A. Hatcher's book [@Hatcher p. 77]. [^13]: If $k\geq 3$ and $G$ is finitely presented, we can further glue in $2$-cells according to the relations in $G$, so that $X^{(2)}$ is the Cayley complex of $G$. In that case the constructed CW complex $X$ satisfies $\pi_1(X)=G$, see for example A. Hatcher's book [@Hatcher p. 77].

arxiv_math

--- abstract: | We begin the study of Lipschitz saturation for germs of toric singularities. By looking at their associated analytic algebras, we prove that if $(X,0)$ is a germ of toric singularity with smooth normalization then its Lipschitz saturation is again toric. Finally we show how to calculate the Lipschitz saturation for some families of toric singularities starting from the semigroup that defines them. author: - Daniel Duarte, Arturo E. Giles Flores title: On the Lipschitz Saturation of Toric Singularities --- # Introduction {#introduction .unnumbered} For a germ $(X,0) \subset (\mathbb C^n,0)$ of reduced complex analytic singularity, the algebra of germs of Lipschitz meromorphic functions is an analytic algebra that sits between $O_{X,0}$ and its normalization $\overline{O_{X,0}}$. It was first studied by Pham and Teissier in [@Ph-Te20] and its associated germ $(X^s,0)$ is important in the study of biLipschitz equisingularity. The case of curves is pretty well understood and described in [@BGG80], [@Fe03], [@GiSiSn22], [@GiNoTe20], [@NePi14], [@Ph-Te20]. However not much is known in higher dimensions. In this work we begin the study of the Lipschitz saturation for toric singularities of arbitrary dimension. Our study is inspired by the fact that the Lipschitz saturation of a germ of irreducible curve is a toric curve, i.e., is parametrized by monomials. Moreover, there exists an explicit procedure for calculating its corresponding numerical semigroup starting from the semigroup of the curve. Although there is no reason to believe that the Lipschitz saturation of a germ of arbitrary dimension has toric structure, we may still ask the following question: starting with a toric singularity, is the Lipschitz saturation also toric? If so, is there a procedure for computing the corresponding affine semigroup from the semigroup of the toric singularity? We answer the first question in the situation that naturally generalizes the case of toric curves, that is, toric varieties having smooth normalization. Regarding the question of the description of the corresponding semigroup, we provide an answer for some families of toric singularities. As a consequence, we exhibit examples showing that some properties of Lipschitz saturation of curves are no longer true in higher dimension. For instance, the embedding dimension has a different behaviour. The paper is divided as follows. In section [1](#secLips){reference-type="ref" reference="secLips"} we recall the general construction of Lipschitz saturation and its main properties. We take a quick look at the case of curves and finish by proving that if a germ $(Y,0)$ can be seen as the generic linear projection of another germ $(X,0)$, then their Lipschitz saturations are isomorphic (proposition [Proposition 4](#ProyGen){reference-type="ref" reference="ProyGen"}). In section [2](#ToricSing){reference-type="ref" reference="ToricSing"} we establish what we mean by a toric singularity $(X,0)$, which is basically taking the germ at the origin of an affine toric variety. We summarize some important facts about the passage from the algebraic toric variety to the germ of analytic space. The results of this section are essentially known (see, for instance, [@GP00b]). We include them here for the sake of completeness. In section [3](#SatofTor){reference-type="ref" reference="SatofTor"} we prove our first main theorem: the Lipschitz saturation $(X^s,0)$ of a toric singularity whose normalization is smooth is again toric (theorem [Theorem 12](#SatTor){reference-type="ref" reference="SatTor"}). The key idea towards this result is that for every $f\in\mathcal{O}_{X,0}^s$ all of its monomials are also in $\mathcal{O}_{X,0}^s$. As a first application of our result, we show that the Whitney Umbrella singularity is Lipschitz saturated (see example [Example 15](#WU){reference-type="ref" reference="WU"}). In section [4](#TorSurf){reference-type="ref" reference="TorSurf"} we use the tools previously developed to give a combinatorial description of the Lipschitz saturation $(X^s,0)$ for some families of toric singularities. More concretely, given the semigroup $\Gamma$ corresponding to the toric singularity $(X,0)$, we explicitly describe the semigroup $\Gamma^s$ of its Lipschitz saturation (see proposition [Proposition 16](#ProdCurv){reference-type="ref" reference="ProdCurv"} and theorem [Theorem 20](#CalculoLip){reference-type="ref" reference="CalculoLip"}). As a byproduct we can compute the embedding dimension of $(X^s,0)$ in these cases (see corollaries [Corollary 17](#edim-mult prod){reference-type="ref" reference="edim-mult prod"} and [Corollary 21](#edim-mult hypersurf){reference-type="ref" reference="edim-mult hypersurf"}). We conclude by illustrating all these results in several examples. # Lipschitz saturation of complex analytic germs {#secLips} The definition of Lipschitz saturation of a reduced complex analytic algebra $\mathcal{O}_{X,0}$ is based on the concept of integral dependence on an ideal. Given an element $r$ and an ideal $I$ in a ring $R$, we say that $\mathit{r}$ *is integral over* $\mathit{I}$ if $r$ satisfies a relation of the form $$r^m + a_1r^{m-1}+ a_2r^{m-2}+\cdots +a_{m-1}r + a_m=0,$$ for some integer $m>0$, with $a_j \in I^j$ for $j=1,2,\ldots,m$. The set $\overline{I}$ consisting of those elements of $R$ which are integral over $I$ is an ideal called the *integral closure of* $I$ in $R$ (see [@HS06], [@LT08]).\ Let us assume for simplicity that $(X,0)$ is irreducible and let $n^*: \mathcal{O}_{X,0}\hookrightarrow \overline{\mathcal{O}_{X,0}}$ be the integral closure of $\mathcal{O}_{X,0}$ in its field of fractions. Geometrically, this corresponds to the normalization map $n:(\overline{X},0) \to (X,0)$ and if we consider the holomorphic map $$\left( \overline{X} \times_X \overline{X}, (0,0)\right) \hookrightarrow \left( \overline{X} \times \overline{X}, (0,0), \right)$$ we get a surjective map of analytic algebras $$\Psi: \overline{\mathcal{O}_{X,0}} \widearc{\otimes}_\mathbb C\overline{\mathcal{O}_{X,0}} \longrightarrow \overline{\mathcal{O}_{X,0}} \widearc{\otimes}_{\mathcal{O}_{X,0}} \overline{\mathcal{O}_{X,0}},$$ where $\widearc{\otimes}$ denotes the analytic tensor product which is the operation on the analytic algebras that corresponds to the fibre product of analytic spaces (for more details see [@Ada12] and [@GiNoTe20]). **Definition 1**. Let $I_\Delta$ be the kernel of the morphism $\Psi$ above. We define the Lipschitz saturation $\mathcal{O}_{X,0}^s$ of $\mathcal{O}_{X,0}$ as the algebra $$\mathcal{O}_{X,0}^s:= \left\{ f \in \overline{\mathcal{O}_{X,0}} \, | \, f \widearc{\otimes}_\mathbb C1 - 1\widearc{\otimes}_\mathbb Cf \in \overline{I_\Delta}\right\}.$$ **Remark 2**. A detailed discussion of the following facts can be found in [@Ph-Te20; @GiNoTe20; @Tei82]: 1. $\mathcal{O}_{X,0}^s$ is an analytic algebra and coincides with the ring of germs of meromorphic functions on $(X,0)$ which are locally Lipschitz with respect to the ambient metric. 2. We have injective ring morphisms $$\mathcal{O}_{X,0}\hookrightarrow \mathcal{O}_{X,0}^s \hookrightarrow \overline{\mathcal{O}_{X,0}}.$$ 3. The corresponding Lipschitz saturation map $$\zeta:(X^s,0) \to (X,0)$$ is a biLipschitz homeomorphism, induces an isomorphism outside the non-normal locus of $X$ and preserves the multiplicity, i.e. $$\textrm{mult }(X^s,0) = \textrm{ mult }(X,0).$$ Moreover, it can be realized as a generic linear projection in the sense of [@GiNoTe20 Def. 8.4.2]. 4. $\overline{\mathcal{O}_{X,0}^s} =\overline{\mathcal{O}_{X,0}}$ and the holomorphic map induced by $\mathcal{O}_{X,0}^s \hookrightarrow \overline{\mathcal{O}_{X,0}}$, $$\overline{X}\stackrel{n_s}{\longrightarrow} X^s,$$ is the normalization map of $X^s$. Moreover the map $$n=\zeta \circ n_s : \overline{X} \to X$$ is the normalization map of $X$. Aside from these facts, little else is known in the general case. However, in the case of curves, the saturation has some very interesting equisingularity properties. First, Pham and Teissier proved that for a *plane curve* $(X,0) \subset (\mathbb C^2,0)$ the Lipschitz saturation $\mathcal{O}_{X,0}^s$ determines and is determined by the characteristic exponents of its branches and their intersection multiplicities, in particular *the curve $(X^s,0)$ is an invariant of the equisingularity class of $(X,0)$* (See [@Ph-Te20] and [@BGG80 Prop. VI.3.2]).\ In the irreducible case (branches) they also prove that the *the curve $(X^s,0)$ is always a toric curve,* (i.e parametrized by monomials), and if we have a parametrization of $(X,0) \subset (\mathbb C^2,0)$ $$t \longmapsto \left(\varphi_1(t),\varphi_2(t)\right); \hspace{0.2in} \varphi_1,\varphi_2 \in \mathbb C\{t\},$$ then there is a simple procedure to calculate a parametrization of $(X^s,0)$. 1. Calculate the set of characteristic exponents $E:=\{\beta_0,\beta_1,\ldots,\beta_g\}$ of $(X,0)$. 2. Calculate the smallest saturated numerical semigroup $\tilde{E} \subset \mathbb N$ containing $E$ as follows [@RG09 Chapter 3, Section 2]: $$\widetilde{E_0}:= E \cup \beta_0 \cdot \mathbb N;$$ $$\widetilde{E_1}:= \widetilde{E_0} \cup \left\{ \beta_1 + ke_1 \,| \, k\in \mathbb N\right\}, \hspace{0.2in} e_1=\textrm{ gcd }\{\beta_0,\beta_1\};$$ $$\widetilde{E_{j+1}}:= \widetilde{E_j} \cup \left\{ \beta_{j+1} + ke_{j+1} \,| \, k\in \mathbb N\right\}, \hspace{0.2in} e_{j+1}=\textrm{ gcd }\{e_j,\beta_{j+1}\};$$ $$\tilde{E}=\widetilde{E_g}.$$ 3. If $\{ a_1, \ldots, a_{\beta_0}\}$ is the minimal system of generators of $\tilde{E}$ then $$t \longmapsto \left(t^{a_1}, \ldots, t^{a_{\beta_0}}\right)$$ is a parametrization of $(X^s,0)$. Recall that in this setting $\beta_0$ is the multiplicity of $(X,0)$ and so we get that *the embedding dimension of $(X^s,0)$ is equal to the multiplicity of $(X,0)$.* These results extend well beyond the plane curve case: in [@BGG80 Thm VI.0.2, Prop. VI.3.1] the authors prove that for a curve $(X,0) \subset (\mathbb C^n,0)$, and a generic lineal projection $\pi: (\mathbb C^n,0) \to (\mathbb C^2,0)$ the Lipschitz saturations $\mathcal{O}_{X,0}$ of $(X,0)$ and $O_{\pi(X),0}^s$ of the plane curve $(\pi(X),0) \subset (\mathbb C^2,0)$ are isomorphic. Even more, by [@GiSiSn22 Thm. 4.12] we get that two germs of curves $(X,0)$ and $(Y,0)$ are bi-Lipschitz equivalent if and only if their Lipschitz saturations are isomorphic. In this sense *the saturated curve $(X^s,0)$ can be seen as a canonical representative of the bi-Lipschitz equivalence class of $(X,0)$.* **Example 3**. Let $(X,0) \subset (\mathbb C^3,0)$ be the space curve with normalization map $$\begin{aligned} \eta: (\mathbb C,0) & \longrightarrow (X,0) \\ t &\longmapsto (t^6,t^{11}-t^9, t^{11}+t^9). \end{aligned}$$ For this curve, the projection on the first two coordinates is generic, giving us the plane curve $$t \longmapsto (t^6,t^{11}-t^9)$$ with characteristic exponents $E=\{6,9,11\}$. Following the procedure described above, we get the saturated numerical semigroup $\tilde{E}$ with minimal system of generators $\{6,9,11,13,14,16\}$. This determines a parametrization of the saturated curve $(X^s,0) \subset (\mathbb C^6,0)$ of the form: $$\begin{aligned} \eta^s:(\mathbb C,0) &\longrightarrow (X^s,0) \\ \tau &\mapsto (\tau^6,\tau^9,\tau^{11},\tau^{13},\tau^{14},\tau^{16}).\end{aligned}$$ Going back to the general case, we can prove that, just as in the case of curves, the Lipschitz saturation remains unchanged under generic linear projections. **Proposition 4**. *Let $(X,0) \subset (\mathbb C^n,0)$ be a germ of reduced and irreducible singularity, and let $\pi: (\mathbb C^n,0) \to (\mathbb C^m,0)$ be a generic linear projection with respect to $(X,0)$. Then $(X,0)$ and its image germ $(\pi(X),0)$ have isomorphic Lipschitz saturations, i.e.: $$\mathcal{O}_{X,0}^s \cong \mathcal{O}_{\pi(X),0}^s.$$* Before going through the proof, recall that the cone $C_5(X,0)$, constructed by taking limits of bi-secants to $X$ at $0$, is an algebraic cone defined by H. Whitney in [@Whi65]. A linear projection $\pi: (\mathbb C^n,0) \to (\mathbb C^m,0)$ with kernel $D$ is called $C_5$-general (or generic) with respect to $(X,0)$ if it is transversal to the cone $C_5(X,0)$, meaning $D \bigcap C_5(X,0) = \{ 0\}$. When $\pi$ is generic, it induces a homeomorphism between $(X,0)$ and its image $(\pi(X),0)$, and these two germs have the same multiplicity, for a detailed explanation see [@GiNoTe20 Section 8.4]. *Proof.* (of proposition [Proposition 4](#ProyGen){reference-type="ref" reference="ProyGen"})\ After a linear change of coordinates we can assume that the linear projection $\pi:(\mathbb C^n,0) \to (\mathbb C^m,0)$ is the projection on the first $m$ coordinates $(z_1,\ldots,z_n) \mapsto (z_1,\ldots,z_m)$. Let $J \subset \mathcal{O}_{X \times X, (0,0)}$ denote the ideal defining the diagonal of $X \times X$ $$J= \left< z_1-w_1, \ldots, z_n-w_n\right>\mathcal{O}_{X \times X, (0,0)}$$ Denote $J_\pi=\left<z_1-w_1,\ldots,z_m-w_m\right>\mathcal{O}_{X \times X, (0,0)}$. Then by [@GiNoTe20 Proposition 8.5.11] the genericity of $\pi$ is equivalent to the equality of integral closures $\overline{J}=\overline{J_\pi}$ in $\mathcal{O}_{X \times X, (0,0)}$. On the other hand, since $\pi:(X,0) \to (\pi(X),0)$ is a finite, generically 1-1 and surjective map, then the morphism $$\pi^*:\mathcal{O}_{\pi(X),0} \to \mathcal{O}_{X,0}$$ is injective, makes $\mathcal{O}_{X,0}$ a finitely generated $\mathcal{O}_{\pi(X),0}$-module and induces an isomorphism of the corresponding field of fractions $Q(\mathcal{O}_{X,0})$. All these together imply that $\mathcal{O}_{X,0}$ and $\mathcal{O}_{\pi(X),0}$ have isomorphic integral closures in $Q(\mathcal{O}_{X,0})$ and so if $\eta:(\overline{X},0) \to (X,0)$ denotes the normalization of $(X,0)$ then the composition $$(\overline{X},0) \stackrel{\eta}{\longrightarrow} (X,0) \stackrel{\pi}{\longrightarrow} (\pi(X),0)$$ is a normalization of $(\pi(X),0)$. Note that the ideal $I_\Delta$ of definition [Definition 1](#DefSat){reference-type="ref" reference="DefSat"} is defined by the "coordinate functions\" of the normalization map $$(\overline{X},0) \stackrel{\eta}{\longrightarrow} (X,0) \stackrel{\pi}{\longrightarrow} (\pi(X),0)$$ $$\underline{y} \mapsto \left( \eta_1(y), \ldots, \eta_n(y) \right) \mapsto \left( \eta_1(y), \ldots, \eta_m(y) \right),$$ that is (see [@GiNoTe20 Section 8.5.2]), $$I_{\Delta_X} = \left< \eta_1(y) -\eta_1(x), \ldots, \eta_n(y)-\eta_n(x)\right>\mathcal{O}_{\overline{X} \times \overline{X}, (0,0)}$$ $$I_{\Delta_{\pi(X)}} = \left< \eta_1(y) -\eta_1(x), \ldots, \eta_m(y)-\eta_m(x)\right>\mathcal{O}_{\overline{X} \times \overline{X}, (0,0)}.$$ To prove the desired result it is enough to prove the equality of the integral closures $\overline{I_{\Delta_X} } = \overline{I_{\Delta_{\pi(X)}} }$. But the germ map: $$\eta \times \eta:\left( \overline{X} \times \overline{X}, (0,0)\right) \longrightarrow \left( X \times X, (0,0) \right)$$ induces a morphism of analytic algebras: $$(\eta \times \eta)^* : \mathcal{O}_{X \times X, (0,0)} \longrightarrow \mathcal{O}_{\overline{X} \times \overline{X}, (0,0)}$$ such that: $$\left< \left( \eta \times \eta\right)^*(J) \right> = I_{\Delta_X}$$ $$\left< \left( \eta \times \eta\right)^*(J_\pi) \right> = I_{\Delta_{\pi(X)}}$$ and since $\overline{J}=\overline{J_\pi}$ in $\mathcal{O}_{X \times X, (0,0)}$ the result follows. ◻ On the downside, it will no longer be true in general that bi-Lipschitz equivalent germs will have isomorphic Lipschitz saturations. This is because a germ $(X,0)$ and its Lipschitz saturation $(X^s,0)$ always have the same multiplicity, however in [@BFSV20] the authors prove that in dimension bigger than two, multiplicity of singularities is not a bi-Lipschitz invariant. # Toric singularities {#ToricSing} In this section we establish what we mean by a toric singularity. We also introduce the notation we use regarding toric varieties. Let $\mathcal{A}=\{\gamma_1,\ldots,\gamma_n\}\subset\mathbb Z^d$, $\Gamma=\mathbb N\mathcal{A}=\big\{\sum_i a_i\gamma_i|a_i\in\mathbb N\big\}$, and $\check{\sigma}=\mathbb R_{\geq0}\mathcal{A}$. Assume that the group generated by $\mathcal{A}$ is $\mathbb Z^d$ and that $\check{\sigma}$ is a strongly convex cone. Consider the following homomorphism of semigroups, $$\begin{aligned} \pi:&\mathbb N^n\to\Gamma\notag\\ &\alpha\mapsto \sum_i \alpha_i\gamma_i,\notag\end{aligned}$$ and the induced $\mathbb C-$algebra homomorphism, $$\begin{aligned} \varphi:&\mathbb C[z_1,\ldots,z_n]\to\mathbb C[t_1^{\pm},\ldots,t_d^{\pm}]\notag\\ &\hspace{1.5cm}z_i \mapsto t^{\gamma_i}=t_1^{\gamma_{i,1}}\cdots t_d^{\gamma_{i,d}}\notag\\ &\hspace{1.5cm}z^{\alpha} \mapsto t^{\pi(\alpha)}.\notag\end{aligned}$$ Let $I_{\Gamma}=\ker\varphi$. Recall that $I_{\Gamma}$ is a prime ideal and $I_{\Gamma}=\langle z^{\alpha}-z^{\beta}|\pi(\alpha)=\pi(\beta)\rangle.$ Let $X\subset\mathbb C^n$ be the affine variety defined by $I_{\Gamma}$. Then $X$ is a $d$-dimensional affine toric variety containing the origin. Let $\mathbb C[X]$ be the ring of regular functions on $X$ and $\mathbb C[\Gamma]$ the $\mathbb C$-algebra of the semigroup $\Gamma$. Recall that $\mathbb C[X]\cong\mathbb C[\Gamma]=\mathbb C[t^{\gamma_1},\ldots,t^{\gamma_n}]\cong\mathbb C[z_1,\ldots,z_n]/I_{\Gamma}$ [@CLS11 Chapter 1]. Next we discuss some basic results on the passage from the algebraic toric variety $X$ to the germ of analytic space $(X,0)$. Throughout this section we use the following notation. - $\mathbb C[[\Gamma]]$ denotes the ring of formal power series with exponents in $\Gamma$. - $\mathbb C\{\Gamma\}$ denotes the subring of $\mathbb C[[\Gamma]]$ consisting of convergent series in a neighborhood of $0\in X$. - $\mathcal{O}_{X,0}$ denotes the algebra of germs of holomorphic functions on $0\in X$. **Remark 5**. Notice that $\mathbb C[[\Gamma]]$ is indeed a ring since $\Gamma$ is contained in a strongly convex cone which implies that every element of $\Gamma$ can be written as a sum of elements of $\Gamma$ in finitely many different ways. **Lemma 6**. *With the previous notation, $$\mathcal{O}_{X,0}\cong\mathbb C\{\Gamma\}\cong\mathbb C\{z_1,\ldots,z_n\}/I_{\Gamma}\mathbb C\{z_1,\ldots,z_n\}.$$* *Proof.* The first isomorphism is proved in [@GP00b Lemme 1] or [@GP00 Lemma 1.1]. The proof given there is for normal toric varieties. However, the same proof holds in the non-normal case. We prove the second isomorphism. Consider the following exact sequence: $$\xymatrix{0\ar[r]&I_{\Gamma}\ar[r]&\mathbb C[z_1,\ldots,z_n]\ar[r]^{\varphi}&\mathbb C[\Gamma]\ar[r]&0}.$$ Let $\mathfrak{m}=\langle z_1,\ldots,z_n \rangle$. Taking completions with respect to $\mathfrak{m}$ we obtain the following exact sequence: $$\xymatrix{0\ar[r]&\hat{I_{\Gamma}}\ar[r]&\mathbb C[[z_1,\ldots,z_n]]\ar[r]^{\varphi_f}&\mathbb C[[\Gamma]]\ar[r]&0}.$$ On the other hand, $\hat{I_{\Gamma}}\cong I_{\Gamma}\mathbb C[[z_1,\ldots,z_n]]$ [@AM Proposition 10.15]. Denote $\varphi_a=\varphi_f|_{\mathbb C\{z_1,\ldots,z_n\}}$. In the proof of [@GP00 Lemma 1.1], alternatively [@GP00b Lemme 1] it is shown that $\mbox{Im }\varphi_a=\mathbb C\{\Gamma\}$. It remains to prove that $\ker\varphi_a=I_{\Gamma}\mathbb C\{z_1,\ldots,z_n\}$. Since $\ker \varphi_f=I_{\Gamma}\mathbb C[[z_1,\ldots,z_n]]$, it follows that $I_{\Gamma}\mathbb C\{z_1,\ldots,z_n\}\subset\ker\varphi_a$. Let $F\in\mathbb C\{z_1,\ldots,z_n\}$ be such that $0=\varphi_a(F)=\varphi_f(F)$. We conclude that $F\in\mathbb C\{z_1,\ldots,z_n\}\cap I_{\Gamma}\mathbb C[[z_1,\ldots,z_n]]=I_{\Gamma}\mathbb C\{z_1,\ldots,z_n\}$ (these ideals are equal by [@dJP Exercise 8.1.5]). ◻ **Lemma 7**. *Let $\bar{X}$ be the (algebraic) normalization of $X$. Then $(\bar{X},0)$ is the (analytic) normalization of $(X,0)$. Moreover, $\mathcal{O}_{\bar{X},0}\cong\mathbb C\{\check{\sigma}\cap\mathbb Z^d\}.$* *Proof.* Let $\eta:\bar{X}\to X$ be the normalization. Recall that $\bar{X}$ is the toric variety defined by the semigroup $\check{\sigma}\cap\mathbb Z^d$ and that $\eta$ is induced by the inclusion of semigroups $\Gamma\subset\check{\sigma}\cap\mathbb Z^d$ [@CLS11 Proposition 1.3.8]. In particular, $\eta$ is a toric morphism. Being the normalization, $\eta$ is an isomorphism on dense open sets. In addition, $\eta^{-1}(0)=0$. Indeed, let $q\in\bar{X}$ be such that $\eta(q)=0$. Recall that points in toric varieties correspond to homomorphisms of semigroups. Hence, the homomorphism corresponding to $\eta(q)$ sends every non-zero element of $\Gamma$ to 0. On the other hand, for every $m\in\check{\sigma}\cap\mathbb Z^d$ there is $k\geq1$ such that $km\in\Gamma$. Since $\eta$ is induced by the inclusion $\Gamma\subset\check{\sigma}\cap\mathbb Z^d$, it follows that $q=0$. By the previous paragraph, the induced germ of an analytic function $\eta:(\bar{X},0)\to (X,0)$ is finite and generically 1-1. On the other hand, it is known that $\bar{X}$ normal implies that $(\bar{X},0)$ normal [@K Satz 4]. By the uniqueness of normalization we conclude that $\eta:(\bar{X},0)\to (X,0)$ is the normalization of $(X,0)$. Finally, $\mathcal{O}_{\bar{X},0}\cong\mathbb C\{\check{\sigma}\cap\mathbb Z^d\}$ follows using lemma [Lemma 6](#lema P){reference-type="ref" reference="lema P"}. ◻ **Corollary 8**. *$(X,0)$ is irreducible as a germ.* *Proof.* By a well-known theorem of Zariski [@Z48], $\bar{X}$ irreducible and normal at $0$ implies that the completion of the local ring at 0 is an integral domain. Hence, $(\bar{X},0)$ is irreducible by lemma [Lemma 6](#lema P){reference-type="ref" reference="lema P"}. Hence, the germ $(X,0)$ must also be irreducible. ◻ **Definition 9**. Let $(X,0)$ be a germ of an analytic space. We say that $(X,0)$ is a germ of a toric singularity if there exists a finitely generated semigroup $\Gamma\subset\mathbb Z^d$ contained in a strongly convex cone such that $\mathcal{O}_{X,0}\cong\mathbb C\{\Gamma\}$. Equivalently, let $X_{\Gamma}$ be the affine toric variety defined by $\Gamma$. Then $(X,0)$ is a toric singularity if it is isomorphic, as germs, to $(X_{\Gamma},0)$. **Example 10**. Let $X\subset\mathbb C^n$ be a toric variety containing the origin. Then $(X,0)$ is a germ of a toric singularity by lemma [Lemma 6](#lema P){reference-type="ref" reference="lema P"}. # Lipschitz saturation of toric singularities {#SatofTor} We have seen that for a toric singularity the analytic algebra $\mathcal{O}_{X,0}$ is generated by monomials. With this in mind, the main idea to prove that the Lipschitz saturation of a toric singularity is again toric, consists of showing that every monomial of an element of the saturation also belongs to the saturation. The first step towards that goal is the study of the integral closure of homogeneous ideals in the ring of power series. It is well known that the integral closure of a homogeneous ideal in a polynomial ring is again a homogeneous ideal ([@Vas05 Prop. (f), pg 38]). However we need this to be true for ideals generated by homogeneous polynomials in power series rings. That is the content of the following proposition, whose proof was kindly communicated to us by Professors Irena Swanson and Craig Huneke. Note that an ideal $\mathcal{I}\subset \mathbb C\{z_1,\ldots,z_n\}$ is generated by homogeneous polynomials if and only if $f \in \mathcal{I}$ implies that every homogeneous component of $f$ is also in $\mathcal{I}$. **Proposition 11**. *Let $\mathcal{I}\subset \mathbb C\{z_1,\ldots,z_n\}$ be an ideal generated by homogeneous polynomials in the ring of convergent power series. Then its integral closure $\overline{\mathcal{I}}$ is also generated by homogeneous polynomials.* *Proof.* Let $R:=\mathbb C\{z_1,\ldots,z_n\}$ denote the ring of convergent power series in $n$-variables over the field of complex numbers with maximal ideal $\mathfrak{m}=\left<z_1,\ldots,z_n\right>$, and let $\mathcal{I}=\left<f_1,\ldots,f_m\right>R$ where each $f_j \in A:=\mathbb C[z_1,\ldots,z_n] \subset R$ is a homogeneous polynomial.\ Let $K=\left<f_1,\ldots, f_m\right>A$ be the ideal generated by the $f_j$'s in the polynomial ring $A$, so $\mathcal{I}=\left<K\right>R$. Suppose that $\mathcal{I}$ contains a power of $\mathfrak{m}$, i.e. there exists $k\geq1$ such that $\mathfrak{m}^k \subset \mathcal{I}$, and let $s \in R$ be in the integral closure of $\mathcal{I}$. We can write $$s = p + s',$$ where $p\in A$ is a polynomial and $s' \in \mathfrak{m}^k \subset \mathcal{I}\subset \overline{\mathcal{I}}$ is a series of order greater than or equal to $k$. Note that $p=s-s' \in \overline{\mathcal{I}}$. Since the ring extension $A_\mathfrak{m}\rightarrow R$ is faithfully flat ([@GLS07 Lemma B.3.4]) then $\overline{\mathcal{I}} \cap A_\mathfrak{m}= \overline{\left<K\right>A_\mathfrak{m}}$ ([@HS06 Prop.1.6.2]), in particular $p/1 \in \overline{\left<K\right>A_\mathfrak{m}}$. This means that in the localized polynomial ring $A_\mathfrak{m}$ we have an equation of integral dependence of the form: $$\frac{p^r}{1}+ \frac{b_1}{c_1} \frac{p^{r-1}}{1} + \cdots + \frac{b_{r-1}}{c_{r-1}}\frac{p}{1} + \frac{b_r}{c_r} = \frac{0}{1},$$ where $b_j \in K^j$ and $c_j \notin \mathfrak{m}$. Letting $u=c_1\cdots c_r \in A$ and multiplying this equation by the unit $\frac{u^r}{1}$ of $A_\mathfrak{m}$ we get the following equality in $A_\mathfrak{m}$: $$\frac{(up)^r + \widetilde{b_1} (up)^{r-1}+ \cdots + \widetilde{b_{r-1}}(up) + \widetilde{b_r} }{1}=\frac{0}{1}.$$ Since A is an integral domain we get an integral dependence equation in $A$ $$(up)^r + \widetilde{b_1} (up)^{r-1}+ \cdots + \widetilde{b_{r-1}}(up) + \widetilde{b_r}=0,$$ that is, $up \in \overline{K} \subset A$. The inclusions $\mathfrak{m}^k \subset K \subset \overline{K}$ imply that $\overline{K}$ is an $\mathfrak{m}$-primary ideal of $A$ and since $u \notin \mathfrak{m}$ then $p \in \overline{K}$. Since $K$ is a homogeneous ideal in the polynomial ring $A$ then $\overline{K}$ is also homogeneous ([@Vas05 Prop. (f), pg 38]) and so each homogeneous component of $p$ is also in $\overline{K} \subset \overline{\mathcal{I}}$. This implies that $\overline{\mathcal{I}}$ is also generated by homogeneous polynomials. Now let $\mathcal{I}\subset R$ be an arbitrary ideal generated by homogeneous polynomials, and let $s$ be in the integral closure $\overline{\mathcal{I}}$ as before. Let $s_0$ be the initial form of $s$, $s_0$ is the non-zero homogeneous component of $s$ of lowest degree. For all $j \in \mathbb N$, $s$ is in the integral closure of $\mathcal{I}+ \mathfrak{m}^j$. We proved in the previous paragraph that the integral closure $\overline{\mathcal{I}+ \mathfrak{m}^j}$ of $\mathcal{I}+ \mathfrak{m}^j$ in $R$ is generated by homogeneous polynomials for all $j \in \mathbb N$. In particular, $s_0$ is in the integral closure $\overline{\mathcal{I}+ \mathfrak{m}^j}$ for all $j$. But by [@HS06 Corollary 6.8.5] this implies that $s_0$ is in the integral closure $\overline{\mathcal{I}}$. Now we start over with $s'=s-s_0 \in \overline{\mathcal{I}}$ and in this way we get that all homogeneous components of $s$ are in $\overline{\mathcal{I}}$ which is what we wanted to prove. ◻ **Theorem 12**. *Let $(X,0)$ be a $d$-dimensional toric singularity with smooth normalization. Then the Lipschitz saturated germ $(X^s,0)$ is also a toric singularity.* We will do this proof in several steps starting with the following lemma. **Lemma 13**. *Let $(X,0)$ be a $d$-dimensional toric singularity with smooth normalization and let $f \in \overline{\mathcal{O}_{X,0}}$ be a homogeneous polynomial such that $f \in \mathcal{O}_{X,0}^s$. Then every monomial of $f$ is also in $\mathcal{O}_{X,0}^s$.* *Proof.* Let $\Gamma\subset\mathbb Z^d$ be the semigroup defining $(X,0)$. Recall that $\check{\sigma}=\mathbb R_{\geq0}\Gamma$ is a strongly convex cone. This fact, together with the condition of smooth normalization allows us to assume, up to a change of coordinates, that $\Gamma\subset\mathbb N^d$ and $\check{\sigma}\cap\mathbb Z^d=\mathbb N^d$ (see lemma [Lemma 7](#normaliz){reference-type="ref" reference="normaliz"}). Let $\mathcal{A}=\{a_1,\ldots,a_n\} \subset \mathbb N^d$ be the minimal generating set of $\Gamma$ defining the toric singularity $(X,0)$. The normalization map can be realized as the monomial morphism: $$\begin{aligned} n: (\mathbb C^d,0) &\longrightarrow (X,0) \\ \left(u_1,\ldots,u_d\right) & \mapsto \left( u^{a_1}, \ldots, u^{a_n} \right).\end{aligned}$$ The ideal $I_\Delta$ of definition [Definition 1](#DefSat){reference-type="ref" reference="DefSat"} is a homogeneous, binomial ideal in the ring of convergent power series $\mathbb C\{x_1,\ldots, x_d, y_1,\ldots,y_d\}$: $$I_\Delta= \left< x^{a_1}-y^{a_1}, \ldots, x^{a_n}-y^{a_n}\right>.$$ For any point $\tau=(t_1,\ldots,t_d) \in \left(\mathbb C^*\right)^d$ we have an automorphism $$\varphi_\tau:\mathbb C\{x_1,\ldots, x_d, y_1,\ldots,y_d\}\circlearrowleft$$ defined by $x_j \mapsto t_jx_j$ and $y_j \mapsto t_jy_j$, such that $\varphi_\tau\left( I_\Delta \right)=I_\Delta$. Now let $f \in \overline{\mathcal{O}_{X,0}} \cong \mathbb C\{u_1,\ldots,u_d\}$ be a homogeneous polynomial such that $f \in \mathcal{O}_{X,0}^s$. Then $f(x)-f(y) \in \overline{I_\Delta}$ and it satisfies an integral dependence equation of the form: $$\left( f(x) - f(y)\right)^m + h_1(x,y)\left( f(x) - f(y)\right)^{m-1} + \cdots + h_m(x,y)=0,$$ where $h_j(x,y) \in I_\Delta^j$. By applying the morphism $\varphi_\tau$ to the previous equation we get: $$\left( f(\tau x) - f(\tau y)\right)^m + h_1(\tau x,\tau y)\left( f(\tau x) - f(\tau y)\right)^{m-1} + \cdots + h_m(\tau x,\tau y)=0,$$ where $h_j(\tau x,\tau y) \in I_\Delta^j$, and so we get that $g_\tau:=f(t_1u_1,\ldots,t_du_d) \in \mathcal{O}_{X,0}^s$. Since $f$ is a homogeneous polynomial, say of order $k$, then we can write it in the form $$f(u_1,\ldots,u_d)= \sum_{\alpha_1+\cdots+\alpha_d=k} b_{\alpha} u^{\alpha},$$ where $b_\alpha \in \mathbb C$ and $\alpha=(\alpha_1,\ldots,\alpha_d)$. Then we have an expression for $g_\tau$ of the form $$\begin{aligned} g_\tau=f(t_1u_1,\ldots,t_du_d)&=\sum_{\alpha_1+\cdots+\alpha_d=k}\tau^\alpha b_{\alpha} u^{\alpha} \\ &= \left< \left(\tau^{\alpha^1},\ldots, \tau^{\alpha^N} \right), \left(b_{\alpha^1} u^{\alpha^1},\ldots, b_{\alpha^N} u^{\alpha^N}\right) \right>. \end{aligned}$$ By choosing $\tau_1, \ldots, \tau_N$ generic points in $\left(\mathbb C^*\right)^d$ we get $$\begin{pmatrix} g_{\tau_1} \\ \vdots \\g_{\tau_N} \end{pmatrix} = \begin{pmatrix} \tau_1^{\alpha^1} & \ldots & \tau_1^{\alpha^N} \\ \vdots & \cdots & \vdots \\ \tau_N^{\alpha^1} & \ldots & \tau_N^{\alpha^N} \end{pmatrix} \begin{pmatrix} b_{\alpha^1} u^{\alpha^1} \\ \vdots \\ b_{\alpha^N} u^{\alpha^N} \end{pmatrix},$$ where the $i$-th row of the matrix corresponds to the image of the point $\tau_i \in \left(\mathbb C^*\right)^d$ of the Veronese map $\nu_k: \mathbb{P}^{d-1} \to \mathbb{P}^{N-1}$ of degree $k$. Since the image of the Veronese map is a nondegenerate projective variety, in the sense that it is not contained in any hyperplane, then these $N$ points are in general position. This implies that the matrix is invertible, and so we have $$\begin{pmatrix} b_{\alpha^1} u^{\alpha^1} \\ \vdots \\ b_{\alpha^N} u^{\alpha^N} \end{pmatrix}= \begin{pmatrix} \tau_1^{\alpha^1} & \ldots & \tau_1^{\alpha^N} \\ \vdots & \cdots & \vdots \\ \tau_N^{\alpha^1} & \ldots & \tau_N^{\alpha^N} \end{pmatrix} ^{-1} \begin{pmatrix} g_{\tau_1} \\ \vdots \\g_{\tau_N} \end{pmatrix}$$ In particular we have that $b_{\alpha^j} u^{\alpha^j} \in \mathcal{O}_{X,0}^s$ which is what we wanted to prove. ◻ *Proof.* (*of theorem [Theorem 12](#SatTor){reference-type="ref" reference="SatTor"}* )\ We want to prove that there exists a finitely generated semigroup $\Gamma^s \subset \mathbb N^d$ such that $$\mathcal{O}_{X,0}^s \cong \mathbb C\{\Gamma^s\}.$$ As we mentioned before, in this setting the ideal $I_\Delta$ is a homogeneous, binomial ideal in the ring of convergent power series $\mathbb C\{x_1,\ldots, x_d, y_1,\ldots,y_d\}$, and by proposition [Proposition 11](#CerraduraEnteraHomogenea){reference-type="ref" reference="CerraduraEnteraHomogenea"} the ideal $\overline{I_\Delta}$ is also generated by homogeneous polynomials. This means that if a series $f\in \mathbb C\{u_1\ldots,u_d\}$ is in $\mathcal{O}_{X,0}^s$, $$f= f_m + f_{m+1}+ \cdots + f_N + \cdots ,$$ then every homogeneous component $f_j$ of $f$ is in $\mathcal{O}_{X,0}^s$. But by lemma [Lemma 13](#Monomial){reference-type="ref" reference="Monomial"} this implies that every monomial of degree $j$ with a non-zero coefficient in $f_j$ is also in $\mathcal{O}_{X,0}^s$. Let $\Gamma^s$ be the semigroup defined by $$\Gamma^s=\{ \alpha \in \mathbb N^d \, | \, u^\alpha \in \mathcal{O}_{X,0}^s \}.$$ We have that $$\mathcal{O}_{X,0}^s \subset \mathbb C\{\Gamma^s\}\subset \mathbb C\{u_1,\ldots,u_d\}.$$ We have to prove that $\Gamma^s$ is finitely generated and the equality $\mathcal{O}_{X,0}^s = \mathbb C\{\Gamma^s\}$.\ To begin with the latter, take $g\in \mathbb C\{\Gamma^s\}$. For every $k\geq 0$, write $g= g_{\leq k} + \widetilde{g_k}$ where $g_{\leq k}$ is the truncation of $g$ to degree $k$, and since it is a finite sum of monomials of $\mathcal{O}_{X,0}^s$, by definition of $\Gamma^s$, we have that $g_{\leq k} \in \mathcal{O}_{X,0}^s$. This means that $$g_{\leq k}(x) -g_{\leq k}(y) \in \overline{I_\Delta}$$ and since $\widetilde{g_k}(x)-\widetilde{g_k}(y) \in \mathfrak{m}^{k+1}\mathbb C\{x,y\}$ we have that $$g(x) -g(y) \in \overline{I_\Delta} + \mathfrak{m}^{k+1} \subset \overline{I_\Delta + \mathfrak{m}^{k+1}}.$$ In particular, $$g(x)-g(y) \in \bigcap_{k \in \mathbb N}\overline{I_\Delta + \mathfrak{m}^{k+1}} = \overline{I_\Delta} \textrm{ by \cite[Corollary 6.8.5]{HS06}}$$ then $g \in \mathcal{O}_{X,0}^s$ and we have the equality we wanted. Now we prove that $\Gamma^s$ is a finitely generated semigroup. Let $\leq$ be a monomial order in $\mathbb C[u_1,\ldots,u_d]$. We order the elements of $\Gamma^s$ and write them as $\{\beta^j\, | \, j \in \mathbb N\}$, where $0=\beta^0 < \beta^1 < \beta^2 <\cdots$. Consider the following ascending chain of ideals in $\mathcal{O}_{X,0}^s$: $$\langle u^{\beta^1} \rangle\subset \langle u^{\beta^1},u^{\beta^2} \rangle \subset \langle u^{\beta^1},u^{\beta^2},u^{\beta^3} \rangle \subset \cdots$$ Since $\mathcal{O}_{X,0}^s$ is Noetherian, we have $u^{\beta^j}\in\langle u^{\beta^1},\ldots,u^{\beta^m} \rangle$, for some $m\in\mathbb N$ and for all $j\in\mathbb N$. We claim that $\Gamma^s=\mathbb N(\beta^1,\ldots,\beta^m)$. Indeed, let $u^{\beta^j}=\sum_{i=1}^m F_iu^{\beta^i}$, for some $F_i\in\mathcal{O}_{X,0}^s$. Since $\mathcal{O}_{X,0}^s\subset\mathbb C\{u_1,\ldots,u_d\}$ it follows that $u^{\beta^j}=u^{\beta}u^{\beta^i}$, for some $i$ and some monomial $u^{\beta}$ of $F_i$. As before, we have $\beta\in\Gamma^s$. Hence, $\beta^j$ is the sum of $\beta^i$ plus an element $\beta\in\Gamma^s$ and $\beta<\beta^j$. Continuing this way, we obtain that $\beta^j\in\mathbb N(\beta^1,\ldots,\beta^m)$. ◻ We now know that for a germ $(X,0)$ of toric singularity with smooth normalization and associated semigroup $\Gamma$, the Lipschitz saturated germ $(X^s,0)$ is again a toric singularity with associated semigroup $\Gamma^s$. In this setting we have $\Gamma\subset \Gamma^s \subset \mathbb N^d$ and we need to determine which elements $\alpha \in \mathbb N^d$ we have to add to $\Gamma$ in order to obtain $\Gamma^s$. Since many properties of a toric variety are encoded in its semigroup, we can use them to start discerning. This is the content of the following proposition. **Proposition 14**. *Let $(X,0)$ be a germ of $d$-dimensional toric singularity with smooth normalization. Let $\Gamma\subset \mathbb N^d$ be the associated semigroup and $K_+(\Gamma)$ the convex hull of $\Gamma\setminus\{0\}$ in $\mathbb R^d$. If $\alpha \in \mathbb N^d \setminus K_+(\Gamma)$ then $\alpha \notin \Gamma^s$.* *Proof.* By [@GKZ08 Chapter 5, theorem 3.14], the multiplicity of a toric germ is determined by the (normalized) volume of the complement of $K_+(\Gamma)$ in $\mathbb N^d$. But we know from [Remark 2](#Rem1){reference-type="ref" reference="Rem1"} that a germ $(X,0)$ and its Lipschitz saturation $(X^s,0)$ have the same multiplicity, and so we must have that $K_+(\Gamma)=K_+(\Gamma^s)$ which finishes the proof. ◻ In the case of curves, this means that if $m$ is the minimal non-zero element of $\Gamma\subset \mathbb N$ then $k\in \mathbb N$, with $k< m$ implies that $k \notin \Gamma^s$ (see section [1](#secLips){reference-type="ref" reference="secLips"}). Recall that in this case $m$ is equal to the multiplicity of the curve. **Example 15**. The Whitney Umbrella.\ The surface singularity $(X,0) \subset (\mathbb C^3,0)$ defined by the equation $y^2-x^2z=0$ is a toric singularity with smooth normalization given by $$(u,v) \mapsto (u, uv,v^2)$$ and associated semigroup $\Gamma\subset \mathbb N^2$ with minimal generating set $\{(1,0),(1,1),(0,2)\}$. This translates to $\mathcal{O}_{X,0}\cong \mathbb C\{u,uv,v^2\} \subset \mathbb C\{u,v\}$ and a point $(a,b) \in \Gamma^s$ is identified with the monomial $u^av^b \in \mathcal{O}_{X,0}^s \subset \mathbb C\{u,v\}$. {width="\\textwidth"} Note that $\mathbb N^2\setminus \Gamma=\left\{ (0,2k+1) \,|\, k\in \mathbb N\right\}$. We will show that none of them are in $\Gamma^s$. Hence, $\Gamma=\Gamma^s$. In the proof of theorem [Theorem 12](#SatTor){reference-type="ref" reference="SatTor"} we showed that $\mathcal{O}_{X,0}^s=\mathbb C\{\Gamma^s\}$. We conclude that $\mathcal{O}_{X,0}=\mathcal{O}_{X,0}^s$ and so the Whitney Umbrella coincides with its Lipschitz saturation $(X^s,0)$. To begin with, the only point in $\mathbb N^2 \setminus K_+(\Gamma)$ is $(0,1)$, and so $(0,1) \notin \Gamma^s$ by the previous proposition. For the points of the form $(0,r)$ with $r>1$ odd, consider the ideal $$I_\Delta=\left< x_1-y_1, x_1x_2-y_1y_2,x_2^2-y_2^2\right> \mathbb C\left\{ x_1,x_2,y_1,y_2\right\}$$ Taking the arc $\varphi:(\mathbb C,0) \to \left(\mathbb C^2 \times \mathbb C^2, 0\right)$ defined by $t \mapsto (t^{r+1},t,t^{r+2},-t)$ we have the corresponding morphism of analytic algebras $\varphi^*: \mathbb C\left\{ x_1,x_2,y_1,y_2\right\} \to \mathbb C\{t\}$ such that $$\varphi^*(x_2^r -y_2^r)=2t^r \notin \left<\varphi^*(I_\Delta)\right>=\left<t^{r+1}\right>.$$ By [@LT08 Thm 2.1] this implies that $x_2^r -y_2^r \notin \overline{I_\Delta}$, i.e. $v^r \notin \mathcal{O}_{X,0}^s$. # Some examples. {#TorSurf} In this section we will show how to calculate the Lipschitz saturation of some families of toric singularities, starting with products of curves.\ Let $(X_1,0)$ and $(X_2,0)$ be two germs of toric singularities of dimension $1$ defined by the semigroups $\Gamma_1$ and $\Gamma_2$ with corresponding minimal generating sets $\mathcal{A}_1=\{\gamma_1,\ldots,\gamma_m\}$ and $\mathcal{A}_2=\{\omega_1,\ldots, \omega_n\}$. The germ $(X,0)=(X_1 \times X_2,0)$ is a toric surface singularity with semigroup $\Gamma\subset \mathbb N^2$ generated by $\mathcal{A}=\left\{(\gamma_i,0), (0,\omega_j)\right\}_{i,j}$, that is $\Gamma=\Gamma_1 \times \Gamma_2$. Note that the normalization of $(X,0)$ is smooth and the normalization map can be written as $$\begin{aligned} \eta: (\mathbb C^2,0) & \longrightarrow (X,0) \subset (\mathbb C^{m+n},0) \\ (u,v) &\mapsto \left( u^{\gamma_1},\ldots,u^{\gamma_m},v^{\omega_1},\ldots, v^{\omega_n}\right). \end{aligned}$$ **Proposition 16**. *For a germ of surface singularity $(X,0)=(X_1 \times X_2,0)$ defined by a product of toric curves, the Lipschitz saturation $(X^s,0)$ is a toric surface singularity with semigroup $$\Gamma^s =\Gamma_1^s \times \Gamma_2^s,$$ where $\Gamma_1^s$ and $\Gamma_2^s$ are the semigroups of the Lipschitz saturated curves $(X_1^s,0)$ and $(X_2^s,0)$ described in section [1](#secLips){reference-type="ref" reference="secLips"}.* *Proof.* We know that $\mathcal{O}_{X,0}^s\subset \mathbb C\{u,v\}$ is an analytic algebra generated by monomials, so we need to characterize them. A monomial $u^\alpha v^\beta \in \mathcal{O}_{X,0}^s$ defines a meromorphic function on a neighborhood $U$ of the origin in $X$ which is locally Lipschitz with respect to the ambient metric. If we consider the normalization of $(X,0)$ as before $$\begin{aligned} \eta: (\mathbb C^2,0) & \longrightarrow (X,0) \subset (\mathbb C^{m+n},0) \\ (u,v) &\mapsto \left( u^{\gamma_1},\ldots,u^{\gamma_m},v^{\omega_1},\ldots, v^{\omega_n}\right), \end{aligned}$$ then for any sufficiently small $v_0$ the restriction of $u^\alpha v^\beta$ to $\left(X_1\times \{v_0^\omega\},(0,v_0^\omega)\right)$ tells us that $u^\alpha v_0^\beta$ defines a meromorphic locally Lipschitz function on $(X_1,0)$ and so $u^\alpha \in \mathcal{O}_{X_1,0}^s$; equivalently $\alpha \in \Gamma_1^s$. The same reasoning with the restriction to $\left( \{u_0^\gamma\} \times X_2, (u_0^\gamma,0)\right)$ tells us that $\beta \in \Gamma_2^s$ and so $(\alpha,\beta) \in \Gamma_1^s \times \Gamma_2^s$. On the other hand, in this setting we have the ideal $I_\Delta$ defined by $$I_\Delta= \left< x_1^{\gamma_1}-y_1^{\gamma_1},\ldots, x_1^{\gamma_m}-y_1^{\gamma_m},x_2^{\omega_1}-y_2^{\omega_1}, \ldots, x_2^{\omega_n}-y_2^{\omega_n} \right>\mathbb C\{x_1,x_2,y_1,y_2\}.$$ Let $\alpha \in \Gamma_1^s$ then by definition we have $$x_1^\alpha - y_1^\alpha \in \overline{\left<x_1^{\gamma_1}-y_1^{\gamma_1},\ldots,x_1^{\gamma_m}-y_1^{\gamma_m}\right>}\mathbb C\{x_1,y_1\}.$$ In particular, $x_1^\alpha - y_1^\alpha \in \overline{I_\Delta}$ and so $u^\alpha \in \mathcal{O}_{X,0}^s \subset \mathbb C\{u,v\}$. Analogously for every $\beta \in \Gamma_2^s$ we have $v^\beta \in \mathcal{O}_{X,0}^s$. Since $\mathcal{O}_{X,0}^s$ is an analytic algebra, this implies that for every $\alpha \in \Gamma_1^s$ and $\beta \in \Gamma_2^s$ the monomial $u^\alpha v^\beta \in \mathcal{O}_{X,0}^s$ which finishes the proof. ◻ **Corollary 17**. *Let $(X,0)=(X_1 \times X_2,0)$ be a product of toric curves. Then $\mathop{\mathrm{edim}}(X^s,0)=\mathop{\mathrm{edim}}(X_1^s,0)+\mathop{\mathrm{edim}}(X_2^s,0)$. In addition, $\mathop{\mathrm{mult}}(X^s,0)=\mathop{\mathrm{mult}}(X_1^s,0)\cdot\mathop{\mathrm{mult}}(X_2^s,0)$.* *Proof.* First, recall that the embedding dimension of the origin of a toric variety i.e., the dimension of its Zariski tangent space, coincides with the cardinality of the minimal generating set of the correspondig semigroup. On the other hand, the multiplicity at the origin of a toric variety is also described combinatorially in terms of the semigroup (see proposition [Proposition 14](#NoAgrego){reference-type="ref" reference="NoAgrego"}). Hence, both assertions follow from proposition [Proposition 16](#ProdCurv){reference-type="ref" reference="ProdCurv"}. ◻ **Remark 18**. Notice that both proposition [Proposition 16](#ProdCurv){reference-type="ref" reference="ProdCurv"} and corollary [Corollary 17](#edim-mult prod){reference-type="ref" reference="edim-mult prod"} hold for the product of any finite number of toric curves, with the same proof. **Example 19**. Starting from the space curves $(X_1,0)$ and $(X_2,0)$ parametrized respectively by $$u \longmapsto (u^4,u^6,v^7), \hspace{1in} v \longmapsto (v^6,v^9,v^{11}),$$ we get the toric surface $(X,0)\subset (\mathbb C^6,0)$ of multiplicity $24$ and embedding dimension $6$ defined by the ideal $$I_X=\left< y^2-x^3, c^3-a^4b,b^2-a^3, z^2-x^2y \right>\mathbb C\{x,y,z,a,b,c\}.$$ The normalization map is given by: $$\begin{aligned} \eta: (\mathbb C^2,0) & \longrightarrow (X,0) \subset (\mathbb C^{6},0) \\ (u,v) &\mapsto \left( u^4,u^6,u^7,v^6,v^9, v^{11}\right). \end{aligned}$$ Following the procedure described in section [1](#secLips){reference-type="ref" reference="secLips"} we obtain that the semigroup $\Gamma_1^s \subset \mathbb N$ is generated by $\mathcal{A}_1=\{4,6,7,9\}$ and the semigroup $\Gamma_2^s \subset \mathbb N$ is generated by $\mathcal{A}_2=\{6,9,11,13,14,16\}$. By proposition [Proposition 16](#ProdCurv){reference-type="ref" reference="ProdCurv"} the Lipschitz saturation $(X^s,0) \subset (\mathbb C^{10},0)$ is the toric singularity defined by the semigroup $\Gamma^s \subset \mathbb N^2$ generated by the set $$\mathcal{A}= \left\{(4,0),(6,0),(7,0),(9,0),(0,6),(0,9),(0,11),(0,13),(0,14),(0,16) \right\},$$ and with normalization map given by: $$\begin{aligned} \eta: (\mathbb C^2,0) & \longrightarrow (X^s,0) \subset (\mathbb C^{6},0) \\ (u,v) &\mapsto \left( u^4,u^6,u^7,u^9,v^6,v^9, v^{11},v^{13},v^{14},v^{16}\right). \end{aligned}$$ $(X^s,0)$ is a toric germ of multiplicity $24$ and embedding dimension $10$. We will now a consider a family of hypersurfaces $(X,0) \subset (\mathbb C^3,0)$ with equation of the form $$y^N-x^{\alpha N}z^\beta=0,$$ where $\alpha,\beta \geq 1$ and $\text{ mcd}(\beta, N)=1$. It is a family of toric surface singularities with semigroup $\Gamma$ generated by $\mathcal{A}=\{(1,0),(\alpha,\beta),(0,N) \}$. **Theorem 20**. *Let $(X,0)\subset (\mathbb C^3,0)$ be the toric hypersurface singularity with normalization map given by $$\begin{aligned} \eta:(\mathbb C^2,0) &\longrightarrow (X,0) \\ (u,v) &\mapsto \left( u, u^\alpha v^\beta, v^N\right), \end{aligned}$$ where $\alpha,\beta \geq 1$ and $\text{ mcd}(\beta, N)=1$. Let $T \subset \mathbb N$ be the numerical semigroup generated by $\{N,\beta\}$, and $T^s \subset \mathbb N$ be its saturation as in section [1](#secLips){reference-type="ref" reference="secLips"}. The monomial $u^av^b \in \mathbb C\{u,v\}$ is in the Lipschitz saturation $\mathcal{O}_{X,0}^s$ if and only if $b=mN$ for some $m \in \mathbb N$ or $a \geq \alpha$ and $b \in T^s$.* *Proof.* In this setting the semigroup $\Gamma$ is generated by $\mathcal{A}=\{(1,0),(\alpha,\beta),(0,N) \}$, and by the proof of theorem [Theorem 12](#SatTor){reference-type="ref" reference="SatTor"} $u^av^b \in \mathcal{O}_{X,0}^s$ is equivalent to $(a,b) \in \Gamma^s$.\ We first show that if $b=mN$ for some $m \in \mathbb N$ or $a \geq \alpha$ and $b \in T^s$ then $(a,b)\in\Gamma^s$. Suppose first that $b=mN$. Since $(1,0),(0,N)\in\Gamma$, it follows that $(a,b)\in\Gamma\subset \Gamma^s$ for all $a\in\mathbb N$. Now suppose that $a \geq \alpha$ and $b \in T^s$. Since $\Gamma\subset \Gamma^s$ we have that $u \in \mathcal{O}_{X,0}^s$ and so it is enough to prove the statement for $a=\alpha$. By definition $u^\alpha v^b \in \mathcal{O}_{X,0}^s$ if and only if $x_1^\alpha x_2^b -y_1^\alpha y_2^b \in \overline{I_\Delta}$ where $$I_\Delta=\left< x_1-y_1, x_1^\alpha x_2^\beta-y_1^\alpha y_2^\beta,x_2^N-y_2^N\right> \mathbb C\left\{ x_1,x_2,y_1,y_2\right\}.$$ The assumption $b \in T^s$ means that $v^b \in \mathcal{O}_{C,0}^s \subset \mathbb C\{v\}$, which can be rephrased in terms of integral closure of ideals by $$x_2^b-y_2^b \in \overline{\left<x_2^\beta-y_2^\beta, x_2^N-y_2^N \right>}\mathbb C\{x_2,y_2\}.$$ And so if we denote $x_2^b-y_2^b$ by $f(x_2,y_2)$ we have an integral dependence equation in $\mathbb C\{x_2,y_2\}$ of the form: $$f^m+g_1(x_2,y_2)f^{m-1} + \cdots + g_m(x_2,y_2) =0,$$ with $g_k(x_2,y_2) \in J^k=\left< (x_2^N-y_2^N)^i(x_2^\beta-y_2^\beta)^j\,|\, i+j=k\right>\mathbb C\{x_2,y_2\}$. Each $g_k$ will then be of the form $$g_k(x_2,y_2)= \sum_{j=0}^k h_j(x_2,y_2)(x_2^N-y_2^N)^{k-j}(x_2^\beta-y_2^\beta)^j.$$ Multiplying the integral dependence equation by $x_1^{\alpha m}$ we obtain: $$\label{DepInt} (x_1^\alpha f)^m + x_1^\alpha g_1\left(x_1^\alpha f \right)^{m-1} + \cdots + x_1^{\alpha(m-1)}g_{m-1} x_1^\alpha f + x_1^{\alpha m}g_m=0.$$ But now $$x_1^{\alpha k}g_k(x_2,y_2)= \sum_{j=0}^k x_1^{\alpha (k-j)}h_j(x_2,y_2)(x_2^N-y_2^N)^{k-j}\left[ x_1^\alpha(x_2^\beta-y_2^\beta)\right]^j.$$ Note that $x_1-y_1, x_1^\alpha x_2^\beta-y_1^\alpha y_2^\beta \in I_\Delta$ implies that $x_1^\alpha(x_2^\beta-y_2^\beta) \in I_\Delta$. In particular we get $$(x_2^N-y_2^N)^{k-j}\left[ x_1^\alpha(x_2^\beta-y_2^\beta)\right]^j \in I_\Delta^k,$$ and so $x_1^{\alpha k}g_k \in I_\Delta^k$. This implies that equation ([\[DepInt\]](#DepInt){reference-type="ref" reference="DepInt"}) is an integral dependence equation for $x_1^\alpha f$ over $I_\Delta$. Finally $x_1^\alpha f=x_1^\alpha \left(x_2^b -y_2^b\right) \in \overline{I_\Delta}$ and $x_1-y_1 \in I_\Delta$ imply $x_1^\alpha x_2^b -y_1^\alpha y_2^b \in \overline{I_\Delta}$ which is what we wanted to prove.\ Now we prove that $(a,b)\in\Gamma^s$ implies $b=mN$ for some $m \in \mathbb N$ or $a \geq \alpha$ and $b \in T^s$. If $b=mN$ for some $m\in\mathbb N$, we are done. Assume that $b\neq mN$ for all $m\in\mathbb N$. We show that $a\geq\alpha$ and $b\in T^s$. Note that for any $u_0 \neq 0$ we have an embedding of the plane toric curve $(C,0)$, with semigroup $T$ generated by $\{N,\beta\}$ and defined by $y^N-z^\beta =0$, via the map $$\begin{aligned} (C,0) &\hookrightarrow \left(X, (u_0,0,0)\right) \\ (y,z) &\mapsto (u_0,u_0^\alpha y, z) \\ (v^\beta, v^N) &\mapsto (u_0, u_0^\alpha v^\beta, v^N). \end{aligned}$$ By remark [Remark 2](#Rem1){reference-type="ref" reference="Rem1"} a monomial $u^av^b \in \mathcal{O}_{X,0}^s$ defines a locally Lipschitz meromorphic function on a small enough neighborhood $U$ of $0$ in $X$. For any small enough $u_0$ it restricts to a locally Lipschitz meromorphic function $u_0^av^b$ on a neighborhood of $(u_0,0,0)$ in the embedded curve $C$. In particular $v^b \in \mathcal{O}_{C,0}^s$, or equivalently $b \in T^s$. To show that $a\geq\alpha$ we prove that $0\leq a<\alpha$ and $b\neq mN$ implies $(a,b)\notin\Gamma^s$. We consider two cases: $a=0$ and $0<a<\alpha$. Following example [Example 15](#WU){reference-type="ref" reference="WU"}, note that all points of the form $(0,k)$ with $k<N$ are in $\mathbb N^2 \setminus K_+(\Gamma)$ and so they are not in $\Gamma^s$ by proposition [Proposition 14](#NoAgrego){reference-type="ref" reference="NoAgrego"}. For the points of the form $(0,k)$ with $k>N$, $k\neq mN$, consider the ideal $$I_\Delta=\left< x_1-y_1, x_1^\alpha x_2^\beta-y_1^\alpha y_2^\beta,x_2^N-y_2^N\right> \mathbb C\left\{ x_1,x_2,y_1,y_2\right\}.$$ Let $\theta \in \mathbb C$ be a primitive $N$-th root of unity and consider the arc $$\begin{aligned} \varphi:(\mathbb C,0) &\to \left(\mathbb C^2 \times \mathbb C^2, 0\right)\\ t &\mapsto (t^{k+1},t,t^{k+2},\theta t). \end{aligned}$$ We have the corresponding morphism of analytic algebras $\varphi^*: \mathbb C\left\{ x_1,x_2,y_1,y_2\right\} \to \mathbb C\{t\}$ such that $$\begin{aligned} \left<\varphi^*(I_\Delta)\right>\mathbb C\{t\}&=\left<\varphi^*(x_1-y_1), \varphi^*(x_1^\alpha x_2^\beta-y_1^\alpha y_2^\beta),\varphi^*(x_2^N-y_2^N) \right> \\ &= \left< (1-t)t^{k+1}, (1-\theta^\beta t^\alpha) t^{\alpha(k+1) + \beta}, 0 \right> \\ &= \left<t^{k+1}\right>\mathbb C\{t\}.\end{aligned}$$ In particular, $$\varphi^*(x_2^k -y_2^k)=(1-\theta^k)t^k \notin \left<\varphi^*(I_\Delta)\right>=\left<t^{k+1}\right>.$$ By [@LT08 Thm 2.1] this impliest that $x_2^k -y_2^k \notin \overline{I_\Delta}$, i.e. $v^k \notin \mathcal{O}_{X,0}^s$. All that is left to prove is that for $0<a< \alpha$ and $b \neq mN$ the monomial $u^av^b$ is not in $\mathcal{O}_{X,0}^s$. We will once again use the arc criterion for integral dependence, but this time with the arc $$\begin{aligned} \psi:(\mathbb C,0) &\to \left(\mathbb C^2 \times \mathbb C^2, 0\right)\\ t &\mapsto (t^{b+1},t,t^{b+1}+t^r,\theta t), \end{aligned}$$ where $\theta \in \mathbb C$ is a primitive $N$-th root of unity. In this setting the image of $I_\Delta$ by the morphism $\psi^*$ in $\mathbb C\{t\}$ is of the form: $$\begin{aligned} \left<\varphi^*(I_\Delta)\right>\mathbb C\{t\}&=\left<\psi^*(x_1-y_1), \psi^*(x_1^\alpha x_2^\beta-y_1^\alpha y_2^\beta),\psi^*(x_2^N-y_2^N) \right> \\ &= \left< -t^r, \left[1-\theta^\beta \left( 1 +t^{r-b-1}\right)^\alpha \right] t^{\alpha(b+1) + \beta}, 0 \right> \\ &= \left<t^{\alpha(b+1) + \beta}\right>\mathbb C\{t\} \hspace{0.1in} \text { for }r\text{ big enough.} \end{aligned}$$ On the other hand we have $$\psi^*\left(x_1^ax_2^b -y_1^ay_2^b\right)=\left[1-\theta^b\left( 1 +t^{r-b-1}\right)^a \right] t^{a(b+1) + b}.$$ Using that $0< a < \alpha$ it is straightforward to verify that for every $b \geq 0$ $$\alpha(b+1) + \beta > a(b+1) +b$$ and so $x_1^ax_2^b -y_1^ay_2^b \notin \overline{I_\Delta}$, i.e. $u^av^b \notin \mathcal{O}_{X,0}^s$. ◻ **Corollary 21**. *Let $(X,0)\subset (\mathbb C^3,0)$ be a germ of toric hypersurface singularity as in theorem [Theorem 20](#CalculoLip){reference-type="ref" reference="CalculoLip"}. Then $(X^s,0)$ is a toric surface singularity of multiplicity $N$ and embedding dimension $N+1$.* *Proof.* By the combinatorial description of multiplicity in toric geometry, it follows that $N=\mathop{\mathrm{mult}}(X,0)=\mathop{\mathrm{mult}}(X^s,0)$. Similarly, the embedding dimension of $(X^s,0)$ corresponds to the cardinality of the minimal set of generators of $\Gamma^s$. We exhibit this minimal set of generators and show that it has cardinality $N+1$. Theorem [Theorem 20](#CalculoLip){reference-type="ref" reference="CalculoLip"} states that $(a,b)\in\Gamma^s$ if and only if $b=mN$ for some $m \in \mathbb N$ or $a \geq \alpha$ and $b \in T^s$. We divide the proof in two cases.\ Case $N<\beta$. Write $\beta=kN+l$, $k\geq1$ and $0<l<N$. By the algorithm for computing $T^s$ we obtain $T^s=\{0,N,2N,\ldots,kN,\beta,\beta+1,\beta+2,\cdots\}$ (see section [1](#secLips){reference-type="ref" reference="secLips"}). Consider the sets $$\begin{aligned} \mathcal{A}&=\{(1,0),(\alpha,\beta),(0,N) \},\notag\\ \mathcal{B}&=\{(\alpha,\beta+1),\ldots,(\alpha,\beta+N-1)\}\setminus\{(\alpha,(k+1)N)\}.\notag\end{aligned}$$ We claim that $\Gamma^s=\mathbb N(\mathcal{A}\cup\mathcal{B})$. Firstly observe that $\mathcal{A}\cup\mathcal{B}\subset\Gamma^s$. Now let $(a,b)\in\Gamma^s$. If $b=mN$ then $(a,b)\in\Gamma=\mathbb N(\mathcal{A})\subset\mathbb N(\mathcal{A}\cup\mathcal{B})$. In particular, this holds for the element $(\alpha,(k+1)N)$. Now assume that $a\geq\alpha$ and $b\in T^s$. Since $(1,0)\in\mathcal{A}$ it is enough to consider $a=\alpha$. - Suppose $b<\beta$. Then $b=mN$ for some $m\in\mathbb N$. Hence $(\alpha,b)\in\mathbb N(\mathcal{A}\cup\mathcal{B})$. - Suppose $b=\beta$. Then $(\alpha,b)\in\mathcal{A}\subset\mathbb N(\mathcal{A}\cup\mathcal{B})$. - Suppose $\beta+1 \leq b\leq \beta+N-1$, $b\neq (\alpha,(k+1)N)$. Then $(\alpha,b)\in\mathcal{B}\subset\mathbb N(\mathcal{A}\cup\mathcal{B})$. - Suppose $\beta+N\leq b$. Write $b-\beta=rN+s$, $r\geq1$, $0\leq s<N$. Then $(\alpha,b)=(\alpha,\beta+rN+s)=(\alpha,\beta+s)+r(0,N)\in\mathbb N(\mathcal{A}\cup\mathcal{B})$. Now we show that $\mathcal{A}\cup\mathcal{B}$ is minimal as a generating set. Indeed, first notice that no element of $\mathcal{A}$ can be generated by $\mathcal{B}$ and viceversa (because of the second entry of the vectors). Similarly, for each $(\alpha,\beta+i)\in\mathcal{B}$ it follows that $(\alpha,\beta+i)\notin\mathbb N(\mathcal{A}\cup\mathcal{B}\setminus\{(\alpha,\beta+i)\})$ (because of the first entry of the vectors). Hence $\mathcal{A}\cup\mathcal{B}$ is the minimal generating set of $\Gamma^s$ and has cardinality $N+1$.\ Case $\beta<N$. Write $N=k\beta+l$, $k\geq1$ and $0<l<\beta$. We compute $T^s$ as before: $T^s=\{0,\beta,\ldots,k\beta,N,N+1,\cdots\}$. Consider the sets $$\begin{aligned} \mathcal{A}=&\{(1,0),(\alpha,\beta),(0,N) \},\notag\\ \mathcal{B}=&\{(\alpha,2\beta),(\alpha,3\beta),\ldots,(\alpha,k\beta)\}\notag\\ &\cup\big[\{(\alpha,N+1),\ldots,(\alpha,N+(N-1))\}\setminus\{(\alpha,N+\beta),\ldots,(\alpha,N+k\beta)\}\big].\notag\end{aligned}$$ We claim that $\Gamma^s=\mathbb N(\mathcal{A}\cup\mathcal{B})$. Firstly observe that $\mathcal{A}\cup\mathcal{B}\subset\Gamma^s$. Now let $(a,b)\in\Gamma^s$. If $b=mN$ then $(a,b)\in\Gamma=\mathbb N(\mathcal{A})\subset\mathbb N(\mathcal{A}\cup\mathcal{B})$. Now assume that $a\geq\alpha$ and $b\in T^s$. As before, it is enough to consider $a=\alpha$. - For each $i\in\{1,\ldots,k\}$ we have $(\alpha,N+i\beta)=(0,N)+(\alpha,i\beta)\in\mathbb N(\mathcal{A}\cup\mathcal{B})$. - Suppose $b<N$. Then $b=m\beta$ for some $m\in\mathbb N$. Hence $(\alpha,b)\in\mathbb N(\mathcal{A}\cup\mathcal{B})$. - Suppose $b=N$. Then $(\alpha,b)\in\mathbb N(\mathcal{A})\subset\mathbb N(\mathcal{A}\cup\mathcal{B})$. - Suppose $N+1 \leq b\leq N+(N-1)$, $b\neq(\alpha,N+i\beta)$, $i\in\{1,\ldots,k\}$. Then $(\alpha,b)\in\mathcal{B}\subset\mathbb N(\mathcal{A}\cup\mathcal{B})$. - Suppose $2N\leq b$. Write $b-N=rN+s$, $r\geq1$, $0\leq s<N$. Then $(\alpha,b)=(\alpha,N+rN+s)=r(0,N)+(\alpha,N+s)\in\mathbb N(\mathcal{A}\cup\mathcal{B})$. A similar analysis as in the previous case shows that $\mathcal{A}\cup\mathcal{B}$ is the minimal generating set of $\Gamma^s$ and has cardinality $N+1$. ◻ **Example 22**. Consider the toric hypersurface singularity $(X,0)\subset (\mathbb C^3,0)$ with normalization map $$\begin{aligned} \eta:(\mathbb C^2,0) &\longrightarrow (X,0) \\ (u,v) &\mapsto \left( u, u^3 v^{11}, v^5 \right). \end{aligned}$$ It is defined by the equation $y^5-x^{15}z^{11}=0$ and so it has multiplicity $5$. Following the notation of theorem [Theorem 20](#CalculoLip){reference-type="ref" reference="CalculoLip"} we have the semigroup $T \subset \mathbb N$ generated by $\{5,11\}$ and its saturation $T^s \subset \mathbb N$ with minimal generating set $\{5,11,12,13,14\}$. {width="\\textwidth"} From the previous corollary we have the semigroup $\Gamma^s$ associated to the Lipschitz saturation $(X^s,0)$ with minimal generating set $$\mathcal{A}^s=\{(1,0), (3,11), (3,12), (3,13), (3,14),(0,5)\},$$ normalization map $$\begin{aligned} \eta:(\mathbb C^2,0) &\longrightarrow (X^s,0)\subset (\mathbb C^6,0) \\ (u,v) &\mapsto \left( u, u^3 v^{11}, u^3 v^{12},u^3 v^{13},u^3 v^{14},v^5 \right), \end{aligned}$$ and embedding dimension $6$. Interchanging the roles of $5$ and $11$ we get the following example. **Example 23**. Consider the toric hypersurface singularity $(X,0)\subset (\mathbb C^3,0)$ with normalization map $$\begin{aligned} \eta:(\mathbb C^2,0) &\longrightarrow (X,0) \\ (u,v) &\mapsto \left( u, u^3 v^5, v^{11} \right). \end{aligned}$$ It is defined by the equation $y^{11}-x^{33}z^{5}=0$ and so it has multiplicity $11$. Following the notation of theorem [Theorem 20](#CalculoLip){reference-type="ref" reference="CalculoLip"} we have again the semigroup $T \subset \mathbb N$ generated by $\{5,11\}$ and its saturation $T^s \subset \mathbb N$ with minimal generating set $\{5,11,12,13,14\}$. {width="\\textwidth"} From the previous corollary we have the semigroup $\Gamma^s$ associated to the Lipschitz saturation $(X^s,0)$ with minimal generating set $$\mathcal{A}^s=\{(1,0), (3,5), (3,10), (3,12), (3,13), (3,14), (3,15), (3,17), (3,18) , (3,19) , (3,20), (0,11)\},$$ normalization map $$\begin{aligned} \eta:(\mathbb C^2,0) &\longrightarrow (X^s,0)\subset (\mathbb C^{12},0) \\ (u,v) &\mapsto \left( u, u^3v^5, u^3v^{10}, u^3 v^{12}, u^3 v^{13},u^3 v^{14},u^3 v^{15}, u^3 v^{17}, u^3 v^{18},u^3 v^{19},u^3 v^{20},v^{11} \right), \end{aligned}$$ and embedding dimension $12$. Note that in both examples we have $(3,10) + \mathbb N^2 \subset \Gamma^s$. **Remark 24**. Recall from section [1](#secLips){reference-type="ref" reference="secLips"} that the Lipschitz saturation of an irreducible curve has multiplicity equal to its embedding dimension. The results and examples from this section shows that there is no general relation among the embedding dimension and the multiplicity of the Lipschitz saturation in higher dimensions. # Acknowledgements {#acknowledgements .unnumbered} The authors would like to thank professors P. Gonzalez Perez, C. Huneke and I. Swanson for the fruitful e-mail exchanges that greatly helped in the preparation of this work. They also acknowledge support by PAPIIT grant IN117523. A. Giles Flores acknowledges support by UAA grant PIM21-1. XXX Atiyah, M. F., Macdonald, I. G.; *Introduction to Commutative Algebra*, Addison-Wesly, 1969. J. Adamus, in *Topics in Complex Analytic Geometry Part II.* Lecture notes <https://www.uwo.ca/math/faculty/adamus/adamus_publications/AGII.pdf> L. Birbrair, A. Fernandes, J.E. Sampaio, M. Verbitsky; Multiplicity of singularities is not a bi-Lipschitz invariant, in *Mathematische Annalen 377*, pp. 115-121, Springer 2020. J. Briancon, A. Galligo, M. Granger; *Déformations équisinguliéres des germes de courbes gauches réduites.* Mém. Soc. Math. France, 2éme serie (1), **69**, 1980. D. Cox, J. Little, H. Schenck; *Toric Varieties*, Graduate Studies in Mathematics, Vol. 124, AMS, 2011. de Jong, T., Pfister, G.; *Local Analytic Geometry*, Vieweg, 2000. A. Fernandes; *Topological equivalence of complex curves and bi-Lipschitz maps*, Michigan Math. J, Vol 51, pp 593-606, 2003. I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky; *Discriminants, Resultants and Multidimensional Determinants*, Modern Birkhäuser Classics, 2008. G.M. Greuel, C. Lossen, E. Shustin; *Introduction to Singularities and Deformations*, Springer Monographs in Mathematics, 2007. A. Giles Flores, O.N. Silva, J. Snoussi; *On the fifth Whitney cone of a complex analytic curve*, Journal of Singularities, vol 24, pp. 96-118, 2022. A. Giles Flores, O.N. Silva, B. Teissier; The biLipschitz Geometry of Complex Curves: an algebraic approach, in *Introduction to Lipschitz Geometry of Singularities*, ed. by W. Neumann, A. Pichon. Lecture Notes in Mathematics, vol 2280, pp 217-271, Springer International Publishing, 2020. P. D. González Perez; *Quasi-ordinary singularities via toric geometry*, Ph. D. Thesis, Universidad de la Laguna, 2000. P. D. González Perez; *Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant.*, in Canadian Journal of Mathematics, 52(2), 348-368. doi:10.4153/CJM-2000-016-8 C. Huneke, I. Swanson, *Integral Closure of Ideals, Rings and Modules*, London Mathematical Society Lecture Note Series, no 336, Cambridge University Press, 2006. N. Kuhlmann; *Die Normalisierung komplexer Räume*, Math. Ann. 144 (1961), pp 110-125. M. Lejeune-Jalabert, B. Teissier; *Clôture intégrale des idéaux et equisingularité*, Ann. Fac. Sci. Tolouse Math. (6) **4**, 781-859, 2008. W.D. Neumann, A. Pichon; *Lipschitz Geometry of Complex Curves*, Journal of Singularities, vol 10, pp 225-234, 2014. F. Pham, B. Teissier; Lipschitz Fractions of a Complex Analytic Algebra and Zariski Saturation, in *Introduction to Lipschitz Geometry of Singularities*, ed. by W. Neumann, A. Pichon. Lecture Notes in Mathematics, vol 2280, pp 309-337, Springer International Publishing, 2020. J.C. Rosales, P.A. García-Sánchez, *Numerical Semigroups*, Developments in Mathematics (Springer Berlin), 2009. American Mathematical Society, Providence, RI, 1996. B.Teissier, Varétés polaires II: Multiplicités polaires, sections planes et conditions de Whitney, in *Algebraic geometry, Proc. int. Conf., La Rábida/Spain 1981*. Lecture Notes in Mathematics, vol. 961 (Springer Berlin, 1982), pp. 314-491. W. Vasconcelos; *Integral Closure: Rees algebras, multiplicities, algorithms*, Springer Monographs in Mathematics, 2005. H. Whitney, Local properties of analytic varieties, in *Differ. and Combinat. Topology, Sympos. Marston Morse, Princeton 1965*, pp. 205-244. O. Zariski; *Analytical irreducibility of normal varieties*, Ann. of Math., Vol. 131, No. 2, (1948), pp 352-361. [D. Duarte, Centro de Ciencias Matemáticas, UNAM.]{.smallcaps}\ E-mail: adduarte\@matmor.unam.mx\ [A. Giles Flores, Universidad Autónoma de Aguascalientes.]{.smallcaps}\ Email: arturo.giles\@edu.uaa.mx

arxiv_math

--- abstract: | In this note we identify the categories of lax and strong monoidal functors from a symmetric monoidal 1-category to a cartesian symmetric monoidal $\infty$-category with full subcategories of categories of hypersheaves, using Voevodsky's theory of cd-structures. In the same context we can characterise those lax or strong monoidal functors that are simultaneously hypersheaves for a topology generated by a sufficiently nice cd-structure. We use this characterisation to prove a lax monoidal version of the Comparison Lemma, our main application of this being a lax monoidal version of our previous result about hypersheaves encoding compactly supported cohomology theories. author: - Josefien Kuijper bibliography: - bibliography.bib title: Monoidal functors and monoidal sheaves --- # Introduction For $(\mathcal{C},\otimes)$ any symmetric monoidal $\infty$-category and $(\mathcal{D},\times)$ a Cartesian symmetric monoidal $\infty$-category, by [@HA Proposition 2.4.1.7] the 1-category of lax symmetric monoidal functors $\mathcal{C}\longrightarrow\mathcal{D}$ is equivalent to the subcategory of $\textup{Fun}(\mathcal{C}^\otimes,\mathcal{D})$ spanned by the *lax Cartesian structures*. These are functors $F:\mathcal{C}^\otimes \longrightarrow\mathcal{D}$ such that for every object $(X_i)_I$ in $\mathcal{C}^\otimes$ the map $$F((X_i)_I)\longrightarrow\prod_I F(X_i)$$ induced by the partial maps $I \dashrightarrow \{i\}$ is an equivalence. This looks a bit like a sheaf condition; in the rest of this note we make this idea precise. It turns out that this perspective can be particularly useful for studying lax monoidal functors which are also sheaves for some Grothendieck topology. **Convention 1**. For us a "sheaf" $F:\mathcal{C}^\textup{op}\longrightarrow\mathcal{D}$ with values in an $\infty$-category will always be a presheaf that satisfies descent for hypercovers, in other words a hypersheaf. We show that for $(\mathcal{C},\otimes)$ an arbitrary symmetric monoidal 1-category, and $\mathcal{D}$ a complete Cartesian symmetric monoidal $\infty$-category, the $\infty$-category of lax Cartesian structures $(\mathcal{C}^\textup{op})^\otimes \longrightarrow\mathcal{D}$ can be expressed as a full subcategory $$\textup{Fun}^{\textup{lax}}(\mathcal{C}^\textup{op},\mathcal{D})\subseteq \textup{Sh}(\mathcal{C}_\otimes;\mathcal{D})$$ of the $\infty$-category of $\mathcal{D}$-valued sheaves for the so-called *monoidal topology* on $\mathcal{C}_\otimes$, where $\mathcal{C}^\textup{op}_\otimes\longrightarrow\textup{Fin}_{\textup{part}}$ is the coCartesian fibration that captures the symmetric monoidal structure on $\mathcal{C}^\textup{op}$ (induced by the symmetric monoidal structure on $\mathcal{C}$). To be exact, $\textup{Fun}^{\textup{lax}}(\mathcal{C}^\textup{op},\mathcal{D})$ is the subcategory $\textup{Sh}(\mathcal{C}_\otimes;\mathcal{D})_0 \subseteq \textup{Sh}(\mathcal{C}_\otimes;\mathcal{D})$ of sheaves for the monoidal topology, which in addition send the zero object $0$ of $\mathcal{C}_\otimes$ to the terminal object $*$ of $\mathcal{D}$. Similarly, we can define a *strong monoidal* on $\mathcal{C}_\otimes$ such that sheaves for this topology, satisfying $F(0)\simeq *$, are strong monoidal functors $\mathcal{C}\longrightarrow\mathcal{D}$. Conveniently, the monoidal and strong monoidal topology are generated by a cd-structures $M_\mathcal{C}$ and $M_\mathcal{C}\cup S_\mathcal{C}$ on $\mathcal{C}_\otimes$. The theory of cd-structures was developed by Voevodsky in [@voevodsky]. The power of cd-structures with sufficiently nice properties, is that they generate topologies with an easy-to-check sheaf condition. For topologies generated by a "good\" cd-structure, the sheaf condition only needs to be checked on a collection of very simple covers (the ones given by a single square). Combining two good cd-structures always gives a good cd-structure, and the sheaf-condition for this combined topology therefore boils down to the sheaf conditions for the original topologies. A consequence of this observation is the following. If the underlying 1-category of $(\mathcal{C},\otimes)$ itself is already endowed with a good cd-structure $P$, then there is an induced good cd-structure $P_\otimes$ on $\mathcal{C}_\otimes$, and it turns out that a lax Cartesian structure $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}$ is a $P_\otimes$-sheaf if and only if the restriction $F|_\mathcal{C}:\mathcal{C}^\textup{op}\longrightarrow\mathcal{D}$ is a $P$-sheaf. Therefore sheaves for the combined topology on $\mathcal{C}_\otimes$, generated by the good cd-structure $M_\mathcal{C}\cup P_\otimes$, are exactly the lax Cartesian structures $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}$ whose restriction to $\mathcal{C}^\textup{op}$ is a $P$-sheaf, and similarly for sheaves for the topology generated by $M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes$. (In contrast, one can combine two arbitrary topologies $\tau_0,\tau_1$ by defining $\langle \tau_0\cup \tau_1 \rangle$ to be the smallest topology containing both $\tau_0$ and $\tau_1$. Then a presheaf that is a sheaf for $\tau_0$ and $\tau_1$ separately, can fail to be a sheaf for $\langle \tau_0\cup \tau_1 \rangle$.) We use this fact to prove a lax monoidal version of the Comparison Lemma [@comparison]. ## Outline and main results In Section [2](#section:preliminaries){reference-type="ref" reference="section:preliminaries"} we recall the necessary preliminaries about symmetric monoidal $\infty$-categories and cd-structures. In Section [3](#section:monodail_cd-struct){reference-type="ref" reference="section:monodail_cd-struct"} we define the *monoidal cd-structure* $M_\mathcal{C}$ and the *strong monoidal cd-structure* $M_\mathcal{C}\cup S_\mathcal{C}$. For $\mathcal{C}$ a 1-category with a Grothendieck topology, $\mathcal{D}$ an $\infty$-category with a terminal object $*$, and $C$ an object in $\mathcal{C}$, let $\textup{Sh}(\mathcal{C};\mathcal{D})_C$ denote the $\infty$-category of sheaves on $\mathcal{C}$ with values on $\mathcal{D}$ satisfying $F(C)\simeq *$. The main results in this sections are the following. **Proposition 1** (Proposition [Proposition 21](#prop:laxmonoidalfunctors){reference-type="ref" reference="prop:laxmonoidalfunctors"} and Proposition [Proposition 25](#prop:strongmonoidalfunctors){reference-type="ref" reference="prop:strongmonoidalfunctors"}). *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category and let $\mathcal{D}$ be a Cartesian symmetric monoidal $\infty$-category. Then* 1. *the $\infty$-category of lax monoidal functors $\textup{Fun}^\textup{lax}(\mathcal{C}^\textup{op},\mathcal{D})$ is equivalent to the $\infty$-category $\textup{Sh}_{M_\mathcal{C}}(\mathcal{C}_\otimes;\mathcal{D})_0$,* 2. *the $\infty$-category of strong monoidal functors $\textup{Fun}^\otimes(\mathcal{C}^\textup{op},\mathcal{D})$ is equivalent to the $\infty$-category $\textup{Sh}_{M_\mathcal{C}\cup S_\mathcal{C}}(\mathcal{C}_\otimes;\mathcal{D})_0$.* In Section [4](#section:monoidal_sheaves){reference-type="ref" reference="section:monoidal_sheaves"} we turn to the case where $\mathcal{C}$ is endowed with a sufficiently nice cd-structure, which generates a topology $\tau_P$ or $\tau^c_P$. We are interested in functors that are both lax/strong monoidal functors and sheaves for $\tau_P$ or $\tau^c_P$. For $\tau$ any Grothendieck topology on $\mathcal{C}$, we denote by $\textup{Fun}_\tau^\textup{lax}(\mathcal{C}^\textup{op}, \mathcal{D})$ the $\infty$-category of lax symmetric monoidal functors whose restriction is a $\tau$-sheaf, and by $\textup{Fun}_\tau^\otimes(\mathcal{C}^\textup{op}, \mathcal{D})$ the $\infty$-category of strong symmetric monoidal functors whose restriction is a $\tau$-sheaf. For $P$ as above, we define a cd-structure $P_\otimes$ on $\mathcal{C}_\otimes$. The main results in this section are the following. **Proposition 2** (Proposition [Proposition 37](#prop:laxmonoidalsheavesinf){reference-type="ref" reference="prop:laxmonoidalsheavesinf"} and Proposition [Proposition 40](#prop:strongmonoidalsheaves){reference-type="ref" reference="prop:strongmonoidalsheaves"}). *For $\mathcal{C}$ a symmetric monoidal 1-category, $P$ sufficiently nice cd-structure on $\mathcal{C}$, and $\mathcal{D}$ a complete Cartesian symmetric monoidal $\infty$-category,* - *we have equivalences of $\infty$-categories $$\textup{Fun}_{\tau_P}^\textup{lax}(\mathcal{C}^\textup{op}, \mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{\emptyset,0}$$ and $$\textup{Fun}_{\tau^c_P}^\textup{lax}(\mathcal{C}^\textup{op}, \mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{0},$$* - *we have equivalences of $\infty$-categories $$\textup{Fun}_{\tau_P}^\otimes(\mathcal{C}^\textup{op}, \mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{\emptyset,0}$$ and $$\textup{Fun}_{\tau^c_P}^\otimes(\mathcal{C}^\textup{op}, \mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{0}.$$* A use for these results is given in Section [5](#section:comparison){reference-type="ref" reference="section:comparison"}. Let $F:(\mathcal{C},\otimes)\longrightarrow(\mathcal{C},\otimes)$ be a strong monoidal functor of symmetric monoidal 1-categories, which are moreover endowed with sufficiently nice cd-structures $P$ and $P'$. **Proposition 3** (Proposition [Proposition 42](#prop:laxcomparison){reference-type="ref" reference="prop:laxcomparison"}). *Let $\tau$ and $\tau'$ be either $\tau_P$ and $\tau_{P'}$ or $\tau^c_P$ and $\tau^c_{P'}$ respectively. If $F$ induces an equivalence of 1-categories $$F:\textup{Sh}_{\tau'}(\mathcal{C}';\textup{Set}) \xrightarrow{\simeq} \textup{Sh}_{\tau}(\mathcal{C};\textup{Set}),$$ then for $\mathcal{D}$ a complete Cartesian symmetric monoidal $\infty$-category, there is an equivalence of $\infty$-categories $$\textup{Fun}_{\tau'}^\textup{lax}(\mathcal{C}^\textup{op},\mathcal{D}) \simeq \textup{Fun}_{\tau}^\textup{lax}(\mathcal{C}^\textup{op},\mathcal{D}).$$* **Remark 2**. In the case that the symmetric monoidal 1-categories $\mathcal{C}$ and $\mathcal{C}'$ are both cartesian, Proposition [Proposition 42](#prop:laxcomparison){reference-type="ref" reference="prop:laxcomparison"} is true when $\tau$, $\tau'$ are arbitrary topologies (not necessarily generated by cd-structures). In fact, under these assumption there is an equivalence of $\infty$-categories of strong monoidal sheaves $$\textup{Fun}_{\tau'}^\otimes(\mathcal{C}^\textup{op},\mathcal{D}) \simeq \textup{Fun}_{\tau}^\otimes(\mathcal{C}^\textup{op},\mathcal{D}),$$ by a generalisation of the proof of [@cirici_horel Proposition 6.6] by Cirici and Horel. Their proof uses Hinich' theory of strict localisations of symmetric monoidal $\infty$-categories [@hinich Section 3.2]. By the same arguments, using Hinich' notion of *right symmetric monoidal localisaiton* ([@hinich Section 3.3]) instead, we get an equivalence of $\infty$-categories of lax monoidal sheaves too. An application is given in Section [6](#section:application){reference-type="ref" reference="section:application"}. In algebraic geometry, cohomology theories of algebraic varieties can often be expressed as sheaves on a 1-category of varieties, where the topology is generated by a cd-structure. An example is the following 1-category. Let $k$ be a field of characteristic zero. **Definition 3**. Let $\mathbf{Span}$ be the 1-category with as objects algebraic varieties over $k$, where a morphism $X\longrightarrow Y$ is a span $$X\hookleftarrow U \xrightarrow{p} Y$$ where $U$ is an open subvariety of $X$ and $p$ is a proper morphism. We compose spans $X\hookleftarrow U \rightarrow Y$ and $Y\hookleftarrow V \rightarrow Y$ by taking the pullback of $U \longrightarrow Y$ along $V \hookrightarrow Y$. On this 1-category we can define a cd-structure $A\cup L$ given by abstract blowup squares, and squares of the form $$\label{eq:localisation_square} \begin{tikzcd} X\setminus U \arrow[r] & X \\ \emptyset \arrow[u,] \arrow[r] & U \arrow[u, hook, "i"] \end{tikzcd}$$ where $i:U\hookrightarrow X$ is an open immersion. If $\mathcal{C}$ is a stable $\infty$-category, then we can consider a sheaf $F$ in $\textup{Sh}(\mathbf{Span};\mathcal{C})_\emptyset$ as encoding a "compactly supported cohomology theory\" by setting $H^n_c(X)=\pi_{-n}F(X)$. Indeed, this description captures contravariance in proper maps, covariance in open immersions, descent for abstract blowups and the existence of localisation sequences, which are common attributes for classical cases of compactly supported cohomology theories, such as compactly supported singular cohomology and compactly supported homotopy algebraic $K$-theory. In earlier work [@kuij_descent] we show that for $\mathcal{C}$ a pointed, complete and cocomplete $\infty$-category, there is an equivalence of $\infty$-categories of sheaves $$\textup{Sh}(\mathbf{Span};\mathcal{C})_\emptyset \simeq \textup{Sh}(\mathbf{Comp};\mathcal{C})\simeq \textup{Sh}(\mathbf{SmComp};\mathcal{C}),$$ where $\mathbf{Comp}$ is the 1-category of complete varieties, endowed with the cd-structure of abstract blowup squares, and $\mathbf{SmComp}$ the 1-category of smooth and complete varieties, with the cd-structure of honest blowup squares. The meaning of this theorem is that any cohomology theory with descent for abstract blowups, has a compactly supported variant; and moreover, this compactly supported cohomology theory is uniquely determined by its restriction to smooth and complete varieties. In Theorem [Theorem 45](#thm:application){reference-type="ref" reference="thm:application"} we refine this result to an equivalence of $\infty$-categories of lax symmetric monoidal functors. A further application of this is given in [@kuij_6ff]. ## Acknowledgements I thank Tim Hosgood and Ivan Di Liberti for interesting conversations, and Dan Petersen for comments on an earlier version of this manuscript. # Preliminaries and conventions {#section:preliminaries} In this section we recall some terminology and notation; no new material is presented. ## Symmetric monoidal $\infty$-categories We recall some basics about symmetric monoidal $\infty$-categories, as treated in [@HA]. However, note that we think as symmetry monoidal $\infty$-categories as coCartesian fibrations over the 1-category of finite sets and partial maps, instead of the equivalent 1-category of pointed sets. **Definition 4**. Let $\textup{Fin}_\textup{part}$ be the 1-category of finite sets and *partial maps*, where a partial map $\alpha:J \dashrightarrow I$ is the data of a subset $\textup{dom}(\alpha)\subseteq J$ and a map of finite sets $\textup{dom}(\alpha)\longrightarrow I$. - A partial map between finite sets $\alpha:J \dashrightarrow I$ is called *inert* if on its domain $\textup{dom}(\alpha)$ it restricts to an isomorphism of finite sets $\textup{dom}(\alpha)\xrightarrow{\cong} I$; in other words $\alpha$ identifies $I$ with a subset of $J$. - A map $I \dashrightarrow J$ in $\textup{Fin}_\textup{part}$ is called *active* if $\textup{dom}(\alpha)=I$. - For $n\in \mathbb{N}$, let $\underline n$ denote the set $\{1,\dots, n\}$. - For $1\leq i\leq n$, let $\rho_i:\underline n \longrightarrow\underline 1$ denote the inert morphism identifying $\underline 1$ with the subset $\{i\}$ of $\underline n$. **Definition 5**. A *symmetric monoidal $\infty$-category* is a coCartesian fibration $$\mathcal{C}^\otimes \longrightarrow\textup{Fin}_\textup{part}$$ such that for each $n$, the maps $\rho_i:\underline n \longrightarrow\underline 1$ induce functors $$\mathcal{C}^\otimes_{\underline n} \longrightarrow\mathcal{C}^\otimes_{\underline 1}$$ which give an equivalence $$\mathcal{C}^\otimes_{\underline n} \simeq \prod_{\underline n} \mathcal{C}_{\underline 1}.$$ For $\mathcal{C}^\otimes$ a symmetric monoidal $\infty$-category, we denote by $\mathcal{C}$ the fibre $\mathcal{C}^\otimes_{\underline 1}$. We call this the underlying $\infty$-category of $\mathcal{C}^\otimes$. We note that $\mathcal{C}$ is embedded in $\mathcal{C}^\otimes$, via the embedding $\mathcal{C}\longrightarrow\mathcal{C}^\otimes$ sending an object $X$ in $\mathcal{C}$ to the object $(X)_{\underline{1}}$ in $\mathcal{C}^\otimes_{\underline 1}$. The object $(X)_{\underline 1}$ in $\mathcal{C}_\otimes$ will hereafter occasionally be denoted by $X$ as well. **Definition 6**. For $p:\mathcal{C}^\otimes\longrightarrow\textup{Fin}_\textup{part}$ and $q:\mathcal{D}^\otimes \longrightarrow\textup{Fin}_\textup{part}$ symmetric monoidal $\infty$-categories, a functor $F:\mathcal{C}^\otimes\longrightarrow\mathcal{D}^\otimes$ over $\textup{Fin}_\textup{part}$ is *lax symmetric monoidal* if it sends $p$-coCartesian morphisms over inert maps in $\textup{Fin}_\textup{part}$ to $q$-coCartesian morphisms. We denote the $\infty$-category of such functors by $$\textup{Fun}^\textup{lax}(\mathcal{C},\mathcal{D}).$$ A functor over $\textup{Fin}_\textup{part}$ is *strong symmetric monoidal* if it sends all $p$-coCartesian morphisms to $q$-coCartesian morphisms, and we denote the $\infty$-category of such functors by $$\textup{Fun}^\otimes(\mathcal{C},\mathcal{D}).$$ For $F:\mathcal{C}^\otimes\longrightarrow\mathcal{D}^\otimes$ a strong/lax monoidal functor, by abuse of notation we denote its restriction to $\mathcal{C}^\otimes_{\underline 1}$ by $$F:\mathcal{C}\longrightarrow\mathcal{D}$$ and call this the *underlying functor* of the strong/lax symmetric monoidal functor $F$. If $\mathcal{C}^\otimes$ and $\mathcal{D}^\otimes$ are symmetric monoidal $\infty$-categories arising from symmetric monoidal 1-categories by [@HA Construction 2.0.0.1], then these notions reduce to the usual notions of of lax and strong monoidal functors of symmetric monoidal 1-categories. In this note, a lax/strong symmetric monoidal functor of 1-categories will always be a functor $$F:\mathcal{C}^\otimes \longrightarrow\mathcal{D}^\otimes$$ over $\textup{Fin}_\textup{part}$. ## Cd-structures The following is after Voevodsky, see [@voevodsky]. A cd-structure on a 1-category $\mathcal{C}$ is a set of commutative squares $$\label{eq:dist_square} \begin{tikzcd} A \arrow[r] \arrow[d] & B \arrow[d,"p"]\\ C \arrow[r, "i"] & D. \end{tikzcd}$$ which we call *distinguished squares*. If $\mathcal{C}$ has an initial object $\emptyset$, then the *topology generated by* $P$ is the coarsest topology $\tau_P$ such that the empty cover covers $\emptyset$, and such that for each distinguished squares of the form ([\[eq:dist_square\]](#eq:dist_square){reference-type="ref" reference="eq:dist_square"}), the set of morphisms $\{i,p\}$ generates a covering sieve. Recall that an initial object $\emptyset$ in $\mathcal{C}$ is a *strict* initial object if any morphism $X \longrightarrow\emptyset$ is an isomorphism. The other extreme is that $\emptyset$ is a zero object, i.e., for any object $X$ in $\mathcal{C}$ there is a (unique) map $X \longrightarrow\emptyset$. In that case, for $\tau_P$ as above, the empty cover covers *any* object. This makes for an uninteresting topology, which is why we modify the definition as follows. The *coarse topology* or *c-topology* generated by $P$ is the coarsest topology $\tau_P^c$ such that for each distinguished squares of the form ([\[eq:dist_square\]](#eq:dist_square){reference-type="ref" reference="eq:dist_square"}), the set of morphisms $\{i,p\}$ generates a covering sieve. See [@kuij_descent Section 2] for more on the c-topology. **Convention 7**. For $P$ a cd-structure on a 1-category $\mathcal{C}$, the (coarse) topology generated by $P$ is equivalent to the (coarse) topology generated by the cd-structure which contains the squares in $P$ and in addition all squares of the form for $X$ in $\mathcal{C}$. Therefore we will assume that any cd-structure contains all such squares, and we call such a squares *degenerate squares*. We recall some additional definitions and results from [@voevodsky] and [@kuij_descent]. **Definition 8**. For $\mathcal{C}$ a 1-category with a cd-structure $P$, the class of simple $P$-coverings $S_P$ is the smallest class of families of morphisms $\{ U_i\longrightarrow X\}_{I}$ such that - $f:Y\longrightarrow X$ an isomorphism, $\{f:Y\longrightarrow X \}$ is in $S_P$, - for $\{ U_i\longrightarrow B\}_I$ and $\{V_j\longrightarrow C \}_J$ in $S_P$, and a distinguished square of the form ([\[eq:dist_square\]](#eq:dist_square){reference-type="ref" reference="eq:dist_square"}), the family of maps $\{U_i\longrightarrow B \xrightarrow{p}D, V_j\longrightarrow C \xrightarrow{i}D \}$ is in $S_P$. **Definition 9**. A cd-structure $P$ on a 1-category $\mathcal{C}$ is *c-complete* if any $\tau^c_P$ covering contains a simple $P$-cover. If $\mathcal{C}$ has an initial object $\emptyset$, then $P$ is *complete* if every $\tau_P$ of an object not isomorphic to $\emptyset$ contains a simple $P$-cover. For $\mathcal{C}$ a 1-category with a cd-structure $P$, let $\rho^c:\mathcal{C}\longrightarrow\textup{Sh}_{\tau^c_P}(\mathcal{C};\textup{Set})$ denote the Yoneda embedding composed with $\tau_P^c$-sheafification. If $\mathcal{C}$ has an initial object, then let $\rho:\mathcal{C}\longrightarrow\textup{Sh}_{\tau_P}(\mathcal{C};\textup{Set})$ denote the Yoneda embedding composed with $\tau_P$-sheafification. **Definition 10**. A cd-structure $P$ on a 1-category $\mathcal{C}$ is called *c-regular* if for every distinguished square of the form ([\[eq:dist_square\]](#eq:dist_square){reference-type="ref" reference="eq:dist_square"}), - the square is a pullback square - the map $i$ is a monomorphism - the map of $\tau^c_P$-sheaves $$\Delta \amalg \rho^c(i)\times \rho^c(i):\rho^c(B) \amalg \rho^c(A)\times_{\rho^c(C)} \rho^c(A) \longrightarrow\rho^c(B)\times_{\rho^c(D)}\rho^c(B)$$ is an epimorphism. If $\mathcal{C}$ has an initial object, then $P$ is called *regular* if (1) and (2) hold, and in addition, - the map of $\tau_P$-sheaves $$\Delta \amalg \rho(i)\times \rho(i):\rho(B) \amalg \rho(A)\times_{\rho(C)} \rho(A) \longrightarrow\rho(B)\times_{\rho(D)}\rho(B)$$ is an epimorphism. We recall the following. **Proposition 11** ([@voevodsky Lemma 2.9 and Proposition 2.15] and [@kuij_descent Proposition 2.10]). *Let $P$ be a cd-structure on a 1-category $\mathcal{C}$. If $P$ is c-complete and c-regular, then a presheaf $$F:\mathcal{C}^ \textup{op}\longrightarrow\textup{Set}$$ is a $\tau^c_P$-sheaf if and only if for every distinguished square of the form ([\[eq:dist_square\]](#eq:dist_square){reference-type="ref" reference="eq:dist_square"}),* *is a pullback square. If $\mathcal{C}$ has an initial object, then $F$ is a $\tau_P$-sheaf if it sends distinguished squares to pullback squares and in addition $F(\emptyset)=*$.* For presheaves with values in an $\infty$-category, the sheaf condition can be checked on distinguished squares too, with an additional requirement. **Definition 12** ([@kuij_descent Definition 3.1]). For $\mathcal{C}$ a 1-category with an initial object, a dimension function is a function $$\dim:\textup{Obj}(\mathcal{C})\longrightarrow\mathbb{Z}_{\geq-1}$$ such that for objects $X$ of $\mathcal{C}$, $\dim(X) = -1$ if and only if $X$ is isomorphic to $\emptyset$. **Definition 13** ([@kuij_descent Definition 3.2]). Let $\mathcal{C}$ be a 1-category with an initial object, a dimension function $\dim$ and a cd-structure $P$. Then the cd-structure and the dimension function are *compatible* if there is cd-structure $P' \subseteq P$ such that for every distinguished square of the form ([\[eq:dist_square\]](#eq:dist_square){reference-type="ref" reference="eq:dist_square"}) in $P'$ we have $\dim(C)\leq \dim(D)$, $\dim(B)\leq \dim(D)$ and $\dim(A)<\dim(D)$, and and such that for every square of the form ([\[eq:dist_square\]](#eq:dist_square){reference-type="ref" reference="eq:dist_square"}) in $P$, the sieve $\langle i,p \rangle$ contains a simple $P'$-cover. **Proposition 14** ([@voevodsky Proposition 3.8] and [@kuij_descent Corollary 3.15 and Proposition 4.7]). *Let $\mathcal{C}$ be a 1-category with an initial object, a dimension function $\dim$ and a compatible a cd-structure $P$. Let $\mathcal{D}$ be a complete $\infty$-category. If $P$ is c-complete and c-regular, then a presheaf $$F:\mathcal{C}^ \textup{op}\longrightarrow\mathcal{D}$$ is a $\tau^c_P$-sheaf if and only if for every distinguished square of the form ([\[eq:dist_square\]](#eq:dist_square){reference-type="ref" reference="eq:dist_square"}),* *is a pullback square in $\mathcal{D}$. If $\mathcal{C}$ has an initial object, then $F$ is a $\tau_P$-sheaf if it sends distinguished squares to pullback squares and in addition $F(\emptyset)$ is isomorphic to the terminal object in $\mathcal{D}$.* # The monoidal cd-structures {#section:monodail_cd-struct} Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category. Then $(\mathcal{C}^\textup{op},\otimes)$ is a symmetric monoidal 1-category as well. In the following definition, we spell out what is the result of applying [@HA Construction 2.0.0.1] to $(\mathcal{C}^\textup{op},\otimes)$ and introduce the following notation for this. **Notation 15** ([@HA Constuction 2.0.0.1]). For $(\mathcal{C},\otimes)$ a symmetric monoidal 1-category, define $\mathcal{C}_\otimes$ to be the 1-category which has as objects tuples $(X_i)_I$ indexed by a finite set, with $X_i$ in $\mathcal{C}$. A morphism $$(X_i)_I \longrightarrow(Y_j)_J$$ is given by a partial map $\alpha:J \dashrightarrow I$, and for all $i\in I$ a map $$X_i \longrightarrow\otimes_{j\in \alpha^{-1}(i)} Y_j.$$ Note that $\mathcal{C}_\otimes$ is actually just a convenient notation for $(\mathcal{C}^\textup{op})^{\otimes,\textup{op}}$; the forgetful functor $$\mathcal{C}_\otimes^\textup{op}\longrightarrow\textup{Fin}_{\textup{part}}$$ sending $(X_i)_I$ to $I$ is a coCartesian fibration, making $\mathcal{C}_\otimes^\textup{op}$ a symmetric monoidal $\infty$-category. ## Lax monoidal functors For $(\mathcal{D},\times)$ an $\infty$-category with finite products, let $\mathcal{D}^\times \longrightarrow\textup{Fin}_\textup{part}$ be the associated symmetric monoidal $\infty$-category as defined in [@HA Construction 2.4.1.14], and $\pi:\mathcal{D}^\times\longrightarrow\mathcal{D}$ the Cartesian structure as in [@HA Proposition 2.4.1.15]. We recall the following proposition in [@HA] which characterises lax monoidal functors $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}^\times$. **Proposition 16** ([@HA Proposition 2.4.1.7]). *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category and $\mathcal{D}$ an $\infty$-category with finite products. Then composition with $\pi$ induces an equivalence between the $\infty$-category $\textup{Fun}^\textup{lax}(\mathcal{C}^\textup{op},\mathcal{D})$ of lax symmetric monoidal functors $\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}^\times$, and the $\infty$-category of lax Cartesian structures, i.e., functors $$F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}$$ such that for $(X_i)_I$ in $\mathcal{C}$, the maps $X_i \longrightarrow(X_i)_I$ over the inert map $I \dashrightarrow \{i\}$ and given by $\textup{id}_{X_i}$, induce an isomorphism $$F((X_i)_I) \xlongrightarrow{\cong} \prod_I F(X_i).$$* As noted in the introduction, the condition on functors $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}$ to be a lax Cartesian structure, is reminiscent of a sheaf condition. Let $0$ denote the empty tuple $()_\emptyset$ in $\mathcal{C}_{\otimes}$, which is both initial and terminal. We define the following cd-structure on $\mathcal{C}_\otimes$. **Definition 17**. Let $M_\mathcal{C}$ be the sets of squares in $\mathcal{C}_\otimes$ of the form $$\label{eq:monoidalsquare} \begin{tikzcd} 0 \arrow[r] \arrow[d] & (X_i)_I \arrow[d, "p_{I,I\sqcup J}"] \\ (X_i)_J \arrow[r, "p_{J,I\sqcup J}"] & (X_i)_{I\sqcup J} \end{tikzcd}$$ where $I, J$ disjoint, and the morphisms $p_{I,I\sqcup J}$ and $p_{J,I\sqcup J}$ given by the inert maps $I\sqcup J \longrightarrow I$ and $I \sqcup J \longrightarrow J$, and the identities $X_i \longrightarrow X_i$ for $i$ in $I$ and $J$ respectively. We call this the *monoidal cd-structure* on $\mathcal{C}_\otimes$, and we call its squares monoidal squares. Since the initial object in $\mathcal{C}_\otimes$ is not a strict initial object, we only consider the coarse topology $\tau^c_{M_\mathcal{C}}$. **Lemma 18**. *For any symmetric monoidal $1$-category $(\mathcal{C},\otimes)$, the monoidal cd-structure $M_\mathcal{C}$ on $\mathcal{C}_\otimes$ is c-complete and c-regular.* *Proof.* To check c-completeness, we use the criterion of [@kuij_descent Lemma 2.9]. For $(X_i)_{I}$ in $\mathcal{C}_\otimes$, it is clear that the maps $p_{\{i\},I}:X_i \longrightarrow(X_i)_I$, induced by the partial map $I \dashrightarrow {i}$ sending $i$ to itself and the identity $X_i\longrightarrow X_i$, together form a simple cover. Let a distinguished square of the form ([\[eq:monoidalsquare\]](#eq:monoidalsquare){reference-type="ref" reference="eq:monoidalsquare"}) and a morphism $f:(Y_k)_K \longrightarrow(X_i)_{I\sqcup J}$ in $\mathcal{C}_\otimes$ be given, and let $\alpha:I\sqcup J\dashrightarrow K$ be the morphism in $\textup{Fin}_\textup{part}$ that lies underneath $f$. The pullback of $f$ along $p_{I,I\sqcup J}$ is the map $$(Y_k)_{(K\setminus \alpha(J))\cup \alpha(I)} \longrightarrow(Y_k)_K$$ given by the partial map that sends elements in $(K\setminus \alpha(J))\cup \alpha(I)$ to itself, and the identities $Y_j\longrightarrow Y_j$. Similarly, the pullback of $f$ along $p_{J,I\sqcup J}$ is $$(Y_k)_{(K\setminus \alpha(I))\cup \alpha(J)} \longrightarrow(Y_k)_K.$$ Since $$(K\setminus \alpha(J))\cup \alpha(I) \cup (K\setminus \alpha(I))\cup \alpha(J) = K,$$ in particular $f^*\langle i,j \rangle$ contains all the maps $p_{\{k\},K}: Y_k \longrightarrow(Y_k)_{K}$, forming a simple cover. This shows that $M_\mathcal{C}$ is c-complete. To show that $M_\mathcal{C}$ is c-regular, we use the criterion of [@voevodsky Lemma 2.11] (it is clear from the proof of [@voevodsky Lemma 2.11] that these conditions suffice to show that a cd-structure is not only regular, but also c-regular). It is easy to check that a square of the form ([\[eq:monoidalsquare\]](#eq:monoidalsquare){reference-type="ref" reference="eq:monoidalsquare"}) is a pullback square, and that $p_{J,I\sqcup J}$ is a monomorphism. The derived square (as in [@voevodsky Lemma 2.11(3)]) of a square of the form ([\[eq:monoidalsquare\]](#eq:monoidalsquare){reference-type="ref" reference="eq:monoidalsquare"}) is which is a distinguished square. This shows that $M_\mathcal{C}$ is c-regular. ◻ **Notation 19**. For $\mathcal{C}$ a 1-category with a Grothendieck topology $\tau$, $C$ an object of $\mathcal{C}$, and $\mathcal{D}$ an $\infty$-category with a terminal object $*$, let $\textup{Sh}(\mathcal{C};\mathcal{D})_C$ denote the full subcategory of $\textup{Sh}(\mathcal{C};\mathcal{D})$ on sheaves $F$ such that $F(C)=*$. In the case that $\mathcal{D}$ is a complete 1-category, Proposition [Proposition 11](#prop:cd_sheaves_of_sets){reference-type="ref" reference="prop:cd_sheaves_of_sets"} and Lemma [Lemma 18](#lem:compreg){reference-type="ref" reference="lem:compreg"} suffices to conclude that $$\textup{Fun}^\textup{lax}(\mathcal{C},\mathcal{D})\simeq \textup{Sh}_{\tau_{M_\mathcal{C}}^c}(\mathcal{C}_\otimes;\mathcal{D})_0.$$ If $\mathcal{D}$ is a complete $\infty$-category but *not* a 1-category, then we need the following. **Definition 20**. For $(\mathcal{C},\otimes)$ a symmetric monoidal 1-category, let $$d_0:\mathrm{Obj}(\mathcal{C}_\otimes)\longrightarrow\mathbb{Z}_{\geq -1}$$ be the function given by $d_0((X_i)_I)= |I|-1$. In particular $d_0((X_i)_I)= -1$ if and only if $(X_i)_I = 0$. It is clear that $M_\mathcal{C}$ is compatible with $d_0$. Now we can show the following. **Proposition 21**. *For $(\mathcal{C},\otimes)$ a symmetric monoidal 1-category and $\mathcal{D}$ a complete $\infty$-category, considered as a symmetric monoidal $\infty$-category $\mathcal{D}^\times$, there is an equivalence of $\infty$-categories $$\textup{Fun}^\textup{lax}(\mathcal{C}^\textup{op},\mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}}}(\mathcal{C}_\otimes;\mathcal{D})_0.$$* *Proof.* Since $M_\mathcal{C}$ is shown to be c-complete and c-regular in Lemma [Lemma 18](#lem:compreg){reference-type="ref" reference="lem:compreg"}, and compatible with a dimension function, Proposition [Proposition 14](#prop:cd_ sheaves){reference-type="ref" reference="prop:cd_ sheaves"} implies that the $\mathcal{D}$-valued sheaves on $\mathcal{C}_\otimes$ with respect to $\tau^c_{M_\mathcal{C}}$ are exactly the functors $$F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}$$ that send monoidal squares to pullback squares. If in addition $F(0)$ is the terminal object in $\mathcal{D}$, then this implies that for an object $(X_i)_{I}$, the natural map $$F((X_i)_I) \longrightarrow\prod_{I} F(X_i)$$ is an isomorphism. By Proposition [Proposition 16](#prop:laxmonoidalfunctorslurie){reference-type="ref" reference="prop:laxmonoidalfunctorslurie"} this 1-category is equivalent to $\textup{Fun}^\textup{lax}(\mathcal{C};\mathcal{D})$. ◻ ## Strong monoidal functors For $(\mathcal{C},\otimes)$ a symmetric monoidal 1-category and $\mathcal{D}$ an $\infty$-category with finite products, strong symmetric monoidal functors can be characterised in a way similar to Proposition [Proposition 16](#prop:laxmonoidalfunctorslurie){reference-type="ref" reference="prop:laxmonoidalfunctorslurie"}. **Proposition 22** ([@HA Proposition 2.4.1.7]). *Let $(C,\otimes)$ be a symmetric monoidal 1-category and $\mathcal{D}$ an $\infty$-category with products. Then composition with $\pi$ induces an equivalence between the $\infty$-category $\textup{Fun}^\otimes(\mathcal{C}^\textup{op};\mathcal{D})$ of strong symmetric monoidal functors $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}^\times$, and the $\infty$-category of functors $$F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}$$ such that for $(X_i)_I$ in $\mathcal{C}$,* - *the maps $X_i \longrightarrow(X_i)_I$ over the inert map $I \dashrightarrow \{i\}$ and given by $\textup{id}_{X_i}$, induce an isomorphism $$F((X_i)_I) \xlongrightarrow{\cong} \prod_I F(X_i),$$* - *and such that for $$(\otimes_I X_i)_{\underline {1}} \longrightarrow(X_i)_I$$ the morphism over the active map $I\dashrightarrow \underline 1$ and given by $\textup{id}_{\otimes_I X_i}$, its image under $F$ $$\prod_I F(X_i)_I \cong F((X_i)_I) \longrightarrow F(\otimes_I X_i)$$ is an equivalence.* As we have seen previously, the first condition on the functor $F:\mathcal{C}^\textup{op}_\otimes \longrightarrow\mathcal{D}$ can (almost) be enforced by a sheaf condition. The second one can be as well: this is just saying that $F$ should send morphisms of the form $(\otimes_I X_i)_{\underline {1}} \longrightarrow(X_i)_I$ to equivalences, i.e, treat them as covers. **Definition 23**. Let $S_\mathcal{C}$ be the cd-structure consisting of squares of the form $$\label{eq:strongmonoidalsquare} \begin{tikzcd} 0 \arrow[d] \arrow[r] & (\otimes_I X_i)_{\underline 1} \arrow[d,"p_\otimes"]\\ 0 \arrow[r,"e"] & (X_i)_I \end{tikzcd}$$ where the right vertical morphism lies over the map $I\dashrightarrow \underline 1$ sending all $i\in I$ to 1 and is given by the identity on $\otimes_I X_i$. We observe that $S_\mathcal{C}$ is compatible with the dimension function $d_0$ defined in Definition [Definition 20](#defn:dimfuncd0){reference-type="ref" reference="defn:dimfuncd0"}. We call $M_\mathcal{C}\cup S_\mathcal{C}$ the *strong monoidal cd-structure*. **Lemma 24**. *The cd-structure $M_\mathcal{C}\cup S_\mathcal{C}$ is c-complete and c-regular.* *Proof.* To show c-completeness, we need to check that for a square of the form ([\[eq:strongmonoidalsquare\]](#eq:strongmonoidalsquare){reference-type="ref" reference="eq:strongmonoidalsquare"}) and a morphism $f:(Y_j)_J \longrightarrow(X_i)_I$, the sieve $f^*\langle e,p_\otimes \rangle$ contains a simple $M_\mathcal{C}\cup S_\mathcal{C}$-cover. Let $\alpha:I\dashrightarrow J$ be the morphism under $f$. If $\textup{dom}(\alpha)=\emptyset$, then the pullback of $f$ along $p_\otimes$ is the identity on $(Y_j)_J$, so $f^*\langle e,p \rangle$ contains a simple cover. If $\textup{dom}(\alpha)\neq \emptyset$, then the pullback is where $Z_j = Y_j$ for $j\in J\setminus \textup{im}(\alpha)$ and $Z_1 = \otimes_{\textup{im}(\alpha)}Y_j$. The morphism $f^*p$ lies over the map $\beta:J\longrightarrow J\setminus \textup{im}(\alpha) \sqcup \underline 1$ sending $j\in J\setminus \textup{im}(\alpha)$ to itself and $j\in \textup{im}(\alpha)$ to 1, and is given by the identities $Z_k \longrightarrow\otimes_{\beta^{-1}(k)} Y_j$ for $k\in J\setminus \textup{im}(\alpha) \sqcup \underline 1$. Precomposing $f^*p$ with the maps $$p_{\{k\},J\setminus \textup{im}(\alpha)\sqcup \underline 1 }: (Y_j)_{\{k\}} \longrightarrow(Z_k)_{J\setminus \textup{im}(\alpha)\sqcup \underline 1}$$ for $k\in J\setminus \textup{im}(\alpha)\sqcup \underline 1$, we get the maps $$p_{\{j\},J}:(Y_j)_{\{j\}} \longrightarrow(Y_j)_J$$ for $j\in J\setminus \textup{im}(\alpha)$, and the map $$(\otimes_{\textup{im}(\alpha)} Y_j)_{\underline 1} \xrightarrow{p_\otimes} (Y_j)_{\textup{im}(\alpha)} \xrightarrow{p_{\textup{im}(\alpha),J}}(Y_j)_J,$$ which form a simple $M_\otimes \cup S_\otimes$-cover that is contained in $f^*\langle e,p_\otimes \rangle$. To show c-regularity we use the criterion of [@voevodsky Lemma 2.11]. It is easily checked that a square of the form ([\[eq:strongmonoidalsquare\]](#eq:strongmonoidalsquare){reference-type="ref" reference="eq:strongmonoidalsquare"}) is a pullback square and that $e$ is a monomorphism. We observe that $$(\otimes_I X_i)_{\underline 1} \times _{(X_i)_I} (\otimes_I X_i)_{\underline 1} = (\otimes_I X_i)_{\underline 1}$$ and therefore the derived square of ([\[eq:strongmonoidalsquare\]](#eq:strongmonoidalsquare){reference-type="ref" reference="eq:strongmonoidalsquare"}) is the square which is a distinguished square. From the proof of [@voevodsky Lemma 2.11] it follows that $S_\mathcal{C}$ is c-regular. By Lemma [Lemma 18](#lem:compreg){reference-type="ref" reference="lem:compreg"} $M_\mathcal{C}$ is c-regular as well, so it follows that $M_\mathcal{C}\cup S_\mathcal{C}$ is c-regular. ◻ Combining Lemma [Lemma 24](#lem:compreg2){reference-type="ref" reference="lem:compreg2"}, Proposition [Proposition 14](#prop:cd_ sheaves){reference-type="ref" reference="prop:cd_ sheaves"} and Proposition [Proposition 22](#prop:strongmonoidalfunctorslurie){reference-type="ref" reference="prop:strongmonoidalfunctorslurie"}, in the same way as in the proof of Proposition [Proposition 21](#prop:laxmonoidalfunctors){reference-type="ref" reference="prop:laxmonoidalfunctors"}, we get the following. **Proposition 25**. *For $(\mathcal{C},\otimes)$ a symmetric monoidal 1-category and $\mathcal{D}$ a complete $\infty$-category, considered as a symmetric monoidal $\infty$-category $\mathcal{D}^\times$, there is an equivalence of $\infty$-categories $$\textup{Fun}^\otimes(\mathcal{C}^\textup{op},\mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup S_\mathcal{C}}}(\mathcal{C}_\otimes;\mathcal{D})_0$$* # Monoidal sheaves {#section:monoidal_sheaves} In this section we study the situation where $(\mathcal{C},\otimes)$ is a symmetric monoidal $1$-category, and the underlying 1-category $\mathcal{C}$ is endowed with a cd-structure $P$. This occurs for example in algebraic geometry, where the 1-categories of (smooth/complete/...) varieties/schemes are often endowed with topologies generated by cd-structures, such as the Zariski topology, the (proper) cdh topology, the Nisnevich topology and more. The goal in this section is to characterise lax and strong monoidal functors $\mathcal{C}^\textup{op}\longrightarrow\mathcal{D}$ for $\mathcal{D}$ a Cartesian symmetric monoidal $\infty$-category, such that moreover the underlying functor $F:\mathcal{C}^\textup{op}\longrightarrow\mathcal{D}$ is a sheaf for the topology on $\mathcal{C}$. ## $\otimes$-stable cd-structures In order to make anything work, the symmetric monoidal structure on $\mathcal{C}$ has to be compatible with the given cd-structure in some way. **Definition 26**. Let $(\mathcal{C},\otimes)$ be a symmetric monoidal $1$-category such that the underlying 1-category $\mathcal{C}$ is endowed with a cd-structure $P$. Then $P$ is called a $\otimes$-*stable* cd-structure if for a distinguished square $$\label{eq:gendistsquare} \begin{tikzcd} B \arrow[r]\arrow[d]&A\arrow[d] \\ Y \arrow[r] & X \end{tikzcd}$$ and an object $Z$ of $\mathcal{C}$, the square is also distinguished **Definition 27**. For $(\mathcal{C},\otimes)$ a symmetric monoidal 1-category and $P$ a cd-structure on $\mathcal{C}$, we define a cd-structure $P_\otimes$ on $\mathcal{C}_\otimes$ as follows. The distinguished squares are the squares of the form $$\label{eq:p_otimessquare} \begin{tikzcd} (B_i)_I \arrow[d] \arrow[r] & (A_i)_I \arrow[d, "p "]\\ (Y_i)_I \arrow[r, "e"] & (X_i)_I \end{tikzcd}$$ over the identity $I \dashrightarrow I$, where for at most one $i_0 \in I$, the square is a non-degenerate distinguished square; for all other $i\neq i_0$, the square is a degenerate square. **Remark 28**. Via the embedding $$\mathcal{C}\longrightarrow\mathcal{C}_{\otimes},\ X \mapsto (X)_{\underline 1}$$ we can consider $P$ as a cd-structure on $\mathcal{C}_\otimes$. The cd-structures $M_\mathcal{C}\cup P$ and $M_\mathcal{C}\cup P_\otimes$ generate the same topology, but we use $P_\otimes$ rather than $P$ to make proofs in the following section easier. Using the inductive nature of the definition of simple covers, one can show the following. **Lemma 29**. *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category, and $P$ a cd-structure on $\mathcal{C}$.* - *A simple $P_\otimes$-cover is a collection of maps $$\{(U ^j_i)_I \longrightarrow(X_i)_I \}_{j\in J}$$ over $\textup{id}:I \dashrightarrow I$, such that for every $i$, the collection $\{U^j_i \longrightarrow X_i\}_{j\in J}$ is a simple $P$-cover.* - *A simple $M_\mathcal{C}\cup P_\otimes$-cover is a collection of maps $$\{ (U^k_i)_{i\in I_j} \longrightarrow(X_i)_{i\in I} \}_{j\in J, k\in K_j}$$ such that the $I_j$ for $j\in J$ form a partition of $I$, the underlying partial maps $I \dashrightarrow I_j$ are inert, and such that for fixed $i$ in $I_j$, the collection $$\{ U^i_k \longrightarrow X_i \}_{k\in K_j }$$ is a simple $P$-cover.* **Lemma 30**. *Let $P$ be a $\otimes$-stable cd-structure on a symmetric monoidal $1$-category $(\mathcal{C},\otimes)$. If $P$ is complete or c-complete, then the cd-structure $P_\otimes$ on $\mathcal{C}_\otimes$ is c-complete.* *Proof.* We use [@kuij_descent Lemma 2.9]. Let us consider a distinguished square $Q$ of the form ([\[eq:p_otimessquare\]](#eq:p_otimessquare){reference-type="ref" reference="eq:p_otimessquare"}) and a morphism $f:(Z_j)_J \longrightarrow(X_i)_I$ over the partial map $\alpha:I\dashrightarrow J$, given by maps $Y_j \longrightarrow\otimes_{\alpha^{-1}(j)} X_i$. Let $i_0\in I$ be the index such that $$\label{eq:distsquareproof1} \begin{tikzcd} B_{i_0} \arrow[d] \arrow[r] & A_{i_0} \arrow[d]\\ Y_{i_0}\arrow[r] & X_{i_0} \end{tikzcd}$$ is a non-degenerate distinguished square. For $j=\alpha(i_0)$, the square $$\label{eq:distsquareproof2} \begin{tikzcd} \otimes_{\alpha^{-1}(j)} B_i \arrow[d] \arrow[r] & \otimes_{\alpha^{-1}(j)} A_i \arrow[d, "p '"] \\ \otimes_{\alpha^{-1}(j)} Y_i \arrow[r, "e'"] & \otimes_{\alpha^{-1}(j)} X_i \end{tikzcd}$$ which is really the square ([\[eq:distsquareproof1\]](#eq:distsquareproof1){reference-type="ref" reference="eq:distsquareproof1"}) tensored with $\otimes_{\alpha^{-1}(j)\setminus \{i_0\}} X_i$, is distinguished since $P$ is a $\otimes$-stable cd-structure. Therefore, by the assumption that $P$ is complete or c-complete, the sieve $f_j^*\langle e', p'\rangle$ contains a simple cover, say $\{U^k\longrightarrow Z_j \}_{k\in K}.$ Consider the maps $$(Z^{k}_j)_J \longrightarrow(Z_j)_J$$ ranging over $k\in K$ , where $Z^{k}_j = Z_j$ for $j\neq \alpha(i_0)$, and $Z^{k}_{\alpha(i_0)} = U^k$. Then this is a simple $P_\otimes$-cover contained in $f^* \langle e,p \rangle$. ◻ We recall that unions of c-complete cd-structures are c-complete; therefore if $P$ is a complete or c-complete cd-structure, then $M_\mathcal{C}\cup P_\otimes$ is c-complete. We cannot quite show that for $P$ a regular or c-regular cd-structure, $P_\otimes$ is c-regular. The following Lemma applies in certain cases when $P$ is regular or c-regular. **Lemma 31**. *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category with a cd-structure $P$, such that for any distinguished square $Q$ of the form ([\[eq:gendistsquare\]](#eq:gendistsquare){reference-type="ref" reference="eq:gendistsquare"}), the following hold:* 1. *$Q$ is a pullback square,* 2. *$e$ is a monomorphism,* 3. *the derived square* *(with the vertical arrows diagonals) exists, and is a distinguished square.* *Then $P_\otimes$ is c-regular.* *Proof.* It is easily seen that for $Q$ a square in $P_\otimes$ of the form ([\[eq:p_otimessquare\]](#eq:p_otimessquare){reference-type="ref" reference="eq:p_otimessquare"}), $Q$ is a pullback square and $e$ is a monomorphism. Let $i_0\in I$ be the index for which the square is non-degenerate. The derived square exists, and is the square which is degenerate for $i\neq i_0$, and by assumption a distinguished square in $P$ for the index $i_0$. By the proof of [@voevodsky Lemma 2.11], this implies that $P_\otimes$ is c-regular. ◻ The following lemma gives another criterion on $P$ which guarantees that $M_\mathcal{C}\cup P_\otimes$ is c-regular. **Lemma 32**. *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category with a $\otimes$-stable cd-structure $P$. Suppose that $P$ is complete (or c-complete), that representable presheaves are separated for the induced topology $\tau_P$ (or for $\tau^c_P$), and that for $Q$ a distinguished square in $P$ of the form ([\[eq:gendistsquare\]](#eq:gendistsquare){reference-type="ref" reference="eq:gendistsquare"}),* - *$e$ is a monomorphism,* - *$Q$ is a pullback square,* - *the morphism $$y_Y \amalg (y_B \times_{y_A} y_B) \longrightarrow y_Y \times_{y_X} y_Y$$ is a locally surjective morphism of presheaves.* *Then $M_\mathcal{C}\cup P_\otimes$ is a c-regular cd-structure on $\mathcal{C}_\otimes$.* *Proof.* We need to check that the distinguished squares in $P_\otimes \cup M_\mathcal{C}$ satisfy the conditions of [@voevodsky Definition 2.10]. Since $M_\mathcal{C}$ is regular, we know that for monoidal squares conditions (1) and (2) hold, and the morphism of $\tau^c_{M_\mathcal{C}}$-sheaves in condition (3) is an epimorphism. Since further sheafification (from $\tau^c_{M_\mathcal{C}}$-sheaves to $\tau^c_{M_\mathcal{C}\cup P_\otimes}$-sheaves) preserves finite limits and colimits, the associated morphism of $\tau^c_{M_\mathcal{C}\cup P_\otimes}$-sheaves is an epimorphism as well. Now we check the conditions for squares in $P_\otimes$. It is easily seen that for a square $Q$ of the form ([\[eq:p_otimessquare\]](#eq:p_otimessquare){reference-type="ref" reference="eq:p_otimessquare"}), $Q$ is a pullback square and $e$ is a monomorphism. To check the third condition for c-regularity on squares in $P_\otimes$, we claim that under the given assumptions, representable presheaves on $\mathcal{C}_\otimes$ are separated with respect to $\tau^c_{M_\mathcal{C}\cup P_\otimes}$; the proof of this is deferred to Lemma [Lemma 33](#lem:representables){reference-type="ref" reference="lem:representables"}. We consider a distinguished $P_\otimes$-square of the form [\[eq:p_otimessquare\]](#eq:p_otimessquare){reference-type="ref" reference="eq:p_otimessquare"}, and let $$F := y_{(Y_i)_I} \amalg (y_{(B_i)_I} \times_{y_{(A_i)_I}} y_{(B_i)_I} )$$ and $$G := y_{(Y_i)_I} \times_{y_{(X_i)_I}} y_{(Y_i)_I}.$$ Both are separated presheaves with respect to $\tau^c_{M_\mathcal{C}\cup P_\otimes}$, since representable presheaves are. In order to show that $M_\mathcal{C}\cup P_\otimes$ is c-regular, we need to show that the induced map on sheafifications $F^\sharp \longrightarrow G^\sharp$ is an epimorphism of sheaves. It is enough to show that $F\longrightarrow G$ is an epimorphism of separated presheaves, and by [@kuij_descent Lemma 6.25] this is the case if it is locally surjective. Consider an element in $G ((Z_j)_J)$. This consists of a pair of maps $$f,f':(Z_j)_J \longrightarrow(Y_i)_I$$ such that $e\circ f=e\circ f'$. This implies that $f,f'$ lie over the same $\alpha:I \dashrightarrow J$, and for $j\in J$, the square commutes, making $(f_j,f_j')$ sections in $y_{\otimes_{\alpha^{-1}(j)}Y_i } \times_{y_{\otimes_{\alpha^{-1}(j)}X_i }}y_{\otimes_{\alpha^{-1}(j)}Y_i } (Z_j)$. Since the morphism of presheaves $$\phi:y_{\otimes_{\alpha^{-1}(j)} Y_j} \amalg y_{\otimes_{\alpha^{-1}(j)}B_i } \times_{y_{\otimes_{\alpha^{-1}(j)}A_i }}y_{\otimes_{\alpha^{-1}(j)}B_i } \longrightarrow y_{\otimes_{\alpha^{-1}(j)}Y_i } \times_{y_{\otimes_{\alpha^{-1}(j)}X_i }}y_{\otimes_{\alpha^{-1}(j)}Y_i }$$ corresponding to the distinguished square ([\[eq:distsquareproof2\]](#eq:distsquareproof2){reference-type="ref" reference="eq:distsquareproof2"}) is locally surjective, there is a simple $P$-cover $$\{U^k_j\longrightarrow Z_j \}_{k\in K_j}$$ of $Z_j$ on which the section $(f_j,f'_j)$ is in the image of $\phi$. Now $$\{(U^k_j)_{\{j \}} \longrightarrow(Y_j)_{\{j\}} \longrightarrow(Y_j)_J \}_{ j\in J,k\in K_j}$$ is a simple $M_\mathcal{C}\cup P_\otimes$-cover on which $f, f'$ are in the image of $F \longrightarrow G$, showing that this is an epimorphism of separated presheaves. ◻ **Lemma 33**. *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category with a $\otimes$-stable cd-structure $P$. Suppose that $P$ is complete (or c-complete) and that representable presheaves are separated for the induced topology $\tau_P$ (or for $\tau^c_P$). Then representable presheaves on $\mathcal{C}_\otimes$ are separated with respect to $\tau^c_{M_\mathcal{C}\cup P_\otimes}$.* *Proof.* Let $(X_i)_I$ and $(Y_j)_J$ be objects in $\mathcal{C}_\otimes$. Let $$f,f':(X_i)_I \longrightarrow(Y_j)_J$$ be elements of $y_{(Y_j)_J}((X_i)_I)$, lying over partial maps $$\alpha,\alpha':J \dashrightarrow I$$ such that $f,f'$ agree on a $\tau^c_{M_\mathcal{C}\cup P_\otimes}$-cover of $(X_i)_I$. Since $M_\mathcal{C}\cup P_\otimes$ is c-complete, this cover contains a simple $M)\mathcal{C}\cup P$-cover, which is of the form $$\{ (U^k_i)_{i\in I_l} \longrightarrow(X_i)_{i\in I} \}_{l\in L, k\in K_l}.$$ by Lemma [Lemma 29](#lem:simplecovers){reference-type="ref" reference="lem:simplecovers"}. Since $f$ and $f'$ agree on this simple cover, in particular for all $l\in L$ the composites of $\alpha$ and $\alpha'$ with the inert map $I \dashrightarrow I_l$ are equal. Since the $I_l$ for $l\in L$ form a partition of $I$, this implies that $\alpha=\alpha'$. Now for each $l$ and $i\in I_l$, we have $$f_i,f'_i:X_i \longrightarrow\otimes_{\alpha^{-1}(i)} Y_j$$ which agree on the simple $P$-cover $$\{U^k_i \longrightarrow X_i \}_{k\in K_l}$$ and since $y_{\otimes_{\alpha^{-1}(i)} Y_j}$ is separated by assumption, this implies $f_i=f'_i$ for all $i$, hence $f=f'$. This shows that $y_{(X_i)_I}$ is separated. ◻ ## Lax monoidal sheaves With the setup from the previous subsection, we can now study functors $\mathcal{C}^\textup{op}\longrightarrow\mathcal{D}$ which are both lax monoidal functors and sheaves for $\tau_P$ or $\tau^c_P$, where $P$ is a cd-structure. **Definition 34**. For $(\mathcal{C},\otimes)$ a symmetric monoidal 1-category, $\tau$ a Grothendieck topology on $\mathcal{C}$, and $\mathcal{D}^\otimes$ a symmetric monoidal $\infty$-category, let $$\textup{Fun}^\textup{lax}_\tau(\mathcal{C}^\textup{op},\mathcal{D})$$ denote the $\infty$-category of lax monoidal functors $F:\mathcal{C}^\textup{op}_\otimes \longrightarrow\mathcal{D}^\otimes$ such that the underlying functor $F:\mathcal{C}^\textup{op}\longrightarrow\mathcal{D}$ is a $\tau$-sheaf. We call such functors *lax monoidal sheaves*. For $P$ a cd-structure, we will occasionally denote $\textup{Fun}^\textup{lax}_{\tau_P}(\mathcal{C}^\textup{op},\mathcal{D})$ or $\textup{Fun}^\textup{lax}_{\tau^c_P}(\mathcal{C}^\textup{op},\mathcal{D})$ by $\textup{Fun}^\textup{lax}_P(\mathcal{C}^\textup{op},\mathcal{D})$ as well, when it is clear from context which of the two is meant. By what we showed about $M_\mathcal{C}\cup P_\otimes$ in the previous section, we can prove the following. **Proposition 35**. *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category, $P$ a $\otimes$-stable cd-structure on $\mathcal{C}$, and $\mathcal{D}$ a complete 1-category, considered as symmetric monoidal 1-category $(\mathcal{D},\times)$. If $P$ is complete and satisfies the assumptions of Lemma [Lemma 31](#lem:Potimesregular0){reference-type="ref" reference="lem:Potimesregular0"} or Lemma [Lemma 32](#lem:Potimesregular){reference-type="ref" reference="lem:Potimesregular"}, then there is an equivalence of 1-categories $$\textup{Fun}^\textup{lax}_{\tau_P}(\mathcal{C}^\textup{op};\mathcal{D}) \simeq \textup{Sh}_{\tau^c_{ M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{\emptyset,0} .$$ If $P$ is c-complete and satisfies the assumptions of Lemma [Lemma 31](#lem:Potimesregular0){reference-type="ref" reference="lem:Potimesregular0"} or Lemma [Lemma 32](#lem:Potimesregular){reference-type="ref" reference="lem:Potimesregular"}, then there is an equivalence of 1-categories $$\textup{Fun}^\textup{lax}_{\tau^c_P}(\mathcal{C}^\textup{op};\mathcal{D}) \simeq \textup{Sh}_{\tau^c_{ M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{0}$$* *Proof.* We show the first equivalence, the proof of the second one is very similar. Recall that by Proposition [Proposition 21](#prop:laxmonoidalfunctors){reference-type="ref" reference="prop:laxmonoidalfunctors"} we have an equivalence $$\pi\circ -:\textup{Fun}^\textup{lax}(\mathcal{C}^\textup{op},\mathcal{D})\simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}}}(\mathcal{C};\mathcal{D})_0.$$ Since $M_\mathcal{C}\cup P_\otimes$ is c-complete and c-regular, by Proposition [Proposition 11](#prop:cd_sheaves_of_sets){reference-type="ref" reference="prop:cd_sheaves_of_sets"} the 1-category $\textup{Sh}_{\tau^c_{M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{\emptyset 0}$ consists of the presheaves $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}$ that send squares in $M_\mathcal{C}\cup P_\otimes$ to distinguished squares, and satisfy $F(0)\cong F(\emptyset)\cong *$. Therefore, restricting the equivalence $\pi\circ -$ on the right to $\tau^c_{M_\mathcal{C}\cup P_\otimes}$-sheaves, yields on the left lax symmetric monoidal functors $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}^\times$ such that the underlying functor $F:\mathcal{C}^\textup{op}\longrightarrow\mathcal{D}$ sends $P$-squares to pullback squares, and $F(\emptyset)$ to $*$. On the other hand, let $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}^\times$ be a lax symmetric monoidal functor such that the underlying functor $F:\mathcal{C}^\textup{op}\longrightarrow\mathcal{D}$ is a $\tau^c_P$-sheaf; equivalently $F$ sends squares in $P$ to pullbacks and $F(\emptyset)\cong *$. Then composing $F$ with $\pi:\mathcal{D}^\times \longrightarrow\mathcal{D}$ gives a sheaf $\pi\circ F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}$ in $\textup{Sh}_{\tau^c_{M_\mathcal{C}}}(\mathcal{C};\mathcal{D})_{\emptyset,0}$. Moreover $\pi\circ F$ sends a square in $P_\otimes$ of the form ([\[eq:p_otimessquare\]](#eq:p_otimessquare){reference-type="ref" reference="eq:p_otimessquare"}) to the square which is the product of pullback squares and therefore a pullback square. This shows that $\pi\circ F$ is in $\textup{Sh}_{\tau^c_{ M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{\emptyset,0}$. ◻ The cd-structure $P_\otimes$ is not necessarily compatible with previously defined dimension function $d_0$; but if $P$ is already compatible with a dimension function on $\mathcal{C}$, then we can define a compatible dimension function on $\mathcal{C}_\otimes$ compatible with $P_\otimes$. **Definition 36**. For $(\mathcal{C},\otimes)$ a symmetric monoidal 1-category and $d:\mathrm{Obj}(\mathcal{C})\longrightarrow\mathbb{Z}_{\geq -1}$ a dimension function, we define $$d_1:\mathrm{Obj}(\mathcal{C}_\otimes)\longrightarrow\mathbb{Z}_{\geq -1}$$ by setting $d_1((X_i)_I)=|I|-1 + \sum_{i\in I}d(X_i)$. It is clear that both $M_\mathcal{C}$ and $P_\otimes$ are compatible with $d_1$. By the same proof as that of Proposition [Proposition 35](#prop:laxmonoidalsheaves1){reference-type="ref" reference="prop:laxmonoidalsheaves1"}, but using Proposition [Proposition 14](#prop:cd_ sheaves){reference-type="ref" reference="prop:cd_ sheaves"} instead of Proposition [Proposition 11](#prop:cd_sheaves_of_sets){reference-type="ref" reference="prop:cd_sheaves_of_sets"}, we can conclude the following. **Proposition 37**. *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category, $P$ a $\otimes$-stable cd-structure on $\mathcal{C}$, and $\mathcal{D}$ a complete $\infty$-category, considered as symmetric monoidal $\infty$-category $\mathcal{D}^\times$. Assume moreover that $P$ is compatible with a dimension function on $\mathcal{C}$. If $P$ is complete and regular, then there is an equivalence of $\infty$-categories $$\textup{Fun}^\textup{lax}_{\tau_P}(\mathcal{C}^\textup{op},\mathcal{D})\simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{\emptyset,0}.$$ If $P$ is c-complete and c-regular, then there is an equivalence of $\infty$-categories $$\textup{Fun}^\textup{lax}_{\tau^c_P}(\mathcal{C}^\textup{op},\mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{0}.$$* ## Strong monoidal sheaves In this section we specialise to functors that are both strong monoidal functors and sheaves. **Notation 38**. For $(\mathcal{C},\otimes)$ a symmetric monoidal 1-category, $\tau$ a Grothendieck topology on $\mathcal{C}$, and $\mathcal{D}^\otimes$ a symmetric monoidal $\infty$-category, let $$\textup{Fun}^\otimes_\tau(\mathcal{C}^\textup{op},\mathcal{D})$$ denote the $\infty$-category of strong monoidal functors $F:\mathcal{C}^\textup{op}_\otimes \longrightarrow\mathcal{D}^\otimes$ such that the underlying functor $F:\mathcal{C}\longrightarrow\mathcal{D}$ is a $\tau$-sheaf. We call such functors *strong monoidal sheaves*. We will denote $\textup{Fun}^\otimes_{\tau_P}(\mathcal{C}^\textup{op},\mathcal{D})$ or $\textup{Fun}^\otimes_{\tau^c_P}(\mathcal{C}^\textup{op},\mathcal{D})$ by $\textup{Fun}^\otimes_P(\mathcal{C}^\textup{op},\mathcal{D})$ as well, when it is clear from context which of the two is meant. As we saw in Lemma [Lemma 30](#lem:potimescomplete){reference-type="ref" reference="lem:potimescomplete"}, if $P$ is complete, then $P_\otimes$ is c-complete. Since $M_\mathcal{C}\cup S_\mathcal{C}$ is c-complete by Lemma [Lemma 24](#lem:compreg2){reference-type="ref" reference="lem:compreg2"}, this implies that $M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes$ is c-complete. On the other hand, from the proof of Lemma [Lemma 24](#lem:compreg2){reference-type="ref" reference="lem:compreg2"} we know that $S_\mathcal{C}$ is c-regular, and if $P$ satisfies the conditions of Lemma [Lemma 31](#lem:Potimesregular0){reference-type="ref" reference="lem:Potimesregular0"} or Lemma [Lemma 32](#lem:Potimesregular){reference-type="ref" reference="lem:Potimesregular"}, then $M_\mathcal{C}\cup P_\otimes$ is c-regular. This implies that $M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes$ is c-regular. Therefore we have an analogue of Proposition [Proposition 35](#prop:laxmonoidalsheaves1){reference-type="ref" reference="prop:laxmonoidalsheaves1"} for strong monoidal sheaves. **Proposition 39**. *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category with a $\otimes$-stable cd-structure $P$, and let $\mathcal{D}$ be a complete 1-category, considered as symmetric monoidal 1-category $(\mathcal{D},\times)$. Then if $P$ is complete and satisfies the conditions of Lemma [Lemma 31](#lem:Potimesregular0){reference-type="ref" reference="lem:Potimesregular0"} or Lemma [Lemma 32](#lem:Potimesregular){reference-type="ref" reference="lem:Potimesregular"}, there is an equivalence of 1-categories $$\textup{Fun}^\otimes_{\tau_P}(\mathcal{C}^\textup{op};\mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{\emptyset,0} .$$ If $P$ is c-complete and satisfies the conditions of Lemma [Lemma 31](#lem:Potimesregular0){reference-type="ref" reference="lem:Potimesregular0"} or Lemma [Lemma 32](#lem:Potimesregular){reference-type="ref" reference="lem:Potimesregular"}, then there is an equivalence of 1-categories $$\textup{Fun}^\otimes_{\tau^c_P}(\mathcal{C}^\textup{op},\mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes }}(\mathcal{C}_\otimes;\mathcal{D})_{0}$$* *Proof.* The proof is very similar to that of Proposition [Proposition 35](#prop:laxmonoidalsheaves1){reference-type="ref" reference="prop:laxmonoidalsheaves1"}, and again we only show the first equivalence. By Proposition [Proposition 25](#prop:strongmonoidalfunctors){reference-type="ref" reference="prop:strongmonoidalfunctors"} we have the equivalence $$\pi\circ-:\textup{Fun}^\otimes(\mathcal{C}^\textup{op},\mathcal{D}) \xrightarrow{\sim} \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup S_\mathcal{C}}}(\mathcal{C}^\textup{op},\mathcal{D})_{0}.$$ It is clear that restricting on the right to $\tau^c_{M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes}$-sheaves which in addition send $\emptyset$ to $*$, on the right this yields strong symmetric monoidal functors $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}^\times$ for which the underlying functor $F:\mathcal{C}^\textup{op}\longrightarrow\mathcal{D}$ sends $P$-squares to pullback squares and $\emptyset$ to $*$, i.e, is a $\tau_P$-sheaf. On the other hand, if $F:\mathcal{C}_\otimes^\textup{op}\longrightarrow\mathcal{D}^\times$ is a strong symmetric monoidal functor sending $\emptyset$ to $*$ and $P$-squares to pullbacks, $\pi\circ F$ sends $P_\otimes$-squares to pullbacks as well, as in the proof of Proposition [Proposition 35](#prop:laxmonoidalsheaves1){reference-type="ref" reference="prop:laxmonoidalsheaves1"}. ◻ Let $d$ is a dimension function on a symmetric monoidal 1-category $(\mathcal{C},\otimes)$. We call $d$ $\otimes$-stable if $d$ satisfies $$d(X\otimes Y)\leq d(X)\cdot d(Y)$$ for all $X,Y$ in $\mathcal{C}$. If this is the case, then $S_\mathcal{C}$ is compatible with $d_1$ as defined in Definition [Definition 36](#defn:dimfuncd1){reference-type="ref" reference="defn:dimfuncd1"}. Therefore we also have an analogue of Proposition [Proposition 37](#prop:laxmonoidalsheavesinf){reference-type="ref" reference="prop:laxmonoidalsheavesinf"} for strong monoidal sheaves. **Proposition 40**. *Let $(\mathcal{C},\otimes)$ be a symmetric monoidal 1-category, $P$ a $\otimes$-stable cd-structure on $P$, and $\mathcal{D}$ a complete $\infty$-category, considered as symmetric monoidal $\infty$-category $\mathcal{D}^\times$. Assume moreover that $P$ is compatible with a dimension function $d$ on $\mathcal{C}$, and that $d$ is $\otimes$-stable. If $P$ is complete and satisfies the conditions of Lemma [Lemma 31](#lem:Potimesregular0){reference-type="ref" reference="lem:Potimesregular0"} or Lemma [Lemma 32](#lem:Potimesregular){reference-type="ref" reference="lem:Potimesregular"}, then there is an equivalence of $\infty$-categories $$\textup{Fun}^\otimes_{\tau_P}(\mathcal{C}^\textup{op},\mathcal{D})\simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{\emptyset,0}.$$ If $P$ is c-complete and satisfies the conditions of Lemma [Lemma 31](#lem:Potimesregular0){reference-type="ref" reference="lem:Potimesregular0"} or Lemma [Lemma 32](#lem:Potimesregular){reference-type="ref" reference="lem:Potimesregular"}, then there is an equivalence of $\infty$-categories $$\textup{Fun}^\otimes_{\tau^c_P}(\mathcal{C}^\textup{op},\mathcal{D}) \simeq \textup{Sh}_{\tau^c_{M_\mathcal{C}\cup S_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{D})_{0}.$$* # The monoidal comparison lemma {#section:comparison} In this section we consider the question: for $F:(\mathcal{C},\otimes,\tau)\longrightarrow(\mathcal{C}',\otimes,\tau')$ a morphism of symmetric monoidal 1-categories equipped with Grothendieck topologies, and $\mathcal{D}$ a Cartesian symmetric monoidal $\infty$-category, when does restriction along $F$ induce an equivalence of $\infty$-categories of lax monoidal sheaves $$F^*:\textup{Fun}^\textup{lax}_{\tau'}(\mathcal{C}'^\textup{op},\mathcal{D}) \xrightarrow{\simeq} \textup{Fun}^\textup{lax}_\tau(\mathcal{C}^\textup{op},\mathcal{D})?$$ We first observe that, for $\mathcal{D}$ any cocomplete $\infty$-category, it suffices to show such an equivalence for the 1-categories of set-valued sheaves. **Lemma 41**. *Let $F:(\mathcal{C},\tau)\longrightarrow(\mathcal{C}',\tau')$ be a morphism of sites such that $f$ induces an equivalence $$F^*:\textup{Sh}_{\tau'}(\mathcal{C}';\textup{Set}) \xrightarrow{\simeq} \textup{Sh}_\tau(\mathcal{C},\textup{Set}).$$ Then for $\mathcal{D}$ any complete $\infty$-category, $F$ induces an equivalence of $\infty$-categories $$F^*:\textup{Sh}_{\tau'}(\mathcal{C}';\mathcal{D}) \xrightarrow{\simeq} \textup{Sh}_\tau(\mathcal{C};\mathcal{D}).$$* *Proof.* See [@kuij_descent Remark 4.3 and Corollary 4.6]. ◻ The Comparison Lemma ([@comparison]) gives sufficient conditions on a functor $F:\mathcal{C}\longrightarrow\mathcal{D}$ between 1-categories equipped with Grothendieck topologies, such that restriction along $F$ induces an equivalence $$F^*:\textup{Sh}(\mathcal{D};\textup{Set}) \xrightarrow{\simeq} \textup{Sh}(\mathcal{C};\textup{Set}).$$ By Lemma [Lemma 41](#lem:sheaves_of_sets){reference-type="ref" reference="lem:sheaves_of_sets"} this implies that for any complete $\infty$-category $\mathcal{E}$, $F$ induces an equivalence of $\infty$-categories $$F^*:\textup{Sh}(\mathcal{D};\mathcal{E}) \longrightarrow\textup{Sh}(\mathcal{C};\mathcal{E}).$$ Now we consider the case where $\mathcal{C}$ and $\mathcal{D}$ are symmetric monoidal 1-categories, and $F:(\mathcal{C},\otimes) \longrightarrow(\mathcal{D}, \otimes)$ a strong monoidal functor. If $\mathcal{C}$ and $\mathcal{D}$ are endowed with $\otimes$-stable and (c-)complete cd-structures $P$ and $P'$, such that the underlying functor $F:\mathcal{C}\longrightarrow\mathcal{D}$ satisfies the comparison lemma, then $F_\otimes:\mathcal{C}_\otimes \longrightarrow\mathcal{D}_\otimes$ satisfies the comparison lemma with respect to the topologies $\tau^c_{M_\mathcal{C}\cup P_\otimes}$ and $\tau^c_{M_\mathcal{D}\cup P'_\otimes}$. **Proposition 42**. *Let $F:\mathcal{C}\longrightarrow\mathcal{D}$ be a strong symmetric monoidal functor of symmetric monoidal 1-categories equipped with (c-)complete and $\otimes$-stable cd-structures $P$ and $P'$, such that $f$ satisfies the condition of the Comparison Lemma ([@comparison]) with respect to the (coarse) topologies generated by $P$ and $P'$. Then the induced map $F_\otimes:\mathcal{C}_\otimes \longrightarrow\mathcal{D}_\otimes$ satisfies the Comparison Lemma with respect to the coarse topologies generated by the cd-structures $M_\mathcal{C}\cup P_\otimes$ and $M_\mathcal{D}\cup P'_\otimes$. If in addition the cd-structures $P$ and $P'$ satisfy the conditions of Lemma [Lemma 31](#lem:Potimesregular0){reference-type="ref" reference="lem:Potimesregular0"} or Lemma [Lemma 32](#lem:Potimesregular){reference-type="ref" reference="lem:Potimesregular"}, then it follows that for $\mathcal{E}$ a complete Cartesian symmetric monoidal $1$-category, there is an equivalence of $\infty$-categories $$\textup{Fun}^\textup{lax}_{P'}(\mathcal{D};\mathcal{E})\cong \textup{Fun}^\textup{lax}_P(\mathcal{C},\mathcal{E}),$$ and if $P$ and $P'$ are compatible with dimension functions, then this also holds if $\mathcal{E}$ is a complete Cartesian symmetric monoidal $\infty$-category.* *Proof.* We check the conditions of the Comparison Lemma with respect to $F_\otimes:\mathcal{C}_\otimes\longrightarrow\mathcal{D}_\otimes$ and the topologies $\tau^c _{M_\mathcal{C}\cup P_\otimes}$ and $\tau^c_{ M_\mathcal{D}\cup P'_\otimes}$. We observe that both $M_\mathcal{C}\cup P_\otimes$ and $M_\mathcal{D}\cup P'_\otimes$ are c-complete. - To see that $F_\otimes$ preserves covers, if suffices to check this for simple $M_\mathcal{C}\cup P_\otimes$-covers. By Lemma [Lemma 29](#lem:simplecovers){reference-type="ref" reference="lem:simplecovers"} such a cover looks like $$\mathcal{U}=\{ (U^j_i)_{i\in I_j} \longrightarrow(X_i)_{i\in I} \}_{j\in J}$$ such that every underlying partial map $I \dashrightarrow I_i$ is inert, and such that for fixed $i$, the collection $$\{ U^i_j \longrightarrow X_i \}_{\{ j\in J \mid i\in I_j \} }$$ is a simple $P$-cover. The image of this cover under $F_\otimes$ is $$F_\otimes \mathcal{U}:= \{ (F(U^j_i))_{i\in I_j} \longrightarrow(F(X_i))_{i\in I} \}_{j\in J}$$ and for fixed $i$, by assumption $$\{ F(U^i_j) \longrightarrow F(X_i) \}_{\{ j\in J \mid i\in I_j \} }$$ contains a simple $P'$-cover, say $$\{V^k_i \longrightarrow F(X_i) \}_{k\in K_i}.$$ Composing with $(F(X_i))_{\{i\}} \longrightarrow(F(X_i))_I$ now gives a simple $M_\mathcal{D}\cup P'_\otimes$ cover $$\mathcal{V}:= \{(V^k_i)_{\{i\}} \longrightarrow(F(X_i))_I \}_{k\in \amalg_I K_i}.$$ Since every $$(V^k_j)_{\{i\}} \longrightarrow(F(X_i))_{\{i\}} \longrightarrow(F(X_i))_I$$ factors as the simple cover $\mathcal{V}$ is contained in the sieve generated by $F_\otimes \mathcal{U}$, so $F_{\otimes }\mathcal{U}$ is a $\tau'_{M_\mathcal{D}\cup P'_\otimes}$-cover. - To show that $F_\otimes$ is locally full, let $$f:(F(Y_j))_J \longrightarrow(F(X_i))_I$$ be a morphism in $\mathcal{D}_\otimes$ between objects in the image of $F$. Let $f$ be given by a partial map $\alpha:I \dashrightarrow J$ and maps $f_j:F(Y_j) \longrightarrow F(\otimes_{i\in \alpha^{-1}(j)} X_i) \cong \otimes_{i\in \alpha^{-1}(j)} F(X_i)$. Since $F$ is locally full, for each $j$ there is a simple $P$-cover $$\{u^j_k:U^j_k \longrightarrow Y_j \}_{k\in K_j}$$ and maps $a^j_k:U^j_k \longrightarrow\otimes_{i\in \alpha^{-1}(j)} X_i$ such that commutes. Now the maps $$\{U_k^j \xrightarrow{u^k_j} Y_j \longrightarrow(Y_j)_J \}_{j\in J, k\in K_j}$$ form a simple $M_\mathcal{C}\cup P_\otimes$-cover. Consider compositions $$U^k_j \xrightarrow{a^k_j} \otimes_{\alpha^{-1}(j)} X_j \longrightarrow(X_i)_I$$ in $\mathcal{C}_\otimes$, where the second morphism is given by the partial map $I \dashrightarrow *$ defined on $\alpha^{-1}(j)$, and the identity on $\otimes_{\alpha^{-1}(j)} X_i$. Now the diagrams commute, showing that $F_\otimes$ is locally full. - To show that $F_\otimes$ is locally faithful, let $f,f':(Y_j)_J \longrightarrow(X_i)_I$ be morphisms in $\mathcal{C}_\otimes$ such that $F(f)=F(j')$. This implies in particular that $f,f'$ lie over the same partial map $\alpha:I \dashrightarrow J$ and for $i\in I$, $$F(f_i)=F(f'_i):F(Y_j) \longrightarrow F(\otimes_{\alpha^{-1}(j)} X_i) \cong \otimes_{\alpha^{-1}(j)} F(X_i).$$ Since $F$ is locally faithful, this implies that for each $j$ there is a simple $P$-cover $$\{u^j_k:U^j_k \longrightarrow Y_j \}_{k\in K_j}$$ such that $f_i\circ u^k_j = f'_i\circ u^k_j$ for all $k$. The maps $$\{U_k^j \xrightarrow{u^k_j} Y_j \longrightarrow(Y_j)_J \}_{j\in J, k\in K_j}$$ now form a simple $M_\mathcal{C}\cup P_\otimes$-cover on which $f$ and $f'$ agree, showing that $F_\otimes$ is locally faithful. - To show that $F_\otimes$ is locally surjective, let $(X_i)_I$ be an object of $\mathcal{D}_\otimes$. Since $F$ is locally surjective, for each $i$ there is a simple $P'$-cover $\{u^j_i: F(U^j_i) \longrightarrow X_i \}_{j\in J_i}$ of $X_i$ by objects in the image of $F$. Now the maps $$\{F(U^j_i) \xrightarrow{u^j_i} X_i \longrightarrow(X_i)_I \}_{i\in I, j\in J_i}$$ form a simple $M_\mathcal{D}\cup P'_\otimes$-cover of $(X_i)_I$ by objects in the image of $F$. - To show that $F_\otimes$ is cocontinuous, let $\{ (U^j_i)_{i\in I_j} \longrightarrow(F(X_i))_I \}_{j\in J}$ be a simple $P'_\otimes \cup M_\mathcal{D}$-cover of an object in the image of $F_\otimes$. Then for $i\in I$ the collection $\{u^j_i: U^j_i \longrightarrow F(X_i) \}_{j \in \{j\in J\mid i\in I_j \}}$ is a simple $P'$-cover. Since $F$ is cocontinuous, the collection of maps $v:V \longrightarrow X_i$ such that $F(v)$ factors through some $u^j_i$ contains some simple $P$-cover $$\{V^k_i \longrightarrow X_i \}_{k\in K_i}.$$ Then the maps $$\{(V^k_i)_{\{i\}} \longrightarrow(X_i)_{\{i\}} \longrightarrow(X_i)_I \}$$ form a simple $M_\mathcal{C}\cup P_\otimes$-cover, such that every map in it factors through some $U^j_i \longrightarrow(U^j_i)_{I_j} \longrightarrow(F(X_i))_I$. This shows that $F_\otimes$ is cocontinuous. From the Comparison Lemma it follows that $$\textup{Sh}_{\tau^c_{M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\textup{Set})\simeq \textup{Sh}_{\tau^c_{ M_\mathcal{D}\cup P'_\otimes}}(\mathcal{D}_\otimes,\textup{Set}),$$ and therefore also $$\textup{Sh}_{\tau^c_{M_\mathcal{C}\cup P_\otimes}}(\mathcal{C}_\otimes;\mathcal{E})\simeq \textup{Sh}_{\tau^c_{ M_\mathcal{D}\cup P'_\otimes}}(\mathcal{D}_\otimes,\mathcal{E})$$ for $\mathcal{E}$ a complete $\infty$-category. This equivalence clearly restricts to the full subcategory of sheaves sending $0$ to $*$, and by Proposition [Proposition 37](#prop:laxmonoidalsheavesinf){reference-type="ref" reference="prop:laxmonoidalsheavesinf"} the result follows. ◻ # An application {#section:application} In this section we use the results in this note to prove a version of [@kuij_descent Theorem 7.2] for lax monoidal sheaves. We recall some notation from[@kuij_descent]. All our varieties are over a fixed field $k$ of characteristic zero. ## Cd-structures on categories of varieties Recall $\mathbf{Span}$ from Definition [Definition 3](#defn:span){reference-type="ref" reference="defn:span"}. As in the introduction, $\mathbf{Comp}$ is the 1-category of complete varieties, and $\mathbf{SmComp}$ the 1-category of smooth and complete varieties. Let $A$ be the set of pullback squares of varieties $$\label{eq:absblowupsquare} \begin{tikzcd} E \arrow[d]\arrow[r]& Y\arrow[d, "p"]\\ C \arrow[r, "i"]& X \end{tikzcd}$$ where $i$ is a closed immersion, $p$ is a proper morphism, and $p$ induces an isomorphism $p: p^{-1}(X \setminus C) \longrightarrow X \setminus C$. We call these *abstract blowup squares*. Note that we can consider this as a cd-structure on $\mathbf{Span}$. Let $L$ be the set of squares of the form ([\[eq:localisation_square\]](#eq:localisation_square){reference-type="ref" reference="eq:localisation_square"}). On $\mathbf{Span}$ we consider the cd-structure $A\cup L$. Let $AC$ be the subset of $A$ consisting of all abstract blowup squares of complete varieties. This is the cd-structure we consider on $\mathbf{Comp}$. Lastly, on $\mathbf{SmComp}$ we consider the cd-structure $B$ of pullback squares in $\mathbf{SmComp}$ where $C \hookrightarrow X$ is a closed immersion and $Bl_C X$ the blowup of $X$ in $C$. We recall the following result. **Theorem 43** ([@kuij_descent Theorem 7.2]). *For $\mathcal{C}$ a complete and cocomplete pointed $\infty$-category, there are equivalences of $\infty$-categories $$\textup{Sh}_{\tau^c_{A\cup L}}(\mathbf{Span};\mathcal{C})_\emptyset \simeq \textup{Sh}_{\tau_{AC}}(\mathbf{Comp};\mathcal{C})\simeq \textup{Sh}_{\tau_B}(\mathbf{SmComp};\mathcal{C}).$$* In the proof of this theorem, an intermediate 1-category $\mathbf{Comp}_0$ is used. We obtain this 1-category from $\mathbf{Comp}$ by the following construction. **Definition 44** ([@kuij_descent Definition 5.1]). Let $\mathcal{X}$ a 1-category with strict initial object $\emptyset$, for every $X$ in $\mathcal{X}$ we denote the unique morphism $\emptyset\longrightarrow X$ by $0$. We define $\mathcal{X}_0$ be the 1-category with the same objects as $\mathcal{X}$, $\textup{Hom}_{\mathcal{X}_0}(\emptyset,X)= \textup{Hom}_\mathcal{X}(\emptyset,X)=\{0\}$, and $$\textup{Hom}_{\mathcal{X}_0}(X,Y) = \textup{Hom}_\mathcal{X}(X,Y) \sqcup \{0 \}$$ if $X$ not isomorphic to $\emptyset$. The composition is determined by the composition in $\mathcal{X}$ and the rule that $f \circ 0 = 0 \circ f = 0$ for all $f$. On $\mathbf{Comp}_0$ we consider the topology $\tau^c_{AC}$. Observe that $\mathbf{Comp}_0$ embeds into $\mathbf{Span}$ by sending a zero morphism $0:X \longrightarrow Y$ to the span $X \hookleftarrow \emptyset \longrightarrow Y$. The Cartesian product of varieties induces a symmetric monoidal structure on $\mathbf{Span}$, $\mathbf{Comp}_0$, $\mathbf{Comp}$, $\mathbf{SmComp}$. However, beware that this monoidal structure is only Cartesian for $\mathbf{Comp}$ and $\mathbf{SmComp}$; the product of varieties is not a categorical product in $\mathbf{Span}$ and $\mathbf{Comp}$. The cd-structures $B$ and $AC$ on $\mathbf{Comp}$ are complete, and $AC$ on $\mathbf{Comp}_0$ and $A\cup L$ are c-complete. Furthermore the cd-structures $B$ and $A\cup L$ satisfy the conditions of Lemma [Lemma 32](#lem:Potimesregular){reference-type="ref" reference="lem:Potimesregular"} (see the proof of [@voevodskyb Lemma 4.5], and the proof of [@kuij_descent Lemma 6.23]), and the cd-structure $A^\mathrm{comp}$ on both $\mathbf{Comp}$ and $\mathbf{Comp}_0$ satisfies the conditions of Lemma [Lemma 31](#lem:Potimesregular0){reference-type="ref" reference="lem:Potimesregular0"} (see the proof of [@voevodskyb Lemma 2.14]). Lastly, all of these cd-structures are compatible with a dimension function (given by the dimension of an algebraic variety). Therefore for each of these sites, Propostion [Proposition 37](#prop:laxmonoidalsheavesinf){reference-type="ref" reference="prop:laxmonoidalsheavesinf"} and Proposition [Proposition 25](#prop:strongmonoidalfunctors){reference-type="ref" reference="prop:strongmonoidalfunctors"} apply. ## Monoidal compactly supported cohomology theories In this section we upgrade Theorem [Theorem 43](#thm:old_theorem){reference-type="ref" reference="thm:old_theorem"} to $\infty$-categories of lax monoidal sheaves. An application is given in [@kuij_6ff]. **Theorem 45**. *For $\mathcal{C}$ a complete and cocomplete pointed Cartesian symmetric monoidal $\infty$-category, there are equivalences of $\infty$-categories of lax monoidal sheaves $$\textup{Fun}^\textup{lax}_{A\cup L}(\mathbf{Span}^\textup{op},\mathcal{C})_\emptyset \simeq \textup{Fun}_{A^\mathrm{comp}}^\textup{lax}(\mathbf{Comp}_0^\textup{op},\mathcal{C})_\emptyset \simeq \textup{Fun}_{A^{\mathrm{comp}}}^\textup{lax}(\mathbf{Comp}^\textup{op},\mathcal{C}) \simeq \textup{Fun}^\textup{lax}_B(\mathbf{SmComp}^\textup{op},\mathcal{C})$$* *Proof.* Theorem 7.2 of l[@kuij_descent] gives, for $\mathcal{C}$ a complete and cocomplete pointed $\infty$-category, equivalences of $\infty$-categories $$\textup{Sh}_{A\cup L}(\mathbf{Span};\mathcal{C})_{\emptyset} \xrightarrow{\sim} \textup{Sh}_{A^\mathrm{comp}}(\mathbf{Comp}_0;\mathcal{C})_\emptyset \xrightarrow{\sim} \textup{Sh}_{A^\mathrm{comp}}(\mathbf{Comp};\mathcal{C}) \xrightarrow{\sim}\textup{Sh}_{B}(\mathbf{SmComp};\mathcal{C}).$$ For each of these three equivalences, we will show that the corresponding map between $\infty$-categories of lax monoidal sheaves is an equivalence. **Step 1.** For the equivalence $\textup{Fun}^\textup{lax}_{A\cup L}(\mathbf{Span}^\textup{op},\mathcal{C})_\emptyset \simeq \textup{Fun}^\textup{lax}_{A^\mathrm{comp}}(\mathbf{Comp}_0^\textup{op},\mathcal{C})_\emptyset$ induced by precomposition with the embedding $i:\mathbf{Comp}_0\longrightarrow\mathbf{Span}$, we already know that $i$ satisfies the conditions of the Comparison Lemma (see [@kuij_descent Lemma 7.8]). Therefore we can apply Propositon [Proposition 42](#prop:laxcomparison){reference-type="ref" reference="prop:laxcomparison"} to conclude that $\textup{Fun}_{A\cup L}^\textup{lax}(\mathbf{Span}^\textup{op},\mathcal{C})\simeq \textup{Fun}_{A^{\mathrm{comp}}}^\textup{lax}(\mathbf{Comp}^\textup{op},\mathcal{C})$, and restricting to lax monoidal sheaves $F$ satisfying $F(\emptyset)=*$, we get $$\textup{Fun}_{A\cup L}^\textup{lax}(\mathbf{Span}^\textup{op},\mathcal{C})_\emptyset \simeq \textup{Fun}_{A^\mathrm{comp}}^\textup{lax}(\mathbf{Comp}_0^\textup{op},\mathcal{C})_\emptyset.$$ **Step 2.** For the equivalence $\textup{Fun}_{A^\mathrm{comp}}^\textup{lax}(\mathbf{Comp}_0^\textup{op},\mathcal{C})_\emptyset \simeq \textup{Fun}_{A^\mathrm{comp}}^\textup{lax}(\mathbf{Comp}^\textup{op},\mathcal{C})$ we know that there is no equivalence of toposes $\textup{Sh}_{A^\mathrm{comp}}(\mathbf{Comp}_0;\textup{Set})\simeq \textup{Sh}_{A^\mathrm{comp}}(\mathbf{Comp};\textup{Set})$, so no version of the comparison lemma is applicable here. However, we claim that there is an equivalence $$\textup{Fun}^\textup{lax}(\mathbf{Comp}_0^\textup{op},\mathcal{C})_\emptyset \simeq \textup{Fun}^\textup{lax}(\mathbf{Comp}^\textup{op},\mathcal{C})_\emptyset$$ between $\infty$-categories of lax monoidal functors sending $\emptyset$ to $*$. It is clear that this equivalence restricts to lax monoidal functors sending squares in $A_\otimes$ to pullback squares, and this gives the equivalence $$\textup{Fun}_{A^\mathrm{comp}}^\textup{lax}(\mathbf{Comp}_0^\textup{op},\mathcal{C})_\emptyset \simeq \textup{Fun}_{A^\mathrm{comp}}^\textup{lax}(\mathbf{Comp}^\textup{op},\mathcal{C}),$$ (note that $\tau_{A^\mathrm{comp}}$-sheaves on $\mathbf{Comp}$ already send $\emptyset$ to $*$, whereas $\tau^c_{A^\mathrm{comp}}$-sheaves on $\mathbf{Comp}_0$ do not). We defer the proof of the claim to Lemma [Lemma 46](#lem:claim){reference-type="ref" reference="lem:claim"}. **Step 3.** For the equivalence $\textup{Fun}_{A^{\mathrm{comp}}}^\textup{lax}(\mathbf{Comp}^\textup{op},\mathcal{C}) \simeq \textup{Fun}^\textup{lax}_B(\mathbf{SmComp}^\textup{op},\mathcal{C})$, we observe that the the inclusion $\mathbf{SmComp}\longrightarrow\mathbf{Comp}$ satisfies the conditions of the Comparison Lemma (see for example the proof of [@voevodskyb Lemma 4.7]). Since in the symmetric monoidal structure on $\mathbf{Comp}$ as well as on $\mathbf{SmComp}$ is Cartesian, we can use Proposition [Proposition 42](#prop:laxcomparison){reference-type="ref" reference="prop:laxcomparison"} to conclude that $$\textup{Fun}_{A^{\mathrm{comp}}}^\textup{lax}(\mathbf{Comp}^\textup{op},\mathcal{C}) \simeq \textup{Fun}^\textup{lax}_B(\mathbf{SmComp}^\textup{op},\mathcal{C}).$$ ◻ The following lemma implies the claim in Step 2 of the proof above, for $\mathcal{X}$ the 1-category $\mathbf{Comp}$. Let $(\mathcal{X},\otimes)$ an arbitrary symmetric monoidal 1-category with a strict initial object $\emptyset$. Let $\mathcal{X}_0$ be defined as in Definiton [Definition 44](#defn:zero_object){reference-type="ref" reference="defn:zero_object"}. Then $(\mathcal{X}_0,\otimes)$ is a symmetric monoidal 1-category as well. **Lemma 46**. *Let $(\mathcal{X},\otimes)$ be a symmetric monoidal 1-category with a strict initial object $\otimes$. Let $\mathcal{C}$ be a pointed, cocomplete $\infty$-category. Restriction along the inclusion $i:\mathcal{X}_\otimes \longrightarrow(\mathcal{X}_0)_\otimes$ induces an equivalence $$i^*:\textup{Fun}^\textup{lax}(\mathcal{X}_0^\textup{op},\mathcal{C})_\emptyset \xrightarrow{\simeq} \textup{Fun}^\textup{lax}(\mathcal{X}^\textup{op}, \mathcal{C})_\emptyset.$$ between the subcategories of lax monoidal functors $F$ such that $F(\emptyset)$ is equivalent to the zero object of $\mathcal{C}$.* *Proof.* It is easily seen that precomposition with $i$ sends $\tau^c_{M_{\mathcal{X}_0}}$-sheaves to $\tau^c_{M_\mathcal{X}}$-sheaves. To show that $i^*$ is an equivalence, we first observe that $i^*$, as defined on all of $\textup{PSh}((\mathcal{X}_0)_\otimes;\mathcal{C})$, is part of an adjunction There is also an adjunction where $L$ denotes the sheafification functor, and $U$ the inclusion of sheaves into presheaves. Since $i^*\circ U$ lands in $\textup{Sh}_{\tau^c_{M_{\mathcal{X}}}}(\mathcal{X}_\otimes;\mathcal{C})$, there is an induced adjunction and we will show this induces an equivalence $\textup{Sh}_{\tau^c_{M_{\mathcal{X}_0}}}((\mathcal{X}_0)_\otimes;\mathcal{C})_{\emptyset,0} \simeq \textup{Sh}_{\tau^c_{M_\mathcal{X}}}(\mathcal{X}_\otimes;\mathcal{C})_{\emptyset,0}$. For $F:\mathcal{X}_\otimes^\textup{op}\longrightarrow\mathcal{C}$ and $(X_i)_I$ in $(X_0)_\otimes$, we observe that $\textup{Lan}_i F((X_i)_I)$ can be computed as the colimit of $\mathcal{X}_\otimes^\textup{op}\times_{(\mathcal{X}_0)_\otimes^\textup{op}} ((\mathcal{X}_{0})_\otimes^\textup{op})_{(X_i)_I/} \rightarrow \mathcal{C}$; in other words, as $$\underset{(X_i)_I\longrightarrow(Y_j)_J}{\textup{colim}}\ F((Y_j)_J)$$ over the diagram of morphisms $(X_i)_I\longrightarrow(Y_j)_J$ in $(\mathcal{X}_0)_\otimes$, where a morphism from $(X_i)_I\longrightarrow(Y_j)_J$ to $(X_i)_I\longrightarrow(Z_k)_K$ in the indexing 1-category is a morphism $(Z_k)_K \longrightarrow(Y_j)_J$ in $\mathcal{X}_\otimes$ that makes the obvious triangle commute. The indexing 1-category splits as a disjoint union of 1-categories which we denote by $\mathcal{I}((X_i)_I;I_0)$ for $I_0\subseteq I$; the 1-category $\mathcal{I}((X_i)_I;I_0) \subseteq \mathcal{X}_\otimes^\textup{op}\times_{(\mathcal{X}_0)_\otimes^\textup{op}} ((\mathcal{X}_{0})_\otimes^\textup{op})_{(X_i)_I/}$ is the full subcategory on objects $f:(X_i)_I \longrightarrow(Y_j)_J$ over $\alpha:$ such that for $i\in I$, $f_i:X_i \longrightarrow\otimes_{\alpha^{-1}(i)} Y_j$ in $\mathcal{X}_0$ is the zero-morphism if and only if $i\in I_0$. Each $\mathcal{I}((X_i)_I;I_0)$ has a terminal object, given by the morphism $(X_i)_I \longrightarrow(X_i^{I_0})_I$ over $\textup{id}_\alpha$, where $X_i^{I_0}$ is $\emptyset$ if $i\in I_0$ and $X_i$ otherwise. Hence we have $$\textup{Lan}_i F((X_i)_I) = \coprod_{I_0\subseteq I} F((X^{I_0}_i)_I).$$ Recalling that for $F\in \textup{Sh}_{\tau^c_{M_{\mathcal{X}}}}(\mathcal{X}_\otimes;\mathcal{C})_{0,\emptyset}$ we have $F(\emptyset)=0$ and $F((X_i)_I)\cong \prod_I F(X_i)$, for such $F$ we have $$\textup{Lan}_i F((X_i)_I) = \coprod_{I_0\subseteq I} \left(\prod_{ I_0} F(X_i)\right).$$ Let $f:(X_i)_I \longrightarrow(Y_j)_J$ be a morphism over $\alpha:J \dashrightarrow I$ in $(\mathcal{X}_0)_\otimes$, and let $\tilde I \subseteq I$ be the subset of $i\in I$ such that $X_i \longrightarrow\otimes_{\alpha^{-1}(i)} Y_j$ is 0. Then the induced map $\textup{Lan}_i F(f)$ is the map $$\coprod_{J_0\subseteq J} \left(\prod_{ J_0} F(Y_j)\right)\longrightarrow\coprod_{I_0\subseteq I} \left(\prod_{I_0} F(X_i)\right)$$ which on the coproduct factor indexed by $J_0\subseteq J$ is given as follows: let $I_0$ be the set $$\{i\in I\setminus \tilde I \mid \alpha^{-1}(i) \subseteq J_0 \}.$$ Then there is a map $$\prod_{ J_0} F(Y_j)\longrightarrow\prod_{ I_0}F(X_i)$$ given by the maps $$\prod_{J_0} F(Y_j)\longrightarrow\prod_{\alpha^{-1}(i)}F(Y_j) \cong F((Y_j)_{\alpha^{-1}(i)}) \longrightarrow F(\otimes_{\alpha^{-1}(i)} Y_j) \xrightarrow{F(f_i)} F(X_i)$$ For $i\in I_0$. Now we consider the sheafification $L\circ \textup{Lan}_i F.$ For an object $(X)_{\underline 1}$ of $(\mathcal{X}_0)_\otimes$ over the one-element set $\underline 1$, it is clear that the only $\tau^c_{M_{\mathcal{X}_0}}$-cover of $(X)_{\underline 1}$ is the maximal sieve containing the identity. Therefore there are no non-trivial covers either, and we have $L\circ \textup{Lan}_i F((X)_{\underline 1}) = \textup{Lan}_i F((X)_{\underline 1}) = F(X)$. Moreover since $L\circ \textup{Lan}_i F$ is in $\textup{Sh}_{\tau^c_{M_{\mathcal{X}_0}}}((\mathcal{X}_0)_\otimes;\mathcal{C})$ by definition, for an arbitary object $(X_i)_I$ it follows that $$L\circ \textup{Lan}_i F((X_i)_I) \cong \prod_I F(X_i) \cong F((X_i)_I).$$ From this it is clear that restricted to $\textup{Sh}_{\tau^c_{M_{\mathcal{X}_0}}}((\mathcal{X}_0)_\otimes;\mathcal{C})_{0,\emptyset}$ and $\textup{Sh}_{\tau^c_{M_\mathcal{X}}}(\mathcal{X}_\otimes;\mathcal{C})_{0,\emptyset}$ respectively, the unit and counit of the adjunction are equivalences, and therefore we have an equivalence of $\infty$-categories. By Proposition [Proposition 21](#prop:laxmonoidalfunctors){reference-type="ref" reference="prop:laxmonoidalfunctors"} the result follows. ◻ **Remark 47**. With some extra work, and using Remark [Remark 2](#rmk:more_comparison){reference-type="ref" reference="rmk:more_comparison"}, it is possible to prove a strong monoidal version of Theorem [Theorem 45](#thm:application){reference-type="ref" reference="thm:application"} as well. This gives equivalences of $\infty$-categories of strong monoidal sheaves $$\textup{Fun}^\otimes_{A\cup L}(\mathbf{Span}^\textup{op},\mathcal{C})_\emptyset \simeq \textup{Fun}_{A^\mathrm{comp}}^\otimes(\mathbf{Comp}_0^\textup{op},\mathcal{C})_\emptyset \simeq \textup{Fun}_{A^{\mathrm{comp}}}^\otimes(\mathbf{Comp}^\textup{op},\mathcal{C}) \simeq \textup{Fun}^\otimes_B(\mathbf{SmComp}^\textup{op},\mathcal{C})$$ for $\mathcal{C}$ a pointed Cartesian symmetric monoidal $\infty$-category. However, all of these are trivial. Indeed, suppose $F:\mathbf{Span}_\times^\textup{op}\longrightarrow\mathcal{C}$ is in $\textup{Fun}^\otimes_{A\cup L}(\mathbf{Span}^\textup{op},\mathcal{C})_\emptyset$. Let $0$ denote the zero object in $\mathcal{C}$ as well as zero morphisms in $\mathcal{C}$. For $\textup{id}:X \longrightarrow X$ and $Y \hookleftarrow \emptyset \longrightarrow Y$ in $\mathbf{Span}$, the induced $\otimes$-product is $$X \times Y \hookleftarrow \emptyset \longrightarrow X \times Y.$$ Hence we obtain a commutative diagram which implies that $\textup{id}_{F(X)}$ must be the zero morphism, so $F(X)=0$ for all $X$.

arxiv_math

--- abstract: | We consider particles that are conditioned to initial and final states. The trajectory of these particles is uniquely shaped by the intricate interplay of internal and external sources of randomness. The internal randomness is aptly modelled through a parameter varying over a deterministic set, thereby giving rise to an ensemble of systems. Concurrently, the external randomness is introduced through the inclusion of white noise. Within this context, our primary objective is to effectively generate the stochastic bridge through the optimization of a random differential equation. As a deviation from the literature, we show that the optimal control mechanism, pivotal in the generation of the bridge, does not conform to the typical Markov strategy. Instead, it adopts a non-Markovian strategy, which can be more precisely classified as a stochastic feedforward control input. This unexpected divergence from the established strategies underscores the complex interrelationships present in the dynamics of the system under consideration. author: - Daniel Owusu Adu and Yongxin Chen title: Stochastic Bridges over Ensemble of Linear Systems --- # INTRODUCTION {#sec:introduction} This paper concerns the problem of conditioning a Markov process at two endpoints. This problem was first studied by Schrodinger in [@SE:31] which we postulate as follows; assume some *fully observed* particles, with density $\rho_0$ at time $t=0$, evolve according to a Markov process in $\mathbb{R}^d$ with density $$\label{eq:Brownian density} q^{B}(s,x,t,y):=\frac{1}{(2\pi(t-s))^{\frac{d}{2}}}\mathrm{exp}\left(-\frac{\|x-y\|^2}{2(t-s)}\right),$$ where $0\leq s\leq t\leq t_f$ and $x,y\in\mathbb{R}^d$. Suppose at time $t=t_f$ the particles are observed to have a distribution $\rho_f$, where $$\rho_f(x_f)\neq\int_{\mathbb{R}^n}q^B(0,x_0,t_f,x_f)\rho_0(x_0)dx_0.$$ Then, $\rho_f$ deviates from the law of large numbers. This means that our assumption of the Markov process is inaccurate. The following question arises: 1. What density $q$ satisfies $$\rho_f(x_f)=\int_{\mathbb{R}^n}q(0,x_0,t_f,x_f)\rho_0(x_0)dx_0.$$ 2. Among such densities $q$, which one is closest to $q^B$ in some suitable sense. Statement 1 and 2 constitute the Schrodinger bridge problem and the most likely stochastic process $\{X(t)\}_{0\leq t\leq t_f}$ such that the densities of the distributions of $X(0)$ and $X(t_f)$ coincides with $\rho_0$ and $\rho_f$, respectively, is called the Schrodinger bridge. An important feature of the Markov process is that it is non-degenerate. That is the stochastic term affects all directions in the coordinate space. Related to our work and motivated by questions regarding the transport of particles having inertia, the case where the Markov process is degenerate has been studied in [@CY-GT:15]. Irrespective of the type of Markov process, it is well-established that stochastic bridges are generated from stochastic optimal control problems (see [@FWH:77; @FWH:05; @FWH-RRW:12; @JB:75; @PPD-PM:90; @DPP:91; @CY-GT:15] and reference therein). Our problem is motivated by ensemble-based Reinforcement Learning [@HC-DL-SR-YC-SC-SI-SY-HC-IH-JK:19]. In ensemble-based Reinforcement Learning, the Ornstein-Uhlenbeck (OU) process is a valuable tool for random exploration when an agent has no prior knowledge of how a system's states transition from one to another [@JN-YK-PS:19]. One can envision a similar scenario where a robot learns to navigate in a new environment. Initially, the robot knows nothing about the environment's dynamics, such as how it moves from one state to another. To effectively learn and make informed decisions, the robot must explore its surroundings randomly. We consider an ensemble of stochastic processes [@QJ-ZA-LJS:13; @BR-KN:00], much like a collection of robots [@LJS-KN:07; @LJS:10] attempting to explore a new environment for the first time (or a robot's various attempts to explore its new environment). Each process, indexed by a parameter, represents a potential trajectory or path the robot might take. Our ultimate goal is to find the paths that are conditioned to meet certain statistical criteria, such as achieving bridging a given state end-points or behaviours. In the context of OU processes, our goal is geared toward understanding its typical behaviour, mean-reverting tendencies, and statistical characteristics which are consistent with the end-states. In this case, averaging the ensemble of OU processes is a practical and effective approach. That is by averaging the ensemble of OU, one can emphasize the mean-reverting behaviour and understand how the system tends to gravitate back to its central trajectory over time. We state here that studying an ensemble of OU processes is not new. In [@DM-VS-SV-SO-PP-CG-PG:18], they provided a mathematical framework to study the statistical properties of the centre-of-mass variable and its relation to the individual processes in the ensemble of OU. In particular, they determined a non-autonomous stochastic differential equation (SDE) satisfied by a process that is equivalent in distribution to the centre-of-mass variable of an ensemble of the OU processes. Furthermore, they established the equivalence in the distribution of the centre-of-mass variable with a randomly scaled Gaussian process (the product of a non-negative random variable and a Gaussian process). We state here that in as much as the centre-of-mass variable can be used to estimate the average concentration over the parameters, our result focuses on the average. Following from [@FWH:77; @FWH:05; @FWH-RRW:12; @JB:75; @PPD-PM:90; @DPP:91; @CY-GT:15], in our case, the ensemble nature of the Markov process in our problem adds its own set of technical challenges in solving the corresponding stochastic optimal control problem. It turns out that averaging an ensemble of Markov processes fails to be a Markov process and seems to represent a more complex stochastic process than is usually encountered in the literature [@NE:67; @VHR:07; @KFC:12; @CY-GT:15]. Therefore, the standard tools in [@JB:75; @PPD-PM:90; @DPP:91; @CY-GT:15] used to generate a bridge will not be applicable in our case. To overcome this challenge, we rely on the equivalent discrete-time stochastic optimal control problem and characterize the optimal control through the approximation of the continuous-time stochastic process. We show that the parameter-independent optimal control that bridges the endpoint condition for an ensemble of Markov processes is a *stochastic feedforward control input*. This deviates from the characterization of the optimal control that induces a stochastic bridge (see [@JB:75; @PPD-PM:90; @DPP:91; @NE:67; @VHR:07; @KFC:12; @CY-GT:15; @OB:03]). The distinction follows from the fact that, in a standard Markov process, it is possible to track that state and feed it back into the system to achieve the bridge. This leads to the optimal control strategy being a Markov Strategy. In our case, as you will see, it is not possible to track the average of an ensemble of a given Markov process. Thus leading to an *stochastic feedforward control*. In stochastic feedforward control, the control input is determined based on past and present values of the noise. Optimal feedforward controllers have been described in [@NH:87; @MPS:82], where it is assumed that the control input is obtained from the output of a linear dynamic system driven by white noise. This characterization of control has applications in flight system control of robotics and crystal growth of power transmission networks (see [@HN-DH-TDB:92; @NH:87; @MPS:82; @HME:89] and reference therein). Secondly, unlike in [@CY-GT:15] where controllability of the system is relevant to establish the Schrodinger bridge for the case of degeneracy, as we showed in [@ADO:22], our result relies on the so-called averaged observability inequality [@LM-ZE:14; @LJ-ZE:17; @LQ-ZE:16; @ZE:14] which is equivalent to the invertibility of a matrix (see [@ADO:22]). This matrix is used to solve both the Schrodinger bridge problem and hence design the optimal control for our problem. We state here that our result is related to ensemble control theory [@LJS-KN:07; @LJS:10; @GB-CX:21; @CX:21; @CX:19] which is motivated by quantum systems [@LJS-KN:06] and also robust control theory [@QJ-ZA-LJS:13; @BR-KN:00] and has applications in a variety of fields including engineering [@VA-WL:78; @HCH-RSJ-GSM:18; @WA-KMV:21] and economics [@HLP-STJ:01; @JA-NAS:11; @OA-SJH:02; @BWA-XA-YAN:14]. The organization of the paper is as follows; We discuss the notion of stochastic averaged control problem in Section [2](#sec: Averaged ensemble control problem){reference-type="ref" reference="sec: Averaged ensemble control problem"}. We state conditions under which this is possible. After that, we state the problem statement and follow with the main result in Section [3](#Sec: Problem Statement and Main Result){reference-type="ref" reference="Sec: Problem Statement and Main Result"}. We conclude with remarks on future work in Section [4](#sec:Conclusion and future work){reference-type="ref" reference="sec:Conclusion and future work"}. # Stochastic averaged ensemble control {#sec: Averaged ensemble control problem} Consider the ensemble of a controlled Markov process defined on a naturally filtered probability space $(\Omega,\mathcal{F},\mathbb{P})$ as follows $$\begin{aligned} \label{eq:ensemble of stochastic system} d X(t,\theta)=&A(\theta)X(t,\theta)d t+B(\theta)u(t)d t+\sqrt{\epsilon}B(\theta)d W(t),\cr X(0,\theta)=&x_0,\end{aligned}$$ where $X(t,\theta)\in\mathbb{R}^d$, is the random state of an individual system at time $t$ indexed by the sample point $\theta\in\Omega$, $A:\Omega \rightarrow \mathbb{R}^{d\times d}$ and $B:\Omega \rightarrow \mathbb{R}^{d\times m}$ are measurable mappings such that $\sup_{\theta\in\Omega}\|A(\theta)\|<\infty$ and $\sup_{\theta\in\Omega}\|B(\theta)\|<\infty$, where the norm here is the Frobenius norm on the space of matrices, $u\in \mathrm{L}^1([0,t_f];\mathbb{R}^m)$ is a *parameter-independent* control input, and $x_0$ is an initial $d$-dimensional deterministic vector and $\{W(t)\}_{t\geq 0}\subset\mathbb{R}^m$ is the Wiener process such that $W(0)=0$. Note that the Markov process indexed by $\theta$ at time $t$ is characterized by $$\label{eq:ensemble of process} X(t,\theta)=e^{A(\theta)t} x_0+\int_{0}^{t}e^{A(\theta)(t-\tau)}B(\theta)u(\tau)d\tau+ \sqrt{\epsilon}\int_{0}^{t}e^{A(\theta)(t-\tau)}B(\theta)d W(\tau).$$ For reasons that will be clear later, for now, we study the controllability of this Markov process in an appropriate sense. Since the system parameter is unknown but belongs to a deterministic set $\Omega$, it is natural to control the average over the parameter. For simplicity of presentation, we assume that the probability space $(\Omega,\mathcal{F},\mathbb{P})$ is a uniform distributed probability space with $\Omega=[0,1]$. To this end, we proceed to the following definition. **Definition 1**. *The ensemble of linear stochastic system [\[eq:ensemble of stochastic system\]](#eq:ensemble of stochastic system){reference-type="eqref" reference="eq:ensemble of stochastic system"} is said to be *averaged controllable* if, for any initial state $x_0\in\mathbb{R}^d$, final state $x_f\in\mathbb{R}^d$, and final time $t_f$, there exists a parameter-independent control input $u\in L^1([0,t_f];\mathbb{R}^m)$ such that the ensemble of states in [\[eq:ensemble of process\]](#eq:ensemble of process){reference-type="eqref" reference="eq:ensemble of process"} satisfies $$\mathbb{E}\int_{0}^{1} X(t_f,\theta)d\theta=x_f.$$* Note that by the linearity of the stochastic system [\[eq:ensemble of stochastic system\]](#eq:ensemble of stochastic system){reference-type="eqref" reference="eq:ensemble of stochastic system"}, the expectation of the control will drive the deterministic part of the dynamics [\[eq:ensemble of stochastic system\]](#eq:ensemble of stochastic system){reference-type="eqref" reference="eq:ensemble of stochastic system"} in the averaged sense. We proceed to the following useful result. **Proposition 1**. *If the matrix $$\label{eq: Gramian} G_{t_f,0}:=\int_{0}^{t_f}\left(\int_{0}^{1}e^{A(\theta)(t_f-\tau)}B(\theta)d\theta\right) \left(\int_{0}^{1}B^{\mathrm{T}}(\theta)e^{A^{\mathrm{T}}(\theta)(t_f-\tau)}d\theta\right) d\tau,$$ is invertible then, the linear stochastic system [\[eq:ensemble of stochastic system\]](#eq:ensemble of stochastic system){reference-type="eqref" reference="eq:ensemble of stochastic system"} is said to be averaged controllable.* *Proof.* Suppose $G_{t_f,0}$ is invertible and for any initial state $x_0\in\mathbb{R}^d$, final state $x_f\in\mathbb{R}^d$ consider $$\label{eq: averaged control} u(t)=\left(\int_{0}^{1}B^{\mathrm{T}}(\theta)e^{A^{\mathrm{T}}(\theta)(t_f-t)}d\theta\right) G_{t_f,0}^{-1} \left(x_f-\left(\int_{0}^{1}e^{A(\theta)t_f}d\theta\right) x_0\right).$$ From [\[eq:ensemble of process\]](#eq:ensemble of process){reference-type="eqref" reference="eq:ensemble of process"}, since $$\label{eq:final state} X(t_f,\theta)=e^{A(\theta)t_f} x_0+\int_{0}^{t_f}e^{A(\theta)(t_f-\tau)}B(\theta)u(\tau)d\tau+ \sqrt{\epsilon}\int_{0}^{t_f}e^{A(\theta)(t_f-\tau)}B(\theta)d W(\tau),$$ by substituting [\[eq: averaged control\]](#eq: averaged control){reference-type="eqref" reference="eq: averaged control"} in [\[eq:final state\]](#eq:final state){reference-type="eqref" reference="eq:final state"}, we obtain $\mathbb{E}\int_{0}^{1} X(t_f,\theta)d\theta=x_f$. This finishes the proof. ◻ # Problem Statement and Main Result {#Sec: Problem Statement and Main Result} Consider an ensemble of processes governed by $$\label{eq:ensemble of linear process} d X(t,\theta)=A(\theta)X(t,\theta)d t+\sqrt{\epsilon}B(\theta)d W(t),$$ with initial condition $$X(0,\theta)=x_0,\quad\text{almost surely (a.s)}.$$ **Problem 1:** *Our goal is to find solutions that are conditioned to have* $$\label{eq:final condition} \int_{0}^{1} X(t_f,\theta)d\theta=x_f, a.s.$$ To characterize such solutions, suppose $\epsilon=0$, then to ensure that [\[eq:final condition\]](#eq:final condition){reference-type="eqref" reference="eq:final condition"} is satisfied, one needs to solve the optimal control problem $$\label{eq:trans_control_cost} c(x_0,x_f):=\min_{u\in\mathcal{U}_{x_0}^{x_f}}\int_0^{t_f}\frac{1}{2}\|u(t)\|^2dt,$$ where $\mathcal{U}_{x_0}^{x_f}$ is the set of control inputs such that $$\begin{aligned} \label{eq:ensem_with stoch_cost} \frac{\partial x}{\partial t}(t,\theta)=&A(\theta)x(t,\theta)+B(\theta)u(t),\cr x(0,\theta)=&x_0\quad\text{and}\quad \int_{0}^{1}x(t_f,\theta)d\theta=x_f,\end{aligned}$$ has a solution. This tends to measure the optimal change in the drift of the ensemble of the autonomous system that ensures that condition [\[eq:final condition\]](#eq:final condition){reference-type="eqref" reference="eq:final condition"} is satisfied. *The fact that the final conditional state in [\[eq:final condition\]](#eq:final condition){reference-type="eqref" reference="eq:final condition"} is parameter-independent motivates the quest to find a parameter-independent control. If the control depends on $\theta\in[0,1]$, it might lead to different behaviours for different realizations, making it challenging to ensure that [\[eq:final condition\]](#eq:final condition){reference-type="eqref" reference="eq:final condition"} is satisfied a.s. Another motivation derived from the condition [\[eq:final condition\]](#eq:final condition){reference-type="eqref" reference="eq:final condition"} is that the natural quantity one observes is the average over the parameter $\theta\in[0,1]$*. A more general problem relating to [\[eq:trans_control_cost\]](#eq:trans_control_cost){reference-type="eqref" reference="eq:trans_control_cost"}-[\[eq:ensem_with stoch_cost\]](#eq:ensem_with stoch_cost){reference-type="eqref" reference="eq:ensem_with stoch_cost"} has been studied in [@ADO:22]. They showed that the optimal value of the control that steers the average of the ensemble of systems in [\[eq:ensem_with stoch_cost\]](#eq:ensem_with stoch_cost){reference-type="eqref" reference="eq:ensem_with stoch_cost"} is characterized by the Euclidean distance $$\label{eq: transport cost} c(x_0,x_f)=\frac{1}{2}\left \|x_f-\left(\int_{0}^{1}e^{A(\theta)t_f}d\theta\right)x_0\right \|^2_{G_{t_f,0}^{-1}},$$ where $\|x\|^2_{G_{t_f,0}^{-1}}=x^{\mathrm{T}}G_{t_f,0}^{-1}x$, for all $x\in\mathbb{R}^d$, whenever $G_{0,t_f}$ in [\[eq: Gramian\]](#eq: Gramian){reference-type="eqref" reference="eq: Gramian"} is invertible. From this observation, let $$\label{eq:Scaled Markov_Kernel} q^{\epsilon, G}(s,x,t,y)=(2\pi\epsilon)^{-\frac{d}{2}}(\mathrm{det}(G_{t,s}))^{-\frac{d}{2}} \exp\left(-\frac{1}{2\epsilon}\left \|y-\left(\int_{0}^{1}e^{A(\theta)(t-s)}d\theta\right)x\right \|^2_{G_{t,s}^{-1}}\right)$$ where $G_{t,s}$ is defined in [\[eq: Gramian\]](#eq: Gramian){reference-type="eqref" reference="eq: Gramian"} with $t=t_f$ and $s=0$, be the transition density of the particles moving independently of each other according to the average diffusion in [\[eq:ensemble of linear process\]](#eq:ensemble of linear process){reference-type="eqref" reference="eq:ensemble of linear process"}. Then, following from [@JB:75; @PPD-PM:90; @DPP:91], the solutions of [\[eq:ensemble of linear process\]](#eq:ensemble of linear process){reference-type="eqref" reference="eq:ensemble of linear process"} condition to be [\[eq:final condition\]](#eq:final condition){reference-type="eqref" reference="eq:final condition"} is characterized by the stochastic optimal control problem $$\label{eq:problem 1} {\bf Problem~2:}\quad\quad\min_{u\in \mathcal{U}}\mathbb{E}\left[\int_{0}^{t_f}\frac{1}{2}\|u(t)\|^2dt\right],$$ subject to $$\begin{aligned} \label{eq:uncertain states} &d X(t,\theta)=A(\theta)X(t,\theta)d t+B(\theta)u(t)d t+\sqrt{\epsilon}B(\theta)d W(t),\cr &X(0,\theta)=x_0\text{ a.s and }\int_{0}^{1} X(t_f,\theta)d\theta=x_f, a.s.\end{aligned}$$ To be more precise, if $u\in\mathcal{U}\subset \mathrm{L}^2([0,t_f];\mathbb{R}^m)$, then; 1. $u(t)$ is $x(t)$-measurable, where $x(t):=\int_{0}^{1} X(t,\theta)d\theta$ with $X(t,\theta)$ characterized in [\[eq:ensemble of process\]](#eq:ensemble of process){reference-type="eqref" reference="eq:ensemble of process"}, for all $t\in [0,t_f]$, 2. $\mathbb{E}\left[\int_{0}^{t_f}\frac{1}{2}\|u(t)\|^2dt\right]<\infty$, 3. $u$ achieves averaged controllability (see Definition [Definition 1](#def:averaged controllability){reference-type="ref" reference="def:averaged controllability"}) for [\[eq:uncertain states\]](#eq:uncertain states){reference-type="eqref" reference="eq:uncertain states"}. Note that in this setting, since we aim to steer the final state to our desired state, the only information available to us is the past and present noise. Here we state our main result. **Theorem 1**. *Suppose $G_{t_f,s}$, for all $0\leq s<t_f$, in [\[eq: Gramian\]](#eq: Gramian){reference-type="eqref" reference="eq: Gramian"} is invertible. Then the optimal control for [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"} subject to [\[eq:uncertain states\]](#eq:uncertain states){reference-type="eqref" reference="eq:uncertain states"} is characterized as $$\label{eq: optimal control} u^*(t)= -\sqrt{\epsilon}\int_0^t\Phi(t_f,t)^T G_{t_f,\tau}^{-1}\Phi(t_f,\tau) dW(\tau)+\Phi(t_f,t)^{T}G_{t_f,0}^{-1}x_f.$$* where $$\label{eq: non-transition matrix} \Phi(t_f,\tau) = \int_0^1e^{A(\theta)(t_f-\tau)}B(\theta)d\theta.$$ Note that re-centring the initial ensemble of systems at the origin $0$ holds no bearing on the system's characterization, given its inherent linearity. However, we see that the characterization of the optimal control is a departure from the conventional stochastic optimal control literature, where the optimal control assumes the form of a Markov strategy [@OB:03; @VHR:07]. In particular, when dealing with a Markov process subject to parameter perturbations, the optimal control that steers the stochastic bridge adopts an approach---*a stochastic feedforward input*, to be precise. This unique characterization emerges because of the intricate presence of parameters within the system, further complicating the endeavour to trace the ensemble's average behaviour. The exhaustive proof is omitted due to spatial constraints, with the subsequent sections devoted to illuminating the rationale behind this assertion. The remainder of this paper articulates the intricate dynamics that lend credence to this phenomenon. **Remark 1**. *To highlight more on the novelty of the above problem, following from [@DPP:91] we have that problem [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"}-[\[eq:uncertain states\]](#eq:uncertain states){reference-type="eqref" reference="eq:uncertain states"}, where $X(0,\theta)\sim\mu_0$ and $\int_{0}^{1} X(t,\theta)d\theta\sim\mu_f$ and $\mu_0,\mu_f$ are given initial and final distributions, is the stochastic control approach to the Schrodinger bridge problem $$\label{eq:regopttranave} \min_{\gamma\in\Pi(\mu_0,\mu_f)}\int_{\mathbb{R}^d\times\mathbb{R}^d}\epsilon\gamma(x_0,x_f) \mathrm{log}\left(\frac{\gamma(x_0,x_f)}{\mathrm{exp}\left(-\frac{c(x_0,x_f)}{\epsilon}\right)}\right)d x_0d x_f,$$ where $c$ is in [\[eq: transport cost\]](#eq: transport cost){reference-type="eqref" reference="eq: transport cost"}. Therefore, aside from the fact that problem [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"}-[\[eq:uncertain states\]](#eq:uncertain states){reference-type="eqref" reference="eq:uncertain states"} is the Dirac extension of [@ADO:22] to include white noise, more importantly, it also extends the works in [@CY-GT:15; @SE:31; @CY-GTT-PM:16; @CY-GTT-PM:15] to the case where the Markov process is generated from a linear diffusion which is submitted to parameter perturbations.* Since we require the control $u(t)$ at time $t$ to be $x(t)$-measurable, our object of interest is the controlled-average process $$\label{eq: output} x(t)=\int_0^t \Phi(t,\tau)(u(\tau)d\tau +\sqrt{\epsilon} dW(\tau)),$$ where we have re-centred the dynamics to initialize at $0$, without any loss and $\Phi(t,\tau)$ is defined in [\[eq: non-transition matrix\]](#eq: non-transition matrix){reference-type="eqref" reference="eq: non-transition matrix"}. Therefore, the optimal control problem [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"}-[\[eq:uncertain states\]](#eq:uncertain states){reference-type="eqref" reference="eq:uncertain states"} is equivalent to the optimal output control problem [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"} subject to [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"}, where the final state is conditioned to be $x(t_f)=x_f$ a.s. Rather than solving problem [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"} subject to [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"} conditioned to satisfy $x(t_f)=x_f$, we consider the corresponding alternative free-endpoint formulation $$\label{eq: free end-point problem} \min_{u} J(u):=\mathbb{E} \bigg[a(x(t_f)-x_f)^T(x(t_f)-x_f) +\int_0^{t_f} \frac{1}{2}\|u(t)\|^2 d\tau\bigg]$$ subject to [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"}, where $a>0$. Note that the optimal solution for [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"} subject to [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"}, is obtained by taking $a\rightarrow\infty$. More precisely, if $$\lim_{a\rightarrow\infty}\|u^*_a-u^*\|^2_{L^2([0,t_f];\mathbb{R}^{m})}=0,$$ where $u^*_a$ is the optimal control for [\[eq: free end-point problem\]](#eq: free end-point problem){reference-type="eqref" reference="eq: free end-point problem"} subject to [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"}, then $u^*$ is the unique optimal solution for [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"} subject to [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"}. We emphasize here that $\Phi(t,\tau)\in\mathbb{R}^{d\times m}$ in [\[eq: non-transition matrix\]](#eq: non-transition matrix){reference-type="eqref" reference="eq: non-transition matrix"} is not a transition matrix in general. The only affirmative case is where $A=A(\theta)$, for all $\theta\in[0,1]$. In the latter case, the average process [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"} satisfies the time-invariant linear diffusion process $$\begin{aligned} \label{eq: time-invariant linear system} d x(t)=&Ax(t)d t+B u(t)d t+\sqrt{\epsilon}Bd W(t),\cr x(0)=&x_0\quad\text{ and }\quad x(t_f)=x_f\end{aligned}$$ where $B=\left(\int_0^1B(\theta)d\theta\right)$ and the controllability of the pair $(A,B)$ plays a major role in establishing results similar to [@CY-GT:15]. In particular, if the system [\[eq:ensemble of linear process\]](#eq:ensemble of linear process){reference-type="eqref" reference="eq:ensemble of linear process"} is submitted to parameter perturbation only in the diffusive coefficient and $(A,B)$ is a controllable pair, then by averaging and then solving the standard stochastic linear-quadratic optimal control problem [\[eq: free end-point problem\]](#eq: free end-point problem){reference-type="eqref" reference="eq: free end-point problem"} subject to [\[eq: time-invariant linear system\]](#eq: time-invariant linear system){reference-type="eqref" reference="eq: time-invariant linear system"} we generate the Brownian bridges with desired statistics (see [@CY-GT:15]). On the other hand, for a fixed $A\in\mathbb{R}^{d\times d}$, one can check that the average process [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"} satisfies the dynamics $$\label{eq:ASDE 0} dx(t)=\bigg(Ax(t)+Bu(t)+\int_0^t F_{A(\theta),B(\theta)}(\tau,t)(u(\tau) d\tau+ \sqrt{\epsilon}dW(\tau))\bigg)dt+\sqrt{\epsilon}BdW(t),$$ where $$%\label{eq: average map of system matrix} F_{A(\theta),B(\theta)}(\tau,t):= \int_0^1\left(A(\theta)-A\right) e^{A(\theta)(t-\tau)}B(\theta)d\theta.$$ In this context, employing the variational approach to optimize [\[eq: free end-point problem\]](#eq: free end-point problem){reference-type="eqref" reference="eq: free end-point problem"} subject to [\[eq:ASDE 0\]](#eq:ASDE 0){reference-type="eqref" reference="eq:ASDE 0"} reveals some significant challenges. The drift term within [\[eq:ASDE 0\]](#eq:ASDE 0){reference-type="eqref" reference="eq:ASDE 0"} assumes the form of a controlled Ito process, causing this equation to deviate from the conventional definition of a stochastic differential equation (SDE), (see for instance [@OB:03; @VHR:07]). Therefore, the average random differential equation [\[eq:ASDE 0\]](#eq:ASDE 0){reference-type="eqref" reference="eq:ASDE 0"} seems to represent a more complex stochastic process than is usually encountered in the literature [@NE:67; @VHR:07; @KFC:12; @CY-GT:15]. However, the real-world significance of [\[eq:ASDE 0\]](#eq:ASDE 0){reference-type="eqref" reference="eq:ASDE 0"} resides in the average process delineated by [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"}. This formulation captures the central tendency behaviour of the system's fluctuations, thereby holding practical importance. Consequently, standard stochastic control techniques, including those rooted in Hamilton-Jacobi Bellman (HJB) conditions [@OB:03; @VHR:07], prove unsuitable for this scenario. An alternative avenue lies in the PDE approach [@FWH:77; @FWH:05; @FWH-RRW:12; @JB:75; @PPD-PM:90; @DPP:91], yet the presence of noise within the drift term presents challenges when adapting the corresponding parabolic PDE. As a result, the methods delineated in [@JB:75; @PPD-PM:90; @DPP:91] and related references are not readily applicable. These observations collectively imply that the optimal control strategy for problem [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"}-[\[eq:uncertain states\]](#eq:uncertain states){reference-type="eqref" reference="eq:uncertain states"}, or its equivalent form involving [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"}, cannot be a Markov strategy. Intriguingly unrelated, this insight also signifies the formidable nature of stabilizing the average process. **Special Case:** Before delving into solution techniques, let us consider the classical case. Consider particles governed by the following equations: $$\begin{aligned} \label{eq:linear process} dx(t)=&\sqrt{\epsilon}d W(t),\cr x(0)=&0,\quad\text{almost surely (a.s)}.\end{aligned}$$ Our primary goal is to ensure that, at the final time $x(t_f)=0$ a.s. In this special case, since $\Phi(t_f,t)=I_{d\times d}$, for all $t\geq t_f$ and $G_{t_f,\tau}=(t_f-\tau)I_{d\times d}$, we have that the stochastic feedforward control input in [\[eq: optimal control\]](#eq: optimal control){reference-type="eqref" reference="eq: optimal control"} reduces to $$\label{eq:simplified feedforward input} u^*(t)=-\sqrt{\epsilon}\int_0^t(t_f-\tau)^{-1} dW(\tau).$$ What is interesting is that under these conditions, this optimal stochastic feedforward control input simplifies into a Markovian control strategy. To get to this point, we follow the approach outlined in [@CY-GT:15]. This involves solving [\[eq:problem 1\]](#eq:problem 1){reference-type="eqref" reference="eq:problem 1"}, which leads us to [\[eq: free end-point problem\]](#eq: free end-point problem){reference-type="eqref" reference="eq: free end-point problem"} subject to [\[eq: time-invariant linear system\]](#eq: time-invariant linear system){reference-type="eqref" reference="eq: time-invariant linear system"}, where $A=0_{d\times d}\in\mathbb{R}^{d\times d}$ and $x_0=x_f=0$. Utilizing the HJB conditions [@OB:03; @VHR:07] and taking limit as $a\rightarrow\infty$, we arrive at the following expression for the optimal control $u^*$: $$\label{eq: Markov strategy} u^*(t)=-(t_f-t)^{-1}x(t),$$ Notably, by substituting $u^*$ in [\[eq: Markov strategy\]](#eq: Markov strategy){reference-type="eqref" reference="eq: Markov strategy"} into [\[eq: time-invariant linear system\]](#eq: time-invariant linear system){reference-type="eqref" reference="eq: time-invariant linear system"}, where $A=0_{d\times d}\in\mathbb{R}^{d\times d}$ and $x_0=x_f=0$, we find that the closed-loop trajectory is: $$x(t)=\sqrt{\epsilon}\int_0^te^{\int_t^{\tau}(t_f-\alpha)^{-1}d\alpha} dW(\tau),$$ thus, $$\label{eq: state solution} x(t)=\sqrt{\epsilon}(t_f-t)\int_0^t(t_f-\tau)^{-1} dW(\tau).$$ By substituting [\[eq: state solution\]](#eq: state solution){reference-type="eqref" reference="eq: state solution"} into [\[eq: Markov strategy\]](#eq: Markov strategy){reference-type="eqref" reference="eq: Markov strategy"} we obtain [\[eq:simplified feedforward input\]](#eq:simplified feedforward input){reference-type="eqref" reference="eq:simplified feedforward input"}. This illustrates that in cases where the system is not an ensemble, the feedforward control input in reduces to the Markovian strategy in [\[eq: Markov strategy\]](#eq: Markov strategy){reference-type="eqref" reference="eq: Markov strategy"}. **Equivalent discrete-time optimal control problem:** To solve problem [\[eq: free end-point problem\]](#eq: free end-point problem){reference-type="eqref" reference="eq: free end-point problem"}-[\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"}, we transform the problem [\[eq: free end-point problem\]](#eq: free end-point problem){reference-type="eqref" reference="eq: free end-point problem"}-[\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"} to an equivalent discrete-time optimal control problem. We partition over time so that it is consistent with the definition of the Ito integral [@NE:67; @VHR:07; @KFC:12]. To this end, let $P:=\{0= t_0<t_2<\dots<t_{k-1}=t_f\}$ be a regular partition with constant step size $\triangle t_k=t_{i+1}-t_i$, for any $i\in\{1,\dots,k\}$ and suppose $u_{k,i}=u(t_i)$ is a constant $x(t_i)$-measurable random variable in $L^2[t_i,t_{i+1})$, where $i\in\{0,\dots,k-1\}$ and consider the discrete-time optimal control problem $$\label{eq: discrete-time free end-point problem} \min_{u_k}J_k(u_k):=\mathbb{E} \bigg[a(x_k-x_f)^T(x_k-x_f) +\frac{1}{2}\sum_{i=0}^{k-1} u_{k,i}^Tu_{k,i}\triangle t_k\bigg],$$ subject to $$\label{eq: discrete-time system} x_k=\sum_{i=0}^{k-1}\Phi_i(t_f)\left(u_{k,i}\triangle t_k+\sqrt{\epsilon}\triangle W_{i} \right)$$ where $x_k:= x(t_k)\in\mathbb{R}^d$, $u_k:=(u_{k,0},\dots,u_{k,k-1})\in(\mathbb{R}^m)^k$, $\Phi_i(t_f):=\Phi(t_f,t_i)\in\mathbb{R}^{d\times m}$ and $\triangle W_{i}:=W(t_{i+1})-W(t_{i})\in\mathbb{R}^m$. We call this problem the equivalent discrete-time optimal control problem because the solution [\[eq: discrete-time free end-point problem\]](#eq: discrete-time free end-point problem){reference-type="eqref" reference="eq: discrete-time free end-point problem"}-[\[eq: discrete-time system\]](#eq: discrete-time system){reference-type="eqref" reference="eq: discrete-time system"} is exactly the same as the solution for [\[eq: free end-point problem\]](#eq: free end-point problem){reference-type="eqref" reference="eq: free end-point problem"} subject to [\[eq: output\]](#eq: output){reference-type="eqref" reference="eq: output"} (see [@TAR-DKWL:84; @TAR-DKWL:00]). We proceed to characterize the optimal control. We omit the proof due to space limitations. **Proposition 2**. *Suppose $G_{t_f,s}$, for all $0\leq s<t_f$, in [\[eq: Gramian\]](#eq: Gramian){reference-type="eqref" reference="eq: Gramian"} is invertible. Then the optimal control for [\[eq: discrete-time free end-point problem\]](#eq: discrete-time free end-point problem){reference-type="eqref" reference="eq: discrete-time free end-point problem"}- [\[eq: discrete-time system\]](#eq: discrete-time system){reference-type="eqref" reference="eq: discrete-time system"} is characterized as $$\begin{gathered} u_{a,k,i}^*=\\ -\sqrt{\epsilon}\sum_{j=0}^i\Phi(t_f,t_i)^T (\sum_{\alpha=j}^{k-1}\Phi(t_f,t_{\alpha})\Phi(t_f,t_\alpha)^T\triangle t_k+\frac{1}{2a}I_{n})^{-1}\Phi(t_f,t_j)\triangle W_j\\+\Phi(t_f,t_i)^{T}(\sum_{\alpha=0}^{k-1}\Phi(t_f,t_{\alpha})\Phi(t_f,t_\alpha)^T\triangle t_k+\frac{1}{2a}I_{n})^{-1}x_f. \end{gathered}$$* From this result, we have that $$\lim_{a\rightarrow\infty}\lim_{k\rightarrow\infty}\|u^*_{a,k}-u^*\|^2_{L^2([0,t_f];\mathbb{R}^{m})}=0,$$ where $u^*$ is in [\[eq: optimal control\]](#eq: optimal control){reference-type="eqref" reference="eq: optimal control"}. # Conclusion and future work {#sec:Conclusion and future work} In this paper, we have discussed the problem of conditioning a Markov process, subjected to parameter perturbations, to initial and final states. The central motivation behind this endeavor lies in our quest to understand and control the dynamics of ensembles of systems characterized by stochastic processes. Specifically, we have explored the problem of steering an ensemble of linear stochastic systems toward average behavior, irrespective of the underlying parameter perturbations. Our investigation has revealed that due to the inherent complexity introduced by parameter perturbations, the optimal control for this problem cannot adopt a traditional Markov strategy. Instead, we've uncovered a unique characterization of the optimal control, involving a stochastic feedforward input that relies on a time-varying drift. One can view the end-point conditions as Dirac distributions for particles emanating and absorbed at particular points in phase space. This characterization provides a powerful tool for controlling and modelling general distributions of particles and interpolation of density functions. 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arxiv_math

--- abstract: | We show that the transition function of the cascaded connection of two FSRs can be viewed as a wreath product element. This allows us to study periods of cascaded connections with algebraic methods, obtaining both a general, nontrivial upper bound on the maximum period of a cascaded connection and a complete, explicit understanding of the periods in the important case of the cascaded connection of an $n$-dimensional De Bruijn sequence into an $m$-dimensional linear FSR. address: School of Mathematics and Statistics, Carleton University, Ontario, Ottawa, K1S 5B6, Canada author: - Alexander Bors, Farzad Maghsoudi, and Qiang Wang bibliography: - paper.bib title: Wreath products and cascaded FSRs --- # Introduction Linear feedback shift registers (LFSRs) have a wide range of applications in coding theory, modern communication systems, and cryptography. There has been substantial use of LFSRs as building blocks in stream ciphers because of their very good statistical properties, efficient implementations, and well-studied algebraic structures. In contrast, stream ciphers based on LFSRs are vulnerable to correlation attacks [@Aumasson] and algebraic attacks [@Cannière]. Consequently, nonlinear feedback shift registers (NFSRs) have attracted increasing attention for their nonlinear update functions. Recently proposed stream ciphers, such as Trivium [@Christophe] and Grain [@Nicolas], use NFSRs as building blocks. Over the past 50 years, NFSRs have been examined. However, some fundamental questions remain unanswered. In particular, there is no efficient way to determine the periods of sequences generated by an arbitrary NFSR. The most important kind of sequence generated by an NFSR, which achieves the maximum period, of $2^n$ in the case of an $n$-stage NFSR, is a *de Bruijn* sequence. LFSRs can also achieve sequences of large periods, namely up to $2^n-1$ when having $n$ stages; the corresponding output sequences are known as $m$-sequences, and the associated transition functions are special cases of Singer cycles. However, each of these two classes of feedback shift registers (FSRs) has drawbacks from an application point of view: 1. De Bruijn sequences with desirable properties are costly to generate for large $n$. 2. Linearity is a property that should be avoided for many applications. It is thus advantageous to have a method of combining smaller FSRs, e.g. a De Bruijn cycle and an LFSR, into larger contraptions which may achieve long cycles in higher dimensions more efficiently than De Bruijn cycles while performing better than LFSRs with respect to certain undesirable properties. This is where cascaded connections of FSRs come into play. The cascaded connection of two FSRs was first introduced in [@Green]. Specifically, the cascaded connection of $\operatorname{FSR}(f)$ into $\operatorname{FSR}(g)$ produces the same family of sequences as the FSR with characteristic function $f\ast g$ [@Green]. This was a motivation for Mykkeltveit, Siu and Tong to study some properties of the cycle structure of the cascaded connection of $\operatorname{FSR}(f)$ into $\operatorname{FSR}(g)$ [@Mykkeltveit]. A Grain-like structure is a cascaded connection of a primitive LFSR into an NFSR. In 2011, Hu and Gong demonstrated that the periods of the sequences generated by an NFSR in a Grain-like structure are multiples of the periods of the sequences generated by its LFSR [@Hu]. They also proposed an open problem whether the sequences generated by the NFSR in a Grain-like structure can achieve the minimum period, i.e., the period of the LFSR. In terms of security, it is clearly undesirable for sequences generated by NFSRs in Grain-like structures to achieve the minimum period. Recently, in 2019, Yang, Zeng and Xu in [@Yang] proved the existence of a Grain-like structure achieving the minimum period by constructing a class of them for the case where the LFSR and NFSR have the same number of stages. Inspired by their work, Wang, Zheng, Zhao and Feng in [@Wang] could improve their result and also prove the existence of such Grain-like structures for the case where the number of stages of the NFSR is larger than the number of stages of the LFSR. They also proved that there are two necessary conditions for Grain-like structures to generate maximal possible period sequences. In 2011, Cheng, Qi and Li proposed a new mathematical tool for calculating matrices called semi-tensor products (STP) [@Analysis]. The STP method has been widely used to study Boolean networks -- see the survey papers [@Li; @Lu] for more information. This method is also used in the study of cascaded connections of FSRs. Particularly, in  [@Liu], using the STP of matrices, Grain-like cascaded FSRs are converted into an equivalent linear equation by declaring them as two Boolean networks. Cascading an NFSR into an LFSR to generate long sequences has been studied as well. In 2020, Chang, Gong and Wang using a linear algebraic approach obtained a description of the cycle structure of a cascaded connection of an arbitrary NFSR generating a de Bruijn sequence into an LFSR [@Chang]. In particular, they showed that the initial state of each cycle can be determined by solving a system of linear equations. Their method also works for the cascaded connection of any NFSR into an LFSR. The wreath product is a special combination of two permutation groups based on the semidirect product in group theory. It is a fundamental concept in permutation group theory [@Dixon Secion 2.6]. There are applications of the wreath product in describing cyclotomic permutations, computing the cycle types of certain permutations, etc. Wan and Lidl in [@Rudolf] used wreath products to study cyclotomic permutations. Based on this earlier work, Bors and Wang expanded those ideas in [@Alexander]. Using the imprimitive wreath product, they were also able to show that some specific functions form a permutation group on $K$ (a finite field) and characterize which of them are complete mappings of $K$ [@Bors]. In this paper, we will use wreath products to study periods of cascaded connections of FSRs. This approach is more conceptual than the one of Mykkeltveit, Siu and Tong [@Mykkeltveit Section 2], and for the special case of the cascaded connection of a De Bruijn cycle into an LFSR, it will yield results that are more explicit/stronger than the ones of Chang, Gong and Wang [@Chang]. In Section [2](#ch:Preliminaries){reference-type="ref" reference="ch:Preliminaries"}, we explain how the transition function of a cascaded connection of two FSRs can be viewed as a wreath product element, and we keep this perspective throughout the rest of the paper. This is followed by Section [3](#ch:Auxiliaries){reference-type="ref" reference="ch:Auxiliaries"}, in which we collect some auxiliary results, of algebraic nature, that are needed for our main results. Those main results are then formulated and proved in Section [4](#results){reference-type="ref" reference="results"}. Specifically, in Theorem [Theorem 17](#thm 3.2){reference-type="ref" reference="thm 3.2"}, we give a nontrivial upper bound on the maximum period that can be achieved by the cascaded connections of two FSRs. Moreover, we give a comprehensive analysis, via algebraic methods, of the possible cycle structures of cascaded connections of a De Bruijn cycle into a linear FSR, leading to Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}, an explicit description of those cycle structures. Finally, in Section [5](#examples){reference-type="ref" reference="examples"}, we go through some computational examples to illustrate our method. # Preliminaries {#ch:Preliminaries} The purpose of this section is to introduce terminology and definitions needed throughout the paper. ## Feedback shift registers and cascaded connections {#subsec2P1} An $n$-stage feedback shift register (FSR) with characteristic function $$\begin{aligned} f(x_0, x_1,\ldots, x_n) = f_1(x_0, x_1,\ldots, x_{n-1}) \oplus x_n \end{aligned}$$ consists of $n$ binary storage devices called stages. Each stage associates with a state variable $x_i \in \{0, 1\}$ which represents the current value of the stage. Moreover, the stages are arranged linearly, say from left to right, and are connected through wires with the neighboring stages as well as with a circuit representing the Boolean function $f_1$, called the *update function* of the FSR, that feeds its output back to the rightmost stage. Finally, the leftmost stage is connected outward via wires; it represents the output bit in each iteration of the FSR. The following picture illustrates the situation. [\[simplefsr\]]{#simplefsr label="simplefsr"} {#simplefsr width="120mm" height="28mm"} In the beginning of the process of creating an output bit sequence with an FSR, its $n$ stages are initialized to certain bit values. Then, in each iteration of the process, the leftmost bit $x_0$ is output, all stages except the rightmost are overwritten with the stored bit value from their right neighbor, and the rightmost stage is overwritten with the bit $f_1(x_0,x_1,\ldots,x_{n-1})$. One may think of each stage of this process as encoded by the vector $(x_0,x_1,\ldots,x_{n-1})\in\mathbb{F}_2^n$ of bits stored in the $n$ stages, and then the passing from one process iteration to the next is represented by an application of the so-called *(state) transition function* $$\tilde{f}:\mathbb{F}_2^n\rightarrow\mathbb{F}_2^n, (x_0,x_1,\ldots,x_{n-1})\mapsto (x_1,x_2,\ldots,f_1(x_0,\ldots,x_{n-1})).$$ Throughout this paper, we will use - lower-case Latin letters such as $f$ or $g$ for the characteristic functions of FSRs; - those same letters with an added subscript $1$, as in $f_1$ or $g_1$, for the associated update function (see also below for the meaning of the corresponding notation with subscript $0$ instead of subscript $1$); and - those same letters with an added tilde, such as $\tilde{f}$ or $\tilde{g}$, to denote the associated transition function. We note that an $n$-stage FSR with characteristic function $f$, which we also denote by $\operatorname{FSR}(f)$ as in the Introduction, is *periodic*, i.e., always returns to any chosen initial state after a suitable number of iterations, if and only if its transition function $\tilde{f}$ is a permutation of $\mathbb{F}_2^n$ if and only if the (Boolean) update function $f_1$ can be written in the form $$f_1(x_0,x_1,\ldots,x_{n-1})=x_0 \oplus f_0(x_1,x_2,\ldots,x_{n-1})$$ for a suitable $(n-1)$-variate Boolean function $f_0$. Next, we recall the definition of a cascaded connection of (two) FSRs. **Definition 1**. [@Wang Definition 2] Let $\operatorname{FSR}(f)$ be an $n$-stage FSR, and let $\operatorname{FSR}(g)$ be an $m$-stage FSR. The *cascaded connection of $\operatorname{FSR}(f)$ into $\operatorname{FSR}(g)$*, denoted by $\operatorname{FSR}(f;g)$, is shown by Figure [2](#Cascaded){reference-type="ref" reference="Cascaded"}. [\[cascaded\]]{#cascaded label="cascaded"} {#Cascaded width="160mm" height="17mm"} Again, the leftmost bit in each iteration is the output bit, and we have a natural notion of transition function for $\operatorname{FSR}(f;g)$, namely the function $$\begin{aligned} &\tilde{f}\ast\tilde{g}:\mathbb{F}_2^{m+n}\rightarrow\mathbb{F}_2^{m+n}, \\ &(x_0,\ldots,x_{m-1},y_0,\ldots,y_{n-1})\mapsto \\ &(x_1,\ldots,x_{m-1},g_1(x_0,\ldots,x_{m-1})+y_0,y_1,\ldots,y_{n-1},f_1(y_0,\ldots,y_{n-1})).\end{aligned}$$ The chosen notation $\tilde{f}\ast\tilde{g}$ makes sense because $f_1$ and $g_1$ can be derived from their associated transition functions $\tilde{f}$ and $\tilde{g}$, whence the transition function of $\operatorname{FSR}(f;g)$ can be viewed as an object depending on $\tilde{f}$ and $\tilde{g}$. It was shown by Green and Dimond in  [@Green] that the cascaded connection $\operatorname{FSR}(f;g)$ is equivalent (in the sense defined below) to the FSR whose characteristic function is the so-called $\ast$-product of $f$ into $g$ (to be distinguished from the $\ast$-product of transition functions introduced in the previous paragraph), which is the function in $n+m$ variables defined by the formula $$\begin{aligned} f\ast g = f(g(x_0, x_1,\ldots, x_m), g(x_1, x_2,\ldots, x_{m+1}),\ldots, g(x_n, x_{n+1},\ldots, x_{m+n})).\end{aligned}$$ Here, equivalence is to be understood as the existence of a bijective function $b:\mathbb{F}_2^{n+m}\rightarrow\mathbb{F}_2^{n+m}$ such that for each $\vec{x}\in\mathbb{F}_2^{n+m}$, the bit sequence produced by $\operatorname{FSR}(f\ast g)$ when initialized with $\vec{x}$ has the same period as the bit sequence produced by $\operatorname{FSR}(f;g)$ when initialized with $b(\vec{x})$. In the next subsection, we provide a description of algebraic nature of the transition function $\tilde{f}\ast\tilde{g}$ of a cascaded connection $\operatorname{FSR}(f;g)$, which will be very useful in the sequel. ## Transition functions of cascaded connections as wreath product elements The goal of this subsection is to explain how to view transition functions of cascaded connections of FSRs as wreath product elements, and how this enables an algebraic approach to study the periods of output sequences of cascaded connections. We start with the following elementary, but important observation, which reduces periods of output sequences to cycle lengths of transition functions: **Lemma 2**. *Let $\operatorname{FSR}(f)$ be an $n$-stage FSR, and let $\operatorname{FSR}(g)$ be an $m$-stage FSR. Moreover, let $\Vec{v}=(x_0,\ldots,x_{m-1},y_0,\ldots,y_{n-1})^T\in\mathbb{F}_2^{m+n}$. The following hold:* 1. *The output sequence produced by the cascaded connection $\operatorname{FSR}(f;g)$ when initialized with $\Vec{v}$ is periodic if and only if $\Vec{v}$ is a periodic point of the transition function $\tilde{f}\ast\tilde{g}$ (i.e., one eventually returns to $\Vec{v}$ after sufficiently many iterations of $\tilde{f}\ast\tilde{g}$).* 2. *Assume that $\Vec{v}$ is a periodic point of $\tilde{f}\ast\tilde{g}$. Then the cycle length of $\Vec{v}$ under $\tilde{f}\ast\tilde{g}$ equals the (least) period of the output sequence produced by $\operatorname{FSR}(f;g)$ when initialized with $\Vec{v}$.* *Proof.* For $t \in \mathbb{N}_0$, let us set $\Vec{v}_t := (\tilde{f}\ast\tilde{g})^{t}(\Vec{v})$. Also, let $b_t$ denote the $t$-th bit in the output sequence. By definition, $b_t$ is the first entry of $\Vec{v}_t$, and the following two statements follow from this: - If $\Vec{v}$ is a periodic point of $\tilde{f}\ast\tilde{g}$, then the associated output sequence $(b_t)_{t\geq0}$ is periodic. - Under the assumption of statement (2), the period length of $(b_t)_{t\geq 0}$ divides that of $(\Vec{v}_t)_{t\geq 0}$, which is the cycle length of $\Vec{v}$ under $\tilde{f}\ast\tilde{g}$. On the other hand, for any $t \in \mathbb{N}_0$, consider the length $m+n$ segment $(b_t,b_{t+1},\ldots,b_{t+m+n-1})$ of $(b_t)_{t\geq 0}$. We prove that $\Vec{v}_t$ can be reconstructed from that segment through applying a certain injective function. Indeed, $(b_t,b_{t+1},\ldots,b_{t+m-1})$ is equal to the length $m$ initial segment of $\Vec{v}_t = (v^{(t)}_0,v^{(t)}_1,\ldots,v^{(t)}_{m+n-1})$, so the first $m$ bits $v^{(t)}_0,v^{(t)}_1,\ldots,v^{(t)}_{m-1}$ in $\Vec{v}_t$ can be directly read off. Moreover, for each $k = 0,1,\ldots,n-1$, it is easy to infer from the definitions of the involved concepts that $$b_{t+m+k} = g_1(b_{t+k},b_{t+k+1},\ldots,b_{t+m+k-1})+ v^{(t)}_{m+k},$$ so that $$v^{(t)}_{m+k} = b_{t+m+k}+ g_1(b_{t+k},b_{t+k+1},\ldots,b_{t+m+k-1})$$ can also be reconstructed. Therefore, there is an injective function $\iota: \mathbb{F}_2^{m+n} \rightarrow \mathbb{F}_2^{m+n}$ such that $\iota(b_t,b_{t+1},\ldots,b_{t+m+n-1}) = \Vec{v}_t$ for all $t \geq 0$. This implies the following two statements and concludes the proof: - If the output sequence $(b_t)_{t\geq0}$ is periodic, then $\Vec{v}$ is a periodic point of $\tilde{f}\ast\tilde{g}$. - Under the assumption of statement (2), the period of $(\Vec{v}_t)_{t\geq 0}$, which is the cycle length of $\Vec{v}$ under $\tilde{f}\ast\tilde{g}$, divides the period of the output sequence $(b_t)_{t\geq 0}$.  ◻ Now we turn to explaining our algebraic approach for studying cycle lengths of transition functions of cascaded connections. First, we need to explain what a wreath product is. Let $X$ and $\Omega$ be sets. We view the Cartesian product $X\times\Omega$ as a disjoint union of $|\Omega|$ copies of $X$, indexed by the elements of $\Omega$. For example, if $\Omega=\{\omega_1, \omega_2, \omega_3\}$, then $$\begin{aligned} X\times\Omega &= (X\times\{\omega_1\})~~\dot{\cup}~~(X\times\{\omega_2\})~~\dot{\cup}~~(X\times\{\omega_3\}) \\& = X_{\omega_1}~\dot{\cup}~X_{\omega_2}~\dot{\cup}~X_{\omega_3}.\end{aligned}$$ In the following definition and beyond, we use the exponent notation for function values, writing $x^f$ instead of $f(x)$. This is a common use of notation in group theory when $f$ is an element from a group that acts on a set of which $x$ is an element. We may still sometimes write $f(x)$ where this improves readability. We also use the notations $\textnormal{Sym}(X)$ to denote the symmetric group over the set $X$ (the group of all permutations of $X$ under function composition). **Definition 3**. Let $G \leq \textnormal{Sym}(X)$ and $P \leq \textnormal{Sym}(\Omega)$ be permutation groups. The *(imprimitive) permutational wreath product* of $G$ and $P$, written $G \wr P$, is the subgroup of $\textnormal{Sym}(X \times \Omega)$ consisting of all permutations that preserve the above partition of $X\times\Omega$, permuting the $\Omega$-indexed blocks according to some element of $P$. That is, each element of $G \wr P$ is the composition of a permutation $\sigma \in P \leq \textnormal{Sym}(\Omega)$ that permutes the blocks according to the rule $(x, \omega) \mapsto (x, \omega^{\sigma})$, and a tuple $(f_{\omega})_{\omega \in \Omega}$ of permutations in $G$ that permutes each block among itself, according to the rule $(x, \omega) \mapsto (x^{f_{\omega}}, \omega)$. The representation of an element of $G \wr P$ in the form $\sigma(f_{\omega})_{\omega \in \Omega}$ is unique. In fact, $\{(x, \omega) \mapsto (x, \omega^{\sigma}): \sigma \in P \}$ and $\{(x, \omega) \mapsto (x^{f_{\omega}}, \omega) : (f_{\omega})_{\omega \in \Omega} \in G^{\Omega} \}$ are subgroups of $G \wr P$, isomorphic to $P$ and (the direct power) $G^{\Omega}$, respectively, and $G \wr P = P \ltimes G^{\Omega}$ (a semidirect product of $P$ and $G^{\Omega}$), where the conjugation action of $P$ on the normal subgroup $G^{\Omega}$ is by permuting coordinates, i.e., $$\begin{aligned} (f_{\omega})_{\omega \in \Omega}^{\sigma} = \sigma^{-1}(f_{\omega})_{\omega \in \Omega}~\sigma = (f_{\sigma^{-1}(\omega)})_{\omega \in \Omega}.\end{aligned}$$ For example, if $\Omega = \{\omega_1, \omega_2, \omega_3\}$ and $\sigma$ is the $3$-cycle $(\omega_1, \omega_2, \omega_3)$, then $$\begin{aligned} (f_{\omega_1}, f_{\omega_2}, f_{\omega_3})^{\sigma} = (f_{\sigma^{-1}(\omega_1)}, f_{\sigma^{-1}(\omega_2)}, f_{\sigma^{-1}(\omega_3)}) = (f_{\omega_3}, f_{\omega_1}, f_{\omega_2}).\end{aligned}$$ We note that the imprimitive wreath product of permutation groups has a natural generalization to transformation semigroups, which will also be needed in the sequel: if $G$ is a subsemigroup of $X^X$, the semigroup of all functions $X\rightarrow X$ under composition, and if $P$ is a subsemigroup of $\Omega^\Omega$, then $G\wr P$ consists of all functions $\Omega\times X \rightarrow \Omega\times X$ of the form $$\sigma(g_{\omega})_{\omega\in\Omega}: (x,\omega)\mapsto (x^{f_{\omega^\sigma}},\omega^\sigma).$$ Before we proceed to explain how the cycle structure of a wreath product element may be determined, we clarify how the concept of a wreath product relates to cascaded connections of FSRs. **Proposition 4**. *Let $\operatorname{FSR}(f)$ be an $n$-stage FSR, and let $\operatorname{FSR}(g)$ be an $m$-stage FSR. Moreover, let $\Vec{t}=(0,\ldots,0,1)^T\in\mathbb{F}_2^m$, and denote by $\rho(\Vec{t}):\mathbb{F}_2^m\rightarrow\mathbb{F}_2^m$ the translation $\Vec{v}\mapsto\Vec{v}+\Vec{t}$. Finally, denote by $\pi_1:\mathbb{F}_2^n\rightarrow\mathbb{F}_2$ the projection onto the first coordinate.* *The transition function $\tilde{f}\ast\tilde{g}$ of the associated cascaded connection is equal to the wreath product element $$\tilde{f}\cdot(\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{y})})_{\Vec{y}\in \mathbb{F}_2^n} \in ({\mathbb{F}_2^m}^{(\mathbb{F}_2^m)}) \wr ({\mathbb{F}_2^n}^{(\mathbb{F}_2^n)}).$$ Moreover, if both $\operatorname{FSR}(f)$ and $\operatorname{FSR}(g)$ are periodic, then so is $\operatorname{FSR}(f;g)$, and $\tilde{f}\ast\tilde{g}$ then lies in the permutational imprimitive wreath product $\operatorname{Sym}(\mathbb{F}_2^m)\wr\operatorname{Sym}(\mathbb{F}_2^n)$.* *Proof.* Let $X:=\mathbb{F}_2^m$ and $\Omega:=\mathbb{F}_2^n$. Then $\mathbb{F}_2^{m+n}=X\times\Omega$, and note the following about $\tilde{f}\ast\tilde{g}$: 1. It maps each copy $$\begin{aligned} X \times \{(y_0, y_1,\ldots, y_{n-1})^{T}\} = \{(x_0, x_1,\ldots, x_{m-1}, y_0,\ldots, y_{n-1})^{T} : x_0,\ldots,x_{m-1} \in \mathbb{F}_2\} \end{aligned}$$ of $X$ into another such copy, namely to $X \times \{\tilde{f}(y_0, y_1,\ldots, y_{n-1})^{T}\}$. Moreover, if $\operatorname{FSR}(f)$ is periodic, then $\tilde{f}$ is surjective onto $\mathbb{F}_2^n$, whence each copy is "hit" in that case. 2. After mapping $X\times\{\Vec{y}\}$ to $X\times\{\tilde{f}(\Vec{y})\}$ without altering the $X$-part, we can get the actual $(\tilde{f}\ast\tilde{g})$-value by applying either $\tilde{g}$ or $\tilde{g}\rho(\Vec{t})$ to the $X$-part, depending on whether or not $y_0=\pi_1(\Vec{y})=0$. Moreover, if $\operatorname{FSR}(g)$ is periodic, then both $\tilde{g}$ and $\rho(\Vec{t})$ are surjective onto $\mathbb{F}_2^m$, whence each point in each block is "hit". Because $\operatorname{Sym}(\mathbb{F}_2^m)\wr\operatorname{Sym}(\mathbb{F}_2^n)$ is the set intersection of $\operatorname{Sym}(\mathbb{F}_2^{m+n})$ and $({\mathbb{F}_2^m}^{(\mathbb{F}_2^m)}) \wr ({\mathbb{F}_2^n}^{(\mathbb{F}_2^n)})$, the proof is complete. ◻ Henceforth, for the sake of simplicity, we restrict ourselves to only considering *periodic* FSRs (and cascaded connections thereof); these are also the most relevant for practical applications. In view of Lemma [Lemma 2](#lemma 2.12){reference-type="ref" reference="lemma 2.12"}, if $\operatorname{FSR}(f)$ and $\operatorname{FSR}(g)$ are periodic FSRs, then understanding the periods of output sequences of the cascaded connection $\operatorname{FSR}(f;g)$ is essentially the same as understanding the cycle lengths of the transition function $\tilde{f}\ast\tilde{g}$. Formally, the cycle structure of a permutation is encoded in its so-called cycle type, defined as follows. **Definition 5**. The *cycle type* of a permutation $\sigma$ of a finite set $X$ is the monomial $$\begin{aligned} \textnormal{CT}(\sigma) := x_1^{e_1}x_2^{e_2}\cdots x_{|X|}^{e_{|X|}} \in \mathbb{Q}[x_n : n \in \mathbb{N}^+] ,\end{aligned}$$ where $e_{\ell}$ for $\ell \in \{1, 2, \ldots, |X|\}$ is the number of cycles of length $\ell$ of $\sigma$. Specifically for understanding the cycle types of elements of imprimitive permutational wreath products, the following concept is crucial. **Definition 6**. [@Alexander Definition 3.4] Let $G \leq \textnormal{Sym}(\Omega)$ be a permutation group on the finite set $\Omega$. Let $d$ be a positive integer, let $\psi \in \textnormal{Sym}(d)$, and let $g_0, g_1,\ldots, g_{d-1} \in G$. Consider the element $g = (\psi, (g_0, g_1,\ldots, g_{d-1}))$ of the imprimitive permutational wreath product $G \wr \textnormal{Sym}(d)$. For each cycle $\zeta = (i_0, i_1,\ldots, i_{\ell-1})$ of $\psi$ we call an element of $G$ of the form $$\begin{aligned} \operatorname{fcp}_{\zeta,i_0}(g) := g_{i_0}g_{i_1}\cdots g_{i_{\ell-1}} \end{aligned}$$ a *forward cycle product* of $g$ with respect to $\zeta$. In the context of Definition [Definition 6](#fcpDef){reference-type="ref" reference="fcpDef"}, we note that there may be several forward cycle products for a given cycle $\zeta$ and tuple $\vec{g}=(g_0,\ldots,g_{d-1})$, depending on the chosen starting point $i_0$ of $\zeta$. However, all forward cycle products for given $\zeta$ and $\vec{g}$ are cyclic shifts of each other and, in particular, they are conjugate elements of $G$ and thus have the same cycle type. The cycle types of elements of (imprimitive) permutational wreath products have been studied by Pólya [@Polya], and we describe the method in the following remark. **Remark 7**. Let $g = \sigma\cdot(f_{\omega})_{\omega\in \Omega} \in G \wr P$, and let $\omega \in \Omega$. The set of all points on the $\sigma$-cycle of $\omega$ is denoted by $\omega^{\langle \sigma \rangle}$. Then the subset $M_{\omega} := \dot{\bigcup}_{\xi \in \omega^{\langle \sigma \rangle}}(X\times\{\xi\})$ of $X \times \Omega$ is mapped to itself by $g$, so it is a disjoint union of cycles of $g$. To determine the cycle type $\operatorname{CT}(g)$, proceed as follows: 1. Writing the cycle of $\omega$ under $\sigma$ as $(\omega_0, \omega_1, \ldots, \omega_{\ell -1})$, where $\omega = \omega_0$, compute the forward cycle product $$\begin{aligned} f := f_{\omega_0}f_{\omega_1}\cdots f_{w_{\ell -1}} \in G \leq \operatorname{Sym}(X) \end{aligned}$$ and its cycle type $\operatorname{CT}(f)$. 2. The cycle type of $g|_{M_{\omega}}$ is the so-called $\ell$-blow-up of $\textnormal{CT}(f)$: $$\begin{aligned} \operatorname{CT}(g|_{M_{\omega}}) = \operatorname{BU}_{\ell}(\operatorname{CT}(f)), \end{aligned}$$ where $\operatorname{BU}_{\ell}$ is the unique $\mathbb{Q}$-algebra endomorphism of $\mathbb{Q}[x_n : n\in \mathbb{N}^+]$ with $\operatorname{BU}_{\ell}(x_n) = x_{\ell\cdot n}$ for all $n \in \mathbb{N}^+$ (all cycle lengths are "blown up" by the factor $\ell$). 3. To get $\operatorname{CT}(g)$ as a whole, take the product of the cycle types computed in point (2) for all the (disjoint) cycles of $\sigma$ on $\Omega$. **Example 8**. Let $X = \{1,2,3,4,5\}$, $\Omega = \{1,2,3\}$ and $g = \sigma\cdot(f_1,f_2,f_3)$, where $\sigma = (1,2,3)$, $f_1 = (1,2,3)(4,5)$, $f_2 = (1,2)(4,5)$, and $f_3 = (1,4,3,2)$. Since $\sigma$ is a long cycle, we have $M_1 = X\times\Omega$ and $\operatorname{CT}(g) = \operatorname{CT}(g|_{M_1})$. So, $$\begin{aligned} \operatorname{CT}(g) = \operatorname{CT}(g|_{M_1}) &= \operatorname{BU}_3(\operatorname{CT}(f_1\cdot f_2\cdot f_3))\\&= \operatorname{BU}_3(\operatorname{CT}((1,2,3)(4,5)\cdot(1,2)(4,5)\cdot(1,4,3,2))) \\&= \operatorname{BU}_3(\operatorname{CT}((1,4,3)))\\&= \operatorname{BU}_3(x_1^2x_3)\\&= x_3^2x_9.\end{aligned}$$ This means that $g$ consists of two cycles of length 3 and one cycle of length 9. At this point, a comparison with the method of Mykkeltveit, Siu and Tong [@Mykkeltveit Section 2] seems appropriate. Please note that each cycle of the block permutation $\sigma$ is associated with a corresponding set of cycles of $g=\sigma(f_{\omega})_{\omega\in\Omega}$, which are just "blown up" cycles of an associated forward cycle product. A close look at [@Mykkeltveit Theorem 2.1] reveals that for a fixed cycle of $\sigma$, the associated set of "blown up" cycles is what Mykkeltveit, Siu and Tong call $C_j$ (when the cycle of $\sigma$ is represented by the periodic bit sequence $a_j$). The fact that the "blow-up factor" is the length of the associated cycle of $\sigma$ is reflected in their observation that the said cycle length divides the length of any associated blown up cycle, see [@Mykkeltveit Theorem 2.1(d)]. Finally, the sequences in Mykkeltveit-Siu-Tong's set $\theta(f)^{-1}(a_j)$ correspond to the cycles of a certain forward cycle product (fixed with the choice of the sequence $a_j$). The upshot of this discussion is that the reduction argument furnished by [@Mykkeltveit Theorem 2.1] can, in fact, be traced back to much older ideas of Pólya, and as we will see in our Section [4](#results){reference-type="ref" reference="results"}, this conceptual algebraic perspective leads to new results on periods of cascaded connections of FSRs, the proofs of which are quite natural but would have been much harder to come up with while following the technical approach of [@Mykkeltveit]. # Auxiliary algebraic results {#ch:Auxiliaries} In this section, we recall a result of the first and third authors' paper [@Bors] and then we state and prove some lemmas which are going to be used in the proofs of our main results in Section [4](#results){reference-type="ref" reference="results"}. **Proposition 9**. *[@Bors Proposition 2.1][\[prop 2.12\]]{#prop 2.12 label="prop 2.12"} Let $q > 1$ be a power of a prime $p$, let $Q, U \in \mathbb{F}_q[X]$ with $Q \neq X$ monic irreducible, and let $e$ be a positive integer. Consider the affine permutation $$\lambda(X,U): R+(Q^e) \mapsto RX+U+(Q^e)$$ of $\mathbb{F}_q[X]/(Q^e)$.* 1. *If $Q \neq X-1$, then $\lambda(X,U)$ has the following cycle count (independently of $U$):* - *1 fixed point;* - *$\frac{q^{\deg(Q)}-1}{\textnormal{ord}(Q)}$ cycles of length $\textnormal{ord}(Q)$;* - *for each $a = 1, 2 ,\ldots, \lceil \log_p(e) \rceil - 1$: $\frac{q^{p^{a-1}\deg(Q)}(q^{\deg(Q)p^{a-1}(p-1)}-1)}{p^a\textnormal{ord}(Q)}$ cycles of length $\textnormal{ord}(Q)p^a$; and* - *$\frac{q^{p^{\lceil \log_p(e)\rceil-1}\deg(Q)}(q^{\deg(Q)(e-p^{\lceil \log_p(e)\rceil-1})}-1)}{p^{\lceil \log_p(e)\rceil}\textnormal{ord}(Q)}$ cycles of length $\textnormal{ord}(Q)p^{\lceil \log_p(e)\rceil}$.* 2. *If $Q = X-1$ and $U + (Q^e)$ is a non-unit in $\mathbb{F}_q[X]/(Q^e)$, then $\lambda(X,U)$ has the following cycle count:* - *$q$ fixed points;* - *for each $a = 1, 2 ,\ldots, \lceil \log_p(e) \rceil - 1$: $\frac{q^{p^{a-1}}(q^{p^{a-1}(p-1)}-1)}{p^a}$ cycles of length $p^a$; and* - *$\frac{q^{p^{\lceil \log_p(e)\rceil-1}}(q^{e-p^{\lceil \log_p(e)\rceil-1}}-1)}{p^{\lceil \log_p(e)\rceil}}$ cycles of length $p^{\lceil \log_p(e)\rceil}$.* 3. *If $Q = X-1$ and $U+(Q^e)$ is a unit in $\mathbb{F}_q[X]/(Q^e)$, then $\lambda(X,U)$ has $\frac{q^e}{p^{\lfloor \log_p(e)\rfloor+1}}$ cycles, all of length $p^{\lfloor \log_p(e)\rfloor+1}$.* We remark that the version of Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"} published in [@Bors] contains a typographical error: in the last bullet point in statement (1), the first exponent of $q$ was wrongly typed as $p^{\lceil\log_p(e)\rceil}\deg(Q)$, as opposed to the correct value of $p^{\lceil\log_p(e)\rceil-1}\deg(Q)$ given here. Moreover, statement (3) in our Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"} was unnecessarily split into two parts in [@Bors Proposition 2.1] using slightly different formulas ($\lceil\cdots\rceil$ instead of $\lfloor\cdots\rfloor$). In what follows, we work with (primary) rational canonical forms of matrices, which we recall briefly. Let $K$ be a field. For a given monic degree $n$ univariate polynomial $P=X^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0\in K[X]$, the *companion matrix of $P$*, written $\textnormal{Comp}(P)$, is the following $(n\times n)$-matrix over $K$: $$\textnormal{Comp}(P)= \begin{pmatrix} 0 & 0 & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & \cdots & 0 & -a_2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -a_{n-1} \end{pmatrix}.$$ Please note that $\textnormal{Comp}(P)$ is the matrix representing the ($K$-linear) modular multiplication by $X$ on the polynomial residue class ring $K[X]/(P)$ with respect to the $K$-basis $1+(P),X+(P),X^2+(P),\ldots,X^{n-1}+(P)$. One can turn the $K$-vector space $K^n$ into a $K[X]$-module by declaring that for all $Q\in K[X]$ and all $v\in K^n$, one has $Q\cdot v:=Q(A)v$. Using an important structural theorem which states that any module over a principal ideal domain (PID), such as $K[X]$, is a direct sum of cyclic modules over that PID, one obtains that the linear function $v\mapsto Av$ can be represented, with respect to a suitable $K$-basis of $K^n$, by a block diagonal matrix each block of which is the companion matrix of a suitable monic polynomial (of degree at most $n$) over $K$. There is a unique (up to reordering of the blocks) such matrix in which the number of diagonal blocks is minimal; this is called the *rational canonical form of $A$* (for more details on the theory of rational canonical forms, see [@Dummit Chapter 12]). On the other hand, if the monic polynomial $P\in K[X]$ admits the factorization $P=Q_1^{e_1}Q_2^{e_2}\cdots Q_r^{e_r}$ into powers of monic irreducible polynomials $Q_i\in K[X]$, then the Chinese Remainder Theorem for PIDs can be used to show that the matrix $\textnormal{Comp}(P)$ is similar to the block diagonal matrix with blocks $\textnormal{Comp}(Q_i^{e_i})$ for $i=1,2,\ldots,r$. Hence, any $(n\times n)$-matrix $A$ over $K$ also has a block diagonal form where each diagonal block is the companion matrix of a power of a monic irreducible polynomial over $K$. This latter block diagonal form is also unique up to reordering of its blocks and is called the *primary rational canonical form of $A$*. Henceforth, we will work with primary rational canonical forms of matrices. We wish to understand, for a fixed automorphism $\alpha$ of $V$, the different possibilities for the cycle types of affine permutations of $V$ of the form $\lambda(\alpha,v): x\mapsto x^{\alpha}+v$, where $v$ ranges over $V$. It turns out that these possibilities are in bijection with the possible $\alpha$-weights of the vectors $v$ in the sense of the following notation; see Proposition [Proposition 12](#prop 2.15){reference-type="ref" reference="prop 2.15"} below. **Notation 10**. Let $q = p^f$ be a prime power, let $V$ be a finite-dimensional $\mathbb{F}_q$-vector space, and let $\alpha$ be an $\mathbb{F}_q$-automorphism of $V$. Assume that the primary rational canonical blocks of $\alpha$ are $\textnormal{Comp}(Q_i^{e_i})$ for $i = 1,2,\ldots,s$, listed with multiplicities, and let $V = \bigoplus_{i=1}^{s}V_i$ be a direct decomposition of $V$ into corresponding block subspaces. For $i = 1,2,\ldots,s$, we define a subspace $W_i$ of $V_i$ as follows. If $Q_i\not=X-1$, then we simply set $W_i:=V_i$. On the other hand, if $Q_i=X-1$, then $W_i:=\operatorname{im}(\alpha_i-\operatorname{id})$, the image of the linear function $\alpha_i-\operatorname{id}$, is the subspace consisting of the so-called *$\alpha_i$-non-units* (an element of $V_i$ that does not lie in $W_i$ is called an *$\alpha_i$-unit*). For $v \in V$, denote by $v_i$ for $i = 1,2,\ldots s$ the projection of $v$ to $V_i$. The $\alpha$-weight of $v$, denoted by $\textnormal{wt}_{\alpha}(v)$, is defined as follows: $$\begin{aligned} \textnormal{wt}_{\alpha}(v):= \textnormal{max}(\{0\}\cup\{1 + \lfloor \log_p(e_i) \rfloor : 1\leq i \leq s, v_i \not\in W_i\}).\end{aligned}$$ Of course, for those $i$ for which $Q_i\not=X-1$, it is impossible to have $v_i\notin W_i$, so only the indices $i$ for which $Q_i=X-1$ are relevant for the computation of the $\alpha$-weight of $v$. In particular, if there are no $i$ such that $Q_i=X-1$, then all $v\in V$ have $\alpha$-weight $0$. Before we can prove Proposition [Proposition 12](#prop 2.15){reference-type="ref" reference="prop 2.15"}, we need some more preparations. Let us use the direct decomposition $V=\oplus_{i=1}^s{V_i}$ from Notation [Notation 10](#notation 2.14){reference-type="ref" reference="notation 2.14"}, and denote by $\alpha_i$, respectively $v_i$, the restriction, respectively projection, of $\alpha$, respectively $v$, to $V_i$. Then we can view $\lambda(\alpha,v)$ as the component-wise application of the affine permutations $\lambda(\alpha_i,v_i)$ of the block subspaces $V_i$. As such, we have $$\begin{aligned} \textnormal{CT}(\lambda(\alpha,v)) = \divideontimes_{i=1}^{s} \textnormal{CT}(\lambda(\alpha_i,v_i))\end{aligned}$$ where $\divideontimes$ is the $\mathbb{Q}$-bilinear product of polynomials introduced by Wei and Xu in [@Wei-xu Definition 2.2]. Specifically, $$\prod_i{x_i^{e_i}}\divideontimes\prod_j{x_j^{\epsilon_j}} = \prod_{i,j}{x_{\operatorname{lcm}(i,j)}^{e_i\epsilon_j\gcd(i,j)}}.$$ For example, $$\begin{aligned} (x_2^2x_3)\divideontimes(x_3^2x_4) &=x_{\operatorname{lcm}(2,3)}^{2\cdot2\cdot\gcd(2,3)}\cdot x_{\operatorname{lcm}(2,4)}^{2\cdot 1\cdot\gcd(2,4)}\cdot x_{\operatorname{lcm}(3,3)}^{1\cdot 2\cdot\gcd(3,3)}\cdot x_{\operatorname{lcm}(3,4)}^{1\cdot 1\cdot\gcd(3,4)} =x_6^4\cdot x_4^4\cdot x_3^6\cdot x_{12} \\ &=x_3^6x_4^4x_6^4x_{12}.\end{aligned}$$ The following result is sometimes useful when trying to simplify a Wei-Xu product, and we will use it in the proof of Proposition [Proposition 12](#prop 2.15){reference-type="ref" reference="prop 2.15"} below. Henceforth, we use the notation $\mathbb{N}^+$ to denote the set of positive integers. **Lemma 11**. *Let $$\gamma=\prod_{\ell\in\mathbb{N}^+}{x_{\ell}^{k_{\ell}}}$$ be the cycle type of a permutation on a finite number $m$ of points, and let $$\delta=\prod_{\ell\in\mathbb{N}^+}{x_{\ell}^{n_{\ell}}}$$ be another cycle type of a finite permutation, not necessarily on $m$ points. Assume that $$\label{absorptionEq} \operatorname{lcm}\{\ell\in\mathbb{N}^+: k_{\ell}\not=0\}\text{ divides }\operatorname{gcd}\{\ell\in\mathbb{N}^+: n_{\ell}\not=0\}.$$ Then $$\gamma\divideontimes\delta=\delta^m.$$* Formulated in words, condition ([\[absorptionEq\]](#absorptionEq){reference-type="ref" reference="absorptionEq"}) says that all cycle lengths of a permutation with cycle type $\gamma$ shall divide all cycle lengths of a permutation with cycle type $\delta$. If this happens, we say that *$\delta$ absorbs $\gamma$*. *Proof of Lemma [Lemma 11](#absorptionLem){reference-type="ref" reference="absorptionLem"}.* Set $$\mathcal{L}_1:=\{\ell\in\mathbb{N}^+: k_{\ell}\not=0\}\text{ and }\mathcal{L}_2:=\{\ell\in\mathbb{N}^+: n_{\ell}\not=0\}.$$ By the definition of $\divideontimes$ and assumption ([\[absorptionEq\]](#absorptionEq){reference-type="ref" reference="absorptionEq"}), we have $$\begin{aligned} \gamma\divideontimes\delta &=(\prod_{\ell_1\in\mathcal{L}_1}{x_{\ell_1}^{k_{\ell_1}}})\divideontimes(\prod_{\ell_2\in\mathcal{L}_2}{x_{\ell_2}^{n_{\ell_2}}})=\prod_{\ell_1\in\mathcal{L}_1,\ell_2\in\mathcal{L}_2}{x_{\operatorname{lcm}(\ell_1,\ell_2)}^{\gcd(\ell_1,\ell_2)k_{\ell_1}n_{\ell_2}}}=\prod_{\ell_1\in\mathcal{L}_1,\ell_2\in\mathcal{L}_2}{x_{\ell_2}^{\ell_1k_{\ell_1}n_{\ell_2}}} \\ &=\prod_{\ell_2\in\mathcal{L}_2}{x_{\ell_2}^{mn_{\ell_2}}}=\delta^m,\end{aligned}$$ as required. ◻ A different way to see that Lemma [Lemma 11](#absorptionLem){reference-type="ref" reference="absorptionLem"} holds is by observing that if $\sigma\in\operatorname{Sym}(\Omega)$ is a permutation of cycle type $\gamma$ and $\psi\in\operatorname{Sym}(\Lambda)$ is a permutation of cycle type $\delta$, and if we consider the component-wise permutation $\sigma\times\psi\in\operatorname{Sym}(\Omega\times\Lambda)$, which is of cycle type $\textnormal{CT}(\sigma)\divideontimes\textnormal{CT}(\psi)$, then the cycle length of a point $(\omega,\lambda)\in\Omega\times\Lambda$ under $\sigma\times\psi$ is the least common multiple of the cycle lengths of $\omega$ under $\sigma$ and $\lambda$ under $\psi$ respectively, which equals the latter by condition ([\[absorptionEq\]](#absorptionEq){reference-type="ref" reference="absorptionEq"}). In other words, the first component $\omega\in\Omega$ has no influence on the cycle length of the pair $(\omega,\lambda)$. At last, we are now ready to understand how weights relate to cycle types. **Proposition 12**. *Under the assumptions of Notation [Notation 10](#notation 2.14){reference-type="ref" reference="notation 2.14"}, the following hold:* 1. *If $v, w \in V$ have the same $\alpha$-weight, then $\textnormal{CT}(\lambda(\alpha,v)) = \textnormal{CT}(\lambda(\alpha,w))$.* 2. *For all $v \in V$ , the shortest cycle length of $\lambda(\alpha,v)$ that is a power of $p$ is $p^{\textnormal{wt}_{\alpha}(v)}$. In particular, if $v, w \in V$ have different $\alpha$-weights, then $\textnormal{CT}(\lambda(\alpha,v)) \neq \textnormal{CT}(\lambda(\alpha, w))$.* *In particular, the number of distinct cycle types of affine permutations of $V$ of the form $\lambda(\alpha,v)$ with $v$ ranging over $V$ is $|\textnormal{im}(\textnormal{wt}_{\alpha})|$, the number of distinct values of the function $\operatorname{wt}_{\alpha}:V\rightarrow\mathbb{N}_0$, where $\mathbb{N}_0$ denotes the set of nonnegative integers.* *Proof.* Let $\mathfrak{m} \in \{0,1,\ldots,\lfloor \log_p(\textnormal{dim}_{\mathbb{F}_q}(V))\rfloor \}$. As observed above, for a given vector $v \in V$, we have $$\begin{aligned} \textnormal{CT}(\lambda(\alpha,v)) = \divideontimes_{i=1}^{s} \textnormal{CT}(\lambda(\alpha_i,v_i)). \end{aligned}$$ For the proof of statement (1), observe that by Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}, the variation in the possible cycle types of $\lambda(\alpha,v)$ with $v \in V$ comes exclusively from the unipotent block subspaces of $\alpha$ in the chosen direct decomposition $V = \bigoplus_{i=0}^{s}V_i$, i.e., from those block subspaces where $Q_i=X-1$. Moreover, by definition of the weight function $\textnormal{wt}_{\alpha}$, if we only consider vectors $v$ such that $\textnormal{wt}_{\alpha}(v) = \mathfrak{m}$, then only the cycle types $\textnormal{CT}(\lambda(\alpha_i,v_i))$ stemming from blocks of the form $\textnormal{Comp}((X-1)^{e_i})$ with $1+\lfloor \log_p(e_i) \rfloor \leq \mathfrak{m}$ can be varied. If $\mathfrak{m} = 0$, then no variation is possible, and the statement is clear. If $\mathfrak{m} > 0$ then at least one of the blocks of $\alpha$ of the form $\textnormal{Comp}((X-1)^{e_i})$ with $1+\lfloor \log_p(e_i) \rfloor = \mathfrak{m}$ yields the cycle type $x_{p^{1+\lfloor \log_{p}(e_i)\rfloor}}^{q^{e_i}/p^{1+\lfloor \log_{p}(e_i)\rfloor}}$, and another look at Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"} shows that this cycle type absorbs all cycle types $\lambda(\alpha_j,v_j)$ coming from blocks $\textnormal{Comp}((X-1)^{e_j})$ with $1+\lfloor \log_p(e_j) \rfloor \leq \mathfrak{m}$, regardless of whether or not $v_j$ is an $\alpha_j$-unit. Therefore, despite possible variation at the individual block level, the overall cycle type of $\lambda(\alpha,v)$ is the same for all $v \in V$ with $\textnormal{wt}_{\alpha}(v) = \mathfrak{m}$. In order to prove statement (2), we note that for each $v \in V$, by Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}(1), all nontrivial cycle lengths in the cycle type $\lambda(\alpha_i,v_i)$ corresponding a non-unipotent block $\textnormal{Comp}(Q_i^{e_i})$ of $\alpha$ are divisible by a prime distinct from $p$. Since the cycle length of a point $u \in V$ under $\lambda(\alpha,v)$ is the least common multiple of the various cycle lengths of the projections $u_i \in V_i$ under $\lambda(\alpha_i,v_i)$, it follows that if the cycle length of $u$ under $V$ is a power of $p$, then the cycle length of $u_i$ under $\lambda(\alpha_i,v_i)$ must be 1 if $Q_i \neq X-1$. On the other hand, for unipotent blocks $\textnormal{Comp}((X-1)^{e_i})$, all cycle lengths of $\lambda(\alpha_i,v_i)$ are (possibly trivial, i.e., equal to $1$) powers of $p$ by Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}(2,3), and we obtain the shortest cycle length of $\lambda(\alpha,v)$ that is a power of $p$ as the least common multiple of the shortest cycle lengths of the affine permutations $\lambda(\alpha_i,v_i)$ for those $i$ where $Q_i = X-1$. But the cycle types for those unipotent blocks where $v_i$ is an $\alpha_i$-non-unit have fixed points, whence the said least common multiple is 1 if there are no unipotent blocks where $v_i$ is an $\alpha_i$-unit. Otherwise, the said least common multiple is the largest cycle length that occurs in the cycle type of a unipotent block where $v_i$ is an $\alpha_i$-unit. In both cases, this least common multiple is represented by the expression $p^{\textnormal{wt}_{\alpha}(v)}$, as required. ◻ We conclude this section with three lemmas, which together can be used to compute the rational canonical forms of powers of invertible matrices (equivalently, of iterates of vector space automorphisms). Of course, it suffices to do this for automorphisms that are given by the companion matrix of a single power of an irreducible polynomial; such automorphisms are called *primary*. We will need the following notations. **Notation 13**. Let $q$ be a prime power, let $Q\in\mathbb{F}_q[X]$ be monic and irreducible, and let $\ell$ be an integer. 1. Let $\xi\in\mathbb{F}_{q^{\deg{Q}}}$ be one of the roots of $Q$. The minimal polynomial over $\mathbb{F}_q$ of the power $\xi^{\ell}$ is independent of the choice of $\xi$, and we denote this minimal polynomial by $\operatorname{pow}_{\ell}(Q)$. 2. We set $\operatorname{ind}_{\ell}(Q):=\frac{\operatorname{deg}{Q}}{\operatorname{deg}{\operatorname{pow}_{\ell}(Q)}}$. Observe that for a given monic irreducible polynomial $Q\in\mathbb{F}_q[X]$ and integers $\ell_1,\ell_2$, one has $$\operatorname{pow}_{\ell_1}(\operatorname{pow}_{\ell_2}(Q))=\operatorname{pow}_{\ell_1\ell_2}(Q).$$ If $\xi$ is a root of the monic irreducible polynomial $Q\in\mathbb{F}_q[X]$, then $\operatorname{ind}_{\ell}(Q)$ is equal to the field extension degree $[\mathbb{F}_q(\xi):\mathbb{F}_q(\xi^{\ell})]$. **Lemma 14**. *Let $q = p^f$ be a prime power, and let $\alpha$ be a primary $\mathbb{F}_q$-automorphism of a finite-dimensional $\mathbb{F}_q$ vector space $V$, with minimal polynomial $Q^e$ where $Q \in \mathbb{F}_q[X]$ is monic and irreducible, and $e \in \mathbb{N}^+$. Moreover, let $\ell$ be an integer that is coprime to $\textnormal{ord}(\alpha) = \textnormal{ord}(Q)\cdot p^{\lceil \log_p(e) \rceil}$. Then $\alpha^{\ell}$ is also a primary $\mathbb{F}_q$-automorphism of $V$, and its minimal polynomial is $\textnormal{pow}_{\ell}(Q)^e$.* *Proof.* Since $\alpha$ and $\alpha^{\ell}$ are powers of each other, they have the same invariant subspaces in $V$. Hence $\alpha^{\ell}$ must be primary, for otherwise, $V$ would admit a nontrivial direct decomposition into $\alpha$-invariant subspaces, contradicting that $\alpha$ is primary. As for the minimal polynomial $M$ of $\alpha^{\ell}$ on $V$, note that by definition of $\textnormal{pow}_{\ell}(Q)$, we have $$\textnormal{pow}_{\ell}(Q)(X^{\ell}) \equiv 0 \pmod{Q},$$ which implies $$\textnormal{pow}_{\ell}(Q)^e(X^{\ell}) \equiv 0 \pmod{Q^e},$$ so say $\textnormal{pow}_{\ell}(Q)^e(X^{\ell}) = R\cdot Q^e$. Since $Q^e(\alpha) = 0_{\textnormal{End}(V)}$, it follows that $$\begin{aligned} \textnormal{pow}_{\ell}(Q)^e(\alpha^{\ell}) = R(\alpha)Q^e(\alpha) = R(\alpha)0_{\textnormal{End}(V)} = 0_{\textnormal{End}(V)},\end{aligned}$$ whence $M$ divides $\textnormal{pow}_{\ell}(Q)^e$. But $$\begin{aligned} \textnormal{deg}~~M = \textnormal{dim}_{\mathbb{F}_q}(V) = e\cdot\textnormal{deg}~~Q = e\cdot\textnormal{deg}(\textnormal{pow}_{\ell}(Q)) = \textnormal{deg}(\textnormal{pow}_{\ell}(Q)^e),\end{aligned}$$ so that $M = \textnormal{pow}_{\ell}(Q)^e$ as required. ◻ **Lemma 15**. *Let $q = p^f$ be a prime power, and let $\alpha$ be a primary $\mathbb{F}_q$-automorphism of a finite-dimensional $\mathbb{F}_q$-vector space $V$, with minimal polynomial $Q^e$ where $Q \in \mathbb{F}_q[X]$ is monic and irreducible, and $e \in \mathbb{N}^+$. Moreover, let $\ell$ be a positive integer that divides $\textnormal{ord}(\alpha)_{p'}$ (the product of all prime power factors in the factorization of $\textnormal{ord}(\alpha)$ except $p^{\nu_p(\textnormal{ord}(\alpha))}$). Then for each fixed $\alpha$-unit $u \in V$, the $\alpha^{\ell}$-cyclic subspaces $U_i:= \mathbb{F}_qu^{\alpha^i\langle \alpha^{\ell}\rangle}$ of $V$ for $i = 0,1,\ldots,\textnormal{ind}_{\ell}(Q)-1$ form an $\alpha^{\ell}$-block subspace decomposition of $V$, and $(\alpha^{\ell})_{|U_i}$ has the minimal polynomial $\textnormal{pow}_{\ell}(Q)^{e}$, independent of $i$.* *Proof.* Without loss of generality, we assume that $V = \mathbb{F}_q[X]/(Q^e)$, that $\alpha$ is the multiplication by $X$ modulo $Q^e$, and that $u = 1 + (Q^e)$. Set $$\begin{aligned} \mathcal{E}:= \sum_{k=0}^{\frac{\textnormal{ord}(Q)}{\ell}-1}\mathbb{F}_{q}X^{k\ell}, \end{aligned}$$ which is an $\mathbb{F}_q$-subspace of $\mathbb{F}_q[X]$. If we view $\mathbb{F}_q[X]/(Q)$ as a copy of the finite field $\mathbb{F}_{q^{\textnormal{deg}Q}}$, then the image $\mathcal{E} \pmod{Q}$ of $\mathcal{E}$ under the canonical projection $\mathbb{F}_q[X] \rightarrow \mathbb{F}_q[X]/(Q)$ corresponds to the degree $\textnormal{ind}_{\ell}(Q)$ subfield generated by $X^{\ell} + (Q)$. In particular, the reductions modulo $Q$ of the $\mathbb{F}_q$-subspaces $X^{i}\mathcal{E}$ of $\mathbb{F}_q[X]$ for $i = 0,1,\ldots,\textnormal{ind}_{\ell}(Q)-1$ form a direct decomposition of $\mathbb{F}_q[X]/(Q)$, because $\{X^i+(Q): i = 0,1,\ldots, \textnormal{ind}_{\ell}(Q)-1\}$ is an ($\mathcal{E} \pmod{Q}$)-basis of $\mathbb{F}_q[X]/(Q)$. For $i = 0,1,\ldots,\textnormal{ind}_{\ell}(Q)-1$ denote by $U_i$ the $\alpha^{\ell}$-cyclic subspace of $V = \mathbb{F}_q[X]/(Q^e)$ generated by $X^i+(Q^e)$. That is, $$\begin{aligned} U_i = \sum_{m\in\mathbb{N}} \mathbb{F}_q(X^{i+\ell m}+(Q^e)). \end{aligned}$$ We observe that $\nu_{Q}(X^{\textnormal{ord}(Q)}-1) = 1$. Indeed, if $Q^2 \divides X^{\textnormal{ord}(Q)}-1$, then $Q^2\cdot R = X^{\textnormal{ord}(Q)}-1$ for some $R\in \mathbb{F}_q[X]$, so $0 \neq \textnormal{ord}(Q)X^{\textnormal{ord}(Q)-1} = 2QQ'R+Q^2R' = Q(2Q'R+QR')$. This implies that $Q \divides X^{\textnormal{ord}(Q)-1}$, hence $Q = X$, a contradiction. Therefore, we can write $X^{\textnormal{ord}(Q)} = 1+AQ$ for some $A \in \mathbb{F}_q[X]$ not divisible by $Q$. We claim that $$\begin{aligned} U_0 &= \mathbb{F}_q\{X^{k\ell}+(Q^e): k=0,1,\ldots,\frac{\textnormal{ord}(Q^e)}{\ell}-1\} \\&= \{\sum_{i=0}^{e-1}A^iE_iQ^i+(Q^e): E_0,E_1,\ldots,E_{e-1}\in \mathcal{E}\} =:U'_{0} \tag{1}\label{eq1}\end{aligned}$$ The first equality is by the definition of $U_0$, using that $$X^{\textnormal{ord}(Q^e)} \equiv 1 \pmod{Q^e},$$ and thus $$\begin{aligned} \mathbb{F}_q(X^{k\ell+t~\textnormal{ord}(Q^e)}+(Q^e)) = \mathbb{F}_q(X^{k\ell}+(Q^e))\end{aligned}$$ for all $k \in \{0,1,\ldots,\frac{\textnormal{ord}(Q^e)}{\ell}-1\}$ and all $t \in \mathbb{N}_0$. To see that the second equality in formula ([\[eq1\]](#eq1){reference-type="ref" reference="eq1"}) holds, we consider a general element $$\begin{aligned} u_0:= \sum_{k=0}^{\frac{\textnormal{ord}(Q^e)}{\ell}-1} a_kX^{k\ell}+(Q^e)\end{aligned}$$ of $U_0$ (with $a_k \in \mathbb{F}_q$ for all $k$), and rewrite it as follows, using that $\textnormal{ord}(Q^e) = \textnormal{ord}(Q)p^{\lceil \log_p(e)\rceil}$ by [@Lidl Chapter 3]: $$\begin{aligned} u_0 &= \sum_{i=0}^{p^{\lceil \log_p(e)\rceil}-1} \sum_{j=0}^{\frac{\textnormal{ord}(Q)}{\ell}-1} a_{i\frac{\textnormal{ord}(Q)}{\ell}+j}X^{(i\frac{\textnormal{ord}(Q)}{\ell}+j)\ell}+(Q^e) \\&= \sum_{i=0}^{p^{\lceil \log_p(e)\rceil}-1} X^{i\cdot\textnormal{ord}(Q)}\sum_{j=0}^{\frac{\textnormal{ord}(Q)}{\ell}-1} a_{i\frac{\textnormal{ord}(Q)}{\ell}+j}X^{j\ell}+(Q^e).\end{aligned}$$ For $i = 0,1,\ldots,p^{\lceil \log_p(e)\rceil}-1$, set $$\begin{aligned} E'_i:= \sum_{j=0}^{\frac{\textnormal{ord}(Q)}{\ell}-1}a_{i\frac{\textnormal{ord}(Q)}{\ell}+j}X^{j\ell}.\end{aligned}$$ Observe that the $E'_i$ are arbitrary and independent elements of $\mathcal{E}$. By the above, we have $$\begin{aligned} u_0 &= \sum_{i=0}^{p^{\lceil\log_p(e)\rceil}-1}X^{i~\textnormal{ord}(Q)}E'_i+(Q^e) = \sum_{i=0}^{p^{\lceil\log_p(e)\rceil}-1} (1+AQ)^{i}E'_i + (Q^e) \\&= \sum_{i=0}^{p^{\lceil\log_p(e)\rceil}-1} \sum_{t=0}^{i} {i\choose t}A^tQ^tE'_i+(Q^e) = \sum_{t=0}^{p^{\lceil\log_p(e)\rceil}-1}A^tQ^t\sum_{i=t}^{p^{\lceil\log_p(e)\rceil}-1}{i\choose t}E'_i+(Q^e). \end{aligned}$$ Hence, setting $$\begin{aligned} E_t:= \sum_{i=t}^{p^{\lceil\log_p(e)\rceil}-1}{i\choose t}E'_i \in \mathcal{E}, \tag{2} \label{eq2} \end{aligned}$$ we obtain the representation $$\begin{aligned} u_0 = \sum_{t=0}^{p^{\lceil\log_p(e)\rceil}-1}A^tQ^tE_t+(Q^e) \in U'_0, \end{aligned}$$ which shows that $U_0 \subseteq U'_0$. Moreover, observe that by formula ([\[eq2\]](#eq2){reference-type="ref" reference="eq2"}), we have $$\begin{aligned} & E_0 \in E'_0+\sum_{i>0}\mathbb{F}_qE'_i \\& E_1 \in E'_1+\sum_{i>1}\mathbb{F}_qE'_i,\\ &\vdots\\& E_{e-1} \in E'_{e-1}+\sum_{i>e-1}\mathbb{F}_qE'_i. \end{aligned}$$ This shows that in order to get a certain tuple of values in $\mathcal{E}$ for $E_0, E_1,\ldots, E_{e-1}$, we can start by choosing $E'_{e-1}, E'_e,\ldots,E'_{p^{\lceil \log_p(e)\rceil}-1} \in \mathcal{E}$ such that $E_{e-1}$ assumes its desired value, and then successively adjust the values of $E_{e-2}, E_{e-3},\ldots, E_1, E_0$ via suitable choices of $E'_{e-2}, E'_{e-3},\ldots,E'_1,E'_0$, respectively. Hence $u_0$ can indeed assume arbitrary values in $U'_0$, and we conclude that $U_0 = U'_0$. Next, we claim that the minimal polynomial $M$ of $\alpha^{\ell}$ on $U_0$ is $\textnormal{pow}_{\ell}(Q)^e$. To see that this holds, observe that as in the proof of Lemma [Lemma 14](#lem 2.16){reference-type="ref" reference="lem 2.16"}, we have $\textnormal{pow}_{\ell}(Q)^e(X^{\ell}) \equiv 0 \pmod{Q^e}$, which shows that $\textnormal{pow}_{\ell}(Q)^e$ annihilates $\alpha^{\ell}$ on all of $\mathbb{F}_q[X]/(Q^e)$, in particular on $U_0$. This shows that $M$ divides $\textnormal{pow}_{\ell}(Q)^e$, and thus $M = \textnormal{pow}_{\ell}(Q)^k$ for some $k \in \{1,2,\ldots,e\}$. In order to show that $k = e$, we need to verify that $\textnormal{pow}_{\ell}(Q)^{e-1}$ does not annihilate $\alpha^{\ell}$ on $U_0$. Assume otherwise. Since $1 + (Q^e) \in U_0$, it follows that $$\begin{aligned} (Q^e) = 0_{U_0} = \textnormal{pow}_{\ell}(Q)^{e-1}(X^{\ell})\cdot(1+(Q^e)) = \textnormal{pow}_{\ell}(Q)^{e-1}(X^{\ell})+(Q^e), \end{aligned}$$ which implies that $Q^e$ divides $\textnormal{pow}_{\ell}(Q)^{e-1}(X^{\ell})$. This is only possible if $\nu_{Q}(\textnormal{pow}_{\ell}(Q)(X^{\ell})) > 1$, i.e., if $Q^2$ divides $\textnormal{pow}_{\ell}(Q)(X^{\ell})$. From this, we conclude that $$\begin{aligned} Q \divides \frac{d}{dX}\textnormal{pow}_{\ell}(Q)(X^{\ell}) = \textnormal{pow}_{\ell}(Q)'(X^{\ell})\cdot \ell X^{\ell-1}. \end{aligned}$$ Since $p \not\divides \ell$ and $Q \neq X$, this implies that $$\begin{aligned} Q \divides \gcd(\textnormal{pow}_{\ell}(Q)(X^{\ell}),\textnormal{pow}_{\ell}(Q)'(X^{\ell})).\end{aligned}$$ An equivalent way of stating this last formula is the following: If $\xi \in \mathbb{F}_{q^{\textnormal{deg}Q}}$ is any of the roots of $Q$, then $\xi^{\ell}$ is a root of both $\textnormal{pow}_{\ell}(Q)$ and $\textnormal{pow}_{\ell}(Q)'$. That $\xi^{\ell}$ is a root of the irreducible polynomial $\textnormal{pow}_{\ell}(Q)$ shows that $\xi^{\ell}$ is algebraic of degree $\textnormal{deg~pow}_{\ell}(Q)$ over $\mathbb{F}_q$, whence it is impossible for it to be also a root of the non-zero polynomial $\textnormal{pow}_{\ell}(Q)'$ of strictly smaller degree, a contradiction. This completes our argument that the minimal polynomial of $\alpha^{\ell}$ on $U_0$ is $\textnormal{pow}_{\ell}(Q)^{e}$. Note that since $U_0$ is by definition an $\alpha^{\ell}$-cyclic subspace of $V$, the $\mathbb{F}_q$-dimension of $U_0$ is equal to the degree of the minimal polynomial of $\alpha^{\ell}$ on $U_0$. Hence $$\begin{aligned} \textnormal{dim}_{\mathbb{F}_q}(U_0) = \textnormal{deg~pow}_{\ell}(Q)^e = e\cdot \frac{\textnormal{deg}(Q)}{\textnormal{ind}_{\ell}(Q)} = \frac{\textnormal{dim}_{\mathbb{F}_q}V}{\textnormal{ind}_{\ell}(Q)}.\end{aligned}$$ Moreover, since each subspace $U_i$ for $i = 0,1,\ldots,\textnormal{ind}_{\ell}(Q)-1$ is an iterated image of $U_0$ under $\alpha$, an $\mathbb{F}_q$-automorphism of $V$ that commutes with $\alpha^{\ell}$, it follows that each $U_i$ is $\alpha^{\ell}$-cyclic and $\alpha^{\ell}$ has the minimal polynomial $\textnormal{pow}_{\ell}(Q)^{e}$ on $U_i$ (in particular, $\textnormal{dim}_{\mathbb{F}_q}U_i = \textnormal{dim}_{\mathbb{F}_q}U_0$). It remains to show that $V = \bigoplus_{i=0}^{\textnormal{ind}_{\ell}(Q)-1} U_i$. Because $\sum_{i=0}^{\textnormal{ind}_{\ell}(Q)-1}{\textnormal{dim}(U_i)}=\textnormal{dim}(V)$, it suffices to prove that $V=\sum_{i=0}^{\textnormal{ind}_{\ell}(Q)-1} U_i$. To see this, observe that by construction, we have $$\begin{aligned} \mathbb{F}_q[X]/(Q) = \sum_{i=0}^{\textnormal{ind}_{\ell}(Q)-1}(U_i\pmod{Q}). \tag{3} \label{eq3}\end{aligned}$$ In order to write an arbitrary element of $V$, $$\begin{aligned} v = \sum_{j=0}^{e-1}v_jQ^j+(Q^e),\end{aligned}$$ where $v_j \in \mathbb{F}_q[X]$ with $\textnormal{deg}~v_j < \textnormal{deg}~Q$, as a sum of elements $$\begin{aligned} u_i = X^i\sum_{j=0}^{e-1}A^jE_{i,j}Q^j+(Q^e) \in U_i\end{aligned}$$ for $i = 0,1,\ldots,\textnormal{ind}_{\ell}(Q)-1$ we take a recursive approach: we successively choose for $j = 0,1,\ldots, e-1$ the values of the "$j$-th digits" $E_{i,j}$ of $v_i$ modulo $Q$ such that the "$j$-th digit" $v_j = (A^j\sum_{i=0}^{\textnormal{ind}(Q)-1}X^iE_{i,j})$ mod $Q$ assumes the desired value (which is possible by formula ([\[eq3\]](#eq3){reference-type="ref" reference="eq3"})). Note that this may cause some constant carry-overs to later "digits" $v_{j'}$ with $j'>j$, which is not a problem as long as those carry-overs are accounted for when adjusting the values of those digits. ◻ **Lemma 16**. *Let $q = p^f$ be a prime power, and let $\alpha$ be a primary $\mathbb{F}_q$-automorphism of a finite-dimensional $\mathbb{F}_q$-vector space $V$, with minimal polynomial $Q^e$ where $Q \in \mathbb{F}_q[X]$ is monic and irreducible, and $e \in \mathbb{N}^+$. Moreover, let $\ell$ be a positive integer that divides $\textnormal{ord}(\alpha)_p$ (in particular, $\ell$ is a power of $p$). Set $\ell':=\min(\ell,e)$, and write $\textnormal{dim}_{\mathbb{F}_q} (V) = e~\textnormal{deg}Q = a\ell'+b$ with $a, b \in \mathbb{Z}$, $a>0$ and $0 \leq b \leq \ell'-1$. The following hold:* 1. *The primary rational canonical form of $\alpha^{\ell}$ on $V$ has $\ell'$ blocks; $b$ of them are of the form $\textnormal{Comp}(\textnormal{pow}_{\ell}(Q)^{a+1})$, and the other $\ell'-b$ blocks are of the form $\textnormal{Comp}(\textnormal{pow}_{\ell}(Q)^{a})$.* 2. *If $Q = X-1$, then for each fixed $\alpha$-unit $u \in V$, the $\mathbb{F}_q$-subspaces $$\begin{aligned} V_i:= \bigoplus_{j=0}^{\lfloor \frac{e-1-i}{\ell}\rfloor} \mathbb{F}_qu^{(\alpha-1)^{i+\ell j}}~~~~for~~~ i=0,1,\ldots,\ell'-1 \end{aligned}$$ form an $\alpha^{\ell}$-block subspace decomposition of $V$ . The minimal polynomial of $(\alpha^{\ell})_{|V_i}$ is $(X-1)^{a+1}$ if $i \in \{0,1,\ldots,b-1\}$, and it is $(X-1)^a$ if $i \in \{b,b+1,\ldots,\ell'-1\}$.* *Proof.* We assume without loss of generality that $V = \mathbb{F}_q[X]/(Q^e)$, that $\alpha$ is the multiplication by $X$ modulo $Q^e$, and (for statement (2)) that $u = 1 + (Q^e)$. We start with the proof of statement (2). Note that under the above assumptions, we have $$\begin{aligned} V_i:= \bigoplus_{j=0}^{\lfloor \frac{e-1-i}{\ell}\rfloor}\mathbb{F}_q((X-1)^{i+\ell j}+((X-1)^e)). \end{aligned}$$ Since $\{(X-1)^{k}+((X-1)^e): k = 0,1,\ldots, e-1 \}$ is an $\mathbb{F}_q$-basis of $V$, it is clear that the subspace sums as which the $V_i$ are defined are indeed direct, and that $V = \bigoplus_{i=0}^{\ell-1}V_i$. Moreover, if $\ell\geq e$ (i.e., if $\ell'=e$), then $\ell=p^{\lceil\log_p(e)\rceil}=\textnormal{ord}(\alpha)_p=\textnormal{ord}(\alpha)$ necessarily, so that $\alpha^{\ell}=\textnormal{id}$, and the rest of statement (2) is thus easily verified under that assumption. For the remainder of the proof of statement (2), we assume that $\ell<e$, which implies that $\ell'=\ell$. Let $i \in \{0,1,\ldots,b-1\}$. Then $$\begin{aligned} (X^{\ell}-1)^{a+1}V_i &= (X-1)^{(a+1)\ell}\bigoplus_{j=0}^{\lfloor \frac{e-1-i}{\ell}\rfloor}\mathbb{F}_q((X-1)^{i+\ell j}+((X-1)^e)) \\&= \bigoplus_{j=0}^{\lfloor \frac{e-1-i}{\ell}\rfloor}\mathbb{F}_q((X-1)^{i+(a+j+1)\ell}+((X-1)^e)) \\&= \bigoplus_{j=0}^{\lfloor \frac{e-1-i}{\ell}\rfloor} \mathbb{F}_q\{((X-1)^e)\} = \{((X-1)^e)\} = \{0_{V}\}, \end{aligned}$$ where the third equality uses that $$i+(a+j+1)\ell\geq(a+1)\ell = a\ell+\ell>a\ell+b = e~\textnormal{deg}(X-1) = e.$$ It follows that the minimal polynomial of $\alpha^{\ell}$ on $V_i$ divides $(X-1)^{a+1}$. On the other hand, $$(X^{\ell}-1)^{a}V_i \supseteq(X^{\ell}-1)^{a}\mathbb{F}_q((X-1)^i+((X-1)^e )) = \mathbb{F}_q((X-1)^{a\ell+i}+((X-1)^e )) \neq \{0_V\}$$ because $a\ell+i\leq a\ell+b-1< e$. This shows that the minimal polynomial of $\alpha^{\ell}$ on $V_i$ is exactly $(X-1)^{a+1}$. By a similar argument, the minimal polynomial of $\alpha^{\ell}$ on $V_i$ is $(X-1)^a$ if $i \in \{b, b + 1,\ldots, \ell-1\}$. In particular, for each $i$, the degree of the minimal polynomial of $\alpha^{\ell}$ on $V_i$ equals the $\mathbb{F}_q$-dimension of $V_i$, whence $V_i$ is $\alpha^{\ell}$-cyclic, and statement (2) is proved. For statement (1), observe that since $\textnormal{pow}_{\ell}(Q)(X^{\ell}) \equiv 0 \pmod{Q}$, we have $\textnormal{pow}_{\ell}(Q)^e(X^{\ell}) \equiv 0 \pmod{Q^e}$, whence the minimal polynomial of $\alpha^{\ell}$ on $V$ divides $\textnormal{pow}_{\ell}(Q)^e$. Let $V = \bigoplus_{i=0}^{s-1}V_i$ be an $\alpha^{\ell}$-block subspace decomposition of $V$. The minimal polynomial of $\alpha^{\ell}$ on $V_i$ divides $\textnormal{pow}_{\ell}(Q)^e$ and thus is of the form $\textnormal{pow}_{\ell}(Q)^{e_i}$ for some $e_i \in \{1,2,\ldots,e\}$. By comparing the $\mathbb{F}_q$-dimensions of $V$ and $\bigoplus_{i=0}^{s-1}V_i$, we have $\sum_{i=0}^{s-1}e_i=e$. According to Lemma [Lemma 15](#lem 2.17){reference-type="ref" reference="lem 2.17"}, when raising $\alpha^{\ell}$ to the $\textnormal{ord}(Q)$-th power, each block\ $\textnormal{Comp}(\textnormal{pow}_{\ell}(Q)^{e_i})$ of $\alpha^{\ell}$ splits into $\textnormal{deg}~Q$ blocks $\textnormal{Comp}((X-1)^{e_i})$. Hence, for each $\epsilon \in \{1,2,\ldots,e\}$, the number of blocks of $\alpha^{\ell~\textnormal{ord}(Q)}$ of the form $\textnormal{Comp}((X-1)^{\epsilon})$ is equal to $$\begin{aligned} \textnormal{deg}~(Q)\cdot|\{i\in\{0,1,\ldots,s-1\}: e_i=\epsilon\}|.\end{aligned}$$ On the other hand, also by Lemma [Lemma 15](#lem 2.17){reference-type="ref" reference="lem 2.17"}, when we raise $\alpha$ to the $\textnormal{ord}(Q)$-th power, its only block $\textnormal{Comp}(Q^e)$ on $V$ splits into $\textnormal{deg}~Q$ blocks $\textnormal{Comp}((X-1)^e)$. Moreover, by the already proved statement (2) of this lemma, if we subsequently raise $\alpha^{\textnormal{ord}(Q)}$ to the $\ell$-th power, each of these blocks $\textnormal{Comp}((X-1)^e)$ splits into $b$ blocks $\textnormal{Comp}((X-1)^{a+1})$ and $\ell'-b$ blocks $\textnormal{Comp}((X-1)^{a})$. It follows that $\alpha^{\ell~\textnormal{ord}(Q)}$ has - $b~~\textnormal{deg}~Q$ blocks of the form $\textnormal{Comp}((X-1)^{a+1})$ and - $(\ell'-b)$ $\textnormal{deg}~Q$ blocks of the form $\textnormal{Comp}((X-1)^a)$. Comparing with the above block count in terms of the $e_i$, we find that $s = \ell'$ and $$|\{i\in\{0,1,\ldots,\ell'-1\}: e_i=\epsilon\}| = \begin{cases} b &\quad\text{if $ \epsilon = a+1 $,}\\ \ell'-b &\quad\text{if $ \epsilon = a $,} \\ 0 &\quad\text{otherwise}. \\ \end{cases}.$$ In other words, the primary rational canonical form of $\alpha^{\ell}$ has $b$ blocks of the form\ $\textnormal{Comp}(\textnormal{pow}_{\ell}(Q)^{a+1})$ and $\ell'-b$ blocks of the form $\textnormal{Comp}(\textnormal{pow}_{\ell}(Q)^a)$, and no other blocks. This is just what we needed to show. ◻ # Cycle structures of cascaded FSRs via wreath products {#results} In this section, we state our main results and prove them. In the following theorem, we give a nontrivial upper bound on the maximum period of a cascaded connection. **Theorem 17**. *Let $n$ and $m$ be positive integers, and let $\tilde{f}\in\operatorname{Sym}(\mathbb{F}_2^n)$ and $\tilde{g}\in\operatorname{Sym}(\mathbb{F}_2^m)$ be transition functions of FSRs (with characteristic functions $f$ and $g$, respectively). Unless $m=n=1$, the maximum period of an output sequence of the cascaded connection $\operatorname{FSR}(f;g)$ is at most $$\begin{aligned} \textnormal{max}\{2^n(2^m-1), 2^m(2^n-1)\} = 2^{\textnormal{min}\{m,n\}}(2^{\textnormal{max}\{m,n\}}-1).\end{aligned}$$* *Proof.* We recall that by Lemma [Lemma 2](#lemma 2.12){reference-type="ref" reference="lemma 2.12"}(2), periods of output sequences of $\operatorname{FSR}(f;g)$ are the same as cycle lengths of the transition function $\tilde{f}\ast\tilde{g}$, so we prove the asserted upper bound for those cycle lengths instead. Moreover, by Proposition [Proposition 4](#prop 2.10){reference-type="ref" reference="prop 2.10"}, $\tilde{f}\ast\tilde{g}$ can be viewed as the wreath product element $\tilde{f}\cdot(\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{y})})_{\Vec{y}\in\mathbb{F}_2^n}$, where $\Vec{t}=(0,\ldots,0,1)^T$. By Remark [Remark 7](#remark 2.6){reference-type="ref" reference="remark 2.6"}, we find that each cycle length of $\tilde{f}\cdot(\tilde{g}\rho(\Vec{t})^{\pi_1(\vec{y})})_{\vec{y}\in\mathbb{F}_2^n}$ is the product of a cycle length of $\tilde{f}$ with the cycle length of a certain permutation of $\mathbb{F}_2^m$ (a certain forward cycle product). Therefore, if $\tilde{f}$ is not a $2^n$-cycle, then the maximum cycle length of $\tilde{f}\ast\tilde{g}$ can be at most $(2^n-1)\cdot2^m$, as desired. We may then assume that $\tilde{f}$ is a $2^n$-cycle (i.e., a De Bruijn cycle). The associated forward cycle product is a product of $2^n$ permutations of $\mathbb{F}_2^m$, of which $2^{n-1}$ are $\tilde{g}$ and $2^{n-1}$ are $\tilde{g}\rho(\Vec{t})$ (there are as many vectors in $\mathbb{F}_2^n$ with the first bit 0 as there are with first bit 1). Denote by $\pi(\sigma)\in\mathbb{F}_2$ the parity of $\sigma\in\operatorname{Sym}(\mathbb{F}_2^m)$, i.e., $\pi(\sigma)=0$ if $\sigma$ is an even permutation (it can be written as a product of an even number of transpositions), and $\pi(\sigma)=1$ if $\sigma$ is odd. Then the parity of the said forward cycle product is $$2^{n-1}\pi(g)+2^{n-1}\pi(g\rho(\Vec{t}))=2^n\cdot\pi(g)+2^{n-1}\cdot\pi(\rho(\Vec{t})) \equiv 0 \pmod{2},$$ using the assumption that $n>1$ or $m>1$ (note that $m>1$ causes $\rho(\Vec{t})$, which is a (disjoint) product of $2^{m-1}$ transpositions, to be an even permutation). The forward cycle product is thus an even permutation of $\mathbb{F}_2^m$. In particular, it cannot be a $2^m$-cycle, so its maximum cycle length is at most $2^m-1$, and the maximum cycle length of $\tilde{f}\ast\tilde{g}$ is at most $2^n(2^m-1)$, as required. ◻ Our second main result is a detailed and explicit description of the periods of output sequences of $\operatorname{FSR}(f;g)$ in the important special case where $\tilde{f}$ is a De Bruijn cycle and $\tilde{g}$ is a linear permutation. We focus on the cycle structure of the transition function $\tilde{f}\ast\tilde{g}$, in view of Lemma [Lemma 2](#lemma 2.12){reference-type="ref" reference="lemma 2.12"}. We also use, again, that by Proposition [Proposition 4](#prop 2.10){reference-type="ref" reference="prop 2.10"}, $\tilde{f}\ast\tilde{g}$ is the wreath product element $$\tilde{f}\cdot(\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{y})})_{\Vec{y}\in\mathbb{F}_2^n} \in \operatorname{Sym}(\mathbb{F}_2^m) \wr \operatorname{Sym}(\mathbb{F}_2^n),$$ where $\Vec{t}=(0,\ldots,0,1)^T$ and $\pi_1$ denotes the projection onto the first coordinate. It is not hard to check that the first $m$ iterates $\tilde{g}^k(\Vec{t})$ for $k=0,1,\ldots,m-1$ form an $\mathbb{F}_2$-basis of $\mathbb{F}_2^m$. Moreover, the representation matrix of $\tilde{g}$ with respect to this basis is $\operatorname{Comp}(P(X))$ for some monic polynomial $P(X)\in\mathbb{F}_2[X]$ of degree $m$. Each composition $\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{y})}$ lies in the group $\operatorname{AGL}_m(2)$ of $\mathbb{F}_2$-affine permutations of $\mathbb{F}_2^m$, and so we can view $\tilde{f}\ast\tilde{g}$ more specifically as an element of $\operatorname{AGL}_m(2)\wr\operatorname{Sym}(\mathbb{F}_2^n)$, which is more advantageous from a computational point of view. In order to describe the cycle type $\operatorname{CT}(\tilde{f}\ast\tilde{g})$, we note that by Remark [Remark 7](#remark 2.6){reference-type="ref" reference="remark 2.6"}, $$\operatorname{CT}(\tilde{f}\ast\tilde{g}) = \operatorname{CT}(\tilde{f}\cdot(\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{y})})_{\Vec{y}\in\mathbb{F}_2^n}) = \operatorname{BU}_{2^n}(\operatorname{CT}(\phi)),$$ where $\operatorname{BU}_{2^n}$ is the $2^n$-blow-up function and $\phi$ is the following forward cycle product of $\tilde{f}\ast\tilde{g}$: $$\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{0})}\cdot \tilde{g}\rho(\Vec{t})^{\pi_1({\tilde{f}(\Vec{0})})}\cdots \tilde{g}\rho(\Vec{t})^{\pi_1({\tilde{f}^{2^n-1}(\Vec{0})})}.$$ Using the semidirect product structure of $\operatorname{AGL}_m(2)$, we can bring this product into the form $$\begin{aligned} \tilde{g}^{2^n}\cdot\rho(\sum_{k=0}^{2^n-1}\pi_1(\tilde{f}^k(\Vec{0})\Vec{t}^{(\tilde{g}^{2^n-1-k})})), \tag{I}\label{I}\end{aligned}$$ which is the standard form of this affine permutation (it is written as the product/composition of the linear permutation $\tilde{g}^{2^n}$ with a translation). We intend to understand the cycle type of ([\[I\]](#I){reference-type="ref" reference="I"}). Now, cycle types of affine permutations of finite vector spaces were described in  [@Bors], but we need to push the theory developed there further to understand the cycle type of the complex expression ([\[I\]](#I){reference-type="ref" reference="I"}) in terms of the involved parameters. First, we apply a certain isomorphism to bring everything into a nicer form. Specifically, we note once more that the representation matrix of $\tilde{g}\in\operatorname{GL}_m(2)$ with respect to the $\mathbb{F}_2$-basis $\tilde{g}^k(\Vec{t})$ for $k=0,1,\ldots,m-1$ is $\operatorname{Comp}(P(X))$ for some monic polynomial $P(X)\in \mathbb{F}_2[X]$ of degree $m$. Under the $\mathbb{F}_2$-vector space isomorphism $\mathbb{F}_2^m \rightarrow \mathbb{F}_2[X]/(P(X))$ with $\tilde{g}^k(\Vec{t}) \mapsto X^k+(P(X))$ for $k=0,1,\ldots,m-1$, the linear automorphism $\tilde{g}$ corresponds to modular multiplication by $X$, and $\Vec{t}=(0,0,\ldots,0,1)^T$ corresponds to $1+(P(X))$. Applying this isomorphism, the problem of determining the cycle type of ([\[I\]](#I){reference-type="ref" reference="I"}) turns into that of determining the cycle type of the affine permutation $$\Gamma:R(X)+(P(X)) \mapsto X^{2^n}R(X)+\sum_{k=0}^{2^n-1}\pi_1(\tilde{f}^k(\Vec{0}))X^{2^n-1-k}\cdot 1+(P(X))$$ of the ring $\mathbb{F}_2[X]/(P(X))$. Now, if we factor $P(X)=Q_1(X)^{e_1}Q_2(X)^{e_2}\cdots Q_r(X)^{e_r}$ into pairwise coprime powers of irreducible polynomials $Q_i(X)$, then the Chinese Remainder Theorem tells us that there is a ring isomorphism $$\mathbb{F}_2[X]/(P(X)) \rightarrow \prod_{i=1}^r\mathbb{F}_2[X]/(Q_i(X)^{e_i}),$$ under which $\Gamma$ corresponds to the component-wise application of its modular reductions $\Gamma\bmod{Q_i(X)^{e_i}}$ for $i=1,\ldots,r$. Therefore, it suffices to study the cycle type of $\Gamma$ modulo each $Q_i^{e_i}$, then take their $\divideontimes$-product as defined by Wei and Xu (see also the text passage before Lemma [Lemma 11](#absorptionLem){reference-type="ref" reference="absorptionLem"}). We fix $i$ and distinguish between two cases: **Case 1**. Assume $Q_i(X) \neq X-1$. By Lemmas [Lemma 14](#lem 2.16){reference-type="ref" reference="lem 2.16"} and [Lemma 16](#lemma 2.18){reference-type="ref" reference="lemma 2.18"}, it can be proved that the automorphism $R(X)+(Q_i(X)^{e_i}) \mapsto X^{2^n}R(X)+(Q_i(X)^{e_i})$ of $\mathbb{F}_2[X]/(Q_i(X)^{e_i})$ has a primary rational canonical form with blocks of the form $\operatorname{Comp}(\operatorname{pow}_{2^n}(Q_i)^{e})$, for some $e \in \mathbb{N}^+$ (not necessarily the same $e$ for each block), where $\operatorname{pow}_{2^n}(Q_i)$ is the minimal polynomial over $\mathbb{F}_2$ of $\xi^{2^n}$, where $\xi$ is any of the roots of $Q_i$ in the algebraic closure $\overline{\mathbb{F}_2}$. Because $\xi^{2^n}$ is the image of $\xi$ under the field automorphism $z\mapsto z^{2^n}$ of $\overline{\mathbb{F}_2}$, it follows that $$\operatorname{ord}(\operatorname{pow}_{2^n}(Q_i))=\operatorname{ord}(\xi^{2^n})=\operatorname{ord}(\xi)=\operatorname{ord}(Q_i)>1.$$ Therefore, we have $\operatorname{pow}_{2^n}(Q_i)\neq X-1$, whence an application of Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"} yields that $$\operatorname{CT}(\Gamma\bmod{Q_i^{e_i}}) = \operatorname{CT}(R(X)+(Q_i(X)^{e_i}) \mapsto X^{2^n}R(X)+(Q_i(X)^{e_i})),$$ the cycle type of the $2^n$-fold iterate of the multiplication by $X$ modulo $Q_i(X)^{e_i}$. Now, the cycle type of the latter can be read off from Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}, and cycle types of iterates can be computed via the following simple formula: if $\gamma=\operatorname{CT}(\sigma)=\prod_{\ell}{x_{\ell}^{e_{\ell}}}$, then for each $n\in\mathbb{Z}$, one has $$\label{itEq} \operatorname{it}_n(\gamma):=\operatorname{CT}(\sigma^n)=\prod_{\ell}{x_{\ell/\gcd(\ell,n)}^{e_{\ell}\cdot\gcd(\ell,n)}}.$$ **Case 2**. Assume $Q_i(X)=X-1$. Set $$\chi_{\tilde{f}}(X):= \sum_{k=0}^{2^n-1}\pi_1(\tilde{f}^k(\Vec{0}))X^{2^n-k-1}\in\mathbb{F}_2[X],$$ a polynomial associated with the De Bruijn cycle $\tilde{f}$, chosen such that $\Gamma(R(X)+(P(X)))=X^{2^n}R(X)+\chi_{\tilde{f}}(X)+(P(X))$. We take note of the following two properties of $\chi_{\tilde{f}}(X):$ 1. $\chi_{\tilde{f}}(X)$ is nonzero with $\deg{\chi_{\tilde{f}}}(X)<2^n$. More specifically, $\chi_{\tilde{f}}(X)$ has exactly $2^{n-1}$ terms, each of degree less than $2^n$, which correspond to the positions on the unique cycle of $\tilde{f}$ where a vector with first bit $1$ sits. 2. $X-1$ divides $\chi_{\tilde{f}}(X)$ if and only if the number $2^{n-1}$ of terms of $\chi_{\tilde{f}}(X)$ is even if and only if $n>1$. In order to describe $\operatorname{CT}(\Gamma\bmod{(X-1)^{e_i}})$, we need to consider the following subcases: **Subcase 1**. Assume $2^n\geq\operatorname{ord}((X-1)^{e_i})$. Then, since $\operatorname{ord}((X-1)^{e_i})$ is a power of $2$, it follows that $\operatorname{ord}((X-1)^{e_i})\divides 2^n$. Hence, the automorphism $R(X)+((X-1)^{e_i}) \mapsto X^{2^n}R(X)+((X-1)^{e_i})$ of $\mathbb{F}_2[X]/((X-1)^{e_i})$ is the identity. It follows that $\Gamma\bmod{(X-1)^{e_i}}$ is the additive shift $R(X)+((X-1)^{e_i}) \mapsto R(X)+\chi_{\tilde{f}}(X)+((X-1)^{e_i})$, whence all of its cycles are of length $$\operatorname{ord}(\chi_{\tilde{f}}(X)+((X-1)^{e_i})) = \begin{cases} 1, &\text{if }(X-1)^{e_i} \mid \chi_{\tilde{f}}(X), \\ 2, &\text{if }(X-1)^{e_i} \nmid \chi_{\tilde{f}}(X). \end{cases}$$ **Subcase 2**. Assume $2^n < \operatorname{ord}((X-1)^{e_i})$. Again, since $\operatorname{ord}((X-1)^{e_i})$ is a power of $2$, this implies that $2^n \mid \operatorname{ord}((X-1)^{e_i})$. Moreover, because $\operatorname{ord}((X-1)^{e_i})=2^{\lceil\log_2(e_i)\rceil}$ is the smallest power of $2$ that is at least $e_i$, the subcase assumption also implies that $2^n<e_i$. The block structure of the automorphism $R(X)+((X-1)^{e_i}) \mapsto X^{2^n}R(X)+((X-1)^{e_i})$ is described in Lemma [Lemma 16](#lemma 2.18){reference-type="ref" reference="lemma 2.18"}(2). Write $e_i=a\cdot 2^n+b$ with $a,b \in \mathbb{N}_0, a>0, b\in\{0,1,\ldots,2^n-1\}$. The said automorphism has $2^n$ primary rational canonical blocks, and a corresponding block subspace decomposition is $$\mathbb{F}_2[X]/((X-1)^{e_i}) = \bigoplus_{k=0}^{2^n-1}{V_k},$$ where $V_k=\bigoplus_{j=0}^{\lfloor\frac{e_i-1-k}{2^n}\rfloor}\mathbb{F}_2((X-1)^{k+j2^n}+(X-1)^{e_i})$ for $k=0,1,\ldots,2^n-1$. Moreover, the restriction of that automorphism to $V_k$ has the minimal polynomial $$\centering \begin{cases} (X-1)^{a+1}, &\text{if }k\in\{0,1,\ldots,b-1\}, \\ (X-1)^a, &\text{if }k\in\{b,b+1,\ldots,2^n-1\}, \end{cases}$$ of degree $$a_k= \begin{cases} a+1, & \text{if }k\in\{0,1,\ldots,b-1\}, \\ a, & \text{if }k\in\{b,b+1,\ldots,2^n-1\}. \end{cases}$$ This allows us to view, via the isomorphism $\mathbb{F}_2 [X]/((X-1)^{e_i}) \rightarrow\prod_{k=0}^{2^n-1}V_k$, the affine permutation $\Gamma\bmod{(X-1)^{e_i}}$ as a component-wise application of affine permutations $A_k\in\operatorname{Sym}(V_k)$. We note that the linear part of $A_k$ is the restriction $\alpha_k$ of $R(X)+((X-1)^{e_i})\mapsto X^{2^n}R(X)+((X-1)^{e_i})$ to $V_k$, and the constant part is the corresponding projection of $\chi_{\tilde{f}}(X)+((X-1)^{e_i})$ to $V_k$. For each fixed $k$, by Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}, there are two possibilities for $\operatorname{CT}(A_k)$, depending on whether or not the said projection of $\chi_{\tilde{f}}(X)+((X-1)^{e_i})$ is a unit (i.e., does not lie in the image of $\alpha_k-\operatorname{id}$). Let us try to understand this more explicitly. Since $\alpha_k-\operatorname{id}$ is the restriction to $V_k$ of modular multiplication by $X^{2^n}-1=(X-1)^{2^n}$, we find, using the definition of $V_k$, that the $\mathbb{F}_2$-subspace of non-units in $$V_k=\bigoplus_{j=0}^{\lfloor\frac{e_i-1-k}{2^n}\rfloor}\mathbb{F}_2((X-1)^{k+j2^n}+(X-1)^{e_i})$$ is $$W_k:=\bigoplus_{j=1}^{\lfloor\frac{e_i-1-k}{2^n}\rfloor}\mathbb{F}_2((X-1)^{k+j2^n}+(X-1)^{e_i}).$$ Now, by Proposition [Proposition 12](#prop 2.15){reference-type="ref" reference="prop 2.15"}, the cycle type of $\Gamma\bmod{(X-1)^{e_i}}$ as a whole is bijectively determined by the maximum element of the set $$\{0\}\cup\{1+\lfloor \log_2(a_k)\rfloor: 0\leq k \leq 2^n-1, (\text{constant part of }A_k)\notin W_k\}$$ To get a more concrete description, we distinguish between two subsubcases: **Subsubcase 1**. Assume $a+1$ is not a power of $2$. Then $\{\lfloor\log_2(a_k)\rfloor: 0\leq k \leq 2^n-1\} = \{\lfloor\log_2(a)\rfloor\}$, so there are at most two distinct possibilities for $\operatorname{CT}(\Gamma\bmod{(X-1)^{e_i}})$, depending on whether or not there exists a $k\in\{0,1,\ldots,2^n-1\}$ such that the constant part of $A_k$ is a unit in $V_k$. Now, no such $k$ exists if and only if for each $k$, the projection of $\chi_{\tilde{f}}(X)+((X-1)^{e_i})$ to $V_k$ has vanishing $\mathbb{F}_2((X-1)^k+(X-1)^{e_i})$-coordinate, i.e., if and only if $(X-1)^{2^n}$ divides $\chi_{\tilde{f}}(X)=\chi_{\tilde{f}}(X)\bmod{(X-1)^{e_i}}$. However, that divisibility cannot hold, because $\chi_{\tilde{f}}(X)$ is a nonzero polynomial in $\mathbb{F}_2[X]$ of degree at most $2^n-1$. We conclude that there is at least one primary rational canonical block of the linear part of $\Gamma\bmod{(X-1)^{e_i}}$ on which the cycle type of the corresponding restricted affine map associated with $\Gamma\bmod{(X-1)^{e_i}}$ is as in statement (3) of Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}, i.e., all cycles on that block are of length $2^{\lfloor\log_2(a)\rfloor+1}$. Moreover, following the proof of Proposition [Proposition 12](#prop 2.15){reference-type="ref" reference="prop 2.15"}, that cycle length is actually a multiple of all cycle lengths occurring in such a block, whence the said block absorbs all other blocks. Hence, in fact, *all* cycles of $\Gamma\bmod{(X-1)^{e_i}}$ are of length $2^{\lfloor\log_2(a)\rfloor+1}$, i.e., $$\operatorname{CT}(\Gamma\bmod{(X-1)^{e_i}}) = x_{2^{\lfloor\log_2(a)\rfloor+1}}^{2^{e_i-\lfloor\log_2(a)\rfloor-1}}.$$ **Subsubcase 2**. Assume $a+1$ is a power of $2$. Then $$\{\lfloor\log_2(a_k)\rfloor: 0\leq k\leq 2^n-1\} = \{\log_2(a+1)-1, \log_2(a+1)\},$$ so a priori, there are three possibilities for $\operatorname{CT}(\Gamma\bmod{(X-1)^{e_i}})$, corresponding to the following cases: 1. There is no $k\in\{0,1,\ldots,2^n-1\}$ at all such that the constant part of $A_k$ is a unit of $V_k$, i.e., $(X-1)^{2^n} \mid \chi_{\tilde{f}}(X)$. As already argued in the previous subsubcase, this is actually impossible. 2. There is a $k\in\{b,b+1,\ldots,2^n-1\}$, but no $k\in\{0,1,\ldots,b-1\}$, such that the constant part of $A_k$ is a unit of $V_k$, i.e., $(X-1)^b \mid \chi_{\tilde{f}}(X)$ but $(X-1)^{2^n} \nmid \chi_{\tilde{f}}(X)$; of course, that indivisibility is always satisfied. Then one of the smaller, $a$-dimensional primary Frobenius blocks of the linear part of $\Gamma\bmod{(X-1)^{e_i}}$ corresponds to a cycle type of the form $x_{2^{\lfloor\log_2(a)\rfloor+1}}^{2^{a-\lfloor\log_2(a)\rfloor-1}}=x_{a+1}^{2^a/(a+1)}$ and absorbs all other $a$-dimensional blocks. Moreover, by Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}(2), the cycle lengths on each $(a+1)$-dimensional block are powers of $2$, and the largest among them is $2^{\lceil\log_2(a+1)\rceil}=a+1$. Hence, the said $a$-dimensional block also absorbs all $(a+1)$-dimensional blocks, and we have $$\operatorname{CT}(\Gamma\bmod{(X-1)^{e_i}})=x_{a+1}^{2^{e_i}/(a+1)}.$$ 3. There is a $k\in\{0,1,\ldots,b-1\}$ such that the constant part of $A_k$ is a unit of $V_k$, i.e., $(X-1)^b \nmid \chi_{\tilde{f}}(X)$. Then one of the larger, $(a+1)$-dimensional primary rational canonical blocks of the linear part of $\Gamma\bmod{(X-1)^{e_i}}$ absorbs all other blocks, whence $$\operatorname{CT}(\Gamma\bmod{(x-1)^{e_i}}) = x_{2^{\lfloor\log_2(a+1)\rfloor+1}}^{2^{e_i-\lfloor\log_2(a+1)\rfloor-1}} = x_{2a+2}^{2^{e_i-1}/(a+1)}.$$ The following theorem summarizes the results of the above discussion. **Theorem 18**. *Let $n$ and $m$ be positive integers, and let $\tilde{f}\in\operatorname{Sym}(\mathbb{F}_2^n)$ and $\tilde{g}\in\operatorname{Sym}(\mathbb{F}_2^m)$ be transition functions of FSRs such that $\tilde{f}$ is a De Bruijn cycle and $\tilde{g}$ is linear. Let $P(X)\in\mathbb{F}_2[X]$ be the unique degree $m$ monic polynomial such that $\tilde{g}$ can be represented, with respect to a suitable $\mathbb{F}_2$-basis of $\mathbb{F}_2^m$, by the companion matrix $\operatorname{Comp}(P(X))$. Write $P(X)=(X-1)^{e_0}\prod_{i=1}^r{Q_i(X)^{e_i}}$ with $e_0\in\mathbb{N}_0$, $e_i\in\mathbb{N}^+$ for $1\leq i\leq r$, and $X-1,Q_1(X),\ldots,Q_r(X)\in\mathbb{F}_2[X]$ being pairwise distinct monic irreducible polynomials. Recall/take note of the following notations.* - *$\pi_1:\mathbb{F}_2^n\rightarrow\mathbb{F}_2$ for the projection to the first coordinate;* - *$\chi_{\tilde{f}}(X):=\sum_{t=0}^{2^n-1}{\pi_1(\tilde{f}^t(\vec{0})X^{2^n-1-t}}\in\mathbb{F}_2[X]$;* - *$\divideontimes$ for the Wei-Xu product of polynomials in $\mathbb{Q}[x_n: n\geq1]$, defined in [@Wei-xu Definition 2.2] (see also the text passage before Lemma [Lemma 11](#absorptionLem){reference-type="ref" reference="absorptionLem"} above);* - *$\operatorname{BU}_{\ell}$, where $\ell$ is a positive integer, for the unique $\mathbb{Q}$-algebra endomorphism of $\mathbb{Q}[x_n: n\geq1]$ such that $\operatorname{BU}_{\ell}(x_n)=x_{\ell\cdot n}$ for all positive integer $n$;* - *$\operatorname{it}_t(\gamma)$ for the cycle type of the $t$-th iterate of any permutation with cycle type $\gamma$ (see also formula ([\[itEq\]](#itEq){reference-type="ref" reference="itEq"}) above);* - *$\Gamma\in\operatorname{Sym}(\mathbb{F}_2[X]/(P(X)))$, $\Gamma(R(X)+(P(X)))=X^{2^n}R(X)+\chi_{\tilde{f}}(X)+(P(X))$;* - *$\Gamma_0$ and $\Gamma_+$ for the reductions of $\Gamma$ modulo $(X-1)^{e_0}$ and $\prod_{i=1}^r{Q_i(X)^{e_i}}$, respectively;* - *$\alpha_0$ for the multiplication by $X$ modulo $(X-1)^{e_0}$. The cycle type of $\alpha_0$ is described explicitly in Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}(2);* - *for $i=1,2,\ldots,r$: $\alpha_i$ for the multiplication by $X$ modulo $Q_i(X)^{e_i}$. The cycle type of $\alpha_i$ is described explicitly in Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}(1);* - *$\alpha_+$ for the multiplication by $X$ modulo $\prod_{i=1}^r{Q_i(X)^{e_i}}$, which satisfies $\operatorname{CT}(\alpha_+)=\divideontimes_{i=1}^r{\operatorname{CT}(\alpha_i)}$;* - *$e_0=a\cdot 2^n+b$ with $a,b\in\mathbb{Z}$, $0\leq b<2^n$.* *The following hold.* 1. *$\operatorname{CT}(\tilde{f}\ast\tilde{g})=\operatorname{BU}_{2^n}(\operatorname{CT}(\Gamma))$.* 2. *$\operatorname{CT}(\Gamma)=\operatorname{CT}(\Gamma_0)\divideontimes\operatorname{CT}(\Gamma_+)$.* 3. *$\operatorname{CT}(\Gamma_+)=\operatorname{CT}(\alpha_+^{2^n})=\operatorname{it}_{2^n}(\operatorname{CT}(\alpha_+))$.* 4. *$$\operatorname{CT}(\Gamma_0)= \begin{cases} x_1, & \text{if }e_0=0; \\ x_1^{2^{e_0}}, & \text{if }e_0>0, n\geq\lceil\log_2(e_0)\rceil,\text{ and }(X-1)^{e_0}\mid\chi_{\tilde{f}}(X); \\ x_2^{2^{e_0-1}}, & \text{if }e_0>0, n\geq\lceil\log_2(e_0)\rceil,\text{ and }(X-1)^{e_0}\nmid\chi_{\tilde{f}}(X); \\ %\ite_{2^n}(\CT(\alpha_0)), & \text{if }e_0>0, n<\lceil\log_2(e_0)\rceil,\text{ and }(X-1)^{2^n}\mid\chi_f(X); \\ x_{2^{\lfloor\log_2(a)\rfloor+1}}^{2^{e_0-\lfloor\log_2(a)\rfloor-1}}, & \text{if }e_0>0, n<\lceil\log_2(e_0)\rceil, \text{ and }\log_2(a+1)\notin\mathbb{Z}; \\ x_{a+1}^{2^{e_0}/(a+1)}, & \text{if }e_0>0, n<\lceil\log_2(e_0)\rceil, \log_2(a+1)\in\mathbb{Z}, \text{ and }(X-1)^b\mid\chi_f(X); \\ x_{2a+2}^{2^{e_0-1}/(a+1)}, & \text{if }e_0>0, n<\lceil\log_2(e_0)\rceil, \log_2(a+1)\in\mathbb{Z},\text{ and }(X-1)^b\nmid\chi_f(X). \end{cases}$$* In addition to the general method of describing the cycle structure of a cascaded connection of FSRs due to Mykkeltveit-Siu-Tong [@Mykkeltveit Theorme 2.1], Chang-Gong-Wang [@Chang] focused on cascaded connections $\operatorname{FSR}(f;g)$ where $f$ is a De Bruijn cycle and $g$ is a linear permutation (the same situation as in Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}). Their general method, represented by [@Chang Theorem 4], consists of reducing the determination of the cycle structure of the transition function of $\operatorname{FSR}(f;g)$ to the solution of a certain system of linear equations over $\mathbb{F}_2$. While this provides an efficient method to determine the cycle structure for each given cascaded connection, it does not lead to explicit formulas as in our Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}. Finally, we note the following theorem as a consequence of Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}. For polynomials $P(X),Q(X)\in\mathbb{F}_2[X]$ where $Q(X)$ is irreducible, we denote by $\nu_{Q(X)}(P(X))$ the $Q(X)$-adic valuation of $P(X)$, i.e., the largest nonnegative integer $v$ such that $Q(X)^v$ divides $P(X)$ (and $\nu_{Q(X)}(0):=\infty$). **Theorem 19**. *Let $n$ and $m$ be positive integers, and let $\tilde{f}\in\operatorname{Sym}(\mathbb{F}_2^n)$ and $\tilde{g}\in\operatorname{Sym}(\mathbb{F}_2^m)$ be transition functions of FSRs such that $\tilde{f}$ is a De Bruijn cycle and $\tilde{g}$ is linear. Let $P(X)\in\mathbb{F}_2[X]$ be the unique degree $m$ monic polynomial such that $\tilde{g}$ can be represented, with respect to a suitable $\mathbb{F}_2$-basis of $\mathbb{F}_2^m$, by the companion matrix $\operatorname{Comp}(P(X))$.* 1. *If $\nu_{X-1}(P(X))\leq 1$ and $n>1$, then $\operatorname{CT}(\tilde{f}\ast\tilde{g}) = \operatorname{BU}_{2^n}(\operatorname{CT}(\tilde{g}^{2^n}))$.* 2. *If $\tilde{g}$ is of odd order and $n>1$, then $\operatorname{CT}(\tilde{f}\ast\tilde{g}) = \operatorname{BU}_{2^n}(\operatorname{CT}(g))$.* *Proof.* For statement (1): Let us use the same notations as in Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}. By that theorem, we have $$\label{ctEq} \operatorname{CT}(\tilde{f}\ast\tilde{g})=\operatorname{BU}_{2^n}(\operatorname{CT}(\Gamma))=\operatorname{BU}_{2^n}(\operatorname{CT}(\Gamma_0)\divideontimes\operatorname{CT}(\Gamma_+))=\operatorname{BU}_{2^n}(\operatorname{CT}(\Gamma_0)\divideontimes\operatorname{CT}(\alpha_+^{2^n})).$$ We argue that under our assumptions here, we have $\operatorname{CT}(\Gamma_0)=\operatorname{CT}(\alpha_0^{2^n})$. Indeed, note that $e_0\in\{0,1\}$. If $e_0=0$, then $$\operatorname{CT}(\Gamma_0)=x_1=\operatorname{CT}(\operatorname{id}_{\mathbb{F}_2^0})=\operatorname{CT}(\operatorname{id}_{\mathbb{F}_2^0}^{2^n})=\operatorname{CT}(\alpha_0^{2^n}).$$ And if $e_0=1$, then we have $e_0>0$, $n\geq\lceil\log_2(e_0)\rceil$ and $(X-1)^{e_0}\mid\chi_f(X)$ (the latter by comment (2) after the definition of $\chi_f(X)$ above). Hence, $$\operatorname{CT}(\Gamma_0)=x_1^2=\operatorname{CT}(\operatorname{id}_{\mathbb{F}_2})=\operatorname{CT}(\operatorname{id}_{\mathbb{F}_2}^{2^n})=\operatorname{CT}(\alpha_0^{2^n}).$$ This concludes the proof of the claim that $\operatorname{CT}(\Gamma_0)=\operatorname{CT}(\alpha_0^{2^n})$. Combining this with formula ([\[ctEq\]](#ctEq){reference-type="ref" reference="ctEq"}), we infer that $$\begin{aligned} \operatorname{CT}(\tilde{f}\ast\tilde{g})&=\operatorname{BU}_{2^n}(\operatorname{CT}(\alpha_0^{2^n})\divideontimes\operatorname{CT}(\alpha_+^{2^n}))=\operatorname{BU}_{2^n}(\operatorname{CT}(\alpha_0^{2^n}\times\alpha_+^{2^n}))=\operatorname{BU}_{2^n}(\operatorname{CT}((\alpha_0\times\alpha_+)^{2^n})) \\ &=\operatorname{BU}_{2^n}(\operatorname{CT}(\tilde{g}^{2^n})).\end{aligned}$$ Here, the last equality uses the observation that because $\tilde{g}$ can be represented by $\operatorname{Comp}(P(X))$, its mapping behavior on $\mathbb{F}_2^m$ corresponds, under a suitable $\mathbb{F}_2$-vector space isomorphism, to that of the multiplication by $X$ modulo $P(X)$, which in turn corresponds (via the Chinese Remainder Theorem) to the component-wise modular multiplication by $X$ modulo $(X-1)^{e_0}$ (i.e., $\alpha_0$) and modulo $\prod_{i=1}^r{Q_i(X)^{e_i}}$ (i.e, $\alpha_+$), respectively. Statement (2) follows as a special case from statement (1). Indeed, $\tilde{g}$ being of odd order (i.e., of order coprime to the characteristic $2$) is equivalent to its characteristic polynomial $P(X)$ being square-free. In particular, we have $\nu_{X-1}(P(X))\leq 1$, whence $\operatorname{CT}(\tilde{f}\ast\tilde{g})=\operatorname{BU}_{2^n}(\operatorname{CT}(\tilde{g}^{2^n}))$ by statement (1). Moreover, because $\tilde{g}$ is of odd order, we have $$\operatorname{CT}(\tilde{g}^{2^n})=\operatorname{it}_{2^n}(\operatorname{CT}(\tilde{g}))=\operatorname{CT}(\tilde{g})$$ by the formula for $\operatorname{it}_t(\gamma)$ from above, which concludes the proof. ◻ We note that [@Chang Corollary 2] is a result that is closely related to, but essentially weaker than, our Theorem [Theorem 19](#shortTheo){reference-type="ref" reference="shortTheo"}; please note that in contrast to that result, we do not require the assumption that $2^n\geq m$, only that $n>1$. # Examples Throughout this section, we provide some examples of calculating the cycle type of $\tilde{f}\ast\tilde{g}$ to clarify our method. **Example 20**. Let $n=2$, $m=3$, and our feedback shift registers be given by the Boolean functions $$f_1(y_0,y_1)=y_0\oplus1\text{ and }g_1(x_0,x_1,x_2)=x_0\oplus x_1 .$$ Now, let $a_{k+2}=a_k+1$, where $k\in\mathbb{N}^{+}$, be the recurrence relation corresponding to $f_1$, $b_{k+3}=b_k+b_{k+1}$, where $k\in\mathbb{N}^{+}$ be the recurrence relation corresponding to $g_1$, and $c_{k+3}=c_k+c_{k+1}+a_k$ the one corresponding to $\operatorname{FSR}(f;g)$. Let the sequence generated by $\operatorname{FSR}(f)$ be $\underline{a}$, which is, up to cyclic shifts, $\underline{a} = (0,0,1,1)$, a De Bruijn sequence. Now we calculate the sequences $\underline{c}$ generated by $\operatorname{FSR}(f;g)$:\ $\underline{a}$: 0011 0011 0011 0011 0011 0011 0011 0011 $\cdots$ -------------------------------------------------------------------------------------------------------- -- -- [c]{.ul}: [0000]{style="color: red"} 0111 0101 1011 1110 0010 1001 [0000]{style="color: red"} $\cdots$ [c]{.ul}:  1100 1100 1100 $\cdots$ As can be seen in the above table, $\operatorname{FSR}(f;g)$ produces exactly two distinct sequences up to cyclic shifts. The first one is of length $28$, and the second one is of length $4$. Hence, $\operatorname{CT}(\tilde{f}\ast\tilde{g})=x_4x_{28}$. Now we use our method to find the cycle type of $\tilde{f}\ast\tilde{g}$. First, we note that $\tilde{f}:\mathbb{F}_2^2 \rightarrow \mathbb{F}_2^2$ is defined by $(y_1,y_2)^T \mapsto (y_2, f_1(y_0,y_1))^T$, and $\tilde{g}: \mathbb{F}_2^3 \rightarrow \mathbb{F}_2^3$, is defined by $(x_0,x_1,x_2)^T \mapsto (x_1,x_2,g_1(x_0,x_1,x_2))^T$. Also, we know, by Proposition [Proposition 4](#prop 2.10){reference-type="ref" reference="prop 2.10"}, that $\tilde{f}\ast\tilde{g}$ can be viewed as the wreath product element $\tilde{f}\cdot(\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{y})})_{\Vec{y}\in \mathbb{F}_2^2}$. Let $\zeta$ be the following De Bruijn cycle corresponding to $\tilde{f}$. Now, by Remark [Remark 7](#remark 2.6){reference-type="ref" reference="remark 2.6"}, we have $$\operatorname{CT}(\tilde{f}\ast\tilde{g}) = \operatorname{CT}(\tilde{f}\cdot(\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{y})})_{\Vec{y}\in\mathbb{F}_2^2}) = \operatorname{BU}_4(\operatorname{CT}(\operatorname{fcp}_{\zeta,(0,0)^T}(\tilde{f}(\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{y})})_{\Vec{y}\in \mathbb{F}_2^2}))).$$ Moreover, $$\begin{aligned} \operatorname{fcp}_{\zeta,(0,0)^T}(\tilde{f}(\tilde{g}\rho(\Vec{t})^{\pi_1(\Vec{y})})_{\Vec{y}\in \mathbb{F}_2^2}) &= \tilde{g}\rho(\Vec{t})^{\pi_1((0,0)^T)}\cdot \tilde{g}\rho(\Vec{t})^{\pi_1((0,1)^T)}\cdot \tilde{g}\rho(\Vec{t})^{\pi_1((1,1)^T)}\cdot \tilde{g}\rho(\Vec{t})^{\pi_1((1,0)^T)} \\&= \tilde{g}\cdot\tilde{g}\cdot \tilde{g}\rho(\Vec{t})\cdot\tilde{g}\rho(\Vec{t}) \\&= \tilde{g}^3\rho(\Vec{t})\tilde{g}\rho(\Vec{t}) = \tilde{g}^4\rho(\tilde{g}(\Vec{t})+\Vec{t}).\end{aligned}$$ Therefore, we intend to find $\operatorname{CT}(\tilde{g}^4\rho(\tilde{g}(\Vec{t})+\Vec{t}))=\operatorname{CT}(\Gamma)$, using the notation of Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}. The representation matrix of $\tilde{g}$ with respect to the $\mathbb{F}_2$-basis $\tilde{g}^k(\Vec{t})$ for $k = 0,1,2$ is $\operatorname{Comp}(X^3+X+1)$. Note that since $X^3+X+1$ is irreducible, and $X^3+X+1 \neq X+1$, **Case [Case 1](#case 1){reference-type="ref" reference="case 1"}** of our discussion leading to Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"} applies, i.e., $\Gamma=\Gamma_+$ in the notation of Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}. An application of that theorem thus yields that $$\operatorname{CT}(\Gamma)=\operatorname{CT}(\Gamma_+)=\operatorname{it}_4(\operatorname{CT}(\alpha_+)),$$ where $\alpha_+$ is the multiplication by $X$ modulo $X^3+X+1$ (the product of all irreducible factors of $P(X)=X^3+X+1$ that are distinct from $X-1$, taken with multiplicity). Because $X^3+X+1$ itself is irreducible, we may read $\operatorname{CT}(\alpha_+)$ off directly from Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"}. Specifically, statement (1) of Proposition [\[prop 2.12\]](#prop 2.12){reference-type="ref" reference="prop 2.12"} with $q=2$, $e=1$ and using that $\operatorname{ord}(X^3+X+1)=7$ yields $\operatorname{CT}(\alpha_+)=x_1x_7^{(2^3-1)/7}=x_1x_7$. We conclude that $$\operatorname{CT}(\Gamma)=\operatorname{it}_4(x_1x_7)=x_{1/\gcd(1,4)}^{1\cdot\gcd(1,4)}x_{7/\gcd(7,4)}^{1\cdot\gcd(7,4)}=x_1x_7$$ and, consequently, $$\operatorname{CT}(\tilde{f}\ast\tilde{g})=\operatorname{BU}_4(x_1x_7)=x_4x_{28}.$$ **Example 21**. Let $n=2$ and $f_1(y_0,y_1)=y_0\oplus 1$ as in Example [Example 20](#ex5.1){reference-type="ref" reference="ex5.1"}, but in this example, let $m=5$ and $g_1(x_0,x_1,x_2,x_3,x_4)=x_0\oplus x_1\oplus x_2$. Let $a_{k+2}=a_k+1$ be the recurrence relation corresponding to $f_1$, and $b_{k+5}=b_k+b_{k+1}+b_{k+2}$ be the recurrence relation corresponding to $g_1$. Moreover, let $c_{k+5}=c_{k}+c_{k+1}+c_{k+2}+a_k$ be the recurrence relation corresponding to $\operatorname{FSR}(f;g)$, where $k\in\mathbb{N}^+$. Like in the previous example, one can directly calculate the sequences generated by $\operatorname{FSR}(f;g)$. It generates the following four sequences, two of length 56 and two of length 8: $$\begin{aligned} &\underline{c}_1 = [0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, \\&1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1]\\& \underline{c}_2 = [0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, \\&0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0]\\& \underline{c}_3 = [0, 0, 1, 1, 1, 1, 0, 0]\\& \underline{c}_4 = [1, 0, 0, 1, 0, 1, 1, 0].\end{aligned}$$ Hence, $\operatorname{CT}(\tilde{f}\ast\tilde{g})=x_8^2x_{56}^2$. Now we use our method to confirm the cycle type of $\tilde{f}\ast\tilde{g}$. Note that $\tilde{f}$ is the same as in Example [Example 20](#ex5.1){reference-type="ref" reference="ex5.1"}, and $\tilde{g}: \mathbb{F}_2^5 \rightarrow \mathbb{F}_2^5$, is defined by $$(x_0,x_1,x_2,x_3,x_4)^T \mapsto (x_1,x_2,x_3,x_4,g_1(x_0,x_1,x_2,x_3,x_4))^T.$$ Since $\tilde{f}$ is the same as in Example [Example 20](#ex5.1){reference-type="ref" reference="ex5.1"}, with the same argument we need to find $\operatorname{CT}(\tilde{g}^4\rho(\tilde{g}(\Vec{t})+\Vec{t}))=\operatorname{CT}(\Gamma)$; then $\operatorname{CT}(\tilde{f}\ast\tilde{g})$ is the $4$-blow-up of that. Note that the representation matrix of $\tilde{g}$ with respect to the $\mathbb{F}_2$-basis $\tilde{g}^{k}(\Vec{t})$ for $k = 0,1,2,3,4$ is $\textnormal{Comp}(P(X))$ with $P(X)=X^5+X^2+X+1$. This time, $P(X)$ is not irreducible, but rather, it admits the factorization $P(X)=(X^3+X+1)(X+1)^2$. Because there are two distinct irreducible factors, we need to compute the cycle type of $\Gamma$ modulo each irreducible power, then take the Wei-Xu product of those two cycle types to obtain $\operatorname{CT}(\Gamma)$. Now, the reduction $\Gamma\bmod{X^3+X+1}=\Gamma_+$ is the $\Gamma$ of the previous example, which lets us conclude without further calculations that $\operatorname{CT}(\Gamma_+)=x_1x_7$. As for $\Gamma\bmod{(X+1)^2}=\Gamma_0$, we use statement (4) of Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"} to compute it. Note that $e_0=2>0$, that $n=2\geq 1=\lceil\log_2(e_0)\rceil$ (which means that we are in Subcase 2.1 of the discussion leading to Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}) and that $$\chi_{\tilde{f}}(X)=\pi_1((0,0)^T)X^3+\pi_1((0,1)^T)X^2+\pi_1((1,1)^T)X+\pi_1((1,0)^T)=X+1$$ is *not* divisible by $(X+1)^{e_0}=(X+1)^2$. Therefore, the third case in statement (4) of Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"} applies, letting us conclude that $\operatorname{CT}(\Gamma_0)=x_2^2$. An application of statement (2) of Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"} now yields that $$\operatorname{CT}(\Gamma)=\operatorname{CT}(\Gamma_0)\divideontimes\operatorname{CT}(\Gamma_+)=x_2^2\divideontimes(x_1x_7)=x_{\operatorname{lcm}(2,1)}^{2\cdot1\cdot\gcd(2,1)}x_{\operatorname{lcm}(2,7)}^{2\cdot1\cdot\gcd(2,7)}=x_2^2x_{14}^2.$$ Finally, $\operatorname{CT}(\tilde{f}\ast\tilde{g})=\operatorname{BU}_4(\operatorname{CT}(\Gamma))=x_8^2x_{56}^2$. **Example 22**. Let $n = 2$, $f_1(y_0,y_1) = y_0\oplus 1$ as in Example [Example 20](#ex5.1){reference-type="ref" reference="ex5.1"}, $m = 8$, and $$g_1(x_0,x_1,\ldots,x_7) = x_0 \oplus x_2\oplus x_3\oplus x_6\oplus x_7.$$ By using a computer, we calculated the output sequences of $\operatorname{FSR}(f;g)$ directly. It generates 16 sequences of length $56$, and 16 sequences of length $8$. Hence, $\operatorname{CT}(\tilde{f}\ast\tilde{g}) = x_8^{16}x_{56}^{16}$. Note that $\tilde{f}$ is the same as in Example [Example 20](#ex5.1){reference-type="ref" reference="ex5.1"}, and $\tilde{g}: \mathbb{F}_2^8 \rightarrow \mathbb{F}_2^8$, is defined by $$(x_0,x_1,\ldots,x_7)^T \mapsto (x_1,\ldots,x_7,g_1(x_0,x_1,\ldots,x_7))^T.$$ Since $\tilde{f}$ is the same as in Example [Example 20](#ex5.1){reference-type="ref" reference="ex5.1"}, with the same argument we need to find $\operatorname{CT}(\tilde{g}^4\rho(\tilde{g}(\Vec{t})+\Vec{t}))=\operatorname{CT}(\Gamma)$. Note that the representation matrix of $\tilde{g}$ with respect to the $\mathbb{F}_2$-basis $\tilde{g}^{k}(\Vec{t})$ for $k=0,1,\ldots,7$ is $\operatorname{Comp}(P(X))$ with $P(X)=X^8+X^7+X^6+X^3+X^2+1$. The factorization of $P(X)$ into powers of irreducible polynomials is $P(X)=(X^3+X+1)(X+1)^5$. Hence, $\Gamma_+=\Gamma\bmod{X^3+X+1}$, which is the $\Gamma_+$ of the previous example (and the $\Gamma$ of the first example), and we conclude immediately that $\operatorname{CT}(\Gamma_+)=x_1x_7$. On the other hand, $\Gamma_0=\Gamma\bmod{(X+1)^5}$. We have $e_0=5$, and thus $n=2<3=\lceil\log_2(e_0)\rceil$, so we are in Subcase 2.2 of the discussion leading to Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}. Because $\tilde{f}$ has not changed from the previous example, we still have $\chi_{\tilde{f}}=X+1$. Integer division of $e_0=5$ by $2^n=4$ yields $5=1\cdot 4+1$, so $a=1$ and $b=1$. Because $\log_2(a+1)=\log_2(2)=1\in\mathbb{Z}$ and $(X+1)^b=(X+1)^1\mid\chi_{\tilde{f}}(X)$, the penultimate case in statement (4) of Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"} applies, letting us conclude that $\operatorname{CT}(\Gamma_0)=x_{a+1}^{2^{e_0}/(a+1)}=x_2^{16}$. It follows that $$\operatorname{CT}(\Gamma)=\operatorname{CT}(\Gamma_0)\divideontimes\operatorname{CT}(\Gamma_+)=x_2^{16}\divideontimes(x_1x_7)=x_{\operatorname{lcm}(2,1)}^{16\cdot1\cdot\gcd(2,1)}x_{\operatorname{lcm}(2,7)}^{16\cdot1\cdot\gcd(2,7)}=x_2^{16}x_{14}^{16}$$ and, finally, $$\operatorname{CT}(\tilde{f}\ast\tilde{g})=\operatorname{BU}_4(\operatorname{CT}(\Gamma))=x_8^{16}x_{56}^{16},$$ confirming our direct calculations. # Concluding remarks {#concRem} We conclude this paper with two related open problems for further research. In the paragraph after Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}, we compared our results on cycle types with Chang-Gong-Wang's method from [@Chang Theorem 4], observing that their result does not lead to explicit formulas for the cycle types as our Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}. An advantage which Chang-Gong-Wang's method does have over ours at the moment is that it allows one to find, in each specific example, explicit representatives for the cycles of the transition function $\tilde{f}\ast\tilde{g}$, and thus explicit bit values with which the stages of the cascaded connection may be initialized to achieve a certain cycle length. However, the wreath product method should allow one to achieve this as well, in the form of a general, parametric description of those cycle representatives. In fact, one should just be able to use an approach analogous to the one outlined in [@AQD Section 3.1], which is for a different class of functions that can also be viewed as wreath product elements. This reduces the problem of finding cycle representatives for $\tilde{f}\ast\tilde{g}$ to that of finding such representatives for a certain affine permutation of $\mathbb{F}_2^m$. We thus pose the following open problem, which we believe to be solvable using linear algebra: **Problem 23**. *Let $q$ be a prime power, $m$ a positive integer, $M$ an invertible $(m\times m)$-matrix over $\mathbb{F}_q$, and $\vec{v}\in\mathbb{F}_q^m$. In terms of $M$ and $\vec{v}$, give an explicit description of a set of representatives for the cycles of the affine permutation $A:\vec{x}\mapsto M\vec{x}+\vec{v}$ of $\mathbb{F}_q^m$.* We note two more things concerning Problem [Problem 23](#openProb1){reference-type="ref" reference="openProb1"}. Firstly, for our application to FSRs, it would suffice to solve this for $q=2$. Secondly, in the desired explicit description, ideally each cycle representative is linked to the length of the associated cycle. That is, one should strive to find an explicit CRL-list for $A$ in the sense of [@AQD Definition 1.2]. To motivate our second open problem, we note that according to Theorem [Theorem 18](#longTheo){reference-type="ref" reference="longTheo"}(4), the cycle type of $\Gamma_0$ often depends on whether a divisibility of the polynomial $\chi_{\tilde{f}}(X)$ by a certain power of $X-1$ holds. For that reason, understanding $\nu_{X-1}(\chi_{\tilde{f}}(X))$, the largest nonnegative integer $v$ such that $(X-1)^v$ divides $\chi_{\tilde{f}}(X)$, is of interest, and we pose the following open problem: **Problem 24**. *For each given positive integer $n$, define a set $M_n$ of nonnegative integers as follows: $$M_n:=\{\nu_{X-1}(\chi_{\tilde{f}}(X)): \tilde{f}\text{ is a De Bruijn cycle of }\mathbb{F}_2^n\}.$$ Describe the sets $M_n$. For example, what are the elements of $M_{100}$?*

arxiv_math

--- author: - "Enrico M. Vitale [^1]" title: Completion under strong homotopy cokernels --- For ${\mathcal A}$ a category with finite colimits, we show that the embedding of ${\mathcal A}$ into the category of arrows $\mathbf{Arr}({\mathcal A})$ determined by the initial object is the completion of ${\mathcal A}$ under strong homotopy cokernels. The nullhomotopy structure of $\mathbf{Arr}({\mathcal A})$ (needed in order to express the notion of homotopy cokernel) is the usual one induced by the canonical string of adjunctions between ${\mathcal A}$ and $\mathbf{Arr}({\mathcal A}).$\ \ nullhomotopy, homotopy cokernel, arrow category, completion.\ *2020 MSC:* 18A30, 18A35, 18N99 # Introduction {#SecIntro} Limits and colimits are a fundamental tool in category theory and its applications. However, these notions are not completely satisfactory in homotopical algebra, and the search for a convenient notion of homotopy limit is a long story, see for example [@BK; @Ma; @TH; @DF; @DHKS]. More recently, (strong) homotopy kernels and (strong) homotopy cokernels in the context of categories equipped with a structure of nullhomotopies have been used in [@SnailEV; @JMMV1; @JMMV2; @MGMV] in order to internalize Gabriel-Zisman [@GZ] and Brown [@BR] exact sequences, and in [@MMMV] to define a general notion of homotopy torsion theory. The aim of the present paper is to exhibit the free completion of a category ${\mathcal A}$ under strong homotopy cokernels. For this, we consider the category $\mathbf{Arr}({\mathcal A})$ of arrows in ${\mathcal A}.$ The canonical embedding of ${\mathcal A}$ in $\mathbf{Arr}({\mathcal A})$ freely adds a factorization system to ${\mathcal A}$ (see [@KT] and also [@GR00; @RV]). If we assume that ${\mathcal A}$ has an initial object $\emptyset,$ we can consider another embedding given by the functor $\Gamma \colon {\mathcal A}\to \mathbf{Arr}({\mathcal A})$ which sends an object $X$ on the unique arrow $\emptyset \to X.$ We prove that, if ${\mathcal A}$ has finite colimits and if we put on $\mathbf{Arr}({\mathcal A})$ the structure of nullhomotopies induced by the canonical string of adjunctions between ${\mathcal A}$ and $\mathbf{Arr}({\mathcal A}),$ then the functor $\Gamma$ is the free completion of ${\mathcal A}$ under strong homotopy cokernels. If ${\mathcal A}$ is finitely complete, by duality we get the free completion of ${\mathcal A}$ under strong homotopy kernels. The layout of the paper is as follows. In Section [2](#SecCatNull){reference-type="ref" reference="SecCatNull"}, we recall the definition of category with nullhomotopies and we complete it with the appropriate notions of morphism and 2-morphism. We introduce also the examples relevant for this paper. More examples can be found in [@MMMV; @DVfree]. Section [3](#SecHCok){reference-type="ref" reference="SecHCok"} is devoted to homotopy cokernels and to the behavior of colimits with respect to nullhomotopies. A particular attention is payed to the category $\mathbf{Arr}({\mathcal A}).$ Part of the material in Sections [2](#SecCatNull){reference-type="ref" reference="SecCatNull"} and [3](#SecHCok){reference-type="ref" reference="SecHCok"} is borrowed from the companion paper [@MMMV]. In Section [4](#SecUniv1){reference-type="ref" reference="SecUniv1"}, we state in a precise way and prove the universality of the full embedding $\Gamma \colon {\mathcal A}\to \mathbf{Arr}({\mathcal A})$ mentioned above. In the last section, we discuss the denormalization functor from the point of view of the universal property of $\mathbf{Arr}({\mathcal A}).$ N.B.: The composition of two arrows $\xymatrix{A \ar[r]^-{f} & B \ar[r]^-{g} & C}$ will be written as $f \cdot g.$ # Categories with nullhomotopies {#SecCatNull} In this section, we fix the terminology and some basic facts concerning nullhomotopies. As far as I know, the notion of category with a structure of nullhomotopies has been introduced in [@GR97]. I follow here the version, a bit stronger, adopted in [@SnailEV; @JMMV1; @MMMV]. **Definition 1**. *A structure of nullhomotopies $\Theta$ on a category ${\mathcal B}$ is given by the following data:* 1. *For every arrow $g$ in ${\mathcal B},$ a set $\Theta(g)$ whose elements are called nullhomotopies on $g.$* 2. *For every triple of composable arrows $\xymatrix{A \ar[r]^f & B \ar[r]^g & C \ar[r]^h & D},$ a map $$f \circ - \circ h \colon \Theta(g) \to \Theta(f \cdot g \cdot h)$$ in such a way that, for every $\varphi \in \Theta(g),$ one has* 1. *$(f' \cdot f) \circ \varphi \circ (h \cdot h') = f' \circ (f \circ \varphi \circ h) \circ h'$ whenever the compositions $f' \cdot f$ and $h \cdot h'$ are defined,* 2. *$\mathrm{id}_B \circ \varphi \circ \mathrm{id}_C = \varphi.$* *When $f=\mathrm{id}_B$ or $h=\mathrm{id}_C,$ we write $\varphi \circ h$ and $f \circ \varphi$ instead of $\mathrm{id}_B \circ \varphi \circ h$ and $f \circ \varphi \circ \mathrm{id}_C.$* **Example 2**. *In this paper, the relevant examples of structures of nullhomotopies are the first and the second example hereunder (and the dual of the first one). The third example is added in order to make clear in which sense a category with a structure of nullhomotopies can be seen as an intermediate notion between that of category and that of 2-category. Some examples having a 2-categorical flavor are discussed in [@DVfree], where the quite involved passage from nullhomotopies to 2-cells in a 2-category is analyzed. Other examples are considered in [@MMMV], where structures of nullhomotopies are obtained from generalized pre-(co)radicals, and where the link between structures of nullhomotopies and ideals of arrows is explained.* 1. *Let ${\mathcal A}$ be a category with an initial object $\emptyset$ and write $\emptyset_C \colon \emptyset \to C$ for the unique arrow. We get a structure of nullhomotopies $\Theta_{\emptyset}$ on ${\mathcal A}$ by taking as set of nullhomotopies on an arrow $g\colon B \to C$ the set $$\Theta_{\emptyset}(g) = \{ \varphi \colon B \to \emptyset \mid \varphi \cdot \emptyset_C = g \}$$ Given arrows $f \colon A \to B$ and $h \colon C \to D,$ we put $f \circ \varphi \circ h = f \cdot \varphi$ for all $\varphi \in \Theta_{\emptyset}(g).$* 2. *Recall that, given a category ${\mathcal A},$ the category $\mathbf{Arr}({\mathcal A})$ has as objects the arrows $b \colon B \to B_0$ of ${\mathcal A}$ and as arrows pairs of arrows $(g,g_0)$ in ${\mathcal A}$ such that $$\xymatrix{B \ar[r]^{g} \ar[d]_{b} & C \ar[d]^{c} \\ B_0 \ar[r]_{g_0} & C_0}$$ commutes. As set of nullhomotopies $\Theta_{\Delta}(g,g_0)$ we take the set of diagonals: $$\Theta_{\Delta}(g,g_0) = \{ \varphi \colon B_0 \to C \mid b \cdot \varphi = g, \; \varphi \cdot c = g_0 \}$$ In the situation of the following diagram $$\xymatrix{A \ar[r]^{f} \ar[d]_{a} & B \ar[r]^{g} \ar[d]_{b} & C \ar[r]^{h} \ar[d]^{c} & D \ar[d]^{d} \\ A_0 \ar[r]_{f_0} & B_0 \ar[r]_{g_0} \ar[ru]^{\varphi} & C_0 \ar[r]_{h_0} & D_0}$$ the composition is given by the formula $$(f,f_0) \circ \varphi \circ (h,h_0) = f_0 \cdot \varphi \cdot h$$ In [@MMMV], it is shown that the structure $\Theta_{\Delta}$ on $\mathbf{Arr}({\mathcal A})$ is the one induced by the string of adjunctions $$\xymatrix{ {\mathcal A}\ar[rr]|-{{\mathcal U}} & & \mathbf{Arr}({\mathcal A}) \ar@<-1.5ex>[ll]_-{{\mathcal C}} \ar@<1.5ex>[ll]^-{{\mathcal D}} } \;\;\;\;\; {\mathcal C}\dashv {\mathcal U}\dashv {\mathcal D}$$ where ${\mathcal C}$ is the codomain finctor, ${\mathcal D}$ is the domain functor and ${\mathcal U}$ is the full and faithful functor which sends an object $X$ on the identity arrow $\mathrm{id}_X.$* 3. *If the underlying category of a 2-category ${\mathcal B}$ has zero object, then ${\mathcal B}$ can be seen as a category with nullhomotopies by taking as nullhomotopies the 2-cells with domain a zero arrow (or the 2-cells with codomain a zero arrow). A relevant example which fits into this situation is discussed in Section [5](#SecDenorm){reference-type="ref" reference="SecDenorm"}.* ** 3**. * The last item of Example [Example 2](#ExNullHom){reference-type="ref" reference="ExNullHom"} justifies the fact that, in a category with nullhomotopies $({\mathcal B},\Theta),$ when a nullhomotopy $\varphi \in \Theta(g)$ is involved in a diagram, it will be depicted as $$\xymatrix{B \ar@/^1.0pc/[rr]^{g} \ar@{-->}@/_1.0pc/[rr]_{0} & \Uparrow \varphi & C}$$ even if the category ${\mathcal B}$ does not have zero arrows. For example, here there are the two ways to depict a nullhomotopy $\varphi \in \Theta_{\Delta}(g,g_0)$ in $\mathbf{Arr}({\mathcal A}) \colon$ $$\xymatrix{B \ar[r]^{g} \ar[d]_{b} & C \ar[d]^{c} \\ B_0 \ar[ru]^{\varphi} \ar[r]_{g_0} & C_0} \;\;\;\;\; \mbox{ or } \;\;\;\;\; \xymatrix{(B,b,B_0) \ar@/^1pc/[rr]^{(g,g_0)} \ar@{-->}@/_1pc/[rr]_{0} & \Uparrow \; \varphi & (C,c,C_0)}$$ * **Definition 4**. *(The 2-category of categories with nullhomotopies) Let $({\mathcal A},\Theta_{{\mathcal A}})$ and $({\mathcal B},\Theta_{{\mathcal B}})$ be two categories with nullhomotopies.* 1. *A morphism ${\mathcal F}\colon ({\mathcal A},\Theta_{{\mathcal A}}) \to ({\mathcal B},\Theta_{{\mathcal B}})$ is a functor ${\mathcal F}\colon {\mathcal A}\to {\mathcal B}$ equipped, for every arrow $g \colon B \to C$ in ${\mathcal A},$ with a map $${\mathcal F}_g \colon \Theta_{{\mathcal A}}(g) \to \Theta_{\cal B}({\mathcal F}(g))$$ such that ${\mathcal F}_{f \cdot g \cdot h}(f \circ \varphi \circ h) = {\mathcal F}(f) \circ {\mathcal F}_g(\varphi) \circ {\mathcal F}(h)$ for all $f \colon A \to B$ and $h \colon C \to D.$\ * 2. *If ${\mathcal G}\colon ({\mathcal A},\Theta_{{\mathcal A}}) \to ({\mathcal B},\Theta_{{\mathcal B}})$ is another morphism, a 2-morphism $\alpha \colon {\mathcal F}\Rightarrow {\mathcal G}$ is a natural transformation such that, for every $g \colon B \to C$ in ${\mathcal A}$ and for every $\varphi \in \Theta_{{\mathcal A}}(g),$ one has $\alpha_B \circ {\mathcal G}_g(\varphi) = {\mathcal F}_g(\varphi) \circ \alpha_C.$* *(I will always omit the suffix $g$ in the map ${\mathcal F}_{g}$ with the only exception of point 2) in the proof of Proposition [Proposition 19](#PropExt){reference-type="ref" reference="PropExt"}.)* **Remark 5**. * Since morphisms compose as functors and since 2-morphisms compose vertically and horizontally as natural transformations, categories with nullhomotopies together with their morphisms and 2-morphisms form a 2-category. Observe also that, if a 2-morphism is invertible as a natural transformation, then the inverse natural transformation is automatically a 2-morphism. * **Example 6**. * If ${\mathcal A}$ is a category with an initial object $\emptyset,$ we get a morphism of categories with nullhomotopies $\Gamma \colon ({\mathcal A},\Theta_{\emptyset}) \to (\mathbf{Arr}({\mathcal A}),\Theta_{\Delta})$ defined on objects, arrows and nullhomotopies by $$\xymatrix{ & \emptyset \ar[d] \\ B_0 \ar[ru]^{\varphi} \ar[r]_{g_0} & C_0} \;\; \mapsto \;\; \xymatrix{\emptyset \ar[r] \ar[d] & \emptyset \ar[d] \\ B_0 \ar[r]_{g_0} \ar[ru]^{\varphi} & C_0}$$ The functor $\Gamma$ is full and faithful. Moreover, for every arrow $g_0 \colon B_0 \to C_0,$ the map $\Gamma_{g_0} \colon \Theta_{\emptyset}(g_0) \to \Theta_{\Delta}(\Gamma(g_0))$ is bijective. * **Condition 7**. * Here we recall a condition crucial in this paper, but which is not always satisfied by a category with nullhomotopies. It has been isolated in [@GR01] under the name of reduced interchange. We say that the reduced interchange holds in a category with nullhomotopies $({\mathcal B},\Theta)$ if, in the situation $$\xymatrix{A \ar@/^1.0pc/[rr]^{f} \ar@{-->}@/_1.0pc/[rr]_{0} & \Uparrow \alpha & B \ar@/^1.0pc/[rr]^{g} \ar@{-->}@/_1.0pc/[rr]_{0} & \Uparrow \beta & C}$$ one has that $\alpha \circ g = f \circ \beta.$ * **Example 8**. *The reduced interchange holds in the examples of categories with nullhomotopies needed in this paper (see below). A more detailed analysis of this condition can be found in [@DVfree], where a simple counterexample is also given.* 1. *In $(\mathbf{Arr}({\mathcal A}),\Theta_{\Delta})$ the reduced interchange holds true. Indeed, given $$\xymatrix{A \ar[d]_{a} \ar[r]^{f} & B \ar[d]_{b} \ar[r]^{g} & C \ar[d]^{c} \\ A_0 \ar[r]_{f_0} \ar[ru]^{\alpha} & B_0 \ar[r]_{g_0} \ar[ru]^{\beta} & C_0}$$ one has $\alpha \circ (g,g_0) = \alpha \cdot g = \alpha \cdot b \cdot \beta = f_0 \cdot \beta = (f,f_0) \circ \beta.$* 2. *Since the reduced interchange holds true in $(\mathbf{Arr}({\mathcal A}),\Theta_{\Delta}),$ the same happens in $({\mathcal A},\Theta_{\emptyset}).$ This follows from the fact that the morphism $\Gamma$ of Example [Example 6](#ExFunctNullHom){reference-type="ref" reference="ExFunctNullHom"} is bijective on nullhomotopies.* 3. *Let me notice here that, if the structure of nullhomotopies $\Theta$ in a category ${\mathcal B}$ is the one induced by the unit of an idempotent monad or by the counit of an idempotent comonad on ${\mathcal B}$ (see [@MMMV]), then the reduced interchange holds true in $({\mathcal B},\Theta).$ The easy proof is left to the reader. The case of $(\mathbf{Arr}({\mathcal A}),\Theta_{\Delta})$ fits into this general remark because $\Theta_{\Delta}$ is induced by ${\mathcal C}\dashv {\mathcal U}$ or by ${\mathcal U}\dashv {\mathcal D},$ as already recalled in Example [Example 2](#ExNullHom){reference-type="ref" reference="ExNullHom"}.* # Homotopy cokernels and strong colimits {#SecHCok} A category with nullhomotopies does not have the 2-dimensional structure needed to express notions like 2-limits or bilimits. The convenient notions in the context of categories with nullhomotopies are those of (strong) homotopy kernels and (strong) homotopy cokernels. We copy the definition and the notation from [@MMMV]. **Definition 9**. *Let $g \colon B \to C$ be an arrow in a category with nullhomotopies $({\mathcal B},\Theta).$* 1. *A homotopy cokernel of $g$ with respect to $\Theta$ (or $\Theta$-cokernel) is a triple $${\mathcal C}(g) \in {\mathcal B}, c_g \colon C \to {\mathcal C}(g), \gamma_g \in \Theta(g \cdot c_g)$$ such that, for any other triple $(D,h,\varphi \in \Theta(g \cdot h)),$ there exists a unique arrow $h'$ such that $c_g \cdot h' = h$ and $\gamma_g \circ h' = \varphi$ $$\xymatrix{ & \ar@{}[d]_{\gamma_g}|{\Downarrow} & {\mathcal C}(g) \ar[dd]^{h'} \\ B \ar@/^1.0pc/@{-->}[rru]^{0} \ar@/_1.0pc/@{-->}[rrd]_{0} \ar[r]^-{g} & C \ar[ru]_{c_g} \ar[rd]^{h} \\ & \ar@{}[u]_{\Uparrow}|{\varphi} & D }$$* 2. *A $\Theta$-cokernel $({\mathcal C}(g),c_g,\gamma_g)$ is strong if, for any triple $(D,h,\varphi \in \Theta(c_g \cdot h)),$ such that $g \circ \varphi = \gamma_g \circ h,$ there exists a unique nullhomotopy $\varphi' \in \Theta(h)$ such that $c_g \circ \varphi' = \varphi$ $$\xymatrix{B \ar[rr]^-{g} \ar@/_1.8pc/@{-->}[rrrr]_{0}^{\gamma_g\; \Uparrow} & & C \ar@/^1.8pc/@{-->}[rrrr]^{0}_{\varphi \; \Downarrow} \ar[rr]^-{c_g} & & {\mathcal C}(g) \ar[rr]^{h} \ar@/_1.5pc/@{-->}[rr]_{0}^{\varphi' \; \Uparrow} & & D }$$* **Remark 10**. *We list here some remarks on the $\Theta$-cokernel of an arrow in a category with nullhomotopies $({\mathcal B},\Theta).$* 1. *Uniqueness: the $\Theta$-cokernel of an arrow is determined by its universal property uniquely up to a unique isomorphism. Moreover, if an arrow has two (necessarily isomorphic) $\Theta$-cokernels and one of them is strong, the other one also is strong.* 2. *Functoriality: in the situation of the following commutative solid diagram $$\xymatrix{A \ar[rr]^{f} \ar[d]_{a} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<<<{\Longrightarrow} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<{\; \gamma_a} & & B \ar[d]^{b} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<<<{\Longleftarrow} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<{\gamma_b \;} \\ A_0 \ar[rr]_{f_0} \ar[d]_{c_a} & & B_0 \ar[d]^{c_b} \\ {\mathcal C}(a) \ar@{..>}[rr]_{{\mathcal C}(f,f_0)} & & {\mathcal C}(b)}$$ there exists a unique arrow ${\mathcal C}(f,f_0) \colon {\mathcal C}(a) \to {\mathcal C}(b)$ such that $c_a \cdot {\mathcal C}(f,f_0) = f_0 \cdot c_b$ and $\gamma_a \circ {\mathcal C}(f,f_0) = f \circ \gamma_b.$* 3. *Behavior with respect to nullhomotopies: in the situation of the following commutative diagram $$\xymatrix{A \ar[rr]^{f} \ar[d]_{a} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<<<{\Longrightarrow} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<{\; \gamma_a} & & B \ar[d]^{b} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<<<{\Longleftarrow} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<{\gamma_b \;} \\ A_0 \ar[rr]_{f_0} \ar[d]_{c_a} \ar[rru]^{d} & & B_0 \ar[d]^{c_b} \\ {\mathcal C}(a) \ar[rr]^{{\mathcal C}(f,f_0)} \ar@{-->}@/_1.5pc/[rr]_{0}^{\Uparrow \; {\mathcal C}(d) } & & {\mathcal C}(b)}$$ if the $\Theta$-cokernel of the arrow $a$ is strong, then there exists a unique nullhomotopy ${\mathcal C}(d) \in \Theta({\mathcal C}(f,f_0))$ such that $c_a \circ {\mathcal C}(d) = d \circ \gamma_b.$* 4. *Cancellation properties:* 1. *In the situation $$\xymatrix{A \ar@/^1.8pc/@{-->}[rrrr]^{0}_{\gamma_f \; \Downarrow} \ar[rr]_{f} && B \ar[rr]_{c_f} && {\mathcal C}(f) \ar@<0.5ex>[rr]^{g} \ar@<-0.5ex>[rr]_{h} && C}$$ if $c_f \cdot g = c_f \cdot h$ and $\gamma_f \circ g = \gamma_f \circ h,$ then $g=h.$* 2. *Assume now that the reduced interchange [Condition 7](#CondRedInter){reference-type="ref" reference="CondRedInter"} holds in $({\mathcal B},\Theta).$ In the situation $$\xymatrix{A \ar[r]^{f} & B \ar[r]^{c_f} & {\mathcal C}(f) \ar@/^1.8pc/@{-->}[rrr]^{0}_{\varphi \, \Downarrow \;\; \Downarrow \, \psi} \ar[rrr]_-{g} &&& C}$$ if the $\Theta$-cokernel is strong and if the nullhomotopies $\varphi, \psi \in \Theta(g)$ are such that $c_f \circ \varphi = c_f \circ \psi,$ then $\varphi = \psi.$* *Proof.* We check point 4.(b) because this is the first place where we use the reduced interchange. Put $\alpha = c_f \circ \varphi.$ By the reduced interchange, we have $\gamma_f \circ g = f \cdot c_f \circ \varphi = f \circ \alpha.$ We can apply the universal property of the $\Theta$-cokernel and we get a unique nullhomotopy $\alpha' \in \Theta(g)$ such that $c_f \circ \alpha' = \alpha.$ Clearly, we can take $\alpha' = \varphi$ but, because of the hypothesis $c_f \circ \varphi = c_f \circ \psi,$ we can take also $\alpha' = \psi.$ By uniqueness of $\alpha',$ we are done. ◻ **Remark 11**. *Let us analyze here objects, arrows and nullhomotopies in $(\mathbf{Arr}({\mathcal A}),\Theta_{\Delta})$ from the point of view of $\Theta_{\Delta}$-cokernels. In fact, the following simple remarks are the starting point to see that $(\mathbf{Arr}({\mathcal A}),\Theta_{\Delta})$ is the completion of ${\mathcal A}$ by strong homotopy cokernels, as we will see in Section [4](#SecUniv1){reference-type="ref" reference="SecUniv1"}.* 1. *Assume that the category ${\mathcal A}$ has an initial object $\emptyset$ and consider the embedding $\Gamma \colon {\mathcal A}\to \mathbf{Arr}({\mathcal A})$ described in Example [Example 6](#ExFunctNullHom){reference-type="ref" reference="ExFunctNullHom"}. For any arrow $a \colon A \to A_0$ in ${\mathcal A},$ the following diagram is a ($\Theta_{\Delta}$-kernel $|$ $\Theta_{\Delta}$-cokernel) diagram in $(\mathbf{Arr}({\mathcal A}),\Theta_{\Delta}) \colon$ $$\xymatrix{\emptyset \ar[r] \ar[d] & \emptyset \ar[r] \ar[d] & A \ar[d]^{a} \\ A \ar[r]_{a} \ar[rru]^<<<<<<{\mathrm{id}} & A_0 \ar[r]_{\mathrm{id}} & A_0} \;\;\;\;\; \mbox{ that is } \;\;\;\;\; \xymatrix{ \\ \Gamma A \ar[rr]_{\Gamma a} \ar@/^1.8pc/@{-->}[rrrr]^{0}_{\mathrm{id}_A \; \Downarrow} & & \Gamma A_0 \ar[rr]_-{(\emptyset_A,\mathrm{id}_{A_0}} & & (A,a,A_0) }$$ In other words, each object $(a \colon A \to A_0)$ of $\mathbf{Arr}({\mathcal A})$ is the $\Theta_{\Delta}$-cokernel of an arrow coming from ${\mathcal A}$ (and each arrow of ${\mathcal A},$ once embedded in $\mathbf{Arr}({\mathcal A}),$ becomes the arrow part of a $\Theta_{\Delta}$-kernel).\ * 2. *More is true: each arrow $(f,f_0) \colon (A,a,A_0) \to (B,b,B_0)$ of $\mathbf{Arr}({\mathcal A})$ is the unique extension to the $\Theta_{\Delta}$-cokernel (in the sense of Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.2) of a commutative square coming from ${\mathcal A},$ as in the following diagram: $$\xymatrix{\Gamma A \ar[rr]^{\Gamma f} \ar[d]_{\Gamma a} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<<<{\Longrightarrow} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<{\mathrm{id}_A} & & \Gamma B \ar[d]^{\Gamma b} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<<<{\Longleftarrow} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<{\mathrm{id}_B} \\ \Gamma A_0 \ar[rr]_{\Gamma f_0} \ar[d]^{(\emptyset_A,\mathrm{id}_{A_0})} & & \Gamma B_0 \ar[d]_{(\emptyset_B,\mathrm{id}_{B_0})} \\ (A,a,A_0) \ar[rr]_{(f,f_0)} & & (B,b,B_0)}$$* 3. *Finally, each nullhomotopy $\varphi \in \Theta_{\Delta}(f,f_0)$ is the unique extension to the $\Theta_{\Delta}$-cokernel (in the sense of Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.3) of a diagonal coming from ${\mathcal A},$ as in the following diagram: $$\xymatrix{\Gamma A \ar[rr]^{\Gamma f} \ar[d]_{\Gamma a} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<<<{\Longrightarrow} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<{\mathrm{id}_A} & & \Gamma B \ar[d]^{\Gamma b} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<<<{\Longleftarrow} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<{\mathrm{id}_B} \\ \Gamma A_0 \ar[rr]_{\Gamma f_0} \ar[d]^{(\emptyset_A,\mathrm{id}_{A_0})} \ar[rru]^{\Gamma \varphi} & & \Gamma B_0 \ar[d]_{(\emptyset_B,\mathrm{id}_{B_0})} \\ (A,a,A_0) \ar[rr]^{(f,f_0)} \ar@{-->}@/_1.5pc/[rr]_{0}^{\Uparrow \; \varphi} & & (B,b,B_0)}$$* The following proposition appears in [@MMMV], where it is deduced from some general results on the existence of homotopy cokernels. **Proposition 12**. *If a category ${\mathcal A}$ has pushouts, then $\mathbf{Arr}({\mathcal A})$ has strong $\Theta_{\Delta}$-cokernels.* ** 13**. * Even if Proposition [Proposition 12](#PropCofNofArr){reference-type="ref" reference="PropCofNofArr"} does not need a proof, I wish to share with the reader the guiding idea to construct $\Theta_{\Delta}$-cokernels in $\mathbf{Arr}({\mathcal A})$ because it seems to me quite easy and instructive in order to understand the arguments behind the proof given in [@MMMV]. The following description already appears in [@SnailEV; @GJ].\ The $\Theta_{\Delta}$-cokernel of an arrow $(f,f_0)$ in $\mathbf{Arr}({\mathcal A})$ must be universal among all diagrams of shape $$\xymatrix{A \ar[d]_{a} \ar[r]^{f} & B \ar[d]_<<<{b} \ar[r]^{g} & C \ar[d]^{c} \\ A_0 \ar[r]_{f_0} \ar[rru]_>>>>>>>>{\varphi} & B_0 \ar[r]_{g_0} & C_0}$$ where the following diagrams commute $$\xymatrix{A \ar[r]^{f} \ar[d]_{a} & B \ar[d]^{g} \\ A_0 \ar[r]_{\varphi} & C} \;\;\;\;\;\;\;\;\;\;\;\;\; \xymatrix{A_0 \ar[d]_{\varphi} \ar[rrd]^<<<<<<{f_0} & & B \ar[lld]_<<<<<<{g} \ar[d]^{b} \\ C \ar[rd]_{c} & & B_0 \ar[ld]^{g_0} \\ & C_0}$$ So, just replace these two diagrams by the corresponding colimits. We get $$\xymatrix{A \ar[rr]^{f} \ar[d]_{a} & & B \ar[d]^{a'} \\ A_0 \ar[rr]_{f'} & & A_0 +_{a,f} B} \;\;\;\;\;\;\;\;\;\;\;\;\; \xymatrix{A_0 \ar[d]_{f'} \ar[rrd]^<<<<<<{f_0} & & B \ar[lld]_<<<<<<{a'} \ar[d]^{b} \\ A_0 +_{a,f} B \ar[rd]_{[f_0,b]} & & B_0 \ar[ld]^{\mathrm{id}} \\ & B_0}$$ where $A_0 +_{a,f} B$ is the pushout of $a$ and $f.$ Finally, the $\Theta_{\Delta}$-cokernel of $(f,f_0)$ is $$\xymatrix{A \ar[rr]^{f} \ar[d]_{a} & & B \ar[d]_<<<{b} \ar[rr]^{a'} & & A_0 +_{a,f} B \ar[d]^{[f_0,b]} \\ A_0 \ar[rrrru]_>>>>>>>>>{f'} \ar[rr]_{f_0} & & B_0 \ar[rr]_{\mathrm{id}} & & B_0}$$ * The interplay between nullhomotopies and colimits will enter in the statement and in the proof of the universal property of $\mathbf{Arr}({\mathcal A}).$ This is why we need the following definitions. **Definition 14**. * Consider two functors ${\mathcal F}, {\mathcal G}\colon {\mathcal D}\to {\mathcal B},$ where $({\mathcal B},\Theta)$ is a category with nullhomotopies. A natural nullhomotopy $$\xymatrix{{\mathcal D}\ar@/^1.0pc/[rr]^{{\mathcal G}} \ar@/_1.0pc/[rr]_{{\mathcal F}} & \Uparrow \, \tau & {\mathcal B}}$$ is given by a family of arrows and a family of nullhomotopies indexed by the objects of ${\mathcal D},$ $$\tau = \{\tau^a_D \colon {\mathcal F}(D) \to {\mathcal G}(D) \;,\;\; \tau^n_D \in \Theta(\tau^a_D) \}_{D \in {\mathcal D}}$$ such that the family of arrows is a natural transformation and the family of nullhomotopies is such that $\tau^n_D \circ {\mathcal G}(g) = {\mathcal F}(g) \circ \tau^n_{D'}$ for all $g \colon D \to D'$ in ${\mathcal D}.$ * **Definition 15**. * Consider a functor ${\mathcal F}\colon {\mathcal D}\to {\mathcal B},$ where $({\mathcal B},\Theta)$ is a category with nullhomotopies, and write $$\{ i_D \colon {\mathcal F}(D) \to \mbox{colim}F \}_{D \in {\mathcal D}}$$ for its colimit. We say that the colimit of ${\mathcal F}$ is strong with respect to nullhomotopies (or $\Theta$-strong) if, for every object $X \in {\mathcal B}$ and for every natural nullhmotopy $$\xymatrix{{\mathcal D}\ar@/^1.0pc/[rr]^{\kappa_X} \ar@/_1.0pc/[rr]_{{\mathcal F}} & \Uparrow \, \tau & {\mathcal B}}$$ ($\kappa_X$ is the constant functor of value $X$) there exists a unique nullhomotopy $t^n \in \Theta(t^a)$ such that $i_D \circ t^n = \tau^n_D$ for all $D \in {\mathcal D},$ where $t^a \colon \mbox{colim}F \to X$ is the unique arrow such that $i_D \cdot t^a = \tau^a_D$ for all $D \in {\mathcal D}.$ * **Remark 16**. *Let us make explicit two special cases of Definition [Definition 15](#DefColimNull){reference-type="ref" reference="DefColimNull"}. The second one appears also in [@MMMV]. Let $({\mathcal B},\Theta)$ be a category with nullhomotopies.* 1. *An initial object $\emptyset$ is $\Theta$-strong if, for every object $X \in {\mathcal B},$ there is a unique nullhomotopy on the unique arrow $\emptyset_X \colon \emptyset \to X.$* 2. *Consider the factorization of a commutative square $f \cdot x = g \cdot y$ through the pushout of $f$ and $g$ as in the following diagram: $$\xymatrix{A \ar[rr]^{g} \ar[d]_{f} & & C \ar[d]_{f'} \ar@/^2pc/[rrdd]^{y} \\ B \ar[rr]^{g'} \ar@/_1.5pc/[rrrrd]_{x} & & B+_{f,g}C \ar[rrd]^{[x,y]} \\ & & & & D}$$ The pushout is $\Theta$-strong if, given two nullhomotopies $\varphi \in \Theta(x)$ and $\psi \in \Theta(y)$ such that $f \circ \varphi = g \circ \psi,$ there exists a unique nullhomotopy $[\varphi,\psi] \in \Theta([x,y])$ such that $g' \circ [\varphi,\psi) = \varphi$ and $f' \circ [\varphi,\psi] = \psi.$* 3. *Clearly, a $\Theta$-strong colimit has a cancellation property with respect to nullhomotopies. Here is the one for a $\Theta$-strong pushout (with the notation of the previous point): given an arrow $h \colon B+_{f,g}C \to D$ and nullhomotopies $\alpha, \beta \in \Theta(h),$ if $g' \circ \alpha = g' \circ \beta$ and $f' \circ \alpha = f' \circ \beta,$ then $\alpha = \beta.$* **Example 17**. * Let $({\mathcal B},\Theta)$ be a category with nullhomotopies and let $\emptyset$ be a $\Theta$-strong initial object in ${\mathcal B}.$ If, for an object $X \in {\mathcal B},$ we call $\gamma_X \in \Theta(\emptyset_X)$ the unique nullhomotopy on $\emptyset_X,$ then the following diagram is a $\Theta$-cokernel: $$\xymatrix{\emptyset \ar[rr]_{\emptyset_X} \ar@/^1.8pc/@{-->}[rrrr]^{0}_{\gamma_X \; \Downarrow} & & X \ar[rr]_-{\mathrm{id}_X} & & X }$$ * Here is the interplay between nullhomotopies and colimits in $\mathbf{Arr}({\mathcal A}).$ **Proposition 18**. *Let ${\mathcal A}$ be a category with finite colimlits.* 1. *Finite colimits in ${\mathcal A}$ are $\Theta_{\emptyset}$-strong.* 2. *$\mathbf{Arr}({\mathcal A})$ has finite colimits and they are $\Theta_{\Delta}$-strong.* 3. *The functor $\Gamma \colon {\mathcal A}\to \mathbf{Arr}({\mathcal A})$ preserves finite colimits.* *Proof.* The first point is an easy exercise. Moreover, colimits in $\mathbf{Arr}({\mathcal A})$ are constructed level-wise from those in ${\mathcal A}$ and obviously $\Gamma$ preserves colimits. Let me check, for example, that pushouts in $\mathbf{Arr}({\mathcal A})$ are $\Theta_{\Delta}$-strong. Consider the following diagrams in $\mathbf{Arr}({\mathcal A}),$ the first one being a pushout and the second one being commutative : $$\xymatrix{& A \ar[rr]^{g} \ar[dd]^<<<<<<{a} \ar[ld]_{f} & & C \ar[ld]_{f'} \ar[dd]^{c} \\ B \ar[rr]^<<<<<<<<<<<<<<<<<<<<{g'} \ar[dd]_{b} & & B+_{f,g}C \ar[dd]^<<<<<<{[b \cdot g_0', c \cdot f_0']} \\ & A_0 \ar[rr]^<<<<<<<<<<{g_0} \ar[ld]_{f_0} & & C_0 \ar[ld]^{f_0'} \\ B_0 \ar[rr]_{g_0'} & & B_0+_{f_0,g_0}C_0} \;\;\;\;\; \xymatrix{& A \ar[rr]^{g} \ar[dd]^<<<<<<{a} \ar[ld]_{f} & & C \ar[ld]_{y} \ar[dd]^{c} \\ B \ar[rr]^>>>>>>>>{x} \ar[dd]_{b} & & D \ar[dd]^<<<<<<{d} \\ & A_0 \ar[rr]^<<<<<<{g_0} \ar[ld]_{f_0} & & C_0 \ar[ld]^{y_0} \\ B_0 \ar[rr]_{x_0} & & D_0}$$ Consider also the unique factorization of the commutative diagram through the pushout: $$\xymatrix{B+_{f,g}C \ar[d]_{[b \cdot g_0', c \cdot f_0']} \ar[rr]^{[x,y]} & & D \ar[d]^{d} \\ B_0+_{f_0,g_0}C_0 \ar[rr]_{[x_0,y_0]} & & D_0}$$ Given two nullhomotopies $$\xymatrix{B \ar[r]^{x} \ar[d]_{b} & D \ar[d]_{d} & C \ar[l]_{y} \ar[d]^{c} \\ B_0 \ar[r]_{x_0} \ar[ru]^{\varphi} & D_0 & C_0 \ar[lu]_{\psi} \ar[l]^{y_0}}$$ the compatibility condition $(f,f_0) \circ \varphi = (g,g_0) \circ \psi$ means that $f_0 \cdot \varphi = g_0 \cdot \psi.$ Therefore, there exists a unique arrow $[\varphi,\psi] \colon B_0+_{f_0,g_0}C_0 \to D$ such that $g_0' \cdot [\varphi,\psi] = \varphi$ and $f_0' \cdot [\varphi,\psi] = \psi.$ It remais to check that $[\varphi,\psi]$ is a nullhomotopy: $$\xymatrix{B+_{f,g}C \ar[d]_{[b \cdot g_0', c \cdot f_0']} \ar[rr]^{[x,y]} & & D \ar[d]^{d} \\ B_0+_{f_0,g_0}C_0 \ar[rr]_{[x_0,y_0]} \ar[rru]^{[\varphi,\psi]} & & D_0}$$ The commutativity of the two triangles follows precomposing with the canonical arrows of the pushout. ◻ # Universality of $\mathbf{Arr}({\mathcal A})$ {#SecUniv1} In this section we show that, if ${\mathcal A}$ has finite colimits, the embedding $\Gamma \colon {\mathcal A}\to \mathbf{Arr}({\mathcal A})$ is the completion of ${\mathcal A}$ by strong homotopy cokernels. We put on $\mathbf{Arr}({\mathcal A})$ the structure of nullhomotopies $\Theta_{\Delta}$ introduced in Example [Example 2](#ExNullHom){reference-type="ref" reference="ExNullHom"}. The main point is to extend a functor ${\mathcal F}\colon {\mathcal A}\to {\mathcal B}$ along $\Gamma \colon {\mathcal A}\to \mathbf{Arr}({\mathcal A}).$ **Proposition 19**. *Consider a category ${\mathcal A}$ with finite colimits, a category with nullhomotopies $({\mathcal B},\Theta)$ satisfying the reduced interchange, and a functor ${\mathcal F}\colon {\mathcal A}\to {\mathcal B}.$ Assume that* 1. *the image by ${\mathcal F}$ of finite colimits are $\Theta$-strong finite colimits, and* 2. *the image by ${\mathcal F}$ of any arrow in ${\mathcal A}$ has a strong $\Theta$-cokernel in ${\mathcal B}.$* *Under these conditions, there exists an essentially unique morphism of categories with nullhomotopies $\widehat {\mathcal F}\colon (\mathbf{Arr}({\mathcal A}),\Theta_{\Delta}) \to ({\mathcal B},\Theta)$ sending $\Theta_{\Delta}$-cokernels to strong $\Theta$-cokernels and such that $\Gamma \cdot \widehat {\mathcal F}\simeq {\mathcal F}.$ $$\xymatrix{{\mathcal A}\ar[r]^<<<<<{\Gamma} \ar[rd]_{{\mathcal F}} & \mathbf{Arr}({\mathcal A}) \ar[d]^{\widehat{\mathcal F}} \\ & {\mathcal B}}$$ Moreover, the image by $\widehat{\mathcal F}$ of finite colimits are $\Theta$-strong finite colimits.* *Proof.* We split the proof into seven steps.\ 1) Construction of $\widehat {\mathcal F}\colon$ start with two objects, an arrow and a nullhomotopy in $\mathbf{Arr}({\mathcal A}) \colon$ $$\xymatrix{A \ar[r]^{f} \ar[d]_{a} & B \ar[d]^{b} \\ A_0 \ar[ru]^{\alpha} \ar[r]_{f_0} & B_0}$$ their images by $\widehat {\mathcal F}$ are depicted in the following commutative diagram, where both columns are $\Theta$-cokernels: $$\xymatrix{{\mathcal F}A \ar[rr]^{{\mathcal F}f} \ar[d]_{{\mathcal F}a} \ar@{-->}@/_3.0pc/[dd]_{0}^<<<<<<<<<<<<<{\Longrightarrow} \ar@{-->}@/_3.0pc/[dd]_{0}^<<<<<<<<<<{\, \gamma_{{\mathcal F}a}} & & {\mathcal F}B \ar[d]^{{\mathcal F}b} \ar@{-->}@/^3.0pc/[dd]^{0}_<<<<<<<<<<<<<{\Longleftarrow} \ar@{-->}@/^3.0pc/[dd]^{0}_<<<<<<<<<<{\gamma_{{\mathcal F}b} \,} \\ {\mathcal F}A_0 \ar[rr]_{{\mathcal F}f_0} \ar[d]_{c_{{\mathcal F}a}} \ar[rru]^{{\mathcal F}\alpha} & & {\mathcal F}B_0 \ar[d]^{c_{{\mathcal F}b}} \\ \widehat {\mathcal F}(A,a,A_0) \ar[rr]^{\widehat {\mathcal F}(f,f_0)} \ar@{-->}@/_1.5pc/[rr]_{0}^{\Uparrow \; \widehat {\mathcal F}\alpha} & & \widehat {\mathcal F}(B,b,B_0)}$$ The arrow $\widehat {\mathcal F}(f,f_0)$ is the unique arrow such that $c_{{\mathcal F}a} \cdot \widehat {\mathcal F}(f,f_0) = {\mathcal F}f_0 \cdot c_{{\mathcal F}b}$ and $\gamma_{{\mathcal F}a} \circ \widehat {\mathcal F}(f,f_0) = {\mathcal F}f \circ \gamma_{{\mathcal F}b},$ see Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.2. The nullhomotopy $\widehat {\mathcal F}\alpha$ is the unique nullhomotopy such that $c_{{\mathcal F}a} \circ \widehat {\mathcal F}\alpha = {\mathcal F}\alpha \circ \gamma_{{\mathcal F}b},$ see Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.3. It is easy to check that $\widehat{\mathcal F}$ is indeed a morphism of categories with nullhomotopies.\ 2) Uniqueness of $\widehat {\mathcal F}$ under the assumptions that $\widehat{\mathcal F}$ preserves homotopy cokernels and extends ${\mathcal F}$ along $\Gamma \colon$ consider once again a nullhomotopy in $\mathbf{Arr}({\mathcal A})$ $$\xymatrix{A \ar[r]^{f} \ar[d]_{a} & B \ar[d]^{b} \\ A_0 \ar[ru]^{\varphi} \ar[r]_{f_0} & B_0}$$ Following Remark [Remark 11](#RemCofCompl){reference-type="ref" reference="RemCofCompl"}, we can present it as $$\xymatrix{\Gamma A \ar[rr]^{\Gamma f} \ar[d]_{\Gamma a} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<<<{\Longrightarrow} \ar@{-->}@/_2.5pc/[dd]_{0}^<<<<<<<<<<<<<<<<<{\gamma_{\Gamma a}} & & \Gamma B \ar[d]^{\Gamma b} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<<<{\Longleftarrow} \ar@{-->}@/^2.5pc/[dd]^{0}_<<<<<<<<<<<<<<<<<{\gamma_{\Gamma b}} \\ \Gamma A_0 \ar[rr]_{\Gamma f_0} \ar[d]_{c_{\Gamma a}} \ar[rru]^{\Gamma \varphi} & & \Gamma B_0 \ar[d]^{c_{\Gamma b}} \\ (A,a,A_0) \ar[rr]^{(f,f_0)} \ar@{-->}@/_1.5pc/[rr]_{0}^{\Uparrow \; \varphi} & & (B,b,B_0)}$$ We have to compare what necessarily is the image by $\widehat{\mathcal F}$ of this diagram with the construction depicted in the first point of the proof.\ (i) On objects: the first equality is due to the fact that $\widehat{\mathcal F}$ preserves homotopy cokernels and the second one to the fact that $\widehat{\mathcal F}$ extends ${\mathcal F}$ along $\Gamma$ $$\resizebox{\displaywidth}{!}{ \xymatrix{ & \widehat{\mathcal F}(\Gamma A_0) \ar[rd]^{\widehat{\mathcal F}(c_{\Gamma a})} \\ \widehat{\mathcal F}(\Gamma A) \ar[ru]^{\widehat{\mathcal F}(\Gamma a)} \ar@{-->}[rr]_0 & \ar@{}[u]|{\Uparrow \; \widehat{\mathcal F}(\gamma_{\Gamma a})} & \widehat{\mathcal F}(A,a,A_0)} = \xymatrix{ & \widehat{\mathcal F}(\Gamma A_0) \ar[rd]^{c_{\widehat{\mathcal F}(\Gamma a)}} \\ \widehat{\mathcal F}(\Gamma A) \ar[ru]^{\widehat{\mathcal F}(\Gamma a)} \ar@{-->}[rr]_0 & \ar@{}[u]|{\Uparrow \; \gamma_{\widehat{\mathcal F}(\Gamma a)}} & {\mathcal C}(\widehat{\mathcal F}(\Gamma a))} = \xymatrix{ & {\mathcal F}A_0 \ar[rd]^{c_{{\mathcal F}a}} \\ {\mathcal F}A \ar[ru]^{{\mathcal F}a} \ar@{-->}[rr]_0 & \ar@{}[u]|{\Uparrow \; \gamma_{{\mathcal F}a}} & {\mathcal C}({\mathcal F}a)}}$$ (ii) On arrows: we have to verify that our assumptions of $\widehat{\mathcal F}$ force the equations $$c_{{\mathcal F}a} \cdot \widehat {\mathcal F}(f,f_0) = {\mathcal F}f_0 \cdot c_{{\mathcal F}b} \;\;\; \mbox{ and } \;\;\; \gamma_{{\mathcal F}a} \circ \widehat {\mathcal F}(f,f_0) = {\mathcal F}f \circ \gamma_{{\mathcal F}b}$$ From Remark [Remark 11](#RemCofCompl){reference-type="ref" reference="RemCofCompl"}, we know that $c_{\Gamma a} \cdot (f,f_0) = \Gamma f_0 \cdot c_{\Gamma b}$ and $\gamma_{\Gamma a} \circ (f,f_0) = \Gamma f \circ \gamma_{\Gamma b}.$ Therefore, by applying $\widehat{\mathcal F}$ and using the conditions of Definition [Definition 4](#DefFunctNullHom){reference-type="ref" reference="DefFunctNullHom"}, we get $$c_{Fa} \cdot \widehat{\mathcal F}(f,f_0) = \widehat{\mathcal F}(c_{\Gamma a}) \cdot \widehat{\mathcal F}(f,f_0) = \widehat{\mathcal F}(\Gamma f_0) \cdot \widehat{\mathcal F}(c_{\Gamma b}) = Ff_0 \cdot c_{Fb}$$ $$\gamma_{Fa} \circ \widehat{\mathcal F}(f,f_0) = \widehat{\mathcal F}_{\Gamma a \cdot c_{\Gamma a}}(\gamma_{\Gamma a}) \circ \widehat{\mathcal F}(f,f_0) = \widehat{\mathcal F}_{\Gamma a \cdot c_{\Gamma a} \cdot (f,f_0)}(\gamma_{\Gamma a} \circ (f,f_0)) =$$ $$= \widehat{\mathcal F}_{\Gamma f \cdot \Gamma b \cdot c_{\Gamma b}}(\Gamma f \circ \gamma_{\Gamma b}) = \widehat{\mathcal F}(\Gamma f) \circ \widehat{\mathcal F}_{\Gamma b \cdot c_{\Gamma b}}(\gamma_{\Gamma b}) = {\mathcal F}f \circ \gamma_{{\mathcal F}b}$$ (iii) On nullhomotopies: we have to verify that our assumptions of $\widehat{\mathcal F}$ force the equation $$c_{{\mathcal F}a} \circ \widehat{\mathcal F}_{(f,f_0)}(\varphi) = {\mathcal F}\varphi \circ \gamma_{{\mathcal F}b}$$ From Remark [Remark 11](#RemCofCompl){reference-type="ref" reference="RemCofCompl"}, we know that $c_{\Gamma a} \circ \varphi = \Gamma \varphi \circ \gamma_{\Gamma b}.$ Therefore, by applying $\widehat{\mathcal F}$ and using the conditions of Definition [Definition 4](#DefFunctNullHom){reference-type="ref" reference="DefFunctNullHom"}, we get $$c_{{\mathcal F}a} \circ \widehat{\mathcal F}_{(f,f_0)}(\varphi) = \widehat{\mathcal F}(c_{\Gamma a}) \circ \widehat{\mathcal F}_{(f,f_0)}(\varphi) = \widehat{\mathcal F}_{c_{\Gamma a} \cdot (f,f_0)}(c_{\Gamma a} \circ \varphi) =$$ $$=\widehat{\mathcal F}_{\Gamma \varphi \cdot \Gamma b \cdot c_{\Gamma b}}(\Gamma \varphi \circ \gamma_{\Gamma b}) = \widehat{\mathcal F}(\Gamma \varphi) \circ \widehat{\mathcal F}_{\Gamma b \cdot c_{\Gamma b}}(\gamma_{\Gamma b}) = {\mathcal F}\varphi \circ \gamma_{{\mathcal F}b}$$ 3) $\widehat{\mathcal F}$ preserves homotopy cokernels: consider a $\Theta_{\Delta}$-cokernel in $\mathbf{Arr}({\mathcal A})$ as in [ 13](#TextCofArr){reference-type="ref" reference="TextCofArr"} $$\xymatrix{A \ar[rr]^{f} \ar[d]_{a} & & B \ar[d]_<<<{b} \ar[rr]^{a'} & & A_0 +_{a,f} B \ar[d]^{[f_0,b]} \\ A_0 \ar[rrrru]_>>>>>>>>>{f'} \ar[rr]_{f_0} & & B_0 \ar[rr]_{\mathrm{id}} & & B_0}$$ and its image by $\widehat{\mathcal F}$ (the three columns are $\Theta$-cokernels, but I omit from the picture the corresponding structural nullhomotopies $\gamma_{{\mathcal F}a}, \gamma_{{\mathcal F}b}$ and $\gamma_{{\mathcal F}[f_0,b]}$): $$\xymatrix{{\mathcal F}A \ar[d]_{{\mathcal F}a} \ar[rr]^{{\mathcal F}f} & & {\mathcal F}B \ar[d]_<<<{{\mathcal F}b} \ar[rr]^{{\mathcal F}a'} & & {\mathcal F}(A_0+_{a,f}B) \ar[d]^{{\mathcal F}[f_0,b]} \\ {\mathcal F}A_0 \ar[d]_{c_{{\mathcal F}a}} \ar[rr]_{{\mathcal F}f_0} \ar[rrrru]_>>>>>>>>>>>>>>{{\mathcal F}f'} & & {\mathcal F}B_0 \ar[d]_{c_{{\mathcal F}b}} \ar[rr]_{\mathrm{id}} & & {\mathcal F}B_0 \ar[d]^{c_{{\mathcal F}[f_0,b]}} \\ \widehat{\mathcal F}(A,a,A_0) \ar[rr]^{\widehat{\mathcal F}(f,f_0)} \ar@{-->}@/_2.0pc/[rrrr]_{0}^{\Uparrow \;\; \widehat{\mathcal F}f'} & & \widehat{\mathcal F}(B,b,B_0) \ar[rr]^{\widehat{\mathcal F}(a',\mathrm{id})} & & \widehat{\mathcal F}[f_0,b]}$$ We have to prove that the bottom row is a $\Theta$-cokernel. For this, consider a nullhomotopy in ${\mathcal B}\colon$ $$\xymatrix{ \widehat{\mathcal F}(A,a,A_0) \ar[rr]_{\widehat{\mathcal F}(f,f_0)} \ar@/^1.8pc/@{-->}[rrrr]^{0}_{\varphi \; \Downarrow} & & \widehat{\mathcal F}(B,b,B_0) \ar[rr]_-{g} & & C }$$ We can construct two nullhomotopies in ${\mathcal B}$ $$c_{{\mathcal F}a} \circ \varphi \in \Theta(c_{{\mathcal F}a} \cdot \widehat{\mathcal F}(f,f_0) \cdot g) = \Theta({\mathcal F}f_0 \cdot c_{{\mathcal F}b} \cdot g) \;\;\mbox{ and }\;\; \gamma_{{\mathcal F}b} \circ g \in \Theta({\mathcal F}b \cdot c_{{\mathcal F}b} \cdot g)$$ which satisfy the following condition (use Condition [Condition 7](#CondRedInter){reference-type="ref" reference="CondRedInter"} for the first equality): $${\mathcal F}a \cdot c_{{\mathcal F}a} \circ \varphi = \gamma_{{\mathcal F}a} \circ \widehat{\mathcal F}(f,f_0) \cdot g = {\mathcal F}f \circ \gamma_{{\mathcal F}b} \circ g$$ Since, by assumption, the image by ${\mathcal F}$ of a pushout is a $\Theta$-strong pushout, we can apply Remark [Remark 16](#RemColimNull){reference-type="ref" reference="RemColimNull"}.2 and we get a unique nullhomotopy $\bar\varphi \in \Theta({\mathcal F}[f_0,b] \cdot c_{{\mathcal F}b} \cdot g)$ such that ${\mathcal F}f' \circ \bar\varphi = c_{{\mathcal F}a} \circ \varphi$ and ${\mathcal F}a' \circ \bar\varphi = \gamma_{{\mathcal F}b} \circ g.$ Now, the existence of $\bar\varphi$ combined with the universal property of the $\Theta$-cokernel $\widehat{\mathcal F}[f_0,b]$ gives a unique arrow $g' \colon \widehat{\mathcal F}[f_0,b] \to C$ such that $c_{{\mathcal F}[f_0,b]} \cdot g' = c_{{\mathcal F}b} \cdot g$ and $\gamma_{{\mathcal F}[f_0,b]} \circ g' = \bar\varphi.$ We have to prove that the arrow $g'$ is the required factorization of $(g,\varphi)$ through $(\widehat{\mathcal F}(a',\mathrm{id}),\widehat{\mathcal F}f'),$ that is, $\widehat{\mathcal F}(a',\mathrm{id}) \cdot g' = g$ and $\widehat{\mathcal F}f' \circ g' = \varphi.$ We use, for both equations, Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.4. For the first one, precompose with $c_{{\mathcal F}b}$ and $\gamma_{{\mathcal F}b} \colon$ $$c_{{\mathcal F}b} \cdot g = c_{{\mathcal F}[f_0,b]} \cdot g' = c_{{\mathcal F}b} \cdot \widehat{\mathcal F}(a',\mathrm{id}) \cdot g'$$ $$\gamma_{{\mathcal F}b} \circ g = {\mathcal F}a' \circ \bar\varphi = {\mathcal F}a' \circ \gamma_{{\mathcal F}[f_0,b]} \circ g' = \gamma_{{\mathcal F}b} \circ \widehat{\mathcal F}(a',\mathrm{id}) \cdot g'$$ For the second one, precompose with $c_{{\mathcal F}a} \colon$ $$c_{{\mathcal F}a} \circ \varphi = {\mathcal F}f' \circ \bar\varphi = {\mathcal F}f' \circ \gamma_{{\mathcal F}[f_0,b]} \circ g' = c_{{\mathcal F}a} \circ \widehat{\mathcal F}f' \circ g'$$ It remains to prove that the factorization $g'$ is unique. For this, assume that there is an arrow $\bar g \colon \widehat{\mathcal F}[f_0,b] \to C$ such that $\widehat{\mathcal F}(a',\mathrm{id}) \cdot \bar g = g$ and $\widehat{\mathcal F}f' \circ \bar g = \varphi.$ To prove that $\bar g = g'$ we have to prove that $c_{{\mathcal F}[f_0,b]} \cdot \bar g = c_{{\mathcal F}b} \cdot g$ and $\gamma_{{\mathcal F}[f_0,b]} \circ \bar g = \bar\varphi.$ The verification of the first equation is direct: $$c_{{\mathcal F}b} \cdot g = c_{{\mathcal F}b} \cdot \widehat{\mathcal F}(a',\mathrm{id}) \cdot \bar g = c_{{\mathcal F}[f_0,b]} \cdot \bar g$$ For the second equation, we go back to the conditions which define $\bar\varphi \colon$ $${\mathcal F}f' \circ \gamma_{{\mathcal F}[f_0,b]} \circ \bar g = c_{{\mathcal F}a} \circ \widehat{\mathcal F}f' \circ \bar g = c_{{\mathcal F}a} \circ \varphi$$ $${\mathcal F}a' \circ \gamma_{{\mathcal F}[f_0,b]} \circ \bar g = \gamma_{{\mathcal F}b} \circ \widehat{\mathcal F}(a',\mathrm{id}) \cdot \bar g = \gamma_{{\mathcal F}b} \circ g$$ 4) The image by $\widehat{\mathcal F}$ of a $\Theta_{\Delta}$-cokernel is a strong $\Theta$-cokernel: consider once again a $\Theta_{\Delta}$-cokernel in $\mathbf{Arr}({\mathcal A})$ and its image by $\widehat{\mathcal F}$ as at the beginning of point 3) of the proof. Consider also a nullhomotopy in ${\mathcal B}$ $$\xymatrix{ \widehat{\mathcal F}(B,b,B_0) \ar[rr]_{\widehat{\mathcal F}(a',\mathrm{id})} \ar@/^1.8pc/@{-->}[rrrr]^{0}_{\varphi \; \Downarrow} & & \widehat{\mathcal F}[f_0,b] \ar[rr]_-{g} & & C }$$ and assume that $\varphi$ is compatible with $\widehat{\mathcal F}f',$ that is, $\widehat{\mathcal F}f' \circ g = \widehat{\mathcal F}(f,f_0) \circ \varphi.$ We get a new nullhomotopy $$c_{{\mathcal F}b} \circ \varphi \in \Theta(c_{{\mathcal F}b} \cdot \widehat{\mathcal F}(a',\mathrm{id}) \cdot g) = \Theta(c_{{\mathcal F}[f_0,b]} \cdot g)$$ In order to prove that $c_{{\mathcal F}b} \circ \varphi$ is compatible with $\gamma_{{\mathcal F}[f_0,b]},$ that is, ${\mathcal F}[f_0,b] \cdot c_{{\mathcal F}b} \circ \varphi = \gamma_{{\mathcal F}[f_0,b]} \circ g,$ we use Remark [Remark 16](#RemColimNull){reference-type="ref" reference="RemColimNull"}.3 once again, because ${\mathcal F}$ sends pushouts to $\Theta$-strong pushouts: $${\mathcal F}f' \cdot {\mathcal F}[f_0,b] \cdot c_{{\mathcal F}b} \circ \varphi = {\mathcal F}f_0 \cdot c_{{\mathcal F}b} \circ \varphi = c_{{\mathcal F}a} \cdot \widehat{\mathcal F}(f,f_0) \circ \varphi = c_{{\mathcal F}a} \circ \widehat{\mathcal F}f' \circ g = {\mathcal F}f' \circ \gamma_{{\mathcal F}[f_0,b]} \circ g$$ $${\mathcal F}a' \cdot {\mathcal F}[f_0,b] \cdot c_{{\mathcal F}b} \circ \varphi = {\mathcal F}b \cdot c_{{\mathcal F}b} \circ \varphi = \gamma_{{\mathcal F}b} \circ \widehat{\mathcal F}(a',\mathrm{id}) \cdot g = {\mathcal F}a' \circ \gamma_{{\mathcal F}[f_0,b]} \circ g$$ Now, the universal property of the $\Theta$-cokernel $\widehat{\mathcal F}[f_0,b]$ gives a unique nullhomotopy $\varphi' \in \Theta(g)$ such that $c_{{\mathcal F}[f_0,b]} \circ \varphi' = c_{{\mathcal F}b} \circ \varphi.$ We still have to check that $\varphi'$ is the required factorization, that is, $\widehat{\mathcal F}(a',\mathrm{id}) \circ \varphi' = \varphi.$ Thanks to Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.4, it is enough to precompose with $c_{{\mathcal F}b} \colon$ $$c_{{\mathcal F}b} \cdot \widehat{\mathcal F}(a',\mathrm{id}) \circ \varphi' = c_{{\mathcal F}[f_0,b]} \circ \varphi' = c_{{\mathcal F}b} \circ \varphi$$ It remains to prove that the factorization $\varphi'$ is unique. For this, assume that there is a nullhomotopy $\bar\varphi \in \Theta(g)$ such that $\widehat{\mathcal F}(a',\mathrm{id}) \circ \bar\varphi = \varphi.$ To prove that $\bar\varphi = \varphi'$ we go back to the condition which defines $\varphi' \colon$ $$c_{{\mathcal F}[f_0,b]} \circ \bar\varphi = c_{{\mathcal F}b} \cdot \widehat{\mathcal F}(a',\mathrm{id}) \circ \bar\varphi = c_{{\mathcal F}b} \circ \varphi$$ 5) $\widehat{\mathcal F}$ extends ${\mathcal F}$ along $\Gamma \colon$ by applying $\Gamma$ to an arrow $f \colon X \to Y$ in ${\mathcal A},$ we get $$\xymatrix{\emptyset \ar[r]^{\mathrm{id}} \ar[d]_{\emptyset_X} & \emptyset \ar[d]^{\emptyset_Y} \\ X \ar[r]_{f} & Y}$$ and we have to compare the two diagrams hereunder, the first one giving the image of $\Gamma f$ by $\widehat{\mathcal F}.$ If we can prove that the second one satisfies the conditions defining the first one, we can conclude that $\Gamma \cdot \widehat{\mathcal F}= {\mathcal F}.$ $$\xymatrix{ & {\mathcal F}\emptyset \ar[rr]^{\mathrm{id}} \ar[d]^{{\mathcal F}\emptyset_X} \ar@{-->}@/_3.2pc/[dd]_{0} & & {\mathcal F}\emptyset \ar[d]_{{\mathcal F}\emptyset_Y} \ar@{-->}@/^3.2pc/[dd]^{0} \\ \ar@{}[r]^{\gamma_{{\mathcal F}\emptyset_X}}|{\Longrightarrow} & {\mathcal F}X \ar[rr]^{{\mathcal F}f} \ar[d]^{c_{{\mathcal F}\emptyset_X}} & & {\mathcal F}Y \ar[d]_{c_{{\mathcal F}\emptyset_Y}}\ar@{}[r]^{\gamma_{{\mathcal F}\emptyset_Y}}|{\Longleftarrow} & \\ & \widehat{\mathcal F}(\Gamma X) \ar[rr]_{\widehat{\mathcal F}(\Gamma f)} & & \widehat{\mathcal F}(\Gamma Y)} \;\;\;\;\; \xymatrix{ & \emptyset \ar[rr]^{\mathrm{id}} \ar[d]^{\emptyset_{{\mathcal F}X}} \ar@{-->}@/_3.0pc/[dd]_{0} & & \emptyset \ar[d]_{\emptyset_{{\mathcal F}Y}} \ar@{-->}@/^3.0pc/[dd]^{0} \\ \ar@{}[r]^{\gamma_{{\mathcal F}X}}|{\Longrightarrow} & {\mathcal F}X \ar[rr]^{{\mathcal F}f} \ar[d]^{\mathrm{id}} & & {\mathcal F}Y \ar[d]_{\mathrm{id}}\ar@{}[r]^{\gamma_{{\mathcal F}Y}}|{\Longleftarrow} & \\ & {\mathcal F}X \ar[rr]_{{\mathcal F}f} & & {\mathcal F}Y}$$ Since, by assumption, ${\mathcal F}$ sends the initial object of ${\mathcal A}$ into a $\Theta$-strong initial object in ${\mathcal B},$ we can use Example [Example 17](#ExTrivCof){reference-type="ref" reference="ExTrivCof"} and the columns of the second diagram are $\Theta$-cokernels. The equation $\mathrm{id}\cdot {\mathcal F}f = {\mathcal F}f \cdot \mathrm{id}$ is trivial. Finally, the equation $\gamma_{{\mathcal F}X} \circ {\mathcal F}f = \mathrm{id}\circ \gamma_{{\mathcal F}Y}$ follows once again from the fact that the initial object in ${\mathcal B}$ is $\Theta$-strong.\ 6) $\widehat{\mathcal F}$ preserves finite colimits: the preservation of the initial object follows from $\Gamma \cdot \widehat{\mathcal F}\simeq {\mathcal F}$ because both $\Gamma$ and ${\mathcal F}$ preserve the initial. Consider now a pushout in $\mathbf{Arr}({\mathcal A})$ (see the proof of Proposition [Proposition 18](#PropEnrLimArr){reference-type="ref" reference="PropEnrLimArr"}) and its image by $\widehat{\mathcal F}$ (I have omitted from the picture the structural nullhomotopies of the four columns, which are $\Theta$-cokernels): $$\xymatrix{& {\mathcal F}A \ar[rr]^{{\mathcal F}g} \ar[dd]^<<<<<<{{\mathcal F}a} \ar[ld]_{{\mathcal F}f} & & {\mathcal F}C \ar[ld]_{{\mathcal F}f'} \ar[dd]^{{\mathcal F}c} \\ {\mathcal F}B \ar[rr]^>>>>>>>>>>>>>>>>>>{{\mathcal F}g'} \ar[dd]_{{\mathcal F}b} & & {\mathcal F}(B+_{f,g}C) \ar[dd]^<<<<<<{{\mathcal F}(b+c)} \\ & {\mathcal F}A_0 \ar[rr]^<<<<<<<<<<<<<<<<<<<<{{\mathcal F}g_0} \ar[ld]_{{\mathcal F}f_0} \ar[dd]_<<<<<<<{c_{{\mathcal F}a}} & & {\mathcal F}C_0 \ar[ld]^{{\mathcal F}f_0'} \ar[dd]^{c_{{\mathcal F}c}} \\ {\mathcal F}B_0 \ar[rr]^>>>>>>>>>>>>>>>>{{\mathcal F}g_0'} \ar[dd]_{c_{{\mathcal F}b}} & & {\mathcal F}(B_0+_{f_0,g_0}C_0) \ar[dd]^<<<<<<<{c_{{\mathcal F}(b+c)}} \\ & \widehat{\mathcal F}(A,a,A_0) \ar[rr]^<<<<<<<<<<<<<<<<{\widehat{\mathcal F}(g,g_0)} \ar[ld]_{\widehat{\mathcal F}(f,f_0)} & & \widehat{\mathcal F}(C,c,C_0) \ar[ld]^<<<<<<<<{\widehat{\mathcal F}(f',f_0')} \\ \widehat{\mathcal F}(B,b,B_0) \ar[rr]_-{\widehat{\mathcal F}(g',g_0')} & & \widehat{\mathcal F}(B+_{f,g}C,b+c,B_0+_{f_0,g_0}C_0)}$$ We have to prove that the ground floor is a pushout in ${\mathcal B}$ and we know, by assumption on ${\mathcal F},$ that the first and the second floor are $\Theta$-strong pushouts. For this, consider two arrows $$h \colon \widehat{\mathcal F}(B,b,B_0) \to X \leftarrow \widehat{\mathcal F}(C,c,C_0) \colon k$$ such that $\widehat{\mathcal F}(f,f_0) \cdot h = \widehat{\mathcal F}(g,g_0) \cdot k.$ Therefore $${\mathcal F}f_0 \cdot c_{{\mathcal F}b} \cdot h = c_{{\mathcal F}a} \cdot \widehat{\mathcal F}(f,f_0) \cdot h = c_{{\mathcal F}a} \cdot \widehat{\mathcal F}(g,g_0) \cdot k = {\mathcal F}g_0 \cdot c_{{\mathcal F}c} \cdot k$$ so that there exists a unique arrow $x \colon {\mathcal F}(B_0+_{f_0,g_0}C_0) \to X$ such that ${\mathcal F}g_0' \cdot x = c_{{\mathcal F}b} \cdot h$ and ${\mathcal F}f_0' \cdot x = c_{{\mathcal F}c} \cdot k.$ We can now costrcut two nullhomotopies $$\gamma_{{\mathcal F}b} \circ h \in \Theta({\mathcal F}b \cdot c_{{\mathcal F}b} \cdot h) = \Theta({\mathcal F}b \cdot {\mathcal F}g_0' \cdot x) = \Theta({\mathcal F}g' \cdot {\mathcal F}(b+c) \cdot x)$$ $$\gamma_{{\mathcal F}c} \circ k \in \Theta({\mathcal F}c \cdot c_{{\mathcal F}c} \cdot k) = \Theta({\mathcal F}c \cdot {\mathcal F}f_0' \cdot x) = \Theta({\mathcal F}f' \cdot {\mathcal F}(b+c) \cdot x)$$ which are compatible, indeed $${\mathcal F}f \circ \gamma_{{\mathcal F}b} \circ h = \gamma_{{\mathcal F}a} \circ \widehat{\mathcal F}(f,f_0) \cdot h = \gamma_{{\mathcal F}a} \circ \widehat{\mathcal F}(g,g_0) \cdot k = {\mathcal F}g \circ \gamma_{{\mathcal F}c} \circ k$$ Since the pushout ${\mathcal F}(B+_{f,g}C)$ is $\Theta$-strong, we get a unique nullhomotopy $\psi \in \Theta({\mathcal F}(b+c) \cdot x)$ such that ${\mathcal F}g' \circ \psi = \gamma_{{\mathcal F}b} \circ h$ and ${\mathcal F}f' \circ \psi = \gamma_{{\mathcal F}c} \circ k.$ By the universal property of the $\Theta$-cokernel, the nullhomotopy $\psi$ produces a unique arrow $$x' \colon \widehat{\mathcal F}(B+_{f,g}C,b+c,B_0+_{f_0,g_0}C_0) \to X$$ such that $c_{{\mathcal F}(b+c)} \cdot x' = x$ and $\gamma_{{\mathcal F}(b+c)} \circ x' = \psi.$ We have to prove that $x'$ is the required factorization, that is, $\widehat{\mathcal F}(g',g_0') \cdot x' = h$ and $\widehat{\mathcal F}(f',f_0') \cdot x' = k.$ We check the first condition (the second one is similar) using Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.4: $$c_{{\mathcal F}b} \cdot \widehat{\mathcal F}(g',g_0') \cdot x' = {\mathcal F}g_0' \cdot c_{{\mathcal F}(b+c)} \cdot x' = {\mathcal F}g_0' \cdot x = c_{{\mathcal F}b} \cdot h$$ $$\gamma_{{\mathcal F}b} \circ \widehat{\mathcal F}(g',g_0') \cdot x' = {\mathcal F}g' \circ \gamma_{{\mathcal F}(b+c)} \circ x' = {\mathcal F}g' \circ \psi = \gamma_{{\mathcal F}b} \circ h$$ It remains to prove that the factorization $x'$ is unique. For this, let $$\bar x \colon \widehat{\mathcal F}(B+_{f,g}C,b+c,B_0+_{f_0,g_0}C_0) \to X$$ be an arrow such that $\widehat{\mathcal F}(g',g_0') \cdot \bar x = h$ and $\widehat{\mathcal F}(f',f_0') \cdot \bar x = k.$ In order to prove that $\bar x = x',$ we have to prove that $c_{{\mathcal F}(b+c)} \cdot \bar x = x$ and $\gamma_{{\mathcal F}(b+c)} \circ \bar x = \psi.$ For the first equation, we check the conditions which define $x \colon$ $${\mathcal F}g_0' \cdot c_{{\mathcal F}(b+c)} \cdot \bar x = c_{{\mathcal F}b} \cdot \widehat{\mathcal F}(g',g_0') \cdot \bar x = c_{{\mathcal F}b} \cdot h$$ $${\mathcal F}f_0' \cdot c_{{\mathcal F}(b+c)} \cdot \bar x = c_{{\mathcal F}c} \cdot \widehat{\mathcal F}(f',f_0') \cdot \bar x = c_{{\mathcal F}c} \cdot k$$ For the second equation, we check the conditions which define $\psi \colon$ $${\mathcal F}g' \circ \gamma_{{\mathcal F}(b+c)} \circ \bar x = \gamma_{{\mathcal F}b} \circ \widehat{\mathcal F}(g',g_0') \cdot \bar x = \gamma_{{\mathcal F}b} \circ h$$ $${\mathcal F}f' \circ \gamma_{{\mathcal F}(b+c)} \circ \bar x = \gamma_{{\mathcal F}c} \circ \widehat{\mathcal F}(f',f_0') \cdot \bar x = \gamma_{{\mathcal F}c} \circ k$$ 7) The image by $\widehat{\mathcal F}$ of finite colimits are $\Theta$-strong finite colimits: the case of the initial object is clear, so we pass to pushouts. We keep the same notations as in point 6). We have to prove that the pushout in ${\mathcal B}$ $$\xymatrix{\widehat{\mathcal F}(A,a,A_0) \ar[rrr]^{\widehat{\mathcal F}(g,g_0)} \ar[d]_{\widehat{\mathcal F}(f,f_0)} & & & \widehat{\mathcal F}(C,c,C_0) \ar[d]^{\widehat{\mathcal F}(f',f_0')} \\ \widehat{\mathcal F}(B,b,B_0) \ar[rrr]_-{\widehat{\mathcal F}(g',g_0')} & & & \widehat{\mathcal F}(B+_{f,g}C,b+c,B_0+_{f_0,g_0}C_0)}$$ is $\Theta$-strong. For this, consider two nullhomotopies $\alpha \in \Theta(h)$ and $\beta \in \Theta(k)$ such that $\widehat{\mathcal F}(f,f_0) \circ \alpha = \widehat{\mathcal F}(g,g_0) \circ \beta.$ It follows that $${\mathcal F}f_0 \cdot c_{{\mathcal F}b} \circ \alpha = c_{{\mathcal F}a} \widehat{\mathcal F}(f,f_0) \circ \alpha = c_{{\mathcal F}a} \cdot \widehat{\mathcal F}(g,g_0) \circ \beta = {\mathcal F}g_0 \cdot c_{{\mathcal F}c} \circ \beta$$ Since the pushout ${\mathcal F}(B_0+_{f_0,g_0}C_0)$ is $\Theta$-strong, we get a unique nullhomotopy $[\alpha,\beta] \in \Theta(x)$ such that ${\mathcal F}g_0' \circ [\alpha,\beta] = c_{{\mathcal F}b} \circ \alpha$ and ${\mathcal F}f_0' \circ [\alpha,\beta] = c_{{\mathcal F}c} \circ \beta.$ Let us check that $\gamma_{{\mathcal F}(b+c)} \circ x' = {\mathcal F}(b+c) \circ [\alpha,\beta] \colon$ since the pushout ${\mathcal F}(B+_{f,g}C)$ is $\Theta$-strong , we can use Remark [Remark 16](#RemColimNull){reference-type="ref" reference="RemColimNull"}.3 and precompose with ${\mathcal F}g'$ and ${\mathcal F}f' \colon$ $${\mathcal F}g' \circ \gamma_{{\mathcal F}(b+c)} \circ x' = {\mathcal F}g' \circ \psi = \gamma_{{\mathcal F}b} \circ h = {\mathcal F}b \cdot c_{{\mathcal F}b} \circ \alpha = {\mathcal F}b \cdot {\mathcal F}g_0' \circ [\alpha,\beta] = {\mathcal F}g' \cdot {\mathcal F}(b+c) \circ [\alpha,\beta]$$ $${\mathcal F}f' \circ \gamma_{{\mathcal F}(b+c)} \circ x' = {\mathcal F}f' \circ \psi = \gamma_{{\mathcal F}c} \circ k = {\mathcal F}c \cdot c_{{\mathcal F}c} \circ \beta = {\mathcal F}c \cdot {\mathcal F}f_0' \circ [\alpha,\beta] = {\mathcal F}f' \cdot {\mathcal F}(b+c) \circ [\alpha,\beta]$$ By the universal property of the $\Theta$-cokernel $\widehat{\mathcal F}(B+_{f,g}C,b+c,B_0+_{f_0,g_0}C_0),$ we get a unique nullhomotopy $\overline{[\alpha,\beta]} \in \Theta(x')$ such that $c_{{\mathcal F}(b+c)} \circ \overline{[\alpha,\beta]} = [\alpha,\beta].$ We have to verify that $\overline{[\alpha,\beta]}$ is the required extension, that is, $\widehat{\mathcal F}(g',g_0') \circ \overline{[\alpha,\beta]} = \alpha$ and $\widehat{\mathcal F}(f',f_0') \circ \overline{[\alpha,\beta]} = \beta.$ We check the first condition (the second one is similar) using Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.4: $$c_{{\mathcal F}b} \cdot \widehat{\mathcal F}(g',g_0') \circ \overline{[\alpha,\beta]} = {\mathcal F}g_0' \cdot c_{{\mathcal F}(b+c)} \circ \overline{[\alpha,\beta]} = {\mathcal F}g_0' \circ [\alpha,\beta] = c_{{\mathcal F}b} \circ \alpha$$ It remains to prove that the extension $\overline{[\alpha,\beta]}$ is unique. For this, let $\psi \in \Theta(x')$ be a nullhomotopy such that $\widehat{\mathcal F}(g',g_0') \circ \psi = \alpha$ and $\widehat{\mathcal F}(f',f_0') \circ \psi = \beta.$ To show that $\psi = \overline{[\alpha,\beta]}$ it suffices to show that $c_{{\mathcal F}(b+c)} \circ \psi = [\alpha,\beta].$ For this, we apply once again Remark [Remark 16](#RemColimNull){reference-type="ref" reference="RemColimNull"}.3 to the $\Theta$-strong pushout ${\mathcal F}(B_0+_{f_0;g_0}C_0) \colon$ $${\mathcal F}g_0' \cdot c_{{\mathcal F}(b+c)} \circ \psi = c_{{\mathcal F}b} \cdot \widehat{\mathcal F}(g',g_0') \circ \psi = c_{{\mathcal F}b} \circ \alpha = {\mathcal F}g_0' \circ [\alpha,\beta]$$ $${\mathcal F}f_0' \cdot c_{{\mathcal F}(b+c)} \circ \psi = c_{{\mathcal F}c} \cdot \widehat{\mathcal F}(f',f_0') \circ \psi = c_{{\mathcal F}c} \circ \beta = {\mathcal F}f_0' \circ [\alpha,\beta]$$ The proof is now complete. ◻ ** 20**. *We restate now Proposition [Proposition 19](#PropExt){reference-type="ref" reference="PropExt"} in terms of an equivalence between hom-categories. Consider a category ${\mathcal A}$ with finite colimits and a category with nullhomotopies $({\mathcal B},\Theta)$ satisfying the reduced interchange. Assume that ${\mathcal B}$ has $\Theta$-strong finite colimits and strong $\Theta$-cokernels. We are going to establish an equivalence between the following categories:* 1. *$\mathrm{Colim}[{\mathcal A},{\mathcal B}] \colon$ objects are functors preserving finite colimits, arrows are natural transformations,* 2. *$\mathrm{HoCok}[\mathbf{Arr}({\mathcal A}),{\mathcal B}] \colon$ objects are those morphisms $(\mathbf{Arr}({\mathcal A}),\Theta_{\Delta}) \to ({\mathcal B},\Theta)$ of Definition [Definition 4](#DefFunctNullHom){reference-type="ref" reference="DefFunctNullHom"}.1 which preserve finite colimits and homotopy cokernels, arrows are the 2-morphisms of Definition [Definition 4](#DefFunctNullHom){reference-type="ref" reference="DefFunctNullHom"}.2.* **Proposition 21**. *Under the assumptions and with the notation of [ 20](#TextPropEquiv){reference-type="ref" reference="TextPropEquiv"}, there is an equivalence of categories $$\xymatrix{\mathrm{HoCok}[\mathbf{Arr}({\mathcal A}),{\mathcal B}] \ar@<0.5ex>[rr]^-{\Gamma \cdot (-)} & & \mathrm{Colim}[{\mathcal A},{\mathcal B}] \ar@<0.5ex>[ll]^-{\widehat{(-)}}}$$* *Proof.* We are going to prove that the functors $\Gamma \cdot (-)$ and $\widehat{(-)}$ are one the quasi-inverse of the other.\ 1) Definition of $\Gamma \cdot (-) \colon$ by Proposition [Proposition 18](#PropEnrLimArr){reference-type="ref" reference="PropEnrLimArr"}, $\Gamma \colon {\mathcal A}\to \mathbf{Arr}({\mathcal A})$ preserves finite colimits, so that $\Gamma \cdot (-)$ is well-defined on objects. Its definition on arrows is obvious.\ 2) Definition of $\widehat{(-)} \colon$ from Proposition [Proposition 19](#PropExt){reference-type="ref" reference="PropExt"}, we already know how $\widehat{(-)}$ is defined on objects. As far as arrows are concerned, consider a natural trasformation $\lambda \colon {\mathcal F}\Rightarrow {\mathcal G}$ in $\mathrm{Colim}[{\mathcal A},{\mathcal B}]$ and a nullhomotopy $$\xymatrix{A \ar[r]^{f} \ar[d]_{a} & B \ar[d]^{b} \\ A_0 \ar[ru]^{\varphi} \ar[r]_{f_0} & B_0}$$ in $\mathbf{Arr}({\mathcal A}).$ The next diagram describes the construction of $\widehat\lambda \colon \widehat{\mathcal F}\Rightarrow \widehat{\mathcal G}$ in $\mathrm{HoCok}[\mathbf{Arr}({\mathcal A}),{\mathcal B}] \colon$ $$\xymatrix{ & {\mathcal F}A \ar[rr]^{\lambda_A} \ar[d]^{{\mathcal F}a} \ar@{-->}@/_3.3pc/[dd]_{0} & & {\mathcal G}A \ar[d]_{{\mathcal G}a} \ar@{-->}@/^3.3pc/[dd]^{0} \\ \ar@{}[r]^{\gamma_{{\mathcal F}a}}|{\Longrightarrow} & {\mathcal F}A_0 \ar[rr]^{\lambda_{A_0}} \ar[d]^{c_{{\mathcal F}a}} & & {\mathcal G}A_0 \ar[d]_{c_{{\mathcal G}a}}\ar@{}[r]^{\gamma_{{\mathcal G}a}}|{\Longleftarrow} & \\ & \widehat{\mathcal F}(A,a,A_0) \ar[rr]_{\widehat\lambda_{(A,a,A_0)}} & & \widehat{\mathcal G}(A,a,A_0)}$$ In other words, $\widehat\lambda_{(A,a,A_0)}$ is the unique arrow such that $c_{{\mathcal F}a} \cdot \widehat\lambda_{(A,a,A_0)} = \lambda_{A_0} \cdot c_{{\mathcal G}a}$ and $\gamma_{{\mathcal F}a} \circ \widehat\lambda_{(A,a,A_0)} = \lambda_A \circ \gamma_{{\mathcal G}a}.$ To check the naturality of $\widehat\lambda,$ precompose with $c_{{\mathcal F}a}$ and $\gamma_{{\mathcal F}a} \colon$ $$c_{{\mathcal F}a} \cdot \widehat{\mathcal F}(f,f_0) \cdot \widehat\lambda_{(B,b,B_0)} = {\mathcal F}f_0 \cdot c_{{\mathcal F}b} \cdot \widehat\lambda_{(B,b,B_0)} = {\mathcal F}f_0 \cdot \lambda_{B_0} \cdot c_{{\mathcal G}b} =$$ $$= \lambda_{A_0} \cdot {\mathcal G}f_0 \cdot c_{{\mathcal G}b} = \lambda_{A_0} \cdot c_{{\mathcal G}a} \cdot \widehat{\mathcal G}(f,f_0) = c_{{\mathcal F}a} \cdot \widehat\lambda_{(A,a,A_0)} \cdot \widehat{\mathcal G}(f,f_0)$$ $$\gamma_{{\mathcal F}a} \circ \widehat{\mathcal F}(f,f_0) \cdot \widehat\lambda_{(B,b,B_0)} = {\mathcal F}f \circ \gamma_{{\mathcal F}b} \circ \widehat\lambda_{(B,b,B_0)} = {\mathcal F}f \cdot \lambda_{B_0} \circ \gamma_{{\mathcal G}b} =$$ $$= \lambda_{A} \cdot {\mathcal G}f \circ \gamma_{{\mathcal G}b} = \lambda_{A} \circ \gamma_{{\mathcal G}a} \circ \widehat{\mathcal G}(f,f_0) = \gamma_{{\mathcal F}a} \circ \widehat\lambda_{(A,a,A_0)} \cdot \widehat{\mathcal G}(f,f_0)$$ To check that $\widehat\lambda$ is compatible with nullhomotopies in the sense of Definition [Definition 4](#DefFunctNullHom){reference-type="ref" reference="DefFunctNullHom"}.2, that is, $\widehat\lambda_{(A,a,A_0)} \circ \widehat{\mathcal G}\varphi = \widehat{\mathcal F}\varphi \circ \widehat\lambda_{(B,b,B_0)},$ precompose with $c_{{\mathcal F}a} \colon$ $$c_{{\mathcal F}a} \cdot \widehat\lambda_{(A,a,A_0)} \circ \widehat{\mathcal G}\varphi = \lambda_{A_0} \cdot c_{{\mathcal G}a} \circ \widehat{\mathcal G}\varphi = \lambda_{A_0} \cdot {\mathcal G}\varphi \circ \gamma_{{\mathcal G}b} =$$ $$= {\mathcal F}\varphi \cdot \lambda_B \circ \gamma_{{\mathcal G}b} = {\mathcal F}\varphi \circ \gamma_{{\mathcal F}b} \circ \widehat\lambda_{(B,b,B_0)} = c_{{\mathcal F}a} \circ \widehat{\mathcal F}\varphi \circ \widehat\lambda_{(B,b,B_0)}$$ 3) Composition $\xymatrix{\mathrm{Colim}[{\mathcal A},{\mathcal B}] \ar[r]^-{\widehat{(-)}} & \mathrm{HoCok}[\mathbf{Arr}({\mathcal A}),{\mathcal B}] \ar[r]^-{\Gamma \cdot (-)} & \mathrm{Colim}[{\mathcal A},{\mathcal B}]} \colon$ from point 5) of the proof of Proposition [Proposition 19](#PropExt){reference-type="ref" reference="PropExt"}, we already know that $\Gamma \cdot \widehat{\mathcal F}= {\mathcal F}$ for any functor ${\mathcal F}\in \mathrm{Colim}[{\mathcal A},{\mathcal B}].$ Consider now a natural transformation $\lambda \colon {\mathcal F}\Rightarrow {\mathcal G}$ in $\mathrm{Colim}[{\mathcal A},{\mathcal B}].$ We have to prove that the restriction along $\Gamma$ of $\widehat\lambda$ is $\lambda.$ This is because, if we start with an object $X \in {\mathcal A},$ the definition of $\widehat\lambda_{\Gamma X}$ reduces to the following diagram (use Remark [Remark 16](#RemColimNull){reference-type="ref" reference="RemColimNull"}.1): $$\xymatrix{ & \emptyset \ar[rr]^{\mathrm{id}} \ar[d]^{\emptyset_{{\mathcal F}X}} \ar@{-->}@/_3.0pc/[dd]_{0} & & \emptyset \ar[d]_{\emptyset_{{\mathcal G}X}} \ar@{-->}@/^3.0pc/[dd]^{0} \\ \ar@{}[r]^{\gamma_{{\mathcal F}X}}|{\Longrightarrow} & {\mathcal F}X \ar[rr]^{\lambda_X} \ar[d]^{\mathrm{id}} & & {\mathcal G}X \ar[d]_{\mathrm{id}}\ar@{}[r]^{\gamma_{{\mathcal G}X}}|{\Longleftarrow} & \\ & {\mathcal F}X \ar[rr]_{\lambda_X} & & {\mathcal G}X}$$ 4) Composition $\xymatrix{\mathrm{HoCok}[\mathbf{Arr}({\mathcal A}),{\mathcal B}] \ar[r]^-{\Gamma \cdot (-)} & \mathrm{Colim}[{\mathcal A},{\mathcal B}] \ar[r]^-{\widehat{(-)}} & \mathrm{HoCok}[\mathbf{Arr}({\mathcal A}),{\mathcal B}]} \colon$ we start with the construction, for any functor ${\mathcal M}\in \mathrm{HoCok}[\mathbf{Arr}({\mathcal A}),{\mathcal B}],$ of an invertible 2-morphism $m \colon \widehat{\Gamma \cdot {\mathcal M}} \to {\mathcal M}.$ Its component at $(A,a,A_0) \in \mathbf{Arr}({\mathcal A})$ is depicted in the following diagram: $$\xymatrix{ & {\mathcal M}\Gamma A \ar[d]_{ {\mathcal M}\Gamma a} \ar@{-->}@/_2.0pc/[ldd]_{0} \ar@{-->}@/^2.0pc/[rdd]^{0} \\ \ar@{}[r]|{\Longrightarrow}^{\gamma_{ {\mathcal M}\Gamma a}} & {\mathcal M}\Gamma A_0 \ar[ld]^{c_{ {\mathcal M}\Gamma a}} \ar[rd]_{ {\mathcal M}(0_A,\mathrm{id}_{A_0})} & \ar@{}[l]|{\Longleftarrow}_{ {\mathcal M}(\mathrm{id}_A)} \\ \widehat{\Gamma \cdot {\mathcal M}}(A,a,A_0) \ar[rr]_{m_{(A,a,A_0)}} & & {\mathcal M}(A,a,A_0)}$$ The triangle on the left is a $\Theta$-cokernel by definition of $\widehat{\Gamma \cdot {\mathcal M}},$ the triangle on the right is a $\Theta$-cokernel by Remark [Remark 11](#RemCofCompl){reference-type="ref" reference="RemCofCompl"} and because ${\mathcal M}$ preserves $\Theta_{\Delta}$-cokernels. So, $m_{(A,a,A_0)}$ is the unique arrow such that $c_{ {\mathcal M}\Gamma a} \cdot m_{(A,a,A_0)} = {\mathcal M}(0_A,\mathrm{id}_{A_0})$ and $\gamma_{ {\mathcal M}\Gamma a} \cdot m_{(A,a,A_0)} = {\mathcal M}(\mathrm{id}_{A}).$ Moreover, $m_{(A,a,A_0)}$ is an isomorphism by Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.1. We have to prove that the family $$m = \{m_{(A,a,A_0)} \mid (A,a,A_0) \in \mathbf{Arr}({\mathcal A})\}$$ is a 2-morphism in the sense of Definition [Definition 4](#DefFunctNullHom){reference-type="ref" reference="DefFunctNullHom"}. For this, consider a nullhomotopy $$\xymatrix{A \ar[r]^{f} \ar[d]_{a} & B \ar[d]^{b} \\ A_0 \ar[ru]^{\varphi} \ar[r]_{f_0} & B_0}$$ in $\mathbf{Arr}({\mathcal A}).$ To check the naturality, precompose with $c_{ {\mathcal M}\Gamma a}$ and $\gamma_{ {\mathcal M}\Gamma a} \colon$ $$c_{ {\mathcal M}\Gamma a} \cdot m_{(A,a,A_0)} \cdot {\mathcal M}(f,f_0) = {\mathcal M}(0_A,\mathrm{id}_{A_0}) \cdot {\mathcal M}(f,f_0) = {\mathcal M}\Gamma f_0 \cdot {\mathcal M}(0_B,\mathrm{id}_{B_0}) =$$ $$= {\mathcal M}\Gamma f_0 \cdot c_{ {\mathcal M}\Gamma b} \cdot m_{(B,b,B_0)} = c_{ {\mathcal M}\Gamma a} \cdot \widehat{\Gamma \cdot {\mathcal M}}(f,f_0) \cdot m_{(B,b,B_0)}$$ $$\gamma_{ {\mathcal M}\Gamma a} \circ m_{(A,a,A_0)} \cdot {\mathcal M}(f,f_0) = {\mathcal M}(\mathrm{id}_A) \circ {\mathcal M}(f,f_0) = {\mathcal M}\Gamma f \circ {\mathcal M}(\mathrm{id}_B) =$$ $$= {\mathcal M}\Gamma f \circ \gamma_{ {\mathcal M}\Gamma b} \circ m_{(B,b,B_0)} = \gamma_{ {\mathcal M}\Gamma a} \circ \widehat{\Gamma \cdot {\mathcal M}}(f,f_0) \cdot m_{(B,b,B_0)}$$ To check the compatibility with nullhomotopies, precompose with $c_{ {\mathcal M}\Gamma a} \colon$ $$c_{ {\mathcal M}\Gamma a} \cdot m_{(A,a,A_0)} \circ {\mathcal M}(\lambda) = {\mathcal M}(0_A,\mathrm{id}_{A_0}) \circ {\mathcal M}(\lambda) = {\mathcal M}\Gamma \lambda \circ {\mathcal M}(\mathrm{id}_B) =$$ $$= {\mathcal M}\Gamma \lambda \circ \gamma_{ {\mathcal M}\Gamma b} \circ m_{(B,b,B_0)} = c_{ {\mathcal M}\Gamma a} \circ \widehat{\Gamma \cdot {\mathcal M}}(\lambda) \circ m_{(B,b,B_0)}$$ It remains to prove that, if $\mu \colon {\mathcal M}\Rightarrow {\mathcal N}$ is a 2-morphism in $\mathrm{HoCok}[\mathbf{Arr}({\mathcal A}),{\mathcal B}],$ then $$\xymatrix{\widehat{\Gamma \cdot {\mathcal M}} \ar[rr]^{m} \ar[d]_{\widehat{\Gamma \cdot \mu}} & & {\mathcal M}\ar[d]^{\mu} \\ \widehat{\Gamma \cdot {\mathcal N}} \ar[rr]_{n} & & {\mathcal N}}$$ commutes. This means that, for any object $(A,a,A_0) \in \mathbf{Arr}({\mathcal A}),$ we have to prove that $\widehat{\Gamma \cdot \mu}_{(A,a,A_0)} \cdot n_{(A,a,A_0)} = m_{(A,a,A_0)} \cdot \mu_{(A,a,A_0)}.$ By Remark [Remark 10](#RemNofCof){reference-type="ref" reference="RemNofCof"}.4, it suffices to chek this equation by precomposing with $c_{ {\mathcal M}\Gamma a}$ and $\gamma_{ {\mathcal M}\Gamma a} \colon$ $$c_{ {\mathcal M}\Gamma a} \cdot \widehat{\Gamma \cdot \mu}_{(A,a,A_0)} \cdot n_{(A,a,A_0)} = \mu_{\Gamma A_0} \cdot c_{ {\mathcal N}\Gamma a} \cdot n_{(A,a,A_0)} =$$ $$= \mu_{\Gamma A_0} \cdot {\mathcal N}(0_A,\mathrm{id}_{A_0}) = {\mathcal M}(0_A,\mathrm{id}_{A_0}) \cdot \mu_{(A,a,A_0)} = c_{ {\mathcal M}\Gamma a} \cdot m_{(A,a,A_0)} \cdot \mu_{(A,a,A_0)}$$ $$\gamma_{ {\mathcal M}\Gamma a} \circ \widehat{\Gamma \cdot \mu}_{(A,a,A_0)} \cdot n_{(A,a,A_0)} = \mu_{\Gamma A} \circ \gamma_{ {\mathcal N}\Gamma a} \circ n_{(A,a,A_0)} =$$ $$= \mu_{\Gamma A} \circ {\mathcal N}(\mathrm{id}_A) = {\mathcal M}(\mathrm{id}_A) \circ \mu_{(A,a,A_0)} = \gamma_{ {\mathcal M}\Gamma a} \circ m_{(A,a,A_0)} \cdot \mu_{(A,a,A_0)}$$ The proof is now complete. ◻ ** 22**. * To end this section, let us point out that the assumptions on $({\mathcal B},\Theta)$ appearing in [ 20](#TextPropEquiv){reference-type="ref" reference="TextPropEquiv"} are not independent. Indeed, we know from [@MMMV] that, if ${\mathcal B}$ has strong $\Theta$-cokernels of identity arrows and $\Theta$-strong pushouts, then it has all the $\Theta$-cokernels and they are strong. Moreover, in the fundamental case where the structure $\Theta$ is induced by a string of adjunction $$\xymatrix{ {\mathcal A}\ar[rr]|-{{\mathcal U}} & & {\mathcal B}\ar@<-1.5ex>[ll]_-{} \ar@<1.5ex>[ll]^-{} }$$ with ${\mathcal U}$ full and faithful and if ${\mathcal B}$ has pushouts, then pushouts are $\Theta$-strong and ${\mathcal B}$ has strong $\Theta$-cokernels. * # The denormalization functor {#SecDenorm} ** 23**. * This short final section is completely devoted to illustrate, on a simple but relevant example, the extension $\widehat{\mathcal F}$ of a functor ${\mathcal F}\colon {\mathcal A}\to {\mathcal B}$ appearing in Proposition [Proposition 19](#PropExt){reference-type="ref" reference="PropExt"}, as well as the dual construction. As far as the dual constriuction is concerned, if we start assuming that ${\mathcal A}$ has finite limits and we write $\ast$ for the terminal object and $\ast^B \colon B \to \ast$ for the unique arrow, the corresponding nullhomotopy structure on ${\mathcal A}$ is $\Theta_{\ast}(g) = \{ \varphi \colon \ast \to C \mid \ast^B \cdot \varphi = g \},$ the embedding $\Lambda \colon ({\mathcal A},\Theta_{\ast}) \to (\mathbf{Arr}({\mathcal A}),\Theta_{\Delta})$ is defined by $$\xymatrix{B \ar[r]^{g} \ar[d] & C \\ \ast \ar[ru]_{\varphi}} \;\; \mapsto \;\; \xymatrix{B \ar[r]^{g} \ar[d] & C \ar[d] \\ \ast \ar[r] \ar[ru]_{\varphi} & \ast}$$ and the extension along $\Lambda$ of a functor ${\mathcal F}\colon {\mathcal A}\to {\mathcal B}$ is denoted by $\widetilde{\mathcal F}\colon \mathbf{Arr}({\mathcal A}) \to {\mathcal B}.$ * ** 24**. * Starting from any category ${\mathcal A},$ we can construct the category $\mathbf{RG}({\mathcal A})$ of reflexive graphs in ${\mathcal A}.$ Objects and arrows are depicted in the following diagram $$\xymatrix{A_1 \ar@<-0.5ex>[d]_{d} \ar@<0.5ex>[d]^{c} \ar[rr]^{f_1} & & B_1 \ar@<-0.5ex>[d]_{d} \ar@<0.5ex>[d]^{c} \\ A_0 \ar[rr]_{f_0} \ar@/^1.5pc/[u]^{i} & & B_0 \ar@/^1.5pc/[u]^{i}}$$ with the conditions $i \cdot d = \mathrm{id}= i \cdot c, \; i \cdot f_1 = f_0 \cdot i, \; d \cdot f_0 = f_1 \cdot d, \; c \cdot f_0 = f_1 \cdot c.$\ If we assume that the category ${\mathcal A}$ has a zero object and kernels, we can construct the so-called normalization functor ${\mathcal K}\colon \mathbf{RG}({\mathcal A}) \to \mathbf{Arr}({\mathcal A})$ defined by $$\xymatrix{A_1 \ar@<-0.5ex>[d]_{d} \ar@<0.5ex>[d]^{c} \ar[rr]^{f_1} & & B_1 \ar@<-0.5ex>[d]_{d} \ar@<0.5ex>[d]^{c} \\ A_0 \ar[rr]_{f_0} \ar@/^1.5pc/[u]^{i} & & B_0 \ar@/^1.5pc/[u]^{i}} \;\;\;\;\;\; \mapsto \;\;\;\;\;\; \xymatrix{\mathrm{Ker}(d) \ar[r]^{K(f_1)} \ar[d]_{k_d} & \mathrm{Ker}(d) \ar[d]^{k_d} \\ A_1 \ar@{.>}[r]^{f_1} \ar[d]_{c} & B_1 \ar[d]^{c} \\ A_0 \ar[r]_{f_0} & B_0}$$ where $K(f_1)$ is the unique arrow such that $K(f_1) \cdot k_d = k_d \cdot f_1.$ A structure of nullhomotopies $\Theta$ on $\mathbf{RG}({\mathcal A})$ can be chosen in such a way that ${\mathcal K}$ is a morphism of categories with nullhomotopies and it is bijective on nullhomotopies. Explicitly, a nullhomotopy on an arrow $(f_1,f_0)$ is an arrow $\varphi \colon A_0 \to B_1$ such that $\varphi \cdot d = 0, \; \varphi \cdot c = f_0, \; k_d \cdot c \cdot \varphi = K(f_1) \cdot k_d.$ * ** 25**. *Now we construct two functors from ${\mathcal A}$ to $\mathbf{RG}({\mathcal A}).$ The first one needs no assumption on ${\mathcal A}.$ For the second one, the existence of a zero object 0 is needed. Here they are:* 1. *$\Gamma' \colon {\mathcal A}\to \mathbf{RG}({\mathcal A}) \;\;\;\;\;\;\;\;\; \Gamma'(\xymatrix{ B_0 \ar[r]^{g_0} & C_0} ) = \xymatrix{& B_0 \ar@<-0.5ex>[d]_{\mathrm{id}} \ar@<0.5ex>[d]^{\mathrm{id}} \ar[rr]^{g_0} & & C_0 \ar@<-0.5ex>[d]_{\mathrm{id}} \ar@<0.5ex>[d]^{\mathrm{id}} \\ & B_0 \ar[rr]_{g_0} \ar@/^1.5pc/[u]^{\mathrm{id}} & & C_0 \ar@/^1.5pc/[u]^{\mathrm{id}}}$* 2. *$\Lambda' \colon {\mathcal A}\to \mathbf{RG}({\mathcal A}) \;\;\;\;\;\;\;\;\; \Lambda'(\xymatrix{B \ar[r]^{g} & C}) = \xymatrix{& B \ar@<-0.5ex>[d] \ar@<0.5ex>[d] \ar[rr]^{g} & & C \ar@<-0.5ex>[d] \ar@<0.5ex>[d] \\ & 0 \ar[rr] \ar@/^1.5pc/[u] & & 0 \ar@/^1.5pc/[u]}$* ** 26**. * Assume now that ${\mathcal A}$ is additive. The main point is to observe that the images by $\Gamma'$ and $\Lambda'$ of any arrow $a \colon A \to A_0$ of ${\mathcal A}$ have, respectively, a $\Theta$-cokernel and a $\Theta$-kernel in $\mathbf{RG}({\mathcal A}).$ Moreover, the $\Theta$-cokernel of $\Gamma'(a)$ coincide with the $\Theta$-kernel of $\Lambda'(a).$ All this is depicted in the following diagram, where the dotted arrows are the structural nullhomotopies of the $\Theta$-cokernel (the one on the left) and of the $\Theta$-kernel (the one on the right): $$\resizebox{\displaywidth}{!}{ \xymatrix{& A \ar@<-0.5ex>[d]_{\mathrm{id}} \ar@<0.5ex>[d]^{\mathrm{id}} \ar[rr]^{a} & & A_0 \ar@<-0.5ex>[d]_{\mathrm{id}} \ar@<0.5ex>[d]^{\mathrm{id}} \ar[rr]^{i_1} & & A_0 \oplus A \ar@<-0.5ex>[d]_{\pi_1} \ar@<0.5ex>[d]^{[\mathrm{id};a]} \ar[rr]^{\pi_2} & & A \ar@<-0.5ex>[d] \ar@<0.5ex>[d] \ar[rr]^{a} & & A_0 \ar@<-0.5ex>[d] \ar@<0.5ex>[d] \\ & A \ar[rr]_{a} \ar@/^1.5pc/[u]^{\mathrm{id}} \ar@{.>}[rrrru]^<<<<<<<<<{i_2} & & A_0 \ar@/^1.5pc/[u]^{\mathrm{id}} \ar[rr]_{\mathrm{id}} & & A_0 \ar@/^1.5pc/[u]^{i_1} \ar[rr] \ar@{.>}[rrrru]^<<<<<<<<<<{\mathrm{id}} & & 0 \ar@/^1.5pc/[u] \ar[rr] & & 0 \ar@/^1.5pc/[u] }}$$ We can therefore extend $\Gamma'$ along $\Gamma$ and $\Lambda'$ along $\Lambda,$ as explained in the proof of Proposition [Proposition 19](#PropExt){reference-type="ref" reference="PropExt"}. In both cases, we get the so-called denormalization functor $${\mathcal D}\colon \mathbf{Arr}({\mathcal A}) \to \mathbf{RG}({\mathcal A})$$ which sends an object $(A,a,A_0)$ on the reflexive graph in the middle of the previous diagram. It is well-known that ${\mathcal D}$ is an equivalence of categories (with nullhomotopies) whose quasi-inverse is the normalization functor ${\mathcal K}$ of [ 24](#TextReflGr){reference-type="ref" reference="TextReflGr"}. To prove this fact, it is enough to check the following isomorphism between a reflexive graph and the denormalization of its normalization: $$\xymatrix{A_1 \ar@<-0.5ex>[d]_{d} \ar@<0.5ex>[d]^{c} \ar@<0.5ex>[rrr]^-{\langle d;\delta \rangle} & & & A_0 \oplus \mathrm{Ker}(d) \ar@<0.5ex>[lll]^-{[i;k_d]} \ar@<-0.5ex>[d]_{\pi_1} \ar@<0.5ex>[d]^{[\mathrm{id};k_d \cdot c]} \\ A_0 \ar@<0.5ex>[rrr]^{\mathrm{id}} \ar@/^1.5pc/[u]^{i} & & & A_0 \ar@<0.5ex>[lll]^{\mathrm{id}} \ar@/^1.5pc/[u]^{i_1} }$$ where $\delta \colon A_1 \to \mathrm{Ker}(d)$ is the unique arrow such that $\delta \cdot k_d = -d \cdot i + \mathrm{id}.$ * ** 27**. * Finally, we know that $\mathbf{RG}({\mathcal A})$ is isomorphic to $\mathbf{Grpd}({\mathcal A}),$ the category of internal groupoids, because there exists a unique composition on the reflexive graphe ${\mathcal D}(A,a,A_0)$ making it an internal category. It is given by $$\mathrm{id}\oplus \nabla \colon A_0 \oplus A \oplus A \to A_0 \oplus A$$ (see [@AC; @PTJ] for a detailed discussion). Transporting the 2-categorical structure of $\mathbf{Grpd}({\mathcal A})$ along $\mathbf{Grpd}({\mathcal A}) \simeq \mathbf{RG}({\mathcal A}) \simeq \mathbf{Arr}({\mathcal A}),$ we get a 2-categorical structure on $\mathbf{Arr}({\mathcal A})$ which extends the structure of nullhomotopies $\Theta_{\Delta} \colon$ for any arrow $(f,f_0) \colon (A,a,A_0) \to (B,b,B_0),$ the set of nullhomotopies $\Theta_{\Delta}((f,f_0))$ coincides with the set of 2-cells from the zero arrow $(0^A_B, 0^{A_0}_{B_0})$ to $(f,f_0).$ * A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizations, *Springer Lecture Notes in Mathematics* 304 (1972) v+348 pp. R. Brown, Fibrations of groupoids, *Journal of Algebra* 15 (1970) 103--132. A. Carboni, Categories of affine spaces, *Journal of Pure and Applied Algebra* 61 (1989) 243--250. E. Dror Farjoun, Homotopy and homology of diagrams of spaces, *Springer Lecture Notes in Mathematics* 1286 (1987) 93--134. M. Dupont and E.M. Vitale, From abelian categories to 2-abelian bicategories, (in preparation). W.G. Dwyer, P.S. Hirschhorn, D.M. Kan and J.H. Smith, Homotopy limit functors on model categories and homotopical categories, *American Mathematical Society* (2004) viii+181 pp. P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, *Springer* (1967) x+168 pp. M. Grandis, Simplicial homotopical algebra and satellites, *Applied Categorical Structures* 5 (1997) 75--97. M. Grandis, Weak subobjects and the epi-monic completion of a category, *Journal of Pure and Applied Algebra* 154 (2000) 193--212. M. Grandis, A note on exactness and stability in homotopical algebra, *Theory and Applications of Categories* 9 (2001) 17--42. M. Grandis, G. Janelidze, From torsion theories to closure operators and factorization systems, *Categories and General Algebraic Structures with Application* 12 (2020) 89--121. P.-A. Jacqmin, S. Mantovani, G. Metere, E.M. Vitale, On fibrations between internal groupoids and their normalizations, *Applied Categorical Structures* 26 (2018) 1015--1039. P.-A. Jacqmin, S. Mantovani, G. Metere, E.M. Vitale, Bipullbacks of fractions and the snail lemma, *Journal of Pure and Applied Algebra* 223 (2019) 5147--5162. P.T. Johnstone, Affine categories and naturally Mal'cev categories, *Journal of Pure and Applied Algebra* 61 (1989) 251--256. M. Korostenski, W. Tholen, Factorization systems as Eilenberg-Moore algebras. *Journal of Pure and Applied Algebra* 85 (1993) 57--72. S. Mantovani, M. Messora, E.M. Vitale, Homotopy torsion theories, *arXiv: 2308.16843* (2023). S. Mantovani, G. Metere, E.M. Vitale, The snail lemma for internal groupoids, *Journal of Algebra* 535 (2019) 1--34. M. Mather, Pull-backs in homotopy theory, *Canadian Journal of Mathematics* 28 (1976) 225--263. J. Rosicky, E.M. Vitale, Exact completion and representations in abelian categories, *Homology, Homotopy and Application* 3 (2001) 453--466. R.W. Thomason, Homotopy colimits in the category of small categories, *Mathematical Proceedings of the Cambridge Philosophical Society* 85 (1979) 91--109. E.M. Vitale, The snail lemma, *Theory and Applications of Categories* 31 (2016) 484--501. [^1]: Institut de recherche en mathématique et physique, Université catholique de Louvain, Chemin du Cyclotron 2, B 1348 Louvain-la-Neuve, Belgique, enrico.vitale\@uclouvain.be

arxiv_math

--- abstract: | Let $(\xi_1, \eta_1)$, $(\xi_2, \eta_2),\ldots$ be independent identically distributed $\mathbb{N}^2$-valued random vectors with arbitrarily dependent components. The sequence $(\Theta_k)_{k\in\mathbb{N}}$ defined by $\Theta_k=\Pi_{k-1}\cdot\eta_k$, where $\Pi_0=1$ and $\Pi_k=\xi_1\cdot\ldots\cdot \xi_{k}$ for $k\in\mathbb{N}$, is called a multiplicative perturbed random walk. We study arithmetic properties of the random sets $\{\Pi_1,\Pi_2,\ldots, \Pi_k\}\subset \mathbb{N}$ and $\{\Theta_1,\Theta_2,\ldots, \Theta_k\}\subset \mathbb{N}$, $k\in\mathbb{N}$. In particular, we derive distributional limit theorems for their prime counts and for the least common multiple. address: - "$^1$National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", 03056 Kyiv, Ukraine. Email: vbogdanskii\\@ukr.net" - "$^2$Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine. Emails: vladyslavbogun\\@gmail.com, marynych\\@unicyb.kiev.ua, isamoil\\@i.ua" author: - Victor Bohdanskyi$^1$ - Vladyslav Bohun$^2$ - Alexander Marynych$^2$ - Igor Samoilenko$^2$ bibliography: - LCM_PRW2023.bib title: Arithmetic properties of multiplicative integer-valued perturbed random walks --- # Introduction Let $(\xi_1, \eta_1)$, $(\xi_2,\eta_2),\ldots$ be independent copies of an $\mathbb{N}^2$-valued random vector $(\xi,\eta)$ with arbitrarily dependent components. Denote by $(\Pi_k)_{k\in\mathbb{N}_0}$ (as usual, $\mathbb{N}_0:=\mathbb{N}\cup\{0\}$) the standard multiplicative random walk defined by $$\Pi_0:=1,\quad \Pi_k=\xi_1\cdot\xi_2\cdots\xi_k,\quad k\in\mathbb{N}.$$ A *multiplicative perturbed random walk* is the sequence $(\Theta_k)_{k\in\mathbb{N}}$ given by $$%\Theta_0:=1,\quad \Theta_k:=\Pi_{k-1}\cdot\eta_k,\quad k\in\mathbb{N}.$$ Note that if $\mathbb{P}\{\eta=\xi\}=1$, then $\Pi_k=\Theta_k$ for all $k\in\mathbb{N}$. If $\mathbb{P}\{\xi=1\}=1$, then $(\Theta_k)_{k\in\mathbb{N}}$ is just a sequence of independent copies of a random variable $\eta$. In this paper we investigate some arithmetic properties of the random sets $(\Pi_k)_{k\in\mathbb{N}}$ and $(\Theta_k)_{k\in\mathbb{N}}$. To set the scene we introduce first some necessary notation. Let $\mathcal{P}$ denote the set of prime numbers. For an integer $n\in\mathbb{N}$ and $p\in\mathcal{P}$, let $\lambda_p(n)$ denote the multiplicity of prime $p$ in the prime decomposition of $n$, that is, $$n=\prod_{p\in\mathcal{P}} p^{\lambda_p(n)}. %,\quad n\in\mn.$$ For every $p\in\mathcal{P}$, the function $\lambda_p:\mathbb{N}\mapsto\mathbb{N}_0$ is totally additive in the sense that $$\lambda_p(mn)=\lambda_p(m)+\lambda_p(n),\quad p\in\mathcal{P},\quad m,n\in\mathbb{N}.$$ The set of functions $(\lambda_p)_{p\in\mathcal{P}}$ is a basic brick from which many other arithmetic functions can be constructed. For example, with ${\rm GCD}\,(A)$ and ${\rm LCM}\,(A)$ denoting the greatest common divisor and the least common multiple of a set $A\subset\mathbb{N}$, respectively, we have $${\rm GCD}\,(A)=\prod_{p\in\mathcal{P}}p^{\min_{n\in A}\lambda_p(n)}\quad\text{and}\quad{\rm LCM}\,(A)=\prod_{p\in\mathcal{P}}p^{\max_{n\in A}\lambda_p(n)}.$$ The listed arithmetic functions applied either to $A=\{\Pi_1,\ldots,\Pi_n\}$ or $A=\{\Theta_1,\ldots,\Theta_n\}$ are the main objects of investigation in the present paper. From the additivity of $\lambda_p$ we infer $$\label{eq:S_k(p)_def} S_k(p):=\lambda_p(\Pi_k)=\sum_{j=1}^{k}\lambda_p(\xi_j),\quad p\in\mathcal{P},\quad k\in\mathbb{N}_0,$$ and $$\label{eq:T_k(p)_def} T_k(p):=\lambda_p(\Theta_k)=\sum_{j=1}^{k-1}\lambda_p(\xi_j)+\lambda_p(\eta_k),\quad p\in\mathcal{P},\quad k\in\mathbb{N}.$$ Fix any $p\in\mathcal{P}$. Formulae [\[eq:S_k(p)\_def\]](#eq:S_k(p)_def){reference-type="eqref" reference="eq:S_k(p)_def"} and [\[eq:T_k(p)\_def\]](#eq:T_k(p)_def){reference-type="eqref" reference="eq:T_k(p)_def"} demonstrate that $S(p):=(S_k(p))_{k\in\mathbb{N}_0}$, is a standard additive random walk with the generic step $\lambda_p(\xi)$, whereas the sequence $T(p):=(T_k(p))_{k\in\mathbb{N}}$, is a particular instance of an *additive perturbed random walk*, see [@Iksanov:2016], generated by the pair $(\lambda_p(\xi),\lambda_p(\eta))$. # Main results ## Distributional properties of the prime counts $(\lambda_p(\xi),\lambda_p(\eta))$ As is suggested by [\[eq:S_k(p)\_def\]](#eq:S_k(p)_def){reference-type="eqref" reference="eq:S_k(p)_def"} and [\[eq:T_k(p)\_def\]](#eq:T_k(p)_def){reference-type="eqref" reference="eq:T_k(p)_def"} the first step in the analysis of $S(p)$ and $T(p)$ should be the derivation of the joint distribution $(\lambda_p(\xi),\lambda_p(\eta))_{p\in\mathcal{P}}$. The next lemma confirms that the finite-dimensional distributions of the infinite vector $(\lambda_p(\xi),\lambda_p(\eta))_{p\in\mathcal{P}}$, are expressible via the probability mass function of $(\xi,\eta)$. However, the obtained formulae are not easy to handle except some special cases. For $i,j\in\mathbb{N}$, put $$u_i:=\mathbb{P}\{\xi=i\},\quad v_j:=\mathbb{P}\{\eta=j\},\quad w_{i,j}:=\mathbb{P}\{\xi=i,\eta=j\}.$$ **Lemma 1**. *Fix $p\in\mathcal{P}$ and nonnegative integers $(k_q)_{q\in\mathcal{P},q\leq p}$ and $(\ell_q)_{q\in\mathcal{P},q\leq p}$. Then $$\mathbb{P}\{\lambda_q(\xi)\geq k_q,\lambda_q(\eta)\geq \ell_q,q\in\mathcal{P},q\leq p\}=\sum_{i,j=1}^{\infty}w_{Ki,Lj},$$ where $K:=\prod_{q\leq p,q\in\mathcal{P}}q^{k_q}$ and $L:=\prod_{q\leq p,q\in\mathcal{P}}q^{\ell_q}$.* *Proof.* This follows from $$\begin{gathered} \mathbb{P}\{\lambda_q(\xi)\geq k_q,\lambda_q(\eta)\geq \ell_q,q\in\mathcal{P},q\leq p\}\\ =\mathbb{P}\left\{\prod_{q\leq p,q\in\mathcal{P}}q^{k_q}\text{ divides }\xi,\prod_{q\leq p,q\in\mathcal{P}}q^{\ell_q}\text{ divides }\eta\right\}=\sum_{i,j=1}^{\infty}w_{Ki,Lj}.\end{gathered}$$ Obviously, if $\xi$ and $\eta$ are independent, then $$\sum_{i,j=1}^{\infty}w_{Ki,Lj}=\left(\sum_{i=1}^{\infty}u_{Ki}\right)\left(\sum_{j=1}^{\infty}v_{Lj}\right).$$ ◻ We proceed with the series of examples. **Example 1**. *For $\alpha>1$, let $\mathbb{P}\{\xi=k\}=(\zeta(\alpha))^{-1} k^{-\alpha}$, $k\in\mathbb{N}$, where $\zeta$ is the Riemann zeta-function. Then, $(\lambda_p(\xi))_{p\in\mathcal{P}}$ are mutually independent and $$\mathbb{P}\{\lambda_p(\xi)\geq k\}=\sum_{i=1}^{\infty}\mathbb{P}\{\xi=p^k i\}=p^{-k\alpha},\quad k\in\mathbb{N}_0,\quad p\in\mathcal{P},$$ which means that $\lambda_p(\xi)$ has a geometric distribution on $\mathbb{N}_0$ with parameter $p^{-\alpha}$.* **Example 2**. *For $\beta\in(0,1)$, let $\mathbb{P}\{\xi=k\}=\beta^{k-1}(1-\beta)$, $k\in\mathbb{N}$. Then $$\mathbb{P}\{\lambda_p(\xi)\geq k\}=\frac{1-\beta}{\beta}\sum_{j=1}^{\infty}\beta^{p^k j}=\frac{(1-\beta)(\beta^{p^k-1})}{1-\beta^{p^k}},\quad k\in\mathbb{N}_0.$$* **Example 3**. *Let ${\rm Poi}(\lambda)$ be a random variable with the Poisson distribution with parameter $\lambda$ and put $$\mathbb{P}\{\xi=k\}=\mathbb{P}\{{\rm Poi}(\lambda)=k|{\rm Poi}(\lambda)\geq 1\}=(e^{\lambda}-1)^{-1}\lambda^k/k!,\quad k\in\mathbb{N}.$$ Then $$\begin{gathered} \mathbb{P}\{\lambda_p(\xi)\geq k\}=(e^{\lambda}-1)^{-1}\sum_{j=1}^{\infty}\lambda^{p^k j}/(p^k j)!\\ =\left(_0 F_{p^k}\left(;\frac{1}{p^k},\frac{2}{p^k},\ldots,\frac{p^k-1}{p^k};\left(\frac{\lambda}{p^k}\right)^{p^k}\right)-1\right),\end{gathered}$$ where $_0 F_{p^k}$ is the generalized hypergeometric function, see Chapter 16 in [@Olver].* In all examples above the distribution of $\lambda_p(\xi)$ for every fixed $p\in\mathcal{P}$, is extremely light-tailed. It is not that difficult to construct 'weird' distributions where all $\lambda_p(\xi)$ have infinite expectations. **Example 4**. *Let $(g_p)_{p\in\mathcal{P}}$ be any probability distribution supported by $\mathcal{P}$, $g_p>0$, and $(t_k)_{k\in\mathbb{N}_0}$ any probability distribution on $\mathbb{N}$ such that $\sum_{k=1}^{\infty} kt_k=\infty$ and $t_k>0$. Define a probability distribution $\mathfrak{h}$ on $\mathcal{Q}:=\bigcup_{p\in\mathcal{P}}\{p,p^2,\ldots\}$ by $$\mathfrak{h}({\{p^k\}})=g_p t_k,\quad p\in\mathcal{P},\quad k\in\mathbb{N}.$$ If $\xi$ is a random variable with distribution $\mathfrak{h}$, then $$\mathbb{P}\{\lambda_p(\xi)\geq k\}=g_p\sum_{j=k}^{\infty}t_j,\quad k\in\mathbb{N},\quad p\in\mathcal{P},$$ which implies $\mathbb{E}[\lambda_p(\xi)]=g_p\sum_{k=1}^{\infty} kt_k=\infty$, $p\in\mathcal{P}$.* *This example can be modified by taking $g:=\sum_{p\in\mathcal{P}}g_p<1$ and charging all points of $\mathbb{N}\setminus \mathcal{Q}$ (this set contains $1$ and all integers having at least two different prime factors) with arbitrary positive masses of the total weight $1-g$. The obtained probability distribution charges all points of $\mathbb{N}$ and still possesses the property that all $\lambda_p$'s have infinite expectations.* Let $X$ be a random variable taking values in $\mathbb{N}$. Since $$\log X=\sum_{p\in\mathcal{P}}\lambda_p(X)\log p,$$ we conclude that $\mathbb{E}[(\lambda_p(X))^k]<\infty$, for all $p\in\mathcal{P}$, whenever $\mathbb{E}[\log^k X]<\infty$, $k\in\mathbb{N}$. It is also clear that the converse implication is false in general. When $k=1$ the inequality $\mathbb{E}[\lambda_p(X)]<\infty$ is equivalent to $\sum_{p\in\mathcal{P}}\mathbb{E}[\lambda_p(X)]\log p<\infty$. As we have seen in the above examples, checking that $\mathbb{E}[(\lambda_p(X))^k]<\infty$ might be a much more difficult task than proving a stronger assumption $\mathbb{E}[\log^k X]<\infty$. Thus, we shall mostly work under moment conditions on $\log\xi$ and $\log\eta$. Our standing assumption throughout the paper is $$\label{eq:finite_mean} \mu_{\xi}:=\mathbb{E} [\log \xi]<\infty,$$ which, by the above reasoning, implies $\mathbb{E} [\lambda_p(\xi)]<\infty$, $p\in\mathcal{P}$. ## Limit theorems for $S(p)$ and $T(p)$ From Donsker's invariance principle we immediately obtain the following proposition. Let $D:=D([0,\infty),\mathbb{R})$ be the Skorokhod space endowed with the standard $J_1$-topology. **Proposition 1**. *Assume that $\mathbb{E}[\log^2 \xi]\in (0,\infty)$. Then, $$\left(\left(\frac{S_{\lfloor ut \rfloor}(p)-ut\mathbb{E}\lambda_p(\xi)}{\sqrt{t}}\right)_{u\geq 0}\right)_{p\in\mathcal{P}}~\Longrightarrow~((W_p(u))_{u\geq 0})_{p\in\mathcal{P}},\quad t\to\infty,$$ on the product space $D^{\mathbb{N}}$, where, for all $n\in\mathbb{N}$ and all $p_1<p_2<\cdots<p_n$, $p_i\in\mathcal{P}$, $i\leq n$, $(W_{p_1}(u),\ldots,W_{p_n}(u))_{u\geq 0}$ is an $n$-dimensional centered Wiener process with covariance matrix $C=||C_{i,\,j}||_{1\leq i,j\leq n}$ given by $C_{i,\,j}=C_{j,\,i}={\rm Cov}\,(\lambda_{p_i}(\xi),\lambda_{p_j}(\xi))$.* According to the proof of Proposition 1.3.13 in [@Iksanov:2016], see pp. 28-29 therein, the following holds true for the perturbed random walks $T(p)$, $p\in\mathcal{P}$. **Proposition 2**. *Assume that $\mathbb{E}[\log^2 \xi]\in (0,\infty)$ and $$\label{eq:eta_p_negligible} \lim_{t\to\infty}t^2\mathbb{P}\{\lambda_p(\eta)\geq t\}=0,\quad p\in\mathcal{P}.$$ Then, $$\left(\left(\frac{T_{\lfloor ut \rfloor}(p)-ut\mathbb{E}\lambda_p(\xi)}{\sqrt{t}}\right)_{u\geq 0}\right)_{p\in\mathcal{P}}~\Longrightarrow~((W_p(u))_{u\geq 0})_{p\in\mathcal{P}},\quad t\to\infty,$$ on the product space $D^{\mathbb{N}}$.* **Remark 1**. *Since $\mathbb{P}\{\lambda_p(\eta)\log p\geq t\}\leq \mathbb{P}\{\log \eta \geq t\}$, the condition $$\label{eq:log_eta_negligible} \lim_{t\to\infty}t^2\mathbb{P}\{\log\eta\geq t\}=0$$ is clearly sufficient for [\[eq:eta_p\_negligible\]](#eq:eta_p_negligible){reference-type="eqref" reference="eq:eta_p_negligible"}.* From the continuous mapping theorem under the assumptions of Proposition [Proposition 2](#prop:MCLT_T){reference-type="ref" reference="prop:MCLT_T"} we infer $$\begin{gathered} \label{eq:sup_t_p_MCLT} \left(\left(\frac{\max_{1\leq k\leq \lfloor ut \rfloor}(T_{k}(p)-k\mathbb{E}\lambda_p(\xi))}{\sqrt{t}}\right)_{u\geq 0}\right)_{p\in\mathcal{P}}\\ \Longrightarrow~((\sup_{0\leq v\leq u}W_p(v))_{u\geq 0})_{p\in\mathcal{P}},\quad t\to\infty,\end{gathered}$$ see Proposition 1.3.13 in [@Iksanov:2016]. Formula [\[eq:sup_t\_p_MCLT\]](#eq:sup_t_p_MCLT){reference-type="eqref" reference="eq:sup_t_p_MCLT"}, for a fixed $p\in\mathcal{P}$, belongs to the realm of limit theorems for the maximum of a single additive perturbed random walk. This circle of problems is well-understood, see Section 1.3.3 in [@Iksanov:2016] and [@Iksanov+Pilipenko+Samoilenko:2017], in the situation when the underlying additive standard random walk is *centered* and attracted to a stable Lévy process. In our setting the perturbed random walks $(T_k(p))_{k\in\mathbb{N}}$ and $(T_k(q))_{k\in\mathbb{N}}$ are dependent whenever $p,q\in\mathcal{P}$, $p\neq q$, which make derivation of the joint limit theorems harder and leads to various asymptotic regimes. Note that [\[eq:eta_p\_negligible\]](#eq:eta_p_negligible){reference-type="eqref" reference="eq:eta_p_negligible"} implies $\mathbb{E}[\lambda_p(\eta)]<\infty$ and [\[eq:log_eta_negligible\]](#eq:log_eta_negligible){reference-type="eqref" reference="eq:log_eta_negligible"} implies $\mathbb{E}[\log \eta]<\infty$. Theorem [Theorem 5](#main1){reference-type="ref" reference="main1"} below tells us that under such moment conditions and assuming also $\mathbb{E}[\log^2 \xi]<\infty$ the maxima $\max_{1\leq k\leq n}\,T_{k}(p)$, $p\in\mathcal{P}$, of *noncentered* perturbed random walks $T(p)$ have the same behavior as $S_{n}(p)$, $p\in\mathcal{P}$ as $n\to\infty$. **Theorem 5**. *Assume that $\mathbb{E}[\log^2 \xi]<\infty$ and $\mathbb{E}[\lambda_p(\eta)]<\infty$, $p\in\mathcal{P}$. Suppose further that $$\label{eq:full_support_assumption} \mathbb{P}\{\xi\text{ is divisible by }p\}=\mathbb{P}\{\lambda_p(\xi)>0\}>0,\quad p\in\mathcal{P}.$$ Then, as $t\to\infty$, $$\label{eq:main1_claim} \left(\left(\frac{\max_{1\leq k\leq \lfloor tu\rfloor}\,T_{k}(p)-\mathbb{E}[\lambda_p(\xi)]tu}{t^{1/2}}\right)_{u\geq 0}\right)_{p\in\mathcal{P}}~\overset{{\rm f.d.d.}}{\longrightarrow}~((W_p(u))_{u\geq 0})_{p\in\mathcal{P}}.$$* **Remark 2**. *If [\[eq:full_support_assumption\]](#eq:full_support_assumption){reference-type="eqref" reference="eq:full_support_assumption"} holds only for some $\mathcal{P}_0\subseteq\mathcal{P}$, then [\[eq:main1_claim\]](#eq:main1_claim){reference-type="eqref" reference="eq:main1_claim"} holds with $\mathcal{P}_0$ instead of $\mathcal{P}$.* In the next result we shall assume that $\eta$ dominates $\xi$ in a sense that the asymptotic behavior of $\max_{1\leq k\leq n}T_{k}(p)$ is regulated by the perturbations $(\lambda_p(\eta_k))_{k\leq n}$ for all $p\in\mathcal{P}_0$, where $\mathcal{P}_0$ is a finite subset of prime numbers and those $p$'s dominate all other primes. **Theorem 6**. *Assume [\[eq:finite_mean\]](#eq:finite_mean){reference-type="eqref" reference="eq:finite_mean"}. Suppose further that there exists a finite set $\mathcal{P}_0\subseteq \mathcal{P}$, $d:=|\mathcal{P}_0|$, such that the distributional tail of $(\lambda_p(\eta))_{p\in\mathcal{P}_0}$ is regularly varying at infinity in the following sense. For some positive function $(a(t))_{t>0}$ and a measure $\nu$ satisfying $\nu(\{x\in \mathbb{R}^d:\|x\|\geq r\})=c\cdot r^{-\alpha}$, $c>0$, $\alpha\in (0,1)$, it holds $$\label{eq:reg_var} t\mathbb{P}\{(a(t))^{-1}(\lambda_p(\eta))_{p\in\mathcal{P}_0}\in\cdot\}~\overset{{\rm v}}{\longrightarrow}~ \nu(\cdot),\quad t\to\infty,$$ on the space of locally finite measures on $(0,\infty]^d$ endowed with the vague topology. Finally, suppose $\mathbb{E}[\lambda_p(\eta)]<\infty$, for $p\in\mathcal{P}\setminus \mathcal{P}_0$. Then $$\label{eq:exteme_dominates1} \left(\left(\frac{\max_{1\leq k\leq \lfloor tu\rfloor}\,T_{k}(p)}{a(t)}\right)_{u\geq 0}\right)_{p\in\mathcal{P}_0}~\overset{{\rm f.d.d.}}{\longrightarrow}~(M_p(u))_{u\geq 0})_{p\in\mathcal{P}_0},\quad t\to\infty,$$ where $(M_p(u))_{u\geq 0})_{p\in\mathcal{P}_0}$ is a multivariate extreme process defined by $$\label{eq:extreme_def} (M_p(u))_{p\in\mathcal{P}_0}=\sup_{k:\,t_k\leq u} y_k,\quad u\geq 0.$$ Here the pairs $(t_k,y_k)$ are the atoms of a Poisson point process on $[0,\infty)\times (0,\infty]^d$ with the intensity measure $\mathbb{LEB}\otimes \nu$ and the supremum is taken coordinatewise. Moreover, $$\label{eq:exteme_dominates2} \left(\left(\frac{\max_{1\leq k\leq \lfloor tu\rfloor}\,T_{k}(p)}{a(t)}\right)_{u\geq 0}\right)_{p\in\mathcal{P}\setminus \mathcal{P}_0}~\overset{{\rm f.d.d.}}{\longrightarrow}~0,\quad t\to\infty.$$* ## Limit theorems for the ${\rm LCM}\,$ The results from the previous section will be applied below to the analysis of $$\mathcal{\Pi}_n:={\rm LCM}\,(\{\Pi_1,\Pi_2,\ldots, \Pi_n\})\quad\text{and}\quad \mathcal{\Theta}_n:={\rm LCM}\,(\{\Theta_1, \Theta_2,\ldots, \Theta_n\}).$$ A moment's reflection shows that the analysis of $\mathcal{\Pi}_n$ is trivial. Indeed, by definition, $\Pi_{n-1}$ divides $\Pi_n$ and thereupon $\mathcal{\Pi}_n=\Pi_n$ for $n\in\mathbb{N}$. Thus, assuming that $\sigma_{\xi}^2:={\rm Var}\,(\log \xi)\in (0,\infty)$, an application of the Donsker functional limit theorem yields $$\label{eq:1} \Big(\frac{\log \mathcal{\Pi}_{\lfloor tu\rfloor}-\mu_{\xi} tu}{ t^{1/2}}\Big)_{u\geq 0}~\Longrightarrow~(\sigma_{\xi} W(u))_{u\geq 0},\quad t\to\infty,$$ on the Skorokhod space $D$, where $(W(u))_{u\geq 0}$ is a standard Brownian motion. A simple structure of the sequence $(\mathcal{\Pi}_n)_{n\in\mathbb{N}}$ breaks down completely upon introducing the perturbations $(\eta_k)$, which makes the analysis of $(\mathcal{\Theta}_n)$ a much harder problem. For instance, it contains as a special case the problem of studying the ${\rm LCM}\,$ of an independent sample, which is itself highly non-trivial. Note that $$\log \mathcal{\Theta}_n=\log \prod_{p\in\mathcal{P}} p^{\max_{1\leq k\leq n}\,(\lambda_p(\xi_1)+\ldots +\lambda_p(\xi_{k-1})+\lambda_p(\eta_k))}=\sum_{p\in\mathcal{P}}\max_{1\leq k\leq n} T_k(p)\log p,$$ which shows that the asymptotic of $\mathcal{\Theta}_n$ is intimately connected with the behavior of $\max_{1\leq k\leq n}T_k(p)$, $p\in\mathcal{P}$. As one can guess from Theorem [Theorem 5](#main1){reference-type="ref" reference="main1"} in a 'typical' situation relation [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"} holds with $\log \mathcal{\Theta}_{\lfloor tu\rfloor}$ replacing $\log \mathcal{\Pi}_{\lfloor tu\rfloor}$. The following heuristics suggest the right form of assumptions ensuring that perturbations $(\eta_k)_{k\in\mathbb{N}}$ have an asymptotically negligible impact on $\log \mathcal{\Theta}_n$. Take a prime $p\in\mathcal{P}$. Its contribution to $\log \mathcal{\Theta}_n$ (up to a factor $\log p$) is $\max_{1\leq k\leq n}T_k(p)$. According to Theorem [Theorem 5](#main1){reference-type="ref" reference="main1"}, this maximum is asymptotically the same as $S_n(p)$. However, as $p$ gets large, the mean $\mathbb{E}[\lambda_p(\xi)]$ of the random walk $S_{n-1}(p)$ becomes small because of the identity $$\sum_{p\in\mathcal{P}}\mathbb{E}[\lambda_p(\xi)] \log p = \mathbb{E}[\log \xi]<\infty.$$ Thus, for large $p\in\mathcal{P}$, the remainder $\max_{1\leq k\leq n}T_k(p)-S_{n-1}(p)$ can, in principle, become larger than $S_{n-1}(p)$ itself if the tail of $\lambda_p(\eta)$ is sufficiently heavy. In order to rule out such a possibility, we introduce the following deterministic sets: $$\label{eq:p_1_p_2_defs} \mathcal{P}_1(n):=\{p\in\mathcal{P}: \mathbb{P}\{\lambda_p(\xi)>0\}\geq n^{-1/2}\}\quad\text{and}\quad \mathcal{P}_2(n):=\mathcal{P}\setminus \mathcal{P}_1(n),$$ and bound the rate of growth of $\max_{1\leq k\leq n}\lambda_p(\eta_k)$ for all $p\in\mathcal{P}_2(n)$. It is important to note that under the assumption [\[eq:full_support_assumption\]](#eq:full_support_assumption){reference-type="eqref" reference="eq:full_support_assumption"} it holds $$\lim_{n\to\infty}\min \mathcal{P}_2(n)=\infty.$$ Therefore, if $\mathbb{E}[\log X]<\infty$ for some random variable $X$, then the relation $$\label{eq:p_2_explanation} \lim_{n\to\infty}\sum_{p\in\mathcal{P}_2(n)} \mathbb{E}[\lambda_p(X)]\log p=0,$$ holds true. **Theorem 7**. *Assume $\mathbb{E}[\log^2 \xi]<\infty$, $\mathbb{E}[\log \eta]<\infty$, [\[eq:full_support_assumption\]](#eq:full_support_assumption){reference-type="eqref" reference="eq:full_support_assumption"} and the following two conditions $$\label{eq:second_moment_diff} \sum_{p\in\mathcal{P}}\mathbb{E}\left[((\lambda_p(\eta)-\lambda_p(\xi))^{+})^2\right]\log p<\infty$$ and $$\label{eq:main2_eta_negligible} \sum_{p\in\mathcal{P}_2(n)}\mathbb{E}[(\lambda_p(\eta)-\lambda_p(\xi))^{+}] \log p=o(n^{-1/2}),\quad n\to\infty.$$ Then $$\label{eq:2} \left(\frac{\log \mathcal{\Theta}_{\lfloor tu\rfloor}-\mu_{\xi} tu}{t^{1/2}}\right)_{u\geq 0}~\overset{{\rm f.d.d.}}{\longrightarrow}~(\sigma_{\xi} W(u))_{u\geq 0},\quad t\to\infty,$$ where $\mu_{\xi}=\mathbb{E}[\log\xi]<\infty$, $\sigma_{\xi}^2={\rm Var}\,[\log \xi]$ and $(W(u))_{u\geq 0}$ is a standard Brownian motion.* **Remark 3**. *If $\mathbb{E}[\log^2 \eta]<\infty$, then [\[eq:second_moment_diff\]](#eq:second_moment_diff){reference-type="eqref" reference="eq:second_moment_diff"} holds true. Indeed, since we assume $\mathbb{E}[\log^2 \xi]<\infty$, $$\begin{aligned} &\hspace{-1cm}\mathbb{E}\left[\sum_{p\in\mathcal{P}}((\lambda_p(\eta)-\lambda_p(\xi))^{+})^2\log p\right]\leq \mathbb{E}\left[\sum_{p\in\mathcal{P}}(\lambda^2_p(\eta)+\lambda_p^2(\xi))\log p\right]\\ &\leq \mathbb{E}\left[\left(\sum_{p\in\mathcal{P}}\lambda_p(\eta)\log p\right)^2\right]+\mathbb{E}\left[\left(\sum_{p\in\mathcal{P}}\lambda_p(\xi)\log p\right)^2\right]\\ &=\mathbb{E}[\log^2 \eta]+\mathbb{E}[\log^2 \xi]<\infty.\end{aligned}$$ The condition [\[eq:main2_eta_negligible\]](#eq:main2_eta_negligible){reference-type="eqref" reference="eq:main2_eta_negligible"} can be replaced by a stronger one which only involves distribution of $\eta$, namely $$\label{eq:main2_eta_negligible_alt} \sum_{p\in\mathcal{P}_2(n)}\mathbb{E}[\lambda_p(\eta)] \log p=o(n^{-1/2}),\quad n\to\infty.$$ Taking into account [\[eq:p_2\_explanation\]](#eq:p_2_explanation){reference-type="eqref" reference="eq:p_2_explanation"} and the fact that $\mathbb{E}[\log \eta]<\infty$, the assumption [\[eq:main2_eta_negligible_alt\]](#eq:main2_eta_negligible_alt){reference-type="eqref" reference="eq:main2_eta_negligible_alt"} is nothing else but a condition of the speed of convergence of the series $$\sum_{p\in\mathcal{P}} \mathbb{E}[\lambda_p(\eta)]\log p=\mathbb{E}[\log \eta].$$* **Example 8**. *In the settings of Example [Example 1](#example1){reference-type="ref" reference="example1"} let $\xi$ and $\eta$ be arbitrarily dependent with $$\mathbb{P}\{\xi=k\}=\frac{1}{\zeta(\alpha)k^{\alpha}},\quad \mathbb{P}\{\eta=k\}=\frac{1}{\zeta(\beta)k^{\beta}},\quad k\in\mathbb{N},$$ for some $\alpha,\beta>1$. Note that $\mathbb{E}[\log^2 \xi]<\infty$ and $\mathbb{E}[\log^2 \eta]<\infty$. Direct calculations show that $$\begin{aligned} \mathcal{P}_1(n)&=\{p\in\mathcal{P}: p^{-\alpha}\geq n^{-1/2}=\{p\in\mathcal{P}: p\leq n^{1/(2\alpha)}\},\\ \mathcal{P}_2(n)&=\{p\in\mathcal{P}: p > n^{1/(2\alpha)}\}.\end{aligned}$$ From the chain of relations $$\mathbb{E}[\lambda_p(\eta)]=\sum_{j\geq 1}\mathbb{P}\{\lambda_p(\eta)\geq j\}=\sum_{j\geq 1}p^{-\beta j}=\frac{p^{-\beta}}{1-p^{-\beta}}\leq 2p^{-\beta},$$ we obtain that $$\begin{gathered} \sum_{p\in\mathcal{P}_2(n)}\mathbb{E}[\lambda_p(\eta)] \log p\leq 2\sum_{p\in\mathcal{P},p>n^{1/(2\alpha)}}p^{-\beta} \log p\\ \sim~2\int_{n^{1/(2\alpha)}}^{\infty}x^{-\beta}\log x \frac{{\rm d}x}{\log x}=\frac{2n^{(1-\beta)/(2\alpha)}}{\beta-1},\quad n\to\infty,\end{gathered}$$ where we have used the prime number theorem for the asymptotic equivalence. Thus, [\[eq:main2_eta_negligible_alt\]](#eq:main2_eta_negligible_alt){reference-type="eqref" reference="eq:main2_eta_negligible_alt"} holds if $$\frac{1}{2}+\frac{1-\beta}{2\alpha}<0~\Longleftrightarrow~\alpha+1<\beta.$$* In the setting of Theorem [Theorem 6](#main11){reference-type="ref" reference="main11"} the situation is much simpler in a sense that almost no extra assumptions are needed to derive a limit theorem for $\mathcal{\Theta}_n$. **Theorem 9**. *Under the same assumptions as in Theorem [Theorem 6](#main11){reference-type="ref" reference="main11"} and assuming additionally that $$\label{eq:eta_without_P0} \sum_{p\in\mathcal{P}\setminus\mathcal{P}_0} \mathbb{E}[\lambda_p(\eta)]\log p<\infty,$$ it holds $$\label{eq:LCM_to_extreme} \left(\frac{\log \mathcal{\Theta}_{\lfloor tu\rfloor}}{a(t)}\right)_{u\geq 0}~\overset{{\rm f.d.d.}}{\longrightarrow}~\left(\sum_{p\in\mathcal{P}_0} M_p(u)\log p\right)_{u\geq 0},\quad t\to\infty.$$* Note that it is allowed to take in Theorem [Theorem 9](#main21){reference-type="ref" reference="main21"} $\xi=1$, which yields the following limit theorem for the ${\rm LCM}\,$ of an independent integer-valued random variables. **Corollary 1**. *Under the same assumptions on $\eta$ as in Theorem [Theorem 6](#main11){reference-type="ref" reference="main11"} it holds $$\left(\frac{\log {\rm LCM}\,(\eta_1,\eta_2,\ldots,\eta_{\lfloor tu\rfloor})}{a(t)}\right)_{u\geq 0}~\overset{{\rm f.d.d.}}{\longrightarrow}~\left(\sum_{p\in\mathcal{P}_0} M_p(u)\log p\right)_{u\geq 0},\quad t\to\infty.$$* **Remark 4**. *The results presented in Theorems [Theorem 7](#main2){reference-type="ref" reference="main2"} and [Theorem 9](#main21){reference-type="ref" reference="main21"} is a contribution to a popular topic in probabilistic number theory, namely, the asymptotic analysis of the ${\rm LCM}\,$ of various random sets. For random sets comprised of independent random variables uniformly distributed on $\{1,2,\ldots,n\}$ this problem has been addressed in [@BosMarRasch:2019; @BurIksMar:2022; @Fernandez+Fernandez:2021; @Hilberdink+Toth:2016; @Kim]. Some models with a more sophisticated dependence structure have been studied [@AlsKabMar:2019] and [@KabMarRasch:2023].* # Limit theorems for coupled perturbed random walks Theorems [Theorem 5](#main1){reference-type="ref" reference="main1"} and [Theorem 6](#main11){reference-type="ref" reference="main11"} will be derived from general limit theorems for the maxima of arbitrary additive perturbed random walks indexed by some parameters ranging in a countable set in the situation when the underlying additive standard random walks are positively divergent and attracted to a Brownian motion. Let $\mathcal{A}$ be a countable or finite set of real numbers and $$((X(r), Y(r)))_{r\in\mathcal{A}},\quad ((X(r),Y(r)))_{r\in \mathcal{A}},\ldots$$ be independent copies of an $\mathbb{R}^{2\times |\mathcal{A}|}$ random vector $(X(r),Y(r))_{r\in \mathcal{A}}$ with arbitrarily dependent components. For each $r\in\mathcal{A}$, the sequence $(S^\ast_k(r))_{k\in\mathbb{N}_0}$ given by $$S^\ast_0(r):=0,\quad S^\ast_k(r):=X_1(r)+\ldots+X_k(r),\quad k\in\mathbb{N},$$ is an additive standard random walk. For each $r\in\mathcal{A}$, the sequence $(T^\ast_k(r))_{k\in\mathbb{N}}$ defined by $$T^\ast_k(r):=S^\ast_{k-1}(r)+Y_k(r),\quad k\in\mathbb{N},$$ is an additive perturbed random walk. The sequence $((T^\ast_k(r))_{k\in\mathbb{N}})_{r\in\mathcal{A}}$ is a collection of (generally) dependent additive perturbed random walks. **Proposition 3**. *Assume that, for each $r\in\mathcal{A}$, $\mu(r):=\mathbb{E}[X(r)]\in (0,\infty)$, ${\rm Var}\,[X(r)]\in [0,\infty)$ and $\mathbb{E}[Y(r)]<\infty$. Then $$\label{eq:main3_FLT} \left(\left(\frac{\max_{1\leq k\leq \lfloor tu\rfloor}\,T^\ast_{k}(r)-\mu(r)tu}{t^{1/2}}\right)_{u\geq 0}\right)_{r\in\mathcal{A}}~\overset{{\rm f.d.d.}}{\longrightarrow}~((W_r(u))_{u\geq 0})_{r\in\mathcal{A}},\quad t\to\infty,$$ where, for all $n\in\mathbb{N}$ and arbitrary $r_1<r_2<\ldots<r_n$ with $r_i\in\mathcal{A}$, $i\leq n$, $(W_{r_1}(u),\ldots, W_{r_n}(u))_{u\geq 0}$ is an $n$-dimensional centered Wiener process with covariance matrix $C=||C_{i,\,j}||_{1\leq i,j\leq n}$ with the entries $C_{i,\,j}=C_{j,\,i}={\rm Cov}\,(X(r_i), X(r_j))$.* *Proof.* We shall prove an equivalent statement that, as $t\to\infty$, $$\left(\left(\frac{\max_{0\leq k\leq \lfloor tu\rfloor}\,T^\ast_{k+1}(r)-\mu(r)tu}{t^{1/2}}\right)_{u\geq 0}\right)_{r\in\mathcal{A}}~\overset{{\rm f.d.d.}}{\longrightarrow}~((W_r(u))_{u\geq 0})_{r\in\mathcal{A}},$$ which differs from [\[eq:main3_FLT\]](#eq:main3_FLT){reference-type="eqref" reference="eq:main3_FLT"} by a shift of the subscript $k$. By the multidimensional Donsker theorem, $$\label{eq:donsker} \left(\left(\frac{S^\ast_{\lfloor tu\rfloor}(r)-\mu(r)tu}{t^{1/2}}\right)_{u\geq 0}\right)_{r\in\mathcal{A}}~\Longrightarrow~\left((W_r(u))_{u\geq 0}\right)_{r\in\mathcal{A}},\quad t\to\infty,$$ in the product topology of $D^{\mathbb{N}}$. Fix any $r\in\mathcal{A}$ and write $$\begin{gathered} \max_{0\leq k\leq \lfloor tu\rfloor}\,T^\ast_{k+1}(r)-\mu(r)tu\\ =\max_{0\leq k\leq \lfloor tu\rfloor}\,(S^\ast_k(r)-S^\ast_{\lfloor tu\rfloor}(r)+Y_{k+1}(r))+S^\ast_{\lfloor tu\rfloor}(r)-\mu(r)tu.\end{gathered}$$ In view of [\[eq:donsker\]](#eq:donsker){reference-type="eqref" reference="eq:donsker"} the proof is complete once we can show that $$\label{eq:proof_prw_joint1} n^{-1/2}\left(\max_{0\leq k\leq n}\,\left(S^\ast_k(r)-S^\ast_{n}(r)+Y_{k+1}(r)\right)\right)~\overset{{\mathbb{P}}}{\to}~0,\quad n\to\infty.$$ Let $(X_0(r), Y_0(r))$ be a copy of $(X(r), Y(r))$ which is independent of $(X_k(r), Y_k(r))_{k\in\mathbb{N}}$. Since the collection $$((X_1(r), Y_1(r)),\ldots, (X_{n+1}(r), Y_{n+1}(r)))$$ has the same distribution as $$((X_{n}(r), Y_{n}(r)),\ldots, (X_0(r),Y_0(r))),$$ the variable $$\max_{0\leq k\leq n}\,(S^\ast_k(r)-S^\ast_{n}(r)+Y_{k+1}(r))$$ has the same distribution as $$\max\big(Y_0(r), \max_{0\leq k\leq n-1}\,(-S^\ast_k(r)+Y_{k+1}(r)-X_{k+1}(r))\big).$$ By assumption, $\mathbb{E}(-S^\ast_1(r))\in (-\infty, 0)$ and $\mathbb{E}(Y(r)-X(r))^+<\infty$. Hence, by Theorem 1.2.1 and Remark 1.2.3 in [@Iksanov:2016], $$\lim_{k\to\infty}(-S^\ast_k(r)+Y_{k+1}(r)-X_{k+1}(r))=-\infty\quad\text{a.s.}$$ As a consequence, the a.s. limit $$\begin{gathered} \lim_{n\to\infty} \max\left(Y_0(r), \max_{0\leq k\leq n-1}\,(-S^\ast_k(r)+Y_{k+1}(r)-X_{k+1}(r)\right)\\ =\max\left(Y_0(r), \max_{k\geq 0}\,(-S^\ast_k(r)+Y_{k+1}(r)-X_{k+1}(r)\right)\end{gathered}$$ is a.s. finite. This completes the proof of [\[eq:proof_prw_joint1\]](#eq:proof_prw_joint1){reference-type="eqref" reference="eq:proof_prw_joint1"}. ◻ *Proof of Theorem [Theorem 5](#main1){reference-type="ref" reference="main1"}.* We apply Proposition [Proposition 3](#main3){reference-type="ref" reference="main3"} with $\mathcal{A}=\mathcal{P}$, $X(p)=\lambda_p(\xi)$ and $Y(p)=\lambda_p(\eta)$. The assumption [\[eq:full_support_assumption\]](#eq:full_support_assumption){reference-type="eqref" reference="eq:full_support_assumption"} in conjunction with $\mathbb{E}[\log^2 \xi]<\infty$ imply that $\mathbb{E}[\lambda_p(\xi)]\in (0,\infty)$ and ${\rm Var}\,[\lambda_p(\xi)]\in [0,\infty)$, for all $p\in\mathcal{P}$. Similarly, $\mathbb{E}[\lambda_p(\eta)]<\infty$ also holds. ◻ **Proposition 4**. *Assume $\mathbb{E}[X(r)]<\infty$, $r\in\mathcal{A}$. Assume further that there exists a finite set $\mathcal{A}_0\subseteq \mathcal{A}$, $d:=|\mathcal{A}_0|$, such that the distributional tail of $(Y(r))_{r\in\mathcal{A}_0}$ is regularly varying at infinity in the following sense. For some positive function $(a(t))_{t>0}$ and a measure $\nu$ satisfying $\nu(\{x\in \mathbb{R}^d:\|x\|\geq r\})=c\cdot r^{-\alpha}$, $c>0$, $\alpha\in (0,1)$, it holds $$\label{eq:reg_varY} t\mathbb{P}\{(a(t))^{-1}(Y(r))_{r\in\mathcal{A}_0}\in\cdot\}~\overset{{\rm v}}{\longrightarrow}~ \nu(\cdot),\quad t\to\infty,$$ on the space of locally finite measures on $(0,\infty]^d$ endowed with the vague topology. If $\mathbb{E}[|Y(r)|]<\infty$, for $r\in\mathcal{A}\setminus \mathcal{A}_0$, then $$\label{eq:exteme_dominates1Y} \left(\left(\frac{\max_{1\leq k\leq \lfloor tu\rfloor}\,T^{\ast}_{k}(r)}{a(t)}\right)_{u\geq 0}\right)_{r\in\mathcal{A}_0}~\overset{{\rm f.d.d.}}{\longrightarrow}~(M_r(u))_{u\geq 0})_{r\in\mathcal{A}_0},\quad t\to\infty,$$ where $(M_r(u))_{u\geq 0})_{r\in\mathcal{A}_0}$ is defined as in [\[eq:extreme_def\]](#eq:extreme_def){reference-type="eqref" reference="eq:extreme_def"}. Moreover, $$\label{eq:exteme_dominates2Y} \left(\left(\frac{\max_{1\leq k\leq \lfloor tu\rfloor}\,T^{\ast}_{k}(r)}{a(t)}\right)_{u\geq 0}\right)_{r\in\mathcal{A}\setminus \mathcal{A}_0}~\overset{{\rm f.d.d.}}{\longrightarrow}~0,\quad t\to\infty.$$* *Proof.* According to Corollary 5.18 in [@Resnick] $$\left(\left(\frac{\max_{1\leq k\leq \lfloor tu\rfloor}Y_{k}(r)}{a(t)}\right)_{u\geq 0}\right)_{r\in\mathcal{A}_0}~\Longrightarrow~\left((M_r(u))_{u\geq 0}\right)_{r\in\mathcal{A}_0},\quad t\to\infty,$$ in the product topology of $D^{\mathbb{N}}$. The function $(a(t))_{t\geq 0}$ is regularly varying at infinity with index $1/\alpha>1$. Thus, by the law of large numbers, for all $r\in\mathcal{A}$, $$\begin{aligned} &\left(\frac{\min_{1\leq k\leq \lfloor tu\rfloor} S^{\ast}_{k-1}(r)}{a(t)}\right)_{u\geq 0}~\overset{{\rm f.d.d.}}{\longrightarrow}~0,\quad t\to\infty,\label{eq:exteme_dominates_proof1-1}\\ &\left(\frac{\max_{1\leq k\leq \lfloor tu\rfloor} S^{\ast}_{k-1}(r)}{a(t)}\right)_{u\geq 0}~\overset{{\rm f.d.d.}}{\longrightarrow}~0,\quad t\to\infty,\label{eq:exteme_dominates_proof1-2}\end{aligned}$$ and [\[eq:exteme_dominates1Y\]](#eq:exteme_dominates1Y){reference-type="eqref" reference="eq:exteme_dominates1Y"} follows from the inequalities $$\begin{gathered} \min_{1\leq k\leq \lfloor tu\rfloor} S^{\ast}_{k-1}(r) +\max_{1\leq k\leq \lfloor tu\rfloor} Y_{k}(r)\leq \max_{1\leq k\leq \lfloor tu\rfloor} T^{\ast}_{k}(r)\\ \leq \max_{1\leq k\leq \lfloor tu\rfloor} S^{\ast}_{k-1}(r) +\max_{1\leq k\leq \lfloor tu\rfloor} Y_{k}(r).\end{gathered}$$ In view of [\[eq:exteme_dominates_proof1-1\]](#eq:exteme_dominates_proof1-1){reference-type="eqref" reference="eq:exteme_dominates_proof1-1"} and [\[eq:exteme_dominates_proof1-2\]](#eq:exteme_dominates_proof1-2){reference-type="eqref" reference="eq:exteme_dominates_proof1-2"} , to prove [\[eq:exteme_dominates2Y\]](#eq:exteme_dominates2Y){reference-type="eqref" reference="eq:exteme_dominates2Y"} it suffices to check that $$\left(\left(\frac{\max_{1\leq k\leq \lfloor tu\rfloor}Y_{k}(r)}{a(t)}\right)_{u\geq 0}\right)~\overset{{\rm f.d.d.}}{\longrightarrow}~0,\quad t\to\infty,$$ for every fixed $r\in\mathcal{A}\setminus\mathcal{A}_0$. This, in turn, follows from $$\frac{Y_{n}(r)}{n}~\overset{{\rm a.s.}}{\longrightarrow}~0,\quad n\to\infty,\quad r\in\mathcal{A}\setminus\mathcal{A}_0,$$ which is a consequence of the assumption $\mathbb{E}[|Y(r)|]<\infty$, $r\in\mathcal{A}\setminus\mathcal{A}_0$ and the Borel-Cantelli lemma. ◻ *Proof of Theorem [Theorem 6](#main11){reference-type="ref" reference="main11"}.* Follows immediately from Proposition [Proposition 4](#prop:eta_dominatesY){reference-type="ref" reference="prop:eta_dominatesY"} applied with $\mathcal{A}=\mathcal{P}$, $X(p)=\lambda_p(\xi)$ and $Y(p)=\lambda_p(\eta)$. ◻ # Proof of Theorem [Theorem 7](#main2){reference-type="ref" reference="main2"} {#proof-of-theorem-main2} We aim at proving that $$\label{eq:main2_proof1} \frac{\sum_{p\in\mathcal{P}}\left(\max_{1\leq k\leq n} T_k(p)-S_{n-1}(p)\right)\log p}{\sqrt{n}}~\overset{{\mathbb{P}}}{\longrightarrow}0,\quad n\to\infty,$$ which together with the relation $$\sum_{p\in\mathcal{P}}S_{n}(p)\log p = \log \Pi_{n}= \log \mathcal{\Pi}_n,\quad n\in\mathbb{N},$$ implies Theorem [Theorem 7](#main2){reference-type="ref" reference="main2"} by Slutskiy's lemma and [\[eq:1\]](#eq:1){reference-type="eqref" reference="eq:1"}. Let $(\xi_0,\eta_0)$ be an independent copy of $(\xi,\eta)$ which is also independent of $(\xi_n,\eta_n)_{n\in\mathbb{N}}$. By the same reasoning as we have used in the proof of [\[eq:proof_prw_joint1\]](#eq:proof_prw_joint1){reference-type="eqref" reference="eq:proof_prw_joint1"} we obtain $$(\max_{1\leq k\leq n} T_k(p)-S_{n-1}(p))_{p\in\mathcal{P}} \overset{d}{=}\left(\max\left(\lambda_p(\eta_0),\max_{1\leq k<n}(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p))\right)\right)_{p\in\mathcal{P}}.$$ Taking into account $$\sum_{p\in\mathcal{P}}\lambda_p(\eta_0)\log p=\log \eta_0,$$ we see that [\[eq:main2_proof1\]](#eq:main2_proof1){reference-type="eqref" reference="eq:main2_proof1"} is a consequence of $$\label{eq:main2_proof2} \frac{\sum_{p\in\mathcal{P}}\max_{1\leq k<n}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p)\right)^{+}\log p}{\sqrt{n}}~\overset{{\mathbb{P}}}{\longrightarrow}0,\quad n\to\infty,$$ Since, for every fixed $p\in\mathcal{P}$, $$\label{eq:maximum_finite} \max_{k\geq 1}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p)\right)^{+}<\infty\quad \text{a.s.}$$ by assumption [\[eq:full_support_assumption\]](#eq:full_support_assumption){reference-type="eqref" reference="eq:full_support_assumption"}, it suffices to check that, for every fixed $\varepsilon>0$, $$\label{eq:main2_proof3} \lim_{M\to\infty}\limsup_{n\to\infty}\mathbb{P}\left\{\sum_{p\in\mathcal{P},p>M}\max_{1\leq k<n}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p)\right)^{+}\log p>\varepsilon\sqrt{n}\right\}.$$ In order to check [\[eq:main2_proof3\]](#eq:main2_proof3){reference-type="eqref" reference="eq:main2_proof3"} we divide the sum into two disjoint parts with summations over $\mathcal{P}_1(n)$ and $\mathcal{P}_2(n)$. For the first sum, by Markov's inequality, we obtain $$\begin{aligned} &\hspace{-0.4cm}\mathbb{P}\left\{\sum_{p\in\mathcal{P}_1(n),p>M}\max_{1\leq k<n}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p)\right)^{+}\log p>\varepsilon\sqrt{n}/2\right\}\\ &\leq \frac{2}{\varepsilon\sqrt{n}}\sum_{p\in\mathcal{P}_1(n),p>M}\mathbb{E}\left(\max_{1\leq k<n}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p)\right)^{+}\right)\log p\\ &\leq \frac{2}{\varepsilon\sqrt{n}}\sum_{p\in\mathcal{P}_1(n),p>M}\log p\sum_{k\geq 1}\mathbb{E}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p)\right)^{+}\\ &= \frac{2}{\varepsilon\sqrt{n}}\sum_{p\in\mathcal{P}_1(n),p>M}\log p\sum_{j\geq 1}\mathbb{P}\{\lambda_p(\eta)-\lambda_p(\xi)=j\}\sum_{k\geq 1}\mathbb{E}(j-S_{k-1}(p))^{+}\\ &\leq \frac{2}{\varepsilon\sqrt{n}}\sum_{p\in\mathcal{P}_1(n),p>M}\log p\sum_{j\geq 1}j\mathbb{P}\{\lambda_p(\eta)-\lambda_p(\xi)=j\}\sum_{k\geq 0}\mathbb{P}\{S_{k}(p)\leq j\}\\ &\leq \frac{2}{\varepsilon\sqrt{n}}\sum_{p\in\mathcal{P}_1(n),p>M}\log p\sum_{j\geq 1}j\mathbb{P}\{\lambda_p(\eta)-\lambda_p(\xi)=j\}\frac{2 j}{\mathbb{E}[(\lambda_p(\xi)\wedge j)]},\end{aligned}$$ where last estimate is a consequence of Erickson's inequality for renewal functions, see Eq. (6.5) in [@Iksanov:2016]. Further, since for $p\in\mathcal{P}_1(n)$, $$\mathbb{E}[(\lambda_p(\xi)\wedge j)]\geq \mathbb{P}\{\lambda_p(\xi)\geq 1\}=\mathbb{P}\{\lambda_p(\xi)>0\}\geq n^{-1/2},$$ we obtain $$\begin{aligned} &\hspace{-1cm}\mathbb{P}\left\{\sum_{p\in\mathcal{P}_1(n),p>M}\max_{1\leq k<n}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p)\right)^{+}\log p>\varepsilon\sqrt{n}/2\right\}\\ &\leq \frac{4}{\varepsilon}\sum_{p\in\mathcal{P}_1(n),p>M}\log p \mathbb{E}\left[ ((\lambda_p(\eta)-\lambda_p(\xi))^{+})^2\right]\\ &\leq \frac{4}{\varepsilon}\sum_{p\in\mathcal{P},p>M}\log p \mathbb{E}\left[((\lambda_p(\eta)-\lambda_p(\xi))^{+})^2\right].\end{aligned}$$ The right-hand side converges to $0$, as $M\to\infty$ by [\[eq:second_moment_diff\]](#eq:second_moment_diff){reference-type="eqref" reference="eq:second_moment_diff"}. For the sum over $\mathcal{P}_2(n)$ the derivation is simpler. By Markov's inequality $$\begin{aligned} &\hspace{-0.4cm}\mathbb{P}\left\{\sum_{p\in\mathcal{P}_2(n),p>M}\max_{1\leq k<n}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p)\right)^{+}\log p>\varepsilon\sqrt{n}/2\right\}\\ &\leq \frac{2}{\varepsilon\sqrt{n}}\mathbb{E}\left[\sum_{p\in\mathcal{P}_2(n),p>M}\max_{1\leq k<n}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)-S_{k-1}(p)\right)^{+}\log p\right]\\ &\leq \frac{2n}{\varepsilon\sqrt{n}}\mathbb{E}\left[\sum_{p\in\mathcal{P}_2(n),p>M}\left(\lambda_p(\eta_k)-\lambda_p(\xi_k)\right)^{+}\log p\right],\end{aligned}$$ and the right-hand side tends to zero as $n\to\infty$ in view of [\[eq:main2_eta_negligible\]](#eq:main2_eta_negligible){reference-type="eqref" reference="eq:main2_eta_negligible"}. The proof is complete. # Proof of Theorem [Theorem 9](#main21){reference-type="ref" reference="main21"} {#proof-of-theorem-main21} From Theorem [Theorem 6](#main11){reference-type="ref" reference="main11"} with the aid of the continuous mapping theorem we conclude that $$\left(\frac{\sum_{p\in\mathcal{P}_0}\max_{1\leq k\leq \lfloor tu\rfloor}T_k(p)\log p}{a(t)}\right)_{u\geq 0}~\overset{{\rm f.d.d.}}{\longrightarrow}~\left(\sum_{p\in\mathcal{P}_0}M_p(u)\log p\right)_{u\geq 0},$$ as $t\to\infty$. It suffices to check $$\label{eq:main21_proof1} \left(\frac{\sum_{p\in\mathcal{P}\setminus\mathcal{P}_0}\max_{1\leq k\leq \lfloor tu\rfloor}T_k(p)\log p}{a(t)}\right)_{u\geq 0}~\overset{{\rm f.d.d.}}{\longrightarrow}~0,\quad t\to\infty.$$ Since $(a(t))$ is regularly varying at infinity, [\[eq:main21_proof1\]](#eq:main21_proof1){reference-type="eqref" reference="eq:main21_proof1"} follows from $$\label{eq:main21_proof2} \frac{\sum_{p\in\mathcal{P}\setminus\mathcal{P}_0}\mathbb{E}[\max_{1\leq k\leq n}T_k(p)]\log p}{a(n)}~\to~0,\quad n\to\infty,$$ by Markov's inequality. To check the latter note that $$\begin{aligned} &\hspace{-1cm}\sum_{p\in\mathcal{P}\setminus\mathcal{P}_0}\mathbb{E}[\max_{1\leq k\leq n}T_k(p)]\log p\leq \sum_{p\in\mathcal{P}\setminus\mathcal{P}_0}\mathbb{E}[S_{n-1}(p)+\max_{1\leq k\leq n}\lambda_p(\eta_k)]\log p\\ &\leq (n-1)\sum_{p\in\mathcal{P}\setminus\mathcal{P}_0}\mathbb{E}[\lambda_p(\xi)]\log p+n\sum_{p\in\mathcal{P}\setminus\mathcal{P}_0}\mathbb{E}[\lambda_p(\eta)]\log p\\ &\leq (n-1)\mathbb{E}[\log\xi]+n\sum_{p\in\mathcal{P}\setminus\mathcal{P}_0}\mathbb{E}[\lambda_p(\eta)]\log p=O(n),\quad n\to\infty,\end{aligned}$$ where we have used that $\mathbb{E}[\log \xi]<\infty$ and the assumption [\[eq:eta_without_P0\]](#eq:eta_without_P0){reference-type="eqref" reference="eq:eta_without_P0"}. Using that $\alpha\in (0,1)$ and $(a(t))$ is regularly varying at infinity with index $1/\alpha$, we obtain [\[eq:main21_proof2\]](#eq:main21_proof2){reference-type="eqref" reference="eq:main21_proof2"}. # Acknowledgment {#acknowledgment .unnumbered} The research was supported by the National Research Foundation of Ukraine (project 2020.02/0014 'Asymptotic regimes of perturbed random walks: on the edge of modern and classical probability').

arxiv_math

--- abstract: | Stabilizer-free $P_k$ virtual elements are constructed on polygonal and polyhedral meshes. Here the interpolating space is the space of continuous $P_k$ polynomials on a triangular-subdivision of each polygon, or a tetrahedral-subdivision of each polyhedron. With such an accurate and proper interpolation, the stabilizer of the virtual elements is eliminated while the system is kept positive-definite. We show that the stabilizer-free virtual elements converge at the optimal order in 2D and 3D. Numerical examples are computed, validating the theory. address: - Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong - Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hongkong, China - Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA. author: - Yanping Lin - Mo Mu - Shangyou Zhang title: Stabilizer-free polygonal and polyhedral virtual elements --- [^1] [^2] # Introduction The virtual element method is proposed and studied in [@Beirao; @Beirao16; @Cao-Chen; @Cao-Chen2; @Chen1; @Chen2; @Chen3; @Feng-Huang; @Feng-Huang2; @Huang1; @Huang2; @Huang3]. In this work, we construct stabilizer-free $P_k$ virtual elements on polygonal and polyhedral meshes. We consider the Poisson equation, $$\begin{aligned} \label{p-e} \begin{aligned} -\Delta u & = f && \hbox{\rm{in }} \ \Omega, \\ u&=0 && \hbox{\rm{on }} \ \partial \Omega, \end{aligned} \end{aligned}$$ where $\Omega\subset \mathbb{R}^d$ is a bounded polygonal or polyhedral domain and $f\in L^2(\Omega)$. The variation form reads: Find $u\in H^1_0(\Omega)$ such that $$\begin{aligned} \label{w-e} \begin{aligned} (\nabla u, \nabla v) & = ( f,v) && \forall v \in H^1_0(\Omega),\end{aligned} \end{aligned}$$ where $(\cdot, \cdot)$ is the $L^2$ inner product on $\Omega$ and we have $|v|_1^2=(\nabla v,\nabla v)$. Let $\mathcal{T}_h=\{ K \}$ be a quasi-uniform polygonal or polyhedral mesh on $\Omega$ with $h$ as the maximum size of the polygons or polyhedrons $K$. Let $\mathcal{E}_h$ denote the set of edges $e$ in $\mathcal{T}_h$. In 3D, let $\mathcal{F}_h$ be the set of all face-polygons $F$ in $\mathcal{T}_h$. For $k\ge 1$, the virtual element space is defined as $$\begin{aligned} \label{t-V-h} \begin{aligned} \tilde V_h=\{ v\in H^1_0(\Omega) & : \tilde v \in \mathbb{B}_k(\mathcal{E}_h ), \Delta \tilde v|_K \in P_{k-2}(K) \} \hbox{\rm{\ in 2D, or as }}\\ \tilde V_h=\{ v\in H^1_0(\Omega) & : \tilde v \in \mathbb{B}_k(\mathcal{E}_h); \Delta_F \tilde v|_F \in P_{k-2}(F), \ F\in \mathcal{F}_h; \\ &\qquad \Delta \tilde v|_K \in P_{k-2}(K) \} \end{aligned} \end{aligned}$$ in 3D, where $P_{-1}=\{0\}$, $\mathbb{B}_k(\mathcal{E}_h)=\{ v\in C^0(\mathcal{E}_h) : v|_e\in P_k(e) \ \forall e\subset \mathcal{E}_h \}$, and $\Delta_F$ is the 2D Laplacian on the flat polygon $F$. In computation, the interpolated virtual finite element space on $\mathcal{T}_h$ is defined by $$\begin{aligned} \label{V-h} V_h = \{ v_h=\Pi_h^\nabla \tilde v \ : \ v_h|_K \in \mathbb{V}_k(K), \ K\in\mathcal{T}_h; \ \tilde v\in \tilde V_h \}, \end{aligned}$$ where $\mathbb{V}_k(K)=P_k(K)$ for the standard virtual elements (and to be defined below in [\[V-k\]](#V-k){reference-type="eqref" reference="V-k"} for the new virtual element method), and $v_h=\Pi_h^\nabla \tilde v$ is the local $H^1$-projection: $$\begin{aligned} \left\{ \begin{aligned} (\nabla(v_h-\tilde v), \nabla w_h)_K&=0\quad \forall w_h=\Pi_h^\nabla \tilde w \in \mathbb{V}_k(K), \\ \langle v_h-\tilde v, w_h\rangle_{\partial K}& =0 \quad \forall w_h\in P_k(K). \end{aligned} \right. \end{aligned}$$ The stabilizer-free virtual element equation reads: Find $u_h=\Pi_h^\nabla\tilde u\in V_h$ such that $$\begin{aligned} \label{f-e} (\nabla u_h,\nabla v_h)_h = (f,v_h) \quad \forall \tilde v\in \tilde V_h, \ v_h=\Pi_h^\nabla\tilde v, \end{aligned}$$ where $(\nabla u_h, \nabla v_h)_h=\sum_{K\in \mathcal{T}_h} (\nabla u_h, \nabla v_h)_K$. In 3D, to find the value of $\tilde v$ inside a face-polygon, we use the moments $\int_F \tilde v_h p_{k-2} dS$ instead of the surface Laplacian values $\Delta_F \tilde v_h\in P_{k-2}(F)$, as the latter uniquely determines $\tilde v_h$ and consequently uniquely determines the $P_{k-2}$ moments of $\tilde v_h$ on $F$. Because the dimension of $V_h$ is less than that of $\tilde V_h$ (equal only when $k=1$ on triangular and tetrahedral meshes), the bilinear form in [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} is not positive-definite and the equation does not have a unique solution. Thus a discrete stabilizer must be added to the equation [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} if the interpolation space $\mathbb{V}_k(K)$ is defined to be $P_k(K)$. With a stabilizer, many degrees of freedom do not fully contribute their approximation power as they are averaged into a smaller dimensional vector space. To be a stabilizer free virtual element, the interpolation space must have at least no less degrees of freedom on each element. But raising the polynomial degree of $\mathbb{V}_k(K)$ in [\[V-h\]](#V-h){reference-type="eqref" reference="V-h"} does not work. It works only for $P_1$ virtual elements in 2D with special treatment, cf. [@Berrone-0; @Berrone-1], where $\mathbb{V}_k(K)=P_{k+l(n)}(K)$ and $n$ is the number of edges of $K$, in the virtual element space [\[V-h\]](#V-h){reference-type="eqref" reference="V-h"}. Another stabilizer-free method for $k=1$ is proposed in [@Berrone] that $\mathbb{V}_k(K)=P_k(K)\cup H_l(K)$, where $H_l(K)$ is the set of 2D harmonic polynomials of degree $l$ or less, and $l$ depends on the number of edges of $K$. This is an excellent idea because the $H_l$ harmonic polynomials may help to gather all boundary edge values while not destroying the gradient approximation, as harmonic polynomials have vanishing Laplacian. The same idea has been implemented in some other finite elements [@Al-Taweel; @Sorokina1; @Sorokina2]. But the method of [@Berrone] is also for 2D $P_1$ polygonal elements, as it is shown numerically not working for $k>3$ in [@Xu-Z]. We propose to use macro-triangles or macro-tetrahedrons $C^0$-$P_k$ spaces as the interpolation space $\mathbb{V}_k(K)$ in [\[V-h\]](#V-h){reference-type="eqref" reference="V-h"}. This method was first used in [@Xu-Z] for $P_k$ triangular virtual elements only. In [@Xu-Z], each triangle $K$ is split into three triangles by connecting its barycenter with the three vertices. $K$ is called a Hsieh-Clough-Tocher macro-triangle [@Clough; @Sorokina2; @Xu-Zhang; @ZhangMG; @Zhang3D]. In this work, we extend the method to polygonal and polyhedral virtual elements. It turns out that the triangular virtual element would be the most complicated case, as we have to introduce a new point in the subdivision in order to get a sufficiently large dimensional vector space. For most polygons and polyhedrons we can subdivide them into triangles and tetrahedrons respectively without adding any new point, when we have enough face-edge and face-polygon degrees of freedom. A different interpolation space $\mathbb{V}_k$ changes the quadrature rule for computing $(\nabla u_h, \nabla v_h)= (\nabla \Pi_h^\nabla \tilde u, \nabla \Pi_h^\nabla \tilde v)$. Such an accurate local interpolation does not increase the computational cost, once the local stiffness matrix is generated. On the other side, eliminating the stabilizer may reduce computational cost, and may improve the condition number of the resulting linear system. More importantly, a stabilizer-free method may utilize fully every degree of freedom in the discrete approximation. Thus stabilizer-free methods may result in superconvergence. The stabilizer is eliminated in the weak Galerkin finite element method [@Al-Taweel-Wang; @Feng-Zhang; @Gao-Z; @Mu1; @Wang-Z20; @Wang-Z21; @Ye-Z20a; @Ye-Z20b; @Ye-Z20c; @Ye-Z21b; @Ye-Z21c; @Ye-Z21d]. It is also eliminated in the $H(\hbox{\rm{div}})$ finite element method [@Mu21; @Ye-Z21a; @Ye-Z21f]. The stabilizer-free $C^0$ or $C^{-1}$ nonconforming finite elements are constructed for the biharmonic equation [@Ye-Z20d; @Ye-Z22a; @Ye-Z22b]. We have stabilizer-free discontinuous Galerkin finite element methods [@Feng-Z21; @Mu23; @Ye-Z20-1; @Ye-Z20-2; @Ye-Z20d]. Without a stabilizer, two-order superconvergent weak Galerkin finite elements are found in [@Al-Taweel-Z21; @Wang-Z23; @Wang-Z23a; @Ye-Z21g; @Ye-Z23b; @Ye-Z23e]. Also two-order superconvergent stabilizer-free discontinuous Galerkin finite elements are constructed in [@Ye-Z22c; @Ye-Z22d; @Ye-Z23c] for second order elliptic equations. One or two-order superconvergent weak Galerkin finite elements are found for the Stokes equations in [@Mu21b; @Ye-Z21e; @Ye-Z22e]. Four-order superconvergent weak Galerkin finite elements [@Ye-Z23d] and four-order superconvergent discontinuous Galerkin finite elements [@Ye-Z22d; @Ye-Z23e] are all stabilizer-free, for the biharmonic equation. For example, a $P_3$ discontinuous finite element solution is as accurate as a $C^1$-$P_7$ finite element solution in solving a 2D biharmonic equation. In this paper, we show that with the macro-triangle/tetrahedron interpolation, the stabilizer-free virtual element equation [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} has a unique and quasi-optimal solution. Numerical examples on the new stabilizer-free virtual elements are computed, verifying the theory. .7cm # The 2D interpolation We define in this section the 2D macro-triangle interpolation space and show that the stabilizer-free virtual element equation has a unique solution. (330,100)(0,0) (0,0) (100,100)(0,0) (20,0)(2,1)60(20,0)(1,1)80 (20,0)(1,2)40(20,0)(0,1)50 (0,80)(a): ( 20.00, 0.00)( 0.250, 0.000)320 ( 100.00, 0.00)( -0.139, 0.208)144 ( 80.00, 30.00)( 0.093, 0.232)215 ( 100.00, 80.00)( -0.250, 0.000)160 ( 60.00, 80.00)( -0.200, -0.150)200 ( 20.00, 50.00)( -0.139, -0.208)144 ( 0.00, 20.00)( 0.177, -0.177)113 (110,0) (100,100)(0,0) (50,30)(-5,-3)50(50,30)(5,-3)50 (50,30)(0,1)60 ( 0.00, 0.00)( 0.250, 0.000)400 ( 100.00, 0.00)( -0.121, 0.219)411 ( 0.00, 0.00)( 0.121, 0.219)411 (0,80)(b): (220,0) (100,100)(0,0) (20,0)(2,1)60(80,30)(-2,5)20 (20,0)(1,2)40(20,0)(0,1)50 (80,30)(-3,1)60 ( 20.00, 0.00)( 0.250, 0.000)320 ( 100.00, 0.00)( -0.139, 0.208)144 ( 80.00, 30.00)( 0.093, 0.232)215 ( 100.00, 80.00)( -0.250, 0.000)160 ( 60.00, 80.00)( -0.200, -0.150)200 ( 20.00, 50.00)( -0.139, -0.208)144 ( 0.00, 20.00)( 0.177, -0.177)113 (0,80)(c): Let $K$ be a 2D polygon. The only requirement is that $K$ is subdivided into more than one tetrahedron. If $K$ has only three sides, i.e., $K$ is a triangle, we add a barycenter point to the triangle, shown as in the Figure [\[f-3-p\]](#f-3-p){reference-type="ref" reference="f-3-p"}(b) macro triangle on $K$. If $K$ is a polygon of four sides or more, we usually can connect some vertices of $K$ to subdivide $K$ into a macro-triangle polygon, cf. Figure [\[f-3-p\]](#f-3-p){reference-type="ref" reference="f-3-p"}(a). If needed, we can add one or two inner points to subdivide $K$, cf. Figure [\[f-3-p\]](#f-3-p){reference-type="ref" reference="f-3-p"}(c), where we intentionally add a new point for the purpose of illustration. With the subdivision $K=\cup_{T_i\subset K} T_i$, we define the interpolation space as, for $k\ge 1$, $$\begin{aligned} \label{V-k} \mathbb{V}_k(K) =\{ v_h \in C^0(K) : v_h|_{T_i} \in P_k(T_i), \ T_i\subset K\}. \end{aligned}$$ One can easily count the internal degrees of freedom of $\mathbb{V}_k(K)$ to get $$\begin{aligned} \dim (\mathbb{V}_k\cap H^1_0(K)) > \dim P_{k-2}(K). \end{aligned}$$ The interpolation operator is defined to be the local $H^1$-projection, i.e., $v_h=\Pi_h^\nabla\tilde v \in \mathbb{V}_k$ such that $v_h|_{\partial K} = \tilde v$ and $$\begin{aligned} \label{l-e} (\nabla (v_h-\tilde v), \nabla w_h)=0 \quad \forall w_h\in \mathbb{V}_k(K). \end{aligned}$$ **Lemma 1**. *The interpolation operator $\Pi_h^\nabla$ is well defined in [\[l-e\]](#l-e){reference-type="eqref" reference="l-e"} and it preserves $P_k$ polynomials, $$\begin{aligned} \label{p-p} \Pi_h^\nabla\tilde v = \tilde v\quad \ \hbox{\rm{if }} \ \tilde v\in P_k(K). \end{aligned}$$* *Proof.* Because $\tilde v|_{\partial K}\in \mathbb{B}_k(\mathcal{E}_h)$, $v_h$ can assume the boundary condition $v_h=\tilde v$ exactly on $\partial K$. The linear system of equations in [\[l-e\]](#l-e){reference-type="eqref" reference="l-e"} is a finite dimensional square system. The existence is implied by the uniqueness. To show the uniqueness, we let $\tilde v=0$ in [\[l-e\]](#l-e){reference-type="eqref" reference="l-e"}. Letting $w_h=v_h$ in [\[l-e\]](#l-e){reference-type="eqref" reference="l-e"}, we get $$\begin{aligned} \nabla v_h ={\bf 0} \quad \hbox{\rm{ on }} \ K. \end{aligned}$$ Thus $v_h=c$ is a constant on $K$. As $v_h$ is continuous on edges, $v_h=c$ is a global constant on the whole domain. By the boundary condition, we get $0=\tilde v|_ {\partial \Omega}=v_h|_{\partial \Omega}=c$. Hence $v_h=0$ and [\[l-e\]](#l-e){reference-type="eqref" reference="l-e"} has a unique solution. If $\tilde v\in P_k(K)\subset \mathbb{V}_k(K)$, defined in [\[V-h\]](#V-h){reference-type="eqref" reference="V-h"}, then the solution of [\[l-e\]](#l-e){reference-type="eqref" reference="l-e"} says, letting $w_h=v_h-\tilde v$, $$\begin{aligned} \nabla (v_h-\tilde v)={\bf 0}. \end{aligned}$$ Thus $v_h-\tilde v$ is a global constant which must be zero as it vanishes at all $\partial K$. [\[p-p\]](#p-p){reference-type="eqref" reference="p-p"} is proved. ◻ .7cm **Lemma 2**. *The stabilizer-free virtual element equation [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} has a unique solution, where the interpolation $\Pi_h^\nabla$ is defined in [\[l-e\]](#l-e){reference-type="eqref" reference="l-e"}.* *Proof.* As both $\tilde u, \tilde v \in \tilde V_h$, [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} is a finite square system of linear equations. The uniqueness of solution implies the existence. To show the uniqueness, we let $f=0$ and $\tilde v=\tilde u$ in [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"}. It follows that $$\begin{aligned} |\Pi_h^\nabla\tilde u|_{1,h} =0. \end{aligned}$$ Thus $\Pi_h^\nabla\tilde u=c$ is constant on each $K$. But $\Pi_h^\nabla\tilde u$ is continuous on the whole domain. By the boundary condition, we get $0=\Pi_h^\nabla\tilde u|_ {\partial \Omega}=c$. That is, $$\begin{aligned} \label{p-u-0} \Pi_h^\nabla\tilde u=0 \ \hbox{\rm{ and }} \ \tilde u|_{\partial K}=\Pi_h^\nabla\tilde u=0. \end{aligned}$$ For $k=1$, $\tilde u$ has no internal degree of freedom, and the lemma is proved by [\[p-u-0\]](#p-u-0){reference-type="eqref" reference="p-u-0"}, $$\begin{aligned} \tilde u=0, \ \hbox{\rm{ if }} \ k=1. \end{aligned}$$ For $k\ge 2$, let $$\begin{aligned} \label{b-K} b_K &=\sum_{i\in \mathcal{N}_2} \phi_i\in H^1_0(K)\cap \mathbb{V}_k(K), \end{aligned}$$ where $\mathcal{N}_2$ is the set of all internal mid-edge points of $\{ T_i\}$, $K=\cup T_i$, and $\phi_i$ is the $P_2$ Lagrange nodal basis at node $i$. Then $b_K>0$ inside polygon $K$ if it does not have any added internal point, cf. Figure [\[f-3-p\]](#f-3-p){reference-type="ref" reference="f-3-p"}. Otherwise, $b_K>0$ inside $K$ except at one or two internal points where $b_K=0$. On one polygon $K$, by [\[l-e\]](#l-e){reference-type="eqref" reference="l-e"}, [\[p-u-0\]](#p-u-0){reference-type="eqref" reference="p-u-0"} and integration by parts, we have $$\begin{aligned} \label{l-e1} (-\Delta \tilde u, w_h) = (\nabla \tilde u, \nabla w_h)=0 \quad \forall w_h\in H^1_0(K)\cap \mathbb{V}_k(K). \end{aligned}$$ By the space $\tilde V_h$ definition [\[t-V-h\]](#t-V-h){reference-type="eqref" reference="t-V-h"}, we denote $$\begin{aligned} \label{p-k-2} p_{k-2} = -\Delta \tilde u \in P_{k-2}(K). \end{aligned}$$ Let the $w_h$ in [\[l-e1\]](#l-e1){reference-type="eqref" reference="l-e1"} be $$\begin{aligned} \label{w-h} w_h= p_{k-2} b_K \in H^1_0(K)\cap \mathbb{V}_k(K), \end{aligned}$$ where the positive $P_2$ bubble $b_K$ is defined in [\[b-K\]](#b-K){reference-type="eqref" reference="b-K"}. With the $w_k$ in [\[w-h\]](#w-h){reference-type="eqref" reference="w-h"}, we get from [\[l-e1\]](#l-e1){reference-type="eqref" reference="l-e1"} and [\[p-k-2\]](#p-k-2){reference-type="eqref" reference="p-k-2"} that $$\begin{aligned} \int_K p_{k-2}^2 b_K d {\bf x} =0. \end{aligned}$$ As $b_K>0$ inside $K$ (other than 1 or 2 possibly internal points), it follows that $$\begin{aligned} p_{k-2}^2 =0 \ \hbox{\rm{ and }} \ p_{k-2} =0 \ \hbox{\rm{ on }} \ K. \end{aligned}$$ By [\[p-u-0\]](#p-u-0){reference-type="eqref" reference="p-u-0"} and [\[p-k-2\]](#p-k-2){reference-type="eqref" reference="p-k-2"}, $\Delta \tilde u=0$ in $K$ and $\tilde u=0$ on $\partial K$. Thus, by the unique solution of the Laplace equation, $\tilde u=0$. The lemma is proved. ◻ .7cm # The 3D interpolation We define in this section the 3D macro-tetrahedron interpolation space and show that the stabilizer-free virtual element equation has a unique solution when using the interpolation. (320,100)(0,0) (0,0) (100,100)(0,0) (0,80)(a): ( 40.00, 100.00)( -0.093, -0.232)430 ( 40.00, 100.00)( 0.129, -0.214)466 ( 40.00, 100.00)( 0.177, -0.177)282 ( 0.00, 0.00)( 0.250, 0.000)400 ( 90.00, 50.00)( 0.049, -0.245)203 ( 88.25, 49.03)( -2.622, -1.457) 34 (110,0) (100,100)(0,0) ( 40.00, 100.00)( -0.093, -0.232)430 ( 40.00, 100.00)( 0.129, -0.214)466 ( 40.00, 100.00)( 0.177, -0.177)282 ( 0.00, 0.00)( 0.250, 0.000)400 ( 90.00, 50.00)( 0.049, -0.245)203 ( 88.25, 49.03)( -2.622, -1.457) 34 ( 46.60, 33.30)( -0.025, 0.249)268 ( 46.60, 33.30)( 0.212, -0.132)251 ( 46.60, 33.30)( -0.203, -0.145)229 ( 76.60, 50.00)( 0.250, 0.000) 53 ( 76.60, 50.00)( 0.106, -0.226)220 ( 76.60, 50.00)( -0.148, 0.202)247 ( 61.37, 16.09)( -2.902, -0.761) 20 ( 65.12, 15.78)( 2.733, -1.236) 12 ( 64.55, 18.16)( 1.873, 2.343) 14 ( 45.30, 50.00)( 3.000, 0.000) 14 ( 41.99, 48.49)( -1.964, -2.268) 22 ( 43.17, 52.00)( -0.198, 2.993) 16 (0,80)(b): (220,0) (100,100)(0,0) ( 40.00, 100.00)( -0.093, -0.232)430 ( 40.00, 100.00)( 0.129, -0.214)466 ( 40.00, 100.00)( 0.177, -0.177)282 ( 0.00, 0.00)( 0.250, 0.000)400 ( 90.00, 50.00)( 0.049, -0.245)203 ( 88.25, 49.03)( -2.622, -1.457) 34 ( 55.82, 36.41)( -2.513, -1.639) 22 ( 56.87, 39.40)( -0.949, 2.846) 18 ( 59.37, 38.22)( 2.800, 1.077) 10 ( 59.00, 36.18)( 2.250, -1.985) 18 ( 55.63, 36.78)( -2.799, -1.079) 2 ( 59.17, 38.60)( 2.510, 1.643) 6 ( 56.00, 38.82)( -2.252, 1.982) 6 ( 58.03, 35.57)( 0.802, -2.891) 6 ( 46.60, 33.30)( -0.025, 0.249)268 ( 46.60, 33.30)( 0.212, -0.132)251 ( 46.60, 33.30)( -0.203, -0.145)229 ( 76.60, 50.00)( 0.250, 0.000) 53 ( 76.60, 50.00)( 0.106, -0.226)220 ( 76.60, 50.00)( -0.148, 0.202)247 ( 61.37, 16.09)( -2.902, -0.761) 20 ( 65.12, 15.78)( 2.733, -1.236) 12 ( 64.55, 18.16)( 1.873, 2.343) 14 ( 45.30, 50.00)( 3.000, 0.000) 14 ( 41.99, 48.49)( -1.964, -2.268) 22 ( 43.17, 52.00)( -0.198, 2.993) 16 (0,80)(c): Let $K$ be a 3D polyhedron. The first requirement is that each face-polygon $F$ must be subdivided into more than one triangle. The subdivision of polygons is defined in the last section. For example, if a face polygon $F$ has only three sides, i.e., $F$ is a triangle, we must add a barycenter point to subdivide it into three triangles, cf. Figure [\[f-2-t\]](#f-2-t){reference-type="ref" reference="f-2-t"}(b). However if a face polygon has more than three edges, we usually can subdivide it into triangles easily, cf. Figure [\[f-3-c\]](#f-3-c){reference-type="ref" reference="f-3-c"}(b). After each face polygon is subdivided into more than one triangle, the next requirement in subdividing $K$ is that every resulting tetrahedron has at least two face-triangles inside $K$. For example, after cutting each face-triangle of a tetrahedron $K$ into three triangles, we add one more internal point to cut $K$ into twelve tetrahedrons, cf. Figure [\[f-2-t\]](#f-2-t){reference-type="ref" reference="f-2-t"}(c). (330,100)(0,0) (0,0) (100,100)(0,0) (0,80)(a): ( 0.00, 0.00)( 0.250, 0.000)240 ( 0.00, 0.00)( 0.000, 0.250)240 ( 60.00, 60.00)( -0.250, 0.000)240 ( 40.00, 90.00)( -0.200, -0.150)200 ( 40.00, 90.00)( 0.250, 0.000)240 ( 100.00, 30.00)( 0.000, 0.250)240 ( 100.00, 30.00)( -0.200, -0.150)200 ( 60.00, 60.00)( 0.000, -0.250)240 ( 60.00, 60.00)( 0.200, 0.150)200 ( 42.00, 30.00)( 3.000, 0.000) 20 ( 38.40, 28.80)( -2.400, -1.800) 16 ( 40.00, 32.00)( 0.000, 3.000) 20 (110,0) (100,100)(0,0) ( 0.00, 0.00)( 0.250, 0.000)240 ( 0.00, 0.00)( 0.000, 0.250)240 ( 60.00, 60.00)( -0.250, 0.000)240 ( 40.00, 90.00)( -0.200, -0.150)200 ( 40.00, 90.00)( 0.250, 0.000)240 ( 100.00, 30.00)( 0.000, 0.250)240 ( 100.00, 30.00)( -0.200, -0.150)200 ( 60.00, 60.00)( 0.000, -0.250)240 ( 60.00, 60.00)( 0.200, 0.150)200 ( 42.00, 30.00)( 3.000, 0.000) 20 ( 38.40, 28.80)( -2.400, -1.800) 16 ( 40.00, 32.00)( 0.000, 3.000) 20 ( 60.00, 60.00)( -0.139, 0.208)144 ( 60.00, 60.00)( -0.177, -0.177)339 ( 60.00, 60.00)( 0.200, -0.150)200 ( 41.11, 28.34)( 1.664, -2.496) 12 ( 38.40, 31.20)( -2.400, 1.800) 16 ( 41.11, 31.66)( 1.664, 2.496) 12 ( 41.41, 31.41)( 2.121, 2.121) 28 (0,80)(b): (220,0) (100,100)(0,0) ( 0.00, 0.00)( 0.250, 0.000)240 ( 0.00, 0.00)( 0.000, 0.250)240 ( 60.00, 60.00)( -0.250, 0.000)240 ( 40.00, 90.00)( -0.200, -0.150)200 ( 40.00, 90.00)( 0.250, 0.000)240 ( 100.00, 30.00)( 0.000, 0.250)240 ( 100.00, 30.00)( -0.200, -0.150)200 ( 60.00, 60.00)( 0.000, -0.250)240 ( 60.00, 60.00)( 0.200, 0.150)200 ( 42.00, 30.00)( 3.000, 0.000) 20 ( 38.40, 28.80)( -2.400, -1.800) 16 ( 40.00, 32.00)( 0.000, 3.000) 20 ( 60.00, 60.00)( -0.139, 0.208)144 ( 60.00, 60.00)( -0.177, -0.177)339 ( 60.00, 60.00)( 0.200, -0.150)200 ( 41.11, 28.34)( 1.664, -2.496) 12 ( 38.40, 31.20)( -2.400, 1.800) 16 ( 41.11, 31.66)( 1.664, 2.496) 12 ( 41.41, 31.41)( 2.121, 2.121) 28 ( 48.51, 43.66)( -2.230, -2.007) 22 ( 48.08, 45.57)( -2.873, 0.862) 16 ( 49.57, 46.95)( -0.651, 2.929) 14 ( 51.49, 46.34)( 2.230, 2.007) 22 ( 51.92, 44.43)( 2.873, -0.862) 16 ( 50.43, 43.05)( 0.651, -2.929) 14 (0,80)(c): For example, for the polyhedron $K$ of a cube in Figure [\[f-3-c\]](#f-3-c){reference-type="ref" reference="f-3-c"}(a), we cut each face-polygon into two triangles without adding any point, and we cut $K$ into six tetrahedrons without adding any internal point, cf. Figure [\[f-3-c\]](#f-3-c){reference-type="ref" reference="f-3-c"}(b). For example, for the polyhedron $K$ of a cube in Figure [\[f-3-c\]](#f-3-c){reference-type="ref" reference="f-3-c"}(a), we can also subdivide it by cutting each face-polygon into two triangles without adding any point, and cutting $K$ into twelve tetrahedrons with one internal point, cf. Figure [\[f-3-c\]](#f-3-c){reference-type="ref" reference="f-3-c"}(c). (230,100)(0,0) (0,0) (100,100)(0,0) (0,80)(a): ( 0.00, 0.00)( 0.250, 0.000)240 ( 0.00, 0.00)( 0.000, 0.250)240 ( 60.00, 60.00)( -0.250, 0.000)240 ( 40.00, 90.00)( -0.200, -0.150)200 ( 40.00, 90.00)( 0.250, 0.000)240 ( 100.00, 30.00)( 0.000, 0.250)240 ( 100.00, 30.00)( -0.200, -0.150)200 ( 60.00, 60.00)( 0.000, -0.250)240 ( 60.00, 60.00)( 0.200, 0.150)200 ( 42.00, 30.00)( 3.000, 0.000) 20 ( 38.40, 28.80)( -2.400, -1.800) 16 ( 40.00, 32.00)( 0.000, 3.000) 20 (130,0) (100,100)(0,0) ( 0.00, 0.00)( 0.250, 0.000)240 ( 0.00, 0.00)( 0.000, 0.250)240 ( 60.00, 60.00)( -0.250, 0.000)240 ( 40.00, 90.00)( -0.200, -0.150)200 ( 40.00, 90.00)( 0.250, 0.000)240 ( 100.00, 30.00)( 0.000, 0.250)240 ( 100.00, 30.00)( -0.200, -0.150)200 ( 60.00, 60.00)( 0.000, -0.250)240 ( 60.00, 60.00)( 0.200, 0.150)200 ( 42.00, 30.00)( 3.000, 0.000) 20 ( 38.40, 28.80)( -2.400, -1.800) 16 ( 40.00, 32.00)( 0.000, 3.000) 20 ( 60.00, 60.00)( -0.139, 0.208)144 ( 60.00, 60.00)( -0.177, -0.177)339 ( 60.00, 60.00)( 0.200, -0.150)200 ( 41.11, 28.34)( 1.664, -2.496) 12 ( 38.40, 31.20)( -2.400, 1.800) 16 ( 41.11, 31.66)( 1.664, 2.496) 12 ( 41.41, 31.41)( 2.121, 2.121) 28 ( 48.51, 43.66)( -2.230, -2.007) 22 ( 48.08, 45.57)( -2.873, 0.862) 16 ( 49.57, 46.95)( -0.651, 2.929) 14 ( 51.49, 46.34)( 2.230, 2.007) 22 ( 51.92, 44.43)( 2.873, -0.862) 16 ( 50.43, 43.05)( 0.651, -2.929) 14 (0,80)(b): ( 0.00, 60.00)( 0.239, 0.072)417 ( 0.00, 60.00)( 0.177, -0.177)339 ( 100.00, 90.00)( -0.102, -0.228)393 ( 41.41, 88.59)( 2.121, -2.121) 28 ( 39.19, 88.17)( -1.218, -2.741) 32 ( 1.92, 0.57)( 2.873, 0.862) 34 ( 50.00, 17.00)( 0.000, 3.000) 22 ( 22.00, 45.00)( 3.000, 0.000) 20 ( 31.60, 31.20)( 2.400, 1.800) 16 For example, for the polyhedron $K$ of a cube in Figure [\[f-2-c\]](#f-2-c){reference-type="ref" reference="f-2-c"}(a), we can also subdivide it by cutting each face-polygon into four triangles with one added point on each face-polygon, and cutting $K$ into twenty four tetrahedrons with one additional point inside $K$, cf. Figure [\[f-2-c\]](#f-2-c){reference-type="ref" reference="f-2-c"}(b). In the analysis, we assume the same face-polygon subdivision on the two polyhedrons sharing the polygon. In the computation, the subdivisions of a shared polygon on the two sides can be different as the interpolation and the computation on the two polyhedrons are independent of each other. We can extend the theory easily to cover the case that different triangulations on a face-polygon of two polyhedrons, as both interpolations are the 2D $H^1$ projection of same $P_{k-2}$ moments. With a proper tetrahedral subdivision of $K=\cup_{T_i\subset K} T_i$, cf. Figures [\[f-2-t\]](#f-2-t){reference-type="ref" reference="f-2-t"}--[\[f-2-c\]](#f-2-c){reference-type="ref" reference="f-2-c"}, we define the interpolation space on $K$ as, for $k\ge 1$, in [\[V-k\]](#V-k){reference-type="eqref" reference="V-k"}, again in 3D. The interpolation operator is defined by two steps. On each face polygon $F\in\mathcal{F}_h$, we solve an $H^1$ projection problem that $v_h|_F=\Pi_h^\nabla\tilde v\in \mathbb{V}_k(F)$ (the restriction of $\mathbb{V}_k(K)$ on $F$) satisfying $$\begin{aligned} \label{v-F} \begin{aligned} (\nabla_F(v_h|_F -\tilde v), \nabla_F w_h) &=0 \quad \forall w_h\in \mathbb{V}_k(F)\cap H^1_0(F), \\ v_h|_F -\tilde v&=0 \quad \hbox{\rm{ on }} \partial F, \end{aligned} \end{aligned}$$ where $\nabla_F$ is the 2D face gradient. This way, the boundary value of $\Pi_h^\nabla\tilde v$ is determined on $\partial K$. The interpolation in 3D is defined as the 3D local $H^1$-projection, i.e., $v_h=\Pi_h^\nabla\tilde v \in \mathbb{V}_k$ such that $$\begin{aligned} \label{l-e-3} \begin{aligned} (\nabla(v_h -\tilde v), \nabla w_h) &=0 \quad \forall w_h\in \mathbb{V}_k(K)\cap H^1_0(K), \\ v_h- v_h|_F&=0 \quad \hbox{\rm{ on all }} F\in \partial K, \end{aligned} \end{aligned}$$ where $v_h|_F$ is defined in [\[v-F\]](#v-F){reference-type="eqref" reference="v-F"}. **Lemma 3**. *The interpolation operator $\Pi_h^\nabla$ is well defined in [\[l-e-3\]](#l-e-3){reference-type="eqref" reference="l-e-3"} and it preserves $P_k$ polynomials, $$\begin{aligned} \label{p-p3} \Pi_h^\nabla\tilde v = \tilde v\quad \ \hbox{\rm{if }} \ \tilde v\in P_k(K). \end{aligned}$$* *Proof.* Because $\tilde v|_{\partial F}\in \mathbb{B}_k(\mathcal{E}_h)$, $v_h$ can assume the boundary condition $v_h=\tilde v$ exactly on $\partial F$, where $F$ is a face-polygon in the polyhedral mesh. As we have proved in Lemma [Lemma 1](#l-2d-w){reference-type="ref" reference="l-2d-w"}, $v_h|_F$ is well-defined in [\[v-F\]](#v-F){reference-type="eqref" reference="v-F"}. Further by Lemma [Lemma 1](#l-2d-w){reference-type="ref" reference="l-2d-w"}, $$\begin{aligned} \label{f-p-k} v_h|_F = p_k|_F \quad\hbox{\rm{ if }} p_k=\tilde v\in P_k(K). \end{aligned}$$ The linear system of equations in [\[l-e-3\]](#l-e-3){reference-type="eqref" reference="l-e-3"} is a finite dimensional square system, after the boundary condition is enforced. The existence is implied by the uniqueness. To show the uniqueness, we let $\tilde v=0$ in [\[l-e-3\]](#l-e-3){reference-type="eqref" reference="l-e-3"}. By Lemma [Lemma 1](#l-2d-w){reference-type="ref" reference="l-2d-w"}, $v_h|_F = 0$. Letting $w_h=v_h$ in [\[l-e-3\]](#l-e-3){reference-type="eqref" reference="l-e-3"}, we get $$\begin{aligned} \nabla v_h ={\bf 0} \quad \hbox{\rm{ on }} \ K. \end{aligned}$$ Thus $v_h=c$ is one constant on all tetrahedrons of $K$. As $v_h$ is continuous, by the zero boundary condition, $v_h=0$ and [\[l-e-3\]](#l-e-3){reference-type="eqref" reference="l-e-3"} has a unique solution. If $\tilde v=p_k\in P_k(K)\subset \mathbb{V}_k(K)$, then [\[l-e-3\]](#l-e-3){reference-type="eqref" reference="l-e-3"} says, letting $w_h=v_h-p_k$, $$\begin{aligned} \nabla (v_h-p_k)={\bf 0}. \end{aligned}$$ Thus $v_h-p_k$ is a constant on $K$, which must be zero by [\[f-p-k\]](#f-p-k){reference-type="eqref" reference="f-p-k"}. [\[p-p3\]](#p-p3){reference-type="eqref" reference="p-p3"} is proved. ◻ .7cm **Lemma 4**. *The stabilizer-free virtual element equation [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} has a unique solution, where the interpolation $\Pi_h^\nabla$ is defined in [\[l-e-3\]](#l-e-3){reference-type="eqref" reference="l-e-3"}.* *Proof.* As both $\tilde u, \tilde v \in \tilde V_h$, [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} is a finite square system of linear equations. We only need to show the uniqueness, by letting $f=0$ and $\tilde v=\tilde u$ in [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"}. As in the 2D proof of Lemma [Lemma 2](#l-2d-e){reference-type="ref" reference="l-2d-e"}, we have $$\begin{aligned} |\Pi_h^\nabla\tilde u|_{1,h} =0 \quad \hbox{\rm{ and }} \ \Pi_h^\nabla\tilde u=0. \end{aligned}$$ For $k=1$, $\tilde u$ has no internal degree of freedom on each face polygon $F$, and $\tilde v|_F=\Pi_h^\nabla\tilde u|_F=0$. Further $\tilde u$ has no internal degree of freedom on each polyhedron $K$, and $\tilde v|_K=\Pi_h^\nabla\tilde u|_K=0$. The lemma is proved. For $k\ge 2$, by the proof of Lemma [Lemma 2](#l-2d-e){reference-type="ref" reference="l-2d-e"}, as each face polygon is subdivided into more than one tetrahedron, we have $\tilde v|_F=\Pi_h^\nabla\tilde u|_F=0$ on every face polygon $F$. Next, as every tetrahedron has at least two internal face triangles, we define an internal $P_2$ bubble by $$\begin{aligned} \label{b-K3} b_K &=\sum_{i\in \mathcal{N}_2} \phi_i\in H^1_0(K)\cap \mathbb{V}_k(K), \end{aligned}$$ where $\mathcal{N}_2$ is the set of all internal mid-edge points of $\{ T_i\}$, $K=\cup T_i$, and $\phi_i$ is the $P_2$ Lagrange nodal basis at node $i$. As every tetrahedron has such an internal $P_2$ node (which is the shared-edge mid-point of two internal face triangles), $b_K>0$ inside polyhedron $K$ if it does not have any added internal point. Otherwise, $b_K>0$ inside $K$ except at one or two internal points of $K$ where $b_K=0$. On one polyhedron $K$, let $$\begin{aligned} \label{w-h3} w_h= p_{k-2} b_K \in H^1_0(K)\cap \mathbb{V}_k(K), \end{aligned}$$ where the positive $P_2$ bubble $b_K$ is defined in [\[b-K3\]](#b-K3){reference-type="eqref" reference="b-K3"}, and $p_{k-2}=-\Delta \tilde u\in P_{k-2}(K)$. With the integration by parts, we get from [\[l-e1\]](#l-e1){reference-type="eqref" reference="l-e1"} and [\[w-h3\]](#w-h3){reference-type="eqref" reference="w-h3"} that $$\begin{aligned} \int_K p_{k-2}^2 b_K d {\bf x} =-\int_K \nabla \tilde u \nabla w_h d {\bf x} =0. \end{aligned}$$ As $b_K>0$ inside $K$ (other than 1 or 2 possibly internal points), it follows that $$\begin{aligned} p_{k-2}^2 =0 \ \hbox{\rm{ and }} \ p_{k-2} =0 \ \hbox{\rm{ on }} \ K. \end{aligned}$$ As $\Delta \tilde u=0$ in $K$ and $\tilde u=0$ on $\partial K$, by the unique solution of the Laplace equation, $\tilde u=0$. The lemma is proved. ◻ .7cm # Convergence We show that the stabilizer-free virtual element solution converges at the optimal order, in this section. **Theorem 5**. *Let the solution of [\[w-e\]](#w-e){reference-type="eqref" reference="w-e"} be $u\in H^{k+1}\cap H^1_0(\Omega)$. Let the stabilizer-free virtual element solution of [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} be $u_h$. Then the discrete solution converges at the optimal order with the following error estimate, $$\begin{aligned} \label{h-1} | u- u_h |_{1} \le Ch^{k} | u|_{k+1}. \end{aligned}$$* *Proof.* As $w_h\in V_h \subset H^1_0(\Omega)$, subtracting [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} from [\[w-e\]](#w-e){reference-type="eqref" reference="w-e"}, we obtain $$\begin{aligned} (\nabla (u-u_h), \nabla w_h) =0\quad \forall w_h\in V_{h }. \end{aligned}$$ By the Schwarz inequality, we get that $$\begin{aligned} | u- u_h|_{1}^2 & = (\nabla( u- u_h), \nabla( u- I_h u))\\ &\le | u- u_h|_{1} | u- I_h u |_{1} \le Ch^{k} |u|_{k+1} | u- u_h|_{1} ,\end{aligned}$$ where $I_h u$ is the Scott-Zhang interpolation on subdivided triangular mesh or tetrahedral mesh, cf. [@Scott-Zhang]. The theorem is proved. ◻ To get the optimal order $L^2$ error estimate, we assume a full regularity for the dual problem that the solution of $$\begin{aligned} \label{dual2} \begin{aligned} -\Delta w &= u-u_h \quad \hbox{\rm{ in }} \ \Omega, \\ w &=0 \quad \hbox{\rm{ on }} \ \partial \Omega, \end{aligned} \end{aligned}$$ satisfies $$\begin{aligned} \label{r} |w|_2 \le C \|u-u_h\|_0. \end{aligned}$$ **Theorem 6**. *Let the solution of [\[w-e\]](#w-e){reference-type="eqref" reference="w-e"} be $u\in H^{k+1}\cap H^1_0(\Omega)$. Let the stabilizer-free virtual element solution of [\[f-e\]](#f-e){reference-type="eqref" reference="f-e"} be $u_h$. Then the discrete solution converges at the optimal order with the following $L^2$ error estimate, assuming [\[r\]](#r){reference-type="eqref" reference="r"}, $$\begin{aligned} \| u- u_h \|_{0} \le Ch^{k+1} | u|_{k+1}. \end{aligned}$$* *Proof.* Let $w_h=\Pi_h^\nabla\tilde w$ be the virtual element solution of [\[dual2\]](#dual2){reference-type="eqref" reference="dual2"}. By [\[dual2\]](#dual2){reference-type="eqref" reference="dual2"}, [\[r\]](#r){reference-type="eqref" reference="r"} and [\[h-1\]](#h-1){reference-type="eqref" reference="h-1"}, we get $$\begin{aligned} \|u-u_h\|_0^2 &=(\nabla w, \nabla (u-u_h) ) = (\nabla (w-w_h), \nabla (u-u_h) ) \\ & \le C h |w|_2 h^{k} |u|_{k+1} \le C h^{k+1} |u|_{k+1}\|u-u_h\|_0. \end{aligned}$$ Canceling a $\|u-u_h\|_0$ on both sides, we proved the optimal-order $L^2$ error bound. ◻ .7cm # Numerical test We solve numerically the Poisson equation [\[p-e\]](#p-e){reference-type="eqref" reference="p-e"} on domain $\Omega=(0,1)\times(0,1)$, where an exact solution is chosen as $$\begin{aligned} \label{s-1} u(x,y)=\sin (\pi x) \sin (\pi y). \end{aligned}$$ We test the $P_k$ ($k=1,2,3,4,5$) stabilizer-free virtual elements on pentagonal meshes shown in Figure [\[mesh2\]](#mesh2){reference-type="ref" reference="mesh2"}. (220,113)(0,0) (0,0)(50,0)1 (0,0)(0,50)1 ( 100., 100.)( 0., 0.) ( 0.00, 0.00)( 0.250, 0.000)200 ( 50.00, 0.00)( 0.000, 0.250)200 ( 25.00, 25.00)( 0.177, 0.177)141 ( 0.00, 50.00)( 0.177, -0.177)141 ( 0.00, 0.00)( 0.000, 0.250)200 ( 100.00, 0.00)( 0.000, 0.250)200 ( 75.00, 75.00)( 0.177, -0.177)141 ( 50.00, 50.00)( 0.177, 0.177)141 ( 50.00, 0.00)( 0.250, 0.000)200 ( 50.00, 100.00)( 0.250, 0.000)200 ( 50.00, 50.00)( 0.000, 0.250)200 ( 100.00, 50.00)( 0.000, 0.250)200 ( 0.00, 50.00)( 0.000, 0.250)200 ( 0.00, 100.00)( 0.250, 0.000)200 (0,103)Grid 1: (120,103)Grid 2: (240,0)(100,0)2 (0,0)(0,100)2 ( 100., 100.)( 0., 0.) ( 0.00, 0.00)( 0.250, 0.000)200 ( 50.00, 0.00)( 0.000, 0.250)200 ( 25.00, 25.00)( 0.177, 0.177)141 ( 0.00, 50.00)( 0.177, -0.177)141 ( 0.00, 0.00)( 0.000, 0.250)200 ( 100.00, 0.00)( 0.000, 0.250)200 ( 75.00, 75.00)( 0.177, -0.177)141 ( 50.00, 50.00)( 0.177, 0.177)141 ( 50.00, 0.00)( 0.250, 0.000)200 ( 50.00, 100.00)( 0.250, 0.000)200 ( 50.00, 50.00)( 0.000, 0.250)200 ( 100.00, 50.00)( 0.000, 0.250)200 ( 0.00, 50.00)( 0.000, 0.250)200 ( 0.00, 100.00)( 0.250, 0.000)200 In Table [1](#ta1){reference-type="ref" reference="ta1"}, we compute the $P_1$--$P_5$ stabilizer-free virtual elements solutions for [\[s-1\]](#s-1){reference-type="eqref" reference="s-1"} on the pentagonal meshes shown in Figure [\[mesh2\]](#mesh2){reference-type="ref" reference="mesh2"}. All virtual element solutions converge at rates of the optimal order in both $L^2$ and $H^1$ norms. Grid $\|\Pi^\nabla_h u-u_h\|_{0}$ $O(h^r)$ $|\Pi^\nabla_h u-u_h|_{1}$ $O(h^r)$ ------ ----------------------------------------------- ---------- ---------------------------- ---------- By the $P_1$ stabilizer-free virtual element. 7 0.4462E-04 2.00 0.5834E-02 1.00 8 0.1116E-04 2.00 0.2916E-02 1.00 9 0.2789E-05 2.00 0.1458E-02 1.00 By the $P_2$ stabilizer-free virtual element. 7 0.1930E-06 3.00 0.1131E-03 2.00 8 0.2413E-07 3.00 0.2826E-04 2.00 9 0.3016E-08 3.00 0.7066E-05 2.00 By the $P_3$ stabilizer-free virtual element. 6 0.2486E-07 4.00 0.8973E-05 3.00 7 0.1554E-08 4.00 0.1122E-05 3.00 8 0.9716E-10 4.00 0.1402E-06 3.00 By the $P_4$ stabilizer-free virtual element. 5 0.1051E-07 5.00 0.1977E-05 4.00 6 0.3286E-09 5.00 0.1236E-06 4.00 7 0.1027E-10 5.00 0.7724E-08 4.00 By the $P_5$ stabilizer-free virtual element. 3 0.7591E-06 6.02 0.4741E-04 4.98 4 0.1181E-07 6.01 0.1488E-05 4.99 5 0.1846E-09 6.00 0.4659E-07 5.00 : The error profile for [\[s-1\]](#s-1){reference-type="eqref" reference="s-1"} on meshes shown in Figure [\[mesh2\]](#mesh2){reference-type="ref" reference="mesh2"}. (220,113)(0,0) (0,0)(50,0)1 (0,0)(0,50)1 ( 100., 100.)( 0., 0.) ( 0.00, 0.00)( 0.250, 0.000)200 ( 44.44, 14.29)( 0.091, -0.233) 61 ( 22.22, 22.22)( 0.235, -0.084) 94 ( 18.18, 44.44)( 0.045, -0.246) 90 ( 0.00, 50.00)( 0.239, -0.073) 76 ( 0.00, 0.00)( 0.000, 0.250)200 ( 100.00, 0.00)( 0.000, 0.250)200 ( 81.82, 50.00)( 0.250, 0.000) 72 ( 72.73, 28.57)( 0.098, 0.230) 93 ( 44.44, 14.29)( 0.223, 0.113)126 ( 50.00, 0.00)( 0.250, 0.000)200 ( 50.00, 100.00)( 0.250, 0.000)200 ( 44.44, 85.71)( 0.091, 0.233) 61 ( 44.44, 85.71)( 0.207, -0.140)136 ( 72.73, 66.67)( 0.120, -0.219) 75 ( 100.00, 50.00)( 0.000, 0.250)200 ( 0.00, 50.00)( 0.000, 0.250)200 ( 18.18, 44.44)( 0.045, 0.246) 90 ( 22.22, 66.67)( 0.190, 0.163)117 ( 0.00, 100.00)( 0.250, 0.000)200 ( 44.44, 85.71)( 0.074, -0.239)149 ( 44.44, 14.29)( 0.074, 0.239)149 (0,103)Grid 1: (120,103)Grid 2: (240,0)(100,0)2 (0,0)(0,100)2 ( 100., 100.)( 0., 0.) ( 0.00, 0.00)( 0.250, 0.000)200 ( 44.44, 14.29)( 0.091, -0.233) 61 ( 22.22, 22.22)( 0.235, -0.084) 94 ( 18.18, 44.44)( 0.045, -0.246) 90 ( 0.00, 50.00)( 0.239, -0.073) 76 ( 0.00, 0.00)( 0.000, 0.250)200 ( 100.00, 0.00)( 0.000, 0.250)200 ( 81.82, 50.00)( 0.250, 0.000) 72 ( 72.73, 28.57)( 0.098, 0.230) 93 ( 44.44, 14.29)( 0.223, 0.113)126 ( 50.00, 0.00)( 0.250, 0.000)200 ( 50.00, 100.00)( 0.250, 0.000)200 ( 44.44, 85.71)( 0.091, 0.233) 61 ( 44.44, 85.71)( 0.207, -0.140)136 ( 72.73, 66.67)( 0.120, -0.219) 75 ( 100.00, 50.00)( 0.000, 0.250)200 ( 0.00, 50.00)( 0.000, 0.250)200 ( 18.18, 44.44)( 0.045, 0.246) 90 ( 22.22, 66.67)( 0.190, 0.163)117 ( 0.00, 100.00)( 0.250, 0.000)200 ( 44.44, 85.71)( 0.074, -0.239)149 ( 44.44, 14.29)( 0.074, 0.239)149 In Table [2](#ta2){reference-type="ref" reference="ta2"}, we compute the $P_1$--$P_5$ stabilizer-free virtual elements solutions for [\[s-1\]](#s-1){reference-type="eqref" reference="s-1"} on the hexagonal meshes shown in Figure [\[mesh1\]](#mesh1){reference-type="ref" reference="mesh1"}. All virtual element solutions converge at rates of the optimal order in both $L^2$ and $H^1$ norms. Grid $\|\Pi^\nabla_h u-u_h\|_{0}$ $O(h^r)$ $|\Pi^\nabla_h u-u_h|_{1}$ $O(h^r)$ ------ ----------------------------------------------- ---------- ---------------------------- ---------- By the $P_1$ stabilizer-free virtual element. 6 0.1142E-03 2.00 0.1255E-01 1.00 7 0.2855E-04 2.00 0.6273E-02 1.00 8 0.7137E-05 2.00 0.3136E-02 1.00 By the $P_2$ stabilizer-free virtual element. 6 0.1011E-05 3.00 0.4338E-03 2.00 7 0.1265E-06 3.00 0.1085E-03 2.00 8 0.1581E-07 3.00 0.2712E-04 2.00 By the $P_3$ stabilizer-free virtual element. 6 0.2132E-07 4.00 0.1081E-04 3.00 7 0.1332E-08 4.00 0.1351E-05 3.00 8 0.8329E-10 4.00 0.1689E-06 3.00 By the $P_4$ stabilizer-free virtual element. 5 0.1200E-07 4.99 0.3029E-05 4.00 6 0.3756E-09 5.00 0.1894E-06 4.00 7 0.1175E-10 5.00 0.1185E-07 4.00 By the $P_5$ stabilizer-free virtual element. 3 0.1177E-05 5.98 0.8909E-04 4.97 4 0.1842E-07 6.00 0.2800E-05 4.99 5 0.2895E-09 5.99 0.8776E-07 5.00 : The error profile for [\[s-1\]](#s-1){reference-type="eqref" reference="s-1"} on meshes shown in Figure [\[mesh1\]](#mesh1){reference-type="ref" reference="mesh1"}. We solve the 3D Poisson equation [\[p-e\]](#p-e){reference-type="eqref" reference="p-e"} on domain $\Omega=(0,1)^3$, where an exact solution is chosen as $$\begin{aligned} \label{s-2} u(x,y,z)= 2^6 (x-x^2)(y-y^2)(z-z^2). \end{aligned}$$ (320,122)(0,3) (0,0) (110,110)(0,0)(0,102)Grid 1: (0,0)(80,0)2(0,1)80 (0,0)(0,80)2(1,0)80 (0,80)(80,0)2(1,1)20 (0,80)(20,20)2(1,0)80 (80,0)(0,80)2(1,1)20 (80,0)(20,20)2(0,1)80 (110,0) (110,110)(0,0)(0,102)Grid 2: (0,0)(40,0)3(0,1)80 (0,0)(0,40)3(1,0)80 (0,80)(40,0)3(1,1)20 (0,80)(10,10)3(1,0)80 (80,0)(0,40)3(1,1)20 (80,0)(10,10)3(0,1)80 (220,0) (110,110)(0,0)(0,102)Grid 3: (0,0)(20,0)5(0,1)80 (0,0)(0,20)5(1,0)80 (0,80)(20,0)5(1,1)20 (0,80)(5,5)5(1,0)80 (80,0)(0,20)5(1,1)20 (80,0)(5,5)5(0,1)80 In Table [3](#ta3){reference-type="ref" reference="ta3"}, we compute the 3D $P_1$--$P_5$ stabilizer-free virtual elements solutions for [\[s-2\]](#s-2){reference-type="eqref" reference="s-2"} on the cubic meshes shown in Figure [\[grid3d\]](#grid3d){reference-type="ref" reference="grid3d"}. All virtual element solutions converge at rates of the optimal order in both $L^2$ and $H^1$ norms. In particular, we have one order superconvergence in $H^1$ semi-norm for the $P_1$ stabilizer-free virtual element solutions. Also, we have one order superconvergence in both $H^1$ semi-norm and $L^2$ for the $P_2$ stabilizer-free virtual element solutions. But we do not have a theory for these superconvergences. It is surprising that the $P_2$ solutions are more accurate than the $P_3$ solutions in Table [3](#ta3){reference-type="ref" reference="ta3"}. Grid $\|\Pi^\nabla_h u-u_h\|_{0}$ $O(h^r)$ $|\Pi^\nabla_h u-u_h|_{1}$ $O(h^r)$ ------ -------------------------------------------------- ---------- ---------------------------- ---------- By the 3D $P_1$ stabilizer-free virtual element. 5 0.4944E-02 1.92 0.2780E-01 1.95 6 0.1254E-02 1.98 0.7011E-02 1.99 7 0.3145E-03 1.99 0.1757E-02 2.00 By the 3D $P_2$ stabilizer-free virtual element. 5 0.1132E-04 3.89 0.1001E-02 2.85 6 0.7294E-06 3.96 0.1313E-03 2.93 7 0.4615E-07 3.98 0.1680E-04 2.97 By the 3D $P_3$ stabilizer-free virtual element. 4 0.4149E-04 3.84 0.1569E-02 2.83 5 0.2688E-05 3.95 0.2053E-03 2.93 6 0.1703E-06 3.98 0.2618E-04 2.97 By the 3D $P_4$ stabilizer-free virtual element. 4 0.7316E-06 4.94 0.4987E-04 3.94 5 0.2331E-07 4.97 0.3169E-05 3.98 6 0.7356E-09 4.99 0.1997E-06 3.99 By the 3D $P_5$ stabilizer-free virtual element. 3 0.2986E-05 5.97 0.7301E-04 4.96 4 0.4707E-07 5.99 0.2318E-05 4.98 5 0.7386E-09 5.99 0.7304E-07 4.99 : The error profile for [\[s-2\]](#s-2){reference-type="eqref" reference="s-2"} on cubic meshes shown in Figure [\[grid3d\]](#grid3d){reference-type="ref" reference="grid3d"}. # Ethical Statement ## Compliance with Ethical Standards   The submitted work is original and is not published elsewhere in any form or language. ## Funding   Yanping Lin is supported in part by HKSAR GRF 15302922 and polyu-CAS joint Lab. Mo Mu is supported in part by Hong Kong RGC CERG HKUST16301218. ## Conflict of Interest   There is no potential conflict of interest . ## Ethical approval   This article does not contain any studies involving animals. This article does not contain any studies involving human participants. ## Informed consent   This research does not have any human participant. ## Availability of supporting data   This research does not use any external or author-collected data. ## Authors' contributions   All authors made equal contribution. ## Acknowledgments   None. 99 A. Al-Taweel, Y. Dong, S. Hussain and X. Wang, A weak Galerkin harmonic finite element method for Laplace equation, Commun. Appl. Math. Comput. 3 (2021), no. 3, 527--543. A. Al-Twaeel, S. Hussian and X. Wang, A stabilizer free weak Galerkin finite element method for parabolic equation, J. Comput. Appl. Math., 392 (2021), 113373. A. Al-Twaeel, X. Wang, X. Ye and S. 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Math. 4 (2023), no. 2, 256--274. X. Ye and S. Zhang, Four-order superconvergent weak Galerkin methods for the biharmonic equation on triangular meshes, Communications on Applied Mathematics and Computation, Doi:10.1007/s42967-022-00201-5, 2023. X. Ye and S. Zhang, Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes, J. Comput. Appl. Math., 2023, doi: 10.1016/j.cam.2023.115516. S. Zhang, An optimal order multigrid method for biharmonic, $C^1$ finite-element equations, Numer. Math. 56 (1989), 613--624. S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comp. 74 (2005), no. 250, 543--554. [^1]: Yanping Lin is supported in part by HKSAR GRF 15302922 and polyu-CAS joint Lab. [^2]: Mo Mu is supported in part by Hong Kong RGC CERG HKUST16301218.

arxiv_math

--- abstract: | We construct a theory of motivic cohomology for quasi-compact, quasi-separated schemes of equal characteristic, which is related to non-connective algebraic $K$-theory via an Atiyah--Hirzebruch spectral sequence, and to étale cohomology in the range predicted by Beilinson and Lichtenbaum. On smooth varieties over a field our theory recovers classical motivic cohomology, defined for example via Bloch's cycle complex. Our construction uses trace methods and (topological) cyclic homology. As predicted by the behaviour of algebraic $K$-theory, the motivic cohomology is in general sensitive to singularities, including non-reduced structure, and is not $\mathbb{A}^1$-invariant. It nevertheless has good geometric properties, satisfying for example the projective bundle formula and pro cdh descent. Further properties of the theory include a Nesterenko--Suslin comparison isomorphism to Milnor $K$-theory, and a vanishing range which simultaneously refines Weibel's conjecture about negative $K$-theory and a vanishing result of Soulé for the Adams eigenspaces of higher algebraic $K$-groups. We also explore the relation of the theory to algebraic cycles, showing in particular that the Levine--Weibel Chow group of zero cycles on a surface arises as a motivic cohomology group. author: - "Elden Elmanto[^1], Matthew Morrow[^2]" bibliography: - ../../Bibliography.bib - bibliography-zar2.bib title: Motivic cohomology of equicharacteristic schemes --- # Introduction The vision of motivic cohomology is due to Beilinson and Lichtenbaum [@Beilinson1987a; @bms-zero; @Lichtenbaum1984]. For a reasonable class of schemes $X$ they predicted the existence of natural complexes of abelian groups $\mathbb{Z}(j)^{\mathrm{mot}}(X)$, for $j\ge0$, satisfying various relations to algebraic $K$-theory and étale cohomology. Perhaps the most important of these relations is a desired *Atiyah--Hirzebruch spectral sequence* $$\label{eq:ahss} E_2^{i,j}=H_{\mathrm{mot}}^{i-j}(X, \mathbb{Z}(-j)) \implies \mathrm{K}_{-i-j}(X),$$ relating the *motivic cohomology groups* $H^i_{\mathrm{mot}}(X, \mathbb{Z}(j)) := H^i(\mathbb{Z}(j)^{\mathrm{mot}}(X))$ to the algebraic $K$-groups of $X$. They asked that this spectral sequence would degenerate rationally and identify the rationalised motivic cohomology $H^i_{\mathrm{mot}}(X, \mathbb{Z}(j))\otimes_\mathbb{Z}\mathbb{Q}$ with the Adams eigenspace $\mathrm{K}_{i-2j}(X)_\mathbb{Q}^{(j)}$. Meanwhile, motivic cohomology with finite coefficients $H^i_{\mathrm{mot}}(X, \mathbb{Z}/\ell (j)) := H^i(\mathbb{Z}(j)^{\mathrm{mot}}(X)/\ell)$ was expected to coincide with étale cohomology $H^i_{\mbox{\rm \scriptsize \'et}}(X,\mu_{\ell}^{\otimes j})$ in the range $i\le j$, whenever $\ell>0$ is invertible on $X$. Note that any such theory of motivic cohomology must necessarily fail to be $\mathbb{A}^1$-invariant for sufficiently singular $X$, i.e., the maps $H^i_{\mbox{\rm \scriptsize mot}}(\mathbb{A}_X^1,\mathbb{Z}(j))\to H^i_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j))$ are not in general isomorphisms, since algebraic $K$-theory also fails to be $\mathbb{A}^1$-invariant on general schemes. In this article, which builds on our joint work with T. Bachmann about cdh-local motivic cohomology [@BachmannElmantoMorrow], we construct such motivic complexes $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)$ whenever $X$ is a quasi-compact, quasi-separated scheme of equal characteristic.[^3] For the rest of the introduction let $\mathbb{F}$ be a prime field, i.e., $\mathbb{Q}$ or $\mathbb{F}_p$ for some prime number $p$. **Theorem 1**. *There exist finitary Nisnevich sheaves $$\mathbb{Z}(j)^{\mathrm{mot}}: \mathrm{S}\mathrm{ch}{}_{\mathbb{F}}^{\mathrm{qcqs},\mathrm{op}} \longrightarrow\rm D(\mathbb{Z})$$ for $j\ge0$, such that the following properties hold for any qcqs $\mathbb{F}$-scheme $X$:* 1. *There exists a functorial, multiplicative, $\mathbb{N}$-indexed filtration $\mathrm{Fil}_{\mbox{\rm \scriptsize mot}}^{\star}\mathrm{K}(X)$ on the non-connective algebraic $K$-theory $K(X)$, such that the graded pieces are naturally given by $$\mathrm{gr}^j_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X) \simeq \mathbb{Z}(j)(X)^{\mathrm{mot}}[2j]$$ for $j\ge0$. In particular, writing $H^i_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j)):=H^i(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X))$ for the corresponding *motivic cohomology groups*, there exists an Atiyah-Hirzebruch spectral sequence $$E_2^{ij}=H_{\mathrm{mot}}^{i-j}(X, \mathbb{Z}(-j)) \implies \mathrm{K}_{-i-j}(X).$$ If $X$ has finite valuative dimension, then the filtration $\mathrm{Fil}_{\mbox{\rm \scriptsize mot}}^{\star}\mathrm{K}(X)$ is complete and the Atiyah--Hirzebruch spectral sequence is convergent.* 2. *Rational structure: the Atiyah--Hirzebruch spectral sequence degenerates rationally and there are natural isomorphisms $$H^i_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j))\otimes_\mathbb{Z}\mathbb{Q}\cong \mathrm{K}_{2j-i}(X)_\mathbb{Q}^{(j)}$$ for all $i\in\mathbb{Z}$ and $j\ge0$, where the right side refers to Adams eigenspaces of rationalised $K$-theory.* 3. *Relation to étale cohomology: for any integer $\ell>0$ invertible in $\mathbb{F}$, there are natural equivalences $$\tau^{\le j}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)/\ell)\simeq\tau^{\le j}R\Gamma_{\mbox{\rm \scriptsize \'et}}(X,\mu_\ell^{\otimes j})$$ for $j\ge0$.* 4. *Relation to syntomic cohomology: if $\mathbb{F}=\mathbb{F}_p$ then for any $r>0$ there are natural equivalences $$\tau^{\leq j}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)/p^r) \simeq \tau^{\leq j}(\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)/p^r)$$ for $j\ge0$, where $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)$ denotes the weight-$j$ *syntomic* cohomology of $X$ in the sense of [@AntieauMathewMorrowNikolaus; @BhattMorrowScholze2].* 5. *Weight zero: there is a natural equivalence $$\mathbb{Z}(0)^{\mbox{\rm \scriptsize mot}}(X)\simeq R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\mathbb{Z})$$ where the right side denotes cdh cohomology with coefficients in the constant sheaf $\mathbb{Z}$.* 6. *Weight one: there is a natural map $$R\Gamma_{\mbox{\rm \scriptsize Nis}}(X,\mathbb{G}_m)[-1]\longrightarrow\mathbb{Z}(1)^{\mbox{\rm \scriptsize mot}}(X),$$ which is an equivalence in degrees $\le 3$.* 7. *Projective bundle formula: For any $j,r \geq 0$, the powers of the first Chern class of the tautological bundle $c_1(\mathcal O(1)) \in \mathrm{Pic}(\mathbb{P}^r_X) \cong H^{2}_{\mbox{\rm \scriptsize mot}}(\mathbb{P}^r_X, \mathbb{Z}(1))$ induce a natural equivalence $$\bigoplus^r_{i=0} \mathbb{Z}(j-i)^{\mathrm{mot}}(X)[-2i] \stackrel{\sim}{\to}\mathbb{Z}(j)^{\mathrm{mot}}(\mathbb{P}^r_X).$$* 8. *Blow-up formula: Given any regular closed immersion $Y\to X$ (i.e., $X$ admits an open affine cover such that, on each such affine, $Y$ is defined by a regular sequence), then $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ carries the cartesian square of schemes $$\xymatrix{ Y\times_X\mathrm{Bl}_Y(X)\ar[d]\ar[r] & \mathrm{Bl}_Y(X)\ar[d]\\ Y\ar[r] & X }$$ to a cartesian square in $\rm D(\mathbb{Z})$.* 9. *Finally, suppose $X$ is a smooth scheme over a field. Then there are equivalences $$\mathbb{Z}(j)^{\mathrm{mot}}(X) \simeq z^j(X,\bullet)[-2j]$$ for $j\ge0$, where $z^j(X,\bullet)$ is Bloch's cycle complex of $X$. Moreover the filtration $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}$ on $K(X)$ is naturally equivalent, as multiplicative filtered spectra, with the filtration coming from Voevodsky's slice filtration as in [@Voevodsky2002a].* The original approach to motivic cohomology was that of Bloch [@Bloch1986b], in terms of his cycle complexes $z^j(X,\bullet)$ for algebraic varieties $X$. Ignoring certain technicalities (such as functoriality, multiplicative structure, quasi-projectivity hypotheses,\...), the work of Bloch, Bloch--Lichtenbaum [@BlochLichtenbaum], Friedlander--Suslin [@friedlander-suslin], and Levine [@levine-tech; @Levine2008] show that the complexes $z^j(X,\bullet)[-2j]$ satisfy a variant of the conjectural framework of Beilinson and Lichtenbaum; the crucial difference is that $z^j(-,\bullet)$ is covariant in the algebraic variety and the Atiyah--Hirzebruch spectral sequence converges not to the $K$-theory of $X$ but rather to the $G$-theory. (In terms of Voevodsky's approach [@lecture-notes-mot-cohom; @voevodsky-triang-motives] via triangulated categories of motives, Bloch's cycle complex appears as Borel--Moore homology.) However, restricting attention to smooth algebraic varieties $X$, the motivic complexes $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X):=z^j(X,\bullet)[-2j]$$ do have all desired properties (and the technicalities can be overcome using motivic stable homotopy theory and the slice filtration); we will call this theory the *classical motivic cohomology* of the smooth algebraic variety $X$. Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}(9) states that the new theory of this paper reduces to the classical theory in the smooth case. **Remark 2**. Although the focus of this article is to extend motivic cohomology beyond smooth algebraic varieties, our results have applications to the smooth case. For example, we will see in Corollary [Corollary 91](#corol_smooth_comparison){reference-type="ref" reference="corol_smooth_comparison"} that Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}(9) implies that the canonical map $R\Gamma_{\mbox{\rm \scriptsize \'et}}(X,\Omega^j_{\mbox{\rm \scriptsize log}})\to R\Gamma_{\mbox{\rm \scriptsize \'eh}}(X,\Omega^j_{\mbox{\rm \scriptsize log}})$ is an equivalence for any smooth variety $X$ over a field of characteristic $p$. The analogous equivalence between the Nisnevich and cdh cohomologies is contained in the joint work with Bachmann [@BachmannElmantoMorrow]. Such results, which are required for example in Geisser's theory of arithmetic cohomology [@Geisser2006], seem to have been previously out of reach without assuming resolution of singularities. ## Relation to $\mathbb{A}^1$-invariant motivic cohomology {#ss_intro_cdh} This article depends on our joint work with Bachmann [@BachmannElmantoMorrow], in which we revisit the theory of $\mathbb{A}^1$-invariant motivic cohomology. Although much of that project works for arbitrary qcqs schemes, we restrict our summary here to the simpler equicharacteristic context. See §[3.3](#ss_cdh_local){reference-type="ref" reference="ss_cdh_local"} for further details. For each $j\ge0$ let $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}:\text{Sch}_\mathbb{F}^{\mbox{\rm \scriptsize qcqs,op}}\longrightarrow\rm D(\mathbb{Z})$$ be the cdh sheafification of the left Kan extension of classical motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}$ from smooth $\mathbb{F}$-schemes to qcqs $\mathbb{F}$-schemes; we call it *cdh-local motivic cohomology* and note that it provides a cdh analogue of Bloch's cycle complex. It was already studied by Friedlander, Suslin, and Voevodsky [@FriedlanderSuslinVoevodsky2000] in the case of singular algebraic varieties assuming resolution of singularities. In [@BachmannElmantoMorrow] we establish various properties of this cdh-local motivic cohomology, without any assumption on resolution of singularities, proving in particular that it is $\mathbb{A}^1$-invariant: namely, for any qcqs equicharacteristic scheme $X$, the canonical maps $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(\mathbb{A}_X^1)\to\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$ are equivalences. Moreover, the cohomology groups $H^i_{\mbox{\rm \scriptsize cdh}}(X,\mathbb{Z}(j)):=H^i(\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X))$ fit into an Atiyah--Hirzebruch spectral sequence converging to the $\mathrm{KH}$-groups of $X$, and with finite coefficients away from the characteristic they are related to étale cohomology. See Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"} for more precise statements. In short, the complexes $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$, for $j\ge0$, satisfy a variant of Beilinson and Lichtenbaum's vision, except that they are $\mathbb{A}^1$-invariant and related not to $\mathrm{K}(X)$ but rather to $\mathrm{KH}(X)$. Although it is not required for the present article, we also prove in [@BachmannElmantoMorrow] that this cdh-local motivic cohomology coincides with the motivic cohomology represented by the zeroth slice of the unit $\pmb1_X$, or equivalently by the motivic Eilenberg--Maclane spectrum $\mathrm{H}\mathbb{Z}_X$, in Morel--Voevodsky's stable homotopy category $\mathcal{SH}(X)$; see Remark [Remark 21](#remark_slice_filtration){reference-type="ref" reference="remark_slice_filtration"} for details. In other words, we have no doubt that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$, for $j\ge0$, provide the "correct" theory of $\mathbb{A}^1$-invariant motivic cohomology. The theory of this paper is designed so that, for any qcqs equicharacteristic scheme $X$, the canonical map $\mathrm{K}(X)\to\mathrm{KH}(X)$ is compatible with the motivic filtrations on each side, thereby inducing comparison maps $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)\label{eqn_nonA1_to_A1}$$ from the new non-$\mathbb{A}^1$-invariant motivic cohomology to the $\mathbb{A}^1$-invariant, cdh-local theory. This comparison map has the following properties, thereby refining to the level of motivic cohomology known comparisons between $\mathrm{K}$-theory and $\mathrm{KH}$-theory: **Theorem 3** (See Thms. [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}, [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}, and [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}). *Let $j\ge0$ and let $\mathbb{F}$ be a prime field.* 1. *The map ([\[eqn_nonA1_to_A1\]](#eqn_nonA1_to_A1){reference-type="ref" reference="eqn_nonA1_to_A1"}) identifies $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$ with the $\mathbb{A}^1$-localisation and the cdh-sheafification of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$. That is, on the category of qcqs $\mathbb{F}$-schemes, there are natural equivalences of $\rm D(\mathbb{Z})$-valued presheaves: $$L_{\mathbb{A}^1}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\simeq \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}} \simeq L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}.$$* 2. *For any qcqs $\mathbb{F}$-scheme and integer $\ell>0$ invertible in $\mathbb{F}$, the map ([\[eqn_nonA1_to_A1\]](#eqn_nonA1_to_A1){reference-type="ref" reference="eqn_nonA1_to_A1"}) is an equivalence mod $\ell$: $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)/\ell\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)/\ell.$$* 3. *For any qcqs $\mathbb{F}_p$-scheme $X$, the map ([\[eqn_nonA1_to_A1\]](#eqn_nonA1_to_A1){reference-type="ref" reference="eqn_nonA1_to_A1"}) is an equivalence after inverting $p$: $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}[\tfrac1p]\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}[\tfrac1p].$$* 4. *For any regular Noetherian $\mathbb{F}$-scheme $X$, the map ([\[eqn_nonA1_to_A1\]](#eqn_nonA1_to_A1){reference-type="ref" reference="eqn_nonA1_to_A1"}) is an equivalence.* Part (1) of the theorem refines the fact that $\mathrm{KH}$-theory is both the $\mathbb{A}^1$-localisation of $K$-theory (by definition) and its cdh-sheafification (as we will recall at the start of §[1.2](#intro_ss_sketch){reference-type="ref" reference="intro_ss_sketch"}). Parts (2) and (3) refine in equicharacteristic results of Weibel [@Weibel1989a] that $\mathrm{K}(A)/\ell \stackrel{\sim}{\to}\mathrm{KH}(A)/\ell$ (resp. $\mathrm{K}(A)[\tfrac1p] \stackrel{\sim}{\to}\mathrm{KH}(A)[\tfrac1p]$) for rings $A$ in which $\ell$ is invertible (resp. in which $p=0$). Finally, part (4) refines in equicharacteristic the equivalence between $\mathrm{K}$-theory and $\mathrm{KH}$-theory for regular Noetherian rings. An input to establishing part (3) of the previous theorem, which is essential to controlling our motivic cohomology in characteristic $p$, is to show that rationalised syntomic cohomology $\mathbb{Q}_p(j)^{\mbox{\rm \scriptsize syn}}:=\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}[\tfrac1p]$ is a cdh sheaf on qcqs $\mathbb{F}_p$-schemes (see Corollary [Corollary 43](#corol_Qpsyn){reference-type="ref" reference="corol_Qpsyn"}). Perhaps this can be proved directly, but our approach is rather to reduce it to the aforementioned fact that $\mathrm{K}[\tfrac1p]=\mathrm{KH}[\tfrac1p]$ on such schemes; the reduction argument passes through the cartesian square ([\[eq:mainsquare\]](#eq:mainsquare){reference-type="ref" reference="eq:mainsquare"}) below and so ultimately depends on trace methods. This extraction of information about cohomology theories from localising invariants is a theme which runs throughout this paper and [@BachmannElmantoMorrow]; we will return to it in Remark [Remark 7](#re_into_pmf){reference-type="ref" reference="re_into_pmf"} when discussing the projective bundle formula. ## The construction of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ via trace methods {#intro_ss_sketch} Our construction of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ is inspired by trace methods in algebraic $K$-theory. For any qcqs scheme $X$ let $\mathrm{TC}(X)$ denote its topological cyclic homology, and $\mathrm{K}^{\mbox{\rm \scriptsize inf}}(X)$ the fibre of the trace map $\mathrm{K}(X)\to\mathrm{TC}(X)$. The presheaf $\mathrm{K}^{\mbox{\rm \scriptsize inf}}$ is nil-invariant by the Dundas--Goodwillie--McCarthy theorem [@Dundas2013], and even a cdh sheaf by Kerz--Strunk--Tamme [@KerzStrunkTamme2018] and Land--Tamme [@LandTamme2019]. Coupled with the surprising fact that Weibel's $\mathrm{KH}$-theory is equivalent to the $\mathrm{cdh}$-sheafification of $K$-theory (first proved by Haesemeyer in characteristic zero [@Haesemeyer2004] and Kerz--Strunk--Tamme [@KerzStrunkTamme2018] in general), we arrive at a cartesian square for any qcqs scheme $$\label{eq:mainsquare} \begin{tikzcd} \mathrm{K}(X) \ar{d} \ar{r} & \mathrm{TC}(X) \ar{d}\\ \mathrm{KH}(X) \ar{r} & L_{\mathrm{cdh}}\mathrm{TC}(X), \end{tikzcd}$$ where the bottom map is the $\mathrm{cdh}$-sheafified trace map. We define the motivic filtration $\mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize mot}}$ on $\mathrm{K}(X)$ by glueing existing filtrations on $\mathrm{KH}(X)$, $\mathrm{TC}(X)$, and $L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X)$: 1. For any qcqs $\mathbb{F}_p$-scheme $X$, Bhatt, Scholze, and the second author [@BhattMorrowScholze2] have defined a filtration on $\mathrm{TC}(X)$ whose graded pieces are $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)[2j]$ for $j\ge0$; here $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)$ is the *syntomic cohomology* of $X$, which modulo $p$ is a derived version of the étale cohomology of Illusie--Milne's sheaves $\Omega^j_{X,\log}$. Cdh sheafifying this filtration over qcqs $\mathbb{F}_p$-schemes induces a filtration on $L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X)$. 2. For any qcqs $\mathbb{Q}$-scheme $X$, its topological cyclic homology $\mathrm{TC}(X)$ identifies with its negative cyclic homology $\mathrm{HC}^-(X/\mathbb{Q})$. Antieau [@antieau-fil] has defined a filtration on $\mathrm{HC}^-(X/\mathbb{Q})$, extending previous work of Loday [@Loday1989] and Weibel [@Weibel1997], whose graded pieces are $R\Gamma(X, \widehat{L\Omega}_{-/\mathbb{Q}}^{\geq j})[2j]$ for $j\in\mathbb{Z}$. Here $\widehat{L\Omega}_{-/\mathbb{Q}}$ is the Hodge-completed *derived de Rham complex* equipped with its Hodge filtration, as studied notably by Illusie [@Illusie1971; @Illusie1972] and Bhatt [@Bhatt2012]. As in characteristic $p$, cdh sheafifying then induces a compatible filtration on $L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X)=L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/\mathbb{Q})$. 3. For any qcqs equicharacteristic scheme $X$, our joint work with Bachmann [@BachmannElmantoMorrow], briefly discussed above in §[1.1](#ss_intro_cdh){reference-type="ref" reference="ss_intro_cdh"}, defines a motivic filtration on $\mathrm{KH}(X)$ by left Kan extending from the smooth context and then cdh sheafifiying. The graded pieces are $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)[2j]$ for $j\ge0$, and it turns out that the filtration agrees with Voevodsky's slice filtration coming from motivic stable homotopy theory. The following compatibility of these filtrations is not difficult to prove but is fundamental to our construction; we refer to Corollary [Corollary 31](#corol_cdh_filtered_trace){reference-type="ref" reference="corol_cdh_filtered_trace"} and Proposition [Proposition 45](#prop_cdh_filtered_trace_p){reference-type="ref" reference="prop_cdh_filtered_trace_p"} for more precise statements: **Proposition 4**. *For any qcqs equicharacteristic scheme $X$, the cdh-sheafified trace map $\mathrm{KH}(X)\to \mathrm{TC}(X)$ respects the filtrations on each side.* We consequently define our motivic filtration $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}$ on $\mathrm{K}(X)$ by glueing the existing filtrations on the three other corners of the square ([\[eq:mainsquare\]](#eq:mainsquare){reference-type="ref" reference="eq:mainsquare"}). Passing to graded pieces yields the following description of our motivic cohomology: **Theorem 5** (See Thms. [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"} and [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}). *For $j\ge0$ and any qcqs scheme $X$ over $\mathbb{Q}$ (resp. $\mathbb{F}_p)$, there is a natural cartesian square in $\textrm{D}(\mathbb{Z})$ $$\label{eq:fundamental_squares} \xymatrix@=1cm{ \mathbb{Z}(j)^{\mathrm{mot}}(X) \ar[r] \ar[d] & R\Gamma(X,\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq j}) \ar[d]\\ \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X) \ar[r] & R\Gamma_{\mathrm{cdh}}(X,\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq j}). } \qquad\text{resp.~} \xymatrix@=1.3cm{ \mathbb{Z}(j)^{\mathrm{mot}}(X) \ar[r] \ar[d] & \mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X) \ar[d]\\ \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X) \ar[r] & L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X). }$$* The cartesian squares ([\[eq:fundamental_squares\]](#eq:fundamental_squares){reference-type="ref" reference="eq:fundamental_squares"}) encapsulate the central idea of our construction of motivic cohomology. They say that the motivic complex $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)$ is a modification of the cdh-local, $\mathbb{A}^1$-invariant theory $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$ (which is governed by algebraic cycles) by derived de Rham/syntomic cohomology. In particular, in characteristic zero the left square of ([\[eq:fundamental_squares\]](#eq:fundamental_squares){reference-type="ref" reference="eq:fundamental_squares"}) yields a fibre sequence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)\longrightarrow{\rm cofib}\left(R\Gamma(X,L\Omega_{-/\mathbb{Q}}^{< j})\to R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\Omega_{-/\mathbb{Q}}^{< j})\right)[-1];$$ this plays the role of the weight-$j$ motivic component of the well-known fibre sequence $$\mathrm{K}(X)\longrightarrow\mathrm{KH}(X)\longrightarrow{\rm cofib}\big(\mathrm{HC}(X)\to L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}(X)\big)[1]$$ arising from ([\[eq:mainsquare\]](#eq:mainsquare){reference-type="ref" reference="eq:mainsquare"}), which was used throughout Cortiñas--Haesemeyer(--Schlichting)--Weibel's work [@Cortinas2008; @Cortinas2008a] on the $K$-theory of singular varieties in characteristic zero. The present paper may in fact be roughly understood as a refinement of their work from the level of $K$-theory to that of motivic cohomology, as well as providing an extension to finite characteristic. The squares ([\[eq:fundamental_squares\]](#eq:fundamental_squares){reference-type="ref" reference="eq:fundamental_squares"}) also provide a refinement of the trace map and its main property to the level of motivic cohomology: **Corollary 6**. *On the category of qcqs schemes over $\mathbb{Q}$ (resp. $\mathbb{F}_p$), there exists for each $j\ge0$ a "weight-$j$ motivic trace map" (namely the top horizontal arrow in ([\[eq:fundamental_squares\]](#eq:fundamental_squares){reference-type="ref" reference="eq:fundamental_squares"})) $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\longrightarrow R\Gamma(-,\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq j}),\qquad\text{resp.~}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\longrightarrow\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}},$$ whose fibre is a cdh sheaf.* **Remark 7** (Projective bundle formula). As already stated in Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}(8), our motivic cohomology satisfies the projective bundle formula. For Grothendieck this was one of the most fundamental desired properties of any cohomology theory, and it means that the motivic cohomology assembles into a motivic spectrum in the sense of Annalla--Iwasa [@AnnalaIwasa2023]. But it is also an essential input into proving the comparison theorems with classical motivic cohomology (Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}(9)) and with the $\mathbb{A}^1$-invariant theory in the regular case (Theorem [Theorem 3](#intro_thm_SH){reference-type="ref" reference="intro_thm_SH"}(4)), as we use Gabber's technique [@Gabber1994; @GrosSuwa1988] axiomatised by Colliot-Thélène--Hoobler--Kahn [@ColliotThelene-Hoobler-Kahn1997]. Remarkably, our proof of the projective bundle formula depends on the theory of localising invariants (at least in characteristic $p$ -- in characteristic zero it is sufficient to use strong resolution of singularities). Indeed, exploiting the fact that cdh-local motivic cohomology and syntomic cohomology are known to have this property (by [@BachmannElmantoMorrow] and [@BhattLurie2022] respectively; the proof in [@BachmannElmantoMorrow] also uses localising invariants), the problem reduces via the right square in ([\[eq:fundamental_squares\]](#eq:fundamental_squares){reference-type="ref" reference="eq:fundamental_squares"}) to showing that cdh-sheafified syntomic cohomology satisfies the projective bundle formula. We prove this in Theorem [Theorem 76](#thm:cdh-syn-pbf){reference-type="ref" reference="thm:cdh-syn-pbf"}, using the fact that the square ([\[eq:mainsquare\]](#eq:mainsquare){reference-type="ref" reference="eq:mainsquare"}) is cartesian and so allows us to upgrade $L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}$ to a localising invariant. ## Relation to Milnor $\mathrm{K}$-theory and lisse motivic cohomology {#ss_milnor} When $F$ is a field, a theorem of Nesterenko--Suslin [@Suslin1989], later reproved by Totaro [@Totaro1992], produces a natural isomorphism between the classical motivic cohomology $H^{j}_{\mathrm{mot}}(F,\mathbb{Z}(j))$ with the Milnor $K$-group $\mathrm{K}^M_{j}(F)$. On the one hand, this gives a "cohomological interpretation\" of the Milnor $K$-groups defined via generators and relations. On the other hand it shows that the graded ring $\bigoplus_{j\ge0}H^{j}_{\mathrm{mot}}(F,\mathbb{Z}(j))$ is generated by elements in degree $1$ and with relations in degree $2$. In his thesis [@Kerz2009], Kerz extended the Nesterenko--Suslin isomorphism to the generality of regular local rings containing an infinite field, thereby settling a conjecture of Beilinson. He later eliminated the hypothesis that the field be infinite, using the improved Milnor $K$-theory $\widehat{\mathrm{K}}^M_j$ which he and Gabber had introduced [@Kerz2010]. Our motivic cohomology satisfies the Nesterenko--Suslin isomorphism for arbitrary local rings containing a field, without any regularity hypotheses: **Theorem 8** (Nesterenko--Suslin isomorphism; see Thm. [Theorem 112](#theorem_NS){reference-type="ref" reference="theorem_NS"}). *Let $A$ be a local ring containing a field. Then the isomorphism $A^\times\cong H^1_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(1))$ of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}(6) induces, by multiplicativity, isomorphisms $$\label{eq:nst} \widehat{\mathrm{K}}^M_{j}(A) \stackrel{\simeq}{\to}H^{j}_{\mathrm{mot}}(A, \mathbb{Z}(j))$$ for all $j\ge1$.* The proof of Theorem [Theorem 8](#thm:nst){reference-type="ref" reference="thm:nst"} is intertwined with a comparison theorem relating our motivic cohomology to a more naive version of the theory obtained by simply left Kan extending classical motivic cohomology. More precisely, for any $\mathbb{F}$-algebra $A$ we define $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)\in \rm D(\mathbb{Z})$ to be the left Kan extension, from smooth $\mathbb{F}$-algebras, of weight-$j$ classical motivic cohomology. More explicitly, there exists a simplicial resolution $P_{\bullet} \stackrel{\sim}{\to}A$ whose terms are ind-smooth $\mathbb{F}$-algebras and whose face maps are henselian surjections; then $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)$ is given by the totalisation of the simplicial complex $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(P_\bullet)$. We call $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)$ the weight-$j$, *lisse motivic cohomology* of $A$ to emphasise the fact that it controlled by smooth algebras. The complexes $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)[2j]$, for $j\ge0$, appear as the graded pieces of a motivic filtration on the connective algebraic $K$-theory $K^{\mbox{\rm \scriptsize cn}}(A)$; see §[3.2](#subsec_lke){reference-type="ref" reference="subsec_lke"} for more details. For any $\mathbb{F}$-algebra $A$ there is a natural comparison map $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A),\label{intro:lke_vs_mot}$$ and we prove the following, which in degree $j$ is the Nesterenko--Suslin isomorphism: **Theorem 9** (see Thm. [Theorem 107](#thm_lke_lej){reference-type="ref" reference="thm_lke_lej"}). *For any local $\mathbb{F}$-algebra $A$ and $j\ge0$, the map ([\[intro:lke_vs_mot\]](#intro:lke_vs_mot){reference-type="ref" reference="intro:lke_vs_mot"}) induces an equivalence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)\stackrel{\sim}{\to}\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A).$$* The theorem states that, in degrees less than or equal to the weight, our motivic cohomology is Zariski locally controlled by classical motivic cohomology; in particular, in this range it is closely related to algebraic cycles. This is the next topic we discuss. ## Relations to algebraic cycles {#sec:cycles-intro} One of the key features of the classical motivic cohomology of smooth algebraic varieties $X$ is its description in terms of algebraic cycles, via Bloch's cycle complex; this yields in particular isomorphisms $$H^{2j}_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j))\cong \textrm{CH}^j(X)\label{eqn_mot_vs_Chow}$$ for each $j\ge0$. In the case of a singular algebraic variety $X$, various definitions of Chow groups have been proposed. A first possibility is Fulton's [@Fulton1998], but his theory is a Borel--Moore homology theory related more to $G(X)$ than $\mathrm{K}(X)$. Another is Baum--Fulton--Macpherson's [@BaumFultonMacPherson1975] theory of cohomological Chow groups, essentially obtained by left Kan extending $\textrm{CH}^j$ from smooth algebraic varieties to arbitrary varieties; it is thus related, at least superficially, to the lisse motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}$ introduced above in §[1.3](#ss_milnor){reference-type="ref" reference="ss_milnor"}. Levine [@Levine1983] refined Baum--Fulton--Macpherson's idea by (roughly speaking) restricting the class of smooth varieties appearing in the left Kan extension procedure to better control the algebraic cycles. The case of zero cycles is particularly well-developed, and the *Levine--Weibel* Chow group $\rm CH^{\mbox{\rm \scriptsize LW}}_0(X)$ of zero cycles [@Levine1985a] on a singular variety $X$ has found concrete applications towards $K$-theoretic problems such as the splitting of vector bundles on affine varieties [@Krishna2002; @Murthy1994]. For a modern text on Levine--Weibel's group, we refer the reader to work of Binda--Krishna [@BindaKrishna2018; @BindaKrishna2022]. In §[9](#section_cf_cycles){reference-type="ref" reference="section_cf_cycles"} we study the relationship between our motivic cohomology and algebraic cycles. We are not sure what to expect in general, but we can show in the case of surfaces that our theory captures the Levine--Weibel group of zero cycles: **Theorem 10** (See Thm. [Theorem 130](#thm:lw-comparison){reference-type="ref" reference="thm:lw-comparison"}). *Let $X$ be a reduced, equi-dimensional, quasi-projective surface over an infinite field $k$; then there is a natural isomorphism $$H^4_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(2))\cong \mathrm{CH}_0^{\mbox{\rm \scriptsize LW}}(X).\label{eqn_mot_vs_Chow2}$$* Whereas one often views ([\[eqn_mot_vs_Chow\]](#eqn_mot_vs_Chow){reference-type="ref" reference="eqn_mot_vs_Chow"}) as a description of motivic cohomology in terms of algebraic cycles, we suggest adopting the alternative point of view on ([\[eqn_mot_vs_Chow2\]](#eqn_mot_vs_Chow2){reference-type="ref" reference="eqn_mot_vs_Chow2"}), namely it provides a new description of zero cycles on singular surfaces. Indeed, bearing in mind the main idea presented after Theorem [Theorem 5](#intro_thm_main_squares){reference-type="ref" reference="intro_thm_main_squares"}, it says that the Levine--Weibel group of zero cycles of a surface is somehow built from cdh-local zero cycles and derived de Rham/syntomic cohomology. **Example 11** (cdh-local zero cycles on surfaces). Let $X$ be as in Theorem [Theorem 10](#thm_intro_LW){reference-type="ref" reference="thm_intro_LW"}. The proof of the Soulé--Weibel vanishing Theorem [Theorem 13](#intro_Weibel_vanishing){reference-type="ref" reference="intro_Weibel_vanishing"} below implies in addition that the canonical map $$\textrm{CH}_0^{\mbox{\rm \scriptsize LW}}(X)=H^{4}_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j))\longrightarrow H^{4}_{\mbox{\rm \scriptsize cdh}}(X,\mathbb{Z}(2))=H^{2}_{\mbox{\rm \scriptsize cdh}}(X,\widehat K_2^M)$$ is surjective, where the right equality follows from a cdh-local version of the Nesterenko--Suslin isomorphism. In other words, any "cdh-local zero cycle" comes from an honest zero cycle. Our results also imply that this map is an isomorphism modulo any integer invertible in the base field $k$, and an isomorphism after inverting $p$ if $k$ has characteristic $p>0$. This may well be known to experts: such comparisons have certainly been established previously for *projective* varieties in arbitrary dimensions [@BindaKrishna2022 Thms. 1.6 & 1.7]. Another context in which algebraic cycles appear in motivic cohomology is the theory of Chow groups with modulus, building on Bloch--Esnault's earlier notion of additive Chow groups [@Bloch2003]. The set-up of the theory varies, but suppose for simplicity that $X$ is a smooth algebraic variety equipped with an effective divisor $D$ such that $D_{\mbox{\rm \scriptsize red}}$ is a simple normal crossing divisor. The theory defines various "Chow groups on $X$ with modulus $D$", which it is hoped will ultimately correspond to a piece of the motivic cohomology of $X$ relative to $D$; these Chow groups with modulus should in particular be related to the $K$-theory of $X$ relative to $D$, but at present the evidence of this relation is limited. A common theme in the subject [@Bloch2003; @Rulling2007; @RullingSaito2018], already present in the original work of Bloch--Esnault, is that Chow groups with modulus, although they are defined purely in terms of algebraic cycles, often contain groups of differential forms, Witt vectors, or more generally de Rham--Witt groups; that is, the theory offers a cycle-theoretic description of the latter objects. Our theory provides a systematic framework to obtain similar descriptions, exemplified as follows, for which the reader should recall that lisse motivic cohomology is described by algebraic cycles: **Example 12** (See Ex. [Example 128](#example_dim_0){reference-type="ref" reference="example_dim_0"}). Let $k$ be a perfect field of characteristic $p>0$, and $j,e\ge0$. Then the lisse motivic cohomology of $k[x]/x^e$ relative to its residue field, i.e., the fibre of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(k[x]/x^e)\to\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(k)$, is naturally equivalent to $(\mathbb{W}_{ej}(k)/V^e\mathbb{W}_j(k))[-1]$. ## Negative $K$-groups and Soulé--Weibel vanishing A major stimulus in the development of the algebraic $K$-theory of singular schemes has been the problem of understanding their negative $K$-groups, in which the central conjecture for many years was Weibel's vanishing conjecture: for a Noetherian scheme $X$ of finite dimension, he predicted that the negative $K$-groups $K_{-n}(X)$ would vanish for $n>\dim X$. Following numerous special cases (see the start of Section [8](#section_Weibel){reference-type="ref" reference="section_Weibel"} for references), the conjecture was proved in general by Kerz--Strunk--Tamme [@KerzStrunkTamme2018]. Meanwhile, concerning the positive $K$-groups of a Noetherian ring $A$, Soulé [@Soule1985 Corol. 1] had proved much earlier the vanishing of the Adams eigenspaces $\mathrm{K}_n(A)_\mathbb{Q}^{(j)}$ whenever $n>0$ and $j>n+\dim A$. The following integral motivic vanishing theorem strenghtens and unifies these two results in the equicharacteristic case: **Theorem 13** (Motivic Soulé--Weibel vanishing; see Thm. [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"}). *Let $j\ge0$ and let $X$ be a Noetherian equicharacteristic scheme of finite dimension. Then $H^i_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j))=0$ for all $i> j+\dim X$.* Kerz--Strunk--Tamme's proof of Weibel vanishing depended on first establishing that $K$-theory satisfied pro cdh descent on Noetherian schemes, again following various special cases which had been proved earlier. It seems in fact that pro cdh descent, which is an analogue of the formal functions theorem from coherent cohomology, is one of the most fundamental properties of algebraic $K$-theory. In any case, as well as its appearance in the proof of Weibel vanishing, it has applications to the study of algebraic cycles on singular varieties [@Krishna2010; @Krishna2002; @Morrow_zero_cycles]. We prove that our motivic cohomology also has this property: **Theorem 14** (Pro cdh descent for motivic cohomology; see Thm. [Theorem 114](#theorem_pro_cdh_descent){reference-type="ref" reference="theorem_pro_cdh_descent"}). *On the category of Noetherian equicharacteristic schemes, the presheaf $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ satisfies pro cdh descent for each $j\ge0$. That is, given any abstract blowup square of Noetherian equicharacteristic schemes $$\xymatrix{ Y'\ar[r]\ar[d] & X'\ar[d] \\ Y\ar[r] & X }$$ the associated square of pro complexes $$\xymatrix{ \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X) \ar[r]\ar[d] & \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X')\ar[d]\\ \{\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(rY)\})_r \ar[r] & \{\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(rY')\})_r }$$ is cartesian.* ## Other recent approaches to motivic cohomology ### Kelly--Saito's pro-cdh-local motivic cohomology Kelly and Saito [@KellySaito2023] have recently defined a Grothendieck topology, called the *pro-cdh topology*, on qcqs schemes with the following property: a presheaf $F$ on qcqs schemes, valued in Sp or $\rm{D}(\mathbb{Z})$, is a pro-cdh sheaf if and only if it is both a Nisnevich sheaf and, for every abstract blow-up square of qcqs schemes denoted as in Theorem [Theorem 14](#intro_pro_cdh_descent){reference-type="ref" reference="intro_pro_cdh_descent"}, the associated square $$\xymatrix{ F(X)\ar[r]\ar[d] & \mathop{\mathrm{lim}}_r F(rY)\ar[d] \\ F(X')\ar[r] & \mathop{\mathrm{lim}}_r F(rY') }$$ is cartesian. They define *pro-cdh-local motivic cohomology* $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize pcdh}}:\text{Sch}^{\mbox{\rm \scriptsize qcqs,op}}\longrightarrow\rm{D}(\mathbb{Z})$$ to be the pro-cdh sheafification of the left Kan extension of classical motivic cohomology from smooth $\mathbb{Z}$-schemes to all qcqs schemes. That is, the definition mimics that of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$, but replacing the cdh topology by their coarser pro-cdh topology. For any Noetherian scheme $X$, its pro-cdh local motivic cohomology fits into an Atiyah--Hirzebruch spectral sequence converging to $\mathrm{K}(X)$; to prove this one uses that, on Noetherian schemes, $K$-theory is the pro-cdh sheafification of connective $\mathrm{K}$-theory. By combining some of Kelly--Saito's main theorems about their topology (in particular the fact that it has enough points, and the description of the points) with some of our own (including Theorems [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}(9), [Theorem 9](#thm_into_lke_lej){reference-type="ref" reference="thm_into_lke_lej"}, and [Theorem 14](#intro_pro_cdh_descent){reference-type="ref" reference="intro_pro_cdh_descent"}), one obtains the following comparison: **Theorem 15** (See [@KellySaito2023]). *For any Noetherian equicharacteristic scheme $X$ and $j\ge0$, there is a natural equivalence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize pcdh}}(X)\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X).\label{pcdh}$$* Thus, on Noetherian equicharacteristic schemes, Kelly--Saito's approach offers an alternative definition of the same motivic cohomology of this paper; their definition is not restricted to equal characteristic and does not require trace methods. On the other hand we are not aware at present whether their approach can be used to establish, for example, the projective bundle formula, the Nesterenko--Suslin isomorphism, the comparisons to $\mathbb{A}^1$-invariant motivic cohomology, or the relation to zero cycles on surfaces. In the generality of non-Noetherian schemes, the two theories differ and pro-cdh-local motivic cohomology is not finitary. The two sides of ([\[pcdh\]](#pcdh){reference-type="ref" reference="pcdh"}) thus seem to have quite different flavours; we hope that the comparison between then will serve as a powerful tool in future work (for example, the pro-cdh approach should ultimately lead to a more conceptual proof of the Soulé--Weibel vanishing bound). ### Annala--Hoyois--Iwasa's non-$\mathbb{A}^1$-invariant motivic homotopy theory Annala, Hoyois, and Iwasa [@AnnalaHoyoisIwasa2023] are currently developing a theory of non-$\mathbb{A}^1$-invariant motivic homotopy theory, building on earlier work of Annala--Iwasa [@AnnalaIwasa2022; @AnnalaIwasa2023]. A theory of motivic cohomology in their framework is provided by forcing the left Kan extension of classical motivic cohomology, from smooth schemes to qcqs schemes, to satisfy the projective bundle bundle and Nisnevich descent. We all hope it coincides in the equicharacteristic case with the motivic cohomology constructed in the present paper. ### Park's yeni higher Chow groups For any algebraic variety $X$, Park [@Park2021] has defined complexes of cycles ${\bf z}^j(X,\bullet)$, for $j\ge0$, which coincide with Bloch's cycle complexes $z^j(X,\bullet)$ when $X$ is smooth. His complexes are Zariski locally supported in negative cohomological degrees and therefore their cohomology groups cannot fit into an Atiyah--Hirzebruch spectral sequence converging to the $K$-groups of $X$. From his construction (by locally embedding $X$ into a smooth variety and looking at certain cycles on the formal completion of the embedding), it seems plausible that ${\bf z}^j(X,\bullet)[-2j]$ is Zariski locally an explicit approximation of the lisse motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(X)$. ## Outline of the paper We briefly summarize the contents of the paper. After reviewing conventions regarding sheaves and filtrations in §[2](#sec:notation){reference-type="ref" reference="sec:notation"} we recall previously known constructions of motivic cohomology in §[3](#sec:recall){reference-type="ref" reference="sec:recall"}. Of note is the lisse version of motivic cohomology reviewed in §[3.2](#subsec_lke){reference-type="ref" reference="subsec_lke"} which is produced simply by left Kan extending and the cdh-local version of motivic cohomology which is jointly produced with Bachmann, recalled in §[3.3](#ss_cdh_local){reference-type="ref" reference="ss_cdh_local"}. Our construction of motivic cohomology is explained in §[4](#s_motivic_def){reference-type="ref" reference="s_motivic_def"}. Specifically, the characteristic zero version is given in Definition [Definition 32](#eq:char0){reference-type="ref" reference="eq:char0"} and the characteristic $p > 0$ version is given in Definition [Definition 46](#def:charp){reference-type="ref" reference="def:charp"}. We briefly discuss an extension of the theory to derived schemes in §[3.3](#ss_cdh_local){reference-type="ref" reference="ss_cdh_local"}. In §[5](#section_pbf){reference-type="ref" reference="section_pbf"} we prove the projective bundle formula and the blowup formula for motivic cohomology. This is the technical heart of the paper. In particular, we prove a $\mathbb{P}^1$-bundle formula for cdh-sheafified syntomic cohomology in §[5.2](#sec:syn-p1){reference-type="ref" reference="sec:syn-p1"}, adopting techniques that we developed in the joint paper with Bachmann in [@BachmannElmantoMorrow]. The projective bundle formula is key in comparing our construction to previous constructions of motivic cohomology in the smooth setting. This is discussed, more generally, under comparison with $\mathbb{A}^1$-invariant versions of motivic cohomology in §[6](#section_smooth){reference-type="ref" reference="section_smooth"}. The last part of the paper is dedicated to deeper properties of motivic cohomology. In §[7](#section_lke){reference-type="ref" reference="section_lke"} we describe a portion of our motivic cohomology using lisse motivic cohomology. Using this, in §[7.2](#sec:singular-nst){reference-type="ref" reference="sec:singular-nst"}, we prove the singular Nesterenko--Suslin isomorphism. In §[8](#section_Weibel){reference-type="ref" reference="section_Weibel"} we prove the motivic Soulé--Weibel vanishing. The key ingredient is a pro cdh descent result which we establish in §[8.1](#sec:pro-cdh){reference-type="ref" reference="sec:pro-cdh"}. We then finish off the paper by examining how our theory relates to algebraic cycles in §[9](#section_cf_cycles){reference-type="ref" reference="section_cf_cycles"}. This paper has two appendices. In Appendix [10](#app:cdh){reference-type="ref" reference="app:cdh"}, we prove a technical result establishing, under certain hypotheses, that the $\mathrm{cdh}$-sheafification of an étale sheaf is an $\text{\'{e}h}$-sheaf. In Appendix [11](#app:chw){reference-type="ref" reference="app:chw"}, we discuss rational motivic cohomology and prove a spectrum-level, multiplicative refinement of a theorem of Cortiñas, Haesemeyer and Weibel on the compatibility between Adams operations on rationalized $K$-theory and negative cyclic homology. This appendix is important in controlling the rational parts of our theory. ## Acknowledgements We are grateful to Ben Antieau, Tom Bachmann, Bhargav Bhatt, Federico Binda, Dustin Clausen, Frédéric Déglise, Christian Haesemeyer, Lars Hesselholt, Marc Hoyois, Ryomei Iwasa, Shane Kelly, Amalendu Krishna, Marc Levine, Akhil Mathew, Jinhyun Park, Arpon Raksit, Shuji Saito, and Peter Scholze for useful discussions without which this project might not have been realised. We also thank Tomer Schlank for suggesting the terminology "deflatable.\" Both the authors would like to take this opportunity to thank especially Chuck Weibel. The first author thanks him for his generosity and encouragement, and lack of pretension in mathematics. He has benefitted from conversations with Chuck on motives throughout the years, starting when he was a first year graduate student. For the second author, Chuck's work on $\mathrm{K}$-theory has been a source of motivation to him for many years and has directly inspired many of the main ideas of this article. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 101001474), and from the NSERC grant "Reimagining motivic cohomology.\" # Some notation and conventions {#sec:notation} In this section, we collect notation and conventions which we will use throughout the paper. We freely use the language of $\infty$-categories as developed in [@LurieHA; @Lurie2009] ## Sheaves Let $(\mathcal C, \tau)$ be an $\infty$-site and $F:\mathcal C^{\mathrm{op}} \rightarrow \mathcal D$ a presheaf, where $\mathcal D$ is a stable, presentable $\infty$-category with a $t$-structure. The examples of $\mathcal D$ that will appear in this paper are mainly the category $\textrm{Sp}$ of spectra and the derived category $\rm D(A)$ where $A$ is a discrete coefficient ring, each equipped with the standard $t$-structure; we will also see filtered variants of these categories. One says that $F$ is *discrete* if it factors through the heart $\mathcal D^{\heartsuit} \subseteq \mathcal D$; in the previous example $\rm{Sp}$ (resp. $\rm D(R)$), a discrete presheaf means a discrete presheaf of abelian groups (resp. of $R$-modules). We will use the following terminology and notation: 1. We write $L_{\tau}$ to be the endofunctor $L_{\tau}: \mathrm{PShv}(\mathcal C) \rightarrow \mathrm{PShv}(\mathcal C)$ reflecting onto the subcategory of $\tau$-sheaves $\mathrm{Shv}_{\tau}(\mathcal C)$; this functor is referred to as *sheafification*. 2. If $F$ is a discrete presheaf, then we write $R\Gamma_{\tau}(-,F)$ then for the $\tau$-cohomology of $F$; in other words we have an equivalence of $\tau$-sheaves $$L_{\tau}F\simeq R\Gamma_{\tau}(-,F).$$ 3. Given another topology $\tau'$ which is finer than $\tau$ then there is an adjunction $$\epsilon^*:\mathrm{Shv}_{\tau} \rightleftarrows \mathrm{Shv}_{\tau'}:\epsilon_*.$$ whose unit gives rise to a canonical map in $L_{\tau}F \rightarrow \epsilon_*L_{\tau'}F$ in $\mathrm{Shv}_{\tau}(\mathcal C)$. (We note that in the case $\mathcal D = \rm D(R)$, then $\epsilon_*$ is often denoted in the literature as $R\epsilon_*$.) Often we regard $L_{\tau}F$ and $L_{\tau'}F$ as presheaves on $\mathcal C$ and simply write the previous map as $L_{\tau}F \rightarrow L_{\tau'}F$; the context should always make it clear how we are viewing the objects. ## Filtrations {#ss_filtrations} For $\mathcal C$ a stable $\infty$-category, the associated stable $\infty$-categories of *filtered objects* and *graded objects* are $$\mathcal C^{\mathbb{Z}^{\mathrm{op}}} := \mathrm{Fun}((\mathbb{Z}, \geq)^{\mathrm{op}}, \mathcal C) \qquad \text{and}\qquad \mathcal C^{\mathbb{Z}^{\delta}} := \mathrm{Fun}(\mathbb{Z}^{\delta}, \mathcal C),$$ where $(\mathbb{Z}, \geq)$ denotes the totally ordered set of the integers and $\mathbb{Z}^{\delta}$ is the discrete category of the integers. Our filtrations are thus, by convention, $\mathbb{Z}$-indexed and always decreasing.[^4] The functor of taking associated graded is written as usual by $\mathrm{gr}^\star:\mathcal C^{\mathbb{Z}^{\mathrm{op}}} \rightarrow \mathcal C^{\mathbb{Z}^{\delta}}$. We tend to write filtered objects of $\mathcal C$ as $\mathrm{Fil}^\star M$, where $M$ is an object of $\mathcal C$; this notation implicitly means that there is a morphism $\mathrm{Fil}^{-\infty}M:=\mathop{\mathrm{colim}}_{j\to\-\infty}\mathrm{Fil}^jM\to M$ in $\mathcal C$. The filtration is said to be *exhaustive* when the latter morphism is an equivalence. The filtration is said to be *$\mathbb{N}$-indexed* when $\mathrm{Fil}^jM\to M$ is an equivalence for all $j\le 0$ (or, equivalently, the filtration is exhaustive and $\mathrm{gr}^jM=0$ for $j<0$). The filtration is said to be *complete* if $\mathop{\mathrm{lim}}_{j\to\infty}\mathrm{Fil}^jM=0$. When $\mathcal C=\text D(\mathbb{Z}),\,\text{Sp}$, etc., then our filtrations are often complete because they satisfy the stronger property of being *bounded* ("uniformly homologically bounded below" to be more precise): for us this means that there exists $d\ge 0$ such that $\mathrm{Fil}^jM$ is supported in cohomological degrees $\le d-j$ for any $j\in\mathbb{Z}$. Then $\mathrm{gr}^jM$ is also supported in cohomological degrees $\le d-j$, and the associated spectral sequence of the filtered complex/spectrum lies in the left half-plane $\{x\le d\}$. Conversely, if the filtration is already known to be complete then boundedness can be checked via the graded pieces: taking the inverse limit, $\mathrm{gr}^jM$ being supported in cohomological degrees $\le d-j$ for all $j\in\mathbb{Z}$ implies the same about all $\mathrm{Fil}^jM$. Assume now that $\mathcal C$ is presentably symmetric monoidal. Then $\mathcal C^{\mathbb{Z}^{\mathrm{op}}}$ and $\mathcal C^{\mathbb{Z}^{\delta}}$ admit canonical symmetric monoidal structures given by Day convolution, which ensures that taking associated graded promotes to a strong symmetric monoidal functor. In particular, we have the $\infty$-category of *filtered $\mathbb{E}_{\infty}$-algebras* $\mathrm{CAlg}(\mathcal C^{\mathbb{Z}^{\mathrm{op}}})$ and *graded $\mathbb{E}_{\infty}$-algebras* $\mathrm{CAlg}(\mathcal C^{\mathbb{Z}^{\delta}})$ such that $\mathrm{gr}^\star$ promotes to a strong symmetric monoidal functor $\mathrm{gr}^\star:\mathrm{CAlg}(\mathcal C^{\mathbb{Z}^{\mathrm{op}}}) \rightarrow \mathrm{CAlg}(\mathcal C^{\mathbb{Z}^{\delta}}).$ Rather abusively, we tend to summarise this wealth of information by speaking simply of *multiplicative filtrations* or *multiplicative graded objects*. We also often consider maps between such structured objects and call them maps which are *multiplicative*. Our main case of interest are when $\mathcal C$ is the $\infty$-category $\text{Sp}$ of spectra or the derived $\infty$-category $\text D(R)$ of modules over some discrete ring $R$. In these cases we write $\text{FSp}$ and $\text{DF}(R)$ for the associated categories of filtered objects.[^5] ## Left Kan extensions Given a fully faithful inclusion of categories $\iota: \mathcal C\subseteq \mathcal C'$ and a functor $F:\mathcal C\to\mathcal D$ valued in a presentable $\infty$-category $\mathcal D$, we write $L_{\mathcal C'/\mathcal C}F:\mathcal C'\to\mathcal D$ for the corresponding left Kan extension. We will use the following standard facts: 1. Left Kan extension provides a left adjoint to the restriction functor $\iota^*:\mathrm{Fun}( \mathcal C', \mathcal D) \rightarrow \mathrm{Fun}( \mathcal C, \mathcal D)$ $$\iota_!:\mathrm{Fun}( \mathcal C, \mathcal D) \rightarrow \mathrm{Fun}( \mathcal C, \mathcal D);$$ which is furthermore fully faithful [@Lurie2009 Prop. 4.3.2.17]. 2. Let $R$ be a commutative base ring. In the special case that $\mathcal C \subset \mathcal C'$ is the inclusion $\mathrm{CAlg}^{{\mathrm{S}\mathrm{m}}}_R \subset \mathrm{CAlg}_R$, and we are given a functor $\mathrm{CAlg}^{\mbox{\rm \scriptsize sm}}_R \rightarrow \mathrm{CAlg}(\mathcal D)$, the left Kan extension $L_{\mathcal C'/\mathcal C}F:\mathcal C'\to\mathcal D$ upgrades to a functor $L_{\mathcal C'/\mathcal C}F:\mathcal C'\to\mathcal\mathrm{CAlg}(\mathcal D)$. The key point here is that, for an $R$-algebra $S$, the diagram $(\mathrm{CAlg}^{\mbox{\rm \scriptsize sm}}_{R})_{/S}$ is *sifted*, whence the colimit computing the left Kan extension in $\mathrm{CAlg}(\mathcal D)$ is computed in $\mathcal D$; see [@LurieHA Corol. 3.2.3.2] fo a reference. This also works for the inclusion $\mathrm{CAlg}^{{\mathrm{S}\mathrm{m}}}_R \subset \mathrm{CAlg}^{\mbox{\rm \scriptsize ani}}_R$ of smooth $R$-algebra into animated $R$-algebras. Therefore, without further comment, we take for granted that the left Kan extensions appearing in this paper preserves multiplicative structures. # Recollections of other cohomologies {#sec:recall} ## Classical motivic cohomology of smooth schemes {#ss_classical} We briefly recall motivic cohomology for smooth schemes over a field $k$; we will often say "classical motivic cohomology" when we wish to draw a comparison with our forthcoming generalisation. For $j\ge0$ and for any given $X\in \text{Sm}_k$, its weight-$j$ motivic complex $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X)$ is given by a shift of Bloch's cycle complex, namely $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X)=z^j(X,\bullet)[-2j].\label{eqn_mot=BL}$$ Bloch's cycle complex is a priori only functorial for flat morphisms between smooth $k$-schemes, which is insufficient for our purposes (notably for left Kan extending beyond smooth $k$-schemes), and its multiplicative properties are unclear (especially in mixed characteristic, although that is irrelevant for the present paper), but these problems are resolved via motivic stable homotopy theory. Indeed, Voevodsky's motivic cohomology (as constructed via the theory of finite correspondences in [@FriedlanderSuslinVoevodsky2000]) is representable in the stable motivic homotopy category $\mathcal{SH}(k)$ via the motivic Eilenberg--Maclane spectrum, thereby defining functorial weight $j$ motivic cohomology as a presheaf $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}:\text{Sm}^{\mathrm{op}}_k\longrightarrow\text D(\mathbb{Z})$$ such that ([\[eqn_mot=BL\]](#eqn_mot=BL){reference-type="ref" reference="eqn_mot=BL"}) holds for any fixed smooth $k$-scheme $X$. The graded object $\mathbb{Z}(\star)^{\mbox{\rm \scriptsize cla}}[2\star]$ is moreover multiplicative. Furthermore, Voevodsky's slice filtration [@Voevodsky2002] equips $K$-theory with a multiplicative, complete $\mathbb{N}$-indexed filtration on smooth $k$-schemes, i.e. $$\label{eq:cla-mot} \mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}: {\mathrm{S}\mathrm{m}}_k^{\mathrm{op}} \longrightarrow\text{FSp}$$ such that $\mathrm{Fil}^0_{\mbox{\rm \scriptsize cla}}K=K$, whose associated graded is $\mathbb{Z}(\star)^{\mbox{\rm \scriptsize cla}}[2\star]$. (Levine's homotopy coniveau tower [@Levine2008] is another approach to defining such a filtration for any given $X$, but there again seem to be technicalities surrounding functoriality and multiplicativity; see however the recent paper [@DegliseFeldJin2023]). This filtration induces the *Atiyah--Hirzebruch*, or *slice*, *spectral sequence* $$E_2^{ij}=H^{i-j}(\mathbb{Z}(-j)^{\mbox{\rm \scriptsize cla}}(X))\implies \mathrm{K}_{-i-j}(X)$$ functorially in $X\in\text{Sm}_k$. The above motivic filtration on $\mathrm{K}(X)$ is bounded in that $\mathrm{Fil}^j_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X)$ is supported in homological degrees $\ge \dim X-j$ (in particular, $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X)$ is supported in cohomological degrees $\le j+\dim X$). Adams operations imply that the motivic filtration splits rationally, i.e., there is a natural equivalence of filtered spectra $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X)_\mathbb{Q}\stackrel{\sim}{\to}\prod_{j\ge0}\mathbb{Q}(j)^{\mbox{\rm \scriptsize cla}}(X)[2j]$, and that the Atiyah--Hirzebruch spectral degenerates rationally; see §[11.2](#sec_adams-slice){reference-type="ref" reference="sec_adams-slice"} for details. ## Lisse motivic cohomology {#subsec_lke} The simplest way to extend motivic cohomology to arbitrary algebras over fields is via left Kan extension of the classical theory: **Definition 16**. Fix a prime field $\mathbb{F}$ (i.e., $\mathbb{F}_p$ for some prime number $p\ge2$ or $\mathbb{Q}$) and $j\ge0$. We define *weight-$j$, lisse motivic cohomology* $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}:=L_{\rm{CAlg}_\mathbb{F}/\rm{CAlg}_\mathbb{F}^{\mbox{\rm \scriptsize sm}}}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}: \text{CAlg}_\mathbb{F} \rightarrow \rm D(\mathbb{Z})$$ as the left Kan extension of classical motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}$ along the inclusion $\rm{CAlg}_\mathbb{F}^{\mbox{\rm \scriptsize sm}} \subseteq \rm{CAlg}_\mathbb{F}$ of smooth $\mathbb{F}$-algebras into all $\mathbb{F}$-algebras. We warn the reader that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}$ is not in general a Zariski sheaf, already in the case $j=1$: **Example 17** ($j=1$). Recall that $\mathbb{Z}(1)^{\mbox{\rm \scriptsize cla}}=R\Gamma_{\mbox{\rm \scriptsize Zar}}(-,\mathbb{G}_m)[-1]$, which is the same as $(\tau^{\le1}R\Gamma_{\mbox{\rm \scriptsize Zar}}(-,\mathbb{G}_m))[-1]$ on smooth $\mathbb{F}$-schemes. Since both units and $\text{Pic}$, as functors $\mathrm{CAlg}_\mathbb{F}\to\text D(\mathbb{Z})$, are left Kan extended from smooth $\mathbb{F}$-algebras, we deduce that there is a natural equivalence $\mathbb{Z}(1)^{\mbox{\rm \scriptsize lse}}(A)\simeq(\tau^{\le1}R\Gamma_{\mbox{\rm \scriptsize Zar}}(A,\mathbb{G}_m))[-1]$ for any $\mathbb{F}$-algebra $A$. However, the truncated presheaf itself is not a Zariski sheaf and the natural map $\tau^{\le1}R\Gamma_{\mbox{\rm \scriptsize Zar}}(-,\mathbb{G}_m)) \rightarrow R\Gamma_{\mbox{\rm \scriptsize Zar}}(-,\mathbb{G}_m)$ witnesses the target as the Zariski sheafification. Lisse motivic cohomology occurs as the graded pieces of a motivic filtration on *connective* $K$-theory $K^{\mbox{\rm \scriptsize cn}}$: **Proposition 18**. *Let $A$ be an $\mathbb{F}$-algebra. Then there exists a natural, $\mathbb{N}$-indexed, multiplicative filtration $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize lse}}\mathrm{K}^{\mbox{\rm \scriptsize cn}}(A)$ on the connective $K$-theory $\mathrm{K}^{\mbox{\rm \scriptsize cn}}(A)$ with graded pieces $$\mathrm{gr}^j_{\mbox{\rm \scriptsize lse}}\mathrm{K}^{\mbox{\rm \scriptsize cn}}(A) \simeq \mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)[2j]$$ for $j\ge0$; moreover $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)$ is supported in cohomological degrees $\le 2j$. If $A$ is local then $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)$ is supported in cohomological degrees $\le j$ and the filtration is bounded.* *Proof.* The desired filtration follows by left Kan extending the classical motivic filtration [\[eq:cla-mot\]](#eq:cla-mot){reference-type="eqref" reference="eq:cla-mot"}, since $\mathrm{K}^{\mbox{\rm \scriptsize cn}}:\text{CAlg}_\mathbb{F}\to\text{Sp}$ is left Kan extended from smooth $\mathbb{F}$-algebras [@EHKSY3 Ex. A.0.6]. The bound holds in the smooth case and is preserved by left Kan extension. If $A$ is local then $\mathrm{Fil}_{\mbox{\rm \scriptsize lse}}^j\mathrm{K}^{\mbox{\rm \scriptsize cn}}(A)$ is supported in homological degrees $\ge j$; indeed, this connectivity bound holds Zariski locally on smooth $\mathbb{F}$-algebras by the Gersten conjecture in motivic cohomology, and is again preserved by left Kan extension. ◻ **Remark 19** ($\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}$ is a cycle complex). Our interest in $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}$ is not just as an intermediate tool nor because it is the "easiest" extension of motivic cohomology beyond smooth schemes, but because it is defined purely in terms of algebraic cycles. This is already clear from the definition, since it is the left Kan extension of the cycle-theoretic $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}=z^j(-,\bullet)[-2j]$ from smooth algebras, but we spell it out more explicitly. Given a $\mathbb{F}$-algebra $A$, we may pick a simplicial resolution $P_\bullet\to A$ where each term $P_m$ is an ind-smooth $\mathbb{F}$-algebra and each face map $P_{m+1}\to P_m$ is a henselian surjection. Then the formalism of left Kan extension from smooth algebras implies that there is a natural equivalence $$\mathop{\mathrm{colim}}_{m\in\Delta^{\mbox{\rm \scriptsize op}}}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(P_m)\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)$$ (in this line and below we implicitly extend $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}$ and $z^j(-,\bullet)$ from smooth to ind-smooth $\mathbb{F}$-algebras, by taking filtered colimits). Expanding each $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(P_m)=z^j(P_m,\bullet)[-2j]$ as a complex of cycles, we see that the left side of the previous line is the $[-2j]$-shift of the totalisation of the bicomplex (really bisimplicial abelian group) $$\label{eq:lke-formula} \xymatrix@=5mm{ &\vdots\ar[d] &\vdots\ar[d] &\vdots\ar[d] \\ \cdots\ar[r] & z^j(P_2,2) \ar[r]\ar[d] & z^j(P_2,1) \ar[r] \ar[d]& z^j(P_2,0) \ar[d]\\ \cdots\ar[r] & z^j(P_1,2) \ar[r]\ar[d] & z^j(P_1,1) \ar[r] \ar[d]& z^j(P_1,0) \ar[d]\\ \cdots\ar[r] & z^j(P_0,2) \ar[r] & z^j(P_0,1) \ar[r] & z^j(P_0,0) }$$ In conclusion $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)$ admits a description in terms of various algebraic cycles on the affine schemes $\mathbb{A}_{P_m}^n$, for $n,m\ge0$ We will discuss comparisons between $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}$ and our new motivic cohomology in §[7](#section_lke){reference-type="ref" reference="section_lke"}. ## cdh-local motivic cohomology {#ss_cdh_local} This paper builds on the $\mathrm{cdh}$-local, $\mathbb{A}^1$-invariant version of motivic cohomology laid out in forthcoming joint work with Bachmann [@BachmannElmantoMorrow]. We apologise for the logical inconsistency of releasing the current paper first, and offer as explanation that the theory in [@BachmannElmantoMorrow] is developed for arbitrary qcqs schemes but simplifies over fields. Here we present a brief overview of the theory in that case. Let $\mathbb{F}$ be a prime field. Recall that an *abstract blowup square* is a cartesian square of qcqs $\mathbb{F}$-schemes $$\label{eqn_cdh_square} \begin{tikzcd} Y' \ar{r} \ar{d} & X' \ar{d}{p} \\ Y \ar[swap]{r}{i} & X, \end{tikzcd}$$ where $i$ is a finitely presented closed immersion and $p$ is a finitely presented, proper morphism inducing an isomorphism $X'\setminus Y'\stackrel{\simeq}{\to}X\setminus Y$. On $\mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}$ the *cdh topology* is the Grothendieck topology generated by the pretopology given by maps $\{ Y \rightarrow X, X' \rightarrow X\}$ for all abstract blow up squares as above and by the Nisnevich pretopology. A result of Voevodsky [@Voevodsky2010], generalized in [@ElmantoHoyoisIwasaKelly2021 Prop. 2.1.5] and [@BachmannHoyois2021 App. A] in the non-noetherian setting, states that cdh sheaves are exactly those presheaves which convert both Nisnevich squares and abstract blowup squares to cartesian squares. The main object of study of [@BachmannElmantoMorrow] is $$\mathbb Z(j)^{\mbox{\rm \scriptsize cdh}}:=L_{\mbox{\rm \scriptsize cdh}}L_{{\mbox{\rm \scriptsize Sch}}_\mathbb{F}^{\mbox{\rm \scriptsize qcqs,op}}/{\mbox{\rm \scriptsize Sm}}_\mathbb{F}^{\mbox{\rm \scriptsize op}}}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}:\operatorname{Sch}_k^{\mbox{\rm \scriptsize qcqs, op}}\longrightarrow\rm D(\mathbb{Z}),\label{eqn:Z(j)cdh}$$ namely the cdh sheafification of the left Kan extension of classical motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}:\operatorname{Sm}_\mathbb{F}^{\mbox{\rm \scriptsize op}}\to\rm D(\mathbb{Z})$ along the inclusion $\operatorname{Sm}_\mathbb{F}^{\mbox{\rm \scriptsize op}}\subseteq \operatorname{Sch}_\mathbb{F}^{\mbox{\rm \scriptsize qcqs, op}}$. In terms of universal properties, $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}:\operatorname{Sch}_k^{\mbox{\rm \scriptsize qcqs, op}}\to\rm D(\mathbb{Z})$ is initial among cdh sheaves on $\operatorname{Sch}_\mathbb{F}^{\mbox{\rm \scriptsize qcqs, op}}$ whose restriction to smooth $k$-schemes is equipped with a map from $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}$. Assuming resolution of singularities, this construction is essentially due to Friedlander, Suslin, and Voevodsky for finite type $k$-schemes [@FriedlanderSuslinVoevodsky2000]. In [@BachmannElmantoMorrow], joint with Bachmann, we establish the following properties of this *cdh-local motivic cohomology*: **Theorem 20** ([@BachmannElmantoMorrow]). *Cdh-local motivic cohomology $\mathbb{Z}(j)^{\mathrm{cdh}}: \mathrm{S}\mathrm{ch}{}_\mathbb{F}^{\mbox{\rm \scriptsize qcqs,op}}\to \rm D(\mathbb{Z})$, for $j \geq 0$, has the following properties for any qcqs $\mathbb{F}$-scheme $X$:* 1. *There exist a functorial, multiplicative, $\mathbb{N}$-indexed filtration $\mathrm{Fil}_{\mbox{\rm \scriptsize cdh}}^{\star}\mathrm{KH}(X)$ on the homotopy invariant $K$-theory $\mathrm{KH}(X)$, such that the graded pieces are naturally given by $\mathrm{gr}_{\mbox{\rm \scriptsize cdh}}^j\mathrm{KH}(X)\simeq \mathbb{Z}(j)^\mathrm{cdh}[2j],$ for $j\ge0$. In particular, writing $H^i_{\mbox{\rm \scriptsize cdh}}(X,\mathbb{Z}(j)):=H^i(\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X))$ for the corresponding *cdh-local motivic cohomology groups*, there exists an Atiyah-Hirzebruch spectral sequence $$\mathrm{H}_{\mathrm{cdh}}^{i-j}(X, \mathbb{Z}(-j)) \implies \mathrm{KH}_{-i-j}(X).$$ If $X$ has finite valuative dimension $\le d$ then this filtration is bounded: more precisely, $\mathrm{Fil}_{\mbox{\rm \scriptsize cdh}}^j\mathrm{KH}(X)$ is supported in cohomological degrees $\le d-j$.* 2. *Finitariness: $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$ is a finitary cdh sheaf.* 3. *Relation to étale cohomology: for any integer $\ell>0$ invertible in $\mathbb{F}$, there are natural equivalences $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}/\ell \simeq L_{\mathrm{cdh}}\tau^{\leq j}R\Gamma_{{\acute{e}t}}(-,\mu_{\ell}^{\otimes j}).$$ for $j\ge0$.* 4. *Relation to syntomic cohomology: if $\mathbb{F}=\mathbb{F}_p$ then for any $r\ge0$ there are natural equivalences $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}/p^r \simeq R\Gamma_{\mathrm{cdh}}(-,W_r\Omega^j_{\log})[-j].$$ for $j\ge0$.* 5. *[\[item-A1-invar\]]{#item-A1-invar label="item-A1-invar"} $\mathbb{A}^1$-invariance: the map $$\mathbb{Z}(j)^{\mathrm{cdh}}(X) \longrightarrow\mathbb{Z}(j)^{\mathrm{cdh}}(X\times \mathbb{A}^1),$$ induced by the projection $X \times \mathbb{A}^1 \rightarrow X$, is an equivalence for each $j\ge0$.* 6. *Weight one: there is a natural equivalence $$\mathbb{Z}(1)^{\mbox{\rm \scriptsize cdh}}(X)\simeq R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\mathbb{G}_m)[-1].$$* 7. *[\[item-PBF\]]{#item-PBF label="item-PBF"} Projective bundle formula: the powers of the first Chern class of the tautological bundle $c_1(\mathcal O(1)) \in \mathrm{Pic}(\mathbb{P}^r_X) \to H^{2}_{\mbox{\rm \scriptsize cdh}}(\mathbb{P}^r_X, \mathbb{Z}(1))$ induce a natural equivalence $$\bigoplus_{i=0}^r\mathbb{Z}(j-i)^{\mbox{\rm \scriptsize cdh}}(X)[-2i]\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(\mathbb{P}^r_X),\label{eqn_intro_PBF}$$* 8. *Comparison to classical motivic cohomology: for any field $k\supseteq\mathbb{F}$ and smooth $k$-scheme $X$, there are equivalences $$\mathbb{Z}(j)^{\mathrm{cdh}}(X) \simeq z^j(X,\bullet)[-2j]$$ for $j\ge0$.* For our purposes, the main point of Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"} is that there is a motivic filtration on $\mathrm{KH}$ of any qcqs equicharacteristic scheme such that its graded pieces are a version of motivic cohomology which is $\mathbb{A}^1$-invariant, satisfies $\mathrm{cdh}$ descent, and agrees with all known definitions of motivic cohomology on smooth $k$-schemes. This filtration on $\mathrm{KH}$ is defined by cdh sheafifying Proposition [Proposition 18](#prop:mot-filt){reference-type="ref" reference="prop:mot-filt"}. The deepest parts of the theorem are the $\mathbb{A}^1$-invariance and projective bundle formula assertions, and the comparison to classical motivic cohomology; these are proved using similar arguments to those of §[5](#section_pbf){reference-type="ref" reference="section_pbf"} and §[6](#section_smooth){reference-type="ref" reference="section_smooth"}. Details will of course appear in [@BachmannElmantoMorrow]. **Remark 21** (Comparison to motivic homotopy theory). We briefly discuss $\mathbb{A}^1$-invariant motivic homotopy theory and the slice filtration, though we stress that the results presented in this remark are not required for the current article. The summary of this remark is that cdh-local motivic cohomology is the same as the $\mathbb{A}^1$-invariant motivic cohomology coming from stable homotopy theory. For a qcqs scheme $X$ let $\mathcal{SH}(X)$ denote its $\infty$-category of motivic spectra, as introduced by Morel and Voevodsky [@Voevodsky1998]; for a modern approach see [@Robalo2015] [@BachmannHoyois2021 §4]. Examples of such motivic spectra include the unit object (or motivic sphere) $\pmb 1_X$, the motivic spectrum $\mathrm{KGL}_X$ representing homotopy invariant $K$-theory of smooth $X$-schemes, and a motivic Eilenberg--Maclane spectrum $\mathrm{H}\mathbb{Z}_X$ constructed by Spitzweck [@Spitzweck2018]. Any motivic spectrum $E\in \mathcal{SH}(X)$ may be equipped with a functorial *slice filtration* $$\cdots \to \mathrm f^{j+1}E\to \mathrm f^{j}E\to\mathrm f^{j-1}E\to \cdots\to E$$ in $\mathcal{SH}(X)$, whose graded pieces are denoted by $s^j E:=\mathrm{cofib}(\mathrm f^{j+1}E\to \mathrm f^j E)$. In [@BachmannElmantoMorrow], using previous work of Bachmann [@Bachmann2022], we establish natural equivalences in $\mathcal{SH}(X)$ $$\mathrm{H}\mathbb{Z}_X\simeq s^0(\pmb 1_X)\simeq s^0(\mathrm{KGL}_X),\label{intro_V_conj}$$ thereby settling Conjectures 1, 7, and 10 of Voevodsky [@Voevodsky2002a] for arbitrary qcqs schemes. The first equivalence of ([\[intro_V\_conj\]](#intro_V_conj){reference-type="ref" reference="intro_V_conj"}) shows that the cohomology theory represented by $\mathrm{H}\mathbb{Z}_X$, which is typically considered to be the correct theory of $\mathbb{A}^1$-invariant motivic cohomology, is the same as the cohomology theory represented by $s^0(\pmb 1_X)$. The second equivalence, combined with the machinery of the slice filtration, justifies this point of view since it implies that this cohomology theory is related to the $\mathrm{KH}$-groups of $X$ via an Atiyah--Hirzebruch spectral sequence. In [@BachmannElmantoMorrow] we also prove, at least for qcqs schemes $X$ of equal characteristic, that the $\mathbb{A}^1$-invariant motivic cohomology represented by $\mathrm{H}\mathbb{Z}_X$ coincides with the cdh-local motivic cohomology, and that the filtration on $\mathrm{KH}(X)$ coming from the slice filtration on $\mathrm{KGL}_X$ coincides with the filtration of Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}. ## (Topological) cyclic homology The constructions of this paper depend on trace methods in algebraic $K$-theory. Recall that if $E$ is a spectrum with $S^1$-action, then we can functorially associate several other spectra: its *homotopy fixed points* $E^{hS^1}$, its *homotopy orbits* $E_{hS^1}$ and its *Tate fixed points* $$E^{tS^1}:= \mathrm{cofib}(\mathrm{Nm}:E_{hS^1}[1] \rightarrow E^{hS^1}).$$ The same formalism exists if we replace $S^1$ by any of its finite subgroups, for example if $p$ is a prime then one has $$E^{tC_p}:= \mathrm{cofib}(\mathrm{Nm}:E_{hC_p} \rightarrow E^{hC_p}).$$ According to [@NikolausScholze2018], a *cyclotomic spectrum* is a spectrum with an $S^1$-action $E$ equipped with $S^1$-equivariant maps $$\phi_p: E \to E^{tC_p}$$ for all primes $p$; here $E^{tC_p}$ is given the residual $S^1/C_p \simeq S^1$-action. In the situation of algebraic geometry, we have the functor of topological Hochschild homology landing in the $\infty$-category of cyclotomic spectra: $$\mathrm{THH}: \mathrm{S}\mathrm{ch}{}^{\mbox{\rm \scriptsize qcqs,op}} \longrightarrow\mathrm{CycSp} \qquad X \mapsto \mathrm{THH}(\mathrm{Perf}(X))=:\mathrm{THH}(X).$$ Let $E$ be a cyclotomic spectrum which is bounded below (which will always be the case in our situations of interest). Firstly, for each prime number $p$ its *$p$-adic topological cyclic homology* is defined to be the $p$-complete spectrum $$\mathrm{TC}(E;\mathbb{Z}_p) := \mathrm{fib}\left( \phi_p^{hS^1} - \mathrm{can}: \left(E^{{\kern -.5pt}\wedge}_p\right)^{hS^1} \longrightarrow\left( E^{tC_p} \right)^{hS^1} \simeq \left( E^{{\kern -.5pt}\wedge}_p \right)^{tS^1} \right).$$ Here we have used that $E^{tC_p}$ is $p$-complete and that $E^{tC_p} \simeq (E^{{\kern -.5pt}\wedge}_p)^{tC_p}$ by [@NikolausScholze2018 Len. I.2.9], and $\mathrm{can}$ is the canonical map from fixed points to the Tate construction. We assemble these $p$-adic constructions to define the *integral topological cyclic homology* $\mathrm{TC}(E)$ of $E$ as the pullback $$\label{eq:integral-tc} \begin{tikzcd} \mathrm{TC}(E) \ar{r} \ar{d} & \left( E_{\mathbb{Q}} \right)^{hS^1}\ar{d}\\ \prod_{p} \mathrm{TC}(E;\mathbb{Z}_p) \ar{r} & \prod_{p} \left( E_p^{{\kern -.5pt}\wedge}[\tfrac1p] \right)^{hS^1}, \end{tikzcd}$$ where the bottom map is the product over $p$ of the compositions $\mathrm{TC}(E;\mathbb{Z}_p)\to \left(E^{{\kern -.5pt}\wedge}_p\right)^{hS^1} \to (E^{{\kern -.5pt}\wedge}_p[\tfrac{1}{p}]))^{hS^1}$. By a standard abuse of notation, for a scheme $X$, we write $\mathrm{TC}(X)$ in place of $\mathrm{TC}(\mathrm{THH}(X))$, and similarly for the $p$-adic variant. **Remark 22**. The square [\[eq:integral-tc\]](#eq:integral-tc){reference-type="eqref" reference="eq:integral-tc"} imitates the original definition of integral topological cyclic homology defined by Goodwillie [@Dundas2013 Lem. 6.4.3.2]. Indeed, [@NikolausScholze2018 Thm. II.4.11] proves that the definitions agree for bounded below cyclotomic spectra. There is a morphism of localizing invariants (in the sense of [@BlumbergGepnerTabuada2013]) called the *cyclotomic trace*, or just *trace map* for short $$\mathrm{tr}: \mathrm{K}\longrightarrow\mathrm{TC}.$$ A major result about this map is the Dundas--Goodwillie--McCarthy theorem [@Dundas2013], stating that its fibre $\mathrm{K}^{\inf}$ is insensitive to nilpotent thickenings. In the language of [@LandTamme2019], $\mathrm{K}^{\mbox{\rm \scriptsize inf}}$ is even *truncating*: for any connective $\mathbb{E}_1$-ring $A$, the map $\mathrm{K}^{\inf}(A) \rightarrow \mathrm{K}^{\inf}(\pi_0A)$ is an equivalence. This property implies not only nil-invariance [@LandTamme2019 Corol. 3.5] but even cdh descent, whence one obtains the following fundamental square: **Theorem 23** (Kerz--Strunk--Tamme, Land--Tamme). *Let $X$ be a qcqs scheme. The the square $$\label{eq:mainsq} \begin{tikzcd} \mathrm{K}(X) \ar{r} \ar{d} & \mathrm{TC}(X) \ar{d} \\ \mathrm{KH}(X) \ar{r} & L_{\mathrm{cdh}}\mathrm{TC}(X). \end{tikzcd}$$ is cartesian.* *Proof.* This follows from the facts that the canonical map $L_{\mathrm{cdh}}\mathrm{K}(X)\to \mathrm{KH}(X)$ is an equivalence [@KerzStrunkTamme2018 Thm. 6.3] (see also [@KellyMorrow2021 Rem. 3.4]) and that $\mathrm{K}^{\mbox{\rm \scriptsize inf}}$ satisfies $\mathrm{cdh}$ descent [@LandTamme2019]. ◻ To be clear, the bottom horizontal arrow in the previous diagram is obtained by cdh sheafifying the trace map $\mathrm{K}\to\mathrm{TC}$. Indeed, as we commented in the proof, we have $\mathrm{KH}\simeq L_{\mbox{\rm \scriptsize cdh}}\mathrm{K}$ and therefore there is an induced *cdh-local trace map* $\mathrm{KH}\to L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}$; it will play an important role in the construction of our motivic cohomology. # Definition of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)$ {#s_motivic_def} In this section we introduce our theory of motivic cohomology and the motivic filtration on algebraic $K$-theory. We also establish a number of other fundamental properties, such as finitariness, to justify that the definition is not unreasonable. ## Characteristic zero {#sec:charzero} We begin with reminders on cyclic homology. First note that for any $X \in \mathrm{S}\mathrm{ch}{}_{\mathbb{Q}}^{\mbox{\rm \scriptsize qcqs}}$ we have $$\mathrm{THH}(X) \simeq \mathrm{THH}(X) \otimes_{\mathrm{THH}(\mathbb{Q})} \mathbb{Q}\simeq \mathrm{HH}(X/\mathbb{Q})$$ where the second equivalence is formal and the first follows from the fact that $\mathrm{THH}(\mathbb{Q})\simeq\mathbb{Q}$. Similarly, the integral topological cyclic homology $\mathrm{TC}(X)$, as defined by the pullback square [\[eq:integral-tc\]](#eq:integral-tc){reference-type="eqref" reference="eq:integral-tc"}, coincides with the *negative cyclic homology* of $X$; indeed, the latter is defined by $\mathrm{HC}^-(X/\mathbb{Q}) := \left( \mathrm{HH}(X/\mathbb{Q}) \right)^{hS^1}$ and the square [\[eq:integral-tc\]](#eq:integral-tc){reference-type="eqref" reference="eq:integral-tc"} collapses to an equivalence $$\mathrm{TC}(X)\stackrel{\sim}{\to}\mathrm{HC}^-(X/\mathbb{Q})$$ (since the bottom terms of the square vanish as $\mathrm{THH}(X)$ has vanishing $p$-completion). The cyclotomic trace becomes the more classical *Goodwillie trace* $$\mathrm{tr}:\mathrm{K}(X)\longrightarrow\mathrm{HC}^-(X/\mathbb{Q}),$$ and Theorem [Theorem 23](#thm:mainsq){reference-type="ref" reference="thm:mainsq"} is rewritten as the cartesian square $$\label{eq:hc-} \begin{tikzcd} \mathrm{K}(X) \ar{d} \ar{r} & \mathrm{HC}^-(X/\mathbb{Q}) \ar{d}\\ \mathrm{KH}(X) \ar{r} & L_{\mathrm{cdh}}\mathrm{HC}^-(X/\mathbb{Q}). \end{tikzcd}$$ We remark that the bottom right term in the previous diagram is poor notation, which we will nevertheless continue to use; it should really be written $(L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(-/\mathbb{Q}))(X)$. **Remark 24** (Replacing $\mathbb{Q}$ by a general case $k$). More generally, let $k$ be a discrete commutative ring. For any qcqs $k$-scheme $X$, let $\mathrm{HC}^-(X/k):=\mathrm{HH}(X/k)^{hS^1}$ denote its negative cyclic homology relative to $k$. Then there is a natural map $\mathrm{TC}(X)\to\mathrm{HC}^-(X/k)$ constructed as follows. Firstly, from square ([\[eq:integral-tc\]](#eq:integral-tc){reference-type="ref" reference="eq:integral-tc"}) we see that $\mathrm{TC}(X)$ naturally maps to the pull back of $$\begin{tikzcd} \ar[dotted]{r} \ar[dotted]{d} & \left( \mathrm{THH}(X)_{\mathbb{Q}} \right)^{hS^1}\ar{d}\\ \prod_{p} (\mathrm{THH}(X)_p^{{\kern -.5pt}\wedge})^{hS^1} \ar{r} & \prod_{p} \left( \mathrm{THH}(X)_p^{{\kern -.5pt}\wedge}[\tfrac1p] \right)^{hS^1}. \end{tikzcd} \label{eqn_fixed_points}$$ Removing the $hS^1$ from the three corners, the pullback of the square is $\mathrm{THH}(X)$; passing to homotopy fixed points preserves pullback squares (and commutes with products), whence the pullback of the square ([\[eqn_fixed_points\]](#eqn_fixed_points){reference-type="ref" reference="eqn_fixed_points"}) is the *negative topological cyclic homology* $\mathrm{TC}^-(X):=\mathrm{THH}(X)^{hS^1}$. This constructs a natural map $\mathrm{TC}(X)\to\mathrm{TC}^-(X)$, which may then be composed with $S^1$-fixed points of $\mathrm{THH}(X)\to \mathrm{HH}(X/k)$. Composing with the cyclotomic trace thereby defines a trace map $\mathrm{K}(X)\to \mathrm{HC}^-(X/k)$ relative to $k$; of course it would have been sufficient to define this in the case $k=\mathbb{Z}$ and then compose with the canonical map $\mathrm{HC}^-(X/\mathbb{Z})\to \mathrm{HC}^-(X/k)$. To construct our motivic filtration on $K$-theory in characteristic zero we first recall the Hochschild--Kostant--Rosenberg filtration on negative cyclic homology, which relies on the theory of derived de Rham cohomology of lllusie [@Illusie1972] and Bhatt [@Bhatt2012; @Bhatt2012a]. Since the following two results do not require any characteristic zero hypothesis,[^6] let $k$ be a discrete commutative ring and recall, for any $k$-algebra $R$, the *Hodge-completed derived de Rham cohomology* $\widehat{L\Omega}_{R/k}\in \text D(k)$ of $R$ and its complete $\mathbb{N}$-indexed *Hodge filtration* $\widehat{L\Omega}_{R/k}^{\ge\star}$. For $j\ge0$ the cofibre of the map $\widehat{L\Omega}_{R/k}^{\ge j}\to \widehat{L\Omega}_{R/k}$ is $L\Omega_{R/k}^{<j}$, which admits a finite decreasing filtration with graded pieces (in increasing order) $$R, L_{R/k}[-1],L^2_{R/k}[-2],\dots,L^{j-1}_{R/k}[-j+1].$$ By fpqc descent of $L_{-/k}$ and its wedge powers on $k$-algebras [@BhattMorrowScholze2 Thm. 3.1], right Kan extension defines a unique fpqc sheaf $$\text{Sch}_k^{\mbox{\rm \scriptsize qcqs,op}}\longrightarrow\text D(k),\qquad X\mapsto R\Gamma(X,\widehat{L\Omega}_{-/k})$$ whose value on affines $\mathrm{Spec}R$ is $\widehat{L\Omega}_{R/k}$; similarly for $\widehat{L\Omega}_{-/k}^{\ge j}$, $L\Omega_{-/k}^{<j}$, and each wedge power of $L_{-/k}$. Alternatively, the fpqc sheaf $R\Gamma(-,\widehat{L\Omega}_{-/k})$ is equivalent to the Nisnevich sheafification of $X\mapsto \widehat{L\Omega}_{\mathcal{O}_X(X)/k}$, and similarly for the variants. The following is the HKR filtration on negative cyclic homology: **Theorem 25** (HKR filtration [@antieau-fil; @raksit-hkr; @mrt-hkr]). *Let $k$ be a discrete commutative ring. For any qcqs $k$-scheme $X$, there exists a functorial, complete, multiplicative filtration $\mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^\star\mathrm{HC}^-(X/k)$ on $\mathrm{HC}^-(X/k)$ whose graded pieces for $j\in\mathbb{Z}$ are given by $$\mathrm{gr}_{\mbox{\rm \scriptsize HKR}}^j\mathrm{HC}^-(X/k) \simeq R\Gamma(X, \widehat{L\Omega}_{-/k}^{\geq j})[2j].$$ Furthermore, if $X$ is quasisyntomic over $k$[^7] then this filtration is exhaustive.* **Remark 26**. If $k$ is a $\mathbb{Q}$-algebra and $X$ is smooth over $k$, then this result is essentially due to Loday [@Loday1989]. Dropping the hypothesis that $X$ be smooth, but remaining in characteristic zero, the product decomposition of the previous theorem is due to Weibel, under the name of the "Hodge decomposition" and written in Adams operator type notation as "$\text{HN}(X/k) \simeq \prod_j\text{HN}^{(j)}(X/k)$" in [@Weibel1997; @Cortinas2008a]. **Remark 27** (Variant: cdh-local HKR filtration). Cdh sheafifying the HKR filtration levelwise we see that, for any qcqs $k$-scheme $X$, there exists a functorial, multiplicative filtration $$\mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{\star}L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/k):= L_{\mathrm{cdh}}\mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{\star}\mathrm{HC}^-(-/k)(X),$$ on $L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/k)$ whose graded pieces for $j\in\mathbb{Z}$ are given by $$\mathrm{gr}_{\mbox{\rm \scriptsize HKR}}^jL_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/k) \simeq R\Gamma_{\mbox{\rm \scriptsize cdh}}(X, \widehat{L\Omega}_{-/k}^{\geq j})[2j].$$ Here we denote by $$\text{Sch}_k^{\mbox{\rm \scriptsize qcqs,op}}\to\text D(k), \qquad X\mapsto R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\widehat{L\Omega}_{-/k})$$ the cdh sheafification of the presheaf $R\Gamma(-,\widehat{L\Omega}_{-/k})$, or equivalently the cdh sheafification of the presheaf $X\mapsto \widehat{L\Omega}_{\mathcal{O}_X(X)/k}$. Similar notation will be used for $L\Omega_{-/k}^{<j}$ and each wedge power of $L_{-/k}$, though we stress that the canonical maps The following is probably known to experts but we could not find a standalone reference in the required degree of generality: **Lemma 28** (Cdh descent of derived de Rham cohomology in characteristic zero). *For any $\mathbb{Q}$-algebra $k$, the two presheaves $$\begin{aligned} \mathrm{S}\mathrm{ch}{}^{\mbox{\rm \scriptsize qcqs,op}}_k&\longrightarrow{\rm D}(k)\\ X&\mapsto R\Gamma(X,\widehat{L\Omega}_{-/k})\\ X&\mapsto\mathrm{HC}^-(X/k)/\mathrm{Fil}^0_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/k)\end{aligned}$$ satisfy cdh descent.* *Proof.* The cited references for Theorem [Theorem 25](#thm:hkr){reference-type="ref" reference="thm:hkr"} also construct an HKR filtration on periodic cyclic homology: for a qcqs $k$-scheme $X$, this is a functorial, complete, multiplicative filtration $\mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^\star\mathrm{HP}(X/k)$ on $\mathrm{HP}(X/k)$ whose graded pieces for $j\in\mathbb{Z}$ are given by $$\mathrm{gr}_{\mbox{\rm \scriptsize HKR}}^j\mathrm{HP}(X/k) \simeq R\Gamma(X, \widehat{L\Omega}_{-/k})[2j].$$ The references show that the canonical map $\mathrm{HC}^-(X/k)\to \mathrm{HP}(X/k)$ respects the HKR filtrations, i.e., naturally upgrades to a filtered map, given on graded pieces by the canonical maps $R\Gamma(X,\widehat{L\Omega}^{\ge j}_{-/k})\to R\Gamma(X,\widehat{L\Omega}_{-/k})$. Since $k$ is a $\mathbb{Q}$-algebra, the HKR filtration on $\mathrm{HP}(X/k)$ is naturally split [@Bals2023], i.e., there is a natural equivalence $\mathrm{HP}(X/k)\simeq\prod_{n\in\mathbb{Z}}R\Gamma(X, \widehat{L\Omega}_{-/k})[2n]$ such that the HKR filtration on the left matches the product filtration $\prod_{n\le -j}$ on the right. The presheaf $R\Gamma(-, \widehat{L\Omega}_{-/k}): \mathrm{S}\mathrm{ch}{}^{\mbox{\rm \scriptsize qcqs,op}}_k\to\text D(k)$ is thus a direct summand of the presheaf $\mathrm{HP}(-/k)$; but the latter is a cdh sheaf thanks to the theory of truncating invariants [@Cortinas2008 Corol. 3.13] [@LandTamme2019 Cor. A.6], so therefore the former is also a cdh sheaf. The cited references for Theorem [Theorem 25](#thm:hkr){reference-type="ref" reference="thm:hkr"} also implicity prove that the canonical map $\mathrm{HC}^{-}(-/k) \rightarrow \mathrm{HP}(-/k)$ induces an equivalence $\mathrm{HC}^-(-/k)/\mathrm{Fil}^0_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/k) \xrightarrow{\simeq} \mathrm{HP}(-/k)/\mathrm{Fil}^0_{\mbox{\rm \scriptsize HKR}}\mathrm{HP}(-/k)$. By the aforementioned splitting the latter is equivalent to $\prod_{n\leq -1}R\Gamma(-, \widehat{L\Omega}_{-/k})[2n]$ and is therefore a cdh sheaf since we have shown that $R\Gamma(-, \widehat{L\Omega}_{-/k})$ is a cdh sheaf. ◻ We next prove the following compatibility, informally stating that for any smooth $k$-scheme $X$ the trace map $\mathrm{K}(X)\to\mathrm{HC}^-(X/k)$ naturally carries the classical motivic filtration on the left to the HKR filtration on the right. In fact, it is rather the cdh-local analogue below (Corollary [Corollary 31](#corol_cdh_filtered_trace){reference-type="ref" reference="corol_cdh_filtered_trace"}) which is crucial to our construction, but the smooth case is required for the proof of the cdh case and also to formulate the comparison map to classical motivic cohomology (Construction [Construction 57](#cons:cla-vs-new){reference-type="ref" reference="cons:cla-vs-new"}): **Proposition 29**. *Let $k_0\to k$ be a quasismooth[^8] map of rings, where $k$ is a field. Then the trace map $\mathrm{K}\to\mathrm{HC}^-(-/k_0)$, viewed as a map between spectra-valued presheaves on ${\mathrm{S}\mathrm{m}}_k$, admits a unique, multiplicative extension to a map of filtered presheaves of spectra $\mathrm{Fil}_{\mbox{\rm \scriptsize cla}}^{\star}\mathrm{K}\to \mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{\star}\mathrm{HC}^-(-/k_0)$.* *Proof.* There is a $t$-structure on $\mathrm{Shv}_{\mathrm{Zar}}({\mathrm{S}\mathrm{m}}_k;\text{Sp})$, which denotes the stable $\infty$-category of Zariski sheaves of spectra on smooth $k$-schemes. This $t$-structure is described as follows: - its non-negative part $\mathrm{Shv}_{\mathrm{Zar}}({\mathrm{S}\mathrm{m}}_k;\text{Sp})_{\geq 0}$ is given by those sheaves of spectra $\mathcal F$ such that the homotopy sheaves $\underline{\pi}_n\mathcal F$ vanish for all $n < 0$; - its non-positive part $\mathrm{Shv}_{\mathrm{Zar}}({\mathrm{S}\mathrm{m}}_k;\text{Sp})_{\le 0}$ is given by those sheaves of spectra $\mathcal F$ such that $\pi_n(\mathcal F(X))$ vanishes for all $X\in{\mathrm{S}\mathrm{m}}_k$ and all $n > 0$. This is a specialization of a much more general result on $\infty$-topoi as in [@LurieSAG Prop. 1.3.2.7]. Now let $j\in\mathbb{Z}$ and observe the following facts about the connectivity of the filtrations on $\mathrm{K}$ and $\mathrm{HC}^-$ with respect to the above $t$-structure: 1. $\mathrm{Fil}^{\geq j}_{\mbox{\rm \scriptsize cla}}\mathrm{K}$ is $j$-connective. This follows from standard vanishing bounds in motivic cohomology, though a little care is required since a priori taking homotopy sheaves might not commute with viewing $K$-theory as a complete filtered object. Let $X$ be a smooth $k$-scheme and set $d:=\dim X$. Recall first that, for any $i\ge0$, the cohomology sheaves $\mathcal H^n(\mathbb{Z}(i)^{\mbox{\rm \scriptsize cla}})=\underline\pi_{-n}\mathbb{Z}(i)^{\mbox{\rm \scriptsize cla}}$ on $X_{\mbox{\rm \scriptsize Zar}}$ vanish for $n>i$ (by Gersten injectivity to reduce to the case of a field); therefore, the motivic complex $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X)$ of $X$ itself vanishes in cohomological degrees $> 2j$ (by the Gersten resolution), and also in degrees $>j+d$ (for dimension reasons). Using the dimension bound we see, for any affine open $\mathrm{Spec}(R)\subseteq X$ in place of $X$ and $i\ge j+d$, that $\mathrm{gr}^i_{\mbox{\rm \scriptsize cla}}\mathrm{K}(R)=\mathbb{Z}(i)^{\mbox{\rm \scriptsize cla}}(R)[2i]$ is supported in cohomological degrees $\le -j$; by completeness of the motivic filtration we deduce the same for $\mathrm{Fil}^{\geq j+d}_{\mbox{\rm \scriptsize cla}}K(R)$, for all open affines $\mathrm{Spec}(R)\subseteq X$. In particular, $\mathrm{Fil}^{\geq j+d}_{\mbox{\rm \scriptsize cla}}\mathrm{K}$ is $j$-connective on $X_{\mbox{\rm \scriptsize Zar}}$. But now the problem reduces, by a finite induction, to checking that $\mathrm{gr}_{\mbox{\rm \scriptsize cla}}^i\mathrm{K}$ is $i$-connective for each $i\ge0$ (in fact, just for $i=j,\dots,j+d-1$), or in other words that the Zariski cohomology sheaves $\mathcal H^n(\mathbb{Z}(i)^{\mbox{\rm \scriptsize cla}})$ vanish for $n>i$. But this was already explained in the previous paragraph and so completes the proof. 2. On the other hand, $\mathrm{Fil}^{<j}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/k_0):=\mathrm{HC}^-(-/k_0)/\mathrm{Fil}^{j}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/k_0)$ is $j-1$-truncated for the $t$-structure. Indeed, for any smooth $k$-algebra $R$ and $i\in\mathbb{Z}$, the $i^{\mbox{\rm \scriptsize th}}$ graded piece of the HKR filtration on $\mathrm{HC}^-(R/k_0)$ is given by $$\mathrm{gr}^i_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(R/k_0) = \widehat{L\Omega}_{R/k_0}^{\geq i}[2i]\simeq \Omega_{R/k_0}^{\geq i}[2i]$$ since the composition $k_0\to k\to R$ is quasismooth; the graded piece therefore vanishes in cohomological degrees $< -i$. By induction it follows, for any $i<j$, that the cofibre of $\mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{j}\mathrm{HC}^-(R/k_0)\to \mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{i}\mathrm{HC}^-(R/k_0)$ vanishes in cohomological degrees $\le -j$. Finally let $i\to\infty$, recalling from Theorem [Theorem 25](#thm:hkr){reference-type="ref" reference="thm:hkr"}(1) that the filtration is exhaustive in this case, to deduce that $\mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{< j}\mathrm{HC}^-(R/k_0)$ vanishes in cohomological degrees $\le - j$ (i.e., homotopical degrees $>j-1$). Therefore, by general results on $t$-structures, the mapping space $$\mathrm{Map}_{\mathrm{Shv}_{\mathrm{Zar}}({\mathrm{S}\mathrm{m}}_k;\text{Sp})}(\mathrm{Fil}^{\geq j}_{\mbox{\rm \scriptsize cla}}\mathrm{K}, \mathrm{Fil}^{<j}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/k_0))$$ is contractible for each $j$. By induction, the trace map $\mathrm{K}\rightarrow \mathrm{HC}^-(-/k_0)$ therefore uniquely refines to compatible maps $\mathrm{Fil}^{j}_{\mbox{\rm \scriptsize cla}}\mathrm{K}\to\mathrm{Fil}^{j}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/k_0)$ for all $j\ge0$, as desired. To ensure multiplicativity, one uses the Postnikov $t$-structure on Zariski sheaves of filtered spectra as introduced in [@raksit-hkr Cons. 3.3.6-7]. This is a $t$-structure which wraps together the $t$-structure on Zariski sheaves and the $t$-structure on the filtered derived category. The (co-)connective part consists of filtered Zariski sheaves $F^{\star}$ such that $F^j \in \mathrm{Shv}_{\mathrm{Zar}}({\mathrm{S}\mathrm{m}}_k;\text{Sp})_{\geq j}$ (resp. $F^j\in \mathrm{Shv}_{\mathrm{Zar}}({\mathrm{S}\mathrm{m}}_k;\text{Sp})_{\leq j}$) for all $j\in\mathbb{Z}$. Furthermore, the truncation functor $\tau^P_{\geq 0}$ admits a lax symmetric monoidal structure such that the counit map $\tau^P_{\geq 0} \rightarrow \operatorname{id}$ is a morphism of lax symmetric monoidal functors. In particular, if $F^\star$ is a filtered, multiplicative sheaf then the map $\tau^P_{\geq 0}F^{\star} \rightarrow F^{\star}$ is multiplicative. Our proof shows that, for the Postnikov $t$-structure, firstly $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}$ is connective, and secondly $\text{cofib}(\mathrm{Fil}^\star_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/k_0)\to\mathrm{HC}^-(-/k_0))$ is $-1$-truncated where the target is given the constant filtration; therefore the map $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/k_0)\to\mathrm{HC}^-(-/k_0)$ is a $\tau^P_{\geq 0}$-equivalence. So we obtain a multiplicative map of filtered objects $$\mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}\xleftarrow{\simeq} \tau^P_{\geq 0}\mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}\rightarrow \tau^P_{\geq 0}\mathrm{HC}^-(-/k_0) \simeq \tau^P_{\geq 0}\mathrm{Fil}^\star_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/k_0) \rightarrow \mathrm{Fil}^\star_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/k_0),$$ as desired. ◻ **Remark 30**. In fact the $t$-structure argument in the above proposition proves slightly more: with the same hypotheses as in Proposition [Proposition 29](#prop:compat){reference-type="ref" reference="prop:compat"}, *any morphism* $K \rightarrow \mathrm{HC}^-(-/k_0)$ promotes uniquely to a filtered map intertwining the classical motivic and HKR filtration. This will be used later to compute Adams operations on rationalized motivic cohomology. It follows that the cdh-local trace map $\mathrm{KH}(X)\to L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/\mathbb{Q})$, for qcqs $\mathbb{Q}$-schemes $X$, also naturally carries the cdh-local motivic filtration on the left to the cdh-local HKR filtration on the right: **Corollary 31**. *The cdh-local trace map $\mathrm{KH}\to L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(-/\mathbb{Q})$, viewed as a map between spectra-valued presheaves on $\mathrm{S}\mathrm{ch}{}_\mathbb{Q}^{\mbox{\rm \scriptsize qcqs}}$, admits a unique multiplicative extension to a map of filtered presheaves $$\mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize mot}}\mathrm{KH}\longrightarrow\mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize HKR}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(-/\mathbb{Q})$$ (the filtration on the left being the cdh-local motivic filtration of Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(1); the filtration on the right is the cdh-local HKR filtration of Remark [Remark 27](#rem_cdh_local_HKR){reference-type="ref" reference="rem_cdh_local_HKR"}).* *Proof.* Given a smooth $\mathbb{Q}$-scheme $X$, we claim that the canonical maps of filtered spectra $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X)\to \mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{KH}(X)$ and $\mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{\star}\mathrm{HC}^-(-/\mathbb{Q})\to \mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{\star}L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(-/\mathbb{Q})$ are equivalences. The first follows from the equivalence $\mathrm{K}(X)\stackrel{\sim}{\to}\mathrm{KH}(X)$ and the equivalences $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X)\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$ of Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(8). The second is a standard consequence of strong resolution of singularities. Consequently, any filtered upgrade of the cdh-local trace map restricts to the unique filtered upgrade of the trace map for smooth $\mathbb{Q}$-schemes (from Proposition [Proposition 29](#prop:compat){reference-type="ref" reference="prop:compat"}), and conversely the filtered upgrade of the cdh-local trace map is then necessarily given by the following composition: $$\mathrm{Fil}_{\mbox{\rm \scriptsize cdh}}^{\star}\mathrm{KH}\longrightarrow L_{\mbox{\rm \scriptsize cdh}}L_{{\mbox{\rm \scriptsize Sch}}_{\mathbb{Q}}^{\mbox{\rm \scriptsize qcqs,op}}/{\mbox{\rm \scriptsize Sm}}_{\mathbb{Q}}^{\mbox{\rm \scriptsize op}}}\mathrm{Fil}^\star_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/\mathbb{Q})\longrightarrow\mathrm{Fil}^\star_{\mbox{\rm \scriptsize HKR}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(-/\mathbb{Q}).$$ Here the first map is given by left Kan extending the filtered trace map for smooth $\mathbb{Q}$-schemes along $\text{Sm}_\mathbb{Q}^{\mbox{\rm \scriptsize op}}\subseteq\text{Sch}_\mathbb{Q}^{\mbox{\rm \scriptsize qcqs, op}}$, then cdh sheafifying. The second map is the cdh sheafification of the canonical map $L_{{\mbox{\rm \scriptsize Sch}}_{\mathbb{Q}}^{\mbox{\rm \scriptsize qcqs,op}}/{\mbox{\rm \scriptsize Sm}}_{\mathbb{Q}}^{\mbox{\rm \scriptsize op}}}\mathrm{Fil}^\star_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/\mathbb{Q})\to \mathrm{Fil}^\star_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/\mathbb{Q})$. Multiplicativity follows from the fact that both left Kan extension and sheafification are multiplicative operations. ◻ We may now construct our motivic cohomology and motivic filtration on qcqs $\mathbb{Q}$-schemes: **Definition 32**. For a qcqs $\mathbb{Q}$-scheme $X$, let $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$ be the filtered spectrum defined as the pullback (in filtered $\mathbb{E}_{\infty}$-algebras) of the diagram $$\label{eq:motfilt} \begin{tikzcd} \mathrm{Fil}^{\star}_{\mathrm{mot}}\mathrm{K}(X) \ar[dotted]{d} \ar[dotted]{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/\mathbb{Q})\ar{d} \\ \mathrm{Fil}^\star_{\mbox{\rm \scriptsize cdh}}\mathrm{KH}(X) \ar{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize HKR}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/\mathbb{Q}). \end{tikzcd}$$ Here the bottom horizontal arrow is the unique filtered upgrade of the cdh-local trace map $\mathrm{KH}(X)\to L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/\mathbb{Q})$ provided by Corollary [Corollary 31](#corol_cdh_filtered_trace){reference-type="ref" reference="corol_cdh_filtered_trace"}, and the right vertical arrow is the canonical map. For $j\in\mathbb{Z}$, define the *weight-$j$ motivic cohomology* of $X$ to be $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X):=(\mathrm{gr}^j_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X))[-2j],$$ which we will see in Theorem [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"} lies in $D(\mathbb{Z})$ and vanishes for $j<0$. The associated motivic cohomology groups, for $i\in\mathbb{Z}$, are $H^i_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j)):=H^i(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X))$. In the following theorem we collect some of the immediate, but fundamental, properties of this motivic cohomology theory for qcqs $\mathbb{Q}$-schemes: **Theorem 33**. *Let $j\in\mathbb{Z}$. For any qcqs $\mathbb{Q}$-scheme $X$, the weight-$j$ motivic cohomology has the following properties:* 1. *$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)=0$ for $j<0$.* 2. *There is a natural pullback square $$\begin{tikzcd} \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X) \ar{r} \ar{d} & R\Gamma(X,\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq j}) \ar{d}\\ \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X) \ar{r} & R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq j}). \end{tikzcd}$$* 3. *Fundamental fibre sequence: there is a natural fibre sequence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)\longrightarrow{\rm cofib}\left(R\Gamma(X,L\Omega_{-/\mathbb{Q}}^{< j})\to R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\Omega_{-/\mathbb{Q}}^{< j})\right)[-1]$$* 4. *For any integer $m\ge0$, the map $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)/m\to\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)/m$ is an equivalence.* 5. *The presheaf $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}:\rm{Sch}_\mathbb{Q}^{\mbox{\rm \scriptsize qcqs}}\to\rm D(\mathbb{Z})$ is a finitary Nisnevich sheaf.* *Proof.* We obtain the pullback square of part (2), even for all $j\in\mathbb{Z}$ if we define $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)=0$ for $j<0$, by taking graded pieces in the cartesian square ([\[eq:motfilt\]](#eq:motfilt){reference-type="ref" reference="eq:motfilt"}). As a reminder, the graded pieces of $\mathrm{KH}(X)$ are described by Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}; those of $\mathrm{HC}^-(X/\mathbb{Q})$ by Theorem [Theorem 25](#thm:hkr){reference-type="ref" reference="thm:hkr"}; and those of its cdh sheafification by Remark [Remark 27](#rem_cdh_local_HKR){reference-type="ref" reference="rem_cdh_local_HKR"}. In particular, when $j<0$ we have established the existence of a cartesian square $$\begin{tikzcd} \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X) \ar{r} \ar{d} & R\Gamma(X,\widehat{L\Omega}_{-/\mathbb{Q}}) \ar{d}\\ 0 \ar{r} & R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\widehat{L\Omega}_{-/\mathbb{Q}}). \end{tikzcd}$$ But the right vertical arrow is an equivalence because Hodge-completed derived de Rham cohomology satisfies cdh descent in characteristic zero by Lemma [Lemma 28](#lemma_cdh_descent_HP){reference-type="ref" reference="lemma_cdh_descent_HP"}. Therefore $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)=0$ for $j<0$. To obtain the fundamental fibre sequence, compute the cofibre of the right vertical arrow in part (2) as follows: compare the fibre sequence $$R\Gamma(X,\widehat{L\Omega}_{-/\mathbb{Q}}^{\ge j})\longrightarrow R\Gamma(X,\widehat{L\Omega}_{-/\mathbb{Q}})\longrightarrow R\Gamma(X,L\Omega_{-/\mathbb{Q}}^{<j})$$ to its cdh sheafified version, and use the following two facts: firstly, cdh sheafifying the middle term of the fibre sequence does not change it, by Lemma [Lemma 28](#lemma_cdh_descent_HP){reference-type="ref" reference="lemma_cdh_descent_HP"}; secondly, the canonical map $R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,L\Omega_{-/\mathbb{Q}}^{<j})\to R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\Omega_{-/\mathbb{Q}}^{<j})$ is an equivalence, either by resolution of singularities or by Gabber--Ramero's results on the cotangent complex of valuation rings [@GabberRamero2003 Thm. 6.5.12 & Corol. 6.5.21]. Part (4) follows from the pullback square of (2), since the complexes on the right side of the square are rational. For part (5), recall that wedge powers $L^i_{-/\mathbb{Q}}$ of the cotangent complex commute with filtered colimits of rings; therefore, by Zariski descent and a finite induction, $R\Gamma(X,L\Omega_{-/\mathbb{Q}}^{<j})$ is finitary. Cdh sheafifying preserves finitariness, so $R\Gamma_{\mbox{\rm \scriptsize cdh}}(-,L\Omega_{-/\mathbb{Q}}^{<j})$ is also finitary. Finally, $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$ is finitary by Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(2). We now deduce finitariness of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ from the fundamental fibre sequence. ◻ **Example 34** (Weight $0$). The right vertical arrow in Theorem [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(2) is an equivalence when $j=0$, by cdh descent of Hodge-completed derived de Rham cohomology; therefore the same is true of the left vertical arrow. That is, there is a natural equivalence $$\mathbb{Z}(0)^{\mbox{\rm \scriptsize mot}}(X)\stackrel{\sim}{\to}\mathbb{Z}(0)^{\mbox{\rm \scriptsize cdh}}(X)=R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\mathbb{Z})$$ for any qcqs $\mathbb{Q}$-scheme $X$ (the equality in the previous line easily following from the definition of $Z(0)^{\mbox{\rm \scriptsize cdh}}$ in §[3.3](#ss_cdh_local){reference-type="ref" reference="ss_cdh_local"}). We will see in Example [Example 49](#example_0p){reference-type="ref" reference="example_0p"} that the same holds in finite characteristic. Next we state some of the fundamental properties of the motivic filtration: namely, $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$ is indeed a filtration on $\mathrm{K}(X)$, as suggested by the notation, and so there is the desired Atiyah--Hirzebruch spectral sequence: **Theorem 35**. *Let $X$ be a qcqs $\mathbb{Q}$-scheme. Then the filtered spectrum $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$ is $\mathbb{N}$-indexed, multiplicative, and satisfies $\mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)=\mathrm{K}(X)$. If $X$ has finite valuative dimension, then:* 1. *the filtration is bounded and so induces a bounded multiplicative Atiyah--Hirzebruch spectral sequence $$E_2^{ij}=H^{i-j}_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(-j))\implies \mathrm{K}_{-i-j}(X);$$* 2. *the filtration is rationally split, i.e., there is a natural, multiplicative equivalence of filtered spectra $$\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)_\mathbb{Q}\stackrel{\sim}{\to}\bigoplus_{j\ge0}\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(X)[2j],$$* 3. *and the spectral sequence degenerates rationally.* *Proof.* Theorem [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(1) already shows that the filtered object $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$ is $\mathbb{N}$-graded. By definition $\mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$ is defined via a pullback square $$\begin{tikzcd} \mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X) \ar{r} \ar{d} & \mathrm{Fil}^0_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/\mathbb{Q}) \ar{d}\\ \mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}\mathrm{KH}(X) \ar{r} & L_{\mbox{\rm \scriptsize cdh}}\mathrm{Fil}^0_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/\mathbb{Q}), \end{tikzcd}$$ which admits a map to the pullback square ([\[eq:hc-\]](#eq:hc-){reference-type="ref" reference="eq:hc-"}). Since $\mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}\mathrm{KH}(X)\stackrel{\sim}{\to}\mathrm{KH}(X)$, the claim reduces to checking that the square $$\begin{tikzcd} \mathrm{Fil}^0_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/\mathbb{Q}) \ar{r} \ar{d} & \mathrm{HC}^-(X/\mathbb{Q}) \ar{d}\\ L_{\mbox{\rm \scriptsize cdh}}\mathrm{Fil}^0_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/\mathbb{Q}) \ar{r} & L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/\mathbb{Q}), \end{tikzcd}$$ is a pullback, i.e., that the cofibre $\mathrm{HC}^-(-/\mathbb{Q})/\mathrm{Fil}^0_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(-/\mathbb{Q})$ satisfies cdh descent on $\text{Sch}_\mathbb{Q}^{\mbox{\rm \scriptsize qcqs}}$. This was explained in Lemma [Lemma 28](#lemma_cdh_descent_HP){reference-type="ref" reference="lemma_cdh_descent_HP"}. Next suppose that $X$ has finite valuative dimension $d$. We know from Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(1) that $\mathrm{Fil}^j_{\mbox{\rm \scriptsize cdh}}\mathrm{KH}(X)$ is supported in cohomological degrees $\le d-j$. Now, $\widehat{L\Omega}^{\geq j}_{-/\mathbb{Q}}$ is supported in cohomological degrees $\le j$; by Zariski or cdh sheafifying, it follows that $\mathrm{Fil}^j_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/\mathbb{Q})$ and $\mathrm{Fil}^j_{\mbox{\rm \scriptsize HKR}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/\mathbb{Q})$ are both supported in homological degrees $\ge d-j$. From the defining pullback square ([\[eq:motfilt_charp\]](#eq:motfilt_charp){reference-type="ref" reference="eq:motfilt_charp"}), we then see that $\mathrm{Fil}^j_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$ is supported in homological degrees $\ge d+1-j$, which is good enough to prove the desired boundedness (but not the optimal bound: see §[8](#section_Weibel){reference-type="ref" reference="section_Weibel"}). The splitting result follows from the result on Adams operations in Proposition [Proposition 36](#prop:adams){reference-type="ref" reference="prop:adams"} below by a standard argument. Indeed, we start with the multiplicative maps of graded objects $$\mathrm{gr}^\star_{\mathrm{mot}}\mathrm{K}(X) \longleftarrow \mathrm{Fil}^{\star}_{\mathrm{mot}}\mathrm{K}(X) \longrightarrow \mathrm{K}(X),$$ where the right-most term is regarded as a constant graded object and the middle term is the graded object $j\mapsto \mathrm{Fil}^{j}_{\mathrm{mot}}\mathrm{K}(X)$. Upon rationalizing and taking eigenspectra, Proposition [Proposition 36](#prop:adams){reference-type="ref" reference="prop:adams"} then produces multiplicative equivalences of graded objects $$\mathrm{gr}^\star_{\mathrm{mot}}\mathrm{K}(X)_{\mathbb{Q}} \simeq (\mathrm{gr}^\star_{\mathrm{mot}}\mathrm{K}(X)_{\mathbb{Q}})^{\psi^{\ell} - \ell^\star} \longleftarrow (\mathrm{Fil}^\star_{\mathrm{mot}}\mathrm{K}(X))^{\psi^{\ell} - \ell^\star} \longrightarrow \mathrm{K}(X)^{\psi^{\ell} - \ell^\star};$$ which proves the claim. The rational degeneration follows. ◻ The splitting and degeneration parts of the parts of the previous theorem were consequences of a finer result, namely the existence of Adams operations acting in the desired way on motivic cohomology: **Proposition 36**. *Let $X$ be a qcqs $\mathbb{Q}$-scheme and $\ell \geq 2$. Then there exists a natural, multiplicative automorphism $\psi^\ell$ of the filtered spectrum $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)_\mathbb{Q}$ such that, for each $j\ge0$, the induced automorphism on $\mathrm{gr}^j_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)_\mathbb{Q}=\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(X)[2j]$ is multiplication by $\ell^j$.* *Proof.* Taking left Kan extension and cdh sheafification extends the compatibility of Corollary [Corollary 148](#cor:adams-compatible){reference-type="ref" reference="cor:adams-compatible"} along the cdh local trace map of Corollary [Corollary 31](#corol_cdh_filtered_trace){reference-type="ref" reference="corol_cdh_filtered_trace"}. Hence, the rationalized version of the diagram [\[eq:motfilt\]](#eq:motfilt){reference-type="eqref" reference="eq:motfilt"} can be promoted to a diagram of filtered spectra with multiplicative endomorphisms $\psi^{\ell}$; these are in fact automorphisms since we can define $\psi^{-\ell}$ which furnishes an inverse. By construction, we then have a filtered, multiplicative automorphism $\psi^{\ell}$ on $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)_\mathbb{Q}$. The action on graded pieces is then determined by the action of $\psi^{\ell}$ on each individual graded pieces of the rationalized filtration which, in turn, can be checked on smooth $\mathbb{Q}$-schemes. This last assertion is Corollary [Corollary 145](#cor:rational){reference-type="ref" reference="cor:rational"} and [@raksit-hkr Prop. 6.4.12]. ◻ ## Characteristic $p > 0$ {#sec:charp} We first give a with a quick overview of syntomic cohomology in characteristic $p$, in the sense of [@BhattMorrowScholze2]. The reader should refer to [@BhattMorrowScholze2 §8] and [@AntieauMathewMorrowNikolaus §6.2] for more details. For any $\mathbb{F}_p$-algebra $A$, let $W_r\Omega^j_{A,{\mbox{\rm \scriptsize log}}}$ denote the global sections of the subsheaf $W_r\Omega^j_{\mbox{\rm \scriptsize log}}$ of the de Rham--Witt sheaf $W_r\Omega^j_{\mathrm{Spec}A}$ which is generated étale locally (or, equivalently, Zariski locally [@Morrow_pro_GL2 Corol. 4.2(i)]) by $\tfrac{d[f_1]}{f_1}\wedge\cdots\wedge \tfrac{d[f_j]}{f_j}$ for units $f_1,\dots,f_j$. Alternatively [@Morrow_pro_GL2 Corol. 4.2(iii)], $W_r\Omega^j_A$ is the kernel of the Artin--Schreier map $$C^{-1}-1:W_r\Omega^j_A\longrightarrow W_r\Omega^j_A/dV^{r-1}\Omega^j_A.\label{eqn_WOmegalog}$$ Since the de Rham-Witt sheaves have no higher cohomology on affines and the Artin--Schreier map is étale locally surjective, the previous observations may alternatively be expressed as a fibre sequence $$R\Gamma_{\mbox{\rm \scriptsize \'et}}(A,W_r\Omega^j_{\mbox{\rm \scriptsize log}})\longrightarrow W_r\Omega^j_A\stackrel{C^{-1}-1}\longrightarrow W_r\Omega^j_A/dV^{r-1}\Omega^j_A;$$ in particular the cohomology of $R\Gamma_{\mbox{\rm \scriptsize \'et}}(A,W_r\Omega^j_{\mbox{\rm \scriptsize log}})$ is concentrated in degrees zero and one. **Definition 37**. *Mod-$p^r$, weight-$j$ syntomic cohomology* of $\mathbb{F}_p$-algebras $$\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(-)/p^r:\text{CAlg}_{\mathbb{F}_p}\to \rm{D}(\mathbb{Z})$$ is defined to be the left Kan extension of $R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j]$ along the inclusion $\text{CAlg}_{\mathbb{F}_p}^\Sigma\subseteq \text{CAlg}_{\mathbb{F}_p}$; here $\text{CAlg}^{\Sigma}_{\mathbb{F}_p}$ denotes the category of finitely generated polynomial $\mathbb{F}_p$-algebras. When $r=1$ we will often write $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}$ to simplify notation. Taking the inverse limit over $r$, the weight-$j$ syntomic cohomology of an $\mathbb{F}_p$-algebra $A$ is defined by $$\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A):=\mathop{\mathrm{lim}}_r\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A)/p^r.$$ **Remark 38**. 1. Taking $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A)$ modulo $p^r$ does recover $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A)/p^r$ as it was initially defined, thanks to Illusie's short exact sequence of étale sheaves $0\to W_{s}\Omega^j_{\mbox{\rm \scriptsize log}}\xrightarrow{p^r}W_{r+s}\Omega^j_{\mbox{\rm \scriptsize log}}\to W_r\Omega^j_{\mbox{\rm \scriptsize log}}\to 0$ on smooth $\mathbb{F}_p$-schemes [@Illusie1979 §I.5.7]. 2. For any $\mathbb{F}_p$-algebra $A$ and $r\ge 1$ there is, by construction, a natural comparison map $$\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A)/p^r\longrightarrow R\Gamma_{\mbox{\rm \scriptsize \'et}}(A,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j].$$ It is an equivalence whenever $A$ is regular Noetherian, or more generally Cartier smooth [@KellyMorrow2021 Prop. 5.1]. Syntomic cohomology can be loosely controlled via the cotangent complex through the following lemma: **Lemma 39**. *For any $\mathbb{F}_p$-algebra $A$, the complex $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}(A)$ admits a natural finite increasing filtration in $\textrm{D}(\mathbb{F}_p)$, of length $2(j+1)$, with graded pieces given in increasing order by $$\begin{aligned} &L_{A/\mathbb{F}_p}^j[-j-1], L_{A/\mathbb{F}_p}^{j-1}[-j],L_{A/\mathbb{F}_p}^{j-2}[-j+1],\dots, L_{A/\mathbb{F}_p}^0[-1],\\ & L_{A/\mathbb{F}_p}^0[0], L_{A/\mathbb{F}_p}^1[-1], L_{A/\mathbb{F}_p}^2[-2],\dots, L_{A/\mathbb{F}_p}^j[-j].\end{aligned}$$* *Proof.* The key is to show the following claim: for $R$ any smooth $\mathbb{F}_p$-algebra, then $\Omega_R^j/d\Omega^{j-1}_R$ admits a natural finite increasing filtration (in $\mathrm D(\mathbb{F}_p)$, not as submodules) of length $2j+1$ with graded pieces in increasing order $$\begin{aligned} &\Omega_R^j,\Omega_R^{j-1}[1],\Omega_R^{j-2}[2],\dots,\Omega_R^0[j],\\ &\Omega_R^0[j+1],\Omega_R^1[j],\Omega_R^2[j-1],\dots,\Omega_R^{j-1}[2].\end{aligned}$$ The case $j=0$ (when the bottom row of the listed graded pieces is empty) is trivial; we proceed by induction to treat the case $j>0$, so assume that we already have the filtration on $\Omega_R^{j-1}/d\Omega^{j-2}_R$, i.e., $$\mathrm{Fil}_0(\Omega_R^{j-1}/d\Omega^{j-2}_R)\to \mathrm{Fil}_1(\Omega_R^{j-1}/d\Omega^{j-2}_R)\to\cdots\to \mathrm{Fil}_{2j-1}(\Omega_R^{j-1}/d\Omega^{j-2}_R)=\Omega_R^{j-1}/d\Omega^{j-2}_R,$$ with the desired graded pieces. Then we define, for $i=1,\dots,2j$, the filtered step $\mathrm{Fil}_i(\Omega_R^j/d\Omega^{j-1}_R)$ to be the pullback $$\xymatrix{ \mathrm{Fil}_{i-1}(\Omega_R^{j-1}/d\Omega^{j-2}_R)[1]\ar[r] & \Omega_R^{j-1}/d\Omega^{j-2}_R[1]\ar[r]^{\pi[1]} & \Omega_R^{j-1}/\ker d[1]\\ \mathrm{Fil}_i(\Omega_R^j/d\Omega^{j-1}_R)\ar@{-->}[u]\ar@{-->}[rr]&&\ar[u]_{\delta}\Omega_R^j/d\Omega^{j-1}_R }$$ Here $\pi$ is the canonical quotient map with kernel $H^{j-1}_{\mbox{\rm \scriptsize dR}}(R)$, and $\delta$ is the connecting map associated to the short exact sequence $$0\longrightarrow\Omega_R^{j-1}/\ker d\xrightarrow{d}\Omega_R^j\longrightarrow\Omega_R^j/d\Omega_R^{j-1}\longrightarrow 0\label{eqn_fil0}.$$ Since pulling back a filtration does not change the graded pieces, we see at once that $\mathrm{gr}_i(\Omega_R^j/d\Omega^{j-1}_R)$ is as desired for $i=1,\dots,2j-1$. We now set $\mathrm{Fil}_0(\Omega_R^j/d\Omega^{j-1}_R):=0$ and $\mathrm{Fil}_{2j+1}(\Omega_R^j/d\Omega^{j-1}_R)=\Omega_R^j/d\Omega^{j-1}_R$; we must show that $\mathrm{gr}_0(\Omega_R^j/d\Omega^{j-1}_R)$ and $\mathrm{gr}_{2j}(\Omega_R^j/d\Omega^{j-1}_R)$ are as desired. Firstly, $\mathrm{gr}_0(\Omega_R^j/d\Omega^{j-1}_R)=\mathrm{Fil}_1(\Omega_R^j/d\Omega^{j-1}_R)$, which was defined to be the pullback of $\mathrm{Fil}_0(\Omega_R^{j-1}/d\Omega^{j-2}_R)=0\to\Omega^{j-1}/\ker d[1]$ along $\delta$; that is, it is given by $\operatorname{fib}(\delta)$, which is indeed $\Omega^j_R$ thanks to ([\[eqn_fil0\]](#eqn_fil0){reference-type="ref" reference="eqn_fil0"}). Secondly, $\mathrm{gr}_{2j}(\Omega_R^j/d\Omega^{j-1}_R)$ is precisely $\operatorname{cofib}(\pi)[1]=H^{j-1}_{\mbox{\rm \scriptsize dR}}(R)[2]$, which identifies with $\Omega_R^{j-1}[2]$ via the Cartier isomorphism. This completes the proof of the existence of the filtration on $\Omega_R^j/d\Omega^{j-1}_R$ when $R$ is smooth. We then obtain the desired filtration on $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}(R)$ by recalling that $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}(R)=\text{fib}(\Omega_R^j\xrightarrow{C^{-1}-1}\Omega_R^j/d\Omega_R^{j-1})[-j]$. Finally the desired filtration on $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}(A)$, for arbitrary $\mathbb{F}_p$-algebras $A$, is obtained by left Kan extension from the smooth case. ◻ As a consequence of the previous lemma and fpqc descent for the cotangent complex, we have that **Corollary 40**. *The presheaves $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ satisfy fpqc descent on the category of $\mathbb{F}_p$-algebras.* Therefore, by right Kan extension, they extend uniquely to fpqc sheaves $$\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}:\text{Sch}_{\mathbb{F}_p}^{\mbox{\rm \scriptsize qcqs}}\longrightarrow\rm D(\mathbb{Z}),$$ thereby defining syntomic cohomology in the non-affine case. Just as derived de Rham cohomology appeared in characteristic zero through the HKR filtration on negative cyclic homology, syntomic cohomology similarly appears through topological cyclic homology: **Theorem 41** (BMS filtration [@BhattMorrowScholze2]). *For any qcqs $\mathbb{F}_p$-scheme $X$, its topological cyclic homology $\mathrm{TC}(X)$ admits a natural, multiplicative, complete, $\mathbb{N}$-indexed filtration $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize BMS}} \mathrm{TC}(X)$ with graded pieces $$\mathrm{gr}^j_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)\simeq\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)[2j]$$ for $j\ge0$. Moreover,* 1. *The filtration is bounded, i.e., there exists $d\ge0$ (depending on $X$) such that, for any $j\ge0$, the filtered step $\mathrm{Fil}^j_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)$ is supported in homological degrees $\ge j-d$ (and so the syntomic cohomology $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)$ is supported in cohomological degrees $\le j+d$).* 2. *The induced filtration on $\mathrm{TC}(X)[\tfrac1p]$ is naturally split, so that $$\mathrm{TC}(X)[\tfrac1p]\simeq\bigoplus_{j\ge 0}\mathbb{Q}_p(j)^{\mbox{\rm \scriptsize syn}}(X)[2j],$$ where $\mathbb{Q}_p(j)^{\mbox{\rm \scriptsize syn}}(X):=\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)[\tfrac1p]$.* *Proof.* The existence of a filtration on $\mathrm{TC}$ with graded pieces given by shifts of syntomic cohomology is one of the main theorems of [@BhattMorrowScholze2], in the case of quasisyntomic $\mathbb{F}_p$-algebras. It was extended, by $p$-completed left Kan extension, to all $\mathbb{F}_p$-algebras in [@AntieauMathewMorrowNikolaus]. It is then obtained for arbitrary qcqs $\mathbb{F}_p$-schemes by right Kan extension. It remains to explain (1) and (2). The proof of part (1) proceeds via several cases. Firstly, for any quasisyntomic $\mathbb{F}_p$-algebra $R$, the BMS filtration on $\mathrm{TC}(R)$ is defined by descent from quasiregular semiperfect rings of the two-speed Postnikov filtration; the latter is manifestly complete, which is preserved by the descent. In particular, for smooth $\mathbb{F}_p$-algebras $R$, the BMS filtration on $\mathrm{TC}(R)$ is complete and each of its graded pieces $\mathrm{gr}_{\mbox{\rm \scriptsize BMS}}^j\mathrm{TC}(R)\simeq \text{lim}_rR\Gamma_{\mbox{\rm \scriptsize et}}(R,W_r\Omega_{\mbox{\rm \scriptsize log}}^r)[j]$ is supported in cohomological degrees $[-j,-(j+1)]$; using the completeness it follows in this case that $\mathrm{Fil}_{\mbox{\rm \scriptsize BMS}}^j\mathrm{TC}(R)$ is supported in homological degrees $\ge j-1$. By left Kan extending we see that $\mathrm{Fil}_{\mbox{\rm \scriptsize BMS}}^j\mathrm{TC}(A)$ is supported in homological degrees $\ge j-1$ for all $\mathbb{F}_p$-algebras $A$; so the filtration is bounded on affines. Finally, right Kan extending preserves completeness, so we deduce that the BMS filtration on $\mathrm{TC}(X)$, for any qcqs $\mathbb{F}_p$-scheme $X$, is at least complete. Therefore it is enough to check boundedness of the filtration on graded pieces, as mentioned in §[2.2](#ss_filtrations){reference-type="ref" reference="ss_filtrations"}; we do this next. Since $X$ is qcqs, it has finite cohomological dimension for quasi-coherent sheaves (even if it does not have finite Krull dimension), and we take $d$ to be one plus this dimension. Then the Zariski sheafification of each graded piece of Lemma [Lemma 39](#lem_fin_fil_on_syn){reference-type="ref" reference="lem_fin_fil_on_syn"} has global sections supported in cohomological degrees $\le j+d$, as required. For part (2) note first that the absolute Frobenius $\phi:X\to X$ induces a natural endomorphism of $\mathrm{TC}(X)$, compatible (by functoriality) with the BMS filtration; its action on the graded piece $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}[2j]$ is as multiplication by $p^j$ (this follows by left Kan extending the same statement for $R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})$ of finitely generated polynomial algebras, for all $r\ge0$). Next observe that the BMS filtration on $\mathrm{TC}(X)[\tfrac1p]$ is bounded thanks to part (1), therefore complete. Since $\phi-p^j$ acts invertibly on $\mathbb{Q}_p(i)^{\mbox{\rm \scriptsize syn}}(X)$ for $i\ge j+1$, we deduce that it acts invertibly on $\mathrm{Fil}^{j+1}_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)[\tfrac1p]$. Similarly it acts invertibly on $\mathrm{TC}(X)[\tfrac1p]/\mathrm{Fil}^{j}\mathrm{TC}(X)[\tfrac1p]$. Taking $\phi-p^j$-fixed points, we have shown that the maps $$\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)[2j]= \mathrm{gr}^j_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)\longleftarrow (\mathrm{Fil}^{j}_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X))^{\phi=p^j}\longrightarrow\mathrm{TC}(X)^{\phi=p^j}$$ are equivalences after inverting $p$. This defines a filtered map $\bigoplus_{j\ge0}\mathbb{Q}_p(j)^{\mbox{\rm \scriptsize syn}}(X)[2j]\to\mathrm{TC}(X)[\tfrac1p]$ which is an equivalence on all graded pieces, therefore an equivalence since the filtrations on both sides are complete (by boundedness). ◻ **Remark 42** (Variant: cdh-local BMS filtration). Cdh sheafifying the BMS filtration levelwise we see that, for any qcqs $\mathbb{F}_p$-scheme $X$, there exists a functorial, multiplicative, $\mathbb{N}$-indexed filtration $$\mathrm{Fil}^\star_{\mbox{\rm \scriptsize BMS}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X):=L_{\mathrm{cdh}}\mathrm{Fil}^\star_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X),$$ on $L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X)$ whose graded pieces for $j\ge0$ are given by $$\mathrm{gr}^j_{\mbox{\rm \scriptsize BMS}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X)\simeq L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)[2j].$$ Warning: here $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ is the cdh sheafification of the presheaf $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}:\mathrm{S}\mathrm{ch}{}^{\mbox{\rm \scriptsize qcqs,op}}_{\mathbb{F}_p}\to\text D(\mathbb{Z})$; but since sheafification does not commute with cofiltered limits in general, there is no reason that $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ should land in $p$-complete complexes.[^9] In fact, such an issue already appeared in characteristic zero: $R\Gamma_{\mbox{\rm \scriptsize cdh}}(-,\widehat{L\Omega}_{-/\mathbb{Q}})$ was not necessarily Hodge complete. We record the following consequence of the fact that the BMS filtration is split after inverting $p$; it will be required to control our motivic cohomology after inverting $p$: **Corollary 43**. *The presheaf $\mathbb{Q}_p(j)^{\mbox{\rm \scriptsize syn}}:\mathrm{Sch}^{\mbox{\rm \scriptsize qcqs}}_{\mathbb{F}_p}\to \mathrm D(\mathbb{Z})$ is a cdh sheaf.* *Proof.* Since $\mathbb{Q}_p(j)^{\mbox{\rm \scriptsize syn}}$ is a direct summand of $\mathrm{TC}[\tfrac1p]$ by Theorem [Theorem 41](#thm_BMS2+){reference-type="ref" reference="thm_BMS2+"}(2), it is sufficient to check that the latter is a cdh sheaf. In other words, since cdh sheafification commutes with inverting $p$, we must show that $\mathrm{TC}(X)[\tfrac1p]\to (L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X))[\tfrac1p]$ is an equivalence for any qcqs $\mathbb{F}_p$-scheme $X$. But this follows from Theorem [Theorem 23](#thm:mainsq){reference-type="ref" reference="thm:mainsq"} and the result of Weibel that $K(X)[\tfrac1p]\stackrel{\sim}{\to}\mathrm{KH}(X)[\tfrac1p]$ [@Weibel1989a]. ◻ We next establish an analogue of Proposition [Proposition 29](#prop:compat){reference-type="ref" reference="prop:compat"}, namely that the trace map in characteristic $p$ is compatible with the motivic and BMS filtrations: **Proposition 44**. *Let $k$ be a field of characteristic $p$. Then the trace map $\mathrm{K}\to\mathrm{TC}$, viewed as a map between spectra-valued presheaves on ${\mathrm{S}\mathrm{m}}_k$, admits a unique, multiplicative extension to a map of filtered presheaves $\mathrm{Fil}_{\mbox{\rm \scriptsize cla}}^{\star}\mathrm{K}\to \mathrm{Fil}_{\mbox{\rm \scriptsize BMS}}^{\star}\mathrm{TC}$.* *Proof.* We apply the same $t$-structure argument as Propositions [Proposition 29](#prop:compat){reference-type="ref" reference="prop:compat"}. Step 1 of that proof was independent of the characteristic, and so shows in the present context that $\mathrm{Fil}^j_{\mbox{\rm \scriptsize cla}}\mathrm{K}$ is $j$-connective for each $j\ge0$. It remains to check that $\mathrm{Fil}^{<j}_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(R)$ vanishes in cohomological degrees $\le -j$ for each smooth $k$-algebra $R$. But for each $i=0,\dots,j-1$ the $i^{\mbox{\rm \scriptsize th}}$ graded piece is $\text{gr}_{\mbox{\rm \scriptsize BMS}}^i\mathrm{TC}(R)\simeq \mathbb{Z}_p(i)^{\mbox{\rm \scriptsize syn}}(R)[2i]$, where $\mathbb{Z}_p(i)^{\mbox{\rm \scriptsize syn}}(R)$ is supported in cohomological degrees $[i,i+1]$; the desired vanishing bound follows by a trivial induction. Multiplicativity follows from the same argument as in characteristic zero using the Postnikov $t$-structure. ◻ As in characteristic zero, we need a cdh-local analogue of the previous proposition; unlike characteristic zero,[^10] it does not formally follow from the previous proposition: **Proposition 45**. *The cdh-local trace map $\mathrm{KH}\to L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}$, viewed as a map between spectra-valued presheaves on $\mathrm{S}\mathrm{ch}{}_{\mathbb{F}_p}^{\mbox{\rm \scriptsize qcqs}}$, admits a unique extension to a multiplicative map of filtered presheaves $$\mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize cdh}}\mathrm{KH}\longrightarrow\mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize BMS}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}$$ (the filtration on the left being the cdh-local motivic filtration of Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(1); the filtration on the right is the cdh-local BMS filtration of Remark [Remark 42](#rem:cdh-local){reference-type="ref" reference="rem:cdh-local"}).* *Proof.* We begin with an argument which is essentially the same as the second half of the proof of Corollary [Corollary 31](#corol_cdh_filtered_trace){reference-type="ref" reference="corol_cdh_filtered_trace"}. Namely, since $L_{\mbox{\rm \scriptsize cdh}}L_{{\mbox{\rm \scriptsize Sch}}_{\mathbb{F}_p}^{\mbox{\rm \scriptsize qcqs,op}}/{\mbox{\rm \scriptsize Sm}}_{\mathbb{F}_p}^{\mbox{\rm \scriptsize op}}}$ is a left adjoint to restricting from cdh sheaves on $\text{Sch}_{\mathbb{F}_p}^{\mbox{\rm \scriptsize qcqs}}$ to presheaves on $\text{Sm}_{\mathbb{F}_p}$, and $\mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize cdh}}\mathrm{KH}=L_{\mbox{\rm \scriptsize cdh}}L_{{\mbox{\rm \scriptsize Sch}}_{\mathbb{F}_p}^{\mbox{\rm \scriptsize qcqs,op}}/{\mbox{\rm \scriptsize Sm}}_{\mathbb{F}_p}^{\mbox{\rm \scriptsize op}}}\mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}$ by definition, the statement of the proposition is equivalent to the following claim: the map of spectra-valued presheaves $\mathrm{K}\to (L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC})|_{{\mbox{\rm \scriptsize Sm}}_{\mathbb{F}_p}}$ on $\text{Sm}_{\mathbb{F}_p}$ admits a unique extension to a multiplicative map of filtered presheaves $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}\to (\mathrm{Fil}^\star_{\mbox{\rm \scriptsize BMS}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC})|_{{\mbox{\rm \scriptsize Sm}}_{\mathbb{F}_p}}$. To prove the claim we apply the same $t$-structure argument as in Propositions [Proposition 29](#prop:compat){reference-type="ref" reference="prop:compat"} and [Proposition 44](#prop:mot-v-bms){reference-type="ref" reference="prop:mot-v-bms"}. We have already noted in the proof of Proposition [Proposition 44](#prop:mot-v-bms){reference-type="ref" reference="prop:mot-v-bms"} that $\mathrm{Fil}^j_{\mbox{\rm \scriptsize cla}}K$ is $j$-connective for any $j\ge0$, so it remains only to show that $L_{\mbox{\rm \scriptsize cdh}}\mathrm{Fil}^{<j}_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)$ vanishes in cohomological degrees $\le -j$ for each smooth $\mathbb{F}_p$-scheme $X$; by induction it is enough to check that $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(i)^{\mbox{\rm \scriptsize syn}}(X)$ is supported in cohomological degrees $\ge i$ for each $i\ge0$. At least modulo $p$ this bound follows from the equivalence $$R\Gamma_{\mbox{\rm \scriptsize eh}}(X,\Omega^i_{\mbox{\rm \scriptsize log}})[-i]\stackrel{\sim}{\to}L_{\mbox{\rm \scriptsize cdh}}\mathbb{F}_p(i)^{\mbox{\rm \scriptsize syn}}(X),$$ which holds in fact for any qcqs $\mathbb{F}_p$-scheme and will be explained in the proof of Theorem [Theorem 51](#theorem_GL){reference-type="ref" reference="theorem_GL"} below. However we must now recall the warning of Remark [Remark 42](#rem:cdh-local){reference-type="ref" reference="rem:cdh-local"}: the presheaf $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(i)^{\mbox{\rm \scriptsize syn}}$ does not take $p$-complete values on arbitrary qcqs $\mathbb{F}_p$-schemes. To complete the proof we must therefore show that it does take a $p$-complete value whenever $X$ is a smooth $\mathbb{F}_p$-scheme (as then the coconnectivity bound modulo $p$ yields the same bound for $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(i)^{\mbox{\rm \scriptsize syn}}(X)$). To prove this $p$-completeness claim we consider the filtered spectrum $\operatorname{fib}(\mathrm{Fil}^\star_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)\to \mathrm{Fil}^\star_{\mbox{\rm \scriptsize BMS}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X))$, whose underlying spectrum is zero since $\mathrm{TC}(X)\stackrel{\sim}{\to}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X)$ (using $\mathrm{K}(X)\stackrel{\sim}{\to}\mathrm{KH}(X)$ and Theorem [Theorem 23](#thm:mainsq){reference-type="ref" reference="thm:mainsq"}) and where the filtration is bounded (see the second paragraph of the proof of Theorem [Theorem 50](#thm:p-ahss){reference-type="ref" reference="thm:p-ahss"} for details). Moreover, the associated bounded spectra sequence $$E_2^{i\,j}=H^{i-j}(\operatorname{fib}(\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)\to L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X))\implies 0$$ degenerates up to bounded denominators thanks to the Frobenius actions, and so each group on the $E_2$ page is annihilated by a bounded power of $p$. Since $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)$ is $p$-complete, we now obtain the same for $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)$, as required to complete the proof. ◻ **Definition 46**. For a qcqs $\mathbb{F}_p$-scheme $X$, let $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$ be the filtered spectrum defined as the pullback (in filtered $\mathbb{E}_{\infty}$-algebras) of the diagram $$\label{eq:motfilt_charp} \begin{tikzcd} \mathrm{Fil}^{\star}_{\mathrm{mot}}\mathrm{K}(X) \ar[dotted]{d} \ar[dotted]{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)\ar{d} \\ \mathrm{Fil}^\star_{\mbox{\rm \scriptsize cdh}}\mathrm{KH}(X) \ar{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize BMS}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X), \end{tikzcd}$$ where the bottom map is given by Proposition [Proposition 45](#prop_cdh_filtered_trace_p){reference-type="ref" reference="prop_cdh_filtered_trace_p"}. For $j\in\mathbb{Z}$, define the *weight-$j$ motivic cohomology* of $X$ to be $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X):=(\mathrm{gr}^j_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X))[-2j],$$ which we will see in Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"} lies in $\rm D(\mathbb{Z})$ and vanishes for $j<0$. The associated motivic cohomology groups, for $i\in\mathbb{Z}$, are $H^i_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j)):=H^i(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X))$. Here are some fundamental properties of our motivic cohomology in characteristic $p$: **Theorem 47**. *Let $j\in \mathbb{Z}$. For any qcqs $\mathbb{F}_p$-scheme $X$, the weight-$j$ motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)$ has the following properties:* 1. *$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)=0$ for $j<0$.* 2. *There is a natural pullback square $$\begin{tikzcd} \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X) \ar{r} \ar{d} & \mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)\ar{d}\\ \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X) \ar{r} & L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X). \end{tikzcd}$$* 3. *The canonical map $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)[\tfrac1p]\to\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)[\tfrac1p]$ is an equivalence. In particular, $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}[\tfrac1p]$ is a cdh sheaf on $\mathrm{S}\mathrm{ch}{}^{\mbox{\rm \scriptsize qcqs,op}}_{\mathbb{F}_p}$.* 4. *The presheaf $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}:\mathrm{S}\mathrm{ch}{}_{\mathbb{F}_p}^{\mbox{\rm \scriptsize qcqs,op}}\to\text D(\mathbb{Z})$ is a finitary Nisnevich sheaf.* 5. *The endomorphism $\phi^*$ of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)$ induced by the absolute Frobenius $\phi:X\to X$ is multiplication by $p^j$.* *Proof.* (1): The three corners in ([\[eq:motfilt_charp\]](#eq:motfilt_charp){reference-type="ref" reference="eq:motfilt_charp"}) used to define the pullback are $\mathbb{N}$-indexed, whence the same is true of the pullback $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$, i.e., the graded pieces vanish in negative weights. Part (2) is obtained by taking graded pieces in the pullback square ([\[eq:motfilt_charp\]](#eq:motfilt_charp){reference-type="ref" reference="eq:motfilt_charp"}). (3): Corollary [Corollary 43](#corol_Qpsyn){reference-type="ref" reference="corol_Qpsyn"} states that $\text{fib}(\mathbb{Z}_p(j)\to L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j))[\tfrac1p] \simeq 0$, whence the result follows from the cartesian square in (2). (4): It suffices to prove that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}[\tfrac1p]$ and $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}/p$ are finitary. The first follows from part (3) and Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(2). The second reduces, via the pullback square of part (2), to finitariness of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}/p$ (which is indeed finitary by another application of Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(2)), of $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}$ (finitary since, on affines, it is left Kan extended from finitely generated polynomial algebras), and of $L_{\mbox{\rm \scriptsize cdh}}\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}$ (finitary since it is the cdh sheafification of a finitary presheaf). (5): As usual it suffices to treat the other corners of the pullback square of part (2). The Frobenius acts on $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$ as multiplication by $p^j$, by left Kan extending and cdh sheafifying the analogous statement for classical motivic cohomology of smooth $\mathbb{F}_p$-schemes [@GeisserLevine2000]. It acts on $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ as multiplication by $p^j$, as we already noted in the proof of Theorem [Theorem 41](#thm_BMS2+){reference-type="ref" reference="thm_BMS2+"}(2), and so similarly for $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ by cdh sheafifying. ◻ **Remark 48**. Neither the presheaf $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ nor $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ is finitary. However, the proof of Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(4) shows that their difference, i.e, the fibre $$X \mapsto \operatorname{fib}\left(\mathbb{Z}_p(j)^{\mathrm{syn}}(X) \rightarrow L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)\right),$$ is a finitary presheaf. **Example 49** (Weight $0$). As in Example [Example 34](#example_00){reference-type="ref" reference="example_00"} in characteristic $0$, the map $$\mathbb{Z}(0)^{\mbox{\rm \scriptsize mot}}(X)\longrightarrow\mathbb{Z}(0)^{\mbox{\rm \scriptsize cdh}}(X)=R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\mathbb{Z})$$ is an equivalence for any qcqs $\mathbb{F}_p$-scheme $X$. Indeed, from the pullback square Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"} it is enough to show that $\mathbb{Z}_p(0)^{\mbox{\rm \scriptsize syn}}$, or equivalently $\mathbb{F}_p(0)^{\mbox{\rm \scriptsize syn}}$, satisfies cdh descent. But it is easily checked from the definitions that $\mathbb{F}_p(0)^{\mbox{\rm \scriptsize syn}}\simeq R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,\mathbb{Z}/p\mathbb{Z})$, which even satisfies cdh descent by Deligne (or even arc descent [@BhattMathew2021]). Here is the analogue, in characteristic $p$, of Theorem [Theorem 35](#theorem_AH_SS_0){reference-type="ref" reference="theorem_AH_SS_0"} about the existence of the Atiyah--Hirzebruch spectral sequence: **Theorem 50**. *Let $X$ be a qcqs $\mathbb{F}_p$-scheme. Then the filtered spectrum $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$ is $\mathbb{N}$-indexed, multiplicative, and satisfies $\mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)=\mathrm{K}(X)$. If $X$ has finite valuative dimension, then:* 1. *the filtration is bounded and so induces a bounded multiplicative Atiyah--Hirzebruch spectral sequence $$E_2^{ij}=H^{i-j}_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(-j))\implies \mathrm{K}_{-i-j}(X),$$* 2. *the filtration is rationally split, i.e., there is a natural equivalence of filtered spectra $$\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)_\mathbb{Q}\stackrel{\sim}{\to}\bigoplus_{j\ge0}\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(X)[2j],$$* 3. *and the spectral sequence degenerates up to bounded denominators.* *Proof.* We have already seen in Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(1) that the filtration is $\mathbb{N}$-indexed; it satisfies $\mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)=\mathrm{K}(X)$ thanks to Theorem [Theorem 23](#thm:mainsq){reference-type="ref" reference="thm:mainsq"}. Now assume $X$ has finite valuative dimension $d$ (hence also finite Krull dimension $\le d$). We already know from Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(1) that $\mathrm{Fil}^j_{\mbox{\rm \scriptsize cdh}}\mathrm{KH}(X)$ is supported in cohomological degrees $\le d-j$. We also saw in the proof of Theorem [Theorem 41](#thm_BMS2+){reference-type="ref" reference="thm_BMS2+"}(1) that, on any affine, $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}$ is supported in cohomological degrees $\le j+1$; by Zariski or cdh sheafifying, it follows that $\mathrm{Fil}^j_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)$ and $\mathrm{Fil}^j_{\mbox{\rm \scriptsize BMS}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X)$ are both supported in homological degrees $\ge d+1-j$. From the defining pullback square ([\[eq:motfilt_charp\]](#eq:motfilt_charp){reference-type="ref" reference="eq:motfilt_charp"}), we then see that $\mathrm{Fil}^j_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$ is supported in homological degrees $\ge d+2-j$, which is good enough to prove the desired boundedness (but not the optimal bound: see §[8](#section_Weibel){reference-type="ref" reference="section_Weibel"}). The filtration is split since the Frobenius actions are incompatible in different weights, thanks to Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(5): argue exactly as in Theorem [Theorem 41](#thm_BMS2+){reference-type="ref" reference="thm_BMS2+"}(2). The Frobenius action also forces the spectral sequence to degenerate up to bounded denominators: each differential $\delta:E_m^{i,j}\to E_m^{i+m,j+1-m}$, where $j\le 0$, is compatible with the Frobenius, which acts as $p^{-j}$ on the domain and $p^{-j-1+m}$ on the codomain, so that $p^{-j}(p^{m-1}-1)\delta=0$. ◻ In the remainder of this section we explicitly describe $p$-adic motivic cohomology in characteristic $p$, analogously to Geisser--Levine's identification [@GeisserLevine2000] of classical mod-$p^r$ motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}/p^r$ as $R\Gamma_{\mbox{\rm \scriptsize Zar}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j]$. More precisely, we show that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}/p^r$ can be obtained by glueing syntomic cohomology to cdh and $\text{\'{e}h}$ cohomologies of $W_r\Omega^j_{\mbox{\rm \scriptsize log}}$: **Theorem 51**. *For any qcqs $\mathbb{F}_p$-scheme $X$ and $j,r\ge0$, there is a natural pullback square in $\rm{D}(\mathbb{Z})$: $$\begin{tikzcd} \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)/p^r \ar{r} \ar{d} & \mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)/p^r\ar{d}\\ R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j] \ar{r} & R\Gamma_{\mbox{\rm \scriptsize eh}}(X,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j] \end{tikzcd}$$* *Proof.* More precisely, we will obtain the square as the mod-$p^r$ reduction of the square of Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(2). The bottom left corner of the square is indeed $R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j]$ by Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(4), or rather by an analogue for mod-$p^r$ rather than mod-$p$. It remains to naturally identify $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}/p^r$ with $R\Gamma_{\mbox{\rm \scriptsize \'eh}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j]$. Globalising Remark [Remark 38](#remark_WOmegalog){reference-type="ref" reference="remark_WOmegalog"}(2) defines a natural comparison map $$\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}/p^r\longrightarrow R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j]$$ which is an isomorphism on any valuation ring since they are Cartier smooth [@KellyMorrow2021 §2] [@LuedersMorrow2023 §5.1]; cdh sheafifying therefore defines an equivalence $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}/p^r\stackrel{\sim}{\to}L_{\mbox{\rm \scriptsize cdh}}R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j]$ of presheaves on $\mathrm{S}\mathrm{ch}{}^{\mbox{\rm \scriptsize qcqs,op}}_{\mathbb{F}_p}$. We finally appeal to Theorem [Theorem 134](#theorem:eh){reference-type="ref" reference="theorem:eh"}: noting that $X \mapsto R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})$ is cohomologically bounded below, we may apply that result to deduce that $L_{\mbox{\rm \scriptsize cdh}}R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})$ is a éh sheaf, and therefore the canonical map $L_{\mbox{\rm \scriptsize cdh}}R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})\to R\Gamma_{\mbox{\rm \scriptsize \'eh}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})$ is an equivalence. That completes the proof. ◻ In practice we often use Theorem [Theorem 51](#theorem_GL){reference-type="ref" reference="theorem_GL"} in the form of a fibre sequence rather than a pullback square. To formulate the statement we need the following invariant: **Definition 52**. The *mod-$p^r$, weight-$j$ Artin--Schreier obstruction* of an $\mathbb{F}_p$-algebra $A$ is the cokernel of the Artin--Schreier map from ([\[eqn_WOmegalog\]](#eqn_WOmegalog){reference-type="ref" reference="eqn_WOmegalog"}) $$\widetilde\nu_r(j)(A):=\text{coker}(C^{-1}-1:W_r\Omega^j_A\longrightarrow W_r\Omega^j_A/dV^{r-1}\Omega^j_A).$$ Given a topology $\tau$ on qcqs $\mathbb{F}_p$-schemes (notably Zariski, Nisnevich, or cdh), then we write $R\Gamma_\tau(X,\widetilde\nu_r(j))$ for the cohomology of the sheafification of $\widetilde\nu_r(j)$ in the topology $\tau$, and similarly $H^i_\tau(X,\widetilde\nu_r(j))$ for the individual cohomology groups. We warn the reader that $\widetilde\nu_r(j)$ is not even a Zariski sheaf on affines, so that in general the map $\widetilde\nu_r(j)(A)\to H^0_{\mbox{\rm \scriptsize Zar}}(A,\widetilde\nu_r(j))$ is not an isomorphism. **Remark 53**. Here are several alternative descriptions of the groups $\widetilde\nu_r(j)(A)$: 1. (Cohomological) Since $C^{-1}-1$ is étale locally surjective and the sheaves $W_r\Omega^j$, $W_r\Omega^j/dV^{r-1}\Omega^{j-1}$ have no higher cohomology on affines, we see that there is a natural isomorphism $$\widetilde\nu_r(j)(A)\cong H^1_{\mbox{\rm \scriptsize \'et}}(A,W_r\Omega^j_{\mbox{\rm \scriptsize log}}).$$ 2. (Syntomic) There is a natural isomorphism $$\widetilde\nu(j)(A)\cong H^{j+1}(\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A)/p^r),$$ and moreover this is the top degree of $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A)/p^r$, by [@AntieauMathewMorrowNikolaus Corol. 5.43]; more precisely, the comparison map of Remark [Remark 38](#remark_WOmegalog){reference-type="ref" reference="remark_WOmegalog"}(2) is an isomorphism in degrees $>j$. 3. ($K$-theoretic) For $A$ local, there are natural isomorphisms$$\widetilde\nu_r(j)(A)\cong\pi_{j-1}\text{cofib}(\mathrm{K}^{\mbox{\rm \scriptsize cn}}(A)/p^r\to\mathrm{TC}(A)/p^r)$$ by [@clausen2018k Thm. 6.11]. At least in the case in which $A=k$ is a field, these invariants have also appeared notably in work of Kato [@Kato1982a], denoted by $H_{p^r}^{j+1}(k)$, and are related to class field theory. **Remark 54** (Rigidity). A key property of $\widetilde\nu_r(j)$ is its *rigidity*, namely whenever $R\to A$ is a Henselian surjection of $\mathbb{F}_p$-algebras, then the induced map $\widetilde\nu_r(j)(R)\to\widetilde\nu_r(j)(A)$ is an isomorphism. This can be deduced directly from the definition and Hensel's lemma [@clausen2018k]. For any qcqs $\mathbb{F}_p$-scheme $X$ there is a natural map $$\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)/p^r\longrightarrow R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\widetilde\nu_r(j))[-j-1],\label{eqn_syn_to_nutilde}$$ defined by Zariski sheafifiying the following composition on affines: $$\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A)/p^r\xrightarrow{{\mbox{\rm \scriptsize Rem.~\ref{remarks_tildenu}(2)}}}\widetilde\nu_r(j)(A)[-j-1]\xrightarrow{{\mbox{\rm \scriptsize can.~map}}} R\Gamma_{\mbox{\rm \scriptsize cdh}}(A,\widetilde\nu_r(j))[-j-1].$$ Our mod-$p^r$ motivic cohomology identifies with the fibre of the map ([\[eqn_syn_to_nutilde\]](#eqn_syn_to_nutilde){reference-type="ref" reference="eqn_syn_to_nutilde"}): **Corollary 55** (Fundamental fibre sequence in characteristic $p$). *For any qcqs $\mathbb{F}_p$-scheme $X$ and $j,r\ge0$, there is a natural fibre sequence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)/p^r\longrightarrow\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)/p^r\stackrel{{\mbox{\rm \scriptsize (\ref{eqn_syn_to_nutilde})}}}{\longrightarrow} R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\widetilde\nu_r(j))[-j-1].$$* *Proof.* In terms of the pullback square of Theorem [Theorem 51](#theorem_GL){reference-type="ref" reference="theorem_GL"}, the map ([\[eqn_syn_to_nutilde\]](#eqn_syn_to_nutilde){reference-type="ref" reference="eqn_syn_to_nutilde"}) is the dotted composition: $$\hspace{-25mm} \xymatrix@C=3mm{ \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}/p^r \ar[r] \ar[d] & \mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}/p^r\ar[d]\ar@{-->}[drrr]&&\\ R\Gamma_{\mbox{\rm \scriptsize cdh}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j] \ar[r] & R\Gamma_{\mbox{\rm \scriptsize eh}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j]\ar@{=}[r]&L_{\mbox{\rm \scriptsize cdh}}R\Gamma_{\mbox{\rm \scriptsize et}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}})[-j]\ar[r] &(L_{\mbox{\rm \scriptsize cdh}}\tau^{\ge1}R\Gamma_{\mbox{\rm \scriptsize et}}(-,W_r\Omega^j_{\mbox{\rm \scriptsize log}}))[-j]\ar@{=}[r] & R\Gamma_{\mbox{\rm \scriptsize cdh}}(-,\widetilde\nu_r(j))[-j-1] }$$ (the middle bottom equality having been explained at the end of the proof of Theorem [Theorem 51](#theorem_GL){reference-type="ref" reference="theorem_GL"}). The claim to be proved is therefore that the bottom row is a fibre sequence; but this follows from exactness of cdh sheafification and the fibre sequence $W_r\Omega^j_{A,{\mbox{\rm \scriptsize log}}}\to R\Gamma_{\mbox{\rm \scriptsize et}}(A,W_r\Omega^j_{\mbox{\rm \scriptsize log}})\to \widetilde\nu_r(j)(A)[-1]$ on affines. ◻ ## A Beilinson--Lichtenbaum equivalence The classical Beilinson--Lichtenbaum conjecture states that motivic cohomology with finite coefficients is given by étale cohomology, in the range where cohomological degree is less than or equal to the weight. We refer to [@HaesemeyerWeibel2019 §2] for a discussion of the conjecture in the smooth case and exactly how it relates to the other main conjectures, such as Bloch--Kato. Here we record that such a Beilinson--Lichtenbaum equivalence holds for our motivic cohomology, including at the characteristic (where the correct replacement for étale cohomology is syntomic cohomology): **Theorem 56**. *Let $X$ be a qcqs $\mathbb{F}$-scheme and $j\ge0$.* 1. *For any integer $\ell>0$ prime to the characteristic of $\mathbb{F}$, there is a natural map $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)/\ell\longrightarrow R\Gamma_{\mbox{\rm \scriptsize \'et}}(X,\mu_\ell^{\otimes j}),$$ whose cofibre is supported in degrees $>j$.* 2. *If $\mathbb{F}=\mathbb{F}_p$ then for any $r\ge0$ there is a natural map $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)/p^r \longrightarrow\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)/p^r,$$ whose cofibre is supported in degrees $>j$.* *Proof.* (1): Recall that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}/\ell\to \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}/\ell$ is an equivalence, by Theorems [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(4) and [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(3), and the latter is given by $L_{\mbox{\rm \scriptsize cdh}}\tau^{\le j}R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,\mu_\ell^{\otimes j})$ by Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(3). It remains only to use that the fiber of the canonical map $L_{\mbox{\rm \scriptsize cdh}}\tau^{\le j}R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,\mu_\ell^{\otimes j})\to R\Gamma_{{\acute{e}t}}(-,\mu_{\ell}^{\otimes j})$ is supported in degrees $>j$, since $R\Gamma_{{\acute{e}t}}(-,\mu_{\ell}^{\otimes j})$ satisfies cdh descent as recalled in Example [Example 49](#example_0p){reference-type="ref" reference="example_0p"}. (2): This is clear from Corollary [Corollary 55](#corol_fundamental_p){reference-type="ref" reference="corol_fundamental_p"}. ◻ ## Comparison maps In this subsection we explicitly record the canonical comparison maps between the classical motivic cohomology of §[3.1](#ss_classical){reference-type="ref" reference="ss_classical"}, the lisse motivic cohomology of §[3.2](#subsec_lke){reference-type="ref" reference="subsec_lke"}, the cdh-local motivic cohomology of §[3.3](#ss_cdh_local){reference-type="ref" reference="ss_cdh_local"} (equivalently, the motivic cohomology of $\mathbb{A}^1$-invariant motivic homotopy theory, as discussed in Remark [Remark 21](#remark_slice_filtration){reference-type="ref" reference="remark_slice_filtration"}), and our new motivic cohomology. These comparisons are induced by various filtered maps between $K$-theory, $\mathrm{KH}$-theory, and connective $K$-theory. **Construction 57** (Classical vs new motivic cohomology of smooth varieties). We claim, for any smooth $\mathbb{F}$-scheme $X$, that there is a natural comparison map of filtered spectra $$\mathrm{Fil}_{\mbox{\rm \scriptsize cla}}^\star \mathrm{K}(X)\longrightarrow\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)\label{eqn_cla_to_mot}$$ given on $\mathrm{Fil}^0$ by $\mathrm{Fil}^0_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X)=\mathrm{K}(X)=\mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X)$. On shifted graded pieces this induces natural maps $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\label{eqn_cla_to_mot_j}$$ for $j\ge0$. We define ([\[eqn_cla_to_mot\]](#eqn_cla_to_mot){reference-type="ref" reference="eqn_cla_to_mot"}) as follows. When $\mathbb{F}=\mathbb{Q}$ (resp. $\mathbb{F}_p$), the filtered cdh-local trace map of Corollary [Corollary 31](#corol_cdh_filtered_trace){reference-type="ref" reference="corol_cdh_filtered_trace"} (resp. Propositoin [Proposition 45](#prop_cdh_filtered_trace_p){reference-type="ref" reference="prop_cdh_filtered_trace_p"}) was designed to fit into a commutative diagram $$\begin{tikzcd} \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X) \ar{d} \ar{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/\mathbb{Q})\ar{d} \\ \mathrm{Fil}^\star_{\mbox{\rm \scriptsize cdh}}\mathrm{KH}(X) \ar{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize HKR}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{HC}^-(X/\mathbb{Q}). \end{tikzcd} \qquad\qquad\mathrm{resp. } \begin{tikzcd} \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X) \ar{d} \ar{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)\ar{d} \\ \mathrm{Fil}^\star_{\mbox{\rm \scriptsize cdh}}\mathrm{KH}(X) \ar{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize BMS}}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X), \end{tikzcd}$$ where the top horizontal arrow is the filtered trace map of Proposition [Proposition 29](#prop:compat){reference-type="ref" reference="prop:compat"} (resp. [Proposition 44](#prop:mot-v-bms){reference-type="ref" reference="prop:mot-v-bms"}) for the smooth $\mathbb{F}$-scheme $X$. From the pullback Definition [Definition 32](#eq:char0){reference-type="ref" reference="eq:char0"} (resp. [Definition 46](#def:charp){reference-type="ref" reference="def:charp"}) of $\mathrm{Fil}_{\mbox{\rm \scriptsize mot}}^\star \mathrm{K}(X)$, there is therefore a natural induced map ([\[eqn_cla_to_mot\]](#eqn_cla_to_mot){reference-type="ref" reference="eqn_cla_to_mot"}) as desired. In Corollary [Corollary 91](#corol_smooth_comparison){reference-type="ref" reference="corol_smooth_comparison"} we will prove that ([\[eqn_cla_to_mot\]](#eqn_cla_to_mot){reference-type="ref" reference="eqn_cla_to_mot"}) is an equivalence for every smooth $\mathbb{F}$-scheme $X$. **Construction 58** (LKE vs new motivic cohomology of affines). Restricting ([\[eqn_cla_to_mot\]](#eqn_cla_to_mot){reference-type="ref" reference="eqn_cla_to_mot"}) to smooth $\mathbb{F}$-algebras and then left Kan extending to all $\mathbb{F}$-algebras defines, for any $\mathbb{F}$-algebra $A$, a natural comparison map of filtered spectra $$\mathrm{Fil}_{\mbox{\rm \scriptsize lse}}^\star \mathrm{K}^{\mbox{\rm \scriptsize cn}}(A)\longrightarrow\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}\mathrm{K}(A)\label{eqn_lke_to_mot}$$ given on $\mathrm{Fil}^0$ by the canonical map $\mathrm{Fil}^0_{\mbox{\rm \scriptsize lse}}\mathrm{K}^{\mbox{\rm \scriptsize cn}}(A)=\mathrm{K}^{\mbox{\rm \scriptsize cn}}(A)\to \mathrm{K}(A)=\mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}\mathrm{K}(A)$. On shifted graded pieces this induces natural comparison maps $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)$$ for $j\ge0$. We will study these comparison maps in detail in Section [7](#section_lke){reference-type="ref" reference="section_lke"}. **Construction 59** (New vs cdh-local motivic cohomology). Tautologically from the pullback definition of $\mathrm{Fil}_{\mbox{\rm \scriptsize mot}}^\star \mathrm{K}(X)$, there is a natural comparison map of filtered spectra $$\mathrm{Fil}_{\mbox{\rm \scriptsize mot}}^\star \mathrm{K}(X)\longrightarrow\mathrm{Fil}_{\mbox{\rm \scriptsize cdh}}^\star \mathrm{KH}(X)$$ for any qcqs $\mathbb{F}$-scheme $X$, given on $\mathrm{Fil}^0$ by the canonical map $\mathrm{Fil}_{\mbox{\rm \scriptsize mot}}^0 \mathrm{K}(X)=\mathrm{K}(X)\to \mathrm{KH}(X)=\mathrm{Fil}_{\mbox{\rm \scriptsize cdh}}^\star \mathrm{KH}(X)$. On shifted graded pieces this induces the natural comparison maps $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\to\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$, for $j\ge0$, which have already appeared in Theorems [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(2) and [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(2). We will study these maps further in Section [6](#section_smooth){reference-type="ref" reference="section_smooth"}. **Remark 60** (([\[eqn_cla_to_mot\]](#eqn_cla_to_mot){reference-type="ref" reference="eqn_cla_to_mot"}) is split). While the various comparison maps are displayed, we point out the following: for any smooth $\mathbb{F}$-scheme $X$ and $j\ge0$, our joint work with Bachmann proves that the composition $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X)\stackrel{{\mbox{\rm \scriptsize Cons.~\ref{cons_lke_to_mot}}}}{\longrightarrow} \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\stackrel{{\mbox{\rm \scriptsize Cons.~\ref{cons_mot_to_cdh}}}}{\longrightarrow}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$$ is an equivalence. (We will see in Corollary [Corollary 91](#corol_smooth_comparison){reference-type="ref" reference="corol_smooth_comparison"} that in fact each map is an equivalence, but the result about the composition will be used in the proof.) Indeed, this is the canonical map obtained by left Kan extending $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}$ to all qcqs schemes, then cdh sheafifying, then restricting back to smooth $\mathbb{F}$-schemes (see ([\[eqn:Z(j)cdh\]](#eqn:Z(j)cdh){reference-type="ref" reference="eqn:Z(j)cdh"})); it is an equivalence by the special case $k=\mathbb{F}$ of Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(8). ## Extension to derived schemes {#sec:derived} We finish this section by briefly explaining that our motivic cohomology extends to derived schemes, though we do not require the theory in such generality in the present article. We write $\mathrm{CAlg}^{\mbox{\rm \scriptsize ani}}_\mathbb{F}$ for the $\infty$-category of *animated $\mathbb{F}$-algebras*, i.e., the subcategory of $\mathrm{Fun}(\mathrm{CAlg}^{\Sigma}_\mathbb{F}, \mathcal{S}\mathrm{pc}{})$ which preserves finite products, where $\mathrm{CAlg}^{\Sigma}_\mathbb{F}$ denotes the category of finitely generated polynomial $\mathbb{F}$-algebras. Animated $\mathbb{F}$-algebras are derived affine schemes, out of which we build the $\infty$-category of *derived $\mathbb{F}$-schemes* $\rm dSch_k$; see [@LurieSAG] for more details. **Construction 61**. Let $\mathbb{F}$ be a prime field and $X$ a qcqs derived $\mathbb{F}$-scheme. Note that the HKR filtration of Theorem [Theorem 25](#thm:hkr){reference-type="ref" reference="thm:hkr"} and the BMS filtration of Theorem [Theorem 41](#thm_BMS2+){reference-type="ref" reference="thm_BMS2+"} extend to the generality of derived $\mathbb{F}$-schemes. Indeed, in the case of the HKR filtration the references [@antieau-fil; @raksit-hkr; @mrt-hkr] work in this degree of generality; for the BMS filtration one $p$-completely left Kan extends the filtration from discrete algebras, as in [@AntieauMathewMorrowNikolaus Cons. 5.33] By naturality of these filtrations, there are natural comparison maps of filtered spectra $\mathrm{Fil}^\star_{\rm HKR}\mathrm{HC}^-(X/\mathbb{Q}) \rightarrow \mathrm{Fil}^\star_{\rm HKR}\mathrm{HC}^-(X^{\mbox{\rm \scriptsize cla}}/\mathbb{Q})$ if $\mathbb{F}=\mathbb{Q}$, and $\mathrm{Fil}^\star_{\mathrm{BMS}}\mathrm{TC}(X) \rightarrow \mathrm{Fil}^\star_{\mathrm{BMS}}\mathrm{TC}(X^{\mbox{\rm \scriptsize cla}})$ and if $\mathbb{F}=\mathbb{F}_p$, where $X^{\mbox{\rm \scriptsize cla}} \hookrightarrow X$ is the classical locus of $X$. The motivic filtration on the $K$-theory of $X$ is then defined by the following cartesian square $$\begin{tikzcd} \mathrm{Fil}^{\star}_{\mathrm{mot}}\mathrm{K}(X) \ar[dotted]{d} \ar[dotted]{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X)\ar{d} \\ \mathrm{Fil}^{\star}_{\mathrm{mot}}\mathrm{K}(X^{\mbox{\rm \scriptsize cla}}) \ar{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X^{\mbox{\rm \scriptsize cla}}), \end{tikzcd} \qquad\text{resp.~} \begin{tikzcd} \mathrm{Fil}^{\star}_{\mathrm{mot}}\mathrm{K}(X) \ar[dotted]{d} \ar[dotted]{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)\ar{d} \\ \mathrm{Fil}^{\star}_{\mathrm{mot}}\mathrm{K}(X^{\mbox{\rm \scriptsize cla}}) \ar{r} & \mathrm{Fil}^{\star}_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X^{\mbox{\rm \scriptsize cla}}), \end{tikzcd}$$ (the first if $\mathbb{F} = \mathbb{Q}$; the second if $\mathbb{F} = \mathbb{F}_p$). In both cases the $\mathrm{Fil}^{\star}_{\mathrm{mot}}\mathrm{K}(X^{\mbox{\rm \scriptsize cla}})$ refers to the motivic filtration which we have defined earlier on the $K$-theory of the classical qcqs $\mathbb{F}$-scheme $X^{\mbox{\rm \scriptsize cla}}$. The *weight-$j$ motivic cohomology* of $X$ is then defined to be $$\mathbb{Z}(j)^{\mathrm{mot}}(X) := \mathrm{gr}^j_{\mathrm{mot}}\mathrm{K}(X)[-2j].$$ It is not our intention to present here an exhaustive account of the motivic filtration on derived schemes; we content ourselves with stating the following summary of the main properties: **Theorem 62** (Motivic filtration for derived schemes). *For any qcqs derived $\mathbb{F}$-scheme, there exists a natural $\mathbb{N}$-indexed, multiplicative filtration $\mathrm{Fil}^\star_{\mathrm{mot}}\mathrm{K}(X)$ on $K(X)$ satisfying $\mathrm{Fil}^0_{\mathrm{mot}}\mathrm{K}(X) \simeq \mathrm{K}(X)$. If $X$ is classical then this filtration agrees with the earlier motivic filtration of Definitions [Definition 32](#eq:char0){reference-type="ref" reference="eq:char0"} and [Definition 46](#def:charp){reference-type="ref" reference="def:charp"}.* *Proof.* We just explain the claim that $\mathrm{Fil}^0_{\mbox{\rm \scriptsize mot}}K(X)=K(X)$, the other statements being clear. By Zariski descent for derived $\mathbb{F}$-schemes, it suffices to prove the result for $X = \mathrm{Spec}(A)$ when $A$ is an animated $\mathbb{F}$-algebra. In this case, the result follows from the same arguments as in the classical case and part of the Dundas--Goodwillie--McCarthy theorem [@Dundas2013], stating that for a simplicial ring $A$ the square of spectra $$\begin{tikzcd} \mathrm{K}(A) \ar{r} \ar{d} & \mathrm{TC}(A) \ar{d} \\ \mathrm{K}(\pi_0A) \ar{r} & \mathrm{TC}(\pi_0A) \end{tikzcd}$$ is cartesian. ◻ # The projective bundle formula and regular blowup squares {#section_pbf} Let $k$ be a field. The following are three important properties concerning classical motivic cohomology of smooth $k$-schemes: 1. ($\mathbb{A}^1$-invariance) If $X$ is a smooth $k$-scheme, then the projection map induces an equivalence $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X) \xrightarrow{\simeq} \mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X \times \mathbb{A}^1)$. 2. ($\mathbb{P}^1$-bundle formula) To each line bundle $\mathcal L$ on $X$, there is natural class $c_1(\mathcal L) \in H^2_{{\mbox{\rm \scriptsize cla}}}(X, \mathbb{Z}(1))$ which induces, using the multiplicative structure on motivic cohomology, an equivalence for $j \geq 1$: $$\mathbb{Z}(j)(X)^{\mbox{\rm \scriptsize cla}} \oplus \mathbb{Z}(j-1)(X)^{\mbox{\rm \scriptsize cla}}[-2] \xrightarrow{\pi^* \oplus c_1(\mathcal O(1))\pi^*} \mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(\mathbb{P}^1_X);$$ 3. (Regular blowup formula) Given a closed immersion of smooth $k$-schemes $Y\to X$ of codimension $c \geq 2$, and letting $\rm Bl_YX \rightarrow X$ denote the corresponding blowup, there is a natural equivalence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(\mathrm{Bl}_YX) \simeq \mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X') \oplus \left(\bigoplus_{1 \leq i \leq c-1} \mathbb{Z}(j-i)(Y)^{\mbox{\rm \scriptsize cla}}[-2i]\right).$$ Since algebraic $K$-theory is not in general $\mathbb{A}^1$-invariant the analog of property (1) fails for our motivic cohomology. On the other hand, the analogues of the other two properties do hold for algebraic $K$-theory, being a direct consequence of additivity. In this section we refine that result by showing that (2) continues to hold for our motivic cohomology, even for non-smooth schemes. Similarly, we also extend property (3) to our motivic cohomology, in the context of blowups along regular immersions of possibly non-smooth schemes; the resulting formula, however, takes the shape of a cartesian square as opposed to a splitting. Property (2) fits within recent developments in the theory of non-$\mathbb{A}^1$-invariant motives as developed by [@AnnalaHoyoisIwasa2023; @AnnalaIwasa2023]. In this theory, the projective bundle formula is isolated as the key property of cohomology theories, in lieu of $\mathbb{A}^1$-invariance. In particular, our results imply that the presheaves $\mathbb{Z}(j)^{\mathrm{mot}}$, for $j\ge0$, assemble into a motivic spectrum in the sense of [@AnnalaIwasa2023]. ## First Chern classes and $\mathbb{P}^1$-bundle formulae {#sec:p1} As usual $\mathbb{F}$ denotes a prime field. To formulate the $\mathbb{P}^1$-bundle formula we need the first Chern class: **Lemma 63**. *There exists a unique map of presheaves on $\mathrm{S}\mathrm{ch}{}_\mathbb{F}^{\mbox{\rm \scriptsize qcqs}}$ $$R\Gamma_{\mbox{\rm \scriptsize Nis}}(-,\mathbb{G}_m)[-1]\longrightarrow\mathbb{Z}(1)^{\mbox{\rm \scriptsize mot}}$$ which is given on smooth $\mathbb{F}$-schemes by the $j=1$ case of ([\[eqn_cla_to_mot_j\]](#eqn_cla_to_mot_j){reference-type="ref" reference="eqn_cla_to_mot_j"}) above (recall that $\mathbb{Z}(1)^{\mbox{\rm \scriptsize cla}}\simeq R\Gamma_{\mbox{\rm \scriptsize Nis}}(-,\mathbb{G}_m)[-1]$ on smooth $\mathbb{F}$-schemes).* *Proof.* As is explained in [\[eqn_cla_to_mot_j\]](#eqn_cla_to_mot_j){reference-type="eqref" reference="eqn_cla_to_mot_j"}, for any smooth $\mathbb{F}$-algebra $R$, we have a comparison map which takes the form of a natural map $\mathbb{Z}(1)^{\mbox{\rm \scriptsize cla}}(R) \simeq R\Gamma_{\mbox{\rm \scriptsize Zar}}(R,\mathbb{G}_m)[-1]\to \mathbb{Z}(1)^{\mbox{\rm \scriptsize mot}}(R)$; the left Kan extension of the left side to all $\mathbb{F}$-algebras is the functor $(\tau^{\le 1}R\Gamma_{\mbox{\rm \scriptsize Zar}}(-,\mathbb{G}_m))[-1]$, so in this way we obtain a functor $(\tau^{\le 1}R\Gamma_{\mbox{\rm \scriptsize Zar}}(-,\mathbb{G}_m))[-1]\to \mathbb{Z}(1)^{\mbox{\rm \scriptsize mot}}$ on affine $\mathbb{F}$-schemes. Nisnevich sheafifying then defines the desired map. Uniqueness follows from the construction. ◻ **Definition 64**. For any qcqs $\mathbb{F}$-scheme $X$, we refer to the map of the previous lemma $$\label{eq:c1-mot} c_1:R\Gamma_{\mbox{\rm \scriptsize Nis}}(X,\mathbb{G}_m)[-1]\longrightarrow\mathbb{Z}(1)^{\mbox{\rm \scriptsize mot}}(X)$$ as the *first Chern class*. We will often refer to the induced natural map on $H^2$, namely $$c_1:\text{Pic}(X)\longrightarrow H^2_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(1)),$$ in the same way. We now formulate the $\mathbb{P}^1$-bundle formula for any $X \in \mathrm{S}\mathrm{ch}{}_{\mathbb{F}}^{\mbox{\rm \scriptsize qcqs}}$. Thanks to the multiplicative structure of $\bigoplus_{j\ge0}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}[2j]$, multiplication by the first Chern class of $\mathcal{O}(1)\in\mathrm{Pic}(\mathbb{P}_X^1)$ defines maps $$\mathbb{Z}(j)^{\mathrm{mot}}(\mathbb{P}^1_X) \xrightarrow{c_1(\mathcal O(1))} \mathbb{Z}(j+1)^{\mathrm{mot}}(\mathbb{P}^1_X)[2]$$ for $j\ge0$. Denoting by $\pi:\mathbb{P}^1_X \rightarrow X$ the canonical projection, we thus have natural maps $$\label{eq:pbf} \mathbb{Z}(j)^{\mathrm{mot}}(X)\oplus \mathbb{Z}(j-1)^{\mathrm{mot}}(X)[-2] \xrightarrow{\pi^* \oplus c_1(\mathcal O(1))\pi^* } \mathbb{Z}(j)^{\mathrm{mot}}(\mathbb{P}^1_X).$$ for $j\ge1$. We can also extend [\[eq:pbf\]](#eq:pbf){reference-type="eqref" reference="eq:pbf"} to $j \leq 0$ by setting $\mathbb{Z}(j)^{\mathrm{mot}} = 0$ for $j < 0$ and so the $c_1(\mathcal O(1))\pi^*$ map has domain $0$ if $j \leq 0$. Our goal in this section is to prove the following $\mathbb{P}^1$-bundle formula for motivic cohomology: **Theorem 65** ($\mathbb{P}^1$-bundle formula). *For any qcqs $\mathbb{F}$-scheme $X$ and $j \geq 0$, the map [\[eq:pbf\]](#eq:pbf){reference-type="eqref" reference="eq:pbf"} is an equivalence.* **Remark 66**. Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"} establishes a computation of motivic cohomology relative to $\mathbb{P}^1$. We will note, in Theorem [Theorem 86](#thm:pbf-blowup){reference-type="ref" reference="thm:pbf-blowup"}, that this promotes quite automatically to a computation relative to $\mathbb{P}^n$ (or even for $\mathbb{P}(\mathcal E)$ for any locally free sheaf $\mathcal E$) for any $n \geq 1$. The key input is an argument from [@AnnalaIwasa2023] which allows one to pass from a $\mathbb{P}^1$-bundle formula to a general projective bundle formula using "elementary blowup excision.\" In this remainder of this subsection we prove Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"} in characteristic zero, as well as a large part in characteristic $p$. An input which we will use is that the projective bundle formula always holds for additive invariants, as we explain in the next construction and lemma. **Construction 67** (Projective bundle formula for additive invariants). Let $k$ be a commutative ring[^11] and $E$ be a $k$-linear additive invariant[^12] in the sense of [@HoyoisScherotzkeSibilla2017 §4], which extends the absolute theory in [@BlumbergGepnerTabuada2013]. Of special interest is the invariant $E = K_{\mathrm{conn}}$. It is corepresented by the unit object in the $\infty$-category of $k$-linear additive invariants [@HoyoisScherotzkeSibilla2017 Thm. 5.24]. Out of this we draw two consequences: first, it acquires a canonical lax monoidal structure and thus its restriction along functor $\mathrm{Perf}: \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs},\mathrm{op}}_k \rightarrow \mathcal{C}\mathrm{at}{}^k_{\infty}$ upgrades to a functor into $\mathbb{E}_{\infty}$-ring spectra: $$K_{\mathrm{conn}} \circ \mathrm{Perf}: \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs},\mathrm{op}}_{k} \rightarrow \mathrm{CAlg}(\text{Sp}).$$ Secondly, any localizing invariant $E$ is canonically a $K_{\mathrm{conn}}$-module[^13]and thus, in the symmetric monoidal $\infty$-category of presheaves on $\mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs},\mathrm{op}}_{k}$, $E \circ \mathrm{Perf}$ is a $K_{\mathrm{conn}} \circ \mathrm{Perf}$-module. From hereon, we suppress the precomposition with $\mathrm{Perf}$ whenever the context is clear. Now, we have a morphism of presheaves of $\mathbb{E}_{\infty}$-monoids on $\mathrm{S}\mathrm{ch}{}^{\mbox{\rm \scriptsize qcqs}}_k$ $$c_1:\mathcal{P}\mathrm{ic}\rightarrow \Omega^{\infty}K_{\mathrm{conn}}.$$ We call this the *first chern class*. By adjunction $\Sigma^{\infty}_+ \dashv \Omega^{\infty}$ there is then an induced map of presheaves of $\mathbb{E}_{\infty}$-rings, i.e., $$c_1: \Sigma^{\infty}_+\mathcal{P}\mathrm{ic}\rightarrow K_{\mathrm{conn}}.$$ which we also denote by $c_1$. This construction (rather, a sheared variant) is discussed more extensively in Appendix [11.1](#sec:ai-thm){reference-type="ref" reference="sec:ai-thm"}. In particular we have morphisms, functorial in $X\in\mathrm{Sch}^{\mbox{\rm \scriptsize qcqs}}_k$, given by $$\label{eq:p1-loc-k} K_{\mathrm{conn}}(X) \oplus K_{\mathrm{conn}}(X) \xrightarrow{\pi^* \oplus c_1(\mathcal O(-1))\pi^* } K_{\mathrm{conn}}(\mathbb{P}^1_X),$$ More generally, using the $K_{\mathrm{conn}}$-module structure on $E$, we have functorial morphisms in $X\in\mathrm{Sch}^{\mbox{\rm \scriptsize qcqs}}_k$, given by $$\label{eq:p1-loc} E(X) \oplus E(X) \xrightarrow{\pi^* \oplus c_1(\mathcal O(-1))\pi^* } E(\mathbb{P}^1_X).$$ **Lemma 68**. *For any additive invariant $E$ as in Construction [Construction 67](#cons_add_invariant){reference-type="ref" reference="cons_add_invariant"} and any qcqs $\mathbb{F}$-scheme $X$, the morphism [\[eq:p1-loc\]](#eq:p1-loc){reference-type="eqref" reference="eq:p1-loc"} is an equivalence.* *Proof.* This follows from the semiorthogonal decomposition of $\mathrm{Perf}(\mathbb{P}^1)$; see, for example, [@CisinskiKhan2020 Thm. 4.25] for a more general result. ◻ We provide a nontrivial example of $E$ as above by observing that $L_{\mathrm{cdh}}\mathrm{TC}$ extends to an additive invariant. **Lemma 69**. *Let $k$ be a commutative ring, then $L_{\mathrm{cdh}}\mathrm{TC}$ extends to an additive invariant on schemes. That is, there exists an additive $k$-linear invariant $$L_{\mathrm{cdh}}\mathrm{TC}: \mathcal{C}\mathrm{at}{}^k_{\infty} \rightarrow \text{Sp};$$ such that we have a natural equivalence $(L_{\mathrm{cdh}}\mathrm{TC}\circ \mathrm{Perf})(X) \simeq L_{\mathrm{cdh}}\mathrm{TC}(X)$ for any qcqs $k$-scheme $X$.* *Proof.* First, we remark that $\mathrm{KH}$ extends to $k$-linear categories by the procedure in [@LandTamme2019 Def. 3.13] which is stated for $\mathbb{Z}$ but works for more general commutative rings. We define $L_{\mathrm{cdh}}\mathrm{TC}$ by taking the following pushout in $\infty$-category of functors $\mathcal{C}\mathrm{at}{}^k_{\infty} \rightarrow \text{Sp}$: $$\label{eq:lax-mon} \begin{tikzcd} \mathrm{K}\ar{r} \ar{d} & \mathrm{TC}\ar[dotted]{d} \\ \mathrm{KH}\ar[dotted]{r} & L_{\mathrm{cdh}}\mathrm{TC}. \end{tikzcd}$$ Since the formation of additive invariants is stable under finite colimits, $L_{\mathrm{cdh}}\mathrm{TC}$ is automatically an additive invariant. Now, by Theorem [Theorem 23](#thm:mainsq){reference-type="ref" reference="thm:mainsq"}, and the fact that $\mathrm{K}, \mathrm{TC}$ and $\mathrm{KH}$ of schemes are constructed by precomposing these additive invariants with $\mathrm{Perf}$, we conclude the result. ◻ Combining Lemmas [Lemma 69](#lem:lcdh-tc){reference-type="ref" reference="lem:lcdh-tc"} and Lemma [Lemma 68](#lem:loc){reference-type="ref" reference="lem:loc"}, we obtain the following non-obvious projective bundle formulas: **Corollary 70**. *If $\mathbb{F} = \mathbb{Q}$, the $\mathrm{K}_{\mathrm{conn}}$-module structure on $L_{\mathrm{cdh}}\mathrm{HC}^-(-/\mathbb{Q})$ induces natural equivalences for any $X \in \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_{\mathbb{Q}}$: $$L_{\mathrm{cdh}}\mathrm{HC}^-(X/\mathbb{Q})\bigoplus L_{\mathrm{cdh}}\mathrm{HC}^-(X/\mathbb{Q}) \xrightarrow{\pi^* \oplus c_1(\mathcal O(-1))\pi^*}L_{\mathrm{cdh}}\mathrm{HC}^-(X/\mathbb{Q}).$$ If $\mathbb{F} = \mathbb{F}_p$, the $\mathrm{K}_{\mathrm{conn}}$-module structure on $L_{\mathrm{cdh}}\mathrm{TC}$ induces natural equivalences for any $X \in \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_{\mathbb{Q}}$: $$L_{\mathrm{cdh}}\mathrm{TC}(X) \bigoplus L_{\mathrm{cdh}}\mathrm{TC}(X) \xrightarrow{\pi^* \oplus c_1(\mathcal O(-1))\pi^*}L_{\mathrm{cdh}}\mathrm{TC}(X).$$* **Remark 71**. The pushout in [\[eq:lax-mon\]](#eq:lax-mon){reference-type="eqref" reference="eq:lax-mon"} has no reason to promote to a pushout of lax monoidal functors. In particular, from its construction, $L_{\mathrm{cdh}}\mathrm{TC}(X)$, for $X$ a qcqs scheme, need not acquire an $\mathbb{E}_{\infty}$-ring structure which is compatible with the map $\mathrm{K}$-theory, by purely formal consideration. Nonetheless, such a structure *does exist* because $L_{\mathrm{cdh}}$ preserves multiplicative structures. With this remark, we freely use that $L_{\mathrm{cdh}}\mathrm{TC}$ is a presheaf on qcqs schemes,valued in $\mathbb{E}_{\infty}$-rings, compatible with the maps from $\mathrm{K}, \mathrm{TC}$ and $\mathrm{KH}$. A second input into proving Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"} is that ([\[eq:p1-loc\]](#eq:p1-loc){reference-type="ref" reference="eq:p1-loc"}) is compatible with filtrations in our cases of interest, at least up to a shearing of the map; this is explained in the next construction and lemma. **Lemma 72**. *Let $\mathbb{F}$ be a prime field. Then the map of spaces $1- c_1:\mathcal{P}\mathrm{ic}\rightarrow \Omega^{\infty}\mathrm{K}$ factors canonically as: $$\mathcal{P}\mathrm{ic}\rightarrow \Omega^{\infty}\mathrm{Fil}^{\geq 1}_{\mbox{\rm \scriptsize lse}}\mathrm{K}_{\mathrm{conn}} \rightarrow \Omega^{\infty}\mathrm{K}_{\mathrm{conn}} \xrightarrow{\simeq} \Omega^{\infty}\mathrm{K}.$$* *Proof.* It suffices to prove the claim when restricted to ${\mathrm{S}\mathrm{m}}_{\mathbb{F}}$, i.e., that the map $\mathcal{P}\mathrm{ic}\rightarrow \Omega^{\infty}\mathrm{K}$ factors through $\Omega^{\infty}\mathrm{Fil}^{\geq \star}_{\mbox{\rm \scriptsize cla}}\mathrm{K}$ on ${\mathrm{S}\mathrm{m}}_{\mathbb{F}}$. In this case, the map $\Omega^{\infty}\mathrm{K}\rightarrow \Omega^{\infty}\mathrm{gr}^0_{\mbox{\rm \scriptsize cla}}\mathrm{K}\simeq \mathbb{Z}$ coincides with the rank map, therefore the composite map of presheaves $1- c_1: \mathcal{P}\mathrm{ic}\rightarrow \Omega^{\infty}\mathrm{gr}^0_{\mbox{\rm \scriptsize cla}}\mathrm{K}$ is contractible and thus factors through the fiber, i.e., the presheaf $\Omega^{\infty}\mathrm{Fil}^{\geq 1}_{\mbox{\rm \scriptsize lse}}\mathrm{K}_{\mathrm{conn}}$. ◻ We now construct a filtered refinement of the map [\[eq:p1-loc\]](#eq:p1-loc){reference-type="eqref" reference="eq:p1-loc"} (up to a shearing), depending only on a multiplicative, filtered refinement of the map $\mathrm{K}_{\mathrm{conn}} \rightarrow E$: **Construction 73**. Let $E$ be a presheaf of spectra on $\mathrm{S}\mathrm{ch}{}_{\mathbb{F}}^{\mathrm{qcqs}}$, which comes equipped with a filtration $\mathrm{Fil}^\star E \rightarrow E$. Suppose further that $E$ is a $K_{\mathrm{conn}}$-module and that the filtration on $E$ promotes to the structure of a filtered $\mathrm{Fil}_{\mbox{\rm \scriptsize lse}}^{\star}\mathrm{K}_{\mathrm{conn}}$-module. Because of Lemma [Lemma 72](#lem:pic-fil1){reference-type="ref" reference="lem:pic-fil1"}, we have a morphism $$\Sigma^{\infty}_+\mathcal{P}\mathrm{ic}\otimes \mathrm{Fil}^{\star}E \xrightarrow{(1 - c_1) \otimes \operatorname{id}} \mathrm{Fil}_{\mbox{\rm \scriptsize lse}}^{1}\mathrm{K}_{\mathrm{conn}} \otimes \mathrm{Fil}^{\star}E \rightarrow \mathrm{Fil}^{\star+1}E$$ where the second map uses the aforementioned module structure. (To clarify: the previous line is a morphism of presheaves of filtered spectra, where $\Sigma_+^\infty\mathcal{P}\mathrm{ic}$ and $\mathrm{Fil}^1_{\mbox{\rm \scriptsize lse}}K^{\mbox{\rm \scriptsize cn}}$ are given constant filtrations.) Using this structure we have, functorially in $X$, a morphism "multiplication by $(1 - c_1)(\mathcal O(-1))$" $$\mathrm{Fil}^{\star}E(\mathbb{P}^1_X) \xrightarrow{(1 - c_1)(\mathcal O(-1))} \mathrm{Fil}^{\star+1}E(\mathbb{P}^1_X).$$ This shows that the composite: $$E(X) \oplus E(X) \xrightarrow{\gamma} E(X) \oplus E(X) \xrightarrow{\pi^* \oplus c_1(\mathcal O(-1))\pi^* } E(\mathbb{P}^1_X),$$ where $\gamma$ is the invertible matrix$\begin{pmatrix} 1 & 1\\ 0 & -1\\ \end{pmatrix},$ and the second map is the induced by the multiplicative map $K_{\mathrm{conn}} \rightarrow E$, is refined by a filtered map $$\label{eq:fil-pbf} \mathrm{Fil}^{\star}E(X) \oplus \mathrm{Fil}^{\star-1}E(X) \xrightarrow{\pi^* \oplus (1 - c_1)(\mathcal O(-1))\pi^*} \mathrm{Fil}^{\star}E(\mathbb{P}^1_X).$$ In turn, ([\[eq:fil-pbf\]](#eq:fil-pbf){reference-type="ref" reference="eq:fil-pbf"}) then induces maps on graded pieces $$\label{eq:gr-pbf} \mathrm{gr}^{j}E(X) \oplus \mathrm{gr}^{j-1}E(X) \xrightarrow{\pi^* \oplus (1 - c_1)(\mathcal O(-1))\pi^*} \mathrm{gr}^{j}E(\mathbb{P}^1_X)$$ for $j\ge0$; in our cases of interest these essentially coincide with [\[eq:pbf\]](#eq:pbf){reference-type="eqref" reference="eq:pbf"}: We next claim that [\[eq:p1-loc\]](#eq:p1-loc){reference-type="eqref" reference="eq:p1-loc"} promotes to a filtered map in our cases of interest. The starting point is the following observation: **Lemma 74**. *Supposing in Construction [Construction 73](#constr:filtp1){reference-type="ref" reference="constr:filtp1"} that $E = \mathrm{K}$ equipped with its motivic filtration (i.e. Definition [Definition 32](#eq:char0){reference-type="ref" reference="eq:char0"} if $\mathbb{F}=\mathbb{Q}$, resp. Definition [Definition 46](#def:charp){reference-type="ref" reference="def:charp"} if $\mathbb{F}=\mathbb{F}_p$). Then, for any $j\ge1$, the map [\[eq:gr-pbf\]](#eq:gr-pbf){reference-type="eqref" reference="eq:gr-pbf"} is homotopic to the map [\[eq:pbf\]](#eq:pbf){reference-type="eqref" reference="eq:pbf"} up to a shift by 2j.* *Proof.* We start with recalling the following observation about classical filtrations on smooth $\mathbb{F}$-schemes. In weight $1$, the second summand of [\[eq:gr-pbf\]](#eq:gr-pbf){reference-type="eqref" reference="eq:gr-pbf"} for $E = K_{\mathrm{conn}}$, i.e., $$\label{eq:classical-c1} \mathrm{gr}^{0}_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X) = \mathbb{Z}(0)^{\mathrm{mot}}(X) \xrightarrow{(1 - c_1)(\mathcal O(-1))\pi^*} \mathrm{gr}^{1}_{\mbox{\rm \scriptsize cla}}\mathrm{K}(\mathbb{P}^1_X) = \mathbb{Z}(1)^{\mathrm{mot}}(\mathbb{P}^1_X)[2]$$ is homotopic to the map $-c_1(\mathcal O(-1))\pi^\star \simeq c_1(\mathcal O(1))\pi^\ast$. Indeed, the map classifying the element $1$ is nullhomotopic on $\mathrm{gr}^1_{\mbox{\rm \scriptsize mot}}$ in classical motivic cohomology. Still restricted on smooth schemes, since the new motivic filtration is a filtered $\mathbb{E}_{\infty}$-algebra under the classical filtration by Construction [Construction 58](#cons_lke_to_mot){reference-type="ref" reference="cons_lke_to_mot"}, the same fact is true the new motivic filtration. Furthermore, on classical motivic cohomology, the map [\[eq:classical-c1\]](#eq:classical-c1){reference-type="eqref" reference="eq:classical-c1"} is equivalent to the map induced by the first chern class on classical motivic cohomology, i.e., the one which induces the equivalence $R\Gamma_{\mathrm{Nis}}(-;\mathbb{G}_m)[-1] \xrightarrow{\simeq} \mathbb{Z}(1)^{\mbox{\rm \scriptsize cla}}$. Therefore, by the construction in Lemma [Lemma 63](#lem:c1-map){reference-type="ref" reference="lem:c1-map"}, the map on the new motivic cohomology for a smooth $X$: $$\mathrm{gr}^{0}_{\mbox{\rm \scriptsize mot}}\mathrm{K}(X) = \mathbb{Z}(0)^{\mathrm{mot}}(X) \xrightarrow{(1 - c_1)(\mathcal O(-1))\pi^*} \mathrm{gr}^{1}_{\mbox{\rm \scriptsize mot}}\mathrm{K}(\mathbb{P}^1_X) = \mathbb{Z}(1)^{\mathrm{mot}}(\mathbb{P}^1_X)[2],$$ is homotopic to $c_1(\mathcal O(1))\pi^{\ast}$. By the uniqueness assertion of Lemma [Lemma 63](#lem:c1-map){reference-type="ref" reference="lem:c1-map"}, the claim follows. ◻ **Remark 75**. For any theory $E(\star)$ admitting a map from motivic cohomology (e.g. $E(\star) = R\Gamma(-,\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq \star})$, $\mathbb{Z}_p(\star)^{\mathrm{syn}}$, and their cdh analogs), precomposing the map from motivic cohomology to $E(\star)$ with the map from Lemma [Lemma 63](#lem:c1-map){reference-type="ref" reference="lem:c1-map"} gives a first chern class map $$c_1:R\Gamma_{\mathrm{Nis}}(-;\mathbb{G}_m)[-1] \rightarrow E(1).$$ Since these invariants are modules over the graded ring $\mathbb{Z}(\star)^{\mathrm{mot}}$, we can define the "multiplication by $c_1(\mathcal O(1))$\" maps: $$E(j-1)[-2](X) \xrightarrow{c_1(\mathcal O(1))\pi^* }E(j)(\mathbb{P}^1_{X}) \qquad j \in \mathbb{Z}.$$ One can check easily that these coincide with other *a priori* constructions of the first chern classes in the literature for these theories. Since the map from $K$-theory to theories "downstream\" of it (e.g. $\mathrm{TC}, \mathrm{HC}^{-}(-/\mathbb{Q})$ and their cdh analogs) are multiplicatively compatible on a filtered level, Lemma [Lemma 74](#lem:gr-compat){reference-type="ref" reference="lem:gr-compat"} proves that the maps $$E(j)(X)\oplus E(j-1)[-2] \xrightarrow{\pi^* \oplus c_1(\mathcal O(1))\pi^* } E(j)(\mathbb{P}^1_{X}).$$ are compatible with the ones induced by taking associated graded pieces of the respective filtrations. We may now prove a large part of Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"}: *Beginning of proof of Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"}.* We first treat the case $\mathbb{F}=\mathbb{Q}$. By Theorem [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(2), it suffices to prove the projective bundle formulae for the theories $$R\Gamma(-,\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq j}), \qquad R\Gamma_{\mbox{\rm \scriptsize cdh}}(-,\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq j}), \qquad \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}.$$ The result for cdh motivic cohomology is in [@BachmannElmantoMorrow] but can be deduced from the smooth case, i.e. the projective bundle formula for $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}$, by resolution of singularities. The results for derived de Rham and its cdh analog follows from the projective bundle formulae for $\mathrm{HC}^-(-/\mathbb{Q})$ (which is an additive invariant) and $L_{\mathrm{cdh}}\mathrm{HC}^-(-/\mathbb{Q})$ respectively (which follows from an analogous argument as in Lemma [Lemma 69](#lem:lcdh-tc){reference-type="ref" reference="lem:lcdh-tc"}) and the rational degeneration result of Theorem [Theorem 25](#thm:hkr){reference-type="ref" reference="thm:hkr"}. Next, assume that $\mathbb{F} = \mathbb{F}_p$. Then, Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(3) reduces the result for $\mathbb{Z}[\tfrac{1}{p}]^{\mathrm{mot}}(-)$ to the one for $\mathrm{cdh}$ motivic cohomology proved in [@BachmannElmantoMorrow], which does not use resolution of singularities. ◻ ## $\mathbb{P}^1$-bundle formula for cdh sheafified syntomic cohomology {#sec:syn-p1} At this juncture, we need only prove Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"} in characteristic $p > 0$ after $p$-adic completion. The key technical input to achieve this is the analogous statement for $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ in characteristic $p > 0$. We note that even though $\mathbb{Z}_p(j)^{\mathrm{syn}}$ has the $\mathbb{P}^1$-bundle formula (see [@BhattLurie2022 Thm. 9.1.1]) there is no reason, in general, for its $\mathrm{cdh}$ sheafification to have the $\mathbb{P}^1$-bundle formula. The technique used to establish this borrows heavily from the ideas in [@BachmannElmantoMorrow]. For this section, we fix $\mathbb{F} = \mathbb{F}_p$. As in Remark [Remark 75](#rem:gr-compat){reference-type="ref" reference="rem:gr-compat"}, composing with the natural maps $\mathbb{Z}(j)^{\mathrm{mot}} \rightarrow \mathbb{Z}_p(j)^{\mathrm{syn}} \rightarrow L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}$, we obtain analogs of the maps [\[eq:pbf\]](#eq:pbf){reference-type="eqref" reference="eq:pbf"} for $L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}$, multiplicatively compatible with the original $c_1$ on $\mathbb{Z}(j)^{\mathrm{mot}}$. Our next goal is to prove the following result: **Theorem 76**. *The maps $$\label{eq:lcdh-pbf} L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}(X)\oplus L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}(X)[-2] \xrightarrow{\pi^* \oplus c_1(\mathcal O(1))\pi^* } L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}(\mathbb{P}^1_X) \qquad j \geq 0,$$ are equivalences for any $X \in \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_{\mathbb{F}_p}$.* In other words, we want to promote the $\mathrm{TC}$ part of Corollary [Corollary 70](#cor:lcdh-pbf){reference-type="ref" reference="cor:lcdh-pbf"} to the filtered level. As noted in Remark [Remark 75](#rem:gr-compat){reference-type="ref" reference="rem:gr-compat"}, the invertible morphism $$L_{\mathrm{cdh}}\mathrm{TC}(-) \bigoplus L_{\mathrm{cdh}}\mathrm{TC}(-) \xrightarrow{\pi^* \oplus (1 - c_1)(\mathcal O(-1))\pi^*}L_{\mathrm{cdh}}\mathrm{TC}(\mathbb{P}^1_{(-)})$$ promotes to a filtered map (as in Construction [Construction 73](#constr:filtp1){reference-type="ref" reference="constr:filtp1"}) such that the induced map on graded pieces are homotopic to the map of [\[eq:lcdh-pbf\]](#eq:lcdh-pbf){reference-type="eqref" reference="eq:lcdh-pbf"}. We set: 1. $C(j)$ to be the cofiber; 2. and write its $p^r$-reductions as $C_r(j):=C(j)/p^r$. By construction and Lemm [Lemma 69](#lem:lcdh-tc){reference-type="ref" reference="lem:lcdh-tc"} we have spectral sequences, natural in $X \in \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_{\mathbb{F}_p}$: $$\label{eq:ss-zero} E_2^{ij}=H^i(C_r(j)(X)) \Rightarrow 0 \qquad E_2^{ij}=H^i(C(j)(X)) \Rightarrow 0.$$ We get some automatic vanishing range of this spectral sequence, at least for the $C_r(j)$ variant, because of cohomological dimension reasons. We record this as Proposition [Proposition 78](#prop:dim){reference-type="ref" reference="prop:dim"} below which will be used throughout the rest of the section. We will ultimately use these spectral sequences to prove that $C(j)\simeq 0$. Our strategy relies on proving that these spectral sequences degenerate for a large enough class of examples, but we first deduce degeneration up to bounded torsion: **Lemma 77** (Theorem [Theorem 76](#thm:cdh-syn-pbf){reference-type="ref" reference="thm:cdh-syn-pbf"} holds up to $p$-torsion). *For each $i,j \geq 0$ and for all $X \in \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_{\mathbb{F}_p}$, we have that the the cohomology groups $H^i(C(j)(X))$ are bounded $p$-torsion. In particular:* 1. *$C(j)(X)$ is derived $p$-complete;* 2. *$C(j)(X)[\tfrac{1}{p}] \simeq 0$.* As explained in the footnote appearing in Lemma [Remark 42](#rem:cdh-local){reference-type="ref" reference="rem:cdh-local"}, $L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}$ need not be derived $p$-complete in general so it is not immediate that the $C(j)$'s are. *Proof.* By construction we get a spectral sequence whose $E_2$-terms consists of the cohomology of $C(j)(X)$ converging to zero (since the $\mathbb{P}^1$-bundle formula holds for $L_{\mathrm{cdh}}\mathrm{TC}$). By the same argument as in Theorem [Theorem 50](#thm:p-ahss){reference-type="ref" reference="thm:p-ahss"} which establishes the degeneration of the motivic spectral sequence up to bounded denominators, we have the same result for this spectral sequence. Therefore, we conclude that each cohomology group $H^i(C(j)(X))$ are bounded $p$-torsion which, in particular, shows that $C(j)(X)$ is derived $p$-complete. This also proves that part (2) of the claim. ◻ In particular, since $C(j)(X)$ is derived $p$-complete, we get an equivalence $$\label{eq:limit} C(j)(X) \simeq \mathop{\mathrm{lim}}C_r(j)(X).$$ In fact, it suffices to prove that $C(j)(X)/p \simeq 0$. However, our argument proceeds by understanding the integral version of $C(j)$ and so we will make claims about this object. To proceed, we need to understand $C(j)(X)$ more explicitly. By Theorem [Theorem 51](#theorem_GL){reference-type="ref" reference="theorem_GL"}, we have that: $$L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}/p^r \simeq R\Gamma_{\text{\'{e}h}}(\mathord-,W_r\Omega^j_{\log})[-j].$$ Based on this identification, the next proposition implies a vanishing range of $C(j)(X)$ based on the valuative dimension of $X$: **Proposition 78**. *Let $X \in \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_{\mathbb{F}_p}$ and assume that it is of valuative dimension $d$, then for all $r \geq 1$, $$H^{> d+1}_{\text{\'{e}h}}(X; W_r\Omega^j_{\log}) = 0.$$* *Proof.* This follows immediately from the fundamental cofiber sequence (which sandwiches $R\Gamma_{\text{\'{e}h}}(X;W_r\Omega^j_{\log})$ between two $\mathrm{cdh}$ cohomologies of discrete sheaves) in Corollary [Corollary 55](#corol_fundamental_p){reference-type="ref" reference="corol_fundamental_p"} and the fact that the valuative dimension coincides with the $\mathrm{cdh}$ cohomological dimension [@ElmantoHoyoisIwasaKelly2021]. ◻ Now, $C_r(j)$ identifies with the cofiber: $$C_r(j)(\mathord-)\simeq\mathrm{cofib}(R\Gamma_{\text{\'{e}h}}(\mathord-;W_r\Omega^j_{\log})\oplus R\Gamma_{\text{\'{e}h}}(\mathord-;W_r\Omega^{j-1}_{\log})[-1] \xrightarrow{\pi^* \oplus c_1(\mathcal O(-1))\pi^*} R\Gamma_{\text{\'{e}h}}(\mathbb{P}^1_{(\mathord-)};W_r\Omega^j_{\log}))[-j].$$ We also write, for any $X \in \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_{\mathbb{F}_p}$: $$R\Gamma_{\text{\'{e}h}}(X;W\Omega^j_{\log}) := \mathop{\mathrm{lim}}R\Gamma_{\text{\'{e}h}}(X, W_r\Omega^j_{\log}).$$ Note that there is always a comparison map $L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}(X) \rightarrow R\Gamma_{\text{\'{e}h}}(X;W\Omega^j_{\log})[-j]$, but this is not always an equivalence because of footnote 8 of Remark [Remark 42](#rem:cdh-local){reference-type="ref" reference="rem:cdh-local"}; indeed the target witnesses the derived $p$-completion of the domain. However, we note that thanks to the derived $p$-completeness assertion and [\[eq:limit\]](#eq:limit){reference-type="eqref" reference="eq:limit"}, we still have an equivalence: $$C(j)(\mathord-) \simeq \mathrm{cofib}(R\Gamma_{\text{\'{e}h}}(\mathord-;W\Omega^j_{\log}) \oplus R\Gamma_{\text{\'{e}h}}(\mathord-;W\Omega^{j-1}_{\log})[-1] \xrightarrow{\pi^* \oplus c_1(\mathcal O(-1))\pi^*}R\Gamma_{\text{\'{e}h}}(\mathbb{P}^1_{(\mathord-)};W\Omega^j_{\log})).$$ One last piece of notation before we embark on the proof: we will reindex $C_r(j)$ and $C(j)$'s up to a shift so that: $$C_r(j):=C_r(j)[j] \qquad C(j) := C(j)[j].$$ This will ensure that $H^0$ of the $C(j)$'s and $C_r(j)$'s are related to zeroth $\text{\'{e}h}$ cohomology of the $W_r\Omega^j_{\log}$ sheaves. We first establish the main result for fields: **Lemma 79** (Theorem [Theorem 76](#thm:cdh-syn-pbf){reference-type="ref" reference="thm:cdh-syn-pbf"} holds for fields). *For any field $F$ of characteristic $p > 0$, we have that $C(j)(F) \simeq 0$.* *Proof.* Setting $H^i(j):=H^i(C(j)(F))$, we claim that the spectral sequence of [\[eq:ss-zero\]](#eq:ss-zero){reference-type="eqref" reference="eq:ss-zero"} for a field takes the following form: $$\label{eq:ss-f} \begin{tikzcd} 0& H^0(0) \ar{rrd} & H^1(0) & H^2(0) & 0\\ 0 & H^0(1) \ar{rrd}& H^1(1) & H^2(1) & 0 \\ 0 & H^0(2) \ar{rrd} & H^1(2) &H^2(2)& 0 \\ \vdots & \vdots & \vdots & \vdots \end{tikzcd}$$ To see this note that the spectral sequence for $C_r(j)$ takes the above form because of the dimension bound in Proposition [Proposition 78](#prop:dim){reference-type="ref" reference="prop:dim"} applied to $\mathbb{P}^1_F$ which is valuative dimension $\leq 1$. To prove the claim for $C(j)(F)$, it suffices to prove that $\mathop{\mathrm{lim}}^1 H^2(C_r(j)(F)) = 0$. By dimension reasons, for all $r \geq 1$, we have isomorphisms $H^2_{\text{\'{e}h}}(\mathbb{P}^1_F; W_r\Omega^j_{\log}) \cong H^2(C_r(j)(F))$. By the Mittag-Leffler condition, it suffices to prove that the maps $H^2_{\text{\'{e}h}}(\mathbb{P}^1_F; W_{r+1}\Omega^j_{\log}) \rightarrow H^2_{\text{\'{e}h}}(\mathbb{P}^1_F; W_r\Omega^j_{\log})$ are surjective. But this follows from the fact that the sheaves $W_{r+1}\Omega^j_{\log} \rightarrow W_{r}\Omega^j_{\log}$ are epimorphisms on strictly henselian valuation rings (hence $\text{\'{e}h}$-locally epimorphisms) [@KellyMorrow2021] and the fact that $H^2_{\text{\'{e}h}}(\mathbb{P}^1_F, -)$ is right exact since it is the top cohomology group. From the claimed form of the spectral sequence, we can read off vanishing of $H^1(j)$'s already, along with vanishing of $H^2(0)$. Furthermore, the differentials induce isomorphisms $H^0(j) \xrightarrow{\cong} H^2(j+1)$. But now, $H^0(j)$ is $p$-torsion-free by Lemma [Lemma 80](#lem:ptf-csm){reference-type="ref" reference="lem:ptf-csm"} and thus $H^2(j+1)$'s are $p$-torsion-free as well. But we know that the groups are $p$-torsion by Lemma [Lemma 77](#lem:rational){reference-type="ref" reference="lem:rational"} and hence $C(j)(F) \simeq 0$. ◻ The following torsion-freeness assertion was used in the lemma above. **Lemma 80**. *Let $X$ be an qcqs $\mathbb{F}_p$-scheme. Then $H^0_{\text{\'{e}h}}(X; W\Omega^j_{\log})$ is $p$-torsion-free.* *Proof.* We have an injection $$H^0_{\text{\'{e}h}}(X; W\Omega^j_{\log}) \hookrightarrow \prod_{\mathrm{Spec}(V) \rightarrow X} H^0(V; W\Omega^j_{\log}) = W\Omega^j_{\log,V};$$ where the product runs along all spectra of strictly henselian local rings mapping into $X$. The target is $p$-torsion-free by [@KellyMorrow2021] and thus we conclude. ◻ Mainly through Gersten injectivity for valuation rings [@KellyMorrow2021], we can boostrap the result for fields to an $H^0$ result for all schemes. **Lemma 81** (Theorem [Theorem 76](#thm:cdh-syn-pbf){reference-type="ref" reference="thm:cdh-syn-pbf"} holds for $H^0$). *Let $X \in \mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_{\mathbb{F}_p}$, then* 1. *for all $j \geq 0$, $H^0(C(j)(X)) = 0$ and,* 2. *for all $j \geq 0, r \geq 1$, $H^0(C_r(j)(X)) = 0$.* *Proof.* First, we note that (2) proves (1). Indeed, we have an exact sequence for all $r \geq 1$: $$\begin{tikzcd} 0 \ar{r} & H^{-1}(C_r(j)(X)) \ar{r} & \mathrm{H}_{\text{\'{e}h}}^0(X; W_r\Omega^j_{\log}) \rar{\pi^*} & H^0_{\mathrm{cdh}}(\mathbb{P}^1_X, W_r\Omega^j_{\log}) \rar \ar[draw=none]{d}[name=X, anchor=center]{} & H^0(C_r(j)(V)) \ar[rounded corners, to path={ -- ([xshift=2ex]\tikztostart.east) |- (X.center) \tikztonodes -| ([xshift=-2ex]\tikztotarget.west) -- (\tikztotarget)}]{dll}[at end]{\delta} \\ & & \mathrm{H}_{\text{\'{e}h}}^1(X; W_r\Omega^j_{\log}) \rar{\pi^*} & \mathrm{H}_{\text{\'{e}h}}^1(\mathbb{P}^1_X; W_r\Omega^j_{\log}) \rar & H^1(C_r(j)(X)) &. \end{tikzcd}$$ But the maps labeled $\pi^*$ are split by the the map induced by the inclusion of the $\infty$-section $\infty: \mathrm{Spec}V \hookrightarrow \mathbb{P}^1_{V}$ so that $H^{-1}(C_r(j)(V))$ vanishes and we have a direct sum decomposition: $H^0_{\text{\'{e}h}}(\mathbb{P}^1_X, W_r\Omega^j) \simeq H^0(C_r(j)(X) \oplus \mathrm{H}_{\text{\'{e}h}}^0(X; W_r\Omega^j)$. In particular the $\mathop{\mathrm{lim}}^1$ term involving $H^{-1}$ vanishes so that $H^0(C(j)(X)) \cong \mathop{\mathrm{lim}}H^0(C_r(j)(X))$ and so (2) proves (1). We now prove (2). We have already proved the result when $X = \mathrm{Spec}(F)$ as a special case of Lemma [Lemma 79](#lem:fields-case){reference-type="ref" reference="lem:fields-case"}. To bootstrap this result, we consider the commutative diagram for all $j \geq 0, r \geq 1$: $$\begin{tikzcd} H^0_{\text{\'{e}h}}(\mathbb{P}_X^1; W_r\Omega^j_{\log}) \ar{r} \ar{d}{\infty^{\ast}} & \prod_{x \in X} H^0_{\text{\'{e}h}}( \mathbb{P}^1_x; W_r\Omega^j_{\log}) \ar{d}{\infty^{\ast}}\\ H^0_{\text{\'{e}h}}(X; W_r\Omega^j_{\log}) \ar{r} & \prod_{x \in X} H^0_{\text{\'{e}h}}(x; W_r\Omega^j_{\log}). \end{tikzcd}$$ It suffices to prove that the left vertical map is injective (since it is already seen to be a split surjection). We have already established that the right vertical map is injective (in fact, an isomorphism), hence it suffices to prove that the top horizontal map is injective. To do so we note that for all $\mathbb{F}_p$-scheme $Y$, by the formula for sheafification, the canonical map $$H^0_{\text{\'{e}h}}(Y; W_r\Omega^j_{\log}) \to \prod_{\mathrm{Spec}(V) \to Y} \mathrm{H}_{\text{\'{e}h}}^0(V; W_r\Omega^j_{\log}) = W_r\Omega^j_{\log,V}$$ is injective where the product runs across all strictly henselian valuation rings (the stalks for the $\text{\'{e}h}$ topology) mapping to $Y$. But now, by the validity of the Gersten injectivity for valuation rings in this context [@KellyMorrow2021], we have a further injection $$H^0_{\text{\'{e}h}}(Y; W_r\Omega^j_{\log}) \to \prod_{\mathrm{Spec}(V) \to Y} W_r\Omega^j_{\log,V} \to \prod_{\mathrm{Spec}(V) \to Y} W_r\Omega^j_{\log,\mathrm{Frac}(V)}.$$ Thus we conclude that the map $$H^0_{\text{\'{e}h}}(Y; W_r\Omega^j_{\log}) \to \prod_{\mathrm{Spec}(F) \rightarrow Y} W_r\Omega^j_{\log,F},$$ is injective where the product runs across all fields mapping to $Y$. We use this to prove that for all $X$, the map $H^0_{\text{\'{e}h}}(\mathbb{P}_X^1; W_r\Omega^j_{\log}) \rightarrow \prod_{x \in X} H^0_{\text{\'{e}h}}( \mathbb{P}^1_x; W_r\Omega^j_{\log})$ is injective. Indeed, let $\alpha \in H^0_{\text{\'{e}h}}(\mathbb{P}_X^1; W_r\Omega^j_{\log})$ be an element such that $\alpha|_{x \times \mathbb{P}^1} = 0$ for all $x \in X$. Now, any map $\mathrm{Spec}(F) \rightarrow \mathbb{P}^1_X$ factors through $x \times \mathbb{P}^1 \rightarrow X \times \mathbb{P}^1$ for some $x \in X$. Therefore, $\alpha|_{\mathrm{Spec}(F)} = 0$, whence we obtain the claim. ◻ **Remark 82**. The last part of the argument imitates the argument, surely well-known to experts, for reducing $\mathbb{A}^1$-invariance over smooth schemes to $\mathbb{A}^1$-invariance over fields for functors satisfying Gersten injectivity. For example, the first author has used this in joint work with Kulkarni and Wendt [@ElmantoKulkarniWendt2022 Prop. 3.5] to prove $\mathbb{A}^1$-invariance of the Nisnevich sheafification of some cohomology sets. We also use this kind of argument in establishing some properties of $\mathrm{cdh}$-motivic cohomology [@BachmannElmantoMorrow]. To proceed, we further appeal to descent properties of $C(j)$. Indeed, we observe that $C(j)$ is a cdh sheaf since it is the cofiber of two cdh sheaves. The next property we use is *henselian $v$-excision* (hv-excision for short). Recall that if $V$ is a valuation ring and $\mathfrak{p}$ is a prime ideal, then we can form the following bicartesian square of rings: $$\label{eq:hv} \begin{tikzcd} V \ar{d} \ar{r} & V_{\mathfrak{p}} \ar{d}\\ V/\mathfrak{p} \ar{r} & \kappa(\mathfrak{p}). \end{tikzcd}$$ If $V$ is a henselian valuation ring, then so are the other vertices in the above square by [@ElmantoHoyoisIwasaKelly2021 Lem. 3.3.5]. A presheaf of spectra or complexes on $\mathbb{F}$-schemes are said to be *hv-excisive* if it converts [\[eq:hv\]](#eq:hv){reference-type="eqref" reference="eq:hv"} to a cartesian square. **Lemma 83**. *For all $j \geq 0$, the presheaves $C(j)$ are* 1. *finitary $\mathrm{cdh}$ sheaves and, in fact, are $\text{\'{e}h}$ sheaves;* 2. *hv-excisive.* *Therefore, $C(j) \simeq 0$ if and only if $C(j)(V) \simeq 0$ for any henselian valuation ring of rank $\leq 1$.* *Proof.* As explained above, $C(j)$ is a $\mathrm{cdh}$ sheaf; it is also an $\text{\'{e}h}$ sheaf because it is a cofiber of a map between $\text{\'{e}h}$ sheaves. The finitary part of $C(j)$ is not quite immediate as $L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}$ is not a finitary sheaf. However, as recorded in the proof of Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}, the fiber of the map $\mathbb{Z}_p(j)^{\mathrm{syn}} \rightarrow L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}$ which we denote by $W(j)$ is finitary. Since the $\mathbb{P}^1$-bundle formula for $\mathbb{Z}_p(j)^{\mathrm{syn}}$ holds by [@BhattLurie2022], $C(j)$ is equivalent to the cofibre of the $\mathbb{P}^1$-bundle maps for $W(j)$ (up to a shift) and is thus finitary. To prove that $C(j)$ is hv-excisive, it suffices to prove the claim for $C(j)[\tfrac{1}{p}]$ and $C(j)/p$; the former is zero so we are left to prove the claim for the latter. For this claim, it suffices to prove that the functors $L_{\mathrm{cdh}}\mathbb{F}_p(j)^{\mathrm{syn}}$ and $L_{\mathrm{cdh}}\mathbb{F}_p(j)^{\mathrm{syn}}(\mathbb{P}^1 \times -)$ are for all $j \geq 0$. Since $L_{\mathrm{cdh}}$ does not change stalks and the terms of the square [\[eq:hv\]](#eq:hv){reference-type="eqref" reference="eq:hv"} are all henselian valuation rings, the claim for $L_{\mathrm{cdh}}\mathbb{F}_p(j)^{\mathrm{syn}}$ follows from the analogous claim for $\mathbb{F}_p(j)^{\mathrm{syn}}$; this is proved in [@BachmannElmantoMorrow]. We note that the argument reduces to the analogous claim for the cotangent complex by the increasing filtration in Lemma [Lemma 39](#lem_fin_fil_on_syn){reference-type="ref" reference="lem_fin_fil_on_syn"} which is what we verify in [@BachmannElmantoMorrow]. To prove the claim for $L_{\mathrm{cdh}}\mathbb{F}_p(j)^{\mathrm{syn}}(\mathbb{P}^1 \times -)$ we use [@ElmantoHoyoisIwasaKelly2021 Lem. 3.3.7] for $\mathcal F = L_{\mathrm{cdh}}\mathbb{F}_p(j)^{\mathrm{syn}}$ and $X = \mathbb{P}^1_V$. Indeed, $\mathcal F$ is a finitary $\mathrm{cdh}$-sheaf; it takes values in the derived category of $\mathbb{F}_p$-vector spaces where compact objects are cotruncated. We have already verified hv-excision and thus the cited result applies. The "therefore\" part of the assertion holds because $C(j)$ is a hypercomplete, finitary $\mathrm{cdh}$ sheaf (since it is a finitary, $\mathrm{cdh}$ sheaf over $\mathbb{F}_p$ we can appeal to [@ElmantoHoyoisIwasaKelly2021 Corol. 2.4.16]) and thus its vanishing is detected on henselian valuatioon rings. Using that it is finitary again, we may reduce to the case that $V$ has finite rank and by hv-excision we reduce, by induction, to $V$ being rank $\leq 1$. ◻ For the rest of the section, we fix $V$ a henselian valuation ring of rank $\leq 1$. This will help us to degenerate enough of the spectral sequences [\[eq:ss-zero\]](#eq:ss-zero){reference-type="eqref" reference="eq:ss-zero"}. The next lemma proves that the spectral sequence $E_2^{ij}=H^i(C_r(j)(V)) \Rightarrow 0$ displays as (after the coconnectedness result of Lemma [Lemma 81](#lem:p-tf-0){reference-type="ref" reference="lem:p-tf-0"}): $$\label{eq:ss} \begin{tikzcd} 0& H^1(0) \ar{rrd} & H^2(0) & H^3(0) & 0\\ 0 & H^1(1) \ar{rrd}& H^2(1) & H^3(1) & 0\\ 0 & H^1(2) \ar{rrd} & H^2(2) & H^3(2) & 0\\ \vdots & \vdots & \vdots & \vdots, \end{tikzcd}$$ and that this patten persists integrally. **Lemma 84**. *Let $V$ be a henselian valuation ring of rank $\leq 1$ over $\mathbb{F}_p$. Then:* 1. *for any $j \geq 0, r \geq 1$, we have that $H^k(C_r(j)(V)) = 0$ whenever $k > 3$;* 2. *for any $j \geq 0, r \geq 1$, $H^2(C_r(j)(V)) = 0$, $H^3(C_r(0)(V)) = 0$, and $H^1(C_r(j)(V)) \xrightarrow{\cong} H^3(C_r(j+1)(V))$.* 3. *for any $j \geq 0$, $H^k(C(j)(V)) = 0$ whenever $k > 3$;* 4. *for any $j \geq 0$, $H^2(C(j)(V)) = 0$, $H^3(C(0)(V)) = 0$, and $H^1(C(j)(V)) \xrightarrow{\cong} H^3(C(j+1)(V))$.* *Proof.* First, note that Proposition [Proposition 78](#prop:dim){reference-type="ref" reference="prop:dim"} tells us that $H^{\geq 4}_{\text{\'{e}h}}(\mathbb{P}^1_V, W_r\Omega^j_{\log}) = 0$ since $\mathbb{P}^1_V$ is valuative dimension $\leq 2$. On the other hand, since $V$ is a stalk for the $\mathrm{cdh}$-topology, we have that $H^{\geq 2}_{\text{\'{e}h}}(V,W_r\Omega^j_{\log}) = 0$. Therefore, by the definition of $C_r(j)$, we have that $H^{\geq 4}(C_r(j)(V)) = 0$. From the pattern of the spectral sequence [\[eq:ss\]](#eq:ss){reference-type="eqref" reference="eq:ss"}, we conclude (2). Statement (3) is the integral version of the statement (1). For this, it suffices to prove that $\mathop{\mathrm{lim}}^1 H^3(C_r(j)(V)) = 0$ for all $j \geq 0$. In turn, it suffices to prove that for each $r \geq 1$, the map $H^3(C_{r+1}(V)) \rightarrow H^3(C_r(V))$ is surjective. By the definition of $C_r(V)$ and the fact that $V$ is valuative dimension $\leq 1$, we have a surjection $H^3_{\text{\'{e}h}}(\mathbb{P}^1_V; W_r\Omega^j_{\log}) \rightarrow H^3(C_r(j)(V)) \rightarrow 0$. Therefore it suffices to prove that $H^3_{\text{\'{e}h}}(\mathbb{P}^1_V; W_{r+1}\Omega^j_{\log}) \rightarrow H^3_{\text{\'{e}h}}(\mathbb{P}^1_V; W_r\Omega^j_{\log})$ is surjective. But this follows from the fact that the sheaves $W_{r+1}\Omega^j_{\log} \rightarrow W_{r}\Omega^j_{\log}$ are epimorphisms on henselian valuation rings (hence $\text{\'{e}h}$-locally epimorphisms) and the fact that $H^3_{\text{\'{e}h}}(\mathbb{P}^1_V, -)$ is right exact since it is the top cohomology group by Proposition [Proposition 78](#prop:dim){reference-type="ref" reference="prop:dim"}. Part (4) immediately follows from the pattern of the spectral sequence [\[eq:ss\]](#eq:ss){reference-type="eqref" reference="eq:ss"}, which is a consequence of (2). ◻ To conclude, we need only prove that, in fact, $H^3(C(j)(V)) = 0$ for all $j \geq 1$. Since we know that this is a $p$-torsion group, we will prove that it is actually $p$-torsion-free though we need to first prove this under a slightly more restricted hypothesis on $V$. **Lemma 85**. *Assume that $V$ is a strictly henselian valuation ring over $\mathbb{F}_p$ of rank $\leq 1$. For any $j \geq 1$, the group $H^3(C(j)(V))$ is $p$-torsion-free. Therefore, it is zero.* *Proof.* First, note that $\mathop{\mathrm{lim}}^1H^2(C_r(j)(V)) = 0$ since the groups vanish by Lemma [Lemma 84](#lem:vanish){reference-type="ref" reference="lem:vanish"}(3). Therefore $$H^3(C(j)(V)) \cong \mathop{\mathrm{lim}}_r H^3(C_r(j)(V)).$$ Therefore, it suffices to prove that the multiplication by $p$ maps $$H^3(C_{r-1}(j)(V)) \xrightarrow{\times p} H^3(C_{r}(j)(V)),$$ are all injective. We claim that there is a canonical isomorphism (this uses that $V$ is strictly henselian): $$H^3_{\text{\'{e}h}}(\mathbb{P}^1_V, W_r\Omega^j_{\log}) \cong H^3(C_r(j)(V))$$ To see this, the vanishing of $H^2$'s of Lemma [Lemma 84](#lem:vanish){reference-type="ref" reference="lem:vanish"} produces a short exact sequece: $$0 \rightarrow H^1_{\text{\'{e}h}}(V; W_r\Omega_{\log}^{j-1}) \xrightarrow{c_1(\mathcal O(1))\pi^{\ast}} H^3_{\text{\'{e}h}}(\mathbb{P}^1_V; W_r\Omega^j_{\log}) \rightarrow H^3(C_r(j)(V)) \rightarrow 0.$$ We have an isomorphism $H^1_{\text{\'{e}h}}(V; W_r\Omega_{\log}^{j-1}) \cong \widetilde{\nu}_{r-1}(j)(V)$, but the rigidity statement of [@clausen2018k Proposition 4.31] shows that $\widetilde{\nu}_{r-1}(j)(V) = \widetilde{\nu}_{r-1}(j)(\kappa)$ where $\kappa$ is the residue field. Since $V$ is assumed to be strictly henselian, this group is zero (it vanishes on any perfect $\mathbb{F}_p$-algebras. Therefore, it suffices to prove that the maps $$H^3_{\text{\'{e}h}}(\mathbb{P}^1_V, W_{r-1}\Omega^j_{\log}) \xrightarrow{\times p} H^3_{\text{\'{e}h}}(\mathbb{P}^1_V, W_{r}\Omega^j_{\log}),$$ are all injective. We have the short exact sequence of presheaves $$0 \rightarrow W_{r-1}\Omega^j_{\log} \xrightarrow{\times p} W_{r}\Omega^j_{\log} \rightarrow \Omega^j_{\log} \rightarrow 0,$$ which induces a long exact sequence $$\cdots H^2_{\text{\'{e}h}}(\mathbb{P}^1_V, W_r\Omega^j_{\log}) \rightarrow H^2_{\text{\'{e}h}}(\mathbb{P}^1_V, \Omega^j_{\log}) \xrightarrow{\delta} H^3_{\text{\'{e}h}}(\mathbb{P}^1_V, W_{r-1}\Omega^j_{\log}) \xrightarrow{\times p} H^3_{\text{\'{e}h}}(\mathbb{P}^1_V, W_{r}\Omega^j_{\log}) \rightarrow \cdots.$$ We claim that the map $\delta$ is zero for which it suffices to prove that $H^2_{\text{\'{e}h}}(\mathbb{P}^1_V, W_r\Omega^j_{\log}) \rightarrow H^2_{\text{\'{e}h}}(\mathbb{P}^1_V, \Omega^j_{\log})$ is surjective. We have a commutative diagram $$\begin{tikzcd} W_{r-1}\Omega^j_{\log,V} \ar{r} \ar[swap]{d}{\cup c_1(\mathcal O(-1))} & \Omega^j_{\log,V} \ar{d}{\cup c_1(\mathcal O(-1))}\\ H^2_{\text{\'{e}h}}(\mathbb{P}^1_V, W_{r-1}\Omega^j_{\log}) \ar{r}{\delta} & H^2_{\text{\'{e}h}}(\mathbb{P}^1_V, \Omega^j_{\log}). \end{tikzcd}$$ The top map is surjective so it suffices to prove that the right vertical map is surjective. But the cokernel of the right vertical map is exactly $H^2(C_1(j)(V)) = 0$ which vanishes by Lemma [Lemma 84](#lem:vanish){reference-type="ref" reference="lem:vanish"}(2). ◻ *Proof of Theorem [Theorem 76](#thm:cdh-syn-pbf){reference-type="ref" reference="thm:cdh-syn-pbf"}.* By Lemma [Lemma 83](#lem:cj-finitary){reference-type="ref" reference="lem:cj-finitary"}, it suffices to prove that $C(j)(V) = 0$ for $V$ a henselian valuation ring of rank $\leq 1$. We have already seen that the cohomology groups of $C(j)$, on any $\mathbb{F}_p$-scheme, are all $p$-torsion by Lemma [Lemma 77](#lem:rational){reference-type="ref" reference="lem:rational"}. We also know that $H^2, H^0$ and $H^{\geq 4}$ and $H^3(C(0)(V))$ are all zero for $V$ a henselian valuation ring of rank $\leq 1$ by Lemma [Lemma 84](#lem:vanish){reference-type="ref" reference="lem:vanish"}. The same lemma also shows that $H^1(C(j)(V)) \cong H^3(C(j+1)(V))$ for a $j \geq 0$ and so it suffices to prove that $H^3(C(j+1)(V))$ is $p$-torsion free. Now $C(j)$ is an $\text{\'{e}h}$ sheaf (by the "in fact\" part of Lemma [Lemma 83](#lem:cj-finitary){reference-type="ref" reference="lem:cj-finitary"}(1)), we may pass to the strict henselization of $V$. The latter does not change the value group, and hence the rank, of the valuation ring [@Stacks Tag 0ASK] and thus we can use Lemma [Lemma 85](#lem:p-tf){reference-type="ref" reference="lem:p-tf"} to conclude. ◻ *Conclusion of Proof of Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"}.* It suffices to prove that $\mathbb{F}_p(j)^{\mathrm{mot}}$ has the $\mathbb{P}^1$-bundle formula for $\mathbb{F} = \mathbb{F}_p$. By Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(2), we need to prove the result for the theories: $$\mathbb{F}_p(j)^{\mathrm{cdh}}, \mathbb{F}_p(j)^{\mathrm{syn}}, L_{\mathrm{cdh}}\mathbb{F}_p(j).$$ The result for $\mathbb{F}_p(j)^{\mathrm{cdh}}$ is verified in [@BachmannElmantoMorrow], while the result for $\mathbb{F}_p(j)^{\mathrm{syn}}$ is verified in [@BhattLurie2022]. The result for last theory follows from Theorem [Theorem 76](#thm:cdh-syn-pbf){reference-type="ref" reference="thm:cdh-syn-pbf"}. ◻ We can now prove a general projective bundle formula for motivic cohomology. To formulate this, let $r \geq 0$ and consider for $0 \leq i \leq r$ the map, induced by multiplicativity of motivic cohomology: $$c_1(\mathcal O(1))^i\pi^{\ast}: \mathbb{Z}(j-i)^{\mbox{\rm \scriptsize mot}}[-2i](X) \rightarrow \mathbb{Z}(j)(\mathbb{P}^r_X).$$ More generally, we let $\pi:\mathbb{P}_X(\mathcal E) \rightarrow X$ be the projectivization of $\mathcal E$, a rank $r+1$ vector bundle on $X$. It classifies subbundles of rank $1$ and thus comes equipped with a tautological bundle $\mathcal O(-1) \subset \pi^*\mathcal E$ whose dual is denoted by $\mathcal O(1)$. Zariski-locally, $\mathbb{P}_X(\mathcal E)$ is isomorphic to $\mathbb{P}^{r}_X$. Then we have an generalization of the previous map: $$c_1(\mathcal O(1))^i\pi^{\ast}: \mathbb{Z}(j-i)^{\mbox{\rm \scriptsize mot}}[-2i](X) \rightarrow \mathbb{Z}(j)(\mathbb{P}(\mathcal E)).$$ **Theorem 86**. *Let $X$ be a qcqs $\mathbb{F}$-scheme and $j\ge 0$.* 1. *Projective bundle formula: for any $r \ge 0$, the map $$\label{eq:pbf-r} \sum c_1(\mathcal O(1))^i\pi^{\ast}\colon\bigoplus_{i=0}^r\mathbb{Z}(j-i)^{\mbox{\rm \scriptsize mot}}[-2i](X)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(\mathbb{P}_X^r)$$ is an equivalence.* 2. *More generally, if $\mathcal E$ is a locally free sheaf on $X$ of rank $r+1$, then the map: $$\sum c_1(\mathcal O(1))^i\pi^{\ast}\colon\bigoplus_{i=0}^r\mathbb{Z}(j-i)^{\mbox{\rm \scriptsize mot}}[-2i](X)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(\mathbb{P}_X(\mathcal E)),$$ is an equivalence.* 3. *Blow-up formula: Let $Y\to X$ be a regular closed immersion (i.e., $X$ admits an open affine cover such that, on each such affine, $Y$ is defined by a regular sequence); then for any $j \geq 0$: we have a cartesian square in $\mathcal{D}(\mathbb{Z})$: $$\begin{tikzcd} \mathbb{Z}(j)^{\mathrm{mot}}(X) \ar{r} \ar{d} &\mathbb{Z}(j)^{\mathrm{mot}}(Y) \ar{d}\\ \mathbb{Z}(j)^{\mathrm{mot}}(\mathrm{Bl}_Y(X)) \ar{r}& \mathbb{Z}(j)^{\mathrm{mot}}(Y\times_X\mathrm{Bl}_Y(X)).\\ \end{tikzcd}$$* *Proof.* We first establish the blowup formula (3). The blowup formula of course holds for any cdh sheaf, since cdh sheaves carry arbitrary abstract blowup squares to cartesian squares. So, using the fundamental fibre sequence of Theorem [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"} in characteristic $0$ (resp. the pullback square of Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"} in characteristic $p$) in characteristic $0$, it remains to check the blowup formula for $R\Gamma(-,L\Omega^{<j}_{-/\mathbb{Q}})$ (resp. $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$). In both cases that reduces to the blowup formula for $R\Gamma(-,\mathbb{L}^i_{-/\mathbb{F}})$ for all $i\ge 0$ (here we use Lemma [Lemma 39](#lem_fin_fil_on_syn){reference-type="ref" reference="lem_fin_fil_on_syn"} in characteristic $p$, which can be proved directly; see [@BhattLurie2022 Lem. 9.4.3]). Knowing the blowup formula, part (1) for arbitrary $j\ge0$ follows from the special case $r=1$, i.e., Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"}. This follows from an argument in [@AnnalaIwasa2023 Lem. 3.3.5]. More precisely, the argument in the cited lemma shows that as soon as $\mathbb{Z}(j)^{\mathrm{mot}}$ converts the blowup square $$\begin{tikzcd} \mathbb{P}^{r-1}_X\ar{d}\ar{r} & \mathrm{Bl}_{\{0\}}(\mathbb{A}^r_X)\ar{d}\\ X\ar{r}{0} & \mathbb{A}_X^r \end{tikzcd} \qquad r \geq 1,$$ to a cartesian square, then the map [\[eq:pbf-r\]](#eq:pbf-r){reference-type="eqref" reference="eq:pbf-r"} is an equivalence. The above blowup square is a special case of the blowup formula in part (3). Part (2) then follows from part (1) by Zariski descent. ◻ **Remark 87**. We note that if $Y \rightarrow X$ is not a regular closed immersion, then the square of Theorem [Theorem 86](#thm:pbf-blowup){reference-type="ref" reference="thm:pbf-blowup"}(3) need not be cartesian. What is more generally true is the pro cdh descent result of Theorem [Theorem 114](#theorem_pro_cdh_descent){reference-type="ref" reference="theorem_pro_cdh_descent"}. Theorem [Theorem 86](#thm:pbf-blowup){reference-type="ref" reference="thm:pbf-blowup"}(3) does admit two other enhancements which we will leave for the reader to supply details: 1. if $Y \rightarrow X$ is the section of a *smooth morphism* $f: Y \rightarrow X$ of relative dimension $r$, then the blowup formula "splits\" to give an equivalence $$\mathbb{Z}(j)^{\mathrm{mot}}(\mathrm{Bl}_Y(X)) \simeq \mathbb{Z}(j)^{\mathrm{mot}}(X) \oplus \bigoplus_{0 < i < r} \mathbb{Z}(j-i)^{\mathrm{mot}}(Y)[-2i];$$ this follows by combining parts (2) and (3) of Theorem [Theorem 86](#thm:pbf-blowup){reference-type="ref" reference="thm:pbf-blowup"}. 2. The extension of $\mathbb{Z}(j)^{\mathrm{mot}}$ to derived schemes, as in §[4.5](#sec:derived){reference-type="ref" reference="sec:derived"}, converts a derived blowup square as in [@KhanRydh2019] to a cartesian square. Indeed, the $\mathrm{cdh}$ parts of the theory does not depend on derived structure and thus converts these derived blowup squares to cartesian squares. On the other hand, we reduce the assertions for $\mathbb{Z}_p(j)^{\mathrm{syn}}$ and filtered derived de Rham cohomology to the case of the cotangent complex. Since derived blowup squares are "pulled back\" from blowups along regular immersions and therefore, the cotangent complex enjoys the same property. # Comparison to $\mathbb{A}^1$-invariant motivic cohomology {#section_smooth} We begin by repeating Construction [Construction 59](#cons_mot_to_cdh){reference-type="ref" reference="cons_mot_to_cdh"} for the sake of clarity: on the category of qcqs $\mathbb{F}$-schemes, there are natural comparison maps of $\mathrm D(\mathbb{Z})$-valued presheaves $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}\label{equation_mot_to_cdh}$$ for $j\ge0$, arising as the shifted graded pieces of a comparison map of filtered presheaves of spectra $\mathrm{Fil}_{\mbox{\rm \scriptsize mot}}^\star \mathrm{K}\to \mathrm{Fil}_{\mbox{\rm \scriptsize cdh}}^\star \mathrm{KH}$. These comparison maps are tautological from the definition of our motivic cohomology, which should be seen a modification of the cdh-local theory. Although it is not strictly necessary for what follows, the reader should also recall from Remark [Remark 21](#remark_slice_filtration){reference-type="ref" reference="remark_slice_filtration"} that, for $X$ any qcqs $\mathbb{F}$-scheme, the cdh-local motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$ is in fact the $\mathbb{A}^1$-invariant motivic cohomology of $X$ which arises from motivic homotopy theory, and $\mathrm{Fil}_{\mbox{\rm \scriptsize cdh}}^\star \mathrm{KH}(X)$ identifies with the slice filtration. The goal of this section is to prove the following comparison equivalences related to the maps ([\[equation_mot_to_cdh\]](#equation_mot_to_cdh){reference-type="ref" reference="equation_mot_to_cdh"}): **Theorem 88**. *Let $\mathbb{F}$ be a prime field and $j\ge0$.* 1. *The maps ([\[equation_mot_to_cdh\]](#equation_mot_to_cdh){reference-type="ref" reference="equation_mot_to_cdh"}) induce equivalences of $\mathrm D(\mathbb{Z})$-valued presheaves on $\mathrm{Sch}^{\mbox{\rm \scriptsize qcqs}}_\mathbb{F}$ $$L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}\qquad\text{and}\qquad L_{\mathbb{A}^1}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}.$$* 2. *For any regular Noetherian $\mathbb{F}$-scheme $X$, the map ([\[equation_mot_to_cdh\]](#equation_mot_to_cdh){reference-type="ref" reference="equation_mot_to_cdh"}) induces an equivalence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X).$$ (Equivalently, using part (1), the maps $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\to \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(\mathbb{A}_X^m)$ are equivalences for all $m\ge0$.)* The maps in part (1) of the theorem are induced by ([\[equation_mot_to_cdh\]](#equation_mot_to_cdh){reference-type="ref" reference="equation_mot_to_cdh"}), recalling that the presheaf $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}:\mathrm{Sch}^{\mbox{\rm \scriptsize qcqs,op}}_\mathbb{F}\to\mathrm D(\mathbb{Z})$ is both a cdh sheaf and $\mathbb{A}^1$-invariant by Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(2)&(5). In case of confusion, here $L_{\mathbb{A}^1}$ denotes the endofunctor of presheaves of complexes (or of spectra) on $\mathrm{S}\mathrm{ch}{}_{\mathbb{F}}^{\mathrm{qcqs}}$ reflecting onto $\mathbb{A}^1$-invariant presheaves; we will recall the explicit formula for $L_{\mathbb{A}^1}$ in the proof of Lemma [Lemma 93](#lem:a10){reference-type="ref" reference="lem:a10"} below. **Remark 89**. Informally, Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(1) says that our motivic cohomology may be viewed as a "de-cdh-sheafification" or "de-$\mathbb{A}^1$-localisation" of $\mathbb{A}^1$-invariant motivic cohomology. More precisely, it states that on equicharacteristic schemes the comparison equivalences $$L_{\mbox{\rm \scriptsize cdh}}\mathrm{K}\stackrel{\sim}{\to}\mathrm{KH}\qquad\text{and}\qquad L_{\mathbb{A}^1}\mathrm{K}\stackrel{\sim}{\to}\mathrm{KH}$$ (the first being part of Theorem [Theorem 23](#thm:mainsq){reference-type="ref" reference="thm:mainsq"}, the second being the definition of $\mathrm{KH}$) upgrade to filtered equivalences, where we equip $\mathrm{KH}$ with the filtration $\mathrm{Fil}_{\mbox{\rm \scriptsize cdh}}^\star$ (or equivalently with the slice filtration), and we equip the left sides with $L_{\mbox{\rm \scriptsize cdh}}$, resp. $L_{\mathbb{A}^1}$, of our motivic filtration $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize mot}}$. That is, the slice filtration on $\mathrm{KH}$-theory can be recovered by cdh sheafifying or $\mathbb{A}^1$-localising our motivic filtration on $\mathrm{K}$-theory. **Remark 90**. Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(2) is a motivic upgrade of the equivalence $\mathrm{K}(X)\stackrel{\sim}{\to}\mathrm{KH}(X)$ for regular Noetherian $\mathbb{F}$-schemes $X$. Indeed, combined with this equivalence, it states that the map of filtered spectra $\mathrm{Fil}_{\mbox{\rm \scriptsize mot}}^\star \mathrm{K}(X)\to \mathrm{Fil}_{\mbox{\rm \scriptsize cdh}}^\star \mathrm{KH}(X)$ is an equivalence. Combined with a comparison isomorphism from the joint project with Bachmann, we obtain the following, starting that our motivic cohomology coincides with the classical theory on smooth $\mathbb{F}$-varieties:[^14] **Corollary 91**. *For any smooth $\mathbb{F}$-scheme $X$ the comparison map ([\[eqn_cla_to_mot\]](#eqn_cla_to_mot){reference-type="ref" reference="eqn_cla_to_mot"}) is an equivalence of filtered spectra, or equivalently the maps $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X)\to \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)$ are equivalences for all $j\ge0$.* *Proof.* As explained in Remark [Remark 60](#remark_split){reference-type="ref" reference="remark_split"}, we show with Bachmann that the composition $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}(X)\to \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\to\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$ is an equivalence. Now apply Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(2) to see that the second map is an equivalence, therefore also the first. ◻ **Corollary 92**. *For any regular Noetherian $\mathbb{F}_p$-scheme $X$ and $j\ge0$, the canonical maps $$\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)\longrightarrow L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)\qquad\text{and}\qquad R\Gamma_{\mbox{\rm \scriptsize \'et}}(X,\Omega^j_{\mbox{\rm \scriptsize log}})\longrightarrow R\Gamma_{\mbox{\rm \scriptsize \'eh}}(X,\Omega^j_{\mbox{\rm \scriptsize log}})$$ are equivalences.* *Proof.* The first follows from Theorem [Theorem 50](#thm:p-ahss){reference-type="ref" reference="thm:p-ahss"}(2) and the cartesian square Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}. The second equivalence follows by taking the first equivalence modulo $p$. ◻ The core of the proof of Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(1) is the fact that derived de Rham and syntomic cohomology are very far from being homotopy invariant [@Elmanto2021; @GellerWeibel1989]: **Lemma 93**. *In the category of $\rm D(\mathbb{Z})$-valued presheaves on $\mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_{\mathbb{F}}$, the following hold for all $j\ge0$:* 1. *$L_{\mathbb{A}^1}R\Gamma(-,L^j_{-/\mathbb{F}}) \simeq 0$;* 2. *if $\mathbb{F}= \mathbb{Q}$ then the map $L_{\mathbb{A}^1}R\Gamma(-,\widehat{L\Omega}^{\geq j}_{-/\mathbb{Q}})\rightarrow L_{\mathbb{A}^1}R\Gamma_{\mathrm{cdh}}(-,\widehat{L\Omega}^{\geq j}_{-/\mathbb{Q}})$ is an equivalence.* 3. *if $\mathbb{F}= \mathbb{F}_p$, then $L_{\mathbb{A}^1}\mathbb{Z}_p(j)^{\mathrm{syn}} \simeq 0$ and $L_{\mathbb{A}^1}L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}} \simeq 0$.* *Proof.* We will use the following explicit formula for the endofunctor $L_{\mathbb{A}^1}$ of presheaves on $\textrm{Sch}^{\mbox{\rm \scriptsize qcqs}}_{\mathbb{F}}$: given a presheaf $\mathcal F$ then $$\label{eq:la1} L_{\mathbb{A}^1}\mathcal F (X) = \mathop{\mathrm{colim}}_{\Delta^{\mathrm{op}}} \mathcal F(X \times \Delta^{\bullet})$$ where $\Delta^{\bullet}$ is the cosimplicial $\mathbb{F}$-scheme built from algebraic $m$-simplices: $$\Delta^m = \mathrm{Spec}(\mathbb{F}[T_0, \cdots, T_m]/(\sum^m_{i=0} T_i = 1).$$ Note that $L_{\mathbb{A}^1}$ preserves Nisnevich and cdh sheaves; indeed, this follows from the description of Nisnevich and cdh descent in terms of cd structures and [\[eq:la1\]](#eq:la1){reference-type="eqref" reference="eq:la1"}. (1): Let $A$ be an $\mathbb{F}$-algebra. By the Künneth formula for the cotangent complex, there is a natural equivalence $$L^j_{A[\Delta^m]/\mathbb{F}} \simeq \bigoplus_{a+b = j} L^a_{A/\mathbb{F}} \otimes \Omega^b_{\mathbb{F}[\Delta^m]/\mathbb{F}}.$$ Therefore $(L_{\mathbb{A}^1}L^j_{-/\mathbb{F}})(A) \simeq \bigoplus_{a+b = j} L^a_{A/\mathbb{F}} \otimes (L_{\mathbb{A}^1}\Omega^b_{-/\mathbb{F}})(\mathbb{F})$, which reduces the problem to showing that $(L_{\mathbb{A}^1}\Omega^b_{-/\mathbb{F}})(\mathbb{F}) \simeq 0$ for all $b \geq 0$. The latter vanishing is due to Geller--Weibel [@GellerWeibel1989]. (2): Since $L_{\mathbb{A}^1}$ preserves $\mathrm{cdh}$ sheaves, it suffices to prove that $L_{\mathbb{A}^1}R\Gamma(-,\widehat{L\Omega}^{\geq j}_{-/\mathbb{Q}})$ is a cdh sheaf. Since $L_{\mathbb{A}^1}$ preserves fibre sequences, we have a fibre sequence for all $j \geq 0$ $$L_{\mathbb{A}^1}R\Gamma(-,L\Omega^{< j}_{-/\mathbb{Q}})[-1] \rightarrow L_{\mathbb{A}^1}R\Gamma(-,\widehat{L\Omega}^{\geq j}_{-/\mathbb{Q}}) \rightarrow L_{\mathbb{A}^1}R\Gamma(-,\widehat{L\Omega}_{-/\mathbb{Q}}).$$ The presheaf $R\Gamma(-,L\Omega^{< j}_{-/\mathbb{Q}})$ is killed by $L_{\mathbb{A}^1}$, thanks to part (1) and induction on $j$. On the other hand, by Lemma [Lemma 28](#lemma_cdh_descent_HP){reference-type="ref" reference="lemma_cdh_descent_HP"}, the last term is a $\mathrm{cdh}$ sheaf since $L_{\mathbb{A}^1}$ preserves $\mathrm{cdh}$ sheaves. In particular, the middle term is a $\mathrm{cdh}$ sheaf, completing the proof. (3): It suffices to prove the vanishings on any affine $\mathbb{F}_p$-scheme $\mathrm{Spec}A$. Firstly, as observed in [@Elmanto2021 Lem. 3.0.3], the complex $L_{\mathbb{A}^1}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A)$ is $p$-complete since we have the universal bound that $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A[\Delta^m])$ is supported in degrees $\le j+1$ for any $m$; so it is enough to prove the vanishing of $L_{\mathbb{A}^1}\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}(A)$, which in turn follows from part (1) and Lemma [Lemma 39](#lem_fin_fil_on_syn){reference-type="ref" reference="lem_fin_fil_on_syn"}. Next we use the multiplicative morphism of presheaves of $\mathbb{E}_\infty$-rings $\bigoplus_{j \geq 0} \mathbb{Z}_p(j)^{\mathrm{syn}}[2j] \to \bigoplus_{j \geq 0} L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}[2j]$; since $L_{\mathbb{A}^1}$ preserves multiplicative structures, we obtain a morphism of $\mathbb{E}_\infty$-rings $\bigoplus_{j \geq 0} \mathbb{Z}_p(j)^{\mathrm{syn}}(A)[2j] \rightarrow \bigoplus_{j \geq 0} L_{\mathrm{cdh}}\mathbb{Z}_p(j)^{\mathrm{syn}}(A)[2j]$. But the domain is $0$ by the first part, so the target is also $0$ as it is receiving a multiplicative map from the zero ring. ◻ *Proof of Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(1).* Cdh sheafifying the pullback squares of Theorems [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(2) or [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(2) shows that $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$. Similarly, $\mathbb{A}^1$-localising the pullback squares and using the previous lemma yields $L_{\mathbb{A}^1}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$. ◻ The remainder of the section is devoted to the proof of Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(2). The key inputs are the $\mathbb{P}^1$-bundle formula for motivic cohomology (Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"}), the already proved Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(1), and an argument of Gabber used to prove Gersten injectivity statements [@Gabber1994; @GrosSuwa1988]. Gabber's argument has been axiomatized by Colliot-Thélène--Hoobler--Kahn [@ColliotThelene-Hoobler-Kahn1997], and we now review their formalism in a more modern language. Let $k$ be any field and suppose that we have a presheaf $\mathcal F: {\mathrm{S}\mathrm{m}}^{\mathrm{op}}_{k} \rightarrow \text{Sp}$; for $X \in {\mathrm{S}\mathrm{m}}_{k}$, we will write $\mathcal F^X$ for the presheaf $U \mapsto \mathcal F(U \times_k X)$. There are two morphisms of presheaves $$j^*, \pi^*\infty^*: \mathcal F^{\mathbb{P}^1} \rightarrow \mathcal F^{\mathbb{A}^1},$$ where: 1. $\pi$ is induced the projection $\mathbb{A}^1 \times_k X \rightarrow X$, 2. $\infty$ is the closed immersion $\mathrm{Spec}k\to\mathbb{P}^1$ of the point at $\infty$, 3. $j: \mathbb{A}^1 \hookrightarrow \mathbb{P}^1$ is the open immersion complementary to the point at $\infty$. In general, there is no reason for the maps $j^*$ and $\pi^*\infty^*$ to be homotopic. This leads to the next definition: **Definition 94**. We say that a presheaf $\mathcal F: {\mathrm{S}\mathrm{m}}^{\mathrm{op}}_{k} \rightarrow \text{Sp}$ is a *deflatable*[^15] if the maps $j^*$ and $\pi^*\infty^*$ are homotopic. **Remark 95**. More precisely, to ask that two morphisms of presheaves are homotopic means that they are identified in the homotopy category of presheaves. This means that there is an $2$-morphism, functorial in smooth $k$-schemes, between these two morphisms of presheaves. We do not keep track of this $2$-morphism, but we note that functoriality in smooth schemes is a substantial amount of extra compatibilities. In fact, calling these $2$-morphisms *deflations*, the space of deflations can be parametrized as following: it is the space of sections $s: \mathcal F^{\mathbb{P}^1} \rightarrow \mathcal E$ of the canonical map $\mathcal E \rightarrow \mathcal F^{\mathbb{P}^1}$, were $\mathcal E$ is the equaliser of the two maps $j^*, \pi^*\infty^*: \mathcal F^{\mathbb{P}^1} \rightrightarrows \mathcal F^{\mathbb{A}^1}$. **Remark 96**. Definition [Definition 94](#def:good){reference-type="ref" reference="def:good"} implies the validity of axiom "SUB 2" of [@ColliotThelene-Hoobler-Kahn1997], which is much weaker than deflatability and instead asks only for scheme-wise homotopy commutativity of a relative variant of this axiom. **Example 97**. If a presheaf $\mathcal F:{\mathrm{S}\mathrm{m}}^{\mathrm{op}}_{k} \rightarrow \text{Sp}$ is $\mathbb{A}^1$-invariant, then it is deflatable. Indeed, the map $\pi^*$ is an equivalence and there is a natural $\mathbb{A}^1$-homotopy between $j^*$ and $\infty^*$. The following lemma is a variant of one of the main results of [@ColliotThelene-Hoobler-Kahn1997], stated in a convenient language for our use. It proves Gersten injectivity for good cohomology theories satisfying Nisnevich descent. We denote by $\mathrm{Reg}_{k}$ the category of regular Noetherian $k$-schemes. **Lemma 98** ([@ColliotThelene-Hoobler-Kahn1997]). *Let $k$ be a perfect field and $\mathcal F: \mathrm{Reg}^{\mathrm{op}}_k \rightarrow \text{Sp}$ be a finitary, Nisnevich sheaf such that $\mathcal F|_{{\mathrm{S}\mathrm{m}}^{\mathrm{op}}_k}$ is deflatable. Then for any $j \in \mathbb{Z}$ and any regular local $k$-algebra $R$ with fraction field $F$, the canonical map $$\pi_j(\mathcal F(R)) \longrightarrow\pi_j(\mathcal F(F))$$ is injective.* *Proof.* For a regular $k$-scheme $X$ and closed immersion $Z\hookrightarrow X$ we will write $$\mathcal F_Z(X):= \mathrm{fibre}(\mathcal F(X) \rightarrow \mathcal F(X \setminus Z)),$$ so that we have a long exact sequence functorial in $X$ and $Z$ $$\cdots \rightarrow \pi_j(\mathcal F(X)) \rightarrow \pi_j(\mathcal F(X \setminus Z)) \rightarrow \pi_{j-1}(\mathcal F_Z(X)) \rightarrow \cdots.$$ By Néron--Popescu it suffices to prove the result when $R=\mathcal{O}_{X,x}$ is the local ring of a closed point $x \in X$ where $X$ is a smooth affine $k$-scheme. Let $s \in \ker(\pi_j(\mathcal F(R))\to\pi_j(\mathcal F(F))$; by possibly shrinking $X$, we may assume that $s$ is defined on $X$ and vanishes away from some closed subscheme $Z \hookrightarrow X$ of positive codimension, i.e., $s$ lifts to an element $\widetilde{s} \in \pi_j(\mathcal F_Z(X))$. To prove the result, it suffices to produce an open neighborhood $U\subseteq X$ of $x$ and a closed subscheme $Z' \hookrightarrow U$ with $Z \cap U \subset Z'$ such that $\widetilde{s}$ vanishes on $\pi_i(\mathcal F_{Z'}(U))$. Gabber's presentation lemma [@ColliotThelene-Hoobler-Kahn1997 Theorem 3.1.1] (see [@gabber-finite] for the case in which $k$ is a finite field) furnishes an open neighborhood $U \subseteq X$ of $x$, a smooth affine $k$-scheme $V$, a morphism $\phi = (\psi, v): U \rightarrow V \times \mathbb{A}^1$ such that $\psi|_{Z \cap U}$ is finite, and a Nisnevich square $$\begin{tikzcd} U \setminus (Z \cap U) \ar{r} \ar{d} & U \ar{d}{\phi}\\ \mathbb{A}^1_V \setminus (\phi(Z \cap U)) \ar{r} & \mathbb{A}^1_V.\\ \end{tikzcd}$$ In particular $\phi(Z \cap U) \hookrightarrow \mathbb{A}^1_V$ is a closed immersion. By Nisnevich excision, we have that $\pi_i(\mathcal F_{Z \cap U}(U)) \cong \pi_i(\mathcal F_{\mathbb{A}^1_V \cap \phi(Z \cap U)}(\mathbb{A}^1_V))$. Now set $F:= \psi(Z \cap U)$ so that $Z \cap U \subset \psi^{-1}(F)=:Z'$. So we have a commutative diagram $$\begin{tikzcd} \pi_j(\mathcal F_{Z\cap U}(U)) \ar{r} & \pi_j(\mathcal F_{Z'}(U)) \\ \pi_j(\mathcal F_{\phi(Z\cap U)}(\mathbb{A}^1_V) ) \ar{r} \ar{u}{\cong}& \pi_j(\mathcal F_{\mathbb{A}^1_F}(\mathbb{A}^1_V))\ar{u}\\ \end{tikzcd}$$ and, to finish the proof, we need only show that the bottom map is zero. The map of interest is the top horizontal map of the following commutative diagram $$\begin{tikzcd} \pi_j(\mathcal F_{\phi(Z\cap U)}(\mathbb{A}^1_V)) \ar{r} & \pi_j(\mathcal F_{\mathbb{A}^1_F}(\mathbb{A}^1_V)) & \\ & & \pi_j(\mathcal F_{F}(V)) \ar{ul}\\ \pi_j(\mathcal F_{\phi(Z\cap U)}(\mathbb{P}^1_V)) \ar{r} \ar{uu}{\simeq} & \pi_j(\mathcal F_{\mathbb{P}^1_F}(\mathbb{P}^1_V))\ar{uu} \ar{ur}&\\ \end{tikzcd}$$ where the triangle commutes exactly because $\mathcal F$ is a deflateable. However, the bottom composite is zero, since $\phi(Z \cap U)$ does not meet the $\infty$-section of $\mathbb{P}^1_V$, and thus the top map is also zero as desired. ◻ The projective bundle formula implies that our motivic cohomology is deflatable: **Lemma 99**. *For any $j\ge0$, the presheaf $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}|_{{\mathrm{S}\mathrm{m}}_{\mathbb{F}}}:{\mathrm{S}\mathrm{m}}^{\mathrm{op}}_\mathbb{F}\longrightarrow\rm D(\mathbb{Z})$$ is deflatable.* *Proof.* This is actually a standard consequence of the $\mathbb{P}^1$-bundle formula. Theorem [Theorem 65](#thm:pbf){reference-type="ref" reference="thm:pbf"} furnishes us with an equivalence $$\mathbb{Z}(j)^{\mathrm{mot}} \oplus \mathbb{Z}(j-1)^{\mathrm{mot}}[-2] \xrightarrow{\pi^* \oplus c_1(\mathcal O(1))\pi^* } (\mathbb{Z}(j)^{\mathrm{mot}})^{\mathbb{P}^1},$$ whence it suffices to explain why the diagram $$\begin{tikzcd} \mathbb{Z}(j)^{\mathrm{mot}} \oplus \mathbb{Z}(j-1)^{\mathrm{mot}}[-2] \ar{rr} \ar{dr} & & (\mathbb{Z}(j)^{\mathrm{mot}})^{\mathbb{A}^1}\\ & \mathbb{Z}(j)^{\mathrm{mot}} \ar{ur} & \end{tikzcd}$$ commutes in the homotopy category of presheaves. On the $\mathbb{Z}(j)^{\mathrm{mot}}$ component, the diagram commutes already at the level of schemes. On the $\mathbb{Z}(j-1)^{\mathrm{mot}}[-2]$ component, the diagram commutes because on $\mathbb{A}^1_\mathbb{F}$ there are natural identifications $\pi^*\infty^*\mathcal O(1) \cong \mathcal O \cong j^*\mathcal O(1)$. ◻ **Remark 100**. More generally, any presheaf of spectra satisfying the $\mathbb{P}^1$-bundle formula as formulated in [@AnnalaIwasa2023] is deflatable. Hence the conclusion of Lemma [Lemma 98](#lem:ColliotThelene-Hoobler-Kahn1997){reference-type="ref" reference="lem:ColliotThelene-Hoobler-Kahn1997"} holds for these theories. We now have all the necessary ingredients to prove Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(2): *Proof of Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(2).* The goal is to prove, for any regular Noetherian $\mathbb{F}$-scheme $X$, that the map $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\to \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$ is an equivalence. Since this factors as $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\to L_{\mathbb{A}^1}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X),\label{eqn_mot_vs_cdh_proof}$$ where the equivalence is Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(1), it is equivalent to show that $\mathcal N(X)\simeq0$ where $\mathcal N:=\mathrm{fib}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\to L_{\mathbb{A}^1}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}})$. Furthermore, since $\mathcal N$ is finitary and satisfies Zariski descent, it is enough to show that $\mathcal N(R)\simeq 0$ for every regular, Noetherian, local $\mathbb{F}$-algebra $R$. But we will show in the next paragraph that $\mathcal N|_{{\mathrm{S}\mathrm{m}}_{\mathbb{F}}}$ is deflatable, whence Lemma [Lemma 98](#lem:ColliotThelene-Hoobler-Kahn1997){reference-type="ref" reference="lem:ColliotThelene-Hoobler-Kahn1997"} implies that $H^n(\mathcal N(R))\to H^n(\mathcal N(F))$ is injective for all $n$, where $F$ is the fraction field of $R$. This therefore reduces the problem to showing that $\mathcal N(F)\simeq0$ for every field extension $F$ of $\mathbb{F}$; appealing again to ([\[eqn_mot_vs_cdh_proof\]](#eqn_mot_vs_cdh_proof){reference-type="ref" reference="eqn_mot_vs_cdh_proof"}), this time for $X=\mathrm{Spec}F$, it is equivalent to show that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(F)\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(F)$. But this follows from the part of Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(1) stating that $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$, as fields are local for the cdh topology. It remains to prove that $\mathcal N|_{{\mathrm{S}\mathrm{m}}_{\mathbb{F}}}$ is deflatable. Firstly, we know from Lemma [Lemma 99](#lem_mot_deflat){reference-type="ref" reference="lem_mot_deflat"} that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}|_{{\mathrm{S}\mathrm{m}}_{\mathbb{F}}}$ is deflatable. Fixing any choice of deflation for it, this deflation induces (using the explicit formula ([\[eq:la1\]](#eq:la1){reference-type="ref" reference="eq:la1"})) a deflation for $L_{\mathbb{A}^1}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}|_{{\mathrm{S}\mathrm{m}}_{\mathbb{F}}}$ which is compatible with the canonical map $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}|_{{\mathrm{S}\mathrm{m}}_{\mathbb{F}}}\to L_{\mathbb{A}^1}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}|_{{\mathrm{S}\mathrm{m}}_{\mathbb{F}}}$. Passing to the fibre induces a deflation for $\mathcal N|_{{\mathrm{S}\mathrm{m}}_{\mathbb{F}}}$, as desired. ◻ # Comparison to lisse motivic cohomology {#section_lke} We continue to fix a prime field $\mathbb{F}$. The goal of this section is to study the comparison map from lisse motivic cohomology to our new motivic cohomology, as discussed in Construction [Construction 58](#cons_lke_to_mot){reference-type="ref" reference="cons_lke_to_mot"}. However, we may now adopt a cleaner point of view on this comparison map. Indeed, we now know from Corollary [Corollary 91](#corol_smooth_comparison){reference-type="ref" reference="corol_smooth_comparison"} that the restriction of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ to smooth $\mathbb{F}$-algebras coincides with classical motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}}$. Therefore we will henceforth view $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}$ as the left Kan extension of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$, restricted to smooth $\mathbb{F}_p$-algebras, back along the inclusion $\text{CAlg}_\mathbb{F}^{\mbox{\rm \scriptsize sm}}\subseteq \text{CAlg}_\mathbb{F}$. For any $\mathbb{F}_p$-algebra, this formally induces the same comparison map $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)\label{eqn:lke_vs_mot}$$ as Construction [Construction 58](#cons_lke_to_mot){reference-type="ref" reference="cons_lke_to_mot"}. This map is certainly not an equivalence in general: the left side is supported in cohomological degree $\le 2j$ (see Proposition [Proposition 18](#prop:mot-filt){reference-type="ref" reference="prop:mot-filt"}) but this bound cannot always be true for the right side: otherwise the Atiyah--Hirzebruch spectral sequence would then imply that $K(A)$ were always connective. In general it seems to be a deep question to what extent the right side of ([\[eqn:lke_vs_mot\]](#eqn:lke_vs_mot){reference-type="ref" reference="eqn:lke_vs_mot"}) of is controlled by the left side. In other words, how much of motivic cohomology can be recovered from that of smooth algebras? In this section we provide some partial answers to this question. In particular we will show that, for $A$ local, ([\[eqn:lke_vs_mot\]](#eqn:lke_vs_mot){reference-type="ref" reference="eqn:lke_vs_mot"}) induces an equivalence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)\stackrel{\sim}{\to}\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A).$$ Note that, in light of Remark [Remark 19](#remark_lke_as_cycles){reference-type="ref" reference="remark_lke_as_cycles"}, this provides a description of $\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)$ purely in terms of algebraic cycles. We will return to the link between motivic cohomology and algebraic cycles in Section [9](#section_cf_cycles){reference-type="ref" reference="section_cf_cycles"}. In this section we also establish some vanishing theorems and prove a Nesterenko--Suslin isomorphism. ## Behaviour of motivic cohomology in degrees $\le 2j$ We begin with the following rational statement, writing $$\mathbb{Q}(j)^{\mbox{\rm \scriptsize lse}}(A):=\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}\otimes_{\mathbb{Z}}\mathbb{Q}, \qquad \mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(A):=\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)\otimes_{\mathbb{Z}}\mathbb{Q}$$ for the rationalisations of our motivic cohomologies. (Note that $\mathbb{Q}(j)^{\mbox{\rm \scriptsize lse}}$ is the left Kan extension of the restriction of $\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}$ to smooth $\mathbb{F}$-algebras, since left Kan extension commutes with filtered colimits of functors.) **Lemma 101**. *For any $\mathbb{F}$-algebra $A$ and $j\ge0$, the map ([\[eqn:lke_vs_mot\]](#eqn:lke_vs_mot){reference-type="ref" reference="eqn:lke_vs_mot"}) induces an equivalence $$\mathbb{Q}(j)^{\mbox{\rm \scriptsize lse}}(A)\stackrel{\sim}{\to}\tau^{\le 2j}\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(A).$$ In other words, the functor $\tau^{\le 2j}\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}:\mathrm{CAlg}_\mathbb{F}\to\text \rm D(\mathbb{Z})$ is left Kan extended from smooth $\mathbb{F}$-algebras.* *Proof.* Rationally, by Theorem [Theorem 35](#theorem_AH_SS_0){reference-type="ref" reference="theorem_AH_SS_0"}(2) (characteristic zero) and Theorem [Theorem 50](#thm:p-ahss){reference-type="ref" reference="thm:p-ahss"}(2) (characteristic $p > 0$), there is a natural isomorphism of filtered spectra $\mathrm{Fil}_{\mbox{\rm \scriptsize mot}}^\star \mathrm{K}(A)_\mathbb{Q}\cong \bigoplus_{j\ge\star}\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(A)[2j]$ for any $\mathbb{F}$-algebra $A$ of finite valuative dimension. Restricting to smooth $\mathbb{F}$-algebras and left Kan extending back identifies the map $$\bigoplus_{j\ge 0}\mathbb{Q}(j)^{\mbox{\rm \scriptsize lse}}(A)[2j]\longrightarrow\bigoplus_{j\ge 0}\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(A)[2j]$$ (obtained by rationalising the direct sum of ([\[eqn:lke_vs_mot\]](#eqn:lke_vs_mot){reference-type="ref" reference="eqn:lke_vs_mot"}) over all weights), for any $\mathbb{F}$-algebra $A$ of finite valuative dimension, with the canonical map $\mathrm{K}^{\mbox{\rm \scriptsize cn}}(A)_\mathbb{Q}\to \mathrm{K}(A)_\mathbb{Q}$. Since the latter map is the connective cover, we deduce the same for the former map, i.e., $\mathbb{Q}(j)^{\mbox{\rm \scriptsize lse}}(A)\stackrel{\sim}{\to}\tau^{\le 2j}\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(A)$. This proves the result for $\mathbb{F}$-algebras of finite valuative dimension; the general case follows by passing to a filtered colimit. ◻ **Corollary 102**. *For any local $\mathbb{F}$-algebra $A$ and $j\ge0$, the map ([\[eqn:lke_vs_mot\]](#eqn:lke_vs_mot){reference-type="ref" reference="eqn:lke_vs_mot"}) induces an equivalence $$\mathbb{Q}(j)^{\mbox{\rm \scriptsize lse}}(A)\stackrel{\sim}{\to}\tau^{\le j}\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(A).$$ In other words, the functor $\tau^{\le j}\mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}:\mathrm{CAlg}_\mathbb{F}^{\mbox{\rm \scriptsize loc}}\to \rm D(\mathbb{Z})$ is left Kan extended from essentially smooth, local $\mathbb{F}$-algebras.* *Proof.* This follows from the previous lemma since, for $A$ local, the lisse motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)$ is supported in cohomological degrees $\le j$. ◻ The proof of the previous corollary also implies the following rational vanishing result: **Corollary 103**. *For any local $\mathbb{F}$-algebra $A$ and $0\le j<i\le 2j$, we have $H^i_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Q}(j))=0$.* **Remark 104** (Rational Drinfeld vanishing). If $A$ is a Henselian local $\mathbb{F}$-algebra, then we can improve the vanishing bound of the previous corollary by $1$; namely we also have $H^{2j+1}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Q}(j))=0$ for all $j\ge0$. Indeed, this follows from the theorem of Drinfeld that $\mathrm{K}_{-1}(A)=0$ and the decomposition $\mathrm{K}(A)_\mathbb{Q}\simeq\bigoplus_{j\ge0} \mathbb{Q}(j)^{\mbox{\rm \scriptsize mot}}(A)[2j]$ (when $A$ has finite valuative dimension, which we may assume by taking a filtered colimit). We now prove an integral version of Corollary [Corollary 102](#corollary_lej_rational){reference-type="ref" reference="corollary_lej_rational"}. By taking $H^1$ of the map [\[eq:c1-mot\]](#eq:c1-mot){reference-type="eqref" reference="eq:c1-mot"}, we get a natural map $$A^\times\longrightarrow H^1_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(1))$$ for any $\mathbb{F}$-algebra $A$; by multiplicativity this induces *symbol maps* $$(A^\times)^{\otimes j}=A^\times\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} A^\times \longrightarrow H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j)\label{eqn_symbol}$$ for all $j\ge1$. **Lemma 105**. *For any local $\mathbb{F}$-algebra $A$ and $j\ge1$, the map ([\[eqn_symbol\]](#eqn_symbol){reference-type="ref" reference="eqn_symbol"}) factors through the Milnor $K$-group $\mathrm{K}_j^M(A)$* *Proof.* We must show that the map respects the Steinberg relation, so may assume $j=2$. Now let $a\in A^\times$ be a unit such that $1-a$ is also a unit; let $\mathbb{F}[t]\to A$, $t\mapsto a$ be the induced map, and $\frak p\subseteq \mathbb{F}[t]$ the pullback of the maximal ideal of $A$. There is a commutative diagram by naturality $$\xymatrix{ \mathbb{F}[t]_\frak p^\times\otimes_{\mathbb{Z}}\mathbb{F}[t]_\frak p^\times\ar[d]\ar[r] & H^2_{\mbox{\rm \scriptsize mot}}(\mathbb{F}[t]_\frak p,\mathbb{Z}(2))\ar[d]\\ A^\times\otimes_{\mathbb{Z}}A^\times\ar[r] & H^2_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(2)) }$$ in which the left vertical arrow sends $t\otimes 1-t$ to $a\otimes 1-a$. The problem therefore reduces to the case of the local ring $\mathbb{F}[t]_\frak p$. But, setting $F:=\text{Frac}(\mathbb{F}[t])$, we have a second commutative diagram by naturality $$\xymatrix{ F^\times\otimes_{\mathbb{Z}}F^\times\ar[r] & H^2_{\mbox{\rm \scriptsize mot}}(F,\mathbb{Z}(2))\\ \mathbb{F}[t]_\frak p^\times\otimes_{\mathbb{Z}}\mathbb{F}[t]_\frak p^\times\ar[u]\ar[r] & H^2_{\mbox{\rm \scriptsize mot}}(\mathbb{F}[t]_\frak p,\mathbb{Z}(2))\ar[u] }$$ in which the right vertical arrow is injective by Gersten injectivity in motivic cohomology (here we are implicitly using that the new motivic cohomology of $\mathbb{F}[t]_\frak p]$ coincides with the classical theory, i.e., ). So the problem finally reduces to the case of the field $F$, in which case it is a theorem of Nesterenko--Suslin and Totaro [@Suslin1989; @Totaro1992] that the symbol map indeed respects the Steinberg relation. ◻ We will repeatedly use the following general observation about functors in what follows. Recall that a functor $F: \mathrm{CAlg}^{{\mbox{\rm \scriptsize loc}}}_{\mathbb{F}} \rightarrow \text{Sp}$ is said to be *rigid* if for any local $\mathbb{F}$-algebra $A$ and henselian ideal $I\subseteq A$, the canonical map is an equivalence $F(A) \xrightarrow{\simeq} F(A/I)$. **Lemma 106**. *Let $F:\mathrm{CAlg}^{{\mbox{\rm \scriptsize loc}}}_{\mathbb{F}} \rightarrow \text{Sp}$ be a rigid functor. Then $F$ is left Kan extended from the subcategory of essentially smooth local $\mathbb{F}$-algebras.* *Proof.* As observed in Remark [Remark 19](#remark_lke_as_cycles){reference-type="ref" reference="remark_lke_as_cycles"}, we can build for any $B \in \mathrm{CAlg}^{{\mbox{\rm \scriptsize loc}}}_{\mathbb{F}}$ a simplicial resolution $P_\bullet\to B$ where each term $P_m$ is an ind-smooth, local $\mathbb{F}$-algebra and each face map $P_{m+1}\to P_m$ is a henselian surjection. Since $F$ is rigid the simplicial spectrum $m \mapsto F(P_{m})$ is equivalent to the constant simplicial diagram at $F(B)$, and so $|F(P_{\bullet})| \stackrel{\sim}{\to}|F(B)|$. Furthermore, since $\Delta^{\mathrm{op}}$ is contractible diagram, we have that $|F(B)|\stackrel{\sim}{\to}F(B)$. ◻ The following is the first main theorem of the section, showing that Zariski locally our weight-$j$ motivic cohomology is left Kan extended from smooth algebras in degrees $\le j$: **Theorem 107**. *For any local $\mathbb{F}$-algebra $A$ and $j\ge0$, the map ([\[eqn:lke_vs_mot\]](#eqn:lke_vs_mot){reference-type="ref" reference="eqn:lke_vs_mot"}) induces an equivalence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)\stackrel{\sim}{\to}\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A).$$ In other words, the functor $\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}:\mathrm{CAlg}_\mathbb{F}^{\mbox{\rm \scriptsize loc}}\to \rm D(\mathbb{Z})$ is left Kan extended from essentially smooth, local $\mathbb{F}$-algebras.* *Proof.* The result is true rationally by Corollary [Corollary 102](#corollary_lej_rational){reference-type="ref" reference="corollary_lej_rational"}, so it suffices to prove the result for $\tau^{\le j}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(-))/\ell$ for all primes $\ell$. We first claim that the result is true for the functor $$\tau^{\le j}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(-)/\ell):\text{CAlg}^{\mbox{\rm \scriptsize loc}}_\mathbb{F}\to\text D(\mathbb{Z}).$$ Indeed, if $\ell$ is invertible in $\mathbb{F}$ then we have an equivalence $\tau^{\le j}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}/\ell)=\tau^{\le j}R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,\mathbb{\mu}_\ell^{\otimes j})$ by Theorem [Theorem 56](#thm_BL){reference-type="ref" reference="thm_BL"}(1). Since étale cohomology is even rigid, it is left Kan extended from smooth $\mathbb{F}$-algebras by Lemma [Lemma 106](#lem:rigid-lke){reference-type="ref" reference="lem:rigid-lke"}. On the other hand, if $\ell=p=\text{char}(\mathbb{F})$ then $\tau^{\le j}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}/p)=\tau^{\le j}(\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}})$ by Corollary [Corollary 55](#corol_fundamental_p){reference-type="ref" reference="corol_fundamental_p"}, which is also left Kan extended from smooth $\mathbb{F}_p$-algebras: indeed, $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}$ is even left Kan extended from finitely generated polynomial $\mathbb{F}_p$-algebras by definition, and $\tau^{>j}\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}$ identifies with $\widetilde\nu(j)[-j-1]$ by Remark [Remark 53](#remarks_tildenu){reference-type="ref" reference="remarks_tildenu"}(2), which is rigid by Remark [Remark 54](#rem_rigidity_of_nutilde){reference-type="ref" reference="rem_rigidity_of_nutilde"} and so left Kan extended from smooth algebras by Lemma [Lemma 106](#lem:rigid-lke){reference-type="ref" reference="lem:rigid-lke"}. We now claim, for all local $\mathbb{F}$-algebras $A$, that the canonical map $$\tau^{\le j}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)/\ell)\to (\tau^{\le j}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)))/\ell$$ is an equivalence for all primes $\ell$; this will complete the proof. For this, it suffices to prove that the map $H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))\to H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j)/\ell)$ is surjective for all local $\mathbb{F}$-algebras $A$. To see this, we pick a Henselian surjection $P\to A$ where $P$ is an ind-smooth (necessarily local) $\mathbb{F}$-algebra. Since $\tau^{\le j}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(-)/\ell)$ is left Kan extended from essentially smooth $\mathbb{F}$-algebras, the induced map in top degree $H^j_{\mbox{\rm \scriptsize mot}}(P,\mathbb{Z}/\ell(j))\to H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}/\ell(j))$ is surjective. By naturality and Lemma [Lemma 105](#lemma_factors_through_Mil){reference-type="ref" reference="lemma_factors_through_Mil"}, this surjective map moreover fits into a commutative diagram $$\begin{tikzcd} \mathrm{K}_j^M(P)\ar[->>]{r} \ar{d} & H^j_{\mathrm{mot}}(P,\mathbb{Z}(j))\ar{d}\ar[->>]{r} & H^j_{\mathrm{mot}}(P,\mathbb{Z}/\ell(j))\ar[->>]{d}\\ \mathrm{K}_j^M(A) \ar{r} & H^j_{\mathrm{mot}}(A,\mathbb{Z}(j))\ar{r} & H^j_{\mathrm{mot}}(A,\mathbb{Z}/\ell(j)). \end{tikzcd}$$ As indicated, the arrows on the top row are also surjective. Indeed, by taking filtered colimits it is enough to prove such surjectivities for an essentially smooth, local $\mathbb{F}$-algebra in place of $P$: then for the first top arrow it is a theorem of Kerz, and for the second arrow it follows from the usual bound that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ is Zariski locally supported in degrees $\le j$ on smooth $\mathbb{F}$-algebras. From this it follows that the map $H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))\to H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}/\ell(j))$ is surjective as required. ◻ The proof of the previous result yields the following vanishing theorem: **Corollary 108** (Hilbert 90). *For any local $\mathbb{F}$-algebra $A$ and $j\ge1$, we have $H^{j+1}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))=0$; if $A$ is henselian then also $H^{1}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(0))=0$.* *Proof.* The surjectivity of the map $H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))\to H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}/\ell(j))$ from the end of the previous proof means that that $H^{j+1}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))$ is torsion-free. But it also a torsion group since it vanishes rationally by Corollary [Corollary 103](#corol_rat_vanishing){reference-type="ref" reference="corol_rat_vanishing"} and Remark [Remark 104](#remark_drinfeld_1){reference-type="ref" reference="remark_drinfeld_1"}. ◻ To summarise the situation so far, we have shown the following Zariski locally about weight-$j$ motivic cohomology: 1. in degrees $\le j$ it is left Kan extended from smooth algebras; 2. in degrees $j+1,\dots,2j$ it vanishes rationally; 3. in degree $j+1$ it vanishes (unless $j=0$, in which case it is only true Nisnevich locally). In general we are not sure what to expect about the behaviour of motivic cohomology in degrees $j+2,\dots,2j$, except that it vanishes rationally. In particular we are uncertain whether the following additional Nisnevich local vanishing should be indicative of a more extensive vanishing range: **Proposition 109** (Hilbert $90+1$). *For any Henselian local $\mathbb{F}$-algebra $A$ and $j\ge1$, we have $H^{j+2}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))=~0$.* *Proof.* The vanishing holds rationally by Corollary [Corollary 103](#corol_rat_vanishing){reference-type="ref" reference="corol_rat_vanishing"}, or Remark [Remark 104](#remark_drinfeld_1){reference-type="ref" reference="remark_drinfeld_1"} if $j=1$. So it remains to show that $H^{j+2}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))$ is torsion-free; we will prove the stronger (actually equivalent, since we already have Corollary [Corollary 108](#corollary_Hilb_90){reference-type="ref" reference="corollary_Hilb_90"}) result that $H^{j+1}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}/\ell(j))=0$ for all prime numbers $\ell$. Let us first suppose that $\ell$ is invertible in $\mathbb{F}$. Consider the fibre sequence $$\tau^{\le j}R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,\mu_{\ell}^{\otimes j})\longrightarrow\tau^{\le j+1}R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,\mu_{\ell}^{\otimes j})\longrightarrow H^{j+1}_{\mbox{\rm \scriptsize \'et}}(-,\mu_\ell^{\otimes j})[-j-1]=:\mathcal F[-j-1]$$ on qcqs $\mathbb{F}$-schemes. Sheafifying this sequence with respect to the cdh topology and using the identification of Theorem [Theorem 56](#thm_BL){reference-type="ref" reference="thm_BL"}(1), we get a fibre sequence: $$\mathbb{Z}^{\mbox{\rm \scriptsize mot}}(j)/\ell\longrightarrow L_{\mbox{\rm \scriptsize cdh}}\tau^{\le j+1}R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,\mu_{\ell}^{\otimes j})\longrightarrow(L_{\mbox{\rm \scriptsize cdh}}\mathcal F)[-j-1].$$ Since étale cohomology satisfies cdh descent, the middle term agrees with $R\Gamma_{\mbox{\rm \scriptsize \'et}}(-,\mu_{\ell}^{\otimes j})$ up to degrees $\leq j+1$, whence we deduce that $$H^{j+1}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}/\ell(j))=\ker\big(H^{j+1}_{\mbox{\rm \scriptsize \'et}}(A,\mu_\ell^{\otimes j})\to H^0(L_{\mbox{\rm \scriptsize cdh}}\mathcal F(A))\big).$$ But now we claim that the map appearing on the right side is injective. Indeed, letting $k$ be the residue field of $A$, by functoriality it fits into a commutative diagram $$\xymatrix{ H^{j+1}_{\mbox{\rm \scriptsize \'et}}(A,\mu_\ell^{\otimes j})\ar[r]\ar[d] & H^0(L_{\mbox{\rm \scriptsize cdh}}\mathcal F(A))\ar[d] \\ H^{j+1}_{\mbox{\rm \scriptsize \'et}}(k,\mu_\ell^{\otimes j})\ar[r] & H^0(L_{\mbox{\rm \scriptsize cdh}}\mathcal F(k)) }$$ where the left vertical arrow is an isomorphism (by rigidity of étale cohomology), and the bottom horizontal arrow is also an isomorphism (since fields are cdh points). This completes the proof that $H^{j+1}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}/\ell(j))=0$. Next suppose that $\mathbb{F}=\mathbb{F}_p$ and $\ell=p$. Then from the fundamental fibre sequence of Corollary [Corollary 55](#corol_fundamental_p){reference-type="ref" reference="corol_fundamental_p"} we see that there is a natural identification $$H^{j+1}_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}/p(j))=\ker\big(\widetilde\nu(j)(A)\to H^0_{\mbox{\rm \scriptsize cdh}}(A,\widetilde\nu(j))\big).$$ Exactly as in the previous case, this vanishes by comparison with the residue field: namely $\widetilde\nu(j)(A)\stackrel{\simeq}{\to}\widetilde\nu(j)(k)\stackrel{\simeq}{\to}H^0_{\mbox{\rm \scriptsize cdh}}(k,\widetilde\nu(j))$, the first isomorphism being rigidity of $\widetilde\nu(j)$ and the second being the fact that $k$ is a point for the cdh topology ◻ For the next corollary, we let $\mathrm{CAlg}_\mathbb{F}^{\mbox{\rm \scriptsize h.loc}}$ be the category of henselian local $\mathbb{F}$-algebras. **Corollary 110**. *For $j\ge1$, the functor $\tau^{\le j+2}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}:\mathrm{CAlg}_\mathbb{F}^{\mbox{\rm \scriptsize h.loc}}\to\text D(\mathbb{Z})$ is left Kan extended from henselisations of essentially smooth, local $\mathbb{F}$-algebras.* *Proof.* Corollary [Corollary 108](#corollary_Hilb_90){reference-type="ref" reference="corollary_Hilb_90"} and Proposition [Proposition 109](#proposition_91){reference-type="ref" reference="proposition_91"} imply that $\tau^{\le j+2}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}=\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ on Henselian local $\mathbb{F}$-algebras, so the claim reduces to Theorem [Theorem 107](#thm_lke_lej){reference-type="ref" reference="thm_lke_lej"}. ◻ An important consequence of the above results is a partial description of weight one motivic cohomology. **Corollary 111** (Weight one motivic cohomology). *For any qcqs $\mathbb{F}$-scheme $X$, the cofibre of the comparison map $R\Gamma_{\mbox{\rm \scriptsize Nis}}(X,\mathbb{G}_m)[-1]\stackrel{\sim}{\to}\mathbb{Z}(1)^{\mbox{\rm \scriptsize mot}}(X)$ from [\[eq:c1-mot\]](#eq:c1-mot){reference-type="eqref" reference="eq:c1-mot"} is supported in degrees $>3$. In particular there are natural isomorphisms $$H^i_{\mathrm{mot}}(X, \mathbb{Z}(1)) \cong \begin{cases} 0 & i \le 0\\ \mathcal O(X)^{\times} & i = 1\\ \mathrm{Pic}(X) & i = 2\\ H^2_{\mathrm{Nis}}(X,\mathbb{G}_m) & i=3. \end{cases}$$* *Proof.* It is enough to prove the claim Nisnevich locally, i.e., that the map $A^\times[-1]\to \tau^{\le 3}\mathbb{Z}(1)^{\mbox{\rm \scriptsize mot}}(A)$ is an equivalence for any Henselian local $\mathbb{F}$-algebra $A$. But that is exactly what the Corollary [Corollary 110](#cor:lke-hloc){reference-type="ref" reference="cor:lke-hloc"} states in the case $j=1$. ◻ ## Singular Nesterenko--Suslin isomorphism {#sec:singular-nst} In the previous subsection we constructed the symbol map $\mathrm{K}_j^M(A)\to H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))$ and used it in the course of the proof of Theorem [Theorem 107](#thm_lke_lej){reference-type="ref" reference="thm_lke_lej"}. We now establish an analogue of the theorem of Nesterenko--Suslin and Totaro, namely the symbol map is essentially an isomorphism; we just need to take care to replace Milnor $K$-theory by the improved variant $\widehat\mathrm{K}_j^M(A)$ of Gabber and Kerz [@Kerz2010]. **Theorem 112** (Singular Nesterenko--Suslin isomorphism). *For any local $\mathbb{F}$-algebra $A$ and $j\ge0$, the symbol map $\mathrm{K}^M_j(A) \rightarrow H^j_{\mathrm{mot}}(A,\mathbb{Z}(j))$ descends to an isomorphism $$\widehat{\mathrm{K}}^M_j(R) \stackrel{\simeq}{\to}H^j_{\mathrm{mot}}(R,\mathbb{Z}(j)).$$* *Proof.* Let $P_\bullet\to A$ be a simplicial resolution as in Remark [Remark 19](#remark_lke_as_cycles){reference-type="ref" reference="remark_lke_as_cycles"}, so that the totalisation of the simplicial complex $m\mapsto \tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(P_m)$ calculates the evaluation on $A$ of the left Kan extension of $\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ from essentially smooth local $\mathbb{F}$-algebras. In light of Theorem [Theorem 107](#thm_lke_lej){reference-type="ref" reference="thm_lke_lej"}, the totalisation is equivalent to $\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)$. Calculating the top degree $H^j$ as a coequaliser, this means that the canonical map $$\text{coeq}\big(H^j_{\mbox{\rm \scriptsize mot}}(P_1,\mathbb{Z}(j))\rightrightarrows H^j_{\mbox{\rm \scriptsize mot}}(P_0,\mathbb{Z}(j))\big)\longrightarrow H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))$$ is an equivalence. The canonical map $$\text{coeq}\big(\mathrm{K}_j^M(P_1)\rightrightarrows \mathrm{K}_j^M(P_1)\big)\longrightarrow\mathrm{K}_j^M(A)$$ is also an equivalence; this is the content of [@LuedersMorrow2023 Prop. 1.17]. Comparing the two coequaliser diagrams via the natural symbol maps we obtain two immediate conclusions. 1. The symbol map $\mathrm{K}_j^M(A)\to H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))$ is surjective. Indeed, as already used in the proof of Theorem [Theorem 107](#thm_lke_lej){reference-type="ref" reference="thm_lke_lej"}, the symbol map $\mathrm{K}_j^M(P_0)\to H^j_{\mbox{\rm \scriptsize mot}}(P_0,\mathbb{Z}(j))$ is surjective by Kerz. 2. Secondly, we may complete the proof in the case that $A$ has big residue field, i.e., its residue field has more than $M_j$ elements in the sense of [@Kerz2010 Prop. 10(5)]. Indeed, in that case the ind-smooth local rings $P_i$, $i=0,1$, also have big residue field and so the symbol maps $\mathrm{K}_j^M(P_i)=\widehat\mathrm{K}_j^M(P_i)\to H^j_{\mbox{\rm \scriptsize mot}}(P_i,\mathbb{Z}(j))$ are isomorphisms by Kerz [@Kerz2010 Prop. 10(11)]. Comparing the two coequaliser diagrams we deduce that the symbol map $\mathrm{K}_j^M(A)=\widehat\mathrm{K}_j^M(A)\to H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))$ is also an isomorphism. It remains to treat the case that $A$ has small (in particular, finite) residue field $\mathbb{F}_q$, which we do by constructing some ad-hoc transfer maps on $H^j_{\mbox{\rm \scriptsize mot}}(-,\mathbb{Z}(j))$. Let $\ell>0$ be an integer prime to $|\mathbb{F}_q:\mathbb{F}_p|$, so that $\mathbb{F}_{q^\ell}=\mathbb{F}_q\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell}$ (this identity holds because the right side is a tensor product of Galois extensions of coprime degree, therefore a field); this also implies that the semi-local ring $A\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell}$ is in fact local, as its quotient by its Jacobson radical is a field. Finally observe that $P_\bullet\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell}\to A\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell}$ is a simplicial resolution satisfying the conditions of Remark [Remark 19](#remark_lke_as_cycles){reference-type="ref" reference="remark_lke_as_cycles"}, and so (replacing $A$ by $A\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell}$ above), we have a coequaliser diagram $$\text{coeq}\big(H^j_{\mbox{\rm \scriptsize mot}}(P_1\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell},\mathbb{Z}(j))\rightrightarrows H^j_{\mbox{\rm \scriptsize mot}}(P_0\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell},\mathbb{Z}(j))\big)\stackrel{\sim}{\to}H^j_{\mbox{\rm \scriptsize mot}}(A\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell},\mathbb{Z}(j))$$ Since classical motivic cohomology of smooth schemes admits functorial transfer maps along finite morphisms, this diagram induces a transfer map $N:H^j_{\mbox{\rm \scriptsize mot}}(A\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell},\mathbb{Z}(j))\to H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))$ such that the pre-composition with the canonical map $H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))\to H^j_{\mbox{\rm \scriptsize mot}}(A\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell},\mathbb{Z}(j))$ is multiplication by $\ell$. We make no claims that this transfer map is natural, independent of the simplicial resolution, compatible with any transfers on Milnor $K$-theory, etc.; in fact, we only care about the resulting fact that therefore $\text{ker}\big(H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))\to H^j_{\mbox{\rm \scriptsize mot}}(A\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell},\mathbb{Z}(j))\big)$ is annihilated by $\ell$. It now follows formally that the symbol map factors through $\widehat\mathrm{K}_j^M(A)$: indeed, given $x\in\text{ker}(\mathrm{K}_j^M(A)\to\widehat\mathrm{K}_j^M(A))$ and any $\ell$ as in the previous paragraph such that $p^\ell>M_j$ then, by functoriality of the symbol map, and the established isomorphism for the local ring $A\otimes_{\mathbb{F}_p}\mathbb{F}_{p^\ell}$, we deduce that $\ell x$ is annihilated by the symbol map. Picking a different value of $\ell$, prime to the first value, shows that $x$ is annihilated by the symbol map, i.e., the latter factors through $\widehat\mathrm{K}_j^M(A)$. The new symbol map $\widehat\mathrm{K}_j^M(A) \rightarrow H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))$ is surjective since the original one was. To prove that the new symbol map $\widehat\mathrm{K}_j^M(A)\to H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}(j))$ is injective, we again use a transfer argument; let $x$ be in the kernel. Then the transfer map for improved Milnor $K$-theory, and the established isomorphism in case of big residue field, shows that $\ell x=0$ for any $\ell$ as above. Again picking coprime values of $\ell$ shows that $x=0$ and so completes the proof. ◻ # Motivic Soulé--Weibel vanishing and pro cdh descent {#section_Weibel} One of the most influential conjectures concerning the algebraic $K$-theory of singular schemes has been Weibel's conjecture [@Weibel1980], now a theorem of Kerz--Strunk--Tamme [@KerzStrunkTamme2018]. It states, in particular, that for a Noetherian scheme $X$ of dimension $\le d$, the negative $K$-groups $\mathrm{K}_{-n}(X)$ vanish for $n>d$. Kerz--Strunk--Tamme's proof proceeds by first establishing pro cdh descent for $K$-theory of Noetherian schemes. For earlier work on special cases on Weibel's conjecture and pro cdh descent, see for example [@Cortinas2008; @GeisserHesselholt2010; @Krishna2006; @Krishna2009a; @Krishna2010; @pro-cdh; @Weibel2001]. Our goal in this section is to prove the following analogous results about our motivic cohomology; as usual, let $\mathbb{F}$ denote a prime field. **Theorem 113** (Motivic Soulé--Weibel vanishing). *Let $j\ge0$ and let $X$ be a Noetherian $\mathbb{F}$-scheme of finite dimension. Then $H^i_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j))=0$ for all $i> j+\dim X$.* **Theorem 114** (Pro cdh descent). *On the category of Noetherian $\mathbb{F}$-schemes, the presheaf $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ satisfies pro cdh descent for each $j\ge0$. That is, given any abstract blowup square of Noetherian $\mathbb{F}$-schemes $$\xymatrix{ Y'\ar[r]\ar[d] & X'\ar[d] \\ Y\ar[r] & X },\label{eqn_blowup}$$ the associated square of pro complexes $$\xymatrix{ \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X) \ar[r]\ar[d] & \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X')\ar[d]\\ \{\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(rY)\})_r \ar[r] & \{\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(rY')\})_r }$$ is cartesian.[^16]* **Remark 115** (Relation to Weibel's $K$-theoretic vanishing conjecture). Let $X$ be a Noetherian $\mathbb{F}$-scheme of dimension $\le d$. Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"} states that the Atiyah--Hirzebruch spectral sequence $E_2^{ij}=H^{i-j}_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j))\Rightarrow \mathrm{K}_{-i-j}(X)$ is supported in the left half plane $x\le d$. From this one immediately reads off parts of Weibel's package conjectures about lower $K$-groups: both the vanishing $\mathrm{K}_{-n}(X)=0$ for $n>d$, and the usual description of $\mathrm{K}_{-d}(X)$ via an edge map isomorphism $$H^{d}_{\mbox{\rm \scriptsize cdh}}(X,\mathbb{Z})=H^{d}_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(0))\cong \mathrm{K}_{-d}(X).$$ Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"} can therefore be seen as a motivic refinement of Weibel's vanishing conjecture. This is moreover reflected in the proof of the theorem, which is based on both the arguments of [@Cortinas2008; @Cortinas2008a], where Cortiñas--Haesemyer--Schlichting--Weibel proved Weibel's vanishing conjecture and Vorst's conjecture for varieties over a characteristic zero field, and of [@KerzStrunkTamme2018], where Kerz--Strunk--Tamme used pro cdh descent to prove Weibel's conjecture in general. We stress however that Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"} is certainly not a new proof of Weibel's vanishing conjecture in $K$-theory, since our theory uses the fundamental square Theorem [Theorem 23](#thm:mainsq){reference-type="ref" reference="thm:mainsq"} which itself relies on the work of Kerz--Strunk--Tamme. We refer the reader to Remark [Example 123](#sec:usual){reference-type="ref" reference="sec:usual"} for more details on this point. **Remark 116** (Applications to Adams eignenspaces). To use the Atiyah--Hirzebruch spectral sequence to deduce the usual $K$-theoretic Weibel conjecture from a vanishing result in motivic cohomology, it would have been sufficient to establish the following weaker diagonal vanishing line: $H^i_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j))=0$ for all $i> 2j+\dim X$. The stronger vertical vanishing line of Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"} is related to a vanishing theorem of Soulé as follows. By rationalising the Atiyah--Hirzebruch spectral sequence and rewriting in terms of Adams eigenspaces, Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"} implies that for any Noetherian $\mathbb{F}$-scheme $X$ of dimension $\le d$ we have the following vanishing for each $n\in\mathbb{Z}$: the Adams eigenspace $\mathrm{K}_n(X)_\mathbb{Q}^{(j)}$ vanishes whenever $j>n+d$. This vanishing is due to SGA6 [@SGA_VI Exp. VI, Thm. 6.9] in the case of $\mathrm{K}_0$ of Noetherian schemes with an ample line bundle, and when $n>0$ to Soulé [@Soule1985 Corol. 1] for the higher algebraic $K$-groups of Noetherian rings; when $n<0$ this vanishing of Adams eigenspaces of negative $K$-groups is new as far as we are aware. In other words, Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"} provides an integral refinement of Soulé's result, as well as an extension beyond the affine case and to negative $K$-groups. ## Pro cdh descent {#sec:pro-cdh} Here we prove Theorem [Theorem 114](#theorem_pro_cdh_descent){reference-type="ref" reference="theorem_pro_cdh_descent"}. We begin by noting a similar pro cdh descent property for the Nisnevich cohomology of wedge powers of the cotangent complex $R\Gamma(-,L^i_{-/\mathbb{F}}): \mathrm{S}\mathrm{ch}{}_\mathbb{F}^{\mathrm{qcqs},\mathrm{op}} \to \rm{Sp}$. The following is a slight generalization of [@pro-cdh Thm. 2.10]. **Lemma 117**. *For any abstract blowup square of Noetherian $\mathbb{F}$-schemes ([\[eqn_blowup\]](#eqn_blowup){reference-type="ref" reference="eqn_blowup"}) and $i\ge0$, the square of pro complexes $$\begin{tikzcd} R\Gamma(X,L^i_{-/\mathbb{F}}) \ar{r} \ar{d} & R\Gamma(X',L^i_{-/\mathbb{F}}) \ar{d}\\ \{ R\Gamma(rY,L^i_{-/\mathbb{F}})\}_r \ar{r} &\{ R\Gamma(rY',L^i_{-/\mathbb{F}})\}_r \end{tikzcd}$$ is cartesian.* *Proof.* The proof works in the exact same way as in [@pro-cdh Thm. 2.10], except that we need to justify why [@pro-cdh Thm. 2.4] does not require the stated finite dimensionality hypothesis; but this follows from the general formal functions theorem of [@LurieSAG Lem. 8.5.1.1]. ◻ Next we establish pro $\mathrm{cdh}$ descent for syntomic cohomology; remarkably, the proof uses algebraic $K$-theory: **Proposition 118**. *For any abstract blowup square of Noetherian $\mathbb{F}_p$-schemes ([\[eqn_blowup\]](#eqn_blowup){reference-type="ref" reference="eqn_blowup"}) and $j\ge0$, the square of pro complexes $$\begin{tikzcd} \mathbb{Z}_p(j)^{\mathrm{syn}}(X) \ar{r} \ar{d} & \mathbb{Z}_p(j)^{\mathrm{syn}}(X') \ar{d}\\ \{ \mathbb{Z}_p(j)^{\mathrm{syn}}(rY)\}_r \ar{r} &\{ \mathbb{Z}_p(j)^{\mathrm{syn}}(rY')\}_r \end{tikzcd}$$ is cartesian.* *Proof.* Since mod-$p$ syntomic cohomology $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}$ admits a finite filtration with graded pieces given by shifts of $R\Gamma(-, L_{-/\mathbb{F}}^i)$ for various $i$ (by sheafifying Lemma [Lemma 39](#lem_fin_fil_on_syn){reference-type="ref" reference="lem_fin_fil_on_syn"}), it satisfies pro cdh descent thanks to Lemma [Lemma 117](#lem_pro_cdh_for_cotangent){reference-type="ref" reference="lem_pro_cdh_for_cotangent"}. The remaining difficulty is to extend the result from $\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}$ to $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$. Fix $n\in\mathbb{Z}$ and set $A_r:=H^n(\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X,X',rY))$ for each $r\ge0$; the goal is to show that the pro abelian group $\{A_r\}_r$ vanishes. We claim that each group $A_r$ is bounded $p$-power torsion. Granting this claim, we may complete the proof as follows. Given $s\ge1$, pick $c>$ such that $p^cA_s=0$. By the previous paragraph and an induction, we see that $\{A_r/p^c\}_r=0$; so there exists $s'> s$ such that the transition map $A_{s'}/p^c\to A_s/p^c$ is zero. But $A_s/p^c=A_s$, so this shows that the transition map $A_{s'}\to A_s$ is zero, as required. It remains to prove that $A_r$ is bounded $p$-power torsion. But it is both derived $p$-complete (since it is $H^n$ of a $p$-complete complex) and satisfies $A_r[\tfrac1p]=0$ (since $\mathbb{Q}_p(j)^{\mbox{\rm \scriptsize syn}}$ satisfies cdh descent by Corollary [Corollary 43](#corol_Qpsyn){reference-type="ref" reference="corol_Qpsyn"}, which we proved using $K$-theory), so it is killed by a power of $p$ by [@Bhatt2019 Thm. 1.1]. ◻ *Proof of Theorem [Theorem 114](#theorem_pro_cdh_descent){reference-type="ref" reference="theorem_pro_cdh_descent"}.* If $\mathbb{F}=\mathbb{F}_p$ then Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(2) shows that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ differs from $\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ by a cdh sheaf; since syntomic cohomology satisfies pro cdh descent by Proposition [Proposition 118](#prop_pro_cdh_syn){reference-type="ref" reference="prop_pro_cdh_syn"}, the same is true for motivic cohomology. If instead $\mathbb{F}=\mathbb{Q}$ then the third term in the fundamental fibre sequence Theorem [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(3) satisfies pro cdh descent by Lemma [Lemma 117](#lem_pro_cdh_for_cotangent){reference-type="ref" reference="lem_pro_cdh_for_cotangent"}; since the middle term in the fibre sequence is a cdh sheaf, it follows that motivic cohomology also satisfies pro cdh descent. ◻ ## Proof of motivic Soulé--Weibel vanishing Fix a weight $j\ge0$. Here we prove Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"}. In fact, we prove the following stronger statement: **Theorem 119**. *Let $X$ be a Noetherian $\mathbb{F}$-scheme of dimension $\le d$. Then the fibre $$W(j)(X):=\operatorname{fib}\big(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)\big)$$ vanishes in degrees $>j+d$.* **Remark 120**. The cohomology theory $W(j)(X)$ are the shifts of graded pieces of a filtration on the fibre of $\mathrm{K}(X) \rightarrow \mathrm{KH}(X)$. In turn each $W(j)(X)$ admits a filtration whose graded pieces are the "$N^r$ of motivic cohomology,\" i.e., the fibres of the maps $\mathbb{Z}(j)^{\mathrm{mot}}(X) \rightarrow \mathbb{Z}(j)^{\mathrm{mot}}(\mathbb{A}^r \times X)$. These groups refine Bass' $N^rK$-groups which measure the failure of algebraic $K$-theory to be $\mathbb{A}^r$-invariant. We intend to explore questions surrounding these groups using motivic methods in the future. Note that Theorem [Theorem 119](#theorem_Weibel_vanishing2){reference-type="ref" reference="theorem_Weibel_vanishing2"} implies Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"}, as we already know from Theorem [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(1) that the cdh-local motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$ is supported in degrees $\le j+d$; but the stronger statement also tells us that the map $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\to \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X)$ is surjective in degree $j+d$. First we quote the following result, whose Zariski version has often appeared in previous work on the subject: **Lemma 121** (Nisnevich vanishing lemma). *Let $\mathcal F$ be a Nisnevich sheaf of abelian groups on a Noetherian scheme $X$, and $d\ge0$ such that the stalk $\mathcal F_x$ vanishes for all $x\in X$ satisfying $\dim\overline{\{x\}}>d$. Then $H^i_{\mbox{\rm \scriptsize Nis}}(X,\mathcal F)=0$ for all $i>d$.* *Proof.* This is proved by induction using the coniveau spectral sequence, as stated in [@KatoSaito1986 (1.2.5)] (see also the proof of [@Nisnevich1989 Thm. 1.32]). ◻ When proving Weibel and Vorst's conjecture for finite type schemes over characteristic zero fields [@Cortinas2008; @Cortinas2008a], Cortiñas--Haesemeyer(--Schlichting)--Weibel analysed the relation between of the top degree Nisnevich and cdh cohomologies of sheaves of differential forms; although pro cdh descent did not appear explicitly, it was implicitly encoded in their use of the formal functions theorem. When proving Weibel's conjecture for $K$-theory [@KerzStrunkTamme2018], Kerz--Strunk--Tamme used pro cdh descent to show that the desired vanishing was of a birational nature. The following proposition may be seen as an axiomatisation of the aforementioned arguments. **Proposition 122**. *Let $k$ be a base ring and $W:\mathrm{S}\mathrm{ch}{}_k^{\mbox{\rm \scriptsize qcqs,op}}\to \rm D(\mathbb{Z})$ a finitary Nisnevish sheaf with the following properties:* 1. *$L_{\mbox{\rm \scriptsize cdh}}W\simeq0$.* 2. *$W$ satisfies pro cdh descent on Noetherian $k$-schemes.* 3. *For any Noetherian, local, henselian $k$-algebra $A$ and nilpotent ideal $I\subseteq A$, the fibre $W(A,I)=\mathrm{fib}(W(A)\to W(A/I))$ is supported in degrees $\le 0$.* *Then, for any Noetherian $k$-scheme $X$ of finite dimension, $W(X)$ is supported in degrees $\le \dim X$.* *Proof.* We begin by globalising hypothesis (3) by noting the following > (3'): for any Noetherian $k$-scheme $X$ of finite dimension and nil immersion $X_0\to X$, the fibre $W(X,X_0)$ is supported in degrees $\le \dim X$ (and so $W(X)\to W(X_0)$ is an equivalence in degrees $>\dim X$). This follows from Nisnevich descent and Nisnevich exactness of the closed embedding $X_0\hookrightarrow X$, more precisely using that $X_{\mbox{\rm \scriptsize Nis}}$ has cohomological dimension $\le\dim X$ and the sheaf $W(-,-\times_XX_0)$ on $X_{\mbox{\rm \scriptsize Nis}}$ has stalks supported in degrees $\le 0$ by (3). Now let $X$ be a Noetherian $k$-scheme. We must show that $W(X)$ is supported in degree $\le\dim X$. Using (3') we may assume that $X$ is reduced. If $\dim X=0$ then $X$ is a finite disjoint union of the spectra of fields. Since spectra of fields are points for the cdh topology we have $W(X)\stackrel{\sim}{\to}L_{\mbox{\rm \scriptsize cdh}}W(X)$, which vanishes by hypothesis (1). We now proceed by induction on $\dim X$, so assume that $d:=\dim X>0$ and that the desired vanishing has been proved for Noetherian $k$-schemes of dimension $<d$. We examine the bounded Nisnevich descent spectral sequence $$E_2^{ab}=H^a_{\mbox{\rm \scriptsize Nis}}(X,\mathcal H^b(W))\implies H^{a+b}(W(X)),$$ where $\mathcal H^b(W)$ is the Nisnevich sheafifcation of the abelian presheaf $Y\mapsto H^b(W(Y))$. The $E_2$ page of this spectral sequence enjoys various vanishings: 1. $E_2^{ab}=0$ if $a>d$ (or if $a<0$), since $X$ has Nisnevich cohomological dimension $\le d$. 2. $E_2^{ab}=0$ if $a>0$ and $b>d$. Indeed, for such $b$ and any $x\in X$ such that $\dim\overline{\{x\}}>0$, then $\dim\mathcal{O}_{X,x}^h<d$ and so the stalk $\mathcal H^b(W)_x=H^b(W(\mathcal{O}_{X_x}^h))$ (the equality is a consequence of $W$ being finitary) vanishes by the inductive hypothesis. Lemma [Lemma 121](#lem:nis-vanish){reference-type="ref" reference="lem:nis-vanish"} now implies that $\mathcal H^b(W(j))$ has no higher cohomology. 3. $E_2^{ab}=0$ if $b\le d$ and $a+b>d$. The proof will be clearest if we start by fixing $b\le d$. Then, for any $x\in X$ such that $\dim\overline{\{x\}}>d-b$, we have that $\dim\mathcal{O}_{X,x}^h<b\le d$, i.e., $b>\dim\mathcal{O}_{X,x}^h$ and $\dim\mathcal{O}_{X,x}^h<d$; so $\mathcal H^b(W(j))_x=0$ by the inductive hypothesis (and again finitariness to compute the stalk in terms of $\mathcal{O}_{X,x}^h$). Lemma [Lemma 121](#lem:nis-vanish){reference-type="ref" reference="lem:nis-vanish"} now implies that $\mathcal H^b(W)$ has no cohomology in degrees $>d-b$, or in other words $H^a(\mathcal H^b(W))=0$ whenever $a+b>d$. Thanks to vanishings (1)--(4), we can read off from the Nisnevich descent spectral sequence edge map isomorphisms $$H^{n}(W(X))\stackrel{\simeq}{\to}H^0_{\mbox{\rm \scriptsize Nis}}(X,\mathcal H^{n}(W))$$ for all $n>d$. For the rest of the proof fix $n>d$. Allowing $X$ to vary, the previous isomorphism may be rephrased as follows: > () On the category of Noetherian $k$-schemes of dimension $\le d$, the abelian presheaf $H^{n}(W(-))$ is a Nisnevich sheaf. In fact, we will only need to know that it is Nisnevich separated. We now return to our fixed $X$ of dimension $\le d$, and pick a cohomology class $\alpha\in H^{n}(W(X))$; we must show that $\alpha=0$. We claim that there exists a modification $f:X'\to X$ (i.e., a proper morphism where $X'$ is also reduced and such there there exists a dense open $U\subseteq X$ satisfying $f^{-1}(U)\stackrel{\simeq}{\to}U$) such that $f^*\alpha=0$ in $H^{n}(W(X'))$. To prove the claim we first use hypothesis (1) to see that, for any $a\in\mathbb{Z}$, the presheaf $H^{a}(W(-))$ vanishes on valuation rings, therefore vanishes after cdh sheafification. In particular there exists a cdh cover $U\to X$ such that $\alpha$ vanishes in $H^{n}(W(U))$; we can then refine $U$ to a cdh cover of the form $X_2\to X_1\xrightarrow{g} X$ where $X_1\to X$ is a proper cdh (often called a cdp) cover and $X_2\to X_1$ is a Nisnevich cover [@SuslinVoevodsky2000 Prop. 5.9]. Next note that there exists a modification $f:X'\to X$ which factors through $X_1$: for example pick a dense open $U\subseteq X$ such that $f^{-1}(U)\stackrel{\simeq}{\to}U$, and define $X'$ to be $(-)_{\mbox{\rm \scriptsize red}}$ of the closure of $U$ in $X'$. By construction $\alpha$ vanishes when we pull back to $X_2$, hence also to $X'\times_X X_2$; but $X'\times_X X_2\to X'$ is a Nisnevich cover of schemes of dimension $\le d$ (since modifications and étale morphisms do not increase dimension), so () implies that $H^{n}(W(X'))\to H^{n}(W(X'\times_XX_2))$ is injective and therefore $\alpha$ already vanished when pulled back to $X'$. This completes the proof of the claim. Our modification $X'\to X$ fits into an abstract blowup square ([\[eqn_blowup\]](#eqn_blowup){reference-type="ref" reference="eqn_blowup"}) in which $Y'$ and $Y$ have dimension $<d$. From hypothesis (2) and the inductive hypothesis applied to the infinitesimal thickenings of $Y$ and $Y'$, we see that $H^{n}(W(X))\to H^{n}(W(X'))$ is an isomorphism. But this map was constructed so as to kill $\alpha$. Therefore $\alpha$ was already zero in $H^{n}(W(X))$, completing the proof. ◻ **Example 123** (Usual Weibel vanishing). Here we present a revisionist version of Kerz--Strunk--Tamme's proof of Weibel vanishing [@KerzStrunkTamme2018]. First note that Proposition [Proposition 122](#prop_axiomatic_Weibel){reference-type="ref" reference="prop_axiomatic_Weibel"} applies verbatim to finitary Nisnevich presheaves of spectra $W:\textrm{Sch}_k^{\mbox{\rm \scriptsize qcqs}}\to\textrm{Sp}$ satisfying the same hypotheses; we stated it for preshaves of complexes only for simplicity. In particular, the proposition applies when $W:=\mathrm{fib}(\mathrm{K}\to\mathrm{KH})$ and $k=\mathbb{Z}$. Indeed, hypothesis (1) follows from the fact that $K(V)\stackrel{\sim}{\to}\mathrm{KH}(V)$ for any valuation ring $V$ [@KerzStrunkTamme2018 Thm. 6.3] [@KellyMorrow2021 Thm. 3.4]; hypothesis (2) follow fom pro cdh descent of $K$-theory and cdh descent of $\mathrm{KH}$-theory [@KerzStrunkTamme2018]; hypothesis (3) follows from nil invariance of negative $K$-theory. We therefore deduce, for any Noetherian scheme $X$, that $\mathrm{fib}(\mathrm{K}(X)\to\mathrm{KH}(X))$ is supported in homological degrees $\ge-\dim X$. Since there are various ways to show that $\mathrm{KH}(X)$ is supported in homological degrees $\ge-\dim X$ [@KerzStrunk2017] [@KellyMorrow2021 Rmk. 3.5(a)], we deduce that $\mathrm{K}(X)$ is also supported in homological degrees $\ge-\dim X$ as required. We now verify that the previous proposition may also be applied in our motivic situation of interest, at least up to a harmless shift: **Proposition 124**. *The presheaf $W(j)=\mathrm{fib}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\to\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}):\mathrm{S}\mathrm{ch}{}_\mathbb{F}^{\mbox{\rm \scriptsize qcqs,op}}\to\textrm D(\mathbb{Z})$ is a finitary Nisnevich sheaf with the following properties:* 1. *$L_{\mbox{\rm \scriptsize cdh}}W(j)=0$.* 2. *$W(j)$ satisfies pro cdh descent on Noetherian $\mathbb{F}$-schemes.* 3. *For any $\mathbb{F}$-algebra $A$ and finitely generated nilpotent ideal $I\subseteq A$, the fibre $W(j)(A,I)$ is supported in degrees $\le j$.* *Proof.* The presheaf $W(j)$ is a finitary Nisnevich sheaf since $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ and $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$ are, by Theorems [Theorem 20](#thm:cdh){reference-type="ref" reference="thm:cdh"}(2), [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(5), and [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(4). (1): $W(j)$ vanishes after cdh sheafification since $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$ by Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}(1). (2): $W(j)$ satisfies pro cdh descent on Noetherian schemes, since the same is true of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ by Theorem [Theorem 114](#theorem_pro_cdh_descent){reference-type="ref" reference="theorem_pro_cdh_descent"}. (3): We first treat the case that $\mathbb{F}=\mathbb{Q}$. From the fundamental fibre sequence of Theorem [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(3), for both $A$ and $A/I$, we have a description of the relative term $W(j)(A,I)$ as $$W(j)(A,I)={\rm fib}\left(L\Omega_{A/\mathbb{Q}}^{< j} \to L\Omega_{(A/I)/\mathbb{Q}}^{< j}\right)[-1].$$ This is supported in degrees $<j$ since $\Omega_{A/\mathbb{Q}}^{j-1}\to\Omega_{(A/I)/\mathbb{Q}}^{j-1}$ is surjective. In the case that $\mathbb{F}=\mathbb{F}_p$, the pullback square of Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(2) shows that $W(j)(A,I)\simeq\mathrm{fib}(\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A)\to\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A/I))$, which is derived $p$-complete; so it is sufficient to prove the claim modulo $p$, namely that $\mathrm{fib}(\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}(A)\to\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}(A/I))$ is supported in degree $<j$. Since nilpotent ideals are Henselian, this is a special case of [@AntieauMathewMorrowNikolaus Thm. 5.2]. ◻ *Proof of Theorem [Lemma 121](#lem:nis-vanish){reference-type="ref" reference="lem:nis-vanish"}.* Apply Proposition [Proposition 122](#prop_axiomatic_Weibel){reference-type="ref" reference="prop_axiomatic_Weibel"} to the presheaf $W:=W(j)[j]$. The hypotheses of the proposition are satisfied thanks to Proposition [Proposition 124](#prop_of_W(j)){reference-type="ref" reference="prop_of_W(j)"}. ◻ Our arguments implicitly reprove the results of [@Cortinas2008; @Cortinas2008a] concerning Nisnevich and cdh cohomology of differential forms, as well as a similar style of result in finite characteristic: **Corollary 125**. 1. *For any Noetherian $\mathbb{F}$-scheme of dimension $d$ and $j \geq 0$, the canonical map $H^{j+d}_{\mathrm{mot}}(X, \mathbb{Z}(d)) \rightarrow H^{j+d}_{\mathrm{cdh}}(X, \mathbb{Z}(d))$ is surjective.* 2. *For $j\ge1$ and any Noetherian $\mathbb{Q}$-scheme $X$ of dimension $\le d$, the canonical map $$H^d_{\mbox{\rm \scriptsize Nis}}(X,\Omega^{j-1}_{-/\mathbb{Q}})\to H^d_{\mbox{\rm \scriptsize cdh}}(X,\Omega^{j-1}_{-/\mathbb{Q}})$$ is surjective.* 3. *For $j\ge0$ and any Noetherian $\mathbb{F}_p$-scheme $X$ of dimension $\le d$, the canonical maps $$H^d_{\mbox{\rm \scriptsize Nis}}(X,\widehat K_j^M/p)\to H^d_{\mbox{\rm \scriptsize cdh}}(X,\widehat K_j^M/p)\quad\mathrm{and}\quad H^{d-1}_{\mbox{\rm \scriptsize Nis}}(X,\widetilde\nu(j))\to H^{d-1}_{\mbox{\rm \scriptsize cdh}}(X,\widetilde\nu(j))$$ are surjective, and the canonical map $$H^{d}_{\mbox{\rm \scriptsize Nis}}(X,\widetilde\nu(j))\longrightarrow H^{d}_{\mbox{\rm \scriptsize cdh}}(X,\widetilde\nu(j))$$ is an isomorphism. Here $\widehat K_j^M/p$ denotes improved Milnor $K$-theory mod $p$, as an abelian Nisnevich or cdh sheaf.* *Proof.* The first claim was explained after Remark [Remark 120](#rem:wj){reference-type="ref" reference="rem:wj"}. The rest are related to $W(j)(X)$ via the following descriptions, which we state in the generality of qcqs schemes for the sake of possible future reference: 1. For any qcqs $\mathbb{Q}$-scheme $X$ of valuation dimension $\le d$, then $W(j)(X)$ vanishes in degrees $>j+d+1$ and there is a natural isomorphism $H^{j+d+1}(W(j)(X)) \cong \operatorname{coker}(H^d_{\mbox{\rm \scriptsize Nis}}(X,\Omega^{j-1}_{-/\mathbb{Q}})\to H^d_{\mbox{\rm \scriptsize cdh}}(X,\Omega^{j-1}_{-/\mathbb{Q}}))$. 2. For any qcqs $\mathbb{F}_p$-scheme $X$ of valuative dimension $\le d$, then $W(j)(X)/p$ vanishes in degrees $>j+d+2$ and there is a natural diagram in which the row and column are exact: $$\xymatrix@=5mm{ &&& H^{d-1}_{\mbox{\rm \scriptsize Nis}}(X,\widetilde\nu(j))\ar[d]&\\ &&& H^{d-1}_{\mbox{\rm \scriptsize cdh}}(X,\widetilde\nu(j))\ar[d]&\\ H^d_{\mbox{\rm \scriptsize Nis}}(X,\widehat K_j^M/p)\ar[r] & H^d_{\mbox{\rm \scriptsize cdh}}(X,\widehat K_j^M/p)\ar[r]^-{\delta} & H^{j+d+1}(W(j)(X)/p)\ar[r] & \operatorname{coker}\delta\ar[r] \ar[d]& 0\\ &&& H^{d}_{\mbox{\rm \scriptsize Nis}}(X,\widetilde\nu(j))\ar[d]&\\ &&& H^{d}_{\mbox{\rm \scriptsize cdh}}(X,\widetilde\nu(j))\ar[d]&\\ &&& H^{j+d+2}(W(j)(X)/p)\ar[d]&\\ &&&0& }$$ These two claims are clearly sufficient to deduce the corollary, since Theorem [Theorem 119](#theorem_Weibel_vanishing2){reference-type="ref" reference="theorem_Weibel_vanishing2"} tells us that $W(j)(X)$ (and so also $W(j)(X)/p$) is supported in degrees $\le j+d$ whenever $X$ is a Noetherian $\mathbb{F}$-scheme of dimension $\le d$. It remains to prove the claims. We first treat the case that $\mathbb{F}=\mathbb{Q}$. From the fundamental fibre sequence Theorem [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(3) we have a description of $W(j)(X)$, for any qcqs $\mathbb{Q}$-scheme $X$, as $$W(j)(X)={\rm fib}\left(R\Gamma(X,L\Omega_{-/\mathbb{Q}}^{< j})\to R\Gamma_{\mbox{\rm \scriptsize cdh}}(A/I,L\Omega_{-/\mathbb{Q}}^{< j})\right)[-1].$$ By cohomological vanishing bounds in the Nisnevich and cdh topologies, this fibre is supported in cohomological degrees $\le j+d+1$ if $X$ has valuative dimension $\le d$, with its $H^{j+d+1}$ being exactly the desired cokernel. Next suppose $\mathbb{F}=\mathbb{F}_p$. From the pullback square of Theorem [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"}(2) we see that $W(j)/p=\operatorname{fib}(\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}\to L_{\mbox{\rm \scriptsize cdh}}\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}})$, which we compute as follows: on the category of qcqs $\mathbb{F}_p$-schemes, Nisnevich sheafifying Remark [Remark 53](#remarks_tildenu){reference-type="ref" reference="remarks_tildenu"} provides us with a fibre sequence $$L_{\mbox{\rm \scriptsize Nis}}\tau^{\le j}\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}\longrightarrow\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}\longrightarrow R\Gamma_{\mbox{\rm \scriptsize Nis}}(-,\widetilde\nu(j))[-j-1],$$ which may be compared to its cdh sheafification to get the following fibre sequence of presheaves on qcqs $\mathbb{F}_p$-schemes: $$\operatorname{fib}\big(L_{\mbox{\rm \scriptsize Nis}}\tau^{\le j}\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}\to L_{\mbox{\rm \scriptsize cdh}}\tau^{\le j}\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}\big)\longrightarrow W(j)/p\longrightarrow\operatorname{fib}\big(R\Gamma_{\mbox{\rm \scriptsize Nis}}(-,\widetilde\nu(j))\to R\Gamma_{\mbox{\rm \scriptsize cdh}}(-,\widetilde\nu(j)))\big[-j-1].\label{eqn_nis_vs_cdh}$$ Moreover, $H^j(\mathbb{F}_p(j)^{\mbox{\rm \scriptsize syn}}(-))$ is Nisnevich locally given by $\widehat K_j^M/p$; this follows from the isomorphisms $\widehat K_j^M(A)/p\stackrel{\simeq}{\to}H^j_{\mbox{\rm \scriptsize mot}}(A,\mathbb{Z}/p\mathbb{Z}(j))\stackrel{\simeq}{\to}H^j_{\mbox{\rm \scriptsize syn}}(\mathbb{F}_p(A))$ for local $\mathbb{F}_p$-algebras $A$, the first being the Nesterenko--Suslin isomorphism of Theorem [Theorem 112](#theorem_NS){reference-type="ref" reference="theorem_NS"} (or rather, the mod-$p$ version obtained using Corollary [Corollary 108](#corollary_Hilb_90){reference-type="ref" reference="corollary_Hilb_90"}) and the second isomorphism coming from the fundamental fibre sequence of Corollary [Corollary 55](#corol_fundamental_p){reference-type="ref" reference="corol_fundamental_p"}. Since the Nisnevich and cdh sites of $X$ have cohomological dimension $\le d$ when $X$ has valuative dimension $\le d$, the claimed vanishing and diagram in (2) can now be read off by by evaluating ([\[eqn_nis_vs_cdh\]](#eqn_nis_vs_cdh){reference-type="ref" reference="eqn_nis_vs_cdh"}) on $X$. ◻ # Some comparisons to algebraic cycles {#section_cf_cycles} We present in this section a variety of contexts in which our motivic cohomology admits a description in terms of algebraic cycles. We do not know what to expect in general. **Definition 126**. For $j\ge0$ and $A$ a local $\mathbb{F}$-algebra, we say that $A$ has *geometric weight-$j$ motivic cohomology* if $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)$ is supported in cohomological degrees $\le j$. If $A$ has geometric weight-$j$ motivic cohomology in the sense of the definition, then Theorem [Theorem 107](#thm_lke_lej){reference-type="ref" reference="thm_lke_lej"} implies that the canonical map $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)\to \mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)$ is an equivalence. But, as explained in Remark [Remark 19](#remark_lke_as_cycles){reference-type="ref" reference="remark_lke_as_cycles"}, the complex $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)$ admits a description purely in terms of algebraic cycles; so in this case we obtain a cycle theoretic description of the whole motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)$. This applies to a surprisingly large class of rings: **Theorem 127**. *Let $j\ge0$ and let $A$ be a local $\mathbb{F}$-algebra.* 1. *If $A$ is regular Noetherian then it has geometric weight-$j$ motivic cohomology.* 2. *If $A$ is a valuation ring then it has geometric weight-$j$ motivic cohomology.* 3. *If there exists a nil ideal $I\subseteq A$ such that $A/I$ has geometric weight-$j$ motivic cohomology, then so does $A$.* 4. *If $A$ is Noetherian, henselian of dimension $1$, then it has geometric weight-$j$ motivic cohomology.* 5. *If $j\ge1$ and $A$ is Noetherian, henselian of dimension $\le 2$, then it has geometric weight-$j$ motivic cohomology.* *Proof.* (1): Using Néron--Popescu we reduce, by taking a filtered colimit, to the case that $A$ is essentially smooth over $\mathbb{F}$; then we apply the usual Gersten vanishing bound for classical motivic cohomology after knowing the classical comparison result, Corollary [Corollary 91](#corol_smooth_comparison){reference-type="ref" reference="corol_smooth_comparison"}. (2): If $A$ is a valuation ring then the canonical map $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A)\to \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(A)$ is an equivalence since the right vertical maps of the fundamental squares of Theorems [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"} and [Theorem 47](#thm:graded-pieces_charp){reference-type="ref" reference="thm:graded-pieces_charp"} are equivalences. But now, $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$ is by definition the cdh sheafification of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}$ on affines, so also $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}(A)\stackrel{\sim}{\to}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(A)$. Finally recall once again from the Gersten bound that $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}$ is Zariski locally supported in cohomological degrees $\le j$. (3): We must show that the relative motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A,I)$ is supported in degrees $\le j$; by finitariness we may assume that $I$ is finitary generated, hence nilpotent. First we treat the case that $\mathbb{F}=\mathbb{Q}$. Since $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$ and $R\Gamma_{\mbox{\rm \scriptsize cdh}}(-,L\Omega^{<j}_{-/\mathbb{Q}})$ are cdh sheaves, they are invariant for the ideal $I$; by taking the horizontal fibres of the fundamental fibre sequence Theorem [Theorem 33](#thm:graded-pieces){reference-type="ref" reference="thm:graded-pieces"}(3) we therefore obtain an equivalence of relative terms $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A,I)\simeq L\Omega^{<j}_{A,I/\mathbb{Q}}[-1].$$ The left side is clearly supported in degrees $\le j$, which completes the proof. Next we assume $\mathbb{F}=\mathbb{F}_p$. Then again $\mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}$ and $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}$ are invariant for $I$, and so we obtain an equivalence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A,I)\simeq\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A,I).$$ In this case it is a non-trivial result that the right side is supported in degrees $\le j$ [@AntieauMathewMorrowNikolaus Thm. 5.2]. (4): Combine the Soulé-Weibel vanishing bound Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"} with Corollary [Corollary 108](#corollary_Hilb_90){reference-type="ref" reference="corollary_Hilb_90"}. Part (5) is proved in the same way, but also using Proposition [Proposition 109](#proposition_91){reference-type="ref" reference="proposition_91"}. ◻ **Example 128** (Dimension $0$). Let $j\ge0$ and let $A$ be a local $\mathbb{F}$-algebra with nil maximal ideal $\frak m$. Then, as we saw in the proof of part (3) of the theorem, in characteristic zero there is an equivalence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A,\frak m)\simeq L\Omega^{<j}_{A,\frak m/\mathbb{Q}}[-1]\label{eqn_cycles=differential}$$ for the relative motivic cohomology, and in characteristic $p$ there is an equivalence $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A,\frak m)\simeq \mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(A,\frak m).\label{eqn_cycles=differential2}$$ These are remarkable equivalences. Indeed, $A$ has geometric weight-$j$ motivic cohomology (combine cases (1) and (3) of the theorem) so, resolving $A$ as in Remark [Remark 19](#remark_lke_as_cycles){reference-type="ref" reference="remark_lke_as_cycles"}, the left sides of ([\[eqn_cycles=differential\]](#eqn_cycles=differential){reference-type="ref" reference="eqn_cycles=differential"}) and ([\[eqn_cycles=differential2\]](#eqn_cycles=differential2){reference-type="ref" reference="eqn_cycles=differential2"}) admit presentations purely in terms of complexes of algebraic cycles. But the right sides are linear invariants ultimately built from differential forms. Isomorphisms between algebraic cycles and differential forms also appear in the theory of Chow groups with modulus [@Bloch2003; @Rulling2007; @RullingSaito2018]. For example, if $A=k[x]/x^e$ where $k$ is a perfect field of characteristic $p$, then ([\[eqn_cycles=differential2\]](#eqn_cycles=differential2){reference-type="ref" reference="eqn_cycles=differential2"}) states that $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(A,\frak m)[1]\simeq \mathbb{W}_{ej}(k)/V^e\mathbb{W}_j(k)$$ (using [@Sulyma2023 Thm. 1.1] to describe the syntomic cohomology), thereby offering a cycle theoretic description of the group $\mathbb{W}(k)/V^e\mathbb{W}(k)$. ## Zero cycles on surfaces In Theorem [Theorem 127](#theorem_geometric_cohomology){reference-type="ref" reference="theorem_geometric_cohomology"} we worked in a local context, but the main ideal globalizes. Suppose that $X$ is a qcqs $\mathbb{F}$-scheme such that, for any $x\in X$, the local ring $\mathcal{O}_{X,x}$ has geometric weight-$j$ motivic cohomology. Then, by checking on stalks, we see that the canonical map $$(L_{\mbox{\rm \scriptsize Zar}}L_{{\mbox{\rm \scriptsize Sch}}_{\mathbb{F}}^{\mbox{\rm \scriptsize qcqs,op}}/{\mbox{\rm \scriptsize Sm}}_{\mathbb{F}}^{\mbox{\rm \scriptsize op}}}\mathbb{Z}(j)^{\mbox{\rm \scriptsize cla}})(X)\longrightarrow\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)$$ is an equivalence; that is, the motivic cohomology $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)$ is given by Zariski sheafifying the left Kan extension of motivic cohomology from smooth $\mathbb{F}$-schemes, or in other words (using Theorem [Theorem 107](#thm_lke_lej){reference-type="ref" reference="thm_lke_lej"}) it is given by Zariski sheafifying $\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ on $X_{\mbox{\rm \scriptsize zar}}$. Similar conclusions holds with Zariski replaced by Nisnevich, if we had instead assumed that each henselian local ring $\mathcal{O}_{X,x}^h$ had geometric weight-$j$ motivic cohomology. These arguments allow us to calculate the motivic cohomology of surfaces in low weights: **Corollary 129**. *Let $X$ be a Noetherian $\mathbb{F}$-scheme of dimension $\le 2$ (e.g., a curve or surface, with arbitrarily bad singularities, over a field extension of $\mathbb{F}$). Then there are natural equivalences $$\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\simeq \begin{cases}R\Gamma_{\mbox{\rm \scriptsize cdh}}(X,\mathbb{Z}) & j=0, \\ R\Gamma_{\mbox{\rm \scriptsize Nis}}(X,\mathbb{G}_m)[-1] & j=1,\\ (L_{\mbox{\rm \scriptsize Nis}}\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}})(X) & j\ge2, \end{cases}$$ and an isomorphism $$H^4_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(2))\cong H^2_{\mbox{\rm \scriptsize Nis}}(X, K_2).$$ (The right side denotes Nisnevich cohomology with coefficients in the Nisnevich sheafification of $\mathrm{K}_2$.)* *Proof.* The description of the weight zero motivic cohomology does not depend on the hypotheses on $X$ and may be founded in Examples [Example 34](#example_00){reference-type="ref" reference="example_00"} and [Example 49](#example_0p){reference-type="ref" reference="example_0p"}. For weight one, we appeal to Corollary [Corollary 111](#corollary_1){reference-type="ref" reference="corollary_1"} and note that $R\Gamma_{\mbox{\rm \scriptsize Nis}}(X,\mathbb{G}_m)$ and $\mathbb{Z}(1)^{\mbox{\rm \scriptsize mot}}(X)$ are supported in degrees $\le 2$ and $\le 3$ respectively; for the cohomology of $\mathbb{G}_m$ this is because $X$ has Krull dimension $\le 2$, and for the motivic cohomology we appeal to the Soulé--Weibel vanishing bound, Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"}. Now let $j\ge 2$ (in fact, the following argument equally works when $j=1$). Then the canonical map $$\label{eq:tau-nis} L_{\mbox{\rm \scriptsize Nis}}\tau^{\le j}\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}\to\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$$ of Nisnevich sheaves is an equivalence at all points of $X_{\mbox{\rm \scriptsize Nis}}$ by Theorem [Theorem 127](#theorem_geometric_cohomology){reference-type="ref" reference="theorem_geometric_cohomology"}(5) (which, notably, uses the Soulé-Weibel vanishing bound), hence is an equivalence when evaluated on $X$. Let us now prove the last statement. We write $\mathcal H^j(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}_X)$ for the sheafification on $X_{\mbox{\rm \scriptsize Nis}}$ of $X_{\mbox{\rm \scriptsize Nis}}\ni U\mapsto H^j_{\mbox{\rm \scriptsize mot}}(U,\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}})$. By the equivalence of [\[eq:tau-nis\]](#eq:tau-nis){reference-type="eqref" reference="eq:tau-nis"} and the fact that $X$ has Nisnevich cohomological dimension $\le 2$, there is a natural edge map isomorphism $$H^{j+2}_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j))\cong H^2_{\mbox{\rm \scriptsize Nis}}(X, \mathcal H^j(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}_X)).$$ But the singular Nesterenko--Suslin Theorem [Theorem 112](#theorem_NS){reference-type="ref" reference="theorem_NS"} defines a symbol isomorphism $\widehat{K}_j^M\stackrel{\simeq}{\to}\mathcal H^j(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}_X))$ where $\widehat{K}_j^M$ is the Nesnevich sheaf of improved Milnor $K$-groups on $X$; in case $j=2$ we moreover have $\widehat{K}_2^M\stackrel{\simeq}{\to}K_2$ [@Kerz2010 Prop. 10(3)], completing the proof. ◻ We are very grateful to F. Binda for help with the following proof: **Theorem 130**. *Let $X$ be a reduced, equi-dimensional, quasi-projective surface over a field $k$; then there is a natural isomorphism $$H^4_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(2))\cong \mathrm{CH}_0(X),$$ where $\mathrm{CH}_0(X)$ denotes the lci Chow group of zero cycles[^17] of [@BindaKrishna2018].* *Proof.* In light of the final isomorphism of Corollary [Corollary 129](#corol_surfaces){reference-type="ref" reference="corol_surfaces"}, we must produce a natural isomorphism $\mathrm{CH}_0(X)\cong H^2_{\mbox{\rm \scriptsize Nis}}(X,\mathcal K_2)$. Such Bloch--Quillen formulae for singular surfaces are are due to Levine [@Levine1985] and Binda--Krishna--Saito [@BindaKrishnaSaito2023]; since the precise form we need does not quite explicitly appear in the papers, we provide the extra details. Firstly, citing from [@BindaKrishnaSaito2023 Lem. 3.4] and the reference there to [@GuptaKrishna2020 Lem. 3.2], there is a commutative diagram $$\xymatrix{ H^2_{\mbox{\rm \scriptsize Zar}}(X, K_2)\ar[r]^{\lambda_X'} & H^2_{\mbox{\rm \scriptsize Nis}}(X, K_2)\ar@/^20mm/[dd]^{{\mbox{\rm \scriptsize edge}}}\\ H^2_{\mbox{\rm \scriptsize Zar}}(X, K_2^M)\ar[u]\ar[r]^{\lambda_X} & H^2_{\mbox{\rm \scriptsize Nis}}(X, K_2^M)\ar[d]^{\gamma_X}\ar[u]\ar[d]\\ \mathrm{CH}_0(X)\ar[u]^{\rho_X}\ar[r]_{{\mbox{\rm \scriptsize cyc}}_X} &\mathrm{K}_0(X) }$$ where - $\text{cyc}_X$ and $\rho_X$ are cycle class maps from the lci Chow group; - $\lambda_X$ and $\lambda_X'$ are change of topology maps; - the two vertical maps at the top are induced by the canonical map $\mathrm{K}_2^M\to \mathrm{K}_2$; - the edge map is the edge map in the Nisnevich descent spectral sequence; and $\gamma_X$ is defined to make the curvy triangle commute. We will explain that the cycle class map $\xi_X:\mathrm{CH}_0(X)\to H^2_{\mbox{\rm \scriptsize Nis}}(X,\mathcal K_2)$, defined to be the composite from the bottom left to the top right of the diagram, is an isomorphism. We first treat the case that $k$ is finite (or more generally perfect). According to [@BindaKrishnaSaito2023 Thm 8.1], the cycle class maps $\text{cyc}_X$ is injective (this does not require the hypothesis on $k$); so $\xi_X$ is also injective. Furthermore, according to [@KatoSaito1986 Thm. 2.5], the Nisnevich cycle class map $\lambda_X\rho_X$ is surjective (this does require the hypothesis on $k$, as it means that the regular locus $X_{\mbox{\rm \scriptsize reg}}$ is "nice" in the terminology of \[op. cit.\]). Finally note that $H^2_{\mbox{\rm \scriptsize Nis}}(X, K_2^M)\to H^2_{\mbox{\rm \scriptsize Nis}}(X, K_2)$ is surjective, because $X_{\mbox{\rm \scriptsize Nis}}$ has cohomological dimension $2$ and the map of Nisnevich sheaves $K_2^M\to K_2$ is surjective. The last two sentences show $\xi_X$ is surjective, completing the proof in this case. Next we treat the case that $k$ is infinite. Then the cycle class map $\mathrm{CH}_0(X)\to H^2_{\mbox{\rm \scriptsize Nis}}(X, K_2)$ (i.e., bottom left to top left of the diagram) is an isomorphism by [@BindaKrishnaSaito2023 Corol. 7.8]. It remains to show that $\lambda_X'$ is an isomorphism (which does not require the hypothesis on $k$); this is well-known to experts but we could not find a reference. This isomorphism is proved by comparing the Zariski descent spectral sequence $E_2^{i,j}=H^i_{\mbox{\rm \scriptsize Zar}}(X, K_{-j})\Rightarrow \mathrm{K}_{-i-j}(X)$ to the analogous Nisnevich descent spectral sequence, as follows. Both spectral sequences are supported in columns $i=0,1,2$ since $X_{\mbox{\rm \scriptsize Zar}}$ and $X_{\mbox{\rm \scriptsize Nis}}$ have cohomological dimension $2$; therefore the only non-zero differentials $\partial$ are on the first page, from the $0^{\mbox{\rm \scriptsize th}}$ column to the $2^{\mbox{\rm \scriptsize ed}}$ column, and so the abutement filtrations on $\mathrm{K}_0(X)$ are described via a commutative diagram $$\xymatrix{ &&&H^0_{\mbox{\rm \scriptsize Zar}}(X,\mathbb{Z})&\\ H^0_{\mbox{\rm \scriptsize Zar}}(X,\mathbb{G}_m)\ar[r]^{\partial} & H^2_{\mbox{\rm \scriptsize Zar}}(X, K_2)\ar[r]^{{\mbox{\rm \scriptsize edge}}} & \mathrm{K}_0(X) \ar[r] & \mathrm{K}_0(X)/\text{edge}(H^2_{\mbox{\rm \scriptsize Zar}}(X, K_2))\ar[r]\ar[u]&0\\ &&&H^1_{\mbox{\rm \scriptsize Zar}}(X,\mathbb{G}_m)\ar[u]&\\ &&&0\ar[u]&\\ }$$ and similarly replacing Zar by Nis everywhere. The Zariski diagram maps to the Nisnevich one, involving in particular the map $\lambda_X'$, and one sees from a diagram chase that $\lambda_X'$ being an isomorphism follows from the following standard facts: 1. $H^0_{\mbox{\rm \scriptsize Zar}}(X,\mathbb{Z})\to H^0_{\mbox{\rm \scriptsize Nis}}(X,\mathbb{Z})$ is injective (it is even an isomorphism); 2. $H^1_{\mbox{\rm \scriptsize Zar}}(X,\mathbb{G}_m)\to H^1_{\mbox{\rm \scriptsize Nis}}(X,\mathbb{G}_m)$ is injective (it is even an isomorphism); 3. the boundary maps $\partial$ in both the Zariski and Nisnevich diagrams is zero because $\mathbb{G}_m$ is representable by a one-dimensional scheme: more precisely, given $f\in H^0_{\mbox{\rm \scriptsize Zar}}(X,\mathbb{G}_m)$ (resp. Nisnevich), let $X\to \mathrm{Spec}(k[t^{\pm1}])$ be the induced map; then the analogous boundary map in the Zariski (resp. Nisnevich) descent spectral sequence for $\mathrm{Spec}(k[t^{\pm1}])$ is zero, simply because $H^2_{\mbox{\rm \scriptsize zar}}(\mathrm{Spec}k[t^{\pm1}],\mathcal K_2)=0$ (resp. Nisnevich) for dimensional reasons; so by functoriality we deduce $\partial(f)=0$, as desired. This completes the proof. ◻ **Remark 131** (Zero cycles). The argument at the end of the proof of Corollary [Corollary 129](#corol_surfaces){reference-type="ref" reference="corol_surfaces"} shows, for any $j\ge0$ and any qcqs $\mathbb{F}$-scheme $X$ of Krull dimension $\le d$, that there is a natural map $$H^d_{\mbox{\rm \scriptsize Nis}}(X,\widehat{K}_j^M) \cong H^{j+d}(L_{\mathrm{Nis}}\tau^{\leq j}\mathbb{Z}(j)^{\mathrm{mot}}(X)) \longrightarrow H^{j+d}_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(j)).$$ Taking $j=d=\text{dim}(X)$, we hope that $H^{2d}_{\mbox{\rm \scriptsize mot}}(X,\mathbb{Z}(d))$ provides a "good" group of zero cycles on $X$. This point of view will be explored further elsewhere. # The $\mathrm{cdh}$-sheafification of an étale sheaf {#app:cdh} In this technical appendix we prove a result about éh sheafification, stating that it is equivalent to étale sheafification followed by cdh sheafification. While this might appear to be merely a curiosity, it is a crucial input into controlling our mod-$p$ motivic cohomology in characteristic $p$. We begin by recalling the definition: **Definition 132**. Letting $R$ be a commutative ring, the *$\text{\'{e}h}$ topology* on $\textrm{Sch}_R^{\mbox{\rm \scriptsize qcqs}}$ is the Grothdendieck topology generated by abstract blowup squares[^18] and étale covers. It is thus relatively formal that a presheaf $\mathcal F:\textrm{Sch}_R^{\mbox{\rm \scriptsize qcqs}}\to \textrm{Sp}$ is an éh sheaf if and only if it is both a cdh and étale sheaf; in particular, assuming $\mathcal F$ is an étale sheaf, then it is an éh sheaf if and only if it sends abstract blowup squares to cartesian squares. **Remark 133**. The $\text{\'{e}h}$ topology is finer than the $\mathrm{cdh}$ topology but coarser than the $\mathrm{h}$ topology; the latter can be defined as the topology generated by abstract blowup squares and fppf covers. It is an insight of Geisser, when defining his *arithmetic cohomology* of separated, finite type schemes over finite fields, that the $\text{\'{e}h}$ topology is better suited to capturing mod-$p$ information than the $h$ topology [@Geisser2006 Page 30, Remark]. Furthermore, the points of the $\text{\'{e}h}$ topology are given by strictly henselian valuation rings. To prove the main result of the appendix, let us recall the following notion introduced in [@ElmantoHoyoisIwasaKelly2021]. We say that an $\infty$-category $\mathcal C$ is *compactly generated by cotruncated objects* if it is compactly generated and each compact object is contruncated. The main point of $\mathcal C$ is that filtered colimits commute with cosimplicial limits as noted in [@ElmantoHoyoisIwasaKelly2021 Lem. 3.1.7(1)]. **Theorem 134**. *Let $R$ be a commutative ring. Let $\mathcal C$ be a stable $\infty$-category which is compactly generated by cotruncated objects. Then for any finitary, étale sheaf of spectra $$F: \mathrm{S}\mathrm{ch}{}_R^{\mathrm{qcqs},\mathrm{op}} \rightarrow \mathcal C,$$ we have that $L_{\mathrm{cdh}}F$ is an $\text{\'{e}h}$-sheaf.* *Proof.* It suffices to prove that $L_{\mathrm{cdh}}F$ satisfies étale descent. Since $\mathrm{cdh}$-sheaves are, in particular, Nisnevich sheaves, it suffices to prove that $L_{\mathrm{cdh}}F$ satifies finite étale descent by the structure result of [@LurieSAG Thm. B.6.4.1]. Let $X \rightarrow Y$ be a finite étale cover and let $C^{\bullet}_Y(X)$ be the Čech nerve; our goal is to prove that the map $L_{\mathrm{cdh}}F(X) \rightarrow \mathop{\mathrm{lim}}_{\Delta} L_{\mathrm{cdh}}F(C^{\bullet}_Y(X))$ is an equivalence. Set $$G: \mathrm{S}\mathrm{ch}{}_X^{\mbox{\rm \scriptsize qcqs,op}} \rightarrow \text{Sp}\qquad U \mapsto \mathrm{Fib}( L_{\mathrm{cdh}}F(U) \rightarrow \mathop{\mathrm{lim}}_{\Delta} L_{\mathrm{cdh}}F(C^{\bullet}_Y(X)\times_X U)).$$ We claim that $G \simeq 0$. Since a $\mathrm{cdh}$ and Nisnevich squares are stable under taking products of schemes, $G$ is a $\mathrm{cdh}$ sheaf on $\mathrm{S}\mathrm{ch}{}_X$. Since totalisations commute with filtered colimits in $\mathcal C$ [@ElmantoHoyoisIwasaKelly2021 Lem. 3.1.7(1)], $G$ is finitary. Furthermore, by part (2) of the same lemma, $G$ is hypercomplete. Hence to prove that $G \simeq 0$, it suffices to argue that for any henselian valuation ring $V$ with a map $X \rightarrow \mathrm{Spec}(V)$, we have that $G(V) \simeq 0$. Now, since finite étale morphisms are stable under base change, the map $C^{\bullet}_Y(X) \times_X \mathrm{Spec}(V) \rightarrow \mathrm{Spec}(V)$ is a cosimplicial diagram such that the face maps are all finite étale. But now, by a theorem of Nagata which we review below, each term of the Čech nerve is a coproduct of henselian valuation rings. Therefore, since $L_{\mathrm{cdh}}F$ does preserve coproduct decompositions and henselian valuation rings are $\mathrm{cdh}$ points, we have an equivalence of cosimplicial objects. $$L_{\mathrm{cdh}}F(C^{\bullet}_Y(X) \times_X V) \simeq F(C^{\bullet}_Y(X) \times_X V ).$$ The limit of the latter is $F(V)$ since $F$ was assumed to be an étale sheaf and we conclude by the fact that $L_{\mathrm{cdh}}F(V) \simeq F(V)$. ◻ **Corollary 135**. *Let $R$ be a commutative ring. With the same hypotheses as in Theorem [Theorem 134](#theorem:eh){reference-type="ref" reference="theorem:eh"} on $\mathcal C$, we have a canonical equivalence of endofucntors on $\mathcal C$-valued presheaves on $\mathrm{S}\mathrm{ch}{}^{\mathrm{qcqs}}_R$: $$L_{\mathrm{cdh}}L_{{\acute{e}t}} \simeq L_{\text{\'{e}h}}.$$* **Lemma 136**. *\[Nagata's Hensel lemma\] Let $R$ be a henselian valuation ring. Then, for any finite étale morphism $R \rightarrow S$, $S$ is a product of henselian valuation rings.* *Proof.* We give a proof using more modern references. Since $S$ is finite over a henselian local ring, by [@Stacks Tag 04GH], $S$ is a finite product of henselian local rings, each of which is finite over $S$. It then suffices to prove to assume furthermore that $S$ is a henselian local ring and prove that $S$ is, in fact, a valuation ring. Since étale morphisms are flat and the diagonal is an open immersion (hence flat), [@HuberKelly2018 Corol. 2.15] asserts that localizations of $S$ at any prime ideal is a valuation ring. But we are done because $S$ is, in fact, a local ring. ◻ **Remark 137**. Let $V$ be a valuation ring and let $\text{\'Et}_V$ be its étale site; the underlying category are étale $V$-schemes. The above proof shows a little more we have an equivalence: $$L_{\mathrm{Zar}}\left( F|_{\text{\'Et}_V} \right) \simeq L_{\mathrm{cdh}}F|_{\text{\'Et}_V}.$$ **Remark 138**. The order of sheafifications in Theorem [Theorem 134](#theorem:eh){reference-type="ref" reference="theorem:eh"} is important: the canonical map $L_{\mbox{\rm \scriptsize \'et}}L_{\mbox{\rm \scriptsize cdh}}\mathcal F\to L_{\mbox{\rm \scriptsize \'eh}}\mathcal F$ need not be an equivalence, because the étale sheafification of a cdh sheaf need not be a cdh sheaf. # A spectrum level Cortiñas--Haesemeyer--Weibel theorem {#app:chw} The goal of this appendix is to prove the following technical enhancement of the main theorem of [@Cortinas2009]. **Theorem 139**. *Let $n \geq 2$ and suppose that $k$ is a $\mathbb{Q}$-algebra. Then for any qcqs $k$-scheme $X$ we have a commutative diagram of $\mathbb{E}_{\infty}$-rings, functorial in $X$: $$\label{eq:chw} \begin{tikzcd} \mathrm{K}(X)[\tfrac{1}{n}] \ar{r}{\psi^n} \ar{d}{ch} & \mathrm{K}(X)[\tfrac{1}{n}] \ar{d}{ch}\\ \mathrm{HC}^-(X/k) \ar{r}{\psi^n} & \mathrm{HC}^-(X/k). \end{tikzcd}$$* Upon taking homotopy groups, we get back the main compatibility result asserted in [@Cortinas2009]. As mentioned in the introduction of [@Cortinas2009], the classical Adams operations on $K$-theory and negative cyclic homology are defined in "very different ways.\" The key point of our proof of Theorem [Theorem 139](#thm:chw-spt){reference-type="ref" reference="thm:chw-spt"} is that, given the new context of [@AnnalaIwasa2023], the two constructions are not so different after all. In particular, our proof is quite different from the one [@Cortinas2009] where the key point is to use Goodwillie's theorem about $\mathrm{K}^{\inf}_{\mathbb{Q}}$ and Cathelineu's result on compatibility of Adams operations in the infinitesimal context (whose proof was repaired in [@Cortinas2009 App. B]). In fact, what we will need in the main body of the paper is a filtered enhancement of Theorem [Corollary 148](#cor:adams-compatible){reference-type="ref" reference="cor:adams-compatible"}, in the context of smooth $k$-schemes when the classical motivic filtration (in the sense of  §[3.1](#ss_classical){reference-type="ref" reference="ss_classical"}) is defined. ## Construction of the Adams operations via the Annala-Iwasa theorem {#sec:ai-thm} To begin, we offer a construction of the Adams operations on $n$-periodic $K$-theory using the technology of [@AnnalaIwasa2023]. Fix a commutative ring $k$ and consider the category of smooth $k$-schemes; as in [@AnnalaIwasa2023] we denote by $$\mathrm{St}_k:=\mathrm{Shv}_{\mathrm{Zar}}({\mathrm{S}\mathrm{m}}_k;\mathcal{S}\mathrm{pc}{}),$$ the $\infty$-category of Zariski sheaves of spaces on smooth $k$-schemes; we call the latter the $\infty$-category of *Zariski $k$-stacks*. We have the *Picard stack*[^19] $\mathcal{P}\mathrm{ic}\in \mathrm{St}_k$[@AnnalaIwasa2023 2.1.4] which comes equipped with a canonical map of Zariski $k$-stacks $$\mathcal{P}\mathrm{ic}\rightarrow \Omega^{\infty}\mathrm{K}.$$ In fact, $\mathcal{P}\mathrm{ic}$ is a grouplike $\mathbb{E}_{\infty}$-monoid in $k$-stacks (under tensor products of line bundles/induced by the group structure on ${\mathbb{G}_m}$); this is equivalent to saying that the functor $\mathcal{P}\mathrm{ic}: {\mathrm{S}\mathrm{m}}^{\mathrm{op}}_k \rightarrow \mathcal{S}\mathrm{pc}{}$ in fact promotes to a functor $\mathcal{P}\mathrm{ic}: {\mathrm{S}\mathrm{m}}^{\mathrm{op}}_k \rightarrow \mathrm{CMon}$ where $\mathrm{CMon}$ denote the $\infty$-category of $\mathbb{E}_{\infty}$-monoids in spaces. The (pointed) suspension spectrum functor $\Sigma^{\infty}_+: \mathcal{S}\mathrm{pc}{}\rightarrow \text{Sp}$ is lax monoidal whence it promotes to a functor $\Sigma^{\infty}_+: \mathrm{CMon}\rightarrow \mathrm{CAlg}(\text{Sp})$. As in [@AnnalaIwasa2023 5.3.1] we set $$\mathbb{S}[\mathcal{P}\mathrm{ic}]:= \Sigma^{\infty}_+\mathcal{P}\mathrm{ic}\in \mathrm{CAlg}(\mathrm{St}_k).$$ Since the canonical map of Zariski stacks $\mathcal{P}\mathrm{ic}\rightarrow \Omega^{\infty}K$ is a morphism of commutative monoids (where the right hand side is given the multiplicative structure), the composite $$\label{eq:spic} \mathbb{S}[\mathcal{P}\mathrm{ic}] \rightarrow \Sigma^{\infty}_+\Omega^{\infty}K \rightarrow K$$ defines a morphism in $\mathrm{CAlg}(\mathrm{St}_k)$. The result [@AnnalaIwasa2023 Thm. 5.3.3] concerns this map, where it is proved that it becomes an equivalence under a certain localization. To make use of this result, we introduce several notation from [@AnnalaIwasa2023]: 1. we have the pointed version of $S$-stacks $\mathrm{St}_{k\ast}$ which is a symmetric monoidal $\infty$-category; it contains the Yoneda image of $\mathbb{P}^1$ pointed at $\infty$. 2. In $\mathrm{St}_{k\ast}$ we have the $\mathbb{E}_{\infty}$-ring $Q(\mathcal{P}\mathrm{ic}):= \Omega^{\infty}\mathbb{S}[\mathcal{P}\mathrm{ic}]$; whence we may speak of $Q(\mathcal{P}\mathrm{ic})$-modules in $\mathrm{St}_{k\ast}$. 3. We have the *Bott element* which is a map in $\mathrm{St}_{k\ast}$ $$\beta:=1 - [\mathcal O(-1)]: \mathbb{P}^1 \rightarrow \Omega^{\infty}\mathbb{S}[\mathcal{P}\mathrm{ic}];$$ classifying the above named bundle. 4. By the previous construction, $Q(\mathcal{P}\mathrm{ic})$-module comes equipped with a canonical map $$\beta:E \rightarrow E^{\mathbb{P}^1};$$ and we say that $E$ is $\mathbb{P}^1$-periodic if this map is an equivalence. 5. we have the stabilized version of $\mathrm{St}_k$ denoted by $\text{Sp}(\mathrm{St}_k)$, modeled by Zariski sheaves of spectra om ${\mathrm{S}\mathrm{m}}_k$; it comes equipped with the functor $\Sigma^{\infty}_+:\mathrm{St}_k \rightarrow \text{Sp}(\mathrm{St}_k)$ with a right adjoint $\Omega^{\infty}$. Since $\Sigma^{\infty}_+$ is lax monoidal, we obtain an induced adjunction: $$Q(\mathcal{P}\mathrm{ic})\text{-}\mathcal{M}\mathrm{od}(\mathrm{St}_k) \leftrightarrows {\mathbb{S}[\mathcal{P}\mathrm{ic}]\text{-}\mathcal{M}\mathrm{od}}(\text{Sp}(\mathrm{St}_k)).$$ 6. the morphism $\mathbb{S}[\mathcal{P}\mathrm{ic}] \rightarrow K$ is automatically a morphism of $\mathbb{E}_{\infty}$-algebra objects in ${\mathbb{S}[\mathcal{P}\mathrm{ic}]}\text{-}\mathcal{M}\mathrm{od}(\text{Sp}(\mathrm{St}_k))$. Here is the main theorem of [@AnnalaIwasa2023]: **Theorem 140** (Annala--Iwasa). *We have an equivalence of $\mathbb{E}_{\infty}$-algebras in ${\mathbb{S}[\mathcal{P}\mathrm{ic}]}\text{-}\mathcal{M}\mathrm{od}(\mathrm{Sp}(\mathrm{St}_k))$: $$\mathbb{S}[\mathcal{P}\mathrm{ic}][\beta^{-1}] \rightarrow \mathrm{K}.$$* With this we can define the Adams operations on the $n$-periodization of non-connective $K$-theory as follows. First, consider the map of group schemes: $${\mathbb{G}_m}\rightarrow {\mathbb{G}_m}\qquad z \mapsto z^n.$$ This induces a map of commutative monoids: $$\left(-\right)^n: \mathcal{P}\mathrm{ic}\rightarrow \mathcal{P}\mathrm{ic};$$ whence a map in $\mathrm{CAlg}(\mathrm{St}_{k})$: $$\psi^n: Q(\mathcal{P}\mathrm{ic}) \rightarrow Q(\mathcal{P}\mathrm{ic}).$$ Composing the Bott element with the map $Q(\mathcal{P}\mathrm{ic}) \rightarrow \Omega^{\infty}K$ we get a map in $\mathrm{St}_{k\ast}$: $$\beta_K:\mathbb{P}^1 \rightarrow \Omega^{\infty}K.$$ By the exact same argument as in [@BachmannHopkins2000 Lem. 3.11] we have: **Lemma 141**. *The map $$\mathbb{P}^1 \rightarrow Q(\mathcal{P}\mathrm{ic}) \xrightarrow{\psi^n} Q(\mathcal{P}\mathrm{ic}) \rightarrow \Omega^{\infty}K$$ is homotopic to $n\beta_K$.* The next construction follows [@BachmannHopkins2000 §3.3.1]. **Construction 142**. Since $\psi^n: Q(\mathcal{P}\mathrm{ic}) \rightarrow Q(\mathcal{P}\mathrm{ic})$ is a morphism of $\mathbb{E}_{\infty}$-rings, we can restrict the $Q(\mathcal{P}\mathrm{ic})$ $\Omega^{\infty}K$ along $\psi^n$; which we denoted by $\Omega^{\infty}K^{[n]}$. We then have a morphism of $Q(\mathcal{P}\mathrm{ic})$-modules $$Q(\mathcal{P}\mathrm{ic}) \xrightarrow{\psi^n} Q(\mathcal{P}\mathrm{ic}) \rightarrow \Omega^{\infty}K^{[n]},$$ whence a morphism of $\mathbb{S}[\mathcal{P}\mathrm{ic}]$-modules: $$\mathbb{S}[\mathcal{P}\mathrm{ic}] \xrightarrow{\psi^n} \mathbb{S}[\mathcal{P}\mathrm{ic}] \rightarrow K^{[n]};$$ here we have abusively denoted the suspension of $\psi^n$ by the same name and $K^{[n]}$ is the $\mathbb{S}[\mathcal{P}\mathrm{ic}]$-module obtained by the same restriction. Inverting $\beta$ along the above morphism and applying Lemma [Lemma 141](#lem:nbeta){reference-type="ref" reference="lem:nbeta"} and Theorem [Theorem 140](#thm:ai-main){reference-type="ref" reference="thm:ai-main"} yields a morphism of $\mathbb{E}_{\infty}$-algebras in $\mathbb{S}[\mathcal{P}\mathrm{ic}]$-modules: $$K \rightarrow K^{[n]}[n\beta^{-1}_{K}] \simeq K[\tfrac{1}{n}],$$ which factors through the $n$-periodization of $K$ and thus gives a map of $\mathbb{E}_{\infty}$-algebras in $\mathbb{S}[\mathcal{P}\mathrm{ic}]$-modules $$\psi^n: K[\tfrac{1}{n}] \rightarrow K[\tfrac{1}{n}].$$ We regard $\psi^n$ as a $\mathbb{E}_{\infty}$-algebra maps in $\text{Sp}(\mathrm{St}_k)$. ## Adams operations and the slice filtration {#sec_adams-slice} We will now study the interaction of the Adams operations, as defined above, with the classical motivic filtration $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X)$ on $K$-theory, at least after rationalization. As reviewied in §[3.1](#ss_classical){reference-type="ref" reference="ss_classical"} the classical motivic filtration is incarnated by Voevodsky's the slice filtration in the sense of stable motivic homotopy theory. This filtration is *a priori* defined on $\mathrm{KH}$ as it relies on $\mathbb{A}^1$-invariance properties. However, in this section, we assume that $k$ is a field so for $X$ a smooth $k$-scheme, we have that $\mathrm{KH}(X) \simeq \mathrm{K}(X)$ and thus $\mathrm{KH}(X)_{\mathbb{Q}} \simeq \mathrm{K}(X)_{\mathbb{Q}}$. To begin, note that the Construction [Construction 142](#constr:adams-k){reference-type="ref" reference="constr:adams-k"} agrees with the construction from motivic homotopy theory. Indeed, our construction manifestly coincides[^20] with the construction from [@BachmannHopkins2000 §3.3] which has been compared to prior constructions due to Riou by [@BachmannHopkins2000 Prop. 3.12]. Therefore, in the context of the motivic stable $\infty$-category over $k$, $\mathcal{SH}(k)$, Construction [Construction 142](#constr:adams-k){reference-type="ref" reference="constr:adams-k"} furnishes multiplicative maps in $\mathcal{SH}(k)$: $$\psi^n: \mathrm{KGL}_{\mathbb{Q}} \rightarrow \mathrm{KGL}_{\mathbb{Q}} \qquad n \geq 2.$$ Here $\mathrm{KGL}$ is the motivic spectrum representing (homotopy invariant) algebraic $K$-theory and $\mathrm{KGL}_{\mathbb{Q}}$ denotes its rationalization. For the purposes of this paper we need to check that $\mathrm{KGL}_{\mathbb{Q}}$ decomposes in a multiplicatively compatible with the slice filtration, which is formal once one gathers some results already available in the literature. The main result of [@Riou2007] furnishes a decomposition in $\mathcal{SH}(S)$[^21]: $$\label{riou-decomp} \mathrm{KGL}_{\mathbb{Q}} \simeq \bigoplus_{j \in \mathbb{Z}} \mathrm{KGL}^{\psi^n-n^j}_{\mathbb{Q}}$$ One then defines the *Beilinson motivic cohomology spectrum* as the piece $$\mathrm{H}\mathbb{Q}:= \mathrm{KGL}^{\psi^n-\operatorname{id}}_{\mathbb{Q}},$$ this construction is independent of $n$. This is a motivic spectrum which represents rationalized motivic cohomology whenever the latter is defined. **Remark 143**. In Spitweck's theory of $\mathbb{A}^1$-invariant motivic cohomology [@Spitzweck2018] $\mathrm{KGL}^{\psi^n-\operatorname{id}}_{\mathbb{Q}}$ is, by construction, the rational part of his motivic cohomology. Furthermore, the Bott element $\beta = 1 - [\mathcal{O}(-1)] \in K_0(\mathbb{P}^1_k)$ induces an invertible map $\beta: \Sigma^{2,1}\mathrm{KGL}\rightarrow \mathrm{KGL}$ in $\mathcal{SH}(k)$ such that the composite $$\Sigma^{2,1}\mathrm{KGL}^{\psi^n-n^j}_{\mathbb{Q}} \rightarrow \Sigma^{2,1}\mathrm{KGL}_{\mathbb{Q}} \xrightarrow{\beta} \mathrm{KGL}_{\mathbb{Q}} \rightarrow \mathrm{KGL}^{\psi^n-n^{j+1}}_{\mathbb{Q}}$$ is an equivalence [@triangulated-mixed-motives Lem. 14.1.4]. Therefore, we have equivalences, for all $j \in \mathbb{Z}$: $$\Sigma^{2j,j}\mathrm{H}\mathbb{Q} \simeq \Sigma^{2,1}\mathrm{KGL}^{\psi^n-n^j}_{\mathbb{Q}}.$$ The interaction between the multiplicative structures of $\mathrm{H}\mathbb{Q}$ and $\mathrm{KGL}_{\mathbb{Q}}$ was examined in [@triangulated-mixed-motives §14.2]. Here are the key points: the inclusion of the summand (as in the decomposition [\[riou-decomp\]](#riou-decomp){reference-type="ref" reference="riou-decomp"}), $\mathrm{H}\mathbb{Q} \rightarrow \mathrm{KGL}_{\mathbb{Q}}$ is an $\mathbb{E}_{\infty}$-map [@triangulated-mixed-motives Corol. 14.2.17]. Furthermore we consider the motivic spectrum $\mathrm{H}\mathbb{Q}[t, t^{-1}]$, the free $\mathbb{E}_{\infty}$-$\mathrm{H}\mathbb{Q}$-algebra generated by a single invertible generator in degree $(2,1)$ (this construction is also studied more explicitly in [@Spitzweck2018 App. C]). The underlying graded spectrum of this object is $\bigoplus_{j \in \mathbb{Z}}\Sigma^{2j,j}\mathrm{H}\mathbb{Q}$. We have a multiplicative filtration[^22] $\mathrm{Fil}^{\star}\mathrm{H}\mathbb{Q}[t, t^{-1}]$ on $\mathrm{H}\mathbb{Q}[t, t^{-1}]$ where, on underlying object, we have $$\mathrm{Fil}^{i}\mathrm{H}\mathbb{Q}[t, t^{-1}] = \bigoplus_{j \geq i} \Sigma^{2j,j}\mathrm{H}\mathbb{Q}.$$ The canonical map $$\label{eq:kq} \mathrm{H}\mathbb{Q}[t, t^{-1}] \rightarrow \mathrm{KGL}_{\mathbb{Q}},$$ induced by the inclusion of the summand $\Sigma^{2,1}\mathrm{H}\mathbb{Q}\simeq \mathrm{KGL}^{\psi^n-n}_{\mathbb{Q}} \rightarrow \mathrm{KGL}_{\mathbb{Q}}$ is an equivalence. We record the following result about the compatibility of this equivalence with the filtration. We use the notation from Remark [Remark 21](#remark_slice_filtration){reference-type="ref" reference="remark_slice_filtration"}: if $E$ is a motivic spectrum then $f^{\star}E$ denotes the slice filtration. **Theorem 144**. *Let $k$ be a field. The equivalence [\[eq:kq\]](#eq:kq){reference-type="eqref" reference="eq:kq"} enhances to a filtered $\mathbb{E}_{\infty}$-equivalence in $\mathcal{SH}(k)$: $$\mathrm{Fil}^{\star} \mathrm{H}\mathbb{Q}[t, t^{-1}] \xrightarrow{\simeq} f^{\star}\mathrm{KGL}_{\mathbb{Q}}.$$* *Proof.* Over any base, by the functoriality and the lax monoidality of the slice filtration [@BachmannHoyois2021], the map $\mathrm{H}\mathbb{Q}[t, t^{-1}] \rightarrow \mathrm{KGL}_{\mathbb{Q}}$ from [\[eq:kq\]](#eq:kq){reference-type="eqref" reference="eq:kq"} induces an $\mathbb{E}_{\infty}$-map of filtered objects: $$f^{\star} \mathrm{H}\mathbb{Q}^{\mathrm{Spi}}[t, t^{-1}] \rightarrow f^{\star}\mathrm{KGL}_{\mathbb{Q}}.$$ To prove the claim, restricting ourselves to the hypothesis on $k$, it suffices to prove that for all $i \in \mathbb{Z}$, we have $f^{\geq i} (\bigoplus_{j < i} \mathrm{H}\mathbb{Q}(i)[2i]) \simeq 0$. Since $f^{\star}$ commutes with colimits [@elso Prop. 3.5] and twists [@rondigs2013slices Lem. 2.1], it suffices to know that $f^{\geq 1}\mathrm{H}\mathbb{Q}^{\mathrm{Spi}} \simeq 0$ which follows from the fact that negative weight motivic cohomology vanishes. ◻ Translating the result above to the classical motivic filtration reviewed in §[3.1](#ss_classical){reference-type="ref" reference="ss_classical"} and Remark [Remark 21](#remark_slice_filtration){reference-type="ref" reference="remark_slice_filtration"}, we have proved: **Corollary 145**. *Let $n \geq 2$ and $k$ a field. For any essentially smooth $k$-scheme $X$, we have that:* 1. *the Adams operations $\psi^{n}: \mathrm{K}(X)_{\mathbb{Q}} \rightarrow \mathrm{K}(X)_{\mathbb{Q}}$ promotes to a multiplicative map of filtered objects $$\psi^{n}:\mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X)_{\mathbb{Q}} \rightarrow \mathrm{Fil}^\star_{\mbox{\rm \scriptsize cla}}\mathrm{K}(X)_{\mathbb{Q}};$$* 2. *on the graded pieces $\mathrm{gr}^{j}_{\mbox{\rm \scriptsize cla}}$, $\psi^n$ acts by $\cdot n^j$;* 3. *there is a natural equivalence of filtered $\mathbb{E}_{\infty}$-rings: $$\bigoplus_{j \in \mathbb{Z}} \mathbb{Q}(j)^{{\mbox{\rm \scriptsize cla}}}(X)[2j] \xrightarrow{\simeq} \mathrm{K}(X)_{\mathbb{Q}}$$ which preserves the natural filtration on the graded object on the domain and the slice filtration on the target.* 4. *The spectral sequence degenerates rationally.* ## Adams operations and the Chern character Let us now recall the Adams operations in negative cyclic homology. While these operations are classically defined for the negative cyclic homology groups and even complexes [@Loday1989; @Weibel1997], we use Raksit's thesis [@raksit-hkr] as our main reference. To the best of our knowledge, it is the first instance where Adams operations on negative cyclic homology was proved to be a functorially a filtered map, on the level of spectra. As in the case of $K$-theory, we write $\mathrm{HH}(X/k)^{[n]}$ be the $S^1$-equivariant $\mathbb{E}_{\infty}$-ring spectrum[^23], obtained by restricting the $S^1$-action along the $n$-power map $[n]:S^1 \rightarrow S^1$. As in [@raksit-hkr Cons. 6.4.3], we obtain an $\mathbb{E}_{\infty}$-ring map $\mathrm{HH}(X/k) \rightarrow \mathrm{HH}(X/k)^{[n]}$. As proved in [@raksit-hkr Prop. 6.4.4], the Adams operations promote to a multiplicative, $S^1$-equvariant, filtered map $$\psi^n: \mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}\mathrm{HH}(X/k) \rightarrow \mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}\mathrm{HH}(X/k)^{[n]}.$$ Passing to fixed points and applying [@raksit-hkr Lems. 6.4.5-6.4.6] we get a the *Adams operation* on $\mathrm{HC}^-$: $$\label{eq:hc-adams} \psi^n: \mathrm{HC}^-(X/k) \rightarrow \mathrm{HC}^-(X/k);$$ as in [@raksit-hkr Cons. 6.4.7]. This refines to a filtered map as verified in [@raksit-hkr Cons. 6.4.8]: $$\label{eq:filtered-hc-} \psi^n: \mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/k) \rightarrow \mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}\mathrm{HC}^-(X/k).$$ **Lemma 146**. *We have the following commutative diagram in $\mathrm{CAlg}(\mathrm{St}_k)$: $$\label{key-hc} \begin{tikzcd} Q(\mathcal{P}\mathrm{ic}) \ar{r} \ar{d} & \Omega^{\infty}\mathrm{HC}^-(-/k) \ar{d}\\ Q(\mathcal{P}\mathrm{ic}) \ar{r} & \Omega^{\infty}\mathrm{HC}^-(-/k). \end{tikzcd}$$* *Proof.* The endofunctor $Q: \mathrm{St}_k \rightarrow \mathrm{St}_k$ is a monad for grouplike $\mathbb{E}_{\infty}$-monoids in $\mathrm{St}_k$ and $Q(\mathcal{P}\mathrm{ic})$ is the free monad on the object $\mathcal{P}\mathrm{ic}$. Since $\Omega^{\infty}\mathrm{HC}^-(-/k)$ is a grouplike $\mathbb{E}_{\infty}$-monoid, to prove commutation of [\[key-hc\]](#key-hc){reference-type="eqref" reference="key-hc"} we need only check commutation of the diagram in $\mathrm{St}_k$: $$\label{key-hc-st} \begin{tikzcd} \mathcal{P}\mathrm{ic}\ar{r} \ar[swap]{d}{\left(-\right)^n} & \Omega^{\infty}\mathrm{HC}^-(-/k) \ar{d}{\psi^n}\\ \mathcal{P}\mathrm{ic}\ar{r} & \Omega^{\infty}\mathrm{HC}^-(-/k). \end{tikzcd}$$ There are several ways to verify this: for example, using (the proof of) [@AnnalaIwasa2023 Corol. 4.3.3] we see that both composites coincide with the element $$nc_1 \in \mathrm{HC}^-_0(k)[[c_1]] = k[[c_1]].$$ ◻ *Proof of Theorem [Theorem 139](#thm:chw-spt){reference-type="ref" reference="thm:chw-spt"}.* It follows immediately from the commutativity of [\[key-hc\]](#key-hc){reference-type="eqref" reference="key-hc"} from Lemma [Lemma 146](#lem:key-q){reference-type="ref" reference="lem:key-q"} that we have the following commutative diagram of $\mathbb{E}_{\infty}$-algebras in $\mathbb{S}[\mathcal{P}\mathrm{ic}]\text{-}\mathcal{M}\mathrm{od}$: $$\label{key-hc-i} \begin{tikzcd} \mathbb{S}[\mathcal{P}\mathrm{ic}] \ar{r} \ar{d} & \mathrm{HC}^-(-/k) \ar{d}\\ \mathbb{S}[\mathcal{P}\mathrm{ic}] \ar{r} & \mathrm{HC}^-(-/k). \end{tikzcd}$$ Since $\mathrm{HC}^-(-/k)$ has the projective bundle formula it is already $\mathbb{P}^1$-periodic, after applying Theorem [Theorem 140](#thm:ai-main){reference-type="ref" reference="thm:ai-main"} and Construction [Construction 142](#constr:adams-k){reference-type="ref" reference="constr:adams-k"} we get a commutative diagram of $\mathbb{E}_{\infty}$-algebras in $\mathbb{S}[\mathcal{P}\mathrm{ic}]\text{-}\mathcal{M}\mathrm{od}(\text{Sp}(\mathrm{St}_k))$ given by $$\label{key-hc-i2} \begin{tikzcd} \mathrm{K}[\tfrac{1}{n}] \ar{r} \ar{d} & \mathrm{HC}^-(-/k) \ar{d}\\ \mathrm{K}[\tfrac{1}{n}] \ar{r} & \mathrm{HC}^-(-/k), \end{tikzcd}$$ as desired. ◻ **Remark 147**. In the main text, we only need Adams operations for $X$ a smooth scheme over a field. Hence the Adams operations on $K$-theory could have been defined using motivic stable homotopy theory as explained in §[11.2](#sec_adams-slice){reference-type="ref" reference="sec_adams-slice"}. However since $\mathrm{HC}^-$ is not $\mathbb{A}^1$-invariant, Theorem [Theorem 139](#thm:chw-spt){reference-type="ref" reference="thm:chw-spt"} could not have been proved within the environment of motivic stable homotopy theory. The main theorem of [@AnnalaIwasa2023] furnishes a universal property of $K$-theory outside of the $\mathbb{A}^1$-invariant setting which is key in our proof of Theorem [Theorem 139](#thm:chw-spt){reference-type="ref" reference="thm:chw-spt"}, even in the smooth setting. **Corollary 148**. *Let $n \geq 2$ and let $k$ be a field of characteristic zero. Then for any essentially smooth $k$-scheme $X$, the commutative diagram [\[eq:chw\]](#eq:chw){reference-type="ref" reference="eq:chw"} canonically enhances to a commutative diagram of filtered $\mathbb{E}_{\infty}$-algebras: $$\label{eq:chw-filt} \begin{tikzcd} \mathrm{Fil}_{\mbox{\rm \scriptsize cla}}^{\star}\mathrm{K}_{\mathbb{Q}} \ar{r}{\psi^n} \ar{d}{ch} & \mathrm{Fil}_{\mbox{\rm \scriptsize cla}}^{\star}\mathrm{K}_{\mathbb{Q}} \ar{d}{ch}\\ \mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{\star}\mathrm{HC}^-(-/k) \ar{r}{\psi^n} &\mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{\star}\mathrm{HC}^-(-/k) . \end{tikzcd}$$* *Proof.* The two circuits of [\[eq:chw\]](#eq:chw){reference-type="eqref" reference="eq:chw"} promote to naturally equivalent, multiplicative filtered morphisms $\mathrm{Fil}_{\mbox{\rm \scriptsize cla}}^{\star}\mathrm{K}_{\mathbb{Q}} \rightarrow \mathrm{Fil}_{\mbox{\rm \scriptsize HKR}}^{\star}\mathrm{HC}^-$ by Remark [Remark 30](#rem:promote-compat){reference-type="ref" reference="rem:promote-compat"}. The commutativity of [\[eq:chw-filt\]](#eq:chw-filt){reference-type="eqref" reference="eq:chw-filt"} then follows from the fact that the Adams operations on both $\mathrm{K}_{\mathbb{Q}}$ and $\mathrm{HC}^-$ admits unique, multiplicative filtered refinement by Corollary [Corollary 145](#cor:rational){reference-type="ref" reference="cor:rational"} and [@raksit-hkr Cons. 6.4.8] respectively. ◻ [^1]: University of Toronto [^2]: CNRS et Laboratoire de Mathématiques d'Orsay [^3]: By *equal characteristic* we mean that the structure map $X\to\operatorname{Spec} \mathbb{Z}$ factors through $\operatorname{Spec}\mathbb{Q}$ or $\operatorname{Spec}\mathbb{F}_p$ for some prime number $p$. The main definition of this paper, namely gluing filtrations on $\mathrm{KH}(X)$ and $\mathrm{TC}(X)$ in order to define motivic cohomology of $X$, can be adapted to work without the equicharacteristic assumption, i.e., on any qcqs scheme, but the resulting theory is incomplete in mixed characteristic. See forthcoming work of Bouis, whose earlier work on the syntomic cohomology of valuation rings [@Bouis2022] means that the theory currently works best in certain highly ramified situations, such as for schemes defined over a perfectoid valuation ring. [^4]: The exception is when we occasionally encounter finite filtrations, in which case we implicitly impose that the filtration be both exhaustive and complete, and we may allow it to be increasing if it makes the indexing easier to follow. [^5]: $\text{DF}$ being standard notation and $\text{SpF}$ looking strange. [^6]: In any case this extra degree of generality should eventually be necessary for extending the theory of this paper to mixed characteristic. [^7]: *i.e., for each affine open $\mathrm{Spec}A\subseteq X$, the cotangent complex $L_{A/k}\in D(A)$ has Tor amplitude in $[-1,0]$.* [^8]: *i.e., the cotangent complex $L_{k/k_0}$ is supported in degree $0$ and $\Omega^1_{k/k_0}$ is a flat $k$-module. In this paper we only require the trivial situation that $\mathbb{Q}=k_0=k$, but we record the more general statement for future use.* [^9]: [\[footnote_LcdhZpsyn\]]{#footnote_LcdhZpsyn label="footnote_LcdhZpsyn"} For any fixed qcqs $\mathbb{F}_p$-scheme $X$, one can show that the following are equivalent: 1. $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)^{\mbox{\rm \scriptsize syn}}(X)$ is derived $p$-complete. 2. Each cohomology group of $\operatorname{fib}(\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}(X)\to \mathbb{Z}(j)^{\mbox{\rm \scriptsize cdh}}(X))$ is bounded $p$-power torsion. If $X$ is quasi-excellent, Noetherian, and of finite Krull dimension, then one can use alterations and the argument of the proof of Proposition [Proposition 45](#prop_cdh_filtered_trace_p){reference-type="ref" reference="prop_cdh_filtered_trace_p"} to check that (1) and (2) are true. However, they definitely do not hold in general. For example, let $A=\mathbb{F}_p[t^{1/p^\infty}]/(t-1)$. Then $H^1$ of $\operatorname{fib}(\mathbb{Z}(1)^{\mbox{\rm \scriptsize mot}}(A)\to \mathbb{Z}(1)^{\mbox{\rm \scriptsize cdh}}(A))$ is the principal units $\operatorname{ker}(A^\times\to\mathbb{F}_p^\times)$, which is not bounded $p$-power torsion. This also shows that the cdh sheaf $L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(1)^{\mbox{\rm \scriptsize syn}}$ is not invariant for the nil (but not nilpotent) ideal $\operatorname{ker}(A\to\mathbb{F}_p)$. [^10]: To be precise it is in fact possible to run the same argument as in Corollary [Corollary 31](#corol_cdh_filtered_trace){reference-type="ref" reference="corol_cdh_filtered_trace"}: we just need to know that the map of filtered spectra $\mathrm{Fil}^\star_{\mbox{\rm \scriptsize BMS}}\mathrm{TC}(X)\to \mathrm{Fil}^\star_{\mbox{\rm \scriptsize BMS}}L_{\mathrm{cdh}}\mathrm{TC}(X)$ is an equivalence for every smooth $\mathbb{F}_p$-scheme $X$. Since $\mathrm{TC}(X)\stackrel{\sim}{\to}L_{\mbox{\rm \scriptsize cdh}}\mathrm{TC}(X)$, using $\mathrm{K}(X)\stackrel{\sim}{\to}\mathrm{KH}(X)$ and Theorem [Theorem 23](#thm:mainsq){reference-type="ref" reference="thm:mainsq"}, the problem reduces to checking that $\mathbb{Z}_p(j)(X)\stackrel{\sim}{\to}L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)(X)$ for all $j\ge0$. We will deduce this in Corollary [Corollary 92](#corol_e_vs_eh){reference-type="ref" reference="corol_e_vs_eh"} using arguments involving motivic cohomology, but the appearance of motivic cohomology is illusory: the core of the proof of Corollary [Corollary 92](#corol_e_vs_eh){reference-type="ref" reference="corol_e_vs_eh"} is really contained in Theorem [Theorem 76](#thm:cdh-syn-pbf){reference-type="ref" reference="thm:cdh-syn-pbf"}, whose proof does not require the new motivic cohomology. However, to avoid any impression of circular logic, in the body of the text we provide a different proof of the proposition, which only requires the weaker result that $\mathbb{Z}_p(j)(X)\to L_{\mbox{\rm \scriptsize cdh}}\mathbb{Z}_p(j)(X)$ is an equivalence up to bounded $p$-power torsion. [^11]: For what follows, we can actually work in the generality of $k$ a $\mathbb{E}_{\infty}$-ring for an appropriate definition of $\mathbb{P}^1$; we refer to [@CisinskiKhan2020] for details. [^12]: In this paper, we do not assume that $E$ is finitary. [^13]: Let us be more precise about this. Connective $K$-theory defines an additive functor from $\mathcal{C}\mathrm{at}{}^k_{\infty} \rightarrow \text{Sp}$; the collection of such functors assemble into a symmetric monoidal $\infty$-category $\mathrm{Fun}_{\mbox{\rm \scriptsize add}}(\mathcal{C}\mathrm{at}{}^k_{\infty},\text{Sp})$ under Day convolution as explained in [@HoyoisScherotzkeSibilla2017 pg. 139]. As explained in [@HoyoisScherotzkeSibilla2017 5.4], connective $K$-theory is the unit object of this symmetric monoidal $\infty$-category and therefore, any other object $E$ acquires a module structure over it. Appealing to Glasman's work identifying lax monoidal functors with commutative monoid objects in functor categories under the Day convolution symmetric monoidal structures [@Glasman2015] lets us translate this into the action of $K$-theory, as a lax monoidal functor, on $E$. We note that the absolute version of these monoidal enhancements of the universal property of $K$-theory is first proved in [@BlumbergGepnerTabuada2014 Thm 5.14]. [^14]: We stress that our motivic cohomology coincides with the classical theory on smooth varieties over any field. This follows from Theorem [Theorem 88](#thm_mot_vs_cdh){reference-type="ref" reference="thm_mot_vs_cdh"}, Remark [Remark 21](#remark_slice_filtration){reference-type="ref" reference="remark_slice_filtration"}, and the fact that motivic homotopy theory recovers classical motivic cohomology in the case of smooth varieties over fields. Here we restrict to the case of varieties over the prime field because that is the important case for our later study of $\mathbb{Z}(j)^{\mbox{\rm \scriptsize lse}}$, and because formulating the comparison maps over arbitrary base fields is not entirely elementary. [^15]: We wish to invoke the picture of deflating a balloon: the Riemann sphere is thought of as a balloon and a presheaf is deflatable if "after puncturing at $\infty$\" the sphere deflates onto a point. [^16]: *Here $rY$ denotes the $r-1^{\mbox{\rm \scriptsize st}}$ infinitesimal thickening of $Y$ inside $X$, and similarly for $rY'$. By "cartesian" we simply mean that all pro cohomology groups of the birelative term are zero as pro abelian groups; since $\mathbb{Z}(j)^{\mbox{\rm \scriptsize mot}}$ of a Noetherian scheme is bounded above depending only on the dimension (this does not require Theorem [Theorem 113](#theorem_Weibel_vanishing){reference-type="ref" reference="theorem_Weibel_vanishing"}, but only the descriptions given in the proof Corollary [Corollary 125](#cor:surjections){reference-type="ref" reference="cor:surjections"}), this is equivalent to being cartesian in the $\infty$-category of pro complexes.* [^17]: *If $k$ is infinite then this is isomorphic to the older Levine--Weibel Chow group of zero cycles $\mathrm{CH}_0^{\mbox{\rm \scriptsize LW}}(X)$.* [^18]: As in §[3.3](#ss_cdh_local){reference-type="ref" reference="ss_cdh_local"}, our convention for abstract blowup squares means that $p$ and $i$ are assumed to be finitely presented. [^19]: For concreteness, we can regard the Picard stack as the functor on ${\mathrm{S}\mathrm{m}}_k$ which assigns $X$ to the groupoid of line bundles on $X$ (regarded as an object of $\mathcal{S}\mathrm{pc}{}$. Alternatively, we may take the connective cover $\tau_{\geq 0}\left( R\Gamma_{\mathrm{Zar}}(-;{\mathbb{G}_m})[1]) \right)$ and apply the Dold-Kan correspondence to get a presheaf of spaces. [^20]: In more details: the construction above naturally takes place in the $\infty$-category $\mathrm{Sp}_{\mathbb{P}^1}(\mathrm{St}_k)$. By [@AnnalaIwasa2023 Rem. 2.2.11] the $\infty$-category $\mathcal{SH}(k)$ embeds fully faithfully inside $\mathrm{Sp}_{\mathbb{P}^1}(\mathrm{St}_k)$. [^21]: There is a standing assumption in [@Riou2007] that $S$ is a regular scheme; this is only used to ensure that $\mathrm{KGL}$ indeed represents algebraic $K$-theory as opposed to homotopy $K$-theory. The arguments work over any qcqs scheme given this caveat. In any event, we only need the relevant compatibility for smooth schemes over a Dedekind domain or a field. [^22]: In more details. If $G^{\ast}$ is a graded object, then applying the functor $\iota_!: \mathcal C^{\mathbb{Z}^{\delta}} \rightarrow \mathcal C^{\mathbb{Z}^{\mathrm{op}}}$ produces a filtered object $\iota_!G^{\ast}$ where $$(\iota_!G^{\ast})_{n} := \bigoplus_{m \geq n} G_m,$$ and the transition maps are given by projections. This is what we refer to as the natural filtration on a graded object $G^{\ast}$. [^23]: The Hochschild complex of discrete commutative rings or, more genereally, animated rings, have a natural structure of a *derived commutative algebra* as in [@raksit-hkr Def. 4.2.22] which we can forget to a $\mathbb{E}_{\infty}$-ring. We will not make use of this richer structure explicitly, but it is used systematically in [@raksit-hkr] to formulate a universal property of Hochschild homology with the HKR filtration.

arxiv_math

--- abstract: | We develop and analyze a mathematical model of oncolytic virotherapy in the treatment of melanoma. We begin with a special, local case of the model, in which we consider the dynamics of the tumour cells in the presence of an oncolytic virus at the primary tumour site. We then consider the more general regional model, in which we incorporate a linear network of lymph nodes through which the tumour cells and the oncolytic virus may spread. The modelling also considers the impact of hypoxia on the disease dynamics. The modelling takes into account both the effects of hypoxia on tumour growth and spreading, as well as the impact of hypoxia on oncolytic virotherapy as a treatment modality. We find that oxygen-rich environments are favourable for the use of adenoviruses as oncolytic agents, potentially suggesting the use of complementary external oxygenation as a key aspect of treatment. Furthermore, the delicate balance between a virus' infection capabilities and its oncolytic capabilities should be considered when engineering an oncolytic virus. If the virus is too potent at killing tumour cells while not being sufficiently effective at infecting them, the infected tumour cells are destroyed faster than they are able to infect additional tumour cells, leading less favourable clinical results. Numerical simulations are performed in order to support the analytic results and to further investigate the impact of various parameters on the outcomes of treatment. Our modelling provides further evidence indicating the importance of three key factors in treatment outcomes: tumour microenvironment oxygen concentration, viral infection rates, and viral oncolysis rates. The numerical results also provide some estimates on these key model parameters which may be useful in the engineering of oncolytic adenoviruses. author: - Tedi Ramaj - Xingfu Zou title: | On the Treatment of Melanoma:\ A Mathematical Model of Oncolytic Virotherapy --- **Keywords**: mathematical modelling, oncolytic virotherapy, disease dynamics, differential equation modelling # Introduction Melanoma is considered the most deadly type of skin cancer. Melanoma begins in melanocytes - the cells responsible for producing melanin - and can develop in various parts of the body [@melanocyte_paper; @melanocyte_paper2]. Melanoma is the fifth most common cancer in adults in the United States [@melanoma_stats1]. While melanoma rates have been steadily rising, mortality has not followed this same trend. This decreased mortality is attributed to various factors such as early detection, increased protection against UV radiation, and improvements in treatment [@melanoma_stats2]. Metastatic melanoma continues to be a major issue contributing the cancer mortality, due to the increased difficulty of treating the disease once it has spread beyond its original site [@melanoma_stats3]. Various forms of therapy, including chemotherapy, immunotherapy, and radiotherapy are used in the treatment of advanced melanoma. Developing new forms of therapy and enhancing existing therapy is always desirable in increasing survival rates of the disease. Oncolytic virotherapy is a method of cancer treatment in which viruses are used to selectively infect and destroy cancer cells via a variety of direct and indirect mechanisms, while leaving surrounding healthy cells unharmed [@OV_paper2; @OV_paper1]. These viruses are called oncolytic viruses (OVs). These therapeutics include both genetically modified viruses and non-modified viruses, such as live attenuated viruses (i.e., the measles virus [@measles_virus_paper]). The genetically modified herpes simplex virus Talimogene laherparepvec (T-VEC) has been used in clinical trials to treat inoperable melanoma [@TVECpaper1; @TVECpaper2]. The treatment is often performed in combination with other therapies, such as being followed up with the use of adjuvant radiotherapy. The oncolytic virus is typically administered via direct subcutaneous injection into the lesion [@TVECpaper3]. The idea is for the virus to selectively infect cancer cells and use them to replicate and perform oncolysis to destroy the neoplasm. The viral infection may also destroy the cancer cells through indirect mechanisms such as activating the immune system and aiding the immune response against the cancer cells [@OV_paper3; @OV_paper2; @OV_paper1]. Other OVs which have been studied (not necessarily in melanoma trials, but in the context of other cancers, such as colorectal cancer) include the adenoviruses ONYX-015 and ZD55-IL-24 [@hu_adenovirus]. ZD55-IL-24's primary mechanism of action is through inducing a systematic anti-tumour cell immune response. There is also evidence that this virus may inhibit tumour cell growth by inhibiting angiogenesis, as was previously observed in an immuno-competent mouse model [@hu_adenovirus]. Such immune-mediated effects of viruses like ZD55-IL-24 are more established in the existing literature. A mechanism of action of ONYX-015 involves replication and lysis of tumour cells that are p-53 deficient [@ONYX015_carlapaper; @ONYX015_RiesPaper]. The importance of this direct mechanism of action of is currently under debate [@davola] and is considered in this present paper via mechanistic modelling. Mathematical modelling of cancer treatment has seen widespread use in the last few decades. These models frequently take the form of ODE, PDE, and delay models in the continuous setting. By studying the effect of disease treatment from a quantitative perspective, based on biological and physical mechanistic modelling, new insights may be obtained to guide future treatment direction. This type of modelling has also been used to study the treatment of cancer via oncolytic virotherapy [@mathmodel_onc1; @mathmodel_onc2]. The recent work of Wang et al [@mathmodel_onc3]. in mathematical modelling of virotherapy as a treatment modality for melanoma, the models were able to provide insights concerning virus treatment thresholds as well as how immunosupperssive drugs may work in tandem with OVs. The work by Urenda-Cazares et al. examined the use of OVs in combination with chemotherapy to treat glioma. As a result of these types of models, some results were obtained on how to optimize treatment in a clinical setting [@mathmodel_onc4]. In this paper, we model the effect of hypoxic environments on oncolytic virotherapy treatment through the use of ordinary differential equation (ODE) modelling. Hypoxic refers to oxygen-poor environments. Typically, viruses which are more efficient at infecting cells in oxygen-rich environments tend to lose their infectivity under hypoxic conditions [@viral_friend_foe]. This is particularly true of adenoviruses such as ONYX-015 [@onyx_adeno1]. Hypoxia has a negative effect on the efficacy of OVs as well as any adjuvant radiotherapy which may be administered [@hypoxiapaper1; @hypoxiapaper2]. In the context of melanoma treatment, hypoxic environments can inhibit the action of OVs, such as their ability to infect cancer cells and their ability to induce the death of cancer cells. Due to the lack of dynamical modelling of this phenomenon, we explore the relationship between tumour microenvironment oxygen concentration and the efficacy of the OV with the objective of contributing to the existing oncology literature from a quantitative perspective. The application of mathematical modelling can capture some elements of the complex interplay between oxygen concentration conditions and OV efficacy. In our model, we study the effect of oxygen concentration when the OV is applied directly to the primary lesion. More specifically, we study the impact which parameters such as the infectiousness of the OV on the efficacy of the treatment under different oxygen conditions. The structure of this paper is organized as follows. In Section 2, we formulate an ODE model and give the assumptions on our functions. We refer to this as our local model, since we are studying the effect of OV directly on the primary tumour. We also perform non-dimensionalization of the model for the purposes of mathematical analysis. We explain the meaning of our model in terms of the biological context. In Section 3, we perform an analysis of the local model. This includes proofs on the well-posedness results. In Subsection 3.1, we first look at the case where we do not take into account the oxygen concentration dependence. In Subsection 3.2, we look at the case of oxygen concentration dependence. We perform an analysis of the stability of the relevant steady states of our system. In Section 4, we perform numerical simulations and give biological interpretations of these results. In Section 5, we extend our model to a regional model, where we take into account the movement of tumour cells into the surrounding lymph nodes. In Section 6, we perform numerical simulations on the regional model. We complete this paper with some conclusions and discuss possible directions for future work in Section 7. # Local oncolytic virotherapy model We begin by considering a melanoma tumour, initially consisting of some initial quantity of proliferating tumour cells. At this initial point in time, a localized treatment of oncolytic virotherapy begins at the tumour site, by introducing the OV via direct injection into the lesion. We consider the use of a virus with oncolytic and replication rates down-regulated by hypoxia. Such a virus shares these features with adenoviruses. The rationale for considering adenoviruses (or OVs with similar hypoxia down-regulating properties as adenoviruses), comes from this consideration of the effects of hypoxia on the action of the OV. Namely, while hypoxic tumour microenvironments reduce the efficacy of adenoviruses, they also promote melanoma tumour progression [@melanoma_progression_paper]. One of the goals of this present work is to mathematically capture and model the dynamics of OVs under the same unfavourable hypoxic conditions which typically have an inverse (favourable) impact on tumour progression. Indeed, the modelling presented in this paper is not only limited to adenoviruses, but to any virus which experiences similar down-regulation in hypoxic tumour microenvironments. The OV then proceeds to infect the tumour cells. The model consists of three variables, the density of uninfected tumour cells, the density of infected tumour cells, and oxygen concentration, at time $t$, respectively represented by $u(t), n(t),$ and $c(t)$. Then, we have the following model: $$\begin{aligned} \dfrac{\mathrm{d}u}{\mathrm{d}t} &= r_1 u \left( 1- \dfrac{u+n}{K} \right) - \dfrac{\theta (c) nu}{\alpha + n}, \label{3eq1}\\ \dfrac{\mathrm{d}n}{\mathrm{d}t} &= r_2 n \left( 1 - \dfrac{u+n}{K} \right) + \dfrac{\theta (c) nu}{\alpha + n} - \gamma (c) n, \label{3eq2} \\ \dfrac{\mathrm{d}c}{\mathrm{d}t} &= \phi - \beta c - q_1uc - q_2nc. \label{3eq3}\end{aligned}$$ Note that we are considering cell-to-cell infections, which have been observed as a mode of infection used by oncolytic viruses [@cell_cell_virus_evidence]. Oncolytic adenoviruses which exhibit cell-to-cell spreading, such as VRX-009, have also previously been constructed [@adenovirus_cell_to_cell]. Importantly, while VRX-009 was not tested as a treatment modality for melanoma, its production provides a proof of concept of the idea of an oncolytic adenovirus with a cell-to-cell spreading mechanism. Our work hence provides a theoretical modelling framework for cell-to-cell spreading of adenoviruses (or adenovirus-like OVs) which may infect melanoma cancer cells. Previous mathematical models of cell-to-cell viral infection made use of a mass-action-like terms to represent infection [@Webb_cell_to_cell; @zou_cell_to_cell] and we adopt a similar approach in our model, but with the infection mechanism of a Holling type II functional response function. We make this consideration to model the saturating effect of melanoma cells that have already been infected by the OV. We prescribe the initial conditions $u(0) = u_0, n(0) = n_0$ and $c(0) = c_0$ to be non-negative quantities. We assume that both classes of tumour cells exhibit logistic growth. The carrying capacity of the tumour cells is given by $K$ and the growth rates of the uninfected tumour cells and the infected tumour cells are given by $r_1$ and $r_2$, respectively. We further assume that $r_1 > r_2$ to reflect that the infected tumour cells are less effective at proliferating due their cell machinery being hijacked by the OV. Following the approach of [@oxygen_con_paper], we use mass-action terms to express the oxygen consumption by the tumour cells. To that end, the parameters $q_1$ and $q_2$ give the oxygen consumption rate by the uninfected tumour cells and the infected tumour cells, respectively. The rate of oxygenation, assumed constant (due to having some control over this parameter, i.e., through certain therapies [@tibbles]), is given by $\phi$ and the rate of oxygen consumption by surrounding *non-cancerous cells (or healthy cells)* is given by $\beta$. We use a Hill function to represent the transition of a tumour cell from uninfected by an OV to infected by an OV. The parameter $\theta \in \mathrm{C}^1 (\mathbb{R_+})$ represents the virus infection rate, which is dependent on available oxygen concentration. The other oxygen dependent parameter $\gamma \in \mathrm{C}^1 (\mathbb{R_+})$ is the virus-induced death rate of the infected tumour cells. Note that the terms *virus-induced death rate* and *oncolysis rate* are used interchangeably. The adenovirus is inhibited by a hypoxic environment and hence we assume that as oxygen concentration is locally decreased, the OV will become less effective, both in infecting the tumour cells and inducing tumour cell death [@onyx_adeno1]. Hence, we have the following conditions on $\theta(c)$ and $\gamma(c)$: $$\begin{dcases} \theta'(c) \geq 0, \quad \gamma'(c) \geq 0, \quad \text{for} \quad c \in (0, \infty), \\[5pt] \theta(0) = \theta_0 \geq 0, \quad \gamma(0) = \gamma_0 \geq 0, \\[5pt] \lim_{c \rightarrow \infty} \theta(c) = \theta_\infty > \theta_0, \quad \lim_{c \rightarrow \infty} \gamma (c) = \gamma_\infty > \gamma_0, \end{dcases} \label{oxygen conditions}$$ where $\theta_\infty$ and $\gamma_\infty$ give the OV efficacy in response to high oxygen environments. Note that in hypoxic environments, oncolytic virotherapy will not be as efficient as an adenovirus is being used. We non-dimensionalize the model by making the following substitutions: $$x:= \dfrac{u}{K}, \quad y:= \dfrac{n}{K}, \quad z := \dfrac{\beta c}{\phi}, \quad \tau := r_1 t.$$ Then we obtain the system, $$\begin{aligned} \dfrac{\mathrm{d}x}{\mathrm{d}\tau} &= x(1-x-y) - \dfrac{\hat{\theta} (z) xy}{\hat{\alpha} + y}, \\ \dfrac{\mathrm{d}y}{\mathrm{d}\tau} &= ry(1-x-y) + \dfrac{\hat{\theta} (z) xy}{\hat{\alpha} + y} - \hat{\gamma} (z) y, \\ \dfrac{\mathrm{d}z}{\mathrm{d}\tau} &= \hat{\beta} (1-z) - \hat{q}_1 xz - \hat{q}_2 yz,\end{aligned}$$ where we define $$\hat{\theta} (z) := \dfrac{1}{r_1} \theta \left(\dfrac{\phi z}{\beta} \right), \quad \hat{\gamma} (z) := \dfrac{1}{r_1} \gamma\left( \dfrac{\phi z}{\beta} \right),$$ $$\hat{r} := \dfrac{r_2}{r_1}, \quad \hat{\alpha} := \dfrac{\alpha}{K}, \quad \hat{\beta} := \dfrac{\beta}{r_1}, \quad \hat{q}_1 := \dfrac{q_1 K}{r_1}, \quad \hat{q}_2 := \dfrac{q_2 K}{r_1}.$$ Note that the properties of $\theta(c)$ and $\gamma(c)$ given in ([\[oxygen conditions\]](#oxygen conditions){reference-type="ref" reference="oxygen conditions"}) are preserved by $\hat{\theta}(z)$ and $\hat{\gamma}(z)$, respectively, up to some scaling. The most notable change is in the long-term behavior: $\hat{\theta}$ will approach $\hat{\theta}_\infty := \theta_\infty /r_1$ and $\hat{\gamma}$ will approach $\hat{\gamma}_\infty := \gamma_\infty / r_1$ as $z \rightarrow \infty$. We now drop the tilde and replace $\tau$ with $t$ for notational convenience and hence, for the subsequent analysis, we consider the following model: $$\begin{aligned} \dfrac{\mathrm{d}x}{\mathrm{d}t} &= x(1-x-y) - \dfrac{{\theta} (z) xy}{{\alpha} + y}, \label{e1} \\ \dfrac{\mathrm{d}y}{\mathrm{d}t} &= ry(1-x-y) + \dfrac{{\theta} (z) xy}{{\alpha} + y} - {\gamma} (z) y, \label{e2} \\ \dfrac{\mathrm{d}z}{\mathrm{d}t} &= \beta (1-z) - q_1 xz - q_2 yz, \label{e3}\end{aligned}$$ with non-negative initial conditions: $$x(0) = x_0 \geq 0, \quad y(0) = y_0 \geq 0, \quad z(0) = z_0 \geq 0. \label{IC}$$ The functions $\theta$ and $\gamma$ once again have the properties given in ([\[oxygen conditions\]](#oxygen conditions){reference-type="ref" reference="oxygen conditions"}). In Section 3, we perform a mathematical analysis of the rescaled model ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}) to explore some predictions related to the effect of available oxygen concentration on the OV treatment. # Analysis of the local model We begin by considering the well-posedness of the model. Existence and uniqueness of the solution of ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}), subject to initial conditions ([\[IC\]](#IC){reference-type="ref" reference="IC"}), follow from the elementary theory of ODEs. We consider the solution of this initial value problem, $(x(t), y(t), z(t)) \in \mathbb{R}^3$. Since the variables represent densities and concentration of physical quantities, the system must remain non-negative for all $t \geq 0$. We begin with equation ([\[e1\]](#e1){reference-type="ref" reference="e1"}). From this equation, it follows that $$x(t) = x_0 \cdot \exp \left[ \int_0^t \left( 1 - x(s) - y(s) - \dfrac{\theta(z(s))y(s)}{\alpha + y(s)} \right) \mathrm{d}s \right],$$ and so, $x(t) \geq 0$ for all $t \geq 0$. Similarly, it follows from equation ([\[e2\]](#e2){reference-type="ref" reference="e2"}) that $$y(t) = y_0 \cdot \exp \left[ \int_0^t \left( r(1-x-y) + \dfrac{\theta(z(s)) x(s)}{\alpha + y(s)} - \gamma(z(s))y(s) \right) \mathrm{d}s \right].$$ Therefore, $y(t) \geq 0$ for all $t \geq 0$. Finally, equation ([\[e3\]](#e3){reference-type="ref" reference="e3"}) gives $$z(t) = z_0 \cdot\exp \left({-\int_0^t (\beta + q_1 x(s) + q_2 y(s)) \mathrm{d}s} \right) + \beta \int_0^t \exp \left({-\int_s^t (\beta + q_1 x(\xi) + q_2 y(\xi)) \mathrm{d} \xi}\right) \mathrm{d}s.$$ This shows that $z(t) \geq 0$ for all $t \geq 0$. In fact, if $t > 0$, then $z$ is strictly positive. Next, we address the boundedness of the solution. To this end, we apply a comparison argument. From equations ([\[e1\]](#e1){reference-type="ref" reference="e1"}) and ([\[e3\]](#e3){reference-type="ref" reference="e3"}), a solution of the system satisfies the inequalities $$\dfrac{\mathrm{d}x}{\mathrm{d}t} \leq x(1-x), \quad \dfrac{\mathrm{d}z}{\mathrm{d}t} \leq \beta (1-z).$$ Then, it follows that $$\limsup_{t \rightarrow \infty} x(t) \leq 1, \quad \limsup_{t \rightarrow \infty} z(t) \leq 1.$$ Hence, $x(t)$ and $z(t)$ are bounded functions. Let $\bar{x}$ be an upper bound for $x(t)$, i.e., $x(t) \leq \bar{x}$ for all $t \geq 0.$ It then follows from equation $(\ref{e2})$ that $$\dfrac{\mathrm{d}y}{\mathrm{d}t} \leq ry(1-y) + \dfrac{\theta_\infty \bar{x} y}{\alpha + y} \implies \dfrac{\mathrm{d}y}{\mathrm{d}t} \leq ry(1-y) + \theta_\infty \bar{x}.$$ Therefore, by a comparison argument, $$\limsup_{t \rightarrow \infty} y(t) \leq \dfrac{r + \sqrt{r^2 + 4r \theta_\infty \bar{x}}}{2r},$$ which shows that $y(t)$ is a bounded function. Summarizing these results, we have the following theorem: **Theorem 1**. *The solution of the initial value problem ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}), satisfying initial conditions ([\[IC\]](#IC){reference-type="ref" reference="IC"}), is non-negative and bounded. [\[theorem_pos_bounded\]]{#theorem_pos_bounded label="theorem_pos_bounded"}* ## Dynamics of the local model -- case I: no oxygen dependence We consider first the case with no oxygen dependence. That is, we set $\theta(z) = \theta$ and $\gamma(z) = \gamma$, where $\theta$ and $\gamma$ are positive constants. In this case, system ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}) reduces to the following two-variable system: $$\begin{aligned} \dfrac{\mathrm{d}x}{\mathrm{d}t} &= x(1-x-y) - \dfrac{{\theta} xy}{{\alpha} + y}, \label{e11} \\ \dfrac{\mathrm{d}y}{\mathrm{d}t} &= ry(1-x-y) + \dfrac{{\theta} xy}{{\alpha} + y} - {\gamma} y. \label{e22} \end{aligned}$$ If we consider system ([\[e11\]](#e11){reference-type="ref" reference="e11"}) - ([\[e22\]](#e22){reference-type="ref" reference="e22"}) over the region $(x,y) \in \mathbb{R}_+^2$, we can rule out the existence of non-constant periodic orbits. **Proposition 1**. *Consider system ([\[e11\]](#e11){reference-type="ref" reference="e11"}) - ([\[e22\]](#e22){reference-type="ref" reference="e22"}) over the region $\mathbb{R}_+^2$. There are no closed orbits contained entirely $\mathbb{R}_+^2$. [\[prop_dulac\]]{#prop_dulac label="prop_dulac"}* #### Proof: Let $S(x,y) = 1/(xy)$ for $x, y > 0$. Then, $$\dfrac{\partial}{\partial x} \left[ S(x,y) \left( x(1-x-y) - \dfrac{{\theta} xy}{{\alpha} + y} \right) \right] + \dfrac{\partial}{\partial y} \left[ S(x,y) \left( ry(1-x-y) + \dfrac{{\theta} xy}{{\alpha} + y} - {\gamma} y \right) \right]$$ may be computed to give $$-\dfrac{1}{y} - \dfrac{r}{x} - \dfrac{\theta}{(\alpha + y)^2} < 0.$$ Since this function does not change sign on $\mathbb{R}_+^2$, we conclude by the Dulac-Bendixson Theorem that there are no closed orbits contained entirely in $\mathbb{R}_+^2$. $\square$ Next, we determine the steady states of system ([\[e11\]](#e11){reference-type="ref" reference="e11"}) - ([\[e22\]](#e22){reference-type="ref" reference="e22"}) by solving the algebraic system $$\begin{aligned} x(1-x-y) - \dfrac{{\theta} xy}{{\alpha} + y} = 0, \label{e11_alg} \\ ry(1-x-y) + \dfrac{{\theta} xy}{{\alpha} + y} - {\gamma} y = 0. \label{e22_alg} \end{aligned}$$ It can be readily seen that $(x,y) = (0,0)$ and $(x,y) = (1,0)$ are solutions of this system for all parameter values. Another solution which may be easily seen is $(x,y) = (0, (r - \gamma)/r)$, which only exists if $r > \gamma$. It can be shown that the remaining steady states (if any exist) are determined by solving the system $$(\theta - r \theta + \gamma)y^2 + (\theta^2 + \alpha \theta + 2\alpha \gamma - r \alpha \theta - \theta)y + \alpha(\alpha \gamma - \theta) = 0, \label{alg1}$$ $$x = 1 - y - \dfrac{\theta y}{\alpha + y}.\label{alg2}$$ We linearize the system at its steady states by first computing the Jacobian matrix $$J(x,y) = \begin{pmatrix} 1 - 2x - y - \dfrac{\theta y}{\alpha + y} & -x - \dfrac{\theta \alpha x}{(\alpha + y)^2} \\[18pt] -ry + \dfrac{\theta y}{\alpha + y} & r - rx - 2ry + \dfrac{\alpha \theta x}{(\alpha + y)^2} - \gamma \end{pmatrix}.$$ We begin with the assumption $r < \gamma$ in order to discount the steady state $(0,(r-\gamma)/r)$. Linearizing at the steady state $(0,0)$ gives $$J(0,0) = \begin{pmatrix} 1 & 0 \\ 0 & r - \gamma \end{pmatrix}.$$ By our assumption that $r < \gamma$, this steady state is a saddle. Linearizing the system at the steady state $(1,0)$ gives $$J(1,0) = \begin{pmatrix} -1 \ & \ -1 - \dfrac{\theta}{\alpha} \\[10pt] 0 & \dfrac{\theta}{\alpha} - \gamma \end{pmatrix}.$$ From $J(1,0)$, we then conclude that $(1,0)$ is locally asymptotically stable if $\theta < \alpha \gamma$ and it is unstable if $\theta > \alpha \gamma$. If we now impose the additional assumption $\theta < \alpha \gamma$, then the system only contains two non-negative steady states: $(0,0)$ and $(1,0)$. To see this, we note that the left-hand side of equation ([\[alg1\]](#alg1){reference-type="ref" reference="alg1"}), as a function of $y$, is a convex parabola (since $r < 1$) with a positive constant term. The coefficient of the $y$ term is also positive, as $$\begin{aligned} \theta^2 + \alpha \theta + 2\alpha \gamma - r \alpha \theta - \theta = \theta^2 + \alpha \gamma + \alpha \theta (1-r) + (\alpha \gamma - \theta) > \theta^2 + \alpha \gamma > 0.\end{aligned}$$ Therefore, the parabola has non-negative roots and system ([\[alg1\]](#alg1){reference-type="ref" reference="alg1"}) - ([\[alg2\]](#alg2){reference-type="ref" reference="alg2"}) has no non-negative solutions. This shows that the only non-negative steady states are $(0,0)$ and $(1,0)$. Next, we consider the case $\theta > \alpha \gamma$. In this case, an additional co-existence steady state, $(x_*, y_*)$, where $x_*, y_* > 0$ may be introduced if system ([\[alg1\]](#alg1){reference-type="ref" reference="alg1"}) - ([\[alg2\]](#alg2){reference-type="ref" reference="alg2"}) has positive solutions. It is clear to see that the parabola on the left-hand side of equation ([\[alg1\]](#alg1){reference-type="ref" reference="alg1"}) is still convex but the constant term is now negative. Hence, this parabola has exactly one positive real root, $y_*$. Then $x_*$ may be obtained from equation ([\[alg2\]](#alg2){reference-type="ref" reference="alg2"}). In order for the steady state to be meaningful, we set $x_*$ must be positive, which is not the case for all values of the model parameters. We impose the following condition to ensure that $x_*$ is positive: $$y_*^2 + (\alpha + \theta)y_* < 1, \label{existence_cond31}$$ where $$y_* = \dfrac{A + \sqrt{A^2 - 4 \alpha (\theta - r \theta + \gamma)(\alpha \gamma - \theta)}}{2(\theta - r \theta + \gamma)},$$ and $$A = r \alpha \theta + \theta - \theta^2 - \alpha \theta - 2 \alpha \gamma.$$ We assume these conditions are satisfied so that the steady state $(x_*, y_*)$ exists and has positive coordinates. In fact, since $r < \gamma$, these conditions are satisfied as the existence of a stable (unique) positive steady state is ensured as a corollary of non-negativity of solutions, boundedness of solutions, and Proposition [\[prop_dulac\]](#prop_dulac){reference-type="ref" reference="prop_dulac"}. As we will now show, the assumption $r < \gamma$ is not necessary for the stability of the steady state $(x_*, y_*)$ -- only its existence is necessary. If it exists, linearizing at this steady state gives the matrix $$J(x_*, y_*) = \begin{pmatrix} -x_* \quad & -x_* - \dfrac{\theta \alpha x_*}{(\alpha + y_*)^2} \\[10pt] -ry_* + \dfrac{\theta y_*}{\alpha + y_*} & -ry_* - \dfrac{\theta x_* y_*}{(\alpha + y_*)^2} \end{pmatrix}.$$ We compute the determinant of this matrix: $$\begin{aligned} \det J(x_*, y_*) &= \left(rx_* y_* + \dfrac{\theta x_*^2 y_*}{(\alpha +y_*)^2} \right) -\left(rx_*y_* + \dfrac{\theta \alpha r x_* y_*}{(\alpha + y_*)^2} - \dfrac{\theta x_* y_*}{\alpha + y_*} - \dfrac{\theta^2 \alpha x_* y_*}{(\alpha + y_*)^3} \right) \\[20pt] &= \dfrac{\theta x_* y_*}{\alpha + y_*} + \dfrac{\theta x_* y_*}{(\alpha + y_*)^2} \left( x_* - ar + \dfrac{\alpha \theta}{\alpha + y_*} \right) \\[20pt] &= \dfrac{\theta x_* y_*}{(\alpha + y_*)^2} \left( \alpha + y_* + x_* - \alpha r + \dfrac{\alpha \theta}{\alpha + y_*} \right) > 0,\end{aligned}$$ where the last inequality follows since $r < 1$. The trace of $J(x_*,y_*)$ is $$\text{tr} \ J(x_*,y_*) = -x_*-ry_* - \dfrac{\theta x_* y_*}{(\alpha + y_*)^2} < 0.$$ Therefore, the steady state $(x_*,y_*)$ is locally asymptotically stable whenever it exists. It is therefore also globally asymptotically stable. By Theorem [\[theorem_pos_bounded\]](#theorem_pos_bounded){reference-type="ref" reference="theorem_pos_bounded"}, Proposition [\[prop_dulac\]](#prop_dulac){reference-type="ref" reference="prop_dulac"}, and the Poincaré-Bendixson Theorem, the local asymptotic stability of the steady state $(1,0)$ implies the global asymptotic stability if $\theta < \alpha \gamma$. Similarly, if $\theta > \alpha \gamma$, then $(x_*, y_*)$ is globally asymptotically stable. We summarize these results as follows. **Theorem 2**. *Consider system ([\[e11\]](#e11){reference-type="ref" reference="e11"}) - ([\[e22\]](#e22){reference-type="ref" reference="e22"}) over the region $\mathbb{R}_+^2$.* 1. *If $\gamma > \max\{r,\theta/\alpha \}$, then the only two non-negative steady states of the system are $(x,y) = (0,0)$ and $(x,y) = (1,0)$. The steady state $(0,0)$ is a saddle and the steady state $(1,0)$ is a stable node. Furthermore, the steady state $(1,0)$ is globally asymptotically stable on $\mathbb{R}_+^2$.\ * 2. *If $r < \gamma < \theta/\alpha$, then there exists an additional, positive, steady state, $(x_*, y_*)$. The steady states $(0,0)$ and $(1,0)$ are unstable (saddles) and $(x_*, y_*)$ is globally asymptotically stable on $\mathbb{R}_+^2$.* *[\[prop_dynamics_2D\]]{#prop_dynamics_2D label="prop_dynamics_2D"}* We numerically illustrate Theorem [\[prop_dynamics_2D\]](#prop_dynamics_2D){reference-type="ref" reference="prop_dynamics_2D"} in Figure [\[phase_portrait_1\]](#phase_portrait_1){reference-type="ref" reference="phase_portrait_1"}. The phase portraits in Figure [\[phase_portrait_1\]](#phase_portrait_1){reference-type="ref" reference="phase_portrait_1"} are produced with all parameters, except for $\alpha$, being assigned (after non-dimensionalization) based on the values in Table 2. Doing so gives the parameter values $\theta = 2.52908, r = 0.531107, \gamma = 1.29362.$ In Figure [\[phase_portrait_1\]](#phase_portrait_1){reference-type="ref" reference="phase_portrait_1"}(a), we set $\alpha = 10.0$ and in Figure [\[phase_portrait_1\]](#phase_portrait_1){reference-type="ref" reference="phase_portrait_1"}(b), we set $\alpha = 1.0$. {#phase_portrait1a width="\\textwidth"} {#phase_portrait1b width="\\textwidth"}\ Clinically, the stability of $(1,0)$ is not a favourable result, representing a failure of the OV treatment. From Proposition [\[prop_dynamics_2D\]](#prop_dynamics_2D){reference-type="ref" reference="prop_dynamics_2D"}, we see that one condition which leads to this occurrence is the virus-induced death rate, $\gamma$, being made sufficiently large. This leads to an idea which will come up again in the case of oxygen dependence: Having a virus-induced death rate which is too large relative to the infection rate will decrease the efficacy of the OV. Instead, it is important to make sure that the tumour cells are not being killed faster than they are able to infect adjacent tumour cells. This suggests that when engineering an OV, it is important to achieve an appropriate balance between the infection rate and oncolysis rate of the virus. Note that the condition $\gamma > \theta/\alpha$ in Proposition ([\[prop_dynamics_2D\]](#prop_dynamics_2D){reference-type="ref" reference="prop_dynamics_2D"}) follows directly from setting $\mathcal{R}_0 < 1$ where $\mathcal{R}_0$ is the basic reproduction number. Following the approach of [@reproduction_number], the basic reproduction number may be computed by using the next generation method. We omit the details here. So far, we have considered the case where $r < \gamma$, i.e., when the growth rate of the infected tumour cells is bounded by their death rate. We have seen that total extinction of the uninfected tumour cells is not possible in this case. We now consider the case $r > \gamma$. In this case, we have an additional non-negative steady state, $(0,(r-\gamma)/r)$. This steady state may represent a semi-successful treatment outcome in the case $\gamma \approx r$. Hence, stability of this steady state is clinically preferable. We note first that if $r > \gamma$, the matrix $J(0,0)$ has two positive eigenvalues and hence, $(0,0)$ is an unstable node. We assume that $\theta < \alpha \gamma$ in order to rule out the existence of a non-negative co-existence steady state. In this case, the eigenvalues of $J(1,0)$ remain negative and so $(1,0)$ remains a stable node. Linearizing system ([\[e11\]](#e11){reference-type="ref" reference="e11"}) - ([\[e22\]](#e22){reference-type="ref" reference="e22"}) at the steady state $(0,(r-\gamma)/r)$: $$J\left(0,\dfrac{r-\gamma}{r} \right) = \begin{pmatrix} \dfrac{r \gamma (\alpha + \theta + 1) - (r^2\theta + \gamma^2)}{r(\alpha r + r - \gamma)} & 0 \\[20pt] \dfrac{(r-\gamma)(\theta + \alpha + \alpha r - r)}{\alpha r + r - \gamma} \quad & \gamma - r \end{pmatrix}.$$ Since this is a lower triangular matrix, the eigenvalues are the elements of the main diagonal. The eigenvalue $\gamma - r$ is negative since $r > \gamma$. The remaining eigenvalue is positive since $$\begin{aligned} r \gamma (\alpha + \theta + 1) &= r (\alpha \gamma) + r \gamma \theta + r \gamma > (r^2)(\theta) + r \gamma \theta + (\gamma)(\gamma) > r^2 \theta + \gamma^2.\end{aligned}$$ Therefore, $(0,(r-\gamma)/r)$ is a saddle and hence unstable. Therefore, even in the case where $r > \gamma$, the only locally stable steady state is $(1,0)$. This also remains true if $r = \gamma$, as can be seen via direct substitution. The steady state $(1,0)$ is unstable if $\theta > \alpha \gamma$ and $(0,0)$ is always unstable. If $r > \gamma$, then the steady state $(0,(r-\gamma)/r)$ is locally asymptotically stable if and only if $$\theta > \gamma \left( \dfrac{\alpha}{r-\gamma} + \dfrac{1}{r} \right). \label{existence_cond32}$$ This condition is obtained by requiring all the eigenvalues of $J(0,(r-\gamma)/r)$ to be negative. Note that since $r < 1$, condition ([\[existence_cond32\]](#existence_cond32){reference-type="ref" reference="existence_cond32"}) implies that $\theta > \alpha \gamma$. Since all solutions are non-negative and bounded, and closed orbits may not exist, it follows that violating condition ([\[existence_cond32\]](#existence_cond32){reference-type="ref" reference="existence_cond32"}) implies the existence and stability of the positive steady state $(x_*, y_*)$. We summarize the results of the case $r > \gamma$ in the following theorem. **Theorem 3**. *Consider system ([\[e11\]](#e11){reference-type="ref" reference="e11"}) - ([\[e22\]](#e22){reference-type="ref" reference="e22"}) over the region $\mathbb{R}_+^2$. If $r > \gamma$, then:* 1. *The steady state $(x,y) = (1,0)$ is globally asymptotically stable on $\mathbb{R}_+^2$ if $$\theta < \alpha \gamma.$$* 2. *The positive steady state $(x,y) = (x_*, y_*)$ is globally asymptotically stable on $\mathbb{R}_+^2$ if $$\alpha \gamma < \theta < \gamma \left( \dfrac{\alpha}{r-\gamma} + \dfrac{1}{r} \right).$$* 3. *The steady state $(x,y) = (0, (r-\gamma)/r)$ is globally asymptotically stable on $\mathbb{R}_+^2$ if $$\theta > \gamma \left(\dfrac{\alpha}{r-\gamma} + \dfrac{1}{r} \right) .$$* *[\[additional_important_theorem_chap3_1\]]{#additional_important_theorem_chap3_1 label="additional_important_theorem_chap3_1"}* {width="\\textwidth"} {width="\\textwidth"}\ {width="\\textwidth"}\ {width="\\textwidth"}\ We numerically illustrate Theorem [\[additional_important_theorem_chap3_1\]](#additional_important_theorem_chap3_1){reference-type="ref" reference="additional_important_theorem_chap3_1"} in Figure [\[profzoussuggestedfigurechap3\]](#profzoussuggestedfigurechap3){reference-type="ref" reference="profzoussuggestedfigurechap3"}. The phase portraits in this figure are produced by using the parameter values given in the code in Appendix A, except for the parameters $r, \theta,$ and $\gamma$. We set $\gamma = 0.3$. In Figure [\[profzoussuggestedfigurechap3\]](#profzoussuggestedfigurechap3){reference-type="ref" reference="profzoussuggestedfigurechap3"} (a), we set $r = 0.5311$ and $\theta = 0.01$. In Figure [\[profzoussuggestedfigurechap3\]](#profzoussuggestedfigurechap3){reference-type="ref" reference="profzoussuggestedfigurechap3"} (b), we set $r = 0.5311$ and $\theta = 0.3$. In Figure [\[profzoussuggestedfigurechap3\]](#profzoussuggestedfigurechap3){reference-type="ref" reference="profzoussuggestedfigurechap3"} (c), we set $r = 0.5311$ and $\theta = 0.9$. In Figure [\[profzoussuggestedfigurechap3\]](#profzoussuggestedfigurechap3){reference-type="ref" reference="profzoussuggestedfigurechap3"} (d), we set $r = 0.4$ and $\theta = 1.4$. Biologically, the case $\theta > \alpha \gamma$ corresponds to a low virus-induced death rate relative to the infection rate (since in practice, $\alpha$ is typically less than 1). This condition leads to a more clinically favourable outcome compared to the condition $\theta < \alpha \gamma$, as the uninfected tumour cell-dominant steady state becomes unstable. If we then consider the additional condition $r > \gamma$, then there exists an infected tumour cell-dominant steady, $(0,(r-\gamma)/\gamma)$, which corresponds to complete eradication of uninfected tumour cells. Biologically, this clinically favourable steady state exists when infected tumour cells can proliferate at a greater rate than they are destroyed by the virus. This (perhaps rather unintuitively) suggests that an OV should not be engineered to hinder the proliferation capability of the cancer cells and, in fact, a greater growth rate of the infected cancer cells can lead to improved clinical outcomes. The idea is to minimize $(r-\gamma)/r$ while also ensuring that the infected tumour cell-dominant steady state is stable, i.e., inequality ([\[existence_cond32\]](#existence_cond32){reference-type="ref" reference="existence_cond32"}) holds. The modelling suggests that the most potent OV is one with a high infection rate, low oncolysis rate, and that minimally inhibits the proliferation rate of the cancer cells. By taking $\gamma \rightarrow r^{-}$, we have $y \rightarrow 0$ as $t \rightarrow \infty$ as long as $\theta$ still satisfies condition ([\[existence_cond32\]](#existence_cond32){reference-type="ref" reference="existence_cond32"}). While this might lead to the naive assumption of simply engineering a virus which has a very large infection rate compared to the proliferate rate of tumour cells, this type of OV may also be associated with increased toxicity [@simpson], adding another layer of complexity. {width="\\textwidth"}\ Figure [\[fig:chap3_extra01\]](#fig:chap3_extra01){reference-type="ref" reference="fig:chap3_extra01"} (a) gives guidance on how to choose the infection rate, $\theta$, given the oncolysis rate, $\gamma$. It can be seen in Figure [\[fig:chap3_extra01\]](#fig:chap3_extra01){reference-type="ref" reference="fig:chap3_extra01"} (b) that if the infection rate is too small, the tumour cell densities will converge to the positive steady state. On the other hand, if $\theta$ is large enough, then all of the tumour cells are eventually infected by the virus. We summarize the existence and stability results of this section in Table 1. ## Dynamics of the local model -- case II: oxygen dependence We now perform a local stability analysis of the relevant steady states of system ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}). We begin by computing the Jacobian matrix of this system, $$J(x,y,z) = \begin{pmatrix} 1 - 2x - y - \dfrac{\theta(z)y}{\alpha + y} & -x - \dfrac{\alpha \theta(z)x}{(\alpha + y)^2} & -\dfrac{\theta'(z)xy}{\alpha + y} \\[18pt] -ry - \dfrac{\theta(z)y}{\alpha + y} & r - rx - 2ry + \dfrac{\alpha \theta(z) x}{(\alpha + y)^2} - \gamma(z) \ \ & \dfrac{\theta'(z)xy}{\alpha + y} - \gamma'(z)y \\[18pt] -q_1z & -q_2z & -\beta - q_1x - q_2 y\end{pmatrix}. \label{Jacobian}$$ We first consider the simplest steady state, the *tumour-free* steady state, $(x,y,z) = (0,0,1)$. Linearizing the system about this point gives $$J(0,0,1) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & r - \gamma(1) & 0 \\ -q_1 & -q_2 & -\beta \end{pmatrix}, \label{tumour-free}$$ which is a lower triangular matrix with eigenvalues $1, r - \gamma(1), -\beta$. Since this matrix will always have a positive eigenvalue, the tumour-free steady state is unstable. The maximum dimension of its stable manifold is 2, which occurs if and only if $r < \gamma(1)$. This corresponds to the fact that if the virus-induced death rate of tumour cells, $\gamma$, is sufficiently large, then there will be larger domain of initial conditions for which the solution will converge to the tumour-free steady state. Next, we consider the case where the uninfected tumour cells dominate, i.e., $x = 1$ and $y = 0$. In this case, we have the following steady state, which corresponds to the failure of OV treatment: $$(x,y,z) = \left( 1, 0, z^* \right), \quad \text{where} \quad z^* := \dfrac{\beta}{\beta + q_1}.$$ Linearizing the system at this steady state gives $$J\left( 1, 0, z^* \right) = \begin{pmatrix} -1 & -1 - \dfrac{\gamma(z^*)}{\alpha} & 0 \\[13pt] 0 & \dfrac{\theta(z^*)}{\alpha} - \gamma(z^*) & 0 \\[13pt] -q_1 z^* & -q_2 z^* & -\beta - q_1 \end{pmatrix}, \label{tumour-dominates}$$ which has eigenvalues $$\lambda_{1}^u = -1, \quad \lambda_2^u = \dfrac{\theta(z^*)}{\alpha} - \gamma(z^*), \quad \lambda_3^u = -\beta - q_1.$$ Considering the conditions for which these eigenvalues are all negative gives the following proposition. **Proposition 2**. *The tumour-dominant steady-state, $(x,y,z) = (1,0,z^*)$, is locally asymptotically stable if $\theta(z^*) < \alpha \gamma(z^*)$. [\[prop2\]]{#prop2 label="prop2"}* From a clinical perspective, the local asymptotic stability of the tumour-dominant steady-state is an unfavourable result. Biologically, this occurs when the infection rate of tumour cells by the OV is too low compared to the virus-induced death rate. This leads to an important insight: engineering a virus which can destroy tumour cells at a fast rate is not useful if the infection rate is too low. It is important to have a virus which is sufficiently effective at infecting cancer cells - not just destroying them. The inequality in Proposition [\[prop2\]](#prop2){reference-type="ref" reference="prop2"} can give an estimate on how large these rates should be for a useful OV. We are also interested in the existence of an uninfected tumour cell-free steady, i.e., one of the form $(0,y_*, z_*)$. From system ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}), it can been seen that such a solution may be determined by solving the system $$\begin{aligned} r(1-y) - \gamma(z) &= 0, \label{alg_sys7} \\ \beta(1-z) - q_2 yz &= 0. \label{alg_sys8}\end{aligned}$$ This system may have no solutions, one solution, or multiple solutions depending on the properties of the oncolysis function, $\gamma(z)$. Stability of this steady state is favourable and hence, we impose the additional condition $\gamma_\infty < r$ so as to ensure the existence of a positive solution of system ([\[alg_sys7\]](#alg_sys7){reference-type="ref" reference="alg_sys7"}) - ([\[alg_sys8\]](#alg_sys8){reference-type="ref" reference="alg_sys8"}). Notice that this condition is similar to the existence of the uninfected tumour cell-free steady state condition in Subsection 3.1. Moreover, it should also be noted that $0 \leq y_*, z_* \leq 1$. Linearizing at $(0,y_*,z_*)$ gives the matrix $$J(0,y_*,z_*) = \begin{pmatrix} 1 - y_* - \dfrac{\theta(z_*)y_*}{\alpha + y_*} & 0 & 0 \\[18pt] -ry_* - \dfrac{\theta(z_*)y_*}{\alpha + y_*} & -ry_* \ \ & - \gamma'(z_*)y_* \\[18pt] -q_1z_* & -q_2z_* & -\dfrac{\beta}{z_*} \end{pmatrix}.$$ The eigenvalues of this matrix are $$\lambda_1^n = 1 - y_* - \dfrac{\theta(z_*)y_*}{\alpha + y_*}, \quad \lambda_{2,3}^n = \dfrac{-(\beta + y_* z_*) \pm \sqrt{(\beta + ry_*z_*)^2 - 4z_* (\beta r y_* - \gamma'(z_*)q_2 y_* z_*^2)}}{2z_*}.$$ It is clear that all of these eigenvalues have no imaginary part. Hence, $(0,y_*,z_*)$ is either a stable node or a three-dimensional saddle. The former case is preferable, as all tumour cells will eventually be infected as $t \rightarrow \infty$. This occurs when the eigenvalues are all negative, leading to the following proposition. **Proposition 3**. *Consider the steady state $(0,y_*,z_*)$, where $y_*$ and $z_*$ satisfy the equations $$\gamma(z_*) = r \left( 1 + \dfrac{\beta}{q_2} -\dfrac{\beta}{q_2 z_*} \right), \quad y_* = \dfrac{\beta}{q_2} \cdot \dfrac{1-z_*}{z_*}. \label{diagram_equations_chap3_1}$$ If $\gamma_\infty < r$, then such $y_*$ and $z_*$ exist and $0 \leq y_* \leq 1$, $\beta/(\beta + q_2) \leq z_* \leq 1$. Moreover, the steady state $(0,y_*,z_*)$ is locally asymptotically stable if and only if $$\begin{aligned} \theta (z_*) > \dfrac{(1-y_*)(\alpha + y_*)}{y_*} \quad \text{and} \quad \gamma'(z_*) < \dfrac{\beta r}{q_2 z_*^2}. \end{aligned}$$[\[prop_gamma_prime\]]{#prop_gamma_prime label="prop_gamma_prime"}* **Remark 1**. *The condition $\gamma_\infty < r$ is a sufficient condition for the existence of the steady state $(0,y_*,z_*)$. A necessary and sufficient condition for the existence of this steady state is $\gamma_\infty < r(1+\beta/q_2)$. The latter condition, however, does not guarantee that $z_* \leq 1$.* Proposition [\[prop_gamma_prime\]](#prop_gamma_prime){reference-type="ref" reference="prop_gamma_prime"} gives some important conditions for constructing an effective OV. The condition $\gamma_\infty < r$, similarly to Subsection 3.1, gives a sufficient existence condition. The first stability condition is consistent with our previous results: namely, a sufficiently large infection rate is an important factor of OV efficacy. The second stability condition is perhaps more interesting: An oncolysis rate which grows *slowly* in response to increases in oxygen concentration of the tumour microenvironment. We now consider system ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}) under certain parameter conditions and establish a global stability result concerning the tumour-dominant steady state, $(1,0,z^*)$. In particular, we consider the case $q_1 = 0$ for the sake of mathematical tractability. Biologically, this corresponds to tumour cells which are unable to consume oxygen. While this condition does not typically represent a biologically realistic situation, it may be considered a *best-case scenario*, as less oxygen is consumed and therefore, more oxygen is available to increase the efficacy of the OV. We begin by proving an auxiliary result for which we do not need the assumption $q_1 = 0$. Consider the following region in the positive octant in $\mathbb{R}^3$: $$\Omega := \left\{ (x,y,z) \in \mathbb{R}^3 : x \geq 0, \ y \geq 0, \ x + y \leq 1, \ 0 \leq z \leq 1 \right\}$$ The idea is to show that this region defines a so-called *trapping region* from which no solution trajectories of system ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}) may exit. We state this in the following lemma. **Lemma 1**. *The region $\Omega \subset \mathbb{R}_+^3$ is a positively invariant set for system ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}). [\[lemma331\]]{#lemma331 label="lemma331"}* #### Proof: Let $(x(t),y(t),z(t))$ denote a solution of system ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}) with initial condition in $\mathcal{U}$. Proving this lemma is equivalent to showing that $\Omega$ defines a trapping region for all $t \geq 0$. First note that by Theorem [\[theorem_pos_bounded\]](#theorem_pos_bounded){reference-type="ref" reference="theorem_pos_bounded"}, $x(t), y(t), z(t) \geq 0$ for all $t \geq 0$. If the trajectory were to exit the region, then by continuity, it would cross either the $z = 1$ boundary or the plane $x+y = 1$ at some time $t^*$. Assume that the trajectory crosses $z=1$ at time $t^*$. Then from equation ([\[e3\]](#e3){reference-type="ref" reference="e3"}), $z'(t^*) = -q_1 x - q_2 y \leq 0$. Therefore, the vector field at this boundary point does not point in the positive $z$ direction, contradicting the assumption since the trajectory may not exit through the $z = 1$ plane. Hence, we have shown that $z \leq 1$. Now we need only establish that no trajectory may exit the region through the plane $x + y = 1$. We do this by showing that the vector field on this plane points into the region $\Omega$. On the plane $x+y = 1$, the sum of equations ([\[e1\]](#e1){reference-type="ref" reference="e1"}) and ([\[e2\]](#e2){reference-type="ref" reference="e2"}) is $$\begin{aligned} \dfrac{\mathrm{d}}{\mathrm{d}t} [x(t) + y(t)] &= (x + ry)(1-x-y) - \gamma (z) y \\ &= (x+ry)(0) - \gamma(z)y < 0.\end{aligned}$$ Hence, $y'(t) < -x'(t)$ which implies that, by chain rule, $\mathrm{d}y/\mathrm{d}x < -1$. Therefore, the vector field on the plane $x+y = 1$ points into the region and no trajectory may exit through this plane. We conclude that no trajectory contained in the region $\Omega$ may exit this region, completing the proof. $\square$ We are now in a position to give the theorem on global stability of the steady state $(1,0,z^*)$. Since we consider the case $q_1 = 0$, we have $z^* = \beta/(\beta+q_1) = 1$. Hence, the steady state becomes $(1,0,1)$. We then have the following theorem: **Theorem 4**. *Consider system ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}) when $q_1 = 0$ and $q_2 > 0$. If $\theta(z) < \alpha \gamma (z)$ for $z \in [0,1]$, then $(x,y,z) = (1,0,1)$ is globally asymptotically stable on $\Omega$. [\[lyapunov_theorem1\]]{#lyapunov_theorem1 label="lyapunov_theorem1"}* #### Proof: Since $[0,1]$ is a compact interval, we can choose $\varepsilon \in (0,1)$ such that $\theta(z) < \varepsilon \alpha \gamma(z)$ for all $z \in [0,1]$. Define the function $V : \text{Int} \ {\Omega} \cup \{ (1,0,1) \} \rightarrow \mathbb{R}$ as $$V(x,y,z) := x -\ln(x) - 1 + y + \dfrac{(1-\varepsilon)\gamma_0}{2q_2} \left[z - \ln\left( {z} \right) - 1\right].$$ We can show that $V$ defines a Lyapunov function. $V$ is positive definite as $V(1,0,1) = 0$ and $V(x,y,z) > 0$ for all $(x,y,z) \in \text{Int} \ \Omega$. Taking the time derivative of $V$ gives $$\begin{aligned} \dfrac{\mathrm{d}V}{\mathrm{d}t} &= \left(\dfrac{x-1}{x}\right) \left[ x(1-x-y) - \dfrac{\theta(z)xy}{\alpha + y} \right] + ry(1-x-y) + \dfrac{\theta(z)xy}{\alpha + y} - \gamma(z)y + \dots \\[20pt] & \hspace{90mm} \dots + \dfrac{(1-\varepsilon)\gamma_0}{2q_2} \cdot \left(\dfrac{z-1}{z} \right)\left[ \beta (1-z) - q_2 yz \right] \\[20pt] &= -(1-x-ry)(1-x-y) - \dfrac{\theta(z)y(x-1)}{\alpha + y} + \dfrac{\theta(z)xy}{\alpha + y} - \gamma(z)y + \dfrac{(1-\varepsilon)\gamma_0}{2q_2} \left[ -\dfrac{\beta}{z} (1-z)^2 - q_2 y (z-1) \right] \\[20pt] &= -(1-x-ry)(1-x-y) + \dfrac{\theta(z)y}{\alpha + y} - \varepsilon \gamma(z)y - (1-\varepsilon) \gamma(z)y + \dfrac{(1-\varepsilon)\gamma_0}{2q_2} \left[ -\dfrac{\beta}{z} (1-z)^2 - q_2 y (z-1) \right] \\[20pt] &\leq -(1-x-ry)(1-x-y) + \dfrac{[\theta(z) - \varepsilon \alpha \gamma(z) ]y}{\alpha+y} - (1-\varepsilon) \gamma(z)y + \dfrac{(1-\varepsilon)\gamma_0 y}{2}.\end{aligned}$$ Note that $\dot{V}(1,0,1) = 0.$ Next, since $r < 1$ and $x+y \leq 1$ by Lemma [\[lemma331\]](#lemma331){reference-type="ref" reference="lemma331"}, we have $1-x-ry \geq 0$. Hence, $-(1-x-ry)(1-x-y) \leq 0$. By our assumption, it follows that $\theta (z) - \varepsilon \alpha \gamma(z) < 0$ since $z$ remains in $[0,1]$ by Lemma [\[lemma331\]](#lemma331){reference-type="ref" reference="lemma331"}. Finally, since $\gamma(z) \geq \gamma_0$, it holds that $$- (1-\varepsilon) \gamma(z)y + \dfrac{(1-\varepsilon)\gamma_0 y}{2} < 0.$$ Therefore, $\dot{V} < 0$ on $\text{Int} \ \Omega$. By LaSalle's invariance principle, we conclude that the tumour-dominant steady state $(x,y,z) = (1,0,1)$ is globally asymptotically stable. $\square$ **Remark 2**. *Theorem [\[lyapunov_theorem1\]](#lyapunov_theorem1){reference-type="ref" reference="lyapunov_theorem1"} assumes that $q_2$ is a positive constant. If $q_2 = 0$, establishing global stability is trivial as equations ([\[e1\]](#e1){reference-type="ref" reference="e1"}) and ([\[e2\]](#e2){reference-type="ref" reference="e2"}) are decoupled from equation ([\[e3\]](#e3){reference-type="ref" reference="e3"}) and global asymptotic stability of the tumour-dominant steady state of the resulting two-variable system follows from Theorem [\[prop_dynamics_2D\]](#prop_dynamics_2D){reference-type="ref" reference="prop_dynamics_2D"}.* As previously stated, the condition $q_1 = 0$ in Theorem [\[lyapunov_theorem1\]](#lyapunov_theorem1){reference-type="ref" reference="lyapunov_theorem1"} biologically represents a best-case scenario in which the uninfected tumour cells are unable to consume oxygen, leading to a more effective adenovirus due to increased oxygen concentration in the tumour microenvironment. In practice $\alpha < 1$ and so the condition $\theta (z) < \alpha \gamma(z)$ reflects a virus which has a significantly larger oncolysis rate compared to its infection rate. This is analogous to the condition required in Theorem [\[prop_dynamics_2D\]](#prop_dynamics_2D){reference-type="ref" reference="prop_dynamics_2D"}, providing further evidence that a very high oncolysis rate is not a favourable characteristic of an oncolytic adenovirus. While we do not analytically consider the case $q_1 > 0$, the numerical simulations in Section 4 lead us to conjecture that the steady state $(1,0,z^*)$ remains globally asymptotically stable in this case, under the condition $\theta(z) < \alpha \gamma(z)$. It is clear that the relationship between the functions $\theta(z)$ and $\gamma(z)$ is an important factor in the dynamics of the system. Biologically, if the infection rate is too low relative to the virus-induced death rate, infected tumour cells may die faster than they are able to infect a sufficient number of uninfected cells, hence leading to an uninfected tumour cell-dominant steady state. On the other hand, if the virus-induced death rate is too low, not enough tumour cells will die for the OV to be an effective therapeutic agent. The interplay between these functions and their effect on the OV efficacy is one of the topics of the next section. # Numerical simulations: local model In this section, we perform numerical simulations of the local model. We perform the simulations using system ([\[3eq1\]](#3eq1){reference-type="ref" reference="3eq1"}) - ([\[3eq3\]](#3eq3){reference-type="ref" reference="3eq3"}). The units of $u$ and $n$ are cells/mm$^3$ and the units of $c$ are millimolars (mM). Unless otherwise stated, we set the initial conditions to be $u_0= 10000$ and $n_0 = 100$, as in [@parameter_esimates_init_cond]. Similarly to [@oxygen_background_concentration], we set $c_0 = 4.3751$. Table 2 gives the parameters value which we use in the case where $\gamma$ and $\theta$ are constants, rather than functions of oxygen concentration. In this case, plotting the tumour cell densities gives Figure [3](#graph00){reference-type="ref" reference="graph00"}. If we consider this to be the standard case, we can test the effect of including oxygen dependence of the functions $\theta$ and $\gamma$. In our simulations, we set $\theta$ and $\gamma$ to be sigmoid functions of $c$. In particular, we have $$\theta(c) = \dfrac{\theta_\infty \theta_0}{\theta_0 + (\theta_\infty - \theta_0)e^{-k_\theta c}}, \quad \gamma(c) = \dfrac{\gamma_\infty \gamma_0}{\gamma_0 + (\gamma_\infty - \gamma_0)e^{-k_\gamma c}}. \label{theta_gamma_equations1}$$ We consider how different parameter values $\theta_0, \theta_\infty, \gamma_0, \gamma_\infty, k_\theta, k_\gamma$ impact the efficacy of the OV. Guided by Proposition [\[prop2\]](#prop2){reference-type="ref" reference="prop2"}, we choose these parameters such that we consider $\theta(c) < (\alpha/K)\gamma(c)$, $\theta(c) > (\alpha/K)\gamma(c)$, etc. We plot these results in Figures [3](#graph00){reference-type="ref" reference="graph00"} - [12](#graph05){reference-type="ref" reference="graph05"}. {#graph00} . [\[graph00\]]{#graph00 label="graph00"} In Figure [3](#graph00){reference-type="ref" reference="graph00"}, we consider the case in which $\theta$ and $\gamma$ are the constants given in Table 2 rather than functions of the oxygen concentration. This is our first numerical exposure to a result which will be echoed throughout this subsection: higher infection rates relative to virus-induced death rates tend to lead to more favourable clinical results. In this case, the tumour cell densities both settle to a steady state well below the carrying capacity, suggesting some inhibition of the growth of the tumour cells. {#fig2_a width="\\textwidth"} {#fig2_b width="\\textwidth"}\ In Figure [\[graph01\]](#graph01){reference-type="ref" reference="graph01"}, we have the case of a high infection rate relative to the virus-induced death rate. The assumption of Proposition [\[prop2\]](#prop2){reference-type="ref" reference="prop2"} is not satisfied and, unsurprisingly, the uninfected cell density is driven below the infected cell density, asymptotically. This case potentially represents a favourable result since in Figure [\[graph01\]](#graph01){reference-type="ref" reference="graph01"} (a), as the tumour cell density approaches a positive stable steady state value below the carrying capacity. In particular, the uninfected tumour cell density remains significantly lower than the infected tumour cell density. This illustrates the importance of the infection rate being sufficiently large. On other hand, as in Figure [\[graph01\]](#graph01){reference-type="ref" reference="graph01"} (b), having the virus-induced death rate be *too low* leads to an unfavourable result in which all the tumour cells are infected but they nevertheless approach a value *near* the carrying capacity -- note that they do not approach the carrying capacity in the case depicted by the figure. This illustrates the delicate balance between viral infection and virus-induced mortality. Furthermore, we note the differences between Figure [\[graph01\]](#graph01){reference-type="ref" reference="graph01"} and Figure [3](#graph00){reference-type="ref" reference="graph00"}: In both cases, $\theta > (\alpha/K)\gamma$, yet the dynamics are qualitatively different. This difference is a result of Figure [\[graph01\]](#graph01){reference-type="ref" reference="graph01"} depending on oxygen concentration; a consideration not made in Figure [3](#graph00){reference-type="ref" reference="graph00"}. {#fig2_a width="\\textwidth"} {#fig2_b width="\\textwidth"}\ {#fig5_a width="\\textwidth"} {#fig5_b width="\\textwidth"}\ On the other hand, Figure [\[graph02\]](#graph02){reference-type="ref" reference="graph02"} shows a clinically unfavourable result. Namely, the uninfected tumour cell density approaches the carrying capacity value while the infected tumour cells die out. In this case, treatment via OV has failed. This occurs when the $\theta$ and $(\alpha/K)\gamma$ curves intersect at some oxygen value, $c^*$. The outcome of the numerics, in this case, directly follows from Proposition [\[prop2\]](#prop2){reference-type="ref" reference="prop2"}. Regardless of the initial density of OV injection, $n_0$, (i.e., Figure [\[graph02\]](#graph02){reference-type="ref" reference="graph02"} (a) vs. Figure [\[graph02\]](#graph02){reference-type="ref" reference="graph02"} (b)) the asymptotic behaviour is the same. Biologically, this gives the following insight: in hypoxic environments, having very low lysis capabilites of the OV yields failure of the treatment regardless of initial density of the OV injection. It is worthwhile to note that the $z^*$ from the steady state considered in Proposition [\[prop2\]](#prop2){reference-type="ref" reference="prop2"} is **NOT** related to the quantity $c^*$, the $c$-coordinate of the intersection point of $\theta$ and $\gamma$. Figure [\[graph03\]](#graph03){reference-type="ref" reference="graph03"} represents the reverse case of Figure [\[graph02\]](#graph02){reference-type="ref" reference="graph02"}, in which the inequalities are reversed and the results are clinically more favourable. This once again illustrates the importance of the virus-induced death rate in hypoxic environments and also shows the importance of the infection rate in oxygen-rich environments. Moving from Figure [\[graph03\]](#graph03){reference-type="ref" reference="graph03"} (a) to Figure [\[graph03\]](#graph03){reference-type="ref" reference="graph03"} (b), we increase the virus-induced death rate, while still maintaining a high viral infection rate in oxygen rich conditions. This further supports the idea of achieving a balance between infection rates and lysis capacity as an OV engineering consideration. {#fig4_a width="\\textwidth"} {#fig4_b width="\\textwidth"}\ In Figure [\[graph04\]](#graph04){reference-type="ref" reference="graph04"}, for low values of oxygen concentration (i.e., hypoxic environments) the infection rate is significantly less than the virus-induced death rate, whereas for high values of oxygen concentration, the virus-induced death rate is reduced. The figure shows that this case also represents a favourable clinical outcome represented by the dampening oscillations in Figure [\[graph04\]](#graph04){reference-type="ref" reference="graph04"} (a). Asymptotically, the tumour cell density approaches a positive steady state value well below the carrying capacity. This (once again) suggests the following insight: in hypoxic environments, it is important that the OV is more efficient at killing cancer cells than infecting them. However, if the oxygen concentration should be large, the OV must be more efficient at infecting tumour cells than inducing their death. In Figure [\[graph04\]](#graph04){reference-type="ref" reference="graph04"} (b), we decrease the growth rate of the $\gamma(c)$ function, leading to near-extinction of all tumour cells. Biologically, this represents an OV which has greater tumour-destroying capabilities over a lesser range of lower oxygen concentrations. Another interpretation is that it would be favourable for the infection rate to surpass the virus-induced death rate at lesser oxygen concentrations as long as the virus-induced death rate does initially dominates under *extremely* hypoxic conditions. Such a virus must be engineered to initially be extremely potent at destroying tumour cells when there is almost no oxygen available in the tumour microenvironment but must quickly be able to adapt by having a much greater infection rate if the available oxygen concentration should increase. These results are consistent with Proposition [\[prop_gamma_prime\]](#prop_gamma_prime){reference-type="ref" reference="prop_gamma_prime"}. {#graph05} Figure [12](#graph05){reference-type="ref" reference="graph05"} shows the case where the infection rate is very low compared to the virus-induced death rate. In this case, the uninfected tumour cell density dominates and approaches the carrying capacity. This result agrees with Proposition [\[prop2\]](#prop2){reference-type="ref" reference="prop2"}. This further supports the idea of a delicate balance between how effective the virus is at infected cancer cells and how potent the virus is at inducing death of tumour cells. In particular, we must ensure that the death rate is not too large compared to the infection rate. These cases illustrate the following point which must be considered when engineering the OV: Having a virus too efficient at destroying and not efficient enough at infecting is not recommended. Perhaps equally importantly, we must also consider the oxygen conditions (i.e., hypoxia) when engineering the OV as the functionality of the virus also depends on whether or not the tumour microenvironment is hypoxic. # Regional oncolytic virotherapy model In this section, we extend our model to the regional setting by considering the case of lymph node invasion by the tumour cells. Since movement through the lymphatic system is one of the main methods through which melanoma tumour cells may spread, it is of vital importance to consider lymph nodes as part of the model system. In the context of hypoxia, there is evidence which suggests that hypoxic conditions contribute to the upregulation of uPAR (a receptor on the surface of melanoma cells), leading to lymph node metastasis of the tumour cells [@Lymph_Hypoxia_Paper]. Hence, it is both of mathematical and biological interest to capture the dynamics of regional (i.e., lymphatic) spread of tumour cells, as well as OV efficacy, under various oxygen conditions. As the thickness of the melanoma tumour increases, there is an increased probability of the tumour spreading to nearest lymph nodes [@lymphprob1]. We model a network of lymph nodes as a one-dimensional lattice, where each node represents a lymph node and the edges represent lymphatic vessels. =\[draw, fill=boxcolor_tumour, minimum size=2em, text width = 2.5cm, align = center, minimum height = 2.5cm\] =\[draw, fill=boxcolor_lymph, minimum size=2em, text width = 2.5cm, align = center, minimum height = 2.5cm\] =\[-\>, red, text = black, line width=0.5mm\] =\[draw, fill=boxcolor_tumour, minimum size=2em, text width = 2.0cm, align = center, minimum height = 2.0cm\] =\[draw, fill=boxcolor_lymph, minimum size=2em, text width = 2.0cm, align = center, minimum height = 2.0cm\] =\[-\>, red, text = black, line width=0.5mm\] The initial concentration of tumour cells at each node is set to $0$. Let $i$ denote the $i^{\text{th}}$ node from the primary tumour for $i = 1, 2, 3, \dots, \ell$ and let $i = 0$ denote the primary tumour. That is, $i = 0$ corresponds to the local case presented in Section 3.2. Note that $u_0, n_0$ and $c_0$ no longer represent initial conditions, but rather the primary tumour. The tumour cells may either travel to the left or to the right of their current position. We assume that the probability of tumour cells spreading to the adjacent lymph nodes depends on the density of the tumour cells at the given node and, hence, on the sum $u_i(t) + n_i(t)$. Let $P_i(u_i+n_i)$ be the spreading rate of some fraction of tumour cells away from node $i$ to an adjacent lymph node in the network. This fraction of cells which leaves a given node is dependent on the tumour cell density at the given node i.e., $u_i(t) + n_i(t)$. The probability of the cells travelling left is $q_{i,L}$ and the probability of travelling right is $q_{i,R}$, where $q_{i,L} + q_{i,R} = 1$ for $i = 1, 2, \dots, \ell-1$. Moreover, $q_{0,R} = 1$. While it has been observed that lymph typically flows only in one direction [@zimmerman_lymphfact], we allow for the possibility of some tumour cells to travel in the reverse direction. We assume that the probability of tumour cells reversing direction is low and therefore, we consider $q_{i,R} >> q_{i,L}$ in the numerical simulations. On each node in the network, we have a system of ODEs which describes the number of tumour cells and the oxygen concentration. We use system ([\[e1\]](#e1){reference-type="ref" reference="e1"}) - ([\[e3\]](#e3){reference-type="ref" reference="e3"}) to model the dynamics of the tumour cells at each individual node. To this end, we propose the following system: $$\begin{aligned} \dfrac{\mathrm{d}u_0}{\mathrm{d}t} &= r_1 u_0 \left( 1- \dfrac{u_0+n_0}{K_0} \right) - \dfrac{\theta (c_0) n_0u_0}{\alpha_0 + n_0} - q_{0,R} P_0 u_0 + q_{1,L} P_1 u_{1},\label{eqlattice1}\\ \dfrac{\mathrm{d}n_0}{\mathrm{d}t} &= r_2 n_0 \left( 1 - \dfrac{u_0+n_0}{K_0} \right) + \dfrac{\theta (c_0) n_0u_0}{\alpha_0 + n_0} - \gamma (c_0) n_0 - q_{0,R} P_0 n_0 + q_{1,L} P_1 n_1, \label{eqlattice2} \\ \dfrac{\mathrm{d}u_i}{\mathrm{d}t} &= r_1 u_i \left( 1- \dfrac{u_i+n_i}{K_i} \right) - \dfrac{\theta (c_i) n_iu_i}{\alpha_i + n_i} + q_{i-1,R} P_{i-1} u_{i-1} + q_{i+1,L} P_{i+1} u_{i+1} - P_i u_{i}, \label{eqlatticei1}\\ \dfrac{\mathrm{d}n_i}{\mathrm{d}t} &= r_2 n_i \left( 1 - \dfrac{u_i+n_i}{K_i} \right) + \dfrac{\theta (c_i) n_iu_i}{\alpha_i + n_i} - \gamma (c_i) n_i + q_{i-1,R} P_{i-1} n_{i-1} + \dots \notag \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \dots + q_{i+1,L}P_{i+1} n_{i+1} - P_i n_{i}, \label{eqlatticei2} \\ \dfrac{\mathrm{d}u_\ell}{\mathrm{d}t} &= r_1 u_\ell \left( 1- \dfrac{u_\ell+n_\ell}{K_\ell} \right) - \dfrac{\theta (c_\ell) n_{\ell}u_{\ell}}{\alpha_{\ell} + n_{\ell}} + q_{\ell-1,R} P_{\ell-1} u_{\ell-1} - q_{\ell,L} P_\ell u_{\ell}, \label{eqlatticel1}\\ \dfrac{\mathrm{d}n_\ell}{\mathrm{d}t} &= r_2 n_\ell \left( 1 - \dfrac{u_\ell+n_\ell}{K_\ell} \right) + \dfrac{\theta (c_\ell) n_\ell u_\ell}{\alpha_\ell + n_\ell} - \gamma (c_\ell) n_\ell + q_{\ell-1,R} P_{\ell-1} n_{\ell-1} - q_{\ell,L} P_{\ell} n_{\ell}, \label{eqlatticel2} \\ \dfrac{\mathrm{d}c_k}{\mathrm{d}t} &= \phi_k - \beta c_k - q_1u_kc_k - q_2n_kc_k, \label{eqlatticel3}\end{aligned}$$ where $i = 1, 2, 3, \dots, \ell-1$, $k = 0, 1, 2, 3, \dots, \ell$, and $P_k := P_k (u_k + n_k)$. Note that $\ell$ is the number of lymph nodes in the network. We set $\phi_0 = \phi$ and $\phi_k = 0$ for $k = 1, 2, 3, \dots, \ell$ for the purpose of following an individual over the course of treatment. The amount of time which tumour cells spend in compartment $i$, in the case of a large density of tumour cells in the compartment, is $1/\eta_i$, in days. We may therefore consider $\eta_i$ to give a per capita *spreading speed* of tumour cells *away* from compartment $i$. We assume that the speed with which tumour cells travel to the next node is equal to the speed with which they travel to the previous node. In practice, $q_{i,R} \approx 1$ and $q_{i,L} \approx 0$, so this assumption is not typically needed. It will be necessary for the purpose of tractability of the subsequent mathematical analysis. In this work, we only consider the case of a linear lymphatic network. With all these considerations, the spreading rate of tumour cells leaving the lymph node and spreading to adjacent nodes is given by $$P_i(x) = \eta_i \left[1 - e^{-\lambda_i x} \right], \quad \lambda_i > 0. \label{P_iprobequation}$$ The rationale behind defining $P_i$ in such a way is based on experimental results relating the size of a primary lesion to the probability of the cancer reaching the sentinel lymph nodes. See, for example, [@lymphprob1]. For the purpose of simulations, we assume that once the carrying capacity is reached, the probability of spreading is 0.7. Hence, we take $(1/\eta_i) P_i(K_i) = 0.7$ and solve this equation to determine the value $\lambda_i$ to be $\lambda_i = -\ln(0.3)/(K_i)$. We can show that the regional model is also well-posed in the sense of existence, uniqueness, non-negativity of the solution of the corresponding initial value problem with non-negative initial conditions, and boundedness of solutions. We summarize this result in the following theorem. **Theorem 5**. *There exists a unique solution of the initial value problem ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}), with non-negative initial conditions, which remains non-negative and bounded for all $t \geq 0$. [\[regional_theorem\]]{#regional_theorem label="regional_theorem"}* #### Proof: Existence and unqiueness of solutions follow directly from the fundamental theory of ODEs. To address the non-negativity of solutions, we apply Theorem 2.1 in Chapter 5 of [@zounotes]. Let $(u_0 (t), n_0(t), u_1 (t), n_1 (t), \dots u_\ell (t), n_\ell (t), c_0 (t), c_1 (t), \dots, c_\ell(t)) \in \mathbb{R}^{3\ell + 3}$ be the solution of the initial value problem consisting of system ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}) with non-negative initial conditions. We begin by showing that none of the components $u_0, u_1, \dots, u_\ell$ become negative. Assume to the contrary that at some time, $t^*$, at least one of the components of the solution becomes negative. By continuity, these components must first cross 0. If $u_0$ is one of these components, then plugging in $u_0 = 0$ into equation ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) gives $$\dfrac{\mathrm{d}u_0}{\mathrm{d}t} = q_{1,L} u_1 P_1 (u_1 + n_1) \geq 0,$$ which implies that $u_0$ is non-decreasing at $t = t^*$. Therefore, $u_0$ cannot become negative, leading to a contradiction. The same argument can be used to show that none of the $u_i$ components may become negative. Similarly, this contradiction argument can be used to conclude the non-negativity of $n_i$ for $i = 0, 1, 2, \dots, \ell$. Finally, it can be seen that for $k = 0, 1, 2, \dots, \ell$, equation ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}) gives $$c_k(t) = c_k (0) \exp \left[ -\int_0^t \beta + q_1 u_k(s) + q_2 n_k(s) \mathrm{d}s \right] + \phi_k \int_0^t \exp \left[ -\int_s^t \left( \beta + q_1 u_k (\xi) + q_2 n_k (\xi) \right) \mathrm{d} \xi \right] \mathrm{d}s,$$ from which non-negativity of $c_k (t)$ follows. Hence, for non-negative initial conditions, $u_i (0), n_i (0), c_i (0) \geq 0$, for $i = 0, 1, 2, \dots, \ell$, it follows that the solution of the initial value problem remains non-negative for all $t \geq 0$. Next, we show that solutions of the regional model remain bounded. Define $U(t) := u_0 (t) + u_1 (t) + \dots + u_\ell (t)$. Then adding equations ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}), ([\[eqlatticei1\]](#eqlatticei1){reference-type="ref" reference="eqlatticei1"}), and ([\[eqlatticel1\]](#eqlatticel1){reference-type="ref" reference="eqlatticel1"}) for $i = 1, 2, \dots, \ell-1$, gives $$\dfrac{\mathrm{d}U}{\mathrm{d}t} \leq \sum_{i=0}^\ell r_1 u_i \left( 1 - \dfrac{u_i}{K} \right),$$ where $K := \max_{i=0,1, \dots, \ell} \{K_i\}$. Then, $$\begin{aligned} \dfrac{\mathrm{d}U}{\mathrm{d}t} &\leq r_1 \left[ (u_0 + u_1 + \dots + u_\ell) - \dfrac{u_0^2 + u_1^2 + \dots + u_\ell^2}{K} \right] \\[12pt] &\leq r_1 \left[ (u_0 + u_1 + \dots + u_\ell) - \dfrac{(u_0 + u_1 + \dots + u_\ell)^2}{(\ell+1)K} \right].\end{aligned}$$ The last inequality follows from the Cauchy-Schwarz inequality, namely, $$(\ell+1)(u_0^2 + u_1^2 + \dots + u_\ell^2) \geq (u_0 + u_1 + \dots + u_\ell)^2.$$ Hence, we have $$\dfrac{\mathrm{d}U}{\mathrm{d}t} \leq r_1 U \left[ 1 - \dfrac{U}{(\ell+1)K} \right] \implies \limsup_{t \rightarrow \infty} U(t) \leq (\ell+1)K.$$ Therefore, the sum $U(t)$ is a bounded function. Since each component of the sum is non-negative, we conclude that each $u_i(t)$ is bounded for each $i = 0, 1, \dots, \ell$. We can similarly show that the infected tumour cells remain bounded at each node by defining $N(t) := n_0 (t) + n_1 (t) + \dots + n_\ell (t)$. Adding equations ([\[eqlattice2\]](#eqlattice2){reference-type="ref" reference="eqlattice2"}), ([\[eqlatticei2\]](#eqlatticei2){reference-type="ref" reference="eqlatticei2"}), and $(\ref{eqlatticel2})$ for $i = 1, 2, \dots, \ell-1$ gives $$\begin{aligned} \dfrac{\mathrm{d}N}{\mathrm{d}t} &\leq \sum_{i=0}^\ell \left[r_2 n_i \left( 1 - \dfrac{n_i}{K} \right) + \theta_\infty u_i \right] \\[12pt] &\leq r_2 N \left[ 1 - \dfrac{N}{(\ell+1)K} \right] + \theta_\infty \bar{U},\end{aligned}$$ where $\bar{U}$ is any upper bound for $U(t)$. The last inequality follows by applying the Cauchy-Schwarz inequality as in the previous case. Hence, $$\limsup_{t \rightarrow \infty} N(t) \leq \dfrac{r_2 (\ell+1)K + \sqrt{r_2^2 (\ell+1)^2K^2 + 4r_2 (\ell+1)K \theta_\infty \bar{U}}}{2r_2}.$$ Since the components of the sum $N(t)$ are non-negative, we conclude that $n_i(t)$ is bounded for each $i = 0, 1, \dots, \ell$. Next, it is clear to see by a comparison argument that $$\limsup_{t \rightarrow \infty} c_i (t) \leq \dfrac{\phi_k}{\beta}, \quad i = 0, 1, 2, \dots, \ell,$$ and so we may conclude that $c_i (t)$ are bounded. We have successfully shown that solutions of the regional model with non-negative initial conditions are non-negative and bounded. $\square$ While an analytic investigation of system ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}) can be challenging to perform due to the potentially large number of equations, we may establish a result which is analogous to Proposition [\[prop2\]](#prop2){reference-type="ref" reference="prop2"} of the local model. In particular, we may establish the a sufficiently large oncolysis rate leads to stability of a tumour-dominant steady state. We begin by showing the existence of this steady state. Since we are also interested in obtaining results related to the spreading speed away from node $i$, i.e., $\eta_i$, we rewrite the function $P_i(x)$ from equation ([\[P_iprobequation\]](#P_iprobequation){reference-type="ref" reference="P_iprobequation"}) by defining the dimensionless quantity $p_i(x) := 1 - e^{-\lambda_i x}$, hence allowing us to formulate $P_i(x)$ in terms of the spreading speed. That is, $$P_i(x) = \eta_i p_i (x).$$ For the remainder of this subsection, we consider only the case where $\phi_k = 0$ for $k = 0, 1, 2, \dots, \ell$. Biologically, this condition corresponds to the case with no external oxygen input. Furthermore, we consider the case where a tumour cell may only travel forward through the network (i.e., in the right, $R$, direction). Hence, we set $q_{j,R} = 1$ and $q_{j,L}= 0$ for $j = 0, 1, 2, \dots, \ell$. This is biologically consistent with the unidirectional flow of tumour cells through the lymphatic system [@zimmerman_lymphfact]. A virus-free or *tumour-dominant* steady state is of the form $$E_u := (u_0, n_0, u_1, n_1, \dots, u_\ell, n_\ell, c_0, c_1, \dots, c_\ell) = (u_0^*, 0, u_1^*, 0, \dots, u_\ell^*, 0, 0, 0, \dots, 0), \label{tumour_dominantsteadystate_regionalmodel}$$ for $i \in \{ 0,1, \dots, \ell \}$, where $u_i^* > 0$. It follows from system ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}) that the components of the tumour-dominant steady state satisfy the equations $$\begin{dcases} r_1 \left( 1 - \dfrac{u_0}{K_0} \right) &= \eta_0 \left(1 - e^{-\lambda_0 u_0}\right), \\[12pt] r_1 u_i\left( 1 - \dfrac{u_i}{K_i} \right) &= -\eta_{i-1} u_{i-1} \left(1-e^{-\lambda_{i-1}u_{i-1}}\right) + \eta_i u_i \left(1-e^{-\lambda_{i}u_i}\right), \quad i \in \{ 1, 2, \dots, \ell - 1 \}, \\[12pt] r_1 u_\ell \left( 1 - \dfrac{u_\ell}{K_\ell} \right) &= -\eta_{\ell-1} u_{\ell - 1}\left( 1 - e^{-\lambda_{\ell-1}u_{\ell-1}} \right) . \end{dcases} \label{341system}$$ The existence of the solution $u_0^* < K_0$ of the first equation of system ([\[341system\]](#341system){reference-type="ref" reference="341system"}) is clear. The solutions of the remaining equations of this system may subsequently be obtained by solving for $u_i^*$ recursively, given $u_{i-1}^*$. Based on a numerical exploration of the system (see Section 6), we also require that $u_i^* > K_i$ for $i \in \{1, 2, \dots, \ell \}$. It is trivial to see that this inequality holds for $i = \ell$. To ensure that this inequality is true for all other values of $i$, it is sufficient to consider the additional condition $$K_i < \dfrac{10 \eta_{i-1}}{7\eta_i}u_{i-1}^* \left( 1 - e^{-\lambda_{i-1}u_{i-1}^*} \right), \quad i \in \{ 1, 2, \dots, \ell - 1 \}.$$ These upper bounds on $K_i$ come from system ([\[341system\]](#341system){reference-type="ref" reference="341system"}) and from $p_i (K_i) = 7/10$. They may be obtained recursively given $u_{i-1}^*$. Let $J = [J_{ij}] \in \mathbb{R}^{(3\ell + 3) \times (3\ell + 3)}$ be the Jacobian matrix of system ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}). We are now in a position to establish the stability of $E_u$. To do so, we make use of the Gershgorin Disc Theorem [@bib_weisstein_chap3] which is stated as follows. **Lemma 2** (Gershgorin Disc Theorem [@bib_weisstein_chap3]). *Consider an $n \times n$ matrix $A = [A_{ij}]$ in $\mathbb{C}^{n \times n}$. Define $$R_i := \sum_{\substack{j=1\\[2pt] j \not= i}}^n \left| A_{ij} \right|, \quad \left| z \right| \ \text{is the modulus of $z \in \mathbb{C}$}.$$ If $\lambda \in \mathbb{C}$ is an eigenvalue of $A$, then $$\lambda \in {\bigcup^n_{i=1}} \left\{ z \in \mathbb{C} : \left| z - A_{ii} \right| \leq R_i \right\}.$$[\[gershgorinlemma\]]{#gershgorinlemma label="gershgorinlemma"}* The circles of the form $\{ z \in \mathbb{C}: \left| z - A_{ii} \right| \leq R_i \} \subset \mathbb{C}$ in Lemma [\[gershgorinlemma\]](#gershgorinlemma){reference-type="ref" reference="gershgorinlemma"} are also called *Gershgorin discs*. Since all eigenvalues of $A$ are contained in these discs, we may bound the real part of these eigenvalues above by $0$ by ensuring that all of the Gershgorin discs lie in the left half of the complex plane. We use the following approach in order to find sufficient conditions for the local asymptotic stability of $E_u$: 1. Linearize system ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}) at the steady state $E_u$. Let $J(E_u) = [J_{ij}(E_u)]$ denote this matrix. 2. For all $i$, compute $R_i$ by adding the absolute value of all of the off-diagonal elements in row $i$ of $J(E_u)$, as in Lemma [\[gershgorinlemma\]](#gershgorinlemma){reference-type="ref" reference="gershgorinlemma"}. 3. Find conditions (if any) such that $\forall i \in \{ 1, 2, \dots, 3\ell + 3 \} : J_{ii}(E_u) + R_i < 0$. If this is possible, then all of the eigenvalues of $J(E_u)$ have negative real part and hence, $E_u$ is locally asymptotically stable. For notational convenience, note that system ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}) may be written in the form $$\begin{aligned} \dfrac{\mathrm{d}u_i}{\mathrm{d}t} &= \mathcal{U}_i (u_0, n_0, u_1, n_1,\dots, u_\ell, n_\ell, c_0, c_1, \dots, c_{\ell}), \\[12pt] \dfrac{\mathrm{d}n_i}{\mathrm{d}t} &= \mathcal{N}_i (u_0, n_0, u_1, n_1,\dots, u_\ell, n_\ell, c_0, c_1, \dots, c_{\ell}), \\[12pt] \dfrac{\mathrm{d}c_i}{\mathrm{d}t} &= \mathcal{C}_i (u_0, n_0, u_1, n_1,\dots, u_\ell, n_\ell, c_0, c_1, \dots, c_{\ell}),\end{aligned}$$ where $i = 0, 1, 2, \dots, \ell$, for appropriately defined functions $\mathcal{U}_i, \mathcal{N}_i,$ and $\mathcal{C}_i$. We begin by noting that the diagonal elements of $J(E_u)$ are $$\begin{aligned} J_{jj}(E_u) &= \begin{dcases} \dfrac{\partial \mathcal{U}_i}{\partial u_i} \bigg|_{E_u} &= r_1 - \dfrac{2r_1}{K_i}u_i^* - \eta_i \left[ p_i(u_i^*) + \lambda_i u_i^* e^{-\lambda_i u_i^*} \right], \quad j = 2i+1, \quad i \in \{ 0,1,\dots,\ell-1 \}, \\[12pt] \dfrac{\partial \mathcal{N}_i}{\partial n_i} \bigg|_{E_u} &= r_2 - \dfrac{r_2}{K_i}u_i^* + \dfrac{\theta_0 u_i^*}{\alpha_i} - \gamma_0 - \eta_i p_i(u_i^*), \quad \quad \ \ j = 2i+2, \quad i \in \{ 0, 1,\dots,\ell-1 \}, \\[12pt] \dfrac{\partial \mathcal{U}_\ell}{\partial u_\ell} \bigg|_{E_u} &= r_1 - \dfrac{2r_1}{K_\ell}u_{\ell}^*, \ \ \qquad \qquad \qquad \qquad \qquad \qquad j = 2\ell + 1,\\[12pt] \dfrac{\partial \mathcal{N}_\ell}{\partial n_\ell} \bigg|_{E_u} &= r_2 - \dfrac{r_2}{K_\ell}u_{\ell}^* + \dfrac{\theta_0 u_{\ell}^*}{\alpha_\ell} - \gamma_0, \quad \qquad \ \ \quad \ \qquad \ j = 2\ell + 2,\\[12pt] \dfrac{\partial \mathcal{C}_i}{\partial c_i} \bigg|_{E_u} &= - \beta - q_1 u_i^*, \qquad \qquad \qquad \qquad \qquad \quad \quad \ \ j = 2\ell + 3 + i, \quad i \in \{ 0, 1, \dots \ell \}, \end{dcases} \end{aligned}$$ To ensure that the eigenvalues lie in the left half of the complex plane, it is sufficient to find conditions such that $J_{ii}(E_u) + R_i < 0$ for all $i$. We begin by computing $J_{11}(E_u) + R_1$ which yields $$\begin{aligned} J_{11}(E_u) + R_1 &= \dfrac{\partial \mathcal{U}_0}{\partial u_0} \bigg|_{E_u} + \left| \dfrac{\partial \mathcal{U}_0}{\partial n_0} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{U}_0}{\partial c_0} \bigg|_{E_u} \right| + \sum_{j=1}^{\ell} \left( \left| \dfrac{\partial \mathcal{U}_0 }{\partial u_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{U}_0 }{\partial n_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{U}_0 }{\partial c_j} \bigg|_{E_u} \right| \right) \\[15pt] &= r_1 - \dfrac{2r_1}{K_0}u_0^* - \eta_0 \left[ p_0(u_0^*) + \lambda_0 u_0^* e^{-\lambda_j u_0^*} \right] + \dfrac{r_1}{K_0}u_0^* + \dfrac{\theta_0 u_0^*}{\alpha_0} + \eta_0 \lambda_0 u_0^* e^{-\lambda_0 u_0^*} \\[15pt] &= r_1 \left(1- \dfrac{u_0^*}{K_0}\right) + \dfrac{\theta_0 u_0^*}{\alpha_0} - \eta_0 p_0 (u_0^*).\end{aligned}$$ This quantity is negative for a sufficiently large spreading speed away from the primary tumour site, $\eta_0$. In particular, this is true if $$\eta_0 > \dfrac{1}{p_0 (u_0^*)}\left[ r_1 \left( 1 - \dfrac{u_0^*}{K_0} \right) + \dfrac{\theta_0 u_0^*}{\alpha_0} \right]. \label{condition_for_eta0}$$ Next, computing $J_{22}(E_u) + R_2$ yields $$\begin{aligned} J_{22}(E_u) + R_2 &= \dfrac{\partial \mathcal{N}_0}{\partial n_0} \bigg|_{E_u} + \left| \dfrac{\partial \mathcal{N}_0}{\partial u_0} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{N}_0}{\partial c_0} \bigg|_{E_u} \right| + \sum_{j=1}^{\ell} \left( \left| \dfrac{\partial \mathcal{N}_0 }{\partial u_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{N}_0 }{\partial n_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{N}_0 }{\partial c_j} \bigg|_{E_u} \right| \right) \\[15pt] &= r_2 - \dfrac{r_2}{K_0}u_0^* + \dfrac{\theta_0 u_0^*}{\alpha_0} - \gamma_0 - \eta_0 p_0(u_0^*).\end{aligned}$$ Condition ([\[condition_for_eta0\]](#condition_for_eta0){reference-type="ref" reference="condition_for_eta0"}) is sufficient for the negativity of this quantity since $r_1 > r_2$ and hence, no additional conditions are necessary. We now consider the case $\ell > 1$, i.e., there are at least two lymph nodes in the network. Let $\mathcal{I} := \{1, 2, \dots, \ell-1\}$. For $i \in \mathcal{I}$, we define $k:= 2i+1$. Then we have $$\begin{aligned} J_{kk}(E_u) + R_k &= \dfrac{\partial \mathcal{U}_i}{\partial u_i} \bigg|_{E_u} + \left| \dfrac{\partial \mathcal{U}_i}{\partial n_i} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{U}_i }{\partial c_i} \bigg|_{E_u} \right| + \sum_{j \in \mathcal{I} \setminus \{i\}} \left( \left| \dfrac{\partial \mathcal{U}_i }{\partial u_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{U}_i }{\partial n_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{U}_i }{\partial c_j} \bigg|_{E_u} \right| \right) \\[15pt] &= r_1 - \dfrac{2r_1}{K_i}u_i^* - \eta_i \left[ p_i(u_i^*) + \lambda_i u_i^* e^{-\lambda_i u_i^*} \right] + \dfrac{r_1}{K_i}u_i^* + \dfrac{\theta_0 u_i^*}{\alpha_i} + \dots \\[15pt] & \quad \quad \quad \dots + \eta_i \lambda_i u_i^* e^{-\lambda_i u_i^*} + \eta_{i-1} \left[ p_{i-1}(u_{i-1}^*) + 2\lambda_{i-1}u_{i-1}^*e^{-\lambda_{i-1}u_{i-1}^*} \right] \\[15pt] &= r_1 \left( 1 - \dfrac{u_i^*}{K_i} \right) + \dfrac{\theta_0 u_i^*}{\alpha_i} - \eta_i p_{i}(u_i^*) + \eta_{i-1} \left[ p_{i-1}(u_{i-1}^*) + 2\lambda_{i-1}u_{i-1}^*e^{-\lambda_{i-1}u_{i-1}^*} \right].\end{aligned}$$ Since $u_i^* > K_i$, the negativity of the above quantity follows given the following condition: $$\begin{aligned} r_1 \left( 1 - \dfrac{u_i^*}{K_i} \right) &< -\dfrac{\theta_0 u_i^*}{\alpha_i} - \eta_{i-1} \left[ p_{i-1}(u_{i-1}^*) + 2 \lambda_{i-1}u_{i-1}^* e^{-\lambda{i-1}u_{i-1}^*} \right] \\[12pt] \iff K_i &< \dfrac{r_1 u_{i}^*}{r_1 + \left[ \dfrac{\theta_0 u_i^*}{\alpha_i} + \eta_{i-1} \left[ p_{i-1}(u_{i-1}^*) + 2 \lambda_{i-1}u_{i-1}^* e^{-\lambda{i-1}u_{i-1}^*} \right] \right]}. \label{K_i_importantcondition}\end{aligned}$$ Biologically, this condition corresponds a sufficiently small carrying capacity of the lymph nodes. Next, for $i \in \mathcal{I}$, let $k = 2i+2$. We have $$\begin{aligned} J_{kk}(E_u) + R_k &= \dfrac{\partial \mathcal{N}_i}{\partial n_i} \bigg|_{E_u} + \left| \dfrac{\partial \mathcal{N}_i}{\partial u_i} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{N}_i}{\partial c_i} \bigg|_{E_u} \right| + \sum_{j \in \mathcal{I} \setminus \{i \}} \left( \left| \dfrac{\partial \mathcal{N}_i }{\partial u_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{N}_i }{\partial n_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{N}_i }{\partial c_j} \bigg|_{E_u} \right| \right) \\[15pt] &= r_2 - \dfrac{r_2}{K_i}u_i^* + \dfrac{\theta_0 u_i^*}{\alpha_i} - \gamma_0 - \eta_i p_i(u_i^*) + \eta_{i-1} p_{i-1}(u_{i-1}^*).\end{aligned}$$ In order to ensure negativity of this quantity, it suffices to impose the condition $$\gamma_0 > \dfrac{\theta_0 u_i^*}{\alpha_i} + \eta_{i-1}p_{i-1}(u_{i-1}^*), \label{gamma0condition_regional}$$ which is a condition for the local asymptotic stability of the tumour-dominant steady state which is similar to that of the local model in Section 3. Finally, we consider the final node in the network, lymph node $\ell$. We have $$\begin{aligned} J_{(2\ell + 1)(2 \ell + 1)}(E_u) + R_{2\ell + 1} &= \dfrac{\partial \mathcal{U}_\ell}{\partial u_\ell} \bigg|_{E_u} + \left| \dfrac{\partial \mathcal{U}_\ell}{\partial n_\ell} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{U}_\ell}{\partial c_\ell} \bigg|_{E_u} \right| + \sum_{j=0}^{\ell-1} \left( \left| \dfrac{\partial \mathcal{U}_\ell }{\partial u_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{U}_\ell }{\partial n_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{U}_\ell }{\partial c_j} \bigg|_{E_u} \right| \right) \\[15pt] &= r_1 - \dfrac{2r_1}{K_\ell}u_{\ell}^* + \dfrac{r_1}{K_\ell}u_{\ell}^* + \dfrac{\theta_0 u_{\ell}^*}{\alpha_\ell} + \eta_{\ell-1} \left[ p_{\ell-1}(u_{\ell-1}^*) + 2\lambda_{\ell-1}u_{\ell-1}^*e^{-\lambda_{\ell-1}u_{\ell-1}^*} \right] \\[15pt] &= r_1 \left( 1 - \dfrac{u_{\ell}^*}{K_\ell} \right) + \dfrac{\theta_0 u_{\ell}^*}{\alpha_\ell} + \eta_{\ell-1} \left[ p_{\ell-1}(u_{\ell-1}^*) + 2\lambda_{\ell-1}u_{\ell-1}^*e^{-\lambda_{\ell-1}u_{\ell-1}^*} \right].\end{aligned}$$ It is clear that this quantity is negative if condition ([\[K_i\_importantcondition\]](#K_i_importantcondition){reference-type="ref" reference="K_i_importantcondition"}) is satisfied for $i = \ell$. Next, $$\begin{aligned} J_{(2\ell + 2)(2 \ell + 2)}(E_u) + R_{2\ell + 2} &= \dfrac{\partial \mathcal{N}_\ell}{\partial n_\ell} \bigg|_{E_u} + \left| \dfrac{\partial \mathcal{N}_\ell}{\partial u_\ell} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{N}_\ell}{\partial c_\ell} \bigg|_{E_u} \right| + \sum_{j=0}^{\ell-1} \left( \left| \dfrac{\partial \mathcal{N}_\ell }{\partial u_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{N}_\ell}{\partial n_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{N}_\ell }{\partial c_j} \bigg|_{E_u} \right| \right) \\[15pt] &= r_2 - \dfrac{r_2}{K_\ell}u_{\ell}^* + \dfrac{\theta_0 u_{\ell}^*}{\alpha_{\ell}} - \gamma_0 + \eta_{\ell - 1} p_{\ell - 1}(u_{\ell-1}^*).\end{aligned}$$ It is again clear that this quantity is negative if condition ([\[gamma0condition_regional\]](#gamma0condition_regional){reference-type="ref" reference="gamma0condition_regional"}) is satisfied for $i = \ell$. Finally, define $\tilde{\mathcal{I}} := \{ 0, 1, 2, \dots, \ell\}$. For $i \in \tilde{\mathcal{I}}$, we define $k:= 2\ell + 3 + i$. It follows that $$\begin{aligned} J_{kk}(E_u) + R_K &= \dfrac{\partial \mathcal{C}_i}{\partial c_i} \bigg|_{E_u} + \left| \dfrac{\partial \mathcal{C}_i}{\partial u_i} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{C}_i}{\partial n_i} \bigg|_{E_u} \right| + \sum_{j \in \tilde{\mathcal{I}}\setminus \{i\}} \left( \left| \dfrac{\partial \mathcal{C}_i }{\partial u_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{C}_i }{\partial n_j} \bigg|_{E_u} \right| + \left| \dfrac{\partial \mathcal{C}_i}{\partial c_j} \bigg|_{E_u} \right| \right) \\[15pt] &= -\beta - q_1 u_i^*.\end{aligned}$$ Hence, $J_{kk}(E_u) + R_K < 0$. By Lemma [\[gershgorinlemma\]](#gershgorinlemma){reference-type="ref" reference="gershgorinlemma"}, we conclude that if all of the above conditions are satisfied, then $E_u$ is locally asymptotically stable. We state this result in the following proposition. **Proposition 4**. *Consider system ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}) when $\phi_k = 0$, $q_{k,R} = 1$, and $q_{k,L} = 0$ for $k = 0, 1, 2, \dots, \ell$. The tumour-dominant steady state $E_u$ is locally asymptotically stable if the following conditions are satisfied:* 1. *$$\eta_0 > \dfrac{1}{p_0 (u_0^*)} \left[ r_1 \left(1 - \dfrac{u_0^*}{K_0} \right) + \dfrac{\theta_0 u_o^*}{\alpha_0} \right].$$* 2. *For $i = 1, 2, \dots, \ell$, $$K_i < \min \left\{ \dfrac{10 \eta_{i-1}}{7\eta_i}u_{i-1}^* \left( 1 - e^{-\lambda_{i-1}u_{i-1}^*} \right), \dfrac{r_1 u_{i}^*}{r_1 + \left[ \dfrac{\theta_0 u_i^*}{\alpha_i} + \eta_{i-1} \left[ p_{i-1}(u_{i-1}^*) + 2 \lambda_{i-1}u_{i-1}^* e^{-\lambda{i-1}u_{i-1}^*} \right] \right]} \right\}.$$* 3. *For $i = 1, 2, \dots, \ell$, $$\gamma_0 > \dfrac{\theta_0 u_i^*}{\alpha_i} + \eta_{i-1}p_{i-1}(u_{i-1}^*).$$* *[\[regional_propchap3\]]{#regional_propchap3 label="regional_propchap3"}* Proposition [\[regional_propchap3\]](#regional_propchap3){reference-type="ref" reference="regional_propchap3"} has some significant biological implications. Since this proposition gives conditions for the stability of the tumour-dominant steady state, the conditions being satisfied represents a clinically unfavourable outcome. The condition $\phi_k = 0$ represents no external oxygenation. Similarly to the local model, we see that hypoxic environments are beneficial to the tumour cells and reduce the efficacy of the adenovirus. Condition 1 of the proposition represents a sufficiently large rate of spreading of tumour cells away from the primary tumour. Condition 2 represents smaller carrying capacities of the lymph nodes -- this is not surprising, as tumour cells will more easily spread away from lymph nodes with lesser carrying capacities, i.e., due to less available resources. Condition 3 once again mirrors an important insight from the local model -- the oncolysis rate must not be too large in relation to the infection rate for an OV to be effective. However, this condition now comes with the additional consideration of incoming tumour cells from the previous lymph node in the network. In general, in a clinical setting, the model suggests effective treatment with an OV requires the engineering of a virus with a sufficiently large infection rate under hypoxic environments, which takes into account the spreading speed of the tumour cells as well as the carrying capacities of the lymph nodes. We further explore the implications of the regional model in the next section. # Numerical simulations: regional model Due to the lack of analytic tractability of system ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"}), we perform simulations to investigate the dynamics of this system. The primary tumour parameters (except for $\theta$ and $\gamma$) are pulled directly from Table 2. The parameters $K_i$ and $\alpha_i$ are estimated by taking into account the corresponding tumour parameters, $K_0$ and $\alpha_0$. In particular, for all $i$, we take $\eta_i = 0.0002$ days$^{-1}$, $K_i = K_0/10$ and $\alpha_i = \alpha_0/10$. Note that these parameters are the same for all lymph nodes. We make the biologically reasonable assumption that cells have a higher probability of migrating away from the primary tumour, i.e., in the direction of increasing node index. Hence, we set $q_{1,L} = q_{2,L} = q_{3,L} = 0.05$ and $q_{1,R} = q_{2,R} = 0.95$. {#fig_lattice1a width="\\textwidth"} {#fig_lattice_1b width="\\textwidth"}\ In Figure [\[3graph10\]](#3graph10){reference-type="ref" reference="3graph10"}, we compare the case of no external oxygen input, $\phi_0 = 0$, to the case of some external oxygen input, $\phi_0 = 10^4$ mM day$^{-1}$. We graph the total number of tumour cells, $u_i (t) + n_i(t)$ over the course of 80 days. The functions $\theta(c)$ and $\gamma(c)$ are given by equations ([\[theta_gamma_equations1\]](#theta_gamma_equations1){reference-type="ref" reference="theta_gamma_equations1"}), where $\theta_0 = 0.005115, \theta_\infty = 1.0, k_\theta = 0.08, \gamma_0 = 0.1, \gamma_\infty = 0.9,$ and $k_\gamma = 0.08$. These parameter values are similar to the ones used in Figure [\[graph04\]](#graph04){reference-type="ref" reference="graph04"} -- they yield a favourable clinical outcome in the local model. The model assumes that external oxygenation may only be performed on the primary tumour site -- not at the lymph nodes. From Figure [\[3graph10\]](#3graph10){reference-type="ref" reference="3graph10"} (a), we see that in the case where no external oxygen is provided, the tumour cells ultimately dominate at the primary tumour site and also approach a value near the carrying capacity at the lymph nodes. This unfavourable result is in stark contrast to the results of Figure [\[graph04\]](#graph04){reference-type="ref" reference="graph04"}, in which the tumour cells are either eradicated or kept under control. On the other hand, in the case of external oxygenation seen in Figure [\[3graph10\]](#3graph10){reference-type="ref" reference="3graph10"} (b), there is a sharp drop in the total tumour cell density. Namely, from a peak value approaching the carrying capacity at the primary site to approximately $4.47 \times 10^5$ cells/mm$^3$. This is a result of the benefit which the OV acquires as a result of an oxygen rich environment. This is consistent with the benefit consistently seen when treating cancer in oxygen-sufficient tumour microenvironments compared to hypoxic microenvironments. Even though the oxygenation occurs only at the primary tumour sites, the model allows for the proliferation of infected tumour cells through the lymphatic vessels into the lymph nodes, and hence the oxygenation also confers an increase in the efficacy of the OV treatment at the lymph nodes. To this end, we turn our attention to the behaviour at the lymph nodes. In Figure [\[3graph10\]](#3graph10){reference-type="ref" reference="3graph10"} (a), the total tumour cell density across all three lymph nodes in the long-term is approximately given by the sum of their carrying capacities. In Figure [\[3graph10\]](#3graph10){reference-type="ref" reference="3graph10"} (b), as a result of external oxygenation, it takes a longer period of time for the tumour cell densities at the lymph nodes to reach their carrying capacities. This is because the benefit of the oxygenation here is less direct -- the oxygenation is only occurring at the primary site. There is still an indirect benefit, however, as a marked decrease of tumour cells at the primary site will result in slower spreading rates. In summary, Figure [\[3graph10\]](#3graph10){reference-type="ref" reference="3graph10"} further illustrates the importance of the oxygen concentration in treatment with adenoviruses, a result which is consistent with the existing oncology literature [@onyx_adeno1]. It may also be worth noting that in contrast to Figure [\[3graph10\]](#3graph10){reference-type="ref" reference="3graph10"} (a), the tumour cell density at lymph node 3 eventually dominates the tumour cell density at lymph node 1 in Figure [\[3graph10\]](#3graph10){reference-type="ref" reference="3graph10"} (b). This may be explained by the fact that oxygenation occurs at the primary tumour site and, therefore, the infected cells are initially closer to the lymph nodes closer to the primary site rather than the subsequent lymph nodes in the network. Hence, lymph node 1 has a slightly greater benefit from the OV treatment than do lymph nodes 2 and 3. From the local model, we found that having a lower virus-induced death rate compared to the infection rate tends to yield more favourable clinical outcomes. To this end, we investigate the dynamics of the regional model in the case where $\theta(c) > (\alpha/K)\gamma(c)$ for all $c \geq 0$. We set $\theta_0 = 0.05115, \theta_\infty = 2.115, k_\theta = 0.016$ and $\gamma(c) = 0.005115$ for all $c\geq 0$. We once again plot the cases $\phi_0 = 0$ and $\phi_0 = 10^4$ mm day$^{-1}$. {#fig_lattice1a width="\\textwidth"} {#fig_lattice_1b width="\\textwidth"}\ Figure [\[3graph11\]](#3graph11){reference-type="ref" reference="3graph11"} shows that the impact of having a sufficiently low virus oncolysis rate in the regional model is consistent with the local model. Once again, the effect of an increased external oxygenation rate is much more pronounced at the primary tumour compared to the lymph nodes. Motivated by Figure [\[3graph11\]](#3graph11){reference-type="ref" reference="3graph11"}, we now consider the impact of the infection rate, $\theta$, and the virus-induced death rate, $\gamma$, on the regional model. These parameters were considered extensively in the the numerical simulations of the local model in Section 4. In this case, we consider keeping $\theta$ and $\gamma$ constant rather than as functions of oxygen concentration. The results of the simulations are plotted in Figure [17](#chap3heatmap){reference-type="ref" reference="chap3heatmap"}. {#chap3heatmap} In Figure [17](#chap3heatmap){reference-type="ref" reference="chap3heatmap"}, we plot the maximum value of the tumour cell density at the primary tumour site over the course of 80 days after OV treatment. That is, we plot $\max \{ u_0(t) + n_0(t) \}$ for different values of $\theta$ and $\gamma$. Consistent with our prior results, we can again visualize the relationship between infection and oncolysis. We see that increasing the virus-induced death rate to a much greater value relative to the infection rate leads to an unfavourable outcome (red region). This also occurs if the virus-induced death rate is too small, regardless of the value of the infection rate. Therefore, this provides further evidence of the importance of a high infection rate and a oncolysis rate that is *not too low* in oder to obtain favourable results (blue region). Finally, we address the case where $\theta$ and $\gamma$ depend on the oxygen concentration. In particular, we assume that we have some mechanism through which to administer external oxygen to the lymph nodes and set $\phi_k = 10^4$ mM day$^{-1}$ for $k = 1, 2, \dots, \ell$. {#fig_lattice1a width="\\textwidth"} {#fig_lattice_1b width="\\textwidth"}\ Figure [\[chap3_everynodeoxygenated\]](#chap3_everynodeoxygenated){reference-type="ref" reference="chap3_everynodeoxygenated"} shows the case of oxygen dependence of the infection rate and the oncolysis rate. In this case, there is a marked reduction in the total tumour cell density in all compartments. Figure [\[chap3_everynodeoxygenated\]](#chap3_everynodeoxygenated){reference-type="ref" reference="chap3_everynodeoxygenated"} (a) shows the uninfected tumour cell density, $u_i$ and Figure [\[chap3_everynodeoxygenated\]](#chap3_everynodeoxygenated){reference-type="ref" reference="chap3_everynodeoxygenated"} (b) shows the infected tumour cell density, $n_i$. In contrast to the case of no oxygen input at the lymph nodes, the infected tumour cell density at each lymph node asymptotically approaches a value below the carrying capacity of its corresponding node. This provides further evidence which supports the lack of efficacy of oncolytic adenoviruses in hypoxic environments and the increased efficacy of these OVs when external oxygenation is provided. # Conclusion and discussion From the mathematical results of this work, as well as the simulations, the importance of the functions $\theta$ and $\gamma$ are emphasized. Biologically, this refers to the interplay between viral infection rate and the virus-induced death rate of the cancer cells. If the virus-induced cancer cell death rate is too large compared to the infection rate, the cancer cells end-up dominating in the long-run. This is a reflection of the virus not being able to infect cells faster than the infected cells are destroyed. On the other-hand, if the infection rate of the OV is significantly large compared to the virus-induced death rate in *all* oxygen environments, the infected tumour cells will dominate in the long-run and will reach some steady state. This steady state may represent the case where we avoid uncontrollable cancer cell growth as the number of cells will not approach the carrying capacity. This translates to a favourable clinical result. On the other hand, it may also represent a state in which the infected tumour cells dominate *at* the carrying capacity if the OV tumour-destroying capabilities are *too* low. For this reason, we suggest that when engineering OVs, it is important to ensure that these viruses have greater infection capabilities than they have oncolytic capabilities while ensuring that the virus-induced death rate is not too low. In particular, our results suggest that maintaining high viral infection rates tends to lead to clinically favourable results regardless of oxygen concentration of the tumour microenvironment. Two approaches which have been identified for the production of efficient OVs under hypoxic conditions are direct genetic engineering and directed evolution [@onyx_adeno1]. While successes from taking a direct engineering approach have not yielded consistently favourable results, the approach of using directed evolution may offer a solution to creating potent OVs. While beyond the scope of this paper, more details on this approach can be found in [@directed_evolution]. Our findings on the importance of the infection rate are consistent with [@Jenner; @more_evidence] and offer further insight on a growing body of literature regarding efficacy of engineered viruses [@Jenner; @more_evidence; @Kuro]. Another important component of this paper is the modelling of the impact of hypoxic conditions on OV treatment efficacy. As previously stated, the modelling suggests that significantly high infection rates are preferable under any oxygen conditions. However, another layer of complexity is added when considering the threat of toxicity which the OV poses toward healthy cells [@simpson]. Furthermore, having a virus-induced death rate which is *too low* will lead to a decreased mortality of cancer cells. Hence, it is not sufficient to simply conclude that engineering extremely infectious viruses is the solution. Instead, we proposed taking into account the effect of different oxygen conditions and hypoxia when constructing the OVs. To address this, we considered the case in which which function dominates, $\theta$ or $\gamma$, depends on the oxygen concentration. A favourable result occurs when $\gamma$ dominates for low oxygen concentrations but $\theta$ dominates for high oxygen concentrations. Hence, we conjecture that another consideration of engineering OVs is whether or not the tumour microenvironment is hypoxic. The preferential virus characteristic would be to have greater oncolytic capabilities in hypoxic environments and greater infectious capabilities in more oxygen-rich environments. According to the modelling, this may lead to stability of a steady tumour load rather than uncontrollable growth. However, we also found that making the virus-induced death rate *too* great under hypoxic conditions also leads to a reduction in the efficacy of the treatment, as the infected tumour cells die faster than they may infect the remaining susceptible tumour cells. We extended the model to a regional model which incorporated spatial structure through considering the axillary lymph nodes. This was done by considering ODEs on a one-dimensional lattice. This natural extension captures the invasive nature of melanoma (and many other invasive cancers). Once again, the importance of considering oxygen cannot be understated. When considering a system with three lymph nodes, we found that providing oxygenation at the site of the primary lesion (through an external oxygen source) yields an approximately 72% decrease in tumour cell density at the site of the primary lesion. Lymph nodes closer to the site of oxygenation similarly obtained benefit from more hyperoxic conditions. This benefit of external oxygenation in (various forms of) the treatment of cancer has also been observed clinically, such as in the use of hyperbaric-oxygen therapy [@tibbles]. Our simulations further support these experimental findings. We also found that the impact of the infection rate, $\theta$, is also present in the regional model and the findings were consistent with those of the local model. This leads us to further stress the importance of oxygen rich microenvironments being used in tandem with highly infectious OVs. This model may be further enhanced by the addition of a variable which accounts for the free virus particles. Although this would increase the complexity of system in terms of mathematical analysis, it would lead to more interesting dynamics, biologically. In terms of the parameters, the growth rate of tumour cells also depends on the available oxygen of the tumour microenvironment [@hypoxia_last]. Hence an important next step is the use of growth rates which depend on the oxygen concentration, i.e., $r_1 (c), r_2(c)$. Future work also includes adding a *continuous* spatial structure to the model, i.e., through the use of PDE modelling. This can take into account the spatial properties of the tumour as well as the efficacy of OV treatment in the context of metastatic disease by modelling cancer cell spreading at the site of the primary lesion. Extending the types of geometry of the lattice representing the lymphatic network is also an important next step. For example, this involves allowing certain lymph nodes in the network to have connections with multiple neighbouring lymph nodes. From a clinical perspective, incorporating the use of conventional chemotherapy along with the virotherapy is also likely to provide potentially useful insights. Finally, the toxic effects of an increased tumour cell infection rate may also be worth considering in order to model a more comprehensive treatment approach. # Acknowledgements {#acknowledgements .unnumbered} This research is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the province of Ontario via the Ontario Graduate Scholarship (OGS). The authors would also like to thank the two anonymous reviewers for their helpful comments, particularly for helping better shape this paper from a biological perspective. # Conflict of interest {#conflict-of-interest .unnumbered} The authors declare there is no conflict of interest. # Appendix The code used to plot the solutions of both the local (([\[3eq1\]](#3eq1){reference-type="ref" reference="3eq1"}) - ([\[3eq3\]](#3eq3){reference-type="ref" reference="3eq3"})) and regional (system ([\[eqlattice1\]](#eqlattice1){reference-type="ref" reference="eqlattice1"}) - ([\[eqlatticel3\]](#eqlatticel3){reference-type="ref" reference="eqlatticel3"})) models is given below. To obtain plots for the local model, we may set the spreading rate of tumour cells away from the primary tumour, $\eta_0$, to $0$. For non-negative $\eta_i$ values, the code produces plots which include lymph node involvement. ``` {.python language="Python"} import numpy as np import matplotlib.pyplot as plt from scipy.integrate import odeint #Parameters: r1, r2 = 0.3954, 0.21; K, alpha = 1e6, 1e5; phi, beta, q1, q2 = 1e4, 5.0976, 5.47e-5, 0.5*(5.47e-5); theta0, thetainf, k_theta = 0.005115, 2.115, 0.016 gamma0, gammainf, k_gamma = 0.1, 0.9, 0.08; K1, K2, K3 = K/10, K/10, K/10; alpha1, alpha2, alpha3 = alpha/10, alpha/10, alpha/10; K_values = [K,K1,K2,K3]; eta = 0.0002; #Comment this out if eta not constant. eta_values = [eta for i in range(4)] #The case with 3 lymph nodes. #Functions: def theta(x): return thetainf*theta0/(theta0 + (thetainf - theta0)*( np.exp((-1)*k_theta*x))) def gamma(x): return gammainf*gamma0/(gamma0 + (gammainf - gamma0)*( np.exp((-1)*k_gamma*x) )) def p(i,x): Lambda = ((-1)*np.log(0.3))/K_values[i] #start at i = 0 return 1 - np.exp((-1)*Lambda*x) def ODEs(x,t): u, n, c, u1, n1, c1 = x[0], x[1], x[2], x[3], x[4], x[5] u2, n2, c2 = x[6], x[7], x[8] u3, n3, c3 = x[9], x[10], x[11] dudt = r1*u*(1 - (u+n)/K) - ((theta(c))*n*u)/(alpha + n) - eta_values[0]*u*p(0,u+n) + 0.05*eta_values[1]*u1*p(1,u1+n1) dndt = r2*n*(1 - (u+n)/K) + ((theta(c))*n*u)/(alpha + n) - (gamma(c))*n - eta_values[0]*n*p(0,u+n) + 0.05*eta_values[1]*n1*p(1,u1+n1) dcdt = phi - beta*c - q1*u*c - q2*n*c du1dt = r1*u1*(1 - (u1+n1)/K1) - ((theta(c1))*n1*u1)/(alpha1 + n1) - eta_values[1]*u1*p(1,u1+n1) + eta_values[0]*u*p(0,u+n) + 0.05*eta_values[2]*u2*p(2,u2+n2) dn1dt = r2*n1*(1 - (u1+n1)/K1) + ((theta(c1))*n1*u1)/(alpha1 + n1) - (gamma(c1))*n1 - eta_values[1]*n1*p(1,u1+n1) + eta_values[0]*n*p(0,u+n) + 0.05*eta_values[2]*n2*p(2,u2 + n2) dc1dt = (-1)*beta*c1 - q1*u1*c1 - q2*n1*c1 du2dt = r1*u2*(1 - (u2+n2)/K2) - ((theta(c2))*n2*u2)/(alpha2 + n2) - eta_values[2]*u2*p(2,u2+n2) + 0.95*eta_values[1]*u1*p(1,u1+n1) + 0.05*eta_values[3]*u3*p(3,u3+n3) dn2dt = r2*n2*(1 - (u2+n2)/K2) + ((theta(c2))*n2*u2)/(alpha2 + n2) - (gamma(c2))*n2 - eta_values[2]*n2*p(2,u2+n2) + 0.95*eta_values[1]*n1*p(1,u1+n1) + 0.05*eta_values[3]*n3*p(3,u3+n3) dc2dt = (-1)*beta*c2 - q1*u2*c2 - q2*n2*c2 du3dt = r1*u3*(1 - (u3+n3)/K3) - ((theta(c3))*n3*u3)/(alpha3 + n3) - 0.05*eta_values[3]*u3*p(3,u3+n3) + 0.95*eta_values[2]*u2*p(2,u2+n2) dn3dt = r2*n3*(1 - (u3+n3)/K3) + ((theta(c3))*n3*u3)/(alpha3 + n3) - (gamma(c3))*n3 - 0.05*eta_values[3]*n3*p(3,u3+n3) + 0.95*eta_values[2]*n2*p(2,u2+n2) dc3dt = (-1)*beta*c3 - q1*u3*c3 - q2*n3*c3 return [dudt, dndt, dcdt, du1dt, dn1dt,dc1dt, du2dt, dn2dt, dc2dt, du3dt, dn3dt, dc3dt] #Initial conditions: u0, n0, c0, u10, n10, c10 = 10000,100, 4.3751, 0, 0, 4.375; u20, n20, c20 = 0, 0, 4.375; u30, n30, c30 = 0, 0, 4.375; init_0 = [u0, n0, c0, u10, n10, c10, u20, n20, c20, u30, n30, c30]; #Numerically solving and plotting the solution of the regional model: t = np.linspace(0,80,10000);#domain x = odeint(ODEs, init_0,t); #integrating u, n, c = x[:,0], x[:,1], x[:,2]; u1, n1, c1 = x[:,3], x[:,4], x[:,5]; u2, n2, c2 = x[:,6], x[:,7], x[:,8]; u3, n3, c3 = x[:,9], x[:,10],x[:,11]; plt.plot(t,u+n,'red',label='Primary Tumour',linewidth=3); plt.plot(t,u1+n1,'green',label='Lymph Node 1',linewidth=3); plt.plot(t,u2+n2,'blue',label='Lymph Node 2',linewidth=3); plt.plot(t,u3+n3,'purple',label='Lymph Node 3',linewidth=3); plt.xlim(0) plt.ylim(0) plt.legend(('Primary Tumour Cell Density', 'Lymph Node 1 Cell Density', 'Lymph Node 2 Cell Density', 'Lymph Node 3 Cell Density'), loc='upper right') plt.ylabel("Cell Density (cells/mm$^3$)") plt.xlabel("Time (days)") ``` The following code produces the heatmap in Figure [17](#chap3heatmap){reference-type="ref" reference="chap3heatmap"}. This multi-parametric analysis shows the peak tumour density value for various values of constant $\theta$ and $\gamma$ over an interval of 100 days after initial OV treatment is administered. ``` {.python language="Python"} t_val = 100; #Solve over this interval. N = 100; #N+1 values of theta and gamma used. theta_values = [0.01*i for i in range(0,N+1)]; gamma_values = [0.01*i for i in range(0,N+1)]; max_cancer_cells = [[] for i in range(0,N+1)]; i = 0; for j in theta_values: theta0 = j; thetainf = j; for k in gamma_values: gamma0 = k; gammainf = k; x = odeint(ODEs, init_0,t); u = x[:,0]; n = x[:,1]; c = x[:,2]; max_cancer_cells[i].append((max(u+n))); i = i + 1; plt.xlabel("Virus-Induced Death Rate ($\gamma$)") plt.ylabel("Infection Rate ($\Theta$)") img = plt.contourf(theta_values,gamma_values,max_cancer_cells,100,cmap='rainbow') plt.colorbar(img) ``` 99 Bauzon, M., & Hermiston, T. W. (2012). Oncolytic viruses: the power of directed evolution. *Advances in Virology*, 2012. 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arxiv_math

--- bibliography: - train_opt.bib --- # Introduction #### Background. As projections indicate that the global urban population is poised to increase from 54% to 66% by the year 2050 [@unitednations2014], the urgency to enhance both the efficiency and sustainability of urban transit systems is escalating. Faced with an estimated influx of 2.5 billion urban inhabitants, there is an impending imperative to restructure existing transportation infrastructure. Rail transit, lauded for its substantial passenger-carrying capacity and safety metrics [@Abril2008], is strategically positioned to play a crucial role in this infrastructural transformation. Notably, Communication-Based Train Control Systems (CBTC) have a significant impact on the effective functioning of these rail networks [@pascoe2009communication]. Utilized by some of the world's busiest metros, CBTC systems offer high-resolution train location determination and continuous data communications. These capabilities significantly improve both safety and efficiency and allow for dynamic adjustments in operational schedules, offering solutions to traffic congestion and fluctuating demand. Thanks to technological advancements like built-in redundancy, modern CBTC systems are also more reliable and easier to maintain. CBTC systems are particularly conducive to developing sustainable and energy-efficient operations, owing to functionalities like the ability to implement automatic driving strategies and the precise implementation of a provided railway timetable [@IRSE2023]. In light of these multi-dimensional advantages, the integration of real-time, energy-efficient timetable optimization within CBTC-enabled railway networks emerges as a cornerstone strategy. This strategic integration not only amplifies operational efficiency and adaptability but also contributes to achieving broader environmental sustainability goals. Hence, it aligns perfectly with the imperatives of modern urban transport, which must be both highly efficient and demonstrably sustainable to meet the complex demands of an increasingly urbanized global landscape. ![This figure graphically illustrates a train's energy consumption and regeneration cycle in CBTC systems. When a train makes a trip from the origin platform to the destination platform, it transitions through four phases - acceleration, speed holding, coasting, and braking, as shown in the speed vs. time graph at the top. The acceleration phase requires the most amount of energy energy, as denoted by the red-shaded area in the power vs. time graph at the bottom. In contrast, the speed holding and coasting phases entail minimal to zero energy usage. Notably, during the braking phase, the train produces regenerative braking energy, represented by the green-shaded area in the power vs. time graph. With appropriate scheduling, this energy can be strategically transferred to nearby accelerating trains.[\[Fig:SpeedProfileOfTrain\]]{#Fig:SpeedProfileOfTrain label="Fig:SpeedProfileOfTrain"}](Figure/power_graph.pdf){#Fig:SpeedProfileOfTrain} #### Motivation. Electricity is the energy source for trains in all modern metro railway networks that utilize CBTC systems. When trains make trips from origin platforms to destination platforms, their speed profile consists of four phases: acceleration, speed holding, coasting, and braking [@Howlett1995], as shown qualitatively in Figure [1](#Fig:SpeedProfileOfTrain){reference-type="ref" reference="Fig:SpeedProfileOfTrain"}. Among these phases, trains consume most of their energy during acceleration. Conversely, when trains brake, the electric motors that make the trains move during the acceleration phase work in reverse, becoming generators. They convert the trains' kinetic energy into electrical energy, referred to as *regenerative braking energy*. CBTC systems are capable of implementing a given timetable precisely, which allows for the transfer of this regenerative energy from braking trains to nearby accelerating trains. For empirical context, the New York City transit railway system, which consumes more than 1,600 GWh of electricity per year, has had all of its trains installed since 2018 capable of producing and transferring regenerative braking energy. Under favorable conditions, the regenerative energy produced can account for up to 50% of the energy consumed [@mohamed2018white]. #### Energy-optimal timetable. The conceptualization and implementation of an energy-optimal timetable offer avenues for substantial energy conservation, obviating the need for infrastructural alterations within CBTC-enabled railway systems. Formally, a railway timetable is a data structure, containing both the arrival and departure times of each train at every platform visited during a designated service period---typically spanning 18 to 24 hours in most operational networks. The core objective of an energy-optimal timetable is to strategically design these temporal decision variables to minimize the network's *effective energy consumption*, which equals the *total energy consumption for all trains during their accelerating phases* minus *the total regenerative braking energy successfully transferred to the accelerating trains from eligible braking counterparts*. #### Related work. Recent years have seen noteworthy contributions to energy-efficient railway timetable computation. For instance, [@Pena-Alcaraz2012] proposes a mixed integer programming (MIP) model limited to single train-lines with successful application on Madrid's Line 3. The work in [@DasGuptaACC15] presents a more tractable MIP model for optimizing regenerative energy transfer between suitable train pairs, applied to the Dockland Light Railway, but does not directly address the actual energy savings. The work in [@Li2014] employs a genetic algorithm to calculate energy-efficient timetables. Their approach seeks to maximize the use of regenerative energy while minimizing the tractive energy of trains. Similarly, [@Le2014] introduces a nonlinear integer programming model that utilizes simulated annealing. The model [@yang2020bi] proposes a non-dominated sorting genetic algorithm for a biobjective timetable optimization model, uniquely incorporating energy allocation and passenger assignment. Continuing in the same vein, Wang and Goverde have conducted studies on multi-train trajectory optimization: their method treats the problem as a multiple-phase optimal control problem, solved by a pseudospectral method [@wang2017multi]. Furthermore, they introduce an innovative approach to energy-efficient timetabling that adjusts the running time allocation of given timetables using train trajectory optimization [@wang2019multi]. From an industry implementation standpoint, the two-stage linear optimization model [@gupta2016two] is, to our knowledge, the sole optimization model incorporated into an industrial timetable compiler. This model presents a two-step linear optimization model for developing an energy-efficient railway timetable. The first stage minimizes total energy consumption for all trains, considering the railway network's constraints. The second stage fixes the trains' trip times to those computed in the first stage and maximizes the transfer of regenerative braking energy between suitable train pairs. However, this model has some limitations. First, although the optimization problems in the first and second stages are solved quickly (in minutes), transitioning between stages requires a time-consuming (in hours) intermediate simulation process, significantly extending the end-to-end runtime. Second, the model in [@gupta2016two] does not directly model the network's actual energy consumption, rather it employs proxy objective functions, the optimization of which might lead to an energy-efficient timetable indirectly. This roundabout approach necessitates extensive physics-based simulations a posteriori to predict the actual reduction in energy consumption. This could limit its applicability in managerial settings, where the model's perceived utility is directly tied to its ability to predict energy consumption levels. Finally, the model requires the precise locations of energy consumption peaks, which are often unavailable, to formulate the optimization problem for the second stage. #### Contribution. We develop a novel single-stage linear optimization model to construct energy-optimal timetables for all the trains in a CBTC-enabled metro railway network for an entire service period. We directly model the total energy consumption, total savings of regenerative energy, and the interaction between regenerative and consumed energy using a data-driven approach. Unlike existing works, our model can predict total energy consumption without requiring time-consuming simulations, making it suitable for widespread use in managerial settings. We also model the transfer of regenerative energy through a set of linear constraints, whereas the existing works use nonconvex constraints. We empirically demonstrate that our model performs extremely well when applied to Metro Line 8 of the Shanghai Railway network, which is one of the busiest railway services in the world in terms of ridership and number of trains. We deploy a warm-started parallel barrier algorithm to solve our linear optimization model, which computes energy-optimal timetables for a full service period of one day with thousands of active trains in real-time (solution time is less than a second on a standard desktop). The optimal timetables exhibit a significant reduction in effective energy consumption in comparison with existing real-world timetables, ranging between approximately 20.93% to 28.68%. Our proposed model offers transformative benefits for managerial decision-making within metro railway networks employing CBTC systems. It not only significantly speeds up the planning process by sidestepping traditional, time-intensive simulations but is also versatile enough for generalized application across any CBTC system. Recognizing these advantages, our model is in the process of being implemented into Thales Canada Inc's industrial timetable compiler. ## Organization The paper is organized as follows. In Section [\[notationAndNotions\]](#notationAndNotions){reference-type="ref" reference="notationAndNotions"}, we present the notation and notions used in this paper. Then in Section [\[sec:Modelling-the-constraints\]](#sec:Modelling-the-constraints){reference-type="ref" reference="sec:Modelling-the-constraints"}, we discuss the constraints that are necessary for proper functioning of a metro railway network. In Section [\[sec:Modeling-the-objective\]](#sec:Modeling-the-objective){reference-type="ref" reference="sec:Modeling-the-objective"}, we present how we model the objective of effective energy consumption. The final optimization model is presented in Section [\[sec:Final-optimization-model\]](#sec:Final-optimization-model){reference-type="ref" reference="sec:Final-optimization-model"}. We present our numerical experiments applied to the Shanghai Railway Network in Section [\[Numerical_Study\]](#Numerical_Study){reference-type="ref" reference="Numerical_Study"}. We describe the architectural framework for industrial integration of our optimization model in Section [7](#sec:archframe){reference-type="ref" reference="sec:archframe"}. # Notation and notions [\[notationAndNotions\]]{#notationAndNotions label="notationAndNotions"} All the sets described in this paper are strictly ordered and finite unless otherwise specified. The cardinality and the $i$-th element of such a set $S$ are denoted by $|S|$ and $S(i)$, respectively. -------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------- **Symbol** **Description** $\mathcal{N}$ The set of all platforms in a railway network $\mathcal{A}$ The set of all tracks $\mathcal{T}$ The set of all trains $\mathcal{N}^{t}$ The set of all platforms visited by a train $t$ in chronological order $\mathcal{A}^{t}$ The set of all tracks visited by a train $t$ in chronological order $a_{i}^{t}$ The arrival time of train $t$ at platform $i$ (decision variable) $d_{i}^{t}$ The departure time of train $t$ from platform $i$ (decision variable) $[\underline{\tau}_{ij}^{t},\overline{\tau}_{ij}^{t}]$ The tack-based trip time window for train $t$ from platform $i$ to platform $j$ $\varphi$ The set of all crossing-overs in the network $\mathcal{B}_{ij}$ The set of all train pairs involved in turn-around events on crossing-over $(i,j)$ $[\underline{\kappa}_{ij}^{tt'},\overline{\kappa}_{ij}^{tt'}]$ The trip time window for train $t$ on the crossing-over $(i,j)$ $[\underline{\delta}_{i}^{t},\overline{\delta}_{i}^{t}]$ The dwell time window for train $t$ at platform $i$ $\chi$ The set of all platform pairs situated at the same interchange stations $\mathcal{C}_{ij}$ The set of connecting train pairs for a platform pair $(i,j)\in\chi$ $[\underline{\chi}_{ij}^{tt'},\overline{\chi}_{ij}^{tt'}]$ The connection window between train $t$ at platform $i$ and and train $t'$ at platform $j$ $\mathcal{H}_{ij}$ The set of train-pairs who move along that track $(i,j)$ $h_{i}^{tt'}$ The headway time between train $t$ and $t'$ at or from platform $i$ $[\underline{\tau}_{\mathcal{P}}^{t},\overline{\tau}_{\mathcal{P}}^{t}]$ The total travel time window for train $t$ to traverse its train path $E_{i,j,t}^{\textrm{con,tr}}$ Energy consumed by train $t$ during acceleration while going from platform $i$ to platform $j$ $E_{i,j,t,t^{\prime}}^{\textrm{con,cr}}$ Energy consumed by train $t$ during acceleration while traversing the crossing-over $(i,j)\in\varphi$ $\Omega$ The set of all platform pairs that are feasible for regenerative energy transfer $\mathcal{T}_{i}$ The set of all trains which arrive at, dwell and then depart from platform $i$ $\overset{\rightharpoonup}{t}$ Temporally close train to the right of train $t$ $\overset{\leftharpoonup}{t}$ Temporally close train to the left of train $t$ $\tilde{t}$ Temporally closes train to train $t$ $\mathcal{E}$ The set of all synchronization processes between suitable train pairs $\overset{\rightharpoonup}{\mathcal{E}}$ A subset of $\mathcal{E}$ containing elements of the form $(i,j,t,\overset{\rightharpoonup}{t})$ $\overset{\leftharpoonup}{\mathcal{E}}$ A subset of $\mathcal{E}$ containing elements of the form $(i,j,t,\overset{\leftharpoonup}{t})$ $E_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ Regenerative energy transferred to accelerating train $t$ on platform $i$ from braking train $\overset{\rightharpoonup}{t}$ on platform $j$ $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ Overlapping time between the effective accelerating phase of train $t$ on platform $i$ and the effective braking phase of train $\overset{\rightharpoonup}{t}$ on platform $j$ $E_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}$ Regenerative energy transferred to accelerating train $\overset{\leftharpoonup}{t}$ on platform $j$ from braking train $t$ on platform $i$ $\sigma_{ij}^{t\overset{\leftharpoonup}{t}}$ Overlapping time between the effective accelerating phase of train $\overset{\leftharpoonup}{t}$ on platform $j$ and the effective braking phase of train $t$ on platform $i$ $[\underline{\beta}_{i}^{t},\overline{\beta}_{i}^{t}]$ The duration of the effective braking phase of train $t$ around platform $i$ $[\underline{\alpha}_{i}^{t},\overline{\alpha}_{i}^{t}]$ The duration of the effective accelerating phase of train $t$ around platform $i$ -------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------- : List of symbols in the order they appear in the paper. Consider a railway network where the set of all stations is denoted by $\mathcal{S}$. The set of all platforms in the railway network is indicated by $\mathcal{N}$. A directed arc between two distinct and non-opposite platforms is called a *track*. The set of all tracks is represented by $\mathcal{A}$. The directed graph of the railway network is expressed by $\mathcal{G}=(\mathcal{N},\mathcal{A})$. A *train-line or line* is a directed path, where the set of nodes represents non-opposite platforms and the set of arcs represents non-opposite tracks. A *crossing-over* is a special type of directed arc that connects two train-lines. If a train arrives at the terminal platform of a train-line, turns around by traversing the crossing-over, and starts traveling through another train-line, then the same physical train is treated and labeled functionally as two different trains by the railway management. The set of all trains to be considered in our problem is denoted by $\mathcal{T}$. The sets of all platforms and all tracks visited by a train $t$ in chronological order are denoted by $\mathcal{N}^{t}\subseteq\mathcal{N}$ and $\mathcal{A}^{t}\subseteq\mathcal{A}$ respectively. The *train-path* of a train is the directed path that contains all platforms and tracks visited by it in chronological order. The decision variables to be determined are the *arrival times* and *departure times* of trains to and from the associated platforms respectively. Let $a_{i}^{t}$ be the arrival time of train $t\in\mathcal{T}$ at platform $i\in\mathcal{N}^{t}$ and $d_{j}^{t}$ be the departure time of train $t$ from platform $j\in\mathcal{N}^{t}$. # Modelling the functional constraints [\[sec:Modelling-the-constraints\]]{#sec:Modelling-the-constraints label="sec:Modelling-the-constraints"} The functional constraints in the railway network show how the events are related and are necessary for the proper functioning of the railway network, which we present in Sections [\[subsec:Trip-time-constraint\]](#subsec:Trip-time-constraint){reference-type="ref" reference="subsec:Trip-time-constraint"}-[\[subsec:Total-travel-time\]](#subsec:Total-travel-time){reference-type="ref" reference="subsec:Total-travel-time"} below. We first discuss how we protect against uncertainty associated with the constraint data in the railway network as follows. #### Incorporating robust constraints through box uncertainty. [\[subsec:Robust-constraints-through\]]{#subsec:Robust-constraints-through label="subsec:Robust-constraints-through"} In practice, there can be some uncertainty in the data associated with the functional constraints, which can cause them to diverge from their intended values. To guard against such uncertainties, we integrate box uncertainty constraints into our model. Abstractly, all the functional constraints can be represented as $\ell\leq x-y\leq u$, where $x$ and $y$ stand for decision variables, and $\ell$ and $u$ refer to the problem data. We then define the following uncertainty set: $\ell\in[\ell_{\textrm{lb}},\ell_{\textrm{ub}}]$ and $u\in[u_{\textrm{lb}},u_{\textrm{ub}}]$. The robust counterpart of the constraint $\ell\leq x-y\leq u,\textrm{ for }\ell\in[\ell_{\textrm{lb}},\ell_{\textrm{ub}}],u\in[u_{\textrm{lb}},u_{\textrm{ub}}]$ can be derived through straightforward computation as: $\ell_{\textrm{ub}}\leq x-y\leq u_{\textrm{lb}}.$ In the interest of brevity, we shall present the robust versions of our constraints directly in the representations that follow. Now we are in a position to present the robustified railway constraints. While from an abstract mathematical point of view, all the constraints have a similar form, from a practical point of view, the constraints are very different. A pictorial representation of what the constraints physically represent is given by Figure [2](#Fig:AllCon){reference-type="ref" reference="Fig:AllCon"}. ![This figure graphically illustrates all the functional constraints present in a metro railway network that the timetable has to satisfy. [\[Fig:AllCon\]]{#Fig:AllCon label="Fig:AllCon"}](Figure/all_constraints.pdf){#Fig:AllCon} ## Trip time constraint [\[subsec:Trip-time-constraint\]]{#subsec:Trip-time-constraint label="subsec:Trip-time-constraint"} The trip time constraints are crucial in determining train energy consumption and regenerative energy production. As discussed previously, while a train is making a trip from the origin platform to the destination platform, almost all the its required energy is consumed during the acceleration phase of its trip and all of the regenerative braking energy is produced during the braking phase. Trip time constraints can be classified into two types described as follows. ### Track-based trip time constraint. Let us consider the trip of any train $t\in\mathcal{T}$ from platform $i$ to platform $j$ along the track $(i,j)\in\mathcal{A}^{t}$. The train departs from platform $i$ at time $d_{i}^{t}$, arrives at platform $j$ at time $a_{j}^{t}$, and its trip time can range from $\underline{\tau}_{ij}^{t}$ to $\overline{\tau}_{ij}^{t}$. We can express the trip time constraint as: $$\begin{aligned} \underline{\tau}_{ij}^{t}\leq a_{j}^{t}-d_{i}^{t}\leq\overline{\tau}_{ij}^{t}, \quad \textrm{for } (i,j)\in\mathcal{A}^{t}, \, t\in\mathcal{T}. & \tag{\texttt{TRACK}}\label{eq:TripTimeConstraint}\end{aligned}$$ This constraint is shown in Figure [2](#Fig:AllCon){reference-type="ref" reference="Fig:AllCon"}(a). ### Crossing-over-based trip time constraint. A crossing-over is a directed arc that connects two train-lines, where a train-line is a directed path consisting of non-opposite platforms and non-opposite tracks. When a train turns around by traversing the crossing-over and starts traveling through another train-line after arriving at the terminal platform of a train-line, it is considered as two different trains by the railway management. Let $\varphi$ be the set of all crossing-overs where turn-around events occur. Suppose we consider any crossing-over $(i,j)\in\varphi$ where the platforms $i$ and $j$ are located on different train-lines. Let $\mathcal{B}_{ij}$ be the set of all train pairs involved in the corresponding turn-around events on the crossing-over $(i,j)$. Let $(t,t')\in\mathcal{B}_{ij}$, where train $t\in\mathcal{T}$ turns around at platform $i$ by traveling through the crossing-over $(i,j)$, and beginning from platform $j$, starts traversing a different train-line as train $t'\in\mathcal{T}\setminus{t}$. A time window $[\underline{\kappa}_{ij}^{tt'},\overline{\kappa}_{ij}^{tt'}]$ must be maintained between the departure of the train from platform $i$ (labeled as train $t$) and the arrival at platform $j$ (labeled as train $t'$). The corresponding trip time constraint can be expressed as: $$\begin{aligned} \underline{\kappa}_{ij}^{tt'}\leq a_{j}^{t'}-d_{i}^{t}\leq\overline{\kappa}_{ij}^{tt'}, \quad \textrm{for } (t,t')\in\mathcal{B}_{ij}, \, (i,j)\in\varphi. & \tag{\texttt{CROSS}}\label{eq:turnAroundConstraint}\end{aligned}$$ This constraint is shown in Figure [2](#Fig:AllCon){reference-type="ref" reference="Fig:AllCon"}(b). ## Dwell time constraint[\[subsec:Dwell-time-constraint\]]{#subsec:Dwell-time-constraint label="subsec:Dwell-time-constraint"} When a train $t\in\mathcal{T}$ arrives at a platform $i\in\mathcal{N}^{t}$, it dwells there for a certain time interval denoted by $[\underline{\delta}_{i}^{t},\overline{\delta}_{i}^{t}]$, during which passengers can embark or disembark. The train departs from the station once the dwell time has elapsed. The difference between the departure time $d_{i}^{t}$ and arrival time $a_{i}^{t}$ due to dwell time lies between $\underline{\delta}_{i}^{t}$ and $\overline{\delta}_{i}^{t}$. We can express the dwell time constraint as: $$\begin{aligned} \underline{\delta}_{i}^{t}\leq d_{i}^{t}-a_{i}^{t}\leq\overline{\delta}_{i}^{t}, \quad \textrm{for } i\in\mathcal{N}^{t}, \, t\in\mathcal{T}. & \tag{\texttt{DWELL}}\label{eq:DwellTimeConstraint}\end{aligned}$$ This constraint is illustrated in Figure [2](#Fig:AllCon){reference-type="ref" reference="Fig:AllCon"}(c). Note that the platform index $i$ is varied over all elements of the set $\mathcal{N}^{t}$ in Equation ([\[eq:DwellTimeConstraint\]](#eq:DwellTimeConstraint){reference-type="ref" reference="eq:DwellTimeConstraint"}). This is so because every train $t\in\mathcal{T}$ arrives at the first platform $\mathcal{N}^{t}(1)$ in its train-path either from the depot or by turning around from some other line, and departs from the final platform $\mathcal{N}^{t}(|\mathcal{N}^{t}|)$ to return to the depot or start as a new train on another line by turning around. Therefore, train $t$ dwells at all the platforms in $\mathcal{N}^{t}$. ## Connection constraint [\[subsec:Connection-constraint\]]{#subsec:Connection-constraint label="subsec:Connection-constraint"} In some cases, there might not be a direct train between the origin and destination of a passenger. To address this issue, the railway management employs connecting trains at interchange stations. Let $\chi\subseteq\mathcal{N}\times\mathcal{N}$ be the set of platform pairs where passengers transfer between trains. If $(i,j)\in\chi$, then both platforms $i$ and $j$ are located at the same station, and there exists a train $t\in\mathcal{T}$ arriving at platform $i$ and another train $t'\in\mathcal{T}$ departing from platform $j$, such that a connection time window must be maintained between trains $t$ and $t'$ for passengers to transfer from the former to the latter. Note that order matters in this context. Let $\mathcal{C}_{ij}$ be the set of train pairs that enable passengers to make the corresponding connection or turn-around event for the platform pair $(i,j)\in\chi$. The connection constraint can be expressed as: $$\begin{aligned} \underline{\chi}_{ij}^{tt'}\leq d_{j}^{t'}-a_{i}^{t}\leq\overline{\chi}_{ij}^{tt'}, \quad \textrm{for } (t,t')\in\mathcal{C}_{ij}, \, (i,j)\in\chi, \tag{\texttt{CONNECT}}\label{eq:ConnectionConstraint}\end{aligned}$$ where $\underline{\chi}_{ij}^{tt'}$ and $\overline{\chi}_{ij}^{tt'}$ are the lower and upper bounds, respectively, of the time window required to achieve the described connection between the associated trains. The connection constraint is shown in Figure [2](#Fig:AllCon){reference-type="ref" reference="Fig:AllCon"}(d). ## Headway constraint [\[subsec:Headway-constraint\]]{#subsec:Headway-constraint label="subsec:Headway-constraint"} {#Headway constraint:} In any railway network, a minimum amount of time is always maintained between the departures of consecutive trains, known as the headway time. Let $(i,j)\in\mathcal{A}$ be the track between two platforms $i$ and $j$, and let $\mathcal{H}_{ij}$ be the set of train pairs that move along that track successively in the order of their departures. Assume that train $t$ and train $t'$ move along this track in the same direction, where $(t,t')\in\mathcal{H}_{ij}$. Let $h_{i}^{tt'}$ and $h_{j}^{tt'}$ be the associated headway times at platforms $i$ and $j$, respectively. The headway constraint can be expressed as follows: $$\begin{aligned} h_{i}^{tt'}\leq d_{i}^{t'}-d_{i}^{t}\;\textrm{ and }\ h_{j}^{tt'}\leq d_{j}^{t'}-d_{j}^{t}, \quad \textrm{for } (t,t')\in\mathcal{H}_{ij}, \, (i,j)\in\mathcal{A}. & \tag{\texttt{HEADWAY}}\label{eq:SafetyConstraint1}\end{aligned}$$ The headway constraint is shown in Figure [2](#Fig:AllCon){reference-type="ref" reference="Fig:AllCon"}(e). To ensure the safety of train operations, we must always maintain the headway constraints between two consecutive trains on the same track. Thus, when a train enters the braking phase and approaches a platform, the platform it enters cannot be immediately occupied by another train, as doing so would result in a temporal distance smaller than the headway time. It follows that selecting two consecutive trains, one accelerating and one braking at the same platform, is not feasible for synchronization from a safety standpoint. This underscores the importance of carefully considering the headway constraints when optimizing train schedules. ## Total travel time constraint [\[subsec:Total-travel-time\]]{#subsec:Total-travel-time label="subsec:Total-travel-time"} In order to provide reliable service in the railway network, it is crucial to ensure that the total travel time for each train $t\in\mathcal{T}$ falls within a specified time window $[\underline{\tau}_{\mathcal{P}}^{t},\overline{\tau}_{\mathcal{P}}^{t}]$, where $\underline{\tau}_{\mathcal{P}}^{t}$ and $\overline{\tau}_{\mathcal{P}}^{t}$ are the corresponding lower and upper bounds, respectively. This means that the time elapsed between the train's departure from its first platform $\mathcal{N}^{t}(1)$ and its arrival at the last platform $\mathcal{N}^{t}(|\mathcal{N}^{t}|)$ must lie within the prescribed time window. Mathematically, we can express this constraint as: $$\begin{aligned} \underline{\tau}_{\mathcal{P}}^{t}\leq a_{\mathcal{N}^{t}(|\mathcal{N}^{t}|)}^{t}-d_{\mathcal{N}^{t}(1)}^{t}\leq\overline{\tau}_{\mathcal{P}}^{t}, \quad \textrm{for } t\in\mathcal{T}. & \tag{\texttt{TRAVEL}}\label{eq:TotalTravelTimeConstraints}\end{aligned}$$ It is worth noting that violating this constraint could result in delayed trains, missed connections, and reduced overall network capacity. This constraint is shown in Figure [2](#Fig:AllCon){reference-type="ref" reference="Fig:AllCon"}(f). ## Domain of the event times {#domain-of-the-event-times .unnumbered} We can define the domain of the decision variables for the railway scheduling problem using the start and end times of the railway service period. To simplify the problem, we set the time of the first event of the service period to zero seconds, which corresponds to the departure of the first train of the day from some platform. We can then obtain an upper bound for the final event of the railway service period, which is the arrival of the last train of the day at some platform, by setting all trip times and dwell times to their maximum possible values. Let this upper bound be denoted by a positive number $m$. Then, the domain of the decision variables can be expressed as follows: $$\begin{aligned} 0\leq a_{i}^{t}\leq m,\;0\leq d_{i}^{t}\leq m, \quad \textrm{for } i\in\mathcal{N}^{t}. & \tag{\texttt{DOMAIN}}\label{eq:domain}\end{aligned}$$ # Modeling the effective energy consumption [\[sec:Modeling-the-objective\]]{#sec:Modeling-the-objective label="sec:Modeling-the-objective"} In this section, we discuss how we model the objective that minimizes the *effective energy consumption* of the railway network. The effective energy consumption is defined as the total energy consumed by all trains during their acceleration phases, denoted by $E^{\textrm{con}}$, minus the total regenerative braking energy transferred to such accelerating trains from eligible braking trains, denoted by $E^{\textrm{reg}}$. This is expressed as $E^{\textrm{con}}-E^{\textrm{reg}}$. We next discuss how we apply data-driven approaches to model $E^{\textrm{con}}$ in Section [4.1](#sec:consum){reference-type="ref" reference="sec:consum"} and $E^{\textrm{reg}}$ in Section [\[sec:regen_descrip\]](#sec:regen_descrip){reference-type="ref" reference="sec:regen_descrip"}. ## Modeling total consumed energy $E^{\textrm{con}}$ {#sec:consum} #### Relationship between energy consumption and the trip time constraint. Trains primarily consume electrical energy during their acceleration phase while going from an origin to a destination platform. As such, trip time constraints critically influence both energy consumption and the production of regenerative energy in trains. Once the trip time gets fixed, the energy-optimal speed profile can be efficiently computed in CBTC systems. A variety of software tools, e.g., **T**rain **K**inetics, **D**ynamics, and **C**ontrol ([KDC]{.sans-serif}) Simulator used in our research, can be employed for this purpose [@selTrac]. The [KDC]{.sans-serif} simulator, based on the strategies of acceleration, speed holding, coasting, and braking, calculates the speed profile. This method aligns with the theoretical optimal speed profile as presented in the monograph [@Howlett1995]. For a deeper dive into the calculation of optimal speed profiles, readers might consider the following papers [@Jiaxin1993; @Howlett2000; @Howlett2009; @Khmelnitsky2000; @Liu2003], with a comprehensive review available in [@Albrecht2015]. The speed profile of a train on a track, e.g., the top subfigure of Figure [1](#Fig:SpeedProfileOfTrain){reference-type="ref" reference="Fig:SpeedProfileOfTrain"}, dictates its electrical power consumption and regeneration, leading to its power versus time graph or *power graph*, as shown in the bottom subfigure of Figure [1](#Fig:SpeedProfileOfTrain){reference-type="ref" reference="Fig:SpeedProfileOfTrain"}. #### Energy consumption of all trains. We denote the total energy consumed by all trains by $$\sum_{t\in\mathcal{T}}\sum_{(i,j)\in\mathcal{A}^{t}}\underbrace{E_{i,j,t}^{\textrm{con,tr}}}_{\substack{\textrm{depends on }\\ \textrm{trip time }(a_{j}^{t}-d_{i}^{t}) } }\quad+\sum_{(i,j)\in\varphi,(t,t^{\prime})\in\mathcal{B}_{i,j}}\underbrace{E_{i,j,t,t^{\prime}}^{\textrm{con,cr}}}_{\substack{\textrm{depends on }\\ \textrm{trip time }(a_{j}^{t'}-d_{i}^{t}) } }$$ where - $E_{i,j,t}^{\textrm{con,tr}}:\mathbb{R}_{++}\to\mathbb{R}_{++}$ is the energy consumed by a train $t\in\mathcal{T}$ during the acceleration phase of the trip from the origin platform $i$ to the destination platform $j$ with $(i,j)\in\mathcal{A}^{t}$. In a CBTC system, this function depends on the trip time $(a_{j}^{t}-d_{i}^{t})$. - $E_{i,j,t,t^{\prime}}^{\textrm{con,cr}}:\mathbb{R}_{++}\to\mathbb{R}_{++}$ is the energy consumed by train $t$ while traversing the crossing-over $(i,j)\in\varphi$; the energy is again consumed during the acceleration phase of $t$'s trip from the origin platform $i$ to destination platform $j$, where it gets labeled as the train $t^{\prime}$ i.e., $(t,t^{\prime})\in\mathcal{B}_{i,j}$. The trip time $a_{j}^{t'}-d_{i}^{t}$ associated with this crossing-over is the argument of $E_{i,j,t,t^{\prime}}^{\textrm{con,cr}}$ in a CBTC system. ![This figure graphically illustrates how the energy consumed by a train $t$ going from platform $i$ to $j$ on track $(i,j)$ varies as the trip time $a_j^t - d_i^t$ is varied. The longer range of trip time, while not practical for a metro railway network, is shown to illustrate the nonlinear nature of the consumed energy versus trip time in long inter-city travels that often span a few hours. On the other hand, the valid range of trip time in a metro network denoted by $[\underline{\tau}_{ij}^{t}, \overline{\tau}_{ij}^{t}]$ is on the order of seconds, and in such a setup an affine approximation for the consumed energy can be reasonable. [\[Fig:ConsumedEngTrain\]]{#Fig:ConsumedEngTrain label="Fig:ConsumedEngTrain"}](Figure/energy_consum_vs_trip_time.pdf){#Fig:ConsumedEngTrain} #### Data-driven approach to model the energy consumption. The exact analytical form of $E_{i,j,t}^{\textrm{con,tr}}$ or $E_{i,j,t,t^{\prime}}^{\textrm{con,cr}}$ might be intractable as suggested by [@Howlett2009; @gupta2016two] and as shown qualitiatively in Figure [1](#Fig:SpeedProfileOfTrain){reference-type="ref" reference="Fig:SpeedProfileOfTrain"}. Regardless, these functions exhibit a consistent characteristic: they are monotonically decreasing with trip time. That is to say, if an optimal speed profile is adhered to, the functions become *non-increasing* as trip time increases, as supported by [@Milroy1980]. Notably, even in scenarios where trains are manually driven, potentially straying from optimal strategies, empirical evidence still supports the monotonic decrease in average energy consumption with increased trip time, as can be observed in [@pena2011approximate Figure 1]. Furthermore, gathering practical measurements for the energy function is straightforward, either by analyzing historical data or employing physics-based simulations. A crucial observation into CBTC-enabled metro railway networks is the tight margin by which trip time is permitted to vary in Equations [\[eq:TripTimeConstraint\]](#eq:TripTimeConstraint){reference-type="eqref" reference="eq:TripTimeConstraint"} and [\[eq:turnAroundConstraint\]](#eq:turnAroundConstraint){reference-type="eqref" reference="eq:turnAroundConstraint"}: such variations are often on the order of seconds. This leads us to the following assumption: [\[assum:triptime\]]{#assum:triptime label="assum:triptime"} The amount by which the trip time is allowed to vary is on the order of seconds, i.e., $\overline{\tau}_{ij}^{t}-\underline{\tau}_{ij}^{t}$ in [\[eq:TripTimeConstraint\]](#eq:TripTimeConstraint){reference-type="eqref" reference="eq:TripTimeConstraint"} and $\overline{\kappa}_{ij}^{tt'} - \underline{\kappa}_{ij}^{tt'}$ in [\[eq:turnAroundConstraint\]](#eq:turnAroundConstraint){reference-type="eqref" reference="eq:turnAroundConstraint"} are on the order of seconds. Given the monotonically decreasing behavior of the energy function and Assumption [\[assum:triptime\]](#assum:triptime){reference-type="ref" reference="assum:triptime"}, we can reasonably approximate the energy consumption functions as affine functions. This is qualitatively depicted in Figure [3](#Fig:ConsumedEngTrain){reference-type="ref" reference="Fig:ConsumedEngTrain"} for track-based trip time constraints. Similar curves can be expected for crossing-over based trip time constraints. Our next objective is to derive the best affine approximation for energy consumed as a function of trip time. We'll achieve this by employing a least-squares method to fit a straight line to the energy versus trip time data, where the data can come from historical records or physics-based simulators, such as the [KDC]{.sans-serif} simulator, which we used in this work [@selTrac]. In other words, we approximate consumed energy as: $$\begin{alignedat}{1}E_{i,j,t}^{\textrm{con,tr}}\triangleq & c_{i,j,t}^{\textrm{con,tr}}(a_{j}^{t}-d_{i}^{t})+b_{i,j,t}^{\textrm{con,tr}},\quad \textrm{for }(i,j)\in\mathcal{A}^{t},\,t\in\mathcal{T},\\ E_{i,j,t,t^{\prime}}^{\textrm{con,cr}}\triangleq & c_{i,j,t,t^{\prime}}^{\textrm{con,cr}}(a_{j}^{t'}-d_{i}^{t})+b_{i,j,t,t^{\prime}}^{\textrm{con,cr}},\quad \textrm{for }(i,j)\in\varphi,(t,t^{\prime})\in\mathcal{B}_{i,j}, \end{alignedat} \label{eq:eng_con_model}$$ where - $(c_{i,j,t}^{\textrm{con,tr}},b_{i,j,t}^{\textrm{con,tr}})$ is computed by solving the following least-squares problem: $$(c_{i,j,t}^{\textrm{con,tr}},b_{i,j,t}^{\textrm{con,tr}})=\textrm{argmin}_{(\tilde{c}_{i,j,t},\tilde{b}_{i,j,t})}\sum_{k=1}^{p}\left(\tilde{c}_{i,j,t}(a_{j}^{t}-d_{i}^{t})^{(k)}+\tilde{b}_{i,j,t}-(\bar{E}_{i,j,t}^{\textrm{con,tr}})^{(k)}\right)^{2},$$ where $(i,j)\in\mathcal{A}^{t},\,t\in\mathcal{T}$, with $(\bar{E}_{i,j,t}^{\textrm{con,tr}})^{(k)}$ being the measured energy consumption associated with the track related trip time $(a_{j}^{t}-d_{i}^{t})^{(k)}$ for observations $k=1,2,\ldots,p$. - $(c_{i,j,t,t^{\prime}}^{\textrm{con,cr}},b_{i,j,t,t^{\prime}}^{\textrm{con,cr}})$ is computed by solving the following least-squares problem: $$(c_{i,j,t,t^{\prime}}^{\textrm{con,cr}},b_{i,j,t,t^{\prime}}^{\textrm{con,cr}})=\textrm{argmin}_{(\tilde{c}_{i,j,t,t^{\prime}},\tilde{b}_{i,j,t,t^{\prime}})}\sum_{k=1}^{q}\left(\tilde{c}_{i,j,t,t^{\prime}}(a_{j}^{t'}-d_{i}^{t})^{(k)}+\tilde{b}_{i,j,t,t^{\prime}}-(\bar{E}_{i,j,t,t^{\prime}}^{\textrm{con,cr}})^{(k)}\right)^{2},$$ where $(i,j)\in \varphi,(t,t^{\prime})\in\mathcal{B}_{i,j}$ with $\bar{E}{}_{i,j,t,t^{\prime}}^{\textrm{con,cr}}{}^{(k)}$ being the measured energy consumption related to crossing-over related trip time $(a_{j}^{t'}-d_{i}^{t})^{(k)}$ for observations $k=1,2,\ldots,q$. ## Modeling transferred regenerative energy $E^{\textrm{reg}}$ [\[sec:regen_descrip\]]{#sec:regen_descrip label="sec:regen_descrip"} #### Robust approximations of the power vs. time graph. Maximizing the transfer of regenerative braking energy between suitable train pairs is equivalent to maximizing the total overlapped area between the power vs. time graphs associated with power consumption and regeneration of those train pairs. The power vs. time graph for a train during its acceleration and braking phases depends on its trip time and is highly nonlinear in nature with no known analytical form in CBTC systems. However, the graph empirically exhibits a region where most of the power is concentrated, which allows us to apply the robust lumping method known as the FWHM (Full Width at Half Maximum) method, approximating the power vs. time graphs as rectangles. The FWHM method can be traced back to the early days of quantum mechanics [@gamow1985thirty], where it was instrumental in approximating the wave absorption vs. wavelength graph of molecules [@diem2021quantum Chapter 5], and since then it has been used successfully in various fields such as numerical analysis, spectroscopy, signal processing, and statistics to approximate complicated graphs, and it's considered a useful, robust metric because it is not sensitive to the exact shape of the peak [@Mahajan2008 page 35-37]. In the context of our problem, the FWHM method is applied by transforming the power vs. time graph into a rectangle (see Figure [4](#Fig:FWHM){reference-type="ref" reference="Fig:FWHM"}) whose height is the height of the peak and whose width is the full width at half maximum. These rectangles allow us to recognize the beginning and end of a train's effective accelerating or braking phases where most of the energy is concentrated, which we illustrate graphically in Figure [4](#Fig:FWHM){reference-type="ref" reference="Fig:FWHM"}. Consider a train $t$ arriving at (i.e., braking), then dwelling, and finally departing (i.e., accelerating) from platform $i$. As shown in Figure [4](#Fig:FWHM){reference-type="ref" reference="Fig:FWHM"}, the effective braking and accelerating phases of the train can be compactly contained in FWHM rectangle. We denote the beginning and end of the effective braking phase of $t$ by $a_{i}^{t}-\underline{\beta}_{i}^{t}$ and $a_{i}^{t}-\overline{\beta}_{i}^{t}$, respectively, and the beginning and end of the effective accelerating phase of the train $t$ by $d_{i}^{t}+\underline{\alpha}_{i}^{t}$ and $d_{i}^{t}+\overline{\alpha}_{i}^{t}$, respectively. Determining the beginning and end of the effective braking and accelerating phases of the trains allows us to model the effective overlapping time in a very compact way. The *effective overlapping time* between two trains is defined as the duration of time during which the effective accelerating phase of the first train and the effective braking phase of the second train overlap, provided that both trains are powered by the same electrical substation, and the transmission loss associated with the transfer of regenerative energy of the braking train to the accelerating train is negligible. Thus, maximizing the transfer of regenerative energy positively correlates with the effective overlapping time between suitable train pairs, which we model next. ![This figure illustrates how to compute the effective braking and acceleration phases of trains using the FWHM (Full Width at Half Maximum) method. Consider a train $t$ arriving (i.e., braking) at, then dwelling, and finally departing (i.e., accelerating) from platform $i$. While the exact power vs. time graph of a train is challenging to model analytically, it is possible to compute a very robust approximation by applying the FWHM method. By doing so, the effective braking and accelerating phase of a train can be compactly represented by these rectangular approximations, which allows us to identify the beginning and end of the effective braking phase of train $t$, denoted by $a_{i}^{t}-\underline{\beta}_{i}^{t}$ and $a_{i}^{t}-\overline{\beta}_{i}^{t}$, respectively, and the beginning and end of the effective accelerating phase of the train $t$, represented by $d_{i}^{t}+\underline{\alpha}_{i}^{t}$ and $d_{i}^{t}+\overline{\alpha}_{i}^{t}$, respectively. [\[Fig:FWHM\]]{#Fig:FWHM label="Fig:FWHM"}](Figure/FWHM_power_graph.pdf){#Fig:FWHM} #### Constructing suitable train pairs for transfer of regenerative braking energy. The set that contains all platform pairs that are feasible for regenerative energy transfer is denoted by $\Omega$. Consider any such platform pair $(i,j)\in\Omega$, and let $\mathcal{T}_{i}\subseteq\mathcal{T}$ be the set of all the trains that arrive at, dwell and then depart from platform $i$. To avoid duplicates in $\Omega$ we construct the elements lexicographically with $i < j$. Suppose, $t\in\mathcal{T}_{i}$. Now, we are interested in finding another train $\tilde{t}$ on platform $j$, i.e., $\tilde{t}\in\mathcal{T}_{j}$, which along with $t$ would form a suitable pair for the transfer of regenerative braking energy. To achieve this, we start with an initial feasible timetable for the railway, which represents the desired service to be delivered. For most of the existing railway networks, the railway management has a feasible timetable. For every train $t$, this feasible timetable provides a feasible arrival time $\bar{a}_{i}^{t}$ and a feasible departure time $\bar{d}_{i}^{t}$ to and from every platform $i\in\mathcal{N}^{t}$ respectively. Intuitively, among all the trains that go through platform $j$, the one that is temporally close to $t$ in the initial timetable would be a good candidate to form a pair with $t$. The temporal proximity can be of two types with respect to $t$, which results in the following definitions. Consider any $(i,j)\in\Omega$. For every train $t\in\mathcal{T}_{i}$, the train $\overset{\rightharpoonup}{t}\in\mathcal{T}_{j}$ is called *temporally close to the right of $t$* if $$\begin{aligned} 0\leq\frac{\bar{a}_{j}^{\overset{\rightharpoonup}{t}}+\bar{d}_{j}^{\overset{\rightharpoonup}{t}}}{2}-\frac{\bar{a}_{i}^{t}+\bar{d}_{i}^{t}}{2}\leq r,\label{eq:temporallyClosestTrainRight}\end{aligned}$$ where $r$ is an empirical parameter determined by the timetable designer and is much smaller than the time horizon of the entire timetable. Consider any $(i,j)\in\Omega$. For every train $t\in\mathcal{T}_{i}$, the train $\overset{\leftharpoonup}{t}\in\mathcal{T}_{j}$ is called *temporally close train to the left of $t$* if $$\begin{aligned} 0<\frac{\bar{a}_{i}^{t}+\bar{d}_{i}^{t}}{2}-\frac{\bar{a}_{j}^{\overset{\leftharpoonup}{t}}+\bar{d}_{j}^{\overset{\leftharpoonup}{t}}}{2}\leq r.\label{eq:temporallyClosestTrainLeft}\end{aligned}$$ Consider any $(i,j)\in\Omega$. For every train $t\in\mathcal{T}_{i}$, the train $\tilde{t}\in\mathcal{T}_{j}$ is called **temporally close to $t$** if it is temporally close to the left or right of $t$. So, any event where transfer of regenerative energy is possible can be described by specifying the corresponding $i$, $j$, $t \in \mathcal{T}_i$ and $\tilde{t}\in \mathcal{T}_j$ by using the definitions above. We construct a set of all such $(i,j,t,\tilde{t})$ with $i<j$ (to avoid duplicates), which we denote by $\mathcal{E}$. Naturally, we can partition $\mathcal{E}$ into two sets denoted by $\overset{\rightharpoonup}{\mathcal{E}}$ and $\overset{\leftharpoonup}{\mathcal{E}}$, which contain elements of the form $(i,j,t,\overset{\rightharpoonup}{t})$ and $(i,j,t,\overset{\leftharpoonup}{t})$ respectively. For every $(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}}$ (called *right event*), our strategy is to synchronize the effective accelerating phase of $t$ with the effective braking phase of $\overset{\rightharpoonup}{t}$. On the other hand, for every $(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}$ (called left event), it is convenient to synchronize the effective accelerating phase of $\overset{\leftharpoonup}{t}$ with the effective braking phase of $t$. For every $(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}}$, the corresponding effective overlapping time is denoted by $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$, (called *right event overlapping time*) and for every $(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}$, the corresponding effective overlapping time is denoted by $\sigma_{ij}^{t\overset{\leftharpoonup}{t}}$ (called *left event overlapping time*). The overlapping time can be: *positive* when there is a positive temporal synchronization between effective acceleration phase and effective braking phase, *zero* where there is no synchronization and the temporal distance between the phases is zero, or *negative* which corresponds to the case where their phases are apart by a certain temporal distance; we will illustrate these different overlapping times graphically later in this section. The total regenerative energy transferred depends on the total effective overlapping time. Our objective is to maximize the transfer of regenerative energy, which we model by: $$\begin{aligned} \sum_{(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}}}\underbrace{E_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}}_{\substack{\textrm{depends on }\\ \textrm{right event }\\ \textrm{overlapping time }\sigma_{ij}^{t\overset{\rightharpoonup}{t}} } }\quad+\sum_{(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}}\underbrace{E_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}}_{\substack{\textrm{depends on }\\ \textrm{left event }\\ \textrm{overlapping time }\sigma_{ij}^{t\overset{\leftharpoonup}{t}} } }, \label{eq:obj_funct-1}\end{aligned}$$ where: - $E_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}:\mathbb{R}\to\mathbb{R}$ measures the transfer of regenerative energy from synchronizing the accelerating phase of $t$ with the braking phase of $\overset{\rightharpoonup}{t}$, with the associated overlapping time $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ being the argument. - $E_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}:\mathbb{R}\to\mathbb{R}$ measures the transfer of regenerative energy from synchronizing the accelerating phase of $\overset{\leftharpoonup}{t}$ with the braking phase of $t$, where $\sigma_{ij}^{t\overset{\leftharpoonup}{t}}$ is the argument. To model [\[eq:obj_funct-1\]](#eq:obj_funct-1){reference-type="eqref" reference="eq:obj_funct-1"} in a tractable manner, we perform three steps as follows. 1. Computing data-driven approximations of $E_{i,j,t,\protect\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ and $E_{i,j,t,\protect\overset{\leftharpoonup}{t}}^{\textrm{reg}}$. 2. Computing data-driven approximations of the effective accelerating and braking phases of trains. 3. Modeling the overlapping times. Now we describe the three aforementioned steps in detail. #### Step 1. Computing data-driven approximations of $E_{i,j,t,\protect\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ and $E_{i,j,t,\protect\overset{\leftharpoonup}{t}}^{\textrm{reg}}$. Using a data-driven approach, we approximate $E_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ and $E_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ as affine functions of effective overlapping time $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ and $\sigma_{ij}^{t\overset{\leftharpoonup}{t}},$ respectively. Due to Assumption [\[assum:triptime\]](#assum:triptime){reference-type="ref" reference="assum:triptime"}, it is again reasonable to approximate the transferred regenerative energy functions as affine functions of the effective overlapping time. In other words, we denote: $$\begin{alignedat}{1}E_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}} & \triangleq c_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\rightharpoonup}{t}}+b_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}},\\ E_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}} & \triangleq c_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\leftharpoonup}{t}}+b_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}, \end{alignedat} \label{eq:regen-energy-approx}$$ where $(c_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}, b_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}})$ and $(c_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}},b_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}})$ are positive coefficients computed via a constrained least squares problem where our input measurement data corresponds to effective overlapping time with output measurement data corresponds to the transferred regenerative braking energy. This measurement data can come from either historical data or physics-based simulation; in our numerical experiments, this data is generated by the [KDC]{.sans-serif} simulator. The positive values of $b_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ and $b_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}$ denote residual transfer of regenerative energy. This may occur due to some energy lying outside of our FWHM rectangles, even when the overlapping time is zero. Our affine modeling of $E_{i,j,t,\protect\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ and $E_{i,j,t,\protect\overset{\leftharpoonup}{t}}^{\textrm{reg}}$ in [\[eq:regen-energy-approx\]](#eq:regen-energy-approx){reference-type="eqref" reference="eq:regen-energy-approx"} leads to the interpretation of the objective in [\[eq:obj_funct-1\]](#eq:obj_funct-1){reference-type="eqref" reference="eq:obj_funct-1"}, which we aim to maximize. Because the coefficients $(c_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}, b_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}})$, $(c_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}, b_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}})$ are nonnegative, positive overlappings would always lead to a positive transfer of regenerative energy. However, a negative or zero overlapping $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ (or $\sigma_{ij}^{t\overset{\leftharpoonup}{t}}$) may have some small transfer of regenerative energy. For $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}\leq-b_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}/c_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ (or $\sigma_{ij}^{t\overset{\leftharpoonup}{t}}\leq-b_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}/c_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}$), there will be little to no transfer of regenerative energy. In those cases, the negative values of $E_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ (or $E_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}$) introduce a *penalty* that reduces the value of the objective [\[eq:obj_funct-1\]](#eq:obj_funct-1){reference-type="eqref" reference="eq:obj_funct-1"} being maximized. We make this modeling decision of introducing this penalty rather than setting the regenerative energy to zero directly because even when the overlapping time is negative, it is preferable to have the absolute value as small as possible. This approach aims to achieve some residual synchronization by reducing the penalty. #### Step 2. Computing data-driven approximations of the effective accelerating and braking phases of trains. To compute the effective overlapping time, we need to compute the beginning and end of the effective accelerating and braking phases of the trains. Consider the trip of any train $t\in\mathcal{T}$ from platform $i$ to platform $j$ along the track $(i,j)\in\mathcal{A}^{t}$. The beginning and end of the effective accelerating phase of $t$ while departing from platform $i$ is denoted by by $d_{i}^{t}+\underline{\alpha}_{i}^{t}$ and $d_{i}^{t}+\overline{\alpha}_{i}^{t}$ respectively. Similarly, the beginning and end of the effective braking phase of $t$ while arriving at platform $j$ by $a_{j}^{t}-\underline{\beta}_{j}^{t}$ and $a_{j}^{t}-\overline{\beta}_{j}^{t}$, respectively. The values for $\underline{\alpha}_{i}^{t},\overline{\alpha}_{i}^{t},\underline{\beta}_{j}^{t},\overline{\beta}_{j}^{t}$ depend on the power graph of the train, which in turn depends on the trip time $a_{j}^{t}-d_{i}^{t}$. Analytical approximations of the aforementioned terms as a function of $a_{j}^{t}-d_{i}^{t}$ for a given train are not known, however, due to Assumption [\[assum:triptime\]](#assum:triptime){reference-type="ref" reference="assum:triptime"}, it is reasonable to work with affine approximations of the trip times. So, for $t\in\mathcal{T}$, and $(i,j)\in\mathcal{A}^{t}$, we define $\underline{\alpha}_{i}^{t},\overline{\alpha}_{i}^{t},\underline{\beta}_{j}^{t},\overline{\beta}_{j}^{t}$ as follows: $$\begin{aligned}\underline{\alpha}_{i}^{t} & \triangleq c_{i,t}^{\underline{\alpha}}(a_{j}^{t}-d_{i}^{t})+b_{i,t}^{\underline{\alpha}},\quad\overline{\alpha}_{i}^{t} \triangleq c_{i,t}^{\overline{\alpha}}(a_{j}^{t}-d_{i}^{t})+b_{i,t}^{\overline{\alpha}},\\ \underline{\beta}_{j}^{t} & \triangleq c_{i,t}^{\underline{\beta}}(a_{j}^{t}-d_{i}^{t})+b_{i,t}^{\underline{\beta}},\quad\overline{\beta}_{j}^{t} \triangleq c_{i,t}^{\overline{\beta}}(a_{j}^{t}-d_{i}^{t})+b_{i,t}^{\overline{\beta}}, \end{aligned} \tag{\texttt{EFF\_ACCEL\_\&\_BRK\_PHASES}} \label{eq:effective_braking_accelearation_compute}$$ where the coefficients $(c_{i,t}^{\underline{\alpha}},b_{i,t}^{\underline{\alpha}})$, $(c_{i,t}^{\overline{\alpha}},b_{i,t}^{\overline{\alpha}})$, $(c_{i,t}^{\underline{\beta}},b_{i,t}^{\underline{\beta}}),$ and $(c_{i,t}^{\overline{\beta}},b_{i,t}^{\overline{\beta}})$ are computed via least-squares with input measurement data corresponds to trip time with output measurement data corresponds to the beginning or end of the effective acceleration and braking phases. #### Step 3. Modeling the overlapping times. Now, we describe how to model the effective overlapping time $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ and $\sigma_{ij}^{t\overset{\leftharpoonup}{t}}$ in terms of the decision variables present in the system. ![This figure illustrates all the possible overlapping times $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$. For the first four cases, $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ shown using the red-shaded region is nonpositive, whereas for the next nine cases $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ is positive shown using the green-shaded region. The overlapping time $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ admits the closed form $\min\left\{ a_{j}^{\overset{\rightharpoonup}{t}}-\overline{\beta}_{j}^{\overset{\rightharpoonup}{t}},d_{i}^{t}+\overline{\alpha}_{i}^{t}\right\} +\min\left\{ -d_{i}^{t}-\underline{\alpha}_{i}^{t},-a_{j}^{\overset{\rightharpoonup}{t}}+\underline{\beta}_{j}^{\overset{\rightharpoonup}{t}}\right\}$. [\[fig:overlap\]]{#fig:overlap label="fig:overlap"}](Figure/all_overlapping.pdf){#fig:overlap} First, we discuss the modeling of right event overlapping time $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$. Using Allen's interval algebra [@Allen1983], we know that there can be thirteen different kinds of overlapping possible between the accelerating phase of train $t$ and the braking phase of train $\overset{\rightharpoonup}{t}$ as shown in Figure [5](#fig:overlap){reference-type="ref" reference="fig:overlap"}. For the first four cases shown in Figure [5](#fig:overlap){reference-type="ref" reference="fig:overlap"}, the overlapping is nonpositive, i.e., the temporal blocks are apart from each other by a nonpositive distance (which is shown in the red-shaded region), whereas for the next nine cases, there is a positive overlapping in time (which is shown using the green-shaded region). Fortunately, we can write all these thirteen possible overlapping times in a closed form: $$\begin{aligned} \sigma_{ij}^{t\overset{\rightharpoonup}{t}} & =\min\left\{ a_{j}^{\overset{\rightharpoonup}{t}}-\overline{\beta}_{j}^{\overset{\rightharpoonup}{t}},d_{i}^{t}+\overline{\alpha}_{i}^{t}\right\} -\max\left\{ d_{i}^{t}+\underline{\alpha}_{i}^{t},a_{j}^{\overset{\rightharpoonup}{t}}-\underline{\beta}_{j}^{\overset{\rightharpoonup}{t}}\right\} \nonumber \\ & =\min\left\{ a_{j}^{\overset{\rightharpoonup}{t}}-\overline{\beta}_{j}^{\overset{\rightharpoonup}{t}},d_{i}^{t}+\overline{\alpha}_{i}^{t}\right\} +\min\left\{ -d_{i}^{t}-\underline{\alpha}_{i}^{t},-a_{j}^{\overset{\rightharpoonup}{t}}+\underline{\beta}_{j}^{\overset{\rightharpoonup}{t}}\right\} \label{eq:sigma_right} \tag{\texttt{RGHT\_EVNT\_OVLP\_TIME}}\end{aligned}$$ which is a concave function in the decision variables $a_{j}^{\overset{\rightharpoonup}{t}}$, $d_{i}^{t},\overline{\alpha}_{i}^{t}$, $\underline{\alpha}_{i}^{t}$, $\underline{\beta}_{j}^{\overset{\rightharpoonup}{t}}$, and $\overline{\beta}_{j}^{\overset{\rightharpoonup}{t}}$, as it is a pointwise minimum of affine functions in the decision variables, which preserves concavity [@Boyd2009 Section 3.2]. Next, we discuss modeling of left event overlapping time $\sigma_{ij}^{t\overset{\leftharpoonup}{t}}$. Similar to the first case, we can write $\sigma_{ij}^{t\overset{\leftharpoonup}{t}}$ as (easily proved by replacing $i$, $j$, $t$ and $\overset{\rightharpoonup}{t}$ in $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ with $j$, $i$, $\overset{\leftharpoonup}{t}$ and $t$ respectively) $$\begin{aligned} \sigma_{ij}^{t\overset{\leftharpoonup}{t}} & =\min\left\{ a_{i}^{t}-\overline{\beta}_{i}^{t},d_{j}^{\overset{\leftharpoonup}{t}}+\overline{\alpha}_{j}^{\overset{\leftharpoonup}{t}}\right\} -\max\left\{ d_{j}^{\overset{\leftharpoonup}{t}}+\underline{\alpha}_{j}^{\overset{\leftharpoonup}{t}},a_{i}^{t}-\underline{\beta}_{i}^{t}\right\} \nonumber \\ & =\min\left\{ a_{i}^{t}-\overline{\beta}_{i}^{t},d_{j}^{\overset{\leftharpoonup}{t}}+\overline{\alpha}_{j}^{\overset{\leftharpoonup}{t}}\right\} +\min\left\{ -d_{j}^{\overset{\leftharpoonup}{t}}-\underline{\alpha}_{j}^{\overset{\leftharpoonup}{t}},-a_{i}^{t}+\underline{\beta}_{i}^{t}\right\} ,\label{eq:sigma_left} \tag{\texttt{LEFT\_EVNT\_OVLP\_TIME}}\end{aligned}$$ which is a again concave function in our decision variables. # Final optimization model [\[sec:Final-optimization-model\]]{#sec:Final-optimization-model label="sec:Final-optimization-model"} Using our developments in the previous sections, we arrive at the following final linear optimization model to minimize the effective energy consumption of the railway network: $$\begin{array}{ll} \underset{}{\mbox{minimize}}\\ \sum_{t\in\mathcal{T},(i,j)\in\mathcal{A}^{t}}\left(c_{i,j,t}^{\textrm{con,tr}}(a_{j}^{t}-d_{i}^{t})+b_{i,j,t}^{\textrm{con,tr}}\right)\;+ \quad \sum_{(i,j)\in\varphi,(t,t^{\prime})\in\mathcal{B}_{i,j}}\left(c_{i,j,t,t^{\prime}}^{\textrm{con,cr}} (a_{j}^{t^{\prime}}- \quad d_{i}^{t})+b_{i,j,t,t^{\prime}}^{\textrm{con,cr}}\right) \quad\rhd\;\textrm{consumption}\\ - \quad \sum_{(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}}}\left(c_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\rightharpoonup}{t}}+b_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}\right)\;-\sum_{(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}}\left(c_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\leftharpoonup}{t}}+b_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}\right) \quad\rhd\;\textrm{transferred regen. energy}\\ \textup{subject to} \\ \begin{rcases} \sigma_{ij}^{t\overset{\rightharpoonup}{t}}\leq\varpi_{ij}^{t\overset{\rightharpoonup}{t}}+\varphi_{ij}^{t\overset{\rightharpoonup}{t}},\quad\textrm{for }(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}}\\ \varpi_{ij}^{t\overset{\rightharpoonup}{t}}\leq a_{j}^{\overset{\rightharpoonup}{t}}-\overline{\beta}_{j}^{\overset{\rightharpoonup}{t}},\quad\textrm{for }(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}}\\ \varpi_{ij}^{t\overset{\rightharpoonup}{t}}\leq d_{i}^{t}+\overline{\alpha}_{i}^{t},\quad\textrm{for }(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}}\\ \varphi_{ij}^{t\overset{\rightharpoonup}{t}}\leq-d_{i}^{t}-\underline{\alpha}_{i}^{t},\quad\textrm{for }(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}}\\ \varphi_{ij}^{t\overset{\rightharpoonup}{t}}\leq-a_{j}^{\overset{\rightharpoonup}{t}}+\underline{\beta}_{j}^{\overset{\rightharpoonup}{t}},\quad\textrm{for }(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}} \end{rcases} \substack{\rhd \; \textrm{hypograph}\\ \textrm{constraints }\\ \textrm{for}\eqref{eq:sigma_right} } \\ \begin{rcases} \sigma_{ij}^{t\overset{\leftharpoonup}{t}}\leq\varpi_{ij}^{t\overset{\leftharpoonup}{t}}+\varphi_{ij}^{t\overset{\leftharpoonup}{t}},\quad\textrm{for }(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}\\ \varpi_{ij}^{t\overset{\leftharpoonup}{t}}\leq a_{i}^{t}-\overline{\beta}_{i}^{t},\quad\textrm{for }(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}\\ \varpi_{ij}^{t\overset{\leftharpoonup}{t}}\leq d_{j}^{\overset{\leftharpoonup}{t}}+\overline{\alpha}_{j}^{\overset{\leftharpoonup}{t}},\quad\textrm{for }(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}\\ \varphi_{ij}^{t\overset{\leftharpoonup}{t}}\leq-d_{j}^{\overset{\leftharpoonup}{t}}-\underline{\alpha}_{j}^{\overset{\leftharpoonup}{t}},\quad\textrm{for }(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}\\ \varphi_{ij}^{t\overset{\leftharpoonup}{t}}\leq-a_{i}^{t}+\underline{\beta}_{i}^{t},\quad\textrm{for }(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}\\ \end{rcases} \substack{\rhd \; \textrm{hypograph}\\ \textrm{constraints }\\ \textrm{for}\eqref{eq:sigma_left} } \\ \textrm{Eq.} \; \eqref{eq:effective_braking_accelearation_compute}, \quad \rhd \; \textrm{beginning and end of acceleration and braking} \\ \textrm{Eq.}\; \eqref{eq:TripTimeConstraint}, \eqref{eq:turnAroundConstraint}, \eqref{eq:DwellTimeConstraint}, \eqref{eq:ConnectionConstraint}, \eqref{eq:SafetyConstraint1}, \eqref{eq:TotalTravelTimeConstraints}, \; \rhd \; \textrm{operational constraints} \\ \textrm{Eq.}\; \eqref{eq:domain}, \; \rhd \; \textrm{domain of decision variable} \end{array} \tag{\texttt{OPT\_MODEL}} \label{eq:final_lp_model}$$ where the decision variables are $a_{i}^{t}$, $d_{i}^{t}$, $\sigma_{ij}^{t\overset{\rightharpoonup}{t}},$$\varpi_{ij}^{t\overset{\rightharpoonup}{t}},$ $\varphi_{ij}^{t\overset{\rightharpoonup}{t}}$, $\sigma_{ij}^{t\overset{\leftharpoonup}{t}},$ $\varpi_{ij}^{t\overset{\leftharpoonup}{t}},$ and $\varphi_{ij}^{t\overset{\leftharpoonup}{t}}$. Note that the first ten constraints recast [\[eq:sigma_right\]](#eq:sigma_right){reference-type="eqref" reference="eq:sigma_right"} and [\[eq:sigma_left\]](#eq:sigma_left){reference-type="eqref" reference="eq:sigma_left"} as linear constraints using the hypograph approach [@Boyd2009 Section 4.1.3]. #### Predicting effective energy consumption from the solution. Once we have computed an optimal solution to [\[eq:final_lp_model\]](#eq:final_lp_model){reference-type="eqref" reference="eq:final_lp_model"}, it provides us with the energy-optimal timetable. An estimate of the effective energy consumption of the final timetable is given by: $$\begin{aligned} & \sum_{t\in\mathcal{T},(i,j)\in\mathcal{A}^{t}}\left(c_{i,j,t}^{\textrm{con,tr}}(a_{j}^{t}-d_{i}^{t})+b_{i,j,t}^{\textrm{con,tr}}\right)\;\quad+\sum_{(i,j)\in\varphi,(t,t^{\prime})\in\mathcal{B}_{i,j}}\left(c_{i,j,t,t^{\prime}}^{\textrm{con,cr}}(a_{j}^{t^{\prime}}-d_{i}^{t})+b_{i,j,t,t^{\prime}}^{\textrm{con,cr}}\right)\nonumber \\ & -\sum_{(i,j,t,\overset{\rightharpoonup}{t})\in\overset{\rightharpoonup}{\mathcal{E}}}\max\left\{ c_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\rightharpoonup}{t}}+b_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}},0\right\} \;\quad-\sum_{(i,j,t,\overset{\leftharpoonup}{t})\in\overset{\leftharpoonup}{\mathcal{E}}}\max\left\{ c_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\leftharpoonup}{t}}+b_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}},0\right\} ,\tag{\texttt{PRED\_EFF\_ENG\_CNSM}}\label{eq:predicted_eng}\end{aligned}$$ where the first line computes the energy consumption during the acceleration, whereas the second line is the negative of the actual transfer of regenerative energy, thus the final expression corresponds to the predicted effective energy consumption. Recall that when $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}\leq0$ and $\sigma_{ij}^{t\overset{\leftharpoonup}{t}}\leq0$, the summands $c_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\rightharpoonup}{t}}+b_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}$ and $c_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\leftharpoonup}{t}}+b_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}$ act as penalty terms in the objective function of [\[eq:final_lp_model\]](#eq:final_lp_model){reference-type="eqref" reference="eq:final_lp_model"}, so $\max\{c_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\rightharpoonup}{t}}+b_{i,j,t,\overset{\rightharpoonup}{t}}^{\textrm{reg}},0\}$ and $\max\{c_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}}\sigma_{ij}^{t\overset{\leftharpoonup}{t}}+b_{i,j,t,\overset{\leftharpoonup}{t}}^{\textrm{reg}},0\}$ correspond to the actual transferred regenerative energy for the individual events, respectively. # Numerical experiment [\[Numerical_Study\]]{#Numerical_Study label="Numerical_Study"} This section has the following organization. In Section [6.1](#sec:sl8){reference-type="ref" reference="sec:sl8"}, we describe the railway network where we apply our model. In Section [6.2](#sec:resultsSl8){reference-type="ref" reference="sec:resultsSl8"}, we present the results of our numerical study. ![Railway network considered for the numerical study.[\[Fig:sl8\]]{#Fig:sl8 label="Fig:sl8"}](Figure/shanghai_line.pdf){#Fig:sl8} ## Railway network in consideration {#sec:sl8} Our empirical analysis focuses on Metro Line 8 of the Shanghai Railway network, a critical component of the world's most extensive metro system by route length. The Shanghai Railway Network also stands second in terms of station count while leading in global metro ridership with approximately 3.88 billion passengers served in 2019 [@ShanghaiMetroWiki]. The Metro Line 8 infrastructure, in particular, traverses some of Shanghai's highest-density residential areas and records an average daily ridership of about 1.08 million. The line spans 37.4 km and encompasses 28 functional stations [@Line8ShanghaiMetroWiki]. Metro Line 8 features three distinct services, each governed by individually optimized timetables generated through communication-based train control systems. In this study, we formulate energy-efficient timetables specifically for the $\textrm{PES}_2$-$\textrm{SFM}_2$ service route within Metro Line 8, as depicted in Figure [6](#Fig:sl8){reference-type="ref" reference="Fig:sl8"}. The said network is composed of two lines---Line 1 and Line 2---with 14 stations represented in all uppercase in the figure. Notably, each station comprises dual platforms catering to trains from both lines; for instance, LXM Station includes platforms $\textrm{LXM}_1$ and $\textrm{LXM}_2$, which serve Line 1 and Line 2, respectively. The line under examination has a total length of 37 km. The stations have an average inter-station distance of 1.4 km, ranging from a minimum of 738 m (between YHR and ZJD) to a maximum of 2.6 km (between PJT and LHR). Operational constraints are subject to well-defined physical limits, including the track slope, which lies within the range of $[-2.00453^{\circ},2.00453^{\circ}]$, and the maximum permissible acceleration and deceleration of trains, specified as 1.04 $\textrm{m/s}^{2}$ and -0.8 $\textrm{m/s}^{2}$, respectively. The conversion efficiency from electrical to kinetic energy is 0.9, and from kinetic to regenerative braking energy is 0.76, as well as the transmission loss factor of regenerative electricity is 0.1. ## Results of the numerical study {#sec:resultsSl8} To compute the energy-efficient timetables for the $\textrm{PES}_2$-$\textrm{SFM}_2$ service on Shanghai Railway Network's Metro Line 8, we consider 11 distinct operational instances. Each instance encompasses unique parameters, including train count, headway durations, train velocities, track gradients, and energy consumption profiles during acceleration and braking phases. Thales Canada Inc. has provided us with existing timetables corresponding to these 11 scenarios. Upon the computation of our energy-optimized timetables, we quantify the improvements by comparing the energy usage against these baseline timetables. C1.7cmC1.7cmC1.7cmC1.9cmC1.9cmC1.9cmC1.9cmC1.9cm **\#Trains** & **\#Variables** & **\#Constraints** & **Solution time (s)** & **Initial effective energy consumption (kWh)** & **Predicted final effective energy consumption (kWh)** & **Reduction predicted by our model (%)** & **Reduction predicted by [SPSIM]{.sans-serif} (%)**\ & 47581 & 151259 & 0.669 & 353721.62 & 252282.87 & 28.68 & 30.27\ 1032 & 49104 & 156102 & 0.674 & 364832.93 & 263288.46 & 27.83 & 31.34\ 1066 & 50726 & 161254 & 0.731 & 376017.98 & 271346.61 & 27.84 & 33.04\ 1100 & 52339 & 166391 & 0.683 & 386087.42 & 284325.21 & 26.36 & 29.10\ 1132 & 53865 & 171239 & 0.760 & 389844.65 & 291664.27 & 25.18 & 26.56\ 1166 & 55487 & 176391 & 0.753 & 407717.20 & 301714.07 & 26.00 & 26.55\ 1198 & 57010 & 181234 & 0.763 & 417767.32 & 330319.76 & 20.93 & 19.27\ 1232 & 58626 & 186376 & 0.855 & 447025.23 & 334638.29 & 25.14 & 25.86\ 1266 & 60242 & 191518 & 0.867 & 438657.08 & 332747.39 & 24.14 & 26.94\ 1298 & 61765 & 196361 & 0.901 & 449379.24 & 333858.16 & 25.71 & 24.83\ 1332 & 63387 & 201513 & 0.906 & 475354.59 & 350642.67 & 26.24 & 25.97\ The computational experiments were conducted on a high-performance computing infrastructure comprising an AMD Ryzen 9 7950X CPU with 16 cores and 32 threads, accompanied by 32 GB of RAM, and operating on a Windows 11 Pro platform. We employed the algebraic modeling language [JuMP]{.sans-serif}, a state-of-the-art, open-source framework, to articulate our optimization problem [@Lubin2023]. Within this framework, we implemented the parallel interior-point algorithm of the [Gurobi Optimizer 10.0]{.sans-serif} [@gurobi2023]. To accelerate the optimization process, we initialized the algorithm using existing timetables generously supplied by Thales Canada Inc, significantly reducing the computational time required to reach an optimal solution. {#fig:a width="\\textwidth"} {#fig:b width="\\textwidth"} The findings of this computational study are encapsulated in Table [\[tab:Results-of-the-numerical-study\]](#tab:Results-of-the-numerical-study){reference-type="ref" reference="tab:Results-of-the-numerical-study"}. Remarkably, in every scenario, the solution time was confined to less than one second, effectively rendering the model amenable to real-time applications. The baseline effective energy consumption metrics, associated with the initial timetables, were ascertained through [SPSIM]{.sans-serif}, a physics-based simulation tool utilized by Thales Canada Inc. Upon obtaining the optimal solution, as formulated in equation [\[eq:final_lp_model\]](#eq:final_lp_model){reference-type="eqref" reference="eq:final_lp_model"}, we proceeded to evaluate the effective energy consumption of the optimized timetables utilizing equation [\[eq:predicted_eng\]](#eq:predicted_eng){reference-type="eqref" reference="eq:predicted_eng"}. The optimized timetables demonstrated a substantial improvement over their real-world counterparts, with the reductions in effective energy consumption spanning from approximately 20.93% in the least favorable instances to as much as 28.68% in the most advantageous cases. This is also graphically illustrated in Figure [7](#fig:a){reference-type="ref" reference="fig:a"}. To corroborate the validity of our model's predictions, we conducted a comparative analysis with the physics-based simulator [SPSIM]{.sans-serif}. The latter's estimates were generated by inputting our optimized timetables into the simulator, calculating the resultant effective energy consumption and subsequently the degree of reduction. The comparative metrics, enumerated in the last two columns of Table [\[tab:Results-of-the-numerical-study\]](#tab:Results-of-the-numerical-study){reference-type="ref" reference="tab:Results-of-the-numerical-study"} and Figure [8](#fig:b){reference-type="ref" reference="fig:b"}, reveal a strong alignment between the two prediction methodologies, although our model tends to yield slightly more conservative estimates on average. The variance can be attributed to the methodology employed by [SPSIM]{.sans-serif} for computing the transfer of regenerative energy, which is based on the precise area of overlap between power-versus-time graphs for pairs of accelerating and decelerating trains. This may occasionally exceed the energy transfer estimates provided by our model, which employs rectangular approximations of these graphs, as delineated in Section [\[sec:regen_descrip\]](#sec:regen_descrip){reference-type="ref" reference="sec:regen_descrip"}. # Architectural framework for industrial integration of the proposed model {#sec:archframe} This section describes the architectural framework that facilitates the incorporation of the optimization model proposed in this paper into Thales Canada Inc's industrial timetable compiler. As shown in Figure [9](#fig:julia_embedd){reference-type="ref" reference="fig:julia_embedd"}, the core architecture has two primary constituents: (i) the existing timetable compiler, showcased on the left, and (ii) the energy-optimal timetable generator, represented on the right. For the purpose of this integration framework, the existing timetable compiler is partitioned into three discrete modules: - *Timetable data:* This module houses the necessary data to solve the optimization problem. Additionally, it includes other information specific to the railway network under consideration, such as gradient attributes and speed limitations in various track segments; all presented in their raw form. - *Optimization class:* This module transforms the raw data into a structure that is compatible with the optimization algorithm we employ. - *[CSCPP]{.sans-serif} class:* This class establishes a bi-directional communication link between the industrial timetable compiler and the energy-optimal timetable generator. It injects the input data into the optimization algorithm, retrieves the optimized timetable, and converts it into a format amenable for deployment in a communication-based train control system. The energy-optimal timetable generator is similarly sub-divided into three modules: - *Type definitions:* This module contains the data structures essential for describing a railway network, as previously elaborated in Sections [\[sec:Modelling-the-constraints\]](#sec:Modelling-the-constraints){reference-type="ref" reference="sec:Modelling-the-constraints"} and [\[sec:Final-optimization-model\]](#sec:Final-optimization-model){reference-type="ref" reference="sec:Final-optimization-model"}. - *Utility functions:* This module contains the critical functions required for the optimization algorithm's operation. Included are the description of the optimization model [\[eq:final_lp_model\]](#eq:final_lp_model){reference-type="eqref" reference="eq:final_lp_model"}, a function to predict effective energy consumption via [\[eq:predicted_eng\]](#eq:predicted_eng){reference-type="eqref" reference="eq:predicted_eng"}, and another to transform the output timetable in a [CSCPP]{.sans-serif}-compatible format. - *Optimization algorithm:* This segment contains the parallel interior-point algorithm used for solving [\[eq:final_lp_model\]](#eq:final_lp_model){reference-type="eqref" reference="eq:final_lp_model"}, supplemented by a warm-start mechanism. Once the final timetable is computed, the algorithm forwards the results to the [CSCPP]{.sans-serif} class through the utility functions. ![The integration architecture for integrating our optimization model with an industrial timetable compiler [\[fig:julia_embedd\]]{#fig:julia_embedd label="fig:julia_embedd"}](Figure/julia_embedd.pdf){#fig:julia_embedd} # Conclusion[\[Conclusion\]]{#Conclusion label="Conclusion"} In this paper, we have proposed a novel single-stage linear optimization model to compute energy-optimal timetables for sustainable communication-based train control systems. Our model simultaneously minimizes the total energy consumption of all trains and maximizes the transfer of regenerative braking energy. Distinct from existing models, that are either NP-hard or require multi-stage simulations, our model facilitates real-time decision-making by producing energy-optimal timetables subject to the inherent functional constraints of metro railway networks. We demonstrated the model's impact via its application to Shanghai Railway Network's Metro Line 8, achieving energy savings between 20.93% and 28.68% in comparison with existing real-world timetables, with sub-second computational times on a standard desktop. Given its managerial implications and computational robustness, our model is poised for global implementation as an integrated component of Thales Canada Inc's timetable compiler.

arxiv_math

--- author: - "Shay Ben-Moshe[^1]" - "Shachar Carmeli[^2]" - Tomer M. Schlank - Lior Yanovski --- [^1]: Einstein Institute of Mathematics, Hebrew University of Jerusalem. [^2]: Department of Mathematics, University of Copenhagen.

arxiv_math

--- abstract: | Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). In this paper, we present a unified approach to constructing stationary measures for most of the known colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (multispecies ASEP, or mASEP); (2) the $q$-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the $q$-Boson particle system; (3) the $q$-deformed Pushing Totally Asymmetric Simple Exclusion Process ($q$-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang--Baxter equation. We express the stationary measures as partition functions of new "queue vertex models" on the cylinder. The stationarity property is a direct consequence of the Yang--Baxter equation. For the mASEP on the ring, a particular case of our vertex model is equivalent to the multiline queues of Martin [@martin2020stationary]. For the colored $q$-Boson process and the $q$-PushTASEP on the ring, we recover and generalize known stationary measures constructed using multiline queues or other methods by Ayyer--Mandelshtam--Martin [@ayyer2022modified], [@Ayyer_2023], and Bukh--Cox [@bukh2019periodic]. Our proofs of stationarity use the Yang--Baxter equation and bypass the Matrix Product Ansatz (used for the mASEP by Prolhac--Evans--Mallick [@Prolhac_2009]). On the line and in a quadrant, we use the Yang--Baxter equation to establish a general colored Burke's theorem, which implies that suitable specializations of our queue vertex models produce stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity. author: - Amol Aggarwal, Matthew Nicoletti, Leonid Petrov bibliography: - bib.bib title: | Colored Interacting Particle Systems on the Ring:\ Stationary Measures from Yang--Baxter Equation --- # Introduction {#sec:intro} ## Background This work connects stationary measures for colored (also referred to as multi-species or multi-type) systems of interacting particles hopping on a one-dimensional lattice (the ring or the whole line) to solvable lattice models. One of the particle systems we consider is the multi-species Asymmetric Simple Exclusion Process (*mASEP*). On the ring with $N$ sites, the $n$-colored mASEP is a continuous time Markov process under which each pair of neighboring particles of colors $0\le i\ne j\le n$ and locations $(k,k+1)$ (mod $N$) swap with rate $1$ if $i<j$, and rate $q\in[0,1)$ if $i>j$ (color $0$ represents holes). See for an illustration.[^1] We also consider the multi-species $q$-TAZRP ($q$-deformed Totally Asymmetric Zero Range Process, also known as the colored stochastic $q$-Boson particle system), and the colored $q$-PushTASEP ($q$-deformed Pushing Totally Asymmetric Simple Exclusion Process). We refer to for definitions of these models on the ring. {#fig:mASEP_intro width=".6\\textwidth"} The stationary measures of these particle systems have been the subject of a systematic investigation recently. On the one hand, their properties and asymptotic behavior highlight interesting physical and probabilistic phenomena (for example, a particle of a different color may follow a "microscopic characteristic"). On the other hand, they have a rich underlying algebraic and combinatorial structure (in particular, deep connections to Macdonald symmetric functions and their nonsymmetric counterparts). The stationary measure for the single-color mASEP (simply called *ASEP*) on the ring with a given number of particles is uniform among all possible arrangements of these particles. On the line, translation invariant stationary measures are all product Bernoulli (that is, each site is independently a particle or a hole with probability $\rho$); see Liggett [@Liggett1985 Ch. VIII]. Prolhac--Evans--Mallick [@Prolhac_2009] constructed stationary measures for mASEP with an arbitrary number of colors using the *Matrix Product Ansatz*, an algebraic formalism utilizing commutation relations of a family of matrices. For previous partial Matrix Product Ansatz results see the references in [@Prolhac_2009]. We discuss the Matrix Product Ansatz approach in the beginning in in the text. Methods for sampling from the mASEP stationary measures using combinatorial structures known as *multiline diagrams* or *multiline queues* were developed by Angel [@angel2006stationary], Ferrari--Martin [@Ferrari_2007], and Martin [@martin2020stationary]. Connections to Macdonald symmetric and nonsymmetric polynomials served as another inspiration for studying stationary measures of interacting particle systems on the ring. Macdonald polynomials are a centerpiece of the symmetric functions theory and have wide applications to representation theory and geometry; see Macdonald [@Macdonald1995 Ch. VI], [@marshall1999symmetric]. We do not focus on symmetric and nonsymmetric functions in this paper, but here we mention the necessary background. A relationship between the mASEP stationary measures and Macdonald polynomials is first observed by Cantini--de Gier--Wheeler [@cantini2015matrix]. Nonsymmetric Macdonald polynomials are constructed via multiline queues by Corteel--Mandelshtam--Williams [@corteel2018multiline]. An integrable vertex model for nonsymmetric Macdonald polynomials is given by Borodin--Wheeler [@borodin2019nonsymmetric]. One can say that the vertex model construction unifies the two points of view. The stochastic single-color $q$-Boson process (also referred to as the $q$-TAZRP) was introduced by Sasamoto--Wadati [@SasamotoWadati1998]. Its dual process, the $q$-TASEP, was extensively studied on the line by Borodin--Corwin [@BorodinCorwin2011Macdonald], Borodin--Corwin--Sasamoto [@BorodinCorwinSasamoto2012], Borodin--Corwin--Petrov--Sasamoto [@BorodinCorwinPetrovSasamoto2013], Barraquand [@barraquand2015phase], and others. On the ring, Wang--Waugh [@Wang2015inhomqTASEP] and Liu--Saenz--Wang [@liu2019integral] obtained integral formulas for transition probabilities and other observables of the $q$-Boson process. The multi-species generalization of the $q$-Boson process is due to Takeyama [@Takeyama2015] and Kuniba--Maruyama--Okado [@kuniba2016multispecies]. Stationary measures for this process on the ring were recently connected to modified Macdonald polynomials by Ayyer--Mandelshtam--Martin [@ayyer2022modified], [@Ayyer_2023]. In contrast with the mASEP, the $q$-Boson process has spectral parameters (rapidities) attached to the sites on the ring, and these parameters enter the modified Macdonald polynomials. Ayyer--Mandelshtam--Martin revealed symmetries of the stationary measures in the parameters and utilized them to compute observables in the stationary model. The colored $q$-PushTASEP on the ring is less studied. On the line, it was introduced by Borodin--Wheeler [@borodin_wheeler2018coloured Section 12.5] as a degeneration of the colored stochastic higher spin six vertex model. Like the $q$-Boson process, it contains spectral parameters, and also the capacity parameter $\mathsf{P}\in \mathbb{Z}_{\ge1}$ which is the maximum number of particles allowed at each site. A discrete time variant of the colored $q$-PushTASEP (with $q=0$ and $\mathsf{P}=1$) on the ring was introduced (under the name "frog model") by Bukh--Cox [@bukh2019periodic] in connection with the problem of the longest common subsequence of a random and a periodic word. In particular, Bukh--Cox [@bukh2019periodic] constructed and investigated stationary measures of the frog model. ## Overview of the main results We unify and generalize existing constructions of stationary measures of the multi-species ASEP, the colored $q$-Boson process, and the colored $q$-PushTASEP on the ring. We utilize *integrable stochastic vertex models*, that is, whose vertex weights have stochastic normalization and satisfy the Yang--Baxter equation. We start from stochastic $U_q(\widehat{\mathfrak{sl}}_{n+1})$ vertex weights whose explicit form was first obtained by Kuniba--Mangazeev--Maruyama--Okado [@kunibaMangazeev2016stochastic] and which were studied in stochastic context by Bosnjak--Mangazeev [@BosnjakMangazeev2016], Garbali--de Gier--Wheeler [@deGierWheeler2016], Kuan [@Kuan2018]. The vertex model foundation of our work was developed by Borodin--Wheeler [@borodin_wheeler2018coloured] who listed various stochastic degenerations of the general $U_q(\widehat{\mathfrak{sl}}_{n+1})$ weights, leading to the mASEP, the colored $q$-Boson process, and the colored $q$-PushTASEP. In a limit when the number of particles (corresponding to vertical arrows in vertex models) of a given color goes to infinity, the $U_q(\widehat{\mathfrak{sl}}_{n+1})$ stochastic weights degenerate into what we call the *queue vertex weights*. Putting them on the cylinder, we obtain our main object --- the *queue vertex model*. Its normalized partition functions serve as stationary measures for our colored stochastic particle systems on the ring. The stationarity of the normalized partition functions is a direct consequence of the Yang--Baxter equation. In detail, for each particle system, we perform the following sequence of steps: 1. We define a colored (higher spin) stochastic transfer matrix $\mathfrak{T}$ on the cylinder. In a Poisson-type limit when the discrete time becomes continuous, large powers of $\mathfrak{T}$ converge to the Markov semigroup (also called the propagator) of the given stochastic process living on configurations of colored particles on the ring. Each of our processes (the mASEP, the colored $q$-Boson process, and the colored $q$-PushTASEP) preserves the number of particles of each color, while $\mathfrak{T}$ before the Poisson-type limit may not. 2. We construct a multiparameter family of transfer matrices $\mathfrak{Q}$ on the cylinder using our queue vertex weights (see for an illustration), such that (a) Their matrix elements $\langle \emptyset| \hspace{1pt}\mathfrak{Q}\hspace{1pt}|\mathbf{V} \rangle$ (which are, by definition, partition functions --- the sums of weights of all allowed arrow configurations) are nonzero for all terminal states $\mathbf{V}$. Here $\langle \emptyset|$ is the empty state (no particles present on the ring), and $\mathbf{V}$ encodes a state with prescribed particle locations and colors. (b) The Yang--Baxter equation implies that $\mathfrak{Q}\hspace{1pt}\mathfrak{T} = \mathfrak{T}\hspace{1pt}\mathfrak{Q}$. In detail, we glue the cylinder with the output $\mathbf{V}$ to a sequence of stochastic $R$ matrices, as shown in . Their composition is the operator $\mathfrak{T}$. Together with the queue weights, these $R$ matrices satisfy the Yang--Baxter equation which allows to commute $\mathfrak{T}$ through the queue weights $\mathfrak{Q}$. Since $\langle \emptyset| \mathfrak{T}=\langle \emptyset|$, after a degeneration of the parameters of $\mathfrak{Q}$ (corresponding to the Poisson-type limit in $\mathfrak{T}$), the quantity $\langle \emptyset| \hspace{1pt}\mathfrak{Q}$ viewed as a row vector (or, equivalently, an unnormalized probability distribution on the space of particle configurations on the ring) becomes stationary under the Markov semigroup. By restricting $\langle \emptyset| \hspace{1pt}\mathfrak{Q}$ to particle configurations with fixed numbers of particles of each color (a *sector*), we obtain a stationary measure for the corresponding particle system. Note that part (a) above seems to violate the conservation of particles (the higher spin analog of the ice rule), which holds for models of six vertex type. It is the limit to the queue vertex weights which allows this violation. 3. For each of the particle systems on the ring, we compute the queue weights explicitly (after the corresponding degeneration of the parameters of $\mathfrak{Q}$), and provide conditions on the remaining complex parameters guaranteeing the positivity of the unnormalized weights. Note that the normalized weights are automatically positive by the stationarity property and the Perron--Frobenius theorem. {#fig:mult_queue_intro width="\\textwidth"} {#fig:twisted_commutation_intro width="\\textwidth"} We call the weights on the cylinder the *queue vertex weights* because each column $(-m)$ of a cylinder resembles a queueing system for which the "unused service times" are assigned the new color $m$. Indeed, for the mASEP and (with certain restrictions) the $q$-Boson process, the output of our vertex model is the same (in distribution) as that of the multiline queues of Martin [@martin2020stationary] and Ayyer--Mandelshtam--Martin [@ayyer2022modified], respectively. For the mASEP, we explain how to set the parameters to exactly match a certain specialization of our stationary measure with the matrix product stationary measure constructed by Prolhac--Evans--Mallick [@Prolhac_2009]. We also show how the underlying algebraic apparatus for the Matrix Product Ansatz can be derived from the Yang--Baxter equation. Furthermore, we describe how more general solutions $A, D, E$ of the quadratic algebra relations of the Matrix Product Ansatz (formula [\[eq:ADE_relations\]](#eq:ADE_relations){reference-type="eqref" reference="eq:ADE_relations"} in the text) can be derived from our queue vertex weights. In , we use similar vertex model techniques to construct stationary distributions for the mASEP, the colored $q$-Boson process, and the colored $q$-PushTASEP on the line $\mathbb{Z}$. In this setting, we again use the Yang--Baxter equation to prove stationarity in a certain quarter plane setting (a colored analog of Burke's theorem), and then we take a bulk limit to obtain the stationarity on $\mathbb{Z}$. The leftover parameters of the queue vertex weights on the line correspond to the densities of particles of each color. Using our quarter plane construction, we also obtain a general description of analogs of the Kardar--Parisi--Zhang (KPZ) pure phases (translation invariant ergodic Gibbs measures) for the colored stochastic six vertex model and compute the corresponding slope relations. In the single-color case, this was done by Aggarwal [@Amol2016Stationary] using the single-color queue vertex weights at the left boundary of the quadrant. Our method of producing stationary distributions as partition functions is conceptually reminiscent of the Bethe Ansatz. In Bethe Ansatz, eigenvectors of a transfer matrix of a quantum integrable model are constructed by applying certain other transfer matrices (with specially chosen parameters satisfying algebraic equations) to the vacuum vector $\langle \emptyset|$. For stationary measures of our Markov processes, we only need the leading (Perron--Frobenius) eigenvector of $\mathfrak{T}$ having the eigenvalue $1$. We do not investigate further connections of our constructions to the Bethe Ansatz or the algebraic equations. ## Related work in progress While preparing the manuscript, we learned about two related works in progress. One by Corteel--Keating [@Corteel_Keating_2023_progress] defines a new particle system with zero-range interactions on the ring. Its stationary distribution is expressed as a queue-like vertex model on the cylinder with fermionic weights related to the algebra $U_q(\widehat{\mathfrak{s l}}(1 | n))$. These weights were recently investigated by Corteel--Gitlin--Keating--Meza [@corteel2022vertex] and Aggarwal--Borodin--Wheeler [@agg-bor-wh2020-sl1n]. In particular, instead of the Macdonald polynomials, they are related to the Lascoux--Leclerc--Thibon (LLT) polynomials [@lascoux1997ribbon]. Another work in progress by Angel--Ayyer--Martin [@Angel_Ayyer_Martin_2023_progress] considers stationary measures for the colored $q$-PushTASEP on the ring (with capacity $\mathsf{P}=1$), and connects them to the multiline queues of Corteel--Mandelshtam--Williams [@corteel2018multiline]. The approach to this proof is different from ours, and it relies on properties of symmetric and nonsymmetric Macdonald polynomials and related functions. ## Outline In , we recall the colored stochastic vertex weights (together with their fully fused version) and the Yang--Baxter equations for them. We define the queue vertex weights, which arise in the limit of the stochastic vertex weights when the number of vertical arrows of a given color goes to infinity. Putting the queue vertex weights on the cylinder, we obtain our main object --- the queue vertex model. In , we employ the Yang--Baxter equation to show that the measure on particle configurations on the ring coming from the queue vertex model is stationary under the twisted and the straight cylinder Markov transition operators. In full generality, these Markov operators are formal (may have negative matrix elements). In , we take a scaling limit under which the twisted cylinder Markov operator converges to the infinitesimal generator of the mASEP (an actual, not formal, Markov operator). The corresponding limit of the queue vertex model yields a stationary measure for the mASEP. Under a specialization of the parameters, we identify our queue vertex model with the multiline queue system of Martin [@martin2020stationary] and connect our constructions to the Matrix Product Ansatz of Prolhac--Evans--Mallick [@Prolhac_2009]. Moreover, a different parameter specialization presumably relates our queue vertex model to the alternative multiline queue system considered also by Martin [@martin2020stationary Section 7]. In , we treat two other colored particle systems --- the $q$-Boson process (also known as the $q$-TAZRP) and the $q$-PushTASEP. For the $q$-Boson process, in a particular case of at most one particle of each color, we identify our queue vertex model with the multiline queueing system recently introduced by Ayyer--Mandelshtam--Martin [@ayyer2022modified]. In , we consider colored stochastic vertex models in the quadrant and use the Yang--Baxter equation to prove a new colored generalization of Burke's theorem. It implies that when put on the infinite vertical strip instead of the cylinder, our queue vertex models produce stationary distributions for the interacting particle systems (mASEP, colored $q$-Boson process, and $q$-PushTASEP) on the line. In , we show that the stationary measures for our colored particle systems on the line respect the operations of color merging (when two or more colors are declared the same). The proof also relies on applying the Yang--Baxter equation to a stochastic vertex model in the quadrant. ## Acknowledgments We are grateful to Alexei Borodin, James Martin, Ananth Sridhar, and Lauren Williams for helpful discussions. Amol Aggarwal was partially supported by a Packard Fellowship for Science and Engineering, a Clay Research Fellowship, by NSF grant DMS-1926686, and by the IAS School of Mathematics. The work of Matthew Nicoletti and Leonid Petrov was partially supported by the NSF grants DMS-1664617 and DMS-2153869, and by the Simons Collaboration Grant for Mathematicians 709055. # Colored vertex weights and Yang--Baxter equation {#sec:colored_YBE} In this section, we collect the necessary formulas for the vertex weights of the colored stochastic vertex model from [@borodin_wheeler2018coloured]. Algebraically, this model is powered by the quantum affine Lie algebra $U_q(\widehat{\mathfrak{sl}}_{n+1})$, where $n$ is the number of colors. A stochastic normalization of the $U_q(\widehat{\mathfrak{sl}}_{n+1})$ vertex weights first appeared in [@kunibaMangazeev2016stochastic]; see also [@BosnjakMangazeev2016], [@deGierWheeler2016], [@Kuan2018]. In certain degenerations, which we recall in below, the colored stochastic vertex model turns into multi-species interacting particle systems. We start with the vertex weights and the Yang--Baxter equation; then, we proceed to the fused weights and define their new *queue specialization*. ## Vertex weights {#sub:vertex_weights_text} Fix the number of colors $n\ge1$. The higher spin $U_q(\widehat{\mathfrak{sl}}_{n+1})$ stochastic vertex weights $L_{s,x}(\mathbf{A},k;\mathbf{B},\ell)$ are indexed by the following data: 1. The quantum parameter $q\in[0,1)$, which is fixed throughout this section; 2. The spectral parameter $x$ and the spin parameter $s$, which may depend on the vertex; 3. The configurations of incoming and outgoing arrows $(\mathbf{A},k;\mathbf{B},\ell)$, where $k,\ell\in\left\{ 0,1,\ldots,n \right\}$, and $\mathbf{A}, \mathbf{B}$ are $n$-tuples $(A_1,\ldots,A_n ),(B_1,\ldots,B_n )$, where $A_i,B_j\in \mathbb{Z}_{\ge0}$. Here $k\ge1$ represents an arrow of color $k$ entering from the left, and $k=0$ corresponds to no arrows entering from the left; similarly, $\ell$ encodes the exiting arrows to the right. Each $A_i$ is the number of arrows of color $i$ entering from the bottom, and $B_j$ is the number of arrows of color $j$ exiting from the top. The arrow counts $(\mathbf{A},k;\mathbf{B},\ell)$ must satisfy the *arrow conservation property* (a higher spin analog of the *ice rule*), with the understanding that all arrows go in the up or right direction. Let $\mathbf{e}_1,\ldots,\mathbf{e}_n$ be the standard basis in $\mathbb{Z}^n$, then, the arrow conservation is $$\mathbf{A}+\mathbf{e}_k\hspace{1pt}\mathbf{1}_{k\ge1}= \mathbf{B}+\mathbf{e}_{\ell}\hspace{1pt}\mathbf{1}_{\ell\ge1}.$$ Here and throughout the paper, $\mathbf{1}_{E}$ denotes the indicator of the event or condition $E$. For $1\le k,\ell \le n$, define $$\mathbf{A}^{+}_{k} \coloneqq \mathbf{A} + \mathbf{e}_k, \hspace{8pt} \mathbf{A}^{-}_{k} \coloneqq \mathbf{A} - \mathbf{e}_k, \hspace{8pt} \mathbf{A}^{+-}_{k\ell} \coloneqq \mathbf{A} + \mathbf{e}_k - \mathbf{e}_\ell, \hspace{8pt} |\mathbf{A}| \coloneqq \sum\nolimits_{k=1}^{n} A_k, \hspace{8pt} A_{[k,\ell]} \coloneqq \sum\nolimits_{i=k}^{\ell} A_k.$$ The values of all the vertex weights $L_{s,x}(\mathbf{A},k;\mathbf{B},\ell)$ are listed in the table in . In [@borodin_wheeler2018coloured Section 2], they are denoted by $\tilde{L}_{s,x}(\mathbf{A},k;\mathbf{B},\ell)$, and they differ from [@borodin_wheeler2018coloured (2.2.2)] by the factor $(-s)^{\mathbf{1}_{\ell >0}}$. However, in this paper, we work with stochastic weights from the beginning, and remove the tilde from the notation. {#fig:L_weights width=".7\\textwidth"} **Proposition 1**. *The vertex weights $L_{s,x}$ satisfy the sum-to-one property $$\label{eq:sum_to_one} \sum_{\mathbf{B}\in \mathbb{Z}_{\ge0}^{n}}\sum_{\ell=0}^{n} L_{s,x}(\mathbf{A},k;\mathbf{B},\ell)=1$$ for any fixed $\mathbf{A},k$. Moreover, if $q\in[0,1)$, $$\label{eq:nonnegativity_of_vertex_weights_condition} -sx>0, \qquad sx\le s^2\le 0,$$ then all the vertex weights $L_{s,x}(\mathbf{A},k;\mathbf{B},\ell)$ are nonnegative.* *Proof.* The sum-to-one property is [@borodin_wheeler2018coloured Proposition 2.5.1]. The nonnegativity under conditions [\[eq:nonnegativity_of_vertex_weights_condition\]](#eq:nonnegativity_of_vertex_weights_condition){reference-type="eqref" reference="eq:nonnegativity_of_vertex_weights_condition"} is straightforward. ◻ Conditions [\[eq:nonnegativity_of_vertex_weights_condition\]](#eq:nonnegativity_of_vertex_weights_condition){reference-type="eqref" reference="eq:nonnegativity_of_vertex_weights_condition"} mean that $s$ and $x$ must be purely imaginary numbers. Observe that the weights $L_{s,x}$ contain $s^2$ and $-sx$ as two independent parameters, and they are more convenient to formulate the nonnegativity. A natural point of view is to interpret the vertex weights $L_{s, x}(\mathbf{A}, k; \mathbf{B}, \ell)$ as matrix elements of a linear operator $\mathbb{C}^{n+1} \otimes V \rightarrow \mathbb{C}^{n+1} \otimes V$, where $\mathbb{C}^{n+1}$ has the standard basis $\{| j \rangle \}_{j=0}^n$, and $V$ has the basis $\{| \mathbf{A} \rangle \}_{\mathbf{A} \in \mathbb{Z}_{\geq 0}^n}$. For either space, we denote vectors of the dual basis by $\langle \mathbf{v} |$, and for tensor products of vectors (or dual vectors), we use the notation $|\mathbf{v}, \mathbf{A} \rangle \coloneqq | \mathbf{v} \rangle \otimes | \mathbf{A} \rangle$. For the tensor product $\mathbb{C}^{n+1} \otimes V$ we use the basis $\{|j, \mathbf{A} \rangle \}_{j \in\{ 0,\dots, n\}, \mathbf{A} \in \mathbb{Z}_{\geq 0}^n}$, and for its dual we use the basis $\{\langle j, \mathbf{A} | \}_{j \in\{ 0,\dots, n\}, \mathbf{A} \in \mathbb{Z}_{\geq 0}^n}$. In these bases the operator $\mathscr{L}_{s, x}$ corresponding to $L_{s, x}$ acts as $$\begin{aligned} \langle k, \mathbf{A} | \hspace{1pt}\mathscr{L}_{s, x} \hspace{1pt}|\ell, \mathbf{B} \rangle = L_{s, x}(\mathbf{A}, k; \mathbf{B}, \ell). \label{eq:L_mat}\end{aligned}$$ In this way, pairs of dual and primal basis vectors of $\mathbb{C}^{n+1} \otimes V$ with nonzero $\mathscr{L}_{s, x}$ matrix entries correspond precisely to the allowed local configurations of paths at a vertex, displayed in . **Remark 2** (Finite-spin reduction). For generic $s \in \mathbb{C}$, the operator $\mathscr{L}_{s, x}$ has infinitely many nonzero matrix entries, so any number of paths can occupy the vertical edges. If, on the other hand, $s = q^{-\frac{\mathsf{M}}{2}}$ for some $\mathsf{M} \in \mathbb{Z}_{\ge1}$, then let us set by definition $L_{s, x}(\mathbf{A}, k; \mathbf{B}, \ell) = 0$ unless $|\mathbf{A}|, |\mathbf{B}| \leq \mathsf{M}$. Note that $L_{s,x}(\mathbf{A},k;\mathbf{A}_k^+,0)$ vanishes automatically if $|\mathbf{A}|=\mathsf{M}$, so vertices with $|\mathbf{A}|>\mathsf{M}$ or $|\mathbf{B}|>\mathsf{M}$ cannot be created from the stochastic evolution started from a configuration where all $|\mathbf{A}|\le \mathsf{M}$. We see that for $s=q^{-\mathsf{M}/2}$, at most $\mathsf{M}$ vertical paths may occupy the vertical edges. In this case, $\mathscr{L}_{s, x}$ acts in the finite-dimensional subspace $\mathbb{C}^{n+1} \otimes V_\mathsf{M}$, with $V_\mathsf{M} \subset V$ spanned by $\{|\mathbf{A}\rangle\}_{|\mathbf{A}| \leq \mathsf{M}}$. ## Yang--Baxter equation {#sub:YBE_text} Let us define the following cross vertex weights $R_{z}(i,j;k,\ell)$, originally due to [@Jimbo:1985ua] and [@bazhanov1985trigonometric]: $$\label{eq:R_matrix_nonfused} \begin{split} &\hspace{6pt} \left. R_z(i,i;i,i) \coloneqq 1, \quad i \in \{0,1,\dots,n \}, \right. \\[5pt] & \left. \begin{array}{ll} R_z(j,i;j,i) \coloneqq \dfrac{q(1-z)}{1-qz}, & \quad R_z(i,j;i,j) \coloneqq \dfrac{1-z}{1-qz} \\[12pt] R_z(j,i;i,j) \coloneqq \dfrac{1-q}{1-qz}, & \quad R_z(i,j;j,i) \coloneqq \dfrac{(1-q)z}{1-qz} \end{array} \right\} \quad i,j \in \{0,1,\dots,n \}, \quad i<j. \end{split}$$ These weights also satisfy the sum-to-one property: $\displaystyle\sum\nolimits_{k,\ell=0}^{n} R_{z}(i,j;k,\ell)=1$ for any $i,j$. They are nonnegative if $0\le z\le 1$. The vertex weights $R_{z}(i,j;k,\ell)$ can also be regarded as matrix elements of an operator $\mathscr{R}_z$ acting in $\mathbb{C}^{n+1} \otimes \mathbb{C}^{n+1}$, namely, $\langle j,i | \mathscr{R}_z | \ell,k \rangle = R_{z}(i,j;k,\ell)$. One can check that $R_z$ is the spin-$\frac{1}{2}$ reduction of $L_{s,x}$ (as in ): $$\label{eq:R_matrix_spin_one_half_reduction} R_{z}(i,j;k,\ell)= L_{q^{-1/2},\hspace{1pt}z^{-1}q^{-1/2}}(\mathbf{e}_i \mathbf{1}_{i\ge1},j;\mathbf{e}_k \mathbf{1}_{k\ge 1},\ell),\qquad i,j,k,\ell\in\left\{ 0,1,\ldots,n \right\}.$$ In the right-hand side, if $i=0$, then $\mathbf{e}_i \mathbf{1}_{i\ge1}=( 0,\ldots,0 )$ ($n$ zeroes), which corresponds to no arrows at an edge. The weights $L_{s,x}$ and $R_{z}$ satisfy the following Yang--Baxter equation: **Proposition 3** ([@borodin_wheeler2018coloured (2.3.1) and Corollary B.4.3]). *For any fixed $i_1,i_2,j_1,j_2\in \left\{ 0,1,\ldots,n \right\}$ and $\mathbf{A}, \mathbf{B}\in \mathbb{Z}_{\ge0}^{n}$, we have $$\label{eq:YBE_nonfused_RLL} \begin{split} &\sum_{\mathbf{K}\in \mathbb{Z}_{\ge0}^{n}} \hspace{1pt} \sum_{k_1,k_2=0}^{n} L_{s,y}(\mathbf{A},i_2;\mathbf{K},k_2)\hspace{1pt} L_{s,x}(\mathbf{K},i_1;\mathbf{B},k_1)\hspace{1pt} R_{y/x}(k_2,k_1;j_2,j_1) \\&\hspace{40pt}= \sum_{\mathbf{K}\in \mathbb{Z}_{\ge0}^{n}} \hspace{1pt} \sum_{k_1,k_2=0}^{n} R_{y/x}(i_2,i_1;k_2,k_1)\hspace{1pt} L_{s,y}(\mathbf{K},k_2;\mathbf{B},j_2)\hspace{1pt} L_{s,x}(\mathbf{A},k_1;\mathbf{K},j_1). \end{split}$$ See for an illustration. Note that the summations in both sides of [\[eq:YBE_nonfused_RLL\]](#eq:YBE_nonfused_RLL){reference-type="eqref" reference="eq:YBE_nonfused_RLL"} are actually finite.* ![An illustration of the Yang--Baxter equation [\[eq:YBE_nonfused_RLL\]](#eq:YBE_nonfused_RLL){reference-type="eqref" reference="eq:YBE_nonfused_RLL"} in , where the sums in both sides are taken over all $k_1,k_2\in \left\{ 0,1,\ldots,n \right\}$ and $\mathbf{K}\in \mathbb{Z}_{\ge0}^n$.](fig_YBE_nonfused_RLL.pdf){#fig:YBE_nonfused_RLL width=".9\\textwidth"} ## Fused weights {#sub:fusion_text} The vertex weights $L_{s,x}(\mathbf{A},k;\mathbf{B},\ell)$ are higher spin (that is, they allow multiple arrows per edge) in the vertical direction. For $s=q^{-1/2}$, they reduce to the fundamental $R$-matrix $R_z(a,k;b,\ell)$, where $a,k,b,\ell\in\left\{ 0,1,\ldots,n \right\}$; see [\[eq:R_matrix_spin_one_half_reduction\]](#eq:R_matrix_spin_one_half_reduction){reference-type="eqref" reference="eq:R_matrix_spin_one_half_reduction"}. The inverse procedure for constructing $L_{s,x}$ from $R_z$ is called *fusion* and dates back to [@KulishReshSkl1981yang]. In the uncolored case, it was put into probabilistic context in [@CorwinPetrov2015arXiv], [@BorodinPetrov2016inhom]. The colored fusion is described in, e.g., [@borodin_wheeler2018coloured Appendix B]. The formula for the fully fused stochastic colored vertex weights $W_{x,\mathsf{L},\mathsf{M}}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$ coming from $U_q(\widehat{\mathfrak{sl}}_{n+1})$ is obtained in [@BosnjakMangazeev2016]; see also [@borodin_wheeler2018coloured Appendix C]. Here we recall the stochastic vertex weights fused in both the horizontal and vertical directions. We need the standard $q$-Pochhammer symbols notation: $$(a ; q)_k\coloneqq (1-a)(1-a q) \ldots(1-a q^{k-1}), \quad k \in \mathbb{Z}_{\geq 0},$$ and $(z ; q)_{\infty}\coloneqq \prod_{i=0}^{\infty}\left(1-z q^i\right)$ is a convergent infinite product because $q\in[0,1)$. For $\mathbf{A},\mathbf{B}\in \mathbb{Z}_{\ge0}^{n}$ such that $A_i\le B_i$ for all $i$, define $$\label{eq:capital_Phi_coloreq_sqW} \Phi(\mathbf{A}, \mathbf{B} ; x, y)\coloneqq \frac{(x ; q)_{|\mathbf{A}|}(y / x ; q)_{|\mathbf{B}-\mathbf{A}|}}{(y ; q)_{|\mathbf{B}|}}\hspace{1pt} (y / x)^{|\mathbf{A}|} \hspace{1pt} q^{\sum_{1\le i<j\le n}\left(B_i-A_i\right) A_j} \prod_{i=1}^n \frac{(q;q)_{B_i}}{(q;q)_{A_i}(q;q)_{B_i-A_i}}.$$ For any fixed $\mathbf{B}\in \mathbb{Z}_{\ge0}^{n}$, we have $$\label{eq:capital_Phi_coloreq_sum_to_one} \sum\nolimits_{\mathbf{A}\in \mathbb{Z}_{\ge0}^n \colon A_i\le B_i\textnormal{ for all $i$}} \Phi(\mathbf{A}, \mathbf{B} ; x, y)=1.$$ With this notation, if $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D} \in\mathbb{Z}_{\ge0}^{n}$, we define the vertex weights $$\label{eq:fully_fused_stochastic_weights} \begin{split} &W_{x,\mathsf{L},\mathsf{M}}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) \coloneqq \mathbf{1}_{\mathbf{A}+\mathbf{B}=\mathbf{C}+\mathbf{D}} \cdot x^{|\mathbf{D}-\mathbf{B}|} \hspace{1pt} (q^{\mathsf{L}})^{|\mathbf{A}|} \hspace{1pt} (q^{\mathsf{M}})^{-|\mathbf{D}|} \\ &\hspace{60pt}\times \sum\nolimits_{\mathbf{P}} \Phi(\mathbf{C}-\mathbf{P}, \mathbf{C}+\mathbf{D}-\mathbf{P} ; q^{\mathsf{L}-\mathsf{M}} x, q^{-\mathsf{M}} x ) \hspace{1pt} \Phi( \mathbf{P}, \mathbf{B} ; q^{-\mathsf{L}} / x, q^{-\mathsf{L}} ). \end{split}$$ The sum in [\[eq:fully_fused_stochastic_weights\]](#eq:fully_fused_stochastic_weights){reference-type="eqref" reference="eq:fully_fused_stochastic_weights"} is finite and is taken over all $\mathbf{P}\in \mathbb{Z}_{\ge0}^{n}$ such that $0\le P_i\le \min\left( B_i,C_i \right)$ for all $i$. **Remark 4**. The parameters $\mathsf{L},\mathsf{M}$ enter the vertex weights [\[eq:fully_fused_stochastic_weights\]](#eq:fully_fused_stochastic_weights){reference-type="eqref" reference="eq:fully_fused_stochastic_weights"} only through the powers $q^{\mathsf{L}},q^{\mathsf{M}}$. Moreover, the weights depend on $q^{\mathsf{L}}$ and $q^{\mathsf{M}}$ in a rational manner. Specializing $\mathsf{L},\mathsf{M}$, or both to positive integers leads to finite-spin reduction as in . The integers $\mathsf{L}$ and $\mathsf{M}$ correspond to the horizontal and the vertical edge capacities, respectively. Outside of the finite-spin specializations, we may view $q^{\mathsf{L}},q^{\mathsf{M}}$ as independent complex parameters of the weights. The weights $W_{x,\mathsf{L},\mathsf{M}}$ [\[eq:fully_fused_stochastic_weights\]](#eq:fully_fused_stochastic_weights){reference-type="eqref" reference="eq:fully_fused_stochastic_weights"} satisfy a version of the Yang--Baxter equation, see formulas (C.1.2)--(C.1.3) in [@borodin_wheeler2018coloured]. Graphically, this equation is similar to the one illustrated in , but the weights $L_{s,x},L_{s,y}$, and $R_{y/x}$ must be replaced with, respectively, $W_{\frac{x}{z},\mathsf{L},\mathsf{N}}$, $W_{\frac{y}{z},\mathsf{M},\mathsf{N}}$, and $W_{\frac{x}{y},\mathsf{L},\mathsf{M}}$. The summation in the Yang--Baxter equation for the $W$'s goes over triples of elements from $\mathbb{Z}_{\ge0}^n$. The weights [\[eq:fully_fused_stochastic_weights\]](#eq:fully_fused_stochastic_weights){reference-type="eqref" reference="eq:fully_fused_stochastic_weights"} sum to one [@borodin_wheeler2018coloured (C.1.5)], $$\label{eq:fully_fused_stochastic_weights_sum_to_one} \sum\nolimits_ {\mathbf{C},\mathbf{D}\in \mathbb{Z}_{\ge 0}^n} W_{x,\mathsf{L},\mathsf{M}}( \mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D} )=1, \qquad \mathbf{A},\mathbf{B}\in \mathbb{Z}_{\ge0}^n,$$ and reduce to the weights $L_{s,x}$ (spin-$\frac{1}{2}$ in the horizontal direction and higher spin in the vertical direction) and $R_z$ (spin-$\frac{1}{2}$ in both directions) from as follows [@borodin_wheeler2018coloured Proposition C.1.4 and formula (C.2.2)]: $$\label{eq:fully_fused_reduction_to_L} \begin{split} R_{z}(i,j;k,\ell) &= W_{z^{-1},1,1} (\mathbf{e}_i\mathbf{1}_{i\ge1}, \mathbf{e}_j\mathbf{1}_{j\ge1}; \mathbf{e}_k\mathbf{1}_{k\ge1}, \mathbf{e}_\ell\mathbf{1}_{\ell\ge1} ), \\ L_{s,x}(\mathbf{A},b;\mathbf{C},d) &= W_{x/s,1,\mathsf{N}} ( \mathbf{A},\mathbf{e}_b\mathbf{1}_{b\ge1}; \mathbf{C},\mathbf{e}_d\mathbf{1}_{d\ge1} ) \big\vert_{q^{\mathsf{N}}=s^{-2}}, \end{split}$$ where $i,j,k,\ell,b,d\in \left\{ 0,1,\ldots,n \right\}$ and $\mathbf{A},\mathbf{C}\in \mathbb{Z}_{\ge0}^{n}$. ## Queue specialization {#sub:mqueue_spec} Here we define a procedure which we call the *queue specialization* of the fully fused stochastic colored vertex weights [\[eq:fully_fused_stochastic_weights\]](#eq:fully_fused_stochastic_weights){reference-type="eqref" reference="eq:fully_fused_stochastic_weights"}. The name "queue" comes from the connection with multiline queues detailed in below. The queue specialization depends on an integer $1\le m\le n$ and on three parameters $u,s_1,s_2$, and proceeds in the following manner: 1. First, encode the parameters $\mathsf{L}$ and $\mathsf{M}$ through $s_1,s_2\in \mathbb{C}$ as $q^{-\mathsf{L}}=s_1^2$, $q^{-\mathsf{M}}=s_2^2$, and let the spectral parameter be $x=s_1s_2^{-1}u$. This is just a change of variables which will be useful in subsequent computations. 2. After that, take the limit of $W_{s_1s_2^{-1}u,\mathsf{L},\mathsf{M}} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$ as $A_m,C_m\to+\infty$ such that all other coordinates of $\mathbf{A},\mathbf{C}$, as well as the whole tuples $\mathbf{B},\mathbf{D}$ are fixed, and $A_m-C_m=D_m-B_m$ is also fixed.[^2] The latter condition follows from the arrow conservation. **Lemma 5**. *The limit of the vertex weights $W_{s_1s_2^{-1}u,\mathsf{L},\mathsf{M}} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$ described above exists. It is given by $$\label{eq:queue_spec_fully_fused} \begin{split} & \lim_{A_m,C_m\to +\infty} W_{s_1s_2^{-1}u,\mathsf{L},\mathsf{M}} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})= \mathbf{1}_{\mathbf{A}+\mathbf{B}=\mathbf{C}+\mathbf{D}} \cdot \mathbf{1}_{D_1=\ldots=D_{m-1}=0} \cdot \frac{(s_1^{-1}s_2 u; q)_{\infty}}{(s_1s_2u ; q)_{\infty}} \\&\hspace{25pt}\times \sum_{\mathbf{P}} \frac{(s_1s_2/u ; q)_{|\mathbf{P}|} (s_1u/s_2 ; q)_{|\mathbf{B}-\mathbf{P}|}} {(s_1^2 ; q)_{|\mathbf{B}|}}\hspace{1pt} q^{\sum_{1\le i<j\le n}\left(B_i-P_i\right) P_j} \prod_{i=1}^n \frac{(q;q)_{B_i}}{(q;q)_{P_i}(q;q)_{B_i-P_i}} \\&\hspace{25pt}\times \Bigl(\frac{s_1s_2}{u} \Bigr)^{|\mathbf{B}|-|\mathbf{P}|} \Bigl(\frac{us_2}{s_1} \Bigr)^{|\mathbf{D}|} % s_1^{|\mathbf{B}|-|\mathbf{D}|-|\mathbf{P}|} % s_2^{|\mathbf{B}|+|\mathbf{D}|-|\mathbf{P}|} % u^{|\mathbf{D}|-|\mathbf{B}|+|\mathbf{P}|} \hspace{1pt} q^{\sum_{m\le i<j\le n} D_i (C_j-P_j)} \hspace{1pt} \frac{ (s_1^2 ; q)_{|\mathbf{D}|}} {(q;q)_{D_m}} \prod_{i=m+1}^n \frac{(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}}, \end{split}$$ where the sum is over $\mathbf{P}\in \mathbb{Z}_{\ge0}^n$ with $0\le P_i\le \min (B_i,C_i)$ for all $i$.* *Proof.* Set $x = s_1 s_2^{-1} u$. Since $\mathbf{B}$ stays fixed, the second factor $$\Phi( \mathbf{P}, \mathbf{B} ; q^{-\mathsf{L}} / x, q^{-\mathsf{L}} ) =\Phi( \mathbf{P}, \mathbf{B} ; s_1s_2 / u, s_1^2 )$$ in the sum in [\[eq:fully_fused_stochastic_weights\]](#eq:fully_fused_stochastic_weights){reference-type="eqref" reference="eq:fully_fused_stochastic_weights"} is also fixed. Moreover, the summation multi-index $\mathbf{P}$ belongs to a fixed finite set where $P_i\le B_i$ for all $i$. Thus, it remains to consider the limit of $$\label{eq:queue_spec_proof} \begin{split} &(q^{\mathsf{L}})^{|\mathbf{A}|}\hspace{1pt} \Phi(\mathbf{C}-\mathbf{P}, \mathbf{C}+\mathbf{D}-\mathbf{P} ; q^{\mathsf{L}-\mathsf{M}} x, q^{-\mathsf{M}} x ) \\&\hspace{20pt}= (q^{\mathsf{L}})^{|\mathbf{A}|}\hspace{1pt} \frac{(q^{\mathsf{L}-\mathsf{M}} x ; q)_{|\mathbf{C}-\mathbf{P}|}(q^{-\mathsf{L}} ; q)_{|\mathbf{D}|}} {(q^{-\mathsf{M}} x ; q)_{|\mathbf{C}-\mathbf{P}+\mathbf{D}|}}\hspace{1pt} (q^{-\mathsf{L}})^{|\mathbf{C}-\mathbf{P}|} q^{\sum_{1\le i<j\le n} D_i (C_j-P_j)} \prod_{i=1}^n \frac{(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}}. \\&\hspace{20pt}= s_1^{-2|\mathbf{D}-\mathbf{B}+\mathbf{P}|}\hspace{1pt} \frac{(s_1^{-1}s_2 u; q)_{|\mathbf{C}-\mathbf{P}|} (s_1^2 ; q)_{|\mathbf{D}|}} {(s_1s_2u ; q)_{|\mathbf{C}-\mathbf{P}+\mathbf{D}|}}\hspace{1pt} q^{\sum_{1\le i<j\le n} D_i (C_j-P_j)} \prod_{i=1}^n \frac{(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}}. \end{split}$$ We have $C_m-P_m\to +\infty$, so [\[eq:queue_spec_proof\]](#eq:queue_spec_proof){reference-type="eqref" reference="eq:queue_spec_proof"} converges to zero unless $D_i=0$ for all $i<m$ (as $q \in [0, 1)$). This leads to the indicator $\mathbf{1}_{D_1=\ldots=D_{m-1}=0}$ in [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"}. Next, if $D_i=0$ for all $i<m$, then all other factors in [\[eq:queue_spec_proof\]](#eq:queue_spec_proof){reference-type="eqref" reference="eq:queue_spec_proof"} behave well, and the desired limit of the whole vertex weight exists. Taking the limit as $C_m \rightarrow +\infty$, we immediately obtain [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"}. This completes the proof. ◻ **Definition 6** (Queue specialization of the vertex weights). Let $1\le m\le n$ and $u,s_1,s_2\in \mathbb{C}$. We denote the limiting vertex weights [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"} in by $\mathbb{W}_{s_1,s_2,u}^{(-m)}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$, and call them the *queue specialization* of the fully fused stochastic colored vertex weights. The term "queue specialization" comes from connections with multiline queues described (in two different degenerations) in below. The label $(-m)$ will be useful when we later place the vertices $\mathbb{W}_{s_1,s_2,u}^{(-m)}$ on a lattice. **Remark 7**. In the queue vertex weights $\mathbb{W}_{s_1,s_2,u}^{(-m)}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$, we abuse the notation of the tuples $\mathbf{A},\mathbf{C}\in \mathbb{Z}_{\ge0}^n$ by setting $A_m,C_m=+\infty$. That is, the tuples with infinitely many arrows of color $m$ are not elements of $\mathbb{Z}_{\ge0}^n$. However, for the uniformity of notation, we will still sometimes treat $\mathbf{A},\mathbf{C}$ as elements of $\mathbb{Z}_{\ge0}^n$, while explicitly stating that $A_m,C_m=+\infty$. **Remark 8**. From [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"} we see that for fixed $B_1,\ldots,B_{m-1}\ge0$, the weights $\mathbb{W}_{s_1,s_2,u}^{(-m)}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$ are *independent* of $A_1,\ldots,A_{m-1},C_1,\ldots, C_{m-1}$ provided that $C_i=A_i+B_i$ for all $i<m$. For example, we can set $A_i=0$ and $C_i=B_i$ for all $i<m$. Note also that since $A_m,C_m=+\infty$, we may have $D_m>0$ even if $B_m=0$. The latter property is essential for our constructions. The next lemma states the independence of the queue vertex weights under $B_m$, too, provided that no lower colors are present: **Lemma 9**. *Let $B_1=\ldots=B_{m-1}=0$. Then the queue vertex weight $\mathbb{W}_{s_1,s_2,u}^{(-m)}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$ is independent of $B_m$.* *Proof.* By , we may set $A_i=C_i=D_i=0$ for all $i<m$. Set, for simplicity, $B_m=b$, $B_{m+1}+\ldots+B_n=b'$, $P_m=p$, $P_{m+1}+\ldots+P_n =p'$, and $D_m=d$. Then the part of [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"} which depends only on $b$ and $p$ has the form $$\begin{split} &\sum_{p=0}^{b} \frac{(s_1s_2/u;q)_{p+p'}(s_1u/s_2;q)_{b+b'-p-p'}} {(s_1^2;q)_{b+b'}} \hspace{1pt} \frac{(q;q)_{b}}{(q;q)_p(q;q)_{b-p}} \hspace{1pt} (s_1s_2/u)^{b-p} q^{(b-p)p'} \\&= \frac{(s_1s_2/u;q)_{p'} (s_1u/s_2;q)_{b'-p'} }{(s_1^2;q)_{b'}} \sum_{p=0}^{b} \frac{(q^{p'} s_1s_2/u;q)_{p} (q^{b'-p'} s_1u/s_2; q)_{b-p} } {(q^{b'} s_1^2;q)_{b}} \hspace{1pt} \frac{(q;q)_{b}}{(q;q)_p(q;q)_{b-p}} \hspace{1pt} (q^{p'} s_1s_2/u )^{b-p} \\&= \frac{(s_1s_2/u;q)_{p'} (s_1u/s_2;q)_{b'-p'} }{(s_1^2;q)_{b'}}. \end{split}$$ In the last equality we used the sum-to-one property [\[eq:capital_Phi_coloreq_sum_to_one\]](#eq:capital_Phi_coloreq_sum_to_one){reference-type="eqref" reference="eq:capital_Phi_coloreq_sum_to_one"} for the single-color case $n=1$ (with $x = q^{b'-p'} s_1u/s_2$, $y = q^{b'} s_1^2$, and $\mathbf{B} = b$). We see that the resulting expression does not depend on $b$, as desired. ◻ **Proposition 10**. *For any $m\in\{1,\ldots,n\}$, the queue vertex weights $\mathbb{W}_{s_1,s_0,\frac{u_1}{u_0}}^{(-m)}$, $\mathbb{W}_{s_2,s_0,\frac{u_2}{u_0}}^{(-m)}$ [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"}, and the fused cross vertex weight $W_{\frac{s_1u_1}{s_2u_2},\mathsf{L},\mathsf{M}}$ [\[eq:fully_fused_stochastic_weights\]](#eq:fully_fused_stochastic_weights){reference-type="eqref" reference="eq:fully_fused_stochastic_weights"}, where $q^{-\mathsf{L}}=s_1^2$ and $q^{-\mathsf{M}}=s_2^2$, satisfy the Yang--Baxter equation given in . In symbols, for all fixed $\mathbf{A},\mathbf{I}_1, \mathbf{I}_2,\mathbf{B}, \mathbf{J}_1,\mathbf{J}_2$ with $A_m,B_m=+\infty$, we have $$\begin{split} & \sum_{\mathbf{K}_1,\mathbf{K}_2,\mathbf{K}_3} \mathbb{W}_{s_2,s_0,\frac{u_2}{u_0}}^{(-m)}(\mathbf{A}, \mathbf{I}_2 ; \mathbf{K}_3, \mathbf{K}_2) \mathbb{W}_{s_1,s_0,\frac{u_1}{u_0}}^{(-m)}(\mathbf{K}_3, \mathbf{I}_1; \mathbf{B}, \mathbf{K}_1) W_{\frac{s_1u_1}{s_2u_2},\mathsf{L},\mathsf{M}} (\mathbf{K}_2, \mathbf{K}_1; \mathbf{J}_2, \mathbf{J}_1) \\ &\hspace{5pt} = \sum_{\mathbf{K}_1,\mathbf{K}_2,\mathbf{K}_3} W_{\frac{s_1u_1}{s_2u_2},\mathsf{L},\mathsf{M}} (\mathbf{I}_2, \mathbf{I}_1; \mathbf{K}_2, \mathbf{K}_1) \mathbb{W}_{s_1,s_0,\frac{u_1}{u_0}}^{(-m)}(\mathbf{A}, \mathbf{K}_1; \mathbf{K}_3, \mathbf{J}_1) \mathbb{W}_{s_2,s_0,\frac{u_2}{u_0}}^{(-m)}(\mathbf{K}_3, \mathbf{K}_2 ; \mathbf{B}, \mathbf{J}_2). \end{split}$$* *Proof.* This is the queue specialization of the Yang--Baxter equation [@borodin_wheeler2018coloured (C.1.2)] for the fully fused stochastic weights. The queue specialization is taken along the vertical line carrying the parameters $z=s_0u_0$ and $\mathsf{N}$ with $q^{-\mathsf{N}}=s_0^2$. ◻ {#fig:YBE_fused_queue width=".9\\textwidth"} Recall that $0\le q<1$. Let us define the following two subsets of the parameters $(s_1,s_2,u)$ in the queue vertex weights: $$\label{eq:queue_general_nonnegative_conditions} \parbox{.88\textwidth}{\begin{enumerate}[$\bullet$] \item (higher horizontal spin) $s_1, s_2 \in [-1,1]$ such that $0 \leq s_1 s_2 < u < \min(\frac{s_1}{s_2}, \frac{s_2}{s_1}, \frac{1}{s_1 s_2})$; \item (finite horizontal spin) $s_1 = q^{-\frac{\mathsf{L}}{2}}$ for some $\mathsf{L} \in\mathbb{Z}_{\ge1}$ and $u = q^{\frac{\mathsf{L}}{2}} u'$, with purely imaginary $u',s_2$ satisfying $s_2 u' \leq s_2^2 \leq 0$ and $s_2/u'\ge q^{\mathsf{L}}$. \end{enumerate}}$$ These subsets present convenient sufficient nonnegativity conditions: **Proposition 11**. *Under [\[eq:queue_general_nonnegative_conditions\]](#eq:queue_general_nonnegative_conditions){reference-type="eqref" reference="eq:queue_general_nonnegative_conditions"}, the queue vertex weights $\mathbb{W}^{(-m)}_{s_1, s_2, u}$ [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"} are nonnegative, and $$\label{eq:queue_general_sum_to_one} \sum_{C_{m+1},C_{m+2},\ldots,C_n, D_{m},D_{m+1},\ldots,D_n=0}^{\infty} \mathbb{W}^{(-m)}_{s_1, s_2, u}(\mathbf{A}, \mathbf{B}; \mathbf{C}, \mathbf{D})=1,$$ where $\mathbf{A},\mathbf{B}$ are fixed, $A_m,C_m=+\infty$, and $C_i=A_i+B_i$, $D_i=0$ for all $i<m$.* *Proof.* Let us first consider the higher horizontal spin case of [\[eq:queue_general_nonnegative_conditions\]](#eq:queue_general_nonnegative_conditions){reference-type="eqref" reference="eq:queue_general_nonnegative_conditions"}. One can check that all arguments of the $q$-Pochhammer symbols in [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"} are in $[0,1)$. Moreover, the total sign coming from the powers $s_1^{|\mathbf{B}|-|\mathbf{D}|-|\mathbf{P}|} s_2^{|\mathbf{B}|+|\mathbf{D}|-|\mathbf{P}|} u^{|\mathbf{D}|-|\mathbf{B}|+|\mathbf{P}|}$ is always nonnegative. Thus, we get the nonnegativity of the queue vertex weights. To see that they sum to one, we need to take the limit of the sum-to-one identity [\[eq:fully_fused_stochastic_weights_sum_to_one\]](#eq:fully_fused_stochastic_weights_sum_to_one){reference-type="eqref" reference="eq:fully_fused_stochastic_weights_sum_to_one"} for the fully fused weights $W_{x,\mathsf{L},\mathsf{M}}$ as $A_m,C_m\to+\infty$. One can check that under our conditions [\[eq:queue_general_nonnegative_conditions\]](#eq:queue_general_nonnegative_conditions){reference-type="eqref" reference="eq:queue_general_nonnegative_conditions"}, the weight $W_{x,\mathsf{L},\mathsf{M}} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$ (with $x=s_1u/s_2$, $q^{-\mathsf{L}}=s_1^2$, $q^{-\mathsf{M}}=s_2^2$) decays exponentially fast when $|\mathbf{D}|\to+\infty$ due to the presence of the power $(us_2/s_1)^{|\mathbf{D}|}$. Thus, since our queue specialization requires $|\mathbf{D}|$ to stay fixed, this decay ensures that identity [\[eq:fully_fused_stochastic_weights_sum_to_one\]](#eq:fully_fused_stochastic_weights_sum_to_one){reference-type="eqref" reference="eq:fully_fused_stochastic_weights_sum_to_one"} yields [\[eq:queue_general_sum_to_one\]](#eq:queue_general_sum_to_one){reference-type="eqref" reference="eq:queue_general_sum_to_one"}. Let us now consider the finite horizontal spin case of [\[eq:queue_general_nonnegative_conditions\]](#eq:queue_general_nonnegative_conditions){reference-type="eqref" reference="eq:queue_general_nonnegative_conditions"}. In this case, $|\mathbf{D}|$ must stay finite (cf. ), so the sum-to-one property [\[eq:queue_general_sum_to_one\]](#eq:queue_general_sum_to_one){reference-type="eqref" reference="eq:queue_general_sum_to_one"} is automatic from [\[eq:fully_fused_stochastic_weights_sum_to_one\]](#eq:fully_fused_stochastic_weights_sum_to_one){reference-type="eqref" reference="eq:fully_fused_stochastic_weights_sum_to_one"}. To see the nonnegativity, observe that since $s_1 = q^{-\frac{\mathsf{L}}{2}} \geq 1$, the term $(s_1^2;q)_{|\mathbf{D}|}$ produces the sign $(-1)^{|\mathbf{D}|}$, which is compensated by $(us_2/s_1)^{|\mathbf{D}|}$. The other combined powers $(s_1s_2/u)^{|\mathbf{B}|-|\mathbf{P}|}$ are always nonnegative. Next, we have $$\frac{(s_1^{-1}s_2 u; q)_{\infty}}{(s_1s_2u ; q)_{\infty}} = \frac{1}{(s_2u';q)_{\mathsf{L}}}\ge 0.$$ In the remaining $q$-Pochhammer symbols, we have $$s_1s_2/u=q^{-\mathsf{L}}s_2/u'\ge1,\qquad s_1u/s_2=u'/s_2\ge1,\qquad s_1^2=q^{-\mathsf{L}}\ge1.$$ Thus, the quantities $(s_1s_2/u ; q)_{|\mathbf{P}|} (s_1u/s_2 ; q)_{|\mathbf{B}-\mathbf{P}|} (s_1^2 ; q)_{|\mathbf{B}|}^{-1}$ are nonnegative for all $\mathbf{P}$ provided that $|\mathbf{B}|\le \mathsf{L}$. This completes the proof. ◻ ## Queue vertex model on the cylinder {#sub:mqueue_states} Let us fix the number $n$ of colors, and another integer $N\ge1$ which is the size of the cylinder. In this subsection we define a linear operator $\mathfrak{Q}$ whose matrix elements $\langle \emptyset | \mathfrak{Q} | \mathbf{V} \rangle$ are partition functions coming from the queue vertex weights on the cylinder $\{-n, \dots, -1\} \times \mathbb{Z}/N \mathbb{Z}$. The vertices on the cylinder are indexed by $(-m,j)$, $m=1,\ldots,n$, $j=1,\ldots,N$, see for an illustration. No paths enter from the left. The paths exiting horizontally from the right are encoded by the integer tuples $\mathbf{V}(1),\dots, \mathbf{V}(N) \in \mathbb{Z}_{\ge0}^n$. Define the vector space for the states on the cylinder, $$\label{eq:queue_state_space_higher_spin} V^{\otimes N} \coloneqq \mathop{\mathrm{Span}} \left( \{ \big| \mathbf{V}\big\rangle =\big| (\mathbf{V}(1),\dots, \mathbf{V}(N)) \big\rangle : \mathbf{V}(j) \in \mathbb{Z}_{\geq 0}^n \ \forall j =1,\dots, N\} \right),$$ and similarly let $\langle \mathbf{V} |$ denote the dual basis in $V^{\otimes N}$. We also need the subspace $V^{\otimes N}_{\mathrm{full}}$ of $V^{\otimes N}$ spanned by the vectors $| \mathbf{V} \rangle$ satisfying $$\sum\nolimits_{j=1}^{N}\mathbf{V}(j)_k > 0 \quad \textnormal{for all $k=1,\ldots,n $}$$ (above, $\mathbf{V}(j)_k$ denotes the $k$-th coordinate of $\mathbf{V}(j)$). In words, each state $| \mathbf{V} \rangle\in V^{\otimes N}_{\mathrm{full}}$ must contain at least one arrow of each of the $n$ colors. **Definition 12**. Fix complex parameters $$\label{eq:queue_state_weights_parameters} \mathbf{u} =(u_1,\dots, u_N) \in \mathbb{C}^N, \quad \mathbf{s}^{(h)} = (s^{(h)}_1,\dots, s^{(h)}_N), \quad \mathbf{v} = (v_1,\dots,v_n), \quad \mathbf{s}^{(v)} = (s^{(v)}_1,\dots, s^{(v)}_n),$$ The linear operator $\mathfrak{Q}=\mathfrak{Q} ( \mathbf{u}; \mathbf{s}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)})$ on $V^{\otimes N}$, called the *queue transfer matrix*, is defined via its matrix elements $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle$ as follows. First, if $| \mathbf{V} \rangle\notin V^{\otimes N}_{\mathrm{full}}$, we set $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle=0$. Otherwise, $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle$ is the partition function of the queue vertex weights on the cylinder $\{-n,\ldots,-1 \}\times \mathbb{Z}/N\mathbb{Z}$ with the following data: 1. The entering arrow configurations $\mathbf{V}'(j)$ along the left horizontal edges $(-n-1,j)\to(-n,j)$, $j=1,\ldots,N$. 2. The terminal arrow configurations $\mathbf{V}(j)$ along the right horizontal edges $(-1,j)\to(0,j)$, $j=1,\ldots,N$. 3. Queue vertex weights $\mathbb{W}^{(-m)}_{s^{(h)}_j, s^{(v)}_m, u_j / v_{m}}$ at each vertex $(-m,j)$ in the cylinder, where $j=1,\ldots,N$, $m=1,\ldots,n$, and $u_j,s_j^{(h)}$ and $v_m,s^{(v)}_m$ are the horizontal and the vertical parameters, respectively. In detail, the partition function $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle$ is equal to the sum $$\sum_{ \mathbf{M} } \hspace{1pt} \sum_{\mathscr{C}\in \mathcal{P}_{\mathbf{M},\mathbf{V}',\mathbf{M},\mathbf{V}}} \hspace{1pt} \prod_{m=1}^{n}\prod_{j=1}^{N} \mathbb{W}^{(-m)}_{s^{(h)}_j, s^{(v)}_m, u_j / v_{m}}(\mathbf{A}(m,j),\mathbf{B}(m,j);\mathbf{C}(m,j),\mathbf{D}(m,j)),$$ where $\mathbf{M}=(\mathbf{M}(-n),\dots,\mathbf{M}(-1))$ encodes the paths winding around the cylinder. The sum over $\mathbf{M}$, by definition, has $$\label{eq:queue_state_trace_conditions} \mathbf{M}(-m)_m=+\infty \quad\textnormal{and}\quad \mathbf{M}(-m)_i=0,\ i<m, \quad \textnormal{for all}\ m=1,\ldots,n.$$ For each $\mathbf{M}$, the sum over $\mathscr{C}$ runs over all path configurations in the rectangle with the boundary conditions ${\mathbf{M},\mathbf{V}',\mathbf{M},\mathbf{V}}$ at the bottom, left, top, and right, respectively. The tuples $\mathbf{A}(m,j)$, $\mathbf{B}(m,j)$, $\mathbf{C}(m,j)$, and $\mathbf{D}(m,j)$ encode the arrow configurations at each vertex $(-m,j)$ of the rectangle. See for an example when $\mathbf{V}' = (\mathbf{0}, \dots, \mathbf{0})$. {#fig:queue_state width=".5\\textwidth"} Observe that the partition function $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle$ in involving the sum over $\mathbf{M}(-n),\ldots,\mathbf{M}(-1)$ cannot be interpreted as a probability in a stochastic vertex model because we are summing over input path configurations at vertices, as well as over the output ones. Moreover, this sum may even be divergent. Therefore, we need to make sure that the quantities $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle$ are well-defined: **Lemma 13**. *For any $| \mathbf{V} \rangle\in V^{\otimes N}_{\mathrm{full}}$, the sum over $\mathbf{M}(-n),\ldots,\mathbf{M}(-1)$ in is convergent for any $|q|<1$.* *Proof.* By , paths of colors $i<m$ cannot leave the column $(-m)$. The fact that the configuration $| \mathbf{V} \rangle\in V^{\otimes N}_{\mathrm{full}}$ contains at least one arrow of each color implies that each column $(-m)$ must horizontally emit at least one path of color $m$. Recall again that after we fix the entering horizontal arrows $\mathbf{B}$ at a lattice site, the summation multi-index $\mathbf{P}$ in each vertex weight belongs to a fixed finite set where $P_i\le B_i$ for all $i$. Furthermore, with $\mathbf{V}$ fixed, the size of tuples $\mathbf{B},\mathbf{D}$ at each vertex is bounded from above. Thus, the factor $q^{-D_i P_j}$ in each vertex weight is bounded from above. Therefore, the presence of the factors $q^{\sum_{m\le i<j\le n} D_i C_j}$ in the vertex weights [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"} implies that the weight of the whole column $(-m)$ with fixed winding path counts $\mathbf{M}(-m)_i$, $i>m$, contains the factor $q^{d\sum_{i=m+1}^n \mathbf{M}(-m)_i}$ for some $d\ge 1$. This implies that for any fixed $m$, the sum over the quantities $\mathbf{M}(-m)_i$, $i>m$, is finite. This completes the proof. ◻ **Remark 14**. The condition that $| \mathbf{V} \rangle\in V^{\otimes N}_{\mathrm{full}}$ is essential for . Indeed, for $n=3$, we have $$\mathbb{W}^{(-2)}_{s_1,s_2,u} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})= \frac{(s_2u/s_1;q)_\infty}{(s_1s_2u;q)_{\infty}},$$ where $\mathbf{A}=\mathbf{C}=(0,\infty,k)$ and $\mathbf{B}=\mathbf{D}=(0,0,0)$. This expression is independent of $k$. For $N=1$, we need to sum it over all $k$, which leads to divergence. However, if $\mathbf{D}=(0,d,0)$, the weight $\mathbb{W}^{(-2)}_{s_1,s_2,u} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$ contains the power $q^{kd}$, eliminating this problem of divergence. The partition functions $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle$ are essentially independent of the entrance state $\langle\mathbf{V}'|$: **Proposition 15**. *Let $| \mathbf{V} \rangle\in V^{\otimes N}_{\mathrm{full}}$. If the entering configuration $\mathbf{V}'$ contains at least one path of color strictly less than $n$, then $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle=0$. Otherwise, if $\mathbf{V}'$ contains only paths of color $n$, then we have $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle= \langle \emptyset | \mathfrak{Q} | \mathbf{V} \rangle$.* *Proof.* guarantees that no paths of color strictly less than $n$ leave column $(-n)$. Since we cannot have paths of color not equal to $n$ infinitely wind in the column $(-n)$, the partition function $\langle \mathbf{V}' | \mathfrak{Q} | \mathbf{V} \rangle$ must vanish if $\mathbf{V}'$ contains at least one path of color strictly less than $n$. This establishes the first claim. The second claim immediately follows from . ◻ **Remark 16** (Trace formula for queue partition functions). The queue vertex model partition function can be interpreted through a product of the following $N$ operators, where $N$ is the size of the ring in the cross-section of the cylinder: $$\label{eq:product_trace_formula} \langle \emptyset | \mathfrak{Q} | \mathbf{V} \rangle= \mathop{\mathrm{Trace}}\nolimits^\bullet \left( \mathscr{X}_{\mathbf{V}(1)}(u_1,s_1^{(h)})\cdots \mathscr{X}_{\mathbf{V}(N)}(u_N,s_N^{(h)}) \right).$$ Here each $\mathscr{X}_{\mathbf{V}}(u,s)$, $\mathbf{V}\in \mathbb{Z}_{\ge0}^n$, acts in the $n$-fold tensor product $V_{-n}\otimes \ldots \otimes V_{-1}$, where $V_{-m}$ has basis $|\mathbf{M}(-m)\rangle$, $\mathbf{M}(-m)\in \mathbb{Z}_{\ge0}^{n}$, $m=1,\ldots,n$. The matrix elements of $\mathscr{X}_{\mathbf{V}}(u,s)$ are partition functions of the queue weights on the lattice $\{1\}\times \{-n,\ldots,-1 \}$. See for an illustration. The operation $\mathop{\mathrm{Trace}}\nolimits^\bullet$ in [\[eq:product_trace_formula\]](#eq:product_trace_formula){reference-type="eqref" reference="eq:product_trace_formula"} means that we restrict the summation to the tuples $\mathbf{M}$ satisfying [\[eq:queue_state_trace_conditions\]](#eq:queue_state_trace_conditions){reference-type="eqref" reference="eq:queue_state_trace_conditions"}. One can turn $\mathop{\mathrm{Trace}}\nolimits^\bullet$ into a genuine trace by suitably modifying the definition of the spaces $V_{-m}$. ![The matrix element $\large\langle \mathbf{M}(-n),\ldots,\mathbf{M}(-1) \large| \hspace{1pt}\mathscr{X}_{\mathbf{V}}(u,s)\hspace{1pt} \large| \mathbf{M}'(-n),\ldots,\mathbf{M}'(-1) \large\rangle$ of one of the operators $\mathscr{X}_{\mathbf{V}}(u,s)$ entering the trace formula [\[eq:product_trace_formula\]](#eq:product_trace_formula){reference-type="eqref" reference="eq:product_trace_formula"}. At the vertex indexed by $-m$, $m=1,\ldots,n$, we have the queue vertex weight $\mathbb{W}^{(-m)}_{s, s^{(v)}_m, u / v_{m}}$.](product_trace_operator.pdf){#fig:product_trace_formula width=".45\\textwidth"} # Stationarity of the queue vertex model {#sec:main_vertex_statement} In this section, we establish two general stationarity properties of the queue vertex model on the cylinder. Specifically, we construct two Markov operators which, when applied to the empty state $\langle\emptyset|$ on the cylinder, commute with the queue transfer matrix $\mathfrak{Q}$ from . Both of these commutation relations follow directly from the Yang--Baxter equation. In this section we work with *formal* Markov operators, that is, we do not assume that their matrix elements are nonnegative. The matrix elements only need to satisfy the corresponding sum-to-one properties. In the future sections we specify the ranges of parameters making the matrix elements nonnegative. ## Twisted cylinder Markov operator {#sub:twisted} Recall that $n$ is the number of colors, and $N$ is the size of the ring in the vertical cross-section of the cylinder $\{-n,\ldots,-1 \}\times \mathbb{Z}/N\mathbb{Z}$ carrying the queue transfer matrix $\mathfrak{Q}=\mathfrak{Q} ( \mathbf{u}; \mathbf{s}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)})$ of . For our first relation, we take the spin-$\frac{1}{2}$ specialization in the horizontal direction. That is, the horizontal spin parameters in [\[eq:queue_state_weights_parameters\]](#eq:queue_state_weights_parameters){reference-type="eqref" reference="eq:queue_state_weights_parameters"} are all equal to $q^{-1/2}$: $$\mathbf{s}^{(h)} = (s^{(h)}_1,\dots, s^{(h)}_N) = \mathbf{s}^{(h)}_{\frac12} \coloneqq (q^{-1/2},\ldots,q^{-1/2} ).$$ We can take each tensor component in the space $V^{\otimes N}$ [\[eq:queue_state_space_higher_spin\]](#eq:queue_state_space_higher_spin){reference-type="eqref" reference="eq:queue_state_space_higher_spin"} to be $V=\mathbb{C}^{n+1}$. We need some extra notation. Let $R_{z}$ be the $R$-matrix [\[eq:R_matrix_nonfused\]](#eq:R_matrix_nonfused){reference-type="eqref" reference="eq:R_matrix_nonfused"}, and denote by $\check{\mathscr{R}}_z$ the operator in $V\otimes V$ with matrix elements (see the end of for basis vector notations) $$\label{eq:swapping_R_check} \langle i,j | \hspace{1pt} \check{\mathscr{R}}_z \hspace{1pt} | \ell,k \rangle \coloneqq R_{z}(i,j;k,\ell).$$ When $\check{\mathscr{R}}_z$ acts on the $k$-th and the $\ell$-th tensor components of $V^{\otimes N}$, we denote it by $\check{\mathscr{R}}_z^{(k\ell)}$. **Definition 17**. Fix two spectral parameters $u,u_1$. Let the *twisted cylinder Markov operator* $\mathfrak{T}(u,u_1)$ be the linear operator on $V^{\otimes N}$ defined as $$\label{eq:twisted_cylinder_Markov_operator} \mathfrak{T}(u, u_1) \coloneqq \check{\mathscr{R}}_{u_1 u^{-1}}^{(1 2)} \hspace{1pt} \check{\mathscr{R}}_{u_1 u^{-1}}^{(2 3)} \cdots \check{\mathscr{R}}_{u_1 u^{-1}}^{(N,1)},$$ Pictorially, $\mathfrak{T}(u, u_1)$ is given in . The product in [\[eq:twisted_cylinder_Markov_operator\]](#eq:twisted_cylinder_Markov_operator){reference-type="eqref" reference="eq:twisted_cylinder_Markov_operator"} is interpreted as a product of Markov operators, that is, we first apply $\check{\mathscr{R}}_{u_1 u^{-1}}^{(1 2)}$ to a fixed configuration of arrows on the cylinder, get a random configuration, then apply $\check{\mathscr{R}}_{u_1 u^{-1}}^{(2 3)}$ to the new configuration, and so on. **Remark 18**. The swapping of the indices in the operator $\check{\mathscr{R}}_z$ compared to the $R$-matrix $R_z$ (see [\[eq:swapping_R\_check\]](#eq:swapping_R_check){reference-type="eqref" reference="eq:swapping_R_check"}) is *purely notational*. We employ it for the following convenience. When is read from left to right, the space $V^{\otimes N}$ encoding configurations on the ring stays the same after every single crossing of the strands. In particular, passing to $\check{\mathscr{R}}_z$ does not affect the random mechanism: the crosses act in in the same way as in the diagram of the Yang--Baxter equation in . {#fig:twisted width=".64\\textwidth"} {#fig:twisted_commutation height=".19\\textwidth"}\ {#fig:twisted_commutation height=".19\\textwidth"} {#fig:twisted_commutation height=".19\\textwidth"} Iterating the sum-to-one property of the $R$-matrix [\[eq:R_matrix_nonfused\]](#eq:R_matrix_nonfused){reference-type="eqref" reference="eq:R_matrix_nonfused"}, we see that for any initial configuration $\mathbf{V}' = (i_1,\ldots,i_N)$, where $i_k \in \{0,\ldots,n\}$, we have $$\label{eq:twisted_cylinder_Markov_operator_sum_to_one} \sum_{\mathbf{V}=(i_1',\ldots,i_N')\in \{0,1,\ldots,n \}^N} \langle \mathbf{V}' | \mathfrak{T}(u, u_1) | \mathbf{V} \rangle=1.$$ If the matrix elements of $\mathfrak{T}(u, u_1)$ are nonnegative, then it is a Markov operator. However, considering $\mathfrak{T}(u, u_1)$ as a formal Markov operator with the sum-to-one property suffices for this section. We have the following stationarity of the queue vertex model under the action of the twisted cylinder Markov operator: **Theorem 19**. *Let the parameters $u,u_1$ of the twisted cylinder Markov operator $\mathfrak{T}$ [\[eq:twisted_cylinder_Markov_operator\]](#eq:twisted_cylinder_Markov_operator){reference-type="eqref" reference="eq:twisted_cylinder_Markov_operator"}, as well as the parameters $\mathbf{v} = (v_1,\dots, v_n)$ and $\mathbf{s}^{(v)} = (s^{(v)}_1,\dots, s^{(v)}_n)$ in the queue transfer matrix $\mathfrak{Q}$ of be arbitrary. Denote $\mathbf{u} \coloneqq (u_1,u,\dots, u)$. Then we have $$\label{eq:twisted_st} \big\langle \emptyset \big| \hspace{1pt} \mathfrak{Q} (\mathbf{u}; \mathbf{s}_{\frac{1}{2}}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)}) \hspace{1pt} \mathfrak{T}(u, u_1) = \big\langle \emptyset \big| \hspace{1pt}\mathfrak{Q} (\mathbf{u}; \mathbf{s}_{\frac{1}{2}}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)}) .$$* *Proof.* We apply the Yang--Baxter equation specialized to the queue vertex weights; see . In this equation, the cross vertex weights have the form $W_{\frac{u}{u_1},1,1}$, which are encoded by the operator $\check{\mathscr{R}}_{u_1u^{-1}}$; see [\[eq:fully_fused_reduction_to_L\]](#eq:fully_fused_reduction_to_L){reference-type="eqref" reference="eq:fully_fused_reduction_to_L"}. We illustrate the argument diagrammatically for $N = 3$ in . Applying the Yang--Baxter equation successively at columns $-n, -n+1, \dots,-2, -1$ of the queue vertex model, we get the intertwining relation $$\label{eq:comm1} \mathfrak{Q}((u_1, u, \dots, u);\mathbf{s}_{\frac{1}{2}}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)})\hspace{1pt} \check{\mathscr{R}}_{u_{1} u^{-1}}^{(1 2)} = \check{\mathscr{R}}_{u_1 u^{-1}}^{(1 2)}\hspace{1pt}\mathfrak{Q}((u, u_1, \dots, u); \mathbf{s}_{\frac{1}{2}}^{(h)}; \mathbf{v} ; \mathbf{s}^{(v)} )$$ Continuing inductively with $\check{\mathscr{R}}_{u_{1} u^{-1}}^{(2 3)}, \ldots, \check{\mathscr{R}}_{u_{1} u^{-1}}^{(N-1,N)}$ turns the horizontal spectral parameter sequence in $\mathfrak{Q}$ into $(u, \dots, u, u_1)$. Finally, after applying the last operator $\check{\mathscr{R}}_{u_{1} u^{-1}}^{(N,1)}$, we get back to the original sequence. Here we used the periodicity in the vertical direction of the vertex model defining $\mathfrak{Q}$, which is crucial to complete the commutation. Therefore, $$\mathfrak{Q} (\mathbf{u}; \mathbf{s}_{\frac{1}{2}}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)}) \hspace{1pt} \mathfrak{T}(u, u_1) = \mathfrak{T}(u, u_1) \hspace{1pt} \mathfrak{Q} (\mathbf{u}; \mathbf{s}_{\frac{1}{2}}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)}).$$ It remains to observe that $\langle \emptyset | \check{\mathscr{R}}_{u_1 u^{-1}}^{(i,j)} = \langle \emptyset |$ for all $i,j$. This completes the proof. ◻ ## Straight cylinder Markov operator {#sub:straight} Here we define another Markov operator on the cylinder of size $N+1$ which commutes with the queue transfer matrix on this cylinder when applied to the empty configuration $\emptyset$ at the left boundary of the cylinder. It acts in the space $\mathbb{C}^{n+1} \otimes V^{\otimes N}$, where the first factor $\mathbb{C}^{n+1}$ is an auxiliary space which we identify with the index $0$ in the superscripts. The tensor components of the space $V^{\otimes N}$ are indexed by $j=1,2,\dots, N$, and have basis vectors $|\mathbf{V}(j)\rangle$, where $\mathbf{V}(j) \in \mathbb{Z}_{\geq 0}^n$ encodes the arrow configuration on site $j$. That is, in contrast with , we return to the higher spin setting. Later in we will take limits in which the marginal process corresponding to the factor $V^{\otimes N}$ will lead to the colored $q$-Boson or the colored $q$-PushTASEP on the ring of size $N$. Recall the Markov operator $\mathscr{L}_{s,z}$ acting in $\mathbb{C}^{n+1}\otimes V$; see [\[eq:L_mat\]](#eq:L_mat){reference-type="eqref" reference="eq:L_mat"}. **Definition 20**. Fix spectral parameters $(x,\mathbf{u})=(x,u_1,u_2,\dots, u_N)\in \mathbb{C}^{N+1}$ and horizontal spin parameters $\mathbf{s}^{(h)}=(s_1^{(h)},\dots, s_N^{(h)})\in \mathbb{C}^N$. The *straight cylinder Markov operator* denoted by $\mathfrak{S}=\mathfrak{S}(x,\mathbf{u};\mathbf{s}^{(h)})$ acts in $\mathbb{C}^{n+1}\otimes V^{\otimes N}$ as follows: $$\label{eq:straight_cylinder_Markov_operator} \mathfrak{S}(x,\mathbf{u};\mathbf{s}^{(h)}) \coloneqq \mathscr{L}^{(0N)}_{s^{(h)}_N,xu_{N}^{-1}}\hspace{1pt} \mathscr{L}^{(0,N-1)}_{s^{(h)}_{N-1},xu_{N-1}^{-1}} \ldots \mathscr{L}^{(01)}_{s^{(h)}_1,xu_{1}^{-1}}.$$ Here by $\mathscr{L}^{(0,j)}_{s,z}$ we denote the operator $\mathscr{L}_{s,z}$ acting on $\mathbb{C}^{n+1}\otimes V^{\otimes N}$ in the auxiliary space $\mathbb{C}^{n+1}$ and the $j$-th tensor component of $V^{\otimes N}$. The operator $\mathfrak{S}$ is illustrated in . **Remark 21**. In and throughout this subsection, we index the tensor factors of the space $\mathbb{C}^{n+1} \otimes V^{\otimes N}$ by $0,1,\ldots,N$, but put the horizontal strand corresponding to the $0$-th factor on top of the cylinder in . This way of indexing is consistent with the action of the operators $\mathscr{L}_{s,z}$ in $\mathbb{C}^{n+1}\otimes V$ in [\[eq:straight_cylinder_Markov_operator\]](#eq:straight_cylinder_Markov_operator){reference-type="eqref" reference="eq:straight_cylinder_Markov_operator"}, while the location of the $0$-th strand on top just below the cylinder's cut is convenient for illustrations. {#fig:straight width=".5\\textwidth"} {#fig:straight_commuted} The operator $\mathfrak{S}(x,\mathbf{u};\mathbf{s}^{(h)})$ satisfies the sum-to-one property similarly to [\[eq:twisted_cylinder_Markov_operator_sum_to_one\]](#eq:twisted_cylinder_Markov_operator_sum_to_one){reference-type="eqref" reference="eq:twisted_cylinder_Markov_operator_sum_to_one"}, and thus is a formal Markov operator. We have the following stationarity of the queue vertex model under $\mathfrak{S}$: **Theorem 22**. *Let us take the parameters $\mathbf{u},\mathbf{s}^{(h)},\mathbf{v},\mathbf{s}^{(v)}$ as in [\[eq:queue_state_weights_parameters\]](#eq:queue_state_weights_parameters){reference-type="eqref" reference="eq:queue_state_weights_parameters"}, and consider the queue vertex transfer matrix on the ring of size $N+1$ with the parameters $$\mathfrak{Q}=\mathfrak{Q}\left( (xq^{\frac12},\mathbf{u});(q^{-\frac{1}{2}},\mathbf{s}^{(h)}) ; \mathbf{v};\mathbf{s}^{(v)}\right).$$ Then we have $$\big\langle \emptyset \big|\hspace{1pt}\mathfrak{Q}\hspace{1pt}\mathfrak{S} = \big\langle \emptyset \big|\hspace{1pt}\mathfrak{Q},$$ where $\mathfrak{S}=\mathfrak{S}(x,\mathbf{u};\mathbf{s}^{(h)})$ has parameters compatible with those in $\mathfrak{Q}$.* *Proof.* We employ the Yang--Baxter equation of . Let us match our parameters to this equation. The queue vertex model at rows $0$ and $j$ has vertex weights $$\mathbb{W}^{(-m)}_{q^{-1/2},s_m^{(v)},\frac{xq^{1/2}}{v_m}} ,\qquad \mathbb{W}^{(-m)}_{s_j^{(h)},s_m^{(v)},\frac{u_j}{v_m}},$$ respectively. The operator $\mathfrak{S}$ has the vertex weight $L_{s_j^{(h)},xu_j^{-1}}$, which, by [\[eq:fully_fused_reduction_to_L\]](#eq:fully_fused_reduction_to_L){reference-type="eqref" reference="eq:fully_fused_reduction_to_L"}, is the same as the fused weight $W_{x u_j^{-1}/s_j^{(h)},1,\mathsf{M}}$, where $q^{-\mathsf{M}/2}=s_j^{(h)}$. We see that these three weights indeed satisfy the Yang--Baxter equation illustrated in . Applying this Yang--Baxter equation with parameters $(v_1,s_1^{(v)}), \ldots,(v_n,s_n^{(v)})$ (that is, at the columns $-1,\ldots,-n$ of the queue vertex model), we obtain the following relation between operators acting in the space $\mathbb{C}^{n+1} \otimes V^{\otimes N}$, applied to the empty configuration $\emptyset$ at the left boundary of the cylinder: $$\label{eq:straight_stat_proof_1} \begin{split} &\big\langle \emptyset \big|\hspace{1pt} \mathfrak{Q}\left( (xq^{\frac12},u_1,\ldots,u_N );(q^{-\frac{1}{2}},s_1^{(h)},\ldots,s_N^{(h)} ) ; \mathbf{v};\mathbf{s}^{(v)}\right) \mathscr{L}^{(0N)}_{s_N^{(h)},xu_{N}^{-1}} \\&\hspace{10pt} = \big \langle \emptyset \big|\hspace{1pt} P^{(0,N)}\hspace{1pt} \mathfrak{Q}\left( (u_N,u_1,\ldots,u_{N-1},xq^{\frac12} );(s_N^{(h)},s_1^{(h)},\ldots,s_{N-1}^{(h)}, q^{-\frac{1}{2}}) ; \mathbf{v};\mathbf{s}^{(v)}\right) P^{(0,N)} . \end{split}$$ Here the permutation operator $P^{(0,N)}$ swaps the $0$-th and the $N$-th tensor components in the space $\mathbb{C}^{n+1}\otimes V\otimes \ldots\otimes V$, and is required since the operator $\mathfrak{Q}$ in the right-hand side acts in $V\otimes \ldots\otimes V\otimes \mathbb{C}^{n+1}$. Identity [\[eq:straight_stat_proof_1\]](#eq:straight_stat_proof_1){reference-type="eqref" reference="eq:straight_stat_proof_1"} represents the first step of the transformations illustrated in . In the next step, the action of $\mathscr{L}^{(0,N-1)}_{s^{(h)}_{N-1},xu_{N-1}^{-1}}$ results in the following identity: $$\begin{split} & \big \langle \emptyset \big|\hspace{1pt} P^{(0,N)}\hspace{1pt} \mathfrak{Q}\left( (u_N,u_1,\ldots,u_{N-1},xq^{\frac12} );(s_N^{(h)},s_1^{(h)},\ldots,s_{N-1}^{(h)}, q^{-\frac{1}{2}}) ; \mathbf{v};\mathbf{s}^{(v)}\right) P^{(0,N)} \mathscr{L}^{(0,N-1)}_{s^{(h)}_{N-1},xu_{N-1}^{-1}} \\&\hspace{5pt} = \big \langle \emptyset \big|\hspace{1pt} P^{(N,N-1)}P^{(0,N)} \mathfrak{Q}\left( (u_N,u_1,\ldots,u_{N-2},xq^{\frac12},u_{N-1} );(s_N^{(h)},s_1^{(h)},\ldots,s_{N-2}^{(h)}, q^{-\frac{1}{2}},s_{N-1}^{(h)}) ; \mathbf{v};\mathbf{s}^{(v)}\right) \\ &\hspace{.79\textwidth} \times P^{(0,N)}P^{(N,N-1)} . \end{split}$$ Iterating the action of the other operators $\mathscr{L}$, after $N$ total steps the horizontal parameter sequences $\mathbf{u}$ and $\mathbf{s}^{(h)}$ return back to their original states $(xq^{\frac12},\mathbf{u});(q^{-\frac{1}{2}},\mathbf{s}^{(h)})$. Here we employed the periodicity of the vertex model defining $\mathfrak{Q}$ to complete the commutation. This establishes the desired stationarity relation. ◻ # Multi-species ASEP from twisted cylinder {#sec:ASEP_matrix_products} In this section we take a continuous time limit of the twisted cylinder Markov operator $\mathfrak{T}$ defined in , and recover the known descriptions of the stationary distribution of the multi-species TASEP and ASEP on the ring from [@FerrariMartin2005], [@Prolhac_2009], and [@martin2020stationary]. ## Multi-species ASEP on the ring {#sub:mASEP} Recall that $n$ is the number of particle species (also called "types" or "colors"), and $N$ is the size of the ring $\mathbb{Z}/N\mathbb{Z}$. The state space of the multi-species ASEP consists of particle configurations $\eta=(\eta_1,\ldots,\eta_N )$ on the ring, where $\eta_k\in \left\{ 0,1,\ldots,n \right\}$ encodes the type of the particle at site $k$. The type $0$ corresponds to the empty site. For a configuration $\eta$ and each pair of neighboring sites $(k,k+1)$ (including $(N,1)$ for $k=N$), denote by $\eta^{k,k+1}$ the configuration $(\eta_1,\ldots,\eta_{k+1},\eta_k ,\ldots,\eta_N )$. That is, $\eta^{k,k+1}$ is obtained from $\eta$ by swapping the types at sites $k$ and $k+1$. **Definition 23**. The *multi-species ASEP* (*mASEP*) is a continuous time Markov chain on the space of particle configurations on the ring, with the following transition rates: $$\label{eq:ASEP_rates} \text{Rate}(\eta \rightarrow \eta^{k,k+1}) = \begin{cases} q, & \eta_k > \eta_{k+1}; \\ 1, & \eta_k < \eta_{k+1}, \end{cases}$$ where $k$ runs over $1,2,\ldots,N$, and in [\[eq:ASEP_rates\]](#eq:ASEP_rates){reference-type="eqref" reference="eq:ASEP_rates"} we assume that $\eta_k\ne \eta_{k+1}$. The multi-species ASEP depends on a single parameter $q\in[0,1)$. **Remark 24**. We use the ordering of colors in which color $n$ has the highest priority (to move from site $j+1$ to site $j$ on the ring), and color $1$ has the lowest priority. Often in the literature on multi-type interacting particle systems, e.g., in [@martin2020stationary], a reverse convention is used, in which type $1$ has the highest priority. In below, when this distinction becomes relevant, we recast all the necessary definitions from the existing literature using our color ordering conventions. Observe that mASEP preserves the number of particles of each type. We denote these type counts by $$\label{eq:mASEP_type_counts} N_m\coloneqq \sum\nolimits_{j=1}^N \mathbf{1}_{\eta_j = m}, \qquad m=1,\ldots,n.$$ We have $N_1+\ldots+N_n\le N$. In addition, throughout this section, we assume that $N_m\ge1$ for all $m=1,\ldots,n$. This assumption is very natural since if there are no particles of a given type, then the evolution of the $n$-type mASEP is the same as that of a $(n-1)$-type mASEP, where the missing type is removed entirely. Note also that at the level of queue vertex models, a violation of the assumption that $N_m\ge1$ for all $m$ leads to problems; see and . **Definition 25** (mASEP stationary distribution). When restricted to a *sector* (namely, the subset of the state space) with fixed type counts $(N_1,\ldots,N_n )$, mASEP becomes an irreducible continuous time Markov chain on a finite state space. Therefore, it admits a *unique stationary distribution* in each sector. We denote this distribution by $\mathop{\mathrm{Prob}}^{\mathrm{mASEP}}\nolimits_{N_1,\ldots,N_n }(\eta)$. It is natural to encode the states $\eta$ as basis vectors in $(\mathbb{C}^{n+1})^{\otimes N}$: $$|\eta\rangle = |\eta_1,\ldots,\eta_N \rangle = |\eta_1 \rangle\otimes \ldots \otimes |\eta_N \rangle.$$ Since all jumps under mASEP are nearest neighbor, the infinitesimal generator of mASEP written as a sum of local rate matrices as follows. Consider two possible configurations of particles $i i'$, $j j'$ at adjacent lattice sites, say, $k$ and $k+1$. Define an operator $\mathscr{M}_{loc}$ in $(\mathbb{C}^{n+1})^{\otimes 2}$ such that $$\label{eq:Mloc} \langle i,i' | \mathscr{M}_{loc} | j,j' \rangle = \big( \mathscr{M}_{loc} \big)_{i i', j j'} \coloneqq \left( \text{jump rate $i i' \rightarrow j j'$} \right).$$ The matrix element [\[eq:Mloc\]](#eq:Mloc){reference-type="eqref" reference="eq:Mloc"} is nonzero if and only if $(i,i')=(j,j')$ or $(i,i')=(j',j)$. The infinitesimal Markov generator of mASEP then has the form $$\label{eq:mASEP_generator} \mathfrak{M}_{\mathrm{mASEP}}= \sum_{j=1}^{N} \big( \mathscr{M}_{loc}\big)^{j,j+1},$$ where $\big( \mathscr{M}_{loc}\big)^{j,j+1}$ acts as $\mathscr{M}_{loc}$ on tensor factors of sites $j, j+1$, and as the identity on all other factors. Denote by $\{\mathfrak{P}_{\mathrm{mASEP}}(t)\}_{t\in \mathbb{R}_{\ge0}}$, the continuous time Markov semigroup generated by [\[eq:mASEP_generator\]](#eq:mASEP_generator){reference-type="eqref" reference="eq:mASEP_generator"}. The passage from the infinitesimal generator to this semigroup is straightforward, as the process lives on a finite state space. See the left side of for an illustration of the process. The next statement identifies mASEP as a Poisson-type limit of the twisted cylinder Markov operators. **Proposition 26**. *For any $u\in \mathbb{R}$, we have the convergence of Markov operators in $(\mathbb{C}^{n+1})^{\otimes N}$: $$\label{eq:limit_twisted_to_mASEP} \lim_{\epsilon\to0} \mathfrak{T}(u,\hspace{1pt}u(1-\epsilon))^{\lfloor (1-q) t/\epsilon \rfloor } = \mathfrak{P}_{\mathrm{mASEP}}(t), \qquad t\in \mathbb{R}_{\ge0}.$$ Here $\mathfrak{T}$ is the twisted cylinder Markov operator [\[eq:twisted_cylinder_Markov_operator\]](#eq:twisted_cylinder_Markov_operator){reference-type="eqref" reference="eq:twisted_cylinder_Markov_operator"}. See the right side of for an illustration of the limiting jump rates.* Recall that $q\in[0,1)$. One readily sees that for any $u\in \mathbb{R}$ and $\epsilon\in(0,1)$, both sides of the limiting relation [\[eq:limit_twisted_to_mASEP\]](#eq:limit_twisted_to_mASEP){reference-type="eqref" reference="eq:limit_twisted_to_mASEP"} are Markov operators with nonnegative matrix elements. *Proof outline of .* This is a standard limit of the stochastic six vertex model leading to the ASEP, see [@BCG6V], [@Aggarwal2017convergence] and also [@borodin_wheeler2018coloured Section 12.3] for its colored version. In short, in the regime $\epsilon\to0$, all paths want to follow a "staircase" motion, with occasional deviations that occur in continuous time according to independent exponential clocks. Because of how the cross vertices are organized to form the twisted cylinder Markov operator (), the staircase motion corresponds to particles staying in place. For convenience, let us reproduce the main computation. First, consider the limit of the local operators $\check{\mathscr{R}}_{z}$ [\[eq:swapping_R\_check\]](#eq:swapping_R_check){reference-type="eqref" reference="eq:swapping_R_check"}, where $z=u_1u^{-1}=1-\epsilon$ because $u_1=u(1-\epsilon)$. We have for the matrix elements [\[eq:R_matrix_nonfused\]](#eq:R_matrix_nonfused){reference-type="eqref" reference="eq:R_matrix_nonfused"}: $$\frac{1-z}{1-qz}=\frac{\epsilon}{1-q}+O(\epsilon^2), \qquad 1-\frac{1-z}{1-qz}=1-O(\epsilon),$$ and similarly for $q\frac{1-z}{1-qz}$ and $1-q\frac{1-z}{1-qz}$. Therefore, the local infinitesimal generator [\[eq:Mloc\]](#eq:Mloc){reference-type="eqref" reference="eq:Mloc"} of mASEP has the following form: $$\label{eq:Mloc_as_derivative} \mathscr{M}_{loc} = (1-q) \left.\frac{\partial}{\partial\epsilon}\right|_{\epsilon=0} \check{\mathscr{R}}_{1-\epsilon}.$$ See , right, for an illustration of how we interpret the operators $\check{\mathscr{R}}_{1-\epsilon}$ in terms of the hopping rates. In the $\epsilon\to0$ limit, the product of the operators $\check{\mathscr{R}}_{1-\epsilon}$ over all pairs of neighboring lattice sites (which is equal to the twisted cylinder operator [\[eq:twisted_cylinder_Markov_operator\]](#eq:twisted_cylinder_Markov_operator){reference-type="eqref" reference="eq:twisted_cylinder_Markov_operator"}) behaves as $Id + \frac{1}{1-q} \epsilon \; \mathfrak{M}_{\mathrm{mASEP}}$, where $Id$ is the identity matrix, and $\mathfrak{M}_{\mathrm{mASEP}}$ is defined in [\[eq:mASEP_generator\]](#eq:mASEP_generator){reference-type="eqref" reference="eq:mASEP_generator"}. This leads to the desired statement about the convergence to the mASEP Markov semigroup. ◻ {#fig:ASEP_dynamics width=".8\\textwidth"} ## Vertex model for the mASEP stationary distribution {#sub:vertex_models_mASEP} Let us now apply to mASEP. Recall that by $\langle \emptyset | \hspace{1pt} \mathfrak{Q} \hspace{1pt} | \mathbf{V} \rangle$ we denote the partition function of the queue vertex model on the cylinder introduced in . For mASEP, the queue transfer matrix $\mathfrak{Q} = \mathfrak{Q}(\mathbf{u};\mathbf{s}^{(h)}_{\frac12} ;\mathbf{v};\mathbf{s}^{(v)})$ has the parameters $$\mathbf{u}=(u,\ldots,u ), \qquad \mathbf{s}^{(h)}_{\frac12}=(q^{-1/2},\ldots,q^{-1/2} ),$$ where $\mathbf{v},\mathbf{s}^{(v)}\in \mathbb{C}^n$ are arbitrary. The $N$-tuple $\mathbf{V}=\mathbf{V}_\eta\coloneqq (\mathbf{e}_{\eta_1} \mathbf{1}_{\eta_1 \ge 1},\dots, \mathbf{e}_{\eta_N} \mathbf{1}_{\eta_N \ge 1})$ encodes the same information as the mASEP state $\eta=(\eta_1,\ldots,\eta_N )$. Indeed, this is because each of the subsets $\mathbf{V}_\eta(j)\subset \mathbb{Z}_{\ge0}^n$, $j=1,\ldots,N$, must have at most one element thanks to the spin-$\frac12$ reduction coming from $\mathbf{s}^{(h)}_{\frac12}$. We split this subsection into two parts. First, we show in that the normalized partition functions of the queue vertex model produce the mASEP stationary distribution. Then we present in a slightly modified vertex model for which all partition functions on the cylinder with right boundary $\eta$ are positive without normalization. **Proposition 27**. *With the above notation and for generic complex parameters $u,\mathbf{v},\mathbf{s}^{(v)}$, the queue vertex model partition function with boundary $\eta$ and vertex weights $\mathbb{W}_{q^{-1/2},s_m^{(v)},u/v_m}^{(-m)}$ is proportional to the stationary probability for the multi-species ASEP on the ring: $$\label{eq:queue_mASEP_stationary_proport} \mathop{\mathrm{Prob}}^{\mathrm{mASEP}}\nolimits_{N_1,\ldots,N_n }(\eta) = \frac{1}{Z^{\mathrm{mASEP}}_{N_1,\ldots,N_n }(u ;\mathbf{v};\mathbf{s}^{(v)})}\hspace{1pt} \langle \emptyset | \hspace{1pt}\mathfrak{Q}(\mathbf{u};\mathbf{s}^{(h)}_{\frac12} ;\mathbf{v};\mathbf{s}^{(v)}) \hspace{1pt}| \mathbf{V}_\eta \rangle.$$ The normalizing constant $Z^{\mathrm{mASEP}}_{N_1,\ldots,N_n }(u ;\mathbf{v};\mathbf{s}^{(v)})$ may depend on the type counts $(N_1,\ldots,N_n)$ [\[eq:mASEP_type_counts\]](#eq:mASEP_type_counts){reference-type="eqref" reference="eq:mASEP_type_counts"}, but not on the state $\eta$ within the sector determined by $(N_1,\ldots,N_n)$.* In , by "generic" we mean that the parameters must ensure that $\langle \emptyset | \hspace{1pt}\mathfrak{Q} \hspace{1pt}| \mathbf{V}_\eta \rangle$ is finite and nonzero for all $\eta$ with type counts $N_m\ge 1$, $m=1,\ldots,n$. For fixed $N$ and $n$, genericity is ensured by excluding zero sets of finitely many polynomials in $u,\mathbf{v}$, and $\mathbf{s}^{(v)}$ from the parameter space. Later in , we present concrete conditions on the parameters producing stationary measures for all $N$ and $n$. *Proof of .* Iterating , we have $$\big\langle \emptyset \big| \hspace{1pt} \mathfrak{Q} \bigl( (u_1,u,\ldots,u ); \mathbf{s}_{\frac{1}{2}}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)}\bigr) \hspace{1pt} \mathfrak{T}(u, u_1)^{\lfloor (1-q)t / \epsilon \rfloor } = \big\langle \emptyset \big| \hspace{1pt} \mathfrak{Q} \bigl( (u_1,u,\ldots,u ); \mathbf{s}_{\frac{1}{2}}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)}\bigr).$$ Setting $u_1=u(1-\epsilon)$ and sending $\epsilon\to0$ turns the power of the twisted cylinder Markov operator in the left-hand side into $\mathfrak{P}_{\mathrm{mASEP}}(t)$; see . The vertex weights in the queue vertex model with horizontal spin $\frac12$ are given in (recall that $\mathfrak{Q}$ involves the weights $\mathbb{W}_{q^{-1/2},s_m^{(v)},u/v_m}^{(-m)}$). These vertex weights are continuous in the spectral parameter $u$ for generic parameters. Therefore, we can simply take the limit $u_1\to u$ in the queue transfer matrix, which implies that $\big\langle \emptyset \big| \hspace{1pt} \mathfrak{Q} \bigl( (u,u,\ldots,u ); \mathbf{s}_{\frac{1}{2}}^{(h)}; \mathbf{v}; \mathbf{s}^{(v)}\bigr)$ is the left (row) eigenvector of the Markov semigroup $\mathfrak{P}_{\mathrm{mASEP}}(t)$ with eigenvalue $1$. Thus, it is proportional to the row vector representing the stationary distribution of mASEP, as desired. ◻ {#fig:ASEP_weights width=".8\\textwidth"} **Remark 28**. expresses the stationary probabilities of mASEP, a system depending on a single parameter $q\in[0,1)$, as normalized partition functions $\langle \emptyset | \hspace{1pt}\mathfrak{Q} \hspace{1pt}| \mathbf{V}_\eta \rangle/ Z^{\mathrm{mASEP}}_{N_1,\ldots,N_n }$ of the queue vertex model. The latter in addition depends on the parameters $u,\mathbf{v}$, and $\mathbf{s}^{(v)}$. However, by [\[eq:queue_mASEP_stationary_proport\]](#eq:queue_mASEP_stationary_proport){reference-type="eqref" reference="eq:queue_mASEP_stationary_proport"}, we see that while $u,\mathbf{v},\mathbf{s}^{(v)}$ enter the weights of the queue vertex model, these parameters do not affect the normalized partition functions. **Definition 29**. For $s\ne 0$ let us define the *mASEP gauge transformation* of the queue vertex weights: $$\label{eq:ASEP_gauge} \mathbb{W}_{q^{-1/2},s,uq^{1/2}}^{(-m),\mathrm{mASEP}_+}(\mathbf{A},k;\mathbf{C},\ell) \coloneqq (-1/s)^{\mathbf{1}_{\ell\ge1}}\hspace{1pt} \mathbb{W}_{q^{-1/2},s,uq^{1/2}}^{(-m)}(\mathbf{A},k;\mathbf{C},\ell).$$ That is, we remove the factor $(-s)$ from the weights in the right three columns in . The resulting weights make sense for $s=0$, too. The notation "$+$" in [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"} indicates that we will impose conditions on the parameters under which these weights are nonnegative. Also, denote by $$\label{eq:ASEP_gauge_normalized} \frac{\langle \emptyset | \hspace{1pt}\mathfrak{Q}^{\mathrm{mASEP}_+} \hspace{1pt}| \mathbf{V}_\eta \rangle}{ Z^{\mathrm{mASEP}_+}_{N_1,\ldots,N_n }}$$ the corresponding normalized partition function of the queue vertex model on the cylinder with the right boundary $\eta$. **Remark 30**. The weights [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"} are the queue limits (as in ) of the non-stochastic higher spin colored vertex weights defined in [@borodin_wheeler2018coloured (2.2.2)]. **Proposition 31**. *Fix the type counts $(N_1,\ldots,N_n)$ with $N_m\ge1$ for all $m$. Let the parameters satisfy $$\label{eq:positivity_parameter_dependence} 0 \le s_m^{(v)} < \frac{u}{v_m q^{1/2}}, \qquad s_m^{(v)} \frac{u}{v_m q^{1/2}} < 1, \qquad m = 1,\ldots,n.$$ Then $$\mathop{\mathrm{Prob}}^{\mathrm{mASEP}}\nolimits_{N_1,\ldots,N_n }(\eta) = \frac{ \langle \emptyset | \hspace{1pt}\mathfrak{Q}^{\mathrm{mASEP}_+} \hspace{1pt}| \mathbf{V}_\eta \rangle}{ Z^{\mathrm{mASEP}_+}_{N_1,\ldots,N_n }},$$ is the mASEP stationary distribution, and $\langle \emptyset | \hspace{1pt}\mathfrak{Q}^{\mathrm{mASEP}_+} \hspace{1pt}| \mathbf{V}_\eta \rangle>0$ for all $\eta$.* Note that conditions [\[eq:positivity_parameter_dependence\]](#eq:positivity_parameter_dependence){reference-type="eqref" reference="eq:positivity_parameter_dependence"} are written for $u/v_m$ entering the vertex weights as $\mathbb{W}_{q^{-1/2},s_m^{(v)},u/v_m}^{(-m),\mathrm{mASEP}_+}$, while for displaying the weights in it was convenient to mulitply the spectral parameter by $q^{1/2}$. *Proof of .* Assume first that $s_m^{(v)}>0$ for all $m$. Replacing the queue vertex weights with their gauge transformed versions [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"} multiplies the partition function $\langle \emptyset | \hspace{1pt}\mathfrak{Q} \hspace{1pt}| \mathbf{V}_\eta \rangle$ by $$\prod\nolimits_{m=1}^{n}(-1/s_m^{(v)})^{N_m+\ldots+N_n },$$ which depends only on the sector, but not on the configuration $\eta$. Therefore, the gauge transformation may be incorporated into the normalizing constant $Z^{\mathrm{mASEP}_+}_{N_1,\ldots,N_n }$. One readily sees that under conditions [\[eq:positivity_parameter_dependence\]](#eq:positivity_parameter_dependence){reference-type="eqref" reference="eq:positivity_parameter_dependence"} and when $s_m^{(v)}>0$ for all $m$, all vertex weights [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"} are positive (see ). This completes the proof in the case when all the $s_m^{(v)}$'s are positive. Setting some (or all) of the $s_m^{(v)}$'s to zero in the weights [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"} is allowed, and leads to a well-defined partition function $\langle \emptyset | \hspace{1pt}\mathfrak{Q}^{\mathrm{mASEP}_+} \hspace{1pt}| \mathbf{V}_\eta \rangle$. In this partition function, some of the vertex weights in vanish. To show that $\langle \emptyset | \hspace{1pt}\mathfrak{Q}^{\mathrm{mASEP}_+} \hspace{1pt}| \mathbf{V}_\eta \rangle$ is still positive and not merely nonnegative for all $\eta$, first notice that there exists $\eta$ for which $\langle \emptyset | \hspace{1pt}\mathfrak{Q}^{\mathrm{mASEP}_+} \hspace{1pt}| \mathbf{V}_\eta \rangle\ne 0$ (this verification is straightforward, and we omit it). Next, observe that $$\label{eq:ASEP_gauge_positive_Perron_Frobenius} \sum\nolimits_{\eta} \langle \emptyset | \hspace{1pt}\mathfrak{Q}^{\mathrm{mASEP}_+} \hspace{1pt}| \mathbf{V}_\eta \rangle \langle \mathbf{V}_\eta |$$ is a nonzero left (row) eigenvector with eigenvalue $1$ of the mASEP semigroup $\mathfrak{P}_{\mathrm{mASEP}}(t)$ corresponding to an irreducible continuous time Markov process on a finite state space. Therefore, [\[eq:ASEP_gauge_positive_Perron_Frobenius\]](#eq:ASEP_gauge_positive_Perron_Frobenius){reference-type="eqref" reference="eq:ASEP_gauge_positive_Perron_Frobenius"} is proportional to the Perron--Frobenius eigenvector of $\mathfrak{P}_{\mathrm{mASEP}}(t)$, which has all components positive. This completes the proof. ◻ ## Matching to multiline queues {#sub:mqueues_martin_matching} ### Original multiline queues {#subsub:mqueue_matching} First, we connect the queue vertex model $\mathfrak{Q}^{\mathrm{mASEP}_+}$ with special parameters with *multiline queues* introduced by Martin [@martin2020stationary]. He showed that the output of the latter produces the stationary distribution of mASEP on the ring. Setting $u=q^{1/2}$ and $s_m^{(v)}=0$, $v_m=1$ for all $m=1,\ldots,N$ in the vertex weights [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"} leads to the weights given in which we denote by $$\label{eq:WQ_mqueue} \mathbb{W}^{(-m),\mathrm{mq}} \coloneqq \mathbb{W}_{q^{-1/2},0,q^{1/2}}^{(-m),\mathrm{mASEP}_+}.$$ By , these weights produce positive partition functions of the queue vertex model on the cylinder. Let us connect the queue vertex model $\mathfrak{Q}^{\mathrm{mASEP}_+}$ with these particular parameters to multiline queue diagrams. These diagrams were defined in [@martin2020stationary Sections 1.1 and 3.6], and the vertex model interpretation follows from formula (3.9) in [@martin2020stationary]. For the reader's convenience, in the rest of we reproduce the main definitions and the matching of queues to our vertex models. ![Weights $\mathbb{W}^{(-m),\mathrm{mq}}$ [\[eq:WQ_mqueue\]](#eq:WQ_mqueue){reference-type="eqref" reference="eq:WQ_mqueue"}. In we match them to probabilities under multiline queues of [@martin2020stationary]. Here $m<k<\ell\le n$, and recall () that our ordering of colors is reversed compared to particle types in [@martin2020stationary].](fig_mqueue_weights.pdf){#fig:mqueue_vertex_weights width=".75\\textwidth"} Let us recast [@martin2020stationary Algorithm 2] (called *the Martin algorithm* in what follows) in the language of one column of a vertex model on the cylinder. Fix $m=1,\ldots,n$, type counts $N_m,N_{m+1},\ldots,N_n$, and assume that we have an arbitrary fixed configuration $\eta$ of paths of colors strictly larger than $m$ entering the column $(-m)$ from the left. The configuration $\eta$ has $N_i$ paths of color $i$, $i>m$. The Martin algorithm (for color $m$) samples a random new configuration $\eta'$ of paths exiting the column $(-m)$ to the right. The configuration $\eta'$ has $N_i$ paths of color $i\ge m$, and is constructed as follows. 1. Start with the empty configuration $\eta'=\{0,\ldots,0 \}$ ($N$ zeros). In addition, sample a uniformly random subset $\mathcal{J}\subset \{1,\ldots,N \}$ of sites on the cylinder of cardinality $N_m+\ldots+N_n$. Next, we randomly update $\eta'$ such that in the end $\eta_j'>0$ if and only if $j \in \mathcal{J}$. 2. For each color $i=n,n-1,\ldots,m+1$ (in this order), let $a_1^i<\ldots<a^i_{N_i}$ be the locations of paths of color $i$ in $\eta$. 1. For $j=1,\ldots,N_i$, if $\eta'_{a_j^i}=0$ and $a_j^i \in \mathcal{J}$, set $\eta'_{a_j^i}=i$ (if a path of color $i$ can come straight through, it does so). 2. Otherwise, the $j$-th path of color $i$ starts from $\eta_{a_j^i}$ and randomly chooses an exit site among yet unoccupied sites in $\mathcal{J}$ as follows. Let $a_j^i<p_1<p_2<\ldots<p_l$ (here, $p < p'$ means that as we read upwards starting from $p$, possibly wrapping around in the vertical direction, we observe $p'$ before getting back to $a_{j}^i$), where $(p_1,\ldots,p_l )$ are all sites in $\mathcal{J}$ for which at this point we have $\eta'_{p_t}=0$, $t=1,\ldots,l$. Then, set $\eta'_{p_t}=i$ with probability proportional to $q^{t-1}$. 3. Equivalently, instead of step **2b**, one can think that the color $i$ path starting from site $a_j^i$ goes up the cylinder and sequentially with probability $1-q$ picks an unoccupied site from $\mathcal{J}$ to exit, or with probability $q$ skips this site (*accepts* or *declines the service*, in queueing terminology). The path continues the motion up the cylinder until its exit, and can go around the cylinder an arbitrary number of times. If the path exits at $p_t$, then after normalization, this produces the same probability proportional to $q^{t-1}$. 3. Once all paths of all colors strictly larger than $m$ are processed, we have $N_m$ unoccupied sites in $\mathcal{J}$ left. We set $\eta'_j=m$ for all these remaining sites. To obtain the mASEP stationary distribution $\mathop{\mathrm{Prob}}^{\mathrm{mASEP}}\nolimits_{N_1,\ldots,N_n }$ (), one needs to apply the Martin algorithm for color $n$ with input $\eta(0)=\emptyset$, and get a random output $\eta(1)$. Then apply the algorithm for color $n-1$ with input $\eta(1)$, get an output $\eta(2)$, and so on. The final output $\eta(n)$ of the algorithm for color $1$ is the random configuration distributed according to the mASEP stationary distribution. ![The Martin algorithm from [@martin2020stationary] with $N=6$, $n=4$, in the column $(-m)$, where $m=2$. The set $\mathcal{J}$ is $\left\{ 1,4,5,6 \right\}$. Given this $\mathcal{J}$, the conditional probability of the configuration in the figure (according to the description with **2b'**) is proportional to $q^5$ (the color $4$ path skips five possibilities) times $q^2$ (the top color $3$ path skips two possibilities) times $1$ (the bottom color $3$ path selects the first available possibility). The boxed numbers indicate the ring sites.](martin_algo.pdf){#fig:martin_algorithm width=".4\\textwidth"} The following statement matches the output of the Martin algorithm to vertex models and essentially coincides with [@martin2020stationary Theorem 3.4]. For convenience, we reproduce it here. **Proposition 32**. *In each sector determined by the fixed type counts $(N_1,\ldots,N_n)$, the output $\eta$ of the Martin algorithm has the same distribution as the output of our queue vertex model on the cylinder with the weights $\mathbb{W}^{(-m),\mathrm{mq}}$ given in .* *Idea of proof.* This follows by matching the vertex weights in to the weights $w_i(Q | A, S)$ given in [@martin2020stationary (3.9)]. The translation from the queueing language to vertex models is straightforward and we omit it. ◻ In [@martin2020stationary], the stationarity of the output $\eta$ of the Martin algorithm under mASEP follows from the Matrix Product Ansatz. The connection between the algorithm and the Matrix Product Ansatz is essentially equivalent to . We link queue vertex models to Matrix Product Ansatz in below. **Remark 33**. While the Martin algorithm and the queue vertex model produce the same output $\eta$ (in distribution, in each sector), it remains unclear whether one can define appropriate "states" of the queueing system under the Martin algorithm such that these states are in a weight-preserving bijection (possibly up to a common proportionality constant) with states of the queue vertex model. Indeed, tracking each particle's choices as in step **2b'** of the Martin algorithm with the input as in involves the following information: 1. Track how many times the path of color $4$ wraps around the cylinder. 2. Pick a bijection between the color $3$ inputs and outputs (there are $2!$ choices in ). 3. For each of the two bijections, track how many times each of the two color $3$ paths wraps around the cylinder. Under the queue vertex model, we do not choose a bijection, and the wrapping arrows of colors $m+1,\ldots,n$ are encoded by the tuple $\mathbf{M}(-m)$. Here $\mathbf{M}(-m)_k$ is the total number of arrows of color $k$ that wrap around the ring (that is, go between the sites $N$ and $1$). To show that summing over all data in the queueing system produces the desired distribution of the output $\eta$, one seems to require the intricate argument with compatible queue-length processes; see [@martin2020stationary Section 4.2] for details. ### Alternative multiline queues and an interpolation {#subsub:mqueue_matching_alternative} Let us consider a different specialization of the queue vertex model $\mathfrak{Q}^{\mathrm{mASEP}_+}$ introduced in . Namely, set $u=1$ and $s = s_m^{(v)}=q$, $v_m=1$ for all $m=1,\ldots,N$ in [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"}, and clear the common denominators $1-q$. The resulting weights $\mathbb{W}^{(-m),\mathrm{mq\,alt}} \coloneqq (1-q)\hspace{1pt} \mathbb{W}_{q^{-1/2},q,q^{1/2}}^{(-m),\mathrm{mASEP}_+}$ are given in . By , they lead to positive partition functions on the cylinder. {#fig:alt_mqueue_weights width=".75\\textwidth"} Just as by , the weights $\mathbb{W}^{(-m),\mathrm{mq}}$ from produce the same output $\eta$ (in distribution) as the Martin algorithm, the new weights $\mathbb{W}^{(-m),\mathrm{mq\,alt}}$ shuold be related to the *alternative multiline queue model* introduced in [@martin2020stationary Section 7]. By definition, the alternative algorithm for color $m$ consists of the same steps as the algorithm described in above, except step **2a**. Instead, if $\eta'_{a^i_j}=0$ and $a^i_j\in \mathcal{J}$, then the entering color $i$ path ($i>m$) from site $a^i_j$ still has probability $q$ to go up the cylinder and not exit immediately through this site. Arguing similarly to [@martin2020stationary Section 4.2], one can show that in each sector determined by the fixed type counts $(N_1,\ldots,N_n)$, the output $\eta$ of the alternative multiline queue algorithm has the same distribution as the output of the queue vertex model on the cylinder with the weights $\mathbb{W}^{(-m),\mathrm{mq\,alt}}$ given in . This identification of the alternative multiline queues with the vertex model would resolve the conjecture from [@martin2020stationary Section 7] that the distribution of $\eta$ is stationary under the mASEP dynamics on the ring. Indeed, this is because the output of the queue vertex model with the weights $\mathbb{W}^{(-m),\mathrm{mq\,alt}}$ is already stationary under the mASEP dynamics on the ring, thanks to our general which ultimately rely on the Yang--Baxter equation for the twisted cylinder (). ![Weights $(1-s)\hspace{1pt}\mathbb{W}_{q^{-1/2},s,q^{1/2}}^{(-m),\mathrm{mASEP}_+}$ interpolating between the vertex weights related to the original and the alternative multiline queues of [@martin2020stationary].](fig_interp_mqueue_weights.pdf){#fig:interpolating_mqueue_weights width=".75\\textwidth"} If we do not specialize the parameter $s$ to $0$ or $q$, we obtain a family of queue vertex weights depending on $q$ and $s$ (see ). The output of the vertex model with these weights produces the mASEP stationary distribution (this again follows from ). The $(q,s)$-dependent weights should be related to a new multiline queue model that interpolates between the original and the alternative multiline queues: **Definition 34** (Interpolating multiline queues). Let $s\in[0,1)$, and modify the Martin algorithm (for a given color $m$) by changing step **2a** as follows. If a color $i$ path ($i>m$) enters at $a^i_j$ and $\eta'_{a^i_j}=0$ (that is, a service is immediately available), then the path exits (accepts the service) with probability $1-s$. With probability $s$, the path turns up the cylinder and skips every successive available exit (service) with probability $q$ (as prescribed by step **2b'**). All other parts of the algorithm remain the same. For $q=0$ (when mASEP becomes the multi-species TASEP), the interpolating model produces a multiline queue model with *random service assignment*. Note that for $q=0$, both the original and the alternative multiline queues become the same and are deterministic. For two colors, this deterministic model was constructed in [@angel2006stationary] to describe the stationary distribution of the two-color TASEP. It was generalized to $n$ colors in [@Ferrari_2007]. Similarly to the argument in [@martin2020stationary Section 4.2], it should be possible to identify the output of the interpolating multiline queues with that of the $(q,s)$-dependent queue vertex model on the cylinder. In below, we outline a possible connection of the latter with the Matrix Product Ansatz. ## Connection to Matrix Product Ansatz {#sub:YBE_MPA} Prior to the multiline queue realization of the mASEP stationary distribution $\mathop{\mathrm{Prob}}^{\mathrm{mASEP}}\nolimits_{N_1,\ldots,N_n }$ in [@martin2020stationary], Prolhac--Evans--Mallick [@Prolhac_2009] showed that $\mathop{\mathrm{Prob}}^{\mathrm{mASEP}}\nolimits_{N_1,\ldots,N_n }$ can be expressed in a *matrix product form*. For processes on the ring, this expression has the same format as our general trace formula [\[eq:product_trace_formula\]](#eq:product_trace_formula){reference-type="eqref" reference="eq:product_trace_formula"}, see [\[eq:product_trace_formula_ASEP\]](#eq:product_trace_formula_ASEP){reference-type="eqref" reference="eq:product_trace_formula_ASEP"} below, and includes matrices $\mathscr{X}^{\mathrm{MPA}}_{m}$, $m=0,1,\ldots,n$, indexed by available colors. Matrix product ansatz representations for stationary probabilities of stochastic interacting particle systems date back to [@Derrida1993solution]. In the single-species case, the stationary distribution on the ring is uniform, and so the Matrix Product Ansatz becomes nontrivial only for ASEP on an open interval, in which particles can hop in and out at the endpoints. This case was considered in [@Derrida1993solution]. For the two- and three-species ASEP on the ring, the matrix product approach was employed, respectively, in [@Derrida1993shock] and [@MallickMallickRajewsky1999]. A full multi-species solution on the ring appeared about ten years later in [@Prolhac_2009]. See also [@BlytheEvansSolverGuide2007] for an earlier survey of Matrix Product Ansatz applications to particle systems. In the multi-species case, the matrices $\mathscr{X}^{\mathrm{MPA}}_{m}$ entering the product ansatz are constructed by recursive tensoring from a few single-species building blocks $A,D$, and $E$ satisfying quadratic relations $$\label{eq:ADE_relations} AD-qDA= EA-qAE=(1-q)A, \qquad ED-qDE=(1-q)(E+D).$$ The matrices $A,D,E$, as well as the $\mathscr{X}_m^{\mathrm{MPA}}$'s, are infinite-dimensional, and their products, as well as the trace of $A$ times a finite product of $D$ and $E$ matrices must be well-defined. The tensoring construction of $\mathscr{X}^{\mathrm{MPA}}_{m}$ resembles the process of horizontally stacking the vertices in columns $-n,-n+1,\ldots,-1$ as in . We refer to [@Prolhac_2009 (24)--(33)] or [@martin2020stationary Section 2] for details on the tensoring construction, and omit them here. Once the matrix product probability distribution is defined through the trace in an appropriate space as $$\label{eq:product_trace_formula_ASEP} \mathop{\mathrm{Prob}}^{\mathrm{mASEP}}\nolimits_{N_1,\ldots,N_n } (\eta)= \frac{\mathop{\mathrm{Trace}} \left( \mathscr{X}^{\mathrm{MPA}}_{\eta_1}\cdots \mathscr{X}^{\mathrm{MPA}}_{\eta_N} \right)}{Z^{\mathrm{MPA}}_{N_1,\ldots,N_n }} , \qquad \eta=(\eta_1,\ldots,\eta_N ),$$ one must independently check that it is stationary under the mASEP dynamics. A key property in the argument is the existence of the so-called *hat matrices* $\widehat{\mathscr{X}}_m^{\mathrm{MPA}}$, $m=0,1,\ldots,N$, satisfying quadratic relations [@Prolhac_2009 (68)]: $$\label{eq:hat_matrices} \sum_{i,i'=0}^{n} \mathscr{X}_i^{\mathrm{MPA}} \mathscr{X}_{i'}^{\mathrm{MPA}} \big( \mathscr{M}_{loc} \big)_{i i', j j'} = \mathscr{X}_j^{\mathrm{MPA}} \widehat{\mathscr{X}}_{j'}^{\mathrm{MPA}} - \widehat{\mathscr{X}}_{j}^{\mathrm{MPA}} \mathscr{X}_{j'}^{\mathrm{MPA}} ,$$ where $\mathscr{M}_{loc}$ are the local infinitesimal rates of the mASEP, see [\[eq:Mloc\]](#eq:Mloc){reference-type="eqref" reference="eq:Mloc"}. These hat matrices are also constructed in [@Prolhac_2009] by recursive tensoring procedures. Note that our notation differs from [@Prolhac_2009] by a transposition (e.g. comparing [\[eq:hat_matrices\]](#eq:hat_matrices){reference-type="eqref" reference="eq:hat_matrices"} with formula (66) in [@Prolhac_2009]). Let us explain how the construction of the hat matrices, and identity [\[eq:hat_matrices\]](#eq:hat_matrices){reference-type="eqref" reference="eq:hat_matrices"} for vertex model partition functions directly follow from the Yang--Baxter equation. We take a concrete realization of the Matrix Product Ansatz matrices $\mathscr{X}_m^{\mathrm{MPA}}$ using our vertex models. Namely, let $\mathscr{X}_m^{\mathrm{MPA}}(u)$, $m=0,1,\ldots,n$, be operators in $V_{-n}\otimes\ldots\otimes V_{-1}$, where $V_{-m}$ has basis $|\mathbf{M}(-m)\rangle$, $\mathbf{M}(-m)\in \mathbb{Z}_{\ge0}^{n}$, $m=1,\ldots,n$. By definition, the matrix elements of $\mathscr{X}^{\mathrm{MPA}}_m$ are partition functions (with weights $\mathbb{W}_{q^{-1/2},s,u}$) on the one-row lattice $\{1\}\times \{-n,\ldots,-1 \}$ with boundary conditions $0$ and $m$ on the left and right, respectively. See for an illustration. Let $\epsilon>0$ be small. By the Yang--Baxter equation (), the matrices $\mathscr{X}^{\mathrm{MPA}}_i(u)$ and $\mathscr{X}^{\mathrm{MPA}}_{i'}(u(1-\epsilon))$ satisfy the following identity involving the elements of $R_{1-\epsilon}$ [\[eq:R_matrix_nonfused\]](#eq:R_matrix_nonfused){reference-type="eqref" reference="eq:R_matrix_nonfused"}: $$\label{eq:hat_eqn_from_R} \sum_{i,i'=0}^{n} \mathscr{X}^{\mathrm{MPA}}_i(u) \hspace{1pt} \mathscr{X}^{\mathrm{MPA}}_{i'}(u(1-\epsilon)) \cdot R_{1-\epsilon}(i,i';j',j)= \mathscr{X}^{\mathrm{MPA}}_{j}(u(1-\epsilon)) \hspace{1pt} \mathscr{X}^{\mathrm{MPA}}_{j'}(u).$$ As in the proof of , let us differentiate [\[eq:hat_eqn_from_R\]](#eq:hat_eqn_from_R){reference-type="eqref" reference="eq:hat_eqn_from_R"} with respect to $\epsilon$ at $\epsilon=0$. Denote $$\widehat{\mathscr{X}}^{\mathrm{MPA}}_j(u)\coloneqq (1-q)\hspace{1pt}u\hspace{1pt}\frac{\partial}{\partial u}\hspace{1pt}\mathscr{X}^{\mathrm{MPA}}_j(u),\qquad j=0,1,\ldots,n .$$ **Proposition 35**. *The matrices $\mathscr{X}_i^{\mathrm{MPA}}(u)$, $\widehat{\mathscr{X}}_j^{\mathrm{MPA}}(u)$ defined above with the help of the queue vertex weights $\mathbb{W}_{q^{-1/2},s,u}$ satisfy the hat matrix identity [\[eq:hat_matrices\]](#eq:hat_matrices){reference-type="eqref" reference="eq:hat_matrices"}.* *Proof.* The $\epsilon$-derivative at $\epsilon=0$ of the right-hand side of [\[eq:hat_eqn_from_R\]](#eq:hat_eqn_from_R){reference-type="eqref" reference="eq:hat_eqn_from_R"} is equal to $$- (1-q)^{-1}\widehat{\mathscr{X}}_j^{\mathrm{MPA}}(u) \mathscr{X}_{j'}^{\mathrm{MPA}}(u).$$ In the left-hand side, differentiating $\mathscr{X}_{i'}^{\mathrm{MPA}}$ and noticing that $R_{1}(i,i';j',j)=\mathbf{1}_{i=j}\mathbf{1}_{i'=j'}$, we obtain $- (1-q)^{-1} \mathscr{X}^{\mathrm{MPA}}_j(u) \widehat{\mathscr{X}}^{\mathrm{MPA}}_{j'}(u)$. This yields the second summand in the right-hand side of [\[eq:hat_matrices\]](#eq:hat_matrices){reference-type="eqref" reference="eq:hat_matrices"}. Finally, differentiating $R_{1-\epsilon}$ and using [\[eq:Mloc_as_derivative\]](#eq:Mloc_as_derivative){reference-type="eqref" reference="eq:Mloc_as_derivative"}, we recover the local infinitesimal rates of mASEP, also multiplied by $(1-q)^{-1}$. This completes the proof. ◻ Let us also note that one can "recognize" the matrix product building blocks $A,D,E$ in the vertex weights of the queue vertex model on the cylinder. This allows to insert two extra parameters into the matrices. Namely, in [@martin2020stationary], two examples of $A,D,E$ are presented, for the original and for the alternative multiline queue models discussed in above. Both these examples satisfy relations [\[eq:ADE_relations\]](#eq:ADE_relations){reference-type="eqref" reference="eq:ADE_relations"}. Informed by these two examples and using the general vertex weights $\mathbb{W}_{q^{-1/2},s,uq^{1/2}}^{(-m),\mathrm{mASEP}_+}$ [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"}, let us define $$\label{eq:ADE_12} \begin{split} A& \coloneqq \left( \begin{array}{ccccc} 1 & s & 0 & \ldots \\ 0 & q & q s & \ldots \\ 0 & 0 & q^2 & \ldots \\ \vdots & \vdots & \vdots & \ddots \\ \end{array} \right), \qquad D\coloneqq u^{-1} \left( \begin{array}{ccccc} u-s & 0 & 0 & \ldots \\ 1-q & u- sq & 0 & \ldots \\ 0 & 1-q^2 & u-sq^2 & \ldots \\ \vdots & \vdots & \vdots & \ddots \\ \end{array} \right), \\ E&\coloneqq \left( \begin{array}{ccccc} 1 & u & 0 & \ldots \\ 0 & 1 & u & \ldots \\ 0 & 0 & 1 & \ldots \\ \vdots & \vdots & \vdots & \ddots \\ \end{array} \right). \end{split}$$ One can check that these matrices also satisfy [\[eq:ADE_relations\]](#eq:ADE_relations){reference-type="eqref" reference="eq:ADE_relations"}. When $u=1$ and $s=0$, the matrices [\[eq:ADE_12\]](#eq:ADE_12){reference-type="eqref" reference="eq:ADE_12"} become the matrix product building blocks of [@Prolhac_2009], and lead to the original multiline queue model [@martin2020stationary (2.5)]. For $u=1$ and $s=q$, they correspond to the alternative model from [@martin2020stationary Section 7]. However, like for the matrices $A,D,E$ for the alternative model, it is not clear how to build product ansatz matrices for mASEP from [\[eq:ADE_12\]](#eq:ADE_12){reference-type="eqref" reference="eq:ADE_12"} by recursive tensoring. We leave this question out of scope of the present work. We remark that matrices similar to [\[eq:ADE_12\]](#eq:ADE_12){reference-type="eqref" reference="eq:ADE_12"} have occurred in the product ansatz for the open ASEP on a bounded interval (for example, see [@Derrida1993solution]). There, the extra parameters are tied to the boundary rates. # Colored stochastic $q$-Boson process from straight cylinder {#sec:qBoson} In this section we consider a specialization of the straight cylinder Markov transition operator leading to the colored stochastic $q$-Boson process on the ring [@Takeyama2015], [@borodin_wheeler2018coloured Section 12.4]. It is also called the multi-species totally asymmetric zero range process (mTAZRP) in [@ayyer2022modified]. The corresponding specialization of the queue vertex model will allow us to recover the stationary distribution of the $q$-Boson process. In the case of at most one particle of each color, we also match path configurations in the vertex model representing this stationary distribution to states of a multiline queue considered in [@ayyer2022modified Section 8]. ## Colored $q$-Boson process on the ring {#sub:qBoson_process} Let us fix the size of the ring $N$ and the number of colors $n$. Also let us fix the type counts $(N_1,\ldots,N_n )$, where $N_i\ge1$ stands for the number of particles of color $i$ in the system. The state space of the colored stochastic $q$-Boson process consists of configurations of particles at sites of the ring, where at each site there can be an arbitrary number of particles. The configurations are encoded by $$\mathbf{V}=(\mathbf{V}(1),\ldots,\mathbf{V}(N) ), \qquad \mathbf{V}(j)\in \mathbb{Z}_{\ge0}^n.$$ Here $\mathbf{V}(j)_i$ denotes the number of particles of color $i$ at site $j$, and $N_i=\sum_{j=1}^{N}\mathbf{V}(j)_i$. **Definition 36**. The *stochastic colored $q$-Boson process* depends on parameters $q\in[0,1)$ and $u_1,\ldots,u_N>0$, and evolves in continuous time as follows. A particle of color $i$ hops from site $k$ to site $k-1$ (cyclically mod $N$) according to an independent exponential clock with rate $$u_k^{-1}(1-q^{\mathbf{V}(k)_i})\hspace{1pt}q^{\mathbf{V}(k)_{[i+1,n]}}.$$ Here we used the usual notation $\mathbf{V}(k)_{[i+1,n]}=\sum_{r=i+1}^n\mathbf{V}(k)_r$. Denote by $\mathfrak{P}_{\mathrm{qBos}}(t)$, $t\in \mathbb{R}_{\ge0}$, the continuous time Markov semigroup of this stochastic process. The colored $q$-Boson process evolution is of *zero range* kind, that is, the jump from site $k$ depends only on the state of the system at site $k$. In [@ayyer2022modified] it is referred to as the *multi-species totally asymmetric zero range process*, or *mTAZRP*. The $q$-Boson process preserves the type counts $(N_1,\ldots,N_n )$. For a fixed vector of type counts, this continuous time Markov chain evolves on a finite state space and is clearly irreducible. Thus, it has a unique stationary distribution. We denote it by $\mathop{\mathrm{Prob}}^{\mathrm{qBos}}\nolimits_{N_1,\ldots,N_n}(\mathbf{V})$. Following [@borodin_wheeler2018coloured Section 12.4.3], we can identify $\mathfrak{P}_{\mathrm{qBos}}(t)$ as a certain Poisson-type continuous time limit of the straight cylinder formal Markov operator $\mathfrak{S}(x,\mathbf{u};\mathbf{s}^{(h)})$ defined in . Recall that $\mathfrak{S}$ has $N+1$ sites on the ring. However, in the degeneration to the $q$-Boson process, the distinguished site corresponding to spectral parameter $x$ (cf. ) will be empty with probability $1$, and the dynamics can be restricted to $N$ sites. Fix small $\epsilon>0$, and set the horizontal spin parameters of $\mathfrak{S}$ to $s_j^{(h)}=\epsilon$. Also let $x=-1$. With this specialization, the matrix elements of the operators $\mathscr{L}_{s_j^{(h)},xu_j^{-1}}$ entering the definition of $\mathfrak{S}$ (formula [\[eq:straight_cylinder_Markov_operator\]](#eq:straight_cylinder_Markov_operator){reference-type="eqref" reference="eq:straight_cylinder_Markov_operator"}, see also ) become as in . The operator $\mathfrak{S}$ is a formal Markov operator acting on states of the form $(c, \mathbf{V})$, where $\mathbf{V}\in \mathbb{Z}_{\ge0}^{n}$ is a state of the color $q$-Boson process, and $c \in\left\{ 0,1,\ldots,n \right\}$ corresponds to the auxiliary line (i.e., the one with the spectral parameter $x$ and the spin parameter $q^{-1/2}$; see ). {#fig:qboson_specialization width=".8\\textwidth"} **Proposition 37**. *Fix $t\in \mathbb{R}_{\ge0}$. With the parameter specialization as above, have $$\lim _{\epsilon\to0}\, \langle 0, \mathbf{V}| \hspace{1pt} \mathfrak{S}(-1,\mathbf{u};(\epsilon, \ldots,\epsilon ))^{\lfloor t/\epsilon \rfloor} \hspace{1pt} |0, \mathbf{V}'\rangle = \langle \mathbf{V}| \hspace{1pt} \mathfrak{P}_{\mathrm{qBos}}(t) \hspace{1pt} |\mathbf{V}'\rangle, \qquad \mathbf{V},\mathbf{V}'\in \mathbb{Z}_{\ge0}^n.$$ Moreover, for any $c\ge 1$ we have $\lim _{\epsilon\to0} \langle 0, \mathbf{V}| \hspace{1pt} \mathfrak{S}(-1,\mathbf{u};(\epsilon, \ldots,\epsilon ))^{\lfloor t/\epsilon \rfloor} \hspace{1pt} |c, \mathbf{V}'\rangle =0$.* Not all matrix elements of $\mathfrak{S}$ are nonnegative before the $\epsilon\to0$ limit. This is not a problem because the limiting semigroup $\mathfrak{P}_{\mathrm{qBos}}(t)$ is a nonnegative Markov semigroup, and the stationarity result (which we prove in below) is a purely algebraic statement. *Proof of .* Both statements follow from the expansions in , after the identification of the vertices in the cylinder (in ) with the stochastic $q$-Boson transitions via $$\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.6] \draw[white!45!black,line width=1.5pt,->] (-5.2,1) --++ (.4,-2); \draw[white!45!black,line width=4pt,->] (-6,0) -- (-4,0); \node[left] at (-6,0) {\small $\mathbf{A}$}; \node[right] at (-4,0) {\small $\mathbf{C}$}; \node[above] at (-5.2,1) {\small $k$}; \node[below] at (-4.8,-1) {\small $\ell$}; \node[left] at (-2,0) {\small $=$}; \draw[white!45!black,line width=1.5pt,->] (-1,0) -- (1,0); \draw[white!45!black,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\small $k$}; \node[right] at (1,0) {\small $\ell$}; \node[below] at (0,-1) {\small $\mathbf{A}$}; \node[above] at (0,1) {\small $\mathbf{C}$}; \node[above] at (2.2,0) { \phantom{.} }; \end{tikzpicture},$$ where in the right-hand side the time is continuous and increases in the upward direction. We see that the auxiliary line may become occupied with probability $O(\epsilon)$, and then instantaneously becomes free again with probability $1+O(\epsilon)$. This means that the vertices of the type $(\mathbf{A},k;\mathbf{A},k)$ are not present in the limit. All other probabilities of order $O(\epsilon)$ in give rise to the corresponding colored $q$-Boson transitions, which leads to the first claim. To get the second claim, observe that with probability going to $1$ in the limit as $\epsilon\to0$, the auxiliary line is not occupied. This completes the proof. ◻ ## Vertex model for the $q$-Boson stationary distribution {#sub:qBoson_stationary} The convergence of together with the general stationarity result () allows us to express the stationary distribution $\mathop{\mathrm{Prob}}^{\mathrm{qBos}}\nolimits_{N_1,\ldots,N_n}(\mathbf{V})$ of the colored $q$-Boson process as a vertex model partition function. To get a queue vertex model on the cylinder with nonnegative vertex weights, we take a certain limit in the vertical parameters $\mathbf{v},\mathbf{s}^{(v)}$. As a first step, let us consider the following degeneration of the queue vertex weights $\mathbb{W}_{s_1,s_2,u}^{(-m)}$ [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"}: **Lemma 38**. *We have $$\label{eq:q_boson_weights_degeneration_vertex_s_dependent} \begin{split} & ( u; q)_{\infty}^{-1}\cdot \lim_{s_1\to 0} \mathbb{W}_{s_1,s,us_1/s}^{(-m)}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) = \mathbf{1}_{\mathbf{A}+\mathbf{B}=\mathbf{C}+\mathbf{D}} \cdot \mathbf{1}_{D_1=\ldots=D_{m-1}=0} \\&\hspace{20pt}\times \sum_{\mathbf{P}} (s^2/u ; q)_{|\mathbf{P}|} (s^2)^{|\mathbf{B}|-|\mathbf{P}|} u^{|\mathbf{D}|-|\mathbf{B}|+|\mathbf{P}|} \hspace{1pt} q^{\sum_{1\le i<j\le n}\left(B_i-P_i\right) P_j} \prod_{i=1}^n \frac{(q;q)_{B_i}}{(q;q)_{P_i}(q;q)_{B_i-P_i}} \\&\hspace{25pt}\times \hspace{1pt}q^{\sum_{m\le i<j\le n} D_i (C_j-P_j)} \hspace{1pt} \frac{ 1} {(q;q)_{D_m}} \prod_{i=m+1}^n \frac{(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}}, \end{split}$$ where the sum is over $\mathbf{P}\in \mathbb{Z}_{\ge0}^n$ with $0\le P_i\le \min (B_i,C_i)$ for all $i$, and $$\label{eq:q_boson_weights_degeneration_vertex} \begin{split} & ( u; q)_{\infty}^{-1}\cdot \lim_{s\to0}\Bigl(\hspace{1pt}\lim_{s_1\to 0} \mathbb{W}_{s_1,s,us_1/s}^{(-m)}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) \Bigr) = \mathbf{1}_{\mathbf{A}+\mathbf{B}=\mathbf{C}+\mathbf{D}} \cdot \mathbf{1}_{D_1=\ldots=D_{m-1}=0}\cdot \prod_{i=1}^m\mathbf{1}_{B_i\le C_i} \\&\hspace{130pt}\times u^{|\mathbf{D}|}\hspace{1pt} q^{\sum_{m\le i<j\le n} D_i (C_j-B_j)} \hspace{1pt} \frac{ 1} {(q;q)_{D_m}} \prod_{i=m+1}^n \frac{(q;q)_{C_i-B_i+D_i}}{(q;q)_{C_i-B_i}(q;q)_{D_i}}. \end{split}$$* *Proof.* We have from [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"}: $$\begin{split} & \mathbb{W}_{s_1,s,us_1/s}^{(-m)}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) = \mathbf{1}_{\mathbf{A}+\mathbf{B}=\mathbf{C}+\mathbf{D}} \cdot \mathbf{1}_{D_1=\ldots=D_{m-1}=0} \cdot \frac{( u; q)_{\infty}}{(s_1^2 u ; q)_{\infty}} \\&\hspace{20pt}\times \sum_{\mathbf{P}} \frac{(s^2/u ; q)_{|\mathbf{P}|} (s_1^2u/s^2 ; q)_{|\mathbf{B}-\mathbf{P}|}} {(s_1^2 ; q)_{|\mathbf{B}|}}\hspace{1pt} \hspace{1pt} q^{\sum_{1\le i<j\le n}\left(B_i-P_i\right) P_j} \prod_{i=1}^n \frac{(q;q)_{B_i}}{(q;q)_{P_i}(q;q)_{B_i-P_i}} \\&\hspace{25pt}\times (s^2)^{|\mathbf{B}|-|\mathbf{P}|} u^{|\mathbf{D}|-|\mathbf{B}|+|\mathbf{P}|} \hspace{1pt}q^{\sum_{m\le i<j\le n} D_i (C_j-P_j)} \hspace{1pt} \frac{ (s_1^2 ; q)_{|\mathbf{D}|}} {(q;q)_{D_m}} \prod_{i=m+1}^n \frac{(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}}. \end{split}$$ Sending $s_1\to0$ immediately leads to [\[eq:q_boson_weights_degeneration_vertex_s\_dependent\]](#eq:q_boson_weights_degeneration_vertex_s_dependent){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex_s_dependent"}. Further letting $s\to0$, we see that $\mathbf{P}=\mathbf{B}$, for otherwise the factor $(s^2)^{|\mathbf{B}|-|\mathbf{P}|}$ vanishes. This eliminates the summation over $\mathbf{P}$ and produces the desired expression [\[eq:q_boson_weights_degeneration_vertex\]](#eq:q_boson_weights_degeneration_vertex){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex"} together with the indicator that $B_i\le C_i$ for all $i$. ◻ We denote the right-hand side of [\[eq:q_boson_weights_degeneration_vertex_s\_dependent\]](#eq:q_boson_weights_degeneration_vertex_s_dependent){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex_s_dependent"} by $\mathbb{W}_{s,u}^{(-m),\mathrm{qBos}}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$. The right-hand side of [\[eq:q_boson_weights_degeneration_vertex\]](#eq:q_boson_weights_degeneration_vertex){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex"} is the $s=0$ degeneration of [\[eq:q_boson_weights_degeneration_vertex_s\_dependent\]](#eq:q_boson_weights_degeneration_vertex_s_dependent){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex_s_dependent"}. The weights $\mathbb{W}_{s,u}^{(-m),\mathrm{qBos}}$ are nonnegative for $q\in[0,1)$ and $u>s^2\ge 0$. **Definition 39**. Fix parameters $\mathbf{u}=(u_1,\ldots,u_N )$, $\mathbf{y}=(y_1,\ldots,y_n)$, and $\mathbf{s}^{(v)}=(s_1^{(v)},\ldots,s_n^{(v)} )$ such that $$\label{eq:s_u_y_conditions_qBoson} 0\le (s_m^{(v)})^2<u_iy_m,\qquad i=1,\ldots,N ,\quad m=1,\ldots,n .$$ Let $\mathfrak{Q}^{\mathrm{qBos}}(\mathbf{u};\mathbf{y};\mathbf{s}^{(v)})$ denote the queue transfer matrix on the $n\times N$ cylinder as in , where the vertex weight at each site $(-m,j)$ is $\mathbb{W}_{s_m^{(v)},u_jy_m}^{(-m),\mathrm{qBos}}$. The vertex model of has nonnegative weights. Note that its partition functions $\langle \emptyset |\hspace{1pt}\mathfrak{Q}^{\mathrm{qBos}}(\mathbf{u};\mathbf{y};\mathbf{s}^{(v)})\hspace{1pt} |\mathbf{V}\rangle$ involve infinite sums over paths winding around the cylinder. Similarly to , we see that these sums are convergent when $\mathbf{V}$ has all type counts $N_i$, $i=1,\ldots,n$, at most $1$. **Proposition 40**. *For any type counts $(N_1,\ldots,N_n )$, $N_i\ge1$, and the parameters $\mathbf{y},\mathbf{s}^{(v)}$ satisfying [\[eq:s_u\_y_conditions_qBoson\]](#eq:s_u_y_conditions_qBoson){reference-type="eqref" reference="eq:s_u_y_conditions_qBoson"}, the stationary distribution of the colored $q$-Boson process with parameters $\mathbf{u}$ has the form $$\label{eq:qBoson_stationary_proposition} \mathop{\mathrm{Prob}}^{\mathrm{qBos}}\nolimits_{N_1,\ldots,N_n}(\mathbf{V}) = \frac{\langle\emptyset | \hspace{1pt} \mathfrak{Q}^{\mathrm{qBos}}(\mathbf{u};\mathbf{y};\mathbf{s}^{(v)}) \hspace{1pt}| \mathbf{V}\rangle}{Z_{N_1,\ldots,N_n }^{\mathrm{qBos}}}.$$ The normalizing constant $Z_{N_1,\ldots,N_n }^{\mathrm{qBos}}$ depends on the parameters and the type counts, but not on the state $\mathbf{V}$ within the sector determined by $(N_1,\ldots,N_n)$.* *Proof.* We use (in particular, recall the queue vertex model on the cylinder interacting with the straight cylinder Markov operator as illustrated in ). Let us choose the parameters of the queue vertex model $$\mathfrak{Q}=\mathfrak{Q}\left( (xq^{\frac12},\mathbf{u});(q^{-\frac{1}{2}},\mathbf{s}^{(h)}) ; \mathbf{v};\mathbf{s}^{(v)}\right)$$ as $$x=-1,\qquad s_j^{(h)}=\epsilon\to 0, \qquad v_m=\frac{s_m^{(v)}}{\epsilon\hspace{1pt}y_m}$$ for all $1\le j\le N$, $1\le m\le n$. By , sending $\epsilon\to0$ turns the weight at each site $(-m,j)$ of this queue vertex model on the cylinder into $(u_jy_m;q)_\infty\mathbb{W}_{s_m^{(v)},u_j y_m}^{(-m),\mathrm{qBos}}$. The overall factor $\prod_{j=1}^{N}\prod_{m=1}^n(u_jy_m;q)_{\infty}$ is absorbed into the normalizing constant, and thus we can ignore it. At the sites $(-m,0)$, before the limit we have the weights $\mathbb{W}_{q^{-1/2},s_m^{(v)},-\epsilon q^{1/2}y_m/s_m^{(v)}}^{(-m)}$. Up to reparametrization, these are the same weights as in . Sending $\epsilon\to0$ (that is, $-su\to 0$ in the notation ), we see that $$\mathbb{W}_{q^{-1/2},s_m^{(v)},-\epsilon q^{1/2}y_m/v_m^{(v)}}^{(-m)} (\mathbf{A},0;\mathbf{A},0)\to 1, \qquad \mathbb{W}_{q^{-1/2},s_m^{(v)},-\epsilon q^{1/2}y_m/v_m^{(v)}}^{(-m)} (\mathbf{A},0;\mathbf{A}_k^{-},k)\to 0.$$ This implies that the auxiliary line (i.e., the one with the spin parameter $q^{-1/2}$) is occupied with weight going to $0$ as $\epsilon\to0$. Therefore, we can remove this auxiliary line from the model on the cylinder as follows: $$\label{eq:going_from_Nplus1_to_N_in_the_model} \lim_{\epsilon\to0}\hspace{1pt} \big\langle 0,\emptyset \big| \hspace{1pt} \mathfrak{Q}\left( (xq^{\frac12},\mathbf{u});(q^{-\frac{1}{2}},\mathbf{s}^{(h)}) ; \mathbf{v};\mathbf{s}^{(v)}\right) \hspace{1pt} \big| 0, \mathbf{V} \big\rangle = \langle \emptyset |\hspace{1pt}\mathfrak{Q}^{\mathrm{qBos}} (\mathbf{u};\mathbf{y};\mathbf{s}^{(v)}) \hspace{1pt} | \mathbf{V} \rangle,$$ where $\mathfrak{Q}^{\mathrm{qBos}}(\mathbf{u};\mathbf{y})$ is defined before the proposition. Arguing as in the proof of , we can take the limit as $\epsilon\to0$ simultaneously in the queue vertex model and in the straight cylinder Markov operator. Before the limit, these operators satisfy the general stationarity relation of . By , the straight cylinder Markov operator converges as $\epsilon\to0$ (in the Poisson-type continuous time limit) to the Markov semigroup $\mathfrak{P}_{\mathrm{qBos}}(t)$. The limit of the general stationarity relation yields $$\begin{aligned} \langle \emptyset |\hspace{1pt}\mathfrak{Q}^{\mathrm{qBos}} (\mathbf{u};\mathbf{y}; & \mathbf{s}^{(v)}) \mathfrak{P}_{\mathrm{qBos}}(t) \hspace{1pt} | \mathbf{V} \rangle \\ &= \lim _{\epsilon\to0}\, \big\langle 0,\emptyset \big| \hspace{1pt} \mathfrak{Q}\left( (xq^{\frac12},\mathbf{u});(q^{-\frac{1}{2}},\mathbf{s}^{(h)}) ; \mathbf{v};\mathbf{s}^{(v)}\right) \hspace{1pt} \mathfrak{S}(-1,\mathbf{u};(\epsilon, \ldots,\epsilon ))^{\lfloor t/\epsilon \rfloor} \hspace{1pt} |0, \mathbf{V}\rangle \\ &= \lim _{\epsilon\to0}\, \big\langle 0,\emptyset \big| \hspace{1pt} \mathfrak{Q}\left( (xq^{\frac12},\mathbf{u});(q^{-\frac{1}{2}},\mathbf{s}^{(h)}); \mathbf{v};\mathbf{s}^{(v)}\right) |0, \mathbf{V}\rangle \\ &= \langle \emptyset |\hspace{1pt}\mathfrak{Q}^{\mathrm{qBos}} (\mathbf{u};\mathbf{y}; \mathbf{s}^{(v)}) | \mathbf{V} \rangle .\end{aligned}$$ This completes the proof. ◻ **Remark 41**. While the quantities in the right-hand side of [\[eq:qBoson_stationary_proposition\]](#eq:qBoson_stationary_proposition){reference-type="eqref" reference="eq:qBoson_stationary_proposition"} seem to depend on $\mathbf{y}$ and $\mathbf{s}^{(v)}$, implies that they are independent of these extra parameters. This observation is parallel to the mASEP situation (see ). ## Matching to multiline queues {#sub:qBoson_multiline} In [@ayyer2022modified Section 8], a multiline queue model for the stationary distribution of the colored $q$-Boson process is presented. Let us match this model to our queue vertex model $\mathfrak{Q}^{\mathrm{qBos}}$ on the cylinder, specialized to $s_m^{(v)}=0$, $1\le m\le n$ (that is, with the simpler product-form weights [\[eq:q_boson_weights_degeneration_vertex\]](#eq:q_boson_weights_degeneration_vertex){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex"}). As in [@ayyer2022modified], we restrict our attention to the simpler *strict* case when, by definition, there is at most one particle of each color. First, we recall the definition of a $q$-Boson multiline queue and its weight. We replace the parameter $t$ from [@ayyer2022modified] by our $q$, and adjust the notation of integer indices, spectral parameters, and the direction of the ring to match the conventions used throughout our paper. ![A multiline diagram () with weight $q^{3}u_1u_2^2u_3^2u_4^2u_5$. Here the refusal statistic $3$ combines $R_{3}=1$ (label $3$ in column $-2$ is "between" the positions of labels $4$ in columns $-3$ and $-2$, in the sense described after Equation [\[eq:refusal_statistic_m\]](#eq:refusal_statistic_m){reference-type="eqref" reference="eq:refusal_statistic_m"}) and $R_{2}=2$ (labels $1$ and $3$ in column $-1$ are between the labels $4$ in columns $-2$ and $-1$). This is the same diagram as in examples in [@ayyer2022modified Section 8], but rotated by $90^\circ$ and with the direction of the ring reversed (to match our vertex model). Here the size of the ring is $N=5$, and the number of colors is $n=4$.](mqueue_qboson.pdf){#fig:mqueue_qboson width=".4\\textwidth"} **Definition 42** ([@ayyer2022modified]). A multiline diagram is an assignment of the labels from $\{1,\ldots,n \}$ to the vertices of a cylinder $\{-n,\ldots,-1 \}\times(\mathbb{Z}/N\mathbb{Z})$, satisfying 1. Each vertex $(-m,j)$ is assigned a multiset of labels. 2. In column $(-m)$, all labels are from $\{m,m+1,\ldots,n \}$. 3. The combined multiset of all labels in column $(-m)$ is obtained from the multiset of labels in column $-(m+1)$, together with some new labels of type $m$. 4. (*strict condition*) Each label $m$, $1\le m\le n$, appears at most once in each of the columns $-m,-(m-1),\ldots,-1$. The weight of a multiline diagram is, by definition, $q^\mathfrak{R} u_1^{c_1}\ldots u_N^{c_N}$, where $c_j$ is the total number of labels assigned to the row $j$, and $\mathfrak{R}$ is the *refusal statistic* defined as follows. Let $$\label{eq:refusal_statistic_m} R_{m}\coloneqq \sum_{m-1\le k<\ell\le n} \mathbf{1}_{p_\ell(-(m-1))> p_k(-(m-1))> p_\ell(-m)},$$ where $p_r(-m)$ is the position of the label $r$ in column $-m$, and the event $a>b>c$ means that, reading along the ring in the downward direction (corresponding to decreasing positions $j$), the label $b$ is strictly between $a$ and $c$. This includes the case $a=c\ne b$; what this means for a corresponding term in the sum is $p_{\ell}(-m) = p_\ell(-(m-1))$, and we think of this as $p_\ell$ making a full loop around the ring to get to its position at $-(m-1)$. Then we set $\mathfrak{R}\coloneqq \sum_{m=2}^{n}R_{m}$. See for an illustration. Given a multiline diagram, associate to it a path configuration on the cylinder with vertex weights $\mathbb{W}_{u_j}^{(-m),\mathrm{qBos}}$ at each vertex $(-m,j)$, and such that the multiset of labels at $(-m,j)$ is exactly the colors of the paths exiting this vertex. Recall that we usually denote the latter multiset of colors by $\mathbf{D}\in \mathbb{Z}_{\ge0}^{n}$. Knowing $\mathbf{D}$ at each vertex is enough to reconstruct the whole path configuration on the cylinder, up to unknown windings of paths around the cylinder. In this way, one multiline diagram corresponds to many configurations of the queue vertex model $\mathfrak{Q}^{\mathrm{qBos}}$ on the cylinder. **Proposition 43**. *Let there be exactly one particle of each color $m$, $m=1,\ldots,n$. Then the mapping between multiline diagrams and configurations of the queue vertex model $\mathfrak{Q}^{\mathrm{qBos}}(\mathbf{u};\mathbf{1};\mathbf{0})$ (that is, $y_i=1$ and $s_i^{(v)}=0$ for all $i$) described before the proposition is weight-preserving. That is, the sum of weights of all vertex model configurations over the winding of the paths around the cylinder is proportional to the weight of the corresponding multiline diagram. The proportionality constant depends on the parameters of the model, but not on the particle configuration.* If there are no particles of some color, then the sum of the vertex model weights might diverge, cf. . On the other hand, we consider the multiline queues for at most one particle of each color. This leads to the restriction in . *Proof of .* It suffices to fix $m$ and consider the behavior in the column $(-m)$. For a configuration of the queue vertex model in this column let the arrow configurations at each vertex $(-m,k)$ be $\mathbf{A}^{(k)},\mathbf{B}^{(k)},\mathbf{C}^{(k)},\mathbf{D}^{(k)}$. The corresponding multiline diagram contains information about $\mathbf{B}^{(k)},\mathbf{D}^{(k)}$, but not about $\mathbf{A}^{(k)},\mathbf{C}^{(k)}$. Let us fix $\mathbf{B}^{(k)},\mathbf{D}^{(k)}$ for all $k=1,\ldots,N$, and sum over $\mathbf{A}^{(k)},\mathbf{C}^{(k)}$, $k=1,\ldots,N$. The resulting sum must be equal to the weight of column $-m$ in the corresponding multiline diagram. With this data fixed, out of all allowed configurations of the vertices in column $(-m)$, there is one in which $C_j^{(N)}$ is minimal for each $j > m$. Fixing $\mathbf{C}^{(N)}$ allows one to reconstruct the whole vertex model configuration in column $(-m)$ in a unique way. Denote this minimal configuration by $\mathbf{C}^{(k), \min}$, and let $E_j^{(k)} \coloneqq C_j^{(k), \min} - B_j^{(k)}$, $j = m+1,\dots, n$. The product of the vertex weights [\[eq:q_boson_weights_degeneration_vertex\]](#eq:q_boson_weights_degeneration_vertex){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex"} in column $(-m)$, summed over all allowed configurations, is proportional to (using the fact that $D_j^{(k)} \in \{0,1\}$ for all $j, k$) $$\label{eq:mqueue_qboson_proof} \begin{split} & \sum_{a_{m+1}=0}^\infty \cdots \sum_{a_n = 0}^{\infty} \prod_{k=1}^N \Biggl(\hspace{1pt} u_k^{|\mathbf{D}^{(k)}|} \hspace{1pt} q^{\sum_{m\le r < s\le n} D_r^{(k)} (E_s^{(k)}+a_s)} \prod_{j=m+1}^n \frac{(q;q)_{a_j+E_j^{(k)}+D^{(k)}_j}} {(q;q)_{a_j+E_j^{(k)}}(q;q)_{D^{(k)}_j}} \Biggr) \\ &\hspace{5pt}= \Biggl(\hspace{1pt}\prod_{k=1}^N u_k^{|\mathbf{D}^{(k)}|} \Biggr) \sum_{a_{m+1}=0}^\infty \cdots \sum_{a_n = 0}^{\infty} \Biggl(\hspace{1pt}\prod_{k=1}^N q^{\sum_{m\le r < s\le n} D_r^{(k)} (E_s^{(k)}+a_s)}\Biggr) \Biggl(\hspace{1pt} \prod_{j=m+1}^n \prod_{k\colon D_j^{(k)}=1} \frac{1-q^{1+a_j+E_j^{(k)}}}{1-q} \Biggr). \end{split}$$ Observe that $B_j^{(k)}\le C_j^{(k)}$ for all $j$ and $k$ (see [\[eq:q_boson_weights_degeneration_vertex\]](#eq:q_boson_weights_degeneration_vertex){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex"}). This implies (by arrow conservation, since $A_j^{(k)}, B_j^{(k)}, C_j^{(k)}, D_j^{(k)} \in \{0, 1\}$) that if $D_j^{(k)}=1$, then $E_j^{(k)}=0$. For each $j=m+1,\ldots,n$, we either have $D_j^{(k)}=0$ or $D_j^{(k)}=1$, and there exists exactly one $k = k_j$ for which $D_j^{(k)}=1$. Thus, inside the summations we have $$\Biggl(\hspace{1pt}\prod_{k=1}^N \prod_{j=m+1}^n q^{\sum_{m\le r < j} D_r^{(k)} (E_j^{(k)}+a_j)}\Biggr) \Biggl(\hspace{1pt} \prod_{j=m+1}^n \frac{1-q^{1+a_j}}{1-q} \Biggr).$$ As a result, the sum over $a_j$ becomes $$q^{ \sum_{k=1}^N( D_m^{(k)}+\ldots+D_{j-1}^{(k)} )\hspace{1pt}E_j^{(k)}} \sum_{a_j=0}^{\infty} q^{a_j \sum_{k=1}^N( D_m^{(k)}+\ldots+D_{j-1}^{(k)} )} \hspace{1pt} \frac{1-q^{1+a_j}}{1-q} = C^{[j]}_{N_1,\ldots,N_n }\hspace{1pt} q^{ \sum_{k=1}^N( D_m^{(k)}+\ldots+D_{j-1}^{(k)} )\hspace{1pt}E_j^{(k)}} ,$$ where $C^{[j]}_{N_1,\ldots,N_n }$ does not depend on the particular multiline diagram but only on the type counts $(N_1,\ldots,N_n )$. Indeed, $\sum_{k=1}^N( D_m^{(k)}+\ldots+D_{j-1}^{(k)} )$ is the total number of colors $i$, $m\le i\le j-1$, leaving column $(-m)$. Thus, we can continue $$\label{eq:mqueue_qboson_proof_2} \eqref{eq:mqueue_qboson_proof}= C_{N_1,\ldots,N_n} \Biggl(\hspace{1pt}\prod_{k=1}^N u_k^{|\mathbf{D}^{(k)}|} \Biggr) \prod_{k=1}^{N} q^{\sum_{m\le i<j\le n}D_i^{(k)}E_j^{(k)}},$$ where $C_{N_1,\ldots,N_n}$ also depends only on the type counts. Note that $D_i^{(k)}, E_j^{(k)}\in\left\{ 0,1 \right\}$. One can readily verify that each pair $m\le i<j\le n$ such that $D_i^{(k)}= E_j^{(k)}=1$ corresponds to an indicator equal to one in the definition of $R_m$ [\[eq:refusal_statistic_m\]](#eq:refusal_statistic_m){reference-type="eqref" reference="eq:refusal_statistic_m"}. In particular, note that the indicator $\mathbf{1}_{B_j \leq C_j}$ in the weights [\[eq:q_boson_weights_degeneration_vertex\]](#eq:q_boson_weights_degeneration_vertex){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex"} prevents a path from passing straight through without any winding. This behavior is accounted for in [\[eq:mqueue_qboson_proof\]](#eq:mqueue_qboson_proof){reference-type="eqref" reference="eq:mqueue_qboson_proof"}, and corresponds to the fact that the case $a = c \neq b$ counts towards the refusal statistic $\mathfrak{R}$ (see the discussion after its definition [\[eq:refusal_statistic_m\]](#eq:refusal_statistic_m){reference-type="eqref" reference="eq:refusal_statistic_m"}). Thus, the power of $q$ in [\[eq:mqueue_qboson_proof_2\]](#eq:mqueue_qboson_proof_2){reference-type="eqref" reference="eq:mqueue_qboson_proof_2"} is exactly the same as the component $R_m$ of the refusal statistic $\mathfrak{R}$. The powers of the $u_j$'s also match the ones for the multiline diagrams. This completes the proof. ◻ Let us make two final remarks in this section. First, [@ayyer2022modified] does not explicitly define the weights of general (not necessarily strict) multiline queue diagrams. We may use $\mathfrak{Q}^{\mathrm{qBos}}(\mathbf{u};\mathbf{1};\mathbf{0})$ and sum over the winding of the paths around the cylinder to define weights of general multiline diagrams. We conjecture that these weights coming from the vertex model should directly correspond to the tableau process of [@ayyer2022modified Section 4], but do not check this here. Second, the Yang--Baxter equation for the queue vertex model () should allow to directly show the symmetry of the stationary distribution in the parameters $u_j$. More precisely [@ayyer2022modified Proposition 7.2], for any $K$, the distribution of the configuration at sites $\{1,\ldots,K \}$ of the ring is symmetric in the parameters $u_{K+1},\ldots,u_N$. Moreover, using the Yang--Baxter equation and couplings similarly to [@petrov2022rewriting], it should be possible to establish the stronger symmetry of the distributions of the whole trajectories of the colored $q$-Boson system. This stronger property is proven only for $q=0$ [@ayyer2022modified Theorem 7.14]. We leave these two questions for future work. # Colored $q$-PushTASEP from straight cylinder {#sec:qPush} This section considers another specialization of the straight cylinder Markov transition operator leading to the colored $q$-PushTASEP. We also present a vertex model on the cylinder producing its stationary distribution. Our argument here is very similar to above. The colored $q$-PushTASEP is a degeneration of the colored stochastic higher spin six vertex model and was introduced in [@borodin_wheeler2018coloured Section 12.5]. Throughout the section, we assume that $q\in (0,1)$ and fix a positive integer $\mathsf{P}$. As usual, let $N$ be the size of the ring, and $n$ be the number of colors. The colored $q$-PushTASEP depends on positive parameters $\mathbf{u} = (u_1,\dots, u_N)$. **Definition 44**. The state space of the colored $q$-PushTASEP is the set of particle configurations on the ring. At any site, there can be at most $\mathsf{P}$ particles. Particles of the same color are indistinguishable. Let $V_{\mathsf{P}}$ be the vector space with the basis $| \mathbf{V}\rangle$, where $\mathbf{V}\in \mathbb{Z}_{\ge0}^n$ with $|\mathbf{V}|\le \mathsf{P}$. The states of the colored $q$-PushTASEP can be identified with the basis vectors of $V_{\mathsf{P}}^{\otimes N}$. The $q$-PushTASEP evolves in continuous time as follows. Let $\mathbf{A}\in \mathbb{Z}_{\ge0}^{n}$ be the configuration of particles at a site $k$. For each $j=1,\ldots,n$, a particle of type $j$ *activates* and instantaneously leaves the site $k$ (moving toward $k+1$) with the rate $u_k^{-1}(q^{-A_j} - 1)\hspace{1pt} q^{\mathsf{P}-A_{[j+1, n]} }$. The active particle triggers other instantaneous updates of the configuration according to the following rules. Let $\mathbf{B}\in \mathbb{Z}_{\ge0}^n$ be the configuration of particles at a site $k'$. Suppose that an activated particle of type $c$ arrives at $k'$. Then the following happens: 1. It deactivates and stays at $k'$ with probability $1- q^{\mathsf{P}-|\mathbf{B}|}$, then the update ends. 2. It deactivates and stays at $k'$, but causes the activation of another particle from $k'$ (which then moves towards $k'+1$) of type $d < c$ with probability $(q^{-B_d}-1) \hspace{1pt}q^{\mathsf{P}-B_{[d+1, n]} }$, and the update continues. 3. It remains active and moves on to site $k' + 1$ with probability $q^{\mathsf{P}- B_{[c,n]} }$, and the update continues. All particle moves from $j$ to $j+1$ are considered cyclically mod $N$. Denote the Markov semigroup of the colored $q$-PushTASEP by $\mathfrak{P}_{\mathrm{qPush}}(t)$, $t\in \mathbb{R}_{\ge0}$. As usual, by $(N_1,\ldots, N_n)$ we denote the type counts in the configuration, which are preserved by the $q$-PushTASEP dynamics. When restricted to a sector determined by $(N_1,\ldots,N_n)$, the colored $q$-PushTASEP is an irreducible continuous time Markov chain on a finite state space. Therefore, it admits a unique stationary distribution which we denote by $\mathop{\mathrm{Prob}}^{\mathrm{qPush}}\nolimits_{N_1,\ldots,N_n }(\mathbf{V})$. **Remark 45** (Frog model). Let us discuss the most degenerate version of the colored $q$-PushTASEP, namely, when $q=0$, $\mathsf{P}=1$, and $u_k=1$ for all $k$. In this case, each particle at any site $k$ can be activated at rate $1$, and moves from $k$ to $k+1$. Then the instantaneous update proceeds as follows: 1. If an active particle arrives at an empty site, it deactivates and stays there, and the update ends. 2. If a particle of type $c$ arrives at a site $k'$ with an existing particle of type $d < c$, then the type $c$ particle stays at $k'$ and displaces the type $d$ particle, which now becomes active. 3. Finally, if a particle of type $c$ arrives at a site $k'$ with an existing particle of type $d \ge c$, then the type $c$ particle moves through to site $k'+1$. The update continues with the type $c$ particle. This process is a particular case of the *frog model* [@bukh2019periodic] related to the problem of the longest common subsequence of a random and a periodic word. Our particular case corresponds to the periodic word with all letters distinct. More general periodic words lead to the simultaneous activation of particles at several sites. The stationary distribution of the frog model was constructed (in the particular case of distinct letters) in [@bukh2019periodic Section 4]. The colored $q$-PushTASEP is a degeneration of the straight cylinder formal Markov operator. Thus, its stationary distribution is accessible through the corresponding limit transition from the queue vertex model on the cylinder. These limits are very similar to the $q$-Boson case (), so we will only provide pictorial illustrations and brief explanations. A queue vertex model leading to the $q$-PushTASEP stationary distribution must have finite spin rows (with the horizontal spin parameters $q^{-\mathsf{P}/2}$). We reverse the direction of the straight cylinder operator to match the direction of the particle jumps from $k$ to $k+1$ (opposite from the $q$-Boson case). That is, consider the queue vertex model $$\label{eq:qpush_stat_vertex_model_Q} \mathfrak{Q}\left( (u_1,\ldots,u_N,u_0); (q^{-\mathsf{P}/2},\ldots,q^{-\mathsf{P}/2},q^{-1/2} ); (v_1,\ldots,v_m );(s_1^{(v)},\ldots, s_n^{(v)} )\right)$$ with $N+1$ sites on the ring indexed by $j=1,\ldots,N,0$. Let the distinguished auxiliary line with $j=0$ be at the bottom; see . In [\[eq:qpush_stat_vertex_model_Q\]](#eq:qpush_stat_vertex_model_Q){reference-type="eqref" reference="eq:qpush_stat_vertex_model_Q"}, the vertex weights at the sites $(-m,j)$, $1\le j\le N$, and at $(-m,0)$ are, respectively, $$\label{eq:qpush_stat_vertex_model_Q_weights_2_cases} \mathbb{W}^{(-m)}_{q^{-\mathsf{P}/2},s_m^{(v)},u_j/v_m} \quad\textnormal{and}\quad \mathbb{W}^{(-m)}_{q^{-1/2},s_m^{(v)},u_0/v_m}.$$ By the Yang--Baxter equation for the queue vertex model (), the vertex weights of the straight cylinder Markov operator must be the fused stochastic weights $W_{q^{1/2-\mathsf{P}/2}u_j/u_0,\mathsf{P}, 1}$ from . They are given in . **Remark 46**. The weights in are matched to transition probabilities of a discrete time particle system on the ring as $$\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.6] \draw[white!45!black,line width=1.5pt,->] (-5.2,-1) --++ (.4,2); \draw[white!45!black,line width=4pt,->] (-6,0) -- (-4,0); \node[left] at (-6,0) {\small $\mathbf{A}$}; \node[right] at (-4,0) {\small $\mathbf{C}$}; \node[above] at (-4.8,1) {\small $k$}; \node[below] at (-5.2,-1) {\small $\ell$}; \node[left] at (-2,0) {\small $=$}; \draw[white!45!black,line width=1.5pt,->] (1,0) -- (-1,0); \draw[white!45!black,line width=4pt,->] (0,-1) -- (0,1); \node[left] at (-1,0) {\small $k$}; \node[right] at (1,0) {\small $\ell$}; \node[below] at (0,-1) {\small $\mathbf{A}$}; \node[above] at (0,1) {\small $\mathbf{C}$}; \node[above] at (2.2,0) { \phantom{.} }; \end{tikzpicture}.$$ The picture in the left-hand side represents vertices in . In the right-hand side, the vertical direction corresponds to time, and the states $\mathbf{A},\mathbf{C}$ encode particle configurations at a given site $j \in \left\{ 1,\ldots,N \right\}$ on the ring. On the right, the horizontal arrow points left because after rotating by $90^\circ$ counterclockwise, the sites on the ring are cyclically ordered as $(N,N-1,\ldots,1 )$. Recall that under the $q$-PushTASEP, particles move in the direction of increasing $j$. This direction of the particle motion is opposite to the $q$-Boson situation, cf. the proof of . {#fig:L_horizontal_weights width=".75\\textwidth"} {#fig:hor_small_eps width=".75\\textwidth"} Now let us pass to a Poisson-type continuous time limit of the straight cylinder Markov operator to get the continuous time Markov semigroup of the colored $q$-PushTASEP. Set $$\label{eq:qpush_cont_lim_parameters} u_0\coloneqq\gamma \epsilon >0,\qquad j=1,\ldots,N; \qquad \gamma\coloneqq q^{\mathsf{P}/2-1/2},$$ where $\epsilon>0$ is small. The $\epsilon\to0$ expansions of the vertex weights from are given in . These expansions imply the convergence as $\epsilon\to0$ of the straight cylinder Markov operators $\mathfrak{S}(\mathbf{u}, \gamma\epsilon; (q^{-\mathsf{P}/2},\ldots,q^{-\mathsf{P}/2},q^{-1/2} ))^{\lfloor t/\epsilon \rfloor }$ to the $q$-PushTASEP semigroup $\mathfrak{P}_{\mathrm{qPush}}(t)$, in the same way as for the $q$-Boson process (). Indeed, the auxiliary spin $1/2$ line becomes occupied at a given instant in time with probability $O(\epsilon)$. Then, with high probability it becomes unoccupied within a finite number of discrete time steps, which corresponds to it becoming unoccupied instantaneously with respect to the macroscopic continuous time $t$. The convergence of the straight cylinder Markov operators to the colored $q$-PushTASEP implies that the stationary distribution of the latter process can be represented as the partition function of a queue vertex model on the cylinder. More precisely, we have the following result: **Proposition 47**. *Let $q\in[0,1)$, $\mathsf{P}\in \mathbb{Z}_{\ge1}$ and $u_1,\ldots,u_N>0$. Fix the type counts $(N_1,\ldots,N_n)$ with $N_i\ge1$ for all $i$. For any $v_1,\ldots,v_n$ and $s_1^{(v)},\ldots,s_n^{(v)}$, the stationary measure of the colored $q$-PushTASEP process on the ring has the form $$\label{eq:qPushTASEP_stationary_distribution_result} \mathop{\mathrm{Prob}}^{\mathrm{qPush}}\nolimits_{N_1,\ldots,N_n }(\mathbf{V}) = \frac{\langle\emptyset | \hspace{1pt} \mathfrak{Q} \bigl( (u_1,\ldots,u_N); (q^{-\mathsf{P}/2},\ldots,q^{-\mathsf{P}/2} ); (v_1,\ldots,v_m ); (s_1^{(v)},\ldots,s_n^{(v)} ) \bigr) \hspace{1pt}| \mathbf{V}\rangle}{Z_{N_1,\ldots,N_n }^{\mathrm{qPush}}}.$$* *Proof outline.* This is proven in the same way as . The queue vertex model [\[eq:qPushTASEP_stationary_distribution_result\]](#eq:qPushTASEP_stationary_distribution_result){reference-type="eqref" reference="eq:qPushTASEP_stationary_distribution_result"} on the cylinder has the weights of the first type in [\[eq:qpush_stat_vertex_model_Q\_weights_2\_cases\]](#eq:qpush_stat_vertex_model_Q_weights_2_cases){reference-type="eqref" reference="eq:qpush_stat_vertex_model_Q_weights_2_cases"}. The weights of the second type have the parameter $u_0 = \gamma \epsilon$. One can check that as $\epsilon\to0$, we have $$\mathbb{W}^{(-m)}_{q^{-1/2},s_m^{(v)},\gamma \epsilon} (\mathbf{A},0;\mathbf{A},0)= 1+O(\epsilon), \qquad \mathbb{W}^{(-m)}_{q^{-1/2},s_m^{(v)},\gamma \epsilon} (\mathbf{A},0 ;\mathbf{A}_k^{-},k)= O(\epsilon).$$ This means that in the limit $\epsilon\to0$, the auxiliary line is unoccupied with probability going to $1$. Thus, the weights of the second type in [\[eq:qpush_stat_vertex_model_Q\_weights_2\_cases\]](#eq:qpush_stat_vertex_model_Q_weights_2_cases){reference-type="eqref" reference="eq:qpush_stat_vertex_model_Q_weights_2_cases"} do not contribute to the queue vertex model, and we may pass from the model [\[eq:qpush_stat_vertex_model_Q\]](#eq:qpush_stat_vertex_model_Q){reference-type="eqref" reference="eq:qpush_stat_vertex_model_Q"} on the cylinder with $N+1$ rows to [\[eq:qPushTASEP_stationary_distribution_result\]](#eq:qPushTASEP_stationary_distribution_result){reference-type="eqref" reference="eq:qPushTASEP_stationary_distribution_result"} with $N$ rows, in the same way as in [\[eq:going_from_Nplus1_to_N\_in_the_model\]](#eq:going_from_Nplus1_to_N_in_the_model){reference-type="eqref" reference="eq:going_from_Nplus1_to_N_in_the_model"}. This completes the proof. ◻ {#fig:qpush_stat width="\\textwidth"} Let us now discuss the nonnegativity of the individual vertex weights in the queue vertex model in [\[eq:qPushTASEP_stationary_distribution_result\]](#eq:qPushTASEP_stationary_distribution_result){reference-type="eqref" reference="eq:qPushTASEP_stationary_distribution_result"}. Note that the normalized partition functions are positive as components of the Perron--Frobenius eigenvector of $\mathfrak{P}_{\mathrm{qPush}}(t)$. Define $$\label{eq:qPushTASEP_stationary_distribution_positive_weights} \mathbb{W}^{(-m),\mathrm{qPush}(\mathsf{P})_+}_{s,u}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) \coloneqq (-1/s)^{|\mathbf{D}|}\hspace{1pt} \mathbb{W}^{(-m)}_{q^{-\mathsf{P}/2},s,u}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}), \qquad \mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}\in \mathbb{Z}_{\ge0}^n.$$ Note that $|\mathbf{B}|,|\mathbf{D}|\le \mathsf{P}$ due to the finite-spin reduction (see ). The multiplication by $(-1/s_m^{(v)})^{|\mathbf{D}|}$ in each column of the queue vertex model [\[eq:qPushTASEP_stationary_distribution_result\]](#eq:qPushTASEP_stationary_distribution_result){reference-type="eqref" reference="eq:qPushTASEP_stationary_distribution_result"} can be absorbed into the normalizing constant, and thus does not affect the normalized partition functions. In other words, we can use the weights [\[eq:qPushTASEP_stationary_distribution_positive_weights\]](#eq:qPushTASEP_stationary_distribution_positive_weights){reference-type="eqref" reference="eq:qPushTASEP_stationary_distribution_positive_weights"} to represent the stationary distribution of the colored $q$-PushTASEP. The weights [\[eq:qPushTASEP_stationary_distribution_positive_weights\]](#eq:qPushTASEP_stationary_distribution_positive_weights){reference-type="eqref" reference="eq:qPushTASEP_stationary_distribution_positive_weights"} arise from the mASEP queue weights [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"} by *fusion*. That is, each weight [\[eq:qPushTASEP_stationary_distribution_positive_weights\]](#eq:qPushTASEP_stationary_distribution_positive_weights){reference-type="eqref" reference="eq:qPushTASEP_stationary_distribution_positive_weights"} is a certain sum of $\mathsf{P}$-fold products of the weights $\mathbb{W}_{q^{-1/2},s,uq^i}^{(-m),\mathrm{mASEP}_+}$, where $i=0,1,\ldots,\mathsf{P}-1$. We refer to [@borodin_wheeler2018coloured Appendix B] and [@borodin2019shift Theorem 8.5] for details. This implies the following nonnegativity of [\[eq:qPushTASEP_stationary_distribution_positive_weights\]](#eq:qPushTASEP_stationary_distribution_positive_weights){reference-type="eqref" reference="eq:qPushTASEP_stationary_distribution_positive_weights"}: **Proposition 48**. *Let $$0\le s_m^{(v)}< \frac{u_j}{v_m}\hspace{1pt}q^{\mathsf{P}-1/2} < \frac{u_j}{v_m}\hspace{1pt}q^{-1/2} \le 1$$ for all $1\le m\le n$, $1\le j\le N$. Then the vertex weights $\mathbb{W}^{(-m),\mathrm{qPush}(\mathsf{P})_+}_{s_m^{(v)},u_j/v_m}$ [\[eq:qPushTASEP_stationary_distribution_positive_weights\]](#eq:qPushTASEP_stationary_distribution_positive_weights){reference-type="eqref" reference="eq:qPushTASEP_stationary_distribution_positive_weights"} (entering the queue vertex model on the cylinder representing the stationary distribution of the $q$-PushTASEP) are nonnegative.* *Proof.* Under the hypotheses, the weights $\mathbb{W}_{q^{-1/2},s_m^{(v)},q^i u_j/v_m }^{(-m),\mathrm{mASEP}_+}$, where $i$ runs from $0$ to $\mathsf{P}-1$, are all nonnegative; see . Together with fusion, this implies the desired nonnegativity of the weights $\mathbb{W}^{(-m),\mathrm{qPush}(\mathsf{P})_+}_{s_m^{(v)},u_j/v_m}$. ◻ **Remark 49**. The stationary distribution for the colored $q$-PushTASEP with equal parameters $u_j = u$, $1\le j\le N$, and with $\mathsf{P}=1$, is the same as for the mASEP. Indeed, this follows by matching the vertex weights (see the discussion before ). On the other hand, the proofs of the stationarity for mASEP and for the $q$-PushTASEP require different Markov operators on the cylinder (the twisted and the straight ones, respectively). # Stationarity in the quarter plane and on the line {#sec:quarter_plane} Here we explain how the queue vertex models on the cylinder from can be used to construct the stationary distributions for mASEP, the colored $q$-Boson and the colored $q$-PushTASEP on the line $\mathbb{Z}$. Instead of passing to the limit as the size of the ring goes to infinity (as in, e.g., [@martin2020stationary Section 5]), our proof of the stationarity on the line passes through applying the Yang--Baxter equation in the quarter plane, which may be viewed as a *colored generalization of Burke's theorem* for stochastic vertex models. Applications of the latter to single-color stochastic integrable systems were the subject of, e.g., [@OConnellYor2001] (semi-discrete Brownian polymer), [@Seppalainen2012] (log-gamma polymer). Particular cases of the colored Burke's theorem (in the language of queues) appeared previously in [@FerrariMartin2005], [@ferrari2009multiclass]. **Remark 50**. We only consider *space-homogeneous* systems on the line ($u_j=u$ for all $j$ for the $q$-Boson and the $q$-PushTASEP; there are no known space-inhomogeneous integrable deformations of the ASEP). In contrast with the ring, stationarity of space-inhomogeneous systems on $\mathbb{Z}$ is much more delicate. If the inhomogeneity is smooth in space, we may locally model stationary distributions by the homogeneous ones. In the non-smooth case, inhomogeneity in the $q$-Boson system may lead to infinite stacks of particles, separating the whole system into independent components. Out-of-equilibrium single-color inhomogeneous models (featuring both smooth and non-smooth inhomogeneity) were considered in, e.g., [@BorodinPetrov2016Exp], [@basu2017invariant], [@SaenzKnizelPetrov2018], [@Petrov2017push], and we refer to those works for further details. A special case of our construction will lead to stationary measures for the stochastic colored six-vertex model, and these will be the same as for the space-homogeneous colored $q$-PushTASEP with $\mathsf{P}=1$. Thus, the stationary measures we get for the for mASEP on the line (which can be obtained as a degeneration of the stochastic colored six vertex model) is the same as for the space-homogeneous colored $q$-PushTASEP with $\mathsf{P}=1$, c.f. . Therefore, in describing the stationary measures on the line, we may restrict our attention to the $q$-Boson and the $q$-PushTASEP. ## Queue steady state {#sub:stationary_queue_regime} Let $n\ge1$ be the number of colors, and fix $1\le m\le n$. Fix parameters $\alpha,\nu$ such that $$\label{eq:alpha_nu_nonneg} 0\le \nu\le \alpha.$$ Consider the queue vertex weights $$\label{eq:WZ_queue_weights} \mathbb{W}^{(-m),\mathrm{line}}_{\alpha,\nu} (\mathbf{A},k;\mathbf{B},\ell) \coloneqq \mathbb{W}^{(-m)}_{q^{-1/2},s,z} (\mathbf{A},k;\mathbf{B},\ell), \qquad \alpha=-szq^{-1/2},\qquad \nu=-s^2.$$ These weights are given in with $z=uq^{1/2}$, and for convenience we reproduce them with the parameters $\alpha,\nu$ in . The next statement is straightforward from these expressions and the sum-to-one property [\[eq:fully_fused_stochastic_weights_sum_to_one\]](#eq:fully_fused_stochastic_weights_sum_to_one){reference-type="eqref" reference="eq:fully_fused_stochastic_weights_sum_to_one"}: **Lemma 51**. *The weights [\[eq:WZ_queue_weights\]](#eq:WZ_queue_weights){reference-type="eqref" reference="eq:WZ_queue_weights"} sum to one over $(\mathbf{B},\ell)$ for any fixed $(\mathbf{A},k)$. Moreover, under [\[eq:alpha_nu_nonneg\]](#eq:alpha_nu_nonneg){reference-type="eqref" reference="eq:alpha_nu_nonneg"}, we have $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha,\nu}(\mathbf{A},k;\mathbf{B},\ell)\ge0$ for all $\mathbf{A},\mathbf{B}\in \mathbb{Z}_{\ge0}^n$ and $k,\ell\in \left\{ 0,1,\ldots,n \right\}$.* ![The weights $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha,\nu}$ [\[eq:WZ_queue_weights\]](#eq:WZ_queue_weights){reference-type="eqref" reference="eq:WZ_queue_weights"}, where $m<k<\ell\le n$, and $A_m=+\infty$.](fig_steady_state_weights.pdf){#fig:steady_state_weights width=".8\\textwidth"} **Remark 52**. On the ring the stochasticity of the queue vertex weights is not essential, and we multiplied them by $(-1/s)^{\mathbf{1}_{\ell >0}}$ to be nonnegative for $s\ge0$; see [\[eq:ASEP_gauge\]](#eq:ASEP_gauge){reference-type="eqref" reference="eq:ASEP_gauge"} and . The factors $(-1/s)^{\mathbf{1}_{\ell >0}}$ were absorbed into the normalizing constant of the stationary distribution on the ring. On the line, this absorption is not possible, so we need to deal with a different range of the parameters $s,z$ as in [\[eq:alpha_nu_nonneg\]](#eq:alpha_nu_nonneg){reference-type="eqref" reference="eq:alpha_nu_nonneg"} and [\[eq:WZ_queue_weights\]](#eq:WZ_queue_weights){reference-type="eqref" reference="eq:WZ_queue_weights"}. Fix parameters $$\label{eq:alpha_nu_parameters_for_steady_queue} \alpha_1>\alpha_2> \ldots> \alpha_n> 0,\qquad \nu_i\in[0,\alpha_i],\quad i=1,\ldots,n .$$ Our first step is to construct a certain queue steady state vertex model. Fix $K\in \mathbb{Z}_{\ge1}$ and consider the rectangle $$\mathrm{R}_{K}\coloneqq \{-n,-n+1,\ldots,-1 \}\times \left\{ -K,-K+1,\ldots,0 \right\}$$ (formed by the bottom $K$ rows and the left $n$ columns in the left side of ). In this rectangle, define a stochastic vertex model with empty inputs from the bottom and the left, and vertex weights $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m}$ at each $(-m,-j)\in \mathrm{R}_K$. Denote the outgoing arrow configuration at the top by $\mathbf{M}_K=(\mathbf{M}_K(-n),\ldots, \mathbf{M}_K(-1))$, where $\mathbf{M}_K(-m)\in \mathbb{Z}_{\ge0}^n$, and the outgoing arrow configuration on the right by $\mathbf{d}_K=(d_K(0),d_K(-1),\ldots,d_K(-K) )$, where $d_K(-j)\in \left\{ 0,1,\ldots,n \right\}$. **Proposition 53**. *Fix arbitrary $c\in \mathbb{Z}_{\ge0}$. As $K\to+\infty$, the random tuples $$\mathbf{M}_K\quad \textnormal{and} \quad (d_K(0),d_K(-1), \ldots,d_K(-c))$$ converge in joint distribution to random tuples $\mathbf{M}$ and $\mathbf{d}^{[-c]}$.* We refer to the $K\to+\infty$ limit of the model in $\mathrm{R}_{K}$ as to the *queue vertex model in steady state*. The limiting tuples $\mathbf{d}^{[-c]}$ are compatible for $c\ge0$. Let us denote the corresponding infinite tuple by $\mathbf{d}= (d(0),d(-1),d(-2),\ldots )$. *Proof of .* View the vertical coordinate in $\mathrm{R}_K$ as discrete time $\mathrm{t}\in\{-K,-K+1,\ldots,0\}$. Observe that $$\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m} (\mathbf{A},k;\mathbf{A}_k^+,0)= \mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m} (\mathbf{A},0;\mathbf{A},0)= \frac{1}{1+\alpha_m}.$$ Therefore, if we do not distinguish the colors $\ge m$, then the arrows leaving the column $(-m)$ form a Bernoulli process (in discrete time $\mathrm{t}$) with probability of success $\alpha_m/(1+\alpha_m)$. This implies that the combined number of arrows of colors $m+1,\ldots,n$ in the column $(-m)$ evolves as a birth and death Markov chain on $\mathbb{Z}_{\ge0}$ starting from $0$, which makes jumps by $-1$, $0$, and $+1$. Denote this chain by $A_{[m+1,n]}(\mathrm{t})$. The jumps by $+1$ and $-1$ have probabilities, respectively, $$\frac{\alpha_{m+1}}{1+\alpha_{m+1}}\cdot \frac{1+\nu_m q^{A_{[m+1,n]}(\mathrm{t})}}{1+\alpha_m} \quad\textnormal{and} \quad \frac{1}{1+\alpha_{m+1}}\cdot \frac{\alpha_m(1- q^{A_{[m+1,n]}(\mathrm{t})})}{1+\alpha_m}.$$ The jump by $0$ occurs with the complementary probability. Since $\alpha_{m+1}<\alpha_m$, for large $A_{[m+1,n]}(\mathrm{t})$ the probability to go down is strictly larger, which implies that the birth and death chain on $\mathbb{Z}_{\ge0}$ is recurrent. Thus, in each column $(-m)$, $1\le m\le n$, the number of arrows of color $>m$ does not grow to infinity. We conclude that the (colored) configurations of arrows in all columns $(-m)$, $1\le m\le n$, jointly form a recurrent Markov chain [@durrett2019probability Chapter 5] --- a system of $n$ queues in tandem. The limiting random tuple $\mathbf{M}$ is its steady state. The limiting configuration $\mathbf{d}=(d(0),d(-1),\dots )$ is the steady state (colored) departure process, with time running from $-\infty$ to $0$. This completes the proof. ◻ **Remark 54**. For $\nu_m=0$, $1\le m\le n$, the system of $n$ queues in tandem in the proof of was considered in [@martin2020stationary Sections 3.4 and 5]. ## Colored Burke's theorem via Yang--Baxter equation {#sub:YBE_quarter_plane} Let us first discuss the general application of the Yang--Baxter equation in the quarter plane without specifying the weights leading to the concrete model. We discuss specializations to our colored stochastic particle systems in below. Assume that there exist stochastic, nonnegative vertex weights $$\label{eq:abstract_weights_for_qp_stat} \mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha,\nu}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) \quad\textnormal{and}\quad W^{\mathrm{qp}}_{\xi}(k,\mathbf{B};\ell,\mathbf{D})$$ which together with the weights $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha,\nu}(\mathbf{A},k;\mathbf{B},\ell)$ [\[eq:alpha_nu_nonneg\]](#eq:alpha_nu_nonneg){reference-type="eqref" reference="eq:alpha_nu_nonneg"} satisfy the Yang--Baxter equation given in . {#fig:abstract_YBE width="\\textwidth"} In above we used the weights $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m}$ to construct the queue steady state vertex model in $\left\{ -n,\ldots,-1 \right\}\times \mathbb{Z}_{\le 0}$ depending on the parameters [\[eq:alpha_nu_parameters_for_steady_queue\]](#eq:alpha_nu_parameters_for_steady_queue){reference-type="eqref" reference="eq:alpha_nu_parameters_for_steady_queue"}. As the output, the steady state model produces the random state $\mathbf{M}=(\mathbf{M}(-n),\ldots,\mathbf{M}(-1))$ at the top, and the random departure process $\mathbf{d}=(d(0),d(-1),\ldots )$ on the right. Let us define two more stochastic vertex models (see the left side of , for an illustration): 1. A queue vertex model in $\left\{ -n,\ldots,1 \right\}\times \left\{ 1,2,\ldots \right\}$, with the weight $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m}$ at each vertex $(-m,j)$. This model has no incoming arrows from the left, and the incoming arrow configuration $\mathbf{M}$ from the bottom. Denote by $\mathbf{V}(j)$, $j\in \mathbb{Z}_{\ge1}$, the outgoing arrow configuration from the $j$-th horizontal line of this model. 2. A vertex model in the quadrant $\mathbb{Z}_{\ge0}\times \mathbb{Z}_{\ge1}$. Let each vertex $(i,j)$ in the quadrant have weight $W^{\mathrm{qp}}_{\xi}$. Let the incoming arrow configurations for this model be $\mathbf{V}(1),\mathbf{V}(2),\ldots$ from the left, and $d(0),d(-1),\ldots$ from the bottom. Let $(i,j)\in \mathbb{Z}_{\ge0}\times \mathbb{Z}_{\ge1}$. Denote by $\mathbf{V}'(j)\in \mathbb{Z}_{\ge0}^{n}$ the arrow configuration at the horizontal edge $(0,j)-(1,j)$, and by $d'(-i)\in\left\{ 0,1,\ldots,n \right\}$ the color of the vertical edge $(i,1)-(i,2)$. **Theorem 55** (Colored Burke's theorem). *We have equalities of joint distributions: $$\Bigl\{ \bigl( d(-i) \bigr)_{i\ge0}, \bigl( \mathbf{V}(j) \bigr)_{j \ge1} \Bigr\} \stackrel{d}{=} \Bigl\{ \bigl( d'(-i) \bigr)_{i\ge0}, \bigl( \mathbf{V}(j) \bigr)_{j \ge2} \Bigr\} \stackrel{d}{=} \Bigl\{ \bigl( d(-i) \bigr)_{i\ge1}, \bigl( \mathbf{V}'(j) \bigr)_{j \ge1} \Bigr\}.$$* In words, the joint distribution of the horizontal and the vertical arrow configurations along the boundary of an arbitrarily shifted quadrant $\mathbb{Z}_{\ge I}\times \mathbb{Z}_{\ge J+1}$ (where $I,J\ge0$) is the same as for the original quadrant $\mathbb{Z}_{\ge0}\times \mathbb{Z}_{\ge1}$. Since by our assumption the weights $W^{\mathrm{qp}}_{\xi}$ are stochastic, can be viewed as the statement that the boundary data given by $(\mathbf{V}, \mathbf{d})$ is stationary for the stochastic vertex model in the quadrant with the weights $W^{\mathrm{qp}}_{\xi}$. *Proof of .* The result follows by repeatedly applying the Yang--Baxter equation from to the combination of the three vertex models in the left side of . Indeed, to shift the index in $\bigl(\mathbf{V}(j)\bigr)_{j\ge1}$ up by one, one needs to drag the crosses from the right to the left, and move the dotted line given by $\{y=1\}$ down to minus infinity. See the top two pictures in the right side of . The cross vertices on the left boundary are empty, have probability weight $1$, and thus can be removed. In the limit as the dotted line goes down to minus infinity, the distribution of the outputs $\mathbf{M}$ and $\mathbf{d}$ of the queue steady state model becomes the same; in fact, the transition matrix for the system of queues in tandem commutes with the transition matrix for the queues obtained from the stochastic vertex weights on the dotted line, and thus, the dotted line preserves $(\mathbf{M},\mathbf{d})$ if $(\mathbf{M},\mathbf{d})$ is stationary. The index shift in $\bigl( d(-i) \bigr)_{i\ge0}$ is performed similarly, but now we need to move the turning line which carried $d(0)$ up to positive infinity (see the bottom picture in the right side of for an illustration). This completes the proof. ◻ {#fig:qp_stationary width=".97\\textwidth"} **Remark 56**. The stochastic vertex model in the quadrant can be made inhomogeneous by letting the parameters $\alpha_m$ and $\xi$ in the queue vertex model part depend on the vertical coordinate $j\in \mathbb{Z}$. This would lead to the weights $W^{\mathrm{qp}}_{v_i\xi_j}$ at each vertex $(i,j)\in\mathbb{Z}_{\ge0}\times \mathbb{Z}_{\ge1}$. One can readily formulate an extension of for this situation, but for simplicity we only discussed the homogeneous setting. See also on stationarity in the presence of space inhomogeneity. Let us record a property of the steady state $\mathbf{M}$ which will be useful in below: **Lemma 57**. *Let $\mathbf{M}=(\mathbf{M}(-n),\ldots,\mathbf{M}(-1))$ be the steady state of the $n$-column queue vertex model with the weights $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m}$ (the bottom $n$ columns in , left). Take a one-row vertex model in $\{-n,\ldots,-1\}\times \{0\}$ with the weight $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m}$ at each vertex $(-m,0)$. If there are no arrows incoming from the left, and the configuration $\mathbf{M}$ of arrows incoming from below into this one-row model, then the distribution of the outgoing arrows from the top is the same as that of $\mathbf{M}$.* *In short, the distribution of $\mathbf{M}$ is preserved by the horizontal action of the $n$-column queue vertex model with the weights $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m}$.* *Proof.* Consider the rectangle $\{-n,\ldots,-1 \}\times \{0,-1,\ldots,-K \}$. Put a single $n$-column layer with the weights $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m}$ at the horizontal coordinate $0$. Below it, let us put $K$ layers of the weights $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m}$. Assume that the incoming arrow configuration at the left boundary is empty, while at the bottom let the input be the random steady state $\mathbf{M}=(\mathbf{M}(-n),\ldots,\mathbf{M}(-1))$. Denote the random output at the top by $\mathbf{M}'=(\mathbf{M}'(-n),\ldots,\mathbf{M}'(-1))$. See , left, for an illustration. Since the solid horizontal lines preserve the distribution of $\mathbf{M}$ (this is the steady state property), it remains to show that $\mathbf{M}'$ and $\mathbf{M}$ have the same distribution. Attach $K$ cross vertices with the stochastic weights $W^{\mathrm{qp}}_{\xi}$ to the rectangle on the right. This does not change the distribution of $\mathbf{M}'$. Then apply the Yang--Baxter equation (as in ) to move these cross vertices to the left. On the left, there are no incoming arrows, so the cross vertices in , right, can be removed. Denote by $\mathbf{M}''$ the random configuration which arises from $\mathbf{M}$ after the single dashed line, that is, the line with the weights $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m}$. The Yang--Baxter equation implies that $\mathbf{M}'$ has the same distribution as the random configuration which arises from $\mathbf{M}''$ after $K$ solid horizontal lines, that is, the lines with the weights $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m}$. Taking $K\to+\infty$, we see that the growing number of solid horizontal lines converges to its steady state distributed as $\mathbf{M}$. In particular, this steady state is independent of the initial configuration $\mathbf{M}''$. Therefore, $\mathbf{M}'$ and $\mathbf{M}$ have the same distribution, as desired. ◻ {#fig:M_is_stationary width=".9\\textwidth"} From , we can construct a stationary random configuration of the vertex model $W^{\mathrm{qp}}_{\xi}$ in the whole plane $\mathbb{Z}^{2}$. The model in the plane is characterized as follows (a similar single-color construction appeared in [@Amol2016Stationary Section A.2]). For any $J\ge 1$, define the map $$\tau_J\colon\mathbb{Z}_{\ge 0}\times \mathbb{Z}_{\ge 1}\to \mathbb{Z}^{2},\qquad \tau_J\colon (i,j)\mapsto (i-J,j-J).$$ By , the random configurations of colored paths in parts of $\mathbb{Z}^2$ coming from shifting the configuration in $\mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 1}$ by $\tau_J$, $J\in \mathbb{Z}_{\ge1}$, are compatible in $J$. Therefore, by the Kolmogorov extension theorem, there exists a distribution on path configurations in the whole plane $\mathbb{Z}^{2}$, which is translation invariant (that is, stationary under the stochastic vertex model $W^{\mathrm{qp}}_{\xi}$). The parameters $(\alpha_m,\nu_m)$ of the queue vertex models attached to the quadrant determine the densities of various types of colors (the "colored slope") of the resulting translation invariant model in $\mathbb{Z}^{2}$. We explore the exact connection between these parameters and the densities of various colors for $q$-Boson and $q$-PushTASEP in below. ## Specialization to stochastic six vertex model and $q$-PushTASEP {#sub:qpush_and_density_app} Let us specialize to the colored $q$-PushTASEP (defined on the line in the same way as in , but with the homogeneous parameters $u_k= u$, $k\in \mathbb{Z}$). Take the weights [\[eq:WZ_queue_weights\]](#eq:WZ_queue_weights){reference-type="eqref" reference="eq:WZ_queue_weights"}, [\[eq:abstract_weights_for_qp_stat\]](#eq:abstract_weights_for_qp_stat){reference-type="eqref" reference="eq:abstract_weights_for_qp_stat"} to be $$\label{eq:qPushTASEP_weights_line_steady} \begin{split} \mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m} (\mathbf{A},k;\mathbf{B},\ell) &= \mathbb{W}^{(-m)}_{q^{-1/2},s_m^{(v)},z/v_m} (\mathbf{A},k;\mathbf{B},\ell), \\ \mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) &= \mathbb{W}^{(-m)}_{q^{-\mathsf{P}/2},s_m^{(v)},u/v_m} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) , \\ W^{\mathrm{qp}}_{\xi}(k,\mathbf{B};\ell,\mathbf{D}) &= W_{q^{1/2-\mathsf{P}/2}u/z,\mathsf{P},1} ( \mathbf{e}_k\mathbf{1}_{k\ge1} ,\mathbf{B}; \mathbf{e}_\ell\mathbf{1}_{\ell\ge1},\mathbf{D}), \\ \alpha_m\coloneqq -s_m^{(v)}q^{-1/2} \hspace{1pt}\frac{z}{v_m}, \qquad \nu_m &\coloneqq -(s_m^{(v)})^2, \qquad \xi\coloneqq u/z. \end{split}$$ Here $\mathsf{P}\in \mathbb{Z}_{\ge1}$, and $\alpha_m,\nu_m$ must satisfy [\[eq:alpha_nu_parameters_for_steady_queue\]](#eq:alpha_nu_parameters_for_steady_queue){reference-type="eqref" reference="eq:alpha_nu_parameters_for_steady_queue"}. The weights in the right-hand sides of [\[eq:qPushTASEP_weights_line_steady\]](#eq:qPushTASEP_weights_line_steady){reference-type="eqref" reference="eq:qPushTASEP_weights_line_steady"} are given, respectively, in , formula [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"}, and . One can check that the weights [\[eq:qPushTASEP_weights_line_steady\]](#eq:qPushTASEP_weights_line_steady){reference-type="eqref" reference="eq:qPushTASEP_weights_line_steady"} are all nonnegative if $$\label{eq:qPushTASEP_steady_state_parameters_nonnegativity} \nu_i\in[0,\alpha_i q^{\mathsf{P}-1}],\quad i=1,\ldots,n, \qquad \xi\ge q^{1/2-\mathsf{P}/2}.$$ Indeed, the first condition corresponds to the fact that the weights $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m}$ come from the fusion of $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m}$ (see for a related nonnegativity property), and the second condition involving the weights $W^{\mathrm{qp}}_{\xi}$ is read off from their explicit form given in . Let us first consider the case $\mathsf{P}=1$. Then the vertex model in the quadrant with the weights $W^{\mathrm{qp}}_{\xi}$ becomes the colored stochastic six vertex model (see [@borodin_wheeler2018coloured Figure 1] for a simulation with non-stationary boundary conditions). and the shifting argument after it allows to construct a translation invariant (stationary) colored stochastic six vertex model in the full plane $\mathbb{Z}^2$. By analogy with [@Neergard1995], [@aggarwal2020nonexistence], let us call this path configuration in $\mathbb{Z}^2$ the *colored KPZ pure phase* of the stochastic six vertex model. The colored KPZ pure phase has a finite number $n$ of colors. From , we see that the probability that no paths leave column $(-m)$ under the stochastic weights $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m}$ and $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m}$, respectively, is equal to $$\label{eq:density_computation_simplest_part} \frac{1}{1+\alpha_m\xi q^{-1/2}} \quad\textnormal{and} \quad \frac{1}{1+\alpha_m q^{-1/2}}.$$ By , the arrow configurations in the columns $-n,\ldots,-1$ are in steady state. This means that the number of colors $i>m$ in the each column $(-m)$ does not grow to infinity. Thus, once a path of color $m$ originates at the column $(-m)$, it must exit the column $(-1)$ at a bounded (random) distance from where it originated. This means that the combined density of paths of colors $\ge m$ exiting the column $(-1)$ (in either the bottom or the top part of the configuration of $n$ vertical columns, see , left) is equal to the complementary probability to [\[eq:density_computation_simplest_part\]](#eq:density_computation_simplest_part){reference-type="eqref" reference="eq:density_computation_simplest_part"}. We conclude that in the colored KPZ pure phase determined by the parameters $\{\alpha_m\}_{m=1}^n$, the horizontal and the vertical densities of each color $m=1,\ldots,n$ are, respectively, $$\label{eq:rho_h_v_for_colored_s6v} \rho_m^{(h)}= \frac{\alpha_m\xi q^{-1/2}}{1+\alpha_m\xi q^{-1/2}}- \frac{\alpha_{m+1}\xi q^{-1/2}}{1+\alpha_{m+1}\xi q^{-1/2}} , \qquad \rho_m^{(v)}= \frac{\alpha_m q^{-1/2}}{1+\alpha_m q^{-1/2}}- \frac{\alpha_{m+1} q^{-1/2}}{1+\alpha_{m+1} q^{-1/2}} .$$ Here $\alpha_{n+1}=0$, by agreement. When $n=1$, we can solve for $\alpha_1$, and get $$\rho_1^{(h)} = \frac{\rho_1^{(v)} \xi}{1 +(\xi-1) \rho_1^{(v)} },$$ which agrees with the slope relation $\rho_1^{(h)}=\varphi( \rho_1^{(v)} )$ in the single-color KPZ phase (for example, see [@aggarwal2020nonexistence (2.6)]). For general $n$, solving for the $\alpha_i$'s in [\[eq:rho_h\_v_for_colored_s6v\]](#eq:rho_h_v_for_colored_s6v){reference-type="eqref" reference="eq:rho_h_v_for_colored_s6v"} yields the following *colored slope relations*: $$\label{eq:colored_KPZ_phase_densities} \rho_m^{(h)} = \frac{\rho_m^{(v)}\xi} { \bigl( 1+(\xi-1)\rho^{(v)}_{[m,n]} \bigr) \bigl( 1+(\xi-1)\rho^{(v)}_{[m+1,n]} \bigr) }, \qquad m=1,\ldots,n,$$ where $\rho^{(v)}_{[a,b]}=\rho^{(v)}_a+ \rho^{(v)}_{a+1}+\ldots+\rho^{(v)}_b$. Under the colored KPZ pure phase, the colors occupying the vertical edges along a given horizontal line induce a random configuration of colors on $\mathbb{Z}$. This random configuration is a stationary distribution for the mASEP. **Remark 58**. As shown in [@ferrari1991microscopic], for given color densities, a translation invariant stationary distribution for mASEP on $\mathbb{Z}$ is unique. We believe that a similar uniqueness holds for the colored stochastic six vertex model, but this statement does not seem to be present in the existing literature. Let us now return to the case of general $\mathsf{P}$, and take a continuous time limit to the colored $q$-PushTASEP. The limit is achieved by setting $z=q^{\mathsf{P}/2-1/2}\epsilon$, and letting $\epsilon\to0$. In this limit, the horizontal direction in the quarter plane, scaled by $1/\epsilon$, turns into continuous time $t \in \mathbb{R}_{\ge0}$. See for the $\epsilon\to0$ expansions of the weights $W^{\mathrm{qp}}_{\xi}$. The continuous time Markov process coming from $W^{\mathrm{qp}}_{\xi}$ lives on configurations on $\mathbb{Z}_{\ge1}$ (where at each site there are at most $\mathsf{P}$ particles), and coincides with the colored $q$-PushTASEP () with homogeneous parameters $u_k\equiv u$. After the rescaling, let us further set $\nu_m=0$, which implies nonnegativity of the remaining weights $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,0}$ and $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,0}$. Indeed, note that as $\alpha_m$ (containing $z$ as a factor) goes to zero, conditions [\[eq:qPushTASEP_steady_state_parameters_nonnegativity\]](#eq:qPushTASEP_steady_state_parameters_nonnegativity){reference-type="eqref" reference="eq:qPushTASEP_steady_state_parameters_nonnegativity"} cannot hold unless $\nu_m=0$. **Remark 59**. In the constructions on the line, we need to set $\nu_m=0$, $m=1,\ldots,n$, from the beginning, to ensure the nonnegativity of the queue vertex weights and the corresponding jump rates in the queue columns (occurring as $\epsilon\to0$). The nonnegativity is required to ensure the existence of the steady state in the queue columns (see ). This should be contrasted to the ring case, where we could initially work with negative probabilities and jump rates formally. Then, when the commutation relation between the queue vertex model transfer matrix and the straight cylinder transfer matrix is established, we can renormalize the queue vertex model on the cylinder to get nonnegative probabilities under the stationary distribution. Thus, we have a whole family of vertex models on the cylinder (depending on the $\nu_m$'s) which produce the same stationary measure, and on the line we have to set $\nu_m=0$ for all $m$. In the remaining vertex models in , the weights $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,0}$ in the top part do not depend on $z$ (see [\[eq:qPushTASEP_weights_line_steady\]](#eq:qPushTASEP_weights_line_steady){reference-type="eqref" reference="eq:qPushTASEP_weights_line_steady"}) and thus do not change in the limit. In the weights $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,0}$ in the bottom part (given in ), we have $$\label{eq:definition_of_y_m_for_qpushTASEP} \alpha_m= -\frac{s_m^{(v)}}{v_m} q^{\mathsf{P}/2-1}\epsilon \eqqcolon y_m \epsilon\to0.$$ Here $y_1> \ldots > y_n>0$ are the new parameters of the continuous time colored $q$-PushTASEP. The fact that the $\alpha_m$'s are proportional to $\epsilon$ corresponds to the scaling of the bottom columns $\{-n,\ldots,-1 \}\times \mathbb{Z}_{\le 0}$ in to continuous ones, $\{-n,\ldots,-1 \}\times \mathbb{R}_{\le 0}$. Note that this scaling does not affect the weights $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m}$ since they do not depend on $z$. The resulting scaled queue steady state model in $\{-n,\ldots,-1 \}\times \mathbb{R}_{\le 0}$ runs in continuous time. Let $\mathbf{M}=(\mathbf{M}(-n),\ldots,\mathbf{M}(-1) )$ be the steady state of these continuous time queues in tandem, and let $\mathbf{d}(t)$, $t\le 0$, be the (continuous time) departure process. Using $\mathbf{d}(t)$ and the output $\mathbf{V}$ of the top columns $\{-n,\ldots,-1 \}\times \mathbb{Z}_{\ge1}$, one can use the Burke's theorem () to construct a stationary version of the colored $q$-PushTASEP on the whole line. Let us compute the densities of the colors under the stationary measure for the colored $q$-PushTASEP. Since more than one arrow may leave the column $(-m)$, we need to take the expectation of the number of arrows. For this expectation, we do not need to distinguish the colors. Employing the color merging discussed in , we may assume that $m=n=1$. By a specialization of [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"}, one can check that the weights have the form (with $a=c=\infty$): $$\begin{split} \mathbb{W}^{(-1),\mathrm{queue}}_{\xi,\alpha_m,0} (a,b;c,d)&= \frac{\mathbf{1}_{a+b=c+d}}{(q^{-\mathsf{P}/2}s_m^{(v)}u/v_m;q)_{\mathsf{P}}} \hspace{1pt} \frac{(q^{\mathsf{P}/2}s_m^{(v)}u/v_m)^d \hspace{1pt}(q^{-\mathsf{P}};q)_d}{(q;q)_d} \\&= \frac{\mathbf{1}_{a+b=c+d}}{(-q^{1-\mathsf{P}}u y_m;q)_{\mathsf{P}}} \hspace{1pt} \frac{(-qu y_m)^d \hspace{1pt}(q^{-\mathsf{P}};q)_d}{(q;q)_d} . \end{split}$$ Note that these weights sum to $1$ over $0\le d\le \mathsf{P}$ by the $q$-binomial theorem [@GasperRahman (1.3.2)]. The expected number of arrows is expressed through the function $$\label{eq:q_digamma} \phi(\zeta)\coloneqq \sum_{k=0}^{\infty} \frac{\zeta q^{k}}{1-\zeta q^{k}}$$ (up to a change of variables and a linear transformation, this is the $q$-digamma function). We have $$\begin{split} \sum_{d=0}^{\mathsf{P}}\frac{\zeta^d(q^{-\mathsf{P}};q)_d}{(q;q)_d}&= \frac{(q^{-\mathsf{P}}\zeta;q)_{\infty}}{(\zeta;q)_{\infty}}= (q^{-\mathsf{P}}\zeta;q)_{\mathsf{P}}, \\ \frac{1}{(q^{-\mathsf{P}}\zeta;q)_{\mathsf{P}}} \sum_{d=0}^{\mathsf{P}} d\cdot \frac{\zeta^d(q^{-\mathsf{P}};q)_d}{(q;q)_d}&= \zeta\frac{\partial}{\partial \zeta}\log (q^{-\mathsf{P}}\zeta;q)_{\mathsf{P}} =-\sum_{i=0}^{\mathsf{P}-1} \frac{q^{i-\mathsf{P}}\zeta}{1-q^{i-\mathsf{P}}\zeta}, \end{split}$$ and the latter sum is a difference of two functions of the form [\[eq:q_digamma\]](#eq:q_digamma){reference-type="eqref" reference="eq:q_digamma"} with the arguments differing by the factor $q^{-\mathsf{P}}$. Therefore, $$\label{eq:qpush_qp_expectation} \sum_{d=0}^{\mathsf{P}} d\cdot \frac{1}{(-q^{1-\mathsf{P}}uy_m;q)_{\mathsf{P}}} \hspace{1pt} \frac{(-qu y_m)^d \hspace{1pt}(q^{-\mathsf{P}};q)_d}{(q;q)_d} = \phi(-qu y_m)- \phi(-q^{1-\mathsf{P}}u y_m).$$ Thus, the horizontal density of the $m$-th color is the difference of the above expressions: $$\label{eq:h_current_qpush} \rho_m^{(h)}= \phi(-qu y_m)- \phi(-q^{1-\mathsf{P}}u y_m)- \phi(-qu y_{m+1})+ \phi(-q^{1-\mathsf{P}}u y_{m+1}) ,$$ where, by agreement, $y_{n+1}=0$. This follows similarly to the case of the colored six vertex model: since the queues are in steady state, a color $m$ cannot accummulate in any column except $(-m)$. This implies that the expectation [\[eq:qpush_qp_expectation\]](#eq:qpush_qp_expectation){reference-type="eqref" reference="eq:qpush_qp_expectation"} is equal to $\rho^{(h)}_{[m,n]}=\rho^{(h)}_{m}+\ldots+\rho^{(h)}_{n}$, which yields [\[eq:h_current_qpush\]](#eq:h_current_qpush){reference-type="eqref" reference="eq:h_current_qpush"}. We can also compute the currents of the colored $q$-PushTASEP in stationarity (that is, the vertical densities of the colors), using $$\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,0} (\mathbf{A},k;\mathbf{B},0)= \frac{1}{1+\alpha_m}= 1-y_m \epsilon+O(\epsilon^2),$$ where $\mathbf{B}=\mathbf{A}+\mathbf{e}_k\mathbf{1}_{k\ge1}$. Therefore, in the continuous time limit, we have $$\label{eq:v_current_qpush} \rho_m^{(v)}= y_{m} - y_{m+1}.$$ Expressing the colored currents $(\rho_1^{(v)}, \ldots, \rho_n^{(v)})$ in terms of the colored densities $(\rho_1^{(h)}, \ldots, \rho_n^{(h)})$ for general $\mathsf{P}$ would require finding the $y_m$'s from [\[eq:h_current_qpush\]](#eq:h_current_qpush){reference-type="eqref" reference="eq:h_current_qpush"}, which is not explicit for general $\mathsf{P}$. However, a reverse expression is essentially given by [\[eq:h_current_qpush\]](#eq:h_current_qpush){reference-type="eqref" reference="eq:h_current_qpush"}--[\[eq:v_current_qpush\]](#eq:v_current_qpush){reference-type="eqref" reference="eq:v_current_qpush"}: $$\rho_m^{(h)}= \phi\bigl(-qu \rho^{(v)}_{[m,n]}\bigr)- \phi\bigl(-q^{1-\mathsf{P}}u \rho^{(v)}_{[m,n]}\bigr)- \phi\bigl(-qu \rho^{(v)}_{[m+1,n]}\bigr)+ \phi\bigl(-q^{1-\mathsf{P}}u \rho^{(v)}_{[m+1,n]}\bigr) .$$ **Remark 60**. We believe that for any $\mathsf{P}$, a translation invariant stationary distribution for the $q$-PushTASEP on $\mathbb{Z}$ with parameter $\mathsf{P}$ and with given densities of the colors is unique. However, this statement does not seem to be present in the existing literature. ## Specialization to $q$-Boson {#sub:qboson_and_density_app} Let us specialize to the stochastic colored $q$-Boson process. It is defined the same way on the line as on the ring (), but we reverse the direction of the particle movement. That is, a particle of color $i$ jumps from $k$ to $k+1$, $k \in \mathbb{Z}$, at the homogeneous rate $u^{-1}(1-q^{\mathbf{V}(k)_i})\hspace{1pt}q^{\mathbf{V}(k)_{[i+1,n]}}$, where $\mathbf{V}(k)\in \mathbb{Z}_{\ge0}^n$ is the arrow configuration at site $k$. To obtain the $q$-Boson process together with its stationary measure from the vertex models in , is it convenient to take horizontally fused weights (meaning multiple paths can occupy a horizontal edge, as compared to , when $\mathbb{W}^{(-m),\mathrm{line}}$ were weights for which at most one path can occupy a horizontal edge) in the bottom part $\{-n,\ldots,-1 \}\times \mathbb{Z}_{\le 0}$. That is, let us take the following pre-limit weights depending on $\epsilon>0$: $$\label{eq:qBoson_weights_line_steady} \begin{split} \mathbb{W}^{(-m),\mathrm{line}}_{\xi,\alpha_m,\nu_m} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) &= \mathbb{W}^{(-m)}_{\epsilon,s,u \epsilon y_m/s} (\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) , \\ \mathbb{W}^{(-m),\mathrm{queue}}_{\alpha_m,\nu_m} (\mathbf{A},k;\mathbf{B},\ell) &= \mathbb{W}^{(-m)}_{q^{-1/2},s,-\epsilon q^{1/2}y_m/s} (\mathbf{A},k;\mathbf{B},\ell) , \\ W^{\mathrm{qp}}_{\xi,\epsilon} (\mathbf{A},k;\mathbf{B},\ell) &= W_{-1 /(u \epsilon),1,\mathsf{N}} (\mathbf{A},\mathbf{e}_k\mathbf{1}_{k\ge1};\mathbf{B},\mathbf{e}_{\ell} \mathbf{1}_{\ell\ge1}) , \\ \alpha_m \coloneqq \epsilon y_m/s , \qquad \nu_m &\coloneqq -s^2 , \qquad \xi\coloneqq u, \qquad q^{-\mathsf{N}^\epsilon} \coloneqq \epsilon^2 . \end{split}$$ Note that here we placed the $\xi$-dependence into the bottom part of the left $n$ columns in , left, instead of the top one. Let us take $\epsilon\to0$ and simultaneously rescale the vertical coordinate of the quadrant by $1/\epsilon$. This turns the vertical coordinate into the continuous time $t\in \mathbb{R}_{\ge0}$. After that, set $s=0$, which would imply the nonnegativity of the jump rates (the restriction $s=0$ is parallel to the case of the $q$-PushTASEP; see ). The results of imply that as $\epsilon\to0$ and $s=0$, the weights [\[eq:qBoson_weights_line_steady\]](#eq:qBoson_weights_line_steady){reference-type="eqref" reference="eq:qBoson_weights_line_steady"} become $$\label{eq:qBoson_weights_line_steady_limiting} \begin{split} \mathbb{W}^{(-m),\mathrm{line}}_{\xi,\alpha_m,\nu_m} &\to (uy_m;q)_\infty\hspace{1pt} \mathbb{W}^{(-m),\mathrm{qBos}}_{0,uy_m} , \\ \mathbb{W}^{(-m),\mathrm{queue}}_{\alpha_m,\nu_m} &\to \textnormal{jump rates in a continuous time queue vertex model (\Cref{fig:qboson_cont_queue_model_s7})} , \\ W^{\mathrm{qp}}_{\xi,\epsilon} &\to \textnormal{colored $q$-Boson jump rates in \Cref{fig:qboson_specialization}}, \end{split}$$ where $\mathbb{W}^{(-m),\mathrm{qBos}}_{0,uy_m}$ is given by the right-hand side of [\[eq:q_boson_weights_degeneration_vertex\]](#eq:q_boson_weights_degeneration_vertex){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex"}, and the limits of $\mathbb{W}^{(-m),\mathrm{queue}}_{\alpha_m,\nu_m}$ to the continuous vertical direction are read off from . These limits are given in . {#fig:qboson_cont_queue_model_s7 width=".8\\textwidth"} One readily sees that the vertex weights [\[eq:qBoson_weights_line_steady_limiting\]](#eq:qBoson_weights_line_steady_limiting){reference-type="eqref" reference="eq:qBoson_weights_line_steady_limiting"} define Markov processes in discrete and continuous time with nonnegative transitions. Similarly to the proof of , for $$y_1 > y_2 > \ldots > y_n > 0$$ one can verify that the queue vertex model with the weights $(uy_m;q)_\infty\hspace{1pt} \mathbb{W}^{(-m),\mathrm{qBos}}_{0,uy_m}$ produces a steady state (that is, it does not run off to infinity). Then, following the proof of and the shifting argument after it, one can construct a stationary version of the colored $q$-Boson process on the whole line. We omit the details of the construction as they are similar to the ones in . **Remark 61**. The results of [@amir2021tazrp] imply that for $q=0$, a translation invariant stationary distribution for the $q$-Boson process on the whole line with given densities of the colors is unique. We believe that a similar result should hold for general $q\in (0,1)$, but its proof does not seem present in the existing literature. Arguing similarly to the case of the $q$-PushTASEP (), we can compute the colored densities and currents in terms of the parameters $y_1>\ldots>y_n>0$. The colored densities are the vertical densities $\rho_m^{(v)}$ under the steady state vertex model with probability weights $(uy_m;q)_\infty\hspace{1pt} \mathbb{W}^{(-m),\mathrm{qBos}}_{0,uy_m}$ [\[eq:q_boson_weights_degeneration_vertex\]](#eq:q_boson_weights_degeneration_vertex){reference-type="eqref" reference="eq:q_boson_weights_degeneration_vertex"}. Let us compute the expected number of arrows leaving the column $(-m)$. By color merging described in , we do not need to distinguish the colors and may assume that $m=n=1$. We have $$(uy_m;q)_\infty \mathbb{W}^{(-m),\mathrm{qBos}}_{0,uy_m} (a,b;c,d)= (uy_m;q)_\infty \cdot\frac{(uy_m)^d}{(q;q)_d} , \qquad d\in \mathbb{Z}_{\ge0}.$$ The expected number of arrows is expressed through the function [\[eq:q_digamma\]](#eq:q_digamma){reference-type="eqref" reference="eq:q_digamma"}: $$\sum_{d=0}^{\infty} d\cdot (uy_m;q)_\infty \cdot\frac{(uy_m)^d}{(q;q)_d} = \phi(uy_m).$$ Thus, we have $$\label{eq:q_boson_rho_v} \rho^{(v)}_m= \phi(uy_m)-\phi(uy_{m+1}).$$ Here $y_{n+1}=0$, by agreement. The colored currents are determined from the continuous time queue vertex model in , and have the form $$\rho_m^{(h)}=y_m-y_{m+1}.$$ Similarly to the $q$-PushTASEP, expressing the currents in terms of the densities is not an explicit operation. On the other hand, a reverse expression is readily available. It is obtained from [\[eq:q_boson_rho_v\]](#eq:q_boson_rho_v){reference-type="eqref" reference="eq:q_boson_rho_v"} by replacing each $y_j$ with $\rho^{(h)}_{[j,n]}$. # Merging of colors in stationary measures on the line {#sec:merging_of_colors} Take a stochastic particle system (mASEP, the $q$-Boson process or the $q$-PushTASEP) with $n$ colors on the line $\mathbb{Z}$. By agreement, holes are viewed as particles of color $0$. If we declare that for some $m=0,1,\ldots,n$, all particles of colors $m$ and $m+1$ are identified, then we get a stochastic process with $n-1$ colors. This operation of identifying two colors is a particular case of *color merging* (see below). A similar color merging can be performed for a stationary distribution of an $n$-colored particle system, and as a result, we should get a stationary distribution of a system with $n-1$ colors and modified densities of the colors. We call this the *color merging property of the stationary distributions*. Here we explain how one can get this property on the line $\mathbb{Z}$ using our queue vertex model constructions from . **Remark 62**. For some (but not all) of our particle systems, it is proven that a translation invariant stationary distribution with given densities of colors is unique (see Remarks [Remark 58](#rmk:uniq_ASEP){reference-type="ref" reference="rmk:uniq_ASEP"}, [Remark 60](#rmk:uniq_push){reference-type="ref" reference="rmk:uniq_push"}, and [Remark 61](#rmk:uniq_boson){reference-type="ref" reference="rmk:uniq_boson"}). When this uniqueness is available, it implies the color merging property of the stationary distributions directly, without reference to queue vertex models. **Remark 63** (Color merging on the ring). For all colored particle systems on the ring (mASEP, the $q$-Boson process, and the $q$-PushTASEP), we readily have uniqueness of the stationary distribution in any given sector determined by the number of particles of each color. Thus, on the ring the color merging property holds automatically. However, it is not clear if this color merging can be seen at the level of queue vertex models on the cylinder. One reason for this is that queue vertex models on a cylinder are *not* stochastic because they involve summing over input and also output path configurations at vertices (see  for more discussion). At the same time, color merging involves summing over outputs only, and the two summations are not readily compatible. In the remainder of , we focus only on systems on the line. **Definition 64** (Color merging). Suppose we have a partition of $\{0,1,\dots,n\}$ into $k$ disjoint intervals $I_0,\dots, I_k \subset \{0,1,\dots, n\}$ which are contiguous (that is, $\max(I_j) = \min(I_{j+1})-1$ for all $j$). The *color merging projection* $\pi = \pi_{I_0,\dots, I_k}$ applied to a state $i\in \{0,1,\ldots,n\}$ or $\mathbf{A}\in \mathbb{Z}_{\ge0}^n$ maps it into $\pi(i)\in\{ 0,1,\ldots,k\}$ or $\pi(\mathbf{A})\in \mathbb{Z}_{\ge0}^k$, respectively, by assigning to all particles (or arrows) with colors in each interval $I_j$ the new color $j$. For a probability measure $\mu$ on $n$-color configurations on $\mathbb{Z}$ (where the maximal number of particles at a site is $1$, $\mathsf{P}$, or $\infty$, depending on the particle system), denote by $\pi_* \mu$ the pushforward of $\mu$ under the color merging projection $\pi$. Fix a partition $\left\{ 0,1,\ldots,n \right\}= I_0\sqcup \ldots \sqcup I_k$ as in . For the densities $\rho_1,\ldots,\rho_n$ of the old colors, denote by $$\label{eq:rho_j_prime_new_densities} \rho_j'\coloneqq \sum_{i\in I_j}\rho_i, \qquad j=1,\ldots,k,$$ the densities of the new colors. In , we showed that the densities $(\rho_1,\ldots,\rho_n )$ are in one-to-one correspondence with ordered $n$-tuples $y_1>\ldots>y_n>0$. More precisely, $y_m$ parametrizes $\sum_{i\ge m}\rho_i$ (the exact parametrization is different for the $q$-PushTASEP and the $q$-Boson process, but for color merging we do not need these exact formulas). Therefore, the new densities $(\rho_1',\ldots,\rho_k' )$ correspond to the ordered $k$-tuple $$\label{eq:y_j_prime_new_densities} y'=(y_{\min(I_1)},\dots, y_{\min(I_k)}).$$ Here, by agreement, if $I_1$ contains the color $0$ (corresponding to the hole), then we need to remove the parameter $y_{\min(I_1)}=y_0$ from the tuple $y'$. Denote by $\mu_y$ and $\mu_{y'}$ the stationary measures, respectively, for the $n$- and the $k$-colored particle systems on the whole line $\mathbb{Z}$. Let us record the color merging properties of the vertex weights which appeared in . We have: $$\begin{aligned} \sum\nolimits_{i_2, j_2 \colon \pi(i_2) = i_2', \pi(j_2) = j_2'} R_z(i_1, j_1; i_2, j_2) &= R_z(\pi(i_1), \pi(j_1); i_2', j_2') \label{eq:R_merge} ,\\ \sum\nolimits_{\mathbf{B}, \ell \colon \pi(\mathbf{B}) = \mathbf{B}', \pi(\ell) = \ell'} L_{s,x}(\mathbf{A}, k; \mathbf{B}, \ell) &= L_{s,x}(\pi(\mathbf{A}), \pi(k); \mathbf{B}', \ell') \label{eq:L_merge} ,\\ \sum\nolimits_{\mathbf{C}, \mathbf{D} \colon \pi(\mathbf{C})= \mathbf{C}', \pi(\mathbf{D}) = \mathbf{D}'} W_{x,\mathsf{L},\mathsf{M}}(\mathbf{A},\mathbf{B};\mathbf{C}, \mathbf{D}) &= W_{x,\mathsf{L},\mathsf{M}}(\pi(\mathbf{A}),\pi(\mathbf{B});\mathbf{C}',\mathbf{D}') \label{eq:W_merge} ,\\ \sum\nolimits_{\mathbf{C}, \mathbf{D} \colon \pi(\mathbf{C})= \mathbf{C}', \pi(\mathbf{D}) = \mathbf{D}'} \mathbb{W}_{s_1,s_2,u}^{(-m)}(\mathbf{A},\mathbf{B};\mathbf{C}, \mathbf{D}) &= \mathbb{W}_{s_1,s_2,u}^{(-\pi(m))}(\pi(\mathbf{A}),\pi(\mathbf{B});\mathbf{C}',\mathbf{D}'). \label{eq:WQ_merge}\end{aligned}$$ Identity [\[eq:R_merge\]](#eq:R_merge){reference-type="eqref" reference="eq:R_merge"} is immediate. Vertical fusion or a direct verification leads to [\[eq:L_merge\]](#eq:L_merge){reference-type="eqref" reference="eq:L_merge"}. Then by horizontal fusion, [\[eq:L_merge\]](#eq:L_merge){reference-type="eqref" reference="eq:L_merge"} leads to [\[eq:W_merge\]](#eq:W_merge){reference-type="eqref" reference="eq:W_merge"}. Finally, we get [\[eq:WQ_merge\]](#eq:WQ_merge){reference-type="eqref" reference="eq:WQ_merge"} from [\[eq:W_merge\]](#eq:W_merge){reference-type="eqref" reference="eq:W_merge"} by the queue specialization defined in . Note that in [\[eq:WQ_merge\]](#eq:WQ_merge){reference-type="eqref" reference="eq:WQ_merge"}, both $m$ and $\pi(m)$ must be strictly positive. In probabilistic language, identity, say, [\[eq:WQ_merge\]](#eq:WQ_merge){reference-type="eqref" reference="eq:WQ_merge"}, can be interpreted as follows. Starting from $(\pi(\mathbf{A}),\pi(\mathbf{B}))$, to sample $(\mathbf{C}',\mathbf{D}')$ under the $k$-color stochastic weight $\mathbb{W}_{s_1,s_2,u}^{(-\pi(m))}$, we may choose any representatives $(\mathbf{A},\mathbf{B})$ for the input, sample $(\mathbf{C},\mathbf{D})$ under the $n$-color stochastic weight $\mathbb{W}_{s_1,s_2,u}^{(-m)}$, and then project the output $(\mathbf{C},\mathbf{D})$ back to $k$ colors using $\pi$. The projection $\pi$ "forgets" some of the information about the colors, and this operation is the same as the summation in the left-hand side of [\[eq:WQ_merge\]](#eq:WQ_merge){reference-type="eqref" reference="eq:WQ_merge"}. The other identities [\[eq:R_merge\]](#eq:R_merge){reference-type="eqref" reference="eq:R_merge"}--[\[eq:W_merge\]](#eq:W_merge){reference-type="eqref" reference="eq:W_merge"} have a similar probabilistic interpretation. Stacking vertices vertically or horizontally results in a Markov mapping which commutes with the projection $\pi$ in the same way as described in [\[eq:R_merge\]](#eq:R_merge){reference-type="eqref" reference="eq:R_merge"}--[\[eq:WQ_merge\]](#eq:WQ_merge){reference-type="eqref" reference="eq:WQ_merge"}. We need an instance of this stacking for queue vertex models on the whole line. Let us take a queue vertex model in $\left\{ -n,\ldots,-1 \right\}\times \mathbb{Z}$ with empty input from the left and vertex weights $\mathbb{W}^{(-m)}_{s_1,s_2,u}$ in the column $-m$, $m=1,\ldots,n$. Assume that the parameters $s_1,s_2,u$ make these weights nonnegative. More precisely, we assume that the weights are of the form $\mathbb{W}^{(-m)}_{q^{-\mathsf{P}/2},s_m^{(v)},u/v_m}$ for the $q$-PushTASEP [\[eq:qPushTASEP_weights_line_steady\]](#eq:qPushTASEP_weights_line_steady){reference-type="eqref" reference="eq:qPushTASEP_weights_line_steady"}, or $(uy_m;q)_\infty\hspace{1pt} \mathbb{W}^{(-m),\mathrm{qBos}}_{0,uy_m}$ for the $q$-Boson process [\[eq:qBoson_weights_line_steady_limiting\]](#eq:qBoson_weights_line_steady_limiting){reference-type="eqref" reference="eq:qBoson_weights_line_steady_limiting"}. In both cases, we can define the steady state distribution of this model as in . Denote the corresponding random tuple by $\mathbf{M}=(\mathbf{M}(-n),\ldots,\mathbf{M}(-1) )$. **Lemma 65**. *Fix a partition $\left\{ 0,1,\ldots,n \right\}= I_0\sqcup \ldots \sqcup I_k$ as in , and let $\pi$ be the corresponding projection.* *Let the random configurations $\mathbf{V} = (\mathbf{V}(1),\mathbf{V}(2),\dots)$ and $\tilde{\mathbf{V}} = (\tilde{\mathbf{V}}(1),\tilde{\mathbf{V}}(2),\dots)$ be sampled from the vertex models shown on the left and right in , respectively. That is, $\mathbf{V}$ is the output of the original $n$-color queue vertex model run in steady state. The model for $\tilde{\mathbf{V}}$ has input $\pi(\mathbf{M})$, and $k$-color queue vertex weights $\mathbb{W}^{(-\pi(m))}_{s_1,s_2,u}$ in the column $-m$, $m=1,\ldots,n$. Then $$\pi(\mathbf{V}) \stackrel{d}{=} \tilde{\mathbf{V}} .$$ Moreover, the distribution of the $n$-tuple $\pi(\mathbf{M})$ is the steady state for the system of $n$ $k$-colored queues in tandem in the right side of .* *Proof.* Both claims follow from an inductive application of the color merging property [\[eq:WQ_merge\]](#eq:WQ_merge){reference-type="eqref" reference="eq:WQ_merge"}. ◻ {#fig:stacking_for_merge} **Lemma 66**. *Let $\pi$ be a projection which merges colors $1$ and $0$. Under $\pi$, the $n$-color queue vertex weights $\mathbb{W}^{(-1)}_{s_1,s_2,u}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D})$ in the rightmost column turn into the $(n-1)$-color fused stochastic weights [\[eq:fully_fused_stochastic_weights\]](#eq:fully_fused_stochastic_weights){reference-type="eqref" reference="eq:fully_fused_stochastic_weights"} $$\label{eq:prelimit_from_merging} W _{s_1s_2^{-1}u,\mathsf{L},\mathsf{M}} (\mathbf{A}',\mathbf{B}';\mathbf{C}',\mathbf{D}') \big\vert_{q^{-\mathsf{L}}=s_1^2,\ q^{-\mathsf{M}}=s_2^{2}} ,$$ where $\mathbf{A}'=(A_2,\ldots,A_n )\in \mathbb{Z}_{\ge0}^{n-1}$, and similarly $\mathbf{B}',\mathbf{C}',\mathbf{D}'$, and $$\mathbf{A}=(\infty,\mathbf{A}'), \qquad \mathbf{B}=(0,\mathbf{B}'), \qquad \mathbf{C}=(\infty,\mathbf{C}'), \qquad \mathbf{D}=(D_1,\mathbf{D}').$$ Here $B_1=0$ because in a queue vertex model, no arrows of color $1$ can enter the column $(-1)$.* *Proof.* We use the explicit expression [\[eq:queue_spec_fully_fused\]](#eq:queue_spec_fully_fused){reference-type="eqref" reference="eq:queue_spec_fully_fused"} for $\mathbb{W}^{(-1)}_{s_1,s_2,u}$. Let $\mathbf{P}'=(P_2,\ldots,P_n )$, which is a part of the summation index in $\mathbb{W}^{(-1)}_{s_1,s_2,u}$. We have $P_1=0$ because $P_1\le B_1$. The projection $\pi$ involves the summation over $D_1$ from $0$ to $\infty$. The latter reduces to the $q$-binomial theorem [@GasperRahman (1.3.2)]: $$\begin{aligned} &\sum_{D_1=0}^{\infty} \mathbb{W}^{(-1)}_{s_1,s_2,u}(\mathbf{A},\mathbf{B};\mathbf{C},\mathbf{D}) = \frac{(s_1^{-1}s_2 u; q)_{\infty}}{(s_1s_2u ; q)_{\infty}} \sum_{\mathbf{P}'} \frac{(s_1s_2/u ; q)_{|\mathbf{P}'|} (s_1u/s_2 ; q)_{|\mathbf{B}'-\mathbf{P}'|}} {(s_1^2 ; q)_{|\mathbf{B}'|}}\hspace{1pt} \\&\hspace{10pt}\times q^{\sum_{2\le i<j\le n}\left(B_i-P_i\right) P_j} (s_1s_2u^{-1})^{|\mathbf{B}'|-|\mathbf{P}'|} (s_1^{-1}s_2u)^{|\mathbf{D}'|} \prod_{i=2}^n \frac{(q;q)_{B_i}}{(q;q)_{P_i}(q;q)_{B_i-P_i}} \frac{(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}} \\&\hspace{10pt}\times \hspace{1pt} q^{\sum_{2\le i<j\le n} D_i (C_j-P_j)} (s_1^2 ; q)_{|\mathbf{D}'|} \sum_{D_1=0}^{\infty} q^{D_1 (|\mathbf{C}'|-|\mathbf{P}'|)} (us_2/s_1)^{D_1} \frac{ (s_1^2 q^{|\mathbf{D}'|} ; q)_{D_1}} {(q;q)_{D_1}} \\& = \frac{(s_1^{-1}s_2 u; q)_{\infty}}{(s_1s_2u ; q)_{\infty}} \sum_{\mathbf{P}'} \frac{(s_1s_2/u ; q)_{|\mathbf{P}'|} (s_1u/s_2 ; q)_{|\mathbf{B}'-\mathbf{P}'|}} {(s_1^2 ; q)_{|\mathbf{B}'|}}\hspace{1pt} \\&\hspace{10pt}\times q^{\sum_{2\le i<j\le n}\left(B_i-P_i\right) P_j} (s_1s_2u^{-1})^{|\mathbf{B}'|-|\mathbf{P}'|} (s_1^{-1}s_2u)^{|\mathbf{D}'|} \prod_{i=2}^n \frac{(q;q)_{B_i}}{(q;q)_{P_i}(q;q)_{B_i-P_i}} \frac{(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}} \\&\hspace{10pt}\times \hspace{1pt} q^{\sum_{2\le i<j\le n} D_i (C_j-P_j)} (s_1^2 ; q)_{|\mathbf{D}'|} \frac{(s_1s_2u q^{|\mathbf{C}'|-|\mathbf{P}'|+|\mathbf{D}'|}; q)_{\infty}} {(s_1^{-1}s_2u q^{|\mathbf{C}'|-|\mathbf{P}'|};q)_{\infty}}. \end{aligned}$$ Canceling out the infinite $q$-Pochhammer symbols, we continue the computation as follows: $$\begin{aligned} &= \sum_{\mathbf{P}'} \frac{(s_1^{-1}s_2 u; q)_{|\mathbf{C}'|-|\mathbf{P}'|} (s_1^2 ; q)_{|\mathbf{D}'|}} {(s_1s_2u ; q)_{|\mathbf{C}'|-|\mathbf{P}'|+|\mathbf{D}'|}} \frac{(s_1s_2/u ; q)_{|\mathbf{P}'|} (s_1u/s_2 ; q)_{|\mathbf{B}'-\mathbf{P}'|}} {(s_1^2 ; q)_{|\mathbf{B}'|}} \hspace{1pt} (s_1s_2u^{-1})^{|\mathbf{B}'|-|\mathbf{P}'|} (s_1^{-1}s_2u)^{|\mathbf{D}'|} \hspace{1pt} \\&\hspace{40pt}\times q^{\sum_{2\le i<j\le n}\left(B_i-P_i\right) P_j} q^{\sum_{2\le i<j\le n} D_i (C_j-P_j)} \prod_{i=2}^n \frac{(q;q)_{B_i}}{(q;q)_{P_i}(q;q)_{B_i-P_i}} \frac{(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}} . \end{aligned}$$ From [\[eq:fully_fused_stochastic_weights\]](#eq:fully_fused_stochastic_weights){reference-type="eqref" reference="eq:fully_fused_stochastic_weights"}, one readily sees that the resulting expression matches [\[eq:prelimit_from_merging\]](#eq:prelimit_from_merging){reference-type="eqref" reference="eq:prelimit_from_merging"}, and we are done. ◻ Now, we can formulate and prove our main statement about the color merging property of the stationary distributions. **Proposition 67**. *We have $\pi_* \mu_y = \mu_{y'}$. Here $\mu_y$ is the stationary distribution of the $n$-colored $q$-PushTASEP or the $q$-Boson process on $\mathbb{Z}$ with the densities of the colors depending on the parameters $y_1>\ldots>y_n>0$ via [\[eq:h_current_qpush\]](#eq:h_current_qpush){reference-type="eqref" reference="eq:h_current_qpush"} or [\[eq:q_boson_rho_v\]](#eq:q_boson_rho_v){reference-type="eqref" reference="eq:q_boson_rho_v"}, respectively. The distribution $\mu_{y'}$ is stationary for the corresponding $k$-colored system, and the parameters $y'$ are obtained from $y$ by merging as in [\[eq:y_j\_prime_new_densities\]](#eq:y_j_prime_new_densities){reference-type="eqref" reference="eq:y_j_prime_new_densities"}.* *Proof.* Arguing inductively, it suffices to consider the merging any two consecutive colors $m$ and $m+1$. Case (1) with $m=0$ is special, we treat it separately first. For $m\ge1$, we need to show that the output of the column $-(m+2)$ (distributed as $\mu_{(y_{m+2},\ldots,y_n)}$), passed through two consecutive columns with infinitely many arrows of color $m$ and parameters $y_{m+1}$ and $y_m$, is distributed as $\mu_{(y_{m},\ldots,y_n)}$. After that, we can pass the output of the column $(-m)$ into further columns, and the final output will be distributed as $\mu_{y'}$ by the very definition. Thus, in Case (2) it suffices to take $m=1$ and merge the colors $1$ and $2$. **Case (1). Step 1.** Consider the system of $n$ columns of queue vertex models ("queues in tandem") which produces the $n$-color stationary distribution $\mu_y$. Let us denote the output of this system by $\mathbf{V}=(\mathbf{V}(1),\mathbf{V}(2),\ldots )$. We know that the color merged output $\pi(\mathbf{V})$ is the same as the output of $n$ queues with $n-1$ colors, where the columns $-2,\ldots,-n$ do not change, and $\pi$ is applied to the rightmost column $(-1)$. Indeed, $\pi$ affects only the column $(-1)$, and erases the color $1$ which does not appear in columns $(-m)$, $m\ge2$. It thus suffices to show that $\pi(\mathbf{V})$ is distributed as $\mu_{y'}$, that is, it is stationary for the $(n-1)$-color interacting particle system. **Case (1). Step 2.** Applying , we see that the weights in the rightmost column $(-1)$ become the $q$-PushTASEP or $q$-Boson specializations of the $(n-1)$-color weights [\[eq:prelimit_from_merging\]](#eq:prelimit_from_merging){reference-type="eqref" reference="eq:prelimit_from_merging"}. More precisely, for the $q$-PushTASEP, we have the following weights in the column $(-2)$ and the column $(-1)$: $$\label{eq:qpush_line_merge_2_weights} \mathbb{W}^{(-2)}_{q^{-\mathsf{P}/2},s,-q^{1-\mathsf{P}/2}s^{-1} uy_2}\Big\vert_{s=0} \quad \textnormal{and} \quad W_{-q^{1-\mathsf{P}}s^{-2}uy_1,\mathsf{P},\mathsf{M}} \Big\vert_{s=0},$$ where $q^{-\mathsf{M}}=s^2$ (see ). Note in particular that for the $q$-PushTASEP, the queue weights $\mathbb{W}^{(-m),\mathrm{queue}}_{\xi,\alpha_m,\nu_m}$ (see [\[eq:qPushTASEP_weights_line_steady\]](#eq:qPushTASEP_weights_line_steady){reference-type="eqref" reference="eq:qPushTASEP_weights_line_steady"}) do not scale with $\epsilon$ which entered the weights $\mathbb{W}^{(-m),\mathrm{line}}_{\alpha_m,\nu_m}$ in the quadrant through the scaling [\[eq:definition_of_y\_m_for_qpushTASEP\]](#eq:definition_of_y_m_for_qpushTASEP){reference-type="eqref" reference="eq:definition_of_y_m_for_qpushTASEP"}. For the $q$-Boson process, these weights are, respectively, $$\label{eq:qBos_line_merge_2_weights} \mathbb{W}^{(-2)}_{\epsilon,s,\epsilon s^{-1}uy_2}\Big\vert_{\epsilon\to0 \text{ then } s=0} \quad \textnormal{and} \quad W_{(\epsilon/s)^2 uy_1,\mathsf{L},\mathsf{M}}\Big\vert_{\epsilon\to0 \text{ then } s=0},$$ where $q^{-\mathsf{L}}=\epsilon^2$ and $q^{-\mathsf{M}}=s^2$ (see ). Let us now choose the auxiliary weights with which [\[eq:qpush_line_merge_2\_weights\]](#eq:qpush_line_merge_2_weights){reference-type="eqref" reference="eq:qpush_line_merge_2_weights"} or [\[eq:qBos_line_merge_2\_weights\]](#eq:qBos_line_merge_2_weights){reference-type="eqref" reference="eq:qBos_line_merge_2_weights"}, respectively, satisfy the Yang--Baxter equation. They are found from . The auxiliary weights for the $q$-PushTASEP and the $q$-Boson are exactly the same, and they have the form $$\label{eq:aux_weights_qpush_qboson} \mathbb{W}^{(-2)}_{s,s,y_2/y_1}\Big\vert_{s=0}.$$ One can check that the weights [\[eq:qpush_line_merge_2\_weights\]](#eq:qpush_line_merge_2_weights){reference-type="eqref" reference="eq:qpush_line_merge_2_weights"}, [\[eq:qBos_line_merge_2\_weights\]](#eq:qBos_line_merge_2_weights){reference-type="eqref" reference="eq:qBos_line_merge_2_weights"}, and [\[eq:aux_weights_qpush_qboson\]](#eq:aux_weights_qpush_qboson){reference-type="eqref" reference="eq:aux_weights_qpush_qboson"} are nonnegative under our restrictions on the parameters (in particular, recall that $y_1>y_2>0$). {#fig:stat_2 width=".85\\textwidth"} **Case (1). Step 3.** Let us show that the random output $\pi(\mathbf{V})$ may be sampled using another vertex model which is displayed in , right. We claim that $$\label{eq:M_pi_M_stationarity} \bigl(\mathbf{M}(-n),\ldots,\mathbf{M}(-2),\pi(\mathbf{M}(-1)) \bigr) \stackrel{d}{=} \bigl(\mathbf{M}'(-n),\ldots,\mathbf{M}'(-2),\mathbf{M}'(-1) \bigr).$$ By , the distribution of $(\mathbf{M}(-n),\ldots,\mathbf{M}(-2))$ in , right, can be generated by infinitely many dashed horizontal lines below the picture (carrying the corresponding queue vertex weights). Using the Yang--Baxter equation, we may bring an arbitrary number, say, $K$, of these dashed horizontal lines into the space between the auxiliary line (solid horizontal line) and the output $(\mathbf{M}'(-n),\ldots,\mathbf{M}'(-2),\mathbf{M}'(-1) )$. See for an illustration. Taking the limit as $K\to\infty$, we get [\[eq:M_pi_M\_stationarity\]](#eq:M_pi_M_stationarity){reference-type="eqref" reference="eq:M_pi_M_stationarity"} because its left-hand side is the steady state of the system of $n$ queues with $(n-1)$ colors. {#fig:step3_merging width=".7\\textwidth"} **Case (1). Step 4.** Now, using the Yang--Baxter equation, we may move the rightmost vertical line in , right, all the way to the left of the column $-n$. There, this vertical line can be removed because it carries no arrows. This application of the Yang--Baxter equation transforms the lattice from the right panel of to the left one. The resulting output configuration from the $(n-1)$-column system is distributed as $\pi(\mathbf{V})$, and we are done. This shows that $\pi(\mathbf{V})$ has the same distribution as the output of the system of $n-1$ queues with $(n-1)$ colors and parameters $y_2> \ldots > y_n>0$. Thus, $\pi(\mathbf{V})$ is distributed as $\mu_{(y_2,\ldots,y_n)}=\mu_{y'}$, as desired. The applications of the Yang--Baxter equation in Steps **3** and **4** are similar to what we used in the colored Burke's theorem in . **Case (2).** Now let us consider the merging of colors $1$ and $2$. Consider the system of $n-1$ queues in tandem, which have $n-1$ colors, and parameters $y_2>\ldots>y_n$. Denote its output by $\mathbf{V}'=(\mathbf{V}'(1),\mathbf{V}'(2),\ldots )$; it is distributed as the $(n-1)$-colored stationary distribution $\mu_{(y_2,\ldots,y_n )}$. We need to replace the color $2$ by $1$ in $\mathbf{V}'$ and pass it as an input into the column $(-1)$ with parameter $y_1$. To complete the proof, it suffices to show that the output $\mathbf{V}$ of the column $(-1)$ in this scenario has the distribution $\mu_{(y_1,y_3,\ldots,y_n )}$, see [\[eq:y_j\_prime_new_densities\]](#eq:y_j_prime_new_densities){reference-type="eqref" reference="eq:y_j_prime_new_densities"}. Notice that by , the queue vertex weights in the column $(-1)$ do not depend on the arrows of color $1$ incoming from the left. Therefore, instead of replacing the color $2$ by $1$ in $\mathbf{V}'$, we may replace the color $2$ by $0$, and pass the result into the column $(-1)$. Denote by $\pi^{\circ}$ the projection which merges the colors $2$ and $0$. By Case (1), $\pi^{\circ}(\mathbf{V}')$ has $n-2$ colors $3,\ldots,n$ and is distributed as $\mu_{(y_3,\ldots,y_n )}$. Passing $\pi^{\circ}(\mathbf{V}')$ through the column $(-1)$ with the parameter $y_1$ adds the color $1$ and, by the very definition of the queue vertex model, produces $\mathbf{V}$ with the distribution $\mu_{(y_1,y_3,\ldots,y_n )}$. This completes the proof. ◻ [A. Aggarwal, Columbia University, New York, NY, USA and Clay Mathematics Institute, Denver, CO, USA]{.smallcaps} E-mail: `[email protected]` [M. Nicoletti, Massachusetts Institute of Technology, Cambridge, MA]{.smallcaps} E-mail: `[email protected]` [L. Petrov, University of Virginia, Charlottesville, VA]{.smallcaps} E-mail: `[email protected]` [^1]: Throughout the paper, we say that an event occurs at rate $\alpha>0$ if the random time $\zeta$ till the occurrence is exponentially distributed with parameter $\alpha$, that is, $\mathbb{P}(\zeta>t)=e^{-\alpha t}$ for $t\ge0$. [^2]: Throughout this limit, we assume that $0<q<1$, and later we will specialize the limit to $q=0$ where needed. In other words, $q^0$ should be treated as $1$ before and after the limit.

arxiv_math

--- abstract: | This paper considers a type of incremental aggregated gradient (IAG) method for large-scale distributed optimization. The IAG method is well suited for the parameter server architecture as the latter can easily aggregate potentially staled gradients contributed by workers. Although the convergence of IAG in the case of deterministic gradient is well known, there are only a few results for the case of its stochastic variant based on streaming data. Considering strongly convex optimization, this paper shows that the streaming IAG method achieves linear speedup when the workers are updating frequently enough, even if the data sample distribution across workers are heterogeneous. We show that the expected squared distance to optimal solution decays at ${\cal O}( (1+T) / (nt) )$, where $n$ is the number of workers, $t$ is the iteration number, and $T/n$ is the update frequency of workers. Our analysis involves careful treatments of the conditional expectations with staled gradients and a recursive system with both delayed and noise terms, which are new to the analysis of IAG-type algorithms. Numerical results are presented to verify our findings. author: - "Xiaolu Wang$^{1}$, Cheng Jin$^{2}$, Hoi-To Wai$^{1}$, Yuantao Gu$^{2}$ [^1]" bibliography: - sIAG.bib title: " **Linear Speedup of Incremental Aggregated Gradient Methods on Streaming Data** " --- # INTRODUCTION Distributed optimization is an important algorithmic paradigm that has received immense attention due to its wide applicability in machine learning, signal processing, control, etc [@yang2019survey; @chang2020distributed; @li2020federated]. It is suitable for a broad range of circumstances where data are dispersed across multiple entities, e.g., CPU cores, computing clusters, wireless sensors, and wearable devices [@yang2019survey]. Classical distributed optimization deals with the case when each worker holds a fixed set of local data samples that is available at any time, which is also referred to as the *batch data learning* setting. However, with the growing scenarios including federated learning [@li2020federated] where the data are acquired in an online fashion (e.g., online review and social network platforms) and each data sample is allowed to be used only once [@chang2020distributed], it is important to adapt the distributed optimization algorithms to the streaming data setting. This paper is concerned with the following stochastic optimization problem: $$\label{eq:opt} \begin{split} \min_{\bm{w}\in\mathbb{R}^{d}}~ & \frac{1}{n} \sum_{i=1}^{n} F_i(\bm{w}),~~ F_i(\bm{w}) := \mathbb{E}_{ \bm{\xi}_i \sim \mathcal{D}_i } \left[ f_i (\bm{w};\bm{\xi}_i) \right], \end{split}$$ where ${\cal D}_i$ represents the data distribution supported on the sample space $\Xi_i$ accessible by worker $i$. The optimization problem shall be solved cooperatively by $n$ workers. For $i \in [n]$, $\bm{\xi}_i \in \Xi_i$, the sampled local loss function $f_i( \cdot ; \bm{\xi}_i )$ is continuously differentiable and is known to the $i$th worker. We denote by $F( \bm{w}) := (1/n) \sum_{i=1}^n F_i(\bm{w})$ the global objective function and assume that $F( \bm{w})$ is strongly convex. Consider solving [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} in a distributed fashion under the coordination of a central server communicating with the $n$ workers. Each of the worker has access to an independent streaming data source ${\cal D}_i$ and the latest iterate $\bm{w}$ stored at the server, with which it computes stochastic estimates of the gradient $\nabla F_i( \bm{w})$. We concentrate on an asynchronous setting where workers can be idle in some iterations, due to, e.g., network connection failure. Notice that using direct average of the stochastic gradients may result in a non-converging algorithm unless the local loss satisfies some form of similarity conditions; see the related studies on FedAvg in [@mcmahan2017communication; @karimireddy2020scaffold]. Remedies such as designing stochastic control variate have been proposed, e.g., [@karimireddy2020scaffold; @mishchenko2022proxskip]. To deal with the worker asynchrony issue over heterogeneous data, this paper utilizes a parameter server (PS) architecture [@assran2020advances; @aytekin2016analysis] for distributed optimization of [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"}. In this setting, the PS maintains a buffer that stores gradient information reported from the workers. The coordinating server then aggregates the information stored in the buffer to update its iterate. While performing gradient aggregation seems to alleviate the reliance on data similarity, we note the aggregated information may contain staled gradients due to worker asynchrony that can affect convergence. To this end, this work inquires the following questions: *Does the above distributed algorithm solve [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"}? Can it achieve linear speedup in convergence rate compared to centralized/sequential SGD solving [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} taking one sample per iteration?* {#fig:ps width="0.8\\columnwidth"} **Our Contributions.** In this paper, we provide an affirmative answer to the above questions. We study a streaming incremental aggregated gradient (sIAG) method adapted from the incremental aggregated gradient (IAG) method [@blatt2007convergent] to handle streaming data. Our key results are as follows: - We show that the sIAG method converges in expectation to the global optimum solution of [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} under mild conditions and *without requiring an explicit similarity condition* between $F_i(\bm{w})$. - Suppose that the maximum stateness of the aggregated gradient is $T$, we show that the expected squared error between the $t$th iterate and optimal solution to [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} is ${\cal O}( \frac{1}{t} \frac{ \sigma^2 (1+T) }{n} )$ for the sIAG method, where $\sigma^2$ is the variance of a gradient sample. As such, the sIAG method achieves linear speedup despite that it utilizes staled gradient information with asynchronous workers. We remark that our analysis utilizes new analysis techniques to yield the tight bound for sIAG; see Sec. [3](#sec:ana){reference-type="ref" reference="sec:ana"}. **Related Works.** The studies of IAG method and its variants based on batch data have generated substantial interests after [@blatt2007convergent]. Notably, [@feyzmahdavian2014delayed; @gurbuzbalaban2017convergence; @vanli2018global] have worked on analyzing the convergence rate of IAG under the bounded delay assumption. Extensions have been considered to speed up convergence, e.g., [@chen2018lag; @wu2022delay] studied adaptive strategies for gradient aggregation, [@wai2020accelerating] utilized local Hessian information. On the other hand, the study of IAG-like methods under the streaming data setting has received attention from the machine learning community. The closest work to ours is [@lian2015asynchronous] that studied an sIAG method with smooth objective function under stronger assumption than ours, e.g., drawing independent samples for staled gradient. As mentioned, federated learning algorithms such as FedAvg [@mcmahan2017communication], FedProx [@li2020federated-a], SCAFFOLD [@karimireddy2020scaffold] adopted similar aggregation technique for the local information reported from the workers. # STREAMING IAG METHOD This section introduces the sIAG method and discusses its implementation in a distributed optimization setting with the PS. To motivate the algorithmic idea of sIAG method, below we first briefly review the IAG method for [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} with batch data [@blatt2007convergent]. Consider a distributed computing architecture with a PS and $n$ workers [@assran2020advances]; see Fig. [1](#fig:ps){reference-type="ref" reference="fig:ps"}. The IAG method assumes that each worker $i \in [n]$ has *full access* to an oracle that queries the gradient of its local loss function $F_i( \bm{w})$ at any point $\bm{w}\in \mathbb{R}^d$, e.g., when the local dataset is fixed. The workers send their computed gradient to the PS that keeps a buffer of $n$ gradient vectors storing the most recent copy of the computed gradient from each worker. Note that during most iterations, this buffer may contain *staled gradient* when the workers is idle at the current iteration. To fix notations, we let ${\cal A}_t \subseteq [n]$ to be set of active workers at iteration $t$ and define $$\label{eq:tau-def} \tau_i(t) = \max\big\{ \tau : \tau \leq t,~i \in {\cal A}_\tau \big\}.$$ In other words, $\tau_i(t) \leq t$ indicates the iteration number in which the gradient stored in the PS from worker $i$ is computed. In iteration $t \geq 0$, the coordinating server performs the update: $$\begin{aligned} \bm{w}^{t+1} = \bm{w}^{t} - \frac{\eta}{n} \sum_{i=1}^{n} \nabla F_i (\bm{w}^{\tau_i(t)}) , \label{eq:iag} \end{aligned}$$ where $\eta > 0$ is the step size. Observe that the server directly aggregates all local (possibly staled) gradients in the PS as the descent direction. Despite that [\[eq:iag\]](#eq:iag){reference-type="eqref" reference="eq:iag"} utilizes some staled gradients in the updates, a key result established in [@gurbuzbalaban2017convergence; @vanli2018global] is that IAG admits *linear convergence* towards the optimal solution of [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} when $F(\bm{w})$ is strongly convex and smooth, similar to a *centralized* gradient method for [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} under the same setting. Especially, this convergence rate holds regardless of the differences between the local loss functions $F_i(\bm{w})$. In comparison, without the aggregation step performed with the PS, the distributed algorithm that relies on aggregating only the current gradients reported by ${\cal A}_t$ may converge sublinearly and requires further modification such as taking a diminishing step size [@bertsekas2011incremental]. Initialization $\bm{w}^0$, step sizes $\{ \eta_t\}_{t\geq 0}$. At the PS, initialize the buffer with $\bm{g}_i^{-1} = {\bm 0}$ for $i=1,\ldots,n$. A set of workers ${\cal A}_t \subseteq [n]$ is selected/active. Take $\bm{w}^t$ from the PS and draw a sample $\bm{\xi}_i^t \sim {\cal D}_i$. Compute the stochastic gradient $\nabla f_i ( \bm{w}^t; \bm{\xi}_i^t )$ and send back to the PS. Update the buffer as $$\begin{aligned} &\bm{g}_i^t \leftarrow \nabla f_i ( \bm{w}^t; \bm{\xi}_i^t )~\text{for}~i \in {\cal A}_t, \\ &\bm{g}_i^t \leftarrow \bm{g}_i^{t-1}~\text{for}~i \notin {\cal A}_t \\[-.9cm] \end{aligned}$$ Compute sIAG update: $\bm{w}^{t+1} \leftarrow \bm{w}^t - (\eta_t/n) \sum_{i=1}^n \bm{g}_i^t$. **Streaming IAG Method.** We are interested in a variant of the IAG method utilizing *streaming data*. We consider the generic form for [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} where each of the local loss function is (possibly) stochastic. Unlike the IAG method, the streaming IAG (sIAG) method considers that each worker only has access to a *stochastic oracle* that queries an unbiased and independent estimate for the gradient of local loss function, denoted by $$\nabla f_i( \bm{w}; \bm{\xi}_i ),~~\bm{\xi}_i \sim {\cal D}_i.$$ Note that $\mathbb{E}[ \nabla f_i( \bm{w}; \bm{\xi}_i ) ] = \nabla F_i( \bm{w})$. The sIAG method reads $$\begin{aligned} \bm{w}^{t+1} = \bm{w}^{t} - (\eta_t/n) \, \bm{g}^t, \label{eq:siag} \end{aligned}$$ for any $t \geq 0$, where $\eta_t > 0$ is a (possibly time varying) step size and $$\begin{aligned} & \bm{g}^t \coloneqq \sum_{i=1}^{n} \nabla f_i ( \bm{w}^{\tau_{i}(t)}; \bm{\xi}_i^{\tau_{i}(t)} ) \label{eq:gt} \end{aligned}$$ denotes the aggregated *stochastic gradients*. Note that in the above, we adopted the same notations as in the IAG method where the index $\tau_i(t)$ was defined in [\[eq:tau-def\]](#eq:tau-def){reference-type="eqref" reference="eq:tau-def"}. The sIAG method [\[eq:siag\]](#eq:siag){reference-type="eqref" reference="eq:siag"}, [\[eq:gt\]](#eq:gt){reference-type="eqref" reference="eq:gt"} is motivated by an instantaneous gradient computation model that can be readily implemented on the parameter server architecture; as summarized in Algorithm [\[alg:siag\]](#alg:siag){reference-type="ref" reference="alg:siag"}. Observe that the algorithm requires active workers to return the stochastic gradient before the current iteration concludes. This restriction, while mild as only stochastic gradients are required, might be relaxed by allowing further delays between the sampled gradient and the iterate in which it is computed. However, in the interest of space, we focus on a current simplified setting. We recall that as $\tau_i(t) = t$ for $i\in\mathcal{A}_t$ and $\tau_i(t) < t$ for $i\notin\mathcal{A}_t$, we may express the aggregated gradient [\[eq:gt\]](#eq:gt){reference-type="eqref" reference="eq:gt"} alternatively as an incremental update via $$\begin{aligned} &\bm{g}^t = \bm{g}^{t-1} \hspace{-1mm}-\hspace{-1.5mm} \sum_{i\in\mathcal{A}_t} \hspace{-1mm}\nabla f_i ( \bm{w}^{\tau_{i}(t-1)}; \bm{\xi}_i^{\tau_{i}(t-1)} ) \hspace{-1mm}+\hspace{-1.5mm} \sum_{i\in\mathcal{A}_t}\hspace{-1mm} \nabla f_i ( \bm{w}^t; \bm{\xi}_i^t ) . \end{aligned}$$ We remark that the above update recursion for $\bm{g}^t$ is related to the famous SAG method [@schmidt2017minimizing]. It is reduced to the SAG method when $f_i \equiv f$, $\bm{\xi}_i^t = \bm{\xi}^t$, and ${\cal D}_i \equiv {\cal D}$ has a finite support for all $i \in [n]$. While convergence guarantees for the IAG method [\[eq:iag\]](#eq:iag){reference-type="eqref" reference="eq:iag"} are well known, the stochastic variant, sIAG, considered in [\[eq:siag\]](#eq:siag){reference-type="eqref" reference="eq:siag"}, [\[eq:gt\]](#eq:gt){reference-type="eqref" reference="eq:gt"} has received less attention. As the first step towards understanding the behavior of sIAG, the next section analyzes the convergence rate of sIAG under the standard setting when $F(\bm{w})$ is strongly convex and smooth. # CONVERGENCE ANALYSIS {#sec:ana} We shall begin by stating some assumptions on [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} and the sIAG method that are necessary for our analysis. First, **Assumption 1**. *Problem [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} satisfies: (i) $F(\bm{w})$ is $\mu$-strongly convex with $\mu > 0$; (ii) for any $i \in [n]$, the gradient $\nabla f_i( \bm{w})$ is $L$-Lipschitz continuous.* The above specifies the function class of interest for [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"}. We define $\bm{w}^\star = \mathop{\mathrm{arg\,min}}_{ \bm{w}\in \mathbb{R}^d} F(\bm{w})$. We also assume: **Assumption 2**. *There exists a constant $\sigma \geq 0$ such that $$\mathbb{E}\| \nabla f_i(\bm{w};\bm{\xi}_i) - \nabla F_i(\bm{w}) \|_2^2 \leq \sigma^2 ( 1 + \| \bm{w}- \bm{w}^\star \|^2 ),$$ for all $i \in [n]$, $\bm{w}\in \mathbb{R}^d$.* **Assumption 3**. *There exists $T \geq 0$ such that $$\tau_i(t) \geq t-T,~\forall~i \in [n], ~t \geq 0.$$* The above assumptions state that the stochastic gradients computed in sIAG has bounded variance, and the staled gradient delay is bounded by $T$. Notice that $n / T$ corresponds roughly to the number of workers that are active at any iteration. We define respectively the following notations for the $t$th suboptimality gap and its delayed version: $$\begin{aligned} {\cal E}_t \coloneqq \mathbb{E}\|\bm{w}^t - \bm{w}^* \|_2^2, \quad {\cal E}^{\max}_t \coloneqq \max_{s\in[ (t-2T)_+,t]} {\cal E}_s, \end{aligned}$$ where $x_+ \coloneqq \max\{x,0\}$ for $x\in\mathbb{R}$. Furthermore, it is instrumental to define the following filtration: $${\cal F}_t \coloneqq \sigma( \bm{\xi}_i^s , i \in [n], s = 0,\ldots, t-1 ),$$ where $\sigma(\cdot)$ denotes the sigma algebra generated by the random variables in the operand. Observe that for any $t \geq 0$, $\bm{w}^t$ is measurable with respect to (w.r.t.) ${\cal F}_t$. We shall use the shorthand notation $\mathbb{E}_t[\cdot] := \mathbb{E}[ \cdot | {\cal F}_t ]$ for conditional expectation. From [\[eq:siag\]](#eq:siag){reference-type="eqref" reference="eq:siag"}, we deduce that for any $t \geq 0$, $$\label{eq:ana-iter} \begin{split} {\cal E}_{t+1} & \hspace{-0.5mm}=\hspace{-0.5mm} {\cal E}_t \hspace{-0.5mm}-\hspace{-0.5mm} 2 \eta_t \mathbb{E} \hspace{-0.5mm}\left[\hspace{-0.5mm} \left\langle\hspace{-1mm} \bm{w}^t \hspace{-0.5mm}-\hspace{-0.5mm} \bm{w}^\star , \frac{1}{n} \bm{g}^t \hspace{-0.6mm}\right\rangle \hspace{-0.5mm}\right] \hspace{-1mm}+\hspace{-0.5mm} \eta_t^2 \mathbb{E} \left[ \left\| \frac{1}{n} \bm{g}^t \right\|^2 \right]\hspace{-0.5mm}. \end{split}$$ We shall control the last two terms in the above decomposition. Observe the following lemmas: **Lemma 1**. *Suppose that Assumptions [Assumption 1](#as:function){reference-type="ref" reference="as:function"}--[Assumption 3](#as:delay){reference-type="ref" reference="as:delay"} hold. Then, $$\begin{aligned} & \mathbb{E}\| (1/n) \bm{g}^t \|_2^2 \leq \frac{ 2 \sigma^2 } {n} + {\rm C}_L \max_{ s \in [ (t-T)_+, t ] } {\cal E}_s, \label{eq:gt-bound} \end{aligned}$$ for any $t \geq 0$, where ${\rm C}_L\coloneqq 20 L^2 + \frac{2\sigma^2}{n}$.* The above is a natural consequence of [\[eq:gt\]](#eq:gt){reference-type="eqref" reference="eq:gt"}, where the bound [\[eq:gt-bound\]](#eq:gt-bound){reference-type="eqref" reference="eq:gt-bound"} depends on both the stochastic gradient variance $\sigma^2$ and the delayed optimality gap $\max_{ s \in [ (t-T)_+, t ] } {\cal E}_s \leq {\cal E}^{\max}_t$. The detailed proof can be found in Appendix [6.1](#ap:aa){reference-type="ref" reference="ap:aa"}. The next lemma controls the inner product term in [\[eq:ana-iter\]](#eq:ana-iter){reference-type="eqref" reference="eq:ana-iter"}: **Lemma 2**. *Suppose that Assumptions [Assumption 1](#as:function){reference-type="ref" reference="as:function"}--[Assumption 3](#as:delay){reference-type="ref" reference="as:delay"} hold. Then $$\begin{aligned} &\mathbb{E}\left\langle \bm{w}^t - \bm{w}^*, (1/n) \, \bm{g}^t \right\rangle \label{eq:innerprod-bound} \\ &\geq \frac{\mu}{4} {\cal E}_t - \left[ {\rm C}_LT \eta_{t-T} + \left( \frac{\mu}{4} + \frac{5L^2}{2 \mu} \right) {\rm C}_LT^2 \eta_{t-T}^2 \right] {\cal E}_{t}^{\max} \nonumber \\ &\quad -\left[ 2T\eta_{t-T} + \left( \frac{\mu}{2} + \frac{5L^2}{\mu} \right) T^2 \eta_{t-T}^2 \right] \frac{\sigma^2}{n},~\forall~t \geq 0. \nonumber \end{aligned}$$* The above lemma shows that the inner product term is *lower bounded* by $(\mu/4) {\cal E}_t$ with (negative) perturbation terms that are controllable by the step sizes. Before substituting [\[eq:gt-bound\]](#eq:gt-bound){reference-type="eqref" reference="eq:gt-bound"}, [\[eq:innerprod-bound\]](#eq:innerprod-bound){reference-type="eqref" reference="eq:innerprod-bound"} into [\[eq:ana-iter\]](#eq:ana-iter){reference-type="eqref" reference="eq:ana-iter"} to derive the convergence rate of sIAG method, we highlight that Lemma [Lemma 2](#lem:bb){reference-type="ref" reference="lem:bb"} deviates from the standard analysis for SGD with strongly convex objective function. In the case of standard SGD where $\bm{g}^t = \nabla f( \bm{w}^t ; \bm{\xi}^t )$ with $\mathbb{E}_t[ \bm{g}^t ] = \nabla F( \bm{w}^t )$, the law of total expectation and the strong convexity of $F$ imply that $$\begin{aligned} &\mathbb{E}\langle \bm{w}^t - \bm{w}^\star , \bm{g}^t \rangle = \mathbb{E}\langle \bm{w}^t - \bm{w}^\star , \mathbb{E}_t [\bm{g}^t] \rangle \nonumber \\ &= \mathbb{E}\langle \bm{w}^t - \bm{w}^\star , \nabla F( \bm{w}^t ) \rangle \geq \mu {\cal E}_t. \end{aligned}$$ However, in the sIAG method, $\bm{g}^t$ is a function of $\bm{\xi}_i^s$ for $i \in [n]$ and $s=0,1,\dots,t$. Note that for any $i \notin \mathcal{A}_t$, $\bm{\xi}_i^{\tau_i(t)}$ is $\mathcal{F}_t$-measurable and thus $\mathbb{E}_t [\nabla f_i(\bm{w}_i^{\tau_i(t)};\bm{\xi}_i^{\tau_i(t)})] \neq \nabla F_i(\bm{w}^{\tau_i(t)})$. This implies that $\bm{g}^t$ is *not* independent of $\mathcal{F}_t$ and thus $$\begin{aligned} \mathbb{E}\left\langle \bm{w}^t - \bm{w}^\star, \bm{g}^t \right\rangle &\neq \mathbb{E}\left\langle \bm{w}^t - \bm{w}^\star , \sum_{i=1}^n \nabla F_i( \bm{w}^{\tau_i(t)} ) \right\rangle. \end{aligned}$$ Our remedy is to consider[^2] the following decomposition for the inner product: $$\label{eq:ana-split} \begin{split} & \langle \bm{w}^t - \bm{w}^*, \nabla f_i( \bm{w}^{\tau_i(t)}; \bm{\xi}_i^{\tau_i(t)} ) \rangle \\ & = \langle \bm{w}^t - \bm{w}^{\tau_i(t)} + \bm{w}^{\tau_i(t)} - \bm{w}^*, \nabla f_i( \bm{w}^{\tau_i(t)}; \bm{\xi}_i^{\tau_i(t)} ) \rangle. \end{split}$$ Observe that $\nabla f_i( \bm{w}^{\tau_i(t)}; \bm{\xi}_i^{\tau_i(t)} )$ is independent of $\bm{w}^{\tau_i(t)}$ and thus the simplification $\mathbb{E}_{ \tau_i(t)} [ \langle \bm{w}^{\tau_i(t)} - \bm{w}^\star, \nabla f_i( \bm{w}^{\tau_i(t)}; \bm{\xi}_i^{\tau_i(t)} ) \rangle ] = \langle \bm{w}^{\tau_i(t)} - \bm{w}^\star, \nabla F_i( \bm{w}^{\tau_i(t)} ) \rangle$. Furthermore, we develop the following bound to control the size of difference $\| \bm{w}^t - \bm{w}^{\tau_i(t)} \|_2^2$: **Lemma 3**. *Suppose that Assumptions [Assumption 1](#as:function){reference-type="ref" reference="as:function"} and [Assumption 2](#as:sigma){reference-type="ref" reference="as:sigma"} holds and $\{\eta_t\}_{t\geq0}$ is a monotonically non-increasing sequence. Then, it holds for all $i \in [n]$ and $t \geq 0$ that $$\begin{aligned} \mathbb{E}[ \| \bm{w}^t - \bm{w}^{\tau_{i}(t)} \|_2^2 ] \leq T^2 \eta_{t-T}^2 \left( \frac{2 \sigma^2}{n} + {\rm C}_L{\cal E}^{\max}_{t} \right). \label{eq:wt-diff-bound} \end{aligned}$$* Observe that the bound is composed of the stochastic gradient's variance and the (delayed) optimality gap. Note that the proof is obtained as a consequence of Lemma [Lemma 1](#lem:aa){reference-type="ref" reference="lem:aa"}. Importantly, [\[eq:wt-diff-bound\]](#eq:wt-diff-bound){reference-type="eqref" reference="eq:wt-diff-bound"} allows us to handle the term $\langle \bm{w}^t - \bm{w}^{\tau_i(t)}, \nabla f_i( \bm{w}^{\tau_i(t)}; \bm{\xi}_i^{\tau_i(t)} ) \rangle$ through applying Cauchy-Schwarz inequality. The detailed proofs of Lemma [Lemma 2](#lem:bb){reference-type="ref" reference="lem:bb"}, [Lemma 3](#lem:diff){reference-type="ref" reference="lem:diff"} are found in Appendix [6.2](#ap:bb){reference-type="ref" reference="ap:bb"}, [6.3](#ap:diff){reference-type="ref" reference="ap:diff"}, respectively. Substituting Lemmas [Lemma 1](#lem:aa){reference-type="ref" reference="lem:aa"} and [Lemma 2](#lem:bb){reference-type="ref" reference="lem:bb"} into [\[eq:ana-iter\]](#eq:ana-iter){reference-type="eqref" reference="eq:ana-iter"} directly yields the following recursive system: for any $t \geq 0$, $$\begin{aligned} & {\cal E}_{t+1} \leq (1 - (\mu/2) \, \eta_t) \, {\cal E}_t \label{eq:ana-recur} \\ & + \left[ 1 + 2 T + \left( \frac{\mu}{2} + \frac{5L^2}{\mu} \right) T^2 \eta_{t-T} \right] {\rm C}_L\eta_{t-T}^2 {\cal E}_{t}^{\max} \nonumber \\ & + \left[ 1 + 2T + \left( \frac{\mu}{2} + \frac{5L^2}{\mu} \right) T^2 \eta_{t-T} \right] \eta_{t-T}^2 \frac{2 \sigma^2}{n}. \nonumber \end{aligned}$$ If we ignore the last term proportional to $\sigma^2/n$, then [\[eq:ana-recur\]](#eq:ana-recur){reference-type="eqref" reference="eq:ana-recur"} reduces into a contracting recursion with delays that has been studied by [@feyzmahdavian2014delayed]. Subsequently, ${\cal E}_t$ converges to zero at a linear rate when a constant step size is used. The introduction of the noise-related terms in [\[eq:ana-recur\]](#eq:ana-recur){reference-type="eqref" reference="eq:ana-recur"} has led to a new recursive system with delayed terms that have not been covered in prior works. As an attempt to derive a tight bound, we fix the step size as $$\eta_t = \beta / (t + \gamma)$$ for some $\gamma, \beta > 0$ and obtain the following convergence rates for ${\cal E}_t$ using induction: **Theorem 1**. *Suppose that $\beta > 2/\mu$ and $$\begin{aligned} \gamma \geq 2T + \max \left\{ \frac{16C_L\beta^2}{\mu\beta-2}\bar{\rho}(T), \sqrt{\frac{8C_L\beta^2}{\mu\beta-4}\bar{\rho}(T)} \right\}, \end{aligned}$$ where $\bar{\rho}(T) \coloneqq 1 + 2 T + \left( \frac{\mu}{2} + \frac{5L^2}{\mu} \right) \beta T$. Let $$\begin{aligned} & \delta_1 \coloneqq \frac{32 \beta^2 \bar{\rho}(T)}{ \mu \beta - 2 } + 1, ~\delta_2 = \gamma^2 {\cal E}_0. \end{aligned}$$ Then, it holds for all $t \geq 0$ that $$\begin{aligned} &{\cal E}_{t} \leq \frac{\delta_1}{\gamma+t} \frac{\sigma^2}{n} + \frac{\delta_2}{(\gamma+t)^2}. \label{eq:rate} \end{aligned}$$* The detailed proof can be found in Appendix [6.4](#ap:thm){reference-type="ref" reference="ap:thm"}. Theorem [Theorem 1](#thm){reference-type="ref" reference="thm"} indicates that the expected squared distance to optimal solution decays at ${\cal O}( (1+T) / (nt) )$. We note that in the interest of space, the constants in the bound are not fully optimized. In general, obtaining tight bounds for recursive system of the form [\[eq:ana-recur\]](#eq:ana-recur){reference-type="eqref" reference="eq:ana-recur"} is an interesting open problem of independent interest. The bound [\[eq:rate\]](#eq:rate){reference-type="eqref" reference="eq:rate"} shows that the sIAG method converges (in expectation) towards the optimal solution of [\[eq:opt\]](#eq:opt){reference-type="eqref" reference="eq:opt"} at the rate of ${\cal O}( \frac{1}{t} \cdot \frac{( 1+T ) \sigma^2 }{n} )$. We notice that: (A) the sublinear rate of ${\cal O}(1/t)$ is similar to that of existing analysis with stochastic gradient methods [@moulines2011non] and is in the same order of the minimax lower bound [@agarwal2012information], (B) the constant factor ${( 1+T ) \sigma^2 } / {n}$ indicates that *linear speedup* can be achieved when the delay satisfies $T = {\cal O}(1)$. This rate is reasonable since the number of samples taken per iteration is approximately $n/T$, the linear speedup ratio should be of the same order. We note that similar slow down due to the delay $T$ is also reported in the analysis for IAG [@gurbuzbalaban2017convergence; @vanli2018global]. # NUMERICAL SIMULATIONS We evaluate the empirical performance of the sIAG algorithm on synthetic data. We independently generate $n$ parameters $\bm{w}_1^*,\bm{w}_2^*,\dots,\bm{w}_n^*$ according to the uniform distribution on $[0,1]^d$. Each data point sampled by worker $i$ takes the form $\bm{\xi}_i=(\bm{A}_i,\bm{y}_i)$, where $\bm{A}_i\in\mathbb{R}^{p\times d}$ and $\bm{y}_i\in\mathbb{R}^p$. The entries of $\bm{A}_i$ are independent and follow the Gaussian distribution ${\cal N}(0,1)$ and $\bm{y}_i \sim \mathcal{N}(\bm{A}_i\bm{w}_i^*,\sigma^2\bm{I}_p)$. The loss functions are defined as $f_i(\bm{w};\bm{A}_i,\bm{y}_i)= \frac{1}{2} \lVert\bm{A}_i\bm{w}-\bm{y}_i\rVert_2^2$ for $i\in[n]$. If $[ \bm{A}_1; \cdots; \bm{A}_n ]$ is full-rank (which holds almost surely when $nd \geq p$), it is obvious that $\bm{w}^*=\sum_{i=1}^n \bm{w}_i^*/{n}$. We compare sIAG with *non-aggregated SGD*, which uses the following descent direction at the $t$th iteration: $$\begin{aligned} \textstyle \bm{g}_{\sf SGD}^t \coloneqq \sum_{ i: \tau_i(t) = t } \nabla f ( \bm{w}^{\tau_{i}(t)}; \bm{\xi}_i^{\tau_{i}(t)} ). \label{eq:sgd} \end{aligned}$$ We simulate three types of worker selection schemes. The first selection scheme chooses one workers at each iteration cyclically, i.e., at iteration $t$, the $(t \, {\rm mod} \, n + 1)$th worker is active. Notice that $T=n$ in this case and there is no linear speedup according to Theorem [Theorem 1](#thm){reference-type="ref" reference="thm"}. The second selection scheme chooses the workers *uniformly at random*. It models the scenario when the workers are equally efficient. Further, the workers will be selected at least once in no more than 15 iterations. The third selection scheme adopts *non-uniform selection*. It models a more realistic scenario when the workers are heterogeneous in terms of efficiency. Specifically, worker $i$ is selected at least once in no more than $T_i$ iterations, where $T_i$ is uniformly distributed on $\{10, \ldots, 20\}$. The faster workers are selected more frequently. Notice that $T=15$ in the second and third scheme. Fig. [3](#fig:random-uniform){reference-type="ref" reference="fig:random-uniform"} presents the convergence of two algorithms with $d=20$, $p=10$, and $\sigma=0.1$ under the uniform and non-uniform selection schemes. Observe that under the uniform selection scheme, the sIAG and SGD achieve comparable convergence performance and both exhibit linear speedup as the number of agents increases. We also observe that there is no linear speedup with the cyclical selection scheme. On the other hand, under the non-uniform worker selection scheme, the sIAG still enjoys linear speedup while SGD is not converging to an optimal solution, as the non-uniform selection scheme has led to biased stochastic gradient. This validates the necessity of aggregating (possibly) staled gradients in [\[eq:siag\]](#eq:siag){reference-type="eqref" reference="eq:siag"} as opposed to using only the latest gradients in [\[eq:sgd\]](#eq:sgd){reference-type="eqref" reference="eq:sgd"} in the presence of system heterogeneity. {#fig:random-uniform width="0.9\\columnwidth"} {#fig:random-uniform width="0.9\\columnwidth"} [\[fig:random-uniform\]]{#fig:random-uniform label="fig:random-uniform"} # CONCLUSION We proposed the sIAG algorithm for distributed optimization over the parameter server architecture with heterogenous streaming data. The sIAG method is adapted from the classical IAG method on batch data. We established that sIAG achieves linear speedup compared to the sequential SGD for strongly convex problems. Our analysis relies on careful treatments of the conditional expectations with staled gradients (see [\[eq:ana-split\]](#eq:ana-split){reference-type="eqref" reference="eq:ana-split"}) and a new recursive system with both delayed and noise-related terms (see [\[eq:ana-recur\]](#eq:ana-recur){reference-type="eqref" reference="eq:ana-recur"}), which can be of independent interest. Numerical results on synthetic data verify our theoretical findings and show significant advantages of sIAG over the non-aggregated SGD method when the workers are not uniformly selected. # Missing Proofs To simplify notations, throughout this appendix, we denote $\bm{g}_i^{t} \coloneqq \nabla f_i(\bm{w}^{t}; \bm{\xi}_i^{t})$ and $\bm{G}_i^{t} \coloneqq \nabla F_i(\bm{w}^{t})$ as respectively the stochastic gradient and exact gradient of the local loss function for any $i \in [n]$ and $t \geq 0$. ## Proof of Lemma [Lemma 1](#lem:aa){reference-type="ref" reference="lem:aa"} {#ap:aa} Observe that $$\begin{aligned} &\mathbb{E}\|(1/n)\bm{g}^t\|_2^2 = \mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} \bm{g}_i^{\tau_{i}(t)} \right\|_2^2 \label{eq:eee} \\ =& \mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} \left( \bm{g}_i^{\tau_{i}(t)} - \bm{G}_i^{\tau_{i}(t)} + \bm{G}_i^{\tau_{i}(t)} \right) \right\|_2^2 \nonumber \\ \leq& 2 \mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} \left( \bm{g}_i^{\tau_{i}(t)} \hspace{-0.5mm}-\textbf{} \bm{G}_i^{\tau_{i}(t)} \right) \right\|_2^2 \hspace{-2mm}+\hspace{-0.5mm} 2 \mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} \bm{G}_i^{\tau_{i}(t)} \right\|_2^2. \notag \end{aligned}$$ We then upper bound the first term in [\[eq:eee\]](#eq:eee){reference-type="eqref" reference="eq:eee"}. Assume without loss of generality that for any $i,j \in [n]$ such that $i \neq j$, we have $\tau_{i}(t) < \tau_{j}(t)$. Then, we have $\mathcal{F}_{\tau_i(t)} \subseteq \mathcal{F}_{\tau_j(t)}$. Hence, using the law of total expectation and the fact that $\mathbb{E}_{ \tau_j(t) } \left[ \bm{g}_j^{\tau_j(t)} \right] = \bm{G}_j^{\tau_j(t)}$, we obtain $$\begin{aligned} &\mathbb{E}\left\langle \bm{g}_i^{\tau_i(t)} - \bm{G}_i^{\tau_i(t)}, \bm{g}_j^{\tau_j(t)} - \bm{G}_j^{\tau_j(t)} \right\rangle \label{eq:0} \\ & = \mathbb{E}\left[ \left\langle \bm{g}_i^{\tau_i(t)} - \bm{G}_i^{\tau_i(t)}, \mathbb{E}_{\tau_j(t)} \left[ \bm{g}_j^{\tau_j(t)} - \bm{G}_j^{\tau_j(t)} \right] \right\rangle \right] = 0 \nonumber. \end{aligned}$$ Then, it follows from [\[eq:0\]](#eq:0){reference-type="eqref" reference="eq:0"} and Assumptions [Assumption 2](#as:sigma){reference-type="ref" reference="as:sigma"}, [Assumption 3](#as:delay){reference-type="ref" reference="as:delay"} that $$\begin{aligned} & \mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} ( \bm{g}_i^{\tau_i(t)} - \bm{G}_i^{\tau_i(t)} ) \right\|_2^2 \label{eq:ggg} \\ & = \frac{1}{n^2} \sum_{i=1}^{n} \mathbb{E}\left[ \| \bm{g}_i^{\tau_i(t)} - \bm{G}_i^{\tau_i(t)} \|_2^2 \right] \nonumber \\ & \leq \frac{1}{n^2} \sum_{i=1}^n \sigma^2 ( 1 + {\cal E}_{\tau_i(t)} ) \leq \frac{\sigma^2}{n} \left( 1 + \max_{s \in [(t-T)_+, t]} {\cal E}_s \right). \nonumber \end{aligned}$$ Besides, let $\bm{G}_i^\star \coloneqq \nabla F_i(\bm{w}^*)$ and observe that $\sum_{i=1}^n \bm{G}_i^\star = {\bm 0}$, then we upper bound the second term in [\[eq:eee\]](#eq:eee){reference-type="eqref" reference="eq:eee"} as follows: $$\begin{aligned} &\mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} \bm{G}_i^{\tau_{i}(t)} \right\|_2^2 = \mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} \left( \bm{G}_i^{\tau_{i}(t)} - \bm{G}_i^t + \bm{G}_i^t \right) \right\|_2^2 \nonumber\\ &\leq \frac{2}{n^2} \mathbb{E}\left\| \sum_{i=1}^{n} \left( \bm{G}_i^{\tau_{i}(t)} - \bm{G}_i^t \right) \right\|_2^2 + \frac{2}{n^2} \mathbb{E}\left\| \sum_{i=1}^{n} \bm{G}_i^t \right\|_2^2 \nonumber\\ &\leq \frac{2 L^2}{n} \sum_{i=1}^{n} \mathbb{E}\| \bm{w}^t - \bm{w}^{\tau_{i}(t)} \|_2^2 + \frac{2 L^2}{n} \sum_{i=1}^{n} \mathbb{E}\| \bm{w}^t - \bm{w}^\star \|_2^2 \nonumber\\ &= 6L^2 \mathbb{E}\| \bm{w}^t - \bm{w}^* \|_2^2 + \frac{4 L^2}{n} \sum_{i=1}^{n} \mathbb{E}\| \bm{w}^{\tau_i(t)} - \bm{w}^\star \|_2^2 \nonumber \\ &\leq 10L^2 \max_{ s \in [ (t-T)_+, t ] } {\cal E}_s, \label{eq:ggg2} \end{aligned}$$ Plugging [\[eq:ggg\]](#eq:ggg){reference-type="eqref" reference="eq:ggg"} and [\[eq:ggg2\]](#eq:ggg2){reference-type="eqref" reference="eq:ggg2"} back into [\[eq:eee\]](#eq:eee){reference-type="eqref" reference="eq:eee"} gives the desired bound. ## Proof of Lemma [Lemma 2](#lem:bb){reference-type="ref" reference="lem:bb"} {#ap:bb} Let $\bar{\tau}(t) \coloneqq \min_{i\in[n]} \tau_i(t)$, then we have $$\begin{aligned} &\mathbb{E}\left\langle \bm{w}^t - \bm{w}^*, (1/n) \bm{g}^t \right\rangle \nonumber \\ =& \underbrace{\mathbb{E}\left\langle \bm{w}^t -\bm{w}^{\bar{\tau}(t)}, \frac{1}{n} \bm{g}^t \right\rangle}_{A}+ \underbrace{\mathbb{E}\left\langle\bm{w}^{\bar{\tau}(t)}-\bm{w}^*, \frac{1}{n} \bm{g}^t \right\rangle}_{B}. \label{eq:club+spade} \end{aligned}$$ Note that as explained previously, $\bm{g}^t$ is independent of $\bm{w}^{\bar{\tau}(t)}$ conditioning on $\mathcal{F}_{\bar{\tau}(t)}$ and thus the $B$ term can be controlled easily. In the sequel, we bound the $A,B$ terms in [\[eq:club+spade\]](#eq:club+spade){reference-type="eqref" reference="eq:club+spade"} to obtain desired result. [i) Bounding $A$:]{.ul} Notice that $\bm{g}^t$ not conditionally *independent* of $\bm{w}^t$. Our remedy is to observe the decomposition $\bm{w}^t - \bm{w}^{\bar{\tau}(t)} = \sum_{s=\bar{\tau}(t)}^{t-1} (\eta_s/n) \bm{g}^s$. We can lower bound $A$ as follows: $$\begin{aligned} &A = \mathbb{E}\left\langle \sum_{s=\bar{\tau}(t)}^{t-1} \eta_s \bm{g}^s, \bm{g}^t \right\rangle = \sum_{s=\bar{\tau}(t)}^{t-1} \eta_s \mathbb{E}\left\langle \bm{g}^s, \bm{g}^t \right\rangle \nonumber\\ &\geq \sum_{s=\bar{\tau}(t)}^{t-1} \eta_s \left( - \frac{1}{2} \mathbb{E}\| \bm{g}^s \|_2^2 - \frac{1}{2} \mathbb{E}\| \bm{g}^t \|_2^2 \right) \nonumber \\ &\geq - T \eta_{t-T} \left( \frac{2 \sigma^2}{n} + {\rm C}_L{\cal E}_t^{\max} \right), \label{eq:club} \end{aligned}$$ where the last inequality uses Lemma [Lemma 1](#lem:aa){reference-type="ref" reference="lem:aa"} and the fact that $t-\bar{\tau}(t) \leq T$, $\eta_{t-T} \geq \eta_s$ for any $s = \bar{\tau}_t, \ldots, t-1$. [ii) Bounding $B$:]{.ul} Since $\bar{\tau}(t) \leq \tau_i(t)$ for all $i \in [n]$, we have $\mathcal{F}_{\bar{\tau}(t)} \subseteq \mathcal{F}_{\tau_i(t)}$. Thus, using $\bm{g}^t = \sum_{i=1}^{n} \bm{g}_i^{\tau_i(t)}$ and the law of total expectation, we have $$\begin{aligned} B &= \mathbb{E}\left[ \mathbb{E}_{\bar{\tau}(t)} \left[ \left\langle \bm{w}^{\bar{\tau}(t)} - \bm{w}^*, \frac{1}{n} \sum_{i=1}^{n} \bm{g}_i^{\tau_i(t)} \right\rangle \right] \right] \nonumber \\ &= \underbrace{\mathbb{E}\left\langle \bm{w}^{\bar{\tau}(t)} - \bm{w}^t, \frac{1}{n} \sum_{i=1}^{n} \bm{G}_i^{\tau_i(t)} \right\rangle}_{B_1} \nonumber\\ &\quad + \underbrace{\mathbb{E}\left\langle \bm{w}^t - \bm{w}^*, \frac{1}{n} \sum_{i=1}^{n} \bm{G}_i^{\tau_i(t)} \right\rangle}_{B_2}. \label{eq:star+tri} \end{aligned}$$ Since $\sum_{i=1}^n \bm{G}_i^* = \bm{0}$, for any $\alpha > 0$, we have $$\begin{aligned} &B_1 = \mathbb{E}\left\langle \bm{w}^{\bar{\tau}(t)} - \bm{w}^t, \frac{1}{n} \sum_{i=1}^{n} \left( \bm{G}_i^{\tau_i(t)} - \bm{G}_i^t \right) \right\rangle \nonumber \\ &\qquad + \mathbb{E}\left\langle \bm{w}^{\bar{\tau}(t)} - \bm{w}^t, \frac{1}{n} \sum_{i=1}^{n} \left( \bm{G}_i^t - \bm{G}_i^* \right) \right\rangle \nonumber \\ &\geq - \frac{\alpha}{2} \mathbb{E}\| \bm{w}^{\bar{\tau}(t)} - \bm{w}^t \|_2^2 - \frac{1}{2\alpha} \mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} ( \bm{G}_i^{\tau_i(t)} - \bm{G}_i^t ) \right\|_2^2 \nonumber \\ &\quad - \frac{\alpha}{2} \mathbb{E}\| \bm{w}^{\bar{\tau}(t)} - \bm{w}^t \|_2^2 - \frac{1}{2\alpha} \mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} \left( \bm{G}_i^t - \bm{G}_i^* \right) \right\|_2^2 \nonumber \\ &\geq -\alpha \mathbb{E}\| \bm{w}^{\bar{\tau}(t)} - \bm{w}^t \|_2^2 \nonumber \\ &\quad - \frac{L^2}{2n\alpha} \sum_{i=1}^{n} \mathbb{E}\| \bm{w}^{\tau_{i}(t)} - \bm{w}^t \|_2^2 - \frac{L^2}{2n\alpha} \sum_{i=1}^{n} \mathbb{E}\| \bm{w}^t - \bm{w}^* \|_2^2. \nonumber \end{aligned}$$ Invoking Lemma [Lemma 3](#lem:diff){reference-type="ref" reference="lem:diff"} allows us to further lower bound the above terms by: $$\begin{aligned} B_1 &\geq -\alpha T^2 \eta_{t-T}^2 \left( \frac{2 \sigma^2}{n} + {\rm C}_L{\cal E}^{\max}_{t} \right) \nonumber \\ &\quad -\frac{L^2 T^2}{2 \alpha} \eta_{t-T}^2 \left( \frac{2 \sigma^2}{n} + {\rm C}_L{\cal E}^{\max}_t \right) - \frac{L^2}{2\alpha} {\cal E}_t \nonumber \\ &= -\left( 2\alpha + \frac{L^2}{\alpha} \right) T^2 \eta_{t-T}^2 \frac{\sigma^2}{n} \nonumber \\ &\quad - \left( \alpha + \frac{L^2}{2 \alpha} \right) {\rm C}_LT^2 \eta_{t-T}^2 {\cal E}_{t}^{\max} - \frac{L^2}{2\alpha} {\cal E}_t, \label{eq:bigstar} \end{aligned}$$ On the other hand, the term $B_2$ can be controlled as $$\begin{aligned} B_2 &= \mathbb{E}\left\langle \bm{w}^t - \bm{w}^*, \frac{1}{n} \sum_{i=1}^{n} ( \bm{G}_i^{\tau_{i}(t)} - \bm{G}_i^t ) \right\rangle \nonumber \\ &\qquad + \mathbb{E}\left\langle \bm{w}^t - \bm{w}^*, \frac{1}{n} \sum_{i=1}^{n} \bm{G}_i^t \right\rangle \nonumber \\ &\geq - \frac{\mu}{2} \mathbb{E}\| \bm{w}^t - \bm{w}^* \|_2^2 - \frac{1}{2\mu} \mathbb{E}\left\| \frac{1}{n} \sum_{i=1}^{n} ( \bm{G}_i^{\tau_{i}(t)} - \bm{G}_i^t ) \right\|_2^2 \nonumber \\ &\quad + \mu \mathbb{E}\| \bm{w}^t - \bm{w}^* \|_2^2 \nonumber \\ &\geq \frac{\mu}{2} \mathbb{E}\| \bm{w}^t - \bm{w}^* \|_2^2 - \frac{L^2}{2\mu n} \sum_{i=1}^{n} \mathbb{E}\| \bm{w}^t - \bm{w}^{\tau_{i}(t)} \|_2^2 \nonumber \\ &\geq \frac{\mu}{2} {\cal E}_t - \frac{L^2T^2}{2 \mu} \eta_{t-T}^2 \left( \frac{2 \sigma^2}{n} + {\rm C}_L{\cal E}_{t}^{\max} \right) \nonumber \\ &= - \frac{L^2 T^2}{\mu} \eta_{t-T}^2 \frac{\sigma^2}{n} - \frac{{\rm C}_LL^2 T^2}{2 \mu} \eta_{t-T}^2 {\cal E}_{t}^{\max} + \frac{\mu}{2} {\cal E}_t, \label{eq:blacktri} \end{aligned}$$ where the last inequality also follows from Lemma [Lemma 3](#lem:diff){reference-type="ref" reference="lem:diff"}. Thus, plugging [\[eq:bigstar\]](#eq:bigstar){reference-type="eqref" reference="eq:bigstar"} and [\[eq:blacktri\]](#eq:blacktri){reference-type="eqref" reference="eq:blacktri"} back into [\[eq:star+tri\]](#eq:star+tri){reference-type="eqref" reference="eq:star+tri"} gives $$\begin{aligned} B \geq& \left( \frac{\mu}{2} \hspace{-0.5mm}-\hspace{-0.5mm} \frac{L^2}{2\alpha} \right) {\cal E}_t - \left( \alpha \hspace{-0.5mm}+\hspace{-0.5mm} \frac{L^2}{2\alpha} \hspace{-0.5mm}+\hspace{-0.5mm} \frac{L^2}{2 \mu} \right) {\rm C}_LT^2 \eta_{t-T}^2 {\cal E}_{t}^{\max} \nonumber \\ & -\left( \alpha + \frac{L^2}{2\alpha} + \frac{L^2}{2\mu} \right) T^2 \eta_{t-T}^2 \frac{2 \sigma^2}{n}. \label{eq:Bgeq} \end{aligned}$$ Setting $\alpha = 2L^2 / \mu$ in [\[eq:Bgeq\]](#eq:Bgeq){reference-type="eqref" reference="eq:Bgeq"} gives $$\begin{aligned} B &\geq \frac{\mu}{4} {\cal E}_t - \left( \frac{\mu}{4} + \frac{5 L^2}{2 \mu} \right) {\rm C}_LT^2 \eta_{t-T}^2 {\cal E}_{t}^{\max} \nonumber \\ &\quad - \left( \frac{\mu}{2} + \frac{5 L^2}{\mu} \right) T^2 \eta_{t-T}^2 \frac{\sigma^2}{n}. \label{eq:spade} \end{aligned}$$ Finally, plugging [\[eq:club\]](#eq:club){reference-type="eqref" reference="eq:club"} and [\[eq:spade\]](#eq:spade){reference-type="eqref" reference="eq:spade"} back into [\[eq:club+spade\]](#eq:club+spade){reference-type="eqref" reference="eq:club+spade"} gives the desired bound. ## Proof of Lemma [Lemma 3](#lem:diff){reference-type="ref" reference="lem:diff"} {#ap:diff} The proof is elementary: $$\begin{aligned} & \mathbb{E}\| \bm{w}^t - \bm{w}^{\tau_{i}(t)} \|_2^2 = \mathbb{E}\left\| \sum_{s = \tau_{i}(t)}^{t-1} \left( \bm{w}^{s+1} - \bm{w}^{s} \right) \right\|_2^2 \nonumber \\ & \leq \left| t - \tau_{i}(t) \right| \sum_{s = \tau_{i}(t)}^{t-1} \mathbb{E}\| \bm{w}^{s+1} - \bm{w}^{s} \|_2^2 \nonumber \\ &\leq T \sum_{s = \tau_{i}(t)}^{t-1} \mathbb{E}\| (\eta_s/n) \bm{g}^s \|_2^2 \nonumber \\ &\leq T \sum_{s = \tau_{i}(t)}^{t-1} \eta_{t-T}^2 \left( \frac{2 \sigma^2}{n} + {\rm C}_L\max_{ s \in [ (t-T)_+, t ] } {\cal E}_s \right) \label{eq:xxx} \\ &\leq T^2 \eta_{t-T}^2 \left( \frac{2 \sigma^2}{n} + {\rm C}_L{\cal E}^{\max}_{t} \right). \nonumber \end{aligned}$$ where [\[eq:xxx\]](#eq:xxx){reference-type="eqref" reference="eq:xxx"} follows from the monotonicity of $\{\eta_t\}_{t\geq0}$ and Lemma [Lemma 1](#lem:aa){reference-type="ref" reference="lem:aa"}. ## Proof of Theorem [Theorem 1](#thm){reference-type="ref" reference="thm"} {#ap:thm} Let $\rho_t \coloneqq 1 + 2 T + \left( \frac{\mu}{2} + \frac{5L^2}{\mu} \right) T^2 \eta_{t-T}$. Since $\gamma \geq 2T$, we have $\eta_{t-T} = \beta/(\gamma+t-T) \leq \beta/T$, which further implies that $$\rho_t \leq 1 + 2 T + \left( \frac{\mu}{2} + \frac{5L^2}{\mu} \right) \beta T = \bar{\rho}(T).$$ Thus, the recurrence relation [\[eq:ana-recur\]](#eq:ana-recur){reference-type="eqref" reference="eq:ana-recur"} implies that $$\begin{aligned} {\cal E}_{t+1} \leq &\left(1 - \frac{\mu}{2} \eta_t \right) {\cal E}_t + \bar{\rho}(T) {\rm C}_L\eta_{t-T}^2 {\cal E}_{t}^{\max} \nonumber \\ & + \bar{\rho}(T) \eta_{t-T}^2 \frac{2 \sigma^2}{n}. \label{eq:recur} \end{aligned}$$ Then, we prove [\[eq:rate\]](#eq:rate){reference-type="eqref" reference="eq:rate"} by induction. [i) Base case:]{.ul} Since $\delta_2 =\gamma^2{\cal E}_0$, we have $$\begin{aligned} &{\cal E}_{0} \leq \frac{\delta_1}{\gamma}\frac{\sigma^2}{n} + \frac{\delta_2}{\gamma^2}. \end{aligned}$$ [ii) Induction step:]{.ul} Suppose that for some $t \geq 0$, it holds that $$\begin{aligned} &{\cal E}_s \leq \frac{\delta_1}{\gamma+s} \frac{\sigma^2}{n} + \frac{\delta_2}{(\gamma+t)^2}, ~s = 0, \dots, t, \nonumber \end{aligned}$$ which implies that $$\begin{aligned} {\cal E}_{t}^{\max} \hspace{-0.5mm}&=\hspace{-0.5mm} \max_{s\in[ (t-2T)_+,t]} {\cal E}_s \leq \frac{\delta_1}{\gamma+t-2T} \frac{\sigma^2}{n} + \frac{\delta_2}{(\gamma+t-2T)^2}. \nonumber \end{aligned}$$ Together with [\[eq:recur\]](#eq:recur){reference-type="eqref" reference="eq:recur"}, this implies that $$\begin{aligned} &{\cal E}_{t+1} \leq \left( 1 - \frac{\mu\beta/2}{\gamma+t} \right) \frac{\delta_1}{\gamma+t} \frac{\sigma^2}{n} \nonumber \\ & \hspace{-0.5mm}+ \frac{\bar{\rho}(T){\rm C}_L\beta^2\delta_1}{(\gamma+t-T)^2 (\gamma+t-2T)} \frac{\sigma^2}{n} + \frac{2\bar{\rho}(T)\beta^2}{(\gamma+t-T)^2} \frac{\sigma^2}{n} \nonumber \\ &\hspace{-0.5mm}+ \hspace{-1mm}\left(\hspace{-1mm} 1 \hspace{-0.5mm}-\hspace{-0.5mm} \frac{\mu\beta/2}{\gamma+t} \hspace{-0.5mm}\right) \hspace{-0.5mm}\frac{\delta_2}{(\gamma+t)^2} \hspace{-0.5mm}+\hspace{-0.5mm} \frac{{\rm C}_L\beta^2\bar{\rho}(T)\delta_2}{(\gamma+t-T)^2 (\gamma\hspace{-0.5mm}+\hspace{-0.5mm}t\hspace{-0.5mm}-\hspace{-0.5mm}2T)^2}. \label{eq:speedup} \end{aligned}$$ We note that $$\begin{aligned} \left( 1 - \frac{\mu\beta/2}{\gamma+t} \right) \frac{1}{\gamma+t} &=\frac{\gamma+t-1}{(\gamma+t)^2} - \frac{\mu\beta/2-1}{(\gamma+t)^2} \nonumber \\ &\leq \frac{1}{\gamma+t+1} - \frac{\mu\beta/2-1}{(\gamma+t)^2}, \label{eq:11} \end{aligned}$$ where the inequality holds because $$\begin{aligned} \frac{\gamma+t-1}{(\gamma+t)^2} \leq \frac{1}{\gamma+t+1} \Leftrightarrow (\gamma+t)^2 \geq (\gamma+t)^2-1. \end{aligned}$$ We also note that $$\begin{aligned} \left( 1 - \frac{\mu\beta/2}{\gamma+t} \right) \frac{1}{(\gamma+t)^2} &=\frac{\gamma+t-2}{(\gamma+t)^3} - \frac{\mu\beta/2-2}{(\gamma+t)^3} \nonumber \\ &\leq \frac{1}{(\gamma+t+1)^2} - \frac{\mu\beta/2-2}{(\gamma+t)^3}, \label{eq:22} \end{aligned}$$ where the inequality holds because $$\begin{aligned} \frac{\gamma+t-2}{(\gamma+t)^3} \leq \frac{1}{(\gamma+t+1)^2} \Leftrightarrow -3 (\gamma+t) \leq 2. \end{aligned}$$ Thus, Plugging [\[eq:11\]](#eq:11){reference-type="eqref" reference="eq:11"} and [\[eq:22\]](#eq:22){reference-type="eqref" reference="eq:22"} into [\[eq:speedup\]](#eq:speedup){reference-type="eqref" reference="eq:speedup"} gives $$\begin{aligned} &{\cal E}_{t+1} \leq \frac{\sigma^2}{n} \left[ \frac{\delta_1}{\gamma+t+1} - \frac{(\mu\beta/2-1)\delta_1}{(\gamma+t)^2} \right. \nonumber \\ &\left. + \frac{{\rm C}_L\beta^2\bar{\rho}(T)\delta_1}{(\gamma+t-T)^2 (\gamma+t-2T)} + \frac{2\beta^2\bar{\rho}(T)}{(\gamma+t-T)^2} \right] \nonumber \\ &+\hspace{-0.7mm} \frac{\delta_2}{(\hspace{-0.5mm}\gamma\hspace{-0.7mm}+\hspace{-0.7mm}t\hspace{-0.7mm}+\hspace{-0.7mm}1\hspace{-0.5mm})^2} \hspace{-0.7mm}-\hspace{-0.7mm} \frac{(\hspace{-0.5mm}\mu\beta/2\hspace{-0.7mm}-\hspace{-0.7mm}2\hspace{-0.5mm})\delta_2}{(\gamma+t)^2} \hspace{-0.7mm}+\hspace{-0.7mm} \frac{{\rm C}_L\beta^2\bar{\rho}(T)\delta_2}{(\hspace{-0.5mm}\gamma\hspace{-0.7mm}+\hspace{-0.7mm}t\hspace{-0.7mm}-\hspace{-0.7mm}T\hspace{-0.3mm})^2 (\hspace{-0.5mm}\gamma\hspace{-0.7mm}+\hspace{-0.7mm}t\hspace{-0.7mm}-\hspace{-0.7mm}2T\hspace{-0.3mm})^2}. \label{eq:Eup} \end{aligned}$$ Moreover, $\gamma\geq2T$ implies that $$\begin{aligned} \frac{1}{(\gamma+t-T)^2} \leq \frac{4}{(\gamma+t)^2}. \label{eq:a1} \end{aligned}$$ Since $\beta > 2/\mu$ and $\gamma \geq 2T + \frac{16C_L\beta^2}{\mu\beta-2}\bar{\rho}(T)$, we have $$\begin{aligned} \frac{{\rm C}_L\beta^2\bar{\rho}(T)\delta_1}{\gamma+t-2T} \leq \left(\frac{\mu\beta}{2}-1\right) \frac{\delta_1}{8}. \label{eq:a2} \end{aligned}$$ Besides, it follows from $\delta_1 = 32\beta^2\bar{\rho}(T)/(\mu\beta-2)$ that $$\begin{aligned} 2\beta^2\bar{\rho}(T) = \left(\frac{\mu\beta}{2}-1\right) \frac{\delta_1}{8}. \label{eq:a3} \end{aligned}$$ Combing [\[eq:a1\]](#eq:a1){reference-type="eqref" reference="eq:a1"}, [\[eq:a2\]](#eq:a2){reference-type="eqref" reference="eq:a2"}, and [\[eq:a3\]](#eq:a3){reference-type="eqref" reference="eq:a3"} gives $$\begin{aligned} &\frac{{\rm C}_L\beta^2\bar{\rho}(T)\delta_1}{(\gamma\hspace{-0.7mm}+\hspace{-0.7mm}t\hspace{-0.7mm}-\hspace{-0.7mm}T)^2 (\gamma\hspace{-0.7mm}+\hspace{-0.7mm}t\hspace{-0.7mm}-\hspace{-0.7mm}2T)} \hspace{-0.7mm}+\hspace{-0.7mm} \frac{2\beta^2\bar{\rho}(T)}{(\gamma\hspace{-0.7mm}+\hspace{-0.7mm}t\hspace{-0.7mm}-\hspace{-0.7mm}T)^2} \hspace{-0.7mm}\leq\hspace{-0.7mm} \frac{(\mu\beta/2\hspace{-0.7mm}-\hspace{-0.7mm}1)\delta_1}{(\gamma+t)^2}. \label{eq:add1} \end{aligned}$$ Since $\beta > 4/\mu$ and $\gamma \geq 2T + \sqrt{\frac{8C_L\beta^2}{\mu\beta-4}\bar{\rho}(T)}$, we have $$\begin{aligned} \frac{{\rm C}_L\beta^2\bar{\rho}(T)}{(\gamma+t-2T)^2} \leq \frac{1}{4} \left(\frac{\mu\beta}{2}-2\right). \label{eq:a4} \end{aligned}$$ Combining [\[eq:a1\]](#eq:a1){reference-type="eqref" reference="eq:a1"} and [\[eq:a4\]](#eq:a4){reference-type="eqref" reference="eq:a4"} gives $$\begin{aligned} &\frac{{\rm C}_L\beta^2\bar{\rho}(T)\delta_2}{(\gamma+t-T)^2 (\gamma+t-2T)^2} \leq \frac{(\mu\beta/2-2)\delta_2}{(\gamma+t)^2}. \label{eq:add2} \end{aligned}$$ Plugging [\[eq:add1\]](#eq:add1){reference-type="eqref" reference="eq:add1"} and [\[eq:add2\]](#eq:add2){reference-type="eqref" reference="eq:add2"} back into [\[eq:Eup\]](#eq:Eup){reference-type="eqref" reference="eq:Eup"} yields $$\begin{aligned} {\cal E}_{t+1} \leq \frac{\delta_1}{\gamma+t+1} \frac{\sigma^2}{n} + \frac{\delta_2}{(\gamma+t+1)^2}, \end{aligned}$$ which completes the proof of the induction step. [^1]: $^{1}$Xiaolu Wang and Hoi-To Wai are with System Engineering & Engineering Management, Faculty of Engineering, The Chinese University of Hong Kong, Hong Kong SAR. Emails: [xwangcu\@gmail.com]([email protected]), [htwai\@se.cuhk.edu.hk]([email protected]). $^{2}$Cheng Jin and Yuantao Gu are with Department of Electronic Engineering, Tsinghua University, Beijing. Emails: [jinc21\@mails.tsinghua.edu.cn]([email protected]), [gyt\@tsinghua.edu.cn]([email protected]). This work is partly supported by CUHK Direct Grant \#4055208. [^2]: We remark that [@lian2018asynchronous] considered an algorithm that involve similar staled aggregation property to sIAG but have employed a simplifying assumption that $\bm{w}^t$ is independent of $\bm{g}^t$. Leveraging this property, their algorithm achieves linear speedup regardless of $T$. We conjecture that such linear speedup cannot be obtained when considering the realistic conditions for sIAG method that $\bm{w}^t$ is not independent of $\bm{g}^t$.

arxiv_math

--- abstract: | In this article we consider the stability threshold of the 2D magnetohydrodynamics (MHD) equations near a combination of Couette flow and large constant magnetic field. We study the partial dissipation regime with full viscous and only horizontal magnetic dissipation. In particular, we show that this regime behaves qualitatively differently than both the fully dissipative and the non-resistive setting. address: - Karlsruhe Institute of Technology, Englerstraße 2, 76131 Karlsruhe, Germany - Karlsruhe Institute of Technology, Englerstraße 2, 76131 Karlsruhe, Germany author: - Niklas Knobel - Christian Zillinger bibliography: - library.bib title: On the Sobolev Stability Threshold for the 2D MHD Equations with Horizontal Magnetic Dissipation --- # Introduction The equations of magnetohydrodynamics (MHD) $$\begin{aligned} \label{aniso} \begin{split} \partial_t V + V\cdot \nabla V+ \nabla \Pi &= (\nu_x\partial_x^2+\nu_y\partial_y^2) V + B\cdot\nabla B, \\ \partial_t B + V\cdot\nabla B &= (\kappa_x\partial_x^2+\kappa _y\partial_y^2) B +B\cdot\nabla V, \\ \nabla\cdot v=\nabla\cdot b &= 0,\\ (t,x,y) &\in \mathbb{R}^+ \times\mathbb{T}\times \mathbb{R}, \end{split}\end{aligned}$$ model the evolution of the velocity $V$ of conducting, non-magnetic fluids interacting with a magnetic field $B$. The MHD equations are commonly used in applications ranging from astrophysics and the description of plasmas to control problems for liquid metals in industrial applications [@davidson_2016]. Similarly to the Navier-Stokes and Euler equations, questions of hydrodynamic stability and the behavior for high Reynolds numbers (that is, for $\nu,\kappa$ tending to zero) are a very active area of research both inner-mathematically and in view of applications. Motivated by stability results for the isotropic full-dissipation case ($\nu_x=\nu_y=\kappa_x=\kappa_y>0$) and instability results for the non-resistive case ($\kappa_x=\kappa_y=0$), we are interested in the behavior of the two-dimensional magnetohydrodynamic (MHD) equations with partial dissipation, where some of the dissipation coefficients $$\begin{aligned} \kappa_y, \kappa_x, \nu_x, \nu_y\ge 0,\end{aligned}$$ are allowed to vanish. More specifically, we study the behavior near the stationary solution given by the combination of Couette flow and a (large) constant magnetic field $$\begin{aligned} \label{eq:Couette} V_s=ye_1, \quad B_s=\alpha e_1,\end{aligned}$$ for the case of vanishing vertical resistivity, $\kappa_y=0$. For the related case of the Navier-Stokes equations (that is, without any magnetic field) the (in)stability of Couette flow at high Reynolds number is known as the Sommerfeld paradox [@Maj] and is related to nonlinear instability of the Euler equations [@bedrossian2015inviscid; @dengmasmoudi2018; @dengZ2019]. However, for the case of sufficiently small data it was proven in [@bedrossian2016sobolev] that (mixing enhanced) dissipation can counteract this instability in the Navier-Stokes equations and that (long time asymptotic) stability holds in Sobolev spaces for initial data with $$\begin{aligned} \|\omega\|_{H^N}\leq \epsilon \ll \nu^{\gamma}\end{aligned}$$ with $\gamma\geq \frac{1}{2}$. Later in [@masmoudi2022stability] this has been improved to $\gamma =\tfrac 13$. This is an example of a stability threshold result, which establishes stability for small data and determines suitable (optimal) exponents $\gamma$ for given norms. Since the addition of the magnetic field is known to possibly destabilize the dynamics (see the following discussion), our main questions concern the MHD equations [\[aniso\]](#aniso){reference-type="eqref" reference="aniso"} in terms of perturbations moving with the underlying shear flow: $$\begin{aligned} v(x,y,t)&= V(x-yt,y,t )- V_s, \\ b(x,y,t)&= B(x-yt,y,t )- B_s.\end{aligned}$$ The corresponding perturbed equations in these new variables read $$\begin{aligned} \begin{split} \partial_t v + v_2 e_1 - 2\partial_x \Delta^{-1}_t \nabla_t v_2 &= \nu \cdot \Delta_t v+ \alpha \partial_x b + b\nabla_t b- v\nabla_t v-\nabla_t \pi , \\ \partial_t b - b_2 e_1 \qquad \qquad \quad \quad \ &= \kappa\cdot \Delta_t b+ \alpha \partial_x v +b\nabla_t v -v\nabla_t b,\\ \nabla_t\cdot v=\nabla_t\cdot b &= 0.\label{anco} \end{split}\end{aligned}$$ Here, we introduce the time-dependent derivatives $\partial_y^t = \partial_y-t\partial_x$, $\nabla_t = (\partial_x , \partial_y^t)$ and $\Delta _t = \partial_x^2+ (\partial_y^t)^2$. Furthermore, we use the following short notation for the dissipation operator: $$\begin{aligned} \nu \cdot \Delta_t &=\nu_x\partial_x^2+\nu_y(\partial_y^t)^2 ,\\ \kappa \cdot \Delta_t &=\kappa_x\partial_x^2+\kappa_y(\partial_y^t)^2.\end{aligned}$$ In this article we aim to establish a Sobolev stability threshold for [\[anco\]](#anco){reference-type="eqref" reference="anco"} for the specific anisotropic, partial dissipation case $$\begin{aligned} \kappa_y=0, \ \kappa_x = \nu_x =\nu_y>0.\end{aligned}$$ In particular, we show that this setting exhibits qualitatively different behavior than the fully dissipative and the non-resistive case. Following a similar notation as [@liss2020sobolev] we make the following definition. **Definition 1** (Stability threshold). *Consider the MHD equations [\[aniso\]](#aniso){reference-type="eqref" reference="aniso"} with anisotropic dissipation $0< \nu_x=\nu_y=\kappa_x=: \mu \ll 1$ and $\kappa_y=0$ and let $X$ be a Banach space with norm $\|(v,b)\|_{X}$. We then say that the exponent $\gamma=\gamma(X)$ is a stability threshold for the space $X$ if for initial data with $$\begin{aligned} \Vert (v_{in},b_{in}) \Vert_X & \leq \epsilon \ll \mu^\gamma, \end{aligned}$$ the corresponding solution of [\[anco\]](#anco){reference-type="eqref" reference="anco"} remains uniformly bounded for all future times with a quantitative control $$\begin{aligned} \sup_{t>0} \Vert (v,b) \Vert_X & \lesssim \epsilon. \end{aligned}$$* We remark that this definition does not require optimality (that is, instability for smaller choices of $\gamma$). Optimal stability thresholds quantify the appearance of instability in the large Reynolds number limit and are an active area of research for many fluid systems. In view of the large literature, the interested reader is referred to the following articles for the Navier-Stokes equations [@bedrossian2016sobolev; @bedrossian2017stability] and the Boussinesq equations [@zhai2022stability; @lai2021optimal; @tao2020stability] for a discussion and further references. For the (isotropic) MHD equations ($\nu:= \nu_x=\nu_y$ and $\kappa:= \kappa_x =\kappa_y$), there exists several results for non-vanishing magnetic dissipation. - When considering full isotropic dissipation $\nu =\kappa>0$, Liss [@liss2020sobolev] established a Sobolev threshold in the 3D case. Under a Diophantine condition on the magnetic field, he establishes stability for $\|(v,b)\|_{H^N}$ with $\gamma= 1$. For the 2D case an improvement to $\gamma = \tfrac{2}{3}$ is expected due to the lack of lift-up instability. Indeed, in a very recent paper, [@Dolce], Dolce establishes such a threshold for the regime $0<C\kappa^3 \leq \nu \leq \kappa$. - In the 2D inviscid case with isotropic magnetic dissipation, $\nu=0$ and $\kappa>0$, in [@knobel2023echoes] the authors established linear instability of nearby (in analytic regularity) so-called traveling wave type solutions in Gevrey $2$ regularity. As an (almost) matching nonlinear result, [@zhao2023asymptotic] established a stability threshold $\gamma\geq 1$ for Gevrey $2-\delta$ regularity for any $0<\delta<1$. - The setting with only an underlying magnetic field but without shear flow exhibits qualitatively different behavior and was studied for the case of the whole space in [@bardos1988longtime; @ren2014global] in the full dissipation case and in [@cao20132d; @ji2019stability] for the partially dissipative case. To the authors' knowledge there are no such results in the literature for the non-resistive case $\kappa=0$ with Couette flow, both for the viscous or inviscid regime $\nu=0$ or $\nu>0$, and neither for partial dissipation regimes. In view of linear instability results [@hussain2018instability] (see also Proposition [Proposition 1](#prop:instability){reference-type="ref" reference="prop:instability"}), for these equations any stability threshold results would need to consider unknowns different from $(v,b)$. As a step towards understanding this non-resistive regime, in this article we consider the 2D MHD equations with isotropic viscosity but only horizontal resistivity (while [@liss2020sobolev; @Dolce] consider full dissipation). In particular, we ask to which extent, as quantified by Sobolev stability thresholds, this partial dissipation regime behaves or does not behave like these extremal cases. In the (ideal) MHD equations ($\nu=\kappa=0$) the interaction of shear flows and the magnetic field has been shown to possibly cause instabilities, with arguments both on physical [@chen1991sufficient; @hirota2005resonance] and mathematical grounds [@hughes2001instability; @zhai2021long]. As our first result, we show that this instability also persists in the viscous but non-resistive MHD. These equations exhibit norm inflation in $H^N$ for all choices of $\nu>0$. **Proposition 1** (Instability for the non-resistive MHD equations). *Consider the isotropic equation with $\nu>0$ and $\kappa=0$ and $N\ge 3$, then the stationary solution [\[eq:Couette\]](#eq:Couette){reference-type="eqref" reference="eq:Couette"} is linearly unstable in $H^N$. More precisely, there exists initial data $(v,b)_{in}\in H^N$ such that the solution to the linearized problem satisfies $$\begin{aligned} \Vert (v,b)\Vert_{H^N}\approx \tfrac \nu {\alpha^2} t\Vert (v,b)_{in} \Vert_{H^N} \end{aligned}$$ as $t\to \infty$.* *As a consequence, the nonlinear equations also exhibit arbitrarily large norm inflation in $H^N$. That is, for any $C=C(\nu) >0$ there exists an $\varepsilon_0>0$ such that for all $\varepsilon<\varepsilon_0$ there exists initial data $(v,b)_{in}$ and a time $T$ such that $$\begin{aligned} \Vert (v,b)_{in} \Vert_{H^N} &=\varepsilon,\\ \Vert (v,b)|_{t=T} \Vert_{H^N}&\ge C \Vert (v,b)_{in} \Vert_{H^N}. \end{aligned}$$ In particular, there cannot exist a Sobolev threshold for $\|(v,b)\|_{H^N}$.* We remark that following the same argument also instability in suitable Gevrey spaces can be established. As mentioned above, the isotropic fully dissipative case is known to be stable in Sobolev regularity [@liss2020sobolev; @Dolce]. For the associated partial dissipation regimes, in view of the underlying shear dynamics the associated vertical dissipation case is expected to behave similarly as the full dissipation case. The effects of partial dissipation are a very actively studied field of research in other fluid systems, such as the Boussinesq equations [@deng2020stability; @cao2013global; @adhikari2022stability]), but, to the authors' knowledge, is largely open in the MHD equations near Couette flow. In the present case of horizontal resistivity, $\kappa_y=0$ and $\nu_x=\nu_y = \kappa_x$, the lack of vertical dissipation leads to stronger instabilities, requiring finer control and use of the coupling by a strong magnetic field. Our main results are summarized in the following theorem. **Theorem 2**. *Consider the MHD equations with horizontal resistivity, $\mu:=\nu_x=\nu_y=\kappa_x>0$ and $\kappa_y=0$, near the stationary solution [\[eq:Couette\]](#eq:Couette){reference-type="eqref" reference="eq:Couette"} with $\alpha >\tfrac 1 2$ and let $N \ge 6$ be given.* *Then there exist constants $c_0=c(\alpha ) >0$, such that for all initial data $(v,b) _{in }$ which satisfy $$\begin{aligned} \Vert (v,b)_{in} \Vert_{ H^N }= \varepsilon\le c_0 \mu^{\frac 32 }\end{aligned}$$ the corresponding solution $(v,b)$ of [\[eq_p\]](#eq_p){reference-type="eqref" reference="eq_p"} satisfies the estimates $$\begin{aligned} \Vert v \Vert_{L^\infty H^N }+\mu^{\frac 1 2 } \Vert \nabla_t v \Vert_{L^2 H^N } &\lesssim \varepsilon, \\ \Vert b \Vert_{L^\infty H^N }+\mu^{\frac 1 2 } \Vert \partial_x b \Vert_{L^2H^N }&\lesssim \varepsilon. \end{aligned}$$* Let us comment on these results: - Proposition [Proposition 1](#prop:instability){reference-type="ref" reference="prop:instability"} shows instability in terms of $(v,b)$ for the non-resistive case. Hence, the (horizontal) magnetic dissipation is shown to be necessary for long-time stability results for $(v,b)$. However, similarly as in the Boussinesq equations [@bedrossian21; @zillinger2021echo], in principle stability results in terms of other unknowns such as the magnetic potential $\phi = (-\Delta_t)^{-1}\nabla^\perp_t b$ could hold for longer or even infinite times, which remains an exciting question for future research. - Theorem [Theorem 2](#thm:anisoThres){reference-type="ref" reference="thm:anisoThres"} establishes a stability threshold $\gamma=\frac{3}{2}$. In particular, we stress that the lack of vertical magnetic dissipation not only poses a key challenge of our analysis but results in a different threshold value than the fully dissipative setting [@liss2020sobolev; @Dolce]. Indeed, the main constraint on our stability threshold is given by the control of the nonlinearity $v \cdot \nabla_t b$ and the reduced decay rates already at the linearized level (see Section [2](#linstab){reference-type="ref" reference="linstab"}). As we show in Section [3.3](#hfw){reference-type="ref" reference="hfw"}, our estimates of the so-called reaction terms [\[est:bvbnR\]](#est:bvbnR){reference-type="eqref" reference="est:bvbnR"} and [\[est:bvbaR\]](#est:bvbaR){reference-type="eqref" reference="est:bvbaR"} require a lower bound on the threshold by $\tfrac 3 2$ and are expected to be optimal for this partial dissipation case. - For simplicity of notation, in Theorem [Theorem 2](#thm:anisoThres){reference-type="ref" reference="thm:anisoThres"} we have stated our result for the case $\mu := \nu_x=\nu_y=\kappa_x$. As we discuss in Sections [2](#linstab){reference-type="ref" reference="linstab"} and [3](#bootHyp){reference-type="ref" reference="bootHyp"}, more generally, instead of equality it suffices to require that $\frac{1}{2\alpha}\nu_y\le \kappa_x\le C \nu_y^{\frac 1 3}$, similarly as in the full dissipation case studied in [@Dolce]. Furthermore, we expect that results can be be extended to the case of purely vertical viscous dissipation with additional technical effort. - Due to missing vertical dissipation, we obtain no decay of the $x$-averaged magnetic field $b_=$ which is forced by the nonlinearity. To prove our results, it is convenient to work with the unknowns $$\begin{aligned} p_1= \Lambda^{-1}_t \nabla^\perp_t \cdot v, \ p_2= \Lambda^{-1}_t \nabla^\perp_t \cdot b; \quad \Lambda_t:= \sqrt{-\Delta_t}. \end{aligned}$$ Similarly to the vorticity and current, the curl operator $\nabla_t^{\perp}$ eliminates the pressure and yields a scalar quantity, while the operator $\Lambda^{-1}_t \nabla^\perp_t \cdot$ is of order $0$. Moreover, since $v$ and $b$ are divergence-free, similarly to viscosity formulations of the 2D Navier-Stokes equations, it can be shown by integration by parts that $$\begin{aligned} \begin{split} \Vert A v\Vert_ {L^2} &= \Vert A p_1 \Vert_{L^2}, \\ \Vert A b\Vert_ {L^2} &= \Vert A p_2 \Vert_{L^2}, \end{split}\end{aligned}$$ for all Fourier multiplier $A$ which commute with $\nabla_t$ and $\Lambda_t$. This, in particular, includes $\langle \nabla\rangle^N$ which corresponds to the Sobolev norm $\| \cdot \|_{H^N}$. In terms of these unknowns our equations read $$\begin{aligned} \label{eq_p} \begin{split} \partial_t p_1 - \partial_x \partial_x^t \Delta^{-1}_t p_1- \alpha \partial_x p_2 &= \nu \cdot \Delta_t p_1 +\Lambda^{-1}_t \nabla^\perp_t (b\nabla_t b- v\nabla_t v), \\ \partial_t p_2 +\partial_x \partial_x^t \Delta^{-1}_t p_2 - \alpha \partial_x p_1 &= \kappa \cdot \Delta_t p_2 +\Lambda^{-1}_t \nabla^\perp_t (b\nabla_t v- v\nabla_t b), \\ \nu= (\mu, \mu), \ \kappa &= (\mu, 0). \end{split}\end{aligned}$$ The remainder of the article is structured as follows: - In Section [2](#linstab){reference-type="ref" reference="linstab"}, as a first step we establish linear stability of the equations [\[eq_p\]](#eq_p){reference-type="eqref" reference="eq_p"}. In view of the lack of vertical resistivity we here crucially rely on the interaction of $p_1$ and $p_2$ due the the underlying constant magnetic field. Moreover, we discuss the effects of partial dissipation and the resulting limited (optimal) decay rates in time. - In Section [3](#bootHyp){reference-type="ref" reference="bootHyp"}, we introduce a bootstrap method for the proof of Theorem [Theorem 2](#thm:anisoThres){reference-type="ref" reference="thm:anisoThres"}. Decomposing into low and high frequency contributions here yields several error terms, which are handled in different subsections. In particular, we need to distinguish between the evolution of the $x$-average (which does not experience enhanced dissipation due to the shear) and its $L^2$-orthogonal complement, as well as different frequency decompositions of the nonlinear terms (called reaction and transport terms in the literature). - More precisely, in Subsection [3.2](#dnl){reference-type="ref" reference="dnl"} we collect all nonlinear terms which can be estimated in a straightforward way. In view of partial magnetic dissipation a main challenge is given by the effect of $v\nabla_t b$ on $p_2$ at high frequencies. Here, we distinguish between terms without $x$-average in Subsection [3.3](#hfw){reference-type="ref" reference="hfw"} and with average in Subsection [3.4](#hfa){reference-type="ref" reference="hfa"} and perform a decomposition into a transport and a reaction term. The low frequency regime is discussed in Subsection [3.5](#lf){reference-type="ref" reference="lf"} and does not require a very precise analysis. - As a complementary result, in Section [4](#instab){reference-type="ref" reference="instab"} we establish instability of the non-resistive, viscous MHD equations and prove Proposition [Proposition 1](#prop:instability){reference-type="ref" reference="prop:instability"}. Here we first prove linear algebraic instability and then deduce a nonlinear norm inflation result as a corollary. ## Notations and conventions {#sec:notation} For two real numbers $a,b \in \mathbb{R}$ we denote the minimum and maximum as $$\begin{aligned} \min(a,b)&=a\wedge b,\\ \max(a,b)&=a\vee b.\end{aligned}$$ We use the notation $f\lesssim g$ if there exists a constant $C$ independent of all relevant parameters such that $|f|\le C |g|$. Furthermore, we write $f\approx g$ if $f\lesssim g$ and $g\lesssim f$. Moreover, for any vector or scalar $v$ we define $$\begin{aligned} \langle v \rangle&= (1+\vert v\vert^2)^{\frac 1 2 }.\end{aligned}$$ For a function $f(x,y) \in L^2(\mathbb{T}\times\mathbb{R})$ we denote the $x$-average and its $L^2$-orthogonal complement as $$\begin{aligned} f_=(y) &= \int_\mathbb{T}f(x,y) dx,\\ f_{\neq }&= f-f_{0}.\end{aligned}$$ Throughout this text, unless noted otherwise, the spatial variables $(x,y)\in \mathbb{T}\times \mathbb{R}$ are periodic in the horizontal direction and the respective Fourier variables are denoted as $$\begin{aligned} (k,\xi)\in (\mathbb{Z}, \mathbb{R})\end{aligned}$$ or $(l,\eta)$. The norms $\Vert \cdot \Vert_{L^p}$ and $\Vert \cdot \Vert_{H^N}$ refer to the standard Lebesgue and Sobolev norms for functions on $\mathbb{T}\times \mathbb{R}$. For time-dependent functions we denote $L^p H^s=L^p_t H^s$ as the space with the norm $$\begin{aligned} \Vert f \Vert_{L^pH^s}&= \left\Vert \Vert f\Vert_{H^s(\mathbb{T}\times \mathbb{R})}\right \Vert_{L^p(0,T)}.\end{aligned}$$ We define the weight $A^N$ and $A^{N'}_\mu$ by the Fourier multipliers $$\begin{aligned} A^N&= M \langle \nabla \rangle^N, \\ A^{N'}_\mu &= M \langle \nabla \rangle^{N'} e^{c\mu t\textbf{1}_{k\neq 0 }},\end{aligned}$$ for $3<N' \le N-2$ and $0<c< \tfrac 12(1-\sqrt{\tfrac 2 3 })$. With slight abuse of notation we identify the multiplier operators with their Fourier symbols. The operator $M$ is a time dependent Fourier multiplier, introduced in [@bedrossian2016sobolev], and is defined to satisfy the following equation: $$\begin{aligned} -\tfrac {\dot M }{M }&= \tfrac {\vert k \vert } {k^2 + \vert \xi -kt\vert^2},\\ M(0,k,\xi)&=1.\end{aligned}$$ That is, $M$ is given as $$\begin{aligned} M(t,k,\xi)&= \exp\left( - \int_0^t d\tau\tfrac {\vert k \vert }{k^2+ (\xi-k\tau )^2}\right).\end{aligned}$$ In particular, the operator $M$ is comparable to the identity in the sense that $$\begin{aligned} 1\ge M(t,k,\xi) \ge c\end{aligned}$$ for some constant $c$ and all $k\neq 0$ (and $M(t,0,\xi):=1$ for $k=0$). The operators $A$ thus define energies comparable to Sobolev (semi)norms: $$\begin{aligned} \Vert A^N \cdot \Vert_{L^2} &\approx \Vert\cdot\Vert_{H^N},\\ \Vert A^{N'}_\mu \cdot \Vert_{L^2} &\approx \Vert e^{c\mu t\textbf{1}_{k\neq 0} } \cdot \Vert_{H^N}.\end{aligned}$$ In particular, since $N$ is sufficiently large, the norm defined by $A^N$ satisfies an algebra property. # Linear stability {#linstab} In this section we study the stability of the linearized version of the equations [\[eq_p\]](#eq_p){reference-type="eqref" reference="eq_p"}: $$\begin{aligned} \label{eq_p_lin} \begin{split} \partial_t p_1 - \partial_x \partial_x^t \Delta^{-1}_t p_1- \alpha \partial_x p_2 &= \nu \cdot \Delta_t p_1,\\ \partial_t p_2 +\partial_x \partial_x^t \Delta^{-1}_t p_2 - \alpha \partial_x p_1 &= \kappa \cdot \Delta_t p_2,\\ \nu= (\mu, \mu), \ \kappa &= (\mu, 0). \end{split}\end{aligned}$$ The ode tools to establish stability of such systems are well-known in related systems such as the Boussinesq equations [@lai2021optimal; @bedrossian21; @bianchini2020linear; @masmoudi2023asymptotic; @zillinger2020boussinesq]. Our main results are summarized in the following proposition. **Proposition 1** (Linear stability). *Let $\mu>0$, $\alpha> \tfrac 1 2$ and $N\geq 6$ be as in Theorem [Theorem 2](#thm:anisoThres){reference-type="ref" reference="thm:anisoThres"}. Then the equations [\[eq_p\_lin\]](#eq_p_lin){reference-type="eqref" reference="eq_p_lin"} are stable in $H^N$ in the sense that for any choice of initial data $p_{in} \in H^N$ the corresponding solution satisfies $$\begin{aligned} \|p\|_{L^\infty H^N} + \mu^{1/2} \|\nabla_t p_1\|_{L^2 H^N} + \mu^{1/2} \|\partial_x p_2\|_{L^2 H^N} \lesssim e^{-C\mu t} \|p_{in}\|_{H^N}. \end{aligned}$$* As we discuss in the proof, in the case $\nu \leq \kappa \leq \nu^{3}$ the optimal decay rate for large times is given by $\mu=\min(\nu^{1/3},\kappa)$. In particular, the coupling induced by the underlying magnetic field cannot yield enhanced dissipation rates for both components once the viscous dissipation becomes too large. *Proof of Proposition [Proposition 1](#prop:lin_stability){reference-type="ref" reference="prop:lin_stability"}.* We note that in this linear evolution equation [\[eq_p\_lin\]](#eq_p_lin){reference-type="eqref" reference="eq_p_lin"} all coefficient functions are independent of both $x$ and $y$. Therefore the equations decouple after a Fourier transform and we may equivalently consider the ode system $$\begin{aligned} \begin{split} \partial_t \hat p_1 - \tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2} \hat p_1- \alpha ik\hat p_2 &=-\nu ( k^2+(\xi-kt)^2)\hat p_1, \\ \partial_t\hat p_2 +\tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2}\hat p_2 - \alpha ik\hat p_1 &= -\kappa k^2\hat p_2, \end{split} \end{aligned}$$ for an arbitrary but fixed frequency $(k,\eta) \in \mathbb{Z}\times \mathbb{R}$. Since the equations are trivial for $k=0$, in the following we further without loss of generality may assume that $k\neq 0$. Furthermore, after shifting $t$ by $\frac{\xi}{k}$, we may assume that $\xi=0$ and thus obtain a system of the form $$\begin{aligned} \label{eq:linodesystem} \begin{split} \partial_t \begin{pmatrix} p_1 \\ p_2 \end{pmatrix} = \begin{pmatrix} -\frac{t}{1+t^2}- \nu k^2 (1+t^2) & i\alpha k \\ i\alpha k & \frac{t}{1+t^2} - \kappa k^2 \end{pmatrix} \begin{pmatrix} p_1\\ p_2 \end{pmatrix}, \end{split} \end{aligned}$$ where we dropped the hats for simplicity of notation. In a first naive estimate, we can test this equations with $(p_1,p_2)$ and obtain that $$\begin{aligned} \partial_t(|p_1|^2 + |p_2|^2) \leq (\frac{|t|}{1+t^2} - \mu k^2) (|p_1|^2 + |p_2|^2),\end{aligned}$$ which already yields the desired decay for times $|t|\gg (\mu k^2)^{-1}$. However, a Gronwall-type estimate on the remaining interval would only yield a very rough upper bound on the possible growth by $$\begin{aligned} \exp\left( \int_{|t|\lesssim (\mu k^2)^{-1}}\frac{|t|}{1+t^2} dt \right) \lesssim (1+\mu k^2)^2.\end{aligned}$$ In order to improve this estimate, a common trick is to make use of the fact that $|\alpha|$ is relatively large and to consider $$\begin{aligned} E= |p_1|^2 + |p_2|^2 + \frac{t}{1+t^2} \Re \left( \frac{1}{i\alpha} p_1 \overline{p_2} \right).\end{aligned}$$ Since $|\alpha|>\frac{1}{2}$ this energy is positive definite and comparable to $|p_1|^2 +|p_2|^2$, with a constant which degenerates as $|\alpha| \downarrow \frac{1}{2}$. Computing the time derivative of the last term, we note that $$\begin{aligned} & \quad \frac{t}{1+t^2} \partial_t \Re \left( \frac{1}{i\alpha} p_1 \overline{p_2} \right)\\ & \leq \frac{t}{1+t^2} (|p_1|^2-|p_2|^2) \\ & \quad + \frac{|t|}{1+t^2} \frac{1}{|\alpha|} \nu k^2 (1+t^2) |p_1||p_2| \\ & \quad + \frac{|t|}{1+t^2} \frac{1}{|\alpha|} \kappa k^2 |p_1||p_2| \\ & \quad + \mathcal{O}(t^{-2})|p_1||p_2|.\end{aligned}$$ The first term exactly cancels out the possibly large contribution in $\partial_t (|p_1|^2+|p_2|^2)$. For the second and third term we use that fact that $\frac{1}{\alpha} < 2$ and that these terms can hence be absorbed into the dissipation terms at the cost of a slight loss of constants, provided that $$\begin{aligned} \frac{1}{2\alpha}\nu \leq \kappa \leq \nu^{1/3}.\end{aligned}$$ Noting that $\partial_t\frac{|t|}{1+t^2}= \mathcal{O}(t^{-2})$ is integrable in time, we thus arrive at $$\begin{aligned} \partial_tE \lesssim \mathcal{O}(t^{-2})E - \nu k^2 (1+t^2)|p_1|^2 - \kappa k^2 |p_2|^2.\end{aligned}$$ Further defining $$\begin{aligned} \tilde{E} = E \exp(\int^t \mathcal{O}(\tau^{-2}) d\tau),\end{aligned}$$ it follows that $\tilde{E}\approx E$ decays exponentially in time and that the damping terms are integrable in time, which yields the desired result. We further remark that for $t$ (corresponding to times $t+\frac{\xi}{k}$) such that $|t|\lesssim (\mu k^2)^{-1/3}$ the system [\[eq:linodesystem\]](#eq:linodesystem){reference-type="eqref" reference="eq:linodesystem"} exhibits mixing enhanced dissipation, even though the dissipation for the magnetic component is only horizontal. Indeed, after relabeling $p_1 \mapsto i p_1$ and introducing the energy $E$ to control contributions by $\frac{t}{1+t^2}$, this follows from the fact that the eigenvalues of the matrix $$\begin{aligned} \begin{pmatrix} -\mu k^2 (1+t^2) & -\alpha \\ \alpha & -\mu k^2 \end{pmatrix}\end{aligned}$$ are given by $$\begin{aligned} \lambda_{1,2} = - \frac{\mu k^2 (2+t ^2)}{2} \pm \sqrt{\frac{1}{4}(\mu k^2 t^2)^2 - \alpha^2}.\end{aligned}$$ Since $|\alpha|>\frac{1}{2}$ by our assumption on $t$ the square root is purely imaginary and hence $\Re(\lambda_1)=\Re(\lambda_2)$ is comparable to the enhanced dissipation term $$\begin{aligned} - \mu k^2 (1+t^2).\end{aligned}$$ However, for times much larger than this (that is, far away from $\frac{\xi}{k}$), the same eigenvalue computation shows that $$\begin{aligned} \lambda_1 \approx -\mu k^2 \langle t\rangle^2, \ \lambda_2 \approx -\mu k^2\end{aligned}$$ and hence enhanced dissipation can only be expected for one of the eigenvalues. ◻ This linear result highlights the effects of the coupling induced by the underlying constant magnetic field and shows which optimal decay estimates can be expected. In particular, it clearly illustrates that the loss of vertical magnetic dissipation incurs a change of decay rate compared to the fully dissipative case. # Bootstrap hypotheses and outline of proof {#bootHyp} We next turn to the full nonlinear problem [\[eq_p\]](#eq_p){reference-type="eqref" reference="eq_p"}, where we intend to treat the nonlinear contributions as errors and make use of the smallness of our initial data. Our approach here follows a bootstrap argument, which is by now standard in the field (see, for instance, [@bedrossian2016sobolev]). In the notation of Section [1.1](#sec:notation){reference-type="ref" reference="sec:notation"} we assume that at the initial time $$\begin{aligned} \label{eq:initialassump} \|A^N p\|_{L^2}^2 + \|A_{\mu}^{N'}p\|_{L^2}^2 \le c_0 \epsilon^2\end{aligned}$$ for $3<N' \le N-2$. The constant $c_0=c_0(\alpha)>0$ will later be chosen small enough and tends to $0$ as $\alpha \to \tfrac 1 2$. Given this estimate at the initial time, our aim in the remainder of this section is to establish the following estimates for the corresponding solution: - **High frequency estimates** $$\begin{aligned} \label{eq:hfassump} \begin{split} \Vert A^N p_1 \Vert^2_{L^\infty L^2 }+\mu \Vert A ^N \nabla_t p_1 \Vert^2_{L^2 L^2 }+ \Vert \sqrt{-\tfrac {\dot M} {M} } A^N p_1 \Vert^2_{L^2L^2 } &<\varepsilon^2, \\ \Vert A^N p_2 \Vert^2_{L^\infty L^2 }+\mu \Vert A^N \partial_x p_2 \Vert^2_{L^2L^2 }+\Vert\sqrt{ -\tfrac {\dot M} {M} } A^N p_2 \Vert^2_{L^2L^2 } &<\varepsilon^2. \end{split}\end{aligned}$$ - **Low frequency estimates** $$\begin{aligned} \label{eq:lfassump} \begin{split} \Vert A^{N'}_\mu p_1 \Vert^2_{L^\infty L^2 }+\mu \Vert A ^{N'}_\mu \nabla_t p_1 \Vert^2_{L^2 L^2 }+\Vert \sqrt{-\tfrac {\dot M} {M} } A^{N'}_\mu p_1 \Vert^2_{L^2L^2 } &<\varepsilon^2, \\ \Vert A^{N'}_\mu p_2 \Vert^2_{L^\infty L^2 }+\mu \Vert A^{N'}_\mu \partial_x p_2 \Vert^2_{L^2L^2 } +\Vert\sqrt{- \tfrac {\dot M} {M} } A^{N'}_\mu p_2 \Vert^2_{L^2L^2 }&<\varepsilon^2. \end{split}\end{aligned}$$ By local well-posedness and our assumptions on the initial data, these estimates are satisfied at least on some (small) time interval $(0,T)$. In our bootstrap approach we assume for the sake of contradiction that the maximal time $T$ with this property is finite. We then show that on that same time interval all estimates hold with improved bounds instead, which however would imply that the estimates could be extended for a small additional time, contradicting the maximality of $T$. With this understanding, we suppress $T$ in our notation (see Section [1.1](#sec:notation){reference-type="ref" reference="sec:notation"}) and all $L^p$ norms in time are understood to be norms on $L^p(0,T)$. The splitting into high and low frequencies is essential to close the estimates in Subsection [3.3](#hfw){reference-type="ref" reference="hfw"} and Subsection [3.4](#hfa){reference-type="ref" reference="hfa"}. In particular, we need the $e^{-c\mu t }$ decay to bound the so-called reaction error. Moreover, we require strong control of commutators involving $A$ in order to control the so-called transport error. Both error terms impose strong restrictions on the energies and do not allow to close estimates in an easy way. We overcome this difficulty by linking separate energy estimates in the high frequency part and the low frequency part. On the one hand, we can use the additional $e^{-c\mu t }$ in the low frequency part to our benefit in the analysis of the high frequency part. On the other hand, the difference in regularity allows us to control derivatives in the low frequency estimate by the using high frequency estimate. Given a solution $(p_1,p_2)$ of [\[eq_p\]](#eq_p){reference-type="eqref" reference="eq_p"} and letting $A= A^N, A^{N'}_\mu$, computing time derivatives we need to control $$\begin{aligned} &\quad \partial_t \Vert A p_1 \Vert^2_{L^2 }+2(1-c)\mu \Vert A \nabla_t p_1 \Vert^2_{L^2 }+2\Vert \sqrt{-\tfrac {\dot M} {M} } A p_1 \Vert^2_{L^2 }\\ &\le 2 \langle A^2 p_1, \partial_x \partial_x^t \Delta^{-1}_t p_1 +\Lambda^{-1}_t \nabla^\perp_t (b\nabla_t b- v\nabla_t v)\rangle =: L[p_1]+NL[p_1], \\ &\quad \partial_t\Vert A p_2 \Vert^2_{ L^2 }+ 2(1-c)\mu \Vert A \partial_x p_2 \Vert^2_{L^2 }+2 \Vert\sqrt{ -\tfrac {\dot M} {M} } A p_2 \Vert^2_{L^2 }\\ &\le 2 \langle A^2 p_2, -\partial_x \partial_x^t \Delta^{-1}_t p_2 +\Lambda^{-1}_t \nabla^\perp_t (b\nabla_t v- v\nabla_t b)\rangle=: L[p_2]+NL[p_2].\end{aligned}$$ Here we have split contributions into linear (that is, quadratic integrals) and nonlinear terms (that is, trilinear integrals). Note that the choice of $0<c< \tfrac 12(1-\sqrt{\tfrac 2 3 })$ is made such that $1-c$ is not too small to absorb linear effects for $\alpha$ close to $\tfrac 1 2$. For later reference, we note the following estimates: $$\begin{aligned} \Vert \partial_x^2 \Lambda_t^{-1} \Lambda^{-1} p\Vert_{H^N }\lesssim \tfrac 1 t \Vert p_{\neq}\Vert_{H^N }\label{damp}\end{aligned}$$ and for $A= A^N, A^{N'}_\mu$ $$\begin{aligned} \label{eq:simpleenergy} \begin{split} \Vert A p_{1,\neq} \Vert_{L^2 L^2 }&\lesssim \mu^{-\frac 1 2 } \varepsilon,\\ \Vert A p_{2,\neq} \Vert_{L^2 L^2 }&\lesssim \mu^{-\frac12} \varepsilon. \end{split}\end{aligned}$$ Furthermore, for the nonlinear terms we will often use the equality $$\begin{aligned} \begin{split} \Vert A v\Vert_ {L^2} &= \Vert A p_1 \Vert_{L^2}, \\ \Vert A b\Vert_ {L^2} &= \Vert A p_2 \Vert_{L^2}. \end{split}\end{aligned}$$ Throughout the following sections, we aim to establish smallness of the contributions by the linear terms $L[\cdot]$ and nonlinear terms $NL[\cdot]$. More precisely, we establish the following proposition. **Proposition 2** (Control of errors). *Under the assumptions of Theorem [Theorem 2](#thm:anisoThres){reference-type="ref" reference="thm:anisoThres"} suppose that the initial data satisfies the smallness condition [\[eq:initialassump\]](#eq:initialassump){reference-type="eqref" reference="eq:initialassump"} and let $T>0$ be such the high and low frequency estimates [\[eq:hfassump\]](#eq:hfassump){reference-type="eqref" reference="eq:hfassump"}, [\[eq:lfassump\]](#eq:lfassump){reference-type="eqref" reference="eq:lfassump"} are satisfied. Then on the same time interval it holds that $$\begin{aligned} \int_0^T L[p_1]+ L[p_2] dt &\leq \tfrac{1}{2\alpha} (c_0+ 1)\varepsilon^2 + O(\mu^{-1}\varepsilon^3) ,\\ \int_0^T NL[p_1] +NL[p_2] dt & \leq \mu^{-\frac 3 2 } \varepsilon^3. \end{aligned}$$* As a consequence, supposing that $\alpha>\frac 1 2$ and $\epsilon\ll \mu^{3/2}$, this implies that both the high frequency and low frequency estimates [\[eq:hfassump\]](#eq:hfassump){reference-type="eqref" reference="eq:hfassump"}, [\[eq:lfassump\]](#eq:lfassump){reference-type="eqref" reference="eq:lfassump"} improve and thus $T$ can only have been maximal if $T=\infty$, which proves Theorem [Theorem 2](#thm:anisoThres){reference-type="ref" reference="thm:anisoThres"}. Thus proving Proposition [Proposition 2](#prop:errors){reference-type="ref" reference="prop:errors"} is our main concern in this section and our proof is split over the following subsections. The most important estimates, highlighting the effects of partial dissipation, are established in Subsections [3.1](#sec:linerror){reference-type="ref" reference="sec:linerror"}, [3.3](#hfw){reference-type="ref" reference="hfw"} and [3.4](#hfa){reference-type="ref" reference="hfa"}. We note that the nonlinear terms $$\begin{aligned} \langle A p_1 , \Lambda^{-1}_t \nabla^\perp_t A (b\nabla_t b- v\nabla_t v)\rangle &=-\langle A v , A (b\nabla_t b- v\nabla_t v)\rangle,\\ \langle A p_2 , \Lambda^{-1}_t \nabla^\perp_t A (b\nabla_t v- v\nabla_t b)\rangle&=-\langle A b , A (b\nabla_t v- v\nabla_t b)\rangle,\end{aligned}$$ for $A= A^N, A^{N'}_\mu$ are all trilinear products involving $$\begin{aligned} a^1a^2a^3\in \{vvv,vbb,bbv,bvb\}\end{aligned}$$ and we will use this notation to refer to the specific terms. Since the $x$-averages do not experience fast (mixing enhanced) decay under the dissipation, we split these products as $$\begin{aligned} \langle A a^1, A(a^2\nabla_t a^3)\rangle &= \langle A a^1_{\neq} , A(a^2_{\neq}\nabla_t a^3_{\neq})_{\neq} \rangle\\ & \quad +\langle A a^1_{\neq} , A(a^2_=\nabla_t a^3_{\neq}) \rangle\\ &\quad +\langle A a^1_{\neq} , A(a^2_{\neq}\nabla_t a^3_=) \rangle\\ &\quad + \langle A a^1_{=} , A(a^2_{\neq}\nabla_t a^3_{\neq})_=,\end{aligned}$$ where the full splitting is only used for the $bvb$ term. ## Estimate of the linear error {#sec:linerror} In this subsection we establish the estimate of the linear terms in Proposition [Proposition 2](#prop:errors){reference-type="ref" reference="prop:errors"}. Here, we use some of the same techniques as in the proof of linear stability in Section [2](#linstab){reference-type="ref" reference="linstab"}, but instead focus on establishing quantitative bounds on the time integral. Taking a Fourier transform of [\[eq_p\]](#eq_p){reference-type="eqref" reference="eq_p"} yields $$\begin{aligned} \begin{split} \partial_t \hat p_1 - \tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2} \hat p_1- \alpha ik\hat p_2 &=-\mu ( k^2+(\xi-kt)^2)\hat p_1 +\mathcal{F}[NL[p_1]], \\ \partial_t\hat p_2 +\tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2}\hat p_2 - \alpha ik\hat p_1 &= - \mu k^2\hat p_2 +\mathcal{F}[NL[p_2]]\label{eq_pf}. \end{split}\end{aligned}$$ Recalling the various contributions, we aim to estimate $$\begin{aligned} & \quad \langle A^2 p_2 , -\partial_x \partial_y^t \Delta^{-1}_t p_2 \rangle + \langle A^2 p_1 , \partial_x \partial_y^t \Delta^{-1}_t p_1\rangle\\ &= \sum_k \int d\xi A^2 \tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2}(\vert \hat p_1 \vert^2-\vert \hat p_2 \vert^2 ) .\end{aligned}$$ In the following, with slight abuse of notation, we omit the hat denoting the Fourier transform and only consider $k\neq 0$, since for $k=0$ this term vanishes. Similarly as in the linear stability results of Section [2](#linstab){reference-type="ref" reference="linstab"}, we note that the Fourier multiplier a priori is not integrable in time and cannot easily be estimated by the partial dissipation. Hence, we rely on the coupling induced by the underlying magnetic field to eliminate some of this contribution and to provide better decay. More precisely, multiplying the equations [\[eq_pf\]](#eq_pf){reference-type="eqref" reference="eq_pf"} with $\hat{p}_2, \hat{p}_1$ and omitting the hats for simplicity of notation, we obtain the following identity: $$\begin{aligned} & \quad \vert p_1 (k) \vert^2-\vert p_2 (k)\vert^2\\ &= -\tfrac 1 {i \alpha k }( p_1 \overline { i \alpha k p_1 } +i \alpha k p_2 \overline { p_2} )\\ &=-\tfrac 1 {i \alpha k } p_1 (\partial_t \overline{p}_2 +\tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2}\ \overline{p}_2 + \mu k^2 \overline{p}_2 -\overline{\mathcal{F}} [{NL}[p_2]]))\\ & \quad -\tfrac 1 {i \alpha k } \overline p_2 (\partial_t p_1 -\tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2} p_1+(\mu k^2+\mu (\xi-kt)^2) p_1 -\mathcal{F}[NL[p_1]]\\ &= \tfrac {-1}{i\alpha k } (\partial_t (p_1 \overline p_2)+ \mu (k^2+(\xi-kt)^2)p_1 \overline p_2+ \mu k^2p_1 \overline p_2) \\ & \quad -\tfrac 1 {\alpha ik } ( p_1, p_2) \cdot \mathcal{F}[\Lambda^{-1}_t \nabla_t^\perp (b\nabla_t b -v\nabla_t v , b\nabla_t v-v\nabla_t b)].\end{aligned}$$ Thus we split $L$ into two linear terms and one nonlinear term: $$\begin{aligned} \label{eq:ONL} \begin{split} L&=2\sum_k \int d\xi A^2 \tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2}\tfrac {-1}{i\alpha k } \partial_t (p_1 \overline p_2) \\ & \quad + 2\sum_k \int d\xi A^2 \tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2}\tfrac {-1}{i\alpha k }(2 \mu k^2+\mu (\xi-kt)^2)p_1 \overline p_2 \\ & \quad -\tfrac 2 {\alpha }\langle A \partial_y^t \Delta_t^{-1} (p_1,p_2)_{\neq} ,A \Lambda^{-1}_t \nabla_t^\perp (b\nabla_t b -v\nabla_t v , b\nabla_t v-v\nabla_t b)_{\neq}\rangle \\ &=L_1 +L_{\mu}+ONL. \end{split}\end{aligned}$$ We estimate $L_{\mu}$ by $$\begin{aligned} L_{\mu}&= \tfrac 2\alpha \mu \sum_{k \neq 0 }\int d\xi A^2 \tfrac {(2 k^2+(\xi-kt )^2 )(\xi-kt )}{k^2+(\xi-kt)^2}p_1 \overline p_2 \\ &= \tfrac 2\alpha \mu \sum_{k \neq 0 }\int d\xi A^2 \tfrac {(2 k^2+(\xi-kt )^2 )(\xi-kt )}{(k^2+(\xi-kt)^2)^{\frac 3 2}} p_1 (k^2+(\xi-kt)^2)^{\frac 1 2 } \overline p_2 \\ &\le \tfrac 2 \alpha \mu \sup_s \left( \tfrac {(2+s^2)s}{(1+s^2)^{\frac 3 2 }} \right) \Vert A \partial_x p_2 \Vert_{L^2} \Vert A \nabla_t p_1 \Vert_{L^2} \\ &\le \sqrt{\tfrac 23 } \tfrac 1 \alpha \mu (\Vert A \partial_x p_2 \Vert_{L^2}^2 +\Vert A \nabla_t p_1 \Vert_{L^2}^2),\end{aligned}$$ where we used that $$\begin{aligned} \left \vert \tfrac {(2 k^2+(\xi-kt )^2 )(\xi-kt )}{(k^2+(\xi-kt)^2)^{\frac 3 2}}\right\vert&= \left\vert \tfrac {(2 +(\frac\xi k -t )^2 )(\frac \xi k-t )}{(1+(\frac \xi k-t)^2)^{\frac 3 2}}\right\vert \\ &\le \sup_s \left( \tfrac {(2+s^2)s}{(1+s^2)^{\frac 3 2 }} \right)\\ &\le \sqrt{\tfrac 23 }.\end{aligned}$$ To estimate $L_1$, we integrate in time and integrate by parts in space to deduce that $$\begin{aligned} & \quad \int d\tau \sum_k \int d\xi A^2 \tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2}\tfrac {-1}{i\alpha k } \partial_t (p_1 \overline p_2)\\ &= \left[ \tfrac {-1}{i\alpha } \sum_k \int d\xi A^2 \tfrac {(\xi-kt )}{k^2+(\xi-kt)^2}p_1 \overline p_2\right]^t_0 \\ & \quad +\int d\tau \tfrac {1}{i\alpha } \sum_k \int d\xi p_1 \overline p_2 \partial_t ( A^2 \tfrac { (\xi-kt )}{k^2+(\xi-kt)^2}) \\ &= \left[ \tfrac {-1}{i\alpha } \sum_k \int d\xi A^2 \tfrac {(\xi-kt )}{k^2+(\xi-kt)^2}p_1 \overline p_2\right]^t_0 \\ &\quad +\int d\tau \tfrac {2}{i\alpha } \sum_k \int d\xi p_1 \overline p_2 \tfrac {\dot M } M A^2 \tfrac { (\xi-kt )}{k^2+(\xi-kt)^2} \\ &\quad +c \mu \textbf{1}_{A=A^{N'}_\mu } \int d\tau \tfrac {2}{i\alpha } \sum_k \int d\xi p_1 \overline p_2 A^2 \tfrac { (\xi-kt )}{k^2+(\xi-kt)^2} \\ &\quad +\int d\tau \tfrac {1}{i\alpha } \sum_k \int d\xi p_1 \overline p_2 A^2 \tfrac { k(k^2-(kt-\xi)^2)}{(k^2+(\xi-kt)^2)^2}.\end{aligned}$$ So we infer by Hölder's inequality that $$\begin{aligned} & \quad \int d\tau \sum_k \int d\xi A^2 \tfrac {k (\xi-kt )}{k^2+(\xi-kt)^2}\tfrac {-1}{i\alpha k } \partial_t (p_1 \overline p_2)\\ &\le \tfrac 1 \alpha (\Vert A p_1 (0)\Vert_{ L^2 } \Vert A p_2(0) \Vert_{L^2 }+ \Vert A p_1 (t)\Vert_{ L^2 } \Vert A p_2 (t)\Vert_{ L^2 }) \\ &\quad + \mu \Vert A \partial_x p_1\Vert_{L^2 L^2 } \Vert A\sqrt{ -\tfrac {\dot M } M } p_2 \Vert_{L^2 L^2 }\\ &\quad + \tfrac 1\alpha \Vert A \sqrt{- \tfrac {\dot M } M } p_1 \Vert_{L^2L^2 } \Vert A \sqrt{ -\tfrac {\dot M } M } p_2 \Vert_{L^2 L^2 }\end{aligned}$$ and thus $$\begin{aligned} & \quad \int L d \tau -\int ONL d\tau \\ &\le \tfrac 1 {2\alpha } (\Vert A p_1 (0)\Vert_{L^2 }^2+ \Vert A p_2(0) \Vert_{ L^2 }^2)\\ &\quad + \tfrac 1 {2\alpha } (\Vert A p_1 \Vert_{ L^\infty L^2 }^2+ \Vert A p_2(t) \Vert_{ L^\infty L^2 }^2)\\ &\quad + \tfrac {1 } \alpha (\mu (1-c) \Vert \partial_x A p\Vert_{L^2}^2+\mu (1-c) \Vert \partial_y^t A p\Vert_{L^2}^2+ \Vert \sqrt{- \tfrac {\dot M }M } A p\Vert_{L^2} ^2) .\end{aligned}$$ Using the dissipation estimates [\[eq:simpleenergy\]](#eq:simpleenergy){reference-type="eqref" reference="eq:simpleenergy"}, we therefore obtain $$\begin{aligned} \int L d \tau &\le \tfrac 1{2\alpha }(c_0+1) \varepsilon^2 + \int ONL d\tau, \label{est:L}\end{aligned}$$ where the $ONL$ part will be estimated at the beginning of the next subsection. ## Immediate nonlinear estimates for $A^N$ {#dnl} In this subsection, we collect some estimates which can be obtained in a straight forward approach using standard techniques (e.g. see [@bedrossian2016sobolev]). In particular, for these terms we are not constrained by the lack of vertical resistivity. For most estimates we do not aim to establish optimal (mixing enhanced) bounds, since these bounds are in any case better than the ones involving horizontal resistivity and hence do not affect the over all stability threshold. In the following we write $A=A^N$. **ONL estimate:** Using integration by parts in space and Hölder's inequality, the nonlinear contribution in [\[eq:ONL\]](#eq:ONL){reference-type="eqref" reference="eq:ONL"} can be estimated by $$\begin{aligned} ONL&= \tfrac 2 {\alpha }\langle A \partial_y^t \Delta_t^{-1} v_{\neq} ,A (b\nabla_t b -v\nabla_t v )_{\neq}\rangle \\ &\quad + \tfrac 2 {\alpha }\langle A \partial_y^t \Delta_t^{-1} b_{\neq} ,A (b\nabla_t v-v\nabla_t b)_{\neq}\rangle \\ &= \tfrac 2 {\alpha }\langle A \partial_y^t \Delta_t^{-1} (\nabla^\perp_t \otimes v_{\neq}) ,A (b\otimes b -v\otimes v )_{\neq}\rangle \\ &\quad + \tfrac 2 {\alpha }\langle A \partial_y^t \Delta_t^{-1} (\nabla^\perp_t \otimes b _{\neq}) ,A (b\otimes v-v\otimes b)_{\neq}\rangle \\ &\lesssim \tfrac 2 \alpha \Vert A (v, b)_{\neq} \Vert^2_{L^2}\Vert A (v, b) \Vert_{L^2}.\end{aligned}$$ Recalling the bounds [\[eq:simpleenergy\]](#eq:simpleenergy){reference-type="eqref" reference="eq:simpleenergy"} and integrating in time we thus obtain that $$\begin{aligned} \int ONL d\tau \lesssim \mu^{-1} \varepsilon^3.\label{est:ONL}\end{aligned}$$ **Estimates with an $x$-average in the second component:** Let $a^1a^2a^3\in \{vvv,vbb,bbv,bvb\}$, then we need to estimate the trilinear products $$\begin{aligned} \langle A a^1_{\neq} , A(a^2_{=} \nabla_t a^3_{\neq} ) \rangle &= \langle A a^1_{\neq} , A(a^2_{=,1} \partial_x a^3_{\neq})\rangle\\ &\lesssim \Vert A a^1_{\neq}\Vert_{L^2}\Vert A a^2_{=,1 } \Vert_{L^2}\Vert A \partial_x a^3_{\neq}\Vert_{L^2}. \end{aligned}$$ Integrating in time and again using the bound [\[eq:simpleenergy\]](#eq:simpleenergy){reference-type="eqref" reference="eq:simpleenergy"} yields a control by $$\begin{aligned} \int d \tau \langle Aa^1 , A(a^2_{=} \nabla_t a^3) \rangle &\lesssim \mu^{-1} \varepsilon^3.\label{est:aver}\end{aligned}$$ The influence of the underlying $x$-averaged velocity and magnetic field on the average-less parts can thus be easily controlled by the dissipation, provided $\epsilon\ll \mu$. In the following we focus on terms involving $a^2_{\neq}$. **vvv estimate:** We first discuss the velocity non-linearity and use the algebra property of $H^N$ and the bounds on $A$ to estimate $$\begin{aligned} \langle A v , A v_{\neq}\nabla_t v\rangle&\le \Vert A v\Vert_{L^2}\Vert A v_{\neq}\Vert_{L^2}\Vert A\nabla_t v\Vert_{L^2}.\end{aligned}$$ Here, the contribution by $\|A \nabla_t v\|_{L^2}$ is square integrable in time due to the dissipation [\[eq:simpleenergy\]](#eq:simpleenergy){reference-type="eqref" reference="eq:simpleenergy"}, while $\|A v_{\neq}\|_{L^2}$ is square integrable in time by the inviscid damping estimates [\[damp\]](#damp){reference-type="eqref" reference="damp"}. Integrating in time thus yields a bound by $$\begin{aligned} \int d \tau \langle A v , A (v_{\neq}\nabla_t v)\rangle &\lesssim \mu^{-1} \varepsilon^3 .\label{est:vvv}\end{aligned}$$ **vbb estimate:** For the contributions by the $vbb$ nonlinearity we intend to argue similarly, but have to account for the lack of vertical magnetic dissipation (which we compensate for by using the full fluid dissipation). We may split the integral as $$\begin{aligned} \langle A v , A (b_{\neq}\nabla_tb)\rangle&= \int A v_1 A(b_{1,\neq}\partial_x +b_{2,\neq}\partial_y^t)b_1\\ &\quad +\int A v_2 A(b_{1,\neq}\partial_x +b_{2,\neq}\partial_y^t) b_2.\end{aligned}$$ For the second term we integrate by parts to obtain $$\begin{aligned} \int A v_1 A(b_{2,\neq}\partial_y^t b_1)&=- \int A \partial_y^t v_1 A( b_{2,\neq} b_1)-\int A v_1 A(\partial_y^t b_{2,\neq}b_1) .\end{aligned}$$ Furthermore, since $b$ is divergence-free, it holds that $\partial_y^t b_2 = -\partial_x b_1$ and hence $$\begin{aligned} \langle A v , A (b_{\neq}\nabla_tb)\rangle &\le \Vert A v\Vert_{L^2} \Vert A b_{\neq}\Vert_{L^2} \Vert A \partial_x b\Vert_{L^2} + \Vert \partial_y^t v\Vert_{L^2} \Vert A b_{\neq}\Vert_{L^2} \Vert A b_2\Vert_{L^2}.\end{aligned}$$ We may therefore estimate this term using the full fluid and horizontal magnetic dissipation [\[eq:simpleenergy\]](#eq:simpleenergy){reference-type="eqref" reference="eq:simpleenergy"} and integrating in time yields a bound by $$\begin{aligned} \label{est:vbb} \int d \tau \langle A v , A (b_{\neq}\nabla_tb)\rangle &\lesssim \mu^{-1}\varepsilon^3.\end{aligned}$$ **bbv estimate:** Finally, for the $bbv$ contribution, we may again use the full fluid dissipation and the algebra property of $A$ (and $H^N$) to obtain a bound $$\begin{aligned} \langle A b , A ( b_{\neq}\nabla_t v)\rangle&\lesssim \Vert Ab \Vert_{L^2}\Vert Ab_{\neq} \Vert_{L^2}\Vert A \nabla_t v\Vert_{L^2}. \end{aligned}$$ Integrating in time and using [\[eq:simpleenergy\]](#eq:simpleenergy){reference-type="eqref" reference="eq:simpleenergy"} we thus obtain a bound by $$\begin{aligned} \label{est:bbv} \int d\tau \langle A b , A ( b_{\neq}\nabla_t v)\rangle&\lesssim \mu^{-1} \varepsilon^3.\end{aligned}$$ ## High frequency $bvb$ term without $x$-average {#hfw} Having established several straightforward estimates using the full fluid dissipation, in this and the following subsections we establish bounds for the high frequency (that is, $A^N$ terms as in [\[eq:hfassump\]](#eq:hfassump){reference-type="eqref" reference="eq:hfassump"}) terms involving $bvb$. For simplicity, we write $A=A^N$ and aim to establish the estimate $$\begin{aligned} \langle A b , A ( v_{\neq}\nabla_t b)\rangle\lesssim \mu^{-\frac 3 2 } \varepsilon^3.\end{aligned}$$ We split the $bvb$ term according to (non)vanishing $x$-averages: $$\begin{aligned} \langle A b , A ( v_{\neq}\nabla_t b)\rangle&= \langle A b_{\neq} , A ( v_{\neq}\nabla_t b_{\neq})_{\neq}\rangle\\ &\quad + \langle A b_{\neq} , A ( v_{\neq}\nabla_t b_{=})_{\neq}\rangle\\ &\quad + \langle A b_= , A ( v_{\neq}\nabla_t b_{\neq})_=\rangle.\end{aligned}$$ Let us first consider the term without any $x$-averages, which can be written as $$\begin{aligned} \langle A b_{\neq} , A ( v_{\neq}\nabla_t b_{\neq})\rangle&= \int A b_{1,\neq} A((v_{1,\neq}\partial_x +v_{2,\neq }\partial_y^t) b_{1,\neq})\\ &\quad +\int A b_{2,\neq} A((v_{1,\neq}\partial_x +v_{2,\neq}\partial_y^t) b_{2,\neq}).\end{aligned}$$ We estimate the second contribution using the algebra property of $H^N$ and that $\partial_y^tb_2=-\partial_x b_1$, since $b$ is divergence-free: $$\begin{aligned} \int d \tau \int & A b_{2,\neq} A(v_{1,\neq}\partial_x +v_{2,\neq}\partial_y^t) b_{2,\neq}\\ &\le \int d \tau \Vert A b_{2,\neq}\Vert_{L^2} (\Vert Av_{1,\neq}\Vert_{L^2}\Vert A \partial_x b_{2,\neq} \Vert_{L^2} +\Vert Av_{2,\neq}\Vert_{L^2}\Vert A\partial_y^t b_{2,\neq}\Vert_{L^2} ) \\ &\le \int d \tau \Vert A b_{2,\neq}\Vert_{L^2} (\Vert Av_{1,\neq}\Vert_{L^2}\Vert \partial_x b_{2,\neq} \Vert_{L^2} +\Vert Av_{2,\neq}\Vert_{L^2}\Vert A \partial_x b_{1,\neq}\Vert_{L^2} ).\end{aligned}$$ Employing Hölder's inequality this contribution can thus be estimated as $$\begin{aligned} \begin{split} \int d \tau \int & A b_{2,\neq} A((v_{1,\neq}\partial_x +v_{2,\neq}\partial_y^t) b_{2,\neq})\\ &\le\int d \tau \Vert A b_{2,\neq}\Vert_{L^2} \Vert Av_{\neq}\Vert_{L^2}\Vert A \partial_x b_{\neq} \Vert_{L^2}\\ &\le \Vert A b_{2,\neq}\Vert_{L^2L^2} \Vert Av_{\neq}\Vert_{L^\infty L^2}\Vert A \partial_x b_{\neq} \Vert_{L^2L^2}\\ &\lesssim \mu^{-1} \varepsilon^3.\label{est:bvbn1} \end{split}\end{aligned}$$ It remains to control the contribution by $b_{1,\neq}$, which in view to the lack of vertical resistivity is the hardest term to control. Since the velocity field $v$ is divergence-free, we observe that $$\begin{aligned} \int A b_{1,\neq }(v_{\neq}\nabla_t A b_{1,\neq })=0.\end{aligned}$$ Therefore, we obtain the following cancellations and introduce a splitting in Fourier space: $$\begin{aligned} \int A b_{1,\neq } A(v_{\neq} \nabla_t b_{1,\neq })&= \int A b_{1,\neq } (A(v\nabla_t b_{1,\neq })-(v_{\neq}\nabla_t A b_{1,\neq }))\\ &= \sum_{k,l,k-l \neq 0} \iint d(\xi,\eta ) A(k,\xi )b_1(k,\xi) \tfrac {(A(k,\xi) -A(l,\eta))(\xi l -\eta k) }{\sqrt{(k-l)^2+(\xi-\eta-(k-l)t)^2}}\\ &\qquad\qquad p_1(k-l ,\xi-\eta)b_1(l,\eta) \\ &=T+R+\mathcal{R}. \end{aligned}$$ Here, the Fourier regions $$\begin{aligned} \Omega_T&=\{\vert k-l,\xi-\eta \vert \le \tfrac 18 \vert l , \eta\vert \}, \\ \Omega_R&=\{\vert l , \eta\vert \le \tfrac 18 \vert k-l,\xi-\eta \vert\},\\ \Omega_\mathcal{R}&=\{\tfrac 18 \vert l , \eta\vert \le \vert k-l,\xi-\eta \vert\le 8\vert l , \eta\vert \},\end{aligned}$$ correspond to the the *transport* ($T$) or low-high term, *reaction* ($R$) or high-low term and the *remainder* ($\mathcal{R}$) or high-high term. In the following we omit the $\neq$ subscripts. **Transport term:** Since $\vert k-l,\xi-\eta \vert \le \tfrac 18 \vert l , \eta\vert$ we obtain that $\vert l,\eta \vert \approx \vert k,\xi \vert$. Without loss of generality we assume that $\xi \le \eta$, since we can use either of the following splittings $$\begin{aligned} \xi l -k \eta &= (\xi-\eta)l -(k-l)\eta \\ &= (\xi-\eta)k - \xi(k-l).\end{aligned}$$ Thus using the second equality we estimate $$\begin{aligned} T &\le \Vert \partial_y\Lambda^{-1}_t p_1 \Vert_{L^\infty } \Vert Ab_1 \Vert_{L^2} \Vert \partial_x A b_1\Vert_{L^2}\\ &\quad + \sum_{k,l\neq 0} \iint d(\xi,\eta ) \textbf{1}_ {\Omega_T } (\textbf{1}_{2\langle t\rangle ( k \vee l)\ge \xi}+\textbf{1}_{2\langle t\rangle ( k \vee l)\le \xi }) \\ &\qquad\qquad \cdot A(k,\xi )b_1(k,\xi) \tfrac {(A(k,\xi) -A(l,\eta))\xi (l - k) }{\sqrt{(k-l)^2+(\xi-\eta-(k-l)t)^2}} p_1(k-l ,\xi-\eta)b_1(l,\eta),\end{aligned}$$ where we distinguished between $2 \langle t\rangle ( k \vee l)\ge \xi$ and $2 \langle t\rangle ( k \vee l)\le \xi$. The first case is estimated by using the dissipation and [\[damp\]](#damp){reference-type="eqref" reference="damp"}: $$\begin{aligned} \sum_{k,l\neq 0} \iint d(\xi,\eta ) &\textbf{1}_{\Omega_T } \textbf{1}_{\xi \le 2 ( k \vee l)\langle t\rangle } A(k,\xi )b_1(k,\xi)\\ &\cdot \tfrac {(A(k,\xi) -A(l,\eta))\xi (l -k) }{\sqrt{(k-l)^2+(\xi-\eta-(k-l)t)^2}} p_1(k-l ,\xi-\eta)b_1(l,\eta) \\ &\lesssim \langle t\rangle \Vert A b_1 \Vert_{L^2} \Vert \Lambda^{-1}_t \partial_x p_1 \Vert_{L^\infty } \Vert \partial_x A b_1\Vert_{L^2 } \\ &\lesssim \Vert A b_1 \Vert_{L^2} \Vert \Lambda \partial_x p_1 \Vert_{L^\infty } \Vert \partial_x A b_1\Vert_{L^2 } \\ &\lesssim \Vert A b_1 \Vert_{L^2} \Vert A p_1 \Vert_{L^2} \Vert \partial_x A b_1\Vert_{L^2 }. \end{aligned}$$ For the second case, $2 \langle t\rangle ( k \vee l)\le \xi$, we need to estimate $$\begin{aligned} (A^N(k,\xi) - A^N(l,\eta)) &= (M(k,\xi)\vert k,\xi\vert^N - M(l,\eta)\vert l,\eta \vert^N )\\ &= M(k,\xi)(\vert k,\xi\vert^N - \vert l,\eta \vert^N)\\ &\quad + M(l,\eta)(\tfrac {M(k,\xi)}{ M(l,\eta)}-1)\vert l,\eta \vert^N.\end{aligned}$$ By the mean value theorem, we obtain $$\begin{aligned} \vert k,\xi\vert^N - \vert l,\eta \vert^N &\le N \vert k-\theta l ,\xi- \theta \eta \vert^{N -1}\vert k-l , \xi- \eta \vert \\ &\lesssim\vert k-l , \xi- \eta \vert ( \vert l,\eta \vert^{N-1} + \vert k-l , \xi- \eta \vert^{N-1} )\\ &\lesssim\vert k-l , \xi- \eta \vert \vert l,\eta \vert^{N-1} . \end{aligned}$$ For the differences in $M$ we use that for $a,b>0$ it holds that $\vert e^{a-b}-1\vert \le e^{a+b}-1$ and hence $$\begin{aligned} \vert \tfrac {M_1(k,\xi)}{ M_1(l,\eta)}-1\vert &= \vert \exp\left( \int_0^t \tfrac {\vert l \vert} {l^2 + (\eta -ls)^2 }-\tfrac {\vert k \vert} {k^2 + (\xi -ks)^2 } ds \right) -1 \vert \\ &\le \vert \exp\left( \int_0^t \tfrac {\vert l \vert} {l^2 + (\eta -ls)^2 }+\tfrac {\vert k \vert} {k^2 + (\xi -ks)^2 } ds \right) -1 \vert. \end{aligned}$$ Thus for $\eta \ge \xi\ge 2 t(k\vee l )$ by integrating we obtain that $$\begin{aligned} \vert \tfrac {M_1(k,\xi)}{ M_1(l,\eta)}-1\vert &\le \exp\left( \tfrac 1 {\vert l \vert } \int_0^t \tfrac {1} {1 + (\frac \eta l -s)^2 }ds + \tfrac 1 {\vert k \vert }\int_0^t \tfrac {1} {1 + (\frac\xi k -s)^2 } ds \right) -1\\ &\le \exp(\tfrac {1} {\eta } +\tfrac {1}\xi ) -1 \\ &\lesssim \tfrac {1} \eta+\tfrac 1\xi .\end{aligned}$$ Therefore, we deduce that $$\begin{aligned} \sum_{k,l,k-l \neq 0} \iint d(\xi,\eta ) &\textbf{1}_ {\Omega_T }\textbf{1}_{\xi \ge 2 (k\vee l) t } A(k,\xi )b_1(k,\xi) \tfrac {(A(k,\xi) -A(l,\eta))\xi(l - k) }{\sqrt{(k-l)^2+(\xi-\eta-(k-l)t)^2}} \\ &\qquad\qquad p_1(k-l ,\xi-\eta)b_1(l,\eta) \\ &\lesssim \Vert A b_1 \Vert_{L^2} \Vert \Lambda_t^{-1} \partial_x p_1 \Vert_{L^\infty } \Vert A b_1\Vert_{L^2 }\\ &\lesssim \langle t\rangle^{-1} \Vert A b_1 \Vert_{L^2} \Vert \Lambda \partial_x p_1 \Vert_{L^\infty } \Vert A b_1\Vert_{L^2 }\\ &\lesssim \langle t\rangle^{-1} \Vert A b_1 \Vert_{L^2} \Vert A p_1 \Vert_{L^2 } \Vert A b_1\Vert_{L^2},\end{aligned}$$ where we used the estimate [\[damp\]](#damp){reference-type="eqref" reference="damp"}. Combining all estimates, we have derived the following estimate of the transport term: $$\begin{aligned} \begin{split} \int Td\tau &\lesssim \Vert A b_1 \Vert_{L^\infty L^2} \Vert A p_1 \Vert_{L^\infty L^2}\Vert A b_1\Vert_{L^2L^2}\\ &\lesssim \mu^{-\frac 1 2 } \varepsilon^3.\label{est:bvbnT} \end{split}\end{aligned}$$ **Reaction term:** Since $\vert l , \eta\vert \le \tfrac 18 \vert k-l,\xi-\eta \vert$ we obtain $\vert k-l,\xi-\eta \vert \approx \vert k,\xi \vert$. With the identity $$\begin{aligned} \xi l -k \eta &= l(\xi-\eta -(k-l) t) - (k-l) (\eta -lt) \end{aligned}$$ and $A(k,\xi) -A(l,\eta)\lesssim A(k-l,\xi-\eta)$ we infer $$\begin{aligned} R&= \sum_{k,l,k-l \neq 0} \iint d(\xi,\eta ) \textbf{1}_{\Omega_R }A(k,\xi )b_1(k,\xi) \tfrac {(A(k,\xi) -A(l,\eta)) ( l(\xi-\eta -(k-l) t) - (k-l) (\eta -lt) ) }{\sqrt{(k-l)^2+(\xi-\eta-(k-l)t)^2}} \\ &\qquad\qquad \cdot p_1(k-l ,\xi-\eta)b_1(l,\eta) \\ &\le \Vert A b_1 \Vert_{L^2} \Vert A \partial_y^t \Lambda_t^{-1} p_1 \Vert_{L^2 } \Vert \partial_x b_1 \Vert_{L^\infty } \\ &\quad +\Vert A b_1 \Vert_{L^2} \Vert A \Lambda_t^{-1} p_1 \Vert_{L^2 }\Vert \partial_y^t\partial_x^2 b_1\Vert_{L^\infty } \\ &\quad +\Vert \partial_x A b_1 \Vert_{L^2} \Vert A \Lambda_t^{-1} p_1 \Vert_{L^2 }\Vert \partial_y^t \partial_x b_1\Vert_{L^\infty } .\end{aligned}$$ We split $\partial_y^t = \partial_y-t\partial_x$ and use the definition of the low-frequency multiplier $A^{N'}_\mu$ to estimate $$\begin{aligned} \Vert \langle \partial_x \rangle^2 \partial_y^t b_1\Vert_{L^\infty }&\le\Vert \langle \partial_x \rangle^2 \partial_y b_1\Vert_{L^\infty }+\Vert \langle \partial_x \rangle^2 t\partial_x b_1\Vert_{L^\infty }\\ &\le t \Vert \Lambda^{N'} b_1\Vert_{L^2} \\ &\lesssim te^{-c\mu t }\Vert A^{N'}_\mu b_1\Vert_{L^2 }\\ &\lesssim \mu^{-1} \Vert A^{N'}_\mu b_1\Vert_{L^2 }.\end{aligned}$$ Therefore, integrating in time yields the estimate $$\begin{aligned} \label{est:bvbnR} \begin{split} \int Rd \tau &\lesssim \Vert A b_1 \Vert_{L^2 L^2} \left(\Vert A \partial_y^t \Lambda_t^{-1} p_1 \Vert_{L^2L^2 } \Vert A b_1 \Vert_{L^\infty L^2 }\right) \\ &\quad +\mu^{-1} \Vert A \partial_x b_1 \Vert_{L^2 L^2}\Vert A \Lambda_t^{-1} p_1 \Vert_{L^2L^2 } \Vert A^{N'}_\mu b_1\Vert_{L^\infty L^2 }\\ &\lesssim \varepsilon^3 \mu^{-\frac 32 }. \end{split}\end{aligned}$$ **$\mathcal{R}$ term:** We consider the Fourier region where $\tfrac 18 \vert l , \eta\vert \le \vert k-l,\xi-\eta \vert\le 8\vert l , \eta\vert$. Thus, we have the bounds $\vert k, \xi \vert \lesssim \vert l, \eta \vert$ and $A( k , \xi )\lesssim A(l,\eta )\approx A(k-l, \xi-\eta)$. Furthermore, we note that $$\begin{aligned} \xi l -\eta k&\le \vert l, \eta \vert^2,\end{aligned}$$ and thus estimate the remainder terms as $$\begin{aligned} \mathcal{R}&= \sum_{k,l,k-l \neq 0} \iint d(\xi,\eta )1_{\Omega_\mathcal{R}} A(k,\xi )b_1(k,\xi) \\ &\qquad\qquad \tfrac {(A(k,\xi) -A(l,\eta))(\xi l -\eta k) }{\sqrt{(k-l)^2+(\xi-\eta-(k-l)t)^2}} p_1(k-l ,\xi-\eta)b_1(l,\eta) \\ &\lesssim \Vert A b_1 \Vert_{L^2} \Vert A \Lambda^{-1}_t p_1 \Vert_{L^2} \Vert \Lambda^2 b_1 \Vert_{L^\infty }\\ &\lesssim \Vert A b_1 \Vert_{L^2} \Vert A \Lambda^{-1}_t p_1 \Vert_{L^2} \Vert A b_1 \Vert_{L^2 }.\end{aligned}$$ Hence after integrating in time, we deduce that $$\begin{aligned} \label{est:bvbncalR} \int\mathcal{R}&\lesssim \ \Vert A b_1 \Vert_{L^2 L^2} \Vert \sqrt{-\tfrac {\dot M} M} A p_1 \Vert_{L^2L^2} \Vert A b_1 \Vert_{L^\infty L^2}\lesssim\mu^{-\frac 1 2 } \varepsilon^3. \end{aligned}$$ Combining the estimates [\[est:bvbn1\]](#est:bvbn1){reference-type="eqref" reference="est:bvbn1"}, [\[est:bvbnT\]](#est:bvbnT){reference-type="eqref" reference="est:bvbnT"}, [\[est:bvbnR\]](#est:bvbnR){reference-type="eqref" reference="est:bvbnR"} and [\[est:bvbncalR\]](#est:bvbncalR){reference-type="eqref" reference="est:bvbncalR"}, we finally conclude that $$\begin{aligned} \label{est:bvbn} \langle A b_{\neq} , A ( v_{\neq}\nabla_t b_{\neq})_{\neq}\rangle\lesssim \mu^{-\frac 3 2 } \varepsilon^3.\end{aligned}$$ ## High frequency estimates for $bvb$ terms with $x$-averages {#hfa} In this subsection we aim to estimate the remaining terms in the $bvb$ integrals, which involve $x$-averages. We consider the two terms $$\begin{aligned} \langle A b_{\neq} &, A ( v_{\neq}\nabla_t b_{=})_{\neq}\rangle+ \langle A b_= , A ( v_{\neq}\nabla_t b_{\neq})_=\rangle\\ &= \langle A b_{1,\neq} , A ( v_{\neq}\nabla_t b_{1,=})_{\neq}\rangle+ \langle A b_{1,=} , A ( v_{\neq}\nabla_t b_{1,\neq})_=\rangle,\end{aligned}$$ where we used that $b_{2,=}=0$, since $b$ is divergence-free. Using integration by parts and the fact that $v$ is divergence-free, we obtain that $$\begin{aligned} \langle A b_{1,\neq} , v_{\neq}\nabla_t A b_{1,=}\rangle& + \langle A b_{1,=} , v_{\neq}\nabla_t A b_{1,\neq}\rangle\\ &= \langle v_{\neq} , \nabla_t (A b_{1,=}A b_{1,\neq})\rangle=0, \end{aligned}$$ and thus $$\begin{aligned} \langle A b_{1,\neq} &, A ( v_{\neq}\nabla_t b_{1,=})\rangle+ \langle A b_{1,=} , A ( v_{\neq}\nabla_t b_{1,\neq})\rangle\\ &=\langle A b_{1,\neq} , A ( v_{\neq}\nabla_t b_{1,=}) - v_{\neq}\nabla_t A b_{1,=}\rangle + \langle A b_{1,=} , A ( v_{\neq}\nabla_t b_{1,\neq})- v_{\neq}\nabla_t A b_{1,\neq}\rangle\\ &= \sum_{k\neq 0} \iint d(\xi,\eta) A(k,\xi) b_1 ( k ,\xi) \tfrac {(A(k,\xi)- A(0,\eta)) (-k\eta)}{\sqrt{k^2+(\xi-\eta -kt)^2} }p_1(k,\xi-\eta ) b_1(0, \eta)\\ &\quad + \sum_{k\neq 0} \iint d(\xi,\eta) A(0,\xi) b_1 ( 0 ,\xi) \tfrac {(A(0,\xi)- A(k,\eta)) (-k\xi) }{\sqrt{k^2+(\xi-\eta -kt)^2} }p_1(k,\xi-\eta ) b_1(-k, \eta).\end{aligned}$$ Again we split this integrals into the transport $T$, reaction $R$ and remainder terms $\mathcal{R}$ with the same definition of sets in Fourier space: $$\begin{aligned} \Omega_T&=\{\vert \xi-\eta \vert \le \tfrac 18 \vert \eta\vert \}, \\ \Omega_R&=\{\vert \eta\vert \le \tfrac 18 \vert \xi-\eta \vert\},\\ \Omega_\mathcal{R}&=\{\tfrac 18 \vert \eta\vert \le \vert \xi-\eta \vert\le 8\vert \eta\vert \}.\end{aligned}$$ **Transport term:** Since $\vert \xi-\eta \vert \le \tfrac 18 \vert \eta\vert$ we obtain that $\vert \eta \vert \approx \vert \xi \vert$. In our estimates, we distinguish the cases $\xi \vee \eta \le 2 k\langle t \rangle$ and $\xi \vee \eta \ge 2 k \langle t \rangle$. In the first case, $\xi \vee \eta \le 2 k\langle t \rangle$ we obtain a bound by $$\begin{aligned} \sum_{k\neq 0} & \iint d(\xi,\eta) \textbf{1}_{\Omega_T}\textbf{1}_ { \xi \vee \eta \le k\langle t \rangle } A(k,\xi) b_1 ( k ,\xi) \tfrac {(A(k,\xi)- A(0,\eta)) k\eta }{\sqrt{k^2+(\xi-\eta -kt)^2} }p_1(k,\xi-\eta ) b_1(0, \eta)\\ + \sum_{k\neq 0} & \iint d(\xi,\eta)\textbf{1}_{\Omega_T} \textbf{1}_ {\xi \vee \eta \le k\langle t \rangle } A(0,\xi) b_1 ( 0 ,\xi) \tfrac {(A(0,\xi)- A(k,\eta)) k\xi }{\sqrt{k^2+(\xi-\eta -kt)^2} }p_1(k,\xi-\eta ) b_1(-k, \eta)\\ &\le t \Vert A b_= \Vert_{L^2 } \Vert \partial_x^2\Lambda_t^{-1} p_{1,\neq} \Vert_{L^\infty } \Vert A b_{1,\neq}\Vert_{L^2}\\ &\lesssim \Vert A b_= \Vert_{L^2 } \Vert A p_{1,\neq} \Vert_{L^2} \Vert Ab_{1,\neq}\Vert_{L^2}.\end{aligned}$$ In the case $\xi \vee \eta \ge 2 k\langle t \rangle$, we instead estimate $$\begin{aligned} A(k,\xi)- A(0,\eta)&\le M(k,\xi) (\xi^2+k^2)^{\frac N 2 } -\eta^N \\ &= (M(k,\xi)-1) (\xi^2+k^2)^{\frac N 2 } +((\xi^2+k^2)^{\frac N 2 }- \eta^N). \end{aligned}$$ Since $\xi \ge 2 k\langle t \rangle$, in the first summand we may bound $$\begin{aligned} M(k,\xi)-1&= \exp\left( - \int_0^t \tfrac {\vert k \vert} {k^2 + (\xi -ks)^2 } ds\right)-1 \\ &\lesssim \tfrac 1 \xi \lesssim \tfrac 1 \eta .\end{aligned}$$ By the mean value theorem we further infer $$\begin{aligned} (\xi^2+k^2)^{\frac N 2 }- \eta^N&\le ((\xi-\theta \eta )^2+k^2)^{\frac {N-1} 2 }\vert k,\xi - \eta \vert \lesssim \vert k,\xi - \eta \vert (\xi^2+k^2)^{\frac {N-1} 2}.\end{aligned}$$ Thus, using that $k\le\xi \lesssim \eta$, we deduce that $$\begin{aligned} A(k,\xi)- A(0,\eta)\lesssim \vert k,\xi - \eta \vert \eta^{N -1},\\ A(k,\eta)- A(0,\xi)\lesssim \vert k,\xi - \eta \vert \eta^{N -1},\end{aligned}$$ where the proof for $A(k,\eta)-A(0,\xi)$ is analogous. Finally, we obtain $$\begin{aligned} \langle A b_{\neq} &,\textbf{1}_{\Omega_T} \textbf{1}_ { \eta \ge kt} A ( v_{\neq}\nabla_t b_{=})\rangle+ \langle A b_= ,\textbf{1}_{\Omega_T} 1_ { \eta \ge kt} A ( v_{\neq}\nabla_t b_{\neq})\rangle\\ &\lesssim \sum_{k\neq 0} \iint d(\xi,\eta) A(k,\xi) b_1 ( k ,\xi) \tfrac {\vert k,\xi - \eta \vert \eta^{N -1} }{\sqrt{k^2+(\xi-\eta -kt)^2} }p_1(k,\xi-\eta ) b_1(0, \eta)\\ &\quad + \sum_{k\neq 0} \iint d(\xi,\eta) A(0,\xi) b_1 ( 0 ,\xi) \tfrac {\vert k,\xi - \eta \vert \eta^{N -1} }{\sqrt{k^2+(\xi-\eta -kt)^2} }p_1(k,\xi-\eta ) b_1(k, \eta)\\ &\lesssim \Vert A b_= \Vert_{L^\infty} \Vert A \Lambda_t^{-1} p_{1,\neq} \Vert_{L^2 } \Vert A b_{1,\neq}\Vert_{L^2}\\ &\lesssim \Vert A b_= \Vert_{L^\infty} \Vert A \Lambda_t^{-1} p_{1,\neq} \Vert_{L^2 } \Vert A b_{1,\neq}\Vert_{L^2},\end{aligned}$$ and integrating in time yields the desired bound: $$\begin{aligned} \begin{split} \int \langle A b_{\neq} &, \textbf{1}_{\Omega_T} A ( v_{\neq}\nabla_t b_{=})\rangle+ \langle A b_= , \textbf{1}_{\Omega_T} A ( v_{\neq}\nabla_t b_{\neq})\rangle d\tau \\ &\lesssim \mu^{-1 } \varepsilon^3 .\label{est:bvbaT} \end{split}\end{aligned}$$ **Reaction term:** Since $\vert \eta \vert \le \tfrac 18 \vert \xi - \eta\vert$ we obtain $\vert \xi- \eta \vert \approx \vert \xi \vert$ and thus $$\begin{aligned} R&=\langle A b_{\neq} , \textbf{1}_{\Omega_R} A( ( v_{\neq}\nabla_t b_{=})- v_{\neq}\nabla_t A b_{=})\rangle+ \langle A b_= , \textbf{1}_{\Omega_R} (A ( v_{\neq}\nabla_t b_{\neq})- v_{\neq}\nabla_t Ab_{\neq})_=\rangle\\ &\le \sum_{k\neq 0} \iint d(\xi,\eta)\textbf{1}_{\Omega_R} A(k,\xi) b_1 ( k ,\xi) \tfrac {(A(k,\xi)- A(0,\eta)) k\eta }{\sqrt{k^2+(\xi-\eta -kt)^2} }p_1(k,\xi-\eta ) b_1(0, \eta)\\ &\quad + \sum_{k\neq 0} \iint d(\xi,\eta)\textbf{1}_{\Omega_R} A(0,\xi) b_1 ( 0 ,\xi) \tfrac {(A(0,\xi)- A(-k,\eta)) k\xi}{\sqrt{k^2+(\xi-\eta -kt)^2} }p_1(k,\xi-\eta ) b_1(-k, \eta)\\ &\lesssim \Vert A b_{1,\neq}\Vert_{L^2} \Vert A \partial_x \Lambda_t^{-1} p_{1,\neq} \Vert_{L^2} \Vert \partial_y b_{1,=}\Vert_{L^\infty } \\ &\quad + \Vert A b_{1,=}\Vert_{L^2} \Vert A \partial_y \partial_x^{-1} \Lambda_t^{-1} p_{1,\neq} \Vert_{L^2} \Vert \partial_x^2 b_{1,\neq}\Vert_{L^\infty }.\end{aligned}$$ Expressing $\partial_y =\partial_y^t+t\partial_x$ in terms of the time-dependent derivatives, at this point we require the splitting into high and low frequency estimates. More precisely, using the additional time decay of the low-frequency part, we estimate $$\begin{aligned} \Vert A \partial_y \partial_x^{-1} \Lambda_t^{-1} p_{1,\neq} \Vert_{L^2}&\le \Vert A \partial_y^t \partial_x^{-1} \Lambda_t^{-1} p_{1,\neq} \Vert_{L^2}+t \Vert A \Lambda_t^{-1} p_{1,\neq} \Vert_{L^2}\\ &\lesssim \Vert A p_{1,\neq} \Vert_{L^2}+t \Vert A\Lambda^{-1}_t p_{1,\neq} \Vert_{L^2}\end{aligned}$$ and using the definition of $A^{N'}_\mu$ we can absorb the growth of the factor $t$ at the cost of a power of $\mu$: $$\begin{aligned} \Vert \partial_x^2 b_{1,\neq}\Vert_{L^\infty }&\le \Vert \Lambda^{N'} b_{1,\neq}\Vert_{L^2 }\\ &\lesssim e^{-c\mu t } \Vert A^{N'}_{\mu } b_{1,\neq}\Vert_{L^2 }\\ &\lesssim \mu^{-1} \langle t \rangle^{-1} \Vert A^{N'}_{\mu } b_{1,\neq}\Vert_{L^2 }.\end{aligned}$$ Thus we obtain $$\begin{aligned} R&\lesssim \Vert A^Np_{1,\neq}\Vert_{L^2} \Vert A^N b_{1,=}\Vert_{L^2 } \Vert A^N b_{1,\neq}\Vert_{L^2} \\ &\quad + \mu^{-1} \Vert A^N b_{1,=}\Vert_{L^2 } \Vert A \Lambda_t^{-1} p_{1,\neq} \Vert_{L^2} \Vert A^{N'}_{\mu } b_{1,\neq}\Vert_{L^2 } .\end{aligned}$$ Integrating in time then yields the estimate $$\begin{aligned} \int R d\tau &\lesssim \mu^{-\frac 3 2 } \varepsilon^3.\label{est:bvbaR}\end{aligned}$$ **$\mathcal{R}$ term:** The remainder term $\mathcal{R}$ can be estimated by the same argument as in the case without $x$-averages in Subsection [3.3](#hfw){reference-type="ref" reference="hfw"}. Combining the estimates [\[est:bvbaT\]](#est:bvbaT){reference-type="eqref" reference="est:bvbaT"}, [\[est:bvbaR\]](#est:bvbaR){reference-type="eqref" reference="est:bvbaR"} and [\[est:bvbn\]](#est:bvbn){reference-type="eqref" reference="est:bvbn"}, we conclude that the $bvb$ term can be controlled as $$\begin{aligned} \langle A b , A ( v_{\neq}\nabla_t b)\rangle\lesssim \mu^{-\frac 3 2 } \varepsilon^3 .\label{est:bvb} \end{aligned}$$ ## Low frequency estimates {#lf} In this subsection we establish the estimates on the low frequency errors. For simplicity of presentation we present the proof of these estimates for the $bvb$ nonlinearity. The estimates with an $x$-average in the second component are analogous to the ones in Subsection [3.2](#dnl){reference-type="ref" reference="dnl"}. The arguments for the $vvv$, $vbb$, $bbv$ or $ONL$ trilinear terms are also analogous. We aim to establish the bound $$\begin{aligned} \langle A^{N'}_\mu b , A^{N'}_\mu ( v_{\neq}\nabla_t b)\rangle&\lesssim \mu^{-\frac 12 }\varepsilon^3,\end{aligned}$$ and, as in the previous section, separately discuss the transport, reaction and remainder term. For the transport term, we note that $$\begin{aligned} v_{\neq}\nabla_t &= \nabla_t^\perp \Lambda^{-1}_t p_1 \nabla_t \\ &= \nabla^\perp \Lambda^{-1}_t p_1 \nabla.\end{aligned}$$ Hence, we may rewrite $$\begin{aligned} \langle A^{N'}_\mu b , A^{N'}_\mu ( v_{\neq}\nabla_t b)\rangle&= \langle A^{N'}_\mu b , A^{N'}_\mu ( \nabla^\perp \Lambda^{-1}_t p_{1,\neq}\nabla b)\rangle. \end{aligned}$$ In a first step, we estimate the $b_{\neq}$ term by using the algebra property of $A^{N'}$: $$\begin{aligned} \langle A^{N'}_\mu b &, A^{N'}_\mu ( \nabla^\perp \Lambda^{-1}_t p_{1,\neq}\nabla b_{\neq})\rangle\\ &\le \Vert A^{N'}_\mu b \Vert_{L^2} e^{c\mu_x t} \big ( \Vert A^{N'} \nabla^\perp \Lambda_t^{-1} p_{1,\neq}\Vert_{L^2} \Vert \nabla b_{\neq}\Vert_{L^\infty }+\\& \qquad \qquad \Vert \nabla^\perp \Lambda_t^{-1} p_{1,\neq}\Vert_{L^\infty } \Vert A^{N'} \nabla b_{\neq}\Vert_{L^2 }\big ) \\ &\le \Vert A^{N'}_\mu b\Vert_{L^2}\big ( \Vert A^{N} \Lambda_t^{-1} p_{1,\neq}\Vert_{L^2} \Vert A^{N'}_\mu b_{\neq}\Vert_{L^2 }+ \Vert A_\mu^{N'} \Lambda_t^{-1} p_{1,\neq}\Vert_{L^2} \Vert A^{N} b_{\neq}\Vert_{L^2}\big). \end{aligned}$$ Integrating in time then yields the estimate $$\begin{aligned} \int d \tau \langle A^{N'}_\mu b , A^{N'}_\mu ( v_{\neq}\nabla_t b_{\neq} )\rangle&\lesssim \mu^{-\frac 1 2 } \varepsilon^3.\label{est:low1}\end{aligned}$$ Furthermore, we estimate the $b_=$ term by partial integration and the algebra property of $A^{N' }$ $$\begin{aligned} &\quad \langle A^{N'}_\mu b , A^{N'}_\mu ( \nabla^\perp \Lambda^{-1}_t p_{1,\neq}\nabla b_=)\rangle\\ & =-\langle A^{N'}_\mu b_{1,\neq} , A^{N'}_\mu ( \partial_x \Lambda^{-1}_t p_{1,\neq}\partial_yb_{1,=} )\rangle\\ &=\langle \partial_x A^{N'}_\mu b_{1,\neq} , A^{N'}_\mu ( \Lambda^{-1}_t p_{1,\neq}\partial_yb_{1,=} )\rangle\\ &\le \Vert \partial_x A^{N'}_\mu b_{1,\neq} \Vert_{L^2} e^{c\mu t } \big( \Vert A^{N'} \Lambda^{-1}_t p_{1,\neq}\Vert_{L^2}\Vert\partial_yb_{1,=} \Vert_{L^\infty }\\ &\qquad \qquad + \Vert \Lambda^{-1}_t p_{1,\neq}\Vert_{L^\infty } \Vert \partial_y^{N'+1} b_{1,=} \Vert_{L^2}\big ) \\ &\lesssim \Vert\partial_x A^{N'}_\mu b_{1,\neq} \Vert_{L^2}\big( \Vert A^{N'}_\mu \Lambda^{-1}_t p_{1, \neq}\Vert_{L^2}\Vert A^{N'}b_{1,=} \Vert_{L^2}\\ &\qquad \qquad + \Vert A^{N'}_\mu \Lambda^{-1}_t p_{1, \neq}\Vert_{L^2} \Vert A^{N}b_{1,=} \Vert_{L^2}\big ).\end{aligned}$$ Integrating in time then yields that $$\begin{aligned} \int d \tau \langle A^{N'}_\mu b_{\neq} , A^{N'}_\mu ( v_{\neq}\nabla_t b_{=} )\rangle &\lesssim \mu^{-\frac 12 }\varepsilon^3 .\label{est:low2}\end{aligned}$$ This concludes our proof of Proposition [Proposition 2](#prop:errors){reference-type="ref" reference="prop:errors"} and hence of Theorem [Theorem 2](#thm:anisoThres){reference-type="ref" reference="thm:anisoThres"}. More precisely, the claimed estimates for both $A^N$ and $A^{N'}_{\mu}$ are obtained by combining the respective linear estimate [\[est:L\]](#est:L){reference-type="eqref" reference="est:L"}, the high frequency nonlinear estimates [\[est:ONL\]](#est:ONL){reference-type="eqref" reference="est:ONL"}, [\[est:aver\]](#est:aver){reference-type="eqref" reference="est:aver"}, [\[est:vvv\]](#est:vvv){reference-type="eqref" reference="est:vvv"}, [\[est:vbb\]](#est:vbb){reference-type="eqref" reference="est:vbb"}, [\[est:bbv\]](#est:bbv){reference-type="eqref" reference="est:bbv"}, [\[est:bvb\]](#est:bvb){reference-type="eqref" reference="est:bvb"}, and the low frequency estimates given in [\[est:low1\]](#est:low1){reference-type="eqref" reference="est:low1"} and [\[est:low2\]](#est:low2){reference-type="eqref" reference="est:low2"}. We emphasize that the stability threshold of $\tfrac 3 2$ is determined by the estimates for the action of the $v\cdot \nabla_t b$ nonlinearity in the estimate [\[est:bvb\]](#est:bvb){reference-type="eqref" reference="est:bvb"} and, in particular, by the estimates of the reaction terms [\[est:bvbnR\]](#est:bvbnR){reference-type="eqref" reference="est:bvbnR"} and [\[est:bvbaR\]](#est:bvbaR){reference-type="eqref" reference="est:bvbaR"}. These estimates are expected to be optimal and together with the linear estimates of Section [2](#linstab){reference-type="ref" reference="linstab"} highlight the effects of the lack of vertical resistivity. The partial dissipation case considered in this article $$\begin{aligned} \kappa_y=0, \ \nu_x=\nu_y=\kappa_x>0,\end{aligned}$$ shows the large impact of (partial) magnetic resistivity on the behavior of the MHD equations and the (de)stabilizing role of the magnetic field. As mentioned following Theorem [Theorem 2](#thm:anisoThres){reference-type="ref" reference="thm:anisoThres"}, more generally our methods of proof extend to the case where $\kappa_x$ is bounded below in terms of $\nu$: $$\begin{aligned} \nu_y^{1/3}\geq \kappa_x \geq \frac{1}{2\alpha} \nu_y. \end{aligned}$$ The complementary regime, where $\kappa_x$ tends to zero quicker than $\nu_{y}$ remains an interesting topic for future work. The limiting case, $\kappa_x=0$, and the associated instability is discussed in the following section. # Instability of the non-resistive MHD system {#instab} As a complementary result, in this section we consider the non-resistive MHD equations and establish the instability estimates of Proposition [Proposition 1](#prop:instability){reference-type="ref" reference="prop:instability"}. ## Linear instability We begin by studying the linearized MHD equations with isotropic viscosity and vanishing resistivity: $$\begin{aligned} \label{eq:liso} \begin{split} \partial_t p_1 - \partial_x \partial_x^t \Delta^{-1}_t p_1- \alpha \partial_x p_2 &= \nu \Delta_t p_1, \\ \partial_t p_2 +\partial_x \partial_x^t \Delta^{-1}_t p_2 - \alpha \partial_x p_1 &= 0. \end{split}\end{aligned}$$ **Lemma 3** (Quantitative linear instability of the non-resistive MHD equations). *For the linearized equations [\[eq:liso\]](#eq:liso){reference-type="eqref" reference="eq:liso"} there exists initial data $p_{in}$ such that $$\begin{aligned} \begin{split} \Vert p(t)\Vert_{H^N } &\ge t\tfrac {\nu }{8\alpha^2 }\Vert p_{in}\Vert_{H^N},\\ \Vert p(t)\Vert_{H^{N-1} } &\ge t\tfrac {\nu^2 }{32\alpha^4 }\Vert p_{in}\Vert_{H^N }.\label{eq:isolow} \end{split} \end{aligned}$$ Furthermore, for all solutions such that at time $\tau$ it holds $p(\tau)\in H^N$, then we obtain $$\begin{aligned} \label{eq:isoup} \Vert p\Vert_{H^N}\le {\langle t\rangle^2 }\Vert p(\tau)\Vert_{H^N}. \end{aligned}$$* *Proof of Lemma [Lemma 3](#lemma:linIns){reference-type="ref" reference="lemma:linIns"}.* After a Fourier transform [\[eq:liso\]](#eq:liso){reference-type="eqref" reference="eq:liso"} yields $$\begin{aligned} \begin{split} \partial_t p_1(k) &= -\tfrac {t-\frac \xi k } {1+(t-\frac \xi k )^2} p_1(k) + \alpha k p_2 (k) -\nu ( k^2 + (\xi-kt)^2) p_1(k), \\ \partial_t p_2 (k) &=\tfrac {t-\frac \xi k } {1+(t-\frac \xi k )^2} p_1(k) - \alpha k p_1 (k).\label{eq_p_FT} \end{split}\end{aligned}$$ We assume that $p_1(0,k,\xi)=0$ and consider variables $k=-1$ and $\xi\ge 2\tfrac {\alpha^2}\nu$ $$\begin{aligned} p_1&= -\alpha \int^t_0 d\tau \sqrt{\tfrac {1+(\tau_1+\xi )^2}{1+(t+\xi )^2}}\exp(-\nu (t-\tau+\tfrac 1 3 ((t+\xi )^3-(\tau_1+\xi )^3)))p_2(\tau_1) \end{aligned}$$ thus we can estimate $p_2$ by $$\begin{aligned} &\quad p_2-\sqrt {\tfrac {1+(t+\xi )^2 } {1+\xi^2 }}p_{2,in}(k)\\ &= -\alpha k \int_0^t d\tau_2\sqrt{\tfrac {1+(t+\xi )^2 } {1+(\tau_2 +\xi )^2 }} p_1(\tau_2,-1) \\ &= -\alpha^2 \int_0^t d\tau_2 \int^{\tau_1}_0 d\tau_1 \tfrac {\sqrt{1+(t+\xi )^2} \sqrt{1+(\tau_1+\xi )^2}}{1+(\tau_2+\xi )^2} p_2(\tau_1) \\ &\qquad \cdot \exp(-\nu (\tau_2-\tau_1+\tfrac 1 3 ((\tau_2+\xi )^3-(\tau_1+\xi )^3)))\\ &\le {\alpha^2}\vert p_2\vert_\infty \int_0^t d\tau_1 \int^{t}_{\tau_2 } d\tau_2 \exp(-\nu (\tau_2-\tau_1+\tfrac 1 3 ((\tau_2+\xi )^3-(\tau_1+\xi )^3))).\end{aligned}$$ We estimate the last integral by $$\begin{aligned} \int_0^t d\tau_1& \int^{t}_{\tau_2 } d\tau_2 \exp(-\nu (\tau_2-\tau_1+\tfrac 1 3 ((\tau_2+\xi )^3-(\tau_1+\xi )^3)))\\ &= \int_0^t d\tau_1 \int^{t}_{\tau_2 } d\tau_2 \tfrac {1+(\tau_2+\xi)^2}{1+(\tau_2+\xi)^2} \exp(-\nu (\tau_2-\tau_1+\tfrac 1 3 ((\tau_2+\xi )^3-(\tau_1+\xi )^3)))\\ &\le \int_0^t d\tau_1 \int^{t}_{\tau_2 } d\tau_2 \tfrac {1+(\tau_2+\xi)^2}{1+(\tau_1+\xi)^2} \exp(-\nu (\tau_2-\tau_1+\tfrac 1 3 ((\tau_2+\xi )^3-(\tau_1+\xi )^3)))\\ &\le \tfrac 1\nu \int_0^t d\tau_1 \tfrac 1{1+(\tau_1+\xi)^2}[\exp(-\nu (\tau_2-\tau_1+\tfrac 1 3 ((\tau_2+\xi )^3-(\tau_1+\xi )^3)))]_{\tau_2=\tau_1}^{\tau_2 =t }\\ &\le \tfrac 1{\nu \xi } \end{aligned}$$ and thus $$\begin{aligned} \vert p_2-\sqrt {\tfrac {1+(t+\xi )^2 } {1+\xi^2 }}p_{2,in}(k)\vert &\le \tfrac {\alpha^2 }{\nu \xi } \vert p_2\vert_\infty.\end{aligned}$$ Since $\xi\ge 2\tfrac {\alpha^2}\nu$ we obtain $$\begin{aligned} p_2(-1) \ge \tfrac 1 2 \sqrt {\tfrac {1+(t+\xi )^2 } {1+\xi^2 }}p_{2,in}(-1)\ge \tfrac t{2\xi}p_{2,in}(-1).\end{aligned}$$ Let $a(\xi )$ be such that $\mathop{\mathrm{supp}}_\xi (a (\xi)) \subset [2\tfrac {\alpha^2}\nu,4\tfrac {\alpha^2}\nu]$ and $\int {(2+\xi^2)^{\frac N 2 }} a^2(\xi)=1$ then we deduce that for the initial data $$\begin{aligned} p_{in} (k, \xi ) &= \textbf{1}_{k=-1} a(\xi )\end{aligned}$$ it holds that $$\begin{aligned} \Vert p_{in} \Vert_{H^N}&=1,\\ \Vert p(t)\Vert_{H^N}&\ge t\tfrac {\nu }{8\alpha^2 },\\ \Vert p(t)\Vert_{H^{N-1}}&\ge t\tfrac {\nu^2 }{32\alpha^4},\end{aligned}$$ which proves [\[eq:isolow\]](#eq:isolow){reference-type="eqref" reference="eq:isolow"}. Furthermore, for all solutions such that $p(\tau)\in H^N$ we estimate $$\begin{aligned} \partial_t \vert p\vert^2(k,\xi,t ) &\le2\tfrac {\vert t-\frac \xi k \vert }{1+\vert t-\frac \xi k \vert^2}\vert p \vert^2(k,\xi,t)\end{aligned}$$ and so $$\begin{aligned} \vert p\vert^2(k,\xi,t )&\le \exp( 2\int_\tau^t \tfrac {\vert s-\frac \xi k \vert }{1+\vert s -\frac \xi k \vert^2} ds ) \vert p_{in} \vert^2(k,\xi )\\ &\le\exp( \int_0^{t } \tfrac { 2\tilde s }{1+ \tilde s^2 }d\tilde s ) \vert p_{in} \vert^2(k,\xi )\\ &\le \langle t\rangle^4 \vert p_{in} \vert^2(k,\xi )\end{aligned}$$ which proves [\[eq:isoup\]](#eq:isoup){reference-type="eqref" reference="eq:isoup"}. ◻ ## Nonlinear norm inflation We next consider the nonlinear non-resistive MHD equations in their perturbative form around the stationary solution [\[eq:Couette\]](#eq:Couette){reference-type="eqref" reference="eq:Couette"}: $$\begin{aligned} \label{NLiso} \begin{split} \partial_t p_1 - \partial_x \partial_x^t \Delta^{-1}_t p_1- \alpha \partial_x p_2 &= \nu \Delta_t p_1 + \nabla^\perp_t\Lambda^{-1}_t (b\nabla_t b - v\nabla_t v ),\\ \partial_t p_2 +\partial_x \partial_x^t \Delta^{-1}_t p_2 - \alpha \partial_x p_1 &= \nabla^\perp_t\Lambda^{-1}_t (b\nabla_t v - v\nabla_t b ). \end{split}\end{aligned}$$ The following lemma establishes the norm inflation result of Proposition [Proposition 1](#prop:instability){reference-type="ref" reference="prop:instability"}. **Lemma 4** (Nonlinear norm inflation for the non-resistive MHD equations). *Consider the non-resistive nonlinear MHD equations [\[NLiso\]](#NLiso){reference-type="eqref" reference="NLiso"}. Then for all $C=C(\nu ) >1$ there exists $\varepsilon_0>0$ such that for all $0<\varepsilon<\varepsilon_0$ there exists initial data $p_{in}$ such that $$\begin{aligned} \Vert p_{in} \Vert_{H^N } =\varepsilon \end{aligned}$$ and $$\begin{aligned} \Vert p\Vert_{L^\infty H^N } \ge \varepsilon C. \end{aligned}$$* *Proof.* For the sake of contradiction we assume that there exists $\varepsilon_0>0$ such that for all $0<\varepsilon\le \varepsilon_0$ and for any choice of initial data with $\|p_{in}\|_{H^N}=\epsilon$ it holds that $$\begin{aligned} \Vert p\Vert_{L^\infty H^N } \le \varepsilon C.\end{aligned}$$ Our plan is to choose initial data such that for a choice of $\varepsilon$ and $t$ we obtain a contradiction to this bound. In particular, we choose $p_{in}$ as the data of the linear instability result, Lemma [Lemma 3](#lemma:linIns){reference-type="ref" reference="lemma:linIns"}, such that the associated linear solution $p_{lin}$ satisfies $$\begin{aligned} \Vert p_{in} \Vert_{H^N } &= \varepsilon, \\ \Vert p_{lin} (t)\Vert_{H^{N-1} } &\ge t \tfrac {\nu^2}{32 \alpha^4}.\end{aligned}$$ Let $S(\tau, t)$ be the solution operator for the linearized system. Then in view of [\[eq:isoup\]](#eq:isoup){reference-type="eqref" reference="eq:isoup"} we have the estimate $$\begin{aligned} \Vert S(\tau, t)\Vert_{H^N\to H^N }\le {\langle t\rangle^2 }.\end{aligned}$$ Thus we deduce that $$\begin{aligned} \partial_t ( p-p_{lin})&\le L( p-p_{lin}) + NL[p]\end{aligned}$$ and therefore $$\begin{aligned} \Vert p-p_{lin}\Vert_{H^{N-1}}^2 &\le \int_0^\tau \Vert S(\tau, t)\Vert_{H^N\to H^N } \Vert p-p_{lin}\Vert_{ H^{N-1 }}\Vert p\Vert_{ H^{N-1 }}\Vert \nabla_t p\Vert_{H^{N-1}}\\ &\lesssim \Vert p-p_{lin}\Vert_{ L^\infty H^{N-1 }}\Vert p\Vert_{L^\infty H^{N-1 }}\Vert p\Vert_{L^\infty H^{N}} 2 \int^t_0 t \langle t \rangle^2 \\ &\lesssim t^2\langle t\rangle^2 \varepsilon^2 C^2 \Vert p-p_{lin}\Vert_{ H^{N-1 }}. \end{aligned}$$ Finally, we obtain $$\begin{aligned} \Vert p\Vert_{H^{N-1} }&\ge \Vert p_{lin} \Vert_{H^{N-1} }-t^2\langle t\rangle^2 \varepsilon^2 C^2\\ &= t \varepsilon( \tfrac {\nu^2 }{32\alpha^4 } - t\langle t\rangle^2 \varepsilon C^2).\end{aligned}$$ This completes our proof by contradiction provided this term is large enough for a given small $\varepsilon$ and suitable time. Indeed for the choice $\varepsilon\le \tfrac {\nu^8} {10^{8}C^5 \alpha^{16}}$ we obtain that at the time $t= 10^2 C \tfrac {\alpha^4} {\nu^2 }$ it holds that $$\begin{aligned} \Vert p\Vert_{H^{N-1} }&\ge t \tfrac {\nu^2 }{10^3\alpha^4 }\varepsilon\ge C \varepsilon.\end{aligned}$$ This concludes our proof of the nonlinear norm inflation and hence completes our proof of Proposition [Proposition 1](#prop:instability){reference-type="ref" reference="prop:instability"}. ◻ The behavior of the MHD equations and, in particular, the interaction of shear flows, the magnetic field and dissipation are an area of current active research [@liss2020sobolev; @Dolce; @zhao2023asymptotic; @knobel2023echoes]. However, prior works have focused on cases where the resistivity is at least as strong as the fluid viscosity and where thus the behavior is closely related to that of the Navier-Stokes equations. In contrast, the non-resistive MHD equations exhibit additional instability, as for instance shown in Proposition [Proposition 1](#prop:instability){reference-type="ref" reference="prop:instability"}. Motivated by this dichotomy, in this article we have studied the anisotropic, partial dissipation regime $$\begin{aligned} \kappa_y=0, \ \kappa_x=\nu_x=\nu_y\end{aligned}$$ and the associated stability threshold in the inviscid limit. As shown in Theorem [Theorem 2](#thm:anisoThres){reference-type="ref" reference="thm:anisoThres"} and highlighted in the estimates of Sections [2](#linstab){reference-type="ref" reference="linstab"}, [3.4](#hfa){reference-type="ref" reference="hfa"} and [3.3](#hfw){reference-type="ref" reference="hfw"}, this partial dissipation regime behaves qualitatively differently than both the fully dissipative case and the non-resistive case. Moreover, our analysis crucially used the coupling of the velocity field and magnetic field induced by the underlying magnetic field, which allowed us to obtain improved estimates for the magnetic field despite the lack of the symmetry of the dissipation. Partial, anisotropic dissipation in the MHD equations is thus shown to give rise to distinct regimes with different (in)stability properties and demonstrates an intricate interplay of shear dynamics, magnetic interaction and anisotropic dissipation. A more complete understanding of all these regimes, the case of resistivity vanishing faster than viscosity and a characterization of the (in)stability properties of the ideal MHD equations remain exciting questions for future research. ## Acknowledgments {#acknowledgments .unnumbered} Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- Project-ID 258734477 -- SFB 1173

arxiv_math

--- abstract: | In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely $$\left\{\begin{array}{lc} -\Delta u={\lambda}e^{u} & \mbox{ in }\Omega\\ u=0 & \mbox{ on }\partial \Omega, \end{array} \right.$$ where $\Omega\subset {\rm I\mskip -3.5mu R}^2$ is a bounded smooth domain and ${\lambda}>0$ is a small parameter. address: - Francesca Gladiali, Dipartimento SCFMN, Università di Sassari - Via Piandanna 4, 07100 Sassari, Italy - Massimo Grossi, Dipartimento di Scienze di Base Applicate per l'Ingegneria, Università degli Studi di Roma *La Sapienza*, P.le Aldo Moro 5, 00185 Roma, Italy author: - F. Gladiali, M. Grossi bibliography: - max-fra-gelfand3.bib title: "On the critical points of solutions of PDE in a non-convex settings: the case of concentrating solutions" --- [^1] [^2] [^3] # Introduction {#s1} The calculation of the number of critical points of solutions of nonlinear elliptic differential equations is an old and fascinating problem. Many questions are still largely unresolved and we are far from a complete description of the problem. In this paper we will limit ourselves to considering $positive$ solutions of the following problem, $$\label{0} \begin{cases} -\Delta u=f(u)&in\ \Omega\\ u=0&on\ \partial\Omega, \end{cases}$$ where $\Omega$ is a bounded smooth domain of ${\rm I\mskip -3.5mu R}^N$, $N\ge2$ and $f$ is a Lipschitz nonlinearity. In this setting the first result to mention is certainly that by Gidas, Ni and Nirenberg (see [@gnn]) where the authors prove, for convex and symmetric domains, the uniqueness of the critical point of positive solutions. An important line of research was that of removing the $symmetry$ assumption and proving the Gidas, Ni and Nirenberg theorem only assuming the convexity of $\Omega$. The answer to this question is thought to be affirmative, indeed no counter-example has been provided. However this seems to be a very difficult open problem. Indeed, if $\Omega$ is a convex (not necessarily symmetric) domain, the uniqueness of the critical point of the solution has been proved as consequence of the strict convexity of the level sets of solutions with appropriate nonlinearities (see [@ml], [@bl], [@app], [@k], [@cf2], [@gm]). Observe that each functions $f$ appearing in the mentioned results is handled differently, actually it is not possible to prove the strict convexity of the level sets of $u$ for generic $f$ (see the example in [@hns]). A result for general nonlinearities $f$ for planar domain with positive curvature was proved by Cabrè and Chanillo, but here the solutions are required to be semi-stable, (see [@dgm] for an extension to $\Omega$ with non-negative curvature). Unfortunately, this assumption is not verified by mountain pass solutions as well as by many nonlinearities in [\[0\]](#0){reference-type="eqref" reference="0"} like $f(s)=e^s$ or $f(u)=u^p$ with $p>1$. The aim of this paper is to deduce some information on the critical points, like the exact number and their index, in planar domains which are not necessarily convex or simply connected. Given the difficulty of the topic, we decided to choose a model problem in the plane and to consider the class of solutions concentrating at a finite number of points. The class of solutions we consider are those that appear in Gelfand's problem, although we believe that these techniques can be adapted to deal with other cases, (such as for example with $f(u)=u^p$ with $p$ large).\ The Gel'fand problem deals with solutions to the problem, $$\label{1} \left\{\begin{array}{lc} -\Delta u={\lambda}e^{u} & \mbox{ in }\Omega\\ u=0 & \mbox{ on }\partial \Omega, \end{array} \right.$$ where $\Omega\subset {\rm I\mskip -3.5mu R}^2$ is a bounded smooth domain and ${\lambda}>0$ is a parameter. This problem is associated with several phenomena in differential geometry, turbulence theory, statistical mechanics and gauge field theory (see the paper of Gel'fand, [@ge9] or [@nasu], [@su2] and the references therein). Problem [\[1\]](#1){reference-type="eqref" reference="1"} has been derived in the context of statistical mechanics in [@clmp1] while the ties with the turbolence theory of an incompressible and homogeneus fluid, via the Euler equation is described in the introduction of [@dem]. This problem can be regarded as a simplified model to describe the steady states of reaction-diffusion equations where the diffusion is of exponential type. Reaction-diffusion equations arise in a wide variety of biological, physiological and chemical contests such as the spreading of a chemical substance or the propagation of a disease, and in the study of complex systems. One can see, as an example the books of Murray [@mu1; @mu2] where spatial models for biomedical applications are considered. In this contest the Laplace operator effectively describes the spatial variation of physiological quantities within a tissue or other (chemical) quantities of the model and the parameter $\lambda$ represents a scaling factor or an intensity parameter that can be related to the diffusion coefficient. The interpretation of the solution properties to stationary problems like [\[1\]](#1){reference-type="eqref" reference="1"}, in the context of physiological models, requires careful evaluation of the involved processes and the integration with other clinical and experimental information. Nevertheless properties of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} can be of great interest for a deeper understanding of the system, its dynamics, and emergent phenomena. The existence and localization of critical points of solutions to reaction-diffusion equations can provide information about the concentration, diffusion, and interaction of substances that play a role. They can be used to study wave propagation or the formation of spatial patterns. It is a fundamental tool for the theoretical study and mathematical modeling of reaction and diffusion processes in various scientific and engineering contexts. Finally, it is interesting to observe the connection between [\[1\]](#1){reference-type="eqref" reference="1"} with the Mean Field equation in a bounded domain of ${\rm I\mskip -3.5mu R}^2$, namely $$\label{MF} \left\{\begin{array}{lc} -\Delta v=\rho \frac{ h(x)e^{v}}{\int_{\Omega} h(x)e^{v}dx} & \mbox{ in }\Omega\\ v=0 & \mbox{ on }\partial \Omega. \end{array} \right.$$ By [@cli], Section 6, the Mean Field equation [\[MF\]](#MF){reference-type="eqref" reference="MF"} with $h(x)=1$ in $\Omega$ is equivalent to [\[1\]](#1){reference-type="eqref" reference="1"} just letting $\lambda=\frac {\rho}{ \int_{\Omega} e^{v}dx}$.   Let us start by recalling some known facts about the structure of the solutions to [\[1\]](#1){reference-type="eqref" reference="1"} (see [@cr2] and also [@cli] for a more detailed construction of the solutions). Let ${\mathcal X}=\{ (\lambda,u)\in{\rm I\mskip -3.5mu R}^+\times C(\Omega) \mid (\ref{1})\hbox{ is satisfied}\}$ be the solutions set to [\[1\]](#1){reference-type="eqref" reference="1"}. The first observation is that ${\mathcal X}=\emptyset$ for $\lambda$ large enough. In particular there exists $\lambda^*=\lambda^*(\Omega)$ such that [\[1\]](#1){reference-type="eqref" reference="1"} admits a unique stable solution $\underline{u}_\lambda$ for every $\lambda \in [0,\lambda^*)$, called $minimal$ solution. Corresponding to $\lambda^*$ there exists a unique solution to [\[1\]](#1){reference-type="eqref" reference="1"}, and the solution curve bends back, so that there exists at least $two$ solutions of [\[1\]](#1){reference-type="eqref" reference="1"} when $\lambda\in (\lambda^*-\delta, \lambda^*)$ if $\delta$ is small enough. This curve of solutions is sufficiently smooth for every domain $\Omega$. Further for every $\lambda\in (0,\lambda^*)$ there exists at least another non stable solution $u_\lambda$. Critical phenomena in fact occur to these (non stable) solutions $u=u_{\lambda}(x)$ to [\[1\]](#1){reference-type="eqref" reference="1"} as $\lambda\downarrow 0$. The first observation is that $\|u_\lambda\|_{\infty}\to \infty$ as $\lambda\to 0$ by [@bremer]. This profile is described by [@nasu] as a quantized blow-up mechanism, and we recall it in details in Section [2](#se:2){reference-type="ref" reference="se:2"}, see [\[2\]](#2){reference-type="eqref" reference="2"}. Moreover sequences of blowing-up solutions $u_{\lambda_n}$ can be construct when $\lambda_n\to 0$ according to the topological and geometrical properties of the domain $\Omega$. In particular in [@we] the author constuct a sequence of solutions on simply connected domains $\Omega$ blowing-up at a critical point $x_0\in \Omega$ of the *Robin* function $\mathcal R(x)$ of $\Omega$, see also [@mo]. Non simply connected domains are considered in [@djlw]. A first complete existence result, for multipeak solutions, was proved in [@bapa] where the authors construct a sequence of solutions that blows-up at $m$ points $\{P_1,\dots, P_m\}$ of $\Omega$ when the point $(P_1, \dots, P_m)$ is a nondegenerate critical point for the *Kirchhoff-Routh* function of $\Omega$, that we denote $\mathcal{KR}_\Omega(x_1,\dots,x_m)$ (see Section [2](#se:2){reference-type="ref" reference="se:2"} for its definition). Observe that these solutions are $never$ semi-stable, so even if $\Omega$ is strictly convex, the Cabrè-Chanillo result is not applicable. In [@egp] and [@dkm] this condition was relaxed requiring that $(P_1, \dots, P_m)$ is a $stable$ critical point of $\mathcal{KR}_\Omega$.\ Moreover the non-degeneracy of $(P_1, \dots, P_m)$ as critical point of $\mathcal{KR}_\Omega$ implies the nondegeneracy of $u_\lambda$ for $\lambda$ small enough. This was proven first for $m=1$ by [@gg1] and then by [@grohsu] in the general case. Let $B_\rho(Q)\subset{\rm I\mskip -3.5mu R}^2$ be the ball centered at $Q$ and radius $\rho$. Our first result is the following, **Theorem 1**. *Let $\Omega$ be a smooth domain with $k\geq 0$ holes and let $u_{\lambda}$ be a family of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} that blow-up at $m\geq 1$ points $\{P_1,\dots,P_m\}$ as ${\lambda}\to 0$. Then, when ${\lambda}$ is small enough $$\label{nc} \sharp\{\hbox{critical point of $u_{\lambda}$ in $\Omega$}\}\ge2m+k-1.$$ More precisely we have that, for ${\lambda}$ small, there exists exactly one critical point (a non-generate maximum) for $u_{\lambda}$ in $B_\rho(P_i)$ $i=1,..,m$ and $\rho$ small. Next, denoting by $D=\Omega\setminus\cup_{i=1}^m B_\rho(P_i)$ and $\mathcal C_{\lambda}$ the set of critical points of $u_{\lambda}$ in $D$ we have that $u_{\lambda}$ admit at least $m+k-1$ nondegenerate saddle points in $D$ and $$\label{nb} \sum_{z_j\in \mathcal C_{\lambda}} index_{z_j} (\nabla u_{\lambda})=1-k-m.$$ Moreover, for any $k,m\ge1$, there exists $\tilde\Omega$ and a corresponding family of solutions $u_{\lambda}$ that blow-up at $m\geq 1$ points $\{P_1,\dots,P_m\}$ such that, again for ${\lambda}$ small enough, $$\label{nc2} \sharp\{\hbox{critical point of $u_{\lambda}$ in $\tilde\Omega$}\}=2m+k-1.$$ Moreover all critical points of $u_{\lambda}$ are nondegenerate, $m$ of them are local maxima and $m+k-1$ saddle points.* The proof of the previous theorem uses two basic tools, - Some delicate estimates on the asymptotic behavior of the solution $u_{\lambda}$ - Techniques of differential topology as the Poincarè-Hopf Theorem and degree theory Observe that the estimates on the asymptotic behavior of $u_{\lambda}$ use quite heavily the shape of the nonlinearity but it is reasonable to conjecture that can be obtained also for other nonlinearities (for example of power type or for Moser-Trudinger problems). The techniques of differential topology mainly concerns the computation of critical points of $harmonic$ functions and are independent of the exponential nonlinearity. A natural question which arises from the previous theorem is the following, *Question $1$.* Let us consider a domain $\Omega$ with $k\ge0$ holes. If we consider a solution to [\[1\]](#1){reference-type="eqref" reference="1"} blowing-up at $m$ points in $\Omega$, for what values $m$ and $k$ does equality in [\[nc\]](#nc){reference-type="eqref" reference="nc"} hold? The rest of the paper is devoted to answer to this question. The first result is, **Theorem 2**. *Assume $\Omega\subset{\rm I\mskip -3.5mu R}^2$ and $u_{\lambda}$ is a family of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} that blow-up at $m\geq 1$ points $\{P_1,\dots,P_m\}$ as ${\lambda}\to 0$. Then we have that* - *If $m=1$ and $k=0$ (i.e. $\Omega$ is simply connected) then any solution $u_{\lambda}$ to [\[1\]](#1){reference-type="eqref" reference="1"} has, for ${\lambda}$ small, only $1$ critical point in $\Omega$, (obviously its maximum) which is nondegenerate.* - *If $m=2$ and $k=0$ (i.e. $\Omega$ is simply connected) then any solution $u_{\lambda}$ to [\[1\]](#1){reference-type="eqref" reference="1"} has, for ${\lambda}$ small, $3$ nondegenerate critical points: the maximum points in the balls $B_\rho(P_1)$, $B_\rho(P_2)$ and a saddle point.* - *If $m=1$ and $k=1$ (i.e $\Omega$ has one hole) then any solution $u_{\lambda}$ to [\[1\]](#1){reference-type="eqref" reference="1"} has, for ${\lambda}$ small, $two$ nondegenerate critical points in $\Omega$; one is the absolute maximum, the second one is a saddle point.* **Remark 3**. *The claim $\bf a)$ in Theorem [Theorem 2](#T1){reference-type="ref" reference="T1"} was proved in [@gg1] under the additional assumption of the $convexity$ of $\Omega$ (see also [@gm] for similar results in higher dimensions). Observe that in [@gg1] it was obtained the stronger claim that the super-level sets of $u_{\lambda}$ are star-shaped but this properties will not be true in our setting of general simply-connected domains.\ * We stress that Theorem [Theorem 2](#T1){reference-type="ref" reference="T1"} claims that, under the assumption $\bf a)$, $\bf b)$ or $\bf c)$ $any$ blowing-up solution admits the same number of critical points.\ This is not true outside of this setting as showed in next Theorem. **Theorem 4**. *For every $m\geq 3$ there exists a simply connected domain $\widehat\Omega$ and a corresponding family of solutions $u_{\lambda}$ to [\[1\]](#1){reference-type="eqref" reference="1"} that blow-up at $m$ points $\{P_1,\dots,P_m\}$ as ${\lambda}\to 0$, such that $u_{\lambda}$ has at least $2m+1$ nondegeneratecritical points, $m+1$ of them are local maxima and $m$ are saddle points.\ Moreover for every $m\geq 3$ and for every $\lambda>0$ small, there exists a domain $\widehat\Omega_{\lambda}$, which has $k\geq 1$ holes and such that problem [\[1\]](#1){reference-type="eqref" reference="1"} admits a solution $\widehat u_{\lambda}$ in $\widehat\Omega_{\lambda}$ with at least $2m+k+1$ nondegenerate critical points, $m+1$ of them are local maxima and $m+k$ are saddle points. As ${\lambda}\to 0$ along a sequence $\widehat\Omega_{\lambda}\to \widehat\Omega$ and $\widehat u_{\lambda}$ blow-up at $m$ points.* **Remark 5**. *In the proof of Theorem [Theorem 4](#ex){reference-type="ref" reference="ex"} we also construct another domain $\Omega$ with $k = mh$ holes, for some $h\in{\rm I\mskip -3.5mu N}$, such that problem [\[1\]](#1){reference-type="eqref" reference="1"} admits a family of solution $u_{\lambda}$ in $\Omega$ that blow-up at $m\ge 1$ points $\{P_1,...,P_m\}$ as ${\lambda}\to 0$, with $2m+k+1$ critical points. In this alternative case the proof is considerably simpler.* **Remark 6**. *Theorems [Theorem 2](#T1){reference-type="ref" reference="T1"} and [Theorem 4](#ex){reference-type="ref" reference="ex"} provide an "almost complete" answer to Question $1$. The cases left open are $m=1$ with $k\ge2$ and $m=2$ with $k\ge1$, where it is not clear if the equality in [\[nc\]](#nc){reference-type="eqref" reference="nc"} always holds.* Another natural question arising by the previous result is the following, *Question $2$.* Let us fix a domain $\Omega$ with $k$ holes and $u_{\lambda}$ be a solution to [\[1\]](#1){reference-type="eqref" reference="1"} with $m$ peaks. Is there an $upper$ bound on the number of critical point of $u_{\lambda}$ depending only by $m$ and $k$? We end this introduction with some words on the techniques used in the proof of our results. The first tool we need is a good $C^1$ estimate of the solution $u_{\lambda}$ as ${\lambda}\to 0$. These results are well known in the literature in the blow-up ball $B_{\delta_{i,{\lambda}}\bar R}(P_i)$, for any $\bar R>0$ ,and in the set $\Omega\setminus\cup _{i=1}^m B_{\rho}(P_i)$ for any $\rho$ small enough(see Section [2](#se:2){reference-type="ref" reference="se:2"}). However no sharp estimate is available in the "annular" region $\{\delta_{i,{\lambda}}\bar R\le|x-P_i|\le \rho\}$. The knowledge of $u_{\lambda}$ in this region is crucial for our computations and it requires delicate estimates (see Section [\[S2\]](#S2){reference-type="ref" reference="S2"}). A second tool is given by refined topological arguments on the index of the critical points of harmonic functions. Here we used both classical arguments like the Poincaré-Hopf theorem and some results by Alessandrini and Magnanini in [@am](see Section [2.2](#ss2){reference-type="ref" reference="ss2"}). Resuming, the computation of the critical point of $u_{\lambda}$ requires to split $\Omega$ in three subdomains, - $B_{\delta_{i,{\lambda}}\bar R}(P_i)$, for some $\bar R>0$ where there is **one** nondegenerate critical point (of course the maximum) - $B_{\rho}(P_i)\setminus B_{\delta_{i,{\lambda}}\bar R}(P_i)$ where there is **no** critical point - $\Omega\setminus\cup_{i=1}^m B_{\rho}(P_i)$ where the number of critical points can vary according with $m$ and $k$. **Remark 7**. *As recalled in the beginning, solutions to [\[1\]](#1){reference-type="eqref" reference="1"} are linked to solutions to [\[MF\]](#MF){reference-type="eqref" reference="MF"}. Then the results in Theorems [Theorem 1](#prop-general){reference-type="ref" reference="prop-general"}, [Theorem 2](#T1){reference-type="ref" reference="T1"}, [Theorem 4](#ex){reference-type="ref" reference="ex"} hold also for solutions $v_\rho$ to [\[MF\]](#MF){reference-type="eqref" reference="MF"} that concentrate at $m$ points of $\Omega$ when $\rho-8\pi m$ is small enough.* The paper is organized as follows: in Section [2](#se:2){reference-type="ref" reference="se:2"} we recall some known facts about the asymptotic behavior of the solution as ${\lambda}\to0$ as well as some classical result about the critical point theory. In Section [\[S2\]](#S2){reference-type="ref" reference="S2"} we refine some asymtotics of the solution $u_{\lambda}$ in an annular set that shrinks slowly. In Section [4](#S3){reference-type="ref" reference="S3"} we apply the previous results to show that, for every $i=1,\dots,m$, in $B_\rho(P_i)$ there is only one nondegnerate critical point. In Section [5](#S4){reference-type="ref" reference="S4"} we analyze the structure of the critical points of $u_{\lambda}$ in the remaining part of $\Omega$ and prove Theorem [Theorem 1](#prop-general){reference-type="ref" reference="prop-general"}. In Section [6](#S5){reference-type="ref" reference="S5"} we prove Theorems [Theorem 2](#T1){reference-type="ref" reference="T1"} and [Theorem 4](#ex){reference-type="ref" reference="ex"}. # Preliminaries {#se:2} ## Asymptotic estimates of the solution $\mathbf{u_{\lambda}}$ {#ss1} In this section we recall some known fact on the asymptotic behavior of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} as ${\lambda}\to0$. Most of the proofs can be found in [@nasu] (see also [@mw2], [@gg1] and the references therein). Let $\{{\lambda}_n\}_{n\in{\rm I\mskip -3.5mu N}}$ be a sequence of positive values such that ${\lambda}_n\to 0$ as $n\to \infty$ and let $u_n=u_n(x)$ be a sequence of solutions of [\[1\]](#1){reference-type="eqref" reference="1"} for ${\lambda}={\lambda}_n$. We have the following quantized blow-up mechanism $$\label{2} {\lambda}_n\int_{\Omega}e^{u_n}\, dx\rightarrow 8\pi m$$ for some $m=0,1,2, \cdots, +\infty$ along a sub-sequence.\ - If $m=0$ the pair $({\lambda_n},u_{n} )\in{\mathcal X}$ converges to $(0,0)$ as ${\lambda_n}\rightarrow0$ and $u_n$ is the minimal solution with Morse index $0$.\ - If $m=+\infty$ there arises the entire blow-up of the solution $u_n$, in the sense that $\inf_C u_n\rightarrow+\infty$ for any $C$ compact, $C\subset \Omega$.\ - If $0<m<\infty$ the solutions $\{ u_{n}\}$ blow-up at $m$-points. - Thus there is a set $$\label{2a} {\mathcal S}=\{ P_1, \cdots,P_m\}\subset \Omega$$ of $m$ distinct points such that $\Arrowvert u_n\Arrowvert_{L^{\infty}(\omega)}=O(1)$ for any $\omega$ compact $\omega\subset \overline{ \Omega}\setminus {\mathcal{S}}$,\ - $u_n{|_{{\mathcal{S}}}}\rightarrow +\infty \quad \hbox{ as }n\to \infty.$ Next we describe the pointwise limit of $u_n$. Here and henceforth, $G(x,y)$ denotes the Green function of $-\Delta$ in $\Omega$ with Dirichlet boundary condition, that is $$\label{4} G(x,y)=\frac 1{2\pi}\log{|x-y|^{-1}}+H(x,y)$$ where $H(x,y)$ is the regular part of $G(x,y)$ and we denote by $\mathcal R(x)=H(x,x)$ the *Robin-function* of $\Omega$. Further we recall the definition the so called *Kirchhoff-Routh* function of $\Omega$ $$\mathcal{KR}_\Omega(x_1,\dots, x_m)=\frac 12 \sum_{j=1}^m \mathcal R(x_j)+\frac 12 \sum_{\substack {1\leq j,h\leq m\\ j\neq h}}G(x_j,x_h).$$ The following results can be found in [@mw2] and [@su3]. **Theorem 8**. *Let $u_n$ be a sequence of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} which blows-up at $\{P_1,\dots,P_m\}$ as $n\to \infty$. Then we have that $$\label{3} u_n \rightarrow 8\pi \sum_{j=1}^m \,G(\cdot, P_j)\quad \hbox{ in }C^{2}_{loc}(\overline{ \Omega} \setminus \{P_1, \dots, P_m\})$$ and $$\label{5} \nabla \mathcal{KR}_\Omega (P_1,\dots ,P_m)=0.$$* Next we describe the $local$ behavior of $u_n$ near the blow up points. Let $\{ P_1, \cdots,P_m\}\in{\mathcal{S}}$ (see [\[2a\]](#2a){reference-type="eqref" reference="2a"}) and take $0<R\ll 1$ satisfying $\bar B_{2R}(P_i)\subset \Omega$ for $i=1,\dots,m$ and $B_R(P_i)\cap B_R(P_j)=\emptyset$ if $i\neq j$. For each $P_j\in {\mathcal{S}}$, $j=1,\dots,m$, there exists a sequence $\{x_{j,n}\}\in B_R(P_j)$ such that - $u_n(x_{j,n})=\sup_{B_R(x_{j,n})}u_n(x)\rightarrow +\infty$, - $x_{j,n}\to P_j$ as $n\to +\infty$. Then we rescale $u_n$ around $x_{j,n}$ as $$\label{2.4} \tilde u_{j,n}(x):= u_n\left( \delta_{j,n} x +x_{j,n}\right)- u_n(x_{j,n})\quad \hbox{ in }B_{\frac{R}{\delta_{j,n}}}(0)$$ where the scaling parameter $\delta_{j,n}$ is determined by $$\label{2.5} {\lambda}_ne^{u_n(x_{j,n})}\delta_{j,n}^2=1.$$ By [@grohsu Corollary 4.3] there exists a constant $d_j>0$ such that, up to a sub-sequence, $$\label{2.6b} \delta_{j,n}=d_j {\lambda}_n^{\frac 12}+o\left( {\lambda}_n^{\frac 12}\right)$$ as $n\to \infty$. Then [\[2.5\]](#2.5){reference-type="eqref" reference="2.5"} and [\[2.6b\]](#2.6b){reference-type="eqref" reference="2.6b"} in turn give $$\label{2.6c} u_n(x_{j,n})=-2 \log {\lambda}_n+O(1)$$ as $n\to \infty$ for any $j=1,\dots,m$.\ The rescaled function $\tilde u_{j,n}$ in ([\[2.4\]](#2.4){reference-type="ref" reference="2.4"}) satisfies $$\nonumber \left\{ \begin{array}{ll} -\Delta \tilde u_{j,n}=e^{\tilde u_{j,n}} & \hbox{ in }B_{\frac{R}{\delta_{j,n}}}(0)\\ \tilde u_{j,n}\leq \tilde u_{j,n}(0)=0& \hbox{ in }B_{\frac{R}{\delta_{j,n}}}(0) \end{array} \right.$$ and then a classification result (see [@cli]) implies $$\label{2.6} \tilde u_{j,n}(x)\rightarrow U(x)=\log \frac 1{\left( 1+\frac{|x|^2}8\right)^2} \quad \hbox{ in }C^{\infty}_{loc}({\rm I\mskip -3.5mu R}^2)$$ where $U(x)$ is the unique solution to the Liouville problem $$\begin{cases} -\Delta U=e^U & \text{ in }{\rm I\mskip -3.5mu R}^2\\ \int_ {{\rm I\mskip -3.5mu R}^2} e^U <\infty. \end{cases}$$ Moreover (see [@liy]), it holds that $$\label{2.6a} \big| \tilde u_{j,n}(x)- U(x)\big|\leq C, \quad \forall x \in B_{\frac{R}{\delta_{j,n}}}(0)$$ with a constant $C>0$. Finally, using [\[2.5\]](#2.5){reference-type="eqref" reference="2.5"} and [\[2.6b\]](#2.6b){reference-type="eqref" reference="2.6b"}, the following estimate holds (see (6.7) in [@cli2] with $h(x)=1$ in $\Omega$ and ${\bm{\lambda_k}}=\underbrace{u_n(x_{j,n})+\log \lambda_n+\log 8\pi m}_{=-\log{\lambda}_n+O(1)}$ for some $j$). $$\label{stima2} \lambda_n\int _{B_R(x_{j,n})}e^{ u_{n}(y)}dy=8\pi+O\big({\lambda}_n\big).$$ We end this section recalling the following nondegeneracy result (see Theorem 1.2 in [@grohsu] and [@gg1] for $m=1$). It will be useful in the proof of Theorem [Theorem 4](#ex){reference-type="ref" reference="ex"}. **Theorem 9**. *Let $u_n$ be a sequence of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} which blows-up at $\{P_1,\dots,P_m\}$ as $n\to \infty$. Suppose that $(P_1,\dots,P_m)$ is a nondegenerate critical point of the Kirchhoff-Routh function $\mathcal{KR}$. Then $u_n$ is nondegenerate, i.e. the linearized problem $$\label{l1} \left\{\begin{array}{lc} -\Delta v={\lambda}_ne^{u_n}v & \mbox{ in }\Omega\\ v=0 & \mbox{ on }\partial \Omega, \end{array} \right.$$ admits only the trivial solution $v\equiv0$ for $n$ large enough.* ## Remarks on differential topology {#ss2} We start this section recalling the celebrated Poincaré-Hopf Theorem which we state in the particular case where $\Omega$ is a bounded domain of ${\rm I\mskip -3.5mu R}^N$. Note that if $v$ is a smooth vector field we denote by $index_{z_j}(v)=deg (v, B_{\delta}(z_j),0)$ the Brower Degree of the vector field $v$ in the ball $B_{\delta}(z_j)$ for some $\delta$ small fixed. **Theorem 10** (Poincaré-Hopf Theorem). *Let $\Omega \subset {\rm I\mskip -3.5mu R}^N$, with $N\geq 2$ be a smooth bounded domain. Let $v$ be a smooth vector field on $\Omega$ with isolated zeroes $z_1,\dots,z_l$ and such that $v(x)\cdot \nu<0$ for any $x\in \partial \Omega$ (here $\nu$ is the outward normal vector to $\partial \Omega$). Then we have the formula $$\sum_{j=1}^l index_{x_j}(v)=(-1)^N\chi(\Omega)$$ where $index_{x_j}(v)=deg (v, B_\delta (x_j),0)$ is the Brower degree of the vector field $v$ in the ball $B_\delta(x_j)$ with a small fixed $\delta>0$ and $\chi(\Omega)$ is the Euler characteristic of $\Omega$.* When $v=\nabla u_n$ this Theorem gives a link between an analytic problem (such as to compute the critical points of solutions to [\[1\]](#1){reference-type="eqref" reference="1"}) and a topological invariant such as $\chi(\Omega)$ that describes the structure of $\Omega$. The Euler characteristic is intrinsic of the manifold or of the domain and it is independent on the triangulation that one can use to reconstruct a manifold in the study of imaging. Another result which plays a crucial role in our computation is the following one, (see [@am]) **Theorem 11**. *Let $\Omega$ be a bounded open set in the plane and let its boundary $\partial\Omega$ be composed of $N$ simple closed curves $\Gamma_1,..,\Gamma_N$, $N\ge1$ of class $C^{1,{\alpha}}$. Consider the solution $u\in C^1(\bar\Omega)\cap C^2(\Omega)$ of the Dirichlet problem $$\begin{cases} \Delta u=0&in\ \Omega\\ u=a_i&on\ \partial\Omega \end{cases}$$ where $a_1, .. , a_N$ are given constants. If $a_1, .. , a_N$ do not all coincide, then $u$ has isolated critical points $z_1, .. , z_k$ in $\bar\Omega$, with finite multiplicities $m_1, .. , m_k$, respectively, and the following identity holds: $$\sum_{z_k\in\Omega}m_k+\frac12\sum_{z_k\in\partial\Omega}m_k=N-2$$* **Remark 12**. *The multiplicity $m_k$ of a critical point $P=(x_0,y_0)$ of an analytic function $u:\Omega\subset{\rm I\mskip -3.5mu R}^2\to{\rm I\mskip -3.5mu R}$ can be easily defined using complex coordinates. Indeed if $z=x+iy$ (and $z_0=x_0+iy_0$) we have that $m_k$ is defined by the following representation formula, $$\partial _z u(z)=(z-z_0)^{m_k}g(z)$$ with $g$ analytic and $g(z_0)\ne0$. Moreover (see pag.569 in [@am]) the following formula holds: $$index_z(\nabla u) =-m_k.$$ A consequence is that the index of a critical point of a harmonic satisfies $$index_z(\nabla u)\le-1.$$ Finally if $index_z(\nabla u)=-1$ (and then $m_k=1$ and there exists a coordinate system in which $z_0$ and the function $u(z)$ can be written as $\partial _z u(z)=a(z-z_0)+o(|z-z_0|^2)$ in a neighborhood of $z_0$. This implies that $z_0$ is a nondegenerate saddle point.* We end this section stating a result on the number of critical points of solutions in domains with small holes. **Theorem 13** (see [@grlu]). *Let $\Omega$ be a smooth bounded set of ${\rm I\mskip -3.5mu R}^2$ with $x_0\in \Omega$. Suppose that $v_\delta$ is a positive solution to $$\label{v-delta} \left\{\begin{array}{lc} -\Delta v_\delta={\lambda}e^{v_\delta}& \mbox{ in }\Omega\setminus B_\delta(x_0)\\ v_\delta=0 & \mbox{ on }\partial \Omega \cup \partial B_\delta(x_0) \end{array} \right.$$ which verifies, for some constant $C$ independent od $\delta$, $$\label{limcr} v_\delta\le C, \text{ in }\Omega\setminus B_\delta(x_0).$$ Denoting by $v_0$ the weak limit of $v_\delta$ as $\delta\to 0$ we get that, if $x_0$ is not a critical point of $v_0$ and all critical points of $v_0$ are nondegenerate, $$\label{mainresult} \sharp\{\hbox{critical points of $v_\delta$ in $\Omega\setminus B_\delta(x_0)$}\}=\sharp\{\hbox{critical points of $v_0$ in $\Omega$}\}+1.$$ Finally the additional critical point $x_\delta$ of $v_\delta$ is a saddle point of index $-1$.* In order to verify the assumption [\[limcr\]](#limcr){reference-type="eqref" reference="limcr"} we will use the following result, **Lemma 14**. *Suppose that $u_\delta$ is a family of solutions to [\[v-delta\]](#v-delta){reference-type="eqref" reference="v-delta"} in $\Omega\setminus B_\delta(x_0)$. Assume $B_\delta(x_0)\subset B_R(x_0) \subset \Omega$ and $$u_\delta\leq C_{R}\hbox{ on }\partial B_R(x_0)$$ where $C_R$ is a constant independent of $\delta$. Extending $u_\delta$ to zero in $B_\delta(x_0)$ suppose that, for every $\delta$, $$\label{epsilon-zero} {\lambda}\int_{B_R(x_0)} e^{u_\delta} dx\leq {\varepsilon}_0<4\pi.$$ Then $\|u_\delta\|_{L^{\infty}(B_R(x_0))}\leq C$ uniformly with respect to $\delta$.* *Proof.* We adapt the proof of Corollary 3 and Theorem 1 in [@bremer] to our case where the domains are moving. For $x,y\in B(x_0,R)$ let us consider the function $$\bar u_\delta(x):=\frac 1{2\pi}\int_{B_R(x_0)}\log \frac {2R}{|x-y|}\left(\lambda e^{u_\delta(y)}\right) dy$$ which solves $$\label{bar-u} \left\{\begin{array}{lc} -\Delta \bar u_\delta={\lambda}e^{u_\delta}& \mbox{ in }{\rm I\mskip -3.5mu R}^2\\ \bar u_\delta\geq 0 & \mbox{ in } B_R(x_0). \end{array} \right.$$ The maximum principle, applied to the function $\bar u_\delta+C_{R}-u_\delta$ in $B_R(x_0)\setminus B_\delta(x_0)$, implies that $$u_\delta\leq \bar u_\delta+C_{R} \ \ \text{ in }B_R(x_0).$$ The Jensen inequality (see the proof of Theorem 1 in [@bremer]) then gives, for every $0<\beta<4\pi$ $$e^{\int _{B_R(x_0)}\frac {{\lambda}e^{u_\delta(y)}}{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}}\frac {\beta}{2\pi} \log \left(\frac {2R}{|x-y|}\right) dy}\leq \int _{B_R(x_0)}\frac {{\lambda}e^{u_\delta(y)}}{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}}e^{\log \left(\frac {2R}{|x-y|}\right) ^{\frac \beta{2\pi}}}dy$$ so that $$e^{\frac \beta{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}}\bar u_\delta(x)}\leq \int _{B_R(x_0)}\frac {{\lambda}e^{u_\delta(y)}}{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}}\left(\frac {2R}{|x-y|}\right) ^{\frac \beta{2\pi}} dy$$ which also implies, $$e^{\frac \beta{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}} u_\delta(x)}\leq \int _{B_R(x_0)}\frac {{\lambda}e^{u_\delta(y)}}{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}}\left(\frac {2R}{|x-y|}\right) ^{\frac \beta{2\pi}} dy+e^{\frac \beta{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}} C_{R}}.$$ Integrating then $$\begin{split} \int _{B_R(x_0)}e^{\frac \beta{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}} u_\delta(x)}dx &\leq \int_{B_R(x_0)}dx \int _{B_R(x_0)}\frac {{\lambda}e^{u_\delta(y)}}{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}}\left(\frac {2R}{|x-y|}\right) ^{\frac \beta{2\pi}} dy\\ &+e^{\frac \beta{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}} C_{R,\delta}}\left| B_R\right|\\ &=\int_{B_R(x_0)} \frac {{\lambda}e^{u_\delta(y)}}{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}}dy \int _{B_R(x_0)}\left(\frac {2R}{|x-y|}\right) ^{\frac \beta{2\pi}} dx\\ &+(\pi R^2)e^{\frac \beta{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}} C_{R}}. \end{split}$$ For $y\in B_R(x_0)$ we have that $$\int _{B_R(x_0)}\left(\frac {2R}{|x-y|}\right) ^{\frac \beta{2\pi}} dx\leq \int _{B_{2R}(y)}\left(\frac {2R}{|x-y|}\right) ^{\frac \beta{2\pi}} dx=\frac{8\pi R^2}{2-\frac \beta{2\pi}}.$$ Then $$\int _{B_R(x_0)}e^{\frac \beta{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}} u_\delta(x)}dx\leq \frac{8\pi R^2}{2-\frac \beta{2\pi}}+(\pi R^2)e^{\frac \beta{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}} C_{R}}\leq C$$ We choose $\beta$ such that $4\pi >\beta>{\varepsilon}_0$ so that $\frac \beta{\|{\lambda}e^{u_{\delta}}\|_{L^1(B_R(x_0))}}>\frac \beta {{\varepsilon}_0}>1$. This means that $\lambda e^{u_\delta}\in L^p$ for some $p>1$. Then we can apply the standard regularity theory to the function $\bar u_\delta$ in [\[bar-u\]](#bar-u){reference-type="eqref" reference="bar-u"}, and we have that $\bar u_\delta\in L^\infty(B_R(x_0))$, uniformly with respect to $\delta$. Then finally also $u_\delta$ is uniformly bounded in $B_R(x_0)$. ◻ # An asymptotic estimate in an annular set that shrinks slowly [\[S2\]]{#S2 label="S2"} From the known results in Section [2](#se:2){reference-type="ref" reference="se:2"} we have good asymptotic for the solutions $u_n$ both in the blow-up balls $B_{\delta_{i_n}\bar R}(x_{i,n})$ for every $\bar R>0$ (formulas [\[2.4\]](#2.4){reference-type="eqref" reference="2.4"}-[\[2.6a\]](#2.6a){reference-type="eqref" reference="2.6a"}) and in the set $\Omega\setminus\cup_{j=1}^kB_R(P_j)$ for any $R>0$ sufficiently small (Theorem [Theorem 8](#T0){reference-type="ref" reference="T0"}). It lasts to consider the case when a sequence of points $x_n\to P_i$ as $n\to \infty$ for some index $i\in \{1,\dots,m\}$ but $x_n$ do not belong to a ball of blow-up. This is the aim of this section, where we prove an asymptotic estimate in annuli centered at $x_{i,n}$ with arbitrary (infinitesimal) radii. Before to give the precise statement, let us give an idea of the proof. If we rescale the solutions $u_n$ near $x_{i,n}$ with respect to a parameter $r_n$ which is slower that $\delta_{i,n}$ in [\[2.5\]](#2.5){reference-type="eqref" reference="2.5"} we obtain, in the limit, a problem that remains singular in the origin. However, some properties of this \"wrong\" scaling will be useful for studying the properties of the solutions $u_n$ in points $x_n$ that converge to some $P_i$ but do no belong to $B_{\delta_{i_n}\bar R}(x_{i,n})$ for every $\bar R$. Here we include one of these results on this scaling of the solution $u_n$ that will be useful later. This idea has been already used in a previous paper [@gg1] in the case of a one-point blowing-up sequence of solutions $u_n$. **Proposition 15**. *Let $u_n$ be a sequence of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} which blows-up at $\{P_1,\dots,P_m\}$ as $n\to \infty$ and let $r_n$ a sequence such that $r_n\to 0$ and $$\label{b10} \frac{r_n}{\delta_{i,n}}\to+\infty\quad\hbox{as }n\to \infty$$ for every $i=1, \dots,m$, where $\delta_{i,n}$ are defined in [\[2.5\]](#2.5){reference-type="eqref" reference="2.5"}.* *Then, for any $i=1,..,m$, and for any $0<\xi_1<\xi_2$, $$\label{convergenza-vi} u_n(x)=-4\log|x-x_{i,n}|+8\pi \mathcal{R}(P_i)+8\pi \sum_{j=1, j\neq i}^{m}G(P_i,P_j)+o(1)\quad\hbox{in }C^1_{loc}(A^i_n)$$ as $n\to \infty$, where $A^i_n=\{\xi_1r_n\le|x-x_{i,n}|\le \xi_2r_n\}$ are shrinking annuli.* *Proof.* We follow an idea already used in [@gg1] in the case $m=1$. For some index $i\in\{1,..,m\}$ let us consider the function $v_{i,n}:\Omega_n=\frac{\Omega-x_{i,n}}{r_n}$ (observe that $\Omega_n\to {\rm I\mskip -3.5mu R}^2$ as $n\to \infty$), $$\label{vi} v_{i,n}(x):= u_n(r_nx+x_{i,n})+4\log r_n.$$ In this setting the estimate [\[convergenza-vi\]](#convergenza-vi){reference-type="eqref" reference="convergenza-vi"} is equivalent to show that $$\label{3.5} v_{i,n}( x)=4\log \frac1{|x|}+8\pi \mathcal{R}(P_i)+8\pi \sum_{j=1, j\neq i}^{m}G(P_i,P_j)+o(1)\quad\hbox{in }C^1_{loc}({\rm I\mskip -3.5mu R}^2\setminus\{0\}).$$$$\label{3.5} v_{i,n}( x)=4\log \frac1{|x|}+8\pi \mathcal{R}(P_i)+8\pi \sum_{j=1, j\neq i}^{m}G(P_i,P_j)+o(1)\quad\hbox{in }C^1_{loc}({\rm I\mskip -3.5mu R}^2\setminus\{0\}).$$ To prove [\[3.5\]](#3.5){reference-type="eqref" reference="3.5"} we use the Green representation formula and we get $$\label{Avi} \begin{split} &v_{i,n}( x)=\lambda_n \int_{\Omega} G(r_n x+x_{i,n},y) e^{u_n(y)}dy+4\log r_n=\\ =&\underbrace{\lambda_n\int_{\Omega\setminus\left(\cup_j B_R(x_{j,n})\right)}G(r_n x+x_{i,n},y) e^{u_n(y)}dy}_{=I_{1,n}}+\underbrace{\lambda_n\sum_{j=1}^n \int_{B_ R(x_{j,n})} G(r_n x+x_{i,n},y) e^{u_n(y)}dy}_{=I_{2,n}}+4\log r_n \end{split}$$ where $R$ is chosen as mentioned just before [\[2.4\]](#2.4){reference-type="eqref" reference="2.4"}. From now we assume that $$\label{g4} \xi_1\le| x|\le \xi_2.$$ We start by proving $C^0$ estimates. We split the proof in some steps. **Step 1: estimate of $I_{1,n}$** In this step we prove that, for $n\to \infty$ $$\label{g2} I_{1,n}=o(1).$$ Indeed in $\Omega\setminus\left(\cup_j B_R(x_{j,n})\right)$, by [\[3\]](#3){reference-type="eqref" reference="3"} $u_n$ is uniformly bounded and since $| x|\le \xi_2$ we have $$I_{1,n}=\lambda_n\int_{\Omega\setminus\left(\cup_j B_R(x_{j,n})\right)}G(r_n x+x_{i,n},y) e^{u_n(y)}dy=O(\lambda_n)=o(1)$$ as $n\to \infty$, which gives [\[g2\]](#g2){reference-type="eqref" reference="g2"}. **Step 2: estimate of $I_{2,n}$** In this step we prove that, for $n\to \infty$ $$\label{g3} I_{2,n}=-4\log r_n+ 4\log \frac 1{| x|}+8\pi \mathcal{R}(P_i)+8\pi \sum_{j=1, j\neq i}^{m}G(P_i,P_j)+o(1).$$ This is the more delicate estimate and we need to consider the different cases $j\ne i$ and $j=i$. **Step 3: case $j\ne i$** When $j\neq i$ it is easy to see that $|G(r_n x+x_{i,n},y)|\leq C$ for $y\in B_R(x_{j,n})$ since $r_n x+x_{i,n}\in B_R(x_{i,n})$ for $n$ large (by $| x|\le \xi _2$) and so $$\begin{split} &\lambda_n\int_{B_R(x_{j,n})} G(r_n x+x_{i,n},y) e^{u_n(y)}dy=\int_{B_{\frac R{\delta_{j,n}}}(0)}G(r_n x+x_{i,n},\delta_{j,n} y+x_{j,n})e^{ \tilde u_{j,n}( y)}d y=\\ & \ \ \ \ 8\pi G(P_i,P_j)+o(1) \end{split}$$ when $n\to \infty$, by [\[2.6a\]](#2.6a){reference-type="eqref" reference="2.6a"}. **Step 4: case $j=i$** Here we have that $$\begin{split} &\lambda_n \int_{B_R(x_{i,n})} G(r_n x+x_{i,n},y) e^{u_n(y)}dy\\ & \ \ \ =\lambda_n\int _{B_R(x_{i,n})} \left[\frac 1{2\pi}\log \frac 1{|r_n x+x_{i,n}-y|}+H(r_n x+x_{i,n},y)\right] e^{u_n(y)}dy\\ & \ \ \ \ \text{ and, letting }y=\delta_{i,n} y+x_{i,n}\\ & \ \ =\frac 1{2\pi}\int_{B_{\frac R{\delta_{i,n}}}(0)}\log \frac 1{|r_n x-\delta_{i,n} y|}e^{ \tilde u_{i,n}( y)}d y+ \int _{B_{\frac R{\delta_{i,n}}}(0)} H(r_n x+x_{i,n},\delta_{i,n} y+x_{i,n})e^{ \tilde u_{i,n}( y)}d y=\\ & \ \ =\frac 1{2\pi}\int_{B_{\frac R{\delta_{i,n}}}(0)}\log \frac 1{| x-\frac{\delta_{i,n}}{r_n} y|}e^{ \tilde u_{i,n}( y)}d y+ \frac 1{2\pi}\log \frac 1{r_n}\int_{B_{\frac R{\delta_{i,n}}}(0)}e^{ \tilde u_{i,n}( y)}d y\\ & \ \ \ \ \ \ \ \ + \int _{B_{\frac R{\delta_{i,n}}}(0)} H(r_n x+x_{i,n},\delta_{i,n} y+x_{i,n})e^{ \tilde u_{i,n}( y)}d y=L_{1,n}+L_{2,n}+L_{3,n} \end{split}$$ The easiest integral to estimate is $L_{3,n}$. Indeed, since $|H(r_n x+x_{i,n},\delta_{i,n} y+x_{i,n})|\leq C$ for $y\in B_{\frac R {\delta_{i,n}}(0)}$ by [\[2.6a\]](#2.6a){reference-type="eqref" reference="2.6a"} then, as $n\to \infty$ $$L_{3,n}=\int _{B_{\frac R{\delta_{i,n}}}(0)} H(r_n x+x_{i,n},\delta_{i,n} y+x_{i,n})e^{ \tilde u_{i,n}( y)}d y= 8\pi H(P_i,P_i)+o(1)=8\pi \mathcal{R}(P_i)+o(1).$$ Next we prove that, as $n\to \infty$ $$\label{g1} L_{2,n}=-4\log r_n+O\big({\lambda}_n^2\log r_n\big)=-4\log r_n+o(1).$$ Indeed by [\[stima2\]](#stima2){reference-type="eqref" reference="stima2"} we have $$\begin{split} L_{2,n}&=\frac 1{2\pi} \log \frac 1{r_n}\int_{B_{\frac R{\delta_{i,n}}}(0)}e^{ \tilde u_{i,n}( y)} d y=\frac{8\pi+O\big({\lambda}_n\big)}{2\pi} \log \frac 1{r_n}\\ &=-4\log r_n+O(\delta_{i,n}^2\log r_n) \end{split}$$ as $n\to \infty$. Finally by [\[b10\]](#b10){reference-type="eqref" reference="b10"} we get $\delta_{i,n}^2\log r_n=o(1)$ and [\[g1\]](#g1){reference-type="eqref" reference="g1"} follows.\ Finally let us estimate the integral $$L_{1,n}:=\frac 1{2\pi}\int_{B_{\frac R{\delta_{i,n}}}(0)}\log \frac 1{| x-\frac{\delta_{i,n}}{r_n} y|}e^{\tilde u_{i,n}( y)}d y.$$ Take $\rho=\frac {\xi _1}2>0$ and let us split $B_{\frac R{\delta_{i,n}}(0)}$ as $$B_{\frac R{\delta_{i,n}}(0)}=B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)\cup\left(B_{\frac R{\delta_{i,n}}(0)}\setminus B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)\right).$$ Observe that by [\[b10\]](#b10){reference-type="eqref" reference="b10"} and [\[g4\]](#g4){reference-type="eqref" reference="g4"} and using that $|x|\geq \xi_1$, if $y\in B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)$ then $| y|=|\frac{r_n}{\delta_{i,n}} x+ y-\frac{r_n}{\delta_{i,n}} x|\geq \frac{r_n}{\delta_{i,n}}\xi _1-\rho \frac{r_n}{\delta_{i,n}}=\frac{r_n}{\delta_{i,n}}\frac{\xi_1}2\to \infty$ if $n\to \infty$. This means that $$B_{\frac R{\delta_{i,n}}(0)}\setminus B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)\to {\rm I\mskip -3.5mu R}^2$$ as $n\to \infty$. Then we write $$\begin{split} L_{1,n}=&\frac 1{2\pi} \int_{B_{\frac R{\delta_{i,n}}(0)}\setminus B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)} \log \frac 1{| x-\frac{\delta_{i,n}}{r_n} y|} e^{ \tilde u_{i,n}( y)}d y\\ &+\frac 1{2\pi} \int_{ B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)} \log \frac 1{| x-\frac{\delta_{i,n}}{r_n} y|} e^{ \tilde u_{i,n}( y)}d y:=J_{1,n}+J_{2,n}. \end{split}$$ In $J_{1,n}$, since $y\notin B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)$ we have that $| x-\frac{\delta_{i,n}}{r_n} y|=\frac{\delta_{i,n}}{r_n}|\frac{r_n}{\delta_{i,n}} x- y|>\rho$. Then $$\left| \log \frac 1{| x-\frac{\delta_{i,n}}{r_n} y|}\right|\leq \max \{|\log \rho|, | x-\frac{\delta_{i,n}}{r_n} y|\}\leq C+| x|+| y|\leq C+| y|.$$ Then, by [\[2.6a\]](#2.6a){reference-type="eqref" reference="2.6a"} we have $$\log \frac 1{| x-\frac{\delta_{i,n}}{r_n} y|} e^{ \tilde u_{i,n}( y)}\leq \frac{ C+| y|}{(1+| y|^2)^2}$$ and passing to the limit we get, as $n\to \infty$ $$J_{1,n}=\frac 1{2\pi}\log \frac {1}{| x|}\int_{{\rm I\mskip -3.5mu R}^2}e^{U( y)}d y+o(1)=4 \log \frac {1}{| x|}+o(1).$$ Next we estimate $J_{2,n}$. We use [\[2.6a\]](#2.6a){reference-type="eqref" reference="2.6a"}, $| y|\geq \frac{r_n}{\delta_{i,n}}\frac{\xi _1}2$ and $\frac {r_n}{\delta_{i,n}}\to+\infty$ to have that $$\label{numero-bis} e^{ \tilde u_{i,n}(y)}\leq \frac C{\left(8+\frac {r_n^2}{\delta_{i,n}^2}\frac{\xi _1^2}{4}\right)^2}\leq C \frac {\delta_{i,n}^4}{r_n^4}.$$ Then $$\begin{split} &|J_{2,n}| \leq\frac 1{2\pi} \int_{ B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)}\left|\log \frac {1}{| x-\frac{\delta_{i,n}}{r_n} y|}\right| e^{ \tilde u_{i,n}( y)}d y\\ & \ \ \ \leq C \frac {\delta_{i,n}^4}{r_n^4}\int_{ B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)} \left|\log \frac {1}{| x-\frac{\delta_{i,n}}{r_n} y|}\right| d y \\ & \ \ \ =C \frac {\delta_{i,n}^4}{r_n^4}\int_0^{ \rho\frac {r_n}{\delta_{i,n}}} r \left|\log\left(r\frac{\delta_{i,n}}{r_n}\right)\right| dr=O\left(\frac {\delta_{i,n}^2}{r_n^2}\log\frac {\delta_{i,n}}{r_n}\right)=o(1) \end{split}$$ as $n\to \infty$. Summarizing, we have as $n\to \infty$ $$\begin{split} I_{2,n}=&\lambda_n\sum_{j=1}^n \int_{B_ R(x_{j,n})} G(r_n x+x_{i,n},y) e^{u_n(y)}dy\\ &=\sum_{j=1, j\neq i}^{m} \underbrace{\lambda_n \int_{B_ R(x_{j,n})} G(r_n x+x_{i,n},y) e^{u_n(y)}dy}_{=8\pi G(P_i,P_j)+o(1)}+\underbrace{L_{1,n}}_{J_{1,n}+J_{2,n}}+\underbrace{L_{2,n}}_{-4\log r_n+o(1)}+\underbrace{L_{3,n}}_{8\pi \mathcal{R}(P_i)+o(1)}\\ &\text{ and using the expansions for $J_{1,n}$ and $J_{2,n}$}\\ &=-4\log r_n+4\log \frac 1{| x|}+8\pi \mathcal{R}(P_i)+8\pi \sum_{j=1, j\neq i}^{m}G(P_i,P_j)+o(1) \end{split}$$ which gives the claim in [\[g3\]](#g3){reference-type="eqref" reference="g3"}. **Step 5, claim of $C^0$-estimate** Putting together all the estimates of the previous steps we get $$\begin{split} v_{i,n}( x) &=\lambda_n\int_{\Omega\setminus\left(\cup_j B_{R}(x_{j,n})\right)}G(r_n x+x_{i,n},y) e^{u_n(y)}dy+\lambda_n\sum_{j=1}^m \int_{B_{R}(x_{j,n})} G(r_n x+x_{i,n},y) e^{u_n(y)}dy \\ &+4\log r_n=I_{1,n}+I_{2,n}+4\log r_n\\ &=4\log \frac 1{| x|}+8\pi \mathcal{R}(P_i)+8\pi \sum_{j=1, j\neq i}^{m}G(P_i,P_j) +o(1) \end{split}$$ as $n\to \infty$ uniformly for $x\in \{x\in {\rm I\mskip -3.5mu R}^2: \xi_1\le x\le\xi_2\}$ which gives the claim of the $C^0$-estimate in [\[3.5\]](#3.5){reference-type="eqref" reference="3.5"} and also in [\[convergenza-vi\]](#convergenza-vi){reference-type="eqref" reference="convergenza-vi"} . **Step 6, claim of $C^1$-estimate** We can do the same computations for the derivatives of $v_{i,n}$. Again from the Green representation formula and equation [\[1\]](#1){reference-type="eqref" reference="1"} we have $$\label{numero} \begin{split} \nabla v_{i,n}(x)&=r_n\lambda_n \int_{\Omega} \nabla_x G(r_n x+x_{i,n},y) e^{u_n(y)}dy=r_n\lambda_n\int_{\Omega\setminus\left(\cup_j B_{R}(x_{j,n})\right)}\nabla_xG(r_n x+x_{i,n},y) e^{u_n(y)}dy+\\ &r_n \lambda_n\sum_{j=1}^n \int_{B_{R}(x_{j,n})} \nabla_x G(r_n x+x_{i,n},y) e^{u_n(y)}dy \end{split}$$ where, as before $$\lambda_n\int_{\Omega\setminus\left(\cup_j B_{R}(x_{j,n})\right)}\nabla_xG(r_n x+x_{i,n},y) e^{u_n(y)}dy=O(\lambda_n)=o(1)$$ as $n\to \infty$ and, for $i\neq j$ $$\begin{split} r_n\lambda_n\int_{B_{R}(x_{j,n})} \nabla_x G(r_n x+x_{i,n},y) e^{u_n(y)}dy&=r_n \int_{B_{\frac R{\delta_{j,n}}(0)}}\nabla_x G(r_n x+x_{i,n},\delta_{j,n}y+x_{j,n}) e^{\tilde u_{j,n}( y)}d y\\ &=r_n 8\pi \nabla _x G(P_i,P_j)+o(r_n)=o(1).\end{split}$$ Next we consider the last term in [\[numero\]](#numero){reference-type="eqref" reference="numero"} $$\begin{split} &r_n\lambda_n\int_{B_{R}(x_{i,n})} \nabla_x G(r_n x+x_{i,n},y) e^{u_n(y)}dy\\ &\ \ \ \ \ \ =r_n\lambda_n \int _{B_{ R}(x_{i,n})}\left(-\frac 1{2\pi}\frac {(r_n x+x_{i,n}-y)}{|r_n x+x_{i,n}-y|^2}+\nabla_x H(r_n x+x_{i,n},y)\right)e^{u_n(y)}dy\\ &\ \ \ \ \ \ =-\frac 1{2\pi} r_n\int_{B_{\frac R{\delta_{i,n}}(0)}}\frac {(r_n x-\delta_{i,n}y)}{|r_n x-\delta_{i,n} y|^2}e^{\tilde u_{i,n}( y)}dy+r_n\int_{B_{\frac R{\delta_{i,n}}(0)}}\nabla_x H(r_n x+x_{i,n},\delta_{i,n}y+x_{i,n})e^{\tilde u_{i,n}( y)}dy \\ &\ \ \ \ \ \ =-\frac 1{2\pi}\int_{B_{\frac R{\delta_{i,n}}(0)}}\frac {( x-\frac {\delta_{i,n}}{r_n} y)}{| x-\frac{\delta_{i,n}}{r_n}y|^2}e^{\tilde u_{i,n}( y)}d y+\underbrace{8\pi r_n \nabla_x H(P_i,P_i)+o(r_n)}_{=o(1)} \end{split}$$ where we used again [\[2.6a\]](#2.6a){reference-type="eqref" reference="2.6a"} to pass to the limit. As we did in the computation of $J_{1,n}$ and $J_{2,n}$ is Step $4$, we split the last integral as follows, $$\begin{split} I_n:=&-\frac 1{2\pi} \int_{B_{\frac R{\delta_{i,n}}(0)}}\frac {( x-\frac {\delta_{i,n}}{r_n} y)}{| x-\frac{\delta_{i,n}}{r_n}y|^2}e^{\tilde u_{i,n}( y)}d y\\ =&-\frac 1{2\pi} \int_{B_{\frac R{4\delta_{i,n}}(0)}\setminus B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)}\frac {( x-\frac {\delta_{i,n}}{r_n} y)}{| x-\frac{\delta_{i,n}}{r_n} y|^2}e^{\tilde u_{i,n}( y)}dy\\ &-\frac 1{2\pi} \int_{ B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)}\frac {( x-\frac {\delta_{i,n}}{r_n} y)}{| x-\frac{\delta_{i,n}}{r_n} y|^2}e^{\tilde u_{i,n}( y)}d y:=\bar I_{1,n}+\bar I_{2,n} \end{split}$$ In $\bar I_{1,n}$ $$\left|\frac {( x-\frac {\delta_{i,n}}{r_n} y)}{| x-\frac{\delta_{i,n}}{r_n} y|^2}\right|= \frac1{| x-\frac{\delta_{i,n}}{r_n} y|}\le\frac1\rho\hbox{ since } y\notin B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)$$ and, by [\[2.6a\]](#2.6a){reference-type="eqref" reference="2.6a"} we get, as $n\to \infty$ $$\bar I_{1,n}= -\frac 1{2\pi}\frac { x}{|x|^2}\int_{{\rm I\mskip -3.5mu R}^2}e^{U(y)}d y+o(1)=-4 \frac { x}{| x|^2}+o(1).$$ In $\bar I_{2,n}$ instead we use [\[numero-bis\]](#numero-bis){reference-type="eqref" reference="numero-bis"} to get $$\begin{split} &|\bar I_{2,n}|=\frac 1{2\pi}\left|\int_{ B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)} \frac {(x-\frac {\delta_{i,n}}{r_n} y)}{| x-\frac{\delta_{i,n}}{r_n} y|^2}e^{\tilde u_{i,n}( y)}dy\right|\\ & \ \ \ \leq C \frac {\delta_{i,n}^4}{r_n^4}\int_{ B_{\rho\frac {r_n}{\delta_{i,n}}}\left(\frac{r_n}{\delta_{i,n}} x\right)}\frac {1}{| x-\frac{\delta_{i,n}}{r_n} y|} d y \\ & \ \ \ =C \frac {\delta_{i,n}^4}{r_n^4} \int_0^{2\pi}d\theta\int_0^{ \rho\frac {r_n}{\delta_{i,n}}} \frac r{r\frac {\delta_{i,n}}{r_n}} dr=2 \pi C \frac {\delta_{i,n}^4}{r_n^4}\frac {r_n^2}{\delta_{i,n}^2}\to 0 \end{split}$$ Putting together all the estimates we get $$\label{numero2} \nabla v_{i,n}( x)=-4\frac{ x}{| x|^2}+o(1)$$ uniformly for $x\in \{x\in {\rm I\mskip -3.5mu R}^2: \xi_1\le x\le\xi_2\}$ as $n\to \infty$ and this concludes the proof. ◻ # Critical points of $u_{\lambda}$ in $B_{\rho}(P_i)$ {#S3} In this section we will prove that any solution $u_{\lambda}$ which blows-up at $m$ points $\{P_1,\dots,P_m\}\in \Omega$, as $\lambda\to 0$, admits $exactly$ one non-degenerate critical point in each ball $B_\rho(P_i)$, for $i=1,..,m$ if $\rho$ is small enough. It is sufficient to prove the result for any sequence of values $\lambda_n$ such that $\lambda_n\to 0$ as $n\to \infty$. **Proposition 16**. *Let $u_n$ be a sequence of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} which blows-up at $\{P_1,\dots,P_m\}$ as $n\to \infty$. Then there exists $\rho>0$ such that $u_n$ has a unique critical point in $\bar B_\rho(P_i)$ for $i=1,\dots,m$ when $n$ is large. This critical point is the maximum $x_{i,n}$ in [\[2.4\]](#2.4){reference-type="eqref" reference="2.4"} and it is nondegenerate.* **Remark 17**. *Actually we will prove something more, precisely that there exists $\bar \rho>0$ such that $$\label{a1} (x-x_{i,n})\cdot \nabla u_n(x)<0 \text{ in } B_{\bar \rho}(x_{i,n})\setminus\{x_{i,n}\}$$ for $i=1,\dots,m$. Then we take $\rho:=\frac{\bar \rho}2$, and we get that $$(x-x_{i,n})\cdot \nabla u_n(x)<0 \text{ in } \bar B_{\rho}(P_i)\setminus\{x_{i,n}\}.$$ This will show that $u_n$ has the unique critical point $x_{i,n}$ in $\bar B_{\rho}(P_i)$.* *Proof.* As pointed out in Remark [Remark 17](#ultimo){reference-type="ref" reference="ultimo"}, we prove [\[a1\]](#a1){reference-type="eqref" reference="a1"} to get that $u_n$ has the unique critical point $x_{i,n}$ in $\bar B_{\rho}(P_i)$.\ We argue by contradiction and we assume there exists an index $i\in \{1,\dots,m\}$ and points $\zeta_n\in B_{\bar \rho_n}(x_{i,n})\setminus\{x_{i,n}\}$ such that $\bar \rho_n\to 0$ and $$(\zeta_n-x_{i,n})\cdot \nabla u_n(\zeta_n)\geq 0 \ \ \text{ for every }n.$$ Up to a subsequence $\zeta_n\to \zeta$ and since $\bar \rho_n\to 0$ then $\zeta=P_i$. Let $\delta_{i,n}$ be the scaling parameter as defined in [\[2.5\]](#2.5){reference-type="eqref" reference="2.5"} related to the index $i$. Up to a subsequence, we have the following alternative, - $\zeta_n\in B_{\bar R\delta_n}(x_{i,n})$ for some $\bar R>0$ for any $n$. - $\zeta_n\notin B_{\delta_{i,n}\bar R}(x_{i,n})$ for any real number $\bar R>0$ for any $n$. [Proof of case a)]{.ul} Here we rescale the solution $u_n$ around $x_{i,n}$ as in [\[2.4\]](#2.4){reference-type="eqref" reference="2.4"} and we let $\tilde \zeta_n:=\frac{\zeta_n-x_{i,n}}{\delta_{i,n}}$. Then $|\tilde \zeta_n|<\bar R$ and so, up to a subsequence $\tilde \zeta_n\to \tilde \zeta\in B_{\bar R}(0)$. From [\[2.4\]](#2.4){reference-type="eqref" reference="2.4"} we have that $\tilde \zeta_n\cdot \nabla \tilde u_{i,n}(\tilde \zeta_n)=(\zeta_n-x_{i,n})\cdot \nabla u_n(\zeta_n)\geq 0$, and, passing to the limit and using [\[2.6\]](#2.6){reference-type="eqref" reference="2.6"} $\tilde \zeta_n\cdot \nabla \tilde u_{i,n}(\tilde \zeta_n)\to\tilde \zeta\cdot \nabla U(\tilde \zeta)=-\frac 12 \frac{|\tilde \zeta|^2}{\left(1+\frac {|\tilde \zeta|^2}8\right)}\leq 0$. If $\tilde \zeta\neq 0$ we reach a contradiction. So suppose that $\tilde \zeta=0$ and consider the function $\varphi_n(t):=\tilde u_{i,n}(t\tilde \zeta_n)$. Since $\varphi_n$ has a maximum at $0$ and $\varphi_n'(1)=\tilde \zeta_n\cdot \nabla \tilde u_{i,n}(\tilde \zeta_n)\ge0$ we deduce that there exists a $minimum$ point $z_n\in(0,\tilde \zeta_n)$. From $\varphi_n''(z_n)\ge0$, passing to the limit we get that $$\frac{\partial^2 U(0)}{\partial y_i\partial y_j}\xi_i\xi_j\ge0$$ with $\xi=\lim\limits_{n\to+\infty}\frac{z_n}{|z_n|}$. This gives a contradiction since $0$ is a $non-degenerate$ maximum of $U$. This proves the uniqueness of the critical point of $u_n$ in $B_{\bar R\delta_n}(x_{i,n})$ and the $C^2$ convergence of $\tilde u_{i,n}$ to $U$ gives also its non-degeneracy. [Proof of case b)]{.ul} In this case we have that $\zeta_n\to0$ with $\lim\limits_{n\to+\infty}\frac{|\zeta_n-x_{i,n}|}{\delta_{i,n}}=+\infty$. Here we use Proposition [Proposition 15](#prop-2.1){reference-type="ref" reference="prop-2.1"} with $r_n:=|\zeta_n-x_{i,n}|$. Let $\tilde\zeta_n:=\frac{\zeta_n-x_{i,n}}{r_n}$. So $|\tilde\zeta_n|=1$ for every $n$ and we have that, if $v_{i,n}$ is the function defined in [\[vi\]](#vi){reference-type="eqref" reference="vi"}, $$\tilde\zeta_n\cdot \nabla v_{i,n}(\tilde \zeta_n)=(\zeta_n-x_{i,n})\cdot \nabla u_n(\zeta_n)\geq 0.$$ Using [\[numero2\]](#numero2){reference-type="eqref" reference="numero2"} we have that, up to a subsequence, $\tilde\zeta_n\to \tilde\zeta\in \partial B_1(0)$ and $\tilde \zeta$ satisfies $$\tilde\zeta\cdot \left(-4\frac{\tilde\zeta}{|\tilde\zeta|^2}\right)=-4 \geq 0$$ which gives a contradiction. So from [\[a1\]](#a1){reference-type="eqref" reference="a1"} we get the claim. ◻ By the previous proposition the problem of the computation of the number of critical points of $u_{\lambda}$ is reduced to understand what happens in the region $\Omega\setminus \cup_{j=1}^m \bar B_\rho(P_j)$. We discuss the different situations in next section. # Critical points of $u_{\lambda}$ in $\Omega\setminus \cup_{j=1}^m \bar B_\rho(P_j)$ {#S4} In this section we study the number of critical points of $u_{\lambda}$ in the set $$D:=\Omega\setminus \cup_{j=1}^m \bar B_\rho(P_j),$$ where $\rho$ is the value given in Proposition [Proposition 16](#lem1){reference-type="ref" reference="lem1"}. As previously pointed out, the geometry and the topology fo $D$ have a great influence. In the set $D$ it will important to consider the harmonic function $$\label{kx} K(x):=8\pi \sum_{j=1}^m G(x, P_j).$$ We start by showing some of the properties of $K(x)$, they are basically a consequence of Theorem [Theorem 11](#AM){reference-type="ref" reference="AM"}. **Proposition 18**. *The function $K(x)$ in $\Omega\setminus\{P_1,\dots,P_m\}$ has only a finite number of critical points $\mathcal C:=\{z_1,\dots, z_l\}$ which are saddle points of finite multiplicity $m_j \geq 1$ and $index_{z_j}(\nabla K) \leq -1$. Moreover whenever $index_{z_j}(\nabla K) =-1$ then $z_j$ is a nondegenerate saddle point.* *Proof.* The function $K(x)$ is harmonic in $\Omega\setminus\{P_1,\dots,P_m\}$. Since - $\lim\limits_{x\to P_j}K(x),|\nabla K(x)|\to +\infty$ for $j=1,\dots,m$ - $K(x), |\nabla K(x)|=O(1)$ in $\omega$ if $\bar \omega\subset \bar \Omega \setminus\{P_1,\dots,P_m\}$ we can select a value of $M$ large in such a way that the level set $K(x)=M$ is given by $m$ curves $\Gamma_1, \dots, \Gamma_m$ such that $\Gamma_i\cap \Gamma_j=\emptyset$ for $i\neq j$. Moreover $M$ can be chosen such that $|\nabla K(x)|>0$ on the curves $\Gamma_j$ for $j=1,\dots,m$. Then these curves are simple, closed, smooth and are the boundary of bounded sets $A_j$ such that $P_j\in A_j$.\ Since $K(x)=M>0$ on $\Gamma_1\cup\dots\cup\Gamma_m$ and $K(x)=0$ on $\partial \Omega$ can apply Theorem [Theorem 11](#AM){reference-type="ref" reference="AM"} getting that $K(x)$ has a finite number of critical points in $\Omega\setminus\cup_{j=1}^m \bar A_j$. Moreover since $K(x)>0$ in $D$ the Hopf Lemma implies that there are not critical points on $\partial \Omega$. The maximum principle for harmonic functions then implies that the critical points $\{z_1,\dots, z_l\}$ are saddle critical points with finite multiplicities $m_1,\dots,m_l$ by the analiticity of harmonic functions. Finally by Remark [Remark 12](#RAM){reference-type="ref" reference="RAM"} we have that if $index_{z_j}(\nabla K)=-1$ then $m_j=1$ and $z_j$ is a nondegenerate saddle point. ◻ As in the previous section it is enough to prove the results for any sequence of values $\lambda_n$ such that $\lambda_n\to 0$. **Proposition 19**. *Let $u_n$ be a sequence of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} which blows-up at $\{P_1,\dots,P_m\}$ as $n\to \infty$. Then $u_n$, for $n$ large enough, has only a finite number of isolated critical points that we denote by $\mathcal C_n:=\{z_{1,n},\dots,z_{l_n,n}\}$. Moreover $$\label{ind} index_{z_{j,n}}(\nabla u_n)\in \{1,0,-1\}$$ and, whenever the index is $1$ then $z_{j,n}$ is a maximum, while whenever the index is $-1$ $z_{j,n}$ is a nondegenerate saddle point.* *Proof.* The proof uses some ideas by [@am]. By classical results $u_n$ is real-analytic in $\Omega$ (see for example [@am], Cor 3.4 and the references therein). By Lemma [Proposition 16](#lem1){reference-type="ref" reference="lem1"} $u_n$ has a unique critical point in $\bar B_\rho(P_i)$ for $i=1,\dots,m$ when $n$ is large enough and it cannot have critical points on $\partial \Omega$ by Hopf lemma. First we show that the critical points of $u_n$ in $D$ are isolated when $n$ is large. By [\[3\]](#3){reference-type="eqref" reference="3"} the critical points of $u_n$ in $D$ should converge to a critical point of $K(x)$ as $n\to \infty$. Let us argue by contradiction and suppose that there exists a critical point $z_0:=(x_0,y_0)\in D$ for the function $K(x)$ which is the limit of a sequence of critical points $z_n$ for $u_n$ and such that the points $z_n$ are not isolated for every $n$. Then there exists sequences of points $z_n^h\in D$ such that $\nabla u_n(z_n^h)=0$ for every $n$ and for every $h$ and such that $z_n^h\to z_n$ as $h\to \infty$ and $z_n^h\to z_0$ as $n\to \infty$. Using that $-\Delta u_n=\lambda_n e^{u_n}>0$ we may assume, up to a rotation, that $(u_n)_{yy}(z_n)\neq 0$. By the implicit function theorem, for every $n$ there exists a neighborhood of $z_n$ in which the set $\sigma_n:=\{x\in D: (u_n)_y=0\}$ is a simple analytic arc that contains infinitely many of the points $z_n^h$. Since $(u_n)_x(z_n^h)=0$ for infinitely many points $z_n^h$ then it should be (by analicity) $(u_n)_x(x)=0$ on $\sigma_n$ meaning that the set of critical points of $u_n$ that passes in $z_n$ is a curve, that we call $\gamma_n$. These curves $\gamma_n$ are closed and contained in $D$ for every $n$ large. (This follows since $u_n$ cannot have critical points on $\partial D$ and a curve of critical points cannot end inside $D$ by the maximum principle).\ We denote by $G_n$ the subset of the plane that is bounded and such that $\partial G_n=\gamma_n$. Observe that $\partial G_n=\gamma_n$ is smooth at least in a neighborhood of $z_n$ by construction. It is not possible that $G_n\subset D$. Indeed $u_n=c_n=costant$ on $\gamma_n$ while $-\Delta u_n>0$ in $G_n$ imply that $u_n(x)>c_n$ in $G_n$. Then the Hopf lemma (that we can apply at least in $z_n$) implies that the normal derivative of $u_n$ on $\gamma_n$ is negative, while $\gamma_n$ being a curve of critical points for $u_n$ forces $\nabla u_n(x)\cdot \nu=0$ for every $x\in \gamma_n$. Then, for every $n$ the set $G_n$ should contain a hole of $D$ (namely either a hole of $\Omega$ or a hole $B_\rho(P_j)$). Up to a subsequence the sets $G_n$ contain the same hole $\mathcal O\subset {\rm I\mskip -3.5mu R}^2$ (that does not depend on $n$) for every $n$. The fact that $\mathcal O\subset G_n$ implies that, in the limit as $n\to \infty$, $\gamma_n$ converges to a closed curve $\gamma$ which is the boundary of the nonempty set $G$, and which, by [\[3\]](#3){reference-type="eqref" reference="3"} is a curve of critical points for the function $K(x)$. But this is not possible by Proposition [Proposition 18](#prop-critical-points-K){reference-type="ref" reference="prop-critical-points-K"}. Then the critical points of $u_n$ are isolated for $n$ large. Finally, since $D$ is compact, then the set of critical points of $u_n$ is finite. The classification of the type of critical points for $u_n$ and formula [\[ind\]](#ind){reference-type="eqref" reference="ind"} then follows by Theorem 3.3 in [@am]. ◻ Now we are able to use the Poincarè Hopf Theorem to get: **Proposition 20**. *Let $u_n$ be a sequence of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} which blows-up at $\{P_1,\dots,P_m\}$ as $n\to \infty$. Then, when $n$ is large $$\label{prima} m+\sum_{z_j\in \mathcal C_n} index_{z_j} (\nabla u_n)=\chi(\Omega)$$ where $\mathcal C_n$ is the set of critical points of $u_n$ in $D$ and $\chi(\Omega)$ is the Euler characteristic of $\Omega$.* *Proof.* Since the critical points of $u_n$ are isolated and finite, when $n$ is large, by Proposition [Proposition 19](#prop2){reference-type="ref" reference="prop2"} we can use the Poincarè Hopf formula (see Theorem [Theorem 10](#teo-hopf){reference-type="ref" reference="teo-hopf"}), with $v=\nabla u_n$ in $\Omega$. Observe that by Hopf Lemma we have that $\nabla u_n \cdot \nu<0$. Then $$\sum_{z_j} index_{z_j}(\nabla u_n)=\chi(\Omega)$$ where the sum is due on all the critical points $z_j$ of $u_n$ in $\Omega$. Next we observe that the points $x_{i,n}$ are the unique critical points of $u_n$ in $B_\rho(P_i)$ for every $i=1,..,m$ by Lemma [Proposition 16](#lem1){reference-type="ref" reference="lem1"} and they are nondegenerate maxima so that $index_{x_{i,n}}(\nabla u_n)=1$. This gives the claim. ◻ **Corollary 21**. *Let $u_n$ be a sequence of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} which blows-up at $\{P_1,\dots,P_m\}$ as $n\to \infty$. Then the critical points of $u_n$ in $D$ converge to the critical points of $K(x)$ (see [\[kx\]](#kx){reference-type="eqref" reference="kx"}) and, for $n$ large, it holds $$\label{3.23} \begin{split} \chi(\Omega)-m&=\sum_{z_j\in \mathcal C_n} index_{z_j} (\nabla u_n) =\sum_{z_j\in \mathcal C} index_{z_j} (\nabla K) \end{split}$$ where $\mathcal C_n$ and $\mathcal C$ are the sets of critical points of $u_n$ and $K$ in the set $D$ respectively.* *Proof.* It is a consequence of the convergence [\[3\]](#3){reference-type="eqref" reference="3"} together with [\[prima\]](#prima){reference-type="eqref" reference="prima"}. ◻ Now we are in position to start the proof of Theorem [Theorem 1](#prop-general){reference-type="ref" reference="prop-general"}. Since the construction of the domain $\tilde\Omega$ is quite lengthy, we will divide the proof into two parts. In the first one we will prove the formulas [\[nc\]](#nc){reference-type="eqref" reference="nc"} and [\[nb\]](#nb){reference-type="eqref" reference="nb"} and subsequently we will construct the domain $\tilde\Omega$ and the corresponding family of solutions $u_{\lambda}$. *Proof of Theorem [Theorem 1](#prop-general){reference-type="ref" reference="prop-general"}.* **Step 1: proof of [\[nb\]](#nb){reference-type="eqref" reference="nb"} and [\[nc\]](#nc){reference-type="eqref" reference="nc"}**\ It is sufficient to prove the result for any sequence of values $\lambda_n\to 0$. Since $\Omega$ has $k$ holes then $\chi(\Omega)=1-k\leq 0$ and [\[3.23\]](#3.23){reference-type="eqref" reference="3.23"} gives that $$\sum_{z_j\in \mathcal C_n} index_{z_j} (\nabla u_n)=1-k-m\leq -1$$ so that the solutions $u_n$ have at least one nondegenerate saddle point in $D$. In the general case, since $index_{z_j}(\nabla u_n)\in \{1,0,-1\}$ we can only say that $u_n$ admits at least $m+k-1$ nondegenerate saddle points in $D$ and, recalling that we have $1$ maximum in each $B_\rho(P_j)$, $j=1,..,m$, we get the existence of at least $2m+k-1$ critical points in $\Omega$ which proves [\[nc\]](#nc){reference-type="eqref" reference="nc"}. ◻ Now we start the construction of the domain $\tilde\Omega$. Let us introduce some notations. **The dumbell domain** Let $\Omega_0:=\Omega_1\cup\dots,\cup\ \Omega_m$, where $\Omega_1,\dots, \Omega_m$ are are $m$ smooth, simply connected, bounded domains in ${\rm I\mskip -3.5mu R}^2$ such that $\Omega_i\cap \Omega_j=\emptyset$ if $i\neq j$. Assume that $\Omega_i\subset\{(x,y)\in {\rm I\mskip -3.5mu R}^2: a_i\leq x\leq b_i \},$ for some $b_i<a_{i+1}$ and $\Omega_i\cap \{y=0\}\neq 0$ for $i=1,\dots,m$. Let $V_\varepsilon:=\{(x,y)\in {\rm I\mskip -3.5mu R}^2: |y|\leq \varepsilon, x\in (a_1,b_m)\}$. Let $\Omega_\varepsilon$ be a smooth simply connected domain such that $\Omega_0\subset \Omega_\varepsilon\subset \Omega_0\cup V_\varepsilon$. We say that $\Omega_\varepsilon$ is a $m-dumbell.$\ In particular each $\Omega_j$ can be a suitable ball of the same radius centered on the $x$-axis. Example of dumbell with $m=3$ Next Lemma was basically proved in [@egp]. We repeat it for reader's convenience. **Lemma 22**. *For any $m\geq 2$ there exists an $m$-dumbell $\Omega_\varepsilon$ such that problem [\[1\]](#1){reference-type="eqref" reference="1"} has one family of solutions $u_{\lambda}$, for $\lambda$ small enough, which blow up at $m$ points $\{P_1,\dots,P_m\}$ in $\Omega_{\varepsilon}$ as $\lambda\to 0$.* *Proof.* The proof follows as in Theorem 5.5 of [@egp] observing that the $Kirkhhoff-Routh$ for the limit domain $\Omega_0$ has a strict local maximum. From this they deduce that when ${\varepsilon}$ is small enough, the $Kirkhhoff-Routh$ function $\mathcal {KR}_{\Omega_{\varepsilon}}(x_1,\dots,x_m)$ also has a strict local maximum in $(P_1,\dots,P_m)$ which is stable. By [@egp] it generates a family of solutions $u_{\lambda}$ to [\[1\]](#1){reference-type="eqref" reference="1"} that blow-up at $\{P_1,\dots,P_m\}$ as ${\lambda}\to 0$. ◻ **Remark 23**. *We can choose the domains $\Omega_j$ that are convex for every $j=1,\dots,m$. In this case by a result of Caffarelli and Friedman in [@cf], the unique critical point of the Robin Function in $\Omega_j$ is nondegenerate. This gives the nondegeneracy of the critical point of the $Kirchhoff-Routh$ function of $\Omega_{\varepsilon}$ at $\{P_1,\dots,P_m\}$. In this case the existence of a family of blowing up solutions $u_{\lambda}$ follows by [@bapa].* **Remark 24**. *The smallness of ${\varepsilon}$ is only need to have the existence of the solution $u$ to [\[1\]](#1){reference-type="eqref" reference="1"}. From now we fix such a ${\varepsilon}$ and denote $\Omega_\varepsilon$ by $\Omega$.* We are in position to construct the domain $\tilde\Omega$ and the family of solutions $u_{\lambda}$ to [\[1\]](#1){reference-type="eqref" reference="1"} when $k=0$. The final example will be obtained modifying it appropriately. **Proposition 25**. *If the dumbell is symmetric with respect to the $x$-axis then, for $\lambda$ small enough, there exists a family of solutions $u_{\lambda}$, which blow up at $m$ points $P_1,\dots,P_m$ as ${\lambda}\to 0$, symmetric with respect to the $x$-axis, with exactly $m-1$ nondegenerate saddle points and $m$ maxima in $\Omega$.* Example of dumbell with $m=3$ *Proof.* If $\Omega$ is symmetric with respect to the $x$-axis we can construct a symmetric family of solutions $u_{\lambda}$ working in the space of functions that are even with respect to the $x$-axis and following [@egp].\ By Proposition [Proposition 16](#lem1){reference-type="ref" reference="lem1"} and the evenness of $u_{\lambda}$ we have that the strict maximum points $x_{1,{\lambda}},\dots, x_{m,{\lambda}}$ are on the $x$-axis. Next, using Rolle's Theorem we get the existence of points $(z_{1,{\lambda}},0),\dots, (z_{m-1,{\lambda}},0)$ such that $\frac{\partial u_{\lambda}}{\partial x}(z_{i,{\lambda}})=0$ for $i=1,..,m-i$. Finally since $\frac{\partial u_{\lambda}}{\partial y}\big|_{y=0}=0$ we get that $z_{1,{\lambda}},\dots, z_{m-1,{\lambda}}$ are saddle points for $u_{\lambda}$ (where we are considering only the first coordinate of the points since the other is always zero). Without loss of generality assume that $x_{1,{\lambda}}<z_{1,{\lambda}}<x_{2,{\lambda}}<z_{2,{\lambda}}<\dots<x_{m,{\lambda}}$.\ Next aim is to show that **no** other critical points occur for $u_{\lambda}$ when $\lambda$ is small enough.\ Let us consider a sequence of values $\lambda_n\to 0$. By Proposition [Proposition 16](#lem1){reference-type="ref" reference="lem1"} and [\[3\]](#3){reference-type="eqref" reference="3"}, up to a subsequence, $z_{i,{\lambda}_n}:= z_{i,n}$ converge to a critical points $z_i$ of $K(x)$ in [\[kx\]](#kx){reference-type="eqref" reference="kx"} verifying $P_1<z_1<P_2<z_2<\dots<P_m$. Then the limit function $K(x)$ has at least $m-1$ critical points $\{z_1,\dots,z_{m-1}\}$ in $\Omega\setminus\{P_1,\dots,P_m\}$.\ By [\[3.23\]](#3.23){reference-type="eqref" reference="3.23"} we know that $$\sum_{z_j\in \mathcal C} index_{z_j} (\nabla K)=1-m$$ if, as before, $\mathcal C$ denotes the set of critical points of $K(x)$ in $\Omega\setminus\{P_1,\dots,P_m\}$. Since, by Proposition [Proposition 18](#prop-critical-points-K){reference-type="ref" reference="prop-critical-points-K"} $index_{z_j}(\nabla K)\leq -1$ for every $z_j\in \mathcal C$ and we have at least $m-1$ critical points, it can only happen that $\mathcal C=\{z_1,\dots,z_{m-1}\}$ and $index_{z_j}(\nabla K)= -1$ for every $j=1,\dots,m-1$. Then each $z_j$ is a nondegenerate saddle point for $K(x)$. This implies in turn that also $u_{{\lambda}_n}:= u_n$ cannot have other critical points in $\Omega$. Since the result holds for any sequence $\lambda_n\to 0$ then it holds for the family $u_{\lambda}$. ◻ Now we are in position to complete the proof of Theorem [Theorem 1](#prop-general){reference-type="ref" reference="prop-general"}. *Proof of existence of $\tilde\Omega$ and $u_{\lambda}$ of Theorem [Theorem 1](#prop-general){reference-type="ref" reference="prop-general"}.* We start considering the case $m=1$. Take a bounded smooth domain $\Omega_1$ which is symmetric with respect to the $x$-axis. (We can consider one of the previous $\Omega_i$). It is well known that, since the Robin function $\mathcal R_{\Omega_1}$ in $\Omega_1$ has a local minimum point $P_1$, there is a family of solutions to [\[1\]](#1){reference-type="eqref" reference="1"} in $\Omega_1$ that blow-up at a point $P_1$. Next we add a handle $\mathcal{C}_1$ which connects two symmetric points with respect to the $x$ axis, (see figure below). We call it a "lateral handle" and the corresponding domain has $one$ hole (namely $k=1$). As in Proposition [Proposition 25](#dumb){reference-type="ref" reference="dumb"}, choosing the lateral handle sufficiently thin, we get the existence of a family of blowing-up symmetric solutions for the new domain $\tilde\Omega$. Next the additional "lateral handle" provides the existence of an additional saddle point $z_{1,{\lambda}}$; this can be easily seen observing that $\frac{\partial u_{\lambda}}{\partial y}=0$ on $\mathcal{C}_1\cap\{x=0\}$ and $u_{\lambda}=0$ on $\partial\mathcal{C}_1\cap\{x=0\}$. Hence again Rolle's Theorem provides the existence of a critical point $z_{1,{\lambda}}$ to $u_{\lambda}$ (for every ${\lambda}$ small enough).\ Then, we get the existence of exactly $2$ critical points, all nondegenerate.\ Iterating this procedure adding $\mathcal{C}_1,\mathcal{C}_2,..,\mathcal{C}_k$ "lateral handles" which are symmetric with respect to the $x$ axis (see figure 1 in the Intoduction) we construct a domain $\tilde\Omega$ with $k$ holes that has a family of $1$-point blowing-up solutions $u_{\lambda}$ that have exactly $k+1$ critical points.\ Next, we turn to the case $m\geq 2$. Let us consider the same symmetric dumbell as in Proposition [Proposition 25](#dumb){reference-type="ref" reference="dumb"} and add a handle $\mathcal{C}_1$ which connects two symmetric points with respect to the $x$ axis, both belonging to the first component $\Omega_1$ (see figure below). As in Proposition [Proposition 25](#dumb){reference-type="ref" reference="dumb"}, choosing the lateral handle sufficiently thin, we get the existence of a sequence of blowing-up symmetric solutions for the new domain $\tilde\Omega$. Of course we again have the existence of at least $m$ strict maximum points $x_{1,{\lambda}},\dots, x_{m,{\lambda}}$ and $m-1$ saddle points $z_{1,{\lambda}},\dots, z_{m-1,{\lambda}}$ as in Proposition [Proposition 25](#dumb){reference-type="ref" reference="dumb"} and, as in the case $m=1$, each "lateral handle" provides an additional saddle point $z_{m,{\lambda}}$ to $u_{\lambda}$ As in Proposition [Proposition 25](#dumb){reference-type="ref" reference="dumb"} we get the existence of exactly $2m$ nondegenerate critical points.\ Iterating this procedure adding $\mathcal{C}_1,\mathcal{C}_2,..,\mathcal{C}_k$ "lateral handles" which are symmetric with respect to the $x$ axis we construct a domain $\tilde\Omega$ with $k$ holes that has a family of blowing-up solutions $u_{\lambda}$ that have exactly $2m+k-1$ critical points. ◻ # Proof of Theorems [Theorem 2](#T1){reference-type="ref" reference="T1"} and [Theorem 4](#ex){reference-type="ref" reference="ex"} {#S5} *Proof of Theorem [Theorem 2](#T1){reference-type="ref" reference="T1"}.* Let $\lambda_n$ be a sequence of values such that $\lambda_n\to 0$, and $u_n$ be the corresponding solution. By Proposition [Proposition 20](#prop-preuguali){reference-type="ref" reference="prop-preuguali"} we have, for $n$ large enough, that $$\sum_{z_j\in \mathcal C_n} index_{z_j} (\nabla u_n)=\chi(\Omega)-m$$ where $\mathcal C_n$ is the sets of critical points of $u_n$ in $D$. Let us consider the following cases, - If $m=1$ and $\Omega$ is simply connected ($\chi(\Omega)=1$) then $$\sum_{z_j\in \mathcal C_n} index_{z_j} (\nabla u_n)=0$$ and, by Corollary [Corollary 21](#prop-uguali){reference-type="ref" reference="prop-uguali"} this implies that $\sum_{z_j\in \mathcal C} index_{z_j} (\nabla K)=0$, which, together with the properties of the critical points of $K(x)$ in $D$ in Proposition [Proposition 19](#prop2){reference-type="ref" reference="prop2"} implies that $K(x)$ has no critical points in $D$. The $C^1$ convergence of $u_n$ to $K$ in $D$ implies that also $u_n$ has no critical points in $D$ for $n$ large enough. Then the unique critical point is the maximum $x_{1,n}$ which is nondegenerate by Lemma [Proposition 16](#lem1){reference-type="ref" reference="lem1"}. - If $m=2$ and $\Omega$ is simply connected we have that $$\sum_{z_j\in \mathcal C_n} index_{z_j} (\nabla u_n)=-1.$$ Then Proposition [Proposition 18](#prop-critical-points-K){reference-type="ref" reference="prop-critical-points-K"} implies that $K(x)$ has a unique nondegenerate critical point $x_0$ and the $C^2$ convergence of $u_n$ to $K$ implies that $u_n$ has a nondegenerate critical point $x_{0,n}\to x_0$. This gives the uniqueness and nondegeneracy of the critical point of $u_n$. - If $m=1$ and $\Omega$ has one hole we have that $\chi(\Omega)=0$ and so $$\sum_{z_j\in \mathcal C_n} index_{z_j} (\nabla u_n)=-1.$$ Arguing as in the previous step we deduce the existence of $exactly$ one nondegenerate critical point in $D$. This fact, jointly with the uniqueness of the maximum point in the ball $B_\rho(P_1)$, gives the claim. Since the result holds for every sequence $\lambda_n$ we get the claim for any ${\lambda}$ small enough. ◻ Now we give the proof of Theorem [Theorem 4](#ex){reference-type="ref" reference="ex"}. As in the Theorem [Theorem 1](#prop-general){reference-type="ref" reference="prop-general"}, the construction of the domain needs some definitions and lemmas. As in the previous section we start by considering a $contractible$ domains. **Case 1: contractible domains** Let us consider a regular polygon with $m$ sides of length $1$ and barycenter at the origin $O$. At each vertex $Q_i$ of the polygon, $i=1,..,m$ we place a ball $B_i$ of radius $r<\frac{1}{4}$ centered at $Q_i$. Then we let $\Omega_0=\cup_{i=1}^mB_i$ and, by Remark [Remark 23](#rem-convesso){reference-type="ref" reference="rem-convesso"}, the point $(Q_1,..,Q_m)$ is nondegenerate for the $Kirkhhoff-Routh$ function in $\Omega_0$. Now, we connect each component $B_i$ of $\Omega_0$ with the barycenter by $m$ straight thin tubes of thickness $\epsilon$ (see figure below for $m=3$). Finally we smooth the corners at the boundary of $B_i$ to obtain a smooth set $\Omega_\epsilon$.\ Alternatively, instead of balls we can consider $m$ copies of a convex domain. The domain $\Omega_{\varepsilon}$ is contractible for every ${\varepsilon}$ and it is invariant by the action of the the group of rotations that fix the barycenter and rotate by an angle of $\theta=\frac {2\pi}m$. As in the previous example then we have **Lemma 26**. *For ${\varepsilon}$ small enough, problem [\[1\]](#1){reference-type="eqref" reference="1"} has in $\Omega_\varepsilon$ at least one family of solutions $u_{\lambda}$ which blow up at the points $\{P_1,..,P_m\}$ as ${\lambda}\to 0$. These solutions $u_{\lambda}$ are invariant by a rotation of angle $\theta=\frac {2\pi}m$.* *Proof.* The proof follows again as in Theorem 5.5 of [@egp]. Here we set our problem in the space of functions invariant by a rotation of angle $\theta=\frac {2\pi}m$ and observe that the $Kirkhhoff-Routh$ function $\mathcal {KR}_{\Omega_{\varepsilon}}$, which is invariant by a rotation of angle $\theta=\frac {2\pi}m$, has a nondegenerate critical point in in $(P_1,..,P_m)$ (which is near $(Q_1,\dots,Q_m)$). ◻ **Remark 27**. *As in Remark [Remark 24](#remeps){reference-type="ref" reference="remeps"} we fix ${\varepsilon}$ small and set $\Omega=\Omega_{\varepsilon}$.* In next proposition we compute the number of critical points. **Proposition 28**. *For every $m\geq 3$ there exists a domain $\Omega$ such that [\[1\]](#1){reference-type="eqref" reference="1"} has at least one family of solutions $u_{\lambda}$ which blow up at $m$ points $\{P_1,\dots,P_m\}$ in $\Omega$ as $\lambda\to 0$. The solutions $u_{\lambda}$ are invariant by a rotation of angle $\theta=\frac {2\pi}m$ and have at least $2m+1$ nondegenerate critical points whose $m$ are saddle points and $m+1$ maxima (one of the maxima coincides with the barycenter $O$). Moreover as ${\lambda}\to 0$ the saddle points converge to the the barycenter $O$ which becomes a degenerate saddle of index $m-1$ for the function $K(x)$.* *Proof.* The existence of the solutions $u_{\lambda}$ is given by Lemma [Lemma 26](#lemma-6.1){reference-type="ref" reference="lemma-6.1"}. We only have to compute the number of the critical points along a sequence $\lambda_n\to 0$. By formula [\[nb\]](#nb){reference-type="eqref" reference="nb"} we get $$\label{fo} \sum_{z_j\in \mathcal C_n} index_{z_j} (\nabla u_n)=1-m$$ where we recall that $\mathcal C_n$ is the set of critical points of $u_n$ in $\Omega\setminus\cup_{i=1}^m B_{\rho}(P_i)$, for $\rho$ small enough. Using [\[ind\]](#ind){reference-type="eqref" reference="ind"} we deduce that $u_n$ has at least $m-1$ saddle points in $\Omega\setminus\cup_{i=1}^m B_{\rho}(P_i)$. Let us show that $u_n$ has at least a $m-th$ saddle points. Indeed, using the simmetry of $u_n$, if no other saddle point occurs, then the $m-1$ saddle points must coincide with the barycenter $O$. This is not possible because we shall have a critical point of index $m-1\ge2$, a contradiction with [\[ind\]](#ind){reference-type="eqref" reference="ind"}. Hence $u_n$ admits at least $m$ saddle points of index $-1$. But if no other critical point occurs, we have that $\{z_{1,n},..,z_{m,n}\}= \mathcal C_n$ and $\sum index_{z_{j,n}} (\nabla u_n)=-m$ a contradiction with [\[fo\]](#fo){reference-type="eqref" reference="fo"}. Therefore, there must be at least one additional critical point of index $1$ (a maximum) to $u_n$ which is necessarily the origin. Otherwise, by symmetry reasons, there would be other $m$ maximum points and the total degree of $\nabla u_n$ should be $0$, again a contradiction. So the maximum point is located at the origin. This proves the claim on the number of critical points. Observe that (although it seems unlikely) we cannot exclude the existence of other $m$ maxima and $m$ saddles of $u_n$. We end the proof discussing the behavior of the saddle points $\{z_{1,n},..,z_{m,n}\}$ when $n\to \infty,$ ( $\lambda_n \to0$). Passing to the limit we get $$\sum_{z_j\in \mathcal C} index_{z_j} (\nabla K)=1-m$$ where $\mathcal C$ is the set of critical points of $K(x)$ in $\Omega\setminus\cup_{i=1}^m B_{\rho}(P_i)$. However, since $K(x)$ is a harmonic function, the origin cannot be a point of maximum, but rather must be a saddle. This means that as ${\lambda}_n \to0$, the $m$ saddles $z_{j,n}$ must collapse to the barycenter, i.e. for any $j=1,..,m$, $z_{j,n} \to O$ as $n \to\infty$, and the point $O$ becomes a degenerate saddle point of index $1-m$ for $K(x)$.\ Since $K(x)$ admits a unique critical point which is the barycenter, then every critical point of $u_n$ in $\mathcal C_n$ must collapse there. ◻ **Case 2: domains with holes, a special case** Before proving the general result for a domain with $k$ holes, let us consider the special case $k=hm$ for some positive integer $h$. Although this is a particular case, there is the advantage that it is much simpler. Indeed this case can be proved straightforwardly, adding $m$ "lateral handles" to the domain constructed in Proposition [Proposition 28](#prop1){reference-type="ref" reference="prop1"}, and reasoning as in the proof of Theorem [Theorem 4](#ex){reference-type="ref" reference="ex"}. **Proposition 29**. *For every $m\geq 3$ there exists a domain $\Omega$ with $hm$ holes such problem [\[1\]](#1){reference-type="eqref" reference="1"} has at least one family of solutions $u_{\lambda}$ which blow up at $m\geq 3$ points $\{P_1,\dots,P_m\}$ in $\Omega$ as ${\lambda}\to 0$. The solutions $u_{\lambda}$ are invariant by a rotation of angle $\theta=\frac {2\pi}m$ and have at least $(h+2)m+1$ critical points whose $(h+1)m$ are nondegenerate saddle points and $m+1$ maxima (one of the maxima coincides with the barycenter $O$). Moreover as ${\lambda}\to0$, $m$ saddle points converge to the the barycenter $O$ which becomes a degenerate saddle point of index $m-1$ for the function $K(x)$.* *Proof.* In proposition [Proposition 28](#prop1){reference-type="ref" reference="prop1"} we considered solutions that are invariant under rotations by an angle $\theta=\frac {2\pi}m$. The same procedure can be applied by additionally requiring that the solutions are also invariant under reflection with respect to the line passing through the center of one of the ball $B_i$ and the barycenter $O$, since the $Kirkhhoff-Routh$ function $\mathcal {KR}_{\Omega}$ has also this symmetry.\ Next, as in the proof of Theorem [Theorem 4](#ex){reference-type="ref" reference="ex"}, we add one or more handles to the balls that make up the domain $\Omega$, as we did in the case of the dumbbell. Due to the invariance by rotation each handle creates $m$ holes, and $h$ handles produce $hm$ holes. Similarly to the dumbbell case, for the invariance under reflection, each handle adds a critical saddle point and the claim follows. ◻ **Case 3: domains with holes, the general case** Here, for every $\lambda$ small fixed, we construct a domain $\widehat \Omega$ which has $k\geq 1$ holes and such that the solution $\widehat u_{\lambda}$ has at least $2m+k+1$ nondegenerate critical points.\ The construction is a little bit delicate because involves different parameter $R,{\lambda},\delta$ which must be fixed independently. **The domain $\Omega_{\delta}$**. Let us consider the contractible domain $\Omega$ of Proposition [Proposition 28](#prop1){reference-type="ref" reference="prop1"} and let $u_{\lambda}$ be the corresponding family of solutions. We choose a point $x_0\in \Omega$ such that $x_0\neq \{O, P_1,\dots,P_m\}$. Then $x_0$ is not a critical point of $u_{\lambda}$ if ${\lambda}$ is small enough.\ We consider a small ball $B_R(x_0)\subset \Omega\setminus\{O,P_1,\dots,P_m\}$ where $R$ is such that $$\lambda^*(\Omega)\pi R^2+R\int_{\partial B_R(x_0)}\left( 12\left|\nabla 8\pi \sum_{i=1}^m G(x,P_i)\right|^2 +\lambda^*(\Omega)\left( e^{4 \sum_{i=1}^m 8\pi G(x,P_i)}+1\right)\right) d\sigma<\frac {{\varepsilon}_0}4$$ where ${\varepsilon}_0$ is as defined in [\[epsilon-zero\]](#epsilon-zero){reference-type="eqref" reference="epsilon-zero"} and $\lambda^*(\Omega)$ is the maximum value such that problem [\[1\]](#1){reference-type="eqref" reference="1"} has solutions.\ This fix the value of $R$. Now we choose $\lambda$ small such that, on $\partial B_R(x_0)$, $$u_{\lambda}(x)\leq 2 \sum_{i=1}^m 8\pi G(x,P_i)$$ and $$|\nabla u_{\lambda}(x)|\leq 2\left|\nabla \sum_{i=1}^m 8\pi G(x,P_i)\right|$$ (see [\[3\]](#3){reference-type="eqref" reference="3"}.) Choosing if necessary $\lambda$ smaller, we can assume that the solutions $u_{\lambda}$ in Proposition [Proposition 28](#prop1){reference-type="ref" reference="prop1"} are nondegenerate and have $2m+1$ nondegenerate critical point in $\Omega$ (see Theorem [Theorem 9](#T5){reference-type="ref" reference="T5"}).\ This fix the value of $\lambda$.\ Next we remove a small ball $B_\delta(x_0)\subset B_R(x_0)\subset \Omega$ (see figure below). We call this new domain $\Omega_{\delta}$. The construction of the solutions $u_\delta$ in $\Omega_\delta$ will be given in the following steps: Step 1. *Existence and nondegeneracy of the solution $u_\delta$ in $\Omega_{\delta}$.* Step 2. *$||u_\delta||_{L^\infty(\Omega)}\le C_0$ with $C$ independent of $\delta$ and $u_\delta\to u_{\lambda}$ uniformly outside of compact sets containing $B_\delta(x_0)$.* Step 3. *Uniqueness of the critical point of $u_{\delta}$ near the hole.* **Proof of Step 1.** By Dancer's results in [@da1; @da2] and the remarks therein, using the nondegeneracy of $u_{\lambda}$ we get that, for $\delta$ small enough, there exists a family of solutions $u_\delta$ in $\Omega_{\delta}$ such that $u_\delta\to u_{\lambda}$ in $L^p(\Omega)$ for every $p>1$. Moreover, again by Theorem $1$ in [@da1], the nondegeneracy of $u_{\lambda}$ implies that of $u_\delta$, again for $\delta$ small. **Proof of Step 2.** By the standard regularity theory we have that $u_\delta\to u_{\lambda}$ in any compact set outside of $B_\delta(x_0)$. Hence it is enough to prove our claim in $B_R(x_0)$. Again using the standard regularity theory we get $u_\delta\to u_{\lambda}$ in $C^1(\partial B_R(x_0))$. In particular $u_\delta\le C_R$ is uniformly bounded on $\partial B_R(x_0)$, where $C_R$ is a positive constant depending only on $R$. We can choose $\delta$ so small that, on $\partial B_R(x_0)$, $$u_\delta (x)<2 u_{\lambda}(x)<4 \sum_{i=1}^m 8\pi G(x,P_i)$$ and $$\left| \nabla u_\delta\right|\leq 2 \left| \nabla u_\lambda\right|<4 \left|\nabla \sum_{i=1}^m 8\pi G(x,P_i)\right|$$ by the previous assumptions on $\lambda$. Now we apply the Pohozaev identity to $u_\delta$ in $B_R(x_0)\setminus B_\delta(x_0)$, (see [@po]) with $F(u)=\lambda\left(e^u-1\right)$. Then we get $$\begin{split} &2\lambda \int _{B_R(x_0)\setminus B_\delta(x_0)}\left( e^{u(x)}-1\right) dx=\\ &\int_{\partial B_R(x_0)} \left(\left[(x-x_0)\cdot \nabla u_\delta\right] \frac{\partial u_\delta}{\partial \nu} -(x-x_0)\cdot \nu \frac {|\nabla u_\delta|^2}2+\lambda (x-x_0)\cdot \nu (e^{u_\delta}-1)\right) d\sigma\\ &\int_{\partial B_\delta(x_0)} \left(\left[(x-x_0)\cdot \nabla u_\delta\right] \frac{\partial u_\delta}{\partial \nu} -(x-x_0)\cdot \nu \frac {|\nabla u_\delta|^2}2\right) d\sigma \end{split}$$ where $\nu$ is the outer normal. Since $u_\delta=0$ on $\partial B_\delta(x_0)$ and $B_\delta(x_0)$ is starshaped then $$\int_{\partial B_\delta(x_0)} \left(\left[(x-x_0)\cdot \nabla u_\delta\right] \frac{\partial u_\delta}{\partial \nu} -(x-x_0)\cdot \nu \frac {|\nabla u_\delta|^2}2\right) d\sigma=\frac 12 \int_{\partial B_\delta(x_0)} (x-x_0)\cdot \nu |\nabla u_\delta|^2 d\sigma\leq 0.$$ Then the previous equality becomes $$\begin{split} &2\lambda \int _{B_R(x_0)\setminus B_\delta(x_0)}e^{u(x)}dx\leq 2\lambda |B_R(x_0)|+\\ &\int_{\partial B_R(x_0)} \left(\left[(x-x_0)\cdot \nabla u_\delta\right] \frac{\partial u_\delta}{\partial \nu} -(x-x_0)\cdot \nu \frac {|\nabla u_\delta|^2}2+\lambda (x-x_0)\cdot \nu (e^{u_\delta}-1)\right) d\sigma\\ &\leq 2\lambda^*(\Omega) \pi R^2+\int_{\partial B_R(x_0)} \left(\frac 32 |x-x_0||\nabla u_\delta|^2+\lambda^*(\Omega)|x-x_0|\left(e^{u_\delta}+1\right)\right)d\sigma\\ &\leq 2\lambda^*(\Omega) \pi R^2+ R \int_{\partial B_R(x_0)}\left(24\left| \nabla 8\pi \sum_{i=1}^m G(x,P_i)\right|^2+\lambda^*(\Omega)\left(e^{4\sum_{i=1}^m 8\pi G(x,P_i)}+1\right)\right)d\sigma \end{split}$$ so that, by the assumption on $R$ we have $$\lambda \int _{B_R(x_0)\setminus B_\delta(x_0)}e^{u(x)}dx\leq \frac {{\varepsilon}_0} 4.$$ Hence Lemma [Lemma 14](#lemma-piccolezza){reference-type="ref" reference="lemma-piccolezza"} implies that $$\| u_\delta\|_{L^{\infty}(B_R(x_0))}\leq C$$ where $C$ is independent on $\delta$ and on $\lambda$. ***Proof of Step 3 and Theorem [Theorem 4](#ex){reference-type="ref" reference="ex"}**.* Once we have the existence of a solution $u_{\delta}$ to [\[1\]](#1){reference-type="eqref" reference="1"} in $\Omega_{\delta}$ which is uniformly bounded in $L^\infty(\Omega)$ we can use Theorem [Theorem 13](#th1-1){reference-type="ref" reference="th1-1"} with $v_\delta=u_{\delta}$ and $v_0=u_{\lambda}$. So the solution $u_{\delta}$ in $\Omega_\delta$ has exactly one more saddle point of index $-1$ than $u_{\lambda}$ and this gives the claim of Theorem [Theorem 4](#ex){reference-type="ref" reference="ex"} when $k=1$ and ${\lambda}$ is fixed. We end the proof showing that the previous procedure can be iterated to handle the case $k\ge1$. We fix the previous domain $\Omega_\delta$ with one hole and remove another ball $B_{\delta_1}(x_1)\subset\Omega\setminus\{O,P_1,\dots,P_m\}$ and such that $B_{\delta_1}(x_1)\cap B_\delta(x_0)=\emptyset$. Proceding as in the case of $k=1$ we construct a solution $u_{\delta_1}$ in a domain with $2$ holes which has one more saddle point of index $-1$ with respect to $u_\delta$, and $two$ more saddle points with respect to $\bar u_{\lambda}$. The procedure can be iterated removing $k$ balls $B_{\delta_i} (x_i)$ obtaining a solution $u:=u_{\lambda}$ in a domain with $k$ holes that has $k$ saddle points more than $\bar u_{\lambda}$. Since this construction can be done for every fixed lambda, we can produce a sequence of domains $\Omega_n$ with $k\ge 1$ holes, and solutions $u_n:=u_{\lambda_n}$ such that $\lambda_n\to 0$ and $u_n$ has at least $2m+k+1$ critical points in $\Omega_n$, $m+1$ of them are local maxima and $m+k$ are saddle points. Finally, by construction, the sequence $u_n$ blows-up at $\{P_1,\dots, P_m\}$ as $n\to+\infty$. ◻ *Acknowledgments.* This work has been developed within the framework of the project e.INS- Ecosystem of Innovation for Next Generation Sardinia (cod. ECS 00000038) funded by the Italian Ministry for Research and Education (MUR) under the National Recovery and Resilience Plan (NRRP) - MISSION 4 COMPONENT 2, \"From research to business\" INVESTMENT 1.5, \"Creation and strengthening of Ecosystems of innovation\" and construction of \"Territorial R&D Leaders\". The first author is funded by Fondazione di Sardegna, Uniss, annual fund installment 2017 and 2020 and by CUP J55F21004240001. The two authors are partially funded by Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). [^1]: 2010 *Mathematics Subject classification:35B09,35B40,35Q* [^2]: *Keywords*: Critical points, multipeaks solutions,index [^3]:

arxiv_math

--- abstract: | A connected graph $G$ of diameter ${\rm diam}(G) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}|=|W_{yx}|$ for every $x,y\in V(G)$ with $d_{G}(x,y)=\ell$, where $W_{xy}$ is the set of vertices of $G$ that are closer to $x$ than to $y$. We prove that the generalized Petersen graph $GP(n,3)$ where $n>16$ is not $\ell$-distance-balanced for any $1\le \ell<{\rm diam}(GP(n,3))$, and $GP(n,4)$ where $n>24$ is not $\ell$-distance-balanced for any $1\le \ell<{\rm diam}(GP(n,4))$. This partially solves a conjecture posed by Š. Miklavič and P. Šparl (Discrete Appl. Math. 244:143--154, 2018). author: - Gang Ma$^{a}$ - "Jianfeng Wang$^{a,}$[^1]" - Sandi Klavžar$^{b,c,d}$ title: Non-$\ell$-distance-balanced generalized Petersen graphs $GP(n,3)$ and $GP(n,4)$ --- $^a$ School of Mathematics and Statistics, Shandong University of Technology\ Zibo, China\ `math\[email protected] [email protected]`\ $^b$ Faculty of Mathematics and Physics, University of Ljubljana, Slovenia\ `[email protected]`\ $^c$ Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia\ $^d$ Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia\ generalized Petersen graph; distance-balanced graph; $\ell$-distance-balanced graph **AMS Subj. Class:** 05C12 # Introduction {#S:intro} If $G = (V(G), E(G))$ is a connected graph and $x, y\in V(G)$, then the *distance* $d_{G}(x, y)$ between $x$ and $y$ is the number of edges on a shortest $x,y$-path. The diameter ${\rm diam}(G)$ of $G$ is the maximum distance between its vertices. The set $W_{xy}$ contains the vertices that are closer to $x$ than to $y$, that is, $$W_{xy}=\{w\in V(G):\ d_{G}(w,x) < d_{G}(w,y)\}\,.$$ Vertices $x$ and $y$ are *balanced* if $|W_{xy}| = |W_{yx}|$. For an integer $\ell \in [{\rm diam}(G)] = \{1,2,\ldots, {\rm diam}(G)\}$ we say that $G$ is $\ell$-*distance-balanced* if each pair of vertices $x,y\in V(G)$ with $d_{G}(x,y) = \ell$ is balanced. $G$ is said to be *highly distance-balanced* if it is $\ell$-distance-balanced for every $\ell\in [{\rm diam}(G)]$. $1$-distance-balanced graphs are simply called *distance-balanced* graphs. Distance-balanced graphs were first considered by Handa [@Handa:1999] back in 1999, while the term "distance-balanced" was proposed a decade later by Jerebic et al. in [@Jerebic:2008]. The latter paper was the trigger for intensive research of distance-balanced graphs, see [@Abiad:2016; @Balakrishnan:2014; @Balakrishnan:2009; @Cabello:2011; @cavaleri-2020; @fernardes-2022; @Ilic:2010; @Kutnar:2006; @Kutnar:2009; @Kutnar:2014; @Miklavic:2012; @YangR:2009]. The study of distance-balanced graphs is interesting from various purely graph-theoretic aspects where one focuses on particular properties of such graphs such as symmetry, connectivity or complexity aspects of algorithms related to such graphs. Moreover, distance-balanced graphs have motivated the introduction of the hitherto much-researched Mostar index [@ali-2021; @doslic-2018] and distance-unbalancedness of graphs [@kr-2021; @miklavic-2021; @xu-2022]. In this context, distance-balanced graphs are the graphs with the Mostar index equal to 0. In his dissertation [@Frelih:2014], Frelih generalized distance-balanced graphs to $\ell$-distance balanced graphs. The special case of $\ell=2$ has been studied in detail in [@Frelih:2018]. Among other results it was demonstrated that there exist $2$-distance-balanced graphs that are not $1$-distance-balanced. $2$-distance-balanced graphs that are not $2$-connected were characterized as well as $2$-distance-balanced Cartesian and lexicographic products. In this direction, $\ell$-distance-balanced corona products and lexicographic products were investigated in [@Jerebic:2021]. In [@Miklavic:2018], Miklavič and Šparl obtained some general results on $\ell$-distance balanced graphs. They studied graphs of diameter at most $3$ and investigated $\ell$-distance-balancedness of cubic graphs, in particular of generalized Petersen graphs. Although generalized Petersen graphs are a family of cubic graphs but it is difficult to determine whether they are $\ell$-distance-balanced or not for some $\ell$. And that is what has stimulated the main interest in this article. Let us define these graphs. If $n\ge 3$ and $1\le k<n/2$, then the *generalized Petersen graph* $GP(n,k)$ is defined by $$\begin{aligned} V(GP(n,k)) & = \{u_i:\ i\in \mathbb{Z}_n\} \cup\{v_i:\ i\in \mathbb{Z}_n\}, \\ E(GP(n,k)) & = \{u_iu_{i+1}:\ i\in \mathbb{Z}_n\} \cup\{v_iv_{i+k}:\ i\in \mathbb{Z}_n\} \cup \{u_iv_i:\ i\in \mathbb{Z}_n\}.\end{aligned}$$ In [@MaG:2023], the authors proved that $GP(n,k)$ is ${\rm diam}(GP(n,k))$-distance-balanced where $n$ is large relative to $k$. The following theorem was proved. **Theorem 1**. *[@MaG:2023] If $n$ and $k$ are integers, where $3\le k< n/2$ and $$n\ge\left\{\begin{array}{ll} 8; & k=3, \\ 10; & k=4, \\ \frac{k(k+1)}{2}; & k \ \text{is odd and}\ k\ge 5,\\ \frac{k^2}{2}; & k \ \text{is even and}\ k\ge 6, \end{array}\right.$$ then $GP(n,k)$ is ${\rm diam}(GP(n,k))$-distance-balanced.* In [@Miklavic:2018], the authors gave the following conjecture and proved that the conjecture was right when $k=2$. **Conjecture 2**. *[@Miklavic:2018][\[C:GP-onlyD-DB\]]{#C:GP-onlyD-DB label="C:GP-onlyD-DB"} Let $k\ge 2$ be an integer and let $$n_k=\left\{\begin{array}{ll} 11, & k=2; \\ (k+1)^2, & k \ \text{odd};\\ k(k+2), & k\ge 4 \ \text{even}. \end{array}\right.$$ Then for any $n>n_k$ the graph $GP(n,k)$ is not $\ell$-distance-balanced for any $1\le \ell<D$, where $D$ is the diameter of $GP(n,k)$. Moreover, $n_k$ is the smallest integer with this property.* In this paper, we study Conjecture [\[C:GP-onlyD-DB\]](#C:GP-onlyD-DB){reference-type="ref" reference="C:GP-onlyD-DB"} and prove that Conjecture [\[C:GP-onlyD-DB\]](#C:GP-onlyD-DB){reference-type="ref" reference="C:GP-onlyD-DB"} is right when $k=3,4$. The following two theorems are the main results of the paper. **Theorem 3**. *For any $n>16$, the generalized Petersen graph $GP(n,3)$ is not $\ell$-distance-balanced for any $1\le \ell<{\rm diam}(GP(n,3))$. Moreover, $16$ is the smallest integer with this property.* **Theorem 4**. *For any $n>24$, the generalized Petersen graph $GP(n,4)$ is not $\ell$-distance-balanced for any $1\le \ell<{\rm diam}(GP(n,4))$. Moreover, $24$ is the smallest integer with this property.* In section [2](#S:GP(n,3)){reference-type="ref" reference="S:GP(n,3)"}, we prove Theorem [Theorem 3](#T:GP(n,3)-onlyD-DB){reference-type="ref" reference="T:GP(n,3)-onlyD-DB"}. In section [3](#S:GP(n,4)){reference-type="ref" reference="S:GP(n,4)"}, we prove Theorem [Theorem 4](#T:GP(n,4)-onlyD-DB){reference-type="ref" reference="T:GP(n,4)-onlyD-DB"}. In section [4](#S:conluding){reference-type="ref" reference="S:conluding"}, we give one problem which is worth studying in the future. # The proof of Theorem [Theorem 3](#T:GP(n,3)-onlyD-DB){reference-type="ref" reference="T:GP(n,3)-onlyD-DB"} {#S:GP(n,3)} From [@Miklavic:2018], the diameter of $GP(16,3)$ is $6$ and $GP(16,3)$ is $\ell$-distance-balanced if and only if $\ell\in\{5,6\}$. So we can suppose $n> 16$ when we prove Theorem [Theorem 3](#T:GP(n,3)-onlyD-DB){reference-type="ref" reference="T:GP(n,3)-onlyD-DB"}. We will prove Theorem [Theorem 3](#T:GP(n,3)-onlyD-DB){reference-type="ref" reference="T:GP(n,3)-onlyD-DB"} via Proposition [Proposition 5](#P:GP(n,3)-1){reference-type="ref" reference="P:GP(n,3)-1"}, [Proposition 6](#P:GP(n,3)-2){reference-type="ref" reference="P:GP(n,3)-2"} and [Proposition 7](#P:GP(n,3)-3){reference-type="ref" reference="P:GP(n,3)-3"}. **Proposition 5**. *For any $n>16$, the generalized Petersen graph $GP(n,3)$ is not $1$-distance-balanced.* *Proof.* In $GP(n,3)$, $d(u_0,v_0)=1$ and we will prove that $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. We divide the discussion into the following six cases according to the size of $n$. \(1\) When $n=6m$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{3t})=1+t$ and $d(v_0,v_{3t})=t$ where $1\le t\le m$. $d(u_0,v_{3t+1})=2+t$ and $d(v_0,v_{3t+1})=3+t$ where $0\le t< m$. $d(u_0,v_{3t+2})=3+t$ and $d(v_0,v_{3t+2})=4+t$ where $0\le t< m$. $d(u_0,u_{3t})=2+t$ and $d(v_0,u_{3t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_0,u_{3t+1})=2+t$ where $1\le t< m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{3t+2})=4+t$ and $d(v_0,u_{3t+2})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(2m+2)+1=4m+5$ and $|W_{v_{0}u_0}|=2(4m-4)+3=8m-5$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(2\) When $n=6m+1$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{3t})=1+t$ and $d(v_0,v_{3t})=t$ where $1\le t\le m$. $d(u_0,v_{3t+1})=2+t$ where $0\le t< m$. $d(v_0,v_{3t+1})=3+t$ where $0\le t\le m-2$, and $d(v_0,v_{3(m-1)+1})=m+1$. $d(u_0,v_{3t+2})=3+t$ and $d(v_0,v_{3t+2})=4+t$ where $0\le t< m$. $d(u_0,u_{3t})=2+t$ and $d(v_0,u_{3t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_0,u_{3t+1})=2+t$ where $1\le t< m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{3t+2})=4+t$ and $d(v_0,u_{3t+2})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(2m+1)+1=4m+3$ and $|W_{v_{0}u_0}|=2(4m-2)+1=8m-3$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(3\) When $n=6m+2$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{3t})=1+t$ and $d(v_0,v_{3t})=t$ where $1\le t\le m$. $d(u_0,v_{3t+1})=2+t$ and $d(v_0,v_{3t+1})=3+t$ where $0\le t\le m$. $d(u_0,v_{3t+2})=3+t$ where $0\le t< m$. $d(v_0,v_{3t+2})=4+t$ where $0\le t\le m-2$, and $d(v_0,v_{3(m-1)+2})=m+1$. $d(u_0,u_{3t})=2+t$ and $d(v_0,u_{3t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_0,u_{3t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{3t+2})=4+t$ and $d(v_0,u_{3t+2})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(2m+1)+2=4m+4$ and $|W_{v_{0}u_0}|=2(4m-1)+2=8m$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(4\) When $n=6m+3$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{3t})=1+t$ and $d(v_0,v_{3t})=t$ where $1\le t\le m$. $d(u_0,v_{3t+1})=2+t$ and $d(v_0,v_{3t+1})=3+t$ where $0\le t\le m$. $d(u_0,v_{3t+2})=3+t$ and $d(v_0,v_{3t+2})=4+t$ where $0\le t< m$. $d(u_0,u_{3t})=2+t$ and $d(v_0,u_{3t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_0,u_{3t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{3t+2})=4+t$ and $d(v_0,u_{3t+2})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(2m+3)+1=4m+7$ and $|W_{v_{0}u_0}|=2(4m-1)+1=8m-1$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(5\) When $n=6m+4$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{3t})=1+t$ and $d(v_0,v_{3t})=t$ where $1\le t\le m$. $d(u_0,v_{3t+1})=2+t$ where $0\le t\le m$. $d(v_0,v_{3t+1})=3+t$ where $0\le t\le m-1$, and $d(v_0,v_{3m+1})=m+1$. $d(u_0,v_{3t+2})=3+t$ and $d(v_0,v_{3t+2})=4+t$ where $0\le t\le m$. $d(u_0,u_{3t})=2+t$ and $d(v_0,u_{3t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_0,u_{3t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{3t+2})=4+t$ and $d(v_0,u_{3t+2})=3+t$ where $1\le t\le m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(2m+2)+2=4m+6$ and $|W_{v_{0}u_0}|=2\times 4m+2=8m+2$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(6\) When $n=6m+5$ where $m\ge 2$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{3t})=1+t$ and $d(v_0,v_{3t})=t$ where $1\le t\le m$. $d(u_0,v_{3t+1})=2+t$ and $d(v_0,v_{3t+1})=3+t$ where $0\le t\le m$. $d(u_0,v_{3t+2})=3+t$ where $0\le t\le m-1$ and $d(u_0,v_{3m+2})=m+2$. $d(v_0,v_{3t+2})=4+t$ where $0\le t\le m-2$, $d(v_0,v_{3(m-1)+2})=m+2$ and $d(v_0,v_{3m+2})=m+1$. $d(u_0,u_{3t})=2+t$ and $d(v_0,u_{3t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_0,u_{3t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{3t+2})=4+t$ and $d(v_0,u_{3t+2})=3+t$ where $1\le t\le m-1$. $d(u_0,u_{3m+2})=m+3$ and $d(v_0,u_{3m+2})=m+2$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(2m+2)+1=4m+5$ and $|W_{v_{0}u_0}|=2(4m+1)+1=8m+3$. Because $m\ge 2$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. ◻ **Proposition 6**. *For any $n>16$, the generalized Petersen graph $GP(n,3)$ is not $2$-distance-balanced.* *Proof.* In $GP(n,3)$, $d(u_0,v_{-3})=2$ and we will prove that $|W_{u_0v_{-3}}|<|W_{v_{-3}u_0}|$. Note that $v_{-3}=v_{n-3}$. Firstly we consider vertices $v_{-1},v_{-2},u_{-1},u_{-2}$. $d(u_0,v_{-1})=2$ and $d(v_{-3},v_{-1})=4$. $d(u_0,v_{-2})=d(v_{-3},v_{-2})=3$. $d(u_0,u_{-1})=1$ and $d(v_{-3},u_{-1})=3$. $d(u_0,u_{-2})=d(v_{-3},u_{-2})=2$. So $u_{-1},v_{-1}\in W_{u_0v_{-3}}$ and no vertex of $\{v_{-1},v_{-2},u_{-1},u_{-2}\}$ is in $W_{v_{-3}u_0}$. Next we consider vertices $v_i$ where $0\le i<n-3$ and $u_j$ where $1\le j\le n-3$. We divide the discussion into the following six cases according to the size of $n$. \(1\) When $n=6m$ where $m\ge 3$. Note that $n-3=6m-3=3(2m-1)$. $d(u_0,v_{3t})=d(v_{6m-3},v_{3t})=1+t$ when $0\le t\le m-1$. $d(v_{6m-3},v_{3t})=2m-1-t$ and $d(u_0,v_{3t})>d(v_{6m-3},v_{3t})$ when $m\le t<2m-1$. $d(u_0,v_{3t+1})=2+t$ and $d(u_0,v_{3t+1})<d(v_{6m-3},v_{3t+1})$ when $0\le t\le m-1$. $d(u_0,v_{3t+1})=d(v_{6m-3},v_{3t+1})=2m-t+2$ when $m\le t< 2m-1$. $d(u_0,v_{3t+2})=3+t$ and $d(u_0,v_{3t+2})<d(v_{6m-3},v_{3t+2})$ when $0\le t\le m-2$. $d(u_0,v_{3t+2})=d(v_{6m-3},v_{3t+2})=2m-t+1$ when $m-1\le t< 2m-1$. $d(u_0,u_{3t})=d(v_{6m-3},u_{3t})=2+t$ when $1\le t\le m-1$. $d(v_{6m-3},u_{3t})=2m-t$ and $d(u_0,u_{3t})>d(v_{6m-3},u_{3t})$ when $m\le t\le 2m-1$. $d(u_0,u_{1})=1$ and $d(v_{6m-3},u_{1})=2m+1$. $d(u_0,u_{3t+1})=d(v_{6m-3},u_{3t+1})=3+t$ when $1\le t\le m-1$. $d(v_{6m-3},u_{3t+1})=2m-t+1$ and $d(u_0,u_{3t+1})>d(v_{6m-3},u_{3t+1})$ when $m\le t< 2m-1$. $d(u_0,u_{2})=2$ and $d(v_{6m-3},u_{2})=2m$. $d(u_0,u_{3t+2})=d(v_{6m-3},u_{3t+2})=4+t$ when $1\le t\le m-2$. $d(v_{6m-3},u_{3t+2})=2m-t$ and $d(u_0,u_{3t+2})>d(v_{6m-3},u_{3t+2})$ when $m-1\le t< 2m-1$. Note that $u_0\in W_{u_0v_{6m-3}}$ and $v_{6m-3}\in W_{v_{6m-3}u_0}$. Combined with the above discussion, $|W_{u_0v_{6m-3}}|=2m+4$ and $|W_{v_{6m-3}u_0}|=4m-1$. Because $m\ge 3$, $|W_{u_0v_{6m-3}}|<|W_{v_{6m-3}u_0}|$. \(2\) When $n=6m+1$ where $m\ge 3$. Note that $n-3=6m-2=3(2m-1)+1$. $d(u_0,v_{3t})=d(v_{6m-2},v_{3t})=1+t$ when $0\le t\le m$. $d(u_0,v_{3t})=d(v_{6m-2},v_{3t})=2m-t+2$ when $m+1\le t\le 2m-1$. $d(u_0,v_{3t+1})=2+t$ and $d(u_0,v_{3t+1})<d(v_{6m-2},v_{3t+1})$ when $0\le t\le m-2$. $d(v_{6m-2},v_{3t+1})=2m-t-1$ and $d(u_0,v_{3t+1})>d(v_{6m-2},v_{3t+1})$ when $m-1\le t< 2m-1$. $d(u_0,v_{3t+2})=3+t$ and $d(u_0,v_{3t+2})<d(v_{6m-2},v_{3t+2})$ when $0\le t\le m-1$. $d(u_0,v_{3t+2})=d(v_{6m-2},v_{3t+2})=2m-t+2$ when $m\le t< 2m-1$. $d(u_0,u_{3t})=d(v_{6m-2},u_{3t})=2+t$ when $1\le t\le m-1$. $d(v_{6m-2},u_{3t})=2m-t+1$ and $d(u_0,u_{3t})>d(v_{6m-2},u_{3t})$ when $m\le t\le 2m-1$. $d(u_0,u_{1})=1$ and $d(v_{6m-2},u_{1})=2m$. $d(u_0,u_{3t+1})=d(v_{6m-2},u_{3t+1})=3+t$ when $1\le t\le m-2$. $d(v_{6m-2},u_{3t+1})=2m-t$ and $d(u_0,u_{3t+1})>d(v_{6m-2},u_{3t+1})$ when $m-1\le t\le 2m-1$. $d(u_0,u_{2})=2$ and $d(v_{6m-2},u_{2})=2m+1$. $d(u_0,u_{3t+2})=d(v_{6m-2},u_{3t+2})=4+t$ when $1\le t\le m-2$. $d(v_{6m-2},u_{3t+2})=2m-t+1$ and $d(u_0,u_{3t+2})>d(v_{6m-2},u_{3t+2})$ when $m-1\le t< 2m-1$. Note that $u_0\in W_{u_0v_{6m-2}}$ and $v_{6m-2}\in W_{v_{6m-2}u_0}$. Combined with the above discussion, $|W_{u_0v_{6m-2}}|=2m+4$ and $|W_{v_{6m-2}u_0}|=4m+2$. Because $m\ge 3$, $|W_{u_0v_{6m-2}}|<|W_{v_{6m-2}u_0}|$. \(3\) When $n=6m+2$ where $m\ge 3$. Note that $n-3=6m-1=3(2m-1)+2$. $d(u_0,v_{3t})=d(v_{6m-1},v_{3t})=1+t$ when $0\le t\le m+1$. $d(u_0,v_{3t})=d(v_{6m-1},v_{3t})=2m-t+3$ when $m+2\le t\le 2m-1$. $d(u_0,v_{3t+1})=2+t$ and $d(u_0,v_{3t+1})<d(v_{6m-1},v_{3t+1})$ when $0\le t\le m-1$. $d(u_0,v_{3t+1})=d(v_{6m-1},v_{3t+1})=2m-t+2$ when $m\le t\le 2m-1$. $d(u_0,v_{3t+2})=3+t$ and $d(u_0,v_{3t+2})<d(v_{6m-1},v_{3t+2})$ when $0\le t\le m-3$. $d(u_0,v_{3(m-2)+2})=d(v_{6m-1},v_{3(m-2)+2})=m+1$. $d(v_{6m-1},v_{3t+2})=2m-t-1$ and $d(u_0,v_{3t+2})>d(v_{6m-1},v_{3t+2})$ when $m-1\le t< 2m-1$. $d(u_0,u_{3t})=d(v_{6m-1},u_{3t})=2+t$ when $1\le t\le m$. $d(v_{6m-1},u_{3t})=2m-t+2$ and $d(u_0,u_{3t})>d(v_{6m-1},u_{3t})$ when $m+1\le t\le 2m-1$. $d(u_0,u_{1})=1$ and $d(v_{6m-1},u_{1})=2m+1$. $d(u_0,u_{3t+1})=d(v_{6m-1},u_{3t+1})=3+t$ when $1\le t\le m-1$. $d(v_{6m-1},u_{3t+1})=2m-t+1$ and $d(u_0,u_{3t+1})>d(v_{6m-1},u_{3t+1})$ when $m\le t\le 2m-1$. $d(u_0,u_{2})=2$ and $d(v_{6m-1},u_{2})=2m$. $d(u_0,u_{3t+2})=d(v_{6m-1},u_{3t+2})=4+t$ when $1\le t\le m-2$. $d(v_{6m-1},u_{3t+2})=2m-t$ and $d(u_0,u_{3t+2})>d(v_{6m-1},u_{3t+2})$ when $m-1\le t\le 2m-1$. Note that $u_0\in W_{u_0v_{6m-1}}$ and $v_{6m-1}\in W_{v_{6m-1}u_0}$. Combined with the above discussion, $|W_{u_0v_{6m-1}}|=2m+3$ and $|W_{v_{6m-1}u_0}|=4m+1$. Because $m\ge 3$, $|W_{u_0v_{6m-1}}|<|W_{v_{6m-1}u_0}|$. \(4\) When $n=6m+3$ where $m\ge 3$. Note that $n-3=6m=3\times 2m$. $d(u_0,v_{3t})=d(v_{6m},v_{3t})=1+t$ when $0\le t\le m-1$. $d(v_{6m},v_{3t})=2m-t$ and $d(u_0,v_{3t})>d(v_{6m},v_{3t})$ when $m\le t< 2m$. $d(u_0,v_{3t+1})=2+t$ and $d(u_0,v_{3t+1})<d(v_{6m},v_{3t+1})$ when $0\le t\le m$. $d(u_0,v_{3t+1})=d(v_{6m},v_{3t+1})=2m-t+3$ when $m+1\le t< 2m$. $d(u_0,v_{3t+2})=3+t$ and $d(u_0,v_{3t+2})<d(v_{6m},v_{3t+2})$ when $0\le t\le m-1$. $d(u_0,v_{3t+2})=d(v_{6m},v_{3t+2})=2m-t+2$ when $m\le t< 2m$. $d(u_0,u_{3t})=d(v_{6m},u_{3t})=2+t$ when $1\le t\le m-1$. $d(v_{6m},u_{3t})=2m-t+1$ and $d(u_0,u_{3t})>d(v_{6m},u_{3t})$ when $m\le t\le 2m$. $d(u_0,u_{1})=1$ and $d(v_{6m},u_{1})=2m+2$. $d(u_0,u_{3t+1})=d(v_{6m},u_{3t+1})=3+t$ when $1\le t\le m-1$. $d(v_{6m},u_{3t+1})=2m-t+2$ and $d(u_0,u_{3t+1})>d(v_{6m},u_{3t+1})$ when $m\le t< 2m$. $d(u_0,u_{2})=2$ and $d(v_{6m},u_{2})=2m+1$. $d(u_0,u_{3t+2})=d(v_{6m},u_{3t+2})=4+t$ when $1\le t\le m-2$. $d(v_{6m},u_{3t+2})=2m-t+1$ and $d(u_0,u_{3t+2})>d(v_{6m},u_{3t+2})$ when $m-1\le t< 2m$. Note that $u_0\in W_{u_0v_{6m}}$ and $v_{6m}\in W_{v_{6m}u_0}$. Combined with the above discussion, $|W_{u_0v_{6m}}|=2m+6$ and $|W_{v_{6m}u_0}|=4m+3$. Because $m\ge 3$, $|W_{u_0v_{6m}}|<|W_{v_{6m}u_0}|$. \(5\) When $n=6m+4$ where $m\ge 3$. Note that $n-3=6m+1=3\times 2m+1$. $d(u_0,v_{3t})=d(v_{6m+1},v_{3t})=1+t$ when $0\le t\le m+1$. $d(u_0,v_{3t})=d(v_{6m+1},v_{3t})=2m-t+3$ when $m+2\le t\le 2m$. $d(u_0,v_{3t+1})=2+t$ and $d(u_0,v_{3t+1})<d(v_{6m+1},v_{3t+1})$ when $0\le t\le m-2$. $d(u_0,v_{3(m-1)+1})=d(v_{6m+1},v_{3(m-1)+1})=m+1$. $d(v_{6m+1},v_{3t+1})=2m-t$ and $d(u_0,v_{3t+1})>d(v_{6m+1},v_{3t+1})$ when $m\le t< 2m$. $d(u_0,v_{3t+2})=3+t$ and $d(u_0,v_{3t+2})<d(v_{6m+1},v_{3t+2})$ when $0\le t\le m-1$. $d(u_0,v_{3t+2})=d(v_{6m+1},v_{3t+2})=2m-t+3$ when $m\le t< 2m$. $d(u_0,u_{3t})=d(v_{6m+1},u_{3t})=2+t$ when $1\le t\le m$. $d(v_{6m+1},u_{3t})=2m-t+2$ and $d(u_0,u_{3t})>d(v_{6m+1},u_{3t})$ when $m+1\le t\le 2m$. $d(u_0,u_{1})=1$ and $d(v_{6m+1},u_{1})=2m+1$. $d(u_0,u_{3t+1})=d(v_{6m+1},u_{3t+1})=3+t$ when $1\le t\le m-1$. $d(v_{6m+1},u_{3t+1})=2m-t+1$ and $d(u_0,u_{3t+1})>d(v_{6m+1},u_{3t+1})$ when $m\le t\le 2m$. $d(u_0,u_{2})=2$ and $d(v_{6m},u_{2})=2m+2$. $d(u_0,u_{3t+2})=d(v_{6m+1},u_{3t+2})=4+t$ when $1\le t\le m-1$. $d(v_{6m+1},u_{3t+2})=2m-t+2$ and $d(u_0,u_{3t+2})>d(v_{6m+1},u_{3t+2})$ when $m\le t< 2m$. Note that $u_0\in W_{u_0v_{6m+1}}$ and $v_{6m+1}\in W_{v_{6m+1}u_0}$. Combined with the above discussion, $|W_{u_0v_{6m+1}}|=2m+4$ and $|W_{v_{6m+1}u_0}|=4m+2$. Because $m\ge 3$, $|W_{u_0v_{6m+1}}|<|W_{v_{6m+1}u_0}|$. \(6\) When $n=6m+5$ where $m\ge 2$. Note that $n-3=6m+2=3\times 2m+2$. $d(u_0,v_{3t})=d(v_{6m+2},v_{3t})=1+t$ when $0\le t\le m+1$. $d(u_0,v_{3t})=d(v_{6m+2},v_{3t})=2m-t+4$ when $m+2\le t\le 2m$. $d(u_0,v_{3t+1})=2+t$ and $d(u_0,v_{3t+1})<d(v_{6m+2},v_{3t+1})$ when $0\le t\le m$. $d(u_0,v_{3t+1})=d(v_{6m+2},v_{3t+1})=2m-t+3$ when $m+1\le t\le 2m$. $d(u_0,v_{3t+2})=3+t$ and $d(u_0,v_{3t+2})<d(v_{6m+2},v_{3t+2})$ when $0\le t\le m-2$. $d(v_{6m+2},v_{3t+2})=2m-t$ and $d(u_0,v_{3t+2})>d(v_{6m+1},v_{3t+2})$ when $m-1\le t< 2m$. $d(u_0,u_{3t})=d(v_{6m+2},u_{3t})=2+t$ when $1\le t\le m$. $d(v_{6m+2},u_{3t})=2m-t+3$ and $d(u_0,u_{3t})>d(v_{6m+2},u_{3t})$ when $m+1\le t\le 2m$. $d(u_0,u_{1})=1$ and $d(v_{6m+2},u_{1})=2m+2$. $d(u_0,u_{3t+1})=d(v_{6m+2},u_{3t+1})=3+t$ when $1\le t\le m-1$. $d(v_{6m+2},u_{3t+1})=2m-t+2$ and $d(u_0,u_{3t+1})>d(v_{6m+2},u_{3t+1})$ when $m\le t\le 2m$. $d(u_0,u_{2})=2$ and $d(v_{6m},u_{2})=2m+1$. $d(u_0,u_{3t+2})=d(v_{6m+2},u_{3t+2})=4+t$ when $1\le t\le m-2$. $d(v_{6m+2},u_{3t+2})=2m-t+1$ and $d(u_0,u_{3t+2})>d(v_{6m+2},u_{3t+2})$ when $m-1\le t\le 2m$. Note that $u_0\in W_{u_0v_{6m+2}}$ and $v_{6m+2}\in W_{v_{6m+2}u_0}$. Combined with the above discussion, $|W_{u_0v_{6m+2}}|=2m+5$ and $|W_{v_{6m+2}u_0}|=4m+5$. Because $m\ge 2$, $|W_{u_0v_{6m+2}}|<|W_{v_{6m+2}u_0}|$. ◻ **Proposition 7**. *For any $n>16$, the generalized Petersen graph $GP(n,3)$ is not $\ell$-distance-balanced for any $3\le \ell<{\rm diam}(GP(n,3))$.* *Proof.* For any $3\le\ell<D$, we first show that there exists $v_j$ such that $d(u_0,v_j)=\ell$ where $6\le j\le n/2$. From [@MaG:2023], there exists $j^*$ such that $d(u_0,u_{j^*})=D$. When $n=6m$ $(m\ge 3)$ or $n=6m+1$ $(m\ge 3)$, from [@MaG:2023] we know that $j^*=3(m-1)+2$ and $D=d(u_0,u_{j^*})=m+3$. Note that $d(u_0,v_{3s+2})=s+3$ where $2\le s\le m-1$ and $d(u_0,v_{3s})=s+1$ where $2\le s\le m$. When $n=6m+2$ $(m\ge 3)$ or $n=6m+3$ $(m\ge 3)$, from [@MaG:2023] we know that $j^*=3m+1$ and $D=d(u_0,u_{j^*})=m+3$. Note that $d(u_0,v_{3s+1})=s+2$ where $2\le s\le m$ and $d(u_0,v_{3s})=s+1$ where $2\le s\le m$. When $n=6m+4$ $(m\ge 3)$, from [@MaG:2023] we know that $j^*=3m+2$ and $D=d(u_0,u_{j^*})=m+4$. Note that $d(u_0,v_{3s+2})=s+3$ where $2\le s\le m$ and $d(u_0,v_{3s})=s+1$ where $2\le s\le m$. When $n=6m+5$ $(m\ge 2)$, from [@MaG:2023] we know that $j^*=3m+1$ and $D=d(u_0,u_{j^*})=m+3$. Note that $d(u_0,v_{3s+1})=s+2$ where $2\le s\le m$ and $d(u_0,v_{3s})=s+1$ where $2\le s\le m$. From the above discussion, there exists $j$ where $6\le j\le n/2$ such that $d(u_0,v_j)=\ell$ for any $3\le\ell<D$. Let $V_1=\{u_i\mid 1\le i\le j-1\}\cup\{v_i\mid 1\le i\le j-1\}$ and $V_2=\{u_i\mid j+1\le i\le n-1\}\cup\{v_i\mid j+1\le i\le n-1\}$. Let $W^1_{u_0v_j}$ be the set of vertices which are in $W_{u_0v_j}$ and also in $V_1\cup\{u_0,v_0,u_j,v_j\}$. Let $W^1_{v_ju_0}$ be the set of vertices which are in $W_{v_ju_0}$ and also in $V_1\cup\{u_0,v_0,u_j,v_j\}$. Let $W^2_{u_0v_j}$ be the set of vertices which are in $W_{u_0v_j}$ and also in $V_2\cup\{u_0,v_0,u_j,v_j\}$. Let $W^2_{v_ju_0}$ be the set of vertices which are in $W_{v_ju_0}$ and also in $V_2\cup\{u_0,v_0,u_j,v_j\}$. Because $6\le j\le n/2$, $|W^2_{u_0v_j}|=|W^1_{u_0v_{n-j}}|$ and $|W^2_{v_ju_0}|=|W^1_{v_{n-j}u_0}|$. So $|W_{u_0v_j}|=|W^1_{u_0v_j}|+|W^2_{u_0v_j}|-2=|W^1_{u_0v_j}|+|W^1_{u_0v_{n-j}}|-2$ and $|W_{v_ju_0}|=|W^1_{v_ju_0}|+|W^2_{v_ju_0}|-2=|W^1_{v_ju_0}|+|W^1_{v_{n-j}u_0}|-2$. In the following we will compute $|W^1_{u_0v_j}|$ and $|W^1_{v_ju_0}|$ where $6\le j\le n-6$. The discussion is divided into the following six cases. (1a) Computation of $|W^1_{u_0v_{3s}}|$ and $|W^1_{v_{3s}u_0}|$ when $s$ is odd and $s\ge 3$. When $s=3$, $d(u_0,v_0)=1$ and $d(v_9,v_0)=3$. $d(u_0,v_3)=d(v_9,v_3)=2$. $d(u_0,v_6)=3$ and $d(v_9,v_6)=1$. $d(u_0,v_1)=2$ and $d(v_9,v_1)=6$. $d(u_0,v_4)=3$ and $d(v_9,v_4)=5$. $d(u_0,v_7)=4$ and $d(v_9,v_7)=4$. $d(u_0,v_2)=3$ and $d(v_9,v_2)=5$. $d(u_0,v_5)=4$ and $d(v_9,v_5)=4$. $d(u_0,v_8)=5$ and $d(v_9,v_8)=3$. So $v_0,v_1,v_4,v_2\in W^1_{u_0v_{9}}$ and $v_6,v_8\in W^1_{v_{9}u_0}$. $d(u_0,u_3)=3$ and $d(v_9,u_3)=3$. $d(u_0,u_6)=4$ and $d(v_9,u_6)=2$. $d(u_0,u_9)=5$ and $d(v_9,u_9)=1$. $d(u_0,u_1)=1$ and $d(v_9,u_1)=5$. $d(u_0,u_4)=4$ and $d(v_9,u_4)=4$. $d(u_0,u_7)=5$ and $d(v_9,u_7)=3$. $d(u_0,u_2)=2$ and $d(v_9,u_2)=4$. $d(u_0,u_5)=5$ and $d(v_9,u_5)=3$. $d(u_0,u_8)=6$ and $d(v_9,u_8)=2$. So $u_1,u_2\in W^1_{u_0v_{9}}$ and $u_6,u_9,u_7,u_5,u_8\in W^1_{v_{9}u_0}$. Note that $u_0\in W^1_{u_0v_{9}}$ and $v_{9}\in W^1_{v_{9}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{9}}|=7$ and $|W^1_{v_{9}u_0}|=8$. When $s\ge 5$, $d(u_0,v_{3t})=1+t$ and $d(v_{3s},v_{3t})=s-t$ where $0\le t<s$. When $0\le t<\frac{s-1}{2}$, $d(u_0,v_{3t})<d(v_{3s},v_{3t})$. When $\frac{s-1}{2}<t<s$, $d(u_0,v_{3t})>d(v_{3s},v_{3t})$. $d(u_0,v_{3t+1})=2+t$ and $d(v_{3s},v_{3t+1})=s-t+3$ where $0\le t<s$. When $0\le t<\frac{s+1}{2}$, $d(u_0,v_{3t+1})<d(v_{3s},v_{3t+1})$. When $\frac{s+1}{2}<t<s$, $d(u_0,v_{3t+1})>d(v_{3s},v_{3t+1})$. $d(u_0,v_{3t+2})=3+t$ and $d(v_{3s},v_{3t+2})=s-t+2$ where $0\le t<s$. When $0\le t<\frac{s-1}{2}$, $d(u_0,v_{3t+2})<d(v_{3s},v_{3t+2})$. When $\frac{s-1}{2}<t<s$, $d(u_0,v_{3t+2})>d(v_{3s},v_{3t+2})$. $d(u_0,u_{3t})=2+t$ and $d(v_{3s},u_{3t})=s-t+1$ where $1\le t\le s$. When $1\le t<\frac{s-1}{2}$, $d(u_0,u_{3t})<d(v_{3s},u_{3t})$. When $\frac{s-1}{2}<t\le s$, $d(u_0,u_{3t})>d(v_{3s},u_{3t})$. $d(u_0,u_{1})=1$ and $d(v_{3s},u_{1})=s+2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_{3s},u_{3t+1})=s-t+2$ where $1\le t< s$. When $1\le t<\frac{s-1}{2}$, $d(u_0,u_{3t+1})<d(v_{3s},u_{3t+1})$. When $\frac{s-1}{2}<t< s$, $d(u_0,u_{3t+1})>d(v_{3s},u_{3t+1})$. $d(u_0,u_{2})=2$ and $d(v_{3s},u_{2})=s+1$. $d(u_0,u_{3t+2})=4+t$ and $d(v_{3s},u_{3t+2})=s-t+1$ where $1\le t< s$. When $1\le t<\frac{s-3}{2}$, $d(u_0,u_{3t+2})<d(v_{3s},u_{3t+2})$. When $\frac{s-3}{2}<t< s$, $d(u_0,u_{3t+2})>d(v_{3s},u_{3t+2})$. Note that $u_0\in W^1_{u_0v_{3s}}$ and $v_{3s}\in W^1_{v_{3s}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{3s}}|=3s-3$ and $|W^1_{v_{3s}u_0}|=3s-1$. (1b) Computation of $|W^1_{u_0v_{3s}}|$ and $|W^1_{v_{3s}u_0}|$ when $s$ is even and $s\ge 2$. When $s=2$, $d(u_0,v_0)=1$ and $d(v_6,v_0)=2$. $d(u_0,v_3)=2$ and $d(v_6,v_3)=1$. $d(u_0,v_1)=2$ and $d(v_6,v_1)=5$. $d(u_0,v_4)=3$ and $d(v_6,v_4)=4$. $d(u_0,v_2)=3$ and $d(v_6,v_2)=4$. $d(u_0,v_5)=4$ and $d(v_6,v_5)=3$. So $v_0,v_1,v_2,v_4\in W^1_{u_0v_{6}}$ and $v_3,v_5\in W^1_{v_{6}u_0}$. $d(u_0,u_3)=3$ and $d(v_6,u_3)=2$. $d(u_0,u_6)=4$ and $d(v_6,u_6)=1$. $d(u_0,u_1)=1$ and $d(v_6,u_1)=4$. $d(u_0,u_4)=4$ and $d(v_6,u_4)=3$. $d(u_0,u_2)=2$ and $d(v_6,u_2)=3$. $d(u_0,u_5)=5$ and $d(v_6,u_5)=2$. So $u_1,u_2\in W^1_{u_0v_{6}}$ and $u_3,u_4,u_5,u_6\in W^1_{v_{6}u_0}$. Note that $u_0\in W^1_{u_0v_{6}}$ and $v_{6}\in W^1_{v_{6}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{6}}|=7$ and $|W^1_{v_{6}u_0}|=7$. When $s\ge 4$, $d(u_0,v_{3t})=1+t$ and $d(v_{3s},v_{3t})=s-t$ where $0\le t<s$. When $0\le t\le\frac{s-2}{2}$, $d(u_0,v_{3t})<d(v_{3s},v_{3t})$. When $\frac{s}{2}\le t<s$, $d(u_0,v_{3t})>d(v_{3s},v_{3t})$. $d(u_0,v_{3t+1})=2+t$ and $d(v_{3s},v_{3t+1})=s-t+3$ where $0\le t<s$. When $0\le t\le\frac{s}{2}$, $d(u_0,v_{3t+1})<d(v_{3s},v_{3t+1})$. When $\frac{s+2}{2}\le t<s$, $d(u_0,v_{3t+1})>d(v_{3s},v_{3t+1})$. $d(u_0,v_{3t+2})=3+t$ and $d(v_{3s},v_{3t+2})=s-t+2$ where $0\le t<s$. When $0\le t\le\frac{s-2}{2}$, $d(u_0,v_{3t+2})<d(v_{3s},v_{3t+2})$. When $\frac{s}{2}\le t<s$, $d(u_0,v_{3t+2})>d(v_{3s},v_{3t+2})$. $d(u_0,u_{3t})=2+t$ and $d(v_{3s},u_{3t})=s-t+1$ where $1\le t\le s$. When $1\le t\le\frac{s-2}{2}$, $d(u_0,u_{3t})<d(v_{3s},u_{3t})$. When $\frac{s}{2}\le t\le s$, $d(u_0,u_{3t})>d(v_{3s},u_{3t})$. $d(u_0,u_{1})=1$ and $d(v_{3s},u_{1})=s+2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_{3s},u_{3t+1})=s-t+2$ where $1\le t< s$. When $1\le t\le\frac{s-2}{2}$, $d(u_0,u_{3t+1})<d(v_{3s},u_{3t+1})$. When $\frac{s}{2}\le t< s$, $d(u_0,u_{3t+1})>d(v_{3s},u_{3t+1})$. $d(u_0,u_{2})=2$ and $d(v_{3s},u_{2})=s+1$. $d(u_0,u_{3t+2})=4+t$ and $d(v_{3s},u_{3t+2})=s-t+1$ where $1\le t< s$. When $1\le t\le\frac{s-4}{2}$, $d(u_0,u_{3t+2})<d(v_{3s},u_{3t+2})$. When $\frac{s-2}{2}\le t< s$, $d(u_0,u_{3t+2})>d(v_{3s},u_{3t+2})$. Note that $u_0\in W^1_{u_0v_{3s}}$ and $v_{3s}\in W^1_{v_{3s}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{3s}}|=3s$ and $|W^1_{v_{3s}u_0}|=3s+2$. (2a) Computation of $|W^1_{u_0v_{3s+1}}|$ and $|W^1_{v_{3s+1}u_0}|$ when $s$ is odd and $s\ge 3$. $d(u_0,v_{3t})=1+t$ and $d(v_{3s+1},v_{3t})=s-t+3$ where $0\le t\le s$. When $0\le t\le\frac{s+1}{2}$, $d(u_0,v_{3t})<d(v_{3s+1},v_{3t})$. When $\frac{s+3}{2}\le t\le s$, $d(u_0,v_{3t})>d(v_{3s+1},v_{3t})$. $d(u_0,v_{3t+1})=2+t$ and $d(v_{3s+1},v_{3t+1})=s-t$ where $0\le t<s$. When $0\le t\le\frac{s-3}{2}$, $d(u_0,v_{3t+1})<d(v_{3s+1},v_{3t+1})$. When $\frac{s-1}{2}\le t<s$, $d(u_0,v_{3t+1})>d(v_{3s+1},v_{3t+1})$. $d(u_0,v_{3t+2})=3+t$ and $d(v_{3s+1},v_{3t+2})=s-t+3$ where $0\le t<s$. When $0\le t\le\frac{s-1}{2}$, $d(u_0,v_{3t+2})<d(v_{3s+1},v_{3t+2})$. When $\frac{s+1}{2}\le t<s$, $d(u_0,v_{3t+2})>d(v_{3s+1},v_{3t+2})$. $d(u_0,u_{3t})=2+t$ and $d(v_{3s+1},u_{3t})=s-t+2$ where $1\le t\le s$. When $1\le t\le\frac{s-1}{2}$, $d(u_0,u_{3t})<d(v_{3s+1},u_{3t})$. When $\frac{s+1}{2}\le t\le s$, $d(u_0,u_{3t})>d(v_{3s+1},u_{3t})$. $d(u_0,u_{1})=1$ and $d(v_{3s+1},u_{1})=s+1$. $d(u_0,u_{3t+1})=3+t$ and $d(v_{3s+1},u_{3t+1})=s-t+1$ where $1\le t\le s$. When $1\le t\le\frac{s-3}{2}$, $d(u_0,u_{3t+1})<d(v_{3s+1},u_{3t+1})$. When $\frac{s-1}{2}\le t\le s$, $d(u_0,u_{3t+1})>d(v_{3s+1},u_{3t+1})$. $d(u_0,u_{2})=2$ and $d(v_{3s+1},u_{2})=s+2$. $d(u_0,u_{3t+2})=4+t$ and $d(v_{3s+1},u_{3t+2})=s-t+2$ where $1\le t< s$. When $1\le t\le\frac{s-3}{2}$, $d(u_0,u_{3t+2})<d(v_{3s+1},u_{3t+2})$. When $\frac{s-1}{2}\le t< s$, $d(u_0,u_{3t+2})>d(v_{3s+1},u_{3t+2})$. Note that $u_0\in W^1_{u_0v_{3s+1}}$ and $v_{3s+1}\in W^1_{v_{3s+1}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{3s+1}}|=3s+1$ and $|W^1_{v_{3s+1}u_0}|=3s+3$. (2b) Computation of $|W^1_{u_0v_{3s+1}}|$ and $|W^1_{v_{3s+1}u_0}|$ when $s$ is even and $s\ge 2$. When $s=2$. $d(u_0,v_0)=1$ and $d(v_7,v_0)=5$. $d(u_0,v_3)=2$ and $d(v_7,v_3)=4$. $d(u_0,v_6)=3$ and $d(v_7,v_6)=3$. $d(u_0,v_1)=2$ and $d(v_7,v_1)=2$. $d(u_0,v_4)=3$ and $d(v_7,v_4)=1$. $d(u_0,v_2)=3$ and $d(v_7,v_2)=5$. $d(u_0,v_5)=4$ and $d(v_7,v_5)=4$. So $v_0,v_2,v_3\in W^1_{u_0v_{7}}$ and $v_4\in W^1_{v_{7}u_0}$. $d(u_0,u_3)=3$ and $d(v_7,u_3)=3$. $d(u_0,u_6)=4$ and $d(v_7,u_6)=2$. $d(u_0,u_1)=1$ and $d(v_7,u_1)=3$. $d(u_0,u_4)=4$ and $d(v_7,u_4)=2$. $d(u_0,u_7)=5$ and $d(v_7,u_7)=1$. $d(u_0,u_2)=2$ and $d(v_7,u_2)=4$. $d(u_0,u_5)=5$ and $d(v_7,u_5)=4$. So $u_1,u_2\in W^1_{u_0v_{7}}$ and $u_4,u_5,u_6,u_7\in W^1_{v_{7}u_0}$. Note that $u_0\in W^1_{u_0v_{7}}$ and $v_{7}\in W^1_{v_{7}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{7}}|=6$ and $|W^1_{v_{7}u_0}|=6$. When $s\ge 4$. $d(u_0,v_{3t})=1+t$ and $d(v_{3s+1},v_{3t})=s-t+3$ where $0\le t\le s$. When $0\le t<\frac{s+2}{2}$, $d(u_0,v_{3t})<d(v_{3s+1},v_{3t})$. When $\frac{s+2}{2}< t\le s$, $d(u_0,v_{3t})>d(v_{3s+1},v_{3t})$. $d(u_0,v_{3t+1})=2+t$ and $d(v_{3s+1},v_{3t+1})=s-t$ where $0\le t<s$. When $0\le t<\frac{s-2}{2}$, $d(u_0,v_{3t+1})<d(v_{3s+1},v_{3t+1})$. When $\frac{s-2}{2}< t<s$, $d(u_0,v_{3t+1})>d(v_{3s+1},v_{3t+1})$. $d(u_0,v_{3t+2})=3+t$ and $d(v_{3s+1},v_{3t+2})=s-t+3$ where $0\le t<s$. When $0\le t<\frac{s}{2}$, $d(u_0,v_{3t+2})<d(v_{3s+1},v_{3t+2})$. When $\frac{s}{2}< t<s$, $d(u_0,v_{3t+2})>d(v_{3s+1},v_{3t+2})$. $d(u_0,u_{3t})=2+t$ and $d(v_{3s+1},u_{3t})=s-t+2$ where $1\le t\le s$. When $1\le t<\frac{s}{2}$, $d(u_0,u_{3t})<d(v_{3s+1},u_{3t})$. When $\frac{s}{2}< t\le s$, $d(u_0,u_{3t})>d(v_{3s+1},u_{3t})$. $d(u_0,u_{1})=1$ and $d(v_{3s+1},u_{1})=s+1$. $d(u_0,u_{3t+1})=3+t$ and $d(v_{3s+1},u_{3t+1})=s-t+1$ where $1\le t\le s$. When $1\le t<\frac{s-2}{2}$, $d(u_0,u_{3t+1})<d(v_{3s+1},u_{3t+1})$. When $\frac{s-2}{2}< t\le s$, $d(u_0,u_{3t+1})>d(v_{3s+1},u_{3t+1})$. $d(u_0,u_{2})=2$ and $d(v_{3s+1},u_{2})=s+2$. $d(u_0,u_{3t+2})=4+t$ and $d(v_{3s+1},u_{3t+2})=s-t+2$ where $1\le t< s$. When $1\le t<\frac{s-2}{2}$, $d(u_0,u_{3t+2})<d(v_{3s+1},u_{3t+2})$. When $\frac{s-2}{2}< t< s$, $d(u_0,u_{3t+2})>d(v_{3s+1},u_{3t+2})$. Note that $u_0\in W^1_{u_0v_{3s+1}}$ and $v_{3s+1}\in W^1_{v_{3s+1}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{3s+1}}|=3s-2$ and $|W^1_{v_{3s+1}u_0}|=3s$. (3a) Computation of $|W^1_{u_0v_{3s+2}}|$ and $|W^1_{v_{3s+2}u_0}|$ when $s$ is odd and $s\ge 3$. When $s=3$. $d(u_0,v_0)=1$ and $d(v_{11},v_0)=7$. $d(u_0,v_3)=2$ and $d(v_{11},v_3)=6$. $d(u_0,v_6)=3$ and $d(v_{11},v_6)=5$. $d(u_0,v_9)=d(v_{11},v_9)=4$. $d(u_0,v_1)=2$ and $d(v_{11},v_1)=6$. $d(u_0,v_4)=3$ and $d(v_{11},v_4)=5$. $d(u_0,v_7)=4$ and $d(v_{11},v_7)=4$. $d(u_0,v_{10})=5$ and $d(v_{11},v_{10})=3$. $d(u_0,v_2)=3$ and $d(v_{11},v_2)=3$. $d(u_0,v_5)=4$ and $d(v_{11},v_5)=2$. $d(u_0,v_8)=5$ and $d(v_{11},v_8)=1$. So $v_0,v_1,v_3,v_4,v_6\in W^1_{u_0v_{11}}$ and $v_5,v_8,v_{10}\in W^1_{v_{11}u_0}$. $d(u_0,u_3)=3$ and $d(v_{11},u_3)=5$. $d(u_0,u_6)=4$ and $d(v_{11},u_6)=4$. $d(u_0,u_9)=5$ and $d(v_{11},u_9)=3$. $d(u_0,u_1)=1$ and $d(v_{11},u_1)=5$. $d(u_0,u_4)=4$ and $d(v_{11},u_4)=4$. $d(u_0,u_7)=5$ and $d(v_{11},u_7)=3$. $d(u_0,u_{10})=6$ and $d(v_{11},u_{10})=2$. $d(u_0,u_2)=2$ and $d(v_{11},u_2)=4$. $d(u_0,u_5)=5$ and $d(v_{11},u_5)=3$. $d(u_0,u_8)=6$ and $d(v_{11},u_8)=2$. $d(u_0,u_{11})=7$ and $d(v_{11},u_{11})=1$. So $u_1,u_2,u_3\in W^1_{u_0v_{11}}$ and $u_5,u_7,u_8,u_9,u_{10},u_{11}\in W^1_{v_{11}u_0}$. Note that $u_0\in W^1_{u_0v_{11}}$ and $v_{11}\in W^1_{v_{11}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{11}}|=9$ and $|W^1_{v_{11}u_0}|=10$. When $s\ge 5$. $d(u_0,v_{3t})=1+t$ and $d(v_{3s+2},v_{3t})=s-t+4$ where $0\le t\le s$. When $0\le t<\frac{s+3}{2}$, $d(u_0,v_{3t})<d(v_{3s+2},v_{3t})$. When $\frac{s+3}{2}<t\le s$, $d(u_0,v_{3t})>d(v_{3s+2},v_{3t})$. $d(u_0,v_{3t+1})=2+t$ and $d(v_{3s+2},v_{3t+1})=s-t+3$ where $0\le t\le s$. When $0\le t<\frac{s+1}{2}$, $d(u_0,v_{3t+1})<d(v_{3s+2},v_{3t+1})$. When $\frac{s+1}{2}<t\le s$, $d(u_0,v_{3t+1})>d(v_{3s+2},v_{3t+1})$. $d(u_0,v_{3t+2})=3+t$ and $d(v_{3s+2},v_{3t+2})=s-t$ where $0\le t<s$. When $0\le t<\frac{s-3}{2}$, $d(u_0,v_{3t+2})<d(v_{3s+2},v_{3t+2})$. When $\frac{s-3}{2}<t<s$, $d(u_0,v_{3t+2})>d(v_{3s+2},v_{3t+2})$. $d(u_0,u_{3t})=2+t$ and $d(v_{3s+2},u_{3t})=s-t+3$ where $1\le t\le s$. When $1\le t<\frac{s+1}{2}$, $d(u_0,u_{3t})<d(v_{3s+2},u_{3t})$. When $\frac{s+1}{2}<t\le s$, $d(u_0,u_{3t})>d(v_{3s+2},u_{3t})$. $d(u_0,u_{1})=1$ and $d(v_{3s+2},u_{1})=s+2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_{3s+2},u_{3t+1})=s-t+2$ where $1\le t\le s$. When $1\le t<\frac{s-1}{2}$, $d(u_0,u_{3t+1})<d(v_{3s+2},u_{3t+1})$. When $\frac{s-1}{2}<t\le s$, $d(u_0,u_{3t+1})>d(v_{3s+2},u_{3t+1})$. $d(u_0,u_{2})=2$ and $d(v_{3s+2},u_{2})=s+1$. $d(u_0,u_{3t+2})=4+t$ and $d(v_{3s+2},u_{3t+2})=s-t+1$ where $1\le t\le s$. When $1\le t<\frac{s-3}{2}$, $d(u_0,u_{3t+2})<d(v_{3s+2},u_{3t+2})$. When $\frac{s-3}{2}<t\le s$, $d(u_0,u_{3t+2})>d(v_{3s+2},u_{3t+2})$. Note that $u_0\in W^1_{u_0v_{3s+2}}$ and $v_{3s+2}\in W^1_{v_{3s+2}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{3s+2}}|=3s-1$ and $|W^1_{v_{3s+2}u_0}|=3s+1$. (3b) Computation of $|W^1_{u_0v_{3s+2}}|$ and $|W^1_{v_{3s+2}u_0}|$ when $s$ is even and $s\ge 2$. When $s=2$. $d(u_0,v_0)=1$ and $d(v_8,v_0)=6$. $d(u_0,v_3)=2$ and $d(v_8,v_3)=5$. $d(u_0,v_6)=3$ and $d(v_8,v_6)=4$. $d(u_0,v_1)=2$ and $d(v_8,v_1)=5$. $d(u_0,v_4)=3$ and $d(v_8,v_4)=4$. $d(u_0,v_7)=4$ and $d(v_8,v_7)=3$. $d(u_0,v_2)=3$ and $d(v_8,v_2)=2$. $d(u_0,v_5)=4$ and $d(v_8,v_5)=1$. So $v_0,v_1,v_3,v_4,v_6\in W^1_{u_0v_{8}}$ and $v_2,v_5,v_7\in W^1_{v_{8}u_0}$. $d(u_0,u_3)=3$ and $d(v_8,u_3)=4$. $d(u_0,u_6)=4$ and $d(v_8,u_6)=3$. $d(u_0,u_1)=1$ and $d(v_8,u_1)=4$. $d(u_0,u_4)=4$ and $d(v_8,u_4)=3$. $d(u_0,u_7)=5$ and $d(v_8,u_7)=2$. $d(u_0,u_2)=2$ and $d(v_8,u_2)=3$. $d(u_0,u_5)=5$ and $d(v_8,u_5)=2$. $d(u_0,u_8)=6$ and $d(v_8,u_8)=1$. So $u_1,u_2,u_3\in W^1_{u_0v_{8}}$ and $u_4,u_5,u_6,u_7,u_8\in W^1_{v_{8}u_0}$. Note that $u_0\in W^1_{u_0v_{8}}$ and $v_{8}\in W^1_{v_{8}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{8}}|=9$ and $|W^1_{v_{8}u_0}|=9$. When $s\ge 4$. $d(u_0,v_{3t})=1+t$ and $d(v_{3s+2},v_{3t})=s-t+4$ where $0\le t\le s$. When $0\le t\le\frac{s+2}{2}$, $d(u_0,v_{3t})<d(v_{3s+2},v_{3t})$. When $\frac{s+4}{2}\le t\le s$, $d(u_0,v_{3t})>d(v_{3s+2},v_{3t})$. $d(u_0,v_{3t+1})=2+t$ and $d(v_{3s+2},v_{3t+1})=s-t+3$ where $0\le t\le s$. When $0\le t\le\frac{s}{2}$, $d(u_0,v_{3t+1})<d(v_{3s+2},v_{3t+1})$. When $\frac{s+2}{2}\le t\le s$, $d(u_0,v_{3t+1})>d(v_{3s+2},v_{3t+1})$. $d(u_0,v_{3t+2})=3+t$ and $d(v_{3s+2},v_{3t+2})=s-t$ where $0\le t<s$. When $0\le t\le\frac{s-4}{2}$, $d(u_0,v_{3t+2})<d(v_{3s+2},v_{3t+2})$. When $\frac{s-2}{2}\le t<s$, $d(u_0,v_{3t+2})>d(v_{3s+2},v_{3t+2})$. $d(u_0,u_{3t})=2+t$ and $d(v_{3s+2},u_{3t})=s-t+3$ where $1\le t\le s$. When $1\le t\le\frac{s}{2}$, $d(u_0,u_{3t})<d(v_{3s+2},u_{3t})$. When $\frac{s+2}{2}\le t\le s$, $d(u_0,u_{3t})>d(v_{3s+2},u_{3t})$. $d(u_0,u_{1})=1$ and $d(v_{3s+2},u_{1})=s+2$. $d(u_0,u_{3t+1})=3+t$ and $d(v_{3s+2},u_{3t+1})=s-t+2$ where $1\le t\le s$. When $1\le t\le\frac{s-2}{2}$, $d(u_0,u_{3t+1})<d(v_{3s+2},u_{3t+1})$. When $\frac{s}{2}\le t\le s$, $d(u_0,u_{3t+1})>d(v_{3s+2},u_{3t+1})$. $d(u_0,u_{2})=2$ and $d(v_{3s+2},u_{2})=s+1$. $d(u_0,u_{3t+2})=4+t$ and $d(v_{3s+2},u_{3t+2})=s-t+1$ where $1\le t\le s$. When $1\le t\le\frac{s-4}{2}$, $d(u_0,u_{3t+2})<d(v_{3s+2},u_{3t+2})$. When $\frac{s-2}{2}\le t\le s$, $d(u_0,u_{3t+2})>d(v_{3s+2},u_{3t+2})$. Note that $u_0\in W^1_{u_0v_{3s+2}}$ and $v_{3s+2}\in W^1_{v_{3s+2}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{3s+2}}|=3s+2$ and $|W^1_{v_{3s+2}u_0}|=3s+4$. When $n\ge 17$, from the above computation of $|W^1_{u_0v_j}|$ and $|W^1_{v_ju_0}|$ where $6\le j\le n-6$, for any $3\le \ell<D$, we know that there exists $j$ where $d(u_0,v_j)=\ell$ and $6\le j\le n/2$ such that $|W_{u_0v_j}|<|W_{v_ju_0}|$. The proof of the theorem completes. ◻ # The proof of Theorem [Theorem 4](#T:GP(n,4)-onlyD-DB){reference-type="ref" reference="T:GP(n,4)-onlyD-DB"} {#S:GP(n,4)} From [@Miklavic:2018], the diameter of $GP(24,4)$ is $6$ and $GP(24,4)$ is $\ell$-distance-balanced if and only if $\ell\in\{1,6\}$. So we can suppose $n> 24$ when we prove Theorem [Theorem 4](#T:GP(n,4)-onlyD-DB){reference-type="ref" reference="T:GP(n,4)-onlyD-DB"}. We will prove Theorem [Theorem 4](#T:GP(n,4)-onlyD-DB){reference-type="ref" reference="T:GP(n,4)-onlyD-DB"} via Proposition [Proposition 8](#P:GP(n,4)-1){reference-type="ref" reference="P:GP(n,4)-1"}, [Proposition 9](#P:GP(n,4)-2){reference-type="ref" reference="P:GP(n,4)-2"} and [Proposition 10](#P:GP(n,4)-3){reference-type="ref" reference="P:GP(n,4)-3"}. **Proposition 8**. *For any $n>24$, the generalized Petersen graph $GP(n,4)$ is not $1$-distance-balanced.* *Proof.* In $GP(n,4)$, $d(u_0,v_0)=1$ and we will prove that $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. We divide the discussion into the following eight cases according to the size of $n$. \(1\) When $n=8m$ where $m\ge 4$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{4t})=1+t$ and $d(v_0,v_{4t})=t$ where $1\le t\le m$. $d(u_0,v_{4t+1})=2+t$ and $d(v_0,v_{4t+1})=3+t$ where $0\le t< m$. $d(u_0,v_{4t+2})=3+t$ and $d(v_0,v_{4t+2})=4+t$ where $0\le t< m$. $d(u_0,v_{4t+3})=3+t$ and $d(v_0,v_{4t+3})=4+t$ where $0\le t< m$. $d(u_0,u_{4t})=2+t$ and $d(v_0,u_{4t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_0,u_{4t+1})=2+t$ where $1\le t< m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{4t+2})=4+t$ and $d(v_0,u_{4t+2})=3+t$ where $1\le t< m$. $d(u_0,u_{3})=3$ and $d(v_0,u_{3})=3$. $d(u_0,u_{4t+3})=4+t$ and $d(v_0,u_{4t+3})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(3m+2)+1=6m+5$ and $|W_{v_{0}u_0}|=2(5m-5)+3=10m-7$. Because $m\ge 4$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(2\) When $n=8m+1$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{4t})=1+t$ and $d(v_0,v_{4t})=t$ where $1\le t\le m$. $d(u_0,v_{4t+1})=2+t$ and $d(v_0,v_{4t+1})=3+t$ where $0\le t\le m-2$. $d(u_0,v_{4(m-1)+1})=d(v_0,v_{4(m-1)+1})=m+1$. $d(u_0,v_{4t+2})=3+t$ and $d(v_0,v_{4t+2})=4+t$ where $0\le t< m$. $d(u_0,v_{4t+3})=3+t$ and $d(v_0,v_{4t+3})=4+t$ where $0\le t< m$. $d(u_0,u_{4t})=2+t$ and $d(v_0,u_{4t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_0,u_{4t+1})=2+t$ where $1\le t< m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{4t+2})=4+t$ and $d(v_0,u_{4t+2})=3+t$ where $1\le t< m$. $d(u_0,u_{3})=3$ and $d(v_0,u_{3})=3$. $d(u_0,u_{4t+3})=4+t$ and $d(v_0,u_{4t+3})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(3m+1)+1=6m+3$ and $|W_{v_{0}u_0}|=2(5m-3)+1=10m-5$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(3\) When $n=8m+2$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{4t})=1+t$ and $d(v_0,v_{4t})=t$ where $1\le t\le m$. $d(u_0,v_{4t+1})=2+t$ and $d(v_0,v_{4t+1})=3+t$ where $0\le t\le m$. $d(u_0,v_{4t+2})=3+t$ and $d(v_0,v_{4t+2})=4+t$ where $0\le t\le m-2$. $d(u_0,v_{4(m-1)+2})=m+2$ and $d(v_0,v_{4(m-1)+2})=m+1$. $d(u_0,v_{4t+3})=3+t$ and $d(v_0,v_{4t+3})=4+t$ where $0\le t< m$. $d(u_0,u_{4t})=2+t$ and $d(v_0,u_{4t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_0,u_{4t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{4t+2})=4+t$ and $d(v_0,u_{4t+2})=3+t$ where $1\le t< m$. $d(u_0,u_{3})=3$ and $d(v_0,u_{3})=3$. $d(u_0,u_{4t+3})=4+t$ and $d(v_0,u_{4t+3})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(3m+1)+2=6m+4$ and $|W_{v_{0}u_0}|=2(5m-2)+2=10m-2$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(4\) When $n=8m+3$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{4t})=1+t$ and $d(v_0,v_{4t})=t$ where $1\le t\le m$. $d(u_0,v_{4t+1})=2+t$ and $d(v_0,v_{4t+1})=3+t$ where $0\le t\le m$. $d(u_0,v_{4t+2})=3+t$ and $d(v_0,v_{4t+2})=4+t$ where $0\le t< m$. $d(u_0,v_{4t+3})=3+t$ and $d(v_0,v_{4t+3})=4+t$ where $0\le t\le m-2$. $d(u_0,v_{4(m-1)+3})=m+2$ and $d(v_0,v_{4(m-1)+3})=m+1$. $d(u_0,u_{4t})=2+t$ and $d(v_0,u_{4t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_0,u_{4t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{4t+2})=4+t$ and $d(v_0,u_{4t+2})=3+t$ where $1\le t< m$. $d(u_0,u_{3})=3$ and $d(v_0,u_{3})=3$. $d(u_0,u_{4t+3})=4+t$ and $d(v_0,u_{4t+3})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(3m+2)+1=6m+5$ and $|W_{v_{0}u_0}|=2(5m-1)+1=10m-1$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(5\) When $n=8m+4$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{4t})=1+t$ and $d(v_0,v_{4t})=t$ where $1\le t\le m$. $d(u_0,v_{4t+1})=2+t$ and $d(v_0,v_{4t+1})=3+t$ where $0\le t\le m$. $d(u_0,v_{4t+2})=3+t$ and $d(v_0,v_{4t+2})=4+t$ where $0\le t\le m$. $d(u_0,v_{4t+3})=3+t$ and $d(v_0,v_{4t+3})=4+t$ where $0\le t< m$. $d(u_0,u_{4t})=2+t$ and $d(v_0,u_{4t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_0,u_{4t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{4t+2})=4+t$ and $d(v_0,u_{4t+2})=3+t$ where $1\le t\le m$. $d(u_0,u_{3})=3$ and $d(v_0,u_{3})=3$. $d(u_0,u_{4t+3})=4+t$ and $d(v_0,u_{4t+3})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(3m+3)+2=6m+8$ and $|W_{v_{0}u_0}|=2(5m-2)+2=10m-2$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(6\) When $n=8m+5$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{4t})=1+t$ and $d(v_0,v_{4t})=t$ where $1\le t\le m$. $d(u_0,v_{4t+1})=2+t$ and $d(v_0,v_{4t+1})=3+t$ where $0\le t\le m-1$. $d(u_0,v_{4m+1})=m+2$ and $d(v_0,v_{4m+1})=m+1$. $d(u_0,v_{4t+2})=3+t$ and $d(v_0,v_{4t+2})=4+t$ where $0\le t\le m$. $d(u_0,v_{4t+3})=3+t$ and $d(v_0,v_{4t+3})=4+t$ where $0\le t< m$. $d(u_0,u_{4t})=2+t$ and $d(v_0,u_{4t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_0,u_{4t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{4t+2})=4+t$ and $d(v_0,u_{4t+2})=3+t$ where $1\le t\le m$. $d(u_0,u_{3})=3$ and $d(v_0,u_{3})=3$. $d(u_0,u_{4t+3})=4+t$ and $d(v_0,u_{4t+3})=3+t$ where $1\le t< m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(3m+3)+1=6m+7$ and $|W_{v_{0}u_0}|=2\times 5m+1=10m+1$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(7\) When $n=8m+6$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{4t})=1+t$ and $d(v_0,v_{4t})=t$ where $1\le t\le m$. $d(u_0,v_{4t+1})=2+t$ and $d(v_0,v_{4t+1})=3+t$ where $0\le t\le m$. $d(u_0,v_{4t+2})=3+t$ and $d(v_0,v_{4t+2})=4+t$ where $0\le t\le m-2$. $d(u_0,v_{4(m-1)+2})=d(v_0,v_{4(m-1)+2})=m+2$. $d(u_0,v_{4m+2})=m+2$ and $d(v_0,v_{4m+2})=m+1$. $d(u_0,v_{4t+3})=3+t$ and $d(v_0,v_{4t+3})=4+t$ where $0\le t\le m$. $d(u_0,u_{4t})=2+t$ and $d(v_0,u_{4t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_0,u_{4t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{4t+2})=4+t$ and $d(v_0,u_{4t+2})=3+t$ where $1\le t\le m-1$. $d(u_0,u_{4m+2})=m+3$ and $d(v_0,u_{4m+2})=m+2$. $d(u_0,u_{3})=3$ and $d(v_0,u_{3})=3$. $d(u_0,u_{4t+3})=4+t$ and $d(v_0,u_{4t+3})=3+t$ where $1\le t\le m$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(3m+2)+2=6m+6$ and $|W_{v_{0}u_0}|=2\times 5m+2=10m+2$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. \(8\) When $n=8m+7$ where $m\ge 3$. By symmetry, we just need to consider vertices $u_i$ and $v_i$ where $1\le i\le \frac{n}{2}$. $d(u_0,v_{4t})=1+t$ and $d(v_0,v_{4t})=t$ where $1\le t\le m$. $d(u_0,v_{4t+1})=2+t$ and $d(v_0,v_{4t+1})=3+t$ where $0\le t\le m$. $d(u_0,v_{4t+2})=3+t$ and $d(v_0,v_{4t+2})=4+t$ where $0\le t\le m$. $d(u_0,v_{4t+3})=3+t$ and $d(v_0,v_{4t+3})=4+t$ where $0\le t\le m-2$. $d(u_0,v_{4(m-1)+3})=d(v_0,v_{4(m-1)+3})=m+2$. $d(u_0,v_{4m+3})=m+2$ and $d(v_0,v_{4m+3})=m+1$. $d(u_0,u_{4t})=2+t$ and $d(v_0,u_{4t})=1+t$ where $1\le t\le m$. $d(u_0,u_{1})=1$ and $d(v_0,u_{1})=2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_0,u_{4t+1})=2+t$ where $1\le t\le m$. $d(u_0,u_{2})=2$ and $d(v_0,u_{2})=3$. $d(u_0,u_{4t+2})=4+t$ and $d(v_0,u_{4t+2})=3+t$ where $1\le t\le m$. $d(u_0,u_{3})=3$ and $d(v_0,u_{3})=3$. $d(u_0,u_{4t+3})=4+t$ and $d(v_0,u_{4t+3})=3+t$ where $1\le t\le m-1$. $d(u_0,u_{4m+3})=m+3$ and $d(v_0,u_{4m+3})=m+2$. Note that $u_0\in W_{u_0v_{0}}$ and $v_{0}\in W_{v_{0}u_0}$. Combined with the above discussion, $|W_{u_0v_{0}}|=2(3m+3)+1=6m+7$ and $|W_{v_{0}u_0}|=2(5m+1)+1=10m+3$. Because $m\ge 3$, $|W_{u_0v_{0}}|<|W_{v_{0}u_0}|$. ◻ **Proposition 9**. *For any $n>24$, the generalized Petersen graph $GP(n,4)$ is not $2$-distance-balanced.* *Proof.* In $GP(n,4)$, $d(u_0,v_{-4})=2$ and we will prove that $|W_{u_0v_{-4}}|<|W_{v_{-4}u_0}|$. Note that $v_{-4}=v_{n-4}$. Firstly we consider vertices $v_{-1},v_{-2},v_{-3},u_{-1},u_{-2},u_{-3}$. $d(u_0,v_{-1})=2$ and $d(v_{-4},v_{-1})=4$. $d(u_0,v_{-2})=3$ and $d(v_{-4},v_{-2})=4$. $d(u_0,v_{-3})=d(v_{-4},v_{-3})=3$. $d(u_0,u_{-1})=1$ and $d(v_{-4},u_{-1})=3$. $d(u_0,u_{-2})=2$ and $d(v_{-4},u_{-2})=3$. $d(u_0,u_{-3})=3$ and $d(v_{-4},u_{-3})=2$. So $u_{-1},u_{-2},v_{-1},v_{-2}\in W_{u_0v_{-4}}$ and $u_{-3}\in W_{v_{-4}u_0}$. Next we consider vertices $v_i$ where $0\le i<n-4$ and $u_j$ where $1\le j\le n-4$. We divide the discussion into the following eight cases according to the size of $n$. \(1\) When $n=8m$ where $m\ge 4$. Note that $n-4=8m-4=4(2m-1)$. $d(u_0,v_{4t})=d(v_{8m-4},v_{4t})=1+t$ when $0\le t\le m-1$. $d(v_{8m-4},v_{4t})=2m-t-1$ and $d(u_0,v_{4t})>d(v_{8m-4},v_{4t})$ when $m\le t<2m-1$. $d(u_0,v_{4t+1})=2+t$ and $d(u_0,v_{4t+1})<d(v_{8m-4},v_{4t+1})$ when $0\le t\le m-1$. $d(u_0,v_{4t+1})=d(v_{8m-4},v_{4t+1})=2m-t+2$ when $m\le t< 2m-1$. $d(u_0,v_{4t+2})=3+t$ and $d(u_0,v_{4t+2})<d(v_{8m-4},v_{4t+2})$ when $0\le t\le m-1$. $d(u_0,v_{4t+2})=d(v_{8m-4},v_{4t+2})=2m-t+2$ when $m\le t< 2m-1$. $d(u_0,v_{4t+3})=3+t$ and $d(u_0,v_{4t+3})<d(v_{8m-4},v_{4t+3})$ when $0\le t\le m-2$. $d(u_0,v_{4t+3})=d(v_{8m-4},v_{4t+3})=2m-t+1$ when $m-1\le t< 2m-1$. $d(u_0,u_{4t})=d(v_{8m-4},u_{4t})=2+t$ when $1\le t\le m-1$. $d(v_{8m-4},u_{4t})=2m-t$ and $d(u_0,u_{4t})>d(v_{8m-4},u_{4t})$ when $m\le t\le 2m-1$. $d(u_0,u_{1})=1$ and $d(v_{8m-4},u_{1})=2m+1$. $d(u_0,u_{4t+1})=d(v_{8m-4},u_{4t+1})=3+t$ when $1\le t\le m-1$. $d(v_{8m-4},u_{4t+1})=2m-t+1$ and $d(u_0,u_{4t+1})>d(v_{8m-4},u_{4t+1})$ when $m\le t< 2m-1$. $d(u_0,u_{2})=2$ and $d(v_{8m-4},u_{2})=2m+1$. $d(u_0,u_{4t+2})=d(v_{8m-4},u_{4t+2})=4+t$ when $1\le t\le m-2$. $d(v_{8m-4},u_{4t+2})=2m-t+1$ and $d(u_0,u_{4t+2})>d(v_{8m-4},u_{4t+2})$ when $m-1\le t< 2m-1$. $d(u_0,u_{3})=3$ and $d(v_{8m-4},u_{3})=2m$. $d(u_0,u_{4t+3})=d(v_{8m-4},u_{4t+3})=4+t$ when $1\le t\le m-2$. $d(v_{8m-4},u_{4t+3})=2m-t$ and $d(u_0,u_{4t+3})>d(v_{8m-4},u_{4t+3})$ when $m-1\le t< 2m-1$. Note that $u_0\in W_{u_0v_{8m-4}}$ and $v_{8m-4}\in W_{v_{8m-4}u_0}$. Combined with the above discussion, $|W_{u_0v_{8m-4}}|=3m+7$ and $|W_{v_{8m-4}u_0}|=5m$. Because $m\ge 4$, $|W_{u_0v_{8m-4}}|<|W_{v_{8m-4}u_0}|$. \(2\) When $n=8m+1$ where $m\ge 3$. Note that $n-4=8m-3=4(2m-1)+1$. $d(u_0,v_{4t})=d(v_{8m-3},v_{4t})=1+t$ when $0\le t\le m$. $d(u_0,v_{4t})=d(v_{8m-3},v_{4t})=2m-t+2$ when $m+1\le t\le 2m-1$. $d(u_0,v_{4t+1})=2+t$ and $d(u_0,v_{4t+1})<d(v_{8m-3},v_{4t+1})$ when $0\le t\le m-2$. $d(v_{8m-3},v_{4t+1})=2m-t-1$ and $d(u_0,v_{4t+1})>d(v_{8m-3},v_{4t+1})$ when $m-1\le t< 2m-1$. $d(u_0,v_{4t+2})=3+t$ and $d(u_0,v_{4t+2})<d(v_{8m-3},v_{4t+2})$ when $0\le t\le m-1$. $d(u_0,v_{4t+2})=d(v_{8m-3},v_{4t+2})=2m-t+2$ when $m\le t< 2m-1$. $d(u_0,v_{4t+3})=3+t$ and $d(u_0,v_{4t+3})<d(v_{8m-3},v_{4t+3})$ when $0\le t\le m-1$. $d(u_0,v_{4t+3})=d(v_{8m-3},v_{4t+3})=2m-t+2$ when $m\le t< 2m-1$. $d(u_0,u_{4t})=d(v_{8m-3},u_{4t})=2+t$ when $1\le t\le m-1$. $d(v_{8m-3},u_{4t})=2m-t+1$ and $d(u_0,u_{4t})>d(v_{8m-3},u_{4t})$ when $m\le t\le 2m-1$. $d(u_0,u_{1})=1$ and $d(v_{8m-3},u_{1})=2m$. $d(u_0,u_{4t+1})=d(v_{8m-3},u_{4t+1})=3+t$ when $1\le t\le m-2$. $d(v_{8m-3},u_{4t+1})=2m-t$ and $d(u_0,u_{4t+1})>d(v_{8m-3},u_{4t+1})$ when $m-1\le t\le 2m-1$. $d(u_0,u_{2})=2$ and $d(v_{8m-3},u_{2})=2m+1$. $d(u_0,u_{4t+2})=d(v_{8m-3},u_{4t+2})=4+t$ when $1\le t\le m-2$. $d(v_{8m-3},u_{4t+2})=2m-t+1$ and $d(u_0,u_{4t+2})>d(v_{8m-3},u_{4t+2})$ when $m-1\le t< 2m-1$. $d(u_0,u_{3})=3$ and $d(v_{8m-3},u_{3})=2m+1$. $d(u_0,u_{4t+3})=d(v_{8m-3},u_{4t+3})=4+t$ when $1\le t\le m-2$. $d(v_{8m-3},u_{4t+3})=2m-t+1$ and $d(u_0,u_{4t+3})>d(v_{8m-3},u_{4t+3})$ when $m-1\le t< 2m-1$. Note that $u_0\in W_{u_0v_{8m-3}}$ and $v_{8m-3}\in W_{v_{8m-3}u_0}$. Combined with the above discussion, $|W_{u_0v_{8m-3}}|=3m+7$ and $|W_{v_{8m-3}u_0}|=5m+3$. Because $m\ge 3$, $|W_{u_0v_{8m-3}}|<|W_{v_{8m-3}u_0}|$. \(3\) When $n=8m+2$ where $m\ge 3$. Note that $n-4=8m-2=4(2m-1)+2$. $d(u_0,v_{4t})=d(v_{8m-2},v_{4t})=1+t$ when $0\le t\le m+1$. $d(u_0,v_{4t})=d(v_{8m-2},v_{4t})=2m-t+3$ when $m+2\le t\le 2m-1$. $d(u_0,v_{4t+1})=2+t$ and $d(u_0,v_{4t+1})<d(v_{8m-2},v_{4t+1})$ when $0\le t\le m-1$. $d(u_0,v_{4t+1})=d(v_{8m-2},v_{4t+1})=2m-t+2$ when $m\le t\le 2m-1$. $d(u_0,v_{4t+2})=3+t$ and $d(u_0,v_{4t+2})<d(v_{8m-2},v_{4t+2})$ when $0\le t< m-2$. $d(u_0,v_{4(m-2)+2})=d(v_{8m-2},v_{4(m-2)+2})=m+1$. $d(v_{8m-2},v_{4t+2})=2m-t-1$ and $d(u_0,v_{4t+2})>d(v_{8m-2},v_{4t+2})$ when $m-2< t< 2m-1$. $d(u_0,v_{4t+3})=3+t$ and $d(u_0,v_{4t+3})<d(v_{8m-2},v_{4t+3})$ when $0\le t\le m-1$. $d(u_0,v_{4t+3})=d(v_{8m-2},v_{4t+3})=2m-t+2$ when $m\le t< 2m-1$. $d(u_0,u_{4t})=d(v_{8m-2},u_{4t})=2+t$ when $1\le t\le m$. $d(v_{8m-2},u_{4t})=2m-t+2$ and $d(u_0,u_{4t})>d(v_{8m-2},u_{4t})$ when $m+1\le t\le 2m-1$. $d(u_0,u_{1})=1$ and $d(v_{8m-2},u_{1})=2m+1$. $d(u_0,u_{4t+1})=d(v_{8m-2},u_{4t+1})=3+t$ when $1\le t\le m-1$. $d(v_{8m-2},u_{4t+1})=2m-t+1$ and $d(u_0,u_{4t+1})>d(v_{8m-2},u_{4t+1})$ when $m\le t\le 2m-1$. $d(u_0,u_{2})=2$ and $d(v_{8m-2},u_{2})=2m$. $d(u_0,u_{4t+2})=d(v_{8m-2},u_{4t+2})=4+t$ when $1\le t\le m-2$. $d(v_{8m-2},u_{4t+2})=2m-t$ and $d(u_0,u_{4t+2})>d(v_{8m-2},u_{4t+2})$ when $m-1\le t\le 2m-1$. $d(u_0,u_{3})=3$ and $d(v_{8m-2},u_{3})=2m+1$. $d(u_0,u_{4t+3})=d(v_{8m-2},u_{4t+3})=4+t$ when $1\le t\le m-2$. $d(v_{8m-2},u_{4t+3})=2m-t+1$ and $d(u_0,u_{4t+3})>d(v_{8m-2},u_{4t+3})$ when $m-1\le t< 2m-1$. Note that $u_0\in W_{u_0v_{8m-2}}$ and $v_{8m-2}\in W_{v_{8m-2}u_0}$. Combined with the above discussion, $|W_{u_0v_{8m-2}}|=3m+6$ and $|W_{v_{8m-2}u_0}|=5m+2$. Because $m\ge 3$, $|W_{u_0v_{8m-2}}|<|W_{v_{8m-2}u_0}|$. \(4\) When $n=8m+3$ where $m\ge 3$. Note that $n-4=8m-1=4(2m-1)+3$. $d(u_0,v_{4t})=d(v_{8m-1},v_{4t})=1+t$ when $0\le t\le m+1$. $d(u_0,v_{4t})=d(v_{8m-1},v_{4t})=2m-t+3$ when $m+2\le t\le 2m-1$. $d(u_0,v_{4t+1})=2+t$ and $d(u_0,v_{4t+1})<d(v_{8m-1},v_{4t+1})$ when $0\le t\le m$. $d(u_0,v_{4t+1})=d(v_{8m-1},v_{4t+1})=2m-t+3$ when $m+1\le t\le 2m-1$. $d(u_0,v_{4t+2})=3+t$ and $d(u_0,v_{4t+2})<d(v_{8m-1},v_{4t+2})$ when $0\le t\le m-1$. $d(u_0,v_{4t+2})=d(v_{8m-1},v_{4t+2})=2m-t+2$ when $m\le t\le 2m-1$. $d(u_0,v_{4t+3})=3+t$ and $d(u_0,v_{4t+3})<d(v_{8m-1},v_{4t+3})$ when $0\le t< m-2$. $d(u_0,v_{4(m-2)+3})=d(v_{8m-1},v_{4(m-2)+3})=m+1$. $d(v_{8m-1},v_{4t+3})=2m-t-1$ and $d(u_0,v_{4t+3})>d(v_{8m-1},v_{4t+3})$ when $m-2< t< 2m-1$. $d(u_0,u_{4t})=d(v_{8m-1},u_{4t})=2+t$ when $1\le t\le m$. $d(v_{8m-1},u_{4t})=2m-t+2$ and $d(u_0,u_{4t})>d(v_{8m-1},u_{4t})$ when $m+1\le t\le 2m-1$. $d(u_0,u_{1})=1$ and $d(v_{8m-1},u_{1})=2m+2$. $d(u_0,u_{4t+1})=d(v_{8m-1},u_{4t+1})=3+t$ when $1\le t\le m-1$. $d(v_{8m-1},u_{4t+1})=2m-t+2$ and $d(u_0,u_{4t+1})>d(v_{8m-1},u_{4t+1})$ when $m\le t\le 2m-1$. $d(u_0,u_{2})=2$ and $d(v_{8m-1},u_{2})=2m+1$. $d(u_0,u_{4t+2})=d(v_{8m-1},u_{4t+2})=4+t$ when $1\le t\le m-2$. $d(v_{8m-1},u_{4t+2})=2m-t+1$ and $d(u_0,u_{4t+2})>d(v_{8m-1},u_{4t+2})$ when $m-1\le t\le 2m-1$. $d(u_0,u_{3})=3$ and $d(v_{8m-1},u_{3})=2m$. $d(u_0,u_{4t+3})=d(v_{8m-1},u_{4t+3})=4+t$ when $1\le t\le m-2$. $d(v_{8m-1},u_{4t+3})=2m-t$ and $d(u_0,u_{4t+3})>d(v_{8m-1},u_{4t+3})$ when $m-1\le t\le 2m-1$. Note that $u_0\in W_{u_0v_{8m-1}}$ and $v_{8m-1}\in W_{v_{8m-1}u_0}$. Combined with the above discussion, $|W_{u_0v_{8m-1}}|=3m+7$ and $|W_{v_{8m-1}u_0}|=5m+3$. Because $m\ge 3$, $|W_{u_0v_{8m-1}}|<|W_{v_{8m-1}u_0}|$. \(5\) When $n=8m+4$ where $m\ge 3$. Note that $n-4=8m=4\times 2m$. $d(u_0,v_{4t})=d(v_{8m},v_{4t})=1+t$ when $0\le t\le m-1$. $d(v_{8m},v_{4t})=2m-t$ and $d(u_0,v_{4t})>d(v_{8m},v_{4t})$ when $m\le t\le 2m-1$. $d(u_0,v_{4t+1})=2+t$ and $d(u_0,v_{4t+1})<d(v_{8m},v_{4t+1})$ when $0\le t\le m$. $d(u_0,v_{4t+1})=d(v_{8m},v_{4t+1})=2m-t+3$ when $m+1\le t\le 2m-1$. $d(u_0,v_{4t+2})=3+t$ and $d(u_0,v_{4t+2})<d(v_{8m},v_{4t+2})$ when $0\le t\le m-1$. $d(u_0,v_{4t+2})=d(v_{8m},v_{4t+2})=2m-t+3$ when $m\le t\le 2m-1$. $d(u_0,v_{4t+3})=3+t$ and $d(u_0,v_{4t+3})<d(v_{8m},v_{4t+3})$ when $0\le t\le m-1$. $d(u_0,v_{4t+3})=d(v_{8m},v_{4t+3})=2m-t+2$ when $m\le t\le 2m-1$. $d(u_0,u_{4t})=d(v_{8m},u_{4t})=2+t$ when $1\le t\le m-1$. $d(v_{8m},u_{4t})=2m-t+1$ and $d(u_0,u_{4t})>d(v_{8m},u_{4t})$ when $m\le t\le 2m$. $d(u_0,u_{1})=1$ and $d(v_{8m},u_{1})=2m+2$. $d(u_0,u_{4t+1})=d(v_{8m},u_{4t+1})=3+t$ when $1\le t\le m-1$. $d(v_{8m},u_{4t+1})=2m-t+2$ and $d(u_0,u_{4t+1})>d(v_{8m},u_{4t+1})$ when $m\le t\le 2m-1$. $d(u_0,u_{2})=2$ and $d(v_{8m},u_{2})=2m+2$. $d(u_0,u_{4t+2})=d(v_{8m},u_{4t+2})=4+t$ when $1\le t\le m-1$. $d(v_{8m},u_{4t+2})=2m-t+2$ and $d(u_0,u_{4t+2})>d(v_{8m},u_{4t+2})$ when $m\le t\le 2m-1$. $d(u_0,u_{3})=3$ and $d(v_{8m},u_{3})=2m+1$. $d(u_0,u_{4t+3})=d(v_{8m},u_{4t+3})=4+t$ when $1\le t\le m-2$. $d(v_{8m},u_{4t+3})=2m-t+1$ and $d(u_0,u_{4t+3})>d(v_{8m},u_{4t+3})$ when $m-1\le t\le 2m-1$. Note that $u_0\in W_{u_0v_{8m}}$ and $v_{8m}\in W_{v_{8m}u_0}$. Combined with the above discussion, $|W_{u_0v_{8m}}|=3m+9$ and $|W_{v_{8m}u_0}|=5m+4$. Because $m\ge 3$, $|W_{u_0v_{8m}}|<|W_{v_{8m}u_0}|$. \(6\) When $n=8m+5$ where $m\ge 3$. Note that $n-4=8m+1=4\times 2m+1$. $d(u_0,v_{4t})=d(v_{8m+1},v_{4t})=1+t$ when $0\le t\le m+1$. $d(u_0,v_{4t})=d(v_{8m+1},v_{4t})=2m-t+3$ when $m+2\le t\le 2m$. $d(u_0,v_{4t+1})=2+t$ and $d(u_0,v_{4t+1})<d(v_{8m+1},v_{4t+1})$ when $0\le t\le m-2$. $d(u_0,v_{4(m-1)+1})=d(v_{8m+1},v_{4(m-1)+1})=m+1$. $d(v_{8m+1},v_{4t+1})=2m-t$ and $d(u_0,v_{4t+1})>d(v_{8m+1},v_{4t+1})$ when $m\le t\le 2m-1$. $d(u_0,v_{4t+2})=3+t$ and $d(u_0,v_{4t+2})<d(v_{8m+1},v_{4t+2})$ when $0\le t\le m-1$. $d(u_0,v_{4t+2})=d(v_{8m+1},v_{4t+2})=2m-t+3$ when $m\le t\le 2m-1$. $d(u_0,v_{4t+3})=3+t$ and $d(u_0,v_{4t+3})<d(v_{8m+1},v_{4t+3})$ when $0\le t\le m-1$. $d(u_0,v_{4t+3})=d(v_{8m+1},v_{4t+3})=2m-t+3$ when $m\le t\le 2m-1$. $d(u_0,u_{4t})=d(v_{8m+1},u_{4t})=2+t$ when $1\le t\le m$. $d(v_{8m+1},u_{4t})=2m-t+2$ and $d(u_0,u_{4t})>d(v_{8m+1},u_{4t})$ when $m+1\le t\le 2m$. $d(u_0,u_{1})=1$ and $d(v_{8m+1},u_{1})=2m+1$. $d(u_0,u_{4t+1})=d(v_{8m+1},u_{4t+1})=3+t$ when $1\le t\le m-1$. $d(v_{8m+1},u_{4t+1})=2m-t+1$ and $d(u_0,u_{4t+1})>d(v_{8m+1},u_{4t+1})$ when $m\le t\le 2m$. $d(u_0,u_{2})=2$ and $d(v_{8m+1},u_{2})=2m+2$. $d(u_0,u_{4t+2})=d(v_{8m+1},u_{4t+2})=4+t$ when $1\le t\le m-1$. $d(v_{8m+1},u_{4t+2})=2m-t+2$ and $d(u_0,u_{4t+2})>d(v_{8m+1},u_{4t+2})$ when $m\le t\le 2m-1$. $d(u_0,u_{3})=3$ and $d(v_{8m+1},u_{3})=2m+2$. $d(u_0,u_{4t+3})=d(v_{8m+1},u_{4t+3})=4+t$ when $1\le t\le m-1$. $d(v_{8m+1},u_{4t+3})=2m-t+2$ and $d(u_0,u_{4t+3})>d(v_{8m+1},u_{4t+3})$ when $m\le t\le 2m-1$. Note that $u_0\in W_{u_0v_{8m+1}}$ and $v_{8m+1}\in W_{v_{8m+1}u_0}$. Combined with the above discussion, $|W_{u_0v_{8m+1}}|=3m+7$ and $|W_{v_{8m+1}u_0}|=5m+3$. Because $m\ge 3$, $|W_{u_0v_{8m+1}}|<|W_{v_{8m+1}u_0}|$. \(7\) When $n=8m+6$ where $m\ge 3$. Note that $n-4=8m+2=4\times 2m+2$. $d(u_0,v_{4t})=d(v_{8m+2},v_{4t})=1+t$ when $0\le t\le m+1$. $d(u_0,v_{4t})=d(v_{8m+2},v_{4t})=2m-t+4$ when $m+2\le t\le 2m$. $d(u_0,v_{4t+1})=2+t$ and $d(u_0,v_{4t+1})<d(v_{8m+2},v_{4t+1})$ when $0\le t\le m$. $d(u_0,v_{4t+1})=d(v_{8m+2},v_{4t+1})=2m-t+3$ when $m+1\le t\le 2m$. $d(u_0,v_{4t+2})=3+t$ and $d(u_0,v_{4t+2})<d(v_{8m+2},v_{4t+2})$ when $0\le t\le m-2$. $d(v_{8m+2},v_{4t+2})=2m-t$ and $d(u_0,v_{4t+2})>d(v_{8m+2},v_{4t+2})$ when $m-1\le t\le 2m-1$. $d(u_0,v_{4t+3})=3+t$ and $d(u_0,v_{4t+3})<d(v_{8m+2},v_{4t+3})$ when $0\le t\le m-1$. $d(u_0,v_{4t+3})=d(v_{8m+2},v_{4t+3})=2m-t+3$ when $m\le t\le 2m-1$. $d(u_0,u_{4t})=d(v_{8m+2},u_{4t})=2+t$ when $1\le t\le m$. $d(v_{8m+2},u_{4t})=2m-t+3$ and $d(u_0,u_{4t})>d(v_{8m+2},u_{4t})$ when $m+1\le t\le 2m$. $d(u_0,u_{1})=1$ and $d(v_{8m+2},u_{1})=2m+2$. $d(u_0,u_{4t+1})=d(v_{8m+2},u_{4t+1})=3+t$ when $1\le t\le m-1$. $d(v_{8m+2},u_{4t+1})=2m-t+2$ and $d(u_0,u_{4t+1})>d(v_{8m+2},u_{4t+1})$ when $m\le t\le 2m$. $d(u_0,u_{2})=2$ and $d(v_{8m+2},u_{2})=2m+1$. $d(u_0,u_{4t+2})=d(v_{8m+2},u_{4t+2})=4+t$ when $1\le t\le m-2$. $d(v_{8m+2},u_{4t+2})=2m-t+1$ and $d(u_0,u_{4t+2})>d(v_{8m+2},u_{4t+2})$ when $m-1\le t\le 2m$. $d(u_0,u_{3})=3$ and $d(v_{8m+2},u_{3})=2m+2$. $d(u_0,u_{4t+3})=d(v_{8m+2},u_{4t+3})=4+t$ when $1\le t\le m-1$. $d(v_{8m+2},u_{4t+3})=2m-t+2$ and $d(u_0,u_{4t+3})>d(v_{8m+2},u_{4t+3})$ when $m\le t\le 2m-1$. Note that $u_0\in W_{u_0v_{8m+2}}$ and $v_{8m+2}\in W_{v_{8m+2}u_0}$. Combined with the above discussion, $|W_{u_0v_{8m+2}}|=3m+8$ and $|W_{v_{8m+2}u_0}|=5m+6$. Because $m\ge 3$, $|W_{u_0v_{8m+2}}|<|W_{v_{8m+2}u_0}|$. \(8\) When $n=8m+7$ where $m\ge 3$. Note that $n-4=8m+3=4\times 2m+3$. $d(u_0,v_{4t})=d(v_{8m+3},v_{4t})=1+t$ when $0\le t\le m+1$. $d(u_0,v_{4t})=d(v_{8m+3},v_{4t})=2m-t+4$ when $m+2\le t\le 2m$. $d(u_0,v_{4t+1})=2+t$ and $d(u_0,v_{4t+1})<d(v_{8m+3},v_{4t+1})$ when $0\le t\le m$. $d(u_0,v_{4t+1})=d(v_{8m+3},v_{4t+1})=2m-t+4$ when $m+1\le t\le 2m$. $d(u_0,v_{4t+2})=3+t$ and $d(u_0,v_{4t+2})<d(v_{8m+3},v_{4t+2})$ when $0\le t\le m-1$. $d(u_0,v_{4t+2})=d(v_{8m+3},v_{4t+2})=2m-t+3$ when $m\le t\le 2m$. $d(u_0,v_{4t+3})=3+t$ and $d(u_0,v_{4t+3})<d(v_{8m+3},v_{4t+3})$ when $0\le t\le m-2$. $d(v_{8m+3},v_{4t+3})=2m-t$ and $d(u_0,v_{4t+3})>d(v_{8m+3},v_{4t+3})$ when $m-1\le t\le 2m-1$. $d(u_0,u_{4t})=d(v_{8m+3},u_{4t})=2+t$ when $1\le t\le m$. $d(v_{8m+3},u_{4t})=2m-t+3$ and $d(u_0,u_{4t})>d(v_{8m+3},u_{4t})$ when $m+1\le t\le 2m$. $d(u_0,u_{1})=1$ and $d(v_{8m+3},u_{1})=2m+3$. $d(u_0,u_{4t+1})=d(v_{8m+3},u_{4t+1})=3+t$ when $1\le t\le m$. $d(v_{8m+3},u_{4t+1})=2m-t+3$ and $d(u_0,u_{4t+1})>d(v_{8m+3},u_{4t+1})$ when $m+1\le t\le 2m$. $d(u_0,u_{2})=2$ and $d(v_{8m+3},u_{2})=2m+2$. $d(u_0,u_{4t+2})=d(v_{8m+3},u_{4t+2})=4+t$ when $1\le t\le m-1$. $d(v_{8m+3},u_{4t+2})=2m-t+2$ and $d(u_0,u_{4t+2})>d(v_{8m+3},u_{4t+2})$ when $m\le t\le 2m$. $d(u_0,u_{3})=3$ and $d(v_{8m+3},u_{3})=2m+1$. $d(u_0,u_{4t+3})=d(v_{8m+3},u_{4t+3})=4+t$ when $1\le t\le m-2$. $d(v_{8m+3},u_{4t+3})=2m-t+1$ and $d(u_0,u_{4t+3})>d(v_{8m+3},u_{4t+3})$ when $m-1\le t\le 2m$. Note that $u_0\in W_{u_0v_{8m+3}}$ and $v_{8m+3}\in W_{v_{8m+3}u_0}$. Combined with the above discussion, $|W_{u_0v_{8m+3}}|=3m+8$ and $|W_{v_{8m+3}u_0}|=5m+6$. Because $m\ge 3$, $|W_{u_0v_{8m+3}}|<|W_{v_{8m+3}u_0}|$. ◻ **Proposition 10**. *For any $n>24$, the generalized Petersen graph $GP(n,4)$ is not $\ell$-distance-balanced for any $3\le \ell<{\rm diam}(GP(n,4))$.* *Proof.* For any $3\le\ell<D$, we first show that there exists $v_j$ such that $d(u_0,v_j)=\ell$ where $8\le j\le n/2$. From [@MaG:2023], there exists $j^*$ such that $d(u_0,u_{j^*})=D$. When $n=8m$ $(m\ge 4)$ or $n=8m+1$ $(m\ge 3)$, from [@MaG:2023] we know that $j^*=4(m-1)+2$ and $D=d(u_0,u_{j^*})=m+3$. Note that $d(u_0,v_{4s+2})=s+3$ where $2\le s\le m-1$ and $d(u_0,v_{4s})=s+1$ where $2\le s\le m$. When $n=8m+2$ $(m\ge 3)$ or $n=8m+3$ $(m\ge 3)$, from [@MaG:2023] we know that $j^*=4m+1$ and $D=d(u_0,u_{j^*})=m+3$. Note that $d(u_0,v_{4s+1})=s+2$ where $3\le s\le m$ and $d(u_0,v_{4s})=s+1$ where $2\le s\le m$. When $n=8m+4$ $(m\ge 3)$ or $n=8m+5$ $(m\ge 3)$, from [@MaG:2023] we know that $j^*=4m+2$ and $D=d(u_0,u_{j^*})=m+4$. Note that $d(u_0,v_{4s+2})=s+3$ where $2\le s\le m$ and $d(u_0,v_{4s})=s+1$ where $2\le s\le m$. When $n=8m+6$ $(m\ge 3)$, from [@MaG:2023] we know that $j^*=4m+3$ and $D=d(u_0,u_{j^*})=m+4$. Note that $d(u_0,v_{4s+3})=s+3$ where $2\le s\le m$ and $d(u_0,v_{4s})=s+1$ where $2\le s\le m$. When $n=8m+7$ $(m\ge 3)$, from [@MaG:2023] we know that $j^*=4m+2$ and $D=d(u_0,u_{j^*})=m+4$. Note that $d(u_0,v_{4s+2})=s+3$ where $2\le s\le m$ and $d(u_0,v_{4s})=s+1$ where $2\le s\le m$. From the above discussion, there exists $j$ where $8\le j\le n/2$ such that $d(u_0,v_j)=\ell$ for any $3\le\ell<D$. Let $V_1=\{u_i\mid 1\le i\le j-1\}\cup\{v_i\mid 1\le i\le j-1\}$ and $V_2=\{u_i\mid j+1\le i\le n-1\}\cup\{v_i\mid j+1\le i\le n-1\}$. Let $W^1_{u_0v_j}$ be the set of vertices which are in $W_{u_0v_j}$ and also in $V_1\cup\{u_0,v_0,u_j,v_j\}$. Let $W^1_{v_ju_0}$ be the set of vertices which are in $W_{v_ju_0}$ and also in $V_1\cup\{u_0,v_0,u_j,v_j\}$. Let $W^2_{u_0v_j}$ be the set of vertices which are in $W_{u_0v_j}$ and also in $V_2\cup\{u_0,v_0,u_j,v_j\}$. Let $W^2_{v_ju_0}$ be the set of vertices which are in $W_{v_ju_0}$ and also in $V_2\cup\{u_0,v_0,u_j,v_j\}$. Because $8\le j\le n/2$, $|W^2_{u_0v_j}|=|W^1_{u_0v_{n-j}}|$ and $|W^2_{v_ju_0}|=|W^1_{v_{n-j}u_0}|$. So $|W_{u_0v_j}|=|W^1_{u_0v_j}|+|W^2_{u_0v_j}|-2=|W^1_{u_0v_j}|+|W^1_{u_0v_{n-j}}|-2$ and $|W_{v_ju_0}|=|W^1_{v_ju_0}|+|W^2_{v_ju_0}|-2=|W^1_{v_ju_0}|+|W^1_{v_{n-j}u_0}|-2$. In the following we will compute $|W^1_{u_0v_j}|$ and $|W^1_{v_ju_0}|$ where $8\le j\le n-8$. The discussion is divided into the following eight cases. (1a) Computation of $|W^1_{u_0v_{4s}}|$ and $|W^1_{v_{4s}u_0}|$ when $s$ is odd and $s\ge 3$. When $s=3$, $d(u_0,v_0)=1$ and $d(v_{12},v_0)=3$. $d(u_0,v_4)=d(v_{12},v_4)=2$. $d(u_0,v_8)=3$ and $d(v_{12},v_8)=1$. $d(u_0,v_1)=2$ and $d(v_{12},v_1)=6$. $d(u_0,v_5)=3$ and $d(v_{12},v_5)=5$. $d(u_0,v_9)=4$ and $d(v_{12},v_9)=4$. $d(u_0,v_2)=3$ and $d(v_{12},v_2)=6$. $d(u_0,v_6)=4$ and $d(v_{12},v_6)=5$. $d(u_0,v_{10})=5$ and $d(v_{12},v_{10})=4$. $d(u_0,v_3)=3$ and $d(v_{12},v_3)=5$. $d(u_0,v_7)=4$ and $d(v_{12},v_7)=4$. $d(u_0,v_{11})=5$ and $d(v_{12},v_{11})=3$. So $v_0,v_1,v_2,v_3,v_5,v_6\in W^1_{u_0v_{12}}$ and $v_8,v_{10},v_{11}\in W^1_{v_{12}u_0}$. $d(u_0,u_4)=3$ and $d(v_{12},u_4)=3$. $d(u_0,u_8)=4$ and $d(v_{12},u_8)=2$. $d(u_0,u_{12})=5$ and $d(v_{12},u_{12})=1$. $d(u_0,u_1)=1$ and $d(v_{12},u_1)=5$. $d(u_0,u_5)=4$ and $d(v_{12},u_5)=4$. $d(u_0,u_9)=5$ and $d(v_{12},u_9)=3$. $d(u_0,u_2)=2$ and $d(v_{12},u_2)=5$. $d(u_0,u_6)=5$ and $d(v_{12},u_6)=4$. $d(u_0,u_{10})=6$ and $d(v_{12},u_{10})=3$. $d(u_0,u_3)=3$ and $d(v_{12},u_3)=4$. $d(u_0,u_7)=5$ and $d(v_{12},u_7)=3$. $d(u_0,u_{11})=6$ and $d(v_{12},u_{11})=2$. So $u_1,u_2,u_3\in W^1_{u_0v_{12}}$ and $u_6,u_7,u_8,u_9,u_{10},u_{11},u_{12}\in W^1_{v_{12}u_0}$. Note that $u_0\in W^1_{u_0v_{12}}$ and $v_{12}\in W^1_{v_{12}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{12}}|=10$ and $|W^1_{v_{12}u_0}|=11$. When $s\ge 5$, $d(u_0,v_{4t})=1+t$ and $d(v_{4s},v_{4t})=s-t$ where $0\le t<s$. When $0\le t<\frac{s-1}{2}$, $d(u_0,v_{4t})<d(v_{4s},v_{4t})$. When $\frac{s-1}{2}<t<s$, $d(u_0,v_{4t})>d(v_{4s},v_{4t})$. $d(u_0,v_{4t+1})=2+t$ and $d(v_{4s},v_{4t+1})=s-t+3$ where $0\le t<s$. When $0\le t<\frac{s+1}{2}$, $d(u_0,v_{4t+1})<d(v_{4s},v_{4t+1})$. When $\frac{s+1}{2}<t<s$, $d(u_0,v_{4t+1})>d(v_{4s},v_{4t+1})$. $d(u_0,v_{4t+2})=3+t$ and $d(v_{4s},v_{4t+2})=s-t+3$ where $0\le t<s$. When $0\le t\le\frac{s-1}{2}$, $d(u_0,v_{4t+2})<d(v_{4s},v_{4t+2})$. When $\frac{s+1}{2}\le t<s$, $d(u_0,v_{4t+2})>d(v_{4s},v_{4t+2})$. $d(u_0,v_{4t+3})=3+t$ and $d(v_{4s},v_{4t+3})=s-t+2$ where $0\le t<s$. When $0\le t<\frac{s-1}{2}$, $d(u_0,v_{4t+3})<d(v_{4s},v_{4t+3})$. When $\frac{s-1}{2}< t<s$, $d(u_0,v_{4t+3})>d(v_{4s},v_{4t+3})$. $d(u_0,u_{4t})=2+t$ and $d(v_{4s},u_{4t})=s-t+1$ where $1\le t\le s$. When $1\le t<\frac{s-1}{2}$, $d(u_0,u_{4t})<d(v_{4s},u_{4t})$. When $\frac{s-1}{2}<t\le s$, $d(u_0,u_{4t})>d(v_{4s},u_{4t})$. $d(u_0,u_{1})=1$ and $d(v_{4s},u_{1})=s+2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_{4s},u_{4t+1})=s-t+2$ where $1\le t< s$. When $1\le t<\frac{s-1}{2}$, $d(u_0,u_{4t+1})<d(v_{4s},u_{4t+1})$. When $\frac{s-1}{2}<t< s$, $d(u_0,u_{4t+1})>d(v_{4s},u_{4t+1})$. $d(u_0,u_{2})=2$ and $d(v_{4s},u_{2})=s+2$. $d(u_0,u_{4t+2})=4+t$ and $d(v_{4s},u_{4t+2})=s-t+2$ where $1\le t< s$. When $1\le t\le\frac{s-3}{2}$, $d(u_0,u_{4t+2})<d(v_{4s},u_{4t+2})$. When $\frac{s-1}{2}\le t< s$, $d(u_0,u_{4t+2})>d(v_{4s},u_{4t+2})$. $d(u_0,u_{3})=3$ and $d(v_{4s},u_{3})=s+1$. $d(u_0,u_{4t+3})=4+t$ and $d(v_{4s},u_{4t+3})=s-t+1$ where $1\le t< s$. When $1\le t<\frac{s-3}{2}$, $d(u_0,u_{4t+3})<d(v_{4s},u_{4t+3})$. When $\frac{s-3}{2}< t< s$, $d(u_0,u_{4t+3})>d(v_{4s},u_{4t+3})$. Note that $u_0\in W^1_{u_0v_{4s}}$ and $v_{4s}\in W^1_{v_{4s}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{4s}}|=4s-3$ and $|W^1_{v_{4s}u_0}|=4s-1$. (1b) Computation of $|W^1_{u_0v_{4s}}|$ and $|W^1_{v_{4s}u_0}|$ when $s$ is even and $s\ge 2$. When $s=2$, $d(u_0,v_0)=1$ and $d(v_{8},v_0)=2$. $d(u_0,v_4)=2$ and $d(v_{8},v_4)=1$. $d(u_0,v_1)=2$ and $d(v_{8},v_1)=5$. $d(u_0,v_5)=3$ and $d(v_{8},v_5)=4$. $d(u_0,v_2)=3$ and $d(v_{8},v_2)=5$. $d(u_0,v_6)=4$ and $d(v_{8},v_6)=4$. $d(u_0,v_3)=3$ and $d(v_{8},v_3)=4$. $d(u_0,v_7)=4$ and $d(v_{8},v_7)=3$. So $v_0,v_1,v_2,v_3,v_5\in W^1_{u_0v_{8}}$ and $v_4,v_{7}\in W^1_{v_{8}u_0}$. $d(u_0,u_4)=3$ and $d(v_{8},u_4)=2$. $d(u_0,u_8)=4$ and $d(v_{8},u_8)=1$. $d(u_0,u_1)=1$ and $d(v_{8},u_1)=4$. $d(u_0,u_5)=4$ and $d(v_{8},u_5)=3$. $d(u_0,u_2)=2$ and $d(v_{8},u_2)=4$. $d(u_0,u_6)=5$ and $d(v_{8},u_6)=3$. $d(u_0,u_3)=3$ and $d(v_{8},u_3)=3$. $d(u_0,u_7)=5$ and $d(v_{8},u_7)=2$. So $u_1,u_2\in W^1_{u_0v_{8}}$ and $u_4,u_5,u_6,u_7,u_{8}\in W^1_{v_{8}u_0}$. Note that $u_0\in W^1_{u_0v_{8}}$ and $v_{8}\in W^1_{v_{8}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{8}}|=8$ and $|W^1_{v_{8}u_0}|=8$. When $s\ge 4$, $d(u_0,v_{4t})=1+t$ and $d(v_{4s},v_{4t})=s-t$ where $0\le t<s$. When $0\le t\le\frac{s-2}{2}$, $d(u_0,v_{4t})<d(v_{4s},v_{4t})$. When $\frac{s}{2}\le t<s$, $d(u_0,v_{4t})>d(v_{4s},v_{4t})$. $d(u_0,v_{4t+1})=2+t$ and $d(v_{4s},v_{4t+1})=s-t+3$ where $0\le t<s$. When $0\le t\le\frac{s}{2}$, $d(u_0,v_{4t+1})<d(v_{4s},v_{4t+1})$. When $\frac{s+2}{2}\le t<s$, $d(u_0,v_{4t+1})>d(v_{4s},v_{4t+1})$. $d(u_0,v_{4t+2})=3+t$ and $d(v_{4s},v_{4t+2})=s-t+3$ where $0\le t<s$. When $0\le t<\frac{s}{2}$, $d(u_0,v_{4t+2})<d(v_{4s},v_{4t+2})$. When $\frac{s}{2}< t<s$, $d(u_0,v_{4t+2})>d(v_{4s},v_{4t+2})$. $d(u_0,v_{4t+3})=3+t$ and $d(v_{4s},v_{4t+3})=s-t+2$ where $0\le t<s$. When $0\le t\le\frac{s-2}{2}$, $d(u_0,v_{4t+3})<d(v_{4s},v_{4t+3})$. When $\frac{s}{2}\le t<s$, $d(u_0,v_{4t+3})>d(v_{4s},v_{4t+3})$. $d(u_0,u_{4t})=2+t$ and $d(v_{4s},u_{4t})=s-t+1$ where $1\le t\le s$. When $1\le t\le\frac{s-2}{2}$, $d(u_0,u_{4t})<d(v_{4s},u_{4t})$. When $\frac{s}{2}\le t\le s$, $d(u_0,u_{4t})>d(v_{4s},u_{4t})$. $d(u_0,u_{1})=1$ and $d(v_{4s},u_{1})=s+2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_{4s},u_{4t+1})=s-t+2$ where $1\le t< s$. When $1\le t\le\frac{s-2}{2}$, $d(u_0,u_{4t+1})<d(v_{4s},u_{4t+1})$. When $\frac{s}{2}\le t< s$, $d(u_0,u_{4t+1})>d(v_{4s},u_{4t+1})$. $d(u_0,u_{2})=2$ and $d(v_{4s},u_{2})=s+2$. $d(u_0,u_{4t+2})=4+t$ and $d(v_{4s},u_{4t+2})=s-t+2$ where $1\le t< s$. When $1\le t<\frac{s-2}{2}$, $d(u_0,u_{4t+2})<d(v_{4s},u_{4t+2})$. When $\frac{s-2}{2}< t< s$, $d(u_0,u_{4t+2})>d(v_{4s},u_{4t+2})$. $d(u_0,u_{3})=3$ and $d(v_{4s},u_{3})=s+1$. $d(u_0,u_{4t+3})=4+t$ and $d(v_{4s},u_{4t+3})=s-t+1$ where $1\le t< s$. When $1\le t\le\frac{s-4}{2}$, $d(u_0,u_{4t+3})<d(v_{4s},u_{4t+3})$. When $\frac{s-2}{2}\le t< s$, $d(u_0,u_{4t+3})>d(v_{4s},u_{4t+3})$. Note that $u_0\in W^1_{u_0v_{4s}}$ and $v_{4s}\in W^1_{v_{4s}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{4s}}|=4s-1$ and $|W^1_{v_{4s}u_0}|=4s+1$. (2a) Computation of $|W^1_{u_0v_{4s+1}}|$ and $|W^1_{v_{4s+1}u_0}|$ when $s$ is odd and $s\ge 3$. $d(u_0,v_{4t})=1+t$ and $d(v_{4s+1},v_{4t})=s-t+3$ where $0\le t\le s$. When $0\le t\le\frac{s+1}{2}$, $d(u_0,v_{4t})<d(v_{4s+1},v_{4t})$. When $\frac{s+3}{2}\le t\le s$, $d(u_0,v_{4t})>d(v_{4s+1},v_{4t})$. $d(u_0,v_{4t+1})=2+t$ and $d(v_{4s+1},v_{4t+1})=s-t$ where $0\le t<s$. When $0\le t\le\frac{s-3}{2}$, $d(u_0,v_{4t+1})<d(v_{4s+1},v_{4t+1})$. When $\frac{s-1}{2}\le t<s$, $d(u_0,v_{4t+1})>d(v_{4s+1},v_{4t+1})$. $d(u_0,v_{4t+2})=3+t$ and $d(v_{4s+1},v_{4t+2})=s-t+3$ where $0\le t<s$. When $0\le t\le\frac{s-1}{2}$, $d(u_0,v_{4t+2})<d(v_{4s+1},v_{4t+2})$. When $\frac{s+1}{2}\le t<s$, $d(u_0,v_{4t+2})>d(v_{4s+1},v_{4t+2})$. $d(u_0,v_{4t+3})=3+t$ and $d(v_{4s+1},v_{4t+3})=s-t+3$ where $0\le t<s$. When $0\le t\le\frac{s-1}{2}$, $d(u_0,v_{4t+3})<d(v_{4s+1},v_{4t+3})$. When $\frac{s+1}{2}\le t<s$, $d(u_0,v_{4t+3})>d(v_{4s+1},v_{4t+3})$. $d(u_0,u_{4t})=2+t$ and $d(v_{4s+1},u_{4t})=s-t+2$ where $1\le t\le s$. When $1\le t\le\frac{s-1}{2}$, $d(u_0,u_{4t})<d(v_{4s+1},u_{4t})$. When $\frac{s+1}{2}\le t\le s$, $d(u_0,u_{4t})>d(v_{4s+1},u_{4t})$. $d(u_0,u_{1})=1$ and $d(v_{4s+1},u_{1})=s+1$. $d(u_0,u_{4t+1})=3+t$ and $d(v_{4s+1},u_{4t+1})=s-t+1$ where $1\le t\le s$. When $1\le t\le\frac{s-3}{2}$, $d(u_0,u_{4t+1})<d(v_{4s+1},u_{4t+1})$. When $\frac{s-1}{2}\le t\le s$, $d(u_0,u_{4t+1})>d(v_{4s+1},u_{4t+1})$. $d(u_0,u_{2})=2$ and $d(v_{4s+1},u_{2})=s+2$. $d(u_0,u_{4t+2})=4+t$ and $d(v_{4s+1},u_{4t+2})=s-t+2$ where $1\le t< s$. When $1\le t\le\frac{s-3}{2}$, $d(u_0,u_{4t+2})<d(v_{4s+1},u_{4t+2})$. When $\frac{s-1}{2}\le t< s$, $d(u_0,u_{4t+2})>d(v_{4s+1},u_{4t+2})$. $d(u_0,u_{3})=3$ and $d(v_{4s+1},u_{3})=s+2$. $d(u_0,u_{4t+3})=4+t$ and $d(v_{4s+1},u_{4t+3})=s-t+2$ where $1\le t< s$. When $1\le t\le\frac{s-3}{2}$, $d(u_0,u_{4t+3})<d(v_{4s+1},u_{4t+3})$. When $\frac{s-1}{2}\le t< s$, $d(u_0,u_{4t+3})>d(v_{4s+1},u_{4t+3})$. Note that $u_0\in W^1_{u_0v_{4s+1}}$ and $v_{4s+1}\in W^1_{v_{4s+1}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{4s+1}}|=4s+1$ and $|W^1_{v_{4s+1}u_0}|=4s+3$. (2b) Computation of $|W^1_{u_0v_{4s+1}}|$ and $|W^1_{v_{4s+1}u_0}|$ when $s$ is even and $s\ge 4$. When $s\ge 4$, $d(u_0,v_{4t})=1+t$ and $d(v_{4s+1},v_{4t})=s-t+3$ where $0\le t\le s$. When $0\le t<\frac{s+2}{2}$, $d(u_0,v_{4t})<d(v_{4s+1},v_{4t})$. When $\frac{s+2}{2}< t\le s$, $d(u_0,v_{4t})>d(v_{4s+1},v_{4t})$. $d(u_0,v_{4t+1})=2+t$ and $d(v_{4s+1},v_{4t+1})=s-t$ where $0\le t<s$. When $0\le t<\frac{s-2}{2}$, $d(u_0,v_{4t+1})<d(v_{4s+1},v_{4t+1})$. When $\frac{s-2}{2}< t<s$, $d(u_0,v_{4t+1})>d(v_{4s+1},v_{4t+1})$. $d(u_0,v_{4t+2})=3+t$ and $d(v_{4s+1},v_{4t+2})=s-t+3$ where $0\le t<s$. When $0\le t<\frac{s}{2}$, $d(u_0,v_{4t+2})<d(v_{4s+1},v_{4t+2})$. When $\frac{s}{2}< t<s$, $d(u_0,v_{4t+2})>d(v_{4s+1},v_{4t+2})$. $d(u_0,v_{4t+3})=3+t$ and $d(v_{4s+1},v_{4t+3})=s-t+3$ where $0\le t<s$. When $0\le t<\frac{s}{2}$, $d(u_0,v_{4t+3})<d(v_{4s+1},v_{4t+3})$. When $\frac{s}{2}< t<s$, $d(u_0,v_{4t+3})>d(v_{4s+1},v_{4t+3})$. $d(u_0,u_{4t})=2+t$ and $d(v_{4s+1},u_{4t})=s-t+2$ where $1\le t\le s$. When $1\le t<\frac{s}{2}$, $d(u_0,u_{4t})<d(v_{4s+1},u_{4t})$. When $\frac{s}{2}< t\le s$, $d(u_0,u_{4t})>d(v_{4s+1},u_{4t})$. $d(u_0,u_{1})=1$ and $d(v_{4s+1},u_{1})=s+1$. $d(u_0,u_{4t+1})=3+t$ and $d(v_{4s+1},u_{4t+1})=s-t+1$ where $1\le t\le s$. When $1\le t<\frac{s-2}{2}$, $d(u_0,u_{4t+1})<d(v_{4s+1},u_{4t+1})$. When $\frac{s-2}{2}< t\le s$, $d(u_0,u_{4t+1})>d(v_{4s+1},u_{4t+1})$. $d(u_0,u_{2})=2$ and $d(v_{4s+1},u_{2})=s+2$. $d(u_0,u_{4t+2})=4+t$ and $d(v_{4s+1},u_{4t+2})=s-t+2$ where $1\le t< s$. When $1\le t<\frac{s-2}{2}$, $d(u_0,u_{4t+2})<d(v_{4s+1},u_{4t+2})$. When $\frac{s-2}{2}< t< s$, $d(u_0,u_{4t+2})>d(v_{4s+1},u_{4t+2})$. $d(u_0,u_{3})=3$ and $d(v_{4s+1},u_{3})=s+2$. $d(u_0,u_{4t+3})=4+t$ and $d(v_{4s+1},u_{4t+3})=s-t+2$ where $1\le t< s$. When $1\le t<\frac{s-2}{2}$, $d(u_0,u_{4t+3})<d(v_{4s+1},u_{4t+3})$. When $\frac{s-2}{2}< t< s$, $d(u_0,u_{4t+3})>d(v_{4s+1},u_{4t+3})$. Note that $u_0\in W^1_{u_0v_{4s+1}}$ and $v_{4s+1}\in W^1_{v_{4s+1}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{4s+1}}|=4s-3$ and $|W^1_{v_{4s+1}u_0}|=4s-1$. (3a) Computation of $|W^1_{u_0v_{4s+2}}|$ and $|W^1_{v_{4s+2}u_0}|$ when $s$ is odd and $s\ge 3$. When $s=3$, $d(u_0,v_0)=1$ and $d(v_{14},v_0)=7$. $d(u_0,v_4)=2$ and $d(v_{14},v_4)=6$. $d(u_0,v_8)=3$ and $d(v_{14},v_8)=5$. $d(u_0,v_{12})=4$ and $d(v_{14},v_{12})=4$. $d(u_0,v_1)=2$ and $d(v_{14},v_1)=6$. $d(u_0,v_5)=3$ and $d(v_{14},v_5)=5$. $d(u_0,v_9)=4$ and $d(v_{14},v_9)=4$. $d(u_0,v_{13})=5$ and $d(v_{14},v_{13})=3$. $d(u_0,v_2)=3$ and $d(v_{14},v_2)=3$. $d(u_0,v_6)=4$ and $d(v_{14},v_6)=2$. $d(u_0,v_{10})=5$ and $d(v_{14},v_{10})=1$. $d(u_0,v_3)=3$ and $d(v_{14},v_3)=6$. $d(u_0,v_7)=4$ and $d(v_{14},v_7)=5$. $d(u_0,v_{11})=5$ and $d(v_{14},v_{11})=4$. So $v_0,v_1,v_3,v_4,v_5,v_7,v_8\in W^1_{u_0v_{14}}$ and $v_6,v_{10},v_{11},v_{13}\in W^1_{v_{14}u_0}$. $d(u_0,u_4)=3$ and $d(v_{14},u_4)=5$. $d(u_0,u_8)=4$ and $d(v_{14},u_8)=4$. $d(u_0,u_{12})=5$ and $d(v_{14},u_{12})=3$. $d(u_0,u_1)=1$ and $d(v_{14},u_1)=5$. $d(u_0,u_5)=4$ and $d(v_{14},u_5)=4$. $d(u_0,u_9)=5$ and $d(v_{14},u_9)=3$. $d(u_0,u_{13})=6$ and $d(v_{14},u_{13})=2$. $d(u_0,u_2)=2$ and $d(v_{14},u_2)=4$. $d(u_0,u_6)=5$ and $d(v_{14},u_6)=3$. $d(u_0,u_{10})=6$ and $d(v_{14},u_{10})=2$. $d(u_0,u_{14})=7$ and $d(v_{14},u_{14})=1$. $d(u_0,u_3)=3$ and $d(v_{14},u_3)=5$. $d(u_0,u_7)=5$ and $d(v_{14},u_7)=4$. $d(u_0,u_{11})=6$ and $d(v_{14},u_{11})=3$. So $u_1,u_2,u_3,u_4\in W^1_{u_0v_{14}}$ and $u_6,u_7,u_9,u_{10},u_{11},u_{12},u_{13},u_{14}\in W^1_{v_{14}u_0}$. Note that $u_0\in W^1_{u_0v_{14}}$ and $v_{14}\in W^1_{v_{14}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{14}}|=12$ and $|W^1_{v_{14}u_0}|=13$. When $s\ge 5$, $d(u_0,v_{4t})=1+t$ and $d(v_{4s+2},v_{4t})=s-t+4$ where $0\le t\le s$. When $0\le t<\frac{s+3}{2}$, $d(u_0,v_{4t})<d(v_{4s+2},v_{4t})$. When $\frac{s+3}{2}< t\le s$, $d(u_0,v_{4t})>d(v_{4s+2},v_{4t})$. $d(u_0,v_{4t+1})=2+t$ and $d(v_{4s+2},v_{4t+1})=s-t+3$ where $0\le t\le s$. When $0\le t<\frac{s+1}{2}$, $d(u_0,v_{4t+1})<d(v_{4s+2},v_{4t+1})$. When $\frac{s+1}{2}< t\le s$, $d(u_0,v_{4t+1})>d(v_{4s+2},v_{4t+1})$. $d(u_0,v_{4t+2})=3+t$ and $d(v_{4s+2},v_{4t+2})=s-t$ where $0\le t<s$. When $0\le t<\frac{s-3}{2}$, $d(u_0,v_{4t+2})<d(v_{4s+2},v_{4t+2})$. When $\frac{s-3}{2}< t<s$, $d(u_0,v_{4t+2})>d(v_{4s+2},v_{4t+2})$. $d(u_0,v_{4t+3})=3+t$ and $d(v_{4s+2},v_{4t+3})=s-t+3$ where $0\le t<s$. When $0\le t\le\frac{s-1}{2}$, $d(u_0,v_{4t+3})<d(v_{4s+2},v_{4t+3})$. When $\frac{s+1}{2}\le t<s$, $d(u_0,v_{4t+3})>d(v_{4s+2},v_{4t+3})$. $d(u_0,u_{4t})=2+t$ and $d(v_{4s+2},u_{4t})=s-t+3$ where $1\le t\le s$. When $1\le t<\frac{s+1}{2}$, $d(u_0,u_{4t})<d(v_{4s+2},u_{4t})$. When $\frac{s+1}{2}< t\le s$, $d(u_0,u_{4t})>d(v_{4s+2},u_{4t})$. $d(u_0,u_{1})=1$ and $d(v_{4s+2},u_{1})=s+2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_{4s+2},u_{4t+1})=s-t+2$ where $1\le t\le s$. When $1\le t<\frac{s-1}{2}$, $d(u_0,u_{4t+1})<d(v_{4s+2},u_{4t+1})$. When $\frac{s-1}{2}< t\le s$, $d(u_0,u_{4t+1})>d(v_{4s+2},u_{4t+1})$. $d(u_0,u_{2})=2$ and $d(v_{4s+2},u_{2})=s+1$. $d(u_0,u_{4t+2})=4+t$ and $d(v_{4s+2},u_{4t+2})=s-t+1$ where $1\le t\le s$. When $1\le t<\frac{s-3}{2}$, $d(u_0,u_{4t+2})<d(v_{4s+2},u_{4t+2})$. When $\frac{s-3}{2}< t\le s$, $d(u_0,u_{4t+2})>d(v_{4s+2},u_{4t+2})$. $d(u_0,u_{3})=3$ and $d(v_{4s+2},u_{3})=s+2$. $d(u_0,u_{4t+3})=4+t$ and $d(v_{4s+2},u_{4t+3})=s-t+2$ where $1\le t< s$. When $1\le t\le\frac{s-3}{2}$, $d(u_0,u_{4t+3})<d(v_{4s+2},u_{4t+3})$. When $\frac{s-1}{2}\le t< s$, $d(u_0,u_{4t+3})>d(v_{4s+2},u_{4t+3})$. Note that $u_0\in W^1_{u_0v_{4s+2}}$ and $v_{4s+2}\in W^1_{v_{4s+2}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{4s+2}}|=4s-1$ and $|W^1_{v_{4s+2}u_0}|=4s+1$. (3b) Computation of $|W^1_{u_0v_{4s+2}}|$ and $|W^1_{v_{4s+2}u_0}|$ when $s$ is even and $s\ge 2$. When $s=2$, $d(u_0,v_0)=1$ and $d(v_{10},v_0)=6$. $d(u_0,v_4)=2$ and $d(v_{10},v_4)=5$. $d(u_0,v_8)=3$ and $d(v_{10},v_8)=4$. $d(u_0,v_1)=2$ and $d(v_{10},v_1)=5$. $d(u_0,v_5)=3$ and $d(v_{10},v_5)=4$. $d(u_0,v_9)=4$ and $d(v_{10},v_9)=3$. $d(u_0,v_2)=3$ and $d(v_{10},v_2)=2$. $d(u_0,v_6)=4$ and $d(v_{10},v_6)=1$. $d(u_0,v_3)=3$ and $d(v_{10},v_3)=5$. $d(u_0,v_7)=4$ and $d(v_{10},v_7)=4$. So $v_0,v_1,v_3,v_4,v_5,v_8\in W^1_{u_0v_{10}}$ and $v_2,v_{6},v_9\in W^1_{v_{10}u_0}$. $d(u_0,u_4)=3$ and $d(v_{10},u_4)=4$. $d(u_0,u_8)=4$ and $d(v_{10},u_8)=3$. $d(u_0,u_1)=1$ and $d(v_{10},u_1)=4$. $d(u_0,u_5)=4$ and $d(v_{10},u_5)=3$. $d(u_0,u_9)=5$ and $d(v_{10},u_9)=2$. $d(u_0,u_2)=2$ and $d(v_{10},u_2)=3$. $d(u_0,u_6)=5$ and $d(v_{10},u_6)=2$. $d(u_0,u_{10})=6$ and $d(v_{10},u_{10})=1$. $d(u_0,u_3)=3$ and $d(v_{10},u_3)=4$. $d(u_0,u_7)=5$ and $d(v_{10},u_7)=3$. So $u_1,u_2,u_3,u_4\in W^1_{u_0v_{10}}$ and $u_5,u_6,u_7,u_{8},u_9,u_{10}\in W^1_{v_{10}u_0}$. Note that $u_0\in W^1_{u_0v_{10}}$ and $v_{8}\in W^1_{v_{10}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{10}}|=11$ and $|W^1_{v_{10}u_0}|=10$. When $s\ge 4$, $d(u_0,v_{4t})=1+t$ and $d(v_{4s+2},v_{4t})=s-t+4$ where $0\le t\le s$. When $0\le t\le\frac{s+2}{2}$, $d(u_0,v_{4t})<d(v_{4s+2},v_{4t})$. When $\frac{s+4}{2}\le t\le s$, $d(u_0,v_{4t})>d(v_{4s+2},v_{4t})$. $d(u_0,v_{4t+1})=2+t$ and $d(v_{4s+2},v_{4t+1})=s-t+3$ where $0\le t\le s$. When $0\le t\le\frac{s}{2}$, $d(u_0,v_{4t+1})<d(v_{4s+2},v_{4t+1})$. When $\frac{s+2}{2}\le t\le s$, $d(u_0,v_{4t+1})>d(v_{4s+2},v_{4t+1})$. $d(u_0,v_{4t+2})=3+t$ and $d(v_{4s+2},v_{4t+2})=s-t$ where $0\le t<s$. When $0\le t\le\frac{s-4}{2}$, $d(u_0,v_{4t+2})<d(v_{4s+2},v_{4t+2})$. When $\frac{s-2}{2}\le t<s$, $d(u_0,v_{4t+2})>d(v_{4s+2},v_{4t+2})$. $d(u_0,v_{4t+3})=3+t$ and $d(v_{4s+2},v_{4t+3})=s-t+3$ where $0\le t<s$. When $0\le t<\frac{s}{2}$, $d(u_0,v_{4t+3})<d(v_{4s+2},v_{4t+3})$. When $\frac{s}{2}< t<s$, $d(u_0,v_{4t+3})>d(v_{4s+2},v_{4t+3})$. $d(u_0,u_{4t})=2+t$ and $d(v_{4s+2},u_{4t})=s-t+3$ where $1\le t\le s$. When $1\le t\le\frac{s}{2}$, $d(u_0,u_{4t})<d(v_{4s+2},u_{4t})$. When $\frac{s+2}{2}\le t\le s$, $d(u_0,u_{4t})>d(v_{4s+2},u_{4t})$. $d(u_0,u_{1})=1$ and $d(v_{4s+2},u_{1})=s+2$. $d(u_0,u_{4t+1})=3+t$ and $d(v_{4s+2},u_{4t+1})=s-t+2$ where $1\le t\le s$. When $1\le t\le\frac{s-2}{2}$, $d(u_0,u_{4t+1})<d(v_{4s+2},u_{4t+1})$. When $\frac{s}{2}\le t\le s$, $d(u_0,u_{4t+1})>d(v_{4s+2},u_{4t+1})$. $d(u_0,u_{2})=2$ and $d(v_{4s+2},u_{2})=s+1$. $d(u_0,u_{4t+2})=4+t$ and $d(v_{4s+2},u_{4t+2})=s-t+1$ where $1\le t\le s$. When $1\le t\le\frac{s-4}{2}$, $d(u_0,u_{4t+2})<d(v_{4s+2},u_{4t+2})$. When $\frac{s-2}{2}\le t\le s$, $d(u_0,u_{4t+2})>d(v_{4s+2},u_{4t+2})$. $d(u_0,u_{3})=3$ and $d(v_{4s+2},u_{3})=s+2$. $d(u_0,u_{4t+3})=4+t$ and $d(v_{4s+2},u_{4t+3})=s-t+2$ where $1\le t< s$. When $1\le t<\frac{s-2}{2}$, $d(u_0,u_{4t+3})<d(v_{4s+2},u_{4t+3})$. When $\frac{s-2}{2}< t< s$, $d(u_0,u_{4t+3})>d(v_{4s+2},u_{4t+3})$. Note that $u_0\in W^1_{u_0v_{4s+2}}$ and $v_{4s+2}\in W^1_{v_{4s+2}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{4s+2}}|=4s+1$ and $|W^1_{v_{4s+2}u_0}|=4s+3$. (4a) Computation of $|W^1_{u_0v_{4s+3}}|$ and $|W^1_{v_{4s+3}u_0}|$ when $s$ is odd and $s\ge 3$. When $s=3$, $d(u_0,v_0)=1$ and $d(v_{15},v_0)=7$. $d(u_0,v_4)=2$ and $d(v_{15},v_4)=6$. $d(u_0,v_8)=3$ and $d(v_{15},v_8)=5$. $d(u_0,v_{12})=4$ and $d(v_{15},v_{12})=4$. $d(u_0,v_1)=2$ and $d(v_{15},v_1)=7$. $d(u_0,v_5)=3$ and $d(v_{15},v_5)=6$. $d(u_0,v_9)=4$ and $d(v_{15},v_9)=5$. $d(u_0,v_{13})=5$ and $d(v_{15},v_{13})=4$. $d(u_0,v_2)=3$ and $d(v_{15},v_2)=6$. $d(u_0,v_6)=4$ and $d(v_{15},v_6)=5$. $d(u_0,v_{10})=5$ and $d(v_{15},v_{10})=4$. $d(u_0,v_{14})=6$ and $d(v_{15},v_{14})=3$. $d(u_0,v_3)=3$ and $d(v_{15},v_3)=3$. $d(u_0,v_7)=4$ and $d(v_{15},v_7)=2$. $d(u_0,v_{11})=5$ and $d(v_{15},v_{11})=1$. So $v_0,v_1,v_2,v_4,v_5,v_6,v_8,v_9\in W^1_{u_0v_{15}}$ and $v_7,v_{10},v_{11},v_{13},v_{14}\in W^1_{v_{15}u_0}$. $d(u_0,u_4)=3$ and $d(v_{15},u_4)=5$. $d(u_0,u_8)=4$ and $d(v_{15},u_8)=4$. $d(u_0,u_{12})=5$ and $d(v_{15},u_{12})=3$. $d(u_0,u_1)=1$ and $d(v_{15},u_1)=6$. $d(u_0,u_5)=4$ and $d(v_{15},u_5)=5$. $d(u_0,u_9)=5$ and $d(v_{15},u_9)=4$. $d(u_0,u_{13})=6$ and $d(v_{15},u_{13})=3$. $d(u_0,u_2)=2$ and $d(v_{15},u_2)=5$. $d(u_0,u_6)=5$ and $d(v_{15},u_6)=4$. $d(u_0,u_{10})=6$ and $d(v_{15},u_{10})=3$. $d(u_0,u_{14})=7$ and $d(v_{15},u_{14})=2$. $d(u_0,u_3)=3$ and $d(v_{15},u_3)=4$. $d(u_0,u_7)=5$ and $d(v_{15},u_7)=3$. $d(u_0,u_{11})=6$ and $d(v_{15},u_{11})=2$. $d(u_0,u_{15})=7$ and $d(v_{15},u_{15})=1$. So $u_1,u_2,u_3,u_4,u_5\in W^1_{u_0v_{15}}$ and $u_6,u_7,u_9,u_{10},u_{11},u_{12},u_{13},u_{14},u_{15}\in W^1_{v_{15}u_0}$. Note that $u_0\in W^1_{u_0v_{15}}$ and $v_{15}\in W^1_{v_{15}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{15}}|=14$ and $|W^1_{v_{15}u_0}|=15$. When $s\ge 5$, $d(u_0,v_{4t})=1+t$ and $d(v_{4s+3},v_{4t})=s-t+4$ where $0\le t\le s$. When $0\le t<\frac{s+3}{2}$, $d(u_0,v_{4t})<d(v_{4s+3},v_{4t})$. When $\frac{s+3}{2}< t\le s$, $d(u_0,v_{4t})>d(v_{4s+3},v_{4t})$. $d(u_0,v_{4t+1})=2+t$ and $d(v_{4s+3},v_{4t+1})=s-t+4$ where $0\le t\le s$. When $0\le t\le\frac{s+1}{2}$, $d(u_0,v_{4t+1})<d(v_{4s+3},v_{4t+1})$. When $\frac{s+3}{2}\le t\le s$, $d(u_0,v_{4t+1})>d(v_{4s+3},v_{4t+1})$. $d(u_0,v_{4t+2})=3+t$ and $d(v_{4s+3},v_{4t+2})=s-t+3$ where $0\le t\le s$. When $0\le t\le\frac{s-1}{2}$, $d(u_0,v_{4t+2})<d(v_{4s+3},v_{4t+2})$. When $\frac{s+1}{2}\le t\le s$, $d(u_0,v_{4t+2})>d(v_{4s+3},v_{4t+2})$. $d(u_0,v_{4t+3})=3+t$ and $d(v_{4s+3},v_{4t+3})=s-t$ where $0\le t<s$. When $0\le t<\frac{s-3}{2}$, $d(u_0,v_{4t+3})<d(v_{4s+3},v_{4t+3})$. When $\frac{s-3}{2}< t<s$, $d(u_0,v_{4t+3})>d(v_{4s+3},v_{4t+3})$. $d(u_0,u_{4t})=2+t$ and $d(v_{4s+3},u_{4t})=s-t+3$ where $1\le t\le s$. When $1\le t<\frac{s+1}{2}$, $d(u_0,u_{4t})<d(v_{4s+3},u_{4t})$. When $\frac{s+1}{2}< t\le s$, $d(u_0,u_{4t})>d(v_{4s+3},u_{4t})$. $d(u_0,u_{1})=1$ and $d(v_{4s+3},u_{1})=s+3$. $d(u_0,u_{4t+1})=3+t$ and $d(v_{4s+3},u_{4t+1})=s-t+3$ where $1\le t\le s$. When $1\le t\le\frac{s-1}{2}$, $d(u_0,u_{4t+1})<d(v_{4s+3},u_{4t+1})$. When $\frac{s+1}{2}\le t\le s$, $d(u_0,u_{4t+1})>d(v_{4s+3},u_{4t+1})$. $d(u_0,u_{2})=2$ and $d(v_{4s+3},u_{2})=s+2$. $d(u_0,u_{4t+2})=4+t$ and $d(v_{4s+3},u_{4t+2})=s-t+2$ where $1\le t\le s$. When $1\le t\le\frac{s-3}{2}$, $d(u_0,u_{4t+2})<d(v_{4s+3},u_{4t+2})$. When $\frac{s-1}{2}\le t\le s$, $d(u_0,u_{4t+2})>d(v_{4s+3},u_{4t+2})$. $d(u_0,u_{3})=3$ and $d(v_{4s+3},u_{3})=s+1$. $d(u_0,u_{4t+3})=4+t$ and $d(v_{4s+3},u_{4t+3})=s-t+1$ where $1\le t\le s$. When $1\le t<\frac{s-3}{2}$, $d(u_0,u_{4t+3})<d(v_{4s+3},u_{4t+3})$. When $\frac{s-3}{2}< t\le s$, $d(u_0,u_{4t+3})>d(v_{4s+3},u_{4t+3})$. Note that $u_0\in W^1_{u_0v_{4s+3}}$ and $v_{4s+3}\in W^1_{v_{4s+3}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{4s+3}}|=4s+1$ and $|W^1_{v_{4s+3}u_0}|=4s+3$. (4b) Computation of $|W^1_{u_0v_{4s+3}}|$ and $|W^1_{v_{4s+3}u_0}|$ when $s$ is even. When $s=2$. $d(u_0,v_0)=1$ and $d(v_{11},v_0)=6$. $d(u_0,v_4)=2$ and $d(v_{11},v_4)=5$. $d(u_0,v_8)=3$ and $d(v_{11},v_8)=4$. $d(u_0,v_1)=2$ and $d(v_{11},v_1)=6$. $d(u_0,v_5)=3$ and $d(v_{11},v_5)=5$. $d(u_0,v_9)=4$ and $d(v_{11},v_9)=4$. $d(u_0,v_2)=3$ and $d(v_{11},v_2)=5$. $d(u_0,v_6)=4$ and $d(v_{11},v_6)=4$. $d(u_0,v_{10})=5$ and $d(v_{11},v_{10})=3$. $d(u_0,v_3)=3$ and $d(v_{11},v_3)=2$. $d(u_0,v_7)=4$ and $d(v_{11},v_7)=1$. So $v_0,v_1,v_2,v_4,v_5,v_8\in W^1_{u_0v_{11}}$ and $v_3,v_7,v_{10}\in W^1_{v_{11}u_0}$. $d(u_0,u_4)=3$ and $d(v_{11},u_4)=4$. $d(u_0,u_8)=4$ and $d(v_{11},u_8)=3$. $d(u_0,u_1)=1$ and $d(v_{11},u_1)=5$. $d(u_0,u_5)=4$ and $d(v_{11},u_5)=4$. $d(u_0,u_9)=5$ and $d(v_{11},u_9)=3$. $d(u_0,u_2)=2$ and $d(v_{11},u_2)=4$. $d(u_0,u_6)=5$ and $d(v_{11},u_6)=3$. $d(u_0,u_{10})=6$ and $d(v_{11},u_{10})=2$. $d(u_0,u_3)=3$ and $d(v_{11},u_3)=3$. $d(u_0,u_7)=5$ and $d(v_{11},u_7)=2$. $d(u_0,u_{11})=6$ and $d(v_{11},u_{11})=1$. So $u_1,u_2,u_4\in W^1_{u_0v_{11}}$ and $u_6,u_7,u_8,u_9,u_{10},u_{11}\in W^1_{v_{11}u_0}$. Note that $u_0\in W^1_{u_0v_{11}}$ and $v_{11}\in W^1_{v_{11}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{11}}|=10$ and $|W^1_{v_{11}u_0}|=10$. When $s\ge 4$. $d(u_0,v_{4t})=1+t$ and $d(v_{4s+3},v_{4t})=s-t+4$ where $0\le t\le s$. When $0\le t\le\frac{s+2}{2}$, $d(u_0,v_{4t})<d(v_{4s+3},v_{4t})$. When $\frac{s+4}{2}\le t\le s$, $d(u_0,v_{4t})>d(v_{4s+3},v_{4t})$. $d(u_0,v_{4t+1})=2+t$ and $d(v_{4s+3},v_{4t+1})=s-t+4$ where $0\le t\le s$. When $0\le t<\frac{s+2}{2}$, $d(u_0,v_{4t+1})<d(v_{4s+3},v_{4t+1})$. When $\frac{s+2}{2}< t\le s$, $d(u_0,v_{4t+1})>d(v_{4s+3},v_{4t+1})$. $d(u_0,v_{4t+2})=3+t$ and $d(v_{4s+3},v_{4t+2})=s-t+3$ where $0\le t\le s$. When $0\le t<\frac{s}{2}$, $d(u_0,v_{4t+2})<d(v_{4s+3},v_{4t+2})$. When $\frac{s}{2}< t\le s$, $d(u_0,v_{4t+2})>d(v_{4s+3},v_{4t+2})$. $d(u_0,v_{4t+3})=3+t$ and $d(v_{4s+3},v_{4t+3})=s-t$ where $0\le t<s$. When $0\le t\le\frac{s-4}{2}$, $d(u_0,v_{4t+3})<d(v_{4s+3},v_{4t+3})$. When $\frac{s-2}{2}\le t<s$, $d(u_0,v_{4t+3})>d(v_{4s+3},v_{4t+3})$. $d(u_0,u_{4t})=2+t$ and $d(v_{4s+3},u_{4t})=s-t+3$ where $1\le t\le s$. When $1\le t\le\frac{s}{2}$, $d(u_0,u_{4t})<d(v_{4s+3},u_{4t})$. When $\frac{s+2}{2}\le t\le s$, $d(u_0,u_{4t})>d(v_{4s+3},u_{4t})$. $d(u_0,u_{1})=1$ and $d(v_{4s+3},u_{1})=s+3$. $d(u_0,u_{4t+1})=3+t$ and $d(v_{4s+3},u_{4t+1})=s-t+3$ where $1\le t\le s$. When $1\le t<\frac{s}{2}$, $d(u_0,u_{4t+1})<d(v_{4s+3},u_{4t+1})$. When $\frac{s}{2}< t\le s$, $d(u_0,u_{4t+1})>d(v_{4s+3},u_{4t+1})$. $d(u_0,u_{2})=2$ and $d(v_{4s+3},u_{2})=s+2$. $d(u_0,u_{4t+2})=4+t$ and $d(v_{4s+3},u_{4t+2})=s-t+2$ where $1\le t\le s$. When $1\le t<\frac{s-2}{2}$, $d(u_0,u_{4t+2})<d(v_{4s+3},u_{4t+2})$. When $\frac{s-2}{2}< t\le s$, $d(u_0,u_{4t+2})>d(v_{4s+3},u_{4t+2})$. $d(u_0,u_{3})=3$ and $d(v_{4s+3},u_{3})=s+1$. $d(u_0,u_{4t+3})=4+t$ and $d(v_{4s+3},u_{4t+3})=s-t+1$ where $1\le t\le s$. When $1\le t\le\frac{s-4}{2}$, $d(u_0,u_{4t+3})<d(v_{4s+3},u_{4t+3})$. When $\frac{s-2}{2}\le t\le s$, $d(u_0,u_{4t+3})>d(v_{4s+3},u_{4t+3})$. Note that $u_0\in W^1_{u_0v_{4s+3}}$ and $v_{4s+3}\in W^1_{v_{4s+3}u_0}$. Combined with the above discussion, $|W^1_{u_0v_{4s+3}}|=4s+1$ and $|W^1_{v_{4s+3}u_0}|=4s+3$. When $n\ge 26$, from the above computation of $|W^1_{u_0v_j}|$ and $|W^1_{v_ju_0}|$ where $8\le j\le n-8$, for any $3\le \ell<D$, we know that there exists $j$ where $d(u_0,v_j)=\ell$ and $8\le j\le n/2$ such that $|W_{u_0v_j}|<|W_{v_ju_0}|$. When $n=25$, $d(u_0,v_8)=3$, $d(u_0,v_{12})=4$, $d(u_0,v_{11})=5$ and $D(GP(25,4))=6$. From the above computation of $|W^1_{u_0v_j}|$ and $|W^1_{v_ju_0}|$, we know that $|W_{u_0v_j}|<|W_{v_ju_0}|$ for any $j=8,11,12$. The proof of the theorem completes. ◻ # Concluding remarks {#S:conluding} In this paper, we prove that $GP(n,3)$ is not $\ell$-distance-balanced where $n>16$ and $1\le\ell<{\rm diam}(GP(n,3))$. We also prove that $GP(n,4)$ is not $\ell$-distance-balanced where $n>24$ and $1\le\ell<{\rm diam}(GP(n,4))$. The authors in [@Miklavic:2018] proved that $GP(n,2)$ is not $\ell$-distance-balanced where $n>11$ and $1\le\ell<{\rm diam}(GP(n,2))$. When $k\ge 5$, Conjecture [\[C:GP-onlyD-DB\]](#C:GP-onlyD-DB){reference-type="ref" reference="C:GP-onlyD-DB"} is worth studying in the future. Combined with the main results of this paper, Theorem [Theorem 1](#T:GP-DDB){reference-type="ref" reference="T:GP-DDB"} and Conjecture [\[C:GP-onlyD-DB\]](#C:GP-onlyD-DB){reference-type="ref" reference="C:GP-onlyD-DB"}, the following problem is worth studying in the future. **Problem 11**. *(1) When $k\ge 5$, $k$ is odd and $\frac{k(k+1)}{2}\le n\le (k+1)^2$, whether $GP(n,k)$ is $\ell$-distance-balanced or not where $1\le\ell<{\rm diam}(GP(n,k))$.\ (2) When $k\ge 6$, $k$ is even and $\frac{k^2}{2}\le n\le k(k+2)$, whether $GP(n,k)$ is $\ell$-distance-balanced or not where $1\le\ell<{\rm diam}(GP(n,k))$.* # Acknowledgments {#acknowledgments .unnumbered} This work was supported by Shandong Provincial Natural Science Foundation of China (ZR2022MA077), the research grant NSFC (11971274) of China and IC Program of Shandong Institutions of Higher Learning For Youth Innovative Talents. 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--- abstract: | With his research on Aristotelian syllogistic, Jan Łukasiewicz [-@Luk29; -@Luk34; -@Luk39; -@Luk51] initiated a branch of logic called the calculus of names. This section deals with axiomatic systems that analyze various fragments of the logic of names, i.e., one that study various forms of names and functors acting on them, as well as logical relationships between sentences in which these names and functors occur. In this work, we want not only to present the genesis of the calculus of names and its first system created by Łukasiewicz. We also want to deliver systems that extend the first. In this work, we will also show that, from the point of view of modern logic, Łukasiewicz's approach to syllogistic is not the only possible one. However, this does not diminish Łukasiewicz's role in the study of syllogism. We believe that the calculus of names is indisputably Łukasiewicz's legacy. address: Department of Logic, Institute of Philosophy, Nicolaus Copernicus University in Toruń author: - Andrzej Pietruszczak title: The calculus of names -- the legacy of Jan Łukasiewicz --- [^1] # Introduction {#introduction .unnumbered} In this work, we want not only to present the genesis of the calculus of names and the first system developed by Łukasiewicz. We also want to present systems that are an extension of that initial one, including those enriched with singular sentences from Stanisław Leśniewski's Ontology, which are not classified as syllogistic. In this work, we will also show that, from the point of view of modern logic, Łukasiewicz's approach to syllogisms is not the only possible one. This in no way diminishes Łukasiewicz's role in the study of syllogistic. We believe that the calculus of names is indisputably Łukasiewicz's legacy. In the first part, we will present the logic of names and so-called traditional logic. We will present various possible interpretations and forms of categorical sentences in modern name logic. The second part will be devoted to the calculus of names as a specific development of traditional logic. We will present the origins of this calculus, as well as other possible approaches to the formalization of Aristotelian syllogistic. We will present Łukasiewicz's original system. Finally, we will present a "competitive approach" in the form of the calculus of sequences. In the third part, we will discuss two modern extensions of the calculus of names that allow it to be applied to null names as well. One---for the so-called weak interpretation of universal affirmative sentences---reported and researched by J.C. @Sh. The second approach---for a strong interpretation of universal affirmative sentences---was initiated by Jerzy @Sl. However, his system is not complete. This was noted in [@ja87], where such complete systems are also given. In the last part, we present and analyze extensions of both types of systems with unit sentences from Leśniewski's Ontology. # The logic of names and traditional logic ## The logic of names It is constructed using the method of logical schemes, which consists in the fact that, based on the analysis of the surface syntactic structure of sentences and expressions of natural language, sentence schemes are introduced in which various types of schematic letters appear instead of names. Name logic is an intermediate link between propositional logic and predicate logic. In propositional logic, we study relations between sentences but are not interested in the syntactic structure of sentences in which there are no propositional conjunctions. We look only at those relationships that depend solely on propositional connectives. In quantifier logic, the opposite is true; we analyze the deep structure of sentences using quantifiers binding variables and additionally introduced sentential connectives. These are ways characteristic of the modern mathematical stage of formal logic. The logic of names can be regarded as a systematic development of specific fragments of traditional, pre-mathematical formal logic. We in-clude to it not only the best-known piece of it, which is Aristotelian syllogistic but also studies on compound names and relative names. The latter, as oblique syllogisms, were already considered by Aristotle in Prior Analysts and Joachim Jungius in Logica Hamburgensis. However, a systemat-ic theory of them appeared only in the nineteenth century in the works of Hamilton, Schröder and de Morgan. The latter analyzed reasoning such as: since every horse is a mammal, then every horse's head is a mammal's head. Aristotelian syllogistic deals with categorical sentences falling into one of four schemes: - Every $S$ is a $P$(*universal affirmative sentence*) - Some S is a P (*particular affirmative sentence*) - No S is a P (*universal denial sentence*) - Some S is not a P (*particular denial sentence*) These schemes can be used for any general names. The letter '$S$' is to be replaced by the name appearing in a subject, and the letter '$P$' is to be replaced by the name appearing in a predicate. The following natural interpretation of these sentences is generally accepted: - A universal affirmative sentence is true if and only if the range of the name from its subject is included in the range of the name of its predicate. - A particular affirmative sentence is true if and only if the names from its subject and predicate have a common referent. - A universal denial sentence is true if and only if the ranges of names of its subject and predicate are disjoint. - A particular denial sentence is true if and only if the name of its subject has a referent that is not a referent of the name of its predicate. The method of logical schemes allows the study of a broad class of natural language sentences. In addition to categorical sentences, we can also study singular sentences of the form '$a$ is a $P$' and '$a$ is not a $P$', in the subject of which there is a name that refers to only one object. We also have a whole spectrum of sentences corresponding to categorical sentences; for example, these are sentences like '$S$ is the same as a $P$' (or otherwise: 'Every $S$ is a $P$ and vice versa', 'All $S$ is a $P$ and vice versa'), 'Exactly one $S$ is a $P$', 'At most one $S$ is a $P$', 'The only $S$ is a $P$', etc. We can analyze plural sentences such as 'Exactly two $S$s are $P$s', 'At least two $S$s are $P$s', 'At most two $S$s are $P$s', etc. We can also treat modal versions of these sentences in which the copula 'is' is replaced by one of the phrases: 'must be', 'may be'; the phrase 'is not' is replaced with one of the phrases 'must not be', 'must not be', 'may not be', 'cannot be'. It is also possible to analyze sentences whose subjects and predicates have compound names of the form: '$S$ and $P$', '$S$ or $P$', 'not-$S$'. The same applies to relative names such as 'friend', 'mother', etc. We can also transform the latter sentences from active to passive (e.g., 'reader' to 'read by') and, from two such names, create a third relative name (e.g., 'mother's father'). Clearly, we do not have to limit ourselves to relative terms but may extend the approach to verbs, e.g., instead of 'is a reader' we take 'reads' [see, e.g., @M; @P-HiM]. Rich metalogical research can be carried out on the material given above: various types of set-theoretic semantics and axiomatizations of different fragments of the logic of names consistent with it can be intro-duced. ## Traditional logic For some reason, Aristotelian syllogistic was applied only to non-empty general names, i.e., those with at least one referent. This limitation was also used by traditional logic, which, using propositional schemes, examined the logical relationships between categorical sentences. The primary connection between them is a logical inference. In traditional logic, this relation was expressed by so-called correct reasoning schemes. If categorical propositions are understood in those mentioned above naturally, when we limit the applications to non-empty names, all the schemes distinguished by traditional logic are valid because they always give a true conclusion with true premises. ## A contemporary approach to the logic of names[\[subsec1.3\]]{#subsec1.3 label="subsec1.3"} In this approach, we allow the use of empty general names (i.e., those that do not apply to anything). In this case, however, we have problems with the validity of some inference schemes distinguished by traditional logic and the interpretation of categorical sentences. Concerning the first of the above problems, it is known that some schemes distinguished by traditional logic will lose their validity when the substitution of empty names is allowed and categorical sentences are still naturally understood. For example, the scheme 'Some $S$ is an $S$' is no longer generally true (i.e., it is not a tautology), and the following consequences from traditional logic no longer apply: - Every $S$ is a $P$ $\vDash$ Some $S$ is a $P$ - Every $S$ is a $P$ $\vDash$ Some $P$ is an $S$ - No $S$ is a $P$ $\vDash$ Some $S$ is not a $P$ - No $S$ is a $P$ $\vDash$ Some $P$ is not an $S$ Concerning the second problem related to the interpretation of categorical sentences, the following question arises: - Can the meaning of categorical sentences be changed so that the validity of traditional logic schemes is preserved even when substituting empty names is allowed? However, this new interpretation is to meet the following two conditions: - when terms are limited to non-empty terms, it coincides with the natural usage, - the whole class of names is within the limits allowed by linguistic usage. Applying the above requirements to "save" the first two consequences mentioned above, the so-called *strong* interpretation of universal affirmative sentences was used, where we require the non-emptiness of the name in its subject for its truth. Thus: - A universal affirmative sentence is true if and only if it has a non-empty name in the subject, the range of which is included in the range of the name from the predicate. An interpretation where we do not apply the added requirement is called *weak*. Of course, for non-empty names, the two interpretations are indistinguishable. Let us note that with the strong interpretation, when we allow the use of empty names, the following consequences of traditional logic no longer apply: - It is not the case that some $S$ is not a $P$ $\vDash$ Every $S$ is a $P$ - It is not the case that every $S$ is a $P$ $\vDash$ Some $S$ is not a $P$ For this reason, Tadeusz Kotarbiński [-@Kot pp. 233--234 in 1961], followed by Czesław Lejewski [-@Lej pp. 128--130 in 1984], proposed the use of additional universal affirmative sentences of the form 'All $S$ is a $P$', which are true for a free empty name standing in the subject (regardless of what name stands in the predicate). These sentences, therefore, have an interpretation of general affirmative sentences in the weak sense. Kotarbiński and Lejewski's proposal indicates that they believed that in the meaning of the phrase 'all $S$', there is no reservation about the non-emptiness of $S$; that is, this reservation is implicitly related to the phrase 'every $S$'. Therefore, the sentences of the form 'Every $S$ is a $P$' are valid only with a strong interpretation. When we limit ourselves to non-empty general names, the interpretations of both types of universal affirmative sentences coincide. Namely, what is supposed to be implicit in the meaning of 'every $S$' is explicitly implied in the assumption imposed on the names. Note that for these new universal affirmative sentences, the following equivalence holds (in the full range of general names): - All $S$ is a $P$ $\equiv$ It is not the case that some $S$ is not a $P$. To "save" the following consequence from traditional logic: - No $S$ is a $P$ ${\vDash}$ Some $S$ is not a $P$ it is enough to use the analogous strong interpretation for universal denial sentences, where we require the non-emptiness of the name standing in the subject for their truth. So, in this interpretation: - A universal denial sentence is true if and only if it has a non-empty name in the subject, the range of which is disjoint with the range of the name from the predicate. Lejewski [-@Lej p. 130 in 1984] proposed the introduction of two functors to construct universal denial sentences. In addition to the functor of weak exclusion 'no ... is', he introduced the functor of strong exclusion 'every ... is not'. To "save" the fourth of the previously given consequence: - No $S$ is a $P$ ${\vDash}$ Some $P$ is not a $S$ we must use an even stronger interpretation of universal denial sentences, where we require both names to be non-empty for their truth. So in this *super strong interpretation*: - A universal denial sentence is true if and only if it has non-empty names in both the subject and the predicate, the ranges of which are disjoint. Neither Kotarbiński nor Lejewski used this interpretation. Furthermore, they did not introduce new sentences expressing it. In [@ja87 p. 163; @ja90; @ja91b; @ja91c], it was proposed that these should be sentences of the form 'Every $S$ is not a $P$ and vice versa'. It was modelled on Kotarbiński's comments on the phrase 'every S' and on the sentences he used in the form 'All $S$ is a $P$ and vice versa' (which state that the ranges of both names are equal). Again, when we limit ourselves to non-empty general names, the interpretations of the three types of universal denial sentences coincide. Indeed, this is implicit in the meaning of 'every $S$' and explicitly implied in the assumption imposed on the names. Moreover, what is expressly contained in the meaning of the phrase 'and vice versa' is implicit in interpreting the functor 'no ... is'. # Calculus of names as an extension of traditional logic ## The genesis of the calculus of names[\[subsec2.1\]]{#subsec2.1 label="subsec2.1"} As we have already stated in the introduction, Łukasiewicz is undoubtedly the creator of the calculus of names. The following words from him [-@Luk34] show the genesis of this calculus:[^2] > **5.** A fundamental difference between a logical thesis and a rule of inference exists. > > A *logical thesis* is a sentence in which, apart from logical constants, there are only sentence or name variables and which is true for all values of the variables that occur in it. An *inference rule* is a provision that authorizes the applicant to derive new theses based on recognized theses. For example, \[principles of identity such as "If $p$, then $p$" and "Every $a$ is an $a$"\] are logical theses, but the rule of inference is the following «rule of detachment»: > > Whoever accepts the implication "If $\alpha$, then $\beta$" and the antecedent of this implication "$\alpha$" as true, has the right to accept as true the consequent of this implication "$\beta$". The problem, however, is that the fact that a given implication is considered to be true can be understood differently. Since the antecedent $\alpha$ and consequent $\beta$ appearing in the implication considered by Łukasiewicz have "variables" (here schematic letters[^3]), two situations can be considered: 1. With a given admissible substitution for variables (schematic letters), the schemes 'If $\alpha$, then $\beta$' and $\alpha$ give true sentences. And then, with this substitution, we also have a true sentence obtained from the consequent $\beta$. 2. The schemes 'If $\alpha$, then $\beta$' and $\alpha$ give true sentences under any admissible substitution for variables, i.e., 'If $\alpha$, then $\beta$' and $\alpha$ are logical tautologies. And then $\beta$ is a logical tautology too. The first of the above points can be summed up in the following words: 1. The detachment rule with any substitution yielding two true premises leads to a true conclusion. Thus, we treat the rule of detachment as a valid argument scheme: $$\frac{\text{If $p$ then $q$\qquad\quad $p$}}{q}$$ However, the second point says: 2. The rule of detachment from two tautologies always leads to a tautology. Here the detachment rule is then something that "produces" a third from two tautologies (the letters '$\varphi$' and '$\psi$' represent arbitrary propositional formulas): $$\frac{\text{If $\varphi$ then $\psi$\qquad\quad$\varphi$}}{\psi}$$ Depending on needs, the rule of detachment can perform both of the above roles. In the logical calculus, including the calculus of names, the detachment rule is used in the latter role, i.e., as a "generator" of logical tautologies. The initial theses (axioms) are tautologies. The rules used, including the rule of detachment, transform tautologies into new tautologies. Therefore, each thesis obtained with their help is also a tautology of a given logic.[^4] However, in some logical calculi, not every rule performs both of the functions indicated above. The primary example is the substitution rule used by Łukasiewicz. It says that from any tautology, for each permissible substitution for "variables", we get a tautology. This rule does not even have a scheme by which to express it. Hence, it cannot be "confused" with the scheme of correct reasoning. Another example, which already has a scheme, is the so-called *necessitation rule*: $$\frac{\varphi}{\text{It is necessary that~} \varphi}$$ It takes us from a tautology to a new tautology. However, it cannot be transformed into a correct inference scheme (where the letter '$p$' stands in the place of a sentence): $$\frac{p}{\text{It is necessary that~} p}$$ Indeed, we have true sentences that are not necessary. For this reason, we cannot reason according to this scheme in modal theories where we are interested in true propositions and not in logical tautologies. The above remarks generally refer to the genesis of various types of logical calculus. The origin of the calculus of names itself can be found by continuing the quote from [@Luk34]: > **6.** The original Aristotelian syllogism is a logical thesis, the traditional syllogism has the meaning of a rule of inference. > > The Barbara mode given above \[see below\], \[...\], is an implication of the type "If $\alpha$ and $\beta$, then $\gamma$", \[...\]. As an implication, an Aristotelian syllogism is a proposition that Aristotle holds to be true, namely that the proposition is true for all values of the variables "$a$", "$b$" and "$c$" that occur in it. Therefore, we get true sentences if we substitute some constant values for these variables. Since in the considered mode, apart from variables, there are only logical constants, namely "if-then", "and" and "every-is", therefore the Aristotelian syllogism is a logical thesis. > > The traditional syllogism: > > $\dfrac{\parbox{20mm}{Every $b$ is an $a$\\ Every $c$ is a $b$}}{\text{Every $c$ is an $a$}}$ > > Is *not* an implication. It consists of three sentence forms, listed one under the other, which do not form a single sentence. Since a traditional syllogism is not a proposition, it cannot be true or false either since, according to the generally accepted view, truth and falsity belonging only to propositions. A traditional syllogism is, therefore, *not* a thesis. If we substitute some constant values for the variables in this syllogism, we do not get a proposition but an *argument*. A traditional syllogism is a scheme of argumentation and has the meaning of a *rule of inference*, which can be more precisely expressed as follows: > > Whoever accepts as true premises such as "Every $b$ is an $a$" and "Every $b$ is a $b$", has the right to accept as true a conclusion like "Every $c$ is an $a$".\[Here is footnote 11: The extent of the inaccuracy of previous historical studies of logic is evidenced by this very characteristic detail that all the authors I know who wrote about Aristotelian logic \[...\] present Aristotelian syllogisms in the traditional form, without realizing that there is a fundamental difference between these forms.\] > > **7.** Thanks to this distinction between logical theses and rules of inference, it became possible to construct logical sciences axiomatically in the form of deductive systems. We omit the dispute between @Cor and Łukasiewicz over whether Aristotelian syllogisms are schemes of inference or implications, as it is irrelevant here. The problem, however, is that Łukasiewicz did not distinguish between two types of "inference rules". Łukasiewicz states that syllogisms cannot be treated as inference rules because only sentences can be transformed using inference rules. However, this issue can be presented differently when we specify the notion of a rule of inference. For example, an appropriate sequent calculus can be used, in which the notations of the form $\pi_1,\ldots,\pi_n\Longrightarrow\omega$ (for $n>0$) and $\Longrightarrow\omega$ are to express the correct argument schemes and tautologies (after this lowercase Greek letters represent arbitrary sentence formulas).[^5] To "generate" sequences, we use derivation rules such as the cut rule [see, e.g., @AI21]. In the considered case, it will have the following form for any $n,m\geqslant 0$: $$\frac{\varphi_1,\ldots,\varphi_n\Longrightarrow\psi\qquad\quad\psi,\pi_1,\ldots,\pi_m \Longrightarrow \omega} {\varphi_1,\ldots,\varphi_n,\pi_1,\ldots,\pi_m\Longrightarrow\omega}$$ Another possible solution is to use an appropriate natural deduction system with inference rules corresponding to acceptable ways of reasoning and the so-called *proof construction rules*. With their help, we derive new inference rules [see, e.g., @AI15]. For example, @Cor proposes an understanding of Aristotelian syllogistic, in which the proof construction rule consists of assumption proofs, and selected correct syllogistic modes are treated as rules of inference. Corcoran presented this approach in his polemic on Łukasiewicz. Let us note, however, that Łukasiewicz rejected both alternative approaches to the reconstruction of syllogistic, as he believed that Aristotle's syllogisms are not schemes of inference but are theses in the form of implications. ## Łukasiewicz's calculus of names Łukasiewicz presented his reconstruction of syllogistic as a calculus of names for the first time in [-@Luk29]. He repeated it in his work [-@Luk34] and then in [-@Luk51 §25 in 1957]. This re-construction is presented in the continuation of the previously quoted text from [-@Luk34]: > **8.** The theory of the Aristotelian syllogism, which Aristotle has already tried to axiomatize, but which has not yet been presented in an axiomatic form, is based on two fundamental con-cepts: "Every $a$ is a $b$", in the signs "$Uab$"[^6], and "Some $a$ is a $b$", in the signs "$Iab$" and on the following axioms: > > 1\. Every $a$ is an $a$. > > 2\. Some $a$ jest an $a$. > > 3\. If every $b$ is an $a$ and every $c$ is a $b$, then every $c$ is an $a$. > > 4\. If every $b$ is an $a$ and some $b$ is a $c$, then some $c$ is an $a$. > > In the signs (the functors "$U$" and "$I$" come before the arguments, and such same the con-junction sign "$K$" = "and")[^7]: > > 1\. $Uaa$. > > 2\. $Iaa$. > > 3\. $CKUbaUcbUca$ (*Barbara*). > > 4\. $CKUbaIbcIca$ (*Datisi*). > > The expressions "Some $a$ is not a $b$", in the signs "$Oab$", and "No $a$ is a $b$", in the signs "$Yab$"[^8], can be defined as follows: > > Df1. $Oac = NUab$. > > Df2. $Yab = NIab$. > > By both rules of substitution and detachment (propositional variables may be substituted with propositional forms of Aristotelian logic, for name variables only other name variables), and with the help of theses of propositional logic, from these axioms and definitions, we can derive all 24 (not 14 nor 19!) the correct modes of Aristotelian syllogistic.\[Footnote 14: The axiomatization of Aristotelian syllogistic presented here, as well as the deduction of all modes, can be found in the script from my lectures, delivered in the autumn trimester of 1928/29 at the University of Warsaw, entitled: *Elementy logiki matematycznej* \[Elements of mathematical logic, i.e., [@Luk29]\], \[...\].\] We will use the following abbreviations for particular schemes of categorical sentences: for 'Every $S$ is a $P$' -- we write: $S\mathbf{a}P$, for 'Some $S$ is a $P$' -- we write: $S\mathbf{i}P$, for 'No $S$ is a $P$' -- we write: $S\mathbf{e}P$, for 'Some $S$ is not a $P$' -- we write: $S\mathbf{o}P$. The abbreviations are derived from the vowels in the Latin words '*affirmo*' and '*nego*'. Let us adopt the following notation of Łukasiewicz's four axioms us-ing the above abbreviations. The first two are the principles of identity, and the next two are syllogisms [@Luk51 p. 88 in 1957]: $$\begin{gathered} S\mathbf{a}S \label{Ia}\tag{I\textbf{a}} \\ S\mathbf{i}S \label{Ii}\tag{I\textbf{i}} \\ (M\mathbf{a}P\:\wedge\: S\mathbf{a}M\:\rightarrow\:S\mathbf{a}P \label{Barbara}\tag{Barbara} \\ (M\mathbf{a}P\:\wedge\: M\mathbf{i}S)\:\rightarrow\:S\mathbf{i}P \label{Datisi}\tag{Datisi} \end{gathered}$$ In the last quote, Łukasiewicz adopted three rules for deriving theses: detaching, substituting and defining. He added fourteen implication-negative tautologies of classical propositional logic (CPL), in which he replaces propositional "variables" with schemes of categorical sentences and their conjunctions [see @Luk51 pp. 88--89 in 1957]. The definition rule does not apply today. Instead, additional axioms are introduced as equivalences (these other axioms are called *definitions*).[^9] In our case, we will assume: $$\begin{gathered} S\ensuremath{\mathbf{e}}P \:\leftrightarrow\:\neg\:S\ensuremath{\mathbf{i}}P\label{dfe}\tag{df\,{\ensuremath{\mathbf{e}}}}\\ S\ensuremath{\mathbf{o}}P \:\leftrightarrow\:\neg\:S\ensuremath{\mathbf{a}}P\label{dfo}\tag{df\,{\ensuremath{\mathbf{o}}}}\end{gathered}$$ To facilitate the derivation of theses, all substitutions of the CPL tautology with formulas of the calculus of names can adopted as axioms. The essence of this calculus is contained in its specific axioms, i.e., in [\[Ia\]](#Ia){reference-type="eqref" reference="Ia"}, [\[Ii\]](#Ii){reference-type="eqref" reference="Ii"}, [\[Barbara\]](#Barbara){reference-type="eqref" reference="Barbara"}, [\[Datisi\]](#Datisi){reference-type="eqref" reference="Datisi"}, [\[dfe\]](#dfe){reference-type="eqref" reference="dfe"} and [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"}. Using CPL and the substitution rule, by [\[Ia\]](#Ia){reference-type="eqref" reference="Ia"} and [\[Datisi\]](#Datisi){reference-type="eqref" reference="Datisi"}, we get the conversion of particular affirmative sentences: $$\label{Ci}\tag{C{\ensuremath{\mathbf{i}}}} P\ensuremath{\mathbf{i}}S\:\rightarrow\: S\ensuremath{\mathbf{i}}P$$ Hence and from [\[dfe\]](#dfe){reference-type="eqref" reference="dfe"}, via CPL, we also have the conversion of universal denial sentences: $$\label{Ce}\tag{C{\ensuremath{\mathbf{e}}}} P\ensuremath{\mathbf{e}}S\:\rightarrow\: S\ensuremath{\mathbf{e}}P$$ From [\[Ii\]](#Ii){reference-type="eqref" reference="Ii"} and [\[Datisi\]](#Datisi){reference-type="eqref" reference="Datisi"} we also get the law of subordination stating that general affirmative sentences can be attenuated to particular affirmative sentences: $$\label{Sa}\tag{S{\ensuremath{\mathbf{a}}}} S\ensuremath{\mathbf{a}}P\:\rightarrow\: S\ensuremath{\mathbf{i}}P$$ Hence and from [\[dfe\]](#dfe){reference-type="eqref" reference="dfe"}, [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"} [\[Ce\]](#Ce){reference-type="eqref" reference="Ce"}, via CPL, we have: $$\begin{gathered} S\ensuremath{\mathbf{e}}P\:\rightarrow\: S\ensuremath{\mathbf{o}}P\label{Se}\tag{S{\ensuremath{\mathbf{e}}}}\\ S\ensuremath{\mathbf{e}}P\:\rightarrow\: P\ensuremath{\mathbf{o}}S \label{CSe}\tag{CS{\ensuremath{\mathbf{e}}}}\end{gathered}$$ The great advantage of the approach to the logic of names proposed in this way by Łukasiewicz is that for the semantic study of a given system, we can use methods known from the metatheory of propositional logic and metatheory of predicate logic [see, e.g., @ja90; @ja91a; @ja91c; @ja92]. Moreover, we can consider a given system one of the open first-order theories and use meta-theorems about such theories [see, e.g., @Sh;; @ja92]. ## The completeness of Łukasiewicz's calculus[\[subsec2.3\]]{#subsec2.3 label="subsec2.3"} We consider Łukasiewicz's system to be traditional because---just like traditional logic---we can apply it only to non-empty names. It is proved that this system is complete in the sense that its theses are all those and only those of its formulas that will give true sentences when any substitution of non-empty general names for name letters is made [see, e.g., @ja90; @ja91c]. From a formal point of view, however, instead of talking about name substitutions for name letters, it is better to use set-theoretic semantics, which uses models of the form $\langle\ensuremath{\mathbbx{D}},\ensuremath{\mathbbx{d}}\rangle$, where $\mathbbx{D}$ is a non-empty set (universe) and $\mathbbx{d}$ is a function of denotation, which assigns to any name letter $\mathcal{S}$ a non-empty subset of $\mathbbx{D}$. Using the previously given interpretation of the functors '$\mathbf{a}$', '$\mathbf{i}$' '$\mathbf{e}$' and '$\mathbf{o}$', we introduce the notions of being a true formula in the model $\ensuremath{\mathfrak{M}}=\langle\ensuremath{\mathbbx{D}},\ensuremath{\mathbbx{d}}\rangle$. For atomic formulas, for any letters $\mathcal{S}$ and $\mathcal{P}$ we assume: - $\ensuremath{\mathcal{S}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}}$ is true in $\mathfrak{M}$ iff every element of $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ is an element of $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$, i.e., $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ is included in $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$; - $\ensuremath{\mathcal{S}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{P}}$ is true in $\mathfrak{M}$ iff the sets $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ and $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$ have a common element; - $\ensuremath{\mathcal{S}}\ensuremath{\mathbf{e}}\ensuremath{\mathcal{P}}$ is true in $\mathfrak{M}$ iff the sets $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ and $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$ have no element in common; - $\ensuremath{\mathcal{S}}\ensuremath{\mathbf{o}}\ensuremath{\mathcal{P}}$ is true in $\mathfrak{M}$ iff $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ has an element which is not an element of $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$. For formulas built with propositional connectives, we use dummy truth tables, i.e., interpretation of connectives as in CPL. Using the above formal semantics, it is provable that Łukasiewicz's calculus is complete in that its theses are all those and only his formulas that are true in every model. This was achieved by @Sh [pp. 145--146], who identified Łukasiewicz's system with an open first-order theory. The representation theorem (equivalent to Stone's theorem for the elementary theory of Boolean algebras) is also provable for this theory. The representation theorem and the Gödel completeness theorem for first-order theories lead to a conclusion which, assuming the identification, corresponds to the statement that the set of theses of this system is equal to the set of formulas true in each model of the class under consideration [see, e.g., @Sh; @ja92]. In [@ja90; @ja91a; @ja91c], the Henkin method was used to demonstrate the completeness of Łukasiewicz's system. For each maximal and consistent set *max* of formulas in this system, we will construct an appropriate model in which all formulas from max are true. The universe, $\mathbbx{D}$, of this model consists of all pairs $\{\ensuremath{\mathcal{M}},\ensuremath{\mathcal{P}}\}$, for which the formula $\ensuremath{\mathcal{M}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{P}}$ belongs to max. Moreover, $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ consists only of those pairs for which the formula $\ensuremath{\mathcal{M}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{S}}\:\vee\:\ensuremath{\mathcal{P}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{S}}$ belongs to *max* (by axiom [\[Ii\]](#Ii){reference-type="eqref" reference="Ii"}, it is a non-empty set). It is easy to see that every set-theoretic tautology is also a tautology in lexical terms (i.e., by name substitutions) because, to each general name, we can assign a set that is its scope. On the other hand, if each set was determined by a name as its range, then without any additional conditions, it could be proved that the opposite is also true, i.e., that every tautology in lexical terms is a set-theoretic tautology. In fact, we lack names; we cannot assign a name to each set that would cover it. However, as we know, in the model theory of predicate logic, and therefore name logic, not all sets are needed. Those that are determined by the formulas of elementary number theory with one free variable will suffice. This theory covers only what can be said about natural numbers using the names of individual numbers, addition, multiplication, equal sign, propositional connectives and quantifiers. The above states that if the natural language, whose general names we substitute for name letters, satisfies the condition: - for each set of natural numbers described above, there is a general name whose scope is this range, then both definitions denote the same set of tautologies. This condition is not too high. After all, we are talking about such general names as 'smallest natural number', 'largest natural number', 'even number', 'number greater than 10', etc. ## A sequent calculus -- an equivalent of the calculus of names As mentioned in point [\[subsec2.1\]](#subsec2.1){reference-type="ref" reference="subsec2.1"}, we can reconstruct traditional logic using a sequent calculus. As axiomatic, we can accept the sequences corresponding to [\[Ia\]](#Ia){reference-type="eqref" reference="Ia"}, [\[Ii\]](#Ii){reference-type="eqref" reference="Ii"}, [\[Barbara\]](#Barbara){reference-type="eqref" reference="Barbara"}, [\[Datisi\]](#Datisi){reference-type="eqref" reference="Datisi"} for any name letters $\mathcal{S}$  $\mathcal{P}$, $\mathcal{M}$: $$\begin{gathered} \Longrightarrow\; S\ensuremath{\mathbf{a}}S \tag*{\eqref{Ia}}\\ \Longrightarrow\; S\ensuremath{\mathbf{i}}S \tag*{\eqref{Ii}}\\ \ensuremath{\mathcal{M}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}},\;\ensuremath{\mathcal{S}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{M}}\:\Longrightarrow\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}}\tag*{\eqref{Barbara}}\\ \ensuremath{\mathcal{M}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}},\;\ensuremath{\mathcal{M}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{S}}\:\Longrightarrow\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{P}}\tag*{\eqref{Datisi}}\end{gathered}$$ We should enter two sequences for each of the formulas of [\[dfe\]](#dfe){reference-type="eqref" reference="dfe"} and [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"}. However, to shorten the notations, let us assume that for any formulas $\varphi$ and $\psi$, the notation $\varphi\:\Longleftrightarrow\: \psi$ yields two sequences: $\varphi\:\Longrightarrow\: \psi$ and $\psi\:\Longrightarrow\: \varphi$. We put: $$\begin{gathered} \ensuremath{\mathcal{S}}\ensuremath{\mathbf{e}}\ensuremath{\mathcal{P}}\:\Longleftrightarrow\: \neg\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{P}}\tag*{\eqref{dfe}}\\ \ensuremath{\mathcal{S}}\ensuremath{\mathbf{o}}\ensuremath{\mathcal{P}}\:\Longleftrightarrow\: \neg\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}}\tag*{\eqref{dfo}}\end{gathered}$$ Moreover, to facilitate the derivation of successive sequences, all sequences corresponding to all substitutions of all consequences and tautologies in CPL with formulas of the calculus of names can be taken as axioms. In other words, if we have the consequence $\varphi_1,\ldots,\varphi_n\vDash\psi$ in CPL, then we take as an axiom the sequent $\pi_1,\ldots,\pi_n\: \Longrightarrow\:\omega$, obtained from this consequence by substituting the propositional letters with formulas of the calculus of names. For example, we have axiomatic sequences obtained from $\varphi\:\Longleftrightarrow\: \neg\:\neg\:\varphi$. To "generate" new sequences we will use the cut rule given in point [\[subsec2.1\]](#subsec2.1){reference-type="ref" reference="subsec2.1"}. Using it, from the pair of sequences $\Longrightarrow\ensuremath{\mathcal{S}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{S}}$ and $\ensuremath{\mathcal{S}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}},\;\ensuremath{\mathcal{S}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{S}}\:\Longrightarrow\: \ensuremath{\mathcal{S}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{P}}$ we get the sequent corresponding to thesis [\[Sa\]](#Sa){reference-type="eqref" reference="Sa"}. Applying the cut to the pair of sequences $\Longrightarrow\ensuremath{\mathcal{P}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}}$ and $\ensuremath{\mathcal{P}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}},\; \ensuremath{\mathcal{P}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{S}}\:\Longrightarrow\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{P}}$ we get a sequence corresponding to thesis [\[Ci\]](#Ci){reference-type="eqref" reference="Ci"}. In addition, two rules must be adopted: *contraposition* and *deduction*: $$\frac{\pi_1,\ldots,\pi_m,\psi\:\Longrightarrow\: \omega}{\pi_1,\ldots,\pi_m, \neg\:\omega\:\Longrightarrow\:\neg\:\psi}\qquad\qquad \frac{\pi_1,\ldots,\pi_m,\psi\:\Longrightarrow\:\omega}{\pi_1,\ldots,\pi_m \:\Longrightarrow\:\psi\:\rightarrow\:\omega}$$ Applying the contraposition to the recently obtained sequences, by cutting, from [\[dfe\]](#dfe){reference-type="eqref" reference="dfe"} and [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"} we get the sequences corresponding to theses [\[Ce\]](#Ce){reference-type="eqref" reference="Ce"}, [\[Se\]](#Se){reference-type="eqref" reference="Se"} and [\[CSe\]](#CSe){reference-type="eqref" reference="CSe"}. Moreover, using "simplifications" of the form $\varphi\:\Longleftrightarrow\:\neg\:\neg\:\varphi$, we get: $$\begin{gathered} \ensuremath{\mathcal{S}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{P}}\:\Longleftrightarrow\:\neg\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{e}}\ensuremath{\mathcal{P}}\\ \ensuremath{\mathcal{S}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}}\:\Longleftrightarrow\:\neg\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{o}}\ensuremath{\mathcal{P}}\end{gathered}$$ Applying the cut and replace of variables from the sequences [\[Barbara\]](#Barbara){reference-type="eqref" reference="Barbara"}, [\[Datisi\]](#Datisi){reference-type="eqref" reference="Datisi"} and previously obtained, we have: $$\begin{gathered} \ensuremath{\mathcal{P}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{M}},\; cS\ensuremath{\mathbf{o}}\ensuremath{\mathcal{M}}\:\Longrightarrow\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{o}}\ensuremath{\mathcal{P}}\label{Baroco}\tag{Baroco}\\ \ensuremath{\mathcal{M}}\ensuremath{\mathbf{o}}\ensuremath{\mathcal{P}},\; \ensuremath{\mathcal{M}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{S}}\:\Longrightarrow\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{o}}\ensuremath{\mathcal{P}}\label{Bocardo}\tag{Bocardo}\\ \ensuremath{\mathcal{P}}\ensuremath{\mathbf{e}}\ensuremath{\mathcal{M}},\; \ensuremath{\mathcal{S}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{M}}\:\Longrightarrow\: \ensuremath{\mathcal{S}}\ensuremath{\mathbf{e}}\ensuremath{\mathcal{P}}\label{Cezare}\tag{Cezare}\\ \ensuremath{\mathcal{M}}\ensuremath{\mathbf{e}}\ensuremath{\mathcal{P}},\; \ensuremath{\mathcal{S}}\ensuremath{\mathbf{i}}\ensuremath{\mathcal{M}}\:\Longrightarrow\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{o}}\ensuremath{\mathcal{P}}\label{Ferio}\tag{Ferio}\\ \ensuremath{\mathcal{M}}\ensuremath{\mathbf{a}}\ensuremath{\mathcal{P}},\;\ensuremath{\mathcal{S}}\ensuremath{\mathbf{e}}\ensuremath{\mathcal{M}}\:\Longrightarrow\:\ensuremath{\mathcal{S}}\ensuremath{\mathbf{e}}\ensuremath{\mathcal{P}}\label{Camestres}\tag{Camestres}\end{gathered}$$ The remaining sequences corresponding to Aristotelian syllogisms will be obtained from those we have so far by converting the particular affirmative and universal denial sentences and subordinating the particular sentences to the universal ones. # Calculi allowing empty names Now we will present calculi which can also be applied to empty general names. As a standard, we will assume that both functors of affirmative sentences will be primitive. The remaining functors will be definable using the original ones. We understand the functor of particular affirmative sentences with its natural interpretation, and we leave the abbreviation '$\mathbf{i}$' for it. The problem, however, is that the functor of universal affirmative sentences has two variations that differ when applied to empty names. As primitive, we will take the weak interpretation in the first two points of this section and the strong one in the rest. ## Shepherdson's axioms[\[subsec3.1\]]{#subsec3.1 label="subsec3.1"} For the interpretation of the functor of universal affirmative sentences, we will leave the abbreviation '$\mathbf{a}$' and---according to the proposal of Kotarbiński and Lejewski---we can read it as 'all ... is'. For the set of the primitive functors '$\mathbf{a}$', '$\mathbf{i}$', in [-@Sh], Shepherdson proposed an axiomatization of the $\mathbf{a}$$\mathbf{i}$-system. Of Łukasiewicz's four axioms, he left three: [\[Ia\]](#Ia){reference-type="eqref" reference="Ia"}, [\[Barbara\]](#Barbara){reference-type="eqref" reference="Barbara"} and [\[Datisi\]](#Datisi){reference-type="eqref" reference="Datisi"}. Shepherdson rejected the principle of identity [\[Ii\]](#Ii){reference-type="eqref" reference="Ii"} because it turns into a false sentence when we substitute an empty name for '$S$'. Instead, he took two axioms weaker than [\[Ii\]](#Ii){reference-type="eqref" reference="Ii"}: $$\begin{gathered} S\ensuremath{\mathbf{i}}P\:\rightarrow\:S\ensuremath{\mathbf{i}}S \label{star}\tag{$\star$}\\ S\ensuremath{\mathbf{i}}S\:\vee\:S\ensuremath{\mathbf{a}}P \label{2star}\tag{$\star\star$}\end{gathered}$$ The last enforces the truth of universal affirmative sentences with an empty subject. All substitutions of all CPL tautologies with propositional formulas of the calculus of names are also accepted as axioms. We have two rules for deriving these: detachment and substitution. We remember that using these means, from [\[Ia\]](#Ia){reference-type="eqref" reference="Ia"} and [\[Datisi\]](#Datisi){reference-type="eqref" reference="Datisi"}, we get [\[Ci\]](#Ci){reference-type="eqref" reference="Ci"}. Let us extend of Shepherdson's system. Firstly, let us define the unary functor '$\mathbf{ex}$' ("exists"), with which we state the non-emptiness of a given name: $$\label{dfex}\tag{df\,{\ensuremath{\mathbf{ex}}}} \ensuremath{\mathbf{ex}}S \:\leftrightarrow\: S\ensuremath{\mathbf{i}}S$$ Hence and from axioms [\[star\]](#star){reference-type="eqref" reference="star"} and [\[2star\]](#2star){reference-type="eqref" reference="2star"} we have, respectively: $$\begin{gathered} S\ensuremath{\mathbf{i}}P\:\rightarrow\:\ensuremath{\mathbf{ex}}S\\ \neg\:\ensuremath{\mathbf{ex}}S\:\rightarrow\:S\ensuremath{\mathbf{a}}P\end{gathered}$$ Moreover, from [\[Datisi\]](#Datisi){reference-type="eqref" reference="Datisi"} and (df ex) we obtain: $$(S\ensuremath{\mathbf{a}}P\:\wedge\:\ensuremath{\mathbf{ex}}S)\:\rightarrow\: S\ensuremath{\mathbf{i}}P$$ Further, applying [\[Ci\]](#Ci){reference-type="eqref" reference="Ci"}, we get: $$(S\ensuremath{\mathbf{a}}P\:\wedge\: \ensuremath{\mathbf{ex}}S)\:\rightarrow\: \ensuremath{\mathbf{ex}}P$$ For the strong interpretation of the functor of universal affirmative sentences, we adopt the abbreviation '$\mathbf{\dot{a}}$' and---following Kotarbiński and Lejewski---we can read it as 'every ... is'. For '$\mathbf{\dot{a}}$', we adopt the following definition: $$\label{dfka}\tag{df\,{\ensuremath{\mathbf{\dot{a}}}}} S\ensuremath{\mathbf{\dot{a}}}P \:\leftrightarrow\: (\ensuremath{\mathbf{ex}}S\:\wedge\:S\ensuremath{\mathbf{a}}P)$$ Now let us---following Kotarbiński---introduce two range equality functors for sentences of the form 'All $S$ is a $P$ and vice versa' and 'Every $S$ is a $P$ and vice versa'.[^10] Let us take the abbreviations '$\circeq$' and '$\doteq$' for these functors and the following definitions: $$\begin{gathered} S\circeq P\:\leftrightarrow\: (S\ensuremath{\mathbf{a}}P\:\wedge\:P\ensuremath{\mathbf{a}}S) \label{dfceq}\tag{df\,$\circeq$}\\ S\doteq P\:\leftrightarrow\:(\ensuremath{\mathbf{ex}}S\:\wedge\:S\ensuremath{\mathbf{a}}P\:\wedge\:P\ensuremath{\mathbf{a}}S) \label{dfdeq}\tag{df\,$\doteq$}\end{gathered}$$ What is more, we have: $$S\doteq P \:\rightarrow\: (\ensuremath{\mathbf{ex}}S\:\wedge\:S\ensuremath{\mathbf{a}}P\:\wedge\:P\ensuremath{\mathbf{a}}S \:\wedge\: \ensuremath{\mathbf{ex}}P)$$ Both of the above functors are, therefore, symmetrical. Moreover, the non-emptiness of one of the names in the true sentence '$S\circeq P$' forces the non-emptiness of the other, i.e., we have: $$(S\circeq P\:\wedge\:\ensuremath{\mathbf{ex}}S)\:\rightarrow\: \ensuremath{\mathbf{ex}}P)$$ We understand the functor of particular denial sentences in the natural sense. We leave the abbreviation '$\mathbf{o}$' and the definition [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"} for it. For universal denial sentences, however, we have three interpretations: weak, strong and super strong. To express the former, we can use the functor 'no ... is', leaving '$\mathbf{e}$' and the definition [\[dfe\]](#dfe){reference-type="eqref" reference="dfe"}. From [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"} and [\[dfe\]](#dfe){reference-type="eqref" reference="dfe"} and the theses already obtained, using CPL, we get [\[Ce\]](#Ce){reference-type="eqref" reference="Ce"} and that only for non-empty names is it allowed to weaken universal denial sentences to particular denial ones: $$(S\ensuremath{\mathbf{e}}P\:\wedge\:\ensuremath{\mathbf{ex}}S)\:\rightarrow\: S\ensuremath{\mathbf{o}}P$$ For the strong and super-strong interpretation, we can use the functors 'every ... is not' and 'every ...is not ... and vice versa' respectively, using the abbreviations '$\mathbf{\dot{e}}$' and '**ë**', and the following definitions: $$\begin{gathered} S\ensuremath{\mathbf{\dot{e}}}P\:\leftrightarrow\:(\ensuremath{\mathbf{ex}}S\:\wedge\:S\ensuremath{\mathbf{e}}P) \label{dfke}\tag{df\,$\ensuremath{\mathbf{\dot{e}}}$}\\ S\textbf{\"{e}}P\:\leftrightarrow\:(\ensuremath{\mathbf{ex}}S\:\wedge\:\ensuremath{\mathbf{ex}}P\:\wedge\:S\ensuremath{\mathbf{e}}P) \label{dfkke}\tag{df\,{\textbf{\"{e}}}}\end{gathered}$$ Universal denial sentences in the strong version thus entail particular denial ones and the non-emptiness of the subject, and in the super-strong version, additionally entail the non-emptiness of the predicate. We also have the following conversion laws: $$\begin{gathered} S\ensuremath{\mathbf{\dot{e}}}P\:\rightarrow\:(\ensuremath{\mathbf{ex}}S\:\wedge\:S\ensuremath{\mathbf{o}}P)\\ S\textbf{\"{e}}P\:\rightarrow\:(\ensuremath{\mathbf{ex}}S \:\wedge\:\ensuremath{\mathbf{ex}}P\:\wedge\:S\ensuremath{\mathbf{o}}P\:\wedge\:P\ensuremath{\mathbf{o}}S)\end{gathered}$$ ## The completeness of Shepherdson's axioms We use similar models as for Łukasiewicz's system. The only difference is that now the denotation function $\mathbbx{d}$ can also assign the empty set to name letters. We re-introduce the notions of being a true formula in the model $\ensuremath{\mathfrak{M}}=\langle\ensuremath{\mathbbx{D}},\ensuremath{\mathbbx{d}}\rangle$ using the previously given interpretation of the functors used. For atomic formulas with '$\mathbf{a}$', '$\mathbf{i}$' '$\mathbf{e}$' and '$\mathbf{o}$' the formal interpretation will be the same as in point [\[subsec2.3\]](#subsec2.3){reference-type="ref" reference="subsec2.3"} (except that now the sets $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ and $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$ may be empty). In addition, for the remaining atomic formulas, we also use the previously given interpretation, i.e., for any letters $\mathcal{S}$ and $\mathcal{P}$, we assume: - $\ensuremath{\mathbf{ex}}\ensuremath{\mathcal{S}}$ is true in $\mathfrak{M}$ iff the set $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ is non-empty; - $\ensuremath{\mathcal{S}}\ensuremath{\mathbf{\dot{a}}}\ensuremath{\mathcal{P}}$ is true in $\mathfrak{M}$ iff the set $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ is non-empty and it is included in $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$; - $\ensuremath{\mathcal{S}}\circeq\ensuremath{\mathcal{P}}$ is true in $\mathfrak{M}$ iff $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})=\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$; - $\ensuremath{\mathcal{S}}\doteq\ensuremath{\mathcal{P}}$ is true in $\mathfrak{M}$ iff $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})=\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$ and the set $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ is non-empty; - $\ensuremath{\mathcal{S}}\ensuremath{\mathbf{\dot{e}}}\ensuremath{\mathcal{P}}$ is true in $\mathfrak{M}$ iff the set $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ is non-empty and is disjoint with $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$; - $\ensuremath{\mathcal{S}}\textbf{\"{e}}\ensuremath{\mathcal{P}}$ is true in $\mathfrak{M}$ iff the sets $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ and $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$ are non-empty and they are disjoint. Moreover, similarly to Łukasiewicz's system, we use the interpretation of CPL connectives for formulas built using them. Using the above formal semantics, @Sh [pp. 144--145] proved that his $\mathbf{a}$$\mathbf{i}$-system is complete in the sense that the theses of this system are all those and only formulas that are true in every model. He achieved this by identifying this system with a first-order open theory. We described this method in point [\[subsec2.3\]](#subsec2.3){reference-type="ref" reference="subsec2.3"} [see also @ja92]. In [@ja90; @ja91c pp. 86--87], the Henkin method was used to obtain the completeness of Shepherdson's system (which we also briefly described in point [\[subsec2.3\]](#subsec2.3){reference-type="ref" reference="subsec2.3"}). ## Słupecki's system[\[subsec3.3\]]{#subsec3.3 label="subsec3.3"} @Sl proposed a calculus of names in which the functors of affirmative sentences were primary, and he adopted a strong interpretation for universal sentences. Therefore, we can abbreviate these functors by '$\mathbf{\dot{a}}$' and '$\mathbf{i}$', respectively. The theses of Słupecki's system can also be applied to empty names. This system includes all correct Aristotelian syllogisms and logical square and conversion laws. Słupecki adopted four $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies as axioms. The first is the law of conversion [\[Ci\]](#Ci){reference-type="eqref" reference="Ci"}; the others are the law of subordination and two syllogisms: $$\begin{gathered} S\ensuremath{\mathbf{\dot{a}}}P\:\rightarrow\: S\ensuremath{\mathbf{i}}P\label{Ska}\tag{S{\ensuremath{\mathbf{\dot{a}}}}}\\ (M\ensuremath{\mathbf{\dot{a}}}P\:\wedge\:S\ensuremath{\mathbf{\dot{a}}}M)\:\rightarrow\:S\ensuremath{\mathbf{\dot{a}}}P \label{Bkarbara}\tag{B\.{a}rbara}\\ (M\ensuremath{\mathbf{\dot{a}}}P\:\wedge\:S\ensuremath{\mathbf{i}}M)\:\rightarrow\: S\ensuremath{\mathbf{i}}P \label{Dkarii}\tag{D\.{a}rii}\end{gathered}$$ Moreover, all substitutions of CPL tautologies with propositional formulas of the calculus of names are accepted as axioms. We also have two derivation rules: detachment and substitution. Słupecki also adopts [\[dfe\]](#dfe){reference-type="eqref" reference="dfe"} and a specific definition of the functor of particular denial sentences. It cannot be [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"} because it uses '$\mathbf{a}$', not $\mathbf{\dot{a}}$'. "Visually", however, the definition adopted by Słupecki corresponded to the formula [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"} because he used the symbol '$\mathbf{a}$', not '$\mathbf{\dot{a}}$', and he understood the former symbol in the strong sense. However, since we have established the meanings of the symbols '$\mathbf{a}$', $\mathbf{\dot{a}}$' and '$\mathbf{o}$', we cannot replace '$\mathbf{a}$' with '$\mathbf{\dot{a}}$' in [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"}, leaving the symbol '$\mathbf{o}$'. We need to replace the latter with another symbol. Let us assume that this symbol is '**õ**' and that the definition adopted for it is: $$\label{deko}\tag{df\,\textbf{\~{o}}} S\textbf{\~{o}} P\:\leftrightarrow\:\neg\:S\ensuremath{\mathbf{\dot{a}}}P$$ Accepting it causes some interpretation complications. According to the adopted interpretation for '$\mathbf{\dot{a}}$', we will get the interpretation of '**õ**', which is not consistent with the linguistic usage for particular denial sentences (cf. point [\[subsec1.3\]](#subsec1.3){reference-type="ref" reference="subsec1.3"}). Namely, it turns out that a sentence of the form '$S\textbf{\~{o}}P$' is to be true iff either the name $S$ is empty or it has a referent which is not a referent of the name $P$. Słupecki himself saw this [-@Sl p. 189]. He, therefore, tried to circumvent the difficulty by advising that "the sentence $Oab$ \[corresponds to our '$S\textbf{\~{o}}P$'\] understand only as an abbreviation of the sentence $NUab$ \[corresponds to our '$\neg\:S\ensuremath{\mathbf{\dot{a}}}P$'\] and read: not true that every $a$ is a $b$."[^11] Thus, we are to reject the original finding that his '$O$' (corresponding to '**õ**') is the symbolic notation of the functor 'some ... is not ...' and assume that it is only the symbolic notation of the phrase 'it is not case that every ... is ...'. The consequences of this are as follows. The meaning of '$S\textbf{\~{o}}P\:\leftrightarrow\:\neg\:S\ensuremath{\mathbf{\dot{a}}}P$' is just an abbreviation of the identity '$\neg\:S\ensuremath{\mathbf{\dot{a}}}P\:\leftrightarrow\:\neg\:S\ensuremath{\mathbf{\dot{a}}}P$'. Similarly, '$(P\ensuremath{\mathbf{\dot{a}}}M\:\wedge\:S\textbf{\~{o}}M)\:\rightarrow\:S\textbf{\~{o}}P$' and '$(M\textbf{\~{o}}P\:\wedge\:M\ensuremath{\mathbf{\dot{a}}}S)\:\rightarrow\:S\textbf{\~{o}}P$' are only abbreviations for '$(P\ensuremath{\mathbf{\dot{a}}}M\:\wedge\:\neg\:S\ensuremath{\mathbf{\dot{a}}}M) \:\rightarrow\:\neg\:S\ensuremath{\mathbf{\dot{a}}}P$' and '$(\neg\:M\ensuremath{\mathbf{\dot{a}}}P\:\wedge\:M\ensuremath{\mathbf{\dot{a}}}S) \:\rightarrow\:\neg\:S\ensuremath{\mathbf{\dot{a}}}P$' obtained from [\[Bkarbara\]](#Bkarbara){reference-type="eqref" reference="Bkarbara"} after substitution and the contraposition of CPL. In the alphabet of Słupecki's calculus, no symbol would represent the functor of particular denial sentences. As already mentioned, in Słupecki's system, all Aristotelian syllogisms, as well as the logical square and conversion laws written with '$\mathbf{\dot{a}}$', '$\mathbf{i}$', $\mathbf{e}$' and '**õ**', obtain. However, as shown in [@ja87], the theses of this system are not, for example, the following $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies: [\[star\]](#star){reference-type="eqref" reference="star"} and $$\begin{gathered} S\ensuremath{\mathbf{i}}S\:\rightarrow\:S\ensuremath{\mathbf{\dot{a}}}S\\ S\ensuremath{\mathbf{i}}P\:\rightarrow\:S\ensuremath{\mathbf{\dot{a}}}\label{I}\tag{I}\\ S\ensuremath{\mathbf{\dot{a}}}P\:\rightarrow\:S\ensuremath{\mathbf{i}}S\label{II}\tag{II}\\ S\ensuremath{\mathbf{\dot{a}}}P\:\rightarrow\:S\ensuremath{\mathbf{\dot{a}}}S\\ S\ensuremath{\mathbf{\dot{a}}}P\:\rightarrow\:P\ensuremath{\mathbf{\dot{a}}}P\end{gathered}$$ However, one cannot claim that Słupecki did not want to obtain the implications that have identities in their consequents because, by [\[Dkarii\]](#Dkarii){reference-type="eqref" reference="Dkarii"}, [\[Ci\]](#Ci){reference-type="eqref" reference="Ci"} and [\[Ska\]](#Ska){reference-type="eqref" reference="Ska"}, we get: $$\label{proc}\tag{\%} P\ensuremath{\mathbf{\dot{a}}}S\:\rightarrow\:S\ensuremath{\mathbf{i}}S$$ ## Complete axiomatizations of $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies In [@ja87; @ja91b; @ja91c] it was shown that the following sets of formulas form full axiomatizations of the set of $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies: - Słupecki's axioms plus formula [\[I\]](#I){reference-type="eqref" reference="I"}; - [\[Ci\]](#Ci){reference-type="eqref" reference="Ci"}, [\[Bkarbara\]](#Bkarbara){reference-type="eqref" reference="Bkarbara"}, [\[Dkarii\]](#Dkarii){reference-type="eqref" reference="Dkarii"} plus formulas [\[I\]](#I){reference-type="eqref" reference="I"} and [\[II\]](#II){reference-type="eqref" reference="II"}; - [\[Ska\]](#Ska){reference-type="eqref" reference="Ska"}, [\[Bkarbara\]](#Bkarbara){reference-type="eqref" reference="Bkarbara"} plus [\[I\]](#I){reference-type="eqref" reference="I"} and the following formula $$\label{Dkatisi}\tag{D\.{a}tisi} (M\ensuremath{\mathbf{\dot{a}}}P\:\wedge\:M\ensuremath{\mathbf{i}}S)\:\rightarrow\: S\ensuremath{\mathbf{i}}P$$ - [\[Bkarbara\]](#Bkarbara){reference-type="eqref" reference="Bkarbara"}, [\[Dkatisi\]](#Dkatisi){reference-type="eqref" reference="Dkatisi"}, [\[I\]](#I){reference-type="eqref" reference="I"} and [\[II\]](#II){reference-type="eqref" reference="II"}. *Remark 1*. Piotr Kulicki [-@Kul00; -@Kul11] attempted to give complete axiomatizations of the set of $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies. In the first of these works, the axioms are: [\[Ska\]](#Ska){reference-type="eqref" reference="Ska"}, [\[Bkarbara\]](#Bkarbara){reference-type="eqref" reference="Bkarbara"}, [\[proc\]](#proc){reference-type="eqref" reference="proc"} and: $$\label{Dimkatis}\tag{{Dim\.{a}tis}} (P\ensuremath{\mathbf{i}}M\:\wedge\:M\ensuremath{\mathbf{\dot{a}}}S)\:\rightarrow\:S\ensuremath{\mathbf{i}}P$$ In the second work, the axioms are [\[Ska\]](#Ska){reference-type="eqref" reference="Ska"}, [\[Bkarbara\]](#Bkarbara){reference-type="eqref" reference="Bkarbara"}, [\[Dkarii\]](#Dkarii){reference-type="eqref" reference="Dkarii"} and [\[proc\]](#proc){reference-type="eqref" reference="proc"}. Both of these attempts, however, were unsuccessful because in both cases, we will not get, for example, [\[star\]](#star){reference-type="eqref" reference="star"} and [\[Ci\]](#Ci){reference-type="eqref" reference="Ci"}, although the author of these works states that is otherwise [@Kul11 p. 56].0◻ To complete the axiomatization of $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies, we can add the following definition of the functor '$\mathbf{a}$': $$\label{dfa}\tag{df\,{\ensuremath{\mathbf{a}}}} S\ensuremath{\mathbf{a}}P\:\leftrightarrow\:(\neg\:S\ensuremath{\mathbf{\dot{a}}}S\:\vee\:S\ensuremath{\mathbf{\dot{a}}}P)$$ It gives us: $$S\ensuremath{\mathbf{a}}P\:\leftrightarrow\:(\neg\:S\ensuremath{\mathbf{i}}S\:\vee\:S\ensuremath{\mathbf{\dot{a}}}P)$$ Having the functor '$\mathbf{a}$', we can introduce [\[dfo\]](#dfo){reference-type="eqref" reference="dfo"} and the definitions of other functors given in point [\[subsec3.1\]](#subsec3.1){reference-type="ref" reference="subsec3.1"}. In [@ja91b], the definitional equivalence of Shepherdson's system for $\mathbf{a}$$\mathbf{i}$-tautologies with the four equivalent systems for $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies given in [@ja87; @ja91b; @ja91c] was demonstrated. The presented axiomatizations of $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies are also complete since they are definitionally equivalent to Shepherdson's system. In [@ja90; @ja91c pp. 87--88], Henkin's method, described in point [\[subsec2.3\]](#subsec2.3){reference-type="ref" reference="subsec2.3"}, proved this. It can also be obtained by identifying the complete systems of $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies with the corresponding first-order open theories (see point [\[subsec2.3\]](#subsec2.3){reference-type="ref" reference="subsec2.3"}). # Syllogistic with Leśniewski's copula The copula 'is' is the only primitive of Leśniewski's Ontology. This theory can be classified as a quantifier calculus of names. In this work, however, we deal only with the standard, i.e., quantifier-free calculus of names. Leśniewski applied his Ontology to all names without dividing them into proper names and explicit or implicit descriptions and distinguishing whether they are general or singular. He had one type of variable for all names. Since, in this paper, we are interested in Ontology only in the context of the logic of names, we will use schematic letters instead of variables. Sentences with the copula 'is' Leśniewski understood as follows: - A sentence '$S$ is a $P$' is true iff the name $S$ has exactly one referent which belongs to the range of the name $P$. Leśniewski's copula 'is' is standardly symbolized by the Greek letter '$\ensuremath{\boldsymbol{\varepsilonup}}$' (which refers to the Latin '*est*' -- is). *Remark 2*. Sentences of the form '$S\ensuremath{\boldsymbol{\varepsilonup}}P$' should be distinguished from singular sentences of the form '$a$ is a $P$'. In the former, any names can be inserted for '$S$'; in the latter, only names with exactly one referent can be inserted in '$a$'. We are not talking about the received sentences' truth but only their syntactic and semantic correctness. 0◻ As in point [\[subsec2.3\]](#subsec2.3){reference-type="ref" reference="subsec2.3"}, we introduce the notion of being a true formula in a model $\ensuremath{\mathfrak{M}}=\langle\ensuremath{\mathbbx{D}},\ensuremath{\mathbbx{d}}\rangle$ using the above interpretation for '$\boldsymbol{\varepsilonup}$'. For any letters $\mathcal{S}$ and $\mathcal{P}$, we assume: - $S\ensuremath{\boldsymbol{\varepsilonup}}P$ is true in $\mathfrak{M}$ iff the set $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{S}})$ is a singleton whose only element belongs to the set $\ensuremath{\mathbbx{d}}(\ensuremath{\mathcal{P}})$. ## The open fragment of Ontology It is the set of its quantifier-free theses. Arata @Ish [theorem 3.4] showed that this set is axiomatizable by the following three theses of Ontology: $$\begin{gathered} S\ensuremath{\boldsymbol{\varepsilonup}}P\:\rightarrow\:S\ensuremath{\boldsymbol{\varepsilonup}}S \label{Ish1}\tag{Ish1}\\ (S\ensuremath{\boldsymbol{\varepsilonup}}M\:\wedge\:M\ensuremath{\boldsymbol{\varepsilonup}}P)\:\rightarrow\:S\ensuremath{\boldsymbol{\varepsilonup}}P \label{Ish2}\tag{Ish2}\\ (S\ensuremath{\boldsymbol{\varepsilonup}}P\:\wedge\:P\ensuremath{\boldsymbol{\varepsilonup}}M)\:\rightarrow\:P\ensuremath{\boldsymbol{\varepsilonup}}S \label{Ish3}\tag{Ish3}\end{gathered}$$ and all substitutions of CPL tautologies with formulas of the form $\ensuremath{\mathcal{S}}\ensuremath{\boldsymbol{\varepsilonup}}\ensuremath{\mathcal{P}}$ plus detachment and substitution rules. It is easy to check that [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"}--[\[Ish3\]](#Ish3){reference-type="eqref" reference="Ish3"} are $\boldsymbol{\varepsilonup}$-tautologies, i.e., true in every model. Furthermore, Mitio @T showed that [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"}--[\[Ish3\]](#Ish3){reference-type="eqref" reference="Ish3"} is a complete axiomatization of the set of $\boldsymbol{\varepsilonup}$-tautologies. ## Shepherdson's system plus '$\varepsilonup$' The copula '$\boldsymbol{\varepsilonup}$' is not definable by the pair of the functors '$\mathbf{a}$' and '$\mathbf{i}$'. Therefore, '$\boldsymbol{\varepsilonup}$' must be added to them as a primitive functor. In [@ja90; @ja91a; @ja91c; @ja92], several complete axiomatizations of the set of $\mathbf{a}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies are given. They all extend Shepherdson's axioms (Sh). In the first, we add the following four $\mathbf{a}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies to Sh: [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"}, $$\begin{gathered} S\ensuremath{\boldsymbol{\varepsilonup}}P\:\rightarrow\:S\ensuremath{\mathbf{a}}P\label{1}\tag{1}\\ S\ensuremath{\boldsymbol{\varepsilonup}}S\:\rightarrow\:S\ensuremath{\mathbf{i}}S \label{2}\tag{2}\\ (S\ensuremath{\mathbf{a}}M\:\wedge\:M\ensuremath{\boldsymbol{\varepsilonup}}M\:\wedge\:S\ensuremath{\mathbf{i}}P)\:\rightarrow\:S\ensuremath{\boldsymbol{\varepsilonup}}P \label{3}\tag{3}\end{gathered}$$ We do not need to take [\[Ish2\]](#Ish2){reference-type="eqref" reference="Ish2"} and [\[Ish3\]](#Ish3){reference-type="eqref" reference="Ish3"} as axioms because we will derive the former from [\[1\]](#1){reference-type="eqref" reference="1"}--[\[3\]](#3){reference-type="eqref" reference="3"}, [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"} and [\[Barbara\]](#Barbara){reference-type="eqref" reference="Barbara"}, and the latter from [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"}, [\[1\]](#1){reference-type="eqref" reference="1"}--[\[3\]](#3){reference-type="eqref" reference="3"}. The second axiomatization is obtained by adding to Sh: [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"}, [\[1\]](#1){reference-type="eqref" reference="1"}, [\[2\]](#2){reference-type="eqref" reference="2"} and $$\begin{gathered} (S\ensuremath{\boldsymbol{\varepsilonup}}S\:\wedge\:S\ensuremath{\mathbf{a}}P)\:\rightarrow\:S\ensuremath{\mathbf{i}}P\label{4}\tag{4}\\ (S\ensuremath{\boldsymbol{\varepsilonup}}S\:\wedge\:S\ensuremath{\mathbf{i}}P)\:\rightarrow\:S\ensuremath{\mathbf{a}}P\label{5}\tag{5}\\ (S\ensuremath{\mathbf{a}}P\:\wedge\:P\ensuremath{\boldsymbol{\varepsilonup}}P\:\wedge\:S\ensuremath{\mathbf{i}}S)\:\rightarrow\: S\ensuremath{\boldsymbol{\varepsilonup}}P \label{6}\tag{6}\end{gathered}$$ The third axiomatization is obtained by adding to Sh: [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"}, [\[1\]](#1){reference-type="eqref" reference="1"}, [\[2\]](#2){reference-type="eqref" reference="2"}, [\[6\]](#6){reference-type="eqref" reference="6"} and $$\label{7}\tag{7} (S\ensuremath{\boldsymbol{\varepsilonup}}S\:\wedge\:S\ensuremath{\mathbf{i}}P)\:\rightarrow\:S\ensuremath{\boldsymbol{\varepsilonup}}P$$ We obtain the fourth axiomatization by adding to Sh: [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"}, [\[1\]](#1){reference-type="eqref" reference="1"}, [\[2\]](#2){reference-type="eqref" reference="2"}, [\[7\]](#7){reference-type="eqref" reference="7"} and $$(S\ensuremath{\boldsymbol{\varepsilonup}}P\:\wedge\:P\ensuremath{\mathbf{a}}S)\:\rightarrow\:P\ensuremath{\boldsymbol{\varepsilonup}}P$$ ## Systems for '$\ensuremath{\mathbf{\dot{a}}}$', '$\ensuremath{\mathbf{i}}$' and '$\varepsilonup$' The copula '$\boldsymbol{\varepsilonup}$' is also not definable by the pair of functors '$\mathbf{\dot{a}}$' and '$\mathbf{i}$'. The complete axiomatizations of $\mathbf{\dot{a}}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies can be obtained by adding to any of the complete axiomatizations of $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies from point [\[subsec3.3\]](#subsec3.3){reference-type="ref" reference="subsec3.3"}: the definition [\[dfa\]](#dfa){reference-type="eqref" reference="dfa"} and any set of $\mathbf{a}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies that we added to Sh in the previous point. In [@ja90; @ja91c p. 107], complete axiomatizations of the set of $\mathbf{\dot{a}}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies are given by adding to any of the complete axiomatizations of $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies three formulas: [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"} and $$\begin{gathered} S\ensuremath{\boldsymbol{\varepsilonup}}P\:\rightarrow\:S\ensuremath{\mathbf{\dot{a}}}P \label{kj}\tag{$\dot{1}$}\\ (S\ensuremath{\mathbf{\dot{a}}}P\:\wedge\:P\ensuremath{\boldsymbol{\varepsilonup}}P\:\wedge\:S\ensuremath{\mathbf{i}}S)\:\rightarrow\:S\ensuremath{\boldsymbol{\varepsilonup}}P \label{ksz}\tag{$\dot{6}$}\end{gathered}$$ *Remark 3*. The book [@Kul11] does not cite the works [@ja90; @ja91c]. Thus, we do not know from this book who first gave the axiomatization of the set of $\mathbf{\dot{a}}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies. The axiomatization related to '$\boldsymbol{\varepsilonup}$' provided by Kulicki consists of four formulas [\[Ish1\]](#Ish1){reference-type="eqref" reference="Ish1"}, [\[kj\]](#kj){reference-type="eqref" reference="kj"} and $$\begin{gathered} (S\ensuremath{\mathbf{i}}P\:\wedge\:P\ensuremath{\boldsymbol{\varepsilonup}}P)\:\rightarrow\:P\ensuremath{\boldsymbol{\varepsilonup}}S\\ (S\ensuremath{\mathbf{\dot{a}}}M\:\wedge\:M\ensuremath{\boldsymbol{\varepsilonup}}P)\:\rightarrow\:S\ensuremath{\boldsymbol{\varepsilonup}}P\end{gathered}$$ The first one corresponds to [\[7\]](#7){reference-type="eqref" reference="7"}. It is also necessary to assume a complete axiomatization of $\mathbf{\dot{a}}$$\mathbf{i}$-tautologies, which is not present in [@Kul11]. 0◻ ## The completeness of axiomatizations of the sets of $\mathbf{a}$$\mathbf{i}$$\varepsilonup$-tautologies and $\mathbf{\dot{a}}$$\mathbf{i}$$\varepsilonup$-tautologies In [@ja90; @ja91c], using Henkin's method, the completeness of the above-discussed axiomatizations for $\mathbf{a}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies was proved. However, due to the copula '$\boldsymbol{\varepsilonup}$', the universes of the constructed models had to be more complicated than those from point [\[subsec2.3\]](#subsec2.3){reference-type="ref" reference="subsec2.3"}. Yet another type of model in Henkin's method for these axiomatizations is given in [@ja91a]. In [@ja92], the completeness of the axiomatization of $\mathbf{a}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautology was proved by identifying the system with an open first-order theory. For this theory, representation theorems have been proved using appropriate filters built on the domain of a given model of an elementary theory. From the representation theorem and the Gödel completeness theorem for first-order theories, a conclusion follows which, assuming the identification, corresponds to the statement that the set of theses of this system is equal to the set of formulas that are true in each model from the class under consideration. It has been proved that the axiomatizations of the set of $\mathbf{a}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies are, by definition, equivalent to the axiomatizations of the set of $\mathbf{\dot{a}}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies. This, in turn, yields completeness to the latter as well. For the second axiomatization, appropriate reconstructions of the methods used to obtain the completeness of the axiomatization of the $\mathbf{a}$$\mathbf{i}$$\boldsymbol{\varepsilonup}$-tautologies set can also be applied. 99 Corcoran, J., 1972, "Aristotle's natural deduction system", pages 85--131 in J. Corcoran (ed.), *Ancient Logic and Its Modern Interpretations*, Dordrecht: Reidel. Indrzejczak, A., 2015, "Natural deduction", in Internet Encyclopedia of Philosophy, J.-Y. Béziau (ed.). URL: <http://www.iep.utm.edu/nat-ded> Indrzejczak, A., 2021, *Sequents and Trees. An Introduction to the Theory and Applications of Propositional Sequent Calculi*, Studies in Universal Logic, Heidelberg: Birkhäuser. Ishimoto, A., 1977, "A propositional fragment of Leśniewski's elementary ontology", *Studia Logica* 36: 285--299. Kotarbiński, T., 1929, *Elementy teorii poznania, logiki formalnej i metodologii nauk* (Elements of the theory of knowledge, formal logic and methodology of the sci-ences), Lwów: Zakład Narodowy im. Ossolińskich. Cited the second revised edition: Wrocław: Ossolineum, 1961. Eng. trans. of 2nd ed., with appendices: *Gnosiology. The Scientific Approach to the Theory of Knowledge*, O. Wojtasiewlcz (trans.), Oxford and Wrocław: Pergamon Press and Ossolineum, 1966. Kulicki, P., 2000, "The use of axiomatic rejection", pages 109--117 in T. Childres (ed.), *The Logica Yearbook 1999*, Praga: Filosophia. Kulicki, P., 2011, *Aksjomatyczne systemy rachunku nazw* (Axiomatic systems of calculus of names), Lublin: Wydawnictwo KUL. Lejewski, C., 1958, 'On Leśniewski's ontology'', *Ratio* 1: 150--176. Reprint: pages 1230--148 in J.T.J. Srzednicki and V.F. Rickey (eds.), *Leśniewski's Systems. Ontology and Mereology*, The Hague and Wrocław: Martinus Nijhoff Publishers (Kluwer) and Ossolineum, 1984. Łukasiewicz, J., 1929, *Elementy logiki matematycznej* (Elements of mathematical logic), Lectures notes, edited by M. Presburger, Warszawa; second edition, Warszawa: PWN, 1958. Łukasiewicz, J., 1934, "Znaczenie analizy logicznej dla poznania" (The importance of logical analysis for cognition), *Przegląd Filozoficzny* 37: 369--377. Reprint: *Filozofia Nauki* 4/3, 1996: 160--167. Łukasiewicz, J., 1939, "O sylogistyce Arystotelesa" (On Aristotle's syllogistic), *Sprawozdania PAU* 44: 220--227. Reprint: pages 220--227 in *J. Łukasiewicz, Z zagadnień logiki i filozofii* (From the issues of logic and philosophy), Warszawa: PWN, 1961. Łukasiewicz, J., 1951/57, *Aristotle's Syllogistic from the Standpoint of Modern Formal Logic*, Oxford University Press; first edition 1951, second edition 1957. Morawiec, A., 1961, "Podstawy logiki nazw", *Studia Logica* 12: 145--161. Summary: "Foundations of the theory of names", *Studia Logica* 12: 167--170. Pietruszczak, A., 1987, "O logice tradycyjnej i rachunku nazw dopuszczającym podstawienia nazw pustych" (On traditional logic and calculus names allowing substitutions of empty names), *Ruch Filozoficzny* 44: 158--166. Pietruszczak, A., 1990, "Logiczno-filozoficzne i formalne problemy w metodologii standardowego rachunku nazw. Systemy i ich metateoria" (Logical-philosophical and formal problems in the methodology of the standard calculus of names. Systems and their metatheory), PhD dissertation, Nicolaus Copernicus University in Toruń. Pietruszczak, A., 1991a, "Standardowe rachunki nazw z funktorem Leśniewskiego" (Standard calculi of names with Leśniewski's functor), *Acta Universitatis Nicolai Copernici*, Logika I: 5--29. Pietruszczak, A., 1991b, "O pewnym ujęciu logiki tradycyjnej" (On a certain approach to traditional logic), *Acta Universitatis Nicolai Copernici*, Logika I: 31--41. Pietruszczak, A., 1991c, *Bezkwantyfikatorowy rachunek nazw. Systemy i ich metateoria* (Quantifier-free calculus of names. Systems and their metatheory), Toruń: Wydawnictwo Adam Marszałek. It is a book edition of a part of the doctoral dissertation [@ja90]. Pietruszczak, A., 1992, "Stała Leśniewskiego w teoriach sylogistycznych. Semantyczne badania pewnych kwantyfikatorowych rachunków nazw" (Leśniewski's functor in syllogistic theories. Semantic studies of certain quantifier calculus of names), *Acta Universitatis Nicolai Copernici*, Logika III: 45---76. Pratt-Hartmann, I., and L.S. Moss, 2009, "Logics for the relational syllogistic", *The Review of Symbolic Logic* 2 (4): 647--683. Quine, W.V., 1953, "Logic and reification of universals", pages 102--129 in *From a Logical Point of View*, Harvard University Press. Shepherdson, J.C., 1956, "On the interpretation of Aristotelian syllogistic", *The Journal of Symbolic Logic* 21: 137--147. Słupecki, J., 1946, "Uwagi o sylogistyce Arystotelesa" (Notes on Aristotle's syllogistic), *Annales UMCS*, sectio F, 1: 187--191. Takano, M., 1985, "A semantical investigation into Leśniewski's axiom of his ontology", *Studia Logica* 44: 71--77. [^1]: The research has been supported by the grant from the National Science Centre (NCN), Poland, project no. 2021/43/B/HS1/03187. [^2]: The Polish text of [@Luk34] is translated by the author of this paper. [^3]: The term 'variable' may be replaced by 'schematic letter'. @Q wrote about the significant difference between these terms. The name 'calculus of names' probably comes from the fact that it is about "variables" for which we substitute names and perform some "calculus" on its formulas. [^4]: The difference between the theses of a given logic, as its tautologies, and the theses of a given formalized theory (e.g., first-order theory) should be emphasized here. The latter are not tautologies. They can only be said to be true in the models of a given theory. [^5]: We assume that repeated premises and their order are unimportant in a given sequence. Therefore, we identify sequences that differ in one or both of these features [^6]: In [@Luk51] the letter '$U$' were replaced by the letter '$A$'. @Sh also uses formulas of the form '$Aab$'. [^7]: Łukasiewicz uses his bracketless notation here. [^8]: In [@Luk51] the letter '$Y$' were replaced by the letter '$E$'. [^9]: It can also be assumed that the notations of the form '$S\ensuremath{\mathbf{e}}P$' and '$S\ensuremath{\mathbf{o}}P$' are only abbreviations for the formulas '$\neg\: S\ensuremath{\mathbf{i}}P$' and '$\neg\:S\ensuremath{\mathbf{a}}P$', respectively. This means that the formers are not among the formulas at all. Such a solution for '$S\ensuremath{\mathbf{o}}P$' was attempted by @Sl; see further point [\[subsec3.3\]](#subsec3.3){reference-type="ref" reference="subsec3.3"}. [^10]: Instead of those used by us, Lejewski [-@Lej p. 130 in 1984] used sentences of the form '$S$ is identical with $P$' ("the functor of weak identity") and 'Only every $S$ is a $P$' ("the functor of strong identity"), respectively. [^11]: The author of this paper translates the Polish text from [@Sl].

arxiv_math

--- abstract: | For a fixed integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin proved that the maximum $\lambda_1(G)$ among all connected outerplanar graphs on $n$ vertices is achieved by the fan graph $K_1\vee P_{n-1}$. In this paper, we consider a similar problem of determining the maximum $\lambda_2$ among all connected outerplanar graphs on $n$ vertices. For $n$ even and sufficiently large, we prove that the maximum $\lambda_2$ is uniquely achieved by the graph $(K_1\vee P_{n/2-1})\!\!-\!\!(K_1\vee P_{n/2-1})$, which is obtained by connecting two disjoint copies of $(K_1\vee P_{n/2-1})$ through a new edge at their ends. When $n$ is odd and sufficiently large, the extremal graphs are not unique. The extremal graphs are those graphs $G$ that contains a cut vertex $u$ such that $G\setminus \{u\}$ is isomorphic to $2(K_1\vee P_{n/2-1})$. We also determine the maximum $\lambda_2$ among all 2-connected outerplanar graphs and asymptotically determine the maximum of $\lambda_k(G)$ among all connected outerplanar graphs for general $k$. author: - "George Brooks [^1]" - "Maggie Gu [^2]" - "Jack Hyatt [^3]" - "William Linz [^4]" - "Linyuan Lu [^5]" title: On the maximum second eigenvalue of outerplanar graphs --- # Introduction Let $G$ be a graph on $n$ vertices, with $n$ a positive integer. The *adjacency matrix* of $G$ is the $n\times n$ matrix $A = (a_{uv})_{u,v \in V(G)}$ with rows and columns indexed by $V(G)$ where $a_{uv} = 1$ if $u\sim v$ and $a_{uv} = 0$ otherwise. The *eigenvalues* of $G$ are the eigenvalues of its adjacency matrix $A$. As $A$ is symmetric, these eigenvalues are real, and we label them in non-increasing order as $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n$. The largest eigenvalue $\lambda_1$ is called the *spectral radius* of a graph and has been extensively studied (see, for example, the survey of Cvetković and Rowlinson [@Cvetkovic-Rowlinson90]). More recently, there has been much interest in finding the maximum spectral radius over a family of graphs, as such results extend Turán-type results via the inequality $\lambda_1 \ge \frac{2m}{n}$, where $m$ is the number of edges in the graph. An archetypal result is the spectral Turán theorem of Nikiforov [@Nik2; @Nik0], which determines the maximum spectral radius of a $K_{r+1}$-free graph. The maximum spectral radius has also been determined for many other classes of graphs, including outerplanar graphs and planar graphs [@TT2017], $K_{s, t}$-minor-free graphs [@Nikiforov2017; @Tait2019; @ZL2022] and graphs forbidding a cycle or path of some prescribed length [@Nik3; @GH2019; @YWZ2012; @WZ2012]. In contrast to the spectral radius, the $k$th largest eigenvalue of a graph has not been well-studied for $k\ge 3$. The second largest eigenvalue of a graph has been reasonably well-studied because of its relation to connectivity and expansion properties of graphs (see, for example, the survey of Cvetković and Simić [@CS95]). However, there are many classes of graphs for which the graph with maximum second largest eigenvalue is unknown, let alone the graph with maximum $k$th largest eigenvalue. Even the graph with maximum $k$th largest eigenvalue over the family of all graphs on $n$ vertices is unknown for any $k\ge 3$ [@Hon; @Nik1; @Linz2023]. The tree with maximum $k$th largest eigenvalue has been determined in a sequence of papers [@Neumaier82; @Yuan86; @Shao95; @Chen07]. In this paper, we study the second eigenvalue $\lambda_2$ of connected outerplanar graphs. A graph is *outerplanar* if it can be drawn in the plane without crossing edges such that each of its vertices lies on the outer face. Equivalently, a graph is outerplanar if and only if it is $K_{2,3}$-minor-free and $K_4$-minor-free. Cvetković and Rowlinson [@Cvetkovic-Rowlinson90] conjectured that the *fan graph* $K_1 \vee P_{n-1}$ (see Figure [\[fig:fangraph\]](#fig:fangraph){reference-type="ref" reference="fig:fangraph"}) is the outerplanar graph with maximum spectral radius. Work was done on this conjecture by Rowlinson [@Rowlinson90] and Zhou, Lin and Hu [@ZLH01]. The conjecture was proved for sufficiently large $n$ by Tait and Tobin [@TT2017]. Subsequently, the outerplanar graph with maximum spectral radius was determined for all values of $n$ by Lin and Ning [@Lin-Ning19]. By contrast, not much seems to be known about the middle eigenvalues of outerplanar graphs. Li and Sun [@LS2023] have studied outerplanar graphs with small $\lambda_2$, but the $k$th largest eigenvalues of outerplanar graphs with $k\ge 2$ has not been previously studied. For a fixed $k$, we define $\lambda_{k,max}(n)$, or $\lambda_{k,max}$ for short, to be the maximum $k$th eigenvalue among all connected outerplanar graphs on $n$ vertices. We have the following general theorem, which determines $\lambda_{k,max}(n)$ up to the second order. **Theorem 1**. *For any fixed integer $k\geq 2$, we have $$\lambda_{k,max}(n)=\sqrt{n/k}+ 1+ O\left(\frac{1}{\sqrt{n}} \right).$$ Moreover, any outerplanar graph $G$ on $n$ vertices with $\lambda_k(G) =\lambda_{k,max}(n)$ contains exactly $k$ vertices of degree $\frac{n}{k} \pm O(1)$.* **Conjecture 1**. *Suppose $n=kq+1$. We conjecture that for a fixed $k$ and sufficiently large $n$, $$\lambda_{k,max}(n)=\lambda_1(K_1\vee P_{q-1}).$$ Moreover, any extremal graph $G$ on $n$ vertices with $\lambda_k(G)=\lambda_{k,max}(n)$ has the following structure: there exist a cut vertex $u$ such that deleting $u$ from $G$ results in $k$ copies of $K_1\vee P_{q-1}$.* We further determine the extremal connected outerplanar graphs which achieve $\lambda_{2,max}$. **Theorem 2**. *For $n$ even and sufficiently large, among all connected outerplanar graphs on $n=2q$ vertices, the graph maximizing $\lambda_2$ is unique and isomorphic to the graph in Figure [\[fig:2qfig\]](#fig:2qfig){reference-type="ref" reference="fig:2qfig"}, denoted by $(K_1\vee P_{q-1})\!\!-\!\!(K_1\vee P_{q-1})$, which is constructed by gluing two disjoint copies of the fan graph $K_1\vee P_{q-1}$ via an edge connecting vertices of smallest degrees.* **Corollary 1**. *For $n=2q$ even and sufficiently large, among all (not necessarily connected) outerplanar graphs on $n$ vertices, the maximum $\lambda_2$ is uniquely achieved by $2(K_1\vee P_{q-1})$.* **Remark 1**. *For $n=12$, $\lambda_{2, max}$ is not achieved by $(K_1\vee P_{5})\!\!-\!\!(K_1\vee P_{5})$. Instead, it is achieved by the graph in Figure [\[fig:12vertexextremal\]](#fig:12vertexextremal){reference-type="ref" reference="fig:12vertexextremal"}. We further conjecture that for all even $n\geq 14$, the graph maximizing $\lambda_2$ is unique and isomorphic to the graph $(K_1\vee P_{q-1})\!\!-\!\!(K_1\vee P_{q-1})$, where $n=2q$.* *As some evidence for this conjecture, note that deleting the bridge in the graph in Figure [\[fig:12vertexextremal\]](#fig:12vertexextremal){reference-type="ref" reference="fig:12vertexextremal"} leaves two copies of the outerplanar graphs on $6$ vertices with maximum spectral radius, whereas Lin and Ning [@Lin-Ning19] showed that the outerplanar graph with maximum spectral radius on $n$ vertices for any $n\ge 7$ is $K_1\vee P_{n-1}$.* **Theorem 3**. *For $n$ odd and sufficiently large, among all connected outerplanar graphs on $n=2q+1$ vertices, the graphs $G$ maximizing $\lambda_2$ have the following structure: $G$ has a unique cut vertex $u$ and $G-u$ consists of two disjoint copies of the fan graph $K_1\vee P_{q-1}$. Moreover, all of these graphs have the same values of $\lambda_2$: $\lambda_2(G)=\lambda_1(K_1\vee P_{q-1})$.* Note that the graphs in Theorem [Theorem 3](#thm:t3){reference-type="ref" reference="thm:t3"} also maximize $\lambda_2$ over all (not necessarily connected) outerplanar graphs on $n=2q+1$ vertices, with $n$ sufficiently large. The extremal graphs $G$ have the same structure, but do not need to be connected. We also determine the maximum $\lambda_2$ among all 2-connected outerplanar graphs on $n$ vertices. **Theorem 4**. *Among all 2-connected outerplanar graphs on $n$ vertices, the maximum $\lambda_2$ is obtained by the graph in Figure [\[fig:2conn\]](#fig:2conn){reference-type="ref" reference="fig:2conn"}, denoted by $(K_1\vee P_{\lfloor \frac{n}{2}\rfloor-1})\!=\!\!(K_1\vee P_{\lceil \frac{n}{2}\rceil-1})$, which is constructed by gluing a fan graph $K_1\vee P_{\lfloor \frac{n}{2}\rfloor-1}$ and another fan graph $K_1\vee P_{\lceil \frac{n}{2}\rceil-1}$ by connecting the first vertex on the path $P_{\lfloor \frac{n}{2}\rfloor-1}$ to the second vertex on the path $P_{\lceil \frac{n}{2}\rceil-1}$, and the second vertex on the path $P_{\lfloor \frac{n}{2}\rfloor-1}$ to the first vertex on the path $P_{\lceil \frac{n}{2}\rceil-1}$.* The paper is organized as follows. In Section 2, we give necessary definitions and lemmas. In Section 3, we study the coarse structure of the outerplanar graphs which have the maximum $\lambda_k$ and prove Theorem [Theorem 1](#thm:t1){reference-type="ref" reference="thm:t1"}. In Section 4, we give the proofs of Theorems [Theorem 2](#thm:t2){reference-type="ref" reference="thm:t2"} and [Theorem 3](#thm:t3){reference-type="ref" reference="thm:t3"}. In Section 5, we prove Theorem [Theorem 4](#thm:t4){reference-type="ref" reference="thm:t4"}. We conclude in Section 6 with remarks and open problems. # Notation and lemmas For a given graph $G = (V,E)$, the neighborhood of a vertex $v \in V$, denoted $N(v)$, is the set of all vertices adjacent to $v$. The degree of a vertex $v \in V$, denoted $d_v$, is $|N(v)|$. A walk in a graph $G$ is a sequence of vertices $v_0v_1 \dots v_{k}$ where $v_iv_{i+1} \in E$ for all $0\leq i \leq k-1$. A path in a graph $G$ is a walk where the vertices are distinct. The path graph $P_n = (V,E)$ is defined by $V = \left\{ v_0, v_1, \dots v_{n-1}\right\}$ and $E = \left\{ v_iv_{i+1} : 0\leq i \leq n-2\right\}$. Let us go over some important tools from linear algebra. (See [@S10], for example, for additional background on matrix theory). Assume $A$ is a $n\times n$ real symmetric matrix (or Hermitian matrix in general). The eigenvalues of $A$ are all real, and we may label them in nonincreasing order as $$\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n.$$ Recall that a sequence $\mu_1 \geq \dotsm \geq \mu_m$ is said to interlace a sequence $\lambda_1 \geq \dotsm \geq \lambda_n$ with $m<n$ when $\lambda_i \geq \mu_i \geq \lambda_{n-m+i}$ for $i = 1 \dots m$. A corollary of Cauchy's Interlacing Theorem states that if $B$ is a principal submatrix of a symmetric matrix $A$, then the eigenvalues of $B$ interlace the eigenvalues of $A$ [@H1995]. In particular, the eigenvalues of a proper induced subgraph $H\subset G$ interlace the eigenvalues of $G$. Tait and Tobin [@TT2017] proved that for sufficiently large $n$, the maximum spectral radius of an outerplanar graph is uniquely achieved by the fan graph $K_1\vee P_{n-1}$. Thus, we have $$\lambda_{1,max}=\lambda_1(K_1\vee P_{n-1}).$$ We need an estimate for the size of the largest eigenvalue $\lambda_1$ of the fan graph $K_1 \vee P_{n-1}$. We use a series expansion proven in [@LLLW2022]. **Lemma 1**. *For a positive integer $n$, let $G$ be the graph $K_1 \vee P_{n-1}$. The largest eigenvalue $\lambda_1$ of $G$ satisfies $$\lambda_1 = \sqrt{n-1}+ 1 + \frac{1}{2\sqrt{n-1}} -\frac{1}{n-1} - \frac{1}{8(n-1)^{3/2}} - \frac{7}{16(n-1)^{5/2}} + O\left(\frac{1}{(n-1)^3}\right). \label{eq:lambda1_fan_graph}$$* *Proof.* From [@LLLW2022], $\lambda_1$ satisfies the following equation. $$\lambda_1^2 = (n-1) + \frac{2n-4}{\lambda_1} + \frac{4n-10}{\lambda_1^2} + \frac{8n-24}{\lambda_1^3} + \frac{16n-54}{\lambda_1^4} + \frac{32n-120}{\lambda_1^5} + \frac{64n-260}{\lambda_1^6} +O\left(\frac{n}{\lambda_1^7}\right).$$ Expanding the largest root of the equation above into a series in terms of $n-1$, we get Equation [\[eq:lambda1_fan_graph\]](#eq:lambda1_fan_graph){reference-type="eqref" reference="eq:lambda1_fan_graph"}. ◻ **Lemma 2**. *Let $G$ be the graph $(K_1\vee P_{q-1})\!\!-\!\!(K_1\vee P_{q-1})$. The second largest eigenvalue $\lambda_2$ of $G$ satisfies the series expansion $$\lambda_2 = \sqrt{q-1} + 1 + \frac{1}{2\sqrt{q-1}} - \frac{3}{2(q-1)} + \frac{7}{8(q-1)^{3/2}} - \frac{2}{(q-1)^2} + O\left(\frac{1}{(q-1)^{5/2}}\right).$$* *Proof.* Assume $n$ is the total number of vertices where $n = 2q$. Let $u_1$ and $u_2$ be the two largest degree vertices in the graph. Let $\bf{x}$ be the eigenvector associated with $\lambda_2$. Because the graph is symmetric, the eigenvector ${\bf x}$ is skew-symmetric. We may assume without loss of generality that ${\bf x}_{u_1} = 1$ and ${\bf x}_{u_2} = -1$. Let $P$ be the path on $2q-2$ vertices obtained after deleting the two large degree vertices, and let $A_p$ be the adjacency matrix of $P$. Let ${\bf y}$ be the restriction of the eigenvector ${\bf x}$ on the path $P$. Let $\beta$ be a $(2q-2)$-dimensional column vector with entries equal to $1$ on vertices in $N(u_1)$ and equal to $-1$ on vertices in $N(u_2)$. The eigen-equation gives $$\begin{aligned} {\bf y} &= (\lambda_2 I -A_p)^{-1}\beta \\ &= \frac{1}{\lambda_2}(I-\lambda_2^{-1}A_p)^{-1}\beta \\ &=\frac{1}{\lambda_2} \sum_{i=0}^\infty \lambda_2^{-i}A_p^i\beta \\ &=\sum_{i=0}^\infty \lambda_2^{-(i+1)}A_p^i\beta. \end{aligned}$$ This series converges because $\lambda_2 > 2 > \lambda_1(A_p)$. Since $\lambda_2(1) = \sum_{i \in N(u_1)} x_i$ and $\lambda_2(-1) = \sum_{i \in N(u_2)} x_i$, we know that $2\lambda_2 = \beta'{\bf y}$. Therefore, $$\label{eq:lambda1_lowerBound} \lambda_2 = \frac{1}{2} \sum_{i=0}^\infty \lambda_2^{-(i+1)}\beta'A_p^i\beta.$$ When $i=0$, we get $$\beta'\beta = 2q-2.$$ When $i=1$, we get $$\beta'A_p\beta = 2(2q-2)- 2(2)- 2(1) = 4q-10.$$ When $i=2$, we get $$\beta'A_p^2\beta = 4(2q-2) - 6(2) - 2(1) = 8q-22.$$ Similarly, for $3\le i\le 5$, we obtain $$\beta'A_p^3\beta = 16q-56,$$ . $$\beta'A_p^4\beta = 32q-118,$$ $$\beta'A_p^5\beta = 64q-272.$$ After multiplying both sides of [\[eq:lambda1_lowerBound\]](#eq:lambda1_lowerBound){reference-type="eqref" reference="eq:lambda1_lowerBound"} by $\lambda_2$ and simplifying, we obtain that $\lambda_2$ is a root of the equation $$\label{eqn:lmb2eqn} \lambda_2^2 = (q-1) + \frac{2q-5}{\lambda_2} + \frac{4q-11}{\lambda_2^2} + \frac{8q-28}{\lambda_2^3} + \frac{16q-59}{\lambda_2^4} + \frac{32q-136}{\lambda_2^5} + O\left(\frac{q}{\lambda_2^6}\right).$$ Using SageMath, we get the following series expansion for $\lambda_2$: $$\lambda_2 = \sqrt{q-1} + 1 + \frac{1}{2\sqrt{q-1}} - \frac{3}{2(q-1)} + \frac{7}{8(q-1)^{3/2}} - \frac{2}{(q-1)^2} + O\left(\frac{1}{(q-1)^{5/2}}\right).$$ ◻ **Corollary 2**. *Let $q=\lfloor \frac{n}{2}\rfloor$. Then we have $$\lambda_{2,max}\geq \sqrt{q-1} +1 + \frac{1}{2\sqrt{q-1}} +O\left(\frac{1}{q-1}\right).$$* For general $k\geq 3$, we get a slightly worse lower bound. **Corollary 3**. *For any fixed $n\geq k\geq 2$, let $q=\lfloor \frac{n-1}{k}\rfloor$. Then we have $$\lambda_{k,max}\geq \sqrt{q-1} +1 + \frac{1}{2\sqrt{q-1}} +O\left(\frac{1}{q-1}\right).$$* *Proof.* Write $n=kq+r$ for $1\leq r\leq k$. Let $H$ be a disjoint union of $r-1$ copies of $K_1\vee P_{q}$ and $k-r+1$ copies of $K_1\vee P_{q-1}$. Add a new vertex $u$ to $H$ and for each connected component of $H$ select one vertex and connect it to $u$. Clearly $G$ is an outerplanar graph. The number of vertices in $G$ is given by $$(r-1)(q+1)+(k-r+1)q+1= kq+r=n.$$ By Cauchy's Interlacing Theorem, we have, $$\lambda_{k,max}\geq \lambda_k(G)\geq \lambda_k(H).$$ Since $H$ is the disjoint union of $k$ fan graphs, $\lambda_k(H)$ is equal to the first eigenvalue of its smallest component. We have $$\begin{aligned} \lambda_{k,max} &\geq \lambda_k(H) \\ &\geq \lambda_{1}(K_1\vee P_{q-1}) \\ &= \sqrt{q-1} + 1 + \frac{1}{2\sqrt{q-1}}+ O\left(\frac{1}{q-1}\right).\end{aligned}$$ Here we apply Lemma [Lemma 1](#lem:specradfangph){reference-type="ref" reference="lem:specradfangph"}. ◻ **Lemma 3**. *For any outerplanar graph $G$, let $h_i(u,v)$ denote the number of $(u,v)$-paths of length i. Then for all $u,v \in V(G)$ we have $$\begin{aligned} h_2(u,v)&\leq 2,\\ h_3(u,v)&\leq 3,\\ h_i(u,v)&= O(1).\\ \end{aligned}$$* *Proof.* Let $G$ be an outerplanar graph and $u,v \in V(G)$. Since $G$ is $K_{2,3}$-minor-free, there cannot be three or more internally disjoint $u,v$-paths. This proves $h_2(u, v) \le 2$. Suppose there are at most two internally disjoint $u,v$-paths of length 3 and let $U \subset V(G)$ be the internal vertices of these paths. Since $G$ is $K_4$-minor-free, there are at most 3 edges in $G\lbrack U \rbrack$ which implies $h_3(u,v) \leq 3.$ Similarly, for paths of length $i$, there can be at most two internally disjoint $(u, v)$-paths, and if $U \subset V(G)$ is the set of internal vertices on these two paths, then $|E(G[U])| \le 2(2i-2) - 3$, so there is only a constant number of paths of length $i$ between $u$ and $v$. ◻ # Proof of Theorem [Theorem 1](#thm:t1){reference-type="ref" reference="thm:t1"} {#proof-of-theorem-thmt1} Let $G$ be a connected outerplanar graph on $n$ vertices with $k$th largest eigenvalue equal to $\lambda_{k,max}$. We denote the adjacency matrix of $G$ by $A(G)$. The aim of this section is to prove Theorem [Theorem 1](#thm:t1){reference-type="ref" reference="thm:t1"}. **Lemma 4**. *Suppose $G$ is an outerplanar graph on $n$ vertices with maximum degree $\Delta$. Then we have $$\label{eq:lambda1_maxdegree} \Delta\geq \lambda_1^2-\frac{4(2n-3)}{\lambda_1}.$$* *Proof.* Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda_1$. Without loss of generality, we normalize $\mathbf{x}$ such that $\left\lVert \mathbf{x}\right\rVert_\infty =1$. We further assume that $x_{u_0}=1$ at some vertex $u_0$. For any vertex $u$, we have $$|\lambda_1 x_u| \leq \sum_{v\in N(u)}|x_v| \le d_{u}.$$ Since any outerplanar graph on $n$ vertices has at most $2n-3$ edges, it follows by summing over all vertices $u\in V(G)$ that, $$\lambda_1\sum_{u\in V(G)}|x_u| \le \sum_{u\in V(G)} d_u \le 2(2n-3).$$ This implies $$\sum_{u\in V(G)}|x_u| \leq \frac{2(2n-3)}{\lambda_1}.$$ We observe $$\begin{aligned} \lambda_1^2 &= \lambda_1^2 x_{u_0}\\ &= \sum_{v \in N(u_0) } \lambda_1 x_v \\ &= \sum_{v \in N(u_0)} \sum_{w\in N(v)} x_w \\ &= d_{u_0} + \sum_{w\in V\setminus \{u_0\}} x_w |N(w)\cap N(u_0)|.\end{aligned}$$ Since $G$ is $K_{2,3}$-free, we have $|N(w)\cap N(u_0)|\leq 2$. Therefore, $$\begin{aligned} \Delta&\geq d_{u_0}\\ &=\lambda_1^2 - \sum_{w\in V\setminus \{u_0\}} x_w |N(w)\cap N(u_0)|\\ &\geq \lambda_1^2 - 2\sum_{u\in V(G)}|x_u| \\ &\geq \lambda_1^2 - \frac{4(2n-3)}{\lambda_1}.\end{aligned}$$ ◻ **Lemma 5**. *There exists $k$ vertices $u_1, \ldots, u_k$ such that for $1\leq i\leq k$ $$d_{u_i} \ge \frac{n}{k} - O(\sqrt{n}).$$* *Proof.* We will find large degree vertices $u_1, u_2,\ldots, u_k$ in sequence. Since $\lambda_1(G)\geq \lambda_k(G)\geq \sqrt{n/k} +1 + O(\frac{1}{\sqrt{n}})$, we have $$\Delta(G)\geq \lambda_1^2- \frac{4(2n-3)}{\lambda_1} \geq \frac{n}{k}-O(\sqrt{n}).$$ Let $u_1$ be the vertex with the largest degree in $G$. Now let $G_1=G-\{u_1\}$. By Cauchy's Interlacing theorem, we have $\lambda_1(G_1)\geq \lambda_2(G)\geq \lambda_k(G)\geq \sqrt{n/k} +1 + O(\frac{1}{\sqrt{n}}).$ Repeat this argument with $G_1$. Let $u_2$ be the vertex with the largest degree in $G_1$. We have $$\Delta(G_1)\geq \lambda_1^2(G_1)- \frac{4(2n-3)}{\lambda_1(G_1)} \geq \frac{n}{k}-O(\sqrt{n}).$$ In general, assume we have already found vertices $u_1,u_2,\ldots, u_i$, for some $i<k$. Consider $G_{i}= G\setminus \{u_1, u_2, \ldots, u_i\}$. We have $\lambda_1(G_i)\geq \lambda_{i+1}(G)\geq \lambda_k(G)\geq \sqrt{n/k} +1 + O(\frac{1}{\sqrt{n}}).$ Thus, there is a vertex $u_{i+1}$ with degree $$d_{u_{i+1}} =\Delta(G_i) \ge \lambda_1^2(G_i)- \frac{4(2n-3)}{\lambda_1(G_i)} \geq \frac{n}{k} - O(\sqrt{n}).$$ ◻ Let $U=\{u_1, u_2,\ldots, u_k\}$ be the set of vertices with large degree, *i.e.*, at least $\frac{n}{k}-O(\sqrt{n})$. Now we show all other vertices not in $U$ have small degree. **Lemma 6**. *For any other vertex $u \not\in \{u_1,\ldots, u_k\}$, we have $$d_u=O(\sqrt{n}).$$* *Proof.* First we show the union of the neighbors of $u_1,\ldots, u_k$ covers almost all vertices in $G$. We have $$\begin{aligned} \left|\bigcup_{i=1}^k N(u_i)\right |& \geq \sum_{i=1}^k |N(u_i)| - \sum_{1\leq i<j\leq k} |N(u_i)\cap N(u_j)|\\ & \geq \sum_{i=1}^k d_{u_i} -2{k\choose 2}\\ & \geq n- O(\sqrt{n}). \end{aligned}$$ This implies $$\left|\bigcap_{i=1}^k \overline{ N(u_i)}\right | = O(\sqrt{n}).$$ For any $u\not\in \{u_1,\ldots, u_k\}$, $u$ can have at most 2 neighbors in $N(u_i)$. Thus, $$d_u\leq 2k + \left|\bigcap_{i=1}^k \overline{ N(u_i)}\right | =O(\sqrt{n}).$$ ◻ For $1\leq i\leq k$, let $\tilde d_{u_i}$ be the number of neighbors in $V(G)\setminus U$, i.e., $$\tilde d_{u_i}=\left| N(u_i)\setminus U\right |.$$ **Lemma 7**. *For sufficiently large $n$ and any $u\in U$, we have $\tilde d_u\geq \frac{n}{k}-O(1).$* *Proof.* Without loss of generality, we assume that $$\tilde d_{u_1}\geq \tilde d_{u_2}\geq \cdots \geq \tilde d_{u_k}.$$ Let $G'=G\setminus \{u_1,\ldots, u_{k-1}\}$. Then $d^{G'}_{u_k}=\tilde d_{u_k}$. In particular, $G'$ has one unique vertex $u_k$ with degree at least $\frac{n}{k}-O(\sqrt{n})$ while all other vertices have degree at most $O(\sqrt{n})$. By Cauchy's Interlacing theorem, we have $$\lambda_1(G')\geq \lambda_{k}(G)\geq \sqrt{\frac{n}{k}}+1 + O\left(\frac{1}{\sqrt{n}}\right).$$ For the rest of the proof, all notations are relative to $G'$ unless stated explicitly otherwise. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda_1$ for $G'$, which is normalized such that $|\mathbf{x}|_\infty = x_{u'} =1$ for some vertex $u'$. The degree of $u'$ is at least $\frac{n}{k}-O(\sqrt{n})$. We have $u'=u_k$.\ For any $u\in V(G')\setminus \{u_k\}$, we have $$\begin{aligned} (\lambda_1^2 -d_{u})|x_u| &= \left|\sum_{w\in V\setminus \{u\}} x_w |N(w)\cap N(u)|\right | \\ &\leq 2\sum_{w\in V\setminus \{u\}}|x_w| \nonumber \\ &\leq \frac{4(|E(G')|)}{\lambda_1}. \label{eq:xu} \end{aligned}$$ Since $d_u=O(\sqrt{n})$, we have $$|x_u|\leq \frac{4|E(G')|}{\lambda_1(\lambda_1^2-d_u)} = O\left(\frac{1}{\sqrt{n}}\right).$$ We have $$\lambda_1^2 -d_{u_k}=\sum_{w\in V\setminus \{u_k\}} x_w |N(w)\cap N(u_k)|.$$ Multiplying by $\lambda_1$ on both sides, we have $$\begin{aligned} \lambda_1(\lambda_1^2 -d_{u_k}) &= \sum_{w\in V\setminus \{u_k\}} \lambda_1 x_w |N(w)\cap N(u_k)| \nonumber\\ &= t_3(u_k)+ \sum_{v\in N(u_k)} x_v (d_v-1) + \sum_{y\in V\setminus \{u_k\}} x_y h(u_k,y). \label{eq:h3}\end{aligned}$$ Here $t_3(u_k)$ is the number of triangles containing $u_k$ while $h(u_k,y)$ denote the number of $u_k,y$-paths of length 3. We have $$t_3(u_k)\leq 2(d_{u_k}-1).$$ By Lemma [Lemma 3](#lem:P3){reference-type="ref" reference="lem:P3"}, we have $$h(u_k,y) \leq 3.$$ Since all other vertices in $G'$ other than $u_k$ have $|x_u|$ at most $O(\frac{1}{\sqrt{n}})$, we have $$\begin{aligned} \sum_{v\in N(u_k)} x_v d_v &\leq O\left(\frac{1}{\sqrt{n}}\right)\sum_{v\not=u_k} d_v\\ &\leq O\left(\frac{1}{\sqrt{n}}\right) 2|E(G')|\\ &=O(\sqrt{n}),\end{aligned}$$ and $$\sum_{y\in V\setminus \{u_k\}} x_y h(u_k,y) \leq 3 \sum_{y} |x_y| \leq O(\sqrt{n}).$$ Plugging to Inequality [\[eq:h3\]](#eq:h3){reference-type="eqref" reference="eq:h3"}, we have $$\label{eq:cubic} \lambda_1(\lambda_1^2-d_{u_k})\leq 2d_{u_k} + O(\sqrt{n}).$$ Therefore, $$\begin{aligned} d_{u_k} &\geq \frac{\lambda_1^3- O(\sqrt{n})}{\lambda_1+2}\\ &= (\lambda_1 -1)^2 -O(1)\\ &\geq \frac{n}{k}-O(1).\end{aligned}$$ Thus, we proved $d_{u_k}\geq \frac{n}{k}-O(1)$. By the construction of $G'$, we conclude that the $k$ largest degree vertices each have degree at least $\frac{n}{k}-O(1)$. Consequently, all vertices except for these $k$ vertices have degree $O(1)$. ◻ *Proof of Theorem [Theorem 1](#thm:t1){reference-type="ref" reference="thm:t1"}.* Rename the vertices $u_1, \ldots, u_k$ such that $$d_{u_1}\geq d_{u_2}\geq \cdots \geq d_{u_k}.$$ We have $$\begin{aligned} d_{u_k}&\leq \frac{1}{k} \sum_{i=1}^k |N({u_i})| \\ &\leq \frac{1}{k} \left( \left| \bigcup_{i=1}^n N({u_i}) \right| + \sum_{1\leq i<j\leq k} \left| N(u_i)\cap N(u_j) \right| \right ) \\ &\leq \frac{1}{k}(n+k^2-k).\end{aligned}$$ This implies $$d^{G'}_{u_0}\leq d_{u_k}\leq \frac{n}{k}+k-1.$$ From Inequality [\[eq:cubic\]](#eq:cubic){reference-type="eqref" reference="eq:cubic"}, we have $$\begin{aligned} \lambda_1(G')^2 &\leq d^{G'}_{u_0} + \frac{2d^{G'}_{u_0} + O(\sqrt{n})}{\lambda_1} \\ &\leq \frac{n}{k}+k-1 + 2\sqrt{\frac{n}{k}} +O(1)\\ &= \frac{n}{k} + 2\sqrt{\frac{n}{k}} +O(1).\end{aligned}$$ $$\lambda_1(G') \leq \sqrt{ \frac{n}{k} + 2\sqrt{\frac{n}{k}} +O(1) } = \sqrt{n/k} + 1+ O\left(\frac{1}{\sqrt{n}}\right).$$ By Cauchy's Interlacing theorem, we have $$\lambda_k(G) \leq \lambda_1(G') \leq \sqrt{n/k} + 1 + O\left(\frac{1}{\sqrt{n}}\right).$$ This matches the lower bound asymptotically. ◻ The following lemma will be useful to determine the extremal graphs for small $k$.\ **Lemma 8**. 1. *For $k\geq 3$, sufficiently large $n$, and any $u\in U$, we have $$\tilde d_u\geq \left\lfloor \frac{n-1}{k} \right\rfloor -1.$$* 2. *For $k=2$, sufficiently large $n$, and any $u\in U$, we have $$\tilde d_u\geq \left\lfloor \frac{n}{k} \right\rfloor -1.$$* *Proof.* Let $q=\lfloor \frac{n-1}{k} \rfloor$ when $k\geq 3$ and $q=\lfloor \frac{n}{k} \rfloor$ when $k=2$. Without loss of generality, we assume that $$\tilde d_{u_1}\geq \tilde d_{u_1}\geq \cdots \geq \tilde d_{u_k}.$$ Let $G'=G\setminus \{u_1,\ldots, u_{k-1}\}$. Then $d^{G'}_{u_k}=\tilde d_{u_k}$. All terminologies below are related to the graph $G'$ if not specified. From Equation [\[eq:h3\]](#eq:h3){reference-type="eqref" reference="eq:h3"}, we have $$\lambda_1(\lambda_1^2 -d_{u_k}) = t_3(u_k)+ \sum_{v\in N(u_k)} x_v (d_v-1) + \sum_{z\in V\setminus \{u_k\}} x_z h(u_k,z). \label{eq:h4}$$ Thus, we have $$\begin{aligned} \lambda_1^2 (\lambda_1^2 -d_{u_k}) &= t_3(u_k)\lambda_1+ \sum_{v\in N(u_k)} \lambda_1 x_v (d_v-1) + \sum_{z\in V\setminus \{u_k\}} \lambda_1x_z h(u_k,z)\\ &= t_3(u_k)\lambda_1+ \sum_{v\in N(u_k)} \sum_{z\in N(v)} x_z (d_v-1) + \sum_{z\in V\setminus \{u_k\}} \sum_{w\in N(z)}x_w h(u_k,z)\\ &= t_3(u_k)\lambda_1 + \sum_{v\in N(u_k)}(d_v-1) + \sum_{v\in N(u_k), z\in N(v)-u_k}x_z (d_v-1) + c_4(u_k) + \sum_{w\not=u_k}x_w h_4(u_k, w)\\ &\hspace*{4mm} + \sum_{v\in N(u_k)} x_v t_3(v) + \sum_{z\not=u_k}x_z h_2(u_k, z)(d_z-1).\end{aligned}$$ Since for any $v\not= u$, $d_v=O(1)$ and $h_i(u_k,v)=O(1)$ by Lemma [Lemma 3](#lem:P3){reference-type="ref" reference="lem:P3"}, we have the following estimates for the lower order terms. $$\begin{aligned} \sum_{v\in N(u_k), z\in N(v)-u_k}x_z (d_v-1) &= O\left(\sum_z |x_z|\right)= O(\sqrt{n}).\\ \sum_{w\not=u_k}x_w h_4(u_k, w) &= O\left(\sum_w |x_w|\right)= O(\sqrt{n}).\\ \sum_{v\in N(u_k)} x_v t_3(v)&=O\left(\sum_v |x_v|\right)= O(\sqrt{n}).\\ \sum_{z\not=u_k}x_z h_2(u_k, z)(d_z-1)&=O\left(\sum_z |x_z|\right)= O(\sqrt{n}).\end{aligned}$$ We also use the following two estimates for the terms $t_3(u_k)$ and $c_4(u_k)$. $$\begin{aligned} t_3(u_k)&\leq 2 \tilde d_{u_k} -2,\\ c_4(u_k)&\leq 2 \tilde d_{u_k} -2 +O(\sqrt{n}).\end{aligned}$$ We have $$\begin{aligned} \sum_{v\in N(u_k)}(d_v-1) + c_4(u_k) &\leq \sum_{v\in N(u_k)}(d_v-1) + 2 \tilde d_{u_k} -2 +O(\sqrt{n}).\\ &= 2|E(G'|_{N(u_k)\setminus U\cup\{u_k\}})|+O(\sqrt{n})\\ &\leq 4 d_{u_k} + O(\sqrt{n}).\end{aligned}$$ Putting everything together, we have $$\lambda_1^2 (\lambda_1^2 -d_{u_k}) \leq \lambda_1 (2 d_{u_k}-2)+ 4 d_{u_k} + O(\sqrt{n})$$ We have $$\begin{aligned} d_{u_k}&\geq \frac{\lambda_1^4+2\lambda_1+O(\sqrt{n})}{\lambda^2+2\lambda+4}\\ &= \lambda^2-2\lambda -O\left(\frac{1}{\sqrt{n}}\right)\\ &= (\lambda-1)^2-1 -O\left(\frac{1}{\sqrt{n}}\right)\\ &=\left(\sqrt{q-1}+\frac{1}{2\sqrt{q-1}} + O\left(\frac{1}{(q-1)}\right)\right)^2 -1-O\left(\frac{1}{\sqrt{n}}\right)\\ &=q-1 -O\left(\frac{1}{\sqrt{q-1}}\right) - O\left(\frac{1}{\sqrt{n}}\right).\end{aligned}$$ Here we applied the lower bound of $\lambda_k$ as in Corollary [Corollary 2](#cor:2lb){reference-type="ref" reference="cor:2lb"} or [Corollary 3](#cor:klb){reference-type="ref" reference="cor:klb"}. Since $d_{u_k}$ is an integer, we have $$d_{u_k}\geq q-1.$$ ◻ # Outerplanar graphs with maximum $\lambda_2$: the exact result Let $G$ be a connected outerplanar graph on $n$ vertices that maximizes $\lambda_2$. Let $\mathbf{x}=(x_1,x_2,\ldots,x_n)$ be the eigenvector for $\lambda_2$. By Lemma [Lemma 8](#lem:degrees){reference-type="ref" reference="lem:degrees"}, $G$ contains exactly two vertices, say $u_1$ and $u_2$, with degree at least $\lfloor \frac{n}{2}\rfloor -1$. Since $G$ is outerplanar, $|N(u_1)\cap N(u_2)|\leq 2$. This implies that there are at most 3 vertices not in $N[u_1]\cup N[u_2]$. Thus, all vertices other than $u_1$ and $u_2$ have degree $O(1)$. Let $V^+=\{v\in V(G)\colon x_v>0\}$, $V^0=\{v\in V(G)\colon x_v=0\}$, and $V^-=\{v\in V(G)\colon x_v<0\}$. For any vertex set $S$, the volume of $S$, denoted by ${\rm Vol}(S)$, is defined as $\sum_{v\in S} |x_v|$. For any vertex $v$, define $N^+(v)=N(v)\cap V^+$, $d^+_v= |N^+(v)|$, $N^-(v)=N(v)\cap V^-$ and $d^-_v=|N^-(v)|$. Let $x^+_{max}=\max \{x_v\colon v\in V^+\}$ and $x^-_{min}=\min \{x_v\colon v\in V^-\}$. **Lemma 9**. *We have* 1. *For any $v\in V^+$, $x_v\leq d^+_v|x^+_{max}|$.* 2. *For any $v\in V^-$, $|x_v|\leq d^-_v|x^-_{min}|$.* 3. *${\rm Vol}(V^+)= O(\sqrt{n})x^+_{max}$ and ${\rm Vol}(V^-)= O(\sqrt{n})|x^-_{min}|$.* 4. *$x^+_{max}$ and $x^-_{min}$ are achieved at $u_1$ and $u_2$.* *Proof.* For any vertex $v\in V^+$, we have $$\lambda_2 x_v = \sum_{u\in N(v)}x_u \le \sum_{u\in N^+(v)} x_u \leq d^+_{v} x^+_{max}.$$ By symmetry, we have for any $v\in V^-$, $|x_v|\leq d^-_v|x^-_{min}|$. Since any outerplanar graph on $n$ vertices has at most $2n-3$ edges, it follows by summing over all vertices $v\in V^+$ that, $$\lambda_2 {\rm Vol}(V^+)= \lambda_2\sum_{v\in V^+}x_v \le x^+_{max}\sum_{v\in V^+} d_v \le 2(2n-3) x^+_{max}.$$ This implies $${\rm Vol}(V^+)\leq \frac{2(2n-3)}{\lambda_2} x^+_{max}.$$ A similar argument implies that $${\rm Vol}(V^{-}) = O\left(\sqrt{n}\right) |x_{min}^{-}|.$$ Assume $x^+_{max}$ is achieved at some vertex $u_0$. We have $$\begin{aligned} \lambda_2^2 x_{u_0} &\leq \sum_{v \in N^+(u_0) } \lambda_2 x_v \nonumber \\ &\leq \sum_{v \in N^+(u_0)} \sum_{w\in N^+(v)} x_w \nonumber\\ &= d^+_{u_0} x_{u_0}+ \sum_{w\in V^+\setminus \{u_0\}} x_w |N^+(w)\cap N^+(u_0)|. \label{eq:quadratic}\end{aligned}$$ Since $G$ is $K_{2,3}$-free, we have $|N(w)\cap N(u_0)|\leq 2$. Therefore, $$\begin{aligned} d^+_{u_0} x_{u_0} &=\lambda_2^2 x_{u_0} - \sum_{w\in V^+\setminus \{u_0\}} x_w |N^+(w)\cap N^+(u_0)|\\ &\geq \lambda_2^2 x_{u_0} - 2\sum_{u\in V^+}|x_u| \\ &\geq \lambda_2^2 x_{u_0} - \frac{4(2n-3)}{\lambda_2}x_{u_0}.\end{aligned}$$ Therefore, $$d^+_{u_0}\geq \lambda_2^2 -\frac{4(2n-3)}{\lambda_2} =\frac{n}{2}-O(\sqrt{n}),$$ which implies that $u_0$ must be one of the vertices $u_1$ and $u_2$. A similar argument shows that $x_{min}^{-}$ is achieved at either $u_1$ or $u_2$. ◻ **Lemma 10**. *We have $u_1u_2\notin E(G)$.* *Proof.* We assume for a contradiction that $u_1u_2 \in E(G)$. Without loss of generality, we assume that $x_{u_1} = 1$ and $|x_{u_2}| \ge |x_{u_1}|$. Since $x_{u_2}$ is negative, we have $$\lambda_2x_{u_1} \leq \sum_{v\in N^{+}(u_1)}x_v - |x_{u_2}|.$$ Multiplying by $\lambda_2$ and using the same steps as in Equation [\[eq:quadratic\]](#eq:quadratic){reference-type="eqref" reference="eq:quadratic"}, we have $$(\lambda_2^2 - d_{u_1}^{+})\le \sum_{v\in V^{+}}x_vh_2(u_1, v) - \lambda_2|x_{u_2}|.$$ Multiplying by $\lambda_2$ again, we obtain $$\lambda_2(\lambda_2^2 - d_{u_1}^{+}) \le t_3^{+}(u_1) + \sum_{v\in N^{+}(u_1)}x_v(d_v-1) + \sum_{w\in V^{+}}x_wh_2(u_1, w) - \lambda_2^2.$$ Therefore, $$\lambda_2(\lambda_2^2 - d_{u_1}^{+}) \le 2d_{u_1}^{+} - \lambda_2^2 + O(\sqrt{n}),$$ as $\sum_{v\in N^{+}(u_1)}x_v(d_v-1) = O(\sqrt{n})$ and $\sum_{w\in V^{+}}x_wh_2(u_1, w) = O(\sqrt{n})$. Rearranging the inequality, we obtain $$(\lambda_2+2)d_{u_1}^{+} \ge \lambda_2^3 + \lambda_2^2 - O(\sqrt{n}),$$ so $$\begin{aligned} d_{u_1}^{+} &\ge \frac{\lambda_2^3+\lambda_2^2}{\lambda_2+2} - O(1)\\ &=\lambda_2^2 - \lambda_2 - O(1)\\ &=\left(\sqrt{q-1}+1 + O\left(\frac{1}{\sqrt{q-1}}\right)\right)^2 - \sqrt{q-1} - O(1)\\ &= q - 1 + \sqrt{q-1} - O(1), \end{aligned}$$ which is a contradiction to the fact that $d_{u_1}^{+} = q-1 + O(1)$. ◻ Assume $G$ is an extremal connected outerplanar graph on $n$ vertices with $\lambda_2(G)=\lambda_{2,max}$. Let $P$ be the induced subgraph obtained from $G$ after deleting $u_1$ and $u_2$. Let $\mathbf{x}$ be the eigenvector of $G$ corresponding to $\lambda_2$. Let $A_P$ be the adjacency matrix of $P$. Let $\mathbf{y}$ be the restriction of $\mathbf{x}$ on the vertex set of $P$. Define $(n-2)$-dimensional column vectors $\beta_1$, $\beta_2$, $\gamma_1$, and $\gamma_2$, indexed by the vertices of $P$, as follows: $$\begin{aligned} \beta_1(v) &=\begin{cases} x_{u_1} & \mbox{ if } v\in N(u_1),\\ 0 & \mbox{ otherwise.} \label{eq:beta1def} \end{cases}\\ \beta_2(v) &=\begin{cases} x_{u_2} & \mbox{ if } v\in N(u_2),\\ 0 & \mbox{ otherwise.} \label{eq:beta2def} \end{cases}\\ \gamma_1(v) &=\begin{cases} \frac{1}{x_{u_1}} & \mbox{ if } v\in N(u_1),\\ 0 & \mbox{ otherwise.} \label{eq:gamma1def} \end{cases}\\ \gamma_2(v) &=\begin{cases} \frac{1}{x_{u_2}} & \mbox{ if } v\in N(u_2),\\ 0 & \mbox{ otherwise.} \label{eq:gamma2def} \end{cases} \end{aligned}$$ Let $\beta=\beta_1+\beta_2$ and $\gamma=\gamma_1+\gamma_2$, so that $$\begin{aligned} \beta(v)=\begin{cases} x_{u_1} & \mbox{ if } v\in N(u_1)\setminus N(u_2),\\ x_{u_2} & \mbox{ if } v\in N(u_2)\setminus N(u_1),\\ x_{u_1}+ x_{u_2} & \mbox{ if } v\in N(u_1)\cap N(u_2),\\ 0 & \mbox{ otherwise.} \end{cases}\\ \gamma(v)=\begin{cases} \frac{1}{x_{u_1}} & \mbox{ if } v\in N(u_1)\setminus N(u_2),\\ \frac{1}{x_{u_2}} & \mbox{ if } v\in N(u_2)\setminus N(u_1),\\ \frac{1}{x_{u_1}}+ \frac{1}{x_{u_2}} & \mbox{ if } v\in N(u_1)\cap N(u_2),\\ 0 & \mbox{ otherwise.} \end{cases}\end{aligned}$$ From the eigen-equation for $\lambda_2$, we have $$\lambda_2\mathbf{y}= A_P\mathbf{y}+ \beta.$$ This implies $$\begin{aligned} \mathbf{y} &= (\lambda_2 I -A_P)^{-1}\beta \\ &= \frac{1}{\lambda_2}(I-\lambda_2^{-1}A_P)^{-1}\beta \\ &=\frac{1}{\lambda_2} \sum_{i=0}^\infty \lambda_2^{-i}A_P^i\beta \\ &=\sum_{i=0}^\infty \lambda_2^{-(i+1)}A_P^i\beta. \end{aligned}$$ Also, by the eigen-equation, we have $$\begin{aligned} \lambda_2 &=x_{u_1}^{-1}\sum_{v\in N(u_1)} x_v = \gamma_1' \mathbf{y}, \\ \lambda_2 &=x_{u_2}^{-1}\sum_{v\in N(u_2)} x_v = \gamma_2' \mathbf{y}.\\\end{aligned}$$ Therefore, $$\begin{aligned} \lambda_2^2 &=\sum_{i=0}^\infty \lambda_2^{-i}\gamma_1' A_P^i\beta, \label{eq:gamma1}\\ \lambda_2^2 &=\sum_{i=0}^\infty \lambda_2^{-i}\gamma_2' A_P^i\beta, \label{eq:gamma2} \end{aligned}$$ Taking the average, we get $$\lambda_2^2 =\frac{1}{2}\sum_{i=0}^\infty \lambda_2^{-i}\gamma' A_P^i\beta. \label{eq:gamma}$$ For $i=0,1,2,\ldots,$ let $a_i=\frac{1}{2} \gamma' A_P^i\beta$. For any two vertices $u$ and $v$ (with $u=v$ allowed), let $w_i(u,v)$ denote the number of $uv$-walks of length $i$. Then we have $$\begin{aligned} \gamma' A_P^i \beta &=\sum_{u,v \in N(u_1)} w_i(u,v) + \sum_{u,v\in N(u_2)} w_i(u,v) + \sum_{u\in N(u_1), v\in N(u_2)} w_i(u,v) \left(\frac{x_{u_2}}{x_{u_1}} + \frac{x_{u_1}}{x_{u_2}}\right).\end{aligned}$$ Since $x_{u_1}$ and $x_{u_2}$ have opposite signs, we have $$\frac{x_{u_2}}{x_{u_1}} + \frac{x_{u_1}}{x_{u_2}} \leq -2.$$ Thus, we have $$\begin{aligned} a_i &\leq \frac{1}{2}\sum_{u,v\in N(u_1)} w_i(u,v) + \frac{1}{2} \sum_{u,v\in N(u_2)} w_i(u,v) -\sum_{u\in N(u_1), v\in N(u_2)} w_i(u,v). %&=\frac{1}{2}\sum_{u\in N(u_1)} w_i(u,u) %+ \sum_{uv\in {N(u_1)\choose 2}} w_i(u,v) %+ \frac{1}{2} %\sum_{u\in N(u_2)} w_i(u,u) %+ \sum_{uv\in {N(u_2)\choose 2}} w_i(u,v) %-\sum_{u\in N(u_1)}\sum_{v\in N(u_2)}w_i(u,v).\end{aligned}$$ In particular, $$\begin{aligned} a_0&= \frac{1}{2} \left(d_{u_1}+d_{u_2}+ |N(u_1)\cap N(u_2)| \left(\frac{x_{u_2}}{x_{u_1}} + \frac{x_{u_1}}{x_{u_2}}\right)\right)\\ &\leq \frac{1}{2} \left(d_{u_1}+d_{u_2}-2|N(u_1)\cap N(u_2)|\right)\\ &= \frac{1}{2} |N(u_1)\Delta N(u_2)|.\end{aligned}$$ **Lemma 11**. *Consider the equation $$\lambda^2 = a_0 + \displaystyle\sum_{i=1}^{\infty} \frac{a_i}{\lambda^i}.$$ The largest root $\lambda$ has the following series expansion: $$\lambda_1 = \sqrt{a_0} + c_1 + \frac{c_2}{\sqrt{a_0}} + \frac{c_3}{a_0} + \frac{c_4}{a_0^{\frac{3}{2}}} + O\left(n^{-2}\right).$$ Here $$\begin{aligned} c_1 &= \frac{a_1}{2a_0}\\ c_2 &= -\frac38 \left(\frac{a_1}{a_0}\right)^2 + \frac12 \frac{a_2}{a_0}, \label{eq:c2} \\ c_3 &= \frac{a_1^3}{2a_0^3} -\frac{a_1a_2}{a_0^2} + \frac{a_3}{2a_0}\\ c_4 &=-\frac{105}{128} \left(\frac{a_1}{a_0}\right)^4 +\frac{35}{16} \left(\frac{a_1}{a_0}\right)^2\frac{a_2}{a_0} -\frac{5}{8}\left(\frac{a_2}{a_0}\right)^2 -\frac{5}{4}\frac{a_1}{a_0}\frac{a_3}{a_0} +\frac{1}{2} \frac{a_4}{a_0}. \label{eq:c4}\end{aligned}$$* The proof of Lemma [Lemma 11](#lem:series){reference-type="ref" reference="lem:series"} is justified by Lemma 17 in [@LLLW2022] and is similar to calculations given in [@LLLW2022] and [@LLW2022 Lemma 9]. The following lemma compares the largest roots. **Lemma 12**. *Let $f(\lambda)$ and $g(\lambda)$ be two decreasing functions on an interval $I$. Suppose $\lambda^2=f(\lambda)$ has a unique positive root $\lambda_f$ in $I$ and and $\lambda^2=g(\lambda)$ has unique root $\lambda_g$ in $I$. If $f(\lambda)>g(\lambda)$ on I, then $\lambda_f>\lambda_g$.* *Proof.* Assume towards a contradiction that $f(\lambda) > g(\lambda)$ on $I$ and $\lambda_f \le \lambda_g$. We observe $$0 = f(\lambda_f) - \lambda_f^2 > g(\lambda_f) - \lambda_f^2 \geq g(\lambda_g) - \lambda_f^2 \geq g(\lambda_g) - \lambda_g^2 = 0.$$ ◻ The basic tools are the same for the cases when $n$ is even and $n$ is odd, but there are various technicalities which make it appropriate to deal with these two cases separately. ## When $n$ is even *Proof of Theorem [Theorem 2](#thm:t2){reference-type="ref" reference="thm:t2"}.* Let $n=2q$ and $G$ be the graph which achieves the maximum $\lambda_2$ among all connected outerplanar graphs on $n$ vertices. We would like to show that $G$ is isomorphic to $G_0=(K_1\vee P_{q-1})\!\!-\!\!(K_1\vee P_{q-1})$. By Lemma [Lemma 2](#lem:lamda2even){reference-type="ref" reference="lem:lamda2even"}, $\lambda(G_0)$ satisfies the equation $\lambda^2=f(\lambda)$, where $$\label{eq:t2f_even} f(\lambda) = (q-1) + \frac{2q-5}{\lambda} + \frac{4q-11}{\lambda^2} + \frac{8q-28}{\lambda^3} + \frac{16q-59}{\lambda^4} + \frac{32q-136}{\lambda^5} + O\left(\frac{q}{\lambda^6}\right).$$ Let $g(\lambda)=\sum_{i=0}^\infty \frac{a_i}{\lambda^i}$. Let $I=(\sqrt{n/2}+1 - \frac{c_1}{\sqrt{n}}, \sqrt{n/2}+1 + \frac{c_2}{\sqrt{n}})$. By Theorem [Theorem 1](#thm:t1){reference-type="ref" reference="thm:t1"} and Lemma [Lemma 2](#lem:lamda2even){reference-type="ref" reference="lem:lamda2even"}, both of the equations $\lambda^2=f(\lambda)$ and $\lambda^2=g(\lambda)$ have a unique root in $I$. We apply Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"} to compare their roots $\lambda_f$ and $\lambda_g$. We first give coarse upper bounds on $a_i$. Since $G$ is an outerplanar graph, for $i=1,2$, $G|_{N(u_i)}$ is a linear forest, which has at most $d_{u_i}-1$ edges. Since $G$ is connected, there is at least one crossing edge connecting the two parts. Thus, $$a_1\leq (d_{u_1}-1) + (d_{u_2}-1) -1 \leq 2q-2 +|N(u_1)\cap N(u_2)| -3 \leq 2q-3.$$ Since $G$ is outerplanar, the number of crossing edges between $N(u_1)$ and $N(u_2)$ is at most $3$. These edges can only contribute $O(1)$ to $a_i$ for any fixed $i$. If we delete these crossing edges from $P$, we get a linear forest. They can contribute at most $2^iq$ to $a_i$. Thus, for $i\geq 2$, we have $$a_i\leq 2^iq + O(1).$$ We have the following claim.\ **Claim 1:** $d_{u_1}=d_{u_2}=q-1$ and $N(u_1)\cap N(u_2)=\emptyset$. Otherwise, we have $a_0\leq \frac{1}{2}(2q-3)$. This is because $$2a_0=d_{u_1}+ d_{u_2}- |N(u_1)\cap N(u_2)|= |N(u_1)\Delta N(u_2)|\leq 2q-2.$$ The equality holds if and only if $N(u_1)\cup N(u_2)$ forms a partition of $V(P)$. Since $d_{u_i}\geq q-1$, equality only holds when $d_{u_1}=d_{u_2}=q-1$ and $N(u_1)\cap N(u_2)=\emptyset$. For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &= a_0 + \frac{a_1}{\lambda} + \frac{a_2}{\lambda^2} + O\left(\frac{q}{\lambda^3}\right)\\ &\leq \frac{1}{2}(2q-3) + \frac{2q-3}{\lambda} + \frac{4q+O(1)}{\lambda^2} + O\left(\frac{q}{\lambda^3}\right)\\ &< (q-1) + \frac{2q-5}{\lambda} + \frac{4q-11}{\lambda^2} + O\left(\frac{q}{\lambda^3}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 1 is proved. **Claim 2:** For $i=1,2$, $G|_{N(u_i)}$ is a path of length $q-2$. There is exactly one edge in $E(N(u_1), N(u_2))$. Otherwise, we have $$a_1< (q-2) + (q-2) -1 = 2q-5.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &= a_0 + \frac{a_1}{\lambda} + \frac{a_2}{\lambda^2} + \frac{a_3}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &\leq q-1 + \frac{2q-6}{\lambda} + \frac{4q+O(1)}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &< (q-1) + \frac{2q-5}{\lambda} + \frac{4q-11}{\lambda^2} + \frac{8q-28}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 2 is proved. **Claim 3**: The unique edge in $E(N(u_1), N(u_2))$ must connect the ending vertices of $P_{q-2}$. Denote this edge by $uv$. The contribution of $uv$ in $a_2$ is given by $$1 -(d_u+d_v)< 1-(1+1)=-1.$$ If either $u$ or $v$ are not ending vertices, we have $$a_2 \leq a_2(G_0)-1 \leq 4q-12.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &= a_0 + \frac{a_1}{\lambda} + \frac{a_2}{\lambda^2} + \frac{a_3}{\lambda^3} ++ \frac{16q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &\leq q-1 + \frac{2q-5}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &< (q-1) + \frac{2q-5}{\lambda} + \frac{4q-11}{\lambda^2} + \frac{8q-28}{\lambda^3} + \frac{16q-59}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 3 is proved. Therefore, $G$ is isomorphic to $G_0$. The proof of Theorem [Theorem 2](#thm:t2){reference-type="ref" reference="thm:t2"} is finished. ◻ We now give the proof of Corollary [Corollary 1](#cor:c1){reference-type="ref" reference="cor:c1"}, which determines the maximum size of $\lambda_2$ over all outerplanar graphs on $n=2q$ vertices with $n$ sufficiently large. *Proof of Corollary [Corollary 1](#cor:c1){reference-type="ref" reference="cor:c1"}.* Theorem [Theorem 2](#thm:t2){reference-type="ref" reference="thm:t2"} determines the unique outerplanar graph with maximum $\lambda_2$ over connected outerplanar graphs, so we only need to consider disconnected outerplanar graphs. Let $G$ be the disconnected outerplanar graph on $n=2q$ vertices with maximum $\lambda_2$ and suppose that $G$ has connected components $G_1, G_2\ldots, G_m$, labelled so that $\lambda_1(G_1) \ge \lambda_1(G_2) \ge \cdots \ge \lambda_1(G_m)$. Note that $\lambda_2(G_i) < \lambda_2((K_1\vee P_{q-1})\!\!-\!\!(K_1\vee P_{q-1}))$ for each $i$ and for $n$ sufficiently large, so we only need to consider the case where $\lambda_2(G) = \lambda_1(G_2)$. By the result of Tait and Tobin on the maximum spectral radius of outerplanar graphs, this is achieved when $G_2 = K_1 \vee P_{q-1}$. Using the series expansions in Lemma [Lemma 1](#lem:specradfangph){reference-type="ref" reference="lem:specradfangph"} and Lemma [Lemma 2](#lem:lamda2even){reference-type="ref" reference="lem:lamda2even"}, we see that $$\lambda_1(K_1 \vee P_{q-1}) = \sqrt{q-1} + 1 + \frac{1}{2\sqrt{q-1}} - \frac{1}{q-1} - \frac{1}{8(q-1)^{3/2}} - \frac{7}{16(q-1)^{5/2}} + O\left(\frac{1}{(q-1)^{3}}\right),$$ while $$\lambda_2((K_1\vee P_{q-1})\!\!-\!\!(K_1\vee P_{q-1})) = \sqrt{q-1} + 1 + \frac{1}{2\sqrt{q-1}} - \frac{3}{2(q-1)} + \frac{7}{8(q-1)^{\frac32}} - \frac{2}{(q-1)^2} + O\left(\frac{1}{(q-1)^{5/2}}\right),$$ so for sufficiently large $n$, $$\lambda_1(K_1\vee P_{q-1}) > \lambda_2((K_1\vee P_{q-1})\!\!-\!\!(K_1\vee P_{q-1})),$$ completing the proof. ◻ ## When $n$ is odd *Proof of Theorem [Theorem 3](#thm:t3){reference-type="ref" reference="thm:t3"}.* Let $n=2q+1$ and $G$ be the graph that reaches the maximum $\lambda_2$ among all connected outerplanar graphs on $n$ vertices. Let $G_0$ be a graph obtained by adding a new vertex connecting two copies of $(K_1\vee P_{q-1})$. By Cauchy's interlacing theorem and Lemma [Lemma 1](#lem:specradfangph){reference-type="ref" reference="lem:specradfangph"}, $\lambda(G_0)$ satisfies the equation $\lambda^2=f(\lambda)$, where $$f(\lambda) = (q-1) + \frac{2q-4}{\lambda} + \frac{4q-10}{\lambda^2} + \frac{8q-24}{\lambda^3} + \frac{16q-54}{\lambda^4} + \frac{32q-120}{\lambda^5} + O\left(\frac{q}{\lambda^6}\right).$$ Let $g(\lambda)=\sum_{i=0}^\infty \frac{a_i}{\lambda^i}$. Let $I=\left(\sqrt{n/2}+1 - \frac{c_1}{\sqrt{n}}, \sqrt{n/2}+1 + \frac{c_2}{\sqrt{n}}\right)$. By Theorem [Theorem 1](#thm:t1){reference-type="ref" reference="thm:t1"} and Lemma [Lemma 2](#lem:lamda2even){reference-type="ref" reference="lem:lamda2even"}, both equations $\lambda^2=f(\lambda)$ and $\lambda^2=g(\lambda)$ have a unique root in $I$. We apply Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"} to compare their roots $\lambda_f$ and $\lambda_g$. Similarly to the even case, for $i\geq 2$, we have $$a_i\leq 2^iq + O(1).$$ We have the following claim.\ **Claim 1:** One of the following four cases must occur: 1. $d_{u_1}=d_{u_2}=q-1$, and $N(u_1)\cap N(u_2)=\emptyset$. 2. $d_{u_1}=d_{u_2}=q$, and $|N(u_1)\cap N(u_2)|=1$. 3. $d_{u_1}=q$, $d_{u_2}=q-1$, and $N(u_1)\cap N(u_2)=\emptyset$. 4. $d_{u_1}=q-1$, $d_{u_2}=q$, and $N(u_1)\cap N(u_2)=\emptyset$. Otherwise, we show $a_0\leq \frac{1}{2}(2q-3)$. This is because $$2a_0=d_{u_1}+ d_{u_2}- |N(u_1)\cap N(u_2)|= |N(u_1)\Delta N(u_2)|\leq 2q-1.$$ If $|N(u_1)\Delta N(u_2)|=2q-1$, we get case 3 or case 4. If $|N(u_1)\Delta N(u_2)|=2q-2$, we get case 1 or case 2. If none of these cases occur, we get $$a_0\leq \frac{2q-3}{2}.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &= a_0 + \frac{a_1}{\lambda} + \frac{a_2}{\lambda^2} + O\left(\frac{q}{\lambda^3}\right)\\ &\leq \frac{1}{2}(2q-3) + \frac{2q+O(1)}{\lambda} + \frac{4q+O(1)}{\lambda^2} + O\left(\frac{q}{\lambda^3}\right)\\ &< (q-1) + \frac{2q-4}{\lambda} + \frac{4q-10}{\lambda^2} + O\left(\frac{q}{\lambda^3}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 1 is proved. **Case 1:** $d_{u_1}=d_{u_2}=q-1$, and $N(u_1)\cap N(u_2)=\emptyset$. Let $u$ be the unique vertex not in $N(u_1)\cap N(u_2)$. **Claim 2:** For $i = 1, 2$, $G|_{N (u_i)}$ is a path of length $q-2$. Moreover, $u$ is a cut vertex of $G$. Observe $u$ has zero contribution to $a_1$. If $u$ is not a cut vertex, then, there is at least an edge in $E(N(u_1), N(u_2))$, which contributes $-1$ to $a_1$. If either $G|_{N (u_1)}$ or $G|_{N (u_2)}$ is not $P_{q-1}$, then it contributes one less in $a_1$. Therefore, if Claim 2 does not hold, we have $$a_1\leq (q-2) + (q-2) -1 = 2q-5.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &= a_0 + \frac{a_1}{\lambda} + \frac{a_2}{\lambda^2} + \frac{a_3}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &\leq q-1 + \frac{2q-5}{\lambda} + \frac{4q+O(1)}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &< (q-1) + \frac{2q-4}{\lambda} + \frac{4q-10}{\lambda^2} + \frac{8q-24}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 2 is proved. **Case 2:** $d_{u_1}=d_{u_2}=q$, and $|N(u_1)\cap N(u_2)|=1$. Let $u$ be the unique vertex in $N(u_1)\cap N(u_2)$. **Claim 3**: For $i = 1, 2$, $G|_{N (u_i)} - \{u\}$ is a path of length $q-2$. Moreover, $u$ is a cut vertex of $G$. Observe $u$ has zero contribution to $a_1$. If $u$ is not a cut vertex, then, there is at least an edge in $E(N(u_1)- \{u\} , N(u_2)-\{u\})$, which contributes $-1$ to $a_1$. If either $G|_{N (u_1)} - \{u\}$ or $G|_{N (u_2)} - \{u\}$ is not $P_{q-1}$, then it contributes one less in $a_1$. Therefore, if Claim 2 does not hold, we have $$a_1\leq (q-2) + (q-2) -1 = 2q-5.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &= a_0 + \frac{a_1}{\lambda} + \frac{a_2}{\lambda^2} + \frac{a_3}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &\leq q-1 + \frac{2q-5}{\lambda} + \frac{4q+O(1)}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &< (q-1) + \frac{2q-4}{\lambda} + \frac{4q-10}{\lambda^2} + \frac{8q-24}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 3 is proved. **Case 3:** $d_{u_1}=q$, $d_{u_2}=q-1$, and $N(u_1)\cap N(u_2)=\emptyset$. From Equation [\[eq:gamma1\]](#eq:gamma1){reference-type="eqref" reference="eq:gamma1"} and [\[eq:gamma2\]](#eq:gamma2){reference-type="eqref" reference="eq:gamma2"}, $\lambda_2$ satisfies two equations. $$\begin{aligned} \lambda^2&= q + \sum_{i=1}^{\infty}\frac{\sum_{u,v \in N(u_1)} w_i(u,v) + \sum_{u\in N(u_1), v\in N(u_2)} w_i(u,v) \frac{x_{u_2}}{x_{u_1}}}{\lambda^i}, \\ \lambda^2&= q-1 + \sum_{i=1}^{\infty}\frac{\sum_{u,v \in N(u_2)} w_i(u,v) + \sum_{u\in N(u_1), v\in N(u_2)} w_i(u,v) \frac{x_{u_1}}{x_{u_2}}}{\lambda^i}. \label{eq:gamma2exp}\end{aligned}$$ Taking the difference and solving for $\frac{x_{u_2}}{x_{u_1}}- \frac{x_{u_1}}{x_{u_2}}$, we have $$\begin{aligned} \frac{x_{u_2}}{x_{u_1}} - \frac{x_{u_1}}{x_{u_2}} &= -\frac{1+ \sum_{i=1}^{\infty}\frac{\sum_{u,v \in N(u_1)} w_i(u,v) - \sum_{u,v \in N(u_2)} w_i(u,v)}{\lambda^i}} {\sum_{i=1}^{\infty} \frac{\sum_{u\in N(u_1), v\in N(u_2)} w_i(u,v)}{\lambda_i} } \\ &= - \frac{(1+O(\frac{1}{\lambda}))}{ \frac{|E(N(u_1), N(u_2))|}{\lambda}}\\ &= -\left(1+O\left(\frac{1}{\lambda}\right)\right) \frac{\lambda}{|E(N(u_1), N(u_2))|}.\end{aligned}$$ Solving for $\frac{x_{u_2}}{x_{u_1}}$, we get $$\frac{x_{u_2}}{x_{u_1}} =-\left(1+O\left(\frac{1}{\lambda}\right)\right) \frac{|E(N(u_1), N(u_2))|}{\lambda}.$$ **Claim 4**: $G|_{N(u_2)}$ forms a path of length $q-2$. Otherwise, we have $$\sum_{u,v \in N(u_2)} w_1(u,v) =2 |E(G|_{N(u_2)})|\leq 2(q-3).$$ Let $g(\lambda)=\sum_{i=0}^\infty \lambda_2^{-i}\gamma_2'A_P^i\beta$. Since $\frac{x_{u_1}}{x_{u_2}}<0$, we have using Equation [\[eq:gamma2exp\]](#eq:gamma2exp){reference-type="eqref" reference="eq:gamma2exp"}, we have $$\begin{aligned} g(\lambda) &\leq q-1 + \frac{2q-6}{\lambda} + \frac{4q+O(1)}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right))\\ &< (q-1) + \frac{2q-4}{\lambda} + \frac{4q-10}{\lambda^2} + \frac{8q-24}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 4 is proved. **Claim 5**: $|E(N(u_1), N(u_2))| \in \{1, 2\}$. If $|E(N(u_1), N(u_2))|=2$, then the two edges in $E(N(u_1), N(u_2))$ share a common vertex in $N(u_1)$. By Claim 4, the induced graph on $N(u_2)$ is a path $P_{q-1}$. Let $c= |E(N(u_1), N(u_2))|$. Let us estimate the contribution of these crossing edges in $\sum_{u,v\in N(u_2)} w_2(u,v)$. Since $G$ is a connected outerplanar graph, we have $c\in\{1,2,3\}$. Let $\eta$ be the contribution of the crossing edges in $\sum_{u,v\in N(u_2)} w_2(u,v)$. We have $$\eta=\begin{cases} 1 & \mbox{ if } c=1,\\ 4 & \mbox{ if } c=2 \mbox{ and two crossing edges share a common vertex in } N(u_1), \\ 2 & \mbox { if } c=2 \mbox { and two crossing edges do not share a common vertex in } N(u_1),\\ 5 & \mbox { if } c=3. \end{cases}$$ When $c=3$, the three crossing edges must form a shape of $N$, thus the contribution is $3+2=5$. Note that the contribution of $P_{q-1}$ to $\sum_{u,v\in N(u_2)} w_2(u,v)$ is $$2(q-1)-2 + 2(q-3)=4q-10.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &=\sum_{i=0}^\infty \lambda_2^{-i}\gamma_2'A_P^i\beta\\ &\leq q-1 + \frac{2q-4 + c \frac{x_{u_1}}{x_{u_2}}}{\lambda} + \frac{4q-10 + \eta +O(1)\cdot \frac{x_{u_1}}{x_{u_2}} }{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &\leq q-1 + \frac{2q-4 - \frac{c^2}{\lambda}}{\lambda} + \frac{4q-10 + \eta} {\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &= (q-1) + \frac{2q-4}{\lambda} + \frac{4q-10 -c^2+\eta}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &< (q-1) + \frac{2q-4}{\lambda} + \frac{4q-10}{\lambda^2} + \frac{8q-28}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &=f(\lambda).\end{aligned}$$ If Claim 5 fails, we have $\eta-c^2\leq -2$. Thus, the last inequality holds. By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 5 is proved. By Claim 5, there exists a cut vertex of $G$ in $N(u_1)$. Call this cut vertex $u$. **Claim 6:** $G\setminus \{u\}= 2 K_1\vee P_{q-1}$. By Claim $4$, one of the components in $G\setminus \{u\}$ is $K_1\vee P_{q-1}$. Call the other component $H$. If $H\not= K_1\vee P_{q-1}$, by Tait and Tobin's result, we have $$\lambda_1(H) <\lambda_1 (K_1\vee P_{q-1}).$$ Thus, we have $$\begin{aligned} \lambda_1(G\setminus \{u\}) & = \lambda_1(K_1\vee P_{q-1}),\\ \lambda_2(G\setminus \{u\}) & = \lambda_1(H).\end{aligned}$$ Since two eigenvalues are not equal, by Cauchy's Interlacing theorem, we have $$\lambda_1(K_1\vee P_{q-1})=\lambda_1(G\setminus \{u\}) > \lambda_2(G)>\lambda_2(G\setminus \{u\})=\lambda_1(H).$$ Thus, $\lambda_2(G)$ is not maximal if $H\neq K_1\vee P_{q-1}$. **Case 4:** $d_{u_1}=q-1$, $d_{u_2}=q$, and $N(u_1)\cap N(u_2)=\emptyset$. This is symmetric to Case 3. Therefore, $G$ always contains a cut vertex $u$ such that $G\setminus \{u\}$ is two disjoint copies of $K_1\vee P_{q-1}$. The proof of Theorem [Theorem 3](#thm:t3){reference-type="ref" reference="thm:t3"} is finished. ◻ # Maximum $\lambda_2$ for 2-connected outerplanar graphs We first compute the series expansion of $\lambda_2$ for the conjectured extremal graph $(K_1\vee P_{\lfloor \frac{n}{2}\rfloor-1})\!\!=\!\!(K_1\vee P_{\lceil \frac{n}{2}\rceil-1})$. **Lemma 13**. *Let $G= (K_1\vee P_{\lfloor \frac{n}{2}\rfloor-1})\!\!=\!\!(K_1\vee P_{\lceil \frac{n}{2}\rceil-1})$. Then:* 1. *For $n=2q$, $\lambda_2(G)$ satisfies the following equation. $$\lambda^2 = (q-1) + \frac{2q-6}{\lambda} + \frac{4q-14}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-70}{\lambda^4}+\frac{32q-152}{\lambda^5} +\frac{64q-324}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right). \label{eq:2conneven}$$ In particular, we have $$\lambda_2 = \sqrt{q-1} + 1 + \frac{1}{2\sqrt{q-1}} - \frac{2}{q-1} + \frac{7}{8(q-1)^{3/2}} + \frac{113}{16(q-1)^{5/2}} + O\left(\frac{1}{(q-1)^{3}}\right).$$* 2. *For $n=2q+1$, $\lambda_2(G)$ satisfies the following equation. $$\lambda^2 = (q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-64}{\lambda^4}+\frac{32q-112}{\lambda^5} +\frac{64q-232}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right). \label{eq:2connodd}$$ In particular, we have $$\lambda_2 = \sqrt{q-1} + 1 + \frac{1}{2\sqrt{q-1}} - \frac{1}{q-1} - \frac{9}{8(q-1)^{3/2}} -\frac{87}{16(q-1)^{5/2}}+ O\left(\frac{1}{(q-1)^{3}}\right)$$* *Proof.* Let $u_1$ be the vertex of the largest degree and $u_2$ be the vertex of the second largest degree in $G$. Let $P$ be the induced subgraph obtained from $G$ after deleting $u_1$ and $u_2$. Let $\mathbf{x}$ be the eigenvector of $G$ corresponding to $\lambda_2$. Let $\beta_1$, $\beta_2$, $\gamma_1$, $\gamma_2$, $\beta$, $\gamma$ defined as in Equations [\[eq:beta1def\]](#eq:beta1def){reference-type="eqref" reference="eq:beta1def"}-[\[eq:gamma\]](#eq:gamma){reference-type="eqref" reference="eq:gamma"}. Equation [\[eq:gamma1\]](#eq:gamma1){reference-type="eqref" reference="eq:gamma1"} can be re-written as $$\lambda^2= F_1(\lambda) + D(\lambda) \frac{x_{u_2}}{x_{u_1}}.$$ Equation [\[eq:gamma2\]](#eq:gamma2){reference-type="eqref" reference="eq:gamma2"} can be re-written as $$\lambda^2= F_2(\lambda) + D(\lambda) \frac{x_{u_1}}{x_{u_2}}.$$ Here $$\begin{aligned} F_1(\lambda)&=\sum_{i=0}^{\infty}\frac{\sum_{u,v \in N(u_1)} w_i(u,v)}{\lambda^i}, \\ F_2(\lambda)&=\sum_{i=0}^{\infty}\frac{\sum_{u,v \in N(u_2)} w_i(u,v)}{\lambda^i}, \\ D(\lambda)&=\sum_{i=1}^{\infty}\frac{\sum_{u\in N(u_1), v\in N(u_2)} w_i(u,v)}{\lambda^i}.\end{aligned}$$ Cancelling $\frac{x_{u_2}}{x_{u_1}}$, we get $$(\lambda^2-F_1(\lambda)) (\lambda^2-F_2(\lambda)) = D(\lambda)^2.$$ We get $$\lambda^2 =\frac{1}{2} \left( (F_1(\lambda) +F_2(\lambda)) - \sqrt{ (F_1(\lambda) -F_2(\lambda))^2 + 4 D(\lambda)^2} \right).$$ Using SageMath, we can calculate $$\begin{aligned} F_2(\lambda) &= (q-1) + \frac{2q-4}{\lambda} + \frac{4q-8}{\lambda^2} + \frac{8q-16}{\lambda^3} + \frac{16q-28}{\lambda^4}+\frac{32q-48}{\lambda^5} +\frac{64q-64}{\lambda^6}+O\left(\frac{q}{\lambda^7}\right).\\ D(\lambda)&= \frac{2}{\lambda} + \frac{6}{\lambda^2} + \frac{16}{\lambda^3} + \frac{42}{\lambda^4}+\frac{104}{\lambda^5} +\frac{260}{\lambda^6}+O\left(\frac{q}{\lambda^7}\right).\end{aligned}$$ When $n=2q$, we have $F_1(\lambda)=F_2(\lambda)$ and $$\lambda^2=F_2(\lambda)-D(\lambda).$$ This implies Equation [\[eq:2conneven\]](#eq:2conneven){reference-type="eqref" reference="eq:2conneven"}. When $n=2q+1$, we have $$\begin{aligned} F_1(\lambda)&= q + \frac{2q-2}{\lambda} + \frac{4q-4}{\lambda^2} + \frac{8q-8}{\lambda^3} + \frac{16q-12}{\lambda^4}+\frac{32q-16}{\lambda^5} +\frac{64q}{\lambda^6}+O\left(\frac{q}{\lambda^7}\right).\end{aligned}$$ Therefore, $$\begin{aligned} \lambda^2 &=\frac{1}{2} \left( (F_1(\lambda) +F_2(\lambda)) - \sqrt{ (F_1(\lambda) -F_2(\lambda))^2 + 4 D(\lambda)^2} \right)\\ &= \frac{1}{2} \left(2q-1 + \frac{4q-6}{\lambda} + \frac{8q-12}{\lambda^2} + \frac{16q-24}{\lambda^3} + \frac{32q-40}{\lambda^4}+\frac{64q-64}{\lambda^5} +\frac{128q-64}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right)\right.\\ &\hspace*{4mm} \left. -\sqrt{ \left(1+ \frac{2}{\lambda} + \frac{4}{\lambda^2} + \frac{8}{\lambda^3} + \frac{16}{\lambda^4}+\frac{32}{\lambda^5} +\frac{64}{\lambda^6}+O\left(\frac{1}{\lambda^7}\right)\right)^2 + 4\left(\frac{2}{\lambda} + \frac{6}{\lambda^2} + \frac{16}{\lambda^3} + \frac{42}{\lambda^4}+\frac{104}{\lambda^5} +\frac{260}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right)\right)^2 } \right) \\ &= (q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-64}{\lambda^4}+\frac{32q-112}{\lambda^5} +\frac{64q-232}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right).\end{aligned}$$ Using SageMath, we can calculate the series expansion of $\lambda$. $$\lambda_2 = \sqrt{q-1} + 1 + \frac{1}{2\sqrt{q-1}} - \frac{1}{q-1} - \frac{9}{8(q-1)^{3/2}} -\frac{87}{16(q-1)^{5/2}}+ O\left(\frac{1}{(q-1)^{3}}\right).$$ ◻ *Proof of Theorem [Theorem 4](#thm:t4){reference-type="ref" reference="thm:t4"}.* Let $G$ be the graph which achieves the maximum $\lambda_2$ among all 2-connected outerplanar graphs on $n$ vertices. Let $G_0$ be the conjectured extremal 2-connected outerplanar graph on $n$ vertices. We follow the proofs of Theorems [Theorem 2](#thm:t2){reference-type="ref" reference="thm:t2"} and [Theorem 3](#thm:t3){reference-type="ref" reference="thm:t3"}. Assume $\lambda_2(G)$ satisfies the equation $$\lambda^2=g(\lambda)$$ while $\lambda_2(G_0)$ satisfies the equation $$\lambda^2=f(\lambda).$$ It is sufficient to compare $f(\lambda)$ and $g(\lambda)$ in a small interval $I= \left(\sqrt{n/2}+1 - \frac{c_1}{\sqrt{n}}, \sqrt{n/2}+1 + \frac{c_2}{\sqrt{n}}\right)$. **Case $n=2q$:** We have $$\label{eq:t2f_even_2_conn} f(\lambda) = (q-1) + \frac{2q-6}{\lambda} + \frac{4q-14}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-70}{\lambda^4} + \frac{32q-152}{\lambda^5} + O\left(\frac{q}{\lambda^6}\right).$$ We have the following claim.\ **Claim 1:** $d_{u_1}=d_{u_2}=q-1$ and $N(u_1)\cap N(u_2)=\emptyset$. The proof is identical to that proof of Claim 1 in Theorem [Theorem 2](#thm:t2){reference-type="ref" reference="thm:t2"}. **Claim 2:** For $i=1,2$, $G|_{N(u_i)}$ is a path of length $q-2$. There are exactly two edges in $E(N(u_1), N(u_2))$. In particular, the two edges form a matching. Otherwise, we have $$a_1< (q-2) + (q-2) -2 = 2q-6.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &= a_0 + \frac{a_1}{\lambda} + \frac{a_2}{\lambda^2} + \frac{a_3}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &\leq q-1 + \frac{2q-7}{\lambda} + \frac{4q+O(1)}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &< (q-1) + \frac{2q-6}{\lambda} + \frac{4q-14}{\lambda^2} + \frac{8q-32}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 2 is proved. Since $G$ is 2-connected, the two crossing edges must form a matching. **Claim 3**: The ending vertices of the two edges in $E(N(u_1), N(u_2))$ have degrees $1,1,2,2$ in $P$. Denote the two edges by $v_1w_1$ and $v_2,w_2$. The contribution of $v_iw_i$ in $a_2$ is given by $$1 -\left(d^P_{v_i}+d^P_{w_i}\right).$$ We have $$a_2 \leq 4q-8 -\left(d^P_{v_1}+d^P_{w_1} +d^P_{v_2}+d^P_{w_2}\right).$$ Note that since $G$ is outerplanar, at most two vertices in $v_1, w_1, v_2, w_2$ are ending vertices. Thus, we have $$a_2 \leq 4q-8 -\left(d^P_{v_1}+d^P_{w_1} +d^P_{v_2}+d^P_{w_2}\right)\leq 4q-14,$$ with equality if and only if the degrees $d^P_{v_1}, d^P_{w_1}, d^P_{v_2}, \text{and } d^P_{w_2}$ are $1,1,2, \text{and }2$, respectively. If Claim 3 fails, we have $$a_2\leq 4q-15.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &= a_0 + \frac{a_1}{\lambda} + \frac{a_2}{\lambda^2} + \frac{a_3}{\lambda^3} + \frac{16q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &\leq q-1 + \frac{2q-6}{\lambda} + \frac{4q-15}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + \frac{16q+O(1)}{\lambda^4} + O\left(\frac{q}{\lambda^4}\right)\\ &< (q-1) + \frac{2q-6}{\lambda} + \frac{4q-14}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-70}{\lambda^4} + O\left(\frac{q}{\lambda^4}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 3 is proved. From Claim 1-3, there are only two graphs left. Call the other graph $G'_0$. [\[fig:enter-label\]]{#fig:enter-label label="fig:enter-label"} Both $G_0$ and $G'_0$ have the same values on $a_0$, $a_1$, and $a_2$. They only differ by $1$ on $a_3$. The graph $G'_0$ contains one more negative $3$-walk, which is highlighted in red in Figure [\[fig2\]](#fig2){reference-type="ref" reference="fig2"}. We have $$a_3(G_0')=a_3(G_0)-1=8q-33.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &= a_0 + \frac{a_1}{\lambda} + \frac{a_2}{\lambda^2} + \frac{a_3}{\lambda^3} + \frac{16q+O(1)}{\lambda^4} + \frac{32q+O(1)}{\lambda^5} + O\left(\frac{q}{\lambda^6}\right)\\ &\leq q-1 + \frac{2q-6}{\lambda} + \frac{4q-14}{\lambda^2} + \frac{8q-33}{\lambda^3} + \frac{16q+O(1)}{\lambda^4} + \frac{32q+O(1)}{\lambda^5} + O\left(\frac{q}{\lambda^6}\right)\\ &< (q-1) + \frac{2q-6}{\lambda} + \frac{4q-14}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-70}{\lambda^4} + \frac{32q-152}{\lambda^5} + O\left(\frac{q}{\lambda^6}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! The proof of the even case is finished. **Case $n=2q+1$:** We have $$f(\lambda)=(q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-64}{\lambda^4}+\frac{32q-112}{\lambda^5} + O\left(\frac{q}{\lambda^6}\right).$$ We have the following claim.\ **Claim 4:** One of the following four cases must occur: 1. $d_{u_1}=d_{u_2}=q-1$, and $N(u_1)\cap N(u_2)=\emptyset$. 2. $d_{u_1}=d_{u_2}=q$, and $|N(u_1)\cap N(u_2)|=1$. 3. $d_{u_1}=q$, $d_{u_2}=q-1$, and $N(u_1)\cap N(u_2)=\emptyset$. 4. $d_{u_1}=q-1$, $d_{u_2}=q$, and $N(u_1)\cap N(u_2)=\emptyset$. The proof is identical to the proof of Theorem [Theorem 3](#thm:t3){reference-type="ref" reference="thm:t3"}. Again, we can show that there is a cut vertex in both Case 1 and Case 2, which cannot occur since $G$ is 2-connected. Cases 3 and 4 are symmetric. From now on, we assume $d_{u_1}=q$, $d_{u_2}=q-1$, and $N(u_1)\cap N(u_2)=\emptyset$. **Claim 5**: $G|_{N(u_2)}$ forms a path of length $q-2$. Otherwise, we have $$\sum_{u,v \in N(u_2)} w_1(u,v) =2 |E(G|_{N(u_2)})|\leq 2(q-3).$$ Let $g(\lambda)=\sum_{i=0}^\infty \lambda_2^{-i}\gamma_2'A_P^i\beta$. Since $\frac{x_{u_1}}{x_{u_2}}<0$, using Equation [\[eq:gamma2exp\]](#eq:gamma2exp){reference-type="eqref" reference="eq:gamma2exp"}, we have $$\begin{aligned} g(\lambda) &\leq q-1 + \frac{2q-6}{\lambda} + \frac{4q+O(1)}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &< (q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-32}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ &=f(\lambda).\end{aligned}$$ By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 5 is proved. **Claim 6**: $E(N(n_1), N(u_2))$ consists of two parallel edges. By Claim 5, the induced graph on $N(u_2)$ is a path $P_{q-1}$. Let $c= |E(N(u_1), N(u_2))|$. Since $G$ is 2-connected outerplanar graph, we have $c=2$ or $c = 3$. When $c=2$, the two crossing edges form a matching. When $c=3$, the three crossing edges must form a shape of $N$. Let us estimate the contribution of these crossing edges in $\sum_{u,v\in N(u_2)} w_2(u,v)$. Let $\eta$ be the contribution of crossing edges in $\sum_{u,v\in N(u_2)} w_2(u,v)$. We have $$\eta=\begin{cases} 2 & \mbox { if } c=2,\\ 5 & \mbox { if } c=3. \end{cases}$$ Note the contribution of $P_{q-1}$ to $\sum_{u,v\in N(u_2)} w_2(u,v)$ is $$2(q-1)-2 + 2(q-3)=4q-10.$$ For any $\lambda\in I$, we have $$\begin{aligned} g(\lambda) &=\sum_{i=0}^\infty \lambda_2^{-i}\gamma_2'A_P^i\beta\\ &\leq q-1 + \frac{2q-4 + c \frac{x_{u_1}}{x_{u_2}}}{\lambda} + \frac{4q-10 + \eta +O(1)\cdot \frac{x_{u_1}}{x_{u_2}} }{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + \frac{16q+O(1)}{\lambda^4} +O\left(\frac{q}{\lambda^5}\right)\\ &\leq q-1 + \frac{2q-4 - \frac{c^2}{\lambda}}{\lambda} + \frac{4q-10 + \eta} {\lambda^2} + \frac{8q+O(1)}{\lambda^3} + \frac{16q+O(1)}{\lambda^4} + O\left(\frac{q}{\lambda^5}\right)\\ &= (q-1) + \frac{2q-4}{\lambda} + \frac{4q-10 -c^2+\eta}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + \frac{16q+O(1)}{\lambda^4} + O\left(\frac{q}{\lambda^5}\right)\\ &< (q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-64}{\lambda^4} + O\left(\frac{q}{\lambda^5}\right)\\ &=f(\lambda).\end{aligned}$$ If Claim 6 fails, we have $\eta-c^2=5-3^2=-4$. Thus, the last inequality holds. By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 6 is proved. **Claim 7**: $G|_{N(u_1)}$ forms a path of length $q-1$. The ending vertices of two edges in $E(N(u_1), N(u_2))$ have degrees $1,1,2,2$ in $P$. Let $|E(G|_{N(u_1))}|=q-1-c'$. Since $G$ is 2-connected and outerplanar, we must have $c'=0$ or $1$. Write the two edges in $E(N(n_1), N(u_2))$ as $v_1w_1$ and $v_2w_2$ with $w_1,w_2\in N(u_2)$. Let $\eta(v_1,v_2)=1$ if $v_1v_2$ is not an edge, and 0 otherwise. We have $$\begin{aligned} F_1(\lambda)&= q + \frac{2q-2-2c'}{\lambda} + \frac{4q+O(1)}{\lambda^2} + \frac{8q+O(1)}{\lambda^3} + O\left(\frac{q}{\lambda^4}\right)\\ F_2(\lambda)&= q -1 + \frac{2q-4}{\lambda} + \frac{4q-8}{\lambda^2} + \frac{8q-10 -2(d^P(w_1)+d^P(w_2))-2\eta(v_1,v_2)}{\lambda^3} + \frac{16q+O(1)}{\lambda^4}+\frac{32q+O(1)}{\lambda^5} \\ &+ O\left(\frac{q}{\lambda^6}\right)\\ D(\lambda) &= \frac{2}{\lambda} +\frac{d^P(v_1)+d^P(w_1)+d^P(v_2)+d^P(w_2) }{\lambda^2} \\\end{aligned}$$ Therefore, $$\begin{aligned} \lambda^2 &=\frac{1}{2} \left( (F_1(\lambda) +F_2(\lambda)) - \sqrt{ (F_1(\lambda) -F_2(\lambda))^2 + 4 D(\lambda)^2} \right)\\ &=\frac{1}{2} \left( (F_1(\lambda) +F_2(\lambda)) - (F_1(\lambda)-F_2(\lambda) \sqrt{ 1+ \frac{4 D(\lambda)^2}{(F_1(\lambda) -F_2(\lambda))^2} } \right)\\ &=\frac{1}{2} \left( (F_1(\lambda) +F_2(\lambda)) - (F_1(\lambda)-F_2(\lambda) \left( 1+ \frac{2 D(\lambda)^2}{(F_1(\lambda) -F_2(\lambda))^2} + O\left( \frac{1}{(q-1)^2}\right) \right) \right)\\ &= F_2(\lambda)- \frac{D(\lambda)^2}{(F_1(\lambda) -F_2(\lambda))}+ O\left( \frac{1}{(q-1)^2}\right)\\ &= q -1 + \frac{2q-4}{\lambda} + \frac{4q-8}{\lambda^2} + \frac{8q-10- 2(d^P(w_1)+d^P(w_2))-2\eta(v_1,v_2)}{\lambda^3} + \frac{16q+O(1)}{\lambda^4}+\frac{32q+O(1)}{\lambda^5} +O\left(\frac{q}{\lambda^6}\right) \\ &\hspace*{2mm} - \left(\frac{2}{\lambda} +\frac{d^P(v_1)+d^P(w_1)+d^P(v_2)+d^P(w_2) }{\lambda^2}\right)^2 \left(1-\frac{2-2c'}{\lambda}\right) + O\left(\frac{1}{(q-1)^2}\right)\\ &= (q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-2 - (4d^P(v_1)+6d^P(w_1)+4d^P(v_2)+6d^P(w_2))-2\eta(v_1,v_2)-8c' }{\lambda^3} \\ &\hspace*{2mm} + \frac{16q+O(1)}{\lambda^4}+\frac{32q+O(1)}{\lambda^5} +O\left(\frac{q}{\lambda^6}\right).\end{aligned}$$ Since $G|_{N(u_2)}=P_{q-1}$, we must have $$d^P(w_1) + d^P(w_2)\geq 1+2=3.$$ When $c'=0$, we have $$4(d^P(v_1)+d^P(v_2))+6(d^P(w_1)+d^P(w_2))\geq 4(1+2)+ 6(1+2)=30,$$ with equality if and only if the ending vertices of two edges in $E(N(u_1), N(u_2))$ have degrees $1,1,2,2$ in $P$. When $c'=1$ and $v_1v_2$ is not an edge, we have $$4(d^P(v_1)+d^P(v_2))+6(d^P(w_1)+d^P(w_2))+2\eta(v_1,v_2)\geq 4(1+0)+ 6(1+2)+2=24.$$ When $c'=1$ and $v_1v_2$ is an edge, we have $$4(d^P(v_1)+d^P(v_2))+6(d^P(w_1)+d^P(w_2))+2\eta(v_1,v_2)\geq 4(1+1)+ 6(1+2)+0=26.$$ Therefore, if Claim 7 fails, then we have $$8q-8 - (6d^P(v_1)+4d^P(w_1)+6d^P(v_2)+4d^P(w_2))-8c'\leq 8q-33.$$ We have $$\begin{aligned} g(\lambda) &\leq (q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-33 }{\lambda^3} + \frac{16q+O(1)}{\lambda^4}+\frac{32q+O(1)}{\lambda^5} +O\left(\frac{q}{\lambda^6}\right)\\ &< (q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-64}{\lambda^4}+\frac{32q-112}{\lambda^5} +O\left(\frac{q}{\lambda^6}\right)\\ &=f(\lambda).\end{aligned}$$ This implies By Lemma [Lemma 12](#lem:compare){reference-type="ref" reference="lem:compare"}, we have $\lambda_{2,max}=\lambda_g<\lambda_f=\lambda_2(G_0).$ Contradiction! Claim 7 is proved. Thus, $G$ must be one of the following graphs. It suffices to show $G$ is not $G'_0$. In fact, for $G'_0$, we can calculate $F_1(\lambda)$, $F_2(\lambda)$, and $D(\lambda)$ as follows: $$\begin{aligned} F_1(\lambda)&= q + \frac{2q-2}{\lambda} + \frac{4q-4}{\lambda^2} + \frac{8q-8}{\lambda^3} + \frac{16q-12}{\lambda^4}+\frac{32q-14}{\lambda^5} + \frac{64q+5}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right),\\ F_2(\lambda) &= (q-1) + \frac{2q-4}{\lambda} + \frac{4q-8}{\lambda^2} + \frac{8q-16}{\lambda^3} + \frac{16q-28}{\lambda^4}+\frac{32q-46}{\lambda^5} + \frac{64q-59}{\lambda^6}+O\left(\frac{q}{\lambda^7}\right).\\ D(\lambda)&= \frac{2}{\lambda} + \frac{6}{\lambda^2} + \frac{17}{\lambda^3} + \frac{44}{\lambda^4}+\frac{112}{\lambda^5} +\frac{276}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right).\end{aligned}$$ Therefore, $$\begin{aligned} g(\lambda) &=\frac{1}{2} \left( (F_1(\lambda) +F_2(\lambda)) - \sqrt{ (F_1(\lambda) -F_2(\lambda))^2 + 4 D(\lambda)^2} \right)\\ &= \frac{1}{2} \left(2q-1 + \frac{4q-6}{\lambda} + \frac{8q-12}{\lambda^2} + \frac{16q-24}{\lambda^3} + \frac{32q-40}{\lambda^4}+\frac{64q-60}{\lambda^5} +\frac{128q-54}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right)\right.\\ &\hspace*{4mm} \left. -\sqrt{ \left(1+ \frac{2}{\lambda} + \frac{4}{\lambda^2} + \frac{8}{\lambda^3} + \frac{16}{\lambda^4}+\frac{32}{\lambda^5} +\frac{64}{\lambda^6}+O\left(\frac{q}{\lambda^7}\right)\right)^2 + 4\left(\frac{2}{\lambda} + \frac{6}{\lambda^2} + \frac{17}{\lambda^3} + \frac{44}{\lambda^4}+\frac{112}{\lambda^5} +\frac{276}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right)\right)^2 } \right) \\ &= (q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-68}{\lambda^4}+\frac{32q-122}{\lambda^5} +\frac{64q-244}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right)\\ &<(q-1) + \frac{2q-4}{\lambda} + \frac{4q-12}{\lambda^2} + \frac{8q-32}{\lambda^3} + \frac{16q-64}{\lambda^4}+\frac{32q-112}{\lambda^5} +\frac{64q-232}{\lambda^6} +O\left(\frac{q}{\lambda^7}\right)\\ &=f(\lambda).\end{aligned}$$ Thus, we have $\lambda_2(G_0')<\lambda_2(G_0)$. Thus $G_0$ is the unique extemal graph among all 2-connected outerplanar graphs on $n$ vertices. ◻ # Future directions In this paper, we have determined the leading order asymptotics of $\lambda_{k, max}$ for any fixed $k$ and also determined the precise extremal graphs for $k=2$. We make the following conjectures for the precise extremal graphs for $k=3$. **Conjecture 2**. *For $n$ sufficiently large, among all connected outerplanar graphs on $n=3q$ vertices, the graph maximizing $\lambda_3$ is unique and isomorphic to the following graph, denoted by $(K_1\vee P_{q-1})\!\!-\!\!(K_1\vee P_{q-1})\!\!-\!\!(K_1\vee P_{q-1})$, which is constructed by gluing three disjoint copies of the fan graph $K_1\vee P_{q-1}$ via edges connecting vertices of smallest degrees to largest degree between all three components.* **Conjecture 3**. *For $n$ sufficiently large, among all connected outerplanar graphs on $n=3q+1$ vertices, the graph maximizing $\lambda_3$ is unique and isomorphic to the following graph, denoted by $3(K_1\vee P_{q-1})\!\!-\!\!K_1$, which is constructed by gluing three disjoint copies of the fan graph $K_1\vee P_{q-1}$ to a $K_1$ via edges connecting any vertices (but maintaining outerplanarity) to that $K_1$.* Note that Conjecture [Conjecture 3](#conj3q+1){reference-type="ref" reference="conj3q+1"} is a special case of Conjecture [Conjecture 1](#conjkq+1){reference-type="ref" reference="conjkq+1"}. **Conjecture 4**. *For $n$ sufficiently large, among all connected outerplanar graphs on $n=3q+2$ vertices, the graph maximizing $\lambda_3$ is unique and isomorphic to the graph that maximizes $\lambda_3$ on $3(q+1)$ vertices after deleting a low degree vertex from one of the fans.* We can ask the following general question. Given a family $\cal F$ of graphs on $n$ vertices, which graph in $\cal F$ has the maximum $\lambda_k$? In particular, we are interested in planar graphs and $K_{s,t}$-minor free graphs. 10 J. Chen, The proof on the conjecture of extremal graphs for the $k$th eigenvalues of trees, *Linear Algebra Appl.* **426** (2007), 12--21. D. Cvetković and P. Rowlinson, The largest eigenvalue of a graph: A survey, *Linear Multilinear Algebra* **28** (1990), 3--33. D. Cvetković and S. Simić, The second largest eigenvalue of a graph (a survey). Algebra, logic and discrete mathematics (Niš 1995). *Filomat*, **9**, part 3 (1995), 449--472. J. Gao and X. Hou, The spectral radius of graphs without long cycles, *Linear Algebra Appl.*, **566** (2019), 17--33. W. Haemers, Interlacing eigenvalues and graphs, *Linear Algebra Appl.*, **226-228** (1995), 593-616. Y. Hong, Bounds of eigenvalues of graphs, *Discrete Math.* **123** (1993), 65--74. S. Li, W. Sun, Characterization of outerplanar graphs whose second eigenvalue is at most 1, *Results Math.*, **78**, no. 3 (2023), Paper No. 104, 19 pp. Z. Li, W. Linz, L. Lu, and Z. Wang, On the maximum spread of planar and outerplanar graphs, available at arXiv:2209.13776. H. Lin and B. Ning, A complete solution to the Cvetković--Rowlinson conjecture, *J. Graph Theory* **97(3)** (2021), 441--450. W. Linz, Improved lower bounds on the extrema of eigenvalues of graphs, *Graphs Combin.*, **39**, no. 4 (2023), 82. W. Linz, L. Lu, and Z. Wang, Maximum spread of $K_{2, t}$-minor-free graphs, *Linear Algebra Appl.*, **676**, (2023), 352--373. A. Neumaier, The second largest eigenvalue of a tree, *Linear Algebra Appl.*, **46** (1982), 9--25. V. Nikiforov, Bounds on graph eigenvalues.II, *Linear Algebra Appl.* **427** (2007), no.2--3, 183--189. V. Nikiforov, Extrema of graph eigenvalues, *Linear Algebra Appl.* **482** (2015), 158--190. V. Nikiforov, Some inequalities for the largest eigenvalue of a graph, *Combin. Probab. Comput.* **11** (2002), no.2, 179--189. V. Nikiforov, The spectral radius of graphs with no $K_{2, t}$-minor, *Linear Algebra Appl.* **531** (2017), 510--515. V. Nikiforov, The spectral radius of graphs without paths and cycles of specified length, *Linear Algebra Appl.* **432** (2010), no.9, 2243--2256 P. Rowlinson, On the index of certain outerplanar graphs, *Ars. Combin.*, **29** (1990), 221--225. D. Serre, *Matrices: Theory and Applications*, 2nd ed., Springer (2010). J. Shao, On the largest $k$th eigenvalues of trees, *Linear Algebra Appl.*, **221** (1995), 131--157. M. Tait, The Colin de Verdière parameter, excluded minors, and the spectral radius, *J. Combin Theory Series A* **166** (2019), 142--158. M. Tait, J. Tobin, Three conjectures in extremal spectral graph theory, *J. Combin Theory Series B*, **126** (2017), 137--161. B. Wang and M. Zhai, Proof of a conjecture on the spectral radius of $C_4$-free graphs, *Linear Algebra Appl.*, **437** (2012) no. 7, 1641--1647. Yuan, Hong, The $k$th largest eigenvalue of a tree, *Linear Algebra Appl.*, **73** (1986), 151--155. W. Yuan, B. Wang, and M. Zhai, On the spectral radii of graphs without given cycles, *Electron. J. Linear Algebra*, **23** (2012), 599--606. M. Zhai and H. Lin, Spectral extrema of $K_{s, t}$-minor-free graphs - On a conjecture of M. Tait, *J. Combin Theory Ser. B*, **157** (2022), 184--215. J. Zhou, C. Lin, and G. Hu, Spectral radius of Hamiltonian planar and outerplanar graphs, *Tsinghua Sci. Technol.*, **6(4)** (2001), 350--354. [^1]: University of South Carolina, Columbia, SC. (`[email protected]`). The author is partially supported by NSF DMS 2038080 grant. [^2]: University of North Carolina. (`[email protected]`). The author is partially supported by NSF DMS 2038080 grant through a summer REU program. [^3]: University of South Carolina. (`[email protected]`). The author is partially supported by NSF DMS 2038080 grant through a summer REU program. [^4]: University of South Carolina, Columbia, SC. (`[email protected]`). The author is partially supported by NSF DMS 2038080 grant. [^5]: University of South Carolina, Columbia, SC. (`[email protected]`). The author is partially supported by NSF DMS 2038080 grant.

arxiv_math

--- abstract: | This paper proposes a mathematical framework for dynamic pricing in an energy community to enable the provision of capacity limitation services to the distribution grid. In this framework, the energy community complies with a time-variant limit on its maximum power import from the distribution grid in exchange for grid tariff discounts. A bi-level optimization model is developed to implicitly coordinate the energy usage of prosumers within the community. In the upper-level problem, the community manager minimizes the total operational cost of the community based on reduced grid tariffs and power capacity limits by setting time-variant and prosumer-specific prices. In the lower-level problem, each prosumer subsequently adjusts their energy usage over a day to minimize their individual operational cost. This framework allows the community manager to maintain central economic market properties such as budget balance and individual rationality for prosumers. We show how the community benefits can be allocated to prosumers either in an equal or a proportional manner. The proposed model is eventually reformulated into a mixed integer second-order cone program and thereafter applied to a distribution grid case study. author: - "Bennevis Crowley, Jalal Kazempour, and Lesia Mitridati [^1] [^2]" bibliography: - bibtex/bib/references.bib title: Dynamic Pricing in an Energy Community Providing Capacity Limitation Services --- Dynamic pricing, energy community, capacity limitation services, grid tariffs, bi-level programming. # Introduction the green transition progresses and more stochastic renewable sources are integrated into the power system, the need for operational flexibility across all levels of the power grid increases. In particular, harnessing the operational flexibility that lies within the demand side can be of great value for multiple stakeholders in the power system [@SaezArmenteros2022Demand-SideEU]. In response to this growing need, the Danish Energy Agency recently released a set of new legislations, the so-called *Market Model 3.0*[@2021MarkedsmodelElnet; @Gade2022EcosystemSolution], aiming at developing flexible electricity markets, with a specific focus on facilitating the market participation of actors of all sizes. This includes allowing residential electricity users to provide flexibility services via flexible production and demand response. These legislations specifically highlight *energy communities* as a promising solution for harnessing and coordinating flexibility from residential electricity users. Energy communities can come in many shapes and sizes [@Capper2022Peer-to-peerModels], but one type is the so-called *citizen energy communities*, which is centered around flexible prosumers, who are consumers with local production (such as rooftop photovoltaic systems), located in close proximity to each other, and hereby connected to the distribution power grid through a joint feeder. Being able to coordinate these prosumers to achieve a common goal could be of great benefit to various power systems stakeholders, but particularly the distribution system operator (DSO). ## Local flexibility: Challenges and current practices One of the main barriers to the large-scale deployment of energy communities is the lack of efficient and practically straightforward coordination mechanisms among community members. The *explicit* control of flexible assets within the community could be a solution [@2021MarkedsmodelElnet], which is based on a direct agreement between prosumers and the so-called *community manager*. Examples of explicit control mechanisms are blackout contracts, local flexibility markets, special regulation services, and capacity limitation services. In the following, we discuss, among others, why this paper focuses on *capacity limitation services*. Blackout contracts are not a feasible long-term solution for a power system that aims to provide security of supply, especially for smaller consumers in a community, who may not be able to afford being subjected to a blackout. Therefore, solutions that circumvent blackouts are preferred. One alternative is local flexibility markets, which are in the implementation phase in some parts of Europe, such as the Piclo Flex Market [@2023PicloServices] in the UK and NorFlex [@Stlsbotn2023NODES2020-23] in southern Norway. These local markets allow for the submission of flexibility provision and purchasing bids, and are subsequently cleared by a market operator, potentially the DSO [@Prat2023Network-AwareMarkets; @Olivella-Rosell2018OptimizationResources]. However, the added burden on DSOs to submit and/or clear bids in these markets remains a major barrier to their implementation. Furthermore, special regulation services are already in use across different countries, such as Denmark, as outlined in [@2023IntroduktionSystemydelser], where down-regulation of renewables is used to manage congestion within bidding zones. However, an independent special regulation market does not exist as such at the moment. Capacity limitation services, which are the focus of this paper, are considered to be the most feasible flexibility contract opportunity by many stakeholders, and have as such been the subject of much research. In such a contract, a portfolio of flexible consumers or prosumers enters an agreement with the DSO to limit their power import from the grid at certain locations and for certain times, when requested, in exchange for some sort of payment or benefit, such as grid tariff discounts. ## Capacity limitation services: Status quo and open questions A major question that has been addressed in the current literature is how to successfully implement contracts, incentivizing flexible assets to provide capacity limitation services. In particular, [@Heinrich2021AServices] investigates how to design a market for a DSO to buy and price those services. Additionally, [@Ziras2020AAggregators] estimates the opportunity cost incurred by a portfolio of flexible assets as a consequence of providing capacity limitation services, which provides a basis to adequately price these flexibility contracts. As energy communities have the capability to access and control flexible assets from a variety of residential consumers and prosumers, they are well suited to enter capacity limitation contracts with a DSO. The energy community manager can take the burden of coordinating the individual community members and their flexible assets off the hand of the DSO while providing the required capacity limitation services. Besides, energy communities allow the community members to gain access to an additional revenue stream stemming from reduced grid tariffs or other benefits shared by the DSO. Therefore, this framework, if successfully implemented, can be a win-win game for both community members and for the DSO. Accordingly, this paper considers the case of an energy community coordinating individual flexible assets to provide flexibility services to the distribution grid, through capacity limitation contracts. A major question not yet addressed is how to control and coordinate individual assets within a community to deliver on a signed capacity limitation contract. As explicit control frameworks are limited by their scalability and the computation and information burden placed on individual community members, *implicit* control strategies may be preferred. Implicit control uses price signals that can vary over both *time* and *location*, so-called *dynamic pricing*, to provoke a reaction in the electrical demand or production of consumers and prosumers to meet the demand for flexibility. In the past, fixed-rate energy prices have been used for consumers throughout Europe, but with increasing price volatility many retailers are moving away from this, and moving to time-of-use prices and other pricing mechanisms of more dynamic nature. At this point, hourly time-differentiated prices are standard throughout Europe, and soon the pricing resolution on wholesale markets will even decrease to 15 minutes [@Fosse2022QuarterlyMarkets]. Furthermore some stakeholders, such as the Danish DSOs, add additional layers of time-differentiated tariffs to encourage consumers to consider the timing of their energy usage [@2023TarifferNetabonnement]. With the growing number of flexible prosumers, the design of efficient price signals will become more vital in harnessing demand-side flexibility. Already in 2010, [@Faruqui2010HouseholdExperiments] completed an analysis of household response programs which showed that dynamic price signals could elicit demand response ranging from 3% (when using simple time-of-use prices) to 44% (when accompanied with enabling assets and more complex time signals). Current research efforts are now investigating how to best utilize the possibilities of dynamic pricing to elicit desirable behaviour from residential end-users. For example, [@Grimm2021OptimalOptimization] studies the optimal design of retailer-prosumer tariffs. In addition, [@Askeland2023AFlexibility] investigates the optimal design of grid tariffs by the DSO to encourage flexible energy consumption. Both [@Grimm2021OptimalOptimization] and [@Askeland2023AFlexibility] look at already established pricing mechanisms, such as price-of-use and capacity based tariffs. Another approach to designing optimal tariffs is to allow for additional degrees of freedom, such as time and spatial differentiation. For example, [@Papavasiliou2018AnalysisPrices] introduces distribution locational marginal pricing, which differentiates prices in a distribution network by node. ## Contributions {#subsec:RQs} Our first contribution is to design a mathematical framework to facilitate the coordination between end-users in an energy community and a DSO, enabling the provision of capacity limitation services. In this setup, the community manager implicitly coordinates the energy use of the community members, via time- and space-differentiated prices, providing a contracted flexibility service at the lowest possible cost for the community. We address how to optimally set dynamic prices to elicit the desired behaviour from individual community members to fulfill a flexibility contract with a DSO, while ensuring desirable economic properties including individual rationality and budget balance. To answer this question, this paper proposes a bi-level optimization program, representing interaction between the community manager and members, in which the community manager sets time- and space-dynamic prices. This mathematical model provides a basis to quantify the potential of a community to offer flexibility services to the distribution grid. As our second contribution, we show how the community manager can ensure that members receive a fair portion of the community benefits. For this purpose, we propose two regularization terms that can be added to the objective function of the community manager, incentivizing the manager to consider distribution of benefit when finding the optimal prices for the community members. These regularisation terms are compared with respect to how much they change the benefit earned by all community members. Lastly, we explore the impact of parameters of the agreed upon capacity limitation contract on the benefit derived by the community. We conduct a numerical analysis of a variety of possible combinations of capacity limitation variation and cost savings offered by the DSO to identify any possible downsides and benefits associated with various parameters stipulated in a capacity limitation contract. This analysis provides valuable insights for both energy communities and DSOs regarding what kind of parameters could be mutually beneficial when designing such a contract in the future. # Proposed Framework {#sec:framework} This section provides details about the proposed coordination framework between the consumers and prosumers inside an energy community and a DSO for the provision of capacity limitation services, via a dynamic pricing scheme. ## Coordination between the DSO and community manager {#sec:coordination} To provide further clarity, an illustrative example of an energy community and its connection to the grid is provided, as depicted in Fig. [\[fig:example\]](#fig:example){reference-type="ref" reference="fig:example"}. Consider an energy community (purple box) with 7 prosumers all located at their own respective node (green circles) in the distribution grid, connected to each other through a radial network within the community. Each prosumer may own local production facilities and/or an energy storage system. The whole community is connected to the DSO-operated upstream distribution grid through a single feeder at the reference node, node $0$ (white circle). The amount of imported/exported power from/to the upstream grid by the community at time $t$ is denoted by $p^{\rm{im}}_t$ / $p^{\rm{ex}}_t$. The community imports power from the upstream grid if the local production of 7 prosumers within the community does not suffice to meet the total community demand (positive residual demand). Otherwise, if the local power production exceeds the total community demand at any point in time (negative residual demand), it will be exported to the upstream grid. Typically, a residential prosumer $i=\{1, 2, ..., 7\}$ pays/is paid not only for their energy consumed/produced $p_{it}$ based on electricity prices (e.g., day-ahead market prices), but also for their usage of grid infrastructures based on grid tariffs. The DSO, sometimes together with the transmission system operator (TSO), sets and collects these grid tariffs. In the proposed framework, the DSO leverages these grid tariffs to incentivize residential prosumers to provide flexibility services. In particular, the DSO offers capacity limitation contracts to a group of prosumers organized in an energy community. The whole community receives a discount factor of $0 \leq \beta_t^{\rm{DSO}} \leq 1$ on the grid tariffs for internal power flows, i.e., downstream from node $0$, as long as the amount of imported/exported power at node $0$ does not exceed a given time-variant cap, say $\bar{P}_t^{\rm{DSO}}$, for each time $t$ of the next day. In addition, the DSO sets a penalty rate $\alpha_t^{\rm{DSO}}$ for violating the capacity limitations. This service could be crucial for the DSO to avoid congestion in the upstream grid. One can see grid tariff discounts as a pragmatic strategy for the DSO to reduce and/or delay the need for grid infrastructure expansion. This type of bilateral agreement between DSOs and energy communities has been identified by [@2021MarkedsmodelElnet] as a future local flexibility solution in the Danish market context. =\[inner sep=0pt\] =\[\|-\>, line width = 1pt\] =\[-, draw=black, fill=rgb,255: red,153; green,255; blue,255\] =\[-\>,line width = 0.6pt\] =\[-, dashed, fill=rgb,255: red,191; green,191; blue,191\] =\[-, fill=rgb,255: red,177; green,156; blue,217\] =\[-, fill=rgb,255: red,153; green,255; blue,153\] =\[-, line width = 1pt\] =\[circle, fill = none, draw = black, minimum size = 0.75cm, line width = 1pt\] =\[circle, fill = white, draw = black, minimum size = 0.75cm\] =\[circle, fill = rgb,255: red,153; green,255; blue,153, draw = black, minimum size = 0.75cm\] =\[-\>, draw = blue, line width = 2pt\] =\[-, draw = blue, line width = 1pt\] ## Coordination between community manager and members The community manager aims to coordinate the energy consumption/production of the community members in order to meet the agreed upon capacity limitation with the DSO. This way, the manager takes advantage of the grid tariff discounts in a way that benefits the community as a whole. The proposed framework achieves this by dynamically sending price signals $x_{it}$, covering both electricity rates and grid tariffs, for each community member $i$ and time $t$ of the next day. Each community member $i$ independently optimizes their own electricity consumption/production $p_{it}$ based on the price signals $x_{it}$, in order to minimize their own energy costs. The timeline of decision-making and information exchange among stakeholders is illustrated in Fig. [\[fig:illustration\]](#fig:illustration){reference-type="ref" reference="fig:illustration"} and summarized as follows: 1. The day-ahead electricity market prices $\lambda^{\rm{spot}}_t$ for each hour of the next day are released by the wholesale market operator (e.g., at 14:00 by Nord Pool). 2. The DSO identifies the expected needs and costs of congestion management services in the distribution grid and calculates the required capacity limitation. 3. The DSO communicates the needed capacity limitations $\bar{P}_t^{\rm{DSO}}$ for each hour $t$ of the next day to the community manager, as well as penalties $\alpha_t^{\rm{DSO}}$ for violating these limitations and the offered grid tariff reduction $\beta_t^{\rm{DSO}}$ for internal power flows within the community. 4. Given the DSO's offer in Step 3, the community manager sets price signals $x_{it}$ for each community member $i$ and hour $t$ of the next day. 5. The community manager communicates the price signals $x_{it}$ to all community members $i$. 6. Given the price signals $x_{it}$, each community member $i$ optimizes their own energy consumption/production profile $p_{it}$ for the next day. The mechanism by which the DSO sets these capacity limitation contracts in Steps 2-3 (i.e., $\bar{P}_t^{\rm{DSO}}, \alpha_t^{\rm{DSO}}, \beta_t^{\rm{DSO}}$) is outside the scope of this paper. The main focus here is to provide a mathematical model for the community manager's dynamic pricing problem in Step 4, subject to a given contract with the DSO, as will be further detailed in Section [3](#sec:models){reference-type="ref" reference="sec:models"}. # Proposed Mathematical Model {#sec:models} First, we introduce the game-theoretical background for the proposed dynamic pricing scheme. Second, we provide a detailed formulation of the proposed bi-level program representing the community manager's problem. Third, we discuss on the distribution of community benefits among prosumers. Finally, we explain the reformulation methods used to ensure efficient solving of this mathematical optimization problem. ## Proposed modeling approach: A Stackelberg game {#sec:stackelberg} A Stackelberg game classically consists of a single leading agent, who needs to make an initial decision which impacts the subsequent decisions of the following agent or agents, whose actions in turn impact the resulting outcome for the leader. This structure of game is represented in Steps 4-6 of the framework described in Section [2](#sec:framework){reference-type="ref" reference="sec:framework"}. The leader (community manager) optimizes their own decisions (price signals) first (Step 4), which then impact (Step 5) the optimal decisions (consumption/production) of multiple followers (community members). In turn, the reaction (Step 6) of the followers impacts the optimal cost of the leader, through the total power imported/exported from the upstream grid. Therefore, the community manager must anticipate the reaction of the community members in order to set the optimal price signals that will elicit the desired reaction. To model this setup, we propose a formulation of the aforementioned community manager's problem as a bi-level program, in which a set of lower-level optimization problems, representing the optimal reaction of the community members, is embedded as constraints of the upper-level optimization problem, representing the optimal decisions of the community manager. A compact formulation of this bi-level program is given in [\[compact1\]](#compact1){reference-type="eqref" reference="compact1"}, including the upper-level problem [\[compact2\]](#compact2){reference-type="eqref" reference="compact2"}-[\[compact3\]](#compact3){reference-type="eqref" reference="compact3"}, and the set of lower-level problems [\[compact4\]](#compact4){reference-type="eqref" reference="compact4"}-[\[compact6\]](#compact6){reference-type="eqref" reference="compact6"}, one per community member $i$, as follows: [\[compact1\]]{#compact1 label="compact1"} $$\begin{aligned} {1} &\underset{\underset{\textbf{p}^{\rm{im}},\textbf{p}^{\rm{ex}}}{\textbf{x}_i,}}{\operatorname{min}} \text{cost}^{\rm{C}}(\textbf{x}_i,\textbf{p}^{\rm{im}} ,\textbf{p}^{\rm{ex}},\textbf{p}_i) \label{compact2} \\ %%%%%%% & \hspace{0.25cm} \text{s.t.} \ g^{\rm{C}}(\textbf{x}_i,\textbf{p}^{\rm{im}} ,\textbf{p}^{\rm{ex}},\textbf{p}_i) = 0 \\ & \hspace{0.25cm} \hspace{0.54cm} h^{\rm{C}}(\textbf{x}_i,\textbf{p}^{\rm{im}} ,\textbf{p}^{\rm{ex}},\textbf{p}_i) \leq 0 \label{compact3} \\ %%%%%% & \hspace{0.25cm} \hspace{0.54cm} \textbf{p}_i \in \underset{\textbf{p}_i}{\text{argmin}} \ \Big\{ \text{cost}_i^{\rm{P}}(\textbf{x}_i,\textbf{p}_i) \label{compact4}\\ & \hspace{0.25cm} \hspace{0.54cm} \hspace{2.12cm} \text{s.t.} \ g_i^{\rm{P}}(\textbf{p}_i) = 0 \label{compact5}\\ & \hspace{0.25cm} \hspace{0.54cm} \hspace{2.12cm} \hspace{0.52cm} h_i^{\rm{P}}(\textbf{p}_i) \leq 0 \Big\} \ \forall i \in \mathcal{I}. \ \label{compact6}\end{aligned}$$ This bi-level program is solved by the community manager, given the day-ahead electricity market prices $\lambda^{\rm{spot}}_t$, the grid tariffs $\textbf{Y}$, as well as the DSO capacity limitation contract parameters ($\bar{P}_t^{\rm{DSO}}, \beta_t^{\rm{DSO}}, \alpha_t^{\rm{DSO}}$). In the upper-level problem, the community manager aims at finding the optimal vectors of price signals $\textbf{x}_i$ which minimize the total community cost, i.e., $\text{cost}^{\rm{C}}(\textbf{x}_i,\textbf{p}^{\rm{im}} ,\textbf{p}^{\rm{ex}},\textbf{p}_i)$ in [\[compact2\]](#compact2){reference-type="eqref" reference="compact2"}. This objective function is subject to community-wide constraints related to limitations on the imported and exported power at node $0$, $\textbf{p}^{\rm{im}}$ and $\textbf{p}^{\rm{ex}}$, feasibility of power flows within the community, and desirable economic properties such as budget balance and individual rationality. Additionally, the community manager anticipates how each community member $i$ optimizes their consumption/production profile $\textbf{p}_i$ in response to the price signals $\textbf{x}_i$. This optimal response of the community members is modeled as the solutions to a set of lower-level optimization problems, one per community member $i$, which are embedded as constraints in the upper-level problem. In each lower-level problem, a given community member $i$ minimizes their daily energy cost, i.e., ${\rm{cost}}_i^{\rm{P}}(.)$, subject to a set of internal constraints [\[compact5\]](#compact5){reference-type="eqref" reference="compact5"}-[\[compact6\]](#compact6){reference-type="eqref" reference="compact6"}, for the given price signals $\textbf{x}_i$ (treated as parameters in the lower-level problems). The implementation of this dynamic pricing scheme requires the community manager to have perfect knowledge of the community members' parameters, which might raise privacy and communication challenges in practice. Additionally, in the case of multiplicity of the solutions to the lower-level problems, the upper-level problem will choose an "optimistic\" solution, which minimizes its own objective. However, in practice, the community manager cannot guarantee which optimal solutions individual members would choose. Therefore, we see the proposed framework as an *ideal benchmark*, which provides an upper-bound on how effective a dynamic pricing scheme can be to incentivize and coordinate prosumers in an energy community to procure capacity limitation services to the DSO. ## Detailed mathematical formulation {#subsec:detailed models} Throughout the paper, lower-case symbols are used for variables, and upper-case symbols for parameters. ### Upper-level problem (dynamic pricing) The full upper-level problem is described in [\[eqs:upper level\]](#eqs:upper level){reference-type="eqref" reference="eqs:upper level"} and makes use of the variable set $\Omega = \{x_{it}$, $p^{\rm{im}}_t$, $p^{\rm{ex}}_t$, $p^{\rm{pen}}_t$, $q^{\rm{im}}_t$, $q^{\rm{ex}}_t$, $f^{\rm{q}}_{lt}$, $f^{\rm{p}}_{lt}$, $u_{nt}$, $\bar{x}$, $w^+_i$, $w^-_i$, $v^+$, $v^-$}. In addition, $\mathcal{T}$ and $\mathcal{I}$ are sets for times and prosumers, respectively. The upper-level objective function [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"}, minimizing the total community cost, reads as [\[eqs:upper level\]]{#eqs:upper level label="eqs:upper level"} $$\begin{aligned} \nonumber \underset{\Omega}{\operatorname{min}} \quad \sum_{t \in \mathcal{T}} &\Bigg[ p^{\rm{im}}_t \Big(\lambda^{\rm{spot}}_t + Y^{\rm{im}}_t\Big) - p^{\rm{ex}}_t \Big(\lambda^{\rm{spot}}_t - Y^{\rm{ex}}_t\Big) \\ \nonumber & + (1-\beta^{\rm{DSO}}_t) Y^{\rm{im}}_t \Big( \sum_{i \in \mathcal{I}} p^+_{it} - p^{\rm{im}}_t \Big) \\ %\sum_{i \in \mathcal{I}}\Big(Y^{\rm{im}}_t p^+_{it} + Y^{\rm{ex}}_t p^-_{it}\Big)\Big(1-\phi_i\Big) \\ & + \alpha^{\rm{DSO}}_t p^{\rm{pen}}_t + \alpha^{\rm{shed}} d^{\, \rm{shed}}_{it} \Bigg] + \rho \bar{x}^2. \label{eq:upper_obj}\end{aligned}$$ The first line corresponds to energy trades and grid tariffs at node $0$. The community pays (is paid) for the imported power $p^{\rm{im}}_t$ from (exported power $p^{\rm{ex}}_t$ to) the upstream grid at given market prices $\lambda^{\rm{spot}}_t$. Additionally, the community pays grid tariffs that are set by the DSO for both import ($Y^{\rm{im}}_t$) and export ($Y^{\rm{ex}}_t$) at node $0$, i.e., for the usage of the feeder between the community and the upstream grid. The second line of [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"} pertains to discounted grid tariffs for internal power flow across the radial grid within the community. The provided DSO discount is implemented by the coefficient $\beta^{\rm{DSO}}_t$, whose value lies between zero and one. Therefore, $(1-\beta^{\rm{DSO}}_t)Y^{\rm{im}}_t$ is the discounted import tariff rate at time $t$ for the usage of lines within the community. Symbols $p^+_{it}$ and $p^-_{it}$ denote the power purchase and sale of prosumer $i$ at time $t$. They are variables to be determined by the prosumer in the lower level. Assuming a lossless linear representation of power flow across the radial system that will be discussed later, $p^+_{it} - p^{\rm{im}}_t$ represents the difference between total purchased power and power imported from the grid, i.e., the internal power flow of the community. The third line of [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"} has three cost elements. The first term accounts for the DSO penalty, such that the parameter $\alpha^{\rm{DSO}}_t$ is the penalty rate set by the DSO for every 1 kW power in the feeder at time $t$ that exceeds the cap $\bar{P}_t^{\rm{DSO}}$, where the variable $p^{\rm{pen}}_t$ is the amount of exceeded power. The second term penalizes any possible load shedding $d^{\, \rm{shed}}_{it}$ in the lower-level problem by prosumer $i$ at time $t$ at a predetermined value of lost load $\alpha^{\rm{shed}}$. This avoids the community manager choosing a solution that prioritizes load shedding over other solutions. Lastly, the third term is a quadratic regularization term, whose purpose is to converge to a unique solution for the price setting. As there is no upper bound for the dynamic prices $x_{it}$, there could be multiple solutions for $x_{it}$, leading to the same value for the objective function of the community. This regularization term prioritizes solutions with lowest maximum price across all prosumers, minimizing the squared maximum price observed $\bar{x}$, multiplied by a small positive weight $\rho$. The upper-level constraints, summarized in [\[compact3\]](#compact3){reference-type="eqref" reference="compact3"}, include community-level constraints [\[eq:budget_balance\]](#eq:budget_balance){reference-type="eqref" reference="eq:budget_balance"} to [\[eq:positive variables\]](#eq:positive variables){reference-type="eqref" reference="eq:positive variables"}. Constraint [\[eq:budget_balance\]](#eq:budget_balance){reference-type="eqref" reference="eq:budget_balance"} ensures that the energy community operator is budget balanced, meaning neither taking a loss nor making a profit: $$\begin{aligned} \nonumber & \sum_{t \in \mathcal{T}} \sum_{i \in \mathcal{I}} \Big( p^+_{it} - p^-_{it} \Big) x_{it} = \sum_{t \in \mathcal{T}} \Bigg[ p^{\rm{im}}_t \Big(\lambda^{\rm{spot}}_t + Y^{\rm{im}}_t\Big) \\ \nonumber & - p^{\rm{ex}}_t \Big(\lambda^{\rm{spot}}_t - Y^{\rm{ex}}_t\Big) + (1-\beta^{\rm{DSO}}_t) Y^{\rm{im}}_t \Big( \sum_{i \in \mathcal{I}} p^+_{it} - p^{\rm{im}}_t \Big) \\ & + \alpha^{\rm{DSO}}_t p^{\rm{pen}}_t \Bigg], \label{eq:budget_balance}\\end{aligned}$$ where the left-hand side is the total payment of community members to the community manager during the day, whereas the right-hand side is identical to [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"} without the regularization, and corresponds to the total payment of the community operator for the electricity and network infrastructure usage. Constraint [\[eq:individual_rationality\]](#eq:individual_rationality){reference-type="eqref" reference="eq:individual_rationality"} guarantees individual rationality for every prosumer, meaning they are better off staying in the community rather than individually trading directly with the grid: $$\begin{aligned} & \sum_{t \in \mathcal{T}} \Big( p^+_{it} - p^-_{it} \Big) x_{it} = C^{\, \rm{ext}}_i + w^+_i - w^-_i \quad \forall i \in \mathcal{I}, \label{eq:individual_rationality}\end{aligned}$$ where $w^+_{i},w^-_{i} \geq 0 \ \forall i \in \mathcal{I}$. The left-hand side calculates the total payment of prosumer $i$ as a community member under the dynamic pricing scheme. The right-hand side includes the pre-calculated value $C^{\, \rm{ext}}_i$, corresponding to the payment of prosumer $i$ in a case under which the prosumer is not part of the community. If so, the prosumer pays the original grid tariffs without any discount and a proportional share of the penalty that would result from all prosumers optimizing their energy costs in an uncoordinated manner, all included in $C^{\, \rm{ext}}_i$. The right-hand side also contains two non-negative slack variables $w^+_i$ and $w^-_i$ to track how much better or worse off each prosumer is individually. These variables will later be used for the formulation of benefit distribution mechanisms among community members in Section [3.3](#sec:distribution){reference-type="ref" reference="sec:distribution"}. In the case $w^-_i$ takes a positive value, it shows the underlying prosumer earns by being in the community, while a positive value of $w^+_i$ indicates the prosumer loses in the community. One may interpret $\sum_{i \in \mathcal{I}}w^-_i$ as the total benefit of forming the community (compared to a case every prosumer acts individually), while $\sum_{i \in \mathcal{I}} w^+_i$ is the total cost in case forming the community is not beneficial. It is desirable that if a community member loses, there is no other member that earns. This is enforced by $$\begin{aligned} & \sum_{i \in \mathcal{I}} w^+_i \leq v^+, \ \sum_{i \in \mathcal{I}} w^-_i \leq v^-, \ \{ v^+, v^- \} \in \text{SOS1}, \label{eq:slack SOS1}\end{aligned}$$ where $v^+$ and $v^-$ are special order set of type 1 (SOS1) variables. By [\[eq:slack SOS1\]](#eq:slack SOS1){reference-type="eqref" reference="eq:slack SOS1"}, the total community benefit $\sum_{i \in \mathcal{I}}w^-_i$ and the total cost $\sum_{i \in \mathcal{I}} w^+_i$ cannot take positive values at the same time, i.e., at least one of these two terms should be zero. Constraint [\[eq:penalty\]](#eq:penalty){reference-type="eqref" reference="eq:penalty"} restricts the amount of power $p^{\rm{pen}}_t$ imported by the community from the upstream grid, which is beyond the capacity limit $\bar{P}^{\rm{DSO}}_t$ imposed by the DSO: $$\begin{aligned} 0 \leq p^{\rm{pen}}_t &\geq p^{\rm{im}}_t - \bar{P}^{\rm{DSO}}_t \quad \forall t \in \mathcal{T}. \label{eq:penalty}\ \end{aligned}$$ To encourage local production injection to the grid, we do not penalize the exported power if it goes beyond the cap. Constraint [\[eq:ul_reg\]](#eq:ul_reg){reference-type="eqref" reference="eq:ul_reg"} determines the maximum price $\bar{x}$ among prosumers and times, used for the regularization while also ensuring that no negative prices are set within the community: $$\begin{aligned} 0 \leq x_{it} &\leq \bar{x} \quad \forall i \in \mathcal{I}, \ t \in \mathcal{T}. \label{eq:ul_reg}\\end{aligned}$$ The remaining upper-level constraints [\[eq:active power balance\]](#eq:active power balance){reference-type="eqref" reference="eq:active power balance"}-[\[eq:positive variables\]](#eq:positive variables){reference-type="eqref" reference="eq:positive variables"} are power flow constraints within the community, following the lossless Linearized Distribution Flow (LinDistFlow) model[^3]. These constraints allow the community manager to ensure that the power flows at any point in time are feasible given the grid constraints of the energy community. Constraints [\[eq:active power balance\]](#eq:active power balance){reference-type="eqref" reference="eq:active power balance"} and [\[eq:reactive power balance\]](#eq:reactive power balance){reference-type="eqref" reference="eq:reactive power balance"} ensure that the total amount of active power and reactive power imported to or exported from the community at time $t$ is equal to the active power flow ($f^{\rm{p}}_{(n=0)t}$) or reactive power flow ($f^{\rm{q}}_{(n=0)t}$), to the reference node $n=0$, i.e., $$\begin{aligned} p^{\rm{im}}_t - p^{\rm{ex}}_t &= f^{\rm{p}}_{(n=0)t} \quad \forall t \in \mathcal{T} \label{eq:active power balance}\ \\ q^{\rm{im}}_t - q^{\rm{ex}}_t &= f^{\rm{q}}_{(n=0)t} \quad \forall t \in \mathcal{T}. \label{eq:reactive power balance}\ %q^{\rm{im}}_t - q^{\rm{ex}}_t &= \sum_{n \in \mathcal{N}}\left( q^+_{i_nt} - q^-_{i_nt} \right) \quad \forall t \in \mathcal{T}. \end{aligned}$$ Constraint [\[eq:active power flow balance\]](#eq:active power flow balance){reference-type="eqref" reference="eq:active power flow balance"} calculates active power flow $f^{\rm{p}}_{nt}$ to every node $n$ except the reference node 0 at time $t$. Note that the set $i \in \mathcal{L}_n$ indicates prosumers $i$ located at node $n$, whereas the set $m \in \mathcal{D}_n$ refers to nodes $m$ downstream from node $n$. Constraint [\[eq:reactive power flow balance\]](#eq:reactive power flow balance){reference-type="eqref" reference="eq:reactive power flow balance"} is similar but for the reactive power flow $f^{\rm{q}}_{nt}$: $$\begin{aligned} &f^{\rm{p}}_{nt} = \sum_{i \in \mathcal{L}_n}\Big( p^+_{it} - p^-_{it} \Big) + \sum_{m \in \mathcal{D}_n} f^{\rm{p}}_{mt} \notag\\ &\hspace{45mm}\forall t \in \mathcal{T}, \ n \in \mathcal{N}\setminus 0 \label{eq:active power flow balance}\\ &f^{\rm{q}}_{nt} = \sum_{i \in \mathcal{L}_n}\Big( q^+_{it} - q^-_{it} \Big) + \sum_{m \in \mathcal{D}_n} f^{\rm{q}}_{mt} \notag\\ &\hspace{45mm}\forall t \in \mathcal{T}, \ n \in \mathcal{N}\setminus 0. \label{eq:reactive power flow balance}\\end{aligned}$$ Constraint [\[eq:line constraint\]](#eq:line constraint){reference-type="eqref" reference="eq:line constraint"} enforces the apparent power flow capacity $\overline{S}_n$ for the line that connects node $n$ to the upstream node in the radial distribution grid of the community: $$\begin{aligned} &(f^{\rm{p}}_{nt})^2 + (f^{\rm{q}}_{nt})^2 \leq \overline{S}_n \quad &&\forall t \in \mathcal{T}, \ n \in \mathcal{N}\setminus 0, \label{eq:line constraint}\ \end{aligned}$$ which is a second-order cone constraint. Constraints [\[eq:base node\]](#eq:base node){reference-type="eqref" reference="eq:base node"}-[\[eq:voltage limits\]](#eq:voltage limits){reference-type="eqref" reference="eq:voltage limits"} bind nodal voltage magnitudes as $$\begin{aligned} &u_{(n=0)t} = 1 \quad \forall t \in \mathcal{T} \label{eq:base node}\\ &u_{nt} = \sum_{m \in \mathcal{U}_{n}} u_{mt} - 2 \Big( f^{\rm{p}}_{nt}R_n + f^{\rm{q}}_{nt}X_n\Big) \quad \notag\\ &\hspace{40mm} \forall t \in \mathcal{T}, \ n \in \mathcal{N}\setminus 0 \label{eq:voltage}\\ & \underbar{$U$} \leq u_{nt} \leq \overline{U} \quad \forall t \in \mathcal{T}, n \in \mathcal{N}\setminus 0, \label{eq:voltage limits}\ \end{aligned}$$ where the squared voltage variable is replaced by the auxiliary variable $u_{nt}$ for modeling convenience. Constraint [\[eq:base node\]](#eq:base node){reference-type="eqref" reference="eq:base node"} sets voltage magnitude of the reference node $n=0$ to 1 per-unit. Constraint [\[eq:voltage\]](#eq:voltage){reference-type="eqref" reference="eq:voltage"} tracks the voltage drop throughout the nodes within the community, taking into account resistance and reactance parameters $R_n$ and $X_n$ for the line connecting node $n$ to the upstream one. Note that the set $m \in \mathcal{U}_{n}$ indicates all upstream nodes from node $n$. Finally, [\[eq:base node\]](#eq:base node){reference-type="eqref" reference="eq:base node"} constrains nodal voltage magnitudes to lie within $\underbar{$U$}$ and $\overline{U}$. Constraints [\[eq:grid active power limit\]](#eq:grid active power limit){reference-type="eqref" reference="eq:grid active power limit"} and [\[eq:grid reactive power limit\]](#eq:grid reactive power limit){reference-type="eqref" reference="eq:grid reactive power limit"} enforce the physical capacity for the active and reactive power trade at the interface between the community and the upstream grid: $$\begin{aligned} &p^{\rm{im}}_t,p^{\rm{ex}}_t \leq \overline{P}^{\rm{grid}} \quad &&\forall t \in \mathcal{T} \label{eq:grid active power limit}\\ &q^{\rm{im}}_t,q^{\rm{ex}}_t \leq \overline{Q}^{\rm{grid}} \quad &&\forall t \in \mathcal{T}. \label{eq:grid reactive power limit}\ \end{aligned}$$ Finally, [\[eq:positive variables\]](#eq:positive variables){reference-type="eqref" reference="eq:positive variables"} declares the non-negativity conditions: $$\begin{aligned} &p^{\rm{im}}_t,p^{\rm{ex}}_t,q^{\rm{im}}_t,q^{\rm{ex}}_t \geq 0 \quad &&\forall t \in \mathcal{T}. \label{eq:positive variables}\\end{aligned}$$ ### Lower-level problem (optimal electricity dispatch) {#subsec:lower level} We consider multiple prosumers in the energy community with flexible production, from rooftop photovoltaic panels and batteries, and/or inflexible consumption. The detailed formulation of the lower-level problems[^4] representing the individual optimal dispatch problem of each prosumer $i$ is given by [\[eqs:lower level\]](#eqs:lower level){reference-type="eqref" reference="eqs:lower level"}. The primal variable set for this problem is given by $\Phi_{i} = \{ p^{+}_{it},p^{-}_{it},q^{+}_{it},q^{-}_{it},p^{\rm{ch}}_{it},p^{\rm{dis}}_{it},e_{it},d^{\, \rm{shed}}_{it}\}$. Additionally, dual variables $\lambda_i^{(.)}$ and $\mu_i^{(.)}$ appear alongside their corresponding constraint as these are necessary when inserting the Karush-Kuhn-Tucker optimality conditions of the lower level into the upper-level optimization problem. Recall that $x_{it}$ is a parameter in the lower-level problem. The objective function of the lower-level problem, minimizing the cost of every prosumer $i$, writes as [\[eqs:lower level\]]{#eqs:lower level label="eqs:lower level"} $$\begin{aligned} \underset{\Phi_i}{\operatorname{min}} \ \sum_{t \in \mathcal{T}} \Big[x_{it} \Big( p^+_{it} - p^-_{it} \Big) + \alpha^{\rm{shed}} d^{\, \rm{shed}}_{it}\Big], \label{eq:lower obj} \end{aligned}$$ where $x_{it} \Big( p^+_{it} - p^-_{it} \Big)$ represents the payment that the prosumer must make to the community manager for their energy use, whereas $\alpha^{\rm{shed}} d^{\, \rm{shed}}_{it}$ is the potential load shedding cost. Without loss of generality, we consider identical value of lost load $\alpha^{\rm{shed}}$ for all prosumers. Constraint [\[eq:ll power balance\]](#eq:ll power balance){reference-type="eqref" reference="eq:ll power balance"} is the power balance equation for each individual prosumer $i$: $$\begin{aligned} & p^+_{it} - p^-_{it} + PV_{it} - D_{it}\notag \\ & \quad +d^{\, \rm{shed}}_{it}- p^{\rm{ch}}_{it} + p^{\rm{dis}}_{it} = 0 && \forall t \in \mathcal{T} &&: \lambda^{(1)}_{it}, \label{eq:ll power balance} \end{aligned}$$ where $PV_{it}$ and $D_{it}$ are deterministic solar production and demand forecasts, respectively. Each prosumer can choose to import or export power ($p^+_{it}$ and $p^-_{it}$, respectively), charge or discharge their battery ($p^{\rm{ch}}_{it}$ and $p^{\rm{dis}}_{it}$), or shed power $d^{\, \rm{shed}}_{it}$. Given battery charging and discharging efficiencies $\eta^{\rm{ch}}_i$ and $\eta^{\rm{dis}}_i$, [\[eq:storage_1\]](#eq:storage_1){reference-type="eqref" reference="eq:storage_1"} ensures that the start and end level of the battery are the same, whereas [\[eq:storage_all\]](#eq:storage_all){reference-type="eqref" reference="eq:storage_all"} is responsible for tracking the state of charge over the course of the day through the variable $e_{it}$: $$\begin{aligned} & e_{i(t=1)} = e_{i(t=24)} + \eta^{\rm{ch}}_i p^{\rm{ch}}_{i(t=1)} - \eta^{\rm{dis}}_i p^{\rm{dis}}_{i(t=1)} &:\lambda^{(2)}_{i} \label{eq:storage_1} \\ & e_{it} = e_{i(t-1)} + \eta^{\rm{ch}}_i p^{\rm{ch}}_{it} - \eta^{\rm{dis}}_i p^{\rm{dis}}_{it} \quad \forall t \in \mathcal{T}\setminus 1 &:\lambda^{(3)}_{it}. \label{eq:storage_all} \end{aligned}$$ Assuming the reactive power demand is determined by a given relationship between the active and reactive power at the prosumers node, we enforce: $$\begin{aligned} & q^+_{it} = \sigma_i p^+_{it} && \forall t \in \mathcal{T} &&: \lambda^{(4)}_{it} \label{eq:PQ positive} \\ & q^-_{it} = \sigma_i p^-_{it} && \forall t \in \mathcal{T} &&: \lambda^{(5)}_{it}, \label{eq:PQ negative} \end{aligned}$$ where $\sigma_i$ is given. We declare non-negativity conditions as $$\begin{aligned} & p^+_{it}, p^-_{it}, q^+_{it}, q^-_{it}, p^{\rm{ch}}_{it}, p^{\rm{dis}}_{it}, e_{it}, d^{\, \rm{shed}}_{it} \geq 0 \ \ \forall t \in \mathcal{T} \notag\\ & \ \ \quad: \mu^{(1)}_{it}, \mu^{(2)}_{it}, \mu^{(3)}_{it}, \mu^{(4)}_{it}, \mu^{(5)}_{it}, \mu^{(6)}_{it},\mu^{(7)}_{it}, \mu^{(8)}_{it}. \label{eq:lower bound nobat} \end{aligned}$$ Finally, we enforce charging and discharging capacity $\bar{P}^{\rm{bat}}_i$ of the battery, energy storage capacity $\bar{E}_{i}$ of the battery, and upper limit of load curtailment: $$\begin{aligned} & p^{\rm{ch}}_{it}, p^{\rm{dis}}_{it} \leq \bar{P}^{\rm{bat}}_i && \forall t \in \mathcal{T} &&: \mu^{(9)}_{it}, \mu^{(10)}_{it} \label{eq:bat_variables_upper1}\\ & e_{it} \leq \bar{E}_{i} && \forall t \in \mathcal{T} &&: \mu^{(11)}_{it} \label{eq:bat_variables_upper2}\\ & d^{\, \rm{shed}}_{it} \leq D_{it} && \forall t \in \mathcal{T} &&: \mu^{(12)}_{it}. \label{eq:upper bound d} %\\ %& && && \hspace{2.17cm} \nonumber \Bigg\} \ \forall i \in \mathcal{I}, \end{aligned}$$ As summary, the resulting bi-level problem is [\[eqs:summary\]]{#eqs:summary label="eqs:summary"} $$\begin{aligned} &\underset{\Omega}{\operatorname{min}} \quad \eqref{eq:upper_obj} \\ %%%%%%% & \text{s.t.} \quad \eqref{eq:budget_balance}-\eqref{eq:positive variables} \\ %%%%%% & p^{+}_{it}, p^{-}_{it}, q^{+}_{it}, q^{-}_{it}, d^{\, \rm{shed}}_{it} \in \underset{\Phi_i}{\text{argmin}} \ \Big\{ \eqref{eq:lower obj} \ \text{s.t.} \ \eqref{eq:ll power balance}-\eqref{eq:upper bound d} \Big\}. %\ \forall i \in \mathcal{I}. \end{aligned}$$ Recall that the upper-level objective function [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"} is quadratic due to the price regularization term, whereas the upper-level constraints are linear, outside of the the conic constraint [\[eq:line constraint\]](#eq:line constraint){reference-type="eqref" reference="eq:line constraint"}. Every lower-level problem is continuous and linear. ## Benefit distribution mechanisms {#sec:distribution} The total benefit $\sum_{i \in \mathcal{I}}w^-_i$ earned by forming a community to provide capacity limitation services (or in an extreme case, the total cost $\sum_{i \in \mathcal{I}}w^+_i$ if the community is unsuccessful) should be systematically distributed among community members. We introduce two distribution mechanisms, namely *equal* and *proportional* distributions. Note that we do not consider any distribution mechanism a posteriori. Instead, both aforementioned mechanisms are based on adding an extra regularization term to the upper-level objective function [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"}, weighted by a positive weight $\gamma$. Assigning a larger value for $\gamma$ further motivates the community manager to follow the underlying distribution, but at the potential expense of a higher cost for the whole community. Both regularization terms preserve the convexity of [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"}. The regularization-based distribution mechanisms do not guarantee that the targeted way of distributing the community benefit will be fully obtained. Rather, it motivates the manager to assign prices accordingly. For example, in the case of equal distribution mechanism, the total benefit may not be shared among members *exactly* equally, however it will be shared in a more uniform way than a case without such a mechanism. ### Equal distribution The equal distribution mechanism motivates (not necessarily restricts) the community manager to allocate an equal amount of benefit among prosumers, irrespective of how much they consume or produce and how much flexibility they provide. This is achieved by penalizing [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"} by an extra regularization term, weighted by $\gamma^{\rm{eq}}$, such that the revised upper-level objective function writes as $$\notag \eqref{eq:upper_obj} + \gamma^{\rm{eq}} \Bigg[\sum_{i\in \mathcal{I}} \Big( w^-_i -\widehat{w}^- \Big)^2 + \sum_{i\in \mathcal{I}} \Big( w^+_i -\widehat{w}^+ \Big)^2 \Bigg],$$ where variables $\widehat{w}^-$ and $\widehat{w}^+$ are the mean values of $w_i^-$ and $w_i^+$ over $i\in \mathcal{I}$, respectively. Note that $\widehat{w}^-$ and $\widehat{w}^+$ should be added to the variable set $\Omega$. This regularizer penalizes any deviation from the mean benefit (if $\sum_{i \in \mathcal{I}}w^-_i$ takes a positive value) and the mean cost (if $\sum_{i \in \mathcal{I}}w^+_i$ takes a positive value). ### Proportional distribution The proportional distribution mechanism motivates (not necessarily restricts) the community manager to allocate all prosumers a proportional amount of benefit (or cost) in line with their share of the total community baseline demand. Let the exogenous value $\Delta_{i}= \sum_t D_{it} - PV_{it}$ represent the total baseline residual load of prosumer $i$ over a day. We can define the demand share of prosumer $i$ as $\Delta_{i}/ \sum_i \Delta_{i}$. The proportional regularization term is then added to [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"}, weighted by $\gamma^{\rm{pro}}$, such that the revised upper-level objective function reads as $$\begin{aligned} \nonumber \eqref{eq:upper_obj} + \gamma^{\rm{pro}} \Bigg[ & \sum_{i\in \mathcal{I}} \Big( w^-_i - \frac{\Delta_i }{\sum\limits_{i \in \mathcal{I}} \Delta_i} \sum_{i \in \mathcal{I}} w^-_i \Big)^2 + \notag\\ % \nonumber & \sum_{i\in \mathcal{I}} \Big( w^+_i - \frac{\Delta_i }{\sum\limits_{i \in \mathcal{I}} \Delta_i} \sum_{i \in \mathcal{I}} w^+_i \Big)^2 \Bigg].\end{aligned}$$ ## Resulting mixed-integer cecond-order cone program In the following we detail the steps used to reformulate the proposed bi-level program as a single-level mixed-integer second order cone program (MISOCP). As the lower-level problems [\[eqs:lower level\]](#eqs:lower level){reference-type="eqref" reference="eqs:lower level"} are convex and satisfy Slater's conditions, their Karush-Kuhn-Tucker (KKT) conditions are both necessary and sufficient optimality conditions, and strong duality holds. Therefore, we replace the lower-level problems [\[eqs:lower level\]](#eqs:lower level){reference-type="eqref" reference="eqs:lower level"} by their equivalent KKT conditions in the upper level, and optimize over the set $\Theta = \big\{ \Omega, \Phi_i, \lambda_i^{(.)}, \mu_i^{(.)} \big\}$ which includes primal variables of the upper- and lower-level problems and dual variables of the lower-level problems. This results in a mathematical program with equilibrium constraints (MPEC). We then address two sources of non-convexity in this MPEC. First, we use the SOS1 reformulation as suggested by [@Askeland2023AFlexibility] and [@Siddiqui2013AnApplication], rather than the traditional Big-M method, to reformulate the complementarity conditions resulting from the lower-level inequality constraints. Second, as $x_{it}$, $p^+_{it}$, and $p^-_{it}$ are all decision variables of the resulting MPEC, [\[eq:budget_balance\]](#eq:budget_balance){reference-type="eqref" reference="eq:budget_balance"} and [\[eq:individual_rationality\]](#eq:individual_rationality){reference-type="eqref" reference="eq:individual_rationality"} contain bilinear terms $x_{it}\left( p^+_{it} - p^-_{it} \right)$, which must be linearized. The application of strong duality theorem to the lower-level problems allows us to replace such bilinear terms by an equivalent linear term, such that $$\begin{aligned} \label{eq:duality} \nonumber & \sum_{t \in \mathcal{T}} x_{it} \Big( p^+_{it} - p^-_{it} \Big) \\ \nonumber & \quad = \sum_{t \in \mathcal{T}} \Bigg( \lambda^{(1)}_{it} \Big( PV_{it} - D_{it} \Big) - \bar{P}_{i}^{\rm{bat}} \Big(\mu^{(9)}_{it} + \mu^{(10)}_{it} \Big) \\ & \quad \quad- \mu^{(12)}_{it}\bar{E}_{i} - \mu^{(12)}_{it} D_{it} - \alpha^{\rm{shed}} d^{\, \rm{shed}}_{it} \Bigg) \ \ \ \forall{i} \in \mathcal{I}.\end{aligned}$$ Finally, the quadratic regularization term $\rho \bar{x}^2$ in the upper-level objective function [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"} can be replaced by an auxiliary variable $z$ and then we can add a second-order cone constraint $z \geq \rho \bar{x}^2$. The regularization terms for the benefit allocation can be treated similarly. By this, [\[eq:upper_obj\]](#eq:upper_obj){reference-type="eqref" reference="eq:upper_obj"} becomes linear, so the resulting problem will be an MISOCP, which can be solved by available commercial solvers for real-life applications. # Numerical Results {#sec:results} The proposed bi-level model is applied to a 14-node radial distribution grid case study, as previously studied in [@Dvorkin2021DifferentiallyGrids] and [@Mieth2018Data-DrivenSystems]. We consider one prosumer per node. Therefore, the community has 14 members. Production and storage assets are arbitrarily distributed within the radial grid. Community members at nodes 1, 4, 7, 8, and 13 own photovoltaic (PV) generation assets, while all but two prosumers (5 and 9) are assumed to have some sort of battery flexibility between 5 and 10 kWh. Demand patterns for all consumers are generated using the residential demand profile model in [@McKenna2020CRESTModel], while PV production profiles are generated for the relevant prosumers using the methodology described in [@Pfenninger2016Long-termData]. Parameters for the capacity limitation services are based on common knowledge from power systems or chosen according to the current Danish legislation. For example, the capacity limitation is set to be inversely proportional to the day-ahead market price. The reason for this choice is that the increased power price is generally correlated with increased demand, which is a time at which the DSO desires a tighter capacity limitation as there is a higher risk of congestion. Values for grid tariffs are taken from [@2023TarifferNetabonnement] and [@2023AktuelleTariffer], whereas the DSO penalty value is assumed based on the recently proposed Danish legislation [@2022BekendtgrelseNetvirksomheder]. All source code and relevant data are publicly available at [@Crowley2023OnlineLimitation]. ## Community cost Our first investigation is to explore the total community cost as a function of the grid tariff discount factor $\beta^{\rm{DSO}}_t$ of the DSO as well as the variation level of capacity limitation $\bar{P}_t^{\rm{DSO}}$ over a day. Recall that $\beta^{\rm{DSO}}_t$ ranges between zero (full tariff) to one (no tariff). We consider the same value for $\beta^{\rm{DSO}}_t$ during all hours of a day. We also define a factor the so-called *capacity limitation variation factor*, varying between zero and one. This factor indicates to what extent the DSO lets $\bar{P}_t^{\rm{DSO}}$ vary over a day. In the case it takes a value of zero, the variation is zero too, meaning $\bar{P}_t^{\rm{DSO}}$ is constant over the day. Fig. [1](#fig:cap vs price){reference-type="ref" reference="fig:cap vs price"} shows $\bar{P}_t^{\rm{DSO}}$ in kW (red Y-axis) when the capacity limitation variation factor is 0, 0.5, and 1. Clearly, $\bar{P}_t^{\rm{DSO}}$ fluctuates more when the factor is 1, compared to other two cases. Fig. [1](#fig:cap vs price){reference-type="ref" reference="fig:cap vs price"} also illustrates the given spot market prices $\lambda^{\rm{spot}}_t$ in DKK/kW (black Y-axis). Recall that these prices are known, as the community manager determines dynamic prices once the spot market is cleared and prices are disseminated. It is evident that $\bar{P}_t^{\rm{DSO}}$ is inversely proportional to $\lambda^{\rm{spot}}_t$, when the capacity limitation variation factor is not zero. When the variation factor is 1, $\bar{P}_t^{\rm{DSO}}$ can reach zero at the highest spot price, and two times the average residual demand in the lowest price hour. The variation factors between zero and one provide a linear interpolation between the largest and smallest possible values of the capacity limitation. {#fig:cap vs price width="0.75\\columnwidth"} Fig. [2](#fig:heatmap){reference-type="ref" reference="fig:heatmap"} depicts the community cost as a function of tariff discount factor $\beta^{\rm{DSO}}_t$ and the capacity limitation variation factor. There is a clear trend that the community cost decreases when both factors increase. When the community has to pay less for internal power flows (grid tariff), the overall cost of the community decreases. Similarly, as the variation factor increases, the opportunity arises for the community to consume more power at hours with comparatively lower spot prices. Therefore, more of the power imported from the grid comes at a cheaper cost, also leading to a lower community cost. Fig. [2](#fig:heatmap){reference-type="ref" reference="fig:heatmap"} shows results for the variation factors above 0.4 only. For any case studies run below this, the community cost increases significantly, as it imposes a tight capacity limit to the community, leading to load shedding in the extreme cases. ## Capacity limitation service delivery We now investigate how successful our proposed bi-level program is in delivering the set capacity limitation services. For this purpose, we consider three cases for a comparison, whose results are given in Fig. [\[fig:delivery\]](#fig:delivery){reference-type="ref" reference="fig:delivery"}. In all three plots of this figure, the capacity limitation $\bar{P}_t^{\rm{DSO}}$ over the day with a variation factor of 1 is provided as a black curve. All three plots show the import power in kW. The blue plot (left) corresponds to the case if there is no community, and meanwhile none of prosumers shifts their load in response to the spot prices. By this, the import power is simply the total residual demand (demand minus renewable production) of the community. It is evident that the community load exceeds $\bar{P}_t^{\rm{DSO}}$ in the later hours of the day. The red plot (middle) shows a case that we refer to it as uncoordinated demand response (DR). There is still no community in this case, however prosumers response to the spot market prices individually, without being aware of the capacity limit $\bar{P}_t^{\rm{DSO}}$ imposed by the DSO. This naive load shifting causes a significant peak load in hours 4 and 5 when the spot price is comparatively low, which is clearly exceeding $\bar{P}_t^{\rm{DSO}}$, leading to a significant load shedding. Finally, the green plot (right) depicts the results obtained from our proposed bi-level program, here referred to as coordinated DR, opposed to the uncoordinated one (the red plot). Here, we consider a grid tariff discount factor of 0.6. We observe that the community reacts optimally, such that the import power is never exceeding $\bar{P}_t^{\rm{DSO}}$. This shows the successful delivery of capacity limitation services to the DSO. {#fig:heatmap width="0.75\\columnwidth"} {width="1.6\\columnwidth"} We then analyze whether the community still successfully provides the capacity limitation services under different variation factors. While keeping the discount tariff factor constant at 0.6, we determine the import power profile of the community for the capacity limitation variation factors of 0 (constant), 0.5 (low variation), and 1 (high variation). The results are provided in Fig. [\[fig:scenarios\]](#fig:scenarios){reference-type="ref" reference="fig:scenarios"}. The left plot of Fig. [\[fig:scenarios\]](#fig:scenarios){reference-type="ref" reference="fig:scenarios"} (high variation) is identical to the right plot of Fig. [\[fig:delivery\]](#fig:delivery){reference-type="ref" reference="fig:delivery"}. We observe that the community is still successful in delivering the capacity limitation services, i.e., not exceeding $\bar{P}_t^{\rm{DSO}}$, in low-variation (middle) and constant (right) cases. Recall that the proposed bi-level program provides an optimistic view, implying that the community manager perfectly knows how every single prosumer reacts to dynamic prices. Therefore, one may see the results as an ideal benchmark, showing the *maximum potential* of a community in the coordination of local flexibility. Although these results look promising, it cannot be guaranteed that a real-life pilot would achieve similar findings. {width="1.6\\columnwidth"} ## Benefit distribution among community members We now provide results for the two proposed regularization-based benefit distribution mechanisms. Recall that by the *community benefit*, we refer to $\sum_{i \in \mathcal{I}}w^-_i$, i.e., the collective earning of the community with respect to a benchmark case under which there is no community and all prosumers act individually. The results for such a benchmark are already provided in the middle plot of Fig. [\[fig:delivery\]](#fig:delivery){reference-type="ref" reference="fig:delivery"} as an uncoordinated DR case. Inspired from the current Danish legislation setting a load shedding penalty for DSOs, we assume the load shedding cost of prosumers is DKK75/kWh. Note that we assume it is prosumers who pay this cost incurred by their naive load shifting, causing a significant peak demand in hours 4 and 5 (see middle plot of Fig. [\[fig:delivery\]](#fig:delivery){reference-type="ref" reference="fig:delivery"}). Hereafter, when it is relevant, we consider a weight $\gamma$ of $10^{-6}$ for the regularizer, motivating a certain distribution mechanism. Fig. [3](#fig:distribution){reference-type="ref" reference="fig:distribution"} shows how the community benefit is shared among 14 members when there is no specific distribution mechanism (blue), i.e., $\gamma=0$, and when the community manager adds a regularizer, motivating an equal distribution (brown) or a proportional distribution (green). When there is no mechanism (blue), the amount of benefit allocated to each prosumer is relatively arbitrary and there does not seem to be a meaningful pattern. Looking at the equal distribution mechanism (brown), we observe that all community members earn almost identically. In the case of the proportional distribution, we observe similar variation of benefit distribution among prosumers as in the case with no distribution.. It may not seem intuitively fair that some prosumers gain very little benefit (prosumer 1) while others gain up to 10 times more (prosumers 11 and 13), but, as intended, the distribution matches up well with the share of residual demand that each member has to meet. ## Dynamic prices within the community We report dynamic prices $x_{it}$ as our final results, since they are impacted by whether a regularizer motivating a benefit distribution mechanism is exploited. Table [\[tab:prices\]](#tab:prices){reference-type="ref" reference="tab:prices"} gives dynamic prices assigned by the community manager to each community member over a day. We only report mean, minimum, and maximum price over 24 hours for every member. We observe that adding a regularizer for the benefit distribution highly impacts prices. In the case with no specific benefit distribution mechanism, the mean price of prosumers vary between zero and DKK2.1/kWh. Both distribution mechanisms drastically change price pattern, such that the manager assigns relatively high prices for certain prosumers, while zero price for others. Note that these prices are still lower than or equal to the load shedding cost of DKK75/kWh, implying that prosumers still prefer these prices over the load shedding alternative. **No mechanism** **Equal** **Proportional** ---- ------------------ ------ ------ ----------- ------- ------- ------------------ ------- ------- \# Mean Min Max Mean Min Max Mean Min Max 1 2.05 1.93 2.13 0.00 0.00 0.00 71.01 66.36 73.35 2 0.00 0.00 0.00 19.55 17.90 19.79 9.46 8.66 9.58 3 2.10 1.93 2.13 0.00 0.00 0.00 0.00 0.00 0.00 4 2.05 1.93 2.13 5.76 0.00 19.74 59.37 55.48 61.32 5 0.36 0.00 2.13 0.00 0.00 0.00 0.00 0.00 0.00 6 2.07 1.93 2.13 19.18 17.70 19.57 7.11 6.56 7.25 7 2.04 1.93 2.13 16.24 15.17 16.77 71.02 66.36 73.35 8 2.05 1.93 2.13 2.75 0.00 7.33 0.00 0.00 0.00 9 0.27 0.00 2.13 0.00 0.00 0.00 0.00 0.00 0.00 10 2.09 1.93 2.13 26.73 24.38 26.95 28.89 26.56 29.35 11 2.09 1.93 2.13 1.91 0.00 2.19 0.00 0.00 0.00 12 2.07 1.93 2.13 0.00 0.00 0.00 0.00 0.00 0.00 13 2.08 1.93 2.13 17.02 15.78 17.44 3.16 2.93 3.23 14 2.08 1.93 2.13 0.00 0.00 0.00 0.00 0.00 0.00 [\[tab:prices\]]{#tab:prices label="tab:prices"} {#fig:distribution width="0.9\\columnwidth"} # Conclusion {#sec:conclusion} We showed how an energy community can be formed based on dynamic pricing, for which we developed a mathematical framework. It enables the provision of capacity limitation services by the community to the DSO, while ensuring the budget balance for the community manager and individual rationality for every community member. We numerically showed the success of the proposed community framework, in comparison to a case under which there is no community and prosumers respond individually to spot prices in an uncoordinated manner. As a degree of freedom to be set by the DSO, we observed that a more time-variant profile for the capacity limit throughout the day results in a higher cost saving for the community, and therefore more willingness to form a community. As another degree of freedom to be again set by the DSO, we observed that a higher grid tariff discount makes the community more successful in cost reduction while meeting the DSO capacity limits. We also looked at mechanisms, to be endogenously implemented within the proposed mathematical framework, for distributing community benefits among community members. While the proposed equal and proportional distribution mechanisms make the manager aware of how to distribute benefits, they may drastically impact the variation of prices assigned over prosumers, although prosumers exposed to high prices still prefer staying in the community over the load shedding alternative. The future work should extend the portfolio of local flexibility resources to thermostatically controlled loads, with their underlying physics, rebound effect, and their stochastic nature. One may also consider additional ways of exploiting the community flexibility, e.g., by selling frequency-supporting ancillary services to the transmission system operator. As already mentioned, this work provided an ideal benchmark with an optimistic view, showing the maximum potential of a community. The more realistic solutions require learning the demand response behaviours, which might be even non-stationary over time. For that, one may explore the applications of multi-armed bandit or similar approaches. It will then be interesting to explore how to derive dynamic prices, and whether the central market properties can still be guaranteed. [^1]: This research was supported by the Danish Energy Technology Development and Demonstration Programme (EUDP) through the Flexible Citizen Energy Communities (FLEX-CEC) project (64021-1090). [^2]: Bennevis Crowley, Jalal Kazempour, and Lesia Mitridati are with the Technical University of Denmark (e-mail: {bewcro, jalal, lemitri}\@dtu.dk). [^3]: LinDistFlow is originally a second-order conic approximation of the AC power flow equations which can be applied in radial distribution networks, as extensively used in the literature, e.g., in [@Taylor2015ConvexSystems], [@Dvorkin2021DifferentiallyGrids], and [@Mieth2018Data-DrivenSystems]. [^4]: We use, without loss of generality, the same generic mathematical formulation for different types of prosumers, and simply set the parameters and decision variables related to the operation of the photovoltaic panels and batteries to zero for prosumers who do not own such assets.

arxiv_math

--- abstract: | I trace the roots of my collaboration with Boris Zilber, which combines categoricity theory, finite model theory, algorithmics, and combinatorics. address: Faculty of Computer Science, Technion--Israel Institute of Technology, Haifa, Israel author: - J.A. Makowsky date: August, 05, 2023 title: How I got to like graph polynomials --- [^1] # Introduction and dedication Boris Zilber played a crucial rôle in my work on graph polynomials. Some of my work he inspired and in which he contributed, is summarized in Kotek et al. [@ar:KotekMakowskyZilber11]. A preprint was posted as Makowsky and Zilber [@makowsky2006polynomial] and a conference paper was published as Kotek et al. [@kotek2008counting]. These are our only published joint papers. Since then a general framework for studying graph polynomials has slowly evolved. It bears witness to the impact of Boris on my own work. In this paper I will sketch how I got to like graph polynomials. Boris and I both started our scientific career in model theory. Boris pursued his highly influential work in various directions of infinte model theory. My path towards graph polynomials took a detour into the foundations of computer science, only to lead me back to model theoretic methods in finite combinatorics. I sketch here how, step by step, I ended up discussing graph polynomials with Boris. Some of those steps owe a lot to serendipity, others were triggered by natural questions arising from previous steps. These steps are described in Sections [2](#se:2){reference-type="ref" reference="se:2"}-[6](#se:6){reference-type="ref" reference="se:6"}. Sections [7](#se:7){reference-type="ref" reference="se:7"}-[8](#se:8){reference-type="ref" reference="se:8"} describe some of the original ideas underlying the model-theoretic approach to graph polynomials. Section [9](#se:9){reference-type="ref" reference="se:9"} summarizes where this encounter with Boris has led me. Ultimately, it looks as if Boris' influence on my path was inevitable, but only in retrospect. Meeting Boris in Oxford was a chance encounter with unexpected consequences. I would like to thanks Boris for giving me an important and fruitful impulse. Happy birthday, and many years of productive mathematics to come, till 120. # Morley's 1965 paper {#se:2} My first attempt to tackle open problems in model theory was a consequence of reading M. Morley's fundamental paper [@ar:Morley65] on categoricity in power, in the *undergraduate seminar* in mathematical logic at ETH Zürich in 1969. The seminar was held by E. Specker and H. Läuchli, and regularly attended by the then still very lucid octogenarian P. Bernays. Building on earlier work by A. Mostowski, A. Ehrenfeucht and R. Vaught, M. Morley proved in 1965 a **truly deep** theorem in model theory: **Theorem 1** (Morley's Theorem). *Let $T$ be a first order theory and assume $T$ has no finite models and is $\kappa$-categorical for **some** uncountable $\kappa$. Then $T$ is $\kappa$-categorical for **every** uncountable $\kappa$.* More importantly, even, the paper ended with a **list of questions** and many logicians and mathematicians were attracted by these. Among them I remember J. Baldwin and A. Lachlan, J.-P. Ressaire, D. Lascar, M. Makkai, V. Harnik, S. Shelah, B. Zilber, M.A. Taitslin and his school, see Taitslin [@Taitslin1970], and myself. In my MSc thesis from 1971 [@makowsky1974some], I managed to prove the following: **Theorem 2**. (i) *A first order theory $T$ which is $\aleph_0$-categorical and strongly minimal (hence categorical in all infinite $\kappa$) cannot be finitely axiomatizable.* (ii) *There is a finitely axiomatizable complete first order theory $T$ without finite models which is superstable.* (iii) *If there is an *infinite, finitely presentable group $G$ with only finitely many conjugacy classes*, there is also a complete finitely axiomatizable $\aleph_1$-categorical theory $T_G$ without finite models.* After I finished my MSc thesis, E. Specker drew my attention to a Soviet paper by Mustafin and Taimanov [@Mustafin1970countable], and as a result of this, I started a correspondence with its second author. Before 1985 there were very few authors citing my work. Among them G. Ahlbrandt, a PhD-student of J. Baldwin, P. Rothmaler, P. Tuschik from the German Democratic Republic, and B. Zilber, M. Peretyat'kin, and A. Slissenko from the Soviet Union. Boris was one of the first to notice and cite my work on categoricity. I soon realized that I could not make any further progress on these questions. I had no new ideas, and competition was overwhelming. S. Shelah's sequence of papers inspired by these open questions led many young researchers to abandon this direction of research in model theory. The finite axiomatizability questions were finally solved by Peretyat'kin [@peretyatkin1980example] and Zilber [@zilber1981totally]. M. Peretyat'kin constructed a finitely axiomatizable theory categorical in $\aleph_1$ but not in $\aleph_0$. B. Zilber showed that no finitely axiomatizable totally categorical first order theory exists. An alternative proof of this was given by G. Cherlin et al. [@cherlin1985omega]. My first acquaintance with Boris Zilber happened via the literature. But our paths diverged (not in the yellow wood), and we did meet personally, but not very often. # From abstract model theory to computer science and graph algorithms {#se:3} After leaving Morley-type model theory, I first worked in abstract model theory, and then in theoretical computer science. In computer science I dealt with the foundations of database theory and logic programming, which led me to finite model theory. My main tools from model theory were pebble games and the Feferman-Vaught theorem and its generalizations. Around this time I met B. Courcelle and became aware of the Robertson-Seymour theorems and their applications to graph algorithms described by Fellows [@fellows1989robertson]. But it was Courcelle [@courcelle1992monadic] who first observed that logical methods would give even more applications, Courcelle's work on the monadic second order theory of graphs is summarized in the monumental monograph from 2012 by Courcelle and Engelfriet [@courcelle2012graph]. Let $d(G)$ be a graph parameter and $P$ be a graph property. If deciding whether a graph $G$ on $n$ vertices with $d(G)=t$ is in $P$ can be done in time $c(t)\cdot n^s$ for some fixed $s$ which does not depend on $d(G)$, nor on the number of vertices of $G$, we say that *$P$ is Fixed Parameter Tractable ($\mathrm{FPT}$)*. This concept was introduced by Fellows and Downey [@downey2013fundamentals]. **Theorem 3** (Courcelle, 1992). *Let $C$ be a class of finite graphs of tree-width at most $t$ and let $P$ be a graph property definable in Monadic Second Order Logic $\mathrm{MSOL}$. Then checking whether a graph $G ,\in C$ with $n$ vertices is in $P$ is in $\mathrm{FPT}$, in fact, it can be solved in linear time $c(t)n$.* In the mid-1990s two students were about to change my research dramatically. My former MSc *Udi Rotics* returned from his experience in industry. His MSc thesis dealt with the logical foundation of databases. However, now he wanted to work for a PhD in *Graph Algorithms* but *without involving Logic*. He proposed to extend the notion of *tree-width* of a class of graphs as a graph parameter in order to get a new width parameter which one can use for *Fixed Parameter Tractability*. Finally, *but still using Logic ($\mathrm{MSOL}$)*, we came up with a notion roughly equivalent to *clique-width*, introduced recently by Courcelle and Olariu [@courcelle1994clique]. This led to my intensive collaboration with Courcelle and Rotics [@courcelle1998linear; @courcelle2000linear; @courcelle2001fixed]. In my own paper [@makowsky2004algorithmic] I examine the algorithmic uses of the Feferman-Vaught theorem for Fixed Parameter Tractability. Applications of my work with B. Courcelle and U. Rotics are well summarized in Downey and Fellows [@downey2013fundamentals]. In 1996 I started to supervise an immigrant student from the former USSR, *Gregory Kogan*, who wanted to work on the complexity of computing the *permanent*. He came with impressive letters of recommendation. He had some spectacular partial results for computing permanents of matrices over a field of characteristic $3$. He was a virtuoso in combinatorial linear algebra. Unfortunately, he dropped out before finishing his PhD. M. Kaminski and I wrote up his results, published under his name alone as Kogan [@DBLP:conf/focs/Kogan96]. # Computing permanents {#se:4} I first came across the problem of computing the permanent at Specker's 60th birthday conference in 1980. The permanent of an $(n \times n)$-matrix $A=(A_{i,j})$ is given as $$per(A) = \sum_{s:[n] \rightarrow [n]} \prod_{i \in [n]} A_{i,s(i)}$$ where $s$ ranges over all permutations of $[n]$. The complexity class $\sharp\mathbf{P}$ is the polynomial time counting class. The class of $\sharp\mathbf{P}$ consists of function problems of the form "compute $f(x)$", where $f$ is the number of accepting paths of a nondeterministic Turing machine running in polynomial time. Unlike most well-known complexity classes, it is not a class of decision problems but a class of function problems. The most difficult representative problems of this class are $\sharp\mathbf{P}$-complete. Counting the number of satisfying assignments for a formula of propositional logic is $\sharp\mathbf{P}$-complete. Typical examples would be described as follows: Let $k$ be a fixed integer. Given an input graph $G$ on $n$ vertices, compute the number of proper $k$-colorings of $G$. For $k=1,2$ this can be computed in polynomial time, but for $k \geq 3$, this is $\sharp\mathbf{P}$-complete with respect to $\mathbf{P}$-time reductions. In general $\sharp\mathbf{P}$ lies between the polynomial hierarchy $\mathbf{PH}$ and $\mathbf{PSpace}$, see Papadimitriou [@bk:papadimitriou94]. Valiant's complexity classes $\mathbf{VP}$ and $\mathbf{VNP}$ are the analogues of $\mathbf{P}$ and $\mathbf{NP}$ in Valiant's model of algebraic computation. Bürgisser's book [@burgisser2000completeness] is entirely dedicated to this model of computation. It is still open whether $\mathbf{P}=\mathbf{NP}$, and also whether $\mathbf{VP}=\mathbf{VNP}$. Valiant presented the complexity classes $\mathbf{VP}$ and $\mathbf{VNP}$ at Specker's 60th birthday conference. **Theorem 4** (L. Valiant [@valiant1979complexity]). *Computing the permanent of a $\{0,1\}$-matrix is hard in the following sense:* (i) *It is $\sharp\mathbf{P}$-complete in the Turing model of computation, and* (ii) *$\mathbf{VNP}$-complete in Valiant's algebraic model of computation.* Valiant [@ar:Valiant80] published in the proceedings of Specker's 60th birthday conference in 1980. Bürgisser's monograph [@burgisser2000completeness] explores Valiant's approach to algebraic complexity further. G. Kogan studied the complexity of computing the permanent over fields of characteristic $3$ for matrices $M$ with $rank(MM^{tr} - \mathbf{1}) = a$. He showed that for $a \leq 1$ this is easy, and for $a \geq 2$ this is hard. I wanted to use results from Courcelle et al. [@courcelle2001fixed] to prove something about permanents G. Kogan could not prove. I looked at adjacency matrices of graphs of fixed tree-width $t$. Barvinok [@barv:96] also studied the complexity of computing the permanent for special matrices. He looked at matrices of fixed rank $r$. Our results were: **Theorem 5** (A. Barvinok, 1996). *Let $\mathcal{M}_r$ be the set of real matrices of fixed rank $r$. There is a polynomial time algorithm $\mathcal{A}_r$ which computes $per(A)$ for every $A \in \mathcal{M}_r$* **Theorem 6** (JAM, 1996). *Let $\mathcal{T}_{w}$ be the set of adjacency matrices of graphs of tree-width at most $w$. There is a polynomial time algorithm $\mathcal{A}_w$ which computes $per(A)$ for every $A \in \mathcal{T}_w$.* The two theorems are incomparable. There are matrices of tree-width $t$ and arbitrary large rank, and there are matrices of rank $r$ and arbitrary large tree-width. However, I realized that the proof of my theorem had nothing to do with permanents. It was much more general and really worked quite generally. It only depended on some *logical restrictions* for polynomials in indeterminates given by the entries of the matrix. If the matrix $A=A_G$ is the adjacency matrix of a graph $G$ where the non-zero entries are $x$, the permanent $per(A_G)$ can be viewed as a graph polynomial in the indeterminate $x$. Alas, at that time I had no clue how to find many interesting examples. # From knot polynomials to graph polynomials {#se:5} During a sabbatical at ETH in Zurich I met *V. Turaev*, who, among other things, is an expert in knot theory. I showed him my Theorem [Theorem 6](#jam-per){reference-type="ref" reference="jam-per"}. He suggested I should try to prove the same for the *Jones polynomial* from *Knot Theory*. So I studied Knot Theory intensively for a few months. While visiting the Fields Institute in 1999 I attended a lecture by J. Mighton[^2] who lectured about the Jones polynomial for series-parallel knot diagrams, cf. his PhD thesis, Mighton [@mighton2000knot]. He showed that in this case the Jones polynomials is computable in polynomial time. Series-parallel graphs are exactly the graphs of tree-width $2$. It seemed reasonable that the same would hold for graphs of tree-width $k$. Indeed, after quite an effort I proved in [@makowsky2001coloured; @makowsky2005coloured]: **Theorem 7**. *Assume $K$ is a knot with knot diagram $D_k$ of tree-width $k$. Then evaluating the Jones polynomial $J(D_k;a,b)$ for fixed complex numbers $a,b \in \mathbb{C}$ and $D_k$ with $n$ vertices is in $\mathbf{FPT}$ with parameter $k$.* Jaeger et al. [@jaeger1990computational]a showed that Without the assumption on tree-width evaluating the Jones polynomial is $\sharp\mathbf{P}$-complete for almost all $a,b \in \mathbb{C}$. Lotz and Makowsky [@lotz2004algebraic] analyze the complexity of the Jones polynomial in Valiant's model of computation. However, again the proof seemed to work for other graph polynomials as well, among them the *Tutte polynomial, chromatic polynomial, characteristic polynomial, matching polynomial*. Univariate graph polynomials are graph invariants which take values in a polynomial ring, usually $\mathbb{Z}[X]$, $\mathbb{R}[X]$ or $\mathbb{C}[X]$. The univariate chromatic polynomial $\chi(G;k)$ of a graph counts the number of proper colorings of a graph with at most $k$ colors. It was introduced by G. Birkhoff in 1912 in a unsuccessful attempt to prove the four colour conjecture. The characteristic polynomial of a graph is the characteristic polynomial (in the sense of linear algebra) of the adjacency matrix $A_G$ of the graph $G$, see the monographs by Chung [@chung1997spectral] and by Brower and Haemer [@brouwer2011spectra]. The coefficients of $X^k$ of the matching polynomial count the number of $k$-matchings of a graph $G$, see Lovász and Plummer [@lovasz2009matching]. Both, the characteristic and the matching polynomial, have found applications in *theoretical chemistry* as described by Trinajstic [@trinajstic2018chemical]. There are also multivariate graph polynomials. The Tutte polynomial is a bivariate generalization of the chromatic polynomial. Both of them are widely studied, see Dong et al. [@dong2005chromatic] and the handbook of the Tutte polynomial edited by Ellis-Monaghan and Moffatt [@ellis2022handbook]. Other widely studied graph polynomials are listed in Makowsky [@makowsky2008zoo]. However, I had no idea, how to find *infinitely many natural and interesting examples*? # Boris, deus ex machina {#se:6} In 2005, while attending CSL, the European Conference in Computer Science Logic in Oxford, I paid a visit to Boris Zilber, whom I knew and had met before due to our work on Morley's problem on finite axiomatizability of totally categorical theories. After a few friendly exchanges the following dialogue evolved: Boris : What do you work on nowadays? Me : Graph polynomials. Boris : What polynomials? It seemed Boris had never heard of graph polynomials. I gave him the standard examples (Tutte, chromatic, matching). He immediately saw them as examples which are interpretable in some totally categorical theory. I could not believe it. We spent the next days together, verifying that all the known graph polynomials fit into Zilber's framework. It was indeed the case. We also produced generalizations of chromatic polynomials, some of which I later called *Harary polynomials* [@herscovici2020harary; @herscovici2020harary-a]. They are generalizations of the chromatic polynomial based on conditional colorings introduced in Harary [@pr:Harary85] in 1985. Conditional colorings are defined using a graph property $P$. A $P$-coloring $f$ of $G$ with at most $k$ colors is a function $f: V \rightarrow [k]$ such that for every color $j \in [k]$ the set $f^{-1}(j)$ induces a graph in $P$. Conditional colorings are studied in the literature, e.g., by Brown and Corneil [@brown1987generalized], mostly in the context of extremal graph theory. However, nobody wrote about the fact that counting the number of such colorings with at most $k$ colors defines a polynomial in $k$. The so called Harary polynomial $\chi_P(G;k)$ counts the number of $P$-colorings of $G$ with at most $k$ colors. # Why is the chromatic polynomial of a graph a polynomial? {#se:7} Let $G=(V,E)$ be a graph. A proper coloring of $G$ with at most $k$ colors is a function $f: V \rightarrow [k]$ such $f(v) =f(v')$ implies that $\neg E(v,v')$. We think of $[k]$ as a set of colors. In other words, if two vertices have the same color they are not adjacent. We denote by $\chi(G;k)$ the number of proper colorings of $G$ with at most $k$ colors. Birkhoff's proof that $\chi(G;k)$ is a polynomial in $\mathbb{Z}[k]$ uses deletion and contraction of edges. Let $e =(u,v)$ be an edge of $G$. $G_{-e}$ is the graph $G_{-e}= (V, E - \{(u,v)\})$ where $e$ is deleted from $E$. $G_{/e}$ is the graph $G_{/e}= (V_{/e}, E|_{V_{/e}})$ where $e$ is contracted to a single vertex to form $V_{/e}$ and $e$ is omitted from $E$. $f$ is a proper coloring of $G_{-e}$ if either it is a proper coloring of $G$ and $f(u) \neq f(v)$ or it is a proper coloring of $G_{/e}$ and $f(u) = f(v)$. Furthermore, $\chi(G;k)$ is multiplicative, i.e., if $G$ is the disjoint union of $G_1$ and $G_2$, then $$\chi(G_1 \sqcup G_2;k) = \chi(G_1;k) \cdot\chi(G_2;k).$$ Let $E_n$ be the edgeless graph with $n$ vertices and $E= \emptyset$. We have $\chi(E_n;k) =k^n$, and $$\chi(G_{-e};k) = \chi(G;k) + \chi(G_{/e};k).$$ By showing that one can compute $\chi(G;k)$ by successively removing edges, and this is independent of the order of the edges, one concludes that $\chi(G;k)$ is a polynomial in $k$. The disadvantage of this elegant proof is, that it does not generalize. Another way of proving that $\chi(G;k)$ is a polynomial in $k$ is by noting that for graphs on $n$ vertices we have $$\chi(G;k) = \sum_{i=1}^n c_i(G) k_{(i)}$$ where the coefficient $c_i(G)$ is the number of proper colorings of $G$ with exactly $i$ colors and $$k_{(i)}= k \cdot (k-1) \cdot \ldots \cdot (k-i+1) = \prod_{i=0}^i (k-i) = {k \choose i}\cdot i!$$ the falling factorial. Note that ${k \choose i} = 0$ for $i > k$. As $k_{(i)}$ is a polynomial in $k$ and $\chi(G; k)$ is a sum of $n$ polynomials in $k$, the result follows. However, this proof *does generalize*, and it works for all Harary polynomials. Later I discussed Zilber's view of graph polynomials with A. Blass. We noted that for most of the graph invariants from the literature, proving that they were polynomial invariants via totally categorical theories was an overkill. This led me to formulate a considerably simplified approach, which indeed covered all the know examples of graph polynomials in the literature. This approach is a simplification of Boris' proof. It generalizes also to other types of graph polynomials such as the bivariate Tutte polynomial and the trivariate edge elimination polynomial from Averbouch et al. [@averbouch2010extension]. More intrinsic examples are also discussed in Makowsky and Zilber [@makowsky2006polynomial] and Kotek et al. [@ar:KotekMakowskyZilber11]. However, the polynomial graph invariants hidden in totally categorical theories are the most general graph invariants which are definable in Second Order Logic $\mathrm{SOL}$, and even in Higher Order Logic $\mathrm{HOL}$, over the graph $G$, see Makowsky and Zilber [@makowsky2006polynomial Corollary C and Theorem 3.15]. Furthermore, it applies to $\mathrm{HOL}$-definable polynomial invariants over arbitrary finite first order structures for finite vocabularies, rather than just to graphs. # The model-theoretic approach to the chromatic polynomial {#se:8} The way Boris looked at the chromatic polynomial was even more general. Given a graph $G$, Boris had in mind an infinite first order structure $\mathfrak{M}(G)$ with universe $M$, and a formula $\phi(x)$ such that (i) The first order theory $T(\mathfrak{M}(G))$ of $\mathfrak{M}(G)$ is totally categorical and strongly minimal with a strongly minimal infinite set $X$ of indiscernibles. (ii) $T(\mathfrak{M}(G))$ has the finite model property, i.e., the algebraic closure $\mathrm{acl}(Y)$ in $\mathfrak{M}(G)$ of a finite subset $Y \subset X$ is finite. (iii) $\mathfrak{M}(G) \models \phi(x)$ iff $x$ is a proper coloring of $G$. Such theories were at the heart of his work [@Zilber-uct]. From Zilber's analysis of totally categorical theories we get in the spirit of [@Zilber-uct Theorem 1.5.5]: **Theorem 8**. *Let $\mathfrak{M}(G)$ and $X$ as above. For every finite set $Y \subset X$ of cardinality $k$ the cardinality of the the set $$%\{ x \in |\mathrm{acl}(Y)| \cap M: \mathrm{Alec}(Y) \models \phi(x) \} \{ x \in \mathrm{acl}(Y) : \mathfrak{M}(G) \models \phi(x) \}$$ is a polynomial in $k$.* In the case of the chromatic polynomial this looks as follows: (i) Let $G = (V,E)$ be a finite graph, with $|V| = n$. (ii) Let $\mathfrak{M}(G) =(V,X;E, \bar{v})$ be a 2-sorted language with sorts $V, X$, a binary relation $E$ on $V$ for the edge relation, and $n$ constant symbols $v_1,\ldots,v_n$ of sort $V$. (iii) The describing axioms state that $(V,E)$ is exactly the finite graph we started with, and the vertices are exactly the $v_i$. Then a model of the axioms is specified up to isomorphism by the cardinality of $X$. (iv) If we add axioms stating that $X$ is infinite then the first order theory $T(\mathfrak{M}$ is totally categorical. (v) We also get finite models $M_k$ where $|X| = k$ for each natural number $k$, and they are algebraically closed subsets of the infinite model $\mathfrak{M}$. (vi) We regard $X$ as a set of colors, and we identify $X^n$ with the colorings of vertices, that is, the set of functions $V \to X$, by identifying $f = (x_1,\ldots,x_n) \in X^n$ with the function $f(v_i) = x_i$. (vii) $f$ is a *proper vertex coloring* if any two adjacent vertices have different colors. So the set of proper vertex colorings is defined as a subset of $X^n$ by the formula $\phi(\bar{x})$ given by $$\phi(\bar{x}): \bigwedge_{\{(i,j) : E(v_i,v_j)\}} x_i \neq x_j.$$ (viii) The chromatic polynomial for the graph $G$ is then $\chi(G;k) = |\phi(M_k)|$. Boris also showed me at our first encounter how the bivariate matching polynomial and the Tutte polynomial can be cast in this framework. For the characteristic polynomial $p(G;x)$ the situation is a bit more complicated, because its original definition uses the characteristic polynomial of the adjacency matrix $A(G)$ of $G$. However, there exists a purely graph theoretic description of the coefficients of $p(G;x)$ by Godsil [@bk:Godsil93], which allows to cast $p(G;x)$ into this framework. We note that it may be unexpectedly tricky to put a graph invariant into Boris' framework, even if one already knows that it is a polynomial invariant. The most general version of this can be found in Kotek et al. [@ar:KotekMakowskyZilber11 Section 8]. Using this method, any multivariate polynomial graph invariant deinable in $\mathrm{HOL}$ can be captured in this way. In the last ten years J. Nešetřil and his various collaborators (A. Goodall, D. Garijo and P. Ossona de Mendez) were exploring various ways to define such graph invariants. However, they did not reach the same generality, cf. [@garijo2011distinguishing; @goodall2016strongly; @garijo2016polynomial]. The potential of the general approach as described in [@ar:KotekMakowskyZilber11 Section 8] still has not been explored in depth. It seems that its abstract generality makes it difficult for the combinatorics community to see through this construction. On the other side, model theorists seemingly are not interested in combinatorial applications of model theory. Exceptions may be in extremal combinatorics, as initiated by Razborov [@razborov2007flag; @razborov2013flag] and surveyed by Coregliano and Razborov [@coregliano2020semantic]. Another direction is counting the number $S_P(n)$ of graphs on $n$ vertices in a hereditary graph property $P$, initiated by Scheinerman [@scheinerman1994size] and further pursued by Balogh et al. [@ar:BaloghBollobasWeinreich2000] and Laskowski and Terry [@laskowski2022jumps]. # Towards a general theory of graph polynomials {#se:9} For the last 20 years I was studying graph polynomials [@makowsky2006zoo; @makowsky2008zoo], aiming to understand what they have in common. - Graph polynomials can be studied to obtain *information on graphs*.\ As an example: Evaluations of the Tutte polynomial encodes many graph invariants. - Graph polynomials can be studied as *polynomials indexed by graphs*.\ As an example: The acyclic matching polynomial of paths, cycles, complete graphs, and complete bipartite graphs are the Chebyshev polynomials of the second and first kinds, Hermite polynomials, and Laguerre polynomials, respectively. - Two graph polynomials have the same *distinctive power* if they distinguish between the same graphs. With my various collaborators I managed to create a new field in graph theory with two Dagstuhl Seminars (16241, 19401), two MATRIX Institute programs, two special sessions at AMS meetings, and one SIAM mini-symposium. 2009 : AMS-ASL Special Session on *Model Theoretic Methods in Finite Combinatorics*, January 2009, Washington DC,\ Organizers: M. Grohe and J.A. Makowsky 2014 : SIAM Conference on Discrete Mathematics,\ Minisymposium: *Graph Polynomials: Towards a General Theory*,\ Minneapolis, June 2014,\ Organizers: Jo Ellis-Monaghan, Andrew Goodall and J.A. Makowsky 2016 : Dagstuhl Seminar 16241:\ *Graph Polynomials: Towards a Comparative Theory*,\ Organizers: Jo Ellis-Monaghan, Andrew Goodall,\ Johann A. Makowsky, Iain Moffatt 2017 : MATRIX Institute program, November 2017: *Tutte Centenary Retreat*\ Organizers: Graham Farr (Chair), Marston Conder, Dillon Mayhew, Kerri Morgan, James Oxley, Gordon Royle 2019 : Dagstuhl Seminar 19401:\ *Comparative Theory for Graph Polynomials*\ Organizers: Jo Ellis-Monaghan, Andrew Goodall,\ Iain Moffatt, Kerri Morgan 2022 : Special Session on *Graph and Matroid Polynomials: Towards a Comparative Theory*, AMS-SMF-EMS Joint International Meeting, Grenoble, France, July 2022\ Organizers: E.Gion, J.A.Makowsky and J.Oxley 2023 : MATRIX Institute program, October 2023: *Workshop on Uniqueness and Discernment in Graph Polynomials*\ Organizers: Jo Ellis-Monaghan, Iain Moffatt, Kerri Morgan, and Graham Farr Without Boris Zilber's eye opener I would not have pursued this line of research as far as I did. Thank you, Boris! 10 Ilia Averbouch, Benny Godlin, and Johann A Makowsky. An extension of the bivariate chromatic polynomial. , 31(1):1--17, 2010. J. Balogh, B. Bollobás, and D. Weinreich. The speed of hereditary properties of graphs. , 79(2):131--156, 2000. A.I. Barvinok. Two algorithmic results for the traveling salesman problem. , 21:65--84, 1996. A. E. Brouwer and W. H. Haemers. . Springer, 2011. J.I. Brown and D. G. Corneil. On generalized graph colorings. , 11(1):87--99, 1987. P. Bürgisser. , volume 7 of *Algorithms and Computations in Mathematics*. Springer, 2000. G. Cherlin, L. Harrington, and A. H. Lachlan. $\omega$-categorical, $\omega$-stable structures. , 28(2), 1985. F. R. K. Chung. , volume 92 of *AMS Regional Conference Series in Mathematics*. American Mathematical Soc., 1997. L. N. Coregliano and A. A. Razborov. Semantic limits of dense combinatorial objects. , 75(4):627, 2020. B. Courcelle. The monadic second-order logic of graphs iii: tree-decompositions, minors and complexity issues. , 26(3):257--286, 1992. B. Courcelle and J. Engelfriet. , volume 138. Cambridge University Press, 2012. B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique width. In *Graph-Theoretic Concepts in Computer Science*, pages 1--16. Springer, 1998. B. Courcelle, J.A. Makowsky, and U. Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. , 33(2):125--150, 2000. B. Courcelle, J.A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. , 108(1-2):23--52, 2001. B. Courcelle and S. Olariu. Clique-width: A graph complexity measure--preliminary results and open problems. In *Proc. 5th Int. Workshop on Graph Grammars and Their Application to Computer Science*, pages 263--270, 1994. F. Dong, K.-M. Koh, and K. L. Teo. . World Scientific, 2005. R. G. Downey and M. R. Fellows. . Springer, 2013. J. A. Ellis-Monaghan and I. Moffatt. . CRC Press, 2022. M. R. Fellows. The Robertson-Seymour theorems: A survey of applications. , 89:1--18, 1989. D. Garijo, A.J. A. Goodall, and J. Nešetřil. Polynomial graph invariants from homomorphism numbers. , 339(4):1315--1328, 2016. D. Garijo, A.J. Goodall, and J. Nešetřil. Distinguishing graphs by their left and right homomorphism profiles. , 32(7):1025--1053, 2011. C.D. Godsil. . Chapman and Hall, 1993. A.J. Goodall, J. Nešetřil, and P. Ossona de Mendez. Strongly polynomial sequences as interpretations. , 18:129--149, 2016. F. Harary. Conditional colorability in graphs. In *Graphs and Applications, Proc. First Col. Symp. Graph Theory (Boulder, Colo., 1982)*, pages 127--136. Wiley-Intersci. Publ., Wiley, New York, 1985. O. Herscovici, J. A. Makowsky, and V. Rakita. Harary polynomials. , 1:2:Article \#S2R13, 2021. O. Herscovici, J.A. Makowsky, and V. Rakita. Harary polynomials. , 2020. F. Jaeger, D. L. Vertigan, and D.J.A. Welsh. On the computational complexity of the jones and tutte polynomials. , 108(1):35--53, 1990. G.P. Kogan. Computing permanents over fields of characteristic 3: Where and why it becomes difficult (extended abstract). In *37th Annual Symposium on Foundations of Computer Science, FOCS '96, Burlington, Vermont, USA, 14-16 October, 1996*, pages 108--114. IEEE Computer Society, 1996. T. Kotek, J. A. Makowsky, and B.I. Zilber. On counting generalized colorings. In *Computer Science Logic: 22nd International Workshop, CSL 2008, 17th Annual Conference of the EACSL, Bertinoro, Italy, September 16-19, 2008. Proceedings 22*, pages 339--353. Springer, 2008. T. Kotek, J. A. Makowsky, and B.I. Zilber. On counting generalized colorings. In M. Grohe and J.A. Makowsky, editors, *Model Theoretic Methods in Finite Combinatorics*, volume 558 of *Contemporary Mathematics*, pages 207--242. American Mathematical Society, 2011. M.C. Laskowski and C.A. Terry. Jumps in speeds of hereditary properties in finite relational languages. , 154:93--135, 2022. M. Lotz and J.A. Makowsky. On the algebraic complexity of some families of coloured tutte polynomials. , 32(1-2):327--349, 2004. L. Lovász and M. D. Plummer. , volume 367. American Mathematical Soc., 2009. J.A. Makowsky. On some conjectures connected with complete sentences. , 81(3):193--202, 1974. J.A. Makowsky. Colored Tutte polynomials and Kaufman brackets for graphs of bounded tree width. In *Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms*, pages 487--495, 2001. J.A. Makowsky. Algorithmic uses of the Feferman--Vaught theorem. , 126(1-3):159--213, 2004. J.A. Makowsky. Coloured tutte polynomials and kauffman brackets for graphs of bounded tree width. , 145(2):276--290, 2005. J.A. Makowsky. From a zoo to a zoology: Descriptive complexity for graph polynomials. In *Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006, Swansea, UK, June 30-July 5, 2006, Proceedings*, volume LNCS 3988, page 330. Springer, 2006. J.A. Makowsky. From a zoo to a zoology: Towards a general theory of graph polynomials. , 43:542--562, 2008. J.A. Makowsky and B.I. Zilber. Polynomial invariants of graphs and totally categorical theories. https://www.logique.jussieu.fr/modnet/Publications/Preprint%20server/papers/21/, 2006. MODNET Preprints, number 21. J. Mighton. . University of Toronto, 2000. M. Morley. Categoricity in power. , 114:514--38, 1965. T.G. Mustafin and A.D. Taimanov. Countable models of $\aleph_1$-categorical but not $\aleph_0$-categorical theories-cagorical theorieses. , 9:338--341, 1970. C. Papadimitriou. . Addison Wesley, 1994. M. G. Peretyatkin. An example for an $\aleph_1$:-categorical complete, finitely axiomatizable theory. , 19(3):317--347, 1980. A.A. Razborov. Flag algebras. , 72(4):1239--1282, 2007. A.A. Razborov. Flag algebras: an interim report. In *The Mathematics of Paul Erdős II*, pages 207--232. Springer, 2013. E.R. Scheinerman and J. Zito. On the size of hereditary classes of graphs. , 61(1):16--39, 1994. M. A. Taitslin. Model theory (russian). , 6, 1970. N. Trinajstic. . CRC press, 2018. L. Valiant. Reducibility by algebraic projections. In *Logic and Arithmetic: An International Symposium held in honour of Ernst Specker*, volume 30 of *L'enseignement Mathématique*, pages 365--380. Université de Genève, 1982. L. G. Valiant. The complexity of computing the permanent. , 8:189--201, 1979. B. I. Zilber. Totally categorical structures and combinatorial geometries. , 259(5):1039--1041, 1981. B.I. Zilber. , volume 117 of *Translations of Mathematical Monographs*. American Mathematical Society, 1993. [^1]: I would like to thank J. Kirby for various suggestion on how to improve the paper. [^2]: John Mighton is a Canadian mathematician, author, and playwright.\ https://en.wikipedia.org/wiki/John_Mighton

arxiv_math

--- abstract: | It is classical that uniform stabilization of solutions to the damped wave equation is equivalent to the geometric control condition The author previously showed that, when the damping depends on time, a generalization of the geometric control condition implies uniform stabilization at an exponential rate. In this paper, it is shown that this generalization of the geometric control condition is necessary for uniform stabilization at an exponential rate. Furthermore, when the damping does not satisfy this generalization, and has some additional structure, upper and lower bounds on non-exponential uniform stabilization are computed. The qualitative behavior of these upper and lower bounds coincide. author: - Perry Kleinhenz bibliography: - mybib.bib title: Sharp conditions for exponential and non-exponential uniform stabilization of the time dependent damped wave equation --- # Introduction Let $(M,g)$ be a smooth compact manifold without boundary and let $\Delta_g$ be the associated Laplace-Beltrami operator. Let $W \in L^{\infty} (M\times \mathbb{R})$ be a nonnegative function. Consider the damped wave equation with time dependent damping $$\label{TDWE} \begin{cases} (\partial_t^2 -\Delta_g + 2W(x,t)\partial_t)u=0 \\ (u,u_t)|_{t=0} =(u_0, u_1) \in H^1(M) \times L^2(M). \end{cases}$$ The standard object of study is the energy of the solution $$E(u,t) = \frac{1}{2} \int_M |\nabla_g u(x,t)|^2 + |\partial_t u(x,t)|^2 dx_g.$$ It is straightforward to compute $$\label{dtE} \frac{d}{dt} E(u,t) =-\int W(x,t) |\partial_t u(x,t)|^2 dx_g \leq 0.$$ Where the sign is guaranteed by $W(x,t) \geq 0$. Because of [\[dtE\]](#dtE){reference-type="eqref" reference="dtE"}, the energy is non-increasing, but there is no indication of a decay rate as $t \rightarrow\infty$. The most straightforward type of decay is uniform stabilization. That is, the existence of a function $r(t)\rightarrow 0$ as $t \rightarrow\infty$ such that $$E(u,t) \leq r(t) E(u,0).$$ When $W$ does not depend on time, uniform stabilization is equivalent to $W$ satisfying the Geometric Control Condition (GCC) [@Ralston1969; @RauchTaylor1975]. The GCC is satisfied if there exists some $L>0,$ such that every geodesic with length at least $L$ intersects the set $\{W>0\}$. There is an equivalent condition to the geometric control condition introduced in [@Lebeau1996]. The GCC is equivalent to the existence of $T_0,\overline{C}>0,$ such that for all unit speed geodesics $\gamma(t)$ and $T \geq T_0$ $$\frac{1}{T}\int_0^T W(\gamma(t)) dt \geq \overline{C}.$$ That is, there is a uniform lower bound on the long time average of the damping along any geodesic. This condition can be generalized when the damping depends on time **Assumption 1**. (Time-dependent geometric control condition or TGCC) Assume there exists $T_0, \overline{C}>0$ such that for all unit speed geodesics $\gamma(t)$, starting times $t_0 \in [0,\infty)$, and $T \geq T_0$ $$\frac{1}{T} \int_0^T W(\gamma(t), t_0 + t) dt \geq \overline{C}.$$ Before stating the result, a necessary technical assumption on the damping **Assumption 2**. Suppose for any $T>0$, there exists $C_T>0$ such that $\left| \left| \nabla^2 W \right| \right|_{L^{\infty}(M \times (0,T))} \leq C_T$ and $\left| \left| \partial_t W \right| \right|_{L^{\infty}(M \times (0,T))} \leq C_T$. It was previously shown in [@Kleinhenz2022a], that a uniformly continuous and uniformly bounded damping, which satisfies Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"} attains exponential uniform stabilization. The main result of this paper is that Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"} is *necessary* for exponential uniform stabilization. **Theorem 1**. Suppose $W(x,t)$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}. If there exists $C,c>0$ such that all solutions to [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} satisfy $$E(u,t+t_0) \leq C e^{-ct} E(u,t_0), \quad t, t_0 \geq 0,$$ then $W$ satisfies Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"}. For $W$ autonomous, solutions are a semigroup. Thus, when uniform stabilization occurs, the rate can always be taken $Ce^{-ct}$ for some $C,c>0$ and so the analogous result totally classifies uniform stabilization for $W$ autonomous. However when $W$ depends on time, solutions no longer form a semigroup and so in the absence of the TGCC uniform stabilization could occur with a non-exponential rate. The other main result of this paper is that non-exponential uniform stabilization indeed occurs. Define the smallest total amount of damping along any trajectory by time $t$, starting from time 0, as $$\Sigma(t) = \inf_{\gamma} \int_0^t W(\gamma(s), s) ds,$$ where the infimum is taken over all unit speed geodesics. When the damping is the product of a damping satisfying the TGCC and a bounded, positive function, the uniform stabilization rate is given by the exponential of the smallest total amount of damping along any trajectory. **Theorem 2**. Suppose $\widetilde{W} \in C^0_u(M \times [0,\infty))$, that is it is uniformly bounded and uniformly continuous, and it satisfies Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"}. Let $f \in L^{\infty}(M \times (0,\infty))$ be positive and let $W(x,t)=\widetilde{W}(x,t) f(x,t)$. Then there exists $C,c>0$, such that, for all $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} with $W=\widetilde{W} f$ $$E(u,t) < C E(u,0) \exp(-c\Sigma(t)).$$ Furthermore, if $\widetilde{W}$ and $f$ satisfy Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}, then $$E(u,t) < C E(u,0) \exp(-c \Sigma(t)),$$ cannot hold for $c> \left| \left| \widetilde{W} \right| \right|_{L^{\infty}} \left| \left| f \right| \right|_{L^{\infty}}$, for some solution. When $\Sigma(t)=o(t)$ this gives non-exponential uniform stabilization, and when $\Sigma(t)\leq C$ there is no uniform stabilization rate. There is a more detailed statement of this result, along with another two classes of examples, in Section [3](#examplesection){reference-type="ref" reference="examplesection"}. **Remark 1**. There are two ways the TGCC can fail: either the average amount of damping goes to 0 as $t$ goes to infinity or a trajectory $(\gamma(t),t)$ never meets $\{W>0\}$. The above result handles the former. In the the latter case there is not a general decay rate. To see this consider $M=\mathbb{S}^1=[0,2\pi)$ with endpoints identified. Let $\chi, \varphi\in C^{\infty}(\mathbb{S}^1),$ with disjoint and nonempty support. Then define $W(x,t) = \chi(x-t)$. Trajectories of the form $(x_0-t, t),$ with $x_0 \in \text{supp }(\varphi),$ never intersect $\{W>0\},$ so the TGCC is not satisfied. Furthermore $u(x,t) =\varphi(x-t)$ solves [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} and has $E(u,t) =E(u,0)$ for all $t$, so no decay occurs. For more details on this example see [@PaunonenSeifert2019]. ## Functional Analysis Framework The equation [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} can be reframed as a system of first order equations $$\partial_t \begin{pmatrix} u \\ v \end{pmatrix} = \mathcal{A}(t) \begin{pmatrix} u \\ v \end{pmatrix} \quad \mathcal{A}(t) = \begin{pmatrix} 0 & 1 \\ \Delta & -W(t) \end{pmatrix} \quad \begin{pmatrix} u(0) \\ v(0) \end{pmatrix} =\begin{pmatrix} u_0 \\ u_1 \end{pmatrix} \in \mathcal{H}$$ where $\mathcal{H}=H^1(M) \times L^2$. When $\mathcal{A}$ is autonomous it is the generator of a $C^0$ contraction semigroup $e^{t\mathcal{A}}$ and uniform stabilization estimates can be obtained by estimating $\|e^{t\mathcal{A}}\|_{\mathcal{L}(\mathcal{H})}$, because $\left| \left| \begin{pmatrix} u \\ v \end{pmatrix} \right| \right|_{\mathcal{H}} \simeq E(u,t)$. When $\mathcal{A}(t)$ is time dependent, there is no longer a semigroup structure, instead solutions are part of a uniform family, also called an evolution family. ## Literature Review When $M$ is a manifold without boundary and $W$ is autonomous, [@RauchTaylor1975] show that the geometric control condition (GCC) implies exponential decay. The techniques of [@Ralston1969] can be used to show the GCC is necessary for exponential decay. The autonomous GCC results were extended to manifolds with boundary in [@BardosLebeauRauch1992; @BurqGerard1997]. When $W$ is time-dependent and time periodic [@LRLTT] showed that a generalization of the GCC implies exponential uniform stabilization. This was generalized to fully time dependent damping in [@Kleinhenz2022a]. This result, together with Theorem [Theorem 1](#GCCnecessary){reference-type="ref" reference="GCCnecessary"}, show that Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"} characterizes exponential uniform stabilization for time-dependent $W$. For $W$ autonomous, the best possible exponential decay rate was computed by [@Lebeau1996], in terms of long time averages of the damping and the spectral abscissa of the stationary operator. This was extended to damping given by a $0$th order pseudodifferential operator in [@KeelerKleinhenz2023]. The challenge in generalizing these results to time-dependent damping is finding a replacement for the stationary operator. Despite this, the methods of those papers are used to show Theorem [Theorem 1](#GCCnecessary){reference-type="ref" reference="GCCnecessary"}. When the TGCC is not satisfied, exponential uniform stabilization cannot occur. If this is because a trajectory $(\gamma(t), t)$ never intersects the positive set of the damping then it is not possible to obtain a uniform stabilization rate, see Remark 1. However, in some cases, as shown in [@PaunonenSeifert2019], the solution decays to a periodic solution. For other time periodic damping see [@Wirthperiodic]. The other way for the TGCC to fail is for the damping to decrease as $t \rightarrow\infty$. This is a natural setting to consider from a physical perspective. Damping materials convert kinetic energy into heat, and this heat can decrease the efficacy of the damping [@lz23; @drda12]. Decreasing damping has been well studied for $M=\mathbb{R}^n$. Results go back to [@Matsumura1977; @Uesaka1980] which considered damping $W(x,t)$ bounded from below by $\frac{1}{1+t}$ and obtained polynomial uniform stabilization rates. See also [@Mochizuki1976]. This was extended to damping $W(x,t)$ bounded from below by $\frac{1}{(1+t) \log(1+t) \log(\log(1+t)) \cdots \log^{[n]}(1+t)},$ and rates of the form $\lim_{t \rightarrow\infty}E(u,t) (\log^{[n]}(1+t))^{C}=0,$ were obtained in [@MochizukiNakazawa1996; @MochizukiNakazawa2001]. Note in this case this is not a uniform stabilization rate. Improvements can be made when the damping is restricted to only depend on time. When $W(t)=\frac{\mu}{1+t},$ [@Wirth2004] obtained the sharp uniform stabilization rate $E(u,t) \leq E(u,0) (1+t)^{-\alpha}$ where $\alpha =\min(\mu,2)$. This rate agrees with the qualitative polynomial decay guaranteed by Theorem [Theorem 2](#decreasingtheorem){reference-type="ref" reference="decreasingtheorem"}. When $\limsup_{t \rightarrow\infty} t W(t) < M,$ and the damping satisfies some symbol style estimates, then [@Wirth2006] obtained a uniform stabilization rate $E(u,t) \leq E(u,0) \exp(-\Sigma(t))$, which agrees with Theorem [Theorem 2](#decreasingtheorem){reference-type="ref" reference="decreasingtheorem"}. On the other hand when $\lim_{t \rightarrow\infty}t W(t) = \infty,$ and the damping satisfies the same symbol estimates, [@Wirth2007] obtained a rate\ $\lim_{t \rightarrow\infty}E(u,t) \int_0^t \frac{ds}{W(s)}= 0$. This is weaker than Theorem [Theorem 2](#decreasingtheorem){reference-type="ref" reference="decreasingtheorem"}; for example with $W=(1+t)^{-\beta}$ with $\beta \in (0,1)$, this provides polynomial decay, while Theorem [Theorem 2](#decreasingtheorem){reference-type="ref" reference="decreasingtheorem"} gives sub-exponential decay. Analogous polynomial rates for $W(x,t) = a(x) b(t)$, with $b$ decreasing and $W(t) \simeq (1+t)^{-\beta}$ with $\beta \in (0,1)$ were proved in [@Kenigson2011]. The symbol assumptions of [@Wirth2006; @Wirth2007] were relaxed in [@HirosawaWirth2008] by considering small oscillations around such functions. This was relaxed further to damping $W$, not necessarily decreasing, satisfying $(1+t)^{-\beta} \simeq W(t)$ for $\beta<1$ in [@vjdl]. Note that Theorem [Theorem 2](#decreasingtheorem){reference-type="ref" reference="decreasingtheorem"} allows for much more general damping functions on manifolds. ## Outline In Section [2](#gaussianbeamsection){reference-type="ref" reference="gaussianbeamsection"}, Gaussian beams are used to obtain lower bounds on uniform stabilization rates, which proves Theorem [Theorem 1](#GCCnecessary){reference-type="ref" reference="GCCnecessary"}. In Section [3](#examplesection){reference-type="ref" reference="examplesection"}, three broad examples of damping which fail to satisfy the TGCC are considered. Uniform stabilization at non-exponential rates are guaranteed and lower bounds on these rates are provided using the results of Section [2](#gaussianbeamsection){reference-type="ref" reference="gaussianbeamsection"}. In two of the three cases these bounds coincide in a manner that indicates the correct qualitative rate has been obtained. Appendix A contains a short-time observability inequality for the wave equation when the observation window is the whole manifold. It could also be described as a short time energy equipartition result as the initial energy is estimated by the kinetic energy up to time $\delta$. **Acknowledgements** I would like to thank Willie Wong for recommending that I study decreasing dampings. I would like to thank Andras Vasy and Jared Wunsch for helpful conversations related to this work. # Lower bounds on uniform stabilization {#gaussianbeamsection} This section uses Gaussian beams to construct solutions of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} with slow uniform stabilization rates. First, these solutions are used to show Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"} is necessary for exponential uniform stabilization. Then, they are used to provide lower bounds for non-exponential rates. ## Proof that Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"} is necessary for Exponential Uniform Stabilization {#proof-that-assumption-tgcc-is-necessary-for-exponential-uniform-stabilization} Let $\gamma(s)$ be a unit speed geodesic. Define $$L(T) = \frac{1}{T} \inf_{\gamma, t_0 \in \mathbb{R}} \int_0^t W(\gamma(s), s+t_0) ds,$$ where the infimum in $\gamma$ is taken over all unit speed geodesics. Define $$L_{\infty} = \lim_{T \rightarrow\infty} L(T).$$ Given a fixed $W \in L^{\infty}(M \times \mathbb{R})$, define the best exponential decay rate as $$\alpha = \sup\{\beta \in \mathbb{R}: \exists C>0 \text{ s.t. } E(u, T+t_0) \leq C e^{-\beta T} E(u,t_0) \text{ for any } u \text { solving } \eqref{TDWE} \text{ and } t_0,T \geq 0\}.$$ When $W$ is autonomous, there is no need to include a start time $t_0$ in this definition. However, tracking the start time is useful in the time-dependent setting. **Proposition 3**. Suppose $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}. The best exponential decay rate is bounded by the long time average of the damping along space time geodesics. That is $$\alpha \leq 2 L_{\infty}.$$ In particular, if Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"} is not satisfied, then $L_{\infty}=0,$ and exponential decay cannot occur. This establishes Theorem 1. The proposition is proved by constructing solutions decaying arbitrarily close to, but slower than, this speed. This is done using the Gaussian beam construction of [@Ralston1982]. To introduce the result, suppose $A(t)$ is a symmetric $n \times n$ matrix-valued function with positive definite imaginary part. Let $\gamma(t)$ be a unit speed geodesic in $\mathbb{R}^n$ and set $$\psi(x,t) = \left\langle\gamma'(t), x-\gamma(t)\right\rangle + \frac{1}{2} \left\langle A(t)(x-\gamma(t)), (x-\gamma(t))\right\rangle.$$ Let $b \in \mathbb{C}^{\infty}(\mathbb{R}^n\times \mathbb{R}^n)$ then define $$\label{gaussianbeam} u_k(x,t) = k^{-1+\frac{n}{4}} b(x,t) e^{ik \psi(x,t)}$$ Given a geodesic $\gamma$, a function of this form can be constructed which is a quasimode of the wave equation. **Lemma 4**. [@Ralston1982] Fix $T>0$ and $(x_0, \xi_0) \in S^* \mathbb{R}^n$. Let $\gamma(t)$ be the geodesic with $(\gamma(0), \gamma'(0)) = (x_0, \xi_0)$. There exists $b \in C^{\infty}(\mathbb{R}^n\times \mathbb{R})$, and an $n \times n$ symmetric matrix-valued function $t \mapsto A(t)$, so that for $u_k$ as in [\[gaussianbeam\]](#gaussianbeam){reference-type="eqref" reference="gaussianbeam"} $$\label{gaussianquasi} \sup_{t \in [0,T]} \left| \left| \partial_t^2 u_k(\cdot, t) - \Delta_g u_k(\cdot, t) \right| \right|_{L^{2}} \leq Ck^{-\frac{1}{2}} \text{ for } k \geq 1.$$ Furthermore, for all $t \in [0,T]$ $$\lim_{k \rightarrow\infty} E(u_k, t) >0,$$ and the limit is always finite and independent of $t$. By the last assertion, it can be assumed that $\lim_{k \rightarrow\infty} E(u_k,t)=1$ for all $t \in [0,T]$. Using coordinate charts and a partition of unity, this construction can be extended to manifolds, which results in a sequence $\{u_k\} \subset C^{\infty}(M \times [0,\infty))$ such that $\lim_{k \rightarrow\infty}E(u_k, t) = 1$ and the appropriate analogue of [\[gaussianquasi\]](#gaussianquasi){reference-type="eqref" reference="gaussianquasi"} holds. Now given $(x_0, \xi_0) \in S^*M$ and $t_0 \in \mathbb{R}$, let $\gamma(s)$ be the unit speed geodesic with $(\gamma(0), \gamma'(0)=(x_0, \xi_0)$ and define $$G(\gamma,t_0,t) = \exp\left( - \int_{t_0}^{t+t_0} W(\gamma(s), s) ds \right).$$ Let $$\label{vkdef} v_k(x,t) = G(\gamma, t_0, t) u_k(x,t+t_0).$$ The following proposition shows that this $v_k$ is a quasisolution of the damped wave equation. The idea is that $u_k$ is concentrated along the geodesic $\gamma$ and to make it a quasisolution of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} it should decay exponentially in proportion to the total amount of damping it encounters along $\gamma$. This construction comes from [@Lebeau1996], where $W$ is autonomous. In this case, $G$ is constructed as the propagator of the defect measure for the damped wave equation, and was used in [@Klein2017; @KeelerKleinhenz2023] to prove analogous statements for matrix valued and pseudodifferential damping respectively. The key difference here is allowing $G$ to depend on the start time $t_0$. **Proposition 5**. Suppose $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}. Given $(x_0, \xi_0) \in S^* M$ and $t_0 \in \mathbb{R}$ let $u_k(t,x)$ be as in [\[gaussianbeam\]](#gaussianbeam){reference-type="eqref" reference="gaussianbeam"} and $v_k$ as in [\[vkdef\]](#vkdef){reference-type="eqref" reference="vkdef"}. For any $T>0$, there exists a constant $C_{T} >0,$ so that $$\sup_{t \in [0,T]} \left| \left| (\partial_t^2 - \Delta_g +2W(x,t+t_0) \partial_t) v_k(\cdot, t) \right| \right|_{L^{2}} \leq C_{T} k^{-\frac{1}{2}}.$$ *Proof.* Directly compute $$\begin{aligned} (\partial_t^2 -\Delta_g +2W(x,t+t_0) \partial_t) G u_k&= G (\partial_t^2 -\Delta_g) u_k\\ &+ 2 \partial_t G \partial_t u_k + (\partial_t^2 G) u_k \\ &+ 2 W(x,t+t_0) \partial_t G u_k+ 2W(x,t+t_0) G \partial_t u_k. %(\p_t^2 -\Delta_g +& 2W(x,t+t_0) \p_t) G(x_0,\xi_0,t_0) u_k(x,t+t_0) = G (\p_t^2 -\Delta_g) u_k(x,t+t_0)\\ %&+ 2 \p_t G \p_t u_k(x,t+t_0) + (\p_t^2 G) u_k(x,t+t_0) \\ %&+ 2 W(x,t+t_0) \p_t G u_k(x,t+t_0) + 2W(x,t+t_0) G \p_t u_k(x,t+t_0).\end{aligned}$$ Then since $\partial_t G(\gamma,t_0, t)=-W(\gamma(t+t_0), t+t_0)) G(\gamma,t_0,t)$, $$\begin{aligned} (\partial_t^2 -\Delta_g +2W(x,t+t_0) \partial_t) G u_k&= G (\partial_t^2 -\Delta_g) u_k \nonumber \\ &+ 2(W(x,t+t_0)-W(\gamma(t+t_0), t+t_0)) G \partial_t u_k \label{quasiest}\\ &+ Gu_k(x,t+t_0) \bigg(-\partial_t W(\gamma(t+t_0), t+t_0) \nonumber\\ &+W(\gamma(t+t_0), t+t_0)^2 \nonumber \\ &-2W(x,t+t_0) W(\gamma(t+t_0), t+t_0)\bigg). \nonumber\end{aligned}$$ By the construction of $u_k$ and the boundedness of $G$ $$\sup_{t \in [0,T]} \left| \left| G (\partial_t^2 -\Delta_g) u_k(\cdot, t+t_0) \right| \right|_{L^{2}} \leq O(k^{-1/2}).$$ Since $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"} and $G$ is bounded $$\begin{aligned} \sup_{t \in [0,T]}& \left| \left| Gu_k(x,t+t_0) \left(-\partial_t W+W^2 -2W(x,t+t_0) W\right) \right| \right|_{L^{2}} \leq C \sup_{t\in [0,T]} \left| \left| u_k(\cdot, t) \right| \right|_{L^{2}} =O(k^{-1}), \nonumber\end{aligned}$$ where the final equality follows from the fact that $\int_{\mathbb{R}^n}k^{\frac{n}{2}} e^{-k|y|^2} dy$ is uniformly bounded in $k$. So it remains to control [\[quasiest\]](#quasiest){reference-type="eqref" reference="quasiest"}. To do so, at each time $t$, Taylor expand $W$ around $\gamma(t+t_0)$ $$W(x,t+t_0) = W(\gamma(t+t_0), t+t_0) + (x-\gamma(t+t_0)) \cdot \nabla_x W(\gamma(t+t_0), t+t_0) + C_2 (x-\gamma(t+t_0))^2,$$ where $C_2 \leq \left| \left| \nabla^2 W \right| \right|_{L^{\infty}}$. So then $$\begin{aligned} 2(W(x,t+t_0)-&W(\gamma(t+t_0), t+t_0)) G \partial_t u_k(x,t+t_0) \\ &= \left( 2(x-\gamma(t+t_0)) \cdot \nabla_x W(\gamma(t+t_0), t+t_0) + C_2 (x-\gamma(t+t_0))^2 \right) G \partial_t u_k\end{aligned}$$ Taking the $L^2$ norm, letting $q=t+t_0$ and $C_1 = \left| \left| \nabla W \right| \right|_{L^{\infty}}$ $$\begin{aligned} & \left| \left| 2(W(x,t+t_0)-W(\gamma(t+t_0), t+t_0) G \partial_t u_k(x,t+t_0) \right| \right|_{L^{2}}^2 \\ &\leq \left| \left| \left( C_1(x-\gamma(q))+C_2(x-\gamma(q))^2 \right) k^{n/4} e^{ik\left\langle x-\gamma(q), \gamma'(q) \right\rangle} e^{ik/2 \left\langle A(x-\gamma(q)), (x-\gamma(q))\right\rangle} b(x) \right| \right|_{L^{2}}^2 \\ &\leq k^{n/2} \int_{\mathbb{R}^n}\left|\left(C_1(x-\gamma(q)) + C_2(x-\gamma(q))^2 \right) e^{ik/2(\left\langle A(x-\gamma(q)), (x-\gamma(q))\right\rangle}\right|^2 dx\end{aligned}$$ Taking a change of variables $k^{1/2} (x-\gamma(q)) = y, dx = k^{-n/2} dy,$ this becomes $$\left| \left| 2(W(x,t+t_0)-W(\gamma(t+t_0), t+t_0) G \partial_t u_k \right| \right|_{L^{2}}^2 \leq C(k^{-1}+ k^{-2}) \int_{\mathbb{R}^n}|y e^{i/2 \left\langle Ay,y\right\rangle} |^2 dy \leq Ck^{-1}.$$ So, taking the supremum over $t \in [0,T],$ each term is $O(k^{-1/2})$ which proves the desired conclusion. ◻ Now, for any geodesic $\gamma$ and starting time $t_0,$ exact solutions to [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} can be constructed with energy approaching $G(\gamma,t_0,t)^2$ **Proposition 6**. Suppose $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}. Given any $T>0$, any $\varepsilon>0$, any unit-speed geodesic $\gamma(t)$, and any $t_0 \in \mathbb{R}$, there exists an exact solution $u$ of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} with $$|E(u,t_0)-1|<\varepsilon,$$ and for any $t \in [0,T]$ $$|E(u,t_0+t) - G(\gamma,t_0,t)^2 | < \varepsilon.$$ Furthermore, for such a solution, for any $t \in [0,T]$ $$E(u,t_0+t) > E(u,t_0) (G(\gamma,t_0,t)^2-2\varepsilon).$$ *Proof.* Let $u_k, v_k$ be as defined above. Then define $\omega_k$ as the unique solution of $$\begin{cases} (\partial_t^2 -\Delta + 2W(x,t+t_0) \partial_t) \omega_k =0, \\ \omega_k(x,0)= v_k(x,0)=u_k(x,t_0),\\ \partial_t \omega_k(x,0) = \partial_t v_k(x,0)=\partial_t u_k(x,t_0). \end{cases}$$ The idea is to choose $k$ large enough so that $\omega_k$ has the desired properties. 1\) It is immediate that $$E(\omega_k, 0) = E(v_k, 0) = E(u_k, t_0) \rightarrow 1 \text{ as } k \rightarrow\infty.$$ 2) To see that, for $t \in [0,T]$, $\lim_{k \rightarrow\infty}E(\omega_k, t)=G(\gamma,t_0,t)^2$, first note by the triangle inequality $$|E(\omega_k, t)^{1/2} - E(v_k, t)^{1/2} | \leq E(\omega_k -v_k, t)^{1/2}.$$ So it suffices to prove the following 1. For $t \in [0,T]$ $$\lim_{k \rightarrow\infty} E(v_k, t) = G(\gamma, t_0, t)^2.$$ 2. For $t \in [0,T]$ $$\lim_{k \rightarrow\infty} E(\omega_k -v_k, t) =0.$$ To see the first note $$E(v_k, t) = \frac{1}{2} \int |G\partial_t u_k(x,t+t_0) - W(\gamma(t+t_0), t+t_0) G u_k(x,t+t_0)|^2 + |G \nabla u(x,t+t_0)|^2 dx_g.$$ Since $W$ and $G$ are bounded, and by construction of $u_k$, $$\int| W(\gamma(t+t_0), t+t_0) G u_k(x,t+t_0)|^2 \leq C \left| \left| u_k(\cdot, t) \right| \right|_{L^{2}}^2 \leq C k^{-2}.$$ And so $$\lim_{k\rightarrow\infty} E(v_k, t) = \lim_{k \rightarrow\infty} \frac{1}{2} \int |G \partial_t u_k|^2 + |G \nabla u_k|^2 dx_g = |G(\gamma, t_0, t)|^2 \lim_{k \rightarrow\infty} E(u_k, t+t_0) = G(\gamma,t_0,t)^2,$$ where in the final equality the fact that $\lim_{k \rightarrow\infty}E(u_k, t)=1$ for $t \in [0,T]$ was used. To control $E(\omega_k- v_k, t),$ let $f_k = (\partial_t^2 -\Delta + 2W(x,t+t_0) \partial_t )v_k$. Then $$(\partial_t^2 -\Delta +2W(x,t+t_0) \partial_t) (\omega_k - v_k) = f_k.$$ Taking the time derivative of $E(\omega_k-v_k,t)$ $$\begin{aligned} \partial_t E(\omega_k -v_k, t) &= \frac{1}{2} \int_M (\partial_t^2 -\Delta_g) (\omega_k-v_k) \overline{\partial_t (\omega_k - v_k)}+ (\partial_t^2 -\Delta_g)\overline{(\omega_k-v_k)} \partial_t(\omega_k-v_k) dx_g \\ &=\text{Re }\int_M (f_k -2W(x,t+t_0) \partial_t(\omega_k -v_k)) \partial_t \overline{(\omega_k-v_k)} dx_g.\end{aligned}$$ Note the second term of this product is non-positive since $W$ is nonnegative. To control the first term compute $$\left| \left| \partial_t(\omega_k-v_k) \right| \right|_{L^{2}} \leq E(\omega_k, t) + E(v_k, t) \leq E(\omega_k, 0) + E(v_k,0) =2E(u_k,t_0) = 2E(u_k,0),$$ which is uniformly bounded by a constant, for $k$ large enough, since $E(u_k,0) \rightarrow 1$ as $k \rightarrow\infty$. This, along with the control over $f_k$ from Proposition [Proposition 5](#quasiprop){reference-type="ref" reference="quasiprop"}, gives $$\sup_{t \in [0,T]} |\partial_t E(\omega_k -v_k, t)| \leq \sup_{t \in [0,T]} \left|\int f_k \partial_t \overline{(\omega_k -v_k)} dx_g \right| \leq \left| \left| f_k(\cdot, t) \right| \right|_{L^{2}} \left| \left| \partial_t (\omega_k -v_k) \right| \right|_{L^{2}} \leq C_T k^{-1/2}.$$ Since $E(\omega_k-v_k,0)=0,$ integrating this time derivative gives $$E(\omega_k-v_k, t) \leq C_{T} T k^{-1/2}.$$ So indeed for any $t \in [0,T]$, $\lim_{k \rightarrow\infty}E(\omega_k, t) = G(\gamma,t_0,t)^2$. 3\) Now let $k$ be large and let $u(x,t)=\omega_k(x,t-t_0)$. So $E(u,t_0)=E(\omega_k,0),$ and for $k$ large enough $$\begin{aligned} |E(u,t_0)-1| &= |E(\omega_k,0)-1|<\varepsilon\\ |E(u,t_0+t)- G(\gamma, t_0, t)^2| &= |E(\omega_k, t) - G(\gamma, t_0, t)^2|<\varepsilon.\end{aligned}$$ For any $t \in [0,T],$ since $1>E(u,t_0)-\varepsilon$ and $E(u,t+t_0) > G(x_0,\xi_0,t_0)^2 -\varepsilon$, $$\begin{aligned} E(u,T+t_0) &> E(u,t+t_0) (E(u,t_0) - \varepsilon) \\ &>E(u,t_0) ( G(\gamma,t_0,t)^2 - \varepsilon) - \varepsilon E(u,t+t_0) \\ &> E(u,t_0) (G(\gamma,t_0,t)^2 -\varepsilon) - \varepsilon E(u,t_0) \\ &>E(u,t_0)(G(\gamma,t_0,t)^2 -2\varepsilon),\end{aligned}$$ where the third inequality followed from the fact that energy of solutions to [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} is non-increasing. ◻ Before proving Proposition [Proposition 3](#bestboundedlongtimeprop){reference-type="ref" reference="bestboundedlongtimeprop"}, it is necessary to see that $t L(t)$ is super additive. This is the crucial step where taking the infimum over starting times in the definition of $L(t)$ is used. Well $$\begin{aligned} (t+r) L(t+r) &= \inf_{\gamma, t_0 \in \mathbb{R}} \int_0^{r+t} W(\gamma(s), t_0+s) ds \\ &= \inf_{\gamma, t_0 \in \mathbb{R}} \left( \int_0^r W(\gamma(s), t_0+s) ds + \int_r^{r+t} W(\gamma(s), t_0+s) ds \right)\\ & \geq \inf_{\gamma, t_0 \in \mathbb{R}} \int_0^r W(\gamma(s), t_0+s) ds + \inf_{\gamma, t_0 \in \mathbb{R}} \int_r^{r+t} W(\gamma(s),t_0+s) ds \\ &= \inf_{\gamma, t_0 \in \mathbb{R}} \int_0^r W(\gamma(s), t_0+s)ds + \inf_{\gamma, t_0 \in \mathbb{R}} \int_0^t W(\gamma(s), t_0+s) ds \\ &= tL(t) + rL(r). \end{aligned}$$ Therefore by Fekete's Lemma $L_{\infty} = \lim_{t \rightarrow\infty} L(t) = \sup_{t \in [0,\infty)} L(t)$ and thus $L(t) \leq L_{\infty}$ for all $t$. Now it can be shown that $\alpha \leq 2 L_{\infty}$. *Proof of Proposition [Proposition 3](#bestboundedlongtimeprop){reference-type="ref" reference="bestboundedlongtimeprop"}.* Assume otherwise, so $\alpha=2 L_{\infty} + 3 \eta$ for some $\eta >0$. Then since $2(L_{\infty} + \eta) < \alpha$, there exists $C>0,$ such that for all $t \geq 0$, all $t_0 \in \mathbb{R},$ and all solutions $u$ of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} $$E(u,t+t_0) \leq C E(u,t_0) e^{-2t(L_{\infty}+\eta)}.$$ To remove the factor of $C$ on the right hand side choose $T>0$ large enough, so that $\max(C,1) < e^{T \eta},$ then $$C e^{-2T(L_{\infty}+\eta)} < e^{-2TL_{\infty}-T\eta}.$$ Since $L(T) \leq L_{\infty}$ for all $T$ $$\label{Linfeq} Ce^{-2T(L_{\infty}+\eta)} < e^{-2TL_{\infty} - T\eta} \leq e^{-2TL(T)- T \eta}.$$ Now note that $$L(T)=-\frac{1}{T} \sup_{\gamma, t_0 \in \mathbb{R}} \ln(G(\gamma, t_0, T)).$$ So there exists a geodesic $\gamma$ and a starting time $t_0 \in \mathbb{R}$ such that $\ln G(\gamma,t_0,T) > -TL(T) - \frac{T \eta}{2}$. Therefore $$\label{GLTeq} e^{-2TL(T)-T \eta} < G(\gamma,t_0,T)^2.$$ Combining [\[Linfeq\]](#Linfeq){reference-type="eqref" reference="Linfeq"} and [\[GLTeq\]](#GLTeq){reference-type="eqref" reference="GLTeq"}, there exists $\delta>0,$ such that $$C e^{-2T(L_{\infty}+\eta)} < G(\gamma, t_0, T)^2 - \delta.$$ Now by Proposition [Proposition 6](#energyapproxprop){reference-type="ref" reference="energyapproxprop"}, there exists an exact solution $u$ of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} such that $$E(u,T+t_0) > E(u,t_0)(G(\gamma, t_0, T)^2-\delta).$$ Therefore $$E(u,T+t_0) > E(u,t_0) \left( G(\gamma,t_0,T)^2 -\delta\right) > C E(u,t_0) e^{-2T(L_{\infty} + \eta)},$$ which is a contradiction. Therefore, $\alpha \leq 2L_{\infty}$. ◻ ## Lower bounds on energy decay for damping not satisfying Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"} {#lower-bounds-on-energy-decay-for-damping-not-satisfying-assumption-tgcc} When Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"} is not satisfied, the same approach can still be used to obtain lower bounds on uniform stabilization rates. In this case it is more convenient to work with $\Sigma(t)=\inf_{\gamma} \int_0^t W(\gamma(s), s) ds$, as it distinguishes how $L(t)$ approaches $0$ as $t \rightarrow\infty$. Note that the infimum is no longer taken over starting times $t_0$. This was necessary to show $tL(t)$ was sub-additive in the previous setting, but here because $\Sigma(t) \rightarrow 0$ as $t \rightarrow\infty$, taking the $\inf$ over starting times would always produce zero. **Lemma 7**. Suppose $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}. For any $\delta, T>0$ there exists a solution $u$ to [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} such that $$E(u,T) > E(u,0) e^{-2\Sigma(T)-\delta}.$$ *Proof.* Well $$\Sigma(T) = - \sup_{\gamma} \ln(G(\gamma,0,T)).$$ Thus there exists a geodesic $\gamma$, such that $$-\Sigma(T) - \frac{\delta}{2} < \ln(G(\gamma,0,T)).$$ Therefore $$\exp(-2\Sigma(T)- \delta) < G(\gamma,0,T)^2$$ By Proposition [Proposition 6](#energyapproxprop){reference-type="ref" reference="energyapproxprop"}, choosing $\varepsilon>0$ small enough, so that $G(\gamma,0,T)^2- 2\varepsilon> \exp(-2\Sigma(T)-\delta)$, there exists $u$ a solution of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} such that $$\begin{aligned} E(u,T) > E(u,0) (G(\gamma,0,T)^2 - 2\varepsilon) \\ > E(u,0) \exp(-2\Sigma(T) - \delta).\end{aligned}$$ ◻ Note, $\Sigma(t)$ is increasing, so either $\Sigma(t) \leq C$ or $\lim_{t \rightarrow\infty} \Sigma(t)=\infty$. When $\Sigma$ is bounded, there is not a uniform stabilization rate. **Lemma 8**. Suppose $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}. Assume $\Sigma(t) \leq C$. Then for all $\delta, t_0>0$ there exists a solution $u$ of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} such that $$E(u,t_0) > E(u,0) \exp(-2C-\delta).$$ *Proof.* By Lemma [Lemma 7](#LTenergy){reference-type="ref" reference="LTenergy"}, given $\delta, t_0>0$ there exists $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} such that $$\begin{aligned} E(u,t_0) &> E(u,0) \exp(-2\Sigma(t_0) -\delta) \\ &> E(u,0) \exp(-2C - \delta).\end{aligned}$$ ◻ On the other hand when $\lim_{t \rightarrow\infty} \Sigma(t) = \infty$, uniform stabilization can occur at a rate proportional to $G(\gamma,0,t)$. This is exactly what one should expect based on the construction and behavior of the quasimodes $v_k$ in the preceeding section. **Proposition 9**. Assume $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"} and suppose $\lim_{t \rightarrow\infty}\Sigma(t)=\infty$. Define $$\sigma = \sup \{\beta \in \mathbb{R}; \exists C>0 \text{ such that } E(u,t) \leq C E(u,0) \exp(-\beta \Sigma(t)) \, t \geq 0, \forall u \text{ solving } \eqref{TDWE} \}.$$ Then $\sigma \leq 2$. *Proof.* Assume otherwise, so there exists $\eta>0$ such that $\sigma = 2 + 3 \eta$. Then since $2 + 2\eta < \sigma$, there exists $C>0,$ such that for all $t \geq 0$ and all solutions $u$ of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} $$E(u,t) \leq C E(u,0) \exp(-(2+2\eta) \Sigma(t)).$$ Since $\lim_{t \rightarrow\infty}\Sigma(t) =\infty$, $T>0$ can be chosen large enough, so that $\exp(\eta \Sigma(T))>\max(C, e^{1/2})$. Then $$C \exp(-(2+2\eta) \Sigma(T)) < \exp(-(2+\eta) \Sigma(T)).$$ Now, by Lemma [Lemma 7](#LTenergy){reference-type="ref" reference="LTenergy"} there exists $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"}, such that $$\begin{aligned} E(u,T) &> E(u,0) \exp\left(-2\Sigma(T) - \frac{1}{2}\right) \\\ &>E(u,0) \exp(-(2+\eta) \Sigma(T)) \\ &> C E(u,0) \exp(-(2+2\eta) \Sigma(T)) \\ &>E(u,T)\end{aligned}$$ This is a contradiction and so the desired conclusion must hold. ◻ **Remark 2**. When $\Sigma(t)=C\ln(t)$, then decay can be no faster than $t^{-2C}$.\ When $\Sigma(t)=\frac{C}{t^{\alpha}}$ and $\alpha \in [0,1)$, then decay can be no faster than $\exp(-2 t^{1-\alpha})$.\ When $\Sigma(t)=\ln(\ln(t)^{\alpha}), \alpha>0$, then decay can be no faster than $\frac{1}{\ln(2+t)^{2\alpha}}$. # Non-exponential uniform stabilization rates {#examplesection} In this section, uniform stabilization rates are proved for three examples. The idea is to obtain quantitative observability estimates for solutions of the damped wave equation and then use the following lemma to obtain decay rates. Lower bounds are obtained by estimating $\Sigma(t)$ and then using Lemma [Lemma 8](#lowerboundlemma){reference-type="ref" reference="lowerboundlemma"} or Proposition [Proposition 9](#lowerboundprop){reference-type="ref" reference="lowerboundprop"}. Lower bounds require that $W$ satisfy Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}. First, I will introduce the notion of observability for wave equations. The wave equation is $$\label{WE} \begin{cases} (\partial_t^2 -\Delta_g )\psi=0 \\ (\psi,\psi_t)|_{t=0} =(\psi_0, \psi_1) \in H^1(M) \times L^2(M). \end{cases}$$ We say [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} (or [\[WE\]](#WE){reference-type="eqref" reference="WE"} resp.) is observable by $V: M \times [0,T] \rightarrow[0, \infty)$ in time $T$, if there exists $C>0$, such that $$E(u,0) \leq C \int_0^T \int_M V(x,t) |\partial_t u|^2 dx dt,$$ for all solutions $u$ of either [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} (or [\[WE\]](#WE){reference-type="eqref" reference="WE"} resp.). Such an inequality is called an observability inequality and $V$ is called the observability operator. The first lemma converts observability inequalities, that depend on the start-time of the observation window, into energy decay rates. **Lemma 10**. Suppose $u$ solves [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"}, let $b: \mathbb{N}_0 \rightarrow[0,\infty)$ and define $B(k)=\sum_{j=0}^{k-1} b(j)$. Assume there exists $T_0>0,$ such that for any $j \in \mathbb{N}_0$ $$b(jT_0) E(u,jT_0) \leq \int_{jT_0}^{(j+1)T_0} \int_M W |\partial_t u|^2 dx dt.$$ Then $$E(u,kT_0) \leq E(u,0) \exp(-B(k)).$$ *Proof.* Integrating [\[dtE\]](#dtE){reference-type="eqref" reference="dtE"} from $jT_0$ to $(j+1)T_0$ and then using the hypothesis $$\begin{aligned} E(u,(j+1)T_0) - E(u,jT_0) &= - \int_{jT_0}^{(j+1)T_0} \int_M W |\partial_t u|^2 dx dt \\ &\leq -b(jT_0) E(u,jT_0).\end{aligned}$$ So $E(u,(j+1)T_0) \leq (1- b(jT_0)) E(u,t_0)$. Apply this iteratively from $j=0$ to $j=k-1$ to obtain $$E(u,kT) \leq E(u,0) \prod_{j=0}^{k-1} (1- b(jT_0)).$$ Now $$\ln \left( \prod_{j=0}^{k-1} (1-b(jT_0)) \right) = \sum_{j=0}^{k-1} \ln(1- b(jT_0)).$$ Since $\ln(x) \leq 1-x$, $\ln(1-b(jT_0)) \leq - b(jT_0)$. Therefore $$\begin{aligned} \sum_{j=0}^{k-1} \ln(1- b(jT_0)) \leq \sum_{j=0}^{k-1} - b(j) = - B(k).\end{aligned}$$ Exponentiating this gives the desired conclusion. ◻ Some mild structure is required for time dependent damping to obtain such an observability estimate. In the autonomous case, this $b$ is related to $\Sigma(t)$ (or $L(t)$), but explicit formulas require taking $t \rightarrow\infty$ [@HumbertPrivatTrelat2019] or give rough bounds like $b(t) \leq e^{\Sigma(t)}$ [@LaurentLeautaud2016]. There are not analogous results in the time dependent setting. The typical way to prove such an estimate is to first prove an observability inequality for the wave equation and then convert it into an observability inequality for the damped wave equation. Before proceeding to the examples, I will introduce an assumption, then state an observability result for the wave equation, and a connection between wave and damped wave observability inequalities. First, a condition introduced in [@LRLTT] that generalizes the geometric control condition and is a finite time version of the TGCC. **Assumption 3**. Fix $T>0$ and consider $Q,$ an open set in $M \times (0,T)$. Let $\gamma(t)$ be a unit speed geodesic. Assume for all such $\gamma$ there exists $t \in (0,T),$ such that $(\gamma(t),t) \in Q$. Next, a wave equation observability inequality with time-dependent observation operator, first proved by [@LRLTT Theorem 1.8], and then relaxed to general initial data by [@Kleinhenz2022a Proposition 2.2]. **Proposition 11**. Suppose $Q$ satisfies Assumption [Assumption 3](#ftGCC){reference-type="ref" reference="ftGCC"} and let $\chi_Q$ be the indicator function on $Q$. Then there exists $C_Q>0,$ such that for all $\psi$ solving [\[WE\]](#WE){reference-type="eqref" reference="WE"} $$\label{eq:observe} \frac{1}{2} \left( \left| \left| \nabla \psi_0 \right| \right|_{L^{2}}^2 + \left| \left| \psi_1 \right| \right|_{L^{2}}^2\right) = E(\psi,0) \leq C_Q \int_0^T \int_M |\chi_Q \partial_t \psi|^2 dx\,dt.$$ There is a connection between observability for the wave equation and observability for the damped wave equation, so long as the observability operator is the damping. The exact statement used here is [@PaunonenSeifert2019 Lemma 3.3]. **Lemma 12**. Let $(u_0, u_1) \in H^1 \times L^2(M)$. Suppose $W \in L^{\infty}(M \times [0,T])$, $u$ solves [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} and $\psi$ solves [\[WE\]](#WE){reference-type="eqref" reference="WE"} with $$(u, \partial_t u)|_{t=0}= (\psi, \partial_t \psi)|_{t=0}=(u_0, u_1).$$ Then $$\int_{0}^T \int_M W |\partial_t \psi|^2 dx\,dt \leq \left(1 +2 T \left| \left| W \right| \right|_{L^{\infty}}\right)^2 \int_0^T \int_M W |\partial_t u|^2 dx\,dt.$$ ## Decreasing Damping Suppose $\widetilde{W}(x,t) \in C^0_u(M \times (0,\infty))$ satisfies Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"}. Suppose $f \in L^{\infty}(M \times (0,\infty))$ is positive and there exists $b \in L^{\infty}([0,\infty))$, decreasing such that $$b(t) \leq f(x,t)$$ Also define $C_W= \left| \left| \widetilde{W} \right| \right|_{L^{\infty}}$. Let $$\label{decreasedampdef} W(x,t)=f(x,t) \widetilde{W}(x,t).$$ In such a setup, the uniform stabilization rate is specified by the exponential of $\Sigma(t)$. The following is a more detailed statement of Theorem [Theorem 2](#decreasingtheorem){reference-type="ref" reference="decreasingtheorem"} **Proposition 13**. With $W$ as in [\[decreasedampdef\]](#decreasedampdef){reference-type="eqref" reference="decreasedampdef"}, there exists $C,c>0$ such that for all $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} $$\label{energysigma} E(u,t) \leq C E(u,0) \exp(-c \Sigma(t)).$$ Furthermore, suppose $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"} 1. If $\lim_{t \rightarrow\infty} \Sigma(t) = \infty$, then [\[energysigma\]](#energysigma){reference-type="eqref" reference="energysigma"} cannot hold with $c>2$. 2. If $\Sigma(t)$ is bounded, then there exists $c \in (0,1)$ such that for all $T$, there exists $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} satisfying $$E(u,T) > cE(u,0).$$ Furthermore, suppose there exists $C_m, C_M>0$ such that $C_m (t+1)^{-\beta} \leq f(x,t) \leq C_M (t+1)^{-\beta}$ with $\beta \geq 0$. Then, for all $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} 1. If $\beta<1$, sub exponential decay occurs that is $E(u,t) < C E(u,0) \exp(-c t^{1-\beta})$ and this cannot hold for $c> \frac{2C_M C_W}{(1-\beta)}$. 2. If $\beta=1$, polynomial decay occurs, that is $E(u,t) < C E(u,0) \left\langle t\right\rangle^{-c}$ and this cannot hold for $c>2C_M C_W$. 3. If $\beta>1$, there is not a uniform stabilization rate. This exactly agrees with the distinction between "effective\" and "non-effective\" dissipation of [@Wirth2004; @Wirth2006; @Wirth2007; @Matsumura1977] but allows for more general damping. For example $f$ need not be decreasing and $W$ can be (not-identically) 0. This proposition is proved by obtaining a damped wave observability inequality that depends on $b$, which is then converted into a uniform stabilization rate by Lemma [Lemma 10](#observetodecay){reference-type="ref" reference="observetodecay"}. **Lemma 14**. Suppose $u$ solves [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} with $W$ as in [\[decreasedampdef\]](#decreasedampdef){reference-type="eqref" reference="decreasedampdef"}. There exists $C >0$, such that for all $s \in [0,\infty)$ $$C b(s+T_0) E(u,t_0) \leq \int_s^{s+T_0} \int_M W|\partial_t u|^2 dx dt.$$ The proof follows that of [@Kleinhenz2022a Proposition 2.1], but is modified to account for $b(s)$. *Proof.* Assume otherwise, so there exist sequences $\{t_j\}, \{u_j\}$ such that $u_j$ solves [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} and $$\label{contra} b(t_j+T_0) E(u_j, t_j) =1, \qquad \lim_{j \rightarrow\infty}\int_{t_j}^{t_j+T_0} \int_M W |\partial_t u_j|^2 dx \, dt =0.$$ Let $v_j(x,t) = u_j(x,t+t_j)$ and $W_j(x,t) = W(x,t+t_j), \widetilde{W}_j(x,t)=\widetilde{W}(x,t+t_j)$, $f_j(x,t)=f(x,t+t_j), b_j(T_0)=b(t_j+T_0)$. So $$\begin{cases} (\partial_t^2 - \Delta + W_j \partial_t ) v_j =0 \\ (v_j, \partial_t v_j)|_{t=0} =: (v_{0,j}, v_{1,j}), \end{cases}$$ and $$b_j(T_0) E(v_j, 0)=1, \qquad \lim_{j \rightarrow\infty}\int_0^{T_0} \int_M W_j|\partial_t v_j|^2=0.$$ Note that $\{\widetilde{W}_j\}$ forms a pointwise bounded family, since $\widetilde{W} \in L^{\infty}(M \times [0,\infty))$, and $\{\widetilde{W}_j\}$ is an equicontinuous family in $C(M \times [0,T])$, by the uniform continuity of $\widetilde{W}$ on $M \times [0,\infty)$. Therefore by Arzelà-Ascoli [@BabyRudin Theorem 7.25] there exists $\widetilde{W}_{\infty} \in C(M \times [0,T])$ such that, after potentially replacing $\widetilde{W}_j$ by a subsequence, $\widetilde{W}_j \rightarrow\widetilde{W}_{\infty}$ in $L^{\infty}(M \times [0,T])$. Recall $\overline{C}$ from Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"}. Now, the claim is that $\left\{\widetilde{W}_{\infty}>\frac{\overline{C}}{2}\right\}:=Q$ satisfies Assumption [Assumption 3](#ftGCC){reference-type="ref" reference="ftGCC"}. To see this, choose $J$ large enough so that $\|\widetilde{W}_{\infty}-\widetilde{W}_j\|_{L^{\infty}} < \frac{\overline{C}}{2}$ for $j \geq J$. For any geodesic $\gamma(t)$, by Assumption [Assumption 1](#TGCC){reference-type="ref" reference="TGCC"} $$\begin{aligned} \frac{1}{T_0} \int_0^{T_0} \widetilde{W}_{\infty} (\gamma(t),t)dt &= \frac{1}{T_0} \int_0^{T_0} \widetilde{W}_j(\gamma(t),t) dt + \frac{1}{T_0} \int_0^{T_0} \widetilde{W}_{\infty} (\gamma(t),t)-\widetilde{W}_j(\gamma(t),t)dt\\ &\geq \frac{1}{T_0} \int_0^{T_0} \widetilde{W}(\gamma(t),t+t_j) dt - \frac{1}{T_0} \int_0^{T_0} \|\widetilde{W}_{\infty}-\widetilde{W}_j\|_{L^{\infty}_0} dt \\ &\geq \overline{C}- \frac{\overline{C}}{2}=\frac{\overline{C}}{2}.\end{aligned}$$ Because the average of $\widetilde{W}_{\infty}$ on $(0,T_0)$ is at least $\frac{\overline{C}}{2}$, $\widetilde{W}_{\infty}$ must be at least $\frac{\overline{C}}{2}$ at some point $(\gamma(t),t)$ for each geodesic $\gamma(t)$. So indeed, $\{\widetilde{W}_{\infty}>\frac{\overline{C}}{2}\}=Q$ satisfies Assumption 2. Now let $\psi_j$ solve [\[WE\]](#WE){reference-type="eqref" reference="WE"} with $(\psi_j, \partial_t \psi_j)|_{t=0} = (v_{0,j}, v_{1,j})$. Then by Proposition [Proposition 11](#observeprop){reference-type="ref" reference="observeprop"}, there exists $C_Q>0$ such that $$E(v_j, 0) = E(\psi_j, 0) \leq C_Q \int_0^{T_0} \int_M \widetilde{W}_{\infty}(x,t) |\partial_t \psi_j|^2 dx \, dt.$$ Note that this $C_Q$ is uniform in $j,$ because the observation set $\{\widetilde{W}_{\infty}>\frac{\overline{C}}{2}\}$ does not change. To make use of Lemma [Lemma 12](#observeconnection){reference-type="ref" reference="observeconnection"}, the $\widetilde{W}_{\infty}$ on the right hand side must be replaced by $W_j$. To do so $$\begin{aligned} \label{wjalign} E(v_j, 0) & \leq C_Q \int_0^{T_0} \int_M \widetilde{W}_{\infty} |\partial_t \psi_j|^2 \nonumber \\ &= C_Q \int_0^{T_0} \int_M (\widetilde{W}_j -\widetilde{W}_j+\widetilde{W}_{\infty})| \partial_t \psi_j|^2 dx\, dt \nonumber \\ & \leq C_Q \int_0^{T_0} \int_M \widetilde{W}_j |\partial_t \psi_j|^2 dx \, dt + C_Q \left| \left| \widetilde{W}_j-\widetilde{W}_{\infty} \right| \right|_{L^{\infty}} \int_0^{T_0} \int_M |\partial_t \psi_j|^2 dx \, dt.\end{aligned}$$ Recall $E(\psi_j,0)=E(\psi_j,t)$ for all $t$. Therefore $$\int_0^{T_0} \int_M |\partial_t \psi_j|^2 dx \, dt \leq \int_0^{T_0} E(\psi_j,t) dt = T_0 E(\psi_j,0) =T_0 E(v_j, 0).$$ Now choosing $J$ large enough so that $\left| \left| \widetilde{W}_j - \widetilde{W}_{\infty} \right| \right|_{L^{\infty}} < \left(2T_0 C_Q \right)^{-1}$ for $j \geq J$ then $$C_Q \left| \left| \widetilde{W}_j - \widetilde{W}_{\infty} \right| \right|_{L^{\infty}} \int_0^{T_0} \int_M |\partial_t \psi_j|^2 dx \, dt \leq \frac{1}{2}E(v_j,0).$$ This term can be absorbed back into the left hand side of [\[wjalign\]](#wjalign){reference-type="eqref" reference="wjalign"} to give $$\begin{aligned} E(v_j,0) &\leq 2 C_Q \int_0^{T_0} \int_M \widetilde{W}_j |\partial_t \psi_j|^2 dx \, dt \\ & = 2C_Q \int_0^{T_0} \int_M \frac{W_j(x,t)}{f_j(x,t)} |\partial_t \psi_j|^2 dx \, dt \leq \frac{2C_Q}{b_j(T_0)} \int_0^{T_0} \int_M W_j |\partial_t \psi_j|^2 dx \, dt.\end{aligned}$$ Now, by Lemma [Lemma 12](#observeconnection){reference-type="ref" reference="observeconnection"} $$\begin{aligned} E(v_j,0) &\leq \frac{C_Q}{b_j(T_0)} \left(1+2 T_0 \left| \left| W_j \right| \right|_{L_0^{\infty}} \right)^2 \int_0^{T_0} \int_M W_j |\partial_t v_j|^2 dx\, dt \\ & \leq \frac{C_Q}{b_j(T_0)} \left(1 + 2 T_0 \left| \left| W \right| \right|_{L^{\infty}} \right)^2 \int_0^{T_0} \int_M W_j |\partial_t v_j|^2 dx \, dt.\end{aligned}$$ So then, for some $C>0$ $$b_j(T_0) E(v_j,0) \leq C \int_0^{T_0} \int_M W_j |\partial_t v_j|^2 dx \, dt.$$ By the second part of [\[contra\]](#contra){reference-type="eqref" reference="contra"}, the term on the right hand side goes to $0$ as $j \rightarrow\infty$, therefore $b_j(T_0) E(v_j,0) \rightarrow 0$ as $j \rightarrow\infty$. This contradicts the first part of [\[contra\]](#contra){reference-type="eqref" reference="contra"}, $b_j(T_0) E(v_j, 0)=1$, so the desired conclusion must hold. ◻ With this observability result, decay rates for damping of the form [\[decreasedampdef\]](#decreasedampdef){reference-type="eqref" reference="decreasedampdef"} can be proved. *Proof of Proposition [Proposition 13](#decreasingdampingdecay){reference-type="ref" reference="decreasingdampingdecay"}.* 1) By Lemma [Lemma 14](#decreasingdwobserve){reference-type="ref" reference="decreasingdwobserve"}, there exists $C>0$ such that $$C b((j+1)T_0) E(u,jT_0) \leq \int_{jT_0}^{(j+1)T_0} \int_M W |\partial_t u|^2 dx dt.$$ Then by Lemma [Lemma 10](#observetodecay){reference-type="ref" reference="observetodecay"}, letting $B(k) = \sum_{j=0}^{k-1} C b((j+1)T_0)$ $$E(u,kT_0) \leq E(u,0) \exp(-B(k)).$$ Now since $b$ is decreasing $$B(k) \geq \frac{1}{T_0} \int_{T_0}^{(k+1)T_0} C b(z) dz \geq \frac{1}{T_0} \int_0^{(k+1)T_0} C b(z) dz - C \left| \left| b \right| \right|_{L^{\infty}}.$$ Since $\left| \left| \widetilde{W} \right| \right|_{L^{\infty}} \leq C_4$ and $f(x,t) \leq C_M b(t)$ $$\int_0^{(k+1)T_0} b(z) dz \geq \frac{1}{C_W C_M} \int_0^{(k+1)T_0} \widetilde{W}(\gamma(t),t) f(\gamma(t),t) dt \geq \frac{\Sigma((k+1)T_0)}{C_W C_M}.$$ Combining the previous two equations, there exists $C,c>0$ such that $$E(u,kT_0) \leq C E(u,0) \exp(-c \Sigma((k+1)T_0).$$ Since $E(u,t)$ is non-increasing, for $t \in [kT_0, (k+1)T_0]$ $$E(u,t) \leq C E(u,0) \exp(-c\Sigma((k+1)T_0)) \leq C E(u,0) \exp(-c\Sigma(t)),$$ as desired. 2\) The next two assertions follow directly from Lemma [Lemma 8](#lowerboundlemma){reference-type="ref" reference="lowerboundlemma"} and Proposition [Proposition 9](#lowerboundprop){reference-type="ref" reference="lowerboundprop"}. 3\) The explicit rates when $b(t)=(t+1)^{-\beta}$ follow from estimating $\Sigma(t)$. $$\begin{aligned} \Sigma(kT_0) &= \inf_{\gamma} \sum_{j=0}^{k-1} \int_{jT_0}^{(j+1)T_0} \widetilde{W}(\gamma(t),t) f(\gamma(t), t) dt \\ &\geq \inf_{\gamma} \sum_{j=0}^{k-1} C_m ((j+1)T_0 +1)^{-\beta} \int_{jT_0}^{(j+1)T_0} \widetilde{W}(\gamma(t), t) dt \\ &= C_m \overline{C}T_0 \sum_{j=0}^{k-1} ((j+1)T_0 +1)^{-\beta} \leq c \int_{T_0}^{(k+1)T_0} (t+1)^{-\beta} dt \\ &= \begin{cases} \frac{c}{1-\beta} \left( ((k+1)T_0)^{1-\beta} -(T_0+1)^{1-\beta} \right) &\beta \neq 1 \\ c \left( \ln((k+1)T_0) - \ln(T_0+1) \right) & \beta=1. \end{cases}\end{aligned}$$ Plugging this into the first part of this proposition, using that $E(u,t)$ is non-increasing and relabeling $c$ gives the desired decay rates. Analogously $$\begin{aligned} \Sigma(t) &= \inf_{\gamma} \int_0^t \widetilde{W}(\gamma(s), s) f(\gamma(s),s) ds \\ &\leq C_W C_M \int_0^t (s+1)^{-\beta} ds = C_W C_M \begin{cases} \frac{1}{1-\beta}\left((t+1)^{1-\beta}-1 \right)& \beta \neq 1\\ \ln(t+1) & \beta =1. \end{cases}\end{aligned}$$ Plugging this into the second part of this proposition gives the desired result. ◻ ## Oscillating Damping, on for shrinking time intervals Let $\chi \in L^{\infty}([0,1])$ be non-negative and $\chi \equiv 1$ on $[\frac{1}{4}, \frac{3}{4}]$. Fix $C_W, S_0>0.$ Consider $f: \mathbb{N}_0 \rightarrow(0,S_0)$, with $C_m, C_M, \beta>0$ such that $C_m (t+1)^{-\beta} \leq f(t) \leq C_M (t+1)^{-\beta}$. For $k \in \mathbb{N}_0$, define $$\label{shrinkingdampdef} W(t) = \begin{cases} C_W \chi\left( \frac{t-k S_0}{f(k)} \right) & kS_0 < t < kS_0 +f(k),\\ 0 & kS_0 + f(k) \leq t \leq (k+1) S_0. \end{cases}$$ That is, the damping is on for an interval of length at most $f(k)$ starting at $kS_0$ and then is off until $(k+1)S_0$. **Proposition 15**. With damping as in [\[shrinkingdampdef\]](#shrinkingdampdef){reference-type="eqref" reference="shrinkingdampdef"}, 1. If $0 \leq \beta<1/3$, there exists $C,c>0$, such that for all solutions $u$,\ $E(u,t) < C E(u,0) \exp(-c t^{1-3\beta})$. 2. If $\beta=1/3$, there exists $C,c>0$, such that for all solutions $u$, $E(u,t) \leq \frac{C}{t^{c}} E(u,0).$ 3. If $\beta>1/3$, there exists $c<1$, such that for $t>f(0),$ for all solutions $u$, $E(u,t) <c E(u,0).$ If in addition $\chi$ is compactly supported in $[0,1]$ and $\chi \in W^{1,\infty}([0,1])$, then $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"} and 1. If $0\leq \beta<1$, $E(u,t) < C E(u,0) \exp(-c t^{1-\beta})$, cannot hold with $c> \frac{2C_M C_W}{(1-\beta)S_0^{1-\beta}}$. 2. If $\beta=1$, $E(u,t) < C E(u,0) \frac{1}{t^{c}} E(u,0)$, cannot hold with $c>2C_M C_W$. 3. If $\beta>1$, there exists $c \in (0,1)$, such that for any $T>0$, there exists a solution $u$ satisfying $E(u,T) > c E(u,0)$. The gap between the upper and lower bounds is due to the power on $f(k)$ in the damped wave observability inequality below. That in turn, comes from the short-time observability constant for the wave equation, Lemma [Lemma 19](#basicwaveobserve){reference-type="ref" reference="basicwaveobserve"}, which cannot be improved in general. This follows from a proof analogous to Proposition [Proposition 13](#decreasingdampingdecay){reference-type="ref" reference="decreasingdampingdecay"}. In particular, a damped wave observability inequality is proved from a wave equation observability inequality, and is then is converted into a uniform stabilization rate using Lemma [Lemma 10](#observetodecay){reference-type="ref" reference="observetodecay"}. **Lemma 16**. Suppose $u$ solves [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} with damping as in [\[shrinkingdampdef\]](#shrinkingdampdef){reference-type="eqref" reference="shrinkingdampdef"}. Then, there exists $C>0$, such that for all $k \in \mathbb{N}_0$ $$C f(k)^3 E(u,kT_0) \leq \int_{kS_0}^{(k+1)S_0} \int_M W|\partial_t u|^2 dx dt.$$ *Proof.* Let $\psi_k$ solve [\[WE\]](#WE){reference-type="eqref" reference="WE"} with $(\psi_k, \partial_t \psi_k)_{t=kS_0}=(u,\partial_t u)|_{t=kS_0}$, so $E(u,kS_0) = E(\psi, kS_0)$. By Lemma [Lemma 19](#basicwaveobserve){reference-type="ref" reference="basicwaveobserve"}, there exists $C>0$, such that for any $\delta>0, s>0$ $$E(\psi_k, k S_0)= E(\psi_k, kS_0+s) \leq \frac{C}{\delta^3}\int_{kS_0+s}^{kS_0+\delta+s} \int_M |\partial_t \psi_k|^2 dx dt.$$ Then, letting $\delta=\frac{f(k)}{2}, s=\frac{f(k)}{4}$ and applying Lemma [Lemma 12](#observeconnection){reference-type="ref" reference="observeconnection"} $$\begin{aligned} E(u,kS_0) &= E(\psi_k, kS_0) \leq \frac{8C}{f(k)^3} \int_{kS_0+\frac{f(k)}{4}}^{kS_0+\frac{3f(k)}{4}} \int_M |\partial_t \psi_k|^2 dx dt \\ &\leq \frac{C}{f(k)^3} \int_{kS_0}^{(k+1)S_0} \int_M W |\partial_t \psi_k|^2 dx dt \leq \frac{C(1+2S_0 C_W)^2}{f(k)^3} \int_{kS_0}^{(k+1)S_0} \int_M W |\partial_t u|^2 dx dt.\end{aligned}$$ Multiplying both sides by $\frac{f(k)^3}{C(1+2S_0 C_W)^2}$ gives the desired conclusion. ◻ *Proof of Proposition [Proposition 15](#shrinkingdecay){reference-type="ref" reference="shrinkingdecay"}.* Since $C_m (1+t)^{-\beta} \leq f(x)$, then by Lemma [Lemma 16](#shrinkingdwobserve){reference-type="ref" reference="shrinkingdwobserve"}, there exists $C>0$ such that $$C \frac{1}{(1+jS_0)^{3\beta}} E(u,jS_0) \leq \int_{jS_0}^{(j+1)S_0} \int_M W |\partial_t u|^2 dx dt.$$ Then by Lemma [Lemma 10](#observetodecay){reference-type="ref" reference="observetodecay"}, letting $B(k) = \sum_{j=0}^{k-1} C (1+jS_0)^{-3\beta}$, $$E(u,kS_0) \leq E(u,0) \exp(-B(k)).$$ Now since $(1+jS_0)^{-3\beta}$ is decreasing $$\begin{aligned} B(k) \geq \int_0^k C (1+ z S_0)^{-3\beta} dz = \begin{cases} \frac{C}{1-3\beta} ((1+kS_0)^{1-3\beta} -1) & \beta \neq 1/3 \\ C \ln(1+kT_0) & \beta = 1/3. \end{cases}\end{aligned}$$ Then for $k$ large enough, there exists $c>0$ such that $$\begin{aligned} B(k) \geq \begin{cases} c(1+kS_0)^{1-3\beta} & \beta \neq 1/3 \\ c \ln(1+kS_0) & \beta =1/3. \end{cases}\end{aligned}$$ Since $E(u,t)$ is non-increasing in $t$, for $t \in [kS_0, (k+1)S_0]$, there exists $C>0$ such that $$\begin{aligned} E(u,t) \leq E(u,kS_0) &\leq C E(u,0) \begin{cases} \exp(-c (1+kS_0)^{1-3\beta}) & \beta \neq 1/3 \\ \exp(-c\ln(1+kS_0)) & \beta =1/3 \end{cases} \\ & \leq C E(u,0) \begin{cases} \exp(-c(1+t)^{1-3\beta}) & \beta \neq 1/3 \\ (1+t)^{-c} & \beta =1/3. \end{cases}\end{aligned}$$ as desired. 2\. Fix $t \in (0,\infty)$ and define $k = \lfloor \frac{t}{S_0} \rfloor$. Then $$\begin{aligned} \Sigma(t)&=\int_0^t W(\gamma(s), s) ds \leq C_W \sum_{j=0}^{k} f(j) \leq C_W \int_0^k \frac{1}{f(x)} dx \leq C_M C_W \int_0^k \frac{1}{(x+1)^{\beta}} dx\\ &= \begin{cases} \frac{C_M C_W}{(1-\beta)} (k^{1-\beta}-1) &\beta \neq 1 \\ C_M C_W \ln(k+1) &\beta=1. \end{cases} \end{aligned}$$ When $\beta<1$, $\Sigma(t) \leq \frac{C_M C_W}{(\beta-1)}(k^{1-\beta}-1)$, and since $k=\lfloor \frac{t}{S_0}\rfloor$, by Proposition [Proposition 9](#lowerboundprop){reference-type="ref" reference="lowerboundprop"}, $$E(u,t) \leq C E(u,0) \exp(-c t^{1-\beta}),$$ cannot hold for $c > \frac{2C_M C_W}{(1-\beta)S_0^{1-\beta}}$.\ When $\beta=1$, $\Sigma(t) \leq C_M C_W \ln(k+1)$, so by Proposition [Proposition 9](#lowerboundprop){reference-type="ref" reference="lowerboundprop"}, $$E(u,t) \leq C E(u,0) \exp(-c \ln(k+1)),$$ cannot hold for $c > 2C_M C_W$. Since $k=\lfloor \frac{t}{S_0}\rfloor$, $$E(u,t) \leq C E(u,0) \frac{1}{t^{c}},$$ cannot hold for $c>2C_M C_W$.\ When $\beta>1$, $\Sigma(t) \leq \frac{C_M C_W}{(\beta-1)}<\infty$, so by Lemma [Lemma 8](#lowerboundlemma){reference-type="ref" reference="lowerboundlemma"}, there exists $c \in (0,1),$ such that for all $t_0$, there exists a solution $u$ of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} such that $$E(u,t_0) \geq c E(u,0).$$ ◻ ## Oscillating damping, off for growing time intervals Fix $L_0$, and suppose $\widetilde{W}(x,t) \in C^0(M \times [0, L_0])$. Assume also that, for some $\varepsilon>0$, $\{\widetilde{W} > \varepsilon\}$ satisfies Assumption [Assumption 3](#ftGCC){reference-type="ref" reference="ftGCC"} on $M \times [0,L_0]$. Let $\overline{C}=\inf_{\gamma} \frac{1}{L_0} \int_0^{L_0} \widetilde{W}(\gamma(t), t) dt$, and let $f:[1,\infty) \rightarrow[C_1, \infty)$ be an increasing function. For $k \in \mathbb{N}_0,$ define $$\label{growingdampdef} W(x,t) = \begin{cases} \widetilde{W}(x,t- kL_0-\sum_{j=1}^k f(j)), & kL_0 + \sum_{j=1}^k f(j) < t < (k+1)L_0 + \sum_{j=1}^k f(j) \\ 0 & (k+1)L_0 + \sum_{j=1}^k f(j) < t < (k+1)L_0 + \sum_{j=1}^{k+1} f(j), \end{cases}$$ That is at the $k$th step, $W$ is on for an time interval of length $T_0$ and then is off for a time interval of length $f(k+1)$. Because $W$ is on for a fixed time interval $L_0$ and $\widetilde{W}$ satisfies Assumption [Assumption 3](#ftGCC){reference-type="ref" reference="ftGCC"} on each of these, it is straightforward to obtain a damped wave observability inequality from Proposition [Proposition 11](#observeprop){reference-type="ref" reference="observeprop"} and Lemma [Lemma 12](#observeconnection){reference-type="ref" reference="observeconnection"}. However, Lemma [Lemma 10](#observetodecay){reference-type="ref" reference="observetodecay"} cannot be used to obtain a uniform stabilization rate, as the observability estimates do not have the appropriate periodic structure. Instead, the number of times the damping has turned on by time $t$ must be estimated. This is done by taking the inverse of an integral of $f$, which, as seen in the examples, provides the appropriate qualitative uniform stabilization rate. **Proposition 17**. Suppose $W$ is as in [\[growingdampdef\]](#growingdampdef){reference-type="eqref" reference="growingdampdef"}. Define $F(k)=\int_1^{k+1} f(z) dz$ and let $F^{-1}(t)$ be the inverse of $F$. Then, there exists $C, c>0$ such that for all $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} $$E(u,t) < C E(u,0) \exp\left(-c F^{-1}\left( \frac{C_1 t}{2L_0+C_1}\right) \right).$$ If in addition, $\widetilde{W}$ is compactly supported on $[0,L_0],$ and $\partial_t \widetilde{W}, \nabla^2 \widetilde{W} \in L^{\infty}(M \times [0,L_0],$ then $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}. In this case, define $B(k)=\int_0^k f(z) dz$ and let $B^{-1}(t)$ be the inverse of $B$. Then $$E(u,t) < C E(u,0) \exp\left(-c B^{-1}(t) \right).$$ cannot hold with $c > 2 \overline{C}L_0$. In particular, suppose $C_1>0, \alpha \geq 0$ and $r>1$ 1. If $f(j)=C_1 j^{\alpha}$, then there exists $C,c>0,$ such that for all $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"}, $E(u,t) < CE(u,0) \exp(-c t^{\frac{1}{\alpha+1}})$.\ Furthermore, if $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}, $E(u,t) < C E(u,0) \exp(-c t^{\frac{1}{\alpha+1}})$, cannot hold with $c > 2\overline{C}T_0 \left( \frac{\alpha+1}{C_1} \right)^{\frac{1}{\alpha+1}}$. 2. If $f(j) = C_1 r^j$, then there exists $C,c>0,$ such that for all $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"}, $E(u,t) < C E(u,0) \frac{1}{\left\langle t\right\rangle^{c}}$.\ Furthermore, if $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}, $E(u,t) < C E(u,0) \frac{1}{\left\langle t\right\rangle^{c}},$ cannot hold with $c > \frac{2 \overline{C}T_0}{\ln(r)}$. 3. If $f(j)=C_1 e^{j+e^{j}}$, then there exists $C,c>0,$ such that for all $u$ solving [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"}, $E(u,t) < C E(u,0) \frac{1}{\ln(2+t)^{c}}$.\ Furthermore, if $W$ satisfies Assumption [Assumption 2](#Wreg){reference-type="ref" reference="Wreg"}, $E(u,t)<CE(u,0) \frac{1}{\ln(2+t)^{c}}$, cannot hold with $c>2\overline{C}T_0$. *Proof.* 1. For $a \in \mathbb{R}$, since $f$ is increasing $f(a) \leq \int_a^{a+1} f(x) dx$. Now, given a large time $t$, estimate $K$, a lower bound on the number of times the damping turns on, and is on for a full $T_0,$ before time $t$. The claim is that $K = F^{-1} \left(\frac{C_1 t}{2L_0+C_1}\right)$. To see this note that $$(K+1)L_0 + \sum_{j=1}^K f(j) \leq (K+1)L_0 + \int_1^{K+1} f(z) dz = (K+1)L_0 + F(K) \leq \left(\frac{2L_0}{C_1} +1\right) F(K) = t.$$ Let $u$ be a solution of [\[TDWE\]](#TDWE){reference-type="eqref" reference="TDWE"} and let $\psi_k$ solve [\[WE\]](#WE){reference-type="eqref" reference="WE"} with $(\psi, \partial_t \psi)=(u,\partial_t u)$ at $t= kL_0+\sum_{j=1}^k f(j)$. Since $W$ satisfies Assumption [Assumption 3](#ftGCC){reference-type="ref" reference="ftGCC"} on the time interval $[kL_0+\sum_{j=1}^k f(j), (k+1)L_0+\sum_{j=1}^k f(j)]$, by Proposition [Proposition 11](#observeprop){reference-type="ref" reference="observeprop"}, there exists $C>0$, uniform in $k$, such that $$E\left(u,kL_0 + \sum_{j=1}^k f(j) \right)= E\left(\psi, kL_0+\sum_{j=1}^k f(j)\right) \leq C \int_{kL_0+\sum f}^{(k+1)L_0 + \sum f} W |\partial_t \psi|^2 dx dt.$$ Then by Lemma [Lemma 12](#observeconnection){reference-type="ref" reference="observeconnection"}, there exists $C>0$, uniform in $k,$ such that $$E\left(u,kL_0+\sum_{j=1}^k f(j) \right) \leq C \int_{kL_0+\sum f}^{(k+1) L_0+ \sum f} W |\partial_t u|^2 dx dt.$$ Now integrating [\[dtE\]](#dtE){reference-type="eqref" reference="dtE"} from $kL_0+\sum f$ to $(k+1)L_0 + \sum f$ $$E\left(u,(k+1)L_0 + \sum_{j=1}^k f(j) \right) - E\left(u,kL_0 + \sum_{j=1}^k f(j)\right) = - \int_{kL_0+\sum f}^{(k+1) L_0 +\sum f} \int_M W|\partial_t u|^2 dx dt.$$ Therefore $$E\left(u,(k+1)L_0 + \sum_{j=1}^k f(j)\right) < (1-C) E\left(u,kL_0+\sum_{j=1}^k f(j)\right).$$ By time $t$, there are at least $K=F^{-1}\left(\frac{C_1t}{2L_0+C_1}\right)$ such intervals, and so $$E(u,t) < (1-C)^K E(u,0).$$ Since the energy of solutions is non-increasing, there exists $, cC>0,$ such that $$E(u,t) < C \exp\left(-c F^{-1}\left(\frac{C_1 t}{2L_0+C_1 } \right) \right) E(u,0).$$ 2. Now, given a large time $t$, estimate $J$, an upper bound on the number of times the damping turns on before time $t$. The claim is that $J \leq B^{-1}(t)$. To see this, note that for $a \in \mathbb{R}$, since $f$ is increasing $f(a) \geq \int_{a-1}^a f(x) dx$. Therefore $$(J+1)L_0 + \sum_{j=1}^J f(j) \geq (J+1)L_0 + \int_0^J f(z) dz \geq B(J) =t.$$ Therefore $$\Sigma(t) = \inf_{\gamma} \int_0^t W(\gamma(s), s) ds \leq B^{-1}(t) \inf \int_0^{L_0} \widetilde{W}(\gamma(s), s) ds = \overline{C}L_0 B^{-1}(t).$$ Proposition [Proposition 9](#lowerboundprop){reference-type="ref" reference="lowerboundprop"} gives the second conclusion. 3\. When $f(j) = C_1 j^{\alpha}$, then $F(k) = C_1 \left( \frac{(k+1)^{\alpha+1}-1}{\alpha+1}\right)$, so $F^{-1}(s) = \left( \frac{(\alpha+1)s}{C_1}+1\right)^{\frac{1}{\alpha+1}}-1$. Thus, there exists $C,c>0,$ such that $$E(u,t) < C \exp\left(-c \left( \frac{(\alpha+1)t}{2L_0+C_1} + 1\right)^{\frac{1}{\alpha+1}} \right) E(u,0).$$ Simplifying and relabeling $C, c$, this becomes $$E(u,t) < C \exp(-c t^{\frac{1}{\alpha+1}} ) E(u,0).$$ On the other hand, $B(k) = \frac{C_1 k^{\alpha+1}}{\alpha+1}$ so $B^{-1}(s) = \left( \frac{(\alpha+1)s}{C_1}\right)^{\frac{1}{\alpha+1}}$. Therefore $$E(u,t) < C E(u,0) \exp\left(-c t^{\frac{1}{\alpha+1}}\right),$$ cannot hold with $c> 2\overline{C}L_0 \left( \frac{\alpha+1}{C_1} \right)^{\frac{1}{\alpha+1}}$.\ 4. When $f(j)=C_1 r^j$, then $F(k)=\frac{C_1 (r^{k+1}-r)}{\ln(r)}$ so $F^{-1}(s) = \frac{\ln\left( \frac{\ln(r)}{C_1} s+ r\right)}{\ln(r)}-1$. Thus, there exists $C, c>0$, such that $$E(u,t) < C \exp\left( - \frac{c}{\ln(r)} \ln\left( \frac{\ln(r)}{2L_0+C_1} t + r\right)\right).$$ Simplifying and relabeling $C, c$, this becomes $$E(u,t) < \frac{C}{\left\langle t\right\rangle^c} E(u,0).$$ On the other hand, $B(k) = \frac{C_1 (r^k-1)}{\ln(r)},$ so $B^{-1}(s) = \frac{\ln\left( \frac{\ln(r)}{C_1} s+ 1\right)}{\ln(r)}$. Therefore $$E(u,t) < C E(u,0) \exp\left(-\frac{c}{\ln(r)} \ln\left( \frac{\ln(r)}{C_1} t + 1\right) \right),$$ cannot hold with $c> 2\overline{C}L_0$. After simplifying, and relabeling $C$, this becomes $$E(u,t) < C E(u,0) \frac{1}{\left\langle t\right\rangle^{c}},$$ cannot hold with $c> \frac{2\overline{C}L_0}{\ln(r)}$. 5\. When $f(j)= C_1 e^{j+e^j}$, then $F(k) = C_1(e^{e^{k+1}}-e^e)$ so $F^{-1}(s)= \ln(\ln( \frac{s}{C_1} + e^e))-1$. Therefore, there exists $C, c>0$ such that $$E(u,t) < C E(u,0) \exp\left(-c\ln\left(\ln\left(\frac{t}{2L_0+C_1}+e^e\right)\right)\right).$$ After simplification and relabeling $C, c$, this gives $$E(u,t) < C E(u,0) \frac{1}{\ln(2+t)^{c}}.$$ On the other hand, $B(k)=C_1(e^{e^k}-e),$ so $B^{-1}(s)= \ln(\ln(\frac{s}{C_1} + e))$. Then $$E(u,t) < C E(u,0) \exp\left(-c \ln\left(\ln\left(\frac{s}{C_1}+e\right)\right)\right),$$ cannot hold with $c>2\overline{C}L_0$. So $$E(u,t) < C E(u,0) \frac{1}{\ln(s+e)^{c}},$$ cannot hold with $c> 2\overline{C}L_0$. ◻ # Short Time Observability This appendix obtains an explicit relationship between the observability constant and the length of the observation window, $\delta$, when the observability operator is the indicator function on the manifold. In general, the observability constant must grow like $1/\delta^3$. This is because solutions of the wave equation which have most of their energy in potential energy, have kinetic energy like $\sin^2(t)$, which for small $t$ is of size $t^2$. Integrating this from $t=0$ to $t=\delta$, gives a quantity of size $\delta^3$, as shown more precisely in Lemma [Lemma 18](#trigintlemma){reference-type="ref" reference="trigintlemma"}. Showing the following lemma holds, regardless of $\lambda$ is, requires some care. **Lemma 18**. Fix $\lambda_0, N$. There exists $C>0$ such that for all $A,B \in \mathbb{C}, \delta\in (0,N]$ and $\lambda \geq \lambda_0$ $$|A|^2 + |B|^2 \leq \frac{C}{\delta^3} \int_0^{\delta} \left|-A \sin(\lambda t) + B \cos(\lambda t)\right|^2 dt.$$ Furthermore, the power on $\delta$ is sharp when $A=1$, $B=0$. *Proof.* Consider two cases 1. $\delta\lambda \leq \frac{1}{2}$, 2. $\delta\lambda \geq \frac{1}{2}$. In case 1, expand, then evaluate the integral to obtain $$\begin{aligned} \label{trigintlemmacase1} \frac{C}{\delta^3} &\int_0^{\delta} |-A \sin(\lambda t) + B \cos(\lambda t)|^2 dt \nonumber \\ &\geq \frac{C}{\delta^3} \int_0^{\delta} |B|^2 \cos^2(\lambda t) + |A|^2 \sin^2(\lambda t) -2|A||B| \sin(\lambda t) \cos(\lambda t) dt \nonumber\\ &= \frac{C}{\delta^2} \left( \frac{2\delta\lambda(|A|^2+|B|^2) +(|B|^2-|A|^2) \sin(2\lambda \delta) + 2|A| |B| (\cos(2\lambda \delta)-1)}{4\lambda \delta} \right).\end{aligned}$$ By Taylor's theorem with remainder, there exists $\zeta \in (0, 2\lambda \delta),$ such that $\sin(2\lambda\delta) = 2\lambda \delta- \frac{4}{3} (\lambda \delta)^3 + \cos(\zeta) \frac{4}{15} (\lambda \delta)^5$. Since $\lambda \delta\leq \frac{1}{2}$, $$2\lambda \delta- \frac{4}{3} \lambda^3 \delta^3 \leq \sin(2\lambda \delta) \leq 2 \lambda \delta- \frac{19}{15} \lambda^3 \delta^3.$$ Similary, there exists $\xi \in (0, 2\lambda \delta),$ such that $\cos(2\lambda \delta) = 1 - 2 (\lambda \delta)^2 + \sin(\xi) \frac{2}{3} (\lambda \delta)^4$. So $$|\cos(2\lambda \delta)-1| \leq 2 \lambda^2 \delta^2.$$ Plugging these back into [\[trigintlemmacase1\]](#trigintlemmacase1){reference-type="eqref" reference="trigintlemmacase1"} $$\begin{aligned} &\geq \frac{C}{\delta^2} \left( \frac{|A|^2+|B|^2}{2} + \frac{|B|^2(2\lambda \delta- \frac{4}{3} \lambda^3\delta^3) - |A|^2 (2\lambda \delta- \frac{19}{15} \lambda^3 \delta^3) - 2|A||B|(2\lambda^2 \delta^2)}{4 \lambda \delta}\right) \\ &= \frac{C}{\delta^2} \left( |B|^2 (1-\frac{\lambda^2 \delta^2}{3}) + \frac{19}{60} |A|^2 \lambda^2 \delta^2 -|A||B| \lambda \delta\right)\end{aligned}$$ Then using that $|A||B|\lambda \delta\leq \frac{1}{2\varepsilon} |B|^2 + \frac{\varepsilon}{2}|A|^2 \lambda^2 \delta^2,$ with $\varepsilon=\frac{18}{30}$, and $\delta\lambda <\frac{1}{2}$ $$\begin{aligned} &\geq \frac{C}{\delta^2}\left( |B|^2\left(1-\frac{\lambda^2\delta^2}{3}-\frac{30}{36}\right) + |A|^2 \frac{ \lambda^2 \delta^2}{60} \right) \\ &\geq \frac{C}{\delta^2}\left(\frac{|B|^2}{12} + \frac{|A|^2 \lambda^2 \delta^2}{60} \right) \\ &=\frac{C|B|^2}{12\delta^2} + \frac{C|A|^2 \lambda^2}{60}\end{aligned}$$ and letting $C>\max\left(12N^2,\frac{60}{\lambda_0^2}\right)$ this is $\geq |A|^2+|B|^2$, as desired. In case 2, use a change of variables $$\frac{C}{\delta^3} \int_0^{\delta} |-A \sin(\lambda t) + B \cos(\lambda t)|^2 dt = \frac{C}{\lambda \delta^3} \int_0^{\lambda \delta} |-A\sin(t) + B \cos(t)|^2 dt.$$ Now, there exists $\phi \in [0,2\pi)$ such that $-A \sin(t) + B \cos(t) = \mathcal{A}\cos(t+\phi)$ where $|\mathcal{A}|^2=|A|^2+|B|^2$. Thus it remains to show that there exists $C>0$ such that $$1 \leq \frac{C}{\lambda \delta^3} \int_0^{\lambda \delta} \cos^2 (t+\phi) dt.$$ When $\lambda \delta=\frac{1}{2}$, there exists some $C_2>0$, such that $\frac{1}{\lambda \delta} \int_0^{\lambda \delta} \cos^2(t+\phi) dt = C_2$. Now for arbitrary $\lambda \delta$, let $k \in \mathbb{N}_0$ be such that $\lambda \delta\in [k \pi, (k+1)\pi)$. Then, since $\cos^2$ is $\pi$-periodic $$\begin{aligned} \frac{1}{\lambda \delta} \int_0^{\lambda \delta} \cos^2(t+\phi) dt &\geq \frac{1}{\lambda \delta} \left( \sum_{j=0}^{k-1} \int_{j\pi}^{j\pi+\frac{1}{2}} \cos^2(t+\phi) dt+ \int_{j\pi+\frac{1}{2}}^{(j+1)\pi} \cos^2(t+\phi) dt\right) \\ &\geq \frac{1}{(k+1)\pi} \frac{k C_2 }{2} \geq \frac{C_2}{4\pi}.\end{aligned}$$ Dividing by $\delta^2$ gives $$\frac{1}{\lambda \delta^3} \int_0^{\lambda \delta} \cos^2(t + \phi) dt \geq \frac{C_2}{4 \pi N^2},$$ so choosing $C> \frac{4 \pi N^2}{C_2}$ gives the desired conclusion. 3\. To see the power on $\delta$ is sharp when $A=1, B=0$ use Taylor's Theorem to expand $\sin^2(\lambda_0 t)$ at $t=0$. Then for small $\delta>0$, compute $$\int_0^\delta\sin^2(\lambda_0 t) dt = \int_0^{\delta} (\lambda_0 t)^2 + O(\lambda_0^4 t^4) dt = C \delta^3.$$ ◻ **Lemma 19**. Suppose $\psi$ solves [\[WE\]](#WE){reference-type="eqref" reference="WE"}. Let $N>0$ There exists $C>0,$ such that for all $\delta\in (0,N]$ $$E(\psi,0) \leq \frac{C}{\delta^3} \int_0^{\delta} \int_M |\partial_t \psi|^2 dx dt,$$ and the power on $\delta$ cannot be reduced. *Proof.* Let $\{v_k\}$ be an orthonormal basis of eigenfunctions of $-\Delta$ on $M,$ with associated eigenvalues $\{\lambda_k^2\}$. That is, $-\Delta v_k = \lambda_k^2 v_k$. Then via separation of variables, c.f. [@Rauchbook Section 5.7], for any $\psi(x,t) \in L^2(M \times [0,\infty))$ solving the wave equation, there exist coefficients $a_k, A_k, B_k \in \mathbb{C}$ such that $$\psi(x,t) = \sum_{k=0}^{\infty} a_k v_k(x) \left(A_k \cos(\lambda_k t) + B_k \sin(\lambda_k t) \right).$$ Therefore $$\left| \left| \nabla \psi(t=0) \right| \right|_{L^{2}}^2 = \sum_{k=0}^{\infty} \lambda_k^2 |a_k|^2 |A_k|^2 \left| \left| v_k \right| \right|_{L^{2}}^2.$$ And $$\begin{aligned} \left| \left| \partial_t \psi \right| \right|_{L^{2}}^2&= \sum_{k=0}^{\infty}\lambda_k^2 |a_k|^2 \left|-A_k \sin(\lambda_k t) + B_k \cos(\lambda_k t)\right|^2 \left| \left| v_k \right| \right|_{L^{2}}^2. \end{aligned}$$ So $$E(\psi, 0) = \frac{1}{2} \sum_{k=0}^{\infty} \lambda_k^2 |a_k|^2 \left| \left| v_k \right| \right|_{L^{2}}^2 (|A_k|^2+|B_k|^2).$$ And $$\sum_{k=0}^{\infty} \lambda_k^2| a_k|^2 \left| \left| v_k \right| \right|_{L^{2}}^2 \frac{C}{\delta^3} \int_0^{\delta} \left|-A_k \sin(\lambda_k t) +B_k\cos(\lambda_k t)\right|^2 dt = \frac{C}{\delta^3} \int_0^\delta\int_M |\partial_t \psi|^2 dx dt$$ Then Lemma [Lemma 18](#trigintlemma){reference-type="ref" reference="trigintlemma"}, applied to each individual term, in the sums, gives the desired inequality. ◻

arxiv_math

--- address: ", , " author: - Taejun Park - Yuji Nakatsukasa bibliography: - wileyNJD-Vancouver.bib title: Approximating Sparse Matrices and their Functions using Matrix-vector products --- # Introduction {#sec:intro} The evaluation of matrix functions is an important task arising frequently in many areas of scientific computing. They appear in the numerical solution of partial differential equations [@pde1; @pde2; @pde3], network analysis [@netanal1; @EstradaHigham2010], electronic structure calculations [@elec2; @elec1] and statistical learning [@CortinovisKressner2021; @Wengeretal2022], among others. In some problems, the matrix of interest has some underlying structure such as bandedness or sparsity. For example, in graphs with community structure one requires functions of sparse matrices [@Newman2006] and for electronic structure calculations one computes functions of banded matrices [@elec2; @elec1]. In this paper, we consider the problem of approximating $f(\bm{A})$ through matrix-vector products $\bm{x}\mapsto\bm{Ax}$ only, assuming $\bm{A}$ is sparse or banded. Let $\bm{A}\in \mathbb{R}^{n\times n}$ and $f$ be a function that is analytic on the closure of a domain $\Omega \subset \mathbb{C}$ that contains the eigenvalues of $\bm{A}$. Then the matrix function $f(\bm{A})$ is defined as $$f(\bm{A}) = \frac{1}{2\pi i} \int_{\partial \Omega} f(z)(z\bm{I}-\bm{A})^{-1} dz,$$ where $\bm{I}$ denotes the identity matrix and $(z\bm{I}-\bm{A})^{-1}$ is called the resolvent matrix of $\bm{A}$. If $\bm{A}\in \mathbb{R}^{n\times n}$ is diagonalizable with $\bm{A} = \bm{X}\bm{D}\bm{X}^{-1}$ where $\bm{X}$ is the eigenvector matrix and $\bm{D}$ is the diagonal matrix containing the eigenvalues $\{\lambda_i\}_{i=1}^n$ of $\bm{A}$, then $$f(\bm{A}) = \bm{X}f(\bm{D})\bm{X}^{-1}, \hspace{1em} f(\bm{D}) = \begin{bmatrix} f(\lambda_1) & & \\ & \ddots & \\ & & f(\lambda_n) \end{bmatrix}.$$ There are many other equivalent ways of defining $f(\bm{A})$, see for example [@matfunchigham]. When the matrix dimension is large, explicitly computing $f(\bm{A})$ using standard algorithms [@matcomp; @matfunchigham] becomes prohibitive in practice, usually requiring $O(n^3)$ operations. Moreover, the matrix $\bm{A}$ may also be too large to store explicitly. In this scenario, $\bm{A}$ is often only available through matrix-vector products and $f(\bm{A})\bm{b}$ [^1] is used to infer information about $f(\bm{A})$ [@Guttel2013; @GuttelKressnerLund2020]. In this paper, we show that the whole matrix $f(\bm{A})$ can still be approximated using $\ll n$ matrix-vector products with $\bm{A}$ if $\bm{A}$ is sparse and $f$ is smooth. We exploit the decay bounds for the entries of $f(\bm{A})$ [@Benzi2007] to recover its large entries using matrix-vector products. We primarily study functions of sparse matrices with general sparsity pattern and matrices with few large entries in each row/column (See Section [2](#sec:sparse){reference-type="ref" reference="sec:sparse"}), but also consider functions of banded matrices and matrices with off-diagonal decay as a special case (See Section [3](#sec:band){reference-type="ref" reference="sec:band"}). Our work is particularly useful if $f(\bm{A})$ is sparse or has only a few entries that are large in each row/column or if $\bm{A}$ is banded with bandwidth $k \ll n$. Let $\bm{A}$ be a diagonalizable, sparse matrix with some sparsity pattern and $f$ (either $f:\mathbb{R}\rightarrow \mathbb{R}$ or $f:\mathbb{C}\rightarrow \mathbb{C}$) a function that is analytic in a region containing the spectrum of $\bm{A}$. Depending on the sparsity pattern of $\bm{A}$ and the smoothness of $f$, some entries of $f(\bm{A})$ can be shown to be exponentially smaller than others [@Benzi2007] and we can view them as noise. For a sparse matrix with a general sparsity pattern, it can be difficult to know the support (location of the large entries) of $f(\bm{A})$ in advance, for example, if the sparsity pattern of $\bm{A}$ is unknown. However, we can write $f(\bm{A}) = \widehat{f(\bm{A})}+ \bm{E}$ with $\left\lVert\bm{E}\right\rVert_{\max} \leq \epsilon$ where $\left\lVert\bm{B}\right\rVert_{\max} = \max\limits_{1\leq i,j \leq n} |B_{ij}|$ and compute $\widehat{f(\bm{A})}$ by viewing $\widehat{f(\bm{A})}$ as signals and $\bm{E}$ as noise using *compressed sensing* row by row or column by column. Compressed sensing is a technique in signal processing for efficiently reconstructing a signal which is sparse in some domain by finding solutions to underdetermined linear systems [@CandesRombergTao06; @CandesTao06; @Donoho06]. As a special case (by taking $f(\bm{A})=\bm{A}$), our algorithm can be used for recovering a sparse matrix $\bm{A}$ itself, where $\bm{A}$ is accessed only through matrix-vector multiplications $\bm{x}\mapsto \bm{A}\bm{x}$. The details will be discussed in Section [2](#sec:sparse){reference-type="ref" reference="sec:sparse"}. Let $\bm{A}$ now be a banded matrix. Then $f(\bm{A})$ can usually be well approximated by a banded matrix [@DMS1984]. There are theoretical results that confirm this property [@BenziGolub], which show that the entries of $f(\bm{A})$ decay exponentially away from the main diagonal. The observation is that $\left|[f(\bm{A})]_{ij}\right| \ll \left|[f(\bm{A})]_{ii}\right|,\left|[f(\bm{A})]_{jj}\right|$ for $|i-j|\gg 1$. Therefore, we can dismiss the exponentially small entries and find a matrix that recovers the large entries of $f(\bm{A})$, i.e., find a matrix $\widehat{f(\bm{A})}$ such that $\left\lVert f(\bm{A})-\widehat{f(\bm{A})}\right\rVert_{\max} \leq \epsilon$ where $\epsilon$ is a tolerance parameter. We can show that the large entries of $f(\bm{A})$ can be captured through matrix-vector products with the following simple deterministic matrix $$\bm{I}_{n}^{(s)}:= \left[\bm{I}_s,\bm{I}_s,...,\bm{I}_s,\bm{I}_s(:,1:\{n/s\}\cdot s)\right]^T \in \mathbb{R}^{n\times s}$$ where $s<n$, $\bm{I}_s$ is the $s\times s$ identity matrix and $\bm{I}_s(:,1:\{n/s\}\cdot s)$ are the first $\{n/s\}\cdot s$ columns of $\bm{I}_s$, which ensures that $\bm{I}_{n}^{(s)}$ has exactly $n$ rows. Here $\{x\}$ is the fractional part of $x$. This is unlike the sparse case where we now have significant information on the support of $f(\bm{A})$. This set of vectors $\bm{I}_{n}^{(s)}$ are identical to the probing vectors proposed in [@FrommerSchnimmelSchweitzer2021] for estimating functions of banded matrices and [@ColemanMore1983; @CPR1974] for estimating sparse, banded Jacobian matrices. If we let $s$ be proportional to the bandwidth of $\bm{A}$, the matrix-vector products $f(\bm{A})\bm{I}_n^{(s)}$ capture the large entries near the main diagonal, giving us a good approximation for $f(\bm{A})$. We discuss the details in Section [3](#sec:band){reference-type="ref" reference="sec:band"}. Both cases require matrix-vector products with $f(\bm{A})$, i.e., $\bm{x}\mapsto f(\bm{A})\bm{x}$. Popular methods for this include the standard (polynomial) Krylov method [@matfunchigham; @Saad92], the rational Krylov method [@Guttel2013] and contour integration [@matveccontour]. In this paper, we discuss three functions: the exponential $e^{\bm{A}}$, the square root function $\sqrt{\bm{A}}$ and the logarithm $\log(\bm{A})$. For the exponential we use the polynomial Krylov method, and for the square root function and the logarithm, we use contour integration as in [@matveccontour method 2] as both functions have a branch cut on the negative real axis $(-\infty,0]$, making them difficult to approximate using polynomials if $\bm{A}$ has eigenvalues close to $0$. If $f$ has poles in the region containing the spectrum of $\bm{A}$ or if $f$ cannot be well-approximated by a low-degree polynomial, then rational Krylov methods can also be used. For Krylov methods, block versions can also be used [@FrommerLundSzyld2018], which compute $f(\bm{A})\bm{B} \in \mathbb{R}^{n\times s}$ at once instead of column by column. We do not discuss the details of computing the matrix-vector product $f(\bm{A})\bm{b}$ further as they are not the focus of this paper. ## Existing methods First, Benzi and Razouk [@Benzi2007] describe an algorithm that exploits the decay property by first finding a good polynomial approximation $p$ to $f$, for example using Chebyshev interpolation, and computing $p(\bm{A})$ while using a dropping strategy to keep the matrix as sparse as possible. However, this method can quickly become infeasible if $f$ cannot be well approximated by a low-degree polynomial or if sparse data structures and arithmetic is not available. Frommer, Schimmel and Schweitzer [@FrommerSchnimmelSchweitzer2021] use probing methods to approximate matrix functions $f(\bm{A})$ and its related quantities, for instance the trace of $f(\bm{A})$. The method finds probing vectors to estimate the entries of $f(\bm{A})$ by partitioning the vertices of the directed graph $G(\bm{A})$ associated with the sparse matrix $\bm{A}$ using graph coloring and the sparsity pattern of $\bm{A}$. Our Section [3](#sec:band){reference-type="ref" reference="sec:band"} on banded matrices can be seen as a special case of [@FrommerSchnimmelSchweitzer2021], but as banded matrices arise naturally in applications and come with strong theoretical results, we make the sensing matrix $\bm{B}$ explicit and study them in detail. Cortinovis, Kressner and Massei [@CortinovisKressnerMassei2022] use a divide-and-conquer approach to approximate matrix functions where the matrix is either banded, hierarchically semiseparable or has a related structure. The divide-and-conquer method is based on (rational) Krylov subspace methods for performing low-rank updates of matrix functions. Under the assumption that the minimal block size in the divide-and-conquer algorithm is $\mathcal{O}(1)$ and the low-rank update in the Krylov method converges in $\mathcal{O}(1)$ iterations, the algorithm runs with complexity $\mathcal{O}(nk^2)$ for $k$-banded matrices. There are other related works that use matrix-vector products to recover matrices with certain properties or using compressed sensing to recover matrices with only few large entries. In [@Townsend22; @LetvittMartinsson2022; @Woodruffetal2021], the authors devise algorithms or strategies based on matrix-vector products for recovering structured matrices such as hierarchical low-rank matrices and Toeplitz matrices or for answering queries such as whether an unknown matrix is symmetric or not. In particular, Curtis, Powell and Reid [@CPR1974] and later Coleman and Moré [@ColemanMore1983] use certain graph coloring of the adjacency graph of a sparse or banded matrix to approximate sparse or banded Jacobian matrices using matrix-vector products, which Bekas, Kokiopoulou and Saad [@BekasKokiopoulouSaad2007] later suggest the same set of coloring as the banded case for matrices with rapid off-diagonal decay. In [@HermanStrohmer2009; @PfranderRauhutTanner2008], the authors formulate an algorithm based on compressed sensing techniques to recover sparse matrices with at most $\mathcal{O}(sn/\log^2 n)$ nonzero entries using $s$ matrix-vector products (See [@PfranderRauhutTanner2008 Thm. 6.3]). This method uses a vectorized form of the matrix in question after a certain transformation, which can become expensive. Lastly, Dasarathy, Shah and Bhaskar [@Dasarathyetal2015] use compressed sensing techniques by applying compression (sensing operators) on both sides, i.e. $\bm{Y}_1 \bm{X}\bm{Y}_2^T$, to recover an unknown sparse matrix $\bm{X}$. The authors use $\mathcal{O}(\sqrt{n}\log n)$ matrix-vector products with $\bm{X}$ where the number of nonzero entries of $\bm{X}$ is $\mathcal{O}(n)$. This can become prohibitive for large $n$ as computing matrix-vector products is usually the dominant cost. ## Contributions Our first contribution is devising strategies for approximating (or sometimes recovering exactly, e.g. for sparse matrices) functions of sparse matrices and matrices with similar structures, e.g., matrices with only few large entries in each row/column or matrices with rapid off-diagonal decay. We provide two strategies, one for functions of sparse matrices or matrices with only few large entries in each row/column and another for functions of banded matrices or matrices with off-diagonal decay. The two strategies are based on approximating the large entries of the matrix in question using matrix-vector products only. This is advantageous if the matrix dimension is too large to store explicitly or the matrix is only available through matrix-vector products. Our work is different from [@FrommerSchnimmelSchweitzer2021] for functions of general sparse matrices as our work uses compressed sensing to estimate the large entries of each row/column, which does not require the sparsity pattern of $\bm{A}$, whereas the work in [@FrommerSchnimmelSchweitzer2021] uses probing vectors obtained by partitioning the indices using graph coloring and the sparsity pattern of $\bm{A}$ to estimate the large entries. For the banded case, our work is analogous to the previous work done in [@ColemanMore1983; @CPR1974; @FrommerSchnimmelSchweitzer2021] but we give a focused treatment of banded matrices. Our second contribution is designing simple algorithms, Algorithm [\[alg:gen\]](#alg:gen){reference-type="ref" reference="alg:gen"} (SpaMRAM) for sparse matrices with general sparsity pattern and Algorithm [\[alg:band\]](#alg:band){reference-type="ref" reference="alg:band"} (BaMRAM) for banded matrices, with an error analysis based on the decay structure of the matrix. We show that SpaMRAM converges quickly as long as the matrix has only few large entries in each row/column and that BaMRAM converges exponentially with the number of matrix-vector products. Assuming that $f$ is sufficiently smooth so that the Krylov method converges in $\mathcal{O}(1)$ iterations when computing $f(\bm{A})\bm{b}$, SpaMRAM runs with complexity $\mathcal{O}(n^2\log n)$ for functions of general sparse matrices and BaMRAM runs linearly in $n$ with complexity $\mathcal{O}(nk^2)$ for functions of $k$-banded matrices. For banded matrices, BaMRAM runs with complexity $\approx 4nk^2$ and for functions of banded matrices, BaMRAM has the same complexity, $\mathcal{O}(nk^2)$, as [@CortinovisKressnerMassei2022]. The output of the algorithm is also sparse and has $\mathcal{O}(nk)$ nonzero entries where $k$ is the number of nonzero entries recovered per row/column, which can be advantageous in many situations as $f(\bm{A})$ need not be exactly sparse even if $\bm{A}$ is sparse. ## Notation Throughout, we write $\left\lfloor {x} \right\rfloor$ and $\{x\}$ to denote the floor function of $x$ and the fractional part of $x$ respectively. We use $\left\lVert\cdot\right\rVert_p$ where $1\leq p\leq \infty$ to denote the $p$-norm for vectors and matrices and define $\left\lVert\cdot\right\rVert_{\max}$ by $\left\lVert\bm{A}\right\rVert_{\max}= \max\limits_{i,j}|A_{ij}|$ where $\bm{A} \in \mathbb{R}^{m\times n}$ is a matrix. We use MATLAB style notation for matrices and vectors. For example, for the $k$th to $(k+j)$th columns of a matrix $\bm{A}$ we write $A(:,k:k+j)$. # Approximation for functions of general sparse matrices {#sec:sparse} Let $\bm{A}$ be a diagonalizable sparse matrix with some sparsity pattern. Suppose that $f$ is analytic in a region containing the spectrum of $\bm{A}$. Then depending on the sparsity pattern of $\bm{A}$, the entries of $f(\bm{A})$ decay exponentially away from certain small regions of the matrix. More formally, associate the unweighted directed graph $G(\bm{A}) = (V,E)$ to $\bm{A} = [A_{ij}]$ so that $\bm{A}$ is the adjacency matrix of $G(\bm{A})$. $V$ is the vertex set consisting of integers from $1$ to $n$ and $E$ is the edge set containing all ordered pairs $(i,j)$ with $A_{ij} \neq 0$. We define the distance $d_G (i,j)$ between vertex $i$ and vertex $j$ to be the shortest directed path from $i$ to $j$.[^2] We then have that there exist constants $C_0>0$ and $0<\lambda_0 <1$ such that $$\label{eq:fsparsedecay} \left|[f(\bm{A})]_{ij}\right| < C_0 \lambda_0^{d_G (i,j)}$$ for all $i,j = 1,2,...,n$. For further details see [@Benzi2007]. This result tells us that if $d_G (i,j)\gg 1$ then $|[f(\bm{A})]_{ij}|\ll 1$, which in turn means that depending on the sparsity pattern of $\bm{A}$, there may only be few large entries in $f(\bm{A})$ with the rest being negligible. With this observation we can aim to recover the dominating entries of each row/column of $f(\bm{A})$. We solve this problem using *compressed sensing* by interpreting small entries as noise and the large entries as signals.[^3] Compressed sensing [@CandesRombergTao06; @Donoho06] is a technique in signal processing for efficiently reconstructing a sparse signal. Let $\bm{Y}\in \mathbb{R}^{n\times s}$ be a so-called *sensing operator* and $\bm{v}\in \mathbb{R}^{n}$ be an unknown sparse vector. Then the problem of recovering $\bm{v}$ using the measurements $\bm{y} = \bm{Y}^T\bm{v}$ is a combinatorial optimization problem given by $$\label{csprob} \min_{\bm{z}\in \mathbb{R}^n} \left\lVert\bm{z}\right\rVert_0 \text{ subject to } \bm{y} = \bm{Y}^T\bm{z}$$ where $\left\lVert\bm{z}\right\rVert_0$ denotes the number of nonzeros in the vector $\bm{z}$. There are two established classes of algorithms, among others, to solve [\[csprob\]](#csprob){reference-type="eqref" reference="csprob"} in the compressed sensing literature, namely basis pursuit or $\ell_1$-minimization using convex optimization [@CandesTao05], and greedy methods [@BlumensathDaviesRiling12]. For large $n$, convex optimization can become infeasible (especially given that we need to solve [\[csprob\]](#csprob){reference-type="eqref" reference="csprob"} $n$ times to recover the whole matrix), so we use the cheaper option, greedy methods. Greedy methods are iterative and at each iteration attempt to find the correct location and the values of the dominant entries. Let $\bm{v}\in \mathbb{R}^n$ be a $k$-sparse vector, that is, a vector with at most $k$ nonzero entries. To recover $\bm{v}$ with theoretical guarantee we typically require at least $s = \mathcal{O}(k\log n)$ measurements. In practice, $s = \mathcal{O}(k)$ measurements seem to work well [@BlanchardTanner15]. Each measurement is taken through vector-vector multiply with $\bm{v}$. The three classes of frequently used sensing operators $\bm{Y}$ are: a Gaussian matrix $\bm{\mathcal{N}}$ with entries drawn from i.i.d. $\mathcal{N}(0,s^{-1})$, $s$ columns of the discrete cosine transform matrix $\bm{\mathcal{C}}$ sampled uniformly at random and a sparse matrix $\bm{\mathcal{S}}_\xi$ with $\xi$ nonzero entries per row with the support set drawn uniformly at random and the nonzero values drawn uniformly at random from $\pm \xi^{-1/2}$. $\xi$ is the sparsity parameter for the sparse sensing operator. There are a number of greedy methods available, including Normalized Iterative Hard Thresholding (NIHT) [@NIHT], Hard Thresholding Pursuit (HTP) [@HTP] and Compressive Sampling Matching Pursuit (CoSaMP) [@CoSaMP]. Among the key differences and the trade-offs between the different methods is the cost per iteration in order to speed up support detection and the number of iterations for better asymptotic convergence rate. Many greedy methods have uniform recovery guarantees if the sensing operator $\bm{Y}$ has sufficiently small restricted isometry constants [@csbook]. This means that if we want to recover a matrix then we can use the same sensing operator $\bm{Y}$ for the recovery of all rows of a matrix, which makes it much more efficient than recovering $n$ rows of a matrix individually using different sensing operators. In this paper, we use NIHT as it is simple and has a low cost per iteration.[^4] NIHT is an iterative hard thresholding algorithm which performs gradient descent and then hard thresholding at each iteration to update the support and the value of the support set: the $(N+1)$th iterate $\bm{v}^{(N+1)}$ of NIHT is $$\label{eq:NIHTeqn} \bm{v}^{(N+1)} = H_k\left(\bm{v}^{(N)} +\mu \bm{Y}\left(\bm{y}-\bm{Y}^T\bm{v}^{(N)}\right)\right).$$ Here $H_k$ is the hard thresholding operator that sends all but the largest $k$ entries (in absolute value) of the input vector to $0$. The stepsize $\mu$ in NIHT is chosen to be optimal when the support of the current iterate is the same as the sparsest solution. Assuming that $\bm{Y}$ satisfies the restricted isometry property (RIP), then at iteration $N$, the output $\bm{v}^{(N)}$ of NIHT satisfies $$\left\lVert\bm{v}-\bm{v}^{(N)}\right\rVert_2 \leq 2^{-N}\left\lVert\bm{v}^{k}\right\rVert_2 + 8 \epsilon_k,$$ where $\bm{v}^k$ is the best $k$-term approximation to $\bm{v}$ and $$\epsilon_k = \left\lVert\bm{v}-\bm{v}^k\right\rVert_2 +\frac{1}{\sqrt{k}} \left\lVert\bm{v}-\bm{v}^k\right\rVert_1,$$ and after at most $N^* = \left\lceil{\log_2 \left(\left\lVert\bm{v}^k\right\rVert_2/\epsilon_k\right)}\right\rceil$ iterations, we have $$\label{eq:NIHTbound} \left\lVert\bm{v}-\bm{v}^{(N^*)}\right\rVert_2 \leq 9\epsilon_k.$$ Further details about NIHT can be found in [@NIHT]. The dominating cost for the greedy methods is the matrix-vector product between the sensing operator or the transpose of the sensing operator and the unknown sparse signal, i.e., the cost of $\bm{v}\mapsto \bm{Y} \bm{v}$ and $\bm{w} \mapsto \bm{Y}^T\bm{w}$ respectively. If the sensing operator is a Gaussian $\bm{\mathcal{N}}$ then the cost is $\mathcal{O}(ns)$, if $\bm{Y}$ is a subsampled discrete cosine matrix $\bm{\mathcal{C}}$ then the cost is $\mathcal{O}(n\log n)$ or $\mathcal{O}(n\log s)$ if we use the subsampled FFT algorithm [@rokhlintygert2008] and lastly, if $\bm{Y}$ is a sparse matrix $\bm{\mathcal{S}}_\xi$ then the cost is $\mathcal{O}(\xi n)$. In our context, we are concerned with recovering the entire matrix. We now use compressed sensing row by row or column by column to approximate functions of sparse matrices. ## Strategy for recovering general sparse matrices In this section, we consider matrices $\bm{B}\in \mathbb{R}^{n\times n}$ that satisfy $$\label{eq:sparsedecay} \left|B_{ij}\right| < C_B \lambda_B^{d(i,j)}$$ where $C_B>0$ and $0<\lambda_B<1$ are constants and $d(i,j) \in \mathbb{R}_{\geq 0} \cup \{+\infty\}$ is some bivariate function that captures the magnitude of the entries of $\bm{B}$. Functions of sparse matrices $\bm{B} = f(\bm{A})$ with a smooth $f$ is a special case of $\bm{B}$ that satisfies [\[eq:fsparsedecay\]](#eq:fsparsedecay){reference-type="eqref" reference="eq:fsparsedecay"}. Another special case is $\bm{B} = \bm{A}$, which recovers the sparse matrix $\bm{A}$ itself. The zero entries of $\bm{B}$ would correspond to $d(i,j) = \infty$ in [\[eq:sparsedecay\]](#eq:sparsedecay){reference-type="eqref" reference="eq:sparsedecay"}. We are concerned with recovering the whole matrix. We approximate the dominant entries of each row of $\bm{B}$ by taking measurements through matrix-vector products $\bm{B}\bm{y}_i$ where $\bm{y}_i$ is the $i$th column of the sensing operator $\bm{Y} \in \mathbb{R}^{n\times s}$.[^5] Let $\texttt{CS}(\bm{Y},\bm{b}_{i}\bm{Y},k)$ be a function that solves [\[csprob\]](#csprob){reference-type="eqref" reference="csprob"} where $\bm{Y} \in \mathbb{R}^{n\times s}$ is the sensing operator, $\bm{b}_{i}\bm{Y} \in \mathbb{R}^{1\times s}$ are the measurements where $\bm{b}_i$ is the $i$th row of $\bm{B}$ (i.e., $\bm{b}_{i}\bm{Y}$ is the $i$th row of $\bm{BY}$) and $k$ is the sparsity parameter. For the compressed sensing algorithm $\texttt{CS}$, we can take, for example $\texttt{CS} =$ NIHT. The algorithm to recover the dominant entries of each row of $\bm{B}$ is given below in Algorithm [\[alg:gen\]](#alg:gen){reference-type="ref" reference="alg:gen"}, which we call SpaMRAM (Sparse Matrix Recovery Algorithm using Matvecs). Draw a sensing operator $\bm{Y}\in \mathbb{R}^{n\times s}$ where $s = \mathcal{O}(k\log n)$ Compute matrix-vector products $\bm{F}_k = \texttt{mvp}(\bm{Y})$ $\displaystyle \widehat{\bm{B}}(i,:) = \texttt{CS}(\bm{Y},\bm{F}_k(i,:),k)$ The algorithm SpaMRAM is a straightforward application of compressed sensing techniques used to recover a sparse matrix instead of a vector, particularly when the goal is to recover $\bm{A}$ itself. We are nonetheless unaware of existing work that spells out explicitly, and SpaMRAM can be significantly more efficient than alternative approaches [@HermanStrohmer2009; @PfranderRauhutTanner2008] that vectorize the matrix. Since most sensing matrices satisfy the RIP with failure probability exponentially decaying in $s$ [@csbook], by the union bound we can conclude that the whole matrix can be recovered with high probability. ### Complexity of SpaMRAM (Alg. [\[alg:gen\]](#alg:gen){reference-type="ref" reference="alg:gen"}) The complexity of SpaMRAM is $s$ matrix-vector products with $\bm{B}$ and $n$ times the complexity of solving the compressed sensing problem. The complexity for solving $n$ compressed sensing problems using NIHT and the subsampled DCT matrix as $\bm{Y}$ is $\mathcal{O}(n^2\log n)$, where we take the number of NIHT iterations to be $\mathcal{O}(1)$. When $\bm{B} = f(\bm{A})$ for sparse $\bm{A}$, we can evaluate $f(\bm{A})\bm{Y}$ using Krylov methods. Assuming that the Krylov method converges (to sufficient accuracy) in $\mathcal{O}(1)$ iterations, the complexity of SpaMRAM is $\mathcal{O}(s\cdot nnz(\bm{A}) + n^2\log n) = \mathcal{O}(k\log n \cdot nnz(\bm{A})+ n^2\log n)$ which consists of $\mathcal{O}( s\cdot nnz(\bm{A})) = \mathcal{O}( k\log n \cdot nnz(\bm{A}))$ flops for computing $s = \mathcal{O}(k\log n)$ matrix-vector products with $f(\bm{A})$ and $\mathcal{O}(n^2 \log n)$ flops for solving $n$ compressed sensing problems. Here $nnz(\bm{A})$ is the number of nonzero entries of $\bm{A}$. ### Adaptive methods There are adaptive methods that reduce the number of matrix-vector products required for sparse recovery, for example [@adaptive1; @adaptive3; @KrahmerWard2014; @adaptive2] by making adaptive measurements based on previous measurements. However, in our context we are concerned with recovering the whole matrix and it is difficult to make adaptive measurements that account for every row/column of $\bm{B}$ when we only have access to $\bm{B}$ using matrix-vector products. A naive strategy is to make adaptive measurements for each row/column, but this requires at least $\mathcal{O}(n)$ matrix-vector products, which is computationally infeasible and futile as we can recover $\bm{B}$ using $n$ measurements using the $n\times n$ identity matrix. ### Shortcomings There are some limitations with SpaMRAM. First, the entries of $f(\bm{A})$ are only negligible when $d(i,j)\gg 1$. Indeed, there are matrices for which the sparsity pattern implies $d(i,j) = \mathcal{O}(1)$ for all or most $i,j$. An example is the arrow matrix which has nonzero entries along the first row, first column and the main diagonal. The sparsity pattern of the arrow matrix makes $d(i,j)\leq 2$, which can make SpaMRAM behave poorly as the a priori estimate from [\[eq:fsparsedecay\]](#eq:fsparsedecay){reference-type="eqref" reference="eq:fsparsedecay"} tells us that $f(\bm{A})$ may have no entries that are significantly larger than others. A similar issue has been discussed in [@Dasarathyetal2015], where the authors assume that the matrix we want to approximate is $d$-distributed, that is, the matrix has at most $d$ nonzero entries along any row or column. We will use a similar notion, which will be described in Subsection [2.2](#subsec:analgen){reference-type="ref" reference="subsec:analgen"} to analyze SpaMRAM. SpaMRAM requires $k$ as an input. This may be straightforward from the sparsity pattern of $\bm{A}$ and the smoothness of $f$, but in some cases a good choice of $k$ can be challenging. Moreover, when most rows of $\bm{B}$ are very sparse but a few are much denser, $k$ would need to be large enough for the densest row to recover the whole matrix $\bm{B}$. However, even with a small $k$, the sparse rows will be recovered. Another downside is in the complexity of SpaMRAM for functions of sparse matrices. The cost of SpaMRAM is $\mathcal{O}(k\log n \cdot nnz(\bm{A})+n^2\log n)$, which is dominated by the cost of solving $n$ compressed sensing problems, $\mathcal{O}(n^2\log n)$ but computing $f(\bm{A})\bm{I}_n$ where $\bm{I}_n$ is the $n\times n$ identity matrix costs $\mathcal{O}(n\cdot nnz(\bm{A}))$, which is cheaper if $nnz(\bm{A})=\mathcal{O}(n)$ and approximates $f(\bm{A})$ with higher accuracy. However, if matrix-vector products with $\bm{B} = f(\bm{A})$ start dominating the cost of SpaMRAM, or when $\bm{A}$ is not so sparse with $nnz(\bm{A})> n\log n$, but $f(\bm{A})$ is approximately sparse, then there are merits in using SpaMRAM. It is also worth noting that storing the measurements $f(\bm{A})\bm{Y}$ and only solving the compressed sensing problem to approximate specific row(s) when necessary would save computational time. This may be useful when we only need to approximate some subset of rows of $f(\bm{A})$, but we do not know the indices in advance. ## Analysis of SpaMRAM (Alg. [\[alg:gen\]](#alg:gen){reference-type="ref" reference="alg:gen"}) {#subsec:analgen} The class of matrices $\bm{B}$ that satisfy the decay bound [\[eq:sparsedecay\]](#eq:sparsedecay){reference-type="eqref" reference="eq:sparsedecay"} can be difficult to analyze as the behaviour of $d(i,j)$ can give non-informative bounds in [\[eq:sparsedecay\]](#eq:sparsedecay){reference-type="eqref" reference="eq:sparsedecay"}, i.e., there may be no entries that are small. For functions of matrices, the arrow matrix is an example, which has the sparsity pattern that makes $d_G(i,j)\leq 2$. When $\bm{B}$ has entries that all have similar magnitudes, SpaMRAM fails to compute a good approximation to $\bm{B}$. We now define a notion similar to $d$-distributed sparse matrices in [@Dasarathyetal2015] to analyze SpaMRAM. Let $d \in \mathbb{Z}^+$ be a distance parameter and define $$k = \max_{1\leq i\leq n} \left\lvert \left\{ j: d(i,j) \leq d\right\} \right\rvert.$$ When $\bm{B} = f(\bm{A})$, $k$ measures the maximum number of vertices that are at most distance $d$ away from any given vertex. $k$ also measures the combined sparsity of $\bm{I}_n,\bm{A},...,\bm{A}^d$ because the cardinality of the union of the support locations in a fixed row in each of $\bm{I}_n,\bm{A},...,\bm{A}^d$ will be at most $k$. If $k = \mathcal{O}(1)$ for $d$ large enough such that $f$ can be approximated by a polynomial of degree $d-1$ on $\bm{A}$'s spectrum, then each row of $f(\bm{A})$ would only have $\mathcal{O}(1)$ dominating entries. Using the definition of $k$, we now analyze SpaMRAM in Theorem [\[thm:gen\]](#thm:gen){reference-type="ref" reference="thm:gen"}. [\[thm:gen\]]{#thm:gen label="thm:gen"} Let $\bm{B}\in\mathbb{R}^{n\times n}$ be a matrix satisfying $$|B_{ij}| < C_B \lambda_B^{d(i,j)} \tag{\ref{eq:sparsedecay}}$$ for constants $C_B>0$ and $0<\lambda_B<1$. Then the output of SpaMRAM, $\widehat{\bm{B}}$, satisfies $$\label{sparsebound2} \left\lVert\widehat{\bm{B}}-\bm{B}\right\rVert_2 \leq 9C_B \left( n + \frac{n^{3/2}}{\sqrt{k}} \right) \lambda_B^{d+1}$$ where $$k = \max_i \left\lvert \left\{ j: d(i,j) \leq d\right\} \right\rvert$$ is the sparsity parameter in Algorithm [\[alg:gen\]](#alg:gen){reference-type="ref" reference="alg:gen"} with the distance parameter $d$ and the compressed sensing problem was solved row by row using NIHT until [\[eq:NIHTbound\]](#eq:NIHTbound){reference-type="eqref" reference="eq:NIHTbound"} was satisfied. *Proof.* Define $\bm{E} = \widehat{\bm{B}}-\bm{B}$. Let $E(i,:)$ be the $i$th row of $\bm{E}$. Then from [\[eq:NIHTbound\]](#eq:NIHTbound){reference-type="eqref" reference="eq:NIHTbound"} we get $$\left\lVert E(i,:)\right\rVert_2 \leq 9\left(\left\lVert\bm{b}_i - \bm{b}_{i}^{k}\right\rVert_2 +\frac{1}{\sqrt{k}} \left\lVert\bm{b}_i - \bm{b}_i^{k}\right\rVert_1\right)$$ where $\bm{b}_{i}$ is the $i$th row of $\bm{B}$ and $\bm{b}_{i}^{k}$ is the best $k$-term approximation to $\bm{b}_{i}$. Since $\bm{b}_{i}^{k}$ is the best $k$-term approximation to $\bm{b}_{i}$, we have $$\left\lVert\bm{b}_{i} - \bm{b}_{i}^{k}\right\rVert_{\max} \leq C_B \lambda_B^{d+1}$$ using [\[eq:sparsedecay\]](#eq:sparsedecay){reference-type="eqref" reference="eq:sparsedecay"}. Therefore $$\left\lVert\bm{b}_{i} - \bm{b}_{i}^{k}\right\rVert_2 \leq \sqrt{n-k} C_B \lambda_B^{d+1}, \qquad \left\lVert\bm{b}_{i} - \bm{b}_{i}^{k}\right\rVert_1 \leq (n-k)C_B\lambda_B^{d+1}.$$ Therefore $$\begin{aligned} \left\lVert\bm{E}\right\rVert_2 &\leq \sqrt{n} \max_{1\leq i \leq n}\left\lVert E(i,:)\right\rVert_2 \\ &\leq 9\sqrt{n} \max_{1\leq i \leq n} \left\lVert\bm{b}_{i} - \bm{b}_{i}^{k}\right\rVert_2 +\frac{1}{\sqrt{k}} \left\lVert\bm{b}_{i} - \bm{b}_{i}^{k}\right\rVert_1 \\ &\leq 9\sqrt{n} \left( C_B\sqrt{n-k} \lambda_B^{d+1} + C_B\frac{n-k}{\sqrt{k}} \lambda_B^{d+1} \right) \\ &< 9C_B \left( n + \frac{n^{3/2}}{\sqrt{k}} \right) \lambda_B^{d+1}. \end{aligned}$$ ◻ 1. The bound in Theorem [\[thm:gen\]](#thm:gen){reference-type="ref" reference="thm:gen"} can be pessimistic as we can replace the $2$-norm bound by the Frobenius norm bound in [\[sparsebound2\]](#sparsebound2){reference-type="eqref" reference="sparsebound2"}. This can be shown by noting $$\begin{aligned} \left\lVert\bm{E}\right\rVert_F &\leq \sqrt{\sum_{1\leq i \leq n}\left\lVert E(i,:)\right\rVert_2^2} \leq \sqrt{n} \max_{1\leq i \leq n}\left\lVert E(i,:)\right\rVert_2 < 9C_B \left( n + \frac{n^{3/2}}{\sqrt{k}} \right) \lambda_B^{d+1}. \end{aligned}$$ 2. The bound in [\[sparsebound2\]](#sparsebound2){reference-type="eqref" reference="sparsebound2"} can be improved if for each $i$, the nonzero entries of $\bm{b}_i - \bm{b}_i^{k}$ decay exponentially. This implies $$\left\lVert\bm{b}_i - \bm{b}_i^{k}\right\rVert_{1} \lesssim \lambda_B^{d+1},\qquad \left\lVert\bm{b}_i - \bm{b}_i^{k}\right\rVert_{2} \lesssim \lambda_B^{d+1}$$and the bound improves to $$\left\lVert\bm{E}\right\rVert_2 \lesssim \sqrt{n}\left(1+ \frac{1}{\sqrt{k}} \right) \lambda^{d+1}.$$ # Approximation for functions of banded matrices {#sec:band} Let $\bm{A}$ be a diagonalizable banded matrix with upper bandwidth $k_1$ ($A_{ij} = 0$ if $j-i>k_1$) and lower bandwidth $k_2$ ($A_{ij} = 0$ if $i-j>k_2$). This is a more restrictive case than Section [2](#sec:sparse){reference-type="ref" reference="sec:sparse"}, and so SpaMRAM can be used; however, in this case the sparsity pattern is predictable, and hence more efficient algorithms are possible. Suppose that $f$ is a function that is analytic in the region containing the spectrum of $\bm{A}$. We then have that the entries of $f(\bm{A})$ decay exponentially away from the main diagonal. More specifically, we have that there exists constants $C_1,C_2>0$ and $0<\lambda_1,\lambda_2<1$ dependent on $\bm{A}$ and $f$ such that for $i< j$, $$\label{eq:ubandbound} \left|[f(\bm{A})]_{ij}\right| < C_1 \lambda_1^{j-i}$$ and for $i\geq j$, $$\label{eq:lbandbound} \left|[f(\bm{A})]_{ij}\right| < C_2 \lambda_2^{i-j}.$$ The proof of this result is based on polynomial approximation of $f$, together with the simple fact that powers of $\bm{A}$ are banded if $\bm{A}$ is. The exponents satisfy $\log \lambda_1^{-1} \propto k_1^{-1}$ and $\log \lambda_2^{-1} \propto k_2^{-1}$. For more details see [@Benzi2007; @BenziSimoncini2015; @PozzaSimoncini2019]. Unlike the general sparse case, here the location of the large entries is known, i.e., near the main diagonal. We will exploit the exponential off-diagonal decay given above in [\[eq:ubandbound\]](#eq:ubandbound){reference-type="eqref" reference="eq:ubandbound"} and [\[eq:lbandbound\]](#eq:lbandbound){reference-type="eqref" reference="eq:lbandbound"} to efficiently recover functions of banded matrices using matrix-vector products. We first design a way to recover banded matrices using matrix-vector products similarly to how banded Jacobian matrices were recovered in [@ColemanMore1983; @CPR1974] and then use the strategy used for the banded matrices along with the exponential off-diagonal decay to approximate functions of banded matrices and matrices with rapid off-diagonal decay using matrix-vector products. The idea of using the same strategy for matrices with rapid off-diagonal decay was suggested in [@BekasKokiopoulouSaad2007] and was later applied to functions of banded matrices in [@FrommerSchnimmelSchweitzer2021]. ## Strategy for recovering banded matrices {#sec:bandrecovery} Let $\bm{A}\in \mathbb{R}^{n\times n}$ be a banded matrix with upper bandwidth $k_1$ and lower bandwidth $k_2$. $\bm{A}$ has at most $1+k_1+k_2$ nonzero entries in each row and column, which lies consecutively near the main diagonal. Therefore if we recover the $1+k_1+k_2$ consecutive nonzero entries near the main diagonal in each row or column then we can recover $\bm{A}$. This motivates us to use a simple set of $s$ vectors, which to the author's knowledge, originates from [@CPR1974] in the context of recovering matrices using matrix-vector products. The $s$ vectors are $$\label{eq:repeatedid} \bm{I}_{n}^{(s)} = [\bm{I}_s,\bm{I}_s,...,\bm{I}_s,\bm{I}_s(:,1:\{n/s\}\cdot s)]^T \in \mathbb{R}^{n\times s}$$ where $s<n$ is a positive integer and $\bm{I}_s(:,1:\{n/s\}\cdot s)$ is the first few columns of $\bm{I}_s$ to ensure $\bm{I}_{n}^{(s)}$ has exactly $n$ rows. This set of $s$ vectors capture $s$ consecutive nonzero entries of $\bm{A}$ in each row when right-multiplied to $\bm{A}$ if the other $(n-s)$ entries are all zeros. By choosing $s = 1+k_1+k_2$ we are able to recover every nonzero entries of $\bm{A}$. We illustrate this idea with an example. #### Example Let $\bm{A}_6 \in \mathbb{R}^{6\times 6}$ be a banded matrix with upper bandwidth $2$ and lower bandwidth $1$. Then the matrix-vector products of $\bm{A}_6$ with $\bm{I}_6^{(1+2+1)}$ gives $$\bm{A}_6 \bm{I}_6^{(4)} = \begin{bmatrix} b_1 & c_1 & d_1 & 0 & 0 & 0 \\ a_1 & b_2 & c_2 & d_2 & 0 & 0 \\ 0 & a_2 & b_3 & c_3 & d_3 & 0 \\ 0 & 0 & a_3 & b_4 & c_4 & d_4 \\ 0 & 0 & 0 & a_4 & b_5 & c_5 \\ 0 & 0 & 0 & 0 & a_5 & b_6 \\ \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix} = \begin{bmatrix} b_1 & c_1 & d_1 & 0 \\ a_1 & b_2 & c_2 & d_2 \\ d_3 & a_2 & b_3 & c_3 \\ c_4 & d_4 & a_3 & b_4 \\ b_5 & c_5 & 0 & a_4 \\ a_5 & b_6 & 0 & 0 \\ \end{bmatrix}.$$ We observe in the example that every nonzero entries of $\bm{A}_6$ in each row has been copied exactly once up to reordering onto the same row of $\bm{A}_6\bm{I}_6^{(4)}$. Since we know $\bm{A}_6$ has upper bandwidth $2$ and lower bandwidth $1$, we can reorganize the entries of $\bm{A}_6\bm{I}_6^{(4)}$ to recover $\bm{A}_6$ exactly. This idea can be generalized as follows [@ColemanMore1983; @CPR1974].[^6] [\[thm:bband\]]{#thm:bband label="thm:bband"} Let $\bm{A} \in \mathbb{R}^{n\times n}$ be a banded matrix with upper bandwidth $k_1$ and lower bandwidth $k_2$. Then $\bm{A}$ can be exactly recovered using $1+k_1+k_2$ matrix-vector products using $\bm{I}_n^{(1+k_1+k_2)}$ as in [\[eq:repeatedid\]](#eq:repeatedid){reference-type="eqref" reference="eq:repeatedid"}. *Proof.* Let $s = 1+k_1+k_2$ and for $1\leq i \leq n$ and $1\leq j \leq s$, define $$\begin{aligned} t_{ij} &= \mathop{\mathrm{arg\,min}}_{\substack{1\leq j+st \leq n \\ t\in \mathbb{Z}}}\left|j+st-\left(i+(s-1)\left(\frac{1}{2}-\frac{k_1}{k_1+k_2}\right)\right)\right| \\ &= \mathop{\mathrm{arg\,min}}_{\substack{1\leq j+st \leq n \\ t\in \mathbb{Z}}}\left|j+st-i-\frac{k_2-k_1}{2}\right|. \end{aligned}$$ Equivalently, $t_{ij}$ is the unique integer satisfying $$k_1 \geq j+st_{ij}-i \geq -k_2, \text{ and } 1\leq j+st_{ij} \leq n.$$ Then the $(i,j)$-entry of $\bm{A}\bm{I}_n^{(s)}$ is given by $$\label{eq:keyterm} \left[\bm{A}\bm{I}_n^{(s)}\right]_{ij} = \sum_{\substack{1\leq j+st \leq n \\ t\in \mathbb{Z}}} A_{i,j+st} = A_{i,j+st_{ij}}$$ since $A_{k\ell} = 0$ for all $k_2 < k-\ell$ and $k_1 < \ell-k$. Now we show that for all $1\leq i,r \leq n$ with $k_1\geq r-i \geq -k_2$, there is a unique entry of $\bm{A}\bm{I}_{n}^{(s)}$ equal to $A_{ir}$, i.e., we can use the entries of $\bm{A}\bm{I}_{n}^{(s)}$ to recover every nonzero entry of $\bm{A}$. Let $1\leq i,r \leq n$ be integers such that $k_1 \geq r-i \geq -k_2$ and $1\leq j \leq s$ be a unique integer such that $r = st^*+j$ for some integer $t^*$. Then $$\left[\bm{A}\bm{I}_n^{(s)}\right]_{ij} = \sum_{\substack{1\leq j+st \leq n\\ t\in \mathbb{Z}}} A_{i,j+st} = A_{i,j+st^{*}} = A_{ir}.$$ Therefore $t^* = t_{ij}$ and for each $k_1 \geq r-i \geq -k_2$, there is a unique entry $\left[\bm{A}\bm{I}_{n}^{(s)}\right]_{ij}$ equal to $A_{ir}$ with $j = r-st_{ij}$. Therefore $\bm{A}$ can be exactly recovered using $1+k_1+k_2$ matrix-vector products using $\bm{I}_n^{(1+k_1+k_2)}$. ◻ We now extend this idea to approximate functions of banded matrices and matrices with off-diagonal decay. ## Strategy for recovering functions of banded matrices To accurately approximate $f(\bm{A})$ using matrix-vector products, we want to find a set of vectors when multiplied with $f(\bm{A})$, approximate the large entries of $f(\bm{A})$. We use the same set of vectors as for the banded case, namely $\bm{I}_n^{(s)}$. We assume that $s$ is an odd integer $s = 2s_0+1$ throughout this section. We will see that this set of simple vectors can approximate functions of banded matrices. This is the same as the observation made in [@BekasKokiopoulouSaad2007] and the work done later in [@FrommerSchnimmelSchweitzer2021], which are based on graph coloring of the adjacency graph of a matrix with off-diagonal decay. In this section, we consider matrices $\bm{B} \in \mathbb{R}^{n\times n}$ with exponential[^7] off-diagonal decay with $\bm{B} = f(\bm{A})$ for banded $\bm{A}$ being the special case. For simplicity we consider matrices with the same decay rate on both sides of the diagonal, that is, $\bm{B}$ satisfies $$\label{eq:symdecayrate} \left|\bm{B}_{ij}\right| < C \lambda^{|i-j|}$$ for some constant $C>0$ and $0<\lambda<1$. The results in this section can easily be extended to the more general case, that is, the class of matrices with entries satisfying [\[eq:ubandbound\]](#eq:ubandbound){reference-type="eqref" reference="eq:ubandbound"} and [\[eq:lbandbound\]](#eq:lbandbound){reference-type="eqref" reference="eq:lbandbound"}, i.e., the decay rate along the two sides of the diagonal are different. See Subsection [3.2.3](#sec:difdecayrate){reference-type="ref" reference="sec:difdecayrate"} for a further discussion. The key idea is to revisit [\[eq:keyterm\]](#eq:keyterm){reference-type="eqref" reference="eq:keyterm"} in the proof of Theorem [\[thm:bband\]](#thm:bband){reference-type="ref" reference="thm:bband"}. For $1\leq i\leq n$ and $1\leq j \leq s$, recall that $t_{ij}$ is defined by $$t_{ij} = \mathop{\mathrm{arg\,min}}_{\substack{1\leq j+st \leq n \\ t\in \mathbb{Z}}}\left|j+st-i\right|$$ when the decay rate is the same on both sides of the diagonal.[^8] $t_{ij}$ is an integer such that $i$ and $j+st_{ij}$ are as close as possible, i.e., the index closest to the main diagonal. [\[eq:keyterm\]](#eq:keyterm){reference-type="eqref" reference="eq:keyterm"} gives us $$\left[\bm{B}\bm{I}_n^{(s)}\right]_{ij} = \sum_{\substack{1\leq j+st \leq n \\ t\in \mathbb{Z}}} B_{i,j+st}.$$ By [\[eq:symdecayrate\]](#eq:symdecayrate){reference-type="eqref" reference="eq:symdecayrate"}, the sum is dominated by $B_{i,j+st_{ij}}$ and the remaining terms decay exponentially. Therefore we can view this as $$\left[\bm{B}\bm{I}_n^{(s)}\right]_{ij} = \underbrace{B_{i,j+st_{ij}}}_{\text{Dominant term}}+\underbrace{\sum_{\substack{1\leq j+st \leq n \\ t\neq t_{ij}\in \mathbb{Z}}} B_{i,j+st}}_{\text{Sum of exponentially decaying terms}},$$ which motivates us to use $\left[\bm{B}\bm{I}_n^{(s)}\right]_{ij}$ as an approximation for $B_{i,j+st_{ij}}$. Now create a matrix $\widehat{\bm{B}}$ by $$\label{reconeq} \widehat{B}_{i,j+st_{ij}} = \left[\bm{B}\bm{I}_n^{(s)}\right]_{ij} = \sum_{\substack{1\leq j+st \leq n \\ t\in \mathbb{Z}}} B_{i,j+st}$$ for $|j+st_{ij}-i| \leq s_0$ where $1\leq i\leq n$ and $1\leq j\leq s$ and zero everywhere else. Notice that by construction, $\widehat{\bm{B}}$ is an $s_0$-banded matrix. Roughly, the nonzero entries of the $i$th row of $\widehat{\bm{B}}$ will be, with some small error, the largest entries of the $i$th row of $\bm{B}$ up to reordering. The entries of $\widehat{\bm{B}}$ approximates $\bm{B}$ well because $$\label{eq:enterr} \left\lvert \widehat{B}_{i,j} - B_{i,j}\right\rvert = \left\lvert\sum_{\substack{1\leq j+st \leq n \\ t \neq 0 \in \mathbb{Z}}} B_{i,j+st}\right\rvert \approx C\lambda^{s_0}$$ for $|j-i| \leq s_0$ and $1\leq i,j \leq n$. The largest error incurred from [\[eq:enterr\]](#eq:enterr){reference-type="eqref" reference="eq:enterr"} is at most $\mathcal{O}\left(\left|B_{i,i\pm s_0}\right|\right)$ for the $i$th row and the zero entries of $\widehat{\bm{B}}$, i.e. the entries not covered by [\[eq:enterr\]](#eq:enterr){reference-type="eqref" reference="eq:enterr"}, incur error of at most $\mathcal{O}\left(\left|B_{i,i\pm s_0}\right|\right)$ also. Therefore, $$\label{erranal} \left\lVert\widehat{\bm{B}}-\bm{B}\right\rVert_{\max} = \mathcal{O}\left(\max_{1\leq i\leq n}\left|B_{i,i\pm s_0}\right|\right).$$ The precise details will be laid out in Theorem [\[thm:band\]](#thm:band){reference-type="ref" reference="thm:band"}. Figure [\[fig:decay\]](#fig:decay){reference-type="ref" reference="fig:decay"} illustrates the error analysis given above for $\bm{B} = f(\bm{A})$ where $f$ is the exponential function $e^x$ and $\bm{A}\in \mathbb{R}^{1000\times 1000}$ is a symmetric $2$-banded matrix with entries drawn from i.i.d. $\mathcal{N}(0,n^{-1})$. The matrix-vector products $f(\bm{A})\bm{I}_n^{(s)}$ were evaluated using the polynomial Krylov method with $4,6$ and $8$ iterations (corresponding to polynomial approximation of degrees $3,5,7$ respectively), which give approximations to $f(\bm{A})\bm{I}_n^{(s)}$ of varying accuracy. In Figure [\[fig1a\]](#fig1a){reference-type="ref" reference="fig1a"}, we see the exponential decay in the entries of $f(\bm{A})$ and in Figure [\[fig1b\]](#fig1b){reference-type="ref" reference="fig1b"}, we see that [\[erranal\]](#erranal){reference-type="eqref" reference="erranal"} captures the decay rate until the error is dominated by the error from computing matrix-vector products using the Krylov method. The stagnation of the $\max\limits_{i = 1,2,...,n} \left|[f(\bm{A})]_{i,i\pm (s-1)/2}\right|$ curve is caused by machine precision. We now devise an algorithm based on the above analysis to approximate matrices that satisfy the exponential off-diagonal decay. The algorithm is given in Algorithm [\[alg:band\]](#alg:band){reference-type="ref" reference="alg:band"}, which we call BaMRAM (Banded Matrix Recovery Algorithm using Matvecs). Compute matrix-vector products $\bm{B}_s = \texttt{mvp}\left(\bm{I}_{n}^{(s)}\right) \in \mathbb{R}^{n\times s}$, where $\bm{I}_n^{(s)}$ is as in [\[eq:repeatedid\]](#eq:repeatedid){reference-type="eqref" reference="eq:repeatedid"} Set the entries of $\widehat{\bm{B}}$ using $\bm{B}_s$ and [\[reconeq\]](#reconeq){reference-type="eqref" reference="reconeq"} ### Complexity of BaMRAM (Alg. [\[alg:band\]](#alg:band){reference-type="ref" reference="alg:band"}) The complexity of BaMRAM is $s$ matrix-vector products with $\bm{B}$. In the case $\bm{B} = f(\bm{A})$ for a $k$-banded matrix $\bm{A}$, we can evaluate $f(\bm{A})\bm{I}_n^{(s)}$ using polynomial or rational Krylov depending on $f$ and and the spectrum of $\bm{A}$. Assuming that the Krylov method converges (to sufficient accuracy) in $\mathcal{O}(1)$ iterations and $s = 2ck+1$ for some constant $c$, the complexity of BaMRAM is $\mathcal{O}(snk) = \mathcal{O}(nk^2)$. If we only have access to the matrix using its transpose, i.e. $\bm{x} \mapsto \bm{A}^T\bm{x}$ (or $\bm{x} \mapsto \bm{B}^T\bm{x}$), then we can take $f(\bm{A}^T)\bm{I}_n^{(s)}$ (or $\bm{B}^T\bm{I}_n^{(s)}$) and use the rows of $f(\bm{A}^T)\bm{I}_n^{(s)}$ (or $\bm{B}^T\bm{I}_n^{(s)}$) to recover the columns of $f(\bm{A})$ (or $\bm{B}$). ### Analysis of BaMRAM(Alg. [\[alg:band\]](#alg:band){reference-type="ref" reference="alg:band"}) {#subsec:bandanal} Here we give theoretical guarantees for BaMRAM. The analysis for BaMRAM is easier and it gives a stronger bound than SpaMRAM as the precise location of the dominant entries and the behaviour of the decay is known. The proof is motivated by Proposition 3.2 in [@Benzi2007]. [\[thm:band\]]{#thm:band label="thm:band"} Let $\bm{B}$ be a matrix satisfying $$\left|\bm{B}_{ij}\right| < C \lambda^{|i-j|} \tag{\ref{eq:symdecayrate}}$$ where $C>0$ and $0<\lambda<1$ are constants. Then the output of BaMRAM, $\widehat{\bm{B}}$, satisfies $$\left\lVert\widehat{\bm{B}} - \bm{B}\right\rVert_2 \leq \frac{4C\lambda}{1-\lambda} \lambda^{s_0}$$ where $s = 2s_0+1$. *Proof.* Define $\bm{E} = \widehat{\bm{B}} - \bm{B}$. We get $$\left\lVert\widehat{\bm{B}} - \bm{B}\right\rVert_2 = \left\lVert\bm{E}\right\rVert_2 \leq \sqrt{\left\lVert\bm{E}\right\rVert_1 \left\lVert\bm{E}\right\rVert_\infty}$$ using a special case of Hölder's inequality [@matcomp §2.3.3]. We now bound $\left\lVert\bm{E}\right\rVert_1$ and $\left\lVert\bm{E}\right\rVert_\infty$ using [\[eq:symdecayrate\]](#eq:symdecayrate){reference-type="eqref" reference="eq:symdecayrate"}. By the construction of $\widehat{\bm{B}}$, the $i$th row of $\bm{E}$ satisfies $$\begin{aligned} \left\lVert E(i,:)\right\rVert_1 &\leq \sum_{\ell = -s_0}^{s_0}\left|\widehat{B}_{i,i+\ell}-B_{i,i+\ell}\right| + \sum_{\ell = s_0+1}^{n-i} |B_{i,i+\ell}| + \sum_{\ell = s_0+1}^{i-1} |B_{i,i-\ell}|\\ &\leq 2\sum_{\ell = s_0+1}^{n-i} |B_{i,i+\ell}| + 2\sum_{\ell = s_0+1}^{i-1} |B_{i,i-\ell}|\\ &< 2C\sum_{\ell = s_0+1}^{n-i} \lambda^\ell + 2C\sum_{\ell = s_0+1}^{i-1} \lambda^\ell \\ &< 4C \lambda^{s_0+1} \sum_{\ell = 0}^{\infty} \lambda^{\ell} = \frac{4C\lambda}{1-\lambda} \lambda^{s_0}. \end{aligned}$$ Similarly, the $j$th column of $\bm{E}$ satisfies $$\left\lVert E(:,j)\right\rVert_1 < \frac{4C\lambda}{1-\lambda} \lambda^{s_0}.$$ Therefore $$\left\lVert\bm{E}\right\rVert_1 = \max_{1\leq j\leq n} \left\lVert E(:,j)\right\rVert_1 < \frac{4C\lambda}{1-\lambda} \lambda^{s_0},\qquad \left\lVert\bm{E}\right\rVert_\infty = \max_{1\leq i \leq n} \left\lVert E(i,:)\right\rVert_1 < \frac{4C\lambda}{1-\lambda} \lambda^{s_0}.$$ We conclude that $$\left\lVert\widehat{\bm{B}} - \bm{B}\right\rVert_2 \leq \frac{4C\lambda}{1-\lambda} \lambda^{s_0}.$$ ◻ 1. When $\bm{B} = f(\bm{A})$ for a $k$-banded matrix $\bm{A}$, we can use $s = 2ck+1$ for some integer constant $c>0$. This value of $s$ is motivated by the fact that if $f$ is analytic then $f(\bm{A})$ can be well approximated by a low degree polynomial $p(\bm{A})$ which has bandwidth deg$(p)k$. 2. In Theorem [\[thm:band\]](#thm:band){reference-type="ref" reference="thm:band"}, if $\lambda \leq 0.9$ then for $\epsilon$ accuracy it suffices to take $$s_0\geq \frac{\log{(36C)}-\log{\epsilon}}{\log{(\lambda^{-1})}}.$$ ### Different decay rates {#sec:difdecayrate} A similar analysis can be performed if the decay rate is different on each side of the diagonal, for example, if the matrix satisfies [\[eq:ubandbound\]](#eq:ubandbound){reference-type="eqref" reference="eq:ubandbound"} and [\[eq:lbandbound\]](#eq:lbandbound){reference-type="eqref" reference="eq:lbandbound"}. The argument is the same with a different definition for $t_{ij}$, which is $$\label{eq:diffdecayt} t_{ij} = \mathop{\mathrm{arg\,min}}\limits_{\substack{1\leq j+st \leq n \\ t\in \mathbb{Z}}}\left|j+st-i-\frac{s-1}{2}\frac{\log(\lambda_1^{-1})-\log(\lambda_2^{-1})}{\log(\lambda_1^{-1})+\log(\lambda_2^{-1})}\right|$$ where $\lambda_1$ is the decay rate above the diagonal and $\lambda_2$ is that below the diagonal. This expression for $t_{ij}$ accounts for the different decay rates and coincides with our analysis in the previous section when $\lambda_1 = \lambda_2$. The algorithm will be the same as BaMRAM with $t_{ij}$ defined as [\[eq:diffdecayt\]](#eq:diffdecayt){reference-type="eqref" reference="eq:diffdecayt"}. For $s = \left\lfloor {c/\log(\lambda_1^{-1})} \right\rfloor+\left\lfloor {c/\log(\lambda_2^{-1})} \right\rfloor+1$ for some constant $c>0$, the algorithm will output $\widehat{\bm{B}}$ with upper bandwidth $\left\lfloor {c/\log(\lambda_1^{-1})} \right\rfloor$ and lower bandwidth $\left\lfloor {c/\log(\lambda_2^{-1})} \right\rfloor$ and $\widehat{\bm{B}}$ will satisfy $$\left\lVert\widehat{\bm{B}}-\bm{B}\right\rVert_2 \lesssim e^{-c}$$ which can easily be shown by following the proof of Theorem [\[thm:band\]](#thm:band){reference-type="ref" reference="thm:band"}. ## Extensions to Kronecker Sum Some of the results for banded matrices and matrices with exponential off-diagonal decay can be naturally extended to Kronecker sums, which arises, for example in the finite difference discretization of the two-dimensional Laplace operator [@LeVequeBook]. Let $\bm{A}^{(1)}, \bm{A}^{(2)} \in \mathbb{R}^{n\times n}$. The Kronecker sum $\mathcal{A} \in \mathbb{R}^{n^2\times n^2}$ of $\bm{A}^{(1)}$ and $\bm{A}^{(2)}$ is defined by $$\mathcal{A} = \bm{A}^{(1)} \oplus \bm{A}^{(2)} = \bm{A}^{(1)} \otimes \bm{I}_n + \bm{I}_n \otimes \bm{A}^{(2)}$$ where $\otimes$ is the Kronecker product, defined by $$\bm{A}\otimes \bm{B} = \begin{bmatrix} A_{11} \bm{B} & A_{12} \bm{B} & ... & A_{1n}\bm{B} \\ A_{21} \bm{B} & A_{22} \bm{B} & ... & A_{2n}\bm{B} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n1} \bm{B} & A_{n2} \bm{B} & ... & A_{nn}\bm{B} \end{bmatrix} \in \mathbb{R}^{n^2\times n^2}$$and $\bm{I}_n$ is the $n\times n$ identity matrix. Suppose $\bm{A}^{(1)}$ and $\bm{A}^{(2)}$ are both banded matrices. Then $\bm{I}_n \otimes \bm{A}^{(2)} \in \mathbb{R}^{n^2\times n^2}$ is a banded matrix with the same bandwidth as $\bm{A}^{(2)}$ and $\bm{A}^{(1)} \otimes \bm{I}_n \in \mathbb{R}^{n^2 \times n^2}$ is permutation equivalent to a banded matrix with the same bandwidth as $\bm{A}^{(1)}$ because $\bm{P}(\bm{A}^{(1)} \otimes \bm{I}_n)\bm{P}^T = \bm{I}_n \otimes \bm{A}^{(1)}$ where $\bm{P}$ is the perfect shuffle matrix [@VanLoan2000]. With this observation, we can use the recovery technique used for banded matrices in Section [3.1](#sec:bandrecovery){reference-type="ref" reference="sec:bandrecovery"} to recover $\mathcal{A}$ for banded matrices $\bm{A}^{(1)}$ and $\bm{A}^{(2)}$. Suppose that $\bm{A}^{(1)}$ and $\bm{A}^{(2)}$ have bandwidth $k_1$ and $k_2$ respectively. Notice that to recover $\mathcal{A}$, we need to recover the off-diagonal entries of $\bm{A}^{(1)}$ and $\bm{A}^{(2)}$ to recover the off-diagonal entries of $\mathcal{A}$ and recover $A_{ii}^{(1)}+A_{jj}^{(2)}$ for all $i,j$ to recover the diagonal entries of $\mathcal{A}$. We use the following $(2+2k_1+2k_2)$ matrix-vector products $$\left[\bm{P}^T\begin{bmatrix} \bm{I}_n^{(1+2k_1)} \\ \bm{0} \end{bmatrix}, \begin{bmatrix} \bm{I}_n^{(1+2k_2)} \\ \bm{0} \end{bmatrix} \right] \in \mathbb{R}^{n^2 \times (2+2k_1+2k_2)}$$ to recover $\mathcal{A}$. Since the top left $n\times n$ block of $\mathcal{A}$ equals $A_{11}^{(1)}\bm{I}_n+\bm{A}^{(2)}$, which is $k_2-$banded, the first $n$ rows of the matrix-vector product with $\begin{bmatrix} \bm{I}_n^{(1+2k_2)} \\ \bm{0} \end{bmatrix}$ recovers $A_{11}^{(1)}\bm{I}_n+\bm{A}^{(2)}$ exactly (Theorem [\[thm:bband\]](#thm:bband){reference-type="ref" reference="thm:bband"}), which in turn recovers all the off-diagonal entries of $\bm{A}^{(2)}$. Similarly, $\bm{P}^T\begin{bmatrix} \bm{I}_n^{(1+2k_1)}\\ \bm{0} \end{bmatrix}$ recovers $\bm{A}^{(1)}+A_{11}^{(2)}\bm{I}_n$ exactly using $\bm{P}(\bm{A}^{(1)} \otimes \bm{I}_n)\bm{P}^T = \bm{I}_n \otimes \bm{A}^{(1)}$. Therefore we get all the off-diagonal entries of $\mathcal{A}$ from recovering $A_{11}^{(1)}\bm{I}_n+\bm{A}^{(2)}$ and $\bm{A}^{(1)}+A_{11}^{(2)}\bm{I}_n$. The diagonal entries of $\mathcal{A}$ can be recovered from the diagonal entries of $A_{11}^{(1)}\bm{I}_n+\bm{A}^{(2)}$ and $\bm{A}^{(1)}+A_{11}^{(2)}\bm{I}_n$ by noting $$A_{ii}^{(1)}+A_{jj}^{(2)} = (A_{ii}^{(1)} + A_{11}^{(2)}) + (A_{11}^{(1)} + A_{jj}^{(2)}) - (A_{11}^{(1)} + A_{11}^{(2)})$$ for all $i,j$. For matrices with rapid off-diagonal decay, we can extend the above idea in certain cases. Focusing on the matrix exponential, we have a special relation [@matfunchigham §10] $$\exp(\bm{A}^{(1)} \oplus \bm{A}^{(2)}) = \exp(\bm{A}^{(1)}) \otimes \exp(\bm{A}^{(2)}).$$ When $\bm{A}^{(1)}$ and $\bm{A}^{(2)}$ are both banded matrices we can use the same procedure above by applying matrix-vector products with $$\left[\bm{P}^T\begin{bmatrix} \bm{I}_n^{(s_1)} \\ \bm{0} \end{bmatrix}, \begin{bmatrix} \bm{I}_n^{(s_2)} \\ \bm{0} \end{bmatrix} \right] \in \mathbb{R}^{n^2 \times (s_1+s_2)}.$$ for sufficiently large $s_1$ and $s_2$ that captures the dominant entries of $\exp(\bm{A}^{(1)})$ and $\exp(\bm{A}^{(2)})$ near the main diagonal respectively. This extension is natural and analogous to the extension from recovering banded matrices to recovering matrices with exponential off-diagonal decay earlier this section. The strategy for recovering banded matrices and approximating matrices with rapid off-diagonal decay can also be used as laid out above in a similar setting involving Kronecker product or sum. Examples include $f(\bm{I} \otimes \bm{A}) = \bm{I} \otimes f(\bm{A})$ and the formulae for the cosine and the sine of the Kronecker sum between two matrices [@matfunchigham §12]. # Numerical Illustrations {#sec:numill} Here we illustrate BaMRAM and SpaMRAM using numerical experiments. In all experiments, we use $20$ iterations of polynomial Krylov method and $50$ sample points for the contour integration from method 2 of [@matveccontour] to evaluate the matrix-vector products with $f(\bm{A})$. In all of the plots, the $x$-axis denotes the number of matrix-vector products $s$, i.e. $\bm{x}\mapsto f(\bm{A})\bm{x}$, and the $y$-axis denotes the relative error in the $2$-norm with respect to $f(\bm{A})$ obtained by dense arithmetic in MATLAB, i.e., `expm`, `sqrtm` and `logm`. For SpaMRAM, the sparsity parameter $k$ is chosen to be $8$ times smaller than the number of matrix-vector products, i.e., $s = 8k$ and the sensing operator $\bm{Y}$ is Gaussian. The experiments were conducted in MATLAB version 2021a using double precision arithmetic. ## Synthetic Examples We illustrate synthetic examples for BaMRAM and SpaMRAM using the functions $e^x, \sqrt{1+x}$ and $\log(1+x)$. We consider a synthetic banded matrix and a synthetic sparse matrix given below. - *Banded case:* symmetric $2$-banded matrix $\bm{A}_B \in \mathbb{R}^{1024\times 1024}$ with its nonzero entries drawn from i.i.d. $\mathcal{N}(0,1)$. $\bm{A}_B$ has been rescaled to $\left\lVert\bm{A}_B\right\rVert_2 = 0.5$. - *General sparse case:* symmetric sparse matrix $\bm{A}_S\in \mathbb{R}^{1024\times 1024}$ created using the MATLAB command `sprandsym` with density $1/1024$. $\bm{A}_S$ has been rescaled to $\left\lVert\bm{A}_S\right\rVert_2 = 0.5$. Symmetry was enforced to ensure that the eigenvalues are real, and $\bm{A}_B$ and $\bm{A}_S$ have been rescaled to ensure that their eigenvalues are sufficiently away from $-1$, which is where $\sqrt{1+x}$ and $\log(1+x)$ have singularities. This was done to ensure that the matrix-vector products $f(\bm{A})\bm{b}$ can be computed with high accuracy with 20 Krylov steps. Figure [\[fig:syneg\]](#fig:syneg){reference-type="ref" reference="fig:syneg"} shows the accuracy of the two algorithms. For the banded case in Figure [\[fig2a\]](#fig2a){reference-type="ref" reference="fig2a"}, we see that the error decays exponentially until the error is dominated by either the machine precision or the error from computing the matrix-vector products. This exponential decay rate is consistent with Theorem [\[thm:band\]](#thm:band){reference-type="ref" reference="thm:band"}. For the general sparse case in Figure [\[fig2b\]](#fig2b){reference-type="ref" reference="fig2b"}, the error decays quickly with the number of nonzero entries recovered in each row. The sparsity pattern influences the decay rate of SpaMRAM, making the behaviour of the decay rate more unexpected. This has been shown partially in Theorem [\[thm:gen\]](#thm:gen){reference-type="ref" reference="thm:gen"} where the bound depends on the sparsity pattern or more specifically the cardinality of the union of the combined support locations of $\bm{I}_n,\bm{A},...,\bm{A}^d$ in each row for $d$ large enough. ## Toy Examples In this section, we manufacture a matrix $\bm{A}$ such that $f(\bm{A})$ has a sparse structure and use our algorithms to approximate $f(\bm{A})$ using matrix-vector products with $\bm{A}$ only. We explore two functions, namely $\sqrt{\bm{A}}$ and $\log{(\bm{A})}$. For the square root function, we set $\sqrt{\bm{A}}$ equal to a sparse positive definite matrix and use our algorithms to recover $\sqrt{\bm{A}}$ via matrix-vector products with $\bm{A}=\left(\sqrt{\bm{A}}\right)^2$ (which here is not very sparse). We follow a similar procedure for the matrix logarithm using the formula $\mbox{exp}(\log(\bm{A}))=\bm{A}$. The aim of this experiment is to show that if $f(\bm{A})$ is sparse then we can efficiently recover $f(\bm{A})$ to high accuracy using our algorithms. We choose a sparse positive definite matrix and a banded positive definite matrix from the SuiteSparse Matrix Collection [@FloridaDataset2011]. - $\mathtt{gr\_ 30 \_ 30}$: $900\times 900$ $31$-banded symmetric positive definite matrix with $7744$ nonzero entries. - $\mathtt{Trefethen\_ 700}$: $700\times 700$ sparse symmetric positive definite matrix with $12654$ nonzero entries and at most $19$ nonzero entries per row and column. In Figure [\[fig:toyeg\]](#fig:toyeg){reference-type="ref" reference="fig:toyeg"}, we see that both matrices are recovered to high accuracy using our algorithms. In Figure [\[fig3a\]](#fig3a){reference-type="ref" reference="fig3a"}, we observe that the matrix $\mathtt{gr\_30\_30}$ is recovered to high accuracy using about $66$ matrix-vector products, which is consistent with the fact that $\mathtt{gr\_30\_30}$ is $31$-banded, and hence has at most $63$ nonzero entries per row and column. In Figure [\[fig3b\]](#fig3b){reference-type="ref" reference="fig3b"}, we see that the matrix $\mathtt{Trefethen\_700}$ has been recovered to high accuracy using about $180$ matrix-vector products. This is fairly consistent with the fact that with $s= 180$ matrix-vector products, we recover about $22$ nonzero entries in each row and $\mathtt{Trefethen\_700}$ has at most $19$ nonzero entries per row. ## Sparse dataset examples We now illustrate BaMRAM and SpaMRAM using the SuiteSparse Matrix Collection [@FloridaDataset2011], using the four matrices below. The first three matrices are sparse and the last matrix is banded. - $\mathtt{west1505}$: $1505\times 1505$ sparse matrix with $5414$ nonzero entries and at most $12$ nonzero entries per row and at most $27$ nonzero entries per column. - $\mathtt{CSphd}$: $1882\times 1882$ sparse matrix with $1740$ nonzero entries and at most $45$ nonzero entries per row and at most $3$ nonzero entries per column. - $\mathtt{tols2000}$: $2000\times 2000$ sparse matrix with $5184$ nonzero entries and at most $90$ nonzero entries per row and at most $22$ nonzero entries per column. - $\mathtt{mhd3200b}$: $3200\times 3200$ $18$-banded symmetric positive definite matrix with $18316$ nonzero entries. For the sparse matrices $\mathtt{west1505}$, $\mathtt{CSphd}$ and $\mathtt{tols2000}$, we use SpaMRAM to recover their matrix exponential. For the banded matrix $\mathtt{mhd3200b}$, we use BaMRAM to recover its matrix exponential, matrix square root and matrix logarithm. The results are illustrated in Figure [\[fig:dataseteg\]](#fig:dataseteg){reference-type="ref" reference="fig:dataseteg"}. In Figure [\[fig:dataseteg\]](#fig:dataseteg){reference-type="ref" reference="fig:dataseteg"}, we observe that our algorithms are able to approximate $f(\bm{A})$ with good accuracy. In Figure [\[fig4a\]](#fig4a){reference-type="ref" reference="fig4a"}, we notice that the error decreases exponentially until the error stagnates at some point. The stagnation is due to the error incurred in the computation of matrix-vector products $f(\bm{A})\bm{b}$, and can be improved by taking more than 20 Krylov steps. For banded matrices, we consistently obtain exponential convergence rate whose convergence rate is dictated by the smoothness of $f$. In Figure [\[fig4b\]](#fig4b){reference-type="ref" reference="fig4b"}, we note that the error decreases at a rapid speed until it stagnates due to machine precision for $\mathtt{CSphd}$ or slows down due to the sparsity pattern in the case of $\mathtt{west1505}$ and $\mathtt{tols2000}$. As observed in prior experiments, the behaviour is more irregular when the matrix has a more general sparsity pattern. The approximation to $f(\bm{A})$ in the above experiments can be highly accurate when the bandwidth of the matrix is small or the matrix has a sparsity pattern that makes $f(\bm{A})$ approximately sparse, in the sense that only few entries in each row/column are large in magnitude. Our algorithms are particularly useful when $\bm{A}$ can only be accessed through matrix-vector products and we require an approximation to the entire matrix function. An example is in network analysis where we are given an adjacency matrix $\bm{A}$ and the off-diagonal entries of $e^{\bm{A}}$ describe the communicability between two vertices and the diagonal entries of $e^{\bm{A}}$ describe the subgraph centrality of a vertex [@BenziEstradaKlymko2013; @EstradaHatano2008; @EstradaHigham2010]. The sparse matrix $\mathtt{CSphd}$ in the above experiment is an adjacency matrix for some underlying graph, and with about $180$ matrix-vector products we can approximate its matrix exponential to $10^{-15}$ accuracy. This means that our algorithm can approximate the communicability between any two vertices and the subgraph centrality of any vertex of $\mathtt{CSphd}$ with $10^{-15}$ accuracy. # Discussion and extensions {#sec:disc} In this paper, we presented two algorithms for approximating functions of sparse matrices and matrices with similar structures using matrix-vector products only. We exploited the decay bound from [@Benzi2007] to recover the entries of $f(\bm{A})$ that are large in magnitude. The task of approximating functions of matrices is not the only application for SpaMRAM and BaMRAM. These two algorithms can be used more generally in applications where we are able to perform matrix-vector products with the matrix in question and the matrix we want to approximate have the decay bound similar to [\[eq:sparsedecay\]](#eq:sparsedecay){reference-type="eqref" reference="eq:sparsedecay"} or [\[eq:symdecayrate\]](#eq:symdecayrate){reference-type="eqref" reference="eq:symdecayrate"}. An example is the decay bound for spectral projectors [@BenziBoitoRazouk2013], computed, for example by a discretized contour integral. Therefore the algorithms can be extended naturally to other similar applications. In many applications, we also want to approximate quantities related to $f(\bm{A})$ such as the trace of $f(\bm{A})$ and the Schatten $p$-norm of $f(\bm{A})$, which for the trace of functions of sparse symmetric matrices has been studied in [@FrommerRinelliSchweitzer2023]. Since SpaMRAM and BaMRAM provide an approximation to $f(\bm{A})$, we can use $\widehat{f(\bm{A})}$ to estimate the quantity of interest. For example, the trace of a matrix function is an important quantity appearing in network analysis through the Estrada index [@EstradaHigham2010] and statistical learning through the log-determinant estimation [@CortinovisKressner2021], among others. Therefore the algorithms can also be used to estimate quantities related to matrix functions. SpaMRAM uses compressed sensing to recover the matrix function row by row. This algorithm is simple, however we may have access to extra information such as the sparsity pattern of the matrix. This information can potentially be used to speed up the algorithm and increase the accuracy of the algorithm. There are techniques in compressed sensing such as variance density sampling [@KrahmerWard2014], which improves signal reconstruction using prior information. The improvement of SpaMRAM and the use of SpaMRAM and BaMRAM in different applications are left for future work. The authors are grateful to Jared Tanner for his helpful discussions and for pointing out useful literature in compressed sensing. TP was supported by the Heilbronn Institute for Mathematical Research. [^1]: We note that the matrix-vector product $f(\bm{A})\bm{b}$ is never computed explicitly by finding $f(\bm{A})$ and then multiplying $\bm{b}$ to $f(\bm{A})$, but with matrix-vector product with $\bm{A}$ only using, for example, the Krylov methods. [^2]: Note that if $\bm{A}$ is not symmetric then $d_G (i,j)\neq d_G (j,i)$ in general. [^3]: Another possibility would be to identify the nonzero structure using the graph structure $(V,E)$ and $f$. This can quickly become intractable, and in some cases $f(\bm{A})$ can be sparse despite the graph structure suggesting otherwise, so here we opt for a generic approach. [^4]: Other greedy methods such as CoSaMP have similar (or slightly stronger) theoretical guarantees as NIHT, but is generally more expensive; hence making NIHT more preferable in our setting ($k\ll n$ and $n$ problems to solve). [^5]: We can also recover the dominant entries of $\bm{B}$ column by column using $\bm{B}^T\bm{Y}$ if the matrix-vector product $\bm{x}\mapsto \bm{B}^T\bm{x}$ is available. [^6]: The work in [@ColemanMore1983; @CPR1974] is based on partitioning/coloring of the adjacency graph of a banded matrix. In this work, we show the vectors more explicitly. [^7]: The analysis can be carried out similarly for any other type of decay, for example, algebraic. [^8]: When the decay rate is different on each side of the diagonal we can carry out the analysis in a similar way by defining $t_{ij}$ differently. See Section [3.2.3](#sec:difdecayrate){reference-type="ref" reference="sec:difdecayrate"}.

arxiv_math

--- author: - | Massimo Lanza de Cristoforis\ Dipartimento di Matematica 'Tullio Levi-Civita',\ Università degli Studi di Padova,\ Via Trieste 63, Padova 35121, Italy.\ E-mail: mldc\@math.unipd.it date:   title: A survey on the boundary behavior of the double layer potential in Schauder spaces in the frame of an abstract approach --- # Introduction In this paper, we consider the double layer potential associated to the fundamental solution of a second order differential operator with constant coefficients. Throughout the paper, we assume that $$n\in {\mathbb{N}}\setminus\{0,1\}\,,$$ where ${\mathbb{N}}$ denotes the set of natural numbers including $0$. Let $\alpha\in]0,1]$, $m\in {\mathbb{N}}$. Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{m,\alpha}$. Here we understand that $C^{m,0}\equiv C^m$. For the notation and standard properties of the (generalized) Schauder spaces and sets of class $C^{m,\alpha}$ we refer for example to Dondi and the author [@DoLa17 §2] and to the reference [@DaLaMu21 §2.11, 2.13] of Dalla Riva, the author and Musolino. Let $\nu \equiv (\nu_{l})_{l=1,\dots,n}$ denote the external unit normal to $\partial\Omega$. If $\Omega$ is an open Lipschitz set $\nu$ is known to exist only for almost all points of $\partial\Omega$. Let $N_{2}$ denote the number of multi-indexes $\gamma\in {\mathbb{N}}^{n}$ with $|\gamma|\leq 2$. For each $$\label{introd0} {\mathbf{a}}\equiv (a_{\gamma})_{|\gamma|\leq 2}\in {\mathbb{C}}^{N_{2}}\,,$$ we set $$a^{(2)}\equiv (a_{lj} )_{l,j=1,\dots,n}\qquad a^{(1)}\equiv (a_{j})_{j=1,\dots,n}\qquad a\equiv a_{0}\,.$$ with $a_{lj} \equiv 2^{-1}a_{e_{l}+e_{j}}$ for $j\neq l$, $a_{jj} \equiv a_{e_{j}+e_{j}}$, and $a_{j}\equiv a_{e_{j}}$, where $\{e_{j}:\,j=1,\dots,n\}$ is the canonical basis of ${\mathbb{R}}^{n}$. We note that the matrix $a^{(2)}$ is symmetric. Then we assume that ${\mathbf{a}}\in {\mathbb{C}}^{N_{2}}$ satisfies the following ellipticity assumption $$\label{ellip} \inf_{ \xi\in {\mathbb{R}}^{n}, |\xi|=1 }{\mathrm{Re}}\,\left\{ \sum_{|\gamma|=2}a_{\gamma}\xi^{\gamma}\right\} >0\,,$$ and we consider the case in which $$\label{symr} a_{lj} \in {\mathbb{R}}\qquad\forall l,j=1,\dots,n\,.$$ Then we introduce the operators $$\begin{aligned} %\label{introd1} P[{\mathbf{a}},D]u&\equiv&\sum_{l,j=1}^{n}\partial_{x_{l}}(a_{lj}\partial_{x_{j}}u) + \sum_{l=1}^{n}a_{l}\partial_{x_{l}}u+au\,, \\ %\label{introd2} B_{\Omega}^{*}v&\equiv&\sum_{l,j=1}^{n} \overline{a}_{jl}\nu_{l}\partial_{x_{j}}v -\sum_{l=1}^{n}\nu_{l}\overline{a}_{l}v\,,\end{aligned}$$ for all $u,v\in C^{2}(\overline{\Omega})$, and a fundamental solution $S_{{\mathbf{a}} }$ of $P[{\mathbf{a}},D]$, and the boundary integral operator corresponding to the double layer potential $$\begin{aligned} \label{introd3} \lefteqn{ W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\mu](x) \equiv {\mathrm{p.v.}}\int_{\partial\Omega}\mu (y)\overline{B^{*}_{\Omega,y}}\left(S_{{\mathbf{a}}}(x-y)\right) \,d\sigma_{y} } \\ \nonumber && \qquad =-{\mathrm{p.v.}}\int_{\partial\Omega}\mu(y)\sum_{l,j=1}^{n} a_{jl}\nu_{l}(y)\frac{\partial S_{ {\mathbf{a}} } }{\partial x_{j}}(x-y)\,d\sigma_{y} \\ \nonumber && \qquad\quad -\int_{\partial\Omega}\mu(y)\sum_{l=1}^{n}\nu_{l}(y)a_{l} S_{ {\mathbf{a}} }(x-y)\,d\sigma_{y} \end{aligned}$$ for almost all $x\in \partial\Omega$, where the density or moment $\mu$ is a function from $\partial\Omega$ to ${\mathbb{C}}$. Here the subscript $y$ of $\overline{B^{*}_{\Omega,y}}$ means that we are taking $y$ as variable of the differential operator $\overline{B^{*}_{\Omega,y}}$, $d\sigma$ is the ordinary $(n-1)$-dimensional measure, and '${\mathrm{p.v.}}$' denotes the principal value of the integral. Thus the kernel of the boundary integral operator corresponding to the double layer potential is the following $$\begin{aligned} \label{eq:tgdlgen} \lefteqn{ K_{\Omega,{\mathbf{a}}}(x,y)\equiv \overline{B^{*}_{\Omega,y}}\left(S_{{\mathbf{a}}}(x-y)\right) } \\ \nonumber &&\qquad\qquad \equiv - \sum_{l,j=1}^{n} a_{jl}\nu_{l}(y)\frac{\partial S_{ {\mathbf{a}} } }{\partial x_{j}}(x-y) - \sum_{l=1}^{n}\nu_{l}(y)a_{l} S_{ {\mathbf{a}} }(x-y) \end{aligned}$$ for all $x\in\partial\Omega$ and for almost all $y\in\partial\Omega$ with $x\neq y$ (cf. ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"})). The role of the double layer potential in the solution of boundary value problems for the operator $P[{\mathbf{a}},D]$ is well known (cf. *e.g.*, Günter [@Gu67], Kupradze, Gegelia, Basheleishvili and Burchuladze [@KuGeBaBu79], Mikhlin [@Mik70], Mikhlin and Prössdorf [@MikPr86], Buchukuri, Chkadua, Duduchava, and Natroshvili [@BuChDuNa12].) Here we provide a summary of the continuity properties of the boundary operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}},\cdot]$ (the so-called Neumann-Poincaré operator in case $P[{\mathbf{a}},D]$ is the Laplace operator) in the frame of Hölder and Schauder spaces. Also, we give references to proofs where a consistent part of the arguments are based on results for integral operators that hold in a metric space with a measure that satisfies certain growth conditions that include non-doubling measures as in a series of papers by Garcı́a-Cuerva and Gatto [@GaGa04], [@GaGa05], Gatto [@Gat09] in the frame of Hölder spaces and that have been further developed in [@La22a]. We now briefly present such abstract setting, that this paper shows to have several applications (see also [@La22b], [@La22d], [@La23b], [@La23c]). Let $(M,d)$ be a metric space and let $X$, $Y$ be subsets of $M$. $$\begin{aligned} \nonumber &&\text{Let}\ {\mathcal{N}} \ \text{ be a $\sigma$-algebra of parts of}\ Y \,, {\mathcal{B}}_Y\subseteq {\mathcal{N}}\,. \\ \label{eq:nu} &&\text{Let}\ \nu\ \text{be measure on}\ {\mathcal{N}} \,. \\ \nonumber &&\text{Let}\ \nu(B(x,r)\cap Y)<+\infty\qquad\forall (x,r)\in X\times ]0,+\infty[\,,\end{aligned}$$ where ${\mathcal{B}}_Y$ denotes the $\sigma$-algebra of the Borel subsets of $Y$ and $$B(\xi,r)\equiv \left\{\eta\in M:\, d(\xi,\eta)<r\right\}\qquad\forall (\xi,r)\in M\times ]0,+\infty[\,.$$ We assume that $\upsilon_Y\in ]0,+\infty[$ and we consider two types of assumptions on $\nu$. The first assumption is that $Y$ is upper $\upsilon_Y$-Ahlfors regular with respect to $X$, *i.e.*, that $$\begin{aligned} \nonumber &&\text{there\ exist}\ r_{X,Y,\upsilon_Y}\in]0,+\infty]\,,\ c_{X,Y,\upsilon_Y}\in]0,+\infty[\ \text{such\ that} \\ \nonumber &&\nu( B(x,r)\cap Y )\leq c_{X,Y,\upsilon_Y} r^{\upsilon_Y} \\ \label{defn:uareg1} &&\text{for\ all}\ x\in X\ \text{and}\ r \in]0,r_{X,Y,\upsilon_Y}[ \,. \end{aligned}$$ In case $X=Y$, we just say that $Y$ is upper $\upsilon_Y$-Ahlfors regular and this is the assumption that has been considered by Garcı́a-Cuerva and Gatto [@GaGa04], [@GaGa05], Gatto [@Gat06], [@Gat09] in case $X=Y=M$. See also Edmunds, Kokilashvili and Meskhi [@EdKoMe02 Chap. 6] in the frame of Lebsgue spaces. An interesting feature of condition ([\[defn:uareg1\]](#defn:uareg1){reference-type="ref" reference="defn:uareg1"}) is that it does not imply any estimate of $\nu( B(x,r)\cap Y )$ from below in terms of $r^{\upsilon_Y}$ as in a so called lower $\upsilon_Y$-Ahlfors regularity condition that together with ([\[defn:uareg1\]](#defn:uareg1){reference-type="ref" reference="defn:uareg1"}) would imply the validity of the so-called $\upsilon_Y$-Ahlfors regularity condition and accordingly the validity of the so-called doubling condition for the measure $\nu$, *i.e.*, the following condition $$\begin{aligned} \nonumber &&\text{there\ exist}\ r_{X,Y}\in]0,+\infty]\,,\ c_{X,Y}\in]0,+\infty[\ \text{such\ that} \\ \nonumber &&\nu( B(x,2r)\cap Y )\leq c_{X,Y} \nu( B(x,r)\cap Y ) \\ \label{defn:doubling} &&\text{for\ all}\ x\in X\ \text{and}\ r \in]0,r_{X,Y}[ \,. \end{aligned}$$ Now condition ([\[defn:uareg1\]](#defn:uareg1){reference-type="ref" reference="defn:uareg1"}) says that one can estimate the $\nu$-measure of a ball $B(x,r)\cap Y$ in $Y$ from above in terms of the measure of a ball of radius $r$ in a Euclidean space of dimension $\upsilon_Y$ (at least for integer values of $\upsilon_Y$). As in [@La22a], we also consider a stronger condition than ([\[defn:uareg1\]](#defn:uareg1){reference-type="ref" reference="defn:uareg1"}) that still does not involve estimates from below for $\nu$ in which we replace the ball $B(x,r)\cap Y$ in $Y$ with an annular domain $(B(x,r_2)\setminus B(x,r_1))\cap Y$ with $0\leq r_1< r_2$ in $Y$ and that says that one can estimate the $\nu$-measure of an annular domain $(B(x,r_2)\setminus B(x,r_1))\cap Y$ in $Y$ from above in terms of the measure of an annular domain of radii $r_1$ and $r_2$ in a Euclidean space of dimension $\upsilon_Y$ (at least for integer values of $\upsilon_Y$). Namely, we assume that $Y$ is strongly upper $\upsilon_Y$-Ahlfors regular with respect to $X$, *i.e.*, that $$\begin{aligned} \nonumber &&\text{there\ exist}\ r_{X,Y,\upsilon_Y}\in]0,+\infty]\,,\ c_{X,Y,\upsilon_Y}\in]0,+\infty[\ \text{such\ that} \\ \nonumber &&\nu( (B(x,r_2)\setminus B(x,r_1))\cap Y )\leq c_{X,Y,\upsilon_Y}(r_2^{\upsilon_Y}-r_1^{\upsilon_Y}) \\ \label{defn:suareg1} &&\text{for\ all}\ x\in X\ \text{and}\ r_1,r_2\in[0,r_{X,Y,\upsilon_Y}[ \ \text{with}\ r_1<r_2\,,\end{aligned}$$ where we understand that $B(x,0)\equiv\emptyset$ (in case $X=Y$, we just say that $Y$ is strongly upper $\upsilon_Y$-Ahlfors regular). So, for example, if $Y$ is the boundary of a bounded open Lipschitz subset of $M={\mathbb{R}}^n$, then $Y$ is upper $(n-1)$-Ahlfors regular with respect to ${\mathbb{R}}^n$ (cf. Proposition [Proposition 5](#prop:lgar){reference-type="ref" reference="prop:lgar"} of the Appendix [9](#appendix){reference-type="ref" reference="appendix"}) and if $Y$ is the boundary of an open bounded subset of $M={\mathbb{R}}^n$ of class $C^1$, then $Y$ is strongly upper $(n-1)$-Ahlfors regular with respect to $Y$ (cf. Proposition [Proposition 5](#prop:lgar){reference-type="ref" reference="prop:lgar"} and Remark [Remark 2](#rem:coc1sa){reference-type="ref" reference="rem:coc1sa"} of the Appendix [9](#appendix){reference-type="ref" reference="appendix"}). One may wonder about the importance of considering only growth conditions from above for the measure as in the upper Ahlfors or strong upper Ahlfors regularity condition and not from below and about the importance of considering measures that do not satisfy the doubling condition. Here we mention the papers of Verdera [@Ve13], [@Ve23 p. 21] in connection with the integral operator of the Cauchy kernel. Also, in section [8](#sec:exavio){reference-type="ref" reference="sec:exavio"}, we present some elementary examples of surfaces in ${\mathbb{R}}^3$ in which such conditions from below and the doubling are actually violated. In the present survey paper $X$ and $Y$ are mainly subsets of the boundary $\partial\Omega$ of some bounded open subset $\Omega$ of $M\equiv {\mathbb{R}}^n$, $d$ is the Euclidean distance and $\nu$ coincides with the ordinary $(n-1)$-dimensional measure on $\partial\Omega$. # Preliminaries and Notation Let $M_n({\mathbb{R}})$ denote the set of $n\times n$ matrices with real entries. $|A|$ denotes the operator norm of a matrix $A$, $A^{t}$ denotes the transpose matrix of $A$. Let $O_{n}({\mathbb{R}})$ denote the set of $n\times n$ orthogonal matrices with real entries. We also set $${\mathbb{B}}_{n}(x,\rho)\equiv \left\{ y\in {\mathbb{R}}^{n}:\,\vert x-y\vert <\rho \right\} \qquad \forall (\xi,\rho)\in {\mathbb{R}}^n\times ]0,+\infty[ \,.$$ Here and in the sequel, $m_n$ denotes the $n$-dimensional Lebesgue measure in ${\mathbb{R}}^n$ and $m_{n-1}$ denotes the ordinary $(n-1)$-dimensional (surface) measure and $$\label{eq:snon} \omega_n\equiv m_n({\mathbb{B}}_n(0,1))\,,\qquad s_n\equiv m_{n-1}(\partial{\mathbb{B}}_n(0,1))\,.$$ For the standard notation of the spaces of Hölder or Lipschitz continuous functions, we refer for example to [@DoLa17 §2], [@DaLaMu21 §2.6]. Let $\Omega$ be an open subset of ${\mathbb{R}}^n$. Let $s\in {\mathbb{N}}\setminus\{0\}$, $f\in \left(C^{1}(\Omega)\right)^{s}$. Then $Df$ denotes the Jacobian matrix of $f$. In order to analyze the kernel of the double layer potential, we need some more information on the fundamental solution $S_{ {\mathbf{a}} }$. To do so, we introduce the fundamental solution $S_{n}$ of the Laplace operator. Namely, we set $$%\label{ups} S_{n}(x)\equiv \left\{ \begin{array}{lll} \frac{1}{s_{n}}\ln |x| \qquad & \forall x\in {\mathbb{R}}^{n}\setminus\{0\},\quad & {\mathrm{if}}\ n=2\,, \\ \frac{1}{(2-n)s_{n}}|x|^{2-n}\qquad & \forall x\in {\mathbb{R}}^{n}\setminus\{0\},\quad & {\mathrm{if}}\ n>2\,. \end{array} \right.$$ and we follow a formulation of Dalla Riva [@Da13 Thm. 5.2, 5.3] and Dalla Riva, Morais and Musolino [@DaMoMu13 Thm. 5.5], that we state as in paper [@DoLa17 Cor. 4.2] of Dondi and the author (see also John [@Jo55], and Miranda [@Mi65] for homogeneous operators, and Mitrea and Mitrea [@MitMit13 p. 203]). **Proposition 1**. *Let ${\mathbf{a}}$ be as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Then there exist an invertible matrix $T\in M_{n}({\mathbb{R}})$ such that $$\label{prop:ourfs0} a^{(2)}=TT^{t}\,,$$ a real analytic function $A_{1}$ from $\partial{\mathbb{B}}_{n}(0,1)\times{\mathbb{R}}$ to ${\mathbb{C}}$ such that $A_{1}(\cdot,0)$ is odd, $b_{0}\in {\mathbb{C}}$, a real analytic function $B_{1}$ from ${\mathbb{R}}^{n}$ to ${\mathbb{C}}$ such that $B_{1}(0)=0$, and a real analytic function $C$ from ${\mathbb{R}}^{n}$ to ${\mathbb{C}}$ such that $$\begin{aligned} %attenzione a formulare diversamente nel lavoro \label{prop:ourfs1} \lefteqn{ S_{ {\mathbf{a}} }(x) = \frac{1}{\sqrt{\det a^{(2)} }}S_{n}(T^{-1}x) } \\ \nonumber &&\qquad +|x|^{3-n}A_{1}(\frac{x}{|x|},|x|) +(B_{1}(x)+b_{0}(1-\delta_{2,n}))\ln |x|+C(x)\,, %\\ \nonumber %&&\qquad %-\frac{\delta_{2,n}}{2\pi}\int_{\partial{\mathbb{B}}_{2}(0,1)}S_{a^{(2)}}-A_{0}\,d\sigma\end{aligned}$$ for all $x\in {\mathbb{R}}^{n}\setminus\{0\}$, and such that both $b_{0}$ and $B_{1}$ equal zero if $n$ is odd. Moreover, $$\frac{1}{\sqrt{\det a^{(2)} }}S_{n}(T^{-1}x)$$ is a fundamental solution for the principal part of $P[{\mathbf{a}},D]$.* In particular for the statement that $A_{1}(\cdot,0)$ is odd, we refer to Dalla Riva, Morais and Musolino [@DaMoMu13 Thm. 5.5, (32)], where $A_{1}(\cdot,0)$ coincides with ${\mathbf{f}}_1({\mathbf{a}},\cdot)$ in that paper. Then for each $\theta\in]0,1]$, we define the function $\omega_{\theta}(\cdot)$ from $[0,+\infty[$ to itself by setting $$\label{omth} \omega_{\theta}(r)\equiv \left\{ \begin{array}{ll} 0 &r=0\,, \\ r^{\theta}\vert \ln r \vert &r\in]0,r_{\theta}]\,, \\ r_{\theta}^{\theta}\vert \ln r_{\theta} \vert & r\in ]r_{\theta},+\infty[\,, \end{array} \right.$$ where $%\label{omth1} r_{\theta}\equiv e^{-1/\theta}$. Next we introduce some notation for the kernels. We do so in the abstract context of metric spaces. If $X$ and $Y$ are subsets of a metric space $M$, we consider off-diagonal kernels $K$ from $(X\times Y)\setminus D_{X\times Y}$ to ${\mathbb{C}}$, where $$D_{X\times Y}\equiv \left\{ (x,y)\in X\times Y:\,x=y \right\}$$ denotes the diagonal set of $X\times Y$ and we introduce the following class of 'potential type' kernels (see also paper [@DoLa17] of the author and Dondi, where such classes have been introduced in a form that generalizes those of Giraud [@Gi34], Gegelia [@Ge67], Kupradze, Gegelia, Basheleishvili and Burchuladze [@KuGeBaBu79 Chap. IV]). **Definition 1**. *Let $(M,d)$ be a metric space. Let $X$, $Y\subseteq M$. Let $s_1$, $s_2$, $s_3\in {\mathbb{R}}$. We denote by the symbol ${\mathcal{K}}_{s_1, s_2, s_3} (X\times Y)$ the set of continuous functions $K$ from $(X\times Y)\setminus D_{X\times Y}$ to ${\mathbb{C}}$ such that $$\begin{aligned} \lefteqn{ \|K\|_{ {\mathcal{K}}_{ s_1, s_2, s_3 }(X\times Y) } \equiv \sup\biggl\{\biggr. d(x,y)^{ s_{1} }\vert K(x,y)\vert :\,(x,y)\in X\times Y, x\neq y \biggl.\biggr\} } \\ \nonumber &&\qquad\qquad\qquad +\sup\biggl\{\biggr. \frac{d(x',y)^{s_{2}}}{d(x',x'')^{s_{3}}} \vert K(x',y)- K(x'',y) \vert :\, \\ \nonumber &&\qquad\qquad\qquad x',x''\in X, x'\neq x'', y\in Y\setminus B(x',2d(x',x'')) \biggl.\biggr\}<+\infty\,.\end{aligned}$$* For $s_2=s_1+s_3$ one has the so-called class of standard kernels that is the case in which Garcı́a-Cuerva and Gatto [@GaGa04], [@GaGa05], Gatto [@Gat09] have proved $T1$ Theorems for the integral operators with kernel $K$ in case of weakly singular, singular and hyper-singular integral operators with $X=Y$. We also consider the following more restrictive class of kernels. **Definition 2**. *Let $(M,d)$ be a metric space. Let $X$, $Y\subseteq M$. Let $\nu$ be as in ([\[eq:nu\]](#eq:nu){reference-type="ref" reference="eq:nu"}). Let $s_1$, $s_2$, $s_3\in {\mathbb{R}}$. We set $$\begin{aligned} \nonumber \lefteqn{ {\mathcal{K}}_{ s_1, s_2, s_3 }^\sharp(X\times Y) \equiv \biggl\{\biggr. K\in {\mathcal{K}}_{ s_1, s_2, s_3 }(X\times Y):\, } \\ \nonumber &&\ \ K(x,\cdot)\ \text{is}\ \nu-\text{integrable\ in}\ Y\setminus B(x,r) \ \text{for\ all}\ (x,r)\in X\times]0,+\infty[\,, \\ \nonumber &&\ \ \sup_{x\in X}\sup_{r\in ]0,+\infty[} \left\vert \int_{Y\setminus B(x,r)}K(x,y)\,d\nu(y)\right\vert<+\infty \biggl.\biggr\} \end{aligned}$$ and $$\begin{aligned} \lefteqn{ \|K\|_{{\mathcal{K}}_{ s_1, s_2, s_3 }^\sharp(X\times Y)} \equiv \|K\|_{{\mathcal{K}}_{ s_1, s_2, s_3 }(X\times Y)} } \\ \nonumber &&+ \sup_{x\in X}\sup_{r\in ]0,+\infty[} \left\vert \int_{Y\setminus B(x,r)}K(x,y)\,d\nu(y) \right\vert\quad\forall K\in {\mathcal{K}}_{ s_1, s_2, s_3 }^\sharp(X\times Y)\,. \end{aligned}$$* Clearly, $({\mathcal{K}}^\sharp_{ s_{1},s_{2},s_{3} }(X\times Y),\|\cdot\|_{ {\mathcal{K}}^\sharp_{s_{1},s_{2},s_{3} }(X\times Y) })$ is a normed space and the space ${\mathcal{K}}^\sharp_{ s_{1},s_{2},s_{3} }(X\times Y)$ is continuously embedded into ${\mathcal{K}}_{ s_{1},s_{2},s_{3} }(X\times Y)$. # The double layer potential on the boundary of Lipschitz and $C^1$ domains {#dolalipc1} For the definition of open Lipschitz subset of ${\mathbb{R}}^n$ and of open subset of ${\mathbb{R}}^n$ of class $C^1$, we refer for example to Dalla Riva, the author and Musolino [@DaLaMu21 §2.9, 2.13]. Salaev [@Sa76] has proved that the Cauchy integral on a rectifiable simple closed curve that satisfies an upper Ahlfors regularity condition is bounded in the space $C^{0,\beta}$ on the curve for $\beta\in]0,1[$. Since the double layer potential is the real part of the Cauchy integral for real densities, it follows that the double layer potential $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}},\cdot]$ is bounded in $C^{0,\beta}(\partial\Omega)$ for $\beta\in]0,1[$ in case $\Omega$ is a Jordan domain bounded by a rectifiable simple closed curve that satisfies an upper Ahlfors regularity condition and $S_{{\mathbf{a}}}$ equals the fundamental solution of the Laplace operator. For the estimate of moduli of continuity of the Cauchy integral on a rectifiable simple closed curve that satisfies an upper Ahlfors regularity condition even in generalized Hölder spaces, we should also mention the papers of Plemelj [@Plemelj-6-2], Privaloff [@Priv16-6-2], Zygmund [@Zyg02-6-2], Magnaradze [@Magn-6-2], Babaev and Salaev [@Bab-Sal73-6-2], Tamrazov [@Tam-mono-6-2], [@Tam78a-6-2], [@Tam78-6-2], Gerus [@G77-6-2], [@G78-6-2], [@G96-6-2], [@Gerus-6-2], Salimov [@Salimov-6-2], Dyn'kin [@Dynkin-6-2], Salaev, Guseı̆nov and Seı̆fullaev [@SaGusSe90], Guseı̆nov [@Gus92]. Then Mitrea, Mitrea and Mitrea [@MitMitMit23a Prop. 25.5.21] have proved that the double layer potential $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}},\cdot]$ is bounded in $C^{0,\beta}(\partial\Omega)$ for $\beta\in]0,1[$ in case $S_{{\mathbf{a}}}$ equals the fundamental solution of the Laplace operator and the ordinary $(n-1)$-dimensional measure on the boundary on $\Omega$ satisfies an upper Ahlfors growth condition, a condition that certainly holds if $\Omega$ is a bounded open Lipschitz subset of ${\mathbb{R}}^n$ with $n\geq 2$. If $\Omega$ is a bounded open Lipschitz subset of ${\mathbb{R}}^n$, then there exists a subset $N$ of measure zero of $\partial\Omega$ such that the outward unit normal $\nu$ exists at all points of $(\partial\Omega)\setminus N$. By Mitrea, Mitrea and Mitrea [@MitMitMit23a Prop. 25.5.21], Mitrea [@Mit23], the function $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\mu]$ is uniformly continuous on $(\partial\Omega)\setminus N$ and thus it admits a unique uniformly continuous extension to the closure of $(\partial\Omega)\setminus N$ in $\partial\Omega$, *i.e.*, to the whole of $\partial\Omega$. However, the principal value of ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) may well exist at some point $x\in N$ and be different from the value of the continuous extension of $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\mu]$ at $x$. For the existence of the principal value of ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) at points of $N$ in dimension 2, we refer to classical books such as that of Gakhov [@Ga66 §4.5, p. 31] and in higher dimensions we refer to Burago and Maz'ya [@BuMa69 Thm. 2, p. 17]. In case $\Omega$ is of class $C^1$, we can take $N=\emptyset$. # The double layer potential on the boundary of domains of class $C^{1,\alpha}$ with $\alpha\in]0,1[$. For the definition of open subset of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1]$, we refer for example to Dalla Riva, the author and Musolino [@DaLaMu21 §2.13]. Here we must say that we only consider the boundary behaviour of the double layer potential. Instead for the regularity properties of the double layer potential in the Schauder space $C^{1,\alpha}$ outside of the boundary we refer to Günter [@Gu67], Kupradze, Gegelia, Basheleishvili and Burchuladze [@KuGeBaBu79], Mikhlin [@Mik70], Mikhlin and Prössdorf [@MikPr86], Miranda [@Mi65], [@Mi70], Wiegner [@Wi93], Dalla Riva [@Da13], Dalla Riva, Morais and Musolino [@DaMoMu13], Mitrea, Mitrea and Verdera [@MitMitVe16] and references therein. We first state the following theorem, that extends a known result of Schauder [@Sc31 p. 614] for the harmonic double layer potential in case $n=3$. For later contributions see also Mitrea [@Mit14]. For the (classical) definition of the generalized Hölder space $C^{0,\omega_1(\cdot)}(\partial\Omega)$ we refer for example to [@DoLa17 §2]. **Theorem 1**. *Let $n\in {\mathbb{N}}\setminus\{0,1\}$. Let ${\mathbf{a}}$ be as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Let $\alpha\in]0,1[$, $\beta\in]0,1]$.* *Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,\alpha}$. Then the following statements hold.* 1. *If $0<\beta<1-\alpha$, then the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\alpha+\beta}(\partial\Omega)$ defined by ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) for all $\mu\in C^{0,\beta}(\partial\Omega)$ is linear and continuous.* 2. *If $\beta=1-\alpha$, then the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\omega_1(\cdot)}(\partial\Omega)$ defined by ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) for all $\mu\in C^{0,\beta}(\partial\Omega)$ is linear and continuous.* **Proof.** We first note that the membership of $1$ in $C^{1,\alpha}(\partial\Omega)$ and Dondi and the author [@DoLa17 Thm 9.2] implies that $$W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,1]\in C^{1,\alpha}(\partial\Omega)\subseteq C^{0,\omega_1(\cdot)}(\partial\Omega)\subseteq C^{0,\theta}(\partial\Omega)\qquad\forall\theta\in]0,1[\,.$$ By [@DoLa17 Rmk 6.1 (ii)], we know that the kernel $\overline{B^{*}_{\Omega,y}}\left(S_{{\mathbf{a}}}(x-y)\right)$ of the double layer potential belongs to the class ${\mathcal{K}}_{n-1-\alpha,n-\alpha,1}((\partial\Omega)\times(\partial\Omega))$. In particular, $\overline{B^{*}_{\Omega,y}}\left(S_{{\mathbf{a}}}(x-y)\right)$ is weakly singular and accordingly $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ defines a linear and continuous map from $C^{0,\beta}(\partial\Omega)$ to $C^{0}(\partial\Omega)$ both in case of statement (i) and of statement (ii) (cf. *e.g.*, [@DoLa17 Prop. 6.1 (i)]). We also note that in case $(n-\alpha)-\beta>n-1$ of statement (i), we also have $1+(n-1)-(n-\alpha-\beta)>0$. Then [@La22a Prop. 5.11] (or [@DoLa17 Lem. 6.1]) implies the existence of $c\in]0,+\infty[$ such that $$\begin{aligned} \label{k0b1} \lefteqn{ | W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\mu](x')-W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\mu](x'')| } \\ \nonumber &&\qquad \leq c\| K \|_{ {\mathcal{K}}_{n-1-\alpha,n-\alpha,1}((\partial\Omega)\times(\partial\Omega)) } \|\mu\|_{ C^{0,\beta}(\partial\Omega) }\omega(|x'-x''|) \\ \nonumber &&\qquad\quad +\|\mu\|_{ C^{0}(\partial\Omega) } | W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,1](x')-W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,1](x'')|\end{aligned}$$ for all $x',x''\in\partial\Omega$ such that $|x'-x''|<e^{-1}$ and for all $\mu\in C^{0,\beta}(\partial\Omega)$, where $$\omega(r)\equiv \left\{ \begin{array}{ll} r^{\min\{\alpha+\beta,1\}} & {\mathrm{if}}\ n-\alpha-\beta<n-1\ \text{as\ in}\ (i)\,, \\ \max\{ r^{\alpha+\beta},\omega_1(r)\} & {\mathrm{if}}\ n-\alpha-\beta=n-1\ \text{as\ in}\ (ii)\,, \end{array} \right. \ \ \forall r\in]0,+\infty[\,.$$ Then under the assumptions of statement (i), we have $$C^{0,r^{\min\{\alpha+\beta,1\}} }(\partial\Omega)=C^{0,\alpha+\beta}(\partial\Omega)$$ and the membership of $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,1]$ in $C^{0,\alpha+\beta}(\partial\Omega)$ and inequality ([\[k0b1\]](#k0b1){reference-type="ref" reference="k0b1"}) imply the validity of statement (i) (see also [@DoLa17 Rmk. 2.2]). Instead, under the assumptions of statement (ii), we have $\alpha+\beta=1$, $$C^{0,\max\{ r^{\alpha+\beta},\omega_1(r)\} }(\partial\Omega)=C^{0,\omega_1(\cdot)}(\partial\Omega)$$ and the membership of $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,1]$ in $C^{0,\omega_1(\cdot)}(\partial\Omega)$ and inequality ([\[k0b1\]](#k0b1){reference-type="ref" reference="k0b1"}) imply the validity of statement (ii) (see also [@DoLa17 Rmk. 2.2]).$\Box$ Next we turn to consider case $\alpha+\beta>1$ and we state the following theorem, that collects and extends results of Fichera and De Vito [@FiDe70 LXXXIII] for the Laplace operator in case $n=2$. See also Miranda [@Mi70 15.VI], where the author mentions a result of Giraud [@Gi32], Mitrea [@Mit14], Dondi and the author [@DoLa17] and [@La22d Thm. 5.1]. For the (classical) definition of the generalized Schauder space $C^{1,\omega_\alpha(\cdot)}(\partial\Omega)$ we refer for example to [@DoLa17 §2]. **Theorem 2**. *Let $n\in {\mathbb{N}}\setminus\{0,1\}$. Let ${\mathbf{a}}$ be as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Let $\alpha\in]0,1[$, $\beta\in]0,1]$, $\alpha+\beta>1$.* *Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,\alpha}$. Then the following statements hold.* 1. *If $\beta<1$, then the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ from $C^{0,\beta}(\partial\Omega)$ to $C^{1,\alpha+\beta-1}(\partial\Omega)$ defined by ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) for all $\mu\in C^{0,\beta}(\partial\Omega)$ is linear and continuous.* 2. *If $\beta=1$, then the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ from $C^{0,\beta}(\partial\Omega)=C^{0,1}(\partial\Omega)$ to $C^{1,\omega_{\alpha+\beta-1}}(\partial\Omega)=C^{1,\omega_\alpha}(\partial\Omega)$ defined by ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) for all $\mu\in C^{0,1}(\partial\Omega)$ is linear and continuous.* For a proof we refer to [@La22d Thm. 5.1]. Here we do not provide a complete proof of Theorem [Theorem 2](#thm:dllregenn){reference-type="ref" reference="thm:dllregenn"}, but we point out that the proof is based on statements that hold in the general setting that we have mentioned in the introduction. Indeed, $\partial\Omega$ is strongly upper $(n-1)$-Ahlfors regular (cf. Proposition [Proposition 5](#prop:lgar){reference-type="ref" reference="prop:lgar"} and Remark [Remark 2](#rem:coc1sa){reference-type="ref" reference="rem:coc1sa"} of the Appendix [9](#appendix){reference-type="ref" reference="appendix"}) and Dondi and the author [@DoLa17 Thm 9.2] implies that $$\label{eq:dllregenn} W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,1]\in C^{1,\alpha}(\partial\Omega)$$ and we note that one can prove classically that $$\begin{aligned} \label{eq:gradformn} \lefteqn{ {\mathrm{grad}}_{\partial\Omega,x}W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\mu](x) = {\mathrm{grad}}_{\partial\Omega,x} \int_{\partial\Omega} \overline{B^{*}_{\Omega,y}}\left(S_{{\mathbf{a}}}(x-y)\right)\mu(y)\,d\sigma_y } \\ \nonumber &&\qquad\qquad\qquad\qquad =\int_{\partial\Omega}[{\mathrm{grad}}_{\partial\Omega,x} \overline{B^{*}_{\Omega,y}}\left(S_{{\mathbf{a}}}(x-y)\right)] (\mu(y)-\mu(x))\,d\sigma_y \\ \nonumber &&\qquad\qquad\qquad\qquad\quad +\mu(x){\mathrm{grad}}_{\partial\Omega} \int_{\partial\Omega} \overline{B^{*}_{\Omega,y}}\left(S_{{\mathbf{a}}}(x-y)\right) \,d\sigma_y \\ \nonumber &&\qquad\qquad\qquad\qquad = Q[K,\mu,1](x) \\ \nonumber &&\qquad\qquad\qquad\qquad\quad +\mu(x){\mathrm{grad}}_{\partial\Omega}W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,1](x) \qquad\forall x\in\partial\Omega \end{aligned}$$ for all $\mu\in C^{0,\beta}(\partial\Omega)$, where ${\mathrm{grad}}_{\partial\Omega}$ denotes the tangental gradient and ${\mathrm{grad}}_{\partial\Omega,x}$ denotes the tangental gradient with respect to the first variable (cf. [@La22b Thm. 6.1]) and $$\begin{aligned} && K(x,y)\equiv -[{\mathrm{grad}}_{\partial\Omega,x} \overline{B^{*}_{\Omega,y}}\left(S_{{\mathbf{a}}}(x-y)\right)]\quad\forall (x,y)\in (\partial\Omega)^2\setminus D_{(\partial\Omega)\times (\partial\Omega)}\,, \\ \nonumber && Q[K,\mu,1](x) \equiv \int_{\partial\Omega} K(x,y) (\mu(x)-\mu(y))\,d\sigma_y \quad\forall x\in\partial\Omega \end{aligned}$$ for all $\mu\in C^{0,\beta}(\partial\Omega)$. Then one can deduce the continuity of the operator $Q[K,\cdot,1]$ from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\alpha+\beta-1}(\partial\Omega)$ if $\beta<1$ and to $C^{0,\omega_\alpha}(\partial\Omega)$ if $\beta=1$ by exploiting the proof of [@La22d Lem. 4.6] on the membership of the tangential gradient of the kernel of the double layer potential in an appropriate class and the abstract result on $Q$ of [@La22a Prop. 6.3 (iii) (c), (cc)] in metric spaces, that generalizes previous work of Gatto [@Gat09]. Then the membership of ([\[eq:dllregenn\]](#eq:dllregenn){reference-type="ref" reference="eq:dllregenn"}) together with the continuity of the pointwise product in generalized Schauder spaces (cf. *e.g.*, [@DoLa17 Lems. 2.4, 2.5]) imply that ${\mathrm{grad}}_{\partial\Omega,x}W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ is bounded from $C^{0,\beta}(\partial\Omega)$ to $C^{1,\alpha+\beta-1}(\partial\Omega)$ if $\beta<1$ and to $C^{1,\omega_\alpha}(\partial\Omega)$ if $\beta=1$ and thus Theorem [Theorem 2](#thm:dllregenn){reference-type="ref" reference="thm:dllregenn"} can be proved to be true. By setting $\beta=\alpha$ in the previous Theorems [Theorem 1](#thm:dllregen){reference-type="ref" reference="thm:dllregen"}, [Theorem 2](#thm:dllregenn){reference-type="ref" reference="thm:dllregenn"}, we immediately deduce the validity of the following corollary that says that in a set of a class $C^{1,\alpha}$ with $\alpha\in]0,1[$. the regularizing effect of boundary double layer potential on the boundary equals $\alpha$, with the only exceptional value $\alpha=1/2$. **Corollary 1**. *Let $n\in {\mathbb{N}}\setminus\{0,1\}$. Let ${\mathbf{a}}$ be as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Let $\alpha\in]0,1[$.* *Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,\alpha}$. Then the following statements hold.* 1. *If $0<\alpha<1/2$, then the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ from $C^{0,\alpha}(\partial\Omega)$ to $C^{0,2\alpha}(\partial\Omega)$ defined by ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) for all $\mu\in C^{0,\alpha}(\partial\Omega)$ is linear and continuous.* 2. *If $\alpha=1/2$, then the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ from $C^{0,\alpha}(\partial\Omega)$ to $C^{0,\omega_{2\alpha}(\cdot)}(\partial\Omega)$ defined by ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) for all $\mu\in C^{0,\alpha}(\partial\Omega)$ is linear and continuous.* 3. *If $1/2<\alpha<1$, then the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ from $C^{0,\alpha}(\partial\Omega)$ to $C^{1,2\alpha-1}(\partial\Omega)$ defined by ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) for all $\mu\in C^{0,\alpha}(\partial\Omega)$ is linear and continuous.* # The double layer potential on the boundary of domains of class $C^{1,1}$ and $C^2$. We first mention some known results in the classical case of the boundary behaviour of the double layer potential in Schauder spaces with $m= 2$. Instead for the regularity properties of the double layer potential in Schauder spaces with $m= 2$ outside of the boundary we refer to Günter [@Gu67], Kupradze, Gegelia, Basheleishvili and Burchuladze [@KuGeBaBu79], Mikhlin [@Mik70], Mikhlin and Prössdorf [@MikPr86], Miranda [@Mi65], [@Mi70], Wiegner [@Wi93], Dalla Riva [@Da13], Dalla Riva, Morais and Musolino [@DaMoMu13], Mitrea, Mitrea and Verdera [@MitMitVe16] and references therein. In case $n=3$ and $\Omega$ is of class $C^{2}$, $\alpha\in]0,1[$ and if $P[{\mathbf{a}},D]$ is the Helmholtz operator, Colton and Kress [@CoKr83] have developed previous work of Günter [@Gu67] and Mikhlin [@Mik70] and proved that the operator $W[\partial\Omega ,{\mathbf{a}},S_{{\mathbf{a}}},\cdot]$ is bounded from $C^{0,\alpha}(\partial\Omega)$ to $C^{1,\alpha}(\partial\Omega)$. In case $n\geq 2$, $\alpha\in]0,1[$ and $\Omega$ is of class $C^{2}$ and if $P[{\mathbf{a}},D]$ is the Laplace operator, Hsiao and Wendland [@HsWe08 Remark 1.2.1, p. 10] deduce that the operator $W[\partial\Omega ,{\mathbf{a}},S_{{\mathbf{a}}},\cdot]$ is bounded from $C^{0,\alpha}(\partial\Omega)$ to $C^{1,\alpha}(\partial\Omega)$ by the work of Mikhlin and Prössdorf [@MikPr86]. We now show that if $\Omega$ is of class $C^{1,1}$, then the double layer potential improves the regularity of one unit if $\beta<1$ (and of less than that if $\beta=1$) Namely, we have the following statement that is a generalization of classical results for the Laplace and Helmoltz operator in case $\Omega$ is of class $C^2$ and $\beta\in]0,1[$ (see Colton and Kress [@CoKr83 Thm. 2.22], Hsiao and Wendland [@HsWe08 Remark 1.2.1, p. 10]). **Theorem 3**. *Let $\beta\in]0,1]$. Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,1}$. Let ${\mathbf{a}}$ be as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Then the following statements hold.* 1. *If $\beta<1$, then the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ from $C^{0,\beta}(\partial\Omega)$ to $C^{1, \beta }(\partial\Omega)$ that is defined by ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) for all $\mu\in C^{0,\beta}(\partial\Omega)$ is linear and continuous.* 2. *If $\beta=1$, then the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ from $C^{0,1}(\partial\Omega)$ to $C^{1, \omega_1(\cdot) }(\partial\Omega)$ that is defined by ([\[introd3\]](#introd3){reference-type="ref" reference="introd3"}) for all $\mu\in C^{0,1}(\partial\Omega)$ is linear and continuous.* For a proof we refer to [@La23b Thm. 1.1]. Here we do not provide a complete proof of Theorem [Theorem 3](#thm:dllreggen){reference-type="ref" reference="thm:dllreggen"}, but we point out that the proof is based on statements that hold in the general setting that we have mentioned in the introduction. Indeed, $\partial\Omega$ is strongly upper $(n-1)$-Ahlfors regular (cf. Proposition [Proposition 5](#prop:lgar){reference-type="ref" reference="prop:lgar"} and Remark [Remark 2](#rem:coc1sa){reference-type="ref" reference="rem:coc1sa"} of the Appendix [9](#appendix){reference-type="ref" reference="appendix"}) and reference [@La22d Lem. 5.4] implies that $$\label{eq:dllreggen} W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,1]\in C^{1,\omega_1(\cdot)}(\partial\Omega)\,.$$ and we note that one can exploit formula ([\[eq:gradformn\]](#eq:gradformn){reference-type="ref" reference="eq:gradformn"}), the proof of [@La22d Lem. 4.6] on the membership of the tangential gradient of the kernel of the double layer potential in an appropriate class and the abstract result of [@La22a Prop. 6.3 (ii) (b), (bb)] in metric spaces, that generalizes previous work of Gatto [@Gat09]. To do so, we deduce the continuity of $Q[K,\cdot,1]$ from $C^{0,\beta}(\partial\Omega)$ to $C^{0, \beta}(\partial\Omega)$ if $\beta<1$ and to $C^{0,\omega_1(\cdot)}(\partial\Omega)$ if $\beta=1$ by the abstract result on $Q$ of [@La22a Prop. 6.3 (ii) (b), (bb)], that generalizes previous work of Gatto [@Gat09]. Indeed, $\partial\Omega$ is strongly upper $(n-1)$-Ahlfors regular (cf. Proposition [Proposition 5](#prop:lgar){reference-type="ref" reference="prop:lgar"} and Remark [Remark 2](#rem:coc1sa){reference-type="ref" reference="rem:coc1sa"} of the Appendix [9](#appendix){reference-type="ref" reference="appendix"}). Then the membership of ([\[eq:dllreggen\]](#eq:dllreggen){reference-type="ref" reference="eq:dllreggen"}) together with the continuity of the pointwise product in generalized Schauder spaces (cf. *e.g.*, [@DoLa17 Lems. 2.4, 2.5]) imply that ${\mathrm{grad}}_{\partial\Omega,x}W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ is bounded from $C^{0,\beta}(\partial\Omega)$ to $C^{1,\beta}(\partial\Omega)$ if $\beta<1$ and to $C^{1,\omega_1(\cdot)}(\partial\Omega)$ if $\beta=1$ and one can deduce the validity of Theorem [Theorem 3](#thm:dllreggen){reference-type="ref" reference="thm:dllreggen"}. # The double layer potential on the boundary of domains of class $C^{m,\alpha}$ with $m\geq 2$, $\alpha\in]0,1[$. We first mention some known results in the classical case of the boundary behaviour of the double layer potential in Schauder spaces with $m\geq 2$. Instead for the regularity properties of the double layer potential in Schauder spaces with $m\geq 2$ outside of the boundary we refer to Günter [@Gu67], Kupradze, Gegelia, Basheleishvili and Burchuladze [@KuGeBaBu79], Mikhlin [@Mik70], Mikhlin and Prössdorf [@MikPr86], Miranda [@Mi65], [@Mi70], Wiegner [@Wi93], Dalla Riva [@Da13], Dalla Riva, Morais and Musolino [@DaMoMu13], Mitrea, Mitrea and Verdera [@MitMitVe16] and references therein. In case $n=3$, $m\geq 2$, $\alpha\in]0,1]$ and $\Omega$ is of class $C^{m,\alpha}$ and if $P[{\mathbf{a}},D]$ is the Laplace operator, Günter [@Gu67 Appendix, § IV, Thm. 3] has proved that $W[\partial\Omega ,{\mathbf{a}},S_{{\mathbf{a}}},\cdot]$ is bounded from $C^{m-2,\alpha}(\partial\Omega)$ to $C^{m-1,\alpha'}(\partial\Omega)$ for $\alpha'\in]0,\alpha[$. In case $n\geq 2$, $\alpha\in]0,1]$, O. Chkadua [@Chk23] has pointed out that one could exploit Kupradze, Gegelia, Basheleishvili and Burchuladze [@KuGeBaBu79 Chap. IV, Sect. 2, Thm 2.9, Chap. IV, Sect. 3, Theorems 3.26 and 3.28] and prove that if $\Omega$ is of class $C^{m,\alpha}$, then $W[\partial\Omega ,{\mathbf{a}},S_{{\mathbf{a}}},\cdot]$ is bounded from $C^{m-1,\alpha'}(\partial\Omega)$ to $C^{m,\alpha'}(\partial\Omega)$ for $\alpha'\in]0,\alpha[$. In case $n=3$, $m\geq 2$, $\alpha\in]0,1[$ and $\Omega$ is of class $C^{m,\alpha}$ and if $P[{\mathbf{a}},D]$ is the Helmholtz operator, Kirsch [@Ki89 Thm. 3.3 (a)] has developed previous work of Günter [@Gu67], Mikhlin [@Mik70] and Colton and Kress [@CoKr83] and has proved that the operator $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ is bounded from $C^{m-1,\alpha}(\partial\Omega)$ to $C^{m,\alpha}(\partial\Omega)$. von Wahl [@vo90] has considered the case of Sobolev spaces and has proved that if $\Omega$ is of class $C^{\infty}$ and if $S_{{\mathbf{a}}}$ is the fundamental solution of the Laplace operator, then the double layer improves the regularity of one unit on the boundary. Then Heinemann [@He92] has developed the ideas of von Wahl in the frame of Schauder spaces and has proved that if $\Omega$ is of class $C^{m+5}$ and if $S_{{\mathbf{a}}}$ is the fundamental solution of the Laplace operator, then the double layer improves the regularity of one unit on the boundary, *i.e.*, $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ is linear and continuous from $C^{m,\alpha}(\partial\Omega)$ to $C^{m+1,\alpha}(\partial\Omega)$. Maz'ya and Shaposhnikova [@MaSh05] have proved that $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ is continuous in fractional Sobolev spaces under sharp regularity assumptions on the boundary and if $P[{\mathbf{a}},D]$ is the Laplace operator. Dondi and the author [@DoLa17] have proved that if $m\geq 2$ and $\Omega$ is of class $C^{m,\alpha}$ with $\alpha\in]0,1[$, then the double layer potential $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ associated to the fundamental solution of a *nonhomogeneous* second order elliptic differential operator with constant coefficients is bounded from $C^{m,\beta}(\partial\Omega)$ to $C^{m,\alpha}(\partial\Omega)$ for all $\beta\in]0,\alpha]$. For corresponding results for the fundamental solution of the heat equation, we refer to the author and Luzzini [@LaLu17], [@LaLu18] and references therein. By exploiting a formula for the tangential derivatives of the double layer potential that involves some auxiliary integral operators of [@DoLa17 Thm. 9.1], which generalizes the corresponding formula of Hofmann, Mitrea and Taylor [@HoMitTa10 (6.2.6)] for homogeneous operators and once more by exploiting the abstract result [@La22a Prop. 6.3] in metric spaces, that generalizes previous work of Gatto [@Gat09], one can prove the following. For the (classical) definition of the generalized Schauder space $C^{m,\omega_1(\cdot)}(\partial\Omega)$ we refer for example to [@DoLa17 §2]. **Theorem 4**. *Let ${\mathbf{a}}$ be as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Let $\alpha\in]0,1]$. Let $m\in{\mathbb{N}}$, $m\geq 2$. Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{m,\alpha}$. Then the following statements hold.* 1. *If $\alpha\in]0,1[$, then $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ is linear and continuous from $C^{m-1,\alpha}(\partial\Omega)$ to $C^{m,\alpha}(\partial\Omega)$.* 2. *If $\alpha=1$, then $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ is linear and continuous from $C^{m-1,1}(\partial\Omega)$ to $C^{m,\omega_1(\cdot)}(\partial\Omega)$.* For a proof, we refer to [@La23c]. Hence, Theorem [Theorem 4](#wreg){reference-type="ref" reference="wreg"} sharpens the work of the above mentioned authors in the sense that if $\Omega$ is of class $C^{m,\alpha}$ with $m\geq 2$, then the class of regularity of the target space of $W_\Omega[{\mathbf{a}},S_{{\mathbf{a}}} ,\cdot]$ is precisely $C^{m,\alpha}$ if $\alpha<1$ and is the generalized Schauder space $C^{m,\omega_1(\cdot)}$ if $\alpha=1$. Moreover, Theorem [Theorem 4](#wreg){reference-type="ref" reference="wreg"} extends the above mentioned result of Kirsch [@Ki89] in the sense that Kirsch [@Ki89] has considered the Helmholtz operator in case $n=3$, $\alpha<1$ and Theorem [Theorem 4](#wreg){reference-type="ref" reference="wreg"} considers a general fundamental solution $S_{{\mathbf{a}}}$ with ${\mathbf{a}}$ as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}), $\alpha\leq 1$ and $n\geq 2$. # An integral operator associated to the conormal derivative of a single layer potential Another relevant layer potential operator associated to the analysis of boundary value problems for the differential operator $P[{\mathbf{a}},D]$ is defined by $$W_{\ast,\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\mu](x)\equiv \int_{\partial\Omega}\mu(y)DS_{ {\mathbf{a}} }(x-y)a^{(2)}\nu(x)\,d\sigma_{y}\qquad\forall x\in\partial\Omega$$ for all $\mu\in C^{0}(\partial\Omega)$, that we consider only in case $\Omega$ is at least of class $C^{1,\alpha}$ for some $\alpha\in ]0,1]$. Now the continuity properties of $W_{\ast,\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ can be deduced by those of $W_{\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ via a simple formula. To do so, we set $$\label{qrs0} Q_j[g,\mu](x) =\int_{\partial\Omega}(g(x)-g(y))\frac{\partial S_{ {\mathbf{a}} }}{\partial x_{j}}(x-y)\mu(y)\,d\sigma_{y}\quad\forall x\in \partial\Omega\,,$$ for all $(g,\mu)\in C^{0,1}(\partial\Omega)\times L^{\infty}(\partial\Omega)$, for all $l\in\{1,\dots,n\}$ and $$\label{eq:silapo} V_\Omega[S_{ {\mathbf{a}} },\mu](x)\equiv \int_{\partial\Omega}S_{ {\mathbf{a}} }(x-y)\mu(y)\,d\sigma_{y} \qquad\forall x\in\partial\Omega$$ for all $\mu\in C^{0,\alpha}(\partial\Omega)$. Then a simple computation shows that $$\begin{aligned} \label{v*regg1} \lefteqn{ W_{\ast,\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\mu](x) =\sum_{b,r=1}^{n}a_{br} Q_b[\nu_{r},\mu](x) } \\ \nonumber &&\qquad\qquad\qquad\qquad - W_\Omega[{\mathbf{a}}, S_{{\mathbf{a}}} ,\mu ](x) - V_\Omega[S_{{\mathbf{a}}} ,(a^{(1)}\nu) \mu](x)\,,\end{aligned}$$ for all $x\in \partial\Omega$ and for all $\mu\in C^{0}(\partial\Omega)$ (cf. [@DoLa17 (10.1)]) and we can prove the following statement. **Theorem 5**. *Let $n\in {\mathbb{N}}\setminus\{0,1\}$. Let ${\mathbf{a}}$ be as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Let $\alpha\in]0,1[$, $\beta\in]0,\alpha]$.* *Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,\alpha}$. Then the operator $W_{\ast,\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ is linear and continuous from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\alpha}(\partial\Omega)$.* **Proof.** Since the components of $\nu$ are of class $C^{0,\alpha}$, Dondi and the author [@DoLa17 Thm. 8.2 (ii)] implies that $Q_b[\nu_{r},\cdot]$ is continuous from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\alpha}(\partial\Omega)$ for all $b$, $r\in\{1,\dots,n\}$. By Theorems [Theorem 1](#thm:dllregen){reference-type="ref" reference="thm:dllregen"}, [Theorem 2](#thm:dllregenn){reference-type="ref" reference="thm:dllregenn"} and by the continuity of the embedding of the target space of $W_{\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ into $C^{0,\alpha}(\partial\Omega)$ in each of the statements of Theorems [Theorem 1](#thm:dllregen){reference-type="ref" reference="thm:dllregen"}, [Theorem 2](#thm:dllregenn){reference-type="ref" reference="thm:dllregenn"}, the operator $W_{\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ is linear and continuous from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\alpha}(\partial\Omega)$. Since the components of $\nu$ are of class $C^{0,\alpha}$, the continuity of the pointwise product in Hölder spaces (cf. *e.g.*, [@DoLa17 Lem. 2.5]) implies that the map from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\beta}(\partial\Omega)$ that takes $\mu$ to $(a^{(1)}\nu) \mu$ is continuous. By [@DoLa17 Th. 7.2], and by the continuity of the embedding of $C^{0,\beta}(\partial\Omega)$ into $L^\infty(\partial\Omega)$, the operator $V_\Omega[ S_{{\mathbf{a}}} ,\cdot]$ is continuous from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\alpha}(\partial\Omega)$. Then formula ([\[v\*regg1\]](#v*regg1){reference-type="ref" reference="v*regg1"}) implies the validity of statement.$\Box$ Similarly, we can consider case $\alpha=1$ and prove the following. **Theorem 6**. *Let $n\in {\mathbb{N}}\setminus\{0,1\}$. Let ${\mathbf{a}}$ be as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Let $\beta\in]0,1]$.* *Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,1}$. Then the operator $W_{\ast,\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ is linear and continuous from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\omega_1(\cdot)}(\partial\Omega)$.* **Proof.** Since the components of $\nu$ are of class $C^{0,1}$, [@DoLa17 Th. 8.2 (i)] and the continuity of the embedding of $C^{0,\beta}(\partial\Omega)$ into $L^\infty(\partial\Omega)$ imply that $Q_b[\nu_{r},\cdot]$ is continuous from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\omega_1(\cdot)}(\partial\Omega)$ for all $b\in\{1,\dots,n\}$. By Theorem [Theorem 3](#thm:dllreggen){reference-type="ref" reference="thm:dllreggen"} (i), (ii) and by the continuity of the embedding of the target space of $W_{\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ into $C^{0, \omega_1(\cdot) }(\partial\Omega)$ in both statements (i) and (ii), the operator $W_{\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ is linear and continuous from $C^{0,\beta}(\partial\Omega)$ to $C^{0, \omega_1(\cdot) }(\partial\Omega)$. Since the components of $\nu$ are of class $C^{0,1}$, the continuity of the pointwise product in Hölder spaces (cf. *e.g.*, [@DoLa17 Lem. 2.5]) implies that the map from $C^{0,\beta}(\partial\Omega)$ to $C^{0,\beta}(\partial\Omega)$ that takes $\mu$ to $(a^{(1)}\nu) \mu$ is continuous. Let $\beta'\in]0,\beta]\cap]0,1[$. By [@DoLa17 Th. 7.1 (i)], $V_\Omega[ S_{{\mathbf{a}}} ,\cdot]$ is continuous from $C^{0,\beta'}(\partial\Omega)$ to $C^{1,\beta'}(\partial\Omega)$, that is continuously embedded into $C^{0, \omega_1(\cdot) }(\partial\Omega)$. Then $V_\Omega[ S_{{\mathbf{a}}} ,\cdot]$ is continuous from $C^{0,\beta}(\partial\Omega)$ to $C^{0, \omega_1(\cdot) }(\partial\Omega)$. Then formula ([\[v\*regg1\]](#v*regg1){reference-type="ref" reference="v*regg1"}) implies the validity of statement.$\Box$ Finally, again by exploiting formula ([\[v\*regg1\]](#v*regg1){reference-type="ref" reference="v*regg1"}), one can prove the validity of the following statement in case $\Omega$ is at least of class $C^{2,\alpha}$ for some $\alpha\in]0,1]$. We also mention that the following statement extends the corresponding result of Kirsch [@Ki89 Thm. 3.3 (b)] who has considered the case in which $S_{ {\mathbf{a}} }$ is the fundamental solution of the Helmholtz operator, $n=3$, $\alpha\in]0,1[$. **Theorem 7**. *Let ${\mathbf{a}}$ be as in ([\[introd0\]](#introd0){reference-type="ref" reference="introd0"}), ([\[ellip\]](#ellip){reference-type="ref" reference="ellip"}), ([\[symr\]](#symr){reference-type="ref" reference="symr"}). Let $S_{ {\mathbf{a}} }$ be a fundamental solution of $P[{\mathbf{a}},D]$. Let $\alpha\in]0,1]$. Let $m\in{\mathbb{N}}$, $m\geq 2$. Let $\Omega$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{m,\alpha}$. Then the following statements hold.* 1. *If $\alpha\in]0,1[$, then the operator $W_{\ast,\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ is linear and continuous from $C^{m-2,\alpha}(\partial\Omega)$ to $C^{m-1,\alpha}(\partial\Omega)$.* 2. *If $\alpha=1$, then the operator $W_{\ast,\Omega}[{\mathbf{a}}, S_{ {\mathbf{a}} },\cdot]$ is linear and continuous from $C^{m-2,1}(\partial\Omega)$ to $C^{m-1,\omega_1(\cdot)}(\partial\Omega)$.* For a proof, we refer to [@La23c] . # Examples of measures in which the lower Ahlfors regularity condition and the doubling condition are violated {#sec:exavio} We plan to present some perhaps known elementary examples of surfaces in ${\mathbb{R}}^3$ in which either the lower Ahlfors regularity inequality or the doubling condition are actually violated. We do so by considering revolution graphs in ${\mathbb{R}}^3$ that are obtained by rotating a curve. To do so, we need some preliminary statement on the curve that we plan to rotate. If $U$ is a subset of ${\mathbb{R}}$ and if $f$ is a function from $U$ to ${\mathbb{R}}$, we say that $f$ is increasing provided that $f(\rho_1)\leq f(\rho_2)$ whenever $\rho_1$, $\rho_2\in U$ and $\rho_1<\rho_2$. Then we say that $f$ is strictly increasing provided that $f(\rho_1)< f(\rho_2)$ whenever $\rho_1$, $\rho_2\in U$ and $\rho_1<\rho_2$. **Proposition 2**. *Let $r_0\in]0,+\infty[$. Let $f$ be a continuous increasing function from $]0,r_0[$ to $]0,+\infty[$ such that $$\label{prop:cufn1} \lim_{x\to 0} f(x)=0\,.$$ Then the following statements hold.* 1. *For each $r\in]0,r_0[$ there exists one and only one $x_r\in]0,r[$ such that $$\label{prop:cufn2} f(x_r)^2+x_r^2=r^2\,.$$* 2. *If $f$ is also continuously differentiable, then the map from $]0,r_0[$ to $]0,r_0[$ that takes $r$ to $x_r$ is continuously differentiable and $$\label{prop:cufn3} \frac{dx_r}{dr}=\frac{r}{f(x_r)f'(x_r)+x_r}\qquad\forall r\in]0,r_0[\,.$$* **Proof.** (i) Since $f$ is increasing and continuous, the function $f(x)^2+x^2$ is strictly increasing and continuous in the variable $x\in]0,r_0[$. Now let $r\in ]0,r_0[$. Since $$\lim_{x\to 0} f(x)^2+x^2=0\,,\qquad f(r)^2+r^2>r^2\,,$$ we conclude that there exists one and only one $x_r\in]0,r[$ such that equality ([\[prop:cufn2\]](#prop:cufn2){reference-type="ref" reference="prop:cufn2"}) holds true. \(ii\) Let $F(x,r)\equiv f(x)^2+x^2-r^2$ for all $(x,r)\in ]0,r_0[^2$. By assumption, $F$ is continuously differentiable. Moreover, (i) implies that $$F(x_r,r)=0\qquad\forall r\in ]0,r_0[\,.$$ Also, $$\frac{\partial F}{\partial x}(x,r)=2f(x) f'(x)+2x\qquad\forall (x,r)\in ]0,r_0[^2 \,.$$ Since $f$ and $f'$ are positive, we have $\frac{\partial F}{\partial x}(x_r,r)>0$ for all $r\in]0,r_0[$ and the Implicit Function Theorem implies the validity of (ii).$\Box$ Next we introduce a surface of revolution that is associated to a function $f\in C^1(]0,r_0[,]0,+\infty[)$. Namely, we set $$\label{eq:gammaf} \gamma_f(\eta_1,\eta_2)\equiv \left\{ \begin{array}{ll} f(\sqrt{\eta_1^2+\eta_2^2}) &\text{if}\ (\eta_1,\eta_2)\in {\mathbb{B}}_2(0,r_0)\setminus\{(0,0)\} \\ 0&\text{if}\ (\eta_1,\eta_2)=(0,0)\,. \end{array} \right.$$ and we plan to consider the area $$\begin{aligned} \label{eq:arf} \lefteqn{ A_f(r)\equiv m_2\left( \left( {\mathbb{B}}_3(0,r)\cap {\mathrm{graph}} (\gamma_f) \right)\setminus\{(0,0,0)\} \right) } \\ \nonumber &&\qquad\qquad\qquad =2\pi\int_0^{x_r}x\sqrt{1+(f'(x))^2}\,dx\qquad\forall r\in]0,r_0[\,.\end{aligned}$$ We first show that under an extra condition on $f$, $\left({\mathrm{graph}} (\gamma_f) \right)\setminus\{(0,0,0)\}$ is strongly upper $2$-Ahlfors regular with respect to $\{(0,0,0)\}$. **Proposition 3**. *Let $r_0\in]0,+\infty[$. Let $f\in C^1(]0,r_0[,]0,+\infty[)$ be increasing and satisfy the following limiting relations $$\label{prop:cufnsua1} \lim_{x\to 0}f(x)=0\,,\qquad \lim_{x\to 0}f'(x)=+\infty\,.$$ Then the following statements hold* 1. *The function $x\sqrt{1+(f'(x))^2}$ is integrable in $x\in]0,r[$ for all $r\in]0,r_0[$. In particular, $A_f(r)<+\infty$ for all $r\in]0,r_0[$.* 2. *$\left({\mathrm{graph}} (\gamma_f) \right)\setminus\{(0,0,0)\}$ with the ordinary $2$-dimensional measure is strongly upper $2$-Ahlfors regular with respect to $\{(0,0,0)\}$.* **Proof.** $f$ satisfies all the assumptions of Proposition [Proposition 2](#prop:cufn){reference-type="ref" reference="prop:cufn"}. Moreover, de l'Hôpital's rule implies that $$\label{prop:cufnsua2} \lim_{x\to 0}\frac{x}{f(x)}=\lim_{x\to 0}\frac{1}{f'(x)}=0\,.$$ By the limiting relations ([\[prop:cufnsua2\]](#prop:cufnsua2){reference-type="ref" reference="prop:cufnsua2"}) and $\lim_{x\to 0}f'(x)=+\infty$, there exists $r_0'\in]0,r_0[$ $$\frac{x}{f(x)}<1\,,\qquad f'(x)>1\qquad\forall x\in]0,r_0'[\,.$$ We now prove statement (i). Let $r\in]0,r_0[$. Since $$\lim_{x\to0}\frac{x\sqrt{1+(f'(x))^2}}{xf'(x)}=\lim_{x\to0}\sqrt{(f'(x))^{-2}+1}=1\,,$$ the function $x\sqrt{1+(f'(x))^2}$ is integrable in $x\in]0,r[$ if and only if $xf'(x)$ is integrable in $]0,r[$. Since $$\int_0^{r}xf'(x)\,dx=\lim_{\epsilon\to 0}\left[xf(x)\right]_{x=\epsilon}^{x=r}-\int_0^{r}f(x)\,dx\leq f(r)r<+\infty,$$ the function $x\sqrt{1+(f'(x))^2}$ is integrable in $x\in]0,r[$ and thus statement (i) holds true. We now prove statement (ii). We note that $$m_2\left( ({\mathbb{B}}_3(x,r_2)\setminus {\mathbb{B}}_3(x,r_1))\cap Y \right) =\int_{r_1}^{r_2}\frac{d}{dr}A_f(r)\,dr$$ for all $r_1,r_2\in ]0,r_0'[$. Thus it suffices to show that $$\label{prop:cufnsua3} \sup_{0<r<r_0'}r^{-1}\frac{d}{dr}A_f(r)<+\infty\,.$$ Since $x_r\in]0,r[$ for all $r\in]0,r_0'[$, Proposition [Proposition 2](#prop:cufn){reference-type="ref" reference="prop:cufn"} (ii) implies that $$\begin{aligned} \lefteqn{ \frac{d}{dr}A_f(r)= 2\pi x_r\sqrt{1+(f'(x_r))^2}\frac{dx_r}{dr} } \\ \nonumber &&\qquad =2\pi x_r\sqrt{1+(f'(x_r))^2}\frac{r}{f(x_r)f'(x_r)+x_r} \\ \nonumber &&\qquad =2\pi \frac{x_r}{f(x_r)}\sqrt{(f'(x_r))^{-2}+1}\frac{r}{ 1+(x_r/f(x_r))(f'(x_r))^{-1}} \\ \nonumber &&\qquad \leq 2\pi \sqrt{1+1}r\qquad\forall r\in]0,r_0'[\end{aligned}$$ and thus inequality ([\[prop:cufnsua3\]](#prop:cufnsua3){reference-type="ref" reference="prop:cufnsua3"}) holds true and the proof of (ii) is complete. $\Box$ Next we prove that by formulating some extra assumption on the function $f$, we can prove an apriori estimate on $x_{2r}$ and $x_r$. **Lemma 1**. *Let $r_0\in]0,+\infty[$. Let $f\in C^1(]0,r_0[,]0,+\infty[)$ be increasing and satisfy the following limiting relations $$\label{lem:cufnd1} \lim_{x\to 0}f(x)=0\,,\qquad \lim_{x\to 0}\frac{x}{f(x)}=0\,.$$ Let $x_r$ be as in Proposition [Proposition 2](#prop:cufn){reference-type="ref" reference="prop:cufn"} for each $r\in]0,r_0[$. Then $$\label{lem:cufnd2} \lim_{r\to 0}\frac{f(x_{2r})}{f(x_r)}=2$$* **Proof.** By the definition of $x_{2r}$, $x_{r}$, we have $$\frac{f(x_{2r})^2+x_{2r}^2}{f(x_{r})^2+x_{r}^2}=\frac{(2r)^2}{r^2}$$ and accordingly $$\left(\frac{f(x_{2r})}{f(x_r)}\right)^2=4\frac{1+\left(x_r/f(x_r)\right)^2}{1+\left(x_{2r}/f(x_{2r})\right)^2} \qquad\forall r\in]0,r_0/2[\,.$$ Since $\lim_{r\to 0}\left(x_r/f(x_r)\right)=0= \lim_{r\to 0}\left(x_{2r}/f(x_{2r})\right)$, then the limiting relation ([\[lem:cufnd2\]](#lem:cufnd2){reference-type="ref" reference="lem:cufnd2"}) holds true.$\Box$ Next we plan to prove a formula in order to compute $$\lim_{r\to 0}\frac{A_f(2r)}{A_f(r)} \,.$$ **Proposition 4**. *Let $r_0\in]0,+\infty[$. Let $f\in C^1(]0,r_0[,]0,+\infty[)$ be increasing and satisfy the following limiting relations $$\label{prop:cufnnr1} \lim_{x\to 0}f(x)=0\,,\qquad \lim_{x\to 0}f'(x)=+\infty\,.$$ Let $x_r$ be as in Proposition [Proposition 2](#prop:cufn){reference-type="ref" reference="prop:cufn"} for each $r\in]0,r_0[$. If $$l\equiv \lim_{r\to 0}x_{2r}/x_r\quad\text{exists\ in}\ [0,+\infty]\,,$$ then $$\label{prop:cufnnr2} \lim_{r\to 0}\frac{A_f(2r)}{A_f(r)}=2l\,,$$ where we understand that $2l=+\infty$ if $l=+\infty$.* **Proof.** By de l'Hôpital's rule, we have $$\label{prop:cufnnr3} \lim_{x\to 0}\frac{x}{f(x)}=\lim_{x\to 0}\frac{1}{f'(x)}=0$$ and thus the assumptions of both Proposition [Proposition 2](#prop:cufn){reference-type="ref" reference="prop:cufn"} and Lemma [Lemma 1](#lem:cufnd){reference-type="ref" reference="lem:cufnd"} are satisfied. In particular, $$\label{prop:cufnnr4} \lim_{r\to 0}\frac{f(x_{2r})}{f(x_{r})}=2\,.$$ Moreover, there exists $r_0'\in ]0,r_0/2[$ such that $f'(x)\neq 0$ for all $x\in ]0,2r_0'[$. By Proposition [Proposition 3](#prop:cufnsua){reference-type="ref" reference="prop:cufnsua"} (i), the function $x\sqrt{1+(f'(x))^2}$ is integrable in $x\in]0,r_0/2[$. Then both the numerator and the denominator of the fraction $$\frac{A_f(2r)}{A_f(r)}=\frac{\int_0^{x_{2r}} x\sqrt{1+(f'(x))^2}\,dx}{ \int_0^{x_{r}} x\sqrt{1+(f'(x))^2}\,dx }$$ tend to $0$ as $r$ tends to $0$. By de l'Hôpital's rule, the limit of $\frac{A_f(2r)}{A_f(r)}$ as $r$ tends to $0$ exists provided that the limit of the following ratio $$\begin{aligned} \lefteqn{ \frac{ x_{2r}\sqrt{1+(f'(x_{2r}))^2} \frac{dx_{2r}}{dr}}{ x_{r}\sqrt{1+(f'(x_{r}))^2} \frac{dx_{r}}{dr}} } \\ \nonumber &&\qquad =\frac{x_{2r}}{x_r}\frac{f'(x_{2r})}{f'(x_{r})} \frac{ \sqrt{(f'(x_{2r}))^{-2}+1} } { \sqrt{(f'(x_{r}))^{-2}+1} } \frac{ \frac{2r}{f(x_{2r})f'(x_{2r})+x_{2r}}2 }{ \frac{r}{f(x_{r})f'(x_{r})+x_{r}} } \\ \nonumber &&\qquad =4\frac{x_{2r}}{x_r} \frac{ \frac{1}{f(x_{2r})} }{ \frac{1}{f(x_{r})} }\frac{ \sqrt{(f'(x_{2r}))^{-2}+1} } { \sqrt{(f'(x_{r}))^{-2}+1} } \frac{ \frac{1}{1+(f'(x_{2r}))^{-1}(x_{2r}/f(x_{2r}) } }{ \frac{1}{1+(f'(x_{r}))^{-1}(x_{r}/f(x_{r})} }\quad\forall r\in]0,r_0'[\end{aligned}$$ exists as $r$ tends to $0$, and if such a limit exists, the two limits are equal. Then the limiting relations ([\[prop:cufnnr1\]](#prop:cufnnr1){reference-type="ref" reference="prop:cufnnr1"}), ([\[prop:cufnnr3\]](#prop:cufnnr3){reference-type="ref" reference="prop:cufnnr3"}), ([\[prop:cufnnr4\]](#prop:cufnnr4){reference-type="ref" reference="prop:cufnnr4"}) imply that $$\lim_{r\to 0}\frac{A_f(2r)}{A_f(r)}=4\lim_{r\to 0}\frac{x_{2r}}{x_r}\frac{1}{2}=2l\,,$$ where we understand that $2l=+\infty$ if $l=+\infty$.$\Box$ We are now ready to present the following example of a graph of a function such that the ordinary $2$-dimensional measure fails to satisfy the doubling condition at a point and that accordingly cannot satisfy a $2$-Ahlfors regularity condition with respect to the set that contains that point. **Example 1**. *Let $r_0\in]0,1[$. Let $$f(x)\equiv \frac{1}{|\log x|}\qquad\forall x\in]0,r_0[\,.$$ Let $\gamma_f$ be as in ([\[eq:gammaf\]](#eq:gammaf){reference-type="ref" reference="eq:gammaf"}). Let $$Y\equiv{\mathrm{graph}}(\gamma_f)\setminus\{(0,0,0)\}$$ be endowed with the ordinary $2$-dimensional measure. Then* 1. *$Y$ is strongly upper $2$-Ahlfors regular with respect to $\{(0,0,0)\}$.* 2. *The ordinary $2$-dimensional measure $m_2$ on $${\mathrm{graph}}(\gamma_f)\setminus\{(0,0,0)\}$$ does not satisfy the doubling condition with respect to the set $\{(0,0,0)\}$. More precisely, $$\lim_{r\to0}\frac{A_f(2r)}{A_f(r)}=+\infty$$ (cf. ([\[eq:arf\]](#eq:arf){reference-type="ref" reference="eq:arf"})).* **Proof.** Since $f'(x)=\frac{1}{x\log^2x}$ for all $x\in]0,r_0[$, $f$ satisfies all the assumptions of Propositions [Proposition 2](#prop:cufn){reference-type="ref" reference="prop:cufn"}, [Proposition 3](#prop:cufnsua){reference-type="ref" reference="prop:cufnsua"} and thus statement (i) holds true. \(ii\) Since $f$ satisfies the assumptions of Propositions [Proposition 2](#prop:cufn){reference-type="ref" reference="prop:cufn"}, Proposition [Proposition 4](#prop:cufnnr){reference-type="ref" reference="prop:cufnnr"}, it suffices to show that $$\label{esem:badloca1} \lim_{r\to 0}\frac{x_{2r}}{x_r}=+\infty\,.$$ By Lemma [Lemma 1](#lem:cufnd){reference-type="ref" reference="lem:cufnd"} there exists $r_1\in]0,r_0/2[$ such that $$\frac{3}{2}<\frac{ \frac{1}{|\log x_{2r}|} }{ \frac{1}{|\log x_r|} }<\frac{5}{2}\qquad\forall r\in]0,r_1[\,,$$ *i.e.*, $$-\frac{3}{2}\log(x_{2r})<-\log (x_r)<-\frac{5}{2}\log(x_{2r})\qquad\forall r\in]0,r_1[\,,$$ or equivalently $$x_{2r}^{-\frac{3}{2}}<x_r^{-1}<x_{2r}^{-\frac{5}{2}} \qquad\forall r\in]0,r_1[\,.$$ Then we have $$\frac{x_{2r}}{x_r}\geq x_{2r} x_{2r}^{-\frac{3}{2}} =x_{2r}^{-\frac{1}{2}} \geq (2r)^{-\frac{1}{2}} \qquad\forall r\in]0,r_1[$$ and thus the limiting relation ([\[esem:badloca1\]](#esem:badloca1){reference-type="ref" reference="esem:badloca1"}) holds true. $\Box$ # Appendix: Conditions of upper Ahlfors regularity on subsets of ${\mathbb{R}}^n$ that are local Lipschitz graphs {#appendix} We first say what we mean by a subset of ${\mathbb{R}}^n$ that is a local graph of a continuous function. **Definition 3**. *Let $n\in {\mathbb{N}}\setminus\{0,1\}$. Let $S$ be a subset of ${\mathbb{R}}^{n}$. Let $p\in S$, $R \in O_{n}({\mathbb{R}})$, $r$, $\delta\in ]0,+\infty[$. We say that the set $$C(p,R,r,\delta)\equiv p+ R^{t}({\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[ )$$ is a coordinate cylinder for $S$ around $p$, provided that the intersection $$R(S-p) \cap ({\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[ )$$ is the graph of a continuous function $\gamma$ from ${\mathbb{B}}_{n-1}(0,r)$ to $]-\delta ,\delta [$, which vanishes at $0$ and such that $|\gamma(\eta)|<\delta/2$ for all $\eta\in {\mathbb{B}}_{n-1}(0,r)$, *i.e.*, provided that there exists $\gamma\in C^{0}({\mathbb{B}}_{n-1}(0,r), ]-\delta ,\delta [)$ such that $$\begin{aligned} \label{prelim.cocylind1} \lefteqn{R(S-p )\cap ({\mathbb{B}}_{n-1}(0,r)\times ]-\delta,\delta[) } \\ \nonumber &&\qquad\ =\left\{ (\eta,y)\in {\mathbb{B}}_{n-1}(0,r)\times ]-\delta,\delta[:\, y=\gamma(\eta) \right\} \equiv{\mathrm{graph}} (\gamma) \,, \\ \nonumber &&|\gamma(\eta)|<\delta/2\qquad\forall \eta\in {\mathbb{B}}_{n-1}(0,r)\,,\qquad \gamma(0)=0\,.\end{aligned}$$* Given a coordinate cylinder $C(p,R,r,\delta)$ for $S$ around $p$, the corresponding function $\gamma$ is uniquely determined. Indeed, if $\eta\in {\mathbb{B}}_{n-1}(0,r)$, then $\gamma(\eta)$ is the unique element $y$ of $]-\delta,\delta[$ such that $$(\eta,y)\in R(S-p )\cap ({\mathbb{B}}_{n-1}(0,r)\times ]-\delta,\delta[)\,.$$ We say that $\gamma$ is the function that represents $S$ in the coordinate cylinder $C(p,R,r,\delta)$ as a graph and that the function $\psi_{p}$ from ${\mathbb{B}}_{n-1}(0,r)$ to ${\mathbb{R}}^{n}$ defined by $$\label{prelim.cocylind3} \psi_{p}(\eta)\equiv p+R^{t}\left( \begin{array}{c} \eta \\ \gamma(\eta) \end{array} \right)\qquad\forall \eta\in {\mathbb{B}}_{n-1}(0,r)\,,$$ is the parametrization of $S$ around $p$ in the coordinate cylinder $C(p,R,r,\delta)$. Since the continuous function $\gamma$ induces the homeomorphism $(\cdot,\gamma(\cdot))$ from its domain ${\mathbb{B}}_{n-1}(0,r)$ onto its graph ${\mathrm{graph}} (\gamma)$, the map $\psi_{p}$ is a homeomorphism from ${\mathbb{B}}_{n-1}(0,r)$ onto $\psi_{p}({\mathbb{B}}_{n-1}(0,r))=S \cap C(p,R,r,\delta)$. It is sometimes useful to know that by shrinking $r$ we still obtain a coordinate cylinder around the point $p$. More precisely, we have the following. **Remark 1**. *If $C(p,R,r,\delta)$ is a coordinate cylinder around the point $p$ of a subset $S$ of ${\mathbb{R}}^n$, then also $C(p,R,\rho,\delta)$ is a coordinate cylinder around the point $p$ of $S$ for each $\rho\in]0,r[$, and the restriction $\gamma_{|{\mathbb{B}}_{n-1}(0,\rho)}$ represents $S$ in $C(p,R,\rho,\delta)$ as a graph.* We also note that ${\mathrm{graph}} (\gamma)$ is easily seen to be path connected and that accordingly $$S\cap C(p,R,r,\delta) =p+R^{t}({\mathrm{graph}} (\gamma))$$ is path connected. Hence, $S\cap C(p,R,r,\delta)$ is contained in at most one connected component of $S$. We are now ready to introduce the following. **Definition 4**. *Let $n\in {\mathbb{N}}\setminus\{0,1\}$. We say that a subset $S$ of ${\mathbb{R}}^{n}$ is a local graph of class $C^{0}$ provided that for every point $p\in S$, there exist $R\in O_{n}({\mathbb{R}})$ and $r,\delta\in]0,+\infty[$ such that $C(p,R,r,\delta)$ is a coordinate cylinder for $S$ around $p$.* If $S$ is a local graph of class $C^{0}$ and if $p\in S$, then possibly shrinking $r$, we can always assume that $r<\delta$ and that the corresponding function $\gamma$, which represents $S$ in the coordinate cylinder $C(p,R,r,\delta)$ as a graph, has a continuous extension to $\overline{{\mathbb{B}}_{n-1}(0,r)}$ (cf. Remark [Remark 1](#prelim.smcyl){reference-type="ref" reference="prelim.smcyl"}). It is also customary to denote such extension by the same symbol $\gamma$. Then we say that $S$ is of class $C^{m}$ or of class $C^{m,\alpha}$ for some $m\in{\mathbb{N}}$, $\alpha\in ]0,1]$ provided that $\gamma$ is of class $C^{m}$ or of class $C^{m,\alpha}$ for all $p\in\partial\Omega$. For the sake of brevity, we set $$\label{cm0} C^{m,0}\equiv C^{m}\,.$$ If $S$ is of class $C^{0,1}$, then we also say that $S$ is a local Lipschitz graph. Since $C^{1}(\overline{{\mathbb{B}}_{n-1}(0,r)})\leq C^{0,\alpha}(\overline{{\mathbb{B}}_{n-1}(0,r)})$ for all $r\in]0,+\infty[$, $\alpha\in [0,1]$, a local graph of class $C^{1}$ is also of class $C^{0,\alpha}$. Then we have the following statement that says that if $S$ is a compact local graph of class $C^{0,\alpha}$, then we can make a uniform choice of the parameters $r$ and $\delta$. For a proof, one can follow line by line the corresponding proof of [@La19 Lemma 10.1] for the case in which $S$ equals the boundary of a bounded open set of class $C^{0,\alpha}$. For the (classical) definition of norm in $C^{0,\alpha}(\overline{{\mathbb{B}}_{n-1}(0,r)})$ we refer for example to [@DoLa17 §2]. **Lemma 2** (of the uniform cylinders for local Hölder graphs). *Let $n\in {\mathbb{N}}$, $n\geq 2$. Let $\alpha\in[0,1]$. Let $S$ be a compact local graph of class $C^{0,\alpha}$. Let $r_\ast$, $\delta_\ast\in]0,+\infty[$. Then there exist $r\in ]0,r_\ast[$, $\delta\in]0,\delta_\ast[$, $r<\delta$ such that for each $x\in S$ there exists $R_x\in O_n({\mathbb{R}})$ such that $C(x,R_x,r,\delta)$ is a coordinate cylinder for $S$ around $x$ and the corresponding function $\gamma_x$ satisfies the inequality $$\sup_{x\in S}\|\gamma_x\|_{ C^{0,\alpha}(\overline{{\mathbb{B}}_{n-1}(0,r)}) }<+\infty\,.$$* Next we prove the following extension of the well-known fact that the boundary of a bounded open Lipschitz subset of ${\mathbb{R}}^n$ is upper $(n-1)$-Ahlfors regular with respect to itself. **Proposition 5**. *Let $n\in {\mathbb{N}}$, $n\geq 2$. Let $S$ be a compact local Lipschitz graph in ${\mathbb{R}}^n$, which we assume to be equipped with the ordinary $(n-1)$-dimensional measure $m_S$. Then $S$ is upper $(n-1)$-Ahlfors regular with respect to ${\mathbb{R}}^n$.* **Proof.** Let $r$, $\delta\in]0,1[$ be as in Lemma [Lemma 2](#lem:unif0cyl){reference-type="ref" reference="lem:unif0cyl"} of the uniform cylinders. Then we know that for each $\xi\in S$ there exist $R_\xi\in O_n({\mathbb{R}})$ such that $C(\xi,R_\xi,r,\delta)$ is a coordinate cylinder for $S$ around $\xi$ and that the corresponding function $\gamma_\xi$ satisfies the inequality $$a\equiv\sup_{\xi\in S}\|\gamma_\xi\|_{ C^{0,\alpha}(\overline{{\mathbb{B}}_{n-1}(0,r)}) }<+\infty\,.$$ If $x\in {\mathbb{R}}^n$ and the distance ${\mathrm{dist}}\,(x,S)$ of $x$ from $S$ is less than $r/4$, then there exists $x_S\in S$ such that $|x-x_S|= {\mathrm{dist}}\,(x,S)$. By the triangular inequality, we have $${\mathbb{B}}_n(x, r/4)\subseteq {\mathbb{B}}_n(x_S, r)\subseteq C(x_S,R_{x_S},r,\delta) \,.$$ In particular, there exists a unique $(\eta_x, y_x)\in {\mathbb{B}}_{n-1}(0,r)\times]-\delta,\delta[$ such that $$x=x_S+R_{x_S}^t(\eta_x, y_x)^t\,, \qquad |\eta_x|^2+|y_x|^2<(r/4)^2 \,.$$ Thus if $\rho\in ]0, r/4[$, we have $$\begin{aligned} \lefteqn{ S\cap {\mathbb{B}}_n(x,\rho) =C(x_S,R_{x_S},r,\delta)\cap S\cap {\mathbb{B}}_n(x,\rho) } \\ \nonumber &&\qquad =x_S+R_{x_S}^t\bigl\{\bigr. (\eta,\gamma_{x_S}(\eta)):\,\eta\in {\mathbb{B}}_{n-1}(0,r)\,, \\ \nonumber &&\qquad\qquad\qquad\qquad\quad |\eta-\eta_x|^2+|\gamma_{x_S}(\eta)-y_x|^2<\rho^2 \bigl.\bigr\} \\ \nonumber &&\qquad \subseteq x_S+R_{x_S}^t\left\{ (\eta,\gamma_{x_S}(\eta)):\,\eta\in {\mathbb{B}}_{n-1}(\eta_x,\rho) \right\}\end{aligned}$$ and thus $$\begin{aligned} \lefteqn{ m_S(S\cap {\mathbb{B}}_n(x,\rho)) } \\ \nonumber &&\qquad \leq \int_{{\mathbb{B}}_{n-1}(\eta_x,\rho)}\sqrt{1+|D\gamma_{x_S}(\eta)|^2}\,d\eta \leq\omega_{n-1}\sqrt{1+a^2}\rho^{n-1}\,.\end{aligned}$$ On the other hand if ${\mathrm{dist}}\,(x,S)\geq r/4$, then we have $S\cap {\mathbb{B}}_n(x,\rho)=\emptyset$ for all $\rho\in ]0, r/4[$ and thus $$m_S(S\cap {\mathbb{B}}_n(x,\rho))=0\leq \omega_{n-1}\sqrt{1+a^2}\rho^{n-1}\qquad\forall \rho\in ]0, r/4[\,.$$ Hence, we conclude that statement (i) holds true and that we can choose $r_{{\mathbb{R}}^n,S,n-1}= r/4$, $c_{{\mathbb{R}}^n,S,n-1}=\omega_{n-1}\sqrt{1+a^2}$.$\Box$ For the strong upper Ahlfors regularity instead, we must formulate some extra assumption and we prove the following statement. **Proposition 6**. *Let $n\in {\mathbb{N}}$, $n\geq 2$. Let $S$ be a compact local Lipschitz graph in ${\mathbb{R}}^n$, which we assume to be equipped with the ordinary $(n-1)$-dimensional measure $m_S$.* *Assume that there exist $r,\delta\in ]0,+\infty[$ such that for each $x\in S$ there exists $R_x\in O_n({\mathbb{R}})$ such that $C(x,R_x,r,\delta)$ is a coordinate cylinder for $S$ around $x$ and the corresponding function $\gamma_x$ satisfies the inequalities $$\begin{aligned} \label{prop:lgar1} &&a\equiv\sup_{x\in S}\|\gamma_x\|_{ C^{0,1}(\overline{{\mathbb{B}}_{n-1}(0,r)}) }<+\infty\,, \\ \nonumber &&b\equiv \inf_{x\in S}{\mathrm{ess\,inf}}_{\eta\in{\mathbb{B}}_{n-1}(0,r)\setminus\{0\}} \frac{(\eta+\gamma_x(\eta)D\gamma_x(\eta))\cdot\eta}{|\eta|^2}>0\,.\end{aligned}$$ Then $S$ is strongly upper $(n-1)$-Ahlfors regular with respect to $S$.* **Proof.** We plan to prove the strong upper $(n-1)$-Ahlfors regularity by estimating the first order derivative of $m_S(S\cap {\mathbb{B}}_n(x,\rho))$ with respect to $\rho$. To do so, we fix $x\in S$ and we note that $$\begin{aligned} \lefteqn{ S\cap {\mathbb{B}}_n(x,\rho) =C(x,R_x,r,\delta)\cap S\cap {\mathbb{B}}_n(x,\rho) } \\ \nonumber && =x+R_x^t\left\{ (\eta,\gamma_{x}(\eta)):\,\eta\in {\mathbb{B}}_{n-1}(0,r)\,, |\eta |^2+|\gamma_{x}(\eta)|^2<\rho^2 \right\} \quad\forall\rho\in]0,r[\,, \end{aligned}$$ and that accordingly $$m_S( S\cap {\mathbb{B}}_n(x,\rho) )= \int_{\{ \eta\in {\mathbb{B}}_{n-1}(0,r):\,|\eta|^2+|\gamma_x(\eta)|^2<\rho^2 \}}\sqrt{1+|D\gamma_x(\eta)|^2}\,d\eta \,,$$ for all $\rho\in]0,r[$. In order to estimate $\frac{d}{d\rho}m_S( S\cap {\mathbb{B}}_n(x,\rho) )$, we plan to compute the integral in the right hand side by exploiting the theorem of integration for functions that are defined on domains that are normal with respect to the unit sphere. To do so however, we need to show that $$A_\rho\equiv\{ \eta\in {\mathbb{B}}_{n-1}(0,r):\,|\eta|^2+|\gamma_x(\eta)|^2<\rho^2 \}$$ is star shaped with respect to $0$ for almost all directions of $\partial {\mathbb{B}}_{n-1}(0,1)$, *i.e.*, that there exists a subset $N$ of measure $0$ of $\partial {\mathbb{B}}_{n-1}(0,1)$ such that $s\eta\in A_\rho$ for all $s\in]0,1]$ and $\eta\in A_\rho$ such that $\frac{\eta}{|\eta|}\in \partial {\mathbb{B}}_{n-1}(0,1)\setminus N$. Since $\gamma_x$ is Lipschitz continuous, the Rademacher Theorem implies that there exists a subset $E$ of measure zero of ${\mathbb{B}}_{n-1}(0,r)$ such that $\gamma_x$ is differentiable at all points of ${\mathbb{B}}_{n-1}(0,r)\setminus E$. Then by applying the Theorem of integration on the spheres to the characteristic function of $E$, we can infer the existence of a subset $N$ of measure $0$ of $\partial {\mathbb{B}}_{n-1}(0,1)$ such that if $u\in \partial {\mathbb{B}}_{n-1}(0,1)\setminus N$, then $\gamma_x$ is differentiable at $su$ for almost all $s\in]0,r[$. Next we plan to show that for each $u\in \partial {\mathbb{B}}_{n-1}(0,1)\setminus N$ and $\rho\in]0,r[$, there exists one and only one $r(\rho,u)\in ]0,r[$ such that $$(r(\rho,u)u,\gamma_x(r(\rho,u)u))\in \partial {\mathbb{B}}_{n}(0,\rho)\,,$$ *i.e.*, such that $$|r(\rho,u)u|^2+|\gamma_x(r(\rho,u)u)|^2=\rho^2\,.$$ To do so, we set $$G(\rho,\eta,s)\equiv |s\eta|^2+|\gamma_x(s\eta)|^2-\rho^2\quad\forall (\rho,\eta,s)\in ]0,r[\times (\partial {\mathbb{B}}_{n-1}(0,1)\setminus N)\times]0,r[ \,.$$ If we fix $(\rho,\eta)\in]0,r[\times (\partial {\mathbb{B}}_{n-1}(0,1)\setminus N)$, the function $G(\rho,\eta,\cdot)$ is differentiable for almost all $s\in]0,r[$ and we have $$\begin{aligned} \label{prop:lgar2} \lefteqn{ \frac{\partial G}{\partial s}(\rho,\eta,s)=2s\eta\cdot\eta+2\gamma_x(s\eta)D\gamma_x(s\eta)\eta } \\ \nonumber &&\qquad =\frac{2}{s}\left[ (s\eta)+\gamma_x(s\eta)D\gamma_x(s\eta) \right]\cdot (s\eta)\geq \frac{2}{s}b|s\eta|^2=2bs|\eta|^2>0\end{aligned}$$ for almost all $\ s\in]0,r[$. Since $G(\rho,\eta,\cdot)$ is Lipschitz continuous in $[0,r]$, $$G(\rho,\eta,0)=-\rho^2<0\,,\qquad G(\rho,\eta,\rho)\geq 0\,,$$ we conclude that $G(\rho,\eta,\cdot)$ is strictly increasing in $[0,r[$ and that there exists one and only one $s\in ]0,\rho]$ such that $G(\rho,\eta,s)=0$ and we set $s\equiv r(\rho,\eta)$. We also note that $$\label{prop:lgar3} |s\eta|^2+|\gamma_x(s\eta)|^2<\rho^2\ \forall s\in ]0,r(\rho,\eta)[\,, \ \ |s\eta|^2+|\gamma_x(s\eta)|^2>\rho^2\ \forall s\in ]r(\rho,\eta),r[\,.$$ Next we turn to show that if $\eta\in (\partial {\mathbb{B}}_{n-1}(0,1))\setminus N$, then the function $r(\cdot,\eta)$ is Lipschitz continuous. Let $\rho_1$, $\rho_2\in ]0,r[$. There is no loss of generality in assuming that $\rho_1<\rho_2$. For the sake of brevity, we set $\varsigma_j\equiv r(\rho_j,\eta)$ for $j\in\{1,2\}$. Since $G(\rho_1,\eta,r(\rho_1,\eta))=0=G(\rho_2,\eta,r(\rho_2,\eta))$, we have $$\begin{aligned} \lefteqn{ \rho_2^2-\rho_1^2= |\varsigma_2\eta|^2+\gamma_x^2(\varsigma_2\eta) -|\varsigma_1\eta|^2-\gamma_x^2(\varsigma_1\eta) } \\ \nonumber &&\qquad =\int_0^1\frac{\partial}{\partial s}_{|s=\varsigma_1+t(\varsigma_2-\varsigma_1)}\left\{ |s\eta|^2+\gamma_x^2(s\eta) \right\}\,dt(\varsigma_2-\varsigma_1) \\ \nonumber &&\qquad =\int_0^1\left\{ 2s\eta\cdot\eta+2\gamma_x(s\eta)D\gamma_x(s\eta)\cdot\eta \right\}_{|s=\varsigma_1+t(\varsigma_2-\varsigma_1)}\,dt(\varsigma_2-\varsigma_1) \\ \nonumber &&\qquad\geq 2b|\eta|^2\int_0^1\varsigma_1+t(\varsigma_2-\varsigma_1)\,dt(\varsigma_2-\varsigma_1) \geq 2b|\eta|^2\frac{\varsigma_1+\varsigma_2}{2}(\varsigma_2-\varsigma_1) \end{aligned}$$ (cf. ([\[prop:lgar2\]](#prop:lgar2){reference-type="ref" reference="prop:lgar2"})). Now by the equalities $|\varsigma_j\eta|^2+\gamma_x^2(\varsigma_j\eta)=\rho_j^2$ for $j\in\{1,2\}$, we obtain $$\rho_j^2\leq \varsigma_j^2+{\mathrm{Lip}}^2(\gamma_x)\varsigma_j^2=\varsigma_j^2(1+{\mathrm{Lip}}^2(\gamma_x))^2 \qquad\forall j\in\{1,2\}$$ and thus $$\varsigma_2-\varsigma_1\leq\frac{1}{b}\frac{\rho_1+\rho_2}{\varsigma_1+\varsigma_2}(\rho_2-\rho_1)\leq\frac{\sqrt{1+{\mathrm{Lip}}^2(\gamma_x)}}{b} (\rho_2-\rho_1) \leq\frac{\sqrt{1+a^2}}{b} (\rho_2-\rho_1)$$ and accordingly, $r(\cdot,\eta)$ is Lipschitz continuous with Lipschitz constant less or equal to $\frac{\sqrt{1+a^2}}{b}$, which is independent of the choice of $\eta$ in $(\partial {\mathbb{B}}_{n-1}(0,1))\setminus N$. Then by integrating on the spheres, we have $$\begin{aligned} \lefteqn{ m_S( S\cap {\mathbb{B}}_{n-1}(x,\rho) ) } \\ \nonumber &&\qquad = \int_{ \partial {\mathbb{B}}_{n-1}(0,1) }\int_0^{r(\rho,\eta)} \sqrt{1+|D\gamma_x(s \eta)|^2} s^{n-2}\, ds\,d\sigma_\eta \quad\forall\rho\in]0,r[\,.\end{aligned}$$ Since $r(\cdot,\eta)$ is Lipschitz continuous with a constant that is independent of $\eta$ in $(\partial {\mathbb{B}}_{n-1}(0,1))\setminus N$ and $$|D\gamma_x(s\eta)|\leq a\qquad \text{a.a.}\ (\eta,s)\in ((\partial {\mathbb{B}}_{n-1}(0,1))\setminus N)\times]0,r[$$ and $r(\rho,\eta)\in]0,\rho]$ for all $(\rho,\eta)\in]0,r[\times((\partial {\mathbb{B}}_{n-1}(0,1))\setminus N)$, we conclude that $m_S( S\cap {\mathbb{B}}_{n-1}(x,\cdot))$ is Lipschitz continuous in $]0,r[$ and that $$\begin{aligned} \lefteqn{ \left|\frac{d}{d\rho}m_S( S\cap {\mathbb{B}}_{n-1}(x,\rho) ) \right| } \\ \nonumber &&\qquad =\left| \int_{ \partial {\mathbb{B}}_{n-1}(0,1) } \sqrt{1+|D\gamma_x(r(\rho,\eta) \eta)|^2}\frac{\partial}{\partial \rho} r(\rho,\eta)r(\rho,\eta)^{n-2}\, d\sigma_\eta\right| \\ \nonumber &&\qquad \leq s_{n-1}\sqrt{1+a^2}\frac{\sqrt{1+a^2}}{b}\rho^{n-2} \quad {\mathrm{a.a.}}\ \rho\in]0,r[ \,,\end{aligned}$$ where $s_{n-1}$ denotes the measure of $\partial{\mathbb{B}}_{n-1}(0,1)$ (cf. ([\[eq:snon\]](#eq:snon){reference-type="ref" reference="eq:snon"})). Hence, $$\begin{aligned} \lefteqn{ m_n( S\cap ({\mathbb{B}}_{n}(x,r_2)\setminus {\mathbb{B}}_n(x,r_1)) ) \leq \int_{r_1}^{r_2} s_{n-1}\frac{(1+a^2)}{b}\rho^{n-2}\,d\rho } \\ &&\qquad\qquad\qquad\qquad\qquad\qquad\qquad \leq\frac{s_{n-1}}{n-1}\frac{(1+a^2)}{b} ( r_2^{n-1}-r_1^{n-1})\end{aligned}$$ for all $x\in S$ and $r_1,r_2\in[0,r[$ with $r_1<r_2$ and thus we can take $$r_{S,S,n-1}=r\,,\qquad c_{S,S,n-1}=\frac{s_{n-1}}{n-1}\frac{(1+a^2)}{b}$$ that are independent of the choice of $x$ in $S$ and the proof is complete. $\Box$ **Remark 2**. *If $S$ is a compact local graph of class $C^1$, then one can follow line by line the proof of [@DaLaMu21 Lemma 2.63], that refers to the case in which $S$ equals the boundary of a bounded open set of class $C^{1}$ and prove prove the existence of $r$ and $\delta$ and of $R_x$, $\gamma_x$ for all $x\in S$ such that $$\sup_{x\in S}\|\gamma_x\|_{ C^{1}(\overline{{\mathbb{B}}_{n-1}(0,r)}) }<+\infty\,, \qquad \sup_{x\in S}\sup_{{\mathbb{B}}_{n-1}(0,r)}|D\gamma_x|<1/2\,$$ and deduce the validity of the conditions in ([\[prop:lgar1\]](#prop:lgar1){reference-type="ref" reference="prop:lgar1"}) of Proposition [Proposition 5](#prop:lgar){reference-type="ref" reference="prop:lgar"} by means of the elementary inequality $$(\eta+\gamma_x(\eta)D\gamma_x(\eta))\cdot\eta\geq \eta\cdot\eta-|\eta|^2\left(\sup_{{\mathbb{B}}_{n-1}(0,r)}|D\gamma_x|\right)^2\geq\frac{3}{4}|\eta|^2$$ for all $\eta\in {\mathbb{B}}_{n-1}(0,r)$. Hence, a compact local graph $S$ of class $C^1$ is strongly upper $(n-1)$-Ahlfors regular (with respect to $S$).* **Acknowledgement** The author acknowledges the support of the Research Project GNAMPA-INdAM $\text{CUP}\_$E53C22001930001 'Operatori differenziali e integrali in geometria spettrale' and is indebted to Prof. Otari Chkadua and Prof. David Natroshvili for a number of references, to Prof. Joan Verdera for references [@Ve13] and [@Ve23] and to Prof. Sergiy Plaksa for the references on the moduli of continuity of the Cauchy integral of section [3](#dolalipc1){reference-type="ref" reference="dolalipc1"}. 11 A.A. Babaev, V.V. Salaev, *One-dimensional singular operator with continuous density along a closed curve* (in Russian). Dokl. Akad. Nauk SSSR., **209** (1973), 1257--1260. T. Buchukuri, O. Chkadua, R. Duduchava, and D. Natroshvili, *Interface Crack Problems for Metallic-Piezoelectric Composite Structures* Mem. Differential Equations Math. Phys., **55** (2012), 1--150. Yu. 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arxiv_math

--- abstract: | We introduce a minimality property for subgroups of topological groups. A subgroup $H$ is a *key subgroup* of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$ the original topology. In fact, this concept appears implicitly in some earlier publications [@MEG95; @DM10; @MS-Fermat]. Every co-minimal subgroup is a key subgroup while the converse is not true even for discrete groups. In this paper, we continue the study of minimality in topological matrix groups initiated in [@MS-Fermat]. Extending some results from [@MEG95; @DM10] concerning the generalized Heisenberg group, we prove that the center of the upper unitriangular group $\operatorname{UT}(n,K)$, defined over a commutative topological unital ring $K$, is a key subgroup. This center is even co-minimal in $\operatorname{UT}(n,K)$ assuming that the multiplication map $m \colon K\times K\to K$ is strongly minimal. We show that the latter holds when $K$ is an archimedean absolute valued field or a local field. address: - Department of Mathematics Bar-Ilan University, 52900 Ramat-Gan Israel - School of Computer Science Reichman University, 4610101 Herzliya Israel author: - M. Megrelishvili - M. Shlossberg date: 2023, September 13 title: Key subgroups and co-minimality in topological groups --- [^1] # Main definition and some related results Unless otherwise is stated the topological groups in this paper are assumed to be Hausdorff. A topological group $G$ is *minimal* [@S71; @Doitch] if every continuous isomorphism $f \colon G\to H$, with $H$ a topological group, is a topological isomorphism (equivalently, if $G$ does not admit a strictly coarser Hausdorff group topology). There are many papers demonstrating that minimal groups are important in algebra, analysis and geometry. See for example, the survey paper [@DM14]. Sometimes, a wider look is required relaxing the definition of minimality. Recall also [@MEG04] (resp., [@DM10]) that a subgroup $H$ of $G$ is said to be *relatively minimal* (resp., *co-minimal*) in $G$ if every coarser Hausdorff group topology on $G$ induces on $H$ (resp., on the coset set $G/H$) the original topology. The next definition gives a new minimality condition which is a further relaxation of co-minimality. **Definition 1**. *Let $H$ be a subgroup of a topological group $(G,\tau)$. We say that $H$ is a key subgroup of $G$ if for every coarser Hausdorff group topology $\tau_1 \subseteq \tau$ on $G$ the coincidence of the two topologies on $H$ (that is, $\tau_1|_H = \tau|_H$) necessarily implies that $\tau_1 = \tau$.* In fact, this concept appears implicitly in some earlier publications [@MEG95; @DM10; @MS-Fermat]. **Lemma 2**. * * 1. *A topological group $G$ is minimal if and only if it contains a relatively minimal key subgroup.* 2. *Let $H$ be a key subgroup of $G.$ If $K$ is a subgroup of $G$ containing $H$, then $K$ is also a key subgroup of $G.$* 3. *Every dense subgroup is a key subgroup.* 4. *$H$ is a key subgroup of $G$ if and only if its closure is a key subgroup.* 5. *Let $G_2 \leq G_1 \leq G$. If $G_2$ is a key for $G_1$ and $G_1$ is a key for $G$, then $G_2$ is a key for $G$. This can be easily extended for every finite chain of subgroups.* 6. *Let $\alpha\colon G \times X \to X$ be a continuous action of $G$ on $X$ by group automorphisms and $X \rtimes_{\alpha} G$ be a topological semidirect product. If $X$ is abelian and the action $\alpha$ is t-exact, then $X$ is a key subgroup of $X \rtimes_{\alpha} G$, [@MEG95 Corollary 2.8].* 7. *For a more general example with a t-exact system of actions and central retractions, see [@MEG95 Proposition 2.7].* The following result is known as *Merson's Lemma* ([@DPS89 Lemma 7.2.3] or [@DM14 Lemma 4.4]). **Fact 3**. *Let $(G,\gamma)$ be a (not necessarily Hausdorff) topological group and $H$ be a subgroup of $G.$ If $\gamma_1\subseteq \gamma$ is a coarser group topology on $G$ such that $\gamma_1|_{H}=\gamma|_{H}$ and $\gamma_1/H=\gamma/H$, then $\gamma_1=\gamma.$* As a corollary one has: **Corollary 4**. *Let $G$ be a topological group. Then every co-minimal subgroup of $G$ is a key subgroup.* *Remark 5*. By Corollary [Corollary 14](#prop: knoco){reference-type="ref" reference="prop: knoco"} and Example [Example 15](#ex:mnosm){reference-type="ref" reference="ex:mnosm"} below, the converse statement is not true. This means that the class of all key subgroups properly contains the class of all co-minimal subgroups. Using [@MEG95 Proposition 2.7] one may show that in Lemma [Lemma 2](#examples1){reference-type="ref" reference="examples1"}.6 the subgroup $X$ is even co-minimal in $X \rtimes_{\alpha} G$. Since the class of minimal groups is closed under taking central subgroups (see, for example, [@DPS89 Proposition 7.2.5]) we immediately obtain: **Lemma 6**. *Let $G$ be a topological group. If the center $Z(G)$ is a key subgroup of $G$, then $G$ is minimal if and only if $Z(G)$ is minimal.* # Generalized Heisenberg groups {#s:GH} We recall a natural generalization of the Heisenberg group (see, for example, [@MEG95; @DM10; @DM14]) which is based on biadditive mappings. Let $E,F,A$ be abelian groups. A map $w: E \times F \to A$ is said to be *biadditive* if the induced mappings $$w_x: F \to A, w_f: E \to A, \ \ w_x(f):=w(x,f)=:w_f(x)$$ are homomorphisms for all $x \in E$ and $f \in F$. We say that $w$ is *separated* if the induced homomorphisms separate points. That is, for every non-zero $x_0 \in E, f_0 \in F$ there exist $f \in F, x \in E$ such that $f(x_0) \neq 0_A, f_0(x) \neq 0_A$, where $0_A$ is the zero element of $A$. **Definition 7**. **Let $E, F$ and $A$ be abelian topological groups and $w: E \times F \to A$ be a continuous biadditive mapping. Define the induced action of $F$ on $A \times E$ by $$w^{\nabla}: F \times (A \times E) \to (A \times E), \ \ \ w^{\nabla}(f,(a,x)) =(a+f(x),x)$$ Every translation under this action is an automorphism of the group $A \times E$. Denote by $$H(w)= (A \oplus E) \rtimes F$$ the semidirect product of $F$ and the group $A \times E$. The resulting group, as a topological space, is the product $A \times E \times F$. This product topology will be denoted by $\gamma$. The group operation is defined by the following rule: for a pair $$u_1=(a_1,x_1,f_1), \hskip 0.4cm u_2=(a_2,x_2,f_2)$$ define $$u_1 \cdot u_2 = (a_1+a_2+f_1(x_2), x_1+x_2, f_1 +f_2)$$ where, $f_1(x_2)=w(x_2,f_1)$.** If $w$ is separated, then $H(w)$ becomes a two-step nilpotent topological group and $Z(H(w))=A$. We call $H(w)$ the *generalized Heisenberg group* induced by $w$. Intuitively we can describe this group in the matrix form $$H(w):= \left( \begin{array}{ccc} 1 & F & A \\ 0 & 1 & E \\ 0 & 0 & 1 \end{array}\right)$$ Elementary computations for the commutator $[u_1,u_2]$ give $$[u_1,u_2] = u_1u_2u_1^{-1}u_2^{-1}= (f_1(x_2)-f_2(x_1),0_E,0_F).$$ Often we will identify $E$ with $\{0_A \} \times E \times \{0_F \}$, $F$ with $\{0_A\} \times \{0_E\} \times F$ and $E \times F$ with $\{0_A\} \times E \times F$ . In the case of a normed space $X$ and the canonical bilinear function $w: X \times X^* \to {\Bbb R}$ we write $H(X)$ instead of $H(w)$. Clearly, the case of $H({\Bbb R}^n)$ (induced by the scalar product $w: {\Bbb R}^n \times {\Bbb R}^n \to {\Bbb R}$) gives the classical 2n+1-dimensional Heisenberg group. **Definition 8**. *Let $(E,\sigma)$, $(F, \tau)$, $(A,\nu)$ be abelian groups such that the separated biadditive mapping $$w: (E,\sigma) \times (F,\tau) \to (A,\nu) \eqno(*)$$ is continuous.* - *A triple $(\sigma_1,\tau_1,\nu_1)$ of coarser Hausdorff group topologies $\sigma_1 \subseteq\sigma$, $\tau_1 \subseteq \tau$, $\nu_1 \subseteq \nu$ on $E, F$ and $A$, respectively, is called *compatible*, if $$w: (E,\sigma_1) \times (F,\tau_1) \to (A,\nu_1)$$ is continuous.* - *[@MEG95] We say that the biadditive mapping is *minimal* if $\sigma_1=\sigma, \tau_1=\tau$ holds for every compatible triple $(\sigma_1,\tau_1, \nu)$ (with $\nu_1:=\nu$).* - *[@DM10] The biadditive mapping (\*) is called *strongly minimal* if for every compatible triple $(\sigma_1,\tau_1,\nu_1)$ it follows that $\sigma_1=\sigma, \tau_1=\tau$.* *Remark 9*. The definition of a compatible triple $(\sigma_1,\tau_1,\nu_1)$ does not require that $\sigma_1$ and $\tau_1$ are Hausdorff. Indeed, since $w$ is separated and $\nu_1$ is Hausdorff it follows that $\sigma_1$ and $\tau_1$ are automatically Hausdorff. **Lemma 10**. * * 1. *Every strongly minimal map is minimal.* 2. *Every compatible triple $(\sigma_1,\tau_1,\nu_1)$ of topologies gives rise to the corresponding (product) topology $\gamma_1$ on the Heisenberg group $H(w)=(A \times E) \rtimes F$ which is a coarser Hausdorff group topology (that is, $\gamma_1 \subseteq \gamma$).* 3. *If the mapping $w$ is minimal and the group $A$ is minimal, then $w$ is strongly minimal.* We now provide some examples of (strongly) minimal biadditive maps. *Example 11*.  [@MEG95; @SH; @DM14] 1. The canonical biadditive mapping $G \times G^* \to {\Bbb T}, \hskip 0.2cm (g,\chi) \mapsto \chi(g)$ is strongly minimal for every locally compact abelian group G. 2. The canonical bilinear mapping $V\times V^* \to {\Bbb R}, \hskip 0.2cm (v,f) \mapsto f(v)$ is strongly minimal for all normed spaces $V$ (where $V$ and its dual $V^*$ carry the norm topology). 3. The multiplication map $m \colon A \times A \to A$ is minimal for every topological unital ring $A$. If $m$ is strongly minimal, then $A$ is necessarily a minimal topological ring. 4. Let $K$ be an archimedean absolute valued (not necessarily associative) division ring. For each $n\in {\Bbb N}$, $w_n\colon K^n\times K^n\to K, \ w_n(\bar{x},\bar{y})= \sum_{i=1}^n x_iy_i$ (where $(\bar{x},\bar{y})=((x_1,\ldots, x_n),(y_1, \ldots, y_n))$ is a strongly minimal biadditive mapping. Regarding assertion (3) recall that every non-discrete locally retrobounded division ring is minimal as a topological ring. Indeed, this follows from [@Warner-R Theorem 13.8]. **Proposition 12**. *Assume that $w \colon E \times F \to A$ is a separated biadditive map. Then the following conditions are equivalent:* 1. *$A$ is a key subgroup of the generalized Heisenberg group $H(w)$.* 2. *$w$ is a minimal biadditive mapping.* *Proof.* (1) $\Rightarrow$ (2): If $w$ is not minimal, then there exist Hausdorff group topologies $\sigma_1 \subseteq\sigma$, $\tau_1 \subseteq \tau$ on $E$ and $F$ respectively such that $$w: (E,\sigma_1) \times (F,\tau_1) \to (A,\nu)$$ is continuous and $\sigma_1 \neq \sigma$ or $\tau_1 \neq \tau$. Then the corresponding Heisenberg group $H(w)= (A \times E) \rtimes F$ is well defined with respect to the product topology $(\nu \times \sigma_1 \times \tau_1)$ which is strictly coarser than the original topology $(\nu \times \sigma\times \tau)$. However, both of them induce the same topology $\nu$ on the subgroup $A$. \(2\) $\Rightarrow$ (1): It is a reformulation of [@MEG95 Proposition 2.9], or of [@DM14 Lemma 5.9]. ◻ **Proposition 13**. *Assume that $w \colon E \times F \to A$ is a separated biadditive map. Then the following conditions are equivalent:* 1. *$A$ is a co-minimal subgroup of $H(w)$.* 2. *$w$ is strongly minimal.* *Proof.* (1) $\Rightarrow$ (2): If $w$ is not strongly minimal then there exist Hausdorff group topologies $\sigma_1 \subseteq\sigma$, $\tau_1 \subseteq \tau$, $\nu_1 \subseteq \nu$ on $E$ and $F$ respectively such that $$w: (E,\sigma_1) \times (F,\tau_1) \to (A,\nu_1)$$ is continuous and $\sigma_1 \neq \sigma$ or $\tau_1 \neq \tau$. Then the corresponding Heisenberg group $(H(w),\gamma_1)= (A \times E) \rtimes F$ is well defined with respect to the product topology $\gamma_1:=(\nu \times \sigma_1 \times \tau_1)$ which is strictly coarser than the original topology $\gamma:=(\nu \times \sigma\times \tau)$. However, the new coset topology on $(H(w),\gamma_1)/A$ is trictly coarser than the original topology $(H(w),\gamma)/A$. \(2\) $\Rightarrow$ (1): See [@DM10 Theorem 5.1]. ◻ **Corollary 14**. *Assume that $w \colon E \times F \to A$ is a separated biadditive map. Then the following conditions are equivalent:* 1. *$A$ is a key subgroup of $H(w)$ but not co-minimal.* 2. *$w$ is minimal but not strongly minimal.* *Example 15*. Consider the multiplication map $m \colon {\Bbb Z}\times {\Bbb Z}\to {\Bbb Z}$, where ${\Bbb Z}$ is the discrete ring of all integers. Clearly, ${\Bbb Z}$ is not a minimal ring ($p$-adic topology is a ring topology). So $m$ is minimal but not strongly minimal. Therefore the center (corner-subgroup) $A:={\Bbb Z}$ is a key subgroup in the discrete Heisenberg group $H(m)=({\Bbb Z}\times {\Bbb Z}) \rtimes {\Bbb Z}$ but this subgroup is not co-minimal. The same idea works for every topological unital ring $M$ which is not a minimal topological ring. **Theorem 16**. *Let ${\Bbb F}$ be a local field. Then the multiplication map $m\colon {\Bbb F}\times {\Bbb F}\to {\Bbb F}$ is strongly minimal.* *Proof.* It is well known that there exists a nontrivial absolute value on ${\Bbb F}$ which generates the topology $\tau$. Assuming that $m$ is not strongly minimal, there exist Hausdorff group topologies $\sigma_1, \tau_1, \nu_1 \subseteq \tau$ on ${\Bbb F}$ such that $$w \colon (F,\sigma_1) \times (F,\tau_1) \to (F,\nu_1)$$ is continuous and $\sigma_1 \neq\ \tau$ or $\tau_1 \neq \tau$. Then the corresponding Heisenberg group $(H(w),\gamma_1)= ({\Bbb F}\times {\Bbb F}) \rtimes {\Bbb F}$ is well defined with respect to the product topology $\gamma_1:=(\nu \times \sigma_1 \times \tau_1)$ which is strictly coarser than the original topology $\gamma:=(\nu \times \sigma\times \tau)$. Without restriction of generality, suppose that $\sigma_1 \neq\ \tau$. We first show that every $\sigma_1$-neighborhood of the zero element is unbounded with respect to the absolute value. Indeed, this follows from the fact that in non-archimedean local fields we have an analogue of the Heine-Borel theorem (see [@V Page 5]). That is, every closed bounded subset in $({\Bbb F}, |\cdot|)$ is $\tau$-compact. Next we show that for every unbounded subset $S \subset {\Bbb F}$ and every $\tau$-neighborhood $V$ of $0_{{\Bbb F}}$ (or, even any nonempty $\tau$-open subset) it holds that $1 \in SV$. Using the continuity of $$w \colon (F,\sigma_1) \times (F,\tau_1) \to (F,\nu_1),$$ we obtain a contradiction to the fact that $\nu_1$ is Hausdorff. Indeed, without restriction of generality, let $V=B(0,{\varepsilon}):=\{x \in {\Bbb F}: |x|<{\varepsilon}\}$. Choose $s \in S$ such that $|s|> \frac{1}{{\varepsilon}}$. Then $|s^{-1}|=|s|^{-1}<{\varepsilon}$ and $s^{-1} \in V$. Hence, $1=s \cdot s^{-1} \in SV.$ ◻ # The upper unitriangular group $\operatorname{UT}(n,K)$ In the sequel $K$ is a commutative topological unital ring and $n\geq 2$ is a positive integer. Let $G=\operatorname{UT}(n,K)$ be the upper unitriangular group over $K$ of degree $n$. Denote by $\operatorname{UT}^{m}(n,K)$ (where $m \in \{0,1,\cdots, n\}$) the subgroup of $\operatorname{UT}(n,K)$ having $m$ consecutive zero-diagonals parallel to the main diagonal. Hence, we have $$\operatorname{UT}(n,K)=\operatorname{UT}^{0}(n,K) \geq \operatorname{UT}^{1}(n,K) \geq \cdots \operatorname{UT}^{n-1}(n,K)=\{I_n\}.$$ It is well known that $Z(\operatorname{UT}(n,K))=\operatorname{UT}^{n-2}(n,K)$ (see, in particular [@KargMerz]). The group $\operatorname{UT}(2,K)$ is an isomorphic copy of the additive group $K$. For $n=3$ we have an important (and motivating) particular case $$H(w)\simeq \operatorname{UT}(3,K):= \Bigg\{\left( \begin{array}{ccc} 1 & a_{1,2} & a_{1,3} \\ 0 & 1 & a_{2,3} \\ 0 & 0 & 1 \end{array}\right)\bigg | \ a_{i,j} \in K \Bigg\}.$$ Below we use several times that $\operatorname{UT}(3,K)$ is naturally isomorphic to the Hesienberg group $H(w) = (K \oplus K) \rtimes K$, modelled by the multiplication map $m \colon K \times K \to K$. Recall that $G=\operatorname{UT}(n,K)$ is nilpotent of class $n-1$. In fact, $$G' = \operatorname{UT}^{1}(n,K)\cong \operatorname{UT}(n-1,K)$$ as every matrix in the derived subgroup $G'$ has the first super diagonal consisting only of zero entries. For $1\leq i<j\leq n,$ let $G_{i,j}$ be the 1-parameter subgroup of $G$ such that for every matrix $X\in G_{i,j}$ we have $p_{k,l}(X)=x_{k,l}=0$ if $k\neq l$ and $(k,l)\neq (i,j),$ where $p_{k,l}\colon G \to K, \ p_{k,l}(X)=x_{k,l}$ is the canonical coordinate projection. Note that $Z(G)=G_{1,n}$. For $i<j$ and $x\in K$ consider also the *transvection matrix* $\overline{x_{i,j}}:=I+x E_{i,j}\in G_{i,j}$, where $E_{i,j}$ is the elementary matrix having $1$ on the $(i,j)$ entry and zeros elsewhere. **Lemma 17**. *Let $\overline{m}\ \in G$, $x\in K$ and $1\leq i<j<k\leq n.$ Then* 1. *$p_{i,k}([\overline{m},\overline{x_{j,k}}])=xp_{i,j}(\overline{m}).$* 2. *$p_{i,k}([\overline{x_{i,j}},\overline{m}])=-xp_{j,k}(\overline{m}^{-1}).$* 3. *$p_{n-2,n}([\overline{x_{n-2,n-1}},\overline{m}])=xp_{n-1,n}(\overline{m}).$* *Proof.* (1) For $1\leq i<j<k\leq n$ we have $$p_{i,k}([\overline{m},\overline{x_{j,k}}])= p_{i,k}(\overline{m}\cdot\overline{x_{j,k}}\cdot \overline{m}^{-1}\cdot\overline{x_{j,k}}^{-1} )=\sum_{s=1}^n p_{i,s}(\overline{m}\cdot\overline{x_{j,k}}) p_{s,k}(\overline{m}^{-1}\cdot\overline{x_{j,k}}^{-1})=$$$$=\sum_{s=1}^n\bigg(\sum_{\ell=1}^n p_{i,\ell}(\overline{m})p_{\ell,s}(\overline{x_{j,k}}) \bigg)\bigg(\sum_{t=1}^n p_{s,t}(\overline{m}^{-1})p_{t,k}(\overline{x_{j,k}}^{-1})\bigg)=$$$$= \bigg(\sum_{s\in \{1,\ldots n\}\setminus \{k\}} p_{i,s}(\overline{m}) (p_{s,k}(\overline{m}^{-1})-x p_{s,j}(\overline{m}^{-1}))\bigg)+(p_{i,k}(\overline{m})+xp_{i,j}(\overline{m})) =$$$$=\bigg(\sum_{s=1}^n ( p_{i,s}(\overline{m}) )(p_{s,k}(\overline{m}^{-1}))\bigg)-\bigg(x\sum_{s=1}^n ( p_{i,s}(\overline{m}) )(p_{s,j}(\overline{m}^{-1}))\bigg)+(xp_{i,j}(\overline{m}))=$$$$=p_{i,k}(I_n)-xp_{i,j}(I_n)+xp_{i,j}(\overline{m})=$$$$0+0+xp_{i,j}(\overline{m})=xp_{i,j}(\overline{m}).$$ (2) We have $$p_{i,k}([\overline{x_{i,j}},\overline{m}])= p_{i,k}(\overline{x_{i,j}}\cdot\overline{m}\cdot \overline{x_{i,j}}^{-1}\cdot\overline{m}^{-1} )=\sum_{s=1}^n p_{i,s}(\overline{x_{i,j}}\cdot\overline{m}) p_{s,k}(\overline{x_{i,j}}^{-1}\cdot\overline{m}^{-1})=$$$$=\sum_{s=1}^n\bigg(\sum_{\ell=1}^n p_{i,\ell}(\overline{x_{i,j}})p_{\ell,s}(\overline{m}) \bigg)\bigg(\sum_{t=1}^n p_{s,t}(\overline{x_{i,j}}^{-1})p_{t,k}(\overline{m}^{-1})\bigg)=$$$$= \bigg(\sum_{s\in \{1,\ldots n\}\setminus \{i\}} (p_{i,s}(\overline{m})+xp_{j,s}(\overline{m})) (p_{s,k}(\overline{m}^{-1})\bigg)+(p_{i,k}(\overline{m}^{-1})-xp_{j,k}(\overline{m})^{-1}) =$$$$=\bigg(\sum_{s=1}^n ( p_{i,s}(\overline{m}) )(p_{s,k}(\overline{m}^{-1}))\bigg)+\bigg(x\sum_{s=1}^n ( p_{j,s}(\overline{m}) )(p_{s,k}(\overline{m}^{-1}))\bigg)-(xp_{j,k}(\overline{m}^{-1}))=$$$$=p_{i,k}(I_n)+xp_{j,k}(I_n)-xp_{j,k}(\overline{m}^{-1})=$$$$0+0-xp_{j,k}(\overline{m}^{-1})=-xp_{j,k}(\overline{m}^{-1}).$$ \(3\) By (2), $p_{n-2,n}([\overline{x_{n-2,n-1}},\overline{m}])=-xp_{n-1,n}(\overline{m}^{-1}).$ As $\overline{m}$ is an upper unitriangular matrix, we have $p_{n-1,n}(\overline{m}^{-1})=-p_{n-1,n}(\overline{m}).$ So, we get $p_{n-2,n}([\overline{x_{n-2,n-1}},\overline{m}])=xp_{n-1,n}(\overline{m}).$ ◻ **Theorem 18**. *Let $K$ be a commutative topological unital ring and $n\geq 2$ be a positive integer. Then the center of $G=\operatorname{UT}(n,K)$ (that is, $\operatorname{UT}^{n-2}(n,K)$) is a key subgroup.* *Proof.* We proceed by induction on $n$. For $n=2$ the assertion is trivial as $\operatorname{UT}(2,K)$ is abelian. The case of $n=3$ follows from Proposition [Proposition 12](#p:AisKey){reference-type="ref" reference="p:AisKey"} and Example [Example 11](#p:s-m-bi){reference-type="ref" reference="p:s-m-bi"}.3, in view of the isomorphism $H(w)\simeq \operatorname{UT}(3,K)$, where $w \colon K\times K\to K$ is the multiplication map. Let $\gamma_1 \subseteq \gamma$ be a coarser Hausdorff group topology on $G=\operatorname{UT}(n,K),$ where $n\geq 4,$ such that $\gamma_1|_{Z(G)}=\gamma|_{Z(G)}$ and recall that $Z(G)=\operatorname{UT}^{n-2}(n,K)=G_{1,n}.$ We have to prove that $\gamma_1=\gamma.$ By the induction hypothesis, $\gamma_1|_{G'}=\gamma|_{G'}$ as $$G' = \operatorname{UT}^{1}(n,K)\cong \operatorname{UT}(n-1,K).$$ In particular, $\gamma_1|_{G_{i,i+2}}=\gamma|_{G_{i,i+2}}$ for every $i\in\{1,\ldots, n-2\}$. We have to show that $\gamma_1=\gamma$. By Merson's Lemma, it is enough to show that $\gamma_1/ {G'}=\gamma / {G'}$. To this aim, it suffices to show that all the projections $$p_{i,i+1} \colon (G,\gamma_1) \to G_{i,i+1}, \ i\in \{1,\ldots n-1\}$$ are continuous (where the one-parameter subgroups $G_{i,i+1}$ carry the original topology $\gamma|_{G_{i,i+1}}=\gamma_1|_{G_{i,i+1}}$). The idea of the multiplication map in the following claim comes from Lemma [Lemma 17](#lem:comandtrans){reference-type="ref" reference="lem:comandtrans"}.1. **Claim 1** The multiplication map $$w \colon (G_{i,i+1}, \gamma_1 / \ker p_{i,i+1})\times (G_{i+1,i+2}, \gamma_1|_{G_{i+1,i+2}}) \to (G_{i,i+2}, \gamma_1|_{G_{i,i+2}}),$$$$w(p_{i,i+1}(\overline{m}),x)=x p_{i,i+1}(\overline{m}), \ \overline{m}\in G, \ x \in G_{i+1,i+2}$$ is continuous for every $i\in \{1,\ldots, n-2\}.$ *Proof.* Fix $i\in \{2,\ldots, n-1\}$ and $(p_{i,i+1}(\overline{m}),x)\in G_{i,i+1}\times G_{i+1,i+2}.$ Let $O$ be a neighborhood of $xp_{i,i+1}(\overline{m})$ in $(G_{i,i+2}, \gamma_1|_{G_{i,i+2}}).$ By Lemma [Lemma 17](#lem:comandtrans){reference-type="ref" reference="lem:comandtrans"}.1, $$p_{i,i+2}([\overline{m},\overline{x_{i+1,i+2}}])=xp_{i,i+1}(\overline{m}).$$ As $i\leq n-2$, it holds that $G_{i,i+2}\leq G'.$ Since $\gamma_1|_{G'}=\gamma|_{G'}$ we can choose a neighborhood $W$ of $[\overline{m},\overline{x_{i+1,i+2}}]$ in $(G', \gamma|_{G'})$ such that $p_{i,i+2}(W) \subseteq O,$ in view of the continuity of the projection $$p_{i,i+2} \colon (G',\gamma|_{G'}) \to (G_{i,i+2},\gamma|_{G_{i,i+2}}).$$ Using the $\gamma_1$-continuity of the commutator function on $G$ as well as the equality $\gamma_1|_{G'}=\gamma|_{G'}$ we can find $\gamma_1$-neighborhoods $U$ and $V$ of $\overline{m}$ and $\overline{x_{i+1,i+2}}$, respectively, such that $[u,v]\in W$ for every $u\in U, \ v\in V.$ In particular, $$y p_{i,i+1}(\overline{n})=p_{i,i+2}([\overline{n},\overline{y_{i+1,i+2}}])\in p_{i+1,i+2}(W) \subseteq O$$ for every $\overline{n}\in U, \ y\in V\cap G_{i+1,i+2}$. This proves the continuity of $$w: (G_{i,i+1}, \gamma_1 / \ker p_{i,i+1})\times (G_{i+1,i+2}, \gamma_1|_{G_{i+1,i+2}}) \to (G_{i,i+2}, \gamma_1|_{G_{1,i+1}})$$ at the arbitrary pair $(p_{i,i+1}(\overline{m}),x)\in G_{i,i+1}\times G_{i+1,i+2}$. ◻ **Claim 2** The multiplication map $$w:(G_{n-2,n-1}, \gamma_1|_{G_{n-2,n-1}}) \times (G_{n-1,n}, \gamma_1 / \ker p_{n-1,n})\to (G_{n-2,n}, \gamma_1|_{G_{n-2,n}}),$$$$w(x,p_{n-1,n}(\overline{m}))=x p_{n-1,n}(\overline{m}), \ \overline{m}\in G, \ x \in G_{n-2,n-1}$$ is continuous. *Proof.* Use the $\gamma_1$-continuity of the commutator function on $G$ as well as Lemma [Lemma 17](#lem:comandtrans){reference-type="ref" reference="lem:comandtrans"}.3. On $G'$ the new topology is the same as the original topology. In particular, on $G_{n-2,n}.$ ◻ By Example [Example 11](#p:s-m-bi){reference-type="ref" reference="p:s-m-bi"}.3, the multiplication map $m:K \times K \to K$ is minimal. It follows from Claim 1 and Claim 2 that $\gamma_1 / \ker p_{i,i+1}=\gamma/ \ker p_{i,i+1}$ for every $i\in \{1,\ldots n-1\}$ and we complete the proof of the theorem. ◻ **Lemma 19**. *[@DM10 Lemma 3.5.1] Let $G_2 \leq G_1 \leq G$. If $G_2$ is co-minimal in $G$, then the bigger subgroup $G_1$ is also co-minimal in $G$.* **Lemma 20**. *Let $G$ be a topological group with two normal subgroups $G_1$ and $G_2.$ If $G_2$ is a co-minimal subgroup of $G_1$ and $G_1$ is co-minimal in $G,$ then $G_2$ is co-minimal in $G.$* *Proof.* If $G$ is a topological group with two normal subgroups $G_1$ and $G_2$ such that $G_2\leq G_1$, then $$G/G_2\cong (G/G_1)/(G_1/G_2)$$ by the third isomorphism theorem for topological groups (see [@AT Theorem 1.5.18]). Now use the co-minimality of $G_2$ in $G_1$ and the co-minimality of $G_1$ in $G.$ ◻ **Theorem 21**. *Let $K$ be a commutative topological unital ring and $n\geq 2$ be a positive integer. If the multiplication map $m \colon K\times K\to K$ is strongly minimal, then $Z(\operatorname{UT}(n,K))$ is co-minimal in $\operatorname{UT}(n,K).$* *Proof.* We prove the theorem using induction on $n$. For $n=2$ the assertion is trivial as $\operatorname{UT}(2,K)$ is abelian. The case of $n=3$ follows from Proposition [Proposition 13](#p:AisCo-Min){reference-type="ref" reference="p:AisCo-Min"}. For $G=\operatorname{UT}(n,K)$ with $n\geq 4$ we have $Z(G)\leq G''\leq G'.$ So, the induction hypothesis and Lemma [Lemma 19](#l:also co-minimal){reference-type="ref" reference="l:also co-minimal"} imply that both $G_{1,n}=Z(G)$ and $G''$ are co-minimal in $G'.$ So by Lemma [Lemma 20](#lem:chainofco){reference-type="ref" reference="lem:chainofco"} it suffices to show that $G'$ is co-minimal in $G.$ Let $\gamma_1 \subseteq \gamma$ be a coarser Hausdorff group topology on $G.$ We need to show that $\gamma_1/ {G'}=\gamma / {G'}$. Now we follow the arguments appearing in the proof of Theorem [Theorem 18](#t:1){reference-type="ref" reference="t:1"} including Claim 1 and Claim 2 with some modifications. This time we do not have the assumption that $\gamma_1|_{G'}=\gamma|_{G'}$ and so we do not know that $\gamma_1|_{G_{i,i+2}}=\gamma|_{G_{i,i+2}}$ for every $i\in\{1,\ldots, n-2\}$. Nevertheless, since the multiplication map is strongly minimal it is enough to assume that $\gamma_1|_{G_{i,i+2}}\subseteq \gamma|_{G_{i,i+2}}.$ Note also that the projections $$p_{i,i+2} \colon (G',\gamma_1|_{G'}) \to (G_{i,i+2},\gamma|_{G_{i,i+2}})$$ are continuous for every $i\in\{1,\ldots, n-2\},$ in view of the co-minimality of $G''$ in $G'.$ ◻ By a theorem of Prodanov--Stoyanov [@PS], minimal abelian groups are precompact. In particular, if $K$ is a topological ring such that the additive group $(K,+)$ is minimal, then $K$ must be precompact. In view of Lemma [Lemma 6](#lem:ceniskey){reference-type="ref" reference="lem:ceniskey"} and Theorem [Theorem 18](#t:1){reference-type="ref" reference="t:1"}, we have the following: **Proposition 22**. *Let $K$ be a commutative topological unital ring. Then the matrix group $\operatorname{UT}(n,K)$ is minimal if and only if the additive group $(K,+)$ is minimal.* **Corollary 23**. *$\operatorname{UT}(n,({\Bbb Z}, \tau_p))$ is a minimal topological group.* **Theorem 24**. *Let ${\Bbb F}$ be a local field or an archimedean absolute valued field. Then the center $Z(\operatorname{UT}(n,{\Bbb F}))$ is co-minimal in $\operatorname{UT}(n,{\Bbb F}).$* *Proof.* Theorems [Theorem 21](#t:2){reference-type="ref" reference="t:2"} and [Theorem 16](#t:newStrMin){reference-type="ref" reference="t:newStrMin"} yield this result for local fields. For archimedean absolute valued field use also Example [Example 11](#p:s-m-bi){reference-type="ref" reference="p:s-m-bi"}.4 which asserts that the multiplication map $m\colon {\Bbb F}\times {\Bbb F}\to {\Bbb F}$ is strongly minimal. ◻ # Some comments Let us say that a topological group $G$ is *center-key* if its center $Z(G)$ is a key subgroup in $G$. According to Theorem [Theorem 18](#t:1){reference-type="ref" reference="t:1"}, $\operatorname{UT}(n,K)$ is center-key. Probably, this remains true for many naturally defined groups (e.g., matrix groups). **Question 25**. *Study natural groups which are center-key.* A (Raikov complete) topological group $G$ is minimal if and only if $Z(G)$ is a minimal (resp., compact) key subgroup. In [@BadLeib], the authors show that groups from a large class of algebraic groups are minimal if and only if the center is compact. Such groups, of course, are center-key. It should be interesting to find a large class of algebraic groups (under natural fields) which are center-key (where the center is not necessarily compact). Lemma [Lemma 2](#examples1){reference-type="ref" reference="examples1"}.6 gives a natural example of a key subgroup which is not the center. Take for example, an infinite dimensional Banach space $X$ with the natural action of $G:={\Bbb R}_{+}$. Then $X$ is a key subgroup in $X \rtimes_{\alpha} {\Bbb R}_{+}$ while the center (which is trivial) is not a key subgroup because $X \rtimes_{\alpha} {\Bbb R}_{+}$ is not minimal (take the weak topology on $X$ and then the corresponding topological semidirect product will be a strictly coarser Hausdorff group topology). The following definition seems promising for further investigation. **Definition 26**. *Let $H$ be a subgroup of a topological group $(G,\tau)$. We say that $H$ is a co-key subgroup of $G$ if for every coarser Hausdorff group topology $\tau_1 \subseteq \tau$ on $G$ satisfying $\tau_1/H=\tau/H$ it holds that $\tau_1 = \tau$.* Observe that a topological group is minimal if and only if it contains a co-minimal co-key subgroup. 10 A. Arhangel'skii, M. Tkachenko, *Topological groups and related structures*, Atlantis Studies in Math. Ed.: J van Mill, World Scientific, 2008. U. Bader, E. Leibtag. *Homomorphic images of algebraic groups*, arXiv:2212.03055. D. Dikranjan, M. Megrelishvili, *Relative minimality and co-minimality of subgroups in topological groups,* Topology Appl. **157** (2010), 62--76. D. Dikranjan, M. Megrelishvili, *Minimality Conditions in Topological Groups,* in: Recent Progress in General Topology III, 229--327, K.P. Hart, J. van Mill, P. Simon (Eds.), Springer, Atlantis Press, 2014. D. Dikranjan, Iv. Prodanov, L. Stoyanov, *Topological groups: characters, dualities and minimal group topologies,* Pure and Appl. 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Shlossberg, *Minimality in topological groups and Heisenberg type groups*, Top. Proc. **35** (2010), 331--344. R.M. Stephenson, Jr., *Minimal topological groups*, Math. Ann. **192** (1971), 193--195. S. Warner, *Topological Rings*, Math. Studies **178** (1993). [^1]: This research was supported by a grant of the Israel Science Foundation (ISF 1194/19)

arxiv_math

--- abstract: | For a hyperbolic knot in $S^3$, Dehn surgery along slope $r \in \mathbb{Q}\cup \{\frac10\}$ is *exceptional* if it results in a non-hyperbolic manifold. We say meridional surgery, $r = \frac10$, is *trivial* as it recovers the manifold $S^3$. We provide evidence in support of two conjectures. The first (inspired by a question of Professor Motegi) states that there are boundary slopes $b_1 < b_2$ such that all non-trivial exceptional surgeries occur, as rational numbers, in the interval $[b_1,b_2]$. We say a boundary slope is *NIT* if it is non-integral or toroidal. Second, when there are non-trivial exceptional surgeries, we conjecture there are NIT boundary slopes $b_1 \leq b_2$ so that the exceptional surgeries lie in $[\left\lfloor b_1\right\rfloor,\left\lceil b_2\right\rceil]$. Moreover, if $\left\lceil b_1\right\rceil \leq \left\lfloor b_2\right\rfloor$, the integers in the interval $[ \left\lceil b_1\right\rceil, \left\lfloor b_2\right\rfloor ]$ are all exceptional surgeries. address: - Department of Mathematics, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan. - Department of Mathematics and Statistics, California State University, Chico, Chico, CA 95929-0525 author: - Kazuhiro Ichihara - Thomas W. Mattman title: Boundary slopes (nearly) bound exceptional slopes --- # Introduction Let $K$ be a hyperbolic knot in $S^3$. Using standard coordinates, we can identify Dehn surgery slopes with elements of $\mathbb{Q}\cup \{ \frac10 \}$, where $\frac10$ corresponds to meridional surgery. We refer to meridional surgery as *trivial* and say that $r \in \mathbb{Q}$ is a non-trivial *exceptional* surgery slope if Dehn surgery along $r$ results in a non-hyperbolic manifold. In this paper, we provide evidence in support of the following conjecture, originally stated as a question by Professor Motegi (see [@MattmanCyclic]). **Conjecture 1**. *For each hyperbolic knot in $S^3$, there exists a pair of boundary slopes $b_1, b_2$ with $b_1 < b_2$ such that all exceptional surgeries occur, as rational numbers, in the interval $[b_1,b_2]$.* Here a *boundary slope* is a slope represented by the boundary curves of an essential embedded surface in the exterior of a knot. As suggested in the earlier paper [@MattmanCyclic], it seems likely that more can be said. For the figure-eight knot, the $(-2,3,7)$-pretzel knot, the twist knots, and the $(-3,3,n)$-pretzel knots, the exceptional surgeries occur as a sequence of rational numbers bounded above and below by boundary slopes. In fact, except for the half-integer toroidal boundary slope of the $(-2,3,7)$-knot, in these examples the non-trivial exceptional slopes are a sequence of consecutive integers bounded between two boundary slopes. Others have made similar observations. Notably, Teragaito [@Teragaito] conjectured that integral exceptional surgeries occur as a sequence of consecutive integers. Dunfield [@DunfieldInvent] showed that, for small knots, if surgery along slope $r$ yields a manifold with cyclic fundamental group, then there is a non-integral boundary slope in the interval $(r-1,r+1)$. This was generalized in [@IMS] where it's shown that surgeries that result in a manifold with finite fundamental group or a Seifert fibered space are also near boundary slopes. Recently Dunfield [@DunfieldCensus] made a complete enumeration of all exceptional fillings of $1$-cusped manifolds with an ideal triangulation of nine or fewer tetrahedra. This includes data for 1267 complements of hyperbolic knots in $S^3$. Making use of this data we propose the following refinement of the earlier conjectures. We will say that a boundary slope is *NIT* if it is non-integral or toroidal. We use $\left\lfloor b\right\rfloor$ (resp., $\left\lceil b\right\rceil$) to denote the floor (resp., ceiling) of $b \in \mathbb{Q}$. If $b \in \mathbb{Z}$, $\left\lfloor b\right\rfloor = \left\lceil b\right\rceil = b$. If not, $b$ is between the consecutive integers $\left\lfloor b\right\rfloor$ and $\left\lceil b\right\rceil$: $\left\lfloor b\right\rfloor < b < \left\lceil b\right\rceil$. **Conjecture 2**. *Let $K$ be a hyperbolic knot in $S^3$ that admits non-trivial exceptional surgeries. There are (possibly equal) NIT boundary slopes $b_1 \leq b_2$ such that all exceptional surgeries occur as rational numbers in the interval $[\left\lfloor b_1\right\rfloor,\left\lceil b_2\right\rceil]$ (or in the set $\{ \left\lfloor b_1\right\rfloor \}$ if $\left\lfloor b_1\right\rfloor = \left\lceil b_2\right\rceil$) and, if $\left\lceil b_1\right\rceil \leq \left\lfloor b_2\right\rfloor$, the integers in the interval $[ \left\lceil b_1\right\rceil, \left\lfloor b_2\right\rfloor ]$ (or the set $\{ \left\lceil b_1\right\rceil \}$ in case $\left\lceil b_1\right\rceil = \left\lfloor b_2\right\rfloor$) are all exceptional surgeries.* Note that, since $b_1 \leq b_2$, $\left\lfloor b_1\right\rfloor = \left\lceil b_2\right\rceil$ implies $b_1 = b_2$ is an integer. For the reader's convenience, here is an equivalent statement of the conjecture that may be easier to parse. **Conjecture 3**. *Let $K$ be a hyperbolic knot in $S^3$ that admits non-trivial exceptional surgeries. One of the following occurs.* 1. *There are (possibly equal) integral toroidal boundary slopes $b_1 \leq b_2$. All exceptional slopes are rational numbers in the interval $[b_1, b_2]$ (or the set $\{b_1\}$ if $b_1 = b_2$) and every integer in that interval (resp., set) is exceptional.* 2. *There is a non-integral boundary slope $b_1$ and an integral toroidal boundary slope $b_2$. If $b_1 < b_2$, (resp. $b_2 < b_1$), every exceptional slope is in the interval $[ \left\lfloor b_1\right\rfloor, b_2 ]$ (resp. $[ b_2, \left\lceil b_1\right\rceil ]$) and every integer in the interval $[ \left\lceil b_1\right\rceil, b_2 ]$ (resp. $[ b_2, \left\lfloor b_1\right\rfloor ]$) is exceptional.* 3. *There are (possibly equal) non-integral boundary slopes $b_1 \leq b_2$ and the exceptional slopes are in the interval $[ \left\lfloor b_1\right\rfloor, \left\lceil b_2\right\rceil ]$. If $\left\lceil b_1\right\rceil = \left\lfloor b_2\right\rfloor$, then that integer is an exceptional surgery. If $\left\lceil b_1\right\rceil < \left\lfloor b_2\right\rfloor$, then every integer in $[\left\lceil b_1\right\rceil, \left\lfloor b_2\right\rfloor]$ is exceptional.* In Section [\[SecCensus\]](#SecCensus){reference-type="ref" reference="SecCensus"} below, we present examples showing that every case of Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} arises. However, as explained further in that section, we do have a question about part (3) of the Conjecture. **Question 1**. *Is there a knot in $S^3$ which satisfies Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} (3), but *only* with the choice $b_1 < b_2$?* Not only does Dunfield's data make these conjectures plausible, we can prove them for alternating knots, Montesinos knots, and some torti-rational knots. (See the next section for the definition of torti-rational knots.) **Theorem 1**. *Conjectures [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} hold for knots that admit a triangulation with at most seven tetrahedra.* **Theorem 2**. *For a hyperbolic alternating knot with non-trivial exceptional slopes, one of the following holds. (i) There is a toroidal boundary slope which is the only exceptional slope. (ii) There are a pair of integral toroidal boundary slopes, and the exceptional slopes for $K$ are the integers contained in the interval bounded by them. Thus Conjectures [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} hold for hyperbolic alternating knots.* **Theorem 3**. *Conjectures [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} hold for hyperbolic Montesinos knots. Moreover, for a hyperbolic Montesinos knot with non-trivial exceptional slopes, one of (1) or (2) stated in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds.* **Theorem 4**. *Let $K$ be a hyperbolic torti-rational knot $K(\beta/\alpha;n)$ with an irreducible fraction $\beta/\alpha$ and $n \ge 4$. Suppose that $K$ admits a non-trivial exceptional slopes. Then one of the following holds. (i) There is a toroidal boundary slope for $K$ which is the only non-trivial exceptional slope. (ii) There are a pair of integral toroidal boundary slopes, and the non-trivial exceptional slopes for $K$ are the integers contained in the interval bounded by them. Thus Conjectures [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} hold for the torti-rational knots.* We say that a Dehn surgery slope $r \in \mathbb{Q}$ is a *cyclic* (respectively *finite*) slope if the surgery results in a manifold having cyclic (resp. finite) fundamental group. Teragaito [@TeragaitoIsol] produced several infinite families of knots that have an integral surgery $m$ such that neither $m-1$ nor $m+1$ is exceptional. For such a knot, Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} would imply that $m$ is the only integral exceptional surgery and either $m$ is a toroidal slope, or else there are (possibly equal) non-integral boundary slopes $b_1 \leq b_2$ such that $m \in [\lfloor b_1 \rfloor, \lceil b_2 \rceil ]$. In particular, we can prove Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} for knots with a single exceptional surgery, which is either toroidal or cyclic. In addition to the infinite families of Teragaito's paper [@TeragaitoIsol], there are many examples of knots whose only non-trivial exceptional surgery is toroidal, including two-bridge knots $K_{[b_1,b_2]}$ with $|b_1|,|b_2| > 2$ and pretzel knots $P(q_1,q_2,q_3)$ with $q_j \neq 0, \pm 1$ for $j = 1,2,3$. (See the proof of Theorem 2 in Section 2.) **Theorem 5**. *Let $K$ be a hyperbolic knot in $S^3$ that has a single non-trivial exceptional surgery, which is toroidal or cyclic. Then $K$ satisfies Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"}* *Proof.* If $K$ has the toroidal slope $m$ as its unique non-trivial exceptional surgery, let $b_1 = b_2 = m$. Suppose the cyclic slope $m$ is the only non-trivial exceptional surgery. Dunfield [@DunfieldInvent] showed that there is a boundary slope $b \in (m-1,m+1)$. Let $b_1 = b_2 = b$. ◻ The figure-eight knot is well-known as the hyperbolic knot in $S^3$ that admits the largest number of exceptional surgeries, including every integer in the interval $[-4,4]$. Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} shows how this is related to the figure-eight knot being amphicheiral. **Theorem 6**. *If $K$ is an amphicheiral knot for which Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} holds and $m$ is an integral exceptional surgery slope, then $|m| \leq 4$.* *Proof.* Since $K$ is amphicheiral, we can assume $m \geq 0$. The conjecture implies the $2m+1$ integers in $[-m, m]$ are all exceptional. If $m > 4$, $K$ would have more than ten exceptional surgeries, contradicting the bound proved by Lackenby and Meyerhoff [@LackenbyMeyerhoff]. ◻ The two-bridge knots $K_{[2b,2b]}$ for $b>1$ are amphicheiral with $0$ as the unique non-trivial exceptional surgery, see the proof of Theorem 2 in Section 2. As far as we know, $0$ is the only non-trivial exceptional surgery that occurs for an amphicheiral knot in $S^3$ that is not the figure-eight knot. In [@MattmanCyclic] we showed a weak form of Conjecture [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} for cyclic surgery slopes and here we prove the analog for finite slopes. **Theorem 7**. *If $t$ is a finite surgery on a hyperbolic knot $K$ in $S^3$ and $r_m$ and $r_M$ are the least and greatest finite boundary slopes of $K$, then $r_m - \frac52 \leq t \leq r_M + \frac52$.* In Section 2, we prove Theorems [Theorem 2](#ThmAlt){reference-type="ref" reference="ThmAlt"}, [Theorem 3](#ThmMont){reference-type="ref" reference="ThmMont"}, and [Theorem 4](#ThmTorti){reference-type="ref" reference="ThmTorti"}. In Section 3, we summarize our analysis of the 1267 knot complements in the census through nine tetrahedra and prove Theorem [Theorem 1](#Thm7tet){reference-type="ref" reference="Thm7tet"}. We also give examples showing that each of the three parts of Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} can occur and discuss Question [Question 1](#queConj53){reference-type="ref" reference="queConj53"}. In Section 4, we prove Theorem [Theorem 7](#ThmFin){reference-type="ref" reference="ThmFin"}. Finally, the appendix includes tables of exceptional surgeries and relevant boundary slopes for the census knots. # Proofs for Theorems [Theorem 2](#ThmAlt){reference-type="ref" reference="ThmAlt"}, [Theorem 3](#ThmMont){reference-type="ref" reference="ThmMont"}, and [Theorem 4](#ThmTorti){reference-type="ref" reference="ThmTorti"} {#proofs-for-theorems-thmalt-thmmont-and-thmtorti} In the following, $r_{(T)}$ (resp., $r_{(S)}$) indicates that the slope $r$ is a toroidal slope (resp., a Seifert slope). *Proof.* (of Theorem [Theorem 2](#ThmAlt){reference-type="ref" reference="ThmAlt"}) Let $K$ be a hyperbolic alternating knot in $S^3$ which admits non-trivial exceptional surgeries. Then, by [@IchiharaMasai2016 Corollary 1.2.], one of the following holds. - $K$ is equivalent to the figure-eight knot, and the exceptional slopes are; $$-4_{(T)}, -3_{(S)}, -2_{(S)},-1_{(S)}, 0_{(T)}, 1_{(S)}, 2_{(S)}, 3_{(S)}, 4_{(T)}.$$ Thus (ii) in the statement of the theorem holds. - $K$ is equivalent to a two bridge knot $K_{[2n, 2]}$ (resp. $K_{[2n, -2]}$) with $| n | > 2$, and the exceptional slopes are; $$-4_{(T)}, -3_{(S)}, -2_{(S)},-1_{(S)}, 0_{(T)}. \qquad (\text{resp. } 0_{(T)}, 1_{(S)}, 2_{(S)}, 3_{(S)}, 4_{(T)} .)$$ Thus (ii) in the statement of the theorem holds. - $K$ is equivalent to a two bridge knot $K_{[b_1,b_2]}$ with $| b_1 | , | b_2 | > 2$ and both $b_1$ and $b_2$ even (resp. $b_1$ is odd and $b_2$ is even), and the exceptional slopes are $0_{(T)}$ (resp. $2 {b_2}_{(T)}$) only. Thus (i) in the statement of the theorem holds. - $K$ is equivalent to a pretzel knot $P(q_1,q_2,q_3)$ with $q_j \ne 0, \pm 1$ for $j = 1,2,3$ and $q_1,q_2,q_3$ are all odd (resp. $q_1$ is even and $q_2,q_3$ are odd), and the exceptional slopes are $0_{(T)}$ (resp. $2(q_2 + q_3)$) only. Thus (i) in the statement of the theorem holds.  ◻ In the following, $r_{(NI)}$ indicates that the slope $r$ is a non-integral boundary slope. *Proof.* (of Theorem [Theorem 3](#ThmMont){reference-type="ref" reference="ThmMont"}). Let $K$ be a hyperbolic Montesinos knot in $S^3$ admitting non-trivial exceptional surgeries. Then, by [@Wu1996 Theorem 3.6], the length of $K$ must be at most three. If $K$ is alternating, then Theorem [Theorem 2](#ThmAlt){reference-type="ref" reference="ThmAlt"} assures the statement of the theorem holds for $K$. Thus, in the following, we assume that $K$ is non-alternating, which implies the length of $K$ is three, i.e., $K = M ( p_1/q_1, p_2/q_2, p_3/q_3 )$ with $q_j \ge 2$ for $j=1,2,3$. Also, since Montesinos knots have no reducible surgeries by [@Wu1996 Corollary 2.6], each exceptional surgery slope for $K$ is a toroidal slope or a Seifert slope. There are no toroidal Seifert slopes for Montesinos knots other than the trefoil [@IchiharaJong2010 Theorem 1.1]. First, suppose that $K$ admits Seifert surgeries. Then, by [@IchiharaMasai2016 Appendix B], together with the program [@DunfieldProgram], which can enumerate the boundary slopes for a given Montesinos knot based on the algorithm developed in [@HatcherOertel1989], one of the following holds. - $K$ is equivalent to $P(-2, 3, 2n+1)$ with $n >3$, and the exceptional slopes and non-integral boundary slopes are; $$4n+6_{(S)}, 4n+6+(1/(n-1))_{(NI)}, 4n+7_{(S)}, 4n+8_{(T)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. Further, there exists boundary slopes 0 and 16 for $K$, and so, Conjecture 1 also holds. - $K$ is equivalent to $P(-2, 3, 2n+1)$ with $n < -1$, and the exceptional slopes and non-integral boundary slopes are; $$4n+6+(2/(2n+1))_{(NI)}, 4n+6_{(S)}, 4n+7_{(S)}, 4n+8_{(T)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. As the exceptional surgeries are between a NI boundary slope and a toroidal boundary slope, Conjecture 1 also holds. - $K$ is equivalent to $P(-2, 3, 7)$, and the exceptional slopes and non-integral boundary slopes are; $$16_{(T)}, 17_{(S)}, 18_{(S)}, 18+1/2_{(T)}, 19_{(S)}, 20_{(T)}.$$ Thus (1) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $P(-3, 3, 3)$, and the exceptional slopes are; $$0_{(T)}, 1_{(S)}, 2_{(T)}.$$ Thus (1) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $P(-3, 3, 4)$, and the exceptional slopes and non-integral boundary slopes are; $$0_{(T)}, 1_{(S)}, 8/5_{(NI)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $P(-3, 3, 5)$, and the exceptional slopes and non-integral boundary slopes are; $$0_{(T)}, 1_{(S)}, 4/3_{(NI)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $P(-3, 3, 6)$, and the exceptional slopes and non-integral boundary slopes are; $$0_{(T)}, 1_{(S)}, 8/7_{(NI)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $M(-1/2, 1/3, 2/5)$, and the exceptional slopes and non-integral boundary slopes are; $$8/3_{(NI)}, 3_{(S)}, 4_{(S)}, 5_{(S)}, 6_{(T)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $M(-1/2, 1/3, 2/7)$, and the exceptional slopes and non-integral boundary slopes are; $$-2_{(T)}, -1_{(S)}, 0_{(S)}, 1_{(S)}, 3/2_{(NI)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $M(-1/2,1/3,2/9)$, and the exceptional slopes and non-integral boundary slopes are; $$3/2_{(NI)}, 2_{(S)}, 3_{(S)}, 4_{(S)}, 5_{(T)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $M(-1/2, 1/3, 2/11)$, and the exceptional slopes and non-integral boundary slopes are; $$-3_{(T)}, -2_{(S)}, -1_{(S)}, 0_{(T)}.$$ Thus (1) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $M(-1/2, 1/5, 2/5)$, and the exceptional slopes and non-integral boundary slopes are; $$32/5_{(NI)}, 7_{(S)}, 8_{(S)}, 9_{(T)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $M(-1/2, 1/7, 2/5)$, and the exceptional slopes and non-integral boundary slopes are; $$72/7_{(NI)}, 11_{(S)}, 12_{(T)}.$$ Thus (2) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $M(-2/3, 1/3, 2/5)$, and the exceptional slopes and non-integral boundary slopes are; $$-6_{(T)}, -5_{(S)}, -4_{(T)}.$$ Thus (1) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. Next, we consider the case that $K$ has only toroidal slopes. If there is a single toroidal slope, then (1) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. The remaining cases are that $K$ has at least two toroidal slopes. Then, by [@Wu2011 (3) in page 308], $K$ admits exactly two toroidal surgeries, and $K$ is equivalent to one of five knots. Three of the five already appear in the above list. The remaining are the following. - $K$ is equivalent to $P(-3,3,7)$ and the exceptional (i.e., toroidal) slopes are; $$0_{(T)}, 1_{(T)}.$$ Thus (1) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. - $K$ is equivalent to $M(-2/3, 1/3, 1/4)$, and the exceptional (i.e., toroidal) slopes are; $$12_{(T)}, 13_{(T)}.$$ Thus (1) in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} holds. This completes the proof. ◻ Let $L_{\beta/\alpha} = K_1 \cup K_2$ be a two-bridge link in $S^3$ associated to an irreducible fraction $\beta/\alpha$. After performing Dehn surgery on the component $K_2$ along the slope $-1/n$ for a positive integer $n$, i.e., after twisting $n$ times along $K_2$, the component $K_1$ becomes a knot in $S^3$. We call such a knot a *torti-rational knot* and denote it by $K(\beta/\alpha;n)$. See [@Eudave-Munozetal2021; @HirasawaMurasugi2010] for properties of torti-rational knots. We will show Theorem [Theorem 4](#ThmTorti){reference-type="ref" reference="ThmTorti"}: certain hyperbolic torti-rational knots satisfy Conjectures [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"}. In the following, let $[a_1,a_2, \dots, a_n]$ denote the continued fraction expansion of a rational number as follows. $$[a_1,a_2, \dots, a_n] = \frac{1}{a_1 + \frac{1}{a_2 + {\dots + \frac{1}{a_n}}}}$$ We follow the convention for two-bridge links used in [@Ichihara2012], which is different (actually, the mirror image) of the standard one used in many knot theory textbooks. See Figure [\[L33\]](#L33){reference-type="ref" reference="L33"} for example. For two-bridge links associated to continued fractions, the following holds by [@KawauchiBook Theorem 2.1.11]. (We remark that the two-bridge link in their notation is the mirror image of ours, but the result still holds.) **Lemma 1**. *Let $[a_1, a_2, \dots, a_n]$ and $[b_1,b_2,\dots, b_m]$ be continued fractions such that all $a_i$'s and $b_j$'s are positive and none of $| a_1 | , | a_n | , | b_1 | , | b_m |$ is one. Then the corresponding two-bridge links $L_{[a_1, a_2, \dots, a_n]}$ and $L_{[b_1, b_2, \dots, b_m]}$ are equivalent (as unoriented links) by an orientation-preserving homeomorphism of $S^3$ if and only if $m = n$ and either $a_i = b_i$ or $a_i = \epsilon b_{m-i}$ with $\epsilon =(-1)^{m-1}$ for all $i$.* We also remark that a two-bridge link $L_{[a_1, a_2, \dots, a_n]}$ is equivalent to $L_{[a_n, a_{n-1}, \dots, a_1]}$ if $n$ is odd (respectively, $L_{[-a_n, -a_{n-1}, \dots, -a_1]}$ if $n$ is even) by an orientation-preserving homeomorphism of $S^3$. Now Theorem [Theorem 4](#ThmTorti){reference-type="ref" reference="ThmTorti"} follows from the following two lemmas. **Lemma 2**. *Let $K = K(\beta/\alpha;n)$ be a hyperbolic torti-rational knot with $n \ge 4$. If a slope $r$ is an exceptional slope for $K$, then the slope $r' = r - n l^2$ is a Seifert slope or a toroidal slope for one component of the hyperbolic two-bridge link $L_{\beta/\alpha} = K_1 \cup K_2$, where $l = lk (K_1, K_2)$. If a slope $r'$ is a Seifert slope (respectively, a toroidal slope) for one component of a hyperbolic two-bridge link $L_{\beta/\alpha} = K_1 \cup K_2$, then $r = r' + n l^2$ is a Seifert slope (resp., toroidal slope) for $K = K(\beta/\alpha;n)$, where $l = lk (K_1, K_2)$.* *Proof.* For $\beta/\alpha$ and $n \ge 4$, let $K = K(\beta/\alpha;n)$ be the torti-rational knot and $L_{\beta/\alpha} = K_1 \cup K_2$ the two-bridge link with $l = lk (K_1, K_2)$. We assume both are hyperbolic. Given a slope $r$ for $K$, we note that the manifold $K(r)$ is also obtained by Dehn surgery on the two-bridge link $L_{\beta/\alpha}$ along the slope $(r', -1/n)$. That is, $K(r) \cong L_{\beta/\alpha} (r',-1/n)$, where $r' = r - n l^2$. For calculation of surgery slopes, see [@RolfsenBook] for example. Suppose first that a slope $r$ is an exceptional slope for $K$. Then, by the result of [@IchiharaJongMasai2019] together with the assumption $n \ge 4$, if 1) $L_{\beta/\alpha} = K_1 \cup K_2$ is hyperbolic, 2) $K$ is hyperbolic, and 3) $K(r) \cong L_{\beta/\alpha} (r',-1/n)$ is non-hyperbolic; then $L_{\beta/\alpha} (r',*)$ is also non-hyperbolic, where $*$ means that $K_2$ remains unfilled and $r' = r - n l^2$. That is, the slope $r'$ is exceptional for a component of $L_{\beta/\alpha} = K_1 \cup K_2$. Such an exceptional slope must be a Seifert slope or a toroidal slope by [@Ichihara2012 Theorem 1.1]. Suppose next that a slope $r'$ is a Seifert slope or a toroidal slope for one component, say $K_1$, of a hyperbolic two-bridge link $L_{\beta/\alpha} = K_1 \cup K_2$. As noted above, $L_{\beta/\alpha} (r',-1/n) \cong K(r)$ holds, where $r = r' + n l^2$. If $r'$ is a Seifert slope, then, by the proof of [@Ichihara2012 Theorem 1.1], $K_2$ becomes a non-trivial non-core torus knot $K'$ in a lens space after $r'$-surgery on $K_1$. This shows that $K(r)$ is obtained by some non-integral surgery on $K'$ and is a small Seifert fibered manifold. If $r'$ is a toroidal slope, then, by the proof of [@Ichihara2012 Theorem 1.1], $K_2$ becomes a non-trivial satellite knot $K'$ in a lens space after $r'$-surgery on $K_1$. Moreover the companion knot of $K_2$ is a torus knot in the lens space, and the pattern of $K'$ is either hyperbolic or a $(2,k)$-cable for some integer $k$. If the pattern of $K$ is a $(2,k)$-cable for some $k$, then $-1/n$-surgery on $K'$ is toroidal for $n \ge 4$ by [@Gordon83 Lemma 7.2]. On the other hand, if the pattern of $K$ is hyperbolic, then $-1/n$-surgery on $K'$ is toroidal for $n \ge 4$ by [@CGLS Theorem 2.0.1], since $1/0$-surgery on $K'$ compresses the satellite torus for $K'$. ◻ **Lemma 3**. *The exceptional slopes of a component, $K$, of a hyperbolic two-bridge link $L$ occur in one of two ways. Either (i) there is a toroidal boundary slope which is the only non-trivial exceptional slope for $K$, or else (ii) there are a pair of integral toroidal boundary slopes, and the non-trivial exceptional slopes for $K$ are the integers in the interval they bound.* *Proof.* Let $L_{\beta/\alpha} = K_1 \cup K_2$ be a hyperbolic two-bridge link. First suppose that there is a Seifert slope $r$ for one component, say $K_1$, of $L_{\beta/\alpha}$. Then, by [@Ichihara2012 Theorem 1.1], the link $L_{\beta/\alpha}$ is equivalent to $L_{[2w+1,2u+1]}$ for $w \ge 1$, $u \ne 0,-1$. There are several cases[^1] depending on the values of $w$ and $u$. If $w \ge 2$ and $u \ne 1,0,-1,-2$, then $L_{[2w+1,2u+1]}$ admits a single Seifert slope $r = -w + u$ on $K_1$ by [@Ichihara2012 Theorem 1.1]. For the case of $w \ge 2$ and $u \ge 2$, by calculating the continued fractions, we see that $[2w+1,2u+1] = [2w,1,2(-u-1)] = [2(w+1),-1,-2u]$. Then, again by [@Ichihara2012 Theorem 1.1], the slopes $-w - (-u-1) = -w+u+1$ and $-(w+1) - (-u) = -w+u-1$ are both toroidal slopes on $K_1 \subset L_{[2w+1,2u+1]}$. That is, $L_{\beta/\alpha} (-w+u+1,*)$ and $L_{\beta/\alpha} (-w+u-1,*)$ both contain essential tori. To complete the proof for the case of $w \ge 2$ and $u \ge 2$, we need to show that there are no other exceptional (i.e., toroidal) slopes for $K_1 \subset L_{[2w+1,2u+1]}$ in this case. We use the following claim that can be shown by direct calculation of continued fraction expansions; we omit the details. **Claim 1**. *Suppose that $w = 1, u = -1, |v| \ge 2$ or $w \ge 2, |u| \ge 2, |v| \ge 1$. Then the simple continued fraction (i.e., all terms positive) for $[2w,v,2u]$ is as follows. $$[2w,v,2u] = \begin{cases} [2w,v,2u] \ %(\mbox{simple continued fraction}) & \mbox{ if } v,u >0 \\ [2w,1,2u]= [2w+1, -2u-1] & \mbox{ if } v=1,u \le -2 \\ [2w,v,-2]= [2, v-1,2] & \mbox{ if } v \ge 2, u =-1, (w=1)\\ [2w,v-1,1,-2u-1] & \mbox{ if } v \ge 2, u \le -2 \\ [2w,-1,2u] = [2w-2,1,2u-2] & \mbox{ if } v = -1, u \ge 2 , (w \ge 2) \\ %[2w,-2,2] = [2w-1,3] & \mbox{ if } v = -2, u = 1 \\ [2w,-2,2u]=[2w-1,2,2u-1] & \mbox{ if } v = -2, u \ge 2, (w \ge 2) \\ %[2w,v,2] = [2w-1,1,-v-2,2] & \mbox{ if } v \le -3, u = 1, (w \ge 2) \\ [2w-1,1,-v-2,1,2u-1] & \mbox{ if } v \le -3, u \ge 2 , (w \ge 2) \\ [2w,-1,2u] = [2w-1,-2u+1] & \mbox{ if } v = -1, u \le -2 , (w \ge 2) \\ [1,1,-v-1,2] & \mbox{ if } v \le -2, u = -1, (w=1)\\ [2w-1,1,-v-1,-2u] & \mbox{ if } v \le -2, u \le -2, (w \ge 2) \end{cases}$$* Assume for a contradiction that $K_1$ admits some other toroidal surgery. This implies that $L_{[2w+1,2u+1]}$ with $w \ge 2$ and $u \ge 2$ admits a toroidal slope on one of its components. Then, by [@Ichihara2012 Theorem 1.1], $L_{[2w+1,2u+1]}$ is equivalent to $L_{[2w',v',2u']}$ with $w' = 1$, $u' = -1$, $|v'| \ge 2$, or $w' \ge 2$, $|u'| \ge 2$, $|v'| \ge 1$. As already seen, $[2w+1,2u+1] = [2w,1,2(-u-1)] = [2(w+1),-1,-2u]$ holds, and so, $(w',v',u')=(w,1,-u-1)$ or $(w',v',u')=(w+1,-1,-u)$ actually happen, but no other cases can happen by Claim [Claim 1](#Clm1){reference-type="ref" reference="Clm1"} together with Lemma [Lemma 1](#Lem1){reference-type="ref" reference="Lem1"}, except for the case $(w',v',u')=(1,v,-1)$ with $v \le -2$. For this case, we see that $$\begin{aligned} L_{[2w+1,2u+1]} & \cong L_{[-2w-1,-2u-1]} \\ & \not\cong L_{[-2,v+1,-2]} \cong L_{[-2,v+1,-1,-1]} \cong L_{[1,1,-v-1,2]} \cong L_{[2,v,-2]}\end{aligned}$$ by Lemma [Lemma 1](#Lem1){reference-type="ref" reference="Lem1"}, where $\cong$ denotes the equivalence of knots by orientation-preserving homeomorphism of $S^3$. This implies that $L_{[2w+1,2u+1]}$ with $w \ge 2$ and $u \ge 2$ only admits toroidal slopes $-w+u-1$ and $-w+u+1$ on $K_1$. That is, the exceptional slopes for a component of the link are; $$-w+u-1_{(T)} , -w+u_{(S)} , -w+u+1_{(T)}.$$ For the case of $w \ge 2$ and $u \le -3$, we can argue similarly. If $L_{[2w+1,2u+1]}$ with $w \ge 2$ and $u \le -3$ admits a toroidal slope on a component, $K$, then $L_{[2w+1,2u+1]}$ is equivalent to $L_{[2w',v',2u']}$ with $w' = 1$, $u' = -1$, $|v'| \ge 2$, or $w' \ge 2$, $|u'| \ge 2$, $|v'| \ge 1$ by [@Ichihara2012 Theorem 1.1]. Since $[2w+1,2u+1]$ has the simple continued fraction $[2w,1,-2u-2]$, the possible cases are only $(w',v',u') =(w,1,-u-1)$ or $(w',v',u') =(w+1,-1,-u)$ by Claim [Claim 1](#Clm1){reference-type="ref" reference="Clm1"} together with Lemma [Lemma 1](#Lem1){reference-type="ref" reference="Lem1"}. In these cases, by [@Ichihara2012 Theorem 1.1], $-w' -u' = -w - (-u-1) = -w+u+1$ and $-w'-u'= -(w+1) - (-u) = -w+u-1$ are both toroidal slopes on the component $K$ of $L_{[2w+1,2u+1]}$ with $w \ge 2$ and $u \le -3$. This implies that $L_{[2w+1,2u+1]}$ with $w \ge 2$ and $u \le -3$ only admits toroidal slopes $-w+u-1$ and $-w+u+1$ on the component $K$. That is, the exceptional slopes for a component of the link are; $$-w+u-1_{(T)} , -w+u_{(S)} , -w+u+1_{(T)}.$$ There remain the cases where $w=1$ or $u=1$ or $u=-2$. In fact, these can be treated simultaneously, for $$L_{[3,2u+1]} \cong L_{[-2u-1,-3]} \cong ( L_{[2w+1,3]} )^*$$ holds by setting $u=w$, and $$L_{[3,2u+1]} \cong L_{[-2u-1,-3]} \cong L_{[2w+1,-3]}$$ holds by setting $u=-w-1$. Thus we only consider the link $L_{[3,2u+1]}$ with $u \ne 0,-1$. Although we can apply the same arguments as above, in this case, we may use a simpler argument due to [@MartelliPetronio2006]. That's because, in this case, $L_{[3,2u+1]}(r,*)$ is homeomorphic to $N(-\frac{1}{u+1},r-u-1)$, where $N$ is the exterior of the minimally twisted 3-chain link, that is, the so-called "magic manifold" [@MartelliPetronio2006]. See Figure [\[L3MT3C\]](#L3MT3C){reference-type="ref" reference="L3MT3C"}. Suppose that $L_{[3,2u+1]}(r,*)$ is non-hyperbolic and $L_{[3,2u+1]}$ is hyperbolic. Note that $u \ne 0,-1$ by the assumption that $L_{[3,2u+1]}$ is hyperbolic. It follows that $-\frac{1}{u+1}$ is not an integer. Then, by [@MartelliPetronio2006 Theorems 1, 2], we have the following cases. If $r-u-1 \in \{ -3,-2,-1,0 \}$, i.e., $r = u-2,u-1,u,$ or $u+1$, then $L_{[3,2u+1]}(r,*)$ is non-hyperbolic. Moreover, by [@MartelliPetronio2006 Tables 1, 2], if $u \ne -2,1$, the exceptional slopes for a component of $L_{[3,2u+1]}$ are; $$u-2_{(T)} , u-1_{(S)} , u_{(S)}, u+1_{(T)}.$$ In the case of $u=-2$, if $L_{[3,-3]}(r,*)$ is non-hyperbolic, then $r=0$ can occur, in addition to $r \in \{ -4,-3,-2,-1 \}$. Thus, by [@MartelliPetronio2006 Tables 1, 2], the exceptional slopes for a component of $L_{[3,-3]}$ are; $$-4_{(T)} , -3_{(S)} , -2_{(S)}, -1_{(S)}, 0_{(T)}.$$ Actually $L_{[3,-3]}$ is the Whitehead link, for which these results are well-known. See [@MartelliPetronio2006 Tables A.1]. In the case of $u=1$, if $L_{[3,3]}(r,*)$ is non-hyperbolic, then $r=-2$ can occur, in addition to $r \in \{ -1,0,1,2 \}$. Thus, by [@MartelliPetronio2006 Tables 1, 2], the exceptional slopes for a component of $L_{[3,3]}$ are; $$-2_{(T)} , -1_{(S)} , 0_{(S)}, 1_{(S)}, 2_{(T)}.$$ Actually $L_{[3,3]} = L_{3/10}$ is also a well-known link. See [@MartelliPetronio2006 Tables A.1]. Next suppose that there are only toroidal slopes for one component, say $K_1$, of $L_{\beta/\alpha}$. We will show that $K_1$ admits only one toroidal slope. In this case, by [@Ichihara2012 Theorem 1.1], the link $L_{\beta/\alpha}$ is equivalent to $L_{[2w,v,2u]}$ with $w = 1$, $u = -1$, $|v| \ge 2$, or $w \ge 2$, $|u| \ge 2$, $|v| \ge 1$. And suppose further that the knot admits another toroidal slope. Then, in the same way, $L_{\beta/\alpha}$ is equivalent to $L_{[2w',v',2u']}$ with $w' = 1$, $u' = -1$, $|v'| \ge 2$, or $w' \ge 2$, $|u'| \ge 2$, $|v'| \ge 1$, and $(w,v,u) \ne (w',v',u')$. Now, by Claim [Claim 1](#Clm1){reference-type="ref" reference="Clm1"} together with Lemma [Lemma 1](#Lem1){reference-type="ref" reference="Lem1"}, it suffices to consider the 10 cases each for $(w,v,u)$ and $(w',v',u')$. However, by Claim [Claim 1](#Clm1){reference-type="ref" reference="Clm1"} together with Lemma [Lemma 1](#Lem1){reference-type="ref" reference="Lem1"}, it is unnecessary to consider cases where the lengths of their simple continued fraction expansions are different unless $v \le -2, u = -1, w = 1$. We look at a limited number of cases in the following. The remaining cases can be shown in the same way. Consider the case that $u,v>0$. Then $[2w,v,2u]$ is a simple continued fraction. If the simple continued fraction associated to $[2w',v',2u']$ is coincident to $[2w,v,2u]$, then we have $(w',v',u')=(w+1,-1,u+1)$ and $v=1$ by Claim [Claim 1](#Clm1){reference-type="ref" reference="Clm1"}. In this case, $[2w,1,2u]=[2w+1,-2u-1]$ also holds. This implies that $L_{[2w,1,2u]} \cong L_{[2(w+1),-1,2(u+1)]} \cong L_{[2w+1,-2u-1]}$ admits a Seifert slope $r=-w+(-u-1)=-w-u-1$ on one component, contradicting that $K_1$ has only toroidal surgeries. The cases $v=1, u \le -2$, $v =-1, u \ge 2$ and $v = -1, u \le -2$ are similar. Consider the case that $v \ge 2, u =-1, (w=1)$. Then $[2w,v,2u]$ has the simple continued fraction $[2,v-1,2]$. Under the conditions for $(w',v',u')$, there are no $[2w',v',2u']$ with the simple continued fraction $[2,v-1,2]$ by Claim [Claim 1](#Clm1){reference-type="ref" reference="Clm1"}. The cases $v =-2, u \ge 2$, $v \ge 2, u\le -2$ and $v \le -3, u \ge 2$ are similar. Consider the case that $v \le -2, u = -1, (w = 1)$. The toroidal slope for a component of $L_{[2,v,-2]}$ is $-1-(-1)=0$. Then $[2w,v,2u]=[2,v,-2]$ ($v \le -2$) has the simple continued fraction $[1, 1, -v-1, 2]$. $$L_{[2,v,-2]} \cong L_{[1,1,-v-1,2]} \cong L_{[-2,v+1,-1,-1]} \cong L_{[-2,v+1,-2]}$$ Thus the link is the mirror image of $L_{[2,-v-1,2]}$ with $v \le -2$. Under the conditions for $(w',v',u')$, there are no $[2w',v',2u']$ with the simple continued fraction $[2,-v-1,2]$ by Claim [Claim 1](#Clm1){reference-type="ref" reference="Clm1"}. Consequently, if there are only toroidal slopes for a component of $L_{\beta/\alpha}$, then, in fact, there is a single toroidal slope. ◻ # Census knots [\[SecCensus\]]{#SecCensus label="SecCensus"} In this section we provide evidence for Conjectures [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} using Dunfield's census of exceptional Dehn fillings [@DunfieldCensus] on knots with complements that have at most nine tetrahedra. We prove Theorem [Theorem 1](#Thm7tet){reference-type="ref" reference="Thm7tet"} that Conjectures [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} hold for knots of at most seven tetrahedra. We give examples that realize each of the three parts of Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} and ask if knots that satisfy part (3) of the conjecture can always do so by way of a single non-integral boundary slope $b_1 = b_2 = b$. Dunfield's census includes the complements of 1267 hyperbolic knots in $S^3$. Of these, 165 have no non-trivial exceptional surgeries and 418 are such that all non-trivial exceptional surgeries are toroidal. In each case, the toroidal surgeries occur as a list of consecutive integers of length between one and three and, therefore, both conjectures are verified for these knots. The remaining knot complements in the census have at least one non-trivial, non-toroidal exceptional surgery. For those with at most seven tetrahedra, the $A$-polynomial calculations of Culler [@CullerApoly] provide a substantially complete list of boundary slopes. (There's no guarantee that the $A$-polynomial detects all boundary slopes. However, in practice, it does seem to find almost all of them; for example see [@Segerman] for a detailed account.) This allows us to prove Theorem [Theorem 1](#Thm7tet){reference-type="ref" reference="Thm7tet"}. (Although it would be enough to rely on the $A$-polynomial calculations, in the data presented in the appendix, for the sake of convenience we also used boundary slopes found using Dunfield's program for two-bridge and Montesinos knots [@DunfieldProgram] and 'Kabaya' boundary slopes found with SnapPy [@SnapPy].) *Proof.* (of Theorem [Theorem 1](#Thm7tet){reference-type="ref" reference="Thm7tet"}) Among the census manifolds with at most seven tetrahedra, there are 201 that are complements of hyperbolic knots in $S^3$. In the appendix we list exceptional and boundary slopes for each of these knot complements, which show that both Conjectures [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} hold. ◻ Note that the complements of hyperbolic knots in $S^3$ with at most seven tetrahedra have names that begin with 'm', 's', or 'v'. The appendix lists all toroidal boundary slopes, but only enough of the other boundary slopes to verify the conjectures. Let us give the details for two examples as an illustration. $$\begin{aligned} \mbox{v0319} \quad & [(-62, \mbox{`T'}), -63, -64, (-194/3, \mbox{`C'}), -65, (-206/3, \mbox{`C'})] \\ & [(-2, \mbox{`T'}), -1, 0, (2/3, \mbox{`C'}), 1, (14/3, \mbox{`C'})] \\ \\ \mbox{v1359} \quad & [(121/2, \mbox{`CK'}), 59, (176/3, \mbox{`C'}), 58, (57, \mbox{`T'})] \\ & [(-5/2, \mbox{`CK'}), -1, (-2/3, \mbox{`C'}), 0, (1, \mbox{`T'})]\end{aligned}$$ Using Dunfield's census and SnapPy coordinates, the exceptional slopes of v0319 are $-2,-1,0,1$, with $-2$ toroidal. For Conjecture [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"}, use $b_1 = -2$ and $b_2 = 14/3$, a slope that we identified using Culler's [@CullerApoly] calculation of the $A$-polynomial. For Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"}, use $b_1 = -2$ and $b_2 = 2/3$, another slope found using the $A$-polynomial. This knot has other boundary slopes besides $-2$, $2/3$, and $14/3$. Here we only mention slopes that are toroidal or useful for verifying the conjectures. For v1359, the exceptional slopes are, $-1,0,1$, and we have $b_2 = 1$, a toroidal slope, for both conjectures. For Conjecture [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"}, $b_1 = -5/2$, a slope identified both using Culler's [@CullerApoly] $A$-polynomial calculation and as a Kabaya boundary slope by SnapPy [@SnapPy]. For Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"}, $b_1 = -2/3$, a slope found using the $A$-polynomial. Similar to Theorem [Theorem 1](#Thm7tet){reference-type="ref" reference="Thm7tet"}, for two bridge knots, which are alternating, or Montesinos knots, we know the boundary slopes by [@HatcherThurston], [@HatcherOertel1989], and [@DunfieldProgram] and we can verify both conjectures for these knots as in Theorems [Theorem 2](#ThmAlt){reference-type="ref" reference="ThmAlt"} and [Theorem 3](#ThmMont){reference-type="ref" reference="ThmMont"}. In other words, in all cases where we have reasonably complete information about the boundary slopes, we can verify both conjectures. Finally, there are many examples of knots where we can verify our two conjectures directly as the set of nontrivial exceptional slopes is bounded between two toroidal boundary slopes. There remain the 336 knot complements that have: a non-trivial, non-toroidal exceptional surgery; a triangulation with eight or nine tetrahedra; are not two bridge or Montesinos; and either the greatest or the least exceptional slope is not toroidal. We have verified both conjectures for many of these knots. However there remain 84 (respectively, 190) knot complements for which we have not been able to verify Conjecture [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} (resp., Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"}). Of the 84 outstanding knots for Conjecture [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} more than twenty satisfy Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"}, leaving 62 knot complements for which we can verify neither conjecture. While there remain many outstanding knot complements for each conjecture, we expect that this mainly reflects our ignorance about the boundary slopes for these knots. To reiterate, in every case where we have substantial knowledge about the boundary slopes, we were able to verify both conjectures. Let's observe that each part of Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} does, in fact, arise. As mentioned in the introduction, two bridge knots $K_{[a_1,a_2]}$ with $|a_1|,|a_2| > 2$ and pretzel knots $P(q_1,q_2,q_3)$ with $q_j \neq 0,\pm 1$ for $j = 1,2,3$ are examples where the only non-trivial exceptional surgery $b$ is toroidal. This means we can choose $b_1 = b_2 = b$ for these knots, as in Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} (1). Examples that realize (1) with with $b_1 < b_2$, both toroidal, include the twist knots $K_{[2n,2]}$ and $K_{[2n,-2]}$ with $|n| > 2$. The pretzel knots $P(-2,3,2n+1)$ with $n > 3$ or $n < -1$ provide examples for Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} (2) with $b_1$ non-integral, $b_2$ toroidal, and $b_1 < b_2$. Examples that realize Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} (3) include s682, s769, v1300, v1628, v1940, v1966, v2217, v2759, v2871, v2925, and v3234. Of these, seven knots: s682, v1300, v1628, v2217, v2759, v2871, and v3234 *could* be explained with a choice of non-integral boundary slopes $b_1 < b_2$. However, these seven also satisfy Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} (3) thanks to a single non-integral slope $b$ with $b_1 = b_2 = b$. For example, s682 has exceptional slopes $-1,0$ and boundary slopes $-3/2$ and $-1/3$. It satisfies Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} both with the choice $b_1 = -3/2$, $b_2 = -1/3$, as well as with $b_1 = b_2 = -1/3$. All the examples that we know of for Conjecture [Conjecture 3](#Conj5){reference-type="ref" reference="Conj5"} (3) can be shown to satisfy the conjecture by the choice of a single non-integral boundary slope $b_1 = b_2 = b$. This is the source of Question [Question 1](#queConj53){reference-type="ref" reference="queConj53"} mentioned in the introduction. In the appendix we provide a listing of the 1267 census knots with their exceptional slopes and relevant boundary slopes. The exceptional slopes come from Dunfield's census. There are several sources for boundary slopes: toroidal slopes, which are exceptional slopes, from Dunfield's census; boundary slopes deduced from Culler's A-polynomial calculations [@CullerApoly]; boundary slopes of two bridge [@HatcherThurston] or Montesinos knots [@HatcherOertel1989]; Kabaya boundary slopes as determined by SnapPy [@SnapPy; @Kabaya]; and the longitude, which is the boundary of a Seifert surface. # Finite slopes In this section we prove Theorem [Theorem 7](#ThmFin){reference-type="ref" reference="ThmFin"}. Let $r_m$ (respectively, $r_M$) denote the smallest (resp., largest) finite boundary slopes, considered as rational numbers. Conjecture [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} states that all non-trivial exceptional surgeries must occur in the interval $[r_m, r_M]$. As progress towards this conjecture, the second author [@MattmanCyclic] showed that all cyclic slopes occur in the interval $(r_m- \frac12, r_M + \frac12)$. Here, we use a similar approach to show that all finite slopes occur in the interval $[r_m - \frac52, r_M + \frac52]$. The argument is similar to that in [@MattmanCyclic] and leverages properties of the Culler-Shalen norm. We refer the reader to that paper for a brief introduction to the relevant properties. See [@CGLS] or [@Boyer] for a more detailed overview. A finite slope is either an integer $n$ or a half integer $\frac{n}{2}$ [@BoyerZhangFiniteFilling]. In their proof of the Finite Filling Conjecture, Boyer and Zhang [@BoyerZhangFiniteFilling] show that the Culler-Shalen norm of a finite slope is bounded by $\max (2s, s+8)$ where $s$ represents the minimal norm. For a hyperbolic knot, $s \geq 4$ [@BoyerZhangFiniteFilling Lemma 9.1], so that $3s$ is always a bound for the norm of a finite filling slope. Note that it follows from [@IMS], that a finite surgery $n \in \mathbb{Z}$ is in $[r_m - 3, r_M + 3]$, while a half-integer slope $\frac{n}{2}$ is in $[r_m - \frac32, r_M + \frac32]$, see [@MattmanCyclic Theorem 1]. In the proof of Theorem [Theorem 7](#ThmFin){reference-type="ref" reference="ThmFin"} below, we show that a half integer finite slope is, in fact, in $[r_m -1, r_M + 1]$. *Proof.* (of Theorem [Theorem 7](#ThmFin){reference-type="ref" reference="ThmFin"}) As in [@MattmanCyclic], the argument comes down to showing: if $r_\gamma > 0$ is a finite slope, then $r_\gamma \leq r_M + \frac52$. We split into two cases depending on whether the finite slope is integer or half-integer. Suppose first that the positive integer $n$ is a finite surgery slope and $\| (n,1) \| = t \leq 3s$. We will argue that $n \leq r_M+\frac52$. For a contradiction, suppose instead $n > r_M+\frac52$. at 42 8 at 250 8 $\frac{s}{t}(r_M,1)$ at 169 82 $\frac{s}{t}(n,1)$ at 212 82 $(r_M,1)$ at 163 97 $V$ at 157 133 ![The triangle $T$ is shaded. [\[figT\]]{#figT label="figT"} ](figT.pdf "fig:"){#figT} Let $P$ denote the fundamental polygon of the Culler-Shalen norm, the set of points of minimal norm $s$. Let $\| \mu \| = \| (1,0) \| = m$. Then $\pm (s/m, 0)$ are points on the boundary of the fundamental polygon $P$ as is the point $\frac{s}{t}(n,1)$. Choose coordinates so that $(r_M,1)$ appears on the $y$-axis. Extend the line through $(s/m,0)$ and $\frac{s}{t}(n,1)$ until it intersects the $y$-axis at $V$, which will be a vertex of $P$. (See Figure [1](#figT){reference-type="ref" reference="figT"}. As in [@MattmanCyclic], we can assume that $P$ has a vertex that corresponds to $r_M$.) By convexity, the segment $(-s/m, 0)$ to $V$ is in $P$ as is the isosceles triangle $T$ with vertices $\pm (s/m,0)$ and $V$. We will argue that the intersection of $T$ with the line $y = 1$ is a segment whose length exceeds one. This implies there is at least one lattice point interior to $T \subset P$, which is a contradiction. Note that if $r_M$ is an integer, then $(r_M,1)$ is itself a lattice point interior to $P$. Let $w(y)$ denote the width of $T$ at height $y$. Note that $w(0) = 2s/m$ and $w(s/t) = 2 \frac{s}{t}( n - r_M)$. Then, since $w(y)$ is a linear function of $y$, $$\begin{aligned} w(1) = & \frac{2s}{m} - \frac{\frac{2s}{m} - \frac{2s}{t} ( n - r_M) }{\frac{s}{t}} \\ = & \frac{2s}{m}(1 - \frac{t}{s}) + 2 (n - r_M) \\ = & 2(n-r_M) - \frac{2}{m} (t-s),\end{aligned}$$ which will be larger than 1. Indeed, $s \leq t \leq 3s$ and $s \leq m$ imply that $\frac{2}{m}(t-s)$ is at most four and we are assuming $2(n - r_M)$ is larger than five. Since $P$ is the polygon defined by the minimal norm $s$, an interior lattice point would be a slope of norm strictly less than the minimal norm, which is absurd. Next assume $\frac{n}{2}$ is a finite slope so that $\| (n,2) \| = t \leq 3s$. The argument is similar where we replace $\frac{s}{t} (n,1)$ as a point on the polygon with $\frac{s}{t}(n,2)$. However, in this case we can argue that $\frac{n}{2} \leq r_M + 1$. Assume, for a contradiction that $\frac{n}{2} > r_M + 1$. Again $w(0) = 2s/m$ and $w(2s/t) = \frac{2s}{t}(n - r_M)$. Then, $$\begin{aligned} w(1) = & \frac{2s}{m} - \frac{\frac{2s}{m} - \frac{2s}{t} ( n - 2r_M) }{\frac{2s}{t}} \\ = & \frac{2s}{m}(1 - \frac{t}{2s}) + (n - 2r_M) \\ = & (n-2r_M) - \frac{t-2s}{m}.\end{aligned}$$ Since $s \leq t \leq 3s$ and $s \leq m$, then $\frac{t-2s}{m}$ is at most one. But $\frac{n}{2} - r_M > 1$ means there will be a lattice point inside the fundamental polygon, which is a contradiction. ◻ # Appendix: Slopes of Census Knots In the appendix we list exceptional and boundary slopes for the 1267 hyperbolic knots in $S^3$ included in Dunfield's [@DunfieldCensus] census. We provide the data in three `.csv` files. Among these knot complements, 165 have no non-trivial exceptional surgeries and 418 are such that all non-trivial exceptional surgeries are toroidal. These 583 knots are gathered in the file `TorOnly.csv`. For each knot, we list the census name, the (possibly empty) list of toroidal slopes both in standard and SnapPy coordinates (as reported in Dunfield's census), and the name of the knot, if we know it. We remark that Baker and Kegel [@BakerKegelpaper; @BakerKegeldata] have determined braid words for every one of the 1267 census knots. As discussed in Section [\[SecCensus\]](#SecCensus){reference-type="ref" reference="SecCensus"}, these knots satisfy both Conjecture [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"}. The remaining census knots each have at least one non-trivial non-toroidal exceptional slope. Of these, the first group of 348 knots presented in the file `Verified348.csv`, are those for which we can easily verify both conjectures. Many of these knots have toroidal slopes as the largest and smallest non-trivial exceptional slopes that we can use as our $b_1$ and $b_2$. For the rest we have substantially complete knowledge of the boundary slopes for one of two reasons: first, if the knot is two bridge or Montesinos, the boundary slopes are determined by Hatcher and Thurston [@HatcherThurston] and Hatcher and Oertel [@HatcherOertel1989]; and second, if the knot has at most seven tetrahedra, we can extract the boundary slopes from Culler's [@CullerApoly] calculation of the $A$-polynomial. For each knot, we list the census name, the list of exceptional and boundary slopes both in standard and SnapPy coordinates, and the name of the knot, if we know it. We include all toroidal boundary slopes, which are exceptional slopes, but otherwise only provide sufficient boundary slopes to verify our two conjectures. We label each boundary slope with one or more certificates C, K, L, M, or T: C if we use Culler's $A$-polynomial calculation, K for Kabaya boundary slopes as determined by SnapPy [@SnapPy; @Kabaya] , L for the longitude, which is the boundary of a Seifert surface, M for boundary slopes of Montesinos or two bridge knots [@DunfieldProgram], and T for the toroidal slopes, which are exceptional slopes identified in Dunfield's census. As discussed in Section [\[SecCensus\]](#SecCensus){reference-type="ref" reference="SecCensus"}, both Conjectures [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"} and [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"} hold for these knots as we can verify using our substantially complete knowledge of the boundary slopes. For a few examples in this and the final `.csv` file, we do not provide the SnapPy coordinates, instead repeating the standard coordinates. These are the knots for which SnapPy does not identify $S^3$ surgery with $\frac10$ and instead uses some other rational number, typically $\frac01$ or $\pm \frac11$, as the trivial slope. This leaves 336 knots, tabulated in `Remaining336.csv`, which have triangulations of eight or nine tetrahedra and non-trivial, non-toroidal exceptional slopes. For many of these, we were not able to verify one or both of our conjectures. For this reason, we have added three columns to the data, indicating whether we have verified Conjecture [Conjecture 1](#Conj1){reference-type="ref" reference="Conj1"}, Conjecture [Conjecture 2](#Conj6){reference-type="ref" reference="Conj6"}, or at least one of the two conjectures. Some of these knots have only one exceptional surgery meaning we cannot determine the linear transformation from SnapPy coordinates to standard coordinates. Even in cases where we found SnapPy coordinates for boundary slopes of these knots, we cannot report the corresponding standard coordinates for those boundary slopes. Note that taking the mirror reflection of a knot changes the sign of all exceptional and boundary slopes. We do not distinguish between a knot and its reflection in our tables of surgery slopes. 99 K. L. Baker and M. Kegel, Census $L$-space knots are braid positive, except for one that is not. to appear in Algebr. Geom. Topol. (preprint) arxiv.org 2203.12013 K. L. Baker and M. Kegel, Code and data to accompany: Census $L$-space knots are braid positive, except for one that is not. 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Dunfield, Program to compute the boundary slopes of a 2-bridge or Montesinos knot, Available online at <http://www.computop.org> N. M. Dunfield, A census of exceptional Dehn fillings, in *Characters in low-dimensional topology*, 143--155, Contemp. Math., 760, Centre Rech. Math. Proc, Amer. Math. Soc.,, RI. MR4193924 N. M. Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. **136** (1999), no. 3, 623--657. MR1695208 M. Eudave-Muñoz,  F. Manjarrez-Gutiérrez, E. Ramı́rez-Losada,  and J. Rodrı́guez-Viorato, Computing genera of satellite tunnel number one knots and torti-rational knots, Osaka J. Math. **58** (2021), no. 2, 383--411. MR4301321 H. Goda, C. Hayashi and H.-J. Song, Dehn surgeries on 2-bridge links which yield reducible 3-manifolds, J. Knot Theory Ramifications **18** (2009), no. 7, 917--956. MR2549476 C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. **275** (1983), no. 2, 687--708. MR0682725 A. Hatcher and U. Oertel, Boundary slopes for Montesinos knots, Topology **28** (1989), no. 4, 453--480. MR1030987 A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements. Invent. Math. **79** (1985), no. 2, 225--246. MR0778125 M. Hirasawa and K. Murasugi, Fibered torti-rational knots, J. Knot Theory Ramifications **19** (2010), no. 10, 1291--1353. MR2735812 K. Ichihara, Exceptional surgeries on components of 2-bridge links, Arch. Math. (Basel) **99** (2012), no. 1, 71--79. MR2948663 K. Ichihara and I. D. Jong, Toroidal Seifert fibered surgeries on Montesinos knots, Comm. Anal. Geom. **18** (2010), no. 3, 579--600. MR2747439 K. Ichihara , I. D. Jong and H. Masai, Complete exceptional surgeries on two-bridge links, to appear in Kodai Math. J. (preprint, arXiv:1909.11319). K. Ichihara and H. Masai, Exceptional surgeries on alternating knots, Comm. Anal. Geom. **24** (2016), no. 2, 337--377. MR3514563 M. Ishikawa, T.W. Mattman and K. Shimokawa, Exceptional surgery and boundary slopes. Osaka J. Math. **43** (2006), no. 4, 807--821. MR2303551 Y. Kabaya, A method to find ideal points from ideal triangulations. J. Knot Theory Ramifications **19** (2010), no. 4, 509--524. MR2646644 A. Kawauchi, *A survey of knot theory*, translated and revised from the 1990 Japanese original by the author, Birkhäuser Verlag, Basel, 1996. MR1417494 M. Lackenby and R. Meyerhoff, The maximal number of exceptional Dehn surgeries. Invent. Math. **191** (2013), no. 2, 341--382. MR3010379 B. Martelli and C. Petronio, Dehn filling of the "magic" 3-manifold, Comm. Anal. Geom. **14** (2006), no. 5, 969--1026. MR2287152 T.W. Mattman, Boundary slopes (nearly) bound cyclic slopes, Algebr. Geom. Topol. **5** (2005), 741--750. MR2153109 D. Rolfsen, *Knots and links*, Math. Lecture Ser., No. 7, Publish or Perish, Inc., Berkeley, CA, 1976. ix+439 pp. MR0515288 H. Segerman, A generalisation of the deformation variety, Algebr. Geom. Topol. **12** (2012), no. 4, 2179---2244. MR3020204 M. Teragaito, Abstract for September 2006 meeting of MSJ. M. Teragaito, Isolated exceptional Dehn surgeries on hyperbolic knots, Topology Appl. **156** (2009), no. 6, 1148--1152. MR2493375 Y.-Q. Wu, Dehn surgery on arborescent knots, J. Differential Geom. **43** (1996), no. 1, 171--197. MR1424423 Y.-Q. Wu, The classification of toroidal Dehn surgeries on Montesinos knots, Comm. Anal. Geom. **19** (2011), no. 2, 305--345. MR2835882 [^1]: There is a typo in the statement of [@Ichihara2012 Theorem 1.1]. There, (b) should be $L_{[2w+1,-3]}$ since the proof of Claim 4 in the paper, which states the corresponding part of the theorem, depends essentially on [@GodaHayashiSong2009 Theorem 11.1(2)]. That theorem says that $L([2w + 1, -3])[-w - 1]$ gives a torus knot in a lens space. However, since $L([2w + 1, -3])[-w - 1]$ is homeomorphic to $L([3, 2(-w-1) + 1)])[-w - 1]$, case (b) is actually included in case (a). Thus we will ignore it in this paper.

arxiv_math

--- abstract: | We consider the block Bregman-Kaczmarz method for finite dimensional linear inverse problems. The block Bregman-Kaczmarz method uses blocks of the linear system and performs iterative steps with these blocks only. We assume a noise model that we call *independent noise*, i.e. each time the method performs a step for some block, one obtains a noisy sample of the respective part of the right-hand side which is contaminated with new noise that is independent of all previous steps of the method. One can view these noise models as making a fresh noisy measurement of the respective block each time it is used. In this framework, we are able to show that a well-chosen adaptive stepsize of the block Bergman-Kaczmarz method is able to converge to the exact solution of the linear inverse problem. The plain form of this adaptive stepsize relies on unknown quantities (like the Bregman distance to the solution), but we show a way how these quantities can be estimated purely from given data. We illustrate the finding in numerical experiments and confirm that these heuristic estimates lead to effective stepsizes. author: - "Lionel Tondji[^1], Idriss Tondji[^2], Dirk Lorenz[^3]" bibliography: - ref.bib title: "Adaptive Bregman-Kaczmarz: An Approach to Solve Linear Inverse Problems with Independent Noise Exactly" --- **Keywords:** Randomized Bregman-Kaczmarz method, adaptive stepsize, inverse problems **AMS Classification:** 65F10, 15A29, 65F20, # Introduction In a finite dimensional linear inverse problem, we have a linear map $\mathbf{A}$ that represents an indirect measurement of some unknown quantity $\hat{x}$. Since the real world measurements always contain noise, one never actually sees $b = \mathbf{A}x$, but always a noisy version of this. The aim is now, to obtain an approximation to $\hat{x}$ only from the knowledge of the noisy data and the linear map $\mathbf{A}$ (see [@engl1996regularization; @mueller2021]). One specific class of iterative methods are row-action or block-action methods that only use single rows or blocks of rows of the linear operator (represented as a matrix) for each iteration. Probably the oldest such method is the Kaczmarz method [@Kac37] that is also known as the algebraic reconstruction technique [@gordon1970algebraic]. In this work, we consider a generalization of the Kaczmarz method, namely the Bregman-Kaczmarz method [@lorenz2014linearized; @LWSM14; @P15] which is a method to solve minimization problems $$\label{eq:PB} \hat x \overset{\text{def}}{=}\operatorname*{arg min}_{x \in \mathbb{R}^n} f(x) \quad \text{subject to} \quad \mathbf{A}x=b,$$ for a strongly convex function $f$ which is finite everywhere. Due to strong convexity, this problem has a unique solution if the system $\mathbf{A}x=b$ is consistent. Our main result in this article is that we show that it is possible to calculate the *exact* solution of [\[eq:PB\]](#eq:PB){reference-type="eqref" reference="eq:PB"} even if one never sees the noise-free right-hand side $b$, but each time that one queries a row or block of the matrix, one obtains the respective entry or block of entries of the right-hand side, contaminated by independent noise. ## Problem statement {#sec:prob-statment} We give a more formal statement of the problem: 1. We are given a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ and a strongly convex function $f:\mathbb{R}^{n}\to\mathbb{R}$. 2. The matrix is partitioned into $M$ blocks of rows, i.e. $$\begin{aligned} \mathbf{A} = \begin{pmatrix} \mathbf{A}_{(1)}\\\vdots\\\mathbf{A}_{(M)} \end{pmatrix} \end{aligned}$$ with $\mathbf{A}_{(i)}\in\mathbb{R}^{m_{i}\times n}$ where, without loss of generality, we assume that the blocks consist of consecutive rows which can always be achieved by reordering the rows. 3. Assume that there exists "true data $b\in\mathbb{R}^{m}$" such that the system $\mathbf{A}x = b$ is consistent, we aim to compute the minimum-$f$-solution of $\mathbf{A}x = b$, i.e. the unique solution $\hat{x}$ of $$\begin{aligned} \mathop{\mathrm{argmin}}_{x\in\mathbb{R}^{n}} f(x) \quad \text{subject to}\quad \mathbf{A}x = b.\end{aligned}$$ 4. We are not given $b$, but each time we query the $i$-th block of $b$, i.e. $b_{(i)}\in\mathbb{R}^{m_{i}}$ we get a perturbation $\tilde{b}_{(i)}\in\mathbb{R}^{m_{i}}$ which is $\tilde{b}_{(i)} = b_{(i)} + \varepsilon_{(i)}$ where $\varepsilon_{(i)}$ is a random vector with zero mean and variance $\sigma_{i}^{2}$, i.e. $$\begin{aligned} \mathbb{E}\left[\varepsilon_{(i)}\right] = 0,\quad \text{and}\quad \mathbb{E}\left[\lVert\varepsilon_{(i)}\rVert^{2}\right] = \sigma_{i}^{2}. \end{aligned}$$ The noise model in the last point is different from other noise models where it is often assumed that a single fixed noisy measurement $b^{\delta}$ is given which fulfills a deterministic error condition $\lVert b-b^{\delta}\rVert\leq \delta$. We call our random noise model *independent noise*. Practically this means that each time we consider the $i$-th block of the right-hand side, we have made a new fresh measurement of $\hat{x}$ with the $i$-th block $\mathbf{A}_{(i)}$ of the measurement matrix $\mathbf{A}$. We denote by $$\begin{aligned} \sigma \overset{\text{def}}{=}(\sum_{i=1}^{M}\sigma_{i}^{2})^{1/2}\end{aligned}$$ the *total noise level*. We also use the notation $$\begin{aligned} \lVert\mathbf{A}\rVert_{\square} \overset{\text{def}}{=}\left( \sum\limits_{i=1}^{M}\lVert\mathbf{A}_{(i)}\rVert_2^2 \right)^{1/2}\end{aligned}$$ for the blockwise mixture of spectral and Frobenius norm (indeed for $M=1$ we have $\lVert\mathbf{A}\rVert_{\square} = \lVert\mathbf{A}\rVert_{2}$ and for $M=m$ we have $\lVert\mathbf{A}\rVert_{\square} = \lVert\mathbf{A}\rVert_{F}$). The strongly convex function $f$ is used to resolve the problem of non-uniqueness in the case that $n>m$. It is also used to impose prior knowledge on the solution, e.g. one can promote sparsity of the solution by using $f(x) = \lambda\lVert x\rVert_{1} + \lVert x\rVert_{2}^{2}$ for $\lambda>0$ [@LWSM14]. ## The Bregman-Kaczmarz method {#sec:bregman-kaczmarz} The Bregman-Kaczmarz method for the solution of [\[eq:PB\]](#eq:PB){reference-type="eqref" reference="eq:PB"} with a fixed given right-hand side $\tilde{b}$ and sampling of single rows of $\mathbf{A}$ works as follows: Given an initialization $x_{0}^{*}\in\mathbb{R}^{n}$, compute $x_{0} = \nabla f^{*}(x_{0}^{*})$ (where $f^{*}$ denotes the convex conjugate of $f$, see [@rockafellar1970]) and with a given sequence $\eta_{k}$ of stepsizes choose in each iteration a (random) row-index $i$ and perform the update $$\begin{aligned} \label{eq:noisy_kaczmarz} x^{*}_{k+1} &= x^{*}_k - \eta_k \frac{\langle a_{i}, x_k \rangle - \Tilde{b}_{i}}{\|a_{i}\|^2_2} \cdot a_{i} \\ x_{k+1} &= \nabla f^*(x^*_{k+1}), \end{aligned}$$ The method is a special case of the (relaxed) randomized Bregman projections [@lorenz2014linearized]. In this paper, we consider a block version where one chooses a random *block* $\mathbf{A}_{(i)}$ from the given partition of $\mathbf{A}$. The choice of a block is random in each iteration and for simplicity we assume that the probability $p_{i}$ to choose the $i$-th block is proportional to the square of the spectral norm of the block matrix, i.e. $p_{i} = \lVert\mathbf{A}_{(i)}\rVert^{2}/\lVert\mathbf{A}\rVert_{\square}^{2}$. The resulting method is detailed in Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"}. Recall that $\tilde{b}_{(i)}$ is a noisy version of $b_{(i)}$ according to the model of independent noise as described in the problem statement in Section [1.1](#sec:prob-statment){reference-type="ref" reference="sec:prob-statment"}. initialize $k=0$ choose a block index $i_k=i\in \left\{ 1,\dots, M \right\}$ with probability $p_i = \frac{\|\mathbf{A}_{(i)}\|^2_2}{\lVert\mathbf{A}\rVert_{\square}^2}$. get a pertubation of the $i$-th block of $b$, $\tilde{b}_{(i)} = b_{(i)} + \varepsilon_{(i)}$ with $\mathbb{E}\left[\varepsilon_{(i)}\right] = 0, \mathbb{E}\left[\lVert\varepsilon_{(i)}\rVert^{2}\right] = \sigma_{i}^{2}.$ update $x^{*}_{k+1} = x^{*}_k - \eta_k \frac{\mathbf{A}_{(i)}^{\top} (\mathbf{A}_{(i)}x_k - \Tilde{b}_{(i)})}{\|\mathbf{A}_{(i)}\|^2_2}$ update $x_{k+1} = \nabla f^*(x^*_{k+1})$ increment $k = k+1$ $x_{k+1}$ ## Related work #### Randomized Kaczmarz. In a large data regime, full matrix operations can be expensive or even infeasible. Then it appears desirable to use iterative algorithms with low computational cost and storage per iteration that produce good approximate solutions of [\[eq:PB\]](#eq:PB){reference-type="eqref" reference="eq:PB"} after relatively few iterations. The Kaczmarz method [@Kac37] and its randomized variants [@SV09; @gower2015randomized] are used to compute the minimum $\ell_2$-norm solutions of consistent linear systems. In each iteration $k$, a row vector $a_i^\top$ of $\mathbf{A}$ is chosen at random from the system $\mathbf{A}x = b$ and the current iterate $x_k$ is projected onto the solution space of that equation to obtain $x_{k+1}$. It has been observed that the convergence of the randomized Kaczmarz (RK) method can be accelerated by introducing relaxation. In a relaxed variant of RK, a step is taken in the direction of this projection with the size of the step depending on a relaxation parameter. Explicitly, the relaxed RK update is given by $$\label{eq:RK} x_{k+1} = x_k - \eta_{k} \frac{\langle a_i, x_k \rangle - b_i}{\|a_i\|^2_2} \cdot a_i$$ with initial values $x_0 = 0$, where the $\eta_{k} \in (0,2)$ is the stepsize. This update rule requires low cost per iteration and storage of order $\mathcal{O}(n)$. The first convergence rate for RK was given in [@SV09] for consistent system and $\eta_k=1$. To further improve the efficiency of the RK method, the idea of parallelism was used, such as in  [@gower2019adaptive; @moorman2021randomized; @necoara2019faster]. The basic idea is to use multiple rows in each step of the iteration. This will increase the cost of each step of the iteration, but it can reduce the number of iterations as expected. In addition to the above work, there are many studies aimed at accelerating the (randomized) Kaczmarz method by the use of block strategies, sampling schemes, or extrapolation; for example, see  [@bai2018greedy; @eldar2011acceleration; @haddock2021greed; @liu2016accelerated; @miao2022greedy; @needell2014paved; @steinerberger2021weighted]. Needell, in [@needell2010randomized] extended the result of [@SV09] to the inconsistent case when only a noisy right-hand side $\Tilde{b}$ is given where $\Tilde{b} = b + \varepsilon$ with arbitrary noise vector $\varepsilon$ such that $|\varepsilon_i| \leq \sigma \|a_i\|$. They proved that in the case $\eta_k = 1$, the iterates in Eq [\[eq:RK\]](#eq:RK){reference-type="eqref" reference="eq:RK"} are expected to reach an error threshold in the order of the noise-level with the same rate as in the noiseless case, cf [@needell2010randomized; @SV09]. An approach was just recently proposed by Haddock et al. [@haddock2022quantile] to recover the true solution, with a high likelihood as long as the right-hand side has sufficiently small corrupted entries, for a certain class of random matrices. They considered the setting where a fraction of the entries have been corrupted (possibly by arbitrarily large numbers). Their result was generalized for any matrix in [@steinerberger2023quantile] and for sparse solutions in [@zhang2022quantile]. Just recently, Marshall et al. in [@marshall2023optimal] proposed an approach that is complementary to the ones in [@haddock2022quantile; @steinerberger2023quantile; @zhang2022quantile]. Instead of considering the setting where a small fraction of the right-hand side entries have been corrupted, they allow for the corruption of all elements of $b$ but assume that corruptions are independent zero-mean random variables so that the noisy right-hand side $\Tilde{b}$ is fixed. They showed that we can recover the solution $\hat x$ to any accuracy if we have access to a sufficient number of equations with independent noise, thanks to the use of an adaptive stepsize. However, their results hold for only one pass over the full matrix, i.e. the number of rows of the matrix, given the fact that they are sampling without replacement. Therefore with this restriction is not possible in practice to recover the true solution $\hat x$ after one epoch only. In the context of inverse problems, Kaczmarz-type methods are often used for non-linear problems [@haltmeier2007kaczmarz; @kaltenbacher2008] but also linear inverse problems have been considered, e.g. in [@elfving2014semi; @jiao2017preasymptotic] and in Hilbert space in [@lu2022stochastic]. #### Randomized sparse Kaczmarz. The randomized Kaczmarz method has been adapted to the randomized sparse Kaczmarz method (RSK) [@LWSM14; @LS19] which has almost the same low cost and storage requirements and has shown good performance in approximating sparse solutions of large consistent linear systems. It belongs to a more general family of methods called Bregman-Kaczmarz (BK) methods. The Bregman-Kaczmarz [@10178390] uses two variables $x^*_{k}$ and $x_k$ and its update is given by : $$\begin{aligned} \label{eq:Bregman_kaczmarz} x^{*}_{k+1} &= x^{*}_k - \eta_k \frac{\langle a_{i}, x_k \rangle - b_{i}}{\|a_{i}\|^2_2} \cdot a_{i} \\ x_{k+1} &= \nabla f^*(x^{*}_{k+1}) \end{aligned}$$ with initial values $x_0 = \nabla f^*(x^*_0)$. The papers [@lorenz2014linearized; @LS19] analyze this method by interpreting it as a sequential, randomized Bregman projection method (where the Bregman projection is done with respect to the function $f$). Variations of RSK including block/averaging variants [@du2020randomized; @P15], averaging methods [@tondji2022faster] and adaptions to least squares problems are given in [@schopfer2022extended; @zouzias2013randomized]. In those variants, one usually needs to have access to more than one row of the matrix $\mathbf{A}$ at the cost of increasing the memory. It has been shown in [@LS19] that for a consistent system $\mathbf{A}x=b$ with an arbitrary matrix $\mathbf{A}$ the iterates $x_k$ of the sparse Kaczmarz $(\eta_k = 1)$ converge linearly to the unique solution $\hat x$ of the regularized Basis Pursuit problem, which is Eq [\[eq:Bregman_kaczmarz\]](#eq:Bregman_kaczmarz){reference-type="eqref" reference="eq:Bregman_kaczmarz"} with $f(x):= \lambda \cdot \|x\|_1 + \tfrac{1}{2}\|x\|_2^2$. More precisely under the assumption that the row index $i_k$ at iteration $k$ is chosen randomly with probability proportional to $\|a_{i_k}\|^2$ they proved that : $$\label{eq:linear_conv} \mathbb{E}[\|x_k -\hat{x}\|^2_2] \leq 2(1 - q)^k f(\hat x)$$ where $q \in (0,1)$. For more details, we refer the reader to [@LS19] and an acceleration for update [\[eq:Bregman_kaczmarz\]](#eq:Bregman_kaczmarz){reference-type="eqref" reference="eq:Bregman_kaczmarz"} was given in [@10178390] for any objective function $f$. In [@LS19], they also consider the case of a noisy linear system: instead of having access to the right-hand side of the consistent linear system $\mathbf{A}x = b$, we are given $\Tilde{b} = b + \varepsilon$, where the entries of $\varepsilon$ satisfy $|\varepsilon_i| \leq \sigma \|a_i\|$. Under these assumptions, they proved that the sparse Kaczmarz with $\eta_k = 1$ satisfies $$\label{eq:linear_conv_noisy} \mathbb{E}[\|x_k -\hat{x}\|^2_2] \leq 2(1 - q)^k f(\hat x) + \frac{\sigma^2}{q}$$ that is, we converge in expectation until we reach some ball of radius $\frac{\sigma}{\sqrt{q}}$ around the solution. ## Contribution In this paper, we make the following contributions: - We show that an adaptive stepsize similar to the one proposed in [@marshall2023optimal] can also be applied in the case of the Bregman-Kaczmarz setting. - We extend the analysis to the block case. - We prove that the method with adaptive stepsize does indeed converge to the true solution if the data obey the independent noise assumption. - We illustrate that this adaptive stepsize can be implemented in practice by using a heuristic similar to the one from [@marshall2023optimal], that estimates the needed hyperparameter only from given data and shows that this heuristic works well in practice. - We investigate the convergence of the method with adaptive stepsize in detail and show that it leads to linear convergence of the iterates at the beginning and an $\mathcal{O}(1/k)$ rate of convergence later on. ## Outline The remainder of the paper is organized as follows. Section [2](#sec:basicnotions){reference-type="ref" reference="sec:basicnotions"} provides notations and a brief overview on convexity and Bregman distances. We state error bound conditions that hold for our objective function. Section [3](#sec:convergence_analysis){reference-type="ref" reference="sec:convergence_analysis"} provides convergence guarantees for our proposed method which depend on some hyperparameters. In Section [4](#sec:heuristic){reference-type="ref" reference="sec:heuristic"}, we derive heuristics estimation of those hyperparameters. In Section [5](#sec:numerical_experiment){reference-type="ref" reference="sec:numerical_experiment"}, numerical experiments demonstrate the effectiveness of our method and provide insight regarding its behavior and its hyperparameters. Finally, Section [6](#sec:conclusion){reference-type="ref" reference="sec:conclusion"} draws some conclusions. # Notation and basic notions {#sec:basicnotions} ## Convex analysis {#sec:convex-analysis} We will analyze the convergence of the Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with the help of the Bregman distance with respect to the objective function $f$. To this end we recall some well-known concepts and properties of convex functions. Let $f:\mathbb{R}^n \to \mathbb{R}$ be convex (note that we assume that $f$ is finite everywhere, hence it is also continuous). The *subdifferential* of $f$ at any $x \in \mathbb{R}^n$ is defined by $$\partial f(x) \overset{\text{def}}{=}\{x^* \in \mathbb{R}^n| f(y) \ge f(x) + \langle x^*, y-x \rangle, \forall\, y \in \mathbb{R}^n \},$$ which is nonempty, compact and convex. The function $f:\mathbb{R}^n \to \mathbb{R}$ is said to be *$\alpha$-strongly convex*, if for all $x,y \in \mathbb{R}^n$ and subgradients $x^* \in \partial f(x)$ we have $$f(y) \geq f(x) + \langle x^*, y-x \rangle + \tfrac{\alpha}{2} \cdot \|y-x\|_2^2 \,.$$ If $f$ is $\alpha$-strongly convex, then $f$ is coercive, i.e. $$\lim_{\|x\|_2 \to \infty} f(x)=\infty \,,$$ and its *Fenchel conjugate* $f^{*}:\mathbb{R}^n \to \mathbb{R}$ given by $$f^*(x^*)\overset{\text{def}}{=}\sup_{y \in \mathbb{R}^n} \langle x^* \,,\, y\rangle - f(y)$$ is also convex, finite everywhere and coercive. Additionally, $f^*$ is differentiable with a *Lipschitz-continuous gradient* with constant $L_{f^*}=\frac{1}{\alpha}$, i.e. for all $x^*,y^* \in \mathbb{R}^n$ we have $$\|\nabla f^*(x^*)-\nabla f^*(y^*)\|_2 \le L_{f^*} \cdot \|x^*-y^*\|_2 \,,$$ which implies the estimate $$\begin{aligned} \label{eq:Lip} f^*(y^*)\! \le \!f^*(x^*)\! -\!\langle \nabla f^*(x^*) \,,\, y^*-x^*\rangle\! +\! \tfrac{L_{f^*}}{2}\|x^{*}-y^{*}\|_2^2. %\nonumber\end{aligned}$$ [\[def:D\]]{#def:D label="def:D"} The *Bregman distance* $D_f^{x^*}(x,y)$ between $x,y \in \mathbb{R}^n$ with respect to $f$ and a subgradient $x^* \in \partial f(x)$ is defined as $$D_f^{x^*}(x,y) \overset{\text{def}}{=}f(y)-f(x) -\langle x^* \,,\, y - x\rangle\,.$$ Fenchel's equality states that $f(x) + f^*(x^*) = \langle x \,,\, x^*\rangle$ if $x^*\in\partial f(x)$ and implies that the Bregman distance can be written as $$D_f^{x^*}(x,y) = f^*(x^*)-\langle x^* \,,\, y\rangle + f(y)\,.$$ (cf. [@LS19 Example 2.3]) [\[exmp:f\]]{#exmp:f label="exmp:f"} The objective function $$\label{eq:spf} f(x) \overset{\text{def}}{=}\lambda\|x\|_1 + \tfrac{1}{2}\|x\|_2^{2}$$ is strongly convex with constant $\alpha=1$ and its conjugate function can be computed with the soft shrinkage operator $S_{\lambda}$ which is defined componentwise by $(S_{\lambda}(x))_j = \max\{|x_j|-\lambda,0\} \cdot \mathop{\mathrm{sign}}(x_j)$ $$f^{*}(x^{*}) = \tfrac{1}{2}\|S_{\lambda}(x^{*})\|_2^{2}, \quad \mbox{with} \quad \nabla f^{*}(x^{*}) = S_{\lambda}(x^{*}) \,.$$ For any $x^*=x+\lambda \cdot s \in \partial f(x)$ we have $$D_f^{x^*}(x,y)=\tfrac{1}{2}\|x-y\|_2^2 + \lambda \cdot(\|y\|_1-\langle s \,,\, y\rangle) \,.$$ which give us $D_f^{x^*}(x,y)=\tfrac{1}{2}\|x-y\|_2^2$ for $\lambda = 0$. The following inequalities are crucial for the convergence analysis of the randomized algorithms. They immediately follow from the definition of the Bregman distance and the assumption of strong convexity of $f$, see [@lorenz2014linearized]. For $x,y \in \mathbb{R}^n$ and $x^* \in \partial f(x)$ we have $$\begin{aligned} \label{eq:D} D_f^{x^*}(x,y) \geq \frac{\alpha}{2} \|x-y\|_2^2, \end{aligned}$$ and hence $D_f^{x^*}(x,y) = 0 \Leftrightarrow x=y.$ ## Error bounds for linearly constrained optimization problems {#sec:error_bound} Considering the feasible, convex and linearly constrained optimization problem [\[eq:PB\]](#eq:PB){reference-type="eqref" reference="eq:PB"}. To obtain convergence rates for the solution algorithm, we will estimate the Bregman distance of the iterates to the solution $\hat x$ by error bounds of the form $D_f^{x^*}(x,\hat x) \leq \theta^{-1} \cdot \|\mathbf{A}x - b\|^2_2$. We will see that such error bounds always hold if $f$ has a Lipschitz-continuous gradient. But they also hold under weaker conditions. For example problem [\[eq:PB\]](#eq:PB){reference-type="eqref" reference="eq:PB"} with objective function defined by eq [\[eq:spf\]](#eq:spf){reference-type="eqref" reference="eq:spf"}, it holds that : $$D_f^{x^*}(x,\hat x) \leq \theta(\hat x)^{-1} \cdot \|\mathbf{A}x - b\|^2_2,$$ we refer the reader to [@schopfer2022extended Lemma 3.1] for more details on the constant $\theta(\hat x)$. To clarify the assumptions under which such error bounds hold for more general objective functions, we are going to define notions [@schopfer2022extended] such as calmness, linearly regularity, and linear growth. Let $B_2$ denote the closed unit ball of the $\ell_2$-norm. The (set-valued) subdifferential mapping $\partial f:\mathbb{R}^n \rightrightarrows \mathbb{R}^n$ is *calm* at $\hat{x}$ if there are constants $\varepsilon,L>0$ such that $$\partial f(x) \subset \partial f(\hat{x}) + L \cdot \|x-\hat{x}\|_2 \cdot B_2 \quad \mbox{for any $x$ with} \quad \|x-\hat{x}\|_2\le \varepsilon\,. \label{eq:calm}$$ Note that calmness is a local growth condition akin to Lipschitz-continuity of a gradient mapping, but for fixed $\hat x$. Of course, any Lipschitz-continuous gradient mapping is calm everywhere. [\[ex:calmness\]]{#ex:calmness label="ex:calmness"} - The subdifferential mapping of any convex piecewise linear quadratic function is calm everywhere. In particular, this holds for the function defined in eq [\[eq:spf\]](#eq:spf){reference-type="eqref" reference="eq:spf"}. - The subdifferential mapping of $f(x) = \lambda \|x\|_2 + \tfrac{1}{2} \|x\|_2^{2}$ is calm everywhere. For more examples on calmness, we refer the reader to [@schopfer2022extended]. Let $\partial f(x) \cap \mathcal{R}(\mathbf{A}^{\top}) \neq \emptyset.$ Then the collection $\{\partial f(\hat x), \mathcal{R}(\mathbf{A}^{\top})\}$ is linearly regular, if there is a constant $\zeta >0$ such that for all $x^{*} \in \mathbb{R}^n$ we have $$\label{eq:linreg} \mathop{\mathrm{dist}}(x^{*}, \partial f(\hat x) \cap \mathcal{R}(\mathbf{A}^{\top})) \leq \zeta \cdot \bigg(\mathop{\mathrm{dist}}(x^{*}, \partial f(\hat x)) + \mathop{\mathrm{dist}}(x^{*}, \mathcal{R}(\mathbf{A}^{\top}))\bigg)$$ The collection $\{\partial f(\hat x), \mathcal{R}(\mathbf{A}^{\top})\}$ is linearly regular, if $\partial f(\hat x)$ is polyhedral (which holds for piecewise linear-quadratic $f$ in particular). We say the subdifferential mapping of $f$ *grows at most linearly*, if there exist $\rho_1,\rho_2 \ge 0$ such that for all $x \in \mathbb{R}^n$ and $x^* \in \partial f(x)$ we have $$\label{eq:lineargrowth} \|x^*\|_2 \le \rho_1 \cdot \|x\|_2 + \rho_2\,.$$ Any Lipschitz-continuous gradient mapping grows at most linearly. Furthermore, the subdifferential mappings of all functions in Example [\[ex:calmness\]](#ex:calmness){reference-type="ref" reference="ex:calmness"} grow at most linearly. (cf. [@schopfer2022extended Theorem 3.9]) [\[th:error_bounds_equality\]]{#th:error_bounds_equality label="th:error_bounds_equality"} Consider the linearly constrained optimization problem [\[eq:PB\]](#eq:PB){reference-type="eqref" reference="eq:PB"} and strongly convex $f:\mathbb{R}^n \to \mathbb{R}$. If its subdifferential mapping grows at most linearly, is calm at the unique solution $\hat x$ of [\[eq:PB\]](#eq:PB){reference-type="eqref" reference="eq:PB"}, and if the collection $\{\partial f(\hat x), \mathcal{R}(\mathbf{A}^{\top})\}$ is linearly regular, then there exists $\theta(\hat x)$ such that for all $x \in \mathbb{R}^n$ and $x^* \in \partial f(x) \cap \mathcal{R}(\mathbf{A}^{\top})$ we have the global error bound : $$\label{eq:EB} D_f^{x^*}(x,\hat x) \leq \dfrac{1}{\theta(\hat x)} \cdot \|\mathbf{A}x-b\|_2^2$$ # Convergence analysis {#sec:convergence_analysis} To start the convergence analysis of the Bregman-Kaczmarz method (Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"}), we first characterize how the Bregman distance of the iterates to the solution $\hat x$ decays from one iteration to the next one. By proper scaling of $f$, we can always assume that $f$ is $1$-strongly convex, i.e. we have $\alpha=1$, and consequently by [\[eq:D\]](#eq:D){reference-type="eqref" reference="eq:D"} we can assume that the Bregman distance is an upper bound to the squared norm, i.e. we always have $$\begin{aligned} \frac{1}{2}\lVert x_{k}-\hat{x}\rVert^{2} \leq D_{f}^{x_{k}^{*}}(x_{k},\hat{x}) .\end{aligned}$$ [\[lem:nWRSK_x\_descent\]]{#lem:nWRSK_x_descent label="lem:nWRSK_x_descent"} Let $(x_k, x_k^*)$ be the two sequences generated by Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"}. Then, it holds that $$\begin{aligned} \label{eq:descent_lem} D_f^{x_{k+1}^*}(x_{k+1},\hat x) &\leq D_f^{x_{k}^*}(x_{k},\hat x) - \frac{\eta_k (2-\eta_k)}{2} \frac{\|\mathbf{A}_{(i_k)}x_k - b_{(i_k)}\|^2}{\|\mathbf{A}_{(i_k)}\|_2^2} \nonumber \\ &+ \frac{\eta_k^2}{2} \frac{\|\varepsilon_{(i_k)}\|^2}{\|\mathbf{A}_{(i_k)}\|^2_2} + \eta_k(1-\eta_k) \varepsilon_{(i_k)}^{\top} \cdot \frac{( \mathbf{A}_{(i_k)}x_k - b_{(i_k)} )}{\|\mathbf{A}_{(i_k)}\|^2_2}\end{aligned}$$ *Proof.* Using $\tilde{b}_{(i)} = b_{(i)} + \varepsilon_{(i)}$ and defining $$\Tilde{x}_k := \hat x + \frac{\mathbf{A}_{(i_k)}^{\top}(\Tilde{b}_{(i_k)} - b_{(i_k)})}{\|\mathbf{A}_{(i_k)}\|^2_2}.$$ Since $\nabla f^*$ is $1$-Lipschitz-continuous, the descent lemma [\[eq:Lip\]](#eq:Lip){reference-type="eqref" reference="eq:Lip"} implies $$\begin{aligned} D_f^{x_{k+1}^*}(x_{k+1},\Tilde{x}_k) &= f^*(x_{k+1}^*) + f(\Tilde{x}_k) - \langle x_{k+1}^*,\Tilde{x}_k\rangle \\ &\leq f^*(x_k^*) + \langle \nabla f^*(x_k^*),x_{k+1}^*-x_k^*\rangle + \frac{1}{2}\|x_{k+1}^*-x_k^*\|_2^2 + f(\Tilde{x}_k) - \langle x_{k+1}^*,\Tilde{x}_k\rangle. \end{aligned}$$ Using the fact that $x_k=\nabla f^*(x_k^*)$ we conclude $$\begin{aligned} D_f^{x_{k+1}^*}(x_{k+1},\Tilde{x}_k) &\leq D_f^{x_k^*}(x_k,\Tilde{x}_k) + \left< x_k, \ - \eta_k \frac{\mathbf{A}_{(i_k)}^{\top} (\mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)})}{\|\mathbf{A}_{(i_k)}\|^2_2} \right> \\ & \hspace{0.5cm} + \frac{1}{2} \Bigl\| - \eta_k \frac{\mathbf{A}_{(i_k)}^{\top} (\mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)})}{\|\mathbf{A}_{(i_k)}\|^2_2} \Bigr\|_2^2 + \Big\langle \eta_k \frac{\mathbf{A}_{(i_k)}^{\top} (\mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)})}{\|\mathbf{A}_{(i_k)}\|^2_2},\Tilde{x}_k \Big\rangle \\ &= D_f^{x_k^*}(x_k,\Tilde{x}_k) - \frac{\eta_k}{\|\mathbf{A}_{(i_k)}\|^2_2} \Big \langle \mathbf{A}_{(i_k)}^{\top}(\mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)}), x_k - \Tilde{x}_k \Big \rangle \\ & \hspace{0.5cm} + \frac{\eta_k^2}{2\|\mathbf{A}_{(i_k)}\|^2_2} \|\mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)} \|^2_2 \\ &= D_f^{x_k^*}(x_k,\Tilde{x}_k) + \langle x_{k+1}^* - x_{k}^*, x_k - \Tilde{x}_k\rangle + \frac{\eta_k^2}{2\|\mathbf{A}_{(i_k)}\|^2_2} \|\mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)} \|^2_2 \end{aligned}$$ Unfolding the expression of $D_f^{x_{k+1}^*}(x_{k+1},\Tilde{x}_k)$ and $D_f^{x_{k}^*}(x_{k},\Tilde{x}_k),$ and adding $f(\hat x)$ to both sides, we obtain the inequality $$\label{eq:nwrsk1} D_f^{x_{k+1}^*}(x_{k+1},\hat x) \leq D_f^{x_{k}^*}(x_{k},\hat x) + \langle x_{k+1}^* - x_{k}^*, \Tilde{x}_k - \hat x\rangle + \langle x_{k+1}^* - x_{k}^*, x_k - \Tilde{x}_k\rangle + \frac{\eta_k^2}{2\|\mathbf{A}_{(i_k)}\|^2_2} \|\mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)} \|^2_2.$$ By observing that $\Big \langle \mathbf{A}_{(i_k)}^{\top}(\mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)}), x_k - \hat x \Big \rangle = \|\mathbf{A}_{(i_k)}x_k - b_{(i_k)}\|^2_2 - \varepsilon_{(i_k)}^{\top} (\mathbf{A}_{(i_k)}x_k - b_{(i_k)})$, we conclude that $$\label{eq:nwrsk2} \langle x_{k+1}^* - x_{k}^*, \Tilde{x}_k - \hat x\rangle + \langle x_{k+1}^* - x_{k}^*, x_k - \Tilde{x}_k\rangle = - \frac{\eta_k}{\|\mathbf{A}_{(i_k)}\|^2_2}\Big \langle \mathbf{A}_{(i_k)}^{\top}(\mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)}), x_k - \hat x \Big \rangle,$$ by rewriting $$\label{eq:nwrsk3} \| \mathbf{A}_{(i_k)}x_k - \Tilde{b}_{(i_k)}\|^2_2 = \|\mathbf{A}_{(i_k)}x_k - b_{(i_k)}\|^2_2 + \|\varepsilon_{(i_k)}\|^2_2 - 2 \varepsilon_{(i_k)}^{\top} (\mathbf{A}_{(i_k)}x_k - b_{(i_k)}).$$ Inserting ([\[eq:nwrsk2\]](#eq:nwrsk2){reference-type="ref" reference="eq:nwrsk2"}) and ([\[eq:nwrsk3\]](#eq:nwrsk3){reference-type="ref" reference="eq:nwrsk3"}) into ([\[eq:nwrsk1\]](#eq:nwrsk1){reference-type="ref" reference="eq:nwrsk1"}), we obtain $$\begin{aligned} D_f^{x_{k+1}^*}(x_{k+1},\hat x) &\leq D_f^{x_{k}^*}(x_{k},\hat x) - \frac{\eta_k (2-\eta_k)}{2} \frac{\|\mathbf{A}_{(i_k)}x_k - b_{(i_k)}\|^2_2}{\|\mathbf{A}_{(i_k)}\|_2^2} \\ &+ \frac{\eta_k^2}{2} \frac{\|\varepsilon_{(i_k)}\|^2}{\|\mathbf{A}_{(i_k)}\|^2_2} + \eta_k(1-\eta_k) \varepsilon_{(i_k)}^{\top} \cdot \frac{(\mathbf{A}_{(i_k)}x_k - b_{(i_k)})}{\|\mathbf{A}_{(i_k)}\|^2_2}. \end{aligned}$$ ◻ The term in [\[eq:descent_lem\]](#eq:descent_lem){reference-type="eqref" reference="eq:descent_lem"} from Lemma [\[lem:nWRSK_x\_descent\]](#lem:nWRSK_x_descent){reference-type="ref" reference="lem:nWRSK_x_descent"} which is linear in $\varepsilon_{(i_k)}$ is $$\begin{aligned} \eta_{k}(1-\eta_{k})\varepsilon_{(i_{k})}^{\top}\cdot\frac{( \mathbf{A}_{(i_k)}x_k - b_{(i_k)} )}{\|\mathbf{A}_{(i_k)}\|^2_2}\end{aligned}$$ and vanishes when the stepsize $\eta_k$ is one (i.e. we do full steps) regardless of the structure of the noise, but does not vanish in the case where we use adaptive stepsizes $\eta_k \neq 1$. However, we assume independent noise, i.e. $\varepsilon_{(i_{k})}$ is independent of all previous randomness in the algorithm. Since we also assume that the noise has zero mean, this term will vanish, when we take the expectation over the noise realization. [\[lem:descent-D\]]{#lem:descent-D label="lem:descent-D"} Assume independent noise (cf. Section [1.1](#sec:prob-statment){reference-type="ref" reference="sec:prob-statment"}) and that $(x_{k},x_{k}^{*})$ are generated by Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} and denote $\gamma = \tfrac{\theta(\hat x)}{\lVert\mathbf{A}\rVert_{\square}^{2}}$ (where $\theta(\hat x)$ comes from Theorem [\[th:error_bounds_equality\]](#th:error_bounds_equality){reference-type="ref" reference="th:error_bounds_equality"}). Then it holds that $$\begin{aligned} \label{eq:des_l} \mathbb{E}_{i_{k}}\mathbb{E}_{\varepsilon}[D_f^{x_{k+1}^*}(x_{k+1},\hat x)] &\leq \bigg( 1 - \frac{\eta_k (2-\eta_k)}{2} \gamma\bigg) D_f^{x_{k}^*}(x_{k},\hat x) + \frac{\eta_k^2}{2}\frac{\sigma^{2}}{\lVert\mathbf{A}\rVert_{\square}^{2}}. \end{aligned}$$ *Proof.* In [\[eq:descent_lem\]](#eq:descent_lem){reference-type="eqref" reference="eq:descent_lem"} in Lemma [\[lem:nWRSK_x\_descent\]](#lem:nWRSK_x_descent){reference-type="ref" reference="lem:nWRSK_x_descent"} we take the expectation with respect to the choice of $i_{k}$ and the noise realization $\varepsilon$ and get (since the noise realization and the choice of the index are independent) $$\begin{aligned} \mathbb{E}_{i_{k}}\mathbb{E}_{\varepsilon}[D_f^{x_{k+1}^*}(x_{k+1},\hat x) ]\leq D_f^{x_{k}^*}(x_{k},\hat x) - \frac{\eta_k (2-\eta_k)}{2} \mathbb{E}_{i_{k}}\left[ \frac{\|\mathbf{A}_{(i_k)}x_k - b_{(i_k)}\|^2}{\|\mathbf{A}_{(i_k)}\|_2^2}\right] \\ + \frac{\eta_k^2}{2} \mathbb{E}_{i_{k}}\mathbb{E}_{\varepsilon}\left[\frac{\|\varepsilon_{(i_k)}\|^2}{\|\mathbf{A}_{(i_k)}\|^2_2}\right] + \eta_k(1-\eta_k) \mathbb{E}_{\varepsilon}\left[\varepsilon_{(i_k)}^{\top}\right] \cdot \mathbb{E}_{i_{k}}\left[\frac{( \mathbf{A}_{(i_k)}x_k - b_{(i_k)} )}{\|\mathbf{A}_{(i_k)}\|^{2}}\right]\end{aligned}$$ We have $\mathbb{E}_{\varepsilon}\left[\varepsilon_{(i_k)}^{\top}\right] = 0$ and $$\begin{aligned} \mathbb{E}_{i_{k}}\mathbb{E}_{\varepsilon}\left[\frac{\|\varepsilon_{(i_k)}\|^2}{\|\mathbf{A}_{(i_k)}\|^2_2}\right] & = \mathbb{E}_{i_{k}}\left[\tfrac{1}{\lVert\mathbf{A}_{(i_{k})}\rVert^{2}}\mathbb{E}_{\varepsilon}\lVert\varepsilon_{(i_{k})}\rVert^{2}\right] = \mathbb{E}_{i_{k}}\left[\frac{\sigma_{i_{k}}^{2}}{\lVert\mathbf{A}_{(i_{k})}\rVert^{2}}\right]\\ & = \sum_{j=1}^{M}p_{j}\frac{\sigma_{j}^{2}}{\lVert\mathbf{A}_{(j)}\rVert^{2}} = \frac{\sum_{j=1}^M\sigma_{j}^{2}}{\sum_{j=1}^M\lVert\mathbf{A}_{(j)}\rVert^{2}} = \frac{\sigma^{2}}{\lVert\mathbf{A}\rVert_{\square}^{2}}.\end{aligned}$$ By Theorem [\[th:error_bounds_equality\]](#th:error_bounds_equality){reference-type="ref" reference="th:error_bounds_equality"} we get $$\begin{aligned} \mathbb{E}_{i_{k}}\left[ \frac{\|\mathbf{A}_{(i_k)}x_k - b_{(i_k)}\|^2}{\|\mathbf{A}_{(i_k)}\|_2^2}\right] & = \sum\limits_{i=1}^M \frac{\|\mathbf{A}_{(i)}\|^2}{\lVert\mathbf{A}\rVert_{\square}^{2}} \frac{\|\mathbf{A}_{(i)}x_k - b_{(i)}\|^2}{\|\mathbf{A}_{(i)}\|_2^2}\\ & = \tfrac1{\lVert\mathbf{A}\rVert_{\square}^{2}}\lVert\mathbf{A}x_{k}-b\rVert^{2} \geq \tfrac{\theta(\hat x)}{\lVert\mathbf{A}\rVert_{\square}^{2}}D_{f}^{x_{k}^{*}}(x_k,\hat x).\end{aligned}$$ Combining these inequalities proves the claim. ◻ Using the rule of total expectation we obtain $$\begin{aligned} \label{eq:des_l_EE} \mathbb{E}\left[D_f^{x_{k+1}^*}(x_{k+1},\hat x)\right] &\leq \bigg( 1 - \frac{\eta_k (2-\eta_k)}{2} \gamma\bigg) \mathbb{E}\left[D_f^{x_{k}^*}(x_{k},\hat x)\right] + \frac{\eta_k^2}{2}\frac{\sigma^{2}}{\lVert\mathbf{A}\rVert_{\square}^{2}}\end{aligned}$$ where the expectation is taken over all randomness involved (i.e. all randomly chosen indices so far and all noise realizations). To make the expectation on the left-hand side of [\[eq:des_l\]](#eq:des_l){reference-type="eqref" reference="eq:des_l"} as small as possible, we aim to minimize the right-hand side and the only thing we have available to do so, is the stepsize $\eta_{k}$. Although the right-hand side is quadratic in $\eta_{k}$, it is not clear how to do that, since we do not have all information that we need available, especially we do not know the distance $D_f^{x_{k}^*}(x_{k},\hat x)$ and the constant $\gamma$. If we had no noise, i.e. $\sigma^{2}=0$, the right-hand side would be minimal for $\eta_{k}=1$, i.e. we would always do full steps. We proceed by assuming that the quantities which are needed to achieve this goal are known (even if they are not in practice) and later in Section [4](#sec:heuristic){reference-type="ref" reference="sec:heuristic"} show how these quantities can be estimated in practice. The derivation follows along the lines presented in [@marshall2023optimal]. We unroll the estimate [\[eq:des_l\_EE\]](#eq:des_l_EE){reference-type="eqref" reference="eq:des_l_EE"} recursively to get $$\begin{aligned} \label{eq:des-l-EE-recursive} \mathbb{E}\left[D_{f}^{x_{k+1}^{*}}(x_{k+1},\hat{x})\right] \leq \prod\limits_{j=0}^{k}\left( 1 - \tfrac{\eta_{j}(2-\eta_{j})}{2} \gamma\right) \mathbb{E}\left[D_{f}^{x_{0}^{*}}(x_{0},\hat{x})\right] + \frac{\sigma^{2}}{\lVert\mathbf{A}\rVert_{\square}^{2}} \sum\limits_{j=0}^{k-1}\frac{\eta_{j}^{2}}{2}\prod\limits_{i=j+1}^{k}\left( 1 - \tfrac{\eta_{i}(2-\eta_{i})}{2}\gamma \right)\end{aligned}$$ (with the convention that the empty product equals one). Let $\eta$ be the vector of all the values $\eta_{j}$ and define $$\begin{aligned} D_k(\eta) \overset{\text{def}}{=}\prod_{j=0}^{k} \bigg( 1 - \frac{\eta_{j} (2-\eta_{j})}{2} \gamma\bigg) \frac{\lVert\mathbf{A}\rVert_{\square}^{2}\,\mathbb{E}[D_f^{x_{0}^*}(x_{0},\hat x)]}{\sigma^2} + \sum_{j=0}^{k}\frac{\eta_{j}^2}{2} \prod_{i=j+1}^{k} \bigg( 1 - \frac{\eta_i (2-\eta_i)}{2} \gamma\bigg).\end{aligned}$$ Then [\[eq:des-l-EE-recursive\]](#eq:des-l-EE-recursive){reference-type="eqref" reference="eq:des-l-EE-recursive"} becomes $$\begin{aligned} \label{eq:des-l-EE-D} \mathbb{E}[D_f^{x_{k+1}^*}(x_{k+1},\hat x)] \leq D_k(\eta)\frac{\sigma^2}{\lVert\mathbf{A}\rVert_{\square}^{2}}.\end{aligned}$$ From [\[eq:des-l-EE-D\]](#eq:des-l-EE-D){reference-type="eqref" reference="eq:des-l-EE-D"} wee see that $D_k(\eta)$ can be written in a recurrence fashion as $$\begin{aligned} \label{eq:recursion-D} D_{k}(\eta) = \bigg( 1 - \frac{\eta_k (2-\eta_k)}{2} \gamma\bigg) D_{k-1}(\eta) + \frac{\eta_k^2}{2}.\end{aligned}$$ From [\[eq:recursion-D\]](#eq:recursion-D){reference-type="eqref" reference="eq:recursion-D"} and [\[eq:des-l-EE-D\]](#eq:des-l-EE-D){reference-type="eqref" reference="eq:des-l-EE-D"} we see that we would like to make $D_{k}(\eta)$ as small as possible by choosing appropriate values for the $\eta_{j}$. From the definition of $D_k(\eta)$, it follows that $D_k$ only depends on $\eta_j$ for $j \leq k$ and hence, we set the partial derivative $\frac{\partial}{\partial \eta_k} D_{k}(\eta)$ of $D_{k}(\eta)$ equal to zero and solve for $\eta_k$ and get $$\begin{aligned} 0 & = \frac{\partial}{\partial \eta_k} D_{k}(\eta) \stackrel{~\eqref{eq:recursion-D}}{=} \frac{\partial}{\partial \eta_k} \bigg( \bigg( 1 - \frac{\eta_k (2-\eta_k)}{2} \gamma\bigg) D_{k-1}(\eta) + \frac{\eta_k^2}{2} \bigg)\\ & = (-1+\eta_{k})\gamma D_{k-1}(\eta) + \eta_{k}\end{aligned}$$ so that $$\begin{aligned} \label{eq:etak-betak} \eta_k = \frac{\gamma D_{k-1}(\eta)}{\gamma D_{k-1}(\eta) + 1} = \frac{\gamma \beta_k}{\gamma \beta_k + 1}\end{aligned}$$ with $\beta_k \overset{\text{def}}{=}D_{k-1}(\eta)$ and $\beta_0 \overset{\text{def}}{=}\frac{\lVert\mathbf{A}\rVert_{\square}^{2}\, D_f^{x_{0}^*}(x_{0},\hat x)}{\sigma^2}$. Moreover, using [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"} we get $$\begin{aligned} \label{eq:etak-betak-aux} - \frac{\eta_k (2-\eta_k)}{2} \gamma \beta_k + \frac{\eta_k^2}{2} = \frac{-\gamma\beta_k\eta_k}{2}.\end{aligned}$$ Using [\[eq:etak-betak-aux\]](#eq:etak-betak-aux){reference-type="eqref" reference="eq:etak-betak-aux"} in the recursion [\[eq:recursion-D\]](#eq:recursion-D){reference-type="eqref" reference="eq:recursion-D"} expressed with $\beta_{k}$ we get $$\begin{aligned} \label{eq:recursion-betak} \beta_{k+1} = D_{k}(\eta)= \bigg( 1 - \frac{\eta_k (2-\eta_k)}{2} \gamma\bigg) \beta_k + \frac{\eta_k^2}{2} = \beta_k(1 - \frac{\gamma \eta_k}{2}).\end{aligned}$$ With this derivation we get from Lemma [\[lem:descent-D\]](#lem:descent-D){reference-type="ref" reference="lem:descent-D"} by lower bounding the Bregman distance with the norm (recall [\[eq:D\]](#eq:D){reference-type="eqref" reference="eq:D"}): [\[cor:decay-norm\]]{#cor:decay-norm label="cor:decay-norm"} Assume that in the context of Lemma [\[lem:descent-D\]](#lem:descent-D){reference-type="ref" reference="lem:descent-D"} we choose $\eta_{k}$ recursively by [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"} and have $\beta_{k}$ defined by $\beta_{0} \overset{\text{def}}{=}\lVert\mathbf{A}\rVert_{\square}^{2}\,D_{f}^{x_0^{*}}(x_0,\hat{x})/\sigma^{2}$ and the recursion [\[eq:recursion-betak\]](#eq:recursion-betak){reference-type="eqref" reference="eq:recursion-betak"}. Then it holds that $$\begin{aligned} \mathbb{E}\left[\lVert x_{k}-\hat{x}\rVert_2^{2}\right] \leq 2\frac{\sigma^2}{\lVert\mathbf{A}\rVert_{\square}^{2}}\beta_{k}.\end{aligned}$$ We see that the decay of the quantity $\beta_{k}$ governs the convergence speed of Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} towards the noise-free solution $\hat{x}$ and hence, we would like to quantify its decay. We have $$\begin{aligned} \frac{\gamma\beta_{k+1}-\gamma\beta_k}{\gamma} & = \beta_{k+1}-\beta_{k} \stackrel{\eqref{eq:recursion-betak}}{=} - \frac{\gamma\beta_{k}}2 \eta_{k} \stackrel{\eqref{eq:etak-betak}}{=} -\frac{\gamma\beta_{k}}{2(\gamma\beta_{k}+1)}\gamma\beta_{k} = -\frac{(\gamma\beta_k)^{2}}{2(\gamma\beta_{k}+1)}.\end{aligned}$$ Setting $v_{k}\overset{\text{def}}{=}\gamma\beta_{k}$ we get the recursion $$\begin{aligned} \label{eq:recursion-v} \frac{v_{k+1}-v_{k}}{\gamma} = -\frac{1}2 \frac{v_{k}^{2}}{v_{k}+1}.\end{aligned}$$   [\[rem:estimate-decay\]]{#rem:estimate-decay label="rem:estimate-decay"} To get the asymptotic decay of $v_{k}$ we make the quick observation that since $v_{k}$ is positive and decreasing, we also have $v_{k+1}-v_{k} \leq -\tfrac{\gamma}{2(v_{0}+1)}v_{k}^{2}$ and from this one can deduce that $$\begin{aligned} \label{eq:estimated-rec} v_{k} \leq \left(\tfrac1{v_{0}} + \tfrac{\gamma}{2(v_{0}+1)}k\right)^{-1} = \mathcal{O}(k^{-1}) \end{aligned}$$ and especially we see that $\beta_{k} = \mathcal{O}(k^{-1})$ and by Corollary [\[cor:decay-norm\]](#cor:decay-norm){reference-type="ref" reference="cor:decay-norm"} that $\lVert x_{k}-\hat{x}\rVert = \mathcal{O}(k^{-1/2})$. This quick estimate is not good for small $k$ and we would like to get a sharper bound in this regime. As observed in[@marshall2023optimal Section 2.5] the recursion [\[eq:recursion-v\]](#eq:recursion-v){reference-type="eqref" reference="eq:recursion-v"} is a forward-Euler approximation to the ordinary differential equation $\dot{u} = - u^{2}/(2u+2)$ which has the solution $u(t) = 1/W(e^{ t/2+c})$ with the Lambert-W function $W$ and $c$ coming from the initial condition $u(0) = u_{0}$, namely $c = \tfrac1{u_{0}} - \ln(u_{0})$. Moreover, it can be shown that $u$ is convex and decreasing and hence, the forward-Euler method is a positive lower bound for the solution. In conclusion, this shows that $$\begin{aligned} \label{eq:lambert-W} v_{k} \leq \frac{1}{W(e^{\tfrac\gamma2 k + c})},\quad c = \tfrac{1}{v_{0}} - \ln(v_{0}).\end{aligned}$$ In the variable $\beta_{k}$ this means that $$\begin{aligned} \label{eq:lambert-W-beta} \beta_{k} \leq \frac{1}{\gamma W(e^{\tfrac\gamma2 k + c})},\quad c = \tfrac{1}{\gamma\beta_{0}} - \ln(\gamma\beta_{0}).\end{aligned}$$ The original recursion from [\[eq:recursion-v\]](#eq:recursion-v){reference-type="eqref" reference="eq:recursion-v"} decays much faster than [\[eq:estimated-rec\]](#eq:estimated-rec){reference-type="eqref" reference="eq:estimated-rec"} initially if $v_{0}$ is large (cf. Figure [\[fig:error_estimates\]](#fig:error_estimates){reference-type="ref" reference="fig:error_estimates"}). However, the $v_{k}$ which solves the recursion [\[eq:recursion-v\]](#eq:recursion-v){reference-type="eqref" reference="eq:recursion-v"} also decay like $\mathcal{O}(1/k)$ asymptotically. The upper estimate from [\[eq:lambert-W\]](#eq:lambert-W){reference-type="eqref" reference="eq:lambert-W"} is usually very close to [\[eq:recursion-v\]](#eq:recursion-v){reference-type="eqref" reference="eq:recursion-v"} in this case and they could not be distinguished in Figure [\[fig:error_estimates\]](#fig:error_estimates){reference-type="ref" reference="fig:error_estimates"}. Next, we give some comments on the hyperparameters of our method. It holds $\eta_0 = \frac{\gamma\beta_0}{\gamma\beta_0 +1} = \frac{\gamma \lVert\mathbf{A}\rVert_{\square}^{2} D_f^{x_{0}^*}(x_{0},\hat x)}{\gamma \lVert\mathbf{A}\rVert_{\square}^{2} D_f^{x_{0}^*}(x_{0},\hat x) + \sigma^2} \approx 1$, since at the beginning of the method we have $\gamma \lVert\mathbf{A}\rVert_{\square}^{2} D_f^{x_{0}^*}(x_{0},\hat x) \gg \sigma^2$. The method initially starts with a stepsize close to 1, so enjoys a linear rate up to some iteration and only decreases the learning rate once the error is smaller than the noise. Furthermore, $\eta_k \in [0, 1]$ is a non-increasing sequence. We introduce the function $$\begin{aligned} \label{eq:gk1} g(k) = \frac{\sigma^{2}}{\gamma W(ce^{\tfrac\gamma2 k})},\quad c = \frac{e^{\tfrac1{\gamma\beta_{0}}}}{\gamma\beta_{0}}\end{aligned}$$ and note that from equation [\[eq:lambert-W-beta\]](#eq:lambert-W-beta){reference-type="eqref" reference="eq:lambert-W-beta"} and Corollary [\[cor:decay-norm\]](#cor:decay-norm){reference-type="ref" reference="cor:decay-norm"} we get that $$\begin{aligned} \mathbb{E}\left[\lVert x_{k}-\hat{x}\rVert_2^{2}\right] \leq \frac{2}{\lVert\mathbf{A}\rVert_{\square}^{2}}g(k).\end{aligned}$$ The following corollaries provide more intuition about the error bound function $g(k)$ and the two asymptotic regimes of the stepsize. First, we consider what happens when the noise vanishes. 1. (limit as $\sigma \to 0$). In the case where the variance of the noise $\sigma^2$ is small and the other parameters are fixed, we have $$g(k) = \lVert\mathbf{A}\rVert_{\square}^{2}e^{-\gamma k/2} \cdot D_f^{x_{0}^*}(x_{0},\hat x) + \mathcal{O}(\frac{\sigma^2}{ D_f^{x_{0}^*}(x_{0},\hat x)^2}) \quad \text{as} \quad \sigma \rightarrow 0$$ where the constant in the big-$\mathcal{O}$ notation depends on $\gamma$ and $k$. In particular, we have $$\eta(k) \to 1 \quad \text{as} \quad \sigma \to 0.$$ 2. (limit as $k \to \infty$). We have $$g(k) = \frac{2\sigma^2}{\gamma^2 k} \bigg(1 + \mathcal{O}(\frac{\text{ln}(k)}{k})\bigg), \quad \eta_k \approx \frac{2}{2 + \gamma k} \quad \text{as} \quad k \to + \infty$$ where the constant in the big-$\mathcal{O}$ notation depends on $\gamma$, $D_f^{x_{0}^*}(x_{0},\hat x)$ and $\sigma^2$. So that from Corollary [\[cor:decay-norm\]](#cor:decay-norm){reference-type="ref" reference="cor:decay-norm"} we should expect $$\begin{aligned} \mathbb{E}\left[\lVert x_{k}-\hat{x}\rVert_2\right] \leq \frac{2}{\gamma \lVert\mathbf{A}\rVert_{\square}} \frac{\sigma}{\sqrt{k}}.\end{aligned}$$ *Proof.* The first order Taylor expansion of series of the Lambert-W function is $W(x) = x + \mathcal{O}(x^2)$ (as $x \rightarrow 0$). Using $\beta_{0} = \lVert\mathbf{A}\rVert_{\square}^{2} D_{f}^{x_{0}^{*}}(x_{0},\hat{x})/\sigma^{2}$ this gives $$\begin{aligned} g(k) &= \frac{\sigma^2}{\gamma} \frac{1}{\frac{\sigma^2}{\gamma \cdot \lVert\mathbf{A}\rVert_{\square}^{2}D_f^{x_{0}^*}(x_{0},\hat x)} e^{\gamma k/2} e^{\frac{\sigma^2}{\gamma \cdot \lVert\mathbf{A}\rVert_{\square}^{2} D_f^{x_{0}^*}(x_{0},\hat x)}} + \mathcal{O}(\frac{\sigma^4}{ D_f^{x_{0}^*}(x_{0},\hat x)^2})} \\ &= \frac{\lVert\mathbf{A}\rVert_{\square}^{2}D_f^{x_{0}^*}(x_{0},\hat x)e^{-\gamma k/2} }{ e^{\frac{\sigma^2}{\gamma \cdot \lVert\mathbf{A}\rVert_{\square}^{2}D_f^{x_{0}^*}(x_{0},\hat x)}} + \mathcal{O}(\frac{\sigma^4}{ D_f^{x_{0}^*}(x_{0},\hat x)^2})}\end{aligned}$$ Canceling terms and using the fact that $e^{\frac{\sigma^2}{\gamma \cdot D_f^{x_{0}^*}(x_{0},\hat x)} } = 1 + \mathcal{O}(\frac{\sigma^2}{ D_f^{x_{0}^*}(x_{0},\hat x)} )$ give $$g(k) = \lVert\mathbf{A}\rVert_{\square}^{2}e^{-\gamma k/2} D_f^{x_{0}^*}(x_{0},\hat x) + \mathcal{O}(\frac{\sigma^2}{ D_f^{x_{0}^*}(x_{0},\hat x)^2}) \quad \text{as} \quad \sigma \rightarrow 0$$ Finally, from [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"} we have $$\begin{aligned} \label{eq:cont-etak-betak} \eta_k = \frac{\gamma \beta_k}{\gamma \beta_k + 1} = \frac{\gamma g(k)/\sigma^2}{\gamma g(k)/\sigma^2 + 1} = \frac{\gamma g(k)}{\gamma g(k) + \sigma^2}\end{aligned}$$ and get $\eta(k) \to 1$ for $\sigma \to 0$. In the standard Bregman-Kaczmarz algorithm with $\eta_{k}=1$ on obtains linear convergence with contraction factor $(1-\gamma/2)^{k}$. Here we get $e^{-\gamma k/2}$ which is just slightly worse. Now we have a look at the case $k\to\infty$. Similarly to [@marshall2023optimal Corollary 1.12] one can obtain that $$g(k) = \frac{2\sigma^2}{\gamma^2 k} \bigg(1 + \mathcal{O}(\frac{\text{ln}(k)}{k})\bigg) \quad \text{as} \quad k \to + \infty$$ and $$\label{eq:eta_limit} \eta(k) = \frac{\gamma g(k)/\sigma^2}{\gamma g(k)/\sigma^2 + 1} \approx \frac{2}{2 + \gamma k} \approx \mathcal{O}(1/k) \quad \text{as} \quad k \to + \infty.$$ ◻ # Heuristic estimation of the hyperparameters {#sec:heuristic} In the previous section, we showed that Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with stepsize $\eta_{k}$ according to [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"} produces iterates that indeed converge to the true noise-free solution $\hat{x}$ without ever using the clean right-hand side $b$, but only using noisy data. However, the algorithm needs to know explicitly the hyperparameters $\gamma$ and $\beta_{0} = \lVert\mathbf{A}\rVert_{\square}^{2}\frac{D(x_0, \hat x)}{\sigma^2}$ in order to calculate the stepsizes. Setting the algorithm with incorrect parameters may result in a slower algorithm or can even destroy convergence towards the true solution. In this part, inspired by [@marshall2023optimal], we introduce some heuristics estimations of those parameters in order to obtain good results in practice. We show how to estimate them from one additional run of the Bregman-Kaczmarz method with stepsize $\eta_k=1$. Let $x_0, x_1, \dots, x_N$ denote the iterates resulting from the additional run of the Bregman-Kaczmarz method with stepsize $\eta_k=1$ for $N$ iterations. We get from [\[eq:des_l\_EE\]](#eq:des_l_EE){reference-type="eqref" reference="eq:des_l_EE"} with $\eta_k=1$ : $$\begin{aligned} \mathbb{E}[D_f^{x_{k+1}^*}(x_{k+1},\hat x)] \leq \bigg( 1 - \frac{\gamma}{2}\bigg) \mathbb{E}[D_f^{x_{k}^*}(x_{k},\hat x)] + \frac{\sigma^2}{2\lVert\mathbf{A}\rVert_{\square}^{2}},\end{aligned}$$ and inductively we infer that $$\label{eq:cvr} \mathbb{E}[D_f^{x_{k}^*}(x_{k},\hat x)] \leq \bigg( 1 - \frac{\gamma}{2}\bigg)^k \cdot D_f^{x_{0}^*}(x_{0},\hat x) + \frac{\sigma^2}{\gamma \lVert\mathbf{A}\rVert_{\square}^{2}}.$$ If we assume that the initial error is above the noise level and assume that eventually, the error stagnates because of the noise, then we expect that, initially, $D_f^{x_{j+1}^*}(x_{j+1},\hat x) \approx (1 - \tfrac{\gamma}{2})D_f^{x_{j}^*}(x_{j},\hat x)$ and eventually $D_f^{x_{j}^*}(x_{j},\hat x) \approx \tfrac{\sigma^2}{\gamma \lVert\mathbf{A}\rVert_{\square}^{2}}$ (see [\[eq:cvr\]](#eq:cvr){reference-type="eqref" reference="eq:cvr"}). Using these estimates and the approximation $x_N \approx \hat x$ yields heuristic estimates for $\gamma$ and $\beta_0 = \tfrac{\lVert\mathbf{A}\rVert_{\square}^{2}D(x_0, \hat x)}{\sigma^2}$ as follows: We choose an index $N_{0}< N$ for which we assume that for $1\leq j\leq N_{0}$ it holds that $D_f^{x_{j+1}^*}(x_{j+1},\hat x) \approx (1 - \tfrac{\gamma}{2})D_f^{x_{j}^*}(x_{j},\hat x)$, i.e. we have $\tfrac\gamma2 \approx 1 - D_f^{x_{j+1}^*}(x_{j+1},x_{N})/D_f^{x_{j}^*}(x_{j},x_{N})$. We replace the quotient of the Bregman distances by their empirical mean over the $j$ and get the estimate $$\label{eq:estimate-gamma} \Tilde{\gamma} \overset{\text{def}}{=}2 \left(1 - \frac{1}{N_0} \sum_{j=1}^{N_0} \frac{D_f^{x_{j}^*}(x_{j},x_N)}{D_f^{x_{j-1}^*}(x_{j-1},x_N)}\right).$$ For $\beta_{0} = \tfrac{\lVert\mathbf{A}\rVert_{\square}^{2} D_{f}^{x_0^{*}}(x_{0},\hat{x})}{\sigma^{2}}$ we choose an $N_{1}<N$ for which we assume that for $N_{1}\leq j<N$ it holds that $D_{f}^{x_{j}^{*}}(x_{j},\hat{x}) \approx \tfrac{\sigma^{2}}{\gamma \lVert\mathbf{A}\rVert_{\square}^{2}}$, i.e. $\tfrac{\lVert\mathbf{A}\rVert_{\square}^{2}}{\sigma^{2}} \approx \tfrac{1}{\gamma D_{f}^{x_j^{*}}(x_j,\hat{x})}$. We replace $\gamma$ by its estimate $\tilde\gamma$ and the Bregman distance by its empirical mean and get the estimate $$\begin{aligned} \label{eq:estimate-beta0} \tilde\beta_{0} \overset{\text{def}}{=}\frac{D_{f}^{x_0^{*}}(x_{0},x_N)}{ \tfrac{\tilde \gamma}{N_{1}}\sum_{j=N-N_{1}}^{N-1} D_{f}^{x_j^{*}}(x_{j},x_{N})} = \left[\frac{\tilde \gamma}{N_1} \sum_{j=N-N_1}^{N-1} \frac{D_f^{x_{j}^*}(x_{j},x_N)}{D_f^{x_0^{*}}(x_0, x_N)} \right]^{-1}.\end{aligned}$$ # Numerical experiments {#sec:numerical_experiment} We present several experiments to demonstrate the effectiveness of Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} under various conditions. The simulations were performed in `Python` on an Intel Core i7 computer with 16GB RAM. The code that produces the figures in this paper is available at <https://github.com/tondji/abk>. ## Synthetic experiments {#sec:synthetic-experiments} In this part, we want to illustrate the result of Corollary [\[cor:decay-norm\]](#cor:decay-norm){reference-type="ref" reference="cor:decay-norm"}. For all experiments we consider $f(x) = \lambda \cdot \|x\|_1 + \frac{1}{2}\|x\|^2_2$, where $\lambda$ is the sparsity parameter which gives us $\nabla f^*(x) = S_{\lambda}(x)$. Note that $f$ is $1$-strongly convex but not smooth. Synthetic data for the experiments are generated as follows: all elements of the data matrix $\mathbf{A} \in \mathbb{R}^{m\times n}$ are chosen independent and identically distributed from the standard normal distribution $\mathcal{N}(0, 1)$. To construct the sparse solution $\hat x \in \mathbb{R}^n$, we generate a random vector with $s$ non-zero entries from the standard normal distribution and we set it as $\hat x$ and the corresponding right-hand side is $b = \mathbf{A}\hat x \in \mathbb{R}^m$. We fixed the number of blocks $M$, all having the same size, and the individual noise variances to $\sigma_{i} = \sigma/\sqrt{M}$. We compared the five following methods: 1. **adaptive RSK**: The adaptive randomized sparse Kaczmarz method, i.e. Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with stepsize $\eta_{k}$ defined by [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"}, $\lambda=0.05$, an exact value for $\beta_{0}$ (using the knowledge of the ground truth solution) and $\gamma=0.1$ from a grid search over a small range of values. 2. **RSK**: The randomized sparse Kaczmarz method, i.e. Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with $\eta_{k}=1$ (hence, neither $\gamma$ nor $\beta_0$ is needed, especially no ground truth knowledge is needed) and $\lambda=0.05$. 3. **heuristic adaptive RSK**: the heuristic adaptive randomized sparse Kaczmarz method, i.e. Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with $\eta_{k}$ from [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"} and $\lambda=0.05$. The parameters $\gamma$ and $\beta_{0}$ we estimated from the run of RSK as described in Section [4](#sec:heuristic){reference-type="ref" reference="sec:heuristic"}. We used $N_{0} = 400$ in the estimate [\[eq:estimate-gamma\]](#eq:estimate-gamma){reference-type="eqref" reference="eq:estimate-gamma"} of $\gamma$ and $N_{1} = 100$ in the estimate [\[eq:estimate-beta0\]](#eq:estimate-beta0){reference-type="eqref" reference="eq:estimate-beta0"} for $\beta_{0}$. 4. **adaptive RK**: The adaptive randomized Kaczmarz method, i.e. Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with $\eta_{k}$ defined by [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"}, $\lambda=0$, an exact value for $\beta_{0}$ (using the knowledge of the ground truth solution) and $\gamma=0.05$ from a grid search over small range values. 5. **RK**: The randomized Kaczmarz method, i.e. Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with $\eta_{k}=1$ (hence, neither $\gamma$ nor $\beta_0$ is needed, especially no ground truth knowledge is needed) and $\lambda=0$. The adaptive methods can be seen as an optimal baseline for the proposed approach: We determined the hyperparameter $\beta_{0}$ exactly and also optimized the performance of $\gamma$ by grid search. On the other hand, the heuristic adaptive RSK does not use any knowledge about the solution or the noise and only relies on the previous run of RSK (again, with no additional parameters). The only parameters that have to be chosen for the heuristic adaptive RSK are the constants $N_{0}$ and $N_{1}$. For each experiment, we run independent trials each starting with the initial iterate $x_0=0$ with the following parameters $m=2000, n=100, \sigma=0.05, s=10$ and $M=200$, i.e. each block is of size $m_i=10$. The best parameter $\gamma$ for every method aRSK and aRK were obtained by grid search over the set $\gamma \in \{0.005, 0.01, 0.05, 0.1, 1, 2\}$. We measured performance by plotting the average relative residual error $\| \mathbf{A}x_k - b\|_2/\|b\|_2$ and the relative error $\|x_k - \hat x \|_2/\|\hat x\|_2$ against the number of iterations in Figure [2](#fig:example1){reference-type="ref" reference="fig:example1"}. The RK and RSK methods will be our baseline comparisons. We did a small grid search and selected the $\lambda$ that gives the smallest relative error for RSK and aRSK. With $\lambda$ fixed for both RSK and aRSK, We used the same procedure to select $\gamma$ for the aRSK and aRK($\lambda=0$). Having obtained the parameter for RSK, we can estimate $\gamma$ and $\beta_0$ with which we run haRSK. Figure [2](#fig:example1){reference-type="ref" reference="fig:example1"} shows in a semilog plot of the relative residuals and the relative reconstruction errors for all methods. Note that the usual RK and RSK methods reduce the error exponentially at the beginning and stagnate when the order of the total noise level is reached as predicted by Eq [\[eq:cvr\]](#eq:cvr){reference-type="eqref" reference="eq:cvr"}, whereas aRK, aRSK and haRSK thanks to their adaptive stepsize are able to bring down the residuals and the errors to zero according to Corollary [\[cor:decay-norm\]](#cor:decay-norm){reference-type="ref" reference="cor:decay-norm"}. The parameters of aRK and aRSK giving the best results are obtained through a small grid search. After choosing $\lambda$ and $\gamma$ for aRK and aRSK we used the same $\lambda$ for the haRSK method and estimate $\gamma$ and $\beta_0$ without knowing the true solution. In Figure [2](#fig:example1){reference-type="ref" reference="fig:example1"} we see that haRSK has a better performance than the other adaptive methods. The reason behind this can be due to the fact that the best parameters were not part of the grid search space. Figure [4](#fig:example1-eta-beta){reference-type="ref" reference="fig:example1-eta-beta"} shows in a semilog plot how the parameters $\eta_{k}$ and $\beta_{k}$ evolve during the iteration for all the adaptive methods. The stepsize for the haRSK has a slow decay compared to other stepsizes. All the stepsizes are starting near 1 and decreasing towards zero. The second plot shows the behavior of $\beta_k$ that governs the reconstruction error for different adaptive methods. {reference-type="ref" reference="sec:synthetic-experiments"} for $m = 2000, n = 100,$ sparsity $s=10$, $\gamma_{aRK}=0.05, \gamma_{aRSK} = 0.1$, $\lambda=0.05$, $N_0=400, N_1=100$. From our heuristics we obtained $\Tilde{\gamma} = 0.0777$ and $\Tilde{\beta}_0 = 1022.60*10^{6}$. Left: Relative residual. Right: Relative error.](figures/residuals_2000_100.pdf "fig:"){#fig:example1 width="0.47 \\textwidth"} {reference-type="ref" reference="sec:synthetic-experiments"} for $m = 2000, n = 100,$ sparsity $s=10$, $\gamma_{aRK}=0.05, \gamma_{aRSK} = 0.1$, $\lambda=0.05$, $N_0=400, N_1=100$. From our heuristics we obtained $\Tilde{\gamma} = 0.0777$ and $\Tilde{\beta}_0 = 1022.60*10^{6}$. Left: Relative residual. Right: Relative error.](figures/errors_2000_100.pdf "fig:"){#fig:example1 width="0.47 \\textwidth"} {reference-type="ref" reference="fig:example1"}.](figures/eta_2000_100.pdf "fig:"){#fig:example1-eta-beta width="0.46 \\textwidth"} {reference-type="ref" reference="fig:example1"}.](figures/beta_2000_100.pdf "fig:"){#fig:example1-eta-beta width="0.46 \\textwidth"} In another experiment, we investigated the influence of the number of blocks on convergence and recovery guarantee. We generated an $m\times n$ matrix with $m=200$ rows and $n=100$ columns and a vector $\hat x\in\mathbb{R}^{n}$ with $s=10$ non-zero entries. The right-hand side $b$ was always evaluated with independent noise with total noise level $\sigma$ such that the relative error was $10\%$. We investigated numbers of blocks $M\in \left\{ 200, 100, 20, 5, 1 \right\}$, each with equally sized blocks, i.e. block sizes of $1,2,10,40,200$. We chose $\lambda=1$. The hyperparameters $\beta_{0}$ and $\gamma$ depend on block size, and for a fair comparison we decided to use "best possible parameters", i.e. we always used the exact value of $\beta_{0}$ (using knowledge of ground truth $\hat x$) and determined the best $\gamma$ for each $M$ by a small grid search. The results are shown in Figure [6](#fig:example3){reference-type="ref" reference="fig:example3"} in a log-log plot. One sees that the residual and error go down well below the noise level and also decay further but very slowly in the end. Both error and residual go down fast in the beginning and do so faster for a larger number of blocks, i.e. for smaller blocks. This phenomenon could be observed for other settings as well but depends on an accurate choice of the parameter $\gamma$. ![The effect of the block number $M$ on the relative error versus epochs for Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} (in the aRSK version) $m = 200, n = 100,$ sparsity $s=10$, $\gamma$ determined by grid search, $\beta_{0}$ exact. Left: Relative residual. Right: Relative error.](figures/blocks_residuals_200_100.pdf "fig:"){#fig:example3 width="0.47 \\textwidth"} ![The effect of the block number $M$ on the relative error versus epochs for Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} (in the aRSK version) $m = 200, n = 100,$ sparsity $s=10$, $\gamma$ determined by grid search, $\beta_{0}$ exact. Left: Relative residual. Right: Relative error.](figures/blocks_errors_200_100.pdf "fig:"){#fig:example3 width="0.47 \\textwidth"} ## Computerized tomography {#sec:ct} As an example on computerized tomography (CT), we used the implementation of the Radon transform from the `Python` library `skimage` and use it to build a system matrix for a parallel beam CT for a phantom of size $N\times N$ with $N=50$ and with $60$ equispaced angles. Hence, the system matrix $\mathbf{A}$ has size $3000\times 2500$, i.e. $m=3000$ and $n=2500$. We interpret the projection for each angle as one block $\mathbf{A}_{(i)}$, i.e. we have $M=60$ and each block has the size $m_{i}=50$. The ground truth solution $\hat x$ is fairly sparse (see below) and we generated the exact right-hand side simply as $b=\mathbf{A}x$. We used the total noise level of $10\%$, i.e. we choose $\sigma = 0.1\lVert b\rVert$ and $\sigma_{i}=\sigma/\sqrt{M}$. We used different methods for reconstruction, each run for $20$ epochs, i.e. for $60.000$ iterations and using the function $f(x)=\lambda\lVert x\rVert_{1} + \tfrac12\lVert x\rVert_2^2$: 1. **adaptive RSK**: The adaptive randomized sparse Kaczmarz method, i.e. Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with stepsize $\eta_{k}$ defined by [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"}, $\lambda=30$, an educated guess for $\gamma$ and exact value for $\beta_{0}$ (using the knowledge of the ground truth solution). 2. **RSK**: The randomized sparse Kaczmarz method, i.e. Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with $\eta_{k}=1$ (hence, neither $\gamma$ nor $\beta_0$ is needed, especially no ground truth knowledge is needed) and $\lambda=30$. 3. **heuristic adaptive RSK**: the heuristic adaptive randomized sparse Kaczmarz method, i.e. Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with $\eta_{k}$ from [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"} and $\lambda=30$. The parameters $\gamma$ and $\beta_{0}$ we estimated from the run of RSK as described in Section [4](#sec:heuristic){reference-type="ref" reference="sec:heuristic"}. We used $N_{0} = 10.000$ in the estimate [\[eq:estimate-gamma\]](#eq:estimate-gamma){reference-type="eqref" reference="eq:estimate-gamma"} of $\gamma$ and $N_{1} = 50.000$ in the estimate [\[eq:estimate-beta0\]](#eq:estimate-beta0){reference-type="eqref" reference="eq:estimate-beta0"} for $\beta_{0}$. 4. **adaptive RK**: The adaptive randomized Kaczmarz method, i.e. Algorithm [\[alg:BK\]](#alg:BK){reference-type="ref" reference="alg:BK"} with $\eta_{k}$ defined by [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"}, $\lambda=0$, an educated guess for $\gamma$ and exact value for $\beta_{0}$ (using the knowledge of the ground truth solution). In other words: The RSK method would be our baseline comparison. It can be applied by just choosing $\lambda$ by trial and error to adapt the sparsity of the outcome. With this, we can estimate the hyperparameter $\gamma$ and $\beta_{0}$ with which we run haRSK. This should produce a reconstruction which is much closer to the ground truth than RSK and even converge to the ground truth if the parameters were estimated exactly. The aRK and the aRSK method are just for further comparison. In Figure [7](#fig:ct-reconstructions){reference-type="ref" reference="fig:ct-reconstructions"} we show the ground truth data and the reconstructions by the adaptive RK, RSK, adaptive RSK and heuristic adaptive RSK. As can be seen, the sparse Kaczmarz methods all produce a clear background due to the sparsity enforcing $f$. The adaptive RSK and the heuristic adaptive RSK also lead to less noise in the non-zero regions. {#fig:ct-reconstructions width="\\textwidth"} Figure [9](#fig:ct-residuals){reference-type="ref" reference="fig:ct-residuals"} shows the relative residuals and the relative reconstruction errors for all methods. While the RSK method brings the residual down to some value in the order of the total noise level, the adaptive methods are able to go way below that and it seems that they even go down further. This indicates that the choice of parameters is quite accurate. We stress that the heuristic adaptive RSK does not need any knowledge of the true solution. The only choice that the user has to made is $\lambda$ (to control the sparsity), and $N_{0}$ and $N_{1}$ for the estimation of $\gamma$ and $\beta_{0}$. We also see that the reconstruction error for adaptive RSK and heuristic adaptive RSK keeps decaying until the last epoch (and is expected to go down even further). We find it quite remarkable that this is possible even though the heuristic adaptive RSK method only sees corrupted data (with independent noise). As expected, the error in RSK and adaptive RK stagnates at some level (RSK because it is not adaptive and adaptive RK because it does not respect the sparsity). {#fig:ct-residuals width="47%"} {#fig:ct-residuals width="47%"} We remark that the choice of $N_{0}$ and $N_{1}$ is crucial for a good estimate of the hyperparameters. The hyperparameter $\gamma$ is related to the asymptotic exponential decay of the Bregman distance. This parameter is quite difficult to estimate since the RSK method converges and reduces the Bregman distance to the solution much faster in the first iterations than in the asymptotic regime. On the other hand, the asymptotic regime is cluttered with effects from the noise. We observed that a fairly large $N_{0}$ lead to good results in most experiments. For the $\beta_{0}$ used an even larger number, i.e. we used $N_{1}$ large but only so large that the pre-asymptotic convergence has already happened. # Conclusion {#sec:conclusion} In this work, we have proposed the block Bregman Kaczmarz using adaptive stepsize for solving finite dimensional linear inverse problems under the assumption of independent noise. In this setting, one never sees the noise-free right-hand side but always a new noisy version with noise being independent of every previous information and identically distributed. We showed that with a well-chosen stepsize [\[eq:etak-betak\]](#eq:etak-betak){reference-type="eqref" reference="eq:etak-betak"} we are able to converge to the exact solution if we have access to a sufficient number of equations with independent noise. We gave a general error bound in terms of total noise variance $\sigma^2$, the block number $M$ and the constant $\theta(\hat x)$. The stepsize depends on two parameters: the signal-to-noise ratio $D_{f}^{x_0^{*}}(x_{0},\hat{x})/\sigma^{2}$ and the rate parameter $\gamma$. However, we showed in Section [4](#sec:heuristic){reference-type="ref" reference="sec:heuristic"} how to estimate these unknown parameters after having chosen $N_0, N_1$ and demonstrated in various scenarios in experiments the effectiveness of these estimations. We have shown in the experiments part that using a block size of 2 can already speed up the convergence a lot but in general taking a bigger block size gives us faster convergence results. It would be interesting to investigate how the actual rate $\mathcal{O}(k^{-1/2})$ to recover the exact solution can be accelerated. The heuristic estimates of the hyperparameters $\gamma$ and $\beta_{0}$ depend on the choices of $N_{0}$ and $N_{1}$, respectively. While these values play a big role, one can usually get good values for these values by inspecting the decay of the residual of a standard Bregman-Kaczmarz run, keeping in mind which regimes should be used for the estimation of the hyperparameters (steady linear decay for $\gamma$ and stagnation for $\beta_{0}$). In this paper, we assumed that the blocks are fixed for the full run of the algorithm. We think that the results can be extended to a framework where the blocks of rows are newly sampled in each iteration (cf. [@necoara2019faster; @tondji2022faster]) and expect that similar results than in this paper can be derived. [^1]: Institute for Analysis and Algebra, TU Braunschweig, 38092 Braunschweig, Germany, `[email protected]`. This work has received funding from the European Union's Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie Grant Agreement No. 861137. [^2]: African Institute for Mathematical Sciences (AIMS), AMMI Senegal, `[email protected]` [^3]: Institute for Analysis and Algebra, TU Braunschweig, 38092 Braunschweig, Germany, `[email protected]`

arxiv_math

--- abstract: | This article investigates the transmission delay of a Low Earth Orbit (LEO) satellite communication system in a bent pipe structure. By employing a stochastic geometry framework, satellites are modeled as spherical binomial point processes (BPP). A suboptimal satellite relay selection strategy is proposed, which achieves optimal conditions through theoretical analysis and numerical exploration. We derive the distance distributions for the uplink and downlink links, and provide corresponding analytical expressions for the transmission delays. author: - | \ bibliography: - references.bib title: | Latency Analysis of LEO Satellite Relay Communication: An Application of Conditional Contact Angle Distribution\ [^1] --- Stochastic geometry, transmission delay, binomial point process, distance distribution, best relay selection. # Introduction Ultra-dense LEO satellite network, because of its seamless global coverage characteristics, is likely to be utilized as a vital part of the future 6G system [@de2021survey; @kodheli2020satellite]. Companies such as SpaceX, Telesat, and OneWeb are accelerating the formation of a network of tens of thousands of LEO satellites [@sheetz2019next]. In real time communication scenarios, satellites fundamentally play the role of a space relay or forwarder to connect two terrestrial stations [@ma2022secure], which can be considered as a terrestrial-satellite-terrestrial unit in long-distance transmission. This leads to a fact that the performance of a terrestrial-satellite-terrestrial unit is worthy to analyse. Stochastic geometry, as an effective mathematical tool, plays a particularly important role in analyzing the performance of satellite networks [@wang2022ultra; @tian2023satellite]. BPP model, which is relatively accurate for closed area networks with a fixed number of satellites, is used in [@9079921] to analyse the coverage and rate of downlink. The user coverage probability for a scenario where satellite gateways (GWs) are deployed on the ground to act as a relay between the users and the LEO satellites is studied in [@talgat2020stochastic]. Most of the authors of the focus on the downlink transmission performance of satellite-terrestrial, while the uplink transmission performance has not been extensively studied. [@wang2022conditional] introduces contact angle distribution to analyze the influences of the number of satellites and the distance between the transmitter and receiver, which does not consider any channel fading [@wang2023reliability]. As the Rician fading model is ubiquitous for the communication links between satellites and ground stations, we adopt shadowed-Rician (SR) fading for the channel between the satellite and the terrestrial station, which is pointed as most accurate channel model. As for relay selection strategy, [@belbase2018coverage] proposes nearly optimal protocol in dual hop scenario. Therefore, based on the existing research, the contributions of this work are summarized as follows. - We give a possible optimal relay selection strategy and explore under what conditions the relay selection strategy is approximately optimal. - Under the certain relay selection strategy, we derive analytical expressions of the distance distribution of uplink and downlink and respectively give expressions for the cumulative distribution function of the signal-to-noise ratio (SNR), which take channel fading into consideration. - Based on above, we derive analytical expressions of the total transmission delay. By simulation, we verify the accuracy of the total transmission delay and investigate the effect of power, number of satellites, and distance between transmitter and receiver on the total transmission delay. - In the simulation, stochastic geometry is used to analyze the performance of satellite communications over multiple hops (or multiple links), which is not found in the existing papers. {#sliding_window width="0.8\\linewidth"} # System Model ## Network Topology In this subsection, we build a terrestrial-satellite-terrestrial relay communication model. $N_{s}$ satellites are distributed on a spherical surface with radius $R_s$ and form a homogeneous BPP [@wang2022evaluating]. Since the BPP distribution remains the same after the rotation, the coordinates of the transmitter and receiver are set to at ($R_e$,0,0) and ($R_e$,$\Theta$,0) for the convenience of calculation. The radius of the Earth is denoted as $R_e$ = 6371km. Here, we consider that the terrestrial stations are fixed and the satellites obey the BPP distribution. Although we analyze only one terrestrial-satellite-terrestrial unit here, in the simulation part we analyse the long-distance transmission with multi units. This implies that the unit model is also applicable to the long-distance transmission models. ## Relay Selection In fixed topology Amplify and Forward (AF) or Decode and Forward (DF) networks, the optimal relay selection criterion is the maximization of end-to-end SNR or the maximum SNR of downlink. However, when the relay satellites form a BPP, analysis of this strategy is intractable. Therefore, we consider a slightly suboptimal but tractable selection strategy [@lou2023coverage]: First, find a set of relays which can provide reliable communication for both the transmitter and receiver. Then, select the relay that has the strongest average received power in downlink. The reason for choosing the the strongest average received power in downlink is that: (i) In AF, the relay satellite amplifies the signal and also amplifies the noise. As a result, the downlink noise must be larger than the uplink. (ii) Considering that satellite energy is expensive, satellite transmission power is relatively low. ## Channel Model Many works have focused on deriving an accurate channel model for the communication links between satellites and ground stations, where it was shown that shadowed-Rician (SR) model proposed in [@1198102] is the most accurate. So, we consider the SR model to calculate fade margins and analyze the performance of communication. For the uplink, the received signal power $p_{(2)}^s$ at relay satellite is $$p_{(2)}^s = p_{(1)}^sA_u \times W_{t}^2,$$ where $A_u$ and $W_{t}^2$ respectively represent the propagation loss and the SR fading. The propagation loss can be calculated by: $$A_u = \frac{G_{(1)}^e G_{(2)}^s \lambda_u^2}{(4 \pi d_{\mathrm{up}})^2L_{(1)}^e L_{(2)}^s L_{\mathrm{add}}},$$ where $G_{(1)}^e$ and $G_{(2)}^s$ denote the transmitter and receiver antenna gain, $\lambda_u$ denotes the carrier wavelength of uplink, $d_{\mathrm{up}}$ is the distance between the transmitter and the relay satellite, $L_{(1)}^e$ and $L_{(2)}^s$ denote the transmitter and receiver antenna feeder loss, $L_{\mathrm{add}}$ denotes the link additional loss, including atmospheric absorption loss, rain attenuation, etc. The probability density function (PDF) of the SR fading power [@1198102] $W_t^2$ is given as follows: $$\begin{split} f_{W_t^2}(t) &= \left(\frac{2b_0m}{2b_0m+\Omega}\right)^m \frac{1}{2b_0} \exp \left(-\frac{t}{2b_0}\right) \\ & \times \sum\limits_{n=0}^{\infty }\frac{(m)_n}{(1)_n n!}{{ \left(\frac{\Omega \,t}{2b_0(2b_0m+\Omega )} \right)}^{n}}, \ \ \ \ t \geq 0, %P_{W_t^2}(t) = \left(\frac{2b_0m}{2b_0m+\Omega}\right)^m \frac{1}{2b_0}\exp(-\frac{t}{2b_0})F_1(m,1,\frac{\Omega \,t}{2b_0(2b_0m+\Omega )} \end{split}$$ where $(\cdot)_{n}$ is the Pochhammer symbol, while $m$, $b_0$ and $\Omega$ are the parameters of the SR fading. With the system model above, the received SNR for a link is given by $$\label{4} {\mathrm{SNR}}_\mathrm{up} = \frac{p_{(2)}^s}{N_u} = \frac{p_{(1)}^eG_{(1)}^e G_{(2)}^s \lambda_u^2W_t^2}{(4 \pi d_{\mathrm{up}})^2L_{(1)}^e L_{(2)}^s L_{\mathrm{add}}N_u} , %= \frac{p_{(2)}^s}{kBT_{u}},$$ where $N_u$ is the noise power of uplink. $N_u$ can be calculated by $N_u = kBT_{u}$, where $k$ is the Boltzmann's constant, $B$ is the bandwidth of transmission and $T_{u}$ denotes the total network noise temperature of the uplink. The received SNR for a link is given by $$\label{5} {\mathrm{SNR}}_\mathrm{down} = \frac{p_{(2)}^e}{N_d} % = \frac{p_{(1)}^s H_{d}^2}{N_d} = \frac{p_{(1)}^sG_{(1)}^s G_{(2)}^e \lambda_d^2W_t^2}{(4 \pi d_{\mathrm{down}})^2L_{(1)}^s L_{(2)}^e L_{\mathrm{add}}N_d}, %= \frac{p_{(1)}^s|A_uW_t|^2}{N_d}$$ where $N_d$ is the noise power of downlink and $p_{(2)}^e$ can be calculated as the same as $p_{(2)}^s$ with downlink parameters. # Time delay analysis ## Distance Distribution In order to contribute expressions for average time delay in the following sections, we first need to characterize some basic distance distributions that stem from the stochastic geometry of the considered system. According to the relay selection strategy mentioned above, the downlink distance $d_{\mathrm{down}}$ is a independent from $\theta_2$. Since the correspondence between the central angle (the angle of the line from two points to the center of the Earth [@wang2022stochastic]) and distance is bijective, the distance distribution can be calculated via calculating the distribution of the central angle (the angle of the line between two points and the center of the Earth). To facilitate the calculation of the uplink distance distribution, we use the CDF of the central angle. The central angle of the receiver and the relay satellite is denoted as $\theta_1$, and the central angle of the transmitter and the relay satellite is denoted as $\theta_2$. **Lemma 1**. *The PDF of the $\theta_1$ from any specific one of the satellites in the constellation to the receiver is given by $$\label{fpsinn} \begin{split} f_{\theta_1}(\theta) = \frac{N_s \sin\theta}{2} \left( \frac{ 1 + \cos\theta }{2} \right)^{N_s-1},0\leq \theta \leq \pi. \end{split}$$* **Proof.* See Appendix [7](#app:lemma1){reference-type="ref" reference="app:lemma1"}. ◻* **Lemma 2**. *The downlink distance distribution $d_{\mathrm{up}}$ is given by $$\label{eq2} \begin{split} F_{d_{\mathrm{down}}}(d_0) &=\left\{ \begin{aligned} &0 , d_0 \leq R_s-R_e\\ & 1-\left(1-\left( \frac{d_0^2-(R_s-R_e)^2}{4R_eR_s} \right)\right)^{ N_{s} }, \mathrm{otherwise}\\ % R_s-R_e\leq d_0 \leq R_s+R_e\\ &1 ,d_0 \geq R_s+R_e\\ \end{aligned} \right. \end{split}$$* **Proof.* See Appendix [8](#app:lemma2){reference-type="ref" reference="app:lemma2"}. ◻* Due to the relay selection strategy, $\theta_1$ is an independent variable while $\theta_2$ is a random variable associated with $\theta_1$. It is important to note that we consider the probability of a satellite appearing in a circle ring with a fixed central angle $\theta_1$ to be uniformly distributed. However, the probability of a satellite appearing in a circle ring with a fixed central angle $\theta_2$ is weighted by $f_{\theta_1}(\theta)$. **Lemma 3**. *Given that the maximum central angle of the transmitter's spherical cap is $\beta$, the approximate CDF of $F_{\theta_2}(\beta)$ is given by $$\begin{split} F_{\theta_2} \left(\beta\right) = \int_0^{2\pi} \int_0^{\beta} \frac{f_{\theta_1}\left( \psi(\theta,\phi) \right)}{2\pi R_s^2 \sin\psi(\theta,\phi)} R_s^2 \sin\theta \mathrm{d}\theta \mathrm{d}\phi, \end{split}$$ where $\psi(\theta,\phi)$ is represented by $$\begin{split} &\psi(\theta,\phi) = 2\arccos \frac{R_s^2+R_e^2-\mathcal{D}^2(\theta,\phi,\Theta,0)}{2R_sR_e}, \end{split}$$ and $\Phi$ is the central angle between the transmitter and receiver. ${\mathcal{D}^2(\theta_1,\phi_1,\theta_2,\phi_2)}$ is an operator and can be expressed by* *$$\begin{split} &{\mathcal{D}^2(\theta_1,\phi_1,\theta_2,\phi_2)}={R_e^2}+{R_s^2} \\ &-2{R_eR_s}(1-(\cos {{\phi }_{1}}\cos {{\phi }_{2}}\cos ({{\theta }_{1}}-{{\theta }_{2}})+\sin {{\theta }_{1}}\sin {{\theta }_{2}))}. \end{split}$$* **Proof.* See Appendix [9](#app:lemma3){reference-type="ref" reference="app:lemma3"}. ◻* Above all, we can obtain the uplink distance distribution $d_{\mathrm{up}}$ in the following lemma. **Lemma 4**. *The uplink distance distribution $d_{\mathrm{up}}$ is given by* *$$\label{Fup} \begin{split} F_{d_{\mathrm{up}}}(d_0) &=\left\{ \begin{aligned} &0 , d_0 \leq R_s-R_e\\ &\frac{d_0}{R_sR_s\sqrt{1-\left(\frac{R_s^2+R_e^2-d_0^2}{2(R_s+R_e)}\right)^2}}\\ &\times F_{\theta_2} \left(\mathrm{arccos}\left(\frac{R_s^2+R_e^2-d_0^2}{2(R_s+R_e)}\right)\right), \mathrm{else}\\ % R_s-R_e\leq d_0 \leq R_s+R_e\\ &1, d_0 \geq R_s+R_e\\ \end{aligned} \right. \end{split}$$* *Proof.* Since the expressions for the central angle $\theta_2$ and downlink distance $d_\mathrm{up}$ is given by $$\begin{split} d_{\mathrm{up}} = \sqrt{R_s^2+R_e^2-2R_sR_e\cos\theta_2}\\ \theta_2 = \mathrm{arccos}\left(\frac{R_s^2+R_e^2-d_{\mathrm{up}}^2}{2R_sR_e}\right), %0 \leq \theta_2 \leq \pi, \end{split}$$ and the uplink distance distribution $d_{\mathrm{up}}$ is given by ([\[Fup\]](#Fup){reference-type="ref" reference="Fup"}). ◻ ## Time Delay We define the transmission time delay of a link as $$\tau_{Q} = \frac{M}{B\log_2({1+\mathrm{SNR}_{Q}})},Q\in \{\mathrm{up},\mathrm{down}\},$$ where $M$ and $B$ respectively denote the size of the packet and the bandwidth of transmission. Thus we can derive the total time delay $\tau_{\mathrm{total}}$ in following theorem. **Theorem 1**. *The average time delay of downlink $\overline{\tau}_{\mathrm{down}}$ is given by $$\overline{\tau}_{\mathrm{down}}=\frac{M}{B\log_2\left({1+\frac{p_{(1)}^sG_{(1)}^s G_{(2)}^e \lambda_d^2(\Omega+2b_0))}{(4 \pi \overline{d}_{\mathrm{down}})^2L_{(1)}^s L_{(2)}^e L_{\mathrm{add}}N_d}}\right)}.$$* **Proof.* See Appendix [10](#app:theorem1){reference-type="ref" reference="app:theorem1"}. ◻* **Theorem 2**. *The average time delay of uplink $\overline{\tau}_{\mathrm{up}}$ is given by $$\overline{\tau}_{\mathrm{up}} = \int_{0}^{\infty} \frac{M}{B\log_2({1+\gamma})} f_{{\mathrm{SNR}}_\mathrm{up}}(\gamma) \mathrm{d}\gamma,$$ where $$\begin{split} f_{{\mathrm{SNR}}_\mathrm{up}}(\gamma) &= \int_{(R_s-R_e)^2}^{(R_s+R_e)^2} \frac{\sqrt{u}}{2} \left(\frac{2b_0m}{2b_0m+\Omega}\right)^m \exp \left(-\frac{z(\gamma)u}{2b_0}\right)\\ & \times \frac{1}{2b_0} \sum\limits_{n=0}^{\infty }\frac{(m)_n}{(1)_n n!}{{ \left(\frac{\Omega \,z(\gamma)u}{2b_0(2b_0m+\Omega )} \right)}^{n}} f_{d_{\mathrm{up}}}(\sqrt{u})\mathrm{d}u, \end{split}$$ and $z(\gamma) = \frac{(4 \pi)^2L_{(1)}^e L_{(2)}^s L_{\mathrm{add}}N_u\gamma}{p_{(1)}^eG_{(1)}^e G_{(2)}^s \lambda_u^2}$.* **Proof.* See Appendix [11](#app:theorem2){reference-type="ref" reference="app:theorem2"}. ◻* ## Optimality Analysis Due to the relay selection strategy and earth blockage, the transmitter and receiver can only communicate with satellites within a maximum distance $L_{\mathrm{max}}$: $$\label{Lmax} L_{\mathrm{max}} = 2R_e\sin\left(\frac{1}{2}\arccos\left(\frac{R_e}{R_s}\right)\right).$$ Under the premise of satisfying the above inequalities, we solve the optimization problem below and explore the optimal relay position of the satellite through numerical results. $$\theta^* = \underset{0 \leq \theta \leq \Theta}{\mathrm{argmin}} \ \overline{\tau}_{\mathrm{up}} + \overline{\tau}_{\mathrm{down}}. % &\underset{N_l,\mathcal{M}_{|N_l}}{\mathrm{maximize}} \ & & \prod_{i=1}^{N_l} P^C_{\rm cond}(l_i), \label{opt1-2} \\ % &\underset{N_l,\mathcal{M}_{|N_l}}{\mathrm{minimize}} \ & \sum_{i=1}^{N_l} & \frac{\varpi }{B \log_{2} (1+{\rm{SNR}}_i) }, \label{opt1-3}$$ {#simufigure1 width="0.7\\linewidth"} In Fig.[2](#simufigure1){reference-type="ref" reference="simufigure1"}, we use markers to denote the polar angle coordinate position of the terrestrial station. The horizontal coordinate is the polar angle of the ideal relay satellite. In the Monte-Carlo simulation, we search for the closest satellite to the ideal satellite coordinates as a relay satellite to calculate the uplink and downlink time delays. It can be seen that choosing the closest satellite to the receiver is a good suboptimal strategy when the ratio of uplink and downlink power differs significantly. # Numerical Results ***Notations*** ***Description*** ***Value(Default)*** -------------------------------------------- -------------------------------------------------------------------- ------------------------------------------ $N_s$ Number of satellites 500 $R_e$; $R_s$ Radius of the Earth; satellites 6371; 6871 (km) $N_u$; $N_d$ Noise power of uplink; downlink $3.6\times10^{-12}$; $3.6\times10^{-12}$ $B_u$; $B_d$ Bandwidth of uplink; downlink $0.5$; $0.25$ (GHz) $p_{(1)}^sG_{(1)}^s$; $p_{(1)}^eG_{(1)}^e$ Effective isotropic radiated power at relay satellite; transmitter 30; 60 (dB) $L_{\mathrm{add}}$ Link additional loss 3 (dB) $\lambda_d$; $\lambda_u$ Carrier wavelength of downlink; uplink 0.0231; 0.015 (m) $\Omega$; $b_0$; $m$ Line-of-sight component; scatter component; Nakagami parameter 1.29; 0.158; 19.4 $M$ The size of the packet 0.5 Gbit In this section, we verify the accurancy of the derived expressions using Monte-Carlo simulations. In addition, we study the influence of various system parameters on the performance of the considered system. In all the figures, markers represent the derived analytical results while the solid lines represent the Monte-Carlo simulations. The system parameters used in the simulations are summarized in Table [\[table_1\]](#table_1){reference-type="ref" reference="table_1"}. {#simufigure2 width="0.65\\linewidth"} {#simufigure3 width="0.65\\linewidth"} {#simufigure4 width="0.65\\linewidth"} In Fig.[3](#simufigure2){reference-type="ref" reference="simufigure2"}, we plot total delay for different fixed number of satellites and study the effect of increasing the altitude of satellite. The results show that transmission delay increase as the the height of satellite and we can observe that for a fixed altitude time delay reduces as we increase the number of satellite. In Fig.[4](#simufigure3){reference-type="ref" reference="simufigure3"}, the time delay is studied under different values for the distance between the transmitter and receiver for companies OneWeb, Telesat and SpaceX with altitudes $h = 1200$ km, $h = 1150$ km and $h = 1110$ km [@del2019technical]. Other parameters such as the number of satellites, effective isotropic radiated power, carrier frequency, bandwidth, etc. are referenced in [@del2019technical]. Fig.[5](#simufigure4){reference-type="ref" reference="simufigure4"} illustrates the scenario of a long distance transmission containing multiple hops. We set the total transmission distance to 15000 km and calculate the minimum number of hops required at different altitude satellites according to ([\[Lmax\]](#Lmax){reference-type="ref" reference="Lmax"}). As the altitude of the satellite increases, it is less affected by ground obscuration and the maximum distance between both sides of communication on the ground is increasing. # Conclusion In this work, we propose a suboptimal satellite relay selection strategy in terrestrial-satellite-terrestrial scenario. Though deriving theoretical expressions of the downlink and uplink distance distribution, we give a expression of total transmission delay. We have verified all the derived expressions using Monte-Carlo simulations and ensured perfect fit. In simulation, we explore the conditions for the proposed strategy to reach optimal and provide the numerical results about the influence of the altitudes of the satellites, their numbers, the distance between two terrestrial stations on the performance of the delay. # Acknowledgement This work was supported by the National Key Research and Development Program of China (No. 2021YFB2900404). # Proof of Lemma [Lemma 1](#lemma1){reference-type="ref" reference="lemma1"} {#app:lemma1} For a homogeneous BPP, the probability of the satellite locates in a spherical cap with central angle $\theta$ is equal to the ratio of the area of $\theta$ to the total surface area of the sphere. So, we can obtain $$\begin{split} \mathbb{P}\left[ \theta_1 \leq \theta\right] &= \frac{\int_0^{2\pi} \int_0^{\theta} R_s sin(\theta_1) \mathrm{d}\theta_1 \mathrm{d}\phi}{4\pi R_s^2}\\ &= \frac{ 2\pi R_s^2 (1-\cos\theta)}{4\pi R_s^2},0\leq \theta \leq \pi. \\ \end{split}$$ Due to the channel assignment by which the serving satellite is the nearest one among all the $N_s$ i.i.d. satellites, the CDF of the $\theta_1$ from any specific one of the satellites in the constellation to the receiver is given by $$\begin{split} F_{\theta_1}(\theta) % & = \mathbb{P}\left[ \theta_1 \leq \theta \right] \\ % & = 1 - \mathbb{P}\left[ {\mathcal{N}\left( {{\mathcal{S}(\theta)}} \right) = 0} \right] \\ & = 1 - \prod_{i=1}^{N_s} \mathbb{P}\left[ \theta_1 \geq \theta\right] \\ & = 1 - \left( 1 - \frac{ 2\pi R_s^2 (1-\cos\theta)}{4\pi R_s^2} \right)^{N_s}\\ & = 1 - \left( \frac{ 1 + \cos\theta }{2} \right)^{N_s},\\ \end{split}$$ The PDF of $\theta_1$ is $$\label{fpsinn} \begin{split} f_{\theta_1}(\theta) = \frac{\mathrm{d}}{\mathrm{d}\theta} F_{\theta_1}(\theta) = \frac{N_s \sin\theta}{2} \left( \frac{ 1 + \cos\theta }{2} \right)^{N_s-1}, \end{split}$$ where $\frac{\mathrm{d}}{\mathrm{d}\theta}$ means take the derivative with respect to $\theta$. # Proof of Lemma [Lemma 2](#lemma2){reference-type="ref" reference="lemma2"} {#app:lemma2} In the authors' work [@9079921], the PDF of $d_{\mathrm{down}}$ was derived as shown in the below lemma. Due to the channel assignment by which the serving satellite is the nearest one among all the $N_s$ i.i.d. satellites, the PDF of the $d_{\mathrm{down}}$ from any specific one of the satellites in the constellation to the user is given by $$\label{fdown} f_{d_{\mathrm{down}}}(d_0) = N_{s} \left( 1- \frac{d_0^2-(R_s-R_e)^2}{4R_eR_s}\right)^{ N_{s}-1 } \frac{d_0}{2R_eR_s},$$ for $R_s-R_e\leq d_0 \leq R_s+R_e$ while $f_{d_{\mathrm{down}}}(d_0)=0$ otherwise. The CDF can be expressed as $$\begin{split} F_{d_{\mathrm{down}}}(d_0) &=\left\{ \begin{aligned} &0 , d_0 \leq R_s-R_e\\ & 1-\left(1-\left( \frac{d_0^2-(R_s-R_e)^2}{4R_eR_s} \right)\right)^{ N_{s} }, \mathrm{else}\\ % R_s-R_e\leq d_0 \leq R_s+R_e\\ &1 ,d_0 \geq R_s+R_e\\ \end{aligned} \right. \end{split}$$ # Proof of Lemma [Lemma 3](#lemma3){reference-type="ref" reference="lemma3"} {#app:lemma3} {#figure2 width="\\linewidth"} To derive the distribution of $\theta_2$, the following steps are taken: (i): assuming that the coordinate position of the satellite is $(R_s,\theta_0,\phi_0)$; (ii): in order to calculate the probability of the satellite appearing in the spherical cap (corresponds to the central angle $\beta$), perform a double integration over the probability of the satellite with respect to $\theta_0$ and $\phi_0$, where $0 \leq \theta_0 \leq \beta$ and $0 \leq \phi_0 \leq 2\pi$; (iii): the probability of occurrence at $(R_s,\theta_0,\phi_0)$ is weighted by $\frac{f_{\theta_1}\left( \psi(\theta,\phi) \right)}{2\pi R_s^2 \sin\psi(\theta,\phi)}$. Given that the maximum central angle of the transmitter's spherical cap is $2\beta$, the approximate CDF of $F_{\theta_2}(\beta)$ is given by $$\begin{split} F_{\theta_2} \left(\beta\right) = \int_0^{2\pi} \int_0^{\beta} \frac{f_{\theta_1}\left( \psi(\theta,\phi) \right)}{2\pi R_s^2 \sin\psi(\theta,\phi)} R_s^2 \sin\theta \mathrm{d}\theta \mathrm{d}\phi. \end{split}$$ And $\psi(\theta,\phi)$ is represented by $$\psi(\theta,\phi) = 2\arccos \left(\sin\theta\sin\Theta\cos\phi+\cos\theta\cos\Theta\right),$$ where $\Theta$ is the central angle between the transmitter and receiver. # Proof of Theorem [Theorem 1](#theorem1){reference-type="ref" reference="theorem1"} {#app:theorem1} The average time delay of uplink $\overline{\tau}_{\mathrm{down}}$ is given by $$\label{27} % \overline{\tau_{\mathrm{up}}} = \mathbb{E}[B\log_2({1+\mathrm{SNR_{\mathrm{up}}}})], \overline{\tau}_{\mathrm{down}} = \frac{M}{B\log_2({1+\overline{\mathrm{SNR}}_{\mathrm{down}}})},$$ where $\mathbb{E}[\cdot]$ denotes taking the mathematical expectation. The average central angle $\overline{\theta}_1$ is given by: $$\begin{split} \label{avertheta1} \overline{\theta}_1 &= \mathbb{E}[\theta_1] = \int_{0}^{\pi}1-F_{\theta_1}(\theta) \mathrm{d}\theta\\ &=\int_{0}^{\pi}\left( \frac{ 1 + \cos\theta }{2} \right)^{N_s}\mathrm{d}\theta =\int_{0}^{\pi} (\cos\frac{\theta}{2})^{2N_s} \mathrm{d}\theta\\ &=2\int_{0}^{\frac{\pi}{2}} (\cos\theta)^{2N_s}\mathrm{d}\theta=\pi\prod_{n=0}^{N_s-1}\frac{2N_s-2n-1}{2N_s-2n} \end{split}$$ The average distance of downlink $\overline{d}_{\mathrm{down}}$ can be calculated by: $$\label{averdowndis} % \overline{d}_{\mathrm{down}} = R_e^2+R_s^2-2R_eR_s\cos(\frac{1}{2}\frac{2N_s-1}{2N_s}\frac{2N_s-3}{2N_s-2}...\frac{1}{2}\frac{\pi}{2}). \overline{d}_{\mathrm{down}} = \sqrt{R_e^2+R_s^2-2R_eR_s\cos\overline{\theta}_1}.$$ According to [@1198102], the moment-generating function (MGF) of the instantaneous power can be shown to be $$M_{W_t^2}(\sigma )= \frac{(2b_0m)^m(1+2b_0\sigma)^{m-1}}{[(2b_0m+\Omega)(1+2b_0\sigma)-\Omega]^m},\sigma \geq 0.$$ The mathematical expectation of $W_t^2$ is given by: $$\label{SRexp} \begin{split} % \mathbb{E}[W_t^2] = M_{W_t^2}^{'}(0) = -\Omega-2b_0, \mathbb{E}[W_t^2] = \frac{\mathrm{d}}{\mathrm{d}\sigma}M_{W_t^2}(0) = \Omega+2b_0. \end{split}$$ Recall ([\[5\]](#5){reference-type="ref" reference="5"}), by substituting ([\[avertheta1\]](#avertheta1){reference-type="ref" reference="avertheta1"}, ([\[averdowndis\]](#averdowndis){reference-type="ref" reference="averdowndis"}) and ([\[SRexp\]](#SRexp){reference-type="ref" reference="SRexp"}) into ([\[27\]](#27){reference-type="ref" reference="27"}), we obtain the average time delay of downlink $\overline{\tau}_{\mathrm{down}}$: $$% \overline{\mathrm{SNR}}_{\mathrm{down}} =\mathbb{E}[\mathrm{SNR_{\mathrm{down}}}] = \frac{p_{(1)}^sG_{(1)}^s G_{(2)}^e \lambda_d^2(-\Omega-2b_0)}{(4 \pi \overline{d}_{\mathrm{down}} )^2L_{(1)}^s L_{(2)}^e L_{\mathrm{add}}N_d}. \overline{\tau}_{\mathrm{down}}=\frac{M}{B\log_2\left({1+\frac{p_{(1)}^sG_{(1)}^s G_{(2)}^e \lambda_d^2(\Omega+2b_0))}{(4 \pi \overline{d}_{\mathrm{down}})^2L_{(1)}^s L_{(2)}^e L_{\mathrm{add}}N_d}}\right)}.$$ # Proof of Theorem [Theorem 2](#theorem2){reference-type="ref" reference="theorem2"} {#app:theorem2} Recall ([\[4\]](#4){reference-type="ref" reference="4"}), since $W_t^2$ and $d_{\mathrm{down}}$ are independent random variables, the PDF of ${\mathrm{SNR}}_\mathrm{up}$ is a two-dimensional random variable and can be given by $$\begin{split} &f_{{\mathrm{SNR}}_\mathrm{up}}(\gamma)\\ &= \mathbb{P}\left(\frac{p_{(1)}^eG_{(1)}^e G_{(2)}^s \lambda_u^2W_t^2}{(4 \pi d_{\mathrm{up}})^2L_{(1)}^e L_{(2)}^s L_{\mathrm{add}}N_u} = \gamma \right)\\ % &= \int_{(R_s-R_e)^2}^{(R_s+R_e)^2} \frac{1}{u}f_{W_t^2}\left(\frac{\gamma u (4 \pi)^2L_{(1)}^s L_{(2)}^e L_{\mathrm{add}}N_d}{p_{(1)}^sG_{(1)}^s G_{(2)}^e \lambda_d^2}\right)f_{d_{\mathrm{down}}^2}(u)\mathrm{d}u &= \int_{(R_s-R_e)^2}^{(R_s+R_e)^2} u f_{W_t^2}\left(z(\gamma)u\right)f_{d_{\mathrm{down}}^2}(u)\mathrm{d}u\ \\ &= \int_{(R_s-R_e)^2}^{(R_s+R_e)^2} \frac{\sqrt{u}}{2}f_{W_t^2}\left(z(\gamma)u\right)f_{d_{\mathrm{down}}}(\sqrt{u})\mathrm{d}u\\ &= \int_{(R_s-R_e)^2}^{(R_s+R_e)^2} \frac{\sqrt{u}}{2} \left(\frac{2b_0m}{2b_0m+\Omega}\right)^m \frac{1}{2b_0} \exp \left(-\frac{z(\gamma)u}{2b_0}\right)\\ &\times \sum\limits_{n=0}^{\infty }\frac{(m)_n}{(1)_n n!}{{ \left(\frac{\Omega \,z(\gamma)u}{2b_0(2b_0m+\Omega )} \right)}^{n}} f_{d_{\mathrm{up}}}(\sqrt{u})\mathrm{d}u,\\ \end{split}$$ where $z(\gamma) = \frac{(4 \pi)^2L_{(1)}^e L_{(2)}^s L_{\mathrm{add}}N_u\gamma}{p_{(1)}^eG_{(1)}^e G_{(2)}^s \lambda_u^2}$. The average time delay of uplink $\overline{\tau}_{\mathrm{up}}$: $$\begin{split} \overline{\tau}_{\mathrm{up}} &= \mathbb{E}\left[\frac{M}{B\log_2({1+\mathrm{SNR}_{\mathrm{up}}})} \right]\\ &= \int_{0}^{\infty} \frac{M}{B\log_2({1+\gamma})} f_{{\mathrm{SNR}}_\mathrm{up}}(\gamma) \mathrm{d}\gamma. \end{split}$$ [^1]: Corresponding author: Xiang Ling, E-mail: xiangling\@uestc.edu.cn. The condensed version of this article has been submitted to 2023 7th International Conference on Communication and Information Systems (ICCIS 2023) held in Chongqing, China.

arxiv_math

--- abstract: | Consider a group acting on a polynomial ring $S$ over a field $\mathbb{K}$ by degree-preserving $\mathbb{K}$-algebra automorphisms. The invariant ring $R$ is a graded subring of $S$; let ${\mathfrak{m}}_R$ denote the homogeneous maximal ideal of $R$. Several key properties of the invariant ring and its embedding in $S$ can be deduced by studying the nullcone $S/{\mathfrak{m}}_R S$ of the group action. This includes, for example, the finite generation of the invariant ring and the purity of the embedding. In this article, we study the nullcones arising from the natural actions of the symplectic and general linear groups. For the natural representation of the symplectic group (via copies of the standard representation), the invariant ring is the ring defined by the principal Pfaffians of a fixed even size of a generic alternating matrix. We show that the nullcone of this embedding is a strongly $F$-regular ring in positive characteristic, and hence in characteristic zero, a ring of strongly $F$-regular type. Independent of characteristic, we give a complete description of the divisor class group of the nullcone and determine precisely when it is Gorenstein. We also show that the nullcone ideal has a squarefree initial ideal. For the natural representation of the general linear group (via copies of the standard representation and copies of its dual), the invariant ring is a generic determinantal ring. The nullcone of this embedding is typically a non-equidimensional ring. The irreducible components of the nullcone are the varieties of complexes of length two, as introduced by Buchsbaum and Eisenbud. We show that each of these irreducible components define strongly $F$-regular rings in positive characteristic. We also show that the Frobenius splitting of the varieties of complexes of length two can be chosen compatibly; it follows that the nullcone is an $F$-pure ring. We also show that the nullcone ideal and the ideals defining the varieties of complexes of length two have squarefree initial ideals with respect to the same monomial order of the ambient polynomial ring. address: - Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA - Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA - Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA author: - Vaibhav Pandey - Yevgeniya Tarasova - Uli Walther bibliography: - nullcone.bib title: | On the natural nullcones of the symplectic\ and general linear groups --- [^1] # Introduction Let $G$ be a group acting on a standard graded polynomial ring $S$ over a field $\mathbb{K}$ via degree-preserving $\mathbb{K}$-algebra automorphisms. Let $R$ denote the graded subring of $S$ consisting of all the polynomials which are invariant under the action of each group element, i.e., $R$ is the invariant ring of this action. Let ${\mathfrak{m}}_R$ denote the homogeneous maximal ideal of $R$. Consider the ideal of $S$ generated by all homogeneous polynomials of positive degree that are invariant under this action. In $1893$, Hilbert famously showed that if $G$ is a linearly reductive group, then a minimal set of homogeneous generators of this ideal also generates the ring $R$ as a $\mathbb{K}$-algebra [@Hilbert]. This settled the finite generation of invariant rings for all linearly reductive groups which was previously known only for the special linear group $\operatorname{SL}_2(\mathbb{C})$. The ideal ${\mathfrak{m}}_RS$ generated by all homogeneous invariants of positive degree is often called *Hilbert's nullcone ideal* and its zero set is called *Hilbert's nullcone*; see [@Kemper-Derksen Chapter 2] for a detailed exposition on Hilbert's nullcone. For more recent work on Hilbert's nullcone, see [@Hesselink; @Wang; @Kraft-Wallach; @Kim; @Kraft-Schwarz; @HJPS]. In this article, we call the coordinate ring $S/{\mathfrak{m}}_RS$ the *nullcone of the embedding $R \subseteq S$*. For example, the nullcone for the action of a finite group is an Artinian ring. For a more illuminating example, let $Y$ be a $d\times n$ matrix of indeterminates over $\mathbb{K}$, and set $S$ to be the polynomial ring $\mathbb{K}[Y]$. Let $R = \mathbb{K}[\{\Delta\}]$ be the $\mathbb{K}$-algebra generated by the size $d$ minors $\{\Delta\}$ of $Y$. Then $R$ is the coordinate ring for the Plücker embedding of the Grassmannian of $d$-dimensional subspaces of an $n$-dimensional vector space. The special linear group $\operatorname{SL}_d(\mathbb{K})$ acts $\mathbb{K}$-linearly on $S$ via the action $$M\colon Y\mapsto MY\qquad\text{ for }\ M\in\operatorname{SL}_d(\mathbb{K}).$$ When $\mathbb{K}$ is an infinite field, the invariant ring is precisely $R$, see [@Igusa] or [@DeConciniProcesi76 §2]. Clearly, the nullcone of the natural embedding $$R = \mathbb{K}[\{\Delta\}] \subseteq \mathbb{K}[Y] = S$$ is the determinantal ring $S/I_d(Y)$ defined by the maximal minors of a generic matrix. When $\mathbb{K}$ has characteristic zero, the group $\operatorname{SL}_d(\mathbb{K})$ is linearly reductive. It follows that $R$ is a direct summand of $S$ as an $R$-module. In contrast, when $\mathbb{K}$ has positive characteristic, it was recently shown in [@HJPS Theorem 1.1] that the above natural embedding typically does *not* split. This non-splitting is due to the Cohen--Macaulay property of the nullcone, when combined with the flatness of the Frobenius map on the ambient polynomial ring $S$. The Cohen--Macaulay property of determinantal rings of maximal minors was proved using the Eagon--Northcott resolution in [@Eagon-Northcott] and later for minors of all sizes by the technique of principal radical systems in [@Hochster-Eagon]. The divisor class group of determinantal rings is the infinite cyclic group by [@Bruns] and they are Gorenstein precisely when the matrix is square by [@Svanes Theorem 5.4.6]. The initial ideals of generic determinantal ideals are squarefree since the natural generators form a Gröbner basis by [@Sturmfels] and [@CGG]. Further, determinantal rings are $F$-regular in positive characteristic by [@HH94 §7] (see also [@Pandey-Tarasova Theorem 4.4]) and therefore have log-terminal, [@HaraWatanabe] and thus rational, singularities in characteristic zero [@Smith Theorem 4.3]. The objective of this paper is to discuss the corresponding results for the nullcones arising from the natural actions of the symplectic and general linear groups. We now describe these group actions, and our main results. Let $Y$ be a $2t\times n$ matrix of indeterminates for positive integers $t$ and $n$. Set $S$ to be the polynomial ring $\mathbb{K}[Y]$ and let $$\Omega\colonequals\begin{pmatrix} 0 & \mathbb{I}\\ -\mathbb{I}& 0 \end{pmatrix}$$ be the size $2t$ standard symplectic block matrix, where $\mathbb{I}$ is the size $t$ identity matrix. The $\mathbb{K}$-algebra $R\colonequals \mathbb{K}[Y^{\mathrm{T}}\Omega Y]$ is canonically isomorphic to $\mathbb{K}[X]/\operatorname{Pf}_{2t+2}(X)$, where $X$ is an $n\times n$ alternating matrix of indeterminates, and $\operatorname{Pf}_{2t+2}(X)$ the ideal generated by its principal size $2t+2$ Pfaffians. The symplectic group $$\operatorname{Sp}_{2t}(\mathbb{K})\colonequals\{M\in\operatorname{GL}_{2t}(\mathbb{K})\ |\ M^{\mathrm{T}}\Omega M=\Omega\}$$ acts $\mathbb{K}$-linearly on $S$ via the action $$M\colon Y\mapsto MY\qquad\text{ for }\ M\in\operatorname{Sp}_{2t}(\mathbb{K}).$$ The invariant ring is precisely $R$ when $\mathbb{K}$ is infinite, see [@DeConciniProcesi76 §6] or [@Hashimoto05 Theorem 5.1]. Notice that the ideal $$\mathfrak{P}(Y):= (Y^{\mathrm{T}}\Omega Y)S$$ generated by the entries of the alternating matrix $Y^{\mathrm{T}}\Omega Y$ is the nullcone ideal for this group action. In this article, we refer to the nullcone of the natural embedding $$R = \mathbb{K}[Y^{\mathrm{T}}\Omega Y] \subseteq \mathbb{K}[Y] = S.$$ as the *(natural) Pfaffian nullcone* $\mathbb{K}[Y]/\mathfrak{P}(Y)$. When $\mathbb{K}$ has characteristic zero, the group $\operatorname{Sp}_{2t}(\mathbb{K})$ is linearly reductive and thus $R$ is a direct summand of $S$ as an $R$-module. When $\mathbb{K}$ has positive characteristic, this embedding typically does *not* split by [@HJPS Theorem 1.1]. As in the case of the Plücker embedding of Grassmannians, this non-splitting is due to the Cohen--Macaulay property of the Pfaffian nullcone, established in [@HJPS Theorem 1.2], in conjunction with the flatness of Frobenius. Earlier, the irreducibility and normality of the Pfaffian nullcone was proved in [@Kraft-Schwarz Theorem 9.1 (3)]. We prove a much stronger result by establishing the $F$-regularity of the Pfaffian nullcone. In addition, we also show that the Pfaffian nullcone ideal has a squarefree initial ideal and is hence ameanable to Gröbner degeneration techniques, for example, as discussed in [@Conca-Varbaro]. Let $Y$ be a matrix of indeterminates of size $2t \times n$ for positive integers $t$ and $n$. Let $\mathbb{K}$ be a field and set $S:=\mathbb{K}[Y]$. 1. The initial ideal of the natural Pfaffian nullcone ideal $\mathfrak{P}(Y)$ is squarefree (with respect to the monomial order $<_B$ constructed in $\S$[\[Subsection PfaffianMonomialOrder\]](#Subsection PfaffianMonomialOrder){reference-type="ref" reference="Subsection PfaffianMonomialOrder"}). 2. If $\mathbb{K}$ is an $F$-finite field of positive characteristic, the natural Pfaffian nullcone $S/\mathfrak{P}(Y)$ is a strongly $F$-regular ring. 3. In consequence, if $\mathbb{K}$ has characteristic zero, the natural Pfaffian nullcone $S/\mathfrak{P}(Y)$ has log-terminal, and hence rational, singularities. Independent of characteristic, we give a complete description of the divisor class group of the Pfaffian nullcone and determine precisely when it is Gorenstein. Let $Y$ be a $2t \times n$ matrix of indeterminates for positive integers $t$ and $n$, and $\mathbb{K}$ be any field. Let $R_{t,n}$ denote the Pfaffian nullcone $\mathbb{K}[Y]/\mathfrak{P}(Y)$. 1. If $n\geq t+1$ then the divisor class group of $R_{t,n}$ is the group $\mathbb{Z}$ of integers. Otherwise, $R_{t,n}$ is a unique factorization domain. 2. Let $Y|_t$ be the submatrix of $Y$ consisting of the first $t$ columns. If $n \geq t+1$, the ideal ${\mathfrak{p}}:= I_{t}(Y|_t)$ of $R_{t,n}$ is prime of height one and generates the divisor class group. Further, the canonical class of $R_{t,n}$ is given by ${\mathfrak{p}}^{(n-t-1)}$. 3. The ring $R_{t,n}$ is Gorenstein precisely if $n \leq t+1$. Next, we discuss the embedding defined by the natural action of the general linear group. For positive integers $m$, $n$, and $t$, let $Y$ and $Z$ be $m\times t$ and $t\times n$ matrices of indeterminates respectively. Set $S$ to be the polynomial ring $\mathbb{K}[Y,Z]$, and take $R$ to be the $\mathbb{K}$-subalgebra generated by the entries of the product matrix $YZ$. Then $R$ is canonically isomorphic to the determinantal ring $\mathbb{K}[X]/I_{t+1}(X)$, where $X$ is an $m\times n$ matrix of indeterminates and $I_{t+1}(X)$ denotes the ideal generated by the size $t+1$ minors of $X$. The general linear group $\operatorname{GL}_t(\mathbb{K})$ acts $\mathbb{K}$-linearly on $S$ via $$M\colon\begin{cases} Y & \mapsto YM^{-1}\\ Z & \mapsto MZ\end{cases}$$ where $M\in\operatorname{GL}_t(\mathbb{K})$. When the field $\mathbb{K}$ is infinite, $R$ is precisely the ring of invariants, see [@DeConciniProcesi76 §3] or [@Hashimoto05 Theorem 4.1]. The nullcone of the natural embedding $$R = \mathbb{K}[YZ] \subseteq \mathbb{K}[Y,Z] = S$$ is the typically non-equidimensional reduced ring $\mathbb{K}[Y,Z]/(YZ)$ defined by the entries of the matrix $YZ$ (see [@Musili-Sheshadri] or [@Mehta-Trivedi2 Theorem 4.1] for the reducedness of the nullcone). In this article, we refer to this ring as the *(natural) determinantal nullcone*. The minimal primes of the determinantal nullcone are the ideals $${\mathfrak{p}}_{r,s}(Y,Z):= I_{r+1}(Y)+I_{s+1}(Z)+\ensuremath{(YZ)}$$ of $S$, for $r+s = t$; see $\S$[5.1](#sec-splitting-generalities){reference-type="ref" reference="sec-splitting-generalities"} for details. These are precisely the ideals defining the varieties of complexes of length two which are exact, as introduced by Buchsbaum--Eisenbud in [@Buchsbaum-Eisenbud]. When $\mathbb{K}$ has characteristic zero, the group $\operatorname{GL}_t(\mathbb{K})$ is linearly reductive and thus the determinantal ring $R$ splits from $S$ as an $R$-module. When $\mathbb{K}$ has positive characteristic, this embedding typically does *not* split by [@HJPS Theorem 1.1]. This is due to the flatness of the Frobenius together with the Cohen--Macaulay property of the varieties of complexes which was proved in [@Huneke Theorem 6.2] using principal radical systems (see also [@Musili-Sheshadri] and [@DeConcini-Lakshmibai]). The divisor class groups of the varieties of complexes are free abelian groups of finite ranks by [@Bruns2 Theorem 3.1] and independently by [@Yoshino Theorem 1.1]. These rings are Gorenstein under certain symmetry conditions on the sizes of the minors involved by [@Bruns2 Theorem 4.3] and by [@Yoshino Theorem 1.2]. Kempf showed in [@Kempf2] that the varieties of complexes have rational singularities, using [@Kempf1]. In positive characteristic, they are $F$-rational relative to the resolution of Kempf, and they are also $F$-split by [@MehtaTrivedi]. We extend these results by showing: Let $Y$ and $Z$ be matrices of indeterminates of sizes $m \times t$ and $t \times n$ respectively for positive integers $m$, $t$, and $n$. Let $\mathbb{K}$ be an $F$-finite field of positive characteristic; set $S :=\mathbb{K}[Y,Z]$ and suppose that $r$ and $s$ are non-negative integers with $r+s \leq t$. 1. The variety of complexes $S/{\mathfrak{p}}_{r,s}(Y,Z)$ is strongly $F$-regular. 2. If $t\le \min(m,n)$ then (for this $m,n,t$) the splittings of the Frobenius map on the varieties of complexes $S/{\mathfrak{p}}_{r,s}(Y,Z)$ can be chosen compatibly. 3. For any triple $(m,n,t)$, the natural determinantal nullcone $S/(YZ)S$ is $F$-pure. In consequence, if $\mathbb{K}$ has characteristic zero, the rings $S/{\mathfrak{p}}_{r,s}(Y,Z)$ have log-terminal (and hence rational) singularities. Recently, Lőrincz has also proved the $F$-regularity of the varieties of complexes using methods from representation theory in [@Lorincz Corollary 4.2]. We also show that the natural determinantal nullcone and the varieties of complexes of length two have squarefree initial ideals. Let $Y$ and $Z$ be matrices of indeterminates of sizes $m \times t$ and $t \times n$ respectively and $\mathbb{K}$ a field; set $S :=\mathbb{K}[Y,Z]$ and assume that $r$ and $s$ are non-negative integers. Let $<_B$ be the monomial irder constructed in $\S$[5.3](#Subsection VoCMonomialOrder){reference-type="ref" reference="Subsection VoCMonomialOrder"}. 1. If $r+s=t$, the ideal ${\mathfrak{p}}_{r,s}(Y,Z)$ defining the variety of exact complexes has a squarefree initial ideal (with respect to the monomial order $<_B$). 2. If $r+s <t$ and $\mathbb{K}$ has positive characteristic, the ideal ${\mathfrak{p}}_{r,s}(Y,Z)$ defining the variety of non-exact complexes and the natural determinantal nullcone ideal $(YZ)S$ have squarefree initial ideals. (with respect to $<_B$). A key tool used in establishing the $F$-regularity of the natural Pfaffian nullcone and the varieties of complexes of length two is the construction of certain very subtle monomial orders; these orders select special lead terms for the generators of the respective nullcone ideal in a characteristic-free manner. Additionally, the nullcone ideals considered have squarefree initial ideals with respect to these monomial orders. We explain these monomial orders with examples in $\S$[\[Subsection PfaffianMonomialOrder\]](#Subsection PfaffianMonomialOrder){reference-type="ref" reference="Subsection PfaffianMonomialOrder"} and $\S$[5.3](#Subsection VoCMonomialOrder){reference-type="ref" reference="Subsection VoCMonomialOrder"}. # Testing for $F$-purity and $F$-regularity We briefly recall some details on the types of singularities in positive characteristic that we discuss in this paper, as well as certain tools to test for them. For a detailed exposition on $F$-singularities, we refer the reader to the book [@Ma-Polstra] of Ma and Polstra. Let $R$ be a reduced Noetherian ring of positive prime characteristic $p$. The letter $e$ denotes a variable nonnegative integer, and $q=p^e$ the $e$-th power. Let $R^{1/q}$ denote the ring obtained by adjoining all $q$-th roots of elements of $R$. The inclusion $R\hookrightarrow R^{1/q}$ can then be identified with the $e$-fold Frobenius endomorphism $F^e\colon R\longrightarrow R$. The ring $R$ is *$F$-finite* if it is a finitely generated $R$-module via the action of the Frobenius endomorphism $F: R \longrightarrow R$; or, equivalently, if $R^{1/p}$ is finitely generated as an $R$-module. A finitely generated algebra over a field $\mathbb{K}$ is $F$-finite if and only if $\mathbb{K}^{1/p}$ is a finite field extension of $\mathbb{K}$. For an ideal $I=(z_1, z_2, \ldots, z_t)$ of $R$, the symbol $I^{[q]}$ denotes the ideal $F(I)R = (z_1^q,z_2^q,\ldots,z_t^q)$ of $R$. The ring $R$ is *$F$-pure* if $F$ is a pure homomorphism, i.e., if the map $R \otimes_R M \longrightarrow R^{1/p}\otimes_R M$ induced by the inclusion of $R$ in $R^{1/p}$ is injective for each $R$-module $M$. We say that $R$ is *$F$-split* if $F$ is a split monomorphism. Clearly, any $F$-split ring is $F$-pure; furthermore, an algebra over an $F$-finite field is $F$-pure if and only if it is $F$-split. An $F$-finite ring $R$ is *strongly $F$-regular* if for each $c$ in $R$ not in any of its minimal primes, there exists an integer $q$ such that the $R$-linear inclusion $R \longrightarrow R^{1/q}$ sending $1$ to $c^{1/q}$ splits as a map of $R$-modules. If the $F$-finite ring $R$ is $\mathbb{N}$-graded, strong $F$-regularity coincides with other similar notions like $F$-regularity and weak $F$-regularity. Before moving forward, we fix some general hypotheses. Unless stated otherwise, all rings in this paper are standard graded algebras over a field. When the field is of positive characteristic, it is assumed to be $F$-finite. Since all rings in this paper are $F$-finite, we may use the words $F$-pure and $F$-split interchangeably. Similarly, since all rings in this paper are standard graded algebras over $F$-finite fields, we sometimes loosely refer to a strongly $F$-regular ring simply as an $F$-regular ring. The following criterion of Fedder is useful for testing $F$-purity in the homogeneous setting: **Theorem 1**. *[@Fedder Theorem 1.12] Let $S = \mathbb{K}[x_1, \ldots, x_n]$ be an $\mathbb{N}$-graded polynomial ring over a field $\mathbb{K}$ of positive characteristic. Let $I$ be a homogeneous ideal of $S$ and let ${\mathfrak{m}}_S$ denote the homogeneous maximal ideal of $S$. Then $S/I$ is $F$-pure if and only if the ideal $I^{[p]}:I$ is not contained in ${{\mathfrak{m}}_S}^{[p]}$.* The following criterion of Glassbrenner is useful for testing $F$-regularity in the graded setting: **Theorem 2**. *[@Glassbrenner Theorem 3.1] Let $\mathbb{K}$ be a field of positive characteristic $p$ and set $S=\mathbb{K}[x_1,\ldots,x_n]$, with homogenesous maximal ideal ${\mathfrak{m}}_S$. Suppose that $R=S/I$ is a finitely generated $\mathbb{N}$-graded domain. Then $R$ is strongly $F$-regular if and only if* - *there exists a homogeneous element $s$ of $S$, not in $I$, for which the ring $R[1/s]$ is strongly $F$-regular, and* - *the ideal $s(I^{[p]}:I)$ is not contained in ${\mathfrak{m}}_S^{[p]}$.0◻* The next two facts are helpful in proving the $F$-purity of a given ring. We shall use the following as an ingredient in establishing the $F$-regularity of the Pfaffian nullcone: **Corollary 3**. *[@Pandey-Tarasova Corollary 3.3] Let $S$ be a polynomial ring over a field of positive prime characteristic $p$ and let $I$ be an equidimensional ideal of $S$. If ${\mathfrak{a}}\subsetneq I$ is an ideal generated by a regular sequence of length equal to the height of $I$, then the ideal ${\mathfrak{a}}^{[p]}:{\mathfrak{a}}$ is contained in $I^{[p]}:I$. In particular, if the ring $S/{\mathfrak{a}}$ is $F$-pure, then so is $S/I$.* *Proof.* Let $J$ be the ideal ${\mathfrak{a}}: I$ of $S$. We have the following chain of containments of ideals in $S$: $$\begin{aligned} {\mathfrak{a}}^{[p]}:{\mathfrak{a}}&\subseteq& {\mathfrak{a}}^{[p]}: IJ \\ &=& ({\mathfrak{a}}^{[p]}: J): I \\ &\subseteq& ({\mathfrak{a}}^{[p]}: J^{[p]}): I \\ &=& ({\mathfrak{a}}: J)^{[p]}: I \\ &=& I^{[p]}: I,\end{aligned}$$ where the second-to-last equality follows from the flatness of the Frobenius map on the regular ring $S$ and the last equality follows from the symmetry of links since $I$ is an equidimensional ideal; see [@PS74]. The result is now immediate from Theorem [Theorem 1](#theoremFedder){reference-type="ref" reference="theoremFedder"}. ◻ **Corollary 4**. *Let $S$ be a polynomial ring over a field of positive characteristic $p$ and let ${\mathfrak{m}}_S$ denote its homogeneous maximal ideal. Let $I$ be a height $h$ prime ideal of $S$. Then, the symbolic power $I^{(h(p-1))}$ is contained in $I^{[p]}: I$. In particular, if $I^{(h(p-1))}$ is not contained in ${\mathfrak{m}}_S^{[p]}$, then $S/I$ is $F$-pure.* *Proof.* By the flatness of the Frobenius map on $S$, the set of associated prime ideals of $S/I^{[p]}$ equals that of $S/I$. So the containment of ideals $$I\cdot I^{(h(p-1))}\subseteq I^{[p]}$$ may be verified locally. In the regular ring $(S_I, IS_I)$, the maximal ideal $IS_I$ is generated by $h$ elements. Recall that the ordinary and symbolic powers of ideals primary to the maximal ideal are equal. Notice that the containment $$I^{h(p-1)+1}S_I \subseteq I^{[p]}S_I$$ immediately follows from the pigeonhole principle. In fact, this result holds more generally when $I$ is a radical ideal and $h$ is the maximum of the heights of the associated prime ideals of $I$. ◻ Corollary [Corollary 4](#CorollarySymbolic){reference-type="ref" reference="CorollarySymbolic"} is especially useful when we understand the primary decomposition of the powers of the ideal of interest. We shall need it in proving the compatible $F$-splittings of varieties of complexes by making use of the primary decomposition of the powers of determinantal ideals. Corollary [Corollary 4](#CorollarySymbolic){reference-type="ref" reference="CorollarySymbolic"} appears implicitly in the proof that Hankel determinantal rings are $F$-pure [@CMSV Theorem 4.1]. In fact, if an ideal $I$ satisfies the conclusion of Corollary [Corollary 4](#CorollarySymbolic){reference-type="ref" reference="CorollarySymbolic"}, then its symbolic powers $\{I^{(n)}\}_{n \geq 0}$ define an $F$-split filtration, i.e., $I$ is a *symbolic $F$-split* ideal---a notion which is stronger than $F$-split, by [@dSMNB21 Corollary 5.10] and [@dSMNB21 Example 5.13]. The following result is used in proving that several ideals considered in this paper have squarefree initial ideals (in any characteristic) with respect to delicate monomial orders $<_B$ constructed in $\S$[\[Subsection PfaffianMonomialOrder\]](#Subsection PfaffianMonomialOrder){reference-type="ref" reference="Subsection PfaffianMonomialOrder"} and $\S$[5.3](#Subsection VoCMonomialOrder){reference-type="ref" reference="Subsection VoCMonomialOrder"}. The following theorem is first proved in positive characteristic by an application of Theorem [Theorem 1](#theoremFedder){reference-type="ref" reference="theoremFedder"}; it is then proved in characteristic zero by using reduction mod $p$ techniques. **Theorem 5**. *[@Varbaro-Koley Theorem 3.13] [\[theorem:sqfreeinitial\]]{#theorem:sqfreeinitial label="theorem:sqfreeinitial"} Let $S$ be a polynomial ring over a field (not necessarily of positive characteristic). Let $I$ be a radical ideal and $<_B$ a monomial order in $S$. Let $h$ be the maximum of the heights of the associated prime ideals of $I$.* *If the initial ideal $\mathop{\mathrm{{\textup{in}_{\it B}}}}(I^{(h)})$ contains a squarefree monomial, then $\mathop{\mathrm{{\textup{in}_{\it B}}}}(I)$ is a squarefree monomial ideal.* # The natural Pfaffian nullcone is strongly $F$-regular The aim of this section is to establish the $F$-regularity of the Pfaffian nullcone. We begin with recalling some known facts. ## Generalities Let $Y$ be a $2t\times n$ matrix of indeterminates and set $S\colonequals \mathbb{K}[Y]$. Let $$\Omega\colonequals\begin{pmatrix} 0 & \mathbb{I}\\ -\mathbb{I}& 0 \end{pmatrix}$$ be the $2t\times 2t$ standard symplectic block matrix, where $\mathbb{I}$ is the size $t$ identity matrix. There is a natural $\mathbb{K}$-algebra isomorphism $$R\colonequals \mathbb{K}[Y^{\mathrm{T}}\Omega Y] \cong \mathbb{K}[X]/\operatorname{Pf}_{2t+2}(X),$$ where $X$ is an $n\times n$ alternating matrix of indeterminates, and $\operatorname{Pf}_{2t+2}(X)$ the ideal generated by its principal Pfaffians of size $2t+2$. This isomorphism is induced by mapping the entries of the matrix $X$ to the corresponding entries of the alternating matrix $Y^{\mathrm{T}}\Omega Y$. The symplectic group $$\operatorname{Sp}_{2t}(\mathbb{K})\colonequals\{M\in\operatorname{GL}_{2t}(\mathbb{K})\ |\ M^{\mathrm{T}}\Omega M=\Omega\}$$ acts $\mathbb{K}$-linearly on $S$ via $$M\colon Y\mapsto MY\qquad\text{ for }\ M\in\operatorname{Sp}_{2t}(\mathbb{K}).$$ The invariant ring is precisely $R$ when $\mathbb{K}$ is infinite, see [@DeConciniProcesi76 §6] or [@Hashimoto05 Theorem 5.1]. The nullcone ideal of this group action is the ideal $$\mathfrak{P}= \mathfrak{P}(Y)\colonequals (Y^{\mathrm{T}}\Omega Y)S,$$ generated by the entries of the matrix $Y^{\mathrm{T}}\Omega Y$ in the polynomial ring $S$. The nullcone for this action of $\operatorname{Sp}_{2t}(\mathbb{K})$ on $S$, the *(natural) Pfaffian nullcone*, is the ring $S/\mathfrak{P}$. The Pfaffian nullcone is a Cohen--Macaulay normal domain with $$\begin{aligned} \label{eqn-S/p-dim} \dim S/\mathfrak{P}&=&\begin{cases} 2nt -\displaystyle{\binom{n}{2}}& \text{if }\ n\le t+1,\\ nt+\displaystyle{\binom{t+1}{2}}& \text{if }\ n\ge t,\\ \end{cases}\end{aligned}$$ according to [@HJPS Theorem 6.8] and [@Kraft-Schwarz Theorem 9.1(3)]. It is a complete intersection ring precisely if $n \leq t+1$ by [@HJPS Theorem 6.3]. The standard monomials for the Pfaffian nullcone are studied in [@Kim §4.2]. [\[RmkGeneratorsofNullconeIdeal\]]{#RmkGeneratorsofNullconeIdeal label="RmkGeneratorsofNullconeIdeal"} Let $Y$ be a size $2t\times n$ matrix of indeterminates over a field $\mathbb{K}$. Notice that when $t=1$, we have $$\begin{aligned} 2 Y^{\mathrm{T}}\Omega Y\ &=\ \begin{pmatrix} y_{1,1} & y_{2,1}\\ \vdots&\vdots\\ y_{1,n} & y_{2,n} \end{pmatrix} \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix} \begin{pmatrix} y_{1,1} & \cdots & y_{1,n}\\ y_{2,1} & \cdots & y_{2,n} \end{pmatrix} =\ \begin{pmatrix} 0 & \Delta_{1,2} & \Delta_{1,3} & \hdots & \Delta_{1,n}\\ -\Delta_{1,2} & 0 & \Delta_{2,3} & \hdots & \Delta_{2,n}\\ -\Delta_{1,3} & -\Delta_{2,3} & 0 & \hdots & \Delta_{3,n}\\ \vdots & \vdots & & \ddots & \vdots\\ -\Delta_{1,n} & -\Delta_{2,n} & -\Delta_{3,n} & \hdots & 0 \end{pmatrix}.\end{aligned}$$ In particular, $Y^{\mathrm{T}}\Omega Y$ is an alternating matrix where, for $i<j$, the matrix entry $(Y^{\mathrm{T}}\Omega Y)_{ij}$ is $$\Delta_{i,j}\colonequals y_{1,i}y_{2,j}-y_{1,j}y_{2,i}.$$ It follows that $\mathfrak{P}$ coincides with the determinantal ideal $I_2(Y)$. So, $\mathfrak{P}$ has height $n-1$ and defines an $F$-regular ring $\mathbb{K}[Y]/\mathfrak{P}$. Note that the ring $\mathbb{K}[Y^{\mathrm{T}}\Omega Y]$ is the homogeneous coordinate ring of the Grassmannian $\operatorname{Gr}_\mathbb{K}(2,n)$ under the Plücker embedding into $\mathbb{P}_\mathbb{K}^{\binom{n}{2}-1}$. More generally, for $t\ge 1$, the ring $\mathbb{K}[Y^{\mathrm{T}}\Omega Y]$ is the homogeneous coordinate ring of the order $t-1$ secant variety $\operatorname{Gr}_\mathbb{K}^{t-1}(2,n)$, the closure of the union of linear spaces spanned by $t$ points of $\operatorname{Gr}_\mathbb{K}(2,n)$ under the Plücker embedding. Moreover, for $1\le i<j\le n$, the entry $(Y^{\mathrm{T}}\Omega Y)_{i,j}$ equals $B(v_i,v_j)$, where $v_i$ and $v_j$ are the $i$-th and $j$-th columns of $Y$, and $B$ is the (nondegenerate) symplectic form $$(v_1,v_2)\mapsto v_1^{\mathrm{T}}\Omega v_2.$$ It follows that the generators of $\mathfrak{P}$ are sums of size two minors given by $$d_{ij} := (Y^{\mathrm{T}}\Omega Y)_{i,j}\ =\ (y_{1,i}y_{t+1,j}-y_{1,j}y_{t+1,i})\ +\ \cdots\ +\ (y_{t,i}y_{2t,j}-y_{t,j}y_{2t,i}).$$ for $1 \leq i<j\leq n$. ## The localization property We show that the Pfaffian nullcone has a localization property analogous to that of the generic determinantal ring, as outlined in [@BrunsVetter Proposition 2.4]. **Lemma 6**. *Let $Y=(y_{i,j})$ be a $2t\times n$ matrix of indeterminates; set $S\colonequals\mathbb{Z}[Y]$. Then there exists a $(2t-2)\times(n-1)$ matrix $Z$ with entries in $S[\frac{1}{y_{1,1}}]$, and elements $f_2,\dots,f_n$ in $S[\frac{1}{y_{1,1}}]$ such that:* 1. *the entries of $Z$ and the elements $f_2,\ldots,f_n, y_{1,1},\ldots,y_{1,n}, y_{2,1},\dots,y_{2t,1}$ taken together are algebraically independent over $\mathbb{Z}$;* 2. *along with $y_{1,1}^{-1}$, the above elements generate $S[\frac{1}{y_{1,1}}]$ as a $\mathbb{Z}$-algebra;* 3. *with $S':=\mathbb{Z}[Y']$, the ideal $\mathfrak{P}(Y)S[\frac{1}{y_{1,1}}]$ equals $\mathfrak{P}(Z)S[\frac{1}{y_{1,1}}]+(f_2,\dots,f_n)S[\frac{1}{y_{1,1}}]$, and we have an isomorphism* *$$\frac{S}{\mathfrak{P}(Y)}[\frac{1}{y_{1,1}}] \cong \frac{S'}{\mathfrak{P}(Z)}[y_{1,1},\dots,y_{1,n}, y_{2,1},\dots,y_{2t,1},\frac{1}{y_{1,1}}].$$* *Proof.* Let us map the entries of the matrix $Y$ to the corresponding entries of $YM$, where $M$ is a matrix with $n$ rows. Clearly, the ideal $(YM)$ generated by the entries of the matrix $YM$ is contained in $(Y)$, hence the ideal $\mathfrak{P}(YM)$ is contained in $\mathfrak{P}(Y)$. It follows that if $M$ is invertible in $S$, then the ideals $\mathfrak{P}(Y)$ and $\mathfrak{P}(YM)$ are equal. In particular, $\mathfrak{P}(Y)$ is unaffected by elementary column operations of the matrix $Y$. After inverting $y_{1,1}$, one may perform elementary column operations to transform $Y$ into a matrix where $y_{1,1}$ is the only nonzero entry in the first row; the resulting matrix is $$\widetilde{Y} \colonequals\begin{pmatrix} y_{1,1} & 0 & 0 & \cdots & 0\\ y_{2,1} & z_{2,2} & z_{2,3} & \cdots & z_{2,n}\\ \vdots & \vdots & \vdots & & \vdots\\ y_{t,1} & z_{t,2} & z_{t,3} & \cdots & z_{t,n}\\ y_{t+1,1} & z_{t+1,2} & z_{t+1,3} & \cdots & z_{t+1,n}\\ y_{t+2,1} & z_{t+2,2} & z_{t+2,3} & \cdots & z_{t+2,n}\\ \vdots & \vdots & \vdots & & \vdots\\ y_{2t,1} & z_{2t,2} & z_{2t,3} & \cdots & z_{2t,n} \end{pmatrix} \quad \text{where} \quad z_{i,j} = y_{i,j} - \frac{y_{i,1}y_{1,j}}{y_{1,1}}.\ $$ By construction, the ideals $\mathfrak{P}(Y)$ and $\mathfrak{P}(\widetilde{Y})$ are equal in the ring $S[\frac{1}{y_{1,1}}]$. Set $Z$ to be the submatrix of $\widetilde{Y}$ obtained by deleting the first column, and rows $1$ and $t+1$. Note that the nonzero entries of the matrix $\widetilde{Y}^{\mathrm{T}}\Omega\widetilde{Y}$ are those of $Z^{\mathrm{T}}\overline{\Omega}Z$, where $\overline{\Omega}$ is the standard symplectic block of size $2t-2$, along with the polynomials $$f_j\colonequals (\widetilde{Y}^{\mathrm{T}}\Omega\widetilde{Y})_{1,j}\ =\ y_{1,1}z_{t+1,j} + (y_{2,1}z_{t+2,j}-y_{t+2,1}z_{2,j}) + \dots + (y_{t,1}z_{2t,j}-y_{2t,1}z_{t,j})$$ for $2 \leq j \leq n$. This proves assertion $(3)$. Assertions $(1)$ and $(2)$ are readily verified since the entries of the matrix $Z$ do not involve the elements $z_{t+1,j}$ which appear (with a unit coefficient) in $f_j$ for $2 \leq j \leq n$. ◻ ## Constructing the monomial order {#Subsection PfaffianMonomialOrder} The aim of this subsection is to describe a recipe for a monomial order $<_B$ that creates special lead terms for the generators of the Pfaffian nullcone ideal. The construction of this monomial order is quite technical; we illustrate it with an example first: Let $Y$ be a $4 \times 4$ matrix of indeterminates and $\mathbb{K}$ be any field; set $S = \mathbb{K}[Y]$. To define a monomial order $<_B$ in the polynomial ring $S$, we first define an order on the variables as follows. Sort the entries of the matrix $Y$ into blocks $B_0, B_1, B_2$, and $B_3$ as suggested in the matrix $$\begin{pmatrix} 1 & 3 & 1 & 0\\ 2 & 2 & 0 & 0\\ 0 & 1 & 3 & 1\\ 0 & 0 & 2 & 2 \end{pmatrix},$$ where the $(i,j)$-entry of the matrix is the block into which $y_{i,j}$ is sorted. Thus, for instance, $y_{1,1}$ is in the block $B_1$, $y_{1,2}$ is in $B_3$, and so on. Now, for $\gamma \in B_\ell$ and $\delta \in B_{k}$, set $\gamma < \delta$ if $\ell <k$. Then, within each set $B_\ell$, fix an arbitrary order among the variables. This gives us a total variable order in the polynomial ring $S$. Our monomial order $<_B$ is the reverse lexicographical order induced by this variable order. For a polynomial $f$, let $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$ denote the initial monomial of $f$ with respect to our monomial order. Then $$\mathfrak{P}= \mathfrak{P}(Y) = (d_{i,j} \; \vert \; 1 \leq i<j\leq 4)$$ is an ideal of height $5$ in $S$; the generators $d_{i,j}$ are as displayed in Remark [\[RmkGeneratorsofNullconeIdeal\]](#RmkGeneratorsofNullconeIdeal){reference-type="ref" reference="RmkGeneratorsofNullconeIdeal"}. One has $$\begin{gathered} \mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{1,2}) = y_{1,1}y_{3,2},\qquad \mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{2,3}) = y_{1,2}y_{3,3},\qquad \mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{3,4}) = y_{1,3}y_{3,4},\\ \mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{1,3}) = y_{2,1}y_{4,3},\qquad \mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{2,4}) = y_{2,2}y_{4,4},\qquad\mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{1,4}) = y_{2,1}y_{4,4}.\end{gathered}$$ Let ${\mathfrak{a}}$ be the ideal of $S$ generated by the elements $d_{1,2},d_{2,3}, d_{3,4}, d_{1,3}$, and $d_{2,4}$. Since the initial terms of the generators of ${\mathfrak{a}}$ are pairwise coprime, the generators of ${\mathfrak{a}}$ form a Gröbner basis and it follows that the ideal ${\mathfrak{a}}$ is generated by a regular sequence of maximum length in $\mathfrak{P}$. The construction of this monomial order is crucial in establishing the $F$-regularity of the Pfaffian nullcone $S/\mathfrak{P}$, as we show next. From now on, asssume that the underlying field $\mathbb{K}$ has positive characteristic $p$. Note that the polynomial $$f := d_{1,2}d_{2,3}d_{3,4}d_{1,3}d_{2,4}$$ is such that $$f^{p-1} \in ({\mathfrak{a}}^{[p]}:{\mathfrak{a}}) \smallsetminus{\mathfrak{m}}_S^{[p]},$$ where ${\mathfrak{m}}_S$ denotes the homogeneous maximal ideal of $S$. This is so because its initial term $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$ is squarefree. It follows from Corollary [Corollary 3](#corMain){reference-type="ref" reference="corMain"} that $S/\mathfrak{P}$ is $F$-pure. In fact, since $y_{1,4}$ does not divide $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$, we get $$y_{1,4}f^{p-1} \in y_{1,4}({\mathfrak{a}}^{[p]}:{\mathfrak{a}}) \subseteq y_{1,4}(\mathfrak{P}^{[p]}:\mathfrak{P}) \quad \text{while}\quad y_{1,4}f^{p-1}\notin {\mathfrak{m}}_S^{[p]}.$$ Since the ring $\frac{S}{\mathfrak{P}}[\frac{1}{y_{1,4}}]$ is a smooth extension of the determinantal ring defined by the size two minors of a $2 \times 3$ matrix of indeterminates by Lemma [Lemma 6](#lemma:matrix:invert){reference-type="ref" reference="lemma:matrix:invert"}, it follows from Theorem [Theorem 2](#theoremGlassbrenner){reference-type="ref" reference="theoremGlassbrenner"} that the Pfaffian nullcone $S/\mathfrak{P}$ is strongly $F$-regular. Finally, since $f$ lies in the ideal $\mathfrak{P}^5$, it is clear that $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$ lies in $\mathop{\mathrm{{\textup{in}_{\it B}}}}(\mathfrak{P}^5)$, and hence in $\mathop{\mathrm{{\textup{in}_{\it B}}}}(\mathfrak{P}^{(5)})$. By Theorem [\[theorem:sqfreeinitial\]](#theorem:sqfreeinitial){reference-type="ref" reference="theorem:sqfreeinitial"}, it immediately follows that the Pfaffian nullcone ideal $\mathfrak{P}$ has a squarefree initial ideal (with respect to the monomial order $<_B$) in any characteristic. We next construct the monomial order $<_B$ illustrated in the above example. For ease of notation, we relabel the entries of lower half of the $2t \times n$ matrix $Y = (y_{i,j})$ as follows: Let $w_{i,j} = y_{i+t,n-j+1}$ for $1 \leq i \leq t$ and $1 \leq j \leq n$. Then we have $$Y =\begin{pmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,n}\\ \vdots & \vdots & & \vdots\\ y_{t,1} & y_{t,2} & \cdots & y_{t,n}\\ w_{1,n} & w_{1,n-1} & \cdots & w_{1,1}\\ \vdots & \vdots & &\vdots\\ w_{t,n} & w_{t,n-1} & \cdots & w_{t,1} \end{pmatrix}.$$ Recall from Remark [\[RmkGeneratorsofNullconeIdeal\]](#RmkGeneratorsofNullconeIdeal){reference-type="ref" reference="RmkGeneratorsofNullconeIdeal"} that the entries of the Pfaffian nullcone ideal $(Y^{\mathrm{T}}\Omega Y)$ in $\mathbb{K}[Y]$ are $$d_{i,j} = \det \begin{pmatrix} y_{1,i} & y_{1,j}\\ w_{1,n-i+1} & w_{1,n-j+1} \end{pmatrix} + \cdots + \det \begin{pmatrix} y_{t,i} & y_{t,j}\\ w_{t,n-i+1} & w_{t,n-j+1} \end{pmatrix},$$ for $1 \leq i<j \leq n$. The following computation is the technical heart of this section: **Lemma 7**. *Let $\mathbb{K}$ be any field. There exists a monomial order $<_B$ in $\mathbb{K}[Y]$ such that for all $1 \leq i < j \leq n$ with $j-i \leq t$, we have $$\mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{i,j}) = y_{j-i,i}w_{j-i,n-j+1},$$ where $\mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{i,j})$ denotes the initial monomial of $d_{i,j}$ with respect to the order $<_B$.* *Proof.* To define a monomial order $<_B$ in the polynomial ring $S=\mathbb{K}[Y]$, we first define an order on the variables as follows. Sort the entries of the matrix $Y$ into blocks $B_0,B_1, \dots, B_{n-1}$ according to the following formula: $$\begin{gathered} \label{eqn-blocks} \text{$y_{i,j},w_{i,j}$ are in block $B_\ell$ where } \ell = \begin{cases} \textrm{(a)}\quad 2j +i -2 & \text{if \; } 1 \leq j< \dfrac{n-i+1}{2}, \\ \textrm{(b)}\quad 2n-2j-i & \text{if \; } \dfrac{n-i+1}{2} \leq j < n-i+1, \\ \textrm{(c)}\quad 0 & \text{otherwise.} \end{cases} \end{gathered}$$ Now, for $\gamma \in B_\ell$ and $\delta \in B_{k}$, set $\gamma < \delta$ if $\ell <k$. Then, within each set $B_\ell$, fix an arbitrary order among the variables. This gives us a total variable order in $S$. Our monomial order $<_B$ is the reverse lexicographical order induced by this variable order in $S$. For a polynomial $f$, let $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$ denote the initial monomial of $f$ with respect to our monomial order. We claim that for $i<j$ and with $$d_{i,j} = \sum_{s=1}^t y_{s,i}w_{s,n-j+1} - \sum_{s=1}^t y_{s,j}w_{s,n-i+1},$$ we have $\mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{i,j}) = y_{j-i,i}w_{j-i,n-j+1}$. To prove our claim, we must show that $$y_{j-i,i}w_{j-i,n-j+1} \geq y_{s,i}w_{s,n-j+1}, \quad \text{and} \quad y_{j-i,i}w_{j-i,n-j+1} \geq y_{s,j}w_{s,n-i+1}$$ for all $1 \leq s \leq t$. As the order is reverse lexicographical, we must show that, for all $1 \leq s \leq t$, $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are greater than or equal to at least one of $y_{s,i}$ and $w_{s,n-j+1}$, as well as at least one of $y_{s,j}$ and $w_{s,n-i+1}$. This can be done by analyzing all possible cases one by one. We include all the details for the sake of completeness. Our proof will proceed as follows: First, we will show that $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are in the same block $B_\ell$. Then we will show that for $s \neq j-i$ at least one of $y_{s,i}$ and $w_{s,n-j+1}$ is in $B_k$ for some $k<\ell$. Lastly, we will show that at least one of $y_{s,j}$ and $w_{s,n-i+1}$ is in $B_k$ for some $k<\ell$. **Part 1**. We claim that $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are both in $B_{i+j-2}$ if $i+j < n+1$, and are in $B_{2n-i-j}$ otherwise. To show this, we first show that $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are not in $B_0$. Suppose $y_{j-i,i} \in B_0$. This requires that $n-(j-i)+1 \leq i$, implying $n+1 \leq j$, which is impossible. Suppose $w_{j-i,n-j+1} \in B_0$. This requires that $n-(j-i)+1 \leq n-j+1$, implying $i \leq 0$, which also impossible. Now, $i+j<n+1$ is equivalent to $i<(n-(j-i)+1)/2$ as well as to $(n-(j-i)+1)/2<n-j+1$. The former means that the block of $y_{j-i,i}$ is decided via formula (a) in [\[eqn-blocks\]](#eqn-blocks){reference-type="eqref" reference="eqn-blocks"} and equals $B_{2i+(j-i)-2}=B_{i+j-2}$; the latter that the block of $w_{j-i,n-j+1}$ comes from formula (b) and is $B_{2n-2(n+1-j)-(j-i)}=B_{i+j-2}$. If on the other hand $i+j>n+1$ then $y_{j-i,i}$ follows case (b) and $w_{j-i,n-j+1}$ follows case (a) by the same computation. The corresponding blocks are computed as $B_{2n-2(i)-(j-i)}=B_{2n-i-j}$ and $B_{2(n-j+1)+(j-i)-2}=B_{2n-j-i}$. In the case $i+j=n+1$, $y_{j-i,i}$ and $w_{j-i,n-j+1}$ have the same label and are in block $B_{i+j-2}=B_{2n-i-j}$. **Part 2**. Here we will show that if $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are in $B_\ell$, then $s \neq j-i$ implies that at least one of $y_{s,i}$ and $w_{s,n-j+1}$ is in $B_k$ where $k<\ell$. If $y_{s,i}$ or $w_{s,n-j+1}$ is in $B_0$, then we are done. So assume that $y_{s,i}$ and $w_{s,n-j+1}$ are not in $B_0$. - Suppose $i+j < n+1$ and $s < j-i$. Then $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are in $B_{i+j-2}$. By the proof of Part 1, $i < \dfrac{n-(j-i)+1}{2} < \dfrac{n-s+1}{2}$, and so $y_{s,i} \in B_{2i+s-2}$. As $s < j-i$ implies that $2i+s-2 < i+j-2$, $y_{s,i}$ is in a lower block than $y_{j-i,i}$. - Suppose $i+j < n+1$ and $s > j-i$. Then, $y_{j-i,i}, w_{j-i,n-j+1}\in B_{i+j-2}$. By the proof of Part 1, $n-j+1 > \dfrac{n-(j-i)+1}{2} > \dfrac{n-s+1}{2}$, and so $w_{s,n-j+1} \in B_{2j-s-2}$. As $s > j-i$ implies that $2j-s-2 < i+j-2$, $w_{s,n-j+1}$ is in a lower block than $w_{j-i,n-j+1}$. - Suppose $i+j \geq n+1$ and $s <j-i$. Then, $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are in $B_{2n-i-j}$. By the proof of Part 1, $n-j+1 \le \dfrac{n-(j-i)+1}{2} < \dfrac{n-s+1}{2}$, and so $w_{s,n-j+1} \in B_{2n-2j+s}$. As $s < j-i$ implies that $2n-2j+s < 2n-i-j$, $w_{s,n-j+1}$ is in a lower block than $w_{j-i,n-j+1}$. - Suppose $i+j \geq n+1$ and $s > j-i$. Then, $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are in $B_{2n-i-j}$. By the proof of Part 1, $i\geq \dfrac{n-(j-i)+1}{2} > \dfrac{n-s+1}{2}$, and so $y_{s,i} \in B_{2n-2i-s}$. As $s > j-i$ implies hat $2n-2i-s< 2n-i-j$, $y_{s,i}$ is in a lower blck than $y_{j-i,i}$. **Part 3**. Here we will show that if $y_{j-i,i},w_{j-i,n-j+1}\in B_\ell$, then for $s\neq j-i$ at least one of $y_{s,j}$ and $w_{s,n-i+1}$ is in $B_k$ for some $k<\ell$. If $y_{s,j}$ or $w_{s,n-i+1}$ is in $B_0$, there is nothing to show by the proof of Part 1. So assume that $y_{s,j}$ and $w_{s,n-i+1}$ are not in $B_0$. - Suppose that $i+j < n+1$. By Part 1, $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are in $B_{i+j-2}$. Since $i+j < n+1$, an inequality $n-i+1 < \frac{n-s+1}{2}$ would imply that $s < 2i-n-1 < i-j<0$, which we know to be false. Hence, $n-i+1 \geq \frac{n-s+1}{2}$, and thus the block of $w_{s,n-i+1}$ is decided by case (b) and equals $B_{2n-2(n-i+1)-s}=B_{2i-s-2}$. As $s > 0 > i-j$ implies that $2i-s-2< i+j-2$, $w_{s,n-i+1}$ is in a lower block than $w_{j-i,n-j+1}$. - Suppose that $i+j \geq n+1$. By Part 2, $y_{j-i,i}$ and $w_{j-i,n-j+1}$ are in $B_{2n-i-j}$. Since $i+j \geq n+1$, an inequality $j < \frac{n-s+1}{2}$ would imply that $s <n-2j+1 \le i-j<0$, which we know to be false. Thus, $j \geq \frac{n-s+1}{2}$, and thus the block of $y_{s,j}$ is decided by case (b) and equals $B_{2n-2j-s}$. As $s > 0 > i-j$ implies that $2n-2j-s < 2n-i-j$, and so $y_{s,j}$ is in a lower block than $y_{j-i,i}$. This finishes the analysis of each possible case. ◻ Having established the monomial order $<_B$, we next exhibit a natural choice of a maximal regular sequence in the Pfaffian nullcone ideal. **Lemma 8**. *Let $\mathbb{K}$ be any field and ${\mathfrak{a}}\subseteq \mathfrak{P}(Y)$ be the ideal of $\mathbb{K}[Y]$ generated by the set $$\alpha = \{d_{i,j} \;\vert \; 1 \leq i < j \leq n \text{ and } j-i \leq t\}.$$ Then the ideal ${\mathfrak{a}}$ has the same height as $\mathfrak{P}(Y)$ and $\alpha$ is a regular sequence.* *Proof.* First, we note that $$\vert \alpha \vert = \begin{cases} \binom{n}{2} & \text{if \;} n \leq t+1,\\ nt - \binom{t+1}{2} & \text{if \:} n \geq t, \end{cases}$$ where $\vert \alpha \vert$ denotes the cardinality of $\alpha$. By [@HJPS Theorem 6.8], we have that $|{\mathfrak{a}}| = \mathop{\mathrm{\textup{ht}}}(\mathfrak{P}(Y))$. By Lemma [Lemma 7](#LemmaMonomialOrder){reference-type="ref" reference="LemmaMonomialOrder"}, there exists a monomial order in $\mathbb{K}[Y]$ such that for all $1 \leq i < j \leq n$ with $j-i \leq t$, we have $$\mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{i,j}) = y_{j-i,i}w_{j-i,n-j+1}.$$ Notice that if $(i,j)\neq (k,\ell)$ then $\mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{i,j})$ and $\mathop{\mathrm{{\textup{in}_{\it B}}}}(d_{l,k})$ are relatively prime. Thus, the elements of $\alpha$ form a Gröbner basis for ${\mathfrak{a}}$ (see, for example, [@HerzogHibi11 Corollary 2.3.4.]), and the initial terms of $\alpha$ form a regular sequence. Therefore, $$\dim(R/{\mathfrak{a}}) = \dim(R/\mathop{\mathrm{{\textup{in}_{\it B}}}}({\mathfrak{a}})) = \mathop{\mathrm{\textup{depth}}}(R/\mathop{\mathrm{{\textup{in}_{\it B}}}}({\mathfrak{a}})) \leq \mathop{\mathrm{\textup{depth}}}(R/{\mathfrak{a}}) \leq \dim(R/{\mathfrak{a}}),$$ so that we must have equality throughout (see [@HerzogHibi11 Corollary 3.3.4]). In particular, $$\vert \alpha \vert \geq \mathop{\mathrm{\textup{ht}}}({\mathfrak{a}}) = \mathop{\mathrm{\textup{ht}}}(\mathop{\mathrm{{\textup{in}_{\it B}}}}({\mathfrak{a}})) = \vert \alpha \vert,$$ and the lemma follows from [@HJPS Theorem 6.8]. ◻ We are now ready to prove the main results of this section. **Theorem 9**. *Let $Y$ be a matrix of indeterminates of size $2t \times n$ for positive integers $t$ and $n$. Let $\mathbb{K}$ be a field and set $S:=\mathbb{K}[Y]$.* 1. *If $\mathbb{K}$ is an $F$-finite field of positive characteristic, the natural Pfaffian nullcone $S/\mathfrak{P}(Y)$ is a strongly $F$-regular ring.* 2. *In consequence, if $\mathbb{K}$ has characteristic zero, the natural Pfaffian nullcone $S/\mathfrak{P}(Y)$ has log-terminal, and hence rational, singularities.* *Proof.* Assertion $(2)$ follows from $(1)$ since rings of characteristic zero of $F$-regular type have log-terminal singularities, which are rational, compare [@Smith Theorem 4.3] and [@HaraWatanabe]. We therefore concentrate on the case where the characteristic of $\mathbb{K}$ is $p>0$. We proceed by induction on $t$. The statement is clear for $t=1$, since then by Remark [\[RmkGeneratorsofNullconeIdeal\]](#RmkGeneratorsofNullconeIdeal){reference-type="ref" reference="RmkGeneratorsofNullconeIdeal"}, the ideal $\mathfrak{P}(Y)$ equals the determinantal ideal $I_2(Y)$ of the size two minors of $Y$. The corresponding ring is strongly $F$-regular as it is a Segre product of standard graded polynomial rings. Now assume that the assertion holds for some $t>1$. By Lemma [Lemma 6](#lemma:matrix:invert){reference-type="ref" reference="lemma:matrix:invert"}, we have $$\frac{S}{\mathfrak{P}(Y)} [\frac{1}{y_{1,n}}] \cong \frac{\mathbb{K}[Z]}{\mathfrak{P}(Z)}[y_{1,1},\dots,y_{1,n}, y_{2,n},\dots,y_{2t,n},\frac{1}{y_{1,n}}]$$ where $Z$ is a matrix of indeterminates of size $(2t-2)\times n$. It follows by induction that the ring $\frac{S}{\mathfrak{P}(Y)} [\frac{1}{y_{1,n}}]$ is strongly $F$-regular. In order to apply Theorem [Theorem 2](#theoremGlassbrenner){reference-type="ref" reference="theoremGlassbrenner"}, we must show that $$y_{1,n}(\mathfrak{P}(Y)^{[p]}:\mathfrak{P}(Y)) \not\subseteq {\mathfrak{m}}_S^{[p]},$$ where ${\mathfrak{m}}_S$ is the homogeneous maximal ideal of $S$. By Corollary [Corollary 3](#corMain){reference-type="ref" reference="corMain"}, it suffices to find an ideal ${\mathfrak{a}}$ in $\mathfrak{P}(Y)$ generated by a regular sequence $\alpha$ of length equal to the height of $\mathfrak{P}(Y)$ such that $$y_{1,n}({\mathfrak{a}}^{[p]}:{\mathfrak{a}}) \not\subseteq {\mathfrak{m}}_S^{[p]}.$$ Let $\alpha = \{d_{i,j} \;\vert \; 1 \leq i < j \leq n \text{ and } j-i \leq t\}$ and ${\mathfrak{a}}=\alpha S$. Consider the polynomial $$f := \prod_{d_{i,j}\in \alpha}d_{i,j}.$$ Clearly $$y_{1,n}f^{p-1} \in y_{1,n}({\mathfrak{a}}^{[p]}:{\mathfrak{a}}).$$ Recall that for polynomials $g$ and $h$ and a fixed monomial order $<_B$ in $S$, we have $\mathop{\mathrm{{\textup{in}_{\it B}}}}(gh) =\mathop{\mathrm{{\textup{in}_{\it B}}}}(g)\mathop{\mathrm{{\textup{in}_{\it B}}}}(h)$. We choose the monomial order $<_B$ as constructed in Lemma [Lemma 7](#LemmaMonomialOrder){reference-type="ref" reference="LemmaMonomialOrder"}. Then we get: $$\begin{aligned} \mathop{\mathrm{{\textup{in}_{\it B}}}}(y_{1,n} f^{p-1}) &=& y_{1,n}\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)^{p-1}\\ % &=&y_{1,n}\prod_{\substack{1\le i<j\le n\\j-i\le t}}(y_{j-i,i}w_{j-i,n-j+1})^{p-1}\\ &=& y_{1,n} \left(\prod_{\substack{1 \leq i <j\le n \\ j-i \leq t}}(y_{j-i,i}w_{j-i,n-j+1})\right)^{p-1} \notin {\mathfrak{m}}_S^{[p]}. \end{aligned}$$ Since ${\mathfrak{m}}_S^{[p]}$ is a monomial ideal, an element lies in ${\mathfrak{m}}_S^{[p]}$ if and only if each of its terms does, and so $$y_{1,n}f^{p-1}\notin{\mathfrak{m}}_S^{[p]}.$$ We are done by Theorem [Theorem 2](#theoremGlassbrenner){reference-type="ref" reference="theoremGlassbrenner"}. ◻ **Corollary 10**. *Let $Y$ be a matrix of indeterminates of size $2t \times n$ for positive integers $t$ and $n$. Let $\mathbb{K}$ be a field (of any characteristic) and set $S:=\mathbb{K}[Y]$. The Pfaffian nullcone ideal $\mathfrak{P}(Y)$ has a squarefree initial ideal.* *Proof.* Let $h$ be the height of $\mathfrak{P}= \mathfrak{P}(Y)$ and the polynomial $f$ be as constructed in the proof of Theorem [Theorem 9](#thm-PfaffianNullcone-Fregular){reference-type="ref" reference="thm-PfaffianNullcone-Fregular"}, i.e., $$f := \prod_{d_{i,j}\in \alpha}d_{i,j}.$$ Then since $f$ lies in the ideal $\mathfrak{P}^h$, it is clear that $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$ lies in $\mathop{\mathrm{{\textup{in}_{\it B}}}}(\mathfrak{P}^h)$, and hence in $\mathop{\mathrm{{\textup{in}_{\it B}}}}(\mathfrak{P}^{(h)})$. By Theorem [\[theorem:sqfreeinitial\]](#theorem:sqfreeinitial){reference-type="ref" reference="theorem:sqfreeinitial"}, it immediately follows that the Pfaffian nullcone ideal $\mathfrak{P}$ has a squarefree initial ideal (with respect to the monomial order $<_B$). ◻ We end this section with the following: Let $Y$ be a $2t \times n$ matrix of indeterminates for positive integers $t$ and $n$, and let $\mathfrak{P}$ denote the natural Pfaffian nullcone ideal in the polynomial ring $\mathbb{K}[Y]$. Denote by $$\mathcal{R}^S(\mathfrak{P}):= \bigoplus_{k \geq 0} \mathfrak{P}^{(k)} \quad \text{and} \quad G^S(\mathfrak{P}):= \bigoplus_{k \geq 0} \mathfrak{P}^{(k)}/\mathfrak{P}^{(k+1)}$$ the *symbolic Rees algebra* and the *symbolic associated graded algebra* of $\mathfrak{P}$ respectively. Are these rings Noetherian? The proof of Theorem [Theorem 9](#thm-PfaffianNullcone-Fregular){reference-type="ref" reference="thm-PfaffianNullcone-Fregular"} shows that the Pfaffian nullcone ring is *symbolic $F$-split* ([@dSMNB21 Corollary 5.10]). It immediately follows by [@dSMNB21 Theorem 4.7] that the symbolic Rees algebra and the symbolic associated graded algebra of the ideal $\mathfrak{P}$ are $F$-split (hence reduced). However, we do not know if either of these blowup algebras are Noetherian. # The divisor class group of the Pfaffian nullcone In this section, we give a characteristic-free description of the divisor class group and the Gorenstein property of the Pfaffian nullcone. This section may be viewed as an application of the technique of principal radical systems introduced by Hochster and Eagon in [@Hochster-Eagon] and studied for the Pfaffian nullcone in [@HJPS Theorem 6.7]. **Theorem 11**. *Let $Y$ be a $2t \times n$ matrix of indeterminates for positive integers $t$ and $n$, and $\mathbb{K}$ be any field. Let $R_{t,n}$ denote the Pfaffian nullcone $\mathbb{K}[Y]/\mathfrak{P}(Y)$.* 1. *If $n\geq t+1$ then the divisor class group of $R_{t,n}$ is the group $\mathbb{Z}$ of integers. Otherwise, $R_{t,n}$ is a unique factorization domain.* 2. *Let $Y|_t$ be the submatrix of $Y$ consisting of the first $t$ columns. If $n \geq t+1$, the ideal ${\mathfrak{p}}:= I_{t}(Y|_t)$ of $R_{t,n}$ is prime of height one and generates the divisor class group. Further, the canonical class of $R_{t,n}$ is given by ${\mathfrak{p}}^{(n-t-1)}$.* 3. *The ring $R_{t,n}$ is Gorenstein precisely if $n \leq t+1$.* *Proof.* Let $W$ be a multiplicatively closed set of $R_{t,n}$. Recall the Nagata exact sequence (see [@Nagata]) of divisor class groups $$0 \longrightarrow U \longrightarrow \mathop{\mathrm{\textup{Cl}}}(R_{t,n}) \longrightarrow \mathop{\mathrm{\textup{Cl}}}(W^{-1}R_{t,n}) \longrightarrow 0,$$ where $U$ is the subgroup of $\mathop{\mathrm{\textup{Cl}}}(R_{t,n})$ consisting of the classes of pure height one ideals which have a nonempty intersection with $W$. Let $W$ be the multiplicatively closed set of $R_{t,n}$ consisting of the powers of $y_{1,1}$. For the remainder of the proof, fix $t \geq 2$. The principal ideal $y_{1,1}R_{t,n}$ is prime by [@HJPS Theorem 6.7(2)]. Thus, the only pure height one ideal containing $y_{1,1}$ is principal and $U$ is the trivial group $\{[y_{1,1}] = 0\}$. Since the class group is unaffected by a polynomial extension or by inverting at a prime element, it follows from Lemma [Lemma 6](#lemma:matrix:invert){reference-type="ref" reference="lemma:matrix:invert"}(3) that the class groups $\mathop{\mathrm{\textup{Cl}}}(W^{-1}R_{t,n})$ and $\mathop{\mathrm{\textup{Cl}}}(R_{t-1,n-1})$ are isomorphic. This gives us an explicit inductive isomorphism of class groups $$\begin{gathered} \label{eqn-Cl-reduction} \mathop{\mathrm{\textup{Cl}}}(R_{t,n}) \cong \mathop{\mathrm{\textup{Cl}}}(R_{t-1,n-1}) \text{ with } [\mathfrak{q}] \mapsto [S^{-1}\mathfrak{q}]. \end{gathered}$$ By Remark [\[RmkGeneratorsofNullconeIdeal\]](#RmkGeneratorsofNullconeIdeal){reference-type="ref" reference="RmkGeneratorsofNullconeIdeal"}, $R_{1,n-t+1}$ is the generic determinantal ring $\mathbb{K}[Y_{2 \times (n-t+1)}]/I_2(Y)$ and its class group can be computed directly using the Künneth formula for local cohomology modules [@GotoWatanabe Theorem 4.1.5] since it is a Segre product of standard graded polynomial rings. Inductively, we get $$\mathop{\mathrm{\textup{Cl}}}(R_{t,n}) \cong \mathop{\mathrm{\textup{Cl}}}(R_{1,n-t+1}) = \begin{cases} \mathbb{Z}& \text{if \;} n-t+1 \geq 2, \\ 0 & \text{otherwise}. \end{cases}$$ Assertion $(1)$ immediately follows. Since the canonical module localizes, Equation [\[eqn-Cl-reduction\]](#eqn-Cl-reduction){reference-type="eqref" reference="eqn-Cl-reduction"} also proves assertion $(3)$. We now prove assertion $(2)$. Let the $(2t-2)\times n$ matrix $Z$ and the elements $f_2, \ldots, f_n$ be as in Lemma [Lemma 6](#lemma:matrix:invert){reference-type="ref" reference="lemma:matrix:invert"}. We have an isomorphism of rings $$\begin{aligned} \mathbb{K}[Y][\frac{1}{y_{1,1}}] &\stackrel{\simeq}{\longrightarrow} \mathbb{K}[Z][y_{1,1}, \ldots, y_{1,n},y_{2,1}, \ldots, y_{2t,n}, f_2, \ldots, f_n, \frac{1}{y_{1,1}}]\\ \intertext{induced by elementary column operations of the matrix $Y$ that sends} \mathfrak{P}(Y) + I_t(Y\vert_t) &\mapsto \mathfrak{P}(Z) + I_{t-1}(Z\vert_{t-1}) +(f_2, \ldots, f_n). \end{aligned}$$ By Lemma [Lemma 6](#lemma:matrix:invert){reference-type="ref" reference="lemma:matrix:invert"}(3), this map descends to the isomorphism $$\begin{aligned} R_{t,n}[\frac{1}{y_{1,1}}] &\stackrel{\simeq}{\longrightarrow} R_{t-1,n-1}[y_{1,1}, \ldots, y_{1,n},y_{2,1}, \ldots, y_{2t,n}, \frac{1}{y_{1,1}}]\\ \intertext{with} {\mathfrak{p}}:= I_t(Y\vert_t) &\mapsto I_{t-1}{(Z\vert_{t-1}}). \end{aligned}$$ Put $t=2$ in the above isomorphism. Clearly the ideal $\mathfrak{q}:= I_1(Z\vert_1)$ is prime of height one in the ring $R_{1,n-1}$ defined by the size two minors of the $2 \times (n-1)$ generic matrix $Z$. Further, the prime ideal $\mathfrak{q}$ generates the class group of $R_{1,n-1}$ and the canonical class is given by $\mathfrak{q}^{{(n-2})}$. This can be computed directly from the fact that $R_{1,n-1}$ is a Segre product of standard graded polynomial rings; alternatively see [@Svanes]. Assertion $(2)$ now follows by induction along the above isomorphism. ◻ Let $Y$ be a $2t \times n$ matrix of indeterminates for positive integers $t$ and $n$. It follows from Theorem [Theorem 11](#TheoremClassGroup){reference-type="ref" reference="TheoremClassGroup"} and [@HJPS Theorem 6.3] that the Pfaffian nullcone $\mathbb{K}[Y]/\mathfrak{P}(Y)$ is Gorenstein if and only if it is $\mathbb{Q}$-Gorenstein if and only if it is a complete intersection ring if and only if $n \leq t+1$. Further, note that $n=t+1$ gives us the only Gorenstein Pfaffian nullcone which is not a unique factorization domain, and that $n=t+2$ gives us the only (non-UFD) Pfaffian nullcone whose class group is generated by the canonical class. # Compatible $F$-splitting of the natural determinantal nullcone, and the varieties of complexes In this section, we prove that the determinantal nullcone is $F$-pure and that each of its irreducible components are $F$-regular. We begin with recalling some known facts about the determinantal nullcone and the varieties of complexes. ## Generalities {#sec-splitting-generalities} For positive integers $m$, $n$, and $t$, let $Y$ and $Z$ be $m\times t$ and $t\times n$ matrices of indeterminates respectively. Set $S$ to be the polynomial ring $\mathbb{K}[Y,Z]$, and take $R$ to be the $\mathbb{K}$-subalgebra generated by the entries of the product matrix $YZ$. There is a natural $\mathbb{K}$-algebra isomorphism $$R := \mathbb{K}[YZ] \cong \mathbb{K}[X]/I_{t+1}(X)$$ where $X$ is an $n \times n$ matrix of indeterminates, and $I_{t+1}(X)$ is the ideal generated by its size $t+1$ minors. This isomorphism is induced by mapping the entries of the matrix $X$ to the corresponding entries of the matrix $YZ$. The general linear group $\operatorname{GL}_t(\mathbb{K})$ acts $\mathbb{K}$-linearly on $S$ via $$M\colon\begin{cases} Y & \mapsto YM^{-1}\\ Z & \mapsto MZ\end{cases}$$ where $M\in\operatorname{GL}_t(\mathbb{K})$. When the field $\mathbb{K}$ is infinite, $R$ is precisely the ring of invariants, see [@DeConciniProcesi76 §3] or [@Hashimoto05 Theorem 4.1]. The nullcone of the natural embedding $$R = \mathbb{K}[YZ] \subseteq \mathbb{K}[Y,Z] = S$$ is the typically non-equidimensional ring $\mathbb{K}[Y,Z]/(YZ)$. In this article, we refer to this ring as the *(natural) determinantal nullcone*. When $\mathbb{K}$ has characteristic zero, the group $\operatorname{GL}_t(\mathbb{K})$ is linearly reductive and thus the determinantal ring $R$ splits from $S$ as an $R$-module. When $\mathbb{K}$ has positive characteristic, this embedding typically does *not* split by [@HJPS Theorem 1.1]. This is due to the Cohen--Macaulay property of the minimal prime ideals of the nullcone ideal ([@Huneke Theorem 6.2]), in conjunction with the flatness of Frobenius. We next discuss the primary decomposition of the determinantal nullcone ideal. Observe that any point $P$ in the zero set of the ideal $(YZ)S$ in $\mathbb{K}^{mt+tn}$ may be regarded as an ordered pair of matrices $P = (A,B)$, where $A$ and $B$ are points in $\mathbb{K}^{mt}$ and $\mathbb{K}^{tn}$ respectively, such that the product matrix $AB$ is zero. Let $r$ and $s$ denote the ranks of the matrices $A$ and $B$ respectively. A point of the zero set of $(YZ)S$ maybe regarded as a complex (of length two) $$\mathbb{K}^n \xlongrightarrow[\text{}]{B} \mathbb{K}^t \xlongrightarrow[\text{}]{A} \mathbb{K}^m.$$ Consequently, we must have that $r+s \leq t$. Consider the ideals $${\mathfrak{p}}_{r,s} = {\mathfrak{p}}_{r,s}(Y,Z):= I_{r+1}(Y)+I_{s+1}(Z)+\ensuremath{(YZ)}$$ of $S$, where $r \leq \min \{m,t\}$, $s \leq \min \{t,n \}$, and $r+s \leq t$. These are precisely the ideals defining the varieties of complexes (of length two) as introduced by Buchsbaum--Eisenbud in [@Buchsbaum-Eisenbud]. The above discussion gives us a one-to-one correspondence between the points of the zero set of $(YZ)S$ with the varieties of complexes of length two. Notice that the ideal ${\mathfrak{p}}_{r,s}$ contains ${\mathfrak{p}}_{r',s'}$ whenever $r \leq r'$ and $s \leq s'$. Therefore, the minimal prime ideals of the determinantal nullcone are the ideals defining the varieties of exact complexes: $$(YZ)S = \bigcap_{r+s=t} {\mathfrak{p}}_{r,s}(Y,Z).$$ The determinantal nullcone $(YZ)S$ is a radical ideal by [@Musili-Sheshadri]; see also [@Mehta-Trivedi2 Theorem 4.1]. It is typically non-equidimensional since, by [@DeConciniStrickland Lemma 2.3], we have $$\dim(S/{\mathfrak{p}}_{r,s})=r(m-r+t)+s(n-s+t)-rs.$$ The varieties of complexes are Cohen--Macaulay normal domains in any characteristic by [@Huneke Theorems 6.2, 7.1] and [@DeConciniStrickland Theorem 2.7]. Their divisor class groups and Gorenstein property are determined independently in [@Bruns2] and [@Yoshino]. Kempf showed that the varieties of complexes $S/{\mathfrak{p}}_{r,s}$ have rational singularities in characteristic zero ([@Kempf2], [@Kempf1]). In [@MehtaTrivedi] it is shown that in positive characteristic, they are $F$-rational relative to the resolution given by Kempf, and that they are also $F$-split. ## The localization property We start with a localization property for the varieties of complexes analogous to that of the Pfaffian nullcone in Lemma [Lemma 6](#lemma:matrix:invert){reference-type="ref" reference="lemma:matrix:invert"}. **Lemma 12**. *Let $Y=(y_{i,j})$ and $Z=(z_{i,j})$ be matrices of indeterminates of sizes $m \times t$ and $t \times n$ respectively; set $S := \mathbb{Z}[Y,Z]$. Let $Z'$ be the submatrix of $Z$ obtained by deleting the first row. Then there exists a matrix $Y'$ of size $(m-1) \times (t-1)$ and elements $f_1, \ldots f_n$ in $S[\frac{1}{y_{1,1}}]$ such that:* 1. *the entries of $Y'$, the entries of $Z'$, and the elements $f_1, \ldots, f_n$ taken together are algebraically independent over $\mathbb{Z}$;* 2. *along with $y_{1,1}$ and $y_{1,1}^{-1}$, the above elements generate $S[\frac{1}{y_{1,1}}]$ as a $\mathbb{Z}$-algebra;* 3. *with $S' := \mathbb{Z}[Y',Z']$, the ideal ${\mathfrak{p}}_{r,s}(Y,Z)S[\frac{1}{y_{1,1}}]$ equals ${\mathfrak{p}}_{r-1,s}(Y',Z')S[\frac{1}{y_{1,1}}] + (f_1, \ldots, f_n)S[\frac{1}{y_{1,1}}]$, and we have an isomorphism $$\frac{S}{{\mathfrak{p}}_{r,s}(Y,Z)}[\frac{1}{y_{1,1}}] \cong \frac{S'}{{\mathfrak{p}}_{r-1,s}(Y',Z')}[y_{1,1}, \ldots, y_{m,1},y_{1,2}, \ldots, y_{1,t}, \frac{1}{y_{1,1}}].$$* *Proof.* Let us map the entries of the matrix $Y$ to the corresponding entries of $MY$, where $M$ is a matrix with $m$ columns. Clearly, the ideal $(MY)$ generated by the entries of the matrix $MY$ is contained in $(Y)$. This maps the ideal $I_{i+1}(Y)$ to $I_{i+1}(MY)$ and the ideal $(YZ)$ to $(MYZ)$. It follows that if $M$ is invertible then the ideals ${\mathfrak{p}}_{r,s}(Y,Z)$ and ${\mathfrak{p}}_{r,s}(MY,Z)$ are equal. In particular, the ideal ${\mathfrak{p}}_{r,s}(Y,Z)$ is unaffected by elementary row operations of the matrix $Y$. After inverting $y_{1,1}$, one may perform elementary row operations to transform $Y$ into a matrix where $y_{1,1}$ is the only nonzero entry in the first column; the resulting matrix then is $$\widetilde{Y} = \begin{pmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,t} \\ 0 & y'_{2,2} & \cdots & y'_{2,t}\\ \vdots & \vdots & & \vdots\\ 0 & y'_{m,2} & \cdots & y'_{m,t} \end{pmatrix} \quad \text{where} \quad y'_{i,j} = y_{i,j} - \frac{y_{i,1}y_{1,j}}{y_{1,1}}.$$ Let $Y'$ be the submatrix of $\widetilde{Y}$ obtained by deleting the first row and column. Note that the ideal $I_{r+1}(Y)S[\frac{1}{y_{1,1}}]$ is generated by the size $r+1$ minors of the matrix $\widetilde{Y}$, and hence equals $I_r(Y')S[\frac{1}{y_{1,1}}]$. As discussed above, in the ring $S[\frac{1}{y_{1,1}}]$, the ideals $(YZ)$ and $(\widetilde{Y}Z)$ are equal. Let $Z'$ be the submatrix of $Z$ obtained by deleting the first row. Note that the entries of the matrix $$\widetilde{Y}Z = \begin{pmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,t} \\ 0 & y'_{2,2} & \cdots & y'_{2,t}\\ \vdots & \vdots & & \vdots\\ 0 & y'_{m,2} & \cdots & y'_{m,t} \end{pmatrix} \begin{pmatrix} z_{1,1} & z_{1,2} & \cdots & z_{1,n} \\ z_{2,1} & z_{2,2} & \cdots & z_{2,n}\\ \vdots & \vdots & & \vdots\\ z_{t,1} & z_{t,2} & \cdots & z_{t,n} \end{pmatrix}$$ are exactly those of the matrix $Y'Z'$ along with the elements $f_1, \ldots f_n$, where $$f_i := y_{1,1}z_{1,i} + \sum_{j=2}^t y_{1,j}z_{j,i}$$ is the dot product of the first row of $\widetilde{Y}$ with the $i$-th column of $Z$. Thus, in the ring $S[\frac{1}{y_{11}}]$, we have $$(YZ) = (\widetilde{Y}Z) = (Y'Z') + (f_1, \ldots, f_n),$$ and the matrix $Z$ can be rewritten as $$Z = \begin{pmatrix} \frac{f_1}{y_{1,1}} - \sum_{j=2}^t \frac{y_{1,j}z_{j,1}}{y_{1,1}} & \cdots & \frac{f_n}{y_{1,1}} - \sum_{j=2}^t \frac{y_{1,j}z_{j,n}}{y_{1,1}} \\ z_{2,1} & \cdots & z_{2,n}\\ \vdots & & \vdots\\ z_{t,1} & \cdots & z_{t,n} \end{pmatrix}.$$ By the additivity of the determinant in any fixed row, a minor of $Z$ of size $s+1$ is exactly the corresponding minor of $$\begin{pmatrix} \frac{f_1}{y_{1,1}} & \cdots & \frac{f_t}{y_{1,1}} \\ z_{2,1} & \cdots & z_{2,n}\\ \vdots & & \vdots\\ z_{t,1} & \cdots & z_{t,n} \end{pmatrix}.$$ Therefore, we get $${\mathfrak{p}}_{r,s}(Y,Z)S[\frac{1}{y_{1,1}}] = {\mathfrak{p}}_{r,s}(\widetilde{Y},Z)S[\frac{1}{y_{1,1}}] = {\mathfrak{p}}_{r-1,s}(Y',Z')S[\frac{1}{y_{1,1}}] + (f_1, \ldots, f_n)S[\frac{1}{y_{1,1}}].$$ The second part of assertion $(3)$ immediately follows. Assertions $(1)$ and $(2)$ are readily verified since the matrices $Y'$ and $Z'$ do not involve the elements $z_{1,j}$ which appear (with unit coefficients) in $f_j$ for $1 \leq j \leq n$. ◻ ## Constructing the monomial order {#Subsection VoCMonomialOrder} In this subsection, we use the following: Let $A$ be an $m\times n$ matrix, and $i,j, k,\ell$ be integers such that $1 \le i \le j \le m$ and $1\le k \le \ell \le n$. We use $A^{[i,j]}_{[k,\ell]}$ to denote the $(j-i+1)\times (\ell-k+1)$ submatrix of $A$ with row indices $i,\dots,j$ and column indices $k,\dots,\ell$. The aim of this subsection is to describe a recipe for a monomial order $<_B$ in the polynomial ring $S$ that creates special lead terms for the generators of the ideals ${\mathfrak{p}}_{r,s}$ defining the varieties of complexes. The construction of this monomial order is quite technical; we illustrate it with an example first. Let $Y$ and $Z$ be matrices of indeterminates of sizes $5 \times 3$ and $3 \times 5$ respectively and let $\mathbb{K}$ be any field; set $S := \mathbb{K}[Y,Z]$. To define a monomial order $<_B$ in the polynomial ring $S$, we first define an order on the variables of $S$. Sort the entries of the matrices $Y$ and $Z$ into blocks $B_1, B_2,\ldots , B_{15}$ as displayed in the respective matrices $$\begin{pmatrix} 12 & 11 & 10 \\ 9 & 14 & 13 \\ 6 & 8 & 15 \\ 3 & 5 & 7 \\ 1 & 2 & 4 \end{pmatrix}, \begin{pmatrix} 12 & 9 & 6 & 3 & 1 \\ 14 & 11 & 8 & 5 & 2 \\ 15 & 13 & 10 & 7 & 4 \end{pmatrix}.$$ Thus, for instance, $y_{1,1}$ is in the block $B_{12}$, $y_{1,2}$ is in $B_{11}$, $z_{1,1}$ is in the block $B_{12}$, and so on. Now, for $\gamma \in B_\ell$ and $\delta \in B_{k}$, set $\gamma < \delta$ if $\ell <k$. Then, within each set $B_\ell$, fix an arbitrary order among the variables. This gives us a total variable order in $S$. Our monomial order $<_B$ is the reverse lexicographical order induced by this variable order in $S$. For a polynomial $f$, let $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$ denote the initial monomial of $f$ with respect to our monomial order. Set $c_{i,j}:=(YZ)_{i,j}$ and let $\alpha$ be the set consisting of the elements $c_{i,j}$ in $YZ$ with $i+j\le 4$, together with the following minors of $Y$ and $Z$: $$\det\Big(Y^{[4,5]}_{[1,2]}\Big),\quad \det\Big(Y^{[3,5]}_{[1,3]}\Big),\quad \det\Big(Y^{[2,4]}_{[1,3]}\Big),\qquad \det\Big(Z^{[1,2]}_{[4,5]}\Big),\quad \det\Big(Z^{[1,3]}_{[3,5}\Big),\quad \det\Big(Z^{[1,3]}_{[2,4]}\Big).$$ Notice that $$\begin{gathered} \mathop{\mathrm{{\textup{in}_{\it B}}}}(c_{1,1}) = y_{1,1}z_{1,1}, \qquad \mathop{\mathrm{{\textup{in}_{\it B}}}}(c_{1,2}) = y_{1,2}z_{2,2}, \qquad \mathop{\mathrm{{\textup{in}_{\it B}}}}(c_{1,3}) = y_{1,3}z_{3,3}, \\ \mathop{\mathrm{{\textup{in}_{\it B}}}}(c_{2,1}) = y_{2,2}z_{2,1}, \qquad \mathop{\mathrm{{\textup{in}_{\it B}}}}(c_{2,2}) = y_{2,3}z_{3,2}, \qquad \mathop{\mathrm{{\textup{in}_{\it B}}}}(c_{3,1}) = y_{3,3}z_{3,1},\end{gathered}$$ and $$\begin{gathered} \mathop{\mathrm{{\textup{in}_{\it B}}}}(\det\Big(Y^{[4,5]}_{[1,2]}\Big)) = y_{4,1}y_{5,2},\quad \mathop{\mathrm{{\textup{in}_{\it B}}}}(\det\Big(Y^{[3,5]}_{[1,3]}\Big)) = y_{3,1}y_{4,2}y_{5,3},\quad \mathop{\mathrm{{\textup{in}_{\it B}}}}(\det\Big(Y^{[2,4]}_{[1,3]}\Big)) = y_{2,1}y_{3,2}y_{4,3},\\ \mathop{\mathrm{{\textup{in}_{\it B}}}}(\det\Big(Z^{[1,2]}_{[4,5]}\Big)) = z_{1,4}z_{2,5},\quad \mathop{\mathrm{{\textup{in}_{\it B}}}}(\det\Big(Z^{[1,3]}_{[3,5]}\Big)) = z_{1,3}z_{2,4}z_{3,5},\quad \mathop{\mathrm{{\textup{in}_{\it B}}}}(\det\Big(Z^{[1,3]}_{[2,4]}\Big)) = z_{1,2}z_{2,3}z_{3,4}.\end{gathered}$$ Let $f$ be the product of the elements of $\alpha$. Since the initial terms of the elements in $\alpha$ are squarefree and pairwise coprime, $f$ has a squarefree initial term. Moreover, $y_{5,1}$ and $z_{1,5}$ do not divide $f$. The construction of this monomial order is crucial in establishing the $F$-regularity of the variety of complex $S/{\mathfrak{p}}_{r,s}(Y,Z)$, as well as the $F$-purity of the determinantal nullcone $S/(YZ)S$, as we show next. From now on, assume that the underlying field $\mathbb{K}$ has positive characteristic $p$. We begin with the case of the varieties of complexes $S/{\mathfrak{p}}_{r,s}$. If $r=0$ or $s=0$, then $S/{\mathfrak{p}}_{r,s}$ is a determinantal ring, and thus we may focus on the case where $(r,s) = (1,2)$ or $(r,s) = (2,1)$. Let $h$ be the height of ${\mathfrak{p}}_{r,s}$; we shall show in the proof of Theorem [Theorem 15](#thm-Complex-Fregular){reference-type="ref" reference="thm-Complex-Fregular"} that $f$ lies in the symbolic power ${\mathfrak{p}}_{r,s}^{(h)}$ whenever $r,s\neq 0$; hence we also have $f^{p-1}\in{\mathfrak{p}}_{r,s}^{(h(p-1))}$ for those $r,s$. As the initial term $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$ is squarefree, it follows from Corollary [Corollary 4](#CorollarySymbolic){reference-type="ref" reference="CorollarySymbolic"} that the ideal ${\mathfrak{p}}_{r,s}$ defines an $F$-pure ring for $r,s\neq 0$. In fact, since $y_{5,1}$ does not divide $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$, we get $$y_{5,1}f^{p-1} \in y_{5,1}({\mathfrak{p}}_{r,s}^{[p]}:{\mathfrak{p}}_{r,s}) \quad \text{while} \quad y_{5,1}f^{p-1} \notin {\mathfrak{m}}_S^{[p]}.$$ The ring $\frac{S}{{\mathfrak{p}}_{r,s}}[\frac{1}{y_{5,1}}]$ is a smooth extension of the determinantal ring defined by the size two minors of a $2 \times 5$ matrix of indeterminates by Lemma [Lemma 12](#lemma:localization of VoC){reference-type="ref" reference="lemma:localization of VoC"}, therefore it is $F$-regular. It follows from Theorem [Theorem 2](#theoremGlassbrenner){reference-type="ref" reference="theoremGlassbrenner"} that the varieties of complexes $S/{\mathfrak{p}}_{r,s}$ are $F$-regular. Now consider the determinantal nullcone $S/(YZ)S$. As $(YZ)S$ is radical, with minimal primes ${\mathfrak{p}}_{r,s}$ with $r+s=3$, we have $$(YZ)S = \bigcap_{r+s=3}{\mathfrak{p}}_{r,s}.$$ Thus, by Lemma [Lemma 14](#lem-colon-contain){reference-type="ref" reference="lem-colon-contain"} and Theorem [Theorem 1](#theoremFedder){reference-type="ref" reference="theoremFedder"}, it suffices to find, for $r+s=3$, a polynomial $g \in {\mathfrak{p}}_{r,s}^{[p]}:{\mathfrak{p}}_{r,s}$ such that $g \notin {\mathfrak{m}}_S^{[p]}$. Let $$g = (y_{5,1}z_{1,5}f)^{p-1}.$$ We will show in the proof of Theorem [Theorem 16](#thm-length-2-F-purity){reference-type="ref" reference="thm-length-2-F-purity"} that $g\in{\mathfrak{p}}_{r,s}^{[p]}:{\mathfrak{p}}_{r,s}$ for *all* $r+s = 3$. As $y_{5,1}$ and $z_{1,5}$ do not divide $f$, we get $g \notin {\mathfrak{m}}_S^{[p]}$, and so we are done by Theorem [Theorem 1](#theoremFedder){reference-type="ref" reference="theoremFedder"}. We now construct the monomial order $<_B$ illustrated in the above example. We start with attaching a natural number to each entry of the matrices $Y$ and $Z$. For $Z$, this process uses exactly the numbers $1,\ldots,t\times n$, starts with 1 in the upper right corner of $Z$, and proceeds along upwards oriented diagonals, in descending order. For $Y$, on and below the main diagonal, and starting at the lower left corner, proceed along upwards oriented diagonals in ascending order. Above the main diagonal of $Y$, associate to $y_{i,j}$ with $i\le j$ the block that belongs to $z_{j,j-i+1}$. In particular, only if $m=n$ will the blocks used for $Y$ be exactly $1,\ldots,m\times t$; otherwise there could be repetitions as well as omissions. The explicit formulæ for the general block numbers are as follows: For $1\le i\le m$ and $1\le j\le n$, the variable $y_{i,j}$ is in block $B_\ell$, where $$\begin{aligned} &&\ell =\left\{ \begin{array}{ccc} \binom{m-i+j+1}{2}-j+1 &\text{ if } &i\geq m-t+1, j\le i-m+t,\\ \binom{t+1}{2}+(t-j)+t(m-t+i-j-1)+1&\text{ if } &i-m+t+1\le j\le i-1,\\ t\cdot n+t-j-\binom{t-i+2}{2}+1&\text{ if } &1\le i\le j\le t. \end{array}\right.\\\end{aligned}$$ The variable $z_{i,j}$ is in block $B_\ell$, where $$\begin{aligned} &\ell =\left\{ \begin{array}{ccc} i-2j+2n+\binom{i-j+n-1}{2}&\text{ if } &1-n\le i-j\le t-n-1,\\ \binom{t}{2}+t(i-j+n-t+1)-i+1&\text{ if } &t-n\le i-j\le -1,\\ t\cdot n-\binom{t-i+j+1}{2}+t-i+1&\text{ if } &0\le i-j\le t-1. \end{array}\right.\end{aligned}$$ As in the example, our monomial order $<_B$ is the reverse lexicographical order induced by this variable order in $S$. For a polynomial $f$, let $\mathop{\mathrm{{\textup{in}_{\it B}}}}(f)$ denote the initial monomial of $f$ with respect to our monomial order. We now consider lead terms of certain elements of the ideal $(YZ)$. **Lemma 13**. *Let $\alpha$ be the set consisting of the elements $c_{i,j}:=(YZ)_{i,j}$ with $i+j\le t+1$, along with the following minors of $Y$ and $Z$:* - *$\det\Big(Y^{[m-i+1,m]}_{[1,i]}\Big)$ for $2\le i \le t-1$,* - *$\det\Big(Y^{[i+1,t+i]}_{[1,t]}\Big)$ for $1 \le i \le m-t$,* - *$\det\Big(Z^{[1,i]}_{[n-i+1,n]}\Big)$ for $2 \le i \le t-1$, and* - *$\det\Big(Z^{[1,t]}_{[i+1,t+i]}\Big)$ for $1 \le i \le n-t$.* *The initial terms of the elements of $\alpha$ are squarefree and pairwise coprime with respect to the monomial order $<_B$.* *Proof.* As all elements considered are sums of square free monomials, their respective lead terms must also be square free. In order to show that the lead terms are also coprime, we first prove the following claim: If $2\le i+j\le t+1$, then - the entry $c_{i,j}=\sum_{k=1}^t y_{i,k}z_{k,j}$ contains a term for which $y_{i,k}$ and $z_{k,j}$ are both in $B_{\ell}$ for the same $\ell$. In fact, this happens for $k=i+j-1$; - the term $y_{i,i+j-1}z_{i+j-1,j}$ is the lead term of $c_{i,j}$ under $\le_B$. The first item is clear, since for $i<k$ we have defined the block numbers of $y_{i,k}$ to agree with the block numbers of $z_{k,k-i+1}$ and so $y_{i,i+j-1}$ and $z_{i+j-1,j}$ are both in $B_{\ell}$ for the same $\ell$. For $i+j-1\le t$, compare $y_{i,i+j-1}z_{i+j-1,j}$ to the other entries $y_{i,k}z_{k,j}$ of $c_{i,j}$. If $k>i+j-1$ then $y_{i,k}$ is in the same row and to the right of $y_{i,i+j-1}$, which in turn is above the main diagonal of $Y$. As the block numbers on and above the diagonal in $Y$ decrease from left to right, $y_{i,k}$ is in $B_{\ell'}$ for $\ell' < \ell$. Since we use revlex on the block value, $y_{i,k}z_{k,j}<_B y_{i,i+j-1}z_{i+j-1,j}$. On the other hand, if $k<i+j-1$, then $z_{k,j}$ is in the same column as, and above, $z_{i+j-1,j}$. As block numbers in $Z$ increase with the row index, $z_{k,j}$ is in $B_{\ell'}$ for $\ell' < \ell$. Since we use revlex on the block value, $y_{i,k}z_{k,j}<_B y_{i,i+j-1}z_{i+j-1,j}$. As the lead terms of the $c_{i,j}$ with $i+j\le t+1$ are the elements $y_{i,i+j-1}z_{i+j-1,j}$ for that range of $i,j$, each $y_{i,j}$ with $i \le j$ and each $z_{k,\ell}$ with $k\geq \ell$ appear exactly once in a lead term of $c_{i,j}$. Now observe that, in $Y$ and $Z$, the lead terms of the relevant minors listed in Lemma [Lemma 13](#length-2-coprime){reference-type="ref" reference="length-2-coprime"} are the main diagonal terms, since any other term of this determinant involves a variable that is under, and one variable that is above its main diagonal, and (depending on whether one looks at minors in $Y$ or $Z$) one or the other of these has smaller block number than any element on the main diagonal of the minor. As the lead terms of the relevant minors of $Y$ and $Z$ are in disjoint diagonals that lie under the diagonal in $Y$, and above the diagonal in $Z$, they are coprime to one another and to the lead terms of the relevant $c_{i,j}$. ◻ ## Main Results We will require the following lemma to prove the main results of this section: **Lemma 14**. *Let $I$ and $J$ be ideals in a regular ring of prime characteristic $p>0$. Then* - *$(I^{[p]}:I)\cap(J^{[p]}:J) \subseteq (I\cap J)^{[p]}:(I\cap J)$,* - *$(I^{[p]}:I)\cap(J^{[p]}:J) \subseteq (I+J)^{[p]}:(I+J).$* *Proof.* By the flatness of the Frobenius map, we have $(I\cap J)^{[p]} = I^{[p]}\cap J^{[p]}$. Thus we get: $$\begin{aligned} (I\cap J)^{[p]}:(I\cap J) &=& \Big(I^{[p]}:(I\cap J)\Big)\cap \Big( J^{[p]}:(I\cap J)\Big) \\ &\supseteq& (I^{[p]}:I)\cap(J^{[p]}:J). \end{aligned}$$ The proof of the second item is immediate (and does not require the flatness of Frobenius). ◻ We are now ready to prove: **Theorem 15**. *Let $Y$ and $Z$ be matrices of indeterminates of sizes $m \times t$ and $t \times n$ respectively for positive integers $m$, $t$, and $n$. Let $\mathbb{K}$ be a field; set $S :=\mathbb{K}[Y,Z]$ and suppose that $r$ and $s$ are non-negative integers with $r+s \leq t$.* 1. *If $\mathbb{K}$ is an $F$-finite field of positive characteristic, the variety of complexes $S/{\mathfrak{p}}_{r,s}(Y,Z)$ is strongly $F$-regular.* 2. *In consequence, if $\mathbb{K}$ has characteristic zero, the variety of complexes $S/{\mathfrak{p}}_{r,s}(Y,Z)$ has log-terminal, and in particular rational, singularities.* *Proof.* Assertion $(2)$ follows from $(1)$ since rings of characteristic zero of $F$-regular type have log-terminal singularities, which are rational, compare [@Smith Theorem 4.3] and [@HaraWatanabe]. We therefore concentrate on the case where the characteristic of $\mathbb{K}$ is $p>0$. We proceed by induction on $r$. The statement is clear for $r=0$, since then the ring $S/{\mathfrak{p}}_{r,s}(Y,Z)$ is isomorphic to the determinantal ring $\mathbb{K}[Z]/I_{s}(Z)$. This ring is strongly $F$-regular by [@HH94 §7]. Now assume that the assertion holds for some $r\geq 1$. By suitably restating Lemma [Lemma 12](#lemma:localization of VoC){reference-type="ref" reference="lemma:localization of VoC"}, we infer $$\frac{S}{{\mathfrak{p}}_{r,s}(Y,Z)}[\frac{1}{y_{m,1}}] \cong \frac{S'}{{\mathfrak{p}}_{r-1,s}(Y',Z')}[y_{1,1}, \ldots, y_{m,1},y_{m,2}, \ldots, y_{m,t}, \frac{1}{y_{m,1}}].$$ where $Y'$ and $Z'$ are matrices of indeterminates of sizes $(m-1) \times (t-1)$ and $(t-1) \times n$ respectively. It follows from induction that the ring $\frac{S}{{\mathfrak{p}}_{r,s}(Y,Z)}[\frac{1}{y_{m,1}}]$ is strongly $F$-regular. If $s = 0$, then the ring $S/{\mathfrak{p}}_{r,s}(Y,Z)$ is isomorphic to the determinantal ring $\mathbb{K}[Y]/I_{r}(Y)$, and we are done. So assume that $s \geq 1$. In order to apply Theorem [Theorem 2](#theoremGlassbrenner){reference-type="ref" reference="theoremGlassbrenner"}, we must show that $$y_{m,1}({\mathfrak{p}}_{r,s}(Y,Z)^{[p]}:{\mathfrak{p}}_{r,s}(Y,Z)) \not\subseteq {\mathfrak{m}}_S^{[p]},$$ where ${\mathfrak{m}}_S$ is the homogeneous maximal ideal of $S$. By Corollary [Corollary 4](#CorollarySymbolic){reference-type="ref" reference="CorollarySymbolic"}, it suffices to find a polynomial $f$ contained in ${\mathfrak{p}}_{r,s}(Y,Z)^{(h)}$, where $h$ is the height of ${\mathfrak{p}}_{r,s}(Y,Z)$, such that $y_{m,1}f^{p-1} \notin {\mathfrak{m}}_S^{[p]}$. We will proceed to find such a polynomial; crucially this polynomial will be *the same* for all $r$ and $s$. However, before finding this polynomial, it is useful to make the following reductions: Firstly, since $F$-regularity is preserved by direct summands, we may assume that $t \leq \min \{m,n\}$ as the following argument shows: Choose two integers $m'\geq m$ and $n'\ge n$. Let $\overline{Y}$ be a generic $m'\times t$ matrix that contains $Y$. Similarly, let $\overline{Z}$ be a generic $t\times n'$ matrix that contains $Z$. Clearly $YZ$ is a submatrix of $\overline{Y}\overline{Z}$. Consider the maps $$\mathbb{K}[Y,Z]\longrightarrow\mathbb{K}[\overline{Y},\overline{Z}]\longrightarrow\mathbb{K}[Y,Z]$$ where the first is the inclusion, and the second is the projection that sends all variables in $\overline{Y}\smallsetminus Y$ and $\overline{Z}\smallsetminus Z$ to zero. The inclusion sends ${\mathfrak{p}}_{r,s}(Y,Z)$ into ${\mathfrak{p}}_{r,s}(\overline{Y},\overline{Z})$ and the projection sends ${\mathfrak{p}}_{r,s}(\overline{Y},\overline{Z})$ onto ${\mathfrak{p}}_{r,s}(Y,Z)$. Since the composition is the identity map, the ring $\mathbb{K}[Y,Z]/{\mathfrak{p}}_{r,s}(Y,Z)$ is a direct-summand of $\mathbb{K}[\overline{Y},\overline{Z}]/{\mathfrak{p}}_{r,s}(\overline{Y},\overline{Z})$. In consequence, $F$-regularity of the ring $\mathbb{K}[\overline{Y},\overline{Z}]/{\mathfrak{p}}_{r,s}(\overline{Y},\overline{Z})$ implies that of $\mathbb{K}[Y,Z]/{\mathfrak{p}}_{r,s}(Y,Z)$. Secondly, it suffices to consider varieties of complexes that are exact; i.e., we may assume that $r+s=t$. Indeed, $${\mathfrak{p}}_{r,s}(Y,Z)+{\mathfrak{p}}_{r',s'}(Y,Z) = {\mathfrak{p}}_{\min(r,r'),\min(s,s')}(Y,Z).$$ Thus, any ideal defining a variety of complexes may be written as the sum of ideals defining varieties of exact complexes with the same $Y$ and $Z$. Thus, by Lemma [Lemma 14](#lem-colon-contain){reference-type="ref" reference="lem-colon-contain"}, if we can find a polynomial $f$ such that $$f^{p-1} \in {\mathfrak{p}}_{r,s}(Y,Z)^{[p]}:{\mathfrak{p}}_{r,s}(Y,Z) \text{ and }f^{p-1} \in {\mathfrak{p}}_{r',s'}(Y,Z)^{[p]}:{\mathfrak{p}}_{r',s'}(Y,Z),$$ then $$f^{p-1} \in {\mathfrak{p}}_{\min(r,r'),\min(s,s')}(Y,Z)^{[p]}:{\mathfrak{p}}_{\min(r,r'),\min(s,s')}(Y,Z).$$ Having made these reductions, we exhibit the desired polynomial $f$ in the remainder of the proof. Choose $f$ to be the product of the elements contained in the set $\alpha$ as in Lemma [Lemma 13](#length-2-coprime){reference-type="ref" reference="length-2-coprime"}. As the lead terms of the elements contained in $\alpha$ are squarefree and coprime with respect to the monomial order constructed in Lemma [Lemma 13](#length-2-coprime){reference-type="ref" reference="length-2-coprime"}, and as $y_{m,1}$ is not a factor of the said lead terms, $y_{m,1}f^{p-1}$ is not contained in ${\mathfrak{m}}_S^{[p]}$. We now show that $f$ is contained in the symbolic power ${\mathfrak{p}}_{r,s}(Y,Z)^{(h)}$. Recall that for $r+s=t$, the height of ${\mathfrak{p}}_{r,s}(Y,Z)$ is $$h=(m-r)(t-r)+(n-s)(t-s)+rs=ms-nr-rs.$$ The set $\alpha$ as defined in Lemma [Lemma 13](#length-2-coprime){reference-type="ref" reference="length-2-coprime"} contains $\binom{t+1}{2}$ elements of the form $c_{i,j}$, and so $f \in (YZ)^{\binom{t+1}{2}}$. Before proceeding further, recall two useful facts about symbolic powers: - Given an $m \times n$ matrix $A$ of indeterminates with $m \le n$, we have $I_{\ell+k-1}(A)\subseteq I_{\ell}(A)^{(k)}$ whenever $1 \leq k \leq m - \ell +1$ by [@BrunsVetter Proposition 10.2]. - Given prime ideals $\mathfrak{p} \subseteq \mathfrak{q}$ in a polynomial ring, we have $\mathfrak{p}^{(k)} \subseteq \mathfrak{q}^{(k)}$ for any $k\geq 0$. Suppose first that $r,s>1$. Then we have $$\begin{aligned} f &\in& (YZ)^{\binom{t+1}{2}}\Bigg(\prod_{k=r+1}^{t-1}I_{k}(Y)\Bigg)I_{t}(Y)^{m-t}\Bigg(\prod_{\ell=s+1}^{t-1}I_{\ell}(Z)\Bigg)I_{t}(Z)^{n-t}\\ &\subseteq& (YZ)^{\binom{t+1}{2}}\Bigg(\prod_{k=r+1}^{t-1}I_{r+1}(Y)^{(k-r)}\Bigg)\Big(I_{r+1}(Y)^{(t-r)}\Big)^{m-t}\Bigg(\prod_{\ell=s+1}^{t-1}I_{s+1}(Z)^{(\ell-s)}\Bigg)\Big(I_{s+1}(Z)^{(t-s)}\Big)^{n-t}\\ &\subseteq&{\mathfrak{p}}_{r,s}(Y,Z)^{\binom{t+1}{2}}\Big(I_{r+1}(Y)^{\big(\binom{t-r}{2}+(m-t)(t-r)\big)}\Big)\Big(I_{s+1}(Z)^{\big(\binom{t-s}{2}+(n-t)(t-s)\big)}\Big)\\ &\subseteq& {\mathfrak{p}}_{r,s}(Y,Z)^{\big(\binom{t+1}{2}+ \binom{t-r}{2}+(m-t)(t-r)+ \binom{t-s}{2}+(n-t)(t-s)\big)}\end{aligned}$$ Since $r+s=t$, an elementary computation shows that $$\begin{aligned} \binom{t+1}{2}+ \binom{t-r}{2}+(m-t)(t-r)+ \binom{t-s}{2}+(n-t)(t-s) &=& % \binom{r+s+1}{2}+ \binom{s}{2}+(m-r-s)s+ \binom{r}{2}+(n-r-s)r &=& \\ % \frac{(r+s+1)(r+s)}{2}+ \frac{s(s-1)}{2}+(m-r-s)s+ \frac{r(r-1)}{2}+(n-r-s)r &=& \\ ms+nr-rs = h.\end{aligned}$$ So, $f$ is contained in ${\mathfrak{p}}_{r,s}(Y,Z)^{(h)}$, as desired. Now assume that $r=1$, and so $h=m(t-1)+n-t+1$ while $s+1>t-1$. Then we have $$\begin{aligned} f &\in& (YZ)^{\binom{t+1}{2}}\Bigg(\prod_{k=2}^{t-1}I_{k}(Y)\Bigg)I_{t}(Y)^{m-t}I_{t}(Z)^{n-t}\\ &\subseteq& (YZ)^{\binom{t+1}{2}}\Bigg(\prod_{k=2}^{t-1}I_{2}(Y)^{(k-1)}\Bigg)\Big(I_{2}(Y)^{(t-1)}\Big)^{m-t}I_{t}(Z)^{n-t}\\ &\subseteq&{\mathfrak{p}}_{r,s}(Y,Z)^{\binom{t+1}{2}}\Big(I_{2}(Y)^{\big(\binom{t-1}{2}+(m-t)(t-1)\big)}\Big)\Big(I_{t}(Z)^{n-t}\Big)\\ &\subseteq& {\mathfrak{p}}_{r,s}(Y,Z)^{\big(\binom{t+1}{2}+ \binom{t-1}{2}+(m-t)(t-1)+(n-t)\big)}.\end{aligned}$$ Notice that $$\begin{aligned} \binom{t+1}{2}+ \binom{t-1}{2}+(m-t)(t-1)+(n-t) &=& % \frac{(t+1)t}{2}+ \frac{(t-1)(t-2)}{2}+(m-t)(t-1)+(n-t) &=& \\ m(t-1)+n-t+1 = h.\end{aligned}$$ So, again, $f$ is contained in ${\mathfrak{p}}_{r,s}(Y,Z)^{(h)}$, as desired. The case $s=1$ is analogous to the case $r=1$, requiring only that we switch the roles of the minors of $Y$ and $Z$. We are done by Theorem [Theorem 2](#theoremGlassbrenner){reference-type="ref" reference="theoremGlassbrenner"}. ◻ **Theorem 16**. *Let $Y$ and $Z$ be matrices of indeterminates of sizes $m \times t$ and $t \times n$ respectively and $\mathbb{K}$ an $F$-finite field of positive characteristic; set $S :=\mathbb{K}[Y,Z]$ and assume that $r$ and $s$ are non-negative integers with $r+s \leq t$.* 1. *If $t\le\min(m,n)$ then the splittings for the Frobenius map on the varieties of complexes $S/{\mathfrak{p}}_{r,s}(Y,Z)$ can be chosen compatibly.* 2. *For any triple $(m,n,t)$, the natural determinantal nullcone $S/(YZ)S$ is $F$-pure.* *Proof.* The ring $S/(YZ)S$ with $t>\min(m,n)$ is a direct summand of the ring $S/(\overline{Y}\overline{Z})S$, where $\overline{Y}$ is an $m' \times t$ matrix which contains $Y$ and $\overline{Z}$ is a $t \times n'$ matrix which contains $Z$ with $m'\geq \max(m,t)$ and $n'\geq \max(n,t)$. Therefore, as in the proof of Theorem [Theorem 15](#thm-Complex-Fregular){reference-type="ref" reference="thm-Complex-Fregular"}, Part (2) of the present theorem then follows from Part (1) and the fact that $F$-purity is inherited by direct summands. Furthermore, for $t\le \min(m,n)$, the $F$-purity of $S/(YZ)S$ will follow once we have shown that the $F$-splittings of each $S/{\mathfrak{p}}_{r,s}(Y,Z)$ are compatible. So, for the remainder of the proof, assume that $t\le\min(m,n)$. Recall that $$(YZ) = \bigcap_{r+s=t}{\mathfrak{p}}_{r,s}(Y,Z).$$ Thus, by Lemma [Lemma 14](#lem-colon-contain){reference-type="ref" reference="lem-colon-contain"} and Theorem [Theorem 1](#theoremFedder){reference-type="ref" reference="theoremFedder"}, it suffices to find a polynomial $g \in {\mathfrak{p}}_{r,s}(Y,Z)^{[p]}:{\mathfrak{p}}_{r,s}(Y,Z)$ for all $r+s = t$ such that $g \notin {\mathfrak{m}}_S^{[p]}$. Let $f$ be the product of the elements of the set $\alpha$ as in Definition [Lemma 13](#length-2-coprime){reference-type="ref" reference="length-2-coprime"}. In the proof of Theorem [Theorem 15](#thm-Complex-Fregular){reference-type="ref" reference="thm-Complex-Fregular"} we showed that, for $r$ and $s$ both nonzero, $f$ is contained in ${\mathfrak{p}}_{r,s}(Y,Z)^{(h)}$ where $h$ is the height of ${\mathfrak{p}}_{r,s}(Y,Z)$, and thus $f^{p-1} \in {\mathfrak{p}}_{r,s}(Y,Z)^{[p]}:{\mathfrak{p}}_{r,s}(Y,Z)$ by Corollary [Corollary 4](#CorollarySymbolic){reference-type="ref" reference="CorollarySymbolic"}. In order to account for the cases where $r$ or $s$ are zero, let $$g := y_{m,1}z_{1,n}f.$$ Clearly $g^{p-1} \in {\mathfrak{p}}_{r,s}(Y,Z)^{[p]}:{\mathfrak{p}}_{r,s}(Y,Z)$ for $r$ and $s$ both nonzero. We claim that $$g \in {\mathfrak{p}}_{r,s}(Y,Z)^{[p]}:{\mathfrak{p}}_{r,s}(Y,Z),$$ when $r$ or $s$ is zero. Assume first that $r = 0$, and so $h = mt$. We have $$\begin{aligned} y_{m,1}z_{1,n}f &\in& (YZ)^{\binom{t+1}{2}}\Bigg(\prod_{k=1}^{t-1}I_{k}(Y)\Bigg)I_{t}(Y)^{m-t}\\ &\subseteq& (YZ)^{\binom{t+1}{2}}\Bigg(\prod_{k=1}^{t-1}I_{1}(Y)^{k}\Bigg)\Big(I_{1}(Y)^{t}\Big)^{m-t}\\ &\subseteq&{\mathfrak{p}}_{r,s}(Y,Z)^{\binom{t+1}{2}}\Big(I_{1}(Y)^{\binom{t}{2}+(m-t)t}\Big)\\ &\subseteq& {\mathfrak{p}}_{r,s}(Y,Z)^{\binom{t+1}{2}+ \binom{t}{2}+(m-t)t}.\end{aligned}$$ Notice that $$\binom{t+1}{2}+ \binom{t}{2}+(m-t)t\,\, =\,\, % \frac{(t+1)t}{2}+ \frac{(t-1)t}{2}+(m-t)t &=& \\ mt\,\, =\,\, h.$$ Therefore $g$ lies in ${\mathfrak{p}}_{r,s}(Y,Z)^{h}\subseteq {\mathfrak{p}}_{r,s}(Y,Z)^{(h)}$, and thus Corollary [Corollary 4](#CorollarySymbolic){reference-type="ref" reference="CorollarySymbolic"} implies the claim. The case $s=0$ is analogous. We conclude the proof of the theorem by observing that the lead terms of the elements of $\alpha$ are squarefree and coprime by Lemma [Lemma 13](#length-2-coprime){reference-type="ref" reference="length-2-coprime"}, and that $y_{m,1}$ and $z_{1,n}$ are not factors of said lead terms. Thus, $g^{p-1}$ is not contained in ${\mathfrak{m}}_S^{[p]}$, and we are done by Corollary [Corollary 4](#CorollarySymbolic){reference-type="ref" reference="CorollarySymbolic"}. ◻ **Corollary 17**. *Let $Y$ and $Z$ be matrices of indeterminates of sizes $m \times t$ and $t \times n$ respectively and $\mathbb{K}$ a field; set $S :=\mathbb{K}[Y,Z]$ and assume that $r$ and $s$ are non-negative integers.* 1. *If $r+s=t$, the ideal ${\mathfrak{p}}_{r,s}(Y,Z)$ defining the variety of exact complexes has a squarefree initial ideal.* 2. *If $r+s <t$ and $\mathbb{K}$ has positive characteristic, the ideal ${\mathfrak{p}}_{r,s}(Y,Z)$ defining the variety of non-exact complexes and the natural determinantal nullcone ideal $(YZ)S$ have squarefree initial ideal.* *Proof.* Let $<_B$ be the monomial order constructed in $\S$[5.3](#Subsection VoCMonomialOrder){reference-type="ref" reference="Subsection VoCMonomialOrder"}. Choose $f$ to be the product of the elements contained in the set $\alpha$ as in Lemma [Lemma 13](#length-2-coprime){reference-type="ref" reference="length-2-coprime"}, and set $$g:=y_{m,1}z_{1,n}f$$ as in the proof of Theorem [Theorem 16](#thm-length-2-F-purity){reference-type="ref" reference="thm-length-2-F-purity"}. By the proof of Theorem [Theorem 16](#thm-length-2-F-purity){reference-type="ref" reference="thm-length-2-F-purity"}, $\mathop{\mathrm{{\textup{in}_{\it B}}}}(g)$ lies in the initial ideal $\mathop{\mathrm{{\textup{in}_{\it B}}}}({\mathfrak{p}}_{r,s}^{(h)})$ for $r+s=t$, where $h$ is the height of ${\mathfrak{p}}_{r,s}(Y,Z)$. We are done by Theorem [\[theorem:sqfreeinitial\]](#theorem:sqfreeinitial){reference-type="ref" reference="theorem:sqfreeinitial"}. For $(2)$, choose the same polynomial $g$; we are done by [@Varbaro-Koley Theorem 3.12], where we additionally need the field to be of positive characteristic. ◻ We end with the following: Let $Y$ and $Z$ be matrices of indeterminates of sizes $m \times t$ and $t \times n$ respectively for positive integers $m$, $t$, and $n$. Let $r$ and $s$ be non-negative integers with $r+s \leq t$ and let ${\mathfrak{p}}_{r,s}$ denote the ideal defining a variety of complexes in the polynomial ring $\mathbb{K}[Y,Z]$. Denote by $$\mathcal{R}^S({\mathfrak{p}}_{r,s}):= \bigoplus_{k \geq 0} {\mathfrak{p}}_{r,s}^{(k)} \quad \text{and} \quad G^S({\mathfrak{p}}_{r,s}):= \bigoplus_{k \geq 0} {\mathfrak{p}}_{r,s}^{(k)}/{\mathfrak{p}}_{r,s}^{(k+1)}$$ the *symbolic Rees algebra* and the *symbolic associated graded algebra* of ${\mathfrak{p}}_{r,s}$ respectively. Are these rings Noetherian? The proof of Theorem [Theorem 15](#thm-Complex-Fregular){reference-type="ref" reference="thm-Complex-Fregular"} shows that the ideals defining the varieties of complexes are *symbolic $F$-split* (see [@dSMNB21 Corollary 5.10]). It immediately follows by [@dSMNB21 Theorem 4.7] that the symbolic Rees algebra and the symbolic associated graded algebra of the ideal ${\mathfrak{p}}_{r,s}$ are $F$-split (hence reduced). However, we do not know if either of these blowup algebras are Noetherian. # Acknowledgements {#acknowledgements .unnumbered} We would like to thank Anurag Singh, Bernd Ulrich, Jack Jeffries, and Mel Hochster for several valuable discussions. We thank Matt Weaver for sharing useful study material with us. Vaibhav Pandey and Yevgeniya Tarasova thank Alessandro De Stefani, Jonathan Montaño, Lisa Seccia, and Luis Núñez-Betancourt for their advice and encouragement. Vaibhav Pandey is especially thankful to Matteo Varbaro for the invitation to the University of Genova, where a part of this work was carried out. [^1]: UW was supported by NSF grant DMS-2100288 and by Simons Foundation Collaboration Grant for Mathematicians \#580839. PV was partially supported by the AMS-Simons Travel Grant.

arxiv_math

--- author: - | Claudia Gariboldi$^{1}$, Anna Ochal$^{2}$,  Mircea Sofonea$^{3}$[^1]\ and\ Domingo A. Tarzia$^{4,5}$\ *$1$ Departamento de Matemática, FCEFQyN*\ *Universidad Nacional de Rio Cuarto*\ *Ruta 36 km 601, 5800 Rio Cuarto, Argentina*\ *$2$ Chair in Optimization and Control*\ *Jagiellonian University in Krakow*\ *Ul. Lojasiewicza 6, 30348 Krakow, Poland*\ *$3$ Laboratoire de Mathématiques et Physique*\ *University of Perpignan Via Domitia*\ *52 Avenue Paul Alduy, 66860 Perpignan, France*\ *$^4$ Departamento de Matemática, FCE*\ *Universidad Austral*\ *Paraguay 1950, S2000FZF Rosario, Argentina*\ *$^5$ CONICET, Argentina* title: "**A Convergence Criterion for Elliptic Variational Inequalities**" --- **Keywords :** Elliptic variational inequality, convergence criterion, convergence results, well-posedness, contact, heat transfer, unilateral constraint.  47J20, 49J40, 40A05, 74M15, 74M10, 35J20. # Introduction {#s1} 0 A large number of mathematical models in Physics, Mechanics and Engineering Science are expressed in terms of strongly nonlinear boundary value problems for partial differential equations which, in a weak formulation, lead to variational inequalities. The theory of variational inequalities was developed based on arguments of monotonicity and convexity, including properties of the subdifferential of a convex function. Because of their importance in partial differential equations theory and engineering applications, a considerable effort has been put into the analysis, the control and the numerical simulations of variational inequalities. Basic references in the field are [@BC; @B; @G; @Kind-St; @Li], for instance. Applications of variational inequalities in Mechanics can be found in the books [@C; @DL; @EJK; @HS; @HHNL; @KO; @P]. In this paper we study the convergence of an arbitrary sequence to the solution of an elliptic variational inequality. Our results below could be extended to more general inequalities in reflexive Banach spaces. Nevertheless, for simplicity, we restrict ourselves to the following functional framework: $X$ is a real Hilbert space endowed with the inner product $(\cdot,\cdot)_X$ and the associated norm $\|\cdot\|_X$, $K\subset X$, $A:X\to X$, $j:X\to{\if mm {\rm I}\mkern -3mu{\rm R}\else \leavevmode \hbox{I}\kern -.17em\hbox{R} \fi}$ and $f\in X$. Then, the inequality problem we consider in this paper is as follows. ${\cal P}$. *Find $u$ such that* $$\label{1}u\in K,\qquad(Au,v-u)_X+j(v)-j(u) \ge(f,v-u)_X \qquad\forall\,v\in K.$$ The unique solvability of Problem $\mbox{{${\cal P}$}}$ follows from well-known results obtained in the literature, under various assumptions on the data. Here, we shall use the existence and uniqueness results that we recall in the next section, Theorem [Theorem 6](#t1){reference-type="ref" reference="t1"}. We also present some convergence results to the solution $u$ of inequality ([\[1\]](#1){reference-type="ref" reference="1"}). Note that a large number of convergence results for inequality [\[1\]](#1){reference-type="eqref" reference="1"} have been obtained in literature. The continuous dependence of the solution with respect to the data, the convergence of the solution to penalty problems when the penalty parameter converges to zero, the convergence of the solutions of discrete numerical schemes, the convergence of the solution of various perturbed problems when some parameters converge are several examples, among others. Note also that the concept of well-posedness (in the sense of Tyknonov or Levitin-Polyak) for inequality ([\[1\]](#1){reference-type="ref" reference="1"}) is also based on the convergence to the solution $u$ of the so-called approximating and generalized approximating sequences, respectively. All these examples, together with various relevant applications in Optimal Control Theory, Physics and Mechanics, lead to the following question: is it possible to describe the convergence of a sequence of $\{u_n\}\subset X$ to the solution $u$ of the variational inequality ([\[1\]](#1){reference-type="ref" reference="1"})? In other words, the question is to provide necessary and sufficient conditions for the convergence $u_n\to u$ in $X$, i.e., to provide a *convergence criterion*. The first aim of this paper is to provide an answer to this question. Here, we state and prove such a criterion of convergence expressed in terms of metric properties. The second aim is to illustrate how this criterion could be used in various examples and applications, in order to deduce some convergence results. A short description of the rest of the manuscript is as follows. First, in Section [2](#s2){reference-type="ref" reference="s2"} we present preliminary results concerning the unique solvability of Problem $\mbox{{${\cal P}$}}$, together with some convergence results. In Section [3](#s3){reference-type="ref" reference="s3"} we state and prove our main result, Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"}, which represents a criterion of convergence to the solution $u$ of inequality ([\[1\]](#1){reference-type="ref" reference="1"}). Section [4](#s4){reference-type="ref" reference="s4"} is devoted to some theoretical applications of Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"}. Here we state and prove two convergence results, introduce a new well-posedness concept and show that it extends the classical Tykhonov and Levitin-Polyak well-posedness concepts for variational inequalities of the form ([\[1\]](#1){reference-type="ref" reference="1"}). Finally, in Sections [5](#s5){reference-type="ref" reference="s5"} and [6](#s6){reference-type="ref" reference="s6"} we present applications of our theoretical results in the study of two specific boundary value problems, which model a static frictionless contact process for elastic materials and a stationary heat transfer problem, respectively. For these problems we state and prove convergence results and provide their physical and mechanical interpretations. # Preliminaries {#s2} 0 Everywhere in this paper, unless it is specified otherwise, we use the functional framework described in Introduction. Notation $0_X$ and $I_X$ will represent the zero element and the identity operator of $X$, respectively. All the limits, upper and lower limits below are considered as $n\to\infty$, even if we do not mention it explicitly. The symbols "$\rightharpoonup$\" and "$\to$\" denote the weak and the strong convergence in various spaces which will be specified, except in the case when these convergence take place in $\mathbb{R}$. For a sequence $\{\mbox{{$\varepsilon$}}_n\}\subset{\if mm {\rm I}\mkern -3mu{\rm R}\else \leavevmode \hbox{I}\kern -.17em\hbox{R} \fi}_+$ that converges to zero we use the short hand notation $0\le\mbox{{$\varepsilon$}}_n\to 0$. Finally, we denote by $d(u,K)$ the distance between an element $u\in X$ to the set $K$, that is $$\label{d} d(u,K)=\inf_{v\in K} \|u-v\|_X.$$ For the convenience of the reader we also recall the following definitions which can be found in many books and surveys, including [@DMP; @Z], for instance. **Definition 1**. *Let $\{K_n\}$ be a sequence of nonempty subsets of  $X$ and let $K$ be a nonempty subset of $X$. We say that the sequence $\{K_n\}$ converges to $K$ *in the sense of Mosco* ([@Mosco]) and we write $K_n\stackrel{M}{\longrightarrow} K$, if the following conditions hold:* *(a) for each $u\in K$, there exists a sequence $\{u_n\}$ such that $u_n\in K_n$ for each $n\in \mathbb{N}$ and $u_n\rightarrow u$ in $X$.* *(b) for each sequence $\{u_n\}$ such that $u_n\in K_n$ for each $n\in \mathbb{N}$ and $u_n\rightharpoonup u$ in $X$, we have $u\in K$.* **Definition 2**. *An operator $A \colon X \to X$ is called:* *${\rm(a)}$  *monotone* if $(Au - A v, u-v)_X \ge 0$  $\forall\, u, v\in X$.* *${\rm(b)}$ *strongly monotone* if there exists $m>0$ such that $$\label{A1} (Au-Av,u-v)_X\ge m\|u-v\|_X^2 \ \quad\forall\, u,v\in X.$$* *${\rm(c)}$ *pseudomonotone*, if it is bounded and the convergence $u_n \rightharpoonup u$ in $X$ together with inequality $\displaystyle \limsup\, (A u_n, u_n -u)_X \le 0$ imply that* *$$\displaystyle \liminf\, (A u_n, u_n - v)_X\ge (A u, u - v)_X\quad\ \forall\,v \in X.$$* *${\rm(d)}$ *hemicontinuous* if for all $u$, $v$, $w \in X$, the function $\lambda \mapsto (A (u + \lambda v), w )_X$ is continuous on $[0, 1]$.* *${\rm(e)}$ *demicontinuous* if $u_n \to u$ in $X$ implies $A u_n\rightharpoonup Au$ in $X$.* *${\rm(d)}$ *Lipschitz continuous* if there exists $M>0$ such that $$\label{A2}\|Au-Av\|_X\le M\,\|u-v\|_X \quad\forall\, u,v\in X.$$* Next, we follow [@PS p.267] and introduce the following notion of a penalty operator. **Definition 3**. *An operator $G:X\to X$ is said to be a penalty operator of $K$ if it is bounded, demicontinuous, monotone, and $Gu=0_X$ if and only if $u\in K$.* Note that if $K$ is a nonempty, closed, convex subset of $X$ and $P_K$ denotes the projection operator on $K$, then it is easy to see that the operator $G=I_X-P_K:X\to X$ is monotone and Lipschitz continuous. Therefore, using Definitions [Definition 2](#AopH){reference-type="ref" reference="AopH"} and [Definition 3](#penalty){reference-type="ref" reference="penalty"} it follows that $G$ is a penalty operator of $K$. Moreover, the proposition below, proved in [@Z], shows that any penalty operator is pseudomonotone. **Proposition 4**. *Let $G:X\to X$ be a bounded, hemicontinuous and monotone operator. Then $G$ is pseudomonotone.* In addition, the following result, stated and proved in [@Z], concerns the sum of two pseudomonotone operators. **Proposition 5**. *Let $A,\ B :X\to X$ be pseudomonotone operators. Then the sum $A+B:X\to X$ is a pseudomonotone operator.* In the study of Problem $\mbox{{${\cal P}$}}$ we consider the following assumptions. $$\begin{aligned} &&\label{K} K \ \mbox{\rm is a nonempty, closed, convex subset of} \ X. %\end{equation} \\[2.5mm] %\begin{equation} &&\label{Ap} \left\{ \begin{array}{l} A:X\to X\ {\rm is\ a\ pseudomonotone\ operator\ and}\\[0mm] {\rm there\ exists}\ m>0\ \ {\rm such\ that\ \eqref{A1}\ holds.} \end{array}\right.\\[2.5mm] %\end{eqnarray} %\begin{equation} &&\label{j}j:X\to \mathbb{R} \mbox{ is convex and lower semicontinuous. }\\[2mm] &&\label{f}f\in X.\end{aligned}$$ We now recall the following well-known existence and uniqueness result. **Theorem 6**. *Assume $(\ref{K})$--$(\ref{f})$. Then, the variational inequality $(\ref{1})$ has a unique solution $u$.* Theorem [Theorem 6](#t1){reference-type="ref" reference="t1"} represents a particular case of Theorem 84 in [@SMBOOK]. We now complete it with some convergence results of the form $$\label{c0} u_n\to u\quad{\rm in}\quad X.$$ Here and everywhere in Sections [2](#s2){reference-type="ref" reference="s2"} and [3](#s3){reference-type="ref" reference="s3"} we keep notation $u$ for the unique solution of Problem $\mbox{{${\cal P}$}}$ obtained in Theorem [Theorem 6](#t1){reference-type="ref" reference="t1"}, even if we do not mention explicitly. Moreover, $\{u_n\}$ represents a sequence of elements of $X$ which will be specified. Consider the sequences $\{K_n\}$, $\{\lambda_n\}$, $\{f_n\}$ such that the following conditions hold, for each $n\in \mathbb{N}$. $$\begin{aligned} &&\label{Kn} K_n \ \mbox{\rm is a nonempty, closed, convex subset of} \ X.\\ [2mm] &&\label{la}\lambda_n>0.\\ [2mm] &&\label{fn}f_n\in X.\end{aligned}$$ Assume also that $$\label{G}G:X\to X \ \ \mbox{is a penalty operator of}\ K.$$ Then, using Propositions [Proposition 4](#1pE1){reference-type="ref" reference="1pE1"} and [Proposition 5](#2pE1){reference-type="ref" reference="2pE1"} it follows that the operator $A+\frac{1}{\lambda_n} G:X\to X$ satisfies condition ([\[Ap\]](#Ap){reference-type="ref" reference="Ap"}), for each $n\in\mathbb{N}$. Therefore, under the previous assumptions, Theorem [Theorem 6](#t1){reference-type="ref" reference="t1"} guarantees the unique solvability of the following three problems. ${\cal P}_n^1$. *Find $u_n$ such that* $$\label{1K}u_n\in K_n,\quad(Au_n,v-u_n)_X+j(v)-j(u_n) \ge(f,v-u_n)_X \quad\forall\,v\in K_n.$$ ${\cal P}_n^2$. *Find $u_n$ such that* $$\begin{aligned} &&\label{1G}u_n\in X,\quad(Au_n,v-u_n)_X+\frac{1}{\lambda_n}(Gu_n,v-u_n)_X+j(v)-j(u_n)\nonumber\\ [0mm] &&\qquad\qquad\qquad\ge(f,v-u_n)_X \quad\forall\,v\in X.\end{aligned}$$ ${\cal P}_n^3$. *Find $u_n$ such that* $$\label{1f}u_n\in K,\quad(Au_n,v-u_n)_X+j(v)-j(u_n) \ge(f_n,v-u_n)_X \quad\forall\,v\in K.$$ Moreover, we have the following result. **Theorem 7**. *Assume $(\ref{K})$--$(\ref{f})$. Then, the convergence $(\ref{c0})$ holds in each of the following three cases.* *a) Condition $(\ref{Kn})$ holds, $K_n\stackrel{M}{\longrightarrow} K$, and $u_n$ denotes the solution to Problem ${\cal P}_n^1$.* *b) Conditions $(\ref{la})$ and $(\ref{G})$ hold, $\lambda_n\to 0$, and $u_n$ denotes the solution to Problem ${\cal P}_n^2$.* *c) Condition $(\ref{fn})$ holds, $f_n\to f$ in $X$, and $u_n$ denotes the solution to Problem ${\cal P}_n^3$.* Theorem [Theorem 7](#t2){reference-type="ref" reference="t2"} represents a version of some convergence results obtained in [@SofMat; @ST2], for instance and, for this reason, we skip its proof. Nevertheless, we mention that a proof of the convergence results in Theorem [Theorem 7](#t2){reference-type="ref" reference="t2"} b), c) will be provided in Section [4](#s4){reference-type="ref" reference="s4"}, under additional assumptions on the operator $A$ and the function $j$. This proof is based on the convergence criterion we state and prove in Section [3](#s3){reference-type="ref" reference="s3"}. Moreover, for the convenience of the reader, we present below a sketch of the proof of the point a) of this theorem, based on standard arguments of compactness, monotonicity, pseudomonotonicity and lower semicontinuity. *Proof of Theorem $\ref{t2}\,a)$*. The proof is structured in three steps, as follows. *Step i)* We use inequality [\[1\]](#1){reference-type="eqref" reference="1"} and the strong monotonicity of the operator $A$ in order to prove that the sequence $\{u_n\}$ is bounded in $X$. Then, using the reflexivity of the space $X$, we deduce that this sequence contains a subsequence, again denoted by $\{u_n\}$, such that $u_n\rightharpoonup\widetilde{u}$ with some $\widetilde{u}\in X$. *Step ii)* We use the pseudomonotonicity of $A$ and the properties of the function $j$ to deduce that the element $\widetilde{u}$ satisfies inequality [\[1\]](#1){reference-type="eqref" reference="1"}. Therefore, by the uniqueness of the solution, we deduce that $\widetilde{u}=u$. Moreover, a careful analysis reveals that any weakly convergent subsequence of the sequence $\{u_n\}$ converges weakly to $u$, in $X$. We then use a standard argument to deduce that the whole sequence $\{u_n\}$ converges weakly in $X$ to $u$. *Step iii)* Finally, we use the strong monotonicity of the operator $A$, and the weak convergence $u_n\rightharpoonup u$ in $X$ to deduce that the strong convergence holds, $(\ref{c0})$, which concludes the proof. $\Box$ We end this section with the remark that Theorem [Theorem 7](#t2){reference-type="ref" reference="t2"} provides relevant examples of sequences which converge to the solution $u$ of the variational inequality ([\[1\]](#1){reference-type="ref" reference="1"}). However, some elementary examples show that, besides the sequences introduced in parts a), b) and c) of this theorem, there exists other sequences $\{u_n\}$ which converge to $u$. A criterion which could identify all such sequences is provided in the next section. # A convergence criterion {#s3} 0 We now consider the following additional condition on the operator $A$ and the function $j$: $$\begin{aligned} &&\hspace{-11mm}\label{A} \left\{ \begin{array}{l} A:X\to X\ {\rm is\ a\ strongly\ monotone\ Lipschitz\ continuous\ operator,}\\[0mm] {\rm i.e.,\ there\ exist}\ m>0\ {\rm and}\ M>0\ {\rm such\ that\ \eqref{A1}\ and\ \eqref{A2}\ hold.} \end{array}\right.\\[2.5mm] &&\hspace{-11mm}\label{jj} \left\{\begin{array}{ll} j:X\to \mathbb{R}\ \mbox{ is convex and for each $D>0$ there exists $L_D>0$ such that}\\ [2mm] |j(u)-j(v)|\le L_D \|u-v\|_X\quad\forall\, u,\ v\in X\ {\rm with}\ \|u\|_X, \ \|v\|_X\le D. \end{array}\right. \end{aligned}$$ Note that conditions [\[A\]](#A){reference-type="eqref" reference="A"} and [\[jj\]](#jj){reference-type="eqref" reference="jj"} imply [\[Ap\]](#Ap){reference-type="eqref" reference="Ap"} and [\[j\]](#j){reference-type="eqref" reference="j"}, respectively. Therefore, Theorems [Theorem 6](#t1){reference-type="ref" reference="t1"} and [Theorem 7](#t2){reference-type="ref" reference="t2"} still hold under assumptions $(\ref{K})$, $(\ref{f})$, $(\ref{A})$ and $(\ref{jj})$. Moreover, condition [\[jj\]](#jj){reference-type="eqref" reference="jj"} shows that the function $j$ is Lipschitz continuous on each bounded sets of $X$. This implies that, in particular, $j$ is locally Lipschitz. In addition, note that any continuous seminorm on the space $X$ satisfies condition [\[jj\]](#jj){reference-type="eqref" reference="jj"}. Our main result in this section is the following. **Theorem 8**. *Assume $(\ref{K})$, $(\ref{f})$, $(\ref{A})$, $(\ref{jj})$ and denote by $u$ the solution of the variational inequality $(\ref{1})$ provided by Theorem $\ref{t1}$. Consider also an arbitrary sequence $\{u_n\}\subset X$, together with the statements [\[c1\]](#c1){reference-type="eqref" reference="c1"} and [\[c2\]](#c2){reference-type="eqref" reference="c2"} below. $$\begin{aligned} &&\label{c1} \quad\ \ u_n\to u\qquad{\rm in}\ X. \\ [4mm] &&\label{c2} \left\{\begin{array}{ll} {\rm (a)}\ d(u_n,K)\to 0\ ;\\[4mm] {\rm (b)}\ \mbox{\rm there exists $0\le\mbox{{$\varepsilon$}}_n\to 0$ such that} \\[3mm] \quad\quad(Au_n,v-u_n)_X+j(v)-j(u_n)+\mbox{{$\varepsilon$}}_n(1+\|v-u_n\|_X)\\ [1mm] \qquad\qquad\qquad\quad \ge(f,v-u_n)_X \quad\forall\,v\in K,\ n\in\mathbb{N}. \end{array}\right. \end{aligned}$$ Then, these statements are equivalent, i.e., [\[c1\]](#c1){reference-type="eqref" reference="c1"} holds if and only if [\[c2\]](#c2){reference-type="eqref" reference="c2"} holds.* The proof of Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} is based on the following result. **Lemma 9**. *Assume $(\ref{K})$, $(\ref{f})$, $(\ref{A})$, $(\ref{jj})$. Then any sequence $\{u_n\}\subset X$ which satisfies condition $(\ref{c2})\,({\rm b)}$ is bounded.* *Proof.* Let $n\in\mathbb{N}$. We test in ([\[c2\]](#c2){reference-type="ref" reference="c2"}) with a fixed element of $K$, say $v=u$. We have $$(Au_n,u-u_n)_X+j(u)-j(u_n)+\mbox{{$\varepsilon$}}_n(1+\|u-u_n\|_X)\ge(f,u-u_n)_X$$ which implies that $$\begin{aligned} &&(Au_n-Au,u_n-u)_X+j(u_n)\\ [2mm] &&\qquad\le (Au,u-u_n)_X+ j(u)+\mbox{{$\varepsilon$}}_n(1+\|u-u_n\|_X)+(f,u_n-u)_X \nonumber\end{aligned}$$ and, moreover, $$\begin{aligned} &&\label{rd1}m\,\|u-u_n\|^2_X+j(u_n)\\ [2mm] &&\qquad\le \|Au\|_X\|u-u_n\|_X+ j(u)+\mbox{{$\varepsilon$}}_n(1+\|u-u_n\|_X)+\|f\|_X\|u_n-u\|_X. \nonumber\end{aligned}$$ On the other hand, assumption ([\[jj\]](#jj){reference-type="ref" reference="jj"}) and a standard result on convex lower semicontinuous functions (see [@KZ p.208], for instance) implies that $j$ is bounded from below by an affine continuous function. Hence, there exist $\alpha\in X$, $\beta\in \mathbb{R}$ such that $$\label{zz} j(v)\ge (\alpha,v)_X+\beta\qquad\forall \,v\in X.$$ This implies that $$\begin{aligned} &&j(u_n)\ge (\alpha,u_n)_X+\beta=(\alpha,u_n-u)_X+(\alpha,u)_X+ \beta\\ [2mm] &&\qquad \ge-\|\alpha\|_X\|u_n-u\|_X-\|\alpha\|_X\|u\|_X-|\beta|\end{aligned}$$ and, substituting this inequality in ([\[rd1\]](#rd1){reference-type="ref" reference="rd1"}) yields $$\begin{aligned} &&m\,\|u-u_n\|^2_X\le (\|\alpha\|_X+\|Au\|_X+\|f\|_X+\mbox{{$\varepsilon$}}_n)\|u-u_n\|_X\\ [2mm] &&\qquad\qquad+j(u)+\|\alpha\|_X\|u\|_X+|\beta|+\mbox{{$\varepsilon$}}_n. \end{aligned}$$ Moreover, since $j(u)\le |j(u)|$, we deduce that $$\begin{aligned} &&m\,\|u-u_n\|^2_X\le (\|\alpha\|_X+\|Au\|_X+\|f\|_X+\mbox{{$\varepsilon$}}_n)\|u-u_n\|_X\\ [2mm] &&\qquad\qquad+|j(u)|+\|\alpha\|_X\|u\|_X+|\beta|+\mbox{{$\varepsilon$}}_n. \end{aligned}$$ Next, we use the implication $$\label{in} x^2\le ax+b\quad \Longrightarrow\quad x\le a+\sqrt{b}\qquad\forall\, x,\, a,\ b\ge 0$$ and the convergence $\mbox{{$\varepsilon$}}_n\to 0$ to see that the sequence $\{\|u-u_n\|_X\}$ is bounded in $\mathbb{R}$. This implies that the sequence $\{u_n\}$ is bounded in $X$, which concludes the proof. $\Box$ We now proceed with the proof of Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"}. *Proof.* Assume that [\[c1\]](#c1){reference-type="eqref" reference="c1"} holds. Then, since $u\in K$ it follows that $d(u_n,K)\le \|u_n-u\|_X$ for each $n\in\mathbb{N}$ which implies that ([\[c2\]](#c2){reference-type="ref" reference="c2"})(a) holds. To prove ([\[c2\]](#c2){reference-type="ref" reference="c2"})(b) we fix $n\in\mathbb{N}$ and $v\in K$. We write $$\begin{aligned} &&(Au_n,v-u_n)_X+j(v)-j(u_n)-(f,v-u_n)_X\\ [2mm] &&\quad=(Au_n-Au,v-u_n)_X+(Au,v-u)_X+(Au,u-u_n)_X\\ [2mm] &&\qquad+j(v)-j(u)+j(u)-j(u_n)-(f,v-u)_X+(f,u_n-u)_X\end{aligned}$$ and, using [\[1\]](#1){reference-type="eqref" reference="1"} we deduce that $$\begin{aligned} &&\label{r23}(Au_n,v-u_n)_X+j(v)-j(u_n)-(f,v-u_n)_X\\ [2mm] &&\quad\ge(Au_n-Au,v-u_n)_X+(Au,u-u_n)_X +j(u)-j(u_n)+(f,u_n-u)_X.\nonumber\end{aligned}$$ We now use [\[r23\]](#r23){reference-type="eqref" reference="r23"} and inequalities $$\begin{aligned} &&(Au_n-Au,v-u_n)_X\ge -\|Au_n-Au\|_X\|v-u_n\|_X\ge-M\|u-u_n\|_X\|v-u_n\|_X,\\ [2mm] &&(Au,u-u_n)_X\ge-\|Au\|_X\|u-u_n\|_X,\\[2mm] &&(f,u_n-u)_X\ge -\|f\|_X\|u-u_n\|_X\end{aligned}$$ to find that $$\begin{aligned} &&(Au_n,v-u_n)_X+j(v)-j(u_n)-(f,v-u_n)_X+j(u_n)-j(u)\\ [2mm] &&\quad+M\|u-u_n\|_X\|v-u_n\|_X+\|Au\|_X\|u-u_n\|_X+ \|f\|_X\|u-u_n\|_X\ge 0.\end{aligned}$$ Therefore, with notation $$\label{ep}\mbox{{$\varepsilon$}}_n={\rm max}\,\{M\|u-u_n\|_X,(\|Au\|_X+ \|f\|_X)\|u-u_n\|_X+j(u_n)-j(u)\}$$ we see that $$\label{e1n} (Au_n,v-u_n)_X+j(v)-j(u_n)+\mbox{{$\varepsilon$}}_n(1+\|v-u_n\|_X)\ge(f,v-u_n)_X.$$ On the other hand, notation [\[ep\]](#ep){reference-type="eqref" reference="ep"}, assumption [\[c1\]](#c1){reference-type="eqref" reference="c1"} and the continuity of the function $j$, guaranteed by hypothesis [\[jj\]](#jj){reference-type="eqref" reference="jj"}, show that $$\label{e2n} \mbox{{$\varepsilon$}}_n\to 0.$$ We now combine ([\[e1n\]](#e1n){reference-type="ref" reference="e1n"}) and ([\[e2n\]](#e2n){reference-type="ref" reference="e2n"}) to see that condition ([\[c2\]](#c2){reference-type="ref" reference="c2"})(b) is satisfied. Conversely, assume now that ([\[c2\]](#c2){reference-type="ref" reference="c2"}) holds. Then, ([\[c2\]](#c2){reference-type="ref" reference="c2"})(a) and definition [\[d\]](#d){reference-type="eqref" reference="d"} of the distance function show that for each $n\in\mathbb{N}$ there exist two elements $v_n$ and $w_n$ such that $$\label{e3} u_n=v_n+w_n,\quad v_n\in K,\quad w_n\in X,\quad \|w_n\|_X\to 0.$$ We fix $n\in\mathbb{N}$ and use ([\[c2\]](#c2){reference-type="ref" reference="c2"})(b) with $v=u\in K$ to see that $$\label{e4} (Au_n,u-u_n)_X+j(u)-j(u_n)+\mbox{{$\varepsilon$}}_n(1+\|u-u_n\|_X)\ge(f,u-u_n)_X.$$ On the other hand, we use the regularity $v_n\in K$ in ([\[e3\]](#e3){reference-type="ref" reference="e3"}) and test with $v=v_n$ in [\[1\]](#1){reference-type="eqref" reference="1"} to find that $$\label{e5} (Au,v_n-u)_X+j(v_n)-j(u)\ge(f,v_n-u)_X.$$ We now add inequalities ([\[e4\]](#e4){reference-type="ref" reference="e4"}), ([\[e5\]](#e5){reference-type="ref" reference="e5"}) to obtain that $$\begin{aligned} &&\label{Z}(Au_n,u-u_n)_X+(Au,v_n-u)_X +j(v_n)-j(u_n)\\ [2mm] &&\quad+\mbox{{$\varepsilon$}}_n(1+\|u-u_n\|_X)\ge(f,v_n-u_n)_X.\nonumber\end{aligned}$$ Next, we use equality $u_n=v_n+w_n$ to see that $$\begin{aligned} &&(Au_n,u-u_n)_X+(Au,v_n-u)_X= (Au,v_n-u)_X-(Av_n,v_n-u)_X\\ [2mm] &&\quad+(Av_n,v_n-u)_X+(Au_n,u-v_n)_X-(Au_n,w_n)_X \\ [2mm] &&\qquad=(Au-Av_n,v_n-u)_X+(Au_n-Av_n,u-v_n)_X-(Au_n,w_n)_X\end{aligned}$$ and, therefore, [\[Z\]](#Z){reference-type="eqref" reference="Z"} implies that $$\begin{aligned} &&(Au-Av_n,v_n-u)_X+(Au_n-Av_n,u-v_n)_X-(Au_n,w_n) +j(v_n)-j(u_n)\\ [2mm] &&\quad+\mbox{{$\varepsilon$}}_n(1+\|u-v_n-w_n\|_X)+ (f,w_n)_X\ge 0.\end{aligned}$$ Using now assumption [\[A\]](#A){reference-type="eqref" reference="A"} and equality $u_n=v_n+w_n$ we deduce that $$\begin{aligned} &&\label{e6}m\|u-v_n\|^2_X\le M\|w_n\|_X\|u-v_n\|_X+ \|Au_n\|_X \|w_n\|_X\\ [2mm] &&\qquad+j(v_n)-j(u_n)+\mbox{{$\varepsilon$}}_n+\mbox{{$\varepsilon$}}_n\|u-v_n\|_X +\mbox{{$\varepsilon$}}_n\|w_n\|_X+\|f\|_X\|w_n\|_X.\nonumber\end{aligned}$$ On the other hand, Lemma [Lemma 9](#lm){reference-type="ref" reference="lm"} guarantees that the sequence $\{u_n\}$ is bounded in $X$. Therefore, [\[e3\]](#e3){reference-type="eqref" reference="e3"} implies that there exists $D>0$ such that $$\label{e10} \|u_n\|_X\le D,\qquad \|v_n\|_X\le D\qquad\forall\, n\in\mathbb{N}.$$ Moreover, using condition ([\[A\]](#A){reference-type="ref" reference="A"}) we may assume that $$\label{e9}\|Au_n\|_X\le D\qquad\forall\, n\in\mathbb{N}.$$ In addition, using ([\[e10\]](#e10){reference-type="ref" reference="e10"}), assumption ([\[jj\]](#jj){reference-type="ref" reference="jj"}) and equality $u_n=v_n+w_n$ in [\[e3\]](#e3){reference-type="eqref" reference="e3"} we find that $$\label{e7} j(v_n)-j(u_n)\le L_D\|w_n\|_X.$$ We now combine the bounds [\[e6\]](#e6){reference-type="eqref" reference="e6"}, [\[e9\]](#e9){reference-type="eqref" reference="e9"} and [\[e7\]](#e7){reference-type="eqref" reference="e7"} to deduce that $$\begin{aligned} &&\label{e8}m\|u-v_n\|^2_X\le (M\|w_n\|_X+\mbox{{$\varepsilon$}}_n)\|u-v_n\|_X\\ [2mm] &&\qquad+ (D +L_D+\mbox{{$\varepsilon$}}_n+\|f\|_X)\|w_n\|_X+\mbox{{$\varepsilon$}}_n.\nonumber\end{aligned}$$ Next, we use ([\[e8\]](#e8){reference-type="ref" reference="e8"}), inequality ([\[in\]](#in){reference-type="ref" reference="in"}) and the convergences $\|w_n\|_X\to 0$, $\mbox{{$\varepsilon$}}_n\to 0$ to find that $\|u-v_n\|_X\to 0$. This implies that $v_n\to u$ in $X$ and, using [\[e3\]](#e3){reference-type="eqref" reference="e3"} we deduce that [\[c1\]](#c1){reference-type="eqref" reference="c1"} holds, which concludes the proof.$\Box$ Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} shows that, under assumptions $(\ref{K})$--$(\ref{f})$, $(\ref{A})$, $(\ref{jj})$, conditions $(\ref{c2})({\rm a})$ and $(\ref{c2})({\rm b})$ represent necessary and sufficient conditions for the convergence $(\ref{c1})$. The two examples below show that, in general, we cannot skip one of these conditions. **Example 10**. *Take $A=I_X$ and $j\equiv 0$. Then the solution of inequality $(\ref{1})$ is $u=P_Kf$ where, recall, $P_K$ represents the projector operator on $K$. Assume now that $f\notin K$ and take $u_n=f$ for each $n\in\mathbb{N}$. It follows from here that $(\ref{c2})({\rm b})$ is satisfied with $\mbox{{$\varepsilon$}}_n=0$. Nevertheless, $(\ref{c2})({\rm a})$ does not hold since $$d(u_n,K)=\|u_n-P_Ku_n\|_X=\|f-P_Kf\|_X>0.$$ Moreover, the convergence $(\ref{c1})$ is not valid. We conclude from here that condition $(\ref{c2})({\rm a})$ cannot be skipped, i.e., condition $(\ref{c2})({\rm b})$ is not a sufficient condition to guarantee the convergence [\[c1\]](#c1){reference-type="eqref" reference="c1"}.* **Example 11**. *Let $X=\mathbb{R}$, $K=[0,1]$, $A=I_X$, $j\equiv 0$ and $f=\frac{1}{2}$. Then, it follows that $u=f=\frac{1}{2}$ and the sequence $\{u_n\}$ with $u_n=0$ satisfies $(\ref{c2})({\rm a})$ but does not satisfy $(\ref{c1})$. We conclude that condition $(\ref{c2})({\rm a})$ is not a sufficient condition to guarantee the convergence $(\ref{c1})$.* We end this section with the remark that Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} was obtained under the assumptions ([\[A\]](#A){reference-type="ref" reference="A"}), ([\[jj\]](#jj){reference-type="ref" reference="jj"}) which, however, are not necessary neither in the statement of Theorem [Theorem 6](#t1){reference-type="ref" reference="t1"} nor in the statement of Theorem [Theorem 7](#t2){reference-type="ref" reference="t2"}. Removing or relaxing these conditions is an interesting problem which clearly deserves to be investigated into future. # Convergence and well-posedness results {#s4} 0 In this section we show how Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} can be used to deduce some theoretical convergence results. We also prove that the well-posedness of inequality ([\[1\]](#1){reference-type="ref" reference="1"}), both in the Tykhonov and Levitin-Polyak sense, can be deduced as a consequence of this theorem. Finally we provide an interpretation of Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} in the context of the $\mbox{{${\cal T}$}}$-well-posedness concept introduced in [@S; @SX16] and used in various papers, including [@ST1; @ST3]. **Convergence results.** A first consequence of Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} is the following continuous dependence result. **Corollary 12**. *Assume $(\ref{K})$, $(\ref{f})$, $(\ref{fn})$, $(\ref{A})$ and $(\ref{jj})$. Also, assume that $f_n\to f$ in $X$ and denote by $u_n$ the solution of Problem ${\cal P}_n^3$. Then, the convergence $(\ref{c0})$ holds.* Let $n\in\mathbb{N}$ and $v\in K$. We use [\[1f\]](#1f){reference-type="eqref" reference="1f"} and write $$u_n\in K,\quad(Au_n,v-u_n)_X+j(v)-j(u_n)+(f-f_n,v-u_n)_X \ge(f,v-u_n)_X,$$ which implies that $$u_n\in K,\quad(Au_n,v-u_n)_X+j(v)-j(u_n)+\|f-f_n\|_X\|v-u_n\|_X \ge(f,v-u_n)_X.$$ It follows from this inequality that conditions ([\[c2\]](#c2){reference-type="ref" reference="c2"})(a) and ([\[c2\]](#c2){reference-type="ref" reference="c2"})(b) are satisfied with $\mbox{{$\varepsilon$}}_n=\|f-f_n\|_X\to 0$. We now use Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} to conclude the proof. $\Box$ A second consequence of Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} concerns the penalty problem $\mbox{{${\cal P}$}}_n^2$ and is as follows. **Corollary 13**. *Assume $(\ref{K})$, $(\ref{f})$, $(\ref{la})$, $(\ref{G})$, $(\ref{A})$ and $(\ref{jj})$. Also, assume that $\lambda_n\to 0$ and denote by $u_n$ the solution of Problem ${\cal P}_n^2$. Then, the convergence $(\ref{c0})$ holds.* The proof is structured in several steps, as follows. Let $n\in\mathbb{N}$ and $v\in K$. Then, ([\[G\]](#G){reference-type="ref" reference="G"}) and Definition $\ref{penalty}$ imply that $Gv=0_X$, $(Gv-Gu_n,v-u_n)_X\ge 0$ and, therefore, $$\label{zzz} (Gu_n,v-u_n)_X\le 0.$$ We now use inequalities ([\[1G\]](#1G){reference-type="ref" reference="1G"}) and ([\[zzz\]](#zzz){reference-type="ref" reference="zzz"}) to see that $$\label{m4} (Au_n,v-u_n)_X+j(v)-j(u_n)\ge(f,v-u_n)_X$$ which shows that ${\rm (\ref{c2})(b)}$ holds with $\mbox{{$\varepsilon$}}_n=0$. Indeed, consider a weakly convergent subsequence of the sequence $\{u_n\}$, again denoted by $\{u_n\}$. Then, there exists an element $\widetilde{u}\in X$ such that $$u_n\rightharpoonup \widetilde{u}\quad{\rm in}\quad X.$$ We shall prove that $\widetilde{u}\in K$ and $u_n\to \widetilde{u}$ in $X$. To this end, we fix $n\in\mathbb{N}$ and $v\in X$. We use ([\[1G\]](#1G){reference-type="ref" reference="1G"}) and ([\[zz\]](#zz){reference-type="ref" reference="zz"}) to write $$\begin{aligned} &&\frac{1}{\lambda_n}( G{u}_n,{u}_n-v)_X\leq ( A{u}_n,v-{u}_n)_X+j(v)-j({u}_n)+(f,{u}_n-v)_X\\ [2mm] &&\quad \leq \|Au_n\|_{X}\|v-u_n\|_X+j(v)+\|\alpha\|_{X}\|u_n\|_{X}+|\beta|+ \|f\|_{X}\|v-u_n\|_X.\end{aligned}$$ On the other hand, Step i) and Lemma [Lemma 9](#lm){reference-type="ref" reference="lm"} imply that the sequence $\{u_n\}$ is bounded. Therefore, from the previous inequality we deduce that there exists a constant $C(v)>0$ which does not depend on $n$ such that $$%\label{e4.8} (G{u}_n,{u}_n-v)_X\leq\lambda_nC(v).$$ Passing to the upper limit in the above inequality and using assumption $\lambda_n\to 0$ we find that $$\label{Ee4.14} \lim\sup\,(G{u}_n,{u}_n-v)_X\leq 0.$$ Taking now $v=\widetilde{u}$ in ([\[Ee4.14\]](#Ee4.14){reference-type="ref" reference="Ee4.14"}) we deduce that $$\limsup\,( G{u}_n,{u}_n-\widetilde{u})_X\leq 0.$$ Then, using the convergence ${u}_n \rightharpoonup {\widetilde{u}}$ in $X$ and the pseudomonotonicity of $G$, guaranteed by assumption ([\[G\]](#G){reference-type="ref" reference="G"}) and Proposition [Proposition 4](#1pE1){reference-type="ref" reference="1pE1"}, we deduce that $$\begin{aligned} \label{Ee4.16} ( G\widetilde{u},\widetilde{u}-v)_X\leq \liminf\,( G{u}_n,{u}_n-v)_X. \end{aligned}$$ We now combine inequalities ([\[Ee4.14\]](#Ee4.14){reference-type="ref" reference="Ee4.14"}) and ([\[Ee4.16\]](#Ee4.16){reference-type="ref" reference="Ee4.16"}) to see that $$%\label{Enb} ( G\widetilde{u},\widetilde{u}-v)_X\leq 0.$$ Recall that this inequality is valid for any $v\in X$. Therefore, we deduce that $G\widetilde{u}=0_X$ and, using assumption ([\[G\]](#G){reference-type="ref" reference="G"}) we find that $$\label{Ereg} \widetilde{u}\in K.$$ Let $n\in\mathbb{N}$. Then, using ([\[m4\]](#m4){reference-type="ref" reference="m4"}) with $v=\widetilde{u}$ we find that $$%\label{Ee4.17} ( A{u}_n,{u}_n-\widetilde{u})_X \le j(\widetilde{u})-j(u_n)+( f,{u}_n-\widetilde{u})_X$$ or, equivalently, $$%\label{Ee4.17} (A{u}_n-A\widetilde{u},{u}_n-\widetilde{u})_X \le (A\widetilde{u},\widetilde{u}-{u}_n)_X+j(\widetilde{u})-j(u_n)+( f,{u}_n-\widetilde{u})_X.$$ Finally, we use the strong monotonicity of the operator $A$, ([\[A1\]](#A1){reference-type="ref" reference="A1"}), to see that $$m\|u_n-\widetilde{u}\|_X^2\le (A\widetilde{u},\widetilde{u}-{u}_n)_X+j(\widetilde{u})-j(u_n)+( f,{u}_n-\widetilde{u})_X.$$ We now pass to the upper limit in this inequality, use the convergence ${u}_n\rightharpoonup \widetilde{u}$ in $X$ and the continuity of $j$ to infer that $$\label{Econv} u_n\to \widetilde{u}\quad{\rm in}\quad X.$$ We now combine [\[Ereg\]](#Ereg){reference-type="eqref" reference="Ereg"} and [\[Econv\]](#Econv){reference-type="eqref" reference="Econv"} to see that $d(u_n,K)\to 0$ which concludes the proof of this step. This step is a direct consequence of Steps i), ii) and Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"}. To this end, we argue by contradiction. Assume that the convergence ([\[c0\]](#c0){reference-type="ref" reference="c0"}) does not hold. Then, there exists $\delta_0>0$ such that for all $k\in\mathbb{N}$ there exists $u_{n_k}\in X$ such that $$\label{m6} \|u_{n_k}-u\|_X\ge \delta_0.$$ Note that the sequence $\{u_{n_k}\}$ is a subsequence of the sequence $\{u_n\}$ and, therefore, Step i) and Lemma [Lemma 9](#lm){reference-type="ref" reference="lm"} imply that it is bounded in $X$. We now use a compactness argument to deduce that there exists a subsequence of the sequence $\{u_{n_k}\}$, again denoted by $\{u_{n_k}\}$, which is weakly convergent in $X$. Then, Step iii) guarantees that $u_{n_k}\to u$ as $k\to \infty$. We now pass to the limit when $k\to \infty$ in [\[m6\]](#m6){reference-type="eqref" reference="m6"} and find that $\delta_0\le 0$. This contradicts inequality $\delta_0>0$ and concludes the proof. $\Box$ Note that Corollaries [Corollary 12](#cor1){reference-type="ref" reference="cor1"} and [Corollary 13](#cor2){reference-type="ref" reference="cor2"} represent a version of Theorem [Theorem 7](#t2){reference-type="ref" reference="t2"} b) and c), obtained under assumption ([\[A\]](#A){reference-type="ref" reference="A"}) and ([\[jj\]](#jj){reference-type="ref" reference="jj"}) instead to ([\[Ap\]](#Ap){reference-type="ref" reference="Ap"}) and ([\[j\]](#j){reference-type="ref" reference="j"}), respectively. Similar arguments can be used to prove the result in Theorem [Theorem 7](#t2){reference-type="ref" reference="t2"} a). **Well-posedness results.** Convergence results for the solution of optimization problems and variational inequalities are strongly related to the well-posedness of these problems. References in the field are [@DZ; @LP; @LP1; @LP2; @Ty] and, more recently [@S]. Here we restrict ourselves to mention only two classical well-posedness concepts in the study of the variational inequality ([\[1\]](#1){reference-type="ref" reference="1"}) and, to this end, we recall the following definitions. **Definition 14**. *a) A sequence $\{u_n\}\subset X$ is called an *approximating sequence* for inequality $(\ref{1})$ if there exists a sequence $0\le\mbox{{$\varepsilon$}}_n\to 0$ such that $$\begin{aligned} &&\label{C42n} u_n\in K,\quad (Au_n,v-u_n)_X+j(v)-j(u_n)+\mbox{{$\varepsilon$}}_n\|v-u_n\|_X\\ [2mm] &&\qquad\qquad\qquad\ge (f,v-u_n)_X\quad\ \forall\, v\in K,\ n\in\mathbb{N}.\nonumber\end{aligned}$$* *b) Inequality $(\ref{1})$ is *well-posed in the sense of Tykhonov* (or, equivalently, is *Tykhonov well-posed*) if it has a unique solution and any approximating sequence converges in $X$ to $u$.* **Definition 15**. *a) A sequence $\{u_n\}\subset X$ is called a *generalized (or $LP$) approximating sequence* for inequality $(\ref{1})$ if there exist two sequences $\{w_n\}\subset X$ and $\{\mbox{{$\varepsilon$}}_n\}\subset\mathbb{R}_+$ such that $w_n\to 0_X$ in $X$, $\mbox{{$\varepsilon$}}_n\to 0$ and, moreover, $$\begin{aligned} &&\hspace{-12mm} \label{C42m} u_n+w_n\in K,\quad (Au_n,v-u_n)_X+j(v)-j(u_n)+\mbox{{$\varepsilon$}}_n\|v-u_n\|_X\\ [2mm] &&\hspace{2mm}\qquad\qquad\ge (f,v-{u_n})_X\quad\ \forall\, v\in K,\ n\in\mathbb{N}.\nonumber\end{aligned}$$* *b) Inequality $(\ref{1})$ is *well-posed in the sense of Levitin-Polyak* (or, equivalently, is *Levitin-Polyak well-posed*) if it has a unique solution and any $LP$-approximating sequence converges in $X$ to $u$.* It is easy to see that if ([\[C42n\]](#C42n){reference-type="ref" reference="C42n"}) holds then both conditions ([\[c2\]](#c2){reference-type="ref" reference="c2"})(a) and ([\[c2\]](#c2){reference-type="ref" reference="c2"})(b) are satisfied. Therefore, using Definition [Definition 14](#Cd41){reference-type="ref" reference="Cd41"} and Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} it is easy to deduce the following result. **Corollary 16**. *Assume $(\ref{K})$, $(\ref{f})$, $(\ref{A})$ and $(\ref{jj})$. Then inequality [\[1\]](#1){reference-type="eqref" reference="1"} is well-posed in the sense of Tykhonov.* The Levitin-Polyak well-posedness of inequality [\[1\]](#1){reference-type="eqref" reference="1"} is a consequence of Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"}, too, as it follows from the following result. **Corollary 17**. *Assume $(\ref{K})$, $(\ref{f})$, $(\ref{A})$ and $(\ref{jj})$. Then inequality [\[1\]](#1){reference-type="eqref" reference="1"} is Levitin-Polyak well-posed.* *Proof.* Let $\{u_n\}\subset X$ be a generalized approximating sequence. Then, Definition [Definition 15](#Cd41n){reference-type="ref" reference="Cd41n"} (a) shows that $d(u_n,K)\le\|w_n\|_X$ for each $n\in\mathbb{N}$ and, since $w_n\to 0_X$, we deduce that condition ([\[c2\]](#c2){reference-type="ref" reference="c2"})(a) is satisfied. Moreover, inequality ([\[C42n\]](#C42n){reference-type="ref" reference="C42n"}) shows that condition ([\[c2\]](#c2){reference-type="ref" reference="c2"})(b) is satisfied, too. We now use Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} to see that $u_n\to u$ in $X$ and conclude the proof by using Definition [Definition 15](#Cd41n){reference-type="ref" reference="Cd41n"} (b). $\Box$ Definitions [Definition 14](#Cd41){reference-type="ref" reference="Cd41"} a) and [Definition 15](#Cd41n){reference-type="ref" reference="Cd41n"} a) show that any approximating sequence is a generalized approximating sequence, too. Therefore, using Corollary [Corollary 17](#cor4){reference-type="ref" reference="cor4"} we obtain the following implications. $$\begin{aligned} &&\mbox{$\{u_n\}$ is an approximating sequence}\\ [2mm] && \Longrightarrow\ \ \mbox{$\{u_n\}$ is a generalized approximating sequence}\\ [2mm] &&\Longrightarrow\ \ u_n\to u\quad{\rm in}\quad X.\end{aligned}$$ The following elementary examples show that the converse of these implications are not valid. **Example 18**. *Consider Problem $\mbox{{${\cal P}$}}$ in the particular case $X=\mathbb{R}$, $K=[0,1]$, $A=I_X$, $j\equiv 0$ and $f=2$. Then $(\ref{1})$ becomes: find $u$ such that $$\label{za} u\in [0,1],\quad u(v-u)\ge f(v-u)\qquad\forall\,v\in[0,1].$$ The solution of this inequality is $u=P_Kf=1$. Let the sequence $\{u_n\}\subset \mathbb{R}$ be given by $u_n=1-\frac{1}{n}$ for all $n\in\mathbb{N}$. Then $u_n\to u$ but $\{u_n\}$ is not a generalized approximating sequence for $(\ref{za})$. Indeed, assume that there exists $0\le\mbox{{$\varepsilon$}}_n\to 0$ such that, for all $n\in\mathbb{N}$ $$\label{zan} u_n(v-u_n)+\mbox{{$\varepsilon$}}_n|v-u_n|\ge f(v-u_n)\qquad\forall\,v\in[0,1].$$ We fix $n\in\mathbb{N}$ and take $v=1-\frac{1}{2n}$ in [\[zan\]](#zan){reference-type="eqref" reference="zan"}. Then, using equalities $u_n=1-\frac{1}{n}$, $f=2$ we deduce that $\mbox{{$\varepsilon$}}_n\ge 1+\frac{1}{n}$. This inequality is valid for each $n\in\mathbb{N}$, which contradicts the convergence $\mbox{{$\varepsilon$}}_n\to 0$.* **Example 19**. *Consider Problem $\mbox{{${\cal P}$}}$ in the particular case $X=\mathbb{R}$, $K=[0,1]$, $A=I_X$, $j\equiv 0$ and $f=1$ and note that the solution of the corresponding inequality $(\ref{1})$ is $u=P_Kf=1$. Let $\{u_n\}\subset \mathbb{R}$ be the sequence given by $u_n=1+\frac{1}{n}$ for all $n\in\mathbb{N}$. Then, $\{u_n\}$ is not an approximating sequence, since condition $u_n\in K$ for each $n\in\mathbb{N}$ is not satisfied. Nevertheless, $\{u_n\}$ is a generalized approximating sequence for $(\ref{za})$. Indeed, it is easy to see that conditions in Definition $\ref{Cd41n}$ a) hold with $w_n=-\frac{1}{n}$ and $\mbox{{$\varepsilon$}}_n=\frac{1}{n}$, for all $n\in\mathbb{N}$.* The examples above show that Tykhonov and Levitin-Polyak well-posedness concepts are not optimal, in the sense that the approximating sequences and the generalized approximate sequences they generate, respectively, do not recover all the sequences of $X$ which converge to the solution $u$ of the variational inequality ([\[1\]](#1){reference-type="ref" reference="1"}). This remark leads in a natural way to the following question: how to identify a class of sequences, say the class of $\mbox{{${\cal T}$}}$-approximating sequences, such that the following equivalence holds: $$\begin{aligned} &&\mbox{$\{u_n\}$ is a $\mbox{{${\cal T}$}}$-approximating sequence} \ \ \Longleftrightarrow\ \ u_n\to u\quad{\rm in}\quad X.\end{aligned}$$ A possible answer to this question is provided by the following definition. **Definition 20**. *A sequence $\{u_n\}\subset X$ is called a *$\mbox{{${\cal T}$}}$-approximating sequence* for inequality $(\ref{1})$ if there exists a sequence $0\le\mbox{{$\varepsilon$}}_n\to 0$ such that $$\begin{aligned} &&\label{C43} d(u_n,K)\le\mbox{{$\varepsilon$}}_n,\quad (Au_n,v-u_n)_X+j(v)-j(u_n)+\mbox{{$\varepsilon$}}_n(1+\|v-u_n\|_X)\\ [2mm] &&\qquad\qquad\qquad\ge (f,v-u_n)_X\quad\ \forall\, v\in K,\ n\in\mathbb{N}.\nonumber \end{aligned}$$* Inspired by Definitions [Definition 14](#Cd41){reference-type="ref" reference="Cd41"} b) and [Definition 15](#Cd41n){reference-type="ref" reference="Cd41n"} b) we complete Definition [Definition 20](#CT){reference-type="ref" reference="CT"} as follows. **Definition 21**. *Inequality $(\ref{1})$ is *$\mbox{{${\cal T}$}}$-well-posed* if it has a unique solution and any $\mbox{{${\cal T}$}}$-approximating sequence converges in $X$ to $u$.* Adopting these definitions, we are in a position to state the following two theorems, which represent equivalent formulations of Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"}. **Theorem 22**. *Assume $(\ref{K})$, $(\ref{f})$, $(\ref{A})$ and $(\ref{jj})$. Then, a sequence $\{u_n\}\subset X$ converges to the solution of inequality $(\ref{1})$ if and only if it is a $\mbox{{${\cal T}$}}$-approximating sequence.* **Theorem 23**. *Assume $(\ref{K})$, $(\ref{f})$, $(\ref{A})$ and $(\ref{jj})$. Then, inequality $(\ref{1})$ has a unique solution if and only if it is $\mbox{{${\cal T}$}}$-well-posed.* Note that Definition [Definition 21](#CTn){reference-type="ref" reference="CTn"} introduces a new concept of well-posedness for the variational inequality ([\[1\]](#1){reference-type="ref" reference="1"}). It can be extended to the study of various nonlinear problems like hemivariational inequalities, inclusions, minimization problems, various classes of time-dependent and evolutionary inequalities. Details can be found in the recent book [@S]. There, the concept of Tykhonov triple, denoted by $\mbox{{${\cal T}$}}$, was introduced. Moreover, the main properties of Tykhonov triples have been stated and proved, together with various examples and counter examples. Then, given a metric space $(X,d)$, Problem $\mbox{{${\cal P}$}}$ and a Tykhonov triple $\mbox{{${\cal T}$}}$, both defined on $X$, the abstract concept of $\mbox{{${\cal T}$}}$-well-posedness for Problem $\mbox{{${\cal P}$}}$ has been introduced, based on two main ingredients: the existence of a unique solution to Problem $\mbox{{${\cal P}$}}$, and the convergence to it to a special kind of sequences, the so-called $\mbox{{${\cal T}$}}$-approximating sequences. Moreover, various applications in Functional Analysis and Contact Mechanics have been provided. We end this section with the remark that well-posedness concepts can be extended to abstract problems for which the set of solutions (assumed to be not empty) is not reduced to a singleton. For such problems various concepts of generalized well-posedness have been introduced in the literature. They are based on the definition of a family of so-called generalized approximating sequences and the condition that every sequence of this family has a subsequence which converges to some point of the solution set. A recent reference on this topic is [@Dey], where the Levitin-Polyak well-posedness of the so-called split equilibrium problems is studied. Additional details can be found in the book [@S]. # A frictionless contact problem {#s5} 0 The abstract results in Sections [3](#s3){reference-type="ref" reference="s3"} and [4](#s4){reference-type="ref" reference="s4"} are useful in the study of various mathematical models which describe the equilibrium of elastic bodies in contact with an obstacle, the so-called foundation. In this section we introduce and study an example of such model and, to this end, we need some notations and preliminaries. Let $d\in\{2,3\}$. We denote by $\mathbb{S}^d$ the space of second order symmetric tensors on $\mathbb{R}^d$ and use the notation $``\cdot"$, $\|\cdot\|$, $\mbox{\boldmath{$0$}}$ for the inner product, the norm and the zero element of the spaces $\mathbb{R}^d$ and $\mathbb{S}^d$, respectively. Let $\Omega\subset\mathbb{R}^d$ be a domain with smooth boundary $\Gamma$ divided into three measurable disjoint parts $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ such that ${ meas}\,(\Gamma_1)>0$. A generic point in $\Omega\cup\Gamma$ will be denoted by $\mbox{\boldmath{$x$}}=(x_i)$. We use the standard notation for Sobolev and Lebesgue spaces associated to $\Omega$ and $\Gamma$. In particular, we use the spaces $L^2(\Omega)^d$, $L^2(\Gamma_2)^d$ and $H^1(\Omega)^d$, endowed with their canonical inner products and associated norms. Moreover, for an element $\mbox{\boldmath{$v$}}\in H^1(\Omega)^d$ we still write $\mbox{\boldmath{$v$}}$ for the trace of $\mbox{\boldmath{$v$}}$ to $\Gamma$. In addition, we consider the space $$\begin{aligned} &&V=\{\,\mbox{\boldmath{$v$}}\in H^1(\Omega)^d\ :\ \mbox{\boldmath{$v$}}=\mbox{\boldmath{$0$}}\ \ {\rm a.e.\ on\ \ }\Gamma_1\,\},\end{aligned}$$ which is a real Hilbert space endowed with the canonical inner product $$(\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}})_V= \int_{\Omega} \mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})\cdot\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})\,dx$$ and the associated norm $\|\cdot\|_V$. Here and below $\mbox{\boldmath{$\varepsilon$}}$ represents the deformation operator, i.e., $$\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})=(\varepsilon_{ij}(\mbox{\boldmath{$u$}})),\quad \varepsilon_{ij}(\mbox{\boldmath{$u$}})=\frac{1}{2}\,(u_{i,j}+u_{j,i}),$$ where an index that follows a comma denotes the partial derivative with respect to the corresponding component of $\mbox{\boldmath{$x$}}$, e.g., $u_{i,j}=\frac{\partial u_i}{\partial x_j}$. The completeness of the space $V$ follows from the assumption ${ meas}\,(\Gamma_1)>0$ which allows us to use Korn's inequality. We denote by $\mbox{\boldmath{$0$}}_V$ the zero element of $V$ and we recall that, for an element $\mbox{\boldmath{$v$}}\in V$, its normal and tangential components on $\Gamma$ are given by $$\mbox{$v_\nu=\mbox{\boldmath{$v$}}\cdot\mbox{\boldmath{$\nu$}}$\ \quad and\ \quad $\mbox{\boldmath{$v$}}_\tau=\mbox{\boldmath{$v$}}-v_\nu\mbox{\boldmath{$\nu$}}$,}$$ respectively. Here and below $\mbox{\boldmath{$\nu$}}$ denote the unitary outward normal to $\Gamma$. We also recall the trace inequality $$\label{trace} \|\mbox{\boldmath{$v$}}\|_{L^2(\Gamma)^d}\leq d_0\|\mbox{\boldmath{$v$}}\|_{V}\qquad \forall\, \mbox{\boldmath{$v$}}\in V$$ in which $d_0$ represents a positive constant. For the inequality problem we consider in this section we use the data ${\cal F}$, $F$, $\mbox{\boldmath{$f$}}_0$, $\mbox{\boldmath{$f$}}_2$ and $k$ which satisfy the following conditions. $$\begin{aligned} &&\label{Fc}\left\{\begin{array}{ll} {\rm (a)}\ {\cal F}\colon \mathbb{S}^d\to \mathbb{S}^d. \\ [1mm] {\rm (b)\ There\ exists}\ M_{\cal F}>0\ {\rm such\ that}\\ {}\qquad \|{\cal F}\mbox{\boldmath{$\varepsilon$}}_1-{\cal F}\mbox{\boldmath{$\varepsilon$}}_2\| \le M_{\cal F} \|\mbox{\boldmath{$\varepsilon$}}_1-\mbox{\boldmath{$\varepsilon$}}_2\|\quad\mbox{for all} \ \ \mbox{\boldmath{$\varepsilon$}}_1,\mbox{\boldmath{$\varepsilon$}}_2 \in \mathbb{S}^d. \\ [1mm] {\rm (c)\ There\ exists}\ m_{\cal F}>0\ {\rm such\ that}\\ {}\qquad ({\cal F}\mbox{\boldmath{$\varepsilon$}}_1-{\cal F}\mbox{\boldmath{$\varepsilon$}}_2) \cdot(\mbox{\boldmath{$\varepsilon$}}_1-\mbox{\boldmath{$\varepsilon$}}_2)\ge m_{\cal F}\, \|\mbox{\boldmath{$\varepsilon$}}_1-\mbox{\boldmath{$\varepsilon$}}_2\|^2\quad \mbox{for all} \ \ \mbox{\boldmath{$\varepsilon$}}_1, \mbox{\boldmath{$\varepsilon$}}_2 \in \mathbb{S}^d.%\\ [1mm] %{\rm (d)}\ {\cal F}(\bx,\bzero)=\bzero\ \ {\rm for\ a.e.}\ \bx\in \Omega. \end{array}\right. \\ [2mm] &&\label{F}F\in L^\infty(\Gamma_3),\qquad F(\mbox{\boldmath{$x$}})\ge 0\ \ {\rm a.e.}\ \mbox{\boldmath{$x$}}\in \Gamma_3.\\ [2mm] &&\label{f0}\mbox{\boldmath{$f$}}_0\in L^2(\Omega)^d,\qquad\mbox{\boldmath{$f$}}_2\in L^2(\Gamma_2)^d.\\ [2mm] &&k>0. \label{k}\end{aligned}$$ Moreover, we use $K$ for the set defined by $$\label{KK}K=\{\,\mbox{\boldmath{$v$}}\in V\ :\ v_\nu \le k\ \ \hbox{a.e. on}\ \Gamma_3\,\}$$ and $r^+$ for the positive part of $r\in\mathbb{R}$, that is $r^+=\max\,\{r,0\}$. Then, the inequality problem we consider is the following. ${\cal P}^c$. *Find $\mbox{\boldmath{$u$}}$ such that* $$\begin{aligned} \label{51} &&\mbox{\boldmath{$u$}}\in K,\quad \int_{\Omega}{\cal F}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})\cdot(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}))\,dx+\int_{\Gamma_3}Fu_\nu^+(v_\nu-u_\nu)\,da\\[2mm] &&\qquad\quad\ge \int_{\Omega}\mbox{\boldmath{$f$}}_0\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\,dx +\int_{\Gamma_2}\mbox{\boldmath{$f$}}_2\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\,da\quad\forall\,\mbox{\boldmath{$v$}}\in K.\nonumber\end{aligned}$$ Following the arguments in [@SofMat; @SMBOOK], it can be shown that Problem $\mbox{{${\cal P}$}}^c$ represents the variational formulation of a mathematical model that describes the equilibrium of an elastic body $\Omega$ which is acted upon by external forces, is fixed on $\Gamma_1$, and is in frictionless contact on $\Gamma_3$. The contact takes place with a rigid foundation covered by a layer of rigid-plastic material of thickness $k$. Here ${\cal F}$ is the elasticity operator, $\mbox{\boldmath{$f$}}_0$ and $\mbox{\boldmath{$f$}}_2$ denote the density of applied body forces and tractions which act on the body and the surface $\Gamma_2$, respectively and $F$ is a given function which describes the yield limit of the rigid-plastic material. Next, consider the sequences $\{F_n\}$, $\{\mu_n\}$, $\{\mbox{\boldmath{$f$}}_{0n}\}$, $\{\mbox{\boldmath{$f$}}_{2n}\}$, $\{k_n\}$ such that, for each $n\in\mathbb{N}$, the following hold. $$\begin{aligned} &&\label{Fn} F_n\in L^\infty(\Gamma_3),\qquad F_n(\mbox{\boldmath{$x$}})\ge 0\ \ {\rm a.e.}\ \mbox{\boldmath{$x$}}\in \Gamma_3.\\ [2mm] &&\label{mu}\mu_n\in L^\infty(\Gamma_3),\qquad \mu_n(\mbox{\boldmath{$x$}})\ge 0\ \ {\rm a.e.}\ \mbox{\boldmath{$x$}}\in \Gamma_3.\\ [2mm] &&\label{f0n}\mbox{\boldmath{$f$}}_{0n}\in L^2(\Omega)^d,\qquad\mbox{\boldmath{$f$}}_{2n}\in L^2(\Gamma_2)^d.\\ [2mm] &&k_n\ge k. \label{kn}\\ [2mm] &&d_0^2\|\mu\|_{L^\infty(\Gamma_3)} \|F_n\|_{L^\infty(\Gamma_3)}<m_{\cal F}. \label{sm}\end{aligned}$$ Recall that in ([\[sm\]](#sm){reference-type="ref" reference="sm"}) and below $d_0$ and $m_{\cal F}$ represent the constants introduced in [\[trace\]](#trace){reference-type="eqref" reference="trace"} and [\[Fc\]](#Fc){reference-type="eqref" reference="Fc"}, respectively. Finally, for each $n\in\mathbb{N}$ define the set $$\label{KKn}K_n=\{\,\mbox{\boldmath{$v$}}\in V\ :\ v_\nu \le k_n\ \ \hbox{a.e. on}\ \Gamma_3\,\}.$$ and consider the following perturbation of Problem $\mbox{{${\cal P}$}}^c$ ${\cal P}^c_n$. *Find $\mbox{\boldmath{$u$}}_n$ such that* $$\begin{aligned} \label{51n} &&\mbox{\boldmath{$u$}}_n\in K_n,\quad \int_{\Omega}{\cal F}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}_n)\cdot(\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})-\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}}_n))\,dx+\int_{\Gamma_3}Fu_{n\nu}^+(v_\nu-u_{n\nu})\,da\\[3mm] &&\qquad+\int_{\Gamma_3}\mu_n\,F_nu_{n\nu}^+(\|{\mbox{\boldmath{$v$}}}_\tau\|-\|{\mbox{\boldmath{$u$}}_n}_\tau\|)\,da\nonumber\\ [3mm] &&\qquad\qquad\ge \int_{\Omega}\mbox{\boldmath{$f$}}_{0n}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})\,dx +\int_{\Gamma_2}\mbox{\boldmath{$f$}}_{2n}\cdot(\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)\,da\quad\forall\,\mbox{\boldmath{$v$}}\in K_n.\nonumber\end{aligned}$$ Problem $\mbox{{${\cal P}$}}_n^c$ represents the variational formulation of a mathematical model of contact, similar to that associated to Problem $\mbox{{${\cal P}$}}^c$. Nevertheless, two differences exist between the corresponding models. The first one arises in the fact that for the model in Problem $\mbox{{${\cal P}$}}_n^c$ the contact is assumed to be frictional and is described with the classical Coulomb law of dry friction, governed by the friction coefficient $\mu_n$. The second difference arises in the fact that in the statement of Problem $\mbox{{${\cal P}$}}^c_n$ we use the data $\mbox{\boldmath{$f$}}_{0n}$, $\mbox{\boldmath{$f$}}_{2n}$ and $k_n$ which represent a perturbation of the data $\mbox{\boldmath{$f$}}_{0}$, $\mbox{\boldmath{$f$}}_{2}$ and $k$, respectively, used in the statement of Problem $\mbox{{${\cal P}$}}^c$. Our main result in this section, is the following. **Theorem 24**. *Assume $(\ref{Fc})$--$(\ref{k})$, $(\ref{Fn})$--$(\ref{sm})$. Then Problem $\mbox{{${\cal P}$}}^c$ has a unique solution and, for each $n\in\mathbb{N}$, Problem $\mbox{{${\cal P}$}}^c_n$ has a unique solution. Moreover, if $$\label{co1c} \left\{\begin{array}{ll} k_n\to k,\quad \mu_n\to 0\ \ {\rm in}\ \ L^\infty(\Gamma_3),\quad F_n\to F\ \ {\rm in}\ \ L^\infty(\Gamma_3),\\ [5mm] \mbox{\boldmath{$f$}}_{0n}\to \mbox{\boldmath{$f$}}_{0}\ \ {\rm in}\ \ L^2(\Omega)^d,\quad \mbox{\boldmath{$f$}}_{2n}\to \mbox{\boldmath{$f$}}_2\ \ {\rm in}\ \ L^2(\Gamma_2)^d\quad{\rm as}\ \ n\to \infty, \end{array}\right.$$ then the solution of Problem $\mbox{{${\cal P}$}}^c_n$ converges to the solution of Problem $\mbox{{${\cal P}$}}^c$, i.e., $$\label{co2c} \mbox{\boldmath{$u$}}_n\to \mbox{\boldmath{$u$}}\quad {\rm in}\ \ V\quad{\rm as}\quad n\to \infty.$$* We start with some additional notation. First, we consider the operator $A:V\to V$, the functions $j,\, j_n:V\to{\if mm {\rm I}\mkern -3mu{\rm R}\else \leavevmode \hbox{I}\kern -.17em\hbox{R} \fi}$, $\varphi_n:V\times V\to{\if mm {\rm I}\mkern -3mu{\rm R}\else \leavevmode \hbox{I}\kern -.17em\hbox{R} \fi}$ and the elements $\mbox{\boldmath{$f$}},\, \mbox{\boldmath{$f$}}_n\in V$ defined as follows: $$\begin{aligned} &&%\hspace{-16mm} \label{8b1}(A\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}})_V =\int_{\Omega}\mbox{{${\cal F}$}}\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$u$}})\cdot\mbox{\boldmath{$\varepsilon$}}(\mbox{\boldmath{$v$}})\,dx,\\ [2mm] &&%\hspace{-16mm} \label{8b3} j(\mbox{\boldmath{$v$}})=\int_{\Gamma_3}Fv_\nu^+\,da,\quad j_n(\mbox{\boldmath{$v$}})=\int_{\Gamma_3}F_nv_\nu^+\,da,\\ [2mm] &&%\hspace{-16mm} \label{8b3b} \varphi_n(\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}})=\int_{\Gamma_3}\mu_n F_nu_\nu^+\|{\mbox{\boldmath{$v$}}}_\tau\|\,da,\\ [2mm] &&%\hspace{-16mm} \label{8b4}\left\{\begin{array}{ll}(\mbox{\boldmath{$f$}},\mbox{\boldmath{$v$}})_V=\displaystyle\int_{\Omega}\mbox{\boldmath{$f$}}_0\cdot\mbox{\boldmath{$v$}}\,dx +\int_{\Gamma_2}\mbox{\boldmath{$f$}}_2\cdot\mbox{\boldmath{$v$}}\,da,\\[4mm](\mbox{\boldmath{$f$}}_n,\mbox{\boldmath{$v$}})_V=\displaystyle\int_{\Omega}\mbox{\boldmath{$f$}}_{0n}\cdot\mbox{\boldmath{$v$}}\,dx +\int_{\Gamma_2}\mbox{\boldmath{$f$}}_{2n}\cdot\mbox{\boldmath{$v$}}\,da \end{array}\right.\end{aligned}$$ for all $\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}}\in V$ and $n\in\mathbb{N}$. Then, it is easy to see that $$\label{e1c} \left\{\begin{array}{l} \mbox{$\mbox{\boldmath{$u$}}$ is a solution of Problem $\mbox{{${\cal P}$}}^c$ if and only if}\\ [4mm] \mbox{\boldmath{$u$}}\in K, \quad (A\mbox{\boldmath{$u$}},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})_V+j(\mbox{\boldmath{$v$}})-j(\mbox{\boldmath{$u$}})\ge (\mbox{\boldmath{$f$}},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}})_V\quad \forall\, \mbox{\boldmath{$v$}}\in K. \end{array}\right.$$ Moreover, for each $n\in\mathbb{N}$, the following equivalence holds: $$\label{e2c} \left\{\begin{array}{l} \mbox{$\mbox{\boldmath{$u$}}_n$ is a solution of Problem $\mbox{{${\cal P}$}}^c_n$ if and only if\qquad}\\ [4mm] \mbox{\boldmath{$u$}}_n\in K_n, \quad (A\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V+j_n(\mbox{\boldmath{$v$}})- j_n(\mbox{\boldmath{$u$}}_n)\\ [2mm] \qquad\qquad\qquad+\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}})-\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$u$}}_n)\ge (\mbox{\boldmath{$f$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V\quad \forall\, \mbox{\boldmath{$v$}}\in K_n. \end{array}\right.$$ Equivalence [\[e1c\]](#e1c){reference-type="eqref" reference="e1c"} suggests us to use the abstract results in Sections [2](#s2){reference-type="ref" reference="s2"} and [3](#s3){reference-type="ref" reference="s3"} with $X=V$, $K$ defined by ([\[KK\]](#KK){reference-type="ref" reference="KK"}), $A$ defined by ([\[8b1\]](#8b1){reference-type="ref" reference="8b1"}), $j$ defined by ([\[8b3\]](#8b3){reference-type="ref" reference="8b3"}) and $\mbox{\boldmath{$f$}}$ given by ([\[8b4\]](#8b4){reference-type="ref" reference="8b4"}). It is easy to see that in this case conditions $(\ref{K})$, $(\ref{f})$, $(\ref{A})$ and $(\ref{jj})$ are satisfied. For instance, using assumption ([\[Fc\]](#Fc){reference-type="ref" reference="Fc"}) we see that $$\begin{aligned} (A\mbox{\boldmath{$u$}}- A\mbox{\boldmath{$v$}},\mbox{\boldmath{$u$}}-\mbox{\boldmath{$v$}})_{V} \geq m_{\cal F} \|\mbox{\boldmath{$u$}}-\mbox{\boldmath{$v$}}\|^{2}_{V},\qquad \|A\mbox{\boldmath{$u$}}-A\mbox{\boldmath{$v$}}\|_V\le M_{\cal F}\, {\|\mbox{\boldmath{$u$}}-\mbox{\boldmath{$v$}}\|_V}\end{aligned}$$ for all $\mbox{\boldmath{$u$}},\, \mbox{\boldmath{$v$}}\in V$. Therefore, conditions ([\[A1\]](#A1){reference-type="ref" reference="A1"}) and ([\[A2\]](#A2){reference-type="ref" reference="A2"}) hold with $m=m_{\cal F}$ and $M=M_{\cal F}$, respectively which shows that $A$ satisfies condition ([\[A\]](#A){reference-type="ref" reference="A"}). Condition ([\[jj\]](#jj){reference-type="ref" reference="jj"}) is also satisfied since $j$ is a continuous seminorm on the space $V$. Therefore, we are in a position to apply Theorem [Theorem 6](#t1){reference-type="ref" reference="t1"} in order to deduce the existence of a unique solution of the variational inequality in ([\[e1c\]](#e1c){reference-type="ref" reference="e1c"}). The unique solvability of the variational inequality in ([\[e2c\]](#e2c){reference-type="ref" reference="e2c"}) follows from a standard argument of quasivariational inequalities. The proof can be found in [@SofMat], for instance and, therefore, we skip it. Note that here, besides the regularities ([\[Fn\]](#Fn){reference-type="ref" reference="Fn"})--([\[f0n\]](#f0n){reference-type="ref" reference="f0n"}) and condition $k_n>0$, we need the smallness assumption ([\[sm\]](#sm){reference-type="ref" reference="sm"}). We now move to the proof of the convergence [\[co2c\]](#co2c){reference-type="eqref" reference="co2c"}. Let $n\in\mathbb{N}$ and $\mbox{\boldmath{$v$}}\in K$. We write $$\begin{aligned} &&\label{w1}(A\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V+j(\mbox{\boldmath{$v$}})-j(\mbox{\boldmath{$u$}}_n)- (\mbox{\boldmath{$f$}},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V \\ [2mm] &&=(A\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V+j_n(\mbox{\boldmath{$v$}})-j_n(\mbox{\boldmath{$u$}}_n) +\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}})-\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$u$}}_n)-(\mbox{\boldmath{$f$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V \nonumber\\ [2mm] &&+\big[j(\mbox{\boldmath{$v$}})-j(\mbox{\boldmath{$u$}}_n)-j_n(\mbox{\boldmath{$v$}})+j_n(\mbox{\boldmath{$u$}}_n)\big] +\big[\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$u$}}_n)-\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}})\big]+(\mbox{\boldmath{$f$}}_n-\mbox{\boldmath{$f$}},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V.\nonumber\end{aligned}$$ Then, we use assumption $k_n\ge k$ to see that $K\subset K_n$ and, therefore, we are allowed to test in ([\[e2c\]](#e2c){reference-type="ref" reference="e2c"}) with $\mbox{\boldmath{$v$}}\in K$. We obtain $$\label{w2} (A\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V+j_n(\mbox{\boldmath{$v$}})-j_n(\mbox{\boldmath{$u$}}_n) +\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}})-\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$u$}}_n)-(\mbox{\boldmath{$f$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V\ge 0$$ and, combining [\[w1\]](#w1){reference-type="eqref" reference="w1"} and [\[w2\]](#w2){reference-type="eqref" reference="w2"} we deduce that $$\begin{aligned} &&\label{w3}(A\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V+j(\mbox{\boldmath{$v$}})-j(\mbox{\boldmath{$u$}}_n)+ \big[j(\mbox{\boldmath{$u$}}_n)-j(\mbox{\boldmath{$v$}})+j_n(\mbox{\boldmath{$v$}})-j_n(\mbox{\boldmath{$u$}}_n)\big]\\ [2mm] &&\qquad+\big[\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}})-\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$u$}}_n)\big]+(\mbox{\boldmath{$f$}}-\mbox{\boldmath{$f$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V\ge (\mbox{\boldmath{$f$}},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V.\nonumber\end{aligned}$$ Next, using definitions ([\[8b3\]](#8b3){reference-type="ref" reference="8b3"})--([\[8b4\]](#8b4){reference-type="ref" reference="8b4"}) and standard embedding and trace arguments we find that $$\begin{aligned} &&\label{w4}j(\mbox{\boldmath{$u$}}_n)-j(\mbox{\boldmath{$v$}})+j_n(\mbox{\boldmath{$v$}})-j_n(\mbox{\boldmath{$u$}}_n)\le c_0\|F_n-F\|_{L^\infty(\Gamma_3)}\|\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n\|_V,\\[3mm] &&\label{w5}(\mbox{\boldmath{$f$}}-\mbox{\boldmath{$f$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V\\ && \qquad\le c_0\big(\|\mbox{\boldmath{$f$}}_{0n}-\mbox{\boldmath{$f$}}_0\|_{L^2(\Omega)^d}+ \|\mbox{\boldmath{$f$}}_{2n}-\mbox{\boldmath{$f$}}_2\|_{L^2(\Gamma_2)^d}\big)\|\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n\|_V,\nonumber\\ [3mm] &&\label{w6}\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}})-\varphi_n(\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$u$}}_n)\le c_0\|\mu_n\|_{L^\infty(\Gamma_3)}\|F_n\|_{L^\infty(\Gamma_3)}\|\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n\|_V,\end{aligned}$$ where $c_0$ represents a positive constant which does not depend on $n$. We now substitute [\[w4\]](#w4){reference-type="eqref" reference="w4"}--[\[w6\]](#w6){reference-type="eqref" reference="w6"} in [\[w3\]](#w3){reference-type="eqref" reference="w3"} and use notation $$\begin{aligned} &&\label{w7}\mbox{{$\varepsilon$}}_n={\rm max}\,\Big\{c_0\|F_n-F\|_{L^\infty(\Gamma_3)},c_0\big(\|\mbox{\boldmath{$f$}}_{0n}-\mbox{\boldmath{$f$}}_0\|_{L^2(\Omega)^d}+ \|\mbox{\boldmath{$f$}}_{2n}-\mbox{\boldmath{$f$}}_2\|_{L^2(\Gamma_2)^d}\big),\nonumber \\ [2mm] &&\qquad\qquad\qquad c_0\|\mu_n\|_{L^\infty(\Gamma_3)}\|F_n\|_{L^\infty(\Gamma_3)}\Big\}\end{aligned}$$ to deduce that $$\label{w8}(A\mbox{\boldmath{$u$}}_n,\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V+j(\mbox{\boldmath{$v$}})-j(\mbox{\boldmath{$u$}}_n)+ \mbox{{$\varepsilon$}}_n\|\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n\|_V\ge (\mbox{\boldmath{$f$}},\mbox{\boldmath{$v$}}-\mbox{\boldmath{$u$}}_n)_V.$$ Note that definition [\[w7\]](#w7){reference-type="eqref" reference="w7"} and assumptions [\[co1c\]](#co1c){reference-type="eqref" reference="co1c"} imply that $\mbox{{$\varepsilon$}}_n\to 0$. Therefore, inequality [\[w8\]](#w8){reference-type="eqref" reference="w8"} shows that condition ([\[c2\]](#c2){reference-type="ref" reference="c2"})(b) is satisfied. We are now in a position to use Lemma [Lemma 9](#lm){reference-type="ref" reference="lm"} to see that $$\label{w9} \mbox{there exists $D>0$ such that}\ \|\mbox{\boldmath{$u$}}_n\|_V\le D\quad\forall\, n\in\mathbb{N}.$$ On the other hand, using definitions [\[KK\]](#KK){reference-type="eqref" reference="KK"} and [\[KKn\]](#KKn){reference-type="eqref" reference="KKn"} it is easy to see that $\frac{k}{k_n}\mbox{\boldmath{$u$}}_n\in K$ and, therefore $$\label{w10}d(\mbox{\boldmath{$u$}}_n,K)\le \Big\|\mbox{\boldmath{$u$}}_n-\frac{k}{k_n}\mbox{\boldmath{$u$}}_n\Big\|_V=\Big|1-\frac{k}{k_n}\Big|\|\mbox{\boldmath{$u$}}_n\|_V \quad\forall\, n\in\mathbb{N}.$$ We now use [\[w10\]](#w10){reference-type="eqref" reference="w10"}, [\[w9\]](#w9){reference-type="eqref" reference="w9"} and assumption $k_n\to k$ to see that $d(\mbox{\boldmath{$u$}}_n,K)\to 0$ which shows that condition ([\[c2\]](#c2){reference-type="ref" reference="c2"})(a) is satisfied, too. It follows from above that we are in a position to use Theorem [Theorem 8](#tm){reference-type="ref" reference="tm"} to deduce the convergence ([\[co2c\]](#co2c){reference-type="ref" reference="co2c"}). These results combined with equivalences ([\[e1c\]](#e1c){reference-type="ref" reference="e1c"}) and ([\[e2c\]](#e2c){reference-type="ref" reference="e2c"}) allows us to conclude the proof of the theorem. $\Box$ We end this section with the following physical interpretation of Theorem [Theorem 24](#t5){reference-type="ref" reference="t5"}. First, the existence and uniqueness part in the theorem proves the unique weak solvability of the contact problems considered: the frictionless contact with a rigid foundation covered by a layer of rigid-plastic material of thickness $k$ and the frictional contact with a rigid foundation covered by a layer of rigid-plastic material of thickness $k_n$. Second, the weak solution of the frictionless contact problem with a rigid foundation covered by a layer of rigid-plastic material depends continuously on the density of body forces and surface tractions as well as on the yield limit and the thickness of the layer. In addition, it can be approached by the solution of the corresponding frictional problem with a small coefficient of friction. # A heat transfer problem {#s6} 0 In this section we apply the abstract results in Sections [2](#s2){reference-type="ref" reference="s2"}--[4](#s4){reference-type="ref" reference="s4"} in the study of a mathematical model which describes a heat transfer phenomenon. The problem we consider represents a version of the problems already considered in [@GMOT; @ST3] and, for this reason, we skip the details. Its classical formulation is the following. . *Find a temperature field $u:\Omega\to{\if mm {\rm I}\mkern -3mu{\rm R}\else \leavevmode \hbox{I}\kern -.17em\hbox{R} \fi}$ such that* $$\begin{aligned} &&\label{d1} -\Delta u=g\qquad\ \ {\rm a.e.\ in\ }\Omega,\\ [3mm] &&\label{d2}u=0\hspace{18mm}{\rm a.e.\ on\ }\Gamma_1,\\ [3mm] &&\label{d3}-\frac{\partial u}{\partial\nu}=q\hspace{11mm}{\rm a.e.\ on\ }\Gamma_2.\\ [3mm] &&\label{d4}u=b\hspace{18mm}{\rm a.e.\ on\ }\Gamma_3. \end{aligned}$$ Here, as in Section [5](#s5){reference-type="ref" reference="s5"}, $\Omega$ is a bounded domain in $\mathbb{R}^d$ ($d=2,3$ in applications) with smooth boundary $\Gamma$, divided into three measurable disjoint parts $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ such that ${ meas}\,(\Gamma_1)>0$. We denote by $\mbox{\boldmath{$\nu$}}$ and outer normal unit to $\Gamma$ and recall that in ([\[d1\]](#d1){reference-type="ref" reference="d1"})--([\[d4\]](#d4){reference-type="ref" reference="d4"}) we do not mention the dependence of the different functions on the spatial variable $\mbox{\boldmath{$x$}}\in\Omega\cup\Gamma$. The functions $g$, $q$ and $b$ are given and will be described below. Here we mention that $g$ represents the internal energy, $q$ is a prescribed heat flux and $b$ denotes a prescribed temperature. Moreover, $\frac{\partial u}{\partial\nu}$ denotes the normal derivative of $u$ on the boundary $\Gamma$. Now, let $\{\lambda_n\}\subset\mathbb{R}$ be a sequence such that $\lambda_n>0$ for each $n\in\mathbb{N}$. Then for each $n\in\mathbb{N}$ we consider the following boundary problem. . *Find a temperature field $u_n:\Omega\to{\if mm {\rm I}\mkern -3mu{\rm R}\else \leavevmode \hbox{I}\kern -.17em\hbox{R} \fi}$ such that* $$\begin{aligned} &&\label{d1n} -\Delta u_n=g\hspace{23mm}{\rm a.e.\ in\ }\Omega,\\ [2mm] &&\label{d2n}u_n=0\hspace{30mm}{\rm a.e.\ on\ }\Gamma_1,\\ [2mm] &&\label{d3n}-\frac{\partial u_n}{\partial\nu}=q\hspace{24mm}{\rm a.e.\ on\ }\Gamma_2,\\ [2mm] &&\label{d4n}-\frac{\partial u_n}{\partial\nu}=\frac{1}{\lambda_n}(u_n-b)\hspace{6mm}{\rm a.e.\ on\ }\Gamma_3. \end{aligned}$$ Note that Problem ${\mbox{{${\cal C}$}}}^t_n$ is obtained from Problem $\mbox{{${\cal C}$}}^t$ by replacing the Dirichlet boundary condition ([\[d4\]](#d4){reference-type="ref" reference="d4"}) with the Neumann boundary condition ([\[d4n\]](#d4n){reference-type="ref" reference="d4n"}). Here $\lambda_n$ is a positive parameter, and its inverse $h_n=\frac{1}{\lambda_n}$ represents the heat transfer coefficient on the boundary $\Gamma_3$. In contrast to Problem $\mbox{{${\cal C}$}}^t$ (in which the temperature is prescribed on $\Gamma_3$), in Problem $\mbox{{${\cal C}$}}^t_n$ this condition is replaced by a condition on the flux of the temperature, governed by a positive heat transfer coefficient. For the variational analysis of Problem $\mbox{{${\cal C}$}}^t$ we consider the space $$V=\ \{\,v\in H^1(\Omega)\ :\ v=0\quad{\rm a.e.\ on}\quad\Gamma_1\}.$$ Note that, here and below in this section, we still write $v$ for trace of the element $v$ to $\Gamma$. Denote in what follows by $(\cdot,\cdot)_V$ the inner product of the space $H^1(\Omega)$ restricted to $V$ and by $\|\cdot\|_V$ the associated norm. Since $meas \,\Gamma_1>0$, it is well known that $(V,(\cdot,\cdot)_V)$ is a real Hilbert space. Next, we assume that $$\begin{aligned} &&\label{dw} g\in L^2(\Omega),\quad q\in L^2(\Gamma_2),\quad b\in H^{\frac{1}{2}}(\Gamma_3),\\ [2mm] &&\label{dwz} \mbox{there exists $v_0\in V$ such that $v_0=b$\ \ a.e.\ on\ \ $\Gamma_3$,}\end{aligned}$$ and, finally, we introduce the set $$\begin{aligned} &&\label{d6} K=\ \{\,v\in V\ :\ v=b\ \ {\rm a.e.\ on}\ \ \Gamma_3\,\}.\end{aligned}$$ Note that assumption ([\[dwz\]](#dwz){reference-type="ref" reference="dwz"}) represents a compatibilty assumption on the data $b$ which guarantees that the set $K$ is not empty. Then, it is easy to see that the variational formulation of problems $\mbox{{${\cal C}$}}^t$ and $\mbox{{${\cal C}$}}^t_n$, obtained by standard arguments, are as follows. ${\cal P}^t$. *Find $u$ such that* $$\label{1d}u\in K,\quad\int_{\Omega}\nabla u\cdot(\nabla v-\nabla u)\,dx+\int_{\Gamma_2}q(v-u)\,da=\int_\Omega g(v-u) \,dx \quad\forall\,v\in K.$$ ${\cal P}^t_n$. *Find $u_n$ such that* $$\begin{aligned} &&\label{1dx}u_n\in V,\qquad\int_{\Omega}\nabla u_n\cdot(\nabla v-\nabla u_n)\,dx+\int_{\Gamma_2}q(v-u_n)\,da\\[2mm] &&\qquad\qquad\ \ +\frac{1}{\lambda_n} \int_{\Gamma_3}(u_n-b)(v-u_n)\,da \ge\int_\Omega g(v-u_n) \,dx \quad\forall\,v\in V.\nonumber\end{aligned}$$ Our main result in this section is the following. **Theorem 25**. *Assume $(\ref{dw})$ and $(\ref{dwz})$. Then, Problem $\mbox{{${\cal P}$}}^t$ has a unique solution and, for each $n\in\mathbb{N}$, Problem $\mbox{{${\cal P}$}}^t_n$ has a unique solution. Moreover, if $\lambda_n\to 0$, then the solution of Problem $\mbox{{${\cal P}$}}^t_n$ converges to the solution of Problem $\mbox{{${\cal P}$}}^t$, i.e., $$\label{cot2} u_n\to u\quad {\rm in}\ \ V\qquad\quad{\rm as}\quad n\to \infty.$$* We consider the operators $A:V\to V$, $G:V\to V$ and the element $f\in V$ defined as follows: $$\begin{aligned} &&\label{d8} \ (Au,v)_V=\int_\Omega \nabla u\cdot\nabla v\,dx\qquad\forall\,u, v\in V,\\ [2mm] &&\label{d9} (Gu,v)_V=\int_{\Gamma_3}(u-b)v\,da\qquad\forall\,u, v\in V,\\ [2mm] &&\label{dd9} \ (f,v)_V=\int_{\Omega}gv\,dx- \int_{\Gamma_2}qv\,da\qquad\forall\, v\in V. \end{aligned}$$ Then, since the set $\{\,v-v_0\ :\ v\in K\,\}$ is a linear subspace on $V$, it is easy to see that $$\label{e1} \left\{\begin{array}{l} \mbox{$u$ is a solution of Problem $\mbox{{${\cal P}$}}^t$ if and only if}\\ [3mm] u\in K, \quad (Au,v-u)_V\ge (f,v-u)_{V}\quad\ \forall\, v\in K. \end{array}\right.$$ Moreover, for each $n\in\mathbb{N}$, $$\label{e2} \left\{\begin{array}{l} \mbox{$u_n$ is a solution of Problem $\mbox{{${\cal P}$}}^t_n$ if and only if\qquad}\\ [3mm] u_n\in V, \quad (Au_n,v-u_n)_V+\frac{1}{\lambda_n}(Gu_n,v-u_n)_V\\ [2mm] \qquad\qquad\qquad\ge (f,v-u_n)_{V}\quad\ \forall\, v\in V. \end{array}\right.$$ We use the abstract results in Sections [2](#s2){reference-type="ref" reference="s2"} and [3](#s3){reference-type="ref" reference="s3"} with $X=V$, $K$ defined by ([\[d6\]](#d6){reference-type="ref" reference="d6"}), $A$ defined by ([\[d8\]](#d8){reference-type="ref" reference="d8"}), $G$ defined by ([\[d9\]](#d9){reference-type="ref" reference="d9"}), $f$ given by [\[dd9\]](#dd9){reference-type="eqref" reference="dd9"}, and $j\equiv 0$. It is easy to see that in this case conditions $(\ref{K})$, $(\ref{f})$, $(\ref{la})$, $(\ref{G})$, $(\ref{A})$ and $(\ref{jj})$ are satisfied. Therefore, we are in a position to apply Theorem [Theorem 6](#t1){reference-type="ref" reference="t1"} in order to deduce the existence of a unique solution of the variational inequalities in ([\[e1\]](#e1){reference-type="ref" reference="e1"}) and ([\[e2\]](#e2){reference-type="ref" reference="e2"}), respectively. Moreover, using Corollary [Corollary 13](#cor2){reference-type="ref" reference="cor2"} we deduce the convergence ([\[cot2\]](#cot2){reference-type="ref" reference="cot2"}). These results combined with ([\[e1\]](#e1){reference-type="ref" reference="e1"}) and ([\[e2\]](#e2){reference-type="ref" reference="e2"}) allow us to conclude the proof. $\Box$ We end this section with the following physical interpretation of Theorem [Theorem 25](#t7){reference-type="ref" reference="t7"}. First, the solutions of Problems $\mbox{{${\cal P}$}}^t$ and $\mbox{{${\cal P}$}}^t_n$ represent weak solutions of the heat transfer Problems $\mbox{{${\cal C}$}}^t$ and $\mbox{{${\cal C}$}}^t_n$, respectively. Therefore, Theorem [Theorem 25](#t7){reference-type="ref" reference="t7"} provides the unique weak solvability of these problems. Second, the convergence result [\[cot2\]](#cot2){reference-type="eqref" reference="cot2"} shows that the weak solution of Problem $\mbox{{${\cal C}$}}^t$ with prescribed temperature on $\Gamma_3$ can be approached by the solution of Problem $\mbox{{${\cal C}$}}_n^t$ with heat transfer on $\Gamma_3$, for a large heat transfer coefficient. # Acknowledgement {#acknowledgement .unnumbered} This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH. 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Lions, *Quelques méthodes de résolution des problèmes aux limites non linéaires*, Gauthiers-Villars, Paris, 1969. R. Lucchetti and F. Patrone, A characterization of Tychonov well-posedness for minimum problems with applications to variational inequalities, *Numer. Funct. Anal. Optim.* **3** (1981), 461--476. R. Lucchetti and F. Patrone, Some properties of "well-posedness\" variational inequalities governed by linear operators, *Numer. Funct. Anal. Optim.* **5** (1983), 349--361. U. Mosco, Convergence of convex sets and of solutions of variational inequalities, *Adv. Math.* **3** (1968), 510--585. P.D. Panagiotopoulos, *Inequality Problems in Mechanics and Applications*, Birkhäuser, Boston, 1985. D. Pascali and S. Sburlan, *Nonlinear Mappings of Monotone Type*, Sijthoff and Noordhoff International Publishers, Alpen aan den Rijn, 1978. M. Sofonea, *Well-posed Nonlinear Problems. A Study of Mathematical Models of Contact*, Springer, 2023, to appear. M. Sofonea and A. Matei, *Mathematical Models in Contact Mechanics*, London Mathematical Society Lecture Note Series **398**, Cambridge University Press, 2012. M. Sofonea and S. Migórski, *Variational-Hemivariational Inequalities with Applications*, Pure and Applied Mathematics, Chapman & Hall/CRC Press, Boca Raton-London, 2018. M. Sofonea and D.A. Tarzia, On the Tykhonov well-posedness of an antiplane shear problem, *Mediterr. J. Math.* **17** (2020), Paper No. 150, 21 pp. M. Sofonea and D.A. Tarzia, Convergence results for optimal control problems governed by elliptic quasivariational inequalities, *Numer. Func. Anal. Optim.* **41** (2020), 1326--1351. M. Sofonea and D.A. Tarzia, Tykhonov well-posedness of a heat transfer problem with unilateral constraints, *Appl. Math.* **67** (2022), 167--197. A.N. Tykhonov, On the stability of functional optimization problems, *USSR Comput. Math. Math. Phys.* **6** (1966), 631--634. Y.B. Xiao and M. Sofonea, Tykhonov triples, well-posedness and convergence results, *Carphatian J. Math.* **37** (2021), 135--143. E. Zeidler, *Nonlinear Functional Analysis and Applications II A/B, Springer*, New York, 1990. [^1]: Corresponding author, E-mail : sofonea\@univ-perp.fr

arxiv_math

--- abstract: | Let $G$ be a connected reductive group, $P$ its parabolic subgroup. We consider the parabolic semi-infinite category of sheaves on the affine Grassmanian of $G$, and construct the parabolic version of the semi-infinite $\operatorname{IC}$-sheaf of each orbit. We establish some of its properties and relate it to sheaves on the Drinfeld compactification $\operatorname{\widetilde\operatorname{Bun}}_P$ of the moduli stack $\operatorname{Bun}_P$ of $P$-torsors on a curve. We also relate the parabolic semi-infinite $\operatorname{IC}$-sheaf with the dual baby Verma object on the spectral side. address: - Department of Mathematics, Yale University 219 Prospect St. New Haven, CT, USA 06511 - Institut Elie Cartan Lorraine, Université de Lorraine, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France author: - G. Dhillon - S. Lysenko title: "Semi-infinite parabolic $\\operatorname{IC}$-sheaf" --- [^1] # Introduction ## Our goals ### Let $G$ be a connected reductive group over an algebraically closed field $k$, $B\subset G$ be a Borel subgroup and $P$ a standard parabolic with standard Levi $M$. In [@Gai19SI] the semi-infinite category of sheaves on $\operatorname{Gr}_G$ and the corresponding semi-infinite intersection cohomology sheaf on $\operatorname{Gr}_G$ was constructed in the Borel case, its properties and applications were studied. In this note we develop the corresponding parabolic version $\operatorname{IC}^{\frac{\infty}{2}}_P$ of the semi-infinte $\operatorname{IC}$-sheaf and its properties. In the case $G=P$ the semi-infinite category of sheaves becomes the usual Satake category, and the semi-infinite $\operatorname{IC}$-sheaves become the $\operatorname{IC}$-sheaves of the closures of $G({\mathcal O})$-orbits. So, our theory interpolates between the Satake case and the semi-infinite Borel case. ### Let ${\mathcal O}=k[[t]]\subset F=k((t))$. Let $U(P)$ be the unipotent radical of $P$. Set $H=M({\mathcal O})U(P)(F)$. The parabolic semi-infinite category of sheaves is defined as $$\operatorname{SI}_P=Shv(\operatorname{Gr}_G)^H$$ (cf. Section [3.1.1](#Sect_Local automorphic side_begins){reference-type="ref" reference="Sect_Local automorphic side_begins"} for details). The $H$-orbits on $\operatorname{Gr}_G$ are indexed by the set $\Lambda^+_M$ of dominant coweight for $M$. Namely, to $\lambda\in\Lambda_M^+$ there corresponds the $H$-orbit $S_P^{\lambda}$ through $t^{\lambda}G({\mathcal O})$. As in the Borel case, the $H$-orbits on $\operatorname{Gr}_G$ (unless $G=P$) are infinite-dimensional and have infinite codimension. For this reason one needs to work on the level of $\operatorname{DG}$-categories to develop this theory. If $\lambda,\nu\in\Lambda^+_M$ then $S_P^{\nu}$ lies in the closure $\bar S_P^{\lambda}$ of $S_P^{\lambda}$ iff $\lambda-\nu\in\Lambda^{pos}$. Here $\Lambda^{pos}$ is the ${\mathbb Z}_+$-span of simple positive coroots. While the definition and some first properties of $\operatorname{SI}_P$ were given in [@LC2; @LC; @BL], we introduce the semi-infinite t-structure on $\operatorname{SI}_P$ and the objects $\operatorname{IC}^{\frac{\infty}{2}}_{P,\lambda}\in \operatorname{SI}_P$ playing the role of the semi-infinite $\operatorname{IC}$-sheaves of $\bar S_P^{\lambda}$. These are our main objects of study. We relate them to the globally defined objects (for a given smooth curve) as well as to the dual baby Verma objects on the spectral side. ## What is done in this paper? Let us describe our results and compare our situation with the Borel case. ### We introduce the semi-infinite parabolic category $Shv(\operatorname{Gr}_G)^H$ and its renormalized version $Shv(\operatorname{Gr}_G)^{H, ren}$. We define actions of $\operatorname{Sph}_G=Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ on these categories, and an action of $\operatorname{Sph}_M=Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$ on $Shv(\operatorname{Gr}_G)^H$. We define a natural semi-infinite t-structure on $\operatorname{SI}_P=Shv(\operatorname{Gr}_G)^H$ and show that the action of ${\operatorname{Rep}}(\check{G})^{\heartsuit}$ and some shifted[^2] action of ${\operatorname{Rep}}(\check{M})^{\heartsuit}$ on $\operatorname{SI}_P$ are t-exact. We also relate $\operatorname{SI}_P$ to the categories $Shv(\operatorname{Gr}_P)^H$ and $Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$, this is analogous to ([@BL], Section 3.1). ### Let $T\subset B$ be a maximal torus. Write $\Lambda_{M, ab}$ for the set of coweights of $T$ orthogonal to the roots of $(M, T)$. Let $\Lambda_{M, ab}^+\subset \Lambda_{M, ab}$ be the subset of those coweights, which are dominant for $G$. Write $\check{M}$ (resp., $\check{G}$) for the Langlands dual group of $M$ (resp., of $G$). Set $\check{M}_{ab}=\check{M}/[\check{M}, \check{M}]$. Let $\check{P}\subset \check{G}$ be the parabolic subgroup dual to $P$. ### As in the Borel case, for $\lambda\in\Lambda^+_M$ we give a local definition of $\operatorname{IC}^{\frac{\infty}{2}}_{P,\lambda}$ by some colimit formula. It differs from the Borel case in two aspects. First, we propose such a definition for any $H$-orbit on $\operatorname{Gr}_G$. Second, the index category over which the colimit is taken changes, instead of being the category of dominant coweights $\Lambda^+$ of $(G, T)$, now it is the category of $\mu\in\Lambda^+_{M, ab}$ such that $\lambda+\mu\in\Lambda^+$. In the case $\lambda=0$ we give a conceptual explanation of this formula. Namely, we propose an analog of the Drinfeld-Plücker formalism (that only applies for $\lambda=0$ for the moment) in the parabolic setting. There are two versions of this formalism corresponding, for historical reasons, to $\operatorname{\widetilde\operatorname{Bun}}_P$ and $\operatorname{\overline{Bun}} _P$ respectively (these are some relative compactifications of the stack $\operatorname{Bun}_P$ of $P$-torsors on a smooth projective curve, cf. Section [3.2](#Sect_Relation between local and global){reference-type="ref" reference="Sect_Relation between local and global"}). It is crucial for this formalism that both $\check{G}/[\check{P},\check{P}]$ and $\check{G}/U(\check{P})$ are quasi-affine, here $U(\check{P})$ is the unipotent radical of $\check{P}$. This formalism is supposed to produce analogs of the dual baby Verma objects in a situation, when we are given a $\operatorname{DG}$-category $C$ equipped with an action of ${\operatorname{Rep}}(\check{M})\otimes{\operatorname{Rep}}(\check{G})$ (resp., of ${\operatorname{Rep}}(\check{M}_{ab})\otimes{\operatorname{Rep}}(\check{G})$) for the $\operatorname{\widetilde\operatorname{Bun}}_P$-version (resp., for $\operatorname{\overline{Bun}} _P$-version). The corresponding colimit for the $\operatorname{\overline{Bun}} _P$-version describes the composition $$C\otimes_{{\operatorname{Rep}}(\check{M}_{ab})\otimes{\operatorname{Rep}}(\check{G})} {\operatorname{Rep}}(\check{P})\to C\otimes_{{\operatorname{Rep}}(\check{M}_{ab})\otimes{\operatorname{Rep}}(\check{G})}{\operatorname{Rep}}(\check{M})\stackrel{\operatorname{oblv}}{\to} C,$$ where the first functor is the pullback along $B(\check{M})\to B(\check{P})$. For the $\operatorname{\widetilde\operatorname{Bun}}_P$-version it describes the similar composition $$C\otimes_{{\operatorname{Rep}}(\check{M})\otimes{\operatorname{Rep}}(\check{G})} {\operatorname{Rep}}(\check{P})\to C\otimes_{{\operatorname{Rep}}(\check{M})\otimes{\operatorname{Rep}}(\check{G})}{\operatorname{Rep}}(\check{M})\stackrel{\operatorname{oblv}}{\to} C$$ We think of the dual baby Verma object as an object of $C$ together with some version of a Hecke property. We then introduce a version of the dual baby Verma object ${\mathcal M}_{\check{G}, \check{P}^-}$ for $$C=\operatorname{IndCoh}((\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-)$$ (as well as its version ${\mathcal M}_{\check{G}, \check{P}}$ with the roles of $P, B$ and $P^-, B^-$ exchanged). Here $\check{\mathfrak{g}}=\operatorname{Lie}\check{G}$ and $\check{\mathfrak{u}}(P^-)$ is the Lie algebra of the unipotent radical of $\check{P}^-$. One of our main results is Theorem [Theorem 1](#Thm_4.5.10){reference-type="ref" reference="Thm_4.5.10"} giving a precise relation between $\operatorname{IC}^{\frac{\infty}{2}}_{P,0}$ and ${\mathcal M}_{\check{G}, \check{P}}$ via an equivalence announced by G. Dhillon and H. Chen (cf. Proposition [Proposition 1](#Pp_Chen_Dhillon){reference-type="ref" reference="Pp_Chen_Dhillon"}) composed with our equivalence ([\[eq_ren_parahoric_versus_H\]](#eq_ren_parahoric_versus_H){reference-type="ref" reference="eq_ren_parahoric_versus_H"}) and the so called *long intertwining operator*, cf. Section [4.5](#Sect_4.5){reference-type="ref" reference="Sect_4.5"}. This also gives some new insight in the structure of the parahoric Hecke $\operatorname{DG}$-categories (cf. Proposition [Proposition 1](#Pp_key_for_baby_Verma_transform_for_P){reference-type="ref" reference="Pp_key_for_baby_Verma_transform_for_P"}). As an aside for our proof of Theorem [Theorem 1](#Thm_4.5.10){reference-type="ref" reference="Thm_4.5.10"}, we obtain a new result about the intertwining functors between two distinct parabolic Hecke categories. Namely, assume given a $\operatorname{DG}$-category $C$ with an action of $Shv(G)$ and two parabolics $P,Q\subset G$. We determine all the pairs $(P, Q)$ for which the composition $$C^P \;\stackrel{\operatorname{oblv}}{\to}\; C^{P\cap Q}\;\stackrel{\operatorname{Av}^{Q/(P\cap Q)}_*}{\to}\; C^Q$$ is an equivalence (cf. Proposition [Proposition 1](#Pp_4.5.3){reference-type="ref" reference="Pp_4.5.3"} and Theorem [Theorem 1](#Thm_B.1.2){reference-type="ref" reference="Thm_B.1.2"}). Here $\operatorname{Av}^{Q/(P\cap Q)}_*$ is the right adjoint to the corresponding oblivion functor. ### We prove a full Hecke property of $\operatorname{IC}^{\frac{\infty}{2}}_{P,0}$. Namely, our Proposition [Proposition 1](#Pp_2.5.18_upgrading_IC_semi-infinite_P){reference-type="ref" reference="Pp_2.5.18_upgrading_IC_semi-infinite_P"} asserts that $\operatorname{IC}^{\frac{\infty}{2}}_{P,0}$ naturally upgrades to an object of $$\operatorname{SI}_P\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})} {\operatorname{Rep}}(\check{M})$$ This is conceptually explained by the Drinfeld-Plücker formalism in its $\operatorname{\widetilde\operatorname{Bun}}_P$-version. ### We show that for $\lambda\in\Lambda^+_M$, $\operatorname{IC}^{\frac{\infty}{2}}_{P,\lambda}$ lies in the heart $\operatorname{SI}_P^{\heartsuit}$ of the t-structure on $\operatorname{SI}_P$. Write $\Lambda_{G,P}$ for the lattice of cocharacters of $M/[M,M]$. For $\theta\in\Lambda_{G,P}$ we consider the usual diagram of affine Grassmanians $$\operatorname{Gr}_M^{\theta}\stackrel{\mathfrak{t}^{\theta}_P}{\gets}\operatorname{Gr}_P^{\theta}\stackrel{v^{\theta}_P}{\to} \operatorname{Gr}_G$$ giving rise to the geometric restriction functor $$(\mathfrak{t}_P^{\theta})_!(v^{\theta}_P)^*: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$$ Recall that $\operatorname{SI}_P\subset Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ is a full subcategory naturally. In Proposition [Proposition 1](#Pp_2.5.15_*-restriction){reference-type="ref" reference="Pp_2.5.15_*-restriction"} for $\eta\in\Lambda^+_M$ we express $(\mathfrak{t}_P^{\theta})_!(v^{\theta}_P)^*\operatorname{IC}^{\frac{\infty}{2}}_{P, \eta}$ in terms of the Satake equivalence for $M$. This answer is to be compared with the main result of [@BFGM]. The advantage is that our isomorphism is canonical, while the related isomorphisms from [@BFGM] are not. The importance of this result comes from the relation with the $\operatorname{IC}$-sheaf of $\operatorname{\widetilde\operatorname{Bun}}_P$, which plays a crucial role in many aspects of the geometric Langlands program. In Proposition [Proposition 1](#Pp_action_of_Rep(checkM)_on_SI-IC_P){reference-type="ref" reference="Pp_action_of_Rep(checkM)_on_SI-IC_P"} we show that if $\eta\in\Lambda^+_M$ then $\operatorname{IC}^{\frac{\infty}{2}}_{P, \eta}$ is obtained from $\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}$ by applying a Hecke functor for $M$ corresponding to the irreducible $\check{M}$-module with highest weight $\eta$. ### Assume in addition that $[G,G]$ is simply-connected[^3]. Pick a smooth projective connected curve $X$ over $k$. Write $\operatorname{Bun}_P$ for the stack of $P$-torsors on $X$, $\operatorname{\widetilde\operatorname{Bun}}_P$ for its Drinfeld compactification. Pick a closed point $x\in G$. We introduce a version $_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P$ of $\operatorname{Bun}_P$ and a natural map $\tilde\pi: \operatorname{Gr}_G\to {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}$, cf. Sections [3.2.2](#Sect_2.3.3_loc_vs_glob){reference-type="ref" reference="Sect_2.3.3_loc_vs_glob"} - [3.2.7](#Sect_2.3.8_local_vs_global){reference-type="ref" reference="Sect_2.3.8_local_vs_global"}. It factors through a morphism $\tilde\pi^M: M({\mathcal O})\backslash \operatorname{Gr}_G\to {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}$, where $M({\mathcal O})\backslash \operatorname{Gr}_G$ denotes the stack quotient of $\operatorname{Gr}_G$. Write $\operatorname{IC}_{\widetilde{glob}}$ for the $\operatorname{IC}$-sheaf of $\operatorname{\widetilde\operatorname{Bun}}_P$. Let $j_{glob}: \operatorname{Bun}_P\stackrel{}{\hookrightarrow}\operatorname{\widetilde\operatorname{Bun}}_P$ be the natural open immersion. Our main results are Theorems [Theorem 1](#Thm_restriction_of_glob_first){reference-type="ref" reference="Thm_restriction_of_glob_first"} and [Theorem 1](#Th_restriction_of_glob_second){reference-type="ref" reference="Th_restriction_of_glob_second"}. They are analogs of the corresponding main results of [@Gai19SI] in the Borel case. Namely, we show that for $\eta\in\Lambda_M^+$, $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ is canonically isomorphic to $(\tilde\pi^M)^!\operatorname{IC}^{\eta}_{\widetilde{glob}}$ up to a cohomological shift, and $\boldsymbol{\vartriangle}^0$ is canonically isomorphic to $(\tilde\pi^M)^!(j_{glob})_!\operatorname{IC}_{\operatorname{Bun}_P}$ up to a cohomological shift. Here for $\eta\in\Lambda^+_M$ we define $\operatorname{IC}^{\eta}_{\widetilde{glob}}$ as the $\operatorname{IC}$-sheaf of the closed substack $$_{x, \ge -w_0^M(\eta)}\operatorname{\widetilde\operatorname{Bun}}_P\stackrel{}{\hookrightarrow} {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P},$$ cf. Sections [3.2.6](#Sect_2.3.7_local_vs_global){reference-type="ref" reference="Sect_2.3.7_local_vs_global"}, [3.2.10](#Sect_2.3.12_local_vs_global){reference-type="ref" reference="Sect_2.3.12_local_vs_global"}. Finally, we establish Theorem [Theorem 1](#Thm_4.1.10){reference-type="ref" reference="Thm_4.1.10"} saying that for $\lambda\in\Lambda_{M, ab}$, the standard object $\boldsymbol{\vartriangle}^{\lambda}\in Shv(\operatorname{Gr}_G)^H$ lies in the heart $Shv(\operatorname{Gr}_G)^{H, \heartsuit}$ of the semi-infinite t-structure. In the case $P=B$ this claim was derived in [@Gai19SI] from a global result going back to [@BG2]. We give a new purely local proof of this relating the question to the t-structure on the spectral side. ## Applications ### All the applications of the semi-infinite $\operatorname{IC}$-sheaves foreseen in [@Gai19SI] are valid also in the parabolic case. We underline that, as in the Borel case, the semi-infinite parabolic $\operatorname{IC}$-sheaf admits a factorizable version (not considered in this note), which is more fundamental than our $\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}$. A metaplectic analog of the semi-infinite sheaf $\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}$ is not difficult to define. It is supposed to play an important role in the metaplectic geometric Langlands program suggested in [@GL]. ### One of our main motivation to study $\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}$ was also its use for constructing local analogs of residues of the geometric Eisenstein series. This project is in preparation. ### We would like to propose, in addition, some relations we expect to representations of quantum groups, which we will return to elsewhere. As setup, consider the equivalence $$Shv(\operatorname{Gr}_G)^{H, ren} \simeq \operatorname{IndCoh}((\check{\mathfrak{u}}(\check{P}^-) \times_{\check{\mathfrak{g}}} 0)/\check{P}^-),$$ which is a combination of ([\[iso_Gurbir_Chen\]](#iso_Gurbir_Chen){reference-type="ref" reference="iso_Gurbir_Chen"}) and ([\[eq_ren_parahoric_versus_H\]](#eq_ren_parahoric_versus_H){reference-type="ref" reference="eq_ren_parahoric_versus_H"}). We expect this is $t$-exact, and hence restricts to an equivalence of abelian categories $$Shv(\operatorname{Gr}_G)^{H, ren, \heartsuit} \simeq {\operatorname{Rep}}(\check{P}^-)^{\heartsuit}.$$ For $P = G$ this is the geometric Satake, and for $P= B$ this follows from ([@Gai19SI], 1.5.7). ### To make contact with quantum groups, let us write $\mathring{I}$ for the prounipotent radical of the Iwahori subgroup, and correspondingly pass from the affine Grassmannian to the enhanced affine flag variety $$\widetilde{{\mathcal F}l}_G := G(F)/\mathring{I}.$$ The semi-infinite category of sheaves $Shv(\widetilde{{\mathcal F}l}_G)^H$ again admits a $t$-structure similar to what is constructed in the present work for the affine Grassmannian. ### To describe the corresponding category of quantum group representations, fix any sufficiently large even root of unity $q$. Associated to our parabolic $P$ is a mixed quantum group $$\mathfrak{U}_q(\mathfrak{g}, P),$$which has divided powers for the simple raising and lowering operators lying in $P$, and no divided powers for the remaining simple lowering operators. For $P = G$, this is the Lusztig form of the quantum group, and for $P = B$ this was considered by Gaitsgory in [@GaiKL]. It has a renormalized derived category of representations $$\operatorname{Rep}_q(\mathfrak{g}, P),$$obtained by ind-completing the pre-triangulated envelope of the parabolic Verma modules in its naive derived category, and we denote its principal block by $$\operatorname{Rep}_q(\mathfrak{g}, P)_\circ \hookrightarrow \operatorname{Rep}_q(\mathfrak{g}, P).$$ ### We conjecture a $t$-exact equivalence $$Shv(\widetilde{{\mathcal F}l}_G)^H \simeq \operatorname{Rep}_q(\mathfrak{g}, P)_\circ.$$ This should match the natural structures of highest weight categories on the hearts, and moreover intertwine the pullback$$Shv(\operatorname{Gr}_G)^{H, \heartsuit} \rightarrow Shv(\widetilde{{\mathcal F}l}_G)^{H, \heartsuit}$$with a suitably defined quantum Frobenius map $$\operatorname{Fr}: \operatorname{Rep}(\check{P}^-)^\heartsuit \rightarrow \operatorname{Rep}_q(\mathfrak{g}, P)_\circ^\heartsuit.$$ For $P=G$, this is a result of Arkhipov--Bezrukavnikov--Ginzburg [@ABG]. For all other cases, to our knowledge this may be new.[^4] ### In addition, to make contact with small quantum groups, as originally envisioned by Feigin-Frenkel and Lusztig [@FeiFr], [@LusICM], we may proceed as follows. Our $Shv(\operatorname{Gr}_G)^H$ is naturally a factorization category, and $\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}$ is upgrades to a factorization algebra in this factorization category. For the factorization modules over the semi-infinite $\operatorname{IC}$ sheaf, we conjecture a $t$-exact equivalence of $\operatorname{DG}$-categories $$\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}-mod^{{fact}}(Shv(\operatorname{Gr}_G)^{H, ren}) \simeq \operatorname{IndCoh}( (0 \underset{\check{\mathfrak{g}}}{\times} 0)/\check{M}),$$ which in particular induces an equivalence of abelian categories $$\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}-mod^{{fact}}(Shv(\operatorname{Gr}_G)^H)^{\heartsuit} \simeq \operatorname{Rep}(\check{M})^{\heartsuit};$$ for $P = B$ this is a forthcoming theorem of Campbell. ### To make contact with quantum groups, we again pass to the enhanced affine flag variety, and consider $$\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}-mod^{{fact}}(Shv(\widetilde{{\mathcal F}l}_G)^{H}).$$ Associated to the Levi factor $M$ of our parabolic $P$ is a version of the small quantum group, which we denote by $$\mathfrak{u}_q(\mathfrak{g}_1M),$$ which contains the small quantum group along with divided powers of the raising and lowering operators corresponding to simple roots of $M$. It has a renormalized derived category of representations $$\operatorname{Rep}_q(\mathfrak{g}_1M),$$ obtained by ind-completing the pre-triangulated envelope of the baby parabolic Verma modules within its naive derived category of representations, and we denote its principal block by $$\operatorname{Rep}_q(\mathfrak{g}_1M)_\circ \hookrightarrow \operatorname{Rep}_q(\mathfrak{g}_1M).$$ ### We conjecture a $t$-exact equivalence $$\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}-mod^{{fact}}(Shv(\widetilde{{\mathcal F}l}_G)^{H}) \simeq \operatorname{Rep}_q(\mathfrak{g}_1M)_\circ.$$ This should match the natural structures of highest weight categories on the hearts, and moreover intertwine the pullback $$\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}-mod^{{fact}}(Shv(\operatorname{Gr}_G)^H)^{\heartsuit} \rightarrow \operatorname{IC}^{\frac{\infty}{2}}_{P, 0}-mod^{{fact}}(Shv(\widetilde{{\mathcal F}l}_G)^H)^{\heartsuit}$$ with a suitably defined quantum Frobenius map $$\operatorname{Fr}: \operatorname{Rep}(\check{M})^\heartsuit \rightarrow \operatorname{Rep}_q(\mathfrak{g}_1M)_\circ.$$ In the case of the Borel, this is closely related to the proposal of Gaitsgory for localization of Kac-Moody modules at critical level introduced in [@Gai19SI], as well as the work of Arkhipov--Bezrukavnikov--Braverman--Gaitsgory--Mirković ([@ABBGM], Theorem 6.1.6). Similarly, when one passes to the enhanced affine flag variety, if we write ${\operatorname{Rep}}_q(\mathfrak{g}_1 M)_\circ$ for the regular block of modules for the $M$-graded small quantum group,[^5] we conjecture a $t$-exact equivalence of $\operatorname{DG}$-categories $$\operatorname{IC}^{\frac{\infty}{2}}_{P, 0}-mod^{{fact}}(Shv(\widetilde{{\mathcal F}l}_G)^H) \simeq \operatorname{Rep}_q( \mathfrak{g}_1 M)_\circ,$$ which again intertwines pullback from the Grassmannian and a quantum Frobenius. ## Conventions and notations ### {#Sect_1.4.1} Work over an algebraically closed field $k$. Write ${\operatorname{Sch}}^{aff}$ for the category of affine schemes, ${\operatorname{Sch}}_{ft}$ for the category of schemes of finite type (over $k$). Let $G$ be a connected reductive group over $k$, $T\subset B\subset G$ be a maximal torus and Borel subgroups, $B^-$ an opposite Borel subgroup with $B\cap B^-=T$. Write $U$ (resp., $U^-$) for the unipotent radical of $B$ (resp., $B^-$). Let $P\subset G$ be a standard parabolic, $P^-$ an opposite parabolic with common Levi subgroup $M=P\cap P^-$. Write $w_0$ for the longest element of the Weyl group $W$, and similarly for $w_0^M\in W_M$, where $W_M$ is the Weyl group of $(M, T)$. Write $U(P)$ (resp., $U(P^-)$) for the unipotent radical of $P$ (resp., of $P^-$). Set $B_M=B\cap M$, $B_M^-=B^-\cap M$. Let ${\mathcal I}$ be the set of vertices of the Dynkin diagram. For $i\in{\mathcal I}$ we write $\alpha_i$ (resp., $\check{\alpha}_i$) for the corresponding simple coroot (resp., simple root). Let ${\mathcal I}_M\subset {\mathcal I}$ correspond to the Dynkin diagram of $M$. Write $\check{P}, \check{B}, \check{T}$, $U(\check{P}), U(\check{P}^-)$ for the corresponding dual objects. Write $\Lambda$ (resp., $\check{\Lambda}$) for the coweights (resp., weights) lattice of $T$, $\Lambda^+$ for the dominant coweights. Write $\Lambda^+_M$ for the dominant coweights of $B_M$. Let $\Lambda^{pos}\subset\Lambda$ be the ${\mathbb Z}_+$-span of positive coroots. Let $\Lambda_M^{pos}\subset\Lambda$ be the ${\mathbb Z}_+$-span of $\alpha_i$, $i\in {\mathcal I}_M$. ### {#section-17} Our conventions about higher categories and sheaf theories are those of [@AGKRRV]. In particular, $\operatorname{Spc}$ denotes the $\infty$-category of spaces, $1-\operatorname{Cat}$ is the $\infty$-category of $(\infty, 1)$-categories ([@G], ch. I.1, 1.1.1). We fix an algebraically closed field $e$ of characteristic zero, the field of coefficients of our sheaf theory. Then $\operatorname{Vect}$ is the $\operatorname{DG}$-category of complexes of $e$-vector spaces defined in ([@G], ch. I.1, 10.1). The categories $\operatorname{DGCat}_{cont}, \operatorname{DGCat}^{non-cocmpl}$ are defined in ([@G], ch. I.1, 10.3). We work in the constructible context. # Analog of the Drinfeld-Plücker formalism ## Case of $\operatorname{\widetilde\operatorname{Bun}}_P$ ### {#section-18} For $\lambda\in\Lambda^+$ write $V^{\lambda}$ for the irreducible $\check{G}$-module with highest weight $\lambda$. We pick vectors $v^{\lambda}\in V^{\lambda}, (v^{\lambda})^*\in (V^{\lambda})^*$ as in ([@Gai19SI], 2.1.2). Namely, $v^{\lambda}$ is a highest weight vector of $V^{\lambda}$. Then $(v^{\lambda})^*$ is characterised by the properties that $\langle v^{\lambda}, (v^{\lambda})^*\rangle=1$, and $(v^{\lambda})^*$ vanished on the weight spaces $V^{\lambda}(\mu)$ for $\mu\ne\lambda$. For $\nu\in\Lambda^+_M$ let $U^{\nu}$ be the irreducible $\check{M}$-module with highest weight $\nu$. If moreover $\nu\in\Lambda^+$ then we simply assume $$U^{\nu}=(V^{\nu})^{U(\check{P})},$$ so we have the highest weight vector $v^{\nu}\in U^{\nu}$. We assume this choice of a highest weight vector $v^{\nu}\in U^{\nu}$ is extended for the whole of $\Lambda^+_M$. It is known that for any finite-dimensional $\check{G}$-module $V$ the natural map $V^{U(\check{P})}\to V\to V_{U(\check{P}^-)}$ is an isomorphism. So, for $\nu\in\Lambda^+$ we get canonically $$((V^{\nu})^*)^{U(\check{P}^-)}\,{\widetilde\to}\, ((V^{\nu})_{U(\check{P}^-)})^*\,{\widetilde\to}\, (U^{\nu})^*$$ This is an irreducible $\check{M}$-module with highest weight $-w_0^M(\nu)$, and $$(v^{\nu})^*\in ((V^{\nu})^*)^{U(\check{P}^-)}\,{\widetilde\to}\,(U^{\nu})^*$$ So, $(U^{\nu})^*$ is equipped with the highest weight vector $(v^{\nu})^*$ with respect to the Borel $B_M^-$. Now define the $B_M^-$-highest weights vectors $(v^{\nu})^*\in (U^{\nu})^*$ for all $\nu\in\Lambda^+_M$ as for $\check{G}$. Namely, they satisfy $\langle v^{\nu},(v^{\nu})^*\rangle=1$, and $(v^{\nu})^*: U^{\nu}\to e^{\nu}$ vanish on all the weight spaces $U^{\nu}(\mu)$ for $\mu\ne\nu$. ### {#section-19} For $\lambda_i\in\Lambda^+$ we denote by $u^{\lambda_1,\lambda_2}: V^{\lambda_1}\otimes V^{\lambda_2}\to V^{\lambda_1+\lambda_2}$ and $v^{\lambda_1,\lambda_2}: V^{\lambda_1+\lambda_2}\to V^{\lambda_1}\otimes V^{\lambda_2}$ the maps fixed in ([@Gai19SI], Section 2.1.4) as well as their duals. For $\lambda_1,\lambda_2\in\Lambda^+_M$ we also denote by $\bar v^{\lambda_1,\lambda_2}: U^{\lambda_1+\lambda_2}\to U^{\lambda_1}\otimes U^{\lambda_2}$ and $\bar u^{\lambda_1,\lambda_2}: U^{\lambda_1}\otimes U^{\lambda_2}\to U^{\lambda_1+\lambda_2}$ the maps defined similarly for $\check{M}$ as well as their duals. For $\lambda_i\in\Lambda^+$ the diagram commutes $$\label{diag_for_Sect_1.0.3} \begin{array}{ccc} (U^{\lambda_1})^*\otimes (U^{\lambda_2})^* & \stackrel{}{\hookrightarrow} & (V^{\lambda_1})^*\otimes (V^{\lambda_2})^*\\ \downarrow\lefteqn{\scriptstyle \bar v^{\lambda_1,\lambda_2}} && \downarrow\lefteqn{\scriptstyle v^{\lambda_1,\lambda_2}}\\ (U^{\lambda_1+\lambda_2})^* & \stackrel{}{\hookrightarrow} & (V^{\lambda_1+\lambda_2})^* \end{array}$$ ### {#section-20} The above gives $$\label{decomp_O(G/U)} {\mathcal O}(\check{G}/U(\check{P}^-))\,{\widetilde\to}\,\mathop{\oplus}\limits_{\lambda\in\Lambda^+} V^{\lambda}\otimes (U^{\lambda})^*\in {\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})$$ For each finite-dimensional representation $V$ of $\check{G}$ we have the matrix coefficient map $V^*\otimes V^{U(\check{P}^-)}\to {\mathcal O}(\check{G}/U(\check{P}^-))$, $u\otimes v\mapsto (g\mapsto \langle u, gv\rangle)$. ### {#section-21} Consider the diagram $$\label{diag_one} \begin{array}{ccc} B(\check{P}^-) & \stackrel{\eta}{\gets} & B(\check{M})\\ \downarrow\lefteqn{\scriptstyle q} & \swarrow\lefteqn{\scriptstyle q_M}\\ B(\check{G}\times\check{M}), \end{array}$$ where the maps come from the diagram $\check{M}\to \check{P}^-\to \check{G}\times\check{M}$, the second map being the diagonal morphism. After the base change $\operatorname{Spec}k\to B(\check{G}\times\check{M})$ the map $\eta$ becomes $\bar\eta: \check{G}\to \check{G}/U(\check{P}^-)$. We get an adjoint pair $$\label{adjoint_pair_eta} \eta^*: \operatorname{QCoh}(B(\check{P}^-))\leftrightarrows \operatorname{QCoh}(B(\check{M})): \eta_*$$ in ${\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})-mod$ by ([@G], ch. I.3, 3.2.4). We have $q_*{\mathcal O}, (q_M)_*{\mathcal O}\in Alg({\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}))$. By ([@G], ch. I.3, 3.3.3) one has $$\operatorname{QCoh}(B(\check{P}^-))\,{\widetilde\to}\, q_*{\mathcal O}-mod({\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}))$$ and $$\operatorname{QCoh}(B(\check{M}))\,{\widetilde\to}\, (q_M)_*{\mathcal O}-mod({\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}))$$ Here $q_*{\mathcal O}\,{\widetilde\to}\, {\mathcal O}(\check{G}/U(\check{P}^-))$ and $(q_M)_*{\mathcal O}\,{\widetilde\to}\, {\mathcal O}(\check{G})$. It is crucial here that $\check{G}/U(\check{P}^-)$ is quasi-affine by ([@BG], 1.1.2). ### {#Sect_2.1.5_now} Let $C\in {\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})-mod(\operatorname{DGCat}_{cont})$. Similarly to [@Gai19SI], set $\operatorname{Hecke}_{\check{G}, \check{M}}(C)=C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})} {\operatorname{Rep}}(\check{M})$. The diagram ([\[diag_one\]](#diag_one){reference-type="ref" reference="diag_one"}) yields an adjoint pair $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})} {\operatorname{Rep}}(\check{P}^-)\leftrightarrows \operatorname{Hecke}_{\check{G}, \check{M}}(C)$$ in $\operatorname{DGCat}_{cont}$. We want to describe the left adjoint $$\label{left_adj_DR-PL_for_P} C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})} {\operatorname{Rep}}(\check{P}^-)\to\operatorname{Hecke}_{\check{G}, \check{M}}(C)$$ in this adjoint pair. First we want to describe the composition $$\label{left_adj_DR-PL_for_P_comoposed} C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})} {\operatorname{Rep}}(\check{P}^-)\to\operatorname{Hecke}_{\check{G}, \check{M}}(C)\stackrel{\operatorname{oblv}}{\to} C$$ Applying ([@G], ch. I.1, 8.5.7), one gets $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})} {\operatorname{Rep}}(\check{P}^-)\,{\widetilde\to}\, {\mathcal O}(\check{G}/U(\check{P}^-))-mod(C)$$ and $\operatorname{Hecke}_{\check{G}, \check{M}}(C)\,{\widetilde\to}\, {\mathcal O}(\check{G})-mod(C)$. Now ([\[left_adj_DR-PL_for_P\]](#left_adj_DR-PL_for_P){reference-type="ref" reference="left_adj_DR-PL_for_P"}) is the functor $${\mathcal O}(\check{G}/U(\check{P}^-))-mod(C)\to {\mathcal O}(\check{G})-mod(C)$$ sending $c$ to ${\mathcal O}(\check{G})\otimes_{{\mathcal O}(\check{G}/U(\check{P}^-))} c$ in the sense of ([@HA], 4.4.2.12). The functor $\operatorname{oblv}: {\mathcal O}(\check{G})-mod(C)\to C$ forgets the ${\mathcal O}(\check{G})$-module structure. ### {#section-22} The category ${\mathcal O}(\check{G}/U(\check{P}^-))-mod(C)$ admits a description in Plücker style as follows. We will write the action of ${\operatorname{Rep}}(\check{G})$ on $c\in C$ on the right, and that of ${\operatorname{Rep}}(\check{M})$ on the left. An object $c\in {\mathcal O}(\check{G}/U(\check{P}^-))-mod(C)$ can be seen as $c\in C$ with the following data. For each $V\in {\operatorname{Rep}}(\check{G})^{\heartsuit}$ finite-dimensional we should be given a map $\kappa_V: V^{U(\check{P}^-)}\ast c\to c\ast V$. For a morphism $V_1\to V_2$ of finite-dimensional $\check{G}$-modules, we are given a commutativity datum for the diagram $$\begin{array}{ccc} V_1^{U(\check{P}^-)}\ast c &\stackrel{\kappa_{V_1}}{\to} & c\ast V_1\\ \downarrow && \downarrow\\ V_2^{U(\check{P}^-)}\ast c& \stackrel{\kappa_{V_2}}{\to}& c\ast V_2 \end{array}$$ Besides, we are given a commutativity datum for the diagram $$\begin{array}{ccc} (V_1^{U(\check{P}^-)}\otimes V_2^{U(\check{P}^-)})\ast c & \stackrel{\kappa_{V_2}}{\to} & V_1^{U(\check{P}^-)}\ast c\ast V_2 \\ \downarrow && \downarrow\lefteqn{\scriptstyle \kappa_{V_1}}\\ (V_1\otimes V_2)^{U(\check{P}^-)}\ast c & \stackrel{\kappa_{V_1\otimes V_2}}{\to} & c\ast (V_1\otimes V_2) \end{array}$$ Besides, for $V$ trivial we are given an identification $\kappa_V\,{\widetilde\to}\,\operatorname{id}$, plus coherent system of compatibilities. **Remark 1**. *For example, if $C$ is equipped with a t-structure, the actions of ${\operatorname{Rep}}(\check{G})^{\heartsuit}$ and ${\operatorname{Rep}}(\check{M})^{\heartsuit}$ on $C$ are t-exact and $c\in C^{\heartsuit}$ then in the above description of a ${\mathcal O}(\check{G}/U(\check{P}^-))$-module structure on $c$ the higher compatibilities are automatic.* ### {#section-23} Let $c\in {\mathcal O}(\check{G}/U(\check{P}^-))-mod(C)$. Then for $\lambda\in\Lambda^+$ we get the action map $\kappa^{\lambda}: (U^{\lambda})^*\ast c\to c\ast (V^{\lambda})^*$. For $\lambda_i\in\Lambda^+$ the above using ([\[diag_for_Sect_1.0.3\]](#diag_for_Sect_1.0.3){reference-type="ref" reference="diag_for_Sect_1.0.3"}) yields the commutativity datum for the diagram $$\label{diag_for_Sect_1.0.8_U(checkP^-)} \begin{array}{ccc} (U^{\lambda_1})^*\otimes (U^{\lambda_2})^*\ast c & \stackrel{\kappa^{\lambda_2}}{\to} (U^{\lambda_1})^*\ast c\ast (V^{\lambda_2})^* \stackrel{\kappa^{\lambda_1}}{\to} & c\ast (V^{\lambda_1})^*\otimes (V^{\lambda_2})^*\\ \downarrow\lefteqn{\scriptstyle \bar v^{\lambda_1,\lambda_2}} && \downarrow\lefteqn{\scriptstyle v^{\lambda_1,\lambda_2}}\\ (U^{\lambda_1+\lambda_2})^*\ast c & \stackrel{\kappa^{\lambda_1+\lambda_2}}{\to} & c\ast (V^{\lambda_1+\lambda_2})^* \end{array}$$ For the convenience of the reader recall the following. If ${\mathcal A}$ is a monoidal $\infty$-category, $D\in 1-\operatorname{Cat}$ then a lax action of ${\mathcal A}$ on the left (resp., on the right) on $D$ is a right lax monoidal functor ${\mathcal A}\to{\operatorname{Fun}}(D,D)$ (resp., ${\mathcal A}^{rm}\to {\operatorname{Fun}}(D,D)$). Here $rm$ stands for the reversed multiplication. Consider the following two lax actions of $\Lambda^+$ on $C$. For the first one $\lambda\in\Lambda^+$ sends $x$ to $(U^{\lambda})^*\ast x$, where the lax structure is given by the morphisms $$(U^{\lambda_1})^*\ast ((U^{\lambda_2})^*\ast x)\,{\widetilde\to}\, ((U^{\lambda_1})^*\otimes (U^{\lambda_2})^*) \ast x\stackrel{\bar v^{\lambda_1,\lambda_2}}{\to} (U^{\lambda_1+\lambda_2})^*\ast x$$ For the second one, $\lambda$ sends $x$ to $x\ast (V^{\lambda})^*$, and the lax structure is given by the morphisms $$(x\ast (V^{\lambda_1})^*)\ast (V^{\lambda_2})^*\,{\widetilde\to}\, x\ast ((V^{\lambda_1})^*\otimes (V^{\lambda_2})^*)\stackrel{v^{\lambda_1,\lambda_2}}{\to} (V^{\lambda_1+\lambda_2})^*$$ Then $c$ inherits a *lax central object* structure in the sense of ([@Gai19SI], 2.7) for these actions. That is, the commutativity datum for ([\[diag_for_Sect_1.0.8_U(checkP\^-)\]](#diag_for_Sect_1.0.8_U(checkP^-)){reference-type="ref" reference="diag_for_Sect_1.0.8_U(checkP^-)"}) is equipped with coherent system of higher compatibilities. This implies that one has a well-defined functor $\Lambda^+\to C$ $$\label{functor_from_Lambda^+} f: \Lambda^+\to C, \; \lambda\mapsto U^{\lambda}\ast c\ast (V^{\lambda})^*$$ Here we consider $\Lambda^+$ with the relation $\lambda_1\le\lambda_2$ iff $\lambda_2-\lambda_1\in\Lambda^+$. This is not a partial order in general, but makes $\Lambda^+$ a filtered category. For $\lambda_i\in\Lambda^+$ with $\lambda_2-\lambda_1=\lambda\in\Lambda^+$ the transition map from $f(\lambda_1)$ to $f(\lambda_2)$ in this diagram is the composition $$\begin{gathered} U^{\lambda_1}\ast c\ast (V^{\lambda_1})^*\stackrel{\bar v^{\lambda, \lambda_1}}{\to} (U^{\lambda_2}\otimes (U^{\lambda})^*)\ast c\ast (V^{\lambda_1})^*\stackrel{\kappa^{\lambda}}{\to} \\ U^{\lambda_2}\ast (c\ast (V^{\lambda})^*)\ast (V^{\lambda_1})^*\stackrel{v^{\lambda, \lambda_1}}{\to} U^{\lambda_2}\ast c\ast (V^{\lambda_2})^*\end{gathered}$$ The higher compatibilities for the above morphisms come automatically from the ${\mathcal O}(\check{G}/U(\check{P}^-))$-module structure on $c$. **Question 1**. *Understand the functor $${\mathcal O}(\check{G}/U(\check{P}^-))-mod(C)\to C,\; c\mapsto \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+}\, U^{\lambda}\ast c\ast (V^{\lambda})^*$$* ### {#section-24} Consider now the restriction of ([\[functor_from_Lambda\^+\]](#functor_from_Lambda^+){reference-type="ref" reference="functor_from_Lambda^+"}) to a full subcategory $$\Lambda^+_{M, ab}\to C, \; \lambda\mapsto e^{\lambda}\ast c\ast (V^{\lambda})^*$$ We give two proofs of the following. **Proposition 1**. *The functor ([\[left_adj_DR-PL_for_P\_comoposed\]](#left_adj_DR-PL_for_P_comoposed){reference-type="ref" reference="left_adj_DR-PL_for_P_comoposed"}) identifies with $$c\mapsto \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} e^{\lambda}\ast c\ast (V^{\lambda})^*$$* *First proof.* By Section [2.1.5](#Sect_2.1.5_now){reference-type="ref" reference="Sect_2.1.5_now"}, it suffices to establish the universal case when $C={\operatorname{Rep}}(\check{G}\times\check{M})$, and $c={\mathcal O}(\check{G}/U(\check{P}^-))\in C$. . First, define a compatible system of morphisms in ${\operatorname{Rep}}(\check{G}\times\check{M})$ $$\label{morphism_for_Pp_2.1.10_colimit_for_Bunt_P} e^{\lambda}\ast {\mathcal O}(\check{G}/U(\check{P}^-))\ast (V^{\lambda})^*\to {\mathcal O}(\check{G})$$ for $\lambda\in\Lambda^+_{M, ab}$ as follows. Let $\check{M}$ act on $\check{G}\times \check{M}$ by right translations via the diagonal homomorphism $\check{M}\to \check{G}\times \check{M}$, form the quotient $(\check{G}\times\check{M})/\check{M}$. Let $\check{G}\times\check{M}$ act by left translations on the latter quotient. We get a $\check{G}\times \check{M}$-equivariant isomorphism $(\check{G}\times\check{M})/\check{M}\,{\widetilde\to}\,\check{G}$, where on the RHS the group $\check{G}$ (resp., $\check{M}$) acts by left (resp., right) translations. By Frobenius reciprocity, a datum of ([\[morphism_for_Pp_2.1.10_colimit_for_Bunt_P\]](#morphism_for_Pp_2.1.10_colimit_for_Bunt_P){reference-type="ref" reference="morphism_for_Pp_2.1.10_colimit_for_Bunt_P"}) is the same as a $\check{M}$-equivariant morphism $$\label{morphism_for_Pp_2.1.10_second} e^{\lambda}\ast {\mathcal O}(\check{G}/U(\check{P}^-))\ast (V^{\lambda})^*\to e,$$ where on the LHS the action is obtained from the $\check{G}\times \check{M}$-action by restricting under the diagonal map $\check{M}\to \check{G}\times \check{M}$. Let $\mathit{ev}: {\mathcal O}(\check{G}/U(\check{P}^-))\to e$ be the evaluation at $U(\check{P}^-)\in \check{G}/U(\check{P}^-)$, it is invariant under the adjoint $\check{M}$-action. Note that $v^{\lambda}: e^{\lambda}\otimes (V^{\lambda})^*\to e$ is $\check{M}$-equivariant, define ([\[morphism_for_Pp_2.1.10_second\]](#morphism_for_Pp_2.1.10_second){reference-type="ref" reference="morphism_for_Pp_2.1.10_second"}) as $\mathit{ev}\otimes v^{\lambda}$. Let $\lambda,\lambda_1\in\Lambda^+_{M, ab}$ and $\lambda_2=\lambda_1+\lambda$. Let us show that the diagram commutes $$\begin{array}{ccc} e^{\lambda_1}\otimes {\mathcal O}(\check{G}/U(\check{P}^-))\otimes (V^{\lambda_1})^*& {\widetilde\to}& e^{\lambda_2}\otimes e^{-\lambda}\otimes {\mathcal O}(\check{G}/U(\check{P}^-))\otimes (V^{\lambda_1})^*\\ \downarrow\lefteqn{\scriptstyle v^{\lambda_1}\otimes \mathit{ev}} && \downarrow\lefteqn{\scriptstyle{\kappa^{\lambda}}}\\ e && e^{\lambda_2}\otimes {\mathcal O}(\check{G}/U(\check{P}^-))\otimes (V^{\lambda})^*\otimes (V^{\lambda_1})^*\\ & \hspace{-1em}\nwarrow\lefteqn{\scriptstyle v^{\lambda_2}\otimes\mathit{ev}}\;\;\;\;\;\;\; & \downarrow\lefteqn{\scriptstyle{v^{\lambda, \lambda_1}}}\\ && e^{\lambda_2}\otimes {\mathcal O}(\check{G}/U(\check{P}^-))\otimes (V^{\lambda_2})^* \end{array}$$ This follows easily from the commutativity of the diagrams $$\begin{array}{ccc} e^{-\lambda} & \stackrel{\mathit{ev}}{\gets} & e^{-\lambda}\otimes{\mathcal O}(\check{G}/U(\check{P}^-))\\ \downarrow\lefteqn{\scriptstyle (v^{\lambda})^*} && \downarrow\lefteqn{\scriptstyle \kappa^{\lambda}}\\ (V^{\lambda})^* & \stackrel{\mathit{ev}}{\gets} &{\mathcal O}(\check{G}/U(\check{P}^-))\otimes (V^{\lambda})^* \end{array}$$ and $$\begin{array}{ccc} e^{\lambda_1}\otimes (V^{\lambda_1})^* & \stackrel{(v^{\lambda})^*}{\to} & e^{\lambda_2}\otimes (V^{\lambda})^*\otimes (V^{\lambda_1})^*\\ \downarrow\lefteqn{\scriptstyle v^{\lambda_1}} && \downarrow\lefteqn{\scriptstyle v^{\lambda, \lambda_1}}\\ e & \stackrel{v^{\lambda_2}}{\gets} & e^{\lambda_2}\otimes (V^{\lambda_2})^* \end{array}$$ Thus, we get a morphism in ${\operatorname{Rep}}(\check{G}\times\check{M})$ $$\label{map_iso_would_be_for_Pp_2.1.10} \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} e^{\lambda}\otimes {\mathcal O}(\check{G}/U(\check{P}^-))\otimes (V^{\lambda})^*\to {\mathcal O}(\check{G})$$ It remains to show it is an isomorphism. . Pick $\nu\in\Lambda^+$. Assuming $\lambda\in\Lambda^+_{M, ab}$ deep enough on the wall of the corresponding Weyl chamber, we get $$\label{exp_for_Step2_Pp2.1.10} \operatorname{Hom}_{\check{G}}(V^{\nu}, e^{\lambda}\otimes {\mathcal O}(\check{G}/U(\check{P}^-))\otimes (V^{\lambda})^*)\,{\widetilde\to}\, \operatorname{Hom}_{\check{G}}(V^{\nu}\otimes V^{\lambda}, e^{\lambda}\otimes {\mathcal O}(\check{G}/U(\check{P}^-)))$$ By Lemma [Lemma 1](#Lm_2.0.15){reference-type="ref" reference="Lm_2.0.15"}, $V^{\nu}\otimes V^{\lambda}\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} V^{\lambda+\mu}\otimes\operatorname{Hom}_{\check{M}}(U^{\mu}, V^{\nu})$. Using ([\[decomp_O(G/U)\]](#decomp_O(G/U)){reference-type="ref" reference="decomp_O(G/U)"}) now ([\[exp_for_Step2_Pp2.1.10\]](#exp_for_Step2_Pp2.1.10){reference-type="ref" reference="exp_for_Step2_Pp2.1.10"}) identifies with $$\begin{gathered} \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} \operatorname{Hom}_{\check{G}}(V^{\lambda+\mu}, e^{\lambda}\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, V^{\nu})^*\otimes V^{\lambda+\mu}\otimes (U^{\lambda+\mu})^*)\,{\widetilde\to}\\ \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} (U^{\mu})^*\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, V^{\nu})^*\,{\widetilde\to}\, (V^{\nu})^*\end{gathered}$$ in ${\operatorname{Rep}}(\check{M})$. We have also in ${\operatorname{Rep}}(\check{M})$ $$\operatorname{Hom}_{\check{G}}(V^{\nu}, {\mathcal O}(\check{G})\,{\widetilde\to}\, (V^{\nu})^*$$ and the map $(V^{\nu})^*\to (V^{\nu})^*$ induced by ([\[map_iso_would_be_for_Pp_2.1.10\]](#map_iso_would_be_for_Pp_2.1.10){reference-type="ref" reference="map_iso_would_be_for_Pp_2.1.10"}) is the identity. ◻ ### Second proof of Proposition [Proposition 1](#Pp_2.1.10_colimit_for_Bunt_P){reference-type="ref" reference="Pp_2.1.10_colimit_for_Bunt_P"} {#second-proof-of-proposition-pp_2.1.10_colimit_for_bunt_p} Let $\check{G}\times\check{M}$ act on $\check{G}/[\check{P}^-, \check{P}^-]$ and on $\check{G}/[\check{M}, \check{M}]$ naturally via its quotient $\check{G}\times\check{M}\to \check{G}\times\check{M}_{ab}$. Then one has a cartesian square in the category of schemes with a $\check{G}\times\check{M}$-action $$\label{diag_main_cartesian_for_2nd_proof} \begin{array}{ccc} \check{G} & \stackrel{\bar\eta}{\to} & \check{G}/U(\check{P}^-)\\ \downarrow &&\downarrow\\ \check{G}/[\check{M}, \check{M}] & \stackrel{\bar\eta_{ab}}{\to} & \check{G}/[\check{P}^-, \check{P}^-]. \end{array}$$ It is obtained as follows. First, consider the diagonal map $\check{P}^-\to \check{G}\times \check{M}_{ab}$ yielding the morphism $B(\check{P}^-)\to B(\check{G}\times \check{M}_{ab})$, which gives in turn $$\label{map_second_for_the_secon_proof_Pp_2.1.10_colimit_for_Bunt_P} \eta\times\operatorname{id}: B(\check{M})\times_{B(\check{G}\times \check{M}_{ab})} \operatorname{Spec}k\to B(\check{P}^-)\times_{B(\check{G}\times \check{M}_{ab})} \operatorname{Spec}k.$$ View ([\[map_second_for_the_secon_proof_Pp_2.1.10_colimit_for_Bunt_P\]](#map_second_for_the_secon_proof_Pp_2.1.10_colimit_for_Bunt_P){reference-type="ref" reference="map_second_for_the_secon_proof_Pp_2.1.10_colimit_for_Bunt_P"}) as a morphism in the category of stacks over $B(\check{G}\times \check{M})\times_{B(\check{G}\times \check{M}_{ab})} \operatorname{Spec}k$, here we use the map $B(\check{P}^-)\to B(\check{G}\times \check{M})$ coming from the diagonal morphism $\check{P}^-\to \check{G}\times\check{M}$. Then ([\[diag_main_cartesian_for_2nd_proof\]](#diag_main_cartesian_for_2nd_proof){reference-type="ref" reference="diag_main_cartesian_for_2nd_proof"}) is obtained from ([\[map_second_for_the_secon_proof_Pp_2.1.10_colimit_for_Bunt_P\]](#map_second_for_the_secon_proof_Pp_2.1.10_colimit_for_Bunt_P){reference-type="ref" reference="map_second_for_the_secon_proof_Pp_2.1.10_colimit_for_Bunt_P"}) by making the base change by $$\operatorname{Spec}k\to B(\check{G}\times \check{M})\times_{B(\check{G}\times \check{M}_{ab})} \operatorname{Spec}k.$$ The diagram ([\[diag_main_cartesian_for_2nd_proof\]](#diag_main_cartesian_for_2nd_proof){reference-type="ref" reference="diag_main_cartesian_for_2nd_proof"}) yields a diagram of affine closures $$\begin{array}{ccc} \check{G} & \stackrel{\bar\eta}{\to} & \overline{\check{G}/U(\check{P}^-)}\\ \downarrow &&\downarrow\\ \check{G}/[\check{M}, \check{M}] & \stackrel{\bar\eta_{ab}}{\to} & \overline{\check{G}/[\check{P}^-, \check{P}^-]}, \end{array}$$ which is also cartesian. This is seen using the Plücker description of points of $\overline{\check{G}/U(\check{P}^-)}$, $\overline{\check{G}/[\check{P}^-, \check{P}^-]}$ given in ([@BG], 1.1.2). So, we have an isomorphism of algebras in ${\operatorname{Rep}}(\check{G}\times\check{M})$ $${\mathcal O}(\check{G}/U(\check{P}^-))\otimes_{{\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-]}{\mathcal O}(\check{G}/[\check{M}, \check{M}])\,{\widetilde\to}\, {\mathcal O}(\check{G}).$$ Now for $c\in {\mathcal O}(\check{G}/U(\check{P}^-))-mod(C)$ one gets $$c\otimes_{{\mathcal O}(\check{G}/U(\check{P}^-))} {\mathcal O}(\check{G})\,{\widetilde\to}\, c\otimes_{{\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-]}{\mathcal O}(\check{G}/[\check{M}, \check{M}])$$ Our claim follows now from Proposition [Proposition 1](#Pp_2.0.11){reference-type="ref" reference="Pp_2.0.11"}. $\square$ ## Case of $\operatorname{\overline{Bun}} _P$ ### {#section-25} Let $\Lambda_{M, ab}=\{\lambda\in \Lambda\mid \langle\lambda, \check{\alpha}_i\rangle=0\;\mbox{for}\; i\in {\mathcal I}_M\}$. This is the lattice of characters of $\check{M}_{ab}:=\check{M}/[\check{M},\check{M}]$. Set for brevity $\Lambda_{M, ab}^+=\Lambda_{M, ab}\cap \Lambda^+$. Given $\lambda\in\Lambda^+$, $(V^{\lambda})^{[\check{P},\check{P}]}$ vanishes unless $\lambda\in\Lambda_{M, ab}^+$, and in the latter case it identifies with the highest weight line $e^{\lambda}\subset V^{\lambda}$ generated by $v^{\lambda}$. So, $${\mathcal O}(\check{G}/[\check{P},\check{P}])\,{\widetilde\to}\, \mathop{\oplus}\limits_{\lambda\in \Lambda_{M, ab}^+} (V^{\lambda})^*\otimes e^{\lambda}\in {\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})$$ The product in this algebra is given for $\lambda, \mu\in\Lambda^+_{M, ab}$ by the maps $$(V^{\lambda})^*\otimes e^{\lambda}\otimes (V^{\mu})^*\otimes e^{\mu}\stackrel{v^{\lambda,\mu}}{\to} (V^{\lambda+\mu})^*\otimes e^{\lambda+\mu}$$ We could replace in the above $e^{\lambda}$ by $U^{\lambda}$, as the corresponding $\check{M}$-module for $\lambda\in \Lambda_{M, ab}^+$ is 1-dimensional. ### {#section-26} One similarly gets $$\label{functions_for_G/P^-} {\mathcal O}(\check{G}/[\check{P}^-,\check{P}^-])\,{\widetilde\to}\, \mathop{\oplus}\limits_{\lambda\in \Lambda_{M, ab}^+} V^{\lambda}\otimes e^{-\lambda}\in {\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})$$ Here $e^{-\lambda}$ coincides with $(U^{\lambda})^*$. The product in this algebra is given for $\lambda,\mu\in \Lambda_{M, ab}^+$ by the maps $$V^{\lambda}\otimes e^{-\lambda}\otimes V^{\mu}\otimes e^{-\mu}\stackrel{u^{\lambda,\mu}}{\to} V^{\lambda+\mu}\otimes e^{-\lambda-\mu}$$ ### {#section-27} Consider the diagram $$\label{diag_two} \begin{array}{ccc} B(\check{P}^-) & \stackrel{\eta}{\gets} & B(\check{M})\\ \downarrow\lefteqn{\scriptstyle q_{ab}} & \swarrow\lefteqn{\scriptstyle q_{M, ab}}\\ B(\check{G}\times \check{M}_{ab}) \end{array}$$ obtained for the diagonal map $\check{P}^-\to \check{G}\times \check{M}_{ab}$. After the base change $\operatorname{Spec}k\to B(\check{G}\times \check{M}_{ab})$ the map $\eta$ becomes $$\bar\eta_{ab}: \check{G}/[\check{M}, \check{M}]\to \check{G}/[\check{P}^-, \check{P}^-]$$ ### {#section-28} One has $${\mathcal O}(\check{G}/[\check{M},\check{M}])\,{\widetilde\to}\, \mathop{\oplus}\limits_{\nu\in\Lambda^+, \; \mu\in\Lambda_{M, ab}} V^{\nu}\otimes e^{-\mu}\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, V^{\nu})^*$$ Indeed, for $\nu\in\Lambda^+$, $$(V^{\nu})_{[\check{M},\check{M}]}\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} (U^{\mu})_{[\check{M},\check{M}]} \otimes \operatorname{Hom}(U^{\mu}, V^{\nu}))$$ Now $(U^{\mu})_{[\check{M},\check{M}]}$ vanishes unless $\mu\in \Lambda_{M, ab}$, in which case it identifies with $e^{\mu}$ as a $\check{M}_{ab}$-module. Finally, $$((V^{\nu})^*)^{[\check{M},\check{M}]}\,{\widetilde\to}\, ((V^{\nu})_{[\check{M},\check{M}]})^*$$ ### {#section-29} View now ([\[adjoint_pair_eta\]](#adjoint_pair_eta){reference-type="ref" reference="adjoint_pair_eta"}) as an adjoint pair in ${\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})$. We get $$(q_{ab})_*{\mathcal O}, (q_{M, ab})_*{\mathcal O}\in {\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})$$ Since $\check{G}/[\check{P}^-, \check{P}^-]$ is quasi-affine by ([@BG], 1.1.2) and $\check{G}/[\check{M}, \check{M}]$ is affine, we similarly get $$\operatorname{QCoh}(B(\check{P}^-))\,{\widetilde\to}\, (q_{ab})_*{\mathcal O}-mod({\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))$$ and $$\operatorname{QCoh}(B(\check{M}))\,{\widetilde\to}\, (q_{M, ab})_*{\mathcal O}-mod({\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))$$ Here $(q_{M, ab})_*{\mathcal O}\,{\widetilde\to}\, {\mathcal O}(\check{G}/[\check{M}, \check{M}])$ and $(q_{ab})_*{\mathcal O}\,{\widetilde\to}\, {\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])$. ### {#section-30} Given $C\in {\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))-mod(\operatorname{DGCat}_{cont})$, set $\operatorname{Hecke}_{\check{G},\check{M}, ab}(C)=C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))}{\operatorname{Rep}}(\check{M})$. The adjoint pair ([\[adjoint_pair_eta\]](#adjoint_pair_eta){reference-type="ref" reference="adjoint_pair_eta"}) gives an adjoint pair $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))}{\operatorname{Rep}}(\check{P}^-)\leftrightarrows \operatorname{Hecke}_{\check{G},\check{M}, ab}(C)$$ We want to better understand the left adjoint $$\label{left_adjoint_ab_case} C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))}{\operatorname{Rep}}(\check{P}^-)\to \operatorname{Hecke}_{\check{G},\check{M}, ab}(C)$$ in this adjoint pair and also the composition $$\label{left_adjoint_ab_case_after_oblv} C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))}{\operatorname{Rep}}(\check{P}^-)\to C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))}{\operatorname{Rep}}(\check{M})\stackrel{\operatorname{oblv}}{\to} C$$ ### {#section-31} By ([@G], ch. I.1, 8.5.7), one gets $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))}{\operatorname{Rep}}(\check{P}^-)\,{\widetilde\to}\, {\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])-mod(C)$$ and $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))}{\operatorname{Rep}}(\check{M}) \,{\widetilde\to}\, {\mathcal O}(\check{G}/[\check{M}, \check{M}])-mod(C)$$ The functor ([\[left_adjoint_ab_case\]](#left_adjoint_ab_case){reference-type="ref" reference="left_adjoint_ab_case"}) is given by $$\label{tens_product_in_C_for_Bunb_P} c\mapsto {\mathcal O}(\check{G}/[\check{M}, \check{M}])\otimes_{{\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])} c$$ in the sense of ([@HA], 4.4.2.12). ### {#section-32} Write the action of ${\operatorname{Rep}}(\check{G})$ on $C$ on the right, and that of ${\operatorname{Rep}}(\check{M}_{ab})$ on the left. The category ${\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])-mod(C)$ is described as the category of $c\in C$ equipped with maps $$\kappa^{\lambda}: c\ast V^{\lambda}\to e^{\lambda}\ast c, \;\; \lambda\in \Lambda_{M, ab}^+$$ with the following additional structures and properties: i) if $\lambda=0$ then $\kappa^{\lambda}$ is identified with the identity map; ii) for $\lambda,\mu\in\Lambda_{M, ab}^+$ we are given a datum of commutativity for the diagram $$\begin{array}{ccc} (c\ast V^{\lambda})\ast V^{\mu} & {\widetilde\to}& c\ast (V^{\lambda}\otimes V^{\mu})\\ \downarrow\lefteqn{\scriptstyle \kappa^{\lambda}} && \downarrow\lefteqn{\scriptstyle u^{\lambda,\mu}}\\ e^{\lambda}\ast c\ast V^{\mu}&& c\ast V^{\lambda+\mu} \\ \downarrow\lefteqn{\scriptstyle \kappa^{\mu}} && \downarrow\lefteqn{\kappa^{\lambda+\mu}}\\ e^{\lambda}\ast (e^{\mu}\ast c) & {\widetilde\to}& e^{\lambda+\mu}\ast c \end{array}$$ iii) a coherent system of higher compatibilities. **Remark 1**. *For example, if $C$ is equipped with a t-structure, and both actions of ${\operatorname{Rep}}(\check{G})$ and of ${\operatorname{Rep}}(\check{M}_{ab})$ are t-exact then for $c\in C^{\heartsuit}$ in the above description of a ${\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])$-module on $c$ the higher compatibilities are automatic.* ### {#Sect_2.2.10_Now} Let $c\in {\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])-mod(C)$. For $\lambda\in\Lambda_{M, ab}^+$ by adjointness (and [@HA], 4.6.2.1), rewrite $\kappa^{\lambda}$ as the map $$\tau^{\lambda}: e^{-\lambda}\ast c\to c\ast (V^{\lambda})^*$$ Consider the following two lax actions of $\Lambda^+_{M, ab}$ on $C$. The left action of $\lambda$ is $c\mapsto e^{-\lambda}\ast c$. The right action of $\lambda$ is $c\ast (V^{\lambda})^*$. For $\lambda,\mu\in\Lambda^+_{M, ab}$ we are using here the lax structure on the right action given by $$(c\ast (V^{\lambda})^*)\ast (V^{\mu})^*\,{\widetilde\to}\, c\ast (V^{\lambda}\otimes V^{\mu})^*\stackrel{v^{\lambda,\mu}}{\to} c\ast (V^{\lambda+\mu})^*$$ **Lemma 1**. *In the situation of Section [2.2.9](#Sect_2.2.10_Now){reference-type="ref" reference="Sect_2.2.10_Now"}, a ${\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])$-module structure on $c$ is the same as a structure of a *lax central object* on $c$ in the sense of ([@Gai19SI], 2.7) with respect to the above lax actions of $\Lambda^+_{M, ab}$ on $C$.* *Proof.* First, for any $\lambda_i\in\Lambda^+$ the composition $V^{\lambda_1+\lambda_2}\stackrel{v^{\lambda_1,\lambda_2}}{\to} V^{\lambda_1}\otimes V^{\lambda_2}\stackrel{u^{\lambda_1,\lambda_2}}{\to} V^{\lambda_1+\lambda_2}$ is $\operatorname{id}$. Second, for any $\lambda_i\in\Lambda^+$ the diagram commutes $$\begin{array}{ccc} V^{\lambda_1}\otimes V^{\lambda_2}\otimes (V^{\lambda_1})^*\otimes (V^{\lambda_2})^* &\stackrel{u\otimes u}{\gets} & e \\ \downarrow\lefteqn{\scriptstyle u^{\lambda_1,\lambda_2}}&& \downarrow\lefteqn{\scriptstyle u}\\ V^{\lambda_1+\lambda_2}\otimes (V^{\lambda_1})^*\otimes (V^{\lambda_2})^* & \stackrel{u^{\lambda_1,\lambda_2}}{\gets} & V^{\lambda_1+\lambda_2}\otimes(V^{\lambda_1+\lambda_2})^*, \end{array}$$ where $u$ every time denotes the unit of the corresponding duality. The desired claim follows. ◻ ### {#Sect_2.0.11_functor_f} Now given $c\in {\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])-mod(C)$, we get a well-defined functor $$f: \Lambda^+_{M, ab}\to C, \;\; \lambda\mapsto e^{\lambda}\ast c\ast (V^{\lambda})^*$$ Here we consider $\Lambda^+_{M, ab}$ with the relation $\lambda_1\le\lambda_2$ iff $\lambda_2-\lambda_1\in \Lambda^+_{M, ab}$. This is not a partial order in general, but $\Lambda^+_{M, ab}$ with the relation $\le$ is a filtered category.[^6] For $\lambda_i\in \Lambda^+_{M, ab}$ the transition map from $f(\lambda_1)$ to $f(\lambda_1+\lambda_2)$ is $$\begin{gathered} e^{\lambda_1}\ast c\ast (V^{\lambda_1})^*\to e^{\lambda_1+\lambda_2}\ast e^{-\lambda_2}\ast c\ast (V^{\lambda_1})^*\stackrel{\tau^{\lambda_2}}{\to} \\ e^{\lambda_1+\lambda_2}\ast (c\ast (V^{\lambda_2})^*)\ast (V^{\lambda_1})^*\stackrel{v^{\lambda_1,\lambda_2}}{\to} e^{\lambda_1+\lambda_2}\ast c\ast (V^{\lambda_1+\lambda_2})^*\end{gathered}$$ **Proposition 1**. *The functor ([\[left_adjoint_ab_case_after_oblv\]](#left_adjoint_ab_case_after_oblv){reference-type="ref" reference="left_adjoint_ab_case_after_oblv"}) identifies with $$c\mapsto \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} e^{\lambda}\ast c\ast (V^{\lambda})^*$$ taken in $C$.* **Remark 1**. *If $G=P$ then $\Lambda_{M, ab}^+$ is the category equivalent to $pt$, and the above colimit identifies with $c$ itself.* *Proof of Proposition [Proposition 1](#Pp_2.0.11){reference-type="ref" reference="Pp_2.0.11"}.* **Step 1** We must show that ([\[tens_product_in_C\_for_Bunb_P\]](#tens_product_in_C_for_Bunb_P){reference-type="ref" reference="tens_product_in_C_for_Bunb_P"}) identifies with the above colimit in $C$. For this it suffices to show that $$\label{colimit_for_functions_over_Lambda^+_Mab} \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} e^{\lambda}\ast {\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])\ast (V^{\lambda})^*\,{\widetilde\to}\, {\mathcal O}(\check{G}/[\check{M}, \check{M}])$$ in ${\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})$. Recall the decomposition ([\[functions_for_G/P\^-\]](#functions_for_G/P^-){reference-type="ref" reference="functions_for_G/P^-"}). Given $\lambda_i\in \Lambda_{M, ab}^+$, the transition map $$e^{\lambda_1}\ast {\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])\ast (V^{\lambda_1})^*\to e^{\lambda_1+\lambda_2}\ast {\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])\ast (V^{\lambda_1+\lambda_2})^*$$ restricts for each $\lambda\in\Lambda_{M, ab}^+$ to a morphism $$e^{\lambda_1}\ast (V^{\lambda}\otimes e^{-\lambda})\ast (V^{\lambda_1})^*\to e^{\lambda_1+\lambda_2}\ast (V^{\lambda+\lambda_2}\otimes e^{-\lambda-\lambda_2}) \ast (V^{\lambda_1+\lambda_2})^*$$ So, the LHS of ([\[colimit_for_functions_over_Lambda\^+\_Mab\]](#colimit_for_functions_over_Lambda^+_Mab){reference-type="ref" reference="colimit_for_functions_over_Lambda^+_Mab"}) identifies with the direct sum over $\nu\in\Lambda_{M, ab}$ of $$\label{mult_space_for_nu_in_LHS_for_Pp2.0.11} \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} e^{\lambda}\ast (V^{\lambda-\nu}\otimes e^{\nu-\lambda})\ast (V^{\lambda})^*,$$ where the colimit is taken over those $\lambda$ satisfying $\lambda-\nu\in \Lambda^+_{M, ab}$. More precisely, ([\[mult_space_for_nu_in_LHS_for_Pp2.0.11\]](#mult_space_for_nu_in_LHS_for_Pp2.0.11){reference-type="ref" reference="mult_space_for_nu_in_LHS_for_Pp2.0.11"}) is the multiplicity space of $e^{\nu}\in{\operatorname{Rep}}(\check{M}_{ab})$ in the LHS of ([\[colimit_for_functions_over_Lambda\^+\_Mab\]](#colimit_for_functions_over_Lambda^+_Mab){reference-type="ref" reference="colimit_for_functions_over_Lambda^+_Mab"}). For each $\lambda\in \Lambda^+_{M, ab}$ such that $\lambda-\nu\in \Lambda^+_{M, ab}$ we define the morphism of $\check{G}$-modules $$\label{map_for_individual_lambda_for_Pp2.0.11} V^{\lambda-\nu}\otimes (V^{\lambda})^*\to {\mathcal O}(\check{G}/[\check{M},\check{M}])$$ as the map that corresponds via the Frobenius reciprocity to the $[\check{M},\check{M}]$-equivariant morphism $(v^{\lambda-\nu})^*\otimes v^{\lambda}: V^{\lambda-\nu}\otimes (V^{\lambda})^*\to e$. It is easy to see that the maps ([\[map_for_individual_lambda_for_Pp2.0.11\]](#map_for_individual_lambda_for_Pp2.0.11){reference-type="ref" reference="map_for_individual_lambda_for_Pp2.0.11"}) are compatible with the transition maps in the diagram ([\[mult_space_for_nu_in_LHS_for_Pp2.0.11\]](#mult_space_for_nu_in_LHS_for_Pp2.0.11){reference-type="ref" reference="mult_space_for_nu_in_LHS_for_Pp2.0.11"}), so define by passing to the colimit the $[\check{M},\check{M}]$-equivariant morphism from ([\[mult_space_for_nu_in_LHS_for_Pp2.0.11\]](#mult_space_for_nu_in_LHS_for_Pp2.0.11){reference-type="ref" reference="mult_space_for_nu_in_LHS_for_Pp2.0.11"}) to $e$. This gives by the Frobenius reciprocity a morphism of $\check{G}$-modules $$\label{map_to_be_proved_iso_Pp2.0.12} \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} e^{\lambda}\ast (V^{\lambda-\nu}\otimes e^{\nu-\lambda})\ast (V^{\lambda})^*\to {\mathcal O}(\check{G}/[\check{M},\check{M}])_{\nu}$$ where the subscript $\nu$ stands for the subspace of ${\mathcal O}(\check{G}/[\check{M},\check{M}])$ on which $\check{M}$ acts by $\nu$. For $v\in V^{\lambda-\nu}, u\in (V^{\lambda})^*$ the map ([\[map_for_individual_lambda_for_Pp2.0.11\]](#map_for_individual_lambda_for_Pp2.0.11){reference-type="ref" reference="map_for_individual_lambda_for_Pp2.0.11"}) sends $v\otimes u$ to the function on $G$ $$g\mapsto \langle(v^{\lambda-v})^*, g^{-1} v\rangle\langle v^{\lambda}, g^{-1}u\rangle$$ **Step 2** Let $\eta\in\Lambda^+$. To finish the proof, it remains to show that for $\lambda\in\Lambda^+_{M, ab}$ large enough with respect to $\eta$ and $\nu$ (that is, $\langle\lambda, \check{\alpha}_i\rangle$ large enough for $i\notin {\mathcal I}_M$), one has naturally $$\operatorname{Hom}_{\check{G}}(V^{\eta}, V^{\lambda-\nu}\otimes (V^{\lambda})^*)\,{\widetilde\to}\,\operatorname{Hom}_{\check{M}}(e^{-\nu}, V^{\eta})^*$$ By Lemma [Lemma 1](#Lm_2.0.15){reference-type="ref" reference="Lm_2.0.15"} below, $$V^{\eta}\otimes V^{\lambda}\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} V^{\lambda+\mu}\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, V^{\eta})$$ Our claim follows. ◻ ### {#section-33} Write $\operatorname{coind}_{\check{P}^-}^{\check{G}}: \operatorname{QCoh}(B(\check{P}^-))\to \operatorname{QCoh}(B(\check{G}))$ for the $*$-direct image map, the right adjoint to the restriction. Then $\operatorname{coind}_{\check{P}^-}^{\check{G}}(U^{\nu})\,{\widetilde\to}\, V^{\nu}$ for $\nu\in \Lambda^+$. These isomorphisms are uniquely normalized by the property that the diagram is required to commute $$\begin{array}{ccc} \operatorname{coind}_{\check{P}^-}^{\check{G}}(U^{\nu}) & {\widetilde\to}& V^{\nu}\\ \downarrow && \downarrow\lefteqn{\scriptstyle (v^{\nu})^*}\\ U^{\nu} & \stackrel{(v^{\nu})^*}{\to} & e^{\nu}, \end{array}$$ where the left vertical arrow comes from adjunction. **Lemma 1**. *Let $V\in{\operatorname{Rep}}(\check{G})^{\heartsuit}$ be finite-dimensional, $\lambda\in\Lambda^+_{M, ab}$. Assume that for any $\nu\in\Lambda_M^+$ appearing in $\operatorname{Res}^{\check{M}} V$, $\nu+\lambda\in\Lambda^+$. Then one has canonically $$V\otimes V^{\lambda}\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} V^{\lambda+\mu}\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, V)$$ in ${\operatorname{Rep}}(\check{G})$.* *Proof.* By the projection formula, $$V\otimes V^{\lambda}\,{\widetilde\to}\, V\otimes \operatorname{coind}_{\check{P}^-}^{\check{G}}(e^{\lambda})\,{\widetilde\to}\, \operatorname{coind}_{\check{P}^-}^{\check{G}}(e^{\lambda}\otimes \operatorname{Res}^{\check{P}^-}(V))$$ Now $\operatorname{Res}^{\check{P}^-}(V)$ is filtered with the associated graded being $\mathop{\oplus}\limits_{\mu\in\Lambda^+_M} U^{\mu}\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, V)$. So, $e^{\lambda}\otimes \operatorname{Res}^{\check{P}^-}(V)$ is filtered with the associated graded being $\mathop{\oplus}\limits_{\mu\in\Lambda^+_M} U^{\mu+\lambda}\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, V)$. For each $\mu$ as above, $\operatorname{coind}_{\check{P}^-}^{\check{G}}(U^{\mu+\lambda})\,{\widetilde\to}\, V^{\mu+\lambda}$. So, the corresponding filtration on $\operatorname{coind}_{\check{P}^-}^{\check{G}}(e^{\lambda}\otimes \operatorname{Res}^{\check{P}^-}(V))$, splits canonically. ◻ If $V\in{\operatorname{Rep}}(\check{G})^{\heartsuit}$ is finite-dimensional, $\lambda\in\Lambda^+_{M, ab}$ is large enough for $V$ then Lemma [Lemma 1](#Lm_2.0.15){reference-type="ref" reference="Lm_2.0.15"} also rewrites as a canonical isomorphism $$\label{Lm__2.0.15_rewritten} V\otimes (V^{\lambda})^*\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} (V^{\lambda+\mu})^*\otimes \operatorname{Hom}_{\check{M}}((U^{\mu})^*, V)$$ ### Version of Hecke property {#Sect_version_of_Hecke_property_2.0.16} Since ${\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))$ and ${\operatorname{Rep}}(\check{M})$ are rigid, by ([@Ly], 9.2.43) we get $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))}{\operatorname{Rep}}(\check{M})\,{\widetilde\to}\,{\operatorname{Fun}}_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab}))}({\operatorname{Rep}}(\check{M}), C)$$ For $c\in {\mathcal O}(\check{G}/[\check{P}^-, \check{P}^-])-mod(C)$, the ${\mathcal O}(\check{G}/[\check{M}, \check{M}])$-action on $$\bar c=\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} e^{\lambda}\ast c\ast (V^{\lambda})^*$$ is as follows. Let $V\in{\operatorname{Rep}}(\check{G})^{\heartsuit}$ finite-dimensional. It suffices to provide the action of $V\otimes (V^*)^{[\check{M}, \check{M}]}$ for each such $V$. It is given by a map $\bar c\ast V\to V_{[\check{M}, \check{M}]}\ast \bar c$. Pick $\lambda\in \Lambda^+_{M, ab}$ large enough for $V$. Note that $$V_{[\check{M}, \check{M}]}\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda_{M, ab}} e^{-\mu}\otimes \operatorname{Hom}_{\check{M}}((U^{\mu})^*, V)$$ Using ([\[Lm\_\_2.0.15_rewritten\]](#Lm__2.0.15_rewritten){reference-type="ref" reference="Lm__2.0.15_rewritten"}), the desired map is the composition $$\begin{gathered} (e^{\lambda}\ast c\ast (V^{\lambda})^*)\ast V\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} (e^{\lambda}\ast c)\ast (V^{\lambda+\mu})^*\otimes \operatorname{Hom}_{\check{M}}((U^{\mu})^*, V)\,{\widetilde\to}\\ \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} (e^{-\mu}\otimes \operatorname{Hom}_{\check{M}}((U^{\mu})^*, V))\ast (e^{\lambda+\mu}\ast c\ast (V^{\lambda+\mu})^*)\to\\ \mathop{\oplus}\limits_{\mu\in\Lambda_{M, ab}} (e^{-\mu}\otimes \operatorname{Hom}_{\check{M}}((U^{\mu})^*, V))\ast (e^{\lambda+\mu}\ast c\ast (V^{\lambda+\mu})^*),\end{gathered}$$ where the latter map is the projection on the corresponding summands. More precisely, when we pass to the colimit over $\lambda$, this becomes the desired morphism. ### {#Sect_2.0.17} For our convenience, we spell a version of the above with $\check{P}^-$ replaced by $\check{P}$. Let $C\in {\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})-mod(\operatorname{DGCat}_{cont})$. As above write ${\operatorname{Rep}}(\check{G})$-action on the right, and ${\operatorname{Rep}}(\check{M}_{ab})$-action on the left. The category of ${\mathcal O}(\check{G}/[\check{P},\check{P}])-mod(C)$ is described as the category of $c\in C$ equipped with maps $$\kappa^{\lambda}: e^{\lambda}\ast c\to c\ast V^{\lambda}, \; \lambda\in\Lambda^+_{M, ab}$$ with the following structures and properties: i) if $\lambda=0$ then $\kappa^{\lambda}$ is identified with the identity map; ii) for $\lambda,\mu\in\Lambda^+_{M, ab}$ we are given a datum of commutativity for the diagram $$\begin{array}{ccc} (c\ast V^{\lambda})\ast V^{\mu} & {\widetilde\to}& c\ast (V^{\lambda}\otimes V^{\mu})\\ \uparrow\lefteqn{\scriptstyle \kappa^{\lambda}} && \downarrow\lefteqn{\scriptstyle u^{\lambda,\mu}}\\ e^{\lambda}\ast c\ast V^{\mu}&& c\ast V^{\lambda+\mu} \\ \uparrow\lefteqn{\scriptstyle \kappa^{\mu}} && \uparrow\lefteqn{\kappa^{\lambda+\mu}}\\ e^{\lambda}\ast (e^{\mu}\ast c) & {\widetilde\to}& e^{\lambda+\mu}\ast c \end{array}$$ iii) a coherent system of higher compatibilities. For $c\in C$ a ${\mathcal O}(\check{G}/[\check{P},\check{P}])$-module structure is the same as the structure of a *lax central object* on $c$ in the sense of ([@Gai19SI], 2.7) with respect to the following actions of $\Lambda^+_{M, ab}$ on $C$. The left action of $\lambda\in\Lambda^+_{M, ab}$ is $c\mapsto e^{\lambda}\ast c$. The right lax action of $\lambda$ is $c\mapsto c\ast V^{\lambda}$. The lax structure on the right action is given by $$(c\ast V^{\lambda})\ast V^{\mu}\,{\widetilde\to}\, c\ast (V^{\lambda}\otimes V^{\mu})\stackrel{u^{\lambda,\mu}}{\to} c\ast V^{\lambda+\mu}$$ for $\lambda,\mu\in \Lambda^+_{M, ab}$. For $c\in {\mathcal O}(\check{G}/[\check{P},\check{P}])-mod(C)$ we get as above a well-defined functor $$f: \Lambda^+_{M, ab}\to C, \;\; \lambda\mapsto e^{-\lambda}\ast c\ast V^{\lambda}$$ For $\lambda_i\in\Lambda^+_{M, ab}$ the transition map from $f(\lambda_1)$ to $f(\lambda_1+\lambda_2)$ is $$\begin{gathered} e^{-\lambda_1}\ast c\ast V^{\lambda_1}\,{\widetilde\to}\, e^{-\lambda_1+\lambda_2}\ast (e^{\lambda_2}\ast c)\ast V^{\lambda_1}\stackrel{\kappa^{\lambda_2}}{\to}\\ e^{-\lambda_1+\lambda_2}\ast c\ast (V^{\lambda_2}\otimes V^{\lambda_1})\stackrel{u^{\lambda_1,\lambda_2}}{\to} e^{-\lambda_1+\lambda_2}\ast c\ast V^{\lambda_1+\lambda_2}\end{gathered}$$ We have similarly $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})} {\operatorname{Rep}}(\check{P})\,{\widetilde\to}\, {\mathcal O}(\check{G}/[\check{P},\check{P}])-mod(C)$$ Consider the composition $$\label{functor_for_2.0.17_for_checkP} C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})} {\operatorname{Rep}}(\check{P})\to C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})} {\operatorname{Rep}}(\check{M})\stackrel{\operatorname{oblv}}{\to} C,$$ where the first functor is the pullback along $B(\check{M})\to B(\check{P})$. A version of Proposition [Proposition 1](#Pp_2.0.11){reference-type="ref" reference="Pp_2.0.11"} in this case affirms that ([\[functor_for_2.0.17_for_checkP\]](#functor_for_2.0.17_for_checkP){reference-type="ref" reference="functor_for_2.0.17_for_checkP"}) identifies with the functor $$c\mapsto \mathop{\operatorname{colim}}\limits_{\lambda\in \Lambda^+_{M, ab}} \; e^{-\lambda}\ast c\ast V^{\lambda},$$ the colimit being taken in $C$. ## A version of dual baby Verma object ### {#section-34} Assume $C\in {\operatorname{Rep}}(\check{P}^-)-mod(\operatorname{DGCat}_{cont})$. We let ${\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})$ act on $C$ via the pull-back along the diagonal map $B(\check{P}^-)\to B(\check{G}\times \check{M}_{ab})$. We get the functor $$\label{functor_from_C_for_Sect_2.1.1} C\to {\mathcal O}(\check{G}/[\check{P}^-,\check{P}^-])-mod(C)$$ sending $c$ to itself with the action maps $$\label{map_action_abstract_coming_from_P^-} e^{-\lambda}\ast c\ast V^{\lambda}\to c$$ for $\lambda\in\Lambda^+_{M, ab}$ given as follows. Consider the morphism $(v^{\lambda})^*: e^{-\lambda}\otimes V^{\lambda}\to e$ in ${\operatorname{Rep}}(\check{P}^-)$. Applying to it the functor $\operatorname{act}(\cdot, c): {\operatorname{Rep}}(\check{P}^-)\to C$ one gets ([\[map_action_abstract_coming_from_P\^-\]](#map_action_abstract_coming_from_P^-){reference-type="ref" reference="map_action_abstract_coming_from_P^-"}). ### {#section-35} Take for a moment $C=\operatorname{QCoh}(B(\check{P}^-))$. Consider the diagram where both squares are cartesian $$\begin{array}{ccccc} B(\check{P}^-) & \gets & \check{P}^-\backslash(\check{M}_{ab}\times\check{G})/\check{P}^- & \gets & \check{M}\backslash (\check{M}_{ab}\times \check{G})/\check{P}^-\\ \downarrow &&\downarrow && \downarrow\\ B(\check{M}_{ab}\times\check{G}) &\gets & B(\check{P}^-) & \gets & B(\check{M}), \end{array}$$ we use here the diagonal maps $\check{P}^-\to \check{M}_{ab}\times\check{G}$ to form the diagram. It gives $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})} {\operatorname{Rep}}(\check{P}^-)\,{\widetilde\to}\, \operatorname{QCoh}(\check{P}^-\backslash(\check{M}_{ab}\times\check{G})/\check{P}^-)$$ Note that $$\check{P}^-\backslash(\check{M}_{ab}\times\check{G})/\check{P}^-\,{\widetilde\to}\, (\check{G}/[\check{P}^-, \check{P}^-])/\operatorname{Ad}(\check{P}^-)$$ Consider the closed immersion $$i: B(\check{P}^-)\stackrel{}{\hookrightarrow} (\check{G}/[\check{P}^-, \check{P}^-])/\operatorname{Ad}(\check{P}^-)$$ The functor ([\[functor_from_C\_for_Sect_2.1.1\]](#functor_from_C_for_Sect_2.1.1){reference-type="ref" reference="functor_from_C_for_Sect_2.1.1"}) in these terms is nothing but $i_*$. Now $$\check{M}\backslash (\check{M}_{ab}\times G)/\check{P}^-\,{\widetilde\to}\, (\check{G}/[\check{P}^-, \check{P}^-])/\operatorname{Ad}(\check{M})$$ Let $$i_M: B(\check{M})\stackrel{}{\hookrightarrow} (\check{G}/[\check{P}^-, \check{P}^-])/\operatorname{Ad}(\check{M})$$ be the closed immersion given by the point $1$. Taking for $c$ the trivial $\check{P}^-$-module $e\in C$, applying ([\[functor_from_C\_for_Sect_2.1.1\]](#functor_from_C_for_Sect_2.1.1){reference-type="ref" reference="functor_from_C_for_Sect_2.1.1"}) and further ([\[tens_product_in_C\_for_Bunb_P\]](#tens_product_in_C_for_Bunb_P){reference-type="ref" reference="tens_product_in_C_for_Bunb_P"}) one gets the direct image $(i_M)_*{\mathcal O}$ of the structure sheaf on $B(\check{M})$. Note that its further direct image to $B(\check{P}^-)$ identifies with ${\mathcal O}(\check{P}^-/\check{M})\in {\operatorname{Rep}}(\check{P}^-)$. So, Proposition [Proposition 1](#Pp_2.0.11){reference-type="ref" reference="Pp_2.0.11"} gives for this $c$ an isomorphism $$\label{iso_in_Rep_checkP-} \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} e^{\lambda}\otimes (V^{\lambda})^*\,{\widetilde\to}\,{\mathcal O}(\check{P}^-/\check{M})$$ in ${\operatorname{Rep}}(\check{P}^-)$, here the colimit is taken in ${\operatorname{Rep}}(\check{P}^-)$. By Proposition [Proposition 1](#Pp_2.1.10_colimit_for_Bunt_P){reference-type="ref" reference="Pp_2.1.10_colimit_for_Bunt_P"}, ([\[iso_in_Rep_checkP-\]](#iso_in_Rep_checkP-){reference-type="ref" reference="iso_in_Rep_checkP-"}) naturally lifts to an object of $C\otimes_{{\operatorname{Rep}}(\check{G}\times\check{M})}{\operatorname{Rep}}(\check{M})$ (that is, has a Hecke property similar to that of $\operatorname{IC}_{\operatorname{\widetilde\operatorname{Bun}}_P}$, cf. Section [3.2.10](#Sect_2.3.12_local_vs_global){reference-type="ref" reference="Sect_2.3.12_local_vs_global"}). **Remark 1**. *One may consider the compactification $U(\check{P}^-)\stackrel{}{\hookrightarrow} \check{G}/\check{P}$ and describe ([\[iso_in_Rep_checkP-\]](#iso_in_Rep_checkP-){reference-type="ref" reference="iso_in_Rep_checkP-"}) as the cohomology of the structure sheaf of $\check{G}/\check{P}$ with prescribed poles along the boundary, as the order of poles goes to infinity. Namely, for $i\in {\mathcal I}$ write $s_i\in W$ for the corresponding simple reflection. Set $W^M=\{w\in W\mid \ell(w s_i)>\ell(w), \; \mbox{for all}\; i\in{\mathcal I}_M\}$. The multiplication $W^M\times W_M\to W$ is bijective by ([@Sw], 2.3.1). If $w\in W$ then $W^M\cap wW_M$ consists of a unique element denoted $w^M$. Then $\check{G}/\check{P}-U(\check{P}^-)$ is a divisor on $\check{G}/\check{P}$, whose irreducible components are the closures of $w_0\check{B}(w_0s_i)^M \check{P}=\check{B}^-s_i \check{P}$ for $i\in {\mathcal I}-{\mathcal I}_M$ by ([@Sw], 3.3.3).* ### {#section-36} Let $\check{\mathfrak{g}}=\operatorname{Lie}\check{G}$, $\check{\mathfrak{u}}(P^-)=\operatorname{Lie}U(\check{P}^-)$. Let ${\mathcal O}=k[[t]]\subset F=k((t))$. Write $\operatorname{Gr}_G$ for the affine Grassmanian of $G$ viewed as the moduli of pairs $({\mathcal F}_G, \beta)$, where ${\mathcal F}_G$ is a $G$-torsor on $D=\operatorname{Spec}{\mathcal O}$ with a trivialization $\beta: {\mathcal F}_G\,{\widetilde\to}\,{\mathcal F}^0_G\mid_{D^*}$, here $D^*=\operatorname{Spec}F$. Write $\operatorname{Sat}: {\operatorname{Rep}}(\check{G})\to Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ for the Satake functor. Write $I_P$ for the preimage of $P$ under $G({\mathcal O})\to G$, this is a parahoric subgroup. Consider the full subcategory $$Shv(\operatorname{Gr}_G)^{I_P, constr}\subset Shv(\operatorname{Gr}_G)^{I_P}$$ of those objects whose image in $Shv(\operatorname{Gr}_G)$ is compact. Then $Shv(\operatorname{Gr}_G)^{I_P, constr}\in\operatorname{DGCat}^{non-cocmpl}$. Set $$Shv(\operatorname{Gr}_G)^{I_P, ren}=\operatorname{Ind}(Shv(\operatorname{Gr}_G)^{I_P, constr})$$ The renormalization is a general procedure, for algebraic stacks locally of finite type with an affine diagonal it is studied in ([@AGKRRV], F.5). As in Section [5.5.4](#Sect_A.5.4){reference-type="ref" reference="Sect_A.5.4"}, we have an adjoint pair $Shv(\operatorname{Gr}_G)^{I_P}\leftrightarrows Shv(\operatorname{Gr}_G)^{I_P, ren}$ in $\operatorname{DGCat}_{cont}$, where the left adjoint is fully faithful. ### {#Sect_2.1.4_objects_cB} For $\mu\in\Lambda^+_M$ write ${\mathcal B}_{\mu, !}, {\mathcal B}_{\mu, *}\in Shv(\operatorname{Gr}_G)^{I_P}$ for the $\operatorname{IC}$-sheaf of $I_Pt^{\mu}G({\mathcal O})/G({\mathcal O})$ extended by zero (resp., by $*$-extension) to $\operatorname{Gr}_G$. If $\mu\in\Lambda_{M, ab}$ then $I_Pt^{\mu}G({\mathcal O})/G({\mathcal O})=It^{\mu}G({\mathcal O})/G({\mathcal O})$, where $I\subset G({\mathcal O})$ is the Iwahori subgroup. If moreover $\mu\in\Lambda^+_{M, ab}$ then $It^{\mu}G({\mathcal O})/G({\mathcal O})=U({\mathcal O})t^{\mu}G({\mathcal O})/G({\mathcal O})=S_B^{\mu}\cap \overline{\operatorname{Gr}}_G^{\mu}$ by ([@MV], proof of Theorem 3.2). In the latter case the open embedding $I_Pt^{\mu}G({\mathcal O})/G({\mathcal O})\stackrel{}{\hookrightarrow} \overline{\operatorname{Gr}}_G^{\mu}$ is affine by ([@MV], 3.1), so that ${\mathcal B}_{\mu, !}, {\mathcal B}_{\mu, *}$ are perverse. Note that for $\mu\in\Lambda^+_{M, ab}$ we have a canonical map $$\label{map_from_Sat(Vmu)_to_cB_mu*} \operatorname{Sat}(V^{\mu})\to {\mathcal B}_{\mu, *}$$ of $I_P$-equivariant perverse sheaves on $\operatorname{Gr}_G$. ### {#section-37} Let ${\mathcal F}l_P=G(F)/I_P$ and ${\mathcal H}_P(G)=Shv({\mathcal F}l_P)^{I_P}$. It is well known that $({\mathcal H}_P(G), *)$ acts on $Shv(\operatorname{Gr}_G)^{I_P}$ by convolutions. It also similarly acts on $Shv(\operatorname{Gr}_G)^{I_P, ren}$. Indeed, ${\mathcal H}_P(G)$ is compactly generated. So, it suffices to show that ${\mathcal H}_P(G)^c$ acts on $Shv(\operatorname{Gr}_G)^{I_P, constr}$ naturally. Given $K\in {\mathcal H}_P(G)^c$, there is a $I_P$-invariant closed subscheme of finite type $Y\subset {\mathcal F}l_P$ such that $\operatorname{oblv}(K)\in Shv({\mathcal F}l_P)$ is the extension by zero from $Y$. Let $\tilde Y\subset G(F)$ be the preimage of $Y$ under $G(F)\to {\mathcal F}l_P$. The desired claim follows now from the fact that the convolution map $\tilde Y\times^{I_P} \operatorname{Gr}_G\to \operatorname{Gr}_G$ is proper. Write $\tilde W$ for the affine extended Weyl group of $(G, T)$. The $I_P$-orbits on ${\mathcal F}l_P$ are indexed by $W_M\backslash \tilde W/W_M$. For $w\in \tilde W$ write $j_{w, !}, j_{w, *}\in {\mathcal H}_P(G)$ for the standard and costandard objects attached to $w\in \tilde W$ and normalized to be perverse on the $I_P$-orbit $I_PwI_P/I_P$. For $\lambda\in\Lambda$ we write for brevity $j_{\lambda, !}=j_{t^{\lambda}, !}$ and $j_{\lambda, *}=j_{t^\lambda, *}$. For the definition of the category $\operatorname{IndCoh}$ on a quasi-smooth Artin stack with a specified singular support condition we refer to ([@AG], Section 8). Gurbir Dhillon and Harrison Chen announced the following, see [@CD] (compare also with Conjecture 3.6.1 in [@BL]). **Proposition 1**. *There is a canonical equivalence $$\label{iso_Gurbir_Chen} \operatorname{IndCoh}((\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-)\,{\widetilde\to}\, Shv(\operatorname{Gr}_G)^{I_P, ren}$$ with the following properties:\ (i) The ${\operatorname{Rep}}(\check{G})$-action on $\operatorname{IndCoh}((\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-)$ arising from the projection $$(\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-\to pt/\check{P}^-\to pt/\check{G}$$ corresponds to the ${\operatorname{Rep}}(\check{G})$-action on $Shv(\operatorname{Gr}_G)^{I_P, ren}$ via $\operatorname{Sat}: {\operatorname{Rep}}(\check{G})\to Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ and the right convolutions.* *(ii) The ${\operatorname{Rep}}(\check{M}_{ab})$-action on $\operatorname{IndCoh}((\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-)$ arising from the projection $$(\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-\to pt/\check{P}^-\to pt/\check{M}\to pt/\check{M}_{ab}$$ corresponds to the ${\operatorname{Rep}}(\check{M}_{ab})$-action on $Shv(\operatorname{Gr}_G)^{I_P, ren}$ such that for $\lambda\in\Lambda_{M, ab}^+$, $e^{\lambda}$ sends $F$ to $j_{\lambda, *}\ast F$. So, it comes from the monoidal functor ([\[mon_functor\_\*\]](#mon_functor_*){reference-type="ref" reference="mon_functor_*"}).* *iii) The object ${\mathcal O}_{pt/\check{P}^-}\in \operatorname{IndCoh}((\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-)$ corresponds under ([\[iso_Gurbir_Chen\]](#iso_Gurbir_Chen){reference-type="ref" reference="iso_Gurbir_Chen"}) to $\delta_{1,\operatorname{Gr}_G}\in Shv(\operatorname{Gr}_G)^{I_P, ren}$.* *iv) For $\lambda\in\Lambda^+_{M, ab}$ the map $(v^{\lambda})^*: V^{\lambda}\to e^{\lambda}$ in ${\operatorname{Rep}}(\check{P}^-)$ corresponds under ([\[iso_Gurbir_Chen\]](#iso_Gurbir_Chen){reference-type="ref" reference="iso_Gurbir_Chen"}) to the morphism $\operatorname{Sat}(V^{\lambda})\to {\mathcal B}_{\lambda,*}$ in $Shv(\operatorname{Gr}_G)^{I_P, ren}$ given by ([\[map_from_Sat(Vmu)\_to_cB_mu\*\]](#map_from_Sat(Vmu)_to_cB_mu*){reference-type="ref" reference="map_from_Sat(Vmu)_to_cB_mu*"}).* *v) The equivalence ([\[iso_Gurbir_Chen\]](#iso_Gurbir_Chen){reference-type="ref" reference="iso_Gurbir_Chen"}) restricts to an equivalence of full subcategories $$\operatorname{IndCoh}_{\operatorname{Nilp}}((\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-)\,{\widetilde\to}\, Shv(\operatorname{Gr}_G)^{I_P},$$ here $\operatorname{Nilp}$ stands for the nilpotent singular support.* **Remark 1**. *We also expect that the ${\operatorname{Rep}}(\check{M})$-action on $Shv(\operatorname{Gr}_G)^{H, ren}$ given below by ([\[action_Rep(checkM)\_shifted\]](#action_Rep(checkM)_shifted){reference-type="ref" reference="action_Rep(checkM)_shifted"}) is related to the ${\operatorname{Rep}}(\check{M})$-action on $\operatorname{IndCoh}((\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-)$ arising from the projection $$(\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-\to pt/\check{P}^-\to pt/\check{M}$$ via ([\[iso_Gurbir_Chen\]](#iso_Gurbir_Chen){reference-type="ref" reference="iso_Gurbir_Chen"}) composed with the equivalence ([\[eq_ren_parahoric_versus_H\]](#eq_ren_parahoric_versus_H){reference-type="ref" reference="eq_ren_parahoric_versus_H"}).* ### {#section-38} Now take $C=\operatorname{IndCoh}((\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-)$ equipped with an action of ${\operatorname{Rep}}(\check{P}^-)$ coming from the projection $$(\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-\to B(\check{P}^-)$$ Let $c\in C$ be the direct image of the structure sheaf ${\mathcal O}$ along the closed embedding $$\{0\}/\check{P}^-\to (\check{\mathfrak{u}}(P^-)\times_{\check{\mathfrak{g}}} 0)/\check{P}^-$$ Applying to $c$ the functors ([\[functor_from_C\_for_Sect_2.1.1\]](#functor_from_C_for_Sect_2.1.1){reference-type="ref" reference="functor_from_C_for_Sect_2.1.1"}) and further ([\[left_adjoint_ab_case\]](#left_adjoint_ab_case){reference-type="ref" reference="left_adjoint_ab_case"}), one gets an object of $\operatorname{Hecke}_{\check{G},\check{M} ,ab}(C)$ denoted ${\mathcal M}_{\check{G},\check{P}^-}$. This is a version of the dual baby Verma object we are interested in. ### {#Sect_2.3.10_almost_final_version} For future applications, we write down an analog of the isomorphism ([\[iso_in_Rep_checkP-\]](#iso_in_Rep_checkP-){reference-type="ref" reference="iso_in_Rep_checkP-"}) with $\check{P}^-$ replaced by $\check{P}$. Take for a moment $C=\operatorname{QCoh}(B(\check{P}))$. Consider the diagram, where both squares are cartesian $$\begin{array}{ccccc} B(\check{P}) & \gets & \check{P}\backslash(\check{M}_{ab}\times\check{G})/\check{P} & \gets & \check{M}\backslash (\check{M}_{ab}\times \check{G})/\check{P}\\ \downarrow &&\downarrow && \downarrow\\ B(\check{M}_{ab}\times\check{G}) &\gets & B(\check{P}) & \gets & B(\check{M}), \end{array}$$ we use here the diagonal maps $\check{P}\to \check{M}_{ab}\times\check{G}$ to form the diagram. It gives $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})} {\operatorname{Rep}}(\check{P})\,{\widetilde\to}\, \operatorname{QCoh}(\check{P}\backslash(\check{M}_{ab}\times\check{G})/\check{P})$$ The analog of ([\[functor_from_C\_for_Sect_2.1.1\]](#functor_from_C_for_Sect_2.1.1){reference-type="ref" reference="functor_from_C_for_Sect_2.1.1"}) for $\check{P}$ is the functor $$\label{functor_from_C_for_checkP_Sect_2.1.6} C\to {\mathcal O}(\check{G}/[\check{P}, \check{P}])-mod(C)$$ sending $c$ to itself with the action maps $e^{\lambda}\ast c\ast (V^{\lambda})^*\to c$ obtained by appling the functor $\operatorname{act}(\cdot, c): {\operatorname{Rep}}(\check{P})\to C$ to $v^{\lambda}: e^{\lambda}\otimes (V^{\lambda})^*\to e$. Taking for $c$ the trivial $\check{P}$-module, applying ([\[functor_from_C\_for_checkP_Sect_2.1.6\]](#functor_from_C_for_checkP_Sect_2.1.6){reference-type="ref" reference="functor_from_C_for_checkP_Sect_2.1.6"}) and further the pullback $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})} {\operatorname{Rep}}(\check{P})\to C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})} {\operatorname{Rep}}(\check{M})$$ along $B(\check{M})\to B(\check{P})$, one gets an object of $\operatorname{QCoh}(\check{M}\backslash (\check{M}_{ab}\times \check{G})/\check{P})$ whose direct image to $B(\check{P})$ identifies with ${\mathcal O}(\check{P}/\check{M})\in {\operatorname{Rep}}(\check{P})$. So, an analog of Proposition [Proposition 1](#Pp_2.0.11){reference-type="ref" reference="Pp_2.0.11"} gives for this $c$ an isomorphism $$\label{iso_O(P/M)_for_Sect_2.1.6} \mathop{\operatorname{colim}}_{\lambda\in\Lambda^+_{M, ab}} e^{-\lambda}\otimes V^{\lambda}\,{\widetilde\to}\, {\mathcal O}(\check{P}/\check{M})$$ in ${\operatorname{Rep}}(\check{P})$. Here the colimit is taken in ${\operatorname{Rep}}(\check{P})$, and the inductive system is described in Section [2.2.13](#Sect_2.0.17){reference-type="ref" reference="Sect_2.0.17"}. Exchanging the roles of $P$ and $P^-$, one gets an analog of the object ${\mathcal M}_{\check{G}, \check{P}^-}$ denoted by ${\mathcal M}_{\check{G}, \check{P}}\in\operatorname{Hecke}_{\check{G}, \check{M}, ab}(C')$, where $C'=\operatorname{IndCoh}((\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})$. ### {#section-39} We need the following generalization of the isomorphism ([\[iso_O(P/M)\_for_Sect_2.1.6\]](#iso_O(P/M)_for_Sect_2.1.6){reference-type="ref" reference="iso_O(P/M)_for_Sect_2.1.6"}). Fix $\eta\in\Lambda^+_M$. Consider the diagram $$\label{diag_for_Sect_2.1.7} \{\lambda\in\Lambda_{M, ab}\mid \lambda+\eta\in\Lambda^+\}\to {\operatorname{Rep}}(\check{P}), \;\lambda\mapsto e^{-\lambda}\otimes V^{\lambda+\eta}$$ Here we consider $\{\lambda\in\Lambda_{M, ab}\mid \lambda+\eta\in\Lambda^+\}$ with the relation $\lambda_1\le \lambda_2$ iff $\lambda_2-\lambda_1\in\Lambda^+$. This is not an order relation, but defines instead a structure of a filtered category on this set. For $\lambda_i\in \Lambda_{M, ab}$ with $\lambda_i+\eta\in\Lambda^+$ and $\lambda=\lambda_2-\lambda_1\in\Lambda^+$ the transition map $$e^{-\lambda_1}\otimes V^{\lambda_1+\eta}\to e^{-\lambda_2}\otimes V^{\lambda_2+\eta}$$ is the composition $$\begin{gathered} e^{-\lambda_1}\otimes V^{\lambda_1+\eta}\,{\widetilde\to}\, e^{-\lambda_2}\otimes e^{\lambda}\otimes V^{\lambda_1+\eta}\stackrel{v^{\lambda}}{\to} e^{-\lambda_2}\otimes V^{\lambda}\otimes V^{\lambda_1+\eta}\stackrel{u^{\lambda, \lambda_1+\eta}}{\to} e^{-\lambda_2}\otimes V^{\lambda_2+\eta},\end{gathered}$$ Write $\operatorname{coind}_{\check{M}}^{\check{P}}: {\operatorname{Rep}}(\check{M})\to {\operatorname{Rep}}(\check{P})$ for the right adjoint to the restriction functor. **Lemma 1**. *Let $\eta\in\Lambda^+_M$. One has canonically in ${\operatorname{Rep}}(\check{P})$ $$\label{iso_for_Lm_2.1.8} \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab},\; \lambda+\eta\in\Lambda^+} \; e^{-\lambda}\otimes V^{\lambda+\eta}\,{\widetilde\to}\, \operatorname{coind}_{\check{M}}^{\check{P}}(U^{\eta})$$* *Proof.* **Step 1**. Let $\lambda\in\Lambda_{M, ab}$ with $\lambda+\eta\in\Lambda^+$. By Frobenius reciprocity, a $\check{P}$-equivariant map $e^{-\lambda}\otimes V^{\lambda+\eta}\to \operatorname{coind}_{\check{M}}^{\check{P}}(U^{\eta})$ is the same as a $\check{M}$-equivariant map $e^{-\lambda}\otimes V^{\lambda+\eta}\to U^{\eta}$. The latter is the same as a $\check{M}$-equivariant morphism $$V^{\lambda+\eta}\to U^{\eta}\otimes e^{\lambda}\,{\widetilde\to}\, U^{\lambda+\eta}\,{\widetilde\to}\, \operatorname{coind}_{\check{B}^-_M}^{\check{M}} e^{\lambda+\eta}$$ The latter comes from the $\check{B}^-_M$-equivariant morphism $(v^{\lambda+\eta})^*: V^{\lambda+\eta}\to e^{\lambda+\eta}$. It is easy to check that the morphism so obtained are compatible with the transition maps in the diagram ([\[diag_for_Sect_2.1.7\]](#diag_for_Sect_2.1.7){reference-type="ref" reference="diag_for_Sect_2.1.7"}). It remains to check that the obtained map ([\[iso_for_Lm_2.1.8\]](#iso_for_Lm_2.1.8){reference-type="ref" reference="iso_for_Lm_2.1.8"}) is an isomorphism. The $\check{P}$-module $\operatorname{coind}_{\check{M}}^{\check{P}}(U^{\eta})$ identifies with ${\mathcal O}(\check{P}/\check{M})\otimes U^{\eta}$, where $\check{P}$ acts diagonally. Here $\check{P}$ acts by left translations on $\check{P}/\check{M}$, by functoriality on ${\mathcal O}(\check{P}/\check{M})$, and via the quotient $\check{P}\to \check{M}$ on $U^{\eta}$. **Step 2** Assume in addition $\eta\in\Lambda^+$. Then we construct a morphism of $\check{P}$-modules $$\label{map_for_proof_of_Lm_2.1.8} \operatorname{coind}_{\check{M}}^{\check{P}}(U^{\eta})\to \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab},\; \lambda+\eta\in\Lambda^+} \; e^{-\lambda}\otimes V^{\lambda+\eta}$$ as follows. For any $\lambda\in\Lambda^+_{M, ab}$ consider the morphism $$e^{-\lambda}\otimes V^{\lambda}\otimes V^{\eta}\stackrel{v^{\lambda,\eta}}{\to} e^{-\lambda}\otimes V^{\lambda+\eta}$$ in ${\operatorname{Rep}}(\check{P})$. These morphisms are compatible with the transition maps in the inductive systems ([\[diag_for_Sect_2.1.7\]](#diag_for_Sect_2.1.7){reference-type="ref" reference="diag_for_Sect_2.1.7"}) and ([\[iso_O(P/M)\_for_Sect_2.1.6\]](#iso_O(P/M)_for_Sect_2.1.6){reference-type="ref" reference="iso_O(P/M)_for_Sect_2.1.6"}). Passing to the colimit, from ([\[iso_O(P/M)\_for_Sect_2.1.6\]](#iso_O(P/M)_for_Sect_2.1.6){reference-type="ref" reference="iso_O(P/M)_for_Sect_2.1.6"}) we get a morphism $${\mathcal O}(\check{P}/\check{M})\otimes V^{\eta}\to \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab},\; \lambda+\eta\in\Lambda^+} \; e^{-\lambda}\otimes V^{\lambda+\eta}$$ in ${\operatorname{Rep}}(\check{P})$. Now ([\[map_for_proof_of_Lm_2.1.8\]](#map_for_proof_of_Lm_2.1.8){reference-type="ref" reference="map_for_proof_of_Lm_2.1.8"}) is defined as the restriction of the latter map under $U^{\eta}\stackrel{}{\hookrightarrow} V^{\eta}$. The two morphisms so obtained are inverse of each other. **Step 3**. Let now $\eta\in\Lambda^+_M$. We reduce our claim to the case of Step 2 as follows. Pick $\lambda_0\in\Lambda^+_{M, ab}$ such that $\eta_0=\eta+\lambda_0\in\Lambda^+$. The LHS of ([\[iso_for_Lm_2.1.8\]](#iso_for_Lm_2.1.8){reference-type="ref" reference="iso_for_Lm_2.1.8"}) identifies with $$\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab},\; \lambda+\lambda_0+\eta\in\Lambda^+} \; e^{-\lambda-\lambda_0}\otimes V^{\lambda+\lambda_0+\eta}\,{\widetilde\to}\, e^{-\lambda_0}\otimes \operatorname{coind}_{\check{M}}^{\check{P}}(U^{\eta_0})$$ by Steps 1 and 2. By the projection formula, $$e^{-\lambda_0}\otimes \operatorname{coind}_{\check{M}}^{\check{P}}(U^{\eta_0})\,{\widetilde\to}\, \operatorname{coind}_{\check{M}}^{\check{P}}(e^{-\lambda_0}\otimes U^{\eta_0})$$ Since $e^{-\lambda_0}\otimes U^{\eta_0}\,{\widetilde\to}\, U^{\eta}$ in ${\operatorname{Rep}}(\check{M})$, we are done. ◻ # Parabolic semi-infinite category of sheaves ## Finite-dimensional counterpart ### {#Sect_Local automorphic side_begins} Let us explain that $H:=U(P)(F)M({\mathcal O})$ is a placid ind-scheme. We equip $\Lambda^+_{M, ab}$ with the relation $\le$ as in Section [2.2.10](#Sect_2.0.11_functor_f){reference-type="ref" reference="Sect_2.0.11_functor_f"}. For $\lambda\in\Lambda^+_{M, ab}$ set $H_{\lambda}=t^{-\lambda}P({\mathcal O})t^{\lambda}$. This is a placid group scheme. If $\lambda\le\mu$ in $\Lambda^+_{M, ab}$ then $H_{\lambda}\subset H_{\mu}$ is a placid closed immersion, and $H\,{\widetilde\to}\, \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} H_{\lambda}$ a placid ind-scheme. So, for $C\in Shv(H)-mod$, $C^H$ makes sense. We will relate the RHS of ([\[iso_Gurbir_Chen\]](#iso_Gurbir_Chen){reference-type="ref" reference="iso_Gurbir_Chen"}) to $Shv(\operatorname{Gr}_G)^{H}$ in way similar to [@Gai19SI]. Note that $H$-orbits on $\operatorname{Gr}_G$ are indexed by $\Lambda^+_M$, to $\mu\in\Lambda^+_M$ we attach the orbit passing through $t^{\mu}$. ### {#section-40} By ([@LyWhit_loc_glob], 1.3.4), we get $Shv(\operatorname{Gr}_G)^{H}\,{\widetilde\to}\,\mathop{\lim}\limits_{\lambda\in (\Lambda^+_{M, ab})^{op}} Shv(\operatorname{Gr}_G)^{H_{\lambda}}$. For each $\lambda$ the functor $$\operatorname{oblv}: Shv(\operatorname{Gr}_G)^{H_{\lambda}}\to Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$$ is a full embedding by Section [5.1.1](#Sect_A.0.2){reference-type="ref" reference="Sect_A.0.2"}. By ([@Ly], 2.7.7), $\mathop{\lim}\limits_{\lambda\in (\Lambda^+_{M, ab})^{op}} Shv(\operatorname{Gr}_G)^{H_{\lambda}}$ is a full subcategory of $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ equal to $$\mathop{\cap}\limits_{\lambda\in \Lambda^+_{M, ab}} Shv(\operatorname{Gr}_G)^{H_{\lambda}}$$ taken in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$. ### {#section-41} Recall that for any smooth affine algebraic group ${\mathcal G}$ of finite type, ${\mathcal G}(F)$ is a placid ind-scheme (cf. [@Ly4], 0.0.51). So, $P(F)$ is a placid ind-scheme, and $$P(F)/H\,{\widetilde\to}\, M(F)/M({\mathcal O})\,{\widetilde\to}\, \operatorname{Gr}_M$$ is an ind-scheme of ind-finite type. As in Section [5.2](#Sect_A.1){reference-type="ref" reference="Sect_A.1"}, one gets an action of $Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$ on $Shv(\operatorname{Gr}_G)^H$. In fact, $Shv(M(F))$ acts naturally on $Shv(\operatorname{Gr}_G)^{U(F)}$, and the desired $Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$-action is obtained by functoriality after passing to $M({\mathcal O})$-invariants, cf. Remark [Remark 1](#Rem_A.1.3_action_of_M(F)){reference-type="ref" reference="Rem_A.1.3_action_of_M(F)"}. Composing ${\operatorname{Rep}}(\check{M})\to Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$ one gets a ${\operatorname{Rep}}(\check{M})$-action on $Shv(\operatorname{Gr}_G)^{H}$. ### {#section-42} Write $I$ for the Iwahori subgroup. Let ${\mathcal F}l=G(F)/I$ be the affine flags. Write ${\mathcal H}(G)=Shv({\mathcal F}l)^I$ for the geometric Iwahori-Hecke algebra. For $\lambda\in\Lambda$ write $j_{\lambda, !}^I, j_{\lambda, *}^I$ for the corresponding objects of ${\mathcal H}(G)$ attached to $t^\lambda$. Write $\ast^I$ for the convolution in ${\mathcal H}(G)$. More generally, for $w\in\tilde W$ we have the standard/costandard objects $j_{w, !}^I, j_{w, *}^I\in {\mathcal H}(G)$. **Lemma 1**. *i) Let $\lambda\in\Lambda^+_{M, ab}$. Then $j_{-\lambda, *}\ast j_{\lambda, !}\,{\widetilde\to}\, \delta_1$ in ${\mathcal H}_P(G)$.\ ii) Let $\lambda,\mu\in \Lambda^+_{M, ab}$. Then $j_{\lambda, !}\ast j_{\mu, !}\,{\widetilde\to}\, j_{\lambda+\mu, !}$ in ${\mathcal H}_P(G)$. More generally, for $\operatorname{oblv}: Shv(\operatorname{Gr}_G)^{I_P}\to Shv(\operatorname{Gr}_G)^I$ and $F\in Shv(\operatorname{Gr}_G)^{I_P}$ one has $$\operatorname{oblv}(j_{\lambda, !}\ast F)\,{\widetilde\to}\, j_{\lambda, !}^I \ast^I \operatorname{oblv}(F),\;\;\;\;\; \operatorname{oblv}(j_{\lambda, *}\ast F)\,{\widetilde\to}\, j_{\lambda, *}^I \ast^I \operatorname{oblv}(F)$$ in $Shv(\operatorname{Gr}_G)^I$.* *Proof.* Let $\lambda\in\Lambda^+_{M, ab}$. Then the natural map $It^{\lambda}I/I\to I_Pt^{\lambda}I_P/I_P$ is an isomorphism, both are affine spaces of dimension $\langle\lambda, 2\check{\rho}\rangle$. For the natural map $\tau: {\mathcal F}l\to {\mathcal F}l_P$ we get $\tau_!(j_{\lambda, !}^I)\,{\widetilde\to}\, j_{\lambda, !}$. Note that $\tau$ is proper. Similarly, the map $It^{-\lambda}I/I\to I_Pt^{-\lambda}I_P/I_P$ is an isomorphism, so that $\tau_!j_{-\lambda, *}^I\,{\widetilde\to}\, j_{-\lambda, *}$. ii\) We have $$\label{equality_of_sets_for_Lm2.2.5} I_Pt^{\lambda}I_P\times^{I_P} {\mathcal F}l_P\,{\widetilde\to}\, It^{\lambda}I\times^{I} {\mathcal F}l_P$$ Recall that $j^I_{\lambda, !}\ast^I j^I_{\mu, !}\,{\widetilde\to}\, j_{\lambda+\mu, !}^I$. Applying $\tau_!$ to this isomorphism, one gets the desired result. More generally, for any $F\in Shv({\mathcal F}l_P)^{I_P}$ let $\operatorname{oblv}(F)\in Shv({\mathcal F}l_P)^I$ then $$\operatorname{oblv}(j_{\lambda, !}\ast F)\,{\widetilde\to}\, j_{\lambda, !}^I \ast^I \operatorname{oblv}(F)$$ in $Shv(\operatorname{Gr}_G)^I$. i\) We have $j_{-\lambda, *}^I\ast^I j_{\lambda, !}^I\,{\widetilde\to}\, \delta_{1, {\mathcal F}l}$ in ${\mathcal H}(G)$. Applying $\tau_*$ this gives $j_{-\lambda, *}^I \ast^I j_{\lambda, !}\,{\widetilde\to}\, \delta_{1, {\mathcal F}l_P}$. Finally $j_{-\lambda, *}^I \ast^I j_{\lambda, !}\,{\widetilde\to}\, j_{-\lambda, *}\ast j_{\lambda, !}$ from ([\[equality_of_sets_for_Lm2.2.5\]](#equality_of_sets_for_Lm2.2.5){reference-type="ref" reference="equality_of_sets_for_Lm2.2.5"}) also. ◻ From this lemma we conclude that there are monoidal functors $\Lambda_{M, ab}\to {\mathcal H}_P(G)$, $$\label{mon_functor_*} \lambda\mapsto j_{\lambda, *}, \;\lambda\in\Lambda^+_{M, ab}$$ and $$\label{mon_functor_!} \lambda\mapsto j_{\lambda, !}, \;\lambda\in\Lambda^+_{M, ab}$$ ### {#section-43} Consider the forgetful functor $\operatorname{oblv}: Shv(\operatorname{Gr}_G)^{I_P}\to Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$. It has a continuous right adjoint denoted $\operatorname{Av}^{I_P/M({\mathcal O})}_*$ by Section [5.3](#Section_A2_some_invarinats){reference-type="ref" reference="Section_A2_some_invarinats"}. Define now $Shv(\operatorname{Gr}_G)^{M({\mathcal O}), ren}$ as follows. Let $Shv(\operatorname{Gr}_G)^{M({\mathcal O}), constr}\subset Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ be the full subcategory of those objects which remain compact in $Shv(\operatorname{Gr}_G)$. Set $$Shv(\operatorname{Gr}_G)^{M({\mathcal O}), ren}=\operatorname{Ind}(Shv(\operatorname{Gr}_G)^{M({\mathcal O}), constr})$$ We have the evident forgetful functor $$\operatorname{oblv}: Shv(\operatorname{Gr}_G)^{I_P, constr}\to Shv(\operatorname{Gr}_G)^{M({\mathcal O}), constr}$$ By construction of $\operatorname{Av}^{I_P/M({\mathcal O})}_*$, we actually get an adjoint pair $$\operatorname{oblv}: Shv(\operatorname{Gr}_G)^{I_P, constr}\leftrightarrows Shv(\operatorname{Gr}_G)^{M({\mathcal O}), constr}: \operatorname{Av}^{I_P/M({\mathcal O})}_*$$ Their ind-extensions also give an adjoint pair $$\operatorname{oblv}^{ren}: Shv(\operatorname{Gr}_G)^{I_P, ren}\leftrightarrows Shv(\operatorname{Gr}_G)^{M({\mathcal O}), ren}: \operatorname{Av}^{I_P/M({\mathcal O}), ren}_*$$ This is a general phenomenon, see Remark [Remark 1](#Rem_A.2.2){reference-type="ref" reference="Rem_A.2.2"}. **Proposition 1**. *The functor $\operatorname{Av}^{I_P/M({\mathcal O})}_*$ restricted to $Shv(\operatorname{Gr}_G)^H\subset Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ defines an equivalence $$\label{equiv_Iwahori_vs_SI} Shv(\operatorname{Gr}_G)^H\to Shv(\operatorname{Gr}_G)^{I_P}$$* ### {#section-44} Now we define the renormalized version $Shv(\operatorname{Gr}_G)^{H, ren}$ as follows. Denote by $$Shv(\operatorname{Gr}_G)^{H, constr}\subset Shv(\operatorname{Gr}_G)^H$$ the full subcategory that corresponds under ([\[equiv_Iwahori_vs_SI\]](#equiv_Iwahori_vs_SI){reference-type="ref" reference="equiv_Iwahori_vs_SI"}) to $Shv(\operatorname{Gr}_G)^{I_P, constr}\subset Shv(\operatorname{Gr}_G)^{I_P}$. Set $Shv(\operatorname{Gr}_G)^{H, ren}=\operatorname{Ind}(Shv(\operatorname{Gr}_G)^{H, constr})$. ### Proof of Proposition [Proposition 1](#Pp_2.2.6){reference-type="ref" reference="Pp_2.2.6"} {#proof-of-proposition-pp_2.2.6} The fully faithful inclusion $Shv(\operatorname{Gr}_G)^{H}\subset Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ admits a left adjoint $\operatorname{Av}_!^{U(P)(F)}: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_G)^{H}$. So, we get adjoint pairs $$Shv(\operatorname{Gr}_G)^{I_P} \leftrightarrows Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\leftrightarrows Shv(\operatorname{Gr}_G)^H$$ where the left composition is $\operatorname{Av}_!^{U(P)(F)}\operatorname{oblv}$, and the right composition is ([\[equiv_Iwahori_vs_SI\]](#equiv_Iwahori_vs_SI){reference-type="ref" reference="equiv_Iwahori_vs_SI"}). **Step 1** We equip $\Lambda^+_{M, ab}$ with the relation $\le$ as in Section [2.2.10](#Sect_2.0.11_functor_f){reference-type="ref" reference="Sect_2.0.11_functor_f"}. For $\lambda\in\Lambda^+_{M, ab}$ set $U_{\lambda}=t^{-\lambda}U(P)({\mathcal O})t^{\lambda}$. This is a placid group scheme. Given $\lambda,\mu\in \Lambda^+_{M, ab}$ with $\lambda\le\mu$ we get a placid closed immersion $U_{\lambda}\subset U_{\mu}$, and $$\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} U_{\lambda}\,{\widetilde\to}\, U(P)(F)$$ is a placid ind-scheme. If $\lambda\in \Lambda^+_{M, ab}$ then $U_{\lambda}G({\mathcal O})/G({\mathcal O})=U_{\lambda}/U_0$ is an affine space of dimension $\langle\lambda, 2\check{\rho}-2\check{\rho}_M\rangle=\langle\lambda, 2\check{\rho}\rangle$. For $\lambda\in\Lambda_{M, ab}$ set $$I_P^{\lambda}=t^{-\lambda}I_Pt^{\lambda}$$ A version of the Iwahori decomposition for $P$ is $$I_P=U(P^-)({\mathcal O})_1M({\mathcal O})U(P)({\mathcal O})$$ with $U(P^-)({\mathcal O})_1=\operatorname{Ker}(U(P^-)({\mathcal O})\to U(P^-))$. For $\lambda\in\Lambda^+_{M, ab}$, $t^{-\lambda}U(P^-)_1({\mathcal O})t^{\lambda}\subset U(P^-)({\mathcal O})_1$, so $$I_P^{\lambda}\subset U_{\lambda}M({\mathcal O})U(P^-)({\mathcal O})_1$$ and $$I^{\lambda}_P I_P/I_P=U_{\lambda}I_P/I_P=U_{\lambda}/U_0$$ Consider the action map $a: I^{\lambda}_PI_P\times^{I_P} \operatorname{Gr}_G\to \operatorname{Gr}_G$. For $F\in Shv(\operatorname{Gr}_G)^{I_P}$ the object $t^{-\lambda}j_{\lambda, !}\ast F[\langle\lambda, 2\check{\rho}\rangle]$ writes as $$a_!(\operatorname{IC}\,\tilde\boxtimes\,F)[\langle\lambda, 2\check{\rho}\rangle],$$ where the functor of the twisted exteriour product $\,\tilde\boxtimes\,$ is normalized to preserve perversity, and $\operatorname{IC}=e[\langle\lambda, 2\check{\rho}\rangle]$ is the $\operatorname{IC}$-sheaf of the affine space $I^{\lambda}_P I_P/I_P$. We see that the composition $$Shv(\operatorname{Gr}_G)^{I_P}\,\stackrel{\operatorname{oblv}}{\to}\, Shv(\operatorname{Gr}_G)^{P({\mathcal O})}\stackrel{\operatorname{Av}^{U_{\lambda}}_!}{\to} Shv(\operatorname{Gr}_G)^{M({\mathcal O})U_{\lambda}}$$ identifies with the functor $F\mapsto t^{-\lambda}j_{\lambda, !}\ast F[\langle\lambda, 2\check{\rho}\rangle]$. The meaning of the shift should be clarified later when considering the t-structure on $Shv(\operatorname{Gr}_G)^H$. **Step 2** Consider the left adjoint $\operatorname{Av}_!^{U(P)(F)}: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_G)^H$ to the inclusion. Recall that it is given as $$\label{functor_Av_!_for_Pp2.2.7} F\mapsto \mathop{\operatorname{colim}}_{\lambda\in\Lambda^+_{M, ab}} \operatorname{Av}^{U_{\lambda}}_!(F)$$ as in Lemma [Lemma 1](#Lm_A.2.3){reference-type="ref" reference="Lm_A.2.3"}. For $K\in Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ and $\lambda\in\Lambda_{M, ab}$ one has canonically $$\label{iso_for_Step2_Av!_versus_t_lambda} t^{-\lambda}\operatorname{Av}_!^{U(P)(F)}(t^{\lambda}K)\,{\widetilde\to}\, \operatorname{Av}_!^{U(P)(F)}(K)$$ Indeed, for $L\in Shv(\operatorname{Gr}_G)^H$ $$\begin{gathered} {{\mathcal H}om}_{Shv(\operatorname{Gr}_G)^{M({\mathcal O})}}(t^{-\lambda}\operatorname{Av}_!^{U(P)(F)}(t^{\lambda}K), L)\,{\widetilde\to}\,{{\mathcal H}om}_{Shv(\operatorname{Gr}_G)^{M({\mathcal O})}}(\operatorname{Av}_!^{U(P)(F)}(t^{\lambda}K), t^{\lambda}L)\\ {\widetilde\to}\, {{\mathcal H}om}_{Shv(\operatorname{Gr}_G)^{M({\mathcal O})}}(t^{\lambda}K, t^{\lambda}L)\,{\widetilde\to}\, {{\mathcal H}om}_{Shv(\operatorname{Gr}_G)^{M({\mathcal O})}}(K, L)\end{gathered}$$ Now from Step 1 we see that for $\lambda\in\Lambda^+_{M, ab}$ and ${\mathcal F}\in Shv(\operatorname{Gr}_G)^{I_P}$ one has $$\label{iso_Av_!_intertwines} t^{\lambda}\operatorname{Av}_!^{U(P)(F)}({\mathcal F})[-\langle\lambda, 2\check{\rho}\rangle]\,{\widetilde\to}\, \operatorname{Av}_!^{U(P)(F)}(j_{\lambda, !}\ast {\mathcal F})$$ **Step 3** Let us show that the unit of the adjunction $${\mathcal F}\to \operatorname{Av}_*^{I_P/M({\mathcal O})}\operatorname{Av}_!^{U(P)(F)}({\mathcal F})$$ is an isomorphism for ${\mathcal F}\in Shv(\operatorname{Gr}_G)^{I_P}$. First, for $\lambda\in\Lambda^+_{M, ab}$ and ${\mathcal F}'\in Shv(\operatorname{Gr}_G)^{I_P}$ we have $t^{-\lambda}{\mathcal F}'\in Shv(\operatorname{Gr}_G)^{I_P^{\lambda}}$ naturally. Now the composition $$Shv(\operatorname{Gr}_G)^{I_P^{\lambda}}\stackrel{\operatorname{oblv}}{\to} Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\,\stackrel{\operatorname{Av}_*^{I_P/M({\mathcal O})}}{\to}Shv(\operatorname{Gr}_G)^{I_P}$$ identifies with the composition $$Shv(\operatorname{Gr}_G)^{I_P^{\lambda}}\stackrel{\operatorname{oblv}}{\to} Shv(\operatorname{Gr}_G)^{I_P^{\lambda}\cap I_P} \,\stackrel{\operatorname{Av}_*^{I_P/I_P^{\lambda}\cap I_P}}{\to}\; Shv(\operatorname{Gr}_G)^{I_P},$$ because for for a prounipotent group the inclusion of invariants is fully faithful. The latter functor writes as $K\mapsto \operatorname{act}_*(\operatorname{IC}\,\tilde\boxtimes\,K)$ for the action map $\operatorname{act}: I_PI^{\lambda}_P\times^{I^{\lambda}_P} \operatorname{Gr}_G\to \operatorname{Gr}_G$. This gives $$\operatorname{act}_*(\operatorname{IC}\,\tilde\boxtimes\,t^{-\lambda}{\mathcal F})\,{\widetilde\to}\, j_{-\lambda, *}\ast {\mathcal F}[-\langle\lambda, 2\check{\rho}\rangle],$$ because $I_P I_P^{\lambda}=I_Pt^{-\lambda}I_Pt^{\lambda}$. So, for ${\mathcal F}\in Shv(\operatorname{Gr}_G)^{I_P}$ one gets canonically $$\label{iso_for_Step3_first} \operatorname{Av}_*^{I_P/M({\mathcal O})}(t^{-\lambda}{\mathcal F})\,{\widetilde\to}\, j_{-\lambda, *}\ast {\mathcal F}[-\langle\lambda, 2\check{\rho}\rangle]$$ Thus, for $\lambda\in\Lambda^+_{M, ab}$ we get $$\operatorname{Av}_*^{I_P/M({\mathcal O})}\operatorname{Av}_!^{U_{\lambda}}({\mathcal F})\,{\widetilde\to}\, j_{-\lambda, *}\ast j_{\lambda, !}\ast {\mathcal F}\,{\widetilde\to}\, {\mathcal F},$$ where the last isomorphism is given by Lemma [Lemma 1](#Lm_2.2.5){reference-type="ref" reference="Lm_2.2.5"}. This gives finally $$\operatorname{Av}_*^{I_P/M({\mathcal O})}\operatorname{Av}_!^{U(P)(F)}({\mathcal F})\,{\widetilde\to}\, \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} \operatorname{Av}_*^{I_P/M({\mathcal O})}\operatorname{Av}_!^{U_{\lambda}}({\mathcal F})\,{\widetilde\to}\, \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}}{\mathcal F}\,{\widetilde\to}{\mathcal F},$$ because $\Lambda^+_{M, ab}$ is filtered. **Step 4** It suffices now to show that $\operatorname{Av}_*^{I_P/M({\mathcal O})}: Shv(\operatorname{Gr}_G)^H\to Shv(\operatorname{Gr}_G)^{I_P}$ is conservative. Let $0\ne {\mathcal F}\in Shv(\operatorname{Gr}_G)^H$. By Section [5.3.2](#Sect_A.2.4){reference-type="ref" reference="Sect_A.2.4"}, there is ${\mathcal F}'\in Shv(\operatorname{Gr}_G)^{K_n}$ for some congruence subgroup $K_n\subset G({\mathcal O})$, $n>0$ such that ${{\mathcal H}om}_{Shv(\operatorname{Gr}_G)}({\mathcal F}', {\mathcal F})\ne 0$, here ${{\mathcal H}om}_{Shv(\operatorname{Gr}_G)}\in\operatorname{Vect}$ denotes the inner hom for the $\operatorname{Vect}$-action on $Shv(\operatorname{Gr}_G)$. By assumption, ${\mathcal F}'$ is equivariant with respect to $t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}$ for $\lambda\in\Lambda^+_{M, ab}$ large enough, so $\operatorname{Av}_*^{t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}}({\mathcal F})\ne 0$. Here $\operatorname{Av}_*^{t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}}: Shv(\operatorname{Gr}_G)\to Shv(\operatorname{Gr}_G)^{t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}}$ is the right adjoint to the inclusion. Now viewing $\operatorname{Av}_*^{t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}}: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_G)^{t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}M({\mathcal O})}$ as the right adjoint to the inclusion, the above also gives $\operatorname{Av}_*^{t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}}({\mathcal F})\ne 0$ in $Shv(\operatorname{Gr}_G)^{t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}M({\mathcal O})}$. Since $I_P=U(P^-)({\mathcal O})_1M({\mathcal O})U(P)({\mathcal O})$, we get $$\operatorname{Av}_*^{t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}}({\mathcal F})\in Shv(\operatorname{Gr}_G)^{I_P^{\lambda}}$$ for $\lambda\in\Lambda^+_{M, ab}$ large enough. For any ${\mathcal F}''\in Shv(\operatorname{Gr}_G)^{I_P^{\lambda}}$ we have $$t^{\lambda}{\mathcal F}''\in Shv(\operatorname{Gr}_G)^{I_P}$$ naturally and $$\operatorname{Av}_*^{U(P^-)({\mathcal O})_1}({\mathcal F}'')\,{\widetilde\to}\, j_{-\lambda, *}\ast (t^{\lambda}{\mathcal F}'')[-\langle\lambda, 2\check{\rho}\rangle]$$ by ([\[iso_for_Step3_first\]](#iso_for_Step3_first){reference-type="ref" reference="iso_for_Step3_first"}). Finally, for ${\mathcal F}$ as above letting ${\mathcal F}''=\operatorname{Av}_*^{t^{-\lambda}U(P^-)({\mathcal O})_1t^{\lambda}}({\mathcal F})$ we get $$\operatorname{Av}_*^{I_P/M({\mathcal O})}({\mathcal F})\,{\widetilde\to}\, \operatorname{Av}_*^{U(P^-)({\mathcal O})_1}({\mathcal F})\,{\widetilde\to}\, \operatorname{Av}_*^{U(P^-)({\mathcal O})_1} ({\mathcal F}'')\,{\widetilde\to}\, j_{-\lambda, *}\ast (t^{\lambda}{\mathcal F}'')[-\langle\lambda, 2\check{\rho}\rangle]$$ Applying again Lemma [Lemma 1](#Lm_2.2.5){reference-type="ref" reference="Lm_2.2.5"}, we see that the latter object is nonzero. Proposition [Proposition 1](#Pp_2.2.6){reference-type="ref" reference="Pp_2.2.6"} is proved. $\square$ ### Actions of $\Lambda_{M, ab}$ {#Sect_2.2.10_action_of_Lambda_Mab} For ${\mathcal F}\in Shv(\operatorname{Gr}_G)$, $\lambda\in\Lambda$ we denote by $t^{\lambda}{\mathcal F}$ the direct image of ${\mathcal F}$ under the multiplication $\operatorname{Gr}_G\to\operatorname{Gr}_G$ by $t^{\lambda}$. Consider $\operatorname{oblv}: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_G)$. We think of $Shv(\operatorname{Gr}_G)^H$ as a full subcategory of $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$. There is an action of $\Lambda_{M, ab}$ on $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ such that $\lambda\in\Lambda_{M, ab}$ sends $K$ to $t^{\lambda}K[-\langle\lambda, 2\check{\rho}\rangle]$. This means by definition that for $\operatorname{oblv}: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_G)$ one has canonically $$\operatorname{oblv}(t^{\lambda}K)\,{\widetilde\to}\, t^{\lambda}(\operatorname{oblv}(K))$$ This action preserves the full subcategory $Shv(\operatorname{Gr}_G)^{H}$. Consider the $\Lambda_{M, ab}$-action on $Shv(\operatorname{Gr}_G)^{I_P}$ given by restricting the action of ${\mathcal H}_P(G)$ via the monoidal functor ([\[mon_functor\_!\]](#mon_functor_!){reference-type="ref" reference="mon_functor_!"}). Proposition [Proposition 1](#Pp_2.2.6){reference-type="ref" reference="Pp_2.2.6"} also shows that the equivalence $\operatorname{Av}^{U(P)(F)}_!: Shv(\operatorname{Gr}_G)^{I_P}\,{\widetilde\to}\, Shv(\operatorname{Gr}_G)^H$ intertwines these two actions of $\Lambda_{M, ab}$. Namely, for $\lambda\in\Lambda_{M, ab}^+$, ${\mathcal F}\in Shv(\operatorname{Gr}_G)^{I_P}$ one has the isomorphism ([\[iso_Av\_!\_intertwines\]](#iso_Av_!_intertwines){reference-type="ref" reference="iso_Av_!_intertwines"}). ### {#Sect_3.1.11_action_of} The equivalence $\operatorname{Av}^{U(P)(F)}_!: Shv(\operatorname{Gr}_G)^{I_P}\,{\widetilde\to}\, Shv(\operatorname{Gr}_G)^H$ commutes with the actions of ${\operatorname{Rep}}(\check{G})$ on both sides. Indeed, this can be seen for example from ([\[functor_Av\_!\_for_Pp2.2.7\]](#functor_Av_!_for_Pp2.2.7){reference-type="ref" reference="functor_Av_!_for_Pp2.2.7"}). We equip $Shv(\operatorname{Gr}_G)^{I_P}$, $Shv(\operatorname{Gr}_G)^{I_P, ren}$ with t-structures as in Section [5.5](#Sect_A.3){reference-type="ref" reference="Sect_A.3"}. So, we have the t-exact oblivion functor $\operatorname{oblv}[\operatorname{dim.rel}]: Shv(\operatorname{Gr}_G)^{I_P}\to Shv(\operatorname{Gr}_G)$. The action of ${\operatorname{Rep}}(\check{G})^c$ on $Shv(\operatorname{Gr}_G)^{I_P}$ preserves the full subcategory $Shv(\operatorname{Gr}_G)^{I_P, constr}$, and the obtained action on $Shv(\operatorname{Gr}_G)^{I_P, constr}$ is t-exact in each variable by ([@G_central], Proposition 6). Passing to the ind-completion this yields a ${\operatorname{Rep}}(\check{G})$-action on $Shv(\operatorname{Gr}_G)^{I_P, ren}$ which is moreover t-exact in each variable. The equivalence of Proposition [Proposition 1](#Pp_2.2.6){reference-type="ref" reference="Pp_2.2.6"} yields an equivalence $$Shv(\operatorname{Gr}_G)^{H, constr}\,{\widetilde\to}\, Shv(\operatorname{Gr}_G)^{I_P, constr}$$ which commutes with the actions of ${\operatorname{Rep}}(G)^c$. Passing to the ind-completion, this gives an equivalence $$\label{eq_ren_parahoric_versus_H} \operatorname{Av}_*^{I_P/M({\mathcal O}), ren}: Shv(\operatorname{Gr}_G)^{H, ren}\,{\widetilde\to}\, Shv(\operatorname{Gr}_G)^{I_P, ren}$$ We equip $Shv(\operatorname{Gr}_G)^{H, ren}$ with ${\operatorname{Rep}}(\check{G})$-action coming from the ind-completion of the ${\operatorname{Rep}}(\check{G})^c$-action on $Shv(\operatorname{Gr}_G)^{H, constr}$. ## Relation between local and global: geometry {#Sect_Relation between local and global} ### {#Sect_2.3.1} Let $\Lambda_{G,P}$ be the lattice of cocharacters of $M/[M,M]$, so $\Lambda_{G,P}$ is the quotient of $\Lambda$ by the span of $\alpha_i, i\in{\mathcal I}_M$. Let $\check{\Lambda}_{G,P}$ denote the dual lattice. Let $\check{\Lambda}^+$ be the dominant weights for $G$. Write $\Lambda_{G,P}^{pos}$ for the ${\mathbb Z}_+$-span of $\alpha_i$, $i\in {\mathcal I}-{\mathcal I}_M$ in $\Lambda_{G,P}$. For $\theta\in\Lambda_{G,P}$ denote by $\operatorname{Gr}_M^{\theta}$ the connected component of the affine Grassmanian $\operatorname{Gr}_M$ containing $t^{\lambda}M({\mathcal O})$ for any $\lambda\in\Lambda$ over $\theta$. As in ([@BG], 4.3.1) for $\theta\in\Lambda_{G,P}$ denote by $\overline{\operatorname{Gr}}_P^{\theta}\subset \operatorname{Gr}_G$ the closed ind-subscheme given by the property that for $\check{\lambda}\in\check{\Lambda}_{G,P}\cap \check{\Lambda}^+_G$ the map $${\mathcal L}^{\check{\lambda}}_{{\mathcal F}^0_{M/[M,M]}}(-\langle\theta, \check{\lambda}\rangle)\to {\mathcal V}^{\check{\lambda}}_{{\mathcal F}_G}$$ is regular on the disk $D$. Let $\operatorname{Gr}^{\theta}_P\subset \overline{\operatorname{Gr}}^{\theta}_P$ be the open subscheme where the above maps have no zeros on $D$. For $\theta,\theta'\in\Lambda_{G,P}$ one has $\operatorname{Gr}_P^{\theta'}\subset \overline{\operatorname{Gr}}^{\theta}_P$ iff $\theta-\theta'\in\Lambda_{G,P}^{pos}$. Consider the natural map $\mathfrak{t}^{\theta}_P: \operatorname{Gr}^{\theta}_P\to \operatorname{Gr}_M^{\theta}$ defined in ([@BG], Pp. 4.3.2). For $\mu\in\Lambda^+_M$ write $\operatorname{Gr}_M^{\mu}$ for the $M({\mathcal O})$-orbit on $\operatorname{Gr}_M$ through $t^{\mu}$. For $\mu\in\Lambda^+_M$ over $\theta\in\Lambda_{G,P}$ write $S^{\mu}_P$ for the preimage of $\operatorname{Gr}_M^{\mu}$ under $\mathfrak{t}^{\theta}_P$. So, $\{S^{\mu}_P\}_{\mu\in\Lambda^+_M}$ are the $H$-orbits on $\operatorname{Gr}_G$. The restriction of $\mathfrak{t}^{\theta}_P$ is denoted $$\mathfrak{t}^{\mu}_P: S^{\mu}_P\to \operatorname{Gr}_M^{\mu}$$ For $\theta\in\Lambda_{G,P}$ let $i^{\theta}_P: \operatorname{Gr}_M^{\theta}\to \operatorname{Gr}^{\theta}_P$ be the natural map, so that $\mathfrak{t}^{\theta}_P i^{\theta}_P=\operatorname{id}$. Write $v^{\theta}_P: \operatorname{Gr}^{\theta}_P\to \operatorname{Gr}_G$ for the natural inclusion. For $\mu\in\Lambda^+_M$ write $\bar S^{\mu}_P$ for the closure of $S^{\mu}_P$ in $\operatorname{Gr}_G$. For $\theta\in\Lambda_{G,P}$ let $\operatorname{Gr}_{P^-}^{\theta}\subset \overline{\operatorname{Gr}}_P^{\theta}\subset \operatorname{Gr}_G$ be the analogs of the corresponding ind-schemes with $P$ replaced by $P^-$. The corresponding morphisms are denoted $\mathfrak{t}^{\theta}_{P^-}: \operatorname{Gr}_{P^-}^{\theta}\to \operatorname{Gr}_M^{\theta}$ and $$\operatorname{Gr}_M^{\theta}\stackrel{i^{\theta}_{P^-}}{\to} \operatorname{Gr}_{P^-}^{\theta}\stackrel{v^{\theta}_{P^-}}{\to} \operatorname{Gr}_G$$ For $\mu\in\Lambda^+_M$ over $\theta\in\Lambda_{G,P}$ write $S_{P^-}^{\mu}$ for the preimage of $\operatorname{Gr}_M^{\mu}$ under $\mathfrak{t}^{\theta}_{P^-}$. Let $\mathfrak{t}^{\mu}_{P^-}: S_{P^-}^{\mu}\to \operatorname{Gr}_M^{\mu}$ denote the restriction of $\mathfrak{t}^{\theta}_{P^-}$. Recall the following consequence of a theorem of Braden ([@DG1], [@Bra]). **Lemma 1**. *Let $\theta\in\Lambda_{G,P}$.\ a) For $K\in Shv(\operatorname{Gr}_G)^T$ one has canonically $$(i^{\theta}_P)^!(v^{\theta}_P)^*K\,{\widetilde\to}\, (i^{\theta}_{P^-})^*(v^{\theta}_{P^-})^!K$$ b) For $K\in Shv(\operatorname{Gr}_P^{\theta})^T$ one has canonically $(\mathfrak{t}^{\theta}_P)_!K\,{\widetilde\to}\,(i^{\theta}_P)^!K$ and $(\mathfrak{t}^{\theta}_P)_*K\,{\widetilde\to}\,(i^{\theta}_P)^*K$ in $Shv(\operatorname{Gr}_M^{\theta})^T$, and similarly for $\operatorname{Gr}_{P^-}^{\theta}$. $\square$* ### {#Sect_2.3.3_loc_vs_glob} From now on we assume $[G,G]$ is simply-connected. Let $X$ be a smooth projective connected curve. Fix a point of our curve $x\in X$. Let $_{x,\infty}\operatorname{\overline{Bun}} _P$ be the stack classifying ${\mathcal F}_{M/[M,M]}, {\mathcal F}_G$ on $X$ and a collection of maps $$\kappa^{\check{\lambda}}: {\mathcal L}^{\check{\lambda}}_{{\mathcal F}_{M/[M,M]}}\to {\mathcal V}^{\check{\lambda}}_{{\mathcal F}_G}(\infty x), \check{\lambda}\in\check{\Lambda}^+\cap \check{\Lambda}_{G,P}$$ satisfying the Plücker relations. Pick a uniformizer $t_x\in {\mathcal O}_x$, hence an isomorphism ${\mathcal O}\,{\widetilde\to}\, {\mathcal O}_x$. This allows to view $\operatorname{Gr}_G$ as the ind-scheme classifying $({\mathcal F}_G, \beta)$, where ${\mathcal F}_G$ is a $G$-torsor on $X$, $\beta: {\mathcal F}_G\,{\widetilde\to}\,{\mathcal F}^0_G$ is trivialization over $X-x$. We get the morphism $\pi: \operatorname{Gr}_G\to {_{x,\infty}\operatorname{\overline{Bun}} _P}$ sending $({\mathcal F}_G, \beta)$ to $({\mathcal F}^0_{M/[M,M]}, {\mathcal F}_G, \kappa)$, where $\kappa$ is induced by the $P$-structure on the trivial $P$-torsor. The preimage $\pi^{-1}\operatorname{\overline{Bun}} _P$ identifies with $\overline{\operatorname{Gr}}^0_P$. We let ${\operatorname{Rep}}(\check{G})$ act on $_{x,\infty}\operatorname{\overline{Bun}} _P$ so that $V\in {\operatorname{Rep}}(\check{G})$ acts as $$_x{\operatorname{H}}^{\rightarrow}_G(\operatorname{Sat}(V), \cdot): Shv(_{x,\infty}\operatorname{\overline{Bun}} _P)\to Shv(_{x,\infty}\operatorname{\overline{Bun}} _P)$$ in the notations of ([@BG], 3.2.4). Since we are in the constructible context, we have the adjoint pair in $\operatorname{DGCat}_{cont}$ $$\pi_!: Shv(\operatorname{Gr}_G)\leftrightarrows Shv(_{x,\infty}\operatorname{\overline{Bun}} _P): \pi^!$$ By $(*, !)$-base change, both these functors commute with ${\operatorname{Rep}}(\check{G})$-actions. ### {#section-45} For $\theta\in\Lambda_{G,P}$ let $_{\le\theta, x}\operatorname{\overline{Bun}} _P\subset {_{x,\infty}\operatorname{\overline{Bun}} _P}$ be the closed substack given by the property that for any $\check{\lambda}\in\check{\Lambda}_{G,P}\cap \check{\Lambda}^+$ the map $$\label{map_for_x_infty_Bunb_P_Sect_2.3.3} {\mathcal L}^{\check{\lambda}}_{{\mathcal F}_{M/[M,M]}}(-\langle\theta, \check{\lambda}\rangle x)\to {\mathcal V}^{\check{\lambda}}_{{\mathcal F}_G}$$ is regular on $X$. Let also $_{=\theta, x}\operatorname{\overline{Bun}} _P\subset {_{\le\theta, x}\operatorname{\overline{Bun}} _P}$ be the open substack given by the property that ([\[map_for_x\_infty_Bunb_P\_Sect_2.3.3\]](#map_for_x_infty_Bunb_P_Sect_2.3.3){reference-type="ref" reference="map_for_x_infty_Bunb_P_Sect_2.3.3"}) have no zeros everywhere on $X$. Note that $$\pi^{-1}(_{\le\theta, x}\operatorname{\overline{Bun}} _P)=\overline{\operatorname{Gr}}^{\theta}_P\;\;\; \mbox{and}\;\;\; \pi^{-1}(_{=\theta, x}\operatorname{\overline{Bun}} _P)=\operatorname{Gr}^{\theta}_P$$ If $\theta,\theta'\in\Lambda_{G,P}$ with $\theta-\theta'\in\Lambda_{G,P}^{pos}$ then $\overline{\operatorname{Gr}}^{\theta'}_P\subset \overline{\operatorname{Gr}}^{\theta}_P$. ### {#section-46} For $\lambda\in\Lambda$ write $\operatorname{Bun}_T^{\lambda}$ for the connected component of $\operatorname{Bun}_T$ classifying ${\mathcal F}_T\in\operatorname{Bun}_T$ such that for any $\check{\lambda}\in\check{\Lambda}$, $\deg{\mathcal L}^{\check{\lambda}}_{{\mathcal F}_T}=-\langle\lambda, \check{\lambda}\rangle$. Similarly, for $\theta\in\Lambda_{G,P}$ let $\operatorname{Bun}_M^{\theta}$ be the preimage of $\operatorname{Bun}_{M/[M,M]}^{\theta}$, this normalization agrees with [@BG]. For $\theta'\in\Lambda_{G,P}$ let $\operatorname{Bun}_P^{\theta'}, \operatorname{\overline{Bun}} _P^{\theta'}$ and so on be the preimage of the component $\operatorname{Bun}_M^{\theta'}$. We have $$\dim\operatorname{Bun}_P^{\theta}=(g-1)\dim P+\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle=\dim(_{=\theta, x}\operatorname{\overline{Bun}} _P^0)$$ This explains the shift in the definition of the t-structure on $Shv(\operatorname{Gr}^{\theta}_P)^H$ in Section [3.3.5](#Sect_t-str_on_S_theta_P){reference-type="ref" reference="Sect_t-str_on_S_theta_P"}. Let $\Lambda_{G,P}$ act on $_{x,\infty}\operatorname{\overline{Bun}} _P$ so that $\theta\in\Lambda_{G,P}$ acts as $$({\mathcal F}_{M/[M,M]}, {\mathcal F}_G,\kappa)\mapsto ({\mathcal F}_{M/[M,M]}(\theta x), {\mathcal F}_G, \kappa)$$ Let now $\Lambda_{M, ab}$ act on $_{x,\infty}\operatorname{\overline{Bun}} _P$ via the inclusion $\Lambda_{M, ab}\stackrel{}{\hookrightarrow} \Lambda_{G,P}$. Then $\pi: \operatorname{Gr}_G\to {_{x,\infty}\operatorname{\overline{Bun}} _P}$ is $\Lambda_{M,ab}$-equivariant, where $\lambda\in\Lambda_{M, ab}$ acts on $\operatorname{Gr}_G$ as $t^{\lambda}$. ### {#Sect_2.3.6_positive part_Gr_M^+} Set $\Lambda_{M,G}^+=\Lambda_M^+\cap w_0^M(\Lambda^{pos})$. We define the positive part of the affine Grassmanian $\operatorname{Gr}_M^+\subset\operatorname{Gr}_M$ as the subscheme of $({\mathcal F}_M, \beta)\in\operatorname{Gr}_M$, where ${\mathcal F}_M$ is a $M$-torsor on the disk $D$, and $\beta: {\mathcal F}_M\,{\widetilde\to}{\mathcal F}^0_M\mid_{D^*}$ such that for any $V\in{\operatorname{Rep}}(G)^{\heartsuit}$ finite-dimensional, the natural map $$V^{U(P)}_{{\mathcal F}_M}\stackrel{\beta}{\to} V^{U(P)}_{{\mathcal F}^0_M}$$ is regular over $D$. Recall that for $\nu\in\Lambda^+_M$ we have $\operatorname{Gr}_M^{\nu}\subset \operatorname{Gr}_M^+$ iff $\nu\in \Lambda_{M,G}^+$ by ([@BG], Proposition 6.2.3). For $\theta\in\Lambda_{G,P}$ we set $\operatorname{Gr}_M^{+,\theta}=\operatorname{Gr}_M^{\theta}\cap \operatorname{Gr}_M^+$. ### {#Sect_2.3.7_local_vs_global} Let $\operatorname{\widetilde\operatorname{Bun}}_P$ and $_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P$ be defined as in ([@BG], 4.1.1). As in *loc.cit*. for $\nu\in\Lambda^+_M$ define the closed substack $_{x,\ge\nu}\operatorname{\widetilde\operatorname{Bun}}_P\subset {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}$ by requiring that for any finite-dimensional $G$-module ${\mathcal V}$ whose weights are $\le \check{\lambda}$, the map $$\label{map_for_2.3.5} {\mathcal V}^{U(P)}_{{\mathcal F}_M}\to {\mathcal V}_{{\mathcal F}_G}(-\langle w_0^M(\nu), \check{\lambda}\rangle x)$$ is regular on $X$. In particular, $\operatorname{\widetilde\operatorname{Bun}}_P={_{x,\ge 0}\operatorname{\widetilde\operatorname{Bun}}_P}$. Let $_{x,\nu}\operatorname{\widetilde\operatorname{Bun}}_P\subset {_{x,\ge\nu}\operatorname{\widetilde\operatorname{Bun}}_P}$ be the open substack defined as in ([@BG], 4.2.2). In fact, it classifies $({\mathcal F}_M, {\mathcal F}_G, \kappa)\in {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}$ such that there is a modification ${\mathcal F}_M\,{\widetilde\to}\, {\mathcal F}'_M\mid_{X-x}$ of $M$-torsors at $x$ for which ${\mathcal F}'_M$ defines a true $P$-structure on ${\mathcal F}_G$ in a neighbourhood of $x$, and such that ${\mathcal F}_M$ is in the position $\nu$ with respect to ${\mathcal F}'_M$ at $\nu$. Recall that by ([@BG], 4.2.3) the stacks $_{x,\nu'}\operatorname{\widetilde\operatorname{Bun}}_P$ for $\nu'\in\Lambda_M^+$ with $w_0^M(\nu'-\nu)\in\Lambda^{pos}$ form a stratification of $_{x,\ge \nu}\operatorname{\widetilde\operatorname{Bun}}_P$. For $\theta\in\Lambda_{G,P}$ denote by $\operatorname{\widetilde\operatorname{Bun}}_P^{\theta}$ be the preimage of $\operatorname{Bun}_M^{\theta}$ under $\operatorname{\widetilde\operatorname{Bun}}_P\to\operatorname{Bun}_M$. ### {#Sect_2.3.8_local_vs_global} Define the morphism $\tilde\pi: \operatorname{Gr}_G\to {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}$ sending $({\mathcal F}_G,\beta)$ to $({\mathcal F}_M^0, {\mathcal F}_G, \kappa)$. The composition $$\operatorname{Gr}_G\stackrel{\tilde\pi}{\to}{_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}\stackrel{\mathfrak{r}}{\to}{_{x,\infty}\operatorname{\overline{Bun}} _P}$$ equals $\pi$. Here $r$ is the map sending $({\mathcal F}_M, {\mathcal F}_G,\kappa)$ to $({\mathcal F}_{M/[M,M]},{\mathcal F}_G, \kappa)$ with ${\mathcal F}_{M/[M,M]}$ induced from ${\mathcal F}_M$. If $\nu\in\Lambda_M^+$ then $\tilde\pi^{-1}(_{x,-w_0^M(\nu)}\operatorname{\widetilde\operatorname{Bun}}_P)$ coincides with $S_P^{\nu}$. This gives the fact that if $\nu\in\Lambda_M^+$ then $\bar S^{\nu}_P$ is stratified by locally closed ind-schemes $S^{\mu}_P$ for $\mu\in\Lambda^+_M$ satisfying $\nu-\mu\in\Lambda^{pos}$. We let $Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ act on $Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)$, so that ${\mathcal S}\in Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ acts as $$_x{\operatorname{H}}^{\rightarrow}_{P, G}({\mathcal S}, \cdot): Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)\to Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)$$ in the notations of ([@BG], 4.1.4). Write for brevity $\_\ast {\mathcal S}={_x{\operatorname{H}}^{\rightarrow}_{P, G}({\mathcal S}, \_)}$. As above, we have the adjoint pair $$\tilde\pi_! : Shv(\operatorname{Gr}_G)\leftrightarrows Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P): \tilde\pi^!,$$ and both these functors commute with ${\operatorname{Rep}}(\check{G})$-actions. Similarly, the functors $$\mathfrak{r}_!: Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)\leftrightarrows Shv(_{x,\infty}\operatorname{\overline{Bun}} _P): \mathfrak{r}^!$$ commute with ${\operatorname{Rep}}(\check{G})$-actions at $x$. Note that $\tilde\pi^{-1}(\operatorname{\widetilde\operatorname{Bun}}_P)=\bar S^0_P$. For $\nu\in\Lambda^+_M$ write $i_{\nu, glob}: {_{x,-w_0^M(\nu)}\operatorname{\widetilde\operatorname{Bun}}_P}\to {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}$ for the natural inclusion and set $$\boldsymbol{\vartriangle}^{\nu}_{\operatorname{glob}}=(i_{\nu, glob})_!\operatorname{IC}(_{x,-w_0^M(\nu)}\operatorname{\widetilde\operatorname{Bun}}_P),\;\;\;\;\; \nabla^{\nu}_{glob}=(i_{\nu, glob})_*\operatorname{IC}(_{x,-w_0^M(\nu)}\operatorname{\widetilde\operatorname{Bun}}_P)$$ ### {#section-47} Denote by $_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_{P, x}$ the stack classifying a point $({\mathcal F}_M, {\mathcal F}_G,\kappa)\in {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}$ together with a trivialization $\epsilon: {\mathcal F}^0_M\,{\widetilde\to}\,{\mathcal F}_M\mid_{D_x}$ of ${\mathcal F}_M$ over the disk $D_x$. The group $M({\mathcal O})$ acts on $_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_{P,x}$ changing the trivialization $\epsilon$. The map $\tilde\pi$ lifts to a $M({\mathcal O})$-equivariant morphism $\operatorname{Gr}_G\to {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_{P,x}}$. We denote by $\tilde\pi^M: M({\mathcal O})\backslash\operatorname{Gr}_G\to {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}$ the morphism obtained by passing to the stack quotient by $M({\mathcal O})$. This gives the ${\operatorname{Rep}}(\check{G})$-linear functor $$\label{functor_tilde_pi^M^!} (\tilde\pi^M)^!: Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)\to Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$$ whose composition with $\operatorname{oblv}: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_G)$ is $\tilde\pi^!$. **Remark 1**. *i) Let $\nu\in\Lambda_M^+$ and $\lambda\in\Lambda^+$. If $S^{\nu}_P\cap \overline{\operatorname{Gr}}_G^{\lambda}\ne\emptyset$ then $\lambda-\nu, \nu+w_0(\lambda)\in \Lambda^{pos}$. Indeed, the map $\mathfrak{t}^{\nu}_P$ is $M({\mathcal O})$-equivariant, so $S^{\nu}\cap \overline{\operatorname{Gr}}_G^{\lambda}\ne\emptyset$, where $S^{\nu}$ is the $U(F)$-orbit through $t^{\nu}$.* *ii) For $\mu\in \Lambda_{M, ab}, \nu\in \Lambda^+_M$ one has $t^{\mu}S^{\nu}_P=S^{\nu+\mu}_P$.* *iii) Let $\mu\in\Lambda_{M, ab}$ over $\bar\mu\in\Lambda_{G,P}$, let $\theta\in\Lambda_{G,P}$. Then $t^{\mu}\operatorname{Gr}_M^{\theta}=\operatorname{Gr}_M^{\theta+\bar\mu}$ and $t^{\mu}\operatorname{Gr}^{\theta}_P=\operatorname{Gr}^{\theta+\bar\mu}_P$. The natural map $\Lambda_{M, ab}\to \Lambda_{G,P}$ is injective.* ### {#section-48} We let ${\operatorname{Rep}}(\check{M})$ act on $Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)$, so that $V\in {\operatorname{Rep}}(\check{M})$ acts as $$_x{\operatorname{H}}^{\leftarrow}_{P, M}(\operatorname{Sat}_M(V), \cdot): Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)\to Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)$$ in the notations of ([@BG], 4.1.2). Then ([\[functor_tilde_pi\^M\^!\]](#functor_tilde_pi^M^!){reference-type="ref" reference="functor_tilde_pi^M^!"}) is ${\operatorname{Rep}}(\check{M})$-linear, where $V\in{\operatorname{Rep}}(\check{M})$ acts on $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ as $K\mapsto \operatorname{Sat}_M(V)\ast K$. For ${\mathcal S}\in Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$, write for brevity ${\mathcal S}\ast \_={_x{\operatorname{H}}^{\leftarrow}_{P, M}({\mathcal S}, \_)}$. ### {#Sect_2.3.12_local_vs_global} Write $\operatorname{IC}_{\widetilde{glob}}$ for the $\operatorname{IC}$-sheaf of $\operatorname{\widetilde\operatorname{Bun}}_P$. Its Hecke property is given by ([@BG], 4.1.5). It says that for $V\in{\operatorname{Rep}}(\check{G})$ one has isomorphisms $$\operatorname{IC}_{\widetilde{glob}}\ast V\,{\widetilde\to}\, \operatorname{Res}(V)\ast \operatorname{IC}_{\widetilde{glob}}$$ in a way compatible with the monoidal structures on ${\operatorname{Rep}}(\check{G}), {\operatorname{Rep}}(\check{M})$. Here $\operatorname{Res}: {\operatorname{Rep}}(\check{G})\to {\operatorname{Rep}}(\check{M})$ is the restriction. This shows that $\operatorname{IC}_{\widetilde{glob}}$ naturally upgrades to an object of $$Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})} {\operatorname{Rep}}(\check{M})$$ For $\nu\in\Lambda^+_M$ write $\operatorname{IC}^{\nu}_{\widetilde{glob}}$ for the $\operatorname{IC}$-sheaf of $_{x,\ge -w_0^M(\nu)}\operatorname{\widetilde\operatorname{Bun}}_P$ extended by zero to $_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P$. ## Structure of $\operatorname{SI}_P$ ### {#Sect_2.3.8} For $\nu\in\Lambda_M^+$ write $j_{\nu}: S^{\nu}_P\stackrel{}{\hookrightarrow} \bar S^{\nu}_P$ for the open immersion. Let $\bar i_{\nu}: \bar S^{\nu}_P\stackrel{}{\hookrightarrow} \operatorname{Gr}_G$ be the closed immersion and $i_{\nu}=\bar i_{\nu}\circ j_{\nu}$. The ind-schemes $S^{\nu}_P$, $\bar S^{\nu}_P$ are acted on by $H$, so we also consider the categories of invariants $$Shv(\bar S^{\nu}_P)^H, \;\; Shv(S^{\nu}_P)^H$$ By restriction we get the adjoint pairs $(\bar i_{\nu})_!: Shv(\bar S^{\nu}_P)^H\leftrightarrows Shv(\operatorname{Gr}_G)^H: (\bar i_{\nu})^!$ and $$j_{\nu}^*: Shv(\bar S^{\nu}_P)^H\leftrightarrows Shv(S^{\nu}_P)^H: (j_{\nu})_*$$ with $(\bar i_{\nu})_!, (j_{\nu})_*$ fully faithful. By Lemma [Lemma 1](#Lm_A.5.3){reference-type="ref" reference="Lm_A.5.3"} and Section [5.6.3](#Sect_A.5.5){reference-type="ref" reference="Sect_A.5.5"}, we have the adjoint pair $$(j_{\nu})_!: Shv(S^{\nu}_P)^H\leftrightarrows Shv(\bar S^{\nu}_P)^H: j_{\nu}^!$$ with $(j_{\nu})_!$ fully faithful. Let $i_P^{\nu}: \operatorname{Gr}_M^{\nu}\to S^{\nu}_P$ be the closed embedding, the $M({\mathcal O})$-orbit through $t^{\nu}$, so that $\mathfrak{t}^{\nu}_Pi_P^{\nu}=\operatorname{id}$. The following is close to ([@LC2], Lemma 2.1.5). **Lemma 1**. *i) Let $\nu\in\Lambda_M^+$. The $!$-restriction under $\mathfrak{t}^{\nu}_P: S^{\nu}_P\to \operatorname{Gr}_M^{\nu}$ yields a fully faithfull embedding $Shv(\operatorname{Gr}_M^{\nu})^{M({\mathcal O})}\stackrel{}{\hookrightarrow} Shv(S^{\nu}_P)^{M({\mathcal O})}$ whose image is $Shv(S^{\nu}_P)^H$. The composition $$\label{equiv_for_Lm_2.3.8} Shv(S^{\nu}_P)^H\stackrel{}{\hookrightarrow} Shv(S^{\nu}_P)^{M({\mathcal O})}\stackrel{(i^{\nu}_P)^!}{\to} Shv(\operatorname{Gr}_M^{\nu})^{M({\mathcal O})}$$ is an equivalence.* *ii) Let $\theta\in\Lambda_{G,P}$. The functor $(\mathfrak{t}^{\theta}_P)^!: Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}\to Shv(\operatorname{Gr}^{\theta}_P)^{M({\mathcal O})}$ is fully faithful, its essential image is $Shv(\operatorname{Gr}^{\theta}_P)^H$. The composition $$\label{composition_for_Lm_2.3.8} Shv(\operatorname{Gr}^{\theta}_P)^H\stackrel{}{\hookrightarrow} Shv(\operatorname{Gr}^{\theta}_P)^{M({\mathcal O})}\stackrel{(i^{\theta}_P)^!}{\to}Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$$ is an equivalence. The natural transformation $(\mathfrak{t}_P^{\theta})_!(\mathfrak{t}_P^{\theta})^!\to \operatorname{id}$ on $Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$ is an equivalence.* *Proof.* i) We have $S^{\nu}_P\,{\widetilde\to}\,\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} H_{\lambda}t^{\nu}$. So, $$Shv(S^{\nu}_P)^H\,{\widetilde\to}\, \mathop{\lim}\limits_{\lambda\le\lambda'\in ({\operatorname{Fun}}([1], \Lambda^+_{M, ab}))^{op}} Shv(H_{\lambda'}t^{\nu})^{H_{\lambda}}$$ However, the diagonal map $\Lambda^+_{M, ab}\to {\operatorname{Fun}}([1], \Lambda^+_{M, ab})$ is cofinal, so the latter limit identifies with $$\mathop{\lim}\limits_{\lambda\in\Lambda^+_{M, ab}} Shv(H_{\lambda}t^{\nu})^{H_{\lambda}}$$ The stabilizor $St_{\nu}$ of $t^{\nu}$ in $H$ is the preimage of $St^M_{\nu}:=M({\mathcal O})\cap t^{\nu}M({\mathcal O})t^{-\nu}$ under $t^{\nu}P({\mathcal O})t^{-\nu}\to t^{\nu}M({\mathcal O})t^{-\nu}$. For for $\lambda$ large enough we have $St_{\nu}\subset H_{\lambda}$ and $$Shv(H_{\lambda}t^{\nu})^{H_{\lambda}}\,{\widetilde\to}\, Shv(B(St_{\nu}))$$ This gives an equivalence $$Shv(S^{\nu}_P)^H\,{\widetilde\to}\, Shv(B(St_{\nu}))$$ The kernel of $St_{\nu}\to St^M_{\nu}$ is prounipotent, so $$Shv(B(St_{\nu}))\,{\widetilde\to}\, Shv(B(St^M_{\nu}))\,{\widetilde\to}\, Shv(\operatorname{Gr}_M^{\nu})^{M({\mathcal O})}$$ ii\) Since $H/M({\mathcal O})$ is ind-pro-unipotent, $\operatorname{oblv}: Shv(\operatorname{Gr}^{\theta}_P)^H\to Shv(\operatorname{Gr}^{\theta}_P)^{M({\mathcal O})}$ is fully faithful. The map $\mathfrak{t}_P^{\theta}$ is $H$-equivariant, so $$\label{functor_gt_P^theta^!_for_Lm_2.4.2} (\mathfrak{t}_P^{\theta})^!: Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}\to Shv(\operatorname{Gr}^{\theta}_P)^{M({\mathcal O})}$$ takes values in $Shv(\operatorname{Gr}^{\theta}_P)^H$. The group $U(P(F))$ acts transitively on the fibres of $\operatorname{Gr}_P^{\theta}\to \operatorname{Gr}_M^{\theta}$. By Lemma [Lemma 1](#Lm_A.3.6_fully_faithful_functors){reference-type="ref" reference="Lm_A.3.6_fully_faithful_functors"}, ([\[functor_gt_P\^theta\^!\_for_Lm_2.4.2\]](#functor_gt_P^theta^!_for_Lm_2.4.2){reference-type="ref" reference="functor_gt_P^theta^!_for_Lm_2.4.2"}) is fully faithful. It remains to show that $$(\mathfrak{t}^{\theta}_P)^!: Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}\to Shv(\operatorname{Gr}^{\theta}_P)^H$$ is essentially surjective. The objects of the form $(i_{\nu})_!F$ for $\nu\in\Lambda_M^+$ over $\theta$, $F\in Shv(S^{\nu}_P)^H$, generate $Shv(\operatorname{Gr}_P^{\theta})^H$. The desired claim follows now from part i). ◻ ### {#Sect_2.3.9_object_cB} By Lemma [Lemma 1](#Lm_2.3.8){reference-type="ref" reference="Lm_2.3.8"}, for each $\nu\in\Lambda^+_M$ we have the object $\omega\in Shv(S^{\nu}_P)^H$ and $(i_{\nu})_!\omega\in Shv(\operatorname{Gr}_G)^H$. For $\lambda\in\Lambda_M^+$ set $$\boldsymbol{\vartriangle}^{\lambda}=(i_{\lambda})_!\omega[-\langle\lambda, 2\check{\rho}\rangle],\;\;\;\; \nabla^{\lambda}=(i_{\lambda})_*\omega[-\langle\lambda, 2\check{\rho}\rangle]$$ in $Shv(\operatorname{Gr}_G)^H$. For $\lambda\in\Lambda_{M, ab}$ the ind-scheme $S^{\lambda}_P$ coincides with the $U(P)(F)$-orbit of $t^{\lambda}\in\operatorname{Gr}_G$. ### {#section-49} For $\lambda\in \Lambda_{M, ab}$ let ${\mathcal W}_{\lambda}\in {\mathcal H}_P(G)$ denote the Wakimoto object defined as follows. Writing $\lambda=\lambda_1-\lambda_2$ with $\lambda_i\in\Lambda_{M, ab}^+$ we set ${\mathcal W}_{\lambda}=j_{\lambda_1, !}\ast j_{-\lambda_2, *}$. **Lemma 1**. *i) For $\lambda\in\Lambda_{M, ab}$ one has naturally $\operatorname{Av}^{U(P)(F)}_!({\mathcal W}_{\lambda}\ast \delta_{1,\operatorname{Gr}_G})\,{\widetilde\to}\, \boldsymbol{\vartriangle}^{\lambda}$ in $Shv(\operatorname{Gr}_G)^H$. Besides, we have canonically $$\operatorname{Av}_!^{U(P)(F)}(\delta_{t^{\lambda},\operatorname{Gr}_G})[-\langle\lambda, 2\check{\rho}\rangle]\,{\widetilde\to}\, \boldsymbol{\vartriangle}^{\lambda}$$ for $\operatorname{Av}_!^{U(P)(F)}: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_G)^H$.\ ii) If $\lambda\in\Lambda^+_{M, ab}$ then $${\mathcal B}_{\lambda, !}\,{\widetilde\to}\, j_{\lambda, !}\ast \delta_{1,\operatorname{Gr}_G}$$ iii) For $\lambda\in\Lambda_{M, ab}$, $\mu\in\Lambda^+_M$, one has canonically $t^{\lambda}\boldsymbol{\vartriangle}^{\mu}[-\langle\lambda, 2\check{\rho}\rangle]\,{\widetilde\to}\, \boldsymbol{\vartriangle}^{\mu+\lambda}$.* *Proof.* i) By ([\[iso_Av\_!\_intertwines\]](#iso_Av_!_intertwines){reference-type="ref" reference="iso_Av_!_intertwines"}) for any $\lambda\in\Lambda_{M, ab}$ one has canonically $$t^{\lambda}\operatorname{Av}_!^{U(P)(F)}(\delta_{1,\operatorname{Gr}_G})[-\langle\lambda, 2\check{\rho}\rangle]\,{\widetilde\to}\, \operatorname{Av}_!^{U(P)(F)}({\mathcal W}_{\lambda}\ast \delta_{1,\operatorname{Gr}_G})$$ One has $t^{\lambda}\boldsymbol{\vartriangle}^0[-\langle\lambda, 2\check{\rho}\rangle]\,{\widetilde\to}\, \boldsymbol{\vartriangle}^{\lambda}$. So, we are reduced to the case $\lambda=0$. For each $\mu\in\Lambda^+_{M, ab}$ consider the embedding $\operatorname{act}: U_{\mu}/U_0\to \operatorname{Gr}_G$, $zU_0\mapsto zG({\mathcal O})$. By definition, $\operatorname{Av}_!^{U_{\mu}}(\delta_{1,\operatorname{Gr}_G})\,{\widetilde\to}\, \operatorname{act}_!\omega$. Now $$\operatorname{Av}_!^{U(P)(F)}(\delta_{1,\operatorname{Gr}_G})\,{\widetilde\to}\,\mathop{\operatorname{colim}}\limits_{\mu\in\Lambda^+_{M, ab}} \operatorname{Av}^{U_{\mu}}_!(\delta_{1,\operatorname{Gr}_G})\,{\widetilde\to}\, \mathop{\operatorname{colim}}\limits_{\mu\in\Lambda^+_{M, ab}} \omega_{U_{\mu}/U_0}\,{\widetilde\to}\,\boldsymbol{\vartriangle}^0$$ For the second claim, by ([\[iso_for_Step2_Av!\_versus_t\_lambda\]](#iso_for_Step2_Av!_versus_t_lambda){reference-type="ref" reference="iso_for_Step2_Av!_versus_t_lambda"}) we have $\operatorname{Av}_!^{U(P)(F)}(\delta_{\lambda,\operatorname{Gr}_G})\,{\widetilde\to}\, t^{\mu}\operatorname{Av}_!^{U(P)(F)}(\delta_{1,\operatorname{Gr}_G})$. The claim follows now from the above. ii\) Let $\operatorname{act}: t^{\lambda}U_{\lambda}/U_0\to\operatorname{Gr}_G$ be the embedding sending $zU_0$ to $zG({\mathcal O})$. By Section [2.3.4](#Sect_2.1.4_objects_cB){reference-type="ref" reference="Sect_2.1.4_objects_cB"}, $j_{\lambda, !}\ast \delta_{1,\operatorname{Gr}_G}\,{\widetilde\to}\, \operatorname{act}_!\operatorname{IC}$. ◻ ### $t$-structure on $Shv(\operatorname{Gr}_G)^H$ First, for $\nu\in\Lambda^+_M$ define a new t-structrure on $Shv(S^{\nu}_P)^H$ by declaring that $K\in Shv(S^{\nu}_P)^H$ lies in $Shv(S^{\nu}_P)^{H,\le 0}$ iff $(i^{\nu}_P)^!(K)$ lies in perverse degrees $\le \langle\nu, 2\check{\rho}-2\check{\rho}_M\rangle$. In fact, $Shv(S^{\nu}_P)^{H,\le 0}\subset Shv(S^{\nu}_P)^H$ is the smallest full subcategory containing $\boldsymbol{\vartriangle}^{\nu}$, stable under extensions and small colimits. Define $$Shv(\operatorname{Gr}_G)^{H, \le 0}\subset Shv(\operatorname{Gr}_G)^H$$ as the smallest full subcategory containing $(i_{\nu})_!F$ for $\nu\in\Lambda_M^+, F\in Shv(S^{\nu}_P)^{H, \le 0}$, closed under extensions and small colimits. By ([@HA], 1.4.4.11), this defines an accessible t-structure on $Shv(\operatorname{Gr}_G)^H$. So, $F\in Shv(\operatorname{Gr}_G)^H$ lies in $Shv(\operatorname{Gr}_G)^{H, >0}$ iff for any $\nu\in\Lambda^+_M$, $i_{\nu}^!F\in Shv(\operatorname{Gr}_G)^{H, >0}$. This shows that the t-structure on $Shv(\operatorname{Gr}_G)^H$ is compatible with filtered colimits. **Remark 1**. *i) The objects of the form $(i_{\nu})_!F$ for $\nu\in\Lambda_M^+, F\in Shv(S^{\nu}_P)^H$ generate $Shv(\operatorname{Gr}_G)^H$.\ ii) The objects of the form $(v^{\theta}_P)_!F$ for $\theta\in\Lambda_{G,P}$, $F\in Shv(\operatorname{Gr}_P^{\theta})^H$ generate $Shv(\operatorname{Gr}_G)^H$.* *Proof.* i) If $S\in{\operatorname{Sch}}_{ft}$ and $S\to \operatorname{Gr}_G$ is a map then the stratification of $\operatorname{Gr}_G$ by $S^{\nu}_P$, $\nu\in\Lambda_M^+$ defines a finite stratification of $S$ by $S\cap S^{\nu}_P$. So, if $i_{\nu}^!F=0$ for all $\nu$ then $F=0$.\ ii) is similar. ◻ **Lemma 1**. *For each $\nu\in\Lambda_M^+$ the adjoint functors $$i_{\nu}^*: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\leftrightarrows Shv(S^{\nu}_P)^{M({\mathcal O})}: (i_{\nu})_*$$ preserve the full subcategories of $H$-invariants and give rise to an adjoint pair $$i_{\nu}^*: Shv(\operatorname{Gr}_G)^{H}\leftrightarrows Shv(S^{\nu}_P)^{H}: (i_{\nu})_*$$ Besides, the diagram canonically commutes $$\begin{array}{ccc} Shv(\operatorname{Gr}_G)^{M({\mathcal O})} & \stackrel{i_{\nu}^*}{\to} &Shv(S^{\nu}_P)^{M({\mathcal O})}\\ \downarrow\lefteqn{\scriptstyle \operatorname{Av}^{U(P)(F)}_!} && \downarrow\lefteqn{\scriptstyle \operatorname{Av}^{U(P)(F)}_!}\\ Shv(\operatorname{Gr}_G)^H & \stackrel{i_{\nu}^*}{\to} &Shv(S^{\nu}_P)^H \end{array}$$* *Proof.* This follows from the results of Section [5.6](#Sect_A.5){reference-type="ref" reference="Sect_A.5"}. ◻ **Remark 1**. *Let $\mu\in\Lambda^+_M$ and $F\in Shv(\operatorname{Gr}_G)^H$ be the extension by zero from $\bar S^{\mu}_P$. Then $F\in Shv(\operatorname{Gr}_G)^{H, \le 0}$ iff $i_{\nu}^*F\in Shv(S^{\nu}_P)^{H, \le 0}$ for all $\nu\in\Lambda_M^+$.* *Proof.* First, let $\lambda\in\Lambda_M^+$, $K\in Shv(S^{\lambda}_P)^{H,\le 0}$. Then for any $\nu\in\Lambda_M^+$, $i_{\nu}^*(i_{\lambda})_!K\in Shv(S^{\nu}_P)^{H, \le 0}$. So, if $F\in Shv(\operatorname{Gr}_G)^{H, \le 0}$ then $i_{\nu}^*F\in Shv(S^{\nu}_P)^{H, \le 0}$ for all $\nu\in\Lambda_M^+$. Conversely, let $0\ne F\in Shv(\operatorname{Gr}_G)^{H, >0}$ be the extension by zero from $\bar S^{\mu}_P$. Assume $i_{\nu}^*F\in Shv(S^{\nu}_P)^{H, \le 0}$ for all $\nu\in\Lambda_M^+$. We must show that $F=0$. Let $\lambda$ be the largest orbit $S^{\lambda}_P$ such that $i_{\lambda}^!F\ne 0$. By definition of the t-structure, $i_{\lambda}^!F\in Shv(S^{\lambda}_P)^{H, >0}$. On the other hand, $i_{\lambda}^!F\,{\widetilde\to}\, i_{\lambda}^*F$, and the assertion follows. ◻ Note that if $G=P$ then the above t-structure on $Shv(\operatorname{Gr}_G)^H\,{\widetilde\to}\, Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ is the perverse t-structure on the latter category. ### {#Sect_t-str_on_S_theta_P} For $\theta\in\Lambda_{G,P}$ define a t-structure on $Shv(\operatorname{Gr}^{\theta}_P)^H$ by declaring that $K\in Shv(\operatorname{Gr}^{\theta}_P)^H$ lies in $Shv(\operatorname{Gr}^{\theta}_P)^{H,\le 0}$ iff $(i^{\theta}_P)^!K\in Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$ lies in perverse degrees $\le \langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$. By Lemma [Lemma 1](#Lm_2.3.8){reference-type="ref" reference="Lm_2.3.8"}, this is equivalent to the property that $$(\mathfrak{t}_P^{\theta})_!K\in Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$$ lies in perverse degrees $\le \langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$. Now $Shv(\operatorname{Gr}_G)^{H,\le 0}$ is the smallest full subcategory containing $(v^{\theta}_P)_!K$ for $\theta\in\Lambda_{G,P}$, $K\in Shv(\operatorname{Gr}^{\theta}_P)^{H,\le 0}$, stable under extensions and small colimits. This implies that $K\in Shv(\operatorname{Gr}_G)^H$ lies in $Shv(\operatorname{Gr}_G)^{H, >0}$ iff for any $\theta\in\Lambda_{G,P}$, $(v^{\theta}_P)^!K\in Shv(\operatorname{Gr}^{\theta}_P)^{H, >0}$. Set $Sph_M=Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$. **Lemma 1**. *For $\theta\in\Lambda_{G,P}$ the adjoint functors $$(v^{\theta}_P)^*: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\leftrightarrows Shv(\operatorname{Gr}^{\theta}_P)^{M({\mathcal O})}: (v^{\theta}_P)_*$$ preserve the full subcategories of $H$-invariants and give rise to an adjoint pair $$(v^{\theta}_P)^*: Shv(\operatorname{Gr}_G)^H\leftrightarrows Shv(\operatorname{Gr}^{\theta}_P)^H: (v^{\theta}_P)_*$$ The functor $(v_P)_*$ is $Sph_M$-linear. Besides, the diagram canonically commutes $$\begin{array}{ccc} Shv(\operatorname{Gr}_G)^{M({\mathcal O})} & \stackrel{(v^{\theta}_P)^*}{\to} & Shv(\operatorname{Gr}^{\theta}_P)^{M({\mathcal O})}\\ \downarrow\lefteqn{\scriptstyle \operatorname{Av}^{U(P)(F)}_!} && \downarrow\lefteqn{\scriptstyle \operatorname{Av}^{U(P)(F)}_!} \\ Shv(\operatorname{Gr}_G)^H & \stackrel{(v^{\theta}_P)^*}{\to} & Shv(\operatorname{Gr}^{\theta}_P)^H \end{array}$$ The functor $\operatorname{Av}^{U(P)(F)}_!: Shv(\operatorname{Gr}^{\theta}_P)^{M({\mathcal O})}\to Shv(\operatorname{Gr}^{\theta}_P)^H$ identifies canonically with $(\mathfrak{t}_P^{\theta})^!(\mathfrak{t}_P^{\theta})_!$.* *Proof.* The first claims are obtained as in Lemma [Lemma 1](#Lm_functor_i^nu^*){reference-type="ref" reference="Lm_functor_i^nu^*"}. Now using Lemma [Lemma 1](#Lm_2.3.8){reference-type="ref" reference="Lm_2.3.8"} for $K\in Shv(\operatorname{Gr}_P)^{M({\mathcal O})}, L\in Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$ we get $$\begin{gathered} {{\mathcal H}om}_{Shv(\operatorname{Gr}_P)^{M({\mathcal O})}}(K, (\mathfrak{t}_P^{\theta})^!L)\,{\widetilde\to}\, {{\mathcal H}om}_{Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}}((\mathfrak{t}_P^{\theta})_! K, L)\,{\widetilde\to}\\ {{\mathcal H}om}_{Shv(\operatorname{Gr}_P)^{M({\mathcal O})}}((\mathfrak{t}_P^{\theta})^!(\mathfrak{t}_P^{\theta})_! K, (\mathfrak{t}_P^{\theta})^!L)\end{gathered}$$ This gives the last claim. ◻ **Remark 1**. *The functor $v_P^*: Shv(\operatorname{Gr}_G)^H\to Shv(\operatorname{Gr}_P)^H$ is left-lax $Sph_M$-linear. We will see in Proposition [Proposition 1](#Pp_2.4.19){reference-type="ref" reference="Pp_2.4.19"} that this left-lax structure is strict (in the case of ${\mathcal D}$-modules this is automatic by ([@GaLocWhit], D.4.4)).* ### {#section-50} Write $\upsilon: Shv(\operatorname{Gr}_G)^{G({\mathcal O})}\to Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ for the functor induced by $G(F)\to G(F), g\mapsto g^{-1}$, and similarly for $\upsilon: Shv(\operatorname{Gr}_M)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$. By Section [5.5](#Sect_A.3){reference-type="ref" reference="Sect_A.3"}, we have canonical Verdier self-dualities $$\operatorname{VD}: Shv(\operatorname{Gr}_G)^{I_P}\,{\widetilde\to}\, (Shv(\operatorname{Gr}_G)^{I_P})^{\vee}$$ and $Shv(\operatorname{Gr}_G)^{I_P, ren}\,{\widetilde\to}\, (Shv(\operatorname{Gr}_G)^{I_P, ren})^{\vee}$. Under this duality, for $K\in Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ the dual of the functor $Shv(\operatorname{Gr}_G)^{I_P}\to Shv(\operatorname{Gr}_G)^{I_P}$, $F\mapsto F\ast K$ identifies with the functor $F\mapsto F\ast (\upsilon K)$. The same holds for the renormalized versions. In view of Proposition [Proposition 1](#Pp_2.2.6){reference-type="ref" reference="Pp_2.2.6"} and the equivalence ([\[eq_ren_parahoric_versus_H\]](#eq_ren_parahoric_versus_H){reference-type="ref" reference="eq_ren_parahoric_versus_H"}), the above yields self-dualities $$\operatorname{VD}_H: Shv(\operatorname{Gr}_G)^H\,{\widetilde\to}\,(Shv(\operatorname{Gr}_G)^H)^{\vee}\;\;\;\mbox{and}\;\;\; Shv(\operatorname{Gr}_G)^{H, ren}\,{\widetilde\to}\, (Shv(\operatorname{Gr}_G)^{H, ren})^{\vee}$$ ### {#section-51} Fix a Chevalley involution $\sigma\in \operatorname{Aut}(G)$ acting as $z\mapsto z^{-1}$ on $T$ and sending $B$ to $B^-$. Then $\sigma(P)=P^-$. Write $I_{P^-}$ for the preimage of $P^-$ under $G({\mathcal O})\to G$. Note that $\sigma(I_P)=I_{P^-}$. By abuse of notations, write also $\sigma: \operatorname{Gr}_G\to\operatorname{Gr}_G$ for the map $zG({\mathcal O})\mapsto \sigma(z)G({\mathcal O})$. It induces an equivalence $$\sigma: Shv(\operatorname{Gr}_G)^{I_P}\,{\widetilde\to}\, Shv(\operatorname{Gr}_G)^{I_{P^-}}$$ Let $H^-=M({\mathcal O})U(P^-)(F)$. The map $\sigma: \operatorname{Gr}_G\to\operatorname{Gr}_G$ intertwines the $H$ and $H^-$-actions via the isomorphism also denoted $\sigma: H\,{\widetilde\to}\, H^-$, hence induces an equivalence $$\sigma: Shv(\operatorname{Gr}_G)^H\,{\widetilde\to}\, Shv(\operatorname{Gr}_G)^{H^-}$$ The following diagram commutes $$\begin{array}{ccccc} Shv(\operatorname{Gr}_G)^H & \stackrel{\operatorname{oblv}}{\to} & Shv(\operatorname{Gr}_G)^{M({\mathcal O})} &\stackrel{\operatorname{Av}_*^{I_P/M({\mathcal O})}}{\to} & Shv(\operatorname{Gr}_G)^{I_P}\\ \downarrow\lefteqn{\scriptstyle \sigma} && \downarrow\lefteqn{\scriptstyle \sigma} &&\downarrow\lefteqn{\scriptstyle \sigma}\\ Shv(\operatorname{Gr}_G)^{H^-} & \stackrel{\operatorname{oblv}}{\to} & Shv(\operatorname{Gr}_G)^{M({\mathcal O})} &\stackrel{\operatorname{Av}_*^{I_{P^-}/M({\mathcal O})}}{\to} & Shv(\operatorname{Gr}_G)^{I_{P^-}} \end{array}$$ The equivalence $\sigma$ is compatible with $\operatorname{VD}$, namely the diagram canonically commutes $$\begin{array}{ccc} Shv(\operatorname{Gr}_G)^{I_P} & \stackrel{\sigma}{\to} & Shv(\operatorname{Gr}_G)^{I_{P^-}}\\ \downarrow\lefteqn{\scriptstyle \operatorname{VD}} && \downarrow\lefteqn{\scriptstyle \operatorname{VD}}\\ (Shv(\operatorname{Gr}_G)^{I_P})^{\vee} & \stackrel{\sigma^{\vee}}{\gets} & (Shv(\operatorname{Gr}_G)^{I_{P^-}})^{\vee} \end{array}$$ This gives the commutativity of the diagram $$\begin{array}{ccc} Shv(\operatorname{Gr}_G)^H & \stackrel{\operatorname{VD}_H}{\to} & ((Shv(\operatorname{Gr}_G)^H)^{\vee}\\ \downarrow\lefteqn{\scriptstyle \sigma} && \uparrow\lefteqn{\scriptstyle \sigma^{\vee}}\\ Shv(\operatorname{Gr}_G)^{H^-} & \stackrel{\operatorname{VD}_{H^-}}{\to} & (Shv(\operatorname{Gr}_G)^{H^-})^{\vee}, \end{array}$$ where $\operatorname{VD}_{H^-}$ is defined similarly to $\operatorname{VD}_H$ (replacing $P$ by $P^-$). The involution $\sigma$ similarly yields a monoidal equivalence $\sigma: \operatorname{Sph}_G\,{\widetilde\to}\,\operatorname{Sph}_G$, for $\lambda\in\Lambda^+$ we have $\sigma\operatorname{Sat}(V^{\lambda})\,{\widetilde\to}\, V^{-w_0(\lambda)}$. For $K\in Shv(\operatorname{Gr}_G)^H$, $F\in \operatorname{Sph}_G$ one has canonically $\sigma(K\ast F)\,{\widetilde\to}\, \sigma(K)\ast \sigma(F)$ in $Shv(\operatorname{Gr}_G)^{H^-}$. **Proposition 1**. *The action of ${\operatorname{Rep}}(\check{G})^{\heartsuit}$ on $Shv(\operatorname{Gr}_G)^H$ is t-exact.* *Proof.* For $G=P$ the claim follows from ([@G_central], Proposition 6). For $P=B$ this is ([@Gai19SI], Proposition 2.8.2). Consider now any $P$. Take $V\in {\operatorname{Rep}}(\check{G})^{\heartsuit}$. It suffices to show that for $V$ finite-dimensional the functor $\_ \ast \operatorname{Sat}(V)$ is t-exact. **Step 1** We show that $\_ \ast \operatorname{Sat}(V)$ is right t-exact. Note that $Shv(\operatorname{Gr}_G)^{H,\le 0}\subset Shv(\operatorname{Gr}_G)^H$ is the smallest full subcategory containing $\boldsymbol{\vartriangle}^{\nu}$ for $\nu\in\Lambda^+_M$, stable under extensions and small colimits. Let $\nu\in \Lambda^+_M$ and $\theta$ be its image in $\Lambda_{G,P}$. Now it suffices to show $\boldsymbol{\vartriangle}^{\nu}\ast \operatorname{Sat}(V)\in Shv(\operatorname{Gr}_G)^{H,\le 0}$. For this by Remark [Remark 1](#Rem_2.3.17_about_t-structure){reference-type="ref" reference="Rem_2.3.17_about_t-structure"} it suffices to show that for any $\theta'\in\Lambda_{G,P}$, $$(v^{\theta'}_P)^*(\boldsymbol{\vartriangle}^{\nu}\ast \operatorname{Sat}(V))\in Shv(\operatorname{Gr}^{\theta'}_P)^{H, \le 0}$$ Recall the convolution diagram $\operatorname{Gr}_G\tilde\times\operatorname{Gr}_G:=\operatorname{Conv}_G=G(F)\times^{G({\mathcal O})}\operatorname{Gr}_G$ with the projections $\operatorname{pr}_i: \operatorname{Gr}_G\tilde\times\operatorname{Gr}_G\to \operatorname{Gr}_G$ on the $i$-th factor, and the product map $\operatorname{act}: \operatorname{Conv}_G\to\operatorname{Gr}_G$. Define the ind-scheme $\operatorname{Gr}^{\theta}_P\tilde\times\operatorname{Gr}^{\theta'-\theta}_P$ as follows. Let $P(F)^{\theta}$ be the preimage of $\operatorname{Gr}_M^{\theta}$ under $P(F)\to M(F)\to \operatorname{Gr}_M$. After passing to reduced ind-schemes, one has an isomorphism $\operatorname{Gr}_P^{\theta}\,{\widetilde\to}\, P(F)^{\theta}/P({\mathcal O})$. We ignore the nilpotents here, as they do not change the category of sheaves on a given ind-scheme of ind-finite type. Set $$\operatorname{Gr}^{\theta}_P\tilde\times\operatorname{Gr}^{\theta'-\theta}_P=P(F)^{\theta}\times^{P({\mathcal O})} \operatorname{Gr}^{\theta'-\theta}_P$$ The base change of $\operatorname{act}: \operatorname{Gr}^{\theta}_P\tilde\times\operatorname{Gr}_G\to \operatorname{Gr}_G$ by $v^{\theta'}_P: \operatorname{Gr}^{\theta'}_P\stackrel{}{\hookrightarrow} \operatorname{Gr}_G$ is the convolution map $$\operatorname{act}': \operatorname{Gr}^{\theta}_P\tilde\times\operatorname{Gr}^{\theta'-\theta}_P\to \operatorname{Gr}^{\theta'}_P.$$ So, we must show that $\operatorname{act}'_!(\boldsymbol{\vartriangle}^{\nu}\,\tilde\boxtimes\,(v^{\theta'-\theta}_P)^*\operatorname{Sat}(V))$ lies in $Shv(\operatorname{Gr}^{\theta'}_P)^{H,\le 0}$ or, equivalently, that $$\label{complex_one_for_Pp_2.3.21} (\mathfrak{t}_P^{\theta'})_!\operatorname{act}'_!(\boldsymbol{\vartriangle}^{\nu}\,\tilde\boxtimes\,(v^{\theta'-\theta}_P)^*\operatorname{Sat}(V))$$ lies in perverse degrees $\le \langle\theta', 2\check{\rho}-2\check{\rho}_M\rangle$. Note that $(\mathfrak{t}^{\theta}_P)_!\boldsymbol{\vartriangle}^{\nu}$ is the extension by zero of $\omega[-\langle\nu, 2\check{\rho}\rangle]$ under $\operatorname{Gr}_M^{\nu}\stackrel{}{\hookrightarrow} \operatorname{Gr}_M^{\theta}$. So, $(\mathfrak{t}^{\theta}_P)_!\boldsymbol{\vartriangle}^{\nu}$ is placed in perverse degrees $\le \langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$. By Proposition [Proposition 1](#Pp_gRes_for_Levi){reference-type="ref" reference="Pp_gRes_for_Levi"} below, $$(\mathfrak{t}_P^{\theta'-\theta})_!(v^{\theta'-\theta}_P)^*\operatorname{Sat}(V)[\langle\theta'-\theta, 2\check{\rho}-2\check{\rho}_M\rangle]\,{\widetilde\to}\, Sat_M(\operatorname{Res}(V)_{\theta'-\theta}),$$ where $\operatorname{Res}(V)$ denotes its restriction under $\check{M}\to \check{G}$, $\operatorname{Sat}_M$ is the Satake functor for $M$, and for ${\mathcal V}\in{\operatorname{Rep}}(\check{M})$ we denote by ${\mathcal V}_{\theta}$ the direct summand on which $Z(\check{M})$ acts by $\theta$. Now ([\[complex_one_for_Pp_2.3.21\]](#complex_one_for_Pp_2.3.21){reference-type="ref" reference="complex_one_for_Pp_2.3.21"}) identifies with $$((\mathfrak{t}^{\theta}_P)_!\boldsymbol{\vartriangle}^{\nu})\ast \operatorname{Sat}_M(\operatorname{Res}(V)_{\theta'-\theta})[\langle\theta-\theta', 2\check{\rho}-2\check{\rho}_M\rangle],$$ where the convolution is for $M$ now. Our claim follows now from ([@G_central], Proposition 6). **Step 2** Recall that $\dim V<\infty$. The left and right adjoint of $Shv(\operatorname{Gr}_G)^H\to Shv(\operatorname{Gr}_G)^H$, $K\mapsto K\ast \operatorname{Sat}(V)$ is the functor $K\mapsto K\ast \operatorname{Sat}(V^*)$. Since the left adjoint if right t-exact, the right adjoint is left t-exact. ◻ ### {#section-52} Recall the following more precise version of ([@BG], Theorem 4.3.4). **Proposition 1**. *Let $\theta\in\Lambda_{G,P}$. The functor $$(\mathfrak{t}_P^{\theta})_!(v_P^{\theta})^*[\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]: \operatorname{Perv}(\operatorname{Gr}_G)^{G({\mathcal O})}\to Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$$ takes values in $\operatorname{Perv}(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$. Let $\operatorname{gRes}: \operatorname{Perv}(\operatorname{Gr}_G)^{G({\mathcal O})}\to \operatorname{Perv}(\operatorname{Gr}_M)^{M({\mathcal O})}$ be the functor $$\mathop{\oplus}\limits_{\theta\in\Lambda_{G,P}} (\mathfrak{t}_P^{\theta})_!(v_P^{\theta})^*[\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$$ The diagram commutes $$\begin{array}{ccc} {\operatorname{Rep}}(\check{G})^{\heartsuit} & \stackrel{\operatorname{Sat}_G}{\to} & \operatorname{Perv}(\operatorname{Gr}_G)^{G({\mathcal O})}\\ \downarrow && \downarrow\lefteqn{\scriptstyle \operatorname{gRes}}\\ {\operatorname{Rep}}(\check{M})^{\heartsuit} & \stackrel{\operatorname{Sat}_M}{\to} & \operatorname{Perv}(\operatorname{Gr}_M)^{M({\mathcal O})}, \end{array}$$ where $\operatorname{Sat}_G, \operatorname{Sat}_M$ denote the Satake functors for $G,M$, and the left vertical arrow is the restriction along $\check{M}\stackrel{}{\hookrightarrow} \check{G}$.* **Remark 1**. *For $K\in \operatorname{Perv}(\operatorname{Gr}_G)^{G({\mathcal O})}$ one has canonically $\upsilon \operatorname{gRes}(K)\,{\widetilde\to}\,\operatorname{gRes}(\upsilon K)$ in $\operatorname{Perv}(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$, as $\mathbb{D}\upsilon =\upsilon \mathbb{D}$ corresponds to the contragredient duality on ${\operatorname{Rep}}(\check{G})^{\heartsuit}$ by ([@FG], 5.6.2).* This gives the following dual version of Proposition [Proposition 1](#Pp_gRes_for_Levi){reference-type="ref" reference="Pp_gRes_for_Levi"}. For $\theta\in\Lambda_{G,P}$ the functor $$(\mathfrak{t}_P^{\theta})_*(v_P^{\theta})^![-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]: \operatorname{Perv}(\operatorname{Gr}_G)^{G({\mathcal O})}\to Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$$ takes values in $\operatorname{Perv}(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$. Let $\operatorname{gRes}^!: \operatorname{Perv}(\operatorname{Gr}_G)^{G({\mathcal O})}\to \operatorname{Perv}(\operatorname{Gr}_M)^{M({\mathcal O})}$ be the functor $$\mathop{\oplus}\limits_{\theta\in\Lambda_{G,P}} (\mathfrak{t}_P^{\theta})_*(v_P^{\theta})^![-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$$ The diagram commutes $$\begin{array}{ccc} {\operatorname{Rep}}(\check{G})^{\heartsuit} & \stackrel{\operatorname{Sat}_G}{\to} & \operatorname{Perv}(\operatorname{Gr}_G)^{G({\mathcal O})}\\ \downarrow && \downarrow\lefteqn{\scriptstyle \operatorname{gRes}^!}\\ {\operatorname{Rep}}(\check{M})^{\heartsuit} & \stackrel{\operatorname{Sat}_M}{\to} & \operatorname{Perv}(\operatorname{Gr}_M)^{M({\mathcal O})}, \end{array}$$ where the left vertical arrow is the restriction along $\check{M}\stackrel{}{\hookrightarrow} \check{G}$. ### {#Sect_true_Rep(checkM)-action_on_SI} The category $Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$ acts on $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ by convolutions on the left. For $F\in Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$, $K\in Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ we denote this (left) action by $(F, K)\mapsto F\ast K$. Consider the action of ${\operatorname{Rep}}(\check{M})$ on $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ such that $V\in{\operatorname{Rep}}(\check{M})$ on which $Z(\check{M})$ acts by $\theta\in\Lambda_{G,P}$ sends $K\in Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ to $$\label{action_Rep(checkM)_shifted} \operatorname{Sat}_M(V)\ast K[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$$ Write $\operatorname{Gr}_M\tilde\times\operatorname{Gr}_G=M(F)\times^{M({\mathcal O})} \operatorname{Gr}_G$ with the action map $\operatorname{act}: \operatorname{Gr}_M\tilde\times\operatorname{Gr}_G\to\operatorname{Gr}_G$ coming from $M(F)\times \operatorname{Gr}_G\to \operatorname{Gr}_G$, $(m, gG({\mathcal O}))\mapsto mgG({\mathcal O})$. More generally, for a $M({\mathcal O})$-invariant ind-subscheme $Y\subset \operatorname{Gr}_G$, one similarly has $\operatorname{Gr}_M\tilde\times\, Y$ and $\operatorname{act}: \operatorname{Gr}_M\tilde\times\, Y\to \operatorname{Gr}_G$. **Proposition 1**. *i) The full subcategory $Shv(\operatorname{Gr}_G)^H\subset Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ is preserved under the action of $Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$ on $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ by convolutions on the left. The functor $$\operatorname{Av}^{U(P)(F)}_!: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_G)^H$$ is ${\operatorname{Rep}}(\check{M})$-linear.\ ii) If $V\in {\operatorname{Rep}}(\check{M})^{\heartsuit}$ and $Z(\check{M})$ acts on $V$ by $\theta'\in\Lambda_{G,P}$ then the functor $$\label{functor_for_shifted_action_of_Rep_checkM} Shv(\operatorname{Gr}_G)^H\to Shv(\operatorname{Gr}_G)^H, \, K\mapsto \operatorname{Sat}_M(V)\ast K[-\langle\theta', 2\check{\rho}-2\check{\rho}_M\rangle]$$ is t-exact.\ iii) Let $\theta,\theta'\in\Lambda_{G,P}$, $F\in Shv(\operatorname{Gr}_M^{\theta'})^{M({\mathcal O})}$, $K\in Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$. One has a canonical isomorphism $$F\ast ((\mathfrak{t}_P^{\theta})_!(v^{\theta}_P)^*K)\,{\widetilde\to}\, (\mathfrak{t}_P^{\theta'+\theta})_!(v^{\theta'+\theta}_P)^*(F\ast K)$$ in $Shv(\operatorname{Gr}_M^{\theta'+\theta})^{M({\mathcal O})}$ functorial in $F$ and $K$. The analog of the latter isomorphism with $P$ replaced by $P^-$ also holds.* *Proof.* i) Let $\theta,\theta'\in \Lambda_{G,P}$, $F\in Shv(\operatorname{Gr}_M^{\theta'})^{M({\mathcal O})}$, and $K\in Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$. By Remark [Remark 1](#Rem_2.3.14){reference-type="ref" reference="Rem_2.3.14"} and Lemma [Lemma 1](#Lm_2.3.8){reference-type="ref" reference="Lm_2.3.8"}, it suffices to show that $F \ast ((\mathfrak{t}^{\theta}_P)^!K)\in Shv(\operatorname{Gr}_G)^H$. The action map $\operatorname{act}: \operatorname{Gr}_M^{\theta'}\tilde\times\operatorname{Gr}_P^{\theta}\to \operatorname{Gr}_G$ factors through $v^{\theta+\theta'}_P: \operatorname{Gr}_P^{\theta+\theta'}\stackrel{}{\hookrightarrow} \operatorname{Gr}_G$, and the square is cartesian $$\label{diag_for_Pp_2.4.19} \begin{array}{ccc} \operatorname{Gr}_M^{\theta'}\tilde\times\operatorname{Gr}_P^{\theta} &\stackrel{\operatorname{act}}{\to} & \operatorname{Gr}_P^{\theta+\theta'}\\ \downarrow\lefteqn{\scriptstyle\operatorname{id}\times \mathfrak{t}^{\theta}_P} && \downarrow\lefteqn{\scriptstyle \mathfrak{t}^{\theta+\theta'}_P}\\ \operatorname{Gr}_M^{\theta'}\tilde\times\operatorname{Gr}_M^{\theta} &\stackrel{\operatorname{act}}{\to} & \operatorname{Gr}_M^{\theta+\theta'} \end{array}$$ We have $$F\ast ((\mathfrak{t}^{\theta}_P)^!K)\,{\widetilde\to}\, \operatorname{act}_*(\operatorname{id}\times\mathfrak{t}_P^{\theta})^!(F\,\tilde\boxtimes\,K)\,{\widetilde\to}\, (\mathfrak{t}^{\theta+\theta'}_P)^!(F\ast K)$$ The first claim follows now from Lemma [Lemma 1](#Lm_2.3.8){reference-type="ref" reference="Lm_2.3.8"}. Since ${\operatorname{Rep}}(\check{M})$ is rigid, the second claim follows from ([@G], ch. I.1, 9.3.6). ii\) We may assume $V\in {\operatorname{Rep}}(\check{M})^{\heartsuit}$ is finite-dimensional. Let us first show that ([\[functor_for_shifted_action_of_Rep_checkM\]](#functor_for_shifted_action_of_Rep_checkM){reference-type="ref" reference="functor_for_shifted_action_of_Rep_checkM"}) is right t-exact. Take $\theta\in\Lambda_{G,P}$. It suffices to show that for any $K\in Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$ placed in perverse degrees $\le \langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$, the object $$\operatorname{Sat}_M(V)\ast ((\mathfrak{t}_P^{\theta})^!K)[-\langle\theta', 2\check{\rho}-2\check{\rho}_M\rangle]$$ lies in $Shv(\operatorname{Gr}_P^{\theta+\theta'})^{H, \le 0}$, that is, $$(\mathfrak{t}_P^{\theta+\theta'})_!(\operatorname{Sat}_M(V)\ast ((\mathfrak{t}_P^{\theta})^!K))$$ is placed in perverse degrees $\le \langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$ over $\operatorname{Gr}_M^{\theta+\theta'}$. The latter complex identifies with $\operatorname{Sat}_M(V)\ast K$. Our claim follows now from ([@G_central], Proposition 6). To see that ([\[functor_for_shifted_action_of_Rep_checkM\]](#functor_for_shifted_action_of_Rep_checkM){reference-type="ref" reference="functor_for_shifted_action_of_Rep_checkM"}) is left t-exact argue as in Step 2 of Proposition [Proposition 1](#Pp_2.3.21){reference-type="ref" reference="Pp_2.3.21"}. Here we use the fact that $Z(\check{M})$ acts on $V^*$ as $-\theta'$. iii\) We may and do assume that $K$ is the extension by zero from a closed $M({\mathcal O})$-invariant subscheme of finite type in $\operatorname{Gr}_G$. **Step 1**. We claim that $$(v^{\theta'+\theta}_P)^*(F\ast K)\,{\widetilde\to}\, F\ast ((v^{\theta}_P)^*K)$$ canonically in $Shv(\operatorname{Gr}_P^{\theta+\theta'})^{M({\mathcal O})}$. Indeed, $K$ in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ admits a finite filtration with succesive quotients $(v^{\mu}_P)_!(v^{\mu}_P)^*K$, $\mu\in \Lambda_{G,P}$. So, $F\ast K$ admits a finite filtration in $Shv(\operatorname{Gr}_P^{\theta+\theta'})^{M({\mathcal O})}$ with succesive quotients $F\ast ((v^{\mu}_P)_!(v^{\mu}_P)^*K)$, $\mu\in \Lambda_{G,P}$. Our claim follows. **Step 2**. From ([\[diag_for_Pp_2.4.19\]](#diag_for_Pp_2.4.19){reference-type="ref" reference="diag_for_Pp_2.4.19"}) we get $$(\mathfrak{t}^{\theta+\theta'}_P)_!(F\ast (v^{\theta}_P)^*K)\,{\widetilde\to}\, F\ast ((\mathfrak{t}_P^{\theta})_!(v^{\theta}_P)^*K)$$ Our claim follows. ◻ ### {#section-53} The action of $\Lambda_{M, ab}$ on $Shv(\operatorname{Gr}_G)^H$ from Section [3.1.8](#Sect_2.2.10_action_of_Lambda_Mab){reference-type="ref" reference="Sect_2.2.10_action_of_Lambda_Mab"} is obtained by restricting the ${\operatorname{Rep}}(\check{M})^{\heartsuit}$-action given by ([\[functor_for_shifted_action_of_Rep_checkM\]](#functor_for_shifted_action_of_Rep_checkM){reference-type="ref" reference="functor_for_shifted_action_of_Rep_checkM"}) to ${\operatorname{Rep}}(\check{M}_{ab})^{\heartsuit}$. ### {#section-54} For $\nu\in\Lambda^+_M$ denote by $j_{\nu}^-: S^{\nu}_{P^-}\stackrel{}{\hookrightarrow} \bar S^{\nu}_{P^-}$ this open immersion. Let $\bar i_{\nu}^-: \bar S^{\nu}_{P^-}\stackrel{}{\hookrightarrow} \operatorname{Gr}_G$ be the closed immersion and $i_{\nu}^-=\bar i_{\nu}^-\circ j_{\nu}^-$. We also have the closed embedding $i^{\nu}_{P^-}: \operatorname{Gr}_M^{\nu}\stackrel{}{\hookrightarrow} S^{\nu}_{P^-}$. ### {#section-55} The following is not used in the paper. To complete the properties of $\operatorname{SI}_P$, we recall the following result of Lin Chen. **Proposition 1** ([@LC], Corollary 1.4.5). *Consider the composition $$Shv(\operatorname{Gr}_G)^H\otimes Shv(\operatorname{Gr}_G)^{H^-}\stackrel{\operatorname{oblv}\otimes\operatorname{oblv}}{\to} Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\otimes Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to\operatorname{Vect}$$ where the right arrow is the Verdier duality from Section [5.5](#Sect_A.3){reference-type="ref" reference="Sect_A.3"}. This is a counit of a duality $Shv(\operatorname{Gr}_G)^H\,{\widetilde\to}\, (Shv(\operatorname{Gr}_G)^{H^-})^{\vee}$.* # The semi-infinite $\operatorname{IC}$-sheaf $\operatorname{IC}^{\frac{\infty}{2}}_P$ ## Definition and first properties ### {#section-56} Recall that $\Lambda^+$ is the set of dominant coweights of $G$. Equip it with the relation $\lambda\le\mu$ iff $\mu-\lambda\in\Lambda^+$. Then $(\Lambda^+, \le)$ is a filtered category. In ([@Gai19SI], Sections 2.3, 2.7) the functor $$\label{functor_initial_for_IC_SI} (\Lambda^+,\le)\to Shv(\operatorname{Gr}_G)$$ given by $\lambda\mapsto t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle]$ was constructed. Here $\operatorname{Sat}: {\operatorname{Rep}}(\check{G})\to Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ is the Satake functor. Recall the construction of the transition maps in ([\[functor_initial_for_IC_SI\]](#functor_initial_for_IC_SI){reference-type="ref" reference="functor_initial_for_IC_SI"}). First, for $\lambda\in\Lambda^+$ one has a canonical map $$\label{map_fibre_of_Sat(V^lambda)} \delta_{1, \operatorname{Gr}_G}\to t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle]$$ If moreover $\lambda\in\Lambda^+_{M, ab}$ then ([\[map_fibre_of_Sat(V\^lambda)\]](#map_fibre_of_Sat(V^lambda)){reference-type="ref" reference="map_fibre_of_Sat(V^lambda)"}) is a map in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ naturally. Now for $\lambda_i\in\Lambda^+$ with $\lambda_2-\lambda_1=\lambda\in\Lambda^+$ the morphism $$\label{transition_map_for_IC_SI} t^{-\lambda_1}\operatorname{Sat}(V^{\lambda_1})[\langle\lambda_1, 2\check{\rho}\rangle]\to t^{-\lambda_2}\operatorname{Sat}(V^{\lambda_2})[\langle\lambda_2, 2\check{\rho}\rangle]$$ is defined as the composition $$\begin{gathered} t^{-\lambda_1}\operatorname{Sat}(V^{\lambda_1})[\langle\lambda_1, 2\check{\rho}\rangle]\,{\widetilde\to}\, t^{-\lambda_1}\delta_{1,\operatorname{Gr}_G} \ast \operatorname{Sat}(V^{\lambda_1})[\langle\lambda_1, 2\check{\rho}\rangle]\stackrel{(\ref{map_fibre_of_Sat(V^lambda)})}{\to}\\ t^{-\lambda_1}t^{-\lambda}\operatorname{Sat}(V^{\lambda})\ast \operatorname{Sat}(V^{\lambda_1})[\langle\lambda+\lambda_1, 2\check{\rho}\rangle]\,{\widetilde\to}\, t^{-\lambda_2}\operatorname{Sat}(V^{\lambda}\otimes V^{\lambda_1})[\langle\lambda_2, 2\check{\rho}\rangle]\stackrel{u^{\lambda, \lambda_1}}{\to}\\ t^{-\lambda_2}\operatorname{Sat}(V^{\lambda_2})[\langle\lambda_2, 2\check{\rho}\rangle]\end{gathered}$$ ### {#section-57} Consider the restriction $$\label{functor_for_IC_SI_parabolic} (\Lambda^+_{M, ab},\le)\to Shv(\operatorname{Gr}_G)$$ of ([\[functor_initial_for_IC_SI\]](#functor_initial_for_IC_SI){reference-type="ref" reference="functor_initial_for_IC_SI"}) under $(\Lambda^+_{M, ab},\le)\to (\Lambda^+,\le)$. If $\lambda\in\Lambda^+_{M, ab}$ then $t^{-\lambda}\operatorname{Sat}(V^{\lambda})$ is naturally $M({\mathcal O})$-equivariant. Besides, ([\[transition_map_for_IC_SI\]](#transition_map_for_IC_SI){reference-type="ref" reference="transition_map_for_IC_SI"}) naturally upgraded to a morphism in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$. Our next purpose is to show that ([\[functor_for_IC_SI_parabolic\]](#functor_for_IC_SI_parabolic){reference-type="ref" reference="functor_for_IC_SI_parabolic"}) naturally upgraded to a functor with valued in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$. The argument of [@Gai19SI] applies in this case, as we are going to explain. Consider a left action of $\Lambda^+_{M, ab}$ on $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ such that $\lambda\in\Lambda^+_{M, ab}$ sends $K$ to $t^{\lambda}K[-\langle\lambda, 2\check{\rho}\rangle]$, this is the action given by ([\[action_Rep(checkM)\_shifted\]](#action_Rep(checkM)_shifted){reference-type="ref" reference="action_Rep(checkM)_shifted"}). Consider also the right lax action of $\Lambda^+_{M, ab}$ on $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ such that $\lambda$ acts on $K$ as $K\ast \operatorname{Sat}(V^{\lambda})$. The right lax structure on this action is given by the morphisms $$(K\ast \operatorname{Sat}(V^{\lambda}))\ast \operatorname{Sat}(V^{\mu})\,{\widetilde\to}\, K\ast \operatorname{Sat}(V^{\lambda}\otimes V^{\mu})\stackrel{u^{\lambda,\mu}}{\to} K\ast \operatorname{Sat}(V^{\lambda+\mu})$$ for $\lambda,\mu\in\Lambda^+_{M, ab}$. We claim that $\delta_{1,\operatorname{Gr}_G}$ acquires a structure of a lax central object with respect to these actions in the sense of ([@Gai19SI], 2.7.1). The corresponding map $$t^{\lambda}\ast \delta_{1,\operatorname{Gr}_G}[-\langle\lambda, 2\check{\rho}\rangle]\to \operatorname{Sat}(V^{\lambda})$$ is ([\[map_fibre_of_Sat(V\^lambda)\]](#map_fibre_of_Sat(V^lambda)){reference-type="ref" reference="map_fibre_of_Sat(V^lambda)"}). Indeed, as in *loc.cit.* it suffices to show that for any $\lambda\in\Lambda^+_M$ the object $$\label{space_is_indeed_discrete_for_Sect_2.5.2} \operatorname{Map}_{Shv(\operatorname{Gr}_G)^{M({\mathcal O})}}(\delta_{t^{\lambda}}[-\langle\lambda, 2\check{\rho}\rangle], \operatorname{Sat}(V^{\lambda}))\in\operatorname{Spc}$$ is discrete. The corresponding internal hom in $\operatorname{Vect}$ $${{\mathcal H}om}_{Shv(\operatorname{Gr}_G)^{M({\mathcal O})}}(\delta_{t^{\lambda}}[-\langle\lambda, 2\check{\rho}\rangle], \operatorname{Sat}(V^{\lambda}))$$ is placed in degrees $\ge 0$. Applying the Dold-Kan functor $\operatorname{Vect}\to\operatorname{Vect}^{\le 0}\to \operatorname{Spc}$, we see that ([\[space_is_indeed_discrete_for_Sect_2.5.2\]](#space_is_indeed_discrete_for_Sect_2.5.2){reference-type="ref" reference="space_is_indeed_discrete_for_Sect_2.5.2"}) is discrete. Thus, ([\[functor_for_IC_SI_parabolic\]](#functor_for_IC_SI_parabolic){reference-type="ref" reference="functor_for_IC_SI_parabolic"}) naturally upgrades to a functor with values in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$. **Definition 1**. *Let $\operatorname{IC}^{\frac{\infty}{2}}_P\in Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ be given by $$\label{Def_IC_semi_inf} \operatorname{IC}^{\frac{\infty}{2}}_P=\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}} t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle]$$* ### {#section-58} Let $\mu\in\Lambda^+_M$. One similarly gets a diagram $$(\{\lambda\in \Lambda^+_{M, ab}, \lambda+\mu\in\Lambda^+\}, \le)\to Shv(\operatorname{Gr}_G)^{M({\mathcal O})}, \lambda\mapsto t^{-\lambda}\operatorname{Sat}(V^{\lambda+\mu})[\langle\lambda, 2\check{\rho}\rangle]$$ Let us only indicate the transition maps. Let $\lambda, \lambda_i\in\Lambda^+_{M, ab}$ with $\lambda_2=\lambda_1+\lambda$ and $\lambda\in\Lambda^+$. The transition morphism $$t^{-\lambda_1}\operatorname{Sat}(V^{\lambda_1+\mu})[\langle\lambda_1, 2\check{\rho}\rangle]\to t^{-\lambda_2}\operatorname{Sat}(V^{\lambda_2+\mu})[\langle\lambda_2, 2\check{\rho}\rangle]$$ is the composition $$\begin{gathered} t^{-\lambda_1}\operatorname{Sat}(V^{\lambda_1+\mu})[\langle\lambda_1, 2\check{\rho}\rangle]\,{\widetilde\to}\, t^{-\lambda_1}\delta_{1,\operatorname{Gr}_G} \ast \operatorname{Sat}(V^{\lambda_1+\mu})[\langle\lambda_1, 2\check{\rho}\rangle]\stackrel{(\ref{map_fibre_of_Sat(V^lambda)})}{\to}\\ t^{-\lambda_1}t^{-\lambda}\operatorname{Sat}(V^{\lambda})\ast \operatorname{Sat}(V^{\lambda_1+\mu})[\langle\lambda+\lambda_1, 2\check{\rho}\rangle]\,{\widetilde\to}\, t^{-\lambda_2}\operatorname{Sat}(V^{\lambda}\otimes V^{\lambda_1+\mu})[\langle\lambda_2, 2\check{\rho}\rangle]\stackrel{u^{\lambda, \lambda_1+\mu}}{\to}\\ t^{-\lambda_2}\operatorname{Sat}(V^{\lambda_2+\mu})[\langle\lambda_2, 2\check{\rho}\rangle]\end{gathered}$$ **Definition 1**. *For $\mu\in\Lambda^+_M$ define $\operatorname{IC}^{\frac{\infty}{2}}_{P,\mu}\in Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ by $$\label{def_SI_IC_for_mu_stratum} \operatorname{IC}^{\frac{\infty}{2}}_{P,\mu}=\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda^+_{M, ab}, \; \lambda+\mu\in\Lambda^+} \, t^{-\lambda}\operatorname{Sat}(V^{\lambda+\mu})[\langle\lambda, 2\check{\rho}\rangle]$$ So, $\operatorname{IC}^{\frac{\infty}{2}}_P=\operatorname{IC}^{\frac{\infty}{2}}_{P,0}$.* ### {#section-59} For $\theta\in\Lambda_{G,P}$, $V\in{\operatorname{Rep}}(\check{M})^{\heartsuit}$ let $V_{\theta}$ be the subspace of $V$ on which the center $Z(\check{M})$ of $\check{M}$ acts by $\theta$. The first properties of $\operatorname{IC}^{\frac{\infty}{2}}_{P,\mu}$ are as follows. **Proposition 1**. *Let $\eta\in\Lambda_M^+$.\ a) The object $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ belongs to $Shv(\operatorname{Gr}_G)^H\subset Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$.\ b) $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ is the extension by zero from $\bar S^{\eta}_P$. The ind-scheme $\bar S^{\eta}_P$ is stratified by $S^{\nu}_P$ with $\nu\in\Lambda^+_M$ such that $\eta-\nu\in\Lambda^{pos}$.* *c) One has $i_{\eta}^*\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\,{\widetilde\to}\, i_{\eta}^!\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\,{\widetilde\to}\, \omega[-\langle\eta, 2\check{\rho}\rangle]$ over $S^{\eta}_P$.* *d) $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ belongs to $Shv(\operatorname{Gr}_G)^{H, \heartsuit}$. It admits no subobjects in $Shv(\operatorname{Gr}_G)^{H, \heartsuit}$, which are extensions by zero from $\bar S^{\eta}_P - S^{\eta}_P$.* *Proof.* a) Pick $\lambda\in\Lambda^+_{M, ab}$. Let us show that $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ is $H_{\lambda}$-equivariant. For any $\mu\ge\lambda$, that is, with $\mu-\lambda\in\Lambda^+_{M, ab}$ the object $t^{-\mu}\operatorname{Sat}(V^{\mu+\eta})$ is naturally $t^{-\mu}G({\mathcal O})t^{\mu}$-equivariant. Since $H_{\lambda}\subset H_{\mu}\subset t^{-\mu}G({\mathcal O})t^{\mu}$, our claim follows, because $\{\mu\in\Lambda^+_{M, ab}\mid\mu\ge\lambda\}\subset \Lambda^+_{M, ab}$ is cofinal. b\) It suffices to show that for any $\lambda\in\Lambda^+_{M, ab}$ one has $t^{-\lambda}\overline{\operatorname{Gr}}_G^{\lambda+\eta}\subset \bar S^{\eta}_P$. Indeed, let $\nu\in\Lambda^+_M, \lambda\in\Lambda^+_{M, ab}$ and $$S^{\nu}_P\cap (t^{-\lambda}\overline{\operatorname{Gr}}_G^{\lambda+\eta})\ne \emptyset$$ By Remark [Remark 1](#Rem_2.3.7){reference-type="ref" reference="Rem_2.3.7"}, $S^{\nu+\lambda}_P\cap \overline{\operatorname{Gr}}_G^{\lambda+\eta}\ne \emptyset$ and $\eta-\nu\in \Lambda^{pos}$. c\) Let $i_{t^{\eta}}: \operatorname{Spec}k\to \operatorname{Gr}_G$ be the point $t^{\eta}$. By Lemma [Lemma 1](#Lm_2.3.8){reference-type="ref" reference="Lm_2.3.8"}, it suffices to show that $$(i_{t^{\eta}})^!\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\,{\widetilde\to}\,e[-\langle\eta, 2\check{\rho}\rangle]$$ We are calculating the colimit of the $!$-fibres at $t^{\eta}\in\operatorname{Gr}_G$ of $$t^{-\lambda}\operatorname{Sat}(V^{\lambda+\eta})[\langle\lambda, 2\check{\rho}\rangle]$$ over $\lambda\in\Lambda^+_{M, ab}$. Each term of this diagram identifies canonically with $e[-\langle\eta, 2\check{\rho}\rangle]$, and the transition maps are the identity. The claim follows. d\) **Step 1**. We show that $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\in Shv(\operatorname{Gr}_G)^{H,\ge 0}$. Let $0\ne \eta-\nu\in\Lambda^{pos}$ with $\nu\in\Lambda^+_M$. We check that $(i_{\nu}(i^{\nu}_P))^!\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ is placed in perverse degrees $> \langle\nu, 2\check{\rho}-2\check{\rho}_M\rangle$ over $\operatorname{Gr}_M^{\nu}$. If $\lambda\in\Lambda_{M, ab}^+$ is large enough for $\nu$ then $\lambda+\nu\in\Lambda^+_G$. It suffices to show that for such $\lambda$, $$(i_{\nu+\lambda}(i^{\nu+\lambda}_P))^!\operatorname{Sat}(V^{\lambda+\eta})[\langle\lambda, 2\check{\rho}\rangle]$$ is placed in perverse degrees $> \langle\nu, 2\check{\rho}-2\check{\rho}_M\rangle$ over $\operatorname{Gr}_M^{\nu+\lambda}$. We have $\operatorname{Gr}_M^{\nu+\lambda}\subset \operatorname{Gr}_G^{\nu+\lambda}$. The !-restriction of $\operatorname{Sat}(V^{\lambda+\eta})$ to $\operatorname{Gr}_G^{\nu+\lambda}$ is placed in perverse degrees $>0$, and has smooth perverse cohomology sheaves. Recall that $\dim\operatorname{Gr}_G^{\nu+\lambda}=\langle\nu+\lambda, 2\check{\rho}\rangle$. For any bounded complex on $\operatorname{Gr}_G^{\nu+\lambda}$ placed in perverse degrees $>0$ and having smooth perverse cohomology sheaves, its !-restriction to $\operatorname{Gr}_M^{\nu+\lambda}$ is placed in perverse degrees $>\operatorname{codim}(\operatorname{Gr}_M^{\nu+\lambda}, \operatorname{Gr}_G^{\nu+\lambda})=\langle\nu+\lambda, 2\check{\rho}-2\check{\rho}_M\rangle$. Since $\langle\lambda, 2\check{\rho}_M\rangle=0$, our claim follows. We proved actually that for $\nu$ as above, $(i_{\nu})^!\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\in Shv(S^{\nu}_P)^{H, >0}$. **Step 2**. Let us show that $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\in Shv(\operatorname{Gr}_G)^{H,\le 0}$. It suffices to show that for $\theta\in\Lambda_{G,P}$, $$(v^{\theta}_P)^*\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\in Shv(\operatorname{Gr}^{\theta}_P)^{H, \le 0}$$ or, equivalently, $(\mathfrak{t}^{\theta}_P)_!(v^{\theta}_P)^*\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ is placed in perverse degrees $\le \langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$ over $\operatorname{Gr}_M^{\theta}$. We have $$(\mathfrak{t}^{\theta}_P)_!(v^{\theta}_P)^*\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\,{\widetilde\to}\,\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M,ab}^+, \; \lambda+\eta\in\Lambda^+} \; (\mathfrak{t}^{\theta}_P)_!(v^{\theta}_P)^*(t^{-\lambda}\operatorname{Sat}(V^{\lambda+\eta}))[\langle\lambda, 2\check{\rho}\rangle]$$ The latter identifies with $$\label{complex_for_d_Pp_2.4.6} \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab}^+, \; \lambda+\eta\in\Lambda^+} \; t^{-\lambda}(\mathfrak{t}^{\theta+\bar\lambda}_P)_!(v^{\theta+\bar\lambda}_P)^*\operatorname{Sat}(V^{\lambda+\eta})[\langle\lambda, 2\check{\rho}\rangle]$$ By Proposition [Proposition 1](#Pp_gRes_for_Levi){reference-type="ref" reference="Pp_gRes_for_Levi"}, $$(\mathfrak{t}^{\theta+\bar\lambda}_P)_!(v^{\theta+\bar\lambda}_P)^*\operatorname{Sat}(V^{\lambda+\eta})[\langle\theta+\bar\lambda, 2\check{\rho}-2\check{\rho}_M\rangle]\,{\widetilde\to}\, \operatorname{Sat}_M((V^{\lambda+\eta})_{\theta+\bar\lambda})$$ So, ([\[complex_for_d\_Pp_2.4.6\]](#complex_for_d_Pp_2.4.6){reference-type="ref" reference="complex_for_d_Pp_2.4.6"}) identifies with $$\label{diag_after_Jacquet_functors_for_theta} \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab}^+, \; \lambda+\eta\in\Lambda^+} \; t^{-\lambda}\operatorname{Sat}_M((V^{\lambda+\eta})_{\theta+\bar\lambda})[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$$ This shows that $(v^{\theta}_P)^*\operatorname{IC}^{\frac{\infty}{2}}_P\in Shv(\operatorname{Gr}^{\theta}_P)^{H, \heartsuit}$ for any $\theta\in\Lambda_{G,P}$. Our claim follows. ◻ **Remark 1**. *i) The object $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ is the extension by zero from the connected component $\operatorname{Gr}_G^{\bar\eta}$ of $\operatorname{Gr}_G$, where $\bar\eta\in \Lambda_{G,G}$ is the image of $\eta$.\ ii) If $P=G$ then $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\,{\widetilde\to}\, \operatorname{Sat}(V^{\eta})$ canonically.* ### Another presentation of $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ The inclusion $Shv(\operatorname{Gr}_G)^H\stackrel{}{\hookrightarrow} Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ commutes with ${\operatorname{Rep}}(\check{G})$-actions. Since ${\operatorname{Rep}}(\check{G})$ is rigid, on the left adjoint $\operatorname{Av}^{U(P)(F)}_!: Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_G)^H$ the left-lax ${\operatorname{Rep}}(\check{G})$-structure is strict by ([@G], ch. I.1, 9.3.6). Let $\eta\in\Lambda^+_M$. Using Lemma [Lemma 1](#Lm_2.3.13_about_Av!){reference-type="ref" reference="Lm_2.3.13_about_Av!"} and applying $\operatorname{Av}^{U(P)(F)}_!$ to ([\[def_SI_IC_for_mu_stratum\]](#def_SI_IC_for_mu_stratum){reference-type="ref" reference="def_SI_IC_for_mu_stratum"}) we get $$\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\,{\widetilde\to}\,\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab}^+,\, \lambda+\eta\in\Lambda^+}\, \boldsymbol{\vartriangle}^{-\lambda}\ast \operatorname{Sat}(V^{\lambda+\eta}),$$ where the colimit is taken in $Shv(\operatorname{Gr}_G)^H$. Generalizing ([@Gai19SI], 1.5.6) we have following. **Theorem 1**. *For any $\lambda\in\Lambda_{M, ab}$, $\boldsymbol{\vartriangle}^{\lambda}\in Shv(\operatorname{Gr}_G)^{H, \heartsuit}$.* The proof is postponed to Section [4.6](#Sect_4.6){reference-type="ref" reference="Sect_4.6"}. ### {#section-60} Let $\theta\in\Lambda_{G,P}, \eta\in\Lambda^+_M$. Consider the diagram $$\label{diag_for_Rep(checkM)} \{\lambda\in\Lambda_{M, ab}, \; \lambda+\eta\in\Lambda^+\} \to {\operatorname{Rep}}(\check{M}), \;\; \lambda\mapsto (e^{-\lambda}\otimes V^{\lambda+\eta})_{\theta}$$ obtained from ([\[diag_for_Sect_2.1.7\]](#diag_for_Sect_2.1.7){reference-type="ref" reference="diag_for_Sect_2.1.7"}) by restricting to $\check{M}$ and imposing on each term the condition that $Z(\check{M})$ acts by $\theta$. Write ${\mathcal O}(U(\check{P}))$ for the ring of regular functions on $U(\check{P})$. By Lemma [Lemma 1](#Lm_2.1.8_some_colimit){reference-type="ref" reference="Lm_2.1.8_some_colimit"}, $$\mathop{\operatorname{colim}}_{\lambda\in \Lambda_{M, ab}, \; \lambda+\eta\in\Lambda^+}\; (e^{-\lambda}\otimes V^{\lambda+\eta})_{\theta}\,{\widetilde\to}\, ({\mathcal O}(U(\check{P}))\otimes U^{\eta})_{\theta}$$ Here $\check{M}$ acts (on the left) diagonally on ${\mathcal O}(U(\check{P}))\otimes U^{\eta}$, the action on the first factor comes from the adjoint $\check{M}$-action on $U(\check{P})$. **Proposition 1**. *Let $\theta\in\Lambda_{G,P}$, $\eta\in\Lambda^+_M$. One has canonically $$(\mathfrak{t}^{\theta}_P)_!(v^{\theta}_P)^*\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}[\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]\;{\widetilde\to}\; Sat_M(({\mathcal O}(U(\check{P}))\otimes U^{\eta})_{\theta})$$ in $Shv(\operatorname{Gr}_M)^{M({\mathcal O})}$. So, $$(v^{\theta}_P)^*\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}[\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]\;{\widetilde\to}\; (\mathfrak{t}^{\theta}_P)^!Sat_M(({\mathcal O}(U(\check{P}))\otimes U^{\eta})_{\theta})$$ in $Shv(\operatorname{Gr}_P^{\theta})^H$.* *Proof.* Applying $\operatorname{Sat}_M: {\operatorname{Rep}}(\check{M})\to \operatorname{Perv}(\operatorname{Gr}_M)^{M({\mathcal O})}$ to ([\[diag_for_Rep(checkM)\]](#diag_for_Rep(checkM)){reference-type="ref" reference="diag_for_Rep(checkM)"}) and further tensoring by $e[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$ one gets the diagram ([\[diag_after_Jacquet_functors_for_theta\]](#diag_after_Jacquet_functors_for_theta){reference-type="ref" reference="diag_after_Jacquet_functors_for_theta"}). The first claim follows as in Step 2 in the proof of Proposition [Proposition 1](#Pp_2.4.7_first_propeties){reference-type="ref" reference="Pp_2.4.7_first_propeties"} d). The second one follows from Lemma [Lemma 1](#Lm_2.3.8){reference-type="ref" reference="Lm_2.3.8"} ii). ◻ **Remark 1**. *Assume that $G\ne P$. It is easy to see that if $\theta\in\Lambda_{G,P}, \eta\in\Lambda^+_M$ then $(\mathfrak{t}_P^{\theta})_*(v^{\theta}_P)^!\operatorname{IC}_{P,\eta}^{\frac{\infty}{2}}$ is infinitely coconnective in the t-structure on $Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$.* **Remark 1**. *Let $\eta\in\Lambda^+_M$. If $G\ne P$ then $\operatorname{IC}^{\frac{\infty}{2}}_{P, \eta}$ is not the intermediate extension under $S^{\eta}_P\stackrel{}{\hookrightarrow} \bar S^{\eta}_P$. Indeed, one can easily find $0\ne \theta'\in -\Lambda_{G,P}^{pos}$ such that ${\mathcal O}(U(\check{P}))_{\theta'}\ne 0$. Let $\theta-\theta'$ be the image of $\eta$ in $\Lambda_{G,P}$. Then $(v^{\theta}_P)^*\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}\in Shv(\operatorname{Gr}_P^{\theta})^{H, \heartsuit}$ is not zero by Proposition [Proposition 1](#Pp_2.5.15_*-restriction){reference-type="ref" reference="Pp_2.5.15_*-restriction"}.* For $\eta\in\Lambda^+$ write $i^{\eta}_M: \overline{\operatorname{Gr}}_M^{\eta}\stackrel{}{\hookrightarrow} \overline{\operatorname{Gr}}_G^{\eta}$ for the natural closed immersion. **Lemma 1**. *Let $\eta\in\Lambda^+$.\ i) The complex $(i^{\eta}_M)^!\operatorname{Sat}(V^{\eta})$ is placed in perverse degrees $\ge \langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle$.\ ii) There is a canonical morphism $$\label{map_for_Lm_2.5.17} \operatorname{Sat}_M(U^{\eta})\ast \delta_{1,\operatorname{Gr}_G}[-\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle]\to \operatorname{Sat}(V^{\eta})$$ in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$, which after applying $(i^{\eta}_M)^!$ becomes an isomorphism on perverse cohomology sheaves in degree $\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle$.* *Proof.* **Step 1** Let us show that $\operatorname{Gr}_G^{\eta}\cap \overline{\operatorname{Gr}}_M^{\eta}=\operatorname{Gr}_M^{\eta}$. Let $\nu\in\Lambda^+_M$ with $0\ne \eta-\nu\in\Lambda^{pos}_M$ such that $\operatorname{Gr}_G^{\eta}\cap\operatorname{Gr}_M^{\nu}\ne\emptyset$. We must get a contradiction. Pick $w\in W$ such that $\nu=w\eta$ and the length of $w$ is minimal with with property. We claim that if $i\in {\mathcal I}_M$ then $w^{-1}(\check{\alpha}_i)$ is a positive root. Indeed, if $w^{-1}(\check{\alpha}_i)$ is negative then $\langle\eta, w^{-1}(\check{\alpha}_i)\rangle\le 0$ on one hand, and on the other hand $\langle\nu, \check{\alpha}_i\rangle=\langle\eta, w^{-1}(\check{\alpha}_i)\rangle\ge 0$. So, $\langle\eta, w^{-1}(\check{\alpha}_i)\rangle=0$, hence $s_{\alpha}w\eta=w\eta$. By ([@Spr], Lemma 8.3.2), $\ell(s_{\alpha}w)<\ell(w)$. This conradicts our choice of $w$. The above implies that for $i\in {\mathcal I}_M$, $w^{-1}(\alpha_i)$ is a positive coroot. Applying $w^{-1}$ to $\eta-w\eta\in \Lambda^{pos}_M$ we get $0\ne w^{-1}\eta-\eta\in \Lambda^{pos}_M$. This is impossible, because $\eta\in\Lambda^+$ and for $w'\in W$, $\eta-w'\eta\in\Lambda^{pos}$. Our claim follows. Note that for $i: \operatorname{Gr}_M^{\eta}\stackrel{}{\hookrightarrow} \operatorname{Gr}_G^{\eta}$ we have canonically $$\label{iso_for_Step1_Lm_2.5.17} i^!\operatorname{Sat}(V^{\eta})\,{\widetilde\to}\, \operatorname{Sat}_M(U^{\eta})[-\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle]\mid_{\operatorname{Gr}_M^{\eta}}$$ **Step 2**. Let $\nu\in\Lambda^+_M$ with $0\ne \eta-\nu\in\Lambda^{pos}_M$. For i) it suffices to show that the $!$-restriction of $\operatorname{Sat}(V^{\eta})$ under $\operatorname{Gr}_M^{\nu}\stackrel{}{\hookrightarrow} \overline{\operatorname{Gr}}_G^{\eta}$ is placed in perverse degrees $\ge \langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle$. Consider the inclusions $$\operatorname{Gr}_M^{\nu}\stackrel{}{\hookrightarrow} \operatorname{Gr}_G^{\nu}\stackrel{}{\hookrightarrow} \overline{\operatorname{Gr}}_G^{\eta}$$ The $!$-restriction of $\operatorname{Sat}(V^{\eta})$ to $\operatorname{Gr}_G^{\nu}$ is placed in perverse degrees $>0$ and has smooth perverse cohomology sheaves. For any bounded complex on $\operatorname{Gr}_G^{\nu}$ placed in perverse degrees $>0$ and having smooth perverse cohomology sheaves, its !-restriction to $\operatorname{Gr}_M^{\nu}$ is placed in perverse degrees $>\operatorname{codim}(\operatorname{Gr}_M^{\nu}, \operatorname{Gr}_G^{\nu})=\langle\nu, 2\check{\rho}-2\check{\rho}_M\rangle=\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle$, because the image of $\eta-\nu$ vanishes in $\Lambda_{G,P}$. Part i) follows. Moreover, the above gives a unique map ([\[map_for_Lm_2.5.17\]](#map_for_Lm_2.5.17){reference-type="ref" reference="map_for_Lm_2.5.17"}) whose restriction to $\operatorname{Gr}_M^{\eta}$ comes from ([\[iso_for_Step1_Lm_2.5.17\]](#iso_for_Step1_Lm_2.5.17){reference-type="ref" reference="iso_for_Step1_Lm_2.5.17"}). ◻ **Proposition 1**. *Let $\eta\in\Lambda^+_M$. One has canonically in $Shv(\operatorname{Gr}_G)^H$ $$\label{iso_M-action_on_IC-SI_Prop_2.5.17} \operatorname{Sat}_M(U^{\eta})\ast \operatorname{IC}^{\frac{\infty}{2}}_P[-\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle]\,{\widetilde\to}\, \operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$$* *Proof.* **Step 1**. In the case $\eta\in\Lambda_{M, ab}, \mu\in\Lambda^+_M$ from ([\[def_SI_IC_for_mu_stratum\]](#def_SI_IC_for_mu_stratum){reference-type="ref" reference="def_SI_IC_for_mu_stratum"}) making the change of variables in the colimit one gets $t^{\eta}\operatorname{IC}^{\frac{\infty}{2}}_{P,\mu}[-\langle\eta, 2\check{\rho}\rangle]\,{\widetilde\to}\, \operatorname{IC}^{\frac{\infty}{2}}_{P,\mu+\eta}$. To establish ([\[iso_M-action_on_IC-SI_Prop_2.5.17\]](#iso_M-action_on_IC-SI_Prop_2.5.17){reference-type="ref" reference="iso_M-action_on_IC-SI_Prop_2.5.17"}) in general, we first reduce to the case $\eta\in\Lambda^+$. For this pick $\lambda\in\Lambda_{M, ab}$ such that $\lambda+\eta\in\Lambda^+$. If $$\operatorname{Sat}_M(U^{\eta+\lambda})\ast \operatorname{IC}^{\frac{\infty}{2}}_P[-\langle\eta+\lambda, 2\check{\rho}-2\check{\rho}_M\rangle]\,{\widetilde\to}\, \operatorname{IC}^{\frac{\infty}{2}}_{P,\eta+\lambda}$$ then applying $t^{-\lambda}[\langle\lambda, 2\check{\rho}\rangle]$ to the latter isomorphism, one gets ([\[iso_M-action_on_IC-SI_Prop_2.5.17\]](#iso_M-action_on_IC-SI_Prop_2.5.17){reference-type="ref" reference="iso_M-action_on_IC-SI_Prop_2.5.17"}) by the above. **Step 2** Assume $\eta\in\Lambda^+$. Let us construct a morphism of functors $$\label{map_of_functors_for_Pp_2.5.18} (\Lambda^+_{M, ab}, \le)\to Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$$ sending $\lambda$ to $$\label{map_first_for_Pp_2.5.18} \operatorname{Sat}_M(U^{\eta})\ast t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle-\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle]\to t^{-\lambda}\operatorname{Sat}(V^{\lambda+\eta})[\langle\lambda, 2\check{\rho}\rangle],$$ here we used the diagram defining $\operatorname{IC}^{\frac{\infty}{2}}_P$ in the LHS and $\operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}$ in the RHS. Since $t^{\lambda}\operatorname{Sat}_M(U^{\eta})\ast \delta_{t^{-\lambda}}\,{\widetilde\to}\, \operatorname{Sat}_M(U^{\eta})$, ([\[map_first_for_Pp_2.5.18\]](#map_first_for_Pp_2.5.18){reference-type="ref" reference="map_first_for_Pp_2.5.18"}) rewrites as $$\operatorname{Sat}_M(U^{\eta})\ast \operatorname{Sat}(V^{\lambda})[-\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle]\to \operatorname{Sat}(V^{\lambda+\eta})$$ We define the latter morphism as the composition $$\begin{gathered} \operatorname{Sat}_M(U^{\eta})\ast \operatorname{Sat}(V^{\lambda})[-\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle]\stackrel{(\ref{map_for_Lm_2.5.17})}{\to} \operatorname{Sat}(V^{\eta})\ast \operatorname{Sat}(V^{\lambda})\\ {\widetilde\to}\,\operatorname{Sat}(V^{\eta}\otimes V^{\lambda})\stackrel{u^{\eta,\lambda}}{\to} \operatorname{Sat}(V^{\lambda+\eta})\end{gathered}$$ These maps naturally upgrade to a morphism of functors ([\[map_of_functors_for_Pp_2.5.18\]](#map_of_functors_for_Pp_2.5.18){reference-type="ref" reference="map_of_functors_for_Pp_2.5.18"}). Passing to the colimit, this gives the morphism ([\[iso_M-action_on_IC-SI_Prop_2.5.17\]](#iso_M-action_on_IC-SI_Prop_2.5.17){reference-type="ref" reference="iso_M-action_on_IC-SI_Prop_2.5.17"}). Let $\theta'$ be the image of $\eta$ in $\Lambda_{G,P}$. To show that ([\[iso_M-action_on_IC-SI_Prop_2.5.17\]](#iso_M-action_on_IC-SI_Prop_2.5.17){reference-type="ref" reference="iso_M-action_on_IC-SI_Prop_2.5.17"}) is an isomorphism, it suffices, in view of Lemma [Lemma 1](#Lm_2.3.8){reference-type="ref" reference="Lm_2.3.8"} ii), to prove that for any $\theta\in\Lambda_{G,P}$ applying the functor $$(\mathfrak{t}_P^{\theta+\theta'})_!(v^{\theta+\theta'}_P)^* : Shv(\operatorname{Gr}_G)^{M({\mathcal O})}\to Shv(\operatorname{Gr}_M^{\theta+\theta'})^{M({\mathcal O})}$$ to ([\[iso_M-action_on_IC-SI_Prop_2.5.17\]](#iso_M-action_on_IC-SI_Prop_2.5.17){reference-type="ref" reference="iso_M-action_on_IC-SI_Prop_2.5.17"}) one gets an isomorphism. By Propositions [Proposition 1](#Pp_2.4.19){reference-type="ref" reference="Pp_2.4.19"} iii) and [Proposition 1](#Pp_2.5.15_*-restriction){reference-type="ref" reference="Pp_2.5.15_*-restriction"} we get $$\begin{gathered} \label{complex_LHS_for_Pp_2.5.18} (\mathfrak{t}_P^{\theta+\theta'})_!(v^{\theta+\theta'}_P)^*(\operatorname{Sat}_M(U^{\eta})\ast \operatorname{IC}^{\frac{\infty}{2}}_P)\,{\widetilde\to}\, \operatorname{Sat}_M(U^{\eta})\ast ((\mathfrak{t}_P^{\theta})_!(v^{\theta}_P)^*\operatorname{IC}^{\frac{\infty}{2}}_P)\,{\widetilde\to}\\ \operatorname{Sat}_M(U^{\eta})\ast \operatorname{Sat}_M(({\mathcal O}(U(\check{P}))_{\theta})[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]\,{\widetilde\to}\\ \operatorname{Sat}_M(({\mathcal O}(U(\check{P})\otimes U^{\eta})_{\theta+\theta'})[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]\end{gathered}$$ and $$\begin{gathered} \label{complex_RHS_for_Pp_2.5.18} (\mathfrak{t}_P^{\theta+\theta'})_!(v^{\theta+\theta'}_P)^* \operatorname{IC}^{\frac{\infty}{2}}_{P, \eta}[\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle]\,{\widetilde\to}\, \operatorname{Sat}_M(({\mathcal O}(U(\check{P})\otimes U^{\eta})_{\theta+\theta'})[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]\end{gathered}$$ One checks that the so obtained map from ([\[complex_LHS_for_Pp_2.5.18\]](#complex_LHS_for_Pp_2.5.18){reference-type="ref" reference="complex_LHS_for_Pp_2.5.18"}) to ([\[complex_RHS_for_Pp_2.5.18\]](#complex_RHS_for_Pp_2.5.18){reference-type="ref" reference="complex_RHS_for_Pp_2.5.18"}) in $Shv(\operatorname{Gr}_M^{\theta+\theta'})^{M({\mathcal O})}$ is an isomorphism. We are done. ◻ ### {#section-61} According to Section [2.2.13](#Sect_2.0.17){reference-type="ref" reference="Sect_2.0.17"}, we interprete Definition [Definition 1](#Def_IC_SI_parabolic){reference-type="ref" reference="Def_IC_SI_parabolic"} as follows. Consider the $\Lambda_{M, ab}$-action on $Shv(\operatorname{Gr}_G)^H$ defined in Section [3.1.8](#Sect_2.2.10_action_of_Lambda_Mab){reference-type="ref" reference="Sect_2.2.10_action_of_Lambda_Mab"}, we also think of it as ${\operatorname{Rep}}(\check{M}_{ab})$-action. It is also equipped with ${\operatorname{Rep}}(\check{G})$-action by right convolutions. Then $\delta_{1,\operatorname{Gr}_G}\in {\mathcal O}(\check{G}/[\check{P},\check{P}])-mod(C)$ for $C=Shv(\operatorname{Gr}_G)^H$, so that $\operatorname{IC}^{\frac{\infty}{2}}_P$ naturally upgrades to an object of $$C\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M}_{ab})} {\operatorname{Rep}}(\check{M}),$$ so has the Hecke property described as in Section [2.2.12](#Sect_version_of_Hecke_property_2.0.16){reference-type="ref" reference="Sect_version_of_Hecke_property_2.0.16"}. In fact, it has a stronger Hecke property given as follows. **Proposition 1**. *$\operatorname{IC}^{\frac{\infty}{2}}_P$ naturally upgrades to an object of $$(Shv(\operatorname{Gr}_G)^H)\otimes_{{\operatorname{Rep}}(\check{G})\otimes{\operatorname{Rep}}(\check{M})} {\operatorname{Rep}}(\check{M}),$$ where we consider the ${\operatorname{Rep}}(\check{M})$-action given by ([\[action_Rep(checkM)\_shifted\]](#action_Rep(checkM)_shifted){reference-type="ref" reference="action_Rep(checkM)_shifted"}), and ${\operatorname{Rep}}(\check{G})$-action by right convolutions.* *Proof.* The structure under consideration is equivalent to the following Hecke property. For $V\in{\operatorname{Rep}}(\check{G})^{\heartsuit}$ one has canonically $$\label{iso_Hecke_propert_of_ICinfty/2_P} \operatorname{IC}^{\frac{\infty}{2}}_P\ast \operatorname{Sat}(V)\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} \operatorname{Sat}_M(U^{\mu})\ast \operatorname{IC}^{\frac{\infty}{2}}_P\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, V)[-\langle\mu, 2\check{\rho}-2\check{\rho}_M\rangle]$$ in a way compatible with the monoidal structures on ${\operatorname{Rep}}(\check{G})^{\heartsuit}$, ${\operatorname{Rep}}(\check{M})^{\heartsuit}$. By Proposition [Proposition 1](#Pp_2.4.19){reference-type="ref" reference="Pp_2.4.19"} the isomorphisms ([\[iso_Hecke_propert_of_ICinfty/2_P\]](#iso_Hecke_propert_of_ICinfty/2_P){reference-type="ref" reference="iso_Hecke_propert_of_ICinfty/2_P"}) take place in the abelian category $Shv(\operatorname{Gr}_G)^{H,\heartsuit}$, so that higher compatibilities will be easy to check. To establish ([\[iso_Hecke_propert_of_ICinfty/2_P\]](#iso_Hecke_propert_of_ICinfty/2_P){reference-type="ref" reference="iso_Hecke_propert_of_ICinfty/2_P"}), we may assume $V$ finite-dimensional. Write $\Lambda^+_{M, ab}(V)$ for the set of $\lambda\in\Lambda^+_{M, ab}$ such that if $\mu\in\Lambda^+_M$ and $U^{\mu}$ appears in $\operatorname{Res}^{\check{M}} V$ then $\mu+\lambda\in\Lambda^+$. Applying Lemma [Lemma 1](#Lm_2.0.15){reference-type="ref" reference="Lm_2.0.15"}, for $\lambda\in \Lambda^+_{M, ab}(V)$ we get $$t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle]\ast \operatorname{Sat}(V)\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} t^{-\lambda}\operatorname{Sat}(V^{\lambda+\mu})\otimes\operatorname{Hom}_{\check{M}}(U^{\mu}, V)[\langle\lambda, 2\check{\rho}\rangle]\ $$ The above isomorphism upgrades to an isomorphism of functors $$(\Lambda^+_{M, ab}(V), \le)\to Shv(\operatorname{Gr}_G)^{M({\mathcal O})},$$ where we used the diagram ([\[Def_IC_semi_inf\]](#Def_IC_semi_inf){reference-type="ref" reference="Def_IC_semi_inf"}) in the LHS, and the diagram ([\[def_SI_IC_for_mu_stratum\]](#def_SI_IC_for_mu_stratum){reference-type="ref" reference="def_SI_IC_for_mu_stratum"}) in the RHS respectively. Passing to the colimit over $(\Lambda^+_{M, ab}(V),\le)$, this gives an isomorphism $$\operatorname{IC}^{\frac{\infty}{2}}_P\ast \operatorname{Sat}(V)\,{\widetilde\to}\, \mathop{\oplus}\limits_{\mu\in\Lambda^+_M} \operatorname{IC}^{\frac{\infty}{2}}_{P,\mu}\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, V)$$ The isomorphism ([\[iso_Hecke_propert_of_ICinfty/2_P\]](#iso_Hecke_propert_of_ICinfty/2_P){reference-type="ref" reference="iso_Hecke_propert_of_ICinfty/2_P"}) follows now from Proposition [Proposition 1](#Pp_action_of_Rep(checkM)_on_SI-IC_P){reference-type="ref" reference="Pp_action_of_Rep(checkM)_on_SI-IC_P"}. The compatibility of ([\[iso_Hecke_propert_of_ICinfty/2_P\]](#iso_Hecke_propert_of_ICinfty/2_P){reference-type="ref" reference="iso_Hecke_propert_of_ICinfty/2_P"}) with the monoidal structure on ${\operatorname{Rep}}(\check{G})^{\heartsuit}$ also follows from the construction. ◻ ## Relation to the $\operatorname{IC}$-sheaf of $\operatorname{\widetilde\operatorname{Bun}}_P$ {#Sect_2.6_Relation} ### {#section-62} Similarly to the case of $B=P$ studied in [@Gai19SI], one has the following. **Theorem 1**. *Let $\eta\in\Lambda^+_M$. There is a canonical isomorphism in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ $$\label{map_for_Thm_restriction_of_glob_first} \operatorname{IC}^{\frac{\infty}{2}}_{P,\eta}[\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle] \,{\widetilde\to}\, (\tilde\pi^M))^!\operatorname{IC}_{\widetilde{\operatorname{glob}}}^{\eta}[(g-1)\dim P]$$* ### {#section-63} By base change, for $\eta\in\Lambda_M^+$ one has canonically $$(\tilde\pi^M)^!\nabla^{\eta}_{glob}[(g-1)\dim P]\,{\widetilde\to}\, \nabla^{\eta}[\langle\eta, 2\check{\rho}-2\check{\rho}_M\rangle]$$ Let $j_{glob}: \operatorname{Bun}_P\stackrel{}{\hookrightarrow} \operatorname{\widetilde\operatorname{Bun}}_P$ be the natural open immersion. Write $\tilde\pi^{M, 0}: M({\mathcal O})\backslash S^0_P\to \operatorname{Bun}_P$ for the restriction of $\tilde\pi^M$. Then $(\tilde\pi^{M,0})_!(\tilde\pi^{M, 0})^!\omega\to\omega$ yields a morphism $$\label{map_for_Th_2.6.4} \boldsymbol{\vartriangle}^0\to (\tilde\pi^M)^!(j_{glob})_!\operatorname{IC}_{\operatorname{Bun}_P}[(g-1)\dim P]$$ in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$. **Theorem 1**. *The map ([\[map_for_Th_2.6.4\]](#map_for_Th_2.6.4){reference-type="ref" reference="map_for_Th_2.6.4"}) is an isomorphism.* ### {#section-64} In the rest of Section [4.2](#Sect_2.6_Relation){reference-type="ref" reference="Sect_2.6_Relation"} we prove Theorems [Theorem 1](#Thm_restriction_of_glob_first){reference-type="ref" reference="Thm_restriction_of_glob_first"} and [Theorem 1](#Th_restriction_of_glob_second){reference-type="ref" reference="Th_restriction_of_glob_second"}. By ([@BG], 4.1.3) for $\eta\in\Lambda^+_M$, one has canonically $$\operatorname{Sat}(U^{\eta})\ast \operatorname{IC}_{\widetilde{glob}}\,{\widetilde\to}\, \operatorname{IC}_{\widetilde{glob}}^{\eta}$$ Since $(\tilde\pi^M)^!$ commutes with ${\operatorname{Rep}}(\check{M})$-actions by left convolutions, Proposition [Proposition 1](#Pp_action_of_Rep(checkM)_on_SI-IC_P){reference-type="ref" reference="Pp_action_of_Rep(checkM)_on_SI-IC_P"} immediately reduces Theorem [Theorem 1](#Thm_restriction_of_glob_first){reference-type="ref" reference="Thm_restriction_of_glob_first"} to its special case $\eta=0$. ### {#section-65} Let us construct the morphism ([\[map_for_Thm_restriction_of_glob_first\]](#map_for_Thm_restriction_of_glob_first){reference-type="ref" reference="map_for_Thm_restriction_of_glob_first"}) for $\eta=0$. It is equivalent to providing a morphism $$\label{map_for_Thm_restriction_of_glob_second} (\tilde\pi^M)_!\operatorname{IC}^{\frac{\infty}{2}}_P\to \operatorname{IC}_{\widetilde{\operatorname{glob}}}[(g-1)\dim P]$$ in $Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)$. We first construct for $\lambda\in\Lambda^+_{M, ab}$ a morphism $$\label{map_for_Sect_2.6.5_from_lambda_to_ICglob} (\tilde\pi^M)_!(t^{-\lambda}\operatorname{Sat}(V^{\lambda}))[\langle\lambda, 2\check{\rho}\rangle]\to \operatorname{IC}_{\widetilde{\operatorname{glob}}}[(g-1)\dim P]$$ in $Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)$. The functor $Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)\to Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)$, $K\mapsto K\ast \operatorname{Sat}(V^{\lambda})$ is both left and right adjoint to the functor $K\mapsto K\ast \operatorname{Sat}((V^{\lambda})^*)$. So, the datum of ([\[map_for_Sect_2.6.5_from_lambda_to_ICglob\]](#map_for_Sect_2.6.5_from_lambda_to_ICglob){reference-type="ref" reference="map_for_Sect_2.6.5_from_lambda_to_ICglob"}) is equivalent to a morphism $$(\tilde\pi^M)_!(\delta_{t^{-\lambda}})[\langle\lambda, 2\check{\rho}\rangle]\to \operatorname{IC}_{\widetilde{\operatorname{glob}}}\ast \operatorname{Sat}((V^{\lambda})^*)[(g-1)\dim P]$$ in $Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)$. Let $\theta\in\Lambda_{G,P}$ be the image of $\lambda$. By the Hecke property of $\operatorname{IC}_{\widetilde{glob}}$, $$\operatorname{IC}_{\widetilde{\operatorname{glob}}}\ast \operatorname{Sat}((V^{\lambda})^*)\,{\widetilde\to}\,\mathop{\oplus}\limits_{\mu\in\Lambda^+_M} \operatorname{IC}_{\widetilde{glob}}^{\mu}\otimes \operatorname{Hom}_{\check{M}}(U^{\mu}, (V^{\lambda})^*)$$ Note that if $\mu\in\Lambda^+_M$ and $\operatorname{Hom}_{\check{M}}(U^{\mu}, (V^{\lambda})^*)\ne 0$ then $\lambda+\mu\in\Lambda^{pos}$ and $$\tilde\pi(t^{-\lambda})\in {_{x, \lambda}\operatorname{\widetilde\operatorname{Bun}}_P}\subset {_{x, \ge -w_0^M(\mu)}\operatorname{\widetilde\operatorname{Bun}}_P}$$ As in [@BG], for $\nu\in\Lambda^+_M$ write $_x{\mathcal H}^{\nu}_M$ for the stack classifying $({\mathcal F}_M, {\mathcal F}'_M,\beta)$, where ${\mathcal F}_M, {\mathcal F}'_M$ are $M$-torsors on $X$ with an isomorphism $\beta: {\mathcal F}_M\,{\widetilde\to}\,{\mathcal F}'_M\mid_{X-x}$ such that ${\mathcal F}'_M$ is in the position $\le \nu$ with respect to ${\mathcal F}_M$ at $x$. Let $$h^{\leftarrow}_M, h^{\rightarrow}_M: {_x{\mathcal H}^{\nu}_M}\to\operatorname{Bun}_M$$ be the map sending the above point to ${\mathcal F}_M$ and ${\mathcal F}'_M$ respectively. Let $\mu\in\Lambda^+_M$ and $\operatorname{Hom}_{\check{M}}(U^{\mu}, (V^{\lambda})^*)\ne 0$. Consider the stack $_x{\mathcal H}^{-\lambda}_M\times_{\operatorname{Bun}_M}\operatorname{Bun}_P^{-\theta}$, where we used the map $h^{\rightarrow}_M$ to form the fibred product. Consider the locally closed immersion $$h: {_x{\mathcal H}^{-\lambda}_M\times_{\operatorname{Bun}_M}\operatorname{Bun}_P^{-\theta}}\stackrel{}{\hookrightarrow} {_{x,\ge -w_0^M(\mu)}\operatorname{\widetilde\operatorname{Bun}}_P}$$ sending $$({\mathcal F}_M, {\mathcal F}'_P, {\mathcal F}'_M={\mathcal F}'_P\times^P M, \beta: {\mathcal F}_M\,{\widetilde\to}\, {\mathcal F}'_M\mid_{X-x})$$ to $({\mathcal F}_M, {\mathcal F}_G, \kappa)$, where ${\mathcal F}_G={\mathcal F}'_P\times^P G$. Then $h^!\operatorname{IC}_{\widetilde{glob}}^{\mu}$ has smooth perverse cohomology sheaves and is placed in perverse degrees $\ge 0$, and the inequality is strict unless $\mu=-\lambda$. Since $\tilde\pi(t^{-\lambda})$ lies in the image of $h$, the $!$-fibre of $\operatorname{IC}_{\widetilde{glob}}^{\mu}$ at $\tilde\pi(t^{-\lambda})$ is placed in degrees $$\ge \dim({_x{\mathcal H}^{-\lambda}_M\times_{\operatorname{Bun}_M}\operatorname{Bun}_P^{-\theta}})=\dim(\operatorname{Bun}_P^{-\theta})= (g-1)\dim P-\langle\lambda, 2\check{\rho}\rangle,$$ and the inequality is strict unless $\mu=-\lambda$. We conclude that the $!$-fibre at $\tilde\pi(t^{-\lambda})$ of $$\operatorname{IC}_{\widetilde{\operatorname{glob}}}\ast \operatorname{Sat}((V^{\lambda})^*)[(g-1)\dim P-\langle\lambda, 2\check{\rho}\rangle]$$ is placed in degrees $\ge 0$, and its 0-th cohomology identifies with the $!$-fibre at $\tilde\pi(t^{-\lambda})$ of $$\operatorname{IC}_{\widetilde{glob}}^{-\lambda}[(g-1)\dim P-\langle\lambda, 2\check{\rho}\rangle]$$ The latter $!$-fibre identifies with $e$, and the corresponding map is $M({\mathcal O})$-equivariant by construction. This gives the desired map ([\[map_for_Sect_2.6.5_from_lambda_to_ICglob\]](#map_for_Sect_2.6.5_from_lambda_to_ICglob){reference-type="ref" reference="map_for_Sect_2.6.5_from_lambda_to_ICglob"}). ### {#section-66} One checks that the maps ([\[map_for_Sect_2.6.5_from_lambda_to_ICglob\]](#map_for_Sect_2.6.5_from_lambda_to_ICglob){reference-type="ref" reference="map_for_Sect_2.6.5_from_lambda_to_ICglob"}) are compatible in the homotopy category with the transition maps in the diagram ([\[Def_IC_semi_inf\]](#Def_IC_semi_inf){reference-type="ref" reference="Def_IC_semi_inf"}). ### {#section-67} Since the internal hom with respect to the $\operatorname{Vect}$-action $${{\mathcal H}om}_{Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)}((\tilde\pi^M)_!(t^{-\lambda}\operatorname{Sat}(V^{\lambda}))[\langle\lambda, 2\check{\rho}\rangle], \operatorname{IC}_{\widetilde{\operatorname{glob}}}[(g-1)\dim P])$$ is placed in degrees $\ge 0$, we conclude applying the Dold-Kan functor $\operatorname{Vect}\to \operatorname{Vect}^{\le 0}\to\operatorname{Spc}$ that $$\operatorname{Map}_{Shv(_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P)}((\tilde\pi^M)_!(t^{-\lambda}\operatorname{Sat}(V^{\lambda}))[\langle\lambda, 2\check{\rho}\rangle], \operatorname{IC}_{\widetilde{\operatorname{glob}}}[(g-1)\dim P])\in\operatorname{Spc}$$ is discrete. So, as in ([@Gai19SI], 3.4.7) the maps ([\[map_for_Sect_2.6.5_from_lambda_to_ICglob\]](#map_for_Sect_2.6.5_from_lambda_to_ICglob){reference-type="ref" reference="map_for_Sect_2.6.5_from_lambda_to_ICglob"}) uniquely combine to the desired morphism ([\[map_for_Thm_restriction_of_glob_second\]](#map_for_Thm_restriction_of_glob_second){reference-type="ref" reference="map_for_Thm_restriction_of_glob_second"}). ### {#Sect_2.6.9_describing_the_composition} As in ([@Gai19SI], 3.4.8), for $\lambda\in\Lambda^+_{M, ab}$ one may describe the composition $$\begin{gathered} \label{map_for_Sect_2.6.8_long_composition} t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle]\to \operatorname{IC}^{\frac{\infty}{2}}_P\,\stackrel{(\ref{map_for_Thm_restriction_of_glob_first})}{\to} \,(\tilde\pi^M)^!\operatorname{IC}_{\widetilde{\operatorname{glob}}}[(g-1)\dim P]\to \\ (\tilde\pi^M)^!\nabla_{\operatorname{glob}}^0[(g-1)\dim P]\,{\widetilde\to}\, \nabla^0\end{gathered}$$ as follows. The base change under $t^{-\lambda}\overline{\operatorname{Gr}}_G^{\lambda}\subset \bar S^0_P$ of $j_0: S^0_P\stackrel{}{\hookrightarrow}\bar S^0_P$ is the open immersion $S^0_P\cap (t^{-\lambda}\overline{\operatorname{Gr}}_G^{\lambda})\subset t^{-\lambda}\overline{\operatorname{Gr}}_G^{\lambda}$, and $\nabla_{\operatorname{glob}}^0$ is the extension by zero under $\bar i_0: \bar S^0_P\stackrel{}{\hookrightarrow}\operatorname{Gr}_G$. Besides, $S^{\lambda}_P\cap \overline{\operatorname{Gr}}_G^{\lambda}\subset \operatorname{Gr}_G^{\lambda}$. As a morphism on $\bar S^0_P$, the map ([\[map_for_Sect_2.6.8_long_composition\]](#map_for_Sect_2.6.8_long_composition){reference-type="ref" reference="map_for_Sect_2.6.8_long_composition"}) equals $$t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle]\to (j_0)_*(j_0)^*(t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle])\,{\widetilde\to}\, (j_0)_*\omega_{(t^{-\lambda}\operatorname{Gr}_G^{\lambda})\cap S^0_P}\to (j_0)_*\omega_{S^0_P}$$ We used the isomorphism $(j_0)^*(t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle])\,{\widetilde\to}\,\omega_{(t^{-\lambda}\operatorname{Gr}_G^{\lambda})\cap S^0_P}$, where the RHS is considered as extended by zero under the closed immersion $$(t^{-\lambda}\operatorname{Gr}_G^{\lambda})\cap S^0_P\stackrel{}{\hookrightarrow} S^0_P$$ ### {#section-68} For $\theta\in\Lambda_{G,P}$ and a $M$-torsor ${\mathcal F}'_M$ on $X$ write $\operatorname{Gr}_{M, x}^{\theta}({\mathcal F}'_M)$ for the ind-scheme classifying $({\mathcal F}_M, \beta_M)$, where ${\mathcal F}_M$ is a $M$-torsor on $X$, $\beta_M: {\mathcal F}_M\,{\widetilde\to}\, {\mathcal F}'_M\mid_{X-x}$ is an isomorphism such that $\beta_M$ induces an isomorphism of $M/[M,M]$-torsors on $X$ $$\bar \beta_M: {\mathcal F}_{M/[M,M]}\,{\widetilde\to}\, {\mathcal F}'_{M/[M,M]}(-\theta x)$$ Write $_{=\theta, x}\operatorname{\widetilde\operatorname{Bun}}_P$ for the preimage of $_{=\theta, x}\operatorname{\overline{Bun}} _P$ under $\mathfrak{r}: {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P}\to {_{x,\infty}\operatorname{\overline{Bun}} _P}$. One has an isomorphism $\operatorname{Bun}_P\,{\widetilde\to}\, {_{=\theta, x}\operatorname{\overline{Bun}} _P}$ sending ${\mathcal F}_P$ to $({\mathcal F}_{M/[M,M]}(\theta x), {\mathcal F}_P\times_P G, \kappa)$, where ${\mathcal F}_{M/[M,M]}$ is obtained from ${\mathcal F}_P$ by the extension of scalars. The stack $_{=\theta, x}\operatorname{\widetilde\operatorname{Bun}}_P$ classifies triples $({\mathcal F}'_P, {\mathcal F}_M,\beta_M)$, where ${\mathcal F}'_P$ is a $P$-torsor on $X$, ${\mathcal F}_M$ is a $M$-torsor on $X$, $\beta_M: {\mathcal F}_M\,{\widetilde\to}\, {\mathcal F}'_M\mid_{X-x}$ is an isomorphism such that $({\mathcal F}'_M, \beta_M)\in \operatorname{Gr}_{M, x}^{\theta}({\mathcal F}_M)$. The map $\tilde\pi$ fits into a cartesian square $$\begin{array}{ccc} \operatorname{Gr}_P^{\theta} & \stackrel{v^{\theta}_P}{\to} & \operatorname{Gr}_G\\ \downarrow && \downarrow\lefteqn{\scriptstyle \tilde\pi}\\ _{=\theta, x}\operatorname{\widetilde\operatorname{Bun}}_P & \to & {_{x,\infty}\operatorname{\widetilde\operatorname{Bun}}_P} \end{array}$$ ### {#section-69} For $\theta\in\Lambda_{G,P}$ and a $M$-torsor ${\mathcal F}'_M$ on $X$ write $\operatorname{Gr}_{M}^{+, \theta}({\mathcal F}'_M)$ for the version of $\operatorname{Gr}_M^{+,\theta}$, where the background torsor ${\mathcal F}^0_M$ is replaced by ${\mathcal F}'_M$. One has $$\operatorname{Gr}_{M}^{+, \theta}({\mathcal F}'_M)\subset \operatorname{Gr}_{M}^{\theta}({\mathcal F}'_M)$$ For $\theta\in\Lambda_{G,P}^{pos}$ write $\operatorname{Gr}_M^{-,-\theta}$ for the scheme of those $({\mathcal F}_M, \beta_M: {\mathcal F}_M\,{\widetilde\to}\, {\mathcal F}^0_M\mid_{X-x})\in\operatorname{Gr}_M^{-\theta}$ for which $({\mathcal F}^0_M, \beta_M)\in \operatorname{Gr}_M^{+,\theta}({\mathcal F}_M)$. ### {#section-70} For $\theta\in \Lambda_{G,P}^{pos}$ write $X^{\theta}$ for the moduli space of $\Lambda_{G,P}^{pos}$-valued divisor of degree $\theta$. For $\theta\in -\Lambda^{pos}_{G,P}$ write $\operatorname{Mod}_M^{-, \theta}$ for the moduli scheme classifying $D\in X^{-\theta}$, a $M$-torsor ${\mathcal F}_M$ on $X$ with a trivialization $\beta_M: {\mathcal F}_M\,{\widetilde\to}\, {\mathcal F}^0_M\mid_{X-\operatorname{supp}(D)}$ such that for any finite-dimensional $G$-module ${\mathcal V}$ the map $${\mathcal V}^{U(P)}_{{\mathcal F}^0_M}\stackrel{\beta_M^{-1}}{\to} {\mathcal V}^{U(P)}_{{\mathcal F}_M}$$ is regular on $X$, and $\beta_M$ induces an isomorphism ${\mathcal F}_{M/[M,M]}\,{\widetilde\to}\, {\mathcal F}^0_{M/[M,M]}(D)$ on $X$. To be precise, here we pick a homomorphism $\Theta: \Lambda_{G,P}^{pos}\to {\mathbb Z}_+$ of semigroups sending each $\alpha_i$, $i\notin {\mathcal I}_M$ to a strictly positive integer, which allows to associate to $D\in X^{-\theta}$ an effective divisor $\Theta(D)$. Then $X-\operatorname{supp}(D)$ is defined as $X-\operatorname{supp}(\Theta(D))$. Let $\pi_M: \operatorname{Mod}_M^{-,\theta}\to X^{-\theta}$ be the projection sending the above point to $D$. ### {#section-71} The following is established exactly as in ([@Gai19SI], Proposition 3.5.2). **Proposition 1**. *Both$(\tilde\pi^M)^!\operatorname{IC}_{\widetilde{glob}}$ and $(\tilde\pi^M)^!(j_{glob})_!\operatorname{IC}_{\operatorname{Bun}_P}$ belong to $Shv(\operatorname{Gr}_G)^H$. $\square$* **Remark 1**. *One has $\operatorname{Gr}_P^0\cap \bar S^0_P=S^0_P$. Indeed, let $\eta\in\Lambda^+_M\cap (-\Lambda^{pos})$ such that $S^{\eta}_P\subset \operatorname{Gr}_P^0$. Then $\eta=0$ in $\Lambda_{G,P}$. Our claim follows from the fact that, since $[M,M]$ is semi-simple and simply-connected, $\Lambda^+_{[M,M]}\subset ({\mathbb Z}_+$-span of $\alpha_i$ for $i\in{\mathcal I}_M$ in $\Lambda)$. Here $\Lambda^+_{[M,M]}$ is the semigroup of dominant coweights of $[M,M]$.* ### {#section-72} In order to prove Theorem [Theorem 1](#Thm_restriction_of_glob_first){reference-type="ref" reference="Thm_restriction_of_glob_first"} it suffices to show that for any $\theta\in -\Lambda_{G,P}^{pos}$ the map $$(v^{\theta}_P)^* \operatorname{IC}^{\frac{\infty}{2}}_{P} \to (v^{\theta}_P)^*(\tilde\pi^M))^!\operatorname{IC}_{\widetilde{\operatorname{glob}}}[(g-1)\dim P]$$ induced by ([\[map_for_Thm_restriction_of_glob_first\]](#map_for_Thm_restriction_of_glob_first){reference-type="ref" reference="map_for_Thm_restriction_of_glob_first"}) is an isomorphism. In view of Remark [Remark 1](#Rem_intersesion_Gr_P^0_with_barS^0_P){reference-type="ref" reference="Rem_intersesion_Gr_P^0_with_barS^0_P"}, to prove Theorem [Theorem 1](#Th_restriction_of_glob_second){reference-type="ref" reference="Th_restriction_of_glob_second"} it suffices to show that for any $0\ne \theta\in -\Lambda^{pos}_{G,P}$ one has $$(v^{\theta}_P)^*(\tilde\pi^M)^!(j_{glob})_!\operatorname{IC}_{\operatorname{Bun}_P}=0$$ ### {#section-73} In view of Lemmas [Lemma 1](#Lm_theorem_of_Braden_for_theta){reference-type="ref" reference="Lm_theorem_of_Braden_for_theta"} and [Lemma 1](#Lm_2.3.8){reference-type="ref" reference="Lm_2.3.8"} ii), Theorems [Theorem 1](#Thm_restriction_of_glob_first){reference-type="ref" reference="Thm_restriction_of_glob_first"} and [Theorem 1](#Th_restriction_of_glob_second){reference-type="ref" reference="Th_restriction_of_glob_second"} are reduced to the following. **Proposition 1**. *Let $\theta\in -\Lambda_{G,P}^{pos}$.\ i) The map $$(\mathfrak{t}^{\theta}_{P^-})_*(v_{P^-}^{\theta})^!\operatorname{IC}^{\frac{\infty}{2}}_{P}\to (\mathfrak{t}^{\theta}_{P^-})_*(v_{P^-}^{\theta})^!(\tilde\pi^M))^!\operatorname{IC}_{\widetilde{\operatorname{glob}}}[(g-1)\dim P]$$ induced by ([\[map_for_Thm_restriction_of_glob_first\]](#map_for_Thm_restriction_of_glob_first){reference-type="ref" reference="map_for_Thm_restriction_of_glob_first"}) is an isomorphism in $Shv(\operatorname{Gr}_M^{\theta})^{M({\mathcal O})}$.* *ii) If $\theta\ne 0$ then $(\mathfrak{t}^{\theta}_{P^-})_*(v_{P^-}^{\theta})^!(\tilde\pi^M)^!(j_{glob})_!\operatorname{IC}_{\operatorname{Bun}_P}=0$.* ### Recollection about the Zastava space For $\theta\in -\Lambda_{G,P}^{pos}$ denote by ${\mathcal Z}^{\theta}$ the moduli stack classifying $({\mathcal F}_M,\beta_M)\in \operatorname{Mod}_M^{-,\theta}$ for which we set $D=\pi_M({\mathcal F}_M,\beta_M)\in X^{-\theta}$, a $G$-torsor ${\mathcal F}_G$ on $X$ together with a trivialization $\beta: {\mathcal F}_G\,{\widetilde\to}\,{\mathcal F}_M\times_M G\mid_{X-\operatorname{supp}(D)}$ such that two conditions hold: - For any finite-dimensional $G$-module ${\mathcal V}$, the map ${\mathcal V}\to {\mathcal V}_{U(P^-)}$ extends to a regular surjective map of vector bundles on $X$ $${\mathcal V}_{{\mathcal F}_G}\stackrel{\beta}{\to}{\mathcal V}_{{\mathcal F}_M}\to ({\mathcal V}_{U(P^-)})_{{\mathcal F}_M}$$ - For any finite-dimensional $G$-module ${\mathcal V}$, the composition $${\mathcal V}^{U(P)}_{{\mathcal F}^0_M}\,\stackrel{\beta_M}{\hookrightarrow}\, {\mathcal V}^{U(P)}_{{\mathcal F}_M}\to {\mathcal V}_{{\mathcal F}_M}\stackrel{\beta^{-1}}{\to} {\mathcal V}_{{\mathcal F}_G}$$ is a regular morphism of coherent sheaves on $X$. ### {#section-74} As in [@BFGM], ${\mathcal Z}^{\theta}$ is representable by an irreducible quasi-projective scheme. Write $\pi_P: {\mathcal Z}^{\theta}\to \operatorname{Mod}_M^{-,\theta}$ for the map sending the above point to $({\mathcal F}_M,\beta_M)$. Let $\mathfrak{s}: \operatorname{Mod}_M^{-,\theta}\to {\mathcal Z}^{\theta}$ denote the natural section of $\pi_P$. Let $\mathfrak{q}_{{\mathcal Z}}: {\mathcal Z}^{\theta}\to \operatorname{\widetilde\operatorname{Bun}}_P$ be the map sending the above point to $({\mathcal F}^0_M, {\mathcal F}_G,\kappa)$. Set $\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}=\mathfrak{q}_{{\mathcal Z}}^{-1}(\operatorname{Bun}_P)$. The scheme $\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}$ is smooth, and $\dim{\mathcal Z}^{\theta}=-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$. Let $\mathfrak{F}^{\theta}$ be the *central fibre* of ${\mathcal Z}^{\theta}$ over $X^{-\theta}$, that is, the preimage of $-\theta x\in X^{-\theta}$ under $\pi_M\circ\pi_P$. Set also $\overset{\scriptscriptstyle\circ}{\mathfrak{F}}{}^{\theta}=\mathfrak{F}^{\theta}\cap \overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}$. Write $i_{\operatorname{Mod}}: \operatorname{Gr}_M^{-,\theta}\stackrel{}{\hookrightarrow} \operatorname{Mod}_M^{-, \theta}$ for the base change of $\operatorname{Mod}_M^{-,\theta}$ under $-\theta x: \operatorname{Spec}k\to X^{-\theta}$. As in [@BFGM], one gets an isomorphism $$\mathfrak{F}^{\theta}\,{\widetilde\to}\, \bar S^0_P\cap \operatorname{Gr}_{P^-}^{\theta},$$ which restricts to an isomorphism of open subshemes $\overset{\scriptscriptstyle\circ}{\mathfrak{F}}{}^{\theta}\,{\widetilde\to}\, S^0_P\cap \operatorname{Gr}_{P^-}^{\theta}$. The diagram commutes $$\begin{array}{ccc} {\mathcal Z}^{\theta} & \xrightarrow{\; \;\;\;\;\;\;\;\;\;\mathfrak{q}_{{\mathcal Z}}\;\;\;\;\;\;\;\;\;\;} & \operatorname{\widetilde\operatorname{Bun}}_P^0\\ \uparrow && \uparrow\lefteqn{\scriptstyle \tilde\pi}\\ \mathfrak{F}^{\theta} & {\widetilde\to}\, \bar S^0_P\cap \operatorname{Gr}_{P^-}^{\theta}\stackrel{}{\hookrightarrow} & \bar S^0_P \end{array}$$ Let $j: \overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}\stackrel{}{\hookrightarrow} {\mathcal Z}^{\theta}$ be the natural open immersions. The following is an analog of ([@Gai19SI], Proposition 3.6.5). **Proposition 1**. *Let $\theta\in -\Lambda_{G,P}^{pos}$.\ a) One has a canonical isomorphism $$\mathfrak{q}_{{\mathcal Z}}^!\operatorname{IC}_{\widetilde{glob}}[(g-1)\dim P]\,{\widetilde\to}\, \operatorname{IC}_{{\mathcal Z}^{\theta}}[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$$ extending the tautological one over $\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}$. Note that $\dim\operatorname{Bun}_P^0=(g-1)\dim P$.* *b) One has a canonical isomorphism $j_!(\omega_{\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}})\,{\widetilde\to}\, \mathfrak{q}_{{\mathcal Z}}^!(j_{glob})_!\omega_{\operatorname{Bun}_P}$* ### {#section-75} The proof of Proposition [Proposition 1](#Pp_2.6.21_Zastava){reference-type="ref" reference="Pp_2.6.21_Zastava"} is given in Section [4.3](#Sect_Proof_Pp_2.6.21_Zastava){reference-type="ref" reference="Sect_Proof_Pp_2.6.21_Zastava"}. Note that $\mathfrak{q}_{{\mathcal Z}}$ naturally extends to a map ${\mathcal Z}^{\theta}\to \operatorname{\widetilde\operatorname{Bun}}_P^0\times_{\operatorname{Bun}_G} \operatorname{Bun}_{P^-}^{\theta}$. Let $\overset{\scriptscriptstyle\circ}{\pi}_{\mathfrak{F}}: \overset{\scriptscriptstyle\circ}{\mathfrak{F}}{}^{\theta}\to \operatorname{Gr}_M^{-, \theta}$ be the restriction of $\pi_P$. Write $U(\mathfrak{u}(\check{P}))$ for the universal enveloping algebra of $\mathfrak{u}(\check{P})$. Recall that $U(\mathfrak{u}(\check{P}))$ is the graded dual of ${\mathcal O}(U(\check{P}))$. The following is a version of ([@BFGM], Proposition 5.9). **Proposition 1**. *The complex $(\overset{\scriptscriptstyle\circ}{\pi}_{\mathfrak{F}})_!e[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$ on $\operatorname{Gr}_M^{-,\theta}$ is placed in perverse degrees $\le 0$, and the $0$-th perverse cohomology sheaf identifies with $$\mathbb{D}\operatorname{Sat}_M({\mathcal O}(U(\check{P}))_{\theta})\,{\widetilde\to}\, \upsilon \operatorname{Sat}_M(U(\mathfrak{u}(\check{P}))_{-\theta})$$* For completeness, we supply a proof of Proposition [Proposition 1](#Pp_2.6.23_about_Gr){reference-type="ref" reference="Pp_2.6.23_about_Gr"} in Section [4.4](#Sect_2.8){reference-type="ref" reference="Sect_2.8"}. The following is a version of ([@BFGM], Proposition 5.7 and 5.8). **Proposition 1**. *a) The complex $i_{\operatorname{Mod}}^!(\pi_P)_*\operatorname{IC}_{{\mathcal Z}^{\theta}}$ is concentrated in perverse cohomological degree zero on $\operatorname{Gr}_M^{-,\theta}$.\ b) The map $$\label{map_for_Pp_2.6.23} i_{\operatorname{Mod}}^!(\pi_P)_*\operatorname{IC}_{{\mathcal Z}^{\theta}}\to i_{\operatorname{Mod}}^!(\pi_P)_*j_*\operatorname{IC}_{\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}}\,{\widetilde\to}\, (\overset{\scriptscriptstyle\circ}{\pi}_{\mathfrak{F}})_*\omega_{\overset{\scriptscriptstyle\circ}{\mathfrak{F}}{}^{\theta}}[\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$$ over $\operatorname{Gr}_M^{-,\theta}$ induces an isomorphism in the (lowest) perverse cohomological degree zero. Here we used the fact that $\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}$ is smooth of dimension $-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$. $\square$* ### Proof of Proposition [Proposition 1](#Pp_2.6.18_main_technical){reference-type="ref" reference="Pp_2.6.18_main_technical"} ii) {#proof-of-proposition-pp_2.6.18_main_technical-ii} In view of ([@BFGM], Proposition 5.2), we are reduced to show that $$i_{\operatorname{Mod}}^!(\pi_P)_*\mathfrak{q}_{{\mathcal Z}}^! (j_{glob})_!\operatorname{IC}_{\operatorname{Bun}_P}=0$$ Applying Proposition [Proposition 1](#Pp_2.6.21_Zastava){reference-type="ref" reference="Pp_2.6.21_Zastava"} b), it suffices to show that $$i_{\operatorname{Mod}}^!(\pi_P)_*j_!(\omega_{\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}})=0$$ As in ([@BFGM], Proposition 5.2), $(\pi_P)_*j_!(\omega_{\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}})\,{\widetilde\to}\, \mathfrak{s}^*j_!(\omega_{\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}})$. If $\theta\ne 0$ then $\mathfrak{s}^*j_!(\omega_{\overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta}})=0$, because the corresponding fibre product is empty. $\square$ ### {#section-76} The following is established as in ([@BFGM], Proposition 6.6). **Lemma 1**. *Let $\nu,\nu'\in\Lambda^+_M$ then $S_P^{\nu}\cap S_{P^-}^{\nu'}$ is a scheme of finite type. If $\lambda\in \Lambda^+_{M, ab}$ is deep enough on the wall of the corresponding Weyl chamber then $$S_P^{\nu+\lambda}\cap S_{P^-}^{\nu'+\lambda}\subset \operatorname{Gr}_G^{\nu+\lambda}$$* *Proof.* Recall the closed immersion $i^{\nu}_P: \operatorname{Gr}_M^{\nu}\stackrel{}{\hookrightarrow}S_P^{\nu}$, so $S_P^{\nu}=U(P)(F)\operatorname{Gr}_M^{\nu}$. Since $S_P^{\nu}\cap S_{P^-}^{\nu'}$ is finite type, there is $\lambda\in \Lambda^+_{M, ab}$ deep enough such that the preimage of $S_P^{\nu}\cap S_{P^-}^{\nu'}$ under the action map $$U(P)(F)\times \operatorname{Gr}_M^{\nu}\to S_P^{\nu}$$ is contained in $\operatorname{Ad}_{t^{-\lambda}}(U(P)({\mathcal O}))\times \operatorname{Gr}_M^{\nu}$. So, the action of $t^{\lambda}$ sends $S_P^{\nu}\cap S_{P^-}^{\nu'}$ inside $U(P)({\mathcal O})\operatorname{Gr}_M^{\nu+\lambda}\subset \operatorname{Gr}_G^{\nu+\lambda}$. ◻ **Corollary 1**. *Let $\theta\in -\Lambda_{G,P}^{pos}$. Then for $\lambda\in\Lambda^+_{M, ab}$ deep enough on the wall of the corresponding Weyl chamber one has $S^0_P\cap \operatorname{Gr}_{P^-}^{\theta}\subset t^{-\lambda}\operatorname{Gr}_G^{\lambda}$.* *Proof.* We know that $\overset{\scriptscriptstyle\circ}{\mathfrak{F}}{}^{\theta}$ is of finite type. It is stratified by locally closed subschemes $S^0_P\cap S_{P^-}^{\nu}$ for $\nu\in\Lambda^+_M$ over $\theta$. Since this stratification is finite, the claim follows from Lemma [Lemma 1](#Lm_2.6.27_inclusion){reference-type="ref" reference="Lm_2.6.27_inclusion"}. ◻ ### Proof of Proposition [Proposition 1](#Pp_2.6.18_main_technical){reference-type="ref" reference="Pp_2.6.18_main_technical"} i) {#proof-of-proposition-pp_2.6.18_main_technical-i} For $\lambda\in\Lambda^+_{M, ab}$ over $\bar\lambda\in\Lambda_{G,P}$ consider the composition $$t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle]\to \operatorname{IC}^{\frac{\infty}{2}}_{P}\to (\tilde\pi^M))^!\operatorname{IC}_{\widetilde{\operatorname{glob}}}[(g-1)\dim P]$$ in $Shv(\operatorname{Gr}_G)^{M({\mathcal O})}$ and the corresponding map $$\begin{gathered} \label{map_for_Sect_2.6.25} (\mathfrak{t}^{\theta}_{P^-})_*(v_{P^-}^{\theta})^!(t^{-\lambda}\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle]\to (\mathfrak{t}^{\theta}_{P^-})_*(v_{P^-}^{\theta})^!(\tilde\pi^M))^!\operatorname{IC}_{\widetilde{\operatorname{glob}}}[(g-1)\dim P]\end{gathered}$$ By Lemma [Lemma 1](#Lm_theorem_of_Braden_for_theta){reference-type="ref" reference="Lm_theorem_of_Braden_for_theta"} and Proposition [Proposition 1](#Pp_gRes_for_Levi){reference-type="ref" reference="Pp_gRes_for_Levi"}, the LHS of ([\[map_for_Sect_2.6.25\]](#map_for_Sect_2.6.25){reference-type="ref" reference="map_for_Sect_2.6.25"}) identifies with $$t^{-\lambda}(\mathfrak{t}_P^{\theta+\bar\lambda})_!(v_P^{\theta+\bar\lambda})^*\operatorname{Sat}(V^{\lambda})[\langle\lambda, 2\check{\rho}\rangle]\,{\widetilde\to}\, t^{-\lambda}\operatorname{Sat}_M((V^{\lambda})_{\theta+\bar\lambda})[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$$ and sits in perverse degree $\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$ on $\operatorname{Gr}_M^{\theta}$. As $\lambda$ runs through $\Lambda^+_{M, ab}$ the corresponding diagram $\Lambda^+_{M, ab}\to Shv(\operatorname{Gr}_M)^{\theta}$ was described in Proposition [Proposition 1](#Pp_2.5.15_*-restriction){reference-type="ref" reference="Pp_2.5.15_*-restriction"}. By Propositions [Proposition 1](#Pp_2.6.21_Zastava){reference-type="ref" reference="Pp_2.6.21_Zastava"} and [Proposition 1](#Pp_2.6.23_citation_from_BFGM){reference-type="ref" reference="Pp_2.6.23_citation_from_BFGM"}, the RHS of ([\[map_for_Sect_2.6.25\]](#map_for_Sect_2.6.25){reference-type="ref" reference="map_for_Sect_2.6.25"}) identifies with $$i_{\operatorname{Mod}}^!(\pi_P)_*\mathfrak{q}_{{\mathcal Z}}^!(\tilde\pi^M)^!\operatorname{IC}_{\widetilde{glob}}[(g-1)\dim P]\,{\widetilde\to}\, i_{\operatorname{Mod}}^!(\pi_P)_* \operatorname{IC}_{{\mathcal Z}^{\theta}}[-\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle]$$ and is concentrated in perverse cohomological degree $\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$. Thus, it suffices to show that the map ([\[map_for_Sect_2.6.25\]](#map_for_Sect_2.6.25){reference-type="ref" reference="map_for_Sect_2.6.25"}), after taking the colimit in the LHS over $\lambda\in\Lambda^+_{M, ab}$, induces an isomorphism on the perverse cohomology sheaves in degree $\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$. ### {#section-77} Write $$\overset{\scriptscriptstyle\circ}{\pi}{}^{\lambda}: S^0_P\cap \operatorname{Gr}_{P^-}^{\theta}\cap (t^{-\lambda}\operatorname{Gr}_G^{\lambda})\to \operatorname{Gr}_M^{-,\theta}$$ for the restriction of $\overset{\scriptscriptstyle\circ}{\pi}_{\mathfrak{F}}$. Since $t^{-\lambda}\overline{\operatorname{Gr}}_G^{\lambda}\subset \bar S^0_P$, we have the open immersion $$S^0_P\cap \operatorname{Gr}_{P^-}^{\theta}\cap (t^{-\lambda}\operatorname{Gr}_G^{\lambda})\subset \operatorname{Gr}_{P^-}^{\theta}\cap (t^{-\lambda}\operatorname{Gr}_G^{\lambda})$$ For $\lambda\in\Lambda^+_{M, ab}$ large enough in the corresponding wall of the Weyl chamber, by Corollary [Corollary 1](#Cor_2.6.28){reference-type="ref" reference="Cor_2.6.28"}, $$S^0_P\cap \operatorname{Gr}_{P^-}^{\theta}\cap (t^{-\lambda}\operatorname{Gr}_G^{\lambda})=S^0_P\cap \operatorname{Gr}_{P^-}^{\theta}=\overset{\scriptscriptstyle\circ}{\mathfrak{F}}{}^{\theta}$$ Now it suffices to show that for $\lambda\in\Lambda^+_{M, ab}$ large enough the diagram commutes $$\begin{array}{ccc} (\mathfrak{t}^{\theta}_{P^-})_*(v^{\theta}_{P^-})^!(t^{-\lambda}\operatorname{Sat}(V^{\lambda}))[\langle\lambda, 2\check{\rho}\rangle] & \to & (\overset{\scriptscriptstyle\circ}{\pi}{}^{\lambda})_*\omega_{S^0_P\cap \operatorname{Gr}_{P^-}^{\theta}\cap (t^{-\lambda}\operatorname{Gr}_G^{\lambda})}\\ \downarrow\lefteqn{\scriptstyle (\ref{map_for_Sect_2.6.25})} && \uparrow\\ (\mathfrak{t}^{\theta}_{P^-})_*(v^{\theta}_{P^-})^!(\tilde\pi^M)^!\operatorname{IC}_{\widetilde{glob}}[(g-1)\dim P] & \stackrel{(\ref{map_for_Pp_2.6.23})}{\to} & (\overset{\scriptscriptstyle\circ}{\pi}_{\mathfrak{F}})_*\omega_{\overset{\scriptscriptstyle\circ}{\mathfrak{F}}{}^{\theta}} \end{array}$$ As in ([@Gai19SI], Section 3.7.3), this follows from the description of the map ([\[map_for_Sect_2.6.8_long_composition\]](#map_for_Sect_2.6.8_long_composition){reference-type="ref" reference="map_for_Sect_2.6.8_long_composition"}) in Section [4.2.7](#Sect_2.6.9_describing_the_composition){reference-type="ref" reference="Sect_2.6.9_describing_the_composition"}. Proposition [Proposition 1](#Pp_2.6.18_main_technical){reference-type="ref" reference="Pp_2.6.18_main_technical"} i) is proved. $\square$ ## Proof of Proposition [Proposition 1](#Pp_2.6.21_Zastava){reference-type="ref" reference="Pp_2.6.21_Zastava"} {#Sect_Proof_Pp_2.6.21_Zastava} ### {#section-78} Write $\operatorname{\widetilde\operatorname{Bun}}_{U(P)}$ for the version of $\operatorname{\widetilde\operatorname{Bun}}_P$, where the corresponding $M$-torsor is fixed and identified with ${\mathcal F}^0_M$. By abuse of notations, the natural map ${\mathcal Z}^{\theta}\to \operatorname{\widetilde\operatorname{Bun}}_{U(P)}$ is also denoted by $\mathfrak{q}_{{\mathcal Z}}$. Let $\mathfrak{u}(P)$ be the Lie algebra of $U(P)$. Write $\operatorname{Bun}_M^{sm}\subset\operatorname{Bun}_M$ for the open substack given by the property that for all $M$-modules $V$ appearing as subquotients of $\mathfrak{u}(P)$ one has ${\operatorname{H}}^1(X, V_{{\mathcal F}_M})=0$. Write $\operatorname{Bun}_{P^-}^{sm}\subset \operatorname{Bun}_{P^-}$ for the preimage of $\operatorname{Bun}_M^{sm}$ under $\operatorname{Bun}_{P^-}\to\operatorname{Bun}_M$. The map $\operatorname{Bun}_{P^-}^{sm}\to\operatorname{Bun}_G$ is smooth. If the image of the projection ${\mathcal Z}^{\theta}\to \operatorname{Bun}_M$, $({\mathcal F}_M, \beta_M,{\mathcal F}_G, \beta)\mapsto {\mathcal F}_M$ is contained in $\operatorname{Bun}_M^{\theta, sm}$ then the claim is easy. It follows in this case as a combination of the fact that the projection ${\mathcal Z}^{\theta}\to \operatorname{\widetilde\operatorname{Bun}}_{U(P)}$ is smooth with the fact that both $\operatorname{IC}_{\widetilde{glob}}$ and $(j_{glob})_!\operatorname{IC}$ are ULA with respect to $\operatorname{\widetilde\operatorname{Bun}}_P\to \operatorname{Bun}_M$ by ([@BG], Theorem 5.1.5). We reduce to this case as in ([@Gai19SI], Propositiion 3.6.5). Set $\operatorname{Bun}_{P^-}^{\theta, sm}=\operatorname{Bun}_{P^-}^{\theta}\cap \operatorname{Bun}_{P^-}^{sm}$. Pick $\theta'\in -\Lambda_{G,P}^{pos}$. Let $$(X^{-\theta}\times X^{-\theta'})_{disj}\subset X^{-\theta}\times X^{-\theta'}$$ denote the open locus of divisors whose supports are disjoint. Recall the factorization property $$\label{fact_property} {\mathcal Z}^{\theta+\theta'}\times_{X^{-\theta-\theta'}} (X^{-\theta}\times X^{-\theta'})_{disj}\,{\widetilde\to}\, ({\mathcal Z}^{\theta}\times {\mathcal Z}^{\theta'})\times_{X^{-\theta}\times X^{-\theta'}} (X^{-\theta}\times X^{-\theta'})_{disj}$$ ### {#section-79} Let $(\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{good}\subset \operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'}$ be the open subscheme given by the property that the maps $$\kappa^{{\mathcal V}}: {\mathcal V}^{U(P)}_{{\mathcal F}^0_M}\stackrel{}{\hookrightarrow} {\mathcal V}_{{\mathcal F}_G}$$ have no zero at the support of the point of $X^{-\theta'}$. Given $S\in{\operatorname{Sch}}^{aff}$ and $D\in X^{-\theta'}(S)$ write $\hat {\mathcal D}_D$ for the formal completion of $\operatorname{supp}(D)$ in $S\times X$. Let ${\mathcal D}_D$ be the affine scheme corresponding to $\hat {\mathcal D}_D$, i.e., the image of $\hat{\mathcal D}_D$ under the functor $$\operatorname{colim}: \operatorname{Ind}({\operatorname{Sch}}^{aff})\to {\operatorname{Sch}}^{aff}$$ Let $\overset{\scriptscriptstyle\circ}{{\mathcal D}}_D\subset {\mathcal D}_D$ be the open subscheme obtained by removing $\operatorname{supp}(D)$. Write $\mathfrak{L}^+(U(P))_{\theta'}$ for the group scheme over $X^{-\theta'}$ classifying a point $D\in X^{-\theta'}$ and a section ${\mathcal D}_D\to U(P)$. We leave it to a reader to formulate a version of this definition with $S$-points for a test scheme $S\in{\operatorname{Sch}}^{aff}$. For a point of $(\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{good}$ one gets a $U(P)$-torsor ${\mathcal F}_{U(P)}$ over ${\mathcal D}_D$. Let $$(\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{level}\to (\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{good}$$ be the stack classifying a point of $(\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{good}$ as above together with an trivialization $${\mathcal F}_{U(P)}\,{\widetilde\to}\, {\mathcal F}^0_{U(P)}$$ of this $U(P)$-torsors on ${\mathcal D}_D$. This is a torsor under the group scheme $\mathfrak{L}^+(U(P))_{\theta'}$. Write $\mathfrak{L}(U(P))_{\theta'}$ for the group ind-scheme over $(\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{good}$ classifying a point $D\in X^{-\theta'}$ and a section $\overset{\scriptscriptstyle\circ}{{\mathcal D}}_D\to U(P)$. The usual gluing procedure allows to extend the action of $\mathfrak{L}^+(U(P))_{\theta'}$ on $(\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{level}$ to that of $\mathfrak{L}(U(P))_{\theta'}$. Pick a group subscheme $$\mathfrak{L}^+(U(P))_{\theta'}\subset U(P)'_{\theta'}\subset \mathfrak{L}(U(P))_{\theta'}$$ pro-smooth over $X^{-\theta'}$, where the first inclusion is a placid closed embedding. Note that the projection $$(\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{good}\to (\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{level}/U(P)'_{good}$$ is smooth, where the RHS denotes the stack quotient. ### {#section-80} Recall that for $\operatorname{Mod}_M^{-,\theta}$ one also has the factrorization isomorphism $$\begin{gathered} \label{fact_property_for_Mod_M^-} (\operatorname{Mod}_M^{-,\theta}\times \operatorname{Mod}_M^{-,\theta'})\times_{(X^{-\theta}\times X^{-\theta'})} (X^{-\theta}\times X^{-\theta'})_{disj}\,{\widetilde\to}\\ \operatorname{Mod}_M^{-, \theta+\theta'}\times_{X^{-\theta-\theta'}} (X^{-\theta}\times X^{-\theta'})_{disj}\end{gathered}$$ Denote by $$(\operatorname{Mod}_M^{-,\theta}\times \operatorname{Mod}_M^{-,\theta'})^{sm}_{disj}$$ the preimage of $\operatorname{Bun}_M^{\theta+\theta', sm}$ under ([\[fact_property_for_Mod_M\^-\]](#fact_property_for_Mod_M^-){reference-type="ref" reference="fact_property_for_Mod_M^-"}) composed with the projection to $\operatorname{Bun}_M^{\theta+\theta'}$. Write $({\mathcal Z}^{\theta}\times \overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta'})^{sm}_{disj}$ for the preimage of $(\operatorname{Mod}_M^{-,\theta}\times \operatorname{Mod}_M^{-,\theta'})^{sm}_{disj}$ under $${\mathcal Z}^{\theta}\times \overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta'}\to {\mathcal Z}^{\theta}\times {\mathcal Z}{}^{\theta'}\to \operatorname{Mod}_M^{-,\theta}\times \operatorname{Mod}_M^{-,\theta'}$$ Write ${\mathcal Z}^{\theta+\theta', sm}$ for he preimage of $\operatorname{Bun}_M^{\theta+\theta', sm}$ under the composition ${\mathcal Z}^{\theta+\theta'}\to \operatorname{Bun}_{P^-}^{\theta+\theta'}\to \operatorname{Bun}_M^{\theta+\theta'}$. ### {#section-81} For $U(P)'_{\theta'}$ large enough the diagram commutes $$\label{diag_for_chasing} \begin{array}{ccc} ({\mathcal Z}^{\theta}\times \overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta'})^{sm}_{disj} & \stackrel{(\ref{fact_property})}{\to} & {\mathcal Z}^{\theta+\theta', sm}\times_{X^{-\theta-\theta'}} (X^{-\theta}\times X^{-\theta'})_{disj}\\ \downarrow && \downarrow\lefteqn{\scriptstyle \mathfrak{q}_{{\mathcal Z}}}\\ ({\mathcal Z}^{\theta}\times X^{-\theta'})\times_{(X^{-\theta}\times X^{-\theta'})} (X^{-\theta}\times X^{-\theta'})_{disj} && (\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{good}\\ \downarrow\lefteqn{\scriptstyle \mathfrak{q}_{{\mathcal Z}}} && \downarrow \\ (\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{good} & \to & (\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{level}/U(P)'_{\theta'} \end{array}$$ **Lemma 1**. *Given $\theta\in -\Lambda_{G,P}^{pos}$, there is $\theta'\in -\Lambda_{G,P}^{pos}$ such that the projection $$(\operatorname{Mod}_M^{-,\theta}\times \operatorname{Mod}_M^{-,\theta'})^{sm}_{disj}\to \operatorname{Mod}_M^{-,\theta}$$ is surjective. $\square$* ### {#section-82} Pick $\theta'$ as in Lemma [Lemma 1](#Lm_2.7.5){reference-type="ref" reference="Lm_2.7.5"}, so that the projection $({\mathcal Z}^{\theta}\times \overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta'})^{sm}_{disj}\to {\mathcal Z}^{\theta}$ is smooth and surjective. Now one finishes te proof of Proposition [Proposition 1](#Pp_2.6.21_Zastava){reference-type="ref" reference="Pp_2.6.21_Zastava"} precisely as ([@Gai19SI], Section 3.9.4) by chasing the diagram ([\[diag_for_chasing\]](#diag_for_chasing){reference-type="ref" reference="diag_for_chasing"}). To get point a), it suffices to show that the $!$-pull-back of the $\operatorname{IC}$-sheaf along the composite left vertical map is isomorphic up to a shift to the $\operatorname{IC}$-sheaf. Since the bottom horizontal arrow is smooth, it suffices to show that the pull-back of the $\operatorname{IC}$-sheaf of $(\operatorname{\widetilde\operatorname{Bun}}_{U(P)}\times X^{-\theta'})^{level}/U(P)'_{\theta'}$ to $({\mathcal Z}^{\theta}\times \overset{\scriptscriptstyle\circ}{{\mathcal Z}}{}^{\theta'})^{sm}_{disj}$ is isomorphic to the $\operatorname{IC}$-sheaf up to a shift. This follows from the fact that both top horizontal arrow and the right vertical arrows in ([\[diag_for_chasing\]](#diag_for_chasing){reference-type="ref" reference="diag_for_chasing"}) are smooth. $\square$ ## Proof of Proposition [Proposition 1](#Pp_2.6.23_about_Gr){reference-type="ref" reference="Pp_2.6.23_about_Gr"} {#Sect_2.8} ### {#section-83} By Section [3.2.5](#Sect_2.3.6_positive part_Gr_M^+){reference-type="ref" reference="Sect_2.3.6_positive part_Gr_M^+"}, given $\nu\in\Lambda^+_M$ over some $\mu\in\Lambda_{G,P}$ one has $\operatorname{Gr}_M^{\nu}\subset\operatorname{Gr}_M^{-,\mu}$ iff $\nu\in -\Lambda^{pos}$. Pick $\nu\in\Lambda^+_M$ over $\theta$ such that $\operatorname{Gr}_M^{\nu}\subset \operatorname{Gr}_M^{-,\theta}$, so $\nu\in -\Lambda^{pos}$. By ([@BFGM], Section 6.6 after Proposition 6.6), $$\dim(S^0_P\cap S_{P^-}^{\nu})\le \langle-w_0^M(\nu), \check{\rho}\rangle$$ So, the fibres of $S^0_P\cap S_{P^-}^{\nu}\to \operatorname{Gr}_M^{\nu}$ are of dimension at most $$\label{dim_fibres_for_Sect_2.8.1} -\langle w_0^M(\nu), \check{\rho}\rangle-\langle\nu, 2\check{\rho}_M\rangle$$ Since $w_0^M(\nu)-\nu$ vanishes in $\Lambda_{G,P}$, $\langle w_0^M(\nu)-\nu, \check{\rho}-\check{\rho}_M\rangle=0$. This implies that ([\[dim_fibres_for_Sect_2.8.1\]](#dim_fibres_for_Sect_2.8.1){reference-type="ref" reference="dim_fibres_for_Sect_2.8.1"}) equals $-\langle\nu, \check{\rho}\rangle$. Since $\dim \operatorname{Gr}_M^{\nu}=\langle\nu, 2\check{\rho}_M\rangle$, this implies that $(\overset{\scriptscriptstyle\circ}{\pi}_{\mathfrak{F}})_!e$ is placed in perverse degrees $\le -\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle$. It remains to show that the set of irreducible components of $S^0_P\cap S_{P^-}^{\nu}$ of (maximal) dimension $\langle-w_0^M(\nu), \check{\rho}\rangle$ naturally form a base in $$\label{vector_space_for_Sect_2.8.1} \operatorname{Hom}_{\check{M}}(U^{-w_0^M(\nu)}, U(\mathfrak{u}(\check{P}))_{-\theta})$$ Pick $\mu'\in\Lambda^+_{M, ab}$ deep enough in the corresponding wall of the Weyl chamber (that is, we require $\langle\mu', \check{\alpha}_i\rangle$ large enough for all $i\in {\mathcal I}-{\mathcal I}_M$). By ([@BFGM], 6.6), one gets $$\label{inclusion_for_Sect_2.8.1} S_P^{-\mu'}\cap S_{P^-}^{\nu-\mu'}\subset S_P^{-\mu'}\cap \operatorname{Gr}_G^{w_0(w_0^M(\nu-\mu'))},$$ and the action of $t^{\mu'}$ gives an isomorphism $S_P^{-\mu'}\cap S_{P^-}^{\nu-\mu'}\,{\widetilde\to}\,S_P^0\cap S_{P^-}^{\nu}$. By ([@MV], Theorem 3.2), $$S_P^{-\mu'}\cap \operatorname{Gr}_G^{w_0(w_0^M(\nu-\mu'))}$$ is of pure dimension $-\langle\check{\rho}, w_0^M(\nu)\rangle$. As in ([@BFGM], Section 6.5) the inclusion ([\[inclusion_for_Sect_2.8.1\]](#inclusion_for_Sect_2.8.1){reference-type="ref" reference="inclusion_for_Sect_2.8.1"}) yields a bijection on the set of irreducible components of (maximal) dimension $-\langle\check{\rho}, w_0^M(\nu)\rangle$ of both sides. By ([@BFGM], Theorem 6.2), the set of ireducible components of $$S_P^{-\mu'}\cap \operatorname{Gr}_G^{w_0(w_0^M(\nu-\mu'))}$$ form a base of $\operatorname{Hom}_{\check{M}}(U^{-\mu'}, V^{w_0(w_0^M(\nu-\mu'))})$ naturally. Under our assumption on $\mu'$ the latter vector space identifies canonically with ([\[vector_space_for_Sect_2.8.1\]](#vector_space_for_Sect_2.8.1){reference-type="ref" reference="vector_space_for_Sect_2.8.1"}). $\square$ ## Relation between the dual baby Verma objects {#Sect_4.5} ### {#Sect_4.5.1} Write $$j_{PP^-}: P\backslash PP^-/P^-\stackrel{}{\hookrightarrow} P\backslash G/P^-,\;\;\;\;\;\;\;\;\; j_{P^-P}: P^-\backslash P^-P/P\stackrel{}{\hookrightarrow} P^-\backslash G/P$$ for the natural open immersions. We get the objects $$j_{*}^P, j_{!}^P\in Shv(P\backslash G/P^-),\;\;\;\;\;\;\; j_{*}^{P^-}, j_{!}^{P^-}\in Shv(P^-\backslash G/P)$$ defined as the corresponding extensions under $j_{PP^-}$ and $j_{P^-P}$ of the $\operatorname{IC}$-sheaves on $P\backslash PP^-/P^-$ and $P^-\backslash P^-P/P$ respectively. One similarly defines $$j_*^B, j_!^B\in Shv(B\backslash G/B^-),\;\;\;\;\;\;\;\; j_*^{B^-}, j_!^{B^-}\in Shv(B^-\backslash G/B)$$ ### {#section-84} For $C\in Shv(G)-mod(\operatorname{DGCat}_{cont})$ we have the functors $\_\overset{P}{\ast}j_*^P: C^P\to C^{P^-}$ and $\_\overset{P^-}{\ast}j_!^{P^-}: C^{P^-}\to C^P$ defined in Section [5.7.4](#Sect_A.7.5){reference-type="ref" reference="Sect_A.7.5"}. The following result is established in Section [6](#Sect_appendixB){reference-type="ref" reference="Sect_appendixB"}. **Proposition 1**. *The diagram commutes $$\begin{array}{ccccc} C^{P} & \stackrel{\_\overset{P}{\ast}j^P_{*}}{\to} & C^{P-} & \stackrel{\_\overset{P^-}{\ast}j_!^{P^-}}{\to} & C^P \\ \downarrow\lefteqn{\scriptstyle\operatorname{oblv}} &&\downarrow\lefteqn{\scriptstyle\operatorname{oblv}} && \downarrow\lefteqn{\scriptstyle\operatorname{oblv}}\\ C^B & \stackrel{\_\overset{B}{\ast}j^B_*}{\to} & C^{B^-} & \stackrel{\_\overset{B^-}{\ast}j_!^{B^-}}{\to} & C^B, \end{array}$$ and the horizontal arrows are equivalences. Besides, the composition in each line is canonically isomorphic to the identity functor.* **Remark 1**. *For $P=B$ this is well-known. Our contribution is to generalize this to an arbitrary standard parabolic $P$.* ### Parahoric version By abuse of notations we also denote by $$j_{PP^-}: I_P\backslash I_PI_{P^-}/I_{P^-}\stackrel{}{\hookrightarrow} I_P\backslash G({\mathcal O})/I_{P^-},\;\;\;\;\;\; j_{P^-P}: I_{P^-}\backslash I_{P^-}I_P/I_P\stackrel{}{\hookrightarrow} I_{P^-}\backslash G({\mathcal O})/I_P$$ the corresponding immersions. We analogously get the objects $$j_{*}^P, j_{!}^P\in Shv(I_P\backslash G({\mathcal O})/I_{P^-}),\;\;\;\;\;\;\; j_{*}^{P^-}, j_{!}^{P^-}\in Shv(I_{P^-}\backslash G({\mathcal O})/I_P)$$ defined as the corresponding extensions under $j_{PP^-}$ and $j_{P^-P}$, of the $\operatorname{IC}$-sheaves on $I_P\backslash I_PI_{P^-}/I_{P^-}$ and $I_{P^-}\backslash I_{P^-}I_P/I_P$ respectively. We similarly have $$j_{*}^B, j_{!}^B\in Shv(I\backslash G({\mathcal O})/I^-),\;\;\;\;\;\;\; j_{*}^{B^-}, j_{!}^{B^-}\in Shv(I^-\backslash G({\mathcal O})/I)$$ The above notations are in ambiguity with those of Section [4.5.1](#Sect_4.5.1){reference-type="ref" reference="Sect_4.5.1"}, the precise sense will be clear from the context. Proposition [Proposition 1](#Pp_4.5.3){reference-type="ref" reference="Pp_4.5.3"} immediately implies the following. **Corollary 1**. *Let $C\in Shv(G({\mathcal O}))-mod(\operatorname{DGCat}_{cont})$. Then the diagram commutes $$\begin{array}{ccccc} C^{I_P} & \stackrel{\_\overset{I_P}{\ast}j^P_{*}}{\to} & C^{I_{P-}} & \stackrel{\_\overset{I_{P^-}}{\ast}j_!^{P^-}}{\to} & C^{I_P} \\ \downarrow\lefteqn{\scriptstyle\operatorname{oblv}} &&\downarrow\lefteqn{\scriptstyle\operatorname{oblv}} && \downarrow\lefteqn{\scriptstyle\operatorname{oblv}}\\ C^I & \stackrel{\_\overset{I}{\ast}j^B_*}{\to} & C^{I^-} & \stackrel{\_\overset{I^-}{\ast}j_!^{B^-}}{\to} & C^I, \end{array}$$ and the horizontal arrows are equivalences. Besides, the composition in each line is canonically isomorphic to the identity functor.* ### {#section-85} Set ${\mathcal F}l_{P^-}=G(F)/I_{P^-}$. For $\lambda\in\Lambda$ we denote by $$j_{\lambda, !}^-,\; j_{\lambda, *}^-\in {\mathcal H}_{P^-}(G):=Shv({\mathcal F}l_{P^-})^{I_{P^-}}$$ the standard and costandard objects attached to $t^{\lambda}\in \tilde W$. Let us reformulate Proposition [Proposition 1](#Pp_Chen_Dhillon){reference-type="ref" reference="Pp_Chen_Dhillon"} with $B$ and $P$ replaced by $B^-$ and $P^-$ respectively. **Proposition 1**. *There is a canonical equivalence $$\label{equivalence_Gurbir_Chen_for_P_instead} \operatorname{IndCoh}((\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})\,{\widetilde\to}\,Shv(\operatorname{Gr}_G)^{I_{P^-}, ren}$$ with the following properties:* *(i) The ${\operatorname{Rep}}(\check{G})$-action on $\operatorname{IndCoh}((\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})$ arising from the projection $$(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P}\to pt/\check{P}\to pt/\check{G}$$ corresponds to the action of ${\operatorname{Rep}}(\check{G})$ on $Shv(\operatorname{Gr}_G)^{I_{P^-}, ren}$ via $\operatorname{Sat}: {\operatorname{Rep}}(\check{G})\to Shv(\operatorname{Gr}_G)^{G({\mathcal O})}$ and the right convolutions.* *(ii) The ${\operatorname{Rep}}(\check{M}_{ab})$-action on $\operatorname{IndCoh}((\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})$ arising from the projection $$(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P}\to pt/\check{M}\to pt/\check{M}_{ab}$$ corresponds to the ${\operatorname{Rep}}(\check{M}_{ab})$-action on $Shv(\operatorname{Gr}_G)^{I_{P^-}, ren}$ such that for $\lambda\in -\Lambda^+_{M, ab}$, $e^{\lambda}$ sends $F$ to $j_{\lambda,*}^-\ast F$.* *(iii) The object ${\mathcal O}_{pt/\check{P}}\in \operatorname{IndCoh}((\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})$ corresponds under ([\[equivalence_Gurbir_Chen_for_P\_instead\]](#equivalence_Gurbir_Chen_for_P_instead){reference-type="ref" reference="equivalence_Gurbir_Chen_for_P_instead"}) to $\delta_{1,\operatorname{Gr}_G}\in Shv(\operatorname{Gr}_G)^{I_{P^-}, ren}$.* *(iv) The equivalence ([\[equivalence_Gurbir_Chen_for_P\_instead\]](#equivalence_Gurbir_Chen_for_P_instead){reference-type="ref" reference="equivalence_Gurbir_Chen_for_P_instead"}) restricts to an equivalence $$\operatorname{IndCoh}_{\operatorname{Nilp}}((\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})\,{\widetilde\to}\,Shv(\operatorname{Gr}_G)^{I_{P^-}}$$ $\square$* ### {#section-86} Recall the fully faithful embedding $\operatorname{ren}: Shv(\operatorname{Gr}_G)^{I_P}\stackrel{}{\hookrightarrow} Shv(\operatorname{Gr}_G)^{I_P, ren}$. Applying ([\[eq_ren_parahoric_versus_H\]](#eq_ren_parahoric_versus_H){reference-type="ref" reference="eq_ren_parahoric_versus_H"}), it gives a full embedding $$\operatorname{ren}: Shv(\operatorname{Gr}_H)^H\stackrel{}{\hookrightarrow} Shv(\operatorname{Gr}_G)^{H, ren}.$$ By abuse of notations, the image of $\operatorname{IC}_P^{\frac{\infty}{2}}$ under the latter functor is also denoted $\operatorname{IC}_P^{\frac{\infty}{2}}$. The following is one of our main results. **Theorem 1**. *The image of $\operatorname{IC}_P^{\frac{\infty}{2}}$ under the composition $$\begin{gathered} Shv(\operatorname{Gr}_G)^{H, ren} \;\stackrel{\operatorname{Av}_*^{I_P/M({\mathcal O}), ren}}{\to} \;Shv(\operatorname{Gr}_G)^{I_P, ren}\;\stackrel{j_*^{P^-}\,\overset{I_P}{\ast}\_}{\to}\\ Shv(\operatorname{Gr}_G)^{I_{P^-}, ren}\stackrel{(\ref{equivalence_Gurbir_Chen_for_P_instead})}{\to} \operatorname{IndCoh}((\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})\end{gathered}$$ is canonically isomorphic to the object ${\mathcal M}_{\check{G}, \check{P}}[\dim U(P^-)]$ defined in Section [2.3.7](#Sect_2.3.10_almost_final_version){reference-type="ref" reference="Sect_2.3.10_almost_final_version"}.* In the rest of Section [4.5](#Sect_4.5){reference-type="ref" reference="Sect_4.5"} we prove Theorem [Theorem 1](#Thm_4.5.10){reference-type="ref" reference="Thm_4.5.10"}. ### {#section-87} Our starting point is the following result from ([@Gai19SI], Section 4.1.3). Given $\lambda\in\Lambda^+$ there is a canonical isomorphism in ${\mathcal H}(G)$ $$\label{iso_original_conjug_by_w_0} j_{w_0,*}^I \overset{I}{\ast}j_{w_0(\lambda),*}^I \,{\widetilde\to}\, j_{\lambda, *}^I \overset{I}{\ast}j_{w_0, *}^I$$ If $\mu\in \Lambda$, $w\in W$ then $wt^{\mu}w^{-1}=t^{w\mu}$. For this reason, $$w_0It^{\mu}Iw_0=I^-w_0t^{\mu}w_0I^-=I^-t^{w_0\mu}I^-$$ Consider the isomorphism $w_0: \operatorname{Gr}_G\,{\widetilde\to}\, \operatorname{Gr}_G, gG({\mathcal O})\mapsto w_0gG({\mathcal O})$. It intertwines the $I$-actions on the source by left translations and the $I$-action on the target such that $i\in I$ acts as $w_0iw_0$. This gives an involution also denoted $$w_0: Shv(I\backslash \operatorname{Gr}_G)\,{\widetilde\to}\, Shv(I^-\backslash \operatorname{Gr}_G)$$ We assume that for an ind-scheme of finite type $Y$ and a placid group scheme ${\mathcal G}$ acting on $Y$ the functor $\operatorname{oblv}: Shv(Y)^G\to Shv(G)$ is related to the !-pullback via our convention of Section [5.5](#Sect_A.3){reference-type="ref" reference="Sect_A.3"}. For $\mu\in\Lambda$ the diagram commutes $$\begin{array}{ccc} Shv(I\backslash \operatorname{Gr}_G) &\stackrel{j_{\mu, *}^I \overset{I}{\ast}}{\to} & Shv(I\backslash\operatorname{Gr}_G)\\ \downarrow\lefteqn{\scriptstyle w_0} && \downarrow\lefteqn{\scriptstyle w_0} \\ Shv(I^-\backslash\operatorname{Gr}_G) & \stackrel{j_{w_0(\mu),*}^{I^-} \overset{I^-}{\ast}}{\to} & Shv(I^-\backslash\operatorname{Gr}_G) \end{array}$$ The composition $$Shv(I^-\backslash \operatorname{Gr}_G)\stackrel{w_0}{\to} Shv(I\backslash \operatorname{Gr}_G)\stackrel{j_{w_0, *}^I \overset{I}{\ast}}{\to} Shv(I\backslash \operatorname{Gr}_G)$$ is $j^B_*\overset{I}{\ast}\_$. So, ([\[iso_original_conjug_by_w\_0\]](#iso_original_conjug_by_w_0){reference-type="ref" reference="iso_original_conjug_by_w_0"}) implies that for $\lambda\in\Lambda^+$ one has an isomorphism $$\label{iso_original_conjug_by_w_0_second} j_*^B\overset{I^-}{\ast}j_{\lambda, *}^{I^-}\,{\widetilde\to}\, j_{\lambda, *}^I \overset{I}{\ast}j_*^B$$ in $Shv(I\backslash G(F)/I^-)$. ### {#section-88} Assume in addition $\lambda\in\Lambda^+_{M, ab}$. Recall that the diagram commutes $$\begin{array}{ccc} Shv(\operatorname{Gr}_G)^{I_P} & \stackrel{j_{\lambda, *}\overset{I_P}{\ast}\_}{\to} & Shv(\operatorname{Gr}_G)^{I_P} \\ \downarrow\lefteqn{\scriptstyle \operatorname{oblv}} && \downarrow\lefteqn{\scriptstyle \operatorname{oblv}}\\ Shv(\operatorname{Gr}_G)^I & \stackrel{j^I_{\lambda, *}\overset{I}{\ast}\_}{\to} & Shv(\operatorname{Gr}_G)^{I} \end{array}$$ **Proposition 1**. *For $\lambda\in\Lambda^+_{M, ab}$ one has an isomorphism $$j_*^P\overset{I_{P^-}}{\ast}j_{\lambda, *}^-\,{\widetilde\to}\, j_{\lambda, *}\overset{I_P}{\ast}j_*^P$$ in $Shv(I_P\backslash G(F)/I_{P^-})$.* *Proof.* It is obtained by applying the direct image under $\eta: I\backslash G(F)/I^-\to I\backslash G(F)/I_{P^-}$ to ([\[iso_original_conjug_by_w\_0_second\]](#iso_original_conjug_by_w_0_second){reference-type="ref" reference="iso_original_conjug_by_w_0_second"}). It turns out that the result is already the pullback from $I_P\backslash G(F)/I_{P^-}$. This is similar to Lemma [Lemma 1](#Lm_2.2.5){reference-type="ref" reference="Lm_2.2.5"}. Namely, recall the natural isomorphisms $$It^{\lambda}I/I\;{\widetilde\to}\; I_Pt^{\lambda}I_P/I_P,\;\;\;\;\;\;\; I^-t^{\lambda}I^-/I^-\;{\widetilde\to}\; I_{P^-}t^{\lambda}I_{P^-}/I_{P^-}$$ from Lemma [Lemma 1](#Lm_2.2.5){reference-type="ref" reference="Lm_2.2.5"}. One has $II_{P^-}=I_PI_{P^-}$. The natural map $$\label{map_to_determine_hope_iso_first} II^-\times^{I^-} (I^-t^{\lambda}I_{P^-})/I_{P^-}\to II_{P^-}\times^{I_{P^-}} (I_{P^-}t^{\lambda}I_{P^-})/I_{P^-}$$ is an isomorphism. Indeed, $I=K_1B, I_{P^-}=K_1P^-$, so the RHS of ([\[map_to_determine_hope_iso_first\]](#map_to_determine_hope_iso_first){reference-type="ref" reference="map_to_determine_hope_iso_first"}) is $$K_1BP^-\times^{I_{P^-}}(I_{P^-}t^{\lambda}I_{P^-})/I_{P^-}\,{\widetilde\to}\, B\times (I_{P^-}t^{\lambda}I_{P^-})/I_{P^-}$$ and the LHS of ([\[map_to_determine_hope_iso_first\]](#map_to_determine_hope_iso_first){reference-type="ref" reference="map_to_determine_hope_iso_first"}) is $$K_1BB^-\times^{I^-} (I^-t^{\lambda}I_{P^-})/I_{P^-} \,{\widetilde\to}\, B\times (I_{P^-}t^{\lambda}I_{P^-})/I_{P^-}$$ So, $$\eta_!(j_*^B\overset{I^-}{\ast}j_{\lambda, *}^{I^-})\,{\widetilde\to}\,\operatorname{oblv}(j_*^P\overset{I_{P^-}}{\ast}j_{\lambda, *}^-)$$ Similarly, one shows that $$\eta_!(j_{\lambda, *}\overset{I}{\ast}j_*^B)\,{\widetilde\to}\, \operatorname{oblv}(j_{\lambda, *}\overset{I_P}{\ast}j_*^P)$$ Our claim follows. ◻ ### {#section-89} Exchanging $B$ with $B^-$ (and $P$ with $P^-$) from Proposition [Proposition 1](#Pp_key_for_baby_Verma_transform_for_P){reference-type="ref" reference="Pp_key_for_baby_Verma_transform_for_P"} one gets the following. **Corollary 1**. *For $\mu\in\Lambda^+_{M, ab}$ one has an isomorphism $$j_*^{P^-}\overset{I_P}{\ast}j_{-\mu, *}\,{\widetilde\to}\, j^-_{-\mu, *}\overset{I_{P^-}}{\ast}j_*^{P^-}$$ in $Shv(I_{P^-}\backslash G(F)/I_P)$. $\square$* ### {#section-90} From the properties of $\operatorname{Av}_*^{I_P/M({\mathcal O})}$ in Sections [3.1.8](#Sect_2.2.10_action_of_Lambda_Mab){reference-type="ref" reference="Sect_2.2.10_action_of_Lambda_Mab"} - [3.1.9](#Sect_3.1.11_action_of){reference-type="ref" reference="Sect_3.1.11_action_of"}, we get $$\label{expression_1_for_dual_baby} \operatorname{Av}^{I_P/M({\mathcal O}), ren}_*(\operatorname{IC}_P^{\frac{\infty}{2}})\,{\widetilde\to}\, \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab}^+} j_{-\lambda, *}\ast \operatorname{Sat}(V^{\lambda}) \in Shv(\operatorname{Gr}_G)^{I_P, ren}$$ where the colimit is taken in $Shv(\operatorname{Gr}_G)^{I_P, ren}$. Note that acting by $j_*^{P^-}\in Shv(I_{P^-}\backslash G({\mathcal O})/I_P)$ on $\delta_{1,\operatorname{Gr}_G}\in Shv(\operatorname{Gr}_G)^{I_P}$ one gets $$j_*^{P^-}\overset{I_P}{\ast}\delta_{1,\operatorname{Gr}_G}\,{\widetilde\to}\, \delta_{1,\operatorname{Gr}_G}[\dim U(P^-)]\in Shv(\operatorname{Gr}_G)^{I_{P^-}}$$ Applying $j_*^{P^-}\overset{I_P}{\ast}\_$ to ([\[expression_1\_for_dual_baby\]](#expression_1_for_dual_baby){reference-type="ref" reference="expression_1_for_dual_baby"}) one gets $$\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab}^+} (j_*^{P^-}\overset{I_P}{\ast}j_{-\lambda, *})\ast \operatorname{Sat}(V^{\lambda}) \in Shv(\operatorname{Gr}_G)^{I_{P^-}, ren},$$ which by Corollary [Corollary 1](#Pp_key_for_baby_Verma_transform_for_Pminus){reference-type="ref" reference="Pp_key_for_baby_Verma_transform_for_Pminus"} identifies with $$\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab}^+} (j^-_{-\lambda, *}\overset{I_{P^-}}{\ast}j_*^{P^-})\ast \operatorname{Sat}(V^{\lambda}) \in Shv(\operatorname{Gr}_G)^{I_{P^-}, ren}$$ The latter object identifies with $$\label{complex_for_Sect_4.5.16} \mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab}^+} j^-_{-\lambda, *}\ast \operatorname{Sat}(V^{\lambda})[\dim U(P^-)]\in Shv(\operatorname{Gr}_G)^{I_{P^-}, ren},$$ where in the latter formula we use the action of ${\mathcal H}_{P^-}(G)$ on $Shv(\operatorname{Gr}_G)^{I_{P^-}, ren}$. By Proposition [Proposition 1](#Pp_Gurbir_Chen_for_P_reformulation){reference-type="ref" reference="Pp_Gurbir_Chen_for_P_reformulation"}, ([\[complex_for_Sect_4.5.16\]](#complex_for_Sect_4.5.16){reference-type="ref" reference="complex_for_Sect_4.5.16"}) under the equivalence ([\[equivalence_Gurbir_Chen_for_P\_instead\]](#equivalence_Gurbir_Chen_for_P_instead){reference-type="ref" reference="equivalence_Gurbir_Chen_for_P_instead"}) identifies with $$\mathop{\operatorname{colim}}\limits_{\lambda\in\Lambda_{M, ab}^+} e^{-\lambda}\otimes V^{\lambda}[\dim U(P^-)]\in \operatorname{IndCoh}((\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})$$ By ([\[iso_O(P/M)\_for_Sect_2.1.6\]](#iso_O(P/M)_for_Sect_2.1.6){reference-type="ref" reference="iso_O(P/M)_for_Sect_2.1.6"}), the latter identifies with the direct image of ${\mathcal O}(\check{P}/\check{M})[\dim U(P^-)]\in{\operatorname{Rep}}(\check{P})$ under $B(\check{P})\to (\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P}$. That is, we get the object ${\mathcal M}_{\check{G},\check{P}}[\dim U(P^-)]$. Theorem [Theorem 1](#Thm_4.5.10){reference-type="ref" reference="Thm_4.5.10"} is proved. $\square$ ## Proof of Theorem [Theorem 1](#Thm_4.1.10){reference-type="ref" reference="Thm_4.1.10"} {#Sect_4.6} ### Semi-infinite relative dimensions It is convenient first to introduce the semi-infinite relative dimensions of orbits. For $\lambda\in \Lambda^+_M$ the relative dimension $\operatorname{dim.rel}(S^{\lambda}_P : S^0_P)$ is defined as follows. Write $H^{\lambda}$ for the stabilizer of $t^{\lambda}G({\mathcal O})\in \operatorname{Gr}_G$ in $H$. Set $$\operatorname{dim.rel}(S^{\lambda}_P: S^0_P)=\operatorname{dim.rel}(H^0: H^{\lambda})=\dim(H^0/H^0\cap H^{\lambda})-\dim(H^{\lambda}/H^0\cap H^{\lambda})$$ This is easy to calculate, one gets $\operatorname{dim.rel}(S^{\lambda}_P: S^0_P)=\langle\lambda, 2\check{\rho}\rangle$. ### {#section-91} For $\theta\in \Lambda_{G,P}$ define the relative dimension $\operatorname{dim.rel}(\operatorname{Gr}_P^{\theta}: \operatorname{Gr}_P^0)$ as follows. Set $M'=[M,M]$. Set ${\mathcal K}=M'(F)U(P)(F)$. Recall that ${\mathcal K}$ acts transitively on $\operatorname{Gr}_P^{\theta}$. Pick any $\lambda\in\Lambda$ over $\theta$. Let ${\mathcal K}^{\lambda}$ be the stabilizer of $t^{\lambda}G({\mathcal O})$ in ${\mathcal K}$. Set $$\operatorname{dim.rel}(\operatorname{Gr}_P^{\theta}: \operatorname{Gr}_P^0)=\operatorname{dim.rel}({\mathcal K}^0: {\mathcal K}^{\lambda})=\dim({\mathcal K}^0/{\mathcal K}^0\cap {\mathcal K}^{\lambda})-\dim({\mathcal K}^{\lambda}/{\mathcal K}^0\cap {\mathcal K}^{\lambda}).$$ One checks that this is independent of a choice of $\lambda$. More precisely, $$\operatorname{dim.rel}(\operatorname{Gr}_P^{\theta}: \operatorname{Gr}_P^0)=\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle.$$ ### Another system of generators Let $\nu\in\Lambda^+_M$ and $\theta\in\Lambda_{G,P}$ its image. Denote by $J_{\nu, !}\in Shv(\operatorname{Gr}_G)^{H, \le 0}$ the object $$J_{\nu, !}=(v_P^{\theta})_!(\mathfrak{t}_P^{\theta})^!\operatorname{Sat}_M(V^{\nu})[\langle\theta, 2\check{\rho}_M-2\check{\rho}\rangle]$$ Note that $Shv(\operatorname{Gr}_G)^{\le 0}$ is the smallest full subcategory containing containing $J_{\nu, !}$ for $\nu\in\Lambda^+_M$, stable under extensions and small colimits. ### {#section-92} For $F\in \operatorname{SI}_P$ the property $F\in Shv(\operatorname{Gr}_G)^{H,\ge 0}$ is equivalent to ${{\mathcal H}om}_{\operatorname{SI}_P}(\boldsymbol{\vartriangle}^{\mu}, F)\in\operatorname{Vect}^{\ge 0}$ for any $\mu\in\Lambda^+_M$. It is easy to see that for $F\in \operatorname{SI}_P$ the property $F\in Shv(\operatorname{Gr}_G)^{H,\ge 0}$ is also equivalent to ${{\mathcal H}om}_{\operatorname{SI}_P}(J_{\nu, !}, F)\in\operatorname{Vect}^{\ge 0}$ for all $\nu\in\Lambda^+_M$. ### {#section-93} Recall that if $V\in{\operatorname{Rep}}(\check{M}), \theta\in\Lambda_{G,P}$ then $V_{\theta}$ denotes the direct summand of $V$, on which $Z(\check{M})$ acts by $\theta$. **Lemma 1**. *Let $\gamma\in\Lambda^+$ and $\theta$ be the image of $w_0(\gamma)$ in $\Lambda_{G, P}$. Then $V^{\gamma}_{\theta}$ is an irreducible $\check{M}$-module with highest weight $w_0^Mw_0(\gamma)$.* *Proof.* Let $\check{\mathfrak{u}}(P^-), \check{\mathfrak{u}}(P), \check{\mathfrak{m}}$ denote the Lie algebras of $U(\check{P}^-), U(\check{P}), \check{M}$ respectively. We have $$U(\check{\mathfrak{g}})\,{\widetilde\to}\, U(\check{\mathfrak{u}}(P))\otimes U(\check{\mathfrak{m}})\otimes U(\check{\mathfrak{u}}(P^-))$$ for the universal enveloping algebras. Let $v\in V^{\gamma}$ be a lowest weight vector. Then $V^{\gamma}=U(\check{\mathfrak{u}}(P))\otimes U(\check{\mathfrak{m}})v$. Moreover, $U(\check{\mathfrak{m}})v\subset V$ is an irreducible $\check{M}$-module. Indeed, otherwise there would exist another nontrivial lowest weight vector $v'\in U(\check{\mathfrak{m}})v$. But then $v'$ would be a lowest vector for $G$ itself, because $U(\check{P}^-)$ is normal in $U(\check{B}^-)$. Here $U(\check{B}^-)$ is the unipotent radical of $\check{B}^-$. Our claim follows now from the observation that the $\check{T}$-weights on $\check{\mathfrak{u}}(P)$ are nonzero elements of $\Lambda_{G,P}^{pos}$. ◻ **Lemma 1**. *Let $\gamma\in\Lambda^+$. There is a canonical fibre sequence in $\operatorname{SI}_P$ $$K\to \boldsymbol{\vartriangle}^0\ast \operatorname{Sat}(V^{\gamma})\to J_{w_0^Mw_0(\gamma), !},$$ where $K$ admits a finite filtration by objects $J_{\mu, !}$ with $\mu\in\Lambda^+_M$ such that $U^{\mu}$ appears in $\operatorname{Res}(V^{\gamma})$ and satisfies $\mu\ne w_0^Mw_0(\gamma)$. For such $\mu$ we have $$\langle\mu-w_0^Mw_0(\gamma), 2\check{\rho}-2\check{\rho}_M\rangle > 0$$* *Proof.* Consider the filtration on $\boldsymbol{\vartriangle}^0\ast \operatorname{Sat}(V^{\gamma})$ coming from the stratification of $\operatorname{Gr}_G$ by $\operatorname{Gr}_P^{\theta}$ first. The successive subquotients of this filtrations are $$\label{object_succ_quotient_for_t-str} (v_P^{\theta})_!(v_P^{\theta})^*(\boldsymbol{\vartriangle}^0\ast \operatorname{Sat}(V^{\gamma})),\;\;\; \mbox{for}\;\;\; \theta\in\Lambda_{G,P}.$$ As we have seen in the proof of Proposition [Proposition 1](#Pp_2.3.21){reference-type="ref" reference="Pp_2.3.21"}, for $\theta\in\Lambda_{G,P}$ one has canonically $$(\mathfrak{t}^{\theta}_P)_!(v_P^{\theta})^*(\boldsymbol{\vartriangle}^0\ast \operatorname{Sat}(V^{\gamma}))\,{\widetilde\to}\, \operatorname{Sat}_M(V^{\gamma}_{\theta})[\langle\theta, 2\check{\rho}_M-2\check{\rho}\rangle]$$ We see that ([\[object_succ_quotient_for_t-str\]](#object_succ_quotient_for_t-str){reference-type="ref" reference="object_succ_quotient_for_t-str"}) is a direct sum of finitely many objects of the form $J_{\nu, !}$ for $\nu\in \Lambda^+_M$ appearing in $V^{\gamma}_{\theta}$. Let now $\theta$ be the image of $w_0(\gamma)$ in $\Lambda_{G,P}$. Write $Y$ for the support of $\boldsymbol{\vartriangle}^0\ast \operatorname{Sat}(V^{\gamma})$. It is easy to see that $\operatorname{Gr}_P^{\theta}\cap Y$ is closed in $Y$. This follows from the fact that for $\theta_1,\theta_2\in\Lambda_{G,P}$ one has $\operatorname{Gr}_P^{\theta_1}\subset \overline{\operatorname{Gr}}_P^{\theta_2}$ iff $\theta_2-\theta_1\in\Lambda_{G,P}^{pos}$. Our claim follows now from Lemma [Lemma 1](#Lm_4.6.6){reference-type="ref" reference="Lm_4.6.6"}. ◻ ### {#section-94} For $\mu\in\Lambda_{M, ab}, \gamma\in\Lambda^+$ set $$\Upsilon^{\mu, \gamma}=\operatorname{Sat}_M(U^{\mu})\ast \boldsymbol{\vartriangle}^0\ast \operatorname{Sat}(V^{\gamma})\in Shv(\operatorname{Gr}_G)^{H,\le 0}$$ It is easy to see that for any $\mu\in\Lambda_{M, ab}$, $\nu\in \Lambda^+_M$ one has canonically $$t^{\mu}J_{\nu, !}[-\langle\mu,2\check{\rho}\rangle]\,{\widetilde\to}\, J_{\nu+\mu, !}$$ Besides, for any $\nu\in\Lambda^+_M$ there is $\mu\in -\Lambda_{M, ab}^+$ such that $w_0w_0^M(\nu+\mu)\in \Lambda^+$. From Lemma [Lemma 1](#Lm_4.6.7){reference-type="ref" reference="Lm_4.6.7"} we immediately derive the following. **Corollary 1**. *For any $\nu\in\Lambda^+_M$ there is $\mu\in\Lambda_{M, ab}^+, \gamma\in\Lambda^+$ and a fibre sequence $$\label{fibre_sequence_for_Gurbir_induction} K\to \Upsilon^{\mu, \gamma}\to J_{\nu, !},$$ where $K$ admits a finite filtration by objects $J_{\nu', !}$ with $\nu'\in\Lambda^+_M$ satisfying $$\langle\nu'-\nu, 2\check{\rho}-2\check{\rho}_M\rangle >0.$$* **Proposition 1**. *Let $F\in Shv(\operatorname{Gr}_G)^H$. Assume there is $N\in{\mathbb Z}$ such that for any $\nu\in\Lambda^+_M$ with $\langle\nu, 2\check{\rho}-2\check{\rho}_M\rangle >N$ one has ${{\mathcal H}om}_{\operatorname{SI}_P}(J_{\nu, !}, F)\in\operatorname{Vect}^{\ge 0}$. Then the following properties are equivalent:* - *$F\in Shv(\operatorname{Gr}_G)^{H,\ge 0}$;* - *if $\mu\in\Lambda_{M, ab}^+, \gamma\in\Lambda^+$ then ${{\mathcal H}om}_{\operatorname{SI}_P}(\Upsilon^{\mu, \gamma}, F)\in \operatorname{Vect}^{\ge 0}$.* *Proof.* i) implies ii), because for $\mu\in\Lambda_{M, ab}, \gamma\in\Lambda^+$ one has $\Upsilon^{\mu, \gamma}\in Shv(\operatorname{Gr}_G)^{H,\le 0}$ by Propositions [Proposition 1](#Pp_2.4.19){reference-type="ref" reference="Pp_2.4.19"} and [Proposition 1](#Pp_2.3.21){reference-type="ref" reference="Pp_2.3.21"}. Assume ii). To get i), it suffices to show that for any $\nu\in\Lambda^+_M$ one has $${{\mathcal H}om}_{\operatorname{SI}_P}(J_{\nu, !}, F)\in\operatorname{Vect}^{\ge 0}.$$ We proceed by the descending induction on $\langle\nu, 2\check{\rho}-2\check{\rho}_M\rangle$. Let $N'\in{\mathbb Z}$. Assume for any $\nu\in \Lambda^+_M$ with $\langle\nu, 2\check{\rho}-2\check{\rho}_M\rangle >N'$ one has ${{\mathcal H}om}_{\operatorname{SI}_P}(J_{\nu, !}, F)\in\operatorname{Vect}^{\ge 0}$. Let $\nu\in \Lambda^+_M$ with $\langle\nu, 2\check{\rho}-2\check{\rho}_M\rangle=N'$. Pick the fibre sequence ([\[fibre_sequence_for_Gurbir_induction\]](#fibre_sequence_for_Gurbir_induction){reference-type="ref" reference="fibre_sequence_for_Gurbir_induction"}). It yields a fibre sequence in $\operatorname{Vect}$ $${{\mathcal H}om}_{\operatorname{SI}_P}(J_{\nu, !}, F)\to {{\mathcal H}om}_{\operatorname{SI}_P}(\Upsilon^{\mu, \gamma}, F)\to {{\mathcal H}om}_{\operatorname{SI}_P}(K, F),$$ which shows that ${{\mathcal H}om}_{\operatorname{SI}_P}(J_{\nu, !}, F)\in\operatorname{Vect}^{\ge 0}$. ◻ *Proof of Theorem [Theorem 1](#Thm_4.1.10){reference-type="ref" reference="Thm_4.1.10"}.* By Lemma [Lemma 1](#Lm_2.3.13_about_Av!){reference-type="ref" reference="Lm_2.3.13_about_Av!"} and Proposition [Proposition 1](#Pp_2.4.19){reference-type="ref" reference="Pp_2.4.19"}, we may and do assume $\mu=0$. If $i\in{\mathcal I}-{\mathcal I}_M$ then $\langle\alpha_i, 2\check{\rho}_M\rangle\le 0$. So, for $\theta\in\Lambda_{G,P}^{pos}$ one has $\langle\theta, 2\check{\rho}-2\check{\rho}_M\rangle\ge 0$. Note that $\boldsymbol{\vartriangle}^0$ is the extension by zero from $\overline{\operatorname{Gr}}_P^0$. So, if $\nu\in\Lambda_M^+$ with $\langle\nu, 2\check{\rho}-2\check{\rho}_M\rangle >0$ then ${{\mathcal H}om}_{\operatorname{SI}_P}(J_{\nu, !},\boldsymbol{\vartriangle}^0)=0$. Thus, By Proposition [Proposition 1](#Pp_induction_step){reference-type="ref" reference="Pp_induction_step"}, it suffices to show that for any $\mu\in\Lambda_{M, ab}^+, \gamma\in\Lambda^+$ one has $${{\mathcal H}om}_{\operatorname{SI}_P}(\Upsilon^{\mu, \gamma}, \boldsymbol{\vartriangle}^0)\in \operatorname{Vect}^{\ge 0}$$ Applying ([\[equiv_Iwahori_vs_SI\]](#equiv_Iwahori_vs_SI){reference-type="ref" reference="equiv_Iwahori_vs_SI"}), we are reduced to show that $$\label{property_for_step_one_Thm_4.1.10} {{\mathcal H}om}_{Shv(\operatorname{Gr}_G)^{I_P}}(j_{\mu, !}\ast \operatorname{Sat}(V^{\gamma}), \delta_{1,\operatorname{Gr}_G})\in \operatorname{Vect}^{\ge 0}.$$ Consider the adjoint pair $$I: \operatorname{IndCoh}_{\operatorname{Nilp}}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})\leftrightarrows \operatorname{IndCoh}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P}): I^!$$ corresponding under ([\[equivalence_Gurbir_Chen_for_P\_instead\]](#equivalence_Gurbir_Chen_for_P_instead){reference-type="ref" reference="equivalence_Gurbir_Chen_for_P_instead"}) to the adjoint pair $\operatorname{ren}: Shv(\operatorname{Gr}_G)^{I_{P^-}}\leftrightarrows Shv(\operatorname{Gr}_G)^{I_{P^-}, ren}: \operatorname{un-ren}$. By Corollary 4.5.15, one has $$j^-_{\mu, !}\overset{I_{P^-}}{\ast}j_*^{P^-}\,{\widetilde\to}\, j_*^{P^-} \overset{I_P}{\ast}j_{\mu, !}$$ Now applying $$Shv(\operatorname{Gr}_G)^{I_P}\,\stackrel{j_*^{P^-}\overset{I_P}{\ast}}{\to}Shv(\operatorname{Gr}_G)^{I_{P^-}}\stackrel{(88)}{\to}\operatorname{IndCoh}_{\operatorname{Nilp}}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P}),$$ the property ([\[property_for_step_one_Thm_4.1.10\]](#property_for_step_one_Thm_4.1.10){reference-type="ref" reference="property_for_step_one_Thm_4.1.10"}) becomes $${{\mathcal H}om}_{\operatorname{IndCoh}_{\operatorname{Nilp}}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})}(I^!i_*(e^{-\mu}\otimes V^{\gamma}), I^!i_*e),$$ where we have denoted by $$i: B(P)\to (\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P}$$ the natural map. Recall that $\operatorname{IndCoh}_{\operatorname{Nilp}}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})$ carries a unique t-structure such that $$i^!: \operatorname{IndCoh}_{\operatorname{Nilp}}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})\to \operatorname{QCoh}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})$$ is t-exact. Moreover, $i^!$ induces an equivalence $$\operatorname{IndCoh}_{\operatorname{Nilp}}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})^+\,{\widetilde\to}\,\operatorname{QCoh}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})^+$$ on the subcategories of eventually coconnective objects. For the t-structures on $\operatorname{IndCoh}$ of an Artin stack we refer to ([@Gai_IndCoh], Proposition 11.7.5). Moreover, the composition $$\begin{gathered} \operatorname{QCoh}(B(P))\stackrel{i_*}{\to} \operatorname{IndCoh}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})\stackrel{I^!}{\to} \operatorname{IndCoh}_{\operatorname{Nilp}}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})\\ \stackrel{i^!}{\to} \operatorname{QCoh}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})\end{gathered}$$ is the usual pushforward $i_*: \operatorname{QCoh}(B(P))\to \operatorname{QCoh}(\check{\mathfrak{u}}(P)\times_{\check{\mathfrak{g}}} 0)/\check{P})$. We are done. ◻ # Generalities ## Some adjoint pairs **Proposition 1**. *Let $H, G$ be placid group schemes, $G\stackrel{}{\hookrightarrow} H$ be a subgroup (not necessarily a placid closed immersion). Assume $G\,{\widetilde\to}\, \lim_{i\in I^{op}} G_i$, where $G_i$ is a smooth group scheme of finite type, $I\in 1-\operatorname{Cat}$ is filtered, and for $i\to j$ in $I$ the map $G_j\to G_i$ is smooth affine surjective homomoprhism of group schemes. Write $K_i=\operatorname{Ker}(G\to G_i)$. Assume $H\,{\widetilde\to}\, \lim_{i\in I^{op}} H/K_i$ in $\operatorname{PreStk}$. Assume $H/G$ is a pro-smooth placid scheme. Consider the projection $p: H/G\to \operatorname{Spec}k$ as $H$-equivariant map. Then* *i) the adjoint pair $p^*: \operatorname{Vect}\leftrightarrows Shv(H/G): p_*$ takes place in $Shv(H)-mod$;\ ii) assume $C\in Shv(H)-mod(\operatorname{DGCat}_{cont})$. The above adjoint pair gives an adjoint pair in $\operatorname{DGCat}_{cont}$ $$\operatorname{oblv}: C^H\leftrightarrows C^G: \operatorname{Av}^{H/G}_*$$* *Proof.* i) Since $p$ is $H$-equivariant map, $p_*: Shv(H/G)\to \operatorname{Vect}$ is a morphism of $Shv(H)$-modules. Now the diagram is cartesian $$\begin{array}{ccc} H\times H/G & \stackrel{act}{\to} & H/G\\ \downarrow\lefteqn{\scriptstyle\operatorname{pr}} && \downarrow\\ H & \to & \operatorname{Spec}k \end{array}$$ So, for $K\in Shv(H)$, $\operatorname{act}_*(K\boxtimes p^*e)\,{\widetilde\to}\, p^*\operatorname{R\Gamma}(H, K)$ canonically. Here we used the base change isomorphism given by Lemma [Lemma 1](#Lm_A.1.3_base_change){reference-type="ref" reference="Lm_A.1.3_base_change"}. ii\) Applying ${\operatorname{Fun}}_{Shv(H)}(\cdot, C)$, we get the adjoint pair $\operatorname{oblv}: {\operatorname{Fun}}_{Shv(H)}(\operatorname{Vect}, C)\leftrightarrows {\operatorname{Fun}}_{Shv(H)}(Shv(H/G), C): \operatorname{Av}^{H/G}_*$. Now using the assumption $H\,{\widetilde\to}\, \lim_{i\in I^{op}} H/K_i$ in $\operatorname{PreStk}$ from ([@Ly4], 0.0.36) we get $Shv(H/G)\,{\widetilde\to}\, Shv(H)^G$ with respect to the $G$-action on $H$ by right translations. Recall that $Shv(H)^G\,{\widetilde\to}\, Shv(H)_G$, because $G$ is placid group scheme. Finally, $${\operatorname{Fun}}_{Shv(H)}(Shv(H/G), C)\,{\widetilde\to}\, {\operatorname{Fun}}_{Shv(H)}(Shv(H)\otimes_{Shv(G)}\operatorname{Vect}, C)\,{\widetilde\to}\, C^G$$ ◻ ### {#Sect_A.0.2} An example of the situation as in Proposition [Proposition 1](#Pp_A.1){reference-type="ref" reference="Pp_A.1"}. Assume $H=G\rtimes \bar H$, where $\bar H\subset H$ is a normal subgroup, $\bar H$ is a placid group scheme, and $G$ acts on $\bar H$ by conjugation. Here $G$ is a placid group scheme. If moreover $\bar H$ is prounipotent then Proposition [Proposition 1](#Pp_A.1){reference-type="ref" reference="Pp_A.1"} also shows that the functor $\operatorname{oblv}: C^H\to C^G$ is fully faithful. Indeed, the natural map $\operatorname{id}\to \operatorname{Av}^{H/G}_*\operatorname{oblv}$ is the identity, because $p_*p^*: \operatorname{Vect}\to \operatorname{Vect}$ identifies with $\operatorname{id}$. ### {#section-95} Consider a cartesian square $$\label{square_1} \begin{array}{ccc} X& \stackrel{f_X}{\to} & X'\\ \downarrow\lefteqn{\scriptstyle g} &&\downarrow\lefteqn{\scriptstyle g'}\\ Y& \stackrel{f_Y}{\to} & Y', \end{array}$$ in $\operatorname{PreStk}$. **Lemma 1** ([@Ly4], Lemma 0.0.20). *Assume $Y'\in Sch_{ft}$, and $Y, X'$ are placid schemes over $Y'$. Then $X$ is also a placid scheme. Assume $Y\,{\widetilde\to}\, \lim_{i\in I^{op}} Y_i$, where $I$ is filtered, $f_{Y,i}: Y_i\to Y'$ is smooth, $Y_i\in Sch_{ft}$, and for $i\to j$ in $I$, $Y_j\to Y_i$ is smooth affine surjective morphism in ${\operatorname{Sch}}_{ft}$. Then one has $f_Y^*g'_*\,{\widetilde\to}\, g_*f_X^*$ as functors $Shv(X')\to Shv(Y)$.* ## Some coinvariants {#Sect_A.1} ### {#section-96} Let $P\subset G$ be a parabolic in a connected split reductive group with Levi $M$ and unipotent radical $U$. Let $F=k((t)), {\mathcal O}=k[[t]]$. Let $H=M({\mathcal O})U(F)$. This is a placid ind-scheme, closed in $P(F)$. We have also $P(F)/H\,{\widetilde\to}\, M(F)/M({\mathcal O})$. Since the object $\delta_1\in Shv(\operatorname{Gr}_M)$ is $H$-invariant, the functor $\operatorname{Vect}\to Shv(\operatorname{Gr}_M)$, $e\mapsto \delta_1$ is $Shv(H)$-linear. Now the $Shv(H)$-action on $Shv(\operatorname{Gr}_M)$ comes as the restriction of a $Shv(P(F))$-action, hence we get by adjointness a canonical functor $$\label{functor_for_Sect_A.1} Shv(P(F))\otimes_{Shv(H)}\operatorname{Vect}\to Shv(\operatorname{Gr}_M)$$ **Lemma 1**. *The functor ([\[functor_for_Sect_A.1\]](#functor_for_Sect_A.1){reference-type="ref" reference="functor_for_Sect_A.1"}) is an equivalence.* *Proof.* Pick a presentation $U(F)=\operatorname{colim}_{n\in{\mathbb N}} U_n$, where $U_n$ is a prounipotent group scheme, for $n\le m$, $U_n\to U_m$ is a placid closed immersion and a homomorphism of group schemes. Assume $M({\mathcal O})$ normalizes each $U_n$, so $M({\mathcal O})U_n=:H_n$ is a placid group scheme, and $H\,{\widetilde\to}\,\operatorname{colim}_{n\in{\mathbb N}} H_n$. We have $P(F)\,{\widetilde\to}\, \operatorname{colim}_{n\in {\mathbb N}} M(F)U_n$ in $\operatorname{PreStk}$, as colimits in $\operatorname{PreStk}$ are universal. Recall that for any morphism $f: Y_1\to Y_2$ of placid ind-schemes the functor $f_*: Shv(Y_1)\to Shv(Y_2)$ is well-defined. Now $$Shv(P(F))\,{\widetilde\to}\, \operatorname{colim}_{n\in {\mathbb N}} Shv(M(F)U_n)$$ with respect to the $*$-push-forwards. Indeed, pick a presentation $M(F)\,{\widetilde\to}\,\operatorname{colim}_{i\in I} M_i$, where $M_i$ is a placid scheme, $I$ is small filtered, for $i\to i'$ the map $M_i\to M_{i'}$ is a placid closed immersion. Then $Shv(M(F)U_n)\,{\widetilde\to}\, \operatorname{colim}_{i\in I} Shv(M_iU_n)$. The above gives $$Shv(P(F))\otimes_{Shv(H)}\operatorname{Vect}\,{\widetilde\to}\,\mathop{\operatorname{colim}}\limits_{(n\le m)\in{\operatorname{Fun}}([1],{\mathbb N})} Shv(M(F)U_m)\otimes_{Shv(H_n)}\operatorname{Vect}$$ The diagonal map ${\mathbb N}\to {\operatorname{Fun}}([1],{\mathbb N})$ is cofinal, so the above identifies with $$\operatorname{colim}_{n\in {\mathbb N}} Shv(M(F)U_n)\otimes_{Shv(H_n)}\operatorname{Vect}$$ Now each term of the latter diagram identifies with $Shv(M(F)/M({\mathcal O}))$ using ([@Ly4], 0.0.36), and we are done. Indeed, for any $I\in 1-\operatorname{Cat}$ the natural map $I\to \mid I\mid$ is cofinal, and for $I$ filtered we get $\mid I\mid\,{\widetilde\to}\, *$. ◻ Further, let $C\in Shv(P(F))-mod(\operatorname{DGCat}_{cont})$. We get $$C^H= {\operatorname{Fun}}_{Shv(H)}(\operatorname{Vect}, C)\,{\widetilde\to}\, {\operatorname{Fun}}_{Shv(P(F))}(Shv(P(F))\otimes_{Shv(H)}\operatorname{Vect}, C)$$ Thus, ${\operatorname{Fun}}_{Shv(P(F))}(Shv(\operatorname{Gr}_M), Shv(\operatorname{Gr}_M))$ acts on $C^H$. Now $${\operatorname{Fun}}_{Shv(P(F))}(Shv(\operatorname{Gr}_M), Shv(\operatorname{Gr}_M))\,{\widetilde\to}Shv(\operatorname{Gr}_M)^H\,{\widetilde\to}\, Shv(\operatorname{Gr}_M)^M$$ acts on $C^H$. **Remark 1**. *As in Lemma [Lemma 1](#Lm_A.1.2_coinvariants){reference-type="ref" reference="Lm_A.1.2_coinvariants"}, one shows that $Shv(P(F))_{U(F)}\,{\widetilde\to}\, Shv(M(F))$ naturally in $Shv(P(F))-mod(\operatorname{DGCat}_{cont})$, where we used the $U(F)$-action on $P(F)$ by right translations. This similarly implies that for any $C\in Shv(P(F))-mod(\operatorname{DGCat}_{cont})$, $C^{U(F)}\in Shv(M(F))-mod(\operatorname{DGCat}_{cont})$ naturally.* ## Some invariants {#Section_A2_some_invarinats} ### {#section-97} An observation about categories of invariants. Let $Y=\operatorname{colim}_{j\in J} Y_j$ be an ind-scheme of ind-finite type, here $J$ is a filtered category, $Y_j$ is a scheme of finite type, and for $j\to j'$ in $J$ the map $Y_j\to Y_{j'}$ is a closed immersion. Let $\alpha: H\to G$ be a homomorphism of group schemes, which are placid schemes. Assume $I$ is a filtered category and $H\,{\widetilde\to}\lim_{i\in I^{op}} H_i$, $G\,{\widetilde\to}\,\lim_{i\in I^{op}} G_i$, where $H_i$, $G_i$ is a smooth group scheme of finite type, for $i\to j$ in $I$ the transition maps $H_j\to H_i$, $G_j\to G_i$ are smooth, affine, surjective homomorphisms. Besides, we are given a diagram $I^{op}\times [1]\to Grp({\operatorname{Sch}}_{ft})$, sending $i$ to $\alpha_i: H_i\to G_i$, where $\alpha_i$ is a closed subgroup. Here $Grp({\operatorname{Sch}}_{ft})$ is the category of groups in ${\operatorname{Sch}}_{ft}$. We assume $\alpha=\lim_{i\in I^{op}} \alpha_i$. We assume for each $i$, $\operatorname{Ker}(H\to H_i)$ and $\operatorname{Ker}(G\to G_i)$ are prounipotent. Assume $G$ acts on $Y$. Moreover, for any $j\in J$, $Y_j$ is $G$-stable, and $G$ acts on $Y_j$ through the quotient $G\to G_i$ for some $i\in I$. We claim that $oblv: Shv(Y)^G\to Shv(Y)^H$ admits a continuous right adjoint $\operatorname{Av}_*$. *Proof.* We have $Shv(Y)^G\,{\widetilde\to}\,\lim_{j\in J^{op}} Shv(Y_j)^G$ with respect to the !-pullbacks, similarly $Shv(Y)^H\,{\widetilde\to}\,\lim_{j\in J^{op}} Shv(Y_j)^H$, and $oblv: Shv(Y)^G\to Shv(Y)^H$ is the limit over $j\in J^{op}$ of $\operatorname{oblv}_j: Shv(Y_j)^G\to Shv(Y_j)^H$. For given $j\in J$ the functor $\operatorname{oblv}_j$ admits a continuous right adjoint $\operatorname{Av}_{j, *}$. Indeed, pick $i\in I$ such that $G$-action on $Y_j$ factors through $G_i$. Then $\operatorname{oblv}_j$ identifies with the functor $f^!: Shv(Y_j/G_i)\to Shv(Y_j/H_i)$ for the projection $f: Y_j/H_i\to Y_j/G_i$. Since $G_i/H_i$ is smooth, $f$ is smooth. So, $f^!$ admits a continuous right adjoint (as $f$ is schematic of finite type). Let now $j\to j'$ be a map in $J$. Pick $i$ such that the $G$-action on $Y_j, Y_{j'}$ factors through $G_i$. Then we get a cartesian square $$\begin{array}{ccc} Y_j/H_i & \stackrel{h}{\to} & Y_{j'}/H_i\\ \downarrow\lefteqn{\scriptstyle f_j}&& \downarrow\lefteqn{\scriptstyle f_{j'}}\\ Y_j/G_i & \stackrel{h'}{\to} & Y_{j'}/G_i, \end{array}$$ where $h, h'$ are closed immersions. We have $(h')^!f_{j', *}\,{\widetilde\to}\, f_{j, *} h^!$. Since $f_j, f_{j'}$ are of the same relative dimension, we see that the diagram commutes $$\begin{array}{ccc} Shv(Y_j)^H & \stackrel{h^!}{\gets} & Shv(Y_{j'})^H\\ \downarrow\lefteqn{\scriptstyle \operatorname{Av}_{j, *}}&&\downarrow\lefteqn{\scriptstyle \operatorname{Av}_{j', *}}\\ Shv(Y_j)^G & \stackrel{(h')^!}{\gets} & Shv(Y_{j'})^G \end{array}$$ By ([@G], ch. I.1, 2.6.4), $\operatorname{oblv}$ admits a right adjoint $\operatorname{Av}_*$, and for the evaliation maps $\mathit{ev}_j: Shv(Y)^G\to Shv(Y_j)^G$, $\mathit{ev}_j: Shv(Y)^H\to Shv(Y_j)^H$ one gets $\mathit{ev}_j\operatorname{Av}_*\,{\widetilde\to}\, \operatorname{Av}_{j, *}\mathit{ev}_j$. So, $\operatorname{Av}_*$ is continuous. ◻ **Remark 1**. *If $L: C\leftrightarrows C': R$ is an adjoint pair in $\operatorname{DGCat}^{non-cocmpl}$ then $\operatorname{Ind}(L): \operatorname{Ind}(C)\leftrightarrows\operatorname{Ind}(C'): \operatorname{Ind}(R)$ is an adjoint pair in $\operatorname{DGCat}_{cont}$.* **Lemma 1**. *Let $C\in \operatorname{DGCat}_{cont}$, $C_i\subset C$ be a full subcategory, this is a map in $\operatorname{DGCat}_{cont}$ for $i\in I$. Here $I\in 1-\operatorname{Cat}$ is filtered. Assume for $i\to j$ in $I$, $C_j\subset C_i$. Set $D=\cap_i C_i=\lim_{i\in I^{op}} C_i$, where the limit is calculated in $\operatorname{DGCat}_{cont}$. Assume $L_i: C\to C_i$ is a left adjoint to the inclusion. Then $D$ is a localization of $C$, and the localization functor $L: C\to D$ is given by $L(c)=\operatorname{colim}_{i\in I} L_i(c)$, where the transition maps are the localization morphisms for $C_j\subset C_i$, and the colimit is calculated in $C$.* *Proof.* For $x\in \cap_i C_i$, $c\in C$ we get $$\begin{gathered} \operatorname{Map}_C(\operatorname{colim}_i L_i(c), x)\,{\widetilde\to}\, \lim_{i\in I^{op}} \operatorname{Map}(L_i(c), x)\,{\widetilde\to}\, \lim_{i\in I^{op}} \operatorname{Map}(c, x)\\ \,{\widetilde\to}\, \operatorname{Map}(c,x)\,{\widetilde\to}\,{\operatorname{Fun}}(I^{op}, \operatorname{Map}(c,x))\end{gathered}$$ For $J\in 1-\operatorname{Cat}$, $Z\in \operatorname{Spc}$ we have ${\operatorname{Fun}}(J, Z)\,{\widetilde\to}\, {\operatorname{Fun}}(\mid J\mid, Z)$, where $\mid J\mid\in\operatorname{Spc}$ is obtained by inverting all arrows. Since a filtered category is contractible, we are done. To explain that $L$ takes values in $\cap C_i$, note that we may equally understand $\operatorname{colim}_i L_i(c)$ as taken in $C_j$ over $i\in I_{j/}$, because the inclusion $C_j\subset C$ is continuous, so the colimit lies in $C_j$ for any $j$. ◻ ### About representations of $G(F)$ {#Sect_A.2.4} Let $G$ be a connected reductive group over $k$. Recall from ([@GaLocWhit], D.1.2) that for any $C\in \operatorname{DGCat}_{cont}$ with an action of $Shv(G(F))$, $C\,{\widetilde\to}\, \mathop{\operatorname{colim}}\limits_{n\in{\mathbb N}} C^{K_n}$, where $K_n=\operatorname{Ker}(G({\mathcal O})\to G({\mathcal O}/t^n))$. So, for any $c\in C$, $$c\,{\widetilde\to}\, \mathop{\operatorname{colim}}\limits_{n\in{\mathbb N}} \operatorname{oblv}_n \operatorname{Av}^{K_n}_*(c),$$ where $\operatorname{oblv}_n: C^{K_n}\to C$ and $\operatorname{Av}^{K_n}_*: C\to C^{K_n}$ are adjoint functors (by [@Ly], 9.2.6). ## Actions ### {#section-98} The theory of placid group ind-schemes acting on categories is developed for ${\mathcal D}$-modules in ([@LC], Appendix B). A version of this theory in the constructible context is developed to some extent in ([@Ly9], Sections 1.3.2 - 1.3.23). Recall the following. If $Y\in\operatorname{PreStk}_{lft}$ then by a placid group (ind)-scheme over $Y$ we mean a group object $G\to Y$ in $\operatorname{PreStk}_{/Y}$ such that for any $S\to Y$ with $S\in{\operatorname{Sch}}_{ft}$, $G\times_Y S$ is a placid group (ind)-scheme over $S$. If $Y\in\operatorname{PreStk}_{lft}$ and $G\to Y$ is a placid group ind-scheme over $Y$, say that $G$ is ind-pro-unipotent if for any $S\in {\operatorname{Sch}}_{ft}$, $G\times_Y S$ is ind-pro-unipotent. In other words, there is a small filtered category $I$, and a presentation $G\times_Y S\,{\widetilde\to}\,\operatorname{colim}_{i\in I} G_{S, i}$, where $G_{S,i}$ is a prounipotent group scheme over $S$, for $i\to j$ in $I$, $G_{S, i}\to G_{S, j}$ is a placid closed immersion and a homomorphism of group schemes over $S$. ### {#section-99} Let $f: Y\to Z$ be a morphism of ind-schemes of ind-finite type, $G\to Z$ be a placid group ind-scheme over $Z$. As in ([@Ly9], 1.3.12) for $S\to Z$ with $S\in{\operatorname{Sch}}_{ft}$ set $G_S=G\times_Z S$, $Y_S=Y\times_Z S$ and view $Shv(G_S)$ as an object of $Alg(Shv(Z)-mod)$. Assume $G$ acts on $Y$ over $Z$. Then set $$Shv(Y)^G=\lim_{(S\to Y)\in ({\operatorname{Sch}}_{ft}/Y)^{op}} Shv(Y_S)^{G_S},$$ see ([@Ly4], 0.0.42) for details. The category of invariants is defined in ([@Ly9], 1.3.2) as $$Shv(Y_S)^{G_S}={\operatorname{Fun}}_{Shv(G_S)}(Shv(S), Shv(Y_S))\in Shv(S)-mod(\operatorname{DGCat}_{cont})$$ ### {#Sect_A.2.5} Let $f: Y\to Z$ be a morphism of ind-schemes of ind-finite type, $G\to Z$ be a placid group ind-scheme over $Z$. Assume $G$ acts on $Y$ over $Z$, and $G$ is ind-pro-unipotent. Let $s: Z\to Y$ be a section of $f$. The stabilizor $St_s$ of $s$ is defined as the fibred product $G\times_{Y\times Z} Z$, that is, by the equation $gs(\bar g)=s(\bar g)$ for $g\in G$. Here $\bar g\in Z$ is the projection of $g$, and the two maps $G\to Y$, $G\to Z$ are $g\mapsto gs(\bar g)$ and $g\mapsto s(\bar g)$. Assume $St_s$ is a placid group scheme over $Z$. Consider the quotient $G/St_s\to Z$ over $Z$ in the sense of stacks. We get a natural map $\bar f: G/St_s\to Y$ over $Z$. Assume that $\bar f$ is an isomorphism, that is, $G$ acts transitively on the fibres of $f$. **Lemma 1**. *In the situation of Section [5.4.3](#Sect_A.2.5){reference-type="ref" reference="Sect_A.2.5"} one has the following.\ i) The composition $$Shv(Y)^G\stackrel{\operatorname{oblv}}{\to} Shv(Y)\stackrel{s^!}{\to} Shv(Z)$$ is an equivalence.\ ii) The functor $f^!: Shv(Z)\to Shv(Y)$ is fully faithful and takes values in the full subcategory $Shv(Y)^G\subset Shv(Y)$.* *Proof.* i) Let $S\in{\operatorname{Sch}}_{ft}$ with a map $S\to Z$. Making the base change by this map, we get a diagram $f_S: Y\times_Z S\to S$ and its section $s_S$. Let $G_S=G\times_Z S$. By ([@Ly9], Lemma 1.3.20), the composition $$Shv(Y\times_Z S)^{G_S}\stackrel{\operatorname{oblv}}{\to} Shv(Y\times_Z S)\stackrel{s_S^!}{\to} Shv(S)$$ is an equivalence. Passing to the limit over $(S\to Z)\in ({\operatorname{Sch}}_{ft}/Z)^{op}$, we conclude. ii\) We may assume $Z\in {\operatorname{Sch}}_{ft}$. Consider the stabilizor $St_s$ of $s$ in $G$ as in Section [5.4.3](#Sect_A.2.5){reference-type="ref" reference="Sect_A.2.5"}. Pick a presentation $G\,{\widetilde\to}\, \operatorname{colim}_{j\in J} G^j$, where $G^j$ is a placid group scheme over $Z$ for $j\in J$, $J$ is a small filtered category, for $j\to j'$ in $J$ the map $G^j\to G^{j'}$ is a placid closed immersion and a homomorphism of group schemes over $Z$. Besides we assume $0\in J$ is an initial object, and $G^0=St_z$. Write $G/G^0$ for the stack quotient, so $Y\,{\widetilde\to}\, G/G^0\,{\widetilde\to}\,\operatorname{colim}_{j\in J} G^j/G^0$. Write $f^j$ for the composition $G^j/G^0\stackrel{}{\hookrightarrow} G/G^0\stackrel{f}{\to} Z$. For each $j$, the functor $(f^j)^!: Shv(Z)\to Shv(G^j/G^0)$ is fully faithful, because $G^j$ is prounipotent. So, $f^!: Shv(Z)\to Shv(Y)$ is fully faithful ◻ **Lemma 1**. *In the situation of Lemma [Lemma 1](#Lm_A.2.6){reference-type="ref" reference="Lm_A.2.6"} assume in addition that $M$ is a placid group scheme acting on $Y, Z$ and $f$ is $M$-equivariant. Then the functor $f^!: Shv(Z)^M\to Shv(Y)^M$ is also fully faithful.* *Proof.* As in Lemma [Lemma 1](#Lm_A.2.6){reference-type="ref" reference="Lm_A.2.6"}, the standard argument reduces our claim to the case when $Z\in{\operatorname{Sch}}_{ft}$, so we assume this. Recall that $f^!: Shv(Z)^M\to Shv(Y)^M$ is obtained by passing to the limit over $[n]\in \bm{\mathit{\Delta}}$ in the diagram $${\operatorname{Fun}}_{e, cont}(Shv(M)^{\otimes n}, Shv(Z))\to {\operatorname{Fun}}_{e, cont}(Shv(M)^{\otimes n}, Shv(Y))$$ For each $[n]\in \bm{\mathit{\Delta}}$ the latter functor is fully faithful, because $f^!: Shv(Z)\to Shv(Y)$ admits a left adjoint. So, passing to the limit we obtain a fully faithful functor. ◻ ## t-structures {#Sect_A.3} ### {#section-100} Let $Y\to S$ be a morphism in ${\operatorname{Sch}}_{ft}$. Equip $Shv(Y)$ with the perverse t-structure. Let $G\to S$ be a placid group scheme over $S$ acting on $Y$ over $S$ through its finite-dimensional quotient group scheme $G\to G_0$. Assume that $\operatorname{Ker}(G\to G_0)$ is a prounipotent group scheme over $S$. We equip $Shv(Y)^G$ with the perverse t-structure as follows. Recall that $Shv(Y)^G\,{\widetilde\to}\, Shv(Y)^{G_0}$ by ([@Ly9], 1.3.21) and the latter identifies with $Shv(Y/G_0)$ in such a way that $\operatorname{oblv}[\dim G_0]: Shv(Y)^{G_0}\to Shv(Y)$ identifies with $q^*[\operatorname{dim.rel}(q)]: Shv(Y/G_0)\to Shv(Y)$ for $q: Y\to Y/G_0$. The latter functor is $t$-exact for the perverse t-structures. So, the perverse t-structure on $Shv(Y/G_0)$ yields one one $Shv(Y)^G$. We denote the resulting t-exact functor by $\operatorname{oblv}[\operatorname{dim.rel}]: Shv(Y)^G\to Shv(Y)$. If $G\to G_1\to G_0$ is another quotient group of finite type then for $a: Y/G_1\to Y/G_0$ we identify $Shv(Y/G_0)$ with $Shv(Y/G_1)$ via $a^*[\operatorname{dim.rel}(a)]$ to obtain the functor $\operatorname{oblv}[\operatorname{dim.rel}]: Shv(Y)^G\to Shv(Y)$ independent of $G_0$. One similarly gets a functor $\operatorname{oblv}: Shv(Y)^G\to Shv(Y)$, which according to oour convention identifies with $q^!$ for $q: Y\to Y/G_0$. Recall that $Y/G_0$ is duality adapted in the sense of ([@AGKRRV], F.2.6). So, the Verdier duality gives an equivalence $$\mathbb{D}: (Shv(Y/G_0)^c)^{op}\,{\widetilde\to}\, (Shv(Y/G_0)^c)$$ This in turn gives an equivalence $Shv(Y/G_0)^{\vee}\,{\widetilde\to}\, Shv(Y/G_0)$ such that the corresponding counit map $Shv(Y/G_0)\otimes Shv(Y/G_0)\to \operatorname{Vect}$ is $$(F_1, F_2)\mapsto \operatorname{R\Gamma}_{\blacktriangle}(Y/G_0, F_1\otimes^! F_2),$$ where $\operatorname{R\Gamma}_{\blacktriangle}: Shv(Y/G_0)\to\operatorname{Vect}$ is the functor dual to $\operatorname{Vect}\to Shv(Y/G_0)$, $e\mapsto \omega_{Y/G_0}$, see ([@AGKRRV], F.2 and [@AGKRRV2], A.4). This counit does not depend on the choice of the above quotient $G\to G_0$, so yields a canonical functor $Shv(Y)^G\otimes Shv(Y)^G\to\operatorname{Vect}$. We have a canonical morphism $$\operatorname{R\Gamma}_{\blacktriangle}(Y/G_0, F_1\otimes^! F_2)\to \operatorname{R\Gamma}(Y/G_0, F_1\otimes^! F_2)$$ If $F_1$ or $F_2$ lies in $Shv(Y/G_0)^c$ then the latter map is an isomorphism by ([@AGKRRV], F.4.5). Define $Shv(Y/G_0)^{constr}\subset Shv(Y/G_0)$ as the full subcategory of those objects whose restriction to $Y$ lies in $Shv(Y)^c$. By definition of the renormalized category of sheaves from ([@AGKRRV], F.5.1), $Shv(Y/G_0)^{ren}=\operatorname{Ind}(Shv(Y/G_0)^{constr})$. This category does not depend on the choice of $G_0$ up to a canonical equivalence, so gives rise to the category $Shv(Y)^{G, ren}$. The Verdier duality extends to an equivalence $$\mathbb{D}: (Shv(Y/G_0)^{constr})^{op}\,{\widetilde\to}\, Shv(Y/G_0)^{constr},$$ see ([@AGKRRV], F.2.5). Passing to the ind-completions, we get $$(Shv(Y/G_0)^{ren})^{\vee}\,{\widetilde\to}\, Shv(Y/G_0)^{ren}$$ As in *loc.cit.*, one has an adjoint pair in $\operatorname{DGCat}_{cont}$ $$\operatorname{ren}: Shv(Y/G_0)\leftrightarrows Shv(Y/G_0)^{ren}: \operatorname{un-ren},$$ where $ren$ is fully faithful. This adjoint pair does not depend on a choice of the above quotient $G\to G_0$, so gives rise to a canonical adjoint pair $$\operatorname{ren}: Shv(Y)^G\leftrightarrows Shv(Y)^{G, ren}: \operatorname{un-ren}.$$ ### {#Sect_A.4.2_on_t-structures} If $Y\to Y'$ is a closed immersion in $({\operatorname{Sch}}_{ft})_{/S}$, assume it is $G$-equivariant, where the $G$-action factors through some finite-dimensional quotient group scheme $G\to G_0$ as above. In this case we get a closed immersion $i: Y/G_0\to Y'/G_0$, hence an adjoint pair $i_!: Shv(Y/G_0)\leftrightarrows Shv(Y'/G_0): i^!$. If $G\to G_1\to G_0$ is another quotient group scheme as above, we get a commutative diagram $$\begin{array}{ccc} Y/G_0 & \stackrel{i}{\to} & Y'/G_0\\ \uparrow\lefteqn{\scriptstyle a} && \uparrow\lefteqn{\scriptstyle a}\\ Y/G_1 & \stackrel{i_1}{\to} & Y'/G_1 \end{array}$$ Then $i_1^!a^*[\operatorname{dim.rel}(a)]\,{\widetilde\to}\, a^*[\operatorname{dim.rel}(a)]i^!$ and $a^*[\operatorname{dim.rel}(a)] i_!\,{\widetilde\to}\, (i_1)_!a^*[\operatorname{dim.rel}(a)]$ canonically. So, we get a well-defined adjoint pair $i_!: Shv(Y)^G\leftrightarrows Shv(Y')^G: i^!$, where $i^!$ is left t-exact, and $i_!$ is t-exact. The functors $i_!, i^!$ commute naturally with both $\operatorname{oblv}, \operatorname{oblv}[\operatorname{dim.rel}]: Shv(Y)^G\to Shv(Y)$. Since we are in the constructible context, the functor $i^!: Shv(Y'/G_0)\to Shv(Y/G_0)$ has a continuous right adjoint, so we get adjoint pairs $$i_!: Shv(Y/G_0)^c\leftrightarrows Shv(Y'/G_0)^c: i^!$$ and $i_!: Shv(Y/G_0)^{constr}\leftrightarrows Shv(Y'/G_0)^{constr}: i^!$. Under the above duality, the dual of $i_!: Shv(Y/G_0)\to Shv(Y'/G_0)$ identifies with $i^!: Shv(Y'/G_0)\to Shv(Y/G_0)$, and similarly for the renormalized version. We similarly get the adjoint pair $$i_!: Shv(Y)^{G, ren}\leftrightarrows Shv(Y')^{G, ren}: i^!,$$ where the left adjoint is fully faithful. ### {#section-101} Let $S\in {\operatorname{Sch}}_{ft}$, let $Y\to S$ be an ind-scheme of ind-finite type over $S$ equipped with a $G$-action over $S$. We assume there is a presentation $Y\,{\widetilde\to}\, \operatorname{colim}_{i\in I} Y_i$ in $\operatorname{PreStk}_{lft}$ such that $I$ is small filtered, $Y_i$ is a scheme of finite type over $S$, for $i\to j$ in $I$ the map $Y_i\to Y_j$ is a closed immersion. Moreover, each $Y_i$ is $G$-stable and $G$-action on $Y_i$ factors through some finite-dimensional quotient scheme $G\to G_i$ such that $\operatorname{Ker}(G\to G_i)$ is a prounipotent group scheme over $S$. In this case $Shv(Y)^G\,{\widetilde\to}\lim_{i\in I^{op}} Shv(Y_i)^G$ with respect to the $!$-inverse images. Passing to the left adjoint, we may also write $Shv(Y)^G\,{\widetilde\to}\,\operatorname{colim}_{i\in I} Shv(Y_i)^G$ with respect to the $!$-direct images. For each $i$ the $!$-extension $Shv(Y_i)^G\to Shv(Y)^G$ is fully faithful (by [@GaiDG], Lemma 1.3.6). We define $(Shv(Y)^G)^{\le 0}$ as the smallest full subcategory containing $(Shv(Y_i)^G)^{\le 0}$ for all $i$, closed under extensions and small colimits. By ([@HA], 1.4.4.11), $(Shv(Y)^G)^{\le 0}$ is presentable and defines an accessible t-structure on $Shv(Y)^G$. Note that $K\in Shv(Y)^G$ lies in $(Shv(Y)^G)^{>0}$ iff for any $i$, its $!$-restriction to $Shv(Y_i)^G$ lies in $(Shv(Y_i)^G)^{>0}$. This shows that this t-structure is compatible with filtered colimits. We write $\operatorname{oblv}[\operatorname{dim.rel}]: Shv(Y)^G\to Shv(Y)$ for the t-exact functor obtained as limit over $i\in I^{op}$ of $\operatorname{oblv}[\operatorname{dim.rel}]: Shv(Y_i)^G\to Shv(Y_i)$. The above self-duality $(Shv(Y_i)^G)^{\vee}\,{\widetilde\to}\, Shv(Y_i)^G$ for each $i$ yields a self-duality $$(Shv(Y)^G)^{\vee}\,{\widetilde\to}\, Shv(Y)^G$$ using ([@GaiDG], Lemma 2.2.2) and Section [5.5.2](#Sect_A.4.2_on_t-structures){reference-type="ref" reference="Sect_A.4.2_on_t-structures"}. ### {#Sect_A.5.4} Define $Shv(Y)^{G, constr}$ as the full subcategory of those $K\in Shv(Y)^G$ such that $\operatorname{oblv}[\operatorname{dim.rel}](K)\in Shv(Y)^c$. Then $Shv(Y)^{G, constr}\in\operatorname{DGCat}^{non-cocmpl}$. Set $Shv(Y)^{G, ren}=\operatorname{Ind}(Shv(Y)^{G, constr})$. Now $Shv(Y)^{G, constr}$ acquires a unique t-structure such that both projections $$Shv(Y)^c\gets Shv(Y)^{G, constr}\to Shv(Y)^G$$ are t-exact. Now by ([@G], ch. II.1, Lemma 1.2.4), $Shv(Y)^{G, ren}$ acquires a unique t-structure compatible with filtered colimits for which the natural map $Shv(Y)^{G, constr}\to Shv(Y)^{G, ren}$ is t-exact. For each $i$ we have a full embedding $Shv(Y_i)^{G, constr}\subset Shv(Y)^{G, constr}$. In fact, $$\mathop{\operatorname{colim}}\limits_{i\in I} Shv(Y_i)^{G, constr}\,{\widetilde\to}\, Shv(Y)^{G, constr},$$ where the colimit is taken in $\operatorname{DGCat}^{non-cocmpl}$. Applying the functor $\operatorname{Ind}$ to the later equivalence, we obtain $Shv(Y)^{G, ren}\,{\widetilde\to}\,\mathop{\operatorname{colim}}\limits_{i\in I} Shv(Y_i)^{G, ren}$, where the colimit is taken in $\operatorname{DGCat}_{cont}$. The self-dualities $(Shv(Y_i)^{G,ren})^{\vee}\,{\widetilde\to}\, Shv(Y_i)^{G, ren}$ yield in the colimit a canonical self-duality $$(Shv(Y)^{G,ren})^{\vee}\,{\widetilde\to}\, Shv(Y)^{G, ren}$$ It actually comes from the Verdier duality $$\mathbb{D}: (Shv(Y)^{G, constr})^{op}\,{\widetilde\to}\, Shv(Y)^{G, constr}$$ Passing to the colimit over $i\in I$ in the adjoint pair $\operatorname{ren}: Shv(Y_i)^G\leftrightarrows Shv(Y_i)^{G, ren}: \operatorname{un-ren}$, one gets the adjoint pair $$\operatorname{ren}: Shv(Y)^G\leftrightarrows Shv(Y)^{G, ren}: \operatorname{un-ren}$$ in $\operatorname{DGCat}_{cont}$, where the left adjoint is fully faithful. ## About averaging functors {#Sect_A.5} ### {#Sect_A.5.1} Let $Y$ be an ind-scheme of ind-finite type. Let $U, G$ be placid group schemes with $U$ prounipotent. Assume $G$ acts on $U$ by congugation, let $H=G\rtimes U$ be the semi-direct product. Assume $H$ acts on $Y$, and $Y\,{\widetilde\to}\,\operatorname{colim}_{i\in I} Y_i$, where $I$ is a small filtered category, if $i\in I$ then $Y_i\stackrel{}{\hookrightarrow} Y$ is a closed $H$-invariant subscheme of finite type, and for $i\to j$ in $I$ the map $Y_i\to Y_j$ is a closed immersion. Since we are in the constructible context, the functor $\operatorname{oblv}: Shv(Y)^H\to Shv(Y)^G$ admits a left adjoint $\operatorname{Av}^U_!: Shv(Y)^G\to Shv(Y)^H$. Since $U$ is prounipotent, $\operatorname{oblv}: Shv(Y)^H\to Shv(Y)^G$ is fully faithful. Assume in addition $Y'$ is another ind-scheme of ind-finite type with a $H$-action satisfying the same assumptions, and $f: Y\to Y'$ is a $H$-equivariant morphism, where $f$ is schematic of finite type. Then $f_*: Shv(Y)^G\to Shv(Y')^G$ admits a left adjoint $$\label{functor_f^*_for_Sect_A.5.1} f^*: Shv(Y')^G\to Shv(Y)^G$$ Both these functors preserve the full subcategories of $H$-invariants, and give rise to an adjoint pair in $\operatorname{DGCat}_{cont}$ $$f^*: Shv(Y')^H\leftrightarrows Shv(Y)^H: f_*$$ Besides, the following diagram canonically commutes $$\label{diag_for_Sect_A.5.1} \begin{array}{ccc} Shv(Y)^G & \stackrel{f^*}{\gets} & Shv(Y')^G\\ \downarrow\lefteqn{\scriptstyle \operatorname{Av}^U_!} &&\downarrow\lefteqn{\scriptstyle \operatorname{Av}^U_!}\\ Shv(Y)^H & \stackrel{f^*}{\gets} & Shv(Y')^H \end{array}$$ The proof is left to a reader, let us only indicate a construction of ([\[functor_f\^\*\_for_Sect_A.5.1\]](#functor_f^*_for_Sect_A.5.1){reference-type="ref" reference="functor_f^*_for_Sect_A.5.1"}). Pick a presentation $Y'\,{\widetilde\to}\, \operatorname{colim}_{j\in J} Y'_j$, where $J$ is small filtered $\infty$-category, $Y'_j\in{\operatorname{Sch}}_{ft}$, for $j\to j'$ in $J$ the map $Y'_j\to Y'_{j'}$ is a $G$-equivariant closed immersion. Let $Y_j=Y'_j\times_{Y'} Y$ for $j\in J$, so $\operatorname{colim}_{j\in J} Y_j\,{\widetilde\to}\, Y$ in $\operatorname{PreStk}_{lft}$. For each $j\in J$ the $G$-action on $Y'_j$ factors through a quotient group scheme of finite type. Let $f_j: Y_j\to Y'_j$ be the restriction of $G$. Then there is a left adjoint $f_j^*: Shv(Y'_j)^G\to Shv(Y_j)^G$ of $(f_j)_*: Shv(Y_j)^G\to Shv(Y'_j)^G$. Besides, $f_j^*$ are compatible with the transition maps given by $!$-extensions for $j\to j'$ in $J$, so that we may pass to the colimit $\operatorname{colim}_{j\in J} f_j^*$. The latter is the desired functor ([\[functor_f\^\*\_for_Sect_A.5.1\]](#functor_f^*_for_Sect_A.5.1){reference-type="ref" reference="functor_f^*_for_Sect_A.5.1"}). ### {#Sect_A.5.2} Assume now $\bar U=\operatorname{colim}_{j\in J} U_j$, where $J$ is a small filtered category, if $j\in J$ then $U_j$ is a placid prounipotent group scheme, and for $j\to j'$ in $J$ the map $U_j\to U_{j'}$ is a homomorphism of group schemes and a placid closed immersion. Let $G, Y$ be as in Section [5.6.1](#Sect_A.5.1){reference-type="ref" reference="Sect_A.5.1"}. Assume $G$ acts by conjugation on each $U_j$ in a way compatible with closed immersions $U_j\to U_{j'}$ for $j\to j'$ in $J$. Write $H_j=G\rtimes U_j$ and $\bar H=G\rtimes \bar U$ for the corresponding semi-direct products. So, $\bar H$ is placid ind-scheme. Assume $H$ acts on $Y$. By Lemma [Lemma 1](#Lm_A.2.3){reference-type="ref" reference="Lm_A.2.3"}, $\operatorname{oblv}: Shv(Y)^{\bar H}\to Shv(Y)^G$ admits a left adjoint denoted $\operatorname{Av}^{\bar U}_!: Shv(Y)^G\to Shv(Y)^{\bar H}$ given as $$\operatorname{Av}^{\bar U}_!(K)\,{\widetilde\to}\, \mathop{\operatorname{colim}}\limits_{j\in J} \operatorname{Av}^{U_j}_!(K),$$ the colimit being taken in $Shv(Y)^G$. Assume in addition $f: Y\to Y'$ is a schematic morphism of finite type, where $Y'$ is an ind-scheme of ind-finite type. Assume $\bar H$ acts on $Y'$, and $f$ is $\bar H$-equivariant. The functors $f^*: Shv(Y')^{H_j}\to Shv(Y)^{H_j}$ yield after passing to the limit over $J^{op}$ the functor $f^*: Shv(Y')^{\bar H}\to Shv(Y)^{\bar H}$. The commutativity of ([\[diag_for_Sect_A.5.1\]](#diag_for_Sect_A.5.1){reference-type="ref" reference="diag_for_Sect_A.5.1"}) implies that the diagram is canonically commutative $$\begin{array}{ccc} Shv(Y)^G & \stackrel{f^*}{\gets} & Shv(Y')^G\\ \downarrow\lefteqn{\scriptstyle \operatorname{Av}^{\bar U}_!} &&\downarrow\lefteqn{\scriptstyle \operatorname{Av}^{\bar U}_!}\\ Shv(Y)^{\bar H} & \stackrel{f^*}{\gets} & Shv(Y')^{\bar H} \end{array}$$ **Lemma 1**. *Keep the assumptions of Section [5.6.1](#Sect_A.5.1){reference-type="ref" reference="Sect_A.5.1"}.\ i) The functor $f^!: Shv(Y')^H\to Shv(Y)^H$ admits a left adjoint $f_!: Shv(Y)^H\to Shv(Y')^H$, and the diagram commutes naturally $$\label{diag_for_Lm_A.5.3} \begin{array}{ccc} Shv(Y) & \stackrel{f_!}{\to} & Shv(Y')\\ \uparrow\lefteqn{\scriptstyle\operatorname{oblv}[\operatorname{dim.rel}]} && \uparrow\lefteqn{\scriptstyle\operatorname{oblv}[\operatorname{dim.rel}]} \\ Shv(Y)^H & \stackrel{f_!}{\to} & Shv(Y')^H \end{array}$$ ii) Assume in addition that $f: Y\to Y'$ is an open immersion. Then $f_!: Shv(Y)^H\to Shv(Y')^H$ is fully faithful.* *Proof.* i) **Step 1** Assume first $Y$ is a scheme of finite type. Then the $H$-action on $Y$ automatically factors through an action of a quotient group scheme $H\to H_0$, where $H_0$ is of finite type, and $\operatorname{Ker}(H\to H_0)$ is prounipotent. We get the cartesian square $$\begin{array}{ccc} Y &\stackrel{f}{\to} & Y'\\ \downarrow && \downarrow\\ Y/H_0 & \stackrel{\bar f}{\to} & Y'/H_0 \end{array}$$ and the desired functor is $\bar f_!: Shv(Y)^H\to Shv(Y')^H$. The commutativity of ([\[diag_for_Lm_A.5.3\]](#diag_for_Lm_A.5.3){reference-type="ref" reference="diag_for_Lm_A.5.3"}) follows from the $(^*, {}_!)$-base change. **Step 2** Pick a presentation $Y'\,{\widetilde\to}\, \operatorname{colim}_{i\in I} Y'_i$, where $I$ is a small filtered $\infty$-category, $Y'_i\in{\operatorname{Sch}}_{ft}$ is $H$-invariant closed subscheme of $Y'$, and for $i\to j$ in $I$, $Y'_i\to Y'_j$ is a $H$-equivariant closed immersion. Set $Y_i=Y'_i\times_{Y'} Y$. Note that $Y\,{\widetilde\to}\,\operatorname{colim}_{i\in I} Y_i$ in $\operatorname{PreStk}_{lft}$. For $i\to j$ in $I$ write $g_{ij}: Y_i\to Y_j$ and $g'_{ij}: Y'_i\to Y'_j$ for the transition maps. Write $f_i: Y_i\to Y'_i$ for the restriction of $f$. Recall that $Shv(Y)^H\,{\widetilde\to}\,\lim_{i\in I^{op}} Shv(Y_i)^H$ with the transition maps being $g_{ij}^!$. Passing to the left adjoints, we get $Shv(Y)^H\,{\widetilde\to}\, \operatorname{colim}_{i\in I} Shv(Y_i)^H$ in $\operatorname{DGCat}_{cont}$ for the transition maps $(g_{ij})_!$. We get a functor ${\mathcal F}: I\times [1]\to \operatorname{DGCat}_{cont}$ sending $i\in I$ to $(Shv(Y_i)^H\stackrel{(f_i)_!}{\to} Shv(Y'_i)^H)$, here for $i\to j$ in $I$ the transition functors are $(g_{ij})_!, (g'_{ij})_!$. Passing to the colimit in $(f_i)_!: Shv(Y_i)^H\to Shv(Y'_i)^H$, one gets the desired functor $f_!: Shv(Y)^H\to Shv(Y')^H$. This functor is the left adjoint to $f^!: Shv(Y')^H\to Shv(Y)^H$ by ([@Ly], 9.2.39). The commutativity of ([\[diag_for_Lm_A.5.3\]](#diag_for_Lm_A.5.3){reference-type="ref" reference="diag_for_Lm_A.5.3"}) is obtained by passing to the colimit over $i\in I$ from the commutativity of $$\begin{array}{ccc} Shv(Y_i) & \stackrel{(f_i)!}{\to} & Shv(Y'_i)\\ \uparrow\lefteqn{\scriptstyle\operatorname{oblv}[\operatorname{dim.rel}]} && \uparrow\lefteqn{\scriptstyle\operatorname{oblv}[\operatorname{dim.rel}]} \\ Shv(Y_i)^H & \stackrel{(f_i)!}{\to} & Shv(Y'_i)^H \end{array}$$ ii\) Keep notations of Step 2. Then for each $i\in I$, $$(f_i)_!: Shv(Y_i)^H\to Shv(Y'_i)^H$$ is fully faithful by coinstruction. Besides, each $Shv(Y_i)^H, Shv(Y'_i)^H$ is compactly generated, and we may pass to right adjoints in the functor ${\mathcal F}$. So, our claim follows from ([@Ly], 9.2.47). ◻ **Remark 1**. *Actually, in the situation of Lemma [Lemma 1](#Lm_A.5.3){reference-type="ref" reference="Lm_A.5.3"} i) the diagram commutes $$\begin{array}{ccc} Shv(Y)^G & \stackrel{f_!}{\to} & Shv(Y')^G\\ \uparrow\lefteqn{\scriptstyle\operatorname{oblv}[\operatorname{dim.rel}]} && \uparrow\lefteqn{\scriptstyle\operatorname{oblv}[\operatorname{dim.rel}]} \\ Shv(Y)^H & \stackrel{f_!}{\to} & Shv(Y')^H, \end{array}$$ and the vertical functors are fully faithful.* ### {#Sect_A.5.5} Assume we are in the situation of Section [5.6.2](#Sect_A.5.2){reference-type="ref" reference="Sect_A.5.2"}. For each $j\in J$ we have the fully functor $\operatorname{oblv}[\operatorname{dim.rel}]: Shv(Y)^{H_i}\to Shv(Y)^G$. Passing to the limit over $j\in J^{op}$, they yield a fully faithful functor $\operatorname{oblv}[\operatorname{dim.rel}]: Shv(Y)^{\bar H}\to Shv(Y)^G$. Since for each $j\in J$ the diagram $$\begin{array}{ccc} Shv(Y)^G & \stackrel{f_!}{\to} & Shv(Y')^G\\ \uparrow\lefteqn{\scriptstyle \operatorname{oblv}[\operatorname{dim.rel}]} && \uparrow\lefteqn{\scriptstyle \operatorname{oblv}[\operatorname{dim.rel}]}\\ Shv(Y)^{H_j} & \stackrel{f_!}{\to} & Shv(Y')^{H_j} \end{array}$$ commutes, passing to the limit over $j\in J$ this gives a commutativity of $$\begin{array}{ccc} Shv(Y)^G & \stackrel{f_!}{\to} & Shv(Y')^G\\ \uparrow\lefteqn{\scriptstyle \operatorname{oblv}[\operatorname{dim.rel}]} && \uparrow\lefteqn{\scriptstyle \operatorname{oblv}[\operatorname{dim.rel}]}\\ Shv(Y)^{\bar H} & \stackrel{f_!}{\to} & Shv(Y')^{\bar H} \end{array}$$ ## Some intertwining functors {#Sect_A.7} ### {#Sect_A.7.1} Let $G$ be a smooth algebraic group of finite type. Let $Y\in \operatorname{PreStk}_{lft}$ with a $G$-action. We take the convention that the natural identification $Shv(Y/G)\,{\widetilde\to}\, Shv(Y)^G$ is such that $\operatorname{oblv}: Shv(Y)^G\to Shv(Y)$ corresponds to $q^!: Shv(Y/G)\to Shv(Y)$ for $q: Y\to Y/G$. ### {#Sect_A.7.2} Let $P,Q\subset G$ be closed subgroups. We define the convolution $Shv(Q\backslash G/P)\otimes Shv(Y/P)\to Shv(Y/Q)$ by $$F\boxtimes K\mapsto F\ast K=\operatorname{act}_*q^!(F\boxtimes K)$$ for the diagram $$(Q\backslash G/P)\times (Y/P)\stackrel{q}{\gets} Q\backslash G\times^P Y\stackrel{\operatorname{act}}{\to} Y/Q.$$ In particular, this is known to be the underlying binary product of a monoidal structure on $Shv(P\backslash G/P)$. ### {#section-102} Let now $C\in Shv(G)-mod(\operatorname{DGCat}_{cont})$. We use the identification $${\operatorname{Fun}}_{Shv(G)}(Shv(G/Q), C)\,{\widetilde\to}\, C^Q$$ coming from $Shv(G/Q)\,{\widetilde\to}\, Shv(G)^Q\,{\widetilde\to}\, Shv(G)_Q$, where the first isomorphism is as in Section [5.7.1](#Sect_A.7.1){reference-type="ref" reference="Sect_A.7.1"}. We write $\delta_1\in Shv(G/Q)$ for the constant sheaf at $1$ extended by zero to $G/Q$. **Lemma 1**. *Consider the equivalence ${\operatorname{Fun}}_{Shv(G)}(Shv(G/Q), Shv(Y))\,{\widetilde\to}\, Shv(Y/Q)$ given by $f\mapsto f(\delta_1)$. The inverse equivalence sends $K\in Shv(Y/Q)$ to the functor $Shv(G/Q)\to Shv(Y)$ given by $F\mapsto m_*q^!(F\boxtimes K)$ for the diagram $$G/Q\times (Y/Q)\stackrel{q}{\gets} G\times^Q Y\stackrel{m}{\to} Y,$$ where $m$ comes from the action map $\operatorname{act}: G\times Y\to Y$. So, $q\in Q$ acts on $(g, y)\in G\times Y$ as $(gq^{-1}, qy)$.* *Proof.* We have the canonical equivalence $Shv(G)\otimes_{Shv(Q)}\operatorname{Vect}\to Shv(G/Q)$ sending $F\boxtimes e$ to $\alpha_*F$, where $\alpha: G\to G/Q$ is the natural map. In the following commutative diagram the square is cartesian $$\begin{array}{ccccc} G\times (Y/Q) & \stackrel{\operatorname{id}\times\beta}{\gets} & G\times Y\\ \downarrow\lefteqn{\scriptstyle \alpha\times\operatorname{id}} && \downarrow\lefteqn{\scriptstyle\bar\alpha} & \searrow\lefteqn{\scriptstyle \operatorname{act}}\\ (G/Q)\times (Y/Q) & \stackrel{q}{\gets} & G\times^Q Y & \stackrel{m}{\to} & Y \end{array}$$ So, for $F\in Shv(G)$, $K\in Shv(Y/Q)$ one has canonically $q^!((\alpha_* F)\boxtimes K)\,{\widetilde\to}\, \bar\alpha_*(F\boxtimes \beta^! K)$, hence also $$F\ast \beta^! K\,{\widetilde\to}\, m_*q^!((\alpha_* F)\boxtimes K)$$ Since the objects $F\boxtimes V$ for $F\in Shv(G), V\in\operatorname{Vect}$ generate $Shv(G)\otimes_{Shv(Q)}\operatorname{Vect}$, our claim follows. ◻ ### {#Sect_A.7.5} Let $\upsilon: Shv(Q\backslash G/P)\,{\widetilde\to}\,Shv(P\backslash G/Q)$ be the equivalence coming from the map $G\,{\widetilde\to}\, G$, $g\mapsto g^{-1}$. Let ${\mathcal K}\in Shv(Q\backslash G/P)$. Denote by $$\label{functor_for_A.7.5} {\mathcal K}\ast\_: C^P\to C^Q$$ and also by $\_\ast \upsilon({\mathcal K})={\mathcal K}\ast\_$ the functor obtained from the $Shv(G)$-linear functor $$\label{functor_for_Sect_A.7.5} \_\ast {\mathcal K}: Shv(G/Q)\to Shv(G/P)$$ by applying ${\operatorname{Fun}}_{Shv(G)}(\_, C)$. In the case of $C=Shv(Y)$ this is unambiguous thanks to the following. **Lemma 1**. *For $C=Shv(Y)$ the functor ([\[functor_for_A.7.5\]](#functor_for_A.7.5){reference-type="ref" reference="functor_for_A.7.5"}) identifies with the convolution functor ${\mathcal K}\ast\_$ from Section [5.7.2](#Sect_A.7.2){reference-type="ref" reference="Sect_A.7.2"}.* *Proof.* This follows from Lemma [Lemma 1](#Lm_2.0.5_inverse_equivalence){reference-type="ref" reference="Lm_2.0.5_inverse_equivalence"}. Namely, let $\tau: G/P\to Q\backslash G/P$ be the natural map. By definition, ([\[functor_for_A.7.5\]](#functor_for_A.7.5){reference-type="ref" reference="functor_for_A.7.5"}) sends $K\in Shv(Y/P)$ to the object of $Shv(Y/Q)$ whose $!$-pullback to $Y$ is $m_*q^!(\tau^!{\mathcal K}\boxtimes K)$ for the diagram $$\begin{array}{ccccc} (G/P)\times (Y/P) & \stackrel{q}{\gets} & G\times^P Y & \stackrel{m}{\to} & Y\\ \downarrow\lefteqn{\scriptstyle \tau\times\operatorname{id}} && \downarrow &&\downarrow\lefteqn{\scriptstyle\beta}\\ (Q\backslash G/P)\times (Y/P) & \stackrel{\bar q}{\gets} & Q\backslash G\times^P Y & \stackrel{\bar m}{\to} & Q\backslash Y, \end{array}$$ where both squares are cartesian. Our claim follows by base change. ◻ Sometimes, we denote ([\[functor_for_A.7.5\]](#functor_for_A.7.5){reference-type="ref" reference="functor_for_A.7.5"}) by ${\mathcal K}\overset{P}{\ast}\_$ to underline that the convolution is calculated with respect to $P$. ### {#section-103} Assume for this subsection that $Q\subset P$. For the natural map $\alpha: G/Q\to G/P$ we have the adjoint pair $$\alpha^*: Shv(G/P)\leftrightarrows Shv(G/Q): \alpha_*$$ in $Shv(G)-mod(\operatorname{DGCat}_{cont})$, as $P/Q$ is smooth. Applying ${\operatorname{Fun}}_{Shv(G)}(\_, C)$, it gives an adjoint pair $\operatorname{oblv}: C^P \leftrightarrows C^Q: \operatorname{Av}^{P/Q}_*$ in $\operatorname{DGCat}_{cont}$. The functor $\alpha^*$ identifies with $$\_\ast i_!e_{P\backslash P/Q}[-2\dim P]: Shv(G/P)\to Shv(G/Q)$$ for the closed immersion $i: P\backslash P/Q\stackrel{}{\hookrightarrow} P\backslash G/Q$. So, $\operatorname{Av}^{P/Q}_*: C^Q\to C^P$ in our notations is the functor $$i_!e_{P\backslash P/Q}[-2\dim P]\ast\_.$$ The functor $\alpha_*$ identifies with $$\_\ast s_!e_{Q\backslash P/P}[-2\dim Q]: Shv(G/Q)\to Shv(G/P)$$ for the closed immersion $s: Q\backslash P/P\stackrel{}{\hookrightarrow} Q\backslash G/P$. So, $\operatorname{oblv}: C^P\to C^Q$ in our notations is the functor $$s_!e_{Q\backslash P/P}[-2\dim Q]\ast\_.$$ The functor $\alpha^*$ has a left adjoint $\alpha_![-2\dim(P/Q)]$. If $P/Q$ is proper then $\alpha$ is proper, and we get an adjoint pair $$\alpha_*[-2\dim(P/Q)]: Shv(G/Q)\leftrightarrows Shv(G/P): \alpha^*$$ in $Shv(G)-mod(\operatorname{DGCat}_{cont})$. It yields an adjoint pair in $\operatorname{DGCat}_{cont}$ $$\operatorname{Av}^{P/Q}_*: C^Q\leftrightarrows C^P: \operatorname{oblv}[-2\dim(P/Q)]$$ ### {#section-104} Let now $P, Q\subset G$ be as in Section [5.7.2](#Sect_A.7.2){reference-type="ref" reference="Sect_A.7.2"}. Define the functor $^Q\operatorname{Av}_*^P: C^P\to C^Q$ as the composition $$C^P\,\stackrel{\operatorname{oblv}}{\to}\, C^{P\cap Q}\,\stackrel{\operatorname{Av}_*^{Q/P\cap Q}}{\to}\, C^Q$$ Write $j: Q\backslash QP/P\to Q\backslash G/P$ for the natural inclusion. Then in our notations $^Q\operatorname{Av}^P_*$ is the functor $j_*e_{Q\backslash QP/P}[-2\dim Q]\ast\_$. Assume in addition that both $G/Q, G/(P\cap Q)$ are proper. Then $^Q\operatorname{Av}^P_*$ admits a right adjoint given as the composition $$C^Q\,\stackrel{\operatorname{oblv}[-2\dim(Q/P\cap Q)]}{\to}\, C^{P\cap Q}\;\stackrel{\operatorname{Av}^{P/(P\cap Q)}_*}{\to} \,C^P$$ That is, $^P\operatorname{Av}^Q_*[-2\dim(Q/P\cap Q)]$ is the right adjoint of $^Q\operatorname{Av}^P_*$. Let $j': P\backslash PQ/Q\to P\backslash G/Q$ be the embedding. Then $^P\operatorname{Av}^Q_*[-2\dim(Q/P\cap Q)]$ identifies with the functor $$j'_*e_{P\backslash PQ/Q}[-2\dim P-2\dim(P\cap Q)].$$ ### {#Sect_A.7.9} Let now $P, Q\subset G$ be as in Section [5.7.2](#Sect_A.7.2){reference-type="ref" reference="Sect_A.7.2"}. Assume $G/P$ proper. Write $j: P\backslash PQ/Q\to P\backslash G/Q$ for the embedding. Recall that the functor $^Q\operatorname{Av}^P_*: C^P\to C^Q$ comes from $\alpha_*\beta^*: Shv(G/Q)\to Shv(G/P)$ for the diagram $$G/P\stackrel{\alpha}{\gets} G/(P\cap Q)\stackrel{\beta}{\to} G/Q$$ The left adjoint to $\alpha_*\beta^*$ is $$\beta_!\alpha^*[2\dim(Q/(P\cap Q)]: Shv(G/P)\to Shv(G/Q)$$ We claim that the latter functor is $Shv(G)$-linear and identifies with $$\label{functor_for_Sect_A.7.9} \_\ast j_!e[-2\dim P+2\dim(Q/(P\cap Q))]: Shv(G/P)\to Shv(G/Q)$$ Indeed, consider the diagram, where the square is cartesian $$\begin{array}{ccccc} (G/P)\times (P\backslash G/Q) & \stackrel{q}{\gets} & G\times^P G/Q & \stackrel{m}{\to} & G/Q\\ \uparrow\lefteqn{\scriptstyle \operatorname{id}\times j} && \uparrow\lefteqn{\scriptstyle \tilde j} & \nearrow\lefteqn{\scriptstyle \tilde m}\\ (G/P)\times (P\backslash PQ/Q) & \stackrel{\tilde q}{\gets} & G\times^P PQ/Q \end{array}$$ Since $G/P$ is proper, $m$ is proper, so for $F\in Shv(G/P)$, $$F\ast j_!e\,{\widetilde\to}\, m_*q^!(F\boxtimes j_!e)\,{\widetilde\to}\, m_! \tilde j_!\tilde q^!(F\boxtimes e)\,{\widetilde\to}\,\beta_!\alpha^*F[2\dim P],$$ because $\tilde m$ identifies with $\beta$. So, $^Q\operatorname{Av}^P_*$ admits a right adjoint denoted $^P\operatorname{Av}^Q_!$ obtained from ([\[functor_for_Sect_A.7.9\]](#functor_for_Sect_A.7.9){reference-type="ref" reference="functor_for_Sect_A.7.9"}) by applying ${\operatorname{Fun}}_{Shv(G)}(\_, C)$. # On the invertibility of some standard objects in parabolic Hecke categories {#Sect_appendixB} ## Associated parabolic subgroups ### {#section-105} Let $T\subset B\subset G$ and $W$ be as in Section [1.4.1](#Sect_1.4.1){reference-type="ref" reference="Sect_1.4.1"}. Let $P, Q\subset G$ be parabolics containing $T$. **Remark 1**. *Any pair of parabolics in $G$ contain a common maximal torus. We fix fix this torus to be $T$ to simplify some notations.* ### {#section-106} Write $L_P\subset P, L_Q\subset Q$ for the unique Levi subgroups containing $T$. Write $W_P, W_Q\subset W$ for the Weyl groups of $L_P, L_Q$. Write also $L_{P\cap Q}$ for the unique Levi subgroup of $P\cap Q$ containing $T$. Let $C\in Shv(G)-mod(\operatorname{DGCat}_{cont})$. In Section [5.7.7](#Sect_A.7.9){reference-type="ref" reference="Sect_A.7.9"} we introduced the adjoint pair $$\label{adj_functors_for_B1} ^Q\operatorname{Av}^P_*: C^P\leftrightarrows C^Q: {^P\operatorname{Av}^Q_!}$$ Our goal here is two determine for which pairs $(P, Q)$ as above these functors are equivalences. **Definition 1**. *Say that $P$ and $Q$ are associated if we have $L_P=L_Q$.* Note that $P$ and $Q$ are associated iff $L_P=L_{P\cap Q}=L_Q$. ### Example The opposite parabolics are associated. **Theorem 1**. *The adjoint functors ([\[adj_functors_for_B1\]](#adj_functors_for_B1){reference-type="ref" reference="adj_functors_for_B1"}) are equivalences (for any $C\in Shv(G)-mod(\operatorname{DGCat}_{cont})$) if and only if $P$ and $Q$ are associated.* **Remark 1**. *We note that the answer given by Theorem [Theorem 1](#Thm_B.1.2){reference-type="ref" reference="Thm_B.1.2"} differs from its function-theoretic counterpart. For parabolic Hecke algebras it is true that if $P$ and $Q$ are associated then the indicator function $T_{PQ}$ of $TP\subset G$ is invertible. This follows by taking the trace of Frobenius.* *However, typically there are more invertible elements. For example, consider $G=GL(V)$ for a finite-dimensional vector space $V$ with $\dim V\ge 3$. Let $P\subset G$ be the parabolic preserving a line $L\subset V$, so $G/P\,{\widetilde\to}\, {\mathbb P}(V)$. There are two $P$-orbits on ${\mathbb P}(V)$, namely $\{L\}$ and its complement. Let $w\in W$ such that $PwP/P\subset G/P$ is open. While $P$ and $wPw^{-1}$ are not associated, the indicator function of the double coset $PwP$ is invertible. The parabolic Hecke algebra here is the usual Hecke algebra for $\operatorname{GL}_2$ with parameter $q+q^2+\ldots+q^{\dim {\mathbb P}(V)}$ (if we work over a finite field of $q$ elements).* **Lemma 1**. *The parabolics $P$ and $Q$ are associated if and only if both $P/(P\cap Q)$ and $Q/(P\cap Q)$ are homologically contractible.* *Proof.* The only if direction is obvious. Assume both $P/(P\cap Q)$ and $Q/(P\cap Q)$ are contractible. Note that $P/(P\cap Q)$ deformation retracts onto $L_P/L_P\cap Q$. Further, $L_P\cap Q$ is a parabolic of $L_P$. Indeed, if $B'\subset Q$ is a Borel subgroup containing $T$ then $L_P\cap B'$ is a Borel subgroup of $L_P$. So, $L_P/(L_P\cap Q)$ is a partial flag variety of $L_P$. In particular, it is homologically contractible iff $L_P\subset Q$, that is, $L_P\subset L_Q$. Interchanging the roles of $P$ and $Q$ we get also $L_Q\subset L_P$. ◻ *Proof of Theorem [Theorem 1](#Thm_B.1.2){reference-type="ref" reference="Thm_B.1.2"}.* **Step 1** Assume $P$ and $Q$ are associated. Pick Borel subgroups $B\subset P$, $B'\subset Q$ containing $T$. Let us show that the diagram canonically commutes $$\begin{array}{ccc} C^P & \stackrel{^Q\operatorname{Av}^P_*}{\to} & C^Q\\ \downarrow\lefteqn{\scriptstyle \operatorname{oblv}} && \downarrow\lefteqn{\scriptstyle\operatorname{oblv}}\\ C^B & \stackrel{^{B'}\operatorname{Av}^B_*}{\to} & C^{B'} \end{array}$$ Let $$j: Q\backslash QP/P\to Q\backslash G/P, \;\; j': B'\backslash B'B/B\to B'\backslash G/B$$ and $$\pi: G/B\to G/P, \;\;\pi': G/B'\to G/Q$$ be the natural maps. By Section [5.7](#Sect_A.7){reference-type="ref" reference="Sect_A.7"}, it suffices to show that the diagram commutes $$\begin{array}{ccc} Shv(G/Q) & \stackrel{\_\ast j_*e_{Q\backslash QP/P}[d]}{\to} & Shv(G/P)\\ \uparrow\lefteqn{\scriptstyle \pi'_*}&& \uparrow\lefteqn{\scriptstyle \pi_*}\\ Shv(G/B') & \stackrel{\_\ast j'_*e_{B'\backslash B'B/B}}{\to} & Shv(G/B) \end{array}$$ for $d=2\dim B'-2\dim Q$. Consider the closed immersions $$s: B\backslash P/P\to B\backslash G/P,\;\; s': B'\backslash Q/Q\to B'\backslash G/Q$$ The functor $\pi_*$ is $\_\ast s_!e_{B\backslash P/P}[-2\dim B]$. The functor $\pi'_*$ is $\_\ast s'_!e_{B'\backslash Q/Q}[-2\dim B']$. So, we must establish an isomorphism $$\label{iso_to_prove_for_ThmB.1.3} s'_!e_{B'\backslash Q/Q}\ast j_*e_{Q\backslash QP/P}[-2\dim Q]\,{\widetilde\to}\, j'_*e_{B'\backslash B'B/B}\ast s_!e_{B\backslash P/P}[-2\dim B]$$ in $Shv(B'\backslash G/P)$. Let $$\bar j: B'\backslash QP/P\to B'\backslash G/P$$ be the natural map. By base change, the LHS of ([\[iso_to_prove_for_ThmB.1.3\]](#iso_to_prove_for_ThmB.1.3){reference-type="ref" reference="iso_to_prove_for_ThmB.1.3"}) identifies with $\bar j_*e$. Let $$m: B'\backslash B'B\times^B P/P\to B'\backslash QP/P$$ be the map induced by the product map $B'B\times^B P\to Q\times P$. The RHS of ([\[iso_to_prove_for_ThmB.1.3\]](#iso_to_prove_for_ThmB.1.3){reference-type="ref" reference="iso_to_prove_for_ThmB.1.3"}) identifies by base change with $\bar j_* m_*e$. In fact, $m_*e\,{\widetilde\to}\, e$. Indeed, consider the diagram $$B'B/B\to B'P/P\to QP/P$$ In this diagram the second map is an isomorphism, because $B'\cap L_P\subset L_Q$, and the first map identifies with the affine fibration $U(B')/U(B')\cap U(B)\to U(B')/U(B')\cap U(P)$. Here $U(B), U(B'), U(P)$ denotes the unipotent radical of the corresponding group. Our claim follows. A dual argument shows that the diagram canonically commutes $$\begin{array}{ccc} C^Q & \stackrel{^P\operatorname{Av}^Q_!}{\to} & C^P\\ \downarrow\lefteqn{\scriptstyle \operatorname{oblv}} && \downarrow\lefteqn{\scriptstyle\operatorname{oblv}}\\ C^{B'} & \stackrel{^B\operatorname{Av}^{B'}_!}{\to} & C^{B} \end{array}$$ Since the functors $\operatorname{oblv}: C^P\to C^B$ and $\operatorname{oblv}: C^{Q}\to C^{B'}$ are conservative, our claim follows from the fact that the adjoint functors $$^{B'}\operatorname{Av}^B_*: C^B\leftrightarrows C^{B'}: {^B\operatorname{Av}^{B'}_!}$$ are equivalences, which is standard. **Step 2** Assume $^Q\operatorname{Av}^P_*$, $^P\operatorname{Av}^Q_!$ are equivalences. Let $\tilde j: P\backslash PQ/Q\stackrel{}{\hookrightarrow} P\backslash G/Q$ be the natural map. We have $$j_*e_{Q\backslash QP/P}[-2\dim Q]\overset{P}{\ast}\tilde j_! e_{P\backslash PQ/Q} [-2\dim P+2\dim(Q/(P\cap Q))]\,{\widetilde\to}\,i_!\omega_{Q\backslash Q/Q},$$ where $i: Q\backslash Q/Q\stackrel{}{\hookrightarrow} Q\backslash G/Q$ is the closed immersion. Let $\operatorname{inv}: Q\backslash QP/P\,{\widetilde\to}\, P\backslash PQ/Q$ be the inversion. We have the cartesian square $$\begin{array}{ccc} Q\backslash (G\times^P G)/Q & \stackrel{m}{\to} & Q\backslash G/Q\\ \uparrow{\lefteqn{\scriptstyle\tilde i}} && \uparrow{\lefteqn{\scriptstyle i}}\\ Q\backslash G/P & \stackrel{\tilde m}{\to} & Q\backslash Q/Q \end{array}$$ So, $$\begin{gathered} i^!(j_*e_{Q\backslash QP/P}\overset{P}{\ast}\tilde j_! e_{P\backslash PQ/Q})\,{\widetilde\to}\, \tilde m_*(j_*e_{Q\backslash QP/P}\otimes^! \operatorname{inv}^!(\tilde j_! e_{P\backslash PQ/Q})))\,{\widetilde\to}\\ \tilde m_*(j_*e_{Q\backslash QP/P}\otimes^! j_!e_{Q\backslash QP/P})\,{\widetilde\to}\, \tilde m_*j_*e_{Q\backslash QP/P}[2\dim(Q\cap P)]\\ {\widetilde\to}\, \omega_{Q\backslash Q/Q}[2\dim P+2\dim(P\cap Q)]\end{gathered}$$ For the map $\eta: \operatorname{Spec}k\to Q\backslash Q/Q$ applying $\eta^!$ this gives $$e[2\dim P-2\dim Q]\,{\widetilde\to}\,\operatorname{R\Gamma}(QP/P, e)$$ Since ${\operatorname{H}}^0(QP/P, e)\,{\widetilde\to}\, e$, this shows that $\dim P=\dim Q$, and $QP/P$ is homologically contractible. Reversing the roles of $P$ and $Q$, one similarly shows that $PQ/P$ is homologically contractible. Our claim follows now from Lemma [Lemma 1](#Lm_B.1.5){reference-type="ref" reference="Lm_B.1.5"}. ◻ **Remark 1**. *Our proof of Theorem [Theorem 1](#Thm_B.1.2){reference-type="ref" reference="Thm_B.1.2"} also shows that if $P$ and $Q$ are associated then $\dim P=\dim Q$.* 99 S. Arkhipov, A. Braverman, R. Bezrukavnikov, D. Gaitsgory, I. Mirković, Modules over the small quantum group and semi-infinite flag manifold, Transform. Groups 10 (2005), no.3-4, 279-362. D. Arinkin, D. Gaitsgory, D. Kazhdan, S. Raskin, N. Rozenblyum, Y. Varshavsky, The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support, arxiv version 2 D. Arinkin, D. Gaitsgory, D. Kazhdan, S. Raskin, N. Rozenblyum, Y. Varshavsky, Duality for automorphic sheaves with nilpotent singular support, arxiv version 2 D. Arinkin, D. Gaitsgory, Singular support of coherent sheaves and the geometric Langlands conjecture, Selecta Math. (N.S.) 21 (2015), no.1, 1 - 199 S. Arkhipov, R. Bezrukavnikov, V. Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution. J. Amer. Math. Soc.17 (2004), no.3, 595 - 678. D. Beraldo, L. Chen, Automorphic Gluing, arXiv:2204.09141 T. Braden, Hyperbolic localization of Intersection Cohomology, Transform. Groups 8 (2003), no. 3, 209 - 216 A. Braverman, M. Finkelberg, D. Gaitsgory, I. Mirkovic, Intersection cohomology of Drinfeld compactifications, Selecta Math. New ser. 8 (2002), 381- 418, erratum in Selecta Math. New Ser. 10 (2004), 429-430 A. Braverman, D. Gaitsgory, Geometric Eisenstein series, Inv. Math. 150 (2002), 287 - 384 A. Braverman, D. Gaitsgory, Deformations of Local Systems and Eisenstein Series, Geom. and Functional Analysis, vol. 17 (2008), 1788 - 1850 L. Chen, Nearby Cycles and Dualities in Geometric Langlands Program, dissertation, Harvard University (2021) L. Chen, Nearby cycles on Drinfeld - Gaitsgory - Vinberg interpolation Grassmanian and long intertwining functor, Duke Math. J. 172(3): 447 - 543 (2023) L. Chen, C. Fu, An Extension of the Kazhdan-Lusztig Equivalence, arXiv:2111.14606 H. Chen, G. Dhillon, A Langlands dual realization of coherent sheaves on the nilpotent cone, in preparation V. Drinfeld, D. Gaitsgory, On a theorem of Braden, Transf. Groups, vol. 19, 313 - 358 (2014) B. Feigin, E. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys.128 (1990), no.1, 161 - 189. E. Frenkel, D. Gaitsgory, Whittaker Patterns in the Geometry of Moduli Spaces of Bundles on Curves, Ann. Math., Second Series, 153, No. 3 (2001), 699 - 748 D. Gaitsgory, The local and global versions of the Whittaker category, Pure and Appl. Math. Quarterly, Volume 16 (2020), Number 3, 775 - 904 D. Gaitsgory, The semi-infinite intersection cohomology sheaf, arxiv version 6 D. Gaitsgory, Ind-coherent sheaves. Mosc. Math. J. (2013), Vol. 13, Nu. 3, 399 - 528 D. Gaitsgory, Notes on geometric Langlands: generalities on $\operatorname{DG}$-categories, available at `https://people.mpim-bonn.mpg.de/gaitsgde/GL` D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Inv. Math. 144, 253 - 280 (2001) D. Gaitsgory, A conjectural extension of the Kazhdan--Lusztig equivalence, Publ. Res. Inst. Math. Sci. 57 (2021), no.3 - 4, 1227 - 1376. D. Gaitsgory, S. Lysenko, Parameters and duality for the metaplectic geometric Langlands theory, Selecta Math., (2018) Vol. 24, Issue 1, 227-301, corrected version is available at `https://lysenko.perso.math.cnrs.fr/papers/twistings_version21dec2022.pdf` D. Gaitsgory, N. Rozenblyum, A study in derived algebraic geometry, book available at `https://people.mpim-bonn.mpg.de/gaitsgde/GL` J. Lurie, Higher algebra, version Sept. 18, 2017, available at `https://www.math.ias.edu/ lurie/` G. Lusztig, Intersection cohomology methods in representation theory. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 155 - 174. Math. Soc. Japan, Tokyo, 1991 S. Lysenko, Assumptions on the sheaf theory for the small FLE paper, available at `https://lysenko.perso.math.cnrs.fr/index_sub.html` S. Lysenko, Comments to Gaitsgory Lurie Tamagawa, available at `https://lysenko.perso.math.cnrs.fr/index_sub.html` S. Lysenko, Comments to \"The local and global versions of the Whittaker category\", available at `https://lysenko.perso.math.cnrs.fr/index_sub.html` S. Lysenko, Comments small FLE, available at\ `https://lysenko.perso.math.cnrs.fr/index_sub.html` I. Mirkovic, K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Annals of Math. 166 (2007), 95 - 143 T. A. Springer, Linear algebraic groups, Birkhäuser, Reprint of the 1998 Second Edition (2009) N. Sweeting, Decomposing flag varieties, expository paper available at `https://scholar.harvard.edu/naomisweeting/home` [^1]: We are grateful to Misha Finkelberg for regular fruitful discussions and to Sam Raskin for answering the second author's questions. We also thank Roma Bezrukavnikov for helpful correspondence. G.D. was supported by an NSF Postdoctoral Fellowship under grant No. 2103387. [^2]: in the sense of ([\[action_Rep(checkM)\_shifted\]](#action_Rep(checkM)_shifted){reference-type="ref" reference="action_Rep(checkM)_shifted"}). [^3]: This is done to simplify the definition of the Drinfeld compactification of $\operatorname{Bun}_P$ [^4]: In the case of $P = B$, this may be proven, along with its variants for singular blocks, using an equivalence between representations of the mixed quantum group and affine Category $\mathcal{O}$ conjectured by Gaitsgory in [@GaiKL] and proven by Chen--Fu [@ChenFu], combined with Kashiwara--Tanisaki localization. We were informed by Losev that he has also obtained a proof by quite different means. [^5]: For the Borel, this is the usual graded small quantum group. For the general case, briefly one takes the graded small quantum group and adjoins divided powers of the simple raising and lowering operators corresponding to $M$. [^6]: If $G$ is semi-simple then this is a partially ordered set.

arxiv_math

--- abstract: | Paul Yiu proved that all Heron triangles are realizable on the integer lattice. We give an analogous result for triangles with vertices on the Eisenstein lattice. address: - Collège Calvin, Geneva, Switzerland 1211 - Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, Australia 3086 author: - Christian Aebi and Grant Cairns title: Following in Yiu's Footsteps but on the Eisenstein Lattice --- A planar triangle is called a *Heron triangle*, or a Heronian triangle, if it has integer side lengths and integer area. They are named after Hero of Alexandria ($\sim$`<!-- -->`{=html}10--75AD), and it would be apt, and much nicer, to call them Hero triangles, but the terms Heron and Heronian are perhaps too well established to change (?). Hero himself is sometimes called Heron, probably in the same way that Plato is called Platon in French and in many other languages. Paul Yiu [@Yiu] proved in this *Monthly* that all Heron triangles are realizable as triangles with vertices on the integer lattice. Considered in the complex plane, the integer lattice is the ring $\mathbb Z[i]$ generated by the elements $1$ and $i$, and the members of $\mathbb Z[i]$ are called *Gaussian integers*. In this note we give an analogous result for the *Eisenstein lattice*, which is the lattice in the complex plane generated by the elements $1$ and $\omega=e^{2\pi i/ 3}=\frac12(-1+i\sqrt3)$. The elements of $\mathbb Z[\omega]$ are called *Eisenstein integers*, named after the German mathematician Gotthold Eisenstein (1823--1852). As usual, we denote the complex conjugate of $z\in \mathbb Z[\omega]$ by $z^*$. Note that $\omega^*=\omega^2=-1-\omega$. For $x,y\in\mathbb N$, the *norm* of the element $z=x+y\omega\in \mathbb Z[\omega]$ is given by $N( z):=zz^*=x^2-xy+y^2$. The key property of the Eisenstein integers we will use below is that, like the Gaussian integers, they enjoy unique factorization (see [@St Chapter 7] and [@IR Chapter 9.1]); that is, every element can be written as a product of primes, and the factors are unique up to multiplication by a unit. The units in $\mathbb Z[\omega]$ are the six elements of norm one: $\{\pm1,\pm\omega,\pm(1+\omega)\}$; see Figure [\[F\]](#F){reference-type="ref" reference="F"}. Since we will be discussing primes in $\mathbb Z$ and in $\mathbb Z[\omega]$, we will refer to the latter as *$\mathbb Z[\omega]$-primes*, and also use *$\mathbb Z[\omega]$-prime* as an adjective. A key fact is that a prime $p\in\mathbb Z$ is $\mathbb Z[\omega]$-prime if and only if $p\equiv 2\pmod 3$ [@IR Chapter 9.1]. If a triangle has its vertices on the Eisenstein lattice, then its area is of the form $\frac{\sqrt3}4 n$, where $n\in\mathbb N$. Indeed, for the triangle with vertices $0,x+y\omega,z+w\omega$, with $x,y,z,w\in\mathbb N$, the area is $\frac{\sqrt3}4 (xw-yz)$. The distance from the point $x+y\omega\in \mathbb Z[w]$ to the origin is $\sqrt{x^2-xy+y^2}$, and it is well known that for integers of the form $x^2-xy+y^2$, all prime divisors congruent to $2 \pmod 3$ have even exponent. Indeed, this follows readily from the description given above of primes that are $\mathbb Z[\omega]$-prime. Alternatively, one can use the fact that an integer can be written in the form $x^2-xy+y^2$ if and only if it can be written in the form $3m^2+n^2$; just consider $$3\left(\frac{x}2\right)^2+\left(\frac{x}2-y\right)^2,\quad3\left(\frac{y}2\right)^2+\left(\frac{y}2-x\right)^2,\quad 3\left(\frac{x-y}2\right)^2+\left(\frac{x+y}2\right)^2,$$ according to whether $x,y$ or $x+y$ is even [@CG p. 223]. The fact that for integers of the form $3m^2+n^2$, all prime divisors congruent to $2 \pmod 3$ have even exponent is a common number theory exercise; see [@AG p. 333]. For our purposes, the important consequence is that the length of the sides of a triangle with vertices on the Eisenstein lattice are of the form $r\sqrt{t}$, where $r,t\in\mathbb N$ and $t$ has no prime divisors congruent to $2 \pmod 3$. **Theorem 1**. *A planar triangle $T$ with side lengths $a,b,c$ is realizable on the Eisenstein lattice if and only if the following three conditions hold:* 1. *the area of $T$ is of the form $\frac{\sqrt3}4 n$, where $n\in\mathbb N$,* 2. *$a^2,b^2,c^2\in\mathbb N$,* 3. *one of the side lengths of $T$ is of the form $r\sqrt{t}$, where $r, t\in\mathbb N$ and $t$ has no prime divisors congruent to $2 \pmod 3$.* *Proof.* Consider a planar triangle $T$ with side lengths $a,b,c$ with $a^2,b^2,c^2\in\mathbb N$ and area $\frac{\sqrt3}4 n$, where $n\in\mathbb N$. Suppose furthermore that $a=r\sqrt{t}$, where $r,t\in\mathbb N$ and $t$ has no prime divisors congruent to $2 \pmod 3$. We will construct a triangle $ABC$, with vertices on the Eisenstein lattice, whose edge lengths agree with those of $T$. Heron's formula [@OW Chapter 6.7] for the area of $T$ gives $$\frac{\sqrt3}4 n=\sqrt{s(s-a )(s-b )(s-c )},$$ where $s=\frac{(a +b +c )}2$ is the semi-perimeter. Hence, $$\label{E:heron0} 3n^2=(a +b +c )(a +b -c )(a -b +c )(-a +b +c ).$$ Expanding and rearranging as a quadratic in $c^2$ gives $$\label{E:heron} c^4-2(a^2+b^2)c^2 +(a^2-b^2)^2 +3n^2=0.$$ As $c^2$ is an integer, the discriminant of [\[E:heron\]](#E:heron){reference-type="eqref" reference="E:heron"} is necessarily a square. Hence $$\label{E:c} c^2=(a^2+b^2)+\Delta,$$ where $\Delta\in\mathbb Z$ and $\Delta^2= (a^2+b^2)^2-((a^2-b^2)^2 +3n^2)$. Thus $$\label{E:dis} \Delta^2 +3n^2=4a^2b^2.$$ From [\[E:dis\]](#E:dis){reference-type="eqref" reference="E:dis"}, the integers $\Delta$ and $n$ have the same parity, so we may set $$\label{E:uv} \Delta=u+v,\qquad n=u-v,$$ for $u,v\in \mathbb Z$. Then $\Delta^2 +3n^2=4(u^2-uv+v^2)$ and so [\[E:dis\]](#E:dis){reference-type="eqref" reference="E:dis"} gives $$\label{E:dis2} u^2-uv+v^2=a^2b^2.$$ Let $z:=(1+\omega)(u+v\omega)$. Then $z$ has norm $zz^*=u^2-uv+v^2=a^2b^2$, by [\[E:dis2\]](#E:dis2){reference-type="eqref" reference="E:dis2"}. Factoring in $\mathbb Z[\omega]$, it is convenient to write $z$ in the form $z=-f\cdot g$, where $N(f)=a^2$ and $N(g)=b^2$. The factors $f,g$ can be found as follows. Notice that because $t$ has no prime divisors congruent to $2 \pmod 3$, its prime divisors are not $\mathbb Z[\omega]$-prime; see [@IR Chapter 9.1]. Hence, in the prime decomposition of $a^2=r^2t$ in $\mathbb Z[\omega]$, the $\mathbb Z[\omega]$-prime factors come in complex conjugate pairs. From each pair of factors, choose a factor which is a divisor of $z$, and let $f$ denote the product of these terms. So $N(f)=a^2$. Then set $g=-z/f$, so $N(g)=b^2$. Now let $f=m+n\omega$ and $g=q+p\omega$, for $m,n,p,q\in \mathbb Z$. Notice that by construction, $m+n\omega$ has norm $$\label{E:mn} m^2-mn+n^2=a^2,$$ $q+p\omega$ has norm $$\label{E:pq} p^2-pq+q^2=b^2,$$ and $z=(1+\omega)(u+v\omega) =-(m+n\omega)(q+p\omega)$. Expanding both sides of the last equation gives $u-v+u\omega=(-m q+ n p)+ (-m p- n q+ n p)\omega$, and so the components $u,v$ are $$\label{E:3uv} u= -m p+ n (p-q),\qquad v=m (q-p)-nq.$$ Thus, from [\[E:c\]](#E:c){reference-type="eqref" reference="E:c"}, [\[E:uv\]](#E:uv){reference-type="eqref" reference="E:uv"}, [\[E:mn\]](#E:mn){reference-type="eqref" reference="E:mn"}, [\[E:pq\]](#E:pq){reference-type="eqref" reference="E:pq"} and [\[E:3uv\]](#E:3uv){reference-type="eqref" reference="E:3uv"}, $$\begin{aligned} c^2&=a^2+b^2+\Delta\notag\\ &=a^2+b^2+(u+v)\notag\\ &=(m^2-mn+n^2)+(p^2-pq+q^2)+m (q-2p)+n(p-2q)\notag\\ &=(m-p)^2-(m-p)(n-q)+(n-q)^2.\label{E:fin}\end{aligned}$$ Consider the triangle $ABC$, where $C$ is the origin, $B=m+n\omega$ and $A=p+q\omega$. Then, as required, $BC$ has length $\sqrt{m^2-mn+n^2}=a$, by [\[E:mn\]](#E:mn){reference-type="eqref" reference="E:mn"}, $AC$ has length $\sqrt{p^2-pq+q^2}=b$, by [\[E:pq\]](#E:pq){reference-type="eqref" reference="E:pq"}, and $AB$ has length $$\sqrt{(m-p)^2-(m-p)(n-q)+(n-q)^2}=c,$$ by [\[E:fin\]](#E:fin){reference-type="eqref" reference="E:fin"}. ◻ Notice that the above theorem has the rather surprising corollary. **Corollary 1**. *Suppose a planar triangle $T$ has area of the form $\frac{\sqrt3}4 n$, where $n\in\mathbb N$, and that the squares of the side lengths of $T$ are integers. If one of its side lengths is of the form $r\sqrt{t}$, where $r,t\in\mathbb N$ and $t$ has no prime divisors congruent to $2 \pmod 3$, then the other two sides have the same form.* 1 Adams, W. W., Goldstein, L. J. (1976). *Introduction to Number Theory*. Englewood Cliffs, N.J.: Prentice-Hall. Conway, J. H., Guy, R. K. (1996). *The Book of Numbers*. New York, NY: Copernicus. Ireland, K., Rosen, M. (1990). *A Classical Introduction to Modern Number Theory*. New York, NY: Springer. Ostermann, A., Wanner, G. (2012). *Geometry by its History*. Heidelberg, Germany: Springer. Stillwell, J. (2003). *Elements of Number Theory*. New York, NY: Springer. Yiu, P. (2001). Heronian triangles are lattice triangles. *Amer. Math. Monthly*. 108(3): 261--263. DOI: 10.1080/00029890.2001.11919751

arxiv_math

--- abstract: | We consider the spatially inhomogeneous Landau equation in the case of very soft and Coulomb potentials, $\gamma \in [-3,-2]$. We show that solutions can be continued as long as the following three quantities remain finite, uniformly in $t$ and $x$: (1) the mass density, (2) the velocity moment of order $s$ for any small $s>0$, and (3) the $L^p_v$ norm for any $p>3/(5+\gamma)$. In particular, we do not require a bound on the energy density. If we specialize our result to the spatially homogeneous case, we recover the best known continuation criterion in that regime. address: - Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901 - Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901 author: - Stanley Snelson - Caleb Solomon bibliography: - landau.bib title: A continuation criterion for the Landau equation with very soft and Coulomb potentials --- # Introduction We consider the Landau equation, a collisional kinetic model from plasma physics. The unknown function $f(t,x,v)\geq 0$ models the density of particles at time $t\geq 0$, location $x\in \mathbb R^3$, and velocity $v\in \mathbb R^3$. The equation reads $$\label{e:landau} \partial_t f + v\cdot\nabla_x f = Q(f,f),$$ where $Q$ is the bilinear Landau collision operator, defined for functions $f,g:\mathbb R^3\to\mathbb R$ by $$\label{e:collision} Q(f,g) = \nabla_v \cdot \left(\int_{\mathbb R^3} a(v-w) [f(w) \nabla_v g(v) - f(v)\nabla_w g(w)] \, \mathrm{d}w\right),$$ Here, the matrix $a$ is defined by $$a(z) = a_\gamma |z|^{\gamma+2} \left( I - \frac{z\otimes z}{|z|^2}\right), \quad z\in \mathbb R^3,$$ for some $\gamma \in [-3,1]$, and $a_\gamma >0$ is a constant depending on $\gamma$. This article is concerned with the case $\gamma \in [-3, -2]$, which is known as *very soft potentials*. This case is the most difficult to analyze mathematically, because the singularity of order $\gamma+2$ in $a(z)$ is the most severe. Included in our analysis is the case $\gamma = -3$ (*Coulomb potentials*), which is the most physically relevant case as a model of plasmas. We are concerned with the *large-data* regime, where $f$ and the initial data are not assumed to be close to an equilibrium state. The equilibrium states for [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"} are known as *Maxwellians* and take the form $c_1 e^{-c_2 |v|^2}$ for $c_1, c_2>0$. It has been known since the work of Guo [@guo2002periodic] in 2002 that a global solution exists if the initial data is sufficiently close to a Maxwellian. (See [@carrapatoso2016cauchy; @CM2017verysoft; @kim2020landau; @duan2019mild; @guo2020landau; @golding2023global] and the references therein for futher results on the close-to-equilibrium regime, and [@luk2019vacuum; @chaturvedi2020vacuum] for global solutions close to the vacuum state $f\equiv 0$.) By contrast, global existence of classical solutions in the large-data regime is a difficult unsolved problem. In recent years, there has been partial progress in the form of conditional regularity results and continuation criteria, see e.g. [@golse2016; @cameron2017landau; @HST2018landau; @HST2019rough]. To discuss these results, let us define for any solution $f$ the densities $$\begin{aligned} M_f(t,x) = \int_{\mathbb R^3} f(t,x,v) \, \mathrm{d}v, \qquad &\text{(mass density)}\\ E_f(t,x) = \int_{\mathbb R^3} |v|^2 f(t,x,v) \, \mathrm{d}v, \qquad&\text{(energy density)}. \end{aligned}$$ The best continuation criterion currently available seems to be [@HST2019rough Theorem 1.3], which says that solutions can be continued for as long as the following quantity remains finite: $$\label{e:old-condition} \begin{cases} \displaystyle\sup_{t\in [0,T],x\in \mathbb R^3} [M_f(t,x) + E_f(t,x)], & \text{ if } \gamma \in (-2,0),\\ \displaystyle\sup_{t\in [0,T],x\in \mathbb R^3} [M_f(t,x) + \|f(t,x,\cdot)\|_{L^q_v(\mathbb R^3)}], & \text{ if } \gamma \in [-3,-2], \end{cases}$$ where $$\begin{cases} q > \dfrac 3 {3+\gamma}, &\gamma\in (-3,-2],\\ q = \infty, &\gamma = -3. \end{cases}$$ The goal of this paper is to improve the continuation criterion [\[e:old-condition\]](#e:old-condition){reference-type="eqref" reference="e:old-condition"} in the case $\gamma \in [-3,-2]$. ## Main results In this paper, we work with classical solutions, which means that $f$ is $C^1$ in $t$ and $x$, $C^2$ in $v$, and satisfies [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"} pointwise. First, we have an upper bound in $L^\infty$ that depends only on the weaker quantities in [\[e:cond\]](#e:cond){reference-type="eqref" reference="e:cond"} and the initial data. **Theorem 1**. *Let $\gamma \in [-3,-2]$, and let $f\geq 0$ be a classical solution to the Landau equation [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"} on $[0,T]\times\mathbb R^6$, for some $T>0$. Assume that the initial data $f_{\rm in}(x,v) = f(0,x,v)$ satisfies the lower bound $$\label{e:fin-lower} f_{\rm in}(x,v) \geq \ell, \quad x\in \mathbb R^3, v\in B_\rho(0),$$ for some $\ell, \rho>0$, as well as the upper bound $$\label{e:fin-upper} f_{\rm in} \leq C_0 e^{-\mu |v|^2},$$ for some $C_0, \mu >0$. Furthermore, assume that $f$ satisfies the upper bounds $$\label{e:cond} M_f(t,x)\leq M_0, \quad \int_{\mathbb R^3} |v|^s f(t,x,v)\, \mathrm{d}v \leq S_0, \quad \|f(t,x,\cdot)\|_{L^{p+\delta}(\mathbb R^3)} \leq P_0,$$ uniformly in $x\in \mathbb R^3$ and $t\in [0,T]$, for some $s \in (0,2)$ and $\delta>0$, where $p= \dfrac 3 {5+\gamma}$.* *Then $f$ satisfies a global upper bound $$f(t,x,v) \leq C,$$ for some $C>0$ depending on $\gamma$, $\ell$, $\rho$, $C_0$, $\mu$, $s$, $\delta$, $T$, and the constants in [\[e:cond\]](#e:cond){reference-type="eqref" reference="e:cond"}.* When combined with the results of [@henderson2017smoothing], our Theorem [Theorem 1](#t:upper){reference-type="ref" reference="t:upper"} implies that $f$ satisfies regularity estimates of all orders on $[t,T]\times\mathbb R^6$ for any $t>0$, with constants depending only on $t$, $T$, the initial data, and the constants in [\[e:cond\]](#e:cond){reference-type="eqref" reference="e:cond"}. Furthermore, the continuation criterion [\[e:old-condition\]](#e:old-condition){reference-type="eqref" reference="e:old-condition"} from [@HST2019rough Theorem 1.3] applies to $f$, because of our assumptions on $f_{\rm in}$. Therefore, bounding the $L^p_v$ norm in [\[e:old-condition\]](#e:old-condition){reference-type="eqref" reference="e:old-condition"} with Theorem [Theorem 1](#t:upper){reference-type="ref" reference="t:upper"}, we immediately obtain: **Corollary 2**. *Let $\gamma \in [-3,-2]$, and let $f$ be a classical solution to the Landau equation, with $f_{\rm in}$ satisfying the hypotheses [\[e:fin-lower\]](#e:fin-lower){reference-type="eqref" reference="e:fin-lower"} and [\[e:fin-upper\]](#e:fin-upper){reference-type="eqref" reference="e:fin-upper"} from Theorem [Theorem 1](#t:upper){reference-type="ref" reference="t:upper"}.* *If $T_*< \infty$ is the maximal time of existence of the solution $f$, i.e. if $f$ cannot be extended to a solution on $[0,T_*+\tau)\times\mathbb R^6$ for any $\tau>0$, then one of the inequalities in [\[e:cond\]](#e:cond){reference-type="eqref" reference="e:cond"} must degenerate as $t\nearrow T_*$, i.e. either $$\sup_{x\in\mathbb R^3} M_f(t,x) \nearrow +\infty \,\, \text{ or } \,\, \sup_{x\in \mathbb R^3} \int_{\mathbb R^3} |v|^s f(t,x,v) \, \mathrm{d}v \nearrow +\infty \,\,\text{ or } \,\, \sup_{x\in\mathbb R^3} \|f(t,x,\cdot)\|_{L^{p+\delta}_v(\mathbb R^3)}\nearrow +\infty,$$ as $t\nearrow T_*$.* Unlike $q$ in [\[e:old-condition\]](#e:old-condition){reference-type="eqref" reference="e:old-condition"}, the critical exponent $p= \dfrac 3 {5+\gamma}$ in Theorem [Theorem 1](#t:upper){reference-type="ref" reference="t:upper"} and Corollary [Corollary 2](#c:cont){reference-type="ref" reference="c:cont"} does not approach $+\infty$ as $\gamma \searrow -3$. We should note that our lower bound condition [\[e:fin-lower\]](#e:fin-lower){reference-type="eqref" reference="e:fin-lower"} on the initial data could be relaxed to allow the presence of vacuum regions, by applying the positivity-spreading result of [@HST2018landau Theorem 1.3]. For the sake of a simple statement of our results, we focus instead on the continuation of solutions with nice (but large) initial data. ## Comparison with homogeneous Landau It is interesting to compare these results to what is known for the spatially homogeneous Landau equation, which arises from assuming the solution of [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"} is constant in $x$. Then $f(t,v)$ satisfies $$\label{e:homogeneous} \partial_t f = Q(f,f).$$ Compared to the full inhomogeneous Landau equation, more results about existence and regularity are available for [\[e:homogeneous\]](#e:homogeneous){reference-type="eqref" reference="e:homogeneous"}, see [@fournier2010uniqueness; @desvillettes2015landau; @silvestre2015landau; @gualdani2017landau; @CDE2017landau; @carillo2020gradient; @GGIV2022landau; @golse2022local; @desvillettes2023mono] and the references therein. In particular, this equation is known to be globally well-posed when $\gamma \geq -2$ [@villani1998landau; @desvillettes2000landau; @alexandre2015apriori; @Wu2014global]. Surprisingly, large-data global existence is unknown for the case $\gamma \in [-3,2)$, even for the homogeneous equation [\[e:homogeneous\]](#e:homogeneous){reference-type="eqref" reference="e:homogeneous"}. The best known continuation criterion for [\[e:homogeneous\]](#e:homogeneous){reference-type="eqref" reference="e:homogeneous"} is as follows: if $f$ is bounded in $L^\infty_t L^{q}_v([0,T]\times\mathbb R^3)$ for some $q>3/(5+\gamma)$, then $f$ can be continued past time $T$ (see, e.g. [@gualdani2017landau; @ABDL; @golding2023local]). If we apply our Corollary [Corollary 2](#c:cont){reference-type="ref" reference="c:cont"} in the homogeneous case, then since the flow of [\[e:homogeneous\]](#e:homogeneous){reference-type="eqref" reference="e:homogeneous"} conserves the mass $\int_{\mathbb R^3} f(t,v)\, \mathrm{d}v$ and energy $\int_{\mathbb R^3} |v|^2 f(t,v) \, \mathrm{d}v$ (which together control the $s$-moment), we recover this continuation criterion. Based on this, we believe that the integrability exponent $3 /(5+\gamma) + \delta$ (with $\delta>0$ arbitrarily small) in Corollary [Corollary 2](#c:cont){reference-type="ref" reference="c:cont"} may be the sharpest available with current techniques. Note that $q>3/(5+\gamma)$ is the minimal condition required so that $\|f\|_{L^q_v}$ controls the convolution $f\ast |v|^{\gamma+2}$ (see [\[e:collision\]](#e:collision){reference-type="eqref" reference="e:collision"} and [\[e:abc\]](#e:abc){reference-type="eqref" reference="e:abc"}) uniformly from above. This condition also appears as a borderline in the result of [@bedrossian2022vpl], which ruled out some approximately self-similar blowup solutions. Recently, Alonso-Bagland-Desvillettes-Lods [@ABDL] have derived a Prodi-Serrin-like condition for homogeneous Landau: if a solution $f$ (up to a polynomial weight) lies in $L^r([0,T],L^q(\mathbb R^3))$ for some $q>1$ and $r\geq 1$ satisfying $$\frac 2 r + \frac 3 q = 5 + \gamma,$$ then $f$ is bounded in $L^\infty$ for positive times and can therefore be continued past time $T$. (Some unweighted Prodi-Serrin conditions were subsequently derived in [@golding2023local].) It would be interesting to derive this kind of condition for the inhomogeneous Landau equation [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"}, using mixed $(t,x,v)$ norms of the form $L^r_t L^q_x L^p_v([0,T]\times\mathbb R^3\times\mathbb R^3)$ for some $r, q, p$. ## Proof ideas The philosophy of our proof is to leverage the diffusive properties of the collision term $Q(f,f)$, while exploiting the *nonlinear structure* of this diffusion more fully, compared to some previous works on the large-data case of the Landau equation. To explain what this means, let us write the bilinear collision operator [\[e:collision\]](#e:collision){reference-type="eqref" reference="e:collision"} in the usual way as a diffusion operator, either in divergence form $$Q(f,g) = \nabla_v\cdot(\bar a^f\nabla_v g) + \bar b^f\cdot \nabla_v g + \bar c^f g,$$ or nondivergence form $$Q(f,g) = \mbox{tr}(\bar a^f D_v^2 g) + \bar c^f g,$$ where $$\label{e:abc} \begin{split} \bar a^f &= a_\gamma\int_{\mathbb R^3} |w|^{\gamma+2}\Pi(w) f(v-w) \, \mathrm{d}w,\\ \bar b^f &= b_\gamma \int_{\mathbb R^3} |w|^\gamma w f(v-w) \, \mathrm{d}w,\\ \bar c^f &= \begin{cases} c_\gamma \int_{\mathbb R^3} |w|^\gamma f(v-w) \, \mathrm{d}w, & \gamma > -3,\\ f, & \gamma = -3, \end{cases} \end{split}$$ where $b_\gamma$ and $c_\gamma >0$ are constants depending on $\gamma$, and $\Pi(w) = \left( I - \dfrac{w\otimes w}{|w|^2}\right)$. Our argument proceeds in the following steps: 1. First, prove a local $L^\infty$ estimate for $f$ by a Moser iteration argument that exploits the gain in regularity/integrability provided by velocity averaging. This argument is inspired by Golse-Imbert-Mouhot-Vasseur [@golse2016 Theorem 12], who considered linear kinetic Fokker-Planck equations of the form $$\label{e:linear} \partial_t f + v\cdot\nabla_x f = \nabla_v\cdot (A\nabla_v f) + B\cdot \nabla_v f + s$$ for general coefficients $A$, $B$, $s$, and then applied their estimate to the Landau equation by placing suitable conditions on $f$ so that the coefficients $\bar a^f$, $\bar b^f$, and $\bar c^f$ in [\[e:abc\]](#e:abc){reference-type="eqref" reference="e:abc"} are well-behaved. This works well when $\gamma \geq -2$, but when $\gamma< -2$, a bound for $f$ in $L^q_v$ is needed to control the coefficients, with $q$ as in [\[e:old-condition\]](#e:old-condition){reference-type="eqref" reference="e:old-condition"}, and we would like to avoid this assumption. We address this problem by "remembering" the coupling between $f$ and the coefficients earlier in the proof, which leads to an estimate whose constant has a less severe dependence on higher integrability norms of $f$.[^1] 2. Next, we improve the local estimate using scaling techniques. It is well known that if $f$ solves the Landau equation [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"}, then for any $r>0$ and $\alpha \in \mathbb R$, the function $$f_r(t,x,v) = r^{\alpha + 3 + \gamma} f(r^\alpha t, r^{1+\alpha} x, r v)$$ is also a solution. By choosing a convenient value of $\alpha$ and a scale $r$ that depends on the $L^\infty$ norm of $f$, we obtain a pointwise upper bound of the form $f(t,x,v) \leq C\|f\|_{L^\infty}^\beta$ for some $\beta < 1$, which would imply an unconditional $L^\infty$ estimate by taking the supremum over $(t,x,v)$. This scaling argument should be compared to [@cameron2017landau], which applied rescaling techniques to estimates for the *linear* equation [\[e:linear\]](#e:linear){reference-type="eqref" reference="e:linear"}. Again, this worked well only when $\gamma \geq -2$. 3. Unfortunately, the constant $C$ in the previous step blows up for large $|v|$, so we cannot naively take the supremum over $v$. We get around this problem using pointwise decay of $f$, which is why we need to assume decay for $f_{\rm in}$ in our main results. It is already well-established that Gaussian decay in $v$ is propagated forward in time by the Landau equation [@cameron2017landau; @henderson2017smoothing], but for our purposes, the key is to obtain *quantitative* dependence of these Gaussian upper bounds on the $L^\infty$ norm of $f$ that is as sharp as possible. We accomplish this via a barrier argument with a barrier of the form $h(v) = K e^{-\mu |v|^2}$. Previous barrier arguments such as [@cameron2017landau; @henderson2017smoothing] derived a contradiction at a first crossing point between $f$ and $h$ by writing $$Q(f,f) \leq Q(f,h) = \mbox{tr}(\bar a^f D_v^2 h) + \bar c^f h,$$ and bounding $\bar a^f$ and $\bar c^f$ using some conditional upper bounds for $f$ like [\[e:old-condition\]](#e:old-condition){reference-type="eqref" reference="e:old-condition"}. By contrast, in our Proposition [Proposition 10](#p:gaussian){reference-type="ref" reference="p:gaussian"}, we get sharper estimates by also using $f\leq h$ in our bounds for the coefficients $\bar a^f$, $\bar c^f$. This kind of "nonlinear barrier argument" has been applied to the Boltzmann equation [@silvestre2016boltzmann; @imbert2018decay] but is apparently new in the study of the Landau equation. 4. Finally, to remove the dependence on the energy density bound, we estimate $\int_{\mathbb R^3} |v|^2 f\, \mathrm{d}v$ from above by interpolating between $\int_{\mathbb R^3} |v|^s f \, \mathrm{d}v$ and the Gaussian upper bound. This interpolation requires a non-obvious argument that previously appeared in [@S2018hardpotentials]. ## Notation We sometimes use the shorthand $z = (t,x,v) \in \mathbb R^7$. To state local estimates, it is convenient to use kinetic cylinders of the form $$Q_r(z_0) = (t_0-r^2, t_0] \times \{ x : |x-x_0 - t v_0|< r^3\} \times B_r(v_0).$$ We also write $Q_1 = Q_1(0)$. The notation $A\lesssim B$ means $A\leq C B$ for a constant $C>0$ depending only on the quantities stated in the given lemma or theorem. The notation $A \approx B$ means $A\lesssim B$ and $B\lesssim A$. ## Outline of the paper In Section [2](#s:prelim){reference-type="ref" reference="s:prelim"}, we review some bounds on the coefficients $\bar a^f$, $\bar b^f$, and $\bar c^f$, as well as some known results on the spreading of positivity. Section [3](#s:upper){reference-type="ref" reference="s:upper"} proves a local $L^\infty$ estimate via Moser iteration, Section [4](#s:gaussian){reference-type="ref" reference="s:gaussian"} establishes quantitative Gaussian upper bounds, and Section [5](#s:improved){reference-type="ref" reference="s:improved"} derives a global $L^\infty$ estimate that depends only on the quantities in Theorem [Theorem 1](#t:upper){reference-type="ref" reference="t:upper"} plus a bound for the energy density. Finally, Section [6](#s:energy){reference-type="ref" reference="s:energy"} removes the dependence on the energy bound. # Preliminaries {#s:prelim} ## Coefficient bounds The following lemma gives upper bounds for the coefficients $\bar a^f$, $\bar b^f$, $\bar c^f$, under the assumption that $L^1_v$ and $L^{p+\delta}_v$ norms of $f$ are bounded, where $p = 3/(5+\gamma)$. The proof is standard, but we need to track the dependence on the $L^\infty$ norm of $f$ precisely. **Lemma 3**. *Let $f:\mathbb R^3 \to \mathbb R$ belong to the space $L^1(\mathbb R^3)\cap L^\infty(\mathbb R^3)$. Let $p = \dfrac{3}{5+\gamma}$, and let $\delta>0$ be an arbitrary small number.* *Then the coefficients $\bar a^f$, $\bar b^f$, and $\bar c^f$ defined in [\[e:abc\]](#e:abc){reference-type="eqref" reference="e:abc"} satisfy, for all $v\in \mathbb R^3$, $$\begin{split} |\bar a^f(v)| &\leq C,\\ |\bar b^f(v)| &\leq C \|f\|_{L^\infty(\mathbb R^3)}^{1-p(\gamma+4)/3},\\ |\bar c^f(v)| &\leq C \|f\|_{L^\infty(\mathbb R^3)}^{1-p(\gamma+3)/3}, \end{split}$$ for a constant $C>0$ depending only on $\gamma$, $\delta$, and the $L^{p+\delta}(\mathbb R^3)$ and $L^1(\mathbb R^3)$ norms of $f$.* *Proof.* The bound for $\bar a^f$ follows from $|(I - |z|^{-2}z\otimes z)| \leq 1$ and the standard convolution estimate $$(|v|^{\gamma+2} \ast f)(v) \leq C \|f\|_{L^{p+\delta}(\mathbb R^3)}^{-(p+\delta)'(\gamma+2)/3} \|f\|_{L^1(\mathbb R^3)}^{1 + (p+\delta)'(\gamma+2)/3}.$$ where $(p+\delta)' = (p+\delta)/(p+\delta - 1)$. The bounds for $\bar b^f$ and $\bar c^f$ follow from $$(|v|^\sigma \ast f)(v) \leq C \|f\|_{L^\infty(\mathbb R^3)}^{1-p(\sigma+3)/3} \|f\|_{L^p(\mathbb R^3)}^{p(\sigma+3)/3}, \quad \text{for $\sigma< - 3\left(1- \frac 1 p\right)$},$$ which holds for both $\sigma = \gamma+1$ and $\sigma = \gamma$, since $p< 3/(3+\gamma)$. Note that we can absorb $\|f\|_{L^p(\mathbb R^3)}$ into the constant $C$ in the statement of the lemma, by interpolation. ◻ Next, we have a lower ellipticity bound for the matrix $\bar a^f$: **Lemma 4**. *[@HST2018landau Lemma 4.3][\[l:a-lower\]]{#l:a-lower label="l:a-lower"} Let $f:\mathbb R^3\to [0,\infty)$ be an integrable function such that $$f(v) \geq \ell, \quad v\in B_\rho(0),$$ for some $\ell, \rho>0$. Then the matrix $\bar a^f$ defined in [\[e:abc\]](#e:abc){reference-type="eqref" reference="e:abc"} satisfies $$\label{e:a-lower} e\cdot( \bar a^f e) \geq c_a \begin{cases} (1+|v|)^\gamma, &e \in \mathbb S^2,\\ (1+|v|)^{\gamma+2}, &e\cdot v = 0. \end{cases}$$ The constant $c_a>0$ depends only on $\gamma$, $\ell$, and $\rho$.* ## Pointwise lower bounds Lower bounds for the solution $f$ will be combined with Lemma [\[l:a-lower\]](#l:a-lower){reference-type="ref" reference="l:a-lower"} to conclude coercivity of the matrix $\bar a^f$, which is essential for the smoothing properties of the equation. These lower bounds for $f$, which are based on propagating lower bounds from time zero to positive times, were first established in [@HST2018landau Theorem 1.3], and a more precise restatement is given in [@HST2019rough Lemma 2.5]. Here, we state the result in a less general form that is tailored to our purposes: **Lemma 5**. *Let $f:[0,T]\times\mathbb R^6\to [0,\infty)$ be a solution of the Landau equation, satisfying $M_f(t,x) \leq M_0$ and $\|f(t,x,\cdot)\|_{L^{p+\delta}_v(\mathbb R^3)} \leq P_0$, uniformly in $(t,x)$, where $p = \dfrac 3 {5+\gamma}$, and $$f_{\rm in}(x,v) \geq \ell, \quad x \in \mathbb R^3, v \in B_\rho(0),$$ for some $\ell, \rho>0$.* *Then $f$ satisfies lower bounds of the form $$f(t,x,v) \geq \ell', \quad x\in \mathbb R^3, v\in B_{\rho/2}(0),$$ where the constant $\ell'>0$ depends only on $\gamma$, $\ell$, $\rho$, $\delta$, $T$, $M_0$, and $P_0$.* # Local $L^\infty$ estimate {#s:upper} In this section, we consider any solution $f$ to the Landau equation on a domain that contains the unit cylinder $Q_1$. We assume that the matrix $\bar a^f$ satisfies some lower ellipticity bound $$e\cdot (\bar a^f(t,x,v) e) \geq \lambda, \quad (t,x,v) \in Q_1, e\in \mathbb S^2,$$ for some $\lambda>0$. Later, we will recenter the estimate around any arbitrary point $(t_0,x_0,v_0)$, and calculate $\lambda$ depending on $v_0$ via Lemma [\[l:a-lower\]](#l:a-lower){reference-type="ref" reference="l:a-lower"}. As discussed in the introduction, the argument of this sectionis a modification of the work in [@golse2016]. The estimate we obtain (Proposition [Proposition 9](#p:inf){reference-type="ref" reference="p:inf"}) is in a form that is convenient to apply scaling techniques and eventually remove any dependence on $\|f\|_{L^\infty}$. **Lemma 6**. *Let $f\geq 0$ be a classical solution of the Landau equation in $(-1,0]\times B_1\times \mathbb R^3$, and let $\chi(t,x,v)$ be any smooth, compactly supported function in $Q_1$. Then for any $q\geq 1$, one has $$(\partial_t + v\cdot \nabla_x) (\chi f^q) \leq \nabla_v \cdot (\bar a^f\nabla_v (\chi f^q)) + H_0 + \nabla_v \cdot H_1,$$ with $$\begin{split} H_0 &= f^q [(\partial_t + v\cdot \nabla_x)\chi + \nabla_v\cdot (\bar a^f\nabla_v \chi) - \nabla_v\cdot(\chi \bar b^f) + q \chi \bar c^f],\\ H_1 &= f^q [-2\bar a^f\nabla_v \chi + \chi \bar b^f]. \end{split}$$* *Proof.* The proof is a direct calculation involving several applications of the product rule. In more detail, using the equation [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"}, we have $$\label{e:abc-eqn} \begin{split} (\partial_t + v\cdot \nabla_x)(\chi f^q) %&= f^q (\partial_t + v\cdot \nabla_x) \chi + q f^{q-1} \chi (\partial_t + v\cdot\nabla_x) f\\ &= f^q (\partial_t + v\cdot \nabla_x) \chi + q f^{q-1} \chi [\nabla_v\cdot(\bar a^f \nabla_v f) + \bar b^f\cdot \nabla_v f + \bar c^f f]. \end{split}$$ For the term on the right involving $\bar a^f$, we have $$\begin{split} q f^{q-1} \chi \nabla_v\cdot(\bar a^f \nabla_v f) &= q\nabla_v\cdot (f^{q-1} \chi \bar a^f \nabla_v f) - q\nabla_v( f^{q-1}\chi)\cdot(\bar a^f \nabla_v f)\\ &= \nabla_v\cdot (\chi \bar a^f \nabla_v(f^q)) - q\chi \nabla_v(f^{q-1})\cdot (\bar a^f \nabla_v f) - qf^{q-1} \nabla_v \chi \cdot (\bar a^f \nabla_v f)\\ &= \nabla_v\cdot (\bar a^f \nabla_v(\chi f^q)) - \nabla_v \cdot ( f^q \bar a^f \nabla_v \chi)\\ &\quad - q(q-1) \chi f^{q-2}\nabla_v f \cdot (\bar a^f \nabla_v f) - \nabla_v \chi\cdot (\bar a^f\nabla_v f^q)\\ &\leq \nabla_v\cdot (\bar a^f \nabla_v(\chi f^q)) - \nabla_v \cdot ( f^q \bar a^f \nabla_v \chi) - \nabla_v \chi\cdot (\bar a^f\nabla_v f^q), \end{split}$$ by the positive-definiteness of $\bar a^f$. Applying the product rule again in the last term, we have $$q f^{q-1} \chi \nabla_v\cdot(\bar a^f \nabla_v f) \leq \nabla_v\cdot (\bar a^f \nabla_v(\chi f^q)) - 2\nabla_v \cdot ( f^q \bar a^f \nabla_v \chi) +f^q \nabla_v\cdot(\bar a^f\nabla_v \chi).$$ For the $\bar b^f$ term in [\[e:abc-eqn\]](#e:abc-eqn){reference-type="eqref" reference="e:abc-eqn"}, we have $$\begin{split} q f^{q-1} \chi \bar b^f\cdot \nabla_v f &= \chi \bar b^f\cdot \nabla_v (f^q) = \nabla_v\cdot(\chi f^q \bar b^f) - f^q\nabla_v\cdot( \chi \bar b^f). \end{split}$$ After collecting terms, we obtain the statement of the lemma. ◻ **Lemma 7**. *With $f$, $\chi$, $H_0$, and $H_1$ as in Lemma [Lemma 6](#l:subsolution){reference-type="ref" reference="l:subsolution"}, the following inequality holds for any $q\geq 1$: $$\|\chi f^q\|_{L^{42/19}(Q_1)}^2 \leq C \left(\frac{1+\|\bar a^f\|_{L^\infty(Q_1)}^2} {\lambda^2}\right) \left( \|H_0\|_{L^2(Q_1)}^2 + \|H_1\|_{L^2(Q_1)}^2\right),$$ for a universal constant $C>0$. In particular, $C>0$ is independent of $q$ and $\chi$.* *Proof.* Let $g$ be the solution to $$(\partial_t +v\cdot\nabla_x) g = \nabla_v \cdot (\bar a^f g) + H_0 + \nabla_v \cdot H_1,$$ in $Q_1$, with $g=0$ on the parabolic boundary of $Q_1$. Here, $H_0$ and $H_1$ are defined in terms of the function $f$. By the comparison principle and Lemma [Lemma 6](#l:subsolution){reference-type="ref" reference="l:subsolution"}, we have $g\geq \chi f^q$ in $Q_1$. Integrating this equation against $g$ over $Q_1 \subset\mathbb R^{7}$, we obtain (writing $\, \mathrm{d}z = \, \mathrm{d}t \, \mathrm{d}x \, \mathrm{d}v$) $$\begin{split} \frac 1 2 \int_{Q_1} \frac d {dt} g^2 \, \mathrm{d}z &\leq - \lambda \int_{Q_1} |\nabla_v g|^2\, \mathrm{d}z + \int_{Q_1} g H_0 \, \mathrm{d}z - \int_{Q_1} \nabla_v g \cdot H_1 \, \mathrm{d}z\\ &\leq \int_{Q_1} g H_0 \, \mathrm{d}z + \frac \lambda 2 \int_{Q_1} |\nabla_v g |^2 \, \mathrm{d}z + \frac 1 {2\lambda} \int_{Q_1} |H_1|^2 \, \mathrm{d}z, \end{split}$$ using the fact that $g=0$ on the parabolic boundary of $Q_1$. The left side of this inequality is nonnegative because $g=0$ on the time slice $\{t=-1\}$, and $g \geq \chi f^q \geq 0$ on the time slice $\{t=0\}$. We now have $$\label{e:g-energy} \int_{Q_1} |\nabla_v g|^2 \, \mathrm{d}z \leq \frac C{\lambda^2} \left(\|H_0\|_{L^2(Q_1)} + \|H_1\|_{L^2(Q_1)}+ \|g\|_{L^2(Q_1)}\right).$$ Next, we apply the hypoelliptic estimate of Bouchut [@bouchut2002hypoelliptic Theorem 1.3] to $g$, yielding $$\begin{split} \|D_t^{1/3}g\|_{L^2(Q_1)}^2 + \|D_x^{1/3}g\|_{L^2(Q_1)}^2 &\lesssim \|g\|_{L^2(Q_1)}^2 + \|\nabla_v g\|_{L^2(Q_1)} \| \langle v\rangle H_0\|_{L^2(Q_1)}\\ &\quad + \|\nabla_v g\|_{L^2(Q_1)}^{4/3}\|\langle v\rangle(H_1+\bar a^f \nabla_v g)\|_{L^2(Q_1)}^{2/3}\\ &\quad + \|\nabla_v g\|_{L^2(Q_1)} \|\langle v\rangle(H_1+\bar a^f \nabla_v g)\|_{L^2(Q_1)}\\ &\lesssim \|g\|_{L^2(Q_1)}^2 + \|\nabla_v g\|_{L^2(Q_1)}^2\\ &\quad+ \| \langle v\rangle H_0\|_{L^2(Q_1)}^2 + \|\langle v\rangle(H_1+\bar a^f \nabla_v g)\|_{L^2(Q_1)}^2. \end{split}$$ By the Poincaré inequality, the term $\|g\|_{L^2(Q_1)}^2$ on the right can be absorbed into $\|\nabla_v g\|_{L^2(Q_1)}^2$. Adding $\|\nabla_v g\|_{L^2(Q_1)}^2$ to both sides and using $\langle v\rangle\leq 1$ as well as the energy estimate [\[e:g-energy\]](#e:g-energy){reference-type="eqref" reference="e:g-energy"}, we obtain $$\begin{split} \|D_t^{1/3}g\|_{L^2(Q_1)}^2 + &\|D_x^{1/3}g\|_{L^2(Q_1)}^2 + \|\nabla_v g\|_{L^2(Q_1)}^2 \\ &\lesssim (1+\|\bar a^f\|_{L^\infty(Q_1)}^2)\|\nabla_v g\|_{L^2(Q_1)}^2 + \|H_0\|_{L^2(Q_1)}^2 + \|H_1\|_{L^2(Q_1)}^2 \\ &\lesssim \frac{1+\|\bar a^f\|_{L^\infty(Q_1)}^2} {\lambda^2} \left( \|H_0\|_{L^2(Q_1)}^2 + \|H_1\|_{L^2(Q_1)}^2\right). \end{split}$$ We now apply the Sobolev embedding $H^{1/3}(\mathbb R^7)\subset L^{42/19}(\mathbb R^7)$ to obtain $$\begin{split} \|g\|_{L^{42/19}(Q_1)} \lesssim \frac{1+\|\bar a^f\|_{L^\infty(Q_1)}^2} {\lambda^2} \left( \|H_0\|_{L^2(Q_1)}^2 + \|H_1\|_{L^2(Q_1)}^2\right). \end{split}$$ With the inequality $\chi f^q \leq g$, the proof is complete. ◻ **Lemma 8**. *Let $f\geq 0$ be a solution of the Landau equation in $Q_1$. For any $0< r_0 < r_1< 1$ and $q\geq 1$, there holds $$\begin{split} \|f\|_{L^{\sigma q}(Q_{r_0})}^{2q} &\leq C \left(\frac {1 + \|\bar a^f\|_{L^\infty(Q_1)}^2} { \lambda^2} \right) \left( 1 + \|\bar a^f\|_{L^\infty(Q_1)}^2 + \|\bar b^f\|_{L^\infty(Q_1)}^2 + \|\bar c^f\|_{L^\infty(Q_1)}^2\right)\\ &\quad \quad\times \left( (r_1-r_0)^{-4} + q^2\right) \|f\|_{L^{2q}(Q_{r_1})}^{2q}, \end{split}$$ with $\sigma = 42/19$.* *Proof.* First, we simplify $H_0$ using the relationships $$\sum_j \cdot \bar a_{ij}^f = -\bar b_i^f, \quad \nabla_v\cdot \bar b^f = -\bar c^f,$$ giving $$H_0 =f^q[(\partial_t + v\cdot\nabla_x)\chi + \mbox{tr}(\bar a^f D_v^2 \chi) - 2\bar b^f\cdot\nabla_v\chi + (q+1) \chi \bar c^f.$$ Next, we choose $\chi \in C_0^\infty(Q_1)$ so that $\chi = 1$ in $Q_{r_0}$ and $\chi = 0$ outside $Q_{r_1}$. Such a $\chi$ can be chosen so that $$|(\partial_t + v\cdot \nabla_x)\chi |\lesssim (r_1-r_0)^{-2}, \quad |\nabla_v\chi|\lesssim (r_1-r_0)^{-1}, \quad |D_v^2 \chi| \lesssim (r_1-r_0)^{-2}.$$ With this choice of $\chi$, note that $H_0$ and $H_1$ are zero outside $Q_{r_1}$. We bound $H_0$ and $H_1$ as follows, using $r_1 - r_0 < 1$: $$\begin{split} \|H_0\|_{L^2(Q_1)} &\lesssim \|f^q\|_{L^2(Q_{r_1})} \left[ (r_1 - r_0)^{-2} \left( 1 + \|\bar a^f\|_{L^\infty(Q_1)} + \|\bar b^f\|_{L^\infty(Q_1)}\right) + q\|\bar c^f\|_{L^\infty(Q_1)} \right]\\ &\lesssim \|f\|_{L^{2q}(Q_{r_1})}^q \left( 1 + \|\bar a^f\|_{L^\infty(Q_1)} + \|\bar b^f\|_{L^\infty(Q_1)} + \|\bar c^f\|_{L^\infty(Q_1)}\right) \left( (r_1 - r_0)^{-2} + q\right), \end{split}$$ and $$\begin{split} \|H_1\|_{L^2(Q_1)} &\lesssim \|f\|_{L^{2q}(Q_{r_1})}^q \left( \|\bar a^f\|_{L^\infty(Q_1)} (r_1 - r_0)^{-1} + \|\bar b^f\|_{L^\infty(Q_1)}\right)\\ &\lesssim \|f\|_{L^{2q}(Q_{r_1})}^q (r_1 - r_0)^{-1} \left(1 + \|\bar a^f\|_{L^\infty(Q_1)} + \|\bar b^f\|_{L^\infty(Q_1)}\right). \end{split}$$ Combining these estimates for $H_0$ and $H_1$ with Lemma [Lemma 7](#l:hypo){reference-type="ref" reference="l:hypo"} yields the conclusion of the lemma. ◻ Now we use Lemma [Lemma 8](#l:gain){reference-type="ref" reference="l:gain"} and a classical Moser iteration procedure to prove a local $L^\infty$ estimate for $f$: **Proposition 9**. *Let $f:(-1,0]\times\mathbb R^3\times\mathbb R^3\to [0,\infty)$ solve the Landau equation [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"} in $Q_1$. Assume that the coefficients $\bar a^f$, $\bar b^f$, and $\bar c^f$ are essentially bounded in $Q_1$, and that the matrix $\bar a^f$ satisfies the lower ellipticity bound $$e\cdot (\bar a^f(t,x,v) e) \geq \lambda, \quad (t,x,v) \in Q_1, e\in \mathbb S^2.$$ Then $$\|f\|_{L^\infty(Q_{1/2})} \leq C \left(\frac {1 + \|\bar a^f\|_{L^\infty(Q_1)}^2} { \lambda^2} \right) \left( 1 + \|\bar a^f\|_{L^\infty(Q_1)}^2 + \|\bar b^f\|_{L^\infty(Q_1)}^2 + \|\bar c^f\|_{L^\infty(Q_1)}^2\right)^{19/4} \|f\|_{L^2(Q_1)},$$ where the constant $C$ depends only on $\gamma$.* *Proof.* Define the radii $r_i$ and exponents $q_i$ by $$r_i := \frac 1 2 + \left(\frac 1 2\right)^i, \quad q_i = \left(\frac \sigma 2\right)^i = \left(\frac {21}{19}\right)^i, \quad i = 1,2,\ldots$$ Since Lemma [Lemma 8](#l:gain){reference-type="ref" reference="l:gain"} holds for any $q>1$ and any concentric cylinders $Q_{r_0} \subset Q_{r_1}$ with $r_0<r_1 <1$, we have for each $i$ the inequality $$\label{e:ineq} \begin{split} \|f\|_{L^{\sigma q_i}(Q_{r_{i+1}})} &\leq K_f^{1/(2q_i)}\left( (r_i - r_{i+1})^{-4} + q^2\right)^{1/(2q_i)} \|f\|_{L^{2q_i}(Q_{r_i})}, \end{split}$$ with $$K_f := C\left(\frac {1 + \|\bar a^f\|_{L^\infty(Q_1)}^2} { \lambda^2} \right) \left( 1 + \|\bar a^f\|_{L^\infty(Q_1)}^2 + \|\bar b^f\|_{L^\infty(Q_1)}^2 + \|\bar c^f\|_{L^\infty(Q_1)}^2\right).$$ Note that $q_i^2 = (21/19)^{2i} \leq (16)^{i+1} = (r_{i+1} - r_i)^{-4}$, so we can rewrite [\[e:ineq\]](#e:ineq){reference-type="eqref" reference="e:ineq"} as $$\begin{split} \|f\|_{L^{\sigma q_i}(Q_{r_{i+1}})} &\leq K_f^{1/(2q_i)}2^{(2i+5/2)/q_i} \|f\|_{L^{2q_i}(Q_{r_i})}. \end{split}$$ Iterating from $i=1,2,\ldots$, we obtain the desired upper bound $$\|f\|_{L^\infty(Q_{1/2})} \leq K_f^{19/4} 2^{893/4} \|f\|_{L^2(Q_{1})},$$ since $$\sum_{i=1}^\infty \frac 1 {2q_i} = \frac {19} 4 \quad \text{and} \quad \sum_{i=1}^\infty \frac{2i+5/2}{q_i} = \frac{893} 4.$$ ◻ # Gaussian bounds {#s:gaussian} This section establishes Gaussian decay estimates in $v$ for the solution $f$. These estimates are needed when applying the local estimate of Proposition [Proposition 9](#p:inf){reference-type="ref" reference="p:inf"} at large velocities, because the lower ellipticity constant $\lambda$ will degenerate to 0 (see Lemma [\[l:a-lower\]](#l:a-lower){reference-type="ref" reference="l:a-lower"}). **Proposition 10**. *Let $f:[0,T]\times\mathbb R^6 \to [0,\infty)$ be a solution of the Landau equation satisfying [\[e:cond\]](#e:cond){reference-type="eqref" reference="e:cond"}, and assume the initial data satisfies $$f_{\rm in}(x,v) \leq C_0 e^{-\mu' |v|^2},$$ for some $C_0, \mu'>0$, as well as the lower bounds $$f_{\rm in}(x,v) \geq \ell, \quad x\in \mathbb R^3, v\in B_\rho(0).$$ for some $\ell, \rho>0$. Assume further that $f$ is bounded and that $$\int_{\mathbb R^3}|v|^2 f(t,x,v) \, \mathrm{d}v \leq E_0, \quad (t,x)\in [0,T]\times\mathbb R^3.$$ Then there exist $c_0>0$ and $C_1>1$, depending on $\gamma$, $C_0$, $\ell$, $\rho$, and the constants in [\[e:cond\]](#e:cond){reference-type="eqref" reference="e:cond"}, such that for $$K = 2\max\{ C_0, C_1\|f\|_{L^\infty([0,T]\times\mathbb R^6)}\}$$ and $$\label{e:mu-def} \mu = \min\left\{\frac {\mu'} 2 , \frac {c_0} {E_0}, \frac {1} {33\log(K/c_0)}\right\},$$ the upper bound $$f(t,x,v) \leq K e^{-\mu |v|^2},$$ holds for all $(t,x,v) \in [0,T]\times\mathbb R^6$.* *Proof.* First, let us reduce to the case where $f$ is periodic in the $x$ variable and decays rapidly in $v$ (in a qualitative sense). These properties will be needed when we find a first crossing point in our barrier argument. For large $R>0$, let $\zeta_R:\mathbb R^3\to [0,1]$ be a smooth function that equals 1 in $B_{R/2}(0)$ and 0 outside $B_{R}(0)$. Let $\mathbb T_R^3\supset B_R(0)$ be the torus of side length $2R$, and let $f^R$ be the solution to the Landau equation [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"} with initial data $$f_{\rm in}^R(x,v) = \zeta_R(x) \zeta(v) f_{\rm in}(x,v), \quad (x,v) \in \mathbb T^3_R \times \mathbb R^3.$$ By the existence theorem [@HST2019rough Theorem 1.2] these solutions exist on $[0,T_0]\times\mathbb T_R^3\times\mathbb R^3$ for some $T_0\leq T$ depending only on $\|\langle v\rangle^5 f_{\rm in}^R\|_{L^\infty(\mathbb R^6)}$, which is bounded independently of $R$. Furthermore, fixing any $t_1>0$ and any bounded domain $\Omega \subset [t_1,T_0]\times\mathbb R^6$, the lower bounds of Lemma [Lemma 5](#l:lower-bound){reference-type="ref" reference="l:lower-bound"} and the smoothing estimates of [@henderson2017smoothing Theorem 1.2] imply the family $\{f^R\}_{R\geq R_0}$ is precompact in $C^k(\Omega)$ for any $k$, for some $R_0>0$ depending on $\Omega$. Therefore, a sequence $R_j\to \infty$ of $f^R$ (extended by periodicity in $x$) converges locally uniformly on $[0,T_0]\times\mathbb R^6$ to a limit, which has initial data $f_{\rm in}$ and therefore must equal $f$ by the uniqueness theorem [@HST2019rough Theorem 1.4]. The initial data $f_{\rm in}^R$ is compactly supported, so it satisfies Gaussian decay in $v$ for any rate $\mu>0$. By [@henderson2017smoothing Theorem 3.4], these upper bounds are propagated to the time interval $[0,T_0]$: $$f(t,x,v) \leq K_{\mu,R,T_0} e^{-\mu|v|^2},$$ for some constant $K_{\mu,R,T_0}>0$. This decay estimate will be used to obtain a first crossing point, but it will not be used quantitatively. It will suffice to prove the conclusion of this proposition for $f^R$, with constants independent of $R$, up to time $T_0$. Indeed, the upper bound $f^R(t,x,v) \leq K e^{-\mu |v|^2}$ and lower bounds of Lemma [Lemma 5](#l:lower-bound){reference-type="ref" reference="l:lower-bound"} imply the solution can be extended to a larger time interval by re-applying the existence theorem [@HST2019rough Theorem 1.2], and this argument can be repeated until $f^R$ exists on the same time interval as $f$. The conclusion of the lemma can then be transferred to $f$ by taking the pointwise limit of $f^R$. For simplicity, we assume $T_0=T$ and omit the dependence on $R$ for the rest of the proof. Throughout this proof, we use the shorthand $\|f\|_{L^\infty} = \|f\|_{L^\infty([0,T]\times\mathbb R^6)}$. Define the barrier $$h(v) = K e^{-\mu|v|^2},$$ with $K$ and $\mu$ as in the statement of the lemma. By construction, the inequality $f<h$ holds at $t=0$. If $f< h$ does not hold in all of $[0,T]\times\mathbb R^6$, we claim there is a point $(t_0,x_0,v_0)$ with $t_0>0$ where $f=h$ for the first time. As discussed at the beginning of this proof, we can assume $f$ is spatially periodic and decays faster than any Gaussian, so continuity in time guarantees the existence of such a point. At the crossing point, since $h$ is constant in $t$ and $x$, we have $$\partial_t f \geq 0, \quad \nabla_x f = 0, \quad D^2_v f \leq D_v^2 h,$$ as well as $$\label{e:less-than-barrier} f(t_0,x_0,v) \leq h(v), \quad v\in \mathbb R^3.$$ From the equation, we then have $$\label{e:crossing-eqn} 0 \leq (\partial_t + v\cdot\nabla_x) f = \mbox{tr}(\bar a^f D_v^2 f) + f^2 \leq \mbox{tr}(\bar a^f D_v^2 h) +\bar c^f h,$$ at $(t_0,x_0,v_0)$, by the positive-definiteness of $\bar a^f$. Our goal is to show the right side of [\[e:crossing-eqn\]](#e:crossing-eqn){reference-type="eqref" reference="e:crossing-eqn"} is negative. We begin by bounding the term $\mbox{tr}(\bar a^f D_v^2 h)$ from above by a negative quantity. To to this, we would like to use the anisotropic upper and lower bounds for the quadratic form $e\mapsto e\cdot (\bar a^f e)$ given by Lemma [\[l:a-lower\]](#l:a-lower){reference-type="ref" reference="l:a-lower"}, so we write $D^2 h(v)$ as a sum of two terms, the first acting on vectors parallel to $v$, and the second acting on vectors perpendicular to $v$. By direct calculation, $$D_v^2 h(v) = 2\mu h(2\mu v\otimes v - I) = 2\mu h( (2\mu - |v|^{-2}) v\otimes v - (I - |v|^{-2} v\otimes v)).$$ Using the positive definiteness of $\bar a^f$, we then have $$\label{e:d2term} \begin{split} \mbox{tr}(\bar a^f D_v^2 h) &=2\mu h\left[ (2\mu - |v|^{-2})\mbox{tr}(\bar a^f v\otimes v) - \mbox{tr}(\bar a^f(I - v\otimes v/|v|^2))\right]\\ &\leq 2\mu h\left[ 2\mu\, \mbox{tr}(\bar a^f v\otimes v) - \mbox{tr}(\bar a^f(I - v\otimes v/|v|^2))\right] \end{split}$$ By direct calculation (see, e.g. [@cameron2017landau Lemma 2.1]) $$\mbox{tr}\left[\Pi(v-z) (v\otimes v)\right] = \mbox{tr}\left[\left(I - \frac{(v-z)\otimes (v-z)}{|v-z|^2} \right)(v\otimes v)\right] = |v|^2 |z|^2\sin^2 \theta_{z,v} |v-z|^{-2},$$ where $\theta_{v,z}$ is the angle between $v$ and $z$. Therefore, $$\begin{split} \mbox{tr}(\bar a^f v\otimes v) &= a_\gamma \int_{\mathbb R^3} \Pi(v-z) |v-z|^{\gamma+2} f(z) \, \mathrm{d}z\\ &= a_\gamma |v|^2 \int_{\mathbb R^3} |z|^2 \sin^2 \theta_{v,z} |v-z|^\gamma f(z) \, \mathrm{d}z. \end{split}$$ To bound this integral (evaluated at $v=v_0$) from above, when $z$ is close to $v_0$, we use the Gaussian upper bound $f\leq h$ as well as the bound on the mass of $f$. In more detail, let $r = |v_0|/2$. When $z\in B_r(v_0)$, one has $|v_0|\approx |z|$, $\sin\theta_{v_0,z} \leq |v_0-z|/|v_0|$, and $f(z) \leq h(z) \leq e^{-|v_0|^2/4}$, which implies $$\begin{split} |v|^2\int_{B_r(v_0)}|z|^2 &\sin^2\theta_{v_0,z} |v_0-z|^{\gamma} f(z) \, \mathrm{d}z\\ &\leq K^{1/2} e^{-\mu |v_0|^2/8} |v|^2 \int_{B_r(v_0)} |v_0-z|^{\gamma+2} f(z)^{1/2} \, \mathrm{d}z \\ &\leq K^{1/2} e^{-\mu|v_0|^2/8} |v_0|^2 \left(\int_{B_r(v_0)} |v_0-z|^{2(\gamma+2)} \, \mathrm{d}z\right)^{1/2} \left(\int_{B_r(v_0)} f(z)\, \mathrm{d}z\right)^{1/2}\\ &\lesssim K^{1/2} e^{-\mu |v_0|^2/8}|v_0|^2 r^{\gamma+7/2} \|f\|_{L^1}^{1/2}\\ &\lesssim K^{1/2} e^{-\mu|v_0|^2/8} |v_0|^{\gamma+11/2}\|f\|_{L^1}^{1/2}. \end{split}$$ Outside of $B_r(v_0)$, we use the energy bound, $\sin^2\theta_{v_0,z} \leq 1$, and $|v_0-z|\geq |v_0|/2$: $$|v_0|^2 \int_{\mathbb R^3\setminus B_r(v_0)} |z|^2 \sin^2\theta_{v_0,z} |v_0-z|^{\gamma} f(z) \, \mathrm{d}z \lesssim |v_0|^{\gamma+2} E_0 .$$ For the last term on the right in [\[e:d2term\]](#e:d2term){reference-type="eqref" reference="e:d2term"}, we use the lower bounds of Lemma [\[l:a-lower\]](#l:a-lower){reference-type="ref" reference="l:a-lower"}. Overall, we have $$\label{e:traf} \begin{split} \mbox{tr}(\bar a^f D_v^2 h) &\lesssim \mu h \left[ 2\mu \left(\sqrt K e^{-\mu |v_0|^2/8} |v_0|^{\gamma+11/2} + E_0 |v_0|^{\gamma+2}\right) - c_a |v_0|^{\gamma+2}\right]\\ &\leq \mu h |v_0|^{\gamma+2} \left[ 2\mu \sqrt K e^{-\mu |v_0|^2/8} |v_0|^{7/2}+2\mu E_0 - c_a \right]. \end{split}$$ Using the inequality $\sup_{x\geq 0} x^m e^{-\mu x^2} \lesssim \mu^{-m/2}$, followed by the general inequality [\[e:calc\]](#e:calc){reference-type="eqref" reference="e:calc"}, we have $$\begin{split} 2\mu \sqrt K e^{-\mu |v_0|^2/8} |v_0|^{7/2} &= 2\mu \sqrt K e^{-\mu |v_0|^2/16} |v_0|^{7/2} e^{-\mu|v_0|^2/16}\\ &\lesssim \mu^{-3/4} \sqrt K e^{-\mu |v_0|^2/16} \\ & \lesssim \mu^{-7/4} e^{-1/(64\mu)} \sqrt K |v_0|^{-1}. \end{split}$$ From the definition [\[e:mu-def\]](#e:mu-def){reference-type="eqref" reference="e:mu-def"} of $\mu$, we obtain $$\mu \leq \frac 1 {33 \log(K/c_0)} < \frac 1 {65\log(\sqrt K/c_0)}.$$ This implies $\mu^{5/4} e^{1/(64\mu)} \gtrsim e^{1/(65\mu)} \geq \sqrt K/c_0$, and if $|v_0|$ is large enough, we have $$|v_0| \geq \sqrt{\frac{\log 2}\mu} \gtrsim \frac{\sqrt K}{c_0 \mu^{7/4} e^{1/(64\mu)}},$$ and therefore, $$2\mu \sqrt K e^{-\mu |v_0|^2/8} |v_0|^{7/2} \lesssim c_0 \leq \frac {c_a} 3,$$ if $c_0$ in [\[e:mu-def\]](#e:mu-def){reference-type="eqref" reference="e:mu-def"} is chosen sufficiently small. Together with $\mu \leq c_a/(3E_0)$, this implies the right-hand side of [\[e:traf\]](#e:traf){reference-type="eqref" reference="e:traf"} is bounded by $$\lesssim -c_a \mu h |v_0|^{\gamma+2},$$ as desired. Our assumption that $|v_0|\geq \sqrt{\log 2/\mu}$ is justified because otherwise, we would have, since $C_1 > 1$, $$h(v_0) = K e^{-\mu |v_0|^2} > \frac 1 2 K \geq \|f\|_{L^\infty} \geq f(t_0,x_0,v_0),$$ a contradiction. Returning to [\[e:crossing-eqn\]](#e:crossing-eqn){reference-type="eqref" reference="e:crossing-eqn"}, we have shown $$0 \leq \left( - c_1 \mu |v_0|^{\gamma+2} + \bar c^f\right) h(v_0),$$ for a constant $c_1>0$ proportional to $c_a$. To bound this right-hand side, we consider the Coulomb ($\gamma = -3$) and non-Coulomb cases separately. In the Coulomb case, we have $$\label{e:contra} 0 \leq \left(-c_1 \mu |v_0|^{-1} + f(t_0,x_0,v_0)\right)h(v_0) . %\leq \left(-C_1 \mu \langle v_0 \rangle^{-1} + K e^{-\mu |v_0|^2}\right) h(v_0).$$ The following inequality for $\mu,s>0$ is easy to prove using calculus: $$\label{e:calc} s e^{-\mu s^2} \leq \frac 1 {2\mu} e^{-1/(4\mu)}.$$ Therefore, we have $$%\label{e:step1} f(t_0,x_0,v_0) \leq K e^{-\mu |v_0|^2} \leq \frac K {2\mu} e^{-1/(4\mu)} |v_0|^{-1}.$$ The function $\mu\mapsto \mu^2e^{1/(8\mu)}$ is uniformly bounded below by a positive constant $c$ on $(0,\infty)$, so if $c_0$ is chosen sufficiently small, our definition [\[e:mu-def\]](#e:mu-def){reference-type="eqref" reference="e:mu-def"} of $\mu$ implies $$\label{e:step2} K \leq c_0 e^{1/(8\mu)} < c c_1 e^{1/(8\mu)} < c_1\mu^2 e^{1/(4\mu)},$$ so that $$\label{e:step3} f(t_0,x_0,v_0) < c_1 \mu |v_0|^{-1},$$ and the right-hand side of [\[e:contra\]](#e:contra){reference-type="eqref" reference="e:contra"} is negative, implying a contradiction. In the non-Coulomb case, instead of [\[e:contra\]](#e:contra){reference-type="eqref" reference="e:contra"}, we have $$\label{e:contra2} 0 \leq \left(-c_1 \mu |v_0 |^{\gamma+2} + \bar c^f(t_0,x_0,v_0)\right) h(v_0).$$ To estimate the integral defining $\bar c^f$, we use [\[e:less-than-barrier\]](#e:less-than-barrier){reference-type="eqref" reference="e:less-than-barrier"} when $w$ is small, and the mass density bound when $w$ is large: $$\begin{split} \bar c^f(t_0,x_0,v_0) &\lesssim \int_{B_{|v_0|/2}} |w|^\gamma K e^{-\mu|v_0-w|^2} \, \mathrm{d}w + \int_{\mathbb R^3\setminus B_{|v_0|/2}} |w|^\gamma f(t_0,x_0,v_0-w) \, \mathrm{d}w\\ &\lesssim K e^{-\mu |v_0|^2/4} |v_0|^{\gamma+3} + M_0 |v_0|^\gamma. \end{split}$$ Using [\[e:calc\]](#e:calc){reference-type="eqref" reference="e:calc"}, we then have $$\begin{split} \bar c^f(t_0,x_0,v_0) &\lesssim |v_0|^\gamma \left( K e^{-\mu |v_0|^2/4} |v_0|^3 + M_0\right)\\ & \lesssim |v_0|^\gamma \left( \frac {2K} \mu e^{-1/\mu} |v_0|^2 +M_0 \right)\\ &\lesssim |v_0|^\gamma \left( \frac {c_1 \mu} 2 |v_0|^2 + M_0\right), \end{split}$$ where in the last line, we used a similar method to [\[e:step2\]](#e:step2){reference-type="eqref" reference="e:step2"} and [\[e:step3\]](#e:step3){reference-type="eqref" reference="e:step3"}. The expression inside parentheses is stricly less than $c_1 \mu |v_0|^2$ so long as $$|v_0| > \sqrt{\frac{2M_0}{c_1 \mu}},$$ which would imply the right side of [\[e:contra2\]](#e:contra2){reference-type="eqref" reference="e:contra2"} is negative, a contradiction. On the other hand if $|v_0| \leq \sqrt{2M_0/(c_1\mu)}$, then we have $$h(v_0) = K e^{-\mu |v_0|^2} \geq K e^{-2M_0/c_1},$$ and we choose $C_1= e^{2M_0/c_1}$ in the definition of $K$, so that this quantity is strictly greater than $\|f\|_{L^\infty}$, which means a crossing cannot occur in this case either. We conclude $f< h$ on $[0,T]\times\mathbb R^6$, as desired. ◻ # Global $L^\infty$ estimate {#s:improved} In this section, we improve the local $L^\infty$ bound of Proposition [Proposition 9](#p:inf){reference-type="ref" reference="p:inf"} via scaling techniques, and incorporate the Gaussian bounds of Proposition [Proposition 10](#p:gaussian){reference-type="ref" reference="p:gaussian"} to obtain an unconditional $L^\infty$ estimate. **Theorem 11**. *Let $f:[0,T]\times\mathbb R^6\to [0,\infty)$ be a solution to the Landau equation [\[e:landau\]](#e:landau){reference-type="eqref" reference="e:landau"} satisfying the hypotheses of Proposition [Proposition 10](#p:gaussian){reference-type="ref" reference="p:gaussian"}, and assume in addition that $$\int_{\mathbb R^3} |v|^2 f(t,x,v) \, \mathrm{d}v \leq E_0, \quad (t,x) \in [0,T]\times\mathbb R^3,$$ for some $E_0>0$. Then $$\|f\|_{L^\infty([0,T]\times\mathbb R^6)}\leq C E_0^{-19\gamma/\delta},$$ for a constant $C$ depending only on $\gamma$, $\delta$, the quantities in [\[e:cond\]](#e:cond){reference-type="eqref" reference="e:cond"}, and the constants $\ell$, $\rho$, $C_0$, and $\mu'$ corresponding to the initial data.* *Proof.* First, for small values of time, the solution $f$ is bounded in $L^\infty$ by some value depending only on the initial data. This can be seen, for example, by applying the existence/uniqueness theorem of [@HST2019rough]: for some $T_*>0$ depending only on $\|\langle v\rangle^5 f_{\rm in}\|_{L^\infty(\mathbb R^6)}$, one has $\|f(t)\|_{L^\infty(\mathbb R^6)} \leq \|\langle v\rangle^5 f(t)\|_{L^\infty(\mathbb R^6)} \leq 2 \|\langle v\rangle^5 f_{\rm in}\|_{L^\infty(\mathbb R^6)} = :L_*$ whenever $t\leq T_*$. Next, let $z_0 = (t_0,x_0,v_0)\in [0,T]\times\mathbb R^6$ be chosen so that $$f(t_0,x_0,v_0) \geq \frac 1 2 \|f\|_{L^\infty([0,T]\times\mathbb R^6)}.$$ We may assume $t_0 > T_*$, since otherwise, $f$ is bounded by $L_*$ in all of $[0,T]\times\mathbb R^6$ and there is nothing left to show. Now, let $r$ be a radius to be chosen later, wth $$0 < r < \min\left\{1, \sqrt{t_0/2}\right\},$$ so that $Q_r(z_0) \subset [0,T]\times\mathbb R^6$. Define the rescaled solution $$f_r(t,x,v) = r^{5+\gamma} f(t_0+r^2 t, x_0 + r^3x + r^2 t v_0, v_0 + rv).$$ By direct calculation, $f_r$ is also a solution to the Landau equation. Applying the $L^\infty$ estimate of Proposition [Proposition 9](#p:inf){reference-type="ref" reference="p:inf"} to $f_r$, we have $$\begin{split} f(t_0,x_0,v_0) &\leq \|f\|_{L^\infty(Q_{r/2}(z_0))}\\ &= \frac{\|f_r\|_{L^\infty(Q_{1/2})}}{r^{5+\gamma}}\\ &\lesssim \left( \frac {\|\bar a^{f_r}\|_{L^\infty(Q_1)}} {\lambda[f_r]}(1+ \|\bar a^{f_r}\|_{L^\infty(Q_1)} + \|\bar b^{f_r}\|_{L^\infty(Q_1)} + \|\bar c^{f_r}\|_{L^\infty(Q_1)})\right)^{19/2} \frac {\|f_r\|_{L^2(Q_1)}} {r^{5+\gamma}} . \end{split}$$ Note that the coefficients appearing in this right-hand side are defined in terms of $f_r$, and $\lambda[f_r]$ is the lower ellipticity constant corresponding to $\bar a^{f_r}$. Calculating these coefficients in terms of $f$, we have $$\begin{split} \bar a^{f_r}(t,x,v) &= \bar a^f(t_0+r^2 t, x_0 + r^3x + r^2 t v_0, v_0 + rv),\\ \bar b^{f_r}(t,x,v) &= r \bar b^f(t_0+r^2 t, x_0 + r^3x + r^2 t v_0, v_0 + rv),\\ \bar c^{f_r}(t,x,v) &= r^2 \bar c^f(t_0+r^2 t, x_0 + r^3x + r^2 t v_0, v_0 + rv), \end{split}$$ and from Lemma [\[l:a-lower\]](#l:a-lower){reference-type="ref" reference="l:a-lower"}, $e\cdot (a^{f_r}(t,x,v) e) \geq \lambda[f_r] \approx (1+|v_0|)^\gamma$ for all $e\in \mathbb S^2$. This yields $$\begin{split} &f(t_0,x_0,v_0)\\ &\lesssim \left( \frac {\|\bar a^{f}\|_{L^\infty(Q_r(z_0))}} {(1+|v_0|)^\gamma}(1+ \|\bar a^{f}\|_{L^\infty(Q_r(z_0))} + r \|\bar b^{f}\|_{L^\infty(Q_r(z_0))} + r^2\|\bar c^{f}\|_{L^\infty(Q_r(z_0))})\right)^{19/2} \frac {\|f_r\|_{L^2(Q_1)}} {r^{5+\gamma}}. \end{split}$$ We analyze the $L^2$ norm on the right as follows: $$\frac{\|f_r\|_{L^2(Q_1)}}{r^{5+\gamma}} \leq \frac{\|f_r\|_{L^\infty_{t,x} L^2_v(Q_1)}}{r^{5+\gamma}} = r^{-3/2} \|f\|_{L^\infty_{t,x} L^2_v(Q_r(z_0))},$$ from the definition of $f_r$. For brevity, let $L_0 = \|f\|_{L^\infty([0,T]\times\mathbb R^6)}$. We can assume $L_0 > 1$ without loss of generality. With this notation, and incorporating the coefficient estimates from Lemma [Lemma 3](#l:coeffs){reference-type="ref" reference="l:coeffs"}, we have $$f(t_0,x_0,v_0) \lesssim (1+|v_0|)^{-19\gamma/2} \left( 1 + rL_0^{1-p(\gamma+4)/3} + r^2 L_0^{1-p(\gamma+3)/3}\right)^{19/2} r^{-3/2} \|f\|_{L^\infty_{t,x} L^2_v(Q_r(z_0))}$$ The optimal scale $r$ is chosen so that the terms inside the parentheses balance: $$r = L_0^{-p/3} \min\left\{1,\sqrt{t_0/2}\right\},$$ and the estimate becomes $$\label{e:19} \begin{split} f(t_0,x_0,v_0) & \lesssim (1+|v_0|)^{-19\gamma/2} \left(1 + L_0^{1-p(\gamma+5)/3}\right)^{19/2} L_0^{p/2} t_0^{-3/4} \|f\|_{L^\infty_{t,x} L^2_v(Q_r(z_0))}\\ & \lesssim (1+|v_0|)^{-19\gamma/2} t_0^{-3/4} L_0^{p/2} \|f\|_{L^\infty_{t,x} L^2_v(Q_r)}, \end{split}$$ where we used $p = \dfrac 3 {5+\gamma}$. Since $t_0> T_*$, and $T_*$ depends only on the initial data, we absorb $t_0^{-3/4}$ into the implied constant. The remainder of the proof proceeds in two cases, depending on whether $|v_0|$ is small or large. Case 1: $|v_0|\leq 2$. Interpolating between $L^\infty$ and $L^{p+\delta}$ (since we can choose $\delta$ small enough that $p+\delta < 2$), we have $$\|f\|_{L^\infty_{t,x}L^2_v(Q_r(z_0))} \leq \|f\|_{L^\infty(Q_r(z_0))}^{(2-p-\delta)/2} \|f\|_{L^\infty_{t,x} L^{p+\delta}_v(Q_r(z_0))}^{(p+\delta)/2} \lesssim L_0^{(2-p-\delta)/2},$$ with implied constant depending on the $L^{p+\delta}$ bound for $f$. We now have $$f(t_0,x_0,v_0) \lesssim L_0^{p/2} L_0^{(2-p-\delta)/2} = L_0^{1-\delta/2}.$$ Since $(t_0,x_0,v_0)$ was chosen so that $f(t_0,x_0,v_0) \geq L_0/2$, this implies $L_0^{\delta/2}$ is bounded above by a constant depending only on $\delta$, the initial data, and the $L^\infty_{t,x} L^1_v$ and $L^\infty_{t.x} L^p_v$ norms of $f$. Case 2: $|v_0|> 2$. In this case, to obtain an upper bound that is independent of $|v_0|$, we need to use the Gaussian decay of $f$. Interpolating as above, and applying the Gaussian upper bound from Proposition [Proposition 10](#p:gaussian){reference-type="ref" reference="p:gaussian"}, we obtain $$\begin{split} \|f\|_{L^\infty_{t,x}L^2_v(Q_r(z_0))} &\leq \|f\|_{L^\infty(Q_r(z_0))}^{(2-p-\delta)/2} \|f\|_{L^\infty_{t,x} L^{p+\delta}_v(Q_r(z_0))}^{(p+\delta)/2} \\ &\lesssim \left( L_0 e^{-\mu |v_0|^2/4} \right)^{(2-p-\delta)/2}. \end{split}$$ with constant depending on the $L^{p+\delta}$ bound for $f$. Returning to [\[e:19\]](#e:19){reference-type="eqref" reference="e:19"}, this gives $$f(t_0,x_0,v_0) %\lesssim |v_0|^{-19\gamma/2} L_0^{p/2} L_0^{(2-p-\delta)/2} e^{-(2-p-\delta)\mu|v_0|^2/8} \lesssim |v_0|^{-19\gamma/2} L_0^{(2 - \delta)/2} e^{-(2-p-\delta)\mu|v_0|^2/8}.$$ Using the inequality $x^m e^{-\mu x^2}\lesssim \mu^{-m/2}$, we then have $$f(t_0,x_0,v_0) \lesssim L_0^{(2-\delta)/2} \mu^{19\gamma/4}.$$ Recalling the definition of $\mu$ in Proposition [Proposition 10](#p:gaussian){reference-type="ref" reference="p:gaussian"}, we may assume $\mu < \mu'/2$ since otherwise, $\mu$ is independent of $E_0$ and $L_0$, and the current proof is easier. We then have $$\mu = \min\left\{\frac {c_0}{E_0} , \frac 1 {33\log(K/c_0)}\right\} \gtrsim \frac 1 {E_0 \log(K)},$$ since we can assume $E_0, K \gtrsim 1$. Similarly, we may assume $K \lesssim L_0$ since the other case, where $K$ is determined only by the initial data, is simpler. We then have $$\mu \gtrsim \frac 1 {E_0 \log(L_0)} \gtrsim \frac{L_0^{-\delta/(19\gamma)}} {E_0},$$ which yields, since $\gamma <0$, $$f(t_0,x_0,v_0) \lesssim L_0^{(2-\delta)/2} E_0^{-19\gamma/4} L_0^{\delta/4} = L_0^{1-\delta/4} E_0^{-19\gamma/4}.$$ By the choice of $(t_0,x_0,v_0)$, this implies $$L_0 \leq C E_0^{-19\gamma/\delta},$$ for a constant $C>0$ depending only on $\delta$, the initial data, and the constants in [\[e:cond\]](#e:cond){reference-type="eqref" reference="e:cond"}. ◻ # Bound for the energy density {#s:energy} In this last section, we show that the upper bound on the energy density in the above estimates can be replaced by a bound on the $s$ moment for some small $s>0$. **Proposition 12**. *Let $f$ be a solution of the Landau equation on $[0,T]\times\mathbb R^6$ satisfying the hypotheses of Theorem [Theorem 1](#t:upper){reference-type="ref" reference="t:upper"}. Let $$E_0 = \sup_{t,x} \int_{\mathbb R^3} |v|^2 f(t,x,v)\, \mathrm{d}v,$$ and let $s\in (0,2)$ be arbitrary.* *Then $E_0$ is bounded above by a constant depending only on $\gamma$, $\delta$, $s$, the initial data, the $L^\infty_{t,x} L^1_v$ and $L^\infty_{t,x}L^{p+\delta}_v$ norms of $f$, and $$\sup_{t,x} \int_{\mathbb R^3} |v|^s f(t,x,v) \, \mathrm{d}v.$$* *Proof.* Combining the $L^\infty$ bound of Theorem [Theorem 11](#t:inf){reference-type="ref" reference="t:inf"} with the Gaussian decay estimate of Proposition [Proposition 10](#p:gaussian){reference-type="ref" reference="p:gaussian"}, we obtain $$\label{e:good-gaussian} f(t,x,v) \leq K e^{-\mu |v|^2}.$$ As in the proof of Theorem [Theorem 11](#t:inf){reference-type="ref" reference="t:inf"}, we can assume $K\lesssim \|f\|_{L^\infty([0,T]\times\mathbb R^6)} \leq C E_0^{-19\gamma/\delta}$, since otherwise $K$ is independent of $E_0$, and the proof becomes simpler. Similarly, for $\mu$ we may assume $$\mu = \min\left\{\frac {c_0} {E_0}, \frac {1} {33\log(K/c_0)}\right\} \gtrsim \frac {1} {E_0},$$ since $\log(K/c_0) \approx \log(E_0) \lesssim E_0$. Here, the implied constants depend only on the quantities in the statement of the lemma. Let $\theta =-19\gamma/\delta$. For any $s\in (0,2)$ and $q>1$, estimate [\[e:good-gaussian\]](#e:good-gaussian){reference-type="eqref" reference="e:good-gaussian"} and Hölder's inequality imply, with $q' = q/(q-1)$, $$\begin{split} \int_{\mathbb R^3} |v|^2 f(t,x,v) \, \mathrm{d}v &\leq K^{1/q} \int_{\mathbb R^3} |v|^{s/q'} |v|^{2-s/q'} f(t,x,v)^{q'} e^{-\mu|v|^2/q} \, \mathrm{d}v\\ &\leq K^{1/q} \left(\int_{\mathbb R^3} |v|^{s} f(t,x,v) \, \mathrm{d}v\right)^{1/q'} \left( \int_{\mathbb R^3} e^{-\mu|v|^2} |v|^{2q - s (q-1)} \, \mathrm{d}v\right)^{1/q}\\ &\lesssim E_0^{\theta/q} \|f\|_{L^\infty_{t,x}(L^1_{s})_v}^{1/q'} \mu^{-1+s/2 - (s+3)/(2q) } C_{q,s}. %&\lesssim C^{1/s} \|f\|_{L^1_q}^{(s/q)(1/2+\theta/s + (m-1)/(2s))} \|f\|_{L^1_{m-1}}^{1/s'} C_s.%\\ %&\lesssim C \|f\|_{L^1_q}^{s/(2s) + (s/q)(-19\gamma/(s s) + 1/(2s'))} \|f\|_{L^1_{q-1}}^{1/s'} C_s. \end{split}$$ Here, we use the notation $\|f\|_{L^\infty_{t,x}(L^1_s)_v} = \sup_{t,x}\int_{\mathbb R^3} |v|^s f(t,x,v)\, \mathrm{d}v$, as well as $$C_{q,s} = \left(\int_{\mathbb R^3} e^{-|w|^2} |w|^{2q - s(q-1)} \, \mathrm{d}w\right)^{1/q}, % = \left(C_d \Gamma\left(\frac{1+m+s} 2\right)\right)^{1/s} \lesssim s^{1/2},$$ which depends only on $s$ and $q$. With $\mu \gtrsim E_0^{-1}$, we have $$\int_{\mathbb R^3} |v|^2 f(t,x,v) \, \mathrm{d}v \lesssim C_{q,s} E_0^{\theta/q + 1 - s/2 +(s+3)/(2q)} \|f\|_{L^\infty_{t,x}(L^1_s)_v}^{1/q'},$$ so we choose $$q = \frac 2 s \left( 2\theta + s + 3\right),$$ so that $\theta/q+ (s+3)/(2q) = s/4$, and the exponent of $E_0$ becomes $1 - s/4$. Taking the supremum over $t$ and $x$, we have $$E_0 \lesssim E_0^{1 - s/4} \|f\|_{L^\infty_{t,x}(L^1_s)_v}^{1/q'},$$ or $$E_0 \lesssim \|f\|_{L^\infty_{t,x}(L^1_s)_v}^{4/(s q')}.$$ ◻ Combining Theorem [Theorem 11](#t:inf){reference-type="ref" reference="t:inf"} with Proposition [Proposition 12](#p:E){reference-type="ref" reference="p:E"}, we obtain Theorem [Theorem 1](#t:upper){reference-type="ref" reference="t:upper"}. [^1]: It turns out to be more convenient to allow this constant to depend on $\|f\|_{L^\infty}$ rather than $\|f\|_{L^q_v}$. The next two steps of the argument will remove the dependence on the $L^\infty$ norm.

arxiv_math

--- abstract: | In this paper, we study the approximation problem for functions in the Gaussian-weighted Sobolev space $W^\alpha_p(\mathbb{R}^d, \gamma)$ of mixed smoothness $\alpha \in \mathbb{N}$ with error measured in the Gaussian-weighted space $L_q(\mathbb{R}^d, \gamma)$. We obtain the exact asymptotic order of pseudo $s$-numbers for the cases $1 \leq q< p < \infty$ and $p=q=2$. Additionally, we also obtain an upper bound and a lower bound for pseudo $s$-numbers of the embedding of $W^\alpha_2(\mathbb{R}^d, \gamma)$ into $L_{\infty}^{\sqrt{g}}({\mathbb R}^d)$. Our result is an extension of that obtained in Dinh Dũng and Van Kien Nguyen (IMA Journal of Numerical Analysis, 2023) for approximation and Kolmogorov numbers. **Keywords and Phrases**: Gaussian-weighted Sobolev space of mixed smoothness; pseudo $s$-numbers; Asymptotic order of convergence. **MSC (2020)**: 41A25; 41A46, 46E35 author: - Van Kien Nguyen bibliography: - AllBib.bib --- # Introduction {#Introduction} Recently there is an increasing interest in the study of multivariate numerical integration and approximation in the context of Gaussian-weighted Sobolev spaces of mixed smoothness [@IKLP2015; @IL2015; @DILP18; @DN23]. This is motivated by the fact that a number of real-world problems in finance, physics, quantum chemistry and machine learning are modeled on function spaces on high-dimensional domains, and often the functions to be integrated or approximated have some Sobolev regularity equipped Gaussian measure. In recent paper [@DN23], Dinh Dũng and the author studied the linear approximation and sampling recovery for functions from Gaussian-weighted Sobolev spaces $W^\alpha_p({\mathbb R}^d,\gamma)$ in the Gaussian-weighted space $L_q({\mathbb R}^d,\gamma)$. The asymptotic optimality in terms of the Kolmogorov, linear and sampling numbers of the approximation was obtained for the cases $1\le q < p <\infty$ and $p=q=2$. The problem of multivariate numerical integration for functions in $W^\alpha_p({\mathbb R}^d,\gamma)$ has been studied in [@IKLP2015; @IL2015; @DILP18; @DN23]. An asymptotically optimal quadrature of numerical integration was constructed recently in [@DN23] based on an assembling method. In the present paper we deal with the approximation problem. We study the asymptotic behavior of pseudo $s$-numbers of the embedding of $W^\alpha_p(\mathbb{R}^d, \gamma)$ into $L_q(\mathbb{R}^d, \gamma)$. Our approach is based on the decomposition technique developed from [@DN23] and on tools from the theory of pseudo $s$-numbers, see [@Pie80B]. We show that in the cases $1 \leq q< p < \infty$ and $p=q=2$, the exact asymptotic order of pseudo $s$-numbers can be obtained. In particular, if $1\leq q<p<\infty$, for injective and additive pseudo $s$-numbers we get $$s_n\big(I_\gamma:W^\alpha_p({\mathbb R}^d,\gamma)\to L_q({\mathbb R}^d,\gamma)\big) \asymp s_n\big(I:\tilde{W}_p^\alpha({\mathbb I}^d) \to \tilde{L}_q({\mathbb I}^d)\big),\ \ n\to \infty\,.$$ Here $I$ is the embedding from periodic Sobolev space with mixed smoothness $\tilde{W}_p^\alpha({\mathbb I}^d)$ into the Lebesgue space $\tilde{L}_q({\mathbb I}^d)$ on the torus ${\mathbb I}^d:=\big[-\frac{1}{2},\frac{1}{2}\big]^d$. Additionally, in this paper we also obtain an upper bound and a lower bound for asymptotic behavior of pseudo $s$-numbers of the embedding of $W^\alpha_2(\mathbb{R}^d, \gamma)$ into $L_{\infty}^{\sqrt{g}}({\mathbb R}^d)$. Note here that we do not have a continuous embedding of $W^\alpha_p({\mathbb R}^d,\gamma)$ into $L_\infty({\mathbb R}^d,\gamma)$ if $1\leq p< \infty$. The space $L_{\infty}^{\sqrt{g}}({\mathbb R}^d)$ is the collection of all functions $f$ on ${\mathbb R}^d$ such that $\big|f({\boldsymbol{x}})\sqrt{g({\boldsymbol{x}})}\big|$ is bounded. Here $g({\boldsymbol{x}})$ is the density of the standard Gaussian measure on ${\mathbb R}^d$. In this context we obtain $$\begin{aligned} n^{-\frac{\alpha}{2}-\frac{d}{4}} (\log n)^{(\frac{\alpha}{2}+\frac{d}{4})(d-1)} & \ll s_n\big(I_\gamma: W_2^\alpha({\mathbb R}^d,\gamma) \to L_{\infty}^{\sqrt{g}}({\mathbb R}^d)\big) \\ &\ll n^{-\frac{\alpha}{2}-\frac{1}{12}+\frac{1}{2}}(\log n)^{(\frac{\alpha}{2}+\frac{1}{12})(d-1)}\,,\qquad \ n\to \infty. \end{aligned}$$ The paper is organized as follows. In Section [2](#sec-pseudo){reference-type="ref" reference="sec-pseudo"}, we recall the notion of pseudo $s$-numbers and some particular pseudo $s$-numbers. Section [3](#Approximation){reference-type="ref" reference="Approximation"} is devoted to the proof of the asymptotic order of pseudo $s$-numbers of the embedding $I_\gamma$ for the case $1\leq q < p<\infty$. In Section [4](#sec-p=2){reference-type="ref" reference="sec-p=2"}, we study approximation problem for functions in $W_2^\alpha({\mathbb R}^d,\gamma)$ with error measured in $L_2({\mathbb R}^d,\gamma)$ or $L_\infty^{\sqrt{g}}({\mathbb R}^d)$. **Notation.** The letter $d$ is always reserved for the underlying dimension of ${\mathbb R}^d$, ${\mathbb N}^d$, etc. Vectors in ${\mathbb R}^d$ are denoted by boldface letters. For ${\boldsymbol{x}}\in {\mathbb R}^d$, $x_i$ denotes the $i$th coordinate, i.e., ${\boldsymbol{x}}:= (x_1,\ldots, x_d)$. For a real number $a$ we denote by $\lfloor a \rfloor$ the greatest integer not larger than $a$. For the quantities $A_n$ and $B_n$ depending on $n$ in an index set $J$ we write $A_n \ll B_n$ if there exists some constant $C >0$ independent of $n$ such that $A_n \leq CB_n$ for all $n \in J$, and $A_n \asymp B_n$ if $A_n \ll B_n$ and $B_n \ll A_n$. General positive constant (may depend on parameters) is denoted by $C$. Values of constant $C$ is not specified and may be different in various places. # Pseudo s-numbers {#sec-pseudo} Let us first recall the definition of pseudo $s$-numbers following Pietsch [@Pie80B Section 12.1.1]. In the paper [@Pie74], see also [@Pie80B Chapter 11], Pietsch introduced the notion of $s$-number. However, dyadic entropy numbers, see the definition below, do not belong to the class of $s$-numbers. To incorporate also dyadic entropy numbers into the framework, Pietsch introduced the notion of pseudo $s$-numbers. Let $X,Y,X_0,Y_0$ be Banach spaces. A pseudo $s$-number is a map $s$ assigning to every linear operator $T\in \mathcal L(X,Y)$ a scalar sequence $(s_n(T))_{n\in {\mathbb N}}$ such that the following conditions are satisfied: 1. $\|T\|=s_1(T)\geq s_2(T)\geq\ldots\geq 0$; 2. $s_{n}(S+T)\leq s_n(S)+ \|T\|$ for all $S\in \mathcal L(X,Y)$ and $m,n\in {\mathbb N}\,$; 3. $s_n(BTA)\leq \|B\| \, \cdot \, s_n(T) \, \cdot \, \|A\|$ for all $A\in \mathcal L(X_0,X)$, $B\in \mathcal L(Y,Y_0)$ . A pseudo $s$-number is called additive if it satisfies 1. $s_{n+m-1}(S+T)\leq s_n(S)+ s_m(T)$ for all $S\in \mathcal L(X,Y)$ and $m,n\in {\mathbb N}\,$. Some of the popular pseudo $s$-numbers are listed below: 1. The $n$th approximation number of the linear operator $T$ is defined as $$a_n(T):=\inf\{\|T-A\|: \ A\in \mathcal L(X,Y),\ \ \text{rank} (A)<n\}\, , \qquad n \in {\mathbb N}\, .$$ 2. The $n$th Kolmogorov number of the linear operator $T \in \mathcal L(X,Y)$ is defined as $$d_n(T)= \inf_{L_{n-1}}\sup_{\|x\|_X\leq 1}\inf_{y\in L_{n-1}}\|Tx-y\|_Y. \label{def1}$$ Here the outer supremum is taken over all linear subspaces $L_{n-1}$ of dimension ($n-1$) in $Y$. 3. The $n$th Gelfand number of the linear operator $T \in \mathcal L(X,Y)$ is defined as $$c_n (T) := \inf\Big\{\|\, T\, J_M^X\, \|: \ {\rm codim\,}(M)< n\Big\},$$ where $J_M^X:M\to X$ refers to the canonical injection of $M$ into $X$. 4. The $n$th Weyl number of $T$ is defined as $$x_n(T):=\sup\{a_n(TA):\ A\in \mathcal L(\ell_2,X),\ \|A\|\leq 1\}\, , \qquad n \in {\mathbb N}\, .$$ 5. The $n$th (dyadic) entropy number of $T$ is defined as $$e_n (T):=\inf\{ \varepsilon >0: T(B_X) \text{ can be covered by } 2^{n-1} \text{ balls in } Y \text{ of radius } \varepsilon\}\, ,$$ where $B_X:= \{x \in X: \: \|x\|_X \le 1\}$ denotes the closed unit ball of $X$. 6. The $n$th Bernstein number of the operator $T\in {\mathcal L}(X,Y)$ is defined as $$b_n(T)= \sup_{L_n}\inf_{\substack{x\in L_n \\ x\not =0}} \dfrac{\|Tx\|_Y}{\| x\|_X} ,$$ where the supremum is taken over all subspaces $L_n$ of $X$ with dimension $n$. Note that approximation, Kolmogorov, Gelfand, Weyl and entropy numbers are additive pseudo $s$-numbers, see [@Pie80B Chapters 11 and 12]. Bernstein number is not an additive pseudo $s$-number [@Pie08]. It is well-known that approximation number is the largest pseudo $s$-number in the set $\{a,b,c,d,x\}$, see [@Pie74; @CaSt90B; @Pie80B]. We also have the following inequalities $$\label{eq-inequality1} b_n(T)\leq \min\{c_n(T),d_n(T)\},$$ see [@Pie74] and $$\label{eq-inequality2} e_n(T)\leq a_n(T),\ \ \ b_n(T)\leq 2\sqrt{2}e_n(T).$$ The first inequality in [\[eq-inequality2\]](#eq-inequality2){reference-type="eqref" reference="eq-inequality2"} can be found in [@CaSt90B], the second one was proved in [@Ng16]. Moreover, if $x_n(T)\asymp n^{-a}(\log n)^{b}$ with $a>0,\ b\geq 0$ then $$\label{eq-inequality3} b_n(T)\leq Cx_n(T)$$ which is a consequence of [@Pie08 Lemma 2]. For a proof, we refer the reader to [@Ng15 Corollary 3.2]. Similarly, if $e_n(T)\asymp n^{-a}(\log n)^{b}$ we get $$\label{eq-inequality4} e_n(T)\leq Cd_n(T),$$ see [@Tem98]. We recall that a metric injection $J$ of a Banach space $Y$ into a Banach space $\tilde{Y}$ is characterized by the property $$\|Jy\|_{\tilde{Y}}=\|y\|_Y,\ \text{for all } y\in Y.$$ A pseudo $s-$number is called weakly injective if, given any metric injection $J\in {\mathcal L}(Y,\tilde{Y})$ we have $$\label{eq-injective} s_n(T)\asymp s_n(JT).$$ In case $s_n(T)= s_n(JT)$ it is called injective. We know that Gelfand, Bernstein, and Weyl numbers are injective pseudo $s$-numbers, while entropy number is weakly injective. Approximation and Kolmogorov numbers are not injective. # The case $1\leq q<p<\infty$ {#Approximation} In this section, we study the asymptotic behavior of pseudo $s$-numbers of the embedding of Gaussian-weighted Sobolev spaces $W^\alpha_p(\mathbb{R}^d, \gamma)$ of mixed smoothness $\alpha \in \mathbb{N}$ into the Gaussian-weighted space $L_q(\mathbb{R}^d, \gamma)$ in the case $1 \leq q< p < \infty$. Let us first introduce Gaussian-weighted Sobolev spaces of mixed smoothness. Denote $$g({\boldsymbol{x}}):=(2\pi)^{-d/2} \exp\left(-|{\boldsymbol{x}}|^2/2\right),\ \ {\boldsymbol{x}}\in {\mathbb R}^d$$ and $\gamma ({\rm d}{\boldsymbol{x}}) := g({\boldsymbol{x}}) {\rm d}{\boldsymbol{x}}$ the $d$-dimensional standard Gaussian measure on ${\mathbb R}^d$ with the density $g({\boldsymbol{x}})$. For $1\leq p<\infty$ and a Lebesgue measurable set $\Omega\subset{\mathbb R}^d$, we define the Gaussian-weighted space $L_p(\Omega,\gamma)$ to be the set of all functions $f$ on $\Omega$ such that the norm $$\|f\|_{L_p(\Omega,\gamma)} : = \bigg( \int_\Omega |f({\boldsymbol{x}})|^p \gamma({\rm d}{\boldsymbol{x}})\bigg)^{1/p} = \bigg( \int_\Omega |f({\boldsymbol{x}})|^p g({\boldsymbol{x}}) {\rm d}{\boldsymbol{x}}\bigg)^{1/p} \ $$ is finite. For $\alpha \in {\mathbb N}$, we define the Gaussian-weighted space $W^\alpha_p(\Omega,\gamma)$ to be the normed space of all functions $f\in L_p(\Omega,\gamma)$ such that the generalized partial derivative $D^{\boldsymbol{r}}f$ of order ${\boldsymbol{r}}$ belongs to $L_p(\Omega,\gamma)$ for all ${\boldsymbol{r}}\in {\mathbb N}_0^d$ satisfying $|{\boldsymbol{r}}|_\infty\leq \alpha$. The norm of a function $f$ in this space is defined by $$\begin{aligned} \label{W-Omega} \|f\|_{W^\alpha_p(\Omega,\gamma)}: = \Bigg(\sum_{|{\boldsymbol{r}}|_\infty \leq \alpha} \|D^{\boldsymbol{r}}f\|_{L_p(\Omega,\gamma)}^p\Bigg)^{1/p}. \end{aligned}$$ The space $W^\alpha_p(\Omega)$ is defined as the classical Sobolev space with mixed smoothness by replacing $L_p(\Omega,\gamma)$ with $L_p(\Omega)$ in [\[W-Omega\]](#W-Omega){reference-type="eqref" reference="W-Omega"}, where as usual, $L_p(\Omega)$ denotes the Lebesgue space of functions on $\Omega$ equipped with the usual $p$-integral norm. For a fixed $\theta>1$ we denote the $d$-cube ${\mathbb I}^d_\theta$ by ${\mathbb I}^d_\theta := \big[-\frac{\theta}{2}, \frac{\theta}{2}\big]^d$, ${\mathbb I}^d_{\theta,{\boldsymbol{k}}}:={\boldsymbol{k}}+{\mathbb I}^d_\theta$ for ${\boldsymbol{k}}\in {\mathbb Z}^d$, and $f_{\theta,{\boldsymbol{k}}}$ the restriction of $f$ on ${\mathbb I}^d_{\theta,{\boldsymbol{k}}}$ for a function $f$ on ${\mathbb R}^d$. Let $(\varphi_{\boldsymbol{k}})_{{\boldsymbol{k}}\in {\mathbb Z}^d}$ be a partition of unity on ${\mathbb R}^d$ satisfying - $\varphi_{\boldsymbol{k}}\in C^\infty_0({\mathbb R}^d)$ and $0 \le \varphi_{\boldsymbol{k}}({\boldsymbol{x}})\le 1$,   ${\boldsymbol{x}}\in {\mathbb R}^d$,   ${\boldsymbol{k}}\in {\mathbb Z}^d$; - the support of $\varphi_{\boldsymbol{k}}$ is contained in the interior of ${\mathbb I}^d_{\theta,{\boldsymbol{k}}}$, ${\boldsymbol{k}}\in {\mathbb Z}^d$; - $\sum_{{\boldsymbol{k}}\in {\mathbb Z}^d}\varphi_{\boldsymbol{k}}({\boldsymbol{x}})= 1$,   ${\boldsymbol{x}}\in {\mathbb R}^d$; - $\left\|{\varphi_{\boldsymbol{k}}}\right\|_{W^\alpha_p({\mathbb I}^d_{\theta,{\boldsymbol{k}}})} \le C_{\alpha,d,\theta}$,   ${\boldsymbol{k}}\in {\mathbb Z}^d$. For a construction of such a system, we refer the reader to [@Stein1970 Chapter VI, 1.3]. Observe that if $f \in W^\alpha_p({\mathbb R}^d,\gamma)$, then we have $$\label{f_theta,bk} f_{\theta,{\boldsymbol{k}}}(\cdot+{\boldsymbol{k}})\varphi_{\boldsymbol{k}}(\cdot+{\boldsymbol{k}}) \in {W}^\alpha_p({\mathbb I}^d_{\theta}),$$ and it holds from algebra property of ${W}^\alpha_p({\mathbb I}^d_\theta)$, see [@NgS17], that $$\label{eq-first} \begin{aligned} \|f_{\theta,{\boldsymbol{k}}}(\cdot+{\boldsymbol{k}})\varphi_{\boldsymbol{k}}(\cdot+{\boldsymbol{k}})\|_{{W}^\alpha_p({\mathbb I}^d_\theta)} & \leq C \|f_{\theta,{\boldsymbol{k}}}(\cdot+{\boldsymbol{k}})\|_{{W}^\alpha_p({\mathbb I}^d_\theta)} \cdot \|\varphi_{{\boldsymbol{k}}}(\cdot+{\boldsymbol{k}})\|_{ {W}^\alpha_p({\mathbb I}^d_\theta)} \\ & = C \|f_{\theta,{\boldsymbol{k}}}(\cdot+{\boldsymbol{k}})\|_{{W}^\alpha_p({\mathbb I}^d_\theta)} \cdot \|\varphi_{{\boldsymbol{k}}} \|_{ {W}^\alpha_p({\mathbb I}^d_{\theta,{\boldsymbol{k}}})} \\ &\leq C\|f_{\theta,{\boldsymbol{k}}}(\cdot+{\boldsymbol{k}})\|_{{W}^\alpha_p({\mathbb I}^d_\theta)}. \end{aligned}$$ Here in the last estimate we used property (iv) of the system $(\varphi_{\boldsymbol{k}})_{{\boldsymbol{k}}\in {\mathbb N}_0^d}$. Note, that when ${\boldsymbol{x}}\in {\mathbb I}^d_{\theta}$ we have $e^{\frac{|{\boldsymbol{x}}+{\boldsymbol{k}}|^2}{2}}\leq e^{\frac{|{\boldsymbol{k}}+ \theta({\rm sign\,}{\boldsymbol{k}})/2 |^2}{2}}$, where ${\rm sign\,}{\boldsymbol{k}}:= \left({\rm sign\,}k_1, \ldots , {\rm sign\,}k_d\right)$ and ${\rm sign\,}x := 1$ if $x \ge 0$, and ${\rm sign\,}x := -1$ otherwise for $x \in {\mathbb R}$. Therefore $$\begin{aligned} \|f_{\theta,{\boldsymbol{k}}}(\cdot+{\boldsymbol{k}})\|_{{W}^\alpha_p({\mathbb I}^d_\theta)} &= \Bigg(\sum_{|{\boldsymbol{r}}|_\infty \leq \alpha} \int_{{\mathbb I}^d_\theta} |D^{\boldsymbol{r}}f_{\boldsymbol{k}}({\boldsymbol{x}}+{\boldsymbol{k}})| ^p {\rm d}{\boldsymbol{x}}\Bigg)^{1/p} \\ &= \Bigg(\sum_{|{\boldsymbol{r}}|_\infty \leq \alpha} (2\pi)^{d/2} \int_{{\mathbb I}^d_\theta}e^{\frac{|{\boldsymbol{x}}+{\boldsymbol{k}}|^2}{2}}|D^{\boldsymbol{r}}f_{\boldsymbol{k}}({\boldsymbol{x}}+{\boldsymbol{k}})| ^pg({\boldsymbol{x}}+{\boldsymbol{k}}){\rm d}{\boldsymbol{x}}\Bigg)^{1/p} \\ & \leq C e^{\frac{|{\boldsymbol{k}}+ (\theta{\rm sign\,}{\boldsymbol{k}})/2 |^2}{2p} }\|f_{\theta,{\boldsymbol{k}}}\|_{ {W}^\alpha_p({\mathbb I}^d_{{\boldsymbol{k}},\theta},\gamma)} \\ & \leq C e^{\frac{|{\boldsymbol{k}}+ (\theta{\rm sign\,}{\boldsymbol{k}})/2 |^2}{2p} }\|f\|_{W^\alpha_p({\mathbb R}^d,\gamma)}. \end{aligned}$$ This and [\[eq-first\]](#eq-first){reference-type="eqref" reference="eq-first"} deduce $$\begin{aligned} \label{multipl-algebra2} \|f_{\theta,{\boldsymbol{k}}}(\cdot+{\boldsymbol{k}})\varphi_{\boldsymbol{k}}(\cdot+{\boldsymbol{k}})\|_{{W}^\alpha_p({\mathbb I}^d_\theta)} \leq C e^{\frac{|{\boldsymbol{k}}+ (\theta{\rm sign\,}{\boldsymbol{k}})/2 |^2}{2p} }\|f\|_{W^\alpha_p({\mathbb R}^d,\gamma)}. \end{aligned}$$ We have $-|x|^2\leq \frac{\theta^2}{2}-\big|k-\frac{\theta({\rm sign\,}k)}{2}\big|^2$ for $x\in [-\frac{\theta}{2},\frac{\theta}{2}]+k,\ k\in {\mathbb Z}$. Hence for any $h\in L_q({\mathbb I}^d_{\theta})$ we obtain $$\label{eq-norm-B} \begin{aligned} \left\|{h(\cdot-{\boldsymbol{k}})}\right\|_{L_q({\mathbb I}^d_{\theta,{\boldsymbol{k}}}, \gamma)} &= \bigg( (2\pi)^{-d/2}\int_{{\mathbb I}^d_{{\boldsymbol{k}},\theta}}\big|h({\boldsymbol{x}}-{\boldsymbol{k}})\big|^q e^{-\frac{|{\boldsymbol{x}}|^2}{2}}{\rm d}{\boldsymbol{x}}\bigg)^{1/q} \\ &\leq C e^{- \frac{|{\boldsymbol{k}}- (\theta{\rm sign\,}{\boldsymbol{k}})/2 |^2}{2q}}\bigg( \int_{{\mathbb I}^d_{{\boldsymbol{k}},\theta}}\big|h({\boldsymbol{x}}-{\boldsymbol{k}})\big|^q {\rm d}{\boldsymbol{x}}\bigg)^{1/q} \\ & \ll e^{- \frac{|{\boldsymbol{k}}- (\theta{\rm sign\,}{\boldsymbol{k}})/2 |^2}{2q}} \left\|{h}\right\|_{{L}_q({\mathbb I}^d_{\theta})} . \end{aligned}$$ If $q<p$, we can choose a fixed $\delta>0$ such that $$\label{<e^{-delta k}} {e^{\frac{|{\boldsymbol{k}}+ (\theta {\rm sign\,}{\boldsymbol{k}})/2 |^2}{2p} -\frac{|{\boldsymbol{k}}- (\theta{\rm sign\,}{\boldsymbol{k}})/2 |^2}{2q}}} \leq C e^{- \delta |{\boldsymbol{k}}|^2}, \ \ {\boldsymbol{k}}\in {\mathbb Z}^d.$$ Denote by $\tilde{L}_q({\mathbb I}^d)$ and $\tilde{W}^\alpha_p({\mathbb I}^d)$ the subspaces of $L_q({\mathbb I}^d)$ and $W^\alpha_p({\mathbb I}^d)$, respectively, of all functions $f$ which can be extended to the whole ${\mathbb R}^d$ as $1$-periodic functions in each variable (denoted again by $f$). Similarly we defined $\tilde{L}_q({\mathbb I}^d_\theta)$ and $\tilde{W}^\alpha_p({\mathbb I}^d_\theta)$. Our main result in this section reads as follows. **Theorem 1**. *Let $\alpha\in {\mathbb N}$, $1\le q < p <\infty$ and $a >0$, $b \ge 0$, $\theta >1$. Let $s$ be a pseudo $s$-number. Assume that $$\label{eq-assumption} s_n\big(I: \tilde{W}^\alpha_p({\mathbb I}^d)\to \tilde{L}_q({\mathbb I}^d)\big)\asymp n^{-a} (\log n)^b\,, \ n\to \infty.$$* 1. *If $s$ is weakly injective in the sense [\[eq-injective\]](#eq-injective){reference-type="eqref" reference="eq-injective"}, then $$\label{A_m-Error-a,b} s_n\big(I_\gamma:W^\alpha_p({\mathbb R}^d,\gamma)\to L_q({\mathbb R}^d,\gamma)\big) \gg n^{-a} (\log n)^b \,, \ n\to \infty .$$* 2. *If $s$ is additive, then $$s_n\big(I_\gamma:W^\alpha_p({\mathbb R}^d,\gamma)\to L_q({\mathbb R}^d,\gamma)\big) \ll n^{-a} (\log n)^b \,, \ n\to \infty.$$* . **Step 1.** *Lower bound*. Consider the commutative diagram $$\begin{CD} W^\alpha_p({\mathbb R}^d,\gamma)@ > I_{\gamma} >> L_q({\mathbb R}^d,\gamma) \\ @AA I_1 A @VV I_2 V \\ \tilde{W}^\alpha_p({\mathbb I}^d) @ > I >> L_q({\mathbb I}^d) \,, \end{CD}$$ where $I_1$ is the embedding operator and $I_2$ is the restriction of functions on ${\mathbb R}^d$ onto ${\mathbb I}^d$. If $f$ is a $1$-periodic function on ${\mathbb R}^d$ and $f\in \tilde{W}^\alpha_p({\mathbb I}^d)$, then $$\label{eq-low01} \begin{aligned} \|f\|_{W^\alpha_p({\mathbb R}^d,\gamma)}&= \Bigg((2\pi)^{-d/2}\sum_{|{\boldsymbol{r}}|_\infty \leq \alpha} \int_{{\mathbb R}^d} |D^{\boldsymbol{r}}f({\boldsymbol{x}})|^p e^{-\frac{|{\boldsymbol{x}}|^2}{2}}{\rm d}{\boldsymbol{x}}\Bigg)^{1/p} \\ & = (2\pi)^{-\frac{d}{2p}}\Bigg(\sum_{|{\boldsymbol{r}}|_\infty \leq \alpha} \sum_{{\boldsymbol{k}}\in {\mathbb Z}^d} \int_{{\mathbb I}^d} |D^{\boldsymbol{r}}f({\boldsymbol{x}}+{\boldsymbol{k}})|^p e^{-\frac{|{\boldsymbol{x}}+{\boldsymbol{k}}|^2}{2}}{\rm d}{\boldsymbol{x}}\Bigg)^{1/p} \\ & \ll \Bigg(\sum_{|{\boldsymbol{r}}|_\infty \leq \alpha} \int_{{\mathbb I}^d} |D^{\boldsymbol{r}}f({\boldsymbol{x}})|^p {\rm d}{\boldsymbol{x}}\sum_{{\boldsymbol{k}}\in {\mathbb Z}^d}e^{-\frac{|{\boldsymbol{k}}- ({\rm sign\,}{\boldsymbol{k}})/2 |^2}{2}}\Bigg)^{1/p} \\ & \ll \|f\|_{\tilde{W}^\alpha_p({\mathbb I}^d)} \end{aligned}$$ which implies that $\|I_1\|\ll 1$. For a function $h\in L_q({\mathbb R}^d,\gamma)$ we have $$\label{eq-low02} \|h\|_{L_q({\mathbb I}^d)}=\bigg((2\pi)^{\frac{d}{2}}\int_{{\mathbb I}^d}|h({\boldsymbol{x}})|^qe^{\frac{|{\boldsymbol{x}}|^2}{2}}g({\boldsymbol{x}}){\rm d}{\boldsymbol{x}}\bigg)^{1/q} \leq (2\pi)^{\frac{d}{2q}}e^{\frac{d}{8q}}\|h\|_{L_q({\mathbb R}^d,\gamma)}.$$ This deduces $\|I_2\|\ll 1$. By the ideal property $(c)$ of pseudo $s$-numbers we get $$\label{eq-low} s_n(I)= s_n(I_2 I_\gamma I_1) \leq \|I_1\| \cdot s_n(I_\gamma)\cdot \|I_2\|\ll s_n(I_\gamma).$$ Since $s$ is an injective pseudo $s$-number in the weak sense [\[eq-injective\]](#eq-injective){reference-type="eqref" reference="eq-injective"}, we get $$s_n\big(I: \tilde{W}^\alpha_p({\mathbb I}^d)\to L_q({\mathbb I}^d)\big)\asymp s_n\big(I: \tilde{W}^\alpha_p({\mathbb I}^d)\to \tilde{L}_q({\mathbb I}^d)\big)\asymp n^{-a} (\log n)^b.$$ This and [\[eq-low\]](#eq-low){reference-type="eqref" reference="eq-low"} give the lower bound.\ **Step 2.** *Upper bound*. First note, that [\[eq-assumption\]](#eq-assumption){reference-type="eqref" reference="eq-assumption"} induces $$\label{eq-induce} s_n\big(I_\theta: \tilde{W}^\alpha_p({\mathbb I}^d_\theta)\to \tilde{L}_q({\mathbb I}^d_\theta)\big)\asymp n^{-a} (\log n)^b\,, \ n\to \infty.$$ We define for $n\in {\mathbb N}$, $$\label{xi-int} \xi_n = \sqrt{\delta^{-1} 2 a(\log n)}\,,$$ and for ${\boldsymbol{k}}\in {\mathbb Z}^d$ with $\ |{\boldsymbol{k}}|< \xi_n$ $$\label{n_bk} n_{{\boldsymbol{k}}}= \lfloor \varrho n e^{-\frac{\delta}{2 a}|{\boldsymbol{k}}|^2} \rfloor,$$ where $\varrho := 2^{-d} \big(1 - e^{-\frac{\delta}{2 a}}\big)^{d}$. We have $$\begin{aligned} \label{<n2} \sum_{|{\boldsymbol{k}}|< \xi_n} n_{\boldsymbol{k}}\le n.\end{aligned}$$ Indeed, $$\begin{aligned} \sum_{|{\boldsymbol{k}}| < \xi_n}n_{\boldsymbol{k}} & \leq \sum_{|{\boldsymbol{k}}|< \xi_n} \varrho n e^{-\frac{\delta}{2\alpha}|{\boldsymbol{k}}|^2} \leq 2^{d} \varrho n \sum_{s=0}^{\lfloor \xi_n \rfloor} \binom{s+d-1}{d-1}e^{-\frac{\delta}{2 a}s^2} \\ & \leq 2^{d}\varrho n \sum_{s=0}^{\infty} \binom{s+d-1}{d-1}e^{-\frac{\delta}{2 a}s} \leq n, \end{aligned}$$ where in the last estimate we used the well-known formula $$\label{eq-auxilary-01} \sum_{j=0}^\infty x^j\binom{j+k}{k}=(1-x)^{-k-1}, \ k\in {\mathbb N}_0, \ x\in (0,1).$$ Let $I_{\theta,{\boldsymbol{k}}}: W^\alpha_p({\mathbb R}^d,\gamma)\to L_q({\mathbb R}^d,\gamma)$ be the operator defined as $$I_{\theta,{\boldsymbol{k}}}(f):= f\varphi_{\boldsymbol{k}}, \ f\in W^\alpha_p({\mathbb R}^d,\gamma).$$ By property (iii) of the system $(\varphi_{\boldsymbol{k}})_{{\boldsymbol{k}}\in {\mathbb Z}^d}$ we have $f = \sum_{{\boldsymbol{k}}\in {\mathbb Z}^d}f_{\theta,{\boldsymbol{k}}}\varphi_{\boldsymbol{k}}$ which implies $I_\gamma=\sum_{{\boldsymbol{k}}\in {\mathbb Z}^d}I_{\theta,{\boldsymbol{k}}}$. In view of the property (b') of additive pseudo $s$-numbers and [\[\<n2\]](#<n2){reference-type="eqref" reference="<n2"} we get $$\begin{aligned} \label{f-A_n'^gf} s_n(I_\gamma) &\leq \sum_{|{\boldsymbol{k}}|< \xi_n} s_{n_{\boldsymbol{k}}}(I_{\theta,{\boldsymbol{k}}}) + \sum_{|{\boldsymbol{k}}|\geq \xi_n} \left\|{I_{\theta,{\boldsymbol{k}}}}\right\|_{}. \end{aligned}$$ For every ${\boldsymbol{k}}\in {\mathbb N}^d_0$, consider the commutative diagram $$\begin{CD} W^\alpha_p({\mathbb R}^d,\gamma)@ > I_{\theta,{\boldsymbol{k}}} >> L_q({\mathbb R}^d,\gamma) \\ @VV A_{\boldsymbol{k}}V @AA B_{\boldsymbol{k}}A\\ \tilde{W}^\alpha_p({\mathbb I}^d_{\theta}) @ > I_{\theta} >> \tilde{L}_q({\mathbb I}^d_{\theta}) \,, \end{CD}$$ where $$\begin{cases} A_{\boldsymbol{k}}(f):=f_{\theta,{\boldsymbol{k}}}(\cdot+{\boldsymbol{k}})\varphi_{\boldsymbol{k}}(\cdot+{\boldsymbol{k}}), & \ f\in W^\alpha_p({\mathbb R}^d,\gamma) \\ B_{\boldsymbol{k}}(h):=h(\cdot-{\boldsymbol{k}}),& \ h\in \tilde{L}_q({\mathbb I}^d_{\theta}). \end{cases}$$ From [\[multipl-algebra2\]](#multipl-algebra2){reference-type="eqref" reference="multipl-algebra2"} and [\[eq-norm-B\]](#eq-norm-B){reference-type="eqref" reference="eq-norm-B"} we deduce that $$\label{eq-norm-A-B} \|A_{\boldsymbol{k}}\| \leq e^{\frac{|{\boldsymbol{k}}+ (\theta {\rm sign\,}{\boldsymbol{k}})/2 |^2}{2p}}\ \ \text{and} \ \ \ \|B_{\boldsymbol{k}}\| \leq e^{- \frac{|{\boldsymbol{k}}- (\theta{\rm sign\,}{\boldsymbol{k}})/2 |^2}{2q}}.$$ By this and the property (c) of pseudo $s$-numbers we obtain $$\begin{aligned} s_{n_{\boldsymbol{k}}}(I_{\theta,{\boldsymbol{k}}}) &=s_{n_{\boldsymbol{k}}}(B_{\boldsymbol{k}}I_\theta A_{\boldsymbol{k}}) \leq \|A_{\boldsymbol{k}}\| \cdot s_{n_{\boldsymbol{k}}}(I_\theta)\cdot \|B_{\boldsymbol{k}}\| \\ & \ll e^{\frac{|{\boldsymbol{k}}+ (\theta {\rm sign\,}{\boldsymbol{k}})/2 |^2}{2p}- \frac{|{\boldsymbol{k}}- (\theta{\rm sign\,}{\boldsymbol{k}})/2 |^2}{2q}} s_{n_{\boldsymbol{k}}}(I_\theta) . \end{aligned}$$ Using [\[\<e\^{-delta k}\]](#<e^{-delta k}){reference-type="eqref" reference="<e^{-delta k}"} and [\[eq-induce\]](#eq-induce){reference-type="eqref" reference="eq-induce"} we get $$\begin{aligned} s_{n_{\boldsymbol{k}}}(I_{\theta,{\boldsymbol{k}}}) & \ll e^{- \delta |{\boldsymbol{k}}|^2} \Big( n e^{-\frac{\delta}{2a}|{\boldsymbol{k}}|^2} \Big)^{-a} (\log n)^b=e^{- \frac{\delta}{2} |{\boldsymbol{k}}|^2} n^{-a} (\log n)^b,\end{aligned}$$ which implies $$\label{eq-sum-1} \begin{aligned} \sum_{|{\boldsymbol{k}}|< \xi_n} s_{n_{\boldsymbol{k}}}(I_{\theta,{\boldsymbol{k}}}) &\ll \sum_{|{\boldsymbol{k}}|< \xi_n} e^{- \frac{ \delta}{2} |{\boldsymbol{k}}|^2} n^{-a} (\log n)^b \ll n^{-a} (\log n)^b. \end{aligned}$$ For a fixed $\varepsilon \in (0,1/2)$, from [\[eq-norm-A-B\]](#eq-norm-A-B){reference-type="eqref" reference="eq-norm-A-B"} we have $$\begin{aligned} \sum_{|{\boldsymbol{k}}|\geq \xi_n} \left\|{I_{\theta,{\boldsymbol{k}}}}\right\|_{} & \leq \sum_{|{\boldsymbol{k}}|\geq \xi_n}\|A_{\boldsymbol{k}}\|\cdot \|I_\theta\|\cdot \|B_{\boldsymbol{k}}\| \\ &\ll \sum_{|{\boldsymbol{k}}|\geq \xi_n} e^{- \frac{|{\boldsymbol{k}}- (\theta {\rm sign\,}{\boldsymbol{k}})/2 |^2}{2q}+ \frac{|{\boldsymbol{k}}+ (\theta{\rm sign\,}{\boldsymbol{k}})/2 |^2}{2p}} \|I_\theta\| \\ & \ll \sum_{|{\boldsymbol{k}}|\geq \xi_n} e^{- \delta |{\boldsymbol{k}}|^2} \ll e^{- \delta (1-\varepsilon) \xi_n^2} \\ & =\, e^{-2 a (1-\varepsilon) \log n} \ll n^{-a} (\log n)^b . \end{aligned}$$ From the last estimate, [\[f-A_n\'\^gf\]](#f-A_n'^gf){reference-type="eqref" reference="f-A_n'^gf"}, and [\[eq-sum-1\]](#eq-sum-1){reference-type="eqref" reference="eq-sum-1"} we obtain the upper bound. Study of asymptotic behavior as well as pre-asymptotic estimate of approximation, Kolmogorov, Gelfand, and entropy numbers has a long history. We refer the reader to the book [@DTU18B] for results and historical commments. Asymptotic behavior of Weyl and Bernstein numbers of the embedding $I: \tilde{W}^\alpha_p({\mathbb I}^d)\to \tilde{L}_q({\mathbb I}^d)$ was investigated in [@Gal91; @Ng16]. As a consequence of Theorem [Theorem 1](#thm:approx-general-theta){reference-type="ref" reference="thm:approx-general-theta"} we have the following result. **Corollary 2**. *Let $\alpha\in {\mathbb N}$ and $d\in {\mathbb N}$. Let $1\leq q<p<\infty$. If $1<q$ then we have $$a_n\big(I_\gamma\big)\asymp d_n(I_\gamma)\asymp c_n(I_\gamma)\asymp e_n(I_\gamma)\asymp n^{-\alpha} (\log n)^{(d-1)\alpha},\ \ n\to \infty;$$ and $$x_n(I_\gamma)\asymp b_n(I_\gamma)\asymp n^{-\beta}(\log n)^{(d-1)\beta},\ \ n\to \infty,$$ where $$\label{eq-beta} \beta = \left\{\begin{array}{lll} \alpha & \text{if}\ \ q<p\leq 2\, ,\ \alpha>0, \\ \alpha-\frac{1}{p}+\frac{1}{2} & \text{if}\ \ q\leq 2<p,\ \alpha> \frac{1}{p},\\ \alpha-\frac{1}{p}+\frac{1}{q} & \text{if}\ \ 2\leq q<p,\ \alpha> \frac{1/q-1/p}{p/2-1},\\ \frac{\alpha p}{2} & \text{if}\ \ 2\leq q <p\,, \ \alpha< \frac{1/q-1/p}{p/2-1} \text{\rm \ or } q\leq 2<p; \alpha < \frac{1}{p}.\ \end{array}\right.$$ When $q=1$, the above results still hold for approximation, Kolmogorov, Weyl and entropy numbers.* . Let $1<q<p<\infty$. We know that $$a_n\big(I\big)\asymp d_n(I)\asymp c_n(I)\asymp e_n(I)\asymp n^{-\alpha} (\log n)^{(d-1)\alpha},\ \ n\to \infty,$$ see [@DTU18B Theorems 4.3.1, 4.5.1, 6.2.1 and Section 9.6] and $$x_n(I_\gamma)\asymp b_n(I_\gamma)\asymp n^{-\beta}(\log n)^{(d-1)\beta},\ \ n\to \infty,$$ where $\beta$ is given in [\[eq-beta\]](#eq-beta){reference-type="eqref" reference="eq-beta"}, see [@Ng16 Theorems 2.1, 2.3] and [@Gal91]. Since approximation, Kolmogorov, Gelfand, Weyl and entropy numbers are additive pseudo $s$-numbers, by Theorem [Theorem 1](#thm:approx-general-theta){reference-type="ref" reference="thm:approx-general-theta"} we obtain the upper bound for these pseudo $s$-numbers of the embedding $I_\gamma$. The upper bound for Bernstein numbers follows from the inequality $b_n(I_\gamma)\ll x_n(I_\gamma)$, see [\[eq-inequality3\]](#eq-inequality3){reference-type="eqref" reference="eq-inequality3"}. By the weakly injective property and Theorem [Theorem 1](#thm:approx-general-theta){reference-type="ref" reference="thm:approx-general-theta"} we get lower bound for Bernstein, Gelfand, entropy, and Weyl numbers of $I_\gamma$. The lower bound for approximation and Kolmogorov numbers follows from the inequalities $e_n(I_\gamma)\leq a_n(I_\gamma)$ and $e_n(I_\gamma)\ll d_n(I_\gamma)$, see [\[eq-inequality3\]](#eq-inequality3){reference-type="eqref" reference="eq-inequality3"} and [\[eq-inequality4\]](#eq-inequality4){reference-type="eqref" reference="eq-inequality4"}. In view of [@DTU18B Theorems 4.3.1, 4.5.1, 6.2.3] and [@Ng16 Theorem 2.6], the argument of the case $q=1$ for Kolmogorov, approximation, entropy and Weyl numbers is carried out similarly. # The case $p = 2$ {#sec-p=2} In this section, we study the approximation of functions from Hilbert spaces ${\mathcal H}^\alpha$ which is an extension of $W_2^\alpha({\mathbb R}^d,\gamma)$ from natural smoothness to the real positive smoothness $\alpha$. For $k\in {\mathbb N}_0$, the normalized probabilistic Hermite polynomial $H_k$ of degree $k$ on ${\mathbb R}$ is defined by $$H_k(x) := \frac{(-1)^k}{\sqrt{k!}} \exp\left(\frac{x^2}{2}\right) \frac{{\rm d}^k}{{\rm d}x^k} \exp\left(-\frac{x^2}{2}\right) .$$ For a multi-index ${\boldsymbol{k}}\in {\mathbb N}^d_0$ we define the $d$-variate Hermite polynomial $H_{\boldsymbol{k}}$ by $$\label{H_bk} H_{\boldsymbol{k}}({\boldsymbol{x}}) :=\prod_{j=1}^d H_{k_j}(x_j), \;\; {\boldsymbol{x}}\in {\mathbb R}^d.$$ It is well-known that the Hermite polynomials $(H_{\boldsymbol{k}})_{{\boldsymbol{k}}\in {\mathbb N}^d_0}$ constitute an orthonormal basis of the Hilbert space $L_2({\mathbb R}^d,\gamma)$ (see, e.g., [@Szego1939 Section 5.5]). In particular, every function $f \in L_2({\mathbb R}^d,\gamma)$ can be represented by the Hermite series $$\label{H-series} f = \sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} \hat{f}({\boldsymbol{k}}) H_{\boldsymbol{k}}\ \ {\rm with} \ \ \hat{f}({\boldsymbol{k}}) := \int_{{\mathbb R}^d} f({\boldsymbol{x}})\, H_{\boldsymbol{k}}({\boldsymbol{x}})\gamma({\rm d}{\boldsymbol{x}})$$ converging in the norm of $L_2({\mathbb R}^d,\gamma)$. In addition, there holds Parseval's identity $$\label{P-id} \left\|{f}\right\|_{L_2({\mathbb R}^d,\gamma)}^2= \sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} |\hat{f}({\boldsymbol{k}})|^2.$$ For $\alpha \in {\mathbb N}_0$ and ${\boldsymbol{k}}\in {\mathbb N}^d_0$, we define the sequence $\rho_{\alpha}:=(\rho_{\alpha,{\boldsymbol{k}}})_{{\boldsymbol{k}}\in {\mathbb N}^d_0}$, where $$\label{rho_bk} \rho_{\alpha,{\boldsymbol{k}}}: = \prod_{j=1}^d \left(k_j + 1\right)^\alpha.$$ The following lemma was proved in [@DILP18]. **Lemma 3**. *Let $\alpha \in {\mathbb N}_0$. Then we have that $$\label{N-eq} \left\|{f}\right\|_{W^\alpha_2({\mathbb R}^d,\gamma)}^2 \asymp \sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} \rho_{\alpha,{\boldsymbol{k}}}|\hat{f}({\boldsymbol{k}})|^2, \quad f \in W^\alpha_2({\mathbb R}^d,\gamma).$$* By this lemma, we extend the space $W^\alpha_2({\mathbb R}^d,\gamma)$ to any $\alpha > 0$. Denote by ${\mathcal H}^\alpha$ the space of all functions $f \in L_2({\mathbb R}^d,\gamma)$ represented by the Hermite series [\[H-series\]](#H-series){reference-type="eqref" reference="H-series"} for which the norm $$\label{Hh-norm} \left\|{f}\right\|_{{\mathcal H}^\alpha} := \Bigg(\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} \rho_{\alpha,{\boldsymbol{k}}}|\hat{f}({\boldsymbol{k}})|^2\Bigg)^{1/2}$$ is finite. With this definition, we can identify $W^\alpha_2({\mathbb R}^d,\gamma)$ with ${\mathcal H}^\alpha$ for $\alpha \in {\mathbb N}$ in the sense of equivalent norms. **Theorem 4**. *Let $\alpha >0$ and $s\in \{a,b,c,d,e,x\}$. Then $$\label{widths:p=q=2} s_n\big(I_\gamma:{{\mathcal H}}^\alpha\to L_2({\mathbb R}^d,\gamma)\big) \asymp n^{-\frac{\alpha}{2}} (\log n)^{\frac{(d-1)\alpha}{2}}, \ \ n\to \infty.$$ Moreover, if $s\in \{a,b,c,d,x\}$ then $$\label{eq-asymptotic} \lim\limits_{n\to \infty} \frac{s_n\big(I_\gamma:{{\mathcal H}}^\alpha\to L_2({\mathbb R}^d,\gamma)\big)}{n^{-\frac{\alpha}{2}}(\ln n)^{\frac{\alpha(d-1)}{2}}}= \bigg( \frac{1}{(d-1)!}\bigg)^{\alpha/2}\,.$$* . It has been proved in [@DN23] that $$a_n\big(I_\gamma\big)= d_n\big(I_\gamma\big) \asymp n^{-\frac{\alpha}{2}} (\log n)^{\frac{(d-1)\alpha}{2}}, \ \ n\to \infty.$$ Since ${{\mathcal H}}^\alpha$ and $L_2({\mathbb R}^d,\gamma)$ are Hilbert spaces, there is only one $s$-number of $I_\gamma$, see [@Pie74; @Pie80B]. This means $$\label{eq-acdx} a_n\big(I_\gamma\big)= d_n\big(I_\gamma\big)=c_n\big(I_\gamma\big)= x_n\big(I_\gamma\big)=b_n\big(I_\gamma\big)\asymp n^{-\frac{\alpha}{2}} (\log n)^{\frac{(d-1)\alpha}{2}}, \ \ n\to \infty.$$ From [\[eq-inequality2\]](#eq-inequality2){reference-type="eqref" reference="eq-inequality2"} we get $b_n(I_\gamma)\ll e_n(I_\gamma)\leq a_n(I_\gamma)$. This together with [\[eq-acdx\]](#eq-acdx){reference-type="eqref" reference="eq-acdx"} implies the result for entropy numbers. We proceed with the proof of [\[eq-asymptotic\]](#eq-asymptotic){reference-type="eqref" reference="eq-asymptotic"}. Consider the diagram $$\begin{CD} {\mathcal H}^\alpha @ > I_\gamma >> L_2({\mathbb R}^d,\gamma) \\ @VV A V @AA B A\\ \ell_2({\mathbb N}^d_0) @ > D_{\sqrt{\rho_{\alpha}}} >> \ell_2({\mathbb N}^d_0) \,, \end{CD}$$ where the linear operators $A$, $D_{\sqrt{\rho_{\alpha}}}$, and $B$ are defined as $$\label{ws-12} \begin{aligned} Af & : = \big( \sqrt{\rho_{\alpha,{\boldsymbol{k}}}}\hat{f}({\boldsymbol{k}})\big)_{{\boldsymbol{k}}\in {\mathbb N}^d_0}\, , \qquad f\in {\mathcal H}^\alpha \\ D_{\sqrt{\rho_\alpha}}\xi & := (\xi_{\boldsymbol{k}}/ \sqrt{\rho_{\alpha,{\boldsymbol{k}}}})_{{\boldsymbol{k}}\in {\mathbb N}^d_0}\, , \qquad \xi=(\xi_{\boldsymbol{k}})_{{\boldsymbol{k}}\in {\mathbb N}^d_0} \\ (B\xi)({\boldsymbol{x}})& := \sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} \xi_{\boldsymbol{k}}\, H_{\boldsymbol{k}}({\boldsymbol{x}})\, , \qquad {\boldsymbol{x}}\in {\mathbb R}^d\, . \end{aligned}$$ It is obvious that $\|A\|=\|B\|=1$. From the ideal property (c) and the identity $I_\gamma = B\,D_{\sqrt{\rho_{\alpha}}} \, A$ it follows $$s_n\big(I_\gamma\big) \le s_n \big(D_{\sqrt{\rho_{\alpha}}} \big) \, .$$ It is easy to see that the operators $A$ and $B$ are invertible and that $\|A^{-1}\|=\|B^{-1}\|=1$. Employing the same type of arguments with respect to the diagram $$\begin{CD} {\mathcal H}^\alpha @ > I_\gamma >> L_2({\mathbb R}^d,\gamma) \\ @AA {A^{-1}} A @VV B^{-1} V\\ \ell_2({\mathbb N}^d_0) @ > D_{\sqrt{\rho_{\alpha}}} >> \ell_2({\mathbb N}^d_0) \, \end{CD}$$ we also get $$s_n\big(I_\gamma\big) \geq s_n \big(D_{\sqrt{\rho_{\alpha}}}\big) \, .$$ Consequently, we obtain $$s_n\big(I_\gamma\big)= s_n\big(D_{\sqrt{\rho_{\alpha}}}\big) .$$ Note, that the sequence $s_n\big(I_\gamma\big)=s_n\big(D_{\sqrt{\rho_{\alpha}}})$ is the non-increasing rearrangement of the sequence $(1/\sqrt{\rho_{\alpha,{\boldsymbol{k}}}} )_{{\boldsymbol{k}}\in {\mathbb N}^d_0}$, see [@Pie80B Theorem 11.11.3]. We put $$c(r,d):= \bigg|\bigg\{{\boldsymbol{k}}\in {\mathbb N}^d_0: ~\prod_{j=1}^d (k_j+1) \leq r \bigg\}\bigg|, \quad r \in {\mathbb N}\, .$$ From [@ChD16 Theorems 3.4 and 3.5] (with $a=1$) we find $$\label{ch} \frac{1}{(d-1)!}\, \frac{r(\ln r)^d}{\ln r+d} < c(r,d) < \frac{1}{(d-1)!} \, \frac{r(\ln r +d\ln 2)^d}{\ln r+d\ln 2 +d-1}$$ for $r>r*>1$. For $n\geq 2$, choose $r\in {\mathbb N}$ such that $c(r-1,d) <n \le c(r,d)$. By the definition of $c(r,d)$ we have $$r^{-\alpha/2}= s_{c(r,d)} (I_\gamma) \leq s_n (I_\gamma) \leq s_{c(r-1,d)} (I_\gamma) =(r-1)^{-\alpha/2}.$$ Clearly, $\lim_{r\to \infty} c(r, d) = \infty$. Moreover, the sequence $n\, (\ln n)^{-(d-1)}$ is increasing for $n > e^{d-1}$. Hence, we obtain for sufficiently large $r \in {\mathbb N}$ the two-sided inequality $$\frac{ c(r - 1, d)}{r (\ln c(r-1,d))^{d-1}} \le \, \frac{s_n(I_\gamma)^{2/\alpha}}{n^{-1}(\ln n)^{d-1}}\le \frac{c(r, d)}{(r-1)\, (\ln c(r, d))^{d-1}}\, .$$ Applying [\[ch\]](#ch){reference-type="eqref" reference="ch"} the claim follows. *Remark 5*. Observe that, we do not have factor $2^d$ in the asymptotic constant in [\[eq-asymptotic\]](#eq-asymptotic){reference-type="eqref" reference="eq-asymptotic"}. This is a difference compared to the asymptotic constant of the embedding $I: \tilde{W}_{2}^\alpha({\mathbb I}^d) \to \title{L}_2({\mathbb I}^d)$, see [@KSU15]. Let us emphasize that for $\alpha>0$, the space ${\mathcal H}^\alpha$ is not embedded in $L_q({\mathbb R}^d)$ with $2<q\leq \infty$. Therefore, in the next step we study the asymptotic behavior of pseudo $s$-numbers of the embedding of ${\mathcal H}^\alpha$ into the space $L_{\infty}^{\sqrt{g}}({\mathbb R}^d)$, where the norm of $f\in L_{\infty}^{\sqrt{g}}({\mathbb R}^d)$ is defined by $$\|f\|_{L_{\infty}^{\sqrt{g}}({\mathbb R}^d)}: = \sup_{{\boldsymbol{x}}\in {\mathbb R}^d}\big|f({\boldsymbol{x}})\sqrt{g({\boldsymbol{x}})}\big|<\infty.$$ We have the following theorem. **Theorem 6**. *Let $\alpha>1-\frac{1}{6}$ and $s\in \{a,b,c,d,e,x\}$. Then we have $$\begin{aligned} n^{-\frac{\alpha}{2}-\frac{d}{4}} (\log n)^{(\frac{\alpha}{2}+\frac{d}{4})(d-1)} & \ll s_n\big(I_\gamma: {\mathcal H}^\alpha\to L_{\infty}^{\sqrt{g}}({\mathbb R}^d)\big) \\ &\ll n^{-\frac{\alpha}{2}-\frac{1}{12}+\frac{1}{2}}(\log n)^{(\frac{\alpha}{2}+\frac{1}{12})(d-1)}\,, \ n\to \infty. \end{aligned}$$* . **Step 1.** We first show that the embedding $I_\gamma:{{\mathcal H}}^\alpha\to L_\infty^{\sqrt{g}}({\mathbb R}^d)$ is continuous. Recall an inequality in [@DILP18] $$H_{\boldsymbol{k}}({\boldsymbol{x}})\sqrt{g({\boldsymbol{x}})}\leq \prod_{j=1}^d \min\bigg(1,\frac{\sqrt{\pi}}{k_j^{1/12}}\bigg)\asymp \prod_{j=1}^d\frac{1}{(1+k_j)^{1/12}} ,\ \ {\boldsymbol{x}}\in {\mathbb R}^d.$$ If $f=\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0}\hat{f}({\boldsymbol{k}})H_{\boldsymbol{k}}\in {\mathcal H}^\alpha$, then we have $$\label{eq-Cauchy} \begin{aligned} \Bigg\|\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0}\hat{f}({\boldsymbol{k}})H_{\boldsymbol{k}}\Bigg\|_{L_\infty^{\sqrt{g}}({\mathbb R}^d)} & = \Bigg\|\sqrt{g({\boldsymbol{x}})}\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} \hat{f}({\boldsymbol{k}}) H_{\boldsymbol{k}}({\boldsymbol{x}})\Bigg\|_{L_\infty({\mathbb R}^d)} \\ & \leq \sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} |\hat{f}({\boldsymbol{k}})| \sup_{{\boldsymbol{x}}\in {\mathbb R}^d} \big|H_{\boldsymbol{k}}({\boldsymbol{x}})\sqrt{g({\boldsymbol{x}})}\big| \\ & \ll \sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} |\hat{f}({\boldsymbol{k}})| \prod_{j=1}^d\frac{1}{(1+k_j)^{1/12}} \\ & \leq \Bigg(\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} \rho_{\alpha,{\boldsymbol{k}}} |\hat{f}({\boldsymbol{k}})|^2\Bigg)^{1/2} \Bigg(\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0}\frac{1}{\rho_{\alpha,{\boldsymbol{k}}}} \prod_{j=1}^d\frac{1}{(1+k_j)^{1/6}} \Bigg)^{1/2}, \end{aligned}$$ where in the last estimate we used the Cauchy--Schwarz inequality. Assumption $\alpha>1-\frac{1}{6}$ implies that $$\begin{aligned} \Bigg(\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0}\frac{1}{\rho_{\alpha,{\boldsymbol{k}}}} \prod_{j=1}^d\frac{1}{(1+k_j)^{1/6}} \Bigg)^{1/2} & = \Bigg(\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} \prod_{j=1}^d\frac{1}{(1+k_j)^{1/6+\alpha}} \Bigg)^{1/2} \\ &= \Bigg( \sum_{k=0}^\infty\frac{1}{(1+k)^{1/6+\alpha}} \Bigg)^{d/2} <\infty.\end{aligned}$$ Consequently, we get $$\|f({\boldsymbol{x}})\|_{L_\infty^{\sqrt{g}}({\mathbb R}^d)} \leq C \Bigg(\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0} \rho_{\alpha,{\boldsymbol{k}}} |\hat{f}({\boldsymbol{k}})|^2\Bigg)^{1/2} =C\|f\|_{{\mathcal H}^\alpha}$$ which means the continuous embedding of ${\mathcal H}^\alpha$ into $L_\infty^{\sqrt{g}}({\mathbb R}^d)$.\ **Step 2.** *Upper bound.* Since approximation number is the largest pseudo $s$-number in the set $\{a,b,c,d,e,x\}$, we prove the upper bound for approximation numbers. Let ${\boldsymbol{s}}= (s_1, \ldots, s_d)$ be a vector whose coordinates are nonnegative integers. We define $$\label{eq-Qxi} \begin{aligned} \delta({\boldsymbol{s}})&:=\big\{{\boldsymbol{k}}\in {\mathbb N}^d_0: \lfloor 2^{s_j-1}\rfloor \leq k_j\leq 2^{s_j}, j=1,\ldots,d\big\} \\ Q_\xi&:=\bigcup_{|{\boldsymbol{s}}|_1\leq \xi}\delta({\boldsymbol{s}}). \end{aligned}$$ For any $\xi \in {\mathbb N}$, denote $A_\xi: {\mathcal H}^\alpha\to L_\infty^{\sqrt{g}}({\mathbb R}^d)$ the operator defined by $$A_\xi(f):=\sum_{{\boldsymbol{k}}\in Q_\xi} \hat{f}({\boldsymbol{k}})H_{\boldsymbol{k}},\ \ f=\sum_{{\boldsymbol{k}}\in {\mathbb N}^d_0}\hat{f}({\boldsymbol{k}})H_{\boldsymbol{k}}\in {\mathcal H}^\alpha.$$ It is well-known that the cardinality of $Q_\xi$ is equivalent to $2^\xi \xi^{d-1}$, see, e.g., [@DTU18B Section 2.3], which implies that ${\rm rank}(A_\xi)\asymp 2^\xi \xi^{d-1}$. As [\[eq-Cauchy\]](#eq-Cauchy){reference-type="eqref" reference="eq-Cauchy"} we have $$\begin{aligned} \|f-A_\xi(f)\|_{L_\infty^{\sqrt{g}}({\mathbb R}^d)} & =\Bigg\|\sum_{{\boldsymbol{k}}\not \in Q_\xi}\hat{f}({\boldsymbol{k}})H_{\boldsymbol{k}}\Bigg\|_{L_\infty^{\sqrt{g}}({\mathbb R}^d)} \\ & \ll \Bigg(\sum_{{\boldsymbol{k}}\not \in Q_\xi} \rho_{\alpha,{\boldsymbol{k}}} |\hat{f}({\boldsymbol{k}})|^2\Bigg)^{1/2} \Bigg(\sum_{{\boldsymbol{k}}\not \in Q_\xi} \prod_{j=1}^d\frac{1}{(1+k_j)^{\alpha+1/6}} \Bigg)^{1/2}.\end{aligned}$$ Observe that $$\begin{aligned} \sum_{{\boldsymbol{k}}\not \in Q_\xi} \prod_{j=1}^d\frac{1}{(1+k_j)^{\alpha+1/6}} & = \sum_{m=\xi+1}^\infty \sum_{ |{\boldsymbol{s}}|_1=m}\sum_{{\boldsymbol{k}}\in \delta({\boldsymbol{s}})}\prod_{j=1}^d\frac{1}{(1+k_j)^{\alpha+1/6}} \\ & \ll \sum_{m=\xi+1}^\infty \sum_{|{\boldsymbol{s}}|_1=m} \frac{2^m}{2^{m(\alpha+1/6)}} \\ & \ll \sum_{m=\xi+1}^\infty 2^{m(-\alpha-\frac{1}{6}+1)}m^{d-1} \\ & \asymp 2^{\xi(-\alpha-\frac{1}{6}+1)}\xi^{d-1}.\end{aligned}$$ Therefore $$\|f-A_\xi(f)\|_{L_\infty^{\sqrt{g}}({\mathbb R}^d)} \ll 2^{\xi(-\alpha-\frac{1}{6}+1)}\xi^{d-1} \|f\|_{{\mathcal H}^\alpha} .$$ For $n\in {\mathbb N}$, $n>2$, we choose $\xi=\xi(n)$ such that $n\asymp 2^\xi \xi^{d-1}$ which implies $\xi \asymp \log n$ and $2^\xi \asymp \frac{n}{(\log n)^{d-1} }$. From this we get $$\begin{aligned} \|f-A_{\xi(n)}(f)\|_{L_\infty^{\sqrt{g}}({\mathbb R}^d)} &\ll \bigg(\frac{n }{(\log n)^{d-1}}\bigg)^{-\frac{\alpha}{2}-\frac{1}{12}+\frac{1}{2}} (\log n)^{\frac{d-1}{2}} \|f\|_{{\mathcal H}^\alpha} \\ & = n^{-\frac{\alpha}{2}-\frac{1}{12}+\frac{1}{2}}(\log n)^{(d-1)(\frac{\alpha}{2}+\frac{1}{12})} \|f\|_{{\mathcal H}^\alpha}.\end{aligned}$$ By the definition of approximation numbers, we obtain the upper bound.\ **Step 3.** *Lower bound.* The inequalities [\[eq-inequality1\]](#eq-inequality1){reference-type="eqref" reference="eq-inequality1"}, [\[eq-inequality2\]](#eq-inequality2){reference-type="eqref" reference="eq-inequality2"} and [\[eq-inequality3\]](#eq-inequality3){reference-type="eqref" reference="eq-inequality3"} induce that Bernstein number is the smallest pseudo $s$-number in the set $\{a,b,c,d,e,x\}$. Therefore, it is enough to prove the lower bound for Bernstein numbers. For $\xi\in {\mathbb N}$, let $L(Q_\xi)$ be the subspace of ${\mathcal H}^\alpha$ defined by $$L(Q_\xi)=\Bigg\{ f=\sum_{{\boldsymbol{k}}\in Q_\xi} a_{\boldsymbol{k}}H_{\boldsymbol{k}},\ a_{\boldsymbol{k}}\in {\mathbb R}\Bigg \},$$ where $Q_\xi$ is given in [\[eq-Qxi\]](#eq-Qxi){reference-type="eqref" reference="eq-Qxi"}. Denote $n:=n(\xi)={\rm dim}(L(Q_\xi))$. By the definition of Bernstein numbers we get $$\begin{aligned} b_n(I_\gamma)\geq \inf_{\substack{f\in L(Q_\xi) \\ f\not =0}} \dfrac{\|f\|_{L_\infty^{\sqrt{g}}({\mathbb R}^d)}}{\| f\|_{{\mathcal H}^\alpha}} &=\inf_{\substack{f\in L(Q_\xi) \\ f\not =0}}\frac{\big\|\sqrt{g}\sum_{{\boldsymbol{k}}\in Q_\xi}a_{\boldsymbol{k}}H_{\boldsymbol{k}}\big\|_{L_\infty({\mathbb R}^d)}}{\big(\sum_{{\boldsymbol{k}}\in Q_\xi} \rho_{\alpha,{\boldsymbol{k}}}|a_{\boldsymbol{k}}|^2\big)^{1/2}}. \end{aligned}$$ Denote by $a_m$ the $m$th Mhaskar-Rakhmanov-Saff number corresponding to $\sqrt{g}$. See [@Lu07 Page 11] for a definition of this number. From [@Lu07 Page 11] we have $$\label{a_m(g)} a_m \ = \ \sqrt{m}.$$ For any polynomial $\varphi$ of degree $\le m$ we have the Nikol'skii-type inequality [@Lu07 Theorem 9.1, Page 61] $$\label{eq-Nilkolski} \|\varphi \sqrt{g} \|_{L_2( {\mathbb R})} \ \le \ C a_m^{\frac{1}{2}}\, \|\varphi \sqrt{g} \|_{L_\infty( {\mathbb R})}\leq C m^{\frac{1}{4}}\, \|\varphi \sqrt{g} \|_{L_\infty( {\mathbb R})}.$$ Observe that, if $f\in L(Q_\xi)$ then $f$ is a polynomial of degree $\leq$ $2^\xi$ with respect to each variable $x_j$, $j=1,\ldots,d$. Applying the inequality [\[eq-Nilkolski\]](#eq-Nilkolski){reference-type="eqref" reference="eq-Nilkolski"} continuously with respect to $x_j$, $j=1,\ldots,d$, we get $$2^{\frac{d\xi}{4}}\Bigg\|\sqrt{g}\sum_{{\boldsymbol{k}}\in Q_\xi}a_{\boldsymbol{k}}H_{\boldsymbol{k}}\Bigg\|_{L_\infty({\mathbb R}^d)} \gg 2^{\frac{d\xi}{4}}\Bigg\|\sqrt{g}\sum_{{\boldsymbol{k}}\in Q_\xi}a_{\boldsymbol{k}}H_{\boldsymbol{k}}\Bigg\|_{L_2({\mathbb R}^d)}=2^{\frac{d\xi}{4}}\Bigg\|\sum_{{\boldsymbol{k}}\in Q_\xi}a_{\boldsymbol{k}}H_{\boldsymbol{k}}\Bigg\|_{L_2({\mathbb R}^d,\gamma)}\,.$$ Consequently, we find $$\begin{aligned} b_n(I_\gamma) & \gg \inf_{(a_{\boldsymbol{k}})_{{\boldsymbol{k}}\in Q_\xi}} \frac{2^{-\frac{d\xi}{4}}\big\|\sum_{{\boldsymbol{k}}\in Q_\xi}a_{\boldsymbol{k}}H_{\boldsymbol{k}}\big\|_{L_2({\mathbb R}^d,\gamma)}}{\big(\sum_{{\boldsymbol{k}}\in Q_\xi} \rho_{\alpha,{\boldsymbol{k}}}|a_{\boldsymbol{k}}|^2\big)^{1/2}} \\ & = \inf_{(a_{\boldsymbol{k}})_{{\boldsymbol{k}}\in Q_\xi}}\frac{2^{-\frac{\xi d}{4}}\big(\sum_{{\boldsymbol{k}}\in Q_\xi} |a_{\boldsymbol{k}}|^2\big)^{1/2}}{2^{\frac{\alpha}{2}\xi}\big(\sum_{{\boldsymbol{k}}\in Q_\xi} \frac{\rho_{\alpha,{\boldsymbol{k}}}}{2^{\alpha\xi }}|a_{\boldsymbol{k}}|^2\big)^{1/2}}\,. \end{aligned}$$ Since $\rho_{\alpha,{\boldsymbol{k}}}\ll 2^{\alpha \xi}$ for ${\boldsymbol{k}}\in Q_\xi$, we obtain $$b_n(I_\gamma)\gg 2^{-\frac{\alpha\xi}{2}-\frac{d\xi }{4}}.$$ Finally from the equivalence $n\asymp 2^\xi \xi^{d-1}$ we get $$b_n(I_\gamma) \gg n^{-\frac{\alpha}{2}-\frac{d}{4}} (\log n)^{(\frac{\alpha}{2}+\frac{d}{4})(d-1)}.$$ The proof is competed.

arxiv_math

--- abstract: | Let $p$ be a prime number and $K$ be a field with embeddings into $\mathbb{R}$ and $\mathbb{Q}_p$. We propose an algorithm that generates continued fraction expansions converging in $\mathbb{Q}_p$ and is expected to simultaneously converge in both $\mathbb{R}$ and $\mathbb{Q}_p$. This algorithm produces finite continued fraction expansions for rational numbers. In the case of $p=2$ and if $K$ is a quadratic field, the continued fraction expansions generated by this algorithm converge in $\mathbb{R}$, and they are eventually periodic or finite. For an element $\alpha$ in $K$, let $p_n/q_n$ denote the $n$-th convergent. There exist constants $u_1$ and $u_2$ in ${\mathbb R}_{>0}$ with $u_1 + u_2 = 2$, and constants $C_1$ and $C_2$ in ${\mathbb R}_{>0}$ such that $|\alpha - p_n/q_n| < C_1/|q_n|^{u_1}$ and $|\alpha - p_n/q_n|_2 < C_2/|q_n|^{u_2}$. Here, $|\cdot|_2$ represents the $2$-adic distance. For prime numbers $p > 2$, we present numerical experiences. author: - Shin-ichi Yasutomi title: Simultaneous Convergent Continued Fraction Algorithm for Real and $p$-adic Fields with Applications to Quadratic Fields --- [^1] [^2] # Introduction The continued fraction algorithm not only provides the best approximation in Diophantine approximation of real numbers, but also possesses various favorable properties and holds significant positions in various mathematical domains. Not only the regular continued fraction algorithm, but also continued fraction algorithms with various features have been proposed. Research on these algorithms remains active up to the present day. Let $p$ be a prime number and $\mathbb{Q}_p$ be the completion of $\mathbb{Q}$ with respect to the $p$-adic topology. For $u\in \mathbb{Q}_p$, let $v_p(u)$ be the valuation of $u$ and $|v|_p:=\frac{1}{p^{v_p(u)}}$. A continued fraction expansion algorithm in $\mathbb{Q}_p$ has yet to be discovered that rivals the regular continued fraction expansion algorithm in the real numbers. Mahler[@M] initiated the first attempt at $p$-adic continued fractions. Schneider [@S] and Ruban [@Ru] independently proposed different algorithms during the same period, both contributing significantly to the field of continued fraction expansion algorithms for $\mathbb{Q}_p$ (see for example [@Ro]). The continued fraction expansion algorithms by Schneider and Ruban are known to reveal that rational numbers do not necessarily possess finite expansions (see [@Bu], [@L], [@W]) and that quadratic irrationals do not always possess periodic expansions (see [@CVZ], [@O], [@We]). The algorithm we propose in this paper follows in the lineage of Ruban's continued fraction expansion algorithm. Let us introduce necessary notation. Let $J$ be a representative system modulo $p$. It is well known that every $u\in {\mathbb Q}_p$ can be written as $$\begin{aligned} u=\sum_{n\in \mathbb{Z}}c_np^n, \ \ c_n\in J,\end{aligned}$$ where $c_k=0$ for $k<v_p(u)$. We define $$\begin{aligned} \lfloor u \rfloor_p^{J}:=\sum_{n\in \mathbb{Z}_{\leq 0}}c_np^n,\quad \lceil u \rceil_p^{J}:=\sum_{n\in \mathbb{Z}_{< 0}}c_np^n.\end{aligned}$$ For the standard representative $J=\{0,1,\ldots,p-1\}$, we denote $\lfloor \cdot \rfloor_p^{J}$ and $\lceil \cdot \rceil_p^{J}$ by $\lfloor \cdot \rfloor_p$ and $\lceil \cdot \rceil_p$ respectively. Ruban's continued fraction algorithm is applied to $\alpha\in {\mathbb Q}_p$ as outlined below. Starting with $\alpha_0=\alpha$, we define sequences $\{a_n\}$ and $\{\alpha_n\}$ as follows: $$\begin{aligned} a_n=\lfloor \alpha_n \rfloor_p,\quad \alpha_{n+1}=\dfrac{1}{\alpha_{n}-a_n}.\end{aligned}$$ Browkin [@Br] defined a following algorithm similar to Ruban's by considering $J=\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\}$ when $p$ is an odd prime, demonstrating finite continued fraction expansions for rational numbers. Starting with $\alpha_0=\alpha$, we define sequences $\{a_n\}$ and $\{\alpha_n\}$ as follows: $$\begin{aligned} \label{Browkin0} a_n=\lfloor \alpha_n \rfloor_p^J,\quad \alpha_{n+1}=\dfrac{1}{\alpha_{n}-a_n}.\end{aligned}$$ In Browkin [@Br2], the following continued fraction algorithm is provided, and the defining expression varies depending on whether $n$ is even or odd. Let $\alpha\in {\mathbb Q}_p$ and $J=\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\}$. Starting with $\alpha_0=\alpha$, The sequences $\{a_n\}$ and $\{\alpha_n\}$ are defined as follows: $$\begin{aligned} \label{Browkin} &a_n=\begin{cases} \lfloor \alpha_n \rfloor_p^J,&\text{if $n$ is even,}\\ \lceil \alpha_n \rceil_p^{J},&\text{if $n$ is odd and $v_p(\alpha_n-\lceil \alpha_n \rceil_p^{J})=0$,}\\ \lceil \alpha_n \rceil_p^{J}-sign(\lceil \alpha_n \rceil_p^{J}),&\text{if $n$ is odd and $v_p(\alpha_n-\lceil \alpha_n \rceil_p^{J})\ne 0$,} \end{cases}\\ &\alpha_{n+1}=\dfrac{1}{\alpha_{n}-a_n}.\nonumber\end{aligned}$$ Browkin [@Br2] showed the experimental results about this algorithm. Barbero, Cerruti and Murru [@BCM] demonstrated that rational numbers have finite continued fraction expansions. Murru and Romeo [@MR] introduced the following modified algorithm, which is based on Algorithm ([\[Browkin\]](#Browkin){reference-type="ref" reference="Browkin"}). They demonstrated that this algorithm improves upon Browkin's algorithms in several aspects. Let $\alpha\in {\mathbb Q}_p$ and $J=\{-\frac{p-1}{2},\ldots,\frac{p-1}{2}\}$. Starting with $\alpha_0=\alpha$, The sequences $\{a_n\}$ and $\{\alpha_n\}$ are defined as follows: $$\begin{aligned} \label{MReq} &a_n=\begin{cases} \lfloor \alpha_n \rfloor_p^J,&\text{if $n$ is even,}\\ \lceil \alpha_n \rceil_p^{J},&\text{if $n$ is odd,} \end{cases}\\ &\alpha_{n+1}=\dfrac{1}{\alpha_{n}-a_n}.\nonumber\end{aligned}$$ Murru, Romeo, and Santilli [@MRS] defined a class including the algorithm mentioned in ([\[MReq\]](#MReq){reference-type="ref" reference="MReq"}), and discussed convergence properties as well as the finite expansion of rational numbers. Let $K$ be a field that has an embedding into $\mathbb{R}$ and $\mathbb{Q}_p$ respectively. Assume $\sigma_{\infty}$ gives an embedding into $\mathbb{R}$ and $\sigma_p$ gives an embedding into $\mathbb{Q}_p$. We define a novel algorithm inspired by the approaches in ([\[Browkin\]](#Browkin){reference-type="ref" reference="Browkin"}) and ([\[MReq\]](#MReq){reference-type="ref" reference="MReq"}), aiming to achieve simultaneous rational approximations in both $\mathbb{R}$ and $\mathbb{Q}_p$ for elements of $K$. While the process involves selecting integers that are close to given numbers, it is designed to maintain proximity within the respective topologies of $\mathbb{R}$ and $\mathbb{Q}_p$. Let $\alpha \in K$. We denote $\sigma_{\infty}(\alpha)$ by $\alpha_{\infty}$ (or alternatively, $(\alpha)_{\infty})$ and $\sigma_{p}(\alpha)$ by $\alpha_{\langle p \rangle}$ (or alternatively, $(\alpha)_{\langle p \rangle})$. For a vector $\mathbf{a}=(x_1,\ldots,x_n)\in K^n$, we denote $((x_1)_{\infty},\ldots,(x_n)_{\infty})$ by $\mathbf{a}_{\infty}$(or alternatively, $(\mathbf{a})_{\infty}$) and $((x_1)_{\langle p \rangle},\ldots,(x_n)_{\langle p \rangle})$ by $\mathbf{a}_{\langle p \rangle}$(or alternatively, $(\mathbf{a})_{\langle p \rangle}$). **Definition 1**. We define a transformation $F_{p,0}$ by for $\alpha\in K$ $$\begin{aligned} F_{p,0}(\alpha):=\dfrac{1}{\alpha-mp-\left\lfloor\alpha_{\langle p \rangle} \right\rfloor_p},\end{aligned}$$ where among $i \in \mathbb{Z}$ for which $|\alpha_{\infty}-ip-\left\lfloor\alpha_{\langle p \rangle} \right\rfloor_p|$ attains its minimum, let $m$ be the smallest value. We define $b^{(0)}(\alpha):=mp+\left\lfloor \alpha_{\langle p \rangle} \right\rfloor_p$. We define a transformation $F_{p,1}$ by for $\alpha\in K$ $$\begin{aligned} F_{p,1}(\alpha):=\dfrac{1}{\alpha-m-\left\lfloor \alpha_{\langle p \rangle} \right\rfloor_p},\end{aligned}$$ where among $i \in \mathbb{Z}$ for which $|\alpha_{\infty}-i-\left\lfloor \alpha_{\langle p \rangle} \right\rfloor_p|$ attains its minimum, let $m$ be the smallest value. We define $b^{(1)}(\alpha):=m+\left\lfloor \alpha_{\langle p \rangle} \right\rfloor_p$. *Remark 1*. We remark that $F_{p,0}$ and $F_{p,1}$ are determined independently of the representative system modulo $p$. **Definition 2**. We define an algorithm as follows: Let $\alpha=\alpha_0 \in K$. We define $\{\alpha_n\}$ and $\{b_n(\alpha)\}$recursively as:\ $$\begin{aligned} &\alpha_{n+1}:=\begin{cases} F_{p,0}(\alpha_{n})& \text{if $n\equiv 0 \mod\ 2$},\\ F_{p,1}(\alpha_{n})& \text{if $n\equiv 1 \mod\ 2$},\\ \end{cases}\\ &\text{and}\\ &b_{n}(\alpha):=\begin{cases} b^{(0)}(\alpha_n)& \text{if $n\equiv 0 \mod\ 2$},\\ b^{(1)}(\alpha_n)& \text{if $n\equiv 1 \mod\ 2$}. \end{cases}\end{aligned}$$ The algorithm halts if $F_{p,0}(\alpha_{n})$ or $F_{p,1}(\alpha_{n})$ is not defined, resulting in $\alpha_{n+1},\ldots$ not being defined. By this algorithm, for $\alpha \in K$, the sequence $\{b_n(\alpha)\}_{n\in \mathbb{Z}{\geq 0}}$ is generated, allowing us to consider its formal continued fraction expansion $\left[b_0(\alpha);b_1(\alpha),\ldots\right]$, namely, $$\begin{aligned} b_0(\alpha)+\cfrac{1}{b_1(\alpha) + \cfrac{1}{b_2(\alpha) + \cfrac{1}{b_3(\alpha)+\cfrac{1}{\ldots}}}}.\end{aligned}$$ We define convergents of continued fraction in the usual manner. **Definition 3**. For $n\in \mathbb{Z}_{> 0}$, $p_n$ and $q_n$ are defined by $$\begin{aligned} \begin{pmatrix} p_{n}&p_{n-1}\\ q_{n}&q_{n-1} \end{pmatrix} = \begin{pmatrix} b_0(\alpha)&1\\ 1&0 \end{pmatrix} \begin{pmatrix} b_1(\alpha)&1\\ 1&0 \end{pmatrix} \cdots \begin{pmatrix} b_n(\alpha)&1\\ 1&0 \end{pmatrix},\end{aligned}$$ where we define $p_{-1}=1$ and $q_{-1}=0$. We note that $p_n$ and $q_n$ are in $\mathbb{Z}[\frac{1}{p}]$ for all $n\in \mathbb{Z}_{\geq 0}$. There are several studies that have dealt with continued fraction expansions that converge in both $\mathbb{Q}_p$ and $\mathbb{R}$ (e.g., [@ABCM], [@BE], [@P]). The discussion on the periodicity of Ruban's continued fraction expansions is effectively utilized in [@CVZ] by embedding them into real numbers. Bekki [@Bekki] proposed a continued fraction algorithm for complex quadratic irrationals, considering reductions in both the $\mathbb{Q}_p$ and $\mathbb{C}$ and demonstrated that these numbers have an eventually periodic continued fraction expansion. It is not difficult to see the convergence of our algorithm in $\mathbb{Q}_p$. We demonstrate that every rational number has a finite continued fraction expansion. We show that for irrational numbers in $K$, the continued fraction expansion of each converges in $\mathbb{R}$ and eventually becomes periodic when $p=2$ and $K$ is a real quadratic field. This corresponds to Lagrange's theorem for regular continued fractions (see for example [@HW]). We also demonstrate that the accuracy of approximation is evaluated by utilizing the denominators of the convergents when $p=2$ and $K$ is a real quadratic field. Ridout [@Ri] extended Roth's theorem to establish limits on simultaneous approximations of algebraic numbers, both in the field of real numbers and across several $p$-adic fields. Our algorithm provide the extent of approximation that approaches the boundary of those limits. Bedocchi [@Be] showed a condition for purely periodic expansions with Algorithm ([\[Browkin0\]](#Browkin0){reference-type="ref" reference="Browkin0"}) for numbers with eventually periodic expansions. for numbers with eventually periodic expansions. Similarly, Murru and Romeo [@MR] showed a condition for purely periodic expansions with respect to Algorithm ([\[MReq\]](#MReq){reference-type="ref" reference="MReq"}) for numbers possessing eventually periodic expansions. These results parallel Galois's theorem for regular continued fractions. We give a necessary condition for the continued fraction expansion of numbers in relation to our algorithm to become purely periodic. In the determination of the periodic points in continued fraction expansions, the natural extension associated with the dynamical system related to the algorithm is commonly employed (see for example [@DK]). Regarding our algorithm, we construct an analogue of the natural extension. Now, it is important to note that our algorithm is defined even when $K$ is embedded into $\mathbb{C}$. From numerical computation examples, it suggests that the algorithm works in such cases as well. We remark that since there exist embeddings from $\mathbb{Q}_p$ to $\mathbb{C}$, our algorithm can be interpreted as an algorithm over $\mathbb{Q}_p$. On the other hand, there are also approaches of deriving the continued fraction expansion using algebraic conditions of elements in $K$ (see [@Br2], [@STY], [@STY2]). # Fundamental properties In this chapter, we outline the fundamental properties of our algorithm. **Lemma 1**. *Let $\alpha\in K$. Then, for even $n\in \mathbb{Z}_{> 0}$, we have $v_p((\alpha_{n})_{\langle p \rangle})\leq 0$. For odd $n\in \mathbb{Z}_{> 0}$, it holds that $v_p((\alpha_{n})_{\langle p \rangle})< 0$. For all $n\in \mathbb{Z}_{>0}$, $v_p((\alpha_{n})_{\langle p \rangle})=v_p(b_n(\alpha))$.* *Proof.* Since it holds that for some integer $m$, $\alpha-b^{(0)}(\alpha)=\alpha-mp-\left\lfloor \alpha_{\langle p \rangle} \right\rfloor_p$, we have $v_p(\alpha_{\langle p \rangle}-b^{(0)}(\alpha))>0$. Therefore, if $\alpha-b^{(0)}(\alpha)\ne 0$, we have $v_p((\alpha_{1})_{\langle p \rangle})=v_p(\frac{1}{\alpha_{\langle p \rangle}-b^{(0)}(\alpha)})<0$. Since it holds that for some integer $m'$, $\alpha_1-b^{(1)}(\alpha_1)=\alpha_1-m'-\left\lfloor (\alpha_1)_{\langle p \rangle} \right\rfloor_p$, we have $v_p(\alpha_1-b^{(1)}(\alpha_1))\geq 0$. Therefore, from the fact that $v_p((\alpha_1)_{\langle p \rangle})<0$, we have $v_p(b^{(1)}(\alpha_1))=v_p((\alpha_1)_{\langle p \rangle})$. Similarly, the proof of the theorem can be established inductively. ◻ **Lemma 2**. *For all $n\in \mathbb{Z}_{> 0}$, $\displaystyle v_p(q_n)=\sum_{k=1}^nv_p(b_k(\alpha))$.* *Proof.* We will prove the claim by induction on $n$. By the definition, we have $v_p(q_1)=v_p(b_1(\alpha))$. From the fact that $q_2=b_2(\alpha)b_1(\alpha)+1$ and Lemma [Lemma 1](#lemord){reference-type="ref" reference="lemord"}, we have $v_p(q_2)=v_p(b_1(\alpha))+v_p(b_2(\alpha))$. Let $n\geq 2$. We assume that the clam holds for $n$ and $n-1$. From Lemma [Lemma 1](#lemord){reference-type="ref" reference="lemord"}, we have $v_p(b_{n+1}(\alpha))+v_p(b_{n}(\alpha))<0$. Therefore, we have $v_p(q_{n+1})=v_p(b_{n+1}(\alpha)q_{n}+q_{n-1})=v_p(b_{n+1}(\alpha)q_{n})=\sum_{k=1}^{n+1}v_p(b_k(\alpha))$. ◻ By Lemmas [Lemma 1](#lemord){reference-type="ref" reference="lemord"} and [Lemma 2](#convergentsp){reference-type="ref" reference="convergentsp"}, we observe that $q_n \ne 0$ for all $n \in \mathbb{Z}_{> 0}$, implying that $\frac{p_n}{q_n}$ are well-defined. Following lemma gives a sufficient condition for the convergence of continued fractions. **Lemma 3** ([@MRS]). *Let $b_0, b_1,\ldots \in \mathbb{Z}[\frac{1}{p}]$ be an infinite sequence such that $$\begin{aligned} v_p(b_nb_{n+1})<0,\end{aligned}$$ for all $n>0$. Then, the continued fraction $\left[b_0; b_1, \ldots\right]$ is convergent to a $p$-adic number.* From Lemma [Lemma 1](#lemord){reference-type="ref" reference="lemord"} and [Lemma 3](#MRS){reference-type="ref" reference="MRS"}, we see that $\left[b_0(\alpha);b_1(\alpha),\ldots\right]$ is convergent to a $p$-adic number. The equality $\alpha_{\langle p \rangle}=\left[b_0(\alpha); b_1(\alpha),\ldots\right]$ can be obtained later. **Lemma 4**. *Let $\alpha\in K$. Then, $\alpha_{\infty}-b^{(0)}(\alpha) \in \left(-\frac{p}{2},\frac{p}{2}\right]$ and $\alpha_{\infty}-b^{(1)}(\alpha) \in \left(-\frac{1}{2},\frac{1}{2}\right]$.* *Proof.* From Definition [Definition 1](#transformation){reference-type="ref" reference="transformation"} we see that $$\begin{aligned} &|\alpha_{\infty}-b^{(0)}(\alpha)|=\min \{|\alpha_{\infty}-ip-\left\lfloor\alpha_{\langle p \rangle} \right\rfloor_p|\mid i\in \mathbb{Z}\},\\ &|\alpha_{\infty}-b^{(1)}(\alpha)|=\min \{|\alpha_{\infty}-i-\left\lfloor\alpha_{\langle p \rangle} \right\rfloor_p|\mid i\in \mathbb{Z}\},\end{aligned}$$ which imply that $|\alpha_{\infty}-b^{(0)}(\alpha)|\leq \frac{p}{2}$ and $|\alpha_{\infty}-b^{(1)}(\alpha)|\leq \frac{1}{2}$. If it holds that $\alpha_{\infty}-b^{(0)}(\alpha)=-\frac{p}{2}$, since it holds that $\alpha_{\infty}-(b^{(0)}(\alpha)-p)=\frac{p}{2}$, which contradicts the definition of $b^{(0)}(\alpha)$. Therefore, we have $\alpha_{\infty}-b^{(0)}(\alpha) \in \left(-\frac{p}{2},\frac{p}{2}\right]$. Similarly, we have $\alpha_{\infty}-b^{(1)}(\alpha) \in \left(-\frac{1}{2},\frac{1}{2}\right]$. ◻ We will present the following theorem using the same argument as in [@MR]. **Theorem 5**. *Let $\alpha\in \mathbb{Q}$. Then, $\{\alpha_n\}$ is a finite sequence.* *Proof.* We assume that $\{\alpha_n\}$ is an infinite sequence. Let $n\in \mathbb{Z}_{> 0}$. From Lemma [Lemma 1](#lemord){reference-type="ref" reference="lemord"}, we can set $$\begin{aligned} &\alpha_{n}=\dfrac{N_{n}}{D_{n}p^{i_n}}, \ \ \text{with $(N_{n},D_{n})=1$ and $p \nmid N_{n}D_{n}$,}\label{ND}\\ &b_n(\alpha)=\dfrac{c_n}{p^{i_n}}, \ \ \text{with $p \nmid c_n$} \nonumber,\end{aligned}$$ where $N_{n}$, $D_{n}$, and $c_n$ are integers. From Lemma [Lemma 1](#lemord){reference-type="ref" reference="lemord"}, we have $i_{2n}\geq 0$, $i_{2n+1}> 0$, and $i_{2n+2}\geq 0$. From the fact that $\alpha_{k+1}=\frac{1}{\alpha_{k}-b_k(\alpha)}$ for $k=2n, 2n+1$, we have $$\begin{aligned} &N_{2n+1}(N_{2n}-c_{2n}D_{2n})=p^{i_{2n}+i_{2n+1}}D_{2n+1}D_{2n},\label{ND2}\\ &N_{2n+2}(N_{2n+1}-c_{2n+1}D_{2n+1})=p^{i_{2n+1}+i_{2n+2}}D_{2n+2}D_{2n+1}.\nonumber\end{aligned}$$ From ([\[ND\]](#ND){reference-type="ref" reference="ND"}) and ([\[ND2\]](#ND2){reference-type="ref" reference="ND2"}), we have $$\begin{aligned} &|N_{2n+1}|=|D_{2n}|,\\ &|N_{2n+2}|=|D_{2n+1}|,\\ &p^{i_{2n}+i_{2n+1}}|D_{2n+1}|=|N_{2n}-c_{2n}D_{2n}|,\\ &p^{i_{2n+1}+i_{2n+2}}|D_{2n+2}|=|N_{2n+1}-c_{2n+1}D_{2n+1}|.\end{aligned}$$ Therefore, using Lemma [Lemma 4](#ineq1){reference-type="ref" reference="ineq1"}, we have $$\begin{aligned} &p^{i_{2n}+i_{2n+1}}|D_{2n+1}|=|N_{2n}-c_{2n}D_{2n}|\\ &=p^{i_{2n}}|D_{2n}|\left|\dfrac{N_{2n}}{D_{2n}p^{i_{2n}}}-\dfrac{c_{2n}}{p^{i_{2n}}}\right|\\ &\leq \dfrac{p^{i_{2n}+1}|D_{2n}|}{2},\end{aligned}$$ which implies $$\begin{aligned} \label{D2n+1} |D_{2n+1}|\leq \dfrac{|D_{2n}|}{2p^{i_{2n+1}-1}}.\end{aligned}$$ Similarly, we have $$\begin{aligned} &p^{i_{2n+1}+i_{2n+2}}|D_{2n+2}|=|N_{2n+1}-c_{2n+1}D_{2n+1}|\\ &=p^{i_{2n+1}}|D_{2n+1}|\left|\dfrac{N_{2n+1}}{p^{i_{2n+1}}D_{2n+1}}-\dfrac{c_{2n+1}}{p^{i_{2n+1}}}\right|\\ &<p^{i_{2n+1}}|D_{2n+1}|,\end{aligned}$$ which implies $$\begin{aligned} \label{D2n+2} |D_{2n+2}|<\dfrac{|D_{2n+1}|}{p^{i_{2n+2}}}.\end{aligned}$$ From Lemma [Lemma 1](#lemord){reference-type="ref" reference="lemord"}, we have $i_{2n+1}\geq 1$ and $i_{2n+2}\geq 0$. Therefore, by inequalities ([\[D2n+1\]](#D2n+1){reference-type="ref" reference="D2n+1"}) and ([\[D2n+2\]](#D2n+2){reference-type="ref" reference="D2n+2"}), we obtain inequalities $$\begin{aligned} |D_{2n+1}|\leq \dfrac{|D_{2n}|}{2} \text{ and }|D_{2n+2}|< |D_{2n+1}|.\end{aligned}$$ Therefore, the sequence $\{|D_n|\}$ is strictly decreasing, which is a contradiction. ◻ **Example 1**. We give the continued fraction expansion of $\dfrac{5}{13}$ for some prime numbers by the algorithm. $$\begin{aligned} \begin{array}{ll} \left[1;-\dfrac{13}{8}\right]&\text{ $p=2$},\\ &\\ \left[-1;\dfrac{2}{9},2\right]&\text{ $p=3$},\\ &\\ \left[0;\dfrac{13}{5}\right]&\text{ $p=5$},\\ &\\ \left[2;-\dfrac{2}{7},-3\right]&\text{ $p=7$}. \end{array}\end{aligned}$$ By considering the continued fraction expansion of $\alpha-b_0(\alpha)$, we are able to confine the initial point to a limited range, thus allowing us to investigate the counterpart of the Gauss map in the regular continued fraction. **Definition 4**. We define a map $T_{p,0}:K^{\times}\to K$ by for $\alpha\in K^{\times}$ $$\begin{aligned} T_{p,0}(\alpha):=\dfrac{1}{\alpha}-mp-\left\lfloor \left(\dfrac{1}{\alpha}\right)_{\langle p \rangle} \right\rfloor_p,\end{aligned}$$ where among $i \in \mathbb{Z}$ for which $|\frac{1}{\alpha}-ip-\lfloor \left(\frac{1}{\alpha}\right)_{\langle p \rangle} \rfloor_p|$ attains its minimum, let $m$ be the smallest value. We define $a^{(0)}(\alpha):=mp+\left\lfloor \frac{1}{\alpha} \right\rfloor_p$. We define a map $T_{p,1}:K^{\times}\to K$ by for $\alpha\in K^{\times}$ $$\begin{aligned} T_{p,1}(\alpha):=\dfrac{1}{\alpha}-m-\left\lfloor \left(\dfrac{1}{\alpha}\right)_{\langle p \rangle} \right\rfloor_p,\end{aligned}$$ where among $i \in \mathbb{Z}$ for which $|\frac{1}{\alpha}-i-\left\lfloor (\frac{1}{\alpha})_{\langle p \rangle} \right\rfloor_p|$ attains its minimum, let $m$ be the smallest value. We define $a^{(1)}(\alpha):=m+\left\lfloor (\frac{1}{\alpha})_{\langle p \rangle} \right\rfloor_p$. From Lemma [Lemma 4](#ineq1){reference-type="ref" reference="ineq1"}, we observe that $\alpha_{\infty}-b^{(0)}(\alpha) \in \left(-\frac{p}{2},\frac{p}{2}\right]$ and it is evident that $v_p(\alpha-b_0(\alpha))>0$. **Definition 5**. We say that $\alpha$ has *property $I$*, if it satisfies $\alpha\in \left(-\frac{p}{2},\frac{p}{2}\right]$ and $v_p(\alpha_{\langle p \rangle})>0$. Let $\alpha\in K/\mathbb{Q}$ with property $I$. **Definition 6**. We define an algorithm as follows: Let $\alpha=\alpha_{(1)} \in K$. We define $\{\alpha_{(n)}\}_{n\in \mathbb{Z}_{> 0}}$ and $\{a_n(\alpha)\}_{n\in \mathbb{Z}_{> 0}}$recursively as follows:\ If $\alpha_{(n)}\ne 0$, $$\begin{aligned} &\alpha_{(n+1)}:=\begin{cases} T_{p,0}(\alpha_{(n)})& \text{if $n\equiv 0 \mod\ 2$},\\ T_{p,1}(\alpha_{(n)})& \text{if $n\equiv 1 \mod\ 2$},\\ \end{cases}\\ &\text{and}\\ &a_{n}(\alpha):=\begin{cases} a^{(0)}(\alpha_n)& \text{if $n\equiv 0 \mod\ 2$},\\ a^{(1)}(\alpha_n)& \text{if $n\equiv 1 \mod\ 2$}. \end{cases}\end{aligned}$$ If $\alpha_{n}=0$, terminate. Namely, we consider the following continued fraction expansion: $$\begin{aligned} \cfrac{1}{a_1(\alpha) + \cfrac{1}{a_2(\alpha) + \cfrac{1}{a_3(\alpha)+\cfrac{1}{\ldots}}}}.\end{aligned}$$ We define convergents of continued fraction in the usual manner. In the sequel, we assume that $\alpha\in K/\mathbb{Q}$ with property $I$. **Definition 7**. For $n\in \mathbb{Z}_{> 0}$, $p_n$ and $q_n$ are defined by $$\begin{aligned} \begin{pmatrix} p_{n-1}&p_{n}\\ q_{n-1}&q_{n} \end{pmatrix} = \begin{pmatrix} 0&1\\ 1&a_1(\alpha) \end{pmatrix} \begin{pmatrix} 0&1\\ 1&a_2(\alpha) \end{pmatrix} \cdots \begin{pmatrix} 0&1\\ 1&a_n(\alpha) \end{pmatrix},\end{aligned}$$ where we define $p_{0}=0$ and $q_{0}=1$. We note that $p_n$ and $q_n$ are in $\mathbb{Z}[\frac{1}{p}]$ for all $n\in \mathbb{Z}_{\geq 0}$. **Lemma 6**. *Let $\alpha\in K$ with property $I$. Then, $b^{(0)}(\alpha)=0$.* *Proof.* From the fact that $v_p(\alpha_{\langle p \rangle})>0$, we have $\left\lfloor \alpha_{\langle p \rangle} \right\rfloor_p=0$. Since $-\frac{p}{2}<\alpha_{\infty}\leq \frac{p}{2}$, we have $b^{(0)}(\alpha)=p\min\{i \mid |\alpha_{\infty}-ip|=\min\{|\alpha_{\infty}-jp|\mid j\in \mathbb{Z}\}\}=0$. ◻ **Corollary 7**. *Let $\alpha\in \mathbb{Q}$ with property $I$. Then, $\{\alpha_{(n)}\}$ is a finite sequence.* *Proof.* From Theorem [Theorem 5](#rational){reference-type="ref" reference="rational"}, we see that $\{\alpha_{n}\}$ is a finite sequence, where the sequence is generated by Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}) to $\alpha$. Let $\{\alpha_{n}\}=\{\alpha_{n}\}_{n=0}^k$. From Lemma [Lemma 6](#twoalgol){reference-type="ref" reference="twoalgol"}, we have $\alpha_1=\frac{1}{\alpha_{0}}=\frac{1}{\alpha_{(1)}}$. Through induction, we establish the following for $n=1,\ldots, k$: $\alpha_k=\frac{1}{\alpha_{(k)}}$. Considering the fact that $\alpha_{k}-b_{k}(\alpha)=0$, we can conclude that $\{\alpha_{(n)}\}=\{(\alpha_{(n)})_{n=1}^k\}$. ◻ The following lemmas are evident from Definitions [Definition 4](#transformationT){reference-type="ref" reference="transformationT"} and [Definition 6](#algol2){reference-type="ref" reference="algol2"}. **Lemma 8**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. Then, for $n\in \mathbb{Z}_{> 0}$, if $n\equiv 0 \mod\ 2$, then $|(\alpha_{(n)})_{\infty}|<\frac{1}{2}$and if $n\equiv 1 \mod\ 2$, then $|(\alpha_{(n)})_{\infty}|<\dfrac{p}{2}$.* **Lemma 9**. *Let $\alpha\in K$. Then, for even $n\in \mathbb{Z}_{> 0}$, we have $v_p((\alpha_{(n)})_{\langle p \rangle})\geq 0$. For odd $n\in \mathbb{Z}_{> 0}$, it holds that $v_p((\alpha_{(n)})_{\langle p \rangle})> 0$. For all $n\in \mathbb{Z}_{> 0}$, $v_p((\alpha_{(n)})_{\langle p \rangle})=-v_p(a_n(\alpha))$.* The following lemma can be proven in a similar manner to Lemma [Lemma 2](#convergentsp){reference-type="ref" reference="convergentsp"}. **Lemma 10**. *For all $n\in \mathbb{Z}_{> 0}$, $\displaystyle v_p(q_n)=\sum_{k=1}^nv_p(a_k(\alpha))$ and $\displaystyle v_p(p_n)=\sum_{k=2}^nv_p(a_k(\alpha))$.* By Lemma [Lemma 10](#convergentsp2adic){reference-type="ref" reference="convergentsp2adic"}, it is established that for all $n\in \mathbb{Z}_{> 0}$, the convergents $\frac{p_n}{q_n}$ are well defined and are not equal to $0$. Therefore, as it is well-known, for all $n\in \mathbb{Z}_{> 0}$, we have $$\begin{aligned} =\dfrac{p_n}{q_n}.\end{aligned}$$ We extend ${T}_{p,\epsilon}$ for $\epsilon\in \{0,1\}$ as a skew product on $K^2$ as follows: **Definition 8**. For $\epsilon\in \{0,1\}$, we define maps $\hat{T}_{p,\epsilon}:(K^{\times})^2\to K^2$ as for $(\alpha,\beta)\in (K^{\times})^2$ $$\begin{aligned} \hat{T}_{p,\epsilon}(\alpha,\beta):=\left(T_{p,\epsilon}(\alpha),\dfrac{1}{\beta}-a^{(\epsilon)}(\alpha)\right).\end{aligned}$$ Using $\hat{T}_{2,\epsilon}$ for $\epsilon\in \{0,1\}$, for $\mathbf{a}=(\alpha,\beta)\in K^2$, we define the following algorithm. **Definition 9**. We define an algorithm as follows: Let $\mathbf{a}=(\alpha,\beta)=\mathbf{a}_{(1)} \in K^2$. We define $\{\mathbf{a}_{(n)}\}_{n\in \mathbb{Z}_{\geq 1}}$ recursively as:\ If $\mathbf{a}_{(n)}\in (K^{\times})^2$, $$\begin{aligned} &\mathbf{a}_{(n+1)}:=\begin{cases} \hat{T}_{p,0}(\mathbf{a}_{(n)})& \text{if $n\equiv 0 \mod\ 2$},\\ \hat{T}_{p,1}(\mathbf{a}_{(n)})& \text{if $n\equiv 1 \mod\ 2$}. \end{cases}\end{aligned}$$ If $\mathbf{a}_{(n)}\notin (K^{\times})^2$, terminate. We note that the first coordinate of $\mathbf{a}_{(n)}$ is $\alpha_{(n)}$. We define $\tau:\mathbb{Z}\to \{0,1\}$ as follows: for $n\in \mathbb{Z}$, $$\begin{aligned} \tau(n):=\begin{cases} 0 & \text{if $n$ is even},\\ 1 & \text{if $n$ is odd}. \end{cases} \end{aligned}$$ **Definition 10**. We define $$\begin{aligned} &D_{p}^0:=\{(x,y)\in \mathbb{Q}_p^2\mid v_p(x)\geq 0, v_p(y)<0\},\\ &D_{p}^1:=\{(x,y)\in \mathbb{Q}_p^2\mid v_p(x)>0, v_p(y)\leq 0\}.\end{aligned}$$ **Lemma 11**. *Let $(\alpha,\beta)\in (K/\mathbb{Q})\times K^{\times}$. Then,* 1. *If $(\alpha_{\langle p \rangle},\beta_{\langle p \rangle})\in D_{p}^0$, then $(\hat{T}_{p,0}(\alpha,\beta))_{\langle p \rangle}\in D_{p}^1$.* 2. *If $(\alpha_{\langle p \rangle},\beta_{\langle p \rangle})\in D_{p}^1$, then $(\hat{T}_{p,1}(\alpha,\beta))_{\langle p \rangle}\in D_{p}^0$.* *Proof.* (1) We assume that $(\alpha_{\langle p \rangle},\beta_{\langle p \rangle})\in D_{p}^0$. Then, $\hat{T}_{p,0}(\alpha,\beta)=(\frac{1}{\alpha}-a^{(0)}(\alpha),\frac{1}{\beta}-a^{(0)}(\alpha))$. By Definition [Definition 4](#transformationT){reference-type="ref" reference="transformationT"}, we see that $v_p((\frac{1}{\alpha}-a^{(0)}(\alpha))_{\langle p \rangle})>0$. Since $v_p(a^{(0)}(\alpha))=v_p((\frac{1}{\alpha})_{\langle p \rangle})\leq 0$ and $v_p((\frac{1}{\beta})_{\langle p \rangle})>0$, we have $v_p((\frac{1}{\beta}-a^{(0)}(\alpha))_{\langle p \rangle})\leq 0$. Thus, we have claim (1).\ (2) We assume that $(\alpha_{\langle p \rangle},\beta_{\langle p \rangle})\in D_{p}^1$. Then, $\hat{T}_{p,1}(\alpha,\beta)=(\frac{1}{\alpha}-a^{(1)}(\alpha),\frac{1}{\beta}-a^{(1)}(\alpha))$. By Definition [Definition 4](#transformationT){reference-type="ref" reference="transformationT"}, we see that $v_p((\frac{1}{\alpha}-a^{(1)}(\alpha))_{\langle p \rangle})\geq 0$. Since $v_p((\frac{1}{\alpha})_{\langle p \rangle})<0$, we have $v_p(a^{(1)}(\alpha))<0$. Therefore, since $v_p((\frac{1}{\beta})_{\langle p \rangle})\geq 0$, we conclude that $v_p((\frac{1}{\beta}-a^{(1)}(\alpha))_{\langle p \rangle})=v_p((a^{(1)}(\alpha))_{\langle p \rangle})<0$. Thus, we have claim (2). ◻ **Theorem 12**. *Let $\alpha\in K$ have property $I$. Then, in $\mathbb{Q}_p$ $$\begin{aligned} \lim_{n\to \infty} \dfrac{p_n}{q_n}=\alpha_{\langle p \rangle}.\end{aligned}$$* *Proof.* If $\alpha\in \mathbb{Q}$, then from Corollary [Corollary 7](#rational2){reference-type="ref" reference="rational2"}, the claim of the theorem holds. We assume that $\alpha\in K/\mathbb{Q}$. Let $n\in \mathbb{Z}_{> 0}$. Let $\mathbf{a}=(\alpha,\frac{p_n}{q_n})$. We apply Algorithm (Definition [Definition 9](#algol3){reference-type="ref" reference="algol3"}) to $\mathbf{a}$. By induction, we obtain the following:\ For $1\leq j \leq n$, $$\begin{aligned} \mathbf{a}_{(j)}=(\alpha_{(j)},[0;a_j(\alpha),a_{j+1}(\alpha),\ldots,a_{n}(\alpha)]).\end{aligned}$$ and $$\begin{aligned} \mathbf{a}_{(n+1)}=(\alpha_{(n+1)},0).\end{aligned}$$ We put $(u_k.v_k)=(\mathbf{a}_{(k)})_{\langle p \rangle}$ for $1\leq k \leq n+1$. For $k\in \mathbb{Z}_{> 0}$, we see that $\displaystyle u_{k+1}=\frac{1}{u_{k}}-a_k(\alpha)$ and $\displaystyle v_{k+1}=\frac{1}{v_{k}}-a_k(\alpha)$ , which implies $$\begin{aligned} |u_k-v_k|_p=|u_kv_k|_p|u_{k+1}-v_{k+1}|_p.\end{aligned}$$ Therefore, for $n\in \mathbb{Z}_{> 0}$, we have $$\begin{aligned} \label{u0v0u0v0} |u_1-v_1|_p=|u_1|_p|v_1|_p\ldots |u_n|_p|v_n|_p|u_{n+1}-v_{n+1}|_p=|u_1|_p|v_1|_p\ldots |u_n|_p|v_n|_p |u_{n+1}|_p.\end{aligned}$$ From Lemma [Lemma 9](#lemord2){reference-type="ref" reference="lemord2"} and [Lemma 10](#convergentsp2adic){reference-type="ref" reference="convergentsp2adic"}, we observe that $|u_k|_p \leq \frac{1}{p}$ and $|v_k|_p \leq \frac{1}{p}$ hold for odd $k$, and $|u_k|_p \leq 1$ and $|v_k|_p \leq 1$ hold for even $k$ for $1\leq k \leq n+1$. Therefore, from the equality ([\[u0v0u0v0\]](#u0v0u0v0){reference-type="ref" reference="u0v0u0v0"}), we have $$\begin{aligned} \left|\alpha_{\langle p \rangle}-\dfrac{p_n}{q_n}\right|_p\leq \dfrac{1}{p^n},\end{aligned}$$ which implies $\lim_{n\to \infty} \dfrac{p_n}{q_n}=\alpha_{\langle p \rangle}$ in $\mathbb{Q}_p$. ◻ **Corollary 13**. *Let $\alpha\in K$. For $n\in \mathbb{Z}_{\geq 0}$, let $\frac{p_n}{q_n}$ be the convergent of $\alpha$ related to Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}). Then, in $\mathbb{Q}_p$ $$\begin{aligned} \lim_{n\to \infty}\dfrac{p_n}{q_n}=\alpha_{\langle p \rangle}.\end{aligned}$$* *Proof.* Since $\beta=\alpha-b_0(\alpha)$ has property $I$, we can deduce that in $\mathbb{Q}_p$ $$\begin{aligned} \displaystyle\lim_{n\to \infty} \dfrac{p'_n}{q'_n}=\beta_{\langle p \rangle},\end{aligned}$$ where $\frac{p'_n}{q'_n}$ represents the $n$-th convergent of $\beta$ obtained using Algorithm (Definition [Definition 6](#algol2){reference-type="ref" reference="algol2"}). Considering that the $\frac{p_n}{q_n}=b_0(\alpha)+\frac{p'_n}{q'_n}$ for $n\in \mathbb{Z}_{> 0}$, we arrive at the theorem's conclusion. ◻ **Example 2**. Let $\alpha$ be the root of $x^2-\dfrac{17}{5}x+\dfrac{63}{25}$ with $\alpha_{\infty}=\dfrac{17+\sqrt{37}}{10}$ and $\alpha_{\langle 7 \rangle}\equiv 2 \mod\ 7$. We give the continued fraction expansion of $\alpha$ for $p=7$ obtained using Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}) as follows: $$\begin{aligned} \left[2; \dfrac{22}{7}, \overline{12, -\dfrac{2}{7}, -5, -\dfrac{29}{7}, -12, \dfrac{2}{7}, 5, \dfrac{29}{7}}\right].\end{aligned}$$ According to Corollary [Corollary 13](#inftyp_nq_ncor){reference-type="ref" reference="inftyp_nq_ncor"}, this expansion converges to $\alpha_{\langle 7 \rangle}$. By utilizing the expansion's periodicity, we can compute the convergents of $\alpha_{\infty}$. This enables us to demonstrate the convergence of this expansion towards $\alpha_{\infty}$. # Case of $p=2$ In this section, we consider the case of $p=2$. Let $\alpha\in K/\mathbb{Q}$ have property $I$. In this chapter, we apply Algorithm (Definition [Definition 6](#algol2){reference-type="ref" reference="algol2"}) to $\alpha$. **Definition 11**. We define $$\begin{aligned} &I_{\infty}^0:=\{x\in \mathbb{R}\mid |x|\leq \frac{1}{2}\},\\ &I_{\infty}^1:=\{x\in \mathbb{R}\mid |x|\leq 1\},\\ &D_{\infty}^0:=\{(x,y)\in \mathbb{R}^2\mid |x|\leq \frac{1}{2},|x-y|> \frac{1}{2}\},\\ &D_{\infty}^1:=\{(x,y)\in \mathbb{R}^2\mid |x|\leq 1,|x-y|> 1\}.\end{aligned}$$ **Lemma 14**. *The following assertions hold:\ Let $(u,v)\in (K^{\times})^2$.* 1. *If $(u_{\infty},v_{\infty})\in D_{\infty}^0$, then $(\hat{T}_{2,0}(u,v))_{\infty}\in D_{\infty}^1.$* 2. *If $(u_{\infty},v_{\infty})\in D_{\infty}^1$, then $(\hat{T}_{2,1}(u,v))_{\infty}\in D_{\infty}^0.$* *Proof.* (1) We assume that $(u_{\infty},v_{\infty})\in D_{\infty}^0$. We assume that $u_{\infty}>0$ without loss of generality. Let $(u',v')=(\hat{T}_{2,0}(u,v))_{\infty}$. By Definition [Definition 4](#transformationT){reference-type="ref" reference="transformationT"} we have $$\begin{aligned} |u'|=\min\{\left|\frac{1}{u_{\infty}}-2i-\left\lfloor \left(\frac{1}{u}\right)_{\langle 2 \rangle} \right\rfloor_2\right|\mid i\in \mathbb{Z} \},\end{aligned}$$ which implies $|u'|\leq 1$. Since $(u',v')=(\frac{1}{u_{\infty}}-a^{(0)}(u),\frac{1}{v_{\infty}}-a^{(0)}(u))$, we have $$\begin{aligned} |u'-v'|=\left|\frac{1}{u_{\infty}}-\frac{1}{v_{\infty}}\right|=\left|\frac{u_{\infty}-v_{\infty}}{u_{\infty}v_{\infty}}\right|.\end{aligned}$$ If it holds that $v_{\infty}<0$, we would have $|u'-v'|>\frac{1}{u_{\infty}}\geq 2$. We assume $v_{\infty}>0$. Since $|u_{\infty}-v_{\infty}|> \frac{1}{2}$ and $u_{\infty},v_{\infty}>0$, there exists $\epsilon>0$ such that $v_{\infty}=u_{\infty}+\dfrac{1}{2}+\epsilon$. We see $$\begin{aligned} \label{uv} u_{\infty}v_{\infty}=u_{\infty}(u_{\infty}+\dfrac{1}{2}+\epsilon)\leq \dfrac{1}{2}(1+\epsilon)<\dfrac{1}{2}+\epsilon.\end{aligned}$$ Therefore, we have $$\begin{aligned} |u'-v'|=\left|\frac{\frac{1}{2}+\epsilon}{u_{\infty}v_{\infty}}\right|>1.\end{aligned}$$ (2) We assume that $(u_{\infty},v_{\infty})\in D_{\infty}^1$. We assume that $u_{\infty}>0$ without loss of generality. Let $(u',v')=\hat{T}_{2,1}(u_{\infty},v_{\infty})$. By Definition [Definition 4](#transformationT){reference-type="ref" reference="transformationT"} we have $$\begin{aligned} |u'|=\min\{\left|\frac{1}{u_{\infty}}-i-\left\lfloor \left(\frac{1}{u}\right)_{\langle 2 \rangle} \right\rfloor_2\right|\mid i\in \mathbb{Z}\},\end{aligned}$$ which implies $|u'|\leq \frac{1}{2}$. Similarly, we have $$\begin{aligned} |u'-v'|=\left|\frac{u_{\infty}-v_{\infty}}{u_{\infty}v_{\infty}}\right|.\end{aligned}$$ If it holds that $|v_{\infty}|\leq 2$, we would have $|u'-v'|=\frac{|u_{\infty}-v_{\infty}|}{|u_{\infty}v_{\infty}|}>\frac{1}{2}$. We assume that $|v_{\infty}|>2$. Then, we have $|u'-v'|=\dfrac{1}{u_{\infty}}-\dfrac{1}{v_{\infty}}>\dfrac{1}{u_{\infty}}-\dfrac{1}{2}\geq \dfrac{1}{2}$. ◻ *Remark 2*. The lemma is extended as follows by adding $\infty$ as a domain. We define $$\begin{aligned} &\mathcal{D}_{\infty}^0:=\{(x,y)\in \mathbb{R}\times (\mathbb{R}\cup \{\infty\})\mid |x|\leq \frac{1}{2},|x-y|> \frac{1}{2}\},\\ &\mathcal{D}_{\infty}^1:=\{(x,y)\in \mathbb{R}\times (\mathbb{R}\cup \{\infty\})\mid |x|\leq 1,|x-y|> 1\}.\end{aligned}$$ The maps $\hat{T}_{2,\epsilon} (\epsilon\in \{0,1\})$ are extended from their domain $(K^{\times})^2$ to $K^{\times}\times (K^{\times}\cup {\infty})$ in the usual manner. Then, we have:\ For $(u,v)\in K^{\times}\times (K^{\times}\cup {\infty})$, 1. If $(u_{\infty},v_{\infty})\in \mathcal{D}_{\infty}^0$, then $(\hat{T}_{2,0}(u,v))_{\infty}\in D_{\infty}^1,$ 2. If $(u_{\infty},v_{\infty})\in \mathcal{D}_{\infty}^1$, then $(\hat{T}_{2,1}(u,v))_{\infty}\in D_{\infty}^0.$ **Lemma 15**. *Let $\mathbf{a}=(\alpha,\beta)\in (K/\mathbb{Q})^2$ with $\alpha\ne\beta$. Then, either (1) or (2) holds.* 1. *There exists $n\in \mathbb{Z}_{> 0}$ such that $(\mathbf{a}_{(2n)})_{\infty}\in D_{\infty}^0$, which is equivalent to saying that there exists $n\in \mathbb{Z}_{> 0}$ such that $(\mathbf{a}_{(2n-1)})_{\infty}\in D_{\infty}^1$.* 2. *For all $n\in \mathbb{Z}_{>0}$, $(\mathbf{a}_{(2n)})_{\infty}\notin D_{\infty}^0$ and one of the following holds:* 1. *$\displaystyle \lim_{m\to \infty} (\mathbf{a}_{(2m)})_{\infty}=\left(\frac{1}{2},1\right)$ and $\displaystyle \lim_{m\to \infty} (\mathbf{a}_{(2m+1)})_{\infty}=(-1,-2)$.* 2. *$\displaystyle \lim_{m\to \infty} (\mathbf{a}_{(2m)})_{\infty}=\left(-\frac{1}{2},-1\right)$ and $\displaystyle \lim_{m\to \infty} (\mathbf{a}_{(2m+1)})_{\infty}=(1,2)$.* *Proof.* For each $n\in \mathbb{Z}_{>0}$, we put $(u_n,v_n):=(\mathbf{a}_{(n)})_{\infty}$. We assume that for all $n\in \mathbb{Z}_{>0}$, $(\mathbf{a}_{(2n)})_{\infty}\notin D_{\infty}^0$. From Lemma [Lemma 14](#Dinfty){reference-type="ref" reference="Dinfty"}, we see that for all $n\in \mathbb{Z}_{\geq 0}$, $(\mathbf{a}_{(2n+1)})_{\infty}\notin D_{\infty}^1$. Therefore, we have $$\begin{aligned} &|u_n-v_n|\leq \frac{1}{2} & \text{if $n\equiv 0 \mod\ 2$},\label{unvne}\\ &|u_n-v_n|\leq 1 & \text{if $n\equiv 1 \mod\ 2$}.\label{unvno}\end{aligned}$$ Since it holds that $|v_n|\leq |u_n|+|v_n-u_n|$ for each $n\in \mathbb{Z}_{> 0}$, we have $$\begin{aligned} \label{vnleq1} &|v_n|\leq 1, &\text{if $n\equiv 0 \mod\ 2$},\\ &|v_n|\leq 2, &\text{if $n\equiv 1 \mod\ 2$}.\label{vnleq2} \end{aligned}$$ For $n\in \mathbb{Z}_{> 0}$, we see that $\displaystyle u_{n+1}=\frac{1}{u_{n}}-a_n(\alpha)$ and $\displaystyle v_{n+1}=\frac{1}{v_{n}}-a_n(\alpha)$ , which implies $$\begin{aligned} |u_n-v_n|=|u_nv_n||u_{n+1}-v_{n+1}|.\end{aligned}$$ Therefore, for $n\in \mathbb{Z}_{> 0}$, we have $$\begin{aligned} \label{u0v0} |u_1-v_1|=|u_1v_1|\ldots |u_nv_n||u_{n+1}-v_{n+1}|\leq |u_1v_1|\ldots |u_nv_n|.\end{aligned}$$ From ([\[vnleq1\]](#vnleq1){reference-type="ref" reference="vnleq1"}), ([\[vnleq2\]](#vnleq2){reference-type="ref" reference="vnleq2"}), and Lemma [Lemma 8](#alphainK){reference-type="ref" reference="alphainK"}, we can derive the inequality for any $m \in \mathbb{Z}_{> 0}$ $$\begin{aligned} \label{u2m} |u_{2m}||v_{2m}||u_{2m+1}||v_{2m+1}| < 1.\end{aligned}$$ We assume that there exists $0<\epsilon<1$ such that for infinitely many $j\in \mathbb{Z}_{>0}$ $|u_{2j}|<\dfrac{1}{2}\epsilon$. Since $|u_{2j}|<\dfrac{1}{2}\epsilon$ implies $|u_{2j}||v_{2j}||u_{2j+1}||v_{2j+1}|<\epsilon$, from ([\[u0v0\]](#u0v0){reference-type="ref" reference="u0v0"}) and ([\[u2m\]](#u2m){reference-type="ref" reference="u2m"}) we have $$\begin{aligned} |u_1-v_1|\leq \lim_{n\to \infty} |u_1v_1|\ldots |u_nv_n|=0.\end{aligned}$$ However, it contradicts $\alpha\ne\beta$. Therefore, we have $$\begin{aligned} \label{limu2n} \lim_{n\to \infty}|u_{2n}|=\frac{1}{2}.\end{aligned}$$ Similarly, we have $$\begin{aligned} \label{limu2n+1} \lim_{n\to \infty}|v_{2n}|=1, \lim_{n\to \infty}|u_{2n+1}|=1, \text{and } \lim_{n\to \infty}|v_{2n+1}|=2.\end{aligned}$$ Therefore, for $\epsilon=\dfrac{1}{8}$, there exists $N\in \mathbb{Z}_{> 0}$ such that for $n\geq N$, the following inequalities hold: $$\begin{aligned} &\left||u_{2n}|-\frac{1}{2}\right|<\epsilon,\hspace{1cm} \left||v_{2n}|-1\right|<\epsilon, \label{u2n}\\ \text{and}\nonumber\\ &\left||u_{2n+1}|-1\right|<\epsilon,\hspace{1cm} \left||v_{2n+1}|-2\right|<\epsilon.\label{u2n1}\end{aligned}$$ We assume that $$\begin{aligned} \label{12u2N} \frac{1}{2}-\epsilon<u_{2N}<\frac{1}{2}.\end{aligned}$$ From ([\[unvne\]](#unvne){reference-type="ref" reference="unvne"}) and ([\[u2n\]](#u2n){reference-type="ref" reference="u2n"}), we have $$\begin{aligned} \label{1-epsilon} 1-\epsilon<v_{2N}<1.\end{aligned}$$ Let us consider the range of $u_{2N+1}$. First, we assume that $1-\epsilon<u_{2N+1}<1$. Since $u_{2N+1}=\dfrac{1}{u_{2N}}-a_{2N}(\alpha)$, we have $$\begin{aligned} \label{-1dfrac1} -1+\dfrac{1}{u_{2N}}<a_{2N}(\alpha)<-\dfrac{7}{8}+\dfrac{1}{u_{2N}}.\end{aligned}$$ From the inequalities ([\[12u2N\]](#12u2N){reference-type="ref" reference="12u2N"}) and ([\[-1dfrac1\]](#-1dfrac1){reference-type="ref" reference="-1dfrac1"}), we have $$\begin{aligned} \label{1<a2N} 1<a_{2N}(\alpha)<\frac{43}{24}.\end{aligned}$$ Since $v_{2N+1}=\frac{1}{v_{2N}}-a_{2N}(\alpha)$, from ([\[1-epsilon\]](#1-epsilon){reference-type="ref" reference="1-epsilon"}) and ([\[1\<a2N\]](#1<a2N){reference-type="ref" reference="1<a2N"}), we have $-\frac{19}{24}<v_{2N+1}<\frac{1}{7}$, which contradicts ([\[u2n1\]](#u2n1){reference-type="ref" reference="u2n1"}). Therefore, we have $$\begin{aligned} \label{-1<u_2N+1} -1<u_{2N+1}<-1+\epsilon.\end{aligned}$$ From ([\[unvno\]](#unvno){reference-type="ref" reference="unvno"}), ([\[u2n1\]](#u2n1){reference-type="ref" reference="u2n1"}), and ([\[-1\<u_2N+1\]](#-1<u_2N+1){reference-type="ref" reference="-1<u_2N+1"}), we have $$\begin{aligned} \label{-1<v_2N+1} -2<v_{2N+1}<-2+\epsilon.\end{aligned}$$ Let us consider the range of $u_{2N+2}$. First, we assume that $-\frac{1}{2}<u_{2N+2}<-\frac{1}{2}+\epsilon$. Since $u_{2N+2}=\dfrac{1}{u_{2N+1}}-a_{2N+1}(\alpha)$, we have $$\begin{aligned} \label{38dfrac1} \dfrac{3}{8}+\dfrac{1}{u_{2N+1}}<a_{2N+1}(\alpha)<\dfrac{1}{2}+\dfrac{1}{u_{2N+1}}.\end{aligned}$$ From the inequalities ([\[-1\<u_2N+1\]](#-1<u_2N+1){reference-type="ref" reference="-1<u_2N+1"}) and ([\[38dfrac1\]](#38dfrac1){reference-type="ref" reference="38dfrac1"}), we have $$\begin{aligned} \label{4356} -\frac{43}{56}<a_{2N+1}(\alpha)<-\frac{1}{2}.\end{aligned}$$ Since $v_{2N+2}=\frac{1}{v_{2N+1}}-a_{2N+1}(\alpha)$, from ([\[-1\<v_2N+1\]](#-1<v_2N+1){reference-type="ref" reference="-1<v_2N+1"}) and ([\[4356\]](#4356){reference-type="ref" reference="4356"}), we have $-\frac{1}{30}<v_{2N+2}<\frac{15}{56}$, which contradicts ([\[u2n\]](#u2n){reference-type="ref" reference="u2n"}). Therefore, we have $$\begin{aligned} \label{-1<u_2N+2} \frac{1}{2}-\epsilon<u_{2N+2}<\frac{1}{2}.\end{aligned}$$ Therefore, we obtain the following inequalities recursively for $n\geq N$: $$\begin{aligned} &\frac{1}{2}-\epsilon<u_{2n}<\frac{1}{2},\ \ 1-\epsilon<v_{2n}<1,\\ &\text{and}\\ &-1<u_{2n+1}<-1+\epsilon,\ \ -2<v_{2N+1}<-2+\epsilon. \end{aligned}$$ By taking into account this fact and considering ([\[limu2n\]](#limu2n){reference-type="ref" reference="limu2n"}) and ([\[limu2n+1\]](#limu2n+1){reference-type="ref" reference="limu2n+1"}), we can derive claim (2)(i) of the lemma. If we assume that $-\frac{1}{2}<u_{2N}<-\frac{1}{2}+\epsilon$, we can derive claim (2)(ii) of the lemma in a similar manner. ◻ **Lemma 16**. *Let $\mathbf{a}=(\alpha,\beta)\in (K/\mathbb{Q})^2$ with $\alpha\ne\beta$. Then, there exists $n\in \mathbb{Z}_{\geq 0}$ such that $(\mathbf{a}_{(2n+1)})_{\langle 2 \rangle}\in D_{2}^1$.* *Proof.* We put $(u_n,v_n):=(\mathbf{a}_{(n)})_{\langle 2 \rangle}$ for $n\in \mathbb{Z}_{> 0}$. According to Lemma [Lemma 1](#lemord){reference-type="ref" reference="lemord"}, for all $n\geq 1$, the following inequalities hold: $$\begin{aligned} \label{v_pu_2n} v_p(u_{2n})\geq 0 \text{ and } v_p(u_{2n-1})> 0.\end{aligned}$$ We assume that for all $n\in \mathbb{Z}_{\geq 0}$, $(u_{2n+1},v_{2n+1})\notin D_{2}^1$. From Lemma [Lemma 11](#Letalpha=){reference-type="ref" reference="Letalpha="}, we see that for all $n\in \mathbb{Z}_{> 0}$, $(u_{2n},v_{2n})\notin D_{2}^0$. Therefore, we have for all $n\in \mathbb{Z}_{\geq 0}$ $$\begin{aligned} \label{v_pv_2n} v_p(v_{2n+2})\geq 0 \text{ and } v_p(v_{2n+1})> 0.\end{aligned}$$ For $n\in \mathbb{Z}_{> 0}$, we see that $\displaystyle u_{n+1}=\frac{1}{u_{n}}-a_n(\alpha)$ and $\displaystyle v_{n+1}=\frac{1}{v_{n}}-a_n(\alpha)$ , which implies $$\begin{aligned} |u_n-v_n|_2=|u_nv_n|_2|u_{n+1}-v_{n+1}|_2.\end{aligned}$$ Therefore, for $n\in \mathbb{Z}_{> 0}$, we have $$\begin{aligned} \label{u0v02} |u_1-v_1|_2=|u_1v_1|_2\ldots |u_nv_n|_2|u_{n+1}-v_{n+1}|_2\leq |u_1v_1|_2\ldots |u_nv_n|_2.\end{aligned}$$ From ([\[v_pu_2n\]](#v_pu_2n){reference-type="ref" reference="v_pu_2n"}), ([\[v_pv_2n\]](#v_pv_2n){reference-type="ref" reference="v_pv_2n"}), and ([\[u0v02\]](#u0v02){reference-type="ref" reference="u0v02"}), as $n \to \infty$, we can conclude that $|u_1 - v_1|_2$ tends to 0, which implies $u_1=v_1$. This contradicts that $\alpha\ne\beta$. ◻ We would like to expect that $\{\frac{p_n}{q_n}\}$ converges to $\alpha_{\infty}$, but for now, let's present a weaker form. **Lemma 17**. *For all $n\in \mathbb{Z}_{> 0}$, it holds that $|\alpha_{\infty}-\frac{p_n}{q_n}|<1$.* *Proof.* Let $n\in \mathbb{Z}_{> 0}$. Let $\mathbf{a}=(\alpha,\frac{p_n}{q_n})$. We apply Algorithm (Definition [Definition 9](#algol3){reference-type="ref" reference="algol3"}) to $\mathbf{a}$. By induction, we obtain the following:\ For $1\leq j \leq n$, $$\begin{aligned} \mathbf{a}_{(j)}=(\alpha_{(j)},[0;a_j(\alpha),a_{j+1}(\alpha),\ldots,a_{n}(\alpha)]).\end{aligned}$$ and $$\begin{aligned} \mathbf{a}_{(n+1)}=(\alpha_{(n+1)},0).\end{aligned}$$ From Lemma [Lemma 8](#alphainK){reference-type="ref" reference="alphainK"}, we see that $(\alpha_{(n+1)})_{\infty}\in I_{\infty}^{\tau(n+1)}$. Therefore, we obtain $(\mathbf{a}_{(n+1)})_{\infty}\notin D_{\infty}^{\tau(n+1)}$. Hence, from Lemma [Lemma 14](#Dinfty){reference-type="ref" reference="Dinfty"}, we have $(\mathbf{a}_{(n)})_{\infty}\notin D_{\infty}^{\tau(n)}$. By induction, we obtain $(\mathbf{a}_{(1)})_{\infty}\notin D_{\infty}^{1}$. Therefore, we have $|\alpha_{\infty}-\frac{p_n}{q_n}|<1$. ◻ The following corollary follows from the proof of Lemma [Lemma 17](#convergentsp2adicda){reference-type="ref" reference="convergentsp2adicda"}. **Corollary 18**. *For all $n,m\in \mathbb{Z}_{> 0}$ with $m\leq n$, it holds that $$\begin{aligned} |(\alpha_{(m)})_{\infty}-\left[0;a_m(\alpha),a_{m+1}(\alpha),\ldots,a_{n}(\alpha)\right]|<\dfrac{1+\tau(m)}{2}.\end{aligned}$$* We observe that, employing the same reasoning as in Lemma [Lemma 10](#convergentsp2adic){reference-type="ref" reference="convergentsp2adic"}, the continued fraction $[a_{n-1}(\alpha);\ldots,a_{1}(\alpha)]$ is well-defined for $n\in \mathbb{Z}_{> 1}$, and this continued fraction is addressed in the following lemma. **Lemma 19**. *For all $n\in \mathbb{Z}_{> 1}$, it holds that $$\begin{aligned} \left|(\alpha_{(n)})_{\infty}+[a_{n-1}(\alpha);\ldots,a_{1}(\alpha)]\right| >\dfrac{1}{2}(1+\tau(n)).\end{aligned}$$* *Proof.* Let $\mathbf{a}=(\alpha,\infty)$. We apply Algorithm (Definition [Definition 9](#algol3){reference-type="ref" reference="algol3"}) to $\mathbf{a}$. By induction, we obtain the following:\ For $n\in \mathbb{Z}_{\geq 2}$ $$\begin{aligned} \mathbf{a}_{(n)}=(\alpha_{(n)},-[a_{n-1}(\alpha);\ldots,a_{1}(\alpha)]).\end{aligned}$$ In fact, we have $\mathbf{a}_{(2)}=(\alpha_{(2)}, -a_{1}(\alpha))=(\alpha_{(2)}, -[a_{1}(\alpha)])$. We assume that the claim holds for $n>1$. Then, we have $$\begin{aligned} \mathbf{a}_{(n+1)}=&\left(\alpha_{(n+1)},-\dfrac{1}{[a_{n-1}(\alpha);\ldots,a_{1}(\alpha)]}-a_{n}(\alpha)\right)\\ =&(\alpha_{(n+1)},-[a_{n}(\alpha);\ldots,a_{1}(\alpha)]).\end{aligned}$$ Since $\mathbf{a}\in \mathcal{D}_{\infty}^{1}$ holds, based on Lemma [Lemma 14](#Dinfty){reference-type="ref" reference="Dinfty"}, [Lemma 8](#alphainK){reference-type="ref" reference="alphainK"}, and Remark [Remark 2](#infinity){reference-type="ref" reference="infinity"}, we can conclude that for $n \in \mathbb{Z}_{\geq 1}$, $\mathbf{a}_{(n)}$ belongs to $\mathcal{D}_{\infty}^{\tau(n)}$. This result implies the statement of the theorem. ◻ Following Lemma is proved easily, so that we omit the proof. **Lemma 20**. *For $n\in \mathbb{Z}_{\geq 1}$, it holds that $$\begin{aligned} \dfrac{q_{n-1}}{q_{n}}=[0;a_{n}(\alpha),\ldots,a_{1}(\alpha)].\end{aligned}$$* **Lemma 21**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. For all $n\in \mathbb{Z}_{\geq 1}$, it holds that $$\begin{aligned} \left|\alpha_{\infty}-\dfrac{p_n}{q_n}\right|<\dfrac{2}{(1+\tau(n))q^2_n}.\end{aligned}$$* *Proof.* We have $$\begin{aligned} &\alpha-\dfrac{p_n}{q_n}=\dfrac{p_{n-1}\alpha_{(n+1)}+p_{n}}{q_{n-1}\alpha_{(n+1)}+q_{n}}-\dfrac{p_n}{q_n}=\dfrac{(-1)^n\alpha_{(n+1)}}{q_n(q_{n-1}\alpha_{(n+1)}+q_{n})}\nonumber\\ &=\dfrac{(-1)^n}{q^2_n\left(\dfrac{q_{n-1}}{q_n}+\dfrac{1}{\alpha_{(n+1)}}\right)}=\dfrac{(-1)^n}{q^2_n\left(\dfrac{q_{n-1}}{q_n}+a_{n+1}(\alpha)+\alpha_{(n+2)}\right)}\label{q2nleft}\\ &=\dfrac{(-1)^n}{q^2_n\left([a_{n+1}(\alpha);a_{n}(\alpha),\ldots,a_{1}(\alpha)]+\alpha_{(n+2)}\right)}.\nonumber\end{aligned}$$ Therefore, from Lemma [Lemma 19](#inversecontinued){reference-type="ref" reference="inversecontinued"}, we have the claim of the lemma. ◻ **Lemma 22**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. For all $n\in \mathbb{Z}_{\geq 1}$, it holds that $$\begin{aligned} v_2\left(\alpha_{\langle 2 \rangle}-\dfrac{p_n}{q_n}\right)=-2v_2(q_n)-v_2(a_{n+1}(\alpha)).\end{aligned}$$* *Proof.* From the equation ([\[q2nleft\]](#q2nleft){reference-type="ref" reference="q2nleft"}), we have $$\begin{aligned} \label{ord2leftal} v_2\left(\alpha_{\langle 2 \rangle}-\dfrac{p_n}{q_n}\right)=-2v_2(q_n)-v_2\left(\dfrac{q_{n-1}}{q_n}+a_{n+1}(\alpha)+(\alpha_{(n+2)})_{\langle 2 \rangle}\right).\end{aligned}$$ From Lemma [Lemma 10](#convergentsp2adic){reference-type="ref" reference="convergentsp2adic"}, $$\begin{aligned} \label{qnright} v_2\left(\dfrac{q_{n-1}}{q_n}\right)=-v_2(a_{n}(\alpha)).\end{aligned}$$ First, we assume that $n$ is even. From Lemma [Lemma 9](#lemord2){reference-type="ref" reference="lemord2"}, we see that $v_2(a_{n}(\alpha))\leq 0$, $v_2(a_{n+1}(\alpha))<0$, and $v_2((\alpha_{(n+2)})_{\langle 2 \rangle})\geq 0$, which implies $$\begin{aligned} v_2\left(\frac{q_{n-1}}{q_n}+a_{n+1}(\alpha)+(\alpha_{(n+2)})_{\langle 2 \rangle}\right) =v_2(a_{n+1}(\alpha)).\end{aligned}$$ Thus, we get the claim of the lemma in this case. Next, we assume that $n$ is odd. From Lemma [Lemma 9](#lemord2){reference-type="ref" reference="lemord2"}, we see that $v_2(a_{n}(\alpha))<0$, $v_2(a_{n+1}(\alpha))\leq 0$, and $v_2((\alpha_{(n+2)})_{\langle 2 \rangle})>0$, which implies $$\begin{aligned} v_2\left(\frac{q_{n-1}}{q_n}+a_{n+1}(\alpha)+(\alpha_{(n+2)})_{\langle 2 \rangle}\right) =v_2(a_{n+1}(\alpha)).\end{aligned}$$ Thus, we get the claim of the lemma. ◻ From now on, let $K$ be a quadratic field that can be embedded into both $\mathbb{R}$ and $\mathbb{Q}_p$. For $\alpha\in K/\mathbb{Q}$, we denote its conjugate by $\bar{\alpha}$. We note that $\bar{\alpha}\in K$. **Lemma 23**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. There exists $C>0$ such that for all $n\in \mathbb{Z}_{\geq 1}$, it holds that $|a_{n}(\alpha)|_{2}<C$.* *Proof.* such that Let $ax^2+bx+c$ be a minimal polynomial of $\alpha$ over $\mathbb{Z}$. We put $g(x,y)=ax^2+bxy+cy^2$. Let $\mathbf{a}=(\alpha,\bar{\alpha})$. From Lemma [Lemma 16](#Letalpha2adic){reference-type="ref" reference="Letalpha2adic"}, there exists $j\in \mathbb{Z}_{>0}$ such that $(\mathbf{a}_{(2j+1)})_{\langle 2 \rangle}\in D_{2}^1$. For the sake of simplicity, we assume that $(\alpha_{\langle 2 \rangle},\bar{\alpha}_{\langle 2 \rangle})\in D_{2}^1$. Let $n\in \mathbb{Z}_{>0}$. Then, from Lemma [Lemma 17](#convergentsp2adicda){reference-type="ref" reference="convergentsp2adicda"} and [Lemma 21](#boundf){reference-type="ref" reference="boundf"}, we have $$\begin{aligned} 0<|g(p_n,q_n)|&=|a||q_n^2|\left|\dfrac{p_n}{q_n}-\alpha_{\infty}\right|\left|\dfrac{p_n}{q_n}-\overline{\alpha}_{\infty}\right|\nonumber\\ &<2|a|\left(\left|\dfrac{p_n}{q_n}\right|+|\overline{\alpha}_{\infty}|\right)\nonumber\\ &<2|a|\left(1+|\alpha_{\infty}|+|\overline{\alpha}_{\infty}|\right)=C_1.\label{2left1+alpha}\end{aligned}$$ We put $m_n=-v_2(q_n)$. From Lemma [Lemma 10](#convergentsp2adic){reference-type="ref" reference="convergentsp2adic"} we see that $-v_2(q_n)\geq -v_2(p_n)$. Therefore, we can conclude that $2^{2m_n}g(p_n,q_n)\in \mathbb{Z}$. Hence, from inequality ([\[2left1+alpha\]](#2left1+alpha){reference-type="ref" reference="2left1+alpha"}), we can conclude: $$\begin{aligned} \label{2m_ng(p_n,q_n)} 0\leq v_2(2^{2m_n}g(p_n,q_n))< 2m_n +\log_2(|C_1|+1). \end{aligned}$$ On the other hand, from Lemma [Lemma 22](#boundf2){reference-type="ref" reference="boundf2"}, we have $$\begin{aligned} \label{2m_n-ord} v_2(2^{m_n}(p_n-q_n\alpha_{\langle 2 \rangle}))=2m_n-v_2(a_{n+1}(\alpha)).\end{aligned}$$ Since $v_2(p_n)=-m_n-v_2(a_1(\alpha))>-m_n$ and $v_2(\overline{\alpha}_{\langle 2 \rangle})\leq 0$, we have $$\begin{aligned} \label{overlinealpha} v_2(2^{m_n}(p_n-q_n\overline{\alpha}_{\langle 2 \rangle}))=v_2(\overline{\alpha}_{\langle 2 \rangle})=C_2.\end{aligned}$$ Therefore, from ([\[2m_n-ord\]](#2m_n-ord){reference-type="ref" reference="2m_n-ord"}) and ([\[overlinealpha\]](#overlinealpha){reference-type="ref" reference="overlinealpha"}), we have $$\begin{aligned} \label{alphaC_2} &v_2(2^{2m_n}g(p_n,q_n))=v_2(2^{m_n}(p_n-q_n\alpha_{\langle 2 \rangle})2^{m_n}(p_n-q_n\overline{\alpha}_{\langle 2 \rangle}))\nonumber\\ &=2m_n-v_2(a_{n+1}(\alpha))+C_2.\end{aligned}$$ From ([\[2m_ng(p_n,q_n)\]](#2m_ng(p_n,q_n)){reference-type="ref" reference="2m_ng(p_n,q_n)"}) and ([\[alphaC_2\]](#alphaC_2){reference-type="ref" reference="alphaC_2"}), we have $$\begin{aligned} -v_2(a_{n+1}(\alpha))<\log_2(|C_1|+1)-C_2,\end{aligned}$$ which prove the claim of the lemma. ◻ **Lemma 24**. *Let $\alpha$ belong to $K/\mathbb{Q}$ with property $I$, and let $\beta$ be an element in $K/\mathbb{Q}$ distinct from $\alpha$. Let $\mathbf{a}=(\alpha,\beta)$. Then, there exists $n\in \mathbb{Z}_{>0}$ such that $(\mathbf{a}_{(2n-1)})_{\infty}\in D_{\infty}^1$.* *Proof.* We put $(u_n.v_n)=(\mathbf{a}_{(n)})_{\infty}$ for $n\in \mathbb{Z}_{>0}$. We assume that for all $n\in \mathbb{Z}_{>0}$, $(u_{2n-1},v_{2n-1})\notin D_{\infty}^1$. Then, according to Lemma [Lemma 15](#Letalpha){reference-type="ref" reference="Letalpha"}, either (2)(i) or (2)(ii) in the lemma holds. We assume that (2)(i) holds: $$\begin{aligned} \label{lim_nto2} \lim_{n\to \infty} (u_{2n},v_{2n})=\left(\frac{1}{2},1\right) \text{ and } \lim_{n\to \infty} (u_{2n-1},v_{2n-1})=(-1,-2).\end{aligned}$$ From the fact that $u_{2n+1}=\dfrac{1}{u_{2n}}-a_{2n}(\alpha)$ for $n\in \mathbb{Z}_{>0}$, we have $$\begin{aligned} \label{lim_nto} \lim_{n\to \infty} a_{2n}(\alpha)=3.\end{aligned}$$ Since $a_{2n}(\alpha)\in \mathbb{Z}[\frac{1}{p}]$ for $n\in \mathbb{Z}_{>0}$ and from Lemma [Lemma 23](#boundpf){reference-type="ref" reference="boundpf"} there exists $C>0$ such that $|a_{n}(\alpha)|_{2}<C$, with equation ([\[lim_nto\]](#lim_nto){reference-type="ref" reference="lim_nto"}), we can conclude that there exists $N_1\in \mathbb{Z}_{>0}$ such that for all $n\in \mathbb{Z}_{>0}$ with $n\geq N_1$ $a_{2n}(\alpha)=3$. Similarly, we have there exists $N_2\in \mathbb{Z}_{>0}$ such that for all $n\in \mathbb{Z}_{>0}$ with $n\geq N_2$ $a_{2n-1}(\alpha)=-\frac{3}{2}$. We set $N=\max\{N_1,N_2\}$. Then, we have for $n\in \mathbb{Z}_{\geq 0}$ $$\begin{aligned} &\begin{pmatrix}0&1\\1&a_{2N-1}(\alpha)\end{pmatrix}\ldots \begin{pmatrix}0&1\\1&a_{2(N+n-1)}(\alpha)\end{pmatrix}= \begin{pmatrix}1&3\\-3/2&-7/2\end{pmatrix}^{n}\\ &=\begin{pmatrix}2(-1/2)^{n}-(-2)^{n}&2(-1/2)^{n}-2(-2)^{n}\\-(-1/2)^{n}+(-2)^{n}&-(-1/2)^{n}+2(-2)^{n}\end{pmatrix}.\end{aligned}$$ Therefore, from Theorem [Theorem 12](#conver){reference-type="ref" reference="conver"}, we obtain that in $2$-adic topology $$\begin{aligned} (\alpha_{(2N-1)})_{\langle 2 \rangle}=\lim_{n\to \infty}\dfrac{2(-1/2)^{n}-2(-2)^{n}}{-(-1/2)^{n}+2(-2)^{n}}=-2,\end{aligned}$$ which contradicts $\alpha\in K/\mathbb{Q}$. We have a contradiction in the case that (2)(ii) holds in a similar manner. ◻ *Remark 3*. When discussing the continued fraction expansion by our algorithm, we assume that the length of its period is limited to even numbers. The reason is that there is a difference in the expressions used to calculate the next term between even and odd indices. We give an example. Let $\alpha=2\sqrt{17}-8$. Then, $\alpha_{\infty}=0.246\cdots$. We assume that $\alpha_{\langle 2 \rangle} \equiv 2 \mod 8$. Then, we have $$\begin{aligned} \alpha=\left[0;\dfrac{9}{2},-\dfrac{3}{2},-\dfrac{3}{2},\overline{5,-\dfrac{5}{2}}\right].\end{aligned}$$ As can be seen from the above, it is evident that $\alpha_{(4)}=\alpha_{(6)}$, thus resulting in a period length of $2$. However, we have $\alpha_{(2)}=\alpha_{(5)}$, which is not reflected in the continued fraction expansion. **Theorem 25**. *Let $\alpha\in K/\mathbb{Q}$ have property $I$ and assume that there exist a positive odd integer $n_1$ such that $\alpha_{(1)}=\alpha_{(n_1)}$. Then, $(\alpha,\bar{\alpha})_{\langle 2 \rangle}\in D_{2}^1$ and $(\alpha,\bar{\alpha})_{\infty}\in D_{\infty}^1$ hold.* *Proof.* Let $\mathbf{a}=(\alpha,\bar{\alpha})$. We apply Algorithm (Definition [Definition 9](#algol3){reference-type="ref" reference="algol3"}) to $\mathbf{a}$. From Lemma [Lemma 16](#Letalpha2adic){reference-type="ref" reference="Letalpha2adic"} and [Lemma 24](#alphainKda){reference-type="ref" reference="alphainKda"}, there exist odd positive integers $N_1, N_2$ such that $(\mathbf{a}_{(N_1)})_{\langle 2 \rangle}\in D_{2}^1$ and $(\mathbf{a}_{(N_2)})_{\infty}\in D_{\infty}^1$. Using Lemma [Lemma 11](#Letalpha=){reference-type="ref" reference="Letalpha="} and [Lemma 14](#Dinfty){reference-type="ref" reference="Dinfty"}, we see that for all odd $n\in \mathbb{Z}_{>0}$ with $n\geq \max\{N_1,N_2\}$ $(\mathbf{a}_{(n)})_{\langle 2 \rangle}\in D_{2}^1$ and $(\mathbf{a}_{(n)})_{\infty}\in D_{\infty}^1$. Therefore, the assumption that $\alpha_{(1)}=\alpha_{(n_1)}$ implies the claim of the theorem. ◻ *Remark 4*. The converse of Theorem [Theorem 25](#Galois){reference-type="ref" reference="Galois"} is generally not true. We give an example. We assume that $(\sqrt{17})_{\langle 2 \rangle} \equiv 1 \mod 8$. Let $\alpha=-\frac{49}{64}+\frac{9\sqrt{17}}{64}$. Then, $\alpha_{\infty}=-0.1858\cdots$. $\bar{\alpha}_{\infty}=-1.3454\cdots$. We obtain that $v_2(\alpha_{\langle 2 \rangle})=3$ and $v_2(\bar{\alpha}_{\langle 2 \rangle})=-5$. Therefore, it holds that $(\alpha,\bar{\alpha})_{\langle 2 \rangle}\in D_{2}^1$ and $(\alpha,\bar{\alpha})_{\infty}\in D_{\infty}^1$. However, we have $$\begin{aligned} \alpha=\left[0;-\dfrac{41}{8},\overline{-3,-\dfrac{3}{2},\dfrac{5}{2},\dfrac{37}{4},-3,\dfrac{5}{4},-5,-\dfrac{33}{8}}\right].\end{aligned}$$ **Theorem 26**. *Let $\alpha\in K/\mathbb{Q}$ have property $I$. Then, $$\begin{aligned} \alpha_{\infty}=\lim_{n\to \infty}\dfrac{p_n}{q_n}.\end{aligned}$$* *Proof.* We assume that $\alpha_{\infty}\ne \lim_{n\to \infty}\frac{p_n}{q_n}$. For $n,m\in \mathbb{Z}_{>0}$ with $m\leq n$, we denote $[0;a_m(\alpha),a_{m+1}(\alpha),\ldots,a_{n}(\alpha)]$ by $v_{m,n}$. We define $v_{n+1,n}=0$. Let $n\in \mathbb{Z}_{> 0}$. Let $\mathbf{a}^{(n)}=(\alpha,\frac{p_n}{q_n})$. From Corollary [Corollary 18](#convergentsp2adicdaco){reference-type="ref" reference="convergentsp2adicdaco"}, for $1\leq m \leq n+1$, we have: $$\begin{aligned} |(\alpha_{(m)})_{\infty}-v_{m,n}|\leq \dfrac{1}{2}(1+\tau(m)).\end{aligned}$$ We set that for $m\in \mathbb{Z}_{>0}$ $u_m=(\alpha_{(m)})_{\infty}$. We obtain the following inequality in a similar manner as inequality ([\[u0v0\]](#u0v0){reference-type="ref" reference="u0v0"}): for $n\in \mathbb{Z}_{>0}$, $$\begin{aligned} \label{u0v00} |u_1-v_{1,n}|=|u_1v_{1,n}|\ldots |u_nv_{n,n}||u_{n+1}-v_{n+1,n}|\leq |u_1v_{1,n}|\ldots |u_nv_{n,n}|.\end{aligned}$$ If we assume that there exists $0<\epsilon<1$ such that for infinitely many $j\in \mathbb{Z}_{>0}$ $|u_{2j}|<\dfrac{1}{2}\epsilon$, a similar argument to Lemma [Lemma 15](#Letalpha){reference-type="ref" reference="Letalpha"} implies that $\lim_{n\to \infty} |u_1-v_{1,n}|=0$, which contradicts the assumption. Therefore, we have $$\begin{aligned} \label{limu2n2n} \lim_{n\to \infty}|u_{2n}|=\frac{1}{2}.\end{aligned}$$ Similarly, we have $$\begin{aligned} \label{limu2n+12n+1} \lim_{n\to \infty}|u_{2n+1}|=1.\end{aligned}$$ We set $$\begin{aligned} &A_{+}=\{n\in \mathbb{Z}_{>0}\mid n \equiv 0 \mod 2, u_{n}\geq 0\},\\ &A_{-}=\{n\in \mathbb{Z}_{>0}\mid n \equiv 0 \mod 2, u_{n}< 0\},\\ &B_{+}=\{n\in \mathbb{Z}_{>0}\mid n \equiv 1 \mod 2, u_{n}\geq 0\},\\ &B_{-}=\{n\in \mathbb{Z}_{>0}\mid n \equiv 1 \mod 2, u_{n}< 0\}.\end{aligned}$$ We assume that $\sharp A_{+}=\infty$. We also assume that there exist infinitely many $j\in \mathbb{Z}_{>0}$ such that $2j\in A_{+}$ and $2j+1\in B_{+}$. Then, we have $$\begin{aligned} \lim_{{\substack{j\to \infty \\ 2j\in A_+,2j+1\in B_+ }}}a_{2j}(\alpha) =\lim_{{\substack{j\to \infty \\ 2j\in A_+,2j+1\in B_+ }}}\left(\dfrac{1}{u_{2j}}-u_{2j+1}\right)=1.\end{aligned}$$ Using the same method as the proof of Lemma [Lemma 24](#alphainKda){reference-type="ref" reference="alphainKda"}, we can conclude that there exists an $N_1 \in \mathbb{Z}_{>0}$ such that for all $n \in \mathbb{Z}_{>0}$ with $n \geq N_1$, if $2n \in A_+$ and $2n+1 \in B_+$, then $a_{2n} = 1$. Let $k \in \mathbb{Z}_{>0}$ satisfy $k \geq N_1$, $2k \in A_+$, and $2k+1 \in B_+$. Then, $u_{2k}$ and $u_{2k+1}$ satisfy following: $$\begin{aligned} &u_{2k}\geq 0, \ \ u_{2k+1}\geq 0,\\ &|u_{2k}|\leq \dfrac{1}{2}, \ \ |u_{2k+1}|\leq 1,\\ &u_{2k+1}=\dfrac{1}{u_{2k}}-1,\end{aligned}$$ which imply that $u_{2k}=\frac{1}{2}$ and $u_{2k+1}=1$. This contradicts that $\alpha\in K/\mathbb{Q}$. Therefore, there exists an $N_2 \in \mathbb{Z}_{>0}$ such that for all $n \in \mathbb{Z}_{>0}$ with $n \geq N_2$, if $2n \in A_+$, then $2n+1 \in B_-$. Next, we assume that there exist infinitely many $j\in \mathbb{Z}_{>0}$ such that $2j\in A_{+}$ and $2j+2\in A_{-}$. Then, using the same method as in the previous argument, we arrive at a contradiction. Therefore, there exists an $N_3 \in \mathbb{Z}_{>0}$ such that for all $n \in \mathbb{Z}_{>0}$ with $n\geq N_3$, $2n \in A_+$ and $2n+1 \in B_-$. Then, employing the same method as the proof of Lemma [Lemma 24](#alphainKda){reference-type="ref" reference="alphainKda"}, we can deduce that there exists an $N_4 \in \mathbb{Z}_{>0}$ such that for all $n \in \mathbb{Z}_{>0}$ with $n\geq N_4$, $a_{2n}(\alpha)=3$ and $a_{2n-1}(\alpha)=-\frac{3}{2}$, leading to a similar contradiction. For the case where $\sharp A_{-}=\infty$, we encounter a similar contradiction. Thus, we obtain the claim of the lemma. ◻ The following corollary immediately follows from Theorem [Theorem 5](#rational){reference-type="ref" reference="rational"} and [Theorem 26](#inftyp_nq_n){reference-type="ref" reference="inftyp_nq_n"}. **Corollary 27**. *Let $\alpha\in K$ and $\{\frac{p_n}{q_n}\}$ be its convergents related to Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}). Then, $$\begin{aligned} \alpha_{\infty}=\lim_{n\to \infty}\dfrac{p_n}{q_n}.\end{aligned}$$* **Lemma 28**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. Then, there exists $C>0$ such that for all $n \in \mathbb{Z}_{>0}$, $$\begin{aligned} \left|\dfrac{q_{n-1}}{q_n}+a_{n+1}(\alpha)+(\alpha_{(n+2)})_{\infty}\right|<C.\end{aligned}$$* *Proof.* Let $ax^2+bx+c$ be a minimal polynomial of $\alpha$ over $\mathbb{Z}$. We put $g(x,y)=ax^2+bxy+cy^2$. From the equation ([\[q2nleft\]](#q2nleft){reference-type="ref" reference="q2nleft"}), we have $$\begin{aligned} &|g(p_n,q_n)|=|a||q_n^2|\left|\dfrac{p_n}{q_n}-\alpha_{\infty}\right|\left|\dfrac{p_n}{q_n}-\overline{\alpha}_{\infty}\right|\nonumber\\ &=\dfrac{|a|}{\left|\dfrac{q_{n-1}}{q_n}+a_{n+1}(\alpha)+(\alpha_{(n+2)})_{\infty}\right|}\left|\dfrac{p_n}{q_n}-\overline{\alpha}_{\infty}\right|. \label{|a||q_n^2|}\end{aligned}$$ From Theorem [Theorem 26](#inftyp_nq_n){reference-type="ref" reference="inftyp_nq_n"}, there exists $C_1>0$ such that for all $n \in \mathbb{Z}_{>0}$, $$\begin{aligned} \label{p_nq_nover} \left|\dfrac{p_n}{q_n}-\overline{\alpha}_{\infty}\right|<C_1.\end{aligned}$$ On the other hand, from Lemma [Lemma 22](#boundf2){reference-type="ref" reference="boundf2"}, we have $$\begin{aligned} \label{|g(p_n,q_n)|_2} &|g(p_n,q_n)|_2=|a|_2|q_n^2|_2\left|\dfrac{p_n}{q_n}-\alpha_{_{\langle 2 \rangle}}\right|_2\left|\dfrac{p_n}{q_n}-\overline{\alpha}_{_{\langle 2 \rangle}}\right|_2 \nonumber\\ &=\dfrac{|a|_2}{|a_{n+1}(\alpha)|_2}\left|\dfrac{p_n}{q_n}-\overline{\alpha}_{_{\langle 2 \rangle}}\right|_2.\end{aligned}$$ From Theorem [Theorem 12](#conver){reference-type="ref" reference="conver"}, there exists $C_2>0$ such that for all $n \in \mathbb{Z}_{>0}$, $$\begin{aligned} \label{leftdfracp_n} 0<\left|\dfrac{p_n}{q_n}-\overline{\alpha}_{\langle 2 \rangle}\right|_2<C_2.\end{aligned}$$ From ([\[\|g(p_n,q_n)\|\_2\]](#|g(p_n,q_n)|_2){reference-type="ref" reference="|g(p_n,q_n)|_2"}), ([\[leftdfracp_n\]](#leftdfracp_n){reference-type="ref" reference="leftdfracp_n"}), we can conclude that for all $n \in \mathbb{Z}_{>0}$, $$\begin{aligned} \label{inequalityC_3} 0<|g(p_n,q_n)|_2<|a|_2C_2,\end{aligned}$$ Considering $g(p_n,q_n)\in \mathbb{Z}[\frac{1}{2}]$, the inequality ([\[inequalityC_3\]](#inequalityC_3){reference-type="ref" reference="inequalityC_3"}) implies that for all $n \in \mathbb{Z}_{>0}$, $$\begin{aligned} \label{inequalityC_32} |g(p_n,q_n)|>\dfrac{1}{|a|_2C_2}.\end{aligned}$$ From ([\[\|a\|\|q_n\^2\|\]](#|a||q_n^2|){reference-type="ref" reference="|a||q_n^2|"}), ([\[p_nq_nover\]](#p_nq_nover){reference-type="ref" reference="p_nq_nover"}), and ([\[inequalityC_32\]](#inequalityC_32){reference-type="ref" reference="inequalityC_32"}), we have for all $n \in \mathbb{Z}_{>0}$, $$\begin{aligned} \left|\dfrac{q_{n-1}}{q_n}+a_{n+1}(\alpha)+(\alpha_{(n+2)})_{\infty}\right|<|a||a|_2C_1C_2.\end{aligned}$$ ◻ **Theorem 29**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. Then, $$\begin{aligned} \lim_{n\to \infty}|q_n|=\infty.\end{aligned}$$* *Proof.* From Theorem [Theorem 26](#inftyp_nq_n){reference-type="ref" reference="inftyp_nq_n"}, Lemma [Lemma 28](#thereexistsC){reference-type="ref" reference="thereexistsC"}, and the equation ([\[q2nleft\]](#q2nleft){reference-type="ref" reference="q2nleft"}), we have $$\begin{aligned} \lim_{n\to \infty} |q^2_n| = \lim_{n\to \infty}\dfrac{|q^2_n|\left|\dfrac{q_{n-1}}{q_n}+a_{n+1}(\alpha)+(\alpha_{(n+2)})_{\infty}\right|}{\left|\dfrac{q_{n-1}}{q_n}+a_{n+1}(\alpha))+(\alpha_{(n+2)})_{\infty}\right|}=\infty.\end{aligned}$$ ◻ **Corollary 30**. *Let $\alpha\in K/\mathbb{Q}$ and $\{\frac{p_n}{q_n}\}$ be its convergents related to Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}).* *Then, $$\begin{aligned} \lim_{n\to \infty}|q_n|=\infty.\end{aligned}$$* **Lemma 31**. *Let $\alpha,\beta\in K/\mathbb{Q}$ with $(\alpha,\beta)_{\langle 2 \rangle}\in D_{2}^1$. Let $\mathbf{a}=(\alpha,\beta)$. We denote $(u_n,v_n)=(\mathbf{a}_{(n)})_{\langle 2 \rangle}$ for $n\in \mathbb{Z}_{>0}$. Then, $v_2(v_n)=v_2(a_{n-1}(\alpha))$ for $n\in \mathbb{Z}_{>1}$.* *Proof.* For $n \in \mathbb{Z}_{>0}$, we observe that: - If $n$ is odd, then $v_2(u_n) > 0$ and $v_2(v_n) \leq 0$. - If $n$ is even, then $v_2(u_n) \geq 0$ and $v_2(v_n) < 0$. According to Lemma [Lemma 9](#lemord2){reference-type="ref" reference="lemord2"}, we have $v_2(u_n) = -v_2(a_{n}(\alpha))$ for $n \in \mathbb{Z}_{>0}$. Thus, we find that $-v_2(v_n) > v_2(a_{n}(\alpha))$ for $n \in \mathbb{Z}_{>0}$. Now let $n \in \mathbb{Z}_{>1}$. Since $v_n = \frac{1}{v_{n-1}} - a_{n-1}(\alpha)$, it follows that $v_2(v_n) = v_2(a_{n-1}(\alpha))$. ◻ Next, we have one of our main theorems. **Theorem 32**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. Then, there exist $n_1, n_2\in \mathbb{Z}_{>0}$ with $n_1<n_2$ such that $\alpha_{(n_1)}=\alpha_{(n_2)}$ and $n_1\equiv n_2 \mod\ 2$.* *Proof.* Let $\mathbf{a}=(\alpha,\bar{\alpha})$. We apply Algorithm (Definition [Definition 9](#algol3){reference-type="ref" reference="algol3"}) to $\mathbf{a}$. From Lemma [Lemma 16](#Letalpha2adic){reference-type="ref" reference="Letalpha2adic"} and [Lemma 24](#alphainKda){reference-type="ref" reference="alphainKda"}, there exists $N_1, N_2\in \mathbb{Z}_{>0}$ such that $(\mathbf{a}_{(2N_1+1)})_{\langle 2 \rangle}\in D_{2}^1$ and $(\mathbf{a}_{(2N_2+1)})_{\infty}\in D_{\infty}^1$. Using Lemma [Lemma 11](#Letalpha=){reference-type="ref" reference="Letalpha="} and [Lemma 14](#Dinfty){reference-type="ref" reference="Dinfty"}, we see that $(\mathbf{a}_{(2N_3+1)})_{\langle 2 \rangle}\in D_{2}^1$ and $(\mathbf{a}_{(2N_3+1)})_{\infty}\in D_{\infty}^1$, where $N_3=\max\{N_1,N_2\}$. For the sake of simplicity, we assume that $(\mathbf{a})_{\langle 2 \rangle}\in D_{2}^1$ and $(\mathbf{a})_{\infty}\in D_{\infty}^1$. We also remark that $\mathbf{a}_{n}=(\alpha_{(n)}, \overline{\alpha_{(n)}})$. Let $p_1(x)=a_1x^2+b_1x+c_1$ be a minimal polynomial of $\alpha_{(1)}$ over $\mathbb{Z}$. For $n\in \mathbb{Z}_{>1}$,we define $p_n(x)\in \mathbb{Z}[\frac{1}{2}][x]$ recursively as: $$\begin{aligned} \label{p_n(x)} p_n(x):=(x+a_{n-1}(\alpha))^2p_{n-1}\left(\dfrac{1}{x+a_{n-1}(\alpha)}\right).\end{aligned}$$ From the fact that $\alpha_{(n)}=\frac{1}{\alpha_{(n-1)}}-a_{n-1}(\alpha)$, we can conclude that $p_n(x)$ is a minimal polynomial of $\alpha_{(n)}$ over $\mathbb{Z}[\frac{1}{2}]$. We set $a_nx^2+b_nx+c_n=p_n(x)$ for $n\in \mathbb{Z}_{>1}$. For $n\in \mathbb{Z}_{>0}$, we define $d_n$ as the discriminant of $p_n(x)$, which is given by $d_n=b_n^2-4a_nc_n$. The relation ([\[p_n(x)\]](#p_n(x)){reference-type="ref" reference="p_n(x)"}) between $p_n(x)$ and $p_{n-1}(x)$ implies $d_n=d_{n-1}$ for $n\in \mathbb{Z}_{>1}$. Let $n\in \mathbb{Z}_{>1}$. We set $d=d_1$. From Lemma [Lemma 31](#2adicalphainK){reference-type="ref" reference="2adicalphainK"}, we have $$\begin{aligned} v_2((\alpha_{(n)}-\overline{\alpha_{(n)}})_{\langle 2 \rangle})=v_2(a_{n-1}(\alpha)).\end{aligned}$$ Therefore, from Lemma [Lemma 23](#boundpf){reference-type="ref" reference="boundpf"}, there exists $C_1>0$ such that $$\begin{aligned} \label{C_1v_2} -C_1<v_2((\alpha_{(n)}-\overline{\alpha_{(n)}})_{\langle 2 \rangle})\leq 0.\end{aligned}$$ Since it holds that $d=d_n=a_n^2(\alpha_{(n)}-\overline{\alpha_{(n)}}))^2$, from the inequality ([\[C_1v_2\]](#C_1v_2){reference-type="ref" reference="C_1v_2"}), we have $$\begin{aligned} \label{dfrac12v_2} \dfrac{1}{2}v_2(d) \leq v_2(a_n)<\dfrac{1}{2}(v_2(d)+C_1).\end{aligned}$$ On the other hand, from the fact that $(\mathbf{a}_{(n)})_{\infty}\in D_{\infty}^{\tau(n)}$, we have $$\begin{aligned} |(\alpha_{(n)}-\overline{\alpha_{(n)}})_{\infty}|> \dfrac{1}{2},\end{aligned}$$ which implies $$\begin{aligned} \label{|a_n|<|d|} |a_n|<2\sqrt{d}.\end{aligned}$$ From the inequalities ([\[dfrac12v_2\]](#dfrac12v_2){reference-type="ref" reference="dfrac12v_2"}) and ([\[\|a_n\|\<\|d\|\]](#|a_n|<|d|){reference-type="ref" reference="|a_n|<|d|"}), there exists $C_2>0$ such that $$\begin{aligned} \sharp\{a_m\mid m\in \mathbb{Z}_{>0}\}<C_2.\end{aligned}$$ Since $b_n=-a_n(\alpha_{(n)}+\overline{\alpha_{(n)}})_{\langle 2 \rangle}$, from the inequalities ([\[C_1v_2\]](#C_1v_2){reference-type="ref" reference="C_1v_2"}) and ([\[dfrac12v_2\]](#dfrac12v_2){reference-type="ref" reference="dfrac12v_2"}), we can conclude that $$\begin{aligned} \label{-C_1+dfrac} -C_1+\dfrac{1}{2}v_2(d) \leq v_2(b_n)<\dfrac{1}{2}(v_2(d)+C_1).\end{aligned}$$ Now, we have $$\begin{aligned} \label{|alpha_(n)} |(\alpha_{(n)}-\overline{\alpha_{(n)}})_{\infty}|=\dfrac{\sqrt{d}}{|a_n|}\leq \dfrac{\sqrt{d}}{C_3},\end{aligned}$$ where $C_3=\min \{|a_m|\mid m\in \mathbb{Z}_{>0}\}$. The inequality ([\[\|alpha\_(n)\]](#|alpha_(n)){reference-type="ref" reference="|alpha_(n)"}) leads to $$\begin{aligned} \label{1+dfrac|d|} |(\overline{\alpha_{(n)}})_{\infty}| \leq |(\alpha_{(n)})_{\infty}|+|(\alpha_{(n)})_{\infty}-(\overline{\alpha_{(n)}})_{\infty}| \leq 1+\dfrac{\sqrt{d}}{C_3}.\end{aligned}$$ Therefore, we have $$\begin{aligned} \label{|b_n|=|a_n|} |b_n|=|a_n||(\alpha_{(n)})_{\infty}+(\overline{\alpha_{(n)}})_{\infty}|<2\sqrt{d}\left(2+\dfrac{\sqrt{d}}{C_3}\right).\end{aligned}$$ From the inequalities ([\[-C_1+dfrac\]](#-C_1+dfrac){reference-type="ref" reference="-C_1+dfrac"}) and ([\[\|b_n\|=\|a_n\|\]](#|b_n|=|a_n|){reference-type="ref" reference="|b_n|=|a_n|"}), there exists $C_4>0$ such that $$\begin{aligned} \sharp\{b_m\mid m\in \mathbb{Z}_{>0}\}<C_4.\end{aligned}$$ Since $c_n=a_n(\alpha_{(n)})_{\langle 2 \rangle}(\overline{\alpha_{(n)}})_{\langle 2 \rangle}$, similarly, we have $$\begin{aligned} \label{v_2(c_n)} \dfrac{1}{2}v_2(d)-C_1 \leq v_2(c_n)<\dfrac{1}{2}(v_2(d)+3C_1).\end{aligned}$$ From the inequality ([\[\|a_n\|\<\|d\|\]](#|a_n|<|d|){reference-type="ref" reference="|a_n|<|d|"}) and ([\[1+dfrac\|d\|\]](#1+dfrac|d|){reference-type="ref" reference="1+dfrac|d|"}), we have $$\begin{aligned} \label{c_n|=|a_n||} |c_n|=|a_n||(\alpha_{(n)})_{\infty}||(\overline{\alpha_{(n)}})_{\infty}|<2\sqrt{d}\left(1+\dfrac{\sqrt{d}}{C_3}\right).\end{aligned}$$ From the inequalities ([\[v_2(c_n)\]](#v_2(c_n)){reference-type="ref" reference="v_2(c_n)"}) and ([\[c_n\|=\|a_n\|\|\]](#c_n|=|a_n||){reference-type="ref" reference="c_n|=|a_n||"}), there exists $C_5>0$ such that $$\begin{aligned} \sharp\{c_m\mid m\in \mathbb{Z}_{>0}\}<C_5.\end{aligned}$$ Therefore, since $\{p_n(x) \mid n \in \mathbb{Z}_{>0}\}$ is a finite set, the theorem's statement can be easily deduced. ◻ **Corollary 33**. *Let $\alpha\in K/\mathbb{Q}$. Then, there exist $n_1, n_2\in \mathbb{Z}_{\geq 0}$ with $n_1<n_2$ such that $\alpha_{n_1}=\alpha_{n_2}$ and $n_1\equiv n_2 \mod\ 2$.* We can represent convergents as a ratio of integers as follows. **Definition 12**. For $n\in \mathbb{Z}_{> 0}$, $p'_n\in\mathbb{Z}$ and $q'_n\in\mathbb{Z}_{> 0}$ are defined by $$\begin{aligned} p'_n=2^{-\sum_{s=1}^{n}v_2(a_{(s)}(\alpha))}p_n\ \ \text{and} \ \ q'_n=2^{-\sum_{s=1}^{n}v_2(a_{(s)}(\alpha))}q_n.\end{aligned}$$ We remark that for $n\in \mathbb{Z}_{> 0}$ $\dfrac{p'_n}{q'_n}=\dfrac{p_n}{q_n}$ and $(p'_n,q'_n)=1$. The following theorems demonstrate the quality of the approximation. **Theorem 34**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. There exists a constant $C>0$ such that for all $n\in\mathbb{Z}_{>0}$, $$\begin{aligned} \dfrac{C}{|q'_n|^{2}}<\left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right| \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2 < \dfrac{2}{(1+\tau(n))|a_{n+1}(\alpha)|_2|q'_n|^2}.\end{aligned}$$* *Proof.* Let $n\in\mathbb{Z}_{>0}$. From Lemma [Lemma 21](#boundf){reference-type="ref" reference="boundf"}, we have $$\begin{aligned} &\left|\alpha_{\infty}-\dfrac{p_n}{q_n}\right|< \dfrac{2}{(1+\tau(n))|q_n|^2},\end{aligned}$$ which implies $$\begin{aligned} \label{qboundf} \left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right|< \dfrac{2\cdot2^{-2\sum_{s=1}^{n}v_2(a_{(s)}(\alpha)) }}{(1+\tau(n))|q'_n|^2}.\end{aligned}$$ From this inequality and Lemma [Lemma 22](#boundf2){reference-type="ref" reference="boundf2"}, we obtain the following. $$\begin{aligned} &\left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right|\left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2 <\dfrac{2}{(1+\tau(n))|a_{n+1}(\alpha)|_2|q'_n|^2}.\end{aligned}$$ Thus, we have the right hand side of the equality of the claim of the theorem. From the equation ([\[q2nleft\]](#q2nleft){reference-type="ref" reference="q2nleft"}) and Lemma [Lemma 22](#boundf2){reference-type="ref" reference="boundf2"}, we have $$\begin{aligned} &\left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right|\left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2=\dfrac{1}{|q'_n|^2|a_{n+1}(\alpha)|_2\left|\dfrac{q_{n-1}}{q_n}+a_{n+1}(\alpha)+(\alpha_{(n+2)})_{\infty}\right|}.\end{aligned}$$ Therefore, from Lemma [Lemma 23](#boundpf){reference-type="ref" reference="boundpf"} and [Lemma 28](#thereexistsC){reference-type="ref" reference="thereexistsC"}, we have the left hand side of the equality of the claim of the theorem. ◻ **Corollary 35**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. Then, for all $n\in\mathbb{Z}_{>0}$, $$\begin{aligned} \left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right| \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2 < \dfrac{1}{|q'_n|^{2}}.\end{aligned}$$* *Proof.* Since if $n$ is even, then $\tau(n)=0$ and $|a_{n+1}(\alpha)|_2\geq 2$ and if $n$ is odd, then $\tau(n)=1$ and $|a_{n+1}(\alpha)|_2\geq 1$, we have $\dfrac{2}{(1+\tau(n))|a_{n+1}(\alpha)|_2}\leq 1$. Thus, from Theorem [Theorem 34](#qualityofapproximation){reference-type="ref" reference="qualityofapproximation"}, we have the claim of the corollary. ◻ **Definition 13**. Let $\alpha\in K/\mathbb{Q}$ and $\{\frac{p_n}{q_n}\}$ be its convergents related to Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}). For $n\in \mathbb{Z}_{> 0}$, $p'_n\in\mathbb{Z}$ and $q'_n\in\mathbb{Z}_{> 0}$ are defined by $$\begin{aligned} p'_n=\begin{cases} 2^{-(\sum_{s=1}^{n}v_2(a_{(s)}(\alpha))}p_n& \text{if } v_2(a_0(\alpha))>0,\\ 2^{-(\sum_{s=0}^{n}v_2(a_{(s)}(\alpha))}p_n& \text{if } v_2(a_0(\alpha))\leq 0,\\ \end{cases}\\ q'_n=\begin{cases} 2^{-(\sum_{s=1}^{n}v_2(a_{(s)}(\alpha))}q_n& \text{if } v_2(a_0(\alpha))>0,\\ 2^{-(\sum_{s=0}^{n}v_2(a_{(s)}(\alpha))}q_n& \text{if } v_2(a_0(\alpha))\leq 0. \end{cases}\end{aligned}$$ We remark that for $n\in \mathbb{Z}_{> 0}$ $\dfrac{p'_n}{q'_n}=\dfrac{p_n}{q_n}$ and $(p'_n,q'_n)=1$. The following corollary follows from Theorem [Theorem 34](#qualityofapproximation){reference-type="ref" reference="qualityofapproximation"}. We will leave the proof to the reader. **Corollary 36**. *Let $\alpha\in K/\mathbb{Q}$ and $\{\frac{p_n}{q_n}\}$ be its convergents related to Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}). There exist $C\in\mathbb{R}_{>0}$such that for all $n\in\mathbb{Z}_{\geq 0}$, $$\begin{aligned} \dfrac{C}{|q'_n|^{2}}<\left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right| \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2 < \dfrac{2\max\{1,(|b_{0}(\alpha)|_2)^2\}}{(1+\tau(n))|b_{n+1}(\alpha)|_2|q'_n|^2}.\end{aligned}$$* The following theorem provides a simultaneous rational approximation to a number in $K$ in both $\mathbb{R}$ and $\mathbb{Q}_2$, making it one of the main results. **Theorem 37**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. There exist $\gamma, C, C' \in\mathbb{R}_{>0}$ with $0<\gamma<1$ such that for all $n\in \mathbb{Z}_{>0}$, $$\begin{aligned} \left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right|\leq \dfrac{C}{|q'_n|^{2-2\gamma}} \text{ and } \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2\leq \dfrac{C'}{|q'_n|^{2\gamma}}.\end{aligned}$$* *Proof.* Let $l\in \mathbb{Z}_{>0}$ be the length of the period of $\{\alpha_{(n)}\}$. We assume that for $n_1, n_2\in \mathbb{Z}_{>0}$ with $n_1<n_2$ such that $\alpha_{(n_1)}=\alpha_{(n_2)}$, $n_1\equiv n_2 \mod\ 2$, and $n_2-n_1=l$. Let $0\leq i\leq l-1$. Let $n>n_2$ for $n\in \mathbb{Z}_{>0}$ with $n\equiv i \mod l$. We take integers $j_1$ and $k$ such that $n_1 \leq j_1 < n_2$ and $n = j_1 + kl$. We put $$\begin{aligned} M=\begin{pmatrix}0&1\\1&a_{j_1+1}(\alpha)\end{pmatrix}\begin{pmatrix}0&1\\1&a_{j_1+2}(\alpha)\end{pmatrix}\ldots \begin{pmatrix}0&1\\1&a_{j_1+l}(\alpha)\end{pmatrix} .\end{aligned}$$ Since ${}^t(\alpha_{(j_1+1)},1)$ and ${}^t(\overline{\alpha_{(j_1+1)}},1)$ are eigenvectors of the matrix $M$, its eigenvalues $\lambda_1, \lambda_2$ belong to $K/\mathbb{Q}$. For the sake of simplicity, we denote $(\lambda_i)_{\infty}$ by $\lambda_i$ for $i=1,2$. Since $|\lambda_1\lambda_2|=1$, we can assume without loss of generality that $|\lambda_1|>1$ and $|\lambda_2|<1$. We note that $\overline{\lambda_1}=\lambda_2$. From the fact that $$\begin{aligned} \begin{pmatrix} p_{n-1}&p_{n}\\ q_{n-1}&q_{n} \end{pmatrix} = \begin{pmatrix} 0&1\\ 1&a_1(\alpha) \end{pmatrix} \cdots \begin{pmatrix} 0&1\\ 1&a_{j_1}(\alpha) \end{pmatrix} M^k,\end{aligned}$$ there exist $\delta_1, \delta_2\in K$ with $\delta_1\delta_2\ne 0$ such that $p_{n}=\delta_1\lambda^k_1+ \overline{\delta_1} \overline{\lambda_1}^k$ and $q_{n}=\delta_2\lambda^k_1+ \overline{\delta_2} \overline{\lambda_1}^k$. Now, we put $$\begin{aligned} &\xi_0=2^{-\sum_{s=1}^{j_1}v_2(a_{(s)}(\alpha))},\\ &\xi_1=2^{-\sum_{s=j_1+1}^{j_1+l}v_2(a_{(s)}(\alpha))},\\ &\gamma=\frac{\log \xi_1}{\log |\lambda_1\xi_1|}.\end{aligned}$$ We remark that $\lambda_1, \xi_1$ and $\gamma$ are constants independent of $n$. Since $|\lambda_1|>1$ and $\xi_1>1$, we see that $0<\gamma<1$. From the inequality ([\[qboundf\]](#qboundf){reference-type="ref" reference="qboundf"}), we have $$\begin{aligned} \label{left|alphainfty} \left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right|< \dfrac{2(\xi_0\xi_1^k)^2}{(1+\tau(n))|q'_n|^2} =\dfrac{2\xi_0^2|\lambda_1\xi_1|^{2k\gamma}}{(1+\tau(n))|q'_n|^2}.\end{aligned}$$ Since $$\begin{aligned} \lim_{k\to \infty}\dfrac{|q'_n|^{\gamma}}{|\lambda_1\xi_1|^{k\gamma}} =\lim_{k\to \infty}\dfrac{|\xi_0\xi_1^k\delta_2\lambda^k_1+ \xi_0\xi_1^k\overline{\delta_2} \overline{\lambda_1}^k|^{\gamma}}{|\lambda_1\xi_1|^{k\gamma}}=|\xi_0\delta_2|^{\gamma},\end{aligned}$$ for sufficiently large $k$, we have $$\begin{aligned} \label{|qngamma} 2|\xi_0\delta_2|^{\gamma}|\lambda_1\xi_1|^{k\gamma}> |q'_n|^{\gamma}>\dfrac{|\xi_0\delta_2|^{\gamma}}{2}|\lambda_1\xi_1|^{k\gamma}.\end{aligned}$$ From the inequalities ([\[left\|alphainfty\]](#left|alphainfty){reference-type="ref" reference="left|alphainfty"}) and ([\[\|qngamma\]](#|qngamma){reference-type="ref" reference="|qngamma"}), for sufficiently large $k$, we have $$\begin{aligned} \left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right|< \dfrac{8\xi_0^2}{|\xi_0\delta_2|^{2\gamma}|q'_n|^{2-2\gamma}}.\end{aligned}$$ From Lemma [Lemma 22](#boundf2){reference-type="ref" reference="boundf2"} and the inequality ([\[\|qngamma\]](#|qngamma){reference-type="ref" reference="|qngamma"}), for sufficiently large $k$, we have $$\begin{aligned} \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2 \leq\dfrac{1}{\xi_0^2\xi_1^{2k}}=\dfrac{1}{\xi_0^2|\lambda_1\xi_1|^{2k\gamma}} <\dfrac{4|\xi_0\delta_2|^{2\gamma}}{\xi_0^2|q'_n|^{2\gamma}}.\end{aligned}$$ Thus, we have the claim of the theorem. ◻ **Corollary 38**. *Let $\alpha\in K/\mathbb{Q}$ and $\{\frac{p_n}{q_n}\}$ be its convergents related to Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}). Let $l\in \mathbb{Z}_{>0}$ be the length of the period of $\{\alpha_{n}\}$. There exist $\gamma, C, C' \in\mathbb{R}_{>0}$ with $0<\gamma<1$ such that for $n\in \mathbb{Z}_{>0}$, $$\begin{aligned} \left|\alpha_{\infty}-\dfrac{p'_n}{q'_n}\right|\leq \dfrac{C}{|q'_n|^{2-2\gamma}} \text{ and } \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2\leq \dfrac{C'}{|q'_n|^{2\gamma}}.\end{aligned}$$* # Complex quadratic number Let $p$ be a prime number. Let $K$ be a quadratic field that has an embedding into $\mathbb{C}$ and $\mathbb{Q}_p$ respectively. We assume that $\sigma_{\infty}$ gives an embedding into $\mathbb{C}$ and $\sigma_p$ gives an embedding into $\mathbb{Q}_p$. We assume that $\sigma_{\infty}(K)\not\subset \mathbb{R}$, i.e., $\sigma_{\infty}(K)$ is a complex quadratic field. Let $\alpha \in K$. We also denote $\sigma_{\infty}(\alpha)$ by $\alpha_{\infty}$ and $\sigma_{p}(\alpha)$ by $\alpha_{\langle p \rangle}$. We denote the real part of $\alpha$ by $Re(\alpha)$ and the imaginary part by $Im(\alpha)$. $F_{p,\epsilon}$ and $T_{p,\epsilon}$ for $\epsilon=0,1$ are applicable to $\alpha$, and therefore, Algorithms (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}, Definition [Definition 6](#algol2){reference-type="ref" reference="algol2"}) are applicable to $\alpha$. We assume that various notations are the same as in the case when $K$ is a real quadratic field. We will extend 'property $I$' slightly. **Definition 14**. We say that $\alpha$ has property $I$, if it satisfies $Re(\alpha)\in \left(-\frac{p}{2},\frac{p}{2}\right]$ and $v_p(\alpha_{\langle p \rangle})>0$. The following lemma can be proven similarly to Lemma [Lemma 22](#boundf2){reference-type="ref" reference="boundf2"}. **Lemma 39**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. For all $n\in \mathbb{Z}_{\geq 1}$, it holds that $$\begin{aligned} v_2\left(\alpha_{\langle 2 \rangle}-\dfrac{p_n}{q_n}\right)=-2v_2(q_n)-v_2(a_{n+1}(\alpha)).\end{aligned}$$* Based on the numerical experiments, we conjecture that the sequence $\{\alpha_n\}$ will eventually become periodic (see section [5](#Numericalexperiments){reference-type="ref" reference="Numericalexperiments"}). **Theorem 40**. *Let $\alpha\in K/\mathbb{Q}$ with property $I$. Let us assume that there exist $n_1, n_2\in \mathbb{Z}_{>0}$ with $n_1<n_2$ such that $\alpha_{(n_1)}=\alpha_{(n_2)}$ and $n_1\equiv n_2 \mod\ 2$. Then, there exists $C\in\mathbb{R}_{>0}$ such that for all $n\in \mathbb{Z}_{>0}$, $$\begin{aligned} \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2\leq \dfrac{C}{\max\{|p'_n|,|q'_n|\}^{2}}.\end{aligned}$$* *Proof.* Let $l=n_2-n_1$. Let $0\leq i\leq l-1$. Let $n>n_2$ for $n\in \mathbb{Z}_{>0}$ with $n\equiv i \mod l$. We take integers $j_1$ and $k$ such that $n_1 \leq j_1 < n_2$ and $n = j_1 + kl$. We put $$\begin{aligned} M=\begin{pmatrix}0&1\\1&a_{j_1+1}(\alpha)\end{pmatrix}\begin{pmatrix}0&1\\1&a_{j_1+2}(\alpha)\end{pmatrix}\ldots \begin{pmatrix}0&1\\1&a_{j_1+l}(\alpha)\end{pmatrix}\end{aligned}$$ Since ${}^t(\alpha_{(j_1+1)},1)$ and ${}^t(\overline{\alpha_{(j_1+1)}},1)$ are eigenvectors of the matrix $M$, its eigenvalues $\lambda_1, \lambda_2$ belong to $K/\mathbb{Q}$. Since $|\lambda_1\lambda_2|=1$ and $\lambda_1$ is a complex quadratic number, we have $|\lambda_1|=|\lambda_2|=1$. We note that $\overline{\lambda_1}=\lambda_2$. From the fact that $$\begin{aligned} \begin{pmatrix} p_{n-1}&p_{n}\\ q_{n-1}&q_{n} \end{pmatrix} = \begin{pmatrix} 0&1\\ 1&a_1(\alpha) \end{pmatrix} \cdots \begin{pmatrix} 0&1\\ 1&a_{j_1}(\alpha) \end{pmatrix} M^k,\end{aligned}$$ there exist $\delta_1, \delta_2\in K$ with $\delta_1\delta_2\ne 0$ such that $p_{n}=\delta_1\lambda^k_1+ \overline{\delta_1} \overline{\lambda_1}^k$ and $q_{n}=\delta_2\lambda^k_1+ \overline{\delta_2} \overline{\lambda_1}^k$. Now, we put $$\begin{aligned} &\xi_0=2^{-\sum_{s=1}^{j_1}v_2(a_{(s)}(\alpha))},\\ &\xi_1=2^{-\sum_{s=j_1+1}^{j_1+l}v_2(a_{(s)}(\alpha))}.\end{aligned}$$ From Lemma [Lemma 39](#boundf3){reference-type="ref" reference="boundf3"}, we have $$\begin{aligned} \label{ineq11} \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2<\dfrac{1}{(\xi_0\xi_1^k)^2}=\dfrac{|\delta_2\lambda^k_1+ \overline{\delta_2} \overline{\lambda_1}^k|^2}{|q'_n|^2}\leq \dfrac{|\delta_2|+|\overline{\delta_2}|}{|q'_n|^2}.\end{aligned}$$ Similarly, we have $$\begin{aligned} \label{ineq2} \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2<\dfrac{1}{(\xi_0\xi_1^k)^2}=\dfrac{|\delta_1\lambda^k_1+ \overline{\delta_1} \overline{\lambda_1}^k|^2}{|p'_n|^2}\leq \dfrac{|\delta_1|+|\overline{\delta_1}|}{|p'_n|^2}.\end{aligned}$$ Inequalities ([\[ineq11\]](#ineq11){reference-type="ref" reference="ineq11"}) and ([\[ineq2\]](#ineq2){reference-type="ref" reference="ineq2"}) lead to the theorem's conclusion. ◻ **Corollary 41**. *Let $\alpha\in K/\mathbb{Q}$ and $\{\frac{p_n}{q_n}\}$ be its convergents related to Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}). Let us assume that there exist $n_1, n_2\in \mathbb{Z}_{>0}$ with $n_1<n_2$ such that $\alpha_{n_1}=\alpha_{n_2}$ and $n_1\equiv n_2 \mod\ 2$. There exists $C\in\mathbb{R}_{>0}$ such that for all $n\in \mathbb{Z}_{>0}$, $$\begin{aligned} \left|\alpha_{\langle 2 \rangle}-\dfrac{p'_n}{q'_n}\right|_2\leq \dfrac{C}{\max\{|p'_n|,|q'_n|\}^{2}}.\end{aligned}$$* # Numerical experiments and conjectures {#Numericalexperiments} We demonstrate in Table [1](#t1){reference-type="ref" reference="t1"} that for $1 < n \leq 200$, the continued fraction expansion of $\sqrt{n}$ obtained using Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}) satisfies the condition $\sqrt{n}=2^m\sqrt{k}$, where $m,k\in \mathbb{Z}{\geq 0}$, and $k$ is not a square of an integer. Additionally, $(\sqrt{k})_{\langle 2 \rangle}\in \mathbb{Q}_2$ and $(\sqrt{k})_{\langle 2 \rangle} \equiv 1 \mod 8$. In this table, let $\gamma$ be the same as that in Theorem [Theorem 37](#qualityofapproximation3){reference-type="ref" reference="qualityofapproximation3"}, so that $\gamma_r = 2 - 2\gamma$ and $\gamma_2 = 2\gamma$. Similarly, we demonstrate in Table [2](#t2){reference-type="ref" reference="t2"} that for $1 < n \leq 200$, the continued fraction expansion of $(\sqrt{-n})_{\langle 2 \rangle}$ obtained using Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}) satisfies the condition $\sqrt{-n}=2^m\sqrt{-k}$, where $m,k\in \mathbb{Z}{\geq 0}$, and $k$ is not a square of an integer. Additionally, $(\sqrt{-k})_{\langle 2 \rangle}\in \mathbb{Q}_2$ and $(\sqrt{-k})_{\langle 2 \rangle} \equiv 1 \mod 8$. As seen in Table [2](#t2){reference-type="ref" reference="t2"}, for all $1 < n \leq 200$, $(\sqrt{-n})_{\langle 2 \rangle}\in \mathbb{Q}_2$ have eventually periodic expansions. Furthermore, we have confirmed that this holds true for all $1 < n \leq 10000$, which amounts to a total of 1665 cases. We give a following conjecture. **Conjecture 42**. *Let $K$ be a quadratic field that has an embedding into $\mathbb{C}$ and $\mathbb{Q}_2$ respectively. Let $\alpha\in K/\mathbb{Q}$ and $\{\alpha_{n}\}$ be the sequence obtained by applying Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}) to $\alpha$. Then, $\{\alpha_{n}\}$ becomes eventually periodic.* In Table [3](#t3){reference-type="ref" reference="t3"} we show the total count when a period is detected in the continued fraction expansion of $\sqrt{n}$ with $1<n\leq 10000$ within $1000$ steps for $p<100$. Here, we have $n = p^{2m}k$, where $m$ is a non-negative integer ($m\in \mathbb{Z}_{\geq 0}$), and $k$ is a positive non-square integer that is also a quadratic residue below $\frac{p}{2}$ modulo $p$. As shown in Table [3](#t3){reference-type="ref" reference="t3"}, for all $3 \leq p \leq 23$, the continued fraction expansions become periodic in every case. Even when $n<20000$, there are a few cases where the period length exceeds 1000. However, the expansions become periodic in all instances. We give a following conjecture. **Conjecture 43**. *Let $p$ be a prime with $3 \leq p \leq 23$. Let $K$ be a quadratic field that has an embedding into $\mathbb{R}$ and $\mathbb{Q}_p$ respectively. Let $\alpha\in K/\mathbb{Q}$ and $\{\alpha_{n}\}$ be the sequence obtained by applying Algorithm (Definition [Definition 2](#algol1){reference-type="ref" reference="algol1"}) to $\alpha$. Then, $\{\alpha_{n}\}$ becomes eventually periodic.* $\sqrt{n}$ continued fraction expansion $\gamma_r$ $\gamma_{2}$ ---------------- ------------------------------------- ------------ -------------- $\sqrt{17}$ $[5; -3/4, -3, \overline{5/2, -5}]$ 1.54328 0.45672 $\sqrt{33}$ $[5;7/4,\overline{-3, 9/4}]$ 1.04289 0.957107 $\sqrt{ 41 }$ $[ 1.20362 0.796382 7 ; -3/2 , \overline{ -5 , -5/4} ]$ $\sqrt{ 57 }$ $[ 0.867752 1.13225 7 ; 3/2 , \overline{ 5/2 , 5/4 } ]$ $\sqrt{ 65 }$ $[ 1.35332 0.646676 9 ; -5/8 , -5/2 , \overline{ 9/2 , -9/2 } ]$ $\sqrt{ 68 }$ $[ 1.54328 0.45672 8 , 9/2 , -3/2 , -3/2 , \overline{ 5 , -5/2 } ]$ $\sqrt{ 73 }$ $[ 0.913949 1.08605 9 ; -7/4 , \overline{ -3 , 13/8 , -3 , -5/4} ]$ $\sqrt{ 89 }$ $[ 1.17433 0.825669 9 ; 11/4 , \overline{ -3 , 3/2 , -13/2 , 3/2 , -3 , 13/4} ]$ $\sqrt{ 97 }$ $[ 1.38203 0.61797 9 ; 13/8 , -3/2 , -5/4 , \overline{-9 , -9/4 } ]$ $\sqrt{ 105 }$ $[ 0.869728 1.13027 11 ; -3/2 , \overline{13/2 , -13/8 } ]$ $\sqrt{ 113 }$ $[ 1.08953 0.910471 11 ; -5/2 , \overline{-9/2 , -9/4} ]$ $\sqrt{ 129 }$ $[ 1.21723 0.78277 11 ; 5/2 , \overline{3 , 9/4 } ]$ $\sqrt{ 132 }$ $[ 1.04289 0.957107 12 ; -3/2 , -3 , \overline{3/2 , -9/2} ]$ $\sqrt{ 137 }$ $[ 1.1728 0.827195 11 ; 3/2 , \overline{-13 , 13/8 } ]$ $\sqrt{ 145 }$ $[ 0.821856 1.17814 13 ; -5/4 , 5 , -13/2 , 5 , \overline{-3/4 , -3 , 3/4 , 3 } ]$ $\sqrt{ 153 }$ $[ 0.396649 1.60335 13 ; -9/8 , \overline{-5/4 , -5/8 } ]$ $\sqrt{ 161 }$ $[ 1.20791 0.792091 13 ; -11/4 , \overline{-3 , 3/2 , -3 , -9/4 } ]$ $\sqrt{ 164 }$ $[ 1.20362 0.796382 12 ; 3/2 , -7/2 , \overline{-5/2 , -5/2 } ]$ $\sqrt{ 177 }$ $[ 0.883413 1.11659 13 ; 15/4 , \overline{-3 , 25/16 , -3 , 17/4 } ]$ $\sqrt{ 185 }$ $[ 0.541126 1.45887 13 ; 17/8 , \overline{-21/8 , 21/8 } ]$ $\sqrt{ 193 }$ $[ 1.03289 0.967111 13 ; 5/4 , \overline{-17/2 , 7/4 , -3 , 3/2 , -7/2 , 17/4 , -7/2 , 3/2 , -3 , 7/4 } ]$ : Continued fraction expansion of $\sqrt{n}$ with $n>0$ $(\sqrt{n})_{\langle 2 \rangle}$ continued fraction expansion --------------------------------------- ------------------------------ $(\sqrt{ -7 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , \overline{-1/2 , 1 } ]$ $(\sqrt{ -15 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -3/2 , \overline{-1/2 , 1/2 } ]$ $(\sqrt{ -23 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -1/4 , \overline{ -3/2 , 3/4 } ]$ $(\sqrt{ -28 })_{\langle 2 \rangle}$ $[ 0 ; -1/2 , 1 , 1/2 , \overline{1 , -1/2 } ]$ $(\sqrt{ -31 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -7/4 , \overline{-1/2 , 1/4 } ]$ $(\sqrt{ -39 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -1/2 , \overline{ -3/2 , 1/2 } ]$ $(\sqrt{ -47 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -5/2 , 3/4 , \overline{ 1 , -1/2 , 1/2 , -1/2 , 1 , -1/4 } ]$ $(\sqrt{ -55 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -5/8 , \overline{ -3/2 , 3/8 } ]$ $(\sqrt{ -60 })_{\langle 2 \rangle}$ $[ 0 ; -1/2 , 1 , 3/4 , \overline{ 1 , -1/4 } ]$ $(\sqrt{ -63 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -15/8 , \overline{-1/2 , 1/8 } ]$ $(\sqrt{ -71 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -1/2 , -1 , \overline{-1/2 , 5/4 , -1/2 , 1 , -5/8 , 1 } ]$ $(\sqrt{ -79 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -3/2 , -5/4 , 1 , -1/8 , \overline{ -7/4 , 7/8 } ]$ $(\sqrt{ -87 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -3/4 , \overline{ -3/2 , 1/4 } ]$ $(\sqrt{ -92 })_{\langle 2 \rangle}$ $[ 0 ; -1/2 , 1 , 1/2 , 1/2 , \overline{ 3/4 , -3/2 } ]$ $(\sqrt{ -95 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -5/4 , -9/8 , \overline{1 , -1/2 , 1 , -1/8 } ]$ $(\sqrt{ -103 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -1/2 , -3/4 , \overline{ -5/8 , 5/4 } ]$ $(\sqrt{ -111 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -5/2 , 3/2 , \overline{-1 , 7/8 , -1 , 1/2 } ]$ $(\sqrt{ -112 })_{\langle 2 \rangle}$ $[ 0 ; 1/4 , -3 , \overline{-1/2 , 1 } ]$ $(\sqrt{ -119 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -13/16 , \overline{-3/2 , 3/16 } ]$ $(\sqrt{ -124 })_{\langle 2 \rangle}$ $[ 0 ; -1/2 , 1 , 7/8 , \overline{1 , -1/8 } ]$ $(\sqrt{ -127 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -31/16 , \overline{-1/2 , 1/16 } ]$ $(\sqrt{ -135 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -1/2 , -1 , \overline{ -5/8 , 1 } ]$ $(\sqrt{ -143 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -3/2 , -3/2 , -1/2 , 7/8 , \overline{ -1/2 , -1/2 , 1 , -1/8 } ]$ $(\sqrt{ -151 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -5/4 , 5/4 , \overline{1/2 , -1 , 3/8 , -1 , 1/2 , -3/4 } ]$ $(\sqrt{ -156 })_{\langle 2 \rangle}$ $[ 0 ; -1/2 , 1 , 1/2 , 1 , \overline{ 3/4 , -1 } ]$ $(\sqrt{ -159 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -11/4 , 3/2 , -1 , \overline{ -1/8 , 1 , -3/4 , 1 } ]$ $(\sqrt{ -167 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -1/2 , -5/2 , \overline{1/4 , 5/4 , 1/4 , -1/2 , -5/8 , -1/2 } ]$ $(\sqrt{ -175 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -5/2 , 15/8 , \overline{ -1 , 7/8 } ]$ $(\sqrt{ -183 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -7/8 , \overline{ -3/2 , 1/8 } ]$ $(\sqrt{ -188 })_{\langle 2 \rangle}$ $[ 0 ; -1/2 , 1 , 5/4 , -3/2 , \overline{-1/2 , 1 , -1/4 , 1 , -1/2 , 1/2 } ]$ $(\sqrt{ -191 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -21/8 , 5/4 , \overline{ 1/4 , 1/4 , -1 , 1/2 , -1/2 , -1/8 , -1/2 , 1/2 , -1 , 1/4 } ]$ $(\sqrt{ -199 })_{\langle 2 \rangle}$ $[ -1 ; 1/2 , -1 , -1/2 , -1 , \overline{-1/2 , 1 , -1/2 , 5/4 , -1/2 , 1 , -1/2 , 1 , -5/8 , 1 } ]$ : Continued fraction expansion of $(\sqrt{n})_{\langle 2 \rangle}$ with $n<0$ $p$ Total Count with Detected Period Total Count with Undetected Period Total Count ----- ---------------------------------- ------------------------------------ ------------- 3 3652 0 3652 5 4067 0 4067 7 4278 0 4278 11 4485 0 4485 13 4543 0 4543 17 4623 0 4623 19 4652 0 4652 23 4696 0 4696 29 4443 291 4734 31 4090 659 4749 37 2964 1806 4770 41 2676 2106 4782 43 2455 2333 4788 47 2219 2582 4801 53 1837 2968 4805 59 1807 3014 4821 61 1753 3066 4819 67 1669 3158 4827 71 1612 3223 4835 73 1562 3271 4833 79 1399 3443 4842 83 1340 3505 4845 89 1288 3557 4845 97 1119 3733 4852 : Periodic Counts in Continued Fractions: $\sqrt{n}$ for $1<n<10000$ within $1000$ steps # Acknowledgements {#acknowledgements .unnumbered} This research was supported by JSPS KAKENHI Grant Number JP22K12197. 9 M. Abrate, S. Barbero, U. Cerruti, N. Murru, Periodic representations and rational approximations of square roots, J. Approx. Theory, 175 (2013), 83-90. S. Barbero, U. Cerruti, N. Murru, Periodic representations for quadratic irrationals in the field of $p$--adic numbers, Math. of Comp., 90(331) (2021), 2267-2280. E. Bedocchi, Nota sulle frazioni continue $p$--adiche, Ann. Mat. Pura Appl., IV. 152 (1988), 197-207. H. Bekki, On periodicity of geodesic continued fractions, J. Number Theory 177(2017), 181--210. R. Belhadef, H. A. Esbelin, On the Limits of Some $p$-adic Schneider Continued Fractions, Advances in Mathematics: Scientific Journal 10, (2021), no.5, 2581--2591. J. Browkin, Continued fractions in local fields. I, Demonstratio Mathematica, 11 (1978), 67-82. J. Browkin, Continued fractions in local fields. II, Math. Comp., 70(235) (2001), 1281-1292. P. Bundschuh, $p$-adische Kettenbrüche und Irrationalität $p$-adischer Zahlen, Elem. Math. 32 (1977), 36--40. L. Capuano, F. Veneziano, U. Zannier, An effective criterion for periodicity of $l$-adic continued fractions, Math. Comp. 88(318) (2019), 1851--1882. K. Dajani and C. Kraaikamp, Ergodic theory of numbers, The Carus Mathematical Monographs, vol. 29, Mathematical Association of America, Washington, DC, (2002). G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford Science Publications, 1979. V. Laohakosol, A characterization of rational numbers by $p$--adic Ruban continued fractions, Austral. Math. Soc. Ser. 39 (1985), no. 3, 300--305. K. Mahler, On a geometrical representation of $p$--adic numbers, Ann. of Math. (2) 41, (1940), 8-56. N. Murru, G. Romeo, A new algorithm for $p$-adic continued fractions, Math. Comp(2023). to appear. N. Murru, G. Romeo, G. Santilli, Convergence conditions for $p$--adic continued fractions, preprint (2022), arXiv:2022.09249 T. Ooto, Trascendental $p$--adic continued fractions, Math. Z. 287 (2017), no. 3-4, 1053-1064. T. Pejković, Schneider's $p$-adic continued fractions. Acta Math. Hung. 169 (2023), No. 1, 191-215. D. Ridout, The $p$-adic generalization of the Thue-Siegel-Roth theorem, Mathematika 5, 40-48 (1958). G. Romeo, Continued fractions in the field of $p$-adic numbers, preprint (2023), arXiv:2306.14837 A. A. Ruban, Certain metric properties of the $p$--adic numbers, Sibirsk Math. Z., 11 (1970), 222-227, English translation: Siberian Math. J 11, 176-180. A. Saito, J. Tamura, S. Yasutomi, Multidimensional $p$--adic continued fraction algorithms, Math. Comp. 89(321) (2020), 351-372. A. Saito, J. Tamura, S. Yasutomi, Continued fraction algorithms and Lagrange's theorem in $\mathbb{Q}_p$, Comment. Math. Univ. St. Pauli 67, No. 1 (2019), 27-48. T. Schneider, Uber $p$-adische Kettenbruche, Symp. Math. , 4 (1969), 181-189. L. Wang, $p$--adic continued fractions, I, II, Scientia Sinica, Ser. A 28 (1985), 1009-1023. B. M. M. de Weger, Periodicity of $p$--adic continued fractions, Elem. Math. 43, No. 4 (1988), 112-116. Shin-ichi Yasutomi: Faculty of Science, Toho University, 2-1 Miyama, Funabashi Chiba, 274-8510, JAPAN\ *E-mail address: shinichi.yasutomi\@sci.toho-u.ac.jp* [^1]: 2020 *Mathematics Subject Classification*. 11J70, 11Y65, 11J61, 11D88. [^2]: *Key words and phrases.* continued fraction, $p$-adic continued fraction, quadratic fields

arxiv_math

--- abstract: | Doubly periodic tangles, or *DP tangles*, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are completely defined by their generating cells, the *flat motifs*, which can be chosen in infinitely many ways. DP tangles are used in modelling materials and physical systems of entangled filaments. In this paper we establish the mathematical framework of the topological theory of DP tangles. We first introduce a formal definition of DP tangles as topological objects and proceed with an exhaustive analysis in order to characterize the notion of *equivalence* between DP tangles and between their flat motifs. We further generalize our results to other diagrammatic categories, such as framed, virtual, singular, pseudo and bonded DP tangles, which could be used in novel applications. address: - Department of Data Analytics and Digitalisation, Maastricht University, School of Business and Economics, P.O.Box 616, 6200 MD, Maastricht, The Netherlands. - School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou campus, GR-15780 Athens, Greece. - Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan; Center for Functional Fabrics, Drexel University, 3101 Market Street, Philadelphia, PA 19104, USA; Department of Physics and Astronomy University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA author: - Ioannis Diamantis - Sofia Lambropoulou - Sonia Mahmoudi title: Equivalence of Doubly Periodic Tangles --- # Introduction {#sec:0} Doubly periodic tangles, *DP tangles*, are complex entanglements of curves embedded in the thickened plane $\mathbb{E}^2 \times I$ that are periodically repeated in two transversal directions. Thus, a DP tangle can be defined as the lift of a knot or link (called *motif*) in the thickened torus, $T^2 \times I$, to the universal cover $\mathbb{E}^2 \times I$. Periodic tangles are appropriate for modelling and studying materials and physical systems of entangled filaments in various scales, such as polymer melts [@Eleni1; @Eleni2; @Eleni3], textiles [@Sabetta; @Sonia1; @Sonia2], cosmic filaments [@Bond; @Hong1; @Hong2], among others. A better understanding of their geometry and topology, often associated to some physical and mechanical properties, could allow the prediction of some of their functions. Following this motivation, Evans, Hyde et al. describe and enumerate periodic tangles using graphs and tilings of the Euclidean and hyperbolic planes [@Evans1; @Evans2; @Evans3; @Hyde]. M. O'Keeffe et al. study periodic tangles based on symmetries assumptions, considering PL embeddings with sticks to model structures in molecular chemistry [@Yaghi; @Treacy]. Yet, there is no universal mathematical study of DP tangles and there are many open questions. Our motivation is the classification of DP tangles using tools from knot theory. The classification is sought via constructing topological invariants for DP tangles, that is, functions that assign the same values to equivalent tangles. It is natural to call two DP tangles *equivalent* if they can be obtained from each other by orientation preserving invertible affine transformations of the plane $\mathbb{E}^2$ carrying along the DP tangles and by ambient isotopies (that is, continuous deformations of the thickened plane carrying along the DP tangles) that preserve the double periodicity. This last condition makes the DP tangle equivalence different from the standard equivalence of links in $T^2 \times I$. By definition, a DP tangle comes equipped with a periodic integer lattice, a generating frame of which is called *flat motif*. The aim is to translate DP tangle equivalence at the level of (flat) motifs for any lattice supporting a given DP tangle. The first effort toward establishing topological equivalence of DP tangles was made by Grishanov et al. in [@Grishanov1], whose prime motivation was the study and classification of textiles. Their definition of equivalence takes into consideration the different transformations mentioned above, while considering only (flat) motifs created by the quotient of a DP tangle by a *fixed minimal point lattice*, namely *minimal motifs*. Based on this definition several numerical, polynomial and finite type invariants were constructed for DP tangles [@Grishanov1; @Grishanov.part1; @Grishanov.part2; @Morton; @Grishanov.Vassiliev1; @Grishanov.Vassiliev2; @Kurlin]. In this paper we establish the mathematical framework of the topological theory of DP tangles and their equivalence in terms of flat motifs. In the first part § [2](#sec:setup){reference-type="ref" reference="sec:setup"}, we set the background on DP tangles and their corresponding motifs. Then, in § [3](#sec:equivalence){reference-type="ref" reference="sec:equivalence"} we introduce the notion of equivalence of DP tangles. DP tangle isotopy can be reduced to the diagrammatic level where isotopies are discretized as sequences of periodic moves on *DP diagrams*, generalizing the classical Reidemeister Theorem. Then, these moves can be translated into diagrammatic moves between flat motifs. DP tangle equivalence can be classified into five categories. Namely, given a motif representing a DP tangle we can have: *local isotopies* of the motif, *global isotopies* of the torus surface corresponding to *re-scalings* of the DP tangle, *rigid motions* of the plane carrying along the DP tangle (translations -that correspond to shiftings of the supporting periodic lattice- and rotations), and torus homeomorphisms, namely *Dehn twists*, that correspond to shearings of the DP tangle. Finally, we have to take into account the equivalence between two motifs of different 'scale' representing the same DP tangle, which corresponds to choosing a different covering map of the plane. This last move is called *scale equivalence*, introduced in [@Sonia3] in the context of weaves. The above lead to the following (Theorem [Theorem 19](#th:equivalence){reference-type="ref" reference="th:equivalence"}): **Theorem (Equivalence of DP tangles)**. *Let $\tau_{1,\infty}$ and $\tau_{2,\infty}$ be two DP tangles in $\mathbb{E}^2 \times I$, with corresponding DP diagrams $d_{1,\infty}$ and $d_{2,\infty}$. Let also $\Lambda_1$ and $\Lambda_2$ be the supporting point lattices such that $d_i = d_{i,\infty} / \Lambda_i$ is a flat motif of $d_{i,\infty}$ for $i \in \{1,2\}$. Then $\tau_{1,\infty}$ and $\tau_{2,\infty}$ are *equivalent* if and only if $d_1$ and $d_2$ are related by a finite sequence of shifts, motif isotopy moves, Dehn twists, orientation preserving affine transformations and scale equivalence.* For proving the theorem we use the theory of mixed links [@LR1], where the fixed sublink representing the 3-manifold is the Hopf link, since its complement in $S^3$ is the thickened torus. In the last section § [4](#sec:othersettings){reference-type="ref" reference="sec:othersettings"}, we generalize our results to other diagrammatic settings. We discuss *regular* and *framed* isotopies as well as *virtual*, *welded*, *singular*, *pseudo*, *tied* and *bonded* DP tangle equivalence. The detailed analysis for DP tangles applies equally to any diagrammatic category. Thus, for the study of DP tangles related to any of the above topological settings, we only need to adapt our analysis in the context of motif isotopy. This observation leads to the Theorems [Theorem 24](#th:regularframedequivalence){reference-type="ref" reference="th:regularframedequivalence"}, [Theorem 28](#th:virtualweldedequivalence){reference-type="ref" reference="th:virtualweldedequivalence"}, [Theorem 31](#th:singularpseudoequivalence){reference-type="ref" reference="th:singularpseudoequivalence"}, and [Theorem 35](#th:tiedbondedequivalence){reference-type="ref" reference="th:tiedbondedequivalence"}, analogues of Theorem [Theorem 19](#th:equivalence){reference-type="ref" reference="th:equivalence"}. Physical applications of the topological classification of DP tangles in these different settings can be taken into consideration. Virtual DP tangles are potentially interesting in materials science, where the prevention of friction between strands is desirable. Pseudo DP tangles can model DNA knots or worn textiles, where distinguishing the relative positions of two entangled strands may not be straightforward. Singular equivalence gives the opportunity to generalize finite type invariants of knots and links to DP tangles. Finally, singular and bonded DP tangles can be relevant in modelling protein chains. Concluding, Theorem [Theorem 19](#th:equivalence){reference-type="ref" reference="th:equivalence"}, along with its analogues related to different diagrammatic settings, will serve as a foundation for future works on the topological study of DP tangles, as for example in the development of new invariants, that advance their classification problem, including our on-going work [@MDL], as well as for potential novel applications. # Topological set-up for DP tangles and motifs {#sec:setup} In this section we introduce the notions of DP tangles and their generating motifs. ## {#sec:DPtangle} Let $\tau$ be a knot or link in the thickened torus $T^2 \times I$, where $I=[0,1]$ the unit interval. A knot is an embedding of a circle into the thickened torus, while a link is an embedding of a finite collection of circles. We shall say 'link' for both knots and links. Let further $\mathbb{E}^2$ denote the Euclidean plane and let $B=\{u,v\}$ be a basis of $\mathbb{E}^2$. We consider the covering map $$\rho: \mathbb{E}^2 \rightarrow{} T^2$$ that assigns a longitude $l$ of $T^2$ to $u$ and a meridian $m$ of $T^2$ to $v$. Note that the covering map $\rho$ extends trivially to a covering map (also denoted) $\rho$: $\mathbb{E}^2 \times I \rightarrow{} T^2 \times I$. ![[\[DP-tangle\]]{#DP-tangle label="DP-tangle"} A link in $T^2 \times I$, a corresponding flat motif and its DP tangle. ](DP-tangle.pdf){#DP-tangle width="5.5in"} **Definition 1**. Let $\tau$ be a link in $T^2 \times I$. A *doubly periodic tangle*, or *DP tangle*, is the lift of $\tau$ under the covering map $\rho$, and is denoted by $\tau_{\infty}$. Moreover, the projection of $\tau$ onto $T^2 \times \{0\}$ is called a *link diagram* of $\tau$, denoted by $d$, and the lift of $d$ under $\rho$ is called a *doubly periodic diagram*, or *DP diagram*, denoted by $d_{\infty}$. In this context, $d$ (resp. $\tau$) is called a *motif* for $d_{\infty}$ (resp. for $\tau_{\infty}$). An example of the above notions is illustrated in Fig. [1](#DP-tangle){reference-type="ref" reference="DP-tangle"}. So, a DP tangle comes equipped with a basis of $\mathbb{E}^2$ and a choice of a longitute-meridian pair $(l,m)$ for $T^2$. By a general position argument, $m$ and $l$ do not intersect crossings of $d$ and no arc of $d$ intersects a corner formed by $m$ and $l$. Note that in [@Grishanov1] a DP tangle is referred to as *doubly periodic structure*, in [@Grishanov.part1; @Grishanov.part2] as a *2-structure*, while in [@Morton] as a *fabric*. ## {#sec:lattice} The set of points $$\Lambda (u,v) = \{xu + yv\, |\, x,y \in \mathbb{Z}\}$$ generated by the basis $B$ of $\mathbb{E}^2$, defines a *periodic lattice*, which can be viewed as a supporting frame for a DP tangle. The parallelograms of the periodic lattice $\Lambda$ are assumed to be of unit area. In fact, we can assume $\Lambda$ to be the usual integer lattice $\mathbb{Z}^2$. This lattice can be generated by the standard orthonormal basis $B_0=\{e_1,e_2\}$ of $\mathbb{E}^2$ but also from different choices of basis. More specifically, two bases $B=\{u,v\}$ and $B'=\{u',v'\}$ generate the same point lattice $\Lambda = \Lambda (u,v) = \Lambda' (u',v')$ if and only if for $x_1, x_2, x_3, x_4 \in \mathbb{Z}$, $$\begin{pmatrix} u' \\ v' \end{pmatrix} = \begin{pmatrix} x_1 & x_2 \\ x_3 & x_4 \end{pmatrix} \cdot \begin{pmatrix} u \\ v \end{pmatrix} \textit{ , where } \mid{x_1 x_4 - x_2 x_3}\mid = \pm 1.$$ ## {#sec:scaling} The torus $T^2$ arises as the identification space, with respect to the boundary, of any one of the parallelograms of the periodic lattice $\Lambda$, each of which shall be called a *flat torus*. Hence, a flat torus contains a tangle diagram which represents the motif $d$ in $T^2$. This leads to the following definition: **Definition 2**. The representation of a motif $d$ on a flat torus is called *flat motif*, also denoted $d = d_{\infty} / \Lambda$. Clearly, a flat motif is a particular case of a tangle diagram in a flat torus. So, seen as a 'tile', $d$ generates the DP diagram $d_{\infty}$ by applying DP conditions (see Fig. [1](#DP-tangle){reference-type="ref" reference="DP-tangle"} for an example). Let us now consider a finite cover $\bar{\rho}$ of the torus $T^2$ supporting the motif $d$: $$\bar{\rho}: \bar{T}^2 \rightarrow{} T^2$$ The lift of the motif $d$ in $T^2$ is a motif $\bar{d}$ in $\bar{T}^2$, which in other words is a finite cover of $d$. Clearly, $\bar{d}$ gives rise to the same DP diagram $d_{\infty}$ by applying DP conditions, so $\bar{d}$ is also a motif for $d_{\infty}$. Moreover, the flat motif $\bar{d}$ is a parallelogram formation made of finitely many adjacent copies of the flat motif $d$. For example, Fig. [12](#Tknot-Tlink){reference-type="ref" reference="Tknot-Tlink"}(c) illustrates a double cover $\bar{d}$ of the flat motif $d$ in Fig. [12](#Tknot-Tlink){reference-type="ref" reference="Tknot-Tlink"}(b). Superimposing the lattice $\Lambda (u,v)$ generated by $d$ in $T^2$ with the lattice, say, $\bar{\Lambda} (u,v)$ associated with $\bar{d}$ in $\bar{T}^2$ (for the same basis $B=\{u,v\}$) of $\mathbb{E}^2$, we observe that $\bar{d} = d_{\infty} / \bar{\Lambda}$ comprises a number of rescaled copies of $d = d_{\infty} / \Lambda$ arranged according to the finite cover. More precisely, consider the following two different lattices, $\Lambda$ and $\bar{\Lambda}$, associated to the same DP diagram $d_{\infty}$: $$\Lambda (u,v) = \{xu + yv \ | \ x,y \in \mathbb{Z}\} \, \textit{ and } \, \bar{\Lambda} (\bar{u},\bar{v}) = \{x \bar{u} + y \bar{v} \ | \ x,y \in \mathbb{Z}\}.$$ Let also $k_u$ and $k_v$ be two positive integers satisfying, $$\bar{u} = k_u \cdot u \, \textit{ and } \, \bar{u} = k_v \cdot v.$$ It follows that $\bar{\Lambda} \subseteq \Lambda$. This inclusion relation between lattices guarantees the existence of a *minimal lattice* for a specific longitude-meridian pair $(l,m)$ of $T^2$. In particular, we have the following definition. **Definition 3**. We define the *minimal lattice* of a DP tangle $\tau_{\infty}$, denoted by $\Lambda_{min}$, to be the lattice satisfying $\Lambda \subseteq \Lambda_{min}$, for all periodic lattices $\Lambda$ of $d_{\infty}$. Accordingly, the motif (resp. flat motif) $d = d_{\infty} / \Lambda_{min}$ is called a *minimal(flat) motif for* $d_{\infty}$. Hence, a minimal lattice is a minimal 'grid' that accommodates a DP tangle and a minimal motif (resp. flat motif) is minimal for generating the DP diagram $d_{\infty}$. See Fig. [17](#crossing){reference-type="ref" reference="crossing"}(d) for an example. Note that in [@Grishanov1; @Grishanov.part1; @Grishanov.part2] a minimal flat motif is referred to as *unit cell*. ## {#sec:mixedlinks} We shall now present another way of representing motifs and DP diagrams. It is well-known that a link in a c.c.o. 3-manifold can be visualized as a mixed link in the 3-sphere $S^3$ (see [@LR1; @DL1] for details), where a (framed) fixed sub-link represents the 3-manifold via surgery; analogously for links in a handlebody (see [@OL] for details). Similarly, a thickened torus $T^2 \times I$ can be viewed as the complement in $S^3$ of the (oriented) Hopf link, $H$. Consequently, a link $\tau$ in $T^2 \times I$ can be represented in $S^3$ uniquely by the mixed link $H\cup \tau$, which is a link in $S^3$ consisting of two sub-links: the *fixed part* $H=X\cup Y$, representing the manifold $T^2 \times I$, and the *moving part* representing the link $\tau$ in $T^2 \times I$. For an illustration see Fig. [2](#mlink1){reference-type="ref" reference="mlink1"}. This point of view is adopted in [@Morton], where a mixed link diagram representing a motif of a DP tangle is called a *kernel*. The simple closed curves $X$ and $Y$ face the front side and the back side of the DP tangle (fabric) respectively. Cf. [@Morton] for details. {#mlink1 width="2.5in"} We note that, following [@Morton], we consider the fixed part $H=X\cup Y$ of the mixed link representing $T^2 \times I$ to be oriented according to the orientation of the given basis of $\mathbb{E}^2$, and the components $X$ and $Y$ ordered. In materials science, textiles for example, DP tangles are classified according to the pattern formed by their crossings at the diagrammatic level, which describes the 'front side' of a material. This pattern may differ from the one on the 'back side', depending on the construction method. Consider for example a particular class of DP tangles called *twill weaves*, where positive (resp. negative) crossings are organized in a diagonal pattern as described in [@Sonia1]. If the diagonal runs in a positive slope, namely from the lower left to the upper right corner, the DP tangle is called a *right-hand* twill, or *Z-twill*. However, the mirror image of the DP diagram of a Z-twill gives rise to a so-called *left-hand* twill, or *S-twill*, where the diagonal runs in a negative slope, as illustrated in Fig. [3](#twill){reference-type="ref" reference="twill"}. These two DP tangles are considered different in materials science. {#twill width="3.2in"} # DP tangle equivalence {#sec:equivalence} Throughout the section $\tau_{\infty}$ denotes a DP tangle and $d_{\infty}$ a diagram of $\tau_{\infty}$. As a topological object $\tau_{\infty}$ is an 1-dimensional manifold embedded in the thickened plane $\mathbb{E}^2 \times I$. As such, it is allowed to undergo isotopies, that is, transformations induced by bi-continuous orientation preserving 'elastic' deformations (i.e. homeomorphisms) of $\mathbb{E}^2 \times I$. Due to the nature of DP tangles, we will restrict our scope only to *DP isotopies* that preserve the double periodicity. Furthermore, any translation and any orientation preserving invertible linear transformation of the Euclidean plane carrying along the DP tangle, clearly give rise to equivalent DP tangles. **Definition 4**. Two DP tangles (accordingly DP diagrams) are said to be *equivalent* if they can be obtained from each other by DP (diagrammatic) isotopies and orientation preserving invertible affine transformations of $\mathbb{E}^2$ carrying along the DP tangles (DP diagrams). **Note 5**. Isotopies that do not preserve the double periodicity of a given DP tangle could be also considered in the general topological theory of DP tangles and shall be considered as *defect isotopies*. For the topological study of DP tangles, we would like to examine DP tangle equivalence and how this shows on the level of motifs and flat motifs along with the supporting lattices. DP isotopies are generated from local motif isotopies and from torus homeomorphisms, applied on any motif of the DP tangle. On the other hand, invertible affine transformations of the plane induce global deformations of DP tangles, and they include re-scalings (stretches, contractions) -which are not area preserving transformations-, translations and rotations of the plane, as well as shear deformations (which are area preserving). DP tangle equivalence comprises also static changes, such as shifts or re-scalings of the underlying lattice (as, for example, motif duplication). In what follows we will discuss all instances of DP tangle equivalence in detail. Throughout the section we denote by $\tau$ a motif for $\tau_{\infty}$ and by $d$ a motif for $d_{\infty}$. Recall that the motif $d$ is framed by a longitude-meridian pair $(l,m)$ of $T^2$ (cutting along which one obtains the flat motif $d$), which is associated to a basis $B=\{u,v\}$ of $\mathbb{E}^2$ generating the point lattice $\Lambda =\Lambda(u,v)$. ## DP tangle equivalence from local motif isotopies {#sec:isotopy} Let $\tau$ be a motif in a fixed thickened torus $T^2 \times I$ that generates $\tau_{\infty}$ and $d$ the corresponding diagram of $\tau$ in the torus surface $T^2$. With $T^2 \times I$ fixed, any isotopy of $\tau$ generates naturally a DP isotopy of $\tau_{\infty}$. Hence it induces a *local isotopy* equivalence relation between the corresponding flat motifs, which are supported by the same fixed lattice $\Lambda(u,v)$. On the diagrammatic level, an isotopy of $\tau$ translates into a finite sequence of local moves on the diagram $d$, comprising *local surface isotopies* (see first two instances of Fig. [6](#planar1){reference-type="ref" reference="planar1"}) and *Reidemeister moves* (see Fig. [4](#Rmoves){reference-type="ref" reference="Rmoves"}). {#Rmoves width="5in"} These moves on the diagram $d$ generate, in turn, on $d_{\infty}$ (local) planar isotopies and Reidemeister moves, that preserve the double periodicity, that is, *DP (local) planar isotopies* and *DP Reidemeister moves*. Clearly the above apply to any finite cover of the motif. An example is illustrated in Fig. [5](#FiniteRmoves){reference-type="ref" reference="FiniteRmoves"}. ![[\[FiniteRmoves\]]{#FiniteRmoves label="FiniteRmoves"} On the top, a flat motif and a corresponding (finite) cover. On the bottom, an isotopic flat motif for the same lattice and the corresponding DP Reidemeister moves.](FiniteRmoves.pdf){#FiniteRmoves width="3.37in"} We now consider the flat motif that corresponds to the motif diagram $d$ with respect to the chosen longitude-meridian pair $(l,m)$ of $T^2$ (also denoted $d$), in order to investigate how motif or DP isotopies show on flat motifs. If the local isotopy move takes place away from $m$ and $l$, then, when 'unfolding' the torus along $(l,m)$, the same move will be visible in the interior of the flat motif (as in the first two instances of Fig. [4](#Rmoves){reference-type="ref" reference="Rmoves"} and Fig. [6](#planar1){reference-type="ref" reference="planar1"} ). We also need to consider local isotopy moves that take place in the motif, so that some arcs involved in the moves cross $m$ or $l$ or even both $m$ and $l$. {#planar1 width="4in"} We first examine local surface isotopies. These on the flat motif level comprise: a) planar isotopies within the flat motif, b) planar isotopies where an arc before lies within the motif but the arc afterwards hits one boundary component (the meridian or the longitude), c) planar isotopies where an arc before and the arc afterwards cross one boundary component, d) planar isotopies where an arc before crosses one boundary component but the arc afterwards crosses both boundary components, and e) the situation where a crossing passes through one boundary component. Moves of types a) and b) are sampled in Fig. [6](#planar1){reference-type="ref" reference="planar1"}, while moves of types c) and d) are sampled in Fig. [7](#planar2){reference-type="ref" reference="planar2"}. In the figures, the top parts illustrate the moves on the motif level and the bottom parts illustrate the corresponding moves on the flat motif level. Moves of type e) are sampled in Fig. [8](#planar3){reference-type="ref" reference="planar3"}, where the left part shows the move of the motif, while the right part shows the corresponding move of the flat motif. {#planar2 width="4in"} {#planar3 width="5.7in"} Any other local surface isotopy can be realized by the above basic moves. As an example, the move illustrated in Fig. [9](#mR3m2b){reference-type="ref" reference="mR3m2b"} can be realized via three moves of type b) and one move of type d). {#mR3m2b width="5.3in"} Consider, now, a Reidemeister move on the motif level, some arcs of which may cross $m$ or $l$ or both $m$ and $l$. Such moves are exemplified in Fig. [10](#mixedreid){reference-type="ref" reference="mixedreid"}. Using surface isotopies, both sides of the Reidemeister move can be pushed away from $m$ and $l$, which results in the move to take place in the interior of the flat motif. After the move is performed we push back all arcs to their original position using surface isotopies. {#mixedreid width="5.5in"} **Remark 6**. The local isotopy moves between DP tangles that are discussed so far, correspond to local moves between motifs on the fixed torus $T^2$, thus between flat motifs supported by the same lattice. Stretches and contractions of flat motifs are not included in the above discussion, as these also transform the underlying lattice. These transformations are addressed further below in this section. Concluding, in local motif isotopy we have a moving link in the thickened torus, different motifs (in general) and different flat motifs before and after the move, yet the same lattice, while the DP diagram undergoes DP isotopy moves. Another way of interpreting local isotopies on the level of flat motifs is by using the correspondence between mixed links and motifs discussed in Subsection [2.4](#sec:mixedlinks){reference-type="ref" reference="sec:mixedlinks"}. We recall from [@LR1 Theorem 5.2] that planar isotopy, the classical Reidemeister moves for the moving part of a mixed link diagram $D$, together with the extended local isotopies that involve the fixed and the moving part of $D$ generate isotopy for links in the thickened torus (see Fig. [11](#isom){reference-type="ref" reference="isom"}, where the fixed components run close and in parallel to the meridian and longitude of $T^2$ respectively). ![[\[isom\]]{#isom label="isom"} Extended local isotopy moves for mixed links.](mReidm.pdf){#isom width="5.8in"} Consequently, local motif isotopy classes correspond bijectively to mixed link isotopy classes. The above, in view of [@LR1 Theorem 5.2], establish the following: **Theorem 7**. *Two flat motifs supported by the same lattice are *isotopic* if and only if they differ by a finite sequence of surface isotopy moves and the classical Reidemeister moves.* ## DP tangle equivalence from torus re-scaling {#sec:rescaling} DP stretches and contractions of the DP tangle $\tau_\infty$ are DP isotopies and as such should be included in DP tangle equivalence. On the level of motifs, stretches and contractions of the plane are induced by analogous re-scaling isotopies of the supporting torus (blow-ups and shrinkings), with homologous longitude-meridian pairs. In turn, the torus re-scaling isotopies re-scale the supporting lattice and, consequently, the flat motifs. For examples, see the pairs (a)-(b) and (c)-(d) in Fig. [12](#Tknot-Tlink){reference-type="ref" reference="Tknot-Tlink"}. So, in re-scaling transformations we have a re-scaled torus, a re-scaled motif with the homologous longitude-meridian pair, and accordingly re-scaled basis vectors of $\mathbb{E}^2$. We shall refer to the equivalence relation among (flat) motifs, generated by local motif isotopies and re-scaling transformations as *(flat) motif isotopy*. ![[\[Tknot-Tlink\]]{#Tknot-Tlink label="Tknot-Tlink"} Motif (a) is a stretching of (b); motif (d) is a contraction of (c); motifs (b) and (c) are scale equivalent.](Tknot-Tlink.pdf){#Tknot-Tlink width="5.5in"} In terms of mixed links, allowing the fixed part to also re-scale, we have that a re-scaling isotopy can be achieved by the local moves of the mixed link isotopy, since a mixed link can be enclosed in a compact region of the plane. Consequently, motif isotopy classes correspond bijectively to mixed link isotopy classes. So, [@LR1 Theorem 5.2] and Theorem [Theorem 7](#thm:localmotifisotopy){reference-type="ref" reference="thm:localmotifisotopy"} establish the following: **Theorem 8**. *Two flat motifs are isotopic if and only if they differ by a finite sequence of surface isotopy moves, the classical Reidemeister moves and re-scaling transformations.* ## DP tangle equivalence from affine transformations of the plane {#planetransformation} Clearly, any invertible orientation preserving affine transformation of the plane $\mathbb{E}^2$ carrying along the DP tangle $\tau_\infty$ results in the same DP tangle or a DP tangle with the same topological properties, i.e. it induces a global isotopy of $\tau_\infty$. So these global deformations must be included in the DP equivalence. An invertible orientation preserving linear transformation of $\mathbb{E}^2$ can be realized by two distinct bases of $\mathbb{E}^2$ related via an invertible $2\times 2$ real matrix with positive determinant. **Remark 9**. If we restrict to affine transformations that preserve the chosen lattice $\Lambda$, then these transformations should be discretized so as to comply with the integral nature of $\Lambda$ supporting $d_\infty$. This means that, in this case, any affine transformation of $\mathbb{E}^2$ should be realized via integer vectors. In particular,an orientation preserving invertible linear transformation of $\mathbb{E}^2$ can be realized by two distinct bases of $\mathbb{E}^2$, $B=\{u,v\}$ and $B^{\prime} =\{u',v'\}$, related via an invertible $2\times 2$ integral matrix: $$\begin{pmatrix} u' \\ v' \end{pmatrix} = \begin{pmatrix} x_1 & x_2 \\ x_3 & x_4 \end{pmatrix} \cdot \begin{pmatrix} u \\ v \end{pmatrix} \textit{ , where } x_1, x_2, x_3, x_4 \in \mathbb{Z} \textit{ and } \mid{x_1 x_4 - x_2 x_3}\mid > 0.$$ Below we single out some special cases of affine transformations. A translation is an affine transformation of $\mathbb{E}^2$ that moves the coordinate system by any vector. On the motif level, a translation shifts the longitude-meridian pair of $T^2$ accordingly, as analyzed further in Subsection [3.4](#sec:shiftequivalence){reference-type="ref" reference="sec:shiftequivalence"}. A plane rotation is an automorphism of $\mathbb{E}^2$ induced by a change of basis by a rotational linear transformation by any angle. On the level of motifs, a rotation preserves the longitude-meridian pair of $T^2$. A horizontal shear (or shear parallel to the x-axis) is an area-preserving automorphism of $\mathbb{E}^2$ that takes a generic point with coordinates $(x,y)$ to the point $(x+my,y)$, where $m$ is a fixed real number. The case of a vertical shear (or shear parallel to the $y$-axis) is analogous. On the torus level, the motif undergoes a Dehn twist, as analyzed in Subsection [3.5](#Dehnequivalence){reference-type="ref" reference="Dehnequivalence"}. A re-scaling sends the basis $\{u,v\}$ of $\mathbb{E}^2$ to a re-scaled basis $\{\lambda u, \lambda' v\}$, where $\lambda, \lambda'$ are fixed positive real numbers. A re-scaling transformation is induced by a re-scaling deformation of the torus that carries along the motif and induces an isotopy of the DP tangle, as analyzed in Subsection [3.2](#sec:rescaling){reference-type="ref" reference="sec:rescaling"}. **Remark 10**. It is worth noting that any orientation preserving invertible affine transformation of $\mathbb{E}^2$ can be realized as a composition of the above. More presisely, to a basis $\{u,v\}$ we can assign any other basis $\{u',v'\}$ of the plane by a finite sequence of operations that preserve the angle between the two vectors, namely translations, rotations and re-scalings (that change the lengths of the vectors), as well as transformations that preserve the area but change the angle between the vectors, namely shears. ## DP tangle equivalence from shift equivalence {#sec:shiftequivalence} Let $d$ be a motif of $d_{\infty}$ in the torus $T^2$. Recall that, given a choice $(l,m)$ of a longitude-meridian pair for $T^2$, a flat motif $d$ is created by cutting $T^2$ along $l$ and $m$. Further, a choice $B=\{u,v\}$ of basis for $\mathbb{E}^2$ associated with the pair $(l,m)$ generates the lattice $\Lambda(u,v)$. Choosing now a different longitude-meridian pair, say $(l',m')$, for cutting along $T^2$ will give rise to a different, in general, flat motif $d'$ and consequently a different integer lattice $\Lambda'(u,v)$. Yet, both flat motifs $d$ and $d'$ generate the same DP diagram $d_{\infty}$. Hence we have: **Definition 11**. Two flat motifs of a DP tangle $\tau_{\infty}$, resp. two integer lattices of $\tau_{\infty}$, are said to be *shift equivalent* if they are related by translated longitude-meridian pairs. More specifically, considering the longitude-meridian pairs for $d$ on $T^2$, $(l,m)$, $(l',m)$ and $(l,m')$ we observe that: the pair $(l',m)$ indicates a vertical shift of the flat motif $d$ created by the pair $(l,m)$, while the pair $(l,m')$ indicates a horizontal shift of the flat motif $d$. The two shifts combined give rise to the flat motif $d'$ created by the longitude-meridian pair $(l',m')$. We note that shift equivalence includes also translations of longitudes or meridians by multiples of $2\pi$. These correspond to *integral translations* of the underlying lattice $\Lambda$, that is, translations fixing $\Lambda$ setwise. Concluding, in shift equivalence we have a fixed link in the thickened torus but a different motif and a shifted flat motif, hence also a shifted lattice, yet the same DP diagram. Moreover, the original basis of $\mathbb{E}^2$ has undergone an affine translation. **Remark 12**. Consider a local isotopy move which is not restricted within the flat motif, so that some arc involved in the move crosses $m$ or $l$ or both $m$ and $l$. Suppose also that we have available shift equivalence. Since the isotopy moves are local, it follows that one or two appropriate shifts of the longitude-meridian pair $(l,m)$ will result in the move to take place in the interior of the new motif. Moreover, note that the composition of a shift move and an isotopy move is commutative. So, this type of isotopy moves rest on the case where the local move takes place in the interior of the motif. Hence, up to shift equivalence, local flat motif isotopy can be reduced to local isotopy moves that take place in the interior of a motif. ## DP tangle equivalence from Dehn twists {#Dehnequivalence} Isotopy of classical links in the thickened torus is not enough to capture the notion of equivalence of DP tangles, as non-isotopic motifs in $T^2 \times I$ may lift to the same DP tangle. This observation highlights the difference between the theory of DP tangles and that of classical links in $T^2\times I$. More precisely, let $\tau$ be a motif in $T^2 \times I$ with $d$ a corresponding diagram motif in $T^2$. $\tau$ resp. $d$ gives rise to the DP tangle $\tau_{\infty}$ resp. the DP diagram $d_{\infty}$. Suppose that $\tau_{\infty}$ undergoes a shearing, which is a particular case of (periodic) planar isotopy, giving rise to a new isotopic DP tangle $\tau^\prime_{\infty}$. Let $d^\prime_{\infty}$ be an associated DP diagram and $\tau^\prime$ and $d^\prime$ a corresponding motif and its diagram. We want to analyze how the motifs (and corresponding flat motifs) $d$ and $d^\prime$ are related. Recall that the motif $d$ comes equipped with a longitude-meridian pair $(l,m)$, which is associated to a basis $B=\{u,v\}$ of $\mathbb{E}^2$ generating the point lattice $\Lambda =\Lambda(u,v)$. We want to compare $d_{\infty}$ and $d^\prime_{\infty}$ with respect to the point lattice $\Lambda$. Recall from Subsection [2.2](#sec:lattice){reference-type="ref" reference="sec:lattice"} that the same lattice $\Lambda$ can be generated by a different basis of $\mathbb{E}^2$, that is, a different flat torus, which in turn will be associated to a different flat motif for the same DP tangle. Let $B'=\{u',v'\}$ be a new basis of $\mathbb{E}^2$ inducing a shearing of the plane. In view of the discussion in Subsection [2.4](#sec:mixedlinks){reference-type="ref" reference="sec:mixedlinks"}, cf. also [@Grishanov1; @Grishanov.part1], we only consider the case of orientation preserving transformations of the plane. Therefore we have, $\Lambda (u,v) = \Lambda' (u',v')$ if and only if, $$\begin{pmatrix} u' \\ v' \end{pmatrix} = \begin{pmatrix} x_1 & x_2 \\ x_3 & x_4 \end{pmatrix} \cdot \begin{pmatrix} u \\ v \end{pmatrix} \textit{ , where } x_1, x_2, x_3, x_4 \in \mathbb{Z} \textit{ and } \mid{x_1 x_4 - x_2 x_3}\mid = 1.$$ The above procedure is exemplified in the left half of Fig. [13](#shearing){reference-type="ref" reference="shearing"}. ![[\[shearing\]]{#shearing label="shearing"} A DP diagram with a fixed lattice and two different bases of $\mathbb{E}^2$, leading to a shearing of the DP tangle.](shearing.pdf){#shearing width="6in"} Let us, now, assign the longitude $l$ of $T^2$ to $u'$ and the meridian $m$ to $v'$ of the basis $B'$. This new covering map, say $\rho'$, creates a new motif $d'$, which differs from the motif $d$ associated to the basis $B$ by a finite sequence of *Dehn twists*, as explained in [@Grishanov1; @Grishanov.part1] and [@Farb]. A Dehn twist gives rise to an orientation preserving self-homeomorphism of the torus. By convention the identity matrix corresponds to the trivial Dehn twist. Fig. [14](#Rtwists){reference-type="ref" reference="Rtwists"} illustrates the two types of Dehn twists. In the universal cover, $d'$ gives rise to a new DP tangle (diagram) $d^\prime_{\infty}$, which is a shearing of the original DP tangle (diagram) $d_{\infty}$. The above procedure is exemplified in the right half of Fig. [13](#shearing){reference-type="ref" reference="shearing"}. ![[\[Rtwists\]]{#Rtwists label="Rtwists"} A longitudinal and a meridional Dehn twist of the torus.](Rtwists.pdf){#Rtwists width="4.5in"} So, we define: **Definition 13**. Two (flat) motifs are said to be *Dehn equivalent* if they are related by a finite sequence of Dehn twists. Fig. [15](#twist){reference-type="ref" reference="twist"} illustrates two motifs related by a meridional Dehn twist and the induced transformation on the associated flat motifs. ![[\[twist\]]{#twist label="twist"} A meridional Dehn twist on a flat motif.](twist.pdf){#twist width="4.5in"} So, in Dehn equivalence we have a twisted motif in the thickened torus and a twisted flat motif, while the DP diagram has undergone a DP shearing, supported by the same point lattice. Some remarks are now due. **Remark 14**. With an eye on applying the above on materials modeled by DP tangles, it should be noted that there is a natural limitation on the amount of shearing a material can undergo (resp. Dehn twists on a motif), due to its geometrical and physical properties. It would be very interesting to investigate this limitation for a given material, and this limitation would comprise a new geometrical invariant for the material. **Remark 15**. A shearing is by definition area preserving. However, the analysis above applies to any change of basis of $\mathbb{E}^2$ with integer vectors (so as to preserve periodicity). Yet, this might involve also scaling (e.g. duplication of the motif), which is not area preserving. Scale equivalence will be analyzed below. ## DP tangle equivalence from lattice re-scaling {#sec:scaleequivalence} It is straightforward that a motif $\bar{\tau}$ formed by, say two, adjacent copies of any given flat motif $\tau$ of a DP tangle $\tau_{\infty}$ is also a motif of $\tau_{\infty}$, as exemplified by the pair (b)-(c) in Fig. [12](#Tknot-Tlink){reference-type="ref" reference="Tknot-Tlink"}, and also in Fig. [17](#crossing){reference-type="ref" reference="crossing"} by the dashed blue lines. The motifs $\tau$ and $\bar{\tau}$ are not shift equivalent, isotopic or Dehn equivalent. As discussed in § [2.3](#sec:scaling){reference-type="ref" reference="sec:scaling"}, they realize two different finite covers of $T^2$ and, consequently, they are associated to different point lattices, say $\Lambda$ and $\bar{\Lambda}$, such that $\bar{\Lambda} \subseteq \Lambda$. We say that each lattice is a re-scaling of the other. In order to accommodate in the DP tangle equivalence the motif reproduction corresponding to lattice re-scaling, we generalize the notion of *scale equivalence* for DP tangles from [@Sonia1], introduced in the context of weave classification. **Definition 16**. Let $d_{\infty}$ be a DP diagram and let $\Lambda_0$, $\Lambda_1$ and $\Lambda_2$ be three (not necessarily distinct) point lattices such that $\Lambda_1 \subseteq \Lambda_0$ and $\Lambda_2 \subseteq \Lambda_0$. Moreover, let $d_0 = d_{\infty} / \Lambda_0$, $d_1 = d_{\infty} / \Lambda_1$ and $d_2 = d_{\infty} / \Lambda_2$ be flat motifs of $d_{\infty}$. Then, $d_1$ and $d_2$ arise as *adjacent* copies of $d_0$, according to the inclusion relations of the lattices. Then $d_1$ and $d_2$ are said to be *scale equivalent*. The notion of scale equivalence extends also to the corresponding motifs, which realize different finite covers of $T^2$. Recalling, further, that the inclusion relation between lattices guarantees the existence of a minimal lattice for a specified longitude-meridian pair $(l,m)$ of $T^2$ (Definition [Definition 3](#def:minimal lattice){reference-type="ref" reference="def:minimal lattice"}), it follows that scale equivalence is an equivalence relation in the sets of motifs and flat motifs, since, for a specified longitude-meridian pair, any flat motif will be scale equivalent to the flat motif associated to the minimal lattice. **Remark 17**. Finding a minimal motif for a periodic tangle can be very tricky. See [@Grishanov1] for a further discussion. In Fig. [16](#R-move-minimal){reference-type="ref" reference="R-move-minimal"} we illustrate a subtle example. In the figure, we start with a minimal motif (a) and we perform an R1 move obtaining a second minimal motif (d) (not necessarily in terms of crossings). Then we take a new motif (e), which is a double (b) of the initial one, so this motif is no longer minimal. On this motif (e) we perform just one of the two R1 moves obtaining (f). The new motif (f) is now minimal, while if we performed both R1 moves the doubled motif (g) would not be minimal and would make a double (c) of the second minimal motif (d). Another subtle situation is presented in Fig. [17](#crossing){reference-type="ref" reference="crossing"}, where from a (seemingly minimal) motif (b) we obtain an actual minimal motif (d), via scale equivalence, Dehn equivalence and shift equivalence. ![[\[R-move-minimal\]]{#R-move-minimal label="R-move-minimal"} A subtle obstruction to motif minimality.](R-move-minimal.pdf){#R-move-minimal width="5.3in"} **Remark 18**. The notion of scale equivalence implies that a motif containing a single component, i.e. a knot, may be scale equivalent to a motif with multiple components, that is, a link. As an example, consider Fig. [12](#Tknot-Tlink){reference-type="ref" reference="Tknot-Tlink"}. On the left (instances (a) and (b)), a motif that corresponds to a knot in $T^2 \times I$ is illustrated, while on the right (instances (c) and (d)), a motif of the same DP tangle is presented, which now corresponds to a link in $T^2 \times I$. ## DP tangle equivalence and flat motif equivalence {#sec:DPequivalent} Equivalence of DP tangles on the level of (flat) motifs has been investigated thoroughly by highlighting two specific cases. On one hand, we described orientation preserving transformations which act on the plane while maintaining fixed the DP tangle (DP diagram). These transformations comprise shifts of the lattice, different choice of basis for the same point lattice, or lattice re-scaling. On the other hand, we distinguished periodic transformations that induce isotopy of the DP tangle (DP diagram) into either local isotopies, or global deformations of the DP tangle, namely rigid planar translations and rotations, shearings, stretchings and contractions. It follows that equivalence of DP tangles can be characterized by any finite sequence of these transformations. The above analysis of the different instances of equivalence between two DP tangles along with Definition [Definition 4](#def:DPequivalence){reference-type="ref" reference="def:DPequivalence"} lead to the following result. **Theorem 19**. *Let $\tau_{1,\infty}$ and $\tau_{2,\infty}$ be two DP tangles in $\mathbb{E}^2 \times I$, with corresponding DP diagrams $d_{1,\infty}$ and $d_{2,\infty}$. Let also $\Lambda_1$ and $\Lambda_2$ be the supporting point lattices such that $d_i = d_{i,\infty} / \Lambda_i$ is a flat motif of $d_{i,\infty}$ for $i \in \{1,2\}$. Then $\tau_{1,\infty}$ and $\tau_{2,\infty}$ are *equivalent* if and only if $d_1$ and $d_2$ are related by a finite sequence of shifts, motif isotopy moves, Dehn twists, orientation preserving affine transformations and scale equivalence.* Fig. [17](#crossing){reference-type="ref" reference="crossing"} captures many instances of Theorem [Theorem 19](#th:equivalence){reference-type="ref" reference="th:equivalence"}, namely: shift equivalence, Dehn equivalence and scale equivalence. Furthermore, in the figure we demonstrate how to obtain from a seemingly minimal motif (b) an actual minimal motif (d). ![[\[crossing\]]{#crossing label="crossing"} Different generating flat motifs for the same DP diagram and transformations among them culminating to the minimal motif (d).](crossing.pdf){#crossing width="5.6in"} **Remark 20**. It is important to note that DP tangle equivalence has been discussed in the literature, mainly in the context of classification of textiles. First Grishanov et al. stated a generalized Reidemeister theorem for the equivalence of DP tangles in [@Grishanov1], which is analogous to our result in the particular case of a fixed (minimal) lattice. This equivalence relation was consequently used for constructing topological invariants for DP tangles, which are mathematical tools for their classification, cf. [@Grishanov1; @Grishanov.part1; @Grishanov.part2; @Morton; @Grishanov.Vassiliev1; @Grishanov.Vassiliev2; @Eleni1; @Eleni2; @Eleni3; @Kurlin; @Sonia1; @Sonia2; @Sonia3]. These invariants give the same values on all minimal motifs of a given DP tangle. Then, more recently, the second author highlighted the importance of including in the DP tangle equivalence a relation between different finite covers that lift to the same doubly periodic structure, namely the notion of scale equivalence, for the particular case of equivalent weaves [@Sonia3]. In the present paper we provide a more detailed and exhaustive analysis, especially in terms of isotopy of DP tangles that comprise both local isotopies and global transformations, including rigid transformations, as well as changes in the supporting lattice, which also include scale equivalence. # DP tangle equivalence in other diagrammatic settings {#sec:othersettings} So far, in Sections [2](#sec:setup){reference-type="ref" reference="sec:setup"} and [3](#sec:equivalence){reference-type="ref" reference="sec:equivalence"}, the links in the thickened torus are meant to be embeddings of (some copies of) the circle undergoing ambient isotopy. Consequently, the diagrams are meant to contain only classical crossings and be related by the discrete isotopy moves presented in Subsection [3.1](#sec:isotopy){reference-type="ref" reference="sec:isotopy"}. Subsequently, the above extend to the generated DP tangles. In this section we discuss DP tangle equivalence when the links in the thickened torus along with their isotopy equivalences belong to other diagrammatic categories. Namely, we first discuss the case of regular isotopy and framed link isotopy. Then we discuss the cases of virtual and welded links, these of pseudolinks and singular links, and these of tied links and bonded links, all forming new classes of knotted objects. We note that the setting for DP tangles described in Section [2](#sec:setup){reference-type="ref" reference="sec:setup"} and the analysis detailed in Section [3](#sec:equivalence){reference-type="ref" reference="sec:equivalence"} apply equally to any diagrammatic category. So, for the study of DP tangles related to any one of the above topological settings, we only need to adapt our analysis in the context of Subsection [3.1](#sec:isotopy){reference-type="ref" reference="sec:isotopy"}. All the rest applies invariantly. We present below each setting separately, focusing especially on its combinatorial isotopy. The central feature here lies in the elements of the diagrams which cross the specified longitude-meridian curves, so as to extend the surface isotopies of Subsection [3.1](#sec:isotopy){reference-type="ref" reference="sec:isotopy"} to our diagrammatic category. Then, using the complete set of surface isotopies, any diagrammatic isotopy move in the category, that crosses the longitude-meridian curves, can be pushed in the interior of a flat motif. ## DP tangle equivalence for regular isotopy and framed isotopy {#sec:DPregularframed} *Regular isotopy* is the equivalence relation between classical link diagrams generated by planar isotopy and only R2 and R3 Reidemeister moves. The move R1 is not allowed in this diagrammatic theory. Regular isotopy is introduced in [@Kauffman1987], see also [@Kauffman1990], where the Kauffman bracket polynomial is constructed as the regular isotopy equivalent of the Jones polynomial invariant for knots and links. An extension of the bracket polynomial for DP tangles has been constructed in [@Grishanov1; @Grishanov.part2]. Regular isotopy projects to regular homotopy of the link projection, equivalently it preserves the Whitney degree of the link diagram. Finally, it has a natural interpretation when considering the link components as flat ribbons. Framed isotopy is the topological equivalence of framed knots and links, see [@Kauffman1990]. A framed knot is a knot endowed with a unit normal vector, hence it can be viewed as an embedded solid torus or, equivalently, an embedded annulus (a 'ribbon') in 3-space, whereas a framed link is an embedding of one or more copies of a solid torus or an annulus. Framed links are employed in the construction of closed, connected, orientable (c.c.o.) 3-manifolds via the surgery technique. The Witten invariant is a homeomorphism invariant of c.c.o. 3-manifolds based on the Jones polynomial. A framing unit (positive or negative) is, roughly speaking, a full twist of the solid torus or the ribbon. Representing a framed link by the central curves of its components, a framing unit is represented by a kink of the curve. So, classical links can represent framed links and the framing is registered as kinks in the diagram. There are two ways of projecting a framing unit on the plane, as exemplified in Fig. [18](#Rframed){reference-type="ref" reference="Rframed"}. So, this move is included in the diagrammatic *framed isotopy*, along with planar isotopy and the Reidemeister moves R2 and R3. Clearly, R1 is excluded from framed isotopy, as indicated in Fig. [18](#Rframed){reference-type="ref" reference="Rframed"}. **Note 21**. It is worth adding that regular isotopy on the surface of the sphere $S^2$ is equivalent to diagrammatic isotopy of framed links, as represented by classical links. This is not the case for the plane, where the framed R1 move cannot be realized by regular isotopy. {#Rframed width="2.5in"} Let now $\tau$ be a motif in the thickened torus and $d$ a diagram of $\tau$ undergoing regular isotopy. This isotopy in the torus is generated by the same local moves described above. This is periodically transmitted to the DP diagram $d_{\infty}$ of the DP tangle $\tau_{\infty}$. On the level of flat motifs, the surface isotopy moves are valid (recall Figs. [6](#planar1){reference-type="ref" reference="planar1"}, [7](#planar2){reference-type="ref" reference="planar2"}, and [8](#planar3){reference-type="ref" reference="planar3"}), so Theorem [Theorem 7](#thm:localmotifisotopy){reference-type="ref" reference="thm:localmotifisotopy"} adapts as follows: **Theorem 22**. *Two flat motifs supported by the same lattice are *regular isotopic* if and only if they differ by a finite sequence of surface isotopy moves and the classical Reidemeister moves R2 and R3.* Suppose now that the diagram $d$ undergoes framed isotopy; then so does the DP diagram $d_{\infty}$. On the level of flat motifs, we want to extend the surface isotopies by the *framed surface isotopy move* indicating the passing of a framing unit through the meridian or longitude, as abstracted in Fig. [19](#planarframed){reference-type="ref" reference="planarframed"}, where a framing unit is represented by a kink. Theorem [Theorem 7](#thm:localmotifisotopy){reference-type="ref" reference="thm:localmotifisotopy"} then adapts as follows: **Theorem 23**. *Two flat motifs supported by the same lattice are *framed isotopic* if and only if they differ by a finite sequence of surface isotopy and framed surface isotopy moves, the framed R1 move and the Reidemeister moves R2 and R3.* The moves in Theorem [Theorem 23](#thm:framedmotifisotopy){reference-type="ref" reference="thm:framedmotifisotopy"} generate the *framed motif isotopy*. Furthermore, by the discussion in the beginning of this section, Theorem [Theorem 19](#th:equivalence){reference-type="ref" reference="th:equivalence"} carries through to the regular/framed isotopy setting, as follows: **Theorem 24**. *Let $\tau_{1,\infty}$ and $\tau_{2,\infty}$ be two DP tangles in $\mathbb{E}^2 \times I$, with corresponding DP diagrams $d_{1,\infty}$ and $d_{2,\infty}$. Let also $\Lambda_1$ and $\Lambda_2$ be the supporting point lattices such that $d_i = d_{i,\infty} / \Lambda_i$ is a flat motif of $d_{i,\infty}$ for $i \in \{1,2\}$. Then $d_{1,\infty}$ and $d_{2,\infty}$ are *regular equivalent* (resp. *framed equivalent*) if and only if $d_1$ and $d_2$ are related by a finite sequence of surface isotopy moves and regular isotopy moves (resp. framed motif isotopy moves), shifts, Dehn twists, orientation preserving affine transformations and scale equivalence.* {#planarframed width="5.7in"} ## Virtual and welded DP tangle equivalence {#sec:DPvirtualwelded} Virtual knot theory is a diagrammatic extension of classical knot theory in the sense that some crossings in a link diagram may represent just the permutation of the two arcs involved, with no further information of 'over' or 'under', and they are called 'virtual' crossings. The theory was introduced by Louis H. Kauffman [@Kau2]. The diagrammatic equivalence in virtual knot theory, called *virtual isotopy*, includes planar isotopy and the classical Reidemeister moves, extended by moves that contain virtual crossings. These moves are exemplified by the moves vR1, vR2, vR3 in Fig. [20](#vmoves){reference-type="ref" reference="vmoves"} (where also the variant with the opposite type of real crossing is assumed) and they are all special cases of the universal 'detour move' whereby an arc containing all virtual crossings can slide across any parts of the diagram. In this theory we also have the virtual forbidden moves vF1, vF2 and vF3 depicted in Fig. [20](#vmoves){reference-type="ref" reference="vmoves"}. {#vmoves width="4in"} Virtual links have an interesting interpretation as embeddings of links in thickened surfaces, taken up to addition and subtraction of empty handles [@CKS]: a virtual crossing is regarded as a detour of one of the arcs in the crossing through a 1-handle that has been attached to the 2-sphere of the original diagram (see middle illustration of Fig. [21](#virtual1){reference-type="ref" reference="virtual1"}). **Remark 25**. Another nice interpretation of a virtual link diagram is obtained by forming a ribbon--neighborhood surface of the diagram, where a virtual crossing is represented by abstract ribbons passing over one another without interacting [@KamadaNS], as in the right part of Fig. [21](#virtual1){reference-type="ref" reference="virtual1"}. This consideration could find a physical application in materials science, where the prevention of friction between strands of an embedded DP tangle may be interesting to consider when defining the energy/relaxing state of the corresponding material. {#virtual1 width="4in"} Virtual knot theory forms a refinement of *welded knot theory*, introduced in [@Fenn]. In this diagrammatic theory the moves generating virtual isotopy are all included but there is additionally the wR3 move (which is the forbidden move vR3 in virtual knot theory), as depicted in Fig. [20](#vmoves){reference-type="ref" reference="vmoves"}. This move contains an over arc and one virtual crossing; in general it enables to detour sequences of classical crossings *over* welded crossings. Hence, welded knot theory can be realized as the quotient of virtual knot theory modulo the wR3 move. The explanation for the choice of moves lies in the fact that the move wR3 preserves the combinatorial fundamental group. This is not the case for the other forbidden move vF2, so it remains forbidden also for welded links. Let now $d$ be a virtual (resp. welded) motif diagram in the torus undergoing virtual (resp. welded) isotopy. This isotopy in the torus is generated by the same local moves described above. This is periodically transmitted to the DP diagram $d_{\infty}$. On the level of flat motifs, the *virtual (resp. welded) surface isotopy move*, whereby a virtual (resp. welded) crossing passes through the meridian or longitude (see Fig. [22](#vsurfacemoves){reference-type="ref" reference="vsurfacemoves"}), extends the usual surface isotopy moves (recall Figs. [6](#planar1){reference-type="ref" reference="planar1"}, [7](#planar2){reference-type="ref" reference="planar2"}, and [8](#planar3){reference-type="ref" reference="planar3"}). {#vsurfacemoves width="5in"} So, as stated above, anyone of the virtual isotopy moves on a motif, which crosses the specified meridian or longitute on the torus, can be pushed to the interior of the flat motif. Hence, Theorem [Theorem 7](#thm:localmotifisotopy){reference-type="ref" reference="thm:localmotifisotopy"} adapts as follows: **Theorem 26**. *Two virtual (resp. welded) flat motifs supported by the same lattice are *virtual (resp. welded) isotopic* if and only if they differ by a finite sequence of surface isotopy moves, virtual (resp. welded) surface isotopy moves, the classical Reidemeister moves and the (allowed) virtual (resp. welded) moves.* The moves in Theorem [Theorem 26](#thm:virtual/weldedmotifisotopy){reference-type="ref" reference="thm:virtual/weldedmotifisotopy"} generate the *virtual (resp. welded) motif isotopy*. **Remark 27**. In the context of DP tangles, a first observation is that any motif, being by definition a link in the thickened torus, can be represented by a virtual link diagram, as in the bottom and left part of Fig. [21](#virtual1){reference-type="ref" reference="virtual1"}. In [@Kawauchi], particular cases of DP diagrams have been studied, which do not contain closed components. Equivalence of the corresponding unit-square flat motifs, called *$(m,n)$-knitting patterns*, is considered up to local isotopy preserving the boundary of the square. Then, by considering the virtual link diagram associated to any such flat motif, a characterization of equivalence of flat motifs is stated up to equivalence of the corresponding virtual link diagrams. Moreover, by the discussion in the beginning of this section, Theorem [Theorem 19](#th:equivalence){reference-type="ref" reference="th:equivalence"} applies for virtual (resp. welded) DP tangles, as follows: **Theorem 28**. *Let $d_{1,\infty}$ and $d_{2,\infty}$ be two virtual (resp. welded) DP tangle diagrams in $\mathbb{E}^2$. Let also $\Lambda_1$ and $\Lambda_2$ be the supporting point lattices such that $d_i = d_{i,\infty} / \Lambda_i$ is a flat motif of $d_{i,\infty}$ for $i \in \{1,2\}$. Then $d_{1,\infty}$ and $d_{2,\infty}$ are *virtual equivalent* (resp. *welded equivalent*) if and only if $d_1$ and $d_2$ are related by a finite sequence of virtual (resp. welded) motif isotopy moves, shifts, Dehn twists, orientation preserving affine transformations and scale equivalence.* ## Singular and pseudo DP tangle equivalence {#sec:DPpseudosingular} *Singular knot theory* appeared in the context of the theory of Vassiliev's finite type knot invariants. Singular knots are knots with finitely many rigid self-intersections, the singular crossings, interpreted as rigid vertices in a spatial graph. So singular link isotopy includes classical link isotopy together with rigid vertex isotopy. Fig. [23](#Rsingularpseudo){reference-type="ref" reference="Rsingularpseudo"} exemplifies the diagrammatic moves in the theory that extend planar isotopy and the classical Reidemeister moves, as well as the singular forbidden moves, SF1, SF2 and sF3 (where the middle crossing could be real or singular). {#Rsingularpseudo width="3.5in"} Another related diagrammatic category is that of *pseudo knots*. Pseudo diagrams of knots, links and spatial graphs were introduce by Hanaki in [@H] as projections on the 2-sphere with missing crossing information on some crossings, called 'pre-crossings'. The theory of singular knots is closely related to the theory of pseudo knots. Namely, the diagrammatic *pseudo link isotopy* is generated by planar isotopy and the classical Reidemeister moves extended by the singular isotopy moves and the move pR1, which is the forbidden move sF1 in singular knot theory (all exemplified in Fig. [23](#Rsingularpseudo){reference-type="ref" reference="Rsingularpseudo"}). Hence, pseudo knot theory can be realized as the quotient of the theory of singular knots, modulo the pseudo-Reidemeister move 1. Several invariants of pseudo knots are constructed in [@HJMR], while in [@D3] and in [@D5], the theories of pseudo links and singular links in the Solid Torus and in handlebodies are introduced. **Remark 29**. Pseudo knots make up a novel and significant model for DNA knots since there are some DNA knots where it is difficult to distinguish (even with electron microscope) the relative positions of the two arcs in some crossings. Analogous situation can certainly occur also in *worn textiles*, which thus can be modelled by pseudo DP tangles. Let now $\tau$ be a singular motif in the thickened torus and $d$ a diagram of $\tau$ undergoing singular isotopy. Respectively, let $d$ be a motif diagram in the torus undergoing pseudo knot isotopy. These isotopies in the torus are generated by the same local moves described above. Each isotopy carries through periodically to the DP diagram $d_{\infty}$ of the DP tangle $\tau_{\infty}$. On the level of flat motifs, the *singular (resp. pseudo) surface isotopy move*, whereby a singular (resp. pre) crossing passes through the meridian or longitude (see Fig. [24](#spsurfacemoves){reference-type="ref" reference="spsurfacemoves"}), extends the usual surface isotopy moves (recall Figs. [6](#planar1){reference-type="ref" reference="planar1"}, [7](#planar2){reference-type="ref" reference="planar2"}, and [8](#planar3){reference-type="ref" reference="planar3"}). {#spsurfacemoves width="5.3in"} So, Theorem [Theorem 7](#thm:localmotifisotopy){reference-type="ref" reference="thm:localmotifisotopy"} adapts as follows: **Theorem 30**. *Two singular (resp. pseudo) flat motifs supported by the same lattice are *singular (resp. pseudo) isotopic* if and only if they differ by a finite sequence of surface isotopy moves, singular (resp. pseudo) surface isotopy moves, the classical Reidemeister moves and the (allowed) singular (resp. pseudo) isotopy moves.* The moves in Theorem [Theorem 30](#thm:singular/pseudomotifisotopy){reference-type="ref" reference="thm:singular/pseudomotifisotopy"} generate the *singular (resp. pseudo) motif isotopy*. Moreover, by the discussion in the beginning of this section, Theorem [Theorem 19](#th:equivalence){reference-type="ref" reference="th:equivalence"} applies for singular (resp. pseudo) DP tangles, as follows: **Theorem 31**. *Let $\tau_{1,\infty}$ and $\tau_{2,\infty}$ be two singular DP tangles in $\mathbb{E}^2 \times I$, with corresponding DP diagrams $d_{1,\infty}$ and $d_{2,\infty}$. Respectively, let $d_{1,\infty}$ and $d_{2,\infty}$ be two pseudo DP tangle diagrams in $\mathbb{E}^2$. Let also $\Lambda_1$ and $\Lambda_2$ be the supporting point lattices such that $d_i = d_{i,\infty} / \Lambda_i$ is a flat motif of $d_{i,\infty}$ for $i \in \{1,2\}$. Then $d_{1,\infty}$ and $d_{2,\infty}$ are *singular equivalent* (resp. *pseudo equivalent*) if and only if $d_1$ and $d_2$ are related by a finite sequence of singular (resp. pseudo) motif isotopy moves, shifts, Dehn twists, orientation preserving affine transformations and scale equivalence.* **Remark 32**. Using the theory of singular motifs, finite type invariants have been constructed in [@Grishanov.Vassiliev1; @Grishanov.Vassiliev2] to distinguish DP tangles which were not differentiated by the bracket-type polynomial invariant constructed in [@Grishanov1; @Grishanov.part2]. The theory of singular DP tangles could also find interesting applications in molecular chemistry. ## Tied and bonded DP tangle equivalence {#sec:DPtied} *Tied links* were introduced in [@AJ1] as generalization of links in $S^3$. A tied link is a classical link equipped with 'ties'. A tie connects two points of the link and behaves like a phantom: its ends can slide along the arcs that it connects, passing across any other parts of the link without obstruction, see left-hand illustration of Fig. [25](#tbmove){reference-type="ref" reference="tbmove"}, where a tie is depicted as red spring. There are further the rules that: two ties joining the same pair of components merge into one tie, and ties joining points of the same component can be deleted or introduced at will. Hence, a set of ties in a link diagram provides a combinatorial structure for defining a partition of the set of components of the link, by considering two components of the tied link to belong to the same partition subset if there is a tie connecting them. Then, *tied link isotopy* is defined as ambient isotopy between the underlying links (ignoring the ties), taking also into consideration that the set of ties in the links define the same partition of the set of components. In terms of diagrams we have planar isotopy and the Reidemeister moves together with local moves involving ties, as illustrated in Figs. [25](#tbmove){reference-type="ref" reference="tbmove"}, [26](#bmoves){reference-type="ref" reference="bmoves"} and [27](#fmoves){reference-type="ref" reference="fmoves"}, where the orange springs represent ties. {#tbmove width="4in"} {#bmoves width="5.5in"} {#fmoves width="1.5in"} In [@F], the author generalizes the concept of tied links in the Solid Torus and in [@D2] the author studies the theory of tied links in other 3-manifolds, including the complement of the $g$-component unlink. **Remark 33**. It is worth mentioning that the above results generalize directly for the diagrammatic theory of *tied singular links*, introduced and studied in [@AJ3] as a generalization of singular links, as well as for *tied pseudo links*, introduced and studied in [@D1]. Considering the ties to be embedded simple arcs in $S^3$, we obtain the theory of *bonded links*. More precisely, a bonded link is a pair $(L, B)$, where $L$ is a link in $S^3$ and $B$ is a set of $k$ pairwise disjoint simple arcs, the 'bonds', properly embedded in the complement $S^3\backslash L$ of the link, such that the boundaries of the bonds intersect the link transversely in $2k$ distinct points. The intersection points are not considered as rigid vertices. Bonds were introduced in [@GGLDSK] in the context of bonded knotoids for modeling open knotted protein chains. See further [@G]. Bonded link isotopy is defined as ambient isotopy between links, taking also into consideration the set of bonds in the links. In terms of diagrams we have planar isotopy and the Reidemeister moves together with local moves involving bonds, as illustrated in Figs. [26](#bmoves){reference-type="ref" reference="bmoves"} and [27](#fmoves){reference-type="ref" reference="fmoves"}, where bonds are depicted as orange line segments. Note that these moves are also valid in tied link isotopy. We also have forbidden moves in the theory, as for example the move depicted in the right-hand side of Fig. [25](#tbmove){reference-type="ref" reference="tbmove"}. Hence, tied knot theory can be realized as the quotient of the bonded knot theory, modulo the tied elementary move, Fig. [25](#tbmove){reference-type="ref" reference="tbmove"} and the tied cancellations/additions. Let now $\tau$ be a tied (resp. bonded) motif in the thickened torus and $d$ a diagram of $\tau$ undergoing tied (resp. bonded) isotopy. This isotopy is generated in the torus by the same diagrammatic local moves described above. Each isotopy carries through periodically to the DP diagram $d_{\infty}$ of the DP tangle $\tau_{\infty}$. On the level of flat motifs, the *tied (resp. bonded) surface isotopy move*, whereby a tie (resp. bond) passes through the meridian or longitude (see Fig. [28](#tbmotifisotopy){reference-type="ref" reference="tbmotifisotopy"}), extends the usual surface isotopy moves (recall Figs. [6](#planar1){reference-type="ref" reference="planar1"}, [7](#planar2){reference-type="ref" reference="planar2"}, and [8](#planar3){reference-type="ref" reference="planar3"}). {#tbmotifisotopy width="5.7in"} So, Theorem [Theorem 7](#thm:localmotifisotopy){reference-type="ref" reference="thm:localmotifisotopy"} adapts as follows: **Theorem 34**. *Two tied (resp. bonded) flat motifs supported by the same lattice are *tied (resp. bonded) isotopic* if and only if they differ by a finite sequence of surface isotopy moves, tied (resp. bonded) surface isotopy moves, the classical Reidemeister moves and the (allowed) tied (resp. bonded) isotopy moves.* The moves in Theorem [Theorem 34](#thm:tied/bondedmotifisotopy){reference-type="ref" reference="thm:tied/bondedmotifisotopy"} generate the *tied (resp. bonded) motif isotopy*. Moreover, by the discussion in the beginning of this section, Theorem [Theorem 19](#th:equivalence){reference-type="ref" reference="th:equivalence"} applies for tied (resp. bonded) DP tangles, as follows: **Theorem 35**. *Let $\tau_{1,\infty}$ and $\tau_{2,\infty}$ be two tied (resp. bonded) DP tangles in $\mathbb{E}^2 \times I$, with corresponding DP diagrams $d_{1,\infty}$ and $d_{2,\infty}$. Let also $\Lambda_1$ and $\Lambda_2$ be the supporting point lattices such that $d_i = d_{i,\infty} / \Lambda_i$ is a flat motif of $d_{i,\infty}$ for $i \in \{1,2\}$. Then $d_{1,\infty}$ and $d_{2,\infty}$ are *tied (resp. bonded) equivalent* if and only if $d_1$ and $d_2$ are related by a finite sequence of tied (resp. bonded) motif isotopy moves, shifts, Dehn twists, orientation preserving affine transformations and scale equivalence.* Theorem [Theorem 19](#th:equivalence){reference-type="ref" reference="th:equivalence"} along with its analogues, Theorems [Theorem 24](#th:regularframedequivalence){reference-type="ref" reference="th:regularframedequivalence"}, [Theorem 28](#th:virtualweldedequivalence){reference-type="ref" reference="th:virtualweldedequivalence"}, [Theorem 31](#th:singularpseudoequivalence){reference-type="ref" reference="th:singularpseudoequivalence"}, and [Theorem 35](#th:tiedbondedequivalence){reference-type="ref" reference="th:tiedbondedequivalence"} related to different diagrammatic settings, will serve as a foundation for future works such as adapting the theory to yet different diagrammatic and combinatorial settings, constructing new topological invariants of DP tangles, which are mathematical tools for their classification (as for example [@MDL]), as well as for novel potential applications. 99 F. 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--- abstract: | We identify a family of $O(|E(G)|^2)$ nontrivial facets of the connected matching polytope of a graph $G$, that is, the convex hull of incidence vectors of matchings in $G$ whose covered vertices induce a connected subgraph. Accompanying software to further inspect the polytope of an input graph is available. author: - Phillippe Samer date: September 23, 2023 title: On a class of strong valid inequalities for the connected matching polytope --- # Introduction {#sec:intro} Our goal with this paper is to bring attention to an interesting polytope, and contribute towards improved algorithms for a recent combinatorial optimization problem defined over it. A [P]{.sans-serif}-matching in a graph $G$ consists of a matching $M$ such that the subgraph induced by vertices covered by $M$ has property [P]{.sans-serif}, *e.g.* being connected. In particular, while finding a maximum cardinality *connected matching* is a well-solved problem, the edge-weighted counterpart is NP-hard even in very restricted graph classes [@gg2023tcs]. We initiate the systematic study of the polytope $\mathfrak{C}(G)$ of connected matchings in a general graph $G$, introducing a relevant class of facet-defining inequalities. Early examples of studies on [P]{.sans-serif}-matching problems include [@StockmeyerVazirani1982] on induced matchings, [@golumbic2001uniquely] on uniquely restricted matchings, and [@Goddard2005disconnected] contemplating acyclic, connected and disconnected matchings. While different [P]{.sans-serif}-matching problems receive increased attention in the recent literature, we highlight [@gg2022latin] and [@gg2023tcs] on weighted connected matchings, who were able to determine several fine-grained complexity results. In particular, they show that it is NP-hard to find a maximum weight connected matching even in bounded-diameter bipartite and planar bipartite graphs. The main argument for our contribution is to bring the machinery of polyhedral studies and mixed-integer programming (MIP) to bear on the investigation of weighted connected matchings in general graphs. In light of decades' worth of progress on matching theory and on the effective use of strong valid inequalities in MIP solvers, the combinatorial analysis of the facial structure of polytope $\mathfrak{C}(G)$ is a natural methodology. On that perspective, the key idea we present next is a powerful ingredient in that direction. We remark that all polyhedra in this work are rational, and that we do not leave the unit hypercube. Nearly all terminology and notation we use are standard in graph theory and polyhedral combinatorics. The following might be worth mentioning. We write $[k] \stackrel{\text{def}}{=}\left\lbrace 1, \ldots, k \right\rbrace$. Given a graph $G$, we denote its *line graph* by $L(G)$, and define the *distance between two edges* in $G$ as $d_L: E(G) \times E(G) \rightarrow \mathbb{Z}_+$ so that $d_L(e_1, e_2)$ equals the number of edges in a shortest path between $e_1$ and $e_2$ in $L(G)$. Given a subset of edges $S \subseteq E(G)$, we denote by $\chi^S$ its *incidence* (or *characteristic*) *vector* in space $\mathbb{Q}^{|E(G)|}$, with $\chi^S_e = 1$ for each $e \in S$, and $\chi^S_e = 0$ otherwise. First, it is convenient to clear implied equations out of systems of inequalities in studies of the connected matching polytope. Using the standard argument that the unit vectors $\left\lbrace \chi^{\lbrace e \rbrace}: e \in E(G) \right\rbrace$ and $\mathbf{0} \in \mathbb{Q}^{|E(G)|}$ are affinely independent and induce trivial incidence vectors of connected matchings in $G$, we have the following result. **Proposition 1**. *The connected matching polytope $\mathfrak{C}(G)$ of an arbitrary graph $G$ is full-dimensional.* Our main result is the following. **Theorem 2**. *Let $G$ be a connected graph and $\left\lbrace e^\prime, e^{\prime \prime} \right\rbrace \subset E(G)$ induce a disconnected matching. Denote by $\Lambda \stackrel{\text{def}}{=}\Lambda(e^\prime, e^{\prime \prime}) = \left\lbrace f \in E(G): d_L(f,e^\prime) = d_L(f, e^{\prime \prime}) = 2 \right\rbrace$ the corresponding set of edges at two hops from both $e^{\prime}$ and $e^{\prime \prime}$. If $\Lambda \neq \varnothing$, the inequality $$\label{eq:vi} x_{e^\prime} + x_{e^{\prime \prime}} - \sum_{f \in \Lambda} x_f \leq 1$$ is valid for $\mathfrak{C}(G)$. Moreover, $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:vi}% \endgroup$ defines a facet when $\Lambda$ induces a clique in $L(G)$ and the subgraph induced by $\left\lbrace e^\prime, e^{\prime \prime}, f \right\rbrace$ is 2-connected for each $f \in \Lambda$.* The proof is saved for the next section. Let us first illustrate how important it may be to consider this small set of facets (note that we have at most one inequality for each pair of edges). There are many options for modelling induced connectivity. Progress in mathematical programming computation of structures like maximum-weight connected subgraphs and Steiner trees build on vertex choosing binary variables $y \in \left\lbrace 0,1 \right\rbrace^{|V(G)|}$ and *minimal separator inequalities* (MSI): $y_a + y_b -\sum_{u \in \mathcal{C}} y_u \leq 1$ for each pair of non-adjacent vertices $a$ and $b$, and each $(a,b)$-separator $\mathcal{C} \subseteq V \backslash \left\lbrace a,b \right\rbrace$, *i.e.* there are no paths connecting $a$ to $b$ if we remove $\mathcal{C}$ from $G$. See [@wang2017imposing] for a thorough polyhedral analysis, and [@fischetti2017MPC] for supporting experimental results of an award-winning solver for Steiner tree problems. In an attempt to build on those results, and impose induced connectivity on a system of inequalities formulating connected matchings while using only natural design variables $x \in \left\lbrace 0,1 \right\rbrace^{|E(G)|}$, as opposed to working with extended formulations, one may use the fact that vertex $u$ belongs to the subgraph induced by matching $M$ if and only if there is exactly one edge in $M$ incident to $u$. That is, projecting MSI onto the space of $x$ variables using $y_u \stackrel{\text{def}}{=}\sum_{e \in \delta(u)} x_e$, we derive a MIP formulation to find maximum weight connected matchings using MSI. We are currently pursuing that endeavour and crafting a branch-and-cut algorithm for weighted connected matchings. In the meantime, we inspected the convex hull $\mathfrak{C}(G)$ for several examples using [polymake]{.sans-serif} [@polymake2000; @polymake2017]. Remarkably, we discovered fine examples where a single inequality in $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:vi}% \endgroup$ dominates several MSI. For instance, taking $G_{J26}$ to be the skeleton graph of Johnson Solid 26, depicted in Figure [1](#fig:JS26){reference-type="ref" reference="fig:JS26"}, and studying the minimal inequality description of $\mathfrak{C}(G_{J26})$, we detect 14 trivial facets from non-negativity bounds, 8 blossom inequalities on handles $H_i=V(G)\backslash \left\lbrace v_i \right\rbrace$, some blossom inequalities on triangles, and 5 facets defined by our inequalities in $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:vi}% \endgroup$. Among the latter, we find $$\label{eq:example_J26} x_5 + x_{13} -x_2 -x_3 \leq 1,$$ which may be interpreted as the lifting of 4 different MSI corresponding to $C \stackrel{\text{def}}{=}\left\lbrace v_1, v_3, v_5, v_7 \right\rbrace$ as a minimal $(v_a, v_b)$-separator for $(a,b) \in \left\lbrace (2,6), (2,8), (4,6), (4,8) \right\rbrace$. In other words, a MIP formulation adding inequality $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:example_J26}% \endgroup$ *a priori* gives a tighter approximation of $\mathfrak{C}(G_{J26})$ than a formulation that depends solely on cutting planes from MSI projected onto $x$ space, which could generate dynamically the cuts $\left\lbrace y_{v_a} +y_{v_b} -\sum_{z \in C} y_{z} \leq 1 \right\rbrace$. For example, when $(a,b) = (2,6)$, the projected inequality is $$\begin{aligned} {2} (x_1 + x_5 + x_6 + x_7) +(x_{11}+x_{12}+x_{13}) & \nonumber \\ -(x_1+x_2+x_3+x_4) -(x_2+x_8+x_9) & \nonumber \\ -(x_3+x_6+x_{11}) -(x_7+x_{10}+x_{12}+x_{14}) & \leq 1, \nonumber\end{aligned}$$ that is, $x_5 +x_{13} -2x_2 -2x_3 -x_4 -x_8 -x_9 -x_{10} -x_{14} \leq 1,$ which is immediately seem to be dominated by $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:example_J26}% \endgroup$. {#fig:JS26} The productive exercise of plugging different input graphs $G$ and inspecting the resulting polytope $\mathfrak{C}(G)$ is made free and open to anyone interested. The code developed to find a $\mathcal{V}$-description of the polytope corresponding to an input graph, and then produce an $\mathcal{H}$-description with [polymake]{.sans-serif}, is available at <https://github.com/phillippesamer/wcm-branch-and-cut/tree/main/polyhedra>, as is the forthcoming branch-and-cut algorithm. Before we resume to the main proof, we remark that it may be straightforward in some particular cases to show that $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:vi}% \endgroup$ induces a facet of the smaller polytope $\mathfrak{C}(G[S])$, with $S = \left\lbrace e^\prime, e^{\prime \prime}, f \right\rbrace$ and $f \in \Lambda$. Nonetheless, this would only be interesting if coupled with a lifting result showing how to derive a facet of our target polytope $\mathfrak{C}(G)$ from the smaller dimensional one. We skip that altogether and give next a direct proof of the general result. # Proof of Theorem [Theorem 2](#thm:facets){reference-type="ref" reference="thm:facets"} {#proof-of-theorem-thmfacets} #### *(i)* The validity of inequality $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:vi}% \endgroup$ follows from simple combinatorial reasoning. Let $\chi^{S}$ be the incidence vector of an arbitrary connected matching $S$. If $e^\prime \not\in S$ or $e^{\prime \prime} \not\in S$, the left-hand side of the expression in $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:vi}% \endgroup$ is at most 1. On the other hand, the fact that $\left\lbrace e^\prime, e^{\prime \prime} \right\rbrace$ induces a disconnected matching implies that, if both $e^\prime \in S$ and $e^{\prime \prime} \in S$, there must be an additional edge $f$ in $S$ to connect their endpoints. From $\left\lbrace e^\prime, e^{\prime \prime}, f \right\rbrace$ being a matching, we have that $d_L(f,e^\prime) \geq 2$ and $d_L(f, e^{\prime \prime}) \geq 2$. For the corresponding vertices in each of $e^\prime$, $e^{\prime \prime}$ and $f$ to induce a connected subgraph, we verify $d_L(f,e^\prime) \leq 2$ and $d_L(f, e^{\prime \prime}) \leq 2$. Hence $f \in \Lambda$, and again the left-hand side of the inequality is bound below $1$ at $\chi^{S}$. Since $\chi^{S}$ is taken as an arbitrary vertex of the polytope, the inequality is valid at all points in $\mathfrak{C}(G)$ by convexity. #### *(ii)* For the facet proof, let $F \stackrel{\text{def}}{=}\left\lbrace x \in \mathfrak{C}(G): x_{e^\prime} + x_{e^{\prime \prime}} - \sum_{f \in \Lambda} x_f = 1 \right\rbrace = \left\lbrace x \in \mathfrak{C}(G): \pi x = \pi_0\right\rbrace$ denote the face corresponding to inequality $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:vi}% \endgroup$; vector $(\pi, \pi_0)$ is just shorthand notation here. By the full-dimensionality observation in Proposition [Proposition 1](#thm:dim){reference-type="ref" reference="thm:dim"}, there is no equation satisfied by all points in the polytope. It thus suffice to show that if $F \subseteq \overline F \stackrel{\text{def}}{=}\left\lbrace x \in \mathfrak{C}(G): \lambda x = \lambda_0 \right\rbrace$, the defining inequalities $\pi x \leq \pi_0$ and $\lambda x \leq \lambda_0$ are actually equivalent (*i.e.* there exists $\rho > 0$ such that $\lambda = \rho \pi$ and $\lambda_0 = \rho \pi_0$), and hence the strict inclusion cannot hold [@NemhauserWolsey1999 Theorem I.4.3.6]. Set $\rho \stackrel{\text{def}}{=}\lambda_0$. To prove the equivalence of the nonzero coefficients, let us denote by $x^1, x^2, x^3$ the incidence vectors of the connected matchings consisting of $\left\lbrace e^\prime \right\rbrace$, $\left\lbrace e^{\prime \prime} \right\rbrace$, and $\left\lbrace e^\prime, e^{\prime \prime}, f_1 \right\rbrace$, respectively, with $f_1 \in \Lambda$ arbitrary. Note that each of these points belong to $F$, and give a single nonzero coefficient when evaluating $\lambda x = \lambda_0$: i. From $x^{1} \stackrel{\text{def}}{=}\left\lbrace x^{1}_{e^\prime} = 1, x^{1}_{*} = 0 \text{ for } * \in E\backslash \left\lbrace e^\prime \right\rbrace \right\rbrace \in F$ we get $\lambda x^{1} = \lambda_{e^{\prime}}$. Together with $\lambda x^{1} = \lambda_0$, we obtain $\lambda_{e^{\prime}} = \lambda_0 \stackrel{\text{def}}{=}\rho = \rho \cdot 1 = \rho \cdot \pi_{e^{\prime}}$. ii. From $x^{2} \stackrel{\text{def}}{=}\left\lbrace x^{2}_{e^{\prime \prime}} = 1, x^{2}_{*} = 0 \text{ for } * \in E\backslash \left\lbrace e^{\prime \prime} \right\rbrace \right\rbrace \in F$ we get $\lambda x^{2} = \lambda_{e^{\prime \prime}}$. Together with $\lambda x^{2} = \lambda_0$, we obtain $\lambda_{e^{\prime \prime}} = \lambda_0 \stackrel{\text{def}}{=}\rho = \rho \cdot 1 = \rho \cdot \pi_{e^{\prime \prime}}$. iii. From $x^{3} \stackrel{\text{def}}{=}\left\lbrace x^{3}_{e^\prime} = x^{3}_{e^{\prime \prime}} = x^{3}_{f_1} = 1, x^{3}_{*} = 0 \text{ for } * \in E\backslash \left\lbrace e^\prime, e^{\prime \prime}, f_1 \right\rbrace \right\rbrace \in F$ we get $\lambda x^{3} = \lambda_{e^{\prime}} + \lambda_{e^{\prime \prime}} + \lambda_{f_1}$. Together with $\lambda x^{3} = \lambda_0$, we have $\lambda_{f_1} = \lambda_0 -\lambda_{e^{\prime}} -\lambda_{e^{\prime \prime}}$. Now, using the coefficients determined in the previous two items, we find that $\lambda_{f_1} = \lambda_0 -\lambda_0 -\lambda_0 = \lambda_0 \cdot (-1) \stackrel{\text{def}}{=}\rho \cdot (-1) = \rho \cdot \pi_{f_1}$. Since the choice of $f_1 \in \Lambda$ is without loss of generality, we repeat the argument in the last item above using the incidence vectors of each of the connected matchings in $\left\lbrace \left\lbrace e^\prime, e^{\prime \prime}, f \right\rbrace : f \in \Lambda \right\rbrace$, to actually determine that $\lambda_{f} = \rho \cdot \pi_f$ for each $f \in \Lambda$. That establishes the equivalence of nonzero coefficients. It remains to show that all other coefficients of $\lambda$ are null. All we need for that are the following two remarks. Fact 1 : Let $\chi^M \in F$ be the incidence vector of matching $M$. By the hypothesis that $\Lambda$ induces a clique in $L(G)$, we have that $M$ contains at most one edge in $\Lambda$. That enables us to judiciously order the edges in $M = \left\lbrace e_1, \ldots, e_m \right\rbrace$ so that (i) the edges in $\overline{\Lambda} \stackrel{\text{def}}{=}\left\lbrace e^\prime, e^{\prime \prime}\right\rbrace \cup \Lambda$ (either one or exactly three) appear first, and (ii) each edge added after we are done with $\overline{\Lambda}$ yields an additional point in our face $F$. Ultimately, we may apply the following simple observation a number of times: $$\begin{aligned} {2} \chi^{\left\lbrace e_1, \ldots, e_k \right\rbrace} \in F \subseteq \overline{F} \implies & \lambda \cdot \chi^{\left\lbrace e_1, \ldots, e_k \right\rbrace} = \overbrace{\sum_{i=1}^{k-1} \lambda_{e_i} \cdot 1}^{= \lambda_0} \ + \lambda_{e_k} \cdot 1 \ + \overbrace{\sum_{\substack{e \in E(G): \\e\not\in\left\lbrace e_1, \ldots, e_k \right\rbrace}} \lambda_e \cdot 0}^{= 0} \ = \ \lambda_0 \nonumber \\ \implies & \lambda_{e_k} = 0 \label{eq:augmenting_lambda}\end{aligned}$$ Fact 2 : For all $e \in E(G)$, there exists a connect matching $M$ including $e$ such that $\chi^M \in F$. Let $S$ be the set of endpoints of edges in $\overline{\Lambda} = \left\lbrace e^\prime, e^{\prime \prime}\right\rbrace \cup \Lambda$, and consider the following two cases. (i) Suppose that $e \in E(G[S])$. Taking $M$ in $\left\lbrace \left\lbrace e^\prime, e^{\prime \prime}, f \right\rbrace : f \in \Lambda \right\rbrace$, the claim follows immediately for $e \in \overline{\Lambda}$, by definition of $F$. Otherwise, if $e = \left\lbrace u,v \right\rbrace$ is an induced edge in $E(G[S])\backslash \overline{\Lambda}$, it joins a vertex from either $e^\prime$ or $e^{\prime \prime}$ to a vertex of some $f \in \Lambda$. Suppose without loss of generality that $u \in e^\prime$ and $v \in f$. Since the subgraph induced by $\left\lbrace e^\prime, e^{\prime \prime}, f \right\rbrace$ is 2-connected by the hypothesis in the theorem, the matching $M \stackrel{\text{def}}{=}\left\lbrace e, e^{\prime \prime} \right\rbrace$ is connected, and $\chi^M \in F$. (ii) If $\hat{e} \not\in E(G[S])$, we use the assumption that $G$ is connected and take a simple path $P = (e_1, \ldots, e_n)$ on $n$ edges beginning at $e_1 = \hat{e}$, and minimal with respect to including an edge in $E(G[S])$, that is, $e_n$ is the only edge in $P$ which is also in the subgraph induced by vertices covered by $\left\lbrace e^\prime, e^{\prime \prime}, \hat{f} \right\rbrace$, for some $\hat{f} \in \Lambda$. Now we let $\hat{M}$ be the matching alternating edges along $P$, *requiring* $\hat{e} \in \hat{M}$, and reason on the parity of the path length. Suppose first that $n$ is odd, so that $e_n$ is also in $\hat{M}$. If $e_n$ is either $e^\prime$ or $e^{\prime \prime}$, we are done. If $e_n = \hat{f}$, we set $M \stackrel{\text{def}}{=}\hat{M} \uplus \left\lbrace e^\prime, e^{\prime \prime}\right\rbrace$, and we are done. Otherwise, $e_n = \left\lbrace u,v \right\rbrace$ as in case (i): without loss of generality, let $e_n$ join $u \in e^\prime$ and $v \in \hat{f}$. Using again the hypothesis that the subgraph induced by $\left\lbrace e^\prime, e^{\prime \prime}, \hat{f} \right\rbrace$ is 2-connected, choose $M \stackrel{\text{def}}{=}\hat{M} \uplus \left\lbrace e^{\prime \prime} \right\rbrace$, and we are done. Suppose now that $n$ is even, and thus $e_n \not\in \hat{M}$. By the 2-connectivity hypothesis, we may augment $\hat{M}$ to get a connected matching with exactly one of $e^\prime$ or $e^{\prime \prime}$, whose characteristic vector is thus in the face $F$. If $e_{n-1}$ is incident to a vertex of $e^\prime$, we set $M \stackrel{\text{def}}{=}\hat{M} \uplus \left\lbrace e^{\prime \prime}, e^\dag \right\rbrace$, where $e^\dag$ joins the endpoint of $e^\prime$ not covered by $\hat{M}$ to a vertex of $\hat{f}$. The analogous argument holds when $e_{n-1}$ is incident to a vertex of $e^{\prime \prime}$. Finally, if $e_{n-1}$ is incident to a vertex of $\hat{f}$, we simply choose $M \stackrel{\text{def}}{=}\hat{M} \uplus \left\lbrace e^\prime \right\rbrace$. To complete the proof, we consider each edge $e_k \not\in \overline{\Lambda}$. By Fact 2, we obtain a connected matching $M$ including $e_k$ such that $\chi^M \in F$. Using $\begingroup \hypersetup{ linkcolor=linkequation, linkbordercolor=linkequation, } \eqref{eq:augmenting_lambda}% \endgroup$ in Fact 1, we conclude that $\lambda_{e_k} = 0$. Hence $(\lambda, \lambda_0) = (\rho \pi, \rho \pi_0)$, and $F$ determines a facet of $\mathfrak{C}(G)$. 0◻ #### Acknowledgement The author is grateful to the support by the Research Council of Norway through the research project 249994 CLASSIS. This work is dedicated to the sweet memory of our department administration member Ingrid Kyllingmark. 10 urlstyle B. Assarf, E. Gawrilow, K. Herr, M. Joswig, B. Lorenz, A. Paffenholz, and T. Rehn. 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Golumbic, T. Hirst, and M. Lewenstein. Uniquely restricted matchings. *Algorithmica*, 31: 139--154, 2001. URL `https://doi.org/10.1007/s00453-001-0004-z`. G. C. M. Gomes, B. P. Masquio, P. E. D. Pinto, V. F. d. Santos, and J. L. Szwarcfiter. Weighted connected matchings. In A. Castañeda and F. Rodríguez-Henríquez, editors, *LATIN 2022: Theoretical Informatics*, pages 54--70, Cham, 2022. Springer International Publishing. ISBN 978-3-031-20624-5. URL `https://doi.org/10.1007/978-3-031-20624-5_4`. G. C. Gomes, B. P. Masquio, P. E. Pinto, V. F. dos Santos, and J. L. Szwarcfiter. Disconnected matchings. *Theoretical Computer Science*, 956: 113821, 2023. ISSN 0304-3975. URL `https://doi.org/10.1016/j.tcs.2023.113821`. G. L. Nemhauser and L. A. Wolsey. *Integer and combinatorial optimization*, volume 55 of *Wiley-Interscience series in discrete mathematics and optimization*. John Wiley & Sons, Inc, 1999. URL `https://doi.org/10.1002/9781118627372`. L. J. Stockmeyer and V. V. 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arxiv_math

--- abstract: | Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincaré duality and the Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein. address: University of Minnesota - Twin Cities, Minneapolis MN 55455 author: - Robert Angarone, Anastasia Nathanson, Victor Reiner bibliography: - references.bib title: Chow rings of matroids as permutation representations --- # Introduction {#intro-section} A *matroid* $\mathcal{M}$ is a combinatorial abstraction of lists of vectors $v_1,v_2,\ldots,v_n$ in a vector space, recording only the information about which subsets of the vectors are linearly independent or dependent, forgetting their coordinates-- see Section [2.1](#matroid-background-section){reference-type="ref" reference="matroid-background-section"} for definitions and references. In groundbreaking work, Adiprasito, Huh and Katz [@AHK] affirmed long-standing conjectures of Rota--Heron--Welsh and Mason about vectors and matroids via a new methodology. Their work employed a certain graded $\mathbb{Z}$-algebra $A(\mathcal{M})$ called the *Chow ring* of $\mathcal{M}$, introduced by Feichtner and Yuzvinsky [@FY] as a generalization of the Chow ring of DeConcini and Procesi's *wonderful compactifications* for hyperplane arrangement complements. A remarkable integral Gröbner basis result proven by Feichtner and Yuzvinsky [@FY Thm. 2] shows that for a matroid $\mathcal{M}$ of rank $r+1$ with Chow ring $A(\mathcal{M})=\bigoplus_{k=0}^r A^k$, each homogeneous component is free abelian: $A^k \cong \mathbb{Z}^{a_k}$ for some Hilbert function $(a_0,a_1,\ldots,a_r)$. A key step in the work of Adiprasito, Huh and Katz shows not only *symmetry* and *unimodality* for the Hilbert function $$\begin{aligned} \label{symmetry} &a_k = a_{r-k}\text{ for } r \leq k/2\\ \label{unimodality} &a_0 \leq a_1 \leq \cdots \leq a_{\lfloor \frac{r}{2} \rfloor} = a_{\lceil \frac{r}{2} \rceil} \geq \cdots \geq a_{r-1} \geq a_r,\end{aligned}$$ but in fact proves that $A(\mathcal{M})$ enjoys a trio of properties referred to as the *Kähler package*, reviewed in Section [2.4](#AHK-section){reference-type="ref" reference="AHK-section"} below. The first of these properties is *Poincaré duality*, proving [\[symmetry\]](#symmetry){reference-type="eqref" reference="symmetry"} via a natural $\mathbb{Z}$-module isomorphism $A^{r-k} \cong \mathrm{Hom}_\mathbb{Z}(A^k,\mathbb{Z})$. The second property, called the *Hard Lefschetz Theorem*, shows that after tensoring over $\mathbb{Z}$ with $\mathbb{R}$ to obtain $A(\mathcal{M})_\mathbb{R}=\bigoplus_{k=0}A^k_\mathbb{R}$, one can find *Lefschetz elements* $\omega$ in $A^1_\mathbb{R}$ such that multiplication by $\omega^{r-2k}$ give $\mathbb{R}$-linear isomorphisms $A^k_\mathbb{R}\rightarrow A^{r-k}_\mathbb{R}$ for $k \leq \frac{r}{2}$. In particular, multiplication by $\omega$ mapping $A^k_\mathbb{R}\rightarrow A_\mathbb{R}^{k+1}$ is *injective* for $k < \frac{r}{2}$, strengthening the unimodality [\[unimodality\]](#unimodality){reference-type="eqref" reference="unimodality"}. We are interested here in how these *Poincaré duality* and *Hard Lefschetz* properties interact with the group $G:=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$ of symmetries of the matroid $\mathcal{M}$. It is not hard to check (see Section [2.1](#matroid-background-section){reference-type="ref" reference="matroid-background-section"} below) that $G$ acts via graded $\mathbb{Z}$-algebra automorphisms on $A(\mathcal{M})$, giving $\mathbb{Z}G$-module structures on each $A^k$, and $\mathbb{R}G$-module structures on each $A^k_\mathbb{R}$. One can also check (see the proof of Corollary [Corollary 16](#integral-equivariant-PD-cor){reference-type="ref" reference="integral-equivariant-PD-cor"} below) that $A^r \cong \mathbb{Z}$ with trivial $G$-action. From this, the Poincaré duality pairing immediately gives rise to a $\mathbb{Z}G$-module isomorphism $$\label{integral-rep-PD-isomorphism} A^{r-k} \cong \mathrm{Hom}_\mathbb{Z}(A^k,\mathbb{Z})$$ where $g$ in $G$ acts on $\varphi$ in $\mathrm{Hom}_\mathbb{Z}(A^k ,\mathbb{Z})$ via $\varphi \mapsto \varphi \circ g^{-1}$; similarly $A^{r-k} \cong \mathrm{Hom}_\mathbb{R}(A^k,\mathbb{R})$ as $\mathbb{R}G$-modules. Furthermore, it is not hard to check (see Corollary [Corollary 17](#AHK-equivariant-Hard-Lefschetz){reference-type="ref" reference="AHK-equivariant-Hard-Lefschetz"} below) that one can pick an explicit Lefschetz element $\omega$ as in [@AHK] which is $G$-fixed, giving $\mathbb{R}G$-module isomorphisms and injections $$\begin{aligned} \label{real-Lefschetz-PD-isomorphism} A_\mathbb{R}^{k} &\overset{\sim}{\longrightarrow} A_\mathbb{R}^{r-k} \quad \text{ for }r \leq \frac{k}{2} \nonumber \\ a &\longmapsto a \cdot \omega^{r-2k}\end{aligned}$$ $$\begin{aligned} \label{real-Lefschetz-injection} A_\mathbb{R}^{k} &\hookrightarrow A_\mathbb{R}^{k+1} \quad \text{ for }r < \frac{k}{2} \nonumber \\ a & \longmapsto a \cdot \omega. \end{aligned}$$ Our goal in this paper is to use Feichtner and Yuzvinsky's Gröbner basis result to prove a combinatorial strengthening of the isomorphisms and injections [\[integral-rep-PD-isomorphism\]](#integral-rep-PD-isomorphism){reference-type="eqref" reference="integral-rep-PD-isomorphism"}, [\[real-Lefschetz-PD-isomorphism\]](#real-Lefschetz-PD-isomorphism){reference-type="eqref" reference="real-Lefschetz-PD-isomorphism"}, [\[real-Lefschetz-injection\]](#real-Lefschetz-injection){reference-type="eqref" reference="real-Lefschetz-injection"}. To this end, recall (or see Section [2.1](#matroid-background-section){reference-type="ref" reference="matroid-background-section"} below) that the matroid $\mathcal{M}$ can be specified by its family $\mathfrak{F}$ of *flats*; in the case where $\mathcal{M}$ is realized by a list of vectors $v_1,v_2,\ldots,v_n$ in a vector space, a subset $F \subseteq \{1,2,\ldots,n\}=:E$ is a flat when $\{v_j\}_{j \in F}$ is linearly closed, meaning that every vector $v_i$ for $i$ in $E$ that lies in the linear span of $\{v_j\}_{j \in F}$ already has $i$ in $F$. Then the *Chow ring* $A(\mathcal{M})$ is presented as a quotient of the polynomial ring $S:=\mathbb{Z}[x_F]$ having one variable $x_F$ for each nonempty flat $F$ in $\mathfrak{F}\setminus \{\varnothing\}$. The presentation takes the form $A(\mathcal{M}) := S / (I + J)$ where $I ,J$ are certain ideals of $S$ defined more precisely in Definition [\[Chow-ring-definition\]](#Chow-ring-definition){reference-type="ref" reference="Chow-ring-definition"} below. Feichtner and Yuzvinsky exhibited (see Theorem [Theorem 10](#FY-GB-theorem){reference-type="ref" reference="FY-GB-theorem"}, Corollary [\[cor: mon_basis\]](#cor: mon_basis){reference-type="ref" reference="cor: mon_basis"} below) a Gröbner basis for $I + J$ that leads to the following standard monomial $\mathbb{Z}$-basis for $A(\mathcal{M})$, which we call the *FY-monomials* of $\mathcal{M}$: $$\mathrm{FY}:=\{ x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell}: (\varnothing =:F_0) \subsetneq F_1 \subsetneq F_2 \subsetneq \cdots \subsetneq F_\ell, \text{ and } m_i \leq \mathop{\mathrm{\mathrm{rk}}}(F_i) - \mathop{\mathrm{\mathrm{rk}}}(F_{i-1})-1\}$$ Here $\mathop{\mathrm{\mathrm{rk}}}(F)$ denotes the matroid rank of the flat $F$ (= the dimension of the linear span of $\{v_j\}_{j \in F}$ when $\mathcal{M}$ is realized by a list of vectors). The subset $\mathrm{FY}^k$ of FY-monomials $x_{F_1}^{m_1} \cdots x_{F_\ell}^{m_\ell}$ of total degree $m_1+\cdots+m_\ell=k$ then gives a $\mathbb{Z}$-basis for $A^k$. One can readily check (see Corollary [Corollary 12](#Chow-ring-carries-perm-reps-cor){reference-type="ref" reference="Chow-ring-carries-perm-reps-cor"} below) that the group $G=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$ permutes the $\mathbb{Z}$-basis $\mathrm{FY}^k$ for $A^k$, endowing $A^k$ with the structure of a *permutation representation*, or *$G$-set*. Our main result, proven in Section [3](#main-theorem-section){reference-type="ref" reference="main-theorem-section"}, is this strengthening of the isomorphisms and injections [\[integral-rep-PD-isomorphism\]](#integral-rep-PD-isomorphism){reference-type="eqref" reference="integral-rep-PD-isomorphism"}, [\[real-Lefschetz-PD-isomorphism\]](#real-Lefschetz-PD-isomorphism){reference-type="eqref" reference="real-Lefschetz-PD-isomorphism"}, [\[real-Lefschetz-injection\]](#real-Lefschetz-injection){reference-type="eqref" reference="real-Lefschetz-injection"}. **Theorem 1**. *For every matroid $\mathcal{M}$ of rank $r+1$, there exist* - *$G$-equivariant bijections $\pi: \mathrm{FY}^k \overset{\sim}{\longrightarrow} \mathrm{FY}^{r-k}$ for $k \leq \frac{r}{2}$, and* - *$G$-equivariant injections $\lambda: \mathrm{FY}^k \hookrightarrow \mathrm{FY}^{k+1}$ for $k < \frac{r}{2}$.* **Example 2**. *Let $\mathcal{M}=U_{4,5}$ be the uniform matroid of rank $4$ on $E=\{1,2,3,4,5\}$, associated to a list of $5$ *generic* vectors $v_1,v_2,v_3,v_4,v_5$ in a $4$-dimensional vector space, so that any quadruple $v_i,v_j,v_k,v_\ell$ is linearly independent. One has these flats of various ranks:* *rank* *flats $F \in \mathfrak{F}$* -------- --------------------------------------------- *$0$* *$\varnothing$* *$1$* *$1,2,3,4,5$* *$2$* *$12,13,14,15,23,24,25,34,35,45$* *$3$* *$123,124,125,134,135,145,234,235,245,345$* *$4$* *$E=12345$* *The Chow ring $A(\mathcal{M})=S/(I+J)$, where $S= \mathbb{Z}[ x_i, x_{jk}, x_{\ell m n}, x_E]$ with $\{i\}, \{j,k\}, \{\ell, m, n\}$ running through all one, two and three-element subsets of $E=\{1,2,3,4,5\}$, and $$I=\Big( x_F x_{F'} \Big)_{F \not\subset F', F' \not \subset F}, \qquad J=\bigg( x_i + \sum_{\substack{1 \leq j < k \leq 5\\i \in \{j,k\}}} x_{jk} + \sum_{\substack{1 \leq \ell < m < n \leq 5\\i \in \{\ell,m,n\} }} x_{\ell m n} \,\,\, + x_E \bigg)_{i=1,2,3,4,5}.$$ The FY-monomial bases for $A^0,A^1,A^2,A^3$ are shown here, together with the $G$-equivariant maps $\lambda$: $$\begin{array}{cccccccc} \bf{FY^0} & & \bf{FY^1}& & \bf{FY^2} & & &\bf{FY^3} \\ & & & & & & & \\ 1 &\overset{\lambda}{\longmapsto}& x_E &\overset{\lambda}{\longmapsto} & x_E^2 & & &x_E^3 \\ & & & & & & &\\ & & x_{ijk} &\overset{\lambda}{\longmapsto} & x_{ijk}^2& & &\\ & & 1 \leq i<j<k \leq 5 & & & & & \\ & & & & & & &\\ & &x_{ij} &\overset{\lambda}{\longmapsto} &x_{ij} \cdot x_E& & &\\ & & 1 \leq i<j \leq 5& & & & &\\ \end{array}$$ Thus $A(\mathcal{M})$ has Hilbert function $(a_0,a_1,a_2,a_3)=(1,21,21,1)$. Here the bijection $\pi: \mathrm{FY}^0 \rightarrow \mathrm{FY}^3$ necessarily maps $1 \longmapsto x_E^3$, and the bijection $\pi: \mathrm{FY}^1 \rightarrow \mathrm{FY}^2$ coincides with the map $\lambda: \mathrm{FY}^1 \rightarrow \mathrm{FY}^2$ above.* Before proving Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"} in Section [3](#main-theorem-section){reference-type="ref" reference="main-theorem-section"}, the background Section [2](#background-section){reference-type="ref" reference="background-section"} reviews matroids and Chow rings, and collects a few simple observations. Section [4](#conjectures-section){reference-type="ref" reference="conjectures-section"} recasts the maps $\lambda$ from Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}(ii) in the language of unimodality within the *Burnside ring* of $G$-permutation representations. This motivates some analogous conjectures about Burnside rings that would extend recent log-concavity and total positivity conjectures for Hilbert functions of Chow rings $A(\mathcal{M})$. Section [5](#sec: further-questions){reference-type="ref" reference="sec: further-questions"} poses some additional questions and conjectures. # Acknowledgements {#acknowledgements .unnumbered} The authors thank Alessio D'Ali, Chris Eur, Luis Ferroni, Matt Larson, Hsin-Chieh Liao, Diane Maclagan, Matt Maestroni, Lorenzo Vecchi and Peter Webb for helpful conversations and references. They thank Trevor Karn for his wonderful `Sage/Cocalc` code that checks whether a symmetric group representation is a permutation representation. The authors received partial support from NSF grants DMS-1949896, DMS-1745638, DMS-2053288. Second author received partial support from PROM project no. POWR.03.03.00-00-PN13/18. # Background {#background-section} ## Matroids {#matroid-background-section} There are many equivalent definitions of a matroid; the one most useful here uses their *flats*. For matroid basics and background, good references are Oxley [@oxley] and Ardila [@icm_matroids]. **Definition 3**. *A matroid $\mathcal{M}= (E, \mathfrak{F})$ consists of a (finite) ground set $E$ and a collection of subsets $\mathfrak{F}=\{F\} \subseteq 2^E$ called *flats*, satisfying these axioms:* 1. *$E\in \mathfrak{F}$.* 2. *If $F, F'\in \mathfrak{F}$ , then $F \cap F' \in \mathfrak{F}$.* 3. *For any $F\in \mathfrak{F}$, and any $i \in E\setminus F$, there is a unique $F' \in \mathfrak{F}$ containing $i$ which *covers $F$* in this sense: $F \subsetneq F'$, and no other $F''$ has $F \subsetneq F'' \subsetneq F$.* These axioms combinatorially abstract properties from the case of a *realizable matroid* $\mathcal{M}$ associated to a list of vectors $v_1,v_2,\ldots,v_n$ in a vector space over some field $\mathbb{F}$. This realizable matroid $\mathcal{M}$ has ground set $E=\{1,2,\ldots,n\}$, and a subset $F \subseteq E$ is a flat in $\mathfrak{F}$ if and only if this inclusion is an equality: $$F \subseteq \{i \in E: v_i \in \mathrm{span}_{\mathbb{F}}\{ v_j\}_{j \in F} \}.$$ Axioms (F1),(F2) imply that the inclusion order $(\mathfrak{F},\subseteq)$ will be a lattice, with *meets* $F \wedge F'=F \cap F'$ and *joins* $F \vee F'=\bigcap_{F'' \supseteq F,F'} F''$. Axiom (F3) further implies that the lattice $\mathfrak{F}$ will be *geometric*, that is, both *atomic* (every $F$ is the join of the atoms below it) and *upper-semimodular*: it has a rank function $\mathop{\mathrm{\mathrm{rk}}}: \mathfrak{F}\rightarrow \{0,1,2,\ldots\}$ satisfying $$\mathop{\mathrm{\mathrm{rk}}}(F \vee F') + \mathop{\mathrm{\mathrm{rk}}}(F \wedge F') \leq \mathop{\mathrm{\mathrm{rk}}}(F) + \mathop{\mathrm{\mathrm{rk}}}(F').$$ The *rank* of the matroid $\mathcal{M}$ itself is defined to be $\mathop{\mathrm{\mathrm{rk}}}(E)$, and we assume throughout that $\mathop{\mathrm{\mathrm{rk}}}(E)=r+1$. An *automorphism* of the matroid $\mathcal{M}$ is any permutation $g: E \to E$ of the ground set $E$ that carries flats to flats: for all $F$ in $\mathfrak{F}$ one has $g(F)$ in $\mathfrak{F}$. Let $G=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$ denote the group of all automorphisms of $\mathcal{M}$. Since such automorphisms respect the partial order via inclusion on $\mathfrak{F}$, they also preserve the rank function: $\mathop{\mathrm{\mathrm{rk}}}(g(F))=\mathop{\mathrm{\mathrm{rk}}}(F)$ for all $g$ in $G$ and $F$ in $\mathfrak{F}$. ## Chow Rings As defined in the Introduction, Feichtner and Yuzvinsky [@FY] introduced the Chow ring $A(\mathcal{M})$ of a matroid $\mathcal{M}$. Their goal was to model the Chow ring of DeConcini and Procesi's *wonderful compactification* of the complement $V \setminus \bigcup_{i=1}^n H_i$ of an arrangement of hyperplanes $H_1,\ldots,H_n$, inside a vector space $V$. Here $\mathcal{M}$ is the matroid in the dual space $V^*$ realized by linear forms $f_1,\ldots,f_n$ in $V^*$ with $H_i=\ker(f_i)$ for $i=1,2,\ldots,n$. **Definition 4**. *[\[Chow-ring-definition\]]{#Chow-ring-definition label="Chow-ring-definition"} The *Chow ring* $A(\mathcal{M})$ of a matroid $\mathcal{M}$ is the quotient $Z$-algebra $$A(\mathcal{M}) := S/(I+J)$$ where $S=\mathbb{Z}[x_F]$ is a polynomial ring having one variable $x_F$ for each nonempty flat $\varnothing \neq F \in \mathfrak{F}$, and where $I,J$ are the following ideals of $S$:* - *$I$ is generated by products $x_F x_{F'}$ for non-nested flats $F, F'$ (neither $F \subseteq F'$ nor $F' \subseteq F$),* - *$J$ is the ideal of $S$ generated by the linear elements $\displaystyle\sum_{a \in F \in \mathfrak{F}} x_F$ for each atom $a$ (= flat of rank $1$) in $\mathfrak{F}$.* The fact that the presentation of the Chow ring $A(\mathcal{M})$ only uses the information about the partial order on the lattice of flats $\mathfrak{F}$ has some consequences. - $A(\mathcal{M})$ depends only upon the associated *simple matroid* of $\mathcal{M}$. That is, one may assume $\mathcal{M}$ has - no *loops*, meaning that one has empty intersection $\varnothing = \bigcap_{F \in \mathfrak{F}} F$, and - no elements $i \neq j$ in $E$ which are *parallel* in the sense $\{F \in \mathfrak{F}: i \in F\}=\{F \in \mathfrak{F}: j \in F\}$. Thus without loss of generality, one may assume that $\mathcal{M}$ is a simple matroid throughout. - Any element $g$ in $G=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$ will send the generators of the ideals $I, J$ to other such generators. Thus $I+J$ is a $G$-stable ideal, and $G$ acts on $A(\mathcal{M})$. **Remark 5**. *A few remarks are in order, comparing the above definition of the Chow ring $A(\mathcal{M})$ to that given by Feichtner--Yuzvinsky [@FY], as well as the one used in Adiprasito-Huh-Katz [@AHK].* *Feichtner and Yuzvinsky in [@FY] consider Chow rings which are more general in two ways: they are associated to the more general class of atomic lattices (not necessarily geometric lattices), and they incorporate the choice of a subset of the lattice called a *building set*. Here we are both assuming that the lattice is geometric, so that it corresponds to a (simple) matroid $\mathcal{M}$, and we are also considering only the case of the *maximal building set*, which is the set of all nonempty flats in $\mathfrak{F}$. A reason for this choice is so that the entire group $G=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$ acts on the Chow ring, which requires the building set to be stable under $G$, which does not always occur.* *In [@AHK], they do consider certain other choices of nonmaximal building sets which we are ignoring. However, they also alter the presentation of $A(\mathcal{M})$ to eliminate the variable $x_E$ from the polynomial ring $S=\mathbb{Z}[x_F]_{\varnothing \subsetneq F \subseteq E}$. Specifically they write $A(\mathcal{M})=\hat{S}/(\hat{I} + \hat{J})$ where $\hat{S}=\mathbb{Z}[x_F]$ has a variable $x_F$ for each nonempty, *proper* flat $F$ with $\varnothing \subsetneq F \subsetneq E$. Then $\hat{I}$ is again the ideal generated by the monomials $x_{F} x_{F'}$ where $F, F'$ are not-nested, but now $\hat{J}$ is generated by the linear elements $$\displaystyle\sum_{a \in F \neq E\in \mathfrak{F}} x_F - \displaystyle\sum_{a' \in F \neq E\in \mathfrak{F}} x_F$$ for each pair of distinct atoms $a \neq a'$ in $\mathfrak{F}$. It is not hard to check that this presentation of $A(\mathcal{M})$ is equivalent to Definition [\[Chow-ring-definition\]](#Chow-ring-definition){reference-type="ref" reference="Chow-ring-definition"}: mutually inverse isomorphisms are induced by the map $\hat{S} \rightarrow S$ sending $x_F \longmapsto x_F$, and the backward map $S \rightarrow \hat{S}$ sending $x_F \longmapsto x_F$ for $F \neq E$ and $x_E \longmapsto -\displaystyle\sum_{a \in F \neq E\in \mathfrak{F}} x_F$ for any atom $a$ in $\mathfrak{F}$.* Note if one considers $S=\mathbb{Z}[x_F]$ as a graded $\mathbb{Z}$-algebra in which $\deg(x_F)=1$ for all nonempty flats $F$, then the ideals $I, J$ are generated by homogeneous elements: all quadratic generators for $I$, all linear generators for $J$. Hence the quotient $A(\mathcal{M})=S/(I+J)$ inherits the structure of a graded $\mathbb{Z}$-algebra $A(\mathcal{M})=\bigoplus_{k=0}^\infty A^k$. Since the action of $G=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$ on $A(\mathcal{M})$ preserves degrees, both $A(\mathcal{M})$ and each homogeneous component $A^k$ become $\mathbb{Z}G$-modules. It is not yet clear that $A(\mathcal{M})$ has only finitely many non-vanishing components $A^k$, nor that it vanishes beyond $A^r$ where $r=\mathop{\mathrm{\mathrm{rk}}}(\mathcal{M})-1$. For this and other purposes, we next consider Feichtner and Yuzvinsky's remarkable Gröbner basis for $I+J$ mentioned in the Introduction. ## In praise of the Feichtner--Yuzvinsky Gröbner basis We recall here one version of Gröbner basis theory with respect to a monomial order on a polynomial ring $\mathbb{Z}[x_1,\ldots,x_n]$, as used in [@FY]. **Definition 6**. *A linear order $\prec$ on the set of all monomials $\{ \mathbf{x}^\alpha:=x_1^{\alpha_1} \cdots x_n^{\alpha_n}\}$ in $\mathbb{Z}[x_1,\ldots,x_n]$ is called a *monomial ordering* if it is a *well-ordering* (every subset of monomials has a $\prec$-minimum element) and $\mathbf{x}^\alpha \prec \mathbf{x}^\beta$ implies $\mathbf{x}^\alpha \cdot \mathbf{x}^\gamma \prec \mathbf{x}^\beta \cdot \mathbf{x}^\gamma$ for all $\alpha,\beta,\gamma$ in $\{0,1,2,\ldots\}^n$.* **Example 7**. *After fixing a linear order on the variables $x_1 < \cdots < x_n$, the associated *lexicographic order* $\prec$ has $\mathbf{x}^\alpha \prec \mathbf{x}^\beta$ if there exists some $k=1,2,\ldots,n$ with $\alpha_1 = \beta_1, \alpha_2=\beta_2,\ldots,\alpha_{k-1} = \beta_{k-1}$, but $\alpha_k < \beta_k$.* **Definition 8**. *For $f=\sum_\alpha c_\alpha \mathbf{x}^\alpha$ in $\mathbb{Z}[x_1,\ldots,x_n]$, let $\mathrm{in}_\prec(f)=\mathbf{x}^\beta$ denote the $\prec$-largest monomial in $f$ having $c_\beta \neq 0$. Say $f$ is *$\prec$-monic* if the initial monomial $\mathrm{in}_\prec(f)=\mathbf{x}^\beta$ has its coefficient $c_\beta=\pm 1$.* *Having fixed a monomial order $\prec$, given an ideal $I \subset \mathbb{Z}[x_1,\ldots,x_n]$, say that a subset $\{g_1,\ldots,g_t\} \subset I$ is a *monic Gröbner basis* for $I$ (with respect to $\prec$) if each $g_i$ is $\prec$-monic, and every $f$ in $I$ has $\mathrm{in}_\prec(f)$ divisible by at least one of the intial monomials $\{ \mathrm{in}_\prec(g_1),\ldots,\mathrm{in}_\prec(g_t)\}$. Call $\mathbf{x}^\alpha$ a *standard monomial* for $\{g_1,\ldots,g_t\}$ with respect to $\prec$ if $\mathbf{x}^\alpha$ is divisible by none of $\{ \mathrm{in}_\prec(g_1),\ldots,\mathrm{in}_\prec(g_t)\}$.* It should be noted that some ideals $I \subset \mathbb{Z}[x_1,\ldots,x_n]$ have *no* monic Gröbner basis, e.g. $I=(2) \subset \mathbb{Z}[x]$. However, whenever $I$ *does* have a monic Gröbner basis, it has the following strong consequence, proven exactly as for Gröbner bases over field cofficients; see, e.g., Cox, Little, O'Shea [@CLO §2.5, 5.3]. **Proposition 9**. *If $\{g_1,\ldots,g_t\}$ is a monic Gröbner basis for $I \subset S=\mathbb{Z}[x_1,\ldots,x_n]$ with respect to $\prec$, then* - *$I=(g_1,\ldots,g_t)$, that is, $\{g_1,\ldots,g_t\}$ generate $I$ as an ideal.* - *The quotient ring $S/I$ is a free $\mathbb{Z}$-module, with a $\mathbb{Z}$-basis given by the standard monomials $\{x^\alpha\}$.* The following crucial result appears as [@FY Thm. 2]. To state it, define an *FY-monomial order* on $S=\mathbb{Z}[x_F]_{\varnothing \neq F \in \mathfrak{F}}$ to be any monomial order based on a linear order of the variables with $x_F > x_{F'}$ if $F\subsetneq F'$. **Theorem 10**. *Given a matroid $\mathcal{M}$ and any FY-monomial order on $S=\mathbb{Z}[x_F]_{\varnothing \neq F \in \mathfrak{F}}$, the ideal $I+J$ presenting $A(\mathcal{M})=S/(I+J)$ has a monic Gröbner basis $\{ g_{F,F'} \}$ indexed by $F \neq F'$ in $\mathfrak{F}$, with $g_{F,F'}$ and their initial terms $\mathrm{in}_\prec(g_{F,F'})$ as shown here:* *condition on $F \neq F'$ in $\mathfrak{F}$* *$g_{F,F'}$* *$\mathrm{in}_\prec(g_{F,F'})$* ---------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------- *$F, F'$ non-nested* *$x_F x_{F'}$* *$x_F x_{F'}$* *$\varnothing \neq F \subsetneq F'$* *$\displaystyle x_F \left( \sum_{\substack{F'' \in \mathfrak{F}: \\ F'' \supseteq F'}} x_{F''} \right)^{\mathop{\mathrm{\mathrm{rk}}}(F')-\mathop{\mathrm{\mathrm{rk}}}(F)}$* *$\displaystyle x_F \cdot x_{F'}^{\mathop{\mathrm{\mathrm{rk}}}(F')-\mathop{\mathrm{\mathrm{rk}}}(F)}$* *$\varnothing = F \subsetneq F'$* *$\displaystyle \left( \sum_{\substack{F'' \in \mathfrak{F}:\\ F'' \supseteq F'}} x_{F''}\right)^{\mathop{\mathrm{\mathrm{rk}}}(F')}$* *$\displaystyle x_{F'}^{\mathop{\mathrm{\mathrm{rk}}}(F')}$* **Corollary 11**. *([@FY Cor. 1]) [\[cor: mon_basis\]]{#cor: mon_basis label="cor: mon_basis"} For a matroid $\mathcal{M}$ of rank $r+1$, the Chow ring $A(\mathcal{M})$ has these properties:* - *$A(\mathcal{M})$ is free as a $\mathbb{Z}$-module, with $\mathbb{Z}$-basis given by the set of FY-monomials $$\label{FY-monomials-definition} \mathrm{FY}:=\{ x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell}: (\varnothing =:F_0) \subsetneq F_1 \subsetneq F_2 \subsetneq \cdots \subsetneq F_\ell, \text{ in }\mathfrak{F}, \text{ and } m_i \leq \mathop{\mathrm{\mathrm{rk}}}(F_i) - \mathop{\mathrm{\mathrm{rk}}}(F_{i-1})-1\}.$$* - *$A(\mathcal{M})$ vanishes in degrees strictly above $r$, that is, $A(\mathcal{M})=\bigoplus_{k=0}^r A^k$.* - *$A^r$ has $\mathbb{Z}$-basis $\{x_E^r\}$, and hence one has a $\mathbb{Z}$-module isomorphism $\deg: A^r \longrightarrow \mathbb{Z}$ sending $x_E^r \longmapsto 1$.* *Proof.* Assertion (i) follows from Theorem [Theorem 10](#FY-GB-theorem){reference-type="ref" reference="FY-GB-theorem"} after checking that the FY-monomials in [\[FY-monomials-definition\]](#FY-monomials-definition){reference-type="eqref" reference="FY-monomials-definition"} are exactly the standard monomials for the $\prec$-monic Gröbner basis $\{g_{F,F'}\}$, that is, the monomials divisible by no $\mathrm{in}_\prec( g_{F,F'} )$. For assertion (ii), note that the typical FY-monomial $x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell}$, has total degree $$\sum_{i=1}^\ell m_i \leq \sum_{i=1}^\ell (\mathop{\mathrm{\mathrm{rk}}}(F_i)-\mathop{\mathrm{\mathrm{rk}}}(F_{i-1})-1) = \mathop{\mathrm{\mathrm{rk}}}(F_\ell) - \ell \leq (r+1) - 1 = r.$$ For assertion (iii), note equality occurs only if $\ell=1$ and $F_\ell=E$, in which case the FY-monomial is $x_E^r$. ◻ Note that for any matroid automorphism $g$, the fact that $\mathop{\mathrm{\mathrm{rk}}}(g(F))=\mathop{\mathrm{\mathrm{rk}}}(F)$ for every flat $F$ in $\mathfrak{F}$ implies that $g$ sends any FY-monomial to another FY-monomial: $$x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell} \,\, \overset{g}{\longmapsto} \,\, x_{g(F_1)}^{m_1} x_{g(F_2)}^{m_2} \cdots x_{g(F_\ell)}^{m_\ell}.$$ This has an immediate corollary, inspired by work of H.-C. Liao on Boolean matroids [@Liao Thm. 2.5]. **Corollary 12**. *For any matroid $\mathcal{M}$, the group $G=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$ permutes the set $\mathrm{FY}$, as well as its subset of degree $k$ monomials $\mathrm{FY}^k \subset \mathrm{FY}$. Consequently, the $\mathbb{Z}G$-modules on the Chow ring $A(\mathcal{M})$ and each of its homogeneous components $A^k$ lift to $G$-permutation representations on $\mathrm{FY}$ and each $\mathrm{FY}^k$.* **Remark 13**. *It is rare to find families of ideals $I$ inside polynomial rings $S$ that are stable under a finite group $G$ acting on $S$, and which *also have a $G$-stable Gröbner basis* $\{g_i\}$ with respect to some monomial order $\prec$. This occurs, for example, with the *Hibi rings* studied in [@Hibi], and the more general $P$-partition rings studied by Feray and the third author in [@FerayR Thm. 6.3]. More often, when starting with a $G$-stable ideal $I$, passing to an intial ideal destroys the $G$-symmetry. One then usually needs alternative techniques to work with the quotient $S/I$, as discussed, e.g., by Faugére [@Faugere] and Sturmfels [@Sturmfels §2.6].* **Remark 14**. *Although the $\mathbb{Z}G$-module structure on $A(\mathcal{M})$ and $A^k$ are canonical, their lifts to permutation representations on the sets $\mathrm{FY}$ and $\mathrm{FY}^k$ are not. In general, one can have two different $G$-permutation representations on sets $X, X'$ (that is, with no $G$-equivariant set bijection $X \overset{\sim}{\longrightarrow} X$) but with a $\mathbb{Z}G$-module isomorphism $\mathbb{Z}[X] \cong \mathbb{Z}[X']$; see, e.g., Conlon [@Conlon] and Scott [@Scott].* ## The Kähler package {#AHK-section} The following theorem on the Kähler package for $A(\mathcal{M})$ compiles some of the main results of the work of Adiprasito, Huh and Katz [@AHK]. **Theorem 15**. *For a matroid $\mathcal{M}$ of rank $r+1$, the Chow ring $A(\mathcal{M})$ satisfies the Kähler package:* - *(Poincaré duality)\ For every $k \leq \frac{r}{2}$, one has a perfect $\mathbb{Z}$-bilinear pairing $$\begin{aligned} A^k \times A^{r-k} &\longrightarrow \mathbb{Z}\\ (a,b) &\longmapsto \deg(a \cdot b)\end{aligned}$$ that is, $b \longmapsto \varphi_b(-)$ defined by $\varphi_b(a)=\deg(a \cdot b)$ is a $\mathbb{Z}$-linear isomorphism $A^{r-k} \cong \mathrm{Hom}_\mathbb{Z}(A^k,\mathbb{Z})$.* - - *(Hard Lefschetz)\ Tensoring over $\mathbb{Z}$ with $\mathbb{R}$, the (real) Chow ring $A_\mathbb{R}(\mathcal{M})=\sum_{k=0}^r A^k_\mathbb{R}$ contains **Lefschetz elements** $\omega$ in $A^1_\mathbb{R}$, meaning that $a \mapsto a \cdot \omega^{r-2k}$ is an $\mathbb{R}$-linear isomorphism $A^k_\mathbb{R}\rightarrow A^{r-k}_\mathbb{R}$ for $k \leq \frac{r}{2}$.* *In particular, multiplication by $\omega$ is an injection $A^k_\mathbb{R}\rightarrow A_\mathbb{R}^{k+1}$ for $k < \frac{r}{2}$.* - - *(Hodge-Riemann-Minkowski inequalities)\ The Lefschetz elements $\omega$ define quadratic forms $a \longmapsto (-1)^k \deg(a \cdot \omega^{r-2k} \cdot a)$ on $A^k_\mathbb{R}$ that become positive definite upon restriction to the kernel of the map $A^k_\mathbb{R}\longrightarrow A^{r-k+1}_\mathbb{R}$ that sends $a \longmapsto a \cdot \omega^{r-2k+1}$.* In fact, they show that one obtains a Lefschetz element $\omega$ whenever $\omega=\sum_{\varnothing \neq F \in \mathfrak{F}} c_F x_F$ has coefficients $c_F$ coming from restricting to $\mathfrak{F}$ any function $A \mapsto c_A$ that maps $2^E \rightarrow \mathbb{R}$ and satisfies these two properties: - the *strict submodular inequality* $c_{A} + c_{B} > c_{A \cap B} + c_{A \cup B}$ for all $A \neq B$, and - $c_{\varnothing}=c_E=0$. This has consequences for $G$ acting on $A(\mathcal{M})$ and each $A^k$. The first will be refined by Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}(i). **Corollary 16**. *For any matroid $\mathcal{M}$, one has an isomorphism of $\mathbb{Z}G$-modules $A^{r-k} \rightarrow A^k$ for each $k \leq \frac{r}{2}$.* *Proof.* Corollary [\[cor: mon_basis\]](#cor: mon_basis){reference-type="ref" reference="cor: mon_basis"}(iii) shows that $A^r$ has only one $\mathbb{Z}$-basis element $x_E^r$, fixed by every $g$ in $G$, so the degree map $\deg: A^r \longrightarrow \mathbb{Z}$ is $G$-equivariant for the trivial $G$-action on the target $\mathbb{Z}$. Thus the Poincaré duality isomorphism $A_{r-k} \longrightarrow \mathrm{Hom}_\mathbb{Z}(A_k,\mathbb{Z})$, sending $b \longmapsto \varphi_b(-)$ with $\varphi_b(a)=\deg(a \cdot b)$, is also $G$-equivariant. It only remains to exhibit a $G$-equivariant isomorphism $\mathrm{Hom}_\mathbb{Z}(A_k,\mathbb{Z}) \rightarrow A_k$. To this end, use Corollary [Corollary 12](#Chow-ring-carries-perm-reps-cor){reference-type="ref" reference="Chow-ring-carries-perm-reps-cor"} to pick a $\mathbb{Z}$-basis $\{e_i\}$ permuted by $G$, so that each element $g$ in $G$ acts by a permutation matrix $P(g)$ in this basis. Letting $\{f_i\}$ be the dual $\mathbb{Z}$-basis for $\mathrm{Hom}_\mathbb{Z}(A_k,\mathbb{Z})$, one finds that $g$ acts via the matrix $P(g^{-1})^T = (P(g)^{-1})^T=P(g)$, since $P(g)$ is a permutation matrix. Hence the map $e_i \longmapsto f_i$ is such a $G$-equivariant isomorphism $\mathrm{Hom}_\mathbb{Z}(A_k,\mathbb{Z}) \rightarrow A_k$. ◻ The next consequence will be refined by Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}(ii). **Corollary 17**. *One has $\mathbb{R}G$-module maps $A^k_\mathbb{R}\rightarrow A_\mathbb{R}^{k+1}$ which are injective for $k < \frac{r}{2}$.* *Proof.* There exist Lefschetz elements $\omega \in A^1_\mathbb{R}$ which are $G$-fixed, such as those exhibited on [@AHK p. 384] having $\omega=\sum_F c_F x_F$ with $c_F=|F| \cdot |E \setminus F|$. Multiplication by such $\omega$ give the asserted $\mathbb{R}G$-module injections. ◻ # Proof of Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"} {#main-theorem-section} We recall the statement of the theorem, involving the FY-monomial $\mathbb{Z}$-basis for $A(\mathcal{M})$ in Corollary [\[cor: mon_basis\]](#cor: mon_basis){reference-type="ref" reference="cor: mon_basis"}: $$\mathrm{FY}:=\{ x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell}: (\varnothing =:F_0) \subsetneq F_1 \subsetneq F_2 \subsetneq \cdots \subsetneq F_\ell \text{ in }\mathfrak{F}, \text{ and } m_i \leq \mathop{\mathrm{\mathrm{rk}}}(F_i) - \mathop{\mathrm{\mathrm{rk}}}(F_{i-1})-1\}$$ This also means that the FY-monomials $\mathrm{FY}^k$ of degree $k$ form a $\mathbb{Z}$-basis for $A^k$ for each $k=0,1,2,\ldots,r$. .1in **Theorem  [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}** *For every matroid $\mathcal{M}$ of rank $r+1$, there exist* - $G$-equivariant bijections $\pi: \mathrm{FY}^k \overset{\sim}{\longrightarrow} \mathrm{FY}^{r-k}$ for $k \leq \frac{r}{2}$, and - $G$-equivariant injections $\lambda: \mathrm{FY}^k \hookrightarrow \mathrm{FY}^{k+1}$ for $k < \frac{r}{2}$. .1in We offer two (related) proofs, the first slightly more conceptual, the second with more explicit maps. ## First proof This proof organizes the FY-monomials according to the fibers of a map $\mathrm{supp}_+: \mathrm{FY}\rightarrow 2^\mathfrak{F}.$ **Definition 18**. *For an FY-monomial $a=x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell}$, define its *extended support* $\mathrm{supp}_+(a) \subset \mathfrak{F}$ by $$\mathrm{supp}_+(a):=\{F_1,\ldots,F_\ell\} \cup \{E\} =\begin{cases} \{F_1,\ldots,F_\ell\} \cup \{E\} & \text{ if }F_\ell \subsetneq E,\\ \{F_1,\ldots,F_\ell\} &\text{ if }F_\ell=E. \end{cases}$$ Define a partial order $<_+$ on the FY-monomials in which $a <_+ b$ if $a$ divides $b$ and $\mathrm{supp}_+(a)=\mathrm{supp}_+(b)$.* For integers $p < q$, let $[p,q]$ denote the usual linear order on the integers $\{p,p+1,\ldots,q-1,q\}$. Given a sequence of such pairs $p_i < q_i$ for $i=1,2,\ldots,m$, let $$\label{typical-product-of-chains} \prod_{i=1}^n [p_i,q_i] =[p_1,q_1] \times [p_2,q_2] \times \cdots \times [p_m,q_m]$$ denote their Cartesian product, partially ordered componentwise. **Proposition 19**. *For any nested flag $\{F_1 \subsetneq \cdots \subsetneq F_\ell \subsetneq E\}$ in $\mathfrak{F}$ containing $E$, with convention $$\begin{aligned} F_0&:=\varnothing,\\ F_{\ell+1}&:=E \end{aligned}$$ the fiber $\mathrm{supp}_+^{-1}\{F_1,\ldots,F_\ell,E\}$ is the set of monomials $\{ x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell} x_E^{m_{\ell+1}} \}$ satisfying these inequalities: $$\begin{aligned} &1 \leq m_{i} \leq \mathop{\mathrm{\mathrm{rk}}}(F_i)-\mathop{\mathrm{\mathrm{rk}}}(F_{i-1})-1 &\text{ for }i=1,2,\ldots,\ell\\ &0 \leq m_{\ell+1} \leq \mathop{\mathrm{\mathrm{rk}}}(E)-\mathop{\mathrm{\mathrm{rk}}}(F_{\ell})-1 =r-\mathop{\mathrm{\mathrm{rk}}}(F_\ell).& \end{aligned}$$* *Consequently, the minimum and maximum degree of monomials in $\mathrm{supp}_+^{-1}\{F_1,\ldots,F_\ell,E\}$ are $\ell$ and $r-\ell$, and one has a poset isomorphism $$\begin{array}{rcl} (\mathrm{supp}_+^{-1}\{F_1,\ldots,F_\ell,E\},<_+) &\longrightarrow & \displaystyle \prod_{i=1}^\ell [1,\mathop{\mathrm{\mathrm{rk}}}(F_i)-\mathop{\mathrm{\mathrm{rk}}}(F_{i-1})-1] \,\, \times \,\, [0,r-\mathop{\mathrm{\mathrm{rk}}}(F_{\ell})]\\ & & \\ x_{F_1}^{m_1} x_{F_2}^{m_2} \cdots x_{F_\ell}^{m_\ell} x_E^{m_{\ell+1}} &\longmapsto & (m_1,m_2,\ldots,m_\ell,m_{\ell+1}). \end{array}$$* *Proof.* Most assertions of the proposition are immediate from the definition of $<_+$ and the map $\mathrm{supp}_+$. The minimum and maximum degrees of monomials in $\mathrm{supp}_+^{-1} \{F_1,\ldots,F_\ell,E\})$ are achieved by $$\begin{aligned} \deg(x_{F_1}^1 x_{F_2}^1 \cdots x_{F_\ell}^1 x_E^0)&=\ell\\ \deg\left(\prod_{i=1}^\ell x_{F_i}^{\mathop{\mathrm{\mathrm{rk}}}(F_i)-\mathop{\mathrm{\mathrm{rk}}}(F_{i})-1} \cdot x_E^{\mathop{\mathrm{\mathrm{rk}}}(E)-\mathop{\mathrm{\mathrm{rk}}}(F_{\ell})-1)}\right) &=\sum_{i=1}^{\ell+1} (\mathop{\mathrm{\mathrm{rk}}}(F_i)-\mathop{\mathrm{\mathrm{rk}}}(F_{i-1}-1) =\mathop{\mathrm{\mathrm{rk}}}(E)-(\ell+1)\ = r-\ell. \qedhere \end{aligned}$$ ◻ The first proof of Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"} stems from the observation that all products of chains, as in [\[typical-product-of-chains\]](#typical-product-of-chains){reference-type="eqref" reference="typical-product-of-chains"}, have *symmetric chain decompositions*, which can then be pulled back to each fiber $\mathrm{supp}_+^{-1}\{F_1,\ldots,F_\ell,E\}.$ **Definition 20**. *A *symmetric chain decomposition (SCD)* of a finite ranked poset $P$ of rank $r$ is a disjoint decomposition $P=\bigsqcup_{i=1}^t C_i$ in which each $C_i$ is a totally ordered subset containing one element of each rank $\{\rho_i,\rho_i+1,\ldots,r-\rho_i-1,r-\rho_i\}$ for some $\rho_i \in \{0,1,2,\ldots,\lfloor \frac{r}{2}\rfloor \}$.* It is not hard to check that when posets $P_1, P_2$ each have an SCD, then so does their Cartesian product. In particular, all products of chains have an SCD; see, e.g., Anderson [@Anderson §3.1]. *Fix one such SCD for each product poset in [\[typical-product-of-chains\]](#typical-product-of-chains){reference-type="eqref" reference="typical-product-of-chains"}, once and for all,* and use the isomorphisms from Proposition [Proposition 19](#iso-to-product-of-chains){reference-type="ref" reference="iso-to-product-of-chains"} to induce an SCD on each fiber $\mathrm{supp}_+^{-1}\{F_1,\ldots,F_\ell,E\}.$ An important point for the proof is that the structure of these symmetric chains will depend only upon the rank sets $\{\mathop{\mathrm{\mathrm{rk}}}(F_i)\}_{i=1}^\ell$. **Example 21**. *[\[extended-support-map-example\]]{#extended-support-map-example label="extended-support-map-example"} Assume $\mathcal{M}$ has $\mathop{\mathrm{\mathrm{rk}}}(E)=10=r+1$ with $r=9$, and one has a pair of nested flats $F \subset F'$ with $\mathop{\mathrm{\mathrm{rk}}}(F)=3, \mathop{\mathrm{\mathrm{rk}}}(F')=7$. Then the poset $\mathrm{supp}_+^{-1}\{F,F',E\}$ and one choice of SCD for it look as follows:* *First proof of Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}..* Let $a$ be any FY-monomial $a$ with $k:=\deg(a)$. Then $a$ lies in a unique fiber $\mathrm{supp}_+^{-1}\{F_1,\ldots,F_\ell,E\}$ of the map $\mathrm{supp}_+$, and on a unique chain $C_i$ in our fixed SCD of this fiber. Writing $$\label{typical-SCD-chain} C_i=\{ a_{\rho} \lessdot a_{\rho+1} \lessdot \cdots \lessdot a_{r-\rho-1} \lessdot a_{r-\rho} \},$$ where $\deg(a_j)=j$ for $j=\rho,\rho+1,\ldots,r-\rho$, so that $a=a_k$, then define the bijection $\pi$ and maps $\lambda$ via $$\begin{aligned} \pi(a)&:=a_{r-k},\\ \lambda(a)&:=a_{k+1} \text{ if }k < \frac{r}{2}. \end{aligned}$$ To see that $\pi,\lambda$ are equivariant, note that having fixed $\{F_1,\ldots,F_\ell,F_{\ell+1}=E\}$, both maps $f=\pi, \lambda$ will send a monomial of the form $a=\prod_{i=1}^{\ell+1} x_{F_i}^{m_i}$ with $m_i \geq 1$ to one of the form $f(a)=\prod_{i=1}^{\ell+1} x_{F_i}^{m'_i}$ in such a way that the exponents $(m'_1,\ldots,m'_\ell,m'_{\ell+1})$ are determined uniquely as a function of the pair of data $$\left( \,\, (m_1,\ldots,m_{\ell},m_{\ell+1}) \,\, , \,\, \{\mathop{\mathrm{\mathrm{rk}}}(F_i)\}_{i=1}^\ell \,\, \right).$$ Thus for any $g$ in $G$, since $\{\mathop{\mathrm{\mathrm{rk}}}(g(F_i))\}_{i=1}^\ell =\{\mathop{\mathrm{\mathrm{rk}}}(F_i)\}_{i=1}^\ell$, either of the maps $f=\pi, \lambda$ will send $$g(a)=\prod_{i=1}^{\ell+1} x_{g(F_i)}^{m_i} \quad \longmapsto \quad f(g(a))= \prod_{i=1}^{\ell+1} x_{g(F_i)}^{m'_i} =g(f(a)). \qedhere$$ ◻ ## Proof via parenthesis pairing This proof borrows an idea from the famous *parenthesis-pairing* SCD of Boolean algebras due to Greene and Kleitman [@GK]. We begin with a two-step encoding of FY-monomials within a fiber of the map $\mathrm{supp}_+$. **Definition 23**. *Consider all FY-monomials $a=x_{F_1}^{m_1} \cdots x_{F_\ell}^{m_\ell} x_E^{m_{\ell+1}}$ having a fixed extended support set $\mathrm{supp}_+(a)=\{F_1 \subsetneq \cdots \subsetneq F_\ell \subsetneq F_{\ell+1}=E\}$, so $m_1,\ldots,m_\ell \geq 1$ and $m_{\ell+1} \geq 0$. In the first step, encode such monomials $a$ via a sequence $\mathfrak{D}(a)$ of length $r$ in three symbols $\times, \bullet$, and a blank space, defined as follows:* - *$\mathfrak{D}(a)$ has $\bullet$ in the positions $\{\mathop{\mathrm{\mathrm{rk}}}(F_1),\ldots,\mathop{\mathrm{\mathrm{rk}}}(F_\ell)\}$.* - *$\mathfrak{D}(a)$ has $\times$ in the first consecutive $m_i$ positions to the left of $\mathop{\mathrm{\mathrm{rk}}}(F_i)$ for each $i=1,2,\ldots,\ell,\ell+1$.* - *$\mathfrak{D}(a)$ has a blank space in the remaining positions.* ***Example 22**. *Continuing with the matroid $\mathcal{M}$ and its flats $F\subset F'$ as discussed in Example [\[extended-support-map-example\]](#extended-support-map-example){reference-type="ref" reference="extended-support-map-example"}. Here the monomials lie in the fiber $\mathrm{supp}_+^{-1}\{F,F',E\}$ where $\{\mathop{\mathrm{\mathrm{rk}}}(F),\mathop{\mathrm{\mathrm{rk}}}(F'),\mathop{\mathrm{\mathrm{rk}}}(E)\}=\{3,7,10\}$, so $r=9$, and the positions $\{\mathop{\mathrm{\mathrm{rk}}}(F),\mathop{\mathrm{\mathrm{rk}}}(F')\}=\{3,7\}$ are shown in green. The monomial $x_F x_{F'}^2$ gets encoded as $$\begin{array}{ccccccccc} 1 & 2 & {\color{green} 3} & 4 & 5 & 6 & {\color{green} 7} & 8 & 9 \\ \\ & \times & {\color{green} \bullet} & & \times & \times & {\color{green} \bullet} & & %) & ( & ) & ) & ( & ( & ) & ) & ) \end{array}$$** Note that one can recover $a$ from $\mathrm{supp}_+(a)=\{F_1,\ldots,F_\ell, E\}$ and $\mathfrak{D}(a)$, since $m_i$ can be read off as the number of $\times$ in $\mathfrak{D}(a)$ between positions $\mathop{\mathrm{\mathrm{rk}}}(F_{i-1})$ and $\mathop{\mathrm{\mathrm{rk}}}(F_i)$, with usual conventions $F_0=\varnothing, F_{\ell+1}:=E$. **Definition 24**. *The second step encodes $\mathfrak{D}(a)$ as a length $r$ parenthesis sequence in $\{(,)\}^r$, having* - *a right parenthesis "$)$\" in the positions of each $\bullet$ and each blank space, and* - *a left parenthesis "(\" in the positions of the $\times$.* **Example 25**. *Continuing Example [Example 22](#dd-example){reference-type="ref" reference="dd-example"}, the diagram $\mathfrak{D}(x_F x_{F'}^2)$ gets encoded as the following sequence of parentheses: $$\begin{array}{ccccccccc} 1 & 2 & {\color{green} 3} & 4 & 5 & 6 & {\color{green} 7} & 8 & 9 \\ \\ & \times & {\color{green} \bullet} & & \times & \times & {\color{green} \bullet} & & \\ ) & ( & ) & ) & ( & ( & ) & ) & ) \end{array}$$* Note that one can recover $\mathfrak{D}(a)$ from this $\{(,)\}^r$ sequence as follows: - $\mathfrak{D}(a)$ has $\times$ occurring in the positions of the left parentheses, and - $\mathfrak{D}(a)$ has the $\bullet$ occurring exactly in the positions of the right parenthesis in a consecutive pair $()$, while blank spaces occur in the position of all other right parentheses. *Second proof of Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}..* Given an FY-monomial $a$ in $\mathrm{FY}^k$ with $k \leq \frac{r}{2}$, we will use its parenthesis sequence in $\{(,)\}^r$ to place $a$ within a chain of monomials as in [\[typical-SCD-chain\]](#typical-SCD-chain){reference-type="eqref" reference="typical-SCD-chain"}, of the form $$C_i=\{ a_{\rho} \lessdot a_{\rho+1} \lessdot \cdots \lessdot a_{r-\rho-1} \lessdot a_{r-\rho} \},$$ where $\deg(a_j)=j$ for $j=\rho,\rho+1,\ldots,r-\rho$, so that $a=a_k$. To this end, define the set of *paired parentheses* in $a$ (shown underlined in Example [\[example-re-revisited\]](#example-re-revisited){reference-type="ref" reference="example-re-revisited"}) by including all consecutive pairs $()$, and after removing these consecutive pairs, including the new $()$ pairs which have become consecutive, repeating this recursively. After removing some number $\rho$ of pairs $()$ via this pairing process, the process ends when one reaches a sequence of $r -2 \rho$ remaining unpaired parentheses of this form: $$\label{unpaired-parentheses} \underbrace{)) \cdots ))}_{k-\rho} \underbrace{{\color{orange}(( \cdots ((}}_{r-\rho-k}.$$ The monomials in the chain $C_i$ are defined to be those whose set of paired parentheses agree exactly with those of $a$, both in their postitions, and left-right pairing structure-- see the underlined parentheses in Example [\[example-re-revisited\]](#example-re-revisited){reference-type="ref" reference="example-re-revisited"}. As in the first proof of Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}, one then defines the bijections $\pi$ and maps $\lambda$ via $$\begin{aligned} \pi(a)&:=a_{r-k},\\ \lambda(a)&:=a_{k+1} \text{ if }k < \frac{r}{2}. \end{aligned}$$ In other words, both maps $\lambda$ and $\pi$ when applied to $a$ will keep all of the paired parentheses fixed, but - $\lambda$ changes the leftmost unpaired left parenthesis ")\" into an unpaired right parenthesis "(\", and - $\pi$ swaps the numbers $k-\rho$ and $r-\rho-k$ of unpaired right and left parentheses in [\[unpaired-parentheses\]](#unpaired-parentheses){reference-type="eqref" reference="unpaired-parentheses"}. The equivariance of these two maps $\pi, \lambda$ is argued exactly as in the first proof of the theorem. ◻ **Example 26**. *[\[example-re-revisited\]]{#example-re-revisited label="example-re-revisited"} Here is an example of a symmetric chain from Example [\[extended-support-map-example\]](#extended-support-map-example){reference-type="ref" reference="extended-support-map-example"}, explained via this two-step encoding with $\mathfrak{D}(a)$ and $\{(,)\}^r$-sequences, as we have seen preceding Examples [Example 22](#dd-example){reference-type="ref" reference="dd-example"} and  [Example 25](#dd-to-()-example){reference-type="ref" reference="dd-to-()-example"}. Paired parentheses are underlined, and are fixed throughout the chain. Moving up the chain, unpaired right parentheses change one-by-one to left parentheses (depicted orange here), in order from right to left:* *$$\begin{array}{ccccccccccl} & 1 & 2 & {\color{green}3} & 4 & 5 & 6 & {\color{green}7} & 8 & 9 \\ & & & & & & & & & \\ x_F^2 x_{F'}^3 x_E & \times & \times & {\color{green}\bullet} & \times & \times & \times &{\color{green}\bullet} & & \times\\ & {\color{orange}(} & \underline{(} & \underline{)} & {\color{orange}(} & \underline{(} & \underline{(} & \underline{)} & \underline{)} & {\color{orange}(}\\ & & & & & & & & & \\ \uparrow & & & & & & & & & \\ & & & & & & & & & \\ x_F x_{F'}^3 x_E & & \times & {\color{green}\bullet} & \times & \times & \times &{\color{green}\bullet} & & \times\\ & ) & \underline{(} & \underline{)} & {\color{orange}(} & \underline{(} & \underline{(} & \underline{)} & \underline{)} & {\color{orange}(}\\ & & & & & & & & & \\ \uparrow & & & & & & & & & \\ & & & & & & & & & \\ x_F x_{F'}^2 x_E & & \times & {\color{green}\bullet} & & \times & \times &{\color{green}\bullet} & & \times\\ & ) & \underline{(} & \underline{)} & ) & \underline{(} & \underline{(} & \underline{)} & \underline{)} & {\color{orange}(}\\ & & & & & & & & & \\ \uparrow & & & & & & & & & \\ & & & & & & & & & \\ x_F x_{F'}^2 & & \times & {\color{green}\bullet} & & \times & \times &{\color{green}\bullet} & & \\ & ) & \underline{(} & \underline{)} & ) & \underline{(} & \underline{(} & \underline{)} & \underline{)} & )\\ \end{array}$$* *Here $r=9$ and the number of parenthesis pairs is $\rho=3$, so that this chain $C_i$ is symmetrically placed within the degrees of $A(\mathcal{M})$, containing monomials of degrees $[\rho,r-\rho]=[3,6]=\{3,4,5,6\}$ out of the list of possible degrees $[0,r]=[0,9]=\{0,1,2,\mathbf{3,4,5,6},7,8,9\}$.* # Conjectures on equivariant and Burnside ring inequalities {#conjectures-section} [\[rep-theoreric-inequalities-section\]]{#rep-theoreric-inequalities-section label="rep-theoreric-inequalities-section"} We have mentioned that the unimodality statement [\[unimodality\]](#unimodality){reference-type="eqref" reference="unimodality"}, asserting for $k < \frac{r}{2}$ that one has $$a_k \leq a_{k+1},$$ is weaker than the statement in Corollary [Corollary 17](#AHK-equivariant-Hard-Lefschetz){reference-type="ref" reference="AHK-equivariant-Hard-Lefschetz"} asserting that there are injective $\mathbb{R}G$-module maps $$A^k_\mathbb{R}\rightarrow A_\mathbb{R}^{k+1},$$ which is weaker than Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}(ii) asserting that there are injective $G$-equivariant maps of the $G$-sets $$\mathrm{FY}^k \hookrightarrow \mathrm{FY}^{k+1}.$$ In this section, we wish to consider not only unimodality for $(a_0,a_1,\ldots,a_r)$, but other properties like *log-concavity*, *the Pólya frequency property*, and how to similarly lift them to statements regarding $\mathbb{R}G$-modules and $G$-permutation representations. In phrasing this, it helps to consider certain algebraic objects. ## Virtual character rings and Burnside rings **Definition 27**. *(Virtual character ring) For a finite group $G$, its *virtual (complex) character ring* $R_\mathbb{C}(G)$ is the free $\mathbb{Z}$-submodule of the ring of (conjugacy) class functions $\{f: G \rightarrow \mathbb{C}\}$ with pointwise addition and multiplication, having as a $\mathbb{Z}$-basis the irreducible complex characters $\{ \chi_1,\ldots,\chi_M\}$, where $M$ is the number of conjugacy classes of $G$. Thus every virtual character $\chi$ in $R_\mathbb{C}(G)$ has a unique expansion $\chi = \sum_{i=1}^M a_i \chi_i$ for some $a_i \in \mathbb{Z}$. If $a_i \geq 0$ for $i=1,2,\ldots,M$, call $\chi$ a *genuine character*, and write $\chi \geq_{R_\mathbb{C}(G)} 0$. Similarly, we write $\chi \geq_{R_\mathbb{C}(G)} \chi'$ when $\chi-\chi' \geq_{R_\mathbb{C}(G)} 0$.* **Definition 28**. *(Burnside ring) For a finite group $G$, to define its *Burnside ring* $B(G)$ one starts with a free $\mathbb{Z}$-module having as basis the isomorphism classes $[X]$ of finite $G$-sets $X$. Then $B(G)$ is the quotient $\mathbb{Z}$-module that mods out by the span of all elements $[X \sqcup Y] - ([X]+[Y])$. Multiplication in $B(G)$ is induced from the rule $[X] \cdot [Y] = [X \times Y]$. It turns out that $B(G)$ has a $\mathbb{Z}$-basis $\{ [G/G_i] \}_{i=1}^N$ as $G_1,\ldots,G_N$ run through representatives of the $G$-conjugacy classes of subgroups of $G$. Thus every element $b$ of $B(G)$ has a unique expansion $b = \sum_{i=1}^N a_i [G/G_i]$ for some $a_i \in \mathbb{Z}$. If $a_i \geq 0$ for $i=1,2,\ldots,N$, call $b$ a *genuine permutation representation*, and write $b \geq_{B(G)} 0$. Similarly, write $b \geq_{B(G)} b'$ when $b-b' \geq_{B(G)} 0$.* Note that there are natural ring maps $$\begin{aligned} B(G) &\longrightarrow R_\mathbb{C}(G),\\ R_\mathbb{C}(G) &\longrightarrow \mathbb{Z}. \end{aligned}$$ The first map sends the class $[X]$ of a $G$-set $X$ to the character $\chi_{\mathbb{C}[X]}$ of its $G$-permutation representation $\mathbb{C}[X]$, having character values $\chi_{\mathbb{C}[X]}(g)=\#\{x \in X:g(x)=x\}$. The second map sends a virtual character $\chi$ to its value $\chi(e)$ on the identity $e$ of $G$. These maps carry genuine elements $b \geq_{B(G)} 0$ in $B(G)$ to genuine characters $\chi \geq_{R_\mathbb{C}(G)} 0$ which are then carried to nonnegative integers $\mathbb{Z}_{\geq 0}$. In this way, inequalities in $B(G)$ lift inequalities in $R_\mathbb{C}(G)$, which lift inequalities in $\mathbb{Z}$. **Example 29**. *We saw that the unimodality inequality [\[unimodality\]](#unimodality){reference-type="eqref" reference="unimodality"} $a_k \leq_\mathbb{Z}a_{k+1}$ lifts to the inequality $A_\mathbb{R}^k \leq_{R_\mathbb{C}(G)} A_\mathbb{R}^{k+1}$ in Corollary [Corollary 17](#AHK-equivariant-Hard-Lefschetz){reference-type="ref" reference="AHK-equivariant-Hard-Lefschetz"}, which lifts to the inequality $[\mathrm{FY}^k] \leq_{B(G)} [\mathrm{FY}^{k+1}]$ in Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}(ii).* It should also be noted that, just as one can multiply inequalities in $\mathbb{Z}$ like $a< b$ and $c < d$ to get new inequalities $ac < bd$, the same works in $R_\mathbb{C}(G)$ and in $B(G)$. This is because $\chi, \chi' \geq_{R_\mathbb{C}(G)} 0$ implies $\chi \cdot \chi' \geq_{R_\mathbb{C}(G)} 0$, and similarly $b, b' \geq_{B(G)} 0$ implies $b \cdot b' \geq_{B(G)} 0$. ## PF sequences and log-concavity For a sequence of *positive* real numbers $(a_0,a_1,\ldots,a_r)$, the property of *unimodality* lies at the bottom of a hierarchy of concepts $$\label{PF-hierarchy} \begin{array}{rccccccccccccccc} \text{unimodal} &\Leftarrow& PF_2 &\Leftarrow& &PF_3& &\Leftarrow& &PF_4& &\Leftarrow& \cdots& \Leftarrow& &PF_\infty\\ & & \Vert & & & & & & & & & & & & &\Vert\\ & & \text{(strongly) log-concave} & & & & & & & & & & & & &PF\\ \end{array}$$ which we next review, along with their equivariant and Burnside ring extensions. For background on the non-equivariant versions, see Brenti [@Brenti] and Stanley [@Stanley-log-concavity]. For the equivariant versions, see Gedeon, Proudfoot and Young [@GPY], Matherne, Miyata, Proudfoot and Ramos [@MMPR], Gui [@Gui2022], Gui and Xiong [@GuiXiong], and Li [@Li2022]. **Definition 30**. *[\[numerical-inequality-conditions\]]{#numerical-inequality-conditions label="numerical-inequality-conditions"} Say a sequence of positive reals $(a_0,a_1,\ldots,a_r)$ is *unimodal* if there is some index $m$ with $$a_0 \leq a_1 \leq \cdots \leq a_{m-1} \leq a_m \geq a_{m+1} \geq \cdots \geq a_{r-1} \geq a_r.$$* *Say the sequence is *strongly[^1] log-concave* (or *$PF_2$*) if $0 \leq i\leq j \leq k \leq \ell\leq r$ and $i+\ell=j+k$ implies $$a_i a_\ell \leq a_j a_k, \text{ or equivalently, } \det\left[ \begin{matrix} a_j & a_\ell \\ a_i & a_k \end{matrix}\right] \geq 0.$$* *For $\ell=2,3,4,\ldots$, say that the sequence is $PF_\ell$ if the associated (infinite) *Toeplitz matrix* $$T(a_0,\ldots,a_r):= \left[ \begin{matrix} a_0 & a_1& a_2& \cdots & a_{r-1} & a_r & 0 & 0 & \cdots \\ 0 & a_0 & a_1 & \cdots & a_{r-2} & a_{r-1} & a_{r} & 0 &\cdots \\ 0 & 0 & a_0 & \cdots & a_{r-3} & a_{r-2} & a_{r-1} & a_{r}& \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \right]$$ has all *nonnegative* square minor subdeterminants of size $m \times m$ for $1\leq m \leq \ell$. Say that the sequence is a *Pólya frequency sequence* (or *$PF_\infty$*, or just *$PF$*) if it is $PF_\ell$ for all $\ell=2,3,\ldots$.* One can check the implication ($PF_2$ implies unimodality) from [\[PF-hierarchy\]](#PF-hierarchy){reference-type="eqref" reference="PF-hierarchy"} using the assumption $a_k >0$ for all $k$. It also turns out that $(a_0,a_1,\ldots,a_r)$ is $PF$ if and only if the polynomial $a_0 + a_1 t+a_2 t^2+ \cdots+ a_r t^r$ has only (negative) real roots; see [@Brenti §2.2, 4.5]. **Definition 31**. *For a finite group $G$ and (genuine, nonzero) $\mathbb{C}G$-modules $(A^0,A^1,\ldots,A^r)$, define the analogous notions of *equivariant unimodality*, *equivariant strong log-concavity*, *equivariant $PF_r$ or $PF_\infty$* by replacing the numerical inequalities in Definition [\[numerical-inequality-conditions\]](#numerical-inequality-conditions){reference-type="ref" reference="numerical-inequality-conditions"} by inequalities in the representation ring $R_\mathbb{C}(G)$.* *Similarly, for (nonempty) $G$-sets $(X_0,X_1,\ldots,X_r)$, define the notions of *Burnside unimodality*, *Burnside strong log-concavity*, *Burnside $PF_r$ or $PF_\infty$* by replacing them with inequalities in the Burnside ring $B(G)$.* **Example 32**. *[\[unimodality-of-various-kinds-example\]]{#unimodality-of-various-kinds-example label="unimodality-of-various-kinds-example"} We've seen for Chow rings $A(\mathcal{M})=\bigoplus_{k=0}^r A^k$ of rank $r+1$ matroids $\mathcal{M}$, and $G=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$,* - *the sequence $(a_0,a_1,\ldots,a_r)$ with $a_k:=\mathop{\mathrm{\mathrm{rk}}}_\mathbb{Z}A_k$ is *unimodal*,* - *after tensoring with $\mathbb{C}$, the sequence of $\mathbb{C}G$-modules $(A^0_\mathbb{C},A^1_\mathbb{C},\ldots,A^r_\mathbb{C})$ is *equivariantly unimodal*, and* - *the sequence of $G$-sets $(\mathrm{FY}^0,\mathrm{FY}^1,\ldots,\mathrm{FY}^r)$ is *Burnside unimodal*.* **Conjecture 33**. *In the same Chow ring context as Example [\[unimodality-of-various-kinds-example\]](#unimodality-of-various-kinds-example){reference-type="ref" reference="unimodality-of-various-kinds-example"}, one has that* - *(Ferroni-Schröter [@FerroniSchroter Conj. 10.19]) $(a_0,\ldots,a_r)$ is $PF_\infty$.* - *$(A^0_\mathbb{C},\ldots,A^r_\mathbb{C})$ is equivariantly $PF_\infty$.* - *$(\mathrm{FY}^0,\ldots,\mathrm{FY}^r)$ is Burnside $PF_2$ (Burnside log-concave), that is, $$[\mathrm{FY}^i][\mathrm{FY}^\ell] \leq_{B(G)} [\mathrm{FY}^j][\mathrm{FY}^k] \qquad \text{for } \text{for } \; i \leq j \leq k \leq \ell \; \text{ with } \; i+\ell=j+k.$$* Of course, in Conjecture [Conjecture 33](#log-concavity-conjectures){reference-type="ref" reference="log-concavity-conjectures"}, assertion (ii) implies assertion (i). However assertion (iii) would only imply the weaker $PF_2$ part of the conjectural assertion (ii), and only imply the $PF_2$ part of Ferroni and Schröter's assertion (i), but not their $PF_\infty$ assertions. Even the $PF_2$ property for $(a_0,\ldots,a_r)$ is still conjectural; see [@FerroniSchroter §10.3] and [@FMSV §3.7] for a discussion of the current evidence for Conjecture [Conjecture 33](#log-concavity-conjectures){reference-type="ref" reference="log-concavity-conjectures"}(i). **Example 34**. * We explain here why Conjecture [Conjecture 33](#log-concavity-conjectures){reference-type="ref" reference="log-concavity-conjectures"} *does not* assert that $(\mathrm{FY}^0,\ldots,\mathrm{FY}^r)$ is Burnside $PF_\infty$. In fact, $(\mathrm{FY}^0,\ldots,\mathrm{FY}^r)$ *fails even to be Burnside $PF_3$*, already when $\mathcal{M}$ is a rank $4$ Boolean matroid. Its Chow ring $A(\mathcal{M})=A^0 \oplus A^1 \oplus A^2 \oplus A^3$ has $A_0,A_3$ carrying the trivial $\mathbb{C}\mathfrak{S}_4$-module, and $A^1,A^2$ carrying isomorphic permutation representations, each having three orbits, whose three $\mathfrak{S}_4$-stabilizer groups are the Young subgroups $\mathfrak{S}_4, \mathfrak{S}_3 \times \mathfrak{S}_1, \mathfrak{S}_2 \times \mathfrak{S}_2$. The red $3 \times 3$ minor of the Toeplitz matrix shown here $$\left[ \begin{matrix} a_0 & {\color{red}a_1} & {\color{red}a_2} & a_3 & {\color{red} 0} & 0 &\cdots \\ 0 & {\color{red}a_0} & {\color{red}a_1}& a_2 & {\color{red} a_3} & 0 & \cdots \\ 0 & {\color{red} 0} & {\color{red} a_0} & a_1 & {\color{red}a_2} & a_3 &\cdots \\ 0 & 0 & 0 & a_0 & a_1 & a_2 &\cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{matrix} \right]$$ has determinant ${\color{red}a_1^2 a_2-a_1 a_3 - a_2^2}.$ Hence the Burnside $PF_3$ condition would require that the following genuine $\mathfrak{S}_4$-character should come from a genuine permutation representation $${\color{red}\left( \chi_{A^1}\right)^2 \cdot \chi_{A^2} - \chi_{A^1} \chi_{A^3} - \left(\chi_{A^2}\right)^2 }= 29 \chi^{(1, 1, 1, 1)} + 124 \chi^{(2, 1, 1)} + 103 \chi^{(2, 2)} + 172 \chi^{(3, 1)} + 76\chi^{(4)},$$ where here $\chi^\lambda$ denotes the irreducible $\mathfrak{S}_n$-representation [@Sagan], [@Stanley-EC2 §7.18] indexed by the partition $\lambda$ of $n$; this expansion was computed using `Sage/Cocalc`. But one can check that this is *not* a permutation representation, as its character value on the conjugacy class of $4$-cycles in $\mathfrak{S}_4$ is $+76-172+124-29=-1<0$.* **Remark 36**. * Although Example [Example 34](#Burnside-PF3-counterexample){reference-type="ref" reference="Burnside-PF3-counterexample"} shows that even the Boolean matroids contradict strengthening Conjecture [Conjecture 33](#log-concavity-conjectures){reference-type="ref" reference="log-concavity-conjectures"}(iii) to Burnside $PF_3$, they seem to satisfy a *different* strengthening of strong log-concavity:* ***Conjecture 35**. *For a Boolean matroid $\mathcal{M}$ of rank $n$ and $i \leq j \leq k \leq \ell$ with $i+\ell=j+k$, not only is the element $[\mathrm{FY}^j][\mathrm{FY}^k]-[\mathrm{FY}^i][\mathrm{FY}^\ell] \geq_{B(\mathfrak{S}_n)} 0,$ so that it is a genuine permutation representation, but furthermore one whose orbit-stabilizers are all Young subgroups $\mathfrak{S}_\lambda:=\mathfrak{S}_{\lambda_1} \times \mathfrak{S}_{\lambda_2} \times \cdots \times \mathfrak{S}_{\lambda_\ell}$.** We note here a small amount of evidence for Conjecture [Conjecture 33](#log-concavity-conjectures){reference-type="ref" reference="log-concavity-conjectures"}(ii), (iii), namely their strong log-concavity assertions hold for the case $i=0$. For assertion (ii), this is an easy consequence of the fact that the Chow ring $A(\mathcal{M})$ is generated by the variables $\{ y_F \}$ spanning its degree one component $A^1$, which shows that this $G$-equivariant multiplication map surjects: $$\label{multiplication-surjects} A^j \otimes A^k \twoheadrightarrow A^{j+k} \left(\cong A^0 \otimes A^{j+k}\right).$$ We next check that the stronger assertion of Conjecture [Conjecture 33](#log-concavity-conjectures){reference-type="ref" reference="log-concavity-conjectures"}(iii) also holds in the special case $i=0$. **Proposition 37**. *For matroids $\mathcal{M}$ and $j,k \geq 0$, one has a $G$-equivariant injection $\mathrm{FY}^{j+k} \hookrightarrow \mathrm{FY}^j \times \mathrm{FY}^k.$* *Proof.* Given $a=x_{F_1}^{m_1} \cdots x_{F_\ell}^{m_\ell}$ in $\mathrm{FY}^{j+k}$, so that $j+k=\sum_{i=1}^\ell m_i$, let $p$ be the smallest index such that $$\label{index-definining-inqualities} \sum_{i=1}^{p-1} m_i < j \leq \sum_{i=1}^{p} m_i$$ and factor the monomial $a=b \cdot c$ where $$a= \underbrace{x_{F_1}^{m_1} \cdots x_{F_{p-1}}^{m_{p-1}} x_{F_p}^\delta}_{b:=} \quad \cdot \quad \underbrace{ x_{F_p}^{m_p-\delta} x_{F_{p+1}}^{m_{p+1}} \cdots x_{F_\ell}^{m_\ell}}_{c:=}$$ with $\delta:=j-\sum_{i=1}^{p-1}m_i$ (so $\delta > 0$ by [\[index-definining-inqualities\]](#index-definining-inqualities){reference-type="eqref" reference="index-definining-inqualities"}), and $m_p-\delta \geq 0$. One can check that, since $a$ lies in $\mathrm{FY}^{j+k}$, one will also have $b, c$ lying in $\mathrm{FY}^j, \mathrm{FY}^k$, respectively. It is easily seen that the map $a \longmapsto (b,c)$ is injective, since its inverse sends $(b,c) \longmapsto bc$. It is also not hard to check that it is $G$-equivariant. ◻ **Remark 38**. *Note that one can iterate the map in the previous proof to construct $G$-equivariant injections $\prod_{i=1}^q \mathrm{FY}^{\alpha_i} \hookrightarrow\prod_{j=1}^p \mathrm{FY}^{\beta_j}$ whenever $\beta = (\beta_1, \beta_2, \ldots, \beta_p)$ is a composition refining $\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_q).$* As another small piece of evidence for Conjecture [Conjecture 33](#log-concavity-conjectures){reference-type="ref" reference="log-concavity-conjectures"} (ii), (iii), we show that $(\mathrm{FY}^0, \ldots, \mathrm{FY}^r)$ is Burnside $PF_2$ for $r \leq 5$, that is, for matroids of rank at most $6$. **Proposition 39**. *For any matroid $\mathcal{M}$ with $\mathop{\mathrm{\mathrm{rk}}}(\mathcal{M}) \leq 6$, the sequence $(\mathrm{FY}^0, \ldots, \mathrm{FY}^r)$ is Burnside $PF_2$.* *Proof sketch..* We check it for $\mathop{\mathrm{\mathrm{rk}}}(\mathcal{M})=6$, and $\mathop{\mathrm{\mathrm{rk}}}(\mathcal{M}) \leq 5$ is similar. Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}(i) shows that in $B(G)$, $$\Big([\mathrm{FY}^0], \, [\mathrm{FY}^1], \, [\mathrm{FY}^2], \, [\mathrm{FY}^3], \,[\mathrm{FY}^4], \, [\mathrm{FY}^5] \Big) = \Big(1, \, [\mathrm{FY}^1], \,[\mathrm{FY}^2], \, [\mathrm{FY}^2], \,[\mathrm{FY}^1], \, 1\Big).$$ Hence one must check nonnegativity in $B(G)$ for all 2 x 2 minors in this infinite Toeplitz matrix: $$\begin{bmatrix} 1 & [\mathrm{FY}^1] & [\mathrm{FY}^2] & [\mathrm{FY}^2] & [\mathrm{FY}^1] & 1 & 0 & 0 & \ldots \\ 0 & 1 & [\mathrm{FY}^1] & [\mathrm{FY}^2] & [\mathrm{FY}^2] & [\mathrm{FY}^1] & 1 & 0 & \ldots \\ 0 & 0 & 1 & [\mathrm{FY}^1] & [\mathrm{FY}^2] & [\mathrm{FY}^2] & [\mathrm{FY}^1] & 1 & \ldots \\ 0 & 0 & 0 & 1 & [\mathrm{FY}^1] & [\mathrm{FY}^2] & [\mathrm{FY}^2] & [\mathrm{FY}^1] & \ldots \\ 0 & 0 & 0 & 0 & 1& [\mathrm{FY}^1] & [\mathrm{FY}^2] & [\mathrm{FY}^2] & \ldots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix}$$ From periodicity of the matrix, one may assume without loss of generality that the $2 \times 2$ minor has its top-left entry in the first row. If the minor has a $0$ as either its lower-left or upper-right entry, then its determinant is a product $[\mathrm{FY}^i] [\mathrm{FY}^j] = [\mathrm{FY}^i \times \mathrm{FY}^j] \geq_{B(G)} 0$. This already leaves only finitely many $2 \times 2$ minors to check. Additionally, if it has $1$ as its lower left entry, then it was shown to be Burnside-nonnegative in Theorem [Proposition 37](#thm:2by2kos){reference-type="ref" reference="thm:2by2kos"}. All of the remaining $2 \times 2$ minors we claim are Burnside-nonnegative because they compare two (possibly non-consecutive) terms in this chain of inequalities: $$1 \overset{(a)}{\leq}_{B(G)}[\mathrm{FY}^1] \overset{(b)}{\leq}_{B(G)} [\mathrm{FY}^2] \overset{(c)}{\leq}_{B(G)} [\mathrm{FY}^1][\mathrm{FY}^1] \overset{(d)}{\leq}_{B(G)} [\mathrm{FY}^1][\mathrm{FY}^2] \overset{(e)}{\leq}_{B(G)} [\mathrm{FY}^2][\mathrm{FY}^2]$$ where inequalities (a),(b) follow from Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"}(ii), inequality (c) follows from Theorem [Proposition 37](#thm:2by2kos){reference-type="ref" reference="thm:2by2kos"}, and inequality (d),(e) come from multiplying inequality (b) by $[\mathrm{FY}^1]$ and multiplying inequality (a) by $[\mathrm{FY}^2]$. ◻ **Remark 40**. *When $\mathop{\mathrm{\mathrm{rk}}}(\mathcal{M}) \geq 7$, one encounters the first $2 \times 2$ minor in $B(G)$ for $G=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$ $$\det \left[ \begin{matrix} [\mathrm{FY}^2] & [\mathrm{FY}^3] \\ [\mathrm{FY}^1] & [\mathrm{FY}^2] \end{matrix} \right] = [\mathrm{FY}^2] [\mathrm{FY}^2] - [\mathrm{FY}^1] [\mathrm{FY}^3] =[ \mathrm{FY}^2 \times \mathrm{FY}^2 ] - [\mathrm{FY}^1 \times \mathrm{FY}^3]$$ whose Burnside nonnegativity does not already follow from our previous results.* ## Koszulity The surjection in [\[multiplication-surjects\]](#multiplication-surjects){reference-type="eqref" reference="multiplication-surjects"} that proved a special case of Conjecture [Conjecture 33](#log-concavity-conjectures){reference-type="ref" reference="log-concavity-conjectures"}(ii) turns out to be the $2 \times 2$ special case of more general equivariant $\ell \times \ell$ Toeplitz minor inequalities for Chow rings $A(\mathcal{M})$. These inequalities follow from general theory of *Koszul algebras*, along with a recent result of Maestroni and McCullough [@MM] showing $A(\mathcal{M})$ is Koszul. After reviewing these results, we state a conjecture that would generalize Proposition [Proposition 37](#thm:2by2kos){reference-type="ref" reference="thm:2by2kos"} and upgrade these Toeplitz minor inequalities from the representation ring $R_\mathbb{C}(G)$ to the Burnside ring $B(G)$. A good reference on Koszul algebras is Polishchuk and Positselski [@PP]. **Definition 41**. *Let $\mathbb{F}$ be a field, and $A$ a *finitely generated standard graded associative $\mathbb{F}$-algebra*. This means $A$ is a quotient $A=T/I$ where $T=\mathbb{F}\langle x_1,\ldots,x_n\rangle$ is the free associative algebra on $n$ noncommuting variables $x_1,\ldots,x_n$, considered to all have $\deg(x_i)=1$, and $I$ is a homogeneous two-sided ideal in $T$.* *Writing[^2] $A=\bigoplus_{k=0}^\infty A_k$, let $A_+:=\bigoplus_{k=1}^\infty A_k$ be the maximal graded two-sided ideal of $A$. Regard the field $\mathbb{F}$ as an $A$-module via the quotient surjection $A \twoheadrightarrow A/A_+ \cong \mathbb{F}$. In other words, each $x_i$ acts as $0$ on $\mathbb{F}$.* *Say that $A$ is a *Koszul algebra* if the above surjection $A \twoheadrightarrow \mathbb{F}$ extends to a *linear* graded free $A$-resolution of $\mathbb{F}$ as an $A$-module, meaning that the $i^{th}$ resolvent $F_i = A(-i)^{\beta_i}$ for some $\beta_i \geq 0$: $$0 \leftarrow \mathbb{F} \leftarrow A \leftarrow A(-1)^{\beta_1} \leftarrow A(-2)^{\beta_2} \leftarrow A(-3)^{\beta_3} \leftarrow \cdots$$* There are several equivalent ways to say when $A$ is Koszul, such as requiring that the polynomial grading of $\mathrm{Tor}^A_i(\mathbb{F},\mathbb{F})$ is concentrated in degree $i$. Equivalently, this means that if one starts with the *bar complex* $\mathcal{B}_A$ as an $A$-resolution of $\mathbb{F}$, and then tensors over $A$ with $\mathbb{F}$, one obtains a complex $\mathbb{F}\otimes_A \mathcal{B}_A$ of graded $\mathbb{F}$-vector spaces whose $i^{th}$ homology is concentrated in degree $i$. The latter characterization leads to the following result of Polishchuk and Positselski. **Theorem 42**. *[@PP Chap. 2, Prop. 8.3] For any Koszul algebra $A$, and any composition $(\alpha_1,\ldots,\alpha_r)$ of $m=\sum_i \alpha_i$, there exists a subcomplex $(C_*,d)$ of $\mathbb{F}\otimes_A \mathcal{B}_A$ of the form $0\rightarrow C_\ell \rightarrow C_{\ell-1} \rightarrow \cdots \rightarrow C_1 \rightarrow 0$ starting with $C_\ell=A_{\alpha_1} \otimes \cdots \otimes A_{\alpha_\ell}$ at left, ending with $C_1=A_{\alpha_1+\cdots+\alpha_\ell}=A_m$ at right, and with $i^{th}$ term $$C_i=\bigoplus_\beta A_{\beta_1} \otimes \cdots \otimes A_{\beta_i}$$ where $\beta$ in the direct sum runs over all compositions with $i$ parts that coarsen $\alpha$. This complex $(C_*,d)$ is exact except at the left end $C_\ell$, meaning that this complex is exact: $$\label{PP-exact=sequence} 0 \rightarrow \ker(d_\ell) \rightarrow C_\ell \rightarrow C_{\ell-1} \rightarrow \cdots \rightarrow C_1 \rightarrow 0$$ The complex $(C_*,d)$ is also $G$-equivariant for any group $G$ of graded $\mathbb{F}$-algebra automorphisms of $A$.* Taking the alternating sum of the Euler characteristics term-by-term in [\[PP-exact=sequence\]](#PP-exact=sequence){reference-type="eqref" reference="PP-exact=sequence"} yields the following, where here we conflate a $\mathbb{F}G$-module $A_k$ with its character $\chi_{A_k}$. **Corollary 43**. *In the above setting, the character of the $\mathbb{F}G$-module $\ker(d_\ell: C_\ell \rightarrow C_{\ell-1})$ has this expression $$\begin{aligned} \chi_{\ker(d_\ell)} =\sum_{i=1}^\ell (-1)^{\ell-i} \chi_{C_i} &=\sum_{i=1}^\ell (-1)^{\ell-i} \sum_{\substack{\beta \colon \ell(\beta)=i \\ \beta \: {\rm coarsens } \: \alpha}} A_{\beta_1} \otimes \cdots \otimes A_{\beta_i}\\ &=\det\left[ \begin{matrix} A_{\alpha_1} & A_{\alpha_1+\alpha_2} & A_{\alpha_1+\alpha_2+\alpha_3} & \cdots &A_m \\ A_0 & A_{\alpha_2} & A_{\alpha_2+\alpha_3} &\cdots &A_{m-\alpha_1} \\ 0 & A_0 & A_{\alpha_3} & \cdots & A_{m-(\alpha_1+\alpha_2)} \\ 0 & 0 & & & \vdots\\ \vdots & \vdots & & & A_{\alpha_{\ell-1}+\alpha_\ell}\\ 0 & 0 & \cdots & A_0 &A_{\alpha_\ell} \end{matrix} \right]\end{aligned}$$ as an $\ell \times \ell$ Toeplitz matrix minor for the sequence of $\mathbb{F}G$-modules $(A_0,A_1,A_2,\ldots)$. In particular, when $\mathbb{F}=\mathbb{C}$, then all Toeplitz minors of this form are genuine characters in $R_\mathbb{C}(G)$.* **Example 44**. *When $\ell=2$ so that $\alpha=(j,k)$, the exact sequence [\[PP-exact=sequence\]](#PP-exact=sequence){reference-type="eqref" reference="PP-exact=sequence"} looks like $$0 \rightarrow \ker(d_2) \rightarrow A_j \otimes A_k \rightarrow A_{j+k} \rightarrow 0$$ giving this character identity $$\det\left[ \begin{matrix} A_j & A_{j+k} \\ A_0 & A_k \end{matrix} \right] =\chi_{\ker(d_2)} \quad (\geq_{R_\mathbb{C}(G)} 0 \text{ if }\mathbb{F}=\mathbb{C}).$$* *When $\ell=3$ so that $\alpha=(a,b,c)$ , the exact sequence [\[PP-exact=sequence\]](#PP-exact=sequence){reference-type="eqref" reference="PP-exact=sequence"} looks like $$0 \rightarrow \ker(d_3) \rightarrow A_a \otimes A_b \otimes A_c \rightarrow \begin{matrix} A_{a+b} \otimes A_c \\ \oplus \\ A_a \otimes A_{b+c} \end{matrix} \rightarrow A_{a+b+c} \rightarrow 0$$ giving this character identity $$\det\left[ \begin{matrix} A_a & A_{a+b} & A_{a+b+c}\\ A_0 & A_b & A_{b+c}\\ 0 & A_0 & A_{c} \end{matrix} \right] =\chi_{\ker(d_3)} \quad (\geq_{R_\mathbb{C}(G)} 0 \text{ if }\mathbb{F}=\mathbb{C}).$$* *When $\ell=4$ so that $\alpha=(a,b,c,d)$, the exact sequence [\[PP-exact=sequence\]](#PP-exact=sequence){reference-type="eqref" reference="PP-exact=sequence"} looks like $$0 \rightarrow \ker(d_4) \rightarrow A_a \otimes A_b \otimes A_c \otimes A_d \rightarrow \begin{matrix} A_{a+b} \otimes A_c \otimes A_d\\ \oplus \\ A_a \otimes A_{b+c} \otimes A_d \\ \oplus\\ A_a \otimes A_b \otimes A_{c+d} \end{matrix} \rightarrow \begin{matrix} A_{a+b+c} \otimes A_d \\ \oplus\\ A_{a+b} \otimes A_{c+d}\\ \oplus\\ A_a \otimes A_{b+c+d} \end{matrix} \rightarrow A_{a+b+c+d} \rightarrow 0$$ giving this character identity $$\det\left[ \begin{matrix} A_a & A_{a+b} & A_{a+b+c}&A_{a+b+c+d}\\ A_0 & A_b & A_{b+c}&A_{b+c+d}\\ 0 & A_0 & A_{c}& A_{c+d}\\ 0 & 0 & A_0 & A_d \end{matrix} \right] =\chi_{\ker(d_4)} \quad (\geq_{R_\mathbb{C}(G)} 0 \text{ if }\mathbb{F}=\mathbb{C}).$$* One can now apply this to the case of Chow rings of matroids, via this result. **Theorem 45**. *(Maestroni and McCullough [@MM]) For any matroid $\mathcal{M}$, the Chow ring $A(\mathcal{M})$ is Koszul.* This gives the following promised generalization of [\[multiplication-surjects\]](#multiplication-surjects){reference-type="eqref" reference="multiplication-surjects"}. **Corollary 46**. *For a matroid $\mathcal{M}$ of rank $r+1$ with Chow ring $A(\mathcal{M})=\bigoplus_{k=0}^r A_k$, and any composition $\alpha=(\alpha_1,\ldots,\alpha_\ell)$ with $m:=\sum_i \alpha \leq r$, the $\ell \times \ell$ Toeplitz minor determinant as shown in in Corollary [Corollary 43](#corollary-of-P-P-exact-sequence){reference-type="ref" reference="corollary-of-P-P-exact-sequence"} is a genuine character in $R_\mathbb{C}(G)$ for $G=\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$.* Here is the conjectural lift of the previous corollary to Burnside rings, whose $2\times 2$-case is Proposition [Proposition 37](#thm:2by2kos){reference-type="ref" reference="thm:2by2kos"}. **Conjecture 47**. *In the same context as Corollary [Corollary 46](#Chow-ring-Koszul-consequence){reference-type="ref" reference="Chow-ring-Koszul-consequence"}, the analogous Toeplitz minors of $G$-sets have $$\det\left[ \begin{matrix} [\mathrm{FY}^{\alpha_1}] & [\mathrm{FY}^{\alpha_1+\alpha_2}] & [\mathrm{FY}^{\alpha_1+\alpha_2+\alpha_3}] & \cdots &[\mathrm{FY}^m] \\ [\mathrm{FY}^0] & [\mathrm{FY}^{\alpha_2}] & [\mathrm{FY}^{\alpha_2+\alpha_3}] &\cdots &[\mathrm{FY}^{m-\alpha_1}] \\ 0 & [\mathrm{FY}^0] & [\mathrm{FY}^{\alpha_3}] & \cdots & [\mathrm{FY}^{m-(\alpha_1+\alpha_2)}] \\ 0 & 0 & & & \vdots\\ \vdots & \vdots & & & [\mathrm{FY}^{\alpha_{\ell-1}+\alpha_\ell}]\\ 0 & 0 & \cdots & [\mathrm{FY}^0] &[\mathrm{FY}^{\alpha_\ell}] \end{matrix} \right] \geq_{B(G)} 0.$$* As a bit of further evidence for Conjecture [Conjecture 47](#Burnside-Koszul-nonnegativity){reference-type="ref" reference="Burnside-Koszul-nonnegativity"}, we check it for the Toeplitz minors with $\alpha=(1,1,1)$. **Theorem 48**. *For any matroid $\mathcal{M}$, the Chow ring $A(\mathcal{M})$ has $$\begin{aligned} \det\left|\begin{matrix} [\mathrm{FY}^1] & [\mathrm{FY}^2] & [\mathrm{FY}^3] \\ [\mathrm{FY}^0] & [\mathrm{FY}^1] & [\mathrm{FY}^2] \\ 0 & [\mathrm{FY}^0] & [\mathrm{FY}^1] \end{matrix}\right| \geq_{B(G)} 0. \end{aligned}$$* *Proof.* Multiplying out the determinant, one needs to prove the following inequality in $B(G)$: $$[\mathrm{FY}^1 \times \mathrm{FY}^1 \times \mathrm{FY}^1] - \left( \begin{matrix} [\mathrm{FY}^2 \times \mathrm{FY}^1]\\ +\\ [\mathrm{FY}^1 \times \mathrm{FY}^2]\end{matrix} \right) + [\mathrm{FY}^3] \,\, \geq_{B(G)} 0,$$ or equivalently, one must show the inequality $$[\quad (\mathrm{FY}^2 \times \mathrm{FY}^1) \,\, \sqcup \,\, (\mathrm{FY}^1 \times \mathrm{FY}^2)\quad ] \,\, \leq_{B(G)} \,\, [\quad (\mathrm{FY}^1 \times \mathrm{FY}^1 \times \mathrm{FY}^1) \,\, \sqcup \,\, \mathrm{FY}^3\quad ].$$ For this, it suffices to provide an injective $G$-equivariant map $$(\mathrm{FY}^1 \times \mathrm{FY}^2) \sqcup (\mathrm{FY}^2 \times \mathrm{FY}^1) \,\, \hookrightarrow\,\, (\mathrm{FY}^1 \times \mathrm{FY}^1 \times \mathrm{FY}^1) \sqcup \mathrm{FY}^3.$$ Such a map is summarized schematically in Figures [\[fig: 3x3a\]](#fig: 3x3a){reference-type="ref" reference="fig: 3x3a"} and [\[fig: 3x3b\]](#fig: 3x3b){reference-type="ref" reference="fig: 3x3b"}, with certain abbreviation conventions: the variables $x,y,z$ always abbreviate the variables $x_{F_1}, x_{F_2},x_{F_3}$ for a generic nested flag of flats $F_1 \subset F_2 \subset F_3$, while the variable $w$ abbreviates $x_F$ for a flat $F$ incomparable to any of $F_1,F_2,F_3$. The Figures [\[fig: 3x3a\]](#fig: 3x3a){reference-type="ref" reference="fig: 3x3a"} and [\[fig: 3x3b\]](#fig: 3x3b){reference-type="ref" reference="fig: 3x3b"} describe for each type of element in $\mathrm{FY}^1 \times \mathrm{FY}^2$ and $\mathrm{FY}^2 \times \mathrm{FY}^1$ an appropriate image in either $\mathrm{FY}^1 \times \mathrm{FY}^1 \times \mathrm{FY}^1$ or $\mathrm{FY}^3$. Loosely speaking, the maps try to send elements to $\mathrm{FY}^3$ whenever possible, that is, whenever their product is a valid element of $\mathrm{FY}^3$. When this fails, we find an image in $\mathrm{FY}^1 \times \mathrm{FY}^1 \times \mathrm{FY}^1$, carefully trying to keep track of which images have been used by noting various conditions on the $x,y,z,$ and $w$, involving their ranks and sometimes their *coranks*, denoted $\mathop{\mathrm{\mathrm{cork}}}(F):=\mathop{\mathrm{\mathrm{rk}}}(E)-\mathop{\mathrm{\mathrm{rk}}}(F)$. Conditions in gray are forced by the form of the given tuple, while conditions in black are assumed to separate the map into disjoint cases.$\qedhere$ ◻ # Further questions and conjectures {#sec: further-questions} In addition to Conjectures [Conjecture 33](#log-concavity-conjectures){reference-type="ref" reference="log-concavity-conjectures"}, [Conjecture 35](#boolean-h-log-concavity-conj){reference-type="ref" reference="boolean-h-log-concavity-conj"}, [Conjecture 47](#Burnside-Koszul-nonnegativity){reference-type="ref" reference="Burnside-Koszul-nonnegativity"} above, we collect here are a few more questions and conjectures. ## Other building sets with symmetry? Here we have focused on the Chow ring of a matroid $\mathcal{M}$ using its *maximal* building set. However, Feichtner and Yuzvinsky [@FY] give a presentation for their Chow ring with respect to *any* building set. Their result [@FY Thm. 2] also provide a Gröbner basis for the ideal presenting the Chow ring that again has the pleasant properties of Theorem [Theorem 10](#FY-GB-theorem){reference-type="ref" reference="FY-GB-theorem"} and Corollary [\[cor: mon_basis\]](#cor: mon_basis){reference-type="ref" reference="cor: mon_basis"}: whenever the building set is stable under some subgroup $G$ of $\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$, the initial ideal for their Gröbner basis is $G$-stable, as is the standard monomial basis for the Chow ring. One relevant example of such a building set is the *minimal* building set, which is stable under the full automorphism group $\mathop{\mathrm{\mathrm{Aut}}}(\mathcal{M})$, and which arises, for example, in the study of the moduli space $\overline{M}_{0,n}$ of genus $0$ curves with $n$ marked points; see, e.g., Dotsenko [@dotsenko], Gibney and Maclagan [@GibneyMaclagan], and Keel [@Keel]. Furthermore, the Chow ring of $\mathcal{M}$ with respect to any building set still satisfies the Kähler package. This follows[^3] combining the results of [@AHK] for the maximal building set, and a theorem of Ardila, Denham and Huh [@ADH Thm. 1.6] asserting that having the Kähler package depends only on the support of the Bergman fan of $\mathcal{M}$, not on how it is subdivided according to the building set. By the same arguments as in Corollaries [Corollary 16](#integral-equivariant-PD-cor){reference-type="ref" reference="integral-equivariant-PD-cor"}, [Corollary 17](#AHK-equivariant-Hard-Lefschetz){reference-type="ref" reference="AHK-equivariant-Hard-Lefschetz"}, the equivariant versions of Poincaré duality and the Hard Lefschetz theorem will also hold. This raises the following question. **Question 49**. *Does the analogue of Theorem [Theorem 1](#main-theorem){reference-type="ref" reference="main-theorem"} hold for the Chow ring of a matroid $\mathcal{M}$ with respect to *any* $G$-stable building set? In particular, what about the *minimal* building set?* ## Explicit formulas for Chow rings as permutation representations? In [@Stembridge Lem. 3.1], Stembridge provides a generating function for the symmetric group representations on each graded component of the Chow ring for all Boolean matroids; see also Liao [@Liao]. Furthermore, Stembridge's expression exhibits them as *permutation representations*, whose orbit-stabilizers are all *Young subgroups* in the symmetric group. **Question 50**. *Can one provide such explicit expressions as permutation representations for other families of matroids with symmetry?* ## Equivariant $\gamma$-positivity? Hilbert functions $(a_0,a_1,\ldots,a_r)$ for Chow rings of rank $r+1$ matroids are not only symmetric and unimodal, but satisfy the stronger condition of *$\gamma$-positivity*: one has *nonnegativity* for all coefficients $\gamma=(\gamma_0,\gamma_1,\ldots,\gamma_{\lfloor \frac{r}{2} \rfloor})$ appearing in the unique expansion $$\label{gamma-defining-relation} \sum_{i=0}^r a_i t^i = \sum_{i=0}^{\lfloor\frac{r}{2}\rfloor} \gamma_i \,\, t^i(1+t)^{r-2i}.$$ See Athanasiadis [@Athanasiadis] for a nice survey on $\gamma$-positivity. It has been shown, independently by Ferroni, Matherne, Schröter and Vecchi [@FMSV Thm. 3.25] and by Wang (see [@FMSV p. 29]), that the $\gamma$-positivity for Hilbert series of Chow rings of matroids follows from results of Braden, Huh, Matherne, Proudfoot and Wang [@BHMPW] on *semismall decompositions*. One also has the notion of *equivariant $\gamma$-positivity* for a sequence of $G$-representations $(A_0,A_1,\ldots,A_r)$, due originally to Shareshian and Wachs [@ShareshianWachs §5] (see also [@Athanasiadis §5.2], [@FMSV Def. 4.13]): upon replacing each $a_i$ in [\[gamma-defining-relation\]](#gamma-defining-relation){reference-type="eqref" reference="gamma-defining-relation"} with the element $[A_i]$ of $R_\mathbb{C}(G)$, one asks that the uniquely defined coefficients $\gamma_i$ in $R_\mathbb{C}(G)$ have $\gamma_i \geq_{R_\mathbb{C}(G)} 0$. Computations suggests the following. **Conjecture 51**. *For any matroid $\mathcal{M}$ of rank $r+1$ and its Chow ring $A(\mathcal{M})=\bigoplus_i A^i$, the sequence of $G$-representations $(A^0_\mathbb{C},A^1_\mathbb{C},\ldots,A^r_\mathbb{C})$ is equivariantly $\gamma$-positive.* For example, [@ShareshianWachs Cor. 5.4] verifies Conjecture [Conjecture 51](#equivariant-gamma-positivity-conj){reference-type="ref" reference="equivariant-gamma-positivity-conj"} for Boolean matroids. However, one can check that the stronger conjecture of *Burnside $\gamma$-nonnegativity* for $(\mathrm{FY}^0,\mathrm{FY}^1,\ldots,\mathrm{FY}^r)$ would *fail* already for the Boolean matroid of rank $3$: here $\mathrm{FY}^0, \mathrm{FY}^2$ carry the trivial $\mathfrak{S}_3$ permutation representation $\mathbf{1}$ , while $\mathrm{FY}^1$ carries the defining $\mathfrak{S}_3$-permutation representation on the set $X=\{1,2,3\}$, so $\gamma_0=[\mathbf{1}]$, but $\gamma_1=[X]-[\mathbf{1}] \not\geq_{B(\mathfrak{S}_3)} 0$. [^1]: *The word "strongly\" here is superfluous, since we assumed each $a_k >0$, so they are strongly log-concave if and only they are weakly log-concave: $a_k^2 \geq a_{k-1} a_{k+1}$. The distinction becomes important for the equivariant analogue; see [@MMPR §2].* [^2]: *Apologies to the reader that we are writing $A_k$ here rather than $A^k$ as we did for the Chow rings.* [^3]: The authors thank Chris Eur for pointing them to this result.

arxiv_math

--- author: - "Styrt O. G.[^1]" title: "Groups $\\Gamma_n^4$[:]{.upright} algebraic properties" --- In the paper, groups $\Gamma_n^4$ closely connected with braid groups are researched from algebraic point of view. More exactly, for $n\geqslant 7$, it is proved that $\Gamma_n^4$ is a nilpotent finite $2$-group with $4$-torsion and that its subgroup $(\Gamma_n^4)'$ is central. *Key words*[:]{.upright} groups $\Gamma_n^4$, braid groups. 5.5ex plus .5ex minus .2ex1.5ex plus .3ex \*The notations used 5.5ex plus .5ex minus .2ex1.5ex plus .3ex Introduction[\[introd\]]{#introd label="introd"} The paper [@Manick] is devoted to researching groups $\Gamma_n^4$ that are closely connected with braid groups. The group $\Gamma_n^4$ is defined in [@Manick [§ ]{.upright}2] by - the generators $d_{(ijkl)}$ for all pairwise distinct $i,j,k,l\in[n]$[;]{.upright} - the relations 1--4. For convenience, denote by $S$ the generator set and by $s_{ik}^{jl}$ the generator $d_{(ijkl)}$. For $P\in C_{[n]}^4$, set $S_P:=\bigl\{s_{ik}^{jl}\colon\{i,j,k,l\}=P\bigr\}\subset S$. Then, the relations [\[inv\]](#inv){reference-type="ref" reference="inv"}--[\[sym\]](#sym){reference-type="ref" reference="sym"} have the following sense[:]{.upright} 1. [\[inv\]]{#inv label="inv"} all elements of $S$ are involutions[;]{.upright} 2. [\[comm\]]{#comm label="comm"} if $P,Q\in C_{[n]}^4$ and $|P\cap Q|\leqslant 2$, then $[S_P,S_Q]=\{e\}$[;]{.upright} 3. [\[sym\]]{#sym label="sym"} $s_{ik}^{jl}=s_{ki}^{jl}=s_{ik}^{lj}=s_{jl}^{ik}$  so, each element $s_{ik}^{jl}$ is uniquely defined by the *unordered* pair $\bigl\{\{i,k\},\{j,l\}\bigr\}$ of the disjoint *unordered* pairs $\{i,k\}$ and $\{j,l\}$[;]{.upright} 4. [\[pent\]]{#pent label="pent"} if $\{i,j,k,l,m\}\in C_{[n]}^5$, then $s_{ik}^{jl}s_{il}^{jm}s_{jl}^{km}s_{ik}^{jm}s_{il}^{km}=e$, i. e. $s_{ki}^{lj}s_{il}^{jm}s_{lj}^{mk}s_{jm}^{ki}s_{mk}^{il}=e$. Hence, the condition [\[pent\]](#pent){reference-type="ref" reference="pent"} means that any pairwise distinct numbers $k_1,\ldots,k_5\in[n]$ satisfy $$\label{pen} s_{k_1k_2}^{k_3k_4}\cdot s_{k_2k_3}^{k_4k_5}\cdot s_{k_3k_4}^{k_5k_1}\cdot s_{k_4k_5}^{k_1k_2}\cdot s_{k_5k_1}^{k_2k_3}=e.$$ The main result of the paper is the following theorem. **Theorem 1**. *If $n\geqslant 7$, then* - *the subgroup $(\Gamma_n^4)'$ of $\Gamma_n^4$ is central[;]{.upright}* - *$(\Gamma_n^4)'\cong(\mathbb{Z}_2)^p$ and $\Gamma_n^4/(\Gamma_n^4)'\cong(\mathbb{Z}_2)^q$, where $p\leqslant C_n^3$ and $q\leqslant 3\cdot C_n^4$.* **Corollary 1**. *If $n\geqslant 7$, then* - *$\Gamma_n^4$ is a nilpotent finite $2$-group[;]{.upright}* - *$\bigl[\Gamma_n^4,(\Gamma_n^4)'\bigr]=\{e\}$[;]{.upright}* - *$a^4=e$ for each $a\in\Gamma_n^4$.* 5.5ex plus .5ex minus .2ex1.5ex plus .3ex Proofs of the results[\[prove\]]{#prove label="prove"} In this section, Theorem [Theorem 1](#main){reference-type="ref" reference="main"} is proved. For $a,x,y\in\Gamma_n^4$, write $x\sim_a y$ if the following (obviously, equivalent) conditions hold[:]{.upright} 1. $x^{-1}y\in Z(a)$[;]{.upright} 2. $xax^{-1}=yay^{-1}$[;]{.upright} 3. $[x,a]=[y,a]$[;]{.upright} 4. $[a,x]=[a,y]$. It is clear that, with $a$ fixed, the *binary* relation $\sim_a$ on *two* elements $x,y$ is an equivalence. From now, we will assume that $n\geqslant 6$. **Lemma 1**. *If $P\in C_{[n]}^4$, $K\in C_P^3$, and $s\in S_P$, then $s_1\sim_s s_2$ for all $s_1,s_2\in\bigcup\limits_{{k'\notin P}}S_{K\sqcup\{k'\}}$.* $\square\quad$ For $k'\notin K$ and $k\in K$, set $t_{k'k}:=s_{k'k}^{ij}$ where $\{i,j\}=K\setminus\{k\}$. Note that $\bigl|[n]\setminus P\bigr|=n-4\geqslant 2$. Thus, it suffices to prove for any distinct $k_1,k_2\in[n]\setminus P$ and distinct $i,j\in K$ the relation $t_{k_1i}\sim_s t_{k_2j}$. In $K$, take arbitrary distinct elements $k_3,k_5$ and denote the rest one by $k_4$. Each subset $Q$ of type $\bigl(K\setminus\{k\}\bigr)\sqcup\{k_1,k_2\}$ ($k\in K$) satisfies $Q\in C_{[n]}^4$ and $P\cap Q\subset Q\setminus\{k_1,k_2\}=K\setminus\{k\}$ that implies $S_Q\subset Z(s)$. By [\[pen\]](#pen){reference-type="eqref" reference="pen"}, $t_{k_2k_3}^{-1}t_{k_1k_5}= s_{k_2k_3}^{k_4k_5}\cdot s_{k_3k_4}^{k_5k_1}=s_{k_1k_2}^{k_3k_4}\cdot s_{k_5k_1}^{k_2k_3}\cdot s_{k_4k_5}^{k_1k_2}\in Z(s)$. **Lemma 2**. *Take any subset $K\in C_{[n]}^3$. Set $S_{(i)}:=S_{K\sqcup\{i\}}$ [(]{.upright}$i\notin K$[)]{.upright}. Then all elements of type $[s_i,s_j]$ [(]{.upright}$s_i\in S_{(i)}$, $s_j\in S_{(j)}$, $i,j\notin K$, $i\ne j$[)]{.upright} are the same involution $c^{(K)}$ that is central for $n\geqslant 7$.* $\square\quad$ Set $M:=[n]\setminus K$. Clearly, $|M|=n-3\geqslant 3$. If $k_0\in M$ and $s\in S_{(k_0)}$, then applying Lemma [Lemma 1](#eqv){reference-type="ref" reference="eqv"} to $P:=K\sqcup\{k_0\}$ gives for any $s_1,s_2\in\bigcup\limits_{{k'\in M\setminus\{k_0\}}}S_{(k')}$ the relations $s_1s_2\in Z(s)$, $[s_1,s]=[s_2,s]$, and $[s,s_1]=[s,s_2]$. Therefore[:]{.upright} - If $\{i,j\}\in C_M^2$, then all elements $[s_i,s_j]$ ($s_i\in S_{(i)}$, $s_j\in S_{(j)}$) are the same element $c^{(K)}_{ij}$. It is obvious that $c^{(K)}_{ji}=(c^{(K)}_{ij})^{-1}$. - If $k,i,j\in M$ and $i,j\ne k$, then $c^{(K)}_{ik}=c^{(K)}_{jk}$ and $c^{(K)}_{ki}=c^{(K)}_{kj}$. Since $|M|\geqslant 3$, all elements $c^{(K)}_{ij}$ ($i,j\in M$, $i\ne j$) are the same element $c^{(K)}=(c^{(K)})^{-1}$. It remains to prove that $[c^{(K)},\Gamma_n^4]=\{e\}$ for $n\geqslant 7$. Suppose that $n\geqslant 7$, $P\in C_{[n]}^4$, and $s\in S_P$[;]{.upright} show that $c^{(K)}\in Z(s)$. There exist distinct numbers $k_1,k_2\in M$ such that $\bigl(|M\setminus P|\geqslant 2\bigr)\Rightarrow(k_1,k_2\notin P)$. Take any elements $s_i\in S_{(k_i)}$ ($i\in[2]$). Then $c^{(K)}=[s_1,s_2]=(s_1s_2)^2$. Assume that $c^{(K)}\notin Z(s)$. We have $s_1s_2\notin Z(s)$ and $s_i\notin Z(s)$ for some $i\in[2]$. Hence, $\Bigl|\bigl(K\sqcup\{k_i\}\bigr)\cap P\Bigr|\geqslant 3$. Thus, - $|K\cap P|\geqslant 2$[;]{.upright} - if $k_i\notin P$, then $|K\cap P|\geqslant 3$, i. e. $P\supset K$. Note that $|M\setminus P|=n-|K\cup P|\geqslant 7-|K\cup P|=|K\cap P|\geqslant 2$. Therefore, $k_1,k_2\notin P$, $P\supset K$. So, $P=K\sqcup\{k_0\}$ ($k_0\in M$, $k_0\ne k_1,k_2$). Hence, $s_1s_2\in Z(s)$, a contradiction. Thus, if $n\geqslant 7$, then $[c^{(K)},S]=\{e\}$ and, therefore, $[c^{(K)},\Gamma_n^4]=\{e\}$. From now, assume that $n\geqslant 7$. Due to Lemma [Lemma 2](#com){reference-type="ref" reference="com"}, the subgroup $H\subset\Gamma_n^4$ generated by the involutions $c^{(K)}$ ($K\in C_{[n]}^3$) is central; hence, $H\lhd\Gamma_n^4$ and $H\cong(\mathbb{Z}_2)^p$, $p\leqslant C_n^3$. Denote by $\pi$ the factoring homomorphism $\Gamma_n^4\twoheadrightarrow\Gamma_n^4/H$[;]{.upright} write $\overline{x}$ for $\pi(x)$. **Lemma 3**. *If $P,Q\in C_{[n]}^4$ and $P\ne Q$, then $[\overline{S}_P,\overline{S}_Q]=\{e\}\subset\Gamma_n^4/H$.* $\square\quad$ We have $[S_P,S_Q]\subset H$ since $$=\begin{cases} \{e\},&|P\cap Q|\leqslant 2;\\ \{c^{(P\cap Q)}\},&|P\cap Q|=3.\end{cases}\qedhere$$ **Lemma 4**. *All elements $\overline{s}\in\Gamma_n^4/H$ [(]{.upright}$s\in S$[)]{.upright} pairwise commute.* $\square\quad$ By Lemma [Lemma 3](#codi){reference-type="ref" reference="codi"}, it suffices to prove that $[\overline{s}_{k_1k_2}^{k_3k_4},\overline{s}_{k_1k_4}^{k_3k_2}]=e$ for any pairwise distinct $k_i$. Consider the set $P\in C_{[n]}^4$ of these $k_i$ and an arbitrary number $k_5\notin P$. For distinct $i',i\in[4]$, set $r_{i'i}:=\overline{s}_{k_5k_i}^{ll'}$ where $\{l,l'\}=P\setminus\{k_i,k_{i'}\}$ (so, $\{k_5,k_i,l,l'\}=\bigl(P\sqcup\{k_5\}\bigr)\setminus\{k_{i'}\}$). It follows from Lemma [Lemma 3](#codi){reference-type="ref" reference="codi"} that $[r_{i'i},r_{j'j}]=e$ whenever $i'\ne j'$. By [\[pen\]](#pen){reference-type="eqref" reference="pen"}, $\overline{s}_{k_1k_2}^{k_3k_4}=r_{14}r_{21}r_{34}r_{41}$. Replace $k_2$ and $k_4$[:]{.upright} $\overline{s}_{k_1k_4}^{k_3k_2}=r_{12}r_{41}r_{32}r_{21}$. Hence, $\overline{s}_{k_1k_2}^{k_3k_4}\cdot\overline{s}_{k_1k_4}^{k_3k_2}=r_{14}r_{12}r_{34}r_{32}$. Now replace $k_1$ and $k_3$[:]{.upright} $\overline{s}_{k_3k_2}^{k_1k_4}\cdot\overline{s}_{k_3k_4}^{k_1k_2}=r_{34}r_{32}r_{14}r_{12}$, i. e. $\overline{s}_{k_1k_4}^{k_3k_2}\cdot\overline{s}_{k_1k_2}^{k_3k_4}=r_{34}r_{32}r_{14}r_{12}=r_{14}r_{12}r_{34}r_{32}= \overline{s}_{k_1k_2}^{k_3k_4}\cdot\overline{s}_{k_1k_4}^{k_3k_2}$. By Lemma [Lemma 4](#coal){reference-type="ref" reference="coal"}, the group $\Gamma_n^4/H$ is generated by pairwise commuting involutions $\overline{s}$ ($s\in S$); hence, $\Gamma_n^4/H\cong(\mathbb{Z}_2)^q$, $q\leqslant|S|\leqslant 3\cdot C_n^4$. In particular, $(\Gamma_n^4/H)'=\{e\}$, i. e. $(\Gamma_n^4)'\subset H$. Since $c^{(K)}\in(\Gamma_n^4)'$ for each $K\in C_{[n]}^3$, then $H=(\Gamma_n^4)'$. Thus, we completely proved Theorem [Theorem 1](#main){reference-type="ref" reference="main"} and, hence, Corollary [Corollary 1](#submain){reference-type="ref" reference="submain"}. 5.5ex plus .5ex minus .2ex1.5ex plus .3ex \*Acknowledgements The author is grateful to Prof. for exciting interest to algebra. The author is grateful to Prof. V. O. Manturov for introducing and attracting to his research area and for useful discussions. The author dedicates the article to E. N. Troshina. 5.5ex plus .5ex minus .2ex1.5ex plus .3ex \*References - '̇ V. O. Manturov, I. M. Nikonov, *The groups $\Gamma_n^4$, braids, and $3$-manifolds*[;]{.upright}\ arXiv: math.GN/2305.06316. [^1]: Russia, MIPT, oleg_styrt\@mail.ru

arxiv_math

--- abstract: | We consider *local* balances of momentum and angular momentum for the incompressible Navier-Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier-Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory. author: - "Maxim A. Olshanskii[^1]" - "Leo G. Rebholz [^2]" title: Local conservation laws of continuous Galerkin method for the incompressible Navier--Stokes equations in EMAC form --- # Introduction We are interested in conservation properties of continuous Galerkin discretizations of the incompressible Navier-Stokes equations (NSE), which are given by $$\label{NSE} \left\{ \begin{aligned} \frac{\partial \mathbf u}{\partial t}+(\mathbf u\cdot \nabla)\mathbf u- \operatorname{div}\boldsymbol{\sigma}&=0 \\ \operatorname{div}\mathbf u&=0 \end{aligned}\right.\quad\text{in}~~\Omega.$$ Here, $\boldsymbol{\sigma}$ is the Cauchy stress tensor, and we restrict to the case of a Newtonian fluid with $\boldsymbol{\sigma}=2\nu\mathbf D(\mathbf u)-p\mathbf I$, where $\mathbf D(\mathbf u)=\tfrac{1}{2}(\nabla\mathbf u+\nabla^T\mathbf u)$ is a rate of deformation tensor. It is well known that the smooth solution to [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} obeys an array of conservation laws, including the conservation of momentum, energy, vorticity, etc., which can be expressed in terms of proper balances for material volumes of fluid. The development of numerical methods that provide discrete counterparts for possibly many of these conservation laws is a long-standing challenge for the computational fluid dynamics community. This challenge has been addressed by numerous authors and from various perspectives; for example, see [@A66; @AM03; @LW04; @OR10b; @evans2013isogeometric; @SCN15; @PG16; @CHOR17; @coppola2019discrete] and references therein. Many of these studies have considered the *global* conservation properties of numerical methods, i.e., balances of physical quantities across the entire computational domain. While properly calibrating these global integral statistics is necessary for a method to be long-time accurate, it is difficult to see how this alone can guarantee the quality of a numerical solution. The proper *local* balances of momentum, energy, vorticity, etc. represent a significantly stronger requirement for a numerical solution. Note that "element-wise conservation\" is a common argument used to motivate the application of discontinuous Galerkin or finite volume discretization techniques (see, for instance, [@cockburn2003discontinuous; @leveque2002finite]). At the same time, there is a widespread belief that continuous (velocity $H^1$-conforming) Galerkin methods inevitably violate local conservation laws; however, see [@HEML00; @HW05] for a different viewpoint. Another obstacle in achieving proper discrete counterparts of both local and global conservation laws for $\mathbf u$ is the fact that continuous Galerkin discretizations of [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} (e.g., conforming finite element methods) typically enforce the divergence-free constraint only weakly [@CHOR17]. The purpose of this paper is to demonstrate that a continuous Galerkin solution, which is only weakly divergence-free, for [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} does satisfy properly formulated local conservation laws for momentum and angular momentum when one applies the so-called EMAC (Energy, Momentum, and Angular Momentum Conserving) formulation of the NSE. The EMAC formulation of the discrete NSE was originally developed in [@CHOR17]. It re-writes the inertia terms as $$\mathbf u\cdot\nabla \mathbf u\rightarrow 2\mathbf D(\mathbf u_h)\mathbf u_h + (\operatorname{div}\mathbf u_h)\mathbf u_h,$$ along with an altered pressure $p_h$ representing $p-\frac12 |\mathbf u|^2$. The motivation for EMAC was that Galerkin schemes using it can be shown to conserve global energy, momentum and angular momentum balances when $\operatorname{div}\mathbf u_h\ne 0$, while schemes using the common nonlinearity formulations such as convective (CONV: $\mathbf u_h\cdot\nabla \mathbf u_h$), skew-symmetric (SKEW: $\mathbf u_h\cdot\nabla \mathbf u_h + \frac12\operatorname{div}\mathbf u_h$) and rotational (ROT: $(\nabla \times \mathbf u_h) \times \mathbf u_h$) do not preserve some or all of these quantities. Perhaps not surprisingly, the use of EMAC has become popular for large scale fluid computations in a wide variety of applications and is shown to give better accuracy especially over longer time intervals e.g. [@PCLRH18; @LHOCR19; @SP18; @SP18b; @LPH19; @KMLP23; @VSA22; @CHOR17; @CHOR19; @OR20; @INRRV23] and is built into Alya which is a massively parallel multiphysics unstructured finite element code [@VHK16]. In addition to the better discrete physics of EMAC discussed above, it was proven in [@OR20] that schemes using EMAC are more long time stable because the Gronwall constant can be shown to be independent of (explicit) dependence on the Reynolds number; such a result does not hold for skew-symmetric, convective, or rotation forms for commonly used velocity-pressure finite elements such as Taylor--Hood elements. The purpose of this paper is to provide more theoretical justification that EMAC is superior compared to other discrete nonlinearity formulations, by proving that continuous Galerkin discretizations using EMAC admit an exact *local* balance of momentum and angular momentum. There are very few results for local conservation properties of continuous finite element methods, with [@HEML00; @HW05] being two fundamental works in this direction. The paper [@HW05] showed that for NSE, typical Galerkin schemes are not generally conservative, although this can be 'fixed' by multiscale formulation and adding a residual term. One observation made in this paper is that although local balances written in different forms -- standard Eulerian, Lagrangian, or weak Eulerian and Lagrangian forms introduced here -- represent the same conservation laws of fluid momentum and angular momentum, after discretization each form can be different. By considering the weak forms, which we refer to as *diffuse-volume forms*, of conservation laws, we can demonstrate that EMAC continuous Galerkin discretizations exactly preserve properly formulated local momentum and angular momentum balances. Furthermore, the discrete balances established here serve as direct analogies to the balances at the partial differential equation (PDE) level, obviating the need for a multiscale approach and additional residual terms to establish this connection. We note also that from the proof construction, it is not possible for SKEW, CONV or ROT to preserve these local balances of momentum and angular momentum in the same manner that EMAC does, since they do not preserve them globally. The rest of the paper is arranged as follows. Section [2](#S2){reference-type="ref" reference="S2"} recalls local conservation laws of momentum and angular momentum. The laws can be equivalently formulated in Eulerian and Lagrangian forms. Section [3](#S3){reference-type="ref" reference="S3"} introduces a different way to formulate local conservation laws, which is given the name diffuse-volume form of the conservation laws due to some similarity with diffuse-interface or phase-field methods in fluid mechanics. We show that this is another equivalent way to formulate the local balances. Here we also distinguish between diffuse-volume Eulerian and Lagrangian forms. Section [4](#S4){reference-type="ref" reference="S4"} demonstrates how the continuous Galerkin method for the NSE in EMAC form satisfies discrete counterparts of the (local) diffuse-volume Eulerian and Lagrangian conservation laws. Section [5](#S5){reference-type="ref" reference="S5"} offers a few illustrative numerical examples. # Eulerian and Lagrangian forms of momentum and angular momentum conservation {#S2} To formulate local conservation laws satisfied by a *smooth* solution to [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"}, we fix some $t$ and let $\omega\subset\Omega$ be a *fixed* subdomain of $\Omega$ with sufficiently smooth boundary $\partial\omega$. For this volume $\omega$, the balance of momentum and angular momentum take the form: $$\begin{aligned} \text{Moment.}\quad & \frac{d}{dt}\int_{\omega}\mathbf u\,dx&=&~ 2\nu\int_{\partial\omega }\mathbf D(\mathbf u)\mathbf n\,ds&-&\quad\int_{\partial\omega }p\mathbf n\,ds&-&\int_{\partial\omega }\mathbf u(\mathbf u\cdot\mathbf n)\,ds, \label{Laws2m}\\ \text{Ang. Moment.}\quad & \underbrace{\frac{d}{dt}\int_{\omega}\mathbf u\times\mathbf x\,dx}_{\scriptsize\begin{array}{c}\text{momentum} \\ \text{rate of change}\end{array}}& =&~\underbrace{2\nu\int_{\partial\omega }(\mathbf D(\mathbf u)\mathbf n)\times\mathbf x\,ds}_{\scriptsize\begin{array}{c}\text{momentum change due}\\ \text{to friction on }\partial\omega \end{array}}&-&\underbrace{\int_{\partial\omega }p(\mathbf n\times\mathbf x)\,ds}_{\scriptsize\begin{array}{c}\text{moment. change due}\\ \text{to pressure on }\partial\omega \end{array}}&-&~\underbrace{\int_{\partial\omega }(\mathbf u\times\mathbf x)(\mathbf u\cdot\mathbf n)\,ds.}_{\scriptsize\begin{array}{c}\text{the flux of}\\ \text{momentum through }\partial\omega \end{array}} \label{Laws2a} \end{aligned}$$ Hereafter $\mathbf n$ is the outward normal vector on $\partial\omega$. Local balances [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"}--[\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"} can be interpreted as *Eulerian form* of the conservation laws, in contrast to the *Lagrangian form* formulated for a material volume below. We now let $\Omega_t\subset\Omega$ be a *material* volume of the fluid. For the material volume, the conservation laws for momentum and angular momentum take the form: $$\begin{aligned} \text{Momentum}\quad & \frac{d}{dt}\int_{\Omega_t}\mathbf u\,dx&=&~ 2\nu\int_{\partial\Omega _t}\mathbf D(\mathbf u)\mathbf n\,ds-\int_{\partial\Omega _t}p\mathbf n\,ds,& \label{Laws1m}\\ \hspace{5ex} \text{Angular Momentum}\quad & \frac{d}{dt}\int_{\Omega_t}\mathbf u\times\mathbf x\,dx&=&~ 2\nu\int_{\partial\Omega _t}(\mathbf D(\mathbf u)\mathbf n)\times\mathbf x\,ds-\int_{\partial\Omega _t}p(\mathbf n\times\mathbf x)\,ds.& \hspace{5ex} \label{Laws1a} \end{aligned}$$ Of course, for smooth solutions to [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} the Eulerian and Lagrangian forms are just two different formulations of the same fundamental laws of continuum mechanics. They both follow from [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"}, and conversely, together with mass conservation they imply [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"}. This equivalence of [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} to the validity of local conservation laws (specifically, those concerning mass and momentum) is textbook material. The standard tools used to verify this equivalence include the divergence theorem, the freedom to choose fluid volumes $\Omega_t$ or $\omega$, and the Reynolds' transport theorem to handle the Lagrangian form, which states $$\frac{d}{dt}\int_{\Omega_t}f\,dx = \int_{\Omega_t}\Big(\frac{\partial f}{\partial t}+\operatorname{div}(f\mathbf u)\Big)\,dx,$$ for a smooth scalar function $f$. Continuous Galerkin methods like the $H^1$-conforming finite element method (FEM) employ finite dimensional subspaces of Sobolev spaces to project [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} and typically do not offer enough flexibility to verify a direct analogue of [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"}--[\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"} or [\[Laws1m\]](#Laws1m){reference-type="eqref" reference="Laws1m"}--[\[Laws1a\]](#Laws1a){reference-type="eqref" reference="Laws1a"}. Below we reformulate local conservation laws in a form more convenient for continuous Galerkin methods. # Weak form of the conservation laws {#S3} The purpose of this section is to derive the conservation laws [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"}--[\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"} and [\[Laws1m\]](#Laws1m){reference-type="eqref" reference="Laws1m"}--[\[Laws1a\]](#Laws1a){reference-type="eqref" reference="Laws1a"} in a form more appropriate for a variational formulation. Let $\omega\subset\Omega$ be an arbitrary subdomain of $\Omega$ with sufficiently smooth $\partial\omega$. Denote by $\phi$ an *arbitrary* smooth function such that $\omega=\mbox{supp}(\phi)$, and set $$\mathbf {\tilde n}:=-\frac{\nabla\phi}{|\nabla\phi|}$$ for $\mathbf x$ such that $\nabla\phi(\mathbf x)\neq0$, and let $\mathbf {\tilde n}(\mathbf x)$ be an arbitrary vector of unit length if $\nabla\phi(\mathbf x)=0$. Note that $\mathbf {\tilde n}(\mathbf x)=\mathbf n(\mathbf x)$ for $\mathbf x\in\partial\omega$. To obtain the weak form of the laws, we multiply the first equation in [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} by $\phi\mathbf e_i$ for momentum conservation and by $\phi\mathbf e_i\times\mathbf x$ for angular momentum conservation. Doing this for $i=1,\dots,d$, integrating over $\omega$ and by parts leads after straightforward computations to the following *weak form of the conservation laws*: $$\begin{aligned} \text{Moment.}\quad & \frac{d}{dt}\int_{\omega}\phi\mathbf u\,dx\hskip4ex = 2\nu\int_{\omega}\mathbf D(\mathbf u)\mathbf {\tilde n}|\nabla\phi|\,dx-\int_{\omega}p\mathbf {\tilde n}|\nabla\phi|\,dx-\int_{\omega}\mathbf u(\mathbf u\cdot\mathbf {\tilde n})|\nabla\phi|\,dx, \label{Laws2phim}\\ \text{Ang. Moment.}\quad & \frac{d}{dt}\int_{\omega}\phi\mathbf u\times\mathbf x\,dx =2\nu\int_{\omega}(\mathbf D(\mathbf u)\mathbf {\tilde n})\times\mathbf x\,|\nabla\phi|\,dx-\int_{\omega}p(\mathbf {\tilde n}\times\mathbf x)|\nabla\phi|\,dx-\int_{\omega}(\mathbf u\times\mathbf x)(\mathbf u\cdot\mathbf {\tilde n})|\nabla\phi|\,dx. \label{Laws2phia} \end{aligned}$$ We note that for all calculations below to make sense it is sufficient to assume $\phi\in W^{1,\infty}(\Omega)$. Given the freedom in choosing $\omega$ and $\phi$ one can show that [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"}--[\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"} and [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"}--[\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"} are *equivalent* if $\mathbf u$ is sufficiently smooth and divergence free. We formulate it as a proposition. **Proposition 1**. *Assume $\mathbf u$ and $p$ are smooth and $\operatorname{div}\mathbf u=0$, then [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"} (or [\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"}) holds for any subdomain $\omega\subset\Omega$ iff [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"} (or [\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"}) holds for any $\phi\in W^{1,\infty}(\Omega)$ with $\mbox{supp}(\phi)\subset\Omega$.* *Proof.* We know that [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"} and $\operatorname{div}\mathbf u=0$ imply [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} by standard arguments, given that $\omega$ can be taken as an arbitrary subdomain of $\Omega$ and for any $t$. Similarly, the fact that [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"} holds for any $\phi\in % \accentset{\mbox{\large\bfseries .}}{C}(\Omega)$ leads to $$\int_\Omega\Big(\frac{\partial \mathbf u}{\partial t}+(\mathbf u\cdot \nabla)\mathbf u- 2\nu\operatorname{div}\mathbf D(\mathbf u) +\nabla p \Big)\phi\,dx=0,\quad\forall\, \phi\in % \accentset{\mbox{\large\bfseries .}}{C}(\Omega),$$ which implies [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} due to the density of smooth compactly supported functions in $L^2(\Omega)$. In turn, both [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"} and [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"} are quick consequences of [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"}. Thus [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"} implies [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"} and vice versa. The same arguments can be applied to show the equivalence of [\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"} and [\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"}. The only difference is that the equivalence of [\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"} is established not to the momentum equation in [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"}, but to the vector product of this equation with $\mathbf x$, and also for [\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"}.\ In addition to the above equivalence result, it is easy to see that each individual term in [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"}--[\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"} can be approximated arbitrarily well by the corresponding term in [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"}--[\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"}. Indeed, fix any $\omega\subset\Omega$ with smooth $\partial\omega$ and for sufficiently small $\varepsilon>0$ define $$\label{eq:phieps} \phi_\varepsilon= \begin{cases} \varepsilon^{-1}\mbox{dist}(\mathbf x,\partial\omega ),& \mathbf x\in\mathcal{O}_\varepsilon(\partial\omega )\cap \omega,\\ 1& \mathbf x\in\omega\setminus\mathcal{O}_\varepsilon(\partial\omega ),\\ 0& \Omega\setminus \omega. \end{cases}$$ We have $\phi_\varepsilon\in W^{1,\infty}(\Omega)$ and one easily checks, letting $\mathbf {\tilde n}=-\nabla\phi_\varepsilon/|\nabla\phi_\varepsilon|$, that $$\frac{d}{dt}\int_{\omega}\phi_\varepsilon\mathbf u\,dx\to \frac{d}{dt}\int_{\omega}\mathbf u\,ds, \quad \int_{\omega}\mathbf D(\mathbf u)\mathbf {\tilde n}|\nabla\phi_\varepsilon|\,dx\to \int_{\partial\omega }\mathbf D(\mathbf u)\mathbf n\,ds,~ \text{for}~\varepsilon\to0,$$ and smooth $\mathbf u$. Similarly, the limit values of other terms in [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"}--[\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"} will be their counterparts in [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"}--[\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"}. Therefore, [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"}--[\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"} can be also interpreted as the *diffuse-volume* version of conservation laws. Eqs. [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"}--[\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"} imply [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"}--[\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"} term by term without the $\operatorname{div}\mathbf u=0$ assumption or equations [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} being invoked. Local conservation laws in the Lagrangian form are written for the time-dependent material volume $\Omega_t$. Denote by $\Omega_0$ the fluid volume at a given initial moment $t=t_0$ and assume $\Omega_t\subset\Omega$ for $t\in [t_0,t_1]$ for some $t_1>t_0$. The evolution of $\Omega_t$ is defined by the Lagrangian mapping $\Phi_t:\Omega_0\to\Omega_t$, i.e. $\mathbf y=\Phi_t(\mathbf x)$ solves the Cauchy problem $$\label{Cauchy} \mathbf y_t=\mathbf u(t,\mathbf y),\quad t\in (t_0,t_1],\quad \mathbf y(t_0)=\mathbf x.$$ To properly reflect this domain evolution in a weak form of [\[Laws1m\]](#Laws1m){reference-type="eqref" reference="Laws1m"}--[\[Laws1a\]](#Laws1a){reference-type="eqref" reference="Laws1a"}, we want $\phi$ to be time dependent and such that $\mbox{supp}(\phi)=\Omega_t$. To this end, consider a smooth function $\phi^0$ such that $\mbox{supp}(\phi^0)=\Omega_0$. We define $\phi=\phi^0\circ\Phi_t^{-1}.$ The constructed $\phi$ is smooth (since $\mathbf u$ is smooth so is the solution to the Cauchy problem [\[Cauchy\]](#Cauchy){reference-type="eqref" reference="Cauchy"}), $\mbox{supp}(\phi)=\Omega_t$, and it satisfies the transport equation $$\label{eq:ls} \frac{\partial \phi}{\partial t}+(\mathbf u\cdot \nabla)\phi =0\quad\text{in}~\Omega,~ t\in (t_0,t_1],\quad \phi(t_0)=\phi^0.$$ Applying the Reynolds' transport theorem and using [\[eq:ls\]](#eq:ls){reference-type="eqref" reference="eq:ls"} and [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} one computes the following weak Lagrangian form of the local balances: $$\begin{aligned} \text{Moment.}\hskip7ex \frac{d}{dt}\int_{\Omega_t}\phi \mathbf u\,dx&~=~ 2\nu\int_{\Omega_t}\mathbf D(\mathbf u)\mathbf {\tilde n}|\nabla\phi|\,dx-\int_{\Omega_t}p\mathbf {\tilde n}|\nabla\phi|\,dx, \label{Laws1phim} \\ \text{Angl. Moment.}\quad \frac{d}{dt}\int_{\Omega_t}\phi\mathbf u\times\mathbf x\,dx& ~=~2\nu\int_{\Omega_t}(\mathbf D(\mathbf u)\mathbf {\tilde n})\times\mathbf x|\nabla\phi|\,dx-\int_{\Omega_t}p(\mathbf {\tilde n}\times\mathbf x)|\nabla\phi|\,dx. \label{Laws1phia} \end{aligned}$$ By the same arguments as we use to prove Proposition [Proposition 1](#pr1){reference-type="ref" reference="pr1"} we prove the following proposition. **Proposition 2**. *Assume $\mathbf u$ and $p$ are smooth and $\operatorname{div}\mathbf u=0$. Then [\[Laws1m\]](#Laws1m){reference-type="eqref" reference="Laws1m"} (or [\[Laws1a\]](#Laws1a){reference-type="eqref" reference="Laws1a"}) holds for any material volume $\Omega_t$ such that $\Omega_t\subset\Omega$ for $t\in [t_0,t_1]$ iff [\[Laws1phim\]](#Laws1phim){reference-type="eqref" reference="Laws1phim"} (or [\[Laws1phia\]](#Laws1phia){reference-type="eqref" reference="Laws1phia"}) holds for any $\phi$ satisfying [\[eq:ls\]](#eq:ls){reference-type="eqref" reference="eq:ls"} with $\phi^0\in W^{1,\infty}(\Omega_{t_0})$, such that $\mbox{supp}(\phi^0)=\Omega_{t_0}$.* Similar to the Eulerian case, it is easy to see that each individual term in [\[Laws1m\]](#Laws1m){reference-type="eqref" reference="Laws1m"}--[\[Laws1a\]](#Laws1a){reference-type="eqref" reference="Laws1a"} can be approximated arbitrarily well by the corresponding term in [\[Laws1phim\]](#Laws1phim){reference-type="eqref" reference="Laws1phim"}--[\[Laws1phia\]](#Laws1phia){reference-type="eqref" reference="Laws1phia"}. This time $\phi_\varepsilon$ is constructed as $\phi_\varepsilon=\phi^0_\varepsilon\circ\Phi_t^{-1}$ with $\phi^0_\varepsilon$ defined by the formula in [\[eq:phieps\]](#eq:phieps){reference-type="eqref" reference="eq:phieps"} with $\omega$ replaced by $\Omega_{t_0}$. It holds $\phi_\varepsilon\in W^{1,\infty}(\Omega\times[t_0,t_1])$ and one verifies, letting $\mathbf {\tilde n}=-\nabla\phi_\varepsilon/|\nabla\phi_\varepsilon|$, that $$\frac{d}{dt}\int_{\Omega_t}\phi_\varepsilon\mathbf u\,dx\to \frac{d}{dt}\int_{\Omega_t}\mathbf u\,ds, \quad \int_{\Omega_t}\mathbf D(\mathbf u)\mathbf {\tilde n}|\nabla\phi_\varepsilon|\,dx\to \int_{\partial\Omega _t}\mathbf D(\mathbf u)\mathbf n\,ds,~ \text{for}~\varepsilon\to0,$$ and smooth $\mathbf u$. The limit values of other terms in [\[Laws1phim\]](#Laws1phim){reference-type="eqref" reference="Laws1phim"}--[\[Laws1phia\]](#Laws1phia){reference-type="eqref" reference="Laws1phia"} will be their counterparts in [\[Laws1m\]](#Laws1m){reference-type="eqref" reference="Laws1m"}--[\[Laws1a\]](#Laws1a){reference-type="eqref" reference="Laws1a"}. Therefore, [\[Laws1phim\]](#Laws1phim){reference-type="eqref" reference="Laws1phim"}--[\[Laws1phia\]](#Laws1phia){reference-type="eqref" reference="Laws1phia"} can be also interpreted as the diffuse-volume version of local conservation laws in the Lagrangian form. In summary, equations [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"}--[\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"} are equivalent formulations of the fundamental (local) conservation laws in the Eulerian formulation, while [\[Laws1phim\]](#Laws1phim){reference-type="eqref" reference="Laws1phim"}--[\[Laws1phia\]](#Laws1phia){reference-type="eqref" reference="Laws1phia"} are equivalent formulations of the fundamental (local) conservation laws in the Lagrangian formulation. We will study the ability of a discretization method to match [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"}--[\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"} and [\[Laws1phim\]](#Laws1phim){reference-type="eqref" reference="Laws1phim"}--[\[Laws1phia\]](#Laws1phia){reference-type="eqref" reference="Laws1phia"} instead of [\[Laws2m\]](#Laws2m){reference-type="eqref" reference="Laws2m"}--[\[Laws2a\]](#Laws2a){reference-type="eqref" reference="Laws2a"} and [\[Laws1m\]](#Laws1m){reference-type="eqref" reference="Laws1m"}--[\[Laws1a\]](#Laws1a){reference-type="eqref" reference="Laws1a"}. # EMAC Galerkin formulation is locally conservative {#S4} As an example of a continuous Galerkin method, we consider a conforming finite element method: Denote by $\mathbf V_h\subset H^1_0(\Omega)^d$ and $Q_h\subset L^2_0(\Omega)$ velocity and pressure finite element spaces with respect to a tessellation $\mathcal T_h$ of $\Omega$ into elements (simplexes or more general polygons or polyhedra). We also need the following auxiliary spaces of continuous finite elements of degree $m+1$ and $m$, with $m\ge1$: $$\label{wbV} \begin{split} V_h&= \{v\in H^1_0(\Omega):\,v\in\mathbb{P}_{m+1}(T)~\forall\,T\in\mathcal T_h\},\\ %\{\phi\in H^1_0(\Omega)\,:\, \be_i\phi\in \bV_h~\text{for}~i=1,\dots,d\},\\ \widetilde{V}_h&= \{v\in H^1_0(\Omega):\,v\in\mathbb{P}_{m}(T)~~~~\forall\,T\in\mathcal T_h\}.%\{\phi\in H^1_0(\Omega)\,:\, (\bx\times\be_i)\phi\in \bV_h~\text{for}~i=1,\dots,d\}. \end{split}$$ We only assume that the velocity space contains all piecewise polynomial continuous functions of degree $m+1$, i.e. $$\label{A1} (V_h)^d\subset \mathbf V_h.$$ We do not have any further assumptions on finite element spaces, and in particular, both LBB stable and stabilized finite elements are admitted. **Remark 1**. *Let $\mathcal T_h$ be a triangulation of $\Omega$ and $m\ge1$ be a polynomial degree. The following examples of LBB stable FE pairs satisfy the assumption: generalized Taylor--Hood $P_{m+1}-P_m$, $P_{m+1}-P_{m-1}^{\rm disc}$ (for $d=2$), $P_{m+1}-P_{m-2}^{\rm disc}$ (for $d=3$, $m>1$), $P_{m+1}^{\rm bubble}-P_{m-1}^{\rm disc}$ (for $d=3$ with face bubbles), generalized conforming Crouzeix--Raviart $P_{m+1}^{\rm bubble}-P_m^{\rm disc}$, Scott-Vogelius $P_{m+1}-P_m^{\rm disc}$ (SV element is LBB stable subject to further assumptions on $\mathcal T_h$ [@guzman2018inf]), as well as LBB unstable equal order $P_{m+1}-P_{m+1}$ elements.* We use $(f,g):=\int_\Omega f\cdot g\,dx$ notation for both scalar and vector functions $f,g$. The EMAC Galerkin formulation of [\[NSE\]](#NSE){reference-type="eqref" reference="NSE"} with $\mathbf u={\bf 0}$ on $\partial\Omega$ reads: Find $\mathbf u_h:(0,T)\to\mathbf V_h$ and $\widehat p_h:(0,T)\to Q_h\cap L^2_0(\Omega)$ $$\label{EMAC} \left\{ \begin{aligned} \Big(\frac{\partial \mathbf u_h}{\partial t},\mathbf v_h\Big)+2(\mathbf D(\mathbf u_h)\mathbf u_h,\mathbf v_h)+ ((\operatorname{div}\mathbf u_h)\mathbf u_h,\mathbf v_h) +2\nu(\mathbf D(\mathbf u_h),\mathbf D(\mathbf v_h))+(\widehat p_h,\operatorname{div}\mathbf v_h) &=0\quad\forall\,\mathbf v_h\in\mathbf V_h, \\ (\operatorname{div}\mathbf u_h,q_h)&=0 \quad\forall\,q_h\in Q_h, \end{aligned}\right.$$ where $\widehat{p}_h$ approximates the EMAC pressure $\widehat{p}=p-\tfrac12|\mathbf u|^2$. The EMAC formulation is equivalent to other commonly used discrete formulations if $\operatorname{div}\mathbf u_h=0$ pointwise. However, $(\operatorname{div}\mathbf u_h,q_h)=0$ does not imply $\operatorname{div}\mathbf u_h=0$ except in special settings. As a consequence, the discrete solution depends on the form of nonlinear terms used (i.e. EMAC, SKEW, CONV, ROT, etc.). Next, we demonstrate that the solution of [\[EMAC\]](#EMAC){reference-type="eqref" reference="EMAC"} satisfies discrete counterparts of local conservation laws in both Eulerian and Lagrangian forms. ## Local conservation in Eulerian form Unlike for the continuous problem, for the discrete case the counterparts of conservation laws in Eulerian and Lagrangian forms do not follow one from another and we have to consider them separately. We start with the Eulerian form. [Conservation of local linear momentum]{.ul}. Consider arbitrary $\phi_h\in V_h$, $\phi_h|_{\partial\Omega }=0$. Then $\phi_h\mathbf e_i\in\mathbf V_h$, for $i=1,\dots,d$, is a legitimate test function in [\[EMAC\]](#EMAC){reference-type="eqref" reference="EMAC"}. Letting $\mathbf v_h=\phi_h\mathbf e_i$ in [\[EMAC\]](#EMAC){reference-type="eqref" reference="EMAC"} we compute for the nonlinear term $$\begin{gathered} 2(\mathbf D(\mathbf u_h)\mathbf u_h,\phi_h\mathbf e_i)= (\mathbf u_h\cdot\nabla\mathbf u_h,\phi_h\mathbf e_i)+((\phi_h\mathbf e_i)\cdot\nabla\mathbf u_h,\mathbf u_h)\\ = -(\mathbf u_h\cdot\nabla(\phi_h\mathbf e_i),\mathbf u_h)-((\operatorname{div}\mathbf u_h)\mathbf u_h,\phi_h\mathbf e_i) -\tfrac12(\operatorname{div}(\phi_h\mathbf e_i)\mathbf u_h,\mathbf u_h) \\ = -(\mathbf u_h \cdot\nabla\phi_h,\mathbf u_h\cdot\mathbf e_i)-((\operatorname{div}\mathbf u_h)\mathbf u_h,\phi_h\mathbf e_i) - \tfrac12(\mathbf e_i\cdot\nabla\phi_h,|\mathbf u_h|^2).\end{gathered}$$ Substituting this in the first equation from [\[EMAC\]](#EMAC){reference-type="eqref" reference="EMAC"} with $\mathbf v_h=\phi_h\mathbf e_i$ we obtain $$\label{aux279} \Big(\frac{\partial \mathbf u_h}{\partial t},\phi_h\mathbf e_i\Big)-(\mathbf u_h \cdot\nabla\phi_h,\mathbf u_h\cdot\mathbf e_i)- \tfrac12(\mathbf e_i\cdot\nabla\phi_h,|\mathbf u_h|^2) +2\nu(\mathbf D(\mathbf u_h),\mathbf D(\phi_h\mathbf e_i))-(\widehat p_h,\operatorname{div}(\phi_h\mathbf e_i)) =0,$$ and after simple re-arrangements, $$\frac{d}{dt}\Big(\mathbf u_h\cdot\mathbf e_i,\phi_h\Big)-(\mathbf u_h \cdot\nabla\phi_h,\mathbf u_h\cdot\mathbf e_i)+2\nu(\mathbf D(\mathbf u_h)\nabla\phi_h,\mathbf e_i)-(\widehat p_h+ \tfrac12|\mathbf u_h|^2,\mathbf e_i\cdot\nabla\phi_h) =0.$$ Let $\mathbf n_h:=-\nabla\phi_h/|\nabla\phi_h|$ for $|\nabla\phi_h|\neq0$ (and arbitrary unit vector otherwise) and define $$\omega_h=\mbox{supp}(\phi_h)\quad \text{and}~~ p_h=\widehat p_h+ \tfrac12|\mathbf u_h|^2,$$ then from equation [\[aux279\]](#aux279){reference-type="eqref" reference="aux279"} for $i=1,\dots,d$ we get, $$\label{EmacLoc} \frac{d}{dt}\int_{\omega_h}\phi_h\mathbf u_h\,dx= 2\nu\int_{\omega_h}\mathbf D(\mathbf u_h)\mathbf n_h|\nabla\phi_h|\,dx-\int_{\omega_h}p_h\mathbf n_h|\nabla\phi_h|\,dx-\int_{\omega_h}\mathbf u_h(\mathbf u_h\cdot\mathbf n_h)|\nabla\phi_h|\,dx,$$ for any $\phi_h\in V_h$. This is the discrete analogue of the local momentum conservation in [\[Laws2phim\]](#Laws2phim){reference-type="eqref" reference="Laws2phim"}. [Conservation of local angular momentum]{.ul}. Consider arbitrary $\phi_h\in \widetilde{V}_h$. Then $\mathbf x\times\phi_h\mathbf e_i\in\mathbf V_h$ for $i=1,\dots,d$ is a legitimate test function. Letting $\mathbf v_h=\mathbf x\times\phi_h\mathbf e_i$ in [\[EMAC\]](#EMAC){reference-type="eqref" reference="EMAC"} we compute for the nonlinear term $$\begin{gathered} 2(\mathbf D(\mathbf u_h)\mathbf u_h,\mathbf x\times\phi_h\mathbf e_i)= (\mathbf u_h\cdot\nabla\mathbf u_h,\mathbf x\times\phi_h\mathbf e_i)+((\mathbf x\times\phi_h\mathbf e_i)\cdot\nabla\mathbf u_h,\mathbf u_h)\\ = -(\mathbf u_h\cdot\nabla(\mathbf x\times\phi_h\mathbf e_i),\mathbf u_h)-((\operatorname{div}\mathbf u_h)\mathbf u_h,\mathbf x\times\phi_h\mathbf e_i) -\tfrac12(\operatorname{div}(\mathbf x\times\phi_h\mathbf e_i)\mathbf u_h,\mathbf u_h) \\ = -(\mathbf u_h \cdot\nabla\phi_h,(\mathbf u_h\times\mathbf x)\cdot\mathbf e_i)-((\operatorname{div}\mathbf u_h)\mathbf u_h,\mathbf x\times\phi_h\mathbf e_i) - \tfrac12(\mathbf x\times\nabla\phi_h,\mathbf e_i|\mathbf u_h|^2).\end{gathered}$$ Substituting this in the first equation from [\[EMAC\]](#EMAC){reference-type="eqref" reference="EMAC"} with $\mathbf v_h=\mathbf x\times\phi_h\mathbf e_i$ we obtain $$\begin{gathered} \label{aux324} \Big(\frac{\partial \mathbf u_h}{\partial t},\mathbf x\times\phi_h\mathbf e_i\Big) -(\mathbf u_h \cdot\nabla\phi_h,(\mathbf u_h\times\mathbf x)\cdot\mathbf e_i)- \tfrac12(\mathbf x\times\nabla\phi_h,\mathbf e_i|\mathbf u_h|^2) \\ +2\nu(\mathbf D(\mathbf u_h),\mathbf D(\mathbf x\times\phi_h\mathbf e_i))-(\widehat p_h,\operatorname{div}(\mathbf x\times\phi_h\mathbf e_i)) =0.\end{gathered}$$ Simple re-arrangements give $$\frac{d}{dt}\Big(\mathbf u_h\times\mathbf x,\phi_h\mathbf e_i\Big)-(\mathbf u_h \cdot\nabla\phi_h,(\mathbf u_h\times\mathbf x)\cdot\mathbf e_i) +2\nu(\mathbf D(\mathbf u_h)\nabla\phi_h,\mathbf x\times\mathbf e_i)-((\widehat p_h+ \tfrac12|\mathbf u_h|^2)\mathbf e_i,\nabla\phi_h\times\mathbf x) =0.$$ From the above equality for $i=1,\dots,d$ we get $$\label{EmacLoc2} \frac{d}{dt}\int_{\omega_h}\phi_h\mathbf u_h\times\mathbf x\,dx =2\nu\int_{\omega_h}(\mathbf D(\mathbf u_h)\mathbf n_h)\times\mathbf x\,|\nabla\phi_h|\,dx-\int_{\omega_h}p_h(\mathbf n_h\times\mathbf x)|\nabla\phi_h|\,dx-\int_{\omega_h}(\mathbf u_h\times\mathbf x)(\mathbf u_h\cdot\mathbf n_h)|\nabla\phi_h|\,dx$$ for any $\phi_h\in \widetilde{V}_h$. This is the discrete analogue of the local angular momentum conservation from [\[Laws2phia\]](#Laws2phia){reference-type="eqref" reference="Laws2phia"}. ## Local conservation in Lagrangian form {#local-conservation-in-lagrangian-form .unnumbered} After discretization, there is no obvious equivalence between the Eulerian and Lagrangian forms of the local balances. Nevertheless, one can show that EMAC form also obeys a discrete counterpart of the linear momentum local conservation in the Lagrangian form. However, we need additional assumption on $V_h$ space. Namely, we assume that the velocity space consists of piecewise polynomial continuous functions of degree $m+1$: $$\label{A2} (V_h)^d =\mathbf V_h.$$ [Conservation of local linear momentum]{.ul}. Consider $\phi_h^0\in V_h$ and $\phi_h:\,[t_0,\hat t_1]\to V_h$ solving $$\label{trans1} \Big(\frac{\partial \phi_h}{\partial t},v_h\Big)+(\mathbf u_h\cdot \nabla\phi_h,v_h)=0\quad\forall~v_h\in V_h,$$ which is the projection of the transport equation [\[eq:ls\]](#eq:ls){reference-type="eqref" reference="eq:ls"} on the finite dimensional space $V_h$ with $\mathbf u$ replaced by $\mathbf u_h$. Letting $\mathbf v_h=\phi_h\mathbf e_i$ in [\[EMAC\]](#EMAC){reference-type="eqref" reference="EMAC"}, we repeat the same calculations as for the Eulerian case and arrive at [\[aux279\]](#aux279){reference-type="eqref" reference="aux279"}. Since $\phi_h$ is time dependent, after re-arrangements [\[aux279\]](#aux279){reference-type="eqref" reference="aux279"} gives $$\frac{d}{dt}\Big(\mathbf u_h\cdot\mathbf e_i,\phi_h\Big)-\Big(\mathbf u_h\cdot\mathbf e_i,\frac{\partial \phi_h}{\partial t}\Big)-(\mathbf u_h \cdot\nabla\phi_h,\mathbf u_h\cdot\mathbf e_i)+2\nu(\mathbf D(\mathbf u_h)\nabla\phi_h,\mathbf e_i)-(\widehat p_h+ \tfrac12|\mathbf u_h|^2,\mathbf e_i\cdot\nabla\phi_h) =0.$$ Thanks to the assumption [\[A2\]](#A2){reference-type="eqref" reference="A2"} and equation [\[trans1\]](#trans1){reference-type="eqref" reference="trans1"}, the second and third terms add to zero. Let $\mathbf n_h:=-\nabla\phi_h/|\nabla\phi_h|$ for $|\nabla\phi_h|\neq0$ (and arbitrary unit vector otherwise) and define $$\Omega_h(t)=\mbox{supp}(\phi_h)\quad \text{and}~~ p_h=\widehat p_h+ \tfrac12|\mathbf u_h|^2,$$ then from equation [\[aux279\]](#aux279){reference-type="eqref" reference="aux279"} for $i=1,\dots,d$ we get, $$\label{EmacLocL} \frac{d}{dt}\int_{\Omega_h(t)}\phi_h\mathbf u_h\,dx= 2\nu\int_{\Omega_h(t)}\mathbf D(\mathbf u_h)\mathbf n_h|\nabla\phi_h|\,dx-\int_{\Omega_h(t)}p_h\mathbf n_h|\nabla\phi_h|\,dx,$$ for any $\phi_h\in V_h$. This is the discrete analogue of the local momentum conservation in [\[Laws1phim\]](#Laws1phim){reference-type="eqref" reference="Laws1phim"}. [Conservation of local angular momentum]{.ul}. Consider $\phi_h^0\in \widetilde{V}_h$ and $\phi_h:\,[t_0,\hat t_1]\to \widetilde{V}_h$ solving $$\Big(\frac{\partial \phi_h}{\partial t},v_h\Big)+(\mathbf u_h\cdot \nabla\phi_h,v_h)=0\quad\forall~v_h\in \widetilde{V}_h.$$ Letting $\mathbf v_h=\mathbf x\times\phi_h\mathbf e_i$ in [\[EMAC\]](#EMAC){reference-type="eqref" reference="EMAC"} we repeat the same calculations as for the Eulerian case and arrive at [\[aux324\]](#aux324){reference-type="eqref" reference="aux324"}. Since $\phi_h$ is time dependent, after re-arrangements [\[aux324\]](#aux324){reference-type="eqref" reference="aux324"} gives $$\begin{gathered} \frac{d}{dt}\Big(\mathbf u_h\times\mathbf x,\phi_h\mathbf e_i\Big)-\Big(\frac{\partial \phi_h}{\partial t},(\mathbf u_h\times\mathbf x)\cdot\mathbf e_i\Big)-(\mathbf u_h \cdot\nabla\phi_h,(\mathbf u_h\times\mathbf x)\cdot\mathbf e_i) \\ +2\nu(\mathbf D(\mathbf u_h)\nabla\phi_h,\mathbf x\times\mathbf e_i)-((\widehat p_h+ \tfrac12|\mathbf u_h|^2)\mathbf e_i,\nabla\phi_h\times\mathbf x) =0.\end{gathered}$$ Denote by $I_m(\mathbf u_h\times\mathbf x)$ a piecewise polynomial of degree $m$ interpolating $\mathbf u_h\times\mathbf x$, i.e. $I_m(\mathbf u_h\times\mathbf x)\in \widetilde{V}_h^3$. Then $I_m(\mathbf u_h\times\mathbf x)\cdot\mathbf e_i\in\widetilde{V}_h$ holds. Therefore, $$\Big(\frac{\partial \phi_h}{\partial t},(\mathbf u_h\times\mathbf x)\cdot\mathbf e_i\Big)+(\mathbf u_h \cdot\nabla\phi_h,(\mathbf u_h\times\mathbf x)\cdot\mathbf e_i) =\left(\frac{d\phi_h}{d t},(\mathbf u_h\times\mathbf x-I_m(\mathbf u_h\times\mathbf x))\cdot\mathbf e_i \right)=: R_i,$$ where $\frac{d\phi_h}{d t}=\frac{\partial \phi_h}{\partial t}+\mathbf u_h \cdot\nabla\phi_h$. Let $\mathbf n_h:=-\nabla\phi_h/|\nabla\phi_h|$ for $|\nabla\phi_h|\neq0$ (and arbitrary unit vector otherwise) and define $$\Omega_h(t)=\mbox{supp}(\phi_h)\quad \text{and}~~ p_h=\widehat p_h+ \tfrac12|\mathbf u_h|^2,$$ then from equation [\[aux279\]](#aux279){reference-type="eqref" reference="aux279"} for $i=1,\dots,d$ we get, $$\label{EmacLoc2L} \frac{d}{dt}\int_{\Omega_h(t)}\phi_h\mathbf u_h\times\mathbf x\,dx =2\nu\int_{\Omega_h(t)}(\mathbf D(\mathbf u_h)\mathbf n_h)\times\mathbf x\,|\nabla\phi_h|\,dx-\int_{\Omega_h(t)}p_h(\mathbf n_h\times\mathbf x)|\nabla\phi_h|\,dx+R$$ for any $\phi_h\in \widetilde{V}_h$. This is the discrete analogue of the local momentum conservation in [\[Laws1phia\]](#Laws1phia){reference-type="eqref" reference="Laws1phia"} *up to the residual term* $R=R_1+\dots+R_d$. If we assume that $\mathbf u_h$ approximates a (smooth) solution to the NSE with order $O(h^{r})$, $r\ge m+1$, in some norm $\|\cdot\|_\ast$ then $\|R\|_\ast=O(h^{m+1})$ once $\frac{d\phi_h}{d t}$ is bounded in the dual norm to $\|\cdot\|_\ast$. According to [\[A2\]](#A2){reference-type="eqref" reference="A2"} the optimal approximation order for $\mathbf u_h$ would be $O(h^{m+2})$ in $L^2(L^2)$ norm. # Numerical Tests {#S5} We now give numerical examples to illustrate the theory above. For these tests, the full Navier--Stokes discretization uses BDF temporal discretizations (and Crank--Nicolson for the initial time steps) and $(\mathbf V_h,Q_h)=( P_2(\tau_h)^d,P_1(\tau_h))$ Taylor--Hood elements on a mesh $\tau_h$. The nonlinear problem at each time step is resolved with Newton's method, and typically it takes just 2 or 3 iterations to resolve. With temporal discretizations, the precise definitions of the discrete local balances will change accordingly, and we derive these now before proceeding to the tests. Denote by $\omega_h$ the approximation of a subdomain $\omega$ whose boundary consists of element edges from the mesh. Define the functions $\phi_h\in V_h$ and $\psi_h \in \tilde V_h$ nodally by $$\begin{aligned} \phi_h(x_j) = \left\{\begin{array}{c} 1 \ \mbox{ if $x_j$ is a node on $P_2(\tau_h)$ in the interior of $\omega_h$} \\0 \ \mbox{ otherwise, } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{array}\right. \label{phi1} \\ \psi_h(x_j) = \left\{\begin{array}{c} 1 \ \mbox{ if $x_j$ is a node on $P_1(\tau_h)$ in the interior of $\omega_h$} \\0 \ \mbox{ otherwise. } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{array}\right. \label{psi1}\end{aligned}$$ In our implementations we apply BDF formulas for the temporal discretization for the momentum and transport equations. For a sequence $\{f^n\}_{n=0,1,\dots}$ of scalar or vector quantities, we use the shortcut notations $$\begin{aligned} \left(\frac{df}{dt}\right)^{n}_{\rm bdf3}& = \frac{\frac{11}{6} f^{n} - 3 f^{n-1} + \frac{3}{2} f^{n-2} - \frac13 f^{n-3} }{\Delta t}, \\ \left(\frac{df}{dt}\right)^{n}_{\rm bdf2}&= \frac{3 f^{n} - 4f^{n-1} + f^{n-2}}{2\Delta t}, \\ \left(\frac{df}{dt}\right)^{n}_{\rm bdf1}&= \frac{f^{n} - f^{n-1}}{\Delta t}. \end{aligned}$$ ### Discrete local conservation in Eulerian form We consider first the discrete Eulerian form of local conservation of momentum and angular momentum. Choosing $\phi_h$ by [\[phi1\]](#phi1){reference-type="eqref" reference="phi1"} and repeating the arguments above that derived [\[EmacLoc\]](#EmacLoc){reference-type="eqref" reference="EmacLoc"} but using the BDFk (k=1,2,3) temporal discretization, we get the following (fully) discrete local momentum balance $$\begin{aligned} %\int_{\omega_h} & \phi_h \frac{\frac{11}{6} \bu_h^{n} - 3 \bu_h^{n} + \frac{3}{2} \bu_h^{n-1} - \frac13 \bu_h^{n-2} }{\Delta t} \,dx \\ &= \left(\frac{d\Big(\int_{\omega_h} \phi_h\mathbf u_h\,dx\Big)}{dt}\right)^{n}_{\rm bdfk} = 2\nu\int_{\omega_h}\mathbf D(\mathbf u_h^{n})\mathbf n_h|\nabla\phi_h|\,dx-\int_{\omega_h}p_h^{n}\mathbf n_h|\nabla\phi_h|\,dx-\int_{\omega_h}\mathbf u_h^{n}(\mathbf u_h^{n}\cdot\mathbf n_h)|\nabla\phi_h|\,dx,\end{aligned}$$ with $\mathbf n_h=-\nabla\phi_h/|\nabla\phi_h|$. Similarly, for discrete local angular momentum conservation we obtain $$\begin{aligned} % \int_{\omega_h}& \psi_h \left( \frac{\frac{11}{6} \bu_h^{n} - 3 \bu_h^{n} + \frac{3}{2} \bu_h^{n-1} - \frac13 \bu_h^{n-2} }{\Delta t} \right) \times\bx\,dx \\ & \left(\frac{d\Big(\int_{\omega_h} \psi_h\mathbf u_h\times\mathbf x\,dx\Big)}{dt}\right)^{n}_{\rm bdfk} =&2\nu\int_{\omega_h}(\mathbf D(\mathbf u_h^{n})\mathbf n_h)\times\mathbf x\,|\nabla\psi_h|\,dx\\ &-\int_{\omega_h}p_h^{n}(\mathbf n_h\times\mathbf x)|\nabla\psi_h|\,dx-\int_{\omega_h}(\mathbf u_h^{n}\times\mathbf x)(\mathbf u_h^{n}\cdot\mathbf n_h)|\nabla\psi_h|\,dx,\end{aligned}$$ where $\psi_h$ is defined by [\[psi1\]](#psi1){reference-type="eqref" reference="psi1"}, and $\mathbf n_h=-\nabla\psi_h/|\nabla\psi_h|$. In our tests, we will show plots of discrete local Eulerian momentum error $$\begin{gathered} e_{E}^{mom} = %\int_{\omega_h}\phi_h &\frac{\frac{11}{6} \bu_h^{n} - 3 \bu_h^{n} + \frac{3}{2} \bu_h^{n-1} - \frac13 \bu_h^{n-2} }{\Delta t} \,dx \left(\frac{d\Big(\int_{\omega_h} \phi_h\mathbf u_h\,dx\Big)}{dt}\right)^{n}_{\rm bdfk} - 2\nu\int_{\omega_h}\mathbf D(\mathbf u_h^{n})\mathbf n_h|\nabla\phi_h|\,dx\\ +\int_{\omega_h}p_h^{n}\mathbf n_h|\nabla\phi_h|\,dx+\int_{\omega_h}\mathbf u_h^{n}(\mathbf u_h^{n}\cdot\mathbf n_h)|\nabla\phi_h|\,dx ,\end{gathered}$$ and discrete local Eulerian angular momentum error $$\begin{gathered} e_{E}^{am} = %\int_{\omega_h}\psi_h \left( \frac{\frac{11}{6} \bu_h^{n} - 3 \bu_h^{n} + \frac{3}{2} \bu_h^{n-1} - \frac13 \bu_h^{n-2} }{\Delta t} \right) \times\bx\,dx \left(\frac{d\Big(\int_{\omega_h} \psi_h\mathbf u_h\times\mathbf x\,dx\Big)}{dt}\right)^{n}_{\rm bdfk} - 2\nu\int_{\omega_h}(\mathbf D(\mathbf u_h^{n})\mathbf n_h)\times\mathbf x\,|\nabla\psi_h|\,dx\\ +\int_{\omega_h}p_h^{n}(\mathbf n_h\times\mathbf x)|\nabla\psi_h|\,dx+\int_{\omega_h}(\mathbf u_h^{n}\times\mathbf x)(\mathbf u_h^{n}\cdot\mathbf n_h)|\nabla\psi_h|\,dx .\end{gathered}$$ ### Discrete local conservation in Lagrangian form Discrete local conservation in Lagrangian form is somewhat more complicated compared to the Eulerian case due to the $\phi_h$ function becoming time dependent in the momentum and angular momentum balances, as well as the transport equations involved in these balances being hyperbolic. As our tests are for illustrative purposes of certain theoretical properties, we approximate the transport equations in the following way for the purpose of ease in computations, even though other approaches to solving the transport equation may be better in practice. Consider $\phi_h^0\in V_h$ to be defined by [\[phi1\]](#phi1){reference-type="eqref" reference="phi1"}, and then define $\phi_h^{n} \in V_h$ (n=1,2,3,\...) via $$\label{trans1h} \Big( \left(\frac{d\phi_h}{dt}\right)^{n}_{\rm bdfj} %\frac{3\phi_h^{n} - 4\phi_h^n + \phi_h^{n-1}}{2\Delta t} ,v_h\Big)+(\mathbf u_h^{n} \cdot \nabla\phi_h^{n} ,v_h) %+ \epsilon (\nabla \phi_h^{n},\nabla v_h) =0\quad\forall~v_h\in V_h,$$ where $j=$`<!-- -->`{=html}1 or 2 in our numerical tests (and if $j=$`<!-- -->`{=html}2 then the first time step is backward Euler). Rederiving the discrete local Lagrangian momentum balance [\[EmacLocL\]](#EmacLocL){reference-type="eqref" reference="EmacLocL"} but now using BDFk ($k=$`<!-- -->`{=html}2 or 3) time stepping for Navier-Stokes together with [\[trans1h\]](#trans1h){reference-type="eqref" reference="trans1h"}, we obtain the balance $$\int_{\Omega} \left( \phi_h \left(\frac{d\mathbf u_h}{dt}\right)^{n}_{\rm bdfk} + \left(\frac{d\phi_h}{dt}\right)^{n}_{\rm bdfj} \mathbf u_h^{n}\right) \ dx = 2\nu\int_{\Omega }\mathbf D(\mathbf u_h^{n})\mathbf n^{n}_h|\nabla\phi^{n}_h|\,dx-\int_{\Omega}p_h^{n} \mathbf n^{n}_h|\nabla\phi^{n}_h|\,dx,$$ and thus define the discrete local Lagrangian momentum error by $$e_L^{mom} = \int_{\Omega} \left( \phi_h^{n} \left(\frac{d\mathbf u_h}{dt}\right)^{n}_{\rm bdfk} + \left(\frac{d\phi_h}{dt}\right)^{n}_{\rm bdfj} \mathbf u_h^{n}\right) \ dx - \left( 2\nu\int_{\Omega }\mathbf D(\mathbf u_h^{n})\mathbf n^{n}_h|\nabla\phi^{n}_h|\,dx-\int_{\Omega}p_h^{n} \mathbf n^{n}_h|\nabla\phi^{n}_h|\,dx \right).$$ Note that if the transport equations are solved in a different way, then these definitions of discrete Lagrangian momentum and angular momentum balances need modified accordingly. For example if an explicit method is used, then the local balance will be defined with some terms at time $t^{n}$ and others and time $t^{n-1}$. For angular momentum, we proceed similarly as for momentum to find a fully discrete analogue to [\[EmacLoc2L\]](#EmacLoc2L){reference-type="eqref" reference="EmacLoc2L"}. Let $\psi_h^0$ be defined by [\[psi1\]](#psi1){reference-type="eqref" reference="psi1"} and find $\psi_h^{n}\in \tilde V_h\cap H^1_0(\Omega)$ for $n=1,2,3,...$ by $$\Big(\left(\frac{d\psi_h}{dt}\right)^{n}_{\rm bdfj},v_h\Big)+(\mathbf u_h^{n} \cdot \nabla\psi_h^{n},v_h) =0\quad\forall~v_h\in \tilde V_h\cap H^1_0(\Omega),$$ and using backward Euler for the first time step if $j=$`<!-- -->`{=html}2. Following similar steps as the theory above, the discrete Lagrangian local angular momentum error is then given by $$\begin{aligned} e_L^{am} = \bigg( \int_{\Omega} \bigg( \psi_h^{n} & \left(\frac{d\mathbf u_h}{dt}\right)^{n}_{\rm bdfk} + \left(\frac{d\psi_h}{dt}\right)^{n}_{\rm bdfj} \mathbf u_h^{n} \bigg) \times \mathbf x \, dx \\ & - 2\nu\int_{\Omega}(\mathbf D(\mathbf u_h^{n})\mathbf n_h^{n})\times\mathbf x\,|\nabla\psi_h^{n}|\,dx+\int_{\Omega}p_h^{n}(\mathbf n_h^{n}\times\mathbf x)|\nabla\psi_h^{n}|\,dx \bigg).\end{aligned}$$ ## Gresho problem For our first test we use a slight variation of the classical Gresho problem on $\Omega=(-0.5,0.5)^2$, which consists r0.37 -2ex {width="35%" height="35%"} -3ex -18ex of a velocity and pressure $$\begin{aligned} \small \mathbf u=\begin{cases} \begin{bmatrix} -5y~5x \end{bmatrix}^T &\text{for }r< 2,\\[1ex] \begin{bmatrix} \frac{2y}{r}+5y~\frac{2x}{r}-5x \end{bmatrix}^T &\text{for }.2 \leq r\leq .4,\\ \begin{bmatrix} 0~0 \end{bmatrix}^T &\text{for }r>.4, \end{cases},\end{aligned}$$ $$\begin{aligned} \small p=\begin{cases} 12.5r^2+C_1 &\text{for }r<.2,\\ 12.5r^2-20r+4\log(r)+C_2 &\text{for }.2 \leq r \leq .4,\\ 0 &\text{for }r>.4, \end{cases}\end{aligned}$$ where $r=\sqrt{x^2+y^2}$ and $$\begin{aligned} &C_2=-12.5(.4)^2+20(.4)^2-4\log(.4),\\ &C_1=C_2-20(.2)+4\log(.2).\end{aligned}$$ This velocity is plotted in figure [\[fig:True_Gresho\]](#fig:True_Gresho){reference-type="ref" reference="fig:True_Gresho"} and is an exact solution of the unforced steady Euler equations, and hence an accurate solver should preserve the initial condition in time. It is shown in [@CHOR17; @OR20] that a NSE solver with EMAC nonlinearity and using Crank-Nicolson time stepping together with Taylor-Hood finite element spatial discretization will preserve pointwise global energy, momentum and angular momentum for this problem while other common nonlinearity formulations such as SKEW, ROT and CONV will not preserve these physical balance laws and moreover will be less accurate in the sense of $L^2(\Omega)$ error. {#Gresho width="30%" height="28%"} {#Gresho width="30%" height="28%"} {#Gresho width="30%" height="28%"} We alter this problem very slightly by changing the viscosity to $\nu=10^{-10}$ so as not to solve the Euler equations but instead the NSE. We note this change of viscosity will (very slightly) change the true solution in time, however this is of no consequence as our interest herein is not the solution but the local conservation of momentum and angular momentum. We choose $\omega$ to be the circle of radius $0.05$ centered at $(0.2,0.09)$, as shown in figure [3](#Gresho){reference-type="ref" reference="Gresho"} at left. Figure [3](#Gresho){reference-type="ref" reference="Gresho"} also shows the mesh $\tau_h$ used for the computations below as well as the mesh zoomed in near $\omega$. We define $\omega_h$ to be the approximation of $\omega$ whose boundary consists of triangle edges from the mesh. ![Shown above is error in discrete local Eulerian (left) and Lagrangian (right) momentum and angular momentum conservation versus time in the (viscous) Gresho problem. [\[EerrG\]]{#EerrG label="EerrG"} ](GEerr.pdf "fig:"){#EerrG width="49%" height="27%"} ![Shown above is error in discrete local Eulerian (left) and Lagrangian (right) momentum and angular momentum conservation versus time in the (viscous) Gresho problem. [\[EerrG\]]{#EerrG label="EerrG"} ](GLerr.pdf "fig:"){#EerrG width="49%" height="27%"} Computations are done using $T=0.5$, $\Delta t=0.005$ and ${\bf f}={\bf 0}$, and we show errors in Eulerian conservation laws in figure [5](#EerrG){reference-type="ref" reference="EerrG"} as absolute values of errors versus time. We observe these quantities are conserved pointwise and are stable in time, just as the theory above predicts. Plots of discrete local Lagrangian momentum and angular momentum versus time are also shown in figure [5](#EerrG){reference-type="ref" reference="EerrG"}, and we observe as expected that momentum is preserved pointwise (up to machine roundoff error). We also observe that Lagrangian angular momentum is not preserved pointwise but instead has values as large as $O(10^{-4})$, which is consistent with the $O(h^2)$ residual our theory above predicts. ## Kelvin-Helmholtz flow ![Shown above are the absolute vorticity contours of the solution velocity at $t=$`<!-- -->`{=html}0, 0.5, 1, 2, 3, 4, 4.5 and 5 (left to right, top to bottom). [\[KHsoln\]]{#KHsoln label="KHsoln"} ](KH0.pdf "fig:"){#KHsoln width="30%" height="27%"} ![Shown above are the absolute vorticity contours of the solution velocity at $t=$`<!-- -->`{=html}0, 0.5, 1, 2, 3, 4, 4.5 and 5 (left to right, top to bottom). [\[KHsoln\]]{#KHsoln label="KHsoln"} ](KHp5.pdf "fig:"){#KHsoln width="30%" height="27%"} ![Shown above are the absolute vorticity contours of the solution velocity at $t=$`<!-- -->`{=html}0, 0.5, 1, 2, 3, 4, 4.5 and 5 (left to right, top to bottom). [\[KHsoln\]]{#KHsoln label="KHsoln"} ](KH1.pdf "fig:"){#KHsoln width="30%" height="27%"}\ ![Shown above are the absolute vorticity contours of the solution velocity at $t=$`<!-- -->`{=html}0, 0.5, 1, 2, 3, 4, 4.5 and 5 (left to right, top to bottom). [\[KHsoln\]]{#KHsoln label="KHsoln"} ](KH2.pdf "fig:"){#KHsoln width="30%" height="27%"} ![Shown above are the absolute vorticity contours of the solution velocity at $t=$`<!-- -->`{=html}0, 0.5, 1, 2, 3, 4, 4.5 and 5 (left to right, top to bottom). [\[KHsoln\]]{#KHsoln label="KHsoln"} ](KH3.pdf "fig:"){#KHsoln width="30%" height="27%"} ![Shown above are the absolute vorticity contours of the solution velocity at $t=$`<!-- -->`{=html}0, 0.5, 1, 2, 3, 4, 4.5 and 5 (left to right, top to bottom). [\[KHsoln\]]{#KHsoln label="KHsoln"} ](KH4.pdf "fig:"){#KHsoln width="30%" height="27%"}\ ![Shown above are the absolute vorticity contours of the solution velocity at $t=$`<!-- -->`{=html}0, 0.5, 1, 2, 3, 4, 4.5 and 5 (left to right, top to bottom). [\[KHsoln\]]{#KHsoln label="KHsoln"} ](KH4p5.pdf "fig:"){#KHsoln width="30%" height="27%"} ![Shown above are the absolute vorticity contours of the solution velocity at $t=$`<!-- -->`{=html}0, 0.5, 1, 2, 3, 4, 4.5 and 5 (left to right, top to bottom). [\[KHsoln\]]{#KHsoln label="KHsoln"} ](KH5.pdf "fig:"){#KHsoln width="30%" height="27%"} For our second test we consider a test problem from [@SJLLLS18] for simulating 2D Kelvin-Helmholtz instability. The domain is the unit square, with periodic boundary conditions at $x=0,1$, representing an infinite extension in the horizontal direction. At $y=0,1$, we enforce for $t>0$ a no slip condition, which differs from [@SJLLLS18] as they use a no penetration and free slip condition. However, as these boundaries are far from the physical behavior of interest, there is little effect on the qualitative behavior of the solution. The initial condition is set by $$\mathbf u_0(x,y) = \left( \begin{array}{c} u_{\infty} \tanh\left( \frac{2y-1}{\delta_0} \right) \\ 0 \end{array} \right) + c_n \left( \begin{array}{c} \partial_y \psi(x,y) \\ -\partial_x \psi(x,y) \end{array} \right),$$ where $\delta_0=\frac{1}{28}$ is the initial vorticity thickness, $u_{\infty}=1$ is a reference velocity, $c_n$ is a noise/scaling factor taken to be $10^{-3}$, and $$\psi(x,y) = u_{\infty} \exp \left( -\frac{(y-0.5)^2 }{\delta_0^2} \right) \left( \cos(8\pi x) + \cos(20\pi x) \right).$$ The Reynolds number is defined by $Re=\frac{\delta_0 u_{\infty}}{\nu} = \frac{1}{28 \nu}$, and $\nu$ is defined by selecting $Re$. We use $Re=$`<!-- -->`{=html}100 for our test. We compute solutions for EMAC discretized with $(P_2,P_1)$ Taylor-Hood elements on a $h=\frac{1}{128}$ uniform mesh, together with BDF2 time stepping and a time step size of $\Delta t=0.01$. Solutions are computed up to $T=$`<!-- -->`{=html}5, with plots of vorticity contours shown in figure [13](#KHsoln){reference-type="ref" reference="KHsoln"} matching those in [@SJLLLS18] qualitatively well. The subdomain $\omega$ is defined to be the square $[\frac18,\frac14] \times [\frac18,\frac14]$. For this domain on this mesh, we have that $\omega_h=\omega$. Plots of discrete Eulerian momentum and angular momentum are shown in figure [15](#KHerr){reference-type="ref" reference="KHerr"} at the top, and we observe that these quantities are conserved exactly, just as the theory above predicts. For discrete Lagrangian momentum and angular momentum, we solve the transport equation using backward Euler (BDF1), and plots of momentum and angular momentum are shown in figure [15](#KHerr){reference-type="ref" reference="KHerr"} at bottom. We observe exact local conservation of discrete Lagrangian momentum, and conservation of discrete Lagrangian angular momentum consistent with the discretization error, as predicted above. ![Shown above is error in discrete local Eulerian (left) and Lagrangian (right) momentum and angular momentum conservation versus time in the Kelvin-Helmholtz problem. [\[KHerr\]]{#KHerr label="KHerr"} ](KHEerr.pdf "fig:"){#KHerr width="49%" height="27%"} ![Shown above is error in discrete local Eulerian (left) and Lagrangian (right) momentum and angular momentum conservation versus time in the Kelvin-Helmholtz problem. [\[KHerr\]]{#KHerr label="KHerr"} ](KHLerr.pdf "fig:"){#KHerr width="49%" height="27%"} ## 2D flow past a cylinder ![The domain for the channel flow past a cylinder numerical experiment.[\[cyldomain\]]{#cyldomain label="cyldomain"} ](cyl.pdf "fig:"){#cyldomain width="98%" height="25%"}\ ![Shown above is error in discrete local Eulerian momentum and angular momentum conservation versus time in the 2D channel flow past a cylinder test.[\[cylerr\]]{#cylerr label="cylerr"} ](CylErrorE.pdf){#cylerr width="70%" height="25%"} For our last test we consider the 2D channel flow past a cylinder problem from [@ST96; @J04]. The domain is the rectangle $[0,2.2]\times [0,0.41]$ as shown in figure [16](#cyldomain){reference-type="ref" reference="cyldomain"}, with a cylinder centered at $(0.2,0.2)$ with radius $0.05.$ We take no external forcing ${\bf f}={\bf 0}$, $\nu=0.001$ (which corresponds to Reynolds number 100), and set inflow/outflow profiles to be $$\begin{aligned} u_1(0,y,t) & = u_1(2.2,y,t) = \frac{6}{0.41^2}y(0.41-y), \\u_2(0,y,t) & = u_2(2.2,y,t) = 0.\end{aligned}$$ We define a subdomain $\omega$ to be a circle radius $0.05$ centered at $(0.35,0.15)$, and $\omega_h$ to be its approximation by the mesh. A plot of $\omega$ is shown in figure [16](#cyldomain){reference-type="ref" reference="cyldomain"}. We use a mesh that provides approximately 60K velocity degrees of freedom (dof) and 7K pressure dof when discretized with Taylor-Hood elements. We compute using BDF3 time stepping to $T=2$ using time step size $\Delta t=0.001$, and start the flow from rest. Errors in discrete local Eulerian momentum and angular momentum conservation are shown in figure [17](#cylerr){reference-type="ref" reference="cylerr"} and we once again observe pointwise local conservation. We do not consider Lagrangian discrete local conservation for this test, since there is an outflow and conservation is therefore not expected except for very short times as the transported quantity will exit the domain through the outflow. # Future directions We have shown that continuous Galerkin discretizations of the Navier-Stokes equations using EMAC nonlinearity form admit (appropriately defined) exact local balances / conservation laws of momentum and angular momentum. These discrete local balances are constructed as weak forms of the momentum and angular momentum conservation laws, and are equivalent to the usual conservation law definitions before discretization. In the discrete case, however, these weak formulations are not equivalent to the usual conservation law definitions and even their Eulerian and Lagrangian constructions are not equivalent. That the discrete schemes admit any exact local balances at all is quite rare, and we note that such an analysis is not possible for such common Navier-Stokes nonlinearity formulations such as convective, skew-symmetric or rotational. We remark that the 'conservative' formulation of nonlinear terms (referred to as CONS, utilizing $\operatorname{div}(\mathbf u_h \mathbf u_h^T)$) also maintains the same conservation properties for momenta as EMAC. However, CONS fails to achieve a proper global energy balance, when $\operatorname{div}\mathbf u_h\neq0$, unlike EMAC. This deficiency leads to unstable finite element schemes using CONS; see examples of CONS underperformance in [@CHOR17; @CHOR19]. Future directions for this work could include an extension of these ideas to other conservation laws of Navier-Stokes such as energy, helicity, enstrophy in 2D, vorticity, and others. It is currently unclear to the authors how to construct appropriate local balances for these quantities for the continuous Galerkin method. # Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Both authors report financial support was provided by National Science Foundation. # Acknowledgements M.O. was partially supported by the NSF grant DMS-2309197. L.R. was partially supported by the NSF grant DMS-2152623. We would like to express our gratitude to Thomas J.R. 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arxiv_math

--- abstract: | In this paper, we investigate some geodesics and $F$-geodesics problems on tangent bundle and on $\varphi$-unit tangent bundle $T^{\varphi}_{1}M$ equipped with the $\varphi$-Sasaki metric over para-Kähler-Norden manifold $(M^{2m}, \varphi, g)$. **2010 Mathematics subject classifications:** Primary 53C22, 58E10 ; Secondary 53C15, 53C07. **Keywords:** Para-Kähler-Norden manifold, $\varphi$-Sasaki metric, F-geodesics. **Corresponding author**: Abderrahim ZAGANE address: RELIZANE University, Faculty of Science and Technology, Department of Mathematics, 48000, RELIZANE-ALGERIA author: - Abderrahim ZAGANE$^{1}$ title: Geodesics and $F$-geodesics on tangent bundle with $\varphi$-Sasaki metric over para-Kähler-Norden manifold --- # Introduction On the tangent bundle of a Riemannian manifold one can define natural Riemannian metrics. Their construction makes use of the Levi-Civita connection. Among them, the so called Sasaki metric [@Sas] is of particular interest. That is why the geometry of tangent bundle equipped with the Sasaki metric has been studied by many authors such as Yano and Ishihara [@Y.I], Dombrowski [@Dom], Salimov and his collaborators [@S.G.A] etc. The rigidity of Sasaki metric has incited some researchers to construct and study other metrics on tangent bundle. This is the reason why they have attempted to search for different metrics on the tangent bundle which are different deformations of the Sasaki metric. Among them, we mention the Cheeger-Gromoll metric [@M.T], the Mus-Sasaki Metric [@Z.D2] and the Berger-type deformed Sasaki metric [@A.S.G; @Yam]. The geometry of tangent bundle remains a rich area of research in differential geometry to this day. Geodesy on the tangent bundle has been studied by many authors. In particular the oblique geodesics, non-vertical geodesics and their projections onto the base manifold. Sasaki [@Sas2] and Sato [@Sat] gave a complete description of the curves and vector fields along them which generated non-vertical geodesics on the tangent bundle and on unit the tangent bundle respectively. They proved that the projected curves have constant geodesic curvatures (Frenet curvatures). Nagy [@Nag] generalized these results to the case of locally symmetric base manifold. Yampolsky [@Yam] also did the same studies on the tangent bundle and on unit the tangent bundle with the Berger-type deformed Sasaki metric over Kählerian manifold, in the cases of locally symmetric base manifold and of the constant holomorphic curvature base manifold. Also, we refer to [@D.Z3; @S.G.A; @S.K; @Zag14]. The notion of $F$-planar curves generalizes the magnetic curves and implicitly the geodesics (see [@H.M; @M.S]), but the notion of $F$-geodesic, which is slightly different from that of $F$-planar curve [@B.D]. Recently, in mathematics literature, a series of papers on magnetic curves, $F$-planar curves and $F$-geodesic (see [@D.G.K; @D.I.M.N; @Nis]). In previous works, [@Zag7; @Zag21], we proposed the $\varphi$-Sasaki metric on the tangent bundle over para-Kähler-Norden manifold $(M^{2m}, \varphi, g)$, where we studied respectively the para-Kähler-Norden properties on the tangent bundle and then Geometry of $\varphi$-Sasaki metric on tangent bundle. In this paper, we investigate some geodesics and $F$-geodesics properties for the $\varphi$-Sasaki metric on the tangent bundle and on $\varphi$-unit tangent bundle. Firstly, we study the geodesics on $\varphi$-unit tangent bundle with respect to the $\varphi$-Sasaki metric, where we establish necessary and sufficient conditions under which a curve be a geodesic with respect to this metric (Theorem [Theorem 5](#th_2){reference-type="ref" reference="th_2"}, Corollary[Corollary 7](#co_1){reference-type="ref" reference="co_1"} and Corollary[Corollary 8](#co_2){reference-type="ref" reference="co_2"}), we then discuss the Frenet curvatures of the projected of the non-vertical geodesic (Theorem [Theorem 11](#th_3){reference-type="ref" reference="th_3"}, Theorem [Theorem 13](#th_4){reference-type="ref" reference="th_4"}, Corollary[Corollary 14](#co_3){reference-type="ref" reference="co_3"} and Theorem [Theorem 15](#th_5){reference-type="ref" reference="th_5"}). Secondly, we study the $F$-geodesics and $F$-planar curves on tangent bundle with respect to the $\varphi$-Sasaki metric (Theorem [Theorem 18](#th_8){reference-type="ref" reference="th_8"}, Theorem [Theorem 20](#th_9){reference-type="ref" reference="th_9"} and Theorem [Theorem 22](#th_10){reference-type="ref" reference="th_10"}). Finally, we study the $F$-geodesics and $F$-planar curves on the $\varphi$-unit tangent bundle with respect to the $\varphi$-Sasaki metric (Theorem [Theorem 26](#th_11){reference-type="ref" reference="th_11"}, Theorem [Theorem 28](#th_12){reference-type="ref" reference="th_12"} and Theorem [Theorem 30](#th_13){reference-type="ref" reference="th_13"}). # Preliminaries Let $TM$ be the tangent bundle over an $m$-dimensional Riemannian manifold $(M^{m},g)$ and the natural projection $\pi: TM \rightarrow M$. A local chart $(U,x^{i})_{i=\overline{1,m}}$ on $M$ induces a local chart $(\pi^{-1}(U),x^{i},\xi^{i})_{i=\overline{1,m}}$ on $TM$. Denote by $\Gamma_{ij}^{k}$ the Christoffel symbols of $g$ and by $\nabla$ the Levi-Civita connection of $g$. The Levi Civita connection $\nabla$ defines a direct sum decomposition $$\begin{aligned} T_{(x,\xi)}TM=V_{(x,\xi)}TM\oplus H_{(x,\xi)}TM.\end{aligned}$$ of the tangent bundle to $TM$ at any $(x,\xi)\in TM$ into vertical subspace $$\begin{aligned} V_{(x,\xi)}TM&=&Ker(d\pi_{(x,\xi)})=\{a^{i}\frac{\partial}{\partial \xi^{i}}|_{(x,\xi)},\, a^{i}\in\mathbb{R}\},\end{aligned}$$ and the horizontal subspace $$\begin{aligned} H_{(x,\xi)}TM&=&\{a^{i}\frac{\partial}{\partial x^{i}}|_{(x,\xi)}-a^{i}\xi^{j}\Gamma_{ij}^{k}\frac{\partial}{\partial \xi^{k}}|_{(x,\xi)},\, a^{i}\in \mathbb{R}\}.\end{aligned}$$. Let $Z=Z^{i}\frac{\partial}{\partial x^{i}}$ be a local vector field on $M$. The vertical and the horizontal lifts of $Z$ are defined by $$\begin{aligned} {}^{V}\!Z&=& Z^{i}\frac{\partial}{\partial \xi^{i}},\\ {}^{H}\!Z&=&Z^{i}(\frac{\partial}{\partial x^{i}}- \xi^{j}\Gamma_{ij}^{k}\frac{\partial}{\partial \xi^{k}}).\end{aligned}$$ We have ${}^{H}\!(\frac{\partial}{\partial x^{i}})=\frac{\partial}{\partial x^{i}}- \xi^{j}\Gamma_{ij}^{k}\frac{\partial}{\partial \xi^{k}}$ and ${}^{V}\!(\frac{\partial}{\partial x^{i}})=\frac{\partial}{\partial \xi^{i}}$, then $({}^{H}\!(\frac{\partial}{\partial x^{i}}),{}^{V}\!(\frac{\partial}{\partial x^{i}}))_{i=\overline{1,m}}$ is a local adapted frame on $TTM$. # $\varphi$-Sasaki metric An almost product structure $\varphi$ on a manifold $M$ is a $(1,1)$-tensor field on $M$ such that $\varphi^{2}=id_{M}$, $\varphi\neq \pm id_{M}$ ($id_{M}$ is the identity tensor field of type $(1,1)$ on $M$). The pair $(M,\varphi)$ is called an almost product manifold. An almost para-complex manifold is an almost product manifold $(M,\varphi)$, such that the two eigenbundles $TM^{+}$ and $TM^{-}$ associated to the two eigenvalues $+1$ and $-1$ of $\varphi$, respectively, have the same rank. Note that the dimension of an almost par-acomplex manifold is necessarily even [@C.F.G]. An almost para-complex structure $\varphi$ is integrable if the Nijenhuis tensor: $$\begin{aligned} N_{\varphi}(X,Y)=[\varphi X,\varphi Y]-\varphi[X,\varphi Y]-\varphi[\varphi X,Y]+[X,Y] \end{aligned}$$ vanishes identically on $M$. On the other hand, in order that an almost para-complex structure be integrable, it is necessary and sufficient that we can introduce a torsion free linear connection $\nabla$ such that $\nabla\varphi=0$ [@S.I.E]. An almost para-complex Norden manifold $(M^{2m}, \varphi, g)$ is a $2m$-dimensional differentiable manifold $M$ with an almost para-complex structure $\varphi$ and a Riemannian metric $g$ such that: $$\begin{aligned} g(\varphi X, Y)= g(X, \varphi Y)&\Leftrightarrow& g(\varphi X, \varphi Y) = g(X, Y),\end{aligned}$$ for any vector fields $X$ and $Y$ on $M$, in this case $g$ is called a pure metric with respect to $\varphi$ or para-Norden metric (B-metric)[@S.I.E]. Also note that $$\begin{aligned} \label{twin} G(X,Y) = g(\varphi X, Y),\end{aligned}$$ is a bilinear, symmetric tensor field of type $(0, 2)$ on $(M, \varphi)$ and pure with respect to the paracomplex structure $\varphi$, which is called the twin (or dual) metric of $g$, and it plays a role similar to the Kähler form in Hermitian Geometry. Some properties of twin Norden metric are investigated in [@I.S; @S.I.E]. A para-Kähler-Norden manifold is an almost para-complex Norden manifold $(M^{2m}, \varphi, g)$ such that $\varphi$ is integrable i.e. $\nabla \varphi = 0$, where $\nabla$ is the Levi-Civita connection of $g$ [@S.G.I; @S.I.E]. It is well known that if $(M^{2m}, \varphi, g)$ is a para-Kähler-Norden manifold, the Riemannian curvature tensor is pure [@S.I.E]. **Definition 1**. [@Zag7] Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold. On the tangent bundle $TM$, we define a $\varphi$-Sasaki metric noted $g^{\varphi}$ by $$\begin{aligned} (1)\quad g^{\varphi}({}^{H}\!X,{}^{H}\!Y)_{(x,\xi)} &=&g_{x}(X,Y),\nonumber\\ (2)\quad g^{\varphi}({}^{H}\!X,{}^{V}\!Y)_{(x,\xi)} &=& 0,\nonumber\\ (3)\quad g^{\varphi}({}^{V}\!X,{}^{V}\!Y)_{(x,\xi)} &=& g_{x}(X,\varphi Y)=G_{x}(X, Y),\nonumber\end{aligned}$$ for any vector fields $X$ and $Y$ on $M$ and $(x,\xi)\in TM$, where $G$ is the twin Norden metric of $g$ defined by [\[twin\]](#twin){reference-type="eqref" reference="twin"}. **Theorem 2**. *[@Zag7] Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $TM$ its tangent bundle equipped with the $\varphi$-Sasaki metric $g^{\varphi}$. If $\nabla$ $($resp $\widetilde{\nabla}$$)$ denote the Levi-Civita connection of $(M^{2m}, \varphi, g)$ $($resp $(TM,g^{\varphi})$ $)$, then we have $$\begin{aligned} (1)\quad (\widetilde{\nabla}_{{}^{H}\!X}{}^{H}\!Y)_{(x,\xi)}&=&{}^{H}\!(\nabla_{X}Y)_{(x,\xi)}-\frac{1}{2}{}^{V}\!(R_{x}(X,Y)\xi),\\ (2)\quad (\widetilde{\nabla}_{{}^{H}\!X}{}^{V}\!Y)_{(x,\xi)}&=&{}^{V}\!(\nabla_{X}Y)_{(x,\xi)}+\dfrac{1}{2}{}^{H}\!(R_{x}(\varphi \xi,Y)X),\\ (3)\quad (\widetilde{\nabla}_{{}^{V}\!X}{}^{H}\!Y)_{(x,\xi)}&=&\dfrac{1}{2}{}^{H}\!(R_{x}(\varphi \xi,X)Y),\\ (4)\quad (\widetilde{\nabla}_{{}^{V}\!X}{}^{V}\!Y)_{(x,\xi)}&=&0,\end{aligned}$$ for all vector fields $X$ and $Y$ on $M$ and $(x,\xi)\in TM$, where $R$ denote the curvature tensor of $(M^{2m}, \varphi, g)$.* The $\varphi$-unit tangent sphere bundle over a para-Kähler-Norden manifold $(M^{2m}, \varphi, g)$, is the hypersurface $$\begin{aligned} T^{\varphi}_{1}M=\big\{(x,\xi)\in TM,\,g(\xi,\varphi \xi)=1\big\}.\end{aligned}$$ The unit normal vector field to $T^{\varphi}_{1}M$ is given by $$\mathcal{N}={}^{V}\!\xi.$$ The tangential lift ${}^{T}\!X$ with respect to $g^{\varphi}$ of a vector $X\in T_{x}M$ to $(x,\xi)\in T^{\varphi}_{1}M$ as the tangential projection of the vertical lift of $X$ to $(x,\xi)$ with respect to $\mathcal{N}$, that is $${}^{T}\!X={}^{V}\!X-g^{\varphi}_{(x,\xi)}({}^{V}\!X,\mathcal{N}_{(x,\xi)})\mathcal{N}_{(x,\xi)}={}^{V}\!X-g_{x}(X,\varphi \xi){}^{V}\!\xi.$$ The tangent space $T_{(x,\xi)}T^{\varphi}_{1}M$ of $T^{\varphi}_{1}M$ at $(x,\xi)$ is given by $$\begin{aligned} T_{(x,\xi)}T^{\varphi}_{1}M= \{{}^{H}\!X+{}^{T}\!Y\,/\, X \in T_{x}M, Y\in \xi^{\bot}\subset T_{x}M\}.\end{aligned}$$ where $\xi^{\bot}=\big\{Y\in T_{x}M,\,g(Y,\varphi \xi)=0\big\}$, see [@Zag21]. **Theorem 3**. *[@Zag21] Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric. If $\widehat{\nabla}$ denote the Levi-Civita connection of $\varphi$-Sasaki metric on $T^{\varphi}_{1}M$, then we have the following formulas. $$\begin{aligned} 1.\,\widehat{\nabla}_{{}^{H}\!X}{}^{H}\!Y&=&{}^{H}\!(\nabla_{X}Y)-\frac{1}{2}{}^{T}\!(R(X,Y)\xi),\\ 2.\,\widehat{\nabla}_{{}^{H}\!X}{}^{T}\!Y&=&{}^{T}\!(\nabla_{X}Y)+\dfrac{1}{2}{}^{H}\!\left(R(\varphi \xi,Y)X\right),\\ 3.\,\widehat{\nabla}_{{}^{T}\!X}{}^{H}\!Y&=&\dfrac{1}{2}{}^{H}\!\left(R(\varphi \xi,X)Y\right),\\ 4.\,\widehat{\nabla}_{{}^{T}\!X}{}^{T}\!Y&=&-g(Y,\varphi \xi){}^{T}\!X,\end{aligned}$$ for all vector fields $X$ and $Y$ on $M$, where $\nabla$ is the Levi-Civita connection and $R$ is its curvature tensor of $(M^{2m}, \varphi, g)$.* # Geodesics on $\varphi$-unit tangent bundle with the $\varphi$-Sasaki metric Let $\Gamma =(\gamma(t),\xi(t))$ be a naturally parameterized curve on the tangent bundle $TM$ (i.e. $t$ is an arc length parameter on $\Gamma$), where $\gamma$ is a curve on $M$ and $\xi$ is a vector field along this curve. Denote $\gamma_{t}^{\prime}=\frac{d\,x}{d\,t}$, $\gamma_{t}^{\prime\prime}=\nabla_{\gamma_{t}^{\prime}}\gamma_{t}^{\prime}$, $\xi_{t}^{\prime}=\nabla_{\gamma_{t}^{\prime}}\xi$, $\xi_{t}^{\prime\prime}=\nabla_{\gamma_{t}^{\prime}}\xi_{t}^{\prime}$ and $\Gamma_{t}^{\prime}=\frac{d\,\Gamma}{d\,t}$. Then $$\label{eq_1} \Gamma_{t}^{\prime}={}^{H}\!\gamma_{t}^{\prime} + {}^{V}\!\xi_{t}^{\prime}.$$ **Lemma 4**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold, $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric and $\Gamma =(\gamma(t),\xi(t))$ be a curve on $T^{\varphi}_{1}M$. Then we have $$\label{eq_2} \Gamma_{t}^{\prime} = {}^{H}\!\gamma_{t}^{\prime} + {}^{T}\!\xi_{t}^{\prime},$$* *Proof.* Using $\eqref{eq_1}$, we have $$\begin{aligned} \Gamma_{t}^{\prime} &=& {}^{H}\!\gamma_{t}^{\prime} + {}^{V}\!\xi_{t}^{\prime}={}^{H}\!\gamma_{t}^{\prime} + {}^{T}\!\xi_{t}^{\prime}+g(\xi_{t}^{\prime},\varphi \xi){}^{V}\!u.\end{aligned}$$ Since $\Gamma =(\gamma(t),\xi(t))\in T^{\varphi}_{1}M$ then $g(\xi,\varphi \xi)=1$, on the other hand $$\begin{aligned} 0&=&\gamma_{t}^{\prime}g(\xi,\varphi \xi)=2g(\xi_{t}^{\prime},\varphi \xi),\end{aligned}$$ i.e. $$\begin{aligned} \label{eq_3} \quad g(\xi_{t}^{\prime},\varphi \xi)=0.\end{aligned}$$ Hence, the proof of the lemma is completed. ◻ From [\[eq_2\]](#eq_2){reference-type="eqref" reference="eq_2"}, we have $$\begin{aligned} \label{eq_4} 1=|\gamma_{t}^{\prime}|^{2}+ g(\xi_{t}^{\prime},\varphi \xi_{t}^{\prime}),\end{aligned}$$ where $|\,.\,|$ mean the norm of vectors with respect to the $(M^{2m}, \varphi, g)$. **Theorem 5**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold, $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric and $\Gamma =(\gamma(t),\xi(t))$ be a curve on $T^{\varphi}_{1}M$. Then $\Gamma$ is a geodesic on $T^{\varphi}_{1}M$ if and only if $$\label{eq_5} \left\{ \begin{array}{lll} \gamma_{t}^{\prime\prime}&=&R(\xi_{t}^{\prime},\varphi \xi)\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}&=&g(\xi_{t}^{\prime\prime}, \varphi \xi) \xi \end{array} \right.$$ Moreover, $$\label{eq_6} \left\{ \begin{array}{lll} |\gamma_{t}^{\prime}|&=&\sqrt{1-\rho^{2}}\\ g(\xi_{t}^{\prime\prime}, \varphi \xi)&=&-g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime})=-\rho^{2} \end{array} \right.$$ where $\rho = const$ and $0\leq \rho \leq1$.* *Proof.* Using formula $\eqref{eq_2}$and Theorem [Theorem 3](#th_1){reference-type="ref" reference="th_1"}, we compute the derivative $\widehat{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime}$. $$\begin{aligned} \widehat{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime} & = &\widehat{\nabla}_{\displaystyle({}^{H}\!\gamma_{t}^{\prime} + {}^{T}\!\xi_{t}^{\prime})}({}^{H}\!\gamma_{t}^{\prime} + {}^{T}\!\xi_{t}^{\prime}) \\ & = &\widehat{\nabla}_{\displaystyle{}^{H}\gamma_{t}^{\prime}}{}^{H}\gamma_{t}^{\prime} +\widehat{\nabla}_{\displaystyle{}^{H}\!\gamma_{t}^{\prime}}{}^{T}\!\xi_{t}^{\prime}+\widehat{\nabla}_{{}^{T}\!\xi_{t}^{\prime}}{}^{H}\!\gamma_{t}^{\prime}+\widehat{\nabla}_{{}^{T}\!\xi_{t}^{\prime}}{}^{T}\!\xi_{t}^{\prime} \\ &=& {}^{H}\!\gamma_{t}^{\prime\prime}+{}^{H}\!(R(\varphi \xi,\xi_{t}^{\prime})\gamma_{t}^{\prime})+{}^{T}\!\xi_{t}^{\prime\prime}\\ &=&{}^{H}\!\big(\gamma_{t}^{\prime\prime}+R(\varphi \xi, \xi_{t}^{\prime})\gamma_{t}^{\prime}\big)+{}^{V}\!(\xi_{t}^{\prime\prime}-g(\xi_{t}^{\prime\prime}, \varphi \xi)\xi).\end{aligned}$$ If we put $\widehat{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime}$ equal to zero, we find $\eqref{eq_5}$.\ From $\eqref{eq_3}$ we get, $0=\gamma_{t}^{\prime}g(\xi_{t}^{\prime}, \varphi \xi)=g(\xi_{t}^{\prime\prime}, \varphi \xi)+g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime})$ then, $$\begin{aligned} g(\xi_{t}^{\prime\prime}, \varphi \xi)=-g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime}),\end{aligned}$$ Using $\eqref{eq_3}$ and the second equation of $\eqref{eq_5}$, we find, $$\gamma_{t}^{\prime}g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime})=2g(\xi_{t}^{\prime\prime}, \varphi \xi_{t}^{\prime})=2g(\xi_{t}^{\prime\prime}, \varphi \xi)g(\xi, \varphi \xi_{t}^{\prime})=0,$$ then $g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime})=\kappa=const$, from $\eqref{eq_4}$, we find, $0\leq \kappa \leq1$,\ hence $|\gamma_{t}^{\prime}|=\sqrt{1-\rho^{2}}$ and $g(\xi_{t}^{\prime\prime}, \varphi \xi)=-g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime})=-\rho^{2}$, where $\rho^{2}=\kappa$. ◻ **Remark 6**. According to [\[eq_6\]](#eq_6){reference-type="eqref" reference="eq_6"}, the geodesics $\Gamma =(\gamma(t),\xi(t))$ of $T^{\varphi}_{1}M$ can be splitted naturally into 3 classes, namely, \(1\) horizontal geodesics, if $\rho=0$, from [\[eq_6\]](#eq_6){reference-type="eqref" reference="eq_6"}, $|\gamma_{t}^{\prime}|=1$, then from [\[eq_4\]](#eq_4){reference-type="eqref" reference="eq_4"}, we have $\xi_{t}^{\prime}=0$ i.e. $\Gamma$ is generated by parallel vector fields $\xi$ along the geodesics $\gamma$ on the base manifold, \(2\) vertical geodesics, if $\rho=1$, from [\[eq_6\]](#eq_6){reference-type="eqref" reference="eq_6"}, $|\gamma_{t}^{\prime}|=0$, then $\gamma(t)$ is a constant i.e. $\Gamma$ is geodesic in Euclidean space, (on a fixed fiber), their equations are  $\xi_{t}^{\prime\prime}=\rho^{2}\xi$. \(3\) umbilical (oblique) geodesics corresponding to $0 < \rho < 1$, In this case, $\Gamma$ can be regarded as a vector field $\xi\neq 0$ along the curve $\gamma$. see [@Y.S]. A curve $\Gamma= (\gamma(t), \xi(t))$ on $TM$ is said to be a horizontal lift of the curve $\gamma$ on $M$ if and only if $\xi_{t}^{\prime}=0$ [@Y.I]. In general, the horizontal lift $\Gamma= (\gamma(t), \xi(t))$ of the curve $\gamma$ on $M$ does not belong to $T^{\varphi}_{1}M$, we have $\xi_{t}^{\prime}=0$, then $0=2g(\xi_{t}^{\prime}, \varphi \xi)=\gamma_{t}^{\prime}g(\xi, \varphi \xi)=0$, hence $g(\xi, \varphi \xi)=const\neq 1$. If $\Gamma\in T^{\varphi}_{1}M$, we get the following Corollary **Corollary 7**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric. If $\Gamma= (\gamma(t), \xi(t))$ is a horizontal lift of $\gamma$ and $\Gamma\in T^{\varphi}_{1}M$, then $\Gamma$ is a geodesic on $T^{\varphi}_{1}M$ if and only if $\gamma$ is a geodesic on $M$.* The curve $\Gamma = (\gamma(t), \gamma_{t}^{\prime}(t))$ is called a natural lift of the curve $\gamma$ on $TM$ [@Y.I]. Thus, we have **Corollary 8**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric. If $\Gamma= (\gamma(t), \gamma_{t}^{\prime}(t))$ is a natural lift of $\gamma$ and $\Gamma\in T^{\varphi}_{1}M$, then $\Gamma$ is a geodesic on $T^{\varphi}_{1}M$ if and only if $\gamma$ is a geodesic on $M$.* **Remark 9**. As a reminder, note that locally we have: $$\begin{aligned} \label{eq_I} \gamma_{t}^{\prime\prime}=\sum_{l=1}^{2m} (\dfrac{d^{2}\gamma^{l}}{dt^{2}}+\sum_{i,j=1}^{2m}\dfrac{d\gamma^{i}}{dt}\dfrac{d\gamma^{j}}{dt}\Gamma_{ij}^{l})\frac{\partial}{\partial x^{l}}, \end{aligned}$$ and $$\begin{aligned} \label{eq_II} \xi_{t}^{\prime}=\sum_{l=1}^{2m} (\dfrac{d\xi^{l}}{dt}+\sum_{i,j=1}^{2m}\frac{d\gamma^{j}}{dt}\xi^{i}\Gamma_{ij}^{l})\frac{\partial}{\partial x^{l}}. \end{aligned}$$ **Example 10**. Let $(\mathbb{R}^{2},\varphi,g)$ be a para-Kähler-Norden manifold such that $$g= e^{2x}dx^{2}+e^{2y}dy^{2},\quad \varphi=\left( \begin{array}{ccc} 0 & e^{y-x}\\ e^{x-y} & 0 \end{array} \right).$$ The non-null Christoffel symbols of the Riemannian connection are: $$\Gamma_{11}^{1} = \Gamma_{22}^{2} = 1.$$ $1)$ Let $\gamma$ be a curve such that $\gamma(t)=(x(t), y(t))$, from [\[eq_I\]](#eq_I){reference-type="eqref" reference="eq_I"}, the geodesics $\gamma$ such that $\gamma(0)= (a, b)\in \mathbb{R}^{2}$ and $\gamma_{t}^{\prime}(0)= (\lambda,\eta)\in \mathbb{R}^{\ast}_{+}\times \mathbb{R}^{\ast}_{+}$ satisfies the system of differential equations, $$\begin{aligned} \frac{d^{2}\gamma^{l}}{dt^{2}} +\sum_{i,j=1}^{2}\frac{d\gamma^{i}}{dt}\frac{d\gamma^{j}}{dt}\Gamma_{ij}^{l}=0&\Leftrightarrow& \left\{\begin{array}{lll} \dfrac{d^{2}x}{dt^{2}} + (\dfrac{dx}{dt})^{2}=0\\\\ \dfrac{d^{2}y}{dt^{2}} + (\dfrac{dy}{dt})^{2}=0 \end{array} \right.\\&\Leftrightarrow& \left\{\begin{array}{lll} x(t)=a+\ln( 1+\lambda t)\\\\ y(t)=b+\ln( 1+\eta t) \end{array} \right.\end{aligned}$$ Hence, $$\gamma_{t}^{\prime}(t)= \dfrac{\lambda}{1 +\lambda t} \dfrac{\partial}{\partial x}+\dfrac{\eta}{1 +\eta t}\dfrac{\partial}{\partial y},\;\gamma(t)= (a+\ln( 1+\lambda t),b+\ln( 1+\eta t)).$$ On the other hand we have $g(\gamma_{t}^{\prime}, \varphi\gamma_{t}^{\prime})= 2\lambda\eta e^{a+b}$, then for $\lambda\eta= \dfrac{1}{2e^{a+b}}$,\ become $\Gamma_{1} =(\gamma(t),\gamma_{t}^{\prime}(t))\in T^{\varphi}_{1}\mathbb{R}^{2}$.\ Hence from Corollary $\ref{co_2}$, the curve $\Gamma_{1}$ is a geodesic on $T^{\varphi}_{1}\mathbb{R}^{2}$.\ $2)$ If $\Gamma_{2} =(\gamma(t),\xi(t))$ is horizontal lift of $\gamma$, such that $\xi(t) =(u(t),v(t))$ i.e. $\xi_{t}^{\prime}=0$, from [\[eq_II\]](#eq_II){reference-type="eqref" reference="eq_II"}, we have $$\dfrac{d\xi^{l}}{dt}+\sum_{i,j=1}^{2}\frac{dx^{j}}{dt}\xi^{i}\Gamma_{ij}^{l}=0 \Leftrightarrow \left\{\begin{array}{lll} \dfrac{du}{dt} + \dfrac{dx}{dt}u=0\\\\ \dfrac{dv}{dt} + \dfrac{dy}{dt}v=0 \end{array} \right.\Leftrightarrow \left\{\begin{array}{lll} u(t)=\dfrac{h_{1}}{1+\lambda t}\\\\ v(t)=\dfrac{h_{2}}{1+\eta t} \end{array} \right.$$ Hence $\xi(t)=\dfrac{h_{1}}{1+\lambda t}\dfrac{\partial}{\partial x}+\dfrac{h_{2}}{1+\eta t}\dfrac{\partial}{\partial y}$, where $h_{1}, h_{2}\in \mathbb{R}.$\ But $g(\xi^{\prime}, \varphi\xi)= 2h_{1}h_{2} e^{a+b}$, then for $h_{1}h_{2}= \dfrac{1}{2e^{a+b}}$,\ become $\Gamma_{2} =(\gamma(t),\xi(t))\in T^{\varphi}_{1}\mathbb{R}^{2}$.\ Hence from Corollary $\ref{co_1}$, the curve $\Gamma_{2}$ is a geodesic on $T^{\varphi}_{1}\mathbb{R}^{2}$. Let $\Gamma$ be a curve on $TM$, the cure $\gamma=\pi\circ \Gamma$ is called the projection (projected curve) of the curve $\Gamma$ on $M$. **Theorem 11**. *Let $(M^{2m}, \varphi, g)$ be a locally symmetric para-Kähler-Norden manifold, $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric and $\Gamma$ be a non-vertical geodesic on $T^{\varphi}_{1}M$, then all Frenet curvatures of the projected curve $\gamma=\pi\circ \Gamma$ are constants.* *Proof.* Using the first equation of $\eqref{eq_5}$, we have $\gamma_{t}^{\prime\prime}=R(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime}.$\ It is easy to see that $$\begin{aligned} \gamma_{t}^{\prime}g(\gamma_{t}^{\prime}, \gamma_{t}^{\prime})=2g(\gamma_{t}^{\prime\prime}, \gamma_{t}^{\prime})=2g(R(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime}, \gamma_{t}^{\prime})=0,\end{aligned}$$ hence, $|\gamma_{t}^{\prime}|= const$.\ Calculate the third derivative, we get $$\begin{aligned} \gamma_{t}^{\prime\prime\prime}&=&(\nabla_{\gamma_{t}^{\prime}}R)(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime}+R(\xi_{t}^{\prime\prime}, \varphi \xi)\gamma_{t}^{\prime}+R(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime})\gamma_{t}^{\prime}+R(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime\prime}\\ &=&R(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime\prime}=R^{2}(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime}.\end{aligned}$$ Since $$\begin{aligned} \gamma_{t}^{\prime}g(\gamma_{t}^{\prime\prime}, \gamma_{t}^{\prime\prime})=2g(\gamma_{t}^{\prime\prime\prime}, \gamma_{t}^{\prime\prime})=2g(R(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime\prime}, \gamma_{t}^{\prime\prime})=0,\end{aligned}$$ hence, $|\gamma_{t}^{\prime\prime}|= const$.\ Continuing the process, we obtain $$\label{eq_7} \gamma_{t}^{(p+1)}=R(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{(p)}=R^{p}(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime},\quad p\geq1$$ and $$\gamma_{t}^{\prime}g(\gamma_{t}^{(p)}, \gamma_{t}^{(p)})=2g(\gamma_{t}^{(p+1)},\gamma_{t}^{(p)})=2g(R(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{(p)},\gamma_{t}^{(p)})=0.$$ Thus, we get $$\label{eq_8} |\gamma_{t}^{(p)}|=const,\quad p\geq 1.$$ Denote by $s$ an arc length parameter on $\gamma$, i.e. $(|\gamma_{s}^{\prime}|=1)$.\ Then, $\gamma_{t}^{\prime}= \gamma_{s}^{\prime}\frac{d\,s}{dt}$, and using $\eqref{eq_6}$, we get $$\label{eq_9} \frac{d\,s}{dt}=\sqrt{1-\rho^{2}}=const,$$ Let $\nu_{1} = \gamma_{s}^{\prime}$ be the first vector in the Frenet frame $\nu_{1}, \ldots, \nu_{2m-1}$ along $\gamma$ and let $k_{1}, \ldots, k_{2m-1}$ the Frenet curvatures of $\gamma$. Then the Frenet formulas verify $$\label{eq_10} \left\{ \begin{array}{lll} (\nu_{1})^{\prime}_{s}&=&k_{1}\nu_{2}\\ (\nu_{i})^{\prime}_{s}&=&-k_{i-1}\nu_{i-1}+k_{i}\nu_{i+1},\quad 2\leq i\leq 2m-2\\ (\nu_{2m-1})^{\prime}_{s}&=&-k_{2m-2}\nu_{2m-2} \end{array} \right.$$ From $\eqref{eq_6}$, we have $$\begin{aligned} \label{eq_11} \gamma_{t}^{\prime}=\gamma_{s}^{\prime}\frac{d\,s}{dt}=\sqrt{1-\rho^{2}}\,\nu_{1}.\end{aligned}$$ Using $\eqref{eq_9}$ and the Frenet formulas $\eqref{eq_10}$, we obtain $$\begin{aligned} \label{eq_12} \gamma_{t}^{\prime\prime}=\sqrt{1-\rho^{2}}\,(\nu_{1})^{\prime}_{t}=\sqrt{1-\rho^{2}}\,(\nu_{1})^{\prime}_{s}\frac{d\,s}{dt}=(1-\rho^{2})k_{1}\nu_{2}.\end{aligned}$$ Now $\eqref{eq_8}$ implies $k_{1} =$ const.\ Next, in a similar way, we have $$\begin{aligned} \label{eq_13} \gamma_{t}^{\prime\prime\prime}&=&(1-\rho^{2})k_{1}(\nu_{2})^{\prime}_{t}=(1-\rho^{2})k_{1}(\nu_{2})^{\prime}_{s}\frac{d\,s}{dt}\nonumber\\ &=&-(1-\rho^{2})^{\frac{3}{2}}k_{1}^{2}\nu_{1}+(1-\rho^{2})^{\frac{3}{2}}k_{1}k_{2}\nu_{3}.\end{aligned}$$ and again $\eqref{eq_8}$ implies $k_{2}=$ const.\ By continuing the process, we finish the proof. ◻ **Lemma 12**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric. If $\Gamma=(\gamma(t),\xi(t))$ is a curve on $T^{\varphi}_{1}M$, then we have* *$(1)$ $\Phi=(\gamma(t),\varphi \xi(t))$ is a curve on $T^{\varphi}_{1}M$.* *$(2)$ $\Phi$ is a geodesic on $T^{\varphi}_{1}M$ if and only if $\Gamma$ is a geodesic on $T^{\varphi}_{1}M$.* *Proof.* $(1)$ We put $\mu(t)=\varphi \xi(t)$, since $\Gamma =(\gamma(t),\xi(t))\in T^{\varphi}_{1}M$, then $g(\xi,\varphi \xi)=1$.\ On the other hand, $g(\mu,\varphi \mu)=g(\varphi \xi,\varphi(\varphi \xi))=g(\varphi \xi,\xi)=1$ i.e. $$\Phi(t) =(\gamma(t),\mu(t))\in T^{\varphi}_{1}M.$$ $(2)$ In a similar way proof of $\eqref{eq_5}$, and using $\mu_{t}^{\prime}=\varphi \xi_{t}^{\prime}$ and $\mu_{t}^{\prime\prime}=\varphi \xi_{t}^{\prime\prime}$, we have $$\begin{aligned} \widehat{\nabla}_{\Phi_{t}^{\prime}}\Phi_{t}^{\prime} &=& {}^{H}\!(\gamma_{t}^{\prime\prime}+R(\varphi \mu,\mu_{t}^{\prime})\gamma_{t}^{\prime})+{}^{V}\!(\mu_{t}^{\prime\prime}+g(\mu_{t}^{\prime}, \varphi \mu_{t}^{\prime})\mu)\\ &=&{}^{H}\!\big(\gamma_{t}^{\prime\prime}+R(\xi,\varphi \xi_{t}^{\prime})\gamma_{t}^{\prime}\big)+{}^{V}\!(\varphi \xi_{t}^{\prime\prime}+g(\varphi \xi_{t}^{\prime}, \xi_{t}^{\prime})\varphi \xi).\end{aligned}$$ Since the Riemannian curvature tensor is pure, we get $$\begin{aligned} \widehat{\nabla}_{\Phi_{t}^{\prime}}\Phi_{t}^{\prime}&=&{}^{H}\!\big(\gamma_{t}^{\prime\prime}+R(\varphi \xi,\xi_{t}^{\prime})\gamma_{t}^{\prime}\big)+{}^{V}\!(\varphi (\xi_{t}^{\prime\prime}+g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime}) \xi)),\end{aligned}$$ hence, $$\begin{aligned} \widehat{\nabla}_{\Phi_{t}^{\prime}}\Phi_{t}^{\prime}=0 &\Leftrightarrow&\left\{ \begin{array}{lll} \gamma_{t}^{\prime\prime}&=&-R(\varphi \xi,\xi_{t}^{\prime})\gamma_{t}^{\prime}\\ \varphi \xi_{t}^{\prime\prime}&=&-g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime})\varphi \xi \end{array}\right.\\ &\Leftrightarrow& \left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}&=&R(\xi_{t}^{\prime},\varphi \xi)\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}&=&-g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime}) \xi \end{array} \right.\\ &\Leftrightarrow& \widehat{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime}=0.\end{aligned}$$ ◻ From Theorem [Theorem 11](#th_3){reference-type="ref" reference="th_3"} and Lemma [Lemma 12](#lem_1){reference-type="ref" reference="lem_1"}, we have the following theorem **Theorem 13**. *Let $(M^{2m}, \varphi, g)$ be a locally symmetric para-Kähler-Norden manifold, $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric and $\Gamma =(\gamma(t), \xi(t))$ be a non-vertical geodesic on $T^{\varphi}_{1}M$ then, all Frenet curvatures of the projected curve $\pi\circ \Phi$ are constants, where $\Phi =(\gamma(t),\varphi \xi(t))$.* Now we study the geodesics on $\varphi$-unit tangent bundle with the $\varphi$-Sasaki metric over para-Kähler-Norden manifold of constant sectional curvature.\ From Theorem [Theorem 5](#th_2){reference-type="ref" reference="th_2"}, we have the following **Corollary 14**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold of constant curvature $c\neq0$, $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric and $\Gamma =(\gamma(t),\xi(t))$ be a curve on $T^{\varphi}_{1}M$. Then $\Gamma$ is a geodesic on $T^{\varphi}_{1}M$ if and only if $$\label{eq_14} \left\{ \begin{array}{lll} \gamma_{t}^{\prime\prime}&=&cg(\varphi \xi, \gamma_{t}^{\prime})\xi_{t}^{\prime}-cg(\xi_{t}^{\prime}, \gamma_{t}^{\prime}) \varphi \xi\\ \xi_{t}^{\prime\prime}&=&-g(\xi_{t}^{\prime}, \varphi \xi_{t}^{\prime}) \xi \end{array} \right.$$* **Theorem 15**. *Let $(\mathbb{R}^{2m}, \varphi, <,>)$ be a para-Kähler-Norden real euclidean space , $T^{\varphi}_{1}\mathbb{R}^{2m}$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric. Any oblique geodesics $\Gamma =(\gamma(t),\xi(t))$ on $T^{\varphi}_{1}\mathbb{R}^{2m}$ is the following form $$\label{eq_15} \left\{ \begin{array}{lll} \gamma(t)&=&c_{1}t+c_{2}\\ \xi(t)&=&c_{3}\cos (\rho t)+c_{4}\sin(\rho t) \end{array} \right.$$ where $\rho = const$ and $0< \rho <1$ and $c_{1}, c_{2}, c_{3}, c_{4}$ are real constants.* *Proof.* In the case of Euclidean space we have $R = 0$, then, using [\[eq_5\]](#eq_5){reference-type="eqref" reference="eq_5"} and [\[eq_6\]](#eq_6){reference-type="eqref" reference="eq_6"} , we get $$\left\{ \begin{array}{lll} \gamma_{t}^{\prime\prime}&=&0\\ \xi_{t}^{\prime\prime}&=&-\rho^{2} \xi \end{array} \right.\Leftrightarrow\left\{ \begin{array}{lll} \gamma(t)&=&c_{1}t+c_{2}\\ \xi(t)&=&c_{3}\cos (\rho t)+c_{4}\sin(\rho t) \end{array} \right.$$ ◻ Define a power of curvature operator $R^{p}(X, Y)$ recurrently in the following way: $$R^{p}(X, Y)Z=R^{p-1}(X, Y)R(X, Y)Z,$$ for any vector fields $X$ and $Y$ on $M$, where $p\geq 2$. **Lemma 16**. *[@Y.S] Let $(M,g)$ be a Riemannian manifold of constant curvature $c$, then we have $$R^{p}(X, Y)=\left\{ \begin{array}{lll} (-b^{2}c^{2})^{k-1}R(X,Y), \;\; \textit{for}\;\; p = 2k-1\\\ (-b^{2}c^{2})^{k-1}R^{2}(X,Y), \;\; \textit{for}\;\; p = 2k \end{array} \right.$$ for any vector fields $X$ and $Y$ on $M$, where $k\geq 2$ and $b^{2}=|X|^{2}|Y|^{2}-g(X,Y)^{2}$.* **Theorem 17**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold of constant curvature $c\neq0$, $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric, $\Gamma$ be a non-vertical geodesic on $T^{\varphi}_{1}M$ and $k_{1}, \ldots, k_{2m-1}$ the Frenet curvatures of the projected curve $\gamma=\pi\circ \Gamma$. If $k_{1}\neq0$ and $k_{2}\neq0$ then $k_{i}=0$ for $i\geq3$.* *Proof.* From the Theorem [Theorem 11](#th_3){reference-type="ref" reference="th_3"}, all Frenet curvatures of the projected curve $\gamma=\pi\circ \Gamma$ are constants, and using [\[eq_13\]](#eq_13){reference-type="eqref" reference="eq_13"} we have $$\begin{aligned} \label{eq_16} \gamma_{t}^{(4)}=-(1-\rho^{2})^{2}k_{1}(k_{1}^{2}+k_{2}^{2})\nu_{2}+(1-\rho^{2})^{2}k_{1}k_{2}k_{3}\nu_{4}.\end{aligned}$$ On the other hand, from [\[eq_7\]](#eq_7){reference-type="eqref" reference="eq_7"}, Lemma [Lemma 16](#lem_2){reference-type="ref" reference="lem_2"} and [\[eq_12\]](#eq_12){reference-type="eqref" reference="eq_12"}, we have $$\label{eq_17} \gamma_{t}^{(4)}=R^{3}(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime}=-b^{2}c^{2}R(\xi_{t}^{\prime}, \varphi \xi)\gamma_{t}^{\prime}=-b^{2}c^{2}\gamma_{t}^{\prime\prime}=-b^{2}c^{2}(1-\rho^{2})k_{1}\nu_{2},$$ If $k_{1}\neq0$ and $k_{2}\neq0$, we get $$\begin{aligned} \label{eq_18} (b^{2}c^{2}-(1-\rho^{2})(k_{1}^{2}+k_{2}^{2}))\nu_{2}+(1-\rho^{2})k_{2}k_{3}\nu_{4}=0.\end{aligned}$$ Therefore, we have $k_{3} = 0$, and $b^{2}c^{2}=(1-\rho^{2})(k_{1}^{2}+k_{2}^{2})$.i.e. $b^{2}=$ const.\ By continuing the process, we obtain $k_{i}=0$ for $i\geq3$. ◻ # $F$-geodesics on tangent bundle with the $\varphi$-Sasaki metric Let $(M^{m}, g)$ be an Riemannian manifold. A magnetic field is a closed $2$-form $\Omega$ on $(M^{m}, g)$ and the Lorentz force of a magnetic field $\Omega$ on $(M^{m},g)$ is a $(1, 1)$-tensor field $\Phi$ given by $$\begin{aligned} \label{eq_19} g(\Phi(X),Y)=\Omega(X,Y),\end{aligned}$$ for any vector fields $X$ and $Y$ on $M$. The magnetic trajectories of $\Omega$ (or simply a magnetic curve) with strength $q\in \mathbb{R}$ are the curves $\gamma$ on $M$ that satisfy the Lorentz equation $$\begin{aligned} \label{eq_20} \gamma_{t}^{\prime\prime}=q\Phi \gamma_{t}^{\prime},\end{aligned}$$ where $\nabla$ is the Levi-Civita connection of $g$. The Lorentz equation generalizes the equation satisfied by the geodesics of $(M^{m}, g)$, namely $\gamma_{t}^{\prime\prime}= 0$. Let $F$ be a $(1,1)$-tensor field on $(M^{m}, g)$. A curve $\gamma$ on $M$ is called $F$-planar if its speed remains, under parallel translation along the curve $\gamma$, in the distribution generated by the vector $\gamma_{t}^{\prime}$ and $F \gamma_{t}^{\prime}$ along $\gamma$. This is equivalent to the fact that the tangent vector $\gamma_{t}^{\prime}$ satisfies $$\begin{aligned} \label{eq_21} \gamma_{t}^{\prime\prime}=\varrho_{1}(t)\gamma_{t}^{\prime} +\varrho_{2}F \gamma_{t}^{\prime},\end{aligned}$$ where $\varrho_{1}$ and $\varrho_{2}$ are some functions of the parameter $t$, see [@M.S; @H.M]. The $F$-planar curves generalize the magnetic curves and therefore, the geodesics. More precisely, when $F=\Phi$ is a Lorentz force on the manifold, $\varrho_{1}=0$ and $\varrho_{2}$ is a constant, we obtain the magnetic trajectories corresponding to $\Phi$ with strength $\varrho_{2}$. In the absence of $F$, one gets the geodesics, see [@Nis].\ We say that a curve $\gamma$ on $M$ is an $F$-geodesic if $\gamma$ satisfies: $$\begin{aligned} \label{eq_22} \gamma_{t}^{\prime\prime}=F \gamma_{t}^{\prime},\end{aligned}$$ One can see that an $F$-geodesic is an $F$-planar curve, but in general an $F$-planar curve is not always an $F$-geodesic, see [@B.D]. According to [\[eq_22\]](#eq_22){reference-type="eqref" reference="eq_22"}, the Lorentz equation [\[eq_20\]](#eq_20){reference-type="eqref" reference="eq_20"} expresses the relation satisfied by an $F$-geodesic on $M$, where $F = q\Phi$, see [@K.Y; @Mik]. ## $F$-geodesics on tangent bundle with the $\varphi$-Sasaki metric  \ Let $\widetilde{\nabla}$ be the Levi-Civita connection of $\varphi$-Sasaki metric on tangent bundle $TM$, given in the Theorem [Theorem 2](#th_0){reference-type="ref" reference="th_0"}. **Theorem 18**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold, $TM$ its tangent bundle equipped with the $\varphi$-Sasaki metric $g^{\varphi}$ and $F$ be a $(1,1)$-tensor field on $M$. A curve $\Gamma =(\gamma(t), \xi(t))$ on $TM$ is an ${}^{H}\!F$-planar with respect to $\widetilde{\nabla}$ if and only if the $$\left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}=R(\xi_{t}^{\prime},\varphi\xi)\gamma_{t}^{\prime}+\varrho_{1}\gamma_{t}^{\prime}+\varrho_{2} F\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=\varrho_{1}\xi_{t}^{\prime}+\varrho_{2} F\xi_{t}^{\prime} \end{array}\right.$$* *Proof.* Using Theorem [Theorem 2](#th_0){reference-type="ref" reference="th_0"} and [\[eq_1\]](#eq_1){reference-type="eqref" reference="eq_1"}, we find $$\begin{aligned} \label{eq_23} \widetilde{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime} &=& \widetilde{\nabla}_{\displaystyle({}^{H}\!\gamma_{t}^{\prime} + {}^{V}\!\xi_{t}^{\prime})}({}^{H}\!\gamma_{t}^{\prime} + {}^{V}\!\xi_{t}^{\prime}) \nonumber\\ &=&\widetilde{\nabla}_{\displaystyle{}^{H}\gamma_{t}^{\prime}}{}^{H}\gamma_{t}^{\prime} +\widetilde{\nabla}_{\displaystyle{}^{H}\!\gamma_{t}^{\prime}}{}^{V}\!\xi_{t}^{\prime}+\widetilde{\nabla}_{{}^{V}\!\xi_{t}^{\prime}}{}^{H}\!\gamma_{t}^{\prime}+\widetilde{\nabla}_{{}^{V}\!\xi_{t}^{\prime}}{}^{V}\!\xi_{t}^{\prime} \nonumber\\ &=&{}^{H}\!(\gamma_{t}^{\prime\prime}+R(\varphi \xi,\xi_{t}^{\prime})\gamma_{t}^{\prime})+{}^{V}\!\xi_{t}^{\prime\prime}\end{aligned}$$ On the other hand, $$\begin{aligned} \label{eq_24} \widetilde{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime}&=&\varrho_{1}\Gamma_{t}^{\prime} +\varrho_{2}{}^{H}\!F \Gamma_{t}^{\prime}\nonumber\\ &=&\varrho_{1}({}^{H}\!\gamma_{t}^{\prime} + {}^{V}\!\xi_{t}^{\prime}) +\varrho_{2}{}^{H}\!F({}^{H}\!\gamma_{t}^{\prime} + {}^{V}\!\xi_{t}^{\prime})\nonumber\\ &=&{}^{V}\!\varrho_{1}{}^{H}\!\gamma_{t}^{\prime} +{}^{V}\!\varrho_{2}{}^{H}\!F {}^{H}\!\gamma_{t}^{\prime}+{}^{V}\!\varrho_{1}{}^{V}\!\xi_{t}^{\prime} +{}^{V}\!\varrho_{2}{}^{H}\!F {}^{V}\!\xi_{t}^{\prime}\nonumber\\ &=&{}^{H}\!(\varrho_{1}\gamma_{t}^{\prime}+\varrho_{2}F \gamma_{t}^{\prime})+{}^{V}\!(\varrho_{1}\xi_{t}^{\prime} +\varrho_{2}F \xi_{t}^{\prime}).\end{aligned}$$ From [\[eq_23\]](#eq_23){reference-type="eqref" reference="eq_23"} and [\[eq_24\]](#eq_24){reference-type="eqref" reference="eq_24"}, the result immediately follows. ◻ **Corollary 19**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $TM$ its tangent bundle equipped with the $\varphi$-Sasaki metric $g^{\varphi}$. A curve $\Gamma =(\gamma(t), \xi(t))$ on $TM$ is an ${}^{H}\!\varphi$-planar with respect to $\widetilde{\nabla}$ if and only if the $$\left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}=R(\xi_{t}^{\prime},\varphi\xi)\gamma_{t}^{\prime}+\varrho_{1}\gamma_{t}^{\prime}+\varrho_{2} \varphi\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=\varrho_{1}\xi_{t}^{\prime}+\varrho_{2} \varphi\xi_{t}^{\prime} \end{array}\right.$$* In the particular case when $\varrho_{1} = 0$ and $\varrho_{2} = 1$ in the Theorem [Theorem 18](#th_8){reference-type="ref" reference="th_8"}, we obtain the following result. **Theorem 20**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold, $TM$ its tangent bundle equipped with the $\varphi$-Sasaki metric $g^{\varphi}$ and $F$ be a $(1,1)$-tensor field on $M$. A curve $\Gamma =(\gamma(t), \xi(t))$ on $TM$ is an ${}^{H}\!F$-geodesic with respect to $\widetilde{\nabla}$ if and only if the $$\left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}=R(\xi_{t}^{\prime},\varphi\xi)\gamma_{t}^{\prime}+ F\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=F\xi_{t}^{\prime} \end{array}\right.$$* **Corollary 21**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $TM$ its tangent bundle equipped with the $\varphi$-Sasaki metric $g^{\varphi}$. A curve $\Gamma =(\gamma(t), \xi(t))$ on $TM$ is an ${}^{H}\!\varphi$-geodesic with respect to $\widetilde{\nabla}$ if and only if the $$\left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}=R(\xi_{t}^{\prime},\varphi\xi)\gamma_{t}^{\prime}+ \varphi\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=\varphi\xi_{t}^{\prime} \end{array}\right.$$* **Theorem 22**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold, $TM$ its tangent bundle equipped with the $\varphi$-Sasaki metric $g^{\varphi}$ and $F$ be a $(1,1)$-tensor field on $M$. If $\Gamma =(\gamma(t), \xi(t))$ is a horizontal lift of a curve $\gamma$, then $\Gamma$ is an ${}^{H}\!F$-planar curve (resp., ${}^{H}\!F$-geodesic) if and only if $\gamma$ is an $F$-planar curve (resp., $F$-geodesic).* *Proof.* Let $\gamma$ be an $F$-planar with respect to $\nabla$ on $M$, i.e. $\gamma$ satisfies $$\begin{aligned} \gamma_{t}^{\prime\prime}=\varrho_{1}\gamma_{t}^{\prime} +\varrho_{2}F \gamma_{t}^{\prime},\end{aligned}$$ where $\varrho_{1}$ and $\varrho_{2}$ are some functions of the parameter $t$. Since $\Gamma =(\gamma(t), \xi(t))$ is a horizontal lift of a curve $\gamma$ then $\xi_{t}^{\prime}=0$ and $\Gamma_{t}^{\prime}={}^{H}\!\gamma_{t}^{\prime}$.\ On the other hand, $$\begin{aligned} \widetilde{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime}&=&{}^{H}\!\gamma_{t}^{\prime\prime}\\ &=&{}^{H}\!(\varrho_{1}\gamma_{t}^{\prime}+\varrho_{2}F \gamma_{t}^{\prime})\\ &=&{}^{V}\!\varrho_{1}{}^{H}\!\gamma_{t}^{\prime} +{}^{V}\!\varrho_{2}{}^{H}\!F {}^{H}\!\gamma_{t}^{\prime}\\ &=&{}^{V}\!\varrho_{1}\Gamma_{t}^{\prime}+{}^{V}\!\varrho_{2}{}^{H}\!F \Gamma_{t}^{\prime}.\end{aligned}$$ i.e. $\Gamma$ be an ${}^{H}\!F$-planar with respect to $\widetilde{\nabla}$. In the case of $\varrho_{1} = 0$ and $\varrho_{2} = 1$, we get that $\Gamma$ is an ${}^{H}\!F$-geodesic if and only $\gamma$ is an $F$-geodesic. ◻ **Corollary 23**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold, $TM$ its tangent bundle equipped with the $\varphi$-Sasaki metric $g^{\varphi}$. If $\Gamma =(\gamma(t), \xi(t))$ is a horizontal lift of a curve $\gamma$, then $\Gamma$ is an ${}^{H}\!\varphi$-planar curve (resp., ${}^{H}\!\varphi$-geodesic) if and only if $\gamma$ is an $\varphi$-planar curve (resp., $\varphi$-geodesic).* **Example 24**. Let $(\mathbb{R}^{2},\varphi,g)$ be a para-Kähler-Norden manifold such that $$g= dx^{2}+dy^{2}, \quad \varphi=\left( \begin{array}{ccc} 1 & 0\\ 0 &-1 \end{array} \right).$$ Let $\Gamma =(\gamma(t), \xi(t))$ such that $\gamma(t)=(x(t), y(t))$ and $\xi(t) =(u(t),v(t))$\ $1)$  $\Gamma$ is an ${}^{H}\!\varphi$-geodesic if and only if the $$\begin{aligned} \left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}= \varphi\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=\varphi\xi_{t}^{\prime} \end{array}\right.\Leftrightarrow \left\{\begin{array}{lll} x^{\prime\prime}= x^{\prime}\\ y^{\prime\prime}= -y^{\prime}\\ u^{\prime\prime}= u^{\prime}\\ v^{\prime\prime}= -v^{\prime} \end{array}\right.\Leftrightarrow \left\{\begin{array}{lll} x(t)= k_{1}e^{t}+k_{2}\\ y(t)= k_{3}e^{-t}+k_{4}\\ u(t)= k_{5}e^{t}+k_{6}\\ v(t)= k_{7}e^{-t}+k_{8} \end{array}\right.\end{aligned}$$ where $k_{i}$, $i=\overline{1,8}$ are real constants.\ $2)$   $\Gamma$ is an ${}^{H}\!\varphi$-planar if and only if the $$\begin{aligned} \left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}=\varrho_{1}\gamma_{t}^{\prime}+ \varrho_{2}\varphi\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=\varrho_{1}\xi_{t}^{\prime}+ \varrho_{2}\varphi\xi_{t}^{\prime} \end{array}\right.&\Leftrightarrow& \left\{\begin{array}{lll} x^{\prime\prime}= (\varrho_{1}+ \varrho_{2})x^{\prime}\\ y^{\prime\prime}= (\varrho_{1}- \varrho_{2})y^{\prime}\\ u^{\prime\prime}= (\varrho_{1}+ \varrho_{2})u^{\prime}\\ v^{\prime\prime}= (\varrho_{1}- \varrho_{2})v^{\prime} \end{array}\right.\\ &\Leftrightarrow&\left\{\begin{array}{lll} x(t)= \varepsilon_{1}\int e^{\int (\varrho_{1}+\varrho_{2})dt}dt\\ y(t)=\varepsilon_{2}\int e^{\int (\varrho_{1}-\varrho_{2})dt}dt\\ u(t)= \varepsilon_{3}\int e^{\int (\varrho_{1}+\varrho_{2})dt}dt\\ v(t)=\varepsilon_{4}\int e^{\int (\varrho_{1}-\varrho_{2})dt}dt \end{array}\right.\end{aligned}$$ where $\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}=\varepsilon_{4}=\pm1$.\ For example: $\varrho_{1}(t)= \dfrac{1}{t+1}$ and $\varrho_{2}(t)= \dfrac{1}{t-1}$, we find $$\left\{\begin{array}{lll} x(t)= a_{1}t^{3}-3a_{1}t+a_{2}\\ y(t)=a_{3}\ln (t+1)^{2}+a_{3}t+a_{4}\\ u(t)= b_{1}t^{3}-3b_{1}t+b_{2}\\ v(t)=b_{3}\ln (t+1)^{2}+b_{3}t+b_{4} \end{array}\right.$$ where $a_{i}$, $b_{i}$, $i=\overline{1,4}$ are real constants, then\ $\Gamma=(a_{1}t^{3}-3a_{1}t+a_{2}, a_{3}\ln (t+1)^{2}+a_{3}t+a_{4}, b_{1}t^{3}-3b_{1}t+b_{2}, b_{3}\ln (t+1)^{2}+b_{3}t+b_{4})$ is an ${}^{H}\!\varphi$-planar on $T\mathbb{R}^{2}$. **Example 25**. Let $(\mathbb{R}^{2},\varphi,g)$ be a para-Kähler-Norden manifold such that $$g= x^{2}dx^{2}+y^{2}dy^{2}, \quad \varphi=\left( \begin{array}{ccc} 0 & \dfrac{y}{x}\\ \dfrac{x}{y} &0 \end{array} \right)\quad and \quad F=\left( \begin{array}{ccc} a & 0\\ 0 & b \end{array} \right),\; a,b\in \mathbb{R}.$$ The non-null Christoffel symbols of the Riemannian connection are: $$\Gamma_{11}^{1} =\dfrac{1}{x}\;,\;\; \Gamma_{22}^{2} = \dfrac{1}{y}.$$ Let $\Gamma =(\gamma(t), \xi(t))$ be a horizontal lift of a curve $\gamma$, such that $\gamma(t)=(x(t), y(t))$ and $\xi(t) =(u(t),v(t))$ then $\xi_{t}^{\prime}=0$, from [\[eq_II\]](#eq_II){reference-type="eqref" reference="eq_II"} we have, $$\dfrac{d\xi^{h}}{dt}+\sum_{i,j=1}^{2}\frac{d\gamma^{j}}{dt}\xi^{i}\Gamma_{ij}^{h}=0 \Leftrightarrow \left\{\begin{array}{lll} u^{\prime} + \dfrac{x^{\prime}}{x}u=0\\ v^{\prime} + \dfrac{y^{\prime}}{y}v=0 \end{array} \right.\Leftrightarrow \left\{\begin{array}{lll} u(t)=\dfrac{k_{1}}{x(t)}\\ v(t)=\dfrac{k_{2}}{y(t)} \end{array} \right.$$ where $k_{1}, k_{2}$ are real constants.\ $\gamma$ is an $F$-geodesic if and only if $\gamma_{t}^{\prime\prime}=F \gamma_{t}^{\prime}$, from [\[eq_I\]](#eq_I){reference-type="eqref" reference="eq_I"} we have $$\left\{\begin{array}{lll} x^{\prime\prime} + \dfrac{(x^{\prime})^{2}}{x}=ax^{\prime}\\ y^{\prime\prime} + \dfrac{(y^{\prime})^{2}}{y}=by^{\prime} \end{array} \right.\Leftrightarrow \left\{\begin{array}{lll} x(t)=\varepsilon_{1}\sqrt{c_{1}e^{at}+c_{2}}\\ y(t)=\varepsilon_{2}\sqrt{c_{3}e^{bt}+c_{4}} \end{array} \right.$$ where $c_{1}, c_{2}, c_{3}, c_{4}$ are real constants and $\varepsilon_{1}=\varepsilon_{2}=\pm1$.\ The horizontal lift $\Gamma =(\varepsilon_{1}\sqrt{c_{1}e^{at}+c_{2}}, \varepsilon_{2}\sqrt{c_{3}e^{bt}+c_{4}}, \dfrac{\varepsilon_{1}k_{1}}{\sqrt{c_{1}e^{at}+c_{2}}}, \dfrac{\varepsilon_{2}k_{2}}{\sqrt{c_{3}e^{bt}+c_{4}}})$ is an ${}^{H}\!F$-geodesic on $T\mathbb{R}^{2}$.\ $\gamma$ is an $F$-planar if and only if $\gamma_{t}^{\prime\prime}=\varrho_{1}\gamma_{t}^{\prime} +\varrho_{2}F \gamma_{t}^{\prime}$, where $\varrho_{1}$ and $\varrho_{2}$ are some functions of the parameter $t$, from [\[eq_I\]](#eq_I){reference-type="eqref" reference="eq_I"} we have $$\left\{\begin{array}{lll} x^{\prime\prime} + \dfrac{(x^{\prime})^{2}}{x}=ax^{\prime}\\ y^{\prime\prime} + \dfrac{(y^{\prime})^{2}}{x}=by^{\prime} \end{array} \right.\Leftrightarrow \left\{\begin{array}{lll} x(t)=\varepsilon_{1}\sqrt{c_{1}e^{(\varrho_{1}+a\varrho_{2}t)}+c_{2}}\\ y(t)=\varepsilon_{1}\sqrt{c_{3}e^{(\varrho_{2}+b\varrho_{2}t)}+c_{4}} \end{array} \right.$$ where $c_{1}, c_{2}, c_{3}, c_{4}$ are real constants and $\varepsilon_{1}=\varepsilon_{2}=\pm1$.\ The horizontal lift $$\Gamma =(\varepsilon_{1}\sqrt{c_{1}e^{(\varrho_{1}+a\varrho_{2}t)}+c_{2}}, \varepsilon_{2}\sqrt{c_{3}e^{(\varrho_{1}+b\varrho_{2}t)}+c_{4}}, \dfrac{\varepsilon_{1}k_{1}}{\sqrt{c_{1}e^{(\varrho_{1}+a\varrho_{2}t)}+c_{2}}}, \dfrac{\varepsilon_{2}k_{2}}{\sqrt{c_{3}e^{(\varrho_{1}+b\varrho_{2}t)}+c_{4}}})$$ is an ${}^{H}\!F$-planar on $T\mathbb{R}^{2}$. ## $F$-geodesics on $\varphi$-unit tangent bundle with the $\varphi$-Sasaki metric  \ Let $\widehat{\nabla}$ be the Levi-Civita connection of $\varphi$-Sasaki metric on $\varphi$-unit tangent bundle $T^{\varphi}_{1}M$, given in the Theorem [Theorem 3](#th_1){reference-type="ref" reference="th_1"}. **Theorem 26**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric and $F$ be a $(1,1)$-tensor field on $M$. A curve $\Gamma =(\gamma(t), \xi(t))$ on $T^{\varphi}_{1}M$ is an ${}^{H}\!F$-planar with respect to $\widehat{\nabla}$ if and only if the $$\left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}=R(\xi_{t}^{\prime},\varphi\xi)\gamma_{t}^{\prime}+\varrho_{1}\gamma_{t}^{\prime}+\varrho_{2} F\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=\varrho_{1}\xi_{t}^{\prime}+\varrho_{2} F\xi_{t}^{\prime}-\rho^{2}\xi \end{array}\right.$$ where $\rho = const$ and $0\leq \rho \leq1$.* *Proof.* Using Theorem [Theorem 3](#th_1){reference-type="ref" reference="th_1"} and [\[eq_2\]](#eq_2){reference-type="eqref" reference="eq_2"}, we find $$\begin{aligned} \label{eq_25} \widehat{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime} &=& \widehat{\nabla}_{\displaystyle({}^{H}\!\gamma_{t}^{\prime} + {}^{V}\!\xi_{t}^{\prime})}({}^{H}\!\gamma_{t}^{\prime} + {}^{T}\!\xi_{t}^{\prime}) \nonumber\\ &=&\widehat{\nabla}_{\displaystyle{}^{H}\gamma_{t}^{\prime}}{}^{H}\gamma_{t}^{\prime} +\widehat{\nabla}_{\displaystyle{}^{H}\!\gamma_{t}^{\prime}}{}^{T}\!\xi_{t}^{\prime}+\widehat{\nabla}_{{}^{T}\!\xi_{t}^{\prime}}{}^{H}\!\gamma_{t}^{\prime}+\widehat{\nabla}_{{}^{T}\!\xi_{t}^{\prime}}{}^{T}\!\xi_{t}^{\prime} \nonumber\\ &=&{}^{H}\!(\gamma_{t}^{\prime\prime}+R(\varphi \xi,\xi_{t}^{\prime})\gamma_{t}^{\prime})+{}^{T}\!\xi_{t}^{\prime\prime} \nonumber\\ &=&{}^{H}\!(\gamma_{t}^{\prime\prime}+R(\varphi \xi,\xi_{t}^{\prime})\gamma_{t}^{\prime})+{}^{V}\!(\xi_{t}^{\prime\prime}-g(\xi_{t}^{\prime\prime}, \varphi \xi)\xi).\end{aligned}$$ On the other hand, $$\begin{aligned} \widehat{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime}&=&\varrho_{1}\Gamma_{t}^{\prime} +\varrho_{2}{}^{H}\!F \Gamma_{t}^{\prime}\nonumber\\ &=&\varrho_{1}({}^{H}\!\gamma_{t}^{\prime} + {}^{T}\!\xi_{t}^{\prime}) +\varrho_{2}{}^{H}\!F({}^{H}\!\gamma_{t}^{\prime} + {}^{T}\!\xi_{t}^{\prime})\nonumber\\ &=&\varrho_{1}({}^{H}\!\gamma_{t}^{\prime} + {}^{V}\!\xi_{t}^{\prime}-g(\xi_{t}^{\prime}, \varphi \xi){}^{V}\!\xi) +\varrho_{2}{}^{H}\!F({}^{H}\!\gamma_{t}^{\prime} + {}^{V}\!\xi_{t}^{\prime}-g(\xi_{t}^{\prime}, \varphi \xi){}^{V}\!\xi).\end{aligned}$$ From [\[eq_3\]](#eq_3){reference-type="eqref" reference="eq_3"}, we have $$\begin{aligned} \label{eq_26} \widehat{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime}&=&{}^{V}\!\varrho_{1}{}^{H}\!\gamma_{t}^{\prime} +{}^{V}\!\varrho_{2}{}^{H}\!F {}^{H}\!\gamma_{t}^{\prime}+{}^{V}\!\varrho_{1}{}^{V}\!\xi_{t}^{\prime} +{}^{V}\!\varrho_{2}{}^{H}\!F {}^{V}\!\xi_{t}^{\prime}\nonumber\\ &=&{}^{H}\!(\varrho_{1}\gamma_{t}^{\prime}+\varrho_{2}F \gamma_{t}^{\prime})+{}^{V}\!(\varrho_{1}\xi_{t}^{\prime} +\varrho_{2}F \xi_{t}^{\prime}).\end{aligned}$$ From [\[eq_25\]](#eq_25){reference-type="eqref" reference="eq_25"}, [\[eq_26\]](#eq_26){reference-type="eqref" reference="eq_26"} and [\[eq_6\]](#eq_6){reference-type="eqref" reference="eq_6"}, the result immediately follows. ◻ **Corollary 27**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric. A curve $\Gamma =(\gamma(t), \xi(t))$ on $T^{\varphi}_{1}M$ is an ${}^{H}\!\varphi$-planar with respect to $\widehat{\nabla}$ if and only if the $$\left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}=R(\xi_{t}^{\prime},\varphi\xi)\gamma_{t}^{\prime}+\varrho_{1}\gamma_{t}^{\prime}+\varrho_{2} \varphi\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=\varrho_{1}\xi_{t}^{\prime}+\varrho_{2} \varphi\xi_{t}^{\prime}-\rho^{2}\xi \end{array}\right.$$* In the particular case when $\varrho_{1} = 0$ and $\varrho_{2} = 1$ in the Theorem [Theorem 26](#th_11){reference-type="ref" reference="th_11"}, we obtain the following result. **Theorem 28**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric and $F$ be a $(1,1)$-tensor field on $M$. A curve $\Gamma =(\gamma(t), \xi(t))$ on $TM$ is an ${}^{H}\!F$-geodesic with respect to $\widehat{\nabla}$ if and only if the $$\left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}=R(\xi_{t}^{\prime},\varphi\xi)\gamma_{t}^{\prime}+ F\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=F\xi_{t}^{\prime}-\rho^{2}\xi \end{array}\right.$$* **Corollary 29**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric. A curve $\Gamma =(\gamma(t), \xi(t))$ on $T^{\varphi}_{1}M$ is an ${}^{H}\!\varphi$-geodesic with respect to $\widehat{\nabla}$ if and only if the $$\left\{\begin{array}{lll} \gamma_{t}^{\prime\prime}=R(\xi_{t}^{\prime},\varphi\xi)\gamma_{t}^{\prime}+ \varphi\gamma_{t}^{\prime}\\ \xi_{t}^{\prime\prime}=\varphi\xi_{t}^{\prime}-\rho^{2}\xi \end{array}\right.$$* **Theorem 30**. *Let $(M^{2m}, \varphi, g)$ be a para-Kähler-Norden manifold and $T^{\varphi}_{1}M$ its $\varphi$-unit tangent bundle equipped with the $\varphi$-Sasaki metric and $F$ be a $(1,1)$-tensor field on $M$. If $\Gamma =(\gamma(t), \xi(t))$ is a horizontal lift of $\gamma$ and $\Gamma\in T^{\varphi}_{1}M$, then $\Gamma$ is an ${}^{H}\!F$-planar curve (resp., ${}^{H}\!F$-geodesic) if and only if $\gamma$ is an $F$-planar curve (resp., $F$-geodesic).* *Proof.* Let $\gamma$ be an $F$-planar with respect to $\nabla$ on $M$, i.e. $\gamma$ satisfies $$\begin{aligned} \gamma_{t}^{\prime\prime}=\varrho_{1}\gamma_{t}^{\prime} +\varrho_{2}F \gamma_{t}^{\prime},\end{aligned}$$ where $\varrho_{1}$ and $\varrho_{2}$ are some functions of the parameter $t$. Since $\Gamma =(\gamma(t), \xi(t))$ is a horizontal lift of a curve $\gamma$ then $\xi_{t}^{\prime}=0$ and $\Gamma_{t}^{\prime}={}^{H}\!\gamma_{t}^{\prime}$.\ On the other hand, $$\begin{aligned} \widehat{\nabla}_{\Gamma_{t}^{\prime}}\Gamma_{t}^{\prime}&=&{}^{H}\!\gamma_{t}^{\prime\prime}\\ &=&{}^{H}\!(\varrho_{1}\gamma_{t}^{\prime}+\varrho_{2}F \gamma_{t}^{\prime})\\ &=&{}^{V}\!\varrho_{1}{}^{H}\!\gamma_{t}^{\prime} +{}^{V}\!\varrho_{2}{}^{H}\!F {}^{H}\!\gamma_{t}^{\prime}\\ &=&{}^{V}\!\varrho_{1}\Gamma_{t}^{\prime}+{}^{V}\!\varrho_{2}{}^{H}\!F \Gamma_{t}^{\prime}.\end{aligned}$$ i.e. $\Gamma$ be an ${}^{H}\!F$-planar with respect to $\widehat{\nabla}$. In the case of $\varrho_{1} = 0$ and $\varrho_{2} = 1$, we get that $\Gamma$ is an ${}^{H}\!F$-geodesic if and only $\gamma$ is an $F$-geodesic. ◻ 9 M. Altunbas, R. Simsek and A. Gezer, *A Study Concerning Berger type deformed Sasaki Metric on the Tangent Bundle*, Zh. Mat. Fiz. 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arxiv_math

--- author: - | Timo Lähivaara${}^{a}$, William Hall${}^{b}$, Matti Malinen${}^{c}$,\ Dale Ota${}^{b}$, Vijaya Shankar${}^{b}$, and Peter Monk${}^{d}$\ ${}^{a}$Department of Technical Physics, University of Eastern Finland, Kuopio, Finland\ ${}^{b}$HyperComp Inc., Westlake Village, USA\ ${}^{c}$Kuava Ltd., Kuopio, Finland\ ${}^{d}$Department of Mathematical Sciences, University of Delaware, Newark, USA title: A High-Order Ultra-Weak Variational Formulation for Electromagnetic Waves Utilizing Curved Elements --- # Abstract {#abstract .unnumbered} The Ultra Weak Variational Formulation (UWVF) is a special Trefftz discontinuous Galerkin method, here applied to the time-harmonic Maxwell's equations. The method uses superpositions of plane waves to represent solutions element by element on a finite element mesh. We discuss the use of our parallel UWVF implementation called *ParMax*, and concentrate on methods for obtaining high order solutions in the presence of scatterers with piecewise smooth boundaries. In particular, we show how curved surface triangles can be incorporated in the UWVF. This requires quadrature to assemble the system matrices. We also show how to implement a total field and scattered field approach, together with the transmission conditions across an interface to handle resistive sheets. We note also that a wide variety of element shapes can be used, that the elements can be large compared to the wavelength of the radiation, and that a matrix free version is easy to implement (although computationally costly). Our contributions are illustrated by several numerical examples showing that curved elements can improve the efficiency of the UWVF, and that the method accurately handles resistive screens as well as PEC and penetrable scatterers. Using large curved elements and the matrix free approach, we are able to simulate scattering from an aircraft at X-band frequencies. The innovations here demonstrate the applicability of the UWVF for industrial examples. # Introduction A Trefftz method [@Trefftz26] for approximating a linear partial differential equation is a numerical method using local solutions of the underlying partial differential equation as basis functions. The version we shall study here, the Ultra Weak Variational Formulation (UWVF) of Maxwell's equations, is a Trefftz type method for approximating the solution of Maxwell's equations on a bounded domain due to O. Cessenat and B. Després [@cessenat_phd; @cessenat03]. The UWVF uses a finite element computational grid, classically composed of tetrahedral elements, and plane wave solutions of Maxwell's equations on each element. A study of this method from the point of view of symmetric hyperbolic systems was presented in [@Huttunen2007] where the inclusion of the Perfectly Matched Layer absorbing condition [@Berenger] into the UWVF was also described. In addition, several computational heuristics relevant to our study were also presented. As a result of this work, a first parallel implementation of the UWVF called *ParMax* was written. This was further developed at Kuava Inc, and the University of Eastern Finland and is the basic software used in this paper. *ParMax* uses MPI and domain decomposition to implement parallelism. From the point of view of theoretical convergence analysis, it was shown in [@buf07] that the UWVF for the related Helmholtz equation is a special discontinuous Galerkin (DG) method, and using duality theory convergence estimates could be proved. A more general DG approach, again for the Helmholtz equation, is taken in [@git07]. Based on these studies, and following the proof of explicit stability bounds for the interior impedance boundary value problem for Maxwell's equations in [@moi11], convergence of Trefftz DG methods (in particular the UWVF) for Maxwell's equations under rather strict geometric assumptions was proved in [@HMP13]. A key point in this analysis is that it is not necessary to use tetrahedral elements, but a wide class of element shapes are theoretically covered and we shall return to this point in Section [3](#TISC){reference-type="ref" reference="TISC"}. A restriction on the use of the UWVF is that the relative electric permittivity denote $\epsilon_{\textrm{r}}$ and relative magnetic permeability $\mu_{\textrm{r}}$ must be piecewise constant (constant on each tetrahedron in the mesh). This assumption could perhaps be weakened using generalised plane waves [@IM-sylvand-21] or embedded Trefftz techniques [@PS_22] but these have not yet been extended to Maxwell's equations (the latter technique has been demonstrated in the open source package [@NGSTrefftz]). The restriction of piecewise constant media is relaxed if a Perfectly Matched Layer (PML) is used. There $\epsilon_{\textrm{r}}$ and $\mu_{\textrm{r}}$ are spatially varying tensors [@Huttunen2007]. The main issue facing the UWVF is that the condition number of the global system rises rapidly as the number of plane wave directions increases. This in turn causes the iterative solution of the linear system to slow down. We adopt the approach of [@Huttunen2007] choosing the number of plane wave directions on a given element on the basis of its geometric size (in wavelengths). An alternative approach is to use an element-wise singular value decomposition to determine a stable basis [@Barucq-conditioning-2021] but this has not been implemented in *ParMax*. We use the same stabilised BiConjugate Gradient (BICGstab) scheme as in [@Huttunen2007] for solving the global linear system resulting from the UWVF. Interesting new results on preconditioned iterative schemes can be found in [@sirdey22]. These results are based on the used of a structured hexahedral grid, and are not implemented in *ParMax* so are not discussed in this paper. The UWVF is by no means the only way to implement a Trefftz method for Maxwell's equations. A least squares approach has been considered in [@HuYuan-14]. This is extended to non-homogeneous problems by using local spectral elements in [@HuYuan-18]. This approach could also be used to extend the UWVF to non-homogeneous problems but has yet to be tried. In this paper, $i=\sqrt{-1}$ and we use the convention that the time variation of the fields and sources is proportional to $\exp(-i\omega t)$ where $\omega$ is the frequency of the radiation, and $t$ is time. All results are then reported in the frequency domain. Bold face quantities are vector valued. The coefficients $\epsilon_0$ and $\mu_0$ are the permittivity and permeability of free space respectively. The wave number $\kappa$ of the radiation is given by $\kappa=\omega\sqrt{\epsilon_0\mu_0}$. To further fix notation and context, we now define the Maxwell system under consideration in this paper. Let $\Omega$ denote the Lipschitz bounded computational domain having unit outward normal $\boldsymbol{\nu}$ and boundary $\Gamma:=\partial\Omega$. For a smooth enough vector field $\mathbf{v}$, we define the tangential component $\mathbf{v}_T$ on $\Gamma$ by $\mathbf{v}_T=(\boldsymbol{\nu}\times\mathbf{v})\times\boldsymbol{\nu}$. Then, given the wave number $\kappa>0$, a tangential boundary vector field $\mathbf{g}$, piecewise constant functions $\epsilon_{\textrm{r}}$ and $\mu_{\textrm{r}}$, and a parameter $Q\in\mathbb{C}$ with $|Q|\leq 1$ we seek the complex valued vector electric field $\mathbf{E}$ that satisfies $$\begin{aligned} \mathop{\mathrm{curl}}\mu_{\textrm{r}}^{-1}\mathop{\mathrm{curl}}\mathbf{E}-\kappa^2\epsilon_{\textrm{r}}\mathbf{E}&=0\mbox{ in }\Omega,\label{eq:maxwell}\\ \boldsymbol{\nu}\times\mu_{\textrm{r}}^{-1}\mathop{\mathrm{curl}}\mathbf{E}+\frac{i\kappa}{Z}\mathbf{E}_T&=Q\left(-\boldsymbol{\nu}\times\mu_{\textrm{r}}^{-1}\mathop{\mathrm{curl}}\mathbf{E}+\frac{i\kappa}{Z}\mathbf{E}_T\right)+\mathbf{g}\mbox{ on }\Gamma. \label{eq:bc}\end{aligned}$$ Here $Z$ is a positive real parameter. The boundary condition ([\[eq:bc\]](#eq:bc){reference-type="ref" reference="eq:bc"}) is of impedance type and is well suited to the UWVF. When $Q=1$ it gives the standard Perfectly Electrically Conducting (PEC) boundary condition while when $Q=0$ it corresponds to an outgoing condition that can be used as a low order radiation condition. This paper presents several novel extensions of the basic UWVF that are of considerable utility in practical applications. In particular: - The basic UWVF uses a tetrahedral grid, however the error estimates in [@HMP13] hold for more general element types. We have implemented hexahedral, and wedge elements. In this paper, hexahedral elements are only used in the PML region. - Very often curved surfaces appear in applications, and we have implemented a mapping technique to approximate smooth curved surfaces. This allows us to use larger elements near a curved boundary. We shall show, using numerical experiments, that this improves the efficiency of the software by decreasing the overall time to compute a solution. Note that we only need to map faces in the mesh which simplifies the implementation, but we have to use quadrature to evaluate integrals on curved faces. - We show how to implement resistive sheet transmission conditions across thin interfaces. Related to this we have also implemented a combined total field and scattered field formulation to allow the solution of problems involving penetrable media. - We point out that a low memory version can be used to solve very large problems by avoiding the storage of the most memory intensive matrix in the algorithm. - We provide numerical results to justify the utility of the above innovations. The paper proceeds as follows. In Section [2](#sec:base){reference-type="ref" reference="sec:base"} we start with a brief derivation of the basic UWVF and describe the plane wave based UWVF. Then in Section [3](#TISC){reference-type="ref" reference="TISC"} we describe the five contributions of this paper. We start with comments on new geometric element types and a brief discussion of implementing an algorithm for using scattered or total fields in different subdomains in the context of scattering from a penetrable object. Then we move to discuss resistive sheets, curved elements and quadrature, and finally a low memory version of the method. Section  [4](#sec:NE){reference-type="ref" reference="sec:NE"} is devoted to numerical examples illustrating the UWVF and the previously mentioned modifications. In Section [4.1](#sec:PEC){reference-type="ref" reference="sec:PEC"} we start with two examples of scattering from a PEC scatterer. The first PEC example is scattering from a sphere for which the Mie series gives an accurate solution for comparison (c.f. [@Monk03]). This example demonstrates the benefits of curved elements and different element types, as well as the use of very coarse meshes (compared to those used by finite element methods). The second example is X-band scattering from an aircraft model. Here again we use curved elements, but in addition use the low-memory version of the software. In Section [4.2](#sec:res){reference-type="ref" reference="sec:res"} we consider two examples involving a resistive sheet The first is a classical example having an exact analytical solution, and the second is a resistive screen surrounding a sphere. Next, in Section [4.3](#sec:het){reference-type="ref" reference="sec:het"}, we consider heterogeneous or penetrable scatterers. The first example is a dielectric sphere where we use the scattered/near field formulation and compare to the Mie series solution. The second example is scattering from a plasma. We end in Section [5](#sec:conc){reference-type="ref" reference="sec:conc"} with our conclusions, and present in Appendix [6](#sec:app){reference-type="ref" reference="sec:app"} an update to the basis selection rule used previously in [@Huttunen2007] for the new element types. # Derivation and properties of the basic UWVF {#sec:base} In this section we provide a sketch of the derivation of the UWVF sufficient to allow us to present the new features of this paper in the following section. For full details see [@cessenat_phd; @Huttunen2007]. ## A brief derivation of the UWVF The version of the UWVF presented here is equivalent to that used in *ParMax* (from [@cessenat_phd]) but with simplified notation. Consider a mesh of $\Omega$ of elements of maximum diameter $h$ denoted by ${\cal T}_h$. A element $K\in {\cal T}_h$ in this mesh is a curvilinear polyhedron (curvilinear tetrahedron, wedge or hexahedron) with boundary $\partial K$ and unit outward norm $\boldsymbol{\nu}^K$. Pyramids are also implemented in *ParMax* but not used in this paper. We now extend the parameter $Z$ to be a piecewise positive constant defined all faces in the grid. Following [@cessenat_phd], we choose $Z$ as follows. Let $$\hat{\epsilon}=\left\{\begin{array}{rl} |\sqrt{\epsilon_{\textrm{r}}|_K\,\epsilon_{\textrm{r}}|_{K'}}|& \mbox{on } K\cap K' \mbox{ for }K,K'\in{\cal T}_h\\ |{\epsilon_{\textrm{r}}}|&\mbox{on boundary faces}\end{array}\right.$$ where $|_K$ denotes the restriction to $K$. The edge function $\hat{\mu}$ is defined in the same way. Then $Z=\sqrt{\hat{\mu}}/\sqrt{\hat{\epsilon}}$. Suppose $\boldsymbol{\xi}$ is a smooth solution of the adjoint Maxwell equation in $K$: $$\mathop{\mathrm{curl}}\overline{\mu_{\textrm{r}}}^{-1}\mathop{\mathrm{curl}}\boldsymbol{\xi}-\kappa^2\overline{\epsilon_{\textrm{r}}}\boldsymbol{\xi}=0. \label{max_adj}$$ Then taking the dot product of ([\[eq:maxwell\]](#eq:maxwell){reference-type="ref" reference="eq:maxwell"}) with $\boldsymbol{\xi}$ (including complex conjugation) and integrating by parts twice provides the following fundamental relation between the electric and magnetic fluxes on $\partial K$: $$\int_{\partial K}{\boldsymbol{\nu}^K\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}}\cdot\overline{\boldsymbol{\xi}}_T+{\boldsymbol{\nu}^K\times\mathbf{E}}\cdot(\overline{\overline{\mu_{\textrm{r}}}^{-1}\nabla\times\boldsymbol{\xi}})_T\,dA=0.\label{fund_eq}$$ Using the above fundamental identity, we can then prove equality ([\[iso\]](#iso){reference-type="ref" reference="iso"}) by expanding both sides of ([\[iso\]](#iso){reference-type="ref" reference="iso"}) and using ([\[fund_eq\]](#fund_eq){reference-type="ref" reference="fund_eq"}). Equality ([\[iso\]](#iso){reference-type="ref" reference="iso"}) gives the conclusion of the "isometry lemma" (c.f. [@cessenat_phd]). $$\begin{aligned} &&\int_{\partial K}Z\left(- \boldsymbol{\nu}^K\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}+\frac{i\kappa{}}{Z} \mathbf{E}_T\right)\cdot\left(-\overline{\boldsymbol{\nu}^K\times{\overline{}}\mu_{\textrm{r}}^{-1}\nabla\times\boldsymbol{\xi}+\frac{i\kappa{}}{Z} \boldsymbol{\xi}_T}\right)\,dA.\nonumber\\ &=&\int_{\partial K}Z\left( \boldsymbol{\nu}^K\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}+\frac{i\kappa{}}{Z} \mathbf{E}_T\right)\cdot\left(\overline{\boldsymbol{\nu}^K\times{}\mu_{\textrm{r}}^{-1}\nabla\times\boldsymbol{\xi}+\frac{i\kappa{}}{Z} \boldsymbol{\xi}_T}\right)\,dA\label{iso}\end{aligned}$$ To simplify the presentation, we define rescaled versions of the unknowns in [@cessenat_phd] as follows: $$\begin{aligned} \boldsymbol{\chi}_K&=&-\boldsymbol{\nu}^K\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}|_{K}+\frac{i\kappa{}}{Z} (\mathbf{E}|_K)_T \label{eq:chi}\\ {\cal Y}_K&=&-\boldsymbol{\nu}^K\times\mu_{\textrm{r}}^{-1}\nabla\times\boldsymbol{\xi}|_{K}+\frac{i\kappa{}}{Z} (\boldsymbol{\xi}|_K)_T.\end{aligned}$$ The next step is to rewrite ([\[iso\]](#iso){reference-type="ref" reference="iso"}) in terms of the above quantities. In doing so we will use the function space $\mathbf{L}_T^2(\partial K):=\{\mathbf{u}\in\mathbf{L}^2(\partial K)\;|\;\mathbf{u}\cdot\boldsymbol{\nu}^K=0\}$. Recalling that $\boldsymbol{\xi}|_K$ satisfies the adjoint Maxwell system in $K$ we can define $\mathbf{F}_K: \mathbf{L}_T^2(\partial K)\to \mathbf{L}_T^2(\partial K)$ by setting $$\mathbf{F}_K{\cal Y}_K=\boldsymbol{\nu}^K\times\mu_{\textrm{r}}^{-1}\nabla\times\boldsymbol{\xi}|_{K}+i\kappa{}Z (\boldsymbol{\xi}|_K)_T.$$ Now suppose elements $K$ and $K'$ meet at a face in the mesh. Then on that face $\boldsymbol{\nu}^K=-\boldsymbol{\nu}^{K'}$. Also we have transmission condition requiring continuity of $\mathbf{E}_T$ and $(\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E})_T$ across faces so $$\boldsymbol{\nu}^K\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}|_K+i\omega{}Z (\mathbf{E}|_K)_T =\boldsymbol{\chi}_{K'}.\label{trans}$$ For a boundary face, we can use the boundary condition to replace the corresponding term on those faces. Using these the above results, we may rewrite ([\[iso\]](#iso){reference-type="ref" reference="iso"}) for every element $K$ in the mesh such that for $\boldsymbol{\chi}_K\in \mathbf{L}_T^2(\partial K)$ the following holds $$\begin{aligned} \int_{\partial K}Z\,\boldsymbol{\chi}_K\cdot{\cal Y}_K\,dA=&\sum_{K'\not= K}\int_{\partial K\cap \partial K'}Z\,\boldsymbol{\chi}_{K'}\cdot \mathbf{F}_K{\cal Y}_K\,dA\nonumber\\ +\int_{\partial K\cap \partial K'}&Z\left[Q\boldsymbol{\chi}_K+\mathbf{g}\right]\cdot \mathbf{F}_K{\cal Y}_K\,dA,\label{eqUWVFeq}\end{aligned}$$ for any ${\cal Y}_K\in \mathbf{L}^2_T(\partial K)$. This equation should hold for every $K\in {\cal T}_h$, and gives the UWVF for Maxwell's equations before discretization. Cessenat [@cessenat_phd] proves the uniqueness of the UWVF solution, and existence follows because ([\[eq:maxwell\]](#eq:maxwell){reference-type="ref" reference="eq:maxwell"})-([\[eq:bc\]](#eq:bc){reference-type="ref" reference="eq:bc"}) is well posed. ## The plane wave UWVF (PW-UWVF) {#Deriv} Now that we have the variational formulation ([\[eqUWVFeq\]](#eqUWVFeq){reference-type="ref" reference="eqUWVFeq"}) we can discretize it by using a subspace of $\mathbf{L}^2_T(\partial K)$ on each element. It is important for efficiency that $\mathbf{F}_K$ be easy to compute and this is where the plane wave basis is useful [@cessenat_phd]. On each element $K$ we choose independent directions $\{\mathbf{d}_{K,j}\}_{j=1}^{p_K}$, $\Vert\mathbf{d}_{K,j}\Vert=1$, using the first $p_K$ Hammersley points [@hardin_16] on the unit sphere. Then we choose $\boldsymbol{\xi}|_K$ to be a linear combination of the $p_K$ plane wave solutions of the adjoint problem $$\boldsymbol{\xi}_{K,\ell,m}=\mathbf{A}_{K,\ell,m} \exp\left(i\kappa\sqrt{\overline{\epsilon_{\textrm{r}}|_K}\overline{\mu_{\textrm{r}}|_K}} \mathbf{d}_{K,\ell}\cdot (\mathbf{x}-\mathbf{x}_{K,0})\right)$$ for $1\leq \ell\leq P_K,$ $1\leq m\leq 2,$ and $K\in {\cal T}_h$. Here $\mathbf{x}_{K,0}$ is the centroid of the element and the polarizations $\mathbf{A}_{K,\ell,m}$ are chosen to be unit vectors such that $\mathbf{d}_{K,\ell}\cdot \mathbf{A}_{K,\ell,m}=0$, $m=1,2$ and $\mathbf{A}_{K,\ell,1}\cdot\mathbf{A}_{K,\ell,2}=0$. No we can define a discrete subspace $\mathbf{W}_{K,h}\subset\mathbf{L}_T^2(\partial K)$ by first defining $$\begin{aligned} &&\mathbf{W}_{K,h,p_K}={\rm span}\{- \boldsymbol{\nu}^K\times\mu_{\textrm{r}}^{-1}\nabla\times\boldsymbol{\xi}_{K,\ell,m}\\&&\qquad+(i\kappa/Z)(\boldsymbol{\xi}_{K,\ell,m})_T, \;m=1,2,\; 1\leq \ell\leq p_K\}.\end{aligned}$$ Then $\mathbf{W}_{h,\mathbf{p}}=\Pi_{K\in{\cal T}_h}\mathbf{W}_{K,h,p_K}$ where $\mathbf{p}$ denotes the vector of number of directions on each element. The dimension of $W_{h,\mathbf{p}}$ is $N_{\mathrm{dof}}=2\sum_{K\in{\cal T}_h}p_K$. In our work $p_K$ is chosen according to the heuristic in [@Huttunen2007] (see Section [6](#sec:app){reference-type="ref" reference="sec:app"} for updates to this formula for larger numbers of directions and new element types). Then $\mathbf{F}_K$ is easy to compute using the definition of the basis functions. The discrete PW-UWVF uses trial and test functions from $\mathbf{W}_{h,\mathbf{p}}$ in place of $\mathbf{W}$ in ([\[eqUWVFeq\]](#eqUWVFeq){reference-type="ref" reference="eqUWVFeq"}). In our implementation, we use domain decomposition by subdividing the mesh according to Metis [@Karypis_metis98] where $p_K$ is used to estimate the work on each element. The elements are sorted using reverse Cuthill-McKee to minimise bandwidth. Then, enumerating the degrees of freedom element by element, the matrices and vectors corresponding to the terms in ([\[eqUWVFeq\]](#eqUWVFeq){reference-type="ref" reference="eqUWVFeq"}) can be computed. The left hand side of ([\[eqUWVFeq\]](#eqUWVFeq){reference-type="ref" reference="eqUWVFeq"}) gives rise to an $N_{\mathrm{dof}}\times N_{\mathrm{dof}}$ block diagonal Hermitian positive-definite matrix $D$, while the remaining sesquilinear forms on the right hand side of ([\[eqUWVFeq\]](#eqUWVFeq){reference-type="ref" reference="eqUWVFeq"}) give rise to a general sparse complex matrix $C$. The data term involving $\mathbf{g}$ gives rise to a corresponding vector $\vec{\mathbf{b}}$. Denoting the vector of unknown degrees of freedom by $\vec{\boldsymbol{\chi}}$, we solve the global matrix equation $$(I-D^{-1}C)\vec{\boldsymbol{\chi}}=D^{-1}\vec{\mathbf{b}}\label{dc}$$ using BICGstab where $D^{-1}$ can be calculated rapidly element by element. Once $\vec{\boldsymbol{\chi}}$ is known, the solution on each element can be reconstructed for post processing. For unbounded scattering problems, we use either the low order absorbing condition ($Q=0$) on the outer boundary or a PML as in [@Huttunen2007]. For a scattering problem the far field pattern can the be calculated using an auxiliary surface containing all the scatterers in its interior in the usual way [@cessenat_phd]. # Towards industrial scale software {#TISC} In this section we discuss the extensions to the basic UWVF in this paper. ## New element types In [@HMP13] error estimates are proved for general elements that have Lipschitz boundaries, are shape regular in the sense of that paper, and are star-shaped with respect to a ball centred at a point in the element. This allows a wide variety of elements. We have implemented curvilinear tetrahedral, pyramid, wedge, and hexahedral elements. In *ParMax* we represent the boundary of each element as a union of possibly curvilinear triangles. The assembly phase is quickest if the triangles are planar since then quadrature can be avoided. For this paper, we rely on COMSOL Multiphysics to generate the meshes. An example of a grid using tetrahedral, wedge, and hexahedral elements is shown in Fig. [3](#fig:pec_sphere_grid){reference-type="ref" reference="fig:pec_sphere_grid"}. Here we use wedge and hexahedral elements in the PML, and tetrahedral elements elsewhere. ## Scattered/total field formulation {#stf} The numerical results we shall present are all of scattering type. The total field $\mathbf{E}$ is composed of an unknown scattered field $\mathbf{E}^s$ and a given incident field $\mathbf{E}^i$ so $\mathbf{E}=\mathbf{E}^s+\mathbf{E}^i$. We will use plane wave incident fields (but point sources or other incident fields can be used) so $$\mathbf{E}^i(\mathbf{x})=\mathbf{p}\exp(i\kappa\mathbf{d}\cdot\mathbf{x}),\label{Einc}$$ where $\mathbf{d}\in \mathbb{R}^3$ is the direction of propagation and $\Vert\mathbf{d}\Vert=1$. The vector polarisation $\mathbf{p}\in\mathbb{C}^3$ is non-zero and satisfies $\mathbf{d}\cdot\mathbf{p}=0$. To allow the use of a PML or other absorbing boundary condition, we need to compute using the scatterered field in the PML. But inside a penetrable scatterer we need to compute with the total field so that there are no current sources in the scatterer. This is standard for finite element methods, but not usual for the UWVF so we outline the process here. Suppose $\Omega$ is partitioned into two subdomains denoted $\Omega_-$ and $\Omega_+$ such that the PML (or a neighbourhood of the absorbing boundary if one is used) is contained in $\Omega_+$ where the scattered field is used and where $\epsilon_{\textrm{r}}=\mu_{\textrm{r}}=1$. The scatterer is contained in $\Omega_-$ where possible $\epsilon_{\textrm{r}}$ or $\mu_{\textrm{r}}$ are no longer unity and the total field is used. Let $\Sigma$ denote the boundary between $\Omega_-$ and $\Omega_+$ and assume that $\Sigma$ is contained in the interior of $\Omega$ and is exactly covered by faces of the mesh. Then suppose that elements $K_-\subset\Omega_-$ and $K_+\subset \Omega_+$ meet at a face $F\subset\Sigma$. Consider first $K_-$. Reviewing the derivation of the UWVF outlined in Section [2.2](#Deriv){reference-type="ref" reference="Deriv"} we see that we must rewrite the term in $\mathbf{E}$ on the right hand side of equation ([\[iso\]](#iso){reference-type="ref" reference="iso"}) in terms of $\boldsymbol{\chi}_{K_-}$ and $\boldsymbol{\chi}_{K_+}$. In particular using the facts that $\boldsymbol{\nu}_{K_-}=-\boldsymbol{\nu}_{K_+}$ and that on $K_+$ the scattered field $\mathbf{E}^s$ is the field to be approximated, so in ([\[eq:chi\]](#eq:chi){reference-type="ref" reference="eq:chi"}) $\mathbf{E}$ is replaced by $\mathbf{E}^s+\mathbf{E}^i$, we obtain $$\begin{aligned} \left( \boldsymbol{\nu}^{K_-}\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}|_{K_-}+\frac{i\kappa{}}{Z} \mathbf{E}|_{K_-,T}\right) &=&-\boldsymbol{\nu}^{K_+}\times\mu_{\textrm{r}}^{-1}\nabla\times(\mathbf{E}^s+\mathbf{E}^i)|_{K_+}\\&& +\frac{i\kappa{}}{Z} (\mathbf{E}^s+\mathbf{E}^i)|_{K_+,T}=\boldsymbol{\chi}_{K_+}+\mathbf{g}_{K_-,F}.\end{aligned}$$ Here we have also used the transmission condition that the tangential components of $\mathbf{E}$ and $\mu_{\textrm{r}}^{-1}\mathop{\mathrm{curl}}\mathbf{E}$ are continuous across $\Sigma$. In this equation, the source function on $F$ associated with $K_-$ is $$\mathbf{g}_{K_-,F}=\left(\boldsymbol{\nu}^{K_-}\times\mu_{\textrm{r}}^{-1}\nabla\times \mathbf{E}^i|_{F}+\frac{i\kappa{}}{Z} \mathbf{E}^i|_{F,T}\right).$$ Carrying out the same procedure on $K_+$ remembering this element supports the scattered field gives another equation and source function $\mathbf{g}_{K_+,F}$ again relating $\mathbf{E}$ and $\mathbf{E}^s$. Thus a combined scattered and total field algorithm can be implemented by allowing for a source function $\mathbf{g}$ on the internal surfaces. ## Resistive sheet A similar procedure to that used to implement the scattered/total field UWVF can be used to derive appropriate modifications to include resistive [@jin_volakis] or conductive sheets [@Senior]. We only consider the resistive sheet. Suppose now that a surface $\Sigma_r\subset\Omega$ is a resistive sheet, and that $\boldsymbol{\nu}_r$ denotes a continuous normal to the sheet. This surface may be open or closed and could intersect the boundary. We just assume that it is the union of a subset of the faces (possibly curvilinear) in the mesh. {#fig:Et width="50%"} Suppose $K_+$ and $K_-$ are two elements that meet at a face $F_r\subset \Sigma_r$ such that $\boldsymbol{\nu}_r$ points into $K_+$. The geometry is shown in Fig. [1](#fig:Et){reference-type="ref" reference="fig:Et"}. The resistive sheet approximation requires us to implement the following transmission conditions across $\Sigma_r$ $$\begin{aligned} \boldsymbol{\nu}_r\times(\mu_{\textrm{r}}^{-1}\mathop{\mathrm{curl}}\mathbf{E}|_{K_+}-\mu_{\textrm{r}}^{-1}\mathop{\mathrm{curl}}\mathbf{E}|_{K_-})&=&i\kappa\sigma d (\boldsymbol{\nu}_r\times \mathbf{E}_{K_+})\times \boldsymbol{\nu}_r,\label{dit1}\\ (\boldsymbol{\nu}_r\times \mathbf{E}|_{K_+})\times \boldsymbol{\nu}_r&=&(\boldsymbol{\nu}_r\times \mathbf{E}_{K_-})\times \boldsymbol{\nu}_r,\label{dit2}\end{aligned}$$ where $\sigma$ is the conductivity of the material in the layer and $d$ is the thickness. Alternatively the resistivity of the sheet is given by $$R=(\sigma d)^{-1}=-\frac{Z_0}{i(\epsilon_{\textrm{r}}-1)k_0d}$$ with $\epsilon_{\textrm{r}}=1+i\sigma /(\epsilon_0\omega)$, $Z_0=\sqrt{\mu_0/\epsilon_0}$ is the impedance of free space and $k_0=\omega\sqrt{\mu_0\epsilon_0}$ is the wave-number. We now set $\eta=\sigma d$. As for the scattered/total field version of UWVF we must write the term in $\mathbf{E}$ on the right hand side of ([\[iso\]](#iso){reference-type="ref" reference="iso"}) using the variables $\boldsymbol{\chi}|_{K_+}$ and $\boldsymbol{\chi}|_{K_-}$ on either side of $F_r$. Adding $\boldsymbol{\chi}|_{K_+}$ to $\boldsymbol{\chi}|_{K_-}$, using the resistive sheet transmission conditions and noting that $\boldsymbol{\nu}^{K_-}=\boldsymbol{\nu}_r=-\boldsymbol{\nu}^{K_+}$ gives $$\begin{aligned} \boldsymbol{\chi}_{K_-}+\boldsymbol{\chi}_{K_+} &=&i\kappa\left(\frac{2}{Z}\mathbf{E}_{K_+,T} + \eta \mathbf{E}_{K_+,T}\right)\\\end{aligned}$$ so $$\mathbf{E}|_{K_+,T}=\mathbf{E}|_{K_-,T}=\frac{\boldsymbol{\chi}_{K_-}+\boldsymbol{\chi}_{K_+}}{2i\kappa/Z+i\kappa\eta }. \label{e1nu1}$$ Now we rewrite the term in $\mathbf{E}$ on the right hand side of ([\[iso\]](#iso){reference-type="ref" reference="iso"}) using the UWVF functions. Using Equation ([\[e1nu1\]](#e1nu1){reference-type="ref" reference="e1nu1"}) we have $$\begin{aligned} \boldsymbol{\nu}^{K_-}\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}|_{K_-}+\frac{i\kappa{}}{Z} \mathbf{E}|_{K_-,T}= \boldsymbol{\nu}^{K_-}\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}|_{K_-}+\frac{1}{Z} \frac{\boldsymbol{\chi}_{K_-}+\boldsymbol{\chi}_{K_+}}{2/Z+\eta }. \label{eqt1}\end{aligned}$$ But, using ([\[dit1\]](#dit1){reference-type="ref" reference="dit1"}), we have $$\boldsymbol{\nu}^{K_-}\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}|_{K_-}=-\boldsymbol{\nu}^{K_+}\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}|_{K_+}-i\kappa\eta\mathbf{E}_{K_+,T}.$$ Then, using the above equality in ([\[eqt1\]](#eqt1){reference-type="ref" reference="eqt1"}) together with ([\[e1nu1\]](#e1nu1){reference-type="ref" reference="e1nu1"}) again we have $$\begin{aligned} \boldsymbol{\nu}^{K_-}\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}|_{K_-}+\frac{i\kappa{}}{Z} \mathbf{E}|_{K_-,T} &=& -\boldsymbol{\nu}^{K_+}\times\mu_{\textrm{r}}^{-1}\nabla\times\mathbf{E}|_{K_+}-i\kappa\eta\mathbf{E}_{K_+,T}+\frac{1}{Z} \frac{\boldsymbol{\chi}_{K_-}+\boldsymbol{\chi}_{K_+}}{2/Z+\eta } \\ &=& \left(\boldsymbol{\chi}_{K_+}-\frac{\eta}{2/Z+\eta }(\boldsymbol{\chi}_{K_-}+\boldsymbol{\chi}_{K_+}) \right).\end{aligned}$$ The new term introduces a new diagonal block into the matrix $C$ (defined before ([\[dc\]](#dc){reference-type="ref" reference="dc"})) and a perturbation to the off diagonal blocks coupling fields on $K_+$ and $K_-$. In the current *ParMax* implementation, $\eta$ is assumed constant on each mesh face of the resistive sheet and may be complex. ## Curved Elements and Quadrature {#CE&Q} We take a straightforward approach to approximating curved boundaries and quadrature. All the elements in *ParMax* have faces that are unions of possibly curvilinear triangles. Suppose a curvilinear face $F$ in the mesh is such that either an edge of $F$, or $F$ itself are entirely contained in a smooth curvilinear subset of the boundary $\Gamma$. We approximate $F$ by a mapping from a reference element $\hat{F}$ (with vertices $\widehat{\mathbf{a}}_1=(0,0)$, $\widehat{\mathbf{a}}_2=(0,1)$, and $\widehat{\mathbf{a}}_3=(1,0)$) in the $(s,t)$ plane to an approximation of $F$ using a degree $\ell$ polynomial map $\mathbf{F}_F:\hat{F}\to F$. In the important case of a quadratic map ($\ell=2$) we choose $\mathbf{a}_{i}$, $i=1,2,3$ to be the vertices of $K$ and take the remaining interpolation points by choosing $\mathbf{a}_{i,j}$ to be a point on the smooth boundary approximately half way between $\mathbf{a}_i$ and $\mathbf{a}_j$ (mapped from $\widehat{\mathbf{a}}_{i,j}=(\widehat{\mathbf{a}}_{i}+\widehat{\mathbf{a}}_j)/2$; see Fig. [2](#fig:quadmap){reference-type="ref" reference="fig:quadmap"}). We use the following nodal basis $$\begin{aligned} \hat{\phi}_1(s,t)&=&(1-s-t)(1-2s-2t)\\ \hat{\phi}_2(s,t)&=&s(2s-1)\\ \hat{\phi}_3(s,t)&=&t(2t-1)\\ \hat{\phi}_{1,2}(s,t)&=&4s(1-s-t)\\ \hat{\phi}_{2,3}(s,t)&=&4st\\ \hat{\phi}_{1,3}(s,t)&=&4t(1-s-t)\end{aligned}$$ and define $\mathbf{F}_F$ by $$\mathbf{F}_F(s,t)=\sum_{i=1}^3\mathbf{a}_{i}\hat{\phi}_i(s,t)+\sum_{i=1}^2\sum_{j=i+1}^3\mathbf{a}_{i,j}\hat{\phi}_{i,j}(s,t).$$ {#fig:quadmap width="60%"} We use $\mathbf{F}_F(\hat{F})\approx F$ for computing the UWVF matrices. Thus we need to evaluate integrals over each face $\mathbf{F}_F(\hat{F})$. Using the reference element, for a smooth function $g$ defined in a neighbourhood of $F$ $$\begin{aligned} \lefteqn{\int_{F}g({\bf y})dS({\mathbf{y}})\approx\int_{\mathbf{F}_F(\hat{F})}g({\bf y})dS({\mathbf{y}})}\\&=&\int_{\hat{F}} g(\mathbf{F}_F(s,t))\left\Vert \frac{\partial {\mathbf{F}_F}}{\partial s}(s,t)\times \frac{\partial {\mathbf{F}_F}}{\partial t}(s,t)\right\Vert\,ds\,dt.\end{aligned}$$ It now suffices to define quadrature on the reference element via the Duffy transform. Suppose $G(s,t)$ is a smooth function on $\hat{F}$ (in particular the integrand above) then setting $t=(1-s)\xi$ we can map to the unit square: $$\begin{aligned} \int_{\hat{F}}G\, d\hat{A}&=&\int_0^1\int_0^{1-s}G(s,t)\,dt\,ds\\&=& \int_0^1(1-s)\int_0^1G(s,(1-s)\xi)\,d\xi\,ds.\end{aligned}$$ Let $(w^g_i,t_i^g)$, $1\leq i\leq N$, denote the $N$-point Gauss-Legendre rule weights and nodes on $(0,1)$. Then for each $s$ $$\int_0^1G(s,(1-s)\xi)\,d\xi\approx \sum_{i=1}^N w_i^g G(s,(1-s)t_i^g).$$ Next, using $N$ point Jacobi quadrature weights and nodes on $(0,1)$ denoted $(w_j^J,x_j^J)$, we obtain finally $$\int_{\hat{F}}G\, dA\approx\sum_{i=1}^N\sum_{j=1}^N w_j^J w_i^g f(x_j^J,t_i^g(1-x_j^J)).$$ Note that the quadrature has positive weight. ## A low memory version {#LoMem} A simple low memory version of *ParMax* is easily available because we use BICGstab to solve the linear system. We compute $D$ as usual (it is block diagonal), and the vector $\vec{\mathbf{b}}$ but do not compute the elements of $C$ in ([\[dc\]](#dc){reference-type="ref" reference="dc"}). Then, as required by BICGstab, to compute $D^{-1}C\vec\mathbf{x}$ for some vector $\vec\mathbf{x}$ we compute the blocks of $C$ element by element and accumulate $C\vec\mathbf{x}$ element by element. Then $D^{-1}$ is computed element by element using precomputed LU decompositions. Obviously computing the entries of $C$ repeatedly greatly increases CPU time but this allows us to compute solutions to problem that would otherwise require very large memory to store $C$. For example, the solution of a scattering problem for a full aircraft at X-band frequencies is shown in Section [4.1.2](#Aircraft){reference-type="ref" reference="Aircraft"}. # Numerical examples {#sec:NE} All results were generated using the computer clusters Puhti and Mahti at the CSC -- IT Center for Science Ltd, Finland. Detailed descriptions of these supercomputers can be found from the CSC's website [@CSC]. Computational grids used in this work were prepared using COMSOL Multiphysics on a personal computer. In addition, the geometry model for aircraft used in Section [4.1.2](#Aircraft){reference-type="ref" reference="Aircraft"} is adapted from the COMSOL's application *Simulating Antenna Crosstalk on an Airplane's Fuselage*. For all numerical experiments, the incident electric field is a plane wave propagating in the direction of the positive $x$-axis polarised in the $y$-direction where the field is given by ([\[Einc\]](#Einc){reference-type="ref" reference="Einc"}). ## Scattering from PEC objects {#sec:PEC} In both experiments, we compute the scattered field and the incident field is used as a source via the PEC condition on the surface of the scatterer. ### A sphere {#scat_sphere} This experiment is intended to demonstrate the advantages of multiple element types, and a cuvilinear approximation to a smooth curved boundary (the surface of the sphere). In particular, we study scattering from a PEC sphere placed in vacuum, $\epsilon_{\textrm{r}}=\mu_{\textrm{r}}=1$, where the frequency of the incident field is $f = 2$ GHz (wavelength in air: $\lambda_0=0.19946$ m). The scatterer is a PEC sphere with radius of 1 meter or approximately $5\lambda_0$. In order to demonstrate the use of several types of large elements and curvilinear grids, we use an unusually large computational domain. In particular, the sphere is placed with its origin at the center of a cube shaped computational domain $[-1-15\lambda_0, 1+15\lambda_0]^3$. An absorbing boundary condition, ([\[eq:bc\]](#eq:bc){reference-type="ref" reference="eq:bc"}) with $Q=0$, is used on the exterior surface geometry. In addition, a PML with thickness of $5\lambda_0$, and constant absorption parameter $\sigma_0=1$ is used within each side of the cube. Two computational grids with different geometric approximations are used (see Fig. [3](#fig:pec_sphere_grid){reference-type="ref" reference="fig:pec_sphere_grid"}). For the first grid (*mesh 1*), we set the requested element size on the PEC sphere to be $h_{\textrm{s}}=\lambda_0/5$ which we will see provides a geometrically accurate surface representation using flat face elements such that the far field pattern predicted by UWVF is in good agreement with the far field computed via Mie scattering. For the second grid (*mesh 2*), the surface grid density is relaxed to $h_{\textrm{s}}=3\lambda_0$. For both cases, the requested element size in the volume is set to $h_{\textrm{v}}=10\lambda_0$. We request wedge and hexahedral elements in the PML region, and tetrahedra elsewhere. We show results for three cases: 1) scattering calculated using *mesh 1*, 2) scattering computed using *mesh 2* with flat facets approximating the surface of the sphere, and 3) the use of *mesh 2* with a quadratic approximation to the boundary of the sphere (see Section [3.4](#CE&Q){reference-type="ref" reference="CE&Q"}). These results are computed on the Puhti system with 5 computing nodes and 10 cores on each. Table [1](#not){reference-type="ref" reference="not"} summarises the notation used in reporting results and Table [3](#tab:pec){reference-type="ref" reference="tab:pec"} gives details of the PEC computations including CPU time. Symbol Definition ------------------------------------------ ----------------------------------------------------- $N_{\textrm{elements}}^{\textrm{tetra}}$ Number of tetrahedral elements $N_{\textrm{elements}}^{\textrm{wedge}}$ Number of wedge elements $N_{\textrm{elements}}^{\textrm{hexa}}$ Number of hexahedral elements $N_{\textrm{vertices}}$ Number of vertices $h_{\min}$ Minimum distance between vertices $h_{\max}$ Maximum distance between vertices $N_{{\rm iter}}^{{\rm BiCG}}$ Number of BiCGstab iterations required to reach the requested tolerance value $10^{-5}$ L2-error Relative L2-error computed from the bistatic RCS CPU-time Elapsed wall-clock time needed to assemble the matrices and reach the solution : Notation used in reporting computational results. [\[not\]]{#not label="not"} ![Cross-sections of the computational grids used to approximate scattering from a PEC sphere. The colorbar shows the number of plane wave directions on each element. The major difference is the grid density on the surface of the sphere. This figure shows how wedge and hexahedral elements can be usefully employed in the outer PML layer. In all figures showing grids, the colorbar shows the number of plane wave direction, see Eq. ([\[eq:Nell\]](#eq:Nell){reference-type="ref" reference="eq:Nell"}), for each element. ](pec_sphere_grid2.png){#fig:pec_sphere_grid width="60%"} Figure [4](#fig:pec_sphere_snap){reference-type="ref" reference="fig:pec_sphere_snap"} shows the modulus of the $y$-component of the scattered electric field $|E_y^s|$ on the $z=0$ plane. There are clear differences between the results for *mesh 1* and *mesh 2* with flat facets. These are caused by the coarse surface grid in the second case. However *mesh 1* with flat facets, and *mesh 2* with a quadratic boundary approximation are in good agreement. As can be seen in Table [3](#tab:pec){reference-type="ref" reference="tab:pec"}, *mesh 2* with quadratic boundary approximation is much cheaper in terms of CPU-time than *mesh 1*. {#fig:pec_sphere_snap width="60%"} Very often the far field pattern of the scattered wave [@Monk03] is the quantity of interest for these calculations, and in particular the Radar Cross Section (RCS) derived from the far field pattern. In this paper, far field directions are defined in terms of the azimuth angle $\phi$ (${}^\circ$) as $(\cos(\phi\pi/180), \sin(\phi\pi/180),0)$. Figure [5](#fig:pec_sphere_mie){reference-type="ref" reference="fig:pec_sphere_mie"} shows a comparison of the bistatic RCS predicted by the computational experiments with the UWVF and one computed by the Mie series. Clearly *mesh 2* with flat facets produces an inaccurate far field pattern, whereas *mesh 1* or *mesh 2* with curved elements produces much more accurate predictions. {#fig:pec_sphere_mie width="60%"} mesh id surface $N_{\textrm{elements}}^{\textrm{tetra}}$ $N_{\textrm{elements}}^{\textrm{wedge}}$ $N_{\textrm{elements}}^{\textrm{hexa}}$ $N_{\textrm{vertices}}$ $h_{\min}$ (cm) $h_{\max}$ (m) --------- --------- ------------------------------------------ ------------------------------------------ ----------------------------------------- ------------------------- ----------------- ---------------- 1 flat 122680 228 56 29890 0.65 1.93 2 flat 1384 \" \" 593 12.79 2.13 2 curved \" \" \" \" \" \" : Table giving details for the computational grids for the PEC sphere, and accuracy and timing. See Table [1](#not){reference-type="ref" reference="not"} for definitions of the reported quantities. mesh id surface $N_{{\rm iter}}^{{\rm BiCG}}$ L2-error (%) CPU-time (s) --------- --------- ------------------------------- -------------- -------------- 1 flat 211 0.21 730 2 flat 73 25.12 355 2 curved 72 0.36 490 : Table giving details for the computational grids for the PEC sphere, and accuracy and timing. See Table [1](#not){reference-type="ref" reference="not"} for definitions of the reported quantities. [\[tab:pec\]]{#tab:pec label="tab:pec"} ### An aircraft at X-band frequency {#Aircraft} The aircraft model used in this section is derived from a model available in COMSOL (application *Simulating Antenna Crosstalk on an Airplane's Fuselage*). We treat the aircraft as a curvilinear perfect conductor. The frequency of the incident field is $f = 8$ GHz so $\lambda_0=0.03737$m. The aircraft is 20.5 m or 547 wavelengths long and 17.8108 m or 475 wavelengths wide. A perfectly matched layer with a thickness of $5\lambda_0$ is added to each side of the cuboid computational domain with side lengths (16.3334, 16.3334, 4.4826) m. For generating the computational grid, curved elements and a mesh size parameter of $h_{\textrm{s}}=3\lambda_0$ were employed on the aircraft's surface. Because we can use 10$\lambda_0$ sized elements away from the boundary, the entire grid can be created on a standard office computer. The grid consists of 697,783 tetrahedral elements with 142,731 vertices covering the computational domain. The surrounding PML layer is discretized using 413 hexahedral and 14,904 wedge elements. In addition, for this grid $h_{\min} = 1.72$ cm and $h_{\max} = 0.55$ m. In Fig. [6](#fig:aircraft_grid){reference-type="ref" reference="fig:aircraft_grid"}, the computational grid on the aircraft's surface is depicted, along with the grid in two planes: $z=-1$ m and $x=0$ m. {#fig:aircraft_grid width="60%"} For this numerical experiment we used the supercomputer Mahti. When we tried this example storing the matrices $C$ and $D$ as usual (see Eq. ([\[dc\]](#dc){reference-type="ref" reference="dc"})), we ran out of memory. So we switched to the low memory version described in Section [3.5](#LoMem){reference-type="ref" reference="LoMem"} to compute the results shown here. In Mahti, we used a total of 200 computing nodes and 100 CPU units / node. The total time for the calculation was 18 hours, including building the system matrix $D$, and then iteratively reaching the requested solution accuracy. The BICGstab algorithm took a total 437 iterations. Figure [7](#fig:aircraft_snap){reference-type="ref" reference="fig:aircraft_snap"} shows the scattered electric field $|E_y^s|$ on the $x=0$, $y=0$, $z=0$ planes. Fig. [8](#fig:aircraft_far){reference-type="ref" reference="fig:aircraft_far"} shows the RCS in a full azimuth angle range $\phi\in[0, 360]^\circ$. {#fig:aircraft_snap width="50%"} {#fig:aircraft_far width="90%"} ## Resistive sheets {#sec:res} ### Salisbury screen We next model a ''Salisbury screen'' (W. W. Salisbury, U. S. Patent US2599944 A 1952). For simplicity, assume $\epsilon_{\textrm{r}}=\mu_{\textrm{r}}=1$. Our standard incident field ([\[Einc\]](#Einc){reference-type="ref" reference="Einc"}) propagates normally to a resistive sheet at $x=-H$, $H>0$ backed by a PEC surface at $x=0$. To the left of the resistive sheet the total electric field is $${\bf E}_{-}=\left(\begin{array}{c}0\\1\\0\end{array}\right) \exp(i \kappa x)+ \left(\begin{array}{c}0\\R_2\\R_3\end{array}\right)\exp(-i\kappa x),\; x<-H,$$ where $(0,R_2,R_3)^T$ is the polarization of the reflected wave. Between the sheet and the PEC surface, where $-H<x<0$, $${\bf E}_+=\left(\begin{array}{c}0\\q_{0,2}\\q_{0,3}\end{array}\right) \exp(i \kappa x)+ \left(\begin{array}{c}0\\q_{1,2}\\q_{1,3}\end{array}\right)\exp(-i\kappa x).$$ Here $(0,q_{0,2},q_{0,3})^T$ and $(0,q_{1,2},q_{1,3})^T$ are the polarizations of the left and right going waves respectively in the gap $-H<x<0$. Imposing the PEC boundary condition at $x=0$ and the resistive sheet transmission conditions ([\[dit1\]](#dit1){reference-type="ref" reference="dit1"})-([\[dit2\]](#dit2){reference-type="ref" reference="dit2"}) at $x=-H$ shows that $R_3=q_{0,3}=q_{1,3}=0$ and $$\begin{aligned} \mathit{R_2} &=& -\frac{\left(i\left( \eta -1\right) \sin \! \left(\kappa H \right)-\cos \! \left(\kappa H \right)\right) \exp\left(-2 i \kappa H\right)}{i\left( \eta +1\right) \sin \! \left(\kappa H \right)-\cos \! \left(\kappa H \right)}, \\\mathit{q_{0,2}} &=& \frac{\exp\left(-i \kappa H\right)}{-i(\eta+1) \sin \! \left(\kappa H \right) +\cos \! \left(\kappa H \right)},\\ \mathit{q_{1,2}} &= & \frac{\exp\left(-i \kappa H\right)}{i\left( \eta +1\right) \sin \! \left(\kappa H \right)-\cos \! \left(\kappa H \right)}.\end{aligned}$$ For given $\kappa$, $H$, and $\eta$ we can compute total field in each region and compare to the analytic solution. As is well known, one choice of $\eta$ gives $R_2=0$: $$\eta=1-i\cot(\kappa H).$$ A particularly interesting case occurs when $\cot(\kappa H)=0$ or $H=\pi/(2\kappa)$. Recalling that the wavelength of the radiation is $\lambda_0=2\pi/\kappa$, we see that zero reflection occurs when $H=\lambda_0/4$. To test the UWVF for resistive sheets, we use a rectangular parallelepiped computational region with faces normal to the coordinate directions (see Fig. [9](#fig:sheetpw_grid){reference-type="ref" reference="fig:sheetpw_grid"}). The rightmost face $x=0$ is PEC ($Q=1$) the left-most face is an ABC with $Q=0$ where we use a non-homogeneous absorbing boundary condition to excite the plane wave. There is no need for a PML since the solution is a wave propagating orthogonally to the absorbing boundary. On the remaining faces $Q=\pm 1$ chosen so that the incident plane wave propagates along the box without distortion. We take the radiation to have frequency $f=2$ GHz and place the resistive sheet one quarter wavelength ($H=3.75$ cm) from the PEC surface. Figure [9](#fig:sheetpw_grid){reference-type="ref" reference="fig:sheetpw_grid"} shows a surface of the grid used in the computations. In this case, the grid consists of 136 tetrahedral elements with 51 vertices. In addition, $h_{\min} = 3.33$ cm and $h_{\max} = 16.18$ cm. Results are show in Fig. [10](#fig:sheetpw_snap){reference-type="ref" reference="fig:sheetpw_snap"}. We plot the magnitude of the $y$-component of the total field as a function of $x$ when $y=z=7.5$ cm. Choosing $\eta=1$ the magnitude of the field is flat to the left of the resistive sheet showing that this choice of $\eta$ gives rise to no reflected wave. However when $\eta=0.5$ the non-constant magnitude of the total field indicates that a reflected wave is present to the left of the sheet. {#fig:sheetpw_grid width="50%"} {#fig:sheetpw_snap width="60%"} ### Sphere with resistive sheet In this second experiment with resistive sheets, a PEC sphere with a radius of 1 meter is placed at the origin of a cube $[-1-15\lambda_0, 1+15\lambda_0]^3$. Surrounding this sphere is a spherical resistive sheet of radius $1+\lambda_0/4$ and surrounding both is an artificial sphere of radius $1+\lambda_0$. Outside this artificial sphere we compute the scattered field, and inside the total field (see Section [3.2](#stf){reference-type="ref" reference="stf"}). The incident field then gives rise to a source on the artificial boundary as detailed in Section [3.2](#stf){reference-type="ref" reference="stf"}. A PML with a thickness of $5\lambda_0$ is applied to the inside each side of the cube, and the frequency of the incident field is set at $f = 2$ GHz. The incident field gives rise to a source located on the sphere. To generate the computational grid, we utilized curved elements and a mesh size parameter of $h_{\textrm{s}}=2\lambda_0$ on the PEC and resistive sheet surfaces. The grid comprises of 428 wedge elements (representing the domain between the resistive sheet and PEC sphere) and 7,589 tetrahedral elements that cover the main domain of interest. Furthermore, the surrounding PML layer is discretized using a combination of 104 hexahedral and 936 wedge elements. The entire grid is composed of 2,550 vertices, with $h_{\min} = 3.75$ cm and $h_{\max} = 1.48$ m. A cross-section of the computational grid is shown in Fig. [11](#fig:sheet_sphere_grid){reference-type="ref" reference="fig:sheet_sphere_grid"}. {#fig:sheet_sphere_grid width="60%"} We used the supercomputer Puhti with a total of 5 computing nodes and 10 CPU units/node to solve the three configurations for different resistive sheet parameters $\eta$. It took 18 ($\eta=0$), 17 ($\eta=0.5$), and 17 ($\eta=1.0$) minutes CPU-time respectively to build the system matrices and then reach the solution with BICGstab. The solution was achieved after 194 iterations for $\eta=0$, 169 iterations for $\eta=0.5$, and 172 iterations for $\eta=1.0$. In Fig. [12](#fig:sheet_sphere_snap){reference-type="ref" reference="fig:sheet_sphere_snap"}, the total field component $\Re(E_y)$ on the plane $z=0$ is shown for three choices of $\eta$. We can no longer expect invisibilty since the screen is curved, but it is evident that backscattering is decreasing as $\eta$ increases. This is evident more clearly in Fig. [13](#fig:sheet_sphere_mie){reference-type="ref" reference="fig:sheet_sphere_mie"} where we show the RCS in each case. {#fig:sheet_sphere_snap width="60%"} {#fig:sheet_sphere_mie width="60%"} ## Heterogeneous models {#sec:het} ### A dielectric sphere {#penet_sphere} In this experiment, a penetrable sphere with a radius of 1 meter is centered at the origin inside the cube $[-1-15\lambda_0, 1+15\lambda_0]^3$, where $\lambda_0$ is the wavelength in vacuum. For the penetrable sphere, we assume $\epsilon_{\textrm{r}}= 1.5+0.5i$ and $\mu_{\textrm{r}}= 1$, while we select vacuum parameters in other domains. The frequency of the incident field is $f = 2$ GHz and a PML with thickness of $5\lambda_0$ is used on each side of the cube. An artificial spherical boundary with radius $1+\lambda_0$ is used to separate a scattered field region outside and a total field region inside this surface. The scattered-total field formulation in Section [3.2](#stf){reference-type="ref" reference="stf"} is used to introduce a source on the artificial boundary. We utilised curved elements and a mesh size parameter of $h_{\textrm{s}}=3\lambda_s$, where $\lambda_s$ is the wavelength in the penetrable sphere, on the material discontinuity surface. Figure [14](#fig:penet_sphere_grid){reference-type="ref" reference="fig:penet_sphere_grid"} shows cross-sections of the computational grids. To solve this problem, we used the supercomputer Puhti with a total of 7 computing nodes and 40 CPU units/node. More detailed information on the computations, including information on the grids and CPU time, is given in Table [5](#tab:penet){reference-type="ref" reference="tab:penet"}. {#fig:penet_sphere_grid width="60%"} In Fig. [15](#fig:penet_sphere_snap){reference-type="ref" reference="fig:penet_sphere_snap"}, we show the total field component $\Re(E_y)$ on the $z=0$-plane for the two meshes. As expected (see also Table [5](#tab:penet){reference-type="ref" reference="tab:penet"}), *mesh 2* with curved face elements produces a solution comparable to *mesh 1*, but faster. The accuracy of the far field pattern is compared to the Mie series solution in Fig. [16](#fig:penet_sphere_mie){reference-type="ref" reference="fig:penet_sphere_mie"}, and again demonstrates that *mesh 1* and *mesh 2* with curved faces give comparable results. {#fig:penet_sphere_snap width="60%"} {#fig:penet_sphere_mie width="60%"} mesh id surface $N_{\textrm{elements}}^{\textrm{tetra}}$ $N_{\textrm{elements}}^{\textrm{wedge}}$ $N_{\textrm{elements}}^{\textrm{hexa}}$ $N_{\textrm{vertices}}$ $h_{\min}$ (cm) $h_{\max}$ (m) --------- --------- ------------------------------------------ ------------------------------------------ ----------------------------------------- ------------------------- ----------------- ---------------- 1 flat 337550 936 104 59053 0.65 1.29 2 flat 7100 \" \" 2161 7.80 1.42 2 curved \" \" \" \" \" \" : Details of the computational results for the penetrable sphere. See Table [1](#not){reference-type="ref" reference="not"} for definitions of the reported quantities. mesh id surface $N_{{\rm iter}}^{{\rm BiCG}}$ L2-error (%) CPU-time (s) --------- --------- ------------------------------- -------------- -------------- 1 flat 187 0.14 346 2 flat 90 7.85 162 2 curved 88 0.14 234 : Details of the computational results for the penetrable sphere. See Table [1](#not){reference-type="ref" reference="not"} for definitions of the reported quantities. [\[tab:penet\]]{#tab:penet label="tab:penet"} ### A plasma sphere In this second experiment for penetrable objects, we assume a penetrable sphere with a radius of 0.25 meter is centred at the origin of a cube $[-0.25-15\lambda_0, 0.25+15\lambda_0]^3$. The material in the sphere is assumed to be a plasma modelled by setting $\epsilon_{\textrm{r}}= -1.5+0.5i$ and $\mu_{\textrm{r}}= 1$. The frequency of the incident field is $f = 2$ GHz and a perfectly matched layer with thickness of $5\lambda_0$ is used inside each side of the cube. The source is introduced on an artificial spherical surface of radius $0.25+\lambda_0$. We utilise curved elements and a mesh size parameter $h_{\textrm{s}}=3\lambda_s$, where $\lambda_s$ denotes the wavelength in the penetrable sphere, on the material discontinuity surface. The grid comprises of 2,959 tetrahedral elements that cover the main domain of interest. Furthermore, the surrounding PML layer is discretized using a combination of 92 hexahedral and 768 wedge elements. The entire grid has 1,304 vertices, with $h_{\min} = 5.88$ cm and $h_{\max} = 1.30$ m. We used the supercomputer Puhti with a total of 5 computing nodes and 10 CPU units/node to solve the problem. It took 427 seconds from each CPU unit to build the system matrices $C$ and $D$ and then reach the solution with 103 bi-conjugate iterations. Snapshots of the total field are shown in Fig. [17](#fig:plasma_sphere_snap){reference-type="ref" reference="fig:plasma_sphere_snap"}. In Fig. [18](#fig:plasma_sphere_mie){reference-type="ref" reference="fig:plasma_sphere_mie"} we show the computed RCS which shows remarkable agreement with the Mie series solution. {#fig:plasma_sphere_snap width="50%"} {#fig:plasma_sphere_mie width="60%"} # Conclusion {#sec:conc} In this study, we explored and extended the Ultra-Weak Variational Formulation (UWVF) applied to the time-harmonic Maxwell's equations. Our research findings led to important contributions that enhance the efficiency and applicability of the UWVF method for solving electromagnetic wave problems. The paper shows a series of numerical examples validating the effectiveness of the new enhancements. Scattering problems from PEC objects were considered, highlighting the benefits of curved elements, different element types, and the low-memory version of the software. The applicability was further demonstrated through simulations of scattering from an full size aircraft, emphasising its potential for real-world industrial scenarios. In conclusion, this paper has not only extended the capabilities of the UWVF for electromagnetic wave problems but has also provided a comprehensive set of numerical results to underline the practical significance of these advancements. The integration of curved elements, various element shapes, and resistive sheets collectively contribute to the method's robustness and utility, making it a valuable tool for addressing complex electromagnetic problems. An important direction for further work would be to refine the heuristics for choosing the number and direction of plane waves on each element. This is particularly needed for elements that might have large aspect ratio such as elongated wedges. # Acknowledgments {#acknowledgments .unnumbered} The research of P. M. is partially supported by the US AFOSR under grant number FA9550-23-1-0256. The research of T. L. is partially supported by the Academy of Finland (the Finnish Center of Excellence of Inverse Modeling and Imaging) and the Academy of Finland project 321761. The authors also wish to acknowledge the CSC -- IT Center for Science, Finland, for generously sharing their computational resources. # The choice of basis {#sec:app} Because we have used new element types and larger numbers of directions in this paper compared to [@Huttunen2007], we need new heuristics for choosing the number of plane wave directions on a particular element. We use the same technique as in [@Huttunen2007]. Computing on the reference element, in Fig. [21](#fig:pw_selection){reference-type="ref" reference="fig:pw_selection"}, the number of plane waves $N_{\ell}$ is plotted as a function of $(k_{\textrm{re}}h^{\textrm{av}})_{\ell}$, when the maximum condition number of the matrix blocks of $D_{\ell}$ is limited by the tolerances $10^5$, $10^7$, and $10^9$. The element size parameter $h^{\textrm{av}}$ is defined as a mean distance of the element vertices from its centroid. This data is fitted by a quadratic polynomial function with the constraint that the polynomial gives at least 4 directions even on the finest grid: $$\label{eq:Nell} N_{\ell} = \left \lceil{a\left(k_{\textrm{abs}}h^{\textrm{av}}\right)_{\ell}^2 + b\left(k_{\textrm{abs}}h^{\textrm{av}}\right)_{\ell} + c}\right \rceil,$$ where $$k_{\textrm{abs}} = \omega\left|\sqrt{\epsilon_{\textrm{r}}\mu_{\textrm{r}}}\right|.$$ Results of this fitting are shown in Table [\[tab:values\]](#tab:values){reference-type="ref" reference="tab:values"}. Although only computed for one element shape, these polynomials are used to set the number of directions for any given mesh. Generally a higher tolerance on the condition number results in more directions per element so greater accuracy, but too high a condition number slows BiCGstab unacceptably. {#fig:pw_selection height="21.5%"} {#fig:pw_selection height="21.5%"} {#fig:pw_selection height="21.5%"} [\[tab:values\]]{#tab:values label="tab:values"} 10 CSC -- IT Center for Science Ltd, Computing environments. <https://docs.csc.fi/computing/available-systems/>. Accessed: Aug. 1, 2023. H. Barucq, A. Bendali, J. Diaz, and S. Tordeux. Local strategies for improving the conditioning of the plane-wave Ultra-Weak Variational Formulation. , 441, 2021. Art. No. 110449. J.-P. Bérenger. A Perfectly Matched Layer for the absorption of electromagnetic waves. , 114(2):185--200, 1994. A. Buffa and P. Monk. Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation. , 42:925--40, 2008. O. Cessenat. PhD thesis, Université Paris IX Dauphine, 1996. O. Cessenat and B. Després. Using plane waves as base functions for solving time harmonic equations with the Ultra Weak Variational Formulation. , 11:227--38, 2003. C. Gittelson, R. Hiptmair, and I. Perugia. Plane wave discontinuous Galerkin methods. , 43:297--331, 2009. D. Hardin, T. Michaels, and E. Saff. A comparison of popular point configurations on ${S}^2$. , 9:16--19, 2016. R. Hiptmair, A. Moiola, and I. Perugia. Stability results for the time-harmonic Maxwell equations with impedance boundary conditions. , 21:2263--87, 2011. R. Hiptmair, A. Moiola, and I. Perugia. Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. , 82:247--268, 2013. Q. Hu and L. Yuan. A plane wave least-squares method for time-harmonic Maxwell's equations in absorbing media. , 36:A1911--A1936, 2014. Q. Hu and L. Yuan. A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. , 44:245--275, 2018. T. Huttunen, M. Malinen, and P. Monk. Solving Maxwell's equations using the ultra weak variational formulation. , 223:731--758, 2007. L. Imbert-Gérard and G. Sylvand. A roadmap for Generalized Plane Waves and their interpolation properties. , 149:87--137, 2021. J. Jin, J. Volakis, C. Yu, and A. Woo. Modeling of resistive sheets in finite element solutions. , 40:727--731, 1992. G. Karypis and V. Kumar. A fast and high quality multilevel scheme for partitioning irregular graphs. , 20(1):359--392, 1998. C. Lehrenfeld and P. Stocker. Embedded Trefftz discontinuous Galerkin methods. , 124(17):3637--3661, 2023. P. Monk. . Oxford University Press, Oxford, 2003. S. Pernet, M. Sirdey, and S. Tordeux. Ultra-weak variational formulation for heterogeneous Maxwell problem in the context of high performance computing. Available as <https://hal.science/hal-03642116> from the Hal preprint server., 2022. T. Senior. Combined resistive and conductive sheets. , AP-33:577--579, 1985. P. Stocker. refftz: Add-on to NGSolve for Trefftz methods. , 7, 2022. Art. No. 4135. E. Trefftz. Ein gegenstück zum Ritz'schen verfahren. In *Proc. 2nd Int. Congr. Appl. Mech.*, pages 131--137, Zurich, 1926.

arxiv_math

--- abstract: | We consider a one-dimensional nonlocal hyperbolic model introduced to describe the formation and movement of self-organizing collectives of animals in homogeneous 1D environments. Previous research has shown that this model exhibits a large number of complex spatial and spatiotemporal aggregation patterns, as evidenced by numerical simulations and weakly nonlinear analysis. In this study, we focus on a particular type of localised patterns with odd/even/no symmetries (which are usually part of snaking solution branches with different symmetries that form complex bifurcation structures called snake-and-ladder bifurcations). To numerically investigate the bifurcating solution branches (to eventually construct the full bifurcating structures), we first need to understand the numerical issues that could appear when using different numerical schemes. To this end, in this study, we consider ten different numerical schemes (the upwind scheme, the MacCormack scheme, the Fractional-Step method, and the Quasi-Steady Wave-Propagation algorithm, combining them with high-resolution methods), while paying attention to the preservation of the solution symmetries with all these schemes. We show several numerical issues: first, we observe the presence of two distinct types of numerical solutions (with different symmetries) that exhibit very small errors, which might initially suggest that we have reached a steady-state solution, but this is not the case (this also implies an extremely slow convergence); second, in some cases, none of the investigated numerical schemes converge, posing a challenge for the development of numerical continuation algorithms for nonlocal hyperbolic systems; lastly, the choice of the numerical schemes, as well as their corresponding parameters such as time-space steps, exert a significant influence on the type and symmetry of bifurcating solutions. To conclude we emphasize that if we want to construct numerically bifurcation diagrams for these localised solutions with different symmetries, the resulting bifurcations may vary when different numerical schemes and/or corresponding parameters are employed. address: - **Thanh Trung Le** Université de Franche-Comté, CNRS, LmB, F-25000 Besançon, France - **Raluca Eftimie** Université de Franche-Comté, CNRS, LmB, F-25000 Besançon, France author: - Thanh Trung Le and Raluca Eftimie bibliography: - References_Postdoc.bib title: Numerical challenges for the understanding of localised solutions with different symmetries in non-local hyperbolic systems --- # Introduction The study of animal aggregations has been investigated intensively over the past fifty years [@Okubo-Grunbaum-Keshet-2001; @Flierl-Grunbaum-Levin-Olson-1999; @Gueron-Levin-Rubenstein-1996; @Mogilner-Keshet-1999; @Reynolds-1987]. One of the most studied aspects of these aggregations is the spatial and spatiotemporal patterns exhibited by them: zigzagging flocks of birds [@Feder-2007], or milling schools of fish, for instance, [@Parrish-Keshet-1999] and [@Lukeman-Li-Keshet-2009]. To investigate the biological mechanisms necessary for the formation and persistence of these patterns, a wide range of mathematical models have been proposed. There are two main classes of mathematical models used for animal aggregations: 1) individual-based models (Lagrangian approach, microscopic models), which track the movements of all individuals in the group, and 2) partial differential equations (PDE) models, formulated as evolution equations for the population density field. Despite the complex group patterns displayed by the individual-based models (e.g., swarms, tori, polarized groups, see [@Couzin-Krause-James-Ruxtion-2002] and the reference therein), the lack of techniques to investigate them causes difficulties in understanding some of these patterns, see [@Eftimie-Vries-Lewis-2009]. Hence, the main method for this approach focuses on numerical simulations with the purpose of comparing the simulated aggregation patterns with the available data, and thus ascertaining the correctness of the micro-scale level assumptions incorporated into the models, for example, see [@Aldana-Dossetti-Huepe-Kenkre-Larralde-2007; @Czirok-Barabasi-Vicsek-1999; @Czirok-Stanley-Vicsek-1997; @Vicsek-Czirok-Jacob-Cohen-Shochet-1995]. The second approach can be classified into two categories: kinetic models (mesoscopic models; see [@Othmer-Dunbar-Alt-1988; @Fetecau-2011], also see [@Eftimie-2012]) and continuum models (Eulerian approach, macroscopic models) which is better represented in the literature on animal aggregations, with a diverse range of parabolic models [@Mogilner-Keshet-1999; @Topaz-Bertozzi-Lewis-2006] and hyperbolic models [@Eftimie-Vries-Lewis-Lutscher-2007; @Eftimie-Vries-Lewis-2007; @Lutscher-2002; @Pfistner-1990]. For continuum models, the main method focuses on the analytical results, e.g., showing the existence of particular solutions exhibited by these models, showing the existence of different types of bifurcations that give rise to different solution types, or trying to connect biological interactions at the micro-scale and macro-scale levels (through various homogenisation approaches). There are, however, models that combine analytical results with numerical simulations, for example, see [@Chuang-Orsogna-Marthaler-Bertozzi-Chayes-2007; @Fetecau-Eftimie-2010; @Buono-Eftimie-2014-SIAM; @Eftimie-Vries-Lewis-2009].\ In [@Eftimie-Vries-Lewis-2007] and [@Eftimie-Vries-Lewis-Lutscher-2007], the authors have proposed a nonlocal hyperbolic model that introduces a general framework to incorporate different communication mechanisms to study the formation of animal groups. In particular, these communication mechanisms influence the social interactions between individuals, namely attraction towards other members of the group that are far away, repulsion from those that are nearby, and a tendency to align with those neighbors that are at intermediate distances. The resulting model, which actually comprises many submodels, is very rich in spatial and spatiotemporal patterns. Numerical simulations have shown at least 10 different patterns, including stationary and traveling pulses, traveling trains, zigzag pulses, breathers, ripples, and feathers. Two particular patterns, traveling trains, and stationary pulses, were shown to arise through subcritical Hopf and Steady-state (codimension-one) bifurcations [@Eftimie-Vries-Lewis-2009]. Other patterns, such as ripples, were shown to arise through codimension-two bifurcations [@Buono-Eftimie-2014-MMMA; @Buono-Eftimie-2014-SIAM]. In this study, we focus on a particular type of localised patterns with odd/even/no symmetries (which are usually part of snaking solution branches with different symmetries that form complex bifurcation structures called snake-and-ladder bifurcations). The snake-and-ladder bifurcations have been observed in various simple fluid dynamics models such as Swift-Hohenberg equation (see [@Avitabile-Lloyd-Burke-Knobloch-Sandstede-2010; @Beck-Knobloch-Lloyd-Sandstede-Wagenknecht-2009; @Burke-Knobloch-2006; @Burke-Knobloch-2007-Chaos; @Burke-Knobloch-2007-PLA; @Liu-Xu-2017] and the reference therein), discrete bistable Allen-Cahn equation [@Taylor-Dawes-2010] and more recently in simple integro-differential equations [@Schmidt-Avitabile-2020]. Normally, the snake-and-ladder bifurcation is structured by two intertwined snaking branches corresponding to even and odd localised solutions. These branches include both stable and unstable branches. Near each fold, the localised patterns undergo a pitchfork bifurcation at which a branch of asymmetric solutions emerges that connects the two snaking branches. The ladder branches may be unstable or stable. For example, we refer to [@Avitabile-Lloyd-Burke-Knobloch-Sandstede-2010 Figure 1].\ Understanding the dynamics of a system describing a physical, biological, or engineering problem, where changes in parameter values can lead to changes in system dynamics, is an important fundamental aspect of applied mathematics. In addition to using analytical approaches to investigate the impact of parameter changes on the number and stability of different model states, numerical simulations are being used to visualize the dynamics of the model as one or multiple parameters are varied. However, such numerical simulations could be very time-consuming due to the manual inspection of solution trajectories to detect transient and asymptotic dynamics, multiple stable steady states, bifurcation points, stable and unstable manifolds, etc. As an alternative, numerical continuation algorithms are being used to automatically detect invariant sets, bifurcations, metastable states, etc. Numerical continuation algorithms for ordinary differential equations (ODEs) are now an established tool for bifurcation analysis in dynamical systems [@Kuehn-2015]. For ODEs, there are several standard good software packages (e.g., XPP [@XXP], AUTO-07p [@AUTO-07p], MatCont [@MATCONT], PyDSTool [@PyDSTool], etc). However, for partial differential equations (PDEs) there are no standard numerical continuation toolboxes that would cover a broad range of different classes of PDEs automatically [@Kuehn-2015]. The toolboxes that exist (e.g., *pde2path* package in [MATLAB]{.smallcaps}) are usually developed for specific classes of partial differential equations (mainly of elliptic or parabolic types), using numerical methods most suitable for those classes. The simplest approach, but also very expensive computationally, is to implement a discretization scheme most suitable for a class of equations, and then add a predictor-corrector continuation algorithm on top of the discretization scheme. However, changing parameters could lead to changing the type and characteristics of the PDEs, which might require also changes in the numerical method used to track the solution branches. One way of solving this particular problem is to glue together different numerical methods developed for different classes of equations [@Kuehn-2015]. Most of these numerical bifurcation studies have been applied to local PDEs of parabolic and elliptic types (mainly arising in fluid dynamics), in which the bifurcations diagrams are obtained with the help of standard continuation software (e.g., XPP, AUTO-07p, MatCont, PyDSTool, etc). More recently, they have started to be applied also to new mathematical models describing various biological and medical problems. Some of these models are also of non-local type, describing long-distance interactions between the various components of the system [@Schmidt-Avitabile-2020]. However, the field of mathematical approaches to biology/medicine is very vast, with new complex nonlinear and non-local mathematical models (sometimes described by PDEs of hyperbolic type) that exhibit interesting patterns and bifurcations being developed continuously. Therefore, new numerical approaches need to be developed to investigate the bifurcations of these patterns in these new PDEs (see for example, the Galerkin schemes and Fourier collocation scheme introduced in [@Schmidt-Avitabile-2020]). This is particularly important for models of hyperbolic type, where the center manifold theorem (required for bifurcation theory) does not always hold true.\ In this paper, we focus on numerical approaches to simulate the localised patterns with odd/even/no symmetry (that likely form the branches of snake-and-ladder bifurcations) in the nonlocal hyperbolic systems for ecological aggregations introduced in [@Eftimie-Vries-Lewis-2007] and [@Eftimie-Vries-Lewis-Lutscher-2007]. Specifically, we will consider here only one of the submodels introduced in [@Eftimie-Vries-Lewis-2007], where information received from all neighbors are taken into account for attractive and repulsive interactions, while for alignment, only information from neighbors moving toward an individual is considered. But for simplicity, we assume that attraction and repulsion are the only possible social interactions (i.e., the magnitude of alignment is zero). Previous research [@Eftimie-Vries-Lewis-2007] has shown that this particular model exhibits stationary localised patterns, characterised by different symmetries. Using a weakly nonlinear analysis combined with numerical simulations, in [@Eftimie-Vries-Lewis-2009] the authors showed that the stationary pulses arise through a real subcritical (i.e., unstable) bifurcation from the spatially homogeneous steady state. In [\[fig:Fi3_WeaklyNonlinear\]](#fig:Fi3_WeaklyNonlinear){reference-type="ref" reference="fig:Fi3_WeaklyNonlinear"} we show the changes in these steady-state homogeneous solutions (with zero amplitudes) and the steady-state heterogeneous solutions (with non-zero amplitudes) as we vary the magnitude of attraction $q_{a}$. We observe here two different types of stable localised solutions on the high-amplitude branch: a stable solution with even symmetry (red circles, for $q_{a}<1.095$) and a stable solution with odd symmetry (green circles, for $q_{a}>1.0.95$). We also note two different types of unstable localised solutions (with odd and even symmetries) on the low-amplitude branch that bifurcates sub-critically at $q_{a}=1.01$. To numerically understand these localised patterns with odd/even/no symmetries (which are probably part of a snake-and-ladder bifurcation structure, formed of two "snaking\" solution branches with odd and even symmetries connected by "ladder\" branches with no symmetry), we employ different numerical schemes. Here we focus on the first-order upwind scheme (see, e.g., [@LeVeque-2002-Finite-volume Chapter 4]), the second-order MacCormack scheme (see [@MacCormack-2003], see also [@Helbing-Treiber-1999]), or the Fractional-Step method (see, e.g., [@LeVeque-2002-Finite-volume Chapter 17]). Our goal is to find a spatially-heterogeneous steady-state solution. However, a numerical challenge arises when approaching the steady state, as the flux gradient becomes almost balanced by the source term. As a result, the magnitude of the time derivative of the solution becomes much smaller than that of the spatial derivative of the flux function which is comparable to the magnitude of the source term. This implies that many numerical methods, such as the Fractional-Step method, have difficulty preserving such steady states and cannot accurately capture small perturbations around them. To overcome that, we consider the Quasi-Steady Wave-Propagation Algorithm (QSA) introduced in [@LeVeque-1998] and combine the algorithm with high-resolution methods. One of the new methods for balance laws (called the f-wave method) introduced by [@Bale-Leveque-Mitran-Rossmanith-2002], is efficient with spatially varying flux functions and/or spatially varying source terms. For our problem, it is equivalent to QSA. In this study, we investigate the impact of different numerical schemes on the type of solutions (even symmetric, odd symmetric, or non-symmetric), which could lead to changes in snake-and-ladder bifurcation diagrams. (Note that here we focus only on the choice of the numerical schemes; the construction of the bifurcation diagrams will be presented in a future paper). We will show here several numerical issues in our study. First, we observe the presence of two distinct types of numerical solutions that exhibit very small errors between subsequent time steps, which might initially suggest that we have reached a steady-state solution, but this is not the case. This also implies an extremely slow convergence. Second, in some cases, none of the investigated numerical schemes converge, posing a challenge to the numerical analysis. Lastly, we have discovered that the choice of numerical schemes, as well as their corresponding parameters such as time-space step and initial conditions, exert a significant influence on the type and symmetry of bifurcating solutions. As a result, the resulting bifurcation diagrams may vary when different numerical schemes and/or corresponding parameters are employed. The paper is structured as follows. In [2](#sec:model){reference-type="ref" reference="sec:model"}, we provide a brief overview of the nonlocal hyperbolic model under analysis, the spatially homogeneous steady states, and the localised heterogeneous solution. [3](#sec:numerical_scheme){reference-type="ref" reference="sec:numerical_scheme"} introduces the numerical schemes that have been investigated in this study. [4](#sec:choice_scheme){reference-type="ref" reference="sec:choice_scheme"} represents the main focus of the paper, where we discuss the numerical challenges encountered. In [5](#sec:conclusion){reference-type="ref" reference="sec:conclusion"}, we summarize our findings and draw conclusions, along with highlighting some open problems for further research. The Appendix includes figures that enhance the clarity of the issues discussed in [4](#sec:choice_scheme){reference-type="ref" reference="sec:choice_scheme"}. # Nonlocal hyperbolic model and Localized spatially-heterogeneous solutions with different symmetries {#sec:model} **Nonlocal hyperbolic model.** Following [@Eftimie-Vries-Lewis-2007] and [@Eftimie-Vries-Lewis-Lutscher-2007], we consider the following one-dimensional hyperbolic model to describe the evolution of densities of right-moving ($u^+$) and left-moving ($u^-$) individuals in a generic ecological population on a 1D spatial domain (i.e., a domain much longer than wide): $$\begin{aligned} %\begin{split} \begin{cases}\label{eqn:main} \partial_t u^+(x,t)+ \partial_x (\gamma u^+(x,t)) &= - \lambda^+(u^+, u^-) u^+(x,t)+ \lambda^-(u^+, u^-) u^-(x,t), \\ \partial_t u^-(x,t)+ \partial_x (- \gamma u^-(x,t)) &= \lambda^+(u^+, u^-) u^+(x,t)- \lambda^-(u^+, u^-) u^-(x,t), \\ u^{\pm}(x, 0) &= u^{\pm}_0 (x), \quad x \in \mathbb{R}, \end{cases} %\end{split} \end{aligned}$$ with the turning rates defined as $$\begin{aligned} \lambda^{\pm}(u^+, u^-) := \lambda_1 + \lambda_2 h (y^{\pm}[u^+, u^-]).\end{aligned}$$ Here $\gamma$ is the constant speed, while the two constants $\lambda_1$ and $\lambda_2$ represent a baseline turning rate and a bias turning rate, respectively. For a biologically realistic case, the turning function $h$ should be a positive, increasing, and bounded functional that depends on the communication signals $y^{\pm}$ perceived from neighbors. As in [@Eftimie-Vries-Lewis-2007] and [@Eftimie-Vries-Lewis-Lutscher-2007] we choose $$\begin{aligned} h (y^{\pm}[u^+, u^-]) = 0.5 + 0.5 \tanh (y^{\pm}[u^+, u^-] - y_0),\end{aligned}$$ where the constant $y_0$ is chosen such that for $y^{\pm}[0] = 0$, the value of $\lambda^{\pm}(0)$ is determined only by $\lambda_1$. These signals $y^{\pm}$ are emitted by neighbors moving to the right ($u^+$) and to the left ($u^-$): $$\begin{aligned} \begin{split} y^{\pm}[u^+, u^-] := &q_r \int_{0}^{\infty} K_r(s) (u(x \pm s, t) - u(x \mp s, t)) ds\\ - &q_a \int_{0}^{\infty} K_a(s) (u(x \pm s, t) - u(x \mp s, t)) ds\\ + &q_{al} \int_{0}^{\infty} K_{al}(s) (u^{\mp}(x \pm s, t) - u^{\pm}(x \mp s, t)) ds. \end{split}\end{aligned}$$ Here we define the total density as $u(x,t)= u^+(x,t)+ u^-(x,t)$. The constants $q_r, q_a$, and $q_{al}$ represent the magnitudes of three social interactions: repulsion, attraction, and alignment, respectively. The interaction kernels $K_j$ (with $j = r, a, al,)$ are described by: $$\begin{aligned} K_j(s) := \dfrac{1}{\sqrt{2 \pi m_j^2}} \exp{\left(-\dfrac{(s -s_j)^2}{2 m_j^2}\right)}, \quad j = r, a, al,\end{aligned}$$ where $s_j, j = r, a, al,$ define the spatial regions for repulsive, alignment, and attractive interactions, while $m_j := s_j/8$ define the width of these regions. We choose the constants $m_j$ such that the support of more than $98\%$ of the mass of the kernels is inside the interval $[0, \infty)$. A more detailed description of this model can be found in [@Eftimie-Vries-Lewis-2007] and [@Eftimie-Vries-Lewis-Lutscher-2007].\ **Spatially homogeneous steady states.** For the rest of the paper (and in particular the numerical simulations), we assume that system [\[eqn:main\]](#eqn:main){reference-type="eqref" reference="eqn:main"} is defined on a bounded domain $[0, L]$ with periodic boundary conditions (to allow us to approximate the infinite domain by a finite domain). The non-local interaction terms are wrapped around the domain (see [@Eftimie-Vries-Lewis-2007] for further discussion). Since system [\[eqn:main\]](#eqn:main){reference-type="eqref" reference="eqn:main"} is conservative, let us define the total population density to be $$\begin{aligned} A := \dfrac{1}{L} \int_0^L (u^+(x,t)+ u^-(x,t)) dx.\end{aligned}$$ The spatially homogeneous steady states of [\[eqn:main\]](#eqn:main){reference-type="eqref" reference="eqn:main"} are the solutions $(u^+, u^-) = (u^*, A - u^*)$ of the steady-state equation $$\begin{aligned} \begin{split}\label{eqn:steady-state} 0 = &- u^* \left( \lambda_1 + 0.5 \lambda_2 + 0.5 \lambda_2 \tanh(A q_{al} - 2q_{al} u^* - y_0) \right) \\ &+ (A - u^*) \left(\lambda_1 + 0.5 \lambda_2 + 0.5 \lambda_2 \tanh(-A q_{al} + 2q_{al} u^* - y_0) \right). \end{split}\end{aligned}$$ We note that $(u^+, u^-) = (A/2, A/2)$ is a solution to [\[eqn:steady-state\]](#eqn:steady-state){reference-type="eqref" reference="eqn:steady-state"} for all $\lambda_1, \lambda_2$ and $q_{al}$. In the case of $q_{al} = 0$, the steady-state equation [\[eqn:steady-state\]](#eqn:steady-state){reference-type="eqref" reference="eqn:steady-state"} has a unique solution $(u^+, u^-) = (A/2, A/2)$, and hence we have the unique spatially homogeneous steady state of [\[eqn:main\]](#eqn:main){reference-type="eqref" reference="eqn:main"}.\ Parameter Description Fixed value ------------- -------------------------------------- ------------------- $\gamma$ Speed $\gamma = 0.1$ $\lambda_1$ Baseline turning rate $\lambda_1 = 0.2$ $\lambda_2$ Bias turning rate $\lambda_2 = 0.9$ $y_0$ Shift of the turning function $y_0 = 2$ $q_a$ Magnitude of attraction $q_a = 1.1$ $q_r$ Magnitude of repulsion $q_r = 2.2$ $q_{al}$ Magnitude of alignment $q_{al} = 0.0$ $s_a$ Attraction range $s_a = 1$ $s_r$ Repulsion range $s_r = 0.25$ $s_{al}$ Alignment range $s_{al} = 0.5$ $m_a$ Width of attraction kernel $m_a = 1/8$ $m_r$ Width of attraction kernel $m_r = 0.25/8$ $m_{al}$ Width of attraction kernel $m_{al} = 0.5/8$ $A$ Total population size $A = 2$ $L$ Size of bounded domain space $[0,L]$ $L = 10$ : A list with the model parameters used during simulations. **Localized spatially-heterogeneous solutions with different symmetries.** In [\[fig:Fi3_WeaklyNonlinear\]](#fig:Fi3_WeaklyNonlinear){reference-type="ref" reference="fig:Fi3_WeaklyNonlinear"}, we observed that the stable branch with odd symmetric localized spatially-heterogeneous solutions can be obtained for $q_a \geq 1.095$, while the stable branch with even symmetric can be obtained for $q_a < 1.095$. Hence, it is possible to fix the value of $q_a = 1.1$, and consider the change of solutions symmetries by changing the initial amplitude. Indeed, for the parameter values listed in [1](#table:parameters){reference-type="ref" reference="table:parameters"} (and explained in more detail in [@Eftimie-Vries-Lewis-Lutscher-2007 Table 1] and [@Eftimie-Vries-Lewis-2009 Fig. 2 and Fig. 3]), we can obtain (see [\[fig:ODD_EVEN_NON\]](#fig:ODD_EVEN_NON){reference-type="ref" reference="fig:ODD_EVEN_NON"}) different localized solutions with different symmetries which depend on the amplitude of perturbations of the spatially homogeneous steady state $(u^+, u^-) = (1, 1)$. For the initial condition $$\begin{aligned} \label{eqn:sin02_model} u(x, 0) = 2 u^+(x, 0) = 2 u^-(x, 0) = 2 + \hat{A} \, (0.5 + 0.5\sin(0.2 \pi x)), \end{aligned}$$ where $\hat{A}$ denotes the initial amplitude of the initial total density $u(x, 0)$, [\[fig:ODD_EVEN_NON\]](#fig:ODD_EVEN_NON){reference-type="ref" reference="fig:ODD_EVEN_NON"} illustrates three types of solutions: **(a)** for $\hat{A}=3.5$ we obtain an even symmetric solution consisting of 10 peaks; **(b)** for $\hat{A}=5.0$ we obtain an odd symmetric solution consisting of 9 peaks; **(c)** for $\hat{A}=10.0$ we obtain a non-symmetric solution. This suggests the possibility that these localized solutions belong to solution branches that are part of a snake-and-ladder bifurcation. To understand better these different types of localised solutions, in the next section, we consider different numerical schemes. # Numerical schemes {#sec:numerical_scheme} Given a finite time horizon $T > 0$, and a bounded domain space $[0, L]$, we consider the computation domain $[0, T] \times [0, L]$. Let $\Delta t$ and $\Delta x$ be the constant time and space steps, respectively. We set $$\begin{aligned} N_t = \floor*{\dfrac{T}{\Delta t}}, \quad \text{and} \quad N_x = \floor*{\dfrac{L}{\Delta x}}. \end{aligned}$$ Then for any $1 \leq i \leq N_x$, $0 \leq n \leq N_t$, we define the discrete mesh points $(x_i, t^n) = (i \Delta x, n \Delta t)$ and the cells $C_i = [x_{i - 1/2}, x_{i + 1/2})$. For $1 \leq i \leq N_x$, $1 \leq n \leq N_t$, we also denote by $(u^+_{i})^n$, $(u^-_{i})^n$ and $(u_{i})^n$ the approximation of the averages of $u^+(x, t^n)$, $u^-(x, t^n)$ and $u(x, t^n)$ on the cells $C_i$, namely $$\begin{aligned} (u^+_{i})^n := \dfrac{1}{\Delta x} \int_{C_i} u^+(x, t^n) dx, \quad (u^-_{i})^n := \dfrac{1}{\Delta x} \int_{C_i} u^-(x, t^n) dx, \quad u_{i}^n := \dfrac{1}{\Delta x} \int_{C_i} u(x, t^n) dx.\end{aligned}$$ As initial conditions we set, for $1 \leq i \leq N_x$, $$\begin{aligned} (u^+_{i})^0 := \dfrac{1}{\Delta x} \int_{C_i} u^+_0(x) dx, \qquad (u^-_{i})^0 := \dfrac{1}{\Delta x} \int_{C_i} u^-_0(x) dx. \end{aligned}$$ At the boundary of the domain, we use periodic boundary conditions, namely $$\begin{aligned} u^{\pm}(x, t) = u^{\pm}(x - L) \quad \text{if } x > L, \quad \text{and} \quad u^{\pm}(x, t) = u^{\pm}(x + L) \quad \text{if } x < 0.\end{aligned}$$ This implies that for $0 \leq n \leq N_t$, $$\begin{aligned} (u^{\pm}_{i})^n = (u^{\pm}_{i - N_x})^n \quad \text{if } i > N_x, \quad \text{and} \quad (u^{\pm}_{i})^n = (u^{\pm}_{i + N_x})^n \quad \text{if } i < 0.\end{aligned}$$ We define the source terms in [\[eqn:main\]](#eqn:main){reference-type="eqref" reference="eqn:main"} as follows: $$\begin{aligned} (s_i^+)^n &:= - \lambda^+((u^+_{i})^n, (u^-_{i})^n) \, (u^+_{i})^n + \lambda^-((u^+_{i})^n, (u^-_{i})^n) \, u^-_{i})^n, \\ (s_i^-)^n &:= \lambda^+((u^+_{i})^n, (u^-_{i})^n) \, (u^+_{i})^n - \lambda^-((u^+_{i})^n, (u^-_{i})^n) \, u^-_{i})^n.\end{aligned}$$ To calculate these source terms, we approximate the infinite integrals by integrals on finite domains: $0 < s < 2s_j, \; j = r, a, al$; after that we approximate these integrals using the composite Simpson's 1/3 rule. This means that, $$\begin{aligned} \begin{split} Y_i^n &:= \int_{0}^{\infty} K_{al}(s) (u^-(x_i + s, t_n) - u^+(x_i - s, t_n)) ds\\ &= \int_{0}^{2s_{al}} K_{al}(s) (u^-(x_i + s, t_n) - u^+(x_i - s, t_n)) ds \\ &= \dfrac{\Delta x}{3} \left[K_{al}(0) \left[(u^-_{i})^n - (u^+_{i})^n\right] + K_{al}(2s_{al})\left[(u^-_{i + N_{al}})^n - (u^+_{i - N_{al}})^n\right] \right] \\ &+ \dfrac{2 \Delta x}{3} \sum_{s = 1}^{N_{al}/2 -1} K_{al}(2s\Delta x)\left[(u^-_{i + 2s})^n - (u^+_{i - 2s})^n\right] \\ &+ \dfrac{4 \Delta x}{3} \sum_{s = 1}^{N_{al}/2} K_{al}((2s-1)\Delta x)\left[(u^-_{i + 2s-1})^n - (u^+_{i - 2s+1})^n\right] \end{split} \label{eqn:Y_i}\end{aligned}$$ where $N_{al} := \frac{2s_{al}}{\Delta x}$. The same formula is applied for the remaining integrals.\ Since the goal of this study is to investigate numerically the localized symmetric/asymmetric patterns that could suggest the presence of a snake-and-ladder bifurcation, in the following we discuss a few numerical schemes that have been used to obtain these localized solutions. To this end, we re-write [\[eqn:main\]](#eqn:main){reference-type="eqref" reference="eqn:main"} as follows: $$\begin{aligned} \label{eqn:main_numerical} \partial_t U + \partial_x F(U) = S(U),\end{aligned}$$ where $U = [u^+, u^-]^{T}$, $F(U) = [\gamma u^+, - \gamma u^-]^{T}$, and $S(U) = [ - \lambda^+u^++ \lambda^-u^-, \lambda^+u^+- \lambda^-u^-]^{T}$. Throughout this study, we consider the following numerical schemes: - **Upwind scheme.** We first apply the upwind scheme, which is equivalent to the Godunov method for our case where the velocity is constant ( see, e.g., [@LeVeque-2002-Finite-volume Chapter 4]). The discretized model reads $$\begin{aligned} (u^+_{i})^{n+1} &= (u^+_{i})^n - \dfrac{\gamma \Delta t}{\Delta x} \left[(u^+_{i})^n - (u^+_{i-1})^n\right] + \Delta t (s_i^+)^n, \\ (u^-_{i})^{n+1} &= (u^-_{i})^n + \dfrac{\gamma \Delta t}{\Delta x} \left[(u^-_{i+1})^n - (u^-_{i})^n\right] + \Delta t (s_i^-)^n.\end{aligned}$$ - **MacCormack scheme.** We consider a two-stage approach known as the MacCormack scheme (see [@MacCormack-2003], see also [@Helbing-Treiber-1999]). The concept behind this scheme is to utilize upwind differences in the first stage and downwind differences in the second stage by using the values in the first stage. The values in the subsequent time step are calculated as the average of the previous step's values and the values from the second stage. The order in which the two directions are used can also be switched, or one can alternate between the two orderings in successive time steps, yielding a more symmetric method. Applying the MacCormack scheme, we obtain the first stage $$\begin{aligned} (u^+_{i})^{*} &= (u^+_{i})^n - \dfrac{\gamma \Delta t}{\Delta x} \left[(u^+_{i})^n - (u^+_{i-1})^n\right] + \Delta t (s_i^+)^n, \\ (u^-_{i})^{*} &= (u^-_{i})^n + \dfrac{\gamma \Delta t}{\Delta x} \left[(u^-_{i+1})^n - (u^-_{i})^n\right] + \Delta t (s_i^-)^n,\end{aligned}$$ the second stage $$\begin{aligned} (u^+_{i})^{**} &= (u^+_{i})^{*} - \dfrac{\gamma \Delta t}{\Delta x} \left[(u^+_{i+1})^{*} - (u^+_{i})^{*} \right] + \Delta t (s_i^+)^{*} , \\ (u^-_{i})^{**} &= (u^-_{i})^{*} + \dfrac{\gamma \Delta t}{\Delta x} \left[(u^-_{i})^{*} - (u^-_{i-1})^{n+1/2}\right] + \Delta t (s_i^-)^{*} ,\end{aligned}$$ and finally: $$\begin{aligned} (u^+_{i})^{n+1} &= \dfrac{1}{2} \left[ (u^+_{i})^n + (u^+_{i})^{**} \right], \\ (u^-_{i})^{n+1} &= \dfrac{1}{2} \left[ (u^-_{i})^n + (u^-_{i})^{**}\right].\end{aligned}$$ - **Fractional-Step Method (FSM).** Next, we explore the Fractional-Step Method, see, e.g., [@LeVeque-2002-Finite-volume Chapter 17]. We first split the system [\[eqn:main_numerical\]](#eqn:main_numerical){reference-type="eqref" reference="eqn:main_numerical"} into two sub-problems: a homogeneous conservation law system and an ordinary differential system (ODEs) as follows $$\begin{aligned} &\text{Problem A:} \quad \partial_t U + \partial_x F(U) = 0, \\ &\text{Problem B:} \quad \partial_t U = S(U).\end{aligned}$$ The idea behind the FSM is to discretize the original system by discretizing the two sub-problems in an alternating manner (by using standard methods for each sub-problem). Here, for the FSM approach, we use the upwind scheme for the homogeneous conservation law in problem A, and a two-stage Runge-Kutta method for ODEs (see, e.g., [@LeVeque-2007-Finite-difference Chapter 5]) in problem B. Hence, we obtain the A-step: $$\begin{aligned} (u^+_{i})^{*} &= (u^+_{i})^n - \dfrac{\gamma \Delta t}{\Delta x} \left[(u^+_{i})^n - (u^+_{i-1})^n\right], \\ (u^-_{i})^{*} &= (u^-_{i})^n + \dfrac{\gamma \Delta t}{\Delta x} \left[(u^-_{i+1})^n - (u^-_{i})^n\right],\end{aligned}$$ the first stage of B-step: $$\begin{aligned} (u^+_{i})^{**} &= (u^+_{i})^{*} + \dfrac{\Delta t}{2} (s_i^+)^{*}, \\ (u^-_{i})^{**} &= (u^-_{i})^{*} + \dfrac{\Delta t}{2} (s_i^-)^{*},\end{aligned}$$ and finally: $$\begin{aligned} (u^+_{i})^{n+1} &= (u^+_{i})^{*} + \Delta t \, (s_i^+)^{**}, \\ (u^-_{i})^{n+1} &= (u^-_{i})^{*} + \Delta t \, (s_i^-)^{**}.\end{aligned}$$ - **Quasi-Steady Wave-Propagation Algorithm (QSA).** We consider here the Quasi-Steady Wave-Propagation Algorithm (QSA) introduced in [@LeVeque-1998]. The basic idea of QSA is to introduce a new discontinuity in the center of each grid cell at the start of each time step, with a value $U_i^{L}$ on the left half of the cell and a value $U_i^{R}$ on the right half. These values are chosen so that $$\begin{aligned} \label{eqn:average} \dfrac{1}{2} (U_i^{L} + U_i^{R}) = U_i,\end{aligned}$$ and also, if possible, that $$\begin{aligned} \label{eqn:discrete_version} \dfrac{F(U_i^{R}) - F(U_i^{L})}{\Delta x} = S(U_i).\end{aligned}$$ The condition [\[eqn:average\]](#eqn:average){reference-type="eqref" reference="eqn:average"} guarantees that the cell average is unchanged by the modification, while [\[eqn:discrete_version\]](#eqn:discrete_version){reference-type="eqref" reference="eqn:discrete_version"}, if satisfied, means that the waves resulting from solving the Riemann problem at this new discontinuity will exactly cancel the effect of the source term in this cell. Note that [\[eqn:discrete_version\]](#eqn:discrete_version){reference-type="eqref" reference="eqn:discrete_version"} is a discrete version of $\partial_x F(U) = S(U)$. So, it is not necessary to apply the source term any longer. One of the possible choices of $(U_i^{L}, U_i^{R})$ is that $$\begin{aligned} U_i^{L} = U_i - \delta_i \quad \text{and} \quad U_i^{R} = U_i + \delta_i,\end{aligned}$$ where $\delta_{i}$ satisfies [\[eqn:average\]](#eqn:average){reference-type="eqref" reference="eqn:average"} and [\[eqn:discrete_version\]](#eqn:discrete_version){reference-type="eqref" reference="eqn:discrete_version"} (for example, see [\[eqn:delta_i\]](#eqn:delta_i){reference-type="eqref" reference="eqn:delta_i"} for a detailed formula of $\delta_{i}$ for our problem [\[eqn:main\]](#eqn:main){reference-type="eqref" reference="eqn:main"}). Applying Godunov's method, we obtain that $$\begin{aligned} U_i^{n+1} = U_i^{n} - \dfrac{\Delta t}{\Delta x} \left[F(U_{i+1/2}^{*}) - F(U_{i-1/2}^{*}) \right],\end{aligned}$$ where the flux $F(U_{i+1/2}^{*})$ is determined by the modified values $(U_i^n)^{R}$ and $(U_{i+1}^n)^{L}$. Then, the high-resolution methods can be applied directly. For our problem, by using QSA combining high-resolution methods, we have that $$\begin{aligned} (u^+_{i})^{n+1} &= (u^+_{i})^n - \dfrac{\gamma \Delta t}{\Delta x} \left[(u^+_{i})^{nL} - (u^+_{i-1})^{nR}\right] - \dfrac{1}{2} \dfrac{\gamma \Delta t}{\Delta x}(\Delta x - \gamma \Delta t) \left[(\sigma_{i}^+)^n - (\sigma_{i-1}^+)^n \right], \\ (u^-_{i})^{n+1} &= (u^-_{i})^n + \dfrac{\gamma \Delta t}{\Delta x} \left[(u^-_{i+1})^{nL} - (u^-_{i})^{nR}\right] - \dfrac{1}{2} \dfrac{\gamma \Delta t}{\Delta x}(\Delta x - \gamma \Delta t) \left[(\sigma_{i+1}^-)^n - (\sigma_{i}^-)^n \right],\end{aligned}$$ where $$\begin{aligned} (u^{\pm}_{i})^{nL} = (u^{\pm}_i)^n - (\delta_i^{\pm})^n \quad \text{and} \quad (u^{\pm}_{i})^{nR} = (u^{\pm}_i)^n + (\delta_i^{\pm})^n\end{aligned}$$ with $$\begin{aligned} \label{eqn:delta_i} (\delta_i^+)^n = \dfrac{\Delta x \, (s_i^+)^n}{2 \gamma} \quad \text{and} \quad (\delta_i^-)^n = - \dfrac{\Delta x \, (s_i^-)^n}{2 \gamma}.\end{aligned}$$ By using different slopes $(\sigma_{i}^{\pm})^n$ (for more information about slopes, please refer to [@LeVeque-2002-Finite-volume Chapter 6]), we obtain different schemes as detailed below. But first we denote $$\begin{aligned} &\text{Centered slope:} &(\sigma_{i}^{\pm})^n &= \dfrac{(u_{i+1}^{\pm})^n - (u_{i-1}^{\pm})^n - (\delta_{i+i}^{\pm})^n - 2 (\delta_{i}^{\pm})^n - (\delta_{i-i}^{\pm})^n}{2 \Delta x}, \label{eqn:centered_slope} \\ &\text{Upwind slope:} &(\sigma_{i}^{\pm})^n &= \dfrac{(u_{i}^{\pm})^n - (u_{i-1}^{\pm})^n - (\delta_{i}^{\pm})^n - (\delta_{i-i}^{\pm})^n}{\Delta x}, \label{eqn:upwind_slope} \\ &\text{Downwind slope:} &(\sigma_{i}^{\pm})^n &= \dfrac{(u_{i+1}^{\pm})^n - (u_{i}^{\pm})^n - (\delta_{i+1}^{\pm})^n - (\delta_{i}^{\pm})^n}{\Delta x}. \label{eqn:downwind_slope}\end{aligned}$$ 1. **QSA scheme.** Here, we use no slope, i.e., $(\sigma_{i}^{\pm})^n = 0$. Hence, we get $$\begin{aligned} (u^+_{i})^{n+1} &= (u^+_{i})^n - \dfrac{\gamma \Delta t}{\Delta x} \left[(u^+_{i})^n - (u^+_{i-1})^n\right] + \dfrac{\Delta t}{2} \left[(s_i^+)^n + (s_{i-1}^+)^n \right], \\ (u^-_{i})^{n+1} &= (u^-_{i})^n + \dfrac{\gamma \Delta t}{\Delta x} \left[(u^-_{i+1})^n - (u^-_{i})^n\right] + \dfrac{\Delta t}{2} \left[ (s_{i+1}^-)^n + (s_{i}^-)^n \right]. \end{aligned}$$ We observe that, for the linear system, QSA is equivalent to applying the upwind method to the original data but with a first-order approximation to the source term also included. 2. **QSA_Center scheme.** Here, we use the centered slope [\[eqn:centered_slope\]](#eqn:centered_slope){reference-type="eqref" reference="eqn:centered_slope"}. 3. **QSA_BW scheme.** Here, we follow the idea of the Beam-Warming method which gives a fully-upwind 3-point method. So, we use the upwind slope [\[eqn:upwind_slope\]](#eqn:upwind_slope){reference-type="eqref" reference="eqn:upwind_slope"} for $(\sigma_{i}^{+})^n$ and the downwind slope [\[eqn:downwind_slope\]](#eqn:downwind_slope){reference-type="eqref" reference="eqn:downwind_slope"} for $(\sigma_{i}^{-})^n$. 4. **QSA_LW scheme.** Here, we follow the idea of the Lax-Wendroff method which gives a centered 3-point method. So, we use the downwind slope [\[eqn:downwind_slope\]](#eqn:downwind_slope){reference-type="eqref" reference="eqn:downwind_slope"} for $(\sigma_{i}^{+})^n$ and the upwind slope [\[eqn:upwind_slope\]](#eqn:upwind_slope){reference-type="eqref" reference="eqn:upwind_slope"} for $(\sigma_{i}^{-})^n$. 5. **QSA_Minmod scheme.** Here, we use the *minmod limiter* as follows $$\begin{aligned} (\sigma_{i}^{\pm})^n = \text{minmod} \left[ \text{upwind slope } \eqref{eqn:upwind_slope}, \text{ downwind slope }\eqref{eqn:downwind_slope} \right], \end{aligned}$$ where $$\begin{aligned} \text{minmod}(a,b) = \begin{cases} \min(\abs{a}, \abs{b}) \quad &\text{if } ab > 0, \\ 0 &\text{if } ab \leq 0. \end{cases} \end{aligned}$$ 6. **QSA_Superbee scheme.** Here, we use the *superbee limiter* as follows $$\begin{aligned} (\sigma_{i}^{\pm})^n = \text{maxmod} \left( (\sigma_{i}^{\pm})^{(1)}, (\sigma_{i}^{\pm})^{(2)} \right), \end{aligned}$$ where $$\begin{aligned} (\sigma_{i}^{\pm})^{(1)} &= \text{minmod} \left[ \text{upwind slope } \eqref{eqn:upwind_slope}, \text{ two times downwind slope }\eqref{eqn:downwind_slope} \right], \\ (\sigma_{i}^{\pm})^{(2)} &= \text{minmod} \left[ \text{two times upwind slope } \eqref{eqn:upwind_slope}, \text{ downwind slope }\eqref{eqn:downwind_slope} \right], \end{aligned}$$ and $$\begin{aligned} \text{maxmod}(a,b) = \begin{cases} \max(\abs{a}, \abs{b}) \quad &\text{if } ab > 0, \\ 0 &\text{if } ab \leq 0. \end{cases} \end{aligned}$$ 7. **QSA_MC scheme.** Here, we use the *monotonized central-difference limiter* (MC limiter) as follows $$\begin{aligned} (\sigma_{i}^{\pm})^n = \text{minmod} \left[ (\sigma_{i}^{\pm})^{(1)}, (\sigma_{i}^{\pm})^{(2)}, (\sigma_{i}^{\pm})^{(3)} \right], \end{aligned}$$ where $$\begin{aligned} (\sigma_{i}^{\pm})^{(1)} &= \text{centered slope } \eqref{eqn:centered_slope},\\ (\sigma_{i}^{\pm})^{(2)} &= \text{two times upwind slope } \eqref{eqn:upwind_slope}, \\ (\sigma_{i}^{\pm})^{(3)} &= \text{two times downwind slope }\eqref{eqn:downwind_slope}. \end{aligned}$$ **Order of accuracy of these schemes.** As known from any numerical analysis textbook, the upwind scheme is first-order accurate, while the MacCormack scheme is second-order accurate. The FSM scheme employed here is combining the first-order upwind scheme and the second-order Runge-Kutta method. In theory, the FSM scheme is first-order accurate. However, since the source terms are complicated, we expect the second-order ODEs solver to maintain overall accuracy. The QSA scheme is also first-order accurate. QSA_Center, QSA_BW, and QSA_LW are second-order schemes. The high-resolution methods such as QSA_Minmod, QSA_Superbee, and QSA_MC are formally not second-order accurate. However, the order of accuracy is not everything, meaning that it is not always true that a method with a higher order of accuracy is more accurate on a particular grid or for a particular problem (for more information refer to [@LeVeque-2002-Finite-volume Chapter 8]). **Calculating the error.** To investigate the convergence of the numerical solution to a steady state solution, we first define the discrete $\mathit{L^1}$ norm: $$\begin{aligned} \norm{u(x, n \Delta t)}_{\mathit{L}^{1}} = \norm{u(x, t^n)}_{\mathit{L}^{1}} := \dfrac{L}{N_x} \sum_{i = 1}^{N_x} \abs{u_i^n}.\end{aligned}$$ Then we denote by $E(t), \, t = 1,2,\ldots T,$ the discrete $\mathit{L^1}$-error between the total density $u$ at two adjacent time steps. To this end we take $t = k \Delta t (\in\{1,2,3...,T\})$ and calculate $$\begin{aligned} \label{eqn:Et} \begin{split} E(t) &:= \norm{u(x,t)-u(x,t-\Delta t)}_{\mathit{L}^{1}}= \norm{u(x, k \Delta t) - u(x, (k-1) \Delta t)}_{\mathit{L}^{1}}\\ &= \norm{u(x, t^k) - u(x, t^{k-1})}_{\mathit{L}^{1}} = \dfrac{L}{N_x} \sum_{i = 1}^{N_x} \abs{u_i^k - u_i^{k-1}}, \quad t = 1,2,\ldots T. \end{split} \end{aligned}$$ **Remark 1**. 1. *This paper employs the $\mathit{L^1}$-norm instead of the $\mathit{L^2}$-norm to quantify the numerical solution. There are several advantages associated with this choice. Firstly, the $\mathit{L^1}$-norm proves more suitable for solutions characterized by large distances between local maxima and minima. In particular, when the $\mathit{L^2}$-norm is utilized, a substantial discrepancy arises during the transition from odd symmetry to even symmetry. Furthermore, the $\mathit{L^2}$-norm of the solution can decrease when switching from even symmetry to odd symmetry, despite the increase in the $\mathit{L^2}$-norm of the initial data. Secondly, the $\mathit{L^1}$-norm of the total density remains independent of time within our model, facilitating control $\mathit{L^1}$-norm of solutions by adjusting the initial data. In fact, the $\mathit{L^1}$-norm of the total density can be approximated based on the initial amplitude $\hat{A}$ in [\[eqn:sin02_model\]](#eqn:sin02_model){reference-type="eqref" reference="eqn:sin02_model"} as $$\begin{aligned} \label{eqn:L1-A} \norm{u(x, t)}_{\mathit{L}^{1}} \approx 20 + 5 \hat{A}, \qquad \forall t \geq 0. \end{aligned}$$* 2. *Since we use periodic boundary conditions and need an approximation of the infinite integrals [\[eqn:Y_i\]](#eqn:Y_i){reference-type="eqref" reference="eqn:Y_i"} at each point on the grid, the adaptive mesh refinement technique seems challenging to apply.* # Numerical aspects related to the identification of solutions with different symmetries {#sec:choice_scheme} In this section, we discuss the advantages and disadvantages of the numerical schemes introduced in [3](#sec:numerical_scheme){reference-type="ref" reference="sec:numerical_scheme"} that are relevant to the localised patterns observed in the non-local hyperbolic systems introduced in [2](#sec:model){reference-type="ref" reference="sec:model"}. Since the localised solutions in Figure [\[fig:ODD_EVEN_NON\]](#fig:ODD_EVEN_NON){reference-type="ref" reference="fig:ODD_EVEN_NON"} exhibit even/odd symmetry and asymmetry, we focus on the dependence of these different types of solutions on the initial condition, the space step, and the time step for each numerical scheme via various test cases. It should be noted that all simulations in this paper were executed using fixed parameters, as outlined in [1](#table:parameters){reference-type="ref" reference="table:parameters"}. We first state the stop criteria for our simulations (to ensure that the solution is approaching a steady state): **Definition 2** (Stop criteria). *A simulation will stop when it reaches* 1. *the given final time $T$, or* 2. *the stop time $T^* := 1.34 \times t_0 < T$ where $t_0$ is the first time that satisfies $E(t_0) < \num{e-14}$, with the error $E(t)$ defined in [\[eqn:Et\]](#eqn:Et){reference-type="eqref" reference="eqn:Et"}.* Next, we define two different types of numerical solutions that have a very small error $E(t)$ (which would suggest that we have reached a steady state, but this is not the case): **Definition 3** (Numerical transient solutions and numerical steady-state solutions). *A numerical solution at time $t^*$ characterized by a minimum error is called* - ***transient solution** if $E(t^*) < \num{e-8}$ and $E(t)$ has a local minimum at $t^*$.* - ***steady-state solution** if $E(t^*) < \num{e-14}$ and $E(t)$ does not change significantly for $t > t^*$ (i.e., $E(t) < \num{e-14},$ $\forall t > t^*$).* **Remark 4**. 1. *Due to slow convergence to a steady-state solution, see [4.1](#subsec:transient_solution){reference-type="ref" reference="subsec:transient_solution"}, as well as the long execution time, see [4.2](#subsec:CPU_time){reference-type="ref" reference="subsec:CPU_time"}, this project requires the use of a fast-computation language. Therefore, we use **C++**, a low-level programming language with the **double** data type, which typically allows for 16 significant digits. In addition, we implemented the shared-memory parallel technique (OpenMP) to compute the integrals in the source terms (see formula [\[eqn:Y_i\]](#eqn:Y_i){reference-type="eqref" reference="eqn:Y_i"}). Besides, **Python**, a high-level programming language, is utilized for the post-processing of the results.* 2. *Numerical simulations are typically stopped at time $t^*$ if $E(t^*) < \num{e-8}$. However, for our specific problem, it is possible that the solution at time $t^*$ is only a transient solution and not the desired steady-state solution. In some cases, the value of $E(t)$ at a transient solution can be less than $\num{e-14}$ (see [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"}). Therefore, we set the stop time $T^* = 1.34 \, t_0$ where $t_0$ is the first time satisfied $E(t_0) < \num{e-14}$ in the stop criteria (ii) to observe the geometric of $E(t)$ for $t > t_0$. This allows us to observe the behavior of $E(t)$ for $t > t_0$, which serves as a basis for determining whether the numerical solution is a transient or steady-state solution. Since the limit of **double** data type in **C++**, it is not possible to use the condition $E(t_0) < \num{e-15}$ or less than.* 3. *Typically, stop criteria (ii) are used to terminate iterative algorithms in our simulation, but in some cases, stop criteria (i) must be applied. These cases include situations where the value of the error $E(t)$ at a transient solution falls below $\num{e-14}$, as illustrated in [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"}, or when it is necessary to observe the behavior of solutions and the geometric of the error $E(t)$ over a prolonged period. In all other cases, stop criteria (ii) are sufficient for determining when to end the simulation.* In [4.1](#subsec:transient_solution){reference-type="ref" reference="subsec:transient_solution"} we will discuss both transient and steady-state solutions, while [4.2](#subsec:CPU_time){reference-type="ref" reference="subsec:CPU_time"} will cover the execution time of simulations. In [4.3](#subsec:initial_data){reference-type="ref" reference="subsec:initial_data"}, [4.4](#subsec:space_step){reference-type="ref" reference="subsec:space_step"}, and [4.5](#subsec:time_step){reference-type="ref" reference="subsec:time_step"} we will explore the impact of initial data, space step, and time step on the types of solutions, respectively. It's important to note that these subsections only address cases where the solution converges; non-convergence solutions will be discussed in [4.6](#subsec:non_convergence){reference-type="ref" reference="subsec:non_convergence"}. Finally, in [4.7](#subsec:overview){reference-type="ref" reference="subsec:overview"}, we will provide the change in the solution symmetry as we vary the amplitude of initial conditions, for different numerical schemes. **Notation:** In the Tables that appear in the rest of this paper, we write, for short, 'ODD' for an odd symmetric solution, 'EVEN' for an even symmetric solution, and 'NON' for a non-symmetric solution. ## Numerical transient solutions vs. numerical steady-state solutions. {#subsec:transient_solution} One of the numerical issues encountered so far is the extremely slow convergence of the transient solution to a steady-state solution that can be followed numerically through the continuation algorithm, see [@Uecker-2022]. We discuss this in more detail for our particular problem, the nonlocal hyperbolic system [\[eqn:main\]](#eqn:main){reference-type="eqref" reference="eqn:main"}. We begin by showing in [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"} a simulation obtained using the upwind scheme, where the amplitude of the initial condition is $\hat{A} = 0.2$ (see sub-panel (a)), the spatial step is $\Delta x = 2^{-7}$, and the time step is $\Delta t = 2^{-6}$. The error $E(t)$ calculated up to time 300000 (see sub-panel (b)) can achieve minimal values at different time points: at an early time $t=6000$ (see the transient solution in [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"}(c)), and at some later times $t>240000$ (see the steady-state solution in [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"}(d)). However, the type of transient solution (odd symmetric) differs from that of a steady-state solution (even symmetric). Note that these two numerical solutions are likely the result of the fact that the spatially heterogeneous steady state (i.e., the localised solution) is an unstable saddle point. The numerical solution trajectories approach quickly the transient solution along the stable manifold (due to a large negative eigenvalue). Then, these solutions move very slowly away from this transient state along the unstable manifold of the saddle point (due to a very small positive eigenvalue). For a very large time $t>200000$ in [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"} we see that the solution eventually approaches a stable spatially heterogenous state (steady-state solution). Besides, the total density at the local maximum of $E(t)$ is non-symmetric, see [\[fig:transient_at_some_time\]](#fig:transient_at_some_time){reference-type="ref" reference="fig:transient_at_some_time"} in Appendix A.1. As shown in [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"}, we have a transition from odd symmetric to even symmetric solutions; we also observe the opposite transition: from even symmetric to odd symmetric solutions in some other cases. Additionally, in certain cases, we could also observe a transition from either odd or even symmetric to non-symmetric and the opposite transition. The transient solutions were observed to exist for all schemes analyzed in this paper, with varying initial amplitudes and different time and space steps. For further information, see Appendix A.1. The presence of the transient solution presents us with two challenges. First, the small error $E(t)$ associated with the transient solutions can cause confusion and misinterpretation of these solutions as steady-state solutions. Hence, using the transient solution results in a distinct branching pattern (with odd symmetry) compared to the branching pattern (with even symmetry) obtained with the steady-state solution. Second, the convergence rate in our simulations is considerably slow, necessitating a significantly higher final time value to attain the steady-state solutions. Notably, the convergence rate is slower when the error of the transient solutions is smaller. It was expected that decreasing the space or time step would eliminate transient solutions or at least improve the convergence rate by increasing the value of $E(t)$ at the transient solutions. However, this was not always the case. In some cases, reducing the space step could lead to the creation of a transient solution (which will be discussed in more detail in [4.4](#subsec:space_step){reference-type="ref" reference="subsec:space_step"}), while decreasing the time step could reduce the value of $E(t)$ at the transient solution, thereby slowing down the convergence (which will be discussed in more detail in [4.5](#subsec:time_step){reference-type="ref" reference="subsec:time_step"}). ## Execution time of different numerical schemes. {#subsec:CPU_time} To compare the results obtained with different numerical schemes, we discuss the execution (CPU) time for our simulations with a fixed final time of $T = 1000$. In [\[fig:CPU_time\]](#fig:CPU_time){reference-type="ref" reference="fig:CPU_time"} we first compare the CPU time for a space step of $\Delta x = 2^{-7}$ and a time step of $\Delta t = 2^{-6}$: the upwind and QSA schemes are the fastest, with a CPU time of around 0.91 hours; the QSA with slope schemes is slower, taking around 1.11 hours; the two-stage schemes, MacCormack and FSM, are the slowest, with a CPU time of around 1.83 hours, see [\[fig:CPU_time\]](#fig:CPU_time){reference-type="ref" reference="fig:CPU_time"}(a). If we fix the value of the space step $\Delta x$, then decreasing the time step $\Delta t$ by a factor of two will increase the CPU time by a factor of two, see [\[fig:CPU_time\]](#fig:CPU_time){reference-type="ref" reference="fig:CPU_time"}(a). On the other hand, if we fix the value of the time step $\Delta t$, decreasing the space step $\Delta x$ by a factor of two will increase the CPU time by a factor of four, see [\[fig:CPU_time\]](#fig:CPU_time){reference-type="ref" reference="fig:CPU_time"}(b). Therefore, if we fix the Courant number at 0.2, i.e., $\Delta t = 2 \Delta x$, then increasing both the space and time steps by a factor of two will increase the CPU time by a factor of eight, see [\[fig:CPU_time\]](#fig:CPU_time){reference-type="ref" reference="fig:CPU_time"}(c). clip,trim=0.8cm 0.7cm 0cm 0.8cm Due to the slow convergence towards steady-state solutions, the final time or stop time required is often large, usually ranging from $4000$ to $16000$ in cases with no transient solution and a space step of $\Delta x = 2^{-7}$. If the space step is smaller, the final time may be even larger, as shown in [\[fig:test_dx_error\]](#fig:test_dx_error){reference-type="ref" reference="fig:test_dx_error"}. In cases where a transient solution exists, the final time required is very large, typically exceeding $50000$. For instance, in [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"}, a final time of $t > 200000$ was necessary to obtain the steady-state solution. Consequently, the CPU time required for our simulations is also very large, it takes 206 hours. Another example, when using the MacCormack scheme with a space step of $\Delta x = 2^{-9}$ and a time step of $\Delta t = 2^{-7}$, it took $2743$ CPU hours to converge to a stop time of $T^* = 49292$. Simulations with the space step of $\Delta x = 2^{-10}$ cannot be executed due to insufficient computational resources (it passes the limit of computation time of eight days for 16 cores on the supercomputer facilities of the Mésocentre de Calcul at the University of Franche-Comté). Since the identification of localised solutions with specific symmetries (that could form snake-and-ladder bifurcation branches) necessitates a significant number of simulations, the selection of numerical schemes as well as their corresponding parameters such as the time and space steps, or the selection of initial conditions, become crucial for the simulations as will be discussed in the following subsections. ## The influence of the initial data. {#subsec:initial_data} Throughout this study we consider initial conditions that are perturbations of the spatially homogeneous steady state $(u^+, u^-) = (1, 1)$. In the following we discuss the following three initial conditions: $$\begin{aligned} \label{eqn:sin02} u(x, 0) = 2 u^+(x, 0) = 2 u^-(x, 0) = 2 + \hat{A} \, (0.5 + 0.5\sin(0.2 \pi x)),\end{aligned}$$ $$\begin{aligned} \label{eqn:sin04} u(x, 0) = 2 u^+(x, 0) = 2 u^-(x, 0) = 2 + \hat{A} \, (0.5 + 0.5\sin(0.4 \pi x)),\end{aligned}$$ $$\begin{aligned} \label{eqn:rand} u(x, 0) = 2 u^+(x, 0) = 2 u^-(x, 0) = 2 + \hat{A} \, rand([0,1)),\end{aligned}$$ where $rand([0,1))$ is a random number in $[0,1)$ and $\hat{A}$ denotes the amplitude of the initial perturbation. [\[fig:different_initial_conditions\]](#fig:different_initial_conditions){reference-type="ref" reference="fig:different_initial_conditions"}(a), (b), and (c) shows these initial conditions for an initial perturbation of amplitude $\hat{A} = 2.5$ (as well as a space step $\Delta x = 2^{-7}$ and a time step $\Delta t = 2^{-6}$). Note that the shape of solutions generated by initial data in the form of $\sin(\cdot)$ or $\cos(\cdot)$ is identical; here we choose the $\sin(\cdot)$ form, as the aggregations of peaks of the solution lie in the middle of the domain, making them easier to visualize. [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"} [\[eqn:sin04\]](#eqn:sin04){reference-type="eqref" reference="eqn:sin04"} [\[eqn:rand\]](#eqn:rand){reference-type="eqref" reference="eqn:rand"} -------------- --------------------------------------------------------------------------- --------------------------------------------------------------------------- ------------------------------------------------------------------------ Upwind EVEN 2-ODD NON MacCormack EVEN 2-ODD NON FSM EVEN 2-ODD NON QSA EVEN 2-ODD NON QSA_Center EVEN 2-ODD NON QSA_BW EVEN 2-ODD NON QSA_LW EVEN 2-ODD NON QSA_Minmod ODD 2-ODD NON QSA_Superbee EVEN 2-ODD EVEN QSA_MC ODD 2-ODD NON : The influence of the initial data on the types of solutions. These simulations were executed with an initial amplitude of $\hat{A} = 2.5$, a space step of $\Delta x = 2^{-7}$, and a time step of $\Delta t = 2^{-6}$. We denote 2-ODD as a solution that includes two odd symmetric aggregations. In [\[fig:different_initial_conditions\]](#fig:different_initial_conditions){reference-type="ref" reference="fig:different_initial_conditions"}(d) we observe that using the initial condition [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"} (depicted in [\[fig:different_initial_conditions\]](#fig:different_initial_conditions){reference-type="ref" reference="fig:different_initial_conditions"}(a)), we obtain an even symmetric solution. However, using the initial condition [\[eqn:sin04\]](#eqn:sin04){reference-type="eqref" reference="eqn:sin04"} (as in [\[fig:different_initial_conditions\]](#fig:different_initial_conditions){reference-type="ref" reference="fig:different_initial_conditions"}(b)), we obtain a solution that includes two odd symmetric aggregations, see [\[fig:different_initial_conditions\]](#fig:different_initial_conditions){reference-type="ref" reference="fig:different_initial_conditions"}(e). [\[fig:different_initial_conditions\]](#fig:different_initial_conditions){reference-type="ref" reference="fig:different_initial_conditions"}(f) shows a non-symmetric solution where the initial condition [\[eqn:rand\]](#eqn:rand){reference-type="eqref" reference="eqn:rand"} is used. For a summary of the results obtained with various numerical schemes, for the initial conditions [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"}, [\[eqn:sin04\]](#eqn:sin04){reference-type="eqref" reference="eqn:sin04"}, and [\[eqn:rand\]](#eqn:rand){reference-type="eqref" reference="eqn:rand"}, with an initial amplitude $\hat{A} = 2.5$, a space step $\Delta x = 2^{-7}$, and a time step $\Delta t = 2^{-6}$, see [2](#table:different_initial_conditions){reference-type="ref" reference="table:different_initial_conditions"}. For the initial condition [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"}, we obtain an even symmetric solution, except for the QSA_Minmod and QSA_MC schemes, which result in an odd symmetric solution. On the other hand, using the initial condition [\[eqn:sin04\]](#eqn:sin04){reference-type="eqref" reference="eqn:sin04"}, we obtain a solution that includes two odd symmetric aggregations. Finally, we obtain a non-symmetric solution using the initial condition [\[eqn:rand\]](#eqn:rand){reference-type="eqref" reference="eqn:rand"}, except for the QSA_Superbee scheme which results in an even symmetric solution. It is worth noting that in the random case, obtaining a steady-state solution takes considerably longer, leading to slow convergence compared to other cases. Therefore, in the rest of this paper, we will use the initial condition [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"} for all simulations. ## The influence of the space step. {#subsec:space_step} clip,trim=1.5cm 0.9cm 0cm 0.8cm Now, we discuss the influence of the space step on the types of solutions. To this end, we perform simulations with different space steps $\Delta x = \{2^{-6}; \, 0.01; \, 2^{-7}; \, 0.005; \, 2^{-8}; \, 2^{-9}\}$, with initial condition [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"} having a fixed initial amplitude of $\hat{A} = 5.0$, and a fixed the Courant number of 0.2, i.e, $\Delta t = 2 \Delta x$. Numerical results show that using different space steps (while keeping all model parameters fixed) could lead to localised solutions with different symmetries. For example, [\[fig:test_dx_MC\]](#fig:test_dx_MC){reference-type="ref" reference="fig:test_dx_MC"} presents three types of solutions obtained with the QSA_MC scheme: even symmetric solutions when $\Delta x = 2^{-6}$, see sub-panel (a); odd symmetric solutions when $\Delta x = 2^{-7}$ and $\Delta x = 2^{-9}$ see sub-panels (c) and (f); non-symmetric solutions when $\Delta x = 0.01$, $\Delta x = 0.005$ and $\Delta x = 2^{-8}$ see sub-panels (b), (d) and (e). We also obtain all three types of solutions using QSA_Minmod. For other schemes, it is possible to obtain two types of solutions, which are odd symmetric solutions and non-symmetric solutions. For more details, please refer to [3](#table:different_space_step){reference-type="ref" reference="table:different_space_step"}. From the [3](#table:different_space_step){reference-type="ref" reference="table:different_space_step"}, it is also observed that different solution symmetries are obtained by different schemes, despite using the same space step in the simulations. For instance, all odd symmetric solutions are obtained for the space step $\Delta x = 2^{-9}$, while all non-symmetric solutions are obtained for the space step $\Delta x = 2^{-8}$. However, for the space steps $\Delta x = 0.005$, $\Delta x = 2^{-7}$, and $\Delta x = 0.01$, both odd symmetric and non-symmetric solutions can be obtained. The space step $\Delta x = 2^{-6}$ yields all three types of solutions. $2^{-6}$ $0.01$ $2^{-7}$ $0.005$ $2^{-8}$ $2^{-9}$ -------------- ---------- -------- ---------- --------- ---------- ---------- Upwind ODD NON ODD NON NON ODD MacCormack ODD ODD ODD NON NON ODD FSM ODD NON ODD NON NON ODD QSA NON ODD ODD NON NON ODD QSA_Center NON ODD ODD NON NON ODD QSA_BW NON ODD ODD NON NON ODD QSA_LW NON ODD ODD NON NON ODD QSA_Minmod EVEN NON NON NON NON ODD QSA_Superbee NON ODD ODD ODD NON ODD QSA_MC EVEN NON ODD NON NON ODD : The influence of the space step on the types of solutions. These simulations were executed by using the initial condition [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"} with an initial amplitude of $\hat{A} = 5.0$, and a time step of $\Delta t = 2 \Delta x$. clip,trim=1.3cm 0.4cm 0cm 0.cm In regard to the impact of the decreasing the space step on the convergence rate of the steady-state solutions, we observe that the smaller space step does not always converge faster than the large space step. In some cases, reducing the space step could lead to the creation of a transient solution. For example, [\[fig:test_dx_error\]](#fig:test_dx_error){reference-type="ref" reference="fig:test_dx_error"} shows the error function $E(t)$ for several numerical schemes: the upwind (sub-panel (a); same properties for the FSM scheme), the MacCormack (sub-panel (b)), the QSA_BW (sub-panel (c); same properties for the QSA, QSA_Center, and QSA_LW schemes), the QSA_Minmod (sub-panel (d)), the QSA_Superbee (sub-panel (e)), and the QSA_MC (sub-panel (f)). In these simulations, it is particularly noticeable that smaller space steps, such as $\Delta x = {0.005; \, 2^{-8}; \, 2^{-9}}$, consistently exhibit slower convergence compared to larger space steps like $\Delta x = {2^{-6}; \, 0.01; \, 2^{-7}}$. Additionally, it is observed that transient solutions do not arise for large space steps, such as $\Delta x = {2^{-6}; \, 0.01; \, 2^{-7}}$, whereas they do occur for smaller space steps. In accordance with the theory of convergent numerical methods (for example, see [@LeVeque-2007-Finite-difference Chapter 2]), decreasing the space step (while maintaining the Courant number) should lead to numerical solutions that approach the exact solution. The observed changes in solution types, as described earlier, raise the question of convergence in the theoretical framework of the investigated numerical schemes for the non-local hyperbolic system. Furthermore, due to the slow convergence rate, selecting an appropriate space step becomes a challenging task in simulations. To strike a balance between achieving a satisfactory convergence rate and maintaining an acceptable computation time, as discussed in Section [4.2](#subsec:CPU_time){reference-type="ref" reference="subsec:CPU_time"}, we opted for a space step of $\Delta x = 2^{-7}$. This choice corresponds to a total of 1281 points on the spatial grid for all subsequent simulations presented in this paper. **Remark 5**. *Note that it is difficult to simulate with smaller space steps, such as $2^{-10} = 0.000976563$, as the execution time would be excessively (at least on the servers of supercomputer facilities of the Mésocentre de Calcul at the University of Franche-Comté), see [4.2](#subsec:CPU_time){reference-type="ref" reference="subsec:CPU_time"}. It is also difficult to use the adaptive mesh refinement technique, due to the changes in the solution symmetries if we change the space step and necessary approximations of the infinite integrals, see [Remark 1](#remrak:L1-L2){reference-type="ref" reference="remrak:L1-L2"}. Therefore, we conclude that such choices regarding the numerical schemes (and corresponding numerical space-time steps) depend also on the local computing environment at different academic institutions.* ## The influence of the time step. {#subsec:time_step} Since we are seeking localised steady-state solutions, we expect that the solutions will remain constant across different time steps, and thus the $\mathit{L}^1$-norm of the difference between two solutions corresponding to consecutive time steps should be small. In this subsection, we discuss the influence of the time steps on such steady-state solutions, when we use the initial condition [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"} and a fixed space step of $\Delta x = 2^{-7}$. In [\[fig:test_dt_0200_upwind\]](#fig:test_dt_0200_upwind){reference-type="ref" reference="fig:test_dt_0200_upwind"} we examine two simulations obtained with the upwind scheme, for an initial amplitude of $\hat{A} = 0.2$. The first simulation is obtained with a time step of $\Delta t = 2^{-6}$ and yields an even symmetric solution ([\[fig:test_dt_0200_upwind\]](#fig:test_dt_0200_upwind){reference-type="ref" reference="fig:test_dt_0200_upwind"}(c)). The second simulation, which employs a time step of $\Delta t = 2^{-5}$, produces an odd symmetric solution ([\[fig:test_dt_0200_upwind\]](#fig:test_dt_0200_upwind){reference-type="ref" reference="fig:test_dt_0200_upwind"}(d)). Note that, for the time step $\Delta t = 2^{-6}$, there exists also a transient solution; see [\[fig:test_dt_0200_upwind\]](#fig:test_dt_0200_upwind){reference-type="ref" reference="fig:test_dt_0200_upwind"}(a). However, it is important to note that differences in the type of solutions generated by varying time steps also exist in cases where there are no transient solutions; see Appendix A.2. For a summary of the localised patterns with different symmetries obtained with two different time steps $\Delta t = 2^{-5}$ and $\Delta t = 2^{-6}$, across all numerical schemes (using 361 different initial amplitude values ranging from $0.001$ to $36$), please see [\[fig:all_schemes_7\_5_6\]](#fig:all_schemes_7_5_6){reference-type="ref" reference="fig:all_schemes_7_5_6"}. There we observe that variations in the symmetry of solutions due to changes in the time step occurs for all the schemes examined in this study, albeit at different values of initial amplitude $\hat{A}$. For further details, please see [4.7](#subsec:overview){reference-type="ref" reference="subsec:overview"}. Also, for a further discussion on the impact of different time steps on model solution, see Appendix A.2. clip,trim=1.0cm 0.6cm 0cm 0.cm Now we discuss the impact of decreasing the time step on the convergence rate of the steady-state solutions. To elaborate further, [\[fig:0300_some_schemes\]](#fig:0300_some_schemes){reference-type="ref" reference="fig:0300_some_schemes"} illustrates the behavior of $E(t)$ with varying time steps $\Delta t = 2^{-5}, \Delta t = 2^{-6}, \Delta t = 2^{-7}$, using an initial amplitude of $\hat{A} = 0.3$ and a space step $\Delta x = 2^{-7}$. This figure reveals that the MacCormack and the FSM schemes converge more slowly as the time steps decrease. These are likely a result of the fact that these schemes are two-stage schemes. Particularly for the upwind and FSM schemes with a time step of $\Delta t = 2^{-7}$, the values of $E(t)$ at transient solutions are less than $\num{e-14}$, and it takes over $t > 100000$ to reach steady-state solutions. However, in the case of the QSA_Superbee scheme, the transient solution only occurs for the time step $\Delta t = 2^{-5}$, making it the most efficient scheme in this regard. For the other schemes, the convergence rate is slightly faster with smaller time steps, but transient solutions still persist. As in the previous subsection, the results here also raise the question of convergence in the theoretical framework of the investigated numerical schemes for nonlocal hyperbolic systems, and the difficulty to choose an appropriate time step for the simulations. ## The non-convergence of numerical schemes. {#subsec:non_convergence} For some specific time steps and some specific initial perturbation amplitudes, we observed the non-convergence of all numerical schemes investigated here. For example, [\[fig:non_convergence_at_some_time\]](#fig:non_convergence_at_some_time){reference-type="ref" reference="fig:non_convergence_at_some_time"} shows a non-convergence of the FSM scheme for the final time $T = 10000$ and initial condition [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"}, with an initial amplitude $\hat{A} = 15.2$, a space step $\Delta x = 2^{-7}$, and a time step $\Delta t = 2^{-7}$. In this case, for $t > 4000$, the error $E(t)$ undergoes oscillations within a narrow range $[\num{2e-4}, \num{8e-4}]$, as shown in sub-panel (a). The inset in sub-panel (a) presents the zoom-in of $E(t)$ in the time interval $[7000, 7300]$, to help us better visualize the variation in this error function. This non-convergence persists if we run simulations for longer time, such as $T = 50000$ or $T = 100000$. [\[fig:non_convergence_at_some_time\]](#fig:non_convergence_at_some_time){reference-type="ref" reference="fig:non_convergence_at_some_time"}(b), (c), and (d) shows total densities at times $t = 4300$, $t = 7000$, and $t = 8500$, respectively. These total densities have different values of $E(t)$ in range $[\num{2e-4}, \num{8e-4}]$, but all patterns are non-symmetric. Scheme used $\Delta t = 2^{-7}$ $\Delta t = 2^{-6}$ $\Delta t = 2^{-5}$ -------------- --------------------- --------------------- --------------------- Upwind Non-convergence Non-convergence Non-convergence MacCormack EVEN EVEN Non-convergence FSM Non-convergence Non-convergence Non-convergence QSA Non-convergence Non-convergence Non-convergence QSA_Center EVEN EVEN Non-convergence QSA_BW Non-convergence Non-convergence EVEN QSA_LW EVEN Non-convergence Non-convergence QSA_Minmod EVEN EVEN Non-convergence QSA_Superbee Non-convergence Non-convergence EVEN QSA_MC EVEN Non-convergence EVEN : Examples of non-convergence of numerical schemes. These simulations were performed using the initial condition [\[eqn:sin02\]](#eqn:sin02){reference-type="eqref" reference="eqn:sin02"}, with an initial amplitude of $\hat{A} = 25.0$ and a space step $\Delta x = 2^{-7}$. For the cases where the solution converges, we also specify the symmetry of this solution, i.e., "EVEN\" symmetry. The non-convergence occurs for all numerical schemes when the initial perturbation amplitude is large enough, typically $\hat{A}>10$. However, this threshold depends on both the time step and the numerical scheme employed; see [\[fig:all_schemes_7\_5_6\]](#fig:all_schemes_7_5_6){reference-type="ref" reference="fig:all_schemes_7_5_6"} discussed in more detail in the next subsection. In [4](#table:non_convergence){reference-type="ref" reference="table:non_convergence"}, we summarise the results of several simulations performed using an initial amplitude $\hat{A}=25.0$, a spatial step $\Delta x=2^{-7}$, and time steps $\Delta t={2^{-7}; \, 2^{-6}; \, 2^{-5}}$. For the upwind, FSM, and QSA schemes, non-convergence occurs at all three time steps. For the QSA_BW, and QSA_Superbee schemes, non-convergence occurs at two smaller time steps $\Delta t=2^{-7}$ and $\Delta t=2^{-6}$, while for the time step $\Delta t=2^{-5}$ the solution converges to an even-symmetric localised pattern. In contrast, the QSA_LW scheme converges to an even symmetric solution for the time step $\Delta t=2^{-7}$, while the non-convergence occurs at two larger time steps $\Delta t=2^{-6}$ and $\Delta t=2^{-5}$. For the MacCormack, the QSA_Center, the QSA_Minmod schemes, convergence occurs only at time step $\Delta t=2^{-5}$, while the QSA_MC scheme is non-convergent at time step $\Delta t=2^{-6}$. Therefore, the non-convergence of the numerical schemes is somewhat random (i.e., we could not identify the exact rules). Increasing or decreasing the time steps or the initial amplitude cannot control the non-convergence, as shown in [4](#table:non_convergence){reference-type="ref" reference="table:non_convergence"} and [\[fig:all_schemes_7\_5_6\]](#fig:all_schemes_7_5_6){reference-type="ref" reference="fig:all_schemes_7_5_6"}. ## Change in the solution symmetry as we vary the amplitude of initial conditions, for different numerical schemes. {#subsec:overview} clip,trim=1.3cm 1.0cm 1.2cm 1.5cm To conclude this section, we now investigate the changes in solution symmetries when the $\mathit{L}^1$-norm of the solutions is increased from 20 to 200. To do this, for each scheme with a fixed space-time step, we perform 361 simulations with different initial amplitudes which belong to the set $K = \{0.001; \, 0.1 \times n\}$ where $n = \overline{1,36}$. From [\[eqn:L1-A\]](#eqn:L1-A){reference-type="eqref" reference="eqn:L1-A"}, we obtain the $\mathit{L}^1$-norm of solutions belongs to $\{20.005; \, 20 + 5 \times n\} \subset (20, 200]$. Further, to show the continuous $\mathit{L}^1$-norm, we assume that the solution obtained with the initial amplitude of $\hat{A} = k \in K$ is actually an average of solutions with initial amplitudes in the interval $\hat{A} \in (k - 0.05, k+0.05]$. This assumption usually holds when the simulation converges to a solution. However, it does not really hold in the non-convergence case: i.e., we could have non-convergence for $\hat{A} = k$, convergence for $\hat{A} = k + 0.01$ and non-convergence again for $\hat{A} = k + 0.02$. But since we can only execute a finite and small number of test cases, we need this assumption to have a continuous $\mathit{L}^1$-norm. [\[fig:all_schemes_7\_5_6\]](#fig:all_schemes_7_5_6){reference-type="ref" reference="fig:all_schemes_7_5_6"} presents the change in the solution symmetries for space step $\Delta x = 2^{-7}$ and two different time steps $\Delta t = 2^{-5}$ and $\Delta t = 2^{-6}$; see columns labeled as **(a)** and **(b)**, respectively. The blue stars on the left side of each scheme in [\[fig:all_schemes_7\_5_6\]](#fig:all_schemes_7_5_6){reference-type="ref" reference="fig:all_schemes_7_5_6"} indicate positions where different solution types (odd symmetric, even symmetric, or non-symmetric) occur between two different time step values. In view of creating (as future work) a bifurcation diagram for the transitions between solutions with different types of symmetries, we note that the transitions between the odd-symmetric (green colour) and even-symmetric (red colour) or non-symmetric (yellow colour) solutions occur for different amplitudes of initial perturbations, and are different for different numerical schemes. This suggests that the corresponding bifurcation diagrams would be slightly different for different schemes. Regarding the non-convergence of numerical schemes (light blue colour in [\[fig:all_schemes_7\_5_6\]](#fig:all_schemes_7_5_6){reference-type="ref" reference="fig:all_schemes_7_5_6"}), the MacCormack scheme exhibits the smallest $\mathit{L}^1$-norm of $73$, and interestingly, it has the fewest instances of non-convergence. On the other hand, the upwind and FSM schemes experience non-convergence at a later stage (around an $\mathit{L}^1$-norm of 94), and they have the highest number of non-convergence cases. In terms of solution types for large $\mathit{L}^1$-norm (ranging from $140$ to $200$), the upwind and FSM schemes yield non-symmetric solutions, while the other schemes produce even symmetric solutions. Additionally, when the $\mathit{L}^1$-norm ranges from 20 to 140, noticeable differences are observed between the various numerical schemes and their corresponding time steps. Despite these differences, the group of QSA, QSA_Center, QSA_BW, and QSA_LW schemes exhibit the smallest discrepancy. Particularly, QSA_Center and QSA_LW demonstrate a very slight distinction between the two time steps of $\Delta t = 2^{-5}$, and $\Delta t = 2^{-6}$. However, in our simulations, the QSA_Center scheme has the advantage of faster convergence compared to the QSA_LW scheme. For more detail, please refer to [\[fig:all_schemes_7\_5_6\]](#fig:all_schemes_7_5_6){reference-type="ref" reference="fig:all_schemes_7_5_6"}. # Summary and Conclusions {#sec:conclusion} In this paper, we studied the effect of different numerical schemes (and their time and space steps), as well as the effect of different initial conditions, on the transitions between different localised solutions characterized by different symmetries, which were previously observed in non-local hyperbolic systems for ecological aggregations [@Eftimie-Vries-Lewis-2009]. We have emphasised here several numerical issues. First, we observed the presence of two distinct types of numerical solutions (transient and steady-state solutions) that exhibited very small errors and could be misleading in terms of stopping times for continuation algorithms (which should stop at steady-state solutions and not transient solutions). This also implies an extremely slow convergence. Second, in some cases, none of the investigated numerical schemes converged, posing a challenge to the numerical analysis. Lastly, we have discovered that the choice of numerical schemes (and the time and space steps used for simulations) as well as the choice of the initial conditions, exert a significant influence on the type and symmetry of bifurcating solutions. In consequence, the resulting bifurcation diagrams (which will be developed in the future) may vary when different numerical schemes and/or corresponding parameters are employed (see [\[fig:all_schemes_7\_5_6\]](#fig:all_schemes_7_5_6){reference-type="ref" reference="fig:all_schemes_7_5_6"}). **Open questions.** In the following, we briefly mention some of the open questions identified through this study, which will have to be addressed in the future: - We need to investigate in more detail (numerically and theoretically) the slow convergence to steady state solutions, and the differences in the types of solutions when we change the time/space steps. - Theoretical investigation of the convergence of all these numerical schemes for nonlocal hyperbolic systems is necessary, to clarify whether: (i) is it the scheme that is not converging for those specific time and/or space steps, or (ii) is it the dynamics of the system that could be exhibiting another type of solution -- heteroclinic orbits connecting different localised solutions -- in that parameter region, and the numerical approach used does not capture these extra dynamics. - The impact of the very small transient error could lead to the stopping of some numerical continuation algorithms, even when the solution did not reach a steady state. Usually, many numerical algorithms stop if the error tolerance (equivalent to $E(t)$) is less than  × 10^−8^ or  × 10^−10^), but as we have seen in this study, even if we chose the tolerance small enough (here  × 10^−14^), in some cases we cannot ensure that we really reach a steady-state solution (see also the discussion in Appendix A.1). For instance, for initial amplitudes $\hat{A} = 1.4, \; 1.5, \; 1.6$, we obtain even symmetric steady-state solutions (see [\[fig:open_question\]](#fig:open_question){reference-type="ref" reference="fig:open_question"}(b)), and in these cases, we also obtain odd symmetric transient solutions (see [\[fig:open_question\]](#fig:open_question){reference-type="ref" reference="fig:open_question"}(a)). We observe that the error of transient solutions is decreasing as we increase the initial amplitudes, as shown by yellow, black, and blue curves in [\[fig:open_question\]](#fig:open_question){reference-type="ref" reference="fig:open_question"}(d). However, if the initial amplitude is $\hat{A}=1.7$, we obtain an odd symmetric steady-state solution (see [\[fig:open_question\]](#fig:open_question){reference-type="ref" reference="fig:open_question"}(c)) and no transient solution (see red curve in [\[fig:open_question\]](#fig:open_question){reference-type="ref" reference="fig:open_question"}(d)). This raises the question of whether this is a stable steady-state solution or a transient solution; but because of the limitations of the numerical scheme, as well as the **double** data type in **C++**, we cannot answer this question. # Appendix A.1. Numerical transient solutions and numerical steady-state solutions {#sec:appendix_A1 .unnumbered} In this Appendix, we show a few more numerical results emphasizing the issue of having transient vs. steady-state solutions with different symmetries presented in [4.1](#subsec:transient_solution){reference-type="ref" reference="subsec:transient_solution"}. As shown in [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"}, there is an odd symmetric transient solution at time $t = 6000$ with $E(t) = \num{1.11 e-15}$ and an even symmetric steady-state solution at time $t = 240 000$ with $E(t) = \num{7.2 e-16}$. To have a better understanding of what happens in this case, in [\[fig:transient_at_some_time\]](#fig:transient_at_some_time){reference-type="ref" reference="fig:transient_at_some_time"} we show the types of solutions observed at different times; see sub-panel (a). Sub-panel (b) shows an odd symmetric solution at time $t = 60 000$ with $E(t) = \num{3.04 e-13}$. At time $t = 150 000$ the total density is still odd symmetric with $E(t) = \num{7.99 e-9}$; see sub-panel (c). At the local maximum of $E(t)$, we obtain a non-symmetric total density solution (at time $t = 198 000$) with $E(t) = \num{9.65 e-6}$; see sub-panel (d). Note that for all simulations performed in this paper, the range of the local maximum of $E(t)$ is between $10^{-6}$ and $10^{-2}$. An even symmetric total density is observed in sub-panel (e) at time $t = 219 000$ with $E(t) = \num{2.13 e-12}$. From time $t > 220 000$, we obtain only even-symmetric steady-state solution with $E(t) = \num{7.2 e-16}$, as shown in sub-panel (f). Moreover, in [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"} we observed a transition from odd-symmetric to even-symmetric solutions; in contrast, in [\[fig:transient_steady_state_solution_QSA_Minmod_5\]](#fig:transient_steady_state_solution_QSA_Minmod_5){reference-type="ref" reference="fig:transient_steady_state_solution_QSA_Minmod_5"} (a simulation of the QSA_Minmod scheme with an initial amplitude of $\hat{A} = 5.0$, a space step of $\Delta x = 2^{-9}$ and a time step $\Delta t = 2^{-7}$) we can show the opposite transition: from even-symmetric to odd-symmetric solutions. In some cases we could also observe a transition from either odd or even symmetric to non-symmetric solutions, or even the opposite transition; see, for example, [\[fig:transient_steady_state_solution_QSA_4\]](#fig:transient_steady_state_solution_QSA_4){reference-type="ref" reference="fig:transient_steady_state_solution_QSA_4"} (a simulation of the QSA scheme with an initial amplitude of $\hat{A} = 5.0$, a space step of $\Delta x = 2^{-8}$ and a time step $\Delta t = 2^{-7}$).\ Overall, transient solutions were observed to exist for all schemes analyzed in this paper, with varying initial amplitudes and different time and space steps; for example, see [\[fig:transient_steady_state_solution\]](#fig:transient_steady_state_solution){reference-type="ref" reference="fig:transient_steady_state_solution"}, [\[fig:test_dx_error\]](#fig:test_dx_error){reference-type="ref" reference="fig:test_dx_error"}, [\[fig:0300_some_schemes\]](#fig:0300_some_schemes){reference-type="ref" reference="fig:0300_some_schemes"}, [\[fig:open_question\]](#fig:open_question){reference-type="ref" reference="fig:open_question"}, [\[fig:transient_at_some_time\]](#fig:transient_at_some_time){reference-type="ref" reference="fig:transient_at_some_time"}, [\[fig:transient_steady_state_solution_QSA_Minmod_5\]](#fig:transient_steady_state_solution_QSA_Minmod_5){reference-type="ref" reference="fig:transient_steady_state_solution_QSA_Minmod_5"}, and [\[fig:transient_steady_state_solution_QSA_4\]](#fig:transient_steady_state_solution_QSA_4){reference-type="ref" reference="fig:transient_steady_state_solution_QSA_4"}. # Appendix A.2. The influence of the time step {#sec:appendix_A2 .unnumbered} In this Appendix, we show a few more numerical results emphasizing the impact of the time step on different symmetries solutions presented in [4.5](#subsec:time_step){reference-type="ref" reference="subsec:time_step"}. In [\[fig:test_dt_0200_upwind\]](#fig:test_dt_0200_upwind){reference-type="ref" reference="fig:test_dt_0200_upwind"}, we have observed differences in the symmetry of localised solution when varying the time steps: an odd-symmetric steady-state solution for $\Delta t = 2^{-5}$, and an even-symmetric steady-state solution for $\Delta t = 2^{-6}$. Note that, for $\Delta t = 2^{-6}$, there exists also an odd symmetric transient solution. However, it is important to note that differences in the type of solutions generated by varying time steps exist also in cases where there is no transient solution. For example, using the QSA_MC scheme with an initial amplitude of $\hat{A} = 5.0$, time steps of $\Delta t = 2^{-8}$ and $\Delta t = 2^{-5}$ produce a non-symmetric solution as shown in [\[fig:test_dt_5000_QSA_MC\]](#fig:test_dt_5000_QSA_MC){reference-type="ref" reference="fig:test_dt_5000_QSA_MC"} (c), and an odd-symmetric solution as shown in [\[fig:test_dt_5000_QSA_MC\]](#fig:test_dt_5000_QSA_MC){reference-type="ref" reference="fig:test_dt_5000_QSA_MC"}(d). In these two simulations, there is no transient solution, see [\[fig:test_dt_5000_QSA_MC\]](#fig:test_dt_5000_QSA_MC){reference-type="ref" reference="fig:test_dt_5000_QSA_MC"}(a) and (b). To compare solutions at different time steps, we have chosen two cases as our reference numerical solutions: (i) $\Delta t = 2^{-8}$ and initial amplitude $\hat{A} = 3.5$; and (ii) $\Delta t = 2^{-9}$ and initial amplitude of $\hat{A} = 5.0$. [\[fig:test_dt_error\]](#fig:test_dt_error){reference-type="ref" reference="fig:test_dt_error"} displays the $\mathit{L}^1$-norm of the difference between these reference solutions and the solutions obtained with other time steps. Although the MacCormack and FSM schemes produce similar solution types, their $\mathit{L}^1$-errors are relatively large, typically greater than $\num{e-2}$ (this is probably caused by the fact that they are two-stage schemes). For the QSA_Superbee scheme with the initial amplitude $\hat{A} = 3.5$ and the QSA_MC scheme with the initial amplitude $\hat{A} = 5.0$, the $\mathit{L}^1$-errors are also large due to changes in the solution types. On the other hand, schemes such as upwind, QSA, QSA_Center, QSA_BW, and QSA_LW produce similar solutions types with very small $\mathit{L}^1$-errors, around $\num{e-10}$. For brevity, in [\[fig:test_dt_error\]](#fig:test_dt_error){reference-type="ref" reference="fig:test_dt_error"} we only display the error of the QSA scheme. Finally, in [\[fig:test_dt_5000_QSA_Minmod\]](#fig:test_dt_5000_QSA_Minmod){reference-type="ref" reference="fig:test_dt_5000_QSA_Minmod"} we show that for the QSA_Minmod scheme with initial amplitude $\hat{A} = 5.0$, all solutions are non-symmetric, but with geometric differences between the peaks of the solutions, resulting in a large error. # Acknowledgments {#acknowledgments .unnumbered} The authors (TTL and RE) acknowledge funding from a Region Bourgogne Franche-Comté "Accueil de Nouvelle Équipe de Recherche (ANER) 2022\" grant number `FC22070.LMB.CL.`, and funding through a University of Franche-Comté \"Chrysalide: soutien aux nouveaux arrivants\" (for RE). Computations have been performed on the supercomputer facilities of the Mésocentre de calcul de Franche-Comté and on the computation server of Laboratoire de Mathématiques de Besançon. The authors would like to thank Julien Yves ROLLAND for his support on the computation servers.

arxiv_math

--- abstract: | We show that $\mathbb{C}[\bar x]^{E}$ is not Noetherian even respect to prime E-ideals. Moreover we give a characterization of exponential radical ideals address: - Università di Firenze - Università degli Studi di Napoli \"Federico II\" author: - Antongiulio Fornasiero - Giuseppina Terzo bibliography: - exponential.bib date: 13/06/23 title: Note on radical and prime E-ideals --- # Introduction The notion of exponential ideal (E-ideal) was introduced in the papers [@van; @macintyre]; it was related to the study of exponential function after the problem posed by Tarski on the decidability of the reals with exponentiation. An exponential ring (E-ring) is a pair $(R, E)$ where R is a commutative ring with 1 and E is a homomorphism $E: (R, +) \rightarrow (R^{*}, \cdot)$. We will always assume that $R$ is a $\mathbb{Q}$-algebra. The classical examples are the reals and the complex numbers with the usual exponentiation. Starting from an E-ring $R$, we can construct the E-polynomial ring in the variables $\bar x$ over $R$ by induction (see [@van; @macintyre]) and we denote it by $R[\bar x]^E$. An E-ideal $I$ of an E-ring $R$ is an ideal with the property that if $\alpha \in I$ then $E(\alpha) - 1 \in I$; they coincide with the possible kernels of homomorphisms which preserve the exponential map $E$. Some further contributions to the study of E-ideals were given in [@manders; @point; @terzo]. In the recent paper [@ideali], together with P. D'Aquino, we studied E-ideals of E-rings, we gave two notions of maximality for E-ideals, and related them to primeness. We proved that the three notions are independent, unlike in the classical case. Moreover, we showed that, for any exponential field $K$, not all maximal E-ideal of $K[\bar x]$ correspond to points of $K^n$. In this paper we further investigate E-ideals of $R[\bar x]^{E}$. It was known that the exponential Zariski topology on $\mathbb{C}^{n}$ is not Noehterian (see [@macintyre]). We show that $\mathbb{C}[\bar x]^{E}$ is not Noetherian even respect to prime E-ideals. Moreover, we give a reasonable notion of E-radical E-ideals, characterize them, and prove some of their properties, using a technique introduced in [@ideali] that permits to extend prime ideals to prime E-ideals. # E-polynomial ring and basic results First to introduce the construction of the E-polynomial ring is useful to recall the notion of partial E-rings and partial E-ideals and some important result related to it proved in [@ideali]. ## Partial E-rings and E-ideals **Definition 1**. A partial E-ring is a triple $D=(D, V, E)$ where 1. $D$ is a $\mathbb{Q}$-algebra; 2. $V$ is a $\mathbb{Q}$-vector subspace of $D$ containing $\mathbb{Q}$; 3. $E : (V, +) \to (D^{*}, \cdot)$ is a group homomorphism. **Definition 2**. A partial E-ideal of $D$ is an ideal $I$ of the ring $D$ such that, for every $v \in I \cap V$, $E(v) - 1 \in I$. If $D=V$ we say that it is an E-ideal. **Remark 3**. If $I$ is an ideal of $D$ with $I\cap V=( 0)$ then $I$ is an E-ideal of $D$. **Definition 4**. An E-ideal I of D is prime if it is prime as ideal, i.e. iff $D/I$ is a domain. An E-ideal I of D is an E-maximal ideal if it is maximal among the E-ideals. It is strongly maximal if it is maximal as ideal. ## Construction of E-polynomial ring The construction of the E-polynomial ring in many variables is well known, see [@van; @macintyre]; for the reader's convenience, we briefly recall it here. Starting from an E-ring $R$, we construct the E-polynomial ring in the variables $\bar x= (x_1, \ldots, x_n)$, denoted by $R[\bar x]^E$, as a union of a chain of partial E-rings equipped with partial E-morphisms. The following three chains are constructed by recursion: $(R_{\scriptsize k}, +, \cdot)_{\scriptsize {k\geq-1}}$, $(B_k, +)_{k\geq0}$, and $(E_k)_{k\geq-1}$ will be rings, abelian groups and partial E-morphisms, respectively. Let $R_{-1} = R$, and $R_0 = R[\overline x]$ as partial exponential ring (where $\exp$ is defined only on $R$). Let $A_0 = (\overline x)$ be the ideal of $R[\overline x]$ generated by $\overline x$. For $k \geq 1$, let $A_{k}$ be the $R_{k-1}$-submodule of $A_{k}$ generated by $t^a$ with $0 \neq a \in A_k$. So, we define $R_k = R_{k-1} \oplus A_k,$ $R_{k+1} = R_{k}[t^{A_k}]$ and $E_{k} : (R_k, +) \rightarrow (R_{k+1}^{*}, \cdot)$ such that $E_k(x) = E_{k-1} (r) \cdot t^b$, for $x = r + b$, $r \in R_{k - 1}$ and $b \in B_k$. Then the E-polynomial ring is the limit of this object, i.e $R[\overline x]^E= \cup R_k.$ Sometimes it is convenient to represents $R[\bar x]^{E}$ as the group ring $R[\bar x][t^{\bigoplus_{i\geq 0} A_i}]$. In [@ideali] we generalized this construction to any partial E-ring R, i.e. we gave a free completion of a partial E-ring R that they denoted by $R^E$. Moreover we gave sufficient conditions on a subring S of $R^E$ such that the free completion of $S,$ $S^E$ is isomorphic to $R^E.$ We recall the result just for E-polynomial rings. **Lemma 5** ([@ideali]\*Lemma 2.10). *Let $R$ be an E-ring and $S$ be a partial subring of $R[\bar x]^E,$ and assume that $S = R[\bar x][e^A]$ for some $\mathbb{Q}$-linear subspace $A$ of $R[\bar x]$ which has trivial intersection with $R$. Then, $S^E = R[\bar x]^E$.* **Lemma 6**. *Let $S \subseteq R[\bar x]^{E}$ be as in Lemma [Lemma 5](#completion){reference-type="ref" reference="completion"}. Let $I$ be an ideal of $S$. If $I$ is a partial prime E-ideal of $R_n$, then $I^E$ (the $E$-ideal of $R[\bar x]^{E}$ generated by $I$) is a prime E-ideal of $R[\bar x]^E$.* # Noetherianity As Macintyre point out in [@macintyre], neither $\Bbb C[x]^E$ nor $\Bbb R[x]^E$ are Noetherian for E-ideals, by considering the E-ideal $I = (E(\frac{x}{2^n}) - 1)_{n \in \Bbb N},$ since it is not finitely generated (for details see also [@terzo]). There is a notion of noetherianity also for topological spaces: **Definition 7**. A topological space $X$ is Noetherian if it satisfies the descending chain condition for closed subsets, i.e. any strictly descending sequence of elements of $X$ is stationary. $\mathbb{C}[\bar x]^{E}$ is very far from being Noetherian for E-ideals, since $\mathbb{C}$ with the exponential Zariski topology is not a Noetherian space, because of the chain: $$\begin{aligned} Z(E(x) -1) & \supset Z(E(\frac{x}{2}) -1) \supset \ldots \supset Z(E(\frac{x}{n!}) -1)\supset \ldots\\ \{2\pi i \mathbb{Z}\} &\supset \{4\pi i \mathbb{Z}\} \supset \ldots \supset \{2 n! \pi i \mathbb{Z}\} \supset \ldots \end{aligned}$$ descendes indefinitely. We show that $\mathbb{C}[\bar x]^E$ is not Noetherian even respect to prime E-ideals. **Theorem 8**. *The ring $\mathbb{C}[\bar x]^E$ doesn't satisfy ACC condition for prime E-ideals.* We consider the subring $S := \mathbb{C}[\bar x, e^{\mathbb{C}\bar x}] = \mathbb{C}[\bar x] [e^{\mathbb{C}\bar x}]$ of the ring $\mathbb{C}[\bar x]^E$. The idea is to construct an ascending chain of prime E-ideals of $S$ and to extend it to be an ascending chain of prime E-ideals of $\mathbb{C}[\bar x]^E$. Let $P_i < S$ be prime ideals of $S$ and we define $Q_i = P_i \mathbb{C}[\bar x]^E.$ For simplicity, we consider the case when we have only one variable $\bar x= x$. We construct prime ideals in the following way: let be $B$ a transcendence basis of $\mathbb{C}$; $B = (b_j : j < 2^{\aleph_0}).$ For all $i \in \Bbb N,$ we define $$p_i := e^{b_{3i}x} +e^{b_{3i + 1}x} + e^{b_{3i +2}x} .$$ Let be $A_n = (p_0, \ldots, p_n)$ as an ideal in $\mathbb{C}[e^{\mathbb{C}x}].$ We need the following result: **Lemma 9**. *The ideal $A_n$ is prime for all $n \in \Bbb N.$* *Proof.* We introduce new variables denoting elements of the form $e^{b_i x}$ for any $b_i \in B$, i.e.$z_i = e^{b_i x}$, so we can denote $A_n = (z_0 + z_1 + z_2, z_3 + z_4 + z_5, \ldots, z_{3n} + z_{3n+1} + z_{3n + 2})$ ad an ideal of $\mathbb{C}[\overline z^{\mathbb{Q}}].$ We prove by induction on $n$ that $A_n$ is prime. For $n=1$ we have that $A_1 = (z_0+z_1+z_2)$ as an ideal of $\Bbb C [z_0^{\Bbb Q}, z_1^{\Bbb Q}, z_2^{\mathbb{Q}}]$. Assume that $p(z_0, z_1, z_2)\cdot q(z_0, z_1, z_2) \in A_1$ then $p(z_0, z_1, z_2)\cdot q(z_0, z_1, z_2) = r(z_0, z_1, z_2) (z_0+z_1+z_2)$; let $k$ be the common denominator of any exponents in $p, q, r$, so we can consider $p, q, r \in \Bbb C [z_0^{\pm \frac{1}{k}}, z_1^{\pm \frac{1}{k}}, z_2^{\pm \frac{1}{k}}]$. By replacing $z_{i}$ with $t_{i}^{k}$, we have that $p,q,r \in \mathbb{C}[t_{0}^{\pm 1}, t_{1}^{\pm 1}, t_{2}^{\pm 1}]$. Notice that $z_{0} + z_{1} + z_{2}$ becomes $s(\bar t) := t_{1}^{k} + t_{2}^{ k} + t_{3}^{k}$, and that $s$ is irreducible in $\mathbb{C}[\bar t]$ and hence the ideal $A_{1}'$ generated by $s$ inside $\mathbb{C}[\bar t]$ is prime. By localization, the ideal $A_{1}''$ generated by $s$ inside $\mathbb{C}[\bar t^{\mathbb{Z}}]$ is also prime; therefore, since $p q \in A_{1}''$, we have either $p \in A_{1}'' \subseteq A_{1}$ or $q \in A_{1}'' \subseteq A_{1}$, proving that $A_1$ is prime. For $n = 2$, $A_2 = (z_0+z_1+z_2, z_3+z_4+z_5)$ this is prime since $$\frac{\mathbb{C}[z_0^{\mathbb{Q}}, \ldots, z_5^{\mathbb{Q}}]}{A_2} \cong \frac{\mathbb{C}[z_0^{\mathbb{Q}}, z_1^{\mathbb{Q}}, z_2^{\mathbb{Q}}]}{A_1} \otimes \frac{\mathbb{C}[z_3^{\mathbb{Q}}, z_4^{\mathbb{Q}}, z_5^{\mathbb{Q}}]}{(p_1)}$$ So $\frac{\Bbb C [z_0^{\Bbb Q}, \ldots, z_5^{\Bbb Q}]}{A_2}$ is a domain, since the tensor product of $\mathbb{C}$-algebrae which are domains is also a domain, see [@bourbaki]\*Chapter V, §17. In a similar way we can prove that $A_n$ is prime for any $n \in \Bbb N.$ ◻ If $A_n$ is prime then $P_n = A_n S$ is partial prime E-ideal in S and so $Q_n$, by Lemma [Lemma 5](#completion){reference-type="ref" reference="completion"} and Lemma [Lemma 6](#idealiprimi){reference-type="ref" reference="idealiprimi"}, will be prime E-ideals in $\Bbb C [\bar x]^E.$ So we have an ascending chain of prime E-ideals. 0◻ Now we give conditions to say when an E-ideal of a particular form is prime. Let $\bar x, \bar y$ be tuples of variables of the same lenght. Given $p(\bar x,\bar y) \in \mathbb{C}[\bar x,\bar y]$, we denote by $\tilde p(\bar x) := p(\bar x, E(\bar x)) \in \mathbb{C}[\bar x]^{E}$. Let $I \subseteq \mathbb{C}[\bar x,\bar y]$ be an ideal, and let $\tilde I := \{\tilde p: p \in I\}$. We denote by $S = \Bbb C[\overline x, e^{\overline Q \overline x}]$, by $J$ the E-ideal of $\mathbb{C}[\bar x]^{E}$ generated by $\tilde I$, and by $H = J \cap S$. **Proposition 10**. *$J$ is prime in $\Bbb C[\overline x]^E$ iff $H$ is prime in $S.$* *Proof.* One direction is trivial. For the other one we assume that $H$ is a prime in $S$ then by Lemma [Lemma 5](#completion){reference-type="ref" reference="completion"} and Lemma [Lemma 6](#idealiprimi){reference-type="ref" reference="idealiprimi"} we have that $J$ is a prime E-ideal of $\Bbb C[\overline x]^E.$ ◻ **Proposition 11**. *If the following hold:* 1. *I is a prime ideal;* 2. *I doesn't contain non zeros elements of the form $a + \overline q \cdot \overline x$ with $\overline q \in \Bbb Q^n$ and $a \in \Bbb C;$* 3. *I doesn't contain any element of the form $\overline y^{\overline q} - a$ where $\overline q \in \Bbb Q^n$ and $a \in \Bbb C;$* 4. *For all $n \in \mathbb{N}$ $I_n$ the ideal in $\mathbb{C}[\bar x, \bar y^{\frac{1}{n}}]$ is prime.* *Then J is a prime E-ideal.* *Proof.* We denote by $K$ the ideal generated by $\tilde I$ in $S.$ If we prove that $K$ is a partial E-ideal of S and it is prime we conclude the proof, because by Corollary 3.13 of [@ideali] we obtain that $J$ is prime and $K = H.$ First we prove that $K$ is a partial E-ideal of $S.$ We note that the domain of the exponential map on $S$ is $\mathbb{Q}\cdot \bar x+ \mathbb{C},$ so we have to prove that $K \cap (\mathbb{Q}\cdot \bar x+ \mathbb{C})= (0).$ By replacing $e^{\mathbb{Q}\bar x}$ with $\bar y^{\mathbb{Q}}$ we have $\mathbb{C}[\bar x, \bar y] \subset \mathbb{C}[\bar x, \bar y^{\mathbb{Q}}]\cong \mathbb{C}[\bar x, e{\mathbb{Q}\bar x}] = S.$ We claim that $$K \cap \mathbb{C}[\bar x, \bar y] =I.$$ If $a \in K \cap \mathbb{C}[\bar x, \bar y]$ then $a = \frac{p}{\bar y^{m}}$ where $p \in I.$ Then $a\cdot\bar y^m \in I,$ but by (2)+(3) we have that $a \in I$ and this implies that $K$ is a partial E-ideal. Now we have to prove that $K$ is prime. We are assuming that the ideal $I$ of $\mathbb{C}[\bar x, \bar y]$ is prime so by commutative algebra (see [@Matsumura]) we know that the ideal $I'$ in the localization $\mathbb{C}[\bar x, \bar y^{\pm 1}]$ is prime or trivial, but it is prime because we are assuming (3). If we consider the ideal $I'' = K$ in the integral extension of $\mathbb{C}[\bar x, \bar y^{\pm 1}]$, i.e. in the extension $\mathbb{C}[\bar x, \bar y^{\mathbb{Q}_{\geq 0}}].$ We first observe that $I'' = \cup_{n} I_n,$ from (4) any $I_n$ is prime and this implies that $I''$ is prime, this conclude the proof. ◻ # Exponential radical ideals We give the notion of E-radical ideal for any E-ring $R$ as follows. **Definition 12**. Let $J$ an E-ideal of an E-ring $R$. We define the E-radical ideal of $J$ as $\mathop{\mathrm{E-rad}}(J) := \cap_{P \supseteq J} P$, where $P$ varies among prime E-ideals. Let $J$ an E-ideal of the E-polynomial ring $K[\bar x]^E$. Let be $F$ an E-ring containing $K$. We can define $\mathcal{I} (V(J))$ as follows:\ $V_F(J) := \{ \overline a \in F^n : f(\overline a) = 0 \mbox{ for all } f(\bar x) \in J \}$,\ $V(J) := \bigcup V_F(J)$ as $F$ varies among all E-fields containing $K$, and\ $\mathcal{I} (V(J)) = \{ p(\bar x) \in K[\bar x]^E : p(\overline a) = 0 \mbox{ for all } \overline a \in V(J) \}$. **Remark 13**. Let $(R,E)$ be an E-domain and $K$ be its fraction field. Then, there exists at least one way to extend the exponential function to all of $K$. *Proof.* Let $A\subset K$ be a complement of $R$ as $\mathbb{Q}$-linear spaces. For every $r \in R$ and $a \in A$, define $E'(a + r) := E(r)$. Then, $E'$ is an exponential function on $K$ extending $E$. ◻ **Corollary 14**. *$\mathcal{I} (V(J)) = \{ p(\bar x) \in K[\bar x]^E : p(\overline a) = 0 \mbox{ for all } \overline a \in V_{F}(J)$ as $F$ varies among all E-domains containing $K\}$.* **Lemma 15**. *Let be $J$ an E-ideal of the E-polynomial ring $K[\bar x]^E.$ Then $\mathop{\mathrm{E-rad}}(J) = \mathcal{I} (V(J))$.* **Remark 16**. The E-ideal $I = (xy)^E$ is not a E-radical ideal, unlike in the classical case. Indeed $I$ is not the intersection of prime E-ideal, since $(xy)^E \neq (x)^E \cap (y)^E,$ because $x(E(y) - 1) \in (x)^E \cap (y)^E \setminus (xy)^E$. **Remark 17**. If $J$ is not contained in any prime E-ideals then $\mathop{\mathrm{E-rad}}(J) = K[\bar x]^E.$ We use the technique introduced in [@ideali] to construct such example. Let be $I = (xy, E(x) + 1, E(y) + 1)$ an ideal of $S = K[x, y, e^{\Bbb Q x}, e^{\Bbb Q y}]$ where $S$ is a subring of $K[\bar x]^E,$ in particular it is a partial $E$-ideal of $S$. $I^E$ is then an E-ideal of $K[\bar x]^E$ which is not contained in any prime E-ideal; indeed, if $I \subseteq P$ where $P$ is E-prime, then $xy \in I \subseteq P.$ Since $P$ is prime, w.l.o.g.$x \in P$, and so $E(x) - 1 \in P$. But $E(x) + 1 \in I \subseteq P$, and thus $2 \in P$, contradiction. # Characterization of radical E-ideals We consider an E-ring $R$ and let be $J$ an E-ideal of $R$. We study prime E-ideals and E-radical ideals, i.e. E-ideals which are equal to their E-radical. We characterize $\mathop{\mathrm{E-rad}}(J)$ using the following theory.\ Let be $\mathcal{L} = \{+, -, \cdot, e^{x}, 0, 1\} \cup \{ V \}$ where $V$ is a unary predicate.\ We recall the following definition: **Definition 18**. Given $p_1, \ldots, p_k, q$ $\mathcal{L}$-terms, we define the associated strict Horn clause the formulas of the type: $$V(p_1) \wedge \ldots \wedge V(p_k) \rightarrow V(q).$$ We denote by $H$ the set of all $\mathcal{L}$-strict Horn clause as above. **Examples 19**. $V(x^2) \rightarrow V(x)$ is in $H$. Also $V(x) \wedge V(y) \rightarrow V(x + y) ,$ $V(x) \rightarrow V(e^x - 1)$ and $V(0)$ are in $H$. In order to lighten the notation, we write $p_1 \wedge \ldots \wedge p_k \rightarrow q$ in place of $V(p_1) \wedge \ldots \wedge V(p_k) \rightarrow V(q)$. We consider the following theories:\ $$\begin{aligned} T_{1} &:= \{ \mbox{ Horn clause } \alpha : \mbox{ for any E-ring R and E-ideal J, } (R, J) \models \alpha \}\\ T_2 &:= \{ \mbox{ Horn clause } \alpha : \mbox{ for any E-ring R and prime E-ideal J, } (R, J) \models \alpha \}. \end{aligned}$$ Clearly, $T_{1}$ is generated by the following Horn clauses: $$0, \quad x \wedge y \rightarrow x-y, \quad x \rightarrow xy, \quad x \rightarrow e^{x}-1.$$ Our aim is to give an explicit description of $T_{2}$ and relate it to the E-radical. In $T_2$ we have, besides the clauses in $T_{1}$, also others; for instance, the following clauses are in $T_{1}$: $x^n \rightarrow x$, $(xy \wedge e^{x} + 1) \rightarrow y$. Let $T$ be a set of Horn clauses. Let $\mathcal M(T)$ be following family of subsets of $R$: $$\mathcal M(T) := \{J \subseteq R: (R, J) \models T\} %\qquad \text{or} \qquad \mathcal F = \set{J \subseteq R: (R, J) \models T_{0}}.$$ **Remark 20**. $\mathcal M(T)$ is closed under arbitrary intersections and under increasing unions. Thus, we can consider the "radical" operator associated to $T$. We will let $X$ vary among subsets of $R$. We define $$\mathop{\mathrm{T-rad}}(X) \coloneqq \bigcap \mathcal M(T).$$ We have that $\mathop{\mathrm{T-rad}}(X)$ is the smallest subset of $R$ containing $X$ and such that $(R, \mathop{\mathrm{T-rad}}(X) ) \models T$. In particular, $(R, X) \models T$ iff $X = \mathop{\mathrm{T-rad}}(X)$. **Example 21**. $\mathop{\mathrm{T_1-rad}}(X) = (X)^{E}$, the E-ideal generated by $X$. **Example 22**. Let $T_{0} := \{0, x \wedge y \rightarrow x - y, x \rightarrow xy\}$. Then, $\mathcal M(T_{0})$ is the family of pairs $(R,J)$ with $J$ ideal of $R$. We have that $\operatorname{{T_{0}}-rad}(X)$ is the ideal generated by $X$. Let $T_{3} := \{0, x \wedge y \rightarrow x - y, x \rightarrow xy, x^{2} \rightarrow x\}$. Then, $\mathcal M(T_{3})$ is the family of pairs $(R,J)$ with $J$ radical ideal of $R$. We have that $\operatorname{{T_{3}}-rad}(X)$ is the radical of the ideal generated by $X$. We can build $\mathop{\mathrm{T-rad}}(J)$ in a "constructive" way: i.e., we have a description of all elements of $\mathop{\mathrm{T-rad}}(J)$. **Definition 23**. Given a family $\mathcal F$ of $\mathcal{L}$-structures, we denote its theory by $$Th(\mathcal F) := \{\alpha \in H: \forall M \in \mathcal F\ M \models \alpha \}.$$ The deductive closure of $T$ is $\overline T:= Th(\mathcal M(T))$. We say that $T$ clauses is deductively closed if $T = Th(\mathcal F)$ for some family $\mathcal F$, or equivalently if $T = \overline T$. An axiomatization of $T$ is a set of Horn clauses $S$ such that $\overline S = \overline T$. **Remark 24**. $$\mathop{\mathrm{T-rad}}(X) = \{q(\bar c): p_{1}(\bar x) \wedge \dots \wedge p_{k}(\bar x) \rightarrow q(\bar x) \in \overline T, \bar c\in R^{< \omega}, p_i(\bar c) \in X, i = 1, \dotsc, k\}.$$ *Proof.* Let us denote $Y := \{q(\bar c): p_{1}(\bar x) \wedge \dots \wedge p_{k}(\bar x) \rightarrow q(\bar x) \in \overline T, \bar c\in R^{< \omega}, p_i(\bar c) \in X, i = 1, \dotsc, k\}$. It is clear that $X \subseteq Y \subseteq \mathop{\mathrm{T-rad}}(X)$. We want to show that $\mathop{\mathrm{T-rad}}(X) \subseteq Y$. It suffices to show that $Y \in \mathcal M(T)$. Let $p_{1}(\bar x) \wedge \dots \wedge p_{k}(\bar x) \rightarrow q(\bar x) \in T$ and $\bar c\in R^{< \omega}$ such that $p_{i}(\bar c) \in Y$, $i = 1, \dotsc, k$. It suffices to show the following: *Claim 1*. $q(\bar c) \in Y$. For simplicity of notation, we assume that $k = 1$ and $p := p_{1}$. Since $p(\bar c) \in Y$, by definition of $Y$, there exist $\beta := r_{1}(\bar x, \bar x') \wedge \dots \wedge r_{\ell}(\bar x, \bar x') \rightarrow p(\bar x) \in T$ and $\bar c' \in R^{< \omega}$ such that $r_{j}(\bar c,\bar c') \in X$, $i = 1, \dotsc, \ell$. Notice that every $M \in \mathcal M(T)$ satisfies $$\beta := \bigwedge_{\ell} r_{\ell}(\bar x, \bar x') \rightarrow q(\bar x).$$ Therefore, $\beta \in \bar T$, and therefore, by definition, $q(\bar c) \in Y$. ◻ **Corollary 25**. *For every $b \in R$, $$\mathop{\mathrm{T-rad}}(Xb) = \{q(\bar c): p_{1}(\bar x) \wedge \dots \wedge p_{k}(\bar x) \rightarrow q(\bar x) \in \overline T, \bar c\in R^{< \omega}, p_i(\bar c) \in X \vee p_{i}(\bar c) = b, i = 1, \dotsc, k\}.$$* The following theorem gives more explicit description of $\mathop{\mathrm{T_2-rad}}(X)$. **Theorem 26**. *For every $n \in N$, we define the operator $\sqrt[n]{\vphantom{x} }$ on subsets of $R$ inductively in the following way:\ $\sqrt[0] X = (X)^E;$\ $\sqrt[1] X = \sqrt[0] { \{ a \in R: %\mbox{ there exist } \exists b_1, b_2 \in R : b_1 \cdot b_2 \in \sqrt[0] X \wedge a \in \sqrt[0]{X b_1} \cap \sqrt[0]{X b_2} \} };$\ $\sqrt[n+1] X = \sqrt[n] { \{ a : %\mbox{ there exists } \exists b_1, b_2 : b_1 \cdot b_2 \in \sqrt[n] X \wedge a \in \sqrt[n] {X b_1} \cap \sqrt[n]{X b_2} \} }.$\ Define $\sqrt[E] X := \bigcup_{n \in \mathbb{N}} \sqrt[n] X$* *Then, $\mathop{\mathrm{E-rad}}X = \mathop{\mathrm{T_2-rad}}(X) = \sqrt[E] X$.* We first need some results. **Lemma 27**. *Let $J \subseteq R$. Let $b_{1} \cdot b_{2} \in J$. Assume that $a \in \mathop{\mathrm{T_2-rad}}(J b_{1}) \cap \mathop{\mathrm{T_2-rad}}(J b_{2})$. Then, $a \in \mathop{\mathrm{T_2-rad}}(J)$.* *Proof.* If $J$ were a prime E-ideal, the result would be clear. In general, there exist Horn clauses $p_{i,1}(\bar x) \wedge \dots \wedge p_{i,k}(\bar x) \wedge z \rightarrow q_{i}(z,\bar x) \in I(T)$, $i = 1,2$, and $\bar c\in R^{< \omega}$, such that $$a = q_{i}(b_{i},\bar c), \quad p_{i,j}(\bar c) \in J, \quad i = 1,2, \quad j = 1, \dotsc, k.$$ Moreover, the Horn clause $$\alpha(w, z_{1}, z_{2}, \bar x) := z_{1} \cdot z_{2} \wedge w - q_{1}(z_{1}, \bar x) \wedge w - q_{2}(z_{2},\bar x) \wedge \bigwedge_{i,k} p_{i,k}(\bar x) \rightarrow w$$ is in $T_{0}$ (since it is satisfied by any prime E-ideal). Thus, $(R, \mathop{\mathrm{T_2-rad}}(J)) \models \alpha$ and the conclusion follows by considering $\alpha(a, b_{1}, b_{2}, \bar c)$. ◻ It is therefore that $\mathop{\mathrm{E-rad}}X \supseteq \mathop{\mathrm{T_2-rad}}(X) \supseteq \sqrt[E] X$. Thus, it suffices to show that $\mathop{\mathrm{E-rad}}X \subseteq \sqrt[E] X$, or, equivalently, that for every $a \in R \setminus \sqrt[E]X$, we have $a \notin \mathop{\mathrm{E-rad}}X$. We need a further result. **Lemma 28**. *Let $a \in R$ and $P$ be an E-ideal of $R$ maximal among the E-ideals $J$of $R$ not containing $a$ and such that $\sqrt[E] J = J$. Then, $P$ is prime.* *Proof.* Suppose $b_1 \cdot b_2 \in P$ and by contradiction we assume that $b_1, b_2 \not \in P$. We define $Q_1 = \sqrt[E]{P b_1}$ and $Q_2 = \sqrt[E]{P b_2}$. We have that $Q_{1},Q_{2} \in \mathcal M(T_{0})$. Notice that $Q_1, Q_2 \supset P$, and therefore the maximality of $P$ implies that $a \in Q_{1}$ and $a \in Q_{2}$. Moreover, $b_1 \cdot b_2 \in P$; since $\sqrt[E] P = P$, we have that $a \in \mathop{\mathrm{T_2-rad}}(P ) = P$, contradiction. ◻ We can now conclude the proof of Thm. [Theorem 26](#thm:Erad-up){reference-type="ref" reference="thm:Erad-up"}. Let $a \notin \sqrt[E]X$. By Lemma [Lemma 28](#lem:Trad-max){reference-type="ref" reference="lem:Trad-max"}, there exists a prime E-ideal $P$ containing $X$ such that $a \notin P$. Thus, $a \notin \mathop{\mathrm{E-rad}}X$, and we are done. 0◻ **Corollary 29**. *$J$ is a $E$-radical iff, for every $a, b_{1}, b_{2} \in R$ $$b_{1} \cdot b_{2} \in J \wedge a \in \sqrt[E]{J b_{1}} \cap \sqrt[E]{J b_{1}} \rightarrow a \in J.$$* From the above Corollary we can extract a recursive axiomatization of $T_{0}$, using Thm. [Theorem 26](#thm:Erad-up){reference-type="ref" reference="thm:Erad-up"} to characterize $\sqrt[E]{J b_{i}}$. We can interpret the discussion above in terms of quasi-varieties. **Definition 30**. An E-ring is E-reduced if $\mathop{\mathrm{E-rad}}(0) = (0)$. By Thm. [Theorem 26](#thm:Erad-up){reference-type="ref" reference="thm:Erad-up"}, the class E-red of E-reduced E-rings is a quasi-variety: i.e., it can be axiomatized via Horn formulae in the language $$(=, + , - ,\cdot, e, 0, 1)$$ (we replace the condition $t \in (0)$ with the condition $t = 0$). By [@Burris-Sankappanavar]\*Thm. 2.25, E-red is closed under isomorphisms, taking substructures, reduced product.\ It is not difficult to see directly that E-red is closed under isomorphisms, substructures, direct products, and ultraproducts: thus, by [@Burris-Sankappanavar]\*Thm. 2.25, E-red is a quasi-variety. From this it is easy to deduce that $\mathop{\mathrm{E-rad}}= \mathop{\mathrm{T_2-rad}}$. With the proof we gave of Thm. [Theorem 26](#thm:Erad-up){reference-type="ref" reference="thm:Erad-up"} we obtained the extra information that $\mathop{\mathrm{E-rad}}= \sqrt[E]{\vphantom{x}}$, and Corollary [Corollary 29](#cor:Erad){reference-type="ref" reference="cor:Erad"}, with the recursive axiomatization of $T_{0}$. 99 S.Burris, H. P. Sankappanavar: *A Course in Universal Algebra*, The Millennium Edition. S. Burris and H.P. Sankappanavar <https://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html> N. Bourbaki: *Elements of Mathematics- Commutative Algebra: Chapters 1-7*, Springer. P. D'Aquino, A. Fornasiero and G. Terzo: *E-ideals in exponential polynomial ring*, Submitted. P. D'Aquino, A. Macintyre and G. Terzo: *Schanuel Nullstellensatz for Zilber fields*, Fundamenta Mathematicae 207, 123-143 (2010). P. D'Aquino, A. Macintyre and G. Terzo: *From Schanuel Conjecture to Shapiro Conjecture*, Commentarii Mathematicae Elvetici 89 (3), 597-616 (2014). L. van den Dries: *Exponential rings, exponential polynomials and exponential functions*, Pacific Journal of Mathematics, 113, (1), 51-66 (1984). G. R. Everest and A. J. van der Poorten: *Factorisation in the ring of exponential polynomials*, of Proceedings of the American Mathematical Society, 125, (5), (1997), 1293-1298. Henson, Rubel and Singer: *Algebraic Properties of the Ring of General Exponential Polynomials* Complex Variables, 13, 1-20 (1989). A. Macintyre: *Exponential Algebra* In: Ursini, A., et al., eds. Logic and Algebra. Proceedings of the international conference dedicated to the memory of Roberto Magari. Lect. Notes Pure Appl. Math 180. M. Dekker, pp. 191--210. K. Manders: *On algebraic geometry over rings with exponentiation* Z. Math. Logik Grundlag. Math. 33 (1987), no. 4, 289-292. D. Marker: *A remark on Zilber's pseudoexponentiation*, The Journal of Symbolic Logic, 71, (3), 791-798 (2006). H. Matsumura: *Commutative Algebra*, W. A. Benjamin, Inc., New York, 1970. F. Point and N. Regnault *Exponential ideals and a Nullstellensatz*, arXiv:2004.10444, 22 April 2020 J.F. Ritt: *A factorization theorem of functions $\sum_{i=1}^na_ie^{\alpha_iz}$*, Transactions of American Mathematical Society 29, (1927), 584-596. G. Terzo: *Some consequences of Schanuel's conjecture in exponential rings.* Communications in Algebra 36, 3, (2008), 1171-1189.

arxiv_math

--- abstract: | We introduce an expansion scheme in reproducing kernel Hilbert spaces, which as a special case covers the celebrated Blaschke unwinding series expansion for analytic functions. The expansion scheme is further generalized to cover Hardy spaces $H^p$, $1<p<\infty$, viewed as Banach spaces of analytic functions with bounded evaluation functionals. In this setting a dichotomy is more transparent: depending on the multipliers used, the expansion of $f \in H^p$ converges either to $f$ in $H^p$-norm or to its projection onto a model space generated by the corresponding multipliers. Some explicit instances of the general expansion scheme, which are not covered by the previously known methods, are also discussed. address: - | Département de Mathématiques et de Statistique\ Université Laval\ Québec\ QC\ G1V 0A6\ Canada - | Département de Mathématiques et de Statistique\ Université Laval\ Québec\ QC\ G1V 0A6\ Canada author: - Javad Mashreghi - William Verreault\* bibliography: - OSC-references.bib title: Nonlinear expansions in reproducing kernel Hilbert spaces --- # Introduction An entire function $f$ has the Taylor series expansion $$\label{E:taylor-expansion} f(z) = \sum_{n=0}^{\infty} a_n z^n,$$ which converges for all values of $z \in \mathbb{C}$. We usually express $a_n$ either by the Brook Taylor formula (1715) $$a_n = \frac{f^{(n)}(0)}{n!},$$ or by the Cauchy integral formula (1875) $$a_n = \frac{1}{2\pi i} \int_{\Gamma} \frac{f(z)}{z^{n+1}} \, dz.$$ In the mid-1990s, R. Coifman had a revolutionary interpretation of the expansion [\[E:taylor-expansion\]](#E:taylor-expansion){reference-type="eqref" reference="E:taylor-expansion"}. The idea was developed in further depth in the doctoral thesis of M. Nahon [@MR2700759]. Briefly speaking, their strategy is as follows. We may write $f(z)$ as $$\label{E:taylor-step1} f(z) = f(0) + \big(f(z)-f(0)\big).$$ The first term on the right side is the constant $f(0)=a_0$. The second term is a function that vanishes at the origin. Hence, we can write $f(z)-f(0) = z f_1(z)$, where $f_1$ is another holomorphic function. Thus, we have extracted a zero of $f(z)-f(0)$ as the multiplicative factor $z$, and then we deal with the new function $f_1$ by iterating the above procedure. More explicitly, we write $$\label{E:taylor-step2} f_1(z) = f_1(0) + \big(f_1(z)-f_1(0)\big),$$ and $f_1(z)-f_1(0) = z f_2(z)$ for yet another holomorphic function $f_2$. Plugging back [\[E:taylor-step2\]](#E:taylor-step2){reference-type="eqref" reference="E:taylor-step2"} into [\[E:taylor-step1\]](#E:taylor-step1){reference-type="eqref" reference="E:taylor-step1"}, we see that $$f(z) = f(0) + f_1(0)z+z^2f_2(z),$$ which at the same times shows $f_1(0)=a_1$. If we continue this procedure infinitely many times, we obtain $$\label{E:taylor-step3} f(z) = f(0) + f_1(0)z+f_2(0)z^2+\cdots,$$ which is a new interpretation of the Taylor series expansion [\[E:taylor-expansion\]](#E:taylor-expansion){reference-type="eqref" reference="E:taylor-expansion"}. The novel vision of the Coifman school was to extract *more* zeros in each step via finite Blaschke products. In the above procedure, instead of factoring out the zero at the origin, we may factor out all the zeros in the open unit disc $\mathbb{D}$. Since $f$ is entire, it has finitely many zeros in $\mathbb{D}$, and thus an appropriate apparatus to extract zeros in $\mathbb{D}$ are finite Blaschke products. Given $\alpha_1,\dots,\alpha_N \in \mathbb{D}$, repetition allowed, the rational function $$B(z) = \prod_{n=1}^{N} \frac{\alpha_n-z}{1-\bar{\alpha}_nz}$$ is called a *finite Blaschke product*. For a treatment of this topic, we refer to the monographs [@MR3793610; @MR2986324]. After the initial step [\[E:taylor-step1\]](#E:taylor-step1){reference-type="eqref" reference="E:taylor-step1"}, we factor $f(z)-f(0)$ as $$f(z)-f(0) = B_1(z) f_1(z),$$ where $f_1$ is analytic on $\overline{\mathbb{D}}$ and has no roots in $\mathbb{D}$. All roots of $f(z)-f(0)$ in $\mathbb{D}$, including the root at the origin, and counting their multiplicities, are gathered in $B_1$. Hence, we can write $$f(z) = f(0) + B_1(z) f_1(z).$$ Since $f_1$ is a holomorphic function on $\overline{\mathbb{D}}$, we can iterate the above procedure with $f_1(z) - f_1(0)$. The outcome, after the second step, is $$f(z) = f(0) + f_1(0)B_1(z) + B_1(z)B_2(z)f_2(z).$$ If we reiterate infinitely many times, we obtain the expansion $$\label{coifman-expansion} f(z) = f(0) + f_1(0)B_1(z) + f_2(0)B_1(z)B_2(z) + \cdots,$$ which is known as the *Blaschke unwinding series expansion* of $f$. The convergence of the Blaschke unwinding series was a major question for a long period. In fact, some of Nahon's numerical experiments suggested that, for functions in $H^2$, the unwinding expansion converges to the original function in mean. However, this fact was only firmly proved later by T. Qian [@MR2662313]. This question was again studied by Coifman and Steinerberger in [@MR3685990], where they obtained more general results for convergence in weighted subspaces of $H^2$. Eventually, Coifman and Peyrière [@MR3953482] proved the convergence of the inner-outer unwinding series for functions in the Hardy spaces $H^p$. In parallel and independently, Qian and collaborators came up with a similar process to obtain a nonlinear phase unwinding of holomorphic functions, which they called *adaptive Fourier decomposition* [@MR2662313; @MR2776445; @MR2907888]. Variants of this type of unwinding have been further investigated in several papers, with a strong emphasis on the applications. Their idea is to start with the Takenaka--Malmquist--Walsh basis. Then, to decompose a function $f\in H^2$ with respect to this basis, they use an adaptive algorithm akin to a greedy algorithm which relies on the existence of a point in $\mathbb{D}$ that minimizes the distance from $f$ to the partial sums of its unwinding series. In this work, we provide a general expansion scheme, which can be viewed as a generalization of the Blaschke unwinding series expansion [\[coifman-expansion\]](#coifman-expansion){reference-type="eqref" reference="coifman-expansion"}. We perform this in two contexts. First, in Section [2](#S:expansion-rkhs){reference-type="ref" reference="S:expansion-rkhs"}, we present a scheme followed by two convergence results in Theorems [Theorem 1](#T:conv-1-rkhs){reference-type="ref" reference="T:conv-1-rkhs"} and [Theorem 1](#T:conv-2-rkhs){reference-type="ref" reference="T:conv-2-rkhs"}, in the general setting of reproducing kernel Hilbert spaces (RKHS). To some extent, these results can be extended to reproducing kernel Banach spaces. However, since in this note, our goal is to extend the above approach to Hardy spaces, we devoted the subsequent section to a more detailed study of $H^p$ spaces. More explicitly, the construction is as follows. We consider a sequence $(b_n)_{n \geq 1}$ of elements in the closed unit ball of $H^\infty$ (not necessarily Blaschke products, or even inner functions). Then using the co-analytic Toeplitz operator $T_{\bar{b}_k}$ and the closely related operators $Q_{b_k}$, we introduce the expansion $$f = Q_{b_1}f + b_1 \cdot Q_{b_2}T_{\bar{b}_1}f + b_1b_2 \cdot Q_{b_3}T_{\bar{b}_1\bar{b}_2}f + b_1b_2b_3 \cdot Q_{b_4}T_{\bar{b}_1\bar{b}_2\bar{b}_3}f +\cdots$$ for an arbitrary element $f \in H^p$. The procedure is explained in Section [3](#S:expansion){reference-type="ref" reference="S:expansion"} and the convergence problem is addressed in Theorems [Theorem 1](#T:conv-1){reference-type="ref" reference="T:conv-1"} and [Theorem 1](#T:conv-2){reference-type="ref" reference="T:conv-2"}. In Section [4](#S:taylor){reference-type="ref" reference="S:taylor"}, we show that as a very special case, Theorem [Theorem 1](#T:conv-1){reference-type="ref" reference="T:conv-1"} leads to the Taylor series expansion. In Section [5](#Blaschke-section){reference-type="ref" reference="Blaschke-section"}, we show that when the $b_k$s are Blaschke factors, the general expansion scheme gives the previously known expansions which were described above, namely the Blaschke unwinding and the adaptive Fourier decomposition. In Section [6](#S:outer-expansion){reference-type="ref" reference="S:outer-expansion"}, as a prototypical example, we study a special expansion created by outer functions, which was not possible with previously known expansions. # The nonlinear expansion in RKHS {#S:expansion-rkhs} Let $\mathcal{H}$ be an RKHS on $X$ with the multiplier algebra $\mathcal{M}(\mathcal{H})$. For a thorough treatment of RKHS, see [@MR3526117]. Let $\phi \in \mathcal{M}(\mathcal{H})$ and define $$\label{E:def-PQ} P_{\phi} := M_{\phi} M_{\phi}^* \quad\mbox{and}\quad Q_{\phi} :=I- M_{\phi} M_{\phi}^*,$$ where $M_\phi(f)=\phi f$ is the multiplication operator on $\mathcal{H}$. While $P_{\phi}$ and $Q_{\phi}$ are certainly bounded operators, in general, they are not necessarily projections (idempotent). As a matter of fact, it is easy to see that they are projections if and only if $$M_{\phi}^{*}(\phi k_x) = k_x, \qquad x \in X,$$ where $k_x$ is the reproducing kernel of $\mathcal{H}$. The scheme is as follows. Trivially $Q_{\phi} + P_{\phi} = I$. Hence, we have $$g = Q_{\phi}g + P_{\phi}g, \qquad g \in \mathcal{H},$$ or equivalently $$\label{E:step0-rkhs} g = Q_{\phi}g + \phi M_{\phi}^{*}g, \qquad g \in \mathcal{H}.$$ This elementary observation is actually the building block in generalizing the Blaschke unwinding series expansion [\[coifman-expansion\]](#coifman-expansion){reference-type="eqref" reference="coifman-expansion"}. To continue, let $(\phi_n)_{n \geq 1}$ be a sequence of elements in the closed unit ball of $\mathcal{M}(\mathcal{H})$. Fix $f \in \mathcal{H}$. Then, by [\[E:step0-rkhs\]](#E:step0-rkhs){reference-type="eqref" reference="E:step0-rkhs"} with $\phi=\phi_1$, $$\label{E:step1-rkhs} f = Q_{\phi_1}f + \phi_1 M_{\phi_1}^{*}f.$$ Then we apply [\[E:step0-rkhs\]](#E:step0-rkhs){reference-type="eqref" reference="E:step0-rkhs"} with $\phi=\phi_2$ to $g=M_{\phi_1}^{*}f$ to obtain $$\label{E:step2-rkhs} M_{\phi_1}^{*}f = Q_{\phi_2}M_{\phi_1}^{*}f + \phi_2 M_{\phi_2}^{*}M_{\phi_1}^{*}f.$$ If we plug [\[E:step2-rkhs\]](#E:step2-rkhs){reference-type="eqref" reference="E:step2-rkhs"} back into [\[E:step1-rkhs\]](#E:step1-rkhs){reference-type="eqref" reference="E:step1-rkhs"}, it gives $$\label{E:step3-rkhs} f = Q_{\phi_1}f + \phi_1 Q_{\phi_2}M_{\phi_1}^{*}f + \phi_1\phi_2 M_{\phi_1\phi_2}^{*}f.$$ Note that we implicitly used $M_{\phi_2}^{*}M_{\phi_1}^{*} = M_{\phi_1\phi_2}^{*}$. We continue and apply again [\[E:step0-rkhs\]](#E:step0-rkhs){reference-type="eqref" reference="E:step0-rkhs"} with $\phi=\phi_3$ and $g=M_{\phi_1\phi_2}^{*}f$ and plug it back into [\[E:step3-rkhs\]](#E:step3-rkhs){reference-type="eqref" reference="E:step3-rkhs"} to obtain $$\label{E:step4-rkhs} f = Q_{\phi_1}f + \phi_1 Q_{\phi_2}M_{\phi_1}^{*}f + \phi_1\phi_2 Q_{\phi_3}M_{\phi_1\phi_2}^{*}f + \phi_1\phi_2\phi_3 M_{\phi_1\phi_2\phi_3}^{*}f.$$ By induction, we can continue this procedure as many times as we wish. The general convergence theorem is as follows. **Theorem 1**. *Let $\mathcal{H}$ be an RKHS on $X$ with the multiplier algebra $\mathcal{M}(\mathcal{H})$. Let $(\phi_n)_{n \geq 1}$ be a sequence of elements in the closed unit ball of $\mathcal{M}(\mathcal{H})$. Write $\Phi_0=1$ and $$\Phi_n := \phi_1\phi_2\cdots \phi_n, \qquad n \geq 1.$$ Assume that $$\lim_{n \to \infty} \Phi_n(x) = 0, \qquad x \in X.$$ Then, for each $f \in \mathcal{H}$, $$f = \sum_{n=1}^{\infty} \Phi_{n-1} \cdot Q_{\phi_n} M_{\Phi_{n-1}}^{*}f,$$ where the series converges in $\mathcal{H}$.* *Proof.* By induction, the general formula for [\[E:step3-rkhs\]](#E:step3-rkhs){reference-type="eqref" reference="E:step3-rkhs"} and [\[E:step4-rkhs\]](#E:step4-rkhs){reference-type="eqref" reference="E:step4-rkhs"} is $$f = \sum_{n=1}^{N} \Phi_{n-1} Q_{\phi_n}M_{\Phi_{n-1}}^{*}f + \Phi_{N} M_{\Phi_{N}}^{*}f.$$ Since $$\|\Phi_{N} M_{\Phi_{N}}^{*} f\|_{\mathcal{H}} \leq \|\Phi_{N}\|_{\mathcal{M}(\mathcal{H})} \|M_{\Phi_{N}}^{*} f\|_{\mathcal{H}} \leq \|M_{\Phi_{N}}^{*} f\|_{\mathcal{H}},$$ it is enough to show that $$\|M_{\Phi_{N}}^{*} f\|_{\mathcal{H}} \to 0$$ as $N \to \infty$. Recall that $$M_{\Phi_{N}}^{*} k_{x} = \overline{\Phi_N(x)}\, k_x, \qquad x \in \mathcal{H}.$$ Hence, $$\|M_{\Phi_{N}}^{*} k_{x}\|_{\mathcal{H}} = |\Phi_N(x)| \, \|k_{x}\|_{\mathcal{H}}.$$ Therefore, according to our main assumption, $$\label{E:lim-kx=0} \lim_{N \to \infty} \|M_{\Phi_{N}}^{*} k_{x}\|_{\mathcal{H}} =0, \qquad x \in X.$$ For a general $f \in \mathcal{H}$, we use two properties of reproducing kernels and multipliers of $\mathcal{H}$. First, the linear span of the kernel functions is dense in $\mathcal{H}$. Second, the operators $M_{\Phi_{N}}^{*}$ are uniformly bounded. More explicitly, given $f \in \mathcal{H}$ and $\varepsilon>0$, there are constants $\alpha_1,\dots,\alpha_m \in \mathbb{C}$ and points $x_1,\dots,x_m \in X$ such that $$\|f-(\alpha_1 k_{x_1} + \cdots + \alpha_m k_{x_m})\|_{\mathcal{H}} < \varepsilon.$$ Hence, $$\begin{aligned} \|M_{\Phi_{N}}^{*}f\|_{\mathcal{H}} &\leq& \|M_{\Phi_{N}}^{*}[f-(\alpha_1 k_{x_1} + \cdots + \alpha_m k_{x_m})]\|_{\mathcal{H}}\\ &+& \|M_{\Phi_{N}}^{*}(\alpha_1 k_{x_1} + \cdots + \alpha_m k_{x_m})\|_{\mathcal{H}}\\ % &\leq& \|\Phi_{N}\|_{\mathcal{M}(\mathcal{H})} \|f-(\alpha_1 k_{x_1} + \cdots + \alpha_m k_{x_m})\|_{\mathcal{H}}\\ &+& |\alpha_1| \, \|M_{\Phi_{N}}^{*}k_{x_1}\|_{\mathcal{H}} + \cdots + |\alpha_m| \, \|M_{\Phi_{N}}^{*}k_{x_m}\|_{\mathcal{H}}\\ % &\leq& \varepsilon+ |\alpha_1| \, \|M_{\Phi_{N}}^{*}k_{x_1}\|_{\mathcal{H}} + \cdots + |\alpha_m| \, \|M_{\Phi_{N}}^{*}k_{x_m}\|_{\mathcal{H}}. %\end{aligned}$$ By [\[E:lim-kx=0\]](#E:lim-kx=0){reference-type="eqref" reference="E:lim-kx=0"}, we see that $$\limsup_{N \to \infty} \|M_{\Phi_{N}}^{*}f\|_{\mathcal{H}} \leq \varepsilon.$$ Since $\varepsilon>0$ is arbitrary, the result follows. ◻ In Theorem [Theorem 1](#T:conv-1-rkhs){reference-type="ref" reference="T:conv-1-rkhs"}, we assume that the multipliers are arranged so that $$\lim_{n \to \infty} \Phi_n(x) = 0, \qquad x \in X.$$ In general, this is not necessarily the case. In fact, quite often we end up with $$\lim_{n \to \infty} \Phi_n(x) = \Phi(x), \qquad x \in X,$$ where $\Phi$ is a non-zero multiplier of $\mathcal{H}$. In this case, an extra term appears which is linked to the model spaces (see next section). In the general setting, the result is as follows. **Theorem 1**. *Let $\mathcal{H}$ be an RKHS on $X$ with the multiplier algebra $\mathcal{M}(\mathcal{H})$. Let $(\phi_n)_{n \geq 1}$ be a sequence of elements in the closed unit ball of $\mathcal{M}(\mathcal{H})$. Write $\Phi_0=1$ and $$\Phi_n := \phi_1\phi_2\cdots \phi_n, \qquad n \geq 1.$$ Assume that there is a multiplier $\Phi \in \mathcal{M}(\mathcal{H})$ such that $$\lim_{n \to \infty} \Phi_n(x) = \Phi(x), \qquad x \in X.$$ Then, for each $f \in \mathcal{H}$, $$f = \Phi M_{\Phi}^{*}f + \sum_{n=1}^{\infty} \Phi_{n-1}\cdot Q_{\phi_n} M_{\Phi_{n-1}}^{*}f,$$ where the series converges in $\mathcal{H}$.* # The expansion in Hardy spaces {#S:expansion} The Hardy spaces $H^p$, $1<p<\infty$, $p \neq 2$, are not Hilbert spaces. However, with appropriate adjustments, the expansions discussed in Theorems [Theorem 1](#T:conv-1-rkhs){reference-type="ref" reference="T:conv-1-rkhs"} and [Theorem 1](#T:conv-2-rkhs){reference-type="ref" reference="T:conv-2-rkhs"} can be extended to this setting. For this purpose, some facts from the theory of Toeplitz operators are needed. For basics of Toeplitz operators, see [@MR3526203 Ch. 4], and for the theory of Hardy spaces, we refer to [@Duren; @MR2500010]. Let $\varphi \in L^\infty(\mathbb{T})$, and let $1<p<\infty$. Then, the Toeplitz operator $T_{\varphi}: H^p \to H^p$ is defined by $$T_{\varphi}f = P_+(\varphi f), \qquad f \in H^p,$$ where $P_+$ is the M. Riesz projection of $L^p$ onto $H^p$. As a consequence of a celebrated result of M. Riesz on the boundedness of the Hilbert transform, $T_{\varphi}$ is also bounded and moreover $$\label{E:toeplitz-0} \|T_{\varphi}\|_{H^p \to H^p} \leq c_p \|\varphi\|_{H^\infty}.$$ It is well-known that if $\varphi$ and $\psi$ are in $H^\infty$, then $$\label{E:toeplitz-1} T_{\varphi}f = \varphi f, \qquad f \in H^p,$$ and $$\label{E:toeplitz-2} T_{\bar{\varphi}} T_{\bar{\psi}} = T_{\bar{\psi}} T_{\bar{\varphi}} = T_{\bar{\varphi}\bar{\psi}}.$$ Let $b$ be an element of the closed unit ball of $H^\infty$. As in [\[E:def-PQ\]](#E:def-PQ){reference-type="eqref" reference="E:def-PQ"}, let $$P_b := T_b T_{\bar{b}} \quad\mbox{and}\quad Q_b:=I-P_b.$$ Here, the bounded operators $P_b$ and $Q_b$ are projections if and only if $b$ is an inner function. We may trivially write $$f = Q_bf + P_bf, \qquad f \in H^p,$$ or equivalently, by [\[E:toeplitz-1\]](#E:toeplitz-1){reference-type="eqref" reference="E:toeplitz-1"}, $$\label{E:step0} f = Q_bf + b T_{\bar{b}}f, \qquad f \in H^p.$$ As before, this identity is actually the basic step in generalizing the expansion scheme to Hardy spaces. Let $(b_n)_{n \geq 1}$ be a sequence of elements in the closed unit ball of $H^\infty$. Fix $f \in H^p$. Then, by [\[E:step0\]](#E:step0){reference-type="eqref" reference="E:step0"} with $b=b_1$, $$\label{E:step1} f = Q_{b_1}f + b_1 T_{\bar{b}_1}f.$$ Then we apply [\[E:step0\]](#E:step0){reference-type="eqref" reference="E:step0"} with $b=b_2$ to $T_{\bar{b}_1}f$ to obtain $$\label{E:step2} T_{\bar{b}_1}f = Q_{b_2}T_{\bar{b}_1}f + b_2 T_{\bar{b}_2}T_{\bar{b}_1}f.$$ If we plug back [\[E:step2\]](#E:step2){reference-type="eqref" reference="E:step2"} into [\[E:step1\]](#E:step1){reference-type="eqref" reference="E:step1"} and also use [\[E:toeplitz-2\]](#E:toeplitz-2){reference-type="eqref" reference="E:toeplitz-2"}, we obtain $$\label{E:step3} f = Q_{b_1}f + b_1 Q_{b_2}T_{\bar{b}_1}f + b_1b_2 T_{\bar{b}_1\bar{b}_2}f.$$ The general convergence theorem is as follows. **Theorem 1**. *Let $(b_n)_{n \geq 1}$ be a sequence of elements in the closed unit ball of $H^\infty$. Write $B_0=1$ and $$B_n := b_1b_2\cdots b_n, \qquad n \geq 1.$$ Assume that $$\lim_{n \to \infty} B_n(z) = 0, \qquad z \in \mathbb{D}.$$ Then, for each $f \in H^p$, $$f = \sum_{n=1}^{\infty} B_{n-1}\cdot Q_{b_n}T_{\bar{B}_{n-1}}f,$$ where the series converges in $H^p$-norm.* *Proof.* By induction, the general formula for [\[E:step3\]](#E:step3){reference-type="eqref" reference="E:step3"} is $$f = \sum_{n=1}^{N} B_{n-1} Q_{b_n}T_{\bar{B}_{n-1}}f + B_{N} T_{\bar{B}_{N}}f.$$ Since $$\|B_{N} T_{\bar{B}_{N}}f\|_{H^p} \leq \|T_{\bar{B}_{N}}f\|_{H^p},$$ it is enough to show that $$\|T_{\bar{B}_{N}}f\|_{H^p} \to 0$$ as $N \to \infty$. Let $k_\lambda$ denote the Cauchy kernel, i.e., $k_{\lambda}(z) = (1-\bar{\lambda}z)^{-1}$. A celebrated property of conjugate-analytic Toeplitz operators is their unusual abundance of eigenvalues and eigenvectors: $$T_{\bar{\varphi}}k_\lambda = \overline{\varphi(\lambda)}\, k_\lambda, \qquad \lambda \in \mathbb{D}, \, \varphi \in H^\infty.$$ Hence, in light of [\[E:toeplitz-2\]](#E:toeplitz-2){reference-type="eqref" reference="E:toeplitz-2"}, for each $\lambda \in \mathbb{D}$, $$T_{\bar{B}_{N}} k_{\lambda} = \overline{B_N(\lambda)}\, k_\lambda,$$ and thus $$\|T_{\bar{B}_{N}} k_{\lambda}\|_{H^p} = |B_N(\lambda)| \, \|k_{\lambda}\|_{H^p}.$$ Therefore, according to our main assumption, $$\lim_{N \to \infty} \|T_{\bar{B}_{N}} k_{\lambda}\|_{H^p} =0.$$ For a general $f \in H^p$, incidentally we know that the span of the Cauchy kernels is also dense in $H^p$. Therefore, given $f \in H^p$ and $\varepsilon>0$, there are constants $\alpha_1,\dots,\alpha_m \in \mathbb{C}$ and points $\lambda_1,\dots,\lambda_m \in \mathbb{D}$ such that $$\|f-(\alpha_1 k_{\lambda_1} + \cdots + \alpha_m k_{\lambda_m})\|_{H^p} < \varepsilon.$$ Hence, by [\[E:toeplitz-0\]](#E:toeplitz-0){reference-type="eqref" reference="E:toeplitz-0"}, $$\begin{aligned} \|T_{\bar{B}_{N}}f\|_{H^p} &\leq& \|T_{\bar{B}_{N}}[f-(\alpha_1 k_{\lambda_1} + \cdots + \alpha_m k_{\lambda_m})]\|_{H^p}\\ &+& \|T_{\bar{B}_{N}}(\alpha_1 k_{\lambda_1} + \cdots + \alpha_m k_{\lambda_m})\|_{H^p}\\ % &\leq& c_p \|\bar{B}_{N}\|_{H^\infty} \|f-(\alpha_1 k_{\lambda_1} + \cdots + \alpha_m k_{\lambda_m})\|_{H^p}\\ &+& |\alpha_1| \, \|T_{\bar{B}_{N}}k_{\lambda_1}\|_{H^p} + \cdots + |\alpha_m| \, \|T_{\bar{B}_{N}}k_{\lambda_m}\|_{H^p}\\ % &\leq& c_p \varepsilon+ |\alpha_1| \, \|T_{\bar{B}_{N}}k_{\lambda_1}\|_{H^p} + \cdots + |\alpha_m| \, \|T_{\bar{B}_{N}}k_{\lambda_m}\|_{H^p}. %\end{aligned}$$ By the preceding paragraph, $$\limsup_{N \to \infty} \|T_{\bar{B}_{N}}f\|_{H^p} \leq c_p \varepsilon.$$ Since $\varepsilon>0$ is arbitrary, the result follows. ◻ Despite the above proof resembling that of Theorem [Theorem 1](#T:conv-1-rkhs){reference-type="ref" reference="T:conv-1-rkhs"}, for just a harmless multiplicative factor $c_p$ shows up at the end of the proof, it is important to note that it heavily rests upon the M. Riesz theorem on the boundedness of the Hilbert transform. One may naturally wonder what happens if $B_N$ does not converge to zero in the topology of $\mbox{Hol}(\mathbb{D})$. In this case, $$B := \prod_{n=1}^{\infty} b_n$$ is a well-defined, not identically zero, function in the closed unit ball of $H^\infty$ and a minor modification of Theorem [Theorem 1](#T:conv-1){reference-type="ref" reference="T:conv-1"} implies the following result. **Theorem 1**. *Let $(b_n)_{n \geq 1}$ be a sequence of elements in the closed unit ball of $H^\infty$. Write $B_0=1$ and $$B_n := b_1b_2\cdots b_n, \qquad n \geq 1.$$ Assume that $$B := \prod_{n=1}^{\infty} b_n,$$ where the product converges uniformly on compact subsets of $\mathbb{D}$ to the element $B \in H^\infty$. Then, for each $f \in H^p$, $$f = BT_{\bar{B}}f + \sum_{n=1}^{\infty} B_{n-1}\cdot Q_{b_n}T_{\bar{B}_{n-1}}f,$$ where the series converges in $H^p$.* # Taylor series {#S:taylor} If $b(z)=z$, then the Toeplitz operator $T_z$ is the *unilateral forward shift* operator. By the same token, $T_{\bar{z}}$ is the *unilateral backward shift* operator. We denote them respectively by $\mathbf{S}$ and $\mathbf{Z}$. In fact, a rather standard notation for the backward shift is $\mathbf{B}$. However, in this note, in order to avoid the consfusion with Blaschke products, we temporarily use $\mathbf{Z}$ for the backward shift. Note that when we consider them as bounded operators on the Hardy--Hilbert space $H^2$, it is legitimate to write $\mathbf{Z}=\mathbf{S}^*$. However, for other Hardy spaces, this identity is meaningless. Using the notation of Theorem [Theorem 1](#T:conv-1){reference-type="ref" reference="T:conv-1"}, let $$b_n(z)=z, \qquad n\geq 1.$$ Then $$\lim_{n \to \infty} B_n(z) = \lim_{n \to \infty} z^n = 0, \qquad z \in \mathbb{D}.$$ Moreover, $$Q_{b_n} = I-T_{b_n}T_{\bar{b}_n} = I-T_{z}T_{\bar{z}} = I-\mathbf{S}\mathbf{Z},$$ and thus, for $f \in H^p$, $$Q_{b_n}T_{\bar{B}_{n-1}}f = (I-\mathbf{S}\mathbf{Z})\mathbf{Z}^{n-1} f = \frac{f^{(n-1)}(0)}{(n-1)!}, \qquad n \geq 1.$$ Since $B_{n-1}(z) = z^{n-1}$, the expansion $$f = \sum_{n=1}^{\infty} B_{n-1}\cdot Q_{b_n}T_{\bar{B}_{n-1}}f$$ reduces to $$f(z) = \sum_{n=1}^{\infty} \frac{f^{(n-1)}(0)}{(n-1)!} \cdot z^{n-1},$$ which is precisely the Taylor series expansion of $f$. # Blaschke unwinding series {#Blaschke-section} In this section and the next, we study some special cases of the expansion described in Theorems [Theorem 1](#T:conv-1){reference-type="ref" reference="T:conv-1"} and [Theorem 1](#T:conv-2){reference-type="ref" reference="T:conv-2"}. Moreover, the decomposition idea for the case treated in this section is borrowed from [@FMN], where the development is studied in the more general setting of $H^p$ spaces. Here, for simplicity, we just treat the case $p=2$ and provide a sketch of proofs to show that our abstract setting implies the classical Blaschke unwinding series as a special case. Let $\lambda \in \mathbb{D}$ and let $$b_{\lambda}(z) := \frac{\lambda-z}{1-\bar{\lambda}z}, \qquad z \in \mathbb{D}.$$ It is well known that $b_{\lambda}$ is an automorphism of the disc $\mathbb{D}$. In this case, $P_{b_\lambda}$ is the orthogonal projection of $H^2$ onto the Beurling subspace $b_{\lambda} H^2$, and $Q_{b_\lambda}$ is the orthogonal projection of $H^2$ onto the model space $$K_{b_\lambda} = \mathbb{C} k_{\lambda}.$$ In this case, we can provide an explicit formula for $Q_{b_\lambda}$. Given $f \in H^2$, put $$g := f - f(\lambda) \frac{k_{\lambda}}{k_{\lambda}(\lambda)}.$$ Then clearly $g \in H^2$ and $g(\lambda)=0$. Hence, by a theorem of F. Riesz [@MR2500010 Page 167], $g= b_{\lambda}h$ for some $h \in H^2$. Therefore, we can write $$f = \frac{f(\lambda)}{k_{\lambda}(\lambda)} k_{\lambda} + b_{\lambda}h.$$ This is precisely the orthogonal decomposition of $f$ with the first component coming from $K_{b_\lambda}$ and the second from $b_{\lambda} H^2$. According to this decomposition, we conclude that $$\label{E:proj-kb} Q_{b_\lambda}f = \frac{f(\lambda)}{k_{\lambda}(\lambda)} k_{\lambda} = (1-|\lambda|^2)f(\lambda) k_{\lambda}.$$ Now let $(\lambda_n)_{n \geq 1}$ be a sequence in $\mathbb{D}$. There are two possibilities, which are described in the following two corollaries. In each case, we are faced with the Takenaka--Malmquist--Walsh basis [@MR3526203 Ch. 5]. See also [@MR2572653; @MR1568210]. **Corollary 1**. *Let $(\lambda_n)_{n \geq 1}$ be a non-Blaschke sequence in $\mathbb{D}$, i.e., $$\sum_{n=1}^{\infty} (1-|\lambda_n|) = \infty.$$ Let $B_0=1$ and $$B_n(z) = \prod_{k=1}^{n} \frac{\lambda_k-z}{1-\bar{\lambda}_kz}, \qquad n \geq 1.$$ Let $f$ be entire. Then $$\label{E:our-B-rep} f = f(\lambda_1) + \sum_{n=1}^{\infty} \big( (T_{\bar{B}_{n}}f)(\lambda_{n+1}) - \bar{\lambda}_n (T_{\bar{B}_{n-1}}f)(\lambda_n) \big) B_n,$$ where the series converges in $H^2$-norm.* *Proof.* In this case, $$\lim_{n \to \infty} B_n(z) = \lim_{n \to \infty} \prod_{k=1}^{n} \frac{\lambda_k-z}{1-\bar{\lambda}_kz} = 0, \qquad z \in \mathbb{D},$$ and thus Theorem [Theorem 1](#T:conv-1){reference-type="ref" reference="T:conv-1"} applies. By [\[E:proj-kb\]](#E:proj-kb){reference-type="eqref" reference="E:proj-kb"}, $$Q_{b_{\lambda_n}}T_{\bar{B}_{n-1}}f = (1-|\lambda_{n}|^2)(T_{\bar{B}_{n-1}}f)(\lambda_n) k_{\lambda_n}, \qquad n \geq 1.$$ Therefore, each $f \in H^2$ has the decomposition $$\label{E:dec-f-ortho1} f = \sum_{n=1}^{\infty} (1-|\lambda_{n}|^2)(T_{\bar{B}_{n-1}}f)(\lambda_n) B_{n-1}k_{\lambda_n}.$$ Note that the orthonormal basis of $H^2$ in this decomposition is $$(1-|\lambda_{n}|^2)^{1/2}B_{n-1}k_{\lambda_n}, \qquad n \geq 1,$$ and the corresponding Fourier coefficients of $f$ with respect to this basis are $$(1-|\lambda_{n}|^2)^{1/2}(T_{\bar{B}_{n-1}}f)(\lambda_n), \qquad n \geq 1.$$ It is easy to see that $b_{\lambda}$ and $k_{\lambda}$ are related via the linear equation $$(1-|\lambda|^2)k_{\lambda} + \bar{\lambda} b_{\lambda} = 1.$$ If we solve for $k_{\lambda}$ and plug it into [\[E:dec-f-ortho1\]](#E:dec-f-ortho1){reference-type="eqref" reference="E:dec-f-ortho1"} (with $\lambda=\lambda_n$) and recalling that $B_{n-1}b_{\lambda_n} = B_{n}$, we obtain $$f = \sum_{n=1}^{\infty} (T_{\bar{B}_{n-1}}f)(\lambda_n)\left( B_{n-1} - \bar{\lambda}_n B_n\right).$$ The convergence is in $H^2$ and all the coefficients of the $B_k$s are scalars. After a rearrangement, the representation [\[E:our-B-rep\]](#E:our-B-rep){reference-type="eqref" reference="E:our-B-rep"} follows. ◻ A remark is in order concerning the last part of the above proof. Even though it is not visible at first glance, a stronger assumption is needed to ensure the convergence after the rearrangement. That $f$ is assumed to be entire is enough for us and covers the classical setting. However, it can be slightly generalized to functions analytic on the closed unit disc. See [@FMN]. The expansion [\[E:our-B-rep\]](#E:our-B-rep){reference-type="eqref" reference="E:our-B-rep"} is a very strong form of the expansion [\[coifman-expansion\]](#coifman-expansion){reference-type="eqref" reference="coifman-expansion"}. It is rather surprising that we do not impose heavy restrictions on $\lambda_n$. Let us explain what happens in the special case of [\[coifman-expansion\]](#coifman-expansion){reference-type="eqref" reference="coifman-expansion"}. Here, in the first step we set $$\label{E:tmp-f0-f0} f(z)-f(0) = \mathbf{B}_1(z) f_1(z),$$ where $\mathbf{B}_1$ is the finite Blaschke product formed with the zeros $$\lambda_1=0, \, \lambda_2, \, \dots, \, \lambda_N$$ of $f(z)-f(0)$ on $\mathbb{D}$. Therefore, respecting the notation of the more general representation [\[E:our-B-rep\]](#E:our-B-rep){reference-type="eqref" reference="E:our-B-rep"}, we have $$\label{E:tmp-f0-f02} \mathbf{B}_1 = B_N = b_{\lambda_1} b_{\lambda_2}\cdots b_{\lambda_N}.$$ Note that $f_1$ is analytic on $\overline{\mathbb{D}}$ and has no roots in $\mathbb{D}$. However, the next set of zeros $$\lambda_{N+1} = 0, \, \lambda_{N+2}, \, \dots, \, \lambda_{M},$$ that we choose are the zeros of $f_1(z)-f_1(0)$. As it is clear now, the origin repeats infinitely many times in this sequence and thus, after all, it is a non-Blaschke sequence. Moreover, by [\[E:tmp-f0-f0\]](#E:tmp-f0-f0){reference-type="eqref" reference="E:tmp-f0-f0"} and [\[E:tmp-f0-f02\]](#E:tmp-f0-f02){reference-type="eqref" reference="E:tmp-f0-f02"}, for $1 \leq n \leq N-1$, $$\bar{B}_{n} f = \bar{B}_{n}f(0) + \bar{B}_{n} \mathbf{B}_1 f_1 = \bar{B}_{n}f(0) + b_{\lambda_{n+1}} \cdots b_{\lambda_N} f_1$$ and, for $n=N$, $$\bar{B}_{N} f = \bar{B}_{N}f(0) + \bar{B}_{N} \mathbf{B}_1 f_1 = \bar{B}_{N}f(0) + f_1.$$ Therefore, $$T_{\bar{B}_{n}} f = b_{\lambda_{n+1}} \cdots b_{\lambda_N} f_1, \qquad 1 \leq n \leq N-1,$$ and $$T_{\bar{B}_{N}} f = f_1.$$ Note that we implicitly used the fact that $\lambda_1=0$ and thus $P_{+}(\bar{B}_{n})=0$. Hence, for $1 \leq n \leq N-1$, $$(T_{\bar{B}_{n}}f)(\lambda_{n+1}) - \bar{\lambda}_n (T_{\bar{B}_{n-1}}f)(\lambda_n) = 0$$ and, for $n=N$, $$(T_{\bar{B}_{N}}f)(\lambda_{N+1}) - \bar{\lambda}_N (T_{\bar{B}_{N-1}}f)(\lambda_N) = f_1(\lambda_{N+1}) = f_1(0).$$ In short, the expansion [\[E:our-B-rep\]](#E:our-B-rep){reference-type="eqref" reference="E:our-B-rep"} becomes $$f = f(0) + f_1(0) B_N + \sum_{n=N+1}^{\infty} \big( (T_{\bar{B}_{n}}f)(\lambda_{n+1}) - \bar{\lambda}_n (T_{\bar{B}_{n-1}}f)(\lambda_n) \big) B_n,$$ which we rewrite as $$\label{E:rep-bwin-2} f = f(0) + f_1(0) \mathbf{B}_1 + \sum_{n=N+1}^{\infty} \big( (T_{\bar{B}_{n}}f)(\lambda_{n+1}) - \bar{\lambda}_n (T_{\bar{B}_{n-1}}f)(\lambda_n) \big) B_n.$$ The first two terms on the right side of [\[E:rep-bwin-2\]](#E:rep-bwin-2){reference-type="eqref" reference="E:rep-bwin-2"} are precisely the first two terms in the classical Blaschke unwinding series [\[coifman-expansion\]](#coifman-expansion){reference-type="eqref" reference="coifman-expansion"}. As a matter of fact, it is now easy to complete the above line of reasoning and see that in [\[E:rep-bwin-2\]](#E:rep-bwin-2){reference-type="eqref" reference="E:rep-bwin-2"}, many terms are zero and it eventually reduces to the classical setting [\[coifman-expansion\]](#coifman-expansion){reference-type="eqref" reference="coifman-expansion"}. **Corollary 1**. *Let $(\lambda_n)_{n \geq 1}$ be a Blaschke sequence in $\mathbb{D}$, i.e., $$\sum_{n=1}^{\infty} (1-|\lambda_n|) < \infty.$$ Let $B_0=1$, $$B_n(z) = \prod_{k=1}^{n} \frac{\lambda_k-z}{1-\bar{\lambda}_kz}, \qquad n \geq 1,$$ and $$B(z) = \prod_{n=1}^{\infty} \frac{|\lambda_n|}{\lambda_n} \, \frac{\lambda_n-z}{1-\bar{\lambda}_nz}.$$ Let $f$ be entire. Then $$f = BT_{\bar{B}}f + \Big( f(\lambda_1) + \sum_{n=1}^{\infty} \big( (T_{\bar{B}_{n}}f)(\lambda_{n+1}) - \bar{\lambda}_n (T_{\bar{B}_{n-1}}f)(\lambda_n) \big) B_n \Big),$$ where the series converges in $H^2$-norm.* *Proof.* In this case, $B$ is a well-defined infinite Blaschke product and thus Theorem [Theorem 1](#T:conv-2){reference-type="ref" reference="T:conv-2"} applies. More explicitly, each $f \in H^2$ has the decomposition $$\label{E:dec-f-ortho2} f = BT_{\bar{B}}f + \sum_{n=1}^{\infty} (1-|\lambda_n|^2) (T_{\bar{B}_{n-1}}f)(\lambda_n) B_{n-1}k_{\lambda_n}.$$ This is precisely the explicit description of the orthogonal decomposition $H^2 = BH^2 \oplus K_B$. The orthonormal basis of $K_B$ in this decomposition is $$(1-|\lambda_n|^2)^{1/2} B_{n-1}k_{\lambda_n}, \qquad n \geq 1,$$ and the corresponding Fourier coefficients of the projection of $f$ onto $K_B$ are $$(1-|\lambda_n|^2)^{1/2} (T_{\bar{B}_{n-1}}f)(\lambda_n), \qquad n \geq 1.$$ The rest of proof is the same as the proof of Corollary [Corollary 1](#Cor:hp-blaschke){reference-type="ref" reference="Cor:hp-blaschke"}. ◻ When $p \ne 2$, the terms in the above expansions are not orthogonal. But, in technical language, Corollaries [Corollary 1](#Cor:hp-blaschke){reference-type="ref" reference="Cor:hp-blaschke"} and [Corollary 1](#Cor:hp-non-blaschke){reference-type="ref" reference="Cor:hp-non-blaschke"} give us Schauder bases for $H^p$ and $K_B$, respectively. Some other Schauder basis of rational functions (not finite Blaschke products) are presented in [@MR3195914] for $H^p$ spaces. # Unwinding series with outer functions {#S:outer-expansion} In this section, we explore the development created by the outer function $$b(z)=\frac{z-1}{2}, \qquad z \in \mathbb{D}.$$ We take $b_1=b_2=\cdots=b$. Hence, clearly $$\lim_{n \to \infty} B_n(z) = \lim_{n \to \infty} b_1(z)b_2(z)\cdots b_n(z) = \lim_{n \to \infty} \left( \frac{z-1}{2} \right)^n = 0, \qquad z \in \mathbb{D}.$$ Therefore, Theorem [Theorem 1](#T:conv-1){reference-type="ref" reference="T:conv-1"} applies and we have, for each $f \in H^p$, $$\begin{aligned} f = \sum_{n=1}^{\infty} B_{n-1} Q_{b_n}T_{\bar{B}_{n-1}}f = \sum_{n=1}^{\infty} \frac{Q_{b}T_{\bar{B}_{n-1}}f}{2^{n-1}} \, (z-1)^{n-1},\end{aligned}$$ where the series converges in $H^p$-norm. We can provide a more familiar formula for $Q_{b}$. Note that $$T_b = \frac{\mathbf{S}-I}{2} \quad \mbox{and} \quad T_{\bar{b}} = \frac{\mathbf{Z}-I}{2}.$$ Recall the definition of $\mathbf{S}$ and $\mathbf{Z}$ from Section [4](#S:taylor){reference-type="ref" reference="S:taylor"}. Therefore, $$\begin{aligned} Q_{b} &=& I - T_b T_{\bar{b}}\\ &=& I - \frac{\mathbf{S}-I}{2} \, \frac{\mathbf{Z}-I}{2}\\ &=& I - \frac{1}{4} (\mathbf{S}\mathbf{Z}-\mathbf{S}-\mathbf{Z}+I)\\ &=& \frac{1}{4} (2I+k_0 \otimes k_0 +\mathbf{S}+\mathbf{Z}).\end{aligned}$$ # Declarations {#declarations .unnumbered} ## Conflict of interest {#conflict-of-interest .unnumbered} On behalf of all authors, the corresponding author states that there is no conflict of interest. ## Competing interests {#competing-interests .unnumbered} On behalf of all authors, the corresponding author states that there are no competing interests. ## Funding information {#funding-information .unnumbered} This work was supported by the NSERC Discovery Grant (Canada), and graduate scholarships from NSERC and FRQNT (Quebec). ## Author contribution {#author-contribution .unnumbered} All authors wrote the manuscript and reviewed the final version.

arxiv_math

--- abstract: | In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1$, $2,\ldots$, with probability $p_k$ of hitting the box $k$. For $j,n\in\mathbb{N}$, denote by $\mathcal{K}^*_j(n)$ the number of boxes containing exactly $j$ balls provided that $n$ balls have been thrown. We call *small counts* the variables $\mathcal{K}^*_j(n)$, with $j$ fixed. Our main result is a law of the iterated logarithm (LIL) for the small counts as the number of balls thrown becomes large. Its proof exploits a Poissonization technique and is based on a new LIL for infinite sums of independent indicators $\sum_{k\geq 1}\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}$ as $t\to\infty$, where the family of events $(A_k(t))_{t\geq 0}$ is not necessarily monotone in $t$. The latter LIL is an extension of a LIL obtained recently by Buraczewski, Iksanov and Kotelnikova (2023+) in the situation that $(A_k(t))_{t\geq 0}$ forms a nondecreasing family of events. author: - "Alexander Iksanov[^1]   and   Valeriya Kotelnikova[^2]" title: A law of the iterated logarithm for small counts in Karlin's occupancy scheme --- Key words: independent indicators; infinite occupancy; law of the iterated logarithm; small counts Mathematics Subject Classification: Primary: 60F15, 60G50 Secondary: 60C05 # Introduction ## Definition of the model {#karlin} Let $(p_k)_{k\in\mathbb{N}}$ be a discrete probability distribution, with $p_k>0$ for infinitely many $k$. The *infinite occupancy scheme* is defined by independent allocation of balls over an infinite array boxes $1$, $2,\ldots$, with probability $p_k$ of hitting the box $k$. The scheme is usually called the *Karlin occupancy scheme* because of Karlin's seminal work [@Karlin:1967]. A survey of the literature on the infinite occupancy up to 2007 is given in [@Gnedin+Hansen+Pitman:2007]. An incomplete list of very recent contributions includes [@Blasi+Mena+Prunster:2022; @Buraczewski+Iksanov+Kotelnikova:2023+; @Chang+Grabchak:2023; @Derbazi+Gnedin+Marynych:2023]. Among other things, the authors of [@Gnedin+Hansen+Pitman:2007] discuss applications of the scheme to ecology, database query optimization and literature. Another portion of possible applications can be found in Section 1.1 of [@Grabchak+Kelbert+Paris:2020]. There are deterministic and Poissonized versions of Karlin's occupancy scheme. In a *deterministic version* the $n$th ball is thrown at time $n\in\mathbb{N}$. For $j,n\in\mathbb{N}$, denote by $\mathcal{K}_j(n)$ and $\mathcal{K}_j^\ast(n)$ the number of boxes hit by at least $j$ balls and exactly $j$ balls, respectively, up to and including time $n$. Observe that $\mathcal{K}_1(n)$ is the number of occupied boxes at time $n$. Sometimes the variables $\mathcal{K}_j^\ast(n)$, with $j$ fixed, are referred to as *small counts*. To define the other version of the scheme we need an additional notation. Let $(S_k)_{k\in\mathbb{N}}$ denote a random walk with independent jumps having an exponential distribution of unit mean. The counting process $\pi:=(\pi(t))_{t\ge 0}$ given by $\pi(t):=\#\{k\in\mathbb{N}: S_k\le t\}$ for $t\geq 0$ is a Poisson process on $[0,\infty)$ of unit intensity. In a *Poissonized version* of Karlin's occupancy scheme the $n$th ball is thrown at time $S_n$, $n\in\mathbb{N}$, and it is assumed that the allocation process is independent of $(S_k)_{k\in\mathbb{N}}$, hence of $\pi$. Thus, in the time interval $[0,t]$ there are $\pi(t)$ balls thrown in the Poissonized version and $\lfloor t\rfloor$ balls thrown in the deterministic version. While the occupancy counts of distinct boxes are dependent in the deterministic version, these are independent in the Poissonized version. The latter fact is a principal advantage of the Poissonized version. It is justified by the thinning property of Poisson processes. For $j\in\mathbb{N}$ and $t\ge 0$, denote by $K_j(t)$ and $K_j^*(t)$ the number of boxes containing at least $j$ balls and exactly $j$ balls, respectively, in the Poissonized scheme at time $t$. The random variables $$K_j(t)=\sum_{k\geq 1}\mathop{\mathrm{\mathbbm{1}}}_{\{\text{the box}~ k~\text{contains at least}~ j~\text{balls}~\text{at time}~t\}}$$ and $$\label{eq:kjast} K_j^*(t)=\sum_{k\geq 1}\mathop{\mathrm{\mathbbm{1}}}_{\{\text{the box}~ k~\text{contains exactly}~ j~\text{balls}~\text{at time}~t\}}$$ are the infinite sums of independent indicators. As a consequence, their analysis is much simpler than that of $\mathcal{K}_j(n)$ and $\mathcal{K}_j^\ast(n)$ which are infinite sums of dependent indicators. ## Main results Put $$\rho(t):=\#\{k\in\mathbb{N}: 1/p_k\le t\},\quad t>0$$ and note that $\rho(t)=0$ for $t\in(0,1]$. Following Karlin [@Karlin:1967] we assume that $\rho$ varies regularly at $\infty$ of index $\alpha\in [0,1]$, that is, $\rho(t)\sim t^\alpha L(t)$ as $t\to\infty$ for some $L$ slowly varying at $\infty$. An encyclopaedic treatment of slowly and regularly varying functions can be found in Section 1 of [@Bingham+Goldie+Teugels:1989]. The function $\rho$ is said to belong to the *de Haan class* $\Pi$ if, for all $\lambda>0$, $$\label{eq:deHaan} \lim_{t\to\infty}\frac{\rho(\lambda t)-\rho(t)}{\ell(t)}=\log \lambda$$ for some $\ell$ slowly varying at $\infty$. The function $\ell$ is called *auxiliary*. According to Theorem 3.7.4 in [@Bingham+Goldie+Teugels:1989], the class $\Pi$ is a subclass of the class of slowly varying functions. Further detailed information regarding the class $\Pi$ is given in Section 3 of [@Bingham+Goldie+Teugels:1989] and in [@Geluk+deHaan:1987]. Denote by $\Pi_{\ell,\,\infty}$ the subclass of the de Haan class $\Pi$ with the auxiliary functions $\ell$ satisfying $\lim_{t\to\infty}\ell(t)=\infty$. In the case $\alpha\in (0,1]$, according to Theorems 3, 5 and 5' in [@Karlin:1967], both $K_j^*(t)$ and $\mathcal{K}_j^\ast(n)$, centered by their means and normalized by their standard deviations, converge in distribution to a random variable with the standard normal distribution. In the case $\rho\in \Pi_{\ell,\,\infty}$, Corollary 1.6 in [@Iksanov+Kotelnikova:2022] provides functional central limit theorems for $K_j^*(t)$ and $\mathcal{K}_j^\ast(n)$, properly scaled. Our purpose is to prove laws of the iterated logarithm (LILs) for $K_j^*(t)$ as $t\to\infty$ and $\mathcal{K}_j^\ast(n)$ as $n\to\infty$. While doing so, we treat the three cases separately: $\alpha=0$, $\alpha\in (0,1)$ and $\alpha=1$. The reason is that the forms of the LILs are slightly or essentially different in these cases. If $\rho$ is slowly varying at $\infty$ and satisfies an additional assumption, then the actual limit relation is either a law of the single logarithm or a LIL. However, to keep the presentation simple we prefer to call LILs all the limit relations involving upper or lower limits which appear in the paper. In Theorems [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"}, [Theorem 4](#thm:Karlin){reference-type="ref" reference="thm:Karlin"} and [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"} we present LILs for the Poissonized variables $K_j^*(t)$ as $t\to\infty$. Theorem [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"} covers a subcase of the case $\alpha=0$ in which $\rho\in\Pi_{\ell,\,\infty}$ with particular $\ell$. **Theorem 1**. *Assume that [\[eq:deHaan\]](#eq:deHaan){reference-type="eqref" reference="eq:deHaan"} holds. If $\ell$ in [\[eq:deHaan\]](#eq:deHaan){reference-type="eqref" reference="eq:deHaan"} satisfies $$\label{eq:slowly} \ell(t)~\sim~ (\log t)^\beta l(\log t),\quad t\to\infty$$ for some $\beta>0$ and $l$ slowly varying at $\infty$, then, for each $j\in\mathbb{N}$, $$\label{eq:LILkar} \limsup_{t\to\infty}\frac{K^*_j(t)-\mathbb{E}K^*_j(t)}{({\rm Var}\,K^*_j(t)\log {\rm Var}\,K^*_j(t))^{1/2}}=\Big(\frac{2}{\beta}\Big)^{1/2}\quad\text{{\rm a.s.}}$$ and $$\label{eq:inf01} \liminf_{t\to\infty}\frac{K^*_j(t)-\mathbb{E}K^*_j(t)}{({\rm Var}\,K^*_j(t)\log {\rm Var}\,K^*_j(t))^{1/2}}=-\Big(\frac{2}{\beta}\Big)^{1/2}\quad\text{{\rm a.s.}}$$ If $\ell$ in [\[eq:deHaan\]](#eq:deHaan){reference-type="eqref" reference="eq:deHaan"} satisfies $$\label{eq:slowly2} \ell(t)~\sim~\exp(\sigma(\log t)^\lambda),\quad t\to\infty$$ for some $\sigma>0$ and $\lambda\in (0,1)$, then, for each $j\in\mathbb{N}$, $$\label{eq:LILkar1} \limsup_{t\to\infty}\frac{K^*_j(t)-\mathbb{E}K^*_j(t)}{({\rm Var}\,K^*_j(t)\log\log {\rm Var}\,K^*_j(t))^{1/2}}=\Big(\frac{2}{\lambda}\Big)^{1/2}\quad\text{{\rm a.s.}}$$ and $$\label{eq:inf02} \liminf_{t\to\infty}\frac{K^*_j(t)-\mathbb{E}K^*_j(t)}{({\rm Var}\,K^*_j(t)\log\log {\rm Var}\,K^*_j(t))^{1/2}}=-\Big(\frac{2}{\lambda}\Big)^{1/2}\quad\text{{\rm a.s.}}$$ In both cases $$\label{eq:meK0} \mathbb{E}K^*_j(t)~\sim~ \frac{\ell(t)}{j},\quad t\to\infty$$ and $$\label{eq:varK0_1} {\rm Var}\,K^*_j(t)~\sim~ \Big(\frac{1}{j}- \frac{(2j-1)!}{(j!)^2 2^{2j}}\Big)\ell(t),\quad t\to\infty.$$* *Remark 2*. Treatment of the situations in which $\rho$ is slowly varying at $\infty$, yet $\rho\notin \Pi$ is beyond our reach. To reveal complications arising in this case we only mention that even the large-time asymptotics of $t\mapsto{\rm Var \,}K^\ast_j(t)$ is not known. To find the asymptotic, a second-order relation for $\rho$ like [\[eq:deHaan\]](#eq:deHaan){reference-type="eqref" reference="eq:deHaan"} seems to be indispensable. If $\alpha\in (0,1]$, then the regular variation of $\rho$ alone ensures that, for all $\lambda>0$, $$\lim_{t\to\infty}\frac{\rho(\lambda t)-\rho(t)}{\rho(t)}=\lambda^\alpha-1.$$ Thus, no extra conditions are needed in this case. *Remark 3*. Our present proof only works provided that, for some $a>0$, $\rho(t)=O((\ell(t))^a)$ as $t\to\infty$. In view of this, Theorem [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"} does not cover the diverging slowly varying functions $\ell$ which grow slower than any positive power of the logarithm, for instance, $\ell(t)\sim\log\log t$ as $t\to\infty$. Indeed, it can be checked that $\lim_{t\to\infty}\ell(t)=\infty$ entails $\lim_{t\to\infty}(\rho(t)/\log t)=\infty$, whence trivially, for all $a>0$, $\lim_{t\to\infty}(\rho(t)/(\ell(t))^a)=\infty$. The following results are concerned with the cases $\alpha\in(0,1)$ and $\alpha=1$, respectively. **Theorem 4**. *Assume that, for some $\alpha\in (0,1)$ and some $L$ slowly varying at $+\infty$, $$\rho(t)~\sim~ t^\alpha L(t),\quad t\to\infty.$$ Then, for each $j\in\mathbb{N}$, $$\label{eq:LILkar2} \limsup_{t\to\infty}\frac{K^*_j(t)-\mathbb{E}K^*_j(t)}{({\rm Var}\,K^*_j(t)\log\log {\rm Var}\,K^*_j(t))^{1/2}}=2^{1/2}\quad\text{{\rm a.s.}}$$ and $$\label{eq:inf1} \liminf_{t\to\infty}\frac{K^*_j(t)-\mathbb{E}K^*_j(t)}{({\rm Var}\,K^*_j(t)\log\log {\rm Var}\,K^*_j(t))^{1/2}}=-2^{1/2}\quad\text{{\rm a.s.}},$$ $$\label{eq:meKgeneral} \mathbb{E}K^*_j(t)~\sim~ \alpha\frac{\Gamma(j-\alpha)}{j!}t^\alpha L(t)$$ and $$\label{eq:varKgeneral} {\rm Var}\,K^*_j(t)~\sim~ c_{j,\,\alpha} t^\alpha L(t),\quad t\to\infty,$$ where $\Gamma$ is the Euler gamma function and $$\label{eq:cj} c_{j,\,\alpha}:=\alpha\Big(\frac{\Gamma(j-\alpha)}{j!}-\frac{2^{\alpha}\Gamma(2j-\alpha)}{2^{2j}(j!)^2}\Big)>0.$$* **Theorem 5**. *Assume that, for some $L$ slowly varying at $+\infty$, $$\label{eqassump1} \rho(t)~\sim~ tL(t),\quad t\to\infty.$$ Then, for each $j\geq 2$, relation [\[eq:LILkar2\]](#eq:LILkar2){reference-type="eqref" reference="eq:LILkar2"} holds, $$\label{eq:meK1} \mathbb{E}K^*_j(t)~\sim~ \frac{1}{j(j-1)}tL(t)$$ and $$\label{eq:cj1} \lim_{t\to\infty}\frac{{\rm Var}\,K^*_j(t)}{tL(t)}= \frac{1}{j(j-1)}-\frac{(2j-2)!}{2^{2j-1}(j!)^2}=c_{j,\,1}.$$* *Assume that, for each small enough $\gamma>0$, $$\label{eq:exotic} \lim_{n\to\infty}\frac{\hat L(\exp((n+1)^{1+\gamma}))}{\hat L(\exp(n^{1+\gamma}))}=0,$$ where $\hat L(t):=\int_t^\infty y^{-1}L(y){\rm d}y$, being well-defined for large $t$, is a function slowly varying at $\infty$ and satisfying $$\label{eq:Lhat} \lim_{t\to\infty} \frac{L(t)}{\hat{L}(t)}=0.$$ Then relation [\[eq:LILkar2\]](#eq:LILkar2){reference-type="eqref" reference="eq:LILkar2"} holds with $j=1$. If [\[eq:exotic\]](#eq:exotic){reference-type="eqref" reference="eq:exotic"} does not hold, then $$\label{eq:LILkar200} \limsup_{t\to\infty}\frac{K^*_1(t)-\mathbb{E}K^*_1(t)}{({\rm Var}\,K^*_1(t)\log\log {\rm Var}\,K^*_1(t))^{1/2}}\leq 2^{1/2}\quad\text{{\rm a.s.}}$$ and $$\label{eq:inf11} \liminf_{t\to\infty}\frac{K^*_1(t)-\mathbb{E}K^*_1(t)}{({\rm Var}\,K^*_1(t)\log\log {\rm Var}\,K^*_1(t))^{1/2}}\geq -2^{1/2}\quad\text{{\rm a.s.}}$$ In any event $$\label{eq:exo} {\rm Var}\,K^*_1(t)~\sim~ \mathbb{E}K^*_1(t)~\sim~ t \hat L(t),\quad t\to\infty.$$* Theorems [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"}, [Theorem 4](#thm:Karlin){reference-type="ref" reference="thm:Karlin"} and [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"} will be deduced in Section [4](#sec:Karlin){reference-type="ref" reference="sec:Karlin"} from the LIL for infinite sums of independent indicators given in Theorem [Theorem 11](#thm:main){reference-type="ref" reference="thm:main"}. Finally, we present LILs for the variables $\mathcal{K}^*_j(n)$. **Theorem 6**. *Under the assumptions of Theorems [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"}, [Theorem 4](#thm:Karlin){reference-type="ref" reference="thm:Karlin"} or [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"}, for $j\in\mathbb{N}$, all the LILs stated there hold true with $\mathcal{K}^*_j(n)$, $\mathbb{E}\mathcal{K}^*_j(n)$ and ${\rm Var \,}\mathcal{K}^*_j(n)$ replacing $K_j^*(t)$, $\mathbb{E}K_j^*(t)$ and ${\rm Var \,}K_j^*(t)$, and $n\to\infty$ replacing $t\to\infty$.* A transfer of results available for the Poissonized version to the deterministic version is called *de-Poissonization*. Theorem [Theorem 6](#thm:depoiss){reference-type="ref" reference="thm:depoiss"} will be deduced in Section [4](#sec:Karlin){reference-type="ref" reference="sec:Karlin"} from Theorems [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"}, [Theorem 4](#thm:Karlin){reference-type="ref" reference="thm:Karlin"} and [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"} with the help of a de-Poissonization technique. # LIL for infinite sums of independent indicators Let $(A_1(t))_{t\geq 0}$, $(A_2(t))_{t\geq 0},\ldots$ be independent families of events defined on a common probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that, for each $t\geq 0$, $\sum_{k\geq 1}\mathbb{P}(A_k(t))<\infty$ and then put $$X(t):=\sum_{k\geq 1}\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)},\quad t\geq 0.$$ Since, for $t\geq 0$, $b(t):=\mathbb{E}X(t)=\sum_{k\geq 1}\mathbb{P}(A_k(t))<\infty$, we infer $X(t)<\infty$ almost surely (a.s.) and further $$a(t):={\rm Var}\,X(t)=\sum_{k\geq 1}\mathbb{P}(A_k(t))(1-\mathbb{P}(A_k(t)))\leq b(t)<\infty.$$ Under the assumption that, for each $k\in\mathbb{N}$ and $0\leq s<t$, $A_k(s)\subseteq A_k(t)$ a LIL for $X(t)$ can be found in Theorem 1.6 of [@Buraczewski+Iksanov+Kotelnikova:2023+]. As an application, LILs for $K_j(t)$ were proved in that paper, see Theorems 3.1, 3.3 and 3.4 therein. According to [\[eq:kjast\]](#eq:kjast){reference-type="eqref" reference="eq:kjast"}, the variable $K^\ast_j(t)$ is a particular instance of $X(t)$. However, for each $k\in\mathbb{N}$, the corresponding events $(A_k(t))_{t\geq 0}$ are not monotone in $t$, which shows that a LIL for $K_j^\ast(t)$ cannot be deduced from Theorem 1.6 of [@Buraczewski+Iksanov+Kotelnikova:2023+]. This serves a motivation for the present section. Here, dropping the monotonicity assumption we provide sufficient conditions under which a LIL for $X(t)$ holds. We shall prove a LIL for $X(t)$ under the following assumptions (A1)-(A5) and (B1)-(B21) or (B22). The lack of monotonicity only affects our proof of $\limsup_{t\to\infty}\leq 1$ a.s. to be done under (A1)-(A5). In view of this, (A2)-(A5) are modified versions of the corresponding assumptions in [@Buraczewski+Iksanov+Kotelnikova:2023+]. (B1), (B21) and (B22) coincide with the corresponding assumptions in [@Buraczewski+Iksanov+Kotelnikova:2023+] under which the relation $\limsup_{t\to\infty}\geq 1$ a.s. was proved in the cited article. (A1) $\lim_{t\to\infty}a(t)=\infty$. (A2) There exist independent a.s. nondecreasing stochastic processes $(\Phi_1(t))_{t\geq 0}$, $(\Phi_2(t))_{t\geq 0},\ldots$ taking values in $\{0, 1, 2, \ldots, M\}$ for some $M\in\mathbb{N}$ and satisfying \(a\) for each $k\in\mathbb{N}$, $0\le s< t$, $|\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}-\mathop{\mathrm{\mathbbm{1}}}_{A_k(s)}|\le \Phi_k(t) - \Phi_k(s)$ a.s.; \(b\) for each $t\geq 0$, $f(t):=\mathbb{E}Y(t)<\infty$, where $Y(t):=\sum_{k\geq 1}\Phi_k(t)$ for $t\ge 0$; \(c\) $b(0)\le f(0)$. *Remark 7*. (A2a) and (A2b) entail, for $0\le s<t$, $$\label{eq:bf_with_s} |b(t)-b(s)|\le \mathbb{E}\sum_{k\geq 1} |\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}-\mathop{\mathrm{\mathbbm{1}}}_{A_k(s)}|\le f(t) - f(s).$$ Inequality [\[eq:bf_with_s\]](#eq:bf_with_s){reference-type="eqref" reference="eq:bf_with_s"} with $s=0$ and (A2c) together imply that $$\label{eq:bf} b(t)\le f(t) - f(0)+b(0)\le f(t),\quad t\geq 0.$$ *Remark 8*. Here is an example of $X$ satisfying (A2) which is motivated by a prospective application of the LIL for $X(t)$ to the variables $K_j^\ast(t)$. Let $(B_1(t))_{t\ge 0}$, $(B_2(t))_{t\ge 0},\ldots$ and $(C_1(t))_{t\ge 0}$, $(C_2(t))_{t\ge 0},\ldots$ be two families of independent events satisfying \(i\) for each $k\in\mathbb{N}$ and $t\ge 0$, $C_k(t)\subseteq B_k(t)$; \(ii\) for each $k\in\mathbb{N}$ and $0\leq s<t$, $B_k(s)\subseteq B_k(t)$ and $C_k(s)\subseteq C_k(t)$; \(iii\) for $t\geq 0$, $\sum_{k\geq 1} \mathbb{P}(B_k(t))<\infty$. For each $k\in\mathbb{N}$ and $t\ge 0$, put $A_k(t):=B_k(t)\setminus C_k(t)$ and $\Phi_k(t):=\mathop{\mathrm{\mathbbm{1}}}_{B_k(t)}+\mathop{\mathrm{\mathbbm{1}}}_{C_k(t)}$. The so defined $\Phi_k$ is a.s. nondecreasing. Since, for $0\leq s<t$, $\mathop{\mathrm{\mathbbm{1}}}_{C_k(s)}\le\mathop{\mathrm{\mathbbm{1}}}_{C_k(t)}$ and $\mathop{\mathrm{\mathbbm{1}}}_{B_k(s)}\le\mathop{\mathrm{\mathbbm{1}}}_{B_k(t)}$ a.s. we conclude that $$|\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}-\mathop{\mathrm{\mathbbm{1}}}_{A_k(s)}|=|\mathop{\mathrm{\mathbbm{1}}}_{B_k(t)}-\mathop{\mathrm{\mathbbm{1}}}_{C_k(t)}-\mathop{\mathrm{\mathbbm{1}}}_{B_k(s)}+\mathop{\mathrm{\mathbbm{1}}}_{C_k(s)}|\le \Phi_k(t)-\Phi_k(s) \quad\text{a.s.}$$ While (A2b) is a consequence of (iii), (A2c) is justified by $\mathbb{P}(A_k(0))=\mathbb{P}(B_k(0))-\mathbb{P}(C_k(0))\le \mathbb{E}\Phi_k(0)$. Putting $C_k(t):=\oslash$ for all $k\in\mathbb{N}$ and $t\geq 0$ we recover the case of monotone in $t$ families $(A_k(t))_{t\geq 0}$ treated in [@Buraczewski+Iksanov+Kotelnikova:2023+]. (A3) Under (A2), there exists $\mu^\ast\geq 1$ such that $f(t)=O((a(t))^{\mu^\ast})$ as $t\to\infty$. In view of [\[eq:bf\]](#eq:bf){reference-type="eqref" reference="eq:bf"} and $a(t)\le b(t)$ for $t\ge 0$, necessarily $\mu^\ast\geq 1$. Put $$\label{eq:infim} \mu:=\inf\{\mu^\ast: f(t)=O((a(t))^{\mu^\ast})\}.$$ If $\mu=1$, we assume additionally that either $f$ is eventually continuous or $${\lim\inf}_{t\to\infty}(\log f(t-1)/\log f(t))>0;$$ and that $$\label{eq:infim2} f(t)/a(t)=O(z_q(a(t))),\quad t\to\infty,$$ where $z_q(t):=(\log t)^{q}\mathcal{L}(\log t)$ for some $q\geq 0$ and $\mathcal{L}$ slowly varying at $\infty$ and, if $q>0$, $f(t)/a(t)\neq O(z_s(a(t)))$ for $s\in (0,q)$. Before introducing our next assumption we need some preparation. In view of (A1) and $a(t)\leq f(t)$ for $t\ge 0$, we infer $\lim_{t\to\infty}f(t)=\infty$. For each $\varrho\in (0,1)$, put $$\label{eq:mutheta} \mu_\varrho:=\mu+\varrho\quad \text{if}~\mu>1\quad\text{and}\quad q_\varrho:=q+\varrho \quad \text{if}~\mu=1.$$ Assuming (A3), fix any $\kappa\in (0,1)$ and $\varrho\in (0,1)$ and put $$\label{tn} t_n=t_n(\kappa, \mu):=\inf\{t>0: f(t)>v_n(\kappa,\mu)\}$$ for $n\in\mathbb{N}$, where $v_n(\kappa, 1)=v_n(\kappa, 1, q, \varrho)=\exp(n^{(1-\kappa)/(q_\varrho+1)})$ and $v_n(\kappa,\mu)=v_n(\kappa, \mu, \varrho)=n^{\mu_\varrho(1-\kappa)/(\mu_\varrho-1)}$ for $\mu>1$. Plainly, the sequence $(t_n)_{n\in\mathbb{N}}$ is nondecreasing with $\lim_{n\to\infty} t_n=+\infty$. (A4) Fix any $\kappa\in (0,1)$ and $\varrho\in (0,1)$. There exists a function $a_0$ satisfying $a(t)\sim a_0(t)$ as $t\to\infty$, and, for each $n$ large enough, there exists $s_n=s_n(\kappa,\mu)\in[t_n(\kappa,\mu),\,t_{n+1}(\kappa,\mu)]$ such that $a_0(t)\geq a_0(s_n)$ for all $t\in [t_n, t_{n+1}]$. *Remark 9*. A sufficient condition for (A4) is either eventual lower semi-continuity or eventual monotonicity of $a_0$. The former means that $\liminf_{y\to x}a_0(y)\geq a_0(x)$, for all large enough $x$. (A5) For each $n$ large enough, there exists $A>1$ and a partition $t_n=t_{0,\,n}<t_{1,\,n}<\ldots<t_{j,\,n}=t_{n+1}$ with $j=j_n$ satisfying $$1\leq f(t_{k,\,n})-f(t_{k-1,\,n})\leq A,\quad 1\leq k\leq j$$ and, for all $\varepsilon>0$, $\big(j_n\exp(-\varepsilon (a(s_n))^{1/2})\big)$ is a summable sequence. *Remark 10*. A sufficient condition for (A5) is that $f$ is eventually strictly increasing and eventually continuous. Indeed, one can then choose a partition that satisfies, for large $n$, $f(t_{k,\,n})-f(t_{k-1,\,n})=1$ for $k\in\mathbb{N}$, $k\leq j-1$ and $f(t_{j,\,n})-f(t_{j-1,\,n})\in [1,2)$. As a consequence, $$j_n=\lfloor v_{n+1}(\kappa, \mu)-v_n(\kappa,\mu)\rfloor=o(a(s_n)),\quad n\to\infty$$ by Lemma [Lemma 13](#lem:aux1){reference-type="ref" reference="lem:aux1"}(b) below, so that the sequence $\big(j_n\exp(-\varepsilon (a(s_n))^{1/2})\big)$ is indeed summable. Assuming (A1) and (A3), fix any $\gamma>0$ and put $$\label{eq:tau} \tau_n=\tau_n(\gamma,\mu):=\inf\{t>0: a(t)>w_n(\gamma,\mu)\}$$ for large $n\in\mathbb{N}$ with $\mu$ as given in [\[eq:infim\]](#eq:infim){reference-type="eqref" reference="eq:infim"}. Here, with $q$ as given in [\[eq:infim2\]](#eq:infim2){reference-type="eqref" reference="eq:infim2"}, $w_n(\gamma, 1)=w_n(\gamma, 1, q)=\exp(n^{(1+\gamma)/(q+1)})$ if $\mu=1$ and $w_n(\gamma, \mu)=n^{(1+\gamma)/(\mu-1)}$ if $\mu>1$. (B1) The function $a$ is eventually continuous or $\lim_{t\to\infty}(\log a(t-1)/\log a(t))=1$ if $\mu=1$ and $\lim_{t\to\infty}(a(t-1)/a(t))=1$ if $\mu>1$. (B21) For sufficiently large $t>0$ and each $\varsigma>0$, let $R_\varsigma(t)$ denote a set of positive integers satisfying the following two conditions: for each $\varsigma>0$ and each $\gamma>0$, both close to $0$ there exists $n_0=n_0(\varsigma, \gamma)\in\mathbb{N}$ such that the sets $R_\varsigma(\tau_{n_0}(\gamma,\mu))$, $R_\varsigma(\tau_{n_0+1}(\gamma,\mu)),\ldots$ are disjoint; and $$\label{eq:onevar} \lim_{t\to\infty}\frac{{\rm Var}\Big(\sum_{k\in R_\varsigma(t)}\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}\Big)}{{\rm Var}\,X(t)}=1-\varsigma.$$ (B22) For sufficiently large $t>0$, let $R_0(t)$ denote a set of positive integers satisfying the following two conditions: for each $\gamma>0$ close to $0$ there exists $n_0=n_0(\gamma)\in\mathbb{N}$ such that the sets $R_0(\tau_{n_0}(\gamma,\mu))$, $R_0(\tau_{n_0+1}(\gamma,\mu)),\ldots$ are disjoint; and $$\label{eq:one} \lim_{t\to\infty}\frac{{\rm Var}\Big(\sum_{k\in R_0(t)}\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}\Big)}{{\rm Var}\,X(t)}=1.$$ Now we are ready to present a LIL for infinite sums of independent indicators. **Theorem 11**. *Suppose (A1)-(A5), (B1) and either (B21) or (B22). Then, with $\mu\geq 1$ and $q\geq 0$ as defined in [\[eq:infim\]](#eq:infim){reference-type="eqref" reference="eq:infim"} and [\[eq:infim2\]](#eq:infim2){reference-type="eqref" reference="eq:infim2"}, respectively, $$\limsup_{t\to\infty}\frac{X(t)-\mathbb{E}X(t)}{(2(q+1){\rm Var}\,X(t)\log\log {\rm Var}\,X(t))^{1/2}}=1\quad\text{{\rm a.s.}}$$ and $$\liminf_{t\to\infty}\frac{X(t)-\mathbb{E}X(t)}{(2(q+1){\rm Var}\,X(t)\log\log {\rm Var}\,X(t))^{1/2}}=-1\quad\text{{\rm a.s.}}$$ if $\mu=1$ and $$\limsup_{t\to\infty}\frac{X(t)-\mathbb{E}X(t)}{(2(\mu-1){\rm Var}\,X(t)\log {\rm Var}\,X(t))^{1/2}}=1\quad\text{{\rm a.s.}}$$ and $$\liminf_{t\to\infty}\frac{X(t)-\mathbb{E}X(t)}{(2(\mu-1){\rm Var}\,X(t)\log {\rm Var}\,X(t))^{1/2}}=-1\quad\text{{\rm a.s.}}$$ if $\mu>1$.* Our proof of Theorem [Theorem 11](#thm:main){reference-type="ref" reference="thm:main"} given in Section [3.2](#sec:main){reference-type="ref" reference="sec:main"} is a modified version of the proof of Theorem 1.6 in [@Buraczewski+Iksanov+Kotelnikova:2023+]. # Proof of Theorem [Theorem 11](#thm:main){reference-type="ref" reference="thm:main"} {#proof-of-theorem-thmmain} ## Auxiliary results We start with a simple inequality which will be used in the last part of the proof of Proposition [Proposition 19](#lilhalf1){reference-type="ref" reference="lilhalf1"}. **Lemma 12**. *Suppose (A2). Then, for $\vartheta \in\mathbb{R}$ and $t> s\ge 0$, $$\mathbb{E}\exp(\vartheta (Y(t)-Y(s)))\le \exp(({\rm e}^{\vartheta M}-1)(f(t)-f(s))).$$* *Proof.* For $k\in\mathbb{N}$ and $t>s\ge 0$, $\mathop{\mathrm{\mathbbm{1}}}_{\{\Phi_k(t)-\Phi_k(s)>0\}}=\mathop{\mathrm{\mathbbm{1}}}_{\{\Phi_k(t)-\Phi_k(s)\ge 1\}}\le \Phi_k(t)-\Phi_k(s)$ a.s. The equality stems from the fact that $\Phi_k$ only takes nonnegative integer values. Hence, for $\vartheta\in\mathbb{R}$ and $0\leq s<t$, $$\begin{gathered} \mathbb{E}\exp(\vartheta (Y(t)-Y(s))=\prod_{k\geq 1}\mathbb{E}\exp (\vartheta (\Phi_k(t)-\Phi_k(s)))\\= \prod_{k\geq 1}\big(1+\mathbb{E}({\rm e}^{\vartheta(\Phi_k(t)-\Phi_k(s))}-1)\mathop{\mathrm{\mathbbm{1}}}_{\{\Phi_k(t)-\Phi_k(s)>0\}}\big)\le \prod_{k\geq 1}\big(1+({\rm e}^{\vartheta M}-1)\mathbb{E}\mathop{\mathrm{\mathbbm{1}}}_{\{\Phi_k(t)-\Phi_k(s)>0\}}\big)\\\leq \exp\Big(({\rm e}^{\vartheta M}-1)\sum_{k\geq 1}\mathbb{E}(\Phi_k(t)-\Phi_k(s))\Big)= \exp(({\rm e}^{\vartheta M}-1)(f(t)-f(s))).\end{gathered}$$ ◻ For each $B\geq 0$ and each $D>1$, put $$\label{eq:defg} g_{1,\,B}(t):=(B+1)\log\log t,\quad t>{\rm e}\quad\text{and}\quad g_D(t):=(D-1)\log t,\quad t>1.$$ Lemma [Lemma 13](#lem:aux1){reference-type="ref" reference="lem:aux1"} does two things. First, it explains the choice of the sequences $(t_n)$ and $(v_n)$ and the functions $g_{1,\,q_\varrho}$ and $g_{\mu_\varrho}$ (even though $(t_n)$ is not present in Lemma [Lemma 13](#lem:aux1){reference-type="ref" reference="lem:aux1"} explicitly, it is of crucial importance for defining the sequence $(s_n)$.) Second, it secures a successful application of the Borel-Cantelli lemma in the proof of Proposition [Proposition 19](#lilhalf1){reference-type="ref" reference="lilhalf1"}. **Lemma 13**. *Suppose (A1), (A3) and (A4). Fix any $\varrho\in (0,1)$, any $\kappa\in (0,1)$ and let $q_\varrho$ and $\mu_\varrho$ be as defined in [\[eq:mutheta\]](#eq:mutheta){reference-type="eqref" reference="eq:mutheta"}.* *(a) If $\mu$ in [\[eq:infim\]](#eq:infim){reference-type="eqref" reference="eq:infim"} is equal to $1$, then $\exp(-g_{1,\,q_\varrho}(a(s_n(\kappa, 1))))=O(n^{-(1-\kappa)})$ as $n\to\infty$, and if $\mu>1$, then $\exp(-g_{\mu_\varrho}(a(s_n(\kappa, \mu))))=O(n^{-(1-\kappa)})$.* *(b) There exists an integer $r\geq 2$ such that $\big(\big((v_{n+1}(\kappa,\mu)-v_n(\kappa, \mu))/a(s_n)\big)^r\big)$ is a summable sequence.* *Proof.* (a) Using the definition of $t_n$, the fact that $f$ is nondecreasing and (A3), we conclude that $$\label{eq:ineq10} \exp(n^{(1-\kappa)/(q_\varrho+1)})\leq f(t_n(\kappa, 1))\le f(s_n(\kappa, 1))=O(a(s_n(\kappa, 1))z_q(a(s_n(\kappa, 1)))),\quad n\to\infty$$ and, for $\mu>1$, $$\label{eq:ineq11} n^{\mu_\varrho(1-\kappa)/(\mu_\varrho-1)}\leq f(t_n(\kappa,\mu))\leq f(s_n(\kappa, \mu))=O((a(s_n(\kappa,\mu)))^{\mu_\varrho}),\quad n\to\infty.$$ Since $\lim_{t\to\infty}(\log z_q(t)/\log t)=0$ we infer $$\exp(-g_{1,\,q_\varrho}(a(s_n(\kappa, 1))))=(\log a(s_n(\kappa, 1)))^{-(q_\varrho+1)}=O(n^{-(1-\kappa)}),\quad n\to\infty.$$ Also, for $\mu>1$, $$\exp(-g_{\mu_\varrho}(a(s_n(\kappa, \mu))))=(a(s_n(\kappa,\mu)))^{-(\mu_\varrho-1)}=O(n^{-(1-\kappa)}),\quad n\to\infty.$$ \(b\) We start by proving that (A3) with $\mu=1$ entails $$\label{eq:ineqlog} \log a(s_n(\kappa, 1))=O(n^{(1-\kappa)/(q_\varrho+1)}),\quad n\to\infty.$$ Assume that $f$ is eventually continuous. Then $f(t_n(\kappa, 1))=v_n(\kappa, 1)$ for large enough $n$ and thereupon $\log a(s_n(\kappa, 1))\leq \log f(s_n(\kappa, 1))\le \log f(t_{n+1}(\kappa, 1))= (n+1)^{(1-\kappa)/(q_\varrho+1)}$ for large $n$. Assuming that ${\lim\inf}_{t\to\infty}(\log f(t-1)/\log f(t))>0$ we obtain [\[eq:ineqlog\]](#eq:ineqlog){reference-type="eqref" reference="eq:ineqlog"} as a consequence of $\log f(t_{n+1}(\kappa, 1)-1) \leq (n+1)^{(1-\kappa)/(q_\varrho+1)}$ and $\log a(s_n(\kappa, 1))\leq \log f(s_n(\kappa, 1))\le \log f(t_{n+1}(\kappa, 1))$. We proceed by noting that, as $n\to\infty$, $$\begin{gathered} v_{n+1}(\kappa,1)-v_n(\kappa, 1)=\exp((n+1)^{(1-\kappa)/(q_\varrho+1)})-\exp(n^{(1-\kappa)/(q_\varrho+1)})\\~\sim~((1-\kappa)/(q_\varrho+1))n^{((1-\kappa)/(q_\varrho+1))-1}\exp(n^{(1-\kappa)/(q_\varrho+1)})\end{gathered}$$ and, for $\mu>1$, $$\begin{gathered} v_{n+1}(\kappa,\mu)-v_n(\kappa, \mu)=(n+1)^{\mu_\varrho(1-\kappa)/(\mu_\varrho-1)}-n^{\mu_\varrho(1-\kappa)/(\mu_\varrho-1)}\\~\sim~ (\mu_\varrho(1-\kappa)/(\mu_\varrho-1))n^{(1-\mu_\varrho\kappa)/(\mu_\varrho-1)}.\end{gathered}$$ Write $$\begin{gathered} \frac{1}{a(s_n(\kappa, 1))}=O((\log a(s_n(\kappa, 1)))^{q_\varrho}\exp(-n^{(1-\kappa)/(q_\varrho+1)}))\\=O(n^{q_\varrho(1-\kappa)/(q_\varrho+1)}\exp(-n^{(1-\kappa)/(q_\varrho+1)})),\quad n\to\infty.\end{gathered}$$ Here, the first equality is implied by $z_q(t)=O((\log t)^{q_\varrho})$ as $t\to\infty$ and [\[eq:ineq10\]](#eq:ineq10){reference-type="eqref" reference="eq:ineq10"}, and the second equality is a consequence of [\[eq:ineqlog\]](#eq:ineqlog){reference-type="eqref" reference="eq:ineqlog"}. In the case $\mu>1$, invoking [\[eq:ineq11\]](#eq:ineq11){reference-type="eqref" reference="eq:ineq11"} we infer $$\frac{1}{a(s_n(\kappa,\mu))}=O(n^{-(1-\kappa)/(\mu_\varrho-1)})),\quad n\to\infty.$$ Thus, we have proved that, for $\mu\geq 1$, $$\frac{v_{n+1}(\kappa,\mu)-v_n(\kappa, \mu)}{a(s_n)}=O(n^{-\kappa}),\quad n\to\infty.$$ Choosing any integer $r\geq 2$ satisfying $r\kappa>1$ completes the proof of part (b). ◻ For $k\in\mathbb{N}$ and $t\geq 0$, put $X^\ast(t):=X(t)-\mathbb{E}X(t)$ and $\eta_k(t):=\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}-\mathbb{P}(A_k(t))$. Note that $$X^\ast(t)=\sum_{k\geq 1} \eta_k(t),\quad t\geq 0$$ and that $\eta_1(t)$, $\eta_2(t),\ldots$ are independent centered random variables. Lemma [Lemma 14](#ineq1){reference-type="ref" reference="ineq1"} provides a uniform bound for higher moments of the increments of $X^\ast$. The bound serves a starting point of the chaining argument in the spirit of Lemma [Lemma 15](#lem:bor){reference-type="ref" reference="lem:bor"}. A result of an application of Lemma [Lemma 15](#lem:bor){reference-type="ref" reference="lem:bor"} to the present setting is given in Lemma [Lemma 16](#billappl){reference-type="ref" reference="billappl"}. **Lemma 14**. *Suppose (A2). Let $r\in\mathbb{N}$ and $t,s\geq 0$. Then $$\label{eq:5} \mathbb{E}(X^\ast(t)-X^\ast(s))^{2r}\leq D_r \max(|f(t)- f(s)|^r, |f(t)-f(s)|)$$ for a positive constant $D_r$ which does not depend on $t$ and $s$.* *Proof.* In view of the representation $$X^\ast(t)-X^\ast(s)=\sum_{k\geq 1}(\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}-\mathbb{P}(A_k(t))-\mathop{\mathrm{\mathbbm{1}}}_{A_k(s)}+\mathbb{P}(A_k(s)))=:\sum_{k\geq 1}\eta_k(s,t),$$ the variable $X^\ast(t)-X^\ast(s)$ is an infinite sum of independent centered random variables with finite moment of order $2r$. Invoking Rosenthal's inequality (Theorem 3 in [@Rosenthal:1970]) in the case $r\geq 2$ we infer $$\mathbb{E}(X^\ast(t)-X^\ast(s))^{2r}\leq C_r \max\Big(\Big(\sum_{k\geq 1}\mathbb{E}(\eta_k(s,t))^2\Big)^r, \sum_{k\geq 1}\mathbb{E}(\eta_k(s,t))^{2r}\Big).$$ In the case $r=1$, the inequality trivially holds with $C_1=1$ as is seen from $$\mathbb{E}(X^\ast(t)-X^\ast(s))^2= \sum_{k\geq 1}\mathbb{E}(\eta_k(s,t))^2.$$ In view of (A2), for $r\in\mathbb{N}$ and $0\le s< t$, $$\begin{gathered} %\label{eq:eta_rth} \sum_{k\geq 1} \mathbb{E}(\eta_k(s,t))^{2r}\le 2^{2r-1} \sum_{k\geq 1} \Big(\mathbb{E}(\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}-\mathop{\mathrm{\mathbbm{1}}}_{A_k(s)})^{2r}+(\mathbb{P}(A_k(t))-\mathbb{P}(A_k(s)))^{2r}\Big)\\\le 2^{2r-1} \sum_{k\in\mathbb{N}} \Big(\mathbb{E}|\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}-\mathop{\mathrm{\mathbbm{1}}}_{A_k(s)}|+|\mathbb{P}(A_k(t))-\mathbb{P}(A_k(s))|\Big)\le 2^{2r}\sum_{k\geq 1} \mathbb{E}(\Phi_k(t)-\Phi_k(s))\\=2^{2r}(f(t)-f(s)).\end{gathered}$$ Here, we have used $(a+b)^{2r}\le 2^{2r-1}(a^{2r}+b^{2r})$, $a,b\in\mathbb{R}$ for the first inequality, the fact that $|\mathop{\mathrm{\mathbbm{1}}}_{A_k(t)}-\mathop{\mathrm{\mathbbm{1}}}_{A_k(s)}|\in \{0,1\}$ a.s. and $|\mathbb{P}(A_k(t))-\mathbb{P}(A_k(s))|\in [0,1]$ for the second and (A2a) for the third. The argument for the case $0\leq t<s$ is analogous. Combining fragments together we conclude that [\[eq:5\]](#eq:5){reference-type="eqref" reference="eq:5"} holds with $D_r:=2^{2r}C_r$. ◻ The next result is borrowed from Lemma 2 in [@Longnecker+Serfling:1977]. **Lemma 15**. *Let $\xi_1$, $\xi_2,\ldots$ be random variables. Fix any $m\in\mathbb{N}$ and assume that $$\mathbb{E}|\xi_{i+1}+\ldots+\xi_k|^{\lambda_1}\leq (u_{i+1}+\ldots+ u_k)^{\lambda_2},\quad 0\leq i<k\leq m$$ for some $\lambda_1>0$, some $\lambda_2>1$ and some nonnegative numbers $u_1,\ldots, u_m$. Then $$\mathbb{E}(\max_{1\leq k\leq m}|\xi_1+\ldots+\xi_k|)^{\lambda_1}\leq A_{\lambda_1,\lambda_2}(u_1+\ldots+u_m)^{\lambda_2}$$ for a positive constant $A_{\lambda_1,\lambda_2}$.* **Lemma 16**. *Suppose (A2) and (A5). Then, for any integer $r\geq 2$, there exists a positive constant $A_r$ such that $$\label{ineq4} \mathbb{E}(\max_{1\leq k\leq j}\,|X^\ast(t_{k-1,\,n})-X^\ast(t_n)|)^{2r}\leq A_r (v_{n+1}(\kappa,\mu)-v_n(\kappa,\mu))^r.$$ Here, $j$ and $(t_{k,\,n})_{0\leq k\leq j}$ are as defined in (A5), and $v_n(\kappa,\mu)$ is as defined in [\[tn\]](#tn){reference-type="eqref" reference="tn"}.* *Proof.* We first show that the assumption of Lemma [Lemma 15](#lem:bor){reference-type="ref" reference="lem:bor"} holds with $\lambda_1=2r$, $\lambda_2=r$, $m=j-1$, $\xi_k:=X^\ast(t_{k,\,n})-X^\ast(t_{k-1,\,n})$ and $u_k:=D_r^{1/r}(f(t_{k,\,n})-f(t_{k-1,\,n}))$ for $k\in\mathbb{N}$, where $D_r$ is the constant defined in Lemma [Lemma 14](#ineq1){reference-type="ref" reference="ineq1"}. Let $0\leq i<k\leq j-1$. By (A5), $f(t_{k,\,n})-f(t_{i,\,n})=\sum_{l=i+1}^k (f(t_{l,\,n})-f(t_{l-1,\,n}))\geq 1$. This in combination with Lemma [Lemma 14](#ineq1){reference-type="ref" reference="ineq1"} yields $$\begin{gathered} \mathbb{E}|\xi_{i+1}+\ldots+\xi_k|^{2r}=\mathbb{E}(X^\ast(t_{k,\,n})-X^\ast(t_{i,\,n}))^{2r}\\\leq D_r \max((f(t_{k,\,n})-f(t_{i,\,n}))^r, f(t_{k,\,n})-f(t_{i,\,n}))=D_r (f(t_{k,\,n})-f(t_{i,\,n}))^r=\Big(\sum_{l=i+1}^k u_l\Big)^r,\end{gathered}$$ thereby proving that the assumption of Lemma [Lemma 15](#lem:bor){reference-type="ref" reference="lem:bor"} does indeed hold. Hence, inequality [\[ineq4\]](#ineq4){reference-type="eqref" reference="ineq4"} follows from Lemma [Lemma 15](#lem:bor){reference-type="ref" reference="lem:bor"} and the definition of $t_n$: $$\begin{gathered} \mathbb{E}(\max_{1\leq k\leq j}\,|X^\ast(t_{k-1,\,n})-X^\ast(t_n)|)^{2r}=\mathbb{E}(\max_{1\leq k\leq j-1}|\xi_1+\ldots+\xi_k|)^{2r}\leq A_{2r,\,r}\Big(\sum_{l=1}^{j-1} u_l\Big)^r\\=A_{2r,\,r} D_r (f(t_{j-1,\,n})-f(t_n))^r\leq A_{2r,\,r} D_r (v_{n+1}(\kappa,\mu)-v_n(\kappa,\mu))^r.\end{gathered}$$ ◻ ## Proof of Theorem [Theorem 11](#thm:main){reference-type="ref" reference="thm:main"} {#sec:main} We start with a lemma and a proposition which are in essence Lemma 4.13 and Proposition 4.7 in [@Buraczewski+Iksanov+Kotelnikova:2023+]. Although our present assumption (A3) is slightly different from the corresponding assumption in [@Buraczewski+Iksanov+Kotelnikova:2023+], we have checked that the proofs of the aforementioned results in [@Buraczewski+Iksanov+Kotelnikova:2023+] go through. **Lemma 17**. *Suppose (A1), (A3), (B1) and either (B21) or (B22) and let $\mu\geq 1$ be as given in [\[eq:infim\]](#eq:infim){reference-type="eqref" reference="eq:infim"}. For sufficiently small $\delta >0$, pick $\gamma\in (0, (\sqrt{5}-1)/2) >0$ satisfying $(1+\gamma)(1-\delta^2/8)<1$. Then $$\limsup_{n\to \infty} (\liminf_{n\to\infty}) \frac 1{(2a(\nu_n )h_0(a(\nu_n))^{1/2}} \sum_{k\ge 1} \eta_k(\nu_n ) \ge 1-\delta~(\leq -(1-\delta)) \quad\text{{\rm a.s.}},$$ where $\nu_n$ is either $\tau_n$ or $\lfloor \tau_n\rfloor$, and $\tau_n=\tau_n(\gamma,\mu)$, with $\gamma$ chosen above, is as defined in [\[eq:tau\]](#eq:tau){reference-type="eqref" reference="eq:tau"}.* **Proposition 18**. *Suppose (A1), (A3), (B1) and either (B21) or (B22). Then, with $\mu\geq 1$ and $q\geq 0$ as defined in [\[eq:infim\]](#eq:infim){reference-type="eqref" reference="eq:infim"} and [\[eq:infim2\]](#eq:infim2){reference-type="eqref" reference="eq:infim2"}, respectively, $$\limsup_{t\to\infty}(\liminf_{t\to\infty})\frac{X^\ast(t)}{(2(q+1)a(t) \log\log a(t))^{1/2}}\geq 1~(\leq -1)\quad\text{{\rm a.s.}}$$ and $$\limsup_{t\to\infty}(\liminf_{t\to\infty})\frac{X^\ast(t)}{(2(\mu-1)a(t)\log a(t))^{1/2}}\geq 1~(\leq -1)\quad \text{{\rm a.s.}}$$ in the cases $\mu=1$ and $\mu>1$, respectively.* Proposition [Proposition 19](#lilhalf1){reference-type="ref" reference="lilhalf1"} is a counterpart of Proposition 4.6 in [@Buraczewski+Iksanov+Kotelnikova:2023+]. Unlike the two previous results it requires a proof. **Proposition 19**. *Suppose (A1)-(A5). Then, with $\mu\geq 1$ and $q\geq 0$ as defined in [\[eq:infim\]](#eq:infim){reference-type="eqref" reference="eq:infim"} and [\[eq:infim2\]](#eq:infim2){reference-type="eqref" reference="eq:infim2"}, respectively, $$\limsup_{t\to\infty}(\liminf_{t\to\infty})\frac{X^\ast(t)}{(2(q+1)a(t) \log\log a(t))^{1/2}}\leq 1~(\geq -1)\quad \text{\rm a.s.}$$ and $$\limsup_{t\to\infty}(\liminf_{t\to\infty})\frac{X^\ast(t)}{(2(\mu-1)a(t)\log a(t))^{1/2}}\leq 1~(\geq -1)\quad \text{{\rm a.s.}}$$ in the cases $\mu=1$ and $\mu>1$, respectively.* *Proof of Proposition [Proposition 19](#lilhalf1){reference-type="ref" reference="lilhalf1"}.* In view of (A4), it is enough to show that, for each $\varrho\in (0,1)$ and each positive $\kappa$ sufficiently close to $0$, $$\label{princip} \limsup_{n\to\infty}\frac{\sup_{u\in [t_n,\,t_{n+1}]}\,X^\ast(u)}{(2 a(s_n) h_\varrho(a(s_n)))^{1/2}}\leq 1+\kappa\quad\text{a.s.},$$ where $t_n=t_n(\kappa,\mu)$ and $s_n=s_n(\kappa,\mu)$ are as defined in [\[tn\]](#tn){reference-type="eqref" reference="tn"} and (A4), respectively, $h_\varrho=g_{1,\,q_\varrho}$ if $\mu$ in [\[eq:infim\]](#eq:infim){reference-type="eqref" reference="eq:infim"} is equal to $1$ and $h_\varrho=g_{\mu_\varrho}$ if $\mu>1$ (see [\[eq:defg\]](#eq:defg){reference-type="eqref" reference="eq:defg"} for the definitions of $g_{1,\,q_\varrho}$ and $g_{\mu_\varrho}$). Indeed, if [\[princip\]](#princip){reference-type="eqref" reference="princip"} holds true, then, for large enough $n$, $$\begin{gathered} \limsup_{t\to\infty}\frac{X^\ast(t)}{(2 a(t) h_\varrho (a(t)))^{1/2}}=\limsup_{t\to\infty}\frac{X^\ast(t)}{(2 a_0(t) h_\varrho (a_0(t)))^{1/2}}\\\leq \limsup_{n\to\infty}\frac{\sup_{u\in [t_n,\,t_{n+1}]}\,X^\ast(u)}{(2 a_0(s_n)h_\varrho (a_0(s_n)))^{1/2}}=\limsup_{n\to\infty}\frac{\sup_{u\in [t_n,\,t_{n+1}]}\,X^\ast(u)}{(2 a(s_n)h_\varrho (a(s_n)))^{1/2}}\leq 1+\kappa\quad \text{a.s.}\end{gathered}$$ The relation $\liminf_{t\to\infty}\frac{X^\ast(t)}{(2 a(t) h_\varrho (a(t)))^{1/2}}\geq -1-\kappa$ a.s. does not require a separate proof. It follows from the argument for $\limsup$ upon replacing $\eta_k(t)$ with $-\eta_k(t)$. To obtain [\[princip\]](#princip){reference-type="eqref" reference="princip"}, we first prove in Lemma [Lemma 20](#inter1){reference-type="ref" reference="inter1"} that $$\label{princip1} {\limsup}_{n\to\infty}\frac{X^\ast(s_n)}{(2 a(s_n)h_\varrho (a(s_n)))^{1/2}}\leq 1+\kappa\quad \text{{\rm a.s.}}$$ and then show that $$\label{princip2} \lim_{n\to\infty}\frac{\sup_{u\in [t_n,\, t_{n+1}]}|X^\ast(u)-X^\ast(s_n)|}{(a(s_n)h_\varrho(a(s_n)))^{1/2}}=0\quad\text{a.s.}$$ **Lemma 20**. *Suppose (A1), (A3) and (A4). Then relation [\[princip1\]](#princip1){reference-type="eqref" reference="princip1"} holds for any $\kappa\in (0, (\sqrt{5}-1)/2)$.* *Proof.* Fix any $\kappa\in (0,(\sqrt{5}-1)/2)$. We first show that there exists $\rho=\rho(\kappa)>0$ satisfying $$\label{eq:choiceofrho} (1-\kappa)(1+\kappa)^2(2-\exp(2(1+\kappa)\rho))>1.$$ To prove this, note that our choice of $\kappa$ ensures $(1-\kappa)(1+\kappa)^2>1$. Observe next that as positive $\rho$ approaches $0$, $2-\exp(2(1+\kappa)\rho)$ becomes arbitrary close to $1$, thereby justifying $2-\exp(2(1+\kappa)\rho)>(1-\kappa)^{-1}(1+\kappa)^{-2}$. By Lemma 4.1 in [@Buraczewski+Iksanov+Kotelnikova:2023+], for $\vartheta\in\mathbb{R}$ and $t\geq 0$, $$\mathbb{E}\exp(\vartheta X^\ast(t))\leq \exp(2^{-1}\vartheta^2\exp(|\vartheta|)a(t)).$$ Fix any $\theta\in\mathbb{R}$ and put $\vartheta=\theta/(2a(s_n)h_\varrho(a(s_n)))^{1/2}$. Observe that, for large enough $n$, $h_\varrho(a(s_n))/a(s_n)\leq2 \rho^2$ for $\rho$ satisfying [\[eq:choiceofrho\]](#eq:choiceofrho){reference-type="eqref" reference="eq:choiceofrho"}. An application of Markov's inequality then yields, for large $n$ as above, $$\begin{gathered} \mathbb{P}\Big\{\frac{X^\ast(s_n)}{(2a(s_n)h_\varrho(a(s_n)))^{1/2}}>1+\kappa\Big\}\leq {\rm e}^{-(1+\kappa)\theta} \mathbb{E}\exp\Big(\theta \frac{X^\ast(s_n)}{(2a(s_n)h_\varrho(a(s_n)))^{1/2}}\Big)\\\leq \exp\Big(-(1+\kappa)\theta+\frac{\theta ^2}{4h_\varrho(a(s_n))}\exp\Big(\frac{\rho |\theta|}{h_\varrho(a(s_n))}\Big)\Big).\end{gathered}$$ Putting $\theta=2(1+\kappa)h_\varrho(a(s_n))$ and then invoking Lemma [Lemma 13](#lem:aux1){reference-type="ref" reference="lem:aux1"}(a) we obtain $$\begin{gathered} \mathbb{P}\Big\{\frac{X^\ast(s_n)}{(2a(s_n)h_\varrho(a(s_n)))^{1/2}}>1+\kappa\Big\}\leq \exp(-(1+\kappa)^2(2-\exp(2(1+\kappa)\rho))h_\varrho(a(s_n)))\\=O\Big(\frac{1}{n^{(1-\kappa)(1+\kappa)^2(2-\exp(2(1+\kappa)\rho))}}\Big),\quad n\to\infty.\end{gathered}$$ According to [\[eq:choiceofrho\]](#eq:choiceofrho){reference-type="eqref" reference="eq:choiceofrho"}, $$\sum_{n\geq n_0}\mathbb{P}\Big\{\frac{X^\ast(t_n)}{(2a(t_n)h_\varrho(a(t_n)))^{1/2}}>1+\kappa\Big\}<\infty$$ for some $n_0\in\mathbb{N}$ large enough. An application of the Borel-Cantelli lemma completes the proof of Lemma [Lemma 20](#inter1){reference-type="ref" reference="inter1"}. ◻ Next, in order to prove [\[princip2\]](#princip2){reference-type="eqref" reference="princip2"} it suffices to show that $$\label{eq:3} \lim_{n\to\infty}\frac{\sup_{u\in [t_n,\, t_{n+1}]}|X^\ast(u)-X^\ast(t_n)|}{(a(s_n))^{1/2}}=0\quad\text{a.s.}$$ and $$\label{eq:4} \lim_{n\to\infty}\frac{|X^\ast(t_n)-X^\ast(s_n)|}{(a(s_n))^{1/2}}=0\quad\text{a.s.}$$ Since [\[eq:4\]](#eq:4){reference-type="eqref" reference="eq:4"} is a consequence of [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"}, we are left with proving [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"}. Let $t_n=t_{0,\,n}<\ldots<t_{j,\,n}=t_{n+1}$ be a partition defined in (A5). With this at hand, write $$\begin{gathered} \sup_{u\in [t_n,\, t_{n+1}]}|X^\ast(u)-X^\ast(t_n)|\\=\max_{1\leq k\leq j}\sup_{v\in [0,\,t_{k,\,n}-t_{k-1,\,n}]}|(X^\ast(t_{k-1,\,n})-X^\ast(t_n))+(X^\ast(t_{k-1,\,n}+v)-X^\ast(t_{k-1,\,n}))|\\\leq \max_{1\leq k\leq j}|X^\ast(t_{k-1,\,n})-X^\ast(t_n)|\\+\max_{1\leq k\leq j}\sup_{v\in [0,\,t_{k,\,n}-t_{k-1,\,n}]}|(X^\ast(t_{k-1,\,n}+v)-X^\ast(t_{k-1,\,n}))|\quad\text{a.s.}\end{gathered}$$ By Markov's inequality and Lemma [Lemma 16](#billappl){reference-type="ref" reference="billappl"}, for any $r>0$ and all $\varepsilon>0$, $$\begin{gathered} \mathbb{P}\Big\{\max_{1\leq k\leq j}|X^\ast(t_{k-1,\,n})-X^\ast(t_n)|>\varepsilon (a(s_n))^{1/2}\Big\}\leq \frac{\mathbb{E}(\max_{1\leq k\leq j}\,|X^\ast(t_{k-1,\,n})-X^\ast(t_n)|)^{2r}}{\varepsilon^{2r}(a(s_n))^r}\\\leq \frac{A_r(v_{n+1}(\kappa, \mu)-v_n(\kappa, \mu))^r}{\varepsilon^{2r}(a(s_n))^r}.\end{gathered}$$ By Lemma [Lemma 13](#lem:aux1){reference-type="ref" reference="lem:aux1"}(b), there exists an integer $r\geq 2$ such that the right-hand side forms a sequence which is summable in $n$. Hence, an application of the Borel-Cantelli lemma yields $$\lim_{n\to\infty}\frac{\max_{1\leq k\leq j}|X^\ast(t_{k-1,\,n})-X^\ast(t_n)|}{(a(s_n))^{1/2}}=0\quad\text{a.s.}$$ Next, we work towards proving that $$\lim_{n\to\infty}\frac{\max_{1\leq k\leq j}\sup_{v\in [0,\, t_{k,\,n}-t_{k-1,\,n}]}|X^\ast(t_{k-1,\,n}+v)-X^\ast(t_{k-1,\,n})|}{(a(s_n))^{1/2}}=0\quad \text{a.s.}$$ According to (A2), for any $0\le s<t$, $|X(t)-X(s)|\le Y(t)-Y(s)$ a.s., where the process $Y$ is a.s. nondecreasing. Taking into account Remark [Remark 7](#rem:bf){reference-type="ref" reference="rem:bf"} we obtain $$\begin{gathered} \sup_{v\in [0,\,t_{k,\,n}-t_{k-1,\,n}]}|X^\ast(t_{k-1,\,n}+v)-X^\ast(t_{k-1,\,n})|\\\leq \sup_{v\in [0,\,t_{k,\,n}-t_{k-1,\,n}]}|X(t_{k-1,\,n}+v)-X(t_{k-1,\,n})|+\sup_{v\in [0,\,t_{k,\,n}-t_{k-1,\,n}]}|b(t_{k-1,\,n}+v)-b(t_{k-1,\,n})|\\ \leq \sup_{v\in [0,\,t_{k,\,n}-t_{k-1,\,n}]}( Y(t_{k-1,\,n}+v)-Y(t_{k-1,\,n})) +\sup_{v\in [0,\,t_{k,\,n}-t_{k-1,\,n}]}( f(t_{k-1,\,n}+v)-f(t_{k-1,\,n})) \\= Y(t_{k,\,n})-Y(t_{k-1,\,n})+f(t_{k,\,n})-f(t_{k-1,\,n})\quad\text{a.s.}\end{gathered}$$ By (A1) and (A5), $$\frac{\max_{1\leq k\leq j}\,(f(t_{k,\,n})-f(t_{k-1,\,n}))}{(a(s_n))^{1/2}}\leq \frac{A}{(a(s_n))^{1/2}}~\to 0,\quad n\to\infty.$$ Finally, for all $\varepsilon>0$, $$\begin{gathered} \mathbb{P}\big\{\max_{1\leq k\leq j}(Y(t_{k,\,n})-Y(t_{k-1,\,n}))>\varepsilon (a(s_n))^{1/2}\big\}\le \sum_{k=1}^{j} \mathbb{P}\big\{Y(t_{k,\,n})-Y(t_{k-1,\,n})>\varepsilon (a(s_n))^{1/2}\big\}\\ \le {\rm e}^{-\varepsilon (a(s_n))^{1/2}} \sum_{k=1}^{j} \mathbb{E}{\rm e}^{Y(t_{k,\,n})-Y(t_{k-1,\,n})}\le {\rm e}^{-\varepsilon (a(s_n))^{1/2}} \sum_{k=1}^{j} \exp(({\rm e}^M-1)(f(t_{k,\,n})-f(t_{k-1,\,n})))\\\leq \exp(A({\rm e}^M-1))j {\rm e}^{-\varepsilon (a(s_n))^{1/2}},\end{gathered}$$ having utilized Markov's inequality for the second inequality, Lemma [Lemma 12](#lem:exp_mom){reference-type="ref" reference="lem:exp_mom"} for the third and (A5) for the fourth. Invoking (A5) once again we conclude that the right-hand side is summable in $n$. Hence, an application of the Borel-Cantelli lemma yields $$\lim_{n\to\infty}\frac{\max_{1\leq k\leq j}(X(t_{k,\,n})-X(t_{k-1,\,n}))}{(a(s_n))^{1/2}}=0\quad\text{a.s.}$$ The proofs of both [\[eq:3\]](#eq:3){reference-type="eqref" reference="eq:3"} and Proposition [Proposition 19](#lilhalf1){reference-type="ref" reference="lilhalf1"} are complete. ◻ # Proofs related to Karlin's occupancy scheme {#sec:Karlin} ## Auxiliary results For ease of reference, we state two known results. The former is an obvious extension of Theorem 1.5.3 in [@Bingham+Goldie+Teugels:1989]. The latter is Lemma 6.2 in [@Buraczewski+Iksanov+Kotelnikova:2023+]. **Lemma 21**. *Let $f$ be a function which varies regularly at $\infty$ of positive index and $g$ a positive nondecreasing function with $\lim_{t\to\infty}g(t)=\infty$. Then there exists a nondecreasing function $h$ satisfying $f(g(t))\sim h(t)$ as $t\to\infty$.* **Lemma 22**. *(a) Conditions [\[eq:deHaan\]](#eq:deHaan){reference-type="eqref" reference="eq:deHaan"} and [\[eq:slowly\]](#eq:slowly){reference-type="eqref" reference="eq:slowly"} entail $$\label{eq:rho} \rho(t)~\sim~(\beta+1)^{-1}(\log t)^{\beta+1}l(\log t),\quad t\to\infty.$$* *(b) Conditions [\[eq:deHaan\]](#eq:deHaan){reference-type="eqref" reference="eq:deHaan"} and [\[eq:slowly2\]](#eq:slowly2){reference-type="eqref" reference="eq:slowly2"} entail $$\label{eq:rho2} \rho(t)~\sim~(\sigma \lambda)^{-1}\exp(\sigma (\log t)^\lambda)(\log t)^{1-\lambda},\quad t\to\infty.$$* ## Asymptotic behavior of $\mathbb{E}K_j(t)$ and ${\rm Var \,}K_j(t)$ Given next is a collection of results on the asymptotics of $\mathbb{E}K_j(t)$ and ${\rm Var \,}K_j(t)$ taken from Lemma 6.5 of [@Buraczewski+Iksanov+Kotelnikova:2023+]. Recall that $\Pi_{\ell,\,\infty}$ denotes the subclass of the de Haan class $\Pi$ with the auxiliary functions $\ell$, see [\[eq:deHaan\]](#eq:deHaan){reference-type="eqref" reference="eq:deHaan"}, satisfying $\lim_{t\to\infty}\ell(t)=\infty$. **Lemma 23**. *Assume that $\rho\in \Pi_{\ell,\,\infty}$. Then, for each $j\in\mathbb{N}$, $$\label{eq:mean0old} \mathbb{E}K_j(t)~\sim~ \rho(t),\quad t\to\infty$$ and $$%\label{eq:var0old} {\rm Var}\,K_j(t)~\sim~ \Big(\log 2- \sum_{k=1}^{j-1}\frac{(2k-1)!}{(k!)^2 2^{2k}}\Big)\ell(t),\quad t\to\infty.$$* *Assume that $\rho(t)\sim t^\alpha L(t)$ as $t\to\infty$ for some $\alpha\in(0,1]$ and some $L$ slowly varying at $\infty$. If $\alpha\in(0,1)$ and $j\in\mathbb{N}$ or $\alpha=1$ and $j\geq 2$, then, as $t\to\infty$, $$\label{eq:oldmomincreas} \mathbb{E}K_j(t)~\sim~\frac{\Gamma(j-\alpha)}{(j-1)!}\rho(t),$$ and $$\label{eq:varold} \lim_{t\to\infty}\frac{{\rm Var}\, K_j(t)}{\rho(t)}=\Big(\sum_{i=0}^{j-1}\frac{\Gamma(i+j-\alpha)}{i!(j-1)!2^{i+j-1-\alpha}}-\frac{\Gamma(j-\alpha)}{(j-1)!}\Big)>0.$$* *If $\alpha=1$, then $$\label{eq:varalone} {\rm Var}\, K_1(t)~\sim~ \mathbb{E}K_1(t)~\sim~ t\hat L(t),\quad t\to\infty.$$* ## Asymptotic behavior of $\mathbb{E}K^*_j(t)$ and ${\rm Var \,}K^*_j(t)$ For $j\in\mathbb{N}$, the asymptotics of $t\mapsto \mathbb{E}K^*_j(t)$ as stated in Theorems [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"}, [Theorem 4](#thm:Karlin){reference-type="ref" reference="thm:Karlin"} and [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"} can be found in Lemma 6.5 of [@Buraczewski+Iksanov+Kotelnikova:2023+]. Next, we show that, for $j\in\mathbb{N}$, the functions $t\mapsto {\rm Var \,}K^*_j(t)$ exhibit the asymptotics given in the aforementioned theorems. **Lemma 24**. *Assume that $\rho\in \Pi_{\ell,\,\infty}$. Then, for each $j\in\mathbb{N}$, $$\label{eq:var0} {\rm Var}\,K^*_j(t)~\sim~ \Big(\frac{1}{j}- \frac{(2j-1)!}{(j!)^2 2^{2j}}\Big)\ell(t),\quad t\to\infty.$$* *Assume that $\rho(t)\sim t^\alpha L(t)$ as $t\to\infty$ for some $\alpha\in(0,1]$ and some $L$ slowly varying at $\infty$. If $\alpha\in(0,1)$ and $j\in\mathbb{N}$ or $\alpha=1$ and $j\geq 2$, then, as $t\to\infty$, $$\label{eq:var} \lim_{t\to\infty}\frac{{\rm Var}\, K^*_j(t)}{t^\alpha L(t)}=c_{j,\,\alpha}>0$$ with $c_{j,\,\alpha}$ as defined in [\[eq:cj\]](#eq:cj){reference-type="eqref" reference="eq:cj"} and [\[eq:cj1\]](#eq:cj1){reference-type="eqref" reference="eq:cj1"}.* *If $\alpha=1$, then $$\label{eq:var1} {\rm Var}\, K^*_1(t)~\sim~ t\hat L(t),\quad t\to\infty.$$* *Proof.* Assume that $\rho\in\Pi_{\ell,\,\infty}$. Putting $u=v=0$ in formula (11) of [@Iksanov+Kotelnikova:2022] we obtain [\[eq:var0\]](#eq:var0){reference-type="eqref" reference="eq:var0"}. According to formula (6) in [@Gnedin+Hansen+Pitman:2007], $$\label{eq:equation_var} {\rm Var \,}K^*_j(t)=\mathbb{E}K^*_j(t)-2^{-2j}\binom{2j}{j}\mathbb{E}K^*_{2j}(2t), \quad t\ge 0, j\in\mathbb{N}.$$ Note that [\[eq:equation_var\]](#eq:equation_var){reference-type="eqref" reference="eq:equation_var"} does not require even regular variation assumption on $\rho$. Assume that $\rho$ is regularly varying at $\infty$ of index $\alpha=1$. We first discuss the properties of the function $\hat L$ stated in Theorem [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"}. By Lemma 3 in [@Karlin:1967], $\lim_{t\to\infty}t^{-1}\rho(t)=0$ and $\int_1^\infty y^{-2}\rho(y){\rm d}y\leq 1$. This implies that the function $\hat L(t)=\int_t^\infty y^{-1}L(y){\rm d}y$ is well-defined for large $t$ and thereupon $\lim_{t\to\infty}\hat L(t)=0$. According to Proposition 1.5.9b [@Bingham+Goldie+Teugels:1989], $\hat{L}$ is slowly varying at $\infty$ and satisfies [\[eq:Lhat\]](#eq:Lhat){reference-type="eqref" reference="eq:Lhat"}. This in combination with [\[eq:meK1\]](#eq:meK1){reference-type="eqref" reference="eq:meK1"}, [\[eq:exo\]](#eq:exo){reference-type="eqref" reference="eq:exo"} and [\[eq:equation_var\]](#eq:equation_var){reference-type="eqref" reference="eq:equation_var"} entails [\[eq:var1\]](#eq:var1){reference-type="eqref" reference="eq:var1"}. Assume now $\alpha\in(0,1)$ and $j\in\mathbb{N}$ or $\alpha=1$ and $j\geq 2$. Then invoking [\[eq:equation_var\]](#eq:equation_var){reference-type="eqref" reference="eq:equation_var"} and either [\[eq:meKgeneral\]](#eq:meKgeneral){reference-type="eqref" reference="eq:meKgeneral"} or [\[eq:meK1\]](#eq:meK1){reference-type="eqref" reference="eq:meK1"} we obtain $$\lim_{t\to\infty}\frac{{\rm Var \,}K^*_j(t)}{t^\alpha L(t)}=\lim_{t\to\infty}\frac{\mathbb{E}K^*_j(t)}{t^\alpha L(t)}-2^{-2j}\binom{2j}{j}\lim_{t\to\infty}\frac{\mathbb{E}K^*_{2j}(2t)}{t^\alpha L(t)}=c_{j,\,\alpha}.$$ We are left with showing that the constants $c_{j,\,\alpha}$ are positive for $\alpha\in (0,1)$ and $j\in\mathbb{N}$ and $\alpha=1$ and $j\geq 2$ or equivalently $$\frac{2^{\alpha}\Gamma(2j-\alpha)}{2^{2j}j!\Gamma(j-\alpha)}<1.$$ This is a consequence of $$\frac{2^{\alpha}\Gamma(2j-\alpha)}{2^{2j}j!\Gamma(j-\alpha)}<\frac{2\,(2j-1)!}{2^{2j}j!(j-1)!}=\frac{(2j-1)!}{(2j)!!(2j-2)!!}<1,$$ where $(2n)!!:=2\cdot 4\cdot\ldots\cdot (2n)$ for $n\in\mathbb{N}$. Here, the last inequality is justified with the help of mathematical induction. The proof of Lemma [Lemma 24](#var){reference-type="ref" reference="var"} is complete. ◻ ## Proof of Theorems [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"}, [Theorem 4](#thm:Karlin){reference-type="ref" reference="thm:Karlin"} and [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"} {#proof-of-theorems-thmkarlin0-thmkarlin-and-thmkarlin1} For $k\in\mathbb{N}$ and $t\geq 0$, denote by $\pi_k(t)$ the number of balls in box $k$ at time $t$ in the Poissonized version. It has already been mentioned in Section [1.1](#karlin){reference-type="ref" reference="karlin"} that the thinning property of Poisson processes implies that the processes $(\pi_1(t))_{t\geq 0}$, $(\pi_2(t))_{t\geq 0},\ldots$ are independent. Moreover, for $k\in\mathbb{N}$, $(\pi_k(t))_{t\geq 0}$ is a Poisson process with intensity $p_k$. As a consequence, both $K_j^*(t)$ and $K_j(t)$ can be represented as the sums of independent indicators $$K_j^*(t)=\sum_{k=1}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)=j\}}\quad\text{and}\quad K_j(t)=\sum_{k=1}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j\}},\quad t\ge 0, j\in\mathbb{N}.$$ Hence, it is reasonable to prove the desired LILs for the small counts by applying Theorem [Theorem 11](#thm:main){reference-type="ref" reference="thm:main"}. As a preparation, we start with a lemma which facilitates checking condition (B22) of Theorem [Theorem 11](#thm:main){reference-type="ref" reference="thm:main"}. **Lemma 25**. *Assume that either $\rho\in \Pi_{\ell,\,\infty}$ or $\rho$ is regularly varying at $\infty$ of index $\alpha\in(0,1]$. If $\rho\in\Pi_{\ell,\,\infty}$ and $j\in\mathbb{N}$ or $\alpha\in (0,1)$ and $j\in\mathbb{N}$ or $\alpha=1$ and $j\geq 2$, then for any positive functions $c$ and $d$ satisfying $\lim_{t\to\infty} c(t)=\infty$, $\lim_{t\to\infty}(c(t)/t)=0$ and $\lim_{t\to\infty}(d(t)/t)=\infty$, $$\label{eq:cutvar} {\rm Var}\Big(\sum_{k\ge 1} \mathop{\mathrm{\mathbbm{1}}}_{\{c(t)<1/p_k\leq d(t)\}}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)= j\}}\Big) ~\sim~ {\rm Var}\,K_j^*(t),\quad t\to\infty.$$* *Proof.* We start by proving a simple but an important inequality. Since $${\rm Cov \,}(\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j\}}, \mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j+1\}})=\mathbb{P}\{\pi_k(t)\ge j+1\}-\mathbb{P}\{\pi_k(t)\ge j\}\mathbb{P}\{\pi_k(t)\ge j+1\}\ge 0,$$ we infer $$\begin{gathered} {\rm Var \,}(\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)=j\}})={\rm Var \,}(\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j\}}-\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j+1\}})\\={\rm Var \,}(\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j+1\}})+{\rm Var \,}(\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j\}})-2{\rm Cov \,}(\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j\}}, \mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j+1\}})\\\le {\rm Var \,}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j+1\}}+{\rm Var \,}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j\}}.\end{gathered}$$ Therefore, it is enough to show that in the setting of the lemma, for all $j\geq 2$ in the case $\alpha=1$ and for all $j\in\mathbb{N}$ in the other cases, $$\label{eq:a} {\rm Var}\,\Big(\sum_{k\ge 1} \mathop{\mathrm{\mathbbm{1}}}_{\{1/p_k> d(t)\}}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j\}}\Big)= o({\rm Var \,}K_j^*(t)), \quad t\to\infty$$ and $$\label{eq:b} {\rm Var}\,\Big(\sum_{k\ge 1} \mathop{\mathrm{\mathbbm{1}}}_{\{1/p_k\leq c(t)\}}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j\}}\Big)=o({\rm Var \,}K_j^*(t)), \quad t\to\infty.$$ According to formulae (86), (87), (79) and (80) in [@Buraczewski+Iksanov+Kotelnikova:2023+], $$\label{eq:a121} {\rm Var}\,\Big(\sum_{k\ge 1} \mathop{\mathrm{\mathbbm{1}}}_{\{1/p_k> d(t)\}}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\geq j\}}\Big)= o(\ell(t)), \quad t\to\infty,$$ $$\label{eq:b121} {\rm Var}\,\Big(\sum_{k\ge 1} \mathop{\mathrm{\mathbbm{1}}}_{\{1/p_k\leq c(t)\}}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\geq j\}}\Big)=o(\ell(t)), \quad t\to\infty,$$ $$\label{eq:aold} {\rm Var}\,\Big(\sum_{k\ge 1} \mathop{\mathrm{\mathbbm{1}}}_{\{1/p_k> d(t)\}}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\geq j\}}\Big)= o(\rho(t)), \quad t\to\infty$$ and $$\label{eq:bold} {\rm Var}\,\Big(\sum_{k\ge 1} \mathop{\mathrm{\mathbbm{1}}}_{\{1/p_k\leq c(t)\}}\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\geq j\}}\Big)=o(\rho(t)), \quad t\to\infty.$$ In view of [\[eq:var0\]](#eq:var0){reference-type="eqref" reference="eq:var0"} or [\[eq:var\]](#eq:var){reference-type="eqref" reference="eq:var"}, depending on the setting, relations [\[eq:a121\]](#eq:a121){reference-type="eqref" reference="eq:a121"} or [\[eq:aold\]](#eq:aold){reference-type="eqref" reference="eq:aold"}, [\[eq:b121\]](#eq:b121){reference-type="eqref" reference="eq:b121"} or [\[eq:bold\]](#eq:bold){reference-type="eqref" reference="eq:bold"} are equivalent to [\[eq:a\]](#eq:a){reference-type="eqref" reference="eq:a"} and [\[eq:b\]](#eq:b){reference-type="eqref" reference="eq:b"}. ◻ *Proof of Theorems [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"}, [Theorem 4](#thm:Karlin){reference-type="ref" reference="thm:Karlin"} and [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"}.* We first prove Theorem [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"} in the case $j=1$. This setting is much simpler than the others, for the LIL for $$\label{eq:equ} K^*_1(t)=K_1(t)-K_2(t)$$ can be derived from the already available LILs for $K_1(t)$ and $K_2(t)$. The statements of Theorem [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"} concerning the function $\hat L$ has already been justified in the proof of Lemma [Lemma 24](#var){reference-type="ref" reference="var"}. According to [\[eq:varalone\]](#eq:varalone){reference-type="eqref" reference="eq:varalone"} and [\[eq:var1\]](#eq:var1){reference-type="eqref" reference="eq:var1"}, ${\rm Var \,}K^*_1(t)\sim{\rm Var \,}K_1(t)\sim t\hat L(t)$ as $t\to\infty$. Invoking the latter relation, [\[eq:varold\]](#eq:varold){reference-type="eqref" reference="eq:varold"} and [\[eq:Lhat\]](#eq:Lhat){reference-type="eqref" reference="eq:Lhat"} we conclude that, ${\rm Var \,}K_2(t)\sim 2^{-1}tL(t)=o({\rm Var \,}K_1(t))$ as $t\to\infty$. By Theorem 3.4 and Remark 1.7 in [@Buraczewski+Iksanov+Kotelnikova:2023+], $$\limsup_{t\to\infty}(\liminf_{t\to\infty})\frac{K_2(t)-\mathbb{E}K_2(t)}{({\rm Var \,}K_2(t)\log\log{\rm Var \,}K_2(t))^{1/2}}=2^{1/2}~(-2^{1/2})\quad\text{a.s.}$$ As a consequence, $K_2(t)-\mathbb{E}K_2(t)=o(({\rm Var \,}K_1(t)\log\log{\rm Var \,}K_1(t))^{1/2})$ a.s. as $t\to\infty$. Now, in view of [\[eq:equ\]](#eq:equ){reference-type="eqref" reference="eq:equ"}, $$\begin{gathered} \limsup_{t\to\infty} (\liminf_{t\to\infty}) \frac{K^*_1(t)-\mathbb{E}K^*_1(t)}{({\rm Var \,}K^*_1(t)\log\log{\rm Var \,}K^*_1(t))^{1/2}}\\=\limsup_{t\to\infty} (\liminf_{t\to\infty})\frac{K_1(t)-\mathbb{E}K_1(t)}{({\rm Var \,}K_1(t)\log\log{\rm Var \,}K_1(t))^{1/2}}\quad\text{a.s.}\end{gathered}$$ Armed with this, the claim of Theorem [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"} in the case $j=1$ is secured by Theorem 3.4 in [@Buraczewski+Iksanov+Kotelnikova:2023+]. Indeed, the theorem states that depending on whether relation [\[eq:exotic\]](#eq:exotic){reference-type="eqref" reference="eq:exotic"} holds or not, the right-hand side is either equal to $2^{1/2}$ ($-2^{1/2}$) or is not larger than $2^{1/2}$ (not smaller than $-2^{1/2}$) a.s. In the remaining part of the proof we treat simultaneously Theorems [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"} and [Theorem 4](#thm:Karlin){reference-type="ref" reference="thm:Karlin"} and the case $j\geq 2$ of Theorem [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"}. It has already been announced that our plan is to derive the LILs from Theorem [Theorem 11](#thm:main){reference-type="ref" reference="thm:main"}. Hence, now we work towards checking the conditions of the aforementioned theorem in the present setting. holds according to [\[eq:var0\]](#eq:var0){reference-type="eqref" reference="eq:var0"} in conjunction with $\lim_{t\to\infty}\ell(t)=\infty$, [\[eq:var\]](#eq:var){reference-type="eqref" reference="eq:var"} and [\[eq:var1\]](#eq:var1){reference-type="eqref" reference="eq:var1"}. is justified by a representation $\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)=j\}}=\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j\}}-\mathop{\mathrm{\mathbbm{1}}}_{\{\pi_k(t)\ge j+1\}}$ a.s., for all $k,j\in\mathbb{N}$ and $t\ge 0$, see Remark [Remark 8](#rem:A2_ex){reference-type="ref" reference="rem:A2_ex"}. The corresponding function $f$ is given by $$\begin{gathered} f_{j,\,\alpha}(t):=\sum_{k\ge 1}(\mathbb{P}\{\pi_k(t)\ge j\}+\mathbb{P}\{\pi_k(t)\ge j+1\})\\=\mathbb{E}K_j(t)+\mathbb{E}K_{j+1}(t)=\sum_{k\ge 1} \Big(1-\sum_{i=0}^{j-1} {\rm e}^{-p_kt}\frac{(p_kt)^i}{i!}\Big)+\sum_{k\ge 1} \Big(1-\sum_{i=0}^{j} {\rm e}^{-p_kt}\frac{(p_kt)^i}{i!}\Big).\end{gathered}$$ We bring out the dependence on $j$ and $\alpha$ to distinguish the so defined functions for the different settings. By the same reasoning, we write $a_{j,\,\alpha}$ instead of $a$, where $a(t)={\rm Var \,}K_j^*(t)$ for $t\geq 0$. Assume first that $\rho\in\Pi_{\ell,\,\infty}$. According to [\[eq:mean0old\]](#eq:mean0old){reference-type="eqref" reference="eq:mean0old"} and [\[eq:var0\]](#eq:var0){reference-type="eqref" reference="eq:var0"}, for each $j\in\mathbb{N}$, $f_{j,\,0}(t)\sim 2\rho(t)$ and $a_{j,\,0}(t)\sim C\ell(t)$ as $t\to\infty$, respectively. Here and hereafter, $C$ denotes a constant whose value is of importance and may vary from formula to formula. Under [\[eq:slowly\]](#eq:slowly){reference-type="eqref" reference="eq:slowly"}, invoking [\[eq:rho\]](#eq:rho){reference-type="eqref" reference="eq:rho"} we conclude that (A3) holds with $\mu=1/\beta+1$. Under [\[eq:slowly2\]](#eq:slowly2){reference-type="eqref" reference="eq:slowly2"}, using [\[eq:rho2\]](#eq:rho2){reference-type="eqref" reference="eq:rho2"} we infer $\mu=1$. Thus, we have to check the additional conditions pertaining to the case $\mu=1$. First, the function $f_{j,\alpha}$ is continuous. Second, $q=1/\lambda-1$ and $\mathcal{L}(t)\equiv 1$ for all $t\geq 0$ by another appeal to [\[eq:rho2\]](#eq:rho2){reference-type="eqref" reference="eq:rho2"}. Assume now that $\alpha\in (0,1)$ and $j\in\mathbb{N}$ or $\alpha=1$ and $j\ge 2$. Then, according to [\[eq:oldmomincreas\]](#eq:oldmomincreas){reference-type="eqref" reference="eq:oldmomincreas"} and [\[eq:var\]](#eq:var){reference-type="eqref" reference="eq:var"}, $f_{j,\,\alpha}(t)\sim C a_{j,\,\alpha}(t)$ as $t\to\infty$, which entails $\mu=1$. Further, $f_{j,\,\alpha}$ is continuous, $q=0$ and $\mathcal{L}(t)\equiv 1$ for $t\geq 0$. Denote by $a_{0;j,\,\alpha}$ a version of $a_0$ for the different settings. Assume first that $\rho\in\Pi_{\ell,\,\infty}$. Then, according to [\[eq:var0\]](#eq:var0){reference-type="eqref" reference="eq:var0"}, $a_{j,\,0}(t)\sim C \ell(t)$ as $t\to\infty$. Therefore, under [\[eq:slowly\]](#eq:slowly){reference-type="eqref" reference="eq:slowly"}, $a_{0;j,\,0}$ can be chosen as a monotone equivalent of $t\mapsto C(\log t)^{\beta}l(\log t)$ which exists by Lemma [Lemma 21](#lem:monotone){reference-type="ref" reference="lem:monotone"}. Under [\[eq:slowly2\]](#eq:slowly2){reference-type="eqref" reference="eq:slowly2"}, $a_{0;j,0}$ can be chosen as $a_{0;j,\,0}(t):=C\exp(\sigma(\log t)^\lambda)$ for all $t\ge 1$. Assume now that $\alpha\in (0,1)$ and $j\in\mathbb{N}$ or $\alpha=1$ and $j\ge 2$. Then, according to [\[eq:var\]](#eq:var){reference-type="eqref" reference="eq:var"}, $a_{0;j,\,\alpha}$ can be chosen as a monotone equivalent of $t\mapsto c_{j,\,\alpha} t^\alpha L(t)$ which exists by Lemma [Lemma 21](#lem:monotone){reference-type="ref" reference="lem:monotone"}. Thus, in all settings (A4) holds according to Remark [Remark 9](#rem:A4){reference-type="ref" reference="rem:A4"}. holds according to Remark [Remark 10](#suff){reference-type="ref" reference="suff"}, for $f_{j,\,\alpha}$ is continuous and strictly increasing. holds in view of $$\label{eq:small_var} a_{j,\,\alpha}(t)={\rm Var \,}K^*_j(t)=\sum_{k\ge 1} {\rm e}^{-p_kt}\frac{(p_kt)^j}{j!}\Big(1-{\rm e}^{-p_kt}\frac{(p_kt)^j}{j!}\Big),\quad t\geq 0$$ which shows that $a_{j,\,\alpha}$ is a continuous function. For $t>1$, put $c(t):=t/\log t$ and $d(t):=t\log t$ and then $$R_0(t):= \{k\in\mathbb{N}: c(t)<1/p_k\leq d(t)\}.$$ By Lemma [Lemma 25](#lem:karl){reference-type="ref" reference="lem:karl"}, in all settings relation [\[eq:one\]](#eq:one){reference-type="eqref" reference="eq:one"} which is the second part of (B22) holds. Passing to the first part of (B22), we are going to refer to the table below which contains all the necessary information. In the first line, we list the values of $\mu$ which have already been found while checking (A3). Recall that the definitions of $w_n(\gamma,\mu)$ and $\tau_n$ can be found right after formula [\[eq:tau\]](#eq:tau){reference-type="eqref" reference="eq:tau"} and in [\[eq:tau\]](#eq:tau){reference-type="eqref" reference="eq:tau"}, respectively. **Setting** **$\rho\in\Pi_{\ell,\,\infty}$, [\[eq:slowly\]](#eq:slowly){reference-type="eqref" reference="eq:slowly"}** $\rho\in\Pi_{\ell,\,\infty}$, [\[eq:slowly2\]](#eq:slowly2){reference-type="eqref" reference="eq:slowly2"} $\alpha\in(0,1)$, $j\in\mathbb{N}$ or $\alpha=1$, $j\ge 2$ ------------------- ------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------- $\mu$ $1/\beta+1$ $1$ $1$ $w_n(\gamma,\mu)$ $n^{\beta(1+\gamma)}$ $\exp(n^{\lambda(1+\gamma)})$ $\exp(n^{1+\gamma})$ $\tau_n\sim$ ${\rm e}^{n^{(1+\gamma)}}o({\rm e}^{n^{(1+\gamma)}})$ ${\rm e}^{\sigma^{-1/\lambda}n^{(1+\gamma)}}(1+o(1))$ ${\rm e}^{\alpha^{-1}n^{(1+\gamma)}}o({\rm e}^{\alpha^{-1}n^{(1+\gamma)}})$ We conclude that in all the settings $\tau_{n+1}/\tau_n$ diverges to $\infty$ superexponentially fast, whereas $\log\tau_n$ only grows polynomially fast. Hence, for large enough $n$, $c(\tau_{n+1})>d(\tau_n)$, which justifies the first part of (B22). The proofs of Theorems [Theorem 1](#thm:Karlin0){reference-type="ref" reference="thm:Karlin0"}, [Theorem 4](#thm:Karlin){reference-type="ref" reference="thm:Karlin"} and [Theorem 5](#thm:Karlin1){reference-type="ref" reference="thm:Karlin1"} are complete. ◻ ## Proofs of Theorem [Theorem 6](#thm:depoiss){reference-type="ref" reference="thm:depoiss"} {#proofs-of-theorem-thmdepoiss} We start with some preparatory work. It is known, see, for instance, Lemma 1 in [@Gnedin+Hansen+Pitman:2007], that for any probability distribution $(p_k)_{k\in\mathbb{N}}$ and $j\in\mathbb{N}$, $$\label{eq:mean_depois} \lim_{n\to\infty} |\mathbb{E}\mathcal{K}_j^*(n)-\mathbb{E}K_j^*(n)|=0.$$ However, we are not aware of a counterpart of this relation for variances. Proposition [Proposition 26](#prop:var_depoiss){reference-type="ref" reference="prop:var_depoiss"} fills up this gap. Recall that $\rho\in\Pi_{\ell,\,\infty}$ means that $\rho\in\Pi$ and that its auxiliary function $\ell$, see [\[eq:deHaan\]](#eq:deHaan){reference-type="eqref" reference="eq:deHaan"}, satisfies $\lim_{t\to\infty}\ell(t)=\infty$. **Proposition 26**. *Assume that either $\rho\in \Pi_{\ell,\,\infty}$ or $\rho$ is regularly varying at $\infty$ of index $\alpha\in (0,1]$. Then, for $j\in\mathbb{N}$, $$\label{eq:ratio} \lim_{n\to\infty} \frac{{\rm Var \,}\mathcal{K}^*_j(n)}{{\rm Var \,}K^*_j(n)}=1.$$* The proof of Proposition [Proposition 26](#prop:var_depoiss){reference-type="ref" reference="prop:var_depoiss"} is partly based on a lemma which is a slight extension of Lemma 6.9 in [@Buraczewski+Iksanov+Kotelnikova:2023+]. The new aspect of the lemma is that unlike the cited result it covers the case where $j=1$ and $l\geq 1$ simultaneously. **Lemma 27**. *Assume that either $\rho\in \Pi_{\ell,\,\infty}$ or $\rho$ is regularly varying at $\infty$ of index $\alpha\in(0,1]$. Then for $l\ge j$, $l,j\in\mathbb{N}$, $$\label{eq:series} \sum_{k\ge 1}\binom{n}{l}p_k^l(1-p_k)^{n}=O({\rm Var \,}K_j^*(n)),\quad n\to\infty.$$* **Proof.* According to the last formula in the proof of Lemma 6.9 in [@Buraczewski+Iksanov+Kotelnikova:2023+], $$\sum_{k\ge 1}\binom{n}{l}p_k^l(1-p_k)^{n}~\sim~\mathbb{E}K_{l}^*(n+l),\quad n\to\infty.$$ According to formulae [\[eq:meK0\]](#eq:meK0){reference-type="eqref" reference="eq:meK0"}, [\[eq:meKgeneral\]](#eq:meKgeneral){reference-type="eqref" reference="eq:meKgeneral"}, [\[eq:meK1\]](#eq:meK1){reference-type="eqref" reference="eq:meK1"} and [\[eq:exo\]](#eq:exo){reference-type="eqref" reference="eq:exo"}, the function $t\mapsto\mathbb{E}K_j^*(t)$ is regularly varying at $\infty$ of index $\alpha\in [0,1]$. This entails $\mathbb{E}K_{l}^*(n+l)\sim\mathbb{E}K_l^*(n)$ as $n\to\infty$.* *If $\rho\in \Pi_{\ell,\,\infty}$ or $\rho$ is regularly varying at $\infty$ of index $\alpha\in(0,1)$, then [\[eq:series\]](#eq:series){reference-type="eqref" reference="eq:series"} is a consequence of [\[eq:meK0\]](#eq:meK0){reference-type="eqref" reference="eq:meK0"} and [\[eq:varK0_1\]](#eq:varK0_1){reference-type="eqref" reference="eq:varK0_1"} or [\[eq:meKgeneral\]](#eq:meKgeneral){reference-type="eqref" reference="eq:meKgeneral"} and [\[eq:varKgeneral\]](#eq:varKgeneral){reference-type="eqref" reference="eq:varKgeneral"}. If $\rho$ is regularly varying at $\infty$ of index $\alpha=1$ and either $j,l\geq 2$ or $j=l=1$, then [\[eq:series\]](#eq:series){reference-type="eqref" reference="eq:series"} follows from [\[eq:meK1\]](#eq:meK1){reference-type="eqref" reference="eq:meK1"} and [\[eq:cj1\]](#eq:cj1){reference-type="eqref" reference="eq:cj1"} or [\[eq:exo\]](#eq:exo){reference-type="eqref" reference="eq:exo"}, respectively. Finally, under the latter regular variation assumption, if $j=1$ and $l\geq 2$, then [\[eq:series\]](#eq:series){reference-type="eqref" reference="eq:series"}, with $o$ replacing $O$, holds true according to [\[eq:meK1\]](#eq:meK1){reference-type="eqref" reference="eq:meK1"}, [\[eq:exo\]](#eq:exo){reference-type="eqref" reference="eq:exo"} and [\[eq:Lhat\]](#eq:Lhat){reference-type="eqref" reference="eq:Lhat"}. ◻* *Proof of Proposition [Proposition 26](#prop:var_depoiss){reference-type="ref" reference="prop:var_depoiss"}.* We start by noting that, in view of [\[eq:var0\]](#eq:var0){reference-type="eqref" reference="eq:var0"}, [\[eq:var\]](#eq:var){reference-type="eqref" reference="eq:var"} or [\[eq:var1\]](#eq:var1){reference-type="eqref" reference="eq:var1"}, for $j\in\mathbb{N}$, $$\label{eq:zero_var} \lim_{t\to\infty}\frac{{\rm Var \,}K^*_j(t)}{t}=0.$$ In the case $\alpha=1$ this is secured by $\lim_{t\to\infty}\hat L(t)=0$, which follows from the definition of $\hat L$. For $k,j,n\in\mathbb{N}$, the event $\{\text{the box}~ k~\text{contains exactly}~j~ \text{balls out of}~ n\}$ will be denoted by $A_k(j,n)$. Then $$\begin{gathered} {\rm Var \,}\mathcal{K}_j^*(n)=\sum_{k\ge 1}\mathbb{P}(A_k(j,n))(1-\mathbb{P}(A_k(j,n)))\\+\sum_{i\neq k}(\mathbb{P}(A_i(j,n)\cap A_k(j,n))-\mathbb{P}(A_i(j,n))\mathbb{P}(A_k(j,n))).\end{gathered}$$ It is enough to prove that $$\label{eq:variance} \lim_{n\to\infty} \frac{\sum_{k\ge 1}\mathbb{P}(A_k(j,n))(1-\mathbb{P}(A_k(j,n)))-{\rm Var \,}K^*_j(n)}{{\rm Var \,}K^*_j(n)}=0$$ and $$\label{eq:12} \lim_{n\to\infty}\frac{\sum_{i\neq k}(\mathbb{P}(A_i(j,n)\cap A_k(j,n))-\mathbb{P}(A_i(j,n))\mathbb{P}(A_k(j,n)))}{{\rm Var \,}K^*_j(n)}=0.$$ . For $k,j,n\in\mathbb{N}$, $$\mathbb{P}(A_k(j,n))=\binom{n}{j}p_k^j(1-p_k)^{n-j}.$$ In view of this and [\[eq:small_var\]](#eq:small_var){reference-type="eqref" reference="eq:small_var"}, the numerator in [\[eq:variance\]](#eq:variance){reference-type="eqref" reference="eq:variance"} is equal to $$\begin{gathered} \sum_{k\ge 1}\Big(\binom{n}{j}p_k^j(1-p_k)^{n-j}-{\rm e}^{-p_kn}\frac{(p_kn)^j}{j!}-\Big(\binom{n}{j}p_k^{j}(1-p_k)^{n-j}\Big)^2-\Big({\rm e}^{-p_kn}\frac{(p_kn)^{j}}{j!}\Big)^2\Big).\end{gathered}$$ According to the penultimate inequality in the proof of Lemma 2.13 in [@Iksanov+Kotelnikova:2022], for large enough $n$ and any $j\le n$, $$\label{eq:ineq1} -B_jp_k\le \binom{n}{i}p_k^j(1-p_k)^{n-j}-{\rm e}^{-p_kn}\frac{(p_kn)^j}{j!}\le A_jp_k$$ for some positive constants $A_j$ and $B_j$. Therefore, $$\sum_{k\ge 1} \Big|\binom{n}{j}p_k^j(1-p_k)^{n-j}-{\rm e}^{-p_kn}\frac{(p_kn)^j}{j!}\Big|\le \max(A_j,B_j)=o({\rm Var \,}K^*_j(n)),\quad n\to\infty$$ since under our assumptions $\lim_{n\to\infty}{\rm Var \,}K^*_j(n)=\infty$. Further, write $$\begin{gathered} \sum_{k\ge 1} \Big|\Big(\binom{n}{j}p_k^j(1-p_k)^{n-j}\Big)^2-\Big({\rm e}^{-p_kn}\frac{(p_kn)^j}{j!}\Big)^2\Big|\\=\sum_{k\ge 1} \Big|\Big(\binom{n}{j}p_k^j(1-p_k)^{n-j}-{\rm e}^{-p_kn}\frac{(p_kn)^j}{j!}\Big)\Big|\Big(\binom{n}{j}p_k^j(1-p_k)^{n-j}+{\rm e}^{-p_kn}\frac{(p_kn)^j}{j!}\Big)\\\le 2\sum_{k\ge 1} \Big|\binom{n}{j}p_k^j(1-p_k)^{n-j}-{\rm e}^{-p_kn}\frac{(p_kn)^j}{j!}\Big|=o({\rm Var \,}K^*_j(n)),\quad n\to\infty.\end{gathered}$$ The proof of [\[eq:variance\]](#eq:variance){reference-type="eqref" reference="eq:variance"} is complete. . For $k,i,j,n\in\mathbb{N}$, $$\begin{gathered} \mathbb{P}(A_i(j,n)\cap A_k(j,n))-\mathbb{P}(A_i(j,n))\mathbb{P}(A_k(j,n))\\=\binom{n}{j}\binom{n-j}{j}p_i^jp_k^j(1-p_i-p_k)^{n-2j}-\binom{n}{j}\binom{n}{j}p_i^jp_k^j(1-p_i)^{n-j}(1-p_k)^{n-j}\\ =:C_j(i,k,n).\end{gathered}$$ We shall use an appropriate decomposition of $C_j$ $$\begin{gathered} C_j(i,k,n)=\binom{n}{j}\binom{n-j}{j}p_i^jp_k^j\Big((1-p_i-p_k)^{n-2j}-(1-p_i)^{n-j}(1-p_k)^{n-j}\Big)\\-\binom{n}{j}\Big(\binom{n}{j}-\binom{n-j}{j}\Big)p_i^jp_k^j(1-p_i)^{n-j}(1-p_k)^{n-j}=: C^{(1)}_j(i,k,n)+C^{(2)}_j(i,k,n).\end{gathered}$$ To analyze $C^{(1)}_j$ we argue as in the proof of Lemma 1 on p. 152 in [@Gnedin+Hansen+Pitman:2007]. Invoking an expansion $$(x-y)^m=x^m+O(mx^{m-1}y),\quad m\to\infty,$$ which holds for positive $x$ and $y$, $x>y$, with $x=(1-p_i)(1-p_k)$, $y=p_ip_k$ and $m=n-2j$, we infer $$\begin{gathered} C^{(1)}_{j}(i,k,n)=\binom{n}{j}\binom{n-j}{j}p_i^jp_k^j\Big((1-p_i)^{n-2j}(1-p_k)^{n-2j}\Big(1-(1-p_i)^{j}(1-p_k)^{j} \Big)\\+O((n-2j)p_ip_k(1-p_i)^{n-2j-1}(1-p_k)^{n-2j-1})\Big)\\=: F_j(i,k,n)+G_j(i,k,n).\end{gathered}$$ Next, we intend to show that the contributions of $F_j(i,k,n)$, $G_j(i,k,n)$ and $C^{(2)}_j(i,k,n)$ to the sum are negligible in comparison to ${\rm Var \,}K^*_j(n)$ as $n\to\infty$. . With Lemma [Lemma 27](#lem:series){reference-type="ref" reference="lem:series"} at hand, we obtain $$\begin{gathered} \sum_{i\ne k}\binom{n}{j}\binom{n-j}{j}(n-2j)p_i^{j+1}p_k^{j+1}(1-p_i)^{n-2j-1}(1-p_k)^{n-2j-1}\\\le \sum_{i\ge 1}n\binom{n}{j}p_i^{j+1}(1-p_i)^{n-2j-1}\sum_{k\ge 1} \binom{n-j}{j}p_k^{j+1}(1-p_k)^{n-2j-1}\\= O\Big({\rm Var \,}K^*_j(n)\frac{{\rm Var \,}K^*_j(n)}{n}\Big)=o({\rm Var \,}K^*_j(n)),\quad n\to\infty\end{gathered}$$ having utilized [\[eq:zero_var\]](#eq:zero_var){reference-type="eqref" reference="eq:zero_var"} for the last limit relation. . For $m\in\mathbb{N}$ and $x\in [0,1]$, $1-x^m\leq m(1-x)$. Using this with $m=j$ and $x=(1-p_i)(1-p_k)$ we conclude that $1-(1-p_i)^j(1-p_k)^j\leq j(p_i+p_k-p_ip_k)\leq j(p_i+p_k)$ and thereupon $$\begin{gathered} F_j(i,k,n)\le j\binom{n}{j}\binom{n-j}{j}p_i^jp_k^j(1-p_i)^{n-2j}(1-p_k)^{n-2j}(p_i+p_k)\\ =:F^{(1)}_j(i,k,n)+F^{(2)}_j(i,k,n).\end{gathered}$$ Further, invoking Lemma [Lemma 27](#lem:series){reference-type="ref" reference="lem:series"} yields $$\begin{gathered} 0\le\sum_{i\neq k}F^{(1)}_j(i,k,n)\le j\sum_{i\ge 1} \binom{n}{j}p_i^{j+1}(1-p_i)^{n-2j}\sum_{k\ge1}\binom{n-j}{j}p_k^j(1-p_k)^{n-2j}\\=O\Big(\frac{{\rm Var \,}{K}^*_{j}(n)}{n}{\rm Var \,}{K}^*_j(n)\Big)=o({\rm Var \,}K^*_j(n)),\quad n\to\infty.\end{gathered}$$ Here, the latter asymptotic relation is a consequence of [\[eq:zero_var\]](#eq:zero_var){reference-type="eqref" reference="eq:zero_var"}. The argument for $F_j^{(2)}$ is analogous, and we omit details. . Notice that $\binom{n}{j}-\binom{n-j}{j}=O(n^{j-1})$ as $n\to\infty$. Hence, mimicking the argument used for the analysis of $F_j^{(1)}$ we conclude that $$\begin{gathered} \sum_{i\ne k} |C^{(2)}_j(i,k,n)| =O\Big(\sum_{i\ne k}n^{2j-1}p_i^jp_k^j(1-p_i)^{n-j}(1-p_k)^{n-j}\Big)\\=O\Big( \sum_{i\ge 1}n^jp_i^j(1-p_i)^{n-j}\sum_{k\ge 1}n^{j-1}p_k^j(1-p_k)^{n-j}\Big)=O\Big({\rm Var \,}K^*_j(n)\frac{{\rm Var \,}K^*_j(n)}{n}\Big)\\=o({\rm Var \,}K_j(n)), \quad n\to\infty.\end{gathered}$$ Combining all the fragments together we arrive at [\[eq:12\]](#eq:12){reference-type="eqref" reference="eq:12"}. ◻ With Proposition [Proposition 26](#prop:var_depoiss){reference-type="ref" reference="prop:var_depoiss"} at hand, we are ready to prove the LIL stated in Theorem [Theorem 6](#thm:depoiss){reference-type="ref" reference="thm:depoiss"}. We argue along the lines of the proof of Theorem 3.7 in [@Buraczewski+Iksanov+Kotelnikova:2023+]. *Proof of Theorem [Theorem 6](#thm:depoiss){reference-type="ref" reference="thm:depoiss"}..* The deterministic and Poissonized schemes discussed in Section [1.1](#karlin){reference-type="ref" reference="karlin"} are not necessarily defined on a common probability space. Our plan is to deduce LILs for $\mathcal{K}_j^\ast(n)$ from the corresponding LILs for $K_j^\ast(t)$. To this end, we need to *couple* the two schemes. Let $X_1$, $X_2,\ldots$ be independent random variables with distribution $(p_k)_{k\in\mathbb{N}}$, which are independent of a Poisson process $\pi$ and particularly its arrival sequence $(S_n)_{n\in\mathbb{N}}$. For all $j,n\in\mathbb{N}$ and $t\geq 0$, we define coupled versions of $\mathcal{K}_j(n)$, $\mathcal{K}_j^*(n)$, $K_j(t)$ and $K_j^\ast(t)$ as follows, keeping the notation for the variables unchanged: $$\mathcal{K}_j(n)=\#~\text{of distinct values that the variables}~ X_1, X_2, \ldots, X_n~\text{take at least $j$ times},$$ $$\mathcal{K}_j^*(n)=\#~ \text{of distinct values that the variables}~ X_1, X_2, \ldots, X_n~\text{take exactly $j$ times},$$ $$K_j(t)=\#~ \text{of distinct values that the variables}~ X_1, X_2, \ldots, X_{\pi(t)}~\text{take at least $j$ times},$$ $$K_j^*(t)=\#~ \text{of distinct values that the variables}~ X_1, X_2, \ldots, X_{\pi(t)}~\text{take exactly $j$ times}.$$ To justify the construction, observe that the variable $X_i$ can be thought of as the index of a box hit by the $i$th ball. The most important conclusion of the preceding discussion is that, for all $j,n\in\mathbb{N}$, $\mathcal{K}_j^*(n)=K_{j}^*(S_n)$ a.s. (for the coupled variables). We prove the result in several steps. . According to Step 2 of the proof of Theorem 3.7 in [@Buraczewski+Iksanov+Kotelnikova:2023+], $$\lim_{n\to\infty}\frac{K_{j}(S_n)-K_j(n)}{({\rm Var \,}K_j(n)m({\rm Var \,}K_j(n)))^{1/2}}=0\quad\text{a.s.},$$ where $m(t)=\log t$ under [\[eq:deHaan\]](#eq:deHaan){reference-type="eqref" reference="eq:deHaan"} and [\[eq:slowly\]](#eq:slowly){reference-type="eqref" reference="eq:slowly"} and $m(t)=\log\log t$ under the other assumptions of Theorem [Theorem 6](#thm:depoiss){reference-type="ref" reference="thm:depoiss"}. By Lemmas [Lemma 23](#lem:old){reference-type="ref" reference="lem:old"} and [Lemma 24](#var){reference-type="ref" reference="var"}, for $j\in\mathbb{N}$, ${\rm Var \,}K^*_j(t)$ and ${\rm Var \,}K_j(t)$ are asymptotically equivalent up to a constant, whence $$\lim_{n\to\infty}\frac{K_{j}(S_n)-K_j(n)}{({\rm Var \,}K^*_j(n)m({\rm Var \,}K^*_j(n)))^{1/2}}=0\quad\text{a.s.}$$ By Lemma [Lemma 24](#var){reference-type="ref" reference="var"}, for $j\in\mathbb{N}$, ${\rm Var \,}K^\ast_{j+1}(t)$ and ${\rm Var \,}K^*_j(t)$ are asymptotically equivalent up to a constant, unless $\alpha=j=1$. In the latter case, invoking in addition [\[eq:Lhat\]](#eq:Lhat){reference-type="eqref" reference="eq:Lhat"} we obtain ${\rm Var \,}K^*_{j+1}(t)=o({\rm Var \,}K^*_j(t))$ as $t\to\infty$. This in combination with the last centered limit relation, in which we replace $j$ with $j+1$, yields $$\lim_{n\to\infty}\frac{K_{j+1}(S_n)-K_{j+1}(n)}{({\rm Var \,}K^*_j(n)m({\rm Var \,}K^*_j(n)))^{1/2}}=0\quad\text{a.s.}$$ Since, for $j\in\mathbb{N}$, $K^*_j(t)=K_j(t)-K_{j+1}(t)$ a.s., subtracting the last two centered limit relations we arrive at $$\lim_{n\to\infty}\frac{K_{j}^*(S_n)-K_j^*(n)}{({\rm Var \,}K_j^*(n)m({\rm Var \,}K_j^*(n)))^{1/2}}=0\quad\text{a.s.}$$ Halves of LILs [\[eq:LILkar\]](#eq:LILkar){reference-type="eqref" reference="eq:LILkar"}, [\[eq:LILkar1\]](#eq:LILkar1){reference-type="eqref" reference="eq:LILkar1"}, [\[eq:LILkar2\]](#eq:LILkar2){reference-type="eqref" reference="eq:LILkar2"} and [\[eq:LILkar200\]](#eq:LILkar200){reference-type="eqref" reference="eq:LILkar200"} ([\[eq:inf01\]](#eq:inf01){reference-type="eqref" reference="eq:inf01"}, [\[eq:inf02\]](#eq:inf02){reference-type="eqref" reference="eq:inf02"}, [\[eq:inf1\]](#eq:inf1){reference-type="eqref" reference="eq:inf1"} and [\[eq:inf11\]](#eq:inf11){reference-type="eqref" reference="eq:inf11"}) read $$\limsup_{n\to\infty}(\liminf_{n\to\infty})\frac{K_j^*(n)-\mathbb{E}K_j^*(n)}{({\rm Var \,}K_j^*(n)m({\rm Var \,}K_j^*(n)))^{1/2}}\le C~ (\ge -C) \quad\text{a.s.},$$ where the case-dependent constant $C$ is equal to the right-hand side of [\[eq:LILkar\]](#eq:LILkar){reference-type="eqref" reference="eq:LILkar"}, [\[eq:LILkar1\]](#eq:LILkar1){reference-type="eqref" reference="eq:LILkar1"}, [\[eq:LILkar2\]](#eq:LILkar2){reference-type="eqref" reference="eq:LILkar2"} or [\[eq:LILkar200\]](#eq:LILkar200){reference-type="eqref" reference="eq:LILkar200"} ([\[eq:inf01\]](#eq:inf01){reference-type="eqref" reference="eq:inf01"}, [\[eq:inf02\]](#eq:inf02){reference-type="eqref" reference="eq:inf02"}, [\[eq:inf1\]](#eq:inf1){reference-type="eqref" reference="eq:inf1"} or [\[eq:inf11\]](#eq:inf11){reference-type="eqref" reference="eq:inf11"}) , respectively. This taken together with the conclusion of Step 1, formula [\[eq:mean_depois\]](#eq:mean_depois){reference-type="eqref" reference="eq:mean_depois"} and Proposition [Proposition 26](#prop:var_depoiss){reference-type="ref" reference="prop:var_depoiss"} enables us to obtain $$\limsup_{n\to\infty} (\liminf_{n\to\infty}) \frac{\mathcal{K}^\ast_j(n)-\mathbb{E}\mathcal{K}^\ast_j(n)}{({\rm Var \,}\mathcal{K}^\ast_j(n)m({\rm Var \,}\mathcal{K}^\ast_j(n)))^{1/2}}\le C~(\ge -C)\quad \text{a.s.}$$ Here, we have used a decomposition $$\mathcal{K}^\ast_j(n)-\mathbb{E}\mathcal{K}^\ast_j(n)=(K_{j}^*(S_n)-K_j^*(n))+(K_j^\ast(n)-\mathbb{E}K_j^\ast(n))+(\mathbb{E}K_j^\ast(n)-\mathbb{E}\mathcal{K}^\ast_j(n))\quad\text{a.s.}$$ This finishes the proof of $$\limsup_{t\to\infty} (\liminf_{n\to\infty})\frac{\mathcal{K}^*_1(n)-\mathbb{E}\mathcal{K}^*_1(n)}{({\rm Var}\,\mathcal{K}^*_1(n)\log\log {\rm Var}\,\mathcal{K}^*_1(n))^{1/2}}\leq 2^{1/2}~(\geq -2^{1/2})\quad\text{{\rm a.s.}}$$ in the situation that $\alpha=1$ and relation [\[eq:exotic\]](#eq:exotic){reference-type="eqref" reference="eq:exotic"} fails to hold. According to Lemma [Lemma 17](#lem:new){reference-type="ref" reference="lem:new"}, for any $\delta>0$ and the deterministic sequence $(\tau_n)$ defined in [\[eq:tau\]](#eq:tau){reference-type="eqref" reference="eq:tau"}, $$\limsup_{n\to\infty} (\liminf_{n\to\infty})\frac{K^*_{j}(\lfloor \tau_n\rfloor)-\mathbb{E}K^*_{j}(\lfloor\tau_n\rfloor)}{C({\rm Var \,}K^*_{j}(\lfloor \tau_n\rfloor)m({\rm Var \,}K^*_j(\lfloor \tau_n\rfloor)))^{1/2}}\ge 1-\delta~(\le -(1-\delta))\quad\text{a.s.}$$ Combining these inequalities with the conclusion of Step 1, formula [\[eq:mean_depois\]](#eq:mean_depois){reference-type="eqref" reference="eq:mean_depois"} and Proposition [Proposition 26](#prop:var_depoiss){reference-type="ref" reference="prop:var_depoiss"} we arrive at $$\begin{gathered} \limsup_{n\to\infty}(\liminf_{n\to\infty})\frac{\mathcal{K}^*_{j}(n)-\mathbb{E}\mathcal{K}^*_{j}(n)}{C({\rm Var \,}\mathcal{K}^*_{j}(n)m({\rm Var \,}\mathcal{K}^*_j(n)))^{1/2}}\\\geq (\leq) \limsup_{n\to\infty} (\liminf_{n\to\infty})\frac{\mathcal{K}^*_{j}(\lfloor \tau_n\rfloor)-\mathbb{E}\mathcal{K}^*_{j}(\lfloor \tau_n\rfloor)}{C({\rm Var \,}\mathcal{K}^*_{j}(\lfloor \tau_n\rfloor)m({\rm Var \,}\mathcal{K}^*_j(\lfloor \tau_n\rfloor)))^{1/2}} \ge 1-\delta ~ (\le -(1-\delta))\quad \text{a.s.}\end{gathered}$$ Sending $\delta\to 0+$ yields $$\limsup_{n\to\infty}(\liminf_{n\to\infty})\frac{\mathcal{K}^*_{j}(n)-\mathbb{E}\mathcal{K}^*_{j}(n)}{({\rm Var \,}\mathcal{K}^*_{j}(n)m({\rm Var \,}\mathcal{K}^*_j(n)))^{1/2}}\geq C~ (\le -C)\quad\text{a.s.},$$ which finishes the proof. ◻ . The research was supported by Applied Probability Trust in the framework of a Ukraine Support Scheme. 30 N. H. Bingham, C. M. Goldie and J. L. Teugels, *Regular variation*. Cambridge University Press, 1989. P. De Blasi, R. H. Mena and I. Prünster, *Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process*. Ann. Inst. Statist. Math. **74** (2022), 143--165. D. Buraczewski, A. Iksanov and V. Kotelnikova, *Limit theorems for sums of independent indicators, with applications to the Ginibre point process and Karlin's occupancy scheme*. Preprint (2023) available at `https://arxiv.org/abs/2306.15027` J. Chang and M. Grabchak, *Necessary and sufficient conditions for the asymptotic normality of higher order Turing estimators*. Bernoulli. **29** (2023), 3369--3395. Z. Derbazi, A. Gnedin and A. Marynych, *Records in the infinite occupancy scheme*. Preprint (2023) available at `https://arxiv.org/abs/2308.01739` J.  L. Geluk and L. de Haan, *Regular variation, extensions and Tauberian theorems*. CWI Tract, 1987. A. Gnedin, A. Hansen, and J. Pitman, *Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws.* Probab. Surveys. **4** (2007), 146--171. M. Grabchak, M. Kelbert and Q. Paris, *On the occupancy problem for a regime-switching model*. J. Appl. Probab. **57** (2020), 53--77. A. Iksanov and V. Kotelnikova, *Small counts in nested Karlin's occupancy scheme generated by discrete Weibull-like distributions*. Stoch. Proc. Appl. **153** (2022), 283--320. S. Karlin, *Central limit theorems for certain infinite urn schemes*. J. Math. Mech. **17** (1967), 373--401. M. Longnecker and R. J. Serfling, *General moment and probability inequalities for the maximum partial sum*. Acta Math. Scient. Hungar. **30** (1977), 129--133. H. P. Rosenthal, *On the subspaces of $L_p$ $(p>2)$ spanned by sequences of independent random variables.* Isr. J. Math. **8** (1970), 273--303. [^1]: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Ukraine; e-mail address: iksan\@univ.kiev.ua [^2]: Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Ukraine; e-mail address: valeria.kotelnikova\@unicyb.kiev.ua

arxiv_math

--- abstract: | We answer the recent problem posed by Baudier, Braga, Farah, Vignati, and Willett that asks whether the $\ell_\infty$-direct sum of the matrix algebras embeds into the uniform Roe algebra or the quasi-local algebra of a uniformly locally finite metric space. The answers are no and yes, respectively. This yields the existence of a quasi-local operator that is not approximable by finite propagation operators. address: RIMS, Kyoto University, 606-8502 Japan author: - Narutaka Ozawa title: Embeddings of matrix algebras into uniform Roe algebras and quasi-local algebras --- # Introduction Throughout this paper, we are interested in a (discrete) metric space $X$ that is *uniformly locally finite*, or *ulf* in short (a.k.a. of bounded geometry), i.e., $$N_X(R) := \sup\{ |\mathop{\mathrm{Ball}}(x,R)| : x \in X \} < \infty$$ for every $R>0$, where $\mathop{\mathrm{Ball}}(x,R) := \{ y\in X : \mathop{\mathrm{dist}}(y,x)\le R\}$. Associated with $X$ are the *uniform Roe algebra* $\mathrm{C}^*_\mathrm{u}[X]$ and the *quasi-local algebra* $\mathrm{C}^*_\mathrm{ql}[X]$, prototypes of which are introduced in [@roe:jdg]. These are $\mathrm{C}^*$-subalgebras of the $\mathrm{C}^*$-algebra $\mathbb{B}(\ell_2X)$ of bounded operators on the Hilbert space $\ell_2X$. For $R>0$, we denote by $$\mathbb{C}_\mathrm{u}^R[X]:=\{ u\in\mathbb{B}(\ell_2X) : \mathopen{\langle}u\delta_x,\delta_y\mathclose{\rangle}=0\mbox{ whenever $\mathop{\mathrm{dist}}(x,y)> R$}\}$$ the set of all operators with *propagation at most $R$*. The uniform Roe algebra $\mathrm{C}^*_\mathrm{u}[X]$ is the norm closure of the $*$-algebra $\bigcup_{R>0}\mathbb{C}_\mathrm{u}^R[X]$ of finite propagation operators on $\ell_2X$. An operator $u$ on $\ell_2X$ is said to have *$\varepsilon$-propagation at most $R$* if it satisfies $\|1_Au1_B\|\le\varepsilon$ whenever $A,B\subset X$ are such that $\mathop{\mathrm{dist}}(A,B)>R$. Here, $1_A \in \mathbb{B}(\ell_2X)$ stands for the orthogonal projection from $\ell_2X$ onto $\ell_2A$ for any $A\subset X$. An operator $u\in\mathbb{B}(\ell_2X)$ is *quasi-local* if it has finite $\varepsilon$-propagation for all $\varepsilon>0$. The quasi-local algebra $\mathrm{C}^*_\mathrm{ql}[X]$ is the $\mathrm{C}^*$-algebra consisting of all quasi-local operators. It is plain to see that $\mathrm{C}^*_\mathrm{u}[X] \subset \mathrm{C}^*_\mathrm{ql}[X]$. The uniform Roe algebras and the quasi-local algebras have different advantages. Generally speaking, an operator in $\mathrm{C}^*_\mathrm{u}[X]$ is easier to handle than that in $\mathrm{C}^*_\mathrm{ql}[X]$, but it is harder to tell if a given operator $u$ belongs to $\mathrm{C}^*_\mathrm{u}[X]$. Thus the problem whether they coincide or not has caught considerable attention (p. 20 in [@roe:cbms], see also [@bbfvw; @engel; @klvz; @lnsz; @st; @sz] and references therein). It is proved in [@sz] that a large class of ulf metric spaces, namely those with property A, satisfy the equality $\mathrm{C}^*_\mathrm{u}[X] = \mathrm{C}^*_\mathrm{ql}[X]$. See Section [4](#sec:propertyA){reference-type="ref" reference="sec:propertyA"} for the definition of property A and an alternative proof of this fact. In this paper, we prove that the inclusion $\mathrm{C}^*_\mathrm{u}[X] \subset \mathrm{C}^*_\mathrm{ql}[X]$ can be proper in general. The proof takes a roundabout way and goes by studying the embeddability of the $\mathrm{C}^*$-algebra $\prod_n \mathbb{M}_n$ of the $\ell_\infty$-direct sum of matrix algebras. Whether embeddings are unital or not will make no essential difference. **Theorem 1**. *The $\mathrm{C}^*$-algebra $\prod_n \mathbb{M}_n$ does not embed into the uniform Roe algebra $\mathrm{C}^*_\mathrm{u}[X]$ of any ulf metric space $X$.* **Theorem 2**. *The $\mathrm{C}^*$-algebra $\prod_n \mathbb{M}_n$ embeds into the quasi-local algebra $\mathrm{C}^*_\mathrm{ql}[X]$ of a ulf metric space $X$, provided that $X$ contains a sequence of expanders.* See Section [3](#sec:B){reference-type="ref" reference="sec:B"} for the definition of expanders. The above results answer the problem posed in [@bbfvw], where it is proved that non-atomic von Neumann algebras do not embed into quasi-local algebras, leaving the possibility for the atomic von Neumann algebra $\prod_n \mathbb{M}_n$ open. **Corollary 3**. *For any ulf metric space $X$ that contains a sequence of expanders, the inclusion $\mathrm{C}^*_\mathrm{u}[X] \subset \mathrm{C}^*_\mathrm{ql}[X]$ is proper. In other words, there exists a quasi-local operator that is not approximable by finite propagation operators.* We remark that this corollary holds as well in the "non-uniform" setting (see Chapter 3 in [@roe:cbms] for the definition and, for a given Hilbert $X$-module $H$, consider an $X$-embedding $\ell_2X \subset H$). Recall that property A is a kind of amenability condition and a sequence of expanders is the most prominent obstruction to it (see e.g., Sections 4 & 5 in [@ny]). It seems natural to expect that $\prod_n \mathbb{M}_n$ embeds into the quasi-local algebra and hence $\mathrm{C}^*_\mathrm{u}[X] \neq \mathrm{C}^*_\mathrm{ql}[X]$ as soon as $X$ does not have property A. ## Acknowledgments {#acknowledgments .unnumbered} The author is grateful to Professor Ilijas Farah for introducing him the problem in [@bbfvw] that led him to the present work. This research was carried out during the author's stay at the Fields Institute for Research in Mathematical Sciences for "Thematic Program on Operator Algebras and Applications" in the Fall 2023. The author acknowledges the kind hospitality, the exciting environment, and the financial support provided by the institute. This research was partially supported by JSPS KAKENHI Grant Numbers 20H01806 and 20H00114. # Proof of Theorem [Theorem 1](#thm:A){reference-type="ref" reference="thm:A"} {#sec:A} The proof of Theorem [Theorem 1](#thm:A){reference-type="ref" reference="thm:A"} is motivated by an operator space theoretic perspective that the matrix algebras are hard to embed completely isomorphically into commutative $\mathrm{C}^*$-algebras. For every Banach space $E$, we denote by $(E)_1$ the closed unit ball of $E$. For every projection $p$, we put $p^\perp := 1-p$. **Lemma 1**. *Let $R>0$ and $n$ be such that $N_X(R) < \sqrt{n}/6$. Then for any possibly non-unital embedding $\mathbb{M}_n\hookrightarrow\mathbb{B}(\ell_2X)$ with the unit $p$, there is $a\in(\mathbb{M}_n)_1$ satisfying that $\mathop{\mathrm{dist}}(a+b,\mathbb{C}_\mathrm{u}^R[X])\geq 1/2$ for every $b\in (p^\perp\mathbb{B}(\ell_2X)p^\perp)_1$.* *Proof of Theorem [Theorem 1](#thm:A){reference-type="ref" reference="thm:A"}.* Suppose for a contradiction that $\prod_n \mathbb{M}_n\hookrightarrow\mathrm{C}^*_\mathrm{u}[X]$. We denote by $p_n \in \mathrm{C}^*_\mathrm{u}[X]$ the unit for $\mathbb{M}_n$. Then Lemma [Lemma 1](#lem:main){reference-type="ref" reference="lem:main"} provides for each $n$ an element $a_n\in(\mathbb{M}_n)_1$ that satisfies $$\inf_{b\in (p_n^\perp\mathbb{B}(\ell_2X)p_n^\perp)_1 } \mathop{\mathrm{dist}}(a_n+b,\mathbb{C}_\mathrm{u}^{R_n}[X]) \geq \frac{1}{2}$$ for $R_n:=\sup\{R>0 : N_X(R) < \sqrt{n}/6\}-1$. Notice that $R_n\nearrow\infty$ by uniform local finiteness. Now $a:=\mathop{\mathrm{diag}}_n (a_n)_n\in(\prod_n \mathbb{M}_n)_1$ satisfies $\mathop{\mathrm{dist}}(a, \mathbb{C}_\mathrm{u}^R[X])\geq1/2$ for all $R>0$, in contradiction with the hypothesis. ◻ The rest of this section is devoted for the proof of Lemma [Lemma 1](#lem:main){reference-type="ref" reference="lem:main"}. The following two lemmas are certainly known to experts, but we put their proofs because they are short. Recall that a *partial translation* on $X$ is a bijection $T$ from $\mathop{\mathrm{dom}}T \subset X$ onto $\mathop{\mathrm{ran}}T \subset X$. **Lemma 2**. *For every $R>0$, there is a family $\{ T_i\}_{i=1}^{2N_X(R)}$ of partial translations that satisfies $\{ (x,y)\in X : \mathop{\mathrm{dist}}(x,y)\le R \} = \bigsqcup_{i=1}^{2N_X(R)}\mathop{\mathrm{graph}}T_i$.* *Proof.* We claim that any maximal (w.r.t. the graph union) family $T_1,\ldots, T_{2N_X(R)}$ of partial translations with mutually disjoint graphs does the job. Suppose this is not the case and $(x_0,y_0)\notin \bigsqcup\mathop{\mathrm{graph}}T_i$. Then for each $i$, either $x_0\in\mathop{\mathrm{dom}}T_i$ and $T_i(x_0)\in \mathop{\mathrm{Ball}}(x_0,R)\setminus\{y_0\}$ or $y_0\in\mathop{\mathrm{ran}}T_i$ and $T_i^{-1}(y_0)\in \mathop{\mathrm{Ball}}(y_0,R)\setminus\{x_0\}$. By the pigeonhole principle, this is impossible. ◻ **Lemma 3**. *For every irreducible unitary representation $\pi\colon\Gamma\to\mathbb{M}_n$ of a finite group $\Gamma$, one has $$\sup_{\alpha\in(\ell_\infty \Gamma)_1} \|\frac{1}{|\Gamma|}\sum_{g\in\Gamma}\alpha_g\pi(g)\|\le\frac{1}{\sqrt{n}}.$$* *Proof.* For any unit vectors $\xi$ and $\eta$, one has $$\begin{aligned} |\mathopen{\langle} \frac{1}{|\Gamma|}\sum_g \alpha_g\pi(g)\xi,\eta\mathclose{\rangle}| \le(\frac{1}{|\Gamma|}\sum_g |\mathopen{\langle}\pi(g)\xi,\eta\mathclose{\rangle}|^2)^{1/2} =\mathopen{\langle} P(\xi\otimes\overline{\xi}),(\eta\otimes\overline{\eta})\mathclose{\rangle}^{1/2},\end{aligned}$$ where $P:=|\Gamma|^{-1}\sum_g(\pi\otimes\overline{\pi})(g)$ is the orthogonal projection onto the space of $(\pi\otimes\overline{\pi})(\Gamma)$ invariant vectors. Since $\pi$ is irreducible, by Schur's lemma, $\mathop{\mathrm{ran}}P = \mathbb{C}(n^{-1/2}\sum_i\zeta_i\otimes\overline{\zeta_i})$, where $\{\zeta_i\}$ is any orthonormal basis. This implies $\mathopen{\langle} P(\xi\otimes\overline{\xi}),(\eta\otimes\overline{\eta})\mathclose{\rangle}=1/n$. ◻ *Proof of Lemma [Lemma 1](#lem:main){reference-type="ref" reference="lem:main"}.* Put $$\varepsilon:=\max_{a\in(\mathbb{M}_n)_1}\min_{b\in (p^\perp\mathbb{B}(\ell_2X)p^\perp)_1} \mathop{\mathrm{dist}}(a+b,\mathbb{C}_\mathrm{u}^R[X]).$$ Take an irreducible unitary representation $\pi\colon\Gamma\to\mathbb{M}_n \subset\mathbb{B}(\ell_2X)$ of a finite group $\Gamma$ (e.g., $\Gamma=\mathfrak{S}_{n+1}$) and choose for each $g\in\Gamma$ elements $b_g\in (p^\perp\mathbb{B}(\ell_2X)p^\perp)_1$ and $c_g\in\mathbb{C}_\mathrm{u}^R[X]$ such that $\| \pi(g)+b_g-c_g \|\le\varepsilon$. One has $$\begin{aligned} \|(\frac{1}{|\Gamma|}\sum_g c_g\otimes\overline{\pi(g)})\| \geq \|\frac{1}{|\Gamma|}\sum_g (\pi(g)+b_g)\otimes\overline{\pi}(g)\| - \varepsilon \geq 1-\varepsilon.\end{aligned}$$ Let $\{ (x,y)\in X : \mathop{\mathrm{dist}}(x,y)\le R \} = \bigsqcup_{i=1}^{2N_X(R)}\mathop{\mathrm{graph}}T_i$ by Lemma [Lemma 2](#lem:graph){reference-type="ref" reference="lem:graph"} and denote by $\Phi_i$ the complete contraction from $\mathbb{B}(\ell_2X)$ onto the space of operators supported on $\mathop{\mathrm{graph}}T_i$. Then, $\sum_{i=1}^{2N_X(R)}\Phi_i$ is the projection onto $\mathbb{C}_\mathrm{u}^R[X]$. It follows that there must be $i$ such that $$\|(\Phi_i\otimes\mathop{\mathrm{id}})(\frac{1}{|\Gamma|}\sum_g c_g\otimes\overline{\pi(g)})\|\geq \frac{1-\varepsilon}{2N_X(R)}.$$ However, since the range of $\Phi_i$ is completely isometric to $\ell_\infty$, one has $$\|(\Phi_i\otimes\mathop{\mathrm{id}})(\frac{1}{|\Gamma|}\sum_g c_g\otimes\overline{\pi(g)})\| \le\sup_{\varphi\in(\mathbb{B}(\ell_2X)^*)_1}\|\frac{1}{|\Gamma|}\sum_g \varphi(c_g)\overline{\pi(g)}\| \le \frac{1+\varepsilon}{\sqrt{n}}$$ by Lemma [Lemma 3](#lem:irr){reference-type="ref" reference="lem:irr"}. Since $N_X(R)<\sqrt{n}/6$, these inequalities imply $\varepsilon>1/2$. ◻ # Proof of Theorem [Theorem 2](#thm:B){reference-type="ref" reference="thm:B"} {#sec:B} The proof of Theorem [Theorem 2](#thm:B){reference-type="ref" reference="thm:B"} uses a similar idea to [@lnsz] and [@klvz]. Recall that a *sequence of expanders* is a sequence $(X_n)_n$ of finite metric spaces (finite graphs in most of the literature) such that $|X_n|\to\infty$ and $$\kappa:=\inf_n \min_{\begin{subarray}{c} A\subset X_n;\\ 0<|A|/|X_n|\le1/2\end{subarray}} \frac{|\{ x \in X_n : \mathop{\mathrm{dist}}(x,A)\le R\}| }{|A|} > 1$$ for some $R>0$. It yields that for any $n$ and any subsets $A,B\subset X_n$ $$\min\{ |A|/|X_n|,\,|B|/|X_n| \}\le\kappa^{-\mathop{\mathrm{dist}}(A,B)/2R}.$$ Hence, the LHS is arbitrarily small if $\mathop{\mathrm{dist}}(A,B)$ is large enough. This property (named *asymptotic expanders* in [@lnsz]) is what we need in this paper. It guarantees that an operator on such a space with "well-spread" matrix coefficients is quasi-local. A new ingredient for constructing quasi-local operators is a random projection of rank $n$. We use the following model of random $n$-dimensional subspaces $V$ in $\mathbb{R}^d = \ell_2([d],\mathbb{R})$. Here $[d]:=\{1,\ldots,d\}$. The difference between real and complex will not matter; if necessary, we view $V$ as its complexification in the complex Hilbert space $\ell_2[d]$. We consider the probability spaces $\mathbb{S}^{d-1}:=\{ x\in\mathbb{R}^d : \|x\|=1\}$ and $(\mathbb{S}^{d-1})^n$ with the probability measures $\mathbb{P}$. For $\mathbf{x}:=(x_1,\ldots,x_n) \in (\mathbb{S}^{d-1})^n$, we put $V(\mathbf{x}):=\mathop{\mathrm{span}}\{ x_1,\ldots,x_n\}$, which is $n$-dimensional with probability $1$. We write $P_V$ for the orthogonal projection onto $V$. **Lemma 4**. *For every $n\in\mathbb{N}$ and $\delta>0$, there are $c>0$ and $D\in\mathbb{N}$ that satisfy the following property. The random $n$-dimensional subspace $V$ in $\mathbb{R}^d$, $d\geq D$, satisfies $$\mathbb{P}\Bigl( \max_{\begin{subarray}{c} E\subset[d];\\ |E|/d\le\delta\end{subarray}} \| P_V |_{\ell_2E} \| <100\sqrt{\delta\log(1/\delta)} \Bigr)\geq 1-e^{-c d}.$$* *Proof of Theorem [Theorem 2](#thm:B){reference-type="ref" reference="thm:B"}.* Assume that $X$ contains a sequence $(X_n)_n$ of expanders. Put $\delta_n:=1/n$ and $\varepsilon_n:=100\sqrt{\delta_n\log(1/\delta_n)}$. For each $n$ find an $n$-dimensional subspace $V(n)$ in $\mathbb{R}^{d(n)}$ that satisfies $$\max\{ \| P_{V(n)} |_{\ell_2E} \| : E\subset[d(n)],\ |E|/d(n)\le\delta_k\}<\varepsilon_k$$ for all $k=1,\ldots,n$. We may assume $|X_n|=d(n)$ and view (the complexification of) $V(n)$ as a subspace of $\ell_2 X_n\subset\ell_2X$. We claim that $\prod_n \mathbb{B}(V(n))$ is contained in $\mathrm{C}^*_{\mathrm{ql}}[X]$. Let $u=\mathop{\mathrm{diag}}_n (u_n)_n \in \prod_n \mathbb{B}(V(n))$ with norm $1$ and $\varepsilon>0$ be given arbitrarily. Fix $k$ with $\varepsilon_k<\varepsilon$ and take $R=R_k>0$ large enough. One has to show $\| 1_A u 1_B\|<\varepsilon$ whenever $A, B\subset X$ are such that $\mathop{\mathrm{dist}}(A,B)>R$. We consider each summand $u_n$ separately. Since $R$ is taken large enough, $A,B\subset X_n$ with $\mathop{\mathrm{dist}}(A,B)>R$ implies that $\min\{|A|/d(n),|B|/d(n)\}<\delta_k$. Thus $$\|1_A u_n 1_B\|\le \min\{ \|1_A P_{V(n)}\|,\,\| P_{V(n)} 1_B\|\} <\varepsilon_k<\varepsilon$$ for all $n\geq k$. This proves $u$ is quasi-local. ◻ The point of Proof of Theorem [Theorem 2](#thm:B){reference-type="ref" reference="thm:B"} is to show the operator $\mathop{\mathrm{diag}}_n (P_{V(n)})_n$ is quasi-local. As mentioned in Introduction, it is harder to tell if it belongs to $\mathrm{C}^*_{\mathrm{u}}[X]$. *Proof of Lemma [Lemma 4](#lem:random){reference-type="ref" reference="lem:random"}.* We may assume $\varepsilon:=25\sqrt{\delta\log(1/\delta)}\le 1/4$. Also for notational simplicity, we assume $d\delta$ is an integer and write $\mathcal{P}(d,\delta):=\{ E\subset [d] : |E|/d=\delta\}$. By the measure concentration phenomenon (Lévy's Lemma, see e.g., 14.3.2, 14.3.3, and 15.2.2 in [@matousek]), every $E\in \mathcal{P}(d,\delta)$ satisfies $$\mathbb{P}( \{ x\in \mathbb{S}^{d-1} : \|1_E x\|>m_\delta+\varepsilon\} ) < 2e^{-\varepsilon^2 d/2}.$$ Here $m_\delta$ is the median of $\|1_E x\|$, which is asymptotically $\delta^{1/2}$. It is important that the estimate is uniform in $\delta>0$. We have $\varepsilon>m_\delta$. Recall $\log \left(\begin{smallmatrix}d\\ \delta d\end{smallmatrix}\right)\le H(\delta)d$, where $H(\delta)=-\delta\log\delta - (1-\delta)\log(1-\delta)$, because $1=(\delta + (1-\delta))^d\geq\left(\begin{smallmatrix}d\\ \delta d\end{smallmatrix}\right)\delta^{\delta d}(1-\delta)^{(1-\delta)d}$. We have $H(\delta)<\varepsilon^2/4$. Thus $$\mathbb{P}( \{ x\in \mathbb{S}^{d-1} : \max_{E\in \mathcal{P}(d,\delta)} \| 1_Ex\| >2\varepsilon\} ) < 2e^{-\varepsilon^2 d /4}.$$ A random $n$-tuple $\mathbf{x}=(x_1,\ldots,x_n) \in (\mathbb{S}^{d-1})^n$ is asymptotically orthonormal as $d\to\infty$. Thus for every $\alpha=(\alpha_k)_{k=1}^n\in \mathbb{S}^{n-1}$, the random vector $\alpha\cdot\mathbf{x}:=\sum_k \alpha_k x_k$ has asymptotically unit norm. Moreover, since the distribution of $\alpha\cdot\mathbf{x}/\| \alpha\cdot\mathbf{x}\|$ is $O(d)$-invariant, one has $$\mathbb{P}( \{ \mathbf{x}\in (\mathbb{S}^{d-1})^n : \max_{E\in \mathcal{P}(d,\delta)} \|1_E \alpha\cdot \mathbf{x}\| > 3\varepsilon\} ) < 2e^{-\varepsilon^2 d /4}$$ for every $\alpha\in \mathbb{S}^{n-1}$ and every $d$ large enough. Considering some $\varepsilon$-dense subset in $\mathbb{S}^{n-1}$, one sees $$\mathbb{P}( \{ \mathbf{x}\in (\mathbb{S}^{d-1})^n : \max_{E\in \mathcal{P}(d,\delta)} \|1_E|_{V(\mathbf{x})}\| > 4\varepsilon\} ) < C(n,\varepsilon) e^{-\varepsilon^2 d /4}$$ for some $C(n,\varepsilon)>0$. Because $\|P_V|_{\ell_2E}\| =\|1_E|_V\|$, this proves the lemma. ◻ # property A implies $\mathrm{C}^*_{\mathrm{u}}[X]=\mathrm{C}^*_{\mathrm{ql}}[X]$ {#sec:propertyA} As mentioned in Introduction, it is proved in [@sz] that property A implies $\mathrm{C}^*_{\mathrm{u}}[X]=\mathrm{C}^*_{\mathrm{ql}}[X]$. The proof in [@sz] is based on the notion of metric sparsification. In this section, we give a more direct and quantitative proof of this fact (in the uniform setting; the proof for the "non-uniform" case is similar, but more bulky). The following fact is proved in Proof of Theorem 2.8 in [@st]. We replicate the proof for completeness. **Lemma 5**. *Let $h\in\ell_\infty X$ be such that $0\le h \le1$ and $|h(x)-h(y)|\le \delta$ for $\mathop{\mathrm{dist}}(x,y)\le R$. If $u \in \mathbb{B}(\ell_2X)$ has $\varepsilon$-propagation at most $R$, then $\|[h,u]\|\le 4\delta^{-2}\varepsilon+ 4\delta\|u\|$.* *Proof.* We may assume $\delta<1$. Put $E(n):=h^{-1}([\delta n,\delta (n+1)))$ for $n=0,\ldots,\lfloor\delta^{-1}\rfloor$. Then $|m-n|>1$ implies that $\mathop{\mathrm{dist}}(E(m),E(n))>R$ and so that $\| u_{m,n}\|<\varepsilon$ for $u_{m,n}:= 1_{E(m)}u1_{E(n)}$. Consider $g := \sum_n \delta n 1_{E(n)}\in\ell_\infty X$. Then $\| g - h \|\le \delta$ and $$\begin{aligned} \| [h,u] \| \approx_{2\delta\| u\|} \| [g,u] \| = \| \sum_{m,n} \delta(m-n) u_{m,n}\| \le(\lfloor\delta^{-1}\rfloor+1)^2\varepsilon+ 2 \delta\|u\|.\end{aligned}$$ ◻ Let $\mathop{\mathrm{Prob}}(X)\subset\ell_1X$ denote the subset of positive elements with norm one. Recall that $X$ has *property A* (see e.g., Section 4 in [@ny]) if for every $\delta>0$ and $R>0$, there are $S>0$ and $\mu\colon X\to\mathop{\mathrm{Prob}}X$ that satisfy $\mathop{\mathrm{\mathop{supp}}}\mu_x \subset\mathop{\mathrm{Ball}}(x,S)$ for every $x$ and $\|\mu_x - \mu_y \|_1<\delta$ whenever $\mathop{\mathrm{dist}}(x,y)\le R$. For the following, we fix such a $\mu\colon X\to\mathop{\mathrm{Prob}}(X)$ and also $T>0$ and $\nu\colon X\to\mathop{\mathrm{Prob}}X$ that satisfy $\mathop{\mathrm{\mathop{supp}}}\nu_x \subset\mathop{\mathrm{Ball}}(x,T)$ for every $x$ and $\|\nu_x - \nu_y \|_1<\delta$ whenever $\mathop{\mathrm{dist}}(x,y)\le S$. The functions $f_z(x) := \nu_x(z)^{1/2}$ satisfy $\sum_z f_z(x)^2 =1$ for every $x$, $\mathop{\mathrm{\mathop{supp}}}f_z \subset \mathop{\mathrm{Ball}}(z,T)$, and $\sum_z |f_z(x)-f_z(y)|^2 < \delta$ whenever $\mathop{\mathrm{dist}}(x,y)\le S$. We view $f_z\in\ell_\infty X\subset\mathbb{B}(\ell_2X)$ and define a unital completely positive map $\Phi_\nu$ on $\mathbb{B}(\ell_2X)$ by $\Phi_\nu(u) := \sum_z f_z u f_z \in \mathbb{C}_{\mathrm{u}}^{2T}[X]$. The RHS is convergent in the strong operator topology. We will prove that if $u$ is a contraction with $\delta^3$-propagation at most $R$, then $\| u - \Phi_\nu(u) \| \le 42\delta^{1/2}$ and hence $\mathop{\mathrm{dist}}(u,\mathbb{C}_{\mathrm{u}}^{2T}[X])\le 42\delta^{1/2}$. This will prove that property A implies $\mathrm{C}^*_{\mathrm{u}}[X]=\mathrm{C}^*_{\mathrm{ql}}[X]$. It is interesting that property A is used twice as in the proof in [@sz]. We consider the probability space $\Omega:=\{\pm1\}^X$ with the uniform probability measure and the i.i.d. Rademacher random variables $r_z\colon\Omega\ni\omega\mapsto\omega_z\in\{\pm1\}$. We define a "random" function $f_\omega\in\ell_\infty X$ by $$f_\omega(x) := \sum_{z\in X} r_z(\omega) f_z(x) = \sum_{z\in \mathop{\mathrm{Ball}}(x,T)} \omega_z f_z(x).$$ Observe that $\| f_\omega\|_\infty \le N_X(T)^{1/2}<\infty$, $\int f_\omega(x)^2\,d\omega=1$ for every $x$, and $$\int |f_\omega(x) - f_\omega(y)|^2\,d\omega = \sum_z |f_z(x)-f_z(y)|^2 < \delta$$ whenever $\mathop{\mathrm{dist}}(x,y)\le S$. Moreover, for every $u\in\mathbb{B}(\ell_2X)$, one has $$\int f_\omega u f_\omega\,d\omega = \sum_z f_z u f_z = \Phi_\nu(u).$$ We will perturb $f_\omega$ to $g_\omega$ with controlled $\| g_\omega\|_\infty$ and then to $h_\omega$ with Lipschitz-type condition for *every* $\omega\in\Omega$. We later use the Cauchy--Schwarz inequality that any strong operator topology measurable operator-valued random variables $a$ and $b$ satisfy $$\| \int a(\omega)^*b(\omega)\,d\omega\| \le \|\int a(\omega)^*a(\omega)\,d\omega\|^{1/2} \|\int b(\omega)^*b(\omega)\,d\omega\|^{1/2}.$$ Put $C:=\delta^{-1/2}$ and $g_\omega(x) := -C \vee f_\omega(x) \wedge C$ so that $\| g_\omega\|_\infty\le C$. Since $$\begin{aligned} \int f_\omega(x)^4\,d\omega &= \sum_{y,z,v,w}r_y(\omega)r_z(\omega)r_v(\omega)r_w(\omega) f_y(x)f_z(x)f_v(x)f_w(x)\\ &=3\sum_{z,w;\ z\neq w}f_z(x)^2f_w(x)^2 + \sum_z f_z(x)^4\\ &=3 - 2 \sum_z f_z(x)^4 \le 3\end{aligned}$$ for every $x\in X$, one has $$\sup_x \int |f_\omega(x)-g_\omega(x)|^2 \,d\omega \le C^{-2}\sup_x\int f_\omega(x)^4\,d\omega \le 3C^{-2}.$$ Put $h_\omega(x) := \sum_z \mu_x(z)g_\omega(z)$. Then one has $\| h_\omega\|_\infty\le C$, $$\begin{aligned} \sup_x \int |g_\omega(x)-h_\omega(x)|^2\,d\omega &= \sup_x \int |\sum_z \mu_x(z) (g_\omega(x)-g_\omega(z))|^2\,d\omega\\ &\le \sup_x \sum_z \mu_x(z) \int |g_\omega(x)-g_\omega(z)|^2\,d\omega < \delta\end{aligned}$$ since $\mathop{\mathrm{\mathop{supp}}}\mu_x \subset\mathop{\mathrm{Ball}}(x,S)$ and $|g_\omega(x)-g_\omega(z)|\le|f_\omega(x)-f_\omega(z)|$, and $$\begin{aligned} |h_\omega(x) - h_\omega(y)| &\le \| \mu_x-\mu_y \|_1\|g_\omega\|_\infty\le C\delta\end{aligned}$$ for every $\omega$ and every $(x,y)$ such that $\mathop{\mathrm{dist}}(x,y)\le R$. Put $\varepsilon:=\delta^3$. By Lemma [Lemma 5](#lem:lip){reference-type="ref" reference="lem:lip"} applied to $(2C)^{-1}(h_\omega+C)$, any contraction $u$ with $\varepsilon$-propagation at most $R$ satisfies $$\|[h_\omega,u]\| \le (16\delta^{-2}\varepsilon+ 2\delta)\cdot 2C \le 36\delta^{1/2}$$ for every $\omega$. Consequently, by the Cauchy--Schwarz inequality, one has $$\begin{aligned} \| u - \Phi_\nu(u) \| &= \|\int f_\omega[f_\omega,u]\,d\omega\|\\ &\le \|\int f_\omega [h_\omega,u]\,d\omega\|+2(3^{1/2}C^{-1}+\delta^{1/2})\\ &\le 36\delta^{1/2} + 6\delta^{1/2}.\end{aligned}$$ This completes the proof. BB+ F. P. Baudier, B. M. Braga, I. Farah, A. Vignati, R. Willett; Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras. *J. Funct. Anal.*, to appear. A. Engel; K-homology classes of elliptic uniform pseudodifferential operators. *Ann. Global Anal. Geom.* **54** (2018), 551--582. A. Khukhro, K. Li, F. Vigolo, J. Zhang; On the structure of asymptotic expanders. *Adv. Math.* **393** (2021), Paper No. 108073, 35 pp. K. Li, P. Nowak, J. Špakula, J. Zhang; Quasi-local algebras and asymptotic expanders. *Groups Geom. Dyn.* **15** (2021), 655--682. J. Matoušek, *Lectures on discrete geometry.* Graduate Texts in Mathematics, 212. Springer-Verlag, New York, 2002. xvi+481 pp. P. W. Nowak, G. Yu; *Large scale geometry.* EMS Textbooks in Mathematics. European Mathematical Society, Zürich, 2012. xiv+189 pp. J. Roe; An index theorem on open manifolds. I, II. *J. Differential Geom.* **27** (1988), 87--113, 115--136. J. Roe; *Index theory, coarse geometry, and topology of manifolds.* CBMS Regional Conf. Ser. in Math., 90. American Mathematical Society, Providence, RI, 1996. x+100 pp. J. Špakula, A. Tikuisis; Relative commutant pictures of roe algebras. *Comm. Math. Phys.* **365** (2019), 1019--1048. J. Špakula, J. Zhang; Quasi-locality and property A. *J. Funct. Anal.* **278** (2020), 108299, 25 pp.

arxiv_math

--- abstract: | We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of circular orders. These results are then applied to the study of arcs and knots, establishing combinatorial properties and lower bounds (in terms of Borel reducibility) for the complexity of some natural relations between these geometrical objects. address: - Dipartimento di scienze matematiche, informatiche e fisiche, Università di Udine, Via delle Scienze 208, 33100 Udine --- Italy - Dipartimento di scienze matematiche, informatiche e fisiche, Università di Udine, Via delle Scienze 208, 33100 Udine --- Italy - Dipartimento di matematica , Università di Torino, Via Carlo Alberto 10, 10121 Torino --- Italy - | Center for Ubiquitous Computing\ Erkki Koiso-Kanttilan katu 3\ door E P.O Box 4500\ FI-90014 University of Oulu author: - Martina Iannella - Alberto Marcone - Luca Motto Ros - Vadim Weinstein title: Convex Embeddability and Knot Theory --- [^1] # Introduction Knots are very familiar and tangible objects in everyday life, and they also play an important role in modern mathematics. A mathematical knot is a homeomorphic copy of $S^1$ embedded in $S^3$. The study of knots and their properties is known as knot theory (see e.g. [@BZ03]). This paper uses discrete objects, such as linear and circular orders, to gain insight into knots. This approach was already exploited in [@Kul17], where it is shown that isomorphism $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ on the Polish space $\mathop{\mathrm{\mathsf{LO}}}$ of linear orders defined on $\mathbb{N}$ strictly Borel reduces to equivalence on knots $\equiv_{\mathop{\mathrm{Kn}}}$. (Recall that Borel reducibility $\leq_B$ provides a hierarchy of complexities for equivalence relations defined on Polish or standard Borel spaces.) The proof in [@Kul17] uses proper arcs (which intuitively are obtained by cutting a knot) and their subarcs, called "components" in [@Kul17], which are the analogues of convex subsets of linear orders. Thus, to expand the previous results it is natural to study the following relation between linear orders. **Definition 1**. *Given linear orders $L$ and $L'$, we set $L \trianglelefteq L'$ if and only if $L$ is isomorphic to a convex subset of $L'$.* We call convex embeddability the relation $\trianglelefteq$, which was already introduced and briefly studied in [@BCP73]. Even if convex embeddability is a very natural relation, as far as we know it has not received much attention in the last 50 years. We first focus on the restriction of $\trianglelefteq$ to $\mathop{\mathrm{\mathsf{LO}}}$, denoted by $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$. We begin establishing that $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ induces a structure on $\mathop{\mathrm{\mathsf{LO}}}$ very different from the one given by the usual embeddability relation: as conjectured by Fraïssé in 1948 ([@Fra00]) and proved by Laver in 1971 ([@La71]), the latter is a well quasi-order (briefly: a wqo), i.e. there are no infinite descending chains and no infinite antichains; in contrast, we show that $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ is not well-founded and has chains and antichains of size continuum (Proposition [Proposition 28](#prop1){reference-type="ref" reference="prop1"}). We prove also other combinatorial properties of $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$, computing in particular its unbounding number $\mathfrak{b}(\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}})$ and its dominating number $\mathfrak{d}(\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}})$ (Propositions [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"} and [Proposition 34](#prop:dom_fam){reference-type="ref" reference="prop:dom_fam"}). We then explore the problem of classifying $\mathop{\mathrm{\mathsf{LO}}}$ under the equivalence relation induced by $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$, which we call convex biembeddability and denote by $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$. We obtain the following results (Corollaries [Corollary 36](#red_iso_cvxeq){reference-type="ref" reference="red_iso_cvxeq"} and [Corollary 47](#cvxeq_baire_iso){reference-type="ref" reference="cvxeq_baire_iso"}): **Theorem 1**. *$\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ is Borel reducible to $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$, in symbols ${\cong_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$;* *[\[thmintro-b\]]{#thmintro-b label="thmintro-b"} $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ is Baire reducible to $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$, in symbols $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}\leq_{\text{\scriptsize\textit{Baire}}} \cong_{\mathop{\mathrm{\mathsf{LO}}}}$.* Actually the reduction of part [\[thmintro-b\]](#thmintro-b){reference-type="ref" reference="thmintro-b"} of the previous theorem is such that the preimage of any Borel set is a Boolean combination of analytic sets, and hence is also universally measurable. Thus, although we are not able to show that ${\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}} \leq_B {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$, the two equivalence relations are similar in some respect, e.g. no turbulent equivalence relation Borel reduces to $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$, and $E_1 \not \leq_B {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$ (Corollaries [Corollary 48](#turb){reference-type="ref" reference="turb"} and [Corollary 50](#E1_cvxeq){reference-type="ref" reference="E1_cvxeq"}). In particular, $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ is not complete for analytic equivalence relations and thus $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ is not complete for analytic quasi-orders. In Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"} we establish a connection between linear orders and the theory of proper arcs, showing that: **Theorem 2**. *${\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\precsim_{\mathop{\mathrm{Ar}}}}$, where $\precsim_{\mathop{\mathrm{Ar}}}$ is the relation of subarc on the standard Borel space $\mathop{\mathrm{Ar}}$ of proper arcs.* This allows us to transfer some properties of $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ and $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ to the corresponding relations on proper arcs (Corollary [Corollary 82](#cor:sumuparcs){reference-type="ref" reference="cor:sumuparcs"}). Moreover, further elaborating on the techniques used to study $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$, we show that $\precsim_{\mathop{\mathrm{Ar}}}$ is very complicated from the combinatorial point of view (Theorem [Theorem 85](#thm:umboundedanddominatingforarcs){reference-type="ref" reference="thm:umboundedanddominatingforarcs"}, Corollary [Corollary 86](#cor:incomparablearcsandunboundedchains){reference-type="ref" reference="cor:incomparablearcsandunboundedchains"}, Theorems [Theorem 88](#thm:basisforarcs){reference-type="ref" reference="thm:basisforarcs"} and [Theorem 89](#thm:antichainsforarcs){reference-type="ref" reference="thm:antichainsforarcs"}) and it shares most of the features of $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ mentioned above, such as the existence of large antichains, the absence of finite bases, and so on. If one wants to study knots (instead of proper arcs), it is more natural to consider orders which are "circular" (instead of linear). The notion of circular order, although not as widespread as that of linear order, is very natural and in fact has been rediscovered several times in different contexts. The oldest mention we found is in Čech's 1936 monograph (see the English version [@Ce69]) and a sample of more recent work is [@Meg76; @KM05; @LM06; @BR16; @CMR18; @PBGGR18; @Mat21; @GM21; @CMMRS23]. There is a natural notion of convex subset of a circular order, but the obvious translation of convex embeddability to circular orders fails to be transitive. We thus consider its "transitivization", called piecewise convex embeddability $\mathrel{\trianglelefteq_{c}^{<\omega}}$, which matches well with the topological notion of piecewise subknot of a knot. Then we study the restriction $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ of $\mathrel{\trianglelefteq_{c}^{<\omega}}$ to the Polish space $\mathop{\mathrm{\mathsf{CO}}}$ of circular orders with domain $\mathbb{N}$, together with its induced equivalence relation $\mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}$. We show that $\mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}$ is strictly more complicated than $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ in terms of Baire reducibility (Corollary [Corollary 69](#cor:piecewiseiscomplex){reference-type="ref" reference="cor:piecewiseiscomplex"}). Indeed, while $E_1$ is not Borel reducible to $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$, we prove in Theorem [Theorem 68](#no_action_group){reference-type="ref" reference="no_action_group"} that **Theorem 3**. *$E_1 \leq_B {\mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}}$.* We then obtain information about the relation of being a piecewise subknot of a knot, denoted by $\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$, proving in Theorem [Theorem 100](#thm:lowerforknots){reference-type="ref" reference="thm:lowerforknots"} that **Theorem 4**. *${\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}} \leq_B {\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}}$.* An interesting consequence of the above results is that the equivalence relation associated to $\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$ is not induced by a Borel action of a Polish group. This is in stark contrast with the relation $\equiv_{\mathop{\mathrm{Kn}}}$ of equivalence on knots, which is induced by a Borel action of the Polish group of homeomorphisms of $S^3$ onto itself (see e.g. [@BZ03 Proposition 1.10]). We also prove a number of combinatorial results concerning $\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$ (see Proposition [Proposition 102](#prop:chains_skr){reference-type="ref" reference="prop:chains_skr"}, Theorem [Theorem 104](#thm:unbounded_and_dominating_for_knots){reference-type="ref" reference="thm:unbounded_and_dominating_for_knots"}, Corollary [Corollary 105](#cor:incomparable_knots_and_unbounded_chains){reference-type="ref" reference="cor:incomparable_knots_and_unbounded_chains"}, Theorems [Theorem 107](#thm:basis_for_knots){reference-type="ref" reference="thm:basis_for_knots"} and [Theorem 108](#thm:antichains_for_knots){reference-type="ref" reference="thm:antichains_for_knots"}), mimicking those obtained for the quasi-order $\precsim_{\mathop{\mathrm{Ar}}}$. Inspired by piecewise convex embeddability on circular orders, in [@IMMRW23] we define the notion of piecewise convex embeddability of linear orders, where the collection of pieces can range in various sets of countable linear orders. In the same paper, we also extend the analysis of $\trianglelefteq$ and of piecewise convex embeddability to uncountable linear orders. The paper is organized as follows. In Section 2 we recall some background about descriptive set theory and Borel reducibility. Moreover, we introduce and prove basic properties of (countable) linear and circular orders. In Section 3 we define and analyse convex embeddability on countable linear orders and countable circular orders, proving some combinatorial properties of these quasi-orders and several results about the complexity of the corresponding equivalence relations with respect to Borel reducibility. In the last section we look at the connections between convex embeddability on $\mathop{\mathrm{\mathsf{LO}}}$ and piecewise convex embeddability on $\mathop{\mathrm{\mathsf{CO}}}$ on one side, and the relations of subarc on proper arcs and piecewise subknot on knots on the other one. We also establish some combinatorial properties of the subarc and piecewise subknot relations. # Preliminaries ## Borel reducibility {#subsection:preliminaries-BR} In this section we introduce some basic definitions and results from descriptive set theory that will be used in the sequel; the standard references are [@Kec95; @Gao09]. A **Polish space** is a separable and completely metrizable topological space. A subset of a Polish space $X$ is **Borel** if it belongs to the smallest $\sigma$-algebra on $X$ containing all open subsets of $X$. Recall that $A \subseteq X$ has the **Baire property** if there exists some open set $U$ such that $A \mathrel{\triangle} U$ is meager, i.e. a countable union of sets whose closure has empty interior. All Borel sets have the Baire property. A **standard Borel space** is a pair $(X, \mathcal{B})$ where $X$ is a set, $\mathcal{B}$ is a $\sigma$-algebra on $X$, and there is a Polish topology on $X$ for which $\mathcal{B}$ is precisely the collection of Borel sets. The elements of $\mathcal{B}$ are called Borel sets of $X$. In particular, every Polish space is standard Borel when equipped with its $\sigma$-algebra of Borel sets. Let $X$ and $Y$ be Polish or standard Borel spaces. A function $\varphi \colon X \to Y$ is **Borel** if the preimage of any Borel subset of $Y$ is Borel in $X$. If $X$ is Polish we say that $\varphi$ is **Baire measurable** if the preimage of any Borel subset of $Y$ has the Baire property. Let $X$ be a standard Borel space. A subset $A\subseteq X$ is **analytic** (or $\mathbf{\Sigma}_1^1$) if it is the Borel image of a standard Borel space, and it is **coanalytic** (or $\mathbf{\Pi}_1^1$) if $X\backslash A$ is analytic. By $D_2(\mathbf{\Pi}_1^1)$ we denote the class of sets which are the intersection of an analytic set and a coanalytic set. Let $X$ and $Y$ be topological spaces and $A \subseteq X$, $B \subseteq Y$. We say that $A$ is **Wadge reducible** to $B$, in symbols $A \leq_W B$, if there is a continuous map $\varphi\colon X \to Y$ such that $x \in A \iff \varphi(x) \in B$, for all $x \in X$. Let $\Gamma$ be a class of sets in Polish spaces. If $Y$ is a Polish space, we say that the subset $A$ of $Y$ is **$\Gamma$-hard** if $B \leq_W A$ for any $B \in \Gamma(X)$ with $X$ a zero-dimensional Polish space. If moreover $A \in \Gamma(Y)$, we say that $A$ is **$\Gamma$-complete**. An important line of research within descriptive set theory is the study of definable equivalence relations, which are typically compared using the next definition. Let $X$ and $Y$ be sets and consider $E$ and $F$ equivalence relations on $X$ and $Y$, respectively. A function $\varphi \colon X \to Y$ is called a **reduction** from $E$ to $F$ if ${x_1}\mathrel{E}{x_2}\iff {\varphi(x_1)}\mathrel{F}{\varphi(x_2)},$ for all $x_1, x_2 \in X$. We say that $E$ is **Borel reducible** to $F$, and write $E \leq_B F$, if $X$ and $Y$ are standard Borel spaces and there exists a Borel map $\varphi$ reducing $E$ to $F$. The equivalence relations $E$ and $F$ are **Borel bireducible**, ${E} \sim_B {F}$ in symbols, if both $E \leq_B F$ and $F \leq_B E$. Finally, we say that $E$ is **Baire reducible** to $F$, and we write ${E} \leq_{\text{\scriptsize\textit{Baire}}} {F}$, if $X$ and $Y$ are topological spaces and there exists a Baire measurable map $\varphi\colon X \to Y$ reducing $E$ to $F$. Let $\Gamma$ be a collection of equivalence relations on standard Borel spaces. We say that an equivalence relation $E$ is **complete for $\Gamma$** (or **$\Gamma$-complete**) if it belongs to $\Gamma$ and any other equivalence relation in $\Gamma$ Borel reduces to $E$. A topological group is **Polish** if its underlying topology is Polish. Examples of Polish groups include the group of permutations of natural numbers $S_\infty$ with the topology inherited as a subspace of the Baire space $\mathbb{N}^\mathbb{N}$, and the group of homeomorphisms of $S^3$ into itself with the topology induced by the uniform metric. If the Polish group $G$ acts on the standard Borel space $X$ we denote by $E_G^X$ the orbit equivalence relation induced by the action. An important class of analytic equivalence relations are those induced by a Borel action of $S^\infty$. Among these are all the isomorphism relations on the countable models of a first-order theory. An analytic equivalence relation is **$S_\infty$-complete** if it is complete for the class of equivalence relations arising from a Borel action of $S_\infty$ on a standard Borel space. We say that an equivalence relation $E$ is **classifiable by countable structures** if $E$ is Borel reducible to some $E_{S_\infty}^Y$. **Theorem 1** (H. Friedman-Stanley, see [@FS89; @Gao09]). *Let $\cong_{\mbox{\tiny \emph{C-GRAPH}}}$ and $\cong_{\mbox{{\tiny \emph{GRAPH}}}}$ be the isomorphism relations on, respectively, the Polish space *C-GRAPH* of countable connected graphs and the Polish space *GRAPH* of countable graphs. Then ${\cong_{\mbox{\tiny \emph{C-GRAPH}}}} \sim_B {\cong_{\mbox{\tiny \emph{GRAPH}}}}$, and both equivalence relations are $S_\infty$-complete.* Let $E$ and $F$ be equivalence relations on standard Borel spaces $X$ and $Y$ respectively. Let $A\subseteq X$. We say that **$E \!\restriction\!A$ is Borel reducible to $F$** if there is a Borel map $\varphi \colon X\to Y$, still called a Borel reduction of $E \!\restriction\!A$ to $F$, such that for every $x,y \in A$, $x\mathrel{E} y \iff \varphi(x)\mathrel{F}\varphi(y)$. This definition is equivalent to the one given in [@CMMR18; @CMMR20] (where $\varphi$ is required to be defined only on $A$) by a theorem of Kuratowski (see [@Kec95 Theorem 12.2]). **Definition 2**. We say that an equivalence relation $E$ on a Polish space $X$ is $\mathbb{\sigma}$**-classifiable by countable structures** if there exists a countable partition $(X_i)_{i \in I}$ of $X$ such that for all $i \in I$: $X_i$ is closed under $E$ (i.e. if $x \in X_i$ and $y \mathrel{E} x$ then $y \in X_i$); $X_i$ has the Baire property; $E \!\restriction\!X_i$ is Borel reducible to ${\cong_{\mbox{{\tiny \emph{GRAPH}}}}}$. Clearly, if an equivalence relation $E$ is classifiable by countable structures then it is $\sigma$-classifiable by countable structures. **Proposition 3**. *Let $E$ be an equivalence relation defined on a Polish space $X$. If $E$ is $\sigma$-classifiable by countable structures, then ${E} \leq_{\text{\scriptsize \textit{Baire}}} {\cong_{\mbox{\tiny \emph{GRAPH}}}}$.* *Proof.* Assume that $E$ is $\sigma$-classifiable by countable structures and fix sets $X_i$ witnessing this. Then by Theorem [Theorem 1](#thm_0.9){reference-type="ref" reference="thm_0.9"} for each $i \in I$ there exists a Borel reduction $\varphi_i$ from $E\!\restriction\!X_i$ to ${\cong_{\mbox{\tiny C-GRAPH}}}$, so that $\varphi_i(x)$ is an infinite connected graph for every $x \in X_i$ (in particular, it is not isomorphic to the graph consisting of a single isolated vertex). Let $\tilde{\varphi}_i \colon X \to \text{GRAPH}$ be defined by $$\tilde{\varphi}_i(x) = \varphi_i(x) \sqcup A_i,$$ where $A_i$ is the graph consisting of $i$-many isolated vertices. It is easy to check that $\tilde{\varphi}_i$ is still a Borel function and it reduces $E \!\restriction\!X_i$ to $\cong_{\mbox{\tiny GRAPH}}$. Finally, define $\varphi \colon X \to \mbox{GRAPH}$ by setting $\varphi(x)=\tilde{\varphi}_i(x)$, where $i$ is the unique index of the subset $X_i$ of $X$ to which $x$ belongs. We first show that $\varphi$ is a reduction. Let $x, y$ be two elements of $X$ such that $x \mathrel{E} y$. Since $X_i$ is closed under $E$ for every $i \in I$, there exists $i_0 \in I$ such that $x,y \in X_{i_0}$. Then $\varphi_{i_0}(x) \cong_{\mbox{\tiny C-GRAPH}} \varphi_{i_0}(y)$, and so $\varphi(x) \cong_{\mbox{\tiny GRAPH}} \varphi(y)$. Conversely, suppose that $\varphi(x)= \varphi_i(x) \sqcup A_i \cong_{\mbox{\tiny GRAPH}} \varphi_j(y) \sqcup A_j =\varphi(y)$, for some $i,j \in I$, $x \in X_i$, and $y \in X_j$. Since isomorphism between graphs preserves connected components, we must have $i=j$ because $\varphi(x)$ contains $i$-many isolated vertices and $\varphi(y)$ contains $j$-many isolated vertices, and moreover $\varphi_i(x) \cong_{\mbox{{\tiny C-GRAPH}}} \varphi_i(y)$ because those are the only infinite connected components in $\varphi(x)$ and $\varphi(y)$, respectively. Since $\varphi_i$ was a reduction we get $x \mathrel{E} y$, as desired. Now take a Borel subset $A$ of $\mbox{GRAPH}$. Then $$\varphi^{-1}(A)= \bigcup_{i \in I} \big(\varphi^{-1}(A) \cap X_i\big) = \bigcup_{i \in I} \big(\tilde{\varphi}_i^{-1}(A) \cap X_i\big).$$ Since $X_i$ has the BP and $\tilde{\varphi}_i$ is Borel for every $i \in I$, we have that $\tilde{\varphi}_i^{-1}(A) \cap X_i$ has the BP for each $i$. Hence also $\varphi^{-1}(A)$ has the BP and $\varphi$ is a Baire measurable reduction. ◻ Not all orbit equivalence relations are Borel reducible, or even Baire reducible, to an $S_\infty$-complete equivalence relation: Hjorth isolated a sufficient condition for this failure, called **turbulence**. **Theorem 4** ([@Hjo00], Corollary 3.19). *There is no Baire measurable reduction of a turbulent orbit equivalence relation to any $E_{S_\infty}^Y$.* Let $E_1$ be the equivalence relation defined on $\mathbb{R}^\mathbb{N}$ by $(x_n)_{n \in \mathbb{N}} \mathrel{E_1} (y_n)_{n \in \mathbb{N}}$ if and only if there exists $m$ such that $x_n = y_n$ for all $n \geq m$. We also use the tail version $E_1^t$, defined by setting $(x_n)_{n \in \mathbb{N}} \mathrel{E_1^t} (y_n)_{n \in \mathbb{N}}$ if and only if there exist $n,m$ such that $x_{n+k} = y_{m+k}$ for all $k$. Notice that $E_1$ and $E_1^t$ are Borel bireducible with the analogous relations defined on $(2^\mathbb{N})^\mathbb{N}$, called $E_0(2^\mathbb{N})$ and $E_t(2^\mathbb{N})$ in [@DJK94]. In the proof of [@DJK94 Theorem 8.1] it is shown that $E_t(2^\mathbb{N}) \leq_B E_0(2^\mathbb{N})$, while the opposite reduction is mentioned in the observation immediately following that proof. This yields: **Proposition 5**. *${E_1} \sim_B {E_1^t}$.* The following result of Shani about $E_1$ generalizes a classical theorem by Kechris and Louveau [@KL97]. (The additional part follows from the fact that by [@Kec95 Theorem 8.38] every Baire measurable map between Polish spaces is actually continuous on a comeager $G_\delta$ set.) **Theorem 6** ([@Sha21 Theorem 4.8]). *The restriction of $E_1$ to any comeager subset of $\mathbb{R}^\mathbb{N}$ is not Borel reducible to an orbit equivalence relation. Thus in particular $E_1 \not\leq_{\text{\scriptsize \textit{Baire}}} {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$.* Let $E$ be an equivalence relation on a standard Borel space $X$. The **Friedman-Stanley jump** of $E$, introduced by Friedman and Stanley in [@FS89] and denoted by $E^{+}$, is the equivalence relation on the space $X^\mathbb{N}=\{(x_n)_{n \in \mathbb{N}} \mid x_n \in X \}$ defined by $$(x_n)_{n \in \mathbb{N}} \mathrel{E^+} (y_n)_{n \in \mathbb{N}} \iff \{[x_n]_E \mid n \in \mathbb{N}\}=\{[y_n]_E \mid n \in \mathbb{N}\}.$$ **Proposition 7** (see [@Gao09]). *Let $E$ and $F$ be equivalence relations on standard Borel spaces. Then $E\leq_B E^+$, and if $E\leq_B F$ then $E^+\leq_B F^+$.* One can transfer many of the above definitions concerning equivalence relations to the wider context of binary relations and, in particular, analytic quasi-orders. We just recall a few results in this direction. **Theorem 8** ([@LR05]). *Every analytic quasi-order Borel reduces to the embeddability relation between countable (connected) graphs, i.e. the latter relation is complete for analytic quasi-orders.* Every analytic quasi-order $R$ on a standard Borel space $X$ canonically induces the analytic equivalence relation $E_R$ on the same space defined by $x \mathrel{E_R} y \iff {x \mathrel{R} y} \wedge {y \mathrel{R} x}$. The complexities of $R$ and $E_R$ are linked by the following result. **Proposition 9** ([@LR05]). *If a quasi-order $R$ on a standard Borel space $X$ is complete for analytic quasi-orders, then $E_R$ is complete for analytic equivalence relations.* ## Countable linear orders {#subsec:ctbllinord} Any $L \in 2^{\mathbb{N}\times\mathbb{N}}$ can be seen as a code of a binary relation on $\mathbb{N}$, namely, the one relating $n$ and $m$ if and only if $L(n,m) = 1$. Denote by $\mathop{\mathrm{\mathsf{LO}}}$ the set of codes for linear orders on $\mathbb{N}$, i.e. $$\mathop{\mathrm{\mathsf{LO}}}= \{L \in 2^{\mathbb{N}\times\mathbb{N}} \mid L \mbox{ codes a reflexive linear order on } \mathbb{N}\}.$$ When $L \in \mathop{\mathrm{\mathsf{LO}}}$ we denote by $\leq_L$ the order on $\mathbb{N}$ coded by $L$, and by $<_L$ its strict part. It is easy to see that $\mathop{\mathrm{\mathsf{LO}}}$ is a closed subset of the Polish space $2^{\mathbb{N}\times\mathbb{N}}$, thus it is a Polish space as well. Given $L \in \mathop{\mathrm{\mathsf{LO}}}$, a neighbourhood base of $L$ in $\mathop{\mathrm{\mathsf{LO}}}$ is determined by the sets $$\{ L' \in \mathop{\mathrm{\mathsf{LO}}}\mid L'\!\restriction\!n = L\!\restriction\!n \}$$ where $n$ varies over $\mathbb{N}$ and $L\!\restriction\!n= L'\!\restriction\!n$ means that $m \leq_L m' \iff m \leq_{L'} m'$ for every $m, m' < n$. We also denote by $\mathop{\mathrm{\mathsf{WO}}}$ the set of all well-orders on $\mathbb{N}$, and recall that it is a proper coanalytic subset of $\mathop{\mathrm{\mathsf{LO}}}$. We denote by $\preccurlyeq$ the quasi-order of embeddability on linear orders, that is: $L \preccurlyeq L'$ if there exists an injection $f$ from $L$ to $L'$, called embedding, such that $n \leq_L m \Rightarrow f(n) \leq_{L'} f(m)$ (equivalently $n \leq_L m \iff f(n) \leq_{L'} f(m)$) for every $n,m \in L$. The restriction $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$ of $\preccurlyeq$ to $\mathop{\mathrm{\mathsf{LO}}}$ is clearly an analytic quasi-order. In contrast with Theorem [Theorem 8](#thm:compl_graph){reference-type="ref" reference="thm:compl_graph"}, the relation $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$ is far from being complete because it is combinatorially simple and it is indeed a wqo. Moreover $\mathop{\mathrm{\mathsf{LO}}}$ has a maximal element under $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$, the equivalence class of non-scattered linear orders (recall that a linear order is scattered if the rationals do not embed into it). The isomorphism relation on $\mathop{\mathrm{\mathsf{LO}}}$ is denoted by $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$, and it is an analytic equivalence relation. **Theorem 10** ([@FS89]). *$\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ is $S_{\infty}$-complete.* Recall that ${E} \leq_B {E^+}$ for any analytic equivalence relation $E$ (Proposition [Proposition 7](#prop 0.10){reference-type="ref" reference="prop 0.10"}). In the case of $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$, we also have the converse. **Proposition 11** (Folklore). *[\[cor 2\]]{#cor 2 label="cor 2"} $(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+ \sim_B {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$.* *Proof.* By Theorem [Theorem 1](#thm_0.9){reference-type="ref" reference="thm_0.9"} and Theorem [Theorem 10](#thm 2){reference-type="ref" reference="thm 2"}, we have that ${\cong_{\mbox{\tiny C-GRAPH}}} \sim_B {\cong_{\mbox{\tiny GRAPH}}} \sim_B {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$, so it is enough to prove that $(\cong_{\mbox{\tiny C-GRAPH}})^+ \leq_B {\cong_{\mbox{\tiny GRAPH}}}$ because $(\cong_{\mbox{\tiny C-GRAPH}})^+ \sim_B (\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+$ by Proposition [Proposition 7](#prop 0.10){reference-type="ref" reference="prop 0.10"}. Given a sequence of countable connected graphs $(A_n)_{n \in \mathbb{N}}$, let $G_A = \bigsqcup_{n, i \in \mathbb{N}} A_{n,i}$ be the disjoint union of the graphs $A_{n,i}$, where $A_{n,i}\cong A_n$ for every $n,i \in \mathbb{N}$. Then the Borel map from $({\mbox{C-GRAPH}})^\mathbb{N}$ to ${{\mbox{GRAPH}}}$ which sends $(A_n)_{n \in \mathbb{N}}$ to $G_A$ is a reduction of $(\cong_{\mbox{\tiny C-GRAPH}})^+$ to ${\cong_{\mbox{\tiny GRAPH}}}$. ◻ We need to deal also with finite linear orders, which are missing in $\mathop{\mathrm{\mathsf{LO}}}$. For this reason, we let $\mathop{\mathrm{\mathsf{Lin}}}$ be the subset of $2^{\mathbb{N}\times\mathbb{N}}$ consisting of all (codes for) linear orders defined either on a finite subset of $\mathbb{N}$ or on the whole $\mathbb{N}$. Thus $\mathop{\mathrm{\mathsf{Lin}}}$ is the union of $\mathop{\mathrm{\mathsf{LO}}}$ and $\mathop{\mathrm{\mathsf{Fin}}}$, where $\mathop{\mathrm{\mathsf{Fin}}}\subset \mathop{\mathrm{\mathsf{Lin}}}$ is the set of (codes for) finite linear orders. It is easy to see that $\mathop{\mathrm{\mathsf{Lin}}}$ is a $F_\sigma$ subset of $2^{\mathbb{N}\times\mathbb{N}}$, and hence it is a standard Borel space, and that isomorphism on $\mathop{\mathrm{\mathsf{Lin}}}$ is induced by a Borel action of $S_{\infty}$. If $L \in \mathop{\mathrm{\mathsf{Lin}}}$ we denote by $L$ also its domain. For convenience, sometimes we use the notation $n_L$ to emphasize that $n$ is an element of the domain of $L$. We recall some isomorphism invariant operations on the class of linear orders that are useful to build Borel reductions. They can all be construed as Borel maps from $\mathop{\mathrm{\mathsf{Lin}}}$, $(\mathop{\mathrm{\mathsf{Lin}}})^n$, or $\mathop{\mathrm{\mathsf{Lin}}}^\mathbb{N}$ to $\mathop{\mathrm{\mathsf{Lin}}}$, and their restriction to $\mathop{\mathrm{\mathsf{LO}}}$ has range contained in $\mathop{\mathrm{\mathsf{LO}}}$. - The **reverse** $L^*$ of a linear order $L$ is the linear order on the domain of $L$ defined by setting $x \leq_{L^*} y \iff y \leq_L x$. - If $L$ and $K$ are linear orders, their **sum** $L+K$ is the linear order defined on the disjoint union of $L$ and $K$ by setting ${x} \leq_{L+K} {y}$ if and only if either $x \in L$ and $y\in K$, or $x, y \in L$ and $x \leq_{L} y$, or $x,y \in K$ and $x \leq_K y$. - In a similar way, given a linear order $K$ and a sequence of linear orders $(L_k)_{k \in K}$ we can define the **$K$-sum** $\sum_{k \in K} L_k$ on the disjoint union of the $L_k$'s by setting $x \leq_{\sum_{k \in K} L_k} y$ if and only if there are $k <_K k'$ such that $x \in L_k$ and $y \in L_{k'}$, or $x,y \in L_k$ for the same $k \in K$ and $x \leq_{L_k} y$. Formally, $\sum_{k \in K} L_k$ is thus defined on the set $\{ (x,k) \mid k \in K , x \in L_k \}$ by stipulating that $(x,k) \leq_{\sum_{k \in K} L_k} (x',k')$ if and only if $k <_K k'$ or else $k = k'$ and $x \leq_{L_k} x'$. - The **product** $LK$ of two linear orders $L$ and $K$ is the cartesian product $L \times K$ ordered antilexicographically. Equivalently, $LK = \sum_{k \in K} L$. For every $n \in \mathbb{N}$, we denote by $\boldsymbol{n}$ the element of $\mathop{\mathrm{\mathsf{Fin}}}$ with domain $\{0,...,n-1\}$ ordered as usual. Similarly, for every infinite ordinal $\alpha < \omega_1$ we fix a well-order $\boldsymbol{\alpha} \in \mathop{\mathrm{\mathsf{LO}}}$ with order type $\alpha$. We also fix computable copies of $(\mathbb{N},{\leq})$, $(\mathbb{Z},{\leq})$ and $(\mathbb{Q},{\leq})$ in $\mathop{\mathrm{\mathsf{LO}}}$, and denote them by $\omega$, $\zeta$ and $\eta$, respectively. We denote by $\min L$ and $\max L$ the minimum and maximum of $L$, if they exist. Finally, we denote by $\mathop{\mathrm{\mathsf{Scat}}}\subseteq \mathop{\mathrm{\mathsf{Lin}}}$ the set of scattered linear orders. **Definition 12**. A subset $I$ of the domain of a linear order $L$ is **($L$-)convex** if $x \leq_L y \leq_L z$ with $x,z \in I$ implies $y \in I$. An $L$-convex set is **proper** if it is neither empty nor the entire $L$. An **initial segment** of a linear order $L$ is a subset $I$ of its domain which is $\leq_L$-downward closed, i.e. $x \in I$ whenever $x \leq_L y$ for some $y \in I$. Dually, $I \subseteq L$ is a **final segment** of $L$ if it is $\leq_L$-upward closed, i.e. if $y \in I$ and $y \leq_L x$ imply $x \in I$. Clearly, initial and final segments are convex of $L$. If $m,n \in L$, we adopt the notations $[m,n]_L$, $(m,n)_L$, $(-\infty,n]_L$, $(-\infty,n)_L$, $[n,+\infty)_L$, and $(n,+\infty)_L$ to indicate the obvious $L$-convex sets. Notice however that not all $L$-convex sets are of one of these forms. Given $L \in \mathop{\mathrm{\mathsf{LO}}}$, we write $L_0 \subseteq L$ (resp. $L_0 \subset L$) if $L_0$ is a (resp. proper) sub-order of $L$, and $L_0 \csube L$ (resp. $L_0 \csub L$) if $L_0$ is a (resp. proper) convex subset of $L$. If $L_0,L_1 \subseteq L$, we write $L_0 \leq_L L_1$ (resp. $L_0 <_L L_1$) iff $n \leq_L m$ (resp. $n <_L m$) for every $n \in L_0$ and $m \in L_1$. Notice that if $L_0 \leq_L L_1$ then either $L_0$ and $L_1$ are disjoint, in which case $L_0 <_L L_1$, or the only element in their intersection is $\max L_0 = \min L_1$. We need to recall some other basic notions about linear orders (see [@Ros82]). Let $L$ be a linear order. The **(finite) condensation** of $L$ is determined by the map $c_F^L \colon L \to \mathscr{P}(L)$ defined by $c_F^L(n)=\{m \mid [n,m]_L \cup [m,n]_L \text{ is finite}\}$ for every $n \in L$. It is immediate that if $m \in c_F^L(n)$ then $c_F^L(m)=c_F^L(n)$, while if $m \notin c_F^L(n)$ then $c_F^L(n) \cap c_F^L(m) = \emptyset$. We call a set $c_F^L(n)$ a **condensation class**. A condensation class may be finite or infinite, and in the latter case its order type is one of $\omega$, $\omega^*$ and $\zeta$. We denote by $L_F$ the set of condensation classes of $L$. In the sequel we use the basic properties of condensation classes which are collected in the following proposition. **Proposition 13**. *Let $L$ be any linear order.* *For every $\ell \in L$, $c_F^L(\ell)$ is convex.* *$\bigcup_{\ell \in L} c_F^L(\ell) = L$, and $c^L_F(\ell) \cap c^L_F(\ell') = \emptyset$ if $c^L_F(\ell) \neq c^L_F(\ell')$; hence $L_F$ is a partition of $L$.* *If $c_F^L(\ell)$ and $c_F^L(\ell')$ are two different condensation classes, then $\ell <_L \ell'$ if and only if $c_F^L(\ell) <_L c_F^L(\ell')$; hence $L_F$ is linearly ordered.* *Let $L, L'$ be linear orders. If $f$ is an isomorphism from $L$ to $L'$ then the restriction of $f$ to each $c_F^L(\ell)$ is an isomorphism between $c_F^L(\ell)$ and $c_F^{L'}(f(\ell))$ and hence $|c_F^L(\ell)|=|c_F^{L'}(f(\ell))|$. Moreover, $L_F \cong L'_F$ via the well-defined map $c^L_F(\ell) \mapsto c^{L'}_F(f(\ell))$.* This condensation is useful to prove results as the next one. **Lemma 14**. *Given two linear orders $L,L'$, $\zeta L \cong \zeta L'$ if and only if $L \cong L'$.* *Proof.* For the nontrivial direction, notice that $c^{\zeta L}_F(i,n) = \zeta\times \{ n \}$, and similarly for the condensation classes of $\zeta L'$. It follows that $(\zeta L)_F \cong L$ and $(\zeta L')_F \cong L'$. By Proposition [Proposition 13](#cond_classes_isom){reference-type="ref" reference="cond_classes_isom"}, if $\zeta L \cong \zeta L'$ then $(\zeta L)_F \cong (\zeta L')_F$, hence $L \cong L'$. ◻ We conclude this section recalling the definition of the powers of $\mathbb{Z}$ and some of their properties. When $\alpha$ is an ordinal we can define $\mathbb{Z}^\alpha$ in two equivalent ways: by induction on $\alpha$ ([@Ros82 Definition 5.34]) and by explicitly defining a linear order on a set ([@Ros82 Definition 5.35]); the latter can actually be used to define $\mathbb{Z}^L$ for any linear order $L$. **Definition 15**. $\mathbb{Z}^0 = \boldsymbol{1}$, $\mathbb{Z}^{\alpha+1}=(\mathbb{Z}^\alpha\omega)^* + \mathbb{Z}^\alpha+ \mathbb{Z}^\alpha\omega$, $\mathbb{Z}^\alpha = \big(\sum_{\beta < \alpha} \mathbb{Z}^\beta \omega\big)^* + \boldsymbol{1} + \sum_{\beta < \alpha} \mathbb{Z}^\beta \omega$ if $\alpha$ is limit. **Definition 16**. Let $L$ be a linear order. For any map $f \colon L \to \mathbb{Z}$, we define the support of $f$ as the set $\mathop{\mathrm{Supp}}(f)=\{n \in L \mid f(n) \neq 0\}$ . The $L$-power of $\mathbb{Z}$, denoted by $\mathbb{Z}^L$, is the linear order on $\{f\colon L \to \mathbb{Z}\mid \mathop{\mathrm{Supp}}(f) \text{ is finite}\}$ defined by the following: if $f,g\colon L \to \mathbb{Z}$ are maps with finite support let $f \leq_{\mathbb{Z}^L}g$ if and only if $f=g$ or $f(n_0) <_{\mathbb{Z}} g(n_0)$ where $n_0 = \max\{n \in \mathop{\mathrm{Supp}}(f) \cup \mathop{\mathrm{Supp}}(g) \mid f(n) \neq g(n)\}$. Sometimes we need the following properties (see [@CCM19 Section 3.2]). **Proposition 17**. *For all ordinals $\beta<\alpha$, we have $$\mathbb{Z}^\alpha \cong \bigg(\sum_{\beta \leq \gamma < \alpha} \mathbb{Z}^\gamma \omega\bigg)^* + \bigg(\sum_{\beta \leq \gamma < \alpha} \mathbb{Z}^\gamma \omega\bigg).$$* **Proposition 18**. *For any linear orders $L$ and $L'$ we have* *[\[power_Z\^L-a\]]{#power_Z^L-a label="power_Z^L-a"} $(\mathbb{Z}^L)^*=\mathbb{Z}^L$,* *$\mathbb{Z}^{L+L'} \cong \mathbb{Z}^L \mathbb{Z}^{L'}$,* *if $L$ is countable and not a well-order then there is a countable ordinal $\alpha$ such that $\mathbb{Z}^L\cong \mathbb{Z}^\alpha\eta$.* ## Circular orders {#sec:circular orders} We now describe the basic notation and notions regarding circular orders. The prototype of a circular order is the unit circle $S^1$ traversed counterclockwise, which we denote by $C_{S^1}$. **Definition 19**. ([@KM05 Definition 2.1]) A ternary relation $C \subset X^3$ on a set $X$ is said to be a **circular order** if the following conditions are satisfied for every $x,y,z,w \in X$: Cyclicity: $(x,y,z) \in C \Rightarrow (y,z,x) \in C$; Antisymmetry and reflexivity: $(x,y,z) \in C \wedge (y,x,z) \in C \iff x=y \lor y=z \lor z=x$; [\[def:co_trans\]]{#def:co_trans label="def:co_trans"} Transitivity: $(x,y,z) \in C \Rightarrow \forall t ((x,y,t) \in C \vee (t,y,z) \in C)$; Totality: $(x,y,z) \in C \lor (y,x,z) \in C$. Notice that, assuming the other conditions, [\[def:co_trans\]](#def:co_trans){reference-type="ref" reference="def:co_trans"} is equivalent to asserting that $(x,y,z) \in C$ and $(x,z,w) \in C$ imply $(x,y,w) \in C$ whenever $x \neq z$. In the sequel we often make use of this reformulation. Definition [Definition 19](#def1){reference-type="ref" reference="def1"} is different from [@Ce69 5.1]: indeed, the latter characterizes the strict relation associated to $C$, i.e. the set of all triples $(x,y,z)$ such that $(x,y,z) \in C$ and $x,y,z$ are all distinct. By abuse of notation, when $C$ is a circular order on $X$ we write $C(x,y,z)$ instead of $(x,y,z) \in C$, for $x, y, z \in X$. The **reverse** $C^*$ of a circular order $C$ on $X$ is the circular order on $X$ defined by $C^*(x,y,z) \iff C(z,y,x)$ for all $x,y,z \in X$. Let $C$ and $C'$ be circular orders on sets $X$ and $X'$, respectively. We say that $C$ is **embeddable** into $C'$, and write $C \preccurlyeq_{c} C'$, if there exists an injective function $f\colon X \to X'$, called embedding, such that for every $x,y,z \in X$, $C(x,y,z) \Rightarrow C'(f(x),f(y),f(z))$ (notice that by totality and antisymmetry, one also has $C'(f(x),f(y),f(z)) \Rightarrow C(x,y,z)$). We say that $C$ and $C'$ are **isomorphic**, and write $C \cong_{c} C'$, if there exists $f$ as above which is a bijection (in which case $f$ is called isomorphism). For a circular order, the notions of successor and predecessor of an element are meaningless. However, we can still define a notion of immediate successor or immediate predecessor. **Definition 20**. Given a circular order $C$ on the set $X$ and $x, y \in X$, we say that $x$ is the **immediate predecessor** (resp. **immediate successor**) of $y$ in $C$ if $x \neq y$ and $C(x,y,z)$ (resp. $C(y,x,z)$) for every $z \in X$. **Definition 21**. Given a linear order $L$, we define a circular order $C[L]$ by setting $C[L](x,y,z)$ if and only if one of the following conditions is satisfied: $$x \leq_L y \leq_L z,\quad y \leq_L z \leq_L x,\quad z \leq_L x \leq_L y.$$ Notice that every circular order $C$ is of the form $C[L]$ for some (in general non unique) linear order $L$. Clearly, for two linear orders $L$ and $L'$ such that $L \preccurlyeq L'$ we have $C[L] \preccurlyeq_{c} C[L']$. Denote by $\mathop{\mathrm{\mathsf{CO}}}$ the set of codes for circular orders on $\mathbb{N}$, i.e. $$\mathop{\mathrm{\mathsf{CO}}}= \{C \in 2^{\mathbb{N}\times \mathbb{N}\times \mathbb{N}} \mid C \text{ codes a circular order on }\mathbb{N}\}.$$ Since $\mathop{\mathrm{\mathsf{CO}}}$ is a closed subset of the Polish space $2^{\mathbb{N}\times \mathbb{N}\times \mathbb{N}}$, we have that it is a Polish space as well. Denote by $\preccurlyeq_{\mathop{\mathrm{\mathsf{CO}}}}$ and $\cong_{\mathop{\mathrm{\mathsf{CO}}}}$ the restriction of the relations of embeddability $\preccurlyeq_c$ and isomorphism $\cong_c$ to $\mathop{\mathrm{\mathsf{CO}}}$, respectively. It is immediate that both $\preccurlyeq_{\mathop{\mathrm{\mathsf{CO}}}}$ and $\cong_{\mathop{\mathrm{\mathsf{CO}}}}$ are analytic. **Proposition 22**. *$\preccurlyeq_{\mathop{\mathrm{\mathsf{CO}}}}$ is a wqo.* *Proof.* Recall that a quasi-order $(X, {\leq_X})$ is a wqo if for every sequence $(x_n)_{n \in \mathbb{N}}$ of elements of $X$, there exist $n<m$ such that $x_n \leq_X x_m$. Suppose that $(C_n)_{n \in \mathbb{N}}$ is a sequence of elements of $\mathop{\mathrm{\mathsf{CO}}}$. For every $n \in \mathbb{N}$ consider the linear order $L_n$ defined by $$x\leq_{L_n} y \iff C_n(0,x,y) \land (y=0 \Rightarrow x=0).$$ Notice that $C[L_n] = C_n$. Since the embeddability relation $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$ on $\mathop{\mathrm{\mathsf{LO}}}$ is wqo, there are $n<m$ such that $L_n\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}} L_m$ and hence $C_n \preccurlyeq_{\mathop{\mathrm{\mathsf{CO}}}} C_m$. ◻ The isomorphism $\cong_{\mathop{\mathrm{\mathsf{CO}}}}$ is an equivalence relation on $\mathop{\mathrm{\mathsf{CO}}}$. Clearly, for $L, L' \in \mathop{\mathrm{\mathsf{LO}}}$, we have that ${{L}\cong_{\mathop{\mathrm{\mathsf{LO}}}}{L'}}$ implies ${C[L]} \cong_{\mathop{\mathrm{\mathsf{CO}}}} {C[L']}$. The converse implication is not true, as showed by $C[\omega+\boldsymbol{1}]$ and $C[\omega]$, for which we have $C[\omega+\boldsymbol{1}] \cong_{\mathop{\mathrm{\mathsf{CO}}}} C[\omega]$, but $\omega+ \boldsymbol{1} \ncong_{\mathop{\mathrm{\mathsf{LO}}}} \omega$. **Theorem 23**. *${{\cong}_{\mathop{\mathrm{\mathsf{CO}}}}} \sim_B {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}.$* *Proof.* For the Borel reduction from $\cong_{\mathop{\mathrm{\mathsf{CO}}}}$ to $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$, it is enough to note that $\cong_{\mathop{\mathrm{\mathsf{CO}}}}$ is an equivalence relation arising from a Borel action of the group $S_\infty$. Then ${{\cong}_{\mathop{\mathrm{\mathsf{CO}}}}} \leq_B {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$ by Theorem [Theorem 10](#thm 2){reference-type="ref" reference="thm 2"}. For the converse, consider the Borel map $\varphi\colon\mathop{\mathrm{\mathsf{LO}}}\to \mathop{\mathrm{\mathsf{CO}}}$ defined by $$\varphi(L) = C[\boldsymbol{1}+\zeta L].$$ If $L\cong_{\mathop{\mathrm{\mathsf{LO}}}}L'$ we have immediately that $\varphi(L) \cong_{\mathop{\mathrm{\mathsf{CO}}}} \varphi(L')$. Suppose now that $\varphi(L) \cong_{\mathop{\mathrm{\mathsf{CO}}}} \varphi(L')$ via the map $f$. Since $\boldsymbol{1}$ is the only element which has no immediate successor in both $\varphi(L)$ and $\varphi(L')$, we have that $f(\boldsymbol{1}) = \boldsymbol{1}$. Thus $\zeta L \cong_{\mathop{\mathrm{\mathsf{LO}}}}\zeta L'$ and by Lemma [Lemma 14](#lem:isom_zetaL){reference-type="ref" reference="lem:isom_zetaL"} we obtain $L \cong_{\mathop{\mathrm{\mathsf{LO}}}}L'$. ◻ # Convex embeddability {#sec:cvx emb} This is the main definition of the paper. **Definition 24** ([@BCP73]). Let $L$ and $L'$ be linear orders. We say that an embedding $f$ from $L$ to $L'$ is a **convex embedding** if $f(L)$ is an $L'$-convex set. We write $L\trianglelefteq L'$ when such $f$ exists, and call **convex embeddability** the resulting binary relation. *Remark 25*. Notice that $L \trianglelefteq L'$ if and only if $$L' \cong L_{l} + L + L_{r},$$ for some (possibly empty) $L_l$ and $L_r$, if and only if $L$ is isomorphic to an $L'$-convex set. While $L \preccurlyeq \eta$, for every countable linear order $L$, we have $L \trianglelefteq \eta$ if and only if $L$ has order type $\boldsymbol{1}$, $\eta$, $\boldsymbol{1}+\eta$, $\eta +\boldsymbol{1}$ or $\boldsymbol{1}+\eta + \boldsymbol{1}$. One easily sees that the restriction of convex embeddability to the Polish space $\mathop{\mathrm{\mathsf{LO}}}$ is an analytic quasi-order, which we denote by $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$. The strict part of $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ is denoted by $\triangleleft_{\mathop{\mathrm{\mathsf{LO}}}}$, that is, $L \triangleleft_{\mathop{\mathrm{\mathsf{LO}}}} L'$ if and only if $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L'$ but $L' \not{\trianglelefteq}_{\mathop{\mathrm{\mathsf{LO}}}} L$. We call **convex biembeddability**, and denote it by $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$, the equivalence relation on $\mathop{\mathrm{\mathsf{LO}}}$ induced by $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$, that is $${L} \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}{L'} \iff {L} \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} {L'} \mbox{ and } {L'} \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} {L} .$$ Clearly, if ${{L}\cong_{\mathop{\mathrm{\mathsf{LO}}}}{L'}}$ then ${{L} \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}{L'}}$. The converse implication does not hold, as witnessed by $\zeta\omega$ and $\omega+ \zeta\omega$. Finally, notice that if $L \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}L'$ then $L \equiv_{\mathop{\mathrm{\mathsf{LO}}}} L'$, where $\equiv_{\mathop{\mathrm{\mathsf{LO}}}}$ is the equivalence relation of biembeddability on $\mathop{\mathrm{\mathsf{LO}}}$ induced by $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$. The converse is not true: the linear orders of the form $\boldsymbol{k}\eta$, for $k>0$, belong to the same $\equiv_{\mathop{\mathrm{\mathsf{LO}}}}$-equivalence class, but they are pairwise $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-incomparable. ## Combinatorial properties {#sec:comb_prop} In this section we explore the combinatorial properties of convex embeddability, pointing out several differences between $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ and the embeddability relation $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$ on $\mathop{\mathrm{\mathsf{LO}}}$. For example, we show that $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ has antichains of size the continuum and chains of order type $(\mathbb{R}, {\leq} )$ (hence descending and ascending chains of arbitrary countable length), that well-orders are unbounded with respect to $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ (hence the unbounding number of $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ is $\aleph_1$), that $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ has dominating number $2^{\aleph_0}$ (thus in particular there is no $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-maximal element), and that all bases for $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ have maximal size $2^{\aleph_0}$. This is in stark contrast with the fact that $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$ is a wqo (and hence has neither infinite antichains nor infinite descending chains), that $\eta$ is the maximum with respect to $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$ (hence there are no $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$-unbounded sets and the dominating number of $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$ is $1$), and that $\{ \omega, \omega^* \}$ is a two-elements basis for $\preccurlyeq_{\mathop{\mathrm{\mathsf{LO}}}}$. Applying Proposition [Proposition 13](#cond_classes_isom){reference-type="ref" reference="cond_classes_isom"} and recalling that a convex embedding $f \colon L \to L'$ is just an isomorphism between $L$ and a convex subset of $L'$, we easily obtain the following useful fact. **Proposition 26**. *Let $L, L'$ be arbitrary linear orders. If $L \trianglelefteq L'$ via some convex embedding $f \colon L \to L'$, then the restriction of $f$ witnesses $c_F^L(\ell) \cong c_F^{L'}(f(\ell)) \cap f(L)$ and hence $|c_F^L(\ell)|=|c_F^{L'}(f(\ell)) \cap f(L)| \leq |c_F^{L'}(f(\ell))|$ for every $\ell \in L$. Moreover, $f(c_F^L(\ell)) = c_F^{L'}(f(\ell))$ for every $\ell \in L$, except for the first and last condensation classes of $L$ (if they exist). Finally, $L_F \trianglelefteq L'_F$ via the well-defined map $c^L_F(\ell) \mapsto c^{L'}_F(f(\ell))$.* Using the previous proposition and arguing as in the proof of Lemma [Lemma 14](#lem:isom_zetaL){reference-type="ref" reference="lem:isom_zetaL"}, it is straightforward to prove that $\zeta L \trianglelefteq \zeta L'$ if and only if $L \trianglelefteq L'$. Given a map $f \colon \mathbb{Q}\to \mathop{\mathrm{\mathsf{Scat}}}$, let $\eta_f \in \mathop{\mathrm{\mathsf{LO}}}$ be (an isomorphic copy on $\mathbb{N}$ of) the linear order $\eta_f = \sum_{q \in \mathbb{Q}} f(q)$. **Lemma 27**. *There is an embedding from the partial order $(\mathop{\mathrm{Int}}(\mathbb{R}),\subseteq)$ into $(\mathop{\mathrm{\mathsf{LO}}},\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}})$, where $\mathop{\mathrm{Int}}(\mathbb{R})$ is the set of the open intervals of $\mathbb{R}$.* *Proof.* Consider an injective map $f \colon \mathbb{Q}\to \{\boldsymbol{n} \mid n \in \mathbb{N}\setminus \{0\}\}$ and consider the resulting linear order $\eta_f$. Notice that for each $(\ell,q) \in \eta_f$, $|c_F^{\eta_f}(\ell,q)|= |f(q)|$ is finite. Moreover if $q$ and $q'$ are distinct rational numbers then $|c_F^{\eta_f}(\ell,q)| \neq |c_F^{\eta_f}(\ell',q')|$ for every $\ell \in f(q)$ and $\ell' \in f(q')$ by injectivity of $f$. An element of $\mathop{\mathrm{Int}}(\mathbb{R})$ is of the form $(x,y)$ where $x \in \{-\infty\} \cup \mathbb{R}$ and $y \in \mathbb{R}\cup \{+\infty\}$ with $x<y$. For such $(x,y)$ we define the linear order $L_{(x,y)} \cong \sum_{q \in \mathbb{Q}\cap (x,y)} f(q)$ as the restriction of $\eta_f$ to $\{(\ell,q) \in \eta_f \mid q \in \mathbb{Q}\cap (x,y)\}$, which is a convex subset of $\eta_f$ with no first and last condensation class. We show that, after canonically coding each $L_{(x,y)}$ as an element of $\mathop{\mathrm{\mathsf{LO}}}$, the map $(x,y) \mapsto L_{(x,y)}$ is an embedding of the partial order $(\mathop{\mathrm{Int}}(\mathbb{R}),\subseteq)$ into $(\mathop{\mathrm{\mathsf{LO}}},\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}})$. Clearly, if $(x,y) \subseteq(x',y')$, then $L_{(x,y)} \csube L_{(x',y')}$ and in particular we have $L_{(x,y)} \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L_{(x',y')}$. Vice versa, take $(x,y),(x',y') \in \mathop{\mathrm{Int}}(\mathbb{R})$, with $(x,y) \not\subseteq (x',y')$ and fix $q \in (x,y) \setminus (x',y')$. The condensation class of $(0,q)$ in $L_{(x,y)}$ has cardinality $f(q)$ and, by injectivity of $f$, no condensation class in $L_{(x',y')}$ has the same cardinality. Since there is no first and last condensation class in $L_{(x,y)}$, we get $L_{(x,y)} \ntrianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L_{(x',y')}$ by Proposition [Proposition 26](#cond_classes){reference-type="ref" reference="cond_classes"}. ◻ **Proposition 28**. *$\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ has chains of order type $(\mathbb{R}, {<} )$, as well as antichains of size $2^{\aleph_0}$.* *Proof.* This is immediate from Lemma [Lemma 27](#open_int){reference-type="ref" reference="open_int"} and the fact that $(\mathop{\mathrm{Int}}(\mathbb{R}), {\subseteq})$ has the same properties: consider e.g. the families $\{ (x,+\infty) \mid x \in \mathbb{R}\}$ and $\{ (x,x+1) \mid x \in \mathbb{R}\}$, respectively. ◻ Let $\mathfrak{b}(\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}})$ be the **unbounding number** of $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$, i.e. the smallest size of a family $\mathcal{F} \subseteq \mathop{\mathrm{\mathsf{LO}}}$ which is unbounded with respect to $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$. Using infinite (countable) sums of linear orders, one can easily prove that $\mathfrak{b}(\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}) > \aleph_0$. The next result thus shows that $\mathfrak{b}(\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}})$ attains the smallest possible value. **Proposition 29**. *$\mathop{\mathrm{\mathsf{WO}}}$ is a maximal $\omega_1$-chain without an upper bound in $\mathop{\mathrm{\mathsf{LO}}}$ with respect to $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$. Hence $\mathfrak{b}(\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}) = \aleph_1$.* *Proof.* Fix $L \in \mathop{\mathrm{\mathsf{LO}}}$ and for every $n \in L$ define $$\alpha_{n,L} = \sup \{\mathop{\mathrm{ot}}(L')\mid L' \text{ is a well-order},\ L'\csube L, \text{ and } n = \min L'\}.$$ Notice that $\alpha_{n,L}$ is actually attained by definition of $\csube$. Therefore, $\alpha_{n,L} < \omega_1$ because $L$ is countable. Let $\alpha_L = \sup_{n \in L} \alpha_{n,L}< \omega_1$. By construction, if $L' \csube L$ and $L$ is well-ordered, then $\mathop{\mathrm{ot}}(L') \leq \alpha_L$, thus $\boldsymbol{\alpha}_L + \boldsymbol{1} \ntrianglelefteq L$. Since $L$ was arbitrary, we showed that for every $L \in \mathop{\mathrm{\mathsf{LO}}}$ there exists $L' \in \mathop{\mathrm{\mathsf{WO}}}$ such that $L'\ntrianglelefteq L$, i.e. that $\mathop{\mathrm{\mathsf{WO}}}$ is $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-unbounded in $\mathop{\mathrm{\mathsf{LO}}}$. Clearly, $\mathop{\mathrm{\mathsf{WO}}}$ is a $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-chain: maximality then follows from unboundedness of $\mathop{\mathrm{\mathsf{WO}}}$, together with the observation that for $\omega \leq \alpha < \omega_1$ and $L \in \mathop{\mathrm{\mathsf{LO}}}$, if $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}\boldsymbol{\alpha}$ and $\boldsymbol{\beta} \triangleleft_{\mathop{\mathrm{\mathsf{LO}}}} L$ for every $\beta < \alpha$, then $L \cong \boldsymbol{\alpha}$. ◻ An easy consequence of Proposition [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"} is that no $L \in \mathop{\mathrm{\mathsf{LO}}}$ is a node with respect to $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$. This will be subsumed by Proposition [Proposition 33](#prop:everylobelongstoantichain){reference-type="ref" reference="prop:everylobelongstoantichain"}. **Corollary 30**. *For every $L \in \mathop{\mathrm{\mathsf{LO}}}$ there is $M \in \mathop{\mathrm{\mathsf{LO}}}$ which is $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-incomparable with $L$, i.e. $L \not\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} M$ and $M \not\trianglelefteq_{LO} L$.* *Proof.* If $L$ is not a well-order, then it is enough to let $M \in \mathop{\mathrm{\mathsf{WO}}}$ be such that $M \not\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L$ (the existence of such an $M$ is granted by Proposition [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"}). If instead $L$ is a well-order, then it is enough to set $M = \omega^*$. ◻ Another easy consequence of Proposition [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"} is that $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ has no maximal element. In fact, much more is true. **Corollary 31**. *Every $L \in \mathop{\mathrm{\mathsf{LO}}}$ is the bottom element of a $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-unbounded chain of length $\omega_1$.* *Proof.* For every $\beta < \omega_1$ set $L_\beta = L+\boldsymbol{\beta}$ (in particular, $L_0 = L$), and consider the (not necessarily strictly) $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-increasing sequence $\langle L_\beta \mid \beta < \omega_1 \rangle$. Since $\boldsymbol{\beta} \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L_\beta$ for every $\beta < \omega_1$, the above sequence is $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-unbounded by Proposition [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"}. Moreover, for every $\beta < \omega_1$ there is $\beta' < \omega_1$ such that $L_\beta \triangleleft_{\mathop{\mathrm{\mathsf{LO}}}} L_{\beta'}$. Indeed, it is enough to set $\beta' = \alpha_{L_\beta} + 1$, where $\alpha_{L_\beta}$ is as in the proof of Proposition [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"}: then $\boldsymbol{\beta}' \not\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L_\beta$, and thus also $L_{\beta'} \not\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L_\beta$. This easily implies that $\langle L_\beta \mid \beta < \omega_1 \rangle$ contains a strictly $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-increasing cofinal (hence $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-unbounded in $\mathop{\mathrm{\mathsf{LO}}}$) chain of length $\omega_1$ beginning with $L_0$, as desired. ◻ We say that a collection $\mathcal{B}$ of (infinite) linear orders on $\mathbb{N}$ is a **basis** for $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ if for every $L \in \mathop{\mathrm{\mathsf{LO}}}$ there is $L' \in \mathcal{B}$ such that $L' \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L$. The next result shows that each basis with respect to $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ is as large as possible. **Proposition 32**. *[\[prop:basisforcvx-1\]]{#prop:basisforcvx-1 label="prop:basisforcvx-1"} There are $2^{\aleph_0}$-many $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-incomparable $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-minimal elements in $\mathop{\mathrm{\mathsf{LO}}}$. In particular, if $\mathcal{B}$ is a basis for $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ then $|\mathcal{B}| = 2^{\aleph_0}$.* *[\[prop:basisforcvx-2\]]{#prop:basisforcvx-2 label="prop:basisforcvx-2"} There is a $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-decreasing $\omega$-sequence in $\mathop{\mathrm{\mathsf{LO}}}$ which is not $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-bounded from below.* *Proof.* [\[prop:basisforcvx-1\]](#prop:basisforcvx-1){reference-type="ref" reference="prop:basisforcvx-1"} Consider an infinite subset $S\subseteq \mathbb{N}$. Let $f_S\colon \mathbb{Q}\to \{ \boldsymbol{n} \mid n \in S \}$ be a map such that $$\forall q,q'(q<q'\rightarrow \forall n \in S\ \exists q''(q<q''<q' \wedge f_S(q'')= \boldsymbol{n})),$$ so that in particular $f_S$ is surjective, and consider the linear order $\eta_{f_S}$. Let $q < q'$ be arbitrary rational numbers. By a back-and-forth argument on the condensation classes, it is easy to see that by choice of $f_S$ the linear order $\eta_{f_S}$ is isomorphic to the restriction $\eta_{f_S} \!\restriction\!(q,q') \cong_{\mathop{\mathrm{\mathsf{LO}}}}\sum_{q'' \in \mathbb{Q}\cap (q,q')} f_S(q'')$ of $\eta_{f_S}$ to its convex subset $\{ ( \ell, q'') \in \eta_{f_S} \mid q < q'' < q' \}$. This implies that each $\eta_{f_S}$ is $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-minimal, because by density of $\eta$ and finiteness of the condensation classes of $\eta_{f_S}$, any infinite convex subset of $\eta_{f_S}$ contains some $\eta_{f_S} \!\restriction\!(q,q')$. Finally, by the choice of $f_S$ for every $n \in S$ there are densely many condensation classes in $(\eta_{f_S})_F$ of size exactly $n$. Thus if $S \neq S'$ we have $\eta_{f_S} \not\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} \eta_{f_{S'}}$ and $\eta_{f_{S'}} \not\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} \eta_{f_{S}}$ by Proposition [Proposition 26](#cond_classes){reference-type="ref" reference="cond_classes"}, as desired. [\[prop:basisforcvx-2\]](#prop:basisforcvx-2){reference-type="ref" reference="prop:basisforcvx-2"} Consider the family $\{ L_{(n,+\infty)} \mid n \in \mathbb{N}\}$, where $L_{(n,+\infty)}$ is as in the proof of Lemma [Lemma 27](#open_int){reference-type="ref" reference="open_int"}. It is a strictly $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-decreasing chain, and we claim that it is $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-unbounded from below. To this aim, it is enough to consider any $L \in \mathop{\mathrm{\mathsf{LO}}}$ with $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L_{(0,+\infty)}$, and show that $L \ntrianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L_{(m,+\infty)}$ for some $m \in \mathbb{N}$. Since $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L_{(0,+\infty)}$, all the condensation classes of $L$ are finite by Proposition [Proposition 26](#cond_classes){reference-type="ref" reference="cond_classes"}. Let $\ell \in L$ be such that $c^L_F(\ell)$ is not the minimum or the maximum of $L_F$, and let $q \in \mathbb{Q}$ be such that $f(q) = |c^L_F(\ell) |$, where $f \colon \mathbb{Q}\to \{\boldsymbol{n} \mid n \in \mathbb{N}\setminus \{0\}\}$ is the function used to define the linear orders $L_{(n,+\infty)}$. Let $m \in \mathbb{N}$ be such that $q < m$. Then $L \ntrianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L_{(m,+\infty)}$ because otherwise by choice of $\ell$ the latter would have a condensation class of size $f(q)$ by Proposition [Proposition 26](#cond_classes){reference-type="ref" reference="cond_classes"}, which is impossible by choice of $m$ and the fact that $f$ is an injection. ◻ Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"} allows us to considerably improve Corollary [Corollary 30](#cor:nonoteincvx){reference-type="ref" reference="cor:nonoteincvx"} as follows. **Proposition 33**. *Every $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-antichain is contained in a $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-antichain of size $2^{\aleph_0}$. In particular, there are no maximal $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-antichains of size smaller than $2^{\aleph_0}$, and every $L \in \mathop{\mathrm{\mathsf{LO}}}$ belongs to a $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-antichain of size $2^{\aleph_0}$.* *Proof.* Let $\mathcal{B}$ be a $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-antichain and assume that $|\mathcal{B}|<2^{\aleph_0}$ (otherwise the statement is trivial). Consider the antichain $\mathcal{A} = \{ \eta_{f_S} \mid S \subseteq \mathbb{N}\wedge S \text{ is infinite} \}$ of size $2^{\aleph_0}$ from Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}. From $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-minimality of $\eta_{f_S}$ it follows that $\mathcal{B} \cup (\mathcal{A} \setminus \bigcup_{L \in \mathcal{B}} \{ K \in \mathcal{A} \mid K \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L \})$ is a $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-antichain. To show that this antichain has size $2^{\aleph_0}$ it suffices to show that **Claim 1**. $\{ K \in \mathcal{A} \mid K \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L \}$ is countable for every $L \in \mathop{\mathrm{\mathsf{LO}}}$, so that $|\bigcup_{L \in \mathcal{B}} \{ K \in \mathcal{A} \mid K \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L \}| \leq \aleph_0 \cdot |\mathcal{B}| = \max \{ \aleph_0, |\mathcal{B}| \} < 2^{\aleph_0}$. To prove the claim, suppose that $S \subseteq \mathbb{N}$ is such that $\eta_{f_S} \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L$, so that without loss of generality we can write $L = L_l + \eta_{f_S} + L_r$. If $f$ were a convex embedding of $\eta_{f_{S'}}$ into $L$ with $f(\eta_{f_{S'}}) \cap \eta_{f_S} \neq \emptyset$, then by density of $\eta$ and finiteness of the condensation classes of $\eta_{f_{S'}}$ there would be rationals $q < q'$ such that $f(\eta_{f_{S'}} \!\restriction\!(q,q')) \subseteq \eta_{f_S}$, and since $\eta_{f_{S'}} \cong \eta_{f_{S'}} \!\restriction\!(q,q')$ we would get $\eta_{f_{S'}} \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}\eta_{f_S}$. Thus if $S \neq S'$, then $f(\eta_{f_{S'}}) \cap \eta_{f_S} = \emptyset$. Since $L$ is countable, this means that there are only countably many distinct $S \subseteq \mathbb{N}$ for which $\eta_{f_S} \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L$ can hold. Finally, the additional part of the statement follows by viewing $L \in \mathop{\mathrm{\mathsf{LO}}}$ as the element of an antichain of size $1$. ◻ We say that a collection $\mathcal{F} \subseteq \mathop{\mathrm{\mathsf{LO}}}$ is a dominating family with respect to $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ if and only if for every $L \in \mathop{\mathrm{\mathsf{LO}}}$ there exists $L' \in \mathcal{F}$ such that $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L'$. Let $\mathfrak{d}(\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}})$ be the **dominating number** of $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$, i.e. the least size of a dominating family with respect to $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$. The next proposition shows that $\mathfrak{d}(\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}})$ is as large as it can be. **Proposition 34**. *$\mathfrak{d}(\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}) = 2^{\aleph_0}$.* *Proof.* Consider again the antichain $\mathcal{A} = \{ \eta_{f_S} \mid S \subseteq \mathbb{N}\}$ from the proof of Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}. If $\mathcal{F}$ were a dominating family with respect to $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ of size $\kappa < 2^{\aleph_0}$, then by $|\mathcal{A}| = 2^{\aleph_0}$ there would be $M \in \mathcal{F}$ such that $\{ K \in \mathcal{A} \mid K \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}M \}$ is uncountable, contradicting Claim [Claim 1](#claim:everylobelongstoantichain){reference-type="ref" reference="claim:everylobelongstoantichain"}. ◻ ## Complexity with respect to Borel reducibility {#sec:complexity of cvx} At the beginning of Section [3](#sec:cvx emb){reference-type="ref" reference="sec:cvx emb"} we introduced the equivalence relation $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ of convex biembeddability on $\mathop{\mathrm{\mathsf{LO}}}$, observing that it is different from both isomorphism and biembeddability. We now focus on determining the complexity of $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ with respect to Borel reducibility. **Theorem 35**. *The map $\varphi$ sending a linear order $L$ to $\varphi(L) = \boldsymbol{1}+\zeta L + \boldsymbol{1}$ is such that* *[\[thm:red_iso_cvxeq-a\]]{#thm:red_iso_cvxeq-a label="thm:red_iso_cvxeq-a"} ${L \cong L'} \iff { \varphi(L) \cong \varphi(L')} \iff {\varphi(L) \mathrel{\underline{\bowtie}} \varphi(L')}\iff {\varphi(L) \trianglelefteq \varphi(L')}$;* *[\[thm:red_iso_cvxeq-b\]]{#thm:red_iso_cvxeq-b label="thm:red_iso_cvxeq-b"} $|\varphi(L)| = \max \{ \aleph_0, |L| \}$.* *Proof.* We claim that $\varphi$ reduces $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ to $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$. The second part is obvious, so let us concentrate on the first one. It is immediate that if $L \cong L'$ then $\varphi(L) \cong \varphi(L')$ and hence $\varphi(L) \mathrel{\underline{\bowtie}} \varphi(L')$, while $\varphi(L) \mathrel{\underline{\bowtie}} \varphi(L')$ clearly implies $\varphi(L) \trianglelefteq \varphi(L')$. Let now $f$ witness $\varphi(L)\trianglelefteq \varphi(L')$. The only elements of $\varphi(L)$ and $\varphi(L')$ without immediate successor and immediate predecessor are their minimum and maximum, respectively. Therefore, we must have $f(\min \varphi(L))=\min \varphi(L')$ and $f(\max \varphi(L))=\max \varphi(L')$. Hence $f$ is also surjective (hence an isomorphism), and $f\restriction (\zeta L)$ witnesses $\zeta L \cong \zeta L'$. Thus $L \cong L'$ by Lemma [Lemma 14](#lem:isom_zetaL){reference-type="ref" reference="lem:isom_zetaL"}. ◻ Noticing that when restricted to $\mathop{\mathrm{\mathsf{LO}}}$ the map from Theorem [Theorem 35](#thm:red_iso_cvxeq){reference-type="ref" reference="thm:red_iso_cvxeq"} is Borel, we immediately get **Corollary 36**. *${\cong_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$.* The main question now becomes whether $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}\leq_B {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$. This is still open and the answer is not obvious because e.g. it is not even clear if $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ is induced by a Borel action of $S_\infty$. We now embark in a deeper analysis of $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$, leading at least to $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}\leq_{\text{\scriptsize \textit{Baire}}} {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$. In the spirit of the definition of convex embeddability and recalling Remark [Remark 25](#def cvx){reference-type="ref" reference="def cvx"}, we introduce the following notions. **Definition 37**. Let $L$ be a linear order. We say that $L$ is **right compressible** if $L = L' + L_{r}$, with $L' \cong L$ and $L_{r} \neq\emptyset$; $L$ is **left compressible** if $L= L_{l} + L'$, with $L' \cong L$ and $L_{l} \neq\emptyset$; $L$ is **bicompressible** if it is both left compressible and right compressible, $L$ is **incompressible** if it is neither left nor right compressible. Notice that the set of right compressible linear orders is invariant with respect to isomorphism. The same holds for the set of left compressible linear orders, the set of bicompressible linear orders, and that of incompressible linear orders. It is clear that $\omega^*$ is right compressible but not left compressible, $\omega$ is left compressible but not right compressible, $\omega + \omega^*$ and $\eta$ are bicompressible, and $\zeta$ is incompressible. The following characterizations of the above notions turn out to be useful. **Lemma 38**. *Let $L$ be a linear order. Then* *[\[strong_right-a\]]{#strong_right-a label="strong_right-a"} $L$ is right compressible if and only if $L = L_{l} + \widetilde{L} + L_{r}$, with $\widetilde{L} \cong L$ and $L_{r} \neq\emptyset$.* *[\[strong_right-b\]]{#strong_right-b label="strong_right-b"} $L$ is left compressible if and only if $L = L_{l} + \widetilde{L} + L_{r}$, with $\widetilde{L} \cong L$ and $L_{l} \neq\emptyset$.* *[\[strong_right-c\]]{#strong_right-c label="strong_right-c"} $L$ is bicompressible if and only if $L=L_{l} + \widetilde{L} + L_{r}$, with $\widetilde{L} \cong L$ and $L_l,L_{r} \neq\emptyset$.* *Proof.* [\[strong_right-a\]](#strong_right-a){reference-type="ref" reference="strong_right-a"} For the non trivial direction, suppose that $L = L_{l} + \widetilde{L} + L_{r}$, with $L \cong \widetilde{L}$ via some $f \colon L \to \widetilde{L}$ and $L_{r} \neq\emptyset$. Let $M_0 = f(L_r)\subseteq \widetilde{L}$ and for every $n \in \mathbb{N}$ define $M_{n+1} = f(M_n)\subseteq \widetilde{L}$. Let $M = \bigcup_{n \in \mathbb{N}} M_n \subseteq \widetilde{L}$ and note that $f\upharpoonright (M + L_r)\colon M + L_r \to M$ is an isomorphism. Then the map $g\colon L\to L_l + \widetilde{L}$ defined by $$g(x) = \begin{cases} f(x) & \mbox{if } x \in M + L_r,\\ x, & \mbox{otherwise } \end{cases}$$ is an isomorphism witnessing $L \cong L_l + \widetilde{L}$. Thus, we can write $L = L' + L_{r}$, with $L' = L_l + \widetilde{L}$. [\[strong_right-b\]](#strong_right-b){reference-type="ref" reference="strong_right-b"} is similar to [\[strong_right-a\]](#strong_right-a){reference-type="ref" reference="strong_right-a"}. [\[strong_right-c\]](#strong_right-c){reference-type="ref" reference="strong_right-c"} If $L= L_{l} + L' + L_{r}$ with $L' \cong L$ and $L_l,L_{r} \neq\emptyset$, then by [\[strong_right-a\]](#strong_right-a){reference-type="ref" reference="strong_right-a"} and [\[strong_right-b\]](#strong_right-b){reference-type="ref" reference="strong_right-b"} we immediately obtain that $L$ is bicompressible. Conversely, suppose that $L$ is bicompressible. Since $L$ is left compressible, then $L=L_{l} + L'$, with $L' \cong L$ and $L_{l} \neq\emptyset$. Since $L'\cong L$ is right compressible, we can write $L'= \widetilde{L} + L_{r}$, with $\widetilde{L} \cong L'$ and $L_{r} \neq\emptyset$. Hence, $L=L_{l} + \widetilde{L} + L_{r}$, with $\widetilde{L} \cong L' \cong L$ and $L_l,L_r\neq\emptyset$. ◻ We denote by $\mathop{\mathrm{\mathsf{LO}}}_{r}\subseteq \mathop{\mathrm{\mathsf{LO}}}$ the set of (codes for) right compressible linear orders on $\mathbb{N}$, and by $\mathop{\mathrm{\mathsf{LO}}}_{l}\subseteq \mathop{\mathrm{\mathsf{LO}}}$ the set of (codes for) left compressible linear orders on $\mathbb{N}$. Note that $\mathop{\mathrm{\mathsf{LO}}}_{r}=\{L \in \mathop{\mathrm{\mathsf{LO}}}\mid L^* \in \mathop{\mathrm{\mathsf{LO}}}_{l}\}$, and vice versa. Moreover, each of the four sets $$\label{subsets of LO} \mathop{\mathrm{\mathsf{LO}}}\setminus (\mathop{\mathrm{\mathsf{LO}}}_{r}\cup \mathop{\mathrm{\mathsf{LO}}}_{l}) \qquad \quad \mathop{\mathrm{\mathsf{LO}}}_{l}\setminus \mathop{\mathrm{\mathsf{LO}}}_{r} \qquad \quad \mathop{\mathrm{\mathsf{LO}}}_{r}\setminus \mathop{\mathrm{\mathsf{LO}}}_{l} \qquad \quad \mathop{\mathrm{\mathsf{LO}}}_{r}\cap \mathop{\mathrm{\mathsf{LO}}}_{l}$$ is closed under isomorphism. The next proposition shows that they are also closed under $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$. **Proposition 39**. *If $L$ is a right compressible linear order and $L' \mathrel{\underline{\bowtie}} L$ (which implies $|L'| = |L|$), then $L'$ is right compressible as well. Similarly, if $L' \mathrel{\underline{\bowtie}} L$ and $L$ is left compressible (respectively: bicompressible, incompressible), then so is $L'$.* *In particular, the four subsets $\mathop{\mathrm{\mathsf{LO}}}\setminus (\mathop{\mathrm{\mathsf{LO}}}_{r}\cup \mathop{\mathrm{\mathsf{LO}}}_{l})$, $\mathop{\mathrm{\mathsf{LO}}}_{l}\setminus \mathop{\mathrm{\mathsf{LO}}}_{r},\mathop{\mathrm{\mathsf{LO}}}_{r}\setminus \mathop{\mathrm{\mathsf{LO}}}_{l}$ and $\mathop{\mathrm{\mathsf{LO}}}_{r}\cap \mathop{\mathrm{\mathsf{LO}}}_{l}$ are invariant with respect to ${\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$.* *Proof.* It is clearly enough to consider the case of right compressible linear orders. Since $L$ is right compressible, then $L = \widetilde{L} + L_{r}$ for some $\widetilde{L} \cong L$ and $L_{r} \neq\emptyset$. Let $f \colon L' \to \widetilde{L}$ and $g \colon L \to L'$ be convex embeddings witnessing $L' \trianglelefteq \widetilde{L}$ and $L \trianglelefteq L'$, respectively, so that $\widetilde{L} = \widetilde{L}_l + f(L') + \widetilde{L}_r$ and $L' = L'_l + g(L) + L'_r$. Then $$\begin{aligned} L' & = L'_l + g(L) + L'_r \\ & = L'_l + g(\widetilde{L}) + g(L_r) + L'_r \\ & = L'_l + g(\widetilde{L}_l) + g(f(L')) + g(\widetilde{L}_r) + g(L_r) + L'_r.\end{aligned}$$ Since $g(f(L')) \cong L'$ and $g(\widetilde{L}_r) + g(L_r) + L'_r \supseteq g(L_r) \neq \emptyset$, by Lemma [Lemma 38](#strong_right){reference-type="ref" reference="strong_right"} we have $L' \in \mathop{\mathrm{\mathsf{LO}}}_r$, as desired. ◻ We are now ready to go back to the study of the complexity of convex biembeddability. We can prove that the restrictions of $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ to each of the four sets in [\[subsets of LO\]](#subsets of LO){reference-type="eqref" reference="subsets of LO"}, which we denote by $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}\setminus (\mathop{\mathrm{\mathsf{LO}}}_{r}\cup \mathop{\mathrm{\mathsf{LO}}}_l)}$, $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{l}\setminus \mathop{\mathrm{\mathsf{LO}}}_r}$, $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\setminus \mathop{\mathrm{\mathsf{LO}}}_l}$ and $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\cap \mathop{\mathrm{\mathsf{LO}}}_l}$, respectively, are Borel bireducible with $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$. To this aim, we first observe that the map $\varphi_0 = \varphi$ from Theorem [Theorem 35](#thm:red_iso_cvxeq){reference-type="ref" reference="thm:red_iso_cvxeq"} reduces isomorphism to convex biembeddability restricted to incompressible linear orders, and that suitable variations of it do the same job but with left compressible (respectively, right compressible, bicompressible) linear orders. **Proposition 40**. *Given a linear order $L$, set $$\begin{aligned} \varphi_0(L) & = \boldsymbol{1}+\zeta L + \boldsymbol{1} \\ \varphi_1(L) & = \eta + \zeta L + \boldsymbol{1}\\ \varphi_2(L) & = \boldsymbol{1} + \zeta L + \eta \\ \varphi_3(L) & = \eta + \zeta L + \eta.\end{aligned}$$ Then $\varphi_0(L)$ is incompressible, $\varphi_1(L)$ is left compressible but not right compressible, $\varphi_2(L)$ is right compressible but not left compressible, and $\varphi_3(L)$ is bicompressible.* *Moreover, Theorem [Theorem 35](#thm:red_iso_cvxeq){reference-type="ref" reference="thm:red_iso_cvxeq"} is still true when $\varphi$ is replaced by any of the above $\varphi_i$'s.* *Proof.* As the minimum and the maximum of $\varphi_0(L)$ are the only elements without immediate predecessor and successor, respectively, we have that $\varphi_0(L)$ is not isomorphic to any of its proper convex subsets, i.e. it is incompressible. Hence we are done with $\varphi_0$ by Theorem [Theorem 35](#thm:red_iso_cvxeq){reference-type="ref" reference="thm:red_iso_cvxeq"}. Using a similar argument, one easily sees that $\varphi_1(L)$ is not right compressible. Indeed, any convex embedding $f$ of $\varphi_1(L)$ into itself cannot send $\max \varphi_1(L)$ into $\zeta L$ (by the argument in the previous paragraph) and cannot send it into $\eta$ either (because otherwise $f(\zeta L) \subseteq \eta$, which is clearly impossible). On the other hand, $\varphi_1(L)$ is trivially left compressible because one can map $\eta$ onto any of its (proper) final segments. Obviously $|\varphi_1(L)| = \max \{ \aleph_0, |L| \}$, ${L \cong L'} \Rightarrow {\varphi_1(L) \cong \varphi_1(L')}$, ${\varphi_1(L) \cong \varphi_1(L')} \Rightarrow {\varphi_1(L) \mathrel{\underline{\bowtie}} \varphi_1(L') }$, and also ${\varphi_1(L) \mathrel{\underline{\bowtie}} \varphi_1(L') } \Rightarrow {\varphi_1(L) \trianglelefteq \varphi_1(L')}$, so it remains to prove that if ${\varphi_1(L) \trianglelefteq \varphi_1(L') }$ then ${L \cong L'}$. Let $f \colon \varphi_1(L) \to \varphi_1(L')$ be a convex embedding. Since the elements of $\eta$ are the unique non-maximal points without immediate predecessor and immediate successor (both in $\varphi_1(L)$ and $\varphi_1(L')$), then $f(\eta) \subseteq \eta$. Similarly, since the elements of $\zeta L$ and $\zeta L'$ are the only elements having both an immediate predecessor and an immediate successor, then $f(\zeta L) \subseteq \zeta L'$. Moreover, the maximal element $\boldsymbol{1}$ has no immediate predecessor, which forbids $f(\boldsymbol{1}) \in \zeta L$, and we cannot have $f(\boldsymbol{1}) \in \eta$ because otherwise $f(\zeta L) \subseteq \eta$: thus $f(\boldsymbol{1}) = \boldsymbol{1}$. Since the range of $f$ is convex, it then follows that $f (\zeta L) = \zeta L'$, hence $\zeta L \cong \zeta L'$ and thus $L \cong L'$ by Lemma [Lemma 14](#lem:isom_zetaL){reference-type="ref" reference="lem:isom_zetaL"}. The cases of $\varphi_2(L)$ and $\varphi_3(L)$ are similar. ◻ When restricted to $\mathop{\mathrm{\mathsf{LO}}}$, the functions $\varphi_i$ are clearly Borel, thus we obtain: **Corollary 41**. *The isomorphism relation $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ is Borel reducible to any of $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}\setminus (\mathop{\mathrm{\mathsf{LO}}}_{r}\cup \mathop{\mathrm{\mathsf{LO}}}_l)}$, $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{l}\backslash \mathop{\mathrm{\mathsf{LO}}}_r}$, $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\backslash \mathop{\mathrm{\mathsf{LO}}}_l}$, and $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\cap \mathop{\mathrm{\mathsf{LO}}}_l}$.* Notice that the ranges of the four reductions used in the proof of Corollary [Corollary 41](#thm 3.2.3){reference-type="ref" reference="thm 3.2.3"} are all Borel, and that on such ranges isomorphism and convex biembeddability coincide. **Theorem 42**. *[\[red_cvxeq_jump_iso-i\]]{#red_cvxeq_jump_iso-i label="red_cvxeq_jump_iso-i"} On the set $\mathop{\mathrm{\mathsf{LO}}}\setminus (\mathop{\mathrm{\mathsf{LO}}}_{r}\cup \mathop{\mathrm{\mathsf{LO}}}_{l})$ the relations $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ and $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ coincide, so that $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}\backslash (\mathop{\mathrm{\mathsf{LO}}}_{r}\cup \mathop{\mathrm{\mathsf{LO}}}_l)}$ is Borel reducible to $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ via the identity map.* *[\[red_cvxeq_jump_iso-ii\]]{#red_cvxeq_jump_iso-ii label="red_cvxeq_jump_iso-ii"} Each of $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\cap \mathop{\mathrm{\mathsf{LO}}}_l}$, $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{l}\setminus \mathop{\mathrm{\mathsf{LO}}}_r}$, and $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\setminus \mathop{\mathrm{\mathsf{LO}}}_l}$ is Borel reducible to $(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+$, and thus to $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$.* *Proof.* [\[red_cvxeq_jump_iso-i\]](#red_cvxeq_jump_iso-i){reference-type="ref" reference="red_cvxeq_jump_iso-i"} Let $L,L' \in \mathop{\mathrm{\mathsf{LO}}}\setminus (\mathop{\mathrm{\mathsf{LO}}}_{r}\cup \mathop{\mathrm{\mathsf{LO}}}_{l})$. It is obvious that if $L \cong_{\mathop{\mathrm{\mathsf{LO}}}}L'$ then $L \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}L'$. For the other direction, assume $L \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}L'$ and let $f$ and $g$ be convex embeddings witnessing $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L'$ and $L' \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L$, respectively. Then $L' = L'_{l} + f(L) + L'_{r}$ and $L= L_{l} + g(L'_l)+ g(f(L))+ g(L'_r) + L_{r}$. Since $L$ is incompressible and $g(f(L)) \cong L$ we have $L_{l} + g(L'_l) = g(L'_r) + L_{r} = \emptyset$ and hence $L'_{l} = L'_{r} = \emptyset$, showing that $f$ is an isomorphism. [\[red_cvxeq_jump_iso-ii\]](#red_cvxeq_jump_iso-ii){reference-type="ref" reference="red_cvxeq_jump_iso-ii"} We start by considering the case of $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\cap \mathop{\mathrm{\mathsf{LO}}}_l}$. Let $\varphi_{r+l}\colon \mathop{\mathrm{\mathsf{LO}}}\setminus [\zeta,\omega,\omega^*]_{\cong} \to \mathop{\mathrm{\mathsf{LO}}}^\mathbb{N}$ be a Borel map such that $\varphi_{r+l}(L)$ is an enumeration (possibly with repetitions) of all the *infinite* subsets of $L$ of the form $[n,m]_L$. Since we are omitting the isomorphism types of $\zeta$, $\omega$, and $\omega^*$ the map is well-defined, i.e. for each $L$ in its domain there is at least one infinite interval $[n,m]_L$, and clearly $\mathop{\mathrm{\mathsf{LO}}}_l \cap \mathop{\mathrm{\mathsf{LO}}}_r \subseteq \mathop{\mathrm{dom}}(\varphi_{r+l})$. By the same reason, its domain is Borel because we are omitting finitely many $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$-classes, which are Borel themselves. We claim that for all $L , L' \in \mathop{\mathrm{\mathsf{LO}}}_l \cap \mathop{\mathrm{\mathsf{LO}}}_r$ $$L \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}L' \iff \varphi_{r+l}(L) \mathrel{(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+} \varphi_{r+l}(L'),$$ so that any Borel extension of $\varphi_{r+l}$ to $\mathop{\mathrm{\mathsf{LO}}}$ witnesses ${\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\cap \mathop{\mathrm{\mathsf{LO}}}_l}} \leq_B {(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+}$, and hence ${\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\cap \mathop{\mathrm{\mathsf{LO}}}_l}} \leq_B {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$ by Theorem [\[cor 2\]](#cor 2){reference-type="ref" reference="cor 2"}. Assume first that $L \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}L'$, and let $f$ be a convex embedding witnessing $L\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L'$. Given any infinite $[n,m]_L$, we have $[n,m]_L \cong_{\mathop{\mathrm{\mathsf{LO}}}}[f(n),f(m)]_{L'}$, so that in particular the latter is infinite and appears among the linear orders in $\varphi_{r+l}(L')$. Symmetrically, if $g$ is a convex embedding witnessing $L' \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L$, then for every infinite $[n,m]_{L'}$ we have $[n,m]_{L'} \cong [g(n),g(m)]_L$. It follows that $\varphi_{r+l}(L) \mathrel{(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+} \varphi_{r+l}(L')$. Conversely, observe that since $L \in \mathop{\mathrm{\mathsf{LO}}}_{l}\cap \mathop{\mathrm{\mathsf{LO}}}_{r}$ then by Lemma [Lemma 38](#strong_right){reference-type="ref" reference="strong_right"} we have $L = L_l+ \widetilde{L} + L_r$, with $\widetilde{L}\cong L$ and both $L_l$ and $L_r$ nonempty. Fix $k \in L_l$ and $m \in L_r$. Then $\widetilde{L} \csube [k,m]_L$, and hence $L \trianglelefteq [k,m]_L$ and $[k,m]_L$ is infinite. Thus if $\varphi_{r+l}(L) \mathrel{(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+} \varphi_{r+l}(L')$, there are $k', m' \in L'$ such that $[k,m]_L\cong [k',m']_{L'}$. But then $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L'$ because $L \trianglelefteq [k,m]_L \cong [k',m']_{L'}\trianglelefteq L'$. The argument to show that if $\varphi_{r+l}(L) \mathrel{(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+} \varphi_{r+l}(L')$ then $L'\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L$ is symmetric. We now move to the case of $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{l}\setminus \mathop{\mathrm{\mathsf{LO}}}_r}$. Let $\varphi_{l}\colon \mathop{\mathrm{\mathsf{LO}}}\setminus [\omega^*]_{\cong} \to \mathop{\mathrm{\mathsf{LO}}}^\mathbb{N}$ be a Borel map such that $\varphi_l(L)$ is an enumeration of all the *infinite* subsets of $L$ of the form $[n,+\infty)_L$, which is well-defined on all $L \not\cong \omega^*$ and such that $\mathop{\mathrm{\mathsf{LO}}}_l \setminus \mathop{\mathrm{\mathsf{LO}}}_r \subseteq \mathop{\mathrm{dom}}(\varphi_l)$. Arguing as above, it is enough to show that for all $L,L' \in \mathop{\mathrm{\mathsf{LO}}}_{l}\setminus \mathop{\mathrm{\mathsf{LO}}}_{r}$ $$L \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}L' \iff \varphi_{l}(L) \mathrel{(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+} \varphi_{l}(L').$$ For the forward direction, let $f$ and $g$ be convex embeddings witnessing $L\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L'$ and $L'\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L$, respectively. We first show that $f(L)$ is a final segment of $L'$. Since $f$ is a convex embedding, $L'=L'_{l} + f(L) + L'_{r}$ with $L'_l$ and $L'_{r}$ possibly empty. Then $L= L_l + g(f(L))+L_r$ with $L_{r}\supseteq g(L'_r)$. Since $g(f(L))\cong L$ and $L \notin \mathop{\mathrm{\mathsf{LO}}}_{r}$, we have $L_r = \emptyset$ and hence $L'_r = \emptyset$, i.e. $L' = L'_l+f(L)$. Thus if $[n, \infty)_L$ is infinite, then $f([n,\infty)_L) = [f(n),\infty)_{L'}$, so that, being infinite, the latter appears in $\varphi_l(L')$ and $[n,\infty)_L \cong [f(n),\infty)_{L'}$. Analogously, $g(L')$ is a final segment of $L$ because $L' \notin \mathop{\mathrm{\mathsf{LO}}}_r$, hence for every infinite $[n,+\infty)_{L'}$, we have $[n,+\infty)_{L'} \cong [g(n),+\infty)_L$. It follows that $\varphi_{l}(L) \mathrel{(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+} \varphi_{l}(L')$. Conversely, assume that $\varphi_{l}(L) \mathrel{(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+} \varphi_{l}(L')$. Using $L \in \mathop{\mathrm{\mathsf{LO}}}_l$, let $L= L_l + \widetilde{L}$ with $L_l \neq \emptyset$ and $\widetilde{L} \cong L$, and fix any $m \in L_l$. Then $\widetilde{L} \csube [m,+\infty)_L$, and thus the latter, being infinite, appears in $\varphi_l(L)$ and $L \trianglelefteq [m,+\infty)_L$. Let $m' \in L'$ be such that $[m,+\infty)_L\cong_{\mathop{\mathrm{\mathsf{LO}}}}[m',+\infty)_{L'}$: then $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} [m,+\infty)_L \cong_{\mathop{\mathrm{\mathsf{LO}}}} [m',+\infty)_{L'}\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L'$. Reversing the role of $L$ and $L'$ we get $L'\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L$ and we are done. The case of $\underline{\bowtie}_{\tiny \mathop{\mathrm{\mathsf{LO}}}_{r}\setminus \mathop{\mathrm{\mathsf{LO}}}_l}$ is symmetric, with the desired Borel reduction be given by any Borel map $\varphi_{r}\colon \mathop{\mathrm{\mathsf{LO}}}\setminus [\omega]_{\cong} \to \mathop{\mathrm{\mathsf{LO}}}^\mathbb{N}$ such that $\varphi_r(L)$ is an enumeration of all the *infinite* subsets of $L$ of the form $(-\infty,n]_L$. ◻ *Remark 43*. The statement and proof of Theorem [Theorem 42](#red_cvxeq_jump_iso){reference-type="ref" reference="red_cvxeq_jump_iso"} can easily be adapted to deal with uncountable linear orders of a given cardinality $\kappa$. However, since we have no use for this in the present paper, for the sake of simplicity we decided to stick to the countable case. If $\mathop{\mathrm{\mathsf{LO}}}_{r}$ and $\mathop{\mathrm{\mathsf{LO}}}_{l}$ were Borel subsets of $\mathop{\mathrm{\mathsf{LO}}}$, then we could glue the reductions from the proof of Theorem [Theorem 42](#red_cvxeq_jump_iso){reference-type="ref" reference="red_cvxeq_jump_iso"} and obtain a Borel reduction from the whole $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ to $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$. Unfortunately, this is not the case: none of the subclasses of $\mathop{\mathrm{\mathsf{LO}}}$ involved in Theorem [Theorem 42](#red_cvxeq_jump_iso){reference-type="ref" reference="red_cvxeq_jump_iso"} is Borel. To prove this, we need the following lemmas. **Lemma 44**. *Let $\alpha > 0$. For any $z \in \mathbb{Z}^{\alpha}$ and $\beta < \alpha$ there exists $\gamma$ such that $\beta \leq \gamma<\alpha$ and ${\mathbb{Z}^{\gamma} \omega} \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}{[z, +\infty)_{\mathbb{Z}^\alpha}}$.* *Proof.* We consider the isomorphic copy of $\mathbb{Z}^\alpha$ given by Proposition [Proposition 17](#power_Z){reference-type="ref" reference="power_Z"}: $$\bigg(\sum_{\beta \leq \gamma< \alpha} \mathbb{Z}^{\gamma} \omega\bigg)^* + \bigg(\sum_{\beta \leq \gamma< \alpha} \mathbb{Z}^{\gamma} \omega\bigg).$$ Without loss of generality we can assume $z \in \sum_{\beta \leq \gamma< \alpha} \mathbb{Z}^{\gamma} \omega$, so that there exists $\gamma$ with $\beta \leq \gamma< \alpha$ such that $z \in \mathbb{Z}^{\gamma} \omega$. Since $\mathbb{Z}^{\gamma} \omega \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}[z, +\infty)_{\mathbb{Z}^{\gamma} \omega} \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}[z, +\infty)_{\mathbb{Z}^{\alpha}}$, this $\gamma$ works. ◻ **Lemma 45**. *For every ordinal $\alpha>0$, $\mathbb{Z}^{\alpha}$ is incompressible.* *Proof.* By induction on $\alpha>0$. We have already noticed that $\mathbb{Z}^1 \cong_{\mathop{\mathrm{\mathsf{LO}}}}\zeta$ is incompressible. Fix $\alpha>1$ and assume that $\mathbb{Z}^\beta$ is incompressible for every $\beta < \alpha$. We consider the isomorphic copy of $\mathbb{Z}^\alpha$ given by Proposition [Proposition 17](#power_Z){reference-type="ref" reference="power_Z"} with $\beta=0$: $$\bigg(\sum_{\gamma< \alpha} \mathbb{Z}^{\gamma} \omega\bigg)^* + \bigg(\sum_{\gamma< \alpha} \mathbb{Z}^{\gamma} \omega\bigg).$$ We just prove that $\mathbb{Z}^\alpha\notin \mathop{\mathrm{\mathsf{LO}}}_r$, as $\mathbb{Z}^\alpha\notin \mathop{\mathrm{\mathsf{LO}}}_l$ can be proved in a symmetric way. Suppose, towards a contradiction, that $\mathbb{Z}^\alpha\in \mathop{\mathrm{\mathsf{LO}}}_{r}$ and let $f$ be a convex embedding of $\mathbb{Z}^\alpha$ into a proper initial segment of $\mathbb{Z}^\alpha$. Assume first that $f(\mathbb{Z}^\alpha) \cap \big(\sum_{\gamma< \alpha} \mathbb{Z}^{\gamma} \omega\big) \neq \emptyset$. Let $\beta<\alpha$ be least such that $\mathbb{Z}^{\gamma} \omega \nsubseteq f(\mathbb{Z}^\alpha)$ for every $\gamma\geq \beta$. (Such a $\beta$ exists by the choice of $f$.) **Claim 2**. $f(\mathbb{Z}^\alpha) \cap \mathbb{Z}^{\gamma} \omega = \emptyset$ for every $\gamma\geq \beta$, so that $f(\mathbb{Z}^\alpha)$ is a final segment of $$\bigg(\sum_{\gamma < \alpha} \mathbb{Z}^\gamma \omega\bigg)^* + \bigg( \sum_{\gamma < \beta} \mathbb{Z}^\gamma \omega\bigg).$$ *Proof of the Claim.* If $\gamma>\beta$ the convexity of $f(\mathbb{Z}^\alpha)$ implies immediately $f(\mathbb{Z}^\alpha) \cap \mathbb{Z}^{\gamma} \omega = \emptyset$, so we only need to consider the case $\gamma=\beta$. Towards a contradiction, assume that $f(\mathbb{Z}^\alpha)$ intersects $\mathbb{Z}^\beta \omega$, and using $f(\mathbb{Z}^\alpha) \nsupseteq \mathbb{Z}^\beta \omega$ let $n$ be maximum such that $f(\mathbb{Z}^\alpha) \cap (\mathbb{Z}^\beta \times \{ n \}) \neq \emptyset$. Pick $z \in \mathbb{Z}^\alpha$ such that $f(z)\in \mathbb{Z}^\beta \times \{ n \}$. By Lemma [Lemma 44](#incomp_z_cl1){reference-type="ref" reference="incomp_z_cl1"} there exists $\beta \leq \gamma < \alpha$ such that $\mathbb{Z}^{\gamma} \omega \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}[z, +\infty)_{\mathbb{Z}^\alpha}$ and hence $\mathbb{Z}^{\gamma} \omega \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}[f(z), +\infty)_{\mathbb{Z}^{\beta} \times \{ n \}}$. But then $\mathbb{Z}^{\gamma} \omega\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}\mathbb{Z}^\beta$, and since $\mathbb{Z}^\beta \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}\mathbb{Z}^{\gamma} \cong \mathbb{Z}^\gamma \times \{ 0 \}$ by $\beta \leq \gamma$ (see Definition [Definition 15](#def:Z^alpha){reference-type="ref" reference="def:Z^alpha"}) this shows that $\mathbb{Z}^\beta$ is right compressible, against the induction hypothesis. ◻ Using Proposition [Proposition 17](#power_Z){reference-type="ref" reference="power_Z"} again, we have $$\begin{aligned} \bigg(\sum_{\gamma < \alpha} \mathbb{Z}^\gamma \omega\bigg)^* + \bigg(\sum_{\gamma < \beta} \mathbb{Z}^\gamma \omega\bigg) & = \bigg(\sum_{\beta \leq \gamma < \alpha} \mathbb{Z}^\gamma \omega\bigg)^* + \bigg(\sum_{\gamma < \beta} \mathbb{Z}^\gamma \omega\bigg)^* + \bigg( \sum_{\gamma < \beta} \mathbb{Z}^\gamma \omega\bigg) \\ & \cong \bigg(\sum_{\beta \leq \gamma < \alpha} \mathbb{Z}^\gamma \omega\bigg)^* + \mathbb{Z}^\beta\end{aligned}$$ Let $g$ be the isomorphism between the first and last element of this chain. Choose $z \in \mathbb{Z}^\alpha$ such that $g(f(z)) \in \mathbb{Z}^\beta$ --- such a $z$ exists because $f(\mathbb{Z}^\alpha)$ is cofinal in $\big(\sum_{\gamma < \alpha} \mathbb{Z}^\gamma \omega\big)^* + \big( \sum_{\gamma < \beta} \mathbb{Z}^\gamma \omega\big)$ by Claim [Claim 2](#claim:Z^alpha){reference-type="ref" reference="claim:Z^alpha"}. Arguing as before, $\mathbb{Z}^{\gamma} \omega \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}[g(f(z)), +\infty)_{ \mathbb{Z}^{\beta}}$ for some $\beta \leq \gamma < \alpha$, contradicting again the incompressibility of $\mathbb{Z}^\beta$. Finally, assume that $f(\mathbb{Z}^\alpha) \cap \big(\sum_{\gamma< \alpha} \mathbb{Z}^{\gamma} \omega\big) = \emptyset$, i.e. $f(\mathbb{Z}^\alpha) \subseteq \big( \sum_{\gamma< \alpha} \mathbb{Z}^{\gamma}\omega\big)^*$. Let $\beta < \alpha$ be smallest such that $f(\mathbb{Z}^\alpha) \cap (\mathbb{Z}^\beta \omega)^* \neq \emptyset$, and let $n$ be smallest such that $f(\mathbb{Z}^\alpha) \cap (\mathbb{Z}^\beta \times \{ n \})^* \neq \emptyset$. Pick $z \in \mathbb{Z}^\alpha$ such that $f(z) \in (\mathbb{Z}^\beta \times \{ n \})^*$. Arguing as before, there is $\beta \leq \gamma < \alpha$ such that $\mathbb{Z}^\gamma \omega \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}(\mathbb{Z}^\beta \times \{ n \})^*$. Since $(\mathbb{Z}^\beta \times \{ n \})^* \cong (\mathbb{Z}^\beta)^* \cong \mathbb{Z}^\beta$ by Proposition [Proposition 18](#power_Z^L){reference-type="ref" reference="power_Z^L"}, this would mean that $\mathbb{Z}^\gamma \omega \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}\mathbb{Z}^\beta$, contradicting again the incompressibility of the latter. ◻ **Theorem 46**. *[\[compl_subsets-a\]]{#compl_subsets-a label="compl_subsets-a"} $\mathop{\mathrm{\mathsf{LO}}}_{l}$, $\mathop{\mathrm{\mathsf{LO}}}_{r}$ and $\mathop{\mathrm{\mathsf{LO}}}_{r} \cap \mathop{\mathrm{\mathsf{LO}}}_l$ are $\mathbf{\Sigma}_1^1$-complete.* *[\[compl_subsets-b\]]{#compl_subsets-b label="compl_subsets-b"} $\mathop{\mathrm{\mathsf{LO}}}\setminus (\mathop{\mathrm{\mathsf{LO}}}_r \cup \mathop{\mathrm{\mathsf{LO}}}_l)$ is $\mathbf{\Pi}_1^1$-complete.* *[\[compl_subsets-c\]]{#compl_subsets-c label="compl_subsets-c"} $\mathop{\mathrm{\mathsf{LO}}}_{r}\setminus \mathop{\mathrm{\mathsf{LO}}}_l$ and $\mathop{\mathrm{\mathsf{LO}}}_{l}\setminus \mathop{\mathrm{\mathsf{LO}}}_r$ are $D_2(\mathbf{\Pi}_1^1)$-complete.* *Proof.* [\[compl_subsets-a\]](#compl_subsets-a){reference-type="ref" reference="compl_subsets-a"} First, we check that $\mathop{\mathrm{\mathsf{LO}}}_{r}$ is $\mathbf{\Sigma}_1^1$. Indeed, $L \in \mathop{\mathrm{\mathsf{LO}}}_{r}$ if and only if $$\begin{aligned} \exists f\colon \mathbb{N}\to \mathbb{N}\big[ & \forall n,m \; (n< _L m \rightarrow f(n)<_L f(m)) \; \wedge\\ & \forall n,m,k \; (f(n)\leq_L k \leq_L f(m)\rightarrow \exists k'(f(k')=k)) \; \wedge\\ & \exists n\; \forall m \; (f(m) <_L n)\big]. \end{aligned}$$ In a similar way, one can prove that $\mathop{\mathrm{\mathsf{LO}}}_{l}$ (and hence also $\mathop{\mathrm{\mathsf{LO}}}_l \cap \mathop{\mathrm{\mathsf{LO}}}_r$) is $\mathbf{\Sigma}_1^1$. We now show that $\mathop{\mathrm{\mathsf{LO}}}_{l}$, $\mathop{\mathrm{\mathsf{LO}}}_{r}$ and $\mathop{\mathrm{\mathsf{LO}}}_{r} \cap \mathop{\mathrm{\mathsf{LO}}}_l$ are $\mathbf{\Sigma}_1^1$-hard by continuously reducing the $\mathbf{\Sigma}^1_1$-complete set $\mathop{\mathrm{\mathsf{LO}}}\setminus \mathop{\mathrm{\mathsf{WO}}}$ to each of them. We can actually use the continuous function $L \mapsto \mathbb{Z}^L$ for all three sets. Indeed, if $L \notin \mathop{\mathrm{\mathsf{WO}}}$, by Proposition [Proposition 18](#power_Z^L){reference-type="ref" reference="power_Z^L"} we have $\mathbb{Z}^L \cong \mathbb{Z}^\alpha \eta$ for some ordinal $\alpha$, and hence $\mathbb{Z}^L$ is obviously bicompressible. If $L \in \mathop{\mathrm{\mathsf{WO}}}$, then $\mathbb{Z}^L$ is incompressible by Lemma [Lemma 45](#incomp_z){reference-type="ref" reference="incomp_z"}. [\[compl_subsets-b\]](#compl_subsets-b){reference-type="ref" reference="compl_subsets-b"} is immediate from the proof of [\[compl_subsets-a\]](#compl_subsets-a){reference-type="ref" reference="compl_subsets-a"}. [\[compl_subsets-c\]](#compl_subsets-c){reference-type="ref" reference="compl_subsets-c"} By [\[compl_subsets-a\]](#compl_subsets-a){reference-type="ref" reference="compl_subsets-a"} it follows that $\mathop{\mathrm{\mathsf{LO}}}_r \setminus \mathop{\mathrm{\mathsf{LO}}}_l$ and $\mathop{\mathrm{\mathsf{LO}}}_l \setminus \mathop{\mathrm{\mathsf{LO}}}_r$ are $D_2(\mathbf{\Pi}_1^1)$. Consider now the set $A = \{(L,L') \in \mathop{\mathrm{\mathsf{LO}}}\times \mathop{\mathrm{\mathsf{LO}}}\mid L \notin \mathop{\mathrm{\mathsf{WO}}}\mbox{ and } L' \in \mathop{\mathrm{\mathsf{WO}}}\}$ and recall that it is $D_2(\mathbf{\Pi}_1^1)$-complete. Define the continuous map $\psi \colon \mathop{\mathrm{\mathsf{LO}}}\times \mathop{\mathrm{\mathsf{LO}}}\to \mathop{\mathrm{\mathsf{LO}}}$ by $\psi(L, L') = \mathbb{Z}^{1+L'} + \eta + \mathbb{Z}^{1+L}$. We claim that $\psi(L,L')$ is left compressible if and only if $L' \notin \mathop{\mathrm{\mathsf{WO}}}$. One direction is obvious: if $L' \notin \mathop{\mathrm{\mathsf{WO}}}$, then $\mathbb{Z}^{1+L'} \cong \mathbb{Z}^\alpha \eta$ for some $\alpha \geq 1$, and thus it has a convex self-embedding onto a proper final segment of it, which can then be naturally extended to a witness of $\psi(L,L') \in \mathop{\mathrm{\mathsf{LO}}}_l$. For the other direction, we use the fact that every convex subset of $\eta$ consists of points which have neither an immediate predecessor nor an immediate successors, while convex subsets of $\mathbb{Z}^{1+L}$ and $\mathbb{Z}^{1+L'}$ with at least two points always contain elements with both an immediate predecessor and an immediate successor in the given linear order. (Here we use again the fact that $\mathbb{Z}^{1+L}$ and $\mathbb{Z}^{1+L'}$ are either of the form $\mathbb{Z}^\alpha$ or $\mathbb{Z}^\alpha \eta$ for some $\alpha\geq 1$, depending on whether $L$ and $L'$ are well-orders or not.) Thus if $f \colon \psi(L,L') \to \psi(L,L')$ is a convex embedding we must have $f(\eta) = \eta$, and hence $f(\mathbb{Z}^{1+L'}) \subseteq \mathbb{Z}^{1+L'}$. Thus if $L' \in \mathop{\mathrm{\mathsf{WO}}}$ then $\mathbb{Z}^{1+L'} \notin \mathop{\mathrm{\mathsf{LO}}}_l$ by Lemma [Lemma 45](#incomp_z){reference-type="ref" reference="incomp_z"}, which implies $f(\mathbb{Z}^{1+L'}) = \mathbb{Z}^{1+L'}$: since $f$ was arbitrary, this shows that $\psi(L,L') \notin \mathop{\mathrm{\mathsf{LO}}}_l$. Analogously, one can check that $\psi(L,L')$ is right compressible if and only if $L \notin \mathop{\mathrm{\mathsf{WO}}}$. Using these facts, it is then easy to prove that $(L,L') \in A$ if and only if $\psi(L,L') \in \mathop{\mathrm{\mathsf{LO}}}_{r}\setminus \mathop{\mathrm{\mathsf{LO}}}_l$, hence $\psi$ witnesses that $\mathop{\mathrm{\mathsf{LO}}}_{r}\setminus \mathop{\mathrm{\mathsf{LO}}}_l$ is $D_2(\mathbf{\Pi}_1^1)$-hard. For $\mathop{\mathrm{\mathsf{LO}}}_{l}\setminus \mathop{\mathrm{\mathsf{LO}}}_r$ it suffices to switch the positions of $\mathbb{Z}^{1+L}$ and $\mathbb{Z}^{1+L'}$ in the definition of $\psi$. ◻ Even if they are not Borel, the sets $\mathop{\mathrm{\mathsf{LO}}}\setminus (\mathop{\mathrm{\mathsf{LO}}}_{r}\cup \mathop{\mathrm{\mathsf{LO}}}_{l})$, $\mathop{\mathrm{\mathsf{LO}}}_{l}\setminus \mathop{\mathrm{\mathsf{LO}}}_{r}$, $\mathop{\mathrm{\mathsf{LO}}}_{r}\setminus \mathop{\mathrm{\mathsf{LO}}}_{l}$ and $\mathop{\mathrm{\mathsf{LO}}}_{r}\cap \mathop{\mathrm{\mathsf{LO}}}_{l}$ belong to the Boolean algebra generated by the analytic sets, and hence have the Baire property and are universally measurable. By Theorem [Theorem 42](#red_cvxeq_jump_iso){reference-type="ref" reference="red_cvxeq_jump_iso"} and Proposition [Proposition 3](#Baire_red_for_qclas_count_str){reference-type="ref" reference="Baire_red_for_qclas_count_str"} we thus obtain the following result. **Corollary 47**. *The equivalence relation $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ is $\sigma$-classifiable by countable structures, and therefore $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}\leq_{\text{\scriptsize \textit{Baire}}} {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$.* Notice that, since the partition of $\mathop{\mathrm{\mathsf{LO}}}$ given by [\[subsets of LO\]](#subsets of LO){reference-type="eqref" reference="subsets of LO"} is finite, we actually have that the preimages of Borel sets via the reduction of $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ to $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ are Boolean combinations of analytic sets. The problem of whether $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ is Borel reducible to $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ remains open. However, from the reductions above we can derive some more information about the complexity of $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$, showing that it shares some important properties with $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$. **Corollary 48**. *If $X$ is a Polish space on which the action of a Polish group $G$ is turbulent, then $E^X_G \nleq_B \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$.* *Proof.* If $E^X_G \leq_B \mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$, then by Corollary [Corollary 47](#cvxeq_baire_iso){reference-type="ref" reference="cvxeq_baire_iso"} we would have that $E^X_G \leq_{\text{\scriptsize \textit{Baire}}} {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$, against Theorem [Theorem 4](#turb_act){reference-type="ref" reference="turb_act"}. ◻ In Proposition [\[cor 2\]](#cor 2){reference-type="ref" reference="cor 2"} we observed that $(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+ \leq_{B} {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$. Replacing Borel reducibility with Baire reducibility, we get an analogous result for $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$. **Corollary 49**. *${(\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}})^+} \leq_{\text{\scriptsize \textit{Baire}}} {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$.* *Proof.* Since ${\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}} \leq_{\text{\scriptsize \textit{Baire}}} {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$, we have that ${(\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}})^+} \leq_{\text{\scriptsize \textit{Baire}}} {(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+}$, but ${(\cong_{\mathop{\mathrm{\mathsf{LO}}}})^+} \leq_B {\cong_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$, so ${(\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}})^+} \leq_{\text{\scriptsize \textit{Baire}}} {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$. ◻ **Corollary 50**. *${E_1} \nleq_{\text{\scriptsize \textit{Baire}}} {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$.* *Proof.* If $E_1 \leq_{\text{\scriptsize \textit{Baire}}} {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$, by Corollary [Corollary 47](#cvxeq_baire_iso){reference-type="ref" reference="cvxeq_baire_iso"} we would have $E_1 \leq_{\text{\scriptsize \textit{Baire}}} {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$, contradicting Theorem [Theorem 6](#E1_orbit){reference-type="ref" reference="E1_orbit"}. ◻ Each one of Corollaries [Corollary 48](#turb){reference-type="ref" reference="turb"} and [Corollary 50](#E1_cvxeq){reference-type="ref" reference="E1_cvxeq"} implies that $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ is not complete for analytic equivalence relations, thus by Proposition [Proposition 9](#compl_qo){reference-type="ref" reference="compl_qo"} we obtain: **Corollary 51**. *$\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ is not complete for analytic quasi-orders.* Recall that by $\mathop{\mathrm{Int}}(\mathbb{R})$ we denote the set of the open intervals of $\mathbb{R}$. We can naturally equip $\mathop{\mathrm{Int}}(\mathbb{R})$ with a Polish topology: indeed, if we extend the usual order on $\mathbb{R}$ to $\mathbb{R}\cup \{\pm \infty\}$ in the obvious way, then $\mathop{\mathrm{Int}}(\mathbb{R})$ is the open subset $\{(x,y) \mid x < y\}$ of the Polish space $(\mathbb{R}\cup \{\pm \infty\})^2$. The inclusion relation on $\mathop{\mathrm{Int}}(\mathbb{R})$ is then closed. Notice now that the embedding from $(\mathop{\mathrm{Int}}(\mathbb{R}), \subseteq)$ to $(\mathop{\mathrm{\mathsf{LO}}},\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}})$ defined in the proof of Lemma [Lemma 27](#open_int){reference-type="ref" reference="open_int"} is actually a Borel reduction. Thus we have the following corollary. **Corollary 52**. *$(\mathop{\mathrm{Int}}(\mathbb{R}),\subseteq) \leq_B (\mathop{\mathrm{\mathsf{LO}}},\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}})$.* ## Convex embeddability between countable circular orders {#sec:cvx_co} Our goal in this section is to define a relation of convex embeddability among circular orders. We first recall the definition of convex subset of a circular order as given by Kulpeshov and Macpherson ([@KM05]). **Definition 53**. Let $C$ be a circular order. The set $A \subseteq C$ is said to be **convex** in $C$, in symbols $A \csube C$, if for any distinct $x,y \in A$ one of the following holds: [\[en1\]]{#en1 label="en1"} for every $c \in C$ with $C(x,c,y)$ we have $c \in A$; [\[en2\]]{#en2 label="en2"} for every $c \in C$ with $C(y,c,x)$ we have $c \in A$. If $A$ is a proper subset of $C$ we write $A \csub C$. Note that if $A \csub C$ then exactly one of [\[en1\]](#en1){reference-type="ref" reference="en1"} and [\[en2\]](#en2){reference-type="ref" reference="en2"} holds for each pair of distinct $x,y \in A$. The following propositions collect some basic properties of convex subsets of circular orders. **Proposition 54**. *If $C$ is a circular order and $A \csube C$ then $C \setminus A$ is a convex subset of $A$ as well.* *Proof.* If $C \setminus A$ is empty or a singleton the result is trivial, so we can assume that $C \setminus A$ contains at least two points. Toward a contradiction, suppose $x, y \in C \setminus A$ are distinct and such that: there exists $c_0 \in A$ with $C(x,c_0,y)$, and there exists $c_1 \in A$ with $C(y,c_1,x)$. By cyclicity and transitivity we obtain $C(c_0,y,c_1)$ and $C(c_1,x,c_0)$, and since $A$ is convex we would have that at least one of $x$ and $y$ belongs to $A$, a contradiction. ◻ The previous proposition highlights a major difference between convex subsets of circular and linear orders: the complement of a convex subset of a linear order is not in general convex. On the other hand, convex subsets of linear orders are closed under intersections, while this is not the case for circular orders: consider the circular order $C[\boldsymbol{4}]$ and its convex subsets $\{0,1,2\}$ and $\{2,3,0\}$. However the intersection of two convex subsets of a circular order is not convex only in some circumstances. **Proposition 55**. *Let $C$ be a circular order. If $A,B\csube C$ then $A \cap B$ is the union of two convex subsets of $C$. Moreover, if $A \cap B$ is not convex then $A \cup B = C$.* *Proof.* If $B \subseteq A$ or $A \subseteq B$, the result is trivial. So, suppose there exist $w \in A \setminus B$ and $z \in B\setminus A$, and consider the partition of $A \cap B$ given by the sets $$A_1 = \{x \in A \cap B \mid C(w,x,z)\} \quad \text{and} \quad A_2 = \{x \in A \cap B \mid C(z,x,w)\}.$$ We claim that $A_1 \csube C$. Let $x,y\in A_1$ be distinct: since $x,y \in A$, without loss of generality we can assume that $u \in A$ for every $u \in C$ such that $C(x,u,y)$. Since $z \notin A$ we have that $C(x,z,y)$ fails and, by totality and cyclicity, we have $C(x,y,z)$. Using cyclicity, transitivity and $C(w,x,z)$ we obtain $C(y,w,x)$. Since $w \notin B$ and $B$ is convex this implies that $u \in B$ for every $u \in C$ such that $C(x,u,y)$. If now $u$ is such that $C(x,u,y)$ we already showed that $u \in A \cap B$. From $C(y,w,x)$ and $C(x,u,y)$ it follows that $C(y,w,u)$ which, combined with $C(w,y,z)$, yields $C(w,u,z)$ and hence $u \in A_1$. The proof that $A_2$ is convex is symmetric. Now assume that $A \cap B$ is not convex, and hence both $A_1$ and $A_2$ are non empty. Fix $x \in A_1$ and $y \in A_2$. From $C(w,x,z)$ and $C(z,y,w)$ it follows that we have $C(x,z,y)$ and $C(y,w,x)$. Since $x,y \in A$ but $z \notin A$ we must have that $C(y,u,x)$ implies $u \in A$. Similarly we obtain that $C(x,u,y)$ implies $u \in B$. By totality it follows that $A \cup B = C$. ◻ **Proposition 56**. *Let $C$ be a circular order. Let $\{A_i \mid i \in I\}$ and $\{B_j \mid j \in J\}$ be two collections of pairwise disjoint convex subsets of $C$. Then there exists at most one pair $(i,j) \in I \times J$ such that $A_i \cap B_j$ is not convex.* *Proof.* Suppose that $A_i \cap B_j$ is not convex. By the second part of Proposition [Proposition 55](#prop:intconvex){reference-type="ref" reference="prop:intconvex"} we have $A_i \cup B_j = C$. Hence for every $i' \neq i$ and $j' \neq j$ we have that $A_{i'} \subseteq B_j$ and $B_{j'} \subseteq A_i$. Therefore $A_{i'} \cap B_j =A_{i'}$, $A_i \cap B_{j'} =B_{j'}$ and $A_{i'} \cap B_{j'} \subseteq A_{i'} \cap A_i = \emptyset$ are all convex. ◻ If $f$ is an embedding between linear orders $L$ and $L'$ and $f(L) \csube L'$, then $f(B) \csube L'$ for every $B \csube L$. This ceases to be true for circular orders, as shown by the following example. The identity map between $C = C[\zeta]$ and $C' = C[\zeta + \boldsymbol{1}]$ has convex range, but the image of the convex set $B = C \setminus \{ 0 \} \csube C$ is no longer convex in $C'$. The following proposition gives a weakening of the above property which is however sufficient for the ensuing proofs. **Proposition 57**. *Let $f$ be an embedding between the circular orders $C$ into $C'$. If $A' \csube C'$, then $f^{-1}(A') \csube C$. Conversely, if $A \csub C$ is such that $f(A) \csube C'$, then $f(B) \csube C'$ for all $B \csube C$ with $B \subseteq A$.* *Proof.* The first part is obvious, so let us consider $A \csub C$ with $f(A) \csube C'$, and fix any $B \csube C$ contained in $A$. Pick distinct points $f(x),f(y) \in f(B) \subseteq f(A)$, so that $x,y \in B$ and $x \neq y$ because $f$ is injective. Since $A \csub C$, without loss of generality we might assume that $c \in A$ for all $c \in C$ with $C(x,c,y)$ and that there is $d \in C$ with $C(y,d,x)$ and $d \notin A$, so that the same is true with $A$ replaced by $B$ because $B \subseteq A$ is convex. Since $f$ is an embedding, $f(d)$ is such that $C'(f(y),f(d),f(x))$ but $f(d) \notin f(A)$. Since $f(A) \csube C'$ by hypothesis, this means that $c' \in f(A)$ for all $c' \in C$ such that $C'(f(x),c',f(y))$. So for such a $c' \in C'$ there is $c \in A$ such that $c' = f(c)$: then $C(x,c,y)$ because $f$ is an embedding, and so $c \in B$ and $f(c)= c' \in f(B)$. This shows that $f(B)$ satisfies [\[en1\]](#en1){reference-type="ref" reference="en1"} of Definition [Definition 53](#def:en){reference-type="ref" reference="def:en"} with respect to $x$ and $y$. Hence $f(B) \csube C'$. ◻ The first natural attempt to define convex embeddability between circular orders is the following. **Definition 58**. Let $C$ and $C'$ be circular orders. We say that $C$ is **convex embeddable** into $C'$, and write $C \trianglelefteq_{c}C'$, if there exists an embedding $f$ from $C$ to $C'$ such that $f(C) \csube C'$. However, $\trianglelefteq_{c}$ is **not** transitive, as witnessed by $C[\zeta] \trianglelefteq_{c}C[\zeta+\boldsymbol{1}]$, $C[\zeta+\boldsymbol{1}] \trianglelefteq_{c}C[\omega+\boldsymbol{1}+\omega^*+\eta]$ (because $C[\zeta+\boldsymbol{1}] \cong_{c} C[\omega+\boldsymbol{1}+\omega^*]$), and $C[\zeta] \ntrianglelefteq_{c}C[\omega+\boldsymbol{1}+\omega^*+\eta]$. Nevertheless, notice that if we partition $C[\zeta]$ into the *two* convex subsets $\omega^*$ and $\omega$ then they are isomorphic to the two convex subsets $\omega^*$ and $\omega$ of $C[\omega+\boldsymbol{1}+\omega^*+\eta]$. By taking the transitive closure of $\trianglelefteq_{c}$ (i.e. the smallest binary relation containing $\trianglelefteq_{c}$) we are naturally led to the following definition. We call **finite convex partition** of the circular order $C$ any finite partition $\{ C_i \mid i < n \}$ of $C$ such that - $C_i \csube C$ for all $i < n$, and - for all $x,y,z \in C$, if $C(x,y,z)$ then $C[\boldsymbol{n}](i,j,k)$ for the unique $i,j,k < n$ such that $x \in C_i$, $y \in C_j$, and $z \in C_k$. Notice that this implies that the $C_i$'s are ordered as $C[\boldsymbol{n}]$, that is: if $i,j,k < n$ are distinct and $C[\boldsymbol{n}](i,j,k)$ then $C(x,y,z)$ for every $x \in C_i$, $y \in C_j$, and $z \in C_k$. Also, the convexity of the $C_i$'s follows from the second condition if $n \geq 3$. **Definition 59**. Let $C$ and $C'$ be circular orders. We say that $C$ is **piecewise convex embeddable** into $C'$, and write $C \mathrel{\trianglelefteq_{c}^{<\omega}}C'$, if there are a finite convex partition $\{ C_i \mid i < n \}$ of $C$ and an embedding $f$ of $C$ into $C'$ such that $f(C_i) \csube C'$ for all $i < n$. We denote by $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ the restriction of $\mathrel{\trianglelefteq_{c}^{<\omega}}$ to the set $\mathop{\mathrm{\mathsf{CO}}}$ of (codes for) circular orders on $\mathbb{N}$. Clearly, $C \trianglelefteq_{c}C'$ implies $C \mathrel{\trianglelefteq_{c}^{<\omega}}C'$. Notice also that when $C$ has at least two elements and $C \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C'$ as witnessed by $\{ C_i \mid i < n \}$ and $f$, without loss of generality we can assume that $n > 1$ and hence $C_i \csub C$. (If not, split $f(C_0) \csube C'$ into two nonempty convex subsets $A, B$ of $C'$, and consider the finite convex partition $\{ f^{-1}(A), f^{-1}(B) \}$ of $C$ together with the same embedding $f$.) **Proposition 60**. *$\mathrel{\trianglelefteq_{c}^{<\omega}}$ is transitive.* *Proof.* Suppose that $C \mathrel{\trianglelefteq_{c}^{<\omega}}C'$, as witnessed by the embedding $f$ and the finite convex partition $\{ C_i \mid i<n \}$ of $C$, and that $C' \mathrel{\trianglelefteq_{c}^{<\omega}}C''$ via the embedding $g$ and the finite convex partition $\{ C'_j \mid j < m \}$ of $C'$. If $C'$ has only one element than so does $C$ and $C \mathrel{\trianglelefteq_{c}^{<\omega}}C''$ is immediate. Thus, without loss of generality, we can assume that $m > 1$, so that $C'_j \csub C'$ for all $j < m$. Notice that $\{f(C_i) \mid i < n\}$ and $\{C'_j \mid j<m\}$ are two collections of pairwise disjoint convex subsets of $C'$. We distinguish two cases. If $C'_{i,j} = f(C_i) \cap C'_j$ is a convex subset of $C'$ for every $i < n$ and $j < m$, then we can order the family of pairwise disjoint convex sets $$\left\{ C'_{i,j} \mid (i,j) \in n \times m \wedge C'_{i,j} \neq \emptyset \right\}$$ following the circular order of $C'$. In this way we obtain a family $\{ D'_k \mid k < \ell \}$, for the suitable $\ell \leq n \cdot m$, such that if $x_0 \in D'_{k_0}$, $x_1 \in D'_{k_1}$, and $x_2 \in D'_{k_2}$ satisfy $C'(x_0,x_1,x_2)$ then $C[\boldsymbol{\ell}](k_0,k_1,k_2)$. Then $\{ f^{-1}(D'_k) \mid k < \ell \}$ is a finite convex partition of $C$ and $g \circ f$ is an embedding of $C$ into $C''$. Moreover, for every $k < \ell$ we have $(g \circ f)(f^{-1}(D'_k)) = g(D'_k) \csube C''$ because $D'_k \subseteq C'_j \csub C'$ for some $j< m$ (Proposition [Proposition 57](#prop:embeddingandconvex){reference-type="ref" reference="prop:embeddingandconvex"}). Thus $C \mathrel{\trianglelefteq_{c}^{<\omega}}C''$. Suppose now that $C'_{i,j} = f(C_i) \cap C'_j$ is not convex for some $(i,j)$. By Proposition [Proposition 56](#prop:partconvex){reference-type="ref" reference="prop:partconvex"} there is at most one such pair $(\bar{\imath},\bar{\jmath})$. By Proposition [Proposition 55](#prop:intconvex){reference-type="ref" reference="prop:intconvex"}, $C'_{\bar{\imath}, \bar{\jmath}}$ is the union of two disjoint convex subsets $A_0$ and $A_1$ of $C'$. Then we can argue as in the previous paragraph but starting with the family $$\left\{ C'_{i,j} \mid (i,j) \in n \times m \wedge (i,j) \neq (\bar{\imath},\bar{\jmath}) \wedge C'_{i,j} \neq \emptyset \right\} \cup \{ A_0,A_1 \}. \qedhere$$ ◻ Thus $\mathrel{\trianglelefteq_{c}^{<\omega}}$ is a quasi-order, and it is easy to see that its restriction $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ to the Polish space $\mathop{\mathrm{\mathsf{CO}}}$ is analytic. We first show that $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ satisfies combinatorial properties similar to those proved for $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ in Section [3.1](#sec:comb_prop){reference-type="ref" reference="sec:comb_prop"}. A key point is that it still makes sense to talk about (finite) condensation in the realm of circular orders. Indeed, given a circular order $C$ the **condensation class** $c_F^C(\ell)$ of $\ell$ is the collection of those $m$ such that either $\{ k \mid C(\ell,k,m) \}$ or $\{ k \mid C(m,k,\ell) \}$ is finite. Each $c_F^C(\ell)$ is convex in $C$, and it again holds that the condensation classes form a partition of $C$. This allows us to define the (**finite**) **condensation** $C_F$ of $C$ in the obvious way. The crucial observation is that we can substitute Proposition [Proposition 26](#cond_classes){reference-type="ref" reference="cond_classes"} with the following lemma. **Lemma 61**. *Let $f$ be an embedding between the circular orders $C$ and $C'$. Fix any $\ell \in C$, and let $A \csub C$ and $a,b \in A \setminus c_F^C(\ell)$ be such that $f(A) \csube C'$, $c_F^C(\ell) \subseteq A$ and $C(a,\ell',b)$ for all $\ell' \in c_F^C(\ell)$. Then the restriction of $f$ to $c_F^C(\ell)$ is an isomorphism between $c_F^C(\ell)$ and $c_F^{C'}(f(\ell))$, and thus $|c_F^C(\ell)|=|c_F^{C'}(f(\ell))|$.* *Proof.* By Proposition [Proposition 57](#prop:embeddingandconvex){reference-type="ref" reference="prop:embeddingandconvex"}, we have $f(c_F^C(\ell)) \csube C'$, which easily implies $f(c^C_F(\ell)) \subseteq c_F^{C'}(f(\ell))$. Conversely, pick any $d' \in c_F^{C'}(f(\ell))$ distinct from $f(\ell)$, and first assume that $\{ k \mid C'(d',k,f(\ell)) \}$ is finite. Consider the set $B = \{ k \in C \mid C(a,k,\ell)\} \subseteq A$. Since $a \notin c_F^C(\ell)$ and $B \csube C$, the set $f(B)$ is infinite and by Proposition [Proposition 57](#prop:embeddingandconvex){reference-type="ref" reference="prop:embeddingandconvex"} $f(B) \csube C'$. We cannot have $C' (d',f(a),f(\ell))$, otherwise $\{ k \mid C'(d',k,f(\ell)) \} \supseteq f(B)$ and the former would be infinite. Thus $C'(f(a),d',f(\ell))$, and so $\{ k \mid C'(d',k,f(\ell)) \} \subseteq f(B)$ because $f(B) \csube C'$. This easily implies that $d' = f(d)$ for some $d \in c_F^C(\ell)$, and we are done. (When $\{ k \mid C'(f(\ell),k,d') \}$ is finite, we work symmetrically on the other side of $\ell$ and use $b$ instead of $a$.) ◻ **Proposition 62**. *[\[prop:antichainsCO-a\]]{#prop:antichainsCO-a label="prop:antichainsCO-a"} There is an embedding from the partial order $(\mathop{\mathrm{Int}}(\mathbb{R}),\subseteq)$ into $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$, and indeed ${(\mathop{\mathrm{Int}}(\mathbb{R}),\subseteq)} \leq_B {\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}}$.* *[\[prop:antichainsCO-b\]]{#prop:antichainsCO-b label="prop:antichainsCO-b"} $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ has chains of order type $(\mathbb{R}, {<} )$, as well as antichains of size $2^{\aleph_0}$.* *Proof.* Given an interval $(x,y) \in \mathop{\mathrm{Int}}(\mathbb{R})$, consider the circular order $C_{(x,y)} \in \mathop{\mathrm{\mathsf{CO}}}$ defined by $C_{(x,y)} = C[L_{(x,y)}]$, where $L_{(x,y)}$ is as in the proof of Lemma [Lemma 27](#open_int){reference-type="ref" reference="open_int"}. We claim that the map $(x,y) \mapsto C_{(x,y)}$ witnesses [\[prop:antichainsCO-a\]](#prop:antichainsCO-a){reference-type="ref" reference="prop:antichainsCO-a"}. Pick two intervals $(x,y) , (x',y') \in \mathop{\mathrm{Int}}(\mathbb{R})$. If $(x,y) \subseteq (x',y')$ then the identity map witnesses $C_{(x,y)} \trianglelefteq_{c}C_{(x',y')}$, hence $C_{(x,y)} \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C_{(x',y')}$. Suppose now that $(x,y) \not\subseteq (x',y')$, and for the sake of definiteness assume that $x < x'$. Towards a contradiction suppose that there are a finite convex partition $\{ C_i \mid i < n \}$ of $C_{(x,y)}$ and an embedding $f$ witnessing $C_{(x,y)} \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C_{(x',y')}$. As usual, we can assume $n > 1$, so that $C_i \csub C_{(x,y)}$ for all $i < n$. Since there are infinitely many rationals between $x$ and $x'$ and all condensation classes of $C_{(x,y)}$ are finite, we can find $i < n$ and $q,q',r \in \mathbb{Q}$ such that $x < q < r < q' < x'$ and the hypothesis of Lemma [Lemma 61](#lem:condensationCO){reference-type="ref" reference="lem:condensationCO"} are satisfied with $A = C_i$, $a = (0,q)$, $b = (0,q')$ and $\ell = (0,r)$. Thus the condensation class of $f(0,r)$ has the same size of the condensation class of $(0,r)$, which by construction can happen only if $r \in (x',y')$, a contradiction. Part [\[prop:antichainsCO-b\]](#prop:antichainsCO-b){reference-type="ref" reference="prop:antichainsCO-b"} is derived from [\[prop:antichainsCO-a\]](#prop:antichainsCO-a){reference-type="ref" reference="prop:antichainsCO-a"} as in Proposition [Proposition 28](#prop1){reference-type="ref" reference="prop1"}. ◻ **Proposition 63**. *$\mathfrak{b}(\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}) = \aleph_1$, and indeed every $C \in \mathop{\mathrm{\mathsf{CO}}}$ is the bottom of a strictly increasing $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-unbounded chain of length $\omega_1$.* *Proof.* For $C \in \mathop{\mathrm{\mathsf{CO}}}$ and $\ell \in C$, let $\alpha_{\ell,C}$ be the sup of those $\omega \leq \alpha < \omega_1$ such that $C[\boldsymbol{\alpha}] \trianglelefteq_{c}C$ via some $f$ satisfying $f(0) = \ell$. Since $\alpha_{\ell,C}$ is attained by definition of convexity, the ordinal $\alpha_C = \sup_{\ell \in C} \alpha_{\ell,C}$ is countable, and by construction $C[\boldsymbol{\alpha}_C+\boldsymbol{1}] \not\trianglelefteq_c C$. Let $\alpha$ be an additively indecomposable [^2] countable ordinal above $\alpha_C+1$: we claim that $C[\boldsymbol{\alpha}] \mathrel{{\ntrianglelefteq}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}C$. Suppose towards a contradiction that $\{ C_i \mid i < n \}$ is a finite convex partition and $f$ an embedding witnessing $C[\boldsymbol{\alpha}] \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C$. As usual we can assume $n> 1$. Then there are $i < n$ and $\gamma < \alpha$ such that $A'_\gamma = \{ \beta \in C[\boldsymbol{\alpha}] \mid \beta \geq \gamma \}$ is contained in $C_i$. Since $\alpha$ is additively indecomposable, the linear order determined by $A_\gamma$ has order type $\alpha \geq \alpha_C+1$, thus we can consider the set $A_\gamma = \{ \beta \in A'_\gamma \mid \beta < \gamma+\alpha_C+1 \}$, which has order type $\alpha_C+1$. Since $A_\gamma \csube C[\boldsymbol{\alpha}]$ and $A_\gamma \subseteq C_i \csub C[\boldsymbol{\alpha}]$, by Proposition [Proposition 57](#prop:embeddingandconvex){reference-type="ref" reference="prop:embeddingandconvex"} the restriction of $f$ to $A_\gamma$ witnesses $C[\boldsymbol{\alpha}_C+\boldsymbol{1}] \trianglelefteq_{c}C$, a contradiction. This shows that the family $\{ C[\boldsymbol{\alpha}] \mid \omega \leq \alpha < \omega_1 \}$ is $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-unbounded in $\mathop{\mathrm{\mathsf{CO}}}$. Since $C[\boldsymbol{\alpha}] \trianglelefteq_{c}C[\boldsymbol{\beta}]$ when $\alpha \leq \beta$, we can extract from it a strictly increasing chain witnessing $\mathfrak{b}(\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}) \leq \aleph_1$. To show that $\mathfrak{b}(\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}) > \aleph_0$, consider a countable family $\{ C_i \mid i \in \mathbb{N}\} \subseteq \mathop{\mathrm{\mathsf{CO}}}$. For each $i \in \mathbb{N}$ pick an arbitrary $\ell_i \in C_i$ and define $L_i \in \mathop{\mathrm{\mathsf{LO}}}$ by setting $x \leq_{L_i} y$ iff $C_i(\ell_i,x,y)$. Then the circular order $C = C \left[ \sum_{i \in \boldsymbol{\mathbb{N}}} L_i \right] \in \mathop{\mathrm{\mathsf{CO}}}$ is such that $C_i \trianglelefteq_{c}C$ for all $i \in \mathbb{N}$, and thus the given family is $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-bounded. For the second part, pick $\ell \in C$ and let $L \in \mathop{\mathrm{\mathsf{LO}}}$ be defined by $x \leq_L y$ iff $C(\ell,x,y)$. Consider the $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-nondecreasing sequence $(C_\alpha)_{\alpha < \omega_1}$ of circular orders defined by $C_0 = C$ and $C_\alpha = C[L+\boldsymbol{\omega}+\boldsymbol{\alpha}]$ when $\alpha > 0$. Since $C[\boldsymbol{\omega}+\boldsymbol{\alpha}] \trianglelefteq_{c}C_\alpha$ for all $\alpha \neq 0$, such a sequence is $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-unbounded. Thus we can extract from it a strictly $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-increasing subsequence of length $\omega_1$ with $C_0$ as first element: being cofinal in the original sequence, it will be $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-unbounded too, as required. ◻ **Proposition 64**. *[\[prop:basisCO-1\]]{#prop:basisCO-1 label="prop:basisCO-1"} There are $2^{\aleph_0}$-many $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-incomparable $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-minimal elements in $\mathop{\mathrm{\mathsf{CO}}}$. In particular, all bases for $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ are of maximal size.* *[\[prop:basisCO-2\]]{#prop:basisCO-2 label="prop:basisCO-2"} There is a $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-decreasing $\omega$-sequence in $\mathop{\mathrm{\mathsf{CO}}}$ which is not $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-bounded from below.* *Proof.* [\[prop:basisCO-1\]](#prop:basisCO-1){reference-type="ref" reference="prop:basisCO-1"} Given an infinite $S \subseteq \mathbb{N}$, let $C_S = C[\eta_{f_S}]$ where $\eta_{f_S}$ is as in the proof of Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}[\[prop:basisforcvx-1\]](#prop:basisforcvx-1){reference-type="ref" reference="prop:basisforcvx-1"}. If $A \csube C_S$ is infinite, then there exist $q,q' \in \mathbb{Q}$ with $q < q'$ such that $\{ (\ell,q'') \in C_S \mid q \leq q'' \leq q' \} \subseteq A$, and thus $C_S$ is convex embeddable into (the circular order determined by) $A$. Let $C \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C_S$, as witnessed by the finite convex partition $\{ C_i \mid i < n \}$ and the embedding $f$. Then there is $i < n$ such that $C_i$, and hence also $f(C_i)$ is infinite. Setting $A = f(C_i)$ in the previous paragraph, we get that $C_S \trianglelefteq_{c}f(C_i) \cong C_i \csube C$, hence $C_S \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C$. This shows that $C_S$ is $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-minimal. Assume now that $C_S \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C_{S'}$ for some infinite $S,S' \subseteq \mathbb{N}$, and let $\{ C_i \mid i < n \}$ be a finite convex partition of $C_S$ and $f \colon C_S \to C_{S'}$ be an embedding witnessing this. As usual, we may assume $n > 1$, so that $C_i \csub C_S$ and $f(C_i) \csub C_{S'}$. Fix any $i < n$ such that $C_i$ is infinite. By the first paragraph, there are $q < q'$ such that $\{ (\ell,q'') \in C_S \mid q \leq q'' \leq q' \} \subseteq C_i$. Given an arbitrary $m \in S$, pick $q'' \in \mathbb{Q}$ such that $q < q'' < q'$ and $f_S(q'') = m$. Then the hypotheses of Lemma [Lemma 61](#lem:condensationCO){reference-type="ref" reference="lem:condensationCO"} are satisfied when we set $A = C_i$, $a = (0,q)$, $b = (0,q')$, and $\ell = (0,q'')$. Thus $C_{S'}$ must contain a condensation class of size $m$, which is possible only if $m \in S'$. This shows that $S \subseteq S'$. Conversely, given $m \in S'$ we work with the infinite set $f(C_i) \csub C_{S'}$ and pick $q,q' \in \mathbb{Q}$ such that $\{ (\ell,q'') \in C_{S'} \mid q \leq q'' \leq q' \} \subseteq f(C_i)$. Then we pick $q'' \in \mathbb{Q}$ such that $q < q'' < q'$ and $f_{S'}(q'') = m$. Applying (the proof of) Lemma [Lemma 61](#lem:condensationCO){reference-type="ref" reference="lem:condensationCO"} we get that the condensation class of $f^{-1}(0,q'')$ has size $m$, hence $m \in S$. Since $m \in S'$ was arbitrary, $S' \subseteq S$, and thus $S = S'$. This shows that $\{ C_S \mid S \subseteq \mathbb{N}\wedge S \text{ is infinite} \}$ is a $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-antichain and we are done. [\[prop:basisCO-2\]](#prop:basisCO-2){reference-type="ref" reference="prop:basisCO-2"} Consider the family $\{ C_{(m,+\infty)} \mid m \in \mathbb{N}\}$, where $C_{(m,+\infty)}$ is as in the proof of Proposition [Proposition 62](#prop:antichainsCO){reference-type="ref" reference="prop:antichainsCO"}[\[prop:antichainsCO-a\]](#prop:antichainsCO-a){reference-type="ref" reference="prop:antichainsCO-a"}. It is a strictly $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-decreasing chain, so we only need to show that it is $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-unbounded from below. Let $C \in \mathop{\mathrm{\mathsf{CO}}}$ be such that $C \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C_{(0,+\infty)}$, as witnessed by the finite convex partition $\{ C_i \mid i < n \}$ (for some $n > 1$) and the embedding $f$. Then there is $i < n$ such that $C_i$ is infinite, which means that $\{ (\ell,q'') \in C_{(0,+\infty)} \mid q < q'' < q' \} \subseteq f(C_i) \csub C_{(0,+\infty)}$ for some rational numbers $0 \leq q < q'$. Thus $C$ contains a convex subset isomorphic to $C_{(q,q')}$ by Lemma [Proposition 57](#prop:embeddingandconvex){reference-type="ref" reference="prop:embeddingandconvex"}. Pick $m \in \mathbb{N}$ with $m > q'$. Then $C \mathrel{{\ntrianglelefteq}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}C_{(m,+\infty)}$ because otherwise $C_{(q,q')} \trianglelefteq_{c}C \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C_{(m,+\infty)}$, contradicting (the proof of) Proposition [Proposition 62](#prop:antichainsCO){reference-type="ref" reference="prop:antichainsCO"}[\[prop:antichainsCO-a\]](#prop:antichainsCO-a){reference-type="ref" reference="prop:antichainsCO-a"}. ◻ **Proposition 65**. *Every $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-antichain is contained in a $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-antichain of size $2^{\aleph_0}$. In particular, there are no maximal $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-antichains of size smaller than $2^{\aleph_0}$, and every $C \in \mathop{\mathrm{\mathsf{CO}}}$ belongs to a $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-antichain of size $2^{\aleph_0}$.* *Proof.* Following the proof of Proposition [Proposition 33](#prop:everylobelongstoantichain){reference-type="ref" reference="prop:everylobelongstoantichain"}, we only need to verify that for every $C \in \mathop{\mathrm{\mathsf{CO}}}$ the set $\{ S \subseteq \mathbb{N}\mid S \text{ is infinite} \wedge C_S \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C \}$ is countable, where the $C_S$'s are defined in the proof Proposition [Proposition 64](#prop:basisCO){reference-type="ref" reference="prop:basisCO"}[\[prop:basisCO-1\]](#prop:basisCO-1){reference-type="ref" reference="prop:basisCO-1"}. First observe that arguing as at the beginning of that proof and using Proposition [Proposition 57](#prop:embeddingandconvex){reference-type="ref" reference="prop:embeddingandconvex"} one can prove that if $C_S \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C$ then $C_S \trianglelefteq_{c}C$. Indeed, let $C_S \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C$ be witnessed by the finite convex partition $\{ C_i \mid i < n \}$ (for some $n > 1$) of $C_S$ and the embedding $f \colon C_S \to C$. Then some $C_i$ must be infinite, so there is an embedding $g \colon C_S \to C_S$ such that $\mathop{\mathrm{Im}}g \csube C_S$ and $\mathop{\mathrm{Im}}g \subseteq C_i \csub C_S$. Hence $f(\mathop{\mathrm{Im}}g) \csube C$, and so $f \circ g$ witnesses $C_S \trianglelefteq_{c}C$. Suppose that $S,S' \subseteq \mathbb{N}$ are distinct infinite sets such that $C_S \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C$ and $C_{S'} \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C$ via corresponding embeddings $f$ and $g$, respectively. Without loss of generality, we may assume that $\mathop{\mathrm{Im}}f \neq C$ and $\mathop{\mathrm{Im}}g \neq C$, as otherwise $C_{S'} \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C_S$ or $C_S \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C_{S'}$, contradicting (the proof of) Proposition [Proposition 64](#prop:basisCO){reference-type="ref" reference="prop:basisCO"}[\[prop:basisCO-1\]](#prop:basisCO-1){reference-type="ref" reference="prop:basisCO-1"}. If $\mathop{\mathrm{Im}}f \cap \mathop{\mathrm{Im}}g \neq \emptyset$, then by Proposition [Proposition 55](#prop:intconvex){reference-type="ref" reference="prop:intconvex"} such intersection is the union of (at most) two proper convex subsets $A_0, A_1$ of $C$, each of which must be infinite by definition of $C_S$ and $C_{S'}$. Thus $f^{-1}(A_0)$ is an infinite convex proper subset of $C_S$, and so $C_S \trianglelefteq_{c}f^{-1}(A_0)$, which in turn implies $C_S \trianglelefteq_{c}A_0$ and $C_S \trianglelefteq_{c}C_{S'}$, a contradiction. Thus $\mathop{\mathrm{Im}}f \cap \mathop{\mathrm{Im}}g = \emptyset$. Since $C$ is countable, there can be at most countably many infinite $S \subseteq \mathbb{N}$ such that $C_S \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C$ and the claim follows. ◻ Once we know that for every $C \in \mathop{\mathrm{\mathsf{CO}}}$ there are at most countably many infinite sets $S \in \mathbb{N}$ such that $C_S \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C$, arguing as in Proposition [Proposition 34](#prop:dom_fam){reference-type="ref" reference="prop:dom_fam"} we easily get **Proposition 66**. *$\mathfrak{d}(\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}) = 2^{\aleph_0}$.* We now move to the study of the (analytic) equivalence relation $\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}$ induced by $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$. Obviously if $C \cong_{\mathop{\mathrm{\mathsf{CO}}}} C'$ then we also have $C \mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} C'$. **Theorem 67**. *${\cong_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}$.* *Proof.* Consider the Borel map $\varphi\colon \mathop{\mathrm{\mathsf{LO}}}\to \mathop{\mathrm{\mathsf{CO}}}$ defined by $$\varphi(L)= C[(\boldsymbol{1}+\zeta L)\omega].$$ We claim that $\varphi$ is a reduction. Clearly, if $L \cong_{\mathop{\mathrm{\mathsf{LO}}}}L'$ then $\varphi(L) \cong_{\mathop{\mathrm{\mathsf{CO}}}} \varphi(L')$ and hence $\varphi(L) \mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} \varphi(L')$. For the converse, let the finite convex partition $\{ C_i \mid i<n \}$ of $\varphi(L)$ and the embedding $f$ of $\varphi(L)$ into $\varphi(L')$ witness $\varphi(L) \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}\varphi(L')$. Without loss of generality $n > 1$, so that $C_i \csub \varphi(L)$ for all $i < n$. Since $n$ is finite, there exists some $j < n$ such that $C_j$ contains at least two copies of $\boldsymbol{1}+\zeta L$, so we can consider a convex set of the form $\boldsymbol{1}+\zeta L+\boldsymbol{1} \subseteq C_j$, so that $f(\boldsymbol{1}+\zeta L+\boldsymbol{1}) \csube \varphi(L')$ by Proposition [Proposition 57](#prop:embeddingandconvex){reference-type="ref" reference="prop:embeddingandconvex"}. Since the $\boldsymbol{1}$'s are the only elements which do not have immediate predecessor and successor both in $\varphi(L)$ and in $\varphi(L')$, and since $f(\boldsymbol{1}+\zeta L+\boldsymbol{1})$ is convex, we have that the images via $f$ of the two $\boldsymbol{1}$'s in $\boldsymbol{1}+\zeta L+\boldsymbol{1} \subseteq C_j$ are two necessarily "consecutive" $\boldsymbol{1}$'s in $\varphi(L')$. It follows that $\boldsymbol{1}+\zeta L+\boldsymbol{1} \subseteq C_j$ is isomorphic to a copy of $\boldsymbol{1}+\zeta L'+\boldsymbol{1}$ in $\varphi(L')$. We thus obtain $\zeta L\cong_{\mathop{\mathrm{\mathsf{LO}}}}\zeta L'$, hence $L\cong_{\mathop{\mathrm{\mathsf{LO}}}}L'$ by Lemma [Lemma 14](#lem:isom_zetaL){reference-type="ref" reference="lem:isom_zetaL"}. ◻ The next results contrasts with Corollary [Corollary 50](#E1_cvxeq){reference-type="ref" reference="E1_cvxeq"}. To simplify the notation, we let $\vec{x}$ and $\vec{y}$ denote the sequences $(x_n)_{n \in \mathbb{N}} , (y_n)_{n \in \mathbb{N}} \in \mathbb{R}^\mathbb{N}$, respectively. **Theorem 68**. *${E_1} \leq_B {\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}$.* *Proof.* By Proposition [Proposition 5](#E1){reference-type="ref" reference="E1"} it suffices to define a Borel reduction from $E_1^t$ to ${\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}$. To this end, fix an injective map $f \colon \mathbb{Q}\to \{\boldsymbol{n} \mid n \in \mathbb{N}\setminus \{ 0,1 \} \}$ and, as in the proofs of Lemma [Lemma 27](#open_int){reference-type="ref" reference="open_int"} and Proposition [Proposition 28](#prop1){reference-type="ref" reference="prop1"}, consider the linear orders $\eta_f$ and $L_{(x,x+1)}$, with $x \in \mathbb{R}$. By Lemma [Lemma 27](#open_int){reference-type="ref" reference="open_int"}, $L_{(x,x+1)}$ and $L_{(x',x'+1)}$ are isomorphic if and only if $x=x'$. Consider the Borel map that sends $\vec{x} = (x_n)_{n \in \mathbb{N}} \in \mathbb{R}^\mathbb{N}$ to the linear order $$L(\vec{x}) = \sum_{n \in \mathbb{Z}} \overline{L}_n,$$ where $\overline{L}_n = \eta_f + \eta$ if $n < 0$ and $\overline{L}_n = L_{(x_n,x_n+1)}+\eta$ if $n \geq 0$. We claim that the Borel map $\varphi \colon \mathbb{R}^\mathbb{N}\to \mathop{\mathrm{\mathsf{CO}}}$ defined by $\varphi(\vec{x}) = C[L(\vec{x})]$ is a reduction from $E_1^t$ to ${\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}$. First suppose that ${\vec{x}} \mathrel{E_1^t} {\vec{y}}$, i.e. that there are $\bar{n},\bar{m} \in \mathbb{N}$ such that $x_{\bar{n}+k}=y_{\bar{m}+k}$ for all $k \in \mathbb{N}$. Consider the finite convex partition $\{ C_i \mid i < 2 \bar{n}+2 \}$ of $\varphi(\vec{x})$ given by setting for $0 \leq j < \bar{n}$ $$\begin{aligned} C_0 & = \sum_{n \in \mathbb{Z}\setminus \mathbb{N}} \overline{L}_n = \{ (\ell,n) \in L(\vec{x}) \mid n < 0 \} \\ C_{2j+1} & = L_{(x_j,x_j+1)} \times \{ j \} \\ C_{2j+2} & = \eta \times \{ j \} \\ C_{2 \bar{n} +1} & = \sum_{n \geq \bar{n}} \overline{L}_{n} = \{ (\ell,n) \in L(\vec{x}) \mid n \geq \bar{n} \}.\end{aligned}$$ Consider the embedding $f$ of $\varphi(\vec{x})$ into $\varphi(\vec{y})$ defined by $$f(\ell,n) = \begin{cases} (\ell, n - \bar{n}) & \text{if } n < \bar{n} \\ (\ell, \bar{m} + (n - \bar{n})) & \text{if } n \geq \bar{n}. \end{cases}$$ By choice of $\bar{n}, \bar{m} \in \mathbb{N}$ and since $L_{(x,x+1)} \csube \eta_f$ for all $x \in \mathbb{R}$, it is easy to verify that $f$ is well-defined and that $f(C_i) \csube \varphi(\vec{y})$ for all $i < 2 \bar{n} + 2$. This witnesses $\varphi(\vec{x}) \mathrel{\trianglelefteq^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} \varphi(\vec{y})$, and since $\varphi(\vec{y}) \mathrel{\trianglelefteq^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} \varphi(\vec{x})$ can be proved symmetrically, we obtain ${\varphi(\vec{x})} \mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} {\varphi(\vec{y})}$. Suppose now that [^3] ${\varphi(\vec{x})} \mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} {\varphi(\vec{y})}$. Let $\{ C_i \mid i<b \}$ with $b \in \mathbb{N}\setminus \{ 0 \}$ be a finite convex partition of $\varphi(\vec{x})$ and $f$ be an embedding of $\varphi(\vec{x})$ into $\varphi(\vec{y})$ witnessing $\varphi(\vec{x}) \mathrel{\trianglelefteq^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} \varphi(\vec{y})$. (As usual, we can assume $b > 1$, so that Proposition [Proposition 57](#prop:embeddingandconvex){reference-type="ref" reference="prop:embeddingandconvex"} can be applied when necessary.) Since $b$ is finite, for some $i < b$ and $\bar{n} \in \mathbb{N}\setminus \{ 0 \}$ we must have $\sum_{n\geq \bar{n}-1} \overline{L}_{n} \subseteq C_i$. Notice that for every $n \geq \bar{n}-1$ and $q \in \eta$, the point $(q,n) \in \eta \times \{ n \} \csub \varphi(\vec{x})$ has no immediate predecessor and immediate successor, while points of the form $(\ell,m)$ for $\ell \in L_{(y_m,y_m+1)}$ and $m \in \mathbb{N}$ or $\ell \in \eta_f$ and $m \in \mathbb{Z}\setminus \mathbb{N}$ have an immediate predecessor or an immediate successor (or both): thus $f(q,n) \in \eta \times m$ for some $m \in \mathbb{Z}$. By a similar argument, $f(L_{(x_n,x_n+1)} \times \{ n \}) \subseteq L_{(y_m,y_{m+1})} \times \{ m \}$ or $f(L_{(x_n,x_n+1)} \times \{ n \}) \subseteq \eta_f \times \{ m \}$ for a suitable $m \in \mathbb{Z}$. This two facts together with the convexity of $f(C_i)$ and the fact that, by the proof of Lemma [Lemma 27](#open_int){reference-type="ref" reference="open_int"}, the only convex subset of $\eta_f$ isomorphic to $L_{(x,x+1)}$ is $L_{(x,x+1)}$ itself, imply that $f(L_{(x_{\bar{n}},x_{\bar{n}+1})} \times \{ \bar{n} \}) = L_{(x_{\bar{m}}, x_{\bar{m}+1})} \times \{ \bar{m} \}$ for some $\bar{m} \in \mathbb{N}$, and in turn $f(L_{(x_{\bar{n}+k},x_{\bar{n}+k+1})} \times \{ \bar{n}+k \}) = L_{(x_{\bar{m}+k}, x_{\bar{m}+k+1})} \times \{ \bar{m}+k \}$ for all $k \in \mathbb{N}$. But by Lemma [Lemma 27](#open_int){reference-type="ref" reference="open_int"} again, this means that $x_{\bar{n} + k}=y_{\bar{m}+k}$ for all $k \in \mathbb{N}$, hence $\vec{x} \mathrel{E^t_1} \vec{y}$. ◻ **Corollary 69**. *${\cong_{\mathop{\mathrm{\mathsf{LO}}}}} <_B {\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}$ and $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}<_{\text{\scriptsize \textit{Baire}}} {\mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}}$. Moreover ${\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}$ is not Baire reducible to an orbit equivalence relation..* *Proof.* All the statements follow from Theorem [Theorem 68](#no_action_group){reference-type="ref" reference="no_action_group"} and some of the previous results. The first two statements need Theorem [Theorem 67](#thm:COpiecewise){reference-type="ref" reference="thm:COpiecewise"} and Corollaries [Corollary 50](#E1_cvxeq){reference-type="ref" reference="E1_cvxeq"} and [Corollary 47](#cvxeq_baire_iso){reference-type="ref" reference="cvxeq_baire_iso"}; the last one follows from Theorem [Theorem 6](#E1_orbit){reference-type="ref" reference="E1_orbit"}. ◻ # Anti-classification results in knot theory In this section we recall the basic notions of knot theory and the relation between proper arcs/knots and linear orders established in [@Kul17]. We analyse the relation of subarc among proper arcs (Definition [\[def 2.1.2\]](#def 2.1.2){reference-type="ref" reference="def 2.1.2"}) using the results and methods developed for convex embeddability -- its counterpart in the realm of linear orders. We then move to knots and define a natural notion of subknot (Definition [Definition 95](#defn0 4){reference-type="ref" reference="defn0 4"}), uncovering a natural connection with circular orders. This allows us to show that the equivalence relation associated to the latter quasi-order is not induced by a Borel action of a Polish group, in strong contrast with knot equivalence. ## Knots and proper arcs: definitions and basic facts {#sec:introknotsarcs} In mathematics, there are essentially two ways to formalize the intuitive concept of a knot: a (mathematical) *knot* is obtained from a real-life knot by joining its ends so that it cannot be undone, while a *proper arc* is obtained by embedding the real-life knot in a closed 3-ball and sticking its ends to the border of the ball, so that again it cannot be undone. The two concepts are strictly related, although not equivalent as there exist knots that cannot be ''cut'' to obtain a proper arc ([@Bi56]). Let us recall the main definitions and related concepts. Depending on the situation, we think of $S^1$ as either the unit circle in $\mathbb{R}^2$ or the one-point compactification of $\mathbb{R}$ obtained by adding $\infty$ to the space. Similarly, $S^3$ can be viewed as the one-point compactification of $\mathbb{R}^3$. **Definition 70**. A **knot** $K$ is a homeomorphic image of $S^1$ in $S^3$, that is, a subspace of $S^3$ of the form $K = \mathop{\mathrm{Im}}f$ for some topological embedding $f \colon S^1 \to S^3$. The collection of all knots is denoted by $\mathop{\mathrm{Kn}}$; as shown in [@Kul17], it can be construed as a standard Borel space. Obviously, if $K \subseteq S^3$ is a knot and $\varphi \colon S^3 \to S^3$ is an embedding, then $\varphi(K)$ is a knot as well. *Remark 71*. Knots can be naturally endowed with a circular order induced by the standard circular order $C_{S^1}$ defined on $S^1$ (see Section [2.3](#sec:circular orders){reference-type="ref" reference="sec:circular orders"}). More precisely, let $f \colon S^1 \to S^3$ be an embedding and $K = \mathop{\mathrm{Im}}f$ be the knot induced by $f$. Then for every $x,y,z \in K$ we can set $$C_f(x,y,z) \iff C_{S^1}(f^{-1}(x),f^{-1}(y),f^{-1}(z)).$$ If $f , f' \colon S^1 \to S^3$ are two embeddings giving rise to the same knot $K = \mathop{\mathrm{Im}}f = \mathop{\mathrm{Im}}f'$, then $f^{-1} \circ f' \colon S^1 \to S^1$ is a homeomorphism, and thus it is either order-preserving or order-reversing with respect to $C_{S^1}$. It follows that either $C_f = C_{f'}$ or $C_f = C_{f'}^*$. Thus a knot $K$ can be endowed with exactly two circular orders, corresponding to the two possible orientations of $K$ sometimes used in knot theory, which are one the reverse of the other one and depend on the specific embedding used to witness $K \in \mathop{\mathrm{Kn}}$. We speak of **oriented** knot $K$ when we single out one specific orientation between the two possibilities. Two knots $K, K' \in \mathop{\mathrm{Kn}}$ are **equivalent**, in symbols ${K} \equiv_{\mathop{\mathrm{Kn}}} {K'},$ if there exists a homeomorphism $\varphi\colon S^3\to S^3$ such that $\varphi(K) = K'$. The relation $\equiv_{\mathop{\mathrm{Kn}}}$ is an analytic equivalence relation on $\mathop{\mathrm{Kn}}$. A knot is **trivial** if it is equivalent to the unit circle $I_{\mathop{\mathrm{Kn}}} = \{(x,y,z) \in S^3 \mid x^2+y^2=1 \wedge z=0\}$. *Remark 72*. In knot theory it is more common to consider the oriented version of $\equiv_{\mathop{\mathrm{Kn}}}$, according to which two knots $K$ and $K'$ are equivalent if there is an *orientation-preserving* homeomorphism $\varphi \colon S^3 \to S^3$ such that $\varphi(K) = K'$ or, equivalently, an ambient isotopy sending $K$ to $K'$. Nevertheless, we are mostly going to prove anti-classification results, and thus they become even stronger if we consider the coarser equivalence relation $\equiv_{\mathop{\mathrm{Kn}}}$. For the interested reader, however, we point out that all our results remain true if we stick to common practice and replace all the relevant equivalence relations and quasi-orders with their oriented versions. Similar considerations apply to the ensuing definitions and results concerning proper arcs. We now move to proper arcs. A proper arc is a special case of an $n$-tangle, namely it is a $1$-tangle. Contrary to most expositions [@Conway1970AnEO; @Khovanov2002], however, we do not assume that the tangles (arcs) are tame. Given $x \in \mathbb{R}^3$ and a positive $r \in \mathbb{R}$, the closed ball with center $x$ and radius $r$ is denoted by $\bar{B}(x,r)$. The origin $(0,0,0)$ of $\mathbb{R}^3$ is sometimes denoted by $\bar{0}$. To avoid repetitions, we convene that from now $\bar{B}$, possibly with subscripts and/or superscripts, is always a closed topological 3-ball, i.e. a homeomorphic copy of a closed ball in $\mathbb{R}^3$. Recall that by compactness of $\bar{B}$ and the invariance of domain theorem, the notion of boundary of $\bar{B}$ as a topological subspace of $\mathbb{R}^3$ and the notion of boundary of $\bar{B}$ as a topological $3$-manifold coincide. Thus we can unambiguously denote by $\partial \bar{B}$ the **boundary** of $\bar{B}$, and set $\mathop{\mathrm{Int}}\bar{B} = \bar{B} \setminus \partial \bar{B}$. Notice also that by the same reasons, if $\varphi \colon \bar{B} \to \mathbb{R}^3$ is an embedding, then $\varphi(\partial \bar{B}) = \partial \, \varphi(\bar{B})$ and $\varphi (\mathop{\mathrm{Int}}\bar{B}) = \mathop{\mathrm{Int}}\varphi(\bar{B})$. **Definition 73**. Given a topological embedding $f \colon [0,1] \to \bar{B}$ we say that the pair $(\bar{B},\mathop{\mathrm{Im}}f)$ is an **proper arc** if $f(x) \in \partial \bar{B} \iff {x = 0} \vee {x =1}$. With an abuse of notation which is standard in knot theory, when there is no danger of confusion *we identify $f$ with its image $\mathop{\mathrm{Im}}f$* and write e.g. $(\bar B, f)$ in place of $(\bar{B},\mathop{\mathrm{Im}}f)$. Any proper arc $(\bar B, f)$ can be canonically turned (up to knot equivalence) into a knot $K_{(\bar B, f)}$ by joining its ends $f(0)$ and $f(1)$ with a simple curve running on the boundary $\partial \bar B$ of its ambient space, assuming, without loss of generality, that $\bar B=\bar B(\bar 0,1)$. The collection of proper arcs is denoted by $\mathop{\mathrm{Ar}}$, and can be construed as a standard Borel subspace of the product $K(\mathbb{R}^3) \times K(\mathbb{R}^3)$ of the Vietoris space $K(\mathbb{R}^3)$ over $\mathbb{R}^3$. This follows as an application of a theorem by Ryll-Nardzewski [@Ryll65] (see [@Kul17] for the analogous construction of the coding space $\mathop{\mathrm{Kn}}$ of knots). Notice that if $(\bar B, f)$ is a proper arc and $\varphi \colon \bar{B} \to \mathbb{R}^3$ is an embedding, then $(\varphi(\bar{B}), \varphi(f)) = (\varphi(\bar{B}), \varphi(\mathop{\mathrm{Im}}f))$ is a proper arc, as witnessed by the embedding $\varphi \circ f \colon [0,1] \to \varphi(\bar{B})$. *Remark 74*. [\[rmk:orientation-1\]]{#rmk:orientation-1 label="rmk:orientation-1"} Every specific embedding $f$ giving rise to an arc $(\bar{B},\mathop{\mathrm{Im}}f)$ induces an orientation on it, namely, the linear order $\leq_f$ on $\mathop{\mathrm{Im}}f$ defined by $$b_0 \leq_f b_1 \iff f^{-1}(b_0) \leq f^{-1}(b_1).$$ If $f,f' \colon [0,1] \to \bar{B}$ are two topological embeddings inducing the same proper arc (that is, $\mathop{\mathrm{Im}}f = \mathop{\mathrm{Im}}f'$), then $f^{-1} \circ f' \colon [0,1] \to [0,1]$ is a homeomorphism, and thus it is either order-preserving or order-reversing. It follows that every proper arc has exactly two orientations. Moreover, the minimum and the maximum of $\leq_f$ always exist; they can be identified, independently of $f$, as the only points of $\mathop{\mathrm{Im}}f$ belonging to $\partial \bar{B}$. We speak of **oriented** proper arc $(\bar B, f)$ when we equip it with the specific orientation given by the displayed $f$. [\[rmk:orientation-2\]]{#rmk:orientation-2 label="rmk:orientation-2"} If $(\bar B, f)$ and $(\bar B', g)$ are proper arcs and $\varphi \colon \bar{B} \to \bar{B}'$ is a topological embedding such that $\varphi(\mathop{\mathrm{Im}}f) \subseteq \mathop{\mathrm{Im}}g$, then $h = g^{-1} \circ \varphi \circ f \colon [0,1] \to [0,1]$ is a topological embedding. It follows that when $(\bar B, f)$ and $(\bar B', g)$ are construed as oriented proper arcs, then $\varphi$ is either order-preserving (that is, $\varphi(b_0) \leq_g \varphi(b_1)$ for all $b_0,b_1 \in \bar{B}$ with $b_0 \leq_f b_1$) or order-reversing (that is, $\varphi(b_1) \leq_g \varphi(b_0)$ for all $b_0,b_1 \in \bar{B}$ with $b_0 \leq_f b_1$). Two proper arcs $(\bar B, f)$ and $(\bar B', g)$ are **equivalent**, in symbols $(\bar B, f)\equiv_{\mathop{\mathrm{Ar}}}(\bar B', g),$ if there exists a homeomorphism $\varphi\colon \bar{B}\to \bar{B}'$ such that $\varphi(\mathop{\mathrm{Im}}f)= \mathop{\mathrm{Im}}g$. The relation $\equiv_{\mathop{\mathrm{Ar}}}$ is an analytic equivalence relation on the standard Borel space $\mathop{\mathrm{Ar}}$. A proper arc $(\bar B, f)$ is **trivial** if it is equivalent to $I_{\mathop{\mathrm{Ar}}} = (\bar{B}(\bar{0}, 1), [-1, 1] \times \{(0, 0)\})$. An important dividing line among knots (respectively, proper arcs) is given by tameness, i.e. the absence of singular points. Given a knot $K \in \mathop{\mathrm{Kn}}$, a **subarc** of $K$ is any proper arc of the form $(\bar B, K \cap \bar{B})$. A point $x \in K$ is called **singular**, or a **singularity**, of $K$ if there is no $\bar B$ such that $x \in \mathop{\mathrm{Int}}\bar B$ and $(\bar B, K \cap \bar B)$ is a trivial proper subarc of $K$. The space of singularities of $K$ is denoted by $\Sigma_K$. An **isolated** singular point of $K$ is an isolated point of the topological space $\Sigma_K$, and the (sub)space of isolated singular points of $K$ is denoted by $I\Sigma_K$. Finally, a knot $K$ is **tame** [^4] if it has no singular points, and **wild** otherwise. Notice also that if $x \in K$ is not a singularity of $K$, then there are arbitrarily small closed topological $3$-balls $\bar B$ witnessing this. The previous definitions can be naturally adapted to proper arcs. Let $(\bar B, f) \in \mathop{\mathrm{Ar}}$. A point $x \in \mathop{\mathrm{Im}}f$ is called **singular**, or a **singularity**, of $(\bar B, f)$ if it belongs to $\Sigma_{K_{(\bar B, f)}}$, while an **isolated** singular point of $(\bar B, f)$ is an an element of $I\Sigma_{K_{(\bar B, f)}}$. Accordingly, the space of singularities of $(\bar B, f)$ is denoted by $\Sigma_{(\bar B, f)}$, while the space of isolated singular points is denoted by $I\Sigma_{(\bar B, f)}$. An arc $(\bar B, f)$ is **tame** if $\Sigma_{(\bar B, f)} = \emptyset$ (equivalently, if $K_{(\bar B, f)}$ is tame), and **wild** otherwise. Notice that if $x \in \mathop{\mathrm{Im}}f \cap \mathop{\mathrm{Int}}\bar B$, then $x \notin \Sigma_{(\bar B, f)}$ if and only if there is $\bar{B}' \subseteq \bar{B}$ such that $x \in \mathop{\mathrm{Int}}\bar{B}'$ and $(\bar B', f \cap \bar{B}')$ is a trivial proper arc. For points on the boundary $\partial \bar B$, instead, it is not enough to consider closed topological $3$-balls $\bar B' \subseteq \bar B$, as we necessarily need to consider a "trivial prolungation" of the curve $\mathop{\mathrm{Im}}f$ beyond its extreme points in order to determine whether they are singular or not. We also introduce the notion of circularization of a proper arc, which generates a knot and gives a characterization of tame knots. **Definition 75**. Let $(\bar B, f) \in \mathop{\mathrm{Ar}}$. Up to equivalence, we can assume that $\bar B =[-1,1]^3$, $f(0)=(-1,0,0)$ and $f(1)=(1,0,0)$. Consider the equivalence relation obtained setting $(-1,y,z) \sim (1,y,z)$ for all $(y,z) \in [-1,1]^2$, so that in the quotient space $T = [-1,1]^3 / {\sim}$ the two lateral faces of the cube $\bar B$ are glued and we have a solid torus. Given a topological embedding $h$ of $T$ into $S^3$, we call **circularitazion of** $(\bar B, f)$, denoted by $C^h[(\bar B, f)]$, the knot which is obtained as the image of $\mathop{\mathrm{Im}}f / {\sim}$ via $h$. Notice that the circularization of a proper arc depends on the topological embedding of the solid torus into $S^3$, hence it is not unique. Moreover, we have that $K \in \mathop{\mathrm{Kn}}$ is tame if and only if $K=C^h[I_{\mathop{\mathrm{Ar}}}]$ for some topological embedding $h \colon T \to S^3$. A substantial part of the analysis of tame knots relies on their prime factorization, which is in turn based on the classical notion of sum (see [@BZ03 Definition 2.7], where the sum is actually called product). We will work with the corresponding sum for proper arcs which is akin to the tangle sum [@Conway1970AnEO] (note that we allow wild arcs). **Definition 76**. Let $(\bar B_0, f_0)$ and $(\bar B_1, f_1)$ be *oriented* proper arcs. Up to equivalence, we may assume that $\bar{B}_0 = [-1,0] \times [-1,1]^2$, $\bar{B}_1 = [0,1] \times [-1,1]^2$, $f_0(0) = (-1,0,0)$, $f_0(1) = f_1(0) = (0,0,0)$, and $f_1(1) = (1,0,0)$. The **sum** $(\bar B_0, f_0) \oplus (\bar B_1, f_1)$ is the proper arc $(\bar B, f)$ where $\bar{B} = [-1,1]^3$ and $f \colon [0,1] \to \bar{B}$ is defined by $f(x) = f_0(2x)$ if $x \leq \frac{1}{2}$ and $f(x) = f_1(2x-1)$ if $x \geq \frac{1}{2}$. By induction on $n \in \mathbb{N}$, one can then define finite sums of proper arcs $(\bar B_0, f_0) \oplus \dots \oplus (\bar B_n, f_n) ,$ abbreviated by $\bigoplus_{i \leq n} \, (\bar B_i, f_i)$. *Remark 77*. Although the sum of two oriented proper arcs is again oriented, in this paper we will tacitly consider it as an unoriented proper arc. Also, we will often sum unoriented proper arcs: what we mean in this case is that the arcs are summed using the natural orientation coming from the way we present them. Tame knots and arcs are in canonical one-to-one correspondence up to equivalence. Given a tame knot $K$ in $S^3$, pick an arbitrary $x\in K$ and a neighbourhood $B_0$ of $x$ such that $(\bar B_0,K\cap B_0)$ is a trivial arc. Then let $\bar B_K=S^3\setminus B_0$ and $f_K=f\setminus B_0$. Then $K_{(\bar B_K,f_K)}$ is equivalent to $K$. Vice versa, for all equivalent arcs $(\bar B, f)$ and $(\bar B', f')$ it is easy to see that $K_{(\bar B, f)} \equiv_{\mathop{\mathrm{Kn}}} K_{(\bar B', f')}$. This transformation commutes with the knot sum $\#$ of tame knots [@BZ03 Definition 7.1]. The latter can actually be defined using $\oplus$. Given two oriented tame knots we then have that, up to equivalence, $$K_0 \mathrel{\#} K_1 = K_{(\bar B_{K_0}, f_{K_0}) \oplus (\bar B_{K_1}, f_{K_1})}.$$ Notice also that if $(\bar B_0, f_0)$ and $(\bar B_1, f_1)$ are (oriented) tame proper arcs, then $$K_{(\bar B_0, f_0) \oplus (\bar B_1, f_1)} \equiv_{\mathop{\mathrm{Kn}}} K_{(\bar B_0, f_0)} \mathrel{\#} K_{(\bar B_1, f_1)}.$$ Recall that a nontrivial tame knot is **prime** if it cannot be written as a sum of nontrivial knots. The prime knot decomposition theorem [@BZ03 Theorem 7.12] states that every tame knot can be expressed as a finite knot sum of prime knots in a unique way up to knot equivalence. We will later consider also prime arcs which are defined in a similar way. ## Proper arcs and their classification {#sec:arcs} The following notion is equivalent to [@Kul17 Definition 2.10]. **Definition 78**. [\[def 2.1.2\]]{#def 2.1.2 label="def 2.1.2"} Let $(\bar B, f), (\bar B', g) \in \mathop{\mathrm{Ar}}$. We say that $(\bar B, f)$ is a **subarc** of $(\bar B', g)$, or that $(\bar B', g)$ has $(\bar B, f)$ as a subarc, if there exists a topological embedding $\varphi\colon \bar{B}\to \bar{B}'$ such that $\varphi(f) = g \cap \mathop{\mathrm{Im}}\ \varphi$. In this case we write $$(\bar B, f)\precsim_{\mathop{\mathrm{Ar}}} (\bar B', g).$$ (Notice that we automatically have that $(\varphi(\bar{B}), g \cap \mathop{\mathrm{Im}}\varphi))$ is a proper arc.) Clearly, the subarc relation $\precsim_{\mathop{\mathrm{Ar}}}$ is an analytic quasi-order on the standard Borel space $\mathop{\mathrm{Ar}}$. We denote by $\prec_{\mathop{\mathrm{Ar}}}$ the strict part of $\precsim_{\mathop{\mathrm{Ar}}}$, i.e. $$(\bar B, f) \prec_{\mathop{\mathrm{Ar}}} (\bar B', g) \iff {(\bar B, f) \precsim_{\mathop{\mathrm{Ar}}} (\bar B', g)} \wedge {(\bar B', g) \not\precsim_{\mathop{\mathrm{Ar}}} (\bar B, f)} .$$ The analytic equivalence relation associated to $\precsim_{\mathop{\mathrm{Ar}}}$ is denoted by $\approx_{\mathop{\mathrm{Ar}}}$, and we say that two proper arcs $(\bar B, f)$ and $(\bar B', g)$ are **mutual subarcs** if $${\label{eq}} (\bar B, f) \approx_{\mathop{\mathrm{Ar}}} (\bar B', g).$$ This may be interpreted as asserting that the two arcs have the "same complexity" because each of them is a subarc of the other one. Notice also that $(\bar B, f)\equiv_{\mathop{\mathrm{Ar}}} (\bar B', g)$ trivially implies $(\bar B, f)\approx_{\mathop{\mathrm{Ar}}} (\bar B', g)$. If $( \bar{B},f), (\bar B', g) \in \mathop{\mathrm{Ar}}$ and $\varphi$ witnesses $(\bar B, f) \equiv_{\mathop{\mathrm{Ar}}} (\bar B', g)$, then $\varphi$ induces a homeomorphism between the spaces $\Sigma_{(\bar B, f)}$ and $\Sigma_{(\bar B', g)}$, and hence also a homeomorphism between $I\Sigma_{(\bar B, f)}$ and $I\Sigma_{(\bar B', g)}$. If instead $\varphi \colon \bar{B} \to \bar{B}'$ is just an embedding witnessing $(\bar B, f) \precsim_{\mathop{\mathrm{Ar}}} (\bar B', g)$, then we still have that $\varphi$ induces an embedding of $\Sigma_{(\bar B, f)}$into $\Sigma_{(\bar B', g)}$, but needs not send isolated singular points into isolated singular points: if $x \in I\Sigma_{(\bar B, f)} \cap \partial \bar{B}$, then it might happen that $\varphi(x) \in \Sigma_{(\bar B', g)} \setminus I\Sigma_{(\bar B', g)}$. However, this is the only exception. **Lemma 79**. *[\[lem:isolatedthroughembeddings-1\]]{#lem:isolatedthroughembeddings-1 label="lem:isolatedthroughembeddings-1"} Let $(\bar B, f) \in \mathop{\mathrm{Ar}}$ and $\bar{B}' \subseteq \bar{B}$ be such that $(\bar B', f \cap \bar{B}') \in \mathop{\mathrm{Ar}}$. Then $\Sigma_{(\bar B', f \cap \bar B')} \subseteq \Sigma_{(\bar B, f)}$, and $\Sigma_{(\bar B', f \cap \bar{B}')} \cap \mathop{\mathrm{Int}}\bar{B}' = \Sigma_{(\bar B, f)} \cap \mathop{\mathrm{Int}}\bar{B}'$.* *[\[lem:isolatedthroughembeddings-2\]]{#lem:isolatedthroughembeddings-2 label="lem:isolatedthroughembeddings-2"} Let $( \bar{B},f), (\bar B', g) \in \mathop{\mathrm{Ar}}$, and let $\varphi \colon \bar{B} \to \bar{B}'$ witness $(\bar B, f) \precsim_{\mathop{\mathrm{Ar}}} (\bar B', g)$. If $x \in I\Sigma_{(\bar B, f)} \cap \mathop{\mathrm{Int}}\bar{B}$, then $\varphi(x) \in I\Sigma_{(\bar B', g)}$.* *Proof.* [\[lem:isolatedthroughembeddings-1\]](#lem:isolatedthroughembeddings-1){reference-type="ref" reference="lem:isolatedthroughembeddings-1"} The first part is easy and is left to the reader. For the nontrivial inclusion of the second part, assume that $x \in \mathop{\mathrm{Int}}\bar B'$ (so that $x \in \mathop{\mathrm{Int}}\bar B$ as well because $\bar B' \subseteq \bar B$) and $x \notin \Sigma_{(\bar B', f \cap \bar{B}')}$. Let $\bar{B}'' \subseteq \bar{B}'$ be a witness of this: then $\bar{B}''$ also witnesses $x \notin \Sigma_{(\bar B, f)}$. [\[lem:isolatedthroughembeddings-2\]](#lem:isolatedthroughembeddings-2){reference-type="ref" reference="lem:isolatedthroughembeddings-2"} By hypothesis and the fact that $\varphi \colon \bar{B} \to \varphi(\bar{B})$ is a homeomorphism, $\varphi(x) \in I\Sigma_{(\varphi(\bar{B}), g \cap \mathop{\mathrm{Im}}\varphi)} \cap \mathop{\mathrm{Int}}\varphi(\bar{B})$. By part [\[lem:isolatedthroughembeddings-1\]](#lem:isolatedthroughembeddings-1){reference-type="ref" reference="lem:isolatedthroughembeddings-1"}, this implies that $\varphi(x) \in \Sigma_{(\bar B', g)}$. Using $\varphi(x) \in \mathop{\mathrm{Int}}\varphi(\bar{B})$, pick a small enough open set $U \subseteq \mathop{\mathrm{Int}}\varphi(\bar{B})$ such that $U \cap \Sigma_{(\varphi(\bar{B}), g \cap \mathop{\mathrm{Im}}\varphi)} = \{ \varphi(x) \}$: then by part [\[lem:isolatedthroughembeddings-1\]](#lem:isolatedthroughembeddings-1){reference-type="ref" reference="lem:isolatedthroughembeddings-1"} again $U \cap \Sigma_{(\bar B', g)} = U \cap \Sigma_{(\varphi(\bar{B}), g \cap \mathop{\mathrm{Im}}\varphi)}$, and thus $U \cap \Sigma_{(\bar B', g)}$ witnesses $\varphi(x) \in I\Sigma_{(\bar B', g)}$. ◻ We now define an infinitary version of the sum operation for (tame) proper arcs introduced in Definition [Definition 76](#def:sumarc){reference-type="ref" reference="def:sumarc"}. Since the ambient space $\bar{B}$ in the definition of a proper arc is a compact space, in order to define such infinitary sums we need the summands to accumulate towards a point $b \in \bar{B}$, which thus becomes a singularity when infinitely many summands are not trivial. **Definition 80**. Let $(\bar B_i, f_i)$ be oriented proper arcs, for $i \in \mathbb{N}$. [^5] The (**infinite**) **sum with limit $b \in \bar{B}$**, denoted by $\bigoplus_{i \in \mathbb{N}}^b \, (\bar B_i, f_i)$, is defined up to equivalence as follows. Without loss of generality, we may assume that $b$ is of the form $(b',0,0)$ for some $b'$ with $0 < b' \leq 1$. Up to equivalence, we may also assume that $\bar{B}_i = [b' - 2^{-i}, b'-2^{-(i+1)}] \times [-2^{-i},2^{-i}]^2$, and that $f_i(0) = (b'-2^{-i},0,0)$ and $f_i(1) = (b'-2^{-(i+1)},0,0)$ for all $i \in \mathbb{N}$. Then $\bigoplus_{i \in \mathbb{N}}^b \, (\bar B_i, f_i)$ is the arc $(\bar B, f)$ where $\bar{B} = [-1,1]^3$ and $\mathop{\mathrm{Im}}f$ is the union of $\bigcup_{i \in \mathbb{N}} \mathop{\mathrm{Im}}f_i$ together with $[-1,b'-1] \times \{ (0,0) \}$ and $[b',1] \times \{ (0,0) \}$ (the latter might reduce to the point $(1,0,0)$ if $b' = 1$ or, equivalently, if $b \in \partial \bar{B}$). Trivially, $(\bar B_j, f_j) \precsim_{\mathop{\mathrm{Ar}}} \bigoplus_{i \leq n} \, (\bar B_i, f_i)$ for all $n \geq j$ and $(\bar B_j, f_j) \precsim_{\mathop{\mathrm{Ar}}} \bigoplus_{i \in \mathbb{N}}^b (\bar B_i, f_i)$ for all $b \in \bar{B}$. Notice that, up to $\equiv_{\mathop{\mathrm{Ar}}}$, Definition [Definition 80](#def:infinitesumarc){reference-type="ref" reference="def:infinitesumarc"} gives rise to precisely two non-equivalent proper arcs, depending on whether $b \in \partial \bar{B}$ or not---besides this dividing line the actual choice of the limit point $b \in \bar{B}$ is completely irrelevant. Therefore we can simplify the notation by denoting with $\bigoplus_{i \in \mathbb{N}} \, (\bar B_i, f_i)$ the infinite sum $\bigoplus^b_{i \in \mathbb{N}} \, (\bar B_i, f_i)$ for some/any $b \in \mathop{\mathrm{Int}}\bar{B}$, and with $\bigoplus^\partial_{i \in \mathbb{N}} \, (\bar B_i, f_i)$ the infinite sum $\bigoplus^b_{i \in \mathbb{N}} \, (\bar B_i, f_i)$ for some/any $b \in \partial \bar{B}$. It is not hard to see that $\bigoplus^\partial_{i \in \mathbb{N}} \, (\bar B_i, f_i) \prec_{\mathop{\mathrm{Ar}}} \bigoplus_{i \in \mathbb{N}} \, (\bar B_i, f_i)$. Finally, if all the proper arcs $(\bar B_i, f_i)$ are equivalent to the same arc $(\bar B', g)$, the two possible infinite sums will be denoted by $\bigoplus_{\mathbb{N}} \, (\bar B', g)$ and $\bigoplus^\partial_{\mathbb{N}} \, (\bar B', g)$, respectively. Obviously, we can also replace $\mathbb{N}$ with any infinite $A \subseteq \mathbb{N}$ and write $\bigoplus_{j \in A}^{(\partial)} \, (\bar B_j, f_j)$ to denote $\bigoplus_{i \in \mathbb{N}}^{(\partial)} \, (\bar B_{r(j)}, f_{r(j)})$, where $r \colon \mathbb{N}\to A$ is the increasing enumeration of $A$; similarly for $\bigoplus_A^{(\partial)} \, (\bar B', g)$. Figure [1](#trefoils){reference-type="ref" reference="trefoils"} presents the arc $\bigoplus_{\mathbb{N}} \, (\bar B', g)$ where $(\bar B', g)$ is the trefoil; its variant $\bigoplus^\partial_{\mathbb{N}} (\bar B', g)$ would be obtained my moving the current limit point $(0,0,0)$ to the point $(1,0,0)$ on $\partial \bar{B}$. ![Infinite sum of trefoils, with limit point internal to the ambient space $\bar{B} = [-1,1]^3$.](im3a.jpg){#trefoils width="90%"} In [@Kul17 Theorem 3.1] it is shown that the isomorphism $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ on countable linear orders Borel reduces to equivalence $\equiv_{\mathop{\mathrm{Kn}}}$ on knots. Employing the same construction, we establish a similar connection between convex embeddability $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$ on linear orders and the subarc relation $\precsim_{\mathop{\mathrm{Ar}}}$ on proper arcs. Fix a proper arc $(\bar B^*, f^*)$ of the form $\bigoplus_\mathbb{N}\, (\bar B_i, f_i)$ with all the proper arcs $(\bar B_i, f_i)$ tame and not trivial. (For the sake of definiteness, one can e.g. assume that $(\bar B^*, f^*)$ is the sum of infinitely many trefoils depicted in Figure [1](#trefoils){reference-type="ref" reference="trefoils"}.) An important feature of such a $(\bar B^*, f^*)$ is that $$\tag{\( \dagger\)} \label{eq:not-inverting} \begin{minipage}{11cm} Any embedding \( \varphi \colon \bar{B^*} \to \bar{B^*}\) with \( \varphi(f^*) = f^* \cap \mathop{\mathrm{Im}}\varphi \) preserves the (natural) orientation of the arc. \end{minipage}$$ Notice also that the only singularity of $(\bar B^*, f^*)$, which is trivially isolated, belongs to $\mathop{\mathrm{Int}}\bar{B^*}$. We first define a Borel map that given $L \in \mathop{\mathrm{\mathsf{LO}}}$ produces an order-embedding $h_L$ of $L$ into $(\mathbb{Q}, \leq)$ and a function $r_L \colon L \to \mathbb{Q}$ such that: [\[Vpwdisj\]]{#Vpwdisj label="Vpwdisj"} the open intervals $V_n^L=(h_L(n)-2r_L(n), h_L(n)+2r_L(n))$ are included in $[-1,1]$ and pairwise disjoint; [\[dense\]]{#dense label="dense"} $\bigcup_{n \in \mathbb{N}} V_n^L$ is dense in $[-1,1]$. To this end, we first establish in a Borel way whether $L$ has extrema, what are they are, and when one element of the linear order is the immediate successor of another. So suppose that the maps $L\mapsto (h_L,r_L)$ are defined. Notice that $\lim_{n \to \infty} r_L(n) = 0$ and that we can assume that $r_L(n+1)<r_L(n)$ for every $n\in \mathbb{N}$. Let $U_n^L = [h_L(n)-r_L(n), h_L(n)+r_L(n)]$. Thinking of $[-1,1]$ as lying on the $x$-axis, we replace $U_n^L$ with the cube $\bar{B}_n^L = U_n^L \times [-r_L(n),r_L(n)]^2$. Let $(\bar B_n^L, f_n^L)$ be equivalent to $(\bar B^*, f^*)$ and such that $f_n^L(0) < f_n^L(1)$ and both belong to the $x$-axis, and set $f_r^L = ([-1,1] \setminus \bigcup_{n \in \mathbb{N}} U_n^L) \times \{(0,0)\}$. Then we define the map $$\label{eq:FfromLOtoA} F \colon \mathop{\mathrm{\mathsf{LO}}}\to \mathop{\mathrm{Ar}}, \qquad L \mapsto (\bar B, f_L)$$ by letting $\bar{B}=[-1,1]^3$ and $f_L = f_r^L \cup \bigcup_{n \in \mathbb{N}} f_n^L$. By construction, every $(h_L(n),0,0)$ is singular and isolated in $\Sigma_{F(L)}$ by (the trace of) $\bar{B}_n^L$, and every other member of $\Sigma_{F(L)}$ is a limit of these singular points. Thus $\Sigma_{F(L)}$ is contained in the $x$-axis and $I\Sigma_{F(L)}=\{(h_L(n),0,0) \mid n \in \mathbb{N}\}$. The latter is naturally ordered by considering the restriction of $\leq_{f_L}$ to $I\Sigma_{F(L)}$, or equivalently, by considering first coordinates ordered as elements of $\mathbb{R}$. Then the map $n \mapsto (h_L(n),0,0)$ is an isomorphism between the linear orders $L$ and $I\Sigma_{F(L)}$. Since the entire construction really depends on the proper arc $(\bar B^*, f^*)$, when relevant we will add this information to the notation and write e.g. $F_{(\bar B^*, f^*)}(L)$. For future reference, we also notice that by construction $I\Sigma_{F(L)} \subseteq \mathop{\mathrm{Int}}\bar{B}$. **Theorem 81**. *The map $F$ from [\[eq:FfromLOtoA\]](#eq:FfromLOtoA){reference-type="eqref" reference="eq:FfromLOtoA"} simultaneously witnesses ${\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\precsim_{\mathop{\mathrm{Ar}}}}$ (hence also ${\underline{\bowtie}}_{\mathop{\mathrm{\mathsf{LO}}}} \leq_B {\approx_{\mathop{\mathrm{Ar}}}}$) and ${\cong_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\equiv_{\mathop{\mathrm{Ar}}}}$.* The lower bound ${\cong_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\equiv_{\mathop{\mathrm{Ar}}}}$ for the relation $\equiv_{\mathop{\mathrm{Ar}}}$ is implicit in (the proof of) [@Kul17 Theorem 3.1]. Notice however that our proof is more natural, as it avoids reducing first $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ to its restriction to linear orders with minimum and without maximum, and then the latter to the relations on arcs and knots, as it is done instead in [@Kul17]. *Proof.* In order to check that $F$ is a Borel function between the Polish space $\mathop{\mathrm{\mathsf{LO}}}$ and the standard Borel space $\mathop{\mathrm{Ar}}$ one can argue as in [@Kul17], so we only need to prove that $F$ is a reduction. Assume first that $L,L' \in \mathop{\mathrm{\mathsf{LO}}}$ are such that $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}L'$, and let $g\colon L\to L'$ witness this. For every $n \in L$ the proper arcs $(\bar B_n^L, f_n^L)$ and $(\bar B_{g(n)}^{L'}, f_{g(n)}^{L'})$ are both equivalent to $(\bar B^*, f^*)$, and hence we can consider a homeomorphism $\varphi_1^n\colon \bar{B}_n^L \to \bar{B}_{g(n)}^{L'}$ witnessing this. Notice that $\varphi_1^n$ is necessarily order-preserving by [\[eq:not-inverting\]](#eq:not-inverting){reference-type="eqref" reference="eq:not-inverting"}. Let $\varphi_1 = \bigcup_{n \in \mathbb{N}} \varphi_1^n$ and notice that $\varphi_1 (h_L(n),0,0) = (h_{L'}(g(n)),0,0)$ for every $n \in L$, so that the restriction of $\varphi_1$ to $I\Sigma_{F(L)}$ is order-preserving into $I\Sigma_{F(L')}$. For every $n \in L$ let $M_n= \max \{2r_L(n), 2r_{L'}(g(n)) \}$ and let $\varphi_2^n \colon \overline{V_n^{L}} \times [-M_n,M_n]^2 \to \overline{V_{g(n)}^{L'}} \times [-M_n,M_n]^2$ be a homeomorphism which extends $\varphi_1^n$ and has the following properties: 1. for all $(y,z) \in [-M_n,M_n]^2$, $\varphi_2^n(h_L(n) \pm 2r_L(n), y,z)=(h_{L'}(g(n)) \pm 2r_{L'}(g(n)), y,z)$; 2. for all $(y,z) \in [-M_n,M_n]^2$ with $\max \{|y|,|z|\}=M_n$ and for all $t \in [-1,1]$ we have $$\varphi_2^n(h_L(n)+ 2r_L(n)t, y,z)=(h_{L'}(g(n)) + 2r_{L'}(g(n))t, y,z)$$ (this condition is missing in [@Kul17]). Let $W_n^{L}= \overline{V_n^{L}} \times [-1,1]^2$ and $W_{g(n)}^{L'} = \overline{V_{g(n)}^{L'}}\times [-1,1]^2$. We can then define a homeomorphism $\varphi_3^n\colon W_n^L\to W_{g(n)}^{L'}$ which extends $\varphi_2^n$ and is such that: 1. [\[lines\]]{#lines label="lines"} for every $(y,z)\in [-1,1]^2$ such that $\max \{|y|, |z|\} \geq M_n$ we have $$\varphi_3^n (h_L(n)+ 2r_L(n)t,y,z) = (h_{L'}(g(n)) + 2r_{L'}(g(n))t,y,z),$$ so that outside $\overline{V_n^{L}} \times [-M_n,M_n]^2$ the lines parallel to the $x$-axis are mapped into themselves. Then $\varphi_3 = \bigcup_{n \in \mathbb{N}} \varphi_3^n$ is a homeomorphism between $\bigcup_{n \in \mathbb{N}} W_n^L$ and $\bigcup_{n \in \mathbb{N}} W_{g(n)}^{L'}$. We finally extend $\varphi_3$ to $\varphi \colon \bar{B} \to \bar{B}$ by looking at each $x_0 \in [-1,1] \setminus \bigcup_{n \in \mathbb{N}} \overline{V_n^L}$ (which is a cluster point of $\mathop{\mathrm{Im}}h_L$) and setting $\varphi (x_0,y,z) = (x'_0,y,z)$ for every $(y,z) \in [-1,1]^2$, where $x_0= \lim_{i \to \infty} h_L(n_i)$ and $x'_0 = \lim_{i \to \infty} h_{L'}(g(n_i))$. Condition [\[lines\]](#lines){reference-type="ref" reference="lines"} ensures that $\varphi$ is continuous and indeed a homeomorphism. It is immediate that $\varphi$ witnesses $F(L) \precsim_{\mathop{\mathrm{Ar}}} F(L')$, and that if $g \colon L \to L'$ was actually an isomorphism, then $\varphi$ witnesses $F(L) \equiv_{\mathop{\mathrm{Ar}}} F(L')$. Conversely, suppose that $\varphi \colon \bar{B}\to \bar{B}$ is an embedding witnessing $F(L)\precsim_{\mathop{\mathrm{Ar}}} F(L')$. Since all isolated points of $F(L)$ belong to $\mathop{\mathrm{Int}}\bar{B}$, by Lemma [Lemma 79](#lem:isolatedthroughembeddings){reference-type="ref" reference="lem:isolatedthroughembeddings"} the map $\varphi \restriction I\Sigma_L$ embeds $I\Sigma_{F(L)}$ into $I\Sigma_{F(L')}$. Furthermore, as explained in [@Kul17], the embedding $\varphi$ preserves the betweenness relation. By [\[eq:not-inverting\]](#eq:not-inverting){reference-type="eqref" reference="eq:not-inverting"}, for any $n \in L$ the restriction of $\varphi$ to the arc $(\bar B_n^L, f_L \cap \bar{B}_n^L)$, which maps it to $(\varphi(\bar{B}_n^L), f_{L'} \cap \varphi(\bar{B}_n^L))$, is order-preserving and hence $\varphi \restriction I\Sigma_{F(L)}$ is order-preserving too. Moreover, since $\varphi$ is continuous and $f_L$ is connected we get that also $\varphi(f_L)$ is connected: it follows that $\varphi(I\Sigma_{F(L)})$ is a convex subset of $I\Sigma_{F(L')}$. Summing up, $\varphi\restriction I\Sigma_{F(L)}$ witnesses that $I\Sigma_{F(L)}\trianglelefteq I\Sigma_{F(L')}$, hence $L \trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}} L'$ because $L\cong I\Sigma_{F(L)} \trianglelefteq I\Sigma_{F(L')}\cong L'$. Obviously, if $\varphi \colon \bar{B} \to \bar{B}$ was actually a homeomorphism, then $\varphi \restriction I\Sigma_{F(L)}$ would be onto $I\Sigma_{F(L')}$, and thus it would witness $I\Sigma_{F(L)}\cong I\Sigma_{F(L')}$, which in turn implies $L \cong_{\mathop{\mathrm{\mathsf{LO}}}}L'$. ◻ By Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"}, the reduction $F \colon \mathop{\mathrm{\mathsf{LO}}}\to \mathop{\mathrm{Ar}}$ allows us to transfer some combinatorial properties of ${\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}}$ discussed in Section [3.1](#sec:comb_prop){reference-type="ref" reference="sec:comb_prop"} to the quasi-order $\precsim_{\mathop{\mathrm{Ar}}}$ (cfr. Lemma [Lemma 27](#open_int){reference-type="ref" reference="open_int"}, Proposition [Proposition 28](#prop1){reference-type="ref" reference="prop1"}, and Corollary [Corollary 52](#cor:intervalsleqBcvx){reference-type="ref" reference="cor:intervalsleqBcvx"}). **Corollary 82**. *[\[cor:sumuparcs-a\]]{#cor:sumuparcs-a label="cor:sumuparcs-a"} There is an embedding from the partial order $(\mathop{\mathrm{Int}}(\mathbb{R}),\subseteq)$ into $\precsim_{\mathop{\mathrm{Ar}}}$, and indeed ${(\mathop{\mathrm{Int}}(\mathbb{R}),\subseteq)} \leq_B {\precsim_{\mathop{\mathrm{Ar}}}}$.* *[\[cor:sumuparcs-b\]]{#cor:sumuparcs-b label="cor:sumuparcs-b"} $\precsim_{\mathop{\mathrm{Ar}}}$ has chains of order type $(\mathbb{R}, {<} )$, as well as antichains of size $2^{\aleph_0}$.* In contrast, the combinatorial properties uncovered in Propositions [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"}, [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}, and [Proposition 34](#prop:dom_fam){reference-type="ref" reference="prop:dom_fam"}, being universal statements, do not transfer through the reduction $F$. We overcome some of these difficulties by using the following construction. Using the orientation induced by $f$, when $(\bar B, f)$ is a proper arc the set $I\Sigma_{(\bar B, f)}$ can naturally be viewed as a linear order $L_{(\bar B, f)} = (I\Sigma_{(\bar B, f)}, \leq_f)$. Since $\Sigma_{(\bar B, f)}$, being a subspace of the Polish space $\bar{B}$, is second-countable, the set $I\Sigma_{(\bar B, f)}$ is (at most) countable and thus up to isomorphism $L_{(\bar B, f)}$ is an element of $\mathop{\mathrm{\mathsf{Lin}}}$. We remark that the linear order $L_{(\bar B, f)}$ really depends on the topological embedding $f$ (or, more precisely, on the orientation it induces) rather than its image. However, if $f$ and $f'$ are two topological embeddings giving rise to the same arc, then either $L_{(\bar B, f)} = L_{(\bar B, f')}$ or $L_{(\bar B, f)} = (L_{(\bar B, f')})^*$ --- indeed the two linear orders correspond to the two possible orientations of the arc $(\bar{B},\mathop{\mathrm{Im}}f)$. Recall that by construction, for proper arcs of the form [^6] $F(L) = (\bar B, f_L)$ we have $I\Sigma_{F(L)} \cong L$. **Lemma 83**. *Let $(\bar B, f), (\bar B', g) \in \mathop{\mathrm{Ar}}$ be such that $(\bar B, f)\precsim_{\mathop{\mathrm{Ar}}} (\bar B', g)$, and let $K = (I\Sigma_{(\bar B, f)} \cap \mathop{\mathrm{Int}}\bar{B},{\leq_f})$. Then either $K \trianglelefteq L_{(\bar B', g)}$ or $K \trianglelefteq (L_{(\bar B', g)})^*$.* *Proof.* Let $\varphi \colon \bar{B} \to \bar{B}'$ be an embedding witnessing $(\bar B, f)\precsim_{\mathop{\mathrm{Ar}}} (\bar B', g)$. By Lemma [Lemma 79](#lem:isolatedthroughembeddings){reference-type="ref" reference="lem:isolatedthroughembeddings"}[\[lem:isolatedthroughembeddings-2\]](#lem:isolatedthroughembeddings-2){reference-type="ref" reference="lem:isolatedthroughembeddings-2"}, $\varphi \restriction (I\Sigma_{(\bar B, f)} \cap \mathop{\mathrm{Int}}\bar{B})$ is an embedding of $I\Sigma_{(\bar B, f)} \cap \mathop{\mathrm{Int}}\bar{B}$ into $I\Sigma_{(\bar B', g)}$, and arguing as in the proof of Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"} we can observe that $\varphi(I\Sigma_{(\bar B, f)} \cap \mathop{\mathrm{Int}}\bar{B})$ is a convex subset of $I\Sigma_{(\bar B', g)}$ with respect to $\leq_g$ because $\mathop{\mathrm{Im}}f \cap \mathop{\mathrm{Int}}\bar{B}$, which is homeomorphic to $(0,1)$, is connected. As already noticed, the embedding $\varphi$ is either order-preserving or order-reversing (Remark [Remark 74](#rmk:orientation){reference-type="ref" reference="rmk:orientation"}[\[rmk:orientation-2\]](#rmk:orientation-2){reference-type="ref" reference="rmk:orientation-2"}): thus $\varphi \restriction (I\Sigma_{(\bar B, f)} \cap \mathop{\mathrm{Int}}\bar{B})$ witnesses $K \trianglelefteq L_{(\bar B', g)}$ in the former case, and $K \trianglelefteq (L_{(\bar B', g)})^*$ in the latter. ◻ *Remark 84*. In the special case where $(\bar B, f)$ is of the form $F(L)$ for some $L \in \mathop{\mathrm{\mathsf{LO}}}$, then all its isolated singular points belong to $\mathop{\mathrm{Int}}\bar{B}$ and $I\Sigma_{F(L)} \cong L$. Thus in this case Lemma [Lemma 83](#lem:froarcstoLin){reference-type="ref" reference="lem:froarcstoLin"} reads as follows: For every $L \in \mathop{\mathrm{\mathsf{LO}}}$ and $(\bar B', g) \in \mathop{\mathrm{Ar}}$ with $F(L) \precsim_{\mathop{\mathrm{Ar}}} (\bar B', g)$, either $L \trianglelefteq L_{(\bar B', g)}$ or $L \trianglelefteq (L_{(\bar B', g)})^*$. Lemma [Lemma 83](#lem:froarcstoLin){reference-type="ref" reference="lem:froarcstoLin"} allows us to prove analogues of Propositions [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"} and [Proposition 34](#prop:dom_fam){reference-type="ref" reference="prop:dom_fam"}. **Theorem 85**. *$\mathfrak{b} (\precsim_{\mathop{\mathrm{Ar}}}) = \aleph_1$ and $\mathfrak{d} (\precsim_{\mathop{\mathrm{Ar}}}) = 2^{\aleph_0}$.* *Proof.* We begin with the unbounding number. First notice that $\mathfrak{b} (\precsim_{\mathop{\mathrm{Ar}}}) > \aleph_0$ because given a countable family of proper arcs $\{ (\bar B_i, f_i) \mid i \in \mathbb{N}\}$, their infinite sum $\bigoplus_\mathbb{N}\, (\bar B_i, f_i)$ is a $\precsim_{\mathop{\mathrm{Ar}}}$-upper bound for them. To show the existence of an $\precsim_{\mathop{\mathrm{Ar}}}$-unbounded family of arcs of size $\aleph_1$ we use Proposition [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"} as follows. Let $F \colon \mathop{\mathrm{\mathsf{LO}}}\to \mathop{\mathrm{Ar}}$ be the reduction introduced in [\[eq:FfromLOtoA\]](#eq:FfromLOtoA){reference-type="eqref" reference="eq:FfromLOtoA"}, and consider the family $\{ F(\boldsymbol{\alpha}) \mid \omega \leq \alpha < \omega_1 \}$. It is strictly $\precsim_{\mathop{\mathrm{Ar}}}$-increasing by Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"}. Suppose towards a contradiction that there is $(\bar B, f) \in \mathop{\mathrm{Ar}}$ such that $F(\boldsymbol{\alpha}) \precsim_{\mathop{\mathrm{Ar}}} (\bar B, f)$ for all $\alpha < \omega_1$. Then $I\Sigma_{(\bar B, f)}$ would be infinite and thus the linear order $L = L_{(\bar B, f)}$ would be, up to isomorphism, an element of $\mathop{\mathrm{\mathsf{LO}}}$. By Lemma [Lemma 83](#lem:froarcstoLin){reference-type="ref" reference="lem:froarcstoLin"} and Remark [Remark 84](#rmk:froarcstoLin){reference-type="ref" reference="rmk:froarcstoLin"}, this would lead to the fact that $L+L^*$ is a $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-upper bound for $\mathop{\mathrm{\mathsf{WO}}}$, contradicting Proposition [Proposition 29](#prop_WO){reference-type="ref" reference="prop_WO"}. We now deal with the dominating number. Consider once again the $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-antichain $\mathcal{A} = \{ L_S \mid S \subseteq \mathbb{N}\}$, where $L_S = \eta_{f_S}$ is as in the proof of Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}[\[prop:basisforcvx-1\]](#prop:basisforcvx-1){reference-type="ref" reference="prop:basisforcvx-1"}, and notice that by the usual back-and-forth argument $L_S \cong (L_S)^*$. We first prove the analogue of Claim [Claim 1](#claim:everylobelongstoantichain){reference-type="ref" reference="claim:everylobelongstoantichain"}. **Claim 3**. For every proper arc $(\bar B, f) \in \mathop{\mathrm{Ar}}$, the collection $$\{ F(L_S) \mid {F(L_S) \precsim_{\mathop{\mathrm{Ar}}} (\bar B, f)} \}$$ is countable. *Proof of the Claim.* By Lemma [Lemma 83](#lem:froarcstoLin){reference-type="ref" reference="lem:froarcstoLin"}, Remark [Remark 84](#rmk:froarcstoLin){reference-type="ref" reference="rmk:froarcstoLin"}, and $L_S \cong (L_S)^*$ we have that $$\{ L_S \in \mathcal{A} \mid F(L_S) \precsim_{\mathop{\mathrm{Ar}}} (\bar B, f) \} \subseteq \{ L_S \in \mathcal{A} \mid {L_S \trianglelefteq L_{(\bar B, f)}} \} .$$ If $L_{(\bar B, f)}$ is finite, then the latter set is empty and so is the set in the claim; if instead $L_{(\bar B, f)}$ is infinite then, up to isomorphism, it is a member of $\mathop{\mathrm{\mathsf{LO}}}$, and thus the result easily follows from Claim [Claim 1](#claim:everylobelongstoantichain){reference-type="ref" reference="claim:everylobelongstoantichain"}. ◻ The proof of the theorem can now be completed using the same argument as the one used in the proof of Proposition [Proposition 34](#prop:dom_fam){reference-type="ref" reference="prop:dom_fam"}: every element of a $\precsim_{\mathop{\mathrm{Ar}}}$-dominating family has only countably many proper arcs of the form $F(L_S)$ below it, and since by Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"} there are $2^{\aleph_0}$-many such arcs the dominating family must have size $2^{\aleph_0}$ too. ◻ As in the case of linear orders, one can then derive the following analogue of Corollary [Corollary 31](#cor_no_max){reference-type="ref" reference="cor_no_max"}. However, the proof is slightly more delicate. **Corollary 86**. *Every proper arc $(\bar B, f)$ is the bottom of an $\precsim_{\mathop{\mathrm{Ar}}}$-unbounded chain of length $\omega_1$.* *Proof.* Consider the sequence of proper arcs $((\bar B_\alpha, f_\alpha)))_{\alpha < \omega_1}$ where $(\bar B_0, f_0) = (\bar B, f)$ and $(\bar B_\alpha, f_\alpha) = (\bar B, f) \oplus F(\boldsymbol{\omega+\alpha})$ if $\alpha \geq 1$. For every $\alpha \leq \beta < \omega_1$ we have $(\bar B_\alpha, f_\alpha) \precsim_{\mathop{\mathrm{Ar}}} (\bar B_\beta, f_\beta)$, and if $\alpha > 0$ we also have $F(\boldsymbol{\omega+\alpha}) \precsim_{\mathop{\mathrm{Ar}}} (\bar B_\alpha, f_\alpha)$. By (the proof of) Theorem [Theorem 85](#thm:umboundedanddominatingforarcs){reference-type="ref" reference="thm:umboundedanddominatingforarcs"}, this implies that the sequence is $\precsim_{\mathop{\mathrm{Ar}}}$-unbounded. Moreover, for every $\alpha < \omega_1$ there is $\beta > \alpha$ such that $(\bar B_\beta, f_\beta) \not\precsim_{\mathop{\mathrm{Ar}}} (\bar B_\alpha, f_\alpha)$ (and hence $(\bar B_\alpha, f_\alpha) \prec_{\mathop{\mathrm{Ar}}} (\bar B_\beta, f_\beta)$), as otherwise $(\bar B_\alpha, f_\alpha)$ would be an upper bound for the sequence $((\bar B_\alpha, f_\alpha))_{\alpha < \omega_1}$. It follows that we can extract from the latter a strictly $\precsim_{\mathop{\mathrm{Ar}}}$-increasing subsequence of length $\omega_1$ with $(\bar B, f)$ as a first element: since such a subsequence is $\precsim_{\mathop{\mathrm{Ar}}}$-cofinal in $((\bar B_\alpha, f_\alpha))_{\alpha < \omega_1}$, it is $\precsim_{\mathop{\mathrm{Ar}}}$-unbounded as well and the proof is complete. ◻ We now move to the possible generalizations of Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}, i.e. we discuss minimal elements and bases for the relation $\precsim_{\mathop{\mathrm{Ar}}}$. If we consider only tame proper arcs, which form a $\precsim_{\mathop{\mathrm{Ar}}}$-downward closed subclass of the collection of all proper arcs, then the situation is pretty clear: the trivial arc $I_{\mathop{\mathrm{Ar}}}$ is the $\precsim_{\mathop{\mathrm{Ar}}}$-minimum within this class. Call **prime arc** any proper arc of the form $(\bar B_K, f_K)$ for $K$ a prime knot: then one can observe that prime arcs are $\precsim_{\mathop{\mathrm{Ar}}}$-minimal above $I_{\mathop{\mathrm{Ar}}}$ (and are the unique such). Indeed, assume that $(\bar B, f)$ is a prime arc and that $(\bar B', g) \precsim_{\mathop{\mathrm{Ar}}} (\bar B, f)$ for some $(\bar B', g) \in \mathop{\mathrm{Ar}}$. Since $(\bar B, f)$ is tame, without loss of generality we can assume that $\bar{B}' \subseteq \mathop{\mathrm{Int}}\bar{B}$ and that both $\bar B$ and $\bar B'$ are smoothly embedded as well as that the witness $\varphi$ for $(\bar B', g) \precsim_{\mathop{\mathrm{Ar}}} (\bar B, f)$ is smooth. Let $K_1 = K_{(\bar B', g)}$, and let $K_2$ be the knot obtained from the remainder $f \setminus \mathop{\mathrm{Int}}\varphi[\bar{B}']$ by connecting $g(0)$ and $g(1)$ with a simple curve lying on $\partial \varphi[\bar B']$ and the extrema $f(0)$ and $f(1)$ with a simple curve on $\partial \bar{B}$. By construction, the prime knot used to construct $(\bar B, f)$ is the sum of $K_1$ and $K_2$, thus one of $K_1$ and $K_2$ is trivial. In the former case $(\bar B', g) \equiv_{\mathop{\mathrm{Ar}}} I_{\mathop{\mathrm{Ar}}}$, while in the latter $(\bar B', g) \equiv_{\mathop{\mathrm{Ar}}} (\bar B, f)$. Prime arcs play the same role in the realm of tame proper arcs as prime knots do in the realm of tame knots: every tame proper arc is of the form $\bigoplus_{i \leq n} \, (\bar B_i^p, f_i^p)$ for some (unique, up to permutations) sequence of prime arcs $(\bar B_i^p, f_i^p)$. This has a number of consequences on the structure of *nontrivial* tame proper arcs under $\precsim_{\mathop{\mathrm{Ar}}}$: There are no infinite descending chains. Since up to equivalence there are only countably many tame proper arcs, and since there are infinitely many prime arcs (consider e.g. the prime arcs obtained from the $(p,q)$ torus knots, where $p,q > 1$), it follows that the collection of prime arcs constitutes a countably infinite antichain basis. In particular, there are no finite bases. If $(\bar B_i^p, f_i^p)$, for $i \in \mathbb{N}$, is an enumeration of the prime arcs, then $(\bigoplus_{i \leq n} \, (\bar B_i^p, f_i^p))_{n \in \mathbb{N}}$ is an unbounded $\omega$-chain. In particular, there is no $\precsim_{\mathop{\mathrm{Ar}}}$-maximal tame proper arc, and the unbounding number of $\precsim_{\mathop{\mathrm{Ar}}}$ restricted to tame proper arcs is $\aleph_0$. Every dominating family is infinite: below every tame proper arc there are only finitely many of the infinitely many pairwise $\precsim_{\mathop{\mathrm{Ar}}}$-incomparable prime arcs, thus no finite family can be dominating with respect to $\precsim_{\mathop{\mathrm{Ar}}}$. Hence the dominating number of $\precsim_{\mathop{\mathrm{Ar}}}$ restricted to tame proper arcs is $\aleph_0$. Having obtained the desired information in the realm of tame proper arcs, it is now natural to move to the wild side and consider the restriction $\precsim_{\mathop{\mathrm{WAr}}}$ of $\precsim_{\mathop{\mathrm{Ar}}}$ to the collection $\mathop{\mathrm{WAr}}$ of wild arcs. By $\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}$-minimality of $\eta_{f_S}$, Lemma [Lemma 83](#lem:froarcstoLin){reference-type="ref" reference="lem:froarcstoLin"} and Remark [Remark 84](#rmk:froarcstoLin){reference-type="ref" reference="rmk:froarcstoLin"}, one may be tempted to conjecture that the proper arcs $F(\eta_{f_S})$ used in the proof of Theorem [Theorem 85](#thm:umboundedanddominatingforarcs){reference-type="ref" reference="thm:umboundedanddominatingforarcs"} are $\precsim_{\mathop{\mathrm{WAr}}}$-minimal. That is not quite true, as the arc $(\bar B^*, f^*) = \bigoplus_\mathbb{N}\ (\bar B_i, f_i)$ used to define the reduction $F = F_{(\bar B^*, f^*)} \colon \mathop{\mathrm{\mathsf{LO}}}\to \mathop{\mathrm{Ar}}$ from [\[eq:FfromLOtoA\]](#eq:FfromLOtoA){reference-type="eqref" reference="eq:FfromLOtoA"} is such that $(\bar B^*, f^*) \prec_{\mathop{\mathrm{WAr}}} F(\eta_{f_S})$, and moreover the proper arc $(\bar B^\partial, f^\partial) = \bigoplus^\partial_\mathbb{N}\, (\bar B_i, f_i)$ is such that $(\bar B^\partial, f^\partial) \prec_{\mathop{\mathrm{WAr}}} (\bar B^*, f^*) \prec_{\mathop{\mathrm{WAr}}} F(\eta_{f_S})$. However, the following lemma allows us to obtain useful information on the $\precsim_{\mathop{\mathrm{WAr}}}$-predecessors of $F(\eta_{f_S})$. **Lemma 87**. *Let $\{ (\bar B_i^p, f_i^p) \mid i \in \mathbb{N}\}$ be a family of (oriented) prime arcs, and let $(\bar B^*, f^*) = \bigoplus_{i \in \mathbb{N}}\, (\bar B_i^p, f_i^p)$ and $(\bar B^\partial, f^\partial) = \bigoplus^\partial_{i \in \mathbb{N}} \, (\bar B_i^p, f_i^p)$. We consider the proper arc $F_{(\bar B^*, f^*)}(L_S)$ for some $S \subseteq \mathbb{N}$, where $L_S = \eta_{f_S}$ is as in the proof of Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}[\[prop:basisforcvx-1\]](#prop:basisforcvx-1){reference-type="ref" reference="prop:basisforcvx-1"}.* 1. *[\[lem:fewpredecessors-1\]]{#lem:fewpredecessors-1 label="lem:fewpredecessors-1"} If $(\bar B', g)$ is a prime arc, then $(\bar B', g) \precsim_{\mathop{\mathrm{Ar}}} (\bar B^\partial, f^\partial)$ if and only if there is $\bar\imath \in \mathbb{N}$ such that $(\bar B', g) \equiv_{\mathop{\mathrm{Ar}}} (\bar B_{\bar\imath}^p, f_{\bar\imath}^p)$. The same is true if $(\bar B^\partial, f^\partial)$ is replaced by $F_{(\bar B^*, f^*)}(L_S)$.* *Let now $(\bar B, f)$ be an arbitrary wild proper arc. Then:* 1. *[\[lem:fewpredecessors-2\]]{#lem:fewpredecessors-2 label="lem:fewpredecessors-2"} $(\bar B, f) \precsim_{\mathop{\mathrm{WAr}}} (\bar B^\partial, f^\partial)$ if and only if $(\bar B, f) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{j \in A}^\partial \, (\bar B_j^p, f_j^p)$ for some infinite set $A \subseteq \mathbb{N}$.* 2. *[\[lem:fewpredecessors-4\]]{#lem:fewpredecessors-4 label="lem:fewpredecessors-4"} If $(\bar B, f) \precsim_{\mathop{\mathrm{WAr}}} F_{(\bar B^*, f^*)}(L_S)$, then there is $\bar B' \subseteq \bar B$ such that $(\bar B', f \cap \bar B') \in \mathop{\mathrm{WAr}}$ and $(\bar B', f \cap \bar B') \precsim_{\mathop{\mathrm{WAr}}} (\bar B^\partial, f^\partial)$.* *Proof.* [\[lem:fewpredecessors-1\]](#lem:fewpredecessors-1){reference-type="ref" reference="lem:fewpredecessors-1"} One direction is obvious. For the other direction, assume that $(\bar B', g) \precsim_{\mathop{\mathrm{Ar}}} (\bar B^\partial, f^\partial)$. Recall that the ambient space $\bar B^\partial$ of $\bigoplus_{i \in \mathbb{N}}^\partial \, (\bar B_i^p, f_i^p)$ is the cube $[-1,1]^3$, and that its only singularity is the point $(1,0,0)$. By the way we defined infinite sums, without loss of generality we may assume that $\bar{B}_i^p = [1-2^{-i},1-2^{-(i+1)}] \times [-1,1]^2$ and that $f_i^p(0) = (1-2^{-i},0,0)$, and $f_i^p(1) = (1-2^{-(i+1)},0,0)$ for all $i \in \mathbb{N}$. Let $\varphi \colon \bar{B}' \to [-1,1]^3$ witness $(\bar B', g) \precsim_{\mathop{\mathrm{Ar}}} (\bar B^\partial, f^\partial)$, and notice that since $(\bar B_0^p, f_0^p)$ is tame we may assume $\mathop{\mathrm{Im}}\varphi \subseteq [0,1] \times [-1,1]^2$. Let $I = \{ i \in \mathbb{N}\mid \varphi(\bar{B}') \cap f_i^p \cap \mathop{\mathrm{Int}}\bar{B}_i^p \neq \emptyset \}$: it is convex (with respect to the usual order $\leq$ on $\mathbb{N}$) because $g$ is connected. Without loss of generality, we may assume that for all $i \in I$ the space $\bar B'_i = \varphi(\bar{B}') \cap \bar{B}_i^p$ is a closed topological 3-ball. Consider the (tame) proper arcs $(\bar B'_i, f_i \cap \bar{B}'_i)$, which by construction are such that either $(\bar B', g) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{i \in I} \, (\bar B'_i, f_i \cap \bar B'_i)$ if $I$ is finite, or $(\bar B', g) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus^\partial_{i \in I} \, (\bar B'_i, f_i \cap \bar B'_i)$ if $I$ is infinite. Moreover each $(\bar B'_i, f_i \cap \bar{B}'_i)$ is either trivial or equivalent to the corresponding $(\bar B_i^p, f_i^p)$ because $(\bar B'_i, f_i \cap \bar{B}'_i) \precsim (\bar B_i^p, f_i^p)$ and the latter is prime. If all the $(\bar B'_i, f_i \cap \bar{B}'_i)$'s were trivial, then $(\bar B', g)$ would be trivial too, a contradiction. Let $\bar \imath \in I$ be such that $(\bar B'_{\bar\imath}, f_{\bar\imath} \cap \bar{B}'_{\bar\imath})$ is not trivial: then $(\bar B'_{\bar\imath}, f_{\bar\imath} \cap \bar{B}'_{\bar\imath}) \equiv_{\mathop{\mathrm{Ar}}} (\bar B_{\bar\imath}^p, f_{\bar\imath}^p)$, and since $(\bar B', g)$ is prime and $\varphi^{-1} \restriction \bar B'_{\bar\imath}$ witnesses $(\bar B'_{\bar\imath}, f_{\bar\imath} \cap \bar{B}'_{\bar\imath}) \precsim_{\mathop{\mathrm{Ar}}} (\bar B', g)$ it follows that $(\bar B_{\bar\imath}^p, f_{\bar\imath}^p) \equiv_{\mathop{\mathrm{Ar}}} (\bar B', g)$, as desired. Suppose now that $(\bar B', g) \precsim_{Ar} F_{(\bar B^*, f^*)}(L_S)$ via some $\varphi$. Recall the notation used in the proof of Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"} and in the discussion preceding it. Since $(\bar B', g)$ is tame and not trivial, $\varphi (g)$ is tame and cannot be contained in $[h_{L_S}(m), h_{L_S}(m) +2r_{L_S}(m)] \times [-1,1]^2$ for any $m \in L_S$; therefore $\varphi (g)$ must be contained either in $[h_{L_S}(n)-2 r_{L_S}(n), h_{L_S}(n) - \varepsilon] \times [-1,1]^2$ or in $[h_{L_S}(m), h_{L_S}(n) - \varepsilon] \times [-1,1]^2$ for some consecutive $m,n \in L_S$ and small enough $\varepsilon > 0$. However, since the part on the right of the singularity $h_{L_S}(m)$ is trivial and $(\bar B_0^p, f_0^p)$ is tame, we can actually assume that we are always in the first case and that $\mathop{\mathrm{Im}}\varphi \subseteq [h_{L_S}(n)-2 r_{L_S}(n), h_{L_S}(n)] \times [-1,1]^2$. Since the subarc of $F_{(\bar B^*, f^*)}(L_S)$ determined by the latter set is equivalent to $\bigoplus_{i \in \mathbb{N}}^{\partial} \, (\bar B_i^p, f_i^p)$, we are done by the first part. [\[lem:fewpredecessors-2\]](#lem:fewpredecessors-2){reference-type="ref" reference="lem:fewpredecessors-2"} Let $\varphi \colon \bar{B} \to [-1,1]^3$ witness $(\bar B, f) \precsim_{\mathop{\mathrm{WAr}}} (\bar B^\partial, f^\partial)$. Being wild and $\precsim_{\mathop{\mathrm{WAr}}}$-below an arc with only one singularity, the proper arc $(\bar B, f)$ has a unique singularity $x \in \bar B$: clearly, $\varphi(x) = (1,0,0)$ by Lemma [Lemma 79](#lem:isolatedthroughembeddings){reference-type="ref" reference="lem:isolatedthroughembeddings"}[\[lem:isolatedthroughembeddings-1\]](#lem:isolatedthroughembeddings-1){reference-type="ref" reference="lem:isolatedthroughembeddings-1"} and thus, necessarily, $x \in \partial \bar{B}$. As before, set $I = \{ i \in \mathbb{N}\mid \varphi(\bar{B}) \cap f_i^p \cap \mathop{\mathrm{Int}}\bar{B}_i^p \neq \emptyset \}$: now, we know that $I$ is a final segment of $(\mathbb{N}, {\leq})$ because $\varphi(x) = (1,0,0)$. We can assume that $\mathop{\mathrm{Im}}\varphi \setminus (\{ 1 \} \times [-1,1]^2) \subseteq \bigcup_{i \in I} \bar{B}_i^p$ and that for all $i \in I$ the space $\bar B'_i = \varphi(\bar{B}) \cap \bar{B}_i^p$ is a closed topological 3-ball. Then $\varphi$ witnesses $(\bar B, f) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{i \in I}^\partial \, (\bar B'_i, f_i^p \cap \bar{B}'_i)$. Each of the proper arcs $(\bar B'_i, f_i^p \cap \bar{B}'_i)$, being a subarc of the prime arc $(\bar B_i^p, f_i^p)$, is either trivial or equivalent to it: set $$A = \{ i \in I \mid (\bar B'_i, f_i^p \cap \bar{B}'_i) \equiv_{\mathop{\mathrm{Ar}}} (\bar B_i^p, f_i^p) \}.$$ The set $A$ is infinite because otherwise $\bigoplus_{i \in I}^\partial \, (\bar B'_i, f_i^p \cap \bar{B}'_i)$, and hence also $(\bar B, f)$, would be tame. Moreover, each of the $(\bar B'_i, f_i^p \cap \bar{B}'_i)$ is tame, so the trivial arcs $(\bar B'_i, f_i^p \cap \bar B'_i)$ occurring in the sequence can be "absorbed" by the next $(\bar B'_j, f_j^p \cap \bar{B}'_j)$ with $j \in A$. Therefore $$(\bar B, f) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{i \in I}^\partial \, (\bar B'_i, f_i^p \cap \bar{B}'_i) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{j \in A}^\partial \, (\bar B'_j, f_j^p \cap \bar{B}'_j) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{j \in A}^\partial \, (\bar B_j^p, f_j^p).$$ Conversely, assume that $(\bar B, f) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus^\partial_{j \in A} \, (\bar B_j^p, f_j^p)$ for some infinite $A \subseteq \mathbb{N}$. For each $i \in \mathbb{N}$, set $(\bar B'_i, f'_i) = (\bar B_i^p, f_i^p)$ if $i \in A$ and $(\bar B'_i, f'_i) = I_{\mathop{\mathrm{Ar}}}$ if $i \notin A$: since all the proper arcs $(\bar B_i^p, f_i^p)$ are tame, we get $\bigoplus^\partial_{j \in A} \, (\bar B_j^p, f_j^p) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus^\partial_{i \in \mathbb{N}} \, (\bar B'_i, f'_i)$, so it is enough to show that the latter is a subarc of $\bigoplus^\partial_{i \in \mathbb{N}} \, (\bar B_i^p, f_i^p)$. Without loss of generality, $\bar B_i = \bar B'_i = [1-2^{-i},1-2^{-(i+1)}] \times [-2^{-i},2^{-i}]^2$, $f_i(0) = f'_i(0) = (1-2^{-i},0,0)$, and $f_i(1) = f'_i(1) = (1-2^{-(i+1)},0,0)$. We can further assume that the ambient space of $\bigoplus^\partial_{i \in \mathbb{N}} \, (\bar B'_i, f'_i)$ is the "step pyramid" $([-1,0] \times [-1,1]^2) \cup {\bigcup_{i \in \mathbb{N}} \bar{B}'_i} \cup \{ (1,0,0) \}$. For each $i \notin A$, fix a tubular neighborhood $\bar{B}''_i \subseteq \bar{B}_i^p$ of $f_i^p$, i.e. a "cylinder" of radius $\varepsilon_i$ with rotation axis given by $f_i^p$ itself --- this is possible because $(\bar B_i^p, f_i^p)$ is tame. Moreover, since each block of consecutive $i \in \mathbb{N}\setminus A$ is finite, we can assume that $\varepsilon_i = \varepsilon_{i+1}$ if $i, i+1 \notin A$. For $i \in A$ pick instead $\bar{B}''_i \subseteq \bar{B}_i^p$ so that: $(\bar B''_i, f_i^p)$ is a proper arc; $\bar{B}''_i$ intersects the left face $\{ 1-2^{-i} \} \times [-2^{-i},2^{-i}]^2$ of $\bar B_i^p$ in a disc of radius $\varepsilon_i$ centered in $(1-2^{-i},0,0)$, where $\varepsilon_i = \varepsilon_{i-1}$ if $i > 0$ and $i -1 \notin A$ and $\varepsilon_i = 2^{-i}$ otherwise; similarly, $\bar{B}''_i$ intersects the right face $\{ 1-2^{-(i+1)} \} \times [-2^{-i},2^{-i}]^2$ of $\bar B_i$ in a disc of radius $\varepsilon_i$ centered in $(1-2^{-(i+1)},0,0)$, where $\varepsilon_i = \varepsilon_{i+1}$ if $i +1 \notin A$ and $\varepsilon_i = 2^{-(i+1)}$ otherwise. Finally, let $\bar B''_{-1} \subseteq [-1,0] \times [-1,1]^2$ be such that $(\bar B''_{-1}, [-1,0] \times \{ (0,0) \})$ is a proper arc and $\bar B''_{-1}$ intersects the left face $\{ 0 \} \times [-1,1]^2$ of $\bar B_0^p$ in a disc of radius $1$ centered in the origin $(0,0,0)$. By construction, $\bar B''_{-1} \cup \bigcup_{i \in \mathbb{N}} \bar B''_i \cup \{ (1,0,0) \}$ is homeomorphic to a (closed) cone, and thus it is a closed topological 3-ball. Moreover, every $(\bar B'_i, f'_i)$ is equivalent to $(\bar B''_i, f_i^p)$ via some $\varphi_i \colon \bar B'_i \to \bar B''_i$. Fix also a homeomorphism $\varphi_{-1} \colon [-1,0] \times [-1,1]^2 \to \bar B''_{-1}$ fixing the interval $[-1,0] \times \{ (0,0)\}$, and let $\varphi_\infty$ be the identity on the singleton $(1,0,0)$. Then $\varphi = \varphi_{-1} \cup {\bigcup_{i \in \mathbb{N}} \varphi_i} \cup \varphi_\infty$ is an embedding witnessing $\bigoplus^\partial_{i \in \mathbb{N}} \, (\bar B'_i, f'_i) \precsim_{\mathop{\mathrm{Ar}}} \bigoplus^\partial_{i \in \mathbb{N}} \, (\bar B_i^p, f_i^p)$, as desired. [\[lem:fewpredecessors-4\]](#lem:fewpredecessors-4){reference-type="ref" reference="lem:fewpredecessors-4"} Let $\varphi \colon \bar B \to [-1,1]^3$ witness $(\bar B, f) \precsim_{\mathop{\mathrm{Ar}}} F_{(\bar B^*, f^*)}(L_S)$. We claim that there is $x \in \Sigma_{(\bar B, f)}$ such that $\varphi(x) \in I\Sigma_{F_{(\bar B^*, f^*)}(L_S)}$. Pick any $x \in \Sigma_{(\bar B, f)}$, so that $\varphi(x) \in \Sigma_{F_{(\bar B^*, f^*)}(L_S)}$ as well by Lemma [Lemma 79](#lem:isolatedthroughembeddings){reference-type="ref" reference="lem:isolatedthroughembeddings"}[\[lem:isolatedthroughembeddings-1\]](#lem:isolatedthroughembeddings-1){reference-type="ref" reference="lem:isolatedthroughembeddings-1"}. If $\varphi(x) \notin I\Sigma_{F_{(\bar B^*, f^*)}(L_S)}$, we use the fact that by construction every singularity $y \in \Sigma_{F_{(\bar B^*, f^*)}(L_S)} \setminus I\Sigma_{F_{(\bar B^*, f^*)}(L_S)}$ is a limit of isolated singularities *from both sides* (unless $y \in \partial \bar B^*$, in which case there is only one side available), and hence for all $\bar B' \subseteq [-1,1]^3$ with $y \in \bar B'$ and $(\bar B', f_{L_S} \cap \bar B') \in \mathop{\mathrm{Ar}}$ the set $I\Sigma_{F_{(\bar B^*, f^*)}(L_S)} \cap \bar B'$ is infinite. Applying this to $\bar B' = \varphi(\bar B)$ and $y = \varphi(x)$, we get that there is some (in fact, infinitely many) $y' \in I\Sigma_{F_{(\bar B^*, f^*)}(L_S)} \cap \mathop{\mathrm{Int}}\bar B'$: by Lemma [Lemma 79](#lem:isolatedthroughembeddings){reference-type="ref" reference="lem:isolatedthroughembeddings"}, replacing $x$ with $\varphi^{-1}(y')$ we are done. Using the same notation as in the proof of Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"}, let $n \in L_S$ be such that $\varphi(x) = (h_{L_S}(n),0,0)$. Without loss of generality, we may assume that $\bar B'' = \mathop{\mathrm{Im}}\varphi \cap [h_{L_S}(n)-2 r_{L_S}(n), h_{L_S}(n)] \times [-1,1]^2$ is a closed topological 3-ball. Moreover, $\varphi(x) = (h_{L_S}(n),0,0) \in \Sigma_{(\bar B'', f_{L_S} \cap \bar B'')}$ because otherwise $x$ would not be a singularity of $(\bar B, f)$ (here we use the fact that on the right of $(h_{L_S}(n),0,0)$ there is a trivial arc), thus $(\bar B'', f_{L_S} \cap \bar B'') \in \mathop{\mathrm{WAr}}$. Since the subarc of $F_{(\bar B^*, f^*)}(L_S)$ determined by $[h_{L_S}(n)-2 r_{L_S}(n), h_{L_S}(n)] \times [-1,1]^2$ is equivalent to $(\bar B^\partial, f^\partial)$, setting $B' = \varphi^{-1}(B'')$ we are done. ◻ We are not able to get a full analogue of Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}, but Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"} allows us to get a similar, although slightly weaker, result. **Theorem 88**. *[\[thm:basisforarcs-1\]]{#thm:basisforarcs-1 label="thm:basisforarcs-1"} There are infinitely many $\precsim_{\mathop{\mathrm{WAr}}}$-incomparable $\precsim_{\mathop{\mathrm{WAr}}}$-minimal elements in $\mathop{\mathrm{WAr}}$.* *[\[thm:basisforarcs-2\]]{#thm:basisforarcs-2 label="thm:basisforarcs-2"} There is a strictly $\precsim_{\mathop{\mathrm{WAr}}}$-decreasing $\omega$-sequence in $\mathop{\mathrm{WAr}}$ which is not $\precsim_{\mathop{\mathrm{WAr}}}$-bounded from below.* *[\[thm:basisforarcs-3\]]{#thm:basisforarcs-3 label="thm:basisforarcs-3"} No basis for $\precsim_{\mathop{\mathrm{WAr}}}$ has size smaller than $2^{\aleph_0}$.* *Proof.* Fix an enumeration without repetitions $\{ (\bar B_i^p, f_i^p) \mid i \in \mathbb{N}\}$ of all prime arcs. [\[thm:basisforarcs-1\]](#thm:basisforarcs-1){reference-type="ref" reference="thm:basisforarcs-1"} For each $k \in \mathbb{N}$ set $(\bar B'_k, g_k) = \bigoplus^\partial_\mathbb{N}\, (\bar B_k^p, f_k^p)$. Every $(\bar B'_k, g_k)$ is $\precsim_{\mathop{\mathrm{WAr}}}$-minimal by Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"}[\[lem:fewpredecessors-2\]](#lem:fewpredecessors-2){reference-type="ref" reference="lem:fewpredecessors-2"} (and the fact that all arcs in the infinitary sum are the same), and if $k \neq k'$ then $(\bar B'_k, g_k) \not\precsim_{\mathop{\mathrm{WAr}}} (\bar B'_{k'}, g_{k'})$ because $(\bar B_k^p, f_k^p) \precsim_{\mathop{\mathrm{Ar}}} (\bar B'_k, g_k)$ but $(\bar B_k^p, f_k^p) \not\precsim_{\mathop{\mathrm{Ar}}} (\bar B'_{k'}, g_{k'})$ by Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"}[\[lem:fewpredecessors-1\]](#lem:fewpredecessors-1){reference-type="ref" reference="lem:fewpredecessors-1"}. [\[thm:basisforarcs-2\]](#thm:basisforarcs-2){reference-type="ref" reference="thm:basisforarcs-2"} Now let $(\bar B'_k, g_k) = \bigoplus_{i \geq k}^\partial \, (\bar B_i^p, f_i^p)$. By parts [\[lem:fewpredecessors-1\]](#lem:fewpredecessors-1){reference-type="ref" reference="lem:fewpredecessors-1"} and [\[lem:fewpredecessors-2\]](#lem:fewpredecessors-2){reference-type="ref" reference="lem:fewpredecessors-2"} of Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"}, if $k < k'$ then $(\bar B'_{k'}, g_{k'}) \prec_{\mathop{\mathrm{Ar}}} (\bar B'_{k}, g_{k})$. Moreover, by Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"}[\[lem:fewpredecessors-2\]](#lem:fewpredecessors-2){reference-type="ref" reference="lem:fewpredecessors-2"} if $(\bar B, f) \in \mathop{\mathrm{WAr}}$ is such that $(\bar B, f) \precsim_{\mathop{\mathrm{WAr}}} (\bar B'_0, g_0)$ then $(\bar B, f) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{j \in A}^\partial \, (\bar B_j^p, f_j^p)$, for some infinite $A \subseteq \mathbb{N}$. Let $k = \min A$: since $(\bar B_k^p, f_k^p) \precsim_{\mathop{\mathrm{WAr}}} \bigoplus_{j \in A}^\partial \, (\bar B_j^p, f_j^p)$ but by Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"}[\[lem:fewpredecessors-1\]](#lem:fewpredecessors-1){reference-type="ref" reference="lem:fewpredecessors-1"} $(\bar B_k^p, f_k^p) \not\precsim_{\mathop{\mathrm{WAr}}} (\bar B'_{k+1}, g_{k+1})$, we have $(\bar B, f) \not\precsim_{\mathop{\mathrm{WAr}}} (\bar B'_{k+1}, g_{k+1})$. Thus the chain formed by the proper arcs $(\bar B'_k, g_k)$ is as required. [\[thm:basisforarcs-3\]](#thm:basisforarcs-3){reference-type="ref" reference="thm:basisforarcs-3"} Let $\{ A_x \mid x \in 2^{\mathbb{N}} \}$ be a family of infinite sets $A_x \subseteq \mathbb{N}$ such that $A_x \cap A_y$ is finite for all distinct $x,y \in 2^{\mathbb{N}}$. (For the sake of definiteness, set $A_x = \{ h(x \restriction n) \mid n \in \mathbb{N}\}$, where $h$ is a bijection from all finite binary sequences to the natural numbers.) Fix a basis $\mathcal{B}$ for $\precsim_{\mathop{\mathrm{WAr}}}$. Then for every $\bigoplus_{j \in A_x}^\partial \, (\bar B_j^p, f_j^p)$ there is some $(\bar B_x, f_x) \in \mathcal{B}$ such that $(\bar B_x, f_x) \precsim_{\mathop{\mathrm{WAr}}} \bigoplus_{j \in A_x}^\partial \, (\bar B_j^p, f_j^p)$. By Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"}[\[lem:fewpredecessors-2\]](#lem:fewpredecessors-2){reference-type="ref" reference="lem:fewpredecessors-2"}, $(\bar B_x, f_x) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{j \in A'_x}^\partial \, (\bar B_j^p, f_j^p)$ for some infinite $A'_x \subseteq A_x$. If there were distinct $x,y \in 2^{\mathbb{N}}$ such that $(\bar B_x, f_x) \equiv_{\mathop{\mathrm{Ar}}} (\bar B_y, f_y)$, then we would get $A'_x= A'_y$ by Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"}[\[lem:fewpredecessors-1\]](#lem:fewpredecessors-1){reference-type="ref" reference="lem:fewpredecessors-1"}, and thus $A'_x \subseteq A_x \cap A_y$, which is impossible because $A'_x$ is infinite. Thus all the proper arcs $(\bar B_x, f_x) \in \mathcal{B}$ are distinct, and thus $|\mathcal{B}| \geq 2^{\aleph_0}$, as desired. ◻ It is plausible that part [\[thm:basisforarcs-1\]](#thm:basisforarcs-1){reference-type="ref" reference="thm:basisforarcs-1"} of the previous theorem can be improved by showing the existence of $2^{\aleph_0}$-many $\precsim_{\mathop{\mathrm{WAr}}}$-incomparable $\precsim_{\mathop{\mathrm{WAr}}}$-minimal elements in $\mathop{\mathrm{WAr}}$. To this end, one could consider proper arcs that are singular everywhere and hence are very different from the arcs constructed in this paper. Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"} is also sufficient to recover an analogue of Proposition [Proposition 33](#prop:everylobelongstoantichain){reference-type="ref" reference="prop:everylobelongstoantichain"} for proper arcs. **Theorem 89**. *Every $\precsim_{\mathop{\mathrm{WAr}}}$-antichain is contained in a $\precsim_{\mathop{\mathrm{WAr}}}$-antichain of size $2^{\aleph_0}$. In particular, there are no maximal $\precsim_{\mathop{\mathrm{WAr}}}$-antichains of size smaller than $2^{\aleph_0}$, and every $(\bar B, f) \in \mathop{\mathrm{WAr}}$ belongs to a $\precsim_{\mathop{\mathrm{WAr}}}$-antichain of size $2^{\aleph_0}$.* *Proof.* Let $\mathcal{A} = \{ (\bar B'_m, g_m) \mid m < \kappa \}$ be a $\precsim_{\mathop{\mathrm{WAr}}}$-antichain, where $\kappa < 2^{\aleph_0}$. Let $(\bar B_i^p, f_i^p)$, for $i \in \mathbb{N}$, be an enumeration without repetitions of all prime arcs, and for $S \subseteq \mathbb{N}$ let $L_S = \eta_{f_S}$ be the linear order from the proof of Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}[\[prop:basisforcvx-1\]](#prop:basisforcvx-1){reference-type="ref" reference="prop:basisforcvx-1"}. Let $\{ A_S \mid S \subseteq \mathbb{N}\}$ be a family of sets $A_S \subseteq \mathbb{N}$ such that $A_S \cap A_{S'}$ is finite for all distinct $S,S' \subseteq \mathbb{N}$. (Such a family can be constructed as in the proof of Theorem [Theorem 88](#thm:basisforarcs){reference-type="ref" reference="thm:basisforarcs"}[\[thm:basisforarcs-3\]](#thm:basisforarcs-3){reference-type="ref" reference="thm:basisforarcs-3"}.) For each $S \subseteq \mathbb{N}$, set $(\bar B^*_S, f^*_S) = \bigoplus_{j \in A_S} \, (\bar B_j^p, f_j^p)$ and $$(\bar B_S, f_S) = F_{(\bar B^*_S, f^*_S)}(L_S).$$ Let $\mathcal{B}$ be the collection of all proper arcs of the form $(\bar B_S, f_S)$ which are $\precsim_{\mathop{\mathrm{WAr}}}$-incomparable with every $(\bar B'_m, g_m) \in \mathcal{A}$. **Claim 4**. $|\mathcal{B}| = 2^{\aleph_0}$. *Proof of the Claim.* By the proof of Claim [Claim 3](#claim:everylobelongstoantichainforarcs){reference-type="ref" reference="claim:everylobelongstoantichainforarcs"} and $\kappa < 2^{\aleph_0}$ there are $2^{\aleph_0}$-many proper arcs $(\bar B_S, f_S)$ such that $(\bar B_S, f_S) \not\precsim_{\mathop{\mathrm{WAr}}} (\bar B'_m, g_m)$ for all $m < \kappa$. On the other hand, we claim that there are at most $\kappa$-many proper arcs $(\bar B_S, f_S)$ such that $(\bar B'_m, g_m) \precsim_{\mathop{\mathrm{WAr}}} (\bar B_S, f_S)$ for some $m < \kappa$, which suffices to prove the claim. Indeed, suppose that $m < \kappa$ and $S \subseteq \mathbb{N}$ are such that $(\bar B'_m, g_m) \precsim_{\mathop{\mathrm{WAr}}} (\bar B_S, f_S)$. Then by parts [\[lem:fewpredecessors-4\]](#lem:fewpredecessors-4){reference-type="ref" reference="lem:fewpredecessors-4"} and [\[lem:fewpredecessors-2\]](#lem:fewpredecessors-2){reference-type="ref" reference="lem:fewpredecessors-2"} of Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"} there is $\bar B'' \subseteq \bar B'_m$ such that $(\bar B'', g_m \cap \bar B'') \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{j \in A}^\partial \, (\bar B_j^p, f_j^p)$ for some infinite $A \subseteq A_S$. Since if $S' \subseteq \mathbb{N}$ is different from $S$ then $A_S \cap A_{S'}$ is finite, there is $\bar \jmath \in A$ such that $\bar \jmath \notin A_{S'}$. If $(\bar B'_m, g_m) \precsim_{\mathop{\mathrm{WAr}}} (\bar B_{S'}, f_{S'})$ then $(\bar B'', g_m \cap \bar B'') \precsim_{\mathop{\mathrm{WAr}}} (\bar B_{S'}, f_{S'})$, and thus $(\bar B_{\bar \jmath}^p, f_{\bar \jmath}^p) \precsim_{\mathop{\mathrm{WAr}}} (\bar B_{S'}, f_{S'})$, which is impossible by Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"}[\[lem:fewpredecessors-1\]](#lem:fewpredecessors-1){reference-type="ref" reference="lem:fewpredecessors-1"} and the choice of $\bar \jmath$. Thus for every $m < \kappa$ there is at most one $S \subseteq \mathbb{N}$ such that $(\bar B'_m, g_m) \precsim_{\mathop{\mathrm{WAr}}} (\bar B_S, f_S)$ and we are done. ◻ By (the proof of) Proposition [Proposition 32](#prop:basisforcvx){reference-type="ref" reference="prop:basisforcvx"}, Lemma [Lemma 83](#lem:froarcstoLin){reference-type="ref" reference="lem:froarcstoLin"} and Remark [Remark 84](#rmk:froarcstoLin){reference-type="ref" reference="rmk:froarcstoLin"} (together with $L_S \cong (L_S)^*$), if $S,S' \subseteq \mathbb{N}$ are distinct then $(\bar B_S, f_S)$ and $(\bar B_{S'}, f_{S'})$ are $\precsim_{\mathop{\mathrm{WAr}}}$-incomparable. Thus $\mathcal{A} \cup \mathcal{B}$ is a $\precsim_{\mathop{\mathrm{WAr}}}$-antichain of size $2^{\aleph_0}$ containing $\mathcal{A}$, as desired. ◻ ## Knots and their classification In the proof of [@Kul17 Theorem 3.1], a function from $\mathop{\mathrm{\mathsf{LO}}}$ to $\mathop{\mathrm{Kn}}$ is defined, call it here $G$, by setting $G(L) = K_{F(\mathbf{1}+L+\mathbf{2}+\eta)}$, where $F$ is the reduction from [\[eq:FfromLOtoA\]](#eq:FfromLOtoA){reference-type="eqref" reference="eq:FfromLOtoA"}. It was claimed that $G$ was a reduction of $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ to $\equiv_{\mathop{\mathrm{Kn}}}$, but this is not the case. Indeed, notice that if $M$ is a linear order, then we have $G(\eta+\mathbf{1}+M) \equiv_{\mathop{\mathrm{Kn}}} G(M)$, essentially because $C[\mathbf{1}+\eta+\mathbf{1}+M+\mathbf{2}+\eta] \cong_{\mathop{\mathrm{\mathsf{CO}}}} C[\mathbf{1}+M+\mathbf{2}+\eta]$; however if $M$ is scattered (and in many other cases) $\eta+\mathbf{1}+M \ncong_{\mathop{\mathrm{\mathsf{LO}}}} M$. One can easily fix this problem by replacing $K_{F(\mathbf{1}+L+\mathbf{2}+\eta)}$ with $K_{F(\mathbf{1}+L+\mathbf{2}+\eta)+\bigoplus_{\mathbb{N}} \, (\bar B^*, f^*)}$, where $(\bar B^*, f^*)$ is a figure-eight arc. More precisely, we can derive [@Kul17 Theorem 3.1] from (the proof of) Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"} connecting the endpoints of each arc $F(L)$ with $\bigoplus_{\mathbb{N}} \, (\bar B^*, f^*)$ and get: **Corollary 90**. *${\cong_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\equiv_{\mathop{\mathrm{Kn}}}}$.* The next result follows from Theorem [Theorem 23](#thm:iso_co_bired_iso_lo){reference-type="ref" reference="thm:iso_co_bired_iso_lo"} and Corollary [Corollary 90](#cor4){reference-type="ref" reference="cor4"}. However, exploiting the obvious analogy between circular orders and knots one obtains a direct and more natural proof. (The reduction $F_{\mathop{\mathrm{Kn}}}$ will be used also in the proof of Theorem [Theorem 100](#thm:lowerforknots){reference-type="ref" reference="thm:lowerforknots"}). **Theorem 91**. *${\cong_{\mathop{\mathrm{\mathsf{CO}}}}} \leq_B {\equiv_{\mathop{\mathrm{Kn}}}}$.* *Proof.* We define a Borel reduction $F_{\mathop{\mathrm{Kn}}} \colon \mathop{\mathrm{\mathsf{CO}}}\to \mathop{\mathrm{Kn}}$ similar to the reduction of the proof of Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"}. Instead of embedding a linear order $L \in \mathop{\mathrm{\mathsf{LO}}}$ into $[-1,1] \subseteq \mathbb{R}$, we embed $C \in \mathop{\mathrm{\mathsf{CO}}}$ into $S^1 = \mathbb{R}\cup \{\infty\}$ by defining a sequence of intervals $(h_C(n)-2r_C(n),h_C(n)+2r_C(n))_{n \in \mathbb{N}}$ of $\mathbb{R}$ denoted by $V_n^C$, satisfying conditions analogous to [\[Vpwdisj\]](#Vpwdisj){reference-type="ref" reference="Vpwdisj"}--[\[dense\]](#dense){reference-type="ref" reference="dense"} of the proof of Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"}. As before, for every $n \in \mathbb{N}$ let $U_n^C = [h_C(n)-r_C(n),h_C(n)+r_C(n)]$, consider $\bar B_n^C = U_n^C \times [-r_C(n),r_C(n)]^2$ and define a proper arc $(\bar B_n^C,f_n^C)$ as in Figure [1](#trefoils){reference-type="ref" reference="trefoils"}. Set $f^C=\{(x,0,0) \mid (x,0,0) \notin \bigcup_{n \in \mathbb{N}} \bar B_n^C\} \cup \{\infty\}$. Finally we consider the knot $F_{\mathop{\mathrm{Kn}}}(C)$ given by $\bigcup_{n \in \mathbb{N}} f_n^C \cup f^C$. The rest of the proof is an adaptation of the proof of Theorem [Theorem 81](#thm:reduc_lo_arcs){reference-type="ref" reference="thm:reduc_lo_arcs"} to this case. ◻ *Remark 92*. Theorem 4.1 of [@Kul17] shows that a certain equivalence relation induced by a turbulent action is Borel reducible to ${\equiv_{\mathop{\mathrm{Kn}}}}$. Therefore, since $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ and $\cong_{\mathop{\mathrm{\mathsf{CO}}}}$ are induced by actions of $S_\infty$ the reductions in Corollary [Corollary 90](#cor4){reference-type="ref" reference="cor4"} and Theorem [Theorem 91](#thm:reduc_co_knots){reference-type="ref" reference="thm:reduc_co_knots"} are actually strict by Theorem [Theorem 4](#turb_act){reference-type="ref" reference="turb_act"}. In order to extend to knots the analysis of $\precsim_{\mathop{\mathrm{Ar}}}$ previously developed, one may be tempted to transfer the subarc relation from proper arcs to knots. To this aim, we first introduce the following notion. **Definition 93**. Let $\{(\bar B_i, f_i) \mid i \leq n\}$ be a collection of oriented proper arcs, $h\colon T \to S^3$ a topological embedding, and $K \in \mathop{\mathrm{Kn}}$. We say that $K$ is the **circular sum of** the $(\bar B_i, f_i)$'s via $h$, if $K=C^h[\bigoplus_{i\leq n}(\bar B_i, f_i)]$ (recall Definitions [Definition 75](#def:circularization_of_arcs){reference-type="ref" reference="def:circularization_of_arcs"} and [Definition 76](#def:sumarc){reference-type="ref" reference="def:sumarc"}). *Remark 94*. A topological embedding $h$ of $T$ in $S^3$ is canonical if the closure of $S^3 \setminus h(T)$ is a solid torus as well (recall the solid torus theorem, see e.g. [@Ro90 p. 107]). For any $(\bar B, f) \in \mathop{\mathrm{Ar}}$, the knot $K_{(\bar B, f)}$ introduced after Definition [Definition 73](#def:arc){reference-type="ref" reference="def:arc"} is equivalent to $C^h[(\bar B, f)\oplus I_{\mathop{\mathrm{Ar}}}]$ for any canonical $h$. Moreover, when $(\bar B, f)$ is tame we have $K_{(\bar B, f)} \equiv_{\mathop{\mathrm{Kn}}} C^h[(\bar B, f)]$ for every such $h$, i.e. the two operations of joining the endpoints of $(\bar B, f)$ with a trivial arc and of circularization of $(\bar B, f)$ yield the same knot (up to equivalence). Intuitively, a knot $K$ is a subknot of $K'$ when $K'$ can be split into two subarcs, one of which (when circularized) is equivalent to $K$. Obviously, when splitting $K'$ we want that its singularities are still singularities of its subarcs, and vice versa. More formally, let $h_0 \colon T \to S^3$ be the canonical embedding. > Given two knots $K, K' \in \mathop{\mathrm{Kn}}$, we say that $K$ is a **subknot** of $K'$ if either [^7] $K \equiv_{\mathop{\mathrm{Kn}}} K'$, or there are proper arcs $(\bar B_0, f_0), (\bar B_1, f_1) \in \mathop{\mathrm{Ar}}$ such that $K' \equiv_{\mathop{\mathrm{Kn}}} C^{h_0}[(\bar B_0, f_0) \oplus (\bar B_1, f_1)]$ and $K \equiv_{\mathop{\mathrm{Kn}}} C^{h_0}[(\bar B_0, f_0)]$. It is easy to see that for tame knots this is equivalent to: $K$ is a subknot of $K'$ if either $K \equiv_{\mathop{\mathrm{Kn}}} K'$ or $K \equiv_{\mathop{\mathrm{Kn}}} K_{(\bar B', K' \cap \bar{B'})}$ for some subarc $(\bar B', K' \cap \bar{B}')$ of $K'$ (see Remark [Remark 94](#rem:tame_equiv_trivial){reference-type="ref" reference="rem:tame_equiv_trivial"}). However, as in the case of convex embeddability for circular orders, this relation is not transitive on $\mathop{\mathrm{Kn}}$ (it is however transitive on the class of tame knots). Since we want to study wild knots, to obtain a transitive relation we define a piecewise version of the subknot relation, which is the analogue for knots of $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ (recall Definition [Definition 59](#cvx_co){reference-type="ref" reference="cvx_co"}). Furthermore, in the following definition we allow arbitrary circularizations: in this way, we work modulo tame variations of the knots at hand (see Remark [Remark 98](#rem:tame_part_irrelevant){reference-type="ref" reference="rem:tame_part_irrelevant"}), so that the anti-classification results we are going to prove are actually stronger than if we allowed only canonical embeddings of $T$ into $S^3$. **Definition 95**. Let $K, K'\in \mathop{\mathrm{Kn}}$. Then $K$ is a **(finite) piecewise subknot** of $K'$, in symbols $$K \mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}K',$$ if and only if either $K \equiv_{\mathop{\mathrm{Kn}}} K'$, or $K'$ can be written as the circular sum of finitely many oriented proper arcs $\{ (\bar B_j, f_j) \mid j \leq k' \}$ (via some embedding $h' \colon T \to S^3$) in such a way that $K \equiv_{\mathop{\mathrm{Kn}}} C^h[\bigoplus_{i \leq k} (\bar B_{j_i}, f_{j_i})]$ for some (increasingly ordered) subsequence $j_0, \dotsc, j_k$ of $0, \dotsc , k'$ and some embedding $h \colon T \to S^3$. The **(finite) piecewise mutual subknot relation** is the relation defined by $K \approx_{\mathop{\mathrm{Kn}}}^{<\omega} K'$ if and only if $K \mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}K'$ and $K' \mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}K$. **Proposition 96**. *$\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$ and $\approx_{\mathop{\mathrm{Kn}}}^{<\omega}$ are an analytic quasi-order and an analytic equivalence relation on $\mathop{\mathrm{Kn}}$, respectively.* *Proof.* It is easy to see that $\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$ is reflexive and analytic. To prove transitivity we can mostly mimic the proof of Proposition [Proposition 60](#prop:pccvx_is_trans){reference-type="ref" reference="prop:pccvx_is_trans"}. ◻ The quasi-order $\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$ is fine enough to distinguish between tame and wild knots, as shown in the next proposition. **Proposition 97**. *Let $K \in \mathop{\mathrm{Kn}}$, and recall that we denote by $I_{\mathop{\mathrm{Kn}}}$ the trivial knot. Then the following are equivalent:* *$K$ is tame;* *$K \approx_{\mathop{\mathrm{Kn}}}^{<\omega} I_{\mathop{\mathrm{Kn}}}$;* *$K \mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}I_{\mathop{\mathrm{Kn}}}$.* *In particular, the $\approx^{<\omega}_{\mathop{\mathrm{Kn}}}$-class of the tame knots is minimal with respect to (the quotient order of) $\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$.* *Proof.* The proof is immediate using the facts that a knot is tame if and only if it is a circularization of the trivial arc, and that the trivial knot can be written as a circular sum only if all summands are trivial arcs and the embedding of the solid torus is canonical. ◻ *Remark 98*. If $(\bar B, f)$ is a proper arc with a tame subarc and $(\bar B', g)$ is a tame arc then it is easy to check that $C^h[(\bar B, f)] \approx^{<\omega}_{\mathop{\mathrm{Kn}}} C^{h'}[(\bar B, f) \oplus (\bar B', g)]$ for any topological embeddings $h$ and $h'$. Notice that the relations $\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$ and $\approx^{<\omega}_{\mathop{\mathrm{Kn}}}$ differ from $\equiv_{\mathop{\mathrm{Kn}}}$ only on the set of knots which are circularizations of proper arcs. For this reason we focus on the following subset of $\mathop{\mathrm{Kn}}$. **Definition 99**. We denote by $\mathop{\mathrm{CKn}}$ and $\mathop{\mathrm{WCKn}}$, respectively, the set of knots which are a circularization of a proper arc (that is, up to knot equivalence, those of the form $C^h[(\bar B, f)]$ for some $(\bar B, f) \in \mathop{\mathrm{Ar}}$ and some embedding $h \colon T \to S^3$), and its subset consisting of wild knots. Let $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$ and $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$ be the restrictions of $\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$ to these sets. Notice that $\mathop{\mathrm{CKn}}$ is a proper subset of $\mathop{\mathrm{Kn}}$: for example, the knot constructed by Bing in [@Bi56] cannot be ''cut'' at any point and thus it does not belong to $\mathop{\mathrm{CKn}}$. However $\mathop{\mathrm{CKn}}$ is quite rich, as it includes any wild knot $K$ satisfying any of the following equivalent conditions: $K$ has at least one isolated singularity (i.e. $I \Sigma_K \neq \emptyset$); the set $\Sigma_K$ of singularities of $K$ is not dense in $K$; there exists a point of $K$ which is not a singularity (i.e. $\Sigma_K \neq K$). Moreover, the wild knots built by Artin and Fox in [@FA48] do not satisfy the previous conditions, yet they belong to $\mathop{\mathrm{CKn}}$. Further evidence of the complexity and richness of $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$ is provided in the results below (see Proposition [Proposition 102](#prop:chains_skr){reference-type="ref" reference="prop:chains_skr"} and Theorems [Theorem 104](#thm:unbounded_and_dominating_for_knots){reference-type="ref" reference="thm:unbounded_and_dominating_for_knots"}--[Theorem 108](#thm:antichains_for_knots){reference-type="ref" reference="thm:antichains_for_knots"}). Since $C^h[(\bar B, f)] \approx_{\mathop{\mathrm{Kn}}}^{<\omega} C^{h'}[(\bar B, f)]$ for any topological embeddings $h$ and $h'$, every $K \in \mathop{\mathrm{CKn}}$ can be assumed to be, up to $\approx_{\mathop{\mathrm{CKn}}}^{<\omega}$, of the form $C^h[(\bar B, f)]$ for some canonical embedding $h\colon T \to S^3$. To simplify the notation we write $C[(\bar B, f)]$ in place of $C^h[(\bar B, f)]$ when $h$ is canonical and we do not mention $h$ and $h'$ witnessing $K \mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}K'$ when they are canonical, which can always be assumed to be the case. The next theorem establishes a lower bound for the complexity of $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$ with respect to Borel reducibility. **Theorem 100**. *${\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}} \leq_B {\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}}$.* *Proof.* We claim that the Borel map $F_{\mathop{\mathrm{Kn}}} \colon \mathop{\mathrm{\mathsf{CO}}}\to \mathop{\mathrm{Kn}}$ from the proof of Theorem [Theorem 91](#thm:reduc_co_knots){reference-type="ref" reference="thm:reduc_co_knots"} is the desired reduction. First of all, notice that by construction $F_{\mathop{\mathrm{Kn}}}(C) \in \mathop{\mathrm{CKn}}$ for every $C \in \mathop{\mathrm{\mathsf{CO}}}$. Fix now $C, C' \in \mathop{\mathrm{\mathsf{CO}}}$. Assume first that ${C}\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}{C'}$, and let the finite convex partition $(C_i)_{i \leq k}$ of $C$ and the embedding $g$ witness this. As usual, we can assume $k>0$. For every $i\leq k$, let $C'_{2i}=g(C_i)$, for every $i<k$ let $C'_{2i+1}$ be the convex subset of $C'$ given by the elements $n$ such that $C'(\ell, n, \ell')$ for every $\ell \in C'_{2i}$ and $\ell' \in C'_{2i+2}$; moreover, let $C'_{2k+1} \csube C'$ be the set of the $n$'s such that $C'(\ell, n, \ell')$ for every $\ell \in C'_{2k}$ and $\ell' \in C'_0$ (if any of these sets is empty just delete it and reindex the remaining convex sets appropriately). For every $j\leq 2k+1$, set $\bar B_j= [a_j,b_j] \times [-1,1]^2$, where $a_j=\inf \bigcup_{n \in C'_j}V^{C'}_n$ and $b_j=\sup\bigcup_{n \in C'_j}V^{C'}_n$, and $f_j = F_{\mathop{\mathrm{Kn}}}(C') \cap \bar B_j$, so that $(\bar B_j, f_j)$ is a proper arc. Notice now that $F_{\mathop{\mathrm{Kn}}}(C')=C[\bigoplus_{j\leq 2k+1}(\bar B_j, f_j)]$. Moreover, since $C \cong_{\mathop{\mathrm{\mathsf{CO}}}} \sum_{i \leq k} C'_{2i}$ we have $F_{\mathop{\mathrm{Kn}}}(C)\equiv_{\mathop{\mathrm{Kn}}} C[\bigoplus_{i\leq k}(\bar B_{2i}, f_{2i})]$. Hence, $F_{\mathop{\mathrm{Kn}}}(C) \mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}F_{\mathop{\mathrm{Kn}}}(C')$. Conversely, suppose that $F_{\mathop{\mathrm{Kn}}}(C)$ and $F_{\mathop{\mathrm{Kn}}}(C')$ (which are elements of $\mathop{\mathrm{CKn}}$) are such that $F_{\mathop{\mathrm{Kn}}}(C) \mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}F_{\mathop{\mathrm{Kn}}}(C')$, and let $\{(\bar{B}_i,f_i) \mid i \leq k'\}$ and the subsequence $(j_i)_{i \leq k}$ of $0,\dots,k'$ witness this. By definition of $F_{\mathop{\mathrm{Kn}}}(C')$, when $\bar B_i \cap \bar B_m$ contains a point $x \in I\Sigma_{F_{\mathop{\mathrm{Kn}}}(C')}$ then $x$ is a singular point of only one of $(\bar{B}_i,f_i)$ and $(\bar{B}_m,f_m)$; by reindexing the sequence $\{(\bar{B}_i,f_i) \mid i \leq k'\}$ we can assume this occurs always for the index which is the immediate predecessor of the other one in $C[\mathbf{k'+1}]$. Since $F_{\mathop{\mathrm{Kn}}}(C) \equiv_{\mathop{\mathrm{Kn}}} C[\bigoplus_{i\leq k}(\bar B_{j_i}, f_{j_i})]$, by an analogue of [\[eq:not-inverting\]](#eq:not-inverting){reference-type="eqref" reference="eq:not-inverting"} we have $I\Sigma_{F_{\mathop{\mathrm{Kn}}}(C)} \cong_{\mathop{\mathrm{\mathsf{CO}}}} I\Sigma_{C[\bigoplus_{i\leq k}(\bar B_{j_i}, f_{j_i})]}$ via some map $g$. Recall that $h_C$ is an isomorphism of circular orders between $C$ and $I\Sigma_{F_{\mathop{\mathrm{Kn}}}(C)}$, so that $g \circ h_C$ is an isomorphism of circular orders as well, and let $C_i = (g \circ h_C)^{-1} (I\Sigma_{F_{\mathop{\mathrm{Kn}}}(C')}\cap\bar{B}_{j_i} \setminus \bar{B}_{j_{i+1}})$. Notice that $(C_i)_{i \leq k}$ is a finite convex partition of $C$. Moreover, since $(\bar B_{j_i}, f_{j_i})$ is a subarc of $F_{\mathop{\mathrm{Kn}}}(C')$ for every $i\leq k$, we have that each $I\Sigma_{F_{\mathop{\mathrm{Kn}}}(C')} \cap \bar{B}'_{j_i}\setminus \bar{B'_m}$ (for $m$ the immediate predecessor of $j_i$ in $C[\mathbf{k'+1}]$) is convex in $I\Sigma_{F_{\mathop{\mathrm{Kn}}}(C')}$. Finally, since $I\Sigma_{F_{\mathop{\mathrm{Kn}}}(C')}\cong C'$ via $h_{C'}^{-1}$, then ${C} \trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C'$, as desired. ◻ **Corollary 101**. *${\mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}} \leq_B {\approx_{\mathop{\mathrm{CKn}}}^{<\omega}}$, hence also ${\cong_{\mathop{\mathrm{\mathsf{LO}}}}}\leq_B {\approx_{\mathop{\mathrm{CKn}}}^{<\omega}}$ and ${E_1} \leq_B {\approx_{\mathop{\mathrm{CKn}}}^{<\omega}}$.* The fact that the isomorphism on linear orders is Borel reducible to $\approx_{\mathop{\mathrm{CKn}}}^{<\omega}$ implies that $\approx_{\mathop{\mathrm{CKn}}}^{<\omega}$ is proper analytic. Moreover, ${\approx_{\mathop{\mathrm{CKn}}}^{<\omega}}$ is not Baire reducible to an orbit equivalence relation because it Borel reduces $E_1$, in stark contrast with knot equivalence $\equiv_{\mathop{\mathrm{Kn}}}$; in particular we have that ${\approx_{\mathop{\mathrm{CKn}}}^{<\omega}}$ is not Borel, or even Baire, reducible to $\equiv_{\mathop{\mathrm{Kn}}}$. Using Theorem [Theorem 100](#thm:lowerforknots){reference-type="ref" reference="thm:lowerforknots"}, we can transfer the combinatorial properties of $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ proved in Proposition [Proposition 62](#prop:antichainsCO){reference-type="ref" reference="prop:antichainsCO"} to $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$. **Proposition 102**. *There is an embedding from the partial order $(\mathop{\mathrm{Int}}(\mathbb{R}),\subseteq)$ into $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$, and indeed ${(\mathop{\mathrm{Int}}(\mathbb{R}),\subseteq)} \leq_B {\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}}$.* *$\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$ has chains of order type $(\mathbb{R}, {<} )$, as well as antichains of size $2^{\aleph_0}$.* To extend the other combinatorial properties of $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ to $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$ we need an analogous of Lemma [Lemma 83](#lem:froarcstoLin){reference-type="ref" reference="lem:froarcstoLin"}. When $K$ is a knot and $f$ is such that $\mathop{\mathrm{Im}}f =K$, the set $I\Sigma_{K}$ can naturally be viewed as a circular order $C^K_f = (I\Sigma_{K}, C_f)$. As it was the case for proper arcs, the set $I\Sigma_{K}$ is (at most) countable and thus $C^K_f$ is either a finite or a countable circular order. If $f,f'\colon S^1 \to S^3$ are topological embeddings giving rise to the same knot, then either $C^K_f = C^K_{f'}$ or $C^K_f = (C^K_{f'})^*$. Recall that by construction, for knots of the form [^8] $F_{\mathop{\mathrm{Kn}}}(C)$ we have $C^{F_{\mathop{\mathrm{Kn}}}(C)}_{f} \cong_{\mathop{\mathrm{\mathsf{CO}}}} C$. **Lemma 103**. *Let $K, K' \in \mathop{\mathrm{CKn}}$ be such that $K\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}K'$ and let $f$ and $f'$ be such that $\mathop{\mathrm{Im}}f =K$ and $\mathop{\mathrm{Im}}f'=K'$. Then there exists a finite set $A \subseteq I\Sigma_K$ such that either $C^K_f \setminus A \mathrel{\trianglelefteq_{c}^{<\omega}}C^{K'}_{f'}$ or $C^K_f \setminus A \mathrel{\trianglelefteq_{c}^{<\omega}}{(C^{K'}_{f'})^*}$.* *Proof.* Let $\{(\bar B_i, f_i) \mid i \leq k'\}$ and the subsequence $(j_i)_{i \leq k}$ of $0,\dots,k'$ witness $K \mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}K'$, and denote by $g$ the homeomorphism witnessing $K \equiv_{\mathop{\mathrm{Kn}}} C[\bigoplus_{i \leq k}(\bar B_{j_i}, f_{j_i})]$. Let $A = \{x \in I\Sigma_K \mid \exists i \leq k \, (g(x) \in \partial \bar{B}_{j_i})\}$, and notice that $A$ contains at most $k+1$ points. We can assume that $g$ agrees with the orientations induced on $K$ and $K'$ by $f$ and $f'$, in which case we show that $C^K_f \setminus A \mathrel{\trianglelefteq_{c}^{<\omega}}C^{K'}_{f'}$ (if $g$ agrees with only one of the orientations we obtain $C^K_f \setminus A \mathrel{\trianglelefteq_{c}^{<\omega}}(C^{K'}_{f'})^*$, and if it disagrees with both it suffices to reverse both orientations). For every $i \leq k$ let $C_i = I\Sigma_K \cap g^{-1}(\mathop{\mathrm{Int}}\bar{B}_{j_i})$. Then each $C_i$ is convex, and $\{ C_i \mid i \leq k \}$ is a finite convex partition of $C^K_f \setminus A$ (some of the $C_i$'s might actually be empty, in which case we would obtain a convex partition with less than $k+1$ sets, but for notational ease we avoid keeping track of this). Since each $(\bar B_{j_i}, f_{j_i})$ is a subarc of $K'$, it is easy to check that $g \!\restriction\!I\Sigma_K$ is an embedding of circular orders and that $g(C_i) \csube C^{K'}_{f'}$ for all $i \leq k$. ◻ Since in $\mathop{\mathrm{Kn}}\setminus \mathop{\mathrm{CKn}}$ the relation $\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}$ is $\equiv_{\mathop{\mathrm{Kn}}}$, it is easy to show that $\mathfrak{b} (\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}) = 2$ and $\mathfrak{d} (\mathop{\mathrm{\precsim_{\mathop{\mathrm{Kn}}}^{<\omega}}}) = 2^{\aleph_0}$. It is therefore more interesting to compute the unbounding and dominating number of $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$. **Theorem 104**. *$\mathfrak{b} (\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}) = \aleph_1$ and $\mathfrak{d} (\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}) = 2^{\aleph_0}$.* *Proof.* We first show the existence of an $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$-unbounded family of knots of size $\aleph_1$. Let $F_{\mathop{\mathrm{Kn}}} \colon \mathop{\mathrm{\mathsf{CO}}}\to \mathop{\mathrm{Kn}}$ be the map defined in the proof of Theorem [Theorem 91](#thm:reduc_co_knots){reference-type="ref" reference="thm:reduc_co_knots"} and used also in the proof of Theorem [Theorem 100](#thm:lowerforknots){reference-type="ref" reference="thm:lowerforknots"}. By (the proof of) Proposition [Proposition 63](#prop:boundingCO){reference-type="ref" reference="prop:boundingCO"} there exists a strictly increasing sequence $\{ C_\alpha\mid \alpha< \omega_1 \} \subseteq \mathop{\mathrm{\mathsf{CO}}}$ without upper bound with respect to $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$. Notice moreover that each $C_\alpha$ has the property that $C_\alpha\cong_{\mathop{\mathrm{\mathsf{CO}}}} C_\alpha\setminus A$ for any finite $A \subseteq C_\alpha$. The sequence $\{ F_{\mathop{\mathrm{Kn}}}(C_\alpha) \mid \alpha< \omega_1 \} \subseteq \mathop{\mathrm{CKn}}$ is then strictly $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$-increasing, and we claim that it is also unbounded in $\mathop{\mathrm{CKn}}$. Suppose towards a contradiction that there is $K \in \mathop{\mathrm{Kn}}$ such that $F_{\mathop{\mathrm{Kn}}}(C_\alpha) \mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}K$ for all $\alpha< \omega_1$. Then $I\Sigma_{K}$ is infinite and thus the circular order $C_f^K$ is, up to isomorphism, an element of $\mathop{\mathrm{\mathsf{CO}}}$. Pick now $\ell \in C_f^K$ and define $L \in \mathop{\mathrm{\mathsf{LO}}}$ by setting $x \leq_L y$ if and only if $C_f^K(\ell,x,y)$ and $x=\ell$ when $y=\ell$. Notice that $C_f^K = C[L]$. Then the circular order $C=C[L+L^*]$ is such that $C_f^K \trianglelefteq_{c}C$ and $(C_f^K)^* \trianglelefteq_{c}C$. By Lemma [Lemma 103](#lem:from_knots_to_CO){reference-type="ref" reference="lem:from_knots_to_CO"} and the fact that each $C^{F_{\mathop{\mathrm{Kn}}}(C_\alpha)}_{f} \cong_{\mathop{\mathrm{\mathsf{CO}}}} C_\alpha$, this would imply that for every $\alpha< \omega_1$ there exists a finite $A_\alpha\subseteq C_\alpha$ such that $C_\alpha\setminus A_\alpha\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}C$. As $C_\alpha\cong_{\mathop{\mathrm{\mathsf{CO}}}} C_\alpha\setminus A$, the circular order $C$ would be a $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-upper bound for $\{ C_\alpha\mid \alpha< \omega_1 \}$, yielding the desired contradiction. We now prove that $\mathfrak{b} (\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}) > \aleph_0$. Let $\{ K_i \mid i \in \mathbb{N}\} \subseteq \mathop{\mathrm{CKn}}$ be a countable family of knots. By definition of $\mathop{\mathrm{CKn}}$, each $K_i$ can be written as $C[(\bar B_i, f_i)]$ for some proper arc $(\bar B_i, f_i)$ (we are using a canonical embedding of the solid torus in $S^3$). Then the knot $C[\bigoplus_\mathbb{N}(\bar B_i, f_i)]$ is a $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$-upper bound for $\{ K_i \mid i \in \mathbb{N}\}$. To prove that $\mathfrak{d} ({\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}}) \geq 2^{\aleph_0}$ we follow the same strategy of the proof of Theorem [Theorem 85](#thm:umboundedanddominatingforarcs){reference-type="ref" reference="thm:umboundedanddominatingforarcs"}. Consider the $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-antichain $\{ C_S \mid S \subseteq \mathbb{N}\}$ defined in the proof of Proposition [Proposition 64](#prop:basisCO){reference-type="ref" reference="prop:basisCO"}[\[prop:basisCO-1\]](#prop:basisCO-1){reference-type="ref" reference="prop:basisCO-1"} and, using Lemma [Lemma 103](#lem:from_knots_to_CO){reference-type="ref" reference="lem:from_knots_to_CO"} (removing finitely many elements from $C_S$ does not affect the argument) and the proof of Proposition [Proposition 65](#prop:antichains2CO){reference-type="ref" reference="prop:antichains2CO"}, prove that for every knot $K \in \mathop{\mathrm{CKn}}$, $\{ F_{\mathop{\mathrm{Kn}}}(C_S) \mid {F_{\mathop{\mathrm{Kn}}}(C_S) \mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}K} \}$ is countable. The proof is then completed using Theorem [Theorem 91](#thm:reduc_co_knots){reference-type="ref" reference="thm:reduc_co_knots"}. ◻ **Corollary 105**. *Every knot $K \in \mathop{\mathrm{CKn}}$ is the bottom of an $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$-unbounded chain of length $\omega_1$.* *Proof.* Given $K \in \mathop{\mathrm{CKn}}$, let $(\bar B, f)$ be a proper arc such that $K=C[(\bar B, f)]$. As in the proof of Theorem [Theorem 104](#thm:unbounded_and_dominating_for_knots){reference-type="ref" reference="thm:unbounded_and_dominating_for_knots"} let $\{ C_\alpha\mid \alpha< \omega_1 \} \subseteq \mathop{\mathrm{\mathsf{CO}}}$ be an unbounded strictly $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$-increasing sequence in $\mathop{\mathrm{\mathsf{CO}}}$, so that $\{ F_{\mathop{\mathrm{Kn}}}(C_\alpha) \mid \alpha< \omega_1 \} \subseteq \mathop{\mathrm{CKn}}$ is unbounded and strictly $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$-increasing in $\mathop{\mathrm{CKn}}$. For every $\alpha<\omega_1$, let $(\bar B_\alpha, f_\alpha)$ be a proper arc obtained by cutting $F_{\mathop{\mathrm{Kn}}}(C_\alpha)$ in a point which is not an isolated singularity, so that in particular $F_{\mathop{\mathrm{Kn}}}(C_\alpha) = C[(\bar B_\alpha, f_\alpha)]$. Let $K_0 = K$ and, for $0<\alpha<\omega_1$, $K_\alpha= C[{(\bar B, f) \oplus (\bar B_\alpha, f_\alpha)}]$. For every $\alpha < \beta < \omega_1$ we have $K_\alpha \mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}K_\beta$ (even though it might happen that $(\bar B_\alpha, f_\alpha) \not\precsim_{\mathop{\mathrm{Ar}}} (\bar B_\beta, f_\beta)$, in which case we need a circular sum of proper arcs with more than one element to witness $K_\alpha \mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}K_\beta$) and $F_{\mathop{\mathrm{Kn}}}(C_\alpha) \mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}K_\alpha$. Hence the sequence $(K_\alpha)_{\alpha<\omega_1}$ is $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$-unbounded. By the same argument used in the proof of Corollary [Corollary 86](#cor:incomparablearcsandunboundedchains){reference-type="ref" reference="cor:incomparablearcsandunboundedchains"} we can extract from $(K_\alpha)_{\alpha<\omega_1}$ a strictly $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$-increasing subsequence of length $\omega_1$ starting with $K$. ◻ We finally deal with minimal elements and bases with respect to $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$. In contrast with the case of proper arcs, it is not interesting to consider the restriction of $\mathop{\mathrm{\precsim_{\mathop{\mathrm{CKn}}}^{<\omega}}}$ to the collection of tame knots because by Proposition [Proposition 97](#prop:tame_knots_for_star){reference-type="ref" reference="prop:tame_knots_for_star"} tame knots are all $\approx_{\mathop{\mathrm{CKn}}}^{<\omega}$-equivalent. Thus let us consider $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$. **Lemma 106**. *Let $\{ (\bar B^p_i, f^p_i) \mid i \in \mathbb{N}\}$ be a family of (oriented) prime arcs, and let $K^*_S = C[\bigoplus_{i \in S} \, (\bar B^p_i, f^p_i)]$ for some infinite $S \subseteq \mathbb{N}$.* *[\[knots_sum_prime_arcs-0\]]{#knots_sum_prime_arcs-0 label="knots_sum_prime_arcs-0"} If $K^*_S = C^h[\bigoplus_{i \leq k} (\bar B_i, f_i)]$ for some $k \in \mathbb{N}$ and $h \colon T \to S^3$, then there is a unique $j \leq k$ such that $(\bar B_j, f_j)$ is wild; moreover, either $(\bar B_j, f_j) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus_{i \in S'} (\bar B^p_i, f^p_i)$ or $(\bar B_j, f_j) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus^\partial_{i \in S'} (\bar B^p_i, f^p_i)$ for some $S' \subseteq S$ with $S \setminus S'$ finite.* *[\[knots_sum_prime_arcs-1\]]{#knots_sum_prime_arcs-1 label="knots_sum_prime_arcs-1"} The knot $K^*_S$ is $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-minimal in $\mathop{\mathrm{WCKn}}$.* *[\[knots_sum_prime_arcs-2\]]{#knots_sum_prime_arcs-2 label="knots_sum_prime_arcs-2"} If $K^*_{S_0} \precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K^*_{S_1}$ then $S_0 =^* S_1$, where $=^*$ is the identity modulo a finite set.* *Proof.* [\[knots_sum_prime_arcs-0\]](#knots_sum_prime_arcs-0){reference-type="ref" reference="knots_sum_prime_arcs-0"} Let $j \leq k$ be such that $(h(\bar B_j), h \circ f_j)$ is wild and contains the unique singularity $x$ of $K^*_S$. There is at least one such $j$ because otherwise $K^*_S$ would be tame, and it is unique because the singularity $x$ is "one-sided", i.e. it is witnessed only on one side while the other side is tame. Thus $(\bar B_j, f_j)$, being equivalent to $(h(\bar B_j), h \circ f_j)$ via $h \restriction \bar B_j$, is wild, while all other $(\bar B_i, f_i)$ with $i \neq j$ are tame because so are the proper arcs $(h(\bar B_i), h \circ f_i)$. Moreover, by construction $(h(\bar B_j), h \circ f_j)$ is either of the form $\bigoplus_{i \in S'} (\bar B^p_i, f^p_i)$ (if $x \in \mathop{\mathrm{Int}}h(\bar B_j)$) or $\bigoplus^\partial_{i \in S'} (\bar B^p_i, f^p_i)$ (if $x \in \partial \, h(\bar B_j)$), for some $S' \subseteq S$ omitting finitely many elements of $S$: since $(\bar B_j, f_j) \equiv_{\mathop{\mathrm{Ar}}} (h(\bar B_j), h \circ f_j)$ we are done. [\[knots_sum_prime_arcs-1\]](#knots_sum_prime_arcs-1){reference-type="ref" reference="knots_sum_prime_arcs-1"} Suppose that $K \in \mathop{\mathrm{WCKn}}$ is such that $K\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K^*_S$ but $K \not\equiv_{\mathop{\mathrm{Kn}}} K^*_S$ (otherwise we are done), and let $\{ (\bar B_j, f_j) \mid j \leq k'\}$ and $(j_i)_{i \leq k}$ witness this. By part [\[knots_sum_prime_arcs-0\]](#knots_sum_prime_arcs-0){reference-type="ref" reference="knots_sum_prime_arcs-0"} there is a unique $\ell$ such that $(\bar B_\ell, f_\ell)$ is wild, and necessarily $\ell$ is one of the $j_i$'s because otherwise $K$ would be tame. Using Remark [Remark 98](#rem:tame_part_irrelevant){reference-type="ref" reference="rem:tame_part_irrelevant"} one easily gets $$K \equiv_{\mathop{\mathrm{Kn}}} C[\bigoplus_{i \leq k}(\bar B_{j_i}, f_{j_i})] \approx_{\mathop{\mathrm{WCKn}}}^{<\omega} C[(\bar B_\ell, f_\ell)] \approx_{\mathop{\mathrm{WCKn}}}^{<\omega} K^*_S.$$ [\[knots_sum_prime_arcs-2\]](#knots_sum_prime_arcs-2){reference-type="ref" reference="knots_sum_prime_arcs-2"} Let $\{ (\bar B_j, f_j) \mid j \leq k'\}$ and the subsequence $(j_i)_{i \leq k}$ of $0, \dots, k'$ witness $K^*_{S_0} \precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K^*_{S_1}$. Apply part [\[knots_sum_prime_arcs-0\]](#knots_sum_prime_arcs-0){reference-type="ref" reference="knots_sum_prime_arcs-0"} to $K^*_{S_1}$ to isolate the unique $\ell \leq k'$ such that the proper arc $(\bar B_\ell, f_\ell)$ is wild. Let also $S'_1 \subseteq S_1$ be such that $(\bar B_\ell, f_\ell) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus\nolimits^{(\partial)}_{i \in S'_1} (\bar B^p_i, f^p_i)$, with $S_1 \setminus S'_1$ is finite. Arguing as in [\[knots_sum_prime_arcs-1\]](#knots_sum_prime_arcs-1){reference-type="ref" reference="knots_sum_prime_arcs-1"}, we have that $\ell$ is one of the $j_i$'s such that $K^*_{S_0} \equiv_{\mathop{\mathrm{Kn}}} C[\bigoplus_{i \leq k} (\bar B_{j_i}, f_{j_i})]$. Then there is a subarc $(\bar B^*, f^*)$ of $K$ such that $(\bar B^*, f^*) \equiv_{\mathop{\mathrm{Ar}}} (\bar B_\ell, f_\ell)$, and since it is wild, by applying [\[knots_sum_prime_arcs-0\]](#knots_sum_prime_arcs-0){reference-type="ref" reference="knots_sum_prime_arcs-0"} we obtain that $(\bar B^*, f^*)$ is of the form $\bigoplus\nolimits^{(\partial)}_{i \in S'_0} (\bar B^p_i, f^p_i)$ with $S_0 \setminus S'_0$ finite. Thus, $\bigoplus^{(\partial)}_{i \in S'_0} (\bar B^p_i, f^p_i) \equiv_{\mathop{\mathrm{Ar}}} \bigoplus^{(\partial)}_{i \in S'_1} (\bar B^p_i, f^p_i)$, whence it follows that $S'_0 = S'_1$. Hence $S_0 =^* S_1$. ◻ **Theorem 107**. *[\[thm:basis_for_knots-1\]]{#thm:basis_for_knots-1 label="thm:basis_for_knots-1"} There are $2^{\aleph_0}$-many $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-incomparable $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-minimal elements in $\mathop{\mathrm{WCKn}}$. In particular, all bases for $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$ are of maximal size.* *[\[thm:basis_for_knots-2\]]{#thm:basis_for_knots-2 label="thm:basis_for_knots-2"} There is a strictly $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-decreasing $\omega$-sequence in $\mathop{\mathrm{WCKn}}$ which is not $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-bounded from below.* *Proof.* Fix an enumeration without repetitions $\{(\bar B_i^p, f_i^p) \mid i \in \mathbb{N}\}$ of all prime arcs. [\[thm:basis_for_knots-1\]](#thm:basis_for_knots-1){reference-type="ref" reference="thm:basis_for_knots-1"} As in the proof of Theorem [Theorem 88](#thm:basisforarcs){reference-type="ref" reference="thm:basisforarcs"}[\[thm:basisforarcs-3\]](#thm:basisforarcs-3){reference-type="ref" reference="thm:basisforarcs-3"}, let $\mathcal{P}$ be a family of size $2^{\aleph_0}$ consisting of infinite subsets of $\mathbb{N}$ with pairwise finite intersections. For every $S \in \mathcal{P}$ consider the knot $K^*_S$ defined in Lemma [Lemma 106](#lem:knots_sum_prime_arcs){reference-type="ref" reference="lem:knots_sum_prime_arcs"}. By Lemma [Lemma 106](#lem:knots_sum_prime_arcs){reference-type="ref" reference="lem:knots_sum_prime_arcs"}[\[knots_sum_prime_arcs-1\]](#knots_sum_prime_arcs-1){reference-type="ref" reference="knots_sum_prime_arcs-1"} each $K^*_S$ is $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-minimal in $\mathop{\mathrm{WCKn}}$, and if $S, S' \in \mathcal{P}$ are distinct then $K^*_S$ and $K^*_{S'}$ are $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-incomparable by Lemma [Lemma 106](#lem:knots_sum_prime_arcs){reference-type="ref" reference="lem:knots_sum_prime_arcs"}[\[knots_sum_prime_arcs-2\]](#knots_sum_prime_arcs-2){reference-type="ref" reference="knots_sum_prime_arcs-2"}. [\[thm:basis_for_knots-2\]](#thm:basis_for_knots-2){reference-type="ref" reference="thm:basis_for_knots-2"} Let $K_n=C[{\bigoplus_{i \geq n} (\bigoplus_{\mathbb{N}}^\partial \, (\bar B_i^p, f_i^p))}]$ (notice that each $i \geq n$ is associated to an element of $I\Sigma_{K_n}$). We prove that $\{K_n \mid n \in \mathbb{N}\}$ is the desired $\omega$-chain. Let $n < n'$. Clearly, $K_{n'} \precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K_n$. Suppose now, towards a contradiction, that $K_n \precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K_{n'}$, as witnessed by $\{(\bar{B}_j,f_j) \mid j \leq k'\}$ and the subsequence $(j_i)_{i \leq k}$ of $0,\dots,k'$. Since $K \equiv_{\mathop{\mathrm{Kn}}} C[\bigoplus_{i \leq k} (\bar B_{j_i}, f_{j_i})]$, there exist $i \leq k$ and $m \in \mathbb{N}$ such that $(\bar B_{j_i}, f_{j_i})$ contains (a proper arc equivalent to) the tail $\bigoplus_{t \geq m}^\partial\, (\bar B_n^p, f_n^p)$ of $\bigoplus_{\mathbb{N}}^\partial\, (\bar B_n^p, f_n^p)$. But $(\bar B_{j_i}, f_{j_i})$ is a subarc of $K'_{n'}$, a contradiction. Hence $K_{n'} \prec_{\mathop{\mathrm{WCKn}}}^{<\omega} K_n$. Suppose now that $K \in \mathop{\mathrm{WCKn}}$ bounds from below $\{K_n \mid n \in \mathbb{N}\}$. Notice that $K \precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K_0$ implies $I\Sigma_K \neq \emptyset$, so that we can fix $x \in I\Sigma_K$. Let $\{(\bar{B}_j,f_j) \mid j \leq k'\}$, and the subsequence $(j_i)_{i \leq k}$ of $0,\dots,k'$ witness $K \precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K_0$. Let $g$ be the homeomorphism witnessing that $K\equiv_{\mathop{\mathrm{Kn}}} C[\bigoplus_{i \leq k} (\bar B_{j_i}, f_{j_i})]$. Then there exists $i \leq k$ such that $g(x) \in \bar B_{j_i}$ and $(\bar B_{j_i}, f_{j_i})$ is wild and contains an element of $I\Sigma_{K_0}$, which belongs to $\bigoplus_{\mathbb{N}}^\partial\, (\bar B_n^p, f_n^p)$ for some $n \geq 0$. This implies that there exists $m \in \mathbb{N}$ such that $C[{\bigoplus^\partial_{t \geq m}\, (\bar B_n^p, f_n^p)}] \precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K$. But by the argument of the previous paragraph $C[{\bigoplus^\partial_{t \geq m}\, (\bar B_n^p, f_n^p)}] \not\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K_{n+1}$, hence $K \not\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K_{n+1}$, which is a contradiction. ◻ When $K \in \mathop{\mathrm{Kn}}$, we say that $x \in K$ is **isolable in $K$** if there exists a subarc $(\bar B, f)$ of $K$ such that $x \in I\Sigma_{(\bar B, f)}$. Notice that every $x \in I\Sigma_K$ is isolable in $K$, but some point which is isolable in $K$ can fail to belong to $I\Sigma_K$ because it is an accumulation point of other singular points only from one side. It is immediate that the set of points isolable in $K$ is countable. **Theorem 108**. *Every $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-antichain is contained in a $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-antichain of size $2^{\aleph_0}$. In particular, there are no maximal $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-antichains of size smaller than $2^{\aleph_0}$, and every $K \in \mathop{\mathrm{WCKn}}$ belongs to a $\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega}$-antichain of size $2^{\aleph_0}$.* *Proof.* Let $\{(\bar B_i^p, f_i^p) \mid i \in \mathbb{N}\}$, $\mathcal{P}$ and $K_S \in \mathop{\mathrm{WCKn}}$, with $S \in\mathcal{P}$, be as in the proof of Theorem [Theorem 107](#thm:basis_for_knots){reference-type="ref" reference="thm:basis_for_knots"}[\[thm:basis_for_knots-1\]](#thm:basis_for_knots-1){reference-type="ref" reference="thm:basis_for_knots-1"}. Following the proof of Proposition [Proposition 33](#prop:everylobelongstoantichain){reference-type="ref" reference="prop:everylobelongstoantichain"}, it is enough to prove that the set $\{ S \in \mathcal{P} \mid K_S \precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K \}$ is countable for each $K \in \mathop{\mathrm{WCKn}}$. Suppose that $S \subseteq \mathbb{N}$ is such that $K_S\precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K$, as witnessed by $\{(\bar{B}_j,f_j) \mid j \leq k'\}$ and the subsequence $(j_i)_{i \leq k}$ of $0,\dots,k'$. There exist $i \leq k$ and $m$ such that $\bigoplus_{i \in S, i \geq m}^\partial\, (\bar B_i^p, f_i^p) \precsim_{\mathop{\mathrm{Ar}}} (\bar B_{j_i}, f_{j_i})$; thus $\bigoplus_{i \in S, i \geq m}^\partial\, (\bar B_i^p, f_i^p)$ is a subarc of $K$. Therefore there exists $x_S \in \Sigma_K$ which is determined by a tail of $\bigoplus_{i \in S}^\partial\, (\bar B_i^p, f_i^p)$. Notice that $x_S$ is isolable in $K$. If $S,S' \in \mathcal{P}$ are distinct and $x_S=x_{S'}$ then by Lemma [Lemma 87](#lem:fewpredecessors){reference-type="ref" reference="lem:fewpredecessors"}[\[lem:fewpredecessors-2\]](#lem:fewpredecessors-2){reference-type="ref" reference="lem:fewpredecessors-2"} (and recalling that $S$ and $S'$ have finite intersection) the images of $\bigoplus_{i \in S, i \geq m}\, (\bar B_i^p, f_i^p)$ and $\bigoplus_{i \in S', i \geq m'}\, (\bar B_i^p, f_i^p)$ approach $x_S$ from opposite sides. Hence, $|\{S \in \mathcal{P} \mid x_S=x\}| \leq 2$ for every $x$ isolable in $K$. Since the set of isolable points in $K$ is countable, $\{ S \in \mathcal{P} \mid K_S \precsim_{\mathop{\mathrm{WCKn}}}^{<\omega} K \}$ is countable as well. ◻ # Open Problems We conclude by discussing some open problems that need more investigation. The main one concerns the classification of $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ with respect to Borel reducibility. In Section [3.2](#sec:complexity of cvx){reference-type="ref" reference="sec:complexity of cvx"} we showed that ${\cong_{\mathop{\mathrm{\mathsf{LO}}}}} \leq_B {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$ and that ${\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}} \leq_{\text{\scriptsize \textit{Baire}}} {\cong_{\mathop{\mathrm{\mathsf{LO}}}}}$. *Open Problem 1*. Is $\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}$ Borel reducible to $\cong_{\mathop{\mathrm{\mathsf{LO}}}}$ or at least to an equivalence relation induced by a continuous action of a Polish group? In Section [3.3](#sec:cvx_co){reference-type="ref" reference="sec:cvx_co"} we introduced the quasi-order $\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}$ of piecewise convex embeddability for countable circular orders and proved that ${\mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}} \nleq_B {\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}}$, so that also ${\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}} \nleq_B {\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}}$. *Open Problem 2*. Is it true that ${\trianglelefteq_{\mathop{\mathrm{\mathsf{LO}}}}} <_B{\trianglelefteq_{\mathop{\mathrm{\mathsf{CO}}}}^{<\omega}}$? What about ${\mathrel{\underline{\bowtie}_{\mathop{\mathrm{\mathsf{LO}}}}}} <_B {\mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}}}$? Considering the combinatorial properties of the quasi-orders defined in the previous sections, many other natural questions can be explored. 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[^3]: Our proof actually shows that $\varphi(\vec{x}) \mathrel{\trianglelefteq^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} \varphi(\vec{y})$ already suffices to obtain $\vec{x} \mathrel{E^t_1} \vec{y}$, so that in particular we get $\varphi(\vec{x}) \mathrel{\trianglelefteq^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} \varphi(\vec{y}) \iff {\varphi(\vec{x})} \mathrel{\underline{\bowtie}^{<\omega}_{\mathop{\mathrm{\mathsf{CO}}}}} {\varphi(\vec{y})}$ for all $\vec{x},\vec{y} \in \mathbb{R}^\mathbb{N}$. [^4]: Our definition of tame knot is equivalent to the classical one, according to which a knot is tame if it is equivalent to a finite polygon (see [@BZ03 Definition 1.3]). [^5]: When summing unoriented proper arcs, if not specified otherwise we use the natural orientation coming from their presentation. [^6]: If not specified otherwise, we always choose the natural orientation of $F(L)$. [^7]: This condition is added to have at least a reflexive relation. [^8]: If not specified otherwise, we always choose the natural orientation of $F_{\mathop{\mathrm{Kn}}}(C)$, witnessed by $f$.

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--- abstract: | We study the Asymmetric Brownian Energy, a model of heat conduction defined on the one-dimensional finite lattice with open boundaries. The system is shown to be dual to the Symmetric inclusion process with absorbing boundaries. The proof relies on a non-local map transformation procedure relating the model to its symmetric version. As an application, we show how the duality relation can be used to analytically compute suitable exponential moments with respect to the stationary measure. author: - Gioia Carinci - Francesco Casini - Chiara Franceschini title: Duality for a boundary driven asymmetric model of energy transport --- **Keywords:** Asymmetric diffusion process; Open boundary; Markov duality; Non-equilibrium steady state. # Introduction The Asymmetric Brownian Energy Processes (ABEP) is an interacting diffusion system describing an asymmetric energy exchange between the sites of a lattice. Its symmetric version (BEP) was originally introduced in [@GKRV] where its symmetries and duality properties where unveiled. These are related to the intrinsic algebraic structure of the infinitesimal generator that can be written in terms of a continuous representation of the non-compact $\mathfrak{su}(1,1)$ Lie algebra. In [@GKRV] the BEP in the closed system (i.e. in absence of external reservoirs) was proven to be dual to the symmetric inclusion process (SIP). This is an interacting particle system modelling particles moving on a lattice with an attractive interaction. The reason behind the above mentioned duality relation lies is in the $\mathfrak{su}(1,1)$ algebraic structure shared by the two processes. BEP and SIP are indeed two elements of a broader class of models all related to the $\mathfrak{su}(1,1)$ Lie algebra and including also other notable models. One of these is the Kipnis-Marchoro-Presutti model (KMP) [@KMP] where the total energy is instantaneously redistributed among sites and that can be recovered as an instantaneous thermalization limit of the BEP. Another model inherently related to the BEP is the Wright-Fisher diffusion [@Ethe] that is the prototype model of mathematical population genetics. Duality between the Wright-fisher and the Moran model can be seen as a particular instance of duality between BEP and SIP (see e.g. [@CGGR1]). In [@CGGR] the analysis was extended to the non-equilibrium situation in which the system is put in contact with two external heat reservoirs imposing two different temperatures $T_\ell\neq T_r$ at the endpoints of the bulk. The corresponding process is also called BEP with open boundaries and has been shown to be dual to the SIP with absorbing boundaries. Not long ago two new models belonging to this class have been introduced in [@Frassek1] via integrable non-compact spin chains and their duality relation shown in [@Frassek2; @FFG]. For these new models a full characterization of the non-equilibrium steady state has been recently proved in [@mixture1; @mixture2], however for an asymmetric dynamics this characterization is still an open problem. The asymmetric version of the model we study (ABEP) was first introduced in [@CGSR2] in a closed boundary setting. This emerged as a scaling limit of the ASIP (an asymmetric version of the inclusion process) in a particular regime of weak asymmetry. In the same work, an alternative construction was proposed for the ABEP, that was shown to be obtainable from BEP, via a non-local transformation $g$ depending on the asymmetry parameter. A duality relation between ABEP and SIP was then deduced in [@CGSR2] as a consequence, independently, of the two above mentioned constructions. The duality function does not have a standard product structure (as is usually the case in the symmetric context) but a nested product structure related to the non-local map $g$. This property is a first instance of a duality relation between a genuinely non-equilibrium asymmetric system (in the sense that it has a non-zero average current) and a symmetric process. This link is made possible by the fact that the dependence on the asymmetry parameter is retained in the duality function, through the map $g$. Here we extend the analysis to the system with open boundaries. In this context the problem becomes the definition of reservoirs itself. The aim is indeed to impose external temperatures $T_\ell\neq T_r$ in such a way as not to alter the condition of existence of a duality relation with the SIP with absorbing boundaries. Our strategy does not directly rely on algebraic considerations on the Markov generator but rather on the link between ABEP and BEP via the non-local map $g$. This transformation procedure allows us to construct reservoirs of the correct form. These turn out to act in a non-standard way. The left reservoir acts only on the left endpoint of the lattice, but its action takes into account the total energy of the system. The right reservoir, instead affects all the sites of the lattice. As a result of this construction we prove a duality relation with the SIP with absorbing boundary by means of two different duality functions. The first one is in a so-said *classical* form whereas the second one is in terms of *generalized Laguerre polynomials*. As far as we know, duality in the presence of an asymmetry together with open boundary condition is still a quite challenging outcome as the classical techniques relying on algebraic considerations do not work. This is due to the fact that the quantum group symmetry needed to construct the duality relation is broken. Results are mainly available for the case of asymmetric simple exclusion process (ASEP). The first attempt is due to Okhubo in [@Ohkubo] where a dual operator has been obtained; however it could not be directly interpreted as a transition matrix for a stochastic process. We mention [@Jeff] where the author generalizes the self-duality of the asymmetric simple exclusion process with an open boundary condition at the left boundary and a closed right boundary. More recent results include [@Guillaume] where a duality relation between an half-line open ASEP and a sub-Markov process where particles perform an asymmetric exclusion dynamics in the bulk and are killed at the boundary is proven. In [@Gunter; @Gunter2] it is shown a reverse duality relation for an open ASEP with open boundary and a shock ASEP with reflecting boundary. .2cm The rest of the paper is organized as follows: in Section [2](#sec2){reference-type="ref" reference="sec2"} we introduce the model of interest, i.e. the ABEP with open boundaries, and show how it can be obtained from its symmetric version via a non-local map transformation. At the end of the section we state some general results which allow to infer properties for a process that can be obtained from another process via a map transformation. In the subsequent two sections these general properties are then specialized to gather information for ABEP starting from known results for BEP: Section [4](#revmes){reference-type="ref" reference="revmes"} is devoted to the study of the case $T_\ell=T_r$ in which the system is proven to be reversible, and the reversible measure is computed; in Section [5](#dualitysection){reference-type="ref" reference="dualitysection"} instead we discuss duality relations. We end with Section [6](#sec6){reference-type="ref" reference="sec6"} where we use the duality results to gather some information on the stationary measure in the general case. In particular we compute what we call the one-point and two-point stationary exponential correlations of the partial energies. # The model {#sec2} The Brownian Energy Process BEP$(\alpha)$ is an interacting diffusion system of continuous spins placed on the sites of a lattice $V$, $\alpha$ is a positive parameter tuning the intensity of the interaction. We consider its asymmetric version ABEP$(\sigma,\alpha)$, $\sigma>0$ the asymmetry parameter, that can be defined when the lattice is one-dimensional $V = \lbrace 1, \ldots, N \rbrace$ and the interaction is nearest-neighbor. To each site of the lattice $i\in V$ is associated an energy $x_i\ge 0$. We denote by $x = (x_1, \ldots, x_N) \in\mathbb R_+^N$ the vector collecting all energies and we call $\Omega :=\mathbb R_+^N$ the state space of the system. When the system is *closed*, or, in other words, in absence of external reservoirs, the dynamics conserves the total energy of the system $E(x):=\sum_{i\in V} x_i$. In this paper we consider the *open system*, i.e. we put the *bulk* lattice $V$ in contact with two external reservoirs placed at artificial sites $V^{{\rm res}} =\left\lbrace 0, N+1 \right\rbrace$. Each reservoir $j \in V^{{\rm res}}$ can be interpreted as a thermal bath characterized by its own fixed temperature $T_j\ge 0$, that is attached to the bulk $V$ only through the boundary sites 1 and $N$. The action of the reservoirs induces an energy exchange between the bulk lattice and the exterior, that destroys the total energy conservation. For simplicity we will also denote by $T_\ell:=T_0$ the temperature of the left reservoir and by $T_r:=T_{N+1}$ the temperature of the right reservoir. .2cm In order to define the model, we need to define two crucial quantities the partial energies $E_i (x)$, $i\in V$, and the non-local map $g$. **Definition 1**. *We define the map $g: \Omega \rightarrow \Omega$ via $$g(x) = \left( g_i \left( x \right) \right)_{i \in V } \qquad \text{with} \qquad g_i(x) := \dfrac{e^{-\sigma E_{i+1}(x)} - e^{-\sigma E_i(x)}}{\sigma }$$ where $E_i (x)$ denotes the energy of the system at the right of site $i \in V$, i.e. $$E_i (x) = \sum_{l = i}^{N} x_{l } \quad \text{for} \quad i=1,\ldots,N \qquad \text{with the convention} \quad E_{N+1}(x) = 0 \;.$$* Notice that the total energy $E(x)$ coincides with the first component $E_1(x)$ of the vector of partial energies. .2cm The stochastic evolution of the collection of energies of the system is governed by a Markov process $\lbrace x(t), t\geq 0 \rbrace$ that we will define by giving its infinitesimal generator $\mathcal{L}^{{\rm ABEP}}$. This acts on smooth functions $f: \Omega\rightarrow \mathbb{R}$ and is given by the sum of three terms, one of them governing the interaction between bulk sites and the other two modelling the action of left and right reservoirs. We define $$\label{abep} \mathcal{L}^{{\rm ABEP}} = \mathcal{L}^{{\rm ABEP}}_{{\rm left}} +\sum_{i=1}^{N-1} \mathcal{L}^{{\rm ABEP}}_{i,i+1} + \mathcal{L}^{{\rm ABEP}}_{{\rm right}}.$$ where, for $i\in \{1, \ldots, N-1\}$, the action on smooth functions $f:\Omega \to \mathbb R$ is $$\begin{aligned} \label{abep-bulk} [\mathcal{L}^{{\rm ABEP}} _{i,i+1} f](x) &= \dfrac{1}{2\sigma ^{2}} \left( 1 - e^{-\sigma x_{i}}\right) \left( e^{\sigma x_{i+1}} -1\right) \left( \dfrac{\partial}{\partial x_{i+1}} - \dfrac{\partial}{\partial x_{i}} \right) ^{2} f(x) \\ & + \dfrac{1}{\sigma } \biggl( \left( 1 - e^{-\sigma x_{i}}\right) \left( e^{\sigma x_{i+1}} -1\right) + \alpha \left(2 - e^{-\sigma x_i} - e^{\sigma x_{i+1}} \right) \biggr) \left( \dfrac{\partial}{\partial x_{i+1}} - \dfrac{\partial}{\partial x_{i}} \right) f(x)\nonumber\end{aligned}$$ whereas $$\label{abep-res} [\mathcal{L}^{{\rm ABEP}}_{{\rm left}}f](x)= T_\ell \left(e^{\sigma E(x) } \left( \alpha -1 + e^{\sigma x_1} \right) \dfrac{\partial}{\partial x_{1} } + \dfrac{e^{\sigma E(x)}}{\sigma }\left(e^{\sigma x_{1}} - 1 \right) \dfrac{\partial^{2}}{\partial x_{1}^{2}} \right) f(x)- \dfrac{e^{\sigma x_{1}} -1}{\sigma } \dfrac{\partial}{\partial x_{1}}f(x)$$ and $$\begin{aligned} \label{abep-res-dx} [\mathcal{L}^{{\rm ABEP}}_{{\rm right}}f](x)&= \left(\alpha T_r- \dfrac{1-e^{-\sigma x_N}}{\sigma } \right) \sum_{l =1}^{N} e^{\sigma E_{l }(x)} \left( \partial_{x_l } - \partial_{x_{l -1}} \right) f(x) + \\ & T_r \dfrac{1-e^{-\sigma x_N}}{\sigma } \sum_{l , j =1}^{N} e^{\sigma \left( E_{l } (x) + E_{j} (x) \right)} \left( \partial_{x_l } - \partial_{x_{l -1}} \right) \left( \partial_{x_j} - \partial_{x_{j-1}} \right)+ T_r \left( 1-e^{-\sigma x_N} \right) \sum_{l = 1}^{N} e^{2\sigma E_{l }(x) } \left( \partial_{x_l } - \partial_{x_{l -1}} \right) f(x) \ . \nonumber\end{aligned}$$ The action of reservoirs is non-local in two different ways. The left reservoir acts only on the left boundary site 1, but its action takes in account the total energy $E(x)$ that is not an invariant of the dynamics. The right reservoir, instead, affects the whole chain. # From BEP to ABEP The BEP$(\alpha)$ on $V$ is the symmetric version of the ABEP$(\sigma, \alpha)$ obtained in the limit as $\sigma \to 0$. As in the previous section we consider the system with nearest-neighbor interaction in contact with two boundary reservoirs kept at temperature $T_\ell$ and $T_r$. We denote by $\{z(t), \ t \ge 0\}$ the Brownian Energy process on the space state $\Omega= \mathbb R_+^N$ describing the evolution of the vectors $z:=(z_1, \ldots, z_N)$ of single-site energies. The infinitesimal generator, acting on smooth functions $f:\Omega \to \mathbb R$, is defined as follows $$\label{bep} \mathcal{L}^{{\rm BEP}} = \mathcal{L}^{{\rm BEP}}_{{\rm left}}+ \sum_{i=1}^{N-1} \mathcal{L}_{i,i+1}^{{\rm BEP}} + \mathcal{L}^{{\rm BEP}}_{{\rm right}}$$ where, for $i\in \{1, \ldots, N-1\}$, $$\label{bulkbep} \mathcal{L}_{i,i+1}^{{\rm BEP}} f (z) = \left[ z_{i}z_{i+1}\left(\partial_{z_{i+1}}-\partial_{z_{i}}\right)^{2}-\alpha(z_{i}-z_{i+1})\left(\partial_{z_{i+1}}-\partial_{z_{i}}\right) \right] f(z)$$ whereas $$\mathcal{L}^{{\rm BEP}}_{{\rm left}} f (z) = \left[ T_\ell \left( \alpha \dfrac{\partial}{\partial_{z_{1}}} + z_{1} \dfrac{\partial^{2}}{\partial_{z_{1}}^{2}} \right) - z_{1} \dfrac{\partial}{\partial_{z_{1}}} \right] f (z)$$ and $$\label{end} \mathcal{L}^{{\rm BEP}}_{{\rm right}} f (z) = \left[ T_r \left( \alpha \dfrac{\partial}{\partial_{z_{N}}} + z_{N} \dfrac{\partial^{2}}{\partial_{z_{N}}^{2}} \right) - z_{N} \dfrac{\partial}{\partial_{z_{N}}} \right] f (z) \ .$$ The latter terms give the action of left and right reservoirs that are attached, respectively, to site $1$ and site $N$. .3cm It can be easily checked that $\mathcal{L}^{{\rm BEP}}$ is recovered from $\mathcal{L}^{{\rm ABEP}}$ by suitably taking the limit as $\sigma \to 0$. On the other hand $\mathcal{L}^{{\rm ABEP}}$ can be constructed from $\mathcal{L}^{{\rm BEP}}$ by acting with the non-local map $g$ introduced in Definition [Definition 1](#mapg){reference-type="ref" reference="mapg"}. This claim has been proven in [@CGSR2] for the closed system. Below we show that such a construction can be extended to the reservoir terms of the generator. **Theorem 1** (From BEP to ABEP). *Let $g$ be the map in Definition [Definition 1](#mapg){reference-type="ref" reference="mapg"}, then for all $f \in {\cal D}({\cal L}^{{\rm BEP}})$ we have $$\label{connect} \mathcal{L}^{{\rm ABEP}} (f \circ g) =\left[ \mathcal{L}^{{\rm BEP}} f \right] \circ g \ .$$* **Proof:** Throughout this proof we will use the alternative notation for the reservoir terms of the generators $\mathcal{L}^{{\rm BEP}}_{0,1}:=\mathcal{L}^{{\rm BEP}}_{{\rm left}}$ and $\mathcal{L}^{{\rm BEP}} _{N,N+1}:=\mathcal{L}^{{\rm BEP}}_{{\rm right}}$, respectively, $\mathcal{L}^{{\rm ABEP}}_{0,1}:=\mathcal{L}^{{\rm ABEP}}_{{\rm left}}$ and $\mathcal{L}^{{\rm ABEP}} _{N,N+1}:=\mathcal{L}^{{\rm ABEP}}_{{\rm right}}$ and write, for $(i,j)\in V \times V^{{\rm res}}$, $$\label{three} \mathcal{L}_{i,j}^{{\rm BEP}} f (z) = \left[ T_{j} \left( \alpha \dfrac{\partial}{\partial_{z_{i}}} + z_{i} \dfrac{\partial^{2}}{\partial_{z_{i}}^{2}} \right) - z_{i} \dfrac{\partial}{\partial_{z_{i}}} \right] f(z)$$ and $$\begin{aligned} \label{abep-res-general} [ \mathcal{L}^{{\rm ABEP}} _{i,j} f](x)&= \left(\alpha T_j - g_i(x) \right) \left[ \sum_{l =1}^{i-1} e^{\sigma E_{l }(x)} \left( 1-e^{-\sigma x_l } \right) \partial_{x_l } + e^{\sigma E_i (x)} \partial_{x_i} \right] f(x)+ \\ \nonumber & T_j g_i(x) \left[ \sum_{l , j =1}^{i-1} e^{\sigma E_{l } (x)} \left( 1- e^{-\sigma x_l } \right) \, e^{\sigma E_{j} (x)} \left( 1- e^{-\sigma x_j} \right) \partial^{2}_{x_l x_j} + e^{2\sigma E_i (x)} \partial^{2}_{x_i} \right. \\ \nonumber & \left. 2 \sum_{l =1}^{i-1} e^{\sigma E_{l }(x)} \left( 1 - e^{-\sigma x_l }\right)e^{\sigma E_i (x)} \partial^{2}_{x_l x_i} + \sum_{l =1}^{i-1} \sigma e^{2\sigma E_{l } (x) } \left( 1- e^{-2\sigma x_l } \right) \partial_{x_l } + \sigma e^{2\sigma E_i (x)} \partial_{x_i} \right] f(x) \;. \end{aligned}$$ In this way we can resume the proof of the theorem in the following two steps: 1. for all $i \in V$, $x \in \Omega$, $$\left[ \mathcal{L}_{i,i+1}^{{\rm BEP}} f \right] (g(x)) = \left[ \mathcal{L}_{i,i+1}^{{\rm ABEP}} f \circ g \right] (x)$$ 2. for all $(i,j)\in V \times V^{{\rm res}}$, $x \in \Omega$, $$\label{compo} \left[ \mathcal{L}_{i,j}^{{\rm BEP}} f \right] (g(x)) = \left[ \mathcal{L}_{i,j}^{{\rm ABEP}} f \circ g \right] (x)$$ Step 1 has been proven in Theorem 3.4 of [@CGSR2]. It remains to prove Step 2. Recalling the definition of $g$: $$\begin{aligned} g:\Omega \;&\rightarrow\; \Omega \\ x\;&\rightarrow\;g(x)=(g_{i}(x))_{i\in V}, \qquad \text{with} \qquad g_i(x)=\frac{e^{-\sigma E_{i+1}(x)}-e^{-\sigma E_{i}(x)}}{\sigma }\end{aligned}$$ with the convention $E_{N+1}(x)=0$ and $E_1(x)=E(x)$. .2cm Notice that the map $g$ is not full range, i.e. $g(\Omega )\neq \Omega$, indeed $$E(g(x))= \frac 1 {\sigma} \left(1-e^{-2\sigma E(x)}\right)\le \frac1{\sigma}$$ so that in particular $g(\Omega)\subseteq \{x\in \Omega : E(x)\le 1/\sigma\}$. Moreover $g$ is a bijection from $\Omega$ to $g(\Omega)$. Indeed, denoting by $g^{\rm{inv}}: g(\Omega) \rightarrow \Omega$ the inverse transform of $g$. In other words, if $z=g(x)\in g(\Omega)$, then $x=g^{\rm{inv}}(z)$ with $i$th component being $$g^{\rm{inv}}_i(z)=\frac 1 {\sigma } \, \ln \left\{ \frac{1-\sigma E_{i+1}(z)}{1-\sigma E_{i}(z)}\right\}$$ Let $F:= f \circ g$, or, equivalently, $f=F \circ g^{\rm{inv}}$ namely $F(x)=f(g(x))$ for $x\in \Omega$ and $f(z)=F(g^{\rm{inv}}(z))$ for $z\in g(\Omega)$, therefore, in order to prove [\[compo\]](#compo){reference-type="eqref" reference="compo"}, it is sufficient to show that, for all $x \in \Omega$, $$\label{compo1} \left[ \mathcal{L}_{i,j}^{{\rm BEP}} (F \circ g^{\rm{inv}} )\right] (g(x)) = \left[ \mathcal{L}_{i,j}^{{\rm ABEP}} F \right] (x)$$ At this aim we compute the first and second derivatives of $f=F \circ g^{\rm{inv}}$. We have $$\begin{aligned} \label{Df} \frac{\partial f}{\partial z_k}(z)= \sum_{l \in V} \frac{\partial F}{\partial x_l }(g^{\rm{inv}}(z)) \cdot \frac{\partial g^{\rm{inv}}_l }{\partial z_k}(z) \qquad \text{for all} \qquad k \in V\end{aligned}$$ and $$\begin{aligned} \label{DDf} \frac{\partial^2 f}{\partial^2 z_k z_m}(z)= \sum_{l ,j\in V} \frac{\partial^2 F}{\partial^2 x_l x_j}(g^{\rm{inv}}(z)) \cdot \frac{\partial g_{l}^{\rm{inv}} }{\partial z_k}(z) \cdot \frac{\partial g_{j}^{\rm{inv}}}{\partial z_m}(z) + \sum_{l \in V} \frac{\partial F}{\partial x_l }(g^{\rm{inv}}(z)) \cdot \frac{\partial^2 g^{\rm{inv}}_{l} }{\partial^2 z_k z_m}(z) \qquad \text{for all} \qquad k,m \in V \;.\end{aligned}$$ We now compute all the first and second derivatives of all the components of the inverse function $g^{\rm{inv}}$, we obtain $$\label{firstDerivativeh} \frac{\partial g^{\rm{inv}}_l }{\partial z_k}(z)=\begin{cases} 0\quad &\text{if}\;k<l \\ \frac{1}{1-\sigma E_{l }(z)}\quad &\text{if}\;k=l \\ \frac{\sigma z_{l }}{(1-\sigma E_{l }(z)) (1-\sigma E_{l +1}(z))}\quad &\text{if}\;k>l \end{cases}$$ and, for $m\leq k$ (it is symmetric in $k$ and $m$), $$\label{secondDerivativeh} \frac{\partial^{2} g^{\rm{inv}}_{l }}{\partial z_{m}\partial z_{k}}(z)= \begin{cases} 0\quad &\text{if}\; m<l \\ \frac{\sigma }{\left(1-\sigma E_{l }(z)\right)^{2}}\quad &l =m\leq k\\ \frac{z_{l }\sigma ^{2}(2-\sigma z_{l }-2\sigma E_{l +1}(z))}{\left[(1-\sigma E_{l }(z)) (1-\sigma E_{l +1}(z))\right]^{2}} \quad &\text{if}\;m>l \end{cases}$$ These derivatives simplify observing that, thanks to telescopicity of the sum, $$\label{energyZenergyX} E_{l }(z)=\sum_{i=l }^{N}z_{i}=\sum_{i=l }^{N}g_{i}(x)=\frac{1}{\sigma }\left(1-e^{-\sigma E_{l }(x)}\right) \;.$$ And then, using [\[energyZenergyX\]](#energyZenergyX){reference-type="eqref" reference="energyZenergyX"} we can simplify the expressions for the derivatives as follows: $$\label{Dh} \frac{\partial g^{\rm{inv}}_{l }}{\partial z_{k}}(z)= \begin{cases} 0\quad &\text{if}\;k<l \\ e^{\sigma E_{l }(x)}\quad &\text{if}\;k=l \\ e^{\sigma E_{l }(x)} \left( 1 - e^{-\sigma x_{l }} \right) \quad &\text{if}\;k>l \end{cases}$$ and $$\label{DDh} \frac{\partial^{2} g^{\rm{inv}}_{l }}{\partial z_{m}\partial z_{k}}(z)= \begin{cases} 0\quad &\text{if}\; m<l \\ \sigma e^{2\sigma E_{l }(x)}\quad &l =m\leq k\\ \sigma e^{2\sigma E_{l }(x)} \left(1-e^{-2\sigma x_{l }}\right) \quad &\text{if}\;m>l \ . \end{cases}$$ Then by substituting the expressions [\[Dh\]](#Dh){reference-type="eqref" reference="Dh"} and [\[DDh\]](#DDh){reference-type="eqref" reference="DDh"} into equations [\[Df\]](#Df){reference-type="eqref" reference="Df"} and [\[DDf\]](#DDf){reference-type="eqref" reference="DDf"} we obtain explicit expressions for the first and second derivatives of $f=F \circ g^{\rm{inv}}$. Finally, the identity [\[compo1\]](#compo1){reference-type="eqref" reference="compo1"} follows by replacing these expressions into the BEP$(\alpha)$ boundary generators given in [\[three\]](#three){reference-type="eqref" reference="three"}. $\square$ ## Some general definitions and properties {#sec3} The construction of ABEP$(\sigma,\alpha)$ as a non-local transformation of BEP$(\alpha)$, allows to derive several fundamental properties of the asymmetric process, such as duality properties or the structure of the stationary measure. These are by starting from the analogous properties of the symmetric process and projecting them via the map $g$. Having this goal in mind, in this section we prove some general results relating two Markov processes that are connected via a map transformation. .2cm We start by recalling the definition duality in terms of infinitesimal generators of two Markov processes. We will denote by ${\cal D}({\cal L})$ the domain of ${\cal L}$. .2cm **Definition 2** (Generator duality). *Let ${\cal L}$ and $L$ be the infinitesimal generators of two Markov processes $\{X(t):t\geq 0\}$ and $\{Y(t): t\geq 0\}$ defined, respectively, on the state spaces $\Omega$ and $\Omega^{{\rm dual}}$. Let $D: \Omega\times {\Omega}^{{\rm dual}}\to\mathbb R$ be a measurable function, such that $D(y, \cdot) \in {\cal D}({\cal L})$ and $D(\cdot, x)\in {\cal D}(L)$. We then say that $D$ is a duality function for generator duality between the processes $\{X(t):t\geq 0\}$ and $\{Y(t): t\geq 0\}$ if for all $x\in \Omega, y\in {\Omega}^{{\rm dual}}$, we have $$\label{gendualfirstdef} \left({\cal L} D(\cdot, x)\right) (y)=\left(L D(y, \cdot) \right)(x)$$* .2cm In the next theorem we will see that if a stationary measure, a reversible measure or a duality function are known for one of the a processes, then the corresponding object can be computed for a process obtained from the original one via a transformation. **Theorem 2**. *Let $g$ be a map $g:\Omega \to \Omega$, with $\Omega \subseteq \mathbb R_+^N$ and let ${\cal L}$ and $\widehat{\cal L}$ be the infinitesimal generators of two Markov processes on the state spaces, respectively $\Omega$ and $\widehat \Omega:=g(\Omega)$. Suppose that $\forall f \in {\mathcal D}({\cal L})$ it holds that $f \circ g \in {\mathcal D}(\widehat {\cal L})$ and $$\label{compo} \widehat{\cal L} (f\circ g)=\left({\cal L}f\right)\circ g$$ then we have the following properties. [\[main\]]{#main label="main"}* - *Let $\mu$ be a measure on $\Omega$ absolutely continuous w.r.t. Lebesgue. Let ${\cal J}$ be the Jacobian matrix of the map $g$. If $\mu$ is a stationary (reversible) measure for ${\cal L}$ then $$\label{compomu} \hat\mu := \left(\mu \cdot \rm{det}{\cal J} \right)\circ g$$ is a stationary (reversible) measure for $\widehat{\cal L}$.* - *Let $L$ be the infinitesimal generators a Markov processes on the state space $\Omega^{{\rm dual}}$. If ${\cal L}$ is dual to $L$ with duality function $D: \Omega \times \Omega^{{\rm dual}} \to \mathbb R$, then $\widehat{\cal L}$ is dual to $L$ with duality function $D: \widehat \Omega \times \Omega^{{\rm dual}} \to \mathbb R$ $$\widehat D(\cdot,\xi):= D(\cdot, \xi) \circ g \ ,\qquad \qquad \xi \in \Omega^{{\rm dual}} \ .$$* **Proof:** .1cm - Due to the absolute continuity of $\mu$ we can write, with a slight abuse of notation, that $\mu(dx)= \mu(x) \ dx$. The stationarity condition for $\mu$ with respect to ${\cal L}$ then reads $$\int[{\cal L}f](z) \, \mu(z) \ dz =0, \qquad \text{for all} \quad f \in {\cal D}({\cal L})$$ that, taking the change of variables $z=g(x)$, gives $$\int [{\cal L}f](g(x)) \, \cdot \mu(g(x)) \cdot \rm{det} {\cal J}(g(x)) \, dx =0, \qquad \text{for all} \quad f \in {\cal D}({\cal L})$$ that, thanks to [\[compo\]](#compo){reference-type="eqref" reference="compo"} and [\[compomu\]](#compomu){reference-type="eqref" reference="compomu"}, is equivalent to $$\int [\widehat{\cal L}(f\circ g)](x) \, \cdot \hat\mu(x) \, dx =0, \qquad \text{for all} \quad f \in {\cal D}({\cal L}) \ .$$ Due to the fact that $D(\widehat{\cal L})= \{F=f\circ g: \: f\in D({\cal L})\}$, the last identity can be rewritten as $$\int [\widehat{\cal L} F](x) \, \cdot \hat\mu(x) \, dx =0, \qquad \text{for all} \quad F\in {\cal D}(\widehat{\cal L})$$ that is the stationary condition of $\hat \mu$ with respect to $\widehat{\cal L}$. The statement regarding reversible measures can be proven in an analogous way. - To prove the second statement we use the duality relation between ${\cal L}$ and $L$ and take the composition of the duality function (as a function of the variable $x$) with the function $g$. For $x\in \Omega$ and $\xi \in \Omega^{{\rm dual}}$, we have $$\begin{aligned} \left[\widehat {\cal L}\widehat D(\cdot,\xi)\right](x) &=&\left[\widehat{\cal L}(D(\cdot,\xi)\circ g)\right](x) \\ & = &\left[{\cal L}D(\cdot,\xi)\right](g(x)) \nonumber\\ &=&\left[LD(g(x),\cdot)\right](\xi)\nonumber\\ & =&\left[L \widehat D(x,\cdot)\right](\xi)\ .\end{aligned}$$ This concludes the proof of the second item. $\square$ In the next two sections we specialize the argument of the above theorem for our model of interest. In Section [4](#revmes){reference-type="ref" reference="revmes"} we focus on the cas in which the external reservoirs impose the same temperatures (i.e. when $T_\ell=T_r=T$). We prove that in this situation ABEP$(\sigma,\alpha)$ is reversible and we find the reversible measure. In Section [5](#dualitysection){reference-type="ref" reference="dualitysection"} we find two duality relations for ABEP$(\sigma,\alpha)$. # Equal temperature reservoirs {#revmes} In this section use item i) of Theorem [Theorem 2](#General_result){reference-type="ref" reference="General_result"} to withdraw some conclusions concerning the case in which the two reservoirs have the same temperature. The idea is to import this property from the reversibility of the corresponding symmetric process. From Section 3 of [@CGGR] we know indeed that, in absence of reservoirs, the BEP$(\alpha)$ is reversible. In particular it admits a one-parameter family of reversible probability measures $\mu_T$, $T\ge 0$, that are products of Gamma distributions of shape parameters $\alpha$ and scale parameter $T$, i.e. $\mu_{T}^{{\rm BEP}} (z) \ dz$ with $$\mu_{T}^{{\rm BEP}} (z)= \prod_{i=1}^{N}\frac{e^{-z_i /T}z_i^{(\alpha -1)}}{\Gamma(\alpha )T^{\alpha }}$$ When the process is in contact with two reservoirs kept at equal temperatures, $T_\ell=T_r=T$, the process remains reversible, admitting $\mu_T$ as the unique stationary probability measure. In the following Theorem we extend the statement to the asymmetric process, for which we prove the existence of a unique reversible probability measure that is in the form of a product measure times a function of the total energy of the system $E(x)$. **Theorem 3** (Reversible measure for ABEP). *The ABEP$(\sigma,\alpha)$ with equal reservoir temperatures $T_\ell = T_r = T$ is reversible with respect to the unique stationary probability measure $\mu_{T}^{{\rm ABEP}} (x) \ dx$, with $$\label{muabep} \mu_{T}^{{\rm ABEP}} (x)= \exp\left\{ \frac{e^{-\sigma E(x)} -1}{\sigma T} \right\}\cdot \prod_{i=1}^{N} \dfrac{(1-e^{-\sigma x_i})^{(\alpha -1)} }{ \Gamma(\alpha ) \sigma^{\alpha -1} T^{\alpha }} \ e^{-\sigma x_i(\alpha(i -1)+1)}$$* **Proof:** We want to use item i) of Theorem [Theorem 2](#General_result){reference-type="ref" reference="General_result"}. To this aim it is enough to compute $(\mu_{T}^{{\rm BEP}}\circ g)(x)$. Indeed, $$\begin{aligned} \mu_{T}^{{\rm ABEP}} (x) &= (\mu_{T}^{{\rm BEP}}\circ g)(x) = \prod_{i=1}^{N}\mu^{{\rm BEP}}_{T}(g_i(x))=\prod_{i=1}^{N}\frac{e^{-g_{i}(x) /T}(g_{i}(x))^{(\alpha -1)}}{\Gamma(\alpha )T^{\alpha }} \: {\rm det} \mathcal{J}(g(x)) \\& = \prod_{i=1}^{N} \dfrac{1}{\Gamma(\alpha ) T^{\alpha }} \cdot \exp\left\{-\frac{e^{-\sigma E_{i+1}(x)}-e^{-\sigma E_{i}(x)}}{\sigma T}\right\} (1-e^{-\sigma x_i})^{(\alpha -1)} \ \dfrac{e^{-\sigma (\alpha -1) E_{i+1}(x)}}{\sigma ^{(\alpha -1)}} \:e^{-\sigma E_i(x)} \\& = \sigma \cdot { e^{\sigma \alpha E(x)} e^{(T-\sigma )E(x)} \exp\left\{ \frac{e^{-\sigma E(x)} -1}{\sigma T} \right\} }\cdot \prod_{i=1}^{N} \frac{(1-e^{-\sigma x_i})^{(\alpha -1)} e^{-(\sigma \alpha i + T)x_i} }{ (\sigma T)^{\alpha }\Gamma(\alpha )} \\& = \exp\left\{ \frac{e^{-\sigma E(x)} -1}{\sigma T} \right\} \cdot \prod_{i=1}^{N} \dfrac{(1-e^{-\sigma x_i})^{(\alpha -1)} }{\sigma ^{\alpha -1} T^{\alpha } \Gamma(\alpha )} e^{-(\sigma \alpha i + T)x_i} e^{(\sigma \alpha + T -\sigma )x_i} \end{aligned}$$ here, by calling again $z=g(x)$, $J$ is the Jacobian $N\times N$ matrix given by $$J(z)=\left(\frac{\partial g_{l}^{\rm{inv}}}{\partial z_{k}}\right)_{k\in\{1,\ldots,N\},\,l\in\{1,\ldots,N\}}$$ where the partial derivative are computed in [\[Dh\]](#Dh){reference-type="eqref" reference="Dh"}. Therefore, [\[muabep\]](#muabep){reference-type="eqref" reference="muabep"} follows. $\square$ **Remark 1**. *In Theorem 3.3 of [@CGSR2]) a family of reversible measures has been found for ABEP$(\sigma,\alpha)$ with closed boundary. This family is labeled by the temperature $T$. The measure corresponding to the temperature $T$ (eq. (3.15 - 3.16) of [@CGSR2]) does not match with $\mu_{T}^{{\rm ABEP}}$ found in [\[muabep\]](#muabep){reference-type="eqref" reference="muabep"}. Indeed it differs from it only for the factor in front of the product in [\[muabep\]](#muabep){reference-type="eqref" reference="muabep"} that is a function of the total energy $E(x)$. This is due to the fact that, in absence of reservoirs, the total energy is an invariant of the dynamics, and then this term becomes a constant that simplifies with the normalizing factor of the probability measure. In the presence of two thermal reservoirs instead, even in the case of equal temperatures $T_\ell=T_r=T$, the system does not conserve the total energy anymore, and the initial factor in [\[muabep\]](#muabep){reference-type="eqref" reference="muabep"} can not be neglected anymore.* # Duality results {#dualitysection} When $T_\ell \neq T_r$ reversibility is lost. Nevertheless there exists a unique stationary measure depending on both temperatures $T_\ell$ and $T_r$. However a full characterization of such a measure is a difficult and still open problem, even for the symmetric case. A tool that has proven to be of great help in the study of the properties of the stationary measure is duality. We will return to the study of steady state in Section [6](#sec6){reference-type="ref" reference="sec6"}. In the next section we prove two duality relations between the Asymmetric Brownian Energy process and the Symmetric Inclusion process with absorbing boundaries. ## Duality between ABEP and SIP The Symmetric Inclusion Process is a system of interacting particles moving in a lattice with attractive, nearest-neighbor interaction. It was originally introduced in [@GKR] as the attractive counterpart of the Simple Symmetric Exclusion Process. Each site can host for an unbounded number of particles, and then the state space of the inclusion process on the lattice $V=\{1, \ldots, N\}$ is $\mathbb N_0^N$. The attraction intensity is tuned by a parameter $\alpha>0$. Each particle may jump to its left or its right with rates proportional to the number of particles in the departure site and to the number of particles in the arrival site plus $\alpha$. We use the acronym SIP$(\alpha)$ for the Symmetric Inclusion process of parameter $\alpha$. Duality between BEP$(\alpha)$ and SIP$(\alpha)$ is well known in the literature. When the BEP system is put in contact with two external reservoirs a duality relation still holds true. The dual process is still a system of inclusion particles inclusion, with the difference that the boundary conditions at the endpoints of the chain are no longer closed but absorbing. We give below the definition of the SIP$(\alpha)$ with absorbing boundaries. Notice that for this process the boundary sites $0$ and $N+1$ are no longer *artificial*, since their state is relevant in the dynamics. Configurations are then $N+2$-dimensional vectors that will be denoted by $\xi:=(\xi_0, \xi_1, \ldots, \xi_N, \xi_{N+1})$, $\xi_i$ being the number of particles at site $i$. The state space of this process is then the set $\Omega^{{\rm dual}}=\mathbb N_0^{V\cup V^{{\rm res}}}$, keeping in mind that, even if we keep the same notation $V^{{\rm res}}$ for the set $\{0,N+1\}$, these sites in the dual process do no longer have the meaning of reservoirs but represent the absorbing sites. These leave eventually the bulk empty by absorbing all the particles. **Definition 3** (SIP with absorbing boundaries). *We denote by $\{\xi(t), \ t \ge 0\}$ the SIP$(\alpha)$ on $V$ with absorbing boundaries $0$ and $N+1$, the Markov process on $\Omega^{{\rm dual}}=\mathbb N_0^{V\cup V^{{\rm res}}}$ whose infinitesimal generator acts on functions $f: \Omega^{{\rm dual}} \rightarrow \mathbb{R}$ and is defined as follows: $$\label{sip} {L}^{{\rm SIP-abs}}= {L}^{\rm{abs}}_{{\rm left}} + \sum_{i =1}^{N-1} {L}_{i,i+1}^{{\rm SIP}} + {L}^{\rm{abs}}_{{\rm right}} \;,$$ where, for all and $i\in \{1, \ldots N-1\}$, $$\begin{aligned} \label{sipbulk} [{L}_{i,i+1}^{{\rm SIP}} \ f ] (\xi )= \sum_{i=1}^{N-1} \xi_i (\alpha +\xi_{i+1}) \left[ f(\xi^{i,i+1}) - f(\xi)\right] + \xi_{i+1} (\alpha +\xi_{i}) \left[ f(\xi^{i+1,i}) - f(\xi)\right] \end{aligned}$$ and $$(\xi) := \xi_1 \left[ f(\xi^{1,0}) - f(\xi)\right] \qquad \text{and}\qquad [ {L}^{\rm{abs}}_{{\rm right}} \ f ] (\xi )= \xi_N \left[ f(\xi^{N,N+1}) - f(\xi)\right] \; .$$* .2cm Besides being dual to the BEP, the Inclusion process with closed boundaries has been proved to be dual to the ABEP. This property has been proved in [@CGSR2] and is the first example of duality between an asymmetric system (i.e. bulk-driven) and a symmetric system (with zero current). This is made possible by the fact that the dependence on the asymmetry parameter $\sigma$ is transferred to the duality function. Here we generalize the result to the ABEP with reservoirs, that will be proven to be dual, again, to the Inclusion process with absorbing boundaries, exactly as its symmetric counterpart. This property will be proven using item 2 of Theorem [Theorem 2](#General_result){reference-type="ref" reference="General_result"} and using the relation [\[General_result\]](#General_result){reference-type="eqref" reference="General_result"} that connects ABEP and BEP through the map $g$. We will prove two different duality relations between the same two processes. The first duality relation is via the so-called *classical duality function* [@CGGR], the second is in terms of a duality function that is a product of Laguerre polynomials, i.e. of the type *orthogonal polynomial duality function* [@simone]. ### Duality properties for the symmetric process. We start by showing two duality relations between BEP$(\alpha)$ with reservoirs and SIP$(\alpha)$ with absorbing boundaries. The relations are given with respect to two different duality functions. The first duality relation is well-known, it is given in terms of the so-called *classical duality* and has been proven in [@CGGR]. The second result instead is given in terms of a duality function belonging to the class of *orthogonal polynomials dualities*, and more precisely it is related to the so-called *generalized Laguerre polynomials*. Differently from the classical one, the orthogonal duality result for the open system is new, being available only for the closed system (see [@FG] for the proof). **Theorem 4** (Duality between open BEP and SIP with absorbing boundaries). *The BEP$(\alpha)$ with an open boundaries, with generator ${\cal L}^{{\rm BEP}}$ defined in [\[bep\]](#bep){reference-type="eqref" reference="bep"}-[\[end\]](#end){reference-type="eqref" reference="end"}, is dual to the SIP$(\alpha)$ with absorbing boundaries defined in Definition [Definition 3](#def-sip){reference-type="ref" reference="def-sip"} with respect to the following duality functions:* 1. ***classical duality:** $$\label{df} D(z, \xi) = T_\ell ^{\xi_0} \cdot \prod_{i=1}^{N} \dfrac{\Gamma(\alpha )}{ \Gamma(\alpha + \xi_i)} z_i^{\xi_i} \cdot T_r^{\xi_{N+1}},$$* 2. ***orthogonal duality:** $$\label{odbepsip} {\mathfrak D}_T(z,\xi) = \left( T_\ell - T \right)^{\xi_0} \cdot \prod_{i=1}^{N} (-T)^{\xi_i}\cdot \mathstrut_1 F_1 \left( {\left. \genfrac{}{}{0pt}{} {-\xi_{i}} { \alpha } \right\vert {\frac{z_{i}}{T}}} \right)\cdot (T_r-T)^{\xi_{N+1}}\;,$$ for all $T >0$.* **Proof:** For the proof of item 1 we refer to Theorem 4.1 of [@CGGR]. In order to prove the second item, we have to show that $$(z) = [{L}^{{\rm SIP}} {\mathfrak D}_T(z, \cdot) ](\xi)$$ Since both $\mathcal{L}^{{\rm BEP}}$ and ${L}^{{\rm SIP}}$ of a bulk term and two reservoir terms, it is sufficient to show that the duality relation for generators holds true term by term. The relation for the bulk terms of the generators has been proved in Section 4.2 of [@FG], where it has been shown that, defining $d(\zeta, k) = (-T)^{k}\mathstrut_1 F_1 \left( {\left. \genfrac{}{}{0pt}{} {-k} { \alpha } \right\vert {\frac{\zeta}{T}}} \right)$, for all $i\in \{1, \ldots, N-1\}$, $$(z_i, z_{i+1}) = [{L}_{i,i+1}^{{\rm SIP}} d(z_i, \cdot) \cdot d(z_{i+1},\cdot)] (\xi_i, \xi_{i+1}) \ .$$ It is remains to show that the duality relation holds for the two boundary terms. i.e. that $$(z) = [{L}_{{\rm left}}^{{\rm abs}} {\mathfrak D}_T(z, \cdot) ](\xi) \qquad \text{and} \qquad [\mathcal{L}_{{\rm right}}^{{\rm BEP}} {\mathfrak D}_T(\cdot, \xi) ](z) = [{L}_{{\rm right}}^{{\rm abs}} {\mathfrak D}_T(z, \cdot) ](\xi) \ .$$ Being the two relations completely analogous, it is sufficient to prove one of them, we prove it for the left boundary. We note that $\mathcal{L}_{{\rm left}}^{{\rm BEP}}$ acts only on site one whereas $[{L}_{{\rm left}}^{{\rm abs}}$ acts only on sites 0 and 1. For this reason it is sufficient to show that, for $d_\ell(k):= (T_\ell - T )^k$, $$(z_1) = [{L}_{{\rm left}}^{{\rm abs}} d_\ell(\cdot) d(z_1, \cdot)](\xi_0,\xi_1)$$ At this aim, using the hypergeometric relation satisfied by Laguerre polynomials (see Section 9.12 in [@Koekoe]), we find that $$\begin{aligned} \label{1} z_1 \partial^{2}_{z_{1}}d(z_1, \xi_1) + \alpha \partial_{z_{1}}d(z_1, \xi_1) = \xi_1 d(z_1, \xi_{1}-1) \\ z_1 \partial_{z_{1}}d(z_1, \xi_1) = \xi_1 d(z_1, \xi_{1}) + \xi _1T d(z_1, \xi_{1}-1) \;. \label{2}\end{aligned}$$ The above identities allow us to write the action of $\mathcal{L}_{{\rm left}}^{{\rm BEP}}$ on $d(z_1, \xi_1)$ as an action on the variable $\xi_1$. $$\begin{aligned} (z_1) = \left( T_\ell - T \right)^{\xi_0} \left[ T_\ell \xi_1 d\left( z_1, \xi_1 -1\right) - \xi_1 d\left( z_1, \xi_{1} \right) - \xi _1 Td \left( z_1, \xi_{1}-1 \right) \right] = \\ \xi_1 \left[ \left( T_\ell - T \right)^{\xi_0 +1} d(z_1, \xi_1 -1) - \left( T_\ell - T \right)^{\xi_0 } d(z_1, \xi_1 ) \right] = [{L}_{{\rm left}}^{{\rm abs}} d_\ell(\cdot) d(z_1, \cdot)](\xi_0,\xi_1)\end{aligned}$$ that concludes the proof. $\square$ **Remark 2**. *The so called orthogonal duality function ${\mathfrak D}_T$ is related to the so-called generalized Laguerre polynomial via a normalizing factor only depending on the variable $\xi$. More precisely, the generalized Laguerre polynomial of degree $n$, variable $x$ and parameter $\beta$ is defined as follows $${\mathfrak L}_{\xi}^{\left( \alpha -1 \right)}(z) =\dfrac{\Gamma(\alpha + \xi) }{\Gamma(\alpha ) \xi!} \mathstrut_1 F_1 \left( {\left. \genfrac{}{}{0pt}{} {- \xi} { \alpha } \right\vert {z}} \right)\;$$ and then the single site duality function $d$ is related to these via the following relation $$d(\zeta, k) = (-T)^{k} \cdot \dfrac{\Gamma(\alpha ) k!} {\Gamma(\alpha +k)}\cdot {\mathfrak L}_{k}^{\left( \alpha -1 \right)}(\zeta) \ .$$* ### Duality properties for the asymmetric process Once the duality relation for the symmetric process is proven we can invoke Theorem [Theorem 2](#General_result){reference-type="ref" reference="General_result"} to extend the result to the ABEP. **Theorem 5** (Duality between open ABEP and SIP with absorbing boundaries). *The ABEP$(\sigma,\alpha)$ with an open boundaries, with generator ${\cal L}^{{\rm ABEP}}$ defined in [\[abep\]](#abep){reference-type="eqref" reference="abep"}-[\[abep-res-dx\]](#abep-res-dx){reference-type="eqref" reference="abep-res-dx"}, is dual to the SIP$(\alpha)$ with absorbing boundaries definded in Definition [Definition 3](#def-sip){reference-type="ref" reference="def-sip"} with respect to the following duality functions:* 1. ***classical duality:** $$\label{dfs} D^\sigma(x, \xi) = T_\ell ^{\xi_0} \cdot \prod_{i=1}^{N} \dfrac{\Gamma(\alpha )}{ \Gamma(\alpha + \xi_i)} (g_{i}(x))^{\xi_i} \cdot T_r^{\xi_{N+1}},$$* 2. ***orthogonal duality:** $$\label{odbepsip} {\mathfrak D}^\sigma_T(x,\xi) = \left( T_\ell - T \right)^{\xi_0} \cdot \prod_{i=1}^{N} (-T)^{\xi_i}\cdot \mathstrut_1 F_1 \left( {\left. \genfrac{}{}{0pt}{} {-\xi_{i}} { \alpha } \right\vert {\frac{g_{i}(x)}{T}}} \right)\cdot (T_r-T)^{\xi_{N+1}}\;,$$ for all $T >0$. Here $g$ is the map given in Definition [Definition 1](#mapg){reference-type="ref" reference="mapg"}.* **Proof:** the result is a natural consequence of Theorem [Theorem 4](#symduality){reference-type="ref" reference="symduality"} and the second item of Theorem [Theorem 2](#General_result){reference-type="ref" reference="General_result"}. $\square$ # Applications of duality {#sec6} Due to irreducibility, the ABEP admits a unique stationary probability measure, that we will also call steady state and we will denote it by $\mu_{ss}$. When $T_\ell = T_r=T$ this is reversible and coincides with the measure $\mu_T$ computed in Theorem [Theorem 3](#rev){reference-type="ref" reference="rev"}. When $T_\ell \neq T_r$, reversibility is lost and $\mu_{ss}$ is no longer easy to compute. We will take advantage of the duality property proven in the previous section to compute some particular observables of $\mu_{ss}$, and more precisely, the one and two-point correlations, with respect to $\mu_{ss}$, of the observables $\{e^{-\sigma E_i(x)}, \ i\in V\}$ that are inherently related to the non-local map $g$. We will informally call these quantities $\sigma$-exponential moments or correlations. The idea is to exploit the simplicity of the dual process that is symmetric interacting particle system. Moreover, the fact that dual particles are eventually absorbed at the boundaries, allow to compute the $\sigma$-exponential moments and correlations in terms of the absorption probabilities of the SIP particles. .2cm To prove our results we use the fact that duality between two Markov generators implies duality in terms of semigroups. This means that, if $\{X_t\}_{t\ge0}$ and $\{Y_t\}_{t\ge0}$ are two Markov processes with state spaces $\Omega$ and $\Omega^{{\rm dual}}$ respectively, whose generators are dual in the sense of Definition [\[gen-daulity-abcd\]](#gen-daulity-abcd){reference-type="eqref" reference="gen-daulity-abcd"} with respect to the duality function $D: \Omega\times \Omega^{{\rm dual}} \to \mathbb{R}$, then for all $x\in\Omega, y\in \Omega^{{\rm dual}}$ and $t>0$, $$\label{standarddualityrelation1} \mathbb{E}_x \big[D(X_t, y)\big]={\mathbf{E}}_{y} \big[D(x, Y_t)\big]\;$$ where $\mathbb{E}_x$ is the expectation with respect to the law of the $\{X_t\}_{t\ge0}$ process started at $x$, while ${\mathbf{E}}_{y}$ denotes expectation with respect to the law of the $\{Y_t\}_{t\ge0}$ process initialized at $y$. **Proposition 1**. *Let $\mu_{ss}$ be the stationary measure of ABEP($\sigma$, $\alpha$) with open boundaries defined in [\[abep\]](#abep){reference-type="eqref" reference="abep"}-[\[abep-res-dx\]](#abep-res-dx){reference-type="eqref" reference="abep-res-dx"}, then $$\mathbb{E}_{\mu_{ss}} \left[ e^{- \sigma E_m (x)}\right] = 1 - \sigma \alpha \, T_\ell (N-m+1) + \dfrac{\sigma \alpha} {N+1} (T_r - T_\ell) \, \dfrac{(m+N)(m-N-1)}{2}\ .$$* **Proof:** Let $\delta_i\in \Omega^{{\rm dual}}$ the SIP($\alpha$) configuration with just one particle at site $i\in V$, then $$D^{\sigma}(x, \delta_i) = \dfrac{\Gamma(\alpha )}{\Gamma(\alpha +1)} \cdot g_i(x) = \dfrac{e^{-\sigma E_{i+1}(x)} - e^{-\sigma E_{i}(x)} }{\sigma \alpha} = \dfrac{e^{-\sigma E_{i+1}(x)} }{\sigma \alpha} \ (1-e^{-\sigma x_i})\label{OO}$$ If we initialize the dual SIP$(\alpha)$ with one particle at site $i \in V$, the dynamics can be described by a continuous time random walk $\{i(t), \ t\ge 0\}$ moving on the lattice $V \cup V^{{\rm res}}$ performing nearest-neighbor jumps at rate $\alpha$ and absorbed at boundary sites $0$ and $N+1$. We will denote by $\mathbb{P}_i$ the probability distribution of this process initialized at time 0 from site $i\in V$. Then the stationary expectation of the quantity in the right hand side of [\[OO\]](#OO){reference-type="eqref" reference="OO"} linearly interpolates between $T_\ell$ and $T_r$: $$\begin{aligned} \label{questa} \mathbb{E}_{\mu_{ss}} \left[ e^{-\sigma E_{i+1}(x)} (1-e^{-\sigma x_i}) \right] & \nonumber= \mathbb{E}_{\mu_{ss}} [D^{\sigma}(x, \delta_i)] = \lim_{t \to \infty} \mathbb{P}_i (i_t =0)D^{\sigma}(x, \delta_0) + \mathbb{P}_i (i_t =N+1)D^{\sigma}(x, \delta_{N+1})\\ & = \sigma \alpha \left( T_\ell + (T_r - T_\ell )\frac{i}{N+1} \right) \ .\end{aligned}$$ We take now the sum from $m$ to $N$ on both sides of equation [\[questa\]](#questa){reference-type="eqref" reference="questa"} to get telescopic cancellation. Since $E_{N+1}=0$, we get $$\mathbb{E}_{\mu_{ss}} \left[ 1-e^{- \sigma E_m (x)}\right] = \sum_{i=m}^{N} \left( \sigma \alpha T_\ell + \sigma \alpha \, (T_r - T_\ell ) \, \frac{i}{N+1} \right)$$ from which follows the result. $\square$ In the next proposition we will show how to relate the above observation to gather information on the stationary ${\sigma }$-exponential expectation of the partial energies. **Remark 3**. *Notice that the observables $\{e^{-\sigma E_i(x)}, \ i\in V\}$ are reminiscent of the microscopic Cole-Hopf transformation (known as the Gärtner transform that has been defined in [@Gartner] for the asymmetric exclusion process). The Cole-Hopf transformation has been used in the literature to connect the KPZ equation for random growing interfaces and the stochastic heat equation. As remarked in [@Corwin], the first hint that such transform is available relies on the existence of a Markov duality relation.* In order to compute the stationary two-point correlation of the exponential observables $\{e^{-\sigma E_i(x)}, \ i\in V\}$ we use the same strategy used in the proof of Proposition [Proposition 1](#p1){reference-type="ref" reference="p1"} to compute the $\sigma$-exponential moments. In this case, though, we initialize the dual system with two (and no longer one) particles. **Proposition 2**. *Let $\mu_{ss}$ be the stationary measure of ABEP($\sigma$, $\alpha$) with open boundaries defined in [\[abep\]](#abep){reference-type="eqref" reference="abep"}-[\[abep-res-dx\]](#abep-res-dx){reference-type="eqref" reference="abep-res-dx"}, then $$\begin{aligned} \mathbb{E}_{\mu_{ss}} \left[ e^{- \sigma E_m (x)} e^{- \sigma E_n (x)} \right] & = 1 - \sigma \alpha T_\ell (2N - m- n +2) + \dfrac{\alpha \sigma }{2(N+1)} (T_r - T_\ell ) [m^2 + n^2 -2N^2 - 2N -m-n] \nonumber\\ & + \dfrac{(\sigma \alpha)^2 (1-m+N)(1-n+N)}{2(N+1)(1+\alpha (N+1))} \left[ T_\ell^2 (N-m+2)(1+\tfrac \alpha 2 (N-n+2)) \right. \nonumber\\ & \left. + T_r^2 (N+n) (1+\tfrac \alpha 2 (N+m)) + T_\ell T_r (m(1-\alpha (n-1)) -n +\alpha (n+N(N+2))) \right] \nonumber\\ & + \dfrac{(2\sigma )^2 \alpha (1-n+N)}{2(N+1)(1+\alpha (N+1))} \left[ T_\ell^2 \left( \dfrac{\alpha }{3} (2n^2 + 2N^2 +2nN -n +N) \right. \right. \nonumber \\ & \left. \left. - (n+N) [2\alpha (N+1) +1]+ 2N+1 +2\alpha (N+1)^2 \right) \right. \nonumber \\ & \left. + T_r^2 \left( \dfrac{\alpha }{3} (2n^2 + 2N^2 +2nN -n +N) + (n+N) -1 \right) \right. \nonumber \\ & \left. + 2T_\ell T_r \left(- \dfrac{\alpha }{3} (2n^2 + 2N^2 +2nN -n +N) + (n+N)(\alpha (N+1) -1) +1 \right) \right] \nonumber\end{aligned}$$ where $m \leq n$.* **Proof:** Let $\xi= \delta_i + \delta_j\in \Omega^{{\rm dual}}$ be the dual configuration with two particles at sites $i,j\in V$, $i \neq j$. The duality function evaluated in $\xi$ is then given by $$D^{\sigma} (x, \delta_i + \delta_j) = \dfrac{e^{-\sigma E_{i+1}(x)} - e^{-\sigma E_{i}(x)} }{\sigma \alpha} \cdot \dfrac{e^{-\sigma E_{j+1}(x)} - e^{-\sigma E_{j}(x)} } {\sigma \alpha} \ .$$ Considering the expectation with respect to the stationary measure: $$\begin{aligned} \label{2punti} & \mathbb{E}_{\mu_{ss}} \left[ \left( e^{-\sigma E_{i+1}(x)} - e^{-\sigma E_{i}(x)} \right) \left(e^{-\sigma E_{j+1}(x)} - e^{-\sigma E_{j}(x) } \right) \right] = ( \sigma \alpha)^2 \cdot \mathbb{E}_{\mu_{ss}}[D^{\sigma}(x, \delta_i + \delta_j) ]\\ & \nonumber = ( \sigma \alpha)^2 \lim_{t \to \infty} \bigg\{\mathbb{P}_{i,j} (i_t =0, j_t = 0 )D^{\sigma}(x, 2\delta_0) + \mathbb{P}_{i,j} (i_t =N+1, j_{t} = N+1)D^{\sigma}(x, 2\delta_{N+1}) + \\ \nonumber& D^{\sigma}(x, \delta_0 + \delta_{N+1}) \left( \mathbb{P}_{i,j} (i_t =0, j_{t} = N+1) + \mathbb{P}_{i,j} (i_t =N+1, j_{t} = 0) \right)\bigg\} \\ & = ( \sigma \alpha)^2 \left\{ T_\ell^2 \dfrac{[1+\alpha (N+1-i)](N+1-j)}{(N+1)(1+\alpha (N+1))} + T_r^2 \dfrac{i(1+\alpha j)}{(N+1)(1+\alpha (N+1))}\right. \\&+\left. T_\ell T_r \dfrac{[\alpha (N+1) -1]i +[1+\alpha (N+1)]j - 2\alpha ij}{(N+1)(1+\alpha (N+1))} \right\}\end{aligned}$$ where $\mathbb{P}_{i,j}$ is the probability distribution associated to two dual SIP$(\alpha)$ particles $\{(i(t),j(t)), \ t\ge 0\}$. On the other hand, if $i=j$ we have: $$D^{\sigma} (x, 2\delta_i ) = \dfrac{ \left(e^{-\sigma E_{i+1}(x)} - e^{-\sigma E_{i}(x)} \right)^2}{\alpha (\alpha +1) \sigma ^2}$$ and considering the expectation with respect to the stationary measure: $$\begin{aligned} \label{2punti} & \mathbb{E}_{\mu_{ss}} \left[ \left( e^{-\sigma E_{i+1}(x)} - e^{-\sigma E_{i}(x)} \right)^2 \right] = \mathbb{E}_{\mu_{ss}} \alpha (\alpha +1) \sigma ^2 D^{\sigma}(x, 2\delta_i ) \\ & \nonumber = \alpha (\alpha +1) \sigma ^2 \lim_{t \to \infty} \bigg\{\mathbb{P}_{i,i} (i_t =0, i_t = 0 )D^{\sigma}(x, 2\delta_0) + \mathbb{P}_{i,i} (i_t =N+1, i_{t} = N+1)D^{\sigma}(x, 2\delta_{N+1}) + \\ \nonumber& D^{\sigma}(x, \delta_0 + \delta_{N+1}) \left( \mathbb{P}_{i,i} (i_t =0, i_{t} = N+1) + \mathbb{P}_{i,i} (i_t =N+1, i_{t} = 0) \right)\bigg\} \\ & = \alpha (\alpha +1) \sigma ^2 \left\{ T_\ell^2 \frac{2(N+1-i) (\alpha (N+1-i)+1) -1}{2(N+1) (\alpha (N+1)+1)} + \right. \\& \left. T_r^2 \dfrac{2i(1+\alpha i) -1}{2(N+1)(\alpha (N+1)+1)} + T_\ell T_r \dfrac{(\alpha (N+1)-1)i + (\alpha (N+1)-1)i -2\alpha i^2 +1}{(N+1)(\alpha (N+1)+1)} \right\} \nonumber \\&= \alpha (\alpha +1) \sigma ^2 \left\{ T_\ell^2 \frac{ 2\alpha i^{2}+(-4\alpha N-4\alpha +2)i+(2\alpha N^2+4\alpha N+2\alpha -2 N-3)}{2(N+1) (\alpha (N+1)+1)} + \right. \\& \left. T_r^2 \dfrac{2\alpha i^2+2 i-1}{2(N+1)(\alpha (N+1)+1)} + T_\ell T_r \dfrac{-2\alpha i^2 + 2i(\alpha N+\alpha -1) + 1}{(N+1)(\alpha (N+1)+1)} \right\} \nonumber\end{aligned}$$ This allows us to gather informations on the two-point $\sigma$-exponential stationary correlations. To achieve this we take a double sum in equation [\[2punti\]](#2punti){reference-type="eqref" reference="2punti"}, one from $m$ to $N$ and one from $n$ to $N$. By telescopic arguments one then gets $$\begin{aligned} \label{ultima} & \mathbb{E}_{\mu_{ss}} \left[e^{- \sigma E_m (x)} e^{- \sigma E_n (x)} \right] = \mathbb{E}_{\mu_{ss}} \left[e^{- \sigma E_m (x)}\right] + \mathbb{E}_{\mu_{ss}} \left[e^{- \sigma E_n (x)}\right] - 1 +\nonumber\\ & \nonumber(\sigma \alpha)^2 \sum_{i=m}^{N} \sum_{j=n}^{N} \left\{ T_\ell^2 \mathbb{P}_{i,j} (i_t =0, j_{t} = 0) + T_r^2 \mathbb{P}_{i,j} (i_t =N+1, j_{t} = N+1) \right.\\&+\left. T_\ell T_r \left[ \mathbb{P}_{i,j} (i_t =0, j_{t} = N+1)+ \mathbb{P}_{i,j} (i_t =N+1, j_{t} = 0) \right] \right\} \nonumber\\ & + (2\sigma )^2 \alpha \sum_{i=n}^{N} \left\{ T_\ell^2 \mathbb{P}_{i,i} (i_t =0, i_{t} = 0) + T_r^2 \mathbb{P}_{i,i} (i_t =N+1, i_{t} = N+1) \right. \nonumber\\&+\left. T_\ell T_r \left[ \mathbb{P}_{i,i} (i_t =0, i_{t} = N+1)+ \mathbb{P}_{i,i} (i_t =N+1, i_{t} = 0) \right] \right\} \end{aligned}$$ where the first two terms on the right hand side have been computed in the previous theorem. To conclude the proof it remains to are plug in the expression above the absorption probabilities of two dual SIP$\alpha$ particles absorbed at the boundaries 0 and $N+1$. These are harmonic function of the two dimensional Laplacian. They solve a systems of discrete equations with appropriate boundary conditions. We show how to get $p_{i,j}:= \mathbb{P}_{i,j} (i_t =0, j_{t} = 0)$ for $i,j \in V$ as the others can be found similarly. $$\begin{cases} 4p_{i,j} = p_{i-1,j} + p_{i+1,j} + p_{i,j-1} + p_{i,j+1} \\ 2 p_{i,i} = p_{i-1,i} + p_{i,i+1} \end{cases}$$ for the first two equations we get that $$p_{i,j}= Ai + Bj + Cij + D \quad \text{for} \quad i \neq j$$ and $$p_{i,i}= (A+B)i + Ci^2 + D + \frac{B-A}{2} \ .$$ Three of the unknown can be found using the boundary conditions: $$\begin{cases} p_{0,0} = D = 1 \\ p_{0,j} = Bj + D = 1- \frac{j}{N+1} \\ p_{N+1,N+1} = A(N+1) + B(N+1) + C(N+1)^2 + D = 0 \end{cases}$$ while the last one can be found conditioning on the first jump, i.e. $$(4\alpha +2) p_{i,i+1} = \alpha p_{i-1,i+1} + \alpha p_{i,i+2} + (\alpha +1) p_{i,i} + (\alpha + 1 )p_{i+1,i+1} \;.$$ This leads to the following solutions for the four unknown $$\begin{cases} A=-\frac{\alpha }{1+\alpha (N+1)}\\ B = - \frac{1}{N+1} \\ C=\frac{\alpha }{(1+N)(1+\alpha (N+1)} \\ D = 1 \end{cases}$$ Finally we obtain $$\mathbb{P}_{i,j} (i_t =0, j_{t} = 0) = p_{i,j}=\frac{(N+1-j) (\alpha (-i+N+1)+1)}{(N+1) (\alpha (N+1)+1)} - \dfrac{1}{2(N+1)(\alpha (N+1)+1)} \mathbbm{1}_{\{ i=j \} }$$ for the absorption probabilities of both particles to the left. Similarly one can get the absorption probabilities of both particles to the right: $$\mathbb{P}_{i,j} (i_t = N+1, j_{t} = N+1) = p_{i,j}=\dfrac{i(1+\alpha j)}{(N+1)(\alpha (N+1)+1)} - \dfrac{1}{2(N+1)(\alpha (N+1)+1)} \mathbbm{1}_{\{ i=j \} }$$ and the absorption probability of one particle to the left and one to the right $$\begin{split} & \mathbb{P}_{i,j} (i_t =0, j_{t} = N+1) + \mathbb{P}_{i,j} (i_t =N+1, j_{t} = 0)\\=& p_{i,j}=\dfrac{(\alpha (N+1)-1)i + (\alpha (N+1)-1)j -2\alpha ij}{(N+1)(\alpha (N+1)+1)} + \dfrac{1}{(N+1)(\alpha (N+1)+1)} \mathbbm{1}_{\{ i=j \} }\ . \end{split}$$ Substituting these expressions in [\[ultima\]](#ultima){reference-type="eqref" reference="ultima"} we obtain the result. $\square$ ## Acknowledgments {#acknowledgments .unnumbered} The authors benefited from inspiring conversation with Leonardo de Carlo, Cristian Giardinà, Claudio Giberti, Seth Lloyd and Frank Redig. C.F. acknowledges hospitality and support from the Galileo Galilei Institute (GGI) during the scientific program 'Randomness, Integrability and Universality' held in Spring 2022. ## Data Avaibility Statement {#data-avaibility-statement .unnumbered} No new data were created or analysed in this study. 99 G. Barraquand, I. Corwin. Markov duality and Bethe ansatz formula for half-line open ASEP. arXiv preprint arXiv:2212.07349 (2022). I. Corwin, T. Li-Cheng. KPZ equation limit of higher-spin exclusion processes. Annals of Probability 45:3 1771--1798 (2017). G. Carinci, C. Franceschini, R. Frassek, C. Giardinà, F. Redig. The open harmonic process: non-equilibrium steady state, pressure, density large deviation and additivity principle. arXiv preprint arXiv:2307.14975 (2023). G. Carinci, C. Franceschini, D. Gabrielli, C. Giardinà, D. Tsagkarogiannis. Solvable stationary non equilibrium states. arXiv preprint arXiv:2307.02793 (2023). G. Carinci, C. Giardinà, C. Giberti, F. Redig. 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--- abstract: | The $\varphi$-divergence-based moment method was recently introduced [@PhiRTE1] for the discretization of the radiative transfer equation. At the continuous level, this method is very close to the entropy-based $M_N$ methods and possesses its main properties, i.e. entropy dissipation, rotational invariance and energy conservation. However, the $\varphi$-divergence based moment systems are easier to resolve numerically due to the improved conditioning of the discrete equations. Moreover, exact quadrature rules can be used to compute moments of the distribution function, which enables the preservation of energy conservation, entropy dissipation and rotational invariants, discretely. In this paper we consider different variants of the $\varphi-$divergence closures that are based on different approximations of the exponential function and the Planck function. We compare the approximation properties of the proposed closures in the numerical benchmarks. author: - "M.R.A. Abdelmalik[^1], Z. Cai[^2]  and T. Pichard[^3]" bibliography: - PhiRTE2.bib title: Some extensions of the $\varphi$-divergence moment closures for the radiative transfer equation --- *Keywords:* Radiative transfer equation, Method of moments, $\varphi$-divergence # Introduction {#sec:intro} This paper is a follow-up to [@PhiRTE1] and is concerned about the discretization of the radiative transfer equation (RTE). This kinetic equation consists in a transport at a velocity $\Omega$ with a constant norm combined with a collision operator. Among the properties that this model satisfies, we want at least to preserve at the discrete level: the conservation of energy, the dissipation of a convex entropy and the rotational invariance. The numerical methods commonly used for this model include the Monte-Carlo solvers (DSMC;[@Carlson_book; @Lewis_Miller_book; @pomraning_book; @mihalas_book]) and the discrete ordinates method (DOM;[@Carlson_book; @Lewis_Miller_book; @pomraning_book; @mihalas_book]), but those are computationnally very expensive, not rotationally invariant and incapable to capture the equilibria distribution the system converges to. Our work falls within the context of the method of moments which is an efficient alternative. The most popular models in this family are the polynomial $P_N$ models ([@Chandrasekhar_book; @pomraning_book; @spectral_meth1; @spectral_meth2]) and the entropy-based $M_N$ models ([@Minerbo; @levermore]) which rely on approximating the $\Omega$-dependencies of the solution respectively by polynomials or by a distribution minimizing the entropy under moment constraints. Those are efficients, but the first fails at modelling beams, the second requires high computational costs due to need to solve large numbers of (potentially ill-conditionned) optimization problems. Those were an inspiration for many other techniques developed in this field, including the simplified models ([@Frank_SPN; @McClaren_SPN]), the flux-limited diffusion ([@Olson_diffusion; @Humbird_diffusion]), the interpolative methods ([@pichard_m2; @Li_B2; @groth_m2]) and others (see e.g. [@Schneider_KN; @PiN]). Recently, the entropy-based method of moment was simplified by exploiting $\varphi$-divergence techniques ([@czizar]) while preserving its main properties, originally in the context of rarefied gases ([@abdelmalik_thesis; @abdelmalik]) and for the RTE in [@PhiRTE1]. This method consists in a Galerkin approximation where the test function space chosen to be polynomials set and the approximation set is chosen to be a non-linear renormalization applied to the same polynomial set. This renormalization is chosen to be a high degree polynomial approximation of the exponential for the final model to dissipate an approximation of Boltzmann entropy. This model was shown to preserve the three aforementionned properties of conservation of energy, rotational invariance and entropy dissipation. Therefore, it possesses the same properties as $M_N$ models but the optimization problem to solve requires a lower cost and they rely on exact quadrature rules and therefore preserves further in the construction of the discretization the rotational invariance. The present work exploits the versatility of the method in order to address two issues: First, when the degree of the renormalization mapping tends to infinity, our method falls back onto the $M_N$ method based on the Boltzmann entropy, it captures therefore exactly the beam distributions but requires to solve worse-conditioned optimization problems. We would like to adapt our method in order to choose the compromise between the acccuracy in the beam regime and the condition of the optimization problems to solve. Second, Boltzmann entrpopy is often considered in radiative transfer by analogy with the rarefied gases models, but other entropies are more physically relevant in this context. We would like to adapt our method such that it converges toward other entropy-based models. These two problems are tackled by constructing other polynomial renormalization mappings, and they eventually preserve the aforementionned properties. The paper is organized as follows. The next section recalls the radiative transfer equation and its properties. The next recalls the construction of our $\varphi$-divergence moment closure. The novel construction and study of polynomial renormalization mappings arises in Section [4](#sec:LCBC){reference-type="ref" reference="sec:LCBC"}. Section [5](#sec:numerics){reference-type="ref" reference="sec:numerics"} is devoted to numerical experiments with the present method and Section [6](#sec:concl){reference-type="ref" reference="sec:concl"} to conclusion. # Radiative transfer equation {#sec:RTE} We aim at solving the radiative transfer equation (RTE) $$\partial_t I + \Omega \cdot \nabla_x I = LI := \sigma\left(\frac{1}{4\pi}\int_{\mathbb{S}^2} I \,\mathrm{d}\Omega - I\right), \label{eq:RTE}$$ where the unknown $I(t,x,\Omega)$ is the radiative intensity depending on $\Omega\in \mathbb{S}^2$, $x\in \mathbb{R}^3$ and $t\in ]0,T[$. This equation is only supplemented with initial condition $$I(t=0,x,\Omega) = I^0(x,\Omega), \label{eq:RTE_IC}$$ and we still consider unbounded spatial domain in order to avoid the difficulties emerging with boundary conditions. [\[sys:RTE\]]{#sys:RTE label="sys:RTE"} This problem is well-posed and preserves the integrability of the initial condition ([@Dautray-Lions]) in the sense: if $I^0 \in \mathcal{L}^p(\mathbb{R}^d\times\mathbb{S}^2)$, then there exists a unique function $I\in \mathcal{C}\left((0,T);\mathcal{L}^p(\mathbb{R}^d\times\mathbb{S}^2)\right)$ satisfying [\[sys:RTE\]](#sys:RTE){reference-type="eqref" reference="sys:RTE"}. Furthermore, if $I^0 \ge 0$, then $I\ge 0$. In the following, we focus on $\mathcal{L}^1$ solutions. This equation is known to dissipate any entropy, i.e. for all convex scalar functions $\eta$, then $$\begin{gathered} \partial_t H(I) + \nabla_x \cdot G(I) = S(I) \le 0, \qquad H(I) = \int_{\mathbb{S}^2} \eta(I(\Omega))d\Omega, \\ G(I) = \int_{\mathbb{S}^2} \Omega \eta(I(\Omega))d\Omega, \qquad S(I) = \int_{\mathbb{S}^2} \eta'(I(\Omega)) LI(\Omega)d\Omega. \end{gathered}$$ [\[eq:entropy\]]{#eq:entropy label="eq:entropy"} In the present case, the space of collisional invariants $$\mathtt{C} = \left\{ f \text{ s.t. } \int_{\mathbb{S}^2} f(\Omega)LI(\Omega)d\Omega = 0\right\}$$ is one-dimensional and composed only of the isotropic functions $$f\in \mathtt{C} \quad\Leftrightarrow\quad f(\Omega) = \frac{1}{4\pi} \int_{\mathbb{S}^2} f(\Omega)d\Omega.$$ Therefore, for all convex function $\eta$, $H$ is minimum when $I$ is isotropic and the system converges toward such equilibria. For the construction of entropy-based moment closure as below, the Boltzmann-Shannon (afterward denoted with the subscript $BS$) entropy $$\eta_{BS}(I) = I\log I$$ is often used as a comparison of the kinetic model [\[eq:RTE\]](#eq:RTE){reference-type="eqref" reference="eq:RTE"} with rarefied gas models. But the Bose-Einstein (afterward denoted with the subscript $BE$) entropy $$\eta_{BE}(I) = I\log I - (I+1)\log(I+1)$$ is more meaningful when considering more physically realistic collision models in [\[sys:RTE\]](#sys:RTE){reference-type="eqref" reference="sys:RTE"}. Typically, interaction of the radiations with matter is often modeled by adding scattering and emission terms in [\[eq:RTE\]](#eq:RTE){reference-type="eqref" reference="eq:RTE"} (see e.g. [@lowrie-morel; @pomraning_book; @mihalas_book]) which are equivalent, using Stefan's law, to a relaxation term toward a Planck function $(\eta_{BE}')^{-1}$ which parameters depend on matter temperature and radiations frequencies. Therefore, dissipating $\eta_{BE}$ is more relevant in this context than $\eta_{BS}$, and moment closures should be adapted to such other types of entropy. Finally, Equation [\[sys:RTE\]](#sys:RTE){reference-type="eqref" reference="sys:RTE"} was shown to be rotationnally invariant, meaning that its solution $I$ satisfies $$(\partial_t I)(\mathcal O \Omega) = \partial_t ( I(\mathcal O \Omega)), \quad (\Omega \cdot \nabla_x I)(\mathcal O \Omega) = (\mathcal O \Omega) \cdot \nabla_x (I(\mathcal O \Omega)), \quad (LI)(\mathcal O \Omega) = L (I(\mathcal O \Omega)),$$ for all rotation matrices $\mathcal{O} \in SO(3)$. # $\varphi$-divergence-based moment equations {#sec:phidiv} We recall here the construction of the moment closure from [@PhiRTE1] and the problems tackled in the present work. ## Construction of the Galerkin framework The moment system is obtained as a Galerkin approximation of [\[eq:RTE\]](#eq:RTE){reference-type="eqref" reference="eq:RTE"} in the $\Omega$ variable. This formulation requires three elements, the choices and properties are recalled here: - A *finite dimensional* test functions space $M$, which must be a *subset of the solution set dual* $\mathcal{L}^\infty(\mathbb{S}^2)$. In order to preserve those properties at the numerical level, $M$ must *contain the collision invariants* $1 \in M$; and $M$ must be *rotational invariant*. The natural choice to satisfy both properties is the set of polynomials up to a certain degree $N$ $$M := \mathbb{P}_N(\mathbb{S}^2).$$ - A renormalization map $\beta$ to account for non-linearity in the approximation. For a convex function $\eta$, choosing $\beta = (\eta')^{-1}$ corresponds to dissipating $\eta$ at the underlying kinetic level (see e.g. [@levermore]). Especially, $\beta$ must be *monotonically increasing* to match such an entropy. Natural choices include $$\beta_{BS}(g) = \exp(g) = (\eta_{BS}')^{-1}(g), \qquad \beta_{BE}(g) = \frac{1}{\exp(g)-1} = (\eta_{BE}')^{-1}(g). \label{eq:beta_BS_BE}$$ In our work, we aim at imposing $\beta \in \mathbb{P}_K(\mathbb{R})$ with $K\ge 1$ for numerical quadrature (up to a sufficient order) to be exact, this provided a rotationally invariant algorithm in [@PhiRTE1]. We chose renormalization of the form $$\beta_K(g) = \left(1+ \frac{g}{K}\right)^K = (\eta_K')^{-1}(g) \quad\text{with}\quad \eta_K(I) = K I \left(\frac{K}{K+1}I^{1/K}-1\right),\label{eq:def_betaK}$$ which are polynomial approximations of $\exp$, and monotonically increasing for odd $K\ge 1$. We exhibit in the next section other choices. - A finite dimensional trial functions space $V$ such that $\beta(V) \subset \mathcal{L}^1$ is a subset of the solution set. Again, $\beta(V)$ must *contain the equilibrium distributions* $\mathtt{C} \subset \beta(V)$; and $V$ must be *rotational invariant*, and we choose again $$V := \mathbb{P}_N(\mathbb{S}^2).$$ Eventually, this yields seeking $g\in \mathbb{P}_N(\mathbb{S}^2)$ such that for all $m\in\mathbb{P}_N(\mathbb{S}^2)$ $$\begin{gathered} \forall (t,x)\in(0,T)\times\mathbb{R}^d,\quad \int_{\mathbb{S}^2} m(\Omega) \left[ (\partial_t + \Omega \cdot \nabla_x) \beta(g) - L\beta(g(t,x,\cdot))(\Omega)\right]d\Omega = 0, \label{eq:moment_system} \\ \forall x\in\mathbb{R}^d,\quad \int_{\mathbb{S}^2} m(\Omega) \left[\beta(g(t=0,x,\Omega)) - I^0(x,\Omega)\right]d\Omega = 0. \label{eq:moment_IC} \end{gathered}$$ [\[eq:mom_gal\]]{#eq:mom_gal label="eq:mom_gal"} Let $\boldsymbol{m}$ denote a vector of all of the basis functions $m(\Omega)\in \mathbb P_N(\mathbb S^2)$, under the constraints mentioned in the last paragraph. Then ([\[eq:mom_gal\]](#eq:mom_gal){reference-type="ref" reference="eq:mom_gal"}) can be expressed as a system of moment equations $$\begin{aligned} \partial_t \mathbf{U} + \operatorname{div}_x(\mathbf{F}(\mathbf{U})) &= \mathbf{L}\mathbf{U}, \label{eq:mom_sym_sys} \\ (\mathbf{U}, \mathbf{F}(\mathbf{U}),\mathbf{L}\mathbf{U}) &= \int_{\mathbb{S}^2} \mathbf{m}(\Omega)\left(\beta(\boldsymbol{\lambda}^T\mathbf{m}(\Omega)),\ \Omega^T\beta(\boldsymbol{\lambda}^T\mathbf{m}(\Omega),\ L\beta(\boldsymbol{\lambda}^T\mathbf{m}(\Omega)\right) \mathrm{d}\Omega, \label{eq:mom_cons}\end{aligned}$$ which possesses a symmetrizer constructed from $\eta$ (the anti-derivative of $\beta^{-1}$) and $\boldsymbol{\lambda}$ are the so-called entropic variables ([@Godlewski_Raviart_book]) in which ([\[eq:mom_sym_sys\]](#eq:mom_sym_sys){reference-type="ref" reference="eq:mom_sym_sys"}) can be written in the so-called symmetric hyperbolic form. Therefore, ([\[eq:mom_sym_sys\]](#eq:mom_sym_sys){reference-type="ref" reference="eq:mom_sym_sys"}) possesses a convenient structure for the study of its well-posedness ([@Kawashima-Yong]). *Remark 1*. The entropic variables $\boldsymbol{\lambda}$ introduced in ([\[eq:mom_cons\]](#eq:mom_cons){reference-type="ref" reference="eq:mom_cons"}) can also be conceived of as Lagrange multipliers that enforce the moment constraints in the so-called entropy minimization problem ([@levermore]) $$\label{eq:entmin} \text{Find } \mathrm{argmin}\left\{ H(h): \int_{\mathbb S^2} \boldsymbol{m} I d\Omega =\int_{\mathbb S^2} \boldsymbol{m} h d\Omega \right\}.$$ ## Position of the problem In this paper, we investigate several modifications of the renormalization map [\[eq:def_betaK\]](#eq:def_betaK){reference-type="eqref" reference="eq:def_betaK"} and its impact on the closure [@PhiRTE1]. The objective is to alter the convergence of the $\varphi$-divergence solution when $K \rightarrow +\infty$ either to accelerate the convergence or to modify the limit value: First, when considering the moments of a Dirac distribution, e.g. $\mathbf{U} = \mathbf{m}(e_1) = \int_{\mathbb{S}^2} \mathbf{m} \delta_{e_1}$, numerical experiments in [@PhiRTE1] showed that the reconstruction $\beta_K$ does converge in $H^{-2}$ toward the distribution $\delta_{e_1}$ when $K \rightarrow +\infty$. However, the rate of convergence is slow. This slow rate of convergence is also illustrated on Fig. [1](#fig:renorm_beta_K){reference-type="ref" reference="fig:renorm_beta_K"} where the function $\beta_K(x)$ is plotted for various $K$ odd together with its limits $\lim\limits_{K\rightarrow \infty} \beta_K = \exp$. {#fig:renorm_beta_K width=".7\\textwidth"} Formally, the Dirac distribution corresponds to the values $x \rightarrow -\infty$ and $x \rightarrow \infty$. Indeed, considering a Gaussian mollifier $\frac{1}{\sqrt{\pi\sigma}}\exp(-\frac{y^2}{\sigma})$, then $x=\frac{-y^2}{\sigma}$ takes for values $\pm\infty$ in the limit. One observes that the sequence $(\beta_K)_{K\in\mathbb{N}}$ indeed converges pointwisely for bounded $x$ toward the exponential function, but this convergence is very slow. Furthermore, the rate of the exponential in the limit $x\rightarrow+\infty$ is not accurately captured and one needs high order $K$, and therefore higher complexity, to capture such large values. Similarly, the zero limit when $x\rightarrow -\infty$ can not be reached by any $\beta_K$ function with finite $K$ as they are polynomials and cannot have bounded value in $-\infty$. Eventually, such Dirac distributions can only be approximated and we only aim at improving the range of accuracy of such approximations. Second, when $K\rightarrow +\infty$, the sequence $(\beta_K)_{K\in\mathbb{N}}$ of approximations may only converge toward the exponential. As mentioned in the previous section, other types of equilibrium can be expected from the solution of the RTE, typically the Planck distribution $\beta_{BE}$ defined in [\[eq:beta_BS_BE\]](#eq:beta_BS_BE){reference-type="eqref" reference="eq:beta_BS_BE"}. The $\beta_K$ approximation does not possess the flexibility to converge toward other $\beta = (\eta')^{-1}$ functions. Therefore, the objective in the next section is to provide another type of approximations which is flexible enough to control the convergence when $K\rightarrow \infty$, i.e. both the convergence rate and the limit function. # Other monotonically increasing polynomial approximations {#sec:LCBC} The solutions considered in this paper consist in *polynomial* approximations. The integral of such polynomial functions can be computed exactly using an appropriate quadrature rule. Therefore, the coefficients $\boldsymbol{\lambda}$ of the approximation in [\[eq:mom_sym_sys\]](#eq:mom_sym_sys){reference-type="eqref" reference="eq:mom_sym_sys"} can be obtained using Newton iterations [@PhiRTE1] that can be computed exactly. Remark that rotation invariance is lost when constructing entropy-based closures (see e.g. [@Hauck]) due to non-exact integration while the present choice of polynomial approximation allows to compute the integrals exactly. For the model to possess a convex entropy dissipated, we still need this approximation to be *monotonically increasing*. Therefore, we require $$\beta' > 0.$$ ## Taylor expansion A first idea originated in the observation that the Taylor expansion of the exponential around zero converges faster (empirically) than the sequence $(\beta_K)_{K\in\mathbb{N}}$. This consists in writing in the shifted monomial basis $\mathbf{b} = \mathbf{b}^K(x) := \left(1,\dots,(x-x_0)^K\right)^T$ $$T_K(x) = \sum\limits_{k=0}^K \alpha_k (x-x_0)^k = \boldsymbol{\alpha}^T \mathbf{b}$$ where $\alpha_k = \frac{\beta^{(k)}(x_0)}{k!}$ for a generic function $\beta$. We provide a simple characterization of monotonically increasing polynomials of this form. **Proposition 1**. *Suppose that $\beta \in \mathcal{C}^{2K+2}$ is such that $\beta^{(i)} \ge 0$ for all $1\le i\le 2K+2$. Then the polynomial $T_{2K+1}$ is monotonically increasing.* *Proof.* This simply follows from the Taylor formula with remainder of the derivative of $f$: there exists $\xi \in [x,x_0]$ such that $$\beta'(x) - T_{2K+1}'(x) = \frac{\beta^{(2K+2)}(\xi)}{(2K+2)!}(x-x_0)^{2K+1},$$ which is negative for $x < x_0$. Therefore, for $x \le x_0$ $$\sum\limits_{i=0}^{2n} \frac{f^{(i+1)}(x_0)}{i!} (x-x_0)^i \ge f'(x) \ge 0.$$ The derivative $T_{2K+1}'(x)$ is also positive for $x>x_0$, then it is monotonically increasing. ◻ All the derivatives of the exponential $\beta_{BS} = \exp$ are positive, then it satisfies this property and the polynomial $$T_{2K+1}(x) = \sum\limits_{k=0}^{2K+1} \frac{e^{x_0}}{k!} (x-x_0)^k$$ is monotonically increasing for odd degree $2K+1$. The convergence radius of this sequence is infinite. Therefore, we have convergence of the approximation toward the desired results for all $x\in\mathbb{R}$ $$T_{2K+1}(x) \underset{K\longrightarrow \infty}{\longrightarrow} \exp(x).$$ Concerning the Planck function $\beta_{BE}$ minimizing the Bose-Einstein entropy, it reads for $x\in\mathbb{R}^{*,-}$ $$\beta_{BE}(x) = \frac{1}{e^{-x}-1} > 0,$$ and one verifies that its derivative satisfies $\beta_{BE}' = (1+\beta_{BE})\beta_{BE}$ such that all the successive derivatives of $\beta_{BE}$ are polynomials in $\beta_{BE}$ with positive coefficients. Especially, those derivatives are all strictly positive for all $x\in\mathbb{R}^{*,-}$ since $\beta_{BE}>0$. Therefore, the polynomial $T_{2K+1}$ of odd degree with the Planck function are also monotonically increasing. The convergence radius $\delta$ of this sequence remains bounded and it depends on the chosen point of expansion $x_0$. This radius $\delta < |x_0|$ since the function $\beta_{BE}$ is singular in zero (and undefined after). Therefore, we only have convergence of the approximation toward the desired results for all $x\in (x_0-\delta,x_0+\delta) \subsetneq \mathbb{R}^{*,-}$ $$T_{2K+1}(x) \underset{K\to \infty}{\to} \beta_{BE}(x).$$ Especially, we do not have convergence $T_{2K+1}(x) \underset{K\to \infty}{\not\to} \beta_{BE}(x)$ for all points $x < 2x_0$. These are illustrated numerically in Section [4.3](#subsec:comparisons_approx){reference-type="ref" reference="subsec:comparisons_approx"} below, which even exhibit divergence for such values of $x$. ## Optimized parameters Another idea is to minimize the $\mathcal{L}^2$ difference between the function $\beta$ to approximate ($\beta_{BS}$ or $\beta_{BE}$) and the polynomials of a given degree, with the constraint that the polynomials must be monotonically increasing. This can be mathematically represented as $$\begin{aligned} O_{2K+1} = & \ \underset{p \in \mathbb{P}_{2K+1}}{\text{argmin}} \ \frac{1}{2} \int_a^b |p(x) - \beta(x)|^2 \,\mathrm{d}x, \label{eq:def_L2_Opt}\\ & \ \text{subject to } p'(x) \geq 0, \quad \forall x \in \mathbb{R}. \nonumber\end{aligned}$$ ### Reformulation of the approximation In order to enforce the constraint, we use the fact that all non-negative one-variable polynomials can be represented as the sum of two squares, one of which has a lower degree than the other ([@Lasserre_book; @Schmuedgen_book; @Szego_book]). Thus, the derivative of the polynomial $O_{2K+1}(x)$ has the form: $$O_{2K+1}'(x) = (a_0 + a_1 x + \cdots + a_K x^K)^2 + (b_0 + b_1 x + \cdots + b_{K-1} x^{K-1})^2.$$ Integrating this equation gives $$\label{eq:admissible_solution} \begin{split} O_{2K+1}(x) = C &+ \sum_{i=0}^K \sum_{j=0}^K (a_i a_j + b_i b_j) \frac{x^{i+j+1}}{i+j+1} \\ = C &+ \sum_{n=1}^{K+1} \sum_{i=0}^{n-1} (a_i a_{n-1-i} + b_i b_{n-1-i}) \frac{x^n}{n} \\ &+ \sum_{n=K+2}^{2K+1} \sum_{i=n-1-K}^{K} (a_i a_{n-1-i} + b_i b_{n-1-i}) \frac{x^n}{n}, \end{split}$$ where we assumed $b_K = 0$. With this form of $O_{2K+1}$, we can turn the optimization problem into an unconstrained optimization problem. For simplicity, we define intermediate parameters $$\alpha_0 = C, \qquad \alpha_n = \frac{1}{n} \sum_{i=\max(0,n-1-k)}^{\min(k,n-1)} (a_i a_{n-1-i} + b_i b_{n-1-i}), \quad n=1,\cdots,2K+1,$$ so that $\alpha_n$ is the $n$-th coefficient of the polynomial $O_{2K+1}$. Let $$\boldsymbol{w} = (C, a_0, \cdots, a_K, b_0, \cdots, b_{K-1})^T, \qquad \boldsymbol{\alpha}(\boldsymbol{w}) = (\alpha_0, \alpha_1, \cdots, \alpha_{2K+1})^T.$$ Then the Jacobian matrix $J = \partial \boldsymbol{\alpha} / \partial \boldsymbol{w}$ has the following form: $$J = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2A & 2B \end{pmatrix},$$ where $$A = \begin{pmatrix} a_0 & 0 & \dots & & \dots & 0\\[5pt] \frac{a_1}{2} & \frac{a_0}{2} & 0 & & & \vdots \\[5pt] \frac{a_2}{3} & \frac{a_1}{3} & \frac{a_0}{3} & \ddots & & \\ \vdots & \frac{a_2}{4} & \frac{a_1}{4} & \ddots & \ddots & \vdots \\ \vdots & \vdots & \frac{a_2}{5} & \ddots & \ddots & 0 \\ \frac{a_{K}}{K+1} & \vdots & \vdots & \ddots & \ddots & \frac{a_0}{K+1} \\ 0 & \frac{a_{K}}{K+2} & \vdots & & \ddots & \frac{a_1}{K+2} \\ \vdots & \ddots & \frac{a_{K}}{K+3} & & & \frac{a_2}{K+3} \\ & & \ddots & \ddots & & \vdots \\ & & & & \ddots & \vdots \\ & & & & \ddots & \frac{a_{K}}{2K+1}\\ & & & & & 0 \\ \vdots & & & & & \vdots \\ 0 & \dots & & & \cdots & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} b_0 & 0 & \dots & & \dots & 0\\[5pt] \frac{b_1}{2} & \frac{b_0}{2} & 0 & & & \vdots \\[5pt] \frac{b_2}{3} & \frac{b_1}{3} & \frac{b_0}{3} & \ddots & & \\ \vdots & \frac{b_2}{4} & \frac{b_1}{4} & \ddots & \ddots & \vdots \\ \vdots & \vdots & \frac{b_2}{5} & \ddots & \ddots & 0 \\ \frac{b_{K-1}}{K} & \vdots & \vdots & \ddots & \ddots & \frac{b_0}{K} \\ 0 & \frac{b_{K-1}}{K+1} & \vdots & & \ddots & \frac{b_1}{K+1} \\ \vdots & \ddots & \frac{b_{K-1}}{K+2} & & & \frac{b_2}{K+2} \\ & & \ddots & \ddots & & \vdots \\ & & & & \ddots & \vdots \\ & & & & \ddots & \frac{b_{K-1}}{2K-1}\\ & & & & & 0 \\ \vdots & & & & & \vdots \\ 0 & \dots & & & \cdots & 0 \end{pmatrix}.$$ To solve the optimization problem, we first reformulate the objective function $\boldsymbol{w} \mapsto f(\boldsymbol{\alpha}(\boldsymbol{w}))$ as a function of the intermediate parameters $\boldsymbol{\alpha}$: $$\begin{split} f(\boldsymbol{\alpha}) & = \frac{1}{2} \int_a^b \left| \sum_{n=0}^{2K+1} \alpha_n x^n - \beta(x) \right|^2 \,\mathrm{d}x \\ &= \frac{1}{2} \boldsymbol{\alpha}^T M \boldsymbol{\alpha} - \boldsymbol{\beta}^T \boldsymbol{\alpha} + \frac{1}{2} \int_a^b \beta(x)^2 \,\mathrm{d}x. \end{split}$$ Note that the value of the last integral does not matter in our optimization problem, and the matrix $M$ and the vector $\boldsymbol{\beta}$ are given by $$\begin{gathered} M_{i,j} = \frac{b^{i+j+1} - a^{i+j+1}}{i+j+1}, \qquad \beta_j = \int_a^b x^j \beta(x) dx.\end{gathered}$$ In the case of the exponential $\beta = \beta_{BS}$, the second coefficient rewrites $$\beta_j = (-1)^j \left[\Gamma(j+1, -b) - \Gamma(j+1,-a)\right],$$ where the incomplete $\Gamma$ function is well-implemented in standard numerical libraries. In the case $\beta=\beta_{BE}$, the integral can be represented using the polylogarithmic function $\operatorname{Li}_s(z)$: $$\beta_j = \sum_{k=0}^j \frac{(-1)^{j+k} j!}{k!} \left[ b^k \operatorname{Li}_{j+1-k} (\mathrm{e}^b) - a^k \operatorname{Li}_{j+1-k} (\mathrm{e}^a) \right].$$ ### Details on the numerical computations The minimum of $f(\boldsymbol{\alpha}(\boldsymbol{w}))$ is attained where the gradient anihilates. Then we need to solve the nonlinear system $\nabla_{\boldsymbol{w}} f(\boldsymbol{\alpha}(\boldsymbol{w})) = 0$, or $$^T M \boldsymbol{\alpha}(\boldsymbol{w}) = 0. \label{eq:gradient_zero_OK}$$ Since $J(\boldsymbol{w})$ is linear and $\boldsymbol{\alpha}(\boldsymbol{w})$ is quadratic, these are actually cubic equations in $\boldsymbol{w}$. In the following, we compute numerically some parameters $\boldsymbol{w}$ and $\boldsymbol{\alpha}(\boldsymbol{w})$ by using Newton's method for [\[eq:gradient_zero_OK\]](#eq:gradient_zero_OK){reference-type="eqref" reference="eq:gradient_zero_OK"}. Note that the $f(\boldsymbol{\alpha}(\boldsymbol{w}))$ is a quartic function which is infinity in infinity, but it might not be convex at all $\boldsymbol{w} \in \mathbb{R}^{2K+2}$, and it may possess several local extrema. Therefore, we cannot guarantee the convergence of Newton's method toward a global minimum. In practice, we choose $500$ initial values of $\boldsymbol{w}$ randomly and pick the final solution with the minimum value of the objective function. Using the density of the (positive) polynomials in $\mathcal{L}^2$, we obtain that this approximation converges in $\mathcal{L}^2(a,b)$ toward the desired $\beta$ function. However, the convergence is again restricted to the chosen interval $(a,b)$ and we can not guarantee convergence out of it. These are illustrated numerically in the next paragraph. ## Comparing the monotonic polynomial approximations {#subsec:comparisons_approx} ### Approximation for the Boltzmann-Shannon entropy The approximations of the exponential $\beta_{BS}$ are compared in Fig. [\[fig:optimized_bs\]](#fig:optimized_bs){reference-type="ref" reference="fig:optimized_bs"} for different degrees $2K+1$. The point of expansion is chosen to be $x_0 = 0$ for the Taylor approximation and the range for the optimized approximation is $[-L,L]$ with $L=1,3,5$. In general, by comparing the $\beta_{2K+1}$ model and the $T_{2K+1}$ model, the $\beta_{2K+1}$ model gives a better approximation for $x \in \mathbb{R}^{*,-}$, while the $T_{2K+1}$ model does better for $x \in \mathbb{R}^{*,+}$. For the optimized approximation, the function depends on the choice of the range $[-L,L]$. As expected, when $K$ is fixed, the function is approximated within the range $[-L,L]$ for smaller values of $L$. It balances the quality of the approximation for both the negative and positive parts, and generally looks better than both $\beta_{2K+1}$ and $T_{2K+1}$ in the plots with linear scales (left panel of Fig. [\[fig:optimized_bs\]](#fig:optimized_bs){reference-type="ref" reference="fig:optimized_bs"}). Similar phenomena can be observed in Fig. [\[fig:comparison_bs\]](#fig:comparison_bs){reference-type="ref" reference="fig:comparison_bs"}, where all the results for the optimized approximation are given for $L = 5$. The log plots show a clear difference between the $O_{2K+1}$ model and the $\beta_{2K+1}$ and $T_{2K+1}$ models: for the latter, the entire approximation is below the exact exponential function, whereas the $O_{2K+1}$ approximation oscillates around the exponential, which is a typical behavior in spectral approximations. Another interesting phenomenon that can be observed from Fig. [\[fig:optimized_bs\]](#fig:optimized_bs){reference-type="ref" reference="fig:optimized_bs"} is that the $O_{2K+1}$ approximation seems to have a good capability in extrapolations. For instance, the red curve in Fig. [\[fig:K5_bs\]](#fig:K5_bs){reference-type="ref" reference="fig:K5_bs"} is computed by minimizing the $\mathcal{L}^2$ distance [\[eq:def_L2_Opt\]](#eq:def_L2_Opt){reference-type="eqref" reference="eq:def_L2_Opt"} only on the interval $[-1,1]$, but this approximation is also quite accurate for $x\in [-3,5]$. Since the choice of the approximation range usually depends on some preliminary estimation of the problem, this property allows us to use the $O_{2K+1}$ approximations without too much worries about the function values out of the chosen range in actual simulations. Here we conjecture that for any fixed interval, the $O_{2K+1}$ model will converge to the function $\beta_{BS}$ as $K$ tends to infinity. \ \ \ \ Fig. [2](#fig:error_bs){reference-type="ref" reference="fig:error_bs"} plots the $\mathcal{L}^2(-L,L)$ difference between the exponential function $\beta_{BS}$ and the $O_{2K+1}$ approximations. Unsurprisingly, the error increases as $L$ increases and decreases as $K$ increases. For a fix $L$, the gaps between lines are nearly the same, showing the spectral convergence rate with respect to $K$. When $L$ gets larger, the gap becomes narrower, indicating slower convergence. For a fixed $K$, the figure implies that the error increases in the form of $L^{\alpha}$ for a certain value of $\alpha$ depending on $K$. {#fig:error_bs width=".8\\textwidth"} ### Approximation for the Bose-Einstein entropy Similar experiments are done for the Planckian $\beta_{BE}$. The results are plotted in Fig. [\[fig:optimized_be\]](#fig:optimized_be){reference-type="ref" reference="fig:optimized_be"} and [\[fig:comparison_be\]](#fig:comparison_be){reference-type="ref" reference="fig:comparison_be"}. Since the function is defined only for negative values, we choose the range of approximation to be $[-L, -1/L]$ with $L = 2,6,10$. In Fig. [\[fig:optimized_be\]](#fig:optimized_be){reference-type="ref" reference="fig:optimized_be"}, three different choices of $x_0$ are considered for the Taylor expansion, and the choices are made with $x_0 = -(L + 1/L)/2$, which is the center of the interval $[-L, -1/L]$ used in the $O_{2K+1}$ approximation. The general behavior is similar to the case of Boltzmann-Shannon entropy: the optimized approximation fits the Planckian better within the range $[-L,-1/L]$, but it may perform worse than the Taylor approximation out of this interval. The convergence with respect to $K$ can be better observed in Fig. [\[fig:comparison_be\]](#fig:comparison_be){reference-type="ref" reference="fig:comparison_be"}, where all Taylor series are expanded about the same point $x_0 = -3.08333$, and the value of $L$ is fixed to be $6$ for all optimized approximations. Compared with Taylor approximations, the optimized approach better approximates the part with larger function values, which suppresses the $\mathcal{L}^2$ error more efficiently. Note that both the $T_{2K+1}$ and $O_{2K+1}$ approximations are increasing functions across the entire real axis $\mathbb{R}$, despite the seemingly oscillatory behavior of $O_{2K+1}$ functions. \ \ \ The $\mathcal{L}^2(-L,-1/L)$ error of the $O_{2K+1}$ approximation for different $K$ and $L$ is given in Fig. [3](#fig:error_be){reference-type="ref" reference="fig:error_be"}, where we can again observe the spectral convergence with respect to $K$ for fixed $L$, and the convergence rates are lower for larger intervals. Comparing Fig. [3](#fig:error_be){reference-type="ref" reference="fig:error_be"} with Fig. [2](#fig:error_bs){reference-type="ref" reference="fig:error_bs"}, we can find that the $\mathcal{L}^2$ error for the Bose-Einstein entropy is significantly larger. This is likely due to the singularity of the Planckian at zero. No polynomial possesses the same property, making the function more difficult to approximate using polynomials. The exponential function, however, tends to infinity only when $x$ tends to infinity, which all polynomials with positive leading coefficients also satisfy. {#fig:error_be width=".8\\textwidth"} # Numerical approximation of specific distributions {#sec:numerics} To complete this study, we reproduce simulations from [@PhiRTE1] with the different approximations and compare their results. These consist in approximating distributions that correspond to physical regime, namely a near-beam distribution, a distribution corresponding to two beams crossing each others, and smooth distributions. Finding the approximation requires solving the moment inversion problem: $$\label{eq:mnt_inv} \int_{\mathbb{S}^2} \mathbf{m}(\Omega) \beta(\boldsymbol{\lambda}^T \mathbf{m}(\Omega)) \,\mathrm{d}\Omega = \mathbf{U},$$ for a given vector of moments $\mathbf{U}$ of specific distributions. The vector function $\mathbf{m}(\Omega)$ is chosen as all real spherical harmonics up to a certain degree $N$. Here the function $\beta(\cdot)$ is either the $\beta_{2K+1}$, $T_{2K+1}$ or $O_{2K+1}$ approximation of the exponential function $\beta_{BS}$, and either the $T_{2K+1}$ or $O_{2K+1}$ approximation of the Planckian $\beta_{BE}$. The right-hand side $\mathbf{U}$ is given by the moments of a given function, which means we first choose a function $I(\Omega)$, and then set $$\mathbf{U} = \int_{\mathbb{S}^2} \mathbf{m}(\Omega) I(\Omega) \,\mathrm{d}\Omega.$$ After solving $\boldsymbol{\lambda}$ from [\[eq:mnt_inv\]](#eq:mnt_inv){reference-type="eqref" reference="eq:mnt_inv"}, the function $\beta(\boldsymbol{\lambda}^T \mathbf{m}(\Omega))$ is regarded as an approximation of $I(\Omega)$. For clarification, we will add the subscript $N$ to the name of the model to denote the moment method, the first subscript $N$ refers to the moment order and the second $2K+1$ to the degree of the polynomial approximation $\beta_{2K+1}$, $T_{2K+1}$ or $O_{2K+1}$. For example, if we use spherical harmonics up to degree $N$ in $\Omega$ and choose $\beta(\cdot)$ to be $T_{2K+1}$, the model is denoted as $T_{N,2K+1}$. Similarly, we will also consider the $\beta_{N,2K+1}$ and $O_{N,2K+1}$ models below. The equation [\[eq:mnt_inv\]](#eq:mnt_inv){reference-type="eqref" reference="eq:mnt_inv"} is solved by Newton's method, for which we need to compute the Jacobian $$\int_{\mathbb{S}^2} \mathbf{m}(\Omega) [\mathbf{m}(\Omega)]^T \beta'(\boldsymbol{\lambda}^T \mathbf{m}(\Omega)) \,\mathrm{d}\Omega.$$ Since the integrand is a polynomial of $\Omega$, the integral can be computed exactly using appropriate integration formulas. Here we adopt the Lebedev quadrature ([@Lebedev1976quadratures; @Lebedev1999quadrature]) as in [@PhiRTE1]. The number of quadrature points is chosen such that the degree of the quadrature is no less than the degree of the polynomial. ## Single beam approximation In this section, we apply these approximate entropy models in the approximation of a single beam. Since all the models are rotationally invariant, the direction of the beam does not affect the result. For simplicity, we consider the approximation of the Dirac-delta function $$I(\Omega) = \delta(\Omega - \Omega_0),$$ where $\Omega_0 = (0,0,1)^T$. Some approximations for the Boltzmann-Shannon entropy with $N = 1$ and $K = 2$ are given in Fig. [\[fig:single_beam_M1_BS\]](#fig:single_beam_M1_BS){reference-type="ref" reference="fig:single_beam_M1_BS"}. The plots show that the $\beta_{1,5}$ model gives a remarkably better result than the $T_{1,5}$. This is not surprising because the value of the approximate function is all below $1.0$, which corresponds to $\exp(x)$ with a negative $x$, where the $\beta_{1,5}$ model can give a better approximation. The quality of the $O_{1,5}$ model shown in Fig. [\[fig:single_beam_M1_BS_O5\]](#fig:single_beam_M1_BS_O5){reference-type="ref" reference="fig:single_beam_M1_BS_O5"} then lies in-between, since it approximates the exponential on the interval $[-5,5]$, which balances both the positive and the negative parts. In order to improve the result, we can shift the domain to the negative side as in Fig. [\[fig:single_beam_M1_BS_O10\]](#fig:single_beam_M1_BS_O10){reference-type="ref" reference="fig:single_beam_M1_BS_O10"}, so that a result similar to the $\beta_{1,5}$ model can be obtained. \ Similar phenomena can be observed when the approximation of the Bose-Einstein entropy is applied. The results are plotted in Fig. [\[fig:single_beam_M1_BE\]](#fig:single_beam_M1_BE){reference-type="ref" reference="fig:single_beam_M1_BE"}. For both $T_{1,2K+1}$ and $O_{1,2K+1}$ models, the approximate intensity function is closer to $I(\Omega)$ if the parameters are chosen to fit the range of the function values. \ All the results above are only for $N = 1$ and $K = 2$. To improve the results, we can increase either $N$ or $K$. Fig. [\[fig:single_beam_M1_K6\]](#fig:single_beam_M1_K6){reference-type="ref" reference="fig:single_beam_M1_K6"} includes some results for $K = 6$. Increasing the value of $K$ from $2$ to $6$ does provide improved result for all other parameters, but the beam is still widely spread for all cases. A more efficient way to get improvement is to increase $N$ from $1$ to $3$. The results shown in Fig. [\[fig:single_beam_M3\]](#fig:single_beam_M3){reference-type="ref" reference="fig:single_beam_M3"} exhibit much sharper beams compared with all previous results, and the functions are mostly positive except the Taylor model with $x_0 = 0$. In this test case, the optimized model shows the highest peak value for both types of entropy. Meanwhile, all these results show that the $\beta_{N,2K+1}$ model studied in [@PhiRTE1] is also a good choice for problems involving beams when the Boltzmann-Shannon entropy is considered. However, this model does not have a counterpart for the Bose-Einstein entropy. \ \ \ \ ## Double beam approximation We now considers the approximation of the following function: $$I(\Omega) = \delta(\Omega - \Omega_1) + \delta(\Omega - \Omega_2),$$ which often occurs when two beams cross each other. Here we focus only on the $T_{N,2K+1}$ and $O_{N,2K+1}$ models, and we refer the readers to [@PhiRTE1] for results of the $\beta_{N,2K+1}$ models. Since there are two beams in the intensity function, a model with $N = 1$ cannot give a meaningful approximation. In our experiments, we choose $\Omega_1 = (0,0,1)^T$, $\Omega_2 = (1,0,0)^T$ and use $N = 3$ and $N = 9$ in the approximation. Other parameters are chosen to be the better combination in the previous subsection. In particular, the value of $K$ is fixed to be $2$ in all our examples. For $N = 3$ (see Fig. [\[fig:double_beam_M3\]](#fig:double_beam_M3){reference-type="ref" reference="fig:double_beam_M3"}), the two Taylor models give quite similar results. The two beams are correctly detected with correction locations, and they are both smeared out due to the smooth approximation. The two optimized results provide sharper beams, as the peak value of the distribution is higher. Negative values can still be spotted near the point $(0,-1,0)^T$, which can be improved by increasing $N$ or $K$. Here we only perform experiments with $N = 9$, which can be found in Fig. [\[fig:double_beam_M9\]](#fig:double_beam_M9){reference-type="ref" reference="fig:double_beam_M9"}. The two bright spots are much more pointy than the results of $N = 3$, and the peak values are now significantly higher. Again, the $O_{N,2K+1}$ models perform slightly better than the $T_{N,2K+1}$ models for both types of entropy. \ \ ## Approximating smooth functions We now consider the the approximation of smooth functions and hope to observe spectral convergence. We take the six-Gaussian function considered in [@PhiRTE1]: $$I(\Omega) = \sum_{k=1}^6 \exp\left(-5\|\Omega - \Omega_k\|^2\right),$$ where $$\begin{gathered} \Omega_1 = (1,0,0)^T, \qquad \Omega_2 = (-1,0,0)^T, \qquad \Omega_3 = (0,1,0)^T, \\ \Omega_4 = (0,-1,0)^T, \qquad \Omega_5 = (0,0,1)^T, \qquad \Omega_6 = (0,0,-1)^T.\end{gathered}$$ This function and its approximation using the $\beta_{5,5}$ model are plotted in Fig. [\[fig:six_Gaussian\]](#fig:six_Gaussian){reference-type="ref" reference="fig:six_Gaussian"}. It can be seen that the $\beta_{5,5}$ approximation overestimates the peak value. Fig. [\[fig:six_Gaussian_BS\]](#fig:six_Gaussian_BS){reference-type="ref" reference="fig:six_Gaussian_BS"} shows some approximations based on the Boltzmann-Shannon entropy. It can be seen by naked eyes that the $O_{5,5}$ model defined by optimization on $[-5,5]$ gives the best result. \ Fig. [\[fig:six_Gaussian_BS_error\]](#fig:six_Gaussian_BS_error){reference-type="ref" reference="fig:six_Gaussian_BS_error"} shows the error decay as $N$ increases. Although all methods provide spectral convergence, the choice of parameters does affect the convergence rate. In this example, the $O_{N,5}$ model optimized on $[-10,0]$ and the $T_{N,5}$ model with $x_0 = -5$ have similar performance, which is worse than the $O_{N,5}$ model optimized on the interval $[-5,5]$ but better the $T_{N,5}$ model with $x_0 = 0$. Note that the convergence rate of these models is not determined by the quality of approximation to the $\beta$ function. This is also observed in [@PhiRTE1], where the $P_N$ (which corresponds to $\beta_{N,1}$) model has the best convergence rate among all $\beta_{N,K}$ models, although the $\beta_{N,1}$ model only approximates the exponential function by $\mathrm{e}^x \approx 1 + x$, which is a poor fit. Here we conjecture that the $O_{N,5}$ model optimized on $[-5,5]$ has a better convergence rate because the function $O_5(x)$ is relatively closer to a linear function with slope $1$ for a certain range on the negative part of the domain (see Fig. [\[fig:TO_BS\]](#fig:TO_BS){reference-type="ref" reference="fig:TO_BS"}). We focus on the negative part because the value of $I(\Omega)$ is mostly between $0$ and $1$. This means the $O_{N,5}$ model optimized on $[-5,5]$ is likely to be closer to the $P_N$ model when approximating this function. Now we test the performance of the methods based on the Bose-Einstein entropy. The results are shown in Fig. [\[fig:six_Gaussian_BE_error\]](#fig:six_Gaussian_BE_error){reference-type="ref" reference="fig:six_Gaussian_BE_error"}. It can be seen that the spectral convergence is observed for three models except the $O_{N,5}$ method optimized on the interval $[-5,-0.2]$. This is likely due to the flatness of the $O_5$ function on the interval from $[-2.5, -1.5]$ caused by enforcing good approximations in the region closer to zero where the Planckian has larger values. This can be improved by slightly shifting the upper bound of the domain. Fig.  [\[fig:error_decay\]](#fig:error_decay){reference-type="ref" reference="fig:error_decay"} gives the result for $O_{N,5}$ method optimized on $[-5, -0.5]$, where a much better convergence rate is obtained. Nevertheless, since the choice of the domain is not optimized, the numerical error is still significantly larger than other lines in Figure [\[fig:error_decay\]](#fig:error_decay){reference-type="ref" reference="fig:error_decay"}. {#fig:six_Gaussian_BE_5_error height=".38\\textwidth"} # Conclusion {#sec:concl} We have shown that the moment approximation developed in [@PhiRTE1] is flexible enough to be adapted to dynamical models dissipating various types of entropies. Especially, this construction is shown to be suitable to construct a closure dissipating an approximation of a chosen physical entropy such as Boltzmann-Shannon's or Bose-Einstein's entropy. In this work, the closure is constructed using polynomials so that the integration can be carried out exactly in the moment inversion problem. Such an extension introduces many parameters to the approximate models. It is demonstrated by numerical tests that the quality of the moment closure depends on the choice of these parameters, but in most cases, decent results can be obtained by optimizing the distance between the polynomial and the physical entropy. In our future work, we are going to further study their performance by applying these moment models to the radiative transfer equation with interaction with matter. # Acknowledgements {#acknowledgements .unnumbered} Zhenning Cai was supported by the Academic Research Fund of the Ministry of Education of Singapore under grant No. A-0004592-00-00. [^1]: Department of Mechanical Engineering, Eindhoven University of Technology, Groene Loper 3, 5612 AE Eindhoven, Netherland [^2]: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076. [^3]: CMAP, CNRS, École polytechnique, Institut polytechnique de Paris, 91120, Palaiseau, France

arxiv_math

--- abstract: | In [@S2], it was shown that, if $M$ is a $3$-dimensional asymptotically harmonic with minimal horospheres, then $M$ is flat. However, there is a gap in the proof of this paper. In this paper, we provide the correct proof of the result. Thus we complete the classification of asymptotically harmonic manifolds of dimension $3$: An asymptotically harmonic manifold of dimension $3$ is either a flat or real hyperbolic space. address: - Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Korea - Harish Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhusi, Allahabad 211019, India author: - Jihun Kim - JeongHyeong Park - Hemangi Madhusudan Shah title: - - Asymptotically harmonic manifolds of dimension $3$ with minimal horospheres --- # Introduction Let $(M,g)$ be a complete, simply connected Riemannian manifold without conjugate points. We denote the unit tangent bundle of $M$ by $SM$. For $v \in SM$, let $\gamma_{v}$ be the geodesic with ${\gamma'_{v}}(0) = v$ and $b_{v} (x) =\displaystyle \lim_{t\to \infty} (d (x,\gamma_v(t)) - t)$, the corresponding *Busemann function* for $\gamma_{v}$. The level sets of the Busemann function are called *horospheres* of $M$. A complete, simply connected Riemannian manifold without conjugate points is called *asymptotically harmonic* if the mean curvature of its horospheres is a universal constant, that is, if its Busemann functions satisfy $\Delta b_v \equiv h,\; \forall v \in SM$, where $h$ is a nonnegative constant. Then $b_v$ is a smooth function on $M$ for all $v$ and all the horospheres of $M$ are smooth, simply connected hypersurfaces in $M$ with constant mean curvature $h$. On the other hand, a Riemannian manifold is called (locally) *harmonic*, if about any point all the geodesic spheres of sufficiently small radii are of constant mean curvature. Since a harmonic manifold is Einstein, harmonic manifolds of dimensions $2$ and $3$ are of constant sectional curvature. In 1944, Lichnerowicz [@Li] showed that a $4$-dimensional harmonic manifold is locally symmetric and conjectured that a harmonic manifold is flat or locally rank-one symmetric space (this conjecture is called *Lichnerowicz's conjecture*). Nikolayevsky [@Ni] proved Lichnerowicz's conjecture in dimension $5$. Szabó [@Sz] proved the conjecture for compact harmonic manifolds. On the other hand, Damek and Ricci [@DR] constructed nonsymmetric harmonic manifolds of dimension $\ge 7$, which are called *Damek-Ricci spaces*. It follows from [@RS1] that every complete, simply connected harmonic manifold without conjugate points is asymptotically harmonic. It is natural to ask whether asymptotically harmonic manifolds are locally symmetric? Heber [@He] proved that for noncompact, simply connected homogeneous space, the manifold is asymptotically harmonic and Einstein if and only if it is flat, or rank-one symmetric space of noncompact type, or a nonsymmetric Damek-Ricci space. For more characterizations of asymptotically harmonic manifolds, we refer to [@KP; @Z]. By the Riccati equation [\[Riccati\]](#Riccati){reference-type="eqref" reference="Riccati"}, one can easily check that a Ricci-flat asymptotically harmonic manifold is also flat. In [@RS], it was shown that harmonic manifolds with minimal horospheres are flat. In [@S2], it was shown that an asymptotically harmonic manifold of dimension $3$ with minimal horospheres is flat. But we found that there is some gap in the proof of [@S2 Lemma 2.2]: $\operatorname{tr}\sqrt{-R(x,v)v}=0$ *does not* imply $R(x,v)v=0$. In this article, we give the correct proof of this theorem. Now we recall some notations about asymptotically harmonic manifolds (see [@H; @S; @S2]). For $v\in SM$ and $x\in v^\perp$, we define $u^{\pm}(v)\in {\rm End}(v^\perp)$ by $$u^+(v)(x)=\nabla_x \nabla b_{-v},\quad u^{-}(v)(x)=-\nabla_x\nabla b_{v}.$$ Also, $u^{\pm}$ satisfies the Riccati equation along the orbits of the geodesic flow $\varphi^t: SM\to SM$. If we let $u^{\pm}(t):=u^{\pm}(\varphi^t v)$ and $R(t):=R\big(~\cdot~, \gamma_v'(t)\big)\gamma_v'(t) \in {\rm End}(\gamma_v'(t)^\perp)$, then $u^{\pm}(t)$ satisfy the following Riccati equation: $$\label{Riccati} (u^{\pm})' + (u^{\pm})^2 + R = 0.$$ Here, $u^{+}(t)$ and $u^{-}(t)$ are called as *unstable* and *stable* Riccati solution, repsectively. We have $\operatorname{tr}u^+ (v) =\Delta b_{-v} = h$ and $\operatorname{tr}u^{-}(v) = -\Delta b_v = -h$ for all $v\in SM$.\ Clearly by the Riccati equation [\[Riccati\]](#Riccati){reference-type="eqref" reference="Riccati"}, we see that any $2$-dimensional asymptotically harmonic manifold, is either a flat or a real hyperbolic plane of constant curvature $-h^2$. This shows that the study of asymptotically harmonic manifold begins with dimension $3$. Towards this, it was shown in [@H] that any Hadamard asymptotically harmonic manifold of bounded sectional curvature, satisfying some mild hypothesis on the curvature tensor is a real hyperbolic space of constant sectional curvature $-\frac{h^2}{4}$. Finally, this result was improved by Schroeder and Shah [@S] by relaxing the hypothesis on the curvature tensor. **Theorem 1** ([@S]). *If $(M,g)$ is a $3$-dimensional asymptotically harmonic manifold with $h>0$, then $M$ is a real hyperbolic space of constant sectional curvature $-\frac{h^2}{4}$.* Our main result is the following. **Theorem 2**. *If $(M,g)$ is a $3$-dimensional asymptotically harmonic manifold with minimal horospheres (i.e., $h=0$), then $M$ is flat.* From Theorem [Theorem 1](#thm:ross){reference-type="ref" reference="thm:ross"} and Theorem [Theorem 2](#thm:mainthm){reference-type="ref" reference="thm:mainthm"}, we obtain the complete classification of asymptotically harmonic manifolds of dimension $3$. **Theorem 3**. *Let $(M,g)$ be a $3$-dimensional asymptotically harmonic manifold. Then $M$ is flat if $h=0$, and $M$ is a real hyperbolic space of constant sectional curvature $-\frac{h^2}{4}$ if $h>0$.* # Proof of Theorem [Theorem 2](#thm:mainthm){reference-type="ref" reference="thm:mainthm"} {#proof-of-theorem-thmmainthm} Let $(M,g)$ be a complete, simply connected Riemannian manifold without conjugate points. To prove our main result, we need the following results. **Lemma 4** ([@Z]). *If $(M, g)$ is an asymptotically harmonic manifold, then the map $v \mapsto u^{\pm}(v)$ is continuous on $SM$.* **Lemma 5**. *Let $(M,g)$ be an $n$-dimensional asymptotically harmonic manifold. Then we have $\operatorname{Ric}(v,v)\le -\frac{h^2}{n-1}$. In particular, if $M$ has minimal horospheres, then $\operatorname{Ric}(v,v)\le 0$.* **Proof.* From the Riccati equation [\[Riccati\]](#Riccati){reference-type="eqref" reference="Riccati"}, we get $\operatorname{Ric}(v,v)=-\operatorname{tr}(u^{+}(v)^2)$. For the eigenvalues of $u^{+}(v)$, the arithmetic-quadratic mean inequality yields $h^2 = (\operatorname{tr}u^{+}(v))^2\le (n-1)\operatorname{tr}(u^{+}(v)^2)$, and hence we have $\operatorname{Ric}(v,v)\le -\frac{h^2}{n-1}$. Note that if $M$ has a minimal horosphere (i.e., $h=0$), then $\operatorname{Ric}(v,v)\le 0$. ◻* **Lemma 6**. *If $(M,g)$ is a $3$-dimensional asymptotically harmonic manifold with minimal horospheres, then for every $p\in M$, there exists an orthonormal basis $\{e_1,e_2,e_3\}$ of $T_p M$ such that the curvature operator $$\mathcal{R}:\wedge^2 T_pM \to \wedge^2 T_p M \;{\textrm{given by}}\;\ g(R(X\wedge Y),Z\wedge W) = g(R(X,Y)W,Z)$$ is diagonal.* **Proof.* As $\dim M =3$, we can identify $S_p M$ with the standard $2$-sphere ${S}^2$, and $v^\perp$ with $T_v S^2$ for $v\in S_p M = {S}^2$. Eigenvalues of $u^{+}(v)$ are $\lambda(v)$ and $-\lambda(v)$. As $TS^2$ is nontrivial, an easy topological argument shows that for every $p\in M$, there exists $v_0 \in S_p M$ such that $u^{+}(v_0 ) =0$ (see [@H p. 848]).* *Let $v_0 = e_1$. Then $u^{+}(e_1)'(x) = -R(x,e_1)e_1$ for $x\in \{e_1\}^\perp$. Since $\operatorname{tr}u^{+}(v) = 0$ for all $v \in SM$, it follows that $\operatorname{tr}u^{+}(v)' =0$. Consequently, we have that Ricci$(e_1, e_1) = 0$ and thus *Ricci curvature attains maximum*. * *Denote the eigenvalues of $u^{+}(e_1)'$ by $\mu$ and $-\mu$. Without loss of generality, we assume that $\mu \ge 0$. If we let $\{e_2, e_3\}\in \{e_1\}^\perp$ be the eigenvectors of $u^{+}(e_1)'$, we have $R(e_2,e_1)e_1 = -\mu e_2$ and $R(e_3,e_1)e_1 = \mu e_3$. We also have $\operatorname{Ric}(e_3,e_3)\le 0$ by Lemma [Lemma 5](#vec){reference-type="ref" reference="vec"}. Hence $K(e_2,e_3):=g(R(e_2,e_3)e_3,e_2)= - \eta \le -\mu \leq 0$.* *Consider for $t\in (-\varepsilon,\varepsilon)$ the vectors $v_t=\operatorname{cos}te_1 + \sin te_2$. Then we let $$f(t):=\operatorname{Ric}(v_t,v_t)=K(e_1,e_2)+\sin^2 t K(e_2,e_3) + \operatorname{cos}^2 t K(e_1,e_3) + \sin 2t\, g(R(e_1,e_3)e_3,e_2).$$ Since $f(0)=0$ and it is maximal, $f'(0)=0$. This implies $g(R(e_1,e_3)e_3,e_2)=0$. If we replace $e_2$ with $e_3$ in the above computation, we also obtain $g(R(e_1,e_2)e_2,e_3)=0$. Therefore, for every point $p$, the curvature operator is diagonal and it is given by $${\mathcal{R}}(p) = \begin{pmatrix} - \mu(p) & 0 & 0\\ 0 & \mu(p) & 0\\ 0 & 0 & { -\eta(p)} \end{pmatrix}$$ in the basis $\{e_1,e_2,e_3\}$ of $T_p M$. ◻* **Remark:** In the above lemma, from the construction of $\mu$ and $\eta$ (using $\operatorname{tr}u^{+}(v)'=0$ and $\operatorname{Ric}(e_3,e_3)\le 0$), we note that the eigenvalues $\mu$ and $\eta$ can be interchanged, but the aforementioned form of the curvature operator is the same. **Proposition 7**. *Let $(M^3, g)$ be an asymptotically harmonic manifold with minimal horospheres. Then around any point $p \in M$, we can find an orthogonal coordinate system $(U, (x^1, x^2, x^3))$ in which the metric is diagonal, that is $g=\sum\limits_{i=1}^{3}a_i(dx^{i})^2$ for some positive function $a_i$. And the orthonormal basis $\{e_1,e_2,e_3\}$ of $T_{p}M$ obtained in Lemma [Lemma 6](#lem:curvdiag){reference-type="ref" reference="lem:curvdiag"} with the associated orthonormal frame with $e_i = \dfrac{1}{\sqrt{a_i}}\dfrac{\partial}{\partial x^i} (p)$ for $i=1,2,3$.* *Proof.* The existence of a required orthogonal coordinate system along with the associated orthonormal frame $\{e_1,e_2,e_3\}$ of $T_{p}M$ is guaranteed by Theorem $4.2$ of [@DY], which says that *every $3$-dimensional Riemannian manifold has orthogonal coordinates, that is, coordinates in which the Riemannian metric has a diagonal form*. ◻ **Corollary 8**. *Let $\{e_1, e_2, e_3\}$ be an orthonormal basis of $T_p M$ obtained from Lemma [Lemma 6](#lem:curvdiag){reference-type="ref" reference="lem:curvdiag"}. Then the Christoffel symbols $\Gamma_{ij}^k=0$ for all distinct indices $i,j,k$.* *Proof.* By the Koszul formula, the Christoffel symbol $\Gamma_{ij}^{k}$ for $\{e_1, e_2, e_3\}$ is given by $$\Gamma_{ij}^{k} = \frac{1}{2}\big(g([e_i,e_j],e_k)-g([e_i,e_k],e_j)-g([e_j,e_k],e_i)\big).$$ Since $e_i = \dfrac{1}{\sqrt{a_i}}\dfrac{\partial}{\partial x^i}$ for orthogonal coordinates $(x^i)$, we have that $\Gamma_{ij}^{k}=0$ for all distinct indices $i,j,k$. ◻ **Lemma 9**. *If $(M,g)$ is a $3$-dimensional asymptotically harmonic manifold with minimal horospheres, then the sectional curvature $K$ of $M$ satisfies $K \le 0$.* *Proof.* Let $\{e_1,e_2,e_3\}$ be as in the Lemma [Lemma 6](#lem:curvdiag){reference-type="ref" reference="lem:curvdiag"}. At fixed point $p\in M$, we can consider a geodesic $\gamma_{e_1}$ with $\gamma_{e_1}(0)=p$, $\gamma_{e_1}'(0)=e_1$ and $b_{e_1}(x)=\displaystyle\lim_{t\to \infty}(d(x,\gamma_{e_1}(t))-t)$. Since $\nabla_{e_1}e_1 = 0$, we get Christoffel symbols $\Gamma_{11}^{1}=\Gamma_{11}^{2}=\Gamma_{11}^{3}=0$. By Corollary [Corollary 8](#cor){reference-type="ref" reference="cor"}, $\Gamma_{ij}^k =0$ for all distinct indices $i,j,k$. So, we need to compute the other Christoffel symbols when at least two indices are the same. From $g(\nabla_{e_i}e_j , e_k) = -g(e_j, \nabla_{e_i} e_k)$ for all $i,j,k$, we get $\Gamma_{ij}^k = -\Gamma_{ik}^j$. [Case 1]{.smallcaps}: All indices are the same. If $1\le i=j=k \le 3$, we have $\Gamma_{11}^1 = \Gamma_{22}^2 = \Gamma_{33}^3=0$. [Case 2]{.smallcaps}: Two indices are the same. From $g(\nabla_{e_i}e_j , e_j)=0$, we obtain $\Gamma_{ij}^j=0$. Therefore, all the Christoffel symbols are zero except $\Gamma_{21}^2=-\Gamma_{22}^1$, $\Gamma_{23}^2=-\Gamma_{22}^3$, $\Gamma_{31}^3=-\Gamma_{33}^1$, $\Gamma_{32}^3=-\Gamma_{33}^2$. Now we compute the curvature tensor $R(e_2,e_1)e_1$. Since $\nabla_{e_2}e_1 = \Gamma_{21}^2 e_2$ and $[e_2,e_1] = \nabla_{e_2}e_1 - \nabla_{e_1}e_2 = \Gamma_{21}^2 e_2$, we obtain $$\begin{aligned} -\mu e_2 &= R(e_2,e_1)e_1\\ &=\nabla_{e_2}\nabla_{e_1}e_1 - \nabla_{e_1}\nabla_{e_2}e_1 - \nabla_{[e_2,e_1]}e_1\\ &=-\left(e_1 \Gamma_{21}^2 +\left(\Gamma_{21}^2\right)^2\right)e_2.\end{aligned}$$ Similarly, we also obtain $\mu e_3 =-\left(e_1 \Gamma_{31}^3 +\left(\Gamma_{31}^3\right)^2 \right)e_3$, as $\mu e_3 = R(e_3 ,e_1)e_1$. Hence, we have $$-\mu = -\left(e_1 \Gamma_{21}^2 +\left(\Gamma_{21}^2\right)^2\right),\quad \mu=-\left(e_1 \Gamma_{31}^3 +\left(\Gamma_{31}^3\right)^2 \right),$$ and hence consequently, $$\label{sum-chri} 0 = e_1 \left(\Gamma_{21}^2 + \Gamma_{31}^3\right) +\left(\Gamma_{21}^2\right)^2+\left(\Gamma_{31}^3\right)^2.$$ We observe that $$\begin{aligned} \Gamma_{21}^2 + \Gamma_{31}^3 &= \Gamma_{11}^1 + \Gamma_{21}^2 + \Gamma_{31}^3 \\ &=g(\nabla_{e_1}e_1, e_1)+g(\nabla_{e_2}e_1, e_2)+g(\nabla_{e_3}e_1, e_3)\\ &=\operatorname{div}(e_1)(p) = \operatorname{div}(\nabla b_{-e_1})(p) = \Delta b_{-e_1}(p) = 0.\end{aligned}$$ From ([\[sum-chri\]](#sum-chri){reference-type="ref" reference="sum-chri"}) this yields that $\Gamma_{21}^2 = \Gamma_{31}^3=0$, so $K(e_1 ,e_2)=K(e_1, e_3)=0$ and $K(e_2, e_3) \le 0$. This implies that the sectional curvature $K$ of $M$ satisfies $K\le 0$. ◻ Now finally, we prove our main result. . From Lemma [Lemma 9](#lem:curvature){reference-type="ref" reference="lem:curvature"}, the sectional curvature $K\le 0$. Using standard comparison geometry, we obtain two eigenvalues $\lambda_1 (v)$ and $\lambda_2 (v)$ of the operator $u^{+}(v)$ satisfy $\lambda_1(v), \lambda_2 (v)\ge 0$ for all $v\in SM$. But $\lambda_1(v)+\lambda_2 (v)=\operatorname{tr}u^{+}(v) = h = 0$, we have $\lambda_1(v)=\lambda_2 (v)=0$. Therefore, $u^{+}(v)\equiv 0$ and so $R(x,v)v = 0$ for all $x\in v^\perp$ by Riccati equation ([\[Riccati\]](#Riccati){reference-type="ref" reference="Riccati"}). This completes the proof.0◻ # Concluding Remarks Let $(M,g)$ be a complete, simply connected Riemannian manifold without conjugate points. In this section, the dimension of $M$ is arbitrary. We define $V(v):=u^{+}(v)-u^{-}(v)$ and $X(v):=-\frac{1}{2}(u^{+}(v)+u^{-}(v))$, correspondingly $V(t):=V(\varphi^t v)$ and $X(t):=X(\varphi^t v)$, respectively, where $\varphi^t :SM \to SM$ is the geodesic flow. Then the Riccati equation [\[Riccati\]](#Riccati){reference-type="eqref" reference="Riccati"} gives $V' = XV+VX$. If we assume that $M$ is an asymptotically harmonic manifold with minimal horospheres and Einstein, then we have that $M$ is flat as follows: **Lemma 10** ([@Z]). *Let $(M,g)$ be an asymptotically harmonic manifold. Then $V(v)\ge 0$ for all $v \in SM$.* **Lemma 11** ([@S1]). *If $(M,g)$ is an asymptotically harmonic manifold, then for every point $p\in M$, there exists $v_0 \in S_p M$ such that $u^{+}(v_0)$ and $u^{+}(-v_0)$ have the same eigenvalues. In matrix sense, for $v_0 \in S_p M$, there exists an orthogonal matrix $p(v_0 )$ such that $p(v_0)^{-1}u^{+}(v_0)p(v_0)=u^{+}(-v_0)$.* **Theorem 12**. *If $(M,g)$ is an asymptotically harmonic manifold with minimal horospheres and Einstein, then $M$ is flat.* **Proof.* Consider $V(v)=u^{+}(v)-u^{-}(v)$. Since $\operatorname{tr}V(v)= 2h = 0$ for all $v\in SM$, from Lemma [Lemma 10](#lem:nonnegativeV){reference-type="ref" reference="lem:nonnegativeV"} we obtain $V(v)\equiv 0$, which implies that $u^{+}(v) = u^{-}(v)$ for all $v\in SM$. Using Lemma [Lemma 11](#lem:u+){reference-type="ref" reference="lem:u+"}, for every point $p\in M$, there exists $v_0\in S_p M$ such that $$p(v_0)^{-1}u^{+}(v_0)p(v_0) = u^{+}(-v_0)= u^{-}(-v_0) = -u^{+}(v_0).$$ It follows that $u^{+}(v_0)$ and $-u^{+}(v_0)$ have the same eigenvalues. Then the corresponding eigenvalues vanish and so $u^{+}(v_0)=0$. Hence, the Riccati equation [\[Riccati\]](#Riccati){reference-type="eqref" reference="Riccati"} yields $\operatorname{Ric}(v_0,v_0)=0$. Since $M$ is Einstein, $M$ is Ricci-flat, which implies that $M$ is flat. ◻* Now we consider the case when $M$ is a symmetric space. **Lemma 13** ([@H]). *Let $(M,g)$ be an asymptotically harmonic manifold. Then $X\equiv 0$ if and only if $M$ is a symmetric space.* **Theorem 14**. *If $(M,g)$ is an asymptotically harmonic with minimal horospheres which is also a symmetric space, then $M$ is flat.* **Proof.* Let us assume that $h=0$. Using $\operatorname{tr}V(v)= 2h = 0$ for all $v\in SM$ and Lemma [Lemma 10](#lem:nonnegativeV){reference-type="ref" reference="lem:nonnegativeV"}, we obtain $V(v)\equiv 0$, which means that $u^{+}(v)=u^{-}(v)$ for all $v\in SM$. Then we get $X(v)=-\frac{1}{2}(u^{+}(v)+u^{-}(v))=-u^{+}(v)$. Since $M$ is symmetric, from Lemma [Lemma 13](#lem:sym){reference-type="ref" reference="lem:sym"} we have $u^{+}(v)=0$ for all $v\in SM$. Hence the Riccati equation [\[Riccati\]](#Riccati){reference-type="eqref" reference="Riccati"} gives us $K\equiv 0$. ◻* On the other hand, it is known that the following relation holds for a Riemannian manifold $M$: $M$ has nonpositive sectional curvature, then $M$ has no focal points which implies that $M$ has no conjugate points. For an asymptotically harmonic manifold, if we strengthen the condition of "without conjugate points\", then we obtain that an asymptotically harmonic manifold with minimal horospheres is flat as follows: If $u^{+}(v)$ is positive semi-definite and $\operatorname{tr}u^{+}(v)=0$, $u^{+}(v)=0$. Then the Riccati equation [\[Riccati\]](#Riccati){reference-type="eqref" reference="Riccati"} implies $K \equiv 0$. **Proposition 15**. *Let $(M,g)$ be a complete, simply connected manifold with nonpositive sectional curvature (i.e. Hadamard manifold). If $M$ is an asymptotically harmonic with minimal horospheres, then $M$ is flat.* **Proof.* As $M$ has nonpositive sectional curvature, using standard comparison geometry we get $u^{+}(v)$ is positive semi-definite. Then by the above argument, we have that $M$ is flat. ◻* **Proposition 16** ([@Z]). *Let $(M,g)$ be a complete, simply connected Riemannian manifold without focal points. If $M$ is an asymptotically harmonic with minimal horospheres, then $M$ is flat. Moreover, asymptotically harmonic manifolds without focal points which is a Riemannian product must be flat.* Besides, when $M$ has no conjugate points, in [@RS2] the authors obtained the following. **Proposition 17** ([@RS2]). *Let $(M,g)$ be a complete, simply connected Riemannian manifold without conjugate points. If $M$ is an asymptotically harmonic with minimal horospheres such that the determinant of the second fundamental form of geodesic spheres is a radial function, then $M$ is flat.* At the end of the paper, we state the following conjecture which is analog of Lichnerowicz's conjecture for asymptotically harmonic manifolds. This conjecture has been partially resolved to date. Including Theorem [Theorem 3](#class){reference-type="ref" reference="class"}, we refer to [@CS; @He; @H; @KP; @S; @Z]. **Conjecture 1**. *Let $(M,g)$ be a complete, simply connected Riemannian manifold without conjugate points. If $M$ is an asymptotically harmonic manifold, then $M$ is either flat or rank-one symmetric space of noncompact type.* # Acknowledgements {#acknowledgements .unnumbered} This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1A2C1083957). The authors thank to Paul-Andi Nagy for the useful discussion on orthogonal coordinates. ccccc P. Castillon, A. Sambusetti, *On asymptotically harmonic manifolds of negative curvature*, Math. Z. **277** (2014), no.3-4, 1049-1072. E. Damek, F. Ricci, *A class of nonsymmetric harmonic Riemannian spaces*, Bull. Amer. Math. Soc. (N.S.) **27** (1992), no.1, 139-142. D. M. DeTurck, D. Yang, *Existence of elastic deformations with prescribed principal strains and triply orthogonal systems*, Duke Math. J. **51** (1984), no.2, 243-260. J. Heber, *On harmonic and asymptotically harmonic homogeneous spaces*, Geom. Funct. Anal. **16** (2006), no.4, 869-890. J. Heber, G. Knieper, H. Shah, *Asymptotically harmonic spaces in dimension $3$*, Proc. Amer. Math. Soc. **135** (2007), 845-849. G. Knieper, N. Peyerimhoff, *Geometric properties of rank one asymptotically harmonic manifolds*, J. Differential Geom. **100** (2015), no.3, 507-532. A. Lichnerowicz, *Sur les espaces riemanniens complètement harmoniques*, Bull. Soc. Math. France **72** (1944), 146-168. Y. Nikolayevsky, *Two theorems on harmonic manifolds*, Comment. Math. Helv. **80** (2005), no.1, 29-50. A. Ranjan, H. Shah, *Convexity of spheres in a manifold without conjugate points*, Proc. Indian Acad. Sci. Math. Sci. **112** (2002), no.4, 595-599. A. Ranjan, H. Shah, *Harmonic manifolds with minimal horospheres*, J. Geom. Anal. **12** (2002), no.4, 683-694. A. Ranjan, H. Shah, *Busemann functions in a harmonic manifold*, Geom. Dedicata **101** (2003), 167--183. H. Shah, *A new proof of the theorem: harmonic manifolds with minimal horospheres are flat*, Proc. Indian Acad. Sci. Math. Sci. **124** (2014), no.3, 419-425. V. Schroeder, H. Shah, *On $3$-dimensional asymptotically harmonic manifolds*, Arch. Math. **90** (2008), 275-278. H. Shah, *$3$-dimensional asymptotically harmonic manifolds with minimal horospheres*, Arch. Math. (Basel) **106** (2016), no.1, 81-84. Z. I. Szabó, *The Lichnerowicz conjecture on harmonic manifolds*, J. Differential Geom. **31** (1990), no.1, 1-28. A. Zimmer, *Compact asymptotically harmonic manifolds*, J. Mod. Dyn. **6** (2012), no.3, 377-403.

arxiv_math

--- abstract: | We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results. author: - "Jan Giesselmann [^1]" - "Kiwoong Kwon[^2]" bibliography: - ./references.bib title: A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model --- **Keywords:** Keller-Segel; chemotaxis; nonlinear diffusion; discontinuous Galerkin scheme; a posteriori error analysis **AMS subject classification** (2020): Primary 65M50, Secondary 65M08, 35K40 # Introduction The Keller-Segel system is a well-known mathematical model in the field of chemotaxis. Introduced by E. Keller and L. Segel [@Keller_1970], it describes the collective motion of cells in response to concentration gradients. It consists of two coupled partial differential equations: $$\label{intro:pp_eqn} \begin{matrix} \begin{aligned} \partial_t\rho +\nabla\cdot(\chi\rho\nabla c - \nabla \rho) &= 0 \\ \epsilon\partial_t c + c - \Delta c - \alpha\rho &= 0 \end{aligned} \end{matrix}$$ and suitable initial and boundary conditions. Here $\rho=\rho(t,x)$ and $c=c(t,x)$ denote the density of the cell population and the concentration of chemically attracting substances, respectively, for any time $t>0$ at location $x \in \Omega$. We will consider the problem on a bounded open set $\Omega$ in $\mathbb{R}^d$ with a piecewise smooth boundary $\partial \Omega$. The parameters $\chi$, $\alpha$, and $\epsilon$ are constant and satisfy $\chi, \alpha >0$ and $\epsilon\geq 0$. The modelling background of this system has been extensively discussed in the literature [@Hillen2008; @Murray2002], and there are large number of extensions, e.g. models accounting for multiple stimuli and multiple species [@2002Wolansky; @2010Horstmann; @2020Karmakar]. In this work, we focus on the parabolic-elliptic form of the Keller-Segel system described by the partial differential equations [\[intro:pp_eqn\]](#intro:pp_eqn){reference-type="eqref" reference="intro:pp_eqn"} when $\epsilon = 0$ (and setting the other parameters to $1$ for simplicity): $$\begin{aligned} \partial_t\rho +\nabla\cdot(\rho\nabla c - \nabla \rho) &= 0 \quad\text{in} \ (0,T)\times\Omega \label{eq:KS1}\\ c - \Delta c &= \rho\quad \text{in} \ (0,T)\times\Omega \label{eq:KS2}\\ \nabla \rho \cdot \mathrm{n} &=0\quad \text{on} \ (0,T) \times \partial \Omega\label{eq:bc1}\\ \nabla c \cdot \mathrm{n} &=0 \quad \text{on} \ (0,T) \times\partial \Omega \label{eq:bc2}\\ \rho(0, \cdot) &= \rho_0 \quad \text{in} \ \Omega \label{eq:ic}\end{aligned}$$ where $\mathrm{n}$ denotes the outward unit normal vector. A striking feature of the Keller-Segel system which motivates detailed investigations is the finite time blow-up of solutions [@Nagai1995]. Such singularities are expected in settings of dimension $d \geq 2$ when the initial cell population surpasses a certain threshold. For $d=2$, this threshold is on the total mass [@Gajewski_1998; @Blanchet2006] and for $d \geq 3$, it depends on the maximal local concentration. For an overview on the Keller-Segel system, we refer to the review papers [@Horstmann2003; @Horstmann2004]. For special initial data explicit formulas for blow up solutions exist but, in general, approximating solutions of this system with blow-up accurately is challenging. Several numerical methods have been developed over the years to tackle this issue. These methods include the fractional step method [@Tyson2000], finite volume schemes [@Filbet2006; @2008Chertock], and conservative upwind finite-element methods [@Saito_2007; @2013Strehl]. Additionally, [@Epshteyn2008/09; @Li2017] employed discontinuous Galerkin methods, and [@2005Budd] utilized a moving mesh method. More generally, solutions to chemotaxis systems frequently display strong growth of cell density close to some point or curve. Indeed, the system may not only exhibit blow-up but it may also develop other 'spiky' structures. Resolving such highly localized structures on uniform meshes is, arguably, inefficient and there has been an intense interest in the development of (mesh) adaptive numerical schemes [@Chertock2019; @Sulman2019] with the goal of increasing accuracy and efficiency. Several heuristic strategies for mesh adaptation can be found in the literature, e.g., [@Sulman2019] presents an adaptive moving mesh finite element method, using a coordinate transformation to concentrate grid nodes in regions of large solution variations, and [@Chertock2019], which proposes a semi-discrete adaptive moving mesh finite-volume upwind method, enhancing resolution in blow-up regions by increasing the density of mesh nodes. To the best of our knowledge, mesh adaptation based on a posteriori error estimates has not been investigated. The goal of this work is to provide a basis for such investigations. At the same time, our results provide rigorous error control of simulations. Mesh adaptation based on a posteriori error estimates has been extremely successful for simpler types of PDEs, even leading to provably optimal meshes in certain cases. For instance, convergence of an adaptive space-time finite element method that relies on an a posteriori error estimator for solving linear parabolic partial differential equations was proven in [@Kreuzer_2012]. A posteriori error estimates come in several varieties: If there is a goal functional of specific interest dual weighted residuals may be used, see e.g. [@Becker2001], which usually leads to very efficient meshes. We follow a different approach that aims at controlling the error in a suitable norm and is nicely explained in [@Makridakis2007]. The main idea is to insert a (sufficiently regular) reconstruction of the numerical solution into the PDE so that a suitable stability theory can be used to bound the difference between the exact solution to the PDE and this reconstruction. For nonlinear equations it may happen that the stability theory allows only *conditional* a posteriori error estimates, e.g. the Allen-Cahn equation [@Bartels], and this is indeed the case here. Different stability frameworks, e.g. ones based on (relative) entropy or negative order norms, could be considered for the Keller-Segel system. We present an a posteriori error estimate using a rather standard $L^2$-based stability framework since this is the framework that, for us, leads to the strongest results. It might seem that using an $L^2$-based stability framework requires unrealistic and unverifiable assumptions on the regularity of exact solutions but it turns out that, for sufficiently regular initial data, weak solutions retain this regularity until blow-up time [@Biler1994] and we provide an a posteriori verifiable condition that guarantees that the exact solution is sufficiently regular. The numerical scheme studied in this manuscript is a straightforward discontinuous Galerkin (dG) scheme. Much more sophisticated schemes have been developed in the literature, e.g. [@Guo2019; @Qiu_2021], offering features such as entropy dissipation and positivity preservation. It is beyond the scope of the current work to extend our error estimator to those schemes but it should be noted that several results from this paper, in particular Theorem [Theorem 8](#thm:stabilityest){reference-type="ref" reference="thm:stabilityest"}, can be directly used for the analysis of those schemes, e.g. the companion paper [@GiesselmannKolbe] provides suitable reconstructions and an $H^{-1}$-norm estimate for the residual for a positivity preserving finite volume scheme. Our analysis is performed in the semi-(spatially)-discrete setting since, arguably, this already covers the key challenges. Our results can be combined with ideas from the study of a posteriori error estimates for temporal discretisation of PDEs with blow-up [@Cangiani_2016; @Kyza_2020]. It is worth noting that the error estimator presented in our work is of optimal order, i.e. the error estimator has the same order of convergence as the norm of the error that it controls. We investigate a semi-discrete discontinuous Galerkin (dG) scheme discretising Laplace operators by the symmetric interior penalty (SIP) bilinear form and the chemotaxis term by the weighted SIP (wSIP) bilinear form treating density as a diffusion coefficient. Our analysis applies to arbitrary polynomial degrees $k \geq 1$. The outline of the paper is as follows: In Section 2, we provide some background on weak solutions for the Keller-Segel system, with a focus on blow-up phenomena. In Section 3, we establish a stability estimate for the strong solution of the system, serving as the foundation for constructing the a posteriori error estimator. It is worth noting that this stability estimate can, in certain situations, a posteriori verify the existence of a sufficiently regular weak solution. In Section 4, we introduce our semi-spatial-discrete discontinuous Galerkin scheme. In Section 5, we define the reconstruction of the numerical solution, utilizing the so-called elliptic reconstruction, and also define the residual. In Section 6, we derive a computable upper bound for the $H^{-1}$-norm of the residual. Section 7, which delineates our main result, presents the full a posteriori error estimator. Finally, in Section 8, we conduct numerical experiments. # Background on weak solutions and blow-up We study the parabolic-elliptic Keller-Segel system with Neumann boundary conditions [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"}--[\[eq:ic\]](#eq:ic){reference-type="eqref" reference="eq:ic"}. The existence of a local-in-time weak solution for this problem is well established [@Biler1994 Section 3]. Let us recall the definition of a weak solution as given in [@Biler1994]: **Definition 1**. *[@Biler1994] [\[def:weaksol\]]{#def:weaksol label="def:weaksol"} A function $\rho \in L^{\infty}\left(0,T ; L^2(\Omega)\right) \cap L^2\left(0,T ; H^1(\Omega)\right)$ is called a *weak solution* of the problem [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"}-[\[eq:ic\]](#eq:ic){reference-type="eqref" reference="eq:ic"} on $\left( 0, T \right)\times \Omega$, if it satisfies $$\begin{gathered} \int_{\Omega} \rho(t, x) \varphi(t, x) d x - \int_0^t \int_{\Omega} \rho \partial_t \varphi - (\nabla\rho - \rho \nabla c) \cdot \nabla \varphi dx ds\\ =\int_{\Omega} \rho_0(x) \varphi(0, x) d x \end{gathered}$$ for every $\varphi \in H^1((0, T)\times\Omega )$ and for a.e. $t \in(0, T)$ and, if, in addition, for a.e. $t \in(0, T)$ $$c(t, \cdot) \in H^1(\Omega) \text { with } c(t, x) = (G * \rho(t,\cdot))\left( x \right) \quad \text{for a.e.}\ x \in \Omega$$ holds, where $G$ is an appropriate Green's function for $\Omega$.* **Theorem 2** (Existence of weak solutions). *[@Biler1994 Theorem 2] [\[thm:weaksol\]]{#thm:weaksol label="thm:weaksol"} Assume that $\Omega$ is a bounded domain in $\mathbb{R}^d$ with piecewise smooth boundary.* 1. *If $d\in\{2,3\}$, and $0 \leq \rho_0 \in L^2(\Omega)$, then there exists $T=$ $T\left( \left\| \rho_0 \right\|_{L^2(\Omega)} \right)$ such that the problem [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"}-[\[eq:ic\]](#eq:ic){reference-type="eqref" reference="eq:ic"} has a unique weak solution $\rho \in$ $L^{\infty}\left(0, T; L^2(\Omega)\right) \cap L^2\left(0, T ; H^1(\Omega)\right)$. Moreover, $\partial_t\rho \in L^2\left(0, T ; H^{-1}(\Omega)\right)$, $\rho(t, x) \geq 0$ for a.e. $x \in \Omega$ and $t \geq 0$, and $\int_{\Omega} \rho(t, x) d x=\int_{\Omega} \rho_0(x) d x$.* 2. *If $d \geq 2$ and $0 \leq \rho_0 \in L^p(\Omega), p>d / 2$, then there exists $T=$ $T\left(p, \left\| \rho_0 \right\|_{L^p(\Omega)}\right)>0$ and a weak solution $\rho$ such that $\rho \in L^{\infty}\left(0, T ; L^p(\Omega)\right)$ and $\rho^{p / 2} \in L^2\left(0, T ; H^1(\Omega)\right)$. These solutions are unique when $p>d$, and regular when $p>d / 2$ in the sense that $\rho \in L_{\mathrm{loc}}^{\infty}\left(0, T ; L^{\infty}(\Omega)\right)$.* **Remark 3**. *With (i) of Theorem [\[thm:weaksol\]](#thm:weaksol){reference-type="ref" reference="thm:weaksol"}, a routine calculation, e.g. [@Biler1998] and references therein, shows that such a weak solution satisfies $\rho \in C\left([0, T] ; L^2(\Omega)\right)$, and the energy balance $$\frac{1}{2} \int_{\Omega} \rho^2(t, x) d x+\int_0^t \int_{\Omega}(\nabla \rho - \rho \nabla c) \cdot \nabla \rho=\frac{1}{2} \int_{\Omega} \rho_0^2(x) d x.$$ Moreover, by the representation $c = G * \rho$, we know that $c \in C([0,T]; H^1(\Omega))$. For notational convenience, we abuse the differential form throughout this paper: $$\frac{1}{2}\frac{d}{dt} \left\| \rho(t, \cdot) \right\|_{L^2(\Omega)}^2 + \left\| \nabla \rho(t, \cdot) \right\|_{L^2(\Omega)}^2 = \int_\Omega \rho(t, \cdot) \nabla c(t, \cdot) \cdot \nabla \rho(t, \cdot).$$ This is obtained through a formal process that involves computing the $L^2$ inner product of the equation with $\rho$, followed by an integration by parts.* We make several observations regarding the characteristics of blow-up of solutions which will play a crucial role in our a posteriori analysis in Section [3](#section:a_stability_framework){reference-type="ref" reference="section:a_stability_framework"}. **Remark 4**. *We remark that (ii) of Theorem [\[thm:weaksol\]](#thm:weaksol){reference-type="eqref" reference="thm:weaksol"} means that if a solution blows up at some time $T>0$ (i.e. $\lim_{t \rightarrow T} \left\| \rho(t, \cdot) \right\|_{L^\infty(\Omega)} = \infty$), then the norms $\left\| \rho(t, \cdot) \right\|_{L^p(\Omega)}$ blow up for all $p \in(d / 2, \infty]$ at the same moment [@Biler1994 p.333].* **Remark 5**. *Let $T_{max} \in (0, +\infty]$ be the maximal existence time of the solution, that is, the supremum of all $T>0$ such that the solution exists on $[0, T]$. If $T_{max}<\infty$, we have $$\label{eq:blow-up} \lim_{t \rightarrow T_{max}}\left\| \rho(t, \cdot) \right\|_{L^\infty(\Omega)} = \infty.$$* The existing literature, e.g. [@Suzuki2005 Theorem 3.2], provides a proof of the blow-up criterion for classical solutions to the Keller-Segel system. However, despite the possibility of establishing this property for weak solutions through the energy method we could not find a detailed proof for weak solutions in the literature. Since this property is crucial for the proof of Corollary [Corollary 9](#thm:guarantee_existence){reference-type="ref" reference="thm:guarantee_existence"}, we present a rigorous proof for the blow-up criterion with respect to weak solutions in this context. Assume that equation [\[eq:blow-up\]](#eq:blow-up){reference-type="eqref" reference="eq:blow-up"} does not hold. We can then extract a sequence $\left\{ t_k \right\}_{k=1}^{\infty}$, such that $\left\| \rho(t_k, \cdot) \right\|_{L^\infty(\Omega)} < C$ for all $k$, and where $t_k$ approaches $T_{max}$ from below as $k$ tends to infinity. Since $\Omega$ is bounded, we have $\left\| \rho(t_k, \cdot) \right\|_{L^2(\Omega)} \leq C$ for all $k$. Now if we use each $t_k$ as an initial time point, the local existence theorem enables us to define an interval $[t_k, t_k + \delta_k)$ with $\delta_k = \delta_k\left( \left\| \rho(t_k, \cdot) \right\|_{L^2(\Omega)} \right) > 0$ on which a solution exists.. We note that all initial data $\left\| \rho(t_k, \cdot) \right\|_{L^2(\Omega)}$ possess a uniform upper bound. As shown in [@Biler1992 the proof of Theorem 1], this means the sequence $\left\{ \delta_k \right\}$ does not shrink to zero, allowing us to define $\delta : = \inf_{k}\delta_k > 0$. Consequently, we can choose $k_0 \in \mathbb{N}$ such that $T_{max} - \frac{1}{2}\delta < t_{k_0}$. By employing the continuation argument, we find that the weak solution exists until $t_{k_0} + \delta > T_{max}$, a contradiction to the definition of $T_{max}$. Thus, [\[eq:blow-up\]](#eq:blow-up){reference-type="eqref" reference="eq:blow-up"} holds true. The stability framework that will be presented in the next section requires that $c$ can be controlled by sufficiently weak norms of $\rho$, i.e. elliptic regularity for [\[eq:KS2\]](#eq:KS2){reference-type="eqref" reference="eq:KS2"}--[\[eq:bc2\]](#eq:bc2){reference-type="eqref" reference="eq:bc2"}. Since [\[eq:KS2\]](#eq:KS2){reference-type="eqref" reference="eq:KS2"} has constant coefficients this depends on properties of $\Omega$. A sufficient condition is that $\Omega$ is convex. **Assumption 6**. *Throughout this paper, we assume $d \in \{2, 3\}$ and that $\Omega$ is such that [\[eq:KS2\]](#eq:KS2){reference-type="eqref" reference="eq:KS2"}, [\[eq:bc2\]](#eq:bc2){reference-type="eqref" reference="eq:bc2"} enjoys elliptic regularity, i.e. there exists a positive constant $C_{ell}>0$, depending only on $\Omega$, such that $\| c(t, \cdot)\|_{H^2} \leq C_{ell} \| \rho \left( t, \cdot \right)\|_{L^2}$ for all $t$.* # A stability framework {#section:a_stability_framework} Let $\left( \rho, c \right)$ be a weak solution to the problem [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"}-[\[eq:ic\]](#eq:ic){reference-type="eqref" reference="eq:ic"} and let $\left( \bar{\rho}, \bar{c} \right)$ with $\bar{\rho} \in C\left( [0,T]; H^1(\Omega) \right)$, $\partial_t \bar{\rho} \in C\left( (0,T]; H^{-1}(\Omega) \right)$, $\bar{c}\in C\left( [0,T]; H^2(\Omega) \right)$ be a *strong* solution to the following perturbed problem: $$\begin{aligned} \partial_t \bar \rho+ \nabla \cdot (\bar \rho\nabla \bar c- \nabla \bar \rho) &=R_\rho \quad \text{ in } (0,T) \times \Omega\label{eq:KS-p1}\\ \bar c- \Delta \bar c&= \bar \rho\quad \text{ in } (0,T) \times \Omega\label{eq:KS-p2}\\ \nabla \rho \cdot \mathrm{n} &=0 \quad \text{ on } (0,T) \times \partial \Omega\\ \nabla c \cdot \mathrm{n} &=0 \quad \text{ on } (0,T) \times \partial \Omega \label{eq:bc-p2}\end{aligned}$$ with some given function $R_\rho \in L^2(0,T; H^{-1}( \Omega))$. We are going to provide an estimate for the difference $(\rho - \bar \rho, c - \bar c)$ in terms of the (possible) difference of initial data and of $R_\rho$. The situation we have in mind is that $\bar \rho, \bar c$ is a reconstruction of a numerical solution as in Section [5](#sec:reconstruction){reference-type="ref" reference="sec:reconstruction"}. However, it is important to note that our stability framework does not depend on how $\bar \rho, \bar c$ was obtained. Taking into account Remark [Remark 3](#rmk:energy_eq){reference-type="ref" reference="rmk:energy_eq"}, we will subsequently provide a formal argument for the derivation of the stability estimate, but it can be made rigorous by interpreting it in the context of appropriate integral formulations. For the sake of brevity, we choose not to explicitly include $\Omega$ in the norm symbols. Furthermore, the time dependency of functions will also not be explicitly denoted. Subtracting [\[eq:KS-p1\]](#eq:KS-p1){reference-type="eqref" reference="eq:KS-p1"} from [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"} and testing with $\rho - \bar{\rho}$ we obtain $$\begin{gathered} \frac{d}{dt} \left[ \int_{\Omega} \frac{1}{2} (\rho - \bar \rho)^2 dx \right] +\int_{\Omega} |\nabla (\rho - \bar \rho)|^2 dx\\ = \int_\Omega \nabla (\rho - \bar \rho) \bar \rho\nabla (c - \bar c) + \nabla (\rho - \bar \rho) (\rho - \bar \rho) \nabla \bar c- R_\rho(\rho - \bar \rho) + \nabla (\rho - \bar \rho) (\rho -\bar \rho) \nabla (c - \bar c) dx, \end{gathered}$$ where we used the homogeneous boundary conditions. Using Cauchy-Schwarz's inequality we obtain $$\begin{gathered} \frac{d}{dt} \left[\frac 12 \| \rho - \bar \rho\|_{L^2}^2 \right] + |\rho - \bar \rho|_{H^1}^2 \\ \leq |\rho - \bar \rho|_{H^1} \|\bar \rho\|_{L^3} |c - \bar c|_{W^{1,6}} + |\rho - \bar \rho|_{H^1} \|\rho - \bar \rho\|_{L^2} \|\nabla \bar c\|_{L^\infty}\\ \quad + \|R_\rho\|_{H^{-1}} \|\rho - \bar \rho\|_{H^1} + |\rho - \bar \rho|_{H^1} \|\rho -\bar \rho\|_{L^3} |c - \bar c|_{W^{1,6}}.\end{gathered}$$ Since the number of space dimensions satisfies $d \leq 3$, we have $\left| c - \bar{c} \right|_{W^{1,6}} \leq C_S \left\| \nabla c - \nabla \bar{c} \right\|_{H^1}$, and $\left\| \rho - \bar{\rho} \right\|_{L^3} \leq C_S' \left\| \rho - \bar{\rho} \right\|_{H^1}$ where $C_S, C_S'$ are the constants of the embedding $H^1 \rightarrow L^6$ and $H^1 \rightarrow L^3$, respectively. Moreover, by elliptic regularity, we have $\left\| c - \bar{c} \right\|_{H^2} \leq C_{ell} \left\| \rho - \bar{\rho} \right\|_{L^2}$. This implies $$\begin{gathered} \frac{d}{dt} \left[\frac 12 \| \rho - \bar \rho\|_{L^2}^2 \right] + |\rho - \bar \rho|_{H^1}^2 \nonumber \\ \leq C_SC_{ell} |\rho - \bar \rho|_{H^1} \|\bar \rho\|_{L^3} \left\| \rho - \bar \rho\right\|_{L^2} + |\rho - \bar \rho|_{H^1} \|\rho - \bar \rho\|_{L^2} \|\nabla \bar c\|_{L^\infty}\nonumber\\ + \|R_\rho\|_{H^{-1}} \|\rho - \bar \rho\|_{H^1} +C_S' C_S C_{ell} |\rho - \bar \rho|_{H^1} \|\rho -\bar \rho\|_{H^1} \left\| \rho - \bar{\rho} \right\|_{L^2}. \end{gathered}$$ Using Young's inequality and gathering terms on the right hand side, we obtain $$\begin{gathered} \label{3} \frac{d}{dt} \left[ \left\| \rho - \bar{\rho} \right\|_{L^2}^2 \right] + \left| \rho - \bar{\rho} \right|_{H^1}^2 \leq \left( 3 C_S^2 C_{ell}^2 \left\| \bar{\rho} \right\|_{L^3}^2 + 3 \left\| \nabla \bar{c} \right\|_{L^\infty}^2 + \frac{1}{3} \right) \left\| \rho - \bar{\rho} \right\|_{L^2}^2 \\ + 3 \left\| R_\rho \right\|_{H^{-1}}^2 + 2C_S'C_S C_{ell} |\rho - \bar \rho|_{H^1} \| \rho - \bar \rho\|_{H^1} \|\rho -\bar \rho\|_{L^2}. \end{gathered}$$ Let us set $$\label{eq:y123a} \begin{split} y_1 (t) &:= \| \rho (t,\cdot) - \bar \rho(t,\cdot)\|_{L^2}^2, \\ y_2(t) &:= | \rho (t,\cdot) - \bar \rho(t,\cdot)|_{H^1}^2, \\ y_3 (t)&:= 2C_S'C_SC_{ell} |\rho(t,\cdot) - \bar \rho(t,\cdot)|_{H^1} \| \rho(t,\cdot) - \bar \rho(t,\cdot)\|_{H^1} \|\rho(t,\cdot) -\bar \rho(t,\cdot)\|_{L^2},\\ a(t) &:= 3 C_S^2C_{ell}^2 \|\bar \rho(t,\cdot)\|_{L^3}^2 + 3 \|\nabla \bar c(t,\cdot)\|_{L^\infty}^2 + \frac{1}{3}. \end{split}$$ Then, we can integrate [\[3\]](#3){reference-type="eqref" reference="3"} in time from $0$ to $T'$ to obtain $$\label{near:gron} y_1(T') + \int_0^{T'} y_2(t) dt \leq y_1(0) + \int_0^{T'}\| R_\rho\|_{H^{-1}}^2 dt + \int_0^{T'} a(t) y_1(t) dt + \int_0^{T'} y_3(t) dt$$ Since $\left\| \rho - \bar{\rho} \right\|_{H^1}^2 = \left\| \rho - \bar{\rho} \right\|_{L^2}^2 + \left| \rho - \bar{\rho} \right|_{H^1}^2$, we have $$\begin{gathered} y_3(t) \leq 2C_S'C_SC_{ell} %\left( |\rho - \brho|_{H^1} \|\rho -\brho\|_{L^2}^2 + \|\rho - \bar \rho\|_{H^1}^2 \|\rho -\bar \rho\|_{L^2} %\right) \\ \leq 2 C_S'C_SC_{ell} \sqrt{y_1(t)} (y_1(t) + y_2(t) ).\end{gathered}$$ In the final inequality, we have used Young's inequality. This leads us to: $$\label{est:y3} \int_0^{T'} y_3(t) dt \leq 2 C_S'C_SC_{ell} \sup_t \sqrt{y_1(t)} \int_0^{T'} y_1(t) + y_2(t) dt .$$ Equations [\[near:gron\]](#near:gron){reference-type="eqref" reference="near:gron"} and [\[est:y3\]](#est:y3){reference-type="eqref" reference="est:y3"} show that our analysis fits into the framework of Proposition [\[prp:GeneralizedGronwall\]](#prp:GeneralizedGronwall){reference-type="ref" reference="prp:GeneralizedGronwall"} with $y_1,y_2,y_3,$ and $a$ as above; $B:= 2 C_S'C_SC_{ell}$, $\beta = \frac 12$, $$\label{eq:AE} A:= y_1(0) + \int_0^{T} \| R_\rho\|_{H^{-1}}^2 dt, \quad \text{and} \quad E := \exp\left( \int_0^{T} a(t) dt\right).$$ **Proposition 7**. *[@Bartels Prop 6.2, Generalized Gronwall lemma] [\[prp:GeneralizedGronwall\]]{#prp:GeneralizedGronwall label="prp:GeneralizedGronwall"} Suppose that nonnegative functions $y_1 \in C([0, T]), y_2, y_3 \in L^1([0, T]), a \in L^{\infty}([0, T])$, and a real number $A \geq 0$ satisfy $$y_1\left(T^{\prime}\right)+\int_0^{T^{\prime}} y_2(t) \mathrm{d} t \leq A+\int_0^{T^{\prime}} a(t) y_1(t) \mathrm{d} t+\int_0^{T^{\prime}} y_3(t) \mathrm{d} t$$ for all $T^{\prime} \in[0, T]$ and that, in addition, for $B \geq 0, \beta>0$, and every $T^{\prime} \in[0, T]$ $$\int_0^{T^{\prime}} y_3(t) \mathrm{d} t \leq B\left(\sup _{t \in\left[0, T^{\prime}\right]} y_1^\beta(t)\right) \int_0^{T^{\prime}}\left(y_1(t)+y_2(t)\right) \mathrm{d} t .$$ Let $E:=\exp \left(\int_0^T a(t) \mathrm{d} t\right)$. Then, provided $8 A E \leq(8 B(1+T) E)^{-1 / \beta}$ is satisfied the following inequality holds: $$\sup _{t \in[0, T]} y_1(t)+\int_0^T y_2(t) \mathrm{d} t \leq 8 A \exp \left(\int_0^T a(s) \mathrm{d} s\right) .$$* Thus, we have the following conditional stability result: **Theorem 8**. *For fixed $T > 0$, let $\left( \rho, c \right)$ be a weak solution to [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"}-[\[eq:ic\]](#eq:ic){reference-type="eqref" reference="eq:ic"} and $\left( \bar{\rho}, \bar{c} \right)$ a strong solution to [\[eq:KS-p1\]](#eq:KS-p1){reference-type="eqref" reference="eq:KS-p1"}--[\[eq:bc-p2\]](#eq:bc-p2){reference-type="eqref" reference="eq:bc-p2"}. Let $y_1,y_2,y_3,$ and $a$ be defined as in [\[eq:y123a\]](#eq:y123a){reference-type="eqref" reference="eq:y123a"} and $B= 2 C_S'C_SC_{ell}$, $\beta = \frac 12$, and $A, \, E$ as in [\[eq:AE\]](#eq:AE){reference-type="eqref" reference="eq:AE"}. Then, provided $$\label{eq:estcondition} 8 A E ( 8B (1 +T) E)^{2} \leq 1,$$ is satisfied, the difference $\rho - \bar \rho$ is controlled as follows: $$\begin{gathered} \label{mainestimate} \sup_{t \in [0,T]} \| \rho (t,\cdot) - \bar \rho(t,\cdot)\|_{L^2(\Omega)}^2 + \int_0^T | \rho (t,\cdot) - \bar \rho(t,\cdot)|_{H^1(\Omega)}^2 dt \\ \leq 8 \left( \| \rho (0,\cdot) - \bar \rho(0,\cdot)\|_{L^2(\Omega)}^2 + \int_0^{T} \| R_\rho\|_{H^{-1}(\Omega)}^2 dt \right) \\ \times \exp \left( \int_0^T 3 C_S^2C_{ell}^2 \|\bar \rho(t,\cdot)\|_{L^3(\Omega)}^2 + 3 \|\nabla \bar c(t,\cdot)\|_{L^\infty(\Omega)}^2 + \frac 1 3 dt\right). \end{gathered}$$* Note that the stability framework presented here is independent of the origin of the approximate solution $(\bar \rho, \bar c)$. Moreover, the assumption that a weak solution $(\rho, c)$ exists until time $T$ is a posteriori verifiable. Indeed, if [\[eq:estcondition\]](#eq:estcondition){reference-type="eqref" reference="eq:estcondition"} holds, the error estimate itself rules out blow-up before time $T$ and thus proves *a posteriori* that a sufficiently regular weak solution exists. More precisely: **Corollary 9**. *Given any initial data $\rho_0 \in L^2(\Omega)$, let $T_{max} > 0$ be the maximal existence time of the weak solution $\rho$ to [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"}-[\[eq:ic\]](#eq:ic){reference-type="eqref" reference="eq:ic"}. Let there exist an approximate strong solution $\bar \rho$, i.e. a solution of [\[eq:KS-p1\]](#eq:KS-p1){reference-type="eqref" reference="eq:KS-p1"}--[\[eq:bc-p2\]](#eq:bc-p2){reference-type="eqref" reference="eq:bc-p2"} such that at some time $T>0$ the condition [\[eq:estcondition\]](#eq:estcondition){reference-type="eqref" reference="eq:estcondition"} is satisfied, $\|\bar \rho\|_{L^\infty(0,T; L^2(\Omega))}$ is finite and the right hand side of [\[mainestimate\]](#mainestimate){reference-type="eqref" reference="mainestimate"} is finite, then $T_{max} > T$, i.e., the weak solution exists at least until time $T$.* *Proof.* If [\[eq:estcondition\]](#eq:estcondition){reference-type="eqref" reference="eq:estcondition"} holds for some time $T>0$ then it also holds for all earlier times, since the left hand side of [\[eq:estcondition\]](#eq:estcondition){reference-type="eqref" reference="eq:estcondition"} is increasing in time. Suppose $T_{max} < T$. Then, [\[mainestimate\]](#mainestimate){reference-type="eqref" reference="mainestimate"} evaluated at time $T$ provides a uniform upper bound for $\|\rho(t, \cdot)\|_{L^2(\Omega)}$ for almost all $t \in [0,T_{max}]$, by bounding its distance from $\bar \rho(t,\cdot)$. In contrast, the blow-up property, explained in Remark [Remark 4](#rmk:uniformbound){reference-type="ref" reference="rmk:uniformbound"} and Remark [Remark 5](#rmk:blow-up){reference-type="ref" reference="rmk:blow-up"}, implies that the $L^2$-norm of $\rho$ must blow up for $t \rightarrow T_{max}$, since $d \leq 3$. This is a contradiction. ◻ # A semi-discrete Discontinuous Galerkin Scheme {#sec:dgscheme} In order to keep our paper self-contained, we begin this section by briefly recalling some basic notations. Details can be found in, e.g. [@Di_Pietro_2012]. We employ a discontinuous Galerkin (dG) finite element method for spatial discretisation. **Definition 10** (Finite element space). *Let $\Omega$ be decomposed into a set $\mathcal{T}$ of disjoint polyhedra $\{T \}$. Let $h_T$ denote the *diameter* of $T$. We use the notation $\mathcal{T}_h$ for a *mesh* $\mathcal{T}$ with mesh size $h$. We define $\mathcal{F}_h^i$ as the set of common *interfaces* of cells in the mesh $\mathcal{T}_h$, and $\mathcal{F}_h^b$ as the set of *boundary faces* on the boundary $\partial \Omega$. We set $$\mathcal{F}_h := \mathcal{F}_h^i \cup \mathcal{F}_h^b,$$ as the set of all faces in the mesh. For all $F \in \mathcal{F}_h$, in dimension $d \geq 2$, we set $h_F$ to be equal to the diameter of the face $F$, while, in dimension 1 , we set $h_F:=\min \left\{ h_{T_1}, h_{T_2} \right\}$ if $F \in \mathcal{F}_h^i$ with $F=\partial T_1 \cap \partial T_2$ and $h_F:=h_T$ if $F \in \mathcal{F}_h^b$ with $F=\partial T \cap \partial \Omega$. Furthermore, for any $T \in \mathcal{T}_h$, we set $$\mathcal{F}_T := \left\{F \in \mathcal{F}_h \mid F \subset \partial T\right\},$$ which collects the mesh faces comprising the boundary of $T$.* *We say that a mesh sequence $\mathcal{T}_h$ is *admissible* if it is shape- and contact-regular and if it has optimal polynomial approximation properties. Definitions of these notions can be found in [@Di_Pietro_2012 Section 1.4]. Throughout this paper, we always assume that the mesh sequence $\mathcal{T}_h$ is *admissible*.* *Let $\mathbb{P}_d^k$ denote the space of polynomials of degree less than or equal to $k$. We define the *discontinuous finite element space* as $$V_h:= \mathbb{P}_d^k (\mathcal{T}_h) := \{v : \Omega \to \mathbb{R} \, : \, v | _{T} \in \mathbb{P}_d^k{(T)} \ \forall\, T \in \mathcal{T}_h\}$$* with polynomial degree $k \geq 1$. **Definition 11** (Broken Sobolev spaces and broken gradients). *We introduce the *broken Sobolev space* $$\begin{aligned} W^{m, p}\left(\mathcal{T}_h\right)&:=\left\{v \in L^p(\Omega) \, : \, v|_T \in W^{m, p}(T), \text{ for each } T \in \mathcal{T}_h \right\}. \end{aligned}$$ For $p=2$, we set $H^m(\mathcal{T}_h): = W^{m, 2}\left(\mathcal{T}_h\right)$. We also make use of functions defined in these broken spaces restricted to the skeleton of the mesh.\ The *broken gradient* $\nabla_h: W^{1, p}\left(\mathcal{T}_h\right) \rightarrow$ $\left[L^p(\Omega)\right]^d$ is defined, by $$\forall v \in W^{1, p}\left(\mathcal{T}_h\right)\quad \forall T \in \mathcal{T}_h,\left.\quad\left(\nabla_h v\right)\right|_T:=\nabla\left(\left.v\right|_T\right).$$* **Definition 12** (Weighted averages and jumps). *For any interface $F \in \mathcal{F}_h^i$ let $\omega^+, \omega^- \in W^{1,\infty}(F)$ be weight functions that are uniformly bounded from below by a positive real number. Let $F = \partial T_1 \cap \partial T_2$ then we define, for a.e. $x \in F$, the weighted average and jump: $$% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}v\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega, F}(x):=\left.\omega^+_{F} v\right|_{T_1}(x)+\left.\omega^-_{F} v\right|_{T_2}(x) ,$$ and $$\llbracket v \rrbracket_F(x):=\left.v\right|_{T_1}(x)-\left.v\right|_{T_2}(x).$$ The weighted averages and jumps are defined for boundary faces by setting, for a. e. $x \in F$ with $F \in \mathcal{F}_h^b$ and $F=\partial T \cap \partial \Omega, % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}v\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega,F}(x)=\llbracket v \rrbracket_F(x):=\left.v\right|_T(x)$. When $v$ is vector-valued, these operators act componentwise. In cases where no confusion arises, the subscript $F$ and the variable $x$ are omitted, and we write $% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}v\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _\omega$ and $\llbracket v \rrbracket$. Moreover, if $\omega^+ = \omega^- \equiv 1/2$, we denote $% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}v\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega}$ as $% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}v\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}%$ and call it the average.* *For each interface we choose $n_F$ as the normal vector pointing from $T_1$ to $T_2$ such that the sign of $\llbracket v \rrbracket_F n_F$ is independent of the labeling of $T_1, T_2$.* Using this notation, we define the symmetric interior penalty bilinear form: $$\label{eq:sip} \begin{aligned} & a^{sip}_h: V_h \times V_h \rightarrow \mathbb{R}\\ & a^{sip}_h(u_h,\phi_h) : = \sum_{T \in \mathcal{T}_h}\int_{T} \nabla u_h \nabla \phi_h\\ & - \sum_{F \in \mathcal{F}_h^i}\int_{F} \Big( \ensuremath{\left[\![ u_h\right]\!]}% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}\nabla \phi_h\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% \cdot \mathrm{n}_F + \ensuremath{\left[\![ \phi_h\right]\!]}% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}\nabla u_h\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% \cdot \mathrm{n}_F - \frac{\eta}{h_F} \ensuremath{\left[\![ u_h\right]\!]}\ensuremath{\left[\![ \phi_h\right]\!]}\Big), \end{aligned}$$ where $\eta>0$ is a parameter which is chosen so large that $a^{sip}_h$ is semi-positive definite. We discretise the chemotaxis term using the weighted symmetric interior penalty bilinear form: $$\label{eq:wsip} \begin{aligned} & a^{wsip}_h: V_h \times V_h \rightarrow \mathbb{R} \\ & a^{wsip}_h(v_h;u_h, \psi_h) = \sum_{T \in \mathcal{T}_h}\int_T \left( v_h \nabla_h u_h \right)\cdot\nabla\psi_h \\ & - \sum_{F \in \mathcal{F}_h^i} \int_F \left( \llbracket u_h \rrbracket % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}v_h \nabla_h \psi_h\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega} \cdot \mathrm{n}_F + \llbracket \psi_h \rrbracket% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}v_h \nabla_h u_h\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega}\cdot \mathrm{n}_F - \sigma \frac{\gamma_{v_h,F}}{h_F} \llbracket u_h \rrbracket \llbracket \psi_h \rrbracket \right), \end{aligned}$$ where $% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}\psi_h\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega}: = \omega^+\psi_h^+ + \omega^-\psi_h^-$ with the weights selected as $$\omega^+ = \frac{v^-_h}{v^+_h + v^-_h}, \ \omega^- = \frac{v^+_h}{v^+_h + v^-_h},$$ $\sigma>0$ is a constant sufficiently large so that $a_h^{wsip}$ is semi-positive definite, and the diffusion-dependent penalty parameter $\gamma_{v_h}$ is set to $$\gamma_{v_h} := \frac{2v_h^+v_h^-}{v_h^+ + v_h^-}.$$ We consider the following DG semi-discretisation: **Definition 13**. *Given $\rho_h^0 \in V_h$ we define $\left( \rho_h, c_h \right) \in C^1(0,T;V_h)\times C(0,T;V_h)$ as the solution of $$\label{def:dg} \begin{aligned} \int_\Omega \left( \partial_t \rho_h \right)\phi_h &= -a^{sip}_h\left( \rho_h, \phi_h \right) + a^{wsip}_h\left( \rho_h; c_h, \phi_h \right) \quad \text{for all} \quad \phi_h \in V_h, \\ a^{sip}_h\left( c_h, \psi_h \right) + \int_\Omega c_h \psi_h &= \int_\Omega \rho_h \psi_h \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \ \text{for all} \quad \psi_h \in V_h, \end{aligned}$$ with $\rho_h(0)=\rho_h^0$.* **Remark 14**. *Neither the symmetry nor the penalty terms of equations [\[eq:sip\]](#eq:sip){reference-type="eqref" reference="eq:sip"} and [\[eq:wsip\]](#eq:wsip){reference-type="eqref" reference="eq:wsip"} contain contributions from boundary faces, since we use homogeneous Neumann boundary data. Additional details can be found in [@Di_Pietro_2012 Section 4.2.2].* # Reconstruction {#sec:reconstruction} In this section, we define reconstructions of the numerical solutions by employing so-called *elliptic reconstruction* [@Makridakis2003; @Makridakis2007]. The reconstruction is needed since inserting the dG solution $(\rho_{h}, c_{h})$ into [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"} and [\[eq:KS2\]](#eq:KS2){reference-type="eqref" reference="eq:KS2"} would result in singular residuals. *Elliptic reconstruction* is a well-known method that leads to residuals that have optimal order in several other problems. The residual serves as a perturbation of the PDE with optimal order. We will make use of this fact to argue that our estimator is of optimal order. For notational convenience, we use the abbreviations $$\left( \cdot, \cdot \right) := \left( \cdot, \cdot \right)_{L^2(\Omega)} \quad \text{and} \quad \left\langle \cdot, \cdot \right\rangle := \left\langle \cdot, \cdot \right\rangle_{H^{-1}(\Omega), H^1(\Omega)}.$$ For the sake of brevity, we will also omit the time dependency of functions when the context makes it clear. Let $A_h:V_h \longrightarrow V_h$ be the following discrete version of the operator $-\Delta$: $$\label{def:linearform} \left( A_hv_h, w_h \right) := a^{sip}_h\left( v_h, w_h \right) \quad \quad \text{for all } v_h, w_h \in V_h.$$ For any fixed time $t \in [0,T]$, we define the elliptic reconstruction $\hat{\rho}(t, \cdot) \in H^1(\Omega)$ of $\rho_h(t, \cdot)$ as the solution of the elliptic problem $$\label{eq:reconstruct} \begin{aligned} -\Delta\hat{\rho} &= A_h\rho_h \ \quad \text{in } \Omega \\ \nabla \hat{\rho}\cdot \mathrm{n} &= 0 \quad \quad \quad \text{on } \partial \Omega \end{aligned}$$ such that $\hat{\rho}\left( t, \cdot \right)$ has the same mean value as the discrete solution, i.e., $$\label{eq:reconstruct_mean} \int_\Omega \hat{\rho}\left( t, \cdot \right) = \int_\Omega \rho_h(t, \cdot).$$ In other words, $\hat{\rho}(t, \cdot)$ satisfies the following: $$a(\hat{\rho}(t), \phi):=(\nabla\hat{\rho}(t), \nabla \phi)= \langle A_h\rho_h(t), \phi \rangle \quad \text{for all} \ \phi \in H^1(\Omega).$$ It is worth noting that $\rho_h(t, \cdot)$ is the SIP-dG solution to [\[eq:reconstruct\]](#eq:reconstruct){reference-type="eqref" reference="eq:reconstruct"}-[\[eq:reconstruct_mean\]](#eq:reconstruct_mean){reference-type="eqref" reference="eq:reconstruct_mean"} due to the definition [\[def:linearform\]](#def:linearform){reference-type="eqref" reference="def:linearform"}. Similarly, for a fixed time $t \in [0,T]$, let us define $\tilde{c}(t, \cdot) \in H^2(\Omega)$ as the elliptic reconstruction of $c_h(t, \cdot)$, i.e., it satisfies $$\label{eq:reconstruct2} a( \tilde{c}(t), \psi ) + \left( \tilde{c}(t), \psi \right) = ( \rho_h(t), \psi ) \quad \text{for all }\psi \in H^1(\Omega).$$ Finally, we define $\hat{c}(t) \in H^1(\Omega)$ as the solution to the elliptic problem: $\hat{c} - \Delta \hat{c} = \hat{\rho} \ \text{ in } \Omega$ with homogeneous Neumann boundary data, i.e. $$a( \hat{c}(t), \chi ) + \left( \hat{c}(t), \chi \right) = ( \hat{\rho}(t), \chi ) \quad \text{for all }\chi \in H^1(\Omega).$$ **Remark 15** (regularity and continuity). *For any fixed time $t \in [0, T]$, we have $$\hat{\rho}(t, \cdot) \in H^2(\Omega), \quad \hat{c}(t, \cdot) \in H^3(\Omega), \quad \tilde{c}(t, \cdot) \in H^2(\Omega),$$ owing to elliptic regularity. Furthermore, continuity (in time) of $\rho_h$ implies that the reconstructions $\hat{\rho}, \hat{c}$, and $\tilde{c}$ are also continuous in time. Moreover, as the bilinear form $a(\cdot, \cdot)$ is time-independent, it follows that $$a(\partial_t\hat{\rho}, \phi) = \langle \partial_t A_h\rho_h(t), \phi \rangle$$ for all $\phi \in H^1(\Omega)$. Thus we have $$\hat{\rho} \in C^1( [0,T]; H^2(\Omega)), \quad \hat{c} \in C ( [0,T]; H^3(\Omega) ), \quad \tilde{c} \in C ( [0,T]; H^2(\Omega) ).$$* **Remark 16** (a posteriori control for elliptic problems). *Reliable and efficient a posteriori error estimators for the $L^2$-norm error associated with continuous finite element discretisations of Poisson's equation are detailed in [@Makridakis2003 (4.4), (4.5)]. These estimators can be adapted to SIP-dG schemes in a straightforward manner. Moreover, [@Karakashian_2003 Theorem 3.1] provides reliable and efficient a posteriori error estimators for the dG-norm error arising from interior penalty dG discretisations of Poisson's equation. We also introduce an a posteriori error estimator in terms of the $H^{-1}$-norm. To state all these estimators concisely, we define $$R_T [\rho_h, f]:= \left\| \Delta \rho_h + f \right\|_{L^2(T)}^2, \ R_F^1 [\rho_h]:= \left\| \llbracket \nabla_h \rho_h \rrbracket \cdot \mathrm{n}_F \right\|_{L^2(F)}^2, \ R_F^0 [\rho_h]:= \left\| \llbracket \rho_h \rrbracket \right\|_{L^2(F)}^2.$$ Then we have the following error estimates: $$\begin{aligned} \left\| \rho_h - \hat\rho \right\|_{L^2(\Omega)} &\leq& C_0 \operatorname{E}_0\left[ \rho_h, A_h\rho_h \right]\label{rhoest1}\\ \| \rho_h - \hat \rho\|_{dG} &\leq& C_1 \operatorname{E}_1[\rho_h, A_h\rho_h]\label{rhoxest1}\\ \left\| \partial_t \rho_h - \partial_t \hat\rho \right\|_{H^{-1}(\Omega)} &\leq& C_{-1} \operatorname{E}_{-1}\left[ \partial_t \rho_h, \partial_t A_h\rho_h \right]\label{rhotest1} \end{aligned}$$ where the dG-norm is defined as $$\|v\|_{\mathrm{dG}}^2 :=\left\|\nabla_h v\right\|_{\left[L^2(\Omega)\right]^d}^2+\sum_{F \in \mathcal{F}_h} \frac{1}{h_F}\|\llbracket v \rrbracket \|_{L^2(F)}^2,$$ and the elliptic error estimators are defined as, for $k\geq1$, $$\begin{aligned} \label{rhoest2} \operatorname{E}_0\left[ \rho_h, f_h \right]^2 := \sum_{T \in \mathcal{T}_h} h^4_T R_T [\rho_h, f_h] + \sum_{F \in \mathcal{F}_h^i} h_F^3 R_F^1 [\rho_h] + \eta^2 \sum_{F \in \mathcal{F}_h^i} h_F R_F^0 [\rho_h], \\ \operatorname{E}_1\left[ \rho_h, f_h \right]^2 :=\sum_{T \in \mathcal{T}_h} h^2_T R_T [\rho_h, f_h] + \sum_{F \in \mathcal{F}_h^i} h_F R_F^1 [\rho_h] + \eta^2 \sum_{F \in \mathcal{F}_h^i} h_F^{-1} R_F^0 [\rho_h], \end{aligned}$$ and for $k \geq 2$, $$\label{rhotest2} \operatorname{E}_{-1}\left[ \rho_h, f_h \right]^2 := \sum_{T \in \mathcal{T}_h} h^6_T R_T [\rho_h, f_h] + \sum_{F \in \mathcal{F}_h^i} h_F^5 R_F^1 [\rho_h] + \eta^2 \sum_{F \in \mathcal{F}_h^i} h_F^{3} R_F^0 [\rho_h].$$ The real numbers $C_0$, $C_1$, and $C_{-1}$ are positive constants dependent only on the regularity of $\Omega$. These can be evaluated on a convex polygonal domain, see [@Di_Pietro_2012 Theorem 5.45]* *For all elliptic error estimators and generic functions $u_h \in V_h$ we subsequently write $E_i[u_h]:= E_i[u_h, A_h u_h]$ for brevity.* *Analogously, we have $$\| c_h - \tilde c\|_{\operatorname{dG}} \leq \tilde{C}_1 \operatorname{\tilde{E}}_1[c_h, \rho_h]$$ with $$\begin{gathered} \label{cest} \operatorname{\tilde{E}}_1[c_h, f_h]^2 := \sum_{T\in\mathcal{T}_h} h_T^2 \|f_h - c_h + \Delta c_h\|_{L^2(T)}^2\\ + \sum_{F\in \mathcal{F}_h} h_F \| \ensuremath{\left[\![ \nabla c_h\right]\!]}\cdot \mathrm{n}_F \|_{L^2(F)}^2 + \sigma^2 \sum_{F\in \mathcal{F}_h} h_F^{-1} \|\ensuremath{\left[\![ c_h\right]\!]}\|_{L^2(F)}^2. \end{gathered}$$ The positive real number $\tilde{C}_1$ also can be evaluated as previously mentioned.* *Finally, $\| \tilde c - \hat c\|_{H^1(\Omega)}$ is controlled by $\| \rho_h - \hat \rho\|_{L_2(\Omega)},$ which is controlled by $\operatorname{E}_0[\rho_h, A_h\rho_h]$. Thus we have $$\label{eq:cest2} \| c_h - \hat c\|_{\operatorname{dG}} \leq C_{ell}C_0\operatorname{E}_0[\rho_h, A_h\rho_h] + \tilde{C}_1\operatorname{\tilde{E}}_1[c_h, \rho_h].$$* For any fixed $t \in [0, T]$, we define the residual $R_\rho(t, \cdot) \in H^{-1}(\Omega)$ as $$\label{eq:residualpair} \left\langle R_\rho, \phi \right\rangle %_{H^{-1}, H^1} := \left\langle \partial_t\hat{\rho}, \phi \right\rangle%_{H^{-1}, H^1} - \left( \hat{\rho}\nabla \hat{c} - \nabla\hat{\rho}, \nabla \phi \right)$$ for all $\phi \in H^1(\Omega)$. The regularities mentioned in Remark [Remark 15](#remark:regularity){reference-type="ref" reference="remark:regularity"} ensure that the above definition is well-defined, as long as $d \leq 3$, and lead to $R_\rho \in L^2(0,T; H^{-1}(\Omega))$. This allows us to interpret $\hat \rho, \hat c$ as a strong solution to the problem outlined in Section [3](#section:a_stability_framework){reference-type="ref" reference="section:a_stability_framework"}: $$\begin{matrix} \begin{aligned} \partial_t\hat{\rho} +\nabla\cdot(\hat{\rho}\nabla \hat{c} - \nabla \hat{\rho}) &=: R_\rho \\ \hat{c} - \Delta \hat{c} &= \hat{\rho} \end{aligned} & \text{in} \ (0,T)\times\Omega. \end{matrix}$$ We remark that the estimates for $\left( \bar{\rho}, \bar{c} \right)$ obtained in Section [3](#section:a_stability_framework){reference-type="ref" reference="section:a_stability_framework"} do not depend on a particular numerical scheme. Thus we will apply them to $\left( \hat{\rho}, \hat{c} \right)$. Since the reconstructions are given as exact solutions to elliptic problems, computing norms of the residual requires some work. # Controlling the $H^{-1}$-norm of the residual {#section:aposteriori} The aim of this section is to establish a computable upper bound for the $H^{-1}$-norm of the residual $R_\rho$ defined in the previous section. Let $\mathcal{T}_{\mathcal{H}}$ be an admissible mesh sequence. We begin by revisiting the $L^2$-orthogonal projection and its approximation properties: **Definition 17** ($L^2$-orthogonal projection). *Let $\pi_h$ denote the $L^2$-orthogonal projection onto $V_h$, that is, $\pi_h: L^2(\Omega) \rightarrow$ $V_h$ is defined such that, for all $v \in L^2(\Omega)$, $\pi_h v$ belongs to $V_h$ and satisfies $$\left(\pi_h v, y_h\right)%_{L^2(\Omega)} =\left(v, y_h\right)%_{L^2(\Omega)} \quad \forall y_h \in V_h .$$* **Lemma 18**. *[@Di_Pietro_2012 Lemma 1.58, Approximation properties] Let $\pi_h$ be the $L^2$-orthogonal projection onto $V_h$, and let $s \in\{0, \ldots, k+1\}$. Then, for all $h \in \mathcal{H}$, all $T \in \mathcal{T}_h$, and all $v \in H^s(T)$, there holds $$\left|v-\pi_h v\right|_{H^m(T)} \leq C_{\text {app }}^{\prime} h_T^{s-m}|v|_{H^s(T)} \quad \forall m \in\{0, \ldots, s\}$$ where the positive real number $C_{\text {app }}^{\prime}$ is independent of both $T$ and $h$.* **Lemma 19**. *[@Di_Pietro_2012 Lemma 1.59, Polynomial approximation on mesh faces] Under the hypotheses of Lemma [Lemma 18](#thm:opL2proj){reference-type="ref" reference="thm:opL2proj"}, assume additionally that $s \geq 1$. Then, for all $h \in \mathcal{H}$, all $T \in \mathcal{T}_h$, and all $F \in \mathcal{F}_T$, there holds $$\left\|v-\pi_h v\right\|_{L^2(F)} \leq C_{\mathrm{app}}^{\prime \prime} h_T^{s-1 / 2}|v|_{H^s(T)},$$ and if $s \geq 2$, $$\left\|\left.\nabla\left(v-\pi_h v\right)\right|_T \cdot \mathrm{n}_T\right\|_{L^2(F)} \leq C_{\mathrm{app}}^{\prime \prime \prime} h_T^{s-3 / 2}|v|_{H^s(T)},$$ where the positive numbers $C_{\mathrm{app}}^{\prime \prime}$ and $C_{\mathrm{app}}^{\prime \prime \prime}$ are independent of both $T$ and $h$.* We now provide an estimate for the $H^{-1}$-norm of $R_\rho$ as follows: **Lemma 20** (a posteriori control on $R_\rho$). *Let $\left( \rho_h, c_h \right) \in C^1\left(0, T ; V_h\right) \times C\left(0, T ; V_h\right)$ solve the problem [\[def:dg\]](#def:dg){reference-type="eqref" reference="def:dg"} and $R_\rho$ be the residual as in [\[eq:residualpair\]](#eq:residualpair){reference-type="eqref" reference="eq:residualpair"}. Then we have the following estimate: $$\label{eq:R_rho} \begin{aligned} &\left\|R_\rho\right\|_{H^{-1}(\Omega)} \\ &\leq C_{-1}\operatorname{E}_{-1}\left[\partial_t \rho_h%, \partial_t\left(A_h \rho_h\right) \right] \\ &\quad + 2C_S'' C_{ell} C_0 \operatorname{E}_0\left[\rho_h%, A_h \rho_h \right] \bigg( C_1^2 \operatorname{E}_1\left[\rho_h%, A_h \rho_h \right]^2 + \sum_{T \in \mathcal{T}_h} \left| \rho_h \right|_{H^1(T)}^2 \bigg)^{1/2} \\ &\quad + \left( \sum_{T \in \mathcal{T}_h} C_{app}'^2 h_T^2 \left\| \nabla_h \cdot \left( \rho_h\nabla_h c_h \right) - \pi_h\bigg( \nabla_h \cdot\left(\rho_h \nabla_h c_h\right) \bigg) \right\|_{L^2(T)}^2 \right)^{1/2}\\ &\quad + \sqrt{2}N_\partial^{1/2}\left\{ \left( C_{app}' + 1 \right)^2 + 1 \right\}^{1/2} C_{max}^{1/2} \left\| \rho_h \right\|_{L^\infty(\Omega)} \left( C_{ell}^2C_0^2\operatorname{E}_0[\rho_h%, A_h\rho_h ]^2 + \tilde{C}_1^2\operatorname{\tilde{E}}_1[c_h, \rho_h]^2 \right)^{1/2} \\ &\quad + N_\partial C_{app}'' \left\| \nabla_h c_h \right\|_{L^\infty(\Omega)} \bigg( \sum_{F \in \mathcal{F}_h^i} h_T \left\| \llbracket \rho_h \rrbracket \right\|_{L^2(F)}^2 \bigg)^{1/2} \\ &=: \operatorname{E}_{R_\rho} \end{aligned}$$ where the estimators $\operatorname{E}_{-1}, \operatorname{E}_0$ and $\operatorname{\tilde{E}}_1$ are defined as [\[rhotest2\]](#rhotest2){reference-type="eqref" reference="rhotest2"}, [\[rhoest2\]](#rhoest2){reference-type="eqref" reference="rhoest2"} and [\[cest\]](#cest){reference-type="eqref" reference="cest"}, respectively, $N_{\partial}:=\max _{T \in \mathcal{T}_h} \operatorname{card}\left(\mathcal{F}_T\right)$ and the constant $C_{max}$ is defined in the proof.* Prior to proving the lemma, there are several remarks to be made. **Remark 21** (Uniform bound of $N_\partial$). *It should be noted that, in the above statement, the constant $N_\partial$ is bounded uniformly in $h$ [@Di_Pietro_2012 Lemma 1.41].* **Remark 22** ($L^2$-orthogonality). *Observe that $\nabla_h \cdot (\rho_h \nabla_h c_h)$ is a polynomial of degree $2k-2$, i.e., the corresponding term in [\[eq:R_rho\]](#eq:R_rho){reference-type="eqref" reference="eq:R_rho"} vanishes for $k \leq 2$, i.e. $$\sum_{T \in \mathcal{T}_h} C_{app}'^2 h_T^2 \left\| \nabla_h \cdot \left( \rho_h\nabla_h c_h \right) - \pi_h\left( \nabla_h \cdot\left(\rho_h \nabla_h c_h\right) \right) \right\|_{L^2(T)}^2=0.$$* **Remark 23** (Optimality of the estimator). *The term $\operatorname{E}_{R_\rho}$ enters linearly into an error estimator for the error of our dG scheme measured in the $L^2$-in-time dG-in-space norm. Thus, optimal scaling of $\operatorname{E}_{R_\rho}$ would be $h^k$ if the exact solution is sufficiently smooth. Indeed, for stable simulations $\left\|\rho_h\right\|_{L^{\infty}(\Omega)}$ and $\left\|\nabla_h c_h\right\|_{L^{\infty}(\Omega)}$ are bounded uniformly in $h$ before blow-up. Moreover, in our numerical experiments the solutions of the corresponding elliptic problems are sufficiently regular for the error estimators $\operatorname{E}_0$ to be of order $h^{k+1}$, and for $\operatorname{E}_1$ and $\tilde{\operatorname{E}}_1$ to be of order $h^{k}$. The error estimator $E_{-1}$ is evaluated at $\partial_t \rho_h$, exhibiting an order between $h^{k}$ and $h^{k+1}$. In addition, $$\Big(\sum_{F \in \mathcal{F}_h^i} h_F \left\| \llbracket \rho_h \rrbracket \right\|_{L^2(F)}^2\Big)^{1/2}$$ is of order $h^{k+1}$.* *Proof of Lemma [Lemma 20](#lem:Erho){reference-type="ref" reference="lem:Erho"}.* Let $\pi_h$ denote the $L^2$-orthogonal projection onto $V_h$. We compute the residual $$\label{eq:residual} \begin{aligned} \left\langle R_\rho, \phi\right\rangle%_{H^{-1}, H^1} =&\left\langle \partial_t\hat{\rho}, \phi\right\rangle%_{{H^{-1}, H^1}} - (\hat{\rho} \nabla \hat{c}, \nabla\phi)%_{L^2(\Omega)} + (\nabla \hat{\rho}, \nabla\phi)%_{L^2(\Omega)} \\ =& \left\langle \partial_t\hat{\rho}, \phi\right\rangle%_{{H^{-1}, H^1}} - (\hat{\rho} \nabla \hat{c}, \nabla\phi)%_{L^2(\Omega)} + \left( A_h\rho_h, \phi \right)%_{L^2(\Omega)}, \\ %=& \left\langle \partial_t\hat{\rho}, \phi\right\rangle_{{H^{-1}, H^1}} - (\hat{\rho} \nabla \hat{c}, \nabla\phi)_{L^2(\Omega)} + \left( A_h\rho_h, \pi_h\phi \right)_{L^2(\Omega)} \\ =& \left\langle \partial_t\hat{\rho}, \phi\right\rangle%_{{H^{-1}, H^1}} - (\hat{\rho} \nabla \hat{c}, \nabla\phi)%_{L^2(\Omega)} + a^{sip}_h(\rho_h, \pi_h\phi) \\ =& \left\langle \partial_t\hat{\rho}, \phi\right\rangle%_{{H^{-1}, H^1}} - (\hat{\rho} \nabla \hat{c}, \nabla\phi)%_{L^2(\Omega)} + a^{wsip}_h(\rho_h; c_h, \pi_h\phi) - (\partial_t \rho_{h}, \pi_h\phi)%_{L^2(\Omega)}. \\ %\text{since $(\rho_{h,t}, \phi - \pi_h\phi)=0$}&\ %\text{owing to $L^2$-orthogonality}, \\ \end{aligned}$$ where we utilized $L^2$-orthogonality in the third equality and $\eqref{def:dg}_1$ in the final equality. By applying $L^2$-orthogonality once more, we obtain: $$\left( \partial_t \rho_h, \pi_h \phi \right)%_{L^2(\Omega)} = \left( \partial_t \rho_h, \phi \right)%_{L^2(\Omega)} = \left\langle \partial_t \rho_h, \phi \right\rangle. %_{H^{-1}, H^1}.$$ Hence we can rewrite [\[eq:residual\]](#eq:residual){reference-type="eqref" reference="eq:residual"} as follows: $$\left\langle R_\rho, \phi\right\rangle%_{H^{-1}, H^1} = \underbrace{\left\langle \partial_t\hat{\rho} - \partial_t\rho_{h}, \phi\right\rangle}_{:=I_1}%_{H^{-1}, H^1} \\ \underbrace{- (\hat{\rho} \nabla \hat{c}, \nabla\phi) + a^{wsip}_h(\rho_h; c_h, \pi_h\phi)}_{:=I_2}$$ We provide separate bounds for $I_1$ and $I_2$ as follows: Firstly, using Cauchy-Schwarz inequality and [\[rhotest2\]](#rhotest2){reference-type="eqref" reference="rhotest2"}, we estimate $I_1$ $$I_1 \leq C_{-1}\operatorname{E}_{-1}\left[\partial_t \rho_h, \partial_t\left(A_h \rho_h\right)\right] \left\| \phi \right\|_{H^1(\Omega)}.$$ Secondly, proceeding from the definition [\[eq:wsip\]](#eq:wsip){reference-type="eqref" reference="eq:wsip"} we write $$\label{eq:compute_residual} \begin{aligned} I_2 &= -\left( \hat{\rho}\nabla\hat{c} - \rho_h\nabla_h c_h, \nabla\phi \right)%_{L^2(\Omega)} \\ &\quad - \sum_{T \in \mathcal{T}_h}\int_T \left( \rho_h \nabla_h c_h \right)\cdot\left( \nabla\phi - \nabla_h \pi_h\phi\right) + \sum_{F \in \mathcal{F}_h^i}\int_F \sigma \frac{\gamma_{\rho_h,F}}{h_F}\llbracket c_h \rrbracket \llbracket \pi_h\phi \rrbracket \\ &\quad -\sum_{F \in \mathcal{F}_h^i}\int_F\llbracket \pi_h\phi \rrbracket% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \rho_h \nabla_h c_h \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega}\cdot \mathrm{n}_F - \sum_{F \in \mathcal{F}_h^i}\int_F \llbracket c_h \rrbracket % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \rho_h \nabla_h \pi_h\phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega}\cdot \mathrm{n}_F, \end{aligned}$$ where the parameter $\gamma_{\rho_h,F}$ is defined as $\frac{2\rho_h|_{T_1}\rho_h|_{T_2}}{\rho_h|_{T_1}+\rho_h|_{T_2}}$ with $F = \partial T_1 \cap \partial T_2$. Next, we consider the first two terms of $I_2$: $$\begin{gathered} -\left( \hat{\rho}\nabla \hat{c} - \rho_h \nabla_h c_h, \nabla \phi \right)%_{L^2(\Omega)} \\ = - \sum_{T \in \mathcal{T}_h} \int_T \left( \hat{\rho} - \rho_h \right)\nabla \hat{c} \cdot \nabla \phi - \sum_{T \in \mathcal{T}_h} \int_T \rho_h \left( \nabla \hat{c} - \nabla_h c_h \right)\cdot \nabla \phi\end{gathered}$$ and $$\begin{gathered} \label{eq:compute_residual2} - \sum_{T \in \mathcal{T}_h}\int_T \left( \rho_h \nabla_h c_h \right)\cdot\left( \nabla\phi - \nabla_h \pi_h\phi\right) \\ = \sum_{T \in \mathcal{T}_h} \int_T \nabla_h \cdot \left( \rho_h \nabla_h c_h \right)\left( \phi -\pi_h\phi \right) - \sum_{F \in \mathcal{F}_h^i} \int_F \llbracket \rho_h \nabla_h c_h \cdot \mathrm{n}_F \left( \phi -\pi_h\phi \right) \rrbracket.\end{gathered}$$ Additionally, utilizing the identity, $$\label{eq:jump_identity} a_1 b_1-a_2 b_2 \\ =\left(\omega_1 a_1+\omega_2 a_2\right)\left(b_1-b_2\right)+\left(a_1-a_2\right)\left(\omega_2 b_1+\omega_1 b_2\right)$$ where $\omega_1$ and $\omega_2$ are positive real numbers such that $\omega_1+\omega_2=1$, we can rewrite the second term on the right-hand side of [\[eq:compute_residual2\]](#eq:compute_residual2){reference-type="eqref" reference="eq:compute_residual2"} as: $$\begin{gathered} - \sum_{F \in \mathcal{F}_h^i} \int_F \llbracket \rho_h \nabla_h c_h \cdot \mathrm{n}_F \left( \phi - \pi_h\phi \right) \rrbracket =\\ - \sum_{F \in \mathcal{F}_h^i} \int_F % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \rho_h \nabla_h c_h \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega}\cdot \mathrm{n}_F \llbracket \phi - \pi_h \phi \rrbracket - \sum_{F \in \mathcal{F}_h^i} \int_F \llbracket \rho_h \nabla_n c_h \rrbracket \cdot \mathrm{n}_F % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \phi - \pi_h \phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\bar{\omega}}. \end{gathered}$$ Combining these results, we can rewrite equation [\[eq:compute_residual\]](#eq:compute_residual){reference-type="eqref" reference="eq:compute_residual"} as: $$\label{eq:residual3} \begin{aligned} I_2 = & -\sum_{T \in \mathcal{T}_h} \int_T \left( \hat{\rho} - \rho_h \right)\nabla \hat{c} \cdot \nabla \phi - \sum_{T \in \mathcal{T}_h} \int_T \rho_h \left( \nabla \hat{c} - \nabla_h c_h \right)\cdot \nabla \phi \\ & + \sum_{T \in \mathcal{T}_h} \int_T \nabla_h \cdot \left( \rho_h \nabla_h c_h \right)\left( \phi -\pi_h\phi \right) \\ & -\sum_{F \in \mathcal{F}_h^i} \int_F \llbracket \rho_h \nabla_n c_h \rrbracket \cdot \mathrm{n}_F % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \phi - \pi_h \phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\bar{\omega}}. \\ & + \sum_{F \in \mathcal{F}_h^i}\sigma \frac{\gamma_{\rho_h,F}}{h_F}\int_F \llbracket c_h \rrbracket \llbracket \pi_h\phi \rrbracket \\ & - \sum_{F \in \mathcal{F}_h^i}\int_F \llbracket c_h \rrbracket % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \rho_h \nabla_h \pi_h\phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega}\cdot \mathrm{n}_F \\ =:& A_1+A_2+A_3+A_4+A_5+A_6. \end{aligned}$$ Owing to $\llbracket \phi \rrbracket = 0$ for $\phi \in H^1(\Omega)$, we have the identity $$\sum_{F \in \mathcal{F}_h^i}\int_F\llbracket \pi_h\phi \rrbracket% \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \rho_h \nabla_h c_h \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega}\cdot \mathrm{n}_F = \sum_{F \in \mathcal{F}_h}\int_F \llbracket \pi_h \phi - \phi \rrbracket % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}\rho_h \nabla_hc_h\mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega}\cdot \mathrm{n}_F.$$ These terms appear twice with opposite signs in the computation of [\[eq:residual3\]](#eq:residual3){reference-type="eqref" reference="eq:residual3"}, thus cancel. Now, we will estimate each term on the right-hand side of [\[eq:residual3\]](#eq:residual3){reference-type="eqref" reference="eq:residual3"}. #### Estimate for $A_1$: We have $$\begin{aligned} \int_\Omega \left( \hat{\rho} - \rho_h \right)\nabla \hat{c} \cdot \nabla \phi \leq \left\|\hat{\rho} - \rho_h\right\|_{L^2(\Omega)} \left\| \nabla \hat{c} \right\|_{L^\infty(\Omega)} |\phi|_{H^1(\Omega)}. \end{aligned}$$ Since $d \leq 3$, we have $%\label{eq:H^2_to_L^infty} \left\| \nabla \hat{c} \right\|_{L^\infty(\Omega)} \leq C_S'' \left\| \nabla \hat{c} \right\|_{H^2(\Omega)}$ where $C_S''$ is the constant of the embedding $H^2 \rightarrow L^\infty$. By employing elliptic regularity, we obtain $$\left\| \nabla \hat{c} \right\|_{H^2(\Omega)} \leq C_{ell} \left\| \nabla \hat{\rho} \right\|_{L^2(\Omega)} \\ \leq 2 C_{ell}\left( \sum_{T \in \mathcal{T}_h} \left| \hat{\rho} - \rho_h \right|_{H^1(T)}^2 + \left| \rho_h \right|_{H^1(T)}^2\right)^{\tfrac12},$$ where $C_{ell}$ is the constant of elliptic regularity, from which we infer, using [\[rhoest1\]](#rhoest1){reference-type="eqref" reference="rhoest1"} and [\[rhoxest1\]](#rhoxest1){reference-type="eqref" reference="rhoxest1"}, $$\begin{aligned} &\int_\Omega \left( \hat{\rho} - \rho_h \right)\nabla \hat{c} \cdot \nabla \phi \\ &\leq 2 C_S'' C_{ell}\left\|\hat{\rho} - \rho_h\right\|_{L^2(\Omega)} \left( \sum_{T \in \mathcal{T}_h} \left| \hat{\rho} - \rho_h \right|_{H^1(T)}^2 + \left| \rho_h \right|_{H^1(T)}^2\right)^{1/2} |\phi|_{H^1(\Omega)} \\ &\leq 2C_S'' C_{ell} C_0 \operatorname{E}_0\left[\rho_h%, A_h \rho_h \right] \left( C_1^2 \operatorname{E}_1\left[\rho_h%, A_h \rho_h \right]^2 + \sum_{T \in \mathcal{T}_h} \left| \rho_h \right|_{H^1(T)}^2 \right)^{1/2} |\phi|_{H^1(\Omega)}. \end{aligned}$$ #### Estimate for $A_2$: We estimate $$\sum_{T \in \mathcal{T}_h} \int_T \rho_h \left( \nabla \hat{c} - \nabla_h c_h \right)\cdot \nabla \phi \leq \left\| \rho_h \right\|_{L^\infty(\Omega)} \sum_{T \in \mathcal{T}_h} \left| \hat{c} - c_h \right|_{H^1(T)} \left| \phi \right|_{H^1(T)} =: A_2'.$$ #### Estimate for $A_3$: We estimate the following expression: $$\begin{aligned} \sum_{T \in \mathcal{T}_h}& \int_T \nabla_h \cdot\left(\rho_h \nabla_h c_h\right)\left(\phi-\pi_h \phi\right) \\ \leq& \sum_{T \in \mathcal{T}_h} \left\| \nabla_h \cdot \left( \rho_h\nabla_h c_h \right) - \pi_h\left( \nabla_h \cdot\left(\rho_h \nabla_h c_h\right) \right) \right\|_{L^2(T)} \left\| \phi - \pi_h \phi \right\|_{L^2(T)} \\ \leq & \sum_{T \in \mathcal{T}_h} C_{app}' h_T \left\| \nabla_h \cdot \left( \rho_h\nabla_h c_h \right) -\pi_h\left( \nabla_h \cdot\left(\rho_h \nabla_h c_h\right) \right)\right\|_{L^2(T)} \left| \phi \right|_{H^1(T)}, \\ %\text{owing to Cauchy-Schwarz inequality},\\ \leq & \left( \sum_{T \in \mathcal{T}_h} C_{app}'^2 h_T^2 \left\| \nabla_h \cdot \left( \rho_h\nabla_h c_h \right) - \pi_h\left( \nabla_h \cdot\left(\rho_h \nabla_h c_h\right) \right) \right\|_{L^2(T)}^2 \right)^{1/2}\left| \phi \right|_{H^1(\Omega)}, \end{aligned}$$ where we utilized the $L^2$-orthogonality of $\pi_h\left( \nabla_h \cdot\left(\rho_h \nabla_h c_h\right) \right)$ and $\phi-\pi_h \phi$ in the first step. Additionally, the Cauchy-Schwarz inequality was employed in the second and third step. #### Estimate for $A_4$: {#estimate-for-a_4 .unnumbered} We first observe that utilizing the identity [\[eq:jump_identity\]](#eq:jump_identity){reference-type="eqref" reference="eq:jump_identity"} with the choice of $$\omega_1 = \frac{\rho^-_h}{\rho^+_h + \rho^-_h}, \omega_2 = \frac{\rho^+_h}{\rho^+_h + \rho^-_h}$$ yields $$\llbracket \rho_h \nabla_h c_h \rrbracket = \gamma_{\rho_h, F} \llbracket \nabla_h c_h \rrbracket + \llbracket \rho_h \rrbracket % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \nabla_h c_h \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\bar{\omega}}.$$ Moreover, we have $\gamma_{\rho_h, F} \leq \left\| \rho_h \right\|_{L^\infty(\Omega)}$ for all $F \in \mathcal{F}_h^i$. By using these inequalities, we estimate $$\begin{aligned} &\sum_{F \in \mathcal{F}^i_h} \int_F \llbracket \rho_h\nabla_h c_h \rrbracket\cdot \mathrm{n}_F % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \phi - \pi_h\phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\bar{\omega}} \\ &= \sum_{F \in \mathcal{F}^i_h} \gamma_{\rho_h, F} \llbracket \nabla_h c_h \rrbracket \cdot \mathrm{n}_F % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \phi - \pi_h\phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\bar{\omega}} + \sum_{F \in \mathcal{F}^i_h} \llbracket \rho_h \rrbracket % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \nabla_h c_h \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\bar{\omega}} \cdot \mathrm{n}_F % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \phi - \pi_h\phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\bar{\omega}} \\ &= \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} \int_F \gamma_{\rho_h, F} \llbracket \nabla_h c_h \rrbracket\cdot \mathrm{n}_F \bar{\omega}_{T,F}\left( \phi -\pi_h \phi \right)\\ &\quad + \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} \int_F \llbracket \rho_h \rrbracket % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \nabla_h c_h \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\bar{\omega}} \cdot \mathrm{n}_F \bar{\omega}_{T,F}\left( \phi -\pi_h \phi \right) \\ &\leq \left\| \rho_h \right\|_{L^\infty(\Omega)} \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} \left\| \llbracket \nabla_h c_h \rrbracket \cdot \mathrm{n}_F \right\|_{L^2(F)} \left\| \phi - \pi_h \phi \right\|_{L^2(F)} \\ &\quad + \left\| \nabla_h c_h \right\|_{L^\infty(\Omega)} \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} \left\| \llbracket \rho_h \rrbracket \right\|_{L^2(F)} \left\| \phi - \pi_h \phi \right\|_{L^2(F)} \end{aligned}$$ so that $$\begin{gathered} \sum_{F \in \mathcal{F}^i_h} \int_F \llbracket \rho_h\nabla_h c_h \rrbracket\cdot \mathrm{n}_F % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \phi - \pi_h\phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\bar{\omega}} \\ \leq \left\| \rho_h \right\|_{L^\infty(\Omega)} \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} C_{app}'' h_T^{1/2} \left\| \llbracket \nabla_h c_h \rrbracket \cdot \mathrm{n}_F \right\|_{L^2(F)} | \phi |_{H^1(T)} \\ + \left\| \nabla_h c_h \right\|_{L^\infty(\Omega)} \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} C_{app}'' h_T^{1/2} \left\| \llbracket \rho_h \rrbracket \right\|_{L^2(F)} | \phi |_{H^1(T)} =: A_4', \end{gathered}$$ where $\bar{\omega}_{T,F}$ is the corresponding weight coefficient on each $T\in \mathcal{T}_h$ and $F \in \mathcal{F}_T$. #### Estimate for $A_5$: Using $\epsilon_{T,F}:= \pm 1$ depending on the edge $F$, we estimate $$\begin{aligned} &\sum_{F \in \mathcal{F}_h^i} \sigma\frac{\gamma_{\rho_h,F}}{h_F}\int_F \llbracket c_h \rrbracket \llbracket \phi - \pi_h \phi \rrbracket \\ &= \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \sigma\frac{\gamma_{\rho_h,F}}{h_F} \epsilon_{T, F}\int_{F} \llbracket c_h \rrbracket \left( \phi - \pi_h \phi|_T \right) \\ &\leq \left\| \rho_h \right\|_{L^\infty(\Omega)}\sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \frac{\sigma}{h_F} \left\| \llbracket c_h \rrbracket \right\|_{L^2(F)} \left\| \phi - \pi_h \phi|_T \right\|_{L^2(F)} \\ &\leq \left\| \rho_h \right\|_{L^\infty(\Omega)}\sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} C_{app}'' h_T^{1/2}\frac{\sigma}{h_F}\left\| \llbracket c_h \rrbracket \right\|_{L^2(F)}\left| \phi \right|_{H^1(T)} =:A_5'. \end{aligned}$$ #### Estimate for $A_6$: We estimate $$\begin{aligned} &\sum_{F \in \mathcal{F}_h^i} \int_F\llbracket c_h \rrbracket % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}}\rho_h\nabla_h \pi_h \phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% _{\omega} \cdot \mathrm{n}_F \\ &= \sum_{F \in \mathcal{F}_h^i} \int_F \gamma_{\rho_h, F} \llbracket c_h \rrbracket % \mathopen{\scalebox{1.0}{$\lbrace\!\mkern-1mu\lbrace$}} \nabla_h\pi_h\phi \mathclose{\scalebox{1.0}{$\rbrace\!\mkern-1mu\rbrace$}}% \cdot \mathrm{n}_F \\ & = \frac{1}{2} \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} \int_F \gamma_{\rho_h, F} \llbracket c_h \rrbracket \nabla_h \pi_h \phi \cdot \mathrm{n}_F \\ &\leq \left\| \rho_h \right\|_{L^\infty(\Omega)}\sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} \left\| \llbracket c_h \rrbracket \right\|_{L^2(F)} \left\| \nabla_h \pi_h \phi \cdot \mathrm{n}_F\right\|_{L^2(F)} \\ &\leq\left\| \rho_h \right\|_{L^\infty(\Omega)}\sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} \frac{C_{tr}}{h_T^{1/2}} \left\| \llbracket c_h \rrbracket \right\|_{L^2(F)} \left\| \nabla_h \pi_h \phi \right\|_{L^2(T)} =:A_6' \end{aligned}$$ where $C_{tr}$ is the constant of the trace inequality. By summing up the terms $A_2', A_4', A_5',$ and $A_6'$, applying the Cauchy-Schwarz inequality, and gathering relevant terms, we obtain: $$\begin{aligned} & A_2' + A_4' + A_5' + A_6' \leq \\& \left\| \rho_h \right\|_{L^\infty(\Omega)} \left( \sum_{T \in \mathcal{T}_h} \left| \hat{c} - c_h \right|_{H^1(T)}^2 + \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} C_{app}''^2 h_T \left\| \llbracket \nabla_h c_h \rrbracket \cdot \mathrm{n}_F \right\|_{L^2(F)}^2 \right.\\ & \left. + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} C_{app}''^2 h_T \frac{\sigma^2}{h_F^2} \left\| \llbracket c_h \rrbracket \right\|_{L^2(F)}^2 + \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} \frac{C_{tr}^2}{h_T} \left\| \llbracket c_h \rrbracket \right\|_{L^2(F)}^2 \right)^{1/2} \\ &\quad \times\left( \sum_{T\in \mathcal{T}_h}\sum_{F\in \mathcal{F}_T} \left| \phi \right|_{H^1(T)}^2 + \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} \left\| \nabla_h \pi_h \phi \right\|_{L^2(T)}^2 \right)^{1/2} \\ & + \left\| \nabla_h c_h \right\|_{L^\infty(\Omega)} \left( \sum_{T \in \mathcal{T}_h}\sum_{F \in \mathcal{F}_T} C_{app}''^2 h_T \left\| \llbracket \rho_h \rrbracket \right\|_{L^2(F)}^2 \right)^{1/2} \left( \sum_{T\in \mathcal{T}_h}\sum_{F\in \mathcal{F}_T} \left| \phi \right|_{H^1(T)}^2 \right)^{1/2}. \end{aligned}$$ Since $\mathcal{T}_{\mathcal{H}}$ is shape- and contact-regular, there exists a mesh regularity parameter $\delta > 0$ such that for any $T \in \mathcal{T}_h$ and any $F \in \mathcal{F}_T$, $\delta h_T \leq h_F$. Further, we can estimate $$\left\| \nabla_h \pi_h \phi \right\|_{L^2(T)} \leq \left( C_{app}' + 1 \right) \left| \phi \right|_{H^1(T)}.$$ Now we can rewrite the expression as: $$\begin{aligned} &A_2' + A_4' + A_5' + A_6' \leq \\& \Bigg[ N_\partial^{1/2}\left\{ \left( C_{app}' + 1 \right)^2 + 1 \right\}^{1/2} \left\| \rho_h \right\|_{L^\infty(\Omega)} \Bigg( \sum_{T \in \mathcal{T}_h} \left| \hat{c} - c_h \right|_{H^1(T)}^2 \\ & + N_\partial C_{app}''^2\sum_{F \in \mathcal{F}_h^i} h_F \left\| \llbracket \nabla_h c_h \rrbracket \right\|_{L^2(F)}^2 + N_\partial \left( C_{app}''^2\sigma^2 \delta^{-1} + C_{tr}^2 \right) \sum_{F \in \mathcal{F}_h^i} h_F^{-1} \left\| \llbracket c_h \rrbracket \right\|_{L^2(F)}^2 \Bigg)^{1/2} \\ & + N_\partial C_{app}'' \left\| \nabla_h c_h \right\|_{L^\infty(\Omega)} \left( \sum_{F \in \mathcal{F}_h^i} h_F \left\| \llbracket \rho_h \rrbracket \right\|_{L^2(F)}^2 \right)^{1/2} \Bigg]\left| \phi \right|_{H^1(\Omega)} \end{aligned}$$ where $N_{\partial}:=\max _{T \in \mathcal{T}_h} \operatorname{card}\left(\mathcal{F}_T\right)$. Utilizing [\[eq:cest2\]](#eq:cest2){reference-type="eqref" reference="eq:cest2"}, we can further estimate the terms relevant to the error in $c_h$: $$\begin{aligned} & \sum_{T \in \mathcal{T}_h} \left| \hat{c} - c_h \right|_{H^1(T)}^2 + N_\partial C_{app}''\sum_{F \in \mathcal{F}_h^i} h_F \left\| \llbracket \nabla_h c_h \rrbracket \right\|_{L^2(F)}^2 \\ &\quad + N_\partial \left( C_{app}''^2\sigma^2 \delta^{-1} + C_{tr}^2 \right) \sum_{F \in \mathcal{F}_h^i} h_F^{-1} \left\| \llbracket c_h \rrbracket \right\|_{L^2(F)}^2 \\ &\leq C_{max} \left( \sum_{T \in \mathcal{T}_h} \left| \hat{c} - c_h \right|_{H^1(T)}^2 + \sum_{F \in \mathcal{F}_h^i} h_F \left\| \llbracket \nabla_h c_h \rrbracket \right\|_{L^2(F)}^2 + \sigma^2\sum_{F \in \mathcal{F}_h^i} h_F^{-1} \left\| \llbracket c_h \rrbracket \right\|_{L^2(F)}^2 \right) \\ &\leq C_{max} \left( C_{ell}^2C_0^2\operatorname{E}_0[\rho_h%, A_h\rho_h ]^2 + \tilde{C}_1^2\operatorname{\tilde{E}}_1[c_h, \rho_h]^2 \right) \end{aligned}$$ where $C_{max}:= \max\left\{ 1, N_\partial C_{app}'', N_\partial \left( C_{app}''^2 \delta^{-1} + C_{tr}^2 \right) \right\}$. Thus, collecting all terms concludes the proof of the Theorem. ◻ # A posteriori error estimator In this section, we state and prove our main result, i.e. the a posteriori error estimates for the dG approximation of the Keller-Segel system given by [\[def:dg\]](#def:dg){reference-type="eqref" reference="def:dg"}. In order to use the stability estimate Theorem [Theorem 8](#thm:stabilityest){reference-type="ref" reference="thm:stabilityest"}, we require computable upper bounds for the quantities $A$ and $E$ in [\[eq:AE\]](#eq:AE){reference-type="eqref" reference="eq:AE"}. We define the following quantities: $$\label{eq:barAbarE} \begin{aligned} \bar{A}:=& 2 \left\| \rho(0,\cdot) - \rho_h(0, \cdot)\right\|_{L^2(\Omega)}^2 + 2\operatorname{E}_0[\rho_h(0, \cdot)]^2 + \int_0^T \operatorname{E}_{R_\rho}^2 dt, \\ \bar{E}:=& \exp{\left( \int_0^T \bar{a} dt \right)}, \end{aligned}$$ where $$\begin{gathered} \bar{a}:= 6C_S^2C_S'^2C_{ell}^2\biggl( C_0^2 \operatorname{E}_0[\rho_h(t, \cdot)]^2 + C_1^2 \operatorname{E}_1[\rho_h(t, \cdot)]^2 + \sum_{T \in \mathcal{T}_h} \left\| \rho_h(t, \cdot) \right\|_{L^2(T)}^2 + \left| \rho_h(t, \cdot) \right|_{H^1(T)}^2 \biggr) \\ + 6C_S''C_{ell}^2 \left( C_1^2 \operatorname{E}_1[\rho_h(t, \cdot)]^2 + \sum_{T \in \mathcal{T}_h}\left| \rho_h(t, \cdot) \right|_{H^1(T)}^2 \right) + 1/3.\end{gathered}$$ It is straightforward that the quantities $\bar{A}$ and $\bar{E}$ bound $A$ and $E$ defined in [\[eq:AE\]](#eq:AE){reference-type="eqref" reference="eq:AE"} from above, i.e., $A \leq \bar{A}$ and $E \leq \bar{E}$. In what follows, the parameter $B= 2 C_S'C_SC_{ell}$ is used as in Theorem [Theorem 8](#thm:stabilityest){reference-type="ref" reference="thm:stabilityest"}. **Theorem 24** (A posteriori error estimate). *Let $\left( \rho, c \right)$ be a weak solution to [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"}-[\[eq:ic\]](#eq:ic){reference-type="eqref" reference="eq:ic"} and let $(\rho_h, c_h)$ be the dG solution to [\[def:dg\]](#def:dg){reference-type="eqref" reference="def:dg"}. If the condition $$2^9\bar{A}\bar{E}^3(B(1+T))^{2} \leq 1$$ holds, we have $$\begin{gathered} \label{eq:fullestimator} \left\| \rho(t, \cdot) - \rho_h(t, \cdot) \right\|_{L^\infty(0,T;L^2(\Omega))}^2 + \int_0^T \left\| \rho(t, \cdot) - \rho_h(t, \cdot) \right\|_{dG}^2 dt \\ \leq 32 \bar{A}\bar{E} + 2 C_0^2 \left\| \operatorname{E}_0\left[ \rho_h(t, \cdot)%, A_h\rho_h \right] \right\|_{L^\infty(0,T)}^2 + 2C_1^2 \left\| \operatorname{E}_1\left[ \rho_h(t, \cdot)%, A_h\rho_h(t, \cdot) \right] \right\|_{L^2(0,T)}^2 \end{gathered}$$ where $\operatorname{E}_{0}$, $\operatorname{E}_{1}$, and $\operatorname{E}_{R_\rho}$ are defined in [\[rhoest2\]](#rhoest2){reference-type="eqref" reference="rhoest2"}, [\[rhoxest1\]](#rhoxest1){reference-type="eqref" reference="rhoxest1"}, and [\[eq:R_rho\]](#eq:R_rho){reference-type="eqref" reference="eq:R_rho"}, respectively.* **Remark 25** (Optimality of the estimator). *Based on the arguments given in Remark [Remark 23](#rmk:optimal){reference-type="ref" reference="rmk:optimal"}, the estimator in Theorem [Theorem 24](#thm:aposteriori){reference-type="ref" reference="thm:aposteriori"} is expected to scale optimally as long as the exact solution is smooth, i.e. with rate $k$. This is confirmed in our numerical experiments which are presented in the next section.* **Remark 26**. *All constants appearing in Theorem [Theorem 24](#thm:aposteriori){reference-type="ref" reference="thm:aposteriori"} can be evaluated on a convex polygonal domain. The Sobolev constants are provided in [@2017Mizuguchi]. For the constant derived from the Poincar'e-type inequality, we refer to [@1960Payne]. The constant of elliptic regularity can also be explicitly computed [@1992Grisvard Chapter 2]. The constants for the discrete trace and inverse inequality can be evaluated as detailed in [@2003Warburton]. We refer to [@Di_Pietro_2012 Theorem 5.45] for the constants of the a posteriori error estimator $E_1$. The constants of the estimators $E_{-1}$ and $E_0$ can be evaluated in a similar manner, see [@Makridakis2003].* *Proof of Theorem [Theorem 24](#thm:aposteriori){reference-type="ref" reference="thm:aposteriori"}.* Let $(\hat{\rho}, \hat{c})$ be the reconstruction of $(\rho_h, c_h)$ as defined in Section [4](#sec:dgscheme){reference-type="ref" reference="sec:dgscheme"}. The triangle inequality implies $$\begin{aligned} \left\| \rho - \rho_h \right\|_{L^\infty(0,T;L^2(\Omega))}^2 &\leq 2 \left\| \rho - \hat{\rho} \right\|_{L^\infty(0,T;L^2(\Omega))}^2 + 2 \left\| \hat{\rho} - \rho_h \right\|_{L^\infty(0,T;L^2(\Omega))}^2 \\ & =: I_1 + I_2 \end{aligned}$$ For the first term $I_1$, using [\[mainestimate\]](#mainestimate){reference-type="eqref" reference="mainestimate"} gives $$\begin{gathered} I_1 \leq 16 \left( \| \rho (0,\cdot) - \hat{\rho} (0,\cdot)\|_{L^2(\Omega)}^2 + \int_0^{T} \| R_\rho\|_{H^{-1}(\Omega)}^2 dt \right) \\ \times \exp \left( \int_0^T 3 C_S^2C_{ell}^2 \|\hat{\rho}(t,\cdot)\|_{L^3(\Omega)}^2 + 3 \|\nabla \hat{c}(t,\cdot)\|_{L^\infty(\Omega)}^2 + \frac 1 3 dt\right), \end{gathered}$$ from which we obtain $$I_1 \leq 16\bar{A}\bar{E}.$$ For the second term $I_2$, we have $$I_2 \leq 2 C_0^2 \left\| \operatorname{E}_0\left[ \rho_h%, A_h\rho_h \right] \right\|_{L^\infty(0,T)}^2.$$ Thus we have the following estimate $$\label{eq:rhoest1} \left\| \rho - \rho_h \right\|_{L^\infty(0,T; L^2(\Omega))}^2 \leq 16 \bar{A}\bar{E} + 2 C_0^2 \left\| \operatorname{E}_0 [\rho_h]\right\|_{L^\infty(0,T)}^2.$$ A similar argument provides the estimate $$\label{eq:rhoest2} \int_0^T \left\| \rho - \rho_h \right\|_{dG}^2 dt \\ \leq 16 \bar{A}\bar{E} + 2C_1^2 \int_0^T \operatorname{E}_1\left[ \rho_h(t, \cdot)%, A_h\rho_h(t, \cdot) \right]^2 dt.$$ Combining [\[eq:rhoest1\]](#eq:rhoest1){reference-type="eqref" reference="eq:rhoest1"} and [\[eq:rhoest2\]](#eq:rhoest2){reference-type="eqref" reference="eq:rhoest2"} gives $$\begin{gathered} \left\| \rho - \rho_h \right\|_{L^\infty(0,T;L^2(\Omega))}^2 +\int_0^T \left\| \rho - \rho_h \right\|_{dG}^2 dt\\ \leq 32 \bar{A}\bar{E} + 2 C_0^2 \left\| \operatorname{E}_0\left[ \rho_h%, A_h\rho_h \right] \right\|_{L^\infty(0,T)}^2 + 2C_1^2 \left\| \operatorname{E}_1\left[ \rho_h(t, \cdot)%, A_h\rho_h(t, \cdot) \right] \right\|_{L^2(0,T)}^2, \end{gathered}$$ which is the assertion of the theorem. ◻ # Numerical Experiments In this section, we present numerical results to assess the performance of the estimator presented in this paper. To accomplish this, we employ a fully discrete scheme for the problem [\[eq:KS1\]](#eq:KS1){reference-type="eqref" reference="eq:KS1"}-[\[eq:ic\]](#eq:ic){reference-type="eqref" reference="eq:ic"}, based on the semi-spatial-discrete scheme [\[def:dg\]](#def:dg){reference-type="eqref" reference="def:dg"}: $$\begin{aligned} \left( \frac{\rho_h^{n+1} - \rho_h^{n}}{t_{n+1} - t_n}, \phi_h \right)_{L^2(\Omega)} + a_h^{sip}(\rho_h^{n+1}, \phi_h) - a_h^{wsip}(\rho_h^n; c_h^{n+1}, \phi_h)&=0 \quad \forall \quad \phi_h \in V_h\\ a_h^{sip}(c_h^{n+1}, \psi_h) + \left( c_h^{n+1}, \psi_h \right)_{L^2(\Omega)} - \left( \rho_h^{n+1}, \psi_h \right)_{L^2(\Omega)} &=0 \quad \forall \quad \psi_h \in V_h \end{aligned}$$ where $0 = t_0 < t_1 < \cdots < t_N = T$ which involves solving a linear system for $\rho_h^{n+1}$ and $c_h^{n+1}$ at each time step. The initial data is given by $\rho_h(0, \cdot) = \pi_h^{ell}\rho_0(\cdot)$, where $\pi^{ell}_h$ denotes the elliptic projection onto $V_h$. That is, it satisfies $a^{sip}_h(\pi^{ell}_h v_h, \phi_h) = a^{sip}_h(v_h, \phi_h)$ for all $\phi_h \in V_h$. The implementation of the tests was carried out using Python. **Definition 27** (Estimated order of convergence). *Given two sequences $a(i)$ and $h(i) \searrow 0$, we define the estimated order of convergence (EOC) to be the local slope of the $\log a(i)$ vs. $\log h(i)$ curve, i.e., $$\operatorname{EOC}(a, h ; i):=\frac{\log (a(i+1) / a(i))}{\log (h(i+1) / h(i))}.$$* Let $\Omega \subset \mathbb{R}^2$ be the unit square $(0,1)\times(0,1)$ and we define $\rho_0(x,y)$ by $$\rho_0(x, y) = 10^{-3} \exp \left( - \frac{(x - x_0)^2 + (y - y_0)^2}{10^{-2}} \right).$$ The solution sequence is calculated using a mesh width of $h = 2^{1/2-i}$ and a time step of $\tau = 2^{2-i}$. Given the center point $x_0 = 0.5$ and $y_0 = 0.5$, the initial data yields a radially symmetric solution $\rho(t, x, y)$, which experiences a blow-up in finite time, as illustrated in Figure [\[fig:blowup\]](#fig:blowup){reference-type="ref" reference="fig:blowup"}. {width="\\linewidth"} {width="\\linewidth"} {width="\\linewidth"} {width="\\linewidth"} Computing the full estimator results in overflows due to the exponential term. Thus, to validate the optimality of the error estimator, we perform experiments on the component estimators $\operatorname{E}_0$, $\operatorname{E}_1$, and $\operatorname{E}_{R_{\rho}}$ as well as the quantities $\bar{A}$ and $\bar{E}$ as specified in [\[eq:barAbarE\]](#eq:barAbarE){reference-type="eqref" reference="eq:barAbarE"}. As indicated in Tables [1](#tab:est_cond_1){reference-type="ref" reference="tab:est_cond_1"} to [4](#tab:2nd_order_Es){reference-type="ref" reference="tab:2nd_order_Es"}, they show the expected EOCs. We observe that since the integrand $\bar{a}$ in $\bar{E}$ incorporates the broken $H^1$-norm of $\rho_h$, it is expected that the complete error estimator will be sensitive to the gradient of solution. Indeed, the full estimator behaves as expected, i.e. it converges with rate $k$ as long as the exact solution is smooth and round-off errors do not dominate. $i$ $\bar{a}$ $E_0[\rho_h(0,\cdot)]$ EOC ----- ----------- ------------------------ ------- $4$ 6.86E+05 3.95E+02 2.017 $5$ 1.59E+05 9.77E+01 2.008 $6$ 3.99E+04 2.43E+01 2.003 $7$ 1.22E+04 6.06E+00 2.001 $8$ 5.48E+03 1.51E+00 2.000 $9$ 3.83E+03 3.78E-01 \- : Tests were conducted for $\bar{a}$ and $E_0[\rho_h(0,\cdot)]$ using different values of $i$, with $k = 1$. Note that the integrand $\bar{a}$ in $\bar{E}$ incorporates the broken $H^1$-norm of $\rho_h$, which is expected to be similar to the $H^1$-norm of $\rho$. The EOC for $E_0[\rho_h(0,\cdot)]$ matches the theoretical order. $i$ $\bar{a}$ $E_0[\rho_h(0,\cdot)]$ EOC ----- ----------- ------------------------ ------- $4$ 7.03E+04 1.11E+02 3.025 $5$ 7.33E+03 1.36E+01 2.946 $6$ 3.54E+03 1.77E+00 2.955 $7$ 3.31E+03 2.28E-01 2.974 $8$ 3.29E+03 2.91E-02 \- : Tests were conducted for $\bar{a}$ and $E_0[\rho_h(0,\cdot)]$ using different values of $i$, with $k = 2$. Note that the integrand $\bar{a}$ in $\bar{E}$ incorporates the broken $H^1$-norm of $\rho_h$, which is expected to be similar to the $H^1$-norm of $\rho$. The EOC for $E_0[\rho_h(0,\cdot)]$ matches the theoretical order. $i$ $\left\| \operatorname{E}_{0} \right\|_{L^\infty(0,T)}$ EOC $\left\| \operatorname{E}_{1} \right\|_{L^2(0,T)}$ EOC $\left\| \operatorname{E}_{R_\rho} \right\|_{L^2(0,T)}$ EOC ----- --------------------------------------------------------- ------- ---------------------------------------------------- ------- --------------------------------------------------------- ------- $4$ 4.30E+02 1.989 5.23E+01 1.068 4.37E+04 2.154 $5$ 1.08E+02 2.002 2.50E+01 1.042 9.81E+03 1.544 $6$ 2.70E+01 2.001 1.21E+01 1.021 3.37E+03 1.182 $7$ 6.75E+00 2.000 5.97E+00 1.011 1.48E+03 1.055 $8$ 1.69E+00 2.000 2.96E+00 1.005 7.14E+02 1.019 $9$ 4.22E-01 \- 1.48E+00 \- 3.52E+02 \- : Polynomial degree $k = 1$. Experimental results for $E_{0}[\rho_h(t, \cdot)]$, $E_{1}[\rho_h(t, \cdot)]$, $E_{R_\rho}[\rho_h(t, \cdot)]$. The sequence of solutions is computed with a mesh size $h = 2^{1/2-i}$, time step $\tau = 2^{2-i}$ and $T = 0.0001$. $i$ $\left\| \operatorname{E}_{0} \right\|_{L^\infty(0,T)}$ EOC $\left\| \operatorname{E}_{1} \right\|_{L^2(0,T)}$ EOC $\left\| \operatorname{E}_{R_\rho} \right\|_{L^2(0,T)}$ EOC ----- --------------------------------------------------------- ------- ---------------------------------------------------- ------- --------------------------------------------------------- ------- $4$ 1.47E+02 2.988 1.64E+01 2.028 5.83E+03 3.042 $5$ 1.85E+01 3.016 4.02E+00 2.024 7.08E+02 2.243 $6$ 2.29E+00 3.005 9.88E-01 2.018 1.50E+02 2.074 $7$ 2.85E-01 3.001 2.44E-01 2.009 3.55E+01 1.728 $8$ 3.56E-02 \- 6.06E-02 \- 1.07E+01 \- : Polynomial degree $k=2$. Experimental results on $E_{0}[\rho_h(t, \cdot)]$, $E_{1}[\rho_h(t, \cdot)]$, $E_{R_\rho}[\rho_h(t, \cdot)]$. The sequence of solutions is computed with a mesh size $h = 2^{1/2-i}$, time step $\tau = 2^{2-i}$ and $T = 0.0001$. **Remark 28**. *We should note that due to the precision limitations of the `float64` data type in the NumPy library, the $L^2$ norm of the jump of $c_h$ cannot decrease below a certain threshold. Consequently, the estimator $\tilde{E}_1[c_h, \rho_h]$ and thus $E_{R_\rho}$ exhibit a deterioration of the EOC for $k=2$ beyond a certain refinement level as round-off errors start to dominate, see Table [5](#tab:2nd_order_Es_tilde){reference-type="ref" reference="tab:2nd_order_Es_tilde"}. Indeed, the squared jump of $c_h$ is bounded from below by $10^{-16}$, corresponding to the machine epsilon for `float64` in NumPy, i.e. $2.22 \times 10^{-16}$.* $i$ $\left\| \operatorname{\tilde{E}}_{1} \right\|_{L^2(0,T)}$ EOC ----- ------------------------------------------------------------ ------- $4$ 3.31E-02 2.071 $5$ 7.87E-03 2.041 $6$ 1.91E-03 2.019 $7$ 4.72E-04 1.696 $8$ 1.46E-04 \- : Polynomial degree $k = 2$. Experimental results on $\tilde{E}_{1}[c_h(t, \cdot)]$. The sequence of solutions is computed with a mesh size $h = 2^{1/2-i}$, time step $\tau = 2^{2-i}$ and $T = 0.0001$. The table shows that the EOC of $\tilde{E}_{1}[c_h(t, \cdot)]$ deteriorates at $i=8$. # Statements and Declarations {#statements-and-declarations .unnumbered} #### Conflict of interest The authors declare no competing interests. # Acknowledgments {#acknowledgments .unnumbered} J.G. is grateful for financial support by the German Science Foundation (DFG) via grant TRR 154 (*Mathematical modelling, simulation and optimization using the example of gas networks*), project C05. The work of J.G. is also supported by the Graduate School CE within Computational Engineering at Technische Universität Darmstadt. K. Kwon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A4A1018190, 2021R1C1C1011867). This work began during the second author's visit to TU Darmstadt in the summer of 2022. The second author would like to express his gratitude to Prof. Min-Gi Lee and Prof. Philsu Kim from Kyungpook National University, Korea, for making the visit possible. [^1]: jan.giesselmann\@tu-darmstadt.de [^2]: Corresponding author; kwkwon\@knu.ac.kr

arxiv_math

--- abstract: | We investigate the problem of computing the operator norm of a matrix with respect to norms induced by linear operators. We show that the dual formulation of this problem can be expressed as a max-min problem which can, for some specific cases, be solved in polynomial time. In other instances, we show that the problem is approximable. Along the way, we develop the concept of push-forward and pull-back of seminorms, and deduce new duality results when optimizing over the unit ball of various norms. author: - "Adrian Kulmburg[^1]" bibliography: - references.bib title: "The Generalized Matrix Norm Problem[^2]" --- operator norm, push-forward norm, pull-back norm, approximation 15A60, 65F35, 68Q25 # Introduction One way to gauge the effect of a linear operator $L : \mathcal{V}\rightarrow \mathcal{W}$ between normed vector spaces $(\mathcal{V}, \norm_\mathcal{V})$ and $(\mathcal{W}, \norm_\mathcal{W})$ is to compute the operator norm $\norm{L}$, which is defined as $$\label{eq:basic_problem} \norm{L} := \max_{\norm{v}_{\mathcal{V}}\leq 1}\norm{L(v)}_{\mathcal{W}}.$$ Computing this quantity has a wide range of applications, depending on the norms chosen on $\mathcal{V}$ and $\mathcal{W}$. One particular class of norms, for which this problem has extensively been studied (see for example [@matrix_p_norms; @bhattiprolu_approximating_2018; @grothendieck_resume_1956; @nesterov_semidefinite_1998; @steinberg_computation_2005]) are the $p$-norms, i.e., $\norm{\vec{x}}_p = \sqrt[p]{|x_1|^p+...+|x_n|^p}$ for $p\in[1,\infty)$, and $\norm{\vec{x}}_{\infty} = \max_i|x_i|$ for $p=\infty$. In that case, computing [\[eq:basic_problem\]](#eq:basic_problem){reference-type="eqref" reference="eq:basic_problem"} can be formulated as $$\label{eq:matrix_problem} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto q} := \max_{\norm{\vec{v}}_p\leq 1} \norm{\bm{\underline{\smash{A}}}\vec{v}}_q,$$ where $\bm{\underline{\smash{A}}} \in \mathbb{R}^{n\times m}$ is some matrix representing a linear operator $\mathbb{R}^{m}\rightarrow \mathbb{R}^{n}$, and $p,q\in[1,\infty]$. Computing the exact value of $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q}$ is known to be NP-hard in many instances, except for example in the cases $p=q=2$, and $p = 1$ or $q = \infty$. Even worse, this problem is often not approximable in polynomial time within arbitrary precision (see [@bhattiprolu_2023] and [@matrix_p_norms]), although constant-ratio approximation algorithms sometimes do exist (see for instance [@bhattiprolu_approximating_2018] and [@steinberg_computation_2005]). However, $p$-norms possess a lot of symmetries, and one might be interested in investigating more complex norms. A simple generalization would be to consider norms of the form $$\vec{v} \mapsto \norm{\bm{\underline{\smash{M}}}\vec{v}}_p,$$ where $\bm{\underline{\smash{M}}}$ is a matrix, and $p\in[1,\infty]$. As we will see, this is a norm if $\bm{\underline{\smash{M}}}$ represents an injective operator, and so a generalization of [\[eq:matrix_problem\]](#eq:matrix_problem){reference-type="eqref" reference="eq:matrix_problem"} would be $$\label{eq:matrix_problem_generalized} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}},\bm{\underline{\smash{C}}}} := \max_{\norm{\bm{\underline{\smash{B}}}\vec{v}}_p\leq 1} \norm{\bm{\underline{\smash{C}}}\,\bm{\underline{\smash{A}}}\vec{v}}_q,$$ for matrices $\bm{\underline{\smash{B}}}$ and $\bm{\underline{\smash{C}}}$. Of course, computing expressions of the form [\[eq:matrix_problem_generalized\]](#eq:matrix_problem_generalized){reference-type="eqref" reference="eq:matrix_problem_generalized"} is as hard as computing $$\label{eq:matrix_problem_calm_down} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}} := \max_{\norm{\bm{\underline{\smash{B}}}\vec{v}}_p\leq 1} \norm{\bm{\underline{\smash{A}}}\vec{v}}_q,$$ since there holds $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}},\bm{\underline{\smash{C}}}} = \norm{\bm{\underline{\smash{C}}}\,\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}}$. Therefore, for our analysis, we can focus on problems of the form [\[eq:matrix_problem_calm_down\]](#eq:matrix_problem_calm_down){reference-type="eqref" reference="eq:matrix_problem_calm_down"}. One key aspect used in [@bhattiprolu_approximating_2018; @steinberg_computation_2005] to analyze the approximability of $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q}$ is the fact that $$\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q} = \norm{\bm{\underline{\smash{A}}}^\top}_{q^*\mapsto p^*},$$ where $p^*$ and $q^*$ are the Hölder conjugates of $p$ and $q$, respectively. This allows one to focus the analysis on cases where $1/p + 1/q \geq 1$. However, such a symmetry does not exist for $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q; \bm{\underline{\smash{B}}}}$, since the dual of the function $\vec{v} \mapsto \norm{\bm{\underline{\smash{B}}}\vec{v}}_p$ can not, in general, be expressed in terms of the $p^*$-norm alone. Instead, we will see in Section [\[sec:push-forward_pull-back\]](#sec:push-forward_pull-back){reference-type="ref" reference="sec:push-forward_pull-back"} that, provided that $\bm{\underline{\smash{B}}}$ represents an injective operator, the dual of $\norm{\bm{\underline{\smash{B}}}\vec{v}}_p$ is the function $$\vec{w} \mapsto \min_{\bm{\underline{\smash{B}}}^\top\vec{\alpha} = \vec{w}}\norm{\vec{\alpha}}_{p^*},$$ and we will use this to show that $$\label{eq:intro_equiv_duality} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}} = \max_{\norm{\vec{\alpha}}_{q^*}\leq 1} \min_{\bm{\underline{\smash{B}}}^\top \vec{\beta} = \bm{\underline{\smash{A}}}^\top \vec{\alpha}} \nnorm{\vec{\beta}}.$$ Analyzing the approximability of expressions of the form $$\max_{\norm{\vec{\alpha}}_{q^*}\leq 1} \; \min_{\bm{\underline{\smash{B}}}^\top \vec{\beta} = \bm{\underline{\smash{A}}}^\top \vec{\alpha} + \vec{c}}\nnorm{\vec{\beta}}_{p^*}$$ has applications on its own, as it is equivalent to solving an ellipsotope containment problem, as shown in [@Kulmburg2021]. The outline of this paper is given as follows: In Section [\[sec:prelims\]](#sec:prelims){reference-type="ref" reference="sec:prelims"} we introduce the notation we will use, as well as some core concepts such as extended seminorms. In Section [\[sec:push-forward_pull-back\]](#sec:push-forward_pull-back){reference-type="ref" reference="sec:push-forward_pull-back"} we will present the new concepts of push-forward and pull-back of a norm, and show that these two notions are dual to each other. We will then turn our attention to the generalized matrix norm problem, by discussing instances where $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}}$ can be computed exactly in polynomial time in Section [\[sec:tractable\]](#sec:tractable){reference-type="ref" reference="sec:tractable"}, whereas in Section [\[sec:approximable\]](#sec:approximable){reference-type="ref" reference="sec:approximable"} we will analyze instances where the problem can only be approximated. Finally, in Section [\[sec:numerical_evaluations\]](#sec:numerical_evaluations){reference-type="ref" reference="sec:numerical_evaluations"} we will verify the accuracy of some of the approximations discovered in Section [\[sec:approximable\]](#sec:approximable){reference-type="ref" reference="sec:approximable"}. # Preliminaries [\[sec:prelims\]]{#sec:prelims label="sec:prelims"} ## Basic Notation A letter with an arrow (e.g., $\vec{v}$) represents a **vector** in $\mathbb{R}^n$, while **matrices** in $\mathbb{R}^{n \times m}$ are denoted by bold, underlined letters (e.g., $\bm{\underline{\smash{M}}}$). The vectors $\vec{e}_i \in \mathbb{R}^n$ for $i=1,...,n$ will denote the **canonical basis vectors** of $\mathbb{R}^n$. For a vector $\vec{v}\in \mathbb{R}^n$, $v_i$ for $i=1,...,n$ refers to the coordinates of $\vec{v}$ with respect to the canonical basis. Similarly, $\bm{\underline{\smash{E}}}_{ij}$ for $i=1,...,n$, $j=1,...,m$ and $n, m\in \mathbb{N}$ denote the **canonical basis matrices** of $\mathbb{R}^{n\times m}$, and for a matrix $\bm{\underline{\smash{M}}}$ the values $M_{ij}$ denote the coordinates of $\bm{\underline{\smash{M}}}$ with respect to this canonical basis. The notation $\bm{\underline{\smash{M}}} \succeq 0$ means that $\bm{\underline{\smash{M}}}$ is positive semi-definite, $\bm{\underline{\smash{M}}}^+$ will denote the Moore-Penrose pseudo-inverse of $\bm{\underline{\smash{M}}}$, and $\mathop{\mathrm{Tr}}(M)$ is the trace of $\bm{\underline{\smash{M}}}$. The matrix $\bm{\underline{\smash{I}}}_n$ will refer to the $n$-dimensional identity matrix, while $\bm{\underline{\smash{0}}}_{n\times m}$ is the $n\times m$-matrix filled with zeros and $\vec{1}_n$ is the $n$-dimensional vector with only ones, $\vec{0}_n$ the $n$-dimensional vector with only zeros. For a vector $\vec{v}\in\mathbb{R}^n$, $\mathop{\mathrm{Diag}}(\vec{v})$ is the diagonal matrix that has the entries of $\vec{v}$ as diagonal elements. On the other hand, for a matrix $\bm{\underline{\smash{M}}}$, $\mathop{\mathrm{diag}}(\bm{\underline{\smash{M}}})$ is the vector corresponding to the diagonal entries of $\bm{\underline{\smash{M}}}$. By a slight abuse of notation, we call a matrix $\bm{\underline{\smash{A}}}$ **injective**/**surjective**/**bijective** if the corresponding linear map is injective/surjective/bijective. For a vector $\vec{v}$ and $p\in[1,\infty]$, $\norm{\vec{v}}_p := \sqrt[p]{|v_1|^p+...+|v_n|^p}$ is the $\bm{p}$**-norm** of $\vec{v}$ (for $p=\infty$, $\norm{\vec{v}}_{\infty} := \max_i |v_i|$). For vectors $\vec{v},\vec{w}\in\mathbb{R}^n$, we denote their (Euclidean) **inner product** as $\vec{v}^\top\vec{w} = \iprod{\vec{v}}{\vec{w}} = v_1w_1+...+v_nw_n$. The (closed) **unit ball** of a normed space $(\mathcal{V}, \norm_{\mathcal{V}})$ will be written as $\mathcal{B}_{\mathcal{V}} := \left\{\vec{x}\;\middle|\;\norm{\vec{x}}_{\mathcal{V}}\leq 1\right\}$. For a random variable $\mathsf{x}\in\Omega$ and a function $f:\Omega \rightarrow \mathbb{R}$, $\mathbb{E}_\mathsf{x}[f(\mathsf{x})]$ will denote the expectation value of $f(\mathsf{x})$ with respect to $\mathsf{x}$. The notation $\mathsf{g}\sim \mathcal{N}(\mu, \sigma)$ will express that $\mathsf{g}$ is a Gaussian random variable with mean $\mu$ and standard deviation $\sigma$. ## Extended Seminorms The definition of a seminorm is well documented in the mathematical literature (see for example [@kubrusly_elements_2011 p. 200]). In contrast, what is less well documented is the concept of *extended* seminorm[^3]: Let $\mathcal{V}$ be a real vector space. A function $\eta : \mathcal{V}\rightarrow [0,\infty]$ is called an *extended seminorm*, if it satisfies the following two conditions: - (Triangle inequality) $$\eta(v+w) \leq \eta(v) + \eta(w), \quad \forall v,w\in \mathcal{V}.$$ - (Absolute homogeneity) $$\eta(cv) = |c|\eta(v), \quad \forall c\in\mathbb{R}\backslash\{0\}, v\in\mathcal{V}.$$ If, in addition, $\eta$ satisfies the condition $$\forall v \in \mathcal{V}, \quad\eta(v) = 0 \;\Rightarrow \; v = 0,$$ then $\eta$ is an *extended norm*. The major difference is that extended (semi)norms may admit a value of $\infty$, which requires slightly modifying the absolute homogeneity requirement to avoid issues arising when multiplying $\infty$ with $0$. ## Duality Let $\mathcal{V}$ be a (real) vector space. We denote by $\mathcal{V}^*$ its **dual vector space**, i.e., $\mathcal{V}^* = \left\{f: \mathcal{V}\rightarrow \mathbb{R}\;\middle|\;f \text{ is linear}\right\}$. If $\norm$ is a norm on $\mathcal{V}$, we denote by $\norm^*$ its **dual norm**, which is defined for $f\in \mathcal{V}^*$ as $$\label{eq:dual_norm_def} \norm{f}^* = \sup_{\norm{\vec{x}} \leq 1} f(\vec{x}).$$ If $\norm$ is only a seminorm, the dual $\norm^*$ can still be defined via [\[eq:dual_norm_def\]](#eq:dual_norm_def){reference-type="ref" reference="eq:dual_norm_def"}, but $\norm^*$ will then be an extended seminorm. For sets $S\subseteq \mathcal{V}$, the **dual set**[^4] is defined as $$S^* := \left\{ f \in V^* \; \middle| \; \sup_{\vec{x} \in S} f(x) \leq 1 \right\}.$$ According to the Riesz representation theorem, for Hilbert spaces such as $\mathbb{R}^n$ there exists an isometric isomorphism between $\mathbb{R}^n$ and $(\mathbb{R}^n)^*$. Consequently, for the sake of simplicity, we see sets $S^* \subseteq (\mathbb{R}^n)^*$ as sets in $\mathbb{R}^n$, i.e., we may write that $S^* \subseteq \mathbb{R}^n$. Similarly, we may see norms on $(\mathbb{R}^n)^*$ as norms on $\mathbb{R}^n$. For the vector $p$-norms, it is a common exercise to prove that $\|\vec{x}\|_{p}^* = \|\vec{x}\|_{p^*}$, where $p^*$ is the Hölder-conjugate of $p$, i.e., $\frac{1}{p} + \frac{1}{p^*} = 1$. ## Matrix Norms A matrix in $\mathbb{R}^{n\times m}$ can be seen as a vector in $\mathbb{R}^{nm}$, or as a linear operator from $\mathbb{R}^m$ to $\mathbb{R}^n$. Each of these representations will lead to a different way of computing the norm of a matrix. The interplay between all these norms will be an important aspect in the next sections, which is why we define these norms properly now. ### Entry-Wise Matrix Norms Based on the vector $p$-norms, we define the $L_{p,q}$- and $L_{p,q}^\top$-norms for matrices: [\[def:matrix_L\_p_q\_norms\]]{#def:matrix_L_p_q_norms label="def:matrix_L_p_q_norms"} For a matrix $\bm{\underline{\smash{A}}}\in\mathbb{R}^{n\times m}$ and numbers $p,q\in[1,\infty]$, the entry-wise $L_{p,q}$-norm is $$\label{eq:L_pq_def} \norm{\bm{\underline{\smash{A}}}}_{L_{p,q}} = \left(\sum_{j=1}^m \left( \sum_{i=1}^n |A_{ij}|^p\right)^{q/p}\right)^{1/q}.$$ For $p=\infty$ or $q=\infty$, the corresponding sum has to be replaced by a maximum. If the columns of $\bm{\underline{\smash{A}}}$ are denoted by $\vec{a}_i$ for $1\leq i \leq m$, the $L_{p,q}$-norm can also be written as follows: $$\norm{\bm{\underline{\smash{A}}}}_{L_{p,q}} = \norm{ \begin{pmatrix}\norm{\vec{a}_1}_p & \cdots & \norm{\vec{a}_m}_p\end{pmatrix}^\top }_q.$$ The transposed $L_{p,q}$-norms, or $L_{p,q}^\top$-norms, are defined as $$\norm{\bm{\underline{\smash{A}}}}_{L_{p,q}^\top} = \nnorm{\bm{\underline{\smash{A}}}^\top}_{L_{p,q}}.$$ For the $L_{p,q}$- and $L_{p,q}^\top$-norms, deducing the dual norm can be done similarly to the dual of the vector $p$-norms: [\[lmm:dual_L\_p_q\]]{#lmm:dual_L_p_q label="lmm:dual_L_p_q"} Let $\bm{\underline{\smash{X}}}\in\mathbb{R}^{n\times m}$. For $p, q\in[1,\infty]$, let $p^*,q^*$ be the Hölder conjugates of $p, q$, respectively. Then $$\|\bm{\underline{\smash{X}}}\|_{L_{p,q}}^* = \|\bm{\underline{\smash{X}}}\|_{L_{p^*,q^*}}.$$ Similarly, for $L_{p,q}^\top$-norms, we have $$\|\bm{\underline{\smash{X}}}\|_{L_{p,q}^\top}^* = \|\bm{\underline{\smash{X}}}\|_{L_{p^*,q^*}^\top}.$$ The proof is left as an easy exercise to the reader. ### Operator Norms Since a matrix can also be seen as a linear operator, one can define the ${p\mapsto q\text{-norms}}$, which are based on the definition of the operator norm: [\[def:matrix_p\_q_norms\]]{#def:matrix_p_q_norms label="def:matrix_p_q_norms"} For a matrix $\bm{\underline{\smash{A}}}\in\mathbb{R}^{n\times m}$ and numbers $p,q\in[1,\infty]$, the operator $p\mapsto q$-norm of $\bm{\underline{\smash{A}}}$ is $$\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q} = \sup_{\norm{\vec{v}}_p\leq1} \norm{\bm{\underline{\smash{A}}}\vec{v}}_q.$$ If $p=q$, we will use $\norm{\bm{\underline{\smash{A}}}}_p$ as a shorthand for $\norm{\bm{\underline{\smash{A}}}}_{p \mapsto p}$. # Push-Forward and Pull-Back Seminorms [\[sec:push-forward_pull-back\]]{#sec:push-forward_pull-back label="sec:push-forward_pull-back"} Now that we have introduced the main norms we will work with, we need to investigate how to deal with those norms in the context of optimization. Specifically, we want to examine problems of the kind $$\min_{L(x) = b}\norm{x},$$ where $\norm$ is some norm, and $L$ some linear map. To analyze this sort of expression, defining the concepts of push-back and pull-forward of seminorms is fruitful. ## Basic Definitions The idea of push-forward or pull-back of a seminorm can be defined similarly as for other areas of mathematics: [\[def:push-foward_pull-back\]]{#def:push-foward_pull-back label="def:push-foward_pull-back"} Let $(\mathcal{V},\norm_\mathcal{V})$ and $(\mathcal{W},\norm_\mathcal{W})$ be normed vector spaces and let $L: \mathcal{V}\rightarrow \mathcal{W}$ be a continuous linear operator. The *push-forward* seminorm on $\mathcal{W}$ induced by $L$ is the extended seminorm $$\label{eq:push-forward} \norm{w}_{L\uparrow\mathcal{V}} := \min_{L(v) = w} \norm{v}_\mathcal{V}.$$ We will denote the unit ball of $\norm_{L\uparrow\mathcal{V}}$ as $\mathcal{B}_{L\uparrow\mathcal{V}}$, and the Banach space $\mathcal{W}$ endowed with that norm as $L\uparrow \mathcal{V}$. On the other hand, the *pull-back* seminorm on $\mathcal{V}$ induced by $L$ is the seminorm $$\label{eq:pull-back} \norm{v}_{L\downarrow\mathcal{W}} := \norm{L(v)}_\mathcal{W}.$$ We will denote the unit ball of $\norm_{L\downarrow\mathcal{W}}$ as $\mathcal{B}_{L\downarrow\mathcal{W}}$, and the Banach space $\mathcal{V}$ endowed with that norm as $L\downarrow \mathcal{W}$. Let us first mention that [\[eq:push-forward\]](#eq:push-forward){reference-type="eqref" reference="eq:push-forward"} is well-defined, i.e., we can take the minimum instead of the infimum: For any $w$, the set $\left\{v\in \mathcal{V}\;\middle|\;L(v) = w\right\}$ is closed, as $L$ is continuous and $\{w\}$ is closed. Consequently, any sequence of elements $v_i\in \left\{v\in \mathcal{V}\;\middle|\;L(v) = w\right\}$ whose norm would converge to the infimum for $i\rightarrow \infty$ converges inside $\left\{v\in \mathcal{V}\;\middle|\;L(v) = w\right\}$, and this limit point is then the minimizer. If there are no solutions such that $L(v) = w$, we are taking the infimum over an empty set, which is equal to $\infty$ by convention. We use the convention that $\infty$ can be a possible value for an extended seminorm, so even in that case, the minimum is reached. The fact that both the push-forward and the pull-back are (extended) seminorms will be covered in the next Lemma: For arbitrary normed vector spaces $(\mathcal{V}, \norm_\mathcal{V})$ and $(\mathcal{W}, \norm_\mathcal{W})$ and a bounded linear transformation $L:\mathcal{V}\rightarrow \mathcal{W}$, the push-forward is an extended seminorm, and the pull-back is a seminorm. Moreover, the push-forward $\norm_{L\uparrow\mathcal{V}}$ is a norm if and only if $L$ is surjective. The pull-back $\norm_{L\downarrow\mathcal{W}}$ is a norm if and only if $L$ is injective. *Proof.* Both $\norm_{L\uparrow\mathcal{V}}$ and $\norm_{L\downarrow\mathcal{W}}$ can easily be seen to be absolutely homogeneous. Using the linearity of $L$ and the triangle inequality for $\norm_\mathcal{W}$, the triangle inequality for the pull-back follows. For the push-forward, we need more refined arguments: If for $w_1, w_2 \in \mathcal{W}$ either $\norm{w_1}_{L\uparrow\mathcal{V}} = \infty$ or $\norm{w_2}_{L\uparrow\mathcal{V}} = \infty$, or both, then the triangle inequality trivially holds. Therefore, assume without loss of generality that $\norm{w_1}_{L\uparrow\mathcal{V}} \neq \infty \neq \norm{w_2}_{L\uparrow\mathcal{V}}$. Then there exist $v_1$ and $v_2$ such that $L(v_i) = w_i$ and $\norm{w_i}_{L\uparrow\mathcal{V}} = \norm{v_i}_\mathcal{V}$ for $i=1,2$. Clearly, $L(v_1+v_2) = w_1+w_2$ by linearity of $L$. As a consequence $$\begin{aligned} \norm{w_1+w_2}_{L\uparrow\mathcal{V}} &= \min_{L(v) = w_1 + w_2} \norm{v}_\mathcal{V} \leq \norm{v_1+v_2}_\mathcal{V} \leq \norm{w_1}_{L\uparrow\mathcal{V}} + \norm{w_2}_{L\uparrow\mathcal{V}} \end{aligned}$$ which proves the triangle inequality for the push-forward. Now that we have proven that the push-forward and the pull-back are seminorms, we prove the second part of the Lemma, describing when they are norms. The remaining property that we need to prove for a seminorm $\norm$ to be an actual norm is to show that it is point-separating, i.e., $\norm{x}=0 \Rightarrow x = 0$. For the pull-back, it is easy to see that this happens if and only if $\text{Ker}(L)=\{0\}$, which is equivalent to $L$ being injective. For the push-forward, note that $\norm_{L\uparrow\mathcal{V}}$ is a seminorm (i.e., it does not output $\infty$) if and only if $L(v) = w$ has at least one solution $v$ for any $w$, which is equivalent to $L$ being surjective. Now, suppose that some $w\in \mathcal{W}$ satisfies $\norm{w}_{L\uparrow\mathcal{V}} = 0$. Then, there exists some $v \in \mathcal{V}$, such that $L(v) = w$ and $\norm{v}_\mathcal{V}= 0$. Since $\norm_\mathcal{V}$ is a norm, it is point-separating, whence $v = 0$ must hold, and thus $w = L(v) = L(0) = 0$ since $L$ is linear. ◻ ## The Dual of the Push-Forward and Pull-Back We now introduce a simple Theorem that will be fundamental for us: [\[thm:push-forward_pull-back_dual\]]{#thm:push-forward_pull-back_dual label="thm:push-forward_pull-back_dual"} Let $(\mathcal{H}_1, \iprod_{\mathcal{H}_1})$ and $(\mathcal{H}_2, \iprod_{\mathcal{H}_2})$ be real Hilbert spaces, and $L : \mathcal{H}_1 \rightarrow \mathcal{H}_2$ a bounded linear operator that is surjective. Let $\norm$ be an arbitrary norm on $\mathcal{H}_1$, turning it into a Banach space that we shall denote as $\mathcal{H}$. Let $\mathcal{H}^*$ denote the dual space of $\mathcal{H}$, endowed with the norm $\norm^*$. Then, for any $y\in\mathcal{H}_2$ there holds $$\label{eq:push-forward_pull-back_dual} (\|y\|_{L\uparrow\mathcal{H}})^* = \|y\|_{L^\top\downarrow\mathcal{H}^*},$$ where $L^\top$ denotes the adjoint operator of $L$. *Proof.* For any $y\in\mathcal{H}_2$, we have $$(\|y\|_{L\uparrow\mathcal{H}})^* =\sup_{\substack{z\in\mathcal{H}_2\\z\neq0}}\frac{\langle y, z\rangle_{\mathcal{H}_2}}{\|z\|_{L\uparrow\mathcal{H}}} =\sup_{\substack{z\in\mathcal{H}_2\\z\neq0}}\max_{L(x)=z}\frac{\langle y, z\rangle_{\mathcal{H}_2}}{\|x\|} =\sup_{\substack{z\in\mathcal{H}_2\\L(x)=z\\z\neq0}}\frac{\langle y, L(x)\rangle_{\mathcal{H}_2}}{\|x\|} .$$ Since $L$ is assumed to be surjective, the supremum can be taken over all $x\in \mathcal{H}_1$, since any $z\in\mathcal{H}_2$ is the image of some $x\in\mathcal{H}_1$ under $L$. Therefore we may write that $$\label{eq:detour} \sup_{\substack{z\in\mathcal{H}_2\\L(x)=z\\z\neq0}}\frac{\langle y, L(x)\rangle_{\mathcal{H}_1}}{\|x\|} = \sup_{\substack{x\in\mathcal{H}_1\\L(x)\neq0}}\frac{\langle y, L(x)\rangle_{\mathcal{H}_1}}{\|x\|} = \sup_{\substack{x\in\mathcal{H}_1\\x\neq0}}\frac{\langle y, L(x)\rangle_{\mathcal{H}_1}}{\|x\|}.$$ The last equality follows from the fact that, unless $y = 0$, there holds $\langle y, L(x)\rangle_{\mathcal{H}_1} > 0$, since $(\|y\|_{L\uparrow\mathcal{H}})^*$ is a norm with respect to $y$, and can thus be zero if and only if $y=0$. This means that either $y=0$, in which case Equation [\[eq:detour\]](#eq:detour){reference-type="ref" reference="eq:detour"} trivially holds, or $y\neq 0$ in which case there must exist some $x\neq 0$ such that $\langle y, L(x)\rangle_{\mathcal{H}_1} > 0$, which can happen if and only if $L(x) \neq 0$ We thus conclude that $$(\|y\|_{L\uparrow\|\cdot\|})^* =\sup_{\substack{x\in\mathcal{H}_1\\x\neq0}}\frac{\langle y, L(x)\rangle_{\mathcal{H}_1}}{\|x\|} =\sup_{\|x\|\leq 1}\langle L^\top(y), x\rangle_{\mathcal{H}_1} =\|L^\top(y)\|^* =\|y\|_{L^\top\downarrow\mathcal{H}^*}$$ ◻ An immediate consequence is the following Corollary, which is a generalization of [@boyd_convex_2004 Eq. (5.12), p. 221-222]: [\[cor:constrained_norm_optimization\]]{#cor:constrained_norm_optimization label="cor:constrained_norm_optimization"} Let $(\mathcal{H}_1, \iprod_{\mathcal{H}_1})$ and $(\mathcal{H}_2, \iprod_{\mathcal{H}_2})$ be Hilbert spaces, $L : \mathcal{H}_1 \rightarrow \mathcal{H}_2$ a bounded linear operator, $b\in \mathcal{H}_2$ a vector, and $\norm$ a norm on $\mathcal{H}_1$. Furthermore, assume that $L$ is surjective. Then, there holds $$\label{eq:constrained_norm_optimization} \min_{L(x)=b} \norm{x} = \max_{\norm{L^\top(y)}^* \leq 1} \iprod{b}{y}_{\mathcal{H}_2}.$$ *Proof.* Let $\mathcal{H}$ denote the space $\mathcal{H}_1$ endowed with the norm $\norm$, and $\mathcal{H}^*$ its dual endowed with the norm $\norm^*$. Using the push-forward and pull-back, we can reformulate [\[eq:constrained_norm_optimization\]](#eq:constrained_norm_optimization){reference-type="eqref" reference="eq:constrained_norm_optimization"} to $$\label{eq:cno_first_formulation} \norm{b}_{L\uparrow \mathcal{H}} = \left(\norm{b}_{L^\top\downarrow \mathcal{H}^*}\right)^*.$$ The bidual of a norm on a Hilbert space coincides (up to an isometric isomorphism) with the original norm, so applying the dual on both sides of [\[eq:cno_second_formulation\]](#eq:cno_second_formulation){reference-type="eqref" reference="eq:cno_second_formulation"} yields $$\label{eq:cno_second_formulation} \left(\norm{b}_{L\uparrow \mathcal{H}}\right)^* = \norm{b}_{L^\top \mathcal{H}^*},$$ which is precisely the statement of Theorem [\[thm:push-forward_pull-back_dual\]](#thm:push-forward_pull-back_dual){reference-type="ref" reference="thm:push-forward_pull-back_dual"}. This concludes the proof since [\[eq:cno_first_formulation\]](#eq:cno_first_formulation){reference-type="eqref" reference="eq:cno_first_formulation"} and [\[eq:cno_second_formulation\]](#eq:cno_second_formulation){reference-type="eqref" reference="eq:cno_second_formulation"} are equivalent. ◻ Using Corollary [\[cor:constrained_norm_optimization\]](#cor:constrained_norm_optimization){reference-type="ref" reference="cor:constrained_norm_optimization"}, we can now prove the dual formulation of the generalized matrix norm problem: [\[cor:matrix_norm_and_optimization\]]{#cor:matrix_norm_and_optimization label="cor:matrix_norm_and_optimization"} Let $\bm{\underline{\smash{A}}} \in \mathbb{R}^{\ell\times n}$ and $\bm{\underline{\smash{B}}} \in \mathbb{R}^{m\times n}$ be two matrices, suppose that $\bm{\underline{\smash{B}}}$ is injective, and let $\norm_{\mathcal{V}}$ be a norm on $\mathbb{R}^{\ell}$ and $\norm_{\mathcal{W}}$ a norm on $\mathbb{R}^m$. Furthermore, let $\vec{c} \in \mathbb{R}^n$ be a vector. Then $$\label{eq:matrix_norm_and_optimization} \max_{\norm{\bm{\underline{\smash{B}}}\vec{x}}_{\mathcal{W}}\leq 1}\norm{\bm{\underline{\smash{A}}}\vec{x}}_{\mathcal{V}}+\vec{c}^\top\vec{x} = \max_{\norm{\vec{\alpha}}_{\mathcal{V}}^*\leq 1}\min_{\bm{\underline{\smash{B}}}^\top\vec{\beta} = \bm{\underline{\smash{A}}}^\top\vec{\alpha} + \vec{c}}\nnorm{\vec{\beta}}_{\mathcal{W}}^*.$$ In particular, for $\vec{c} = \vec{0}$, there holds $$\label{eq:matrix_norm_and_optimization_c=0} \max_{\norm{\bm{\underline{\smash{B}}}\vec{x}}_{\mathcal{W}}\leq 1}\norm{\bm{\underline{\smash{A}}}\vec{x}}_{\mathcal{V}} = \max_{\norm{\vec{\alpha}}_{\mathcal{V}}^*\leq 1}\min_{\bm{\underline{\smash{B}}}^\top\vec{\beta} = \bm{\underline{\smash{A}}}^\top\vec{\alpha}}\nnorm{\vec{\beta}}_{\mathcal{W}}^*.$$ *Proof.* By duality, we have that $$\begin{aligned} \max_{\norm{\bm{\underline{\smash{B}}}\vec{x}}_{\mathcal{W}}\leq 1}\norm{\bm{\underline{\smash{A}}}\vec{x}}_{\mathcal{V}}+\vec{c}^\top\vec{x} &= \max_{\norm{\bm{\underline{\smash{B}}}\vec{x}}_{\mathcal{W}}\leq 1}\max_{\norm{\vec{\alpha}}_{\mathcal{V}}^*\leq 1}\vec{\alpha}^\top\bm{\underline{\smash{A}}}\vec{x}+\vec{c}^\top\vec{x}\\ &= \max_{\norm{\vec{\alpha}}_{\mathcal{V}}^* \leq 1}\max_{\norm{\bm{\underline{\smash{B}}}\vec{x}}_{\mathcal{W}}\leq 1}\left(\vec{c}^\top + \vec{\alpha}^\top\bm{\underline{\smash{A}}}\right)\vec{x}. \end{aligned}$$ If $\bm{\underline{\smash{B}}}$ is injective, $\bm{\underline{\smash{B}}}^\top$ is surjective, therefore a simple application of Corollary [\[cor:constrained_norm_optimization\]](#cor:constrained_norm_optimization){reference-type="ref" reference="cor:constrained_norm_optimization"} yields [\[eq:matrix_norm_and_optimization\]](#eq:matrix_norm_and_optimization){reference-type="eqref" reference="eq:matrix_norm_and_optimization"}. ◻ # Tractable Matrix Norms [\[sec:tractable\]]{#sec:tractable label="sec:tractable"} We now begin to explore for which values of $p$ and $q$ the expression $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q; \bm{\underline{\smash{B}}}}$ can be computed exactly, or whether there exist approximations in the case where the problem is $\mathcal{NP}$-hard. Before we begin our actual analysis, we want to mention that we may, without loss of generality, assume that $\bm{\underline{\smash{B}}}$ is always injective. Indeed, if that is not the case, there holds $\text{Ker}(\bm{\underline{\smash{B}}}) \neq \{\vec{0}\}$, and so one of two things can happen: - If $\text{Ker}(\bm{\underline{\smash{B}}}) \subseteq \text{Ker}(\bm{\underline{\smash{A}}})$, let $k = \text{dim}(\text{Ker}(\bm{\underline{\smash{B}}}))$. Through elementary linear algebra (for example, by computing the singular value decompositions of $\bm{\underline{\smash{A}}}$ and $\bm{\underline{\smash{B}}}$), one can show that there exist invertible matrices $\bm{\underline{\smash{P}}} \in \mathbb{R}^{m\times m}$ and $\bm{\underline{\smash{Q}}} \in \mathbb{R}^{\ell \times \ell}$, as well as matrices $\bm{\underline{\smash{A}}}' \in \mathbb{R}^{(\ell-k)\times n}$ and $\bm{\underline{\smash{B}}}'\in \mathbb{R}^{(m-k)\times n}$ such that $$\bm{\underline{\smash{A}}} = \bm{\underline{\smash{Q}}}\begin{pmatrix} \bm{\underline{\smash{A}}}'\\ \bm{\underline{\smash{0}}}_{k\times n} \end{pmatrix} , \quad \bm{\underline{\smash{B}}} = \bm{\underline{\smash{P}}}\begin{pmatrix} \bm{\underline{\smash{B}}}'\\ \bm{\underline{\smash{0}}}_{k\times n} \end{pmatrix} .$$ By further separating $\bm{\underline{\smash{P}}}$ and $\bm{\underline{\smash{Q}}}$ into $\bm{\underline{\smash{P}}} = \begin{pmatrix}\bm{\underline{\smash{P}}}_1 & \bm{\underline{\smash{P}}}_2\end{pmatrix}$ and $\bm{\underline{\smash{Q}}} = \begin{pmatrix}\bm{\underline{\smash{Q}}}_1 & \bm{\underline{\smash{Q}}}_2\end{pmatrix}$, it easily follows that $$\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q; \bm{\underline{\smash{B}}}} = \norm{\bm{\underline{\smash{Q}}}_1\bm{\underline{\smash{A}}}'}_{p\mapsto q; \bm{\underline{\smash{P}}}_1\bm{\underline{\smash{B}}}'},$$ where now $\bm{\underline{\smash{P}}}_1\bm{\underline{\smash{B}}}'$ is injective. The matrices $\bm{\underline{\smash{A}}}', \bm{\underline{\smash{B}}}', \bm{\underline{\smash{P}}}, \bm{\underline{\smash{Q}}}$ can all be computed in polynomial time. - If $\text{Ker}(\bm{\underline{\smash{B}}}) \not\subseteq \text{Ker}(\bm{\underline{\smash{A}}})$, there exists a vector $\vec{v} \in \mathbb{R}^{n}$ such that $\bm{\underline{\smash{B}}}\vec{v} = \vec{0}$, but $\bm{\underline{\smash{A}}}\vec{v} \neq \vec{0}$. This means that for any $\lambda > 0$, we have $\norm{\bm{\underline{\smash{B}}}\lambda \vec{v}}_p = 0\leq 1$, but $\norm{\bm{\underline{\smash{A}}}\lambda \vec{v}}_q = \lambda \norm{\bm{\underline{\smash{A}}}\vec{v}}_q = \lambda c$ for some fixed constant $c > 0$. Letting $\lambda \rightarrow \infty$ then shows that $$\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q; \bm{\underline{\smash{B}}}} = \infty,$$ i.e., the problem is unbounded. We conclude that if $\bm{\underline{\smash{B}}}$ is not injective, either the problem is unbounded, or we can reduce the problem to the case where $\bm{\underline{\smash{B}}}$ is injective. ## The Case $p\in[1,\infty], q=\infty$ As we have discussed in Corollary [\[cor:matrix_norm_and_optimization\]](#cor:matrix_norm_and_optimization){reference-type="ref" reference="cor:matrix_norm_and_optimization"}, if $\bm{\underline{\smash{B}}}$ is injective, $\norm{\bm{\underline{\smash{A}}}}_{p,\infty;\bm{\underline{\smash{B}}}}$ can also be written as $$\label{eq:q=infty_reformulation_first} \norm{\bm{\underline{\smash{A}}}}_{p,\infty;\bm{\underline{\smash{B}}}} = \max_{\norm{\vec{\alpha}}_1\leq 1} \min_{\bm{\underline{\smash{B}}}^\top\vec{\beta} = \bm{\underline{\smash{A}}}^\top\vec{\alpha}}\nnorm{\vec{\beta}}_{\infty},$$ and the term $$\label{eq:reformulation_case_q=infty} \min_{\bm{\underline{\smash{B}}}^\top\vec{\beta} = \bm{\underline{\smash{A}}}^\top\vec{\alpha}}\nnorm{\vec{\beta}}_{\infty}$$ is a norm with respect to $\vec{\alpha}$, which means in particular that it is convex. By the Bauer maximum principle, the maximum in [\[eq:q=infty_reformulation_first\]](#eq:q=infty_reformulation_first){reference-type="eqref" reference="eq:q=infty_reformulation_first"} over $\vec{\alpha}$ is thus attained at $\vec{\alpha} = \pm \vec{e}_i$, for some $i=1,...,\ell$. Furthermore, the term [\[eq:reformulation_case_q=infty\]](#eq:reformulation_case_q=infty){reference-type="eqref" reference="eq:reformulation_case_q=infty"} can be evaluated in polynomial time, as it is the minimum of a convex function over a linear subspace (which is convex as well). Consequently, to compute $\norm{\bm{\underline{\smash{A}}}}_{p,\infty;\bm{\underline{\smash{B}}}}$, it suffices to compute [\[eq:reformulation_case_q=infty\]](#eq:reformulation_case_q=infty){reference-type="eqref" reference="eq:reformulation_case_q=infty"} for the $2\ell$ values $\vec{\alpha} = \pm \vec{e}_i$ (in fact, $\vec{\alpha} = \vec{e}_i$ is sufficient, as [\[eq:reformulation_case_q=infty\]](#eq:reformulation_case_q=infty){reference-type="eqref" reference="eq:reformulation_case_q=infty"} is a norm, and is thus symmetric), which can be done in polynomial time: $$\label{eq:final_q=infty} \norm{\bm{\underline{\smash{A}}}}_{p,\infty;\bm{\underline{\smash{B}}}} = \max_{i}\min_{\bm{\underline{\smash{B}}}^\top\vec{\beta} = \bm{\underline{\smash{A}}}^\top\vec{e}_i}\nnorm{\vec{\beta}}_{\infty}.$$ Using the convention that, for any function $f: D \rightarrow \mathbb{R}$, we have $\min_{x \in D} f(x) = \infty$ if $D = \emptyset$, we leave it to the reader to verify that [\[eq:final_q=infty\]](#eq:final_q=infty){reference-type="eqref" reference="eq:final_q=infty"} still holds even when $\bm{\underline{\smash{B}}}$ is not injective. We summarize our findings in the following theorem: Let $\ell, m, n\in\mathbb{N}$, and $\bm{\underline{\smash{A}}} \in \mathbb{R}^{\ell\times n}$, $\bm{\underline{\smash{B}}}\in \mathbb{R}^{m\times n}$. Then, for any $p\in[1,\infty]$, $$\max_{\norm{\bm{\underline{\smash{B}}}\vec{v}}_p\leq 1}\norm{\bm{\underline{\smash{A}}}\vec{v}}_{\infty} = \max_{i}\min_{\bm{\underline{\smash{B}}}^\top\vec{\beta} = \bm{\underline{\smash{A}}}^\top\vec{e}_i}\nnorm{\vec{\beta}}_{\infty},$$ which can be evaluated in polynomial time with respect to $\ell, m,$ and $n$. ## The Case $p = q = 2$ In general, when $p=2$, the generalized matrix norm problem can be simplified: [\[lmm:p=2\]]{#lmm:p=2 label="lmm:p=2"} Let $\ell, m, n\in\mathbb{N}$, and $\bm{\underline{\smash{A}}} \in \mathbb{R}^{\ell\times n}$, $\bm{\underline{\smash{B}}}\in \mathbb{R}^{m\times n}$. Then for any $q\in[1,\infty]$ $$\label{eq:p=2} \norm{\bm{\underline{\smash{A}}}}_{2\mapsto q;\bm{\underline{\smash{B}}}} = \norm{\bm{\underline{\smash{A}}}\bm{\underline{\smash{B}}}^+}_{2\mapsto q},$$ where $\bm{\underline{\smash{B}}}^+$ denotes the Moore-Penrose pseudoinverse of $\bm{\underline{\smash{B}}}$. *Proof.* We will only cover the case where $\bm{\underline{\smash{B}}}$ is injective. The general case is left as an easy exercise to the reader. Let $\bm{\underline{\smash{B}}} = \bm{\underline{\smash{U}}}\,\bm{\underline{\smash{\Sigma}}}\,\bm{\underline{\smash{V}}}$ be a singular value decomposition of $\bm{\underline{\smash{B}}}$. Since the 2-norm is invariant under orthogonal transformations, we may write $$\norm{\bm{\underline{\smash{A}}}}_{2\mapsto q;\bm{\underline{\smash{B}}}} = \max_{\norm{\bm{\underline{\smash{B}}}\vec{x}}_2\leq1}\norm{\bm{\underline{\smash{A}}}\vec{x}}_q = \max_{\norm{\bm{\underline{\smash{\Sigma}}}\vec{y}}_2\leq1}\norm{\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{V}}}\vec{y}}_q,$$ where we used the variable transformation $\vec{y} = \bm{\underline{\smash{V}}}^\top\vec{x}$. Since $\bm{\underline{\smash{B}}}$ is injective, $\bm{\underline{\smash{\Sigma}}}$ has the form $$\bm{\underline{\smash{\Sigma}}}_{\bm{\underline{\smash{B}}}} = \begin{pmatrix} \bm{\underline{\smash{D}}}\\\bm{\underline{\smash{0}}}_{(m-n)\times n}\end{pmatrix}$$ for some diagonal, invertible matrix $\bm{\underline{\smash{D}}} \in \mathbb{R}^{n\times n}$. The constraint $\nnorm{\bm{\underline{\smash{\Sigma}}}\vec{y}}_2\leq 1$ then simplifies to $\norm{\bm{\underline{\smash{D}}}\vec{y}}_2\leq 1$, and we have, using the variable transformation $\vec{z} = \bm{\underline{\smash{D}}}\vec{y}$: $$\norm{\bm{\underline{\smash{A}}}}_{2\mapsto q;\bm{\underline{\smash{B}}}} = \max_{\norm{\vec{z}}_2\leq 1}\norm{\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{V}}}\,\bm{\underline{\smash{D}}}^{-1}\vec{z}}_q %=\max_{\norm{\v{z}'}_2\leq 1}\norm{\mat{A}\,\mat{V}\begin{pmatrix} \mat{D}^{-1} & \mat{0}_{n\times (m-n)}\end{pmatrix}\v{z}'}_{q} =\max_{\norm{\vec{w}}_2\leq 1}\norm{\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{V}}}\begin{pmatrix} \bm{\underline{\smash{D}}}^{-1} & \bm{\underline{\smash{0}}}_{n\times (m-n)}\end{pmatrix}\bm{\underline{\smash{U}}}^\top\vec{w}}_{q},$$ where we have extended the vector $\vec{z}\in\mathbb{R}^{n}$ to a vector $\vec{z}'\in\mathbb{R}^{m}$, and used the variable transformation $\vec{w} = \bm{\underline{\smash{U}}}\vec{z}'$. According to [@noauthor_generalized_2003 p. 207, Corollary 1], the $\bm{\underline{\smash{B}}}^+$ coincides with $\bm{\underline{\smash{V}}}\begin{pmatrix} \bm{\underline{\smash{D}}}^{-1} & \bm{\underline{\smash{0}}}_{n\times (m-n)}\end{pmatrix}\bm{\underline{\smash{U}}}^\top$, which yields [\[eq:p=2\]](#eq:p=2){reference-type="eqref" reference="eq:p=2"}. ◻ We immediately conclude the following result for $p=2$ and $q=2$: Let $\ell, m, n\in\mathbb{N}$, and $\bm{\underline{\smash{A}}} \in \mathbb{R}^{\ell\times n}$, $\bm{\underline{\smash{B}}}\in \mathbb{R}^{m\times n}$. Then, if $\bm{\underline{\smash{B}}}$ is injective, or more generally if $\text{Ker}(\bm{\underline{\smash{B}}}) \subseteq \text{Ker}(\bm{\underline{\smash{A}}})$, there holds $$\max_{\norm{\bm{\underline{\smash{B}}}\vec{v}}_2\leq 1}\norm{\bm{\underline{\smash{A}}}\vec{v}}_2 = \norm{\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{B}}}^+}_{2\mapsto 2}.$$ Instead, if $\text{Ker}(\bm{\underline{\smash{B}}}) \not\subseteq \text{Ker}(\bm{\underline{\smash{A}}})$, there holds $$\max_{\norm{\bm{\underline{\smash{B}}}\vec{v}}_2\leq 1}\norm{\bm{\underline{\smash{A}}}\vec{v}}_2 = \infty.$$ # Approximable Matrix Norms [\[sec:approximable\]]{#sec:approximable label="sec:approximable"} We now turn towards cases where $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q}$ can not necessarily be computed exactly in polynomial time, but where approximations exist. We will present two main methods, one for the cases where $1 < q \leq 2 \leq p < \infty$, and the other for the cases where $q = 1$ and $p\in [1,\infty]$. We will also present a more efficient method for the case where $q=1$ and $p=2$. ## The Case $1 < q \leq 2 \leq p < \infty$ If $1 < q \leq 2 \leq p < \infty$, [@bhattiprolu_2023] showed that computing $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q}$ is $\mathcal{APX}$-hard, so it follows that evaluating $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}}$ is also $\mathcal{APX}$-hard. Consequently, the best we can hope for is to find an approximation. We shall need the following elementary Lemma: [\[lmm:E_ugamma\]]{#lmm:E_ugamma label="lmm:E_ugamma"} Let $\vec{\mathsf{g}}$ denote a standard Gaussian random variable on $\mathbb{R}^k$, $\vec{u} \in \mathbb{R}^k$ an arbitrary but fixed vector, and $r \in [0,\infty)$. Then $$\mathbb{E}_{\vec{\mathsf{g}}}\left[\absiprod{\vec{u}}{\vec{\mathsf{g}}}^r\right] = \gamma_r\norm{\vec{u}}_2^r,$$ where $\gamma_r$ denotes the $r$-th moment of a standard Gaussian random variable, i.e., $\gamma_r = \frac{2^{r/2}}{\sqrt{\pi}}\Gamma\left(\frac{r+1}{2}\right)$, where $\Gamma$ is the Euler Gamma function. The proof is left as an easy exercise for the reader. We will also need a method to transform a maximization problem into the computation of certain expectation values. We will achieve this by utilizing the following result from [@bhattiprolu_approximating_2018]: [\[lmm:vijay\]]{#lmm:vijay label="lmm:vijay"} Let $\Omega$ be a probability space, $\mathsf{x}\in \Omega$ a random variable, and $f_1 : \Omega \rightarrow \mathbb{R}$, $f_2 : \Omega \rightarrow (0,\infty)$ two functions. Then $$\sup_{\omega\in\Omega}\frac{f_1(\omega)}{f_2(\omega)} \geq \frac{\mathbb{E}_{\mathsf{x}}\left[f_1(\mathsf{x})\right]}{\mathbb{E}_{\mathsf{x}}\left[f_2(\mathsf{x})\right]}$$ A proof of Lemma [\[lmm:vijay\]](#lmm:vijay){reference-type="ref" reference="lmm:vijay"} can be found in [@bhattiprolu_approximating_2018 p. 10]. Finally, we need one last Lemma concerning the dual of certain semi-definite optimization problems: [\[lmm:semi-definite_dual\]]{#lmm:semi-definite_dual label="lmm:semi-definite_dual"} Let $\bm{\underline{\smash{M}}} \in \mathbb{R}^{n\times n}$, $\bm{\underline{\smash{Q}}} \in \mathbb{R}^{n\times \ell}$, and $\bm{\underline{\smash{R}}} \in \mathbb{R}^{n\times m}$. Furthermore, let $\vec{s} \in \mathbb{R}^\ell$ and $\vec{t} \in \mathbb{R}^m$ be non-negative vectors, i.e., $s_i\geq 0$ and $t_j \geq 0$ for all $i,j$. Then there holds $$\max_{\substack{\bm{\underline{\smash{Z}}}\succeq 0\\\mathop{\mathrm{diag}}(\bm{\underline{\smash{R}}}^\top\bm{\underline{\smash{Z}}}\,\bm{\underline{\smash{R}}}) = \vec{t}\\\mathop{\mathrm{diag}}(\bm{\underline{\smash{Q}}}^\top\bm{\underline{\smash{Z}}}\,\bm{\underline{\smash{Q}}}) = \vec{s}}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{M}}}\,\bm{\underline{\smash{Z}}}) = \min_{\vec{v}, \vec{w}} \left\{\vec{v}^\top\vec{t} + \vec{w}^\top\vec{s}\;\middle|\;\bm{\underline{\smash{R}}}\mathop{\mathrm{Diag}}(\vec{v})\bm{\underline{\smash{R}}}^\top + \bm{\underline{\smash{Q}}}\mathop{\mathrm{Diag}}(\vec{w})\bm{\underline{\smash{Q}}}^\top \preceq \bm{\underline{\smash{M}}}\right\}$$ *Proof.* Following the arguments from [@nesterov_semidefinite_2000 Lemma 13.2.2], we have that $$\begin{aligned} &\max_{\substack{\bm{\underline{\smash{Z}}}\succeq 0\\\mathop{\mathrm{diag}}(\bm{\underline{\smash{R}}}^\top\bm{\underline{\smash{Z}}}\,\bm{\underline{\smash{R}}})= \vec{t}\\\mathop{\mathrm{diag}}(\bm{\underline{\smash{Q}}}^\top\bm{\underline{\smash{Z}}}\,\bm{\underline{\smash{Q}}}) = \vec{s}}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{M}}}\,\bm{\underline{\smash{Z}}})\\ &\quad=\max_{\bm{\underline{\smash{Z}}}\succeq 0} \min_{\vec{v},\vec{w}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{M}}}\,\bm{\underline{\smash{Z}}}) + \vec{v}^\top(\vec{t} - \mathop{\mathrm{diag}}(\bm{\underline{\smash{R}}}^\top\bm{\underline{\smash{Z}}}\,\bm{\underline{\smash{R}}})) + \vec{w}^\top(\vec{s}-\mathop{\mathrm{diag}}(\bm{\underline{\smash{Q}}}^\top\bm{\underline{\smash{Z}}}\,\bm{\underline{\smash{Q}}}))\\ &\quad=\min_{\vec{v},\vec{w}} \max_{\bm{\underline{\smash{Z}}}\succeq 0} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{M}}}\,\bm{\underline{\smash{Z}}}) - \mathop{\mathrm{Tr}}(\bm{\underline{\smash{R}}}\mathop{\mathrm{Diag}}(\vec{v})\bm{\underline{\smash{R}}}^\top\bm{\underline{\smash{Z}}}) - \mathop{\mathrm{Tr}}(\bm{\underline{\smash{Q}}}\mathop{\mathrm{Diag}}(\vec{w})\bm{\underline{\smash{Q}}}^\top\bm{\underline{\smash{Z}}}) + \vec{v}^\top\vec{t} + \vec{w}^\top\vec{s}\\ &\quad=\min_{\vec{v},\vec{w}} \max_{\bm{\underline{\smash{Z}}}\succeq 0} \mathop{\mathrm{Tr}}\left(\left(\bm{\underline{\smash{M}}} - \bm{\underline{\smash{R}}}\mathop{\mathrm{Diag}}(\vec{v})\bm{\underline{\smash{R}}}^\top - \bm{\underline{\smash{Q}}}\mathop{\mathrm{Diag}}(\vec{w})\bm{\underline{\smash{Q}}}^\top\right)\bm{\underline{\smash{Z}}}\right) + \vec{v}^\top\vec{t} + \vec{w}^\top\vec{s}\\ &\quad=\min_{\vec{v}, \vec{w}} \left\{\vec{v}^\top\vec{t} + \vec{w}^\top\vec{s}\;\middle|\;\bm{\underline{\smash{R}}}\mathop{\mathrm{Diag}}(\vec{v})\bm{\underline{\smash{R}}}^\top + \bm{\underline{\smash{Q}}}\mathop{\mathrm{Diag}}(\vec{w})\bm{\underline{\smash{Q}}}^\top \preceq \bm{\underline{\smash{M}}}\right\}\\ \end{aligned}$$ ◻ We now have all the necessary tools to construct an approximation: [\[thm:main_q\<2\<p\]]{#thm:main_q<2<p label="thm:main_q<2<p"} Let $\ell, m, n\in\mathbb{N}$, and $\bm{\underline{\smash{A}}} \in \mathbb{R}^{\ell\times n}$, $\bm{\underline{\smash{B}}}\in \mathbb{R}^{m\times n}$. Then, for any $p,q \in [1,\infty]$ such that $q \leq 2 \leq p$, $$\label{eq:main_thm_q<2<p} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}} \leq \frac{1}{2}\min_{(\vec{v},\vec{w})\in K} \norm{\vec{v}}_{\frac{q^*}{2-q^*}}+\norm{\vec{w}}_{\frac{p}{2-p}} \leq \gamma_p^{1/p} \gamma_{q^*}^{1/q^*}\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}},$$ where $$\begin{gathered} K := \left\{(\vec{v},\vec{w}) \in \mathbb{R}^{\ell}\times \mathbb{R}^{m}\;\middle|\;\bm{\underline{\smash{P}}}_{\bm{\underline{\smash{V}}}}\mathop{\mathrm{Diag}}(\vec{v})\bm{\underline{\smash{P}}}_{\bm{\underline{\smash{V}}}}^\top + \bm{\underline{\smash{P}}}_{\bm{\underline{\smash{U}}}}\bm{\underline{\smash{B}}}^\top\mathop{\mathrm{Diag}}(\vec{w})\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{P}}}_{\bm{\underline{\smash{U}}}}^\top \preceq \begin{bmatrix}\bm{\underline{\smash{0}}} & \bm{\underline{\smash{A}}}\\\bm{\underline{\smash{A}}}^\top & \bm{\underline{\smash{0}}}\end{bmatrix}\right\},\\ \bm{\underline{\smash{P}}}_{\bm{\underline{\smash{V}}}} := \begin{bmatrix}\bm{\underline{\smash{I}}}_{\ell}\\\bm{\underline{\smash{0}}}_{n\times \ell}\end{bmatrix}, \qquad \bm{\underline{\smash{P}}}_{\bm{\underline{\smash{U}}}} := \begin{bmatrix}\bm{\underline{\smash{0}}}_{\ell\times n}\\\bm{\underline{\smash{I}}}_n\end{bmatrix}. \end{gathered}$$ and $\gamma_r = \frac{2^{r/2}}{\sqrt{\pi}}\Gamma\left(\frac{r+1}{2}\right)$ for $r\in[1,\infty)$. *Proof.* Before we prove [\[eq:main_thm_q\<2\<p\]](#eq:main_thm_q<2<p){reference-type="eqref" reference="eq:main_thm_q<2<p"}, we first need to show the following inequalities: $$\label{eq:main_thm_q<2<p_original} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}} \leq \max_{\substack{\bm{\underline{\smash{X}}} = \bm{\underline{\smash{U}}}^\top\bm{\underline{\smash{V}}}\\\norm{\bm{\underline{\smash{V}}}}_{L_{2,q^*}}\leq 1\\\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top}_{L_{2,p}^\top}\leq 1}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\bm{\underline{\smash{X}}}) \leq \gamma_p^{1/p}\gamma_{q^*}^{1/q^*}\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}}.$$ where the second term in [\[eq:main_thm_q\<2\<p_original\]](#eq:main_thm_q<2<p_original){reference-type="eqref" reference="eq:main_thm_q<2<p_original"} is taken over all decompositions $\bm{\underline{\smash{X}}} = \bm{\underline{\smash{U}}}^\top\bm{\underline{\smash{V}}}$, with $\bm{\underline{\smash{U}}} \in \mathbb{R}^{k\times n}$ and $\bm{\underline{\smash{V}}} \in \mathbb{R}^{k \times \ell}$ for some $k\in \mathbb{N}$. **Step 1: The first inequality** We begin with the first inequality of [\[eq:main_thm_q\<2\<p_original\]](#eq:main_thm_q<2<p_original){reference-type="eqref" reference="eq:main_thm_q<2<p_original"}. This easily follows by choosing $k=1$, so that $\bm{\underline{\smash{U}}} = \vec{\mu}^\top$ and $\bm{\underline{\smash{V}}} = \vec{\nu}^\top$ for some row vectors $\vec{\mu} \in \mathbb{R}^{n}$ and $\vec{\nu} \in \mathbb{R}^{\ell}$. In that case we obtain $$\begin{aligned} \max_{\substack{\bm{\underline{\smash{X}}} = \bm{\underline{\smash{U}}}^\top\bm{\underline{\smash{V}}}\\\norm{\bm{\underline{\smash{V}}}}_{L_{2,q^*}}\leq 1\\\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top}_{L_{2,p}^\top}\leq 1}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{X}}}) \geq \max_{\substack{\norm{\vec{\nu}}_{q^*}\leq 1\\\norm{\bm{\underline{\smash{B}}}\vec{\mu}}_{p}\leq 1}} \vec{\nu}^\top\bm{\underline{\smash{A}}}\vec{\mu} = \norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}} \end{aligned}$$ **Step 2: Probabilistic relaxation** We now consider the second inequality of [\[eq:main_thm_q\<2\<p_original\]](#eq:main_thm_q<2<p_original){reference-type="eqref" reference="eq:main_thm_q<2<p_original"}. Choose an arbitrary $k\in \mathbb{N}$, and let $\vec{u}_i, \vec{v}_j \in \mathbb{R}^k$ be vectors for $i=1,...,n$ and $j=1,...,\ell$. Let $\bm{\underline{\smash{U}}}$ and $\bm{\underline{\smash{V}}}$ be the matrices with columns $\vec{u}_i$ and $\vec{v}_j$, respectively. For the rest of this step, the $\vec{u}_i$ and $\vec{v}_j$ can be arbitrary, but can not be chosen such that all $\vec{u}_i$ are zero or all $\vec{v}_j$ are zero (in other words, $\bm{\underline{\smash{U}}}$ and $\bm{\underline{\smash{V}}}$ may not be the zero-matrix). We first transform $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto 1;\bm{\underline{\smash{B}}}}$ using duality: $$\begin{aligned} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}} &= \max_{\norm{\bm{\underline{\smash{B}}}\vec{x}}_p\leq 1}\nnorm{\bm{\underline{\smash{A}}}\vec{x}}_q =\max_{\substack{\norm{\vec{y}}_{q^*}\leq 1\\\norm{\bm{\underline{\smash{B}}}\vec{x}}_p\leq 1}}\vec{y}^\top\bm{\underline{\smash{A}}}\vec{x} =\max_{\vec{x}\neq \vec{0}, \vec{y}\neq \vec{0}}\frac{\sum_{ij}A_{ji}x_iy_j}{\norm{\bm{\underline{\smash{B}}}\vec{x}}_p\norm{\vec{y}}_{q^*}} \end{aligned}$$ We now use a similar technique to [@guruswami_bypassing_2016] and [@bhattiprolu_approximating_2018]: Let $\vec{\mathsf{g}}$ denote a standard Gaussian random variable in $\mathbb{R}^{k}$. We replace $\vec{x}$ by $\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}$ and $\vec{y}$ by $\bm{\underline{\smash{V}}}^\top\vec{\mathsf{g}}$ according to Lemma [\[lmm:vijay\]](#lmm:vijay){reference-type="ref" reference="lmm:vijay"} to obtain $$\label{eq:numerator_denumerator_q<2<p} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}} \geq \frac{\mathbb{E}_{\vec{\mathsf{g}}}\left[\sum_{ij}A_{ji}\vec{u}_i^\top\vec{\mathsf{g}}\vec{\mathsf{g}}^\top\vec{v}_j\right]}{\mathbb{E}_{\vec{\mathsf{g}}}\left[\norm{\bm{\underline{\smash{B}}}\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}}_p \norm{\bm{\underline{\smash{V}}}^\top\vec{\mathsf{g}}}_{q^*}\right]}.$$ We now compute the numerator and the denominator of [\[eq:numerator_denumerator_q\<2\<p\]](#eq:numerator_denumerator_q<2<p){reference-type="eqref" reference="eq:numerator_denumerator_q<2<p"} separately. The numerator is simple, since $\mathbb{E}_{\vec{\mathsf{g}}}\left[\vec{\mathsf{g}}\vec{\mathsf{g}}^\top\right] = \bm{\underline{\smash{I}}}_k$, so $$\mathbb{E}_{\vec{\mathsf{g}}}\left[\sum_{ij}A_{ji}\vec{u}_i^\top\vec{\mathsf{g}}\vec{\mathsf{g}}^\top\vec{v}_j\right] = \sum_{ij}A_{ji}\vec{u}_i^\top\vec{v}_j = \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{U}}}^\top\bm{\underline{\smash{V}}}).$$ For the denominator, note that since $q \leq 2 \leq p$, there holds $1/p + 1/q^* \leq 1$, so we may use the Hölder inequality to get $$\mathbb{E}_{\vec{\mathsf{g}}}\left[\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}}_p \norm{\bm{\underline{\smash{V}}}^\top\vec{\mathsf{g}}}_{q^*}\right] \leq \left(\mathbb{E}_{\vec{\mathsf{g}}}\left[\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}}_p^p\right]\right)^{1/p} \left(\mathbb{E}_{\mathsf{g}}\left[\norm{\bm{\underline{\smash{V}}}^\top\vec{\mathsf{g}}}_{q^*}^{q^*}\right]\right)^{1/q^*}.$$ Let $\vec{\beta}_i$ denote the rows of $\bm{\underline{\smash{B}}}$ for $i=1,\cdots,m$. Using Lemma [\[lmm:E_ugamma\]](#lmm:E_ugamma){reference-type="ref" reference="lmm:E_ugamma"} we can deduce $$\begin{aligned} \mathbb{E}_{\vec{\mathsf{g}}}\left[\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}}_p^p\right] &= \sum_i\mathbb{E}_{\vec{\mathsf{g}}}\left[|\vec{\beta}_i^\top\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}|^p\right] =\sum_i \gamma_p\norm{\bm{\underline{\smash{U}}}\vec{b}_i}_2^p =\gamma_p\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top}_{L_{2,p}^\top}^p \end{aligned}$$ and similarly $$\begin{aligned} \mathbb{E}_{\mathsf{g}}\left[\norm{\bm{\underline{\smash{V}}}^\top\vec{\mathsf{g}}}_{q^*}^{q^*}\right] &= \sum_j \gamma_{q^*}\norm{\vec{v}_j}_2^{q^*} = \gamma_{q^*}\norm{\bm{\underline{\smash{V}}}}_{L_{2,q^*}}^{q*}. \end{aligned}$$ Putting everything together, we obtain the second inequality of [\[eq:main_thm_q\<2\<p_original\]](#eq:main_thm_q<2<p_original){reference-type="eqref" reference="eq:main_thm_q<2<p_original"}. **Step 3: Positive Semi-Definite Reformulation** As in the previous step, let $\vec{\beta}_i$ denote the $i$-th row of $\bm{\underline{\smash{B}}}$ for $i=1,\cdots,m$, and $\vec{v}_j$ the $j$-th column of $\bm{\underline{\smash{V}}}$ for $j=1,\cdots,\ell$: $$\begin{aligned} \max_{\substack{\bm{\underline{\smash{X}}} = \bm{\underline{\smash{U}}}^\top\bm{\underline{\smash{V}}}\\\norm{\bm{\underline{\smash{V}}}}_{L_{2,q^*}}\leq 1\\\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top}_{L_{2,p}^\top}\leq 1}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{X}}}) %&=\max_{\substack{\norm{\mat{V}}_{L_{2,q^*}}\leq 1\\\norm{\mat{B}\,\mat{U}^\top}_{L_{2,p}^\top}\leq 1}} \frac{1}{2}\trace(\begin{bmatrix}\mat{0} & \mat{A}\\\mat{A}^\top & \mat{0}\end{bmatrix}\begin{bmatrix}\mat{V}^\top\\\mat{U}^\top\end{bmatrix}\begin{bmatrix}\mat{V} & \mat{U}\end{bmatrix})\\ &=\max_{\substack{\left(\sum_j (\vec{v}_j^\top\vec{v}_j)^{q^*/2}\right)^{1/q^*}\leq 1\\\left(\sum_i(\vec{b}_i^\top\bm{\underline{\smash{U}}}^\top\bm{\underline{\smash{U}}}\vec{\beta}_i)^{p/2}\right)^{1/p}\leq 1}} \frac{1}{2}\mathop{\mathrm{Tr}}(\begin{bmatrix}\bm{\underline{\smash{0}}} & \bm{\underline{\smash{A}}}\\\bm{\underline{\smash{A}}}^\top & \bm{\underline{\smash{0}}}\end{bmatrix}\begin{bmatrix}\bm{\underline{\smash{V}}}^\top\\\bm{\underline{\smash{U}}}^\top\end{bmatrix}\begin{bmatrix}\bm{\underline{\smash{V}}} & \bm{\underline{\smash{U}}}\end{bmatrix})\\ %&=\max_{\substack{\nnorm{\v{t}}_{q^*/2}^2 \leq 1\\ \nnorm{\v{s}}_{p/2}^2 \leq 1\\\v{s}\geq \v{0},\v{t}\geq \v{0}}}\max_{\substack{\v{v}_j^\top\v{v}_j= t_j\\\v{\beta}_i^\top\mat{U}^\top\mat{U}\v{b}_i= s_i}} \frac{1}{2}\trace(\begin{bmatrix}\mat{0} & \mat{A}\\\mat{A}^\top & \mat{0}\end{bmatrix}\begin{bmatrix}\mat{V}^\top\\\mat{U}^\top\end{bmatrix}\begin{bmatrix}\mat{V} & \mat{U}\end{bmatrix})\\ &=\max_{\substack{\nnorm{\vec{t}}_{q^*/2} \leq 1\\ \nnorm{\vec{s}}_{p/2} \leq 1}}\max_{\substack{Z\succeq0\\\mathop{\mathrm{diag}}\left(\bm{\underline{\smash{P}}}_{\bm{\underline{\smash{V}}}}^\top\bm{\underline{\smash{Z}}}\,\bm{\underline{\smash{P}}}_{\bm{\underline{\smash{V}}}}\right) = t\\\mathop{\mathrm{diag}}\left(\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{P}}}_{\bm{\underline{\smash{U}}}}^\top\bm{\underline{\smash{Z}}}\,\bm{\underline{\smash{P}}}_{\bm{\underline{\smash{U}}}}\bm{\underline{\smash{B}}}^\top\right) = s}} \frac{1}{2}\mathop{\mathrm{Tr}}(\begin{bmatrix}\bm{\underline{\smash{0}}} & \bm{\underline{\smash{A}}}\\\bm{\underline{\smash{A}}}^\top & \bm{\underline{\smash{0}}}\end{bmatrix}\bm{\underline{\smash{Z}}}) \end{aligned}$$ We can now use Lemma [\[lmm:semi-definite_dual\]](#lmm:semi-definite_dual){reference-type="ref" reference="lmm:semi-definite_dual"}, which yields $$\label{eq:almost_final_q<2<p} \max_{\substack{\bm{\underline{\smash{X}}} = \bm{\underline{\smash{U}}}^\top\bm{\underline{\smash{V}}}\\\norm{\bm{\underline{\smash{V}}}}_{L_{2,q^*}}\leq 1\\\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top}_{L_{2,p}^\top}\leq 1}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{X}}}) =\frac{1}{2}\max_{\substack{\nnorm{\vec{t}}_{q^*/2} \leq 1\\ \nnorm{\vec{s}}_{p/2} \leq 1}} \min_{(\vec{v},\vec{w})\in K} \vec{v}^\top\vec{t}+\vec{w}^\top\vec{s}.$$ Since $q \leq 2 \leq p$, the functions $\norm{\vec{t}}_{q^*/2}$ and $\norm{\vec{s}}_{p/2}$ are convex, and thus by a simple application of the minimax theorem, we can swap the maximum and minimum in [\[eq:almost_final_q\<2\<p\]](#eq:almost_final_q<2<p){reference-type="eqref" reference="eq:almost_final_q<2<p"} and obtain [\[eq:main_thm_q\<2\<p\]](#eq:main_thm_q<2<p){reference-type="eqref" reference="eq:main_thm_q<2<p"}. ◻ ## The Special Case $p=2, q=1$ Before we move on to the more general case of $p\in[1,\infty)$ and $q=1$, we consider the special case of $p=2$ and $q=1$, as a better approximation than the one from Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} can be found, using Nesterov's $\frac{\pi}{2}$-Theorem: Let $\bm{\underline{\smash{M}}} \in \mathbb{R}^{m\times m}$ be a symmetric, positive semi-definite matrix. Then there holds $$\max_{\norm{\vec{x}}_{\infty}\leq 1} \vec{x}^\top\bm{\underline{\smash{M}}}\vec{x} \leq \max_{\substack{\bm{\underline{\smash{X}}}\succeq 0\\X_{ii} = 1, \forall i}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{M}}}\,\bm{\underline{\smash{X}}}) \leq \frac{\pi}{2}\max_{\norm{\vec{x}}_{\infty}\leq 1} \vec{x}^\top\bm{\underline{\smash{M}}}\vec{x}.$$ This leads to the following result for $p=2$ and $q=1$: [\[thm:main_nesterov\]]{#thm:main_nesterov label="thm:main_nesterov"} Let $\ell, m, n\in\mathbb{N}$, and $\bm{\underline{\smash{A}}} \in \mathbb{R}^{\ell\times n}$, $\bm{\underline{\smash{B}}}\in \mathbb{R}^{m\times n}$. Assume that $\bm{\underline{\smash{B}}}$ is injective. Then $$\label{eq:main_nesterov} \norm{\bm{\underline{\smash{A}}}}_{2\mapsto1;\bm{\underline{\smash{B}}}} \leq \sqrt{\max_{\substack{\bm{\underline{\smash{X}}}\succeq 0\\X_{ii}=1, \forall i}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{B}}}^+\bm{\underline{\smash{B}}}^{+\top}\bm{\underline{\smash{A}}}^\top\bm{\underline{\smash{X}}})} \leq \sqrt{\frac{\pi}{2}}\cdot\norm{\bm{\underline{\smash{A}}}}_{2\mapsto1;\bm{\underline{\smash{B}}}}.$$ *Proof.* By Lemma [\[lmm:p=2\]](#lmm:p=2){reference-type="ref" reference="lmm:p=2"}, there holds $$\norm{\bm{\underline{\smash{A}}}}_{2\mapsto1;\bm{\underline{\smash{B}}}} = \max_{\norm{\vec{x}}_2\leq 1} \norm{\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{B}}}^+\vec{x}}_1 =\max_{\norm{\vec{y}}_{\infty}\leq 1} \norm{\bm{\underline{\smash{B}}}^{+\top}\bm{\underline{\smash{A}}}^\top\vec{y}}_2 = \sqrt{\max_{\norm{\vec{y}}_{\infty}\leq 1} \vec{y}^\top\bm{\underline{\smash{M}}}\vec{y}},$$ where the matrix $\bm{\underline{\smash{M}}} = \bm{\underline{\smash{A}}}\,\bm{\underline{\smash{B}}}^+\bm{\underline{\smash{B}}}^{+\top}\bm{\underline{\smash{A}}}^\top$ is positive semi-definite. Therefore we may use Nesterov's $\frac{\pi}{2}$-theorem, which yields [\[eq:main_nesterov\]](#eq:main_nesterov){reference-type="eqref" reference="eq:main_nesterov"}. ◻ ## The Case $p\in [1,\infty), q = 1$ For $p=\infty$ and $q \leq 2$, [@bhattiprolu_2023] showed that computing $\norm{\bm{\underline{\smash{A}}}}_{\infty \mapsto 2}$ is $\mathcal{APX}$-hard. Moreover, it was proven in [@matrix_p_norms Theorem 6.4.] that computing $\norm{\bm{\underline{\smash{A}}}}_{\infty \mapsto q}$ is not even in $\mathcal{APX}$ for $q\in (2,\infty)$ (i.e., there can not exist any constant-ratio approximations, unless $\mathcal{P}=\mathcal{NP}$). Since $\norm{\bm{\underline{\smash{A}}}}_{\infty \mapsto q} = \norm{\bm{\underline{\smash{A}}}^\top}_{q^* \mapsto 1}$, it follows that computing $\norm{\bm{\underline{\smash{A}}}^\top}_{p \mapsto 1}$ is $\mathcal{APX}$-hard for $p\in (1,\infty]$, which entails that $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto 1;\bm{\underline{\smash{B}}}}$ is $\mathcal{APX}$-hard to compute for $p\in (1,\infty]$. Additionally, for $p=q=1$ we indirectly proved in [@Kulmburg2021] that computing $\norm{\bm{\underline{\smash{A}}}}_{1\mapsto 1;\bm{\underline{\smash{B}}}}$ is $\mathcal{NP}$-hard. This suggests, again, that the best we can hope for is to find an approximation. Our main tool will be the Kahane contraction principle: [\[thm:kahane\]]{#thm:kahane label="thm:kahane"} For $k\in\mathbb{N}$ let $\mathsf{v}_1,...,\mathsf{v}_k$ be symmetric, independent random variables in some Banach space $\mathcal{V}$ with norm $\norm$, and let $c_1,\cdots, c_k \in \mathbb{R}$ be some constants. Then, for any $p\geq 1$ $$\mathbb{E}_{\mathsf{v}_1,...,\mathsf{v}_k}\left[\norm{c_1\mathsf{v}_1+\cdots+c_k\mathsf{v}_k}^p\right] \leq \max_i|c_i|^p\mathbb{E}_{\mathsf{v}_1,...,\mathsf{v}_k}\left[\norm{\mathsf{v}_1+\cdots+\mathsf{v}_k}^p\right].$$ Using the Kahane contraction principle, we can construct an efficient approximation scheme, even though it is not a constant-ratio approximation: [\[thm:main\]]{#thm:main label="thm:main"} Let $\ell, m, n\in\mathbb{N}$, and $\bm{\underline{\smash{A}}} \in \mathbb{R}^{\ell\times n}$, $\bm{\underline{\smash{B}}}\in \mathbb{R}^{m\times n}$. Then, for any $p\in[1,\infty)$, $$\label{eq:main_thm_q=1} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto1;\bm{\underline{\smash{B}}}} \leq \min_{\bm{\underline{\smash{B}}}^\top\bm{\underline{\smash{Y}}} = \bm{\underline{\smash{A}}}^\top} \norm{\bm{\underline{\smash{Y}}}}_{L_{1,p^*}^\top} \leq \frac{\gamma_p^{1/p}\sqrt{\ell}}{\gamma_1}\norm{\bm{\underline{\smash{A}}}}_{p\mapsto1;\bm{\underline{\smash{B}}}},$$ where $\gamma_r = \frac{2^{r/2}}{\sqrt{\pi}}\Gamma\left(\frac{r+1}{2}\right)$ for $r\in[1,\infty)$. *Proof.* **Step 1: The first inequality** We begin the proof of [\[eq:main_thm_q=1\]](#eq:main_thm_q=1){reference-type="eqref" reference="eq:main_thm_q=1"} by showing the first inequality. By looking more closely at the $L_{1,p^*}^\top$-norm, we notice that for any matrix $\bm{\underline{\smash{Y}}}$ with rows $\vec{\upsilon}_i$, there holds $$\begin{aligned} \norm{\bm{\underline{\smash{Y}}}}_{L_{1,p^*}^\top} = \left(\sum_i\max_{\norm{\vec{\sigma}}_{\infty}\leq 1}\absiprod{\vec{\upsilon}_i}{\vec{\sigma}}^{p^*}\right)^{1/p^*} \geq\max_{\norm{\vec{\sigma}}_{\infty}\leq 1}\norm{\bm{\underline{\smash{Y}}}\vec{\sigma}}_{p^*} \end{aligned}$$ where for the inequality, we have used the fact that, for any family of functions $f_i(x)$, there always holds $\max_x \sum_i f_i(x) \leq \sum_i \max_x f_i(x)$. Using the fact that for any function $f(x,y)$, there holds $\max_x\min_y f(x,y) \leq \min_y \max_x f(x,y)$, we conclude $$\min_{\bm{\underline{\smash{B}}}^\top\bm{\underline{\smash{Y}}} = \bm{\underline{\smash{A}}}^\top} \max_{\norm{\vec{\sigma}}_{\infty}\leq 1}\norm{\bm{\underline{\smash{Y}}}\vec{\sigma}}_{p^*} \geq \max_{\norm{\vec{\sigma}}_{\infty}\leq 1} \min_{\bm{\underline{\smash{B}}}^\top\bm{\underline{\smash{Y}}} = \bm{\underline{\smash{A}}}^\top}\norm{\bm{\underline{\smash{Y}}}\vec{\sigma}}_{p^*}.$$ It then suffices to apply Corollary [\[cor:matrix_norm_and_optimization\]](#cor:matrix_norm_and_optimization){reference-type="ref" reference="cor:matrix_norm_and_optimization"} to get $$\min_{\bm{\underline{\smash{B}}}^\top\bm{\underline{\smash{Y}}} = \bm{\underline{\smash{A}}}^\top} \norm{\bm{\underline{\smash{Y}}}}_{L_{1,p^*}^\top} \geq \max_{\norm{\vec{\sigma}}_{\infty}\leq1}\max_{\norm{\bm{\underline{\smash{B}}}\vec{x}}_p\leq 1}\vec{\sigma}^\top\bm{\underline{\smash{A}}}\vec{x} = \max_{\norm{\bm{\underline{\smash{B}}}\vec{x}}_p\leq 1}\norm{\bm{\underline{\smash{A}}}\vec{x}}_1 = \norm{\bm{\underline{\smash{A}}}}_{p\mapsto1;\bm{\underline{\smash{B}}}}.$$ **Step 2: Probabilistic relaxation** We now turn to the second inequality of [\[eq:main_thm_q=1\]](#eq:main_thm_q=1){reference-type="eqref" reference="eq:main_thm_q=1"}. For $i=1,\cdots,n$, let $\vec{u}_i \in \mathbb{R}^{\ell}$ be arbitrary but fixed vectors that are not all zero, and let $\bm{\underline{\smash{U}}}$ be the matrix with columns $\vec{u}_i$. As in the proof of Theorem [\[thm:main_q\<2\<p\]](#thm:main_q<2<p){reference-type="ref" reference="thm:main_q<2<p"}, we transform $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto 1;\bm{\underline{\smash{B}}}}$ using duality: $$\begin{aligned} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto 1;\bm{\underline{\smash{B}}}} =\max_{\substack{\norm{\vec{\sigma}}_{\infty}\leq 1\\\vec{x}\neq \vec{0}}}\frac{\sum_{ij}A_{ji}x_i\sigma_j}{\norm{\bm{\underline{\smash{B}}}\vec{x}}_p} \end{aligned}$$ Again, we use a similar idea to that of [@guruswami_bypassing_2016] and [@bhattiprolu_approximating_2018]: Let $\vec{\mathsf{g}}$ denote a standard Gaussian random variable in $\mathbb{R}^{\ell}$. We replace $\vec{x}$ by $\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}$ according to Lemma [\[lmm:vijay\]](#lmm:vijay){reference-type="ref" reference="lmm:vijay"} to obtain $$\label{eq:numerator_denumerator} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto 1;\bm{\underline{\smash{B}}}} \geq \frac{\mathbb{E}_{\vec{\mathsf{g}}}\max_{\norm{\vec{\sigma}}_{\infty}\leq 1}\left[\sum_{ij}A_{ji}\vec{u}_i^\top\vec{\mathsf{g}}\sigma_j\right]}{\mathbb{E}_{\vec{\mathsf{g}}}\left[\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}}_p\right]}.$$ We compute the numerator and the denominator of [\[eq:numerator_denumerator\]](#eq:numerator_denumerator){reference-type="eqref" reference="eq:numerator_denumerator"} separately: **Step 3: Bounding the numerator** For the numerator, we can bound from below the maximum with respect to $\vec{\sigma}$ by replacing it with $\text{sign}(\vec{\mathsf{g}})$: $$\begin{aligned} \mathbb{E}_{\vec{\mathsf{g}}}\left[\max_{\norm{\vec{\sigma}}_{\infty}\leq 1}\sum_{ij}A_{ji}\vec{u}_i^\top\vec{\mathsf{g}}\sigma_j\right] &\geq\sum_{ijk}A_{ji}\mathbb{E}_{\vec{\mathsf{g}}}[u_{i,k}\mathsf{g}_k\mathop{\mathrm{sign}}(\mathsf{g}_j)] \end{aligned}$$ The random variables $\mathsf{g}_k$ are symmetric, thus $\mathbb{E}_{\vec{\mathsf{g}}}\left[\mathsf{g}_k\mathop{\mathrm{sign}}(\mathsf{g}_j)\right] = 0$ if $k\neq j$, and $\mathbb{E}_{\vec{\mathsf{g}}}\left[\mathsf{g}_k\mathop{\mathrm{sign}}(\mathsf{g}_j)\right] = \mathbb{E}_{\vec{\mathsf{g}}}\left[|\mathsf{g}_k|\right]$ if $k=j$. We thus conclude that the numerator satisfies the bound $$\mathbb{E}_{\vec{\mathsf{g}}}\left[\max_{\norm{\vec{\sigma}}_{\infty}\leq 1}\sum_{ij}A_{ji}\vec{u}_i^\top\vec{\mathsf{g}}\sigma_j\right] \geq \gamma_1\sum_{ij}A_{ji}u_{i,j} = \gamma_1\cdot\mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{U}}}^\top).$$ **Step 4: Bounding the denominator** We start with a standard application of Jensen's inequality since the function $x \rightarrow x^{1/p}$ is concave for $p\geq 1$: $$\begin{aligned} \mathbb{E}_{\vec{\mathsf{g}}}\left[\nnorm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}}_p\right] = \mathbb{E}_{\vec{\mathsf{g}}} \left[ \sum_i\big| \sum_j B_{ij}\iprod{\vec{u}_j}{\vec{\mathsf{g}}} \big|^p \right]^{1/p} \leq \left[\sum_i\mathbb{E}_{\vec{\mathsf{g}}}\big|\langle\sum_j B_{ij}\vec{u}_j,\vec{\mathsf{g}}\rangle\big|^p\right]^{1/p}. \end{aligned}$$ We now use the Kahane contraction principle on the term $\mathbb{E}_{\vec{\mathsf{g}}}\big|\langle\sum_j B_{ij}\vec{u}_j,\vec{\mathsf{g}}\rangle\big|^p$: $$\mathbb{E}_{\vec{\mathsf{g}}}\big|\langle\sum_j B_{ij}\vec{u}_j,\vec{\mathsf{g}}\rangle\big|^p \leq \max_k\big|\sum_jB_{ij}u_{j,k}\big|^p\mathbb{E}_{\vec{\mathsf{g}}}|\mathsf{g}_1+\cdots+\mathsf{g}_{\ell}|^p.$$ Since $\mathsf{g}_i \sim \mathcal{N}(0,1)$, there holds $\mathsf{g}_1+\cdots\mathsf{g}_{\ell} \sim \mathcal{N}(0,\sqrt{\ell})$, and thus $\mathbb{E}_{\vec{\mathsf{g}}}|\mathsf{g}_1+\cdots+\mathsf{g}_{\ell}|^p = \sqrt{\ell}^p\gamma_p$. Overall, we obtain $$\mathbb{E}_{\vec{\mathsf{g}}}\left[\nnorm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{U}}}^\top\vec{\mathsf{g}}}_p\right] \leq \gamma_p^{1/p}\sqrt{l}\cdot\left(\sum_i \max_k \left|\sum_j B_{ij}u_{j,k}\right| \right)^{1/p} = \gamma_p^{1/p}\sqrt{l}\cdot\norm{\bm{\underline{\smash{B}}}\bm{\underline{\smash{U}}}^\top}_{L_{\infty,p}^\top}.$$ **Step 5: Final transformation** Putting the results from Steps 3 and 4 together, we end up with $$\max_{\nnorm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{X}}}}_{L_{\infty,p}^\top}\leq1} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\bm{\underline{\smash{X}}}) \leq \frac{\gamma_p^{1/p}\sqrt{\ell}}{\gamma_1}\cdot \norm{\bm{\underline{\smash{A}}}}_{p\mapsto1;\bm{\underline{\smash{B}}}}$$ It now suffices to use duality to transform the left-hand side: Since the bilinear function $(\bm{\underline{\smash{X}}}, \bm{\underline{\smash{Y}}}) \mapsto \mathop{\mathrm{Tr}}(\bm{\underline{\smash{X}}}^\top\bm{\underline{\smash{Y}}})$ defines an inner product on $\mathbb{R}^{n\times \ell}$, using Corollary [\[eq:constrained_norm_optimization\]](#eq:constrained_norm_optimization){reference-type="ref" reference="eq:constrained_norm_optimization"} and Lemma [\[lmm:dual_L\_p_q\]](#lmm:dual_L_p_q){reference-type="ref" reference="lmm:dual_L_p_q"} we get $$\max_{\norm{\bm{\underline{\smash{B}}}\,\bm{\underline{\smash{X}}}}_{L_{\infty,p}^\top}\leq 1} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{X}}}) = \min_{\bm{\underline{\smash{B}}}^\top\bm{\underline{\smash{Z}}} = \bm{\underline{\smash{A}}}^\top} \norm{\bm{\underline{\smash{Z}}}}_{L_{1,p^*}^\top}.$$ ◻ Alongside the result from Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"}, a different approximation is available for $p\in[1,2]$ and $q=1$, which can easily be derived from Theorem [\[thm:main_nesterov\]](#thm:main_nesterov){reference-type="ref" reference="thm:main_nesterov"}: [\[cor:main_nesterov\]]{#cor:main_nesterov label="cor:main_nesterov"} Let $\ell, m, n\in\mathbb{N}$, and $\bm{\underline{\smash{A}}} \in \mathbb{R}^{\ell\times n}$, $\bm{\underline{\smash{B}}}\in \mathbb{R}^{m\times n}$. Then, for any $p\in[1,2]$, $$\label{eq:main_cor_q=1} \norm{\bm{\underline{\smash{A}}}}_{p\mapsto1;\bm{\underline{\smash{B}}}} \leq \sqrt{\max_{\substack{\bm{\underline{\smash{X}}}\succeq 0\\X_{ii}=1, \forall i}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{B}}}^+\bm{\underline{\smash{B}}}^{+\top}\bm{\underline{\smash{A}}}^\top\bm{\underline{\smash{X}}})} \leq \sqrt{\frac{\pi}{2}}\cdot m^{1/p-1/2}\norm{\bm{\underline{\smash{A}}}}_{p\mapsto1;\bm{\underline{\smash{B}}}}.$$ On the other hand, if $p\in[2,\infty]$, there holds $$\label{eq:main_cor_q=1_p_in_2_infty} \sqrt{\frac{2}{\pi}}\cdot m^{1/p-1/2}\norm{\bm{\underline{\smash{A}}}}_{p\mapsto1;\bm{\underline{\smash{B}}}} \leq \sqrt{\max_{\substack{\bm{\underline{\smash{X}}}\succeq 0\\X_{ii}=1, \forall i}} \mathop{\mathrm{Tr}}(\bm{\underline{\smash{A}}}\,\bm{\underline{\smash{B}}}^+\bm{\underline{\smash{B}}}^{+\top}\bm{\underline{\smash{A}}}^\top\bm{\underline{\smash{X}}})} \leq\norm{\bm{\underline{\smash{A}}}}_{p\mapsto1;\bm{\underline{\smash{B}}}}.$$ *Proof.* If $p\in[1,2]$, [\[eq:main_cor_q=1\]](#eq:main_cor_q=1){reference-type="eqref" reference="eq:main_cor_q=1"} follows directly from the fact that, for any vector $\vec{v}\in\mathbb{R}^{m}$, we have $\norm{\vec{v}}_2 \leq \norm{\vec{v}}_p \leq m^{1/p-1/2} \norm{\vec{v}}_2$. The proof of [\[eq:main_cor_q=1_p\_in_2\_infty\]](#eq:main_cor_q=1_p_in_2_infty){reference-type="eqref" reference="eq:main_cor_q=1_p_in_2_infty"} is similar. ◻ The approximations from Corollary [\[cor:main_nesterov\]](#cor:main_nesterov){reference-type="ref" reference="cor:main_nesterov"} should be handled with care; while they may scale better than the result from Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} for $p\in(1,2)$, if $\ell=m$ and $p=1$ the approximation ratio from Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} is always better than that of Corollary [\[cor:main_nesterov\]](#cor:main_nesterov){reference-type="ref" reference="cor:main_nesterov"}. For $p\in(1,2)$, this depends on the values of $m$ and $\ell$. Additionally, the approximations of Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} can usually be computed much faster than that of Corollary [\[cor:main_nesterov\]](#cor:main_nesterov){reference-type="ref" reference="cor:main_nesterov"}, since they can be computed using gradient descent methods, instead of positive semi-definite programming. # Experimental Results [\[sec:numerical_evaluations\]]{#sec:numerical_evaluations label="sec:numerical_evaluations"} In order to verify that the approximation ratios we have uncovered in Section [\[sec:approximable\]](#sec:approximable){reference-type="ref" reference="sec:approximable"} are accurate, we will examine their performance numerically. Specifically, we will only analyze the performance of the methods described in Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} and Theorem [\[thm:main_nesterov\]](#thm:main_nesterov){reference-type="ref" reference="thm:main_nesterov"} for the case $q=1$. We will not investigate the approximation given in Theorem [\[thm:main_q\<2\<p\]](#thm:main_q<2<p){reference-type="ref" reference="thm:main_q<2<p"} for $q\in(1,2]$, for the following reason: by Corollary [\[cor:matrix_norm_and_optimization\]](#cor:matrix_norm_and_optimization){reference-type="ref" reference="cor:matrix_norm_and_optimization"} we have $$\norm{\bm{\underline{\smash{A}}}}_{p\mapsto 1;\bm{\underline{\smash{B}}}} = \max_{\norm{\vec{\alpha}}_{\infty}\leq 1} \min_{\bm{\underline{\smash{B}}}^\top\vec{\beta} = \bm{\underline{\smash{A}}}^\top\vec{\alpha}}\nnorm{\vec{\beta}}_{p^*}.$$ The term $$\label{eq:loupe_term} \min_{\bm{\underline{\smash{B}}}^\top\vec{\beta} = \bm{\underline{\smash{A}}}^\top\vec{\alpha}}\nnorm{\vec{\beta}}_{p^*}$$ can be evaluated in polynomial time using gradient descent methods, since it is a simple equality-constrained convex minimization problem. Furthermore, [\[eq:loupe_term\]](#eq:loupe_term){reference-type="eqref" reference="eq:loupe_term"} is convex in $\vec{\alpha}$, so by the Bauer maximum principle the exact value of $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto 1;\bm{\underline{\smash{B}}}}$ can be computed by taking the maximum of [\[eq:loupe_term\]](#eq:loupe_term){reference-type="eqref" reference="eq:loupe_term"} over all $\vec{\alpha}\in \{-1,+1\}^{\ell}$. A similar approach would not work for $q > 1$, since the set $\{\vec{\alpha}\in\mathbb{R}^{\ell}\,|\,\norm{\vec{\alpha}}_{q^*}\leq 1\}$ has infinitely many extreme points. Let $\mathtt{approx}_p(\bm{\underline{\smash{A}}}, \bm{\underline{\smash{B}}})$ denote one of the approximations from Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} or Theorem [\[thm:main_nesterov\]](#thm:main_nesterov){reference-type="ref" reference="thm:main_nesterov"} for given matrices $\bm{\underline{\smash{A}}}$ and $\bm{\underline{\smash{B}}}$. Consider the quantity $$\rho(\bm{\underline{\smash{A}}},\bm{\underline{\smash{B}}}) := \frac{\mathtt{approx}_p(\bm{\underline{\smash{A}}},\bm{\underline{\smash{B}}})}{\norm{\bm{\underline{\smash{A}}}}_{p\mapsto 1;\bm{\underline{\smash{B}}}}},$$ which gives a lower bound for the actual approximation ratio. Using a global optimization solver (in our case, we used the surrogate optimization solver from MATLAB, see [@surrogate]), one can search for the maximum of $\rho(\bm{\underline{\smash{A}}}, \bm{\underline{\smash{B}}})$ as a function of the variables $\bm{\underline{\smash{A}}}$ and $\bm{\underline{\smash{B}}}$. Concretely, we searched for the maximal value $\rho_{\max}$ of $\rho(\bm{\underline{\smash{A}}}, \bm{\underline{\smash{B}}})$ over the set $$\left\{(\bm{\underline{\smash{A}}},\bm{\underline{\smash{B}}})\in\mathbb{R}^{\ell\times n}\times \mathbb{R}^{m\times n}\;\middle|\;|A_{i,k}|\leq 10, |B_{j,k}|\leq 10, \forall i,j,k\right\}.$$ For each iteration, we verified that $\bm{\underline{\smash{B}}}$ was injective; if it was not, $\rho(\bm{\underline{\smash{A}}}, \bm{\underline{\smash{B}}})$ was manually set to 1, so as to not impact the final result. For the cases where $p\neq 1$, we used 1000 iterations for each choice of $\ell,m,$ and $n$, while for $p=1$ we used 10000 iterations. This is due to the fact that, for $p=1$, computing the exact value of $\norm{\bm{\underline{\smash{A}}}}_{1\mapsto 1;\bm{\underline{\smash{B}}}}$ can be reformulated as a zonotope containment problem (see [@Kulmburg2021]), for which there exist faster algorithms. In particular, we used the $\verb|polymax|$ algorithm from [@Kulmburg2021], which requires the CORA toolbox [@cora]. For $p>1$, we chose $\ell, m,$ and $n$ such that $\ell = m = 2n$, and performed the experiment for different values of $n=1,\cdots, 7$. For $p=1$, we chose to repeat the experiment for each $\ell = n,\cdots,20$, with $n=1,\cdots,7$ and $m = \ell$. In Figure [\[fig:p_equals_1\]](#fig:p_equals_1){reference-type="ref" reference="fig:p_equals_1"} we displayed the maximal value of $\rho(\bm{\underline{\smash{A}}}, \bm{\underline{\smash{B}}})$ for fixed values of $\ell$, over multiple values of $n$. Finally, for $p=2$ we repeated the same series of experiments both for the approximation from Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"}, as well as the approximation from Theorem [\[thm:main_nesterov\]](#thm:main_nesterov){reference-type="ref" reference="thm:main_nesterov"} based on Nesterov's theorem. In order to solve the positive semi-definite optimization problem, we used the YALMIP toolbox (see [@Lofberg2004]) combined with the SeDuMi solver [@sedumi]. The results of the numerical experiments can be found in Figures [\[fig:p_equals_1.5\]](#fig:p_equals_1.5){reference-type="ref" reference="fig:p_equals_1.5"}-[\[fig:p_equals_1\]](#fig:p_equals_1){reference-type="ref" reference="fig:p_equals_1"}; the constant $C_p$ stands for the constant in Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"}, i.e., $C_p = \gamma_p^{1/p}/\gamma_1$. The code used to generate these figures can be found at the URL <https://github.com/AdrianKulmburg/generalizedMatrixNorms> ## Discussion As can be seen in Figures [\[fig:p_equals_1.5\]](#fig:p_equals_1.5){reference-type="ref" reference="fig:p_equals_1.5"}-[\[fig:p_equals_1\]](#fig:p_equals_1){reference-type="ref" reference="fig:p_equals_1"}, the approximation ratios we have deduced in Theorem [\[thm:main\]](#thm:main){reference-type="ref" reference="thm:main"} and Theorem [\[thm:main_nesterov\]](#thm:main_nesterov){reference-type="ref" reference="thm:main_nesterov"} hold for all tested samples (i.e., the theoretical worst-case approximation ratio always over-approximated $\rho_{\max}$). However, in some cases our bounds seem to be too conservative. This is particularly apparent for the case $p=1$, but even for $p=2$ with the algorithm from Theorem [\[thm:main_nesterov\]](#thm:main_nesterov){reference-type="ref" reference="thm:main_nesterov"} based on Nesterov's $\pi/2$-Theorem, the theoretical approximation ratio is larger than the values of $\rho_{\max}$ we found out numerically. In fact, for higher values of $\ell$, the value of $\rho_{\max}$ even seems to *drop*, contrary to our expectations. This can partly be explained by the fact that, for higher $\ell,m,$ and $n$, the size of the space of matrices for $\bm{\underline{\smash{A}}}$ and $\bm{\underline{\smash{B}}}$ greatly increases, so the solver computing $\rho_{\max}$ does not have enough iterations to converge to the actual maximum for $\rho_{\max}$. Furthermore, due to the high cost of computing the exact value of $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto 1;\bm{\underline{\smash{B}}}}$, we could only perform our experiments on relatively low values for $\ell,m,$ and $n$. It is possible that for higher values of these three quantities, one could find higher values for $\rho_{\max}$. # Conclusions The concepts of push-forward and pull-back of norms, as well as the fact that these are dual to each other, allowed us to reformulate the generalized matrix norm problem. In particular, we discovered that max-min problems involving vector norms could sometimes be reformulated, via duality, as generalized matrix norm problems. This kind of max-min problems occur naturally in the context of containment problems (i.e., when checking whether a set is contained in another, see [@Kulmburg2021]), but it would be interesting to investigate whether such problems can also be encountered in other contexts, for example as approximations to more general max-min problems. We also analyzed the computability of the matrix norm $\norm{\bm{\underline{\smash{A}}}}_{p\mapsto q;\bm{\underline{\smash{B}}}}$ for different values of $p$ and $q$. In some instances, we found exact algorithms that run in polynomial time with respect to the representation size of $\bm{\underline{\smash{A}}}$ and $\bm{\underline{\smash{B}}}$. For cases were such exact algorithms were not available, we discovered approximations that run in polynomial time. However, in some instances, these approximations seem to perform significantly better than the error bounds we deduced, which leaves open the possibility that more accurate error bounds could be found in the future, especially for the case $q=1$, $p\in[1,\infty)$. [^1]: Department of Robotics, Artificial Intelligence and Embedded Systems, Technical University of Munich (, <https://www.ce.cit.tum.de/air/people/adrian-kulmburg-msc/>). [^2]: [^3]: To the best of our knowledge, the only explicit mention of this concept can be found on the following website, but only for extended norms: <https://planetmath.org/extendednorm> [^4]: The more common denomination in the geometric literature is \"polar set\", but the term \"dual set\" is more adequate for our purpose.

arxiv_math

--- abstract: | The continuum and discrete fractional nonlinear Schrödinger equations (fDNLS) represent new models in nonlinear wave phenomena with unique properties. In this paper, we focus on various aspects of localization associated to fDNLS featuring modulational instability, asymptotic construction of onsite and offsite solutions, and the role of Peierls-Nabarro barrier. In particular, the localized onsite and offsite solutions are constructed using the map approach. Under the long-range interaction characterized by the Lévy index $\alpha>0$, the phase space of solutions is infinite-dimensional unlike that of the well-studied nearest-neighbor interaction. We show that an orbit corresponding to this spatial dynamics translates to an approximate solution that decays algebraically. We also show as $\alpha \rightarrow \infty$, the discrepancy between local and nonlocal dynamics becomes negligible on a compact time interval, but persists on a global time scale. Moreover it is shown that data of small mass scatter to free solutions under a sufficiently high nonlinearity, which proves the existence of strictly positive excitation threshold for ground state solutions of fDNLS. author: - "Brian Choi[^1], Austin Marstaller[^2], Alejandro Aceves[^3]" bibliography: - ref.bib title: On Localization of the Fractional Discrete Nonlinear Schrödinger Equation --- **Keywords.** nonlinear modes, localization, stability, asymptotic approximation, nonlocal dynamics **MSC 2020** 34A08, 34A12, 34A34, 37K45, 37K60, 78M35 # Introduction. Research in arrays of coupled nonlinear oscillators remains a field of intense research whether it aims at understanding thermalization or the emergence of localized coherent structures, or global synchronization as exemplified by the FPUT or the Kuramoto oscillator model, respectively. Their relevance goes beyond theory as they model systems in a wide range of applications including photonics, plasmonics [@eisenberg1998discrete; @morandotti1999dynamics; @aceves1996discrete], Bose-Einstein condensates [@anderson1998macroscopic; @kalosakas2002delocalizing; @smerzi1997quantum; @trombettoni2001discrete], biological/chemical phenomena [@Mingaleev1999; @PhysRevE.55.6141], neuroscience [@neuro] and the power grid [@powergrid]. For more comprehensive surveys, see [@kevrekidis; @kivshar2003optical; @SulemSulem:NLS_book; @RevModPhys.83.247]. In all these cases, the emerging models are large systems of coupled ordinary differential equations. Similarly for continuous fields, universal equations such as the nonlinear Schrödinger equation (NLSE) or the sine-Gordon equation have provided a platform to study many features in nonlinear wave phenomena including the existence and interactions of solitons, and the formation of coherent structures and singular blow-up dynamics. Specific to nonlinear photonics and its applications, two universal models are the NLSE and the discrete nonlinear Schrödinger equation (DNLS) [\[dnls\]](#dnls){reference-type="eqref" reference="dnls"}. In the discrete case, the model describes waveguide or resonator arrays with the nearest-neighbor coupling, whereas the continuum version models, for example, pulses in optical fibers or light filaments propagating in air. In this paper, we present results for nonlocal, nonlinear discrete systems, departing from the nearest-neighbor coupling, whose long-range interaction is described by a nonlocal interaction kernel whose coupling strength decays algebraically. This model is known as the fractional discrete nonlinear Schrödinger equation (fDNLS) given by $$\label{fdnls} i \dot{u}_n = \epsilon \sum_{m\in\mathbb{Z} \setminus \{n\} } \frac{u_n - u_m}{|n - m|^{1+\alpha}} - |u_n|^2 u_n,\ (n,t) \in \mathbb{Z} \times \mathbb{R},\ \epsilon>0,\ \alpha > 0,$$ where our interest goes beyond this specific model to study a general long-range interaction given by [\[generalmodel\]](#generalmodel){reference-type="eqref" reference="generalmodel"}. Such nonlocal model was previously studied [@molina2020two; @johansson1998switching; @rasmussen1998localized; @christiansen2001multi; @korabel2007transition] in various contexts including stability, chaos, and higher-dimensional dynamics. Unlike the vast amount of experimental results on waveguide arrays modeled by the DNLS, at time there is no concrete photonics-based array for which the fDNLS is an experimentally-verifiable model. We do believe, however, the results could pave the way to future realizations where non-locality presents a new degree of freedom. It should be noted that the continuum version of the fDNLS $$\label{continuum_fnls} i\partial_t U = (-\Delta)^{\frac{\alpha}{2}} U - |U|^2 U,\ (x,t) \in \mathbb{R}^{d+1},$$ where $U(x,t) \in \mathbb{C}$ models, for example, the envelope dynamics of an electric field, was proposed as a model for an optical cavity [@longhi2015fractional] whereby a proper design of lenses allows one to engineer diffraction to behave as $|k|^{\alpha},\ 0 < \alpha < 2$ in the Fourier representation. The suggested benefit of such a design is to produce laser beam outputs with unique (Airy-like) profiles as opposed to the classical Gaussian-like outputs. The continuum and discrete dynamics, either viewed as a discretization of a continuum model, or in reverse, the continuum limit of an intrinsically discrete model, exhibit distinct features, one of them being the Peierls-Nabarro Barrier (PNB) in the discrete model [@KivCam; @bang1996exploiting; @morandotti1999dynamics]. Of central concern in the theory of nonlinear lattices is the propagation of localized waves in discrete media. A motion along the lattice requires the localized wave to switch its structure from onsite to offsite, and vice versa, interacting with the effective energy potential barrier, or the PNB, rising from the inherent discreteness of the lattice. The continuum power-type NLSE, say [\[continuum_fnls\]](#continuum_fnls){reference-type="eqref" reference="continuum_fnls"} with $\alpha = 2$, satisfies the Galilean invariance: if $u(x,t)$ is a solution for $x \in \mathbb{R}^d$, then so is $e^{i \frac{v}{2}\cdot (x-\frac{tv}{2})} u(x-vt,t)$ for any $v \in \mathbb{R}^d$. In particular, a ground state solution can be boosted to yield a family of solutions whose amplitudes are traveling waves. Moreover the mass-critical NLSE is invariant under the pseudoconformal symmetry, which could be used to show the existence of excitation threshold of mass [@Weinstein1983NonlinearSE; @weinstein1989nonlinear] for certain NLSE with sufficiently high nonlinearity. On the other hand, the lattice structure lacks these symmetries rising from the smooth structure of the Euclidean space, giving rise to, among many others, a non-zero PNB. We confirm the presence of nonlocal PNB in our asymptotic analysis where the energy difference between onsite and offsite solutions is computed explicitly. Our theory is supported by numerical simulations of the dynamics of [\[fdnls\]](#fdnls){reference-type="eqref" reference="fdnls"} for varying degrees of non-locality where localized solutions with an initial boost are eventually pinned at a lattice site. On the other hand, the role of non-locality and discreteness on the formation of localized states by modulational instability (MI) is investigated. A periodic train of pulses, under a small perturbation in its spectrum that originates from nonlinearity, localizes as a breather-like excitation driven by MI. The literature on MI is vast, and for our purposes, readers are directed to [@ZAKHAROV2009540; @dauxois1993energy; @daumont1997modulational; @zakharov2006freak; @zakharov2013nonlinear; @gelash2019bound] for survey results with applications in nonlinear optics or fluid dynamics. However a detailed analysis of MI of nonlocal dynamics is still at its infancy. See [@zhang2017modulational] for MI in relation to the fractional NLSE. Furthermore see [@Copeland:20] that investigates the mixed-fractional NLSE in the context of MI. While this paper only discusses the discrete model, observe in [\[fdnls\]](#fdnls){reference-type="eqref" reference="fdnls"} that by considering $\epsilon = c(d,\alpha) h^{-\alpha}$ where $c(d,\alpha)$ is given by [\[frac_lap\]](#frac_lap){reference-type="eqref" reference="frac_lap"} and $\alpha \in (0,2)$, [\[fdnls\]](#fdnls){reference-type="eqref" reference="fdnls"} could be understood as a finite difference scheme of the continuum fNLSE [\[continuum_fnls\]](#continuum_fnls){reference-type="eqref" reference="continuum_fnls"}. The well-posedness theory of [\[continuum_fnls\]](#continuum_fnls){reference-type="eqref" reference="continuum_fnls"} requires a more delicate analysis than its discrete version (see ) given that blow-up phenomena could occur [@DINH2019117]. See [@1534-0392_2015_6_2265; @dinh:hal-01426761] for the rigorous analysis of the well-posedness of fNLSE in Sobolev spaces. The functional-analytic tools of the aforementioned references do not directly apply when the non-locality, represented by the linear dispersion relation, is not uniform in space (which occurs when the spatial dimension is at least two), and thus to account for such non-homogeneity that rises in the mixed-fractional NLSE, the non-smooth analogue of the Littlewood-Paley theory was developed to establish the well-posedness in anisotropic Sobolev spaces [@choi2022well]. Even with the continuum model being well-posed for data of sufficiently high regularity, it is not trivial to show the continuum limit as $h \rightarrow 0$, if it holds at all. For example, the mass-critical cubic NLSE on $\mathbb{R}^2$ contains solutions that blow up in finite time whereas the lattice dynamics is always global. It was shown rigorously in [@kirkpatrick2013continuum] that the weak convergence of nonlocal discrete to continuum dynamics of the Schrödinger evolution holds. A further research done by [@hong2019strong] (see references therein) refined the weak to strong convergence by employing discrete Strichartz estimates uniform in the discreteness parameter. Furthermore note [@choi2023continuum] for the first result in the continuum limit of fDNLS to fNLSE in dimension two. In the context of onsite/offsite solutions, it is expected that the PNB would tend to zero in the continuum limit since the Galilean invariance needs to be recovered. This is indeed true, and moreover the quantitative bound that PNB vanishes exponentially fast as $h \rightarrow 0$ was shown in [@JenWeinLocal; @jenkinson_weinstein_2017] for DNLS and fDNLS, respectively. This paper is organized as follows. In , the main model and its properties are introduced. In , the global dynamics corresponding to $\alpha \gg 1$ is investigated. In the formal limit as $\alpha \rightarrow \infty$, [\[fdnls\]](#fdnls){reference-type="eqref" reference="fdnls"} converges to the DNLS $$\label{dnls} i \dot{u}_n = -\epsilon \delta^2 u_n - |u_n|^2 u_n,\ (n,t) \in \mathbb{Z}^d \times \mathbb{R},\ \epsilon>0,$$ with $d=1$, where $\delta^2 u_n := \sum\limits_{|j-n|=1} u_j - 2du_n$ and $|\cdot|$ defined with the $l^1$ norm on $\mathbb{Z}^d$. The nature of this limit, whether it is regular or singular, is subtle as it relates to the Soliton Resolution Conjecture and the long-time dynamics of Hamiltonian systems whose solutions do not dissipate. In , linear stability analysis on the CW solution of fDNLS is shown with explicit regions of MI. The onset of nonlinear bound states resulting from MI is shown via numerical simulations. In , the family of onsite and offsite solutions and the corresponding PNB are constructed asymptotically for DNLS and fDNLS, respectively. Although the rigorous aspects of numerical analysis (consistency, convergence, etc) are not our main focus, we provide the numerical discretization of the fractional Laplacian in . # Background. {#background} In this paper, we consider the generalized nonlocal model, also considered in [@kirkpatrick2013continuum], where the infinitesimal generator is given by $$\label{inf} \mathscr{L}_\alpha f_n = \sum_{m \neq n} J_{|n-m|}(f_n - f_m),\ (f_n) \in l^2(\mathbb{Z}),\ \alpha >0,$$ where $J = (J_{n})_{n=1}^\infty,\ J_n \geq 0$ is the $\alpha$-kernel satisfying the limit property $$\lim\limits_{n \rightarrow \infty} n^{1+\alpha} J_n = A_\alpha \in (0,\infty),$$ and when $\alpha = \infty$, define $J$ as an $\infty$-kernel if $\lim\limits_{n \rightarrow \infty} n^{1+\alpha} J_n = 0$ for all $\alpha >0$; assume that $(J_n)$ is not identically zero. Note that $\mathscr{L}_\alpha$ defines a family of self-adjoint, bounded linear operators on $l^2(\mathbb{Z})$. In applications, our focus lies in specific long-range interaction kernels defined by $J_n = |n|^{-(1+\alpha)}$, but assume the general form unless otherwise specified. A particular nonlinear model generated by [\[inf\]](#inf){reference-type="eqref" reference="inf"} is given by $$\label{generalmodel} i \dot{u}_n = \epsilon\mathscr{L}_\alpha u_n - |u_n|^2 u_n,\ u(0) = f \in l^2(\mathbb{Z}),\ \epsilon > 0,$$ and the stationary model, by taking the ansatz $u_n(t) = e^{iwt} Q_n$ where $w>0,\ Q_n \in \mathbb{R}$, is given by $$\label{generalmodel_stationary} -w Q_n = \epsilon\mathscr{L}_\alpha Q_n - Q_n^3.$$ Since the dynamics of interest is posed on $\mathbb{Z}^d$, it is natural to use Fourier analysis. For $f \in l^1(\mathbb{Z}^d)$, define $\mathcal{F}[f](k) = \sum\limits_{n \in \mathbb{Z}^d} f_n e^{in \cdot k}$ for $k \in \mathbb{T}^d = (-\pi,\pi]^d$, after which $\mathcal{F}$ extends uniquely to an isomorphism $l^2(\mathbb{Z}^d) \xrightarrow[]{\cong} L^2(\mathbb{T}^d)$ by the standard density argument. Under the flow [\[generalmodel\]](#generalmodel){reference-type="eqref" reference="generalmodel"}, there are at least two conserved quantities given by $$\label{energy} N[u(t)] = \sum_{n} |u_n|^2,\ E[u(t)] = \frac{\epsilon}{2} \langle \mathscr{L}_\alpha u_n , u_n \rangle_{l^2} - \frac{1}{4} \sum_{n}|u_n|^4,$$ representing the particle number (or mass) and energy, respectively. The kinetic energy terms corresponding to DNLS [\[dnls\]](#dnls){reference-type="eqref" reference="dnls"} and fDNLS [\[fdnls\]](#fdnls){reference-type="eqref" reference="fdnls"} are given by $\frac{\epsilon}{2}\sum\limits_{n \in \mathbb{Z}} |u_{n+1} - u_n|^2$ and $\frac{\epsilon}{4}\sum\limits_{n,m: n\neq m} \frac{|u_n - u_{m}|^2}{|n-m|^{1 + \alpha}}$, respectively. For $\alpha \in (0,2),\ x \in \mathbb{R}^d$, the singular integral representation of the fractional Laplacian [@lischke2019fractional] is defined as $$\label{frac_lap} (-\Delta)^{\frac{\alpha}{2}}u(x) = c(d,\alpha) \text{p.v.} \int\limits_{\mathbb{R}^{d}} \frac{u(x) - u(y)}{|x-y|^{d+\alpha}}dy := \frac{2^{\alpha}\Gamma(\frac{\alpha}{2} + \frac{d}{2})}{\pi^{\frac{d}{2}}|\Gamma(-\frac{\alpha}{2})|}\text{p.v.}\int\limits_{\mathbb{R}^{d}} \frac{u(x) - u(y)}{|x-y|^{d+\alpha}}dy.$$ In fact, there are many inequivalent definitions of the fractional Laplacian on bounded domains, and hence a particular numerical discretization of $(-\Delta)^{\frac{\alpha}{2}}$ needs to be defined since numerical simulations depend on an appropriate spatial truncation. In where the evolution of $u_n(0) = A > 0,\ -N \leq n \leq N$ is studied, the periodic boundary condition is imposed whereas the zero exterior Dirichlet boundary condition is imposed to simulate localized wave solutions in . # Global Well-posedness and Small Data Scattering. {#wellposedness} By the contraction mapping argument and the embedding $l^p(\mathbb{Z}^d) \hookrightarrow l^q(\mathbb{Z}^d)$ whenever $p \leq q$, the well-posedness of [\[fdnls\]](#fdnls){reference-type="eqref" reference="fdnls"}, [\[dnls\]](#dnls){reference-type="eqref" reference="dnls"} is established. As long as the long-range interaction is described by a self-adjoint operator and the nonlinear interaction, by a local nonlinearity, the following well-posedness result is proved similarly as [@kirkpatrick2013continuum Proposition 4.1]. **Proposition 1**. *Let $L$ be a bounded, self-adjoint operator on $l^2(\mathbb{Z}^d)$ and $N:\mathbb{C} \rightarrow \mathbb{C}$ such that $$N(0) = 0,\ |N(z_1) - N(z_2)| \leq C\left(\max(|z_1|,|z_2|)\right)|z_1 - z_2|,$$ where $C:[0,\infty) \rightarrow [0,\infty)$ is increasing. Then the initial-value problem $$\label{ivp} i\dot{u}_n = L u_n + N(u_n),\ u(0) = f \in l^2(\mathbb{Z}^d),$$ is globally well-posed; for any $f \in l^2(\mathbb{Z}^d)$, there exists a unique solution $u \in C^1_{loc}(\mathbb{R};l^2(\mathbb{Z}^d))$ to [\[ivp\]](#ivp){reference-type="eqref" reference="ivp"} such that $u(0) = f$ and the data-to-solution map $f \mapsto u$ is Lipschitz continuous.* **Proposition 2**. *Assume the hypotheses of hold. Let $\{L_\alpha\}_{\alpha>0}$ and $L$ be bounded, self-adjoint operators on $l^2(\mathbb{Z}^d)$ such that $L_\alpha \xrightarrow[\alpha \rightarrow \infty]{} L$ in norm. Let $u^{(\alpha)},v \in C^1_{loc}(\mathbb{R};l^2(\mathbb{Z}^d))$ be the well-posed solutions to [\[ivp\]](#ivp){reference-type="eqref" reference="ivp"} given by $L_\alpha, L$, respectively, satisfying $u^{(\alpha)}(0) = f^{(\alpha)} \in l^2(\mathbb{Z}^d),\ v(0) = g \in l^2(\mathbb{Z}^d)$. Assume $\sup\limits_{\alpha > 0}\| f^{(\alpha)} \|_{l^2} \leq M$. Then there exists $C = C(M, \| g \|_{l^2})>0$ such that for all $t \geq 0$, $$\label{exp_growth} \| u^{(\alpha)}(t) - v(t) \|_{l^2} \leq e^{Ct}\Big(\| f^{(\alpha)}-g \|_{l^2} + t \| L_\alpha - L\| \cdot \| g \|_{l^2}\Big).$$* *Proof.* For notational brevity, say $u = u^{(\alpha)},\ f = f^{(\alpha)}$. The well-posedness of $u,v$ follows by . Setting $\phi = u - v$, we have $$i\dot{\phi}_n = L_\alpha \phi_n + (L_\alpha - L)v_n + N(u_n) - N(v_n).$$ By integrating, it follows that $$\phi_n(t) = e^{-it L_\alpha} \phi_n(0) - i \int_0^t e^{-i(t-t^\prime)L_\alpha} \bigl\{(L_\alpha - L)v_n(t^\prime) + N(u_n(t^\prime)) - N(v_n(t^\prime))\bigl\}dt^\prime,$$ and by the triangle inequality, the unitarity of $e^{-it L_\alpha}$, the conservation of particle numbers, and the embedding $l^2(\mathbb{Z}^d)\hookrightarrow l^\infty(\mathbb{Z}^d)$, we have $$\| \phi(t) \|_{l^2} \leq \| \phi(0) \|_{l^2} + t \| L_\alpha - L \| \cdot \| g \|_{l^2} + C(\max(M,\| g \|_{l^2})) \int_0^t \| \phi(t^\prime )\|_{l^2} dt^\prime,$$ where $C>0$ is the (local) Lipschitz constant of the nonlinearity $N$. The proof follows from the Gronwall's inequality. ◻ **Remark 1**. *In the context of long-range interaction and DNLS, the operators defined by $$L_\alpha f_n = \epsilon\sum\limits_{m \neq n} \frac{f_n - f_m}{|n-m|^{d+\alpha}},\ L = -\epsilon \delta^2,$$ on $l^2(\mathbb{Z}^d)$ are Fourier multipliers with the symbols given by $$\begin{split} \sigma_\alpha(k) &= 2\epsilon \sum_{n \neq 0} \frac{\sin^2\left(\frac{n \cdot k}{2}\right)}{|n|^{d+\alpha}},\\ \sigma(k) &= 2\epsilon \sum_{|n|=1}\sin^2\left(\frac{n\cdot k}{2}\right) = 2d - \sum_{|n|=1} e^{i n \cdot k}, \end{split}$$ respectively, for $k \in \mathbb{T}^d$ and $|(z_1,\dots,z_d)| = \sum\limits_{j=1}^d |z_j|$. Then the norm-convergence hypothesis of is satisfied since $$|\sigma_\alpha(k) - \sigma(k)| = 2\epsilon \left|\sum_{|n| \geq 2} \frac{\sin^2\left(\frac{nk}{2}\right)}{|n|^{d+\alpha}}\right| \leq \sum_{|n| \geq 2} \frac{2\epsilon}{|n|^{d+\alpha}} \lesssim_d \epsilon \int_2^\infty \frac{dr}{r^{1+\alpha}} = \frac{\epsilon}{\alpha 2^\alpha} \xrightarrow[\alpha \to \infty]{} 0,$$ uniformly in $k$. Hence the short-time dynamics of fDNLS is well-approximated by that of the DNLS for large $\alpha$ on $\mathbb{Z}^d$.* The previous remark on the approximation of DNLS via fDNLS for large $\alpha$ need not hold for the long-time dynamics, and the exponential bound [\[exp_growth\]](#exp_growth){reference-type="eqref" reference="exp_growth"} provides no coercive growth-in-time estimates. Moreover the emerging pattern in with different wavenumbers corresponding to various $\alpha$ is a numerical evidence that $u^{(\alpha)} \nrightarrow v$ in $C(\mathbb{R};l^2(\mathbb{Z}^d))$ as $\alpha \rightarrow \infty$ where $u^{(\alpha)},v$ are solutions in $C^1_{loc}(\mathbb{R};l^2(\mathbb{Z}^d))$, that follow from , with $u^{(\alpha)}(0) = v(0) \in l^2(\mathbb{Z}^d) \setminus \{0\}$. The long-time dynamics of an extended Hamiltonian system exhibits a rich structure featuring multi-breathers, transition into chaos, and small data scattering, just to name a few, where such variety of features rises from conservation laws in stark contrast to dissipative systems. The following proposition is an example of small data scattering uniform in $\alpha$ for sufficiently high nonlinearity. As a corollary, this proves the existence of strictly positive excitation threshold of ground state solutions for fDNLS. Note that [@weinstein1999excitation] showed that $p \geq 5$ for DNLS is the sufficient and necessary condition for the positive excitation threshold. By the following result, we leave it as a future work to investigate the case $5 \leq p < 7$ for fDNLS. **Proposition 3**. *Let $\mathscr{L}_\infty := \mathscr{L} = -\delta^2$ and $\mathscr{L}_\alpha$ be defined by $J_n = |n|^{-(1+\alpha)}$, and define $U(t) = e^{-it\mathscr{L}}$ and $U^{(\alpha)}(t) = e^{-it\mathscr{L}_\alpha}$. By , let $u^{(\alpha)}, v := u^{(\infty)} \in C^1_{loc}(\mathbb{R};l^2(\mathbb{Z}))$ be the well-posed solutions to $$\label{gwp_divergence2} \begin{split} i \dot{u}_n^{(\alpha)} &= \mathscr{L}_\alpha u_n^{(\alpha)} + \mu |u_n^{(\alpha)}|^{p-1} u_n^{(\alpha)},\ u^{(\alpha)}(0) = f \in l^2(\mathbb{Z}),\\ i \dot{v}_n &= \mathscr{L} v_n + \mu |v_n|^{p-1} v_n,\ v(0) = f \in l^2(\mathbb{Z}), \end{split}$$ where $\mu = \pm 1,\ p \geq 7$. Then there exists $\Tilde{\alpha} > 0$ such that for all $\Tilde{\alpha} < \alpha < \infty$, the data-to-asymptotic-state map is well-defined as an isometric homeomorphism in a small neighborhood of $l^2(\mathbb{Z})$ uniformly in $\Tilde{\alpha} < \alpha \leq \infty$; more precisely, there exists $\delta_1(\Tilde{\alpha}) > 0$ such that whenever $\| f \|_{l^2} < \delta_1$, there exists unique $u_+^{(\alpha)} \in l^2(\mathbb{Z})$ such that $$\label{scattering} \| u^{(\alpha)}(t) - U^{(\alpha)}(t) u_+^{(\alpha)}\|_{l^2} \xrightarrow[t \rightarrow \infty]{} 0.$$ Furthermore there exists $\delta_2(\Tilde{\alpha})>0$ such that whenever $0 < \| f \|_{l^{\frac{5}{4}}} < \delta_2$, we have $$\label{dispersive_est} \| u^{(\alpha)}(t) \|_{l^5} \leq C(\Tilde{\alpha}) (1+|t|)^{-\frac{1}{5}} \| f \|_{l^{\frac{5}{4}}},$$ and $$\label{singular_limit} \varlimsup_{\alpha \rightarrow \infty}\sup_{t \in [0,\infty)}\| u^{(\alpha)}(t) - v(t) \|_{l^2} > 0.$$* **Lemma 1**. *There exists $\alpha_0^* > 0$ such that for all $\alpha > \alpha_0^*$, there exists $t_\alpha>2^\alpha$ and $$\| (U^{(\alpha)}(t_\alpha) - U(t_\alpha))f \|_{l^2} \geq c > 0,$$ for all $f \in l^2(\mathbb{Z}) \setminus \{0\}$ where $c>0$ is independent of $\alpha$ and $f$.* *Proof.* By the Plancherel Theorem and the reverse Hölder inequality, $$\begin{aligned} \label{int} \| (U^{(\alpha)}(t/4) - U(t/4))f\|_{l^2}^2 &= \frac{1}{2\pi} \bigintsss_{-\pi}^{\pi} \left|\exp \left(-it \sum_{m=1}^\infty\frac{\sin^2\left(\frac{mk}{2}\right)}{m^{1+\alpha}}\right) - \exp\left(-it \sin^2\left(\frac{k}{2}\right)\right)\right|^2 \cdot|\widehat{f}(k)|^2 dk\nonumber\\ &\gtrsim \| \widehat{f} \|_{L^{1/3}}^2 \left(\bigintsss_{-\pi}^{\pi} \left|\exp \left(-it \sum_{m=2}^\infty\frac{\sin^2\left(\frac{mk}{2}\right)}{m^{1+\alpha}}\right) - 1\right|^{-\frac{2}{5}} dk\right)^{-5}\nonumber\\ &\gtrsim \left(\bigintsss_{0}^{\pi} \left(1-\cos\left(t \sum\limits_{m=2}^\infty\frac{\sin^2\left(\frac{mk}{2}\right)}{m^{1+\alpha}}\right)\right)^{-\frac{1}{5}} dk\right)^{-5},\end{aligned}$$ since $0 < \| \widehat{f} \|_{L^{1/3}} \lesssim \| \widehat{f} \|_{L^2} < \infty$. For the convenience of notation, let $$X(k) = t \sum\limits_{m=2}^\infty\frac{\sin^2\left(\frac{mk}{2}\right)}{m^{1+\alpha}}.$$ Then the integral above on $k \in [0,\pi]$ is split into $$\int_0^{\pi} (1-\cos X)^{-\frac{1}{5}} dk = \int_{\{1-\cos X < c\}} (1-\cos X)^{-\frac{1}{5}} dk + \int_{\{1-\cos X \geq c\}} (1-\cos X)^{-\frac{1}{5}} dk =: I + II,$$ where $c>0$ is sufficiently small. Then $II$ is $O(1)$ depending only on $c$, and therefore it suffices to estimate $I$. First consider $0 \leq X < X_0$ where $X_0$ is the smallest positive real such that $1-\cos X_0 = c$, and suppose $t = 2^{1+\alpha}(2\pi + X_0)$. Then $$\label{admissible} 0 \leq \sin^2 k < \frac{X_0}{2\pi + X_0} - 2^{1+\alpha}\sum_{m \geq 3} \frac{\sin^2(\frac{mk}{2})}{m^{1+\alpha}},$$ and $k \in [0,\pi]$ that satisfies [\[admissible\]](#admissible){reference-type="eqref" reference="admissible"} are near $k=0,\pi$ since the series term is negligible for all $\alpha>0$ sufficiently large due to the uniform estimate $$\label{technical} 2^{1+\alpha}\sum_{m \geq 3} \frac{\sin^2(\frac{mk}{2})}{m^{1+\alpha}} < 2\left(\frac{1}{3}+\frac{1}{\alpha}\right) \left(\frac{2}{3} \right)^{\alpha},$$ that follows from majorizing the series into an appropriate integral. The arguments for the estimation near $k = 0,\pi$ are similar, and therefore assume $k$ is near $0$. Note that [\[admissible\]](#admissible){reference-type="eqref" reference="admissible"} is satisfied for $0 \leq k < k_0$ where $k_0$ is the smallest positive real root of $$\sin^2(k_0) = \frac{X_0}{2\pi + X_0} - 2^{1+\alpha}\sum_{m \geq 3} \frac{\sin^2(\frac{mk_0}{2})}{m^{1+\alpha}}.$$ Then for all $\alpha>0$ sufficiently large depending on $X_0$, we have $k_0 < k_0^*$ where $k_0^*$ is the smallest positive real root of $\sin^2(k_0^*) = \frac{X_0}{2\pi + X_0}$. By the Taylor expansion, $$k_0 < k_0^* \lesssim X_0^{\frac{1}{2}},$$ and $$\label{technical3} \begin{split} \int_{\{0 \leq X < X_0\}} (1-\cos X)^{-\frac{1}{5}} dk &\lesssim \int_{\{0 \leq X < X_0\}} X^{-\frac{2}{5}} dk \lesssim \int_{\{0 \leq X < X_0\}} (2\pi + X_0)^{-\frac{2}{5}} \sin^{-\frac{4}{5}}(k) dk\\ &\lesssim \int_{0}^{k_0} k^{-\frac{4}{5}} dk \lesssim X_0^{\frac{1}{10}}, \end{split}$$ where the implicit constants depend only on $c,X_0$. Now consider $|X-2\pi| < X_0$. Since the analysis for $0 < X - 2\pi < X_0$ and $-X_0 < X - 2\pi <0$ are similar, we focus on the former. Then $$\begin{aligned} \label{sin} 2\pi < 2^{1+\alpha}(2\pi + X_0) \left(\frac{\sin^2 k}{2^{1+\alpha}} + \sum_{m \geq 3} \frac{\sin^2(\frac{mk}{2})}{m^{1+\alpha}}\right) &< 2\pi + X_0\nonumber\\ \frac{2\pi}{2\pi + X_0} - 2^{1+\alpha}\sum_{m \geq 3} \frac{\sin^2(\frac{mk}{2})}{m^{1+\alpha}} &< \sin^2 k < 1- 2^{1+\alpha}\sum_{m \geq 3} \frac{\sin^2(\frac{mk}{2})}{m^{1+\alpha}} < 1. \end{aligned}$$ Define $k_1(\alpha),k_1^* \in (0,\frac{\pi}{2})$ such that $$\begin{split} \sin^2 k_1 &= \frac{2\pi}{2\pi + X_0} - 2^{1+\alpha}\sum_{m \geq 3} \frac{\sin^2(\frac{mk}{2})}{m^{1+\alpha}}\\ \sin^2 k_1^* &= \frac{2\pi}{2\pi + X_0}. \end{split}$$ The Taylor expansion of $\cos X$ near $X = 2\pi$ yields $$\begin{split} \int_{\{0<X-2\pi<X_0\}} (1-\cos X)^{-\frac{1}{5}}dk &\lesssim \int_{E} |X-2\pi|^{-\frac{2}{5}}dk\\ &= \int_{k_1}^{\frac{\pi}{2}} |X-2\pi|^{-\frac{2}{5}}dk + \int_{\frac{\pi}{2}}^{\pi - k_1} |X-2\pi|^{-\frac{2}{5}}dk =: III+ IV \end{split}$$ where $$E = \{k \in [0,\pi]:\frac{2\pi}{2\pi + X_0} - 2^{1+\alpha}\sum\limits_{m \geq 3} \frac{\sin^2(\frac{mk}{2})}{m^{1+\alpha}} < \sin^2 k < 1\} = (k_1,\pi - k_1).$$ The term $IV$ can be estimated similarly as $III$, and therefore the work for $III$ is shown. For sufficiently large $\alpha>0$, the Taylor expansion of $X(k)$ near $k = k_1$ yields $$\begin{split} X(k) &= X(k_1) + X^\prime(k_1) (k-k_1) + O(|k-k_1|^2)\\ &= 2\pi + 2^\alpha (2\pi + X_0) \sum_{m \geq 2} \frac{\sin (mk_1)}{m^\alpha} (k-k_1) + O(|k-k_1|^2). \end{split}$$ By arguing as [\[technical\]](#technical){reference-type="eqref" reference="technical"}, the coefficient of $k-k_1$ is bounded above by a constant independent of $\alpha$. Conversely, since $k_1 < k_1^*$ and $k_1 \xrightarrow[\alpha \rightarrow \infty]{}k_1^*$, we have $2k_1 \in (2k_1^* - \delta,2k_1^*)$ for some $\delta>0$ and $\sin (2k_1) >0$. The lower bound is given by $$2^\alpha \sum_{m \geq 2} \frac{\sin (mk_1)}{m^\alpha} \geq \sin (2k_1) - \sum_{m \geq 3} \left(\frac{2}{m}\right)^\alpha > \sin (2k_1^*) - \sum_{m \geq 3} \left(\frac{2}{m}\right)^\alpha \gtrsim 1,$$ where the implicit constant is independent of $\alpha$. Hence the linear coefficient is bounded above and below by a positive constant independent of $\alpha$, and similarly, the quadratic coefficient is bounded above uniformly in $\alpha$. Then $$\label{technical2} III \lesssim \int_{k_1}^{\frac{\pi}{2}} |k-k_1|^{-\frac{2}{5}}dk = O(1).$$ Hence by [\[technical3\]](#technical3){reference-type="eqref" reference="technical3"}, [\[technical2\]](#technical2){reference-type="eqref" reference="technical2"}, and $t = 2^{1+\alpha} (2\pi + X_0)$, there exists $M>0$ independent of $\alpha$ such that $$\int_{\{1- \cos X <c\}} (1-\cos X)^{-\frac{1}{5}} dk \leq \int_{\{0 < X < X_0\}} (1-\cos X)^{-\frac{1}{5}} dk + \int_{\{|X-2\pi| < X_0\}} (1-\cos X)^{-\frac{1}{5}} dk \leq M,$$ and this shows the lower bound of [\[int\]](#int){reference-type="eqref" reference="int"}. ◻ *Proof of .* The proof strategy is as follows. As [@stefanov2005asymptotic] in their analysis of DNLS, derive linear dispersive estimates (Strichartz estimates) uniform in $\alpha$ and apply them to establish the small data scattering [\[scattering\]](#scattering){reference-type="eqref" reference="scattering"}. By the fixed point argument in the Strichartz space, derive [\[dispersive_est\]](#dispersive_est){reference-type="eqref" reference="dispersive_est"}. Lastly estimate $\| U^{(\alpha)}(t) - U(t)\|$ for $t \gg 1$ to obtain [\[singular_limit\]](#singular_limit){reference-type="eqref" reference="singular_limit"}. By the discrete Fourier transform, $U^{(\alpha)}(t) f_n = \left(K_{t,\alpha} \ast f\right)_n$ where $$K_{t,\alpha}(n) := \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{-it\phi(k)}dk,\ \phi(k) = 4\sum_{m=1}^\infty \frac{\sin^2\left(\frac{mk}{2}\right)}{m^{1+\alpha}} + \frac{nk}{t}.$$ By the Dominated Convergence Theorem, $$\phi^{\prime\prime}(k) \xrightarrow[\alpha \rightarrow \infty]{} 2\cos(k),\ \phi^{\prime\prime\prime}(k) \xrightarrow[\alpha \rightarrow \infty]{} -2\sin(k),$$ and therefore $\max\limits_{k \in [-\pi,\pi]} \left(|\phi^{\prime\prime}(k)|,|\phi^{\prime\prime\prime}(k)|\right) \gtrsim 1$ for $\alpha$ sufficiently large. In fact, the bound holds as long as $\alpha \geq \alpha_0 > 3$ under which the term-by-term differentiation of $\phi^{\prime\prime\prime}$ holds uniformly. By the Van der Corput Lemma [@stein1993harmonic p.334], this implies $$\sup_{n \in \mathbb{Z}}|K_{t,\alpha}(n)| \lesssim_{\alpha_0} (1+|t|)^{-\frac{1}{3}},$$ where the inhomogeneous bound follows from applying the triangle inequality to $K_{t,\alpha}$. As a corollary, we obtain $$\label{strichartz2} \| u^{(\alpha)}(t) \|_{l^s} \lesssim (1+|t|)^{-\frac{s-2}{3s}} \| f \|_{l^{s^\prime}},$$ by the complex interpolation of linear operators where $s \in [2,\infty]$ and $s^\prime := \frac{s}{s-1}$. Let $(q,r) \in [2,\infty]^2$ satisfy $\frac{3}{q} + \frac{1}{r} = \frac{1}{2}$, and say that such pair is DNLS-admissible. A further corollary [@keel1998endpoint Theorem 1.2] implies $$\label{strichartz} \begin{split} \| U^{(\alpha)}(t) f \|_{L^q_t l^r(I\times \mathbb{Z})} &\lesssim \| f \|_{l^2},\\ \left|\left| \int_0^\infty U^{(\alpha)}(-t^\prime) N(t^\prime)dt^\prime \right|\right|_{l^2} &\lesssim \| N \|_{L^{\Tilde{q}^\prime}_t l^{\Tilde{r}^\prime}(\mathbb{R}\times \mathbb{Z})},\\ \left|\left| \int_0^t U^{(\alpha)}(t-t^\prime) N(t^\prime)dt^\prime \right|\right|_{L^q_t l^r(I\times \mathbb{Z})} &\lesssim \| N \|_{L^{\Tilde{q}^\prime}_t l^{\Tilde{r}^\prime}(I\times \mathbb{Z})}, \end{split}$$ where $(\Tilde{q},\Tilde{r})$ is DNLS-admissible and $I \subseteq \mathbb{R}$. Note that the implicit constants depend on the DNLS-admissible pairs and $\alpha_0$. The solution in the integral form satisfies $$\label{duhamel} u^{(\alpha)}(t) = U^{(\alpha)}(t)f - i\mu \int_0^t U^{(\alpha)}(t-t^\prime)\left(|u^{(\alpha)}(t^\prime)|^{p-1}u^{(\alpha)}(t^\prime)\right)dt^\prime.$$ Let $X = C(\mathbb{R};l^2(\mathbb{Z})) \cap L^{\frac{6p}{5}}_t l^{\frac{2p}{p-5}}(\mathbb{R}\times\mathbb{Z})$ and let $\Gamma u$ be the RHS of [\[duhamel\]](#duhamel){reference-type="eqref" reference="duhamel"}. By the fixed point argument, $\Gamma$ has a unique fixed point in $X$ if $\| f \|_{l^2} = O(1)$ where the bound depends only on $\alpha_0$ since the Strichartz constant of [\[strichartz2\]](#strichartz2){reference-type="eqref" reference="strichartz2"} can be chosen uniformly. Then $$\label{scattering2} \begin{split} \left|\left|\int_{t_1}^{t_2} U^{(\alpha)}(-t^\prime)\left(|u^{(\alpha)}(t^\prime)|^{p-1}u^{(\alpha)}(t^\prime)\right)dt^\prime \right|\right|_{l^2} \lesssim \| u^{(\alpha)}\|_{L^{\frac{6p}{5}}_t l^{p}([t_1,t_2]\times\mathbb{Z})}^p\leq \| u^{(\alpha)}\|_{L^{\frac{6p}{5}}_t l^{\frac{2p}{p-5}}([t_1,t_2]\times\mathbb{Z})}^p \xrightarrow[t_1,t_2 \rightarrow \infty]{} 0, \end{split}$$ where the first inequality is by [\[strichartz\]](#strichartz){reference-type="eqref" reference="strichartz"}, the second inequality by the Hölder's inequality and $p \geq 7$, and the last limit by $\| u^{(\alpha)}\|_{L^{\frac{6p}{5}}_t l^{\frac{2p}{p-5}}([t_1,t_2]\times\mathbb{Z})} \leq \| u^{(\alpha)}\|_{L^{\frac{6p}{5}}_t l^{\frac{2p}{p-5}}(\mathbb{R}\times\mathbb{Z})} \lesssim \| f \|_{l^2} < \infty$ with $t_1 \leq t_2$ without loss of generality. This shows the existence of $u_+^{(\alpha)} = \lim\limits_{t\rightarrow\infty} U^{(\alpha)}(-t)u^{(\alpha)}(t) \in l^2(\mathbb{Z})$ and by [\[duhamel\]](#duhamel){reference-type="eqref" reference="duhamel"}, $$\label{duhamel2} u^{(\alpha)}(t) = U^{(\alpha)}(t) u^{(\alpha)}_+ + i\mu \int_t^{\infty} U^{(\alpha)}(t-t^\prime)\left(|u^{(\alpha)}(t^\prime)|^{p-1}u^{(\alpha)}(t^\prime)\right)dt^\prime.$$ To show that the inverse of data-to-asymptotic-state (wave operator) is continuous, we argue by the fixed point theorem on $X_T = C([T,\infty);l^2(\mathbb{Z})) \cap L^{\frac{6p}{5}}_t l^{\frac{2p}{p-5}}([T,\infty)\times\mathbb{Z})$ where $\Gamma^\prime u$ is defined as the RHS of [\[duhamel2\]](#duhamel2){reference-type="eqref" reference="duhamel2"} and $T>0$ is chosen such that $\| U^{(\alpha)}(t) u_+^{(\alpha)}\|_{L^{\frac{6p}{5}}_t l^{\frac{2p}{p-5}}([T,\infty)\times\mathbb{Z})}$ is small; this proof is standard in the literature and can be consulted in, say, [@tao2006nonlinear Chapter 3]. Let $u^{(\alpha)} \in X_T$ be the unique fixed point. Flowing $u^{(\alpha)}$ backward in time, one obtains $f := u^{(\alpha)}(0)$. Hence the desired homeomorphism follows where the isometry is due to the conservation of $l^2$ norm. Here it is shown that the asymptotic-state map $\alpha \mapsto u_+^{(\alpha)} \in l^2(\mathbb{Z})$, with a fixed initial datum, is continuous. From [\[duhamel2\]](#duhamel2){reference-type="eqref" reference="duhamel2"}, $$u_+^{(\alpha)} = f - i\mu \int_0^{\infty} U^{(\alpha)}(-t^\prime)\left(|u^{(\alpha)}(t^\prime)|^{p-1}u^{(\alpha)}(t^\prime)\right)dt^\prime,$$ and hence it suffices to show the continuity of the integral. For a fixed $t^\prime \in (0,\infty)$, the integrand converges to $U(-t^\prime)\left(|v(t^\prime)|^{p-1}v(t^\prime)\right) \in l^2(\mathbb{Z})$ as $\alpha \rightarrow \infty$; the argument of may be modified to show pointwise (in $t^\prime$) continuity in $\alpha$. One may follow the argument of [@stefanov2005asymptotic Theorem 4] verbatim to derive the decay estimate [\[dispersive_est\]](#dispersive_est){reference-type="eqref" reference="dispersive_est"}. The fDNLS obeys the same Strichartz estimates as DNLS, and therefore the nonlinear decay for $\| f \|_{l^{\frac{5}{4}}} < \delta_2$, for some $\delta_2(\alpha_0)>0$, follows where $\alpha \geq \alpha_0$ guarantees the uniformity in $\alpha$. Applying [\[dispersive_est\]](#dispersive_est){reference-type="eqref" reference="dispersive_est"} and the unitarity, we have $$\int_0^\infty \| |u^{(\alpha)}|^{p-1}u^{(\alpha)}\|_{l^2} dt^\prime = \int_0^\infty \| u^{(\alpha)} \|_{l^{2p}}^p dt^\prime \leq \int_0^\infty \| u^{(\alpha)} \|_{l^{5}}^p dt^\prime \lesssim_{\alpha_0} \int_0^\infty (1+|t^\prime|)^{-\frac{p}{5}} \| f\|_{l^{\frac{5}{4}}}^p dt^\prime < \infty,$$ and therefore $\alpha \mapsto u_+^{\alpha}$ is continuous by the Dominated Convergence Theorem. Lastly [\[singular_limit\]](#singular_limit){reference-type="eqref" reference="singular_limit"} is proved by contradiction. Let $0 < \| f \|_{l^{\frac{5}{4}}} < \delta_2$. Let $v_+ \in l^2(\mathbb{Z})\setminus \{0\}$ be the asymptotic state corresponding to $f$ under the DNLS flow. For any $\epsilon > 0$, there exists $\alpha_1>0$ such that $\| u^{(\alpha)}(t) - v(t)\|_{l^2} < \epsilon$ for all $t \in [0,\infty)$ for $\alpha \geq \alpha_1$. By continuity, we have $\| u_+^{(\alpha)} - v_+\|_{l^2} < \epsilon$ for $\alpha \geq \alpha_2 >0$, and let $\alpha > \max (\alpha_0^*,\alpha_0,\alpha_1,\alpha_2)$ where $\alpha_0^* > 0$ is from . The scattering results imply that there exists $T>0$ such that if $t \geq T$, then $$\| u^{(\alpha)}(t) - U^{(\alpha)}(t)u_+^{(\alpha)}\|_{l^2}, \| v(t) - U(t)v_+\|_{l^2} < \epsilon.$$ The triangle inequality yields $$\| (U^{(\alpha)}(t) - U(t))v_+\|_{l^2} - \| u_+^{(\alpha)} - v_+\|_{l^2} - \| u^{(\alpha)}(t) - U^{(\alpha)}(t)u_+^{(\alpha)}\|_{l^2} - \| v(t) - U(t)v_+\|_{l^2} \leq \| u^{(\alpha)}(t) - v(t)\|_{l^2},$$ and therefore $$\sup_{t \in [T,\infty)}\| (U^{(\alpha)}(t) - U(t))v_+\|_{l^2} < 4\epsilon,$$ a contradition by . ◻ # Modulational Instability of CW Solutions. {#MI} Here we focus on periodic solutions in the direction of propagation (in $t$) that do not decay in space. Hence there is no localization, initially, but MI triggers the formation of coherent states, a process that requires a further study. For $A>0$, define $u_{n}^{cw} = Ae^{iA^2 t}$ as the continuous wave (CW) solution and consider $u_n = (A + \mu v_n (t))e^{iA^2 t}$ where $v_n(t) \in \mathbb{C}$ and $|\mu|\ll 1$. **Proposition 4**. *The CW solution is linearly unstable if and only if $\Phi :=$ $\epsilon\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km)) - A^2 < 0$. The zero set $\{\Phi = 0\}$ is the graph of $A = A(k,\alpha,\epsilon)$ where $A$ is analytic on $\mathbb{T} \setminus \{0\} \times (0,\infty) \times (0,\infty)$. Furthermore there exists $C=C(\alpha)>0$ such that $$A \sim \left(\epsilon C \delta(k)\right)^{\frac{1}{2}} \text{as}\ k \rightarrow 0,$$ where $$\delta(k)= \begin{cases} |k|^\alpha, &\alpha \in (0,2),\\ (-\log |k|)|k|^2, &\alpha = 2,\\ |k|^2, &\alpha \in (2,\infty]. \end{cases}$$* {width="\\linewidth"} {width="\\linewidth"} {width="\\linewidth"} {width="\\linewidth"} *Proof.* The $\mathcal{O}(\mu)$ expansion yields $$\label{perturb} i\partial_t v_n = \epsilon\mathscr{L}_\alpha v_n - A^{2}(v_n + \overline{v_n}),$$ and setting $v_n = f_n + ig_n := \Re v_n + i \Im v_n$, [\[perturb\]](#perturb){reference-type="eqref" reference="perturb"} is equivalent to $$\label{perturb2} \frac{d}{dt} \begin{pmatrix} f_n \\ g_n \end{pmatrix} = \begin{pmatrix} 0 & \epsilon \mathscr{L}_\alpha\\ -\epsilon \mathscr{L}_\alpha + 2A^2 & 0 \end{pmatrix} \begin{pmatrix} f_n \\ g_n \end{pmatrix}.$$ Taking the Fourier transform $F(k,t) = \mathcal{F}[f_n(t)](k),\ G(k,t) = \mathcal{F}[g_n(t)](k)$ and the ansatz $$\begin{pmatrix} F\\G \end{pmatrix} = \begin{pmatrix} P(k)\\Q(k) \end{pmatrix} e^{-i \Omega t},$$ [\[perturb2\]](#perturb2){reference-type="eqref" reference="perturb2"} reduces to $$\label{perturb3} \begin{pmatrix} i\Omega & 2\epsilon \sum\limits_{m=1} J_m (1-\cos km)\\ -\left(2\epsilon \sum\limits_{m=1} J_m (1-\cos km) - 2A^2\right) & i\Omega \end{pmatrix} \begin{pmatrix} P\\Q \end{pmatrix} = \begin{pmatrix} 0\\0 \end{pmatrix},$$ where a nontrivial solution to [\[perturb3\]](#perturb3){reference-type="eqref" reference="perturb3"} exists if and only if $\Omega$ satisfies the dispersion relation $$\label{dispersion_relation} \Omega^2 = 4 \left (\epsilon \sum_{m=1}^{\infty}J_m (1-\cos(km)) \right ) \left (\epsilon\sum_{m=1}^{\infty}J_{m}(1-\cos(km)) - A^2 \right ).$$ The frequency $\Omega$ is purely imaginary if and only if $\epsilon\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km)) < A^2$, leading to modulational instability. Suppose $\Phi = 0$ and note that $\partial_A \Phi = -2A$. Since $A = 0$ if and only if $k=0$, $A(k,\alpha,\epsilon)$ is analytic whenever $k \neq 0$ by the analytic Implicit Function Theorem. In the neighborhood of $k=0$, [@kirkpatrick2013continuum Lemma A.1] implies that there exists $C(\alpha)>0$ such that $$\frac{A^2}{\epsilon C \delta(k)} = \frac{A^2}{\epsilon C \delta(k)}\cdot \frac{\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km))}{\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km))} = \frac{\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km))}{C \delta(k)} \xrightarrow[k\rightarrow 0]{}1.$$ ◻ When $J_n = n^{-(1+\alpha)}$, shows the region of instability. Note the non-analytic dependence of $A,\alpha$ on $k$ near the zero wavenumber. The instability region expands for bigger values of $A$, which is consistent with the $-A^2$ term in [\[dispersion_relation\]](#dispersion_relation){reference-type="eqref" reference="dispersion_relation"}. To compute the value(s) of $k \in [-\pi,\pi]$ that maximizes the exponential gain of the modulational instability, the RHS of [\[dispersion_relation\]](#dispersion_relation){reference-type="eqref" reference="dispersion_relation"} needs to be minimized under the condition $\Phi < 0$; that a minimum exists follows from the Extreme Value Theorem since $\Omega(k)^2$ is continuous by the definition of $(J_n)_{n=1}^\infty$. An explicit computation that minimizes $\Omega(k)^2$ is shown for a specific interaction kernel. ![$k_{max}$ as a function of $\alpha$ when $J_n = n^{-(1+\alpha)}$. The values $A=1, \epsilon =1$ are used to generate this plot. For such parameters, $A_0 \geq 2 > A$ where $A_0$ is defined in , and therefore [\[wavenumber\]](#wavenumber){reference-type="eqref" reference="wavenumber"} is used to solve $k_{max}$ in $\alpha$. Observe that small $\alpha$ gives small $k_{max}$, manifested in the top left plot of . That $k_{max}$ has an upper bound is numerically verified in for increasing values of $\alpha$.](fig/k_vs_alpha.pdf){#fig:k_vs_alpha width="30%"} **Corollary 1**. *For $\alpha,\epsilon >0$, let $J_n = n^{-(1+\alpha)},\ \Tilde{w}(k) = \sum\limits_{m=1}^{\infty} \frac{1-\cos km}{m^{1+\alpha}},$ and $A_0 := \left(4\epsilon (1-2^{-(1+\alpha)})\zeta(1+\alpha)\right)^{\frac{1}{2}}$. Then for any $A \geq A_0$, $$\label{min_largeA} \min_{k \in [-\pi,\pi]}\Omega(k)^2 = \Omega(\pm \pi)^2 = -A_0^2 (2A^2 - A_0^2),$$ and if $0 < A < A_0$, then there exists a unique $k_{max} \in (0,\pi)$ that minimizes $\Omega^2$ where $k_{max}$ satisfies $$\label{wavenumber} \Tilde{w}(k_{max}) = \frac{A^2}{2\epsilon}$$ and $$\label{min_smallA} \min_{k \in [-\pi,\pi]}\Omega(k)^2 = \Omega(\pm k_{max})^2 = -A^4.$$* *Proof.* Since $\Omega(k)^2$ is even and $\Omega(0) = 0$, let $k \in (0,\pi]$ without loss of generality. By direct computation, it can be shown that $\Tilde{w}(0)=0,\ \Tilde{w} \in C^1(0,\pi)$, and $\Tilde{w}$ is increasing on $(0,\pi)$. From the derivative, we have $$\label{deriv} \begin{split} \frac{d}{dk} \left(\Omega(k)^2\right) &= 4\epsilon \frac{d}{dk}\Tilde{w}(k) \left( 2\epsilon \overset{\sim}{w}(k) - A^2 \right ), \end{split}$$ and therefore $\Omega^2$ is decreasing if and only if $2\epsilon \Tilde{w}(k) - A^2 \leq 0$. Since $$\label{zeta_odd} \max_{k \in [-\pi,\pi]} \Tilde{w}(k) = \Tilde{w}(\pm\pi) = \sum_{m \geq 1,\ m\ \text{odd}} \frac{2}{m^{1+\alpha}}= 2(1-2^{-(1+\alpha)})\zeta(1+\alpha),$$ if $A \geq A_0$, then $2\epsilon \Tilde{w}(k) - A^2 \leq 0$ for all $k \in (0,\pi]$. Hence $\Omega^2$ is minimized at $k = \pm \pi$ and the value of $\Omega(\pm \pi)^2$ follows from [\[zeta_odd\]](#zeta_odd){reference-type="eqref" reference="zeta_odd"}. If $0 < A < A_0$, then there exists a unique $k_{max} \in (0,\pi)$ satisfying $\Tilde{w}(k_{max}) = \frac{A^2}{2\epsilon}$ by the Intermediate Value Theorem and $$\Omega(k_{max})^2 = 4\epsilon \Tilde{w}(k_{max}) \left(\epsilon\Tilde{w}(k_{max}) - A^2\right) = -A^4.$$ ◻ What dynamics emerges after the initial linear growth due to MI is a question that has been posed since the studies of the FPUT chain. Based on many studies in discrete and continuous models, it is expected that the discrete modulational instability is the potential mechanism for the formation of nonlinear localized modes, such as discrete breathers and envelope solitons. What is not known is which type emerges in the fNLSE and what is the role of the parameter $\alpha$. Figure 3 shows a sample of emerging patterns triggered by numerical noise. While at this time we will not examine these emerging patterns, it is clear that $\alpha$-dependent coherent and robust patterns develop. {#fig:5 width="\\linewidth"} {#fig:5 width="\\linewidth"} {#fig:5 width="\\linewidth"} {#fig:5 width="\\linewidth"} # Asymptotic Construction of Stationary Solutions. {#solution_DNLS} An important class of solutions for nonlinear coupled oscillators, are localized states carrying finite energy. It is expected as shown in the previous section, that stable localized modes are indeed coherent structures the system selects to allocate energy. Generically, there are no known analytic solutions, so typically one relies on numerical or asymptotic approximations. In order to highlight similarities and differences between the local and nonlocal coupling, we include known results from the DNLS. In one dimensional arrays, solutions are onsite if their intensity is centered at a grid site, and offsite if centered between two consecutive grid sites. To avoid confusions in notation, we denote $q_n$ by the onsite solution (mode $A$) satisfying $q_{n} = q_{-n}$ and $g_n$, the offsite solution (mode $B$) satisfying $g_{n} = g_{-(n-1)}$. In , the stationary equation of DNLS is recasted as a discrete recurrence relation similar to [@kevrekidis Chapter 11]. In where the interaction kernel is of long range, the recurrence relation does not define a Markov chain; instead the forward map depends on all previous iterates and the phase space of solutions is infinite dimensional. Our asymptotic method has a disadvantage that the orbit of the recurrence relation does not give an exact solution, but has the advantage that it yields a localized sequence that satisfies the stationary equation in an appropriate limit. ## Localized Solutions for DNLS. {#localized} We revisit the derivation of stationary solutions in [@KivCam Section II] while relaxing the hypotheses that $\mathrm{supp} \{q_n\}$ and $\mathrm{supp} \{g_n\}$ are compact. To motivate , consider [\[generalmodel_stationary\]](#generalmodel_stationary){reference-type="eqref" reference="generalmodel_stationary"} with $J_n = 1$, if $n=1$, and $J_n = 0$ otherwise (nearest-neighbor coupling). Assume $q_n \geq 0$ and $q_{n+1} \ll q_n$ for all $n \geq 0$; see for a precise statement. Neglecting $q_{\pm 1}$ yields $$wq_0 = -2\epsilon q_0 + q_0^3,$$ and therefore $q_0 = \sqrt{w+2\epsilon}$. For $n \geq 1$, further assume $q_n \ll 1$, after which [\[generalmodel_stationary\]](#generalmodel_stationary){reference-type="eqref" reference="generalmodel_stationary"} neglecting $q_{n+1}$ and the nonlinear term yields $$wq_n = \epsilon (q_{n-1} -2q_n),$$ and therefore $q_n = \frac{\epsilon}{(w+2\epsilon)} q_{n-1}$. By this method, a forward map is defined where $q_n$ depends on $q_{n-1}$. A similar computation can be done with $\{g_n\}$. This explicit construction yields a sequence in $l^2(\mathbb{Z})$ that satisfies [\[dnls\]](#dnls){reference-type="eqref" reference="dnls"} asymptotically as $\rho := \frac{\epsilon}{w} \rightarrow 0$. **Proposition 5**. *For $w,\epsilon>0$, let $\{q_n\}$ satisfy $$\label{dnls4} q_n = \left(\frac{\epsilon}{w+2\epsilon}\right)^{|n|} \sqrt{w+2\epsilon},\ n \in \mathbb{Z}.$$ Then $q_{n+1} = o(q_n)$ as $\rho := \frac{\epsilon}{w} \rightarrow 0$. Moreover $\{q_n\}$ satisfies [\[generalmodel_stationary\]](#generalmodel_stationary){reference-type="eqref" reference="generalmodel_stationary"}, with the nearest-neighbor coupling, asymptotically as $\rho \rightarrow 0$ uniformly in $n \in \mathbb{Z}$, or more precisely, $$\label{dnls_asymp} \lim_{\rho \rightarrow 0} \sup_{n \in \mathbb{Z}}\left|\frac{wq_n}{\epsilon(q_{n+1}+q_{n-1}-2q_{n}) + q_n^3} - 1\right|= 0.$$* *Proof.* By symmetry, let $n \geq 0$. An explicit computation yields, $$\frac{q_{n+1}}{q_n} = \frac{\rho}{1+2\rho} \xrightarrow[\rho \rightarrow 0]{} 0,$$ and $$\label{dnls_limit} \frac{wq_n}{\epsilon(q_{n+1}+q_{n-1}-2q_{n}) + q_n^3}= \begin{cases} \frac{1}{1+\rho^2 (1+2\rho)^{-1} + \rho^{2n}(1+2\rho)^{1-2n}},\ n > 0,\\ \frac{1}{1+2\rho^2(1+2\rho)^{-1}},\ n=0. \end{cases}$$ Taking the limit of the RHS of [\[dnls_limit\]](#dnls_limit){reference-type="eqref" reference="dnls_limit"} as $\rho \rightarrow 0$, [\[dnls_asymp\]](#dnls_asymp){reference-type="eqref" reference="dnls_asymp"} is shown. ◻ **Remark 2**. *For $n \geq 1$, define $\{g_n\}$ as $$\label{dnls5} g_n = \left(\frac{\epsilon}{w+2\epsilon}\right)^{n-1} \sqrt{w+\epsilon},$$ and extend to $\mathbb{Z}$ by $g_{n} = g_{-(n-1)}$. Then $g_{n+1} = o(g_n)$ as $\rho \rightarrow 0$ for all $n \geq 1$ and satisfies [\[dnls_asymp\]](#dnls_asymp){reference-type="eqref" reference="dnls_asymp"} with $q_n$ replaced by $g_n$.* ## Peierls-Nabarro Barrier for DNLS. {#pnb} A direct computation via geometric series and [\[energy\]](#energy){reference-type="eqref" reference="energy"} yields an explicit expression for $E_{DNLS}$. **Proposition 6**. *Let $q_n,g_n$ be given by [\[dnls4\]](#dnls4){reference-type="eqref" reference="dnls4"}, [\[dnls5\]](#dnls5){reference-type="eqref" reference="dnls5"}, respectively. Then, $$\label{energy_dnls} \begin{split} E_{A}(\epsilon,w) &= \frac{\epsilon (w+\epsilon) (w+2\epsilon)}{w+3\epsilon} - \frac{(w+2\epsilon)^2 ((w+2\epsilon)^4+\epsilon^4)}{4((w+2\epsilon)^4-\epsilon^4)},\\ E_{B}(\epsilon,w) &= \frac{\epsilon (w+\epsilon)^2}{w+3\epsilon} - \frac{(w+\epsilon)^2}{2\left(1-\frac{\epsilon^4}{(w+2\epsilon)^4}\right)}. \end{split}$$* {#fig width="65%"} Suppose $q_n$ oscillates in time at $w_A$ and $g_n$ at $w_B$. If $q_n,g_n$ are two modes of the same traveling wave solution, then $N_A = N_B$. Since $N_A \sim w_A,\ N_B \sim 2w_B$ for $w_A,w_B \gg 1$, assume $w_A = 2w_B$. As [@KivCam Equation 9], the Peierls-Nabarro barrier is defined as the energy difference of the two modes at $w_A, w_B$, respectively, which can be computed explicitly by [\[energy_dnls\]](#energy_dnls){reference-type="eqref" reference="energy_dnls"}. See . **Corollary 2**. *Let $\Delta E_{AB} = E_A(\epsilon,w_A) - E_B(\epsilon,\frac{w_A}{2})$. Setting $\epsilon = k w_A$ for $k>0$, we have $$\Delta E_{AB} = -\gamma(k) w_A^2,$$ where $\gamma(k)$ is a strictly positive rational function satisfying $\gamma(k) \xrightarrow[k\rightarrow 0+]{}\frac{1}{8}$ and $\inf\limits_{k \geq 0} \gamma(k) >0$. Therefore $\Delta E_{AB}<0$ for any $\epsilon,w_A>0$ and $\lim\limits_{(\epsilon,w_A)\rightarrow (\epsilon^*,w_A^{*})} \Delta E_{AB} = 0$ if and only if $(\epsilon^*,w_A^{*}) = (0,0)$. Furthermore as $w_A \rightarrow \infty$, $$\Delta E_{AB} \sim_\epsilon -\frac{w_A^2}{8},\ \frac{E_{A}(\epsilon,w_A)}{E_{B}(\epsilon,w_B)} \sim_\epsilon 2.$$* **Remark 3**. *As $h = \epsilon^{-\frac{1}{2}}\rightarrow 0$, we have $\Delta E_{AB} \sim -\frac{4}{15 h^4}$ by direct computation via [\[energy_dnls\]](#energy_dnls){reference-type="eqref" reference="energy_dnls"}, which seems inconsistent with the exponential smallness of PNB $$|\Delta E_{AB}| \lesssim (\sqrt{w}h)^{2-d} e^{-\frac{C}{\sqrt{w}h}},\ d=1,2,3,$$ that recovers the Galilean invariance of NLS as $h \rightarrow 0$; see [@JenWeinLocal Theorem 3.3]. However, since $q_n,g_n$ do not satisfy the DNLS asymptotically as $\epsilon\rightarrow \infty$, $\Delta E_{AB}$ for $\epsilon \gg 1$ given in , or equivalently $h \ll 1$ on $h\mathbb{Z}$, does not accurately describe the PNB.* ## Localized Solutions for fDNLS. {#fDNLS} Localized on and offsite solutions for the fDNLS were studied in [@PhysRevE.55.6141]. The authors correctly point out that the long-distance behavior of the intrinsically localized states depends on the rate of decay $1 + \alpha$ and that as $\alpha$ gets larger, localization resembles that of the DNLS. As it relates to the asymptotic behavior ($n$ large), for the stationary solutions of $$-\omega {q}_n = \epsilon \sum_{m\in\mathbb{Z} \setminus \{n\} } \frac{q_n - q_m}{|n - m|^{1+\alpha}} - q_n^3,\ (n,t) \in \mathbb{Z} \times \mathbb{R},\ \epsilon>0,\ \alpha > 0,$$ and using the Green's function ($G_{n-m}$) formalism so that $q_n = \sum_{m}G_{n-m}q_m^3$, the authors in [@PhysRevE.55.6141] conclude that the tail of the mode decays algebraically if $2 < 1+ \alpha < 3$ and exponentially in $n$ for $1+ \alpha > 3$. Our approach is different and instead we show below that in fact in all instances the tail decays algebraically. Define q_0 = , q_n = , & $n \geq 1$ [\[dnls2\]]{#dnls2 label="dnls2"}\ g_0 = , g_n = , & $n \geq 2$. [\[fdnls5\]]{#fdnls5 label="fdnls5"} To provide an insight for the definitions above, the first two terms of $q_n$ are shown explicitly. For $q_0$, neglect $q_n$ for $|n| \geq 1$. Then $$-w q_0 = 2\epsilon J q_0 - q_0^3,$$ and hence $q_0$. For $q_1$, neglect the nonlinear term $q_1^3$ and $q_n$ where $|n| \geq 2$. Since $q_1 = q_{-1}$, $$-w q_1 = \epsilon\sum_{m \neq 1} J_{|1-m|}(q_1 - q_m) = 2\epsilon J q_1 - \epsilon\left(J_1 q_0 + J_2 q_1\right),$$ and so follows $q_1$. Although the analytic description of the asymptotic sequences is not immediately tractable, they are given by the power series expansion in the high-frequency regime. **Proposition 7**. *Let $\rho = \frac{\epsilon}{w}$ and $0 < \rho < \frac{1}{2J}$. Then there exist sequences $\{c_{nk}\}_{k \geq 1},\{d_{nk}\}_{k \geq 1} \subseteq \mathbb{R}$ such that $$\label{induction} \begin{split} \frac{q_n}{q_0} &= \sum_{k=1}^\infty c_{nk}\rho^k,\ n \geq 1\\ \frac{g_n}{g_0} &= \sum_{k=1}^\infty d_{nk}\rho^k,\ n\geq 2, \end{split}$$ where the series are absolutely convergent and $c_{n1} = J_n,\ d_{n1} = J_{n-1} + J_n$.* *Proof.* For brevity, our proof concerns $q_n$ only. Let $$\gamma_n = \frac{\rho}{1+\rho(2J-J_{2n})}.$$ Since $\rho$ is sufficiently small by hypothesis, $\gamma_n = \rho + O_\alpha(\rho^2)$ by series expansion where the error term is uniform in $n$ since $|\rho(2J-J_{2n})| < 2 \rho J <1$. Let $n=1$. From [\[dnls2\]](#dnls2){reference-type="eqref" reference="dnls2"}, $$\frac{q_1}{q_0} = \gamma_1 J_1 = \frac{\rho J_1}{1+\rho (2J - J_1)},$$ and hence [\[induction\]](#induction){reference-type="eqref" reference="induction"} with $c_{n1} = J_1$. Suppose the claim holds for $m = 1,\dots,n-1$. Then $$\begin{aligned} \label{taylorseries} \frac{q_n}{q_0} &= \gamma_n J_n + \gamma_n \sum\limits_{m=1}^{n-1} \left(J_{n-m} + J_{n+m}\right)\frac{q_m}{q_0}\nonumber\\ &= \gamma_n J_n + \gamma_n \sum_{k=1}^\infty \sum_{m=1}^{n-1} \left(J_{n-m} + J_{n+m}\right) c_{mk} \rho^k, \end{aligned}$$ where the series is absolutely convergent since $$\sum_{k=1}^\infty \sum_{m=1}^{n-1} \left(J_{n-m} + J_{n+m}\right) |c_{mk}| \rho^k \leq J \cdot \max_{1 \leq m \leq n-1} \left(\sum_{k=1}^\infty |c_{mk}| \rho^k \right) < \infty.$$ The second term of [\[taylorseries\]](#taylorseries){reference-type="eqref" reference="taylorseries"} is $O(\rho^2)$, and hence the dominant term of $\frac{q_n}{q_0}$ is $\rho J_n$. ◻ By , $$\left|\frac{q_{n}/q_{0}}{\rho J_n} - 1 \right| = J_n^{-1} \left| \sum_{k=2}^\infty c_{nk} \rho^{k-1}\right| \xrightarrow[\rho \rightarrow 0]{} 0,$$ and hence for any $n \in \mathbb{Z} \setminus \{0\}$, $$\label{convergence_rate} \frac{q_n}{q_0} \sim \rho J_n,\ \rho \rightarrow 0.$$ In fact, the convergence rate of [\[convergence_rate\]](#convergence_rate){reference-type="eqref" reference="convergence_rate"} is uniform in $n$. **Proposition 8**. *For all $\epsilon_1,\alpha >0$, there exists $\rho_* = \rho_*(\epsilon_1,\alpha) > 0$ such that for any $0< \rho < \rho_*$, $$\label{uniformbound} \begin{split} \frac{1}{1+\epsilon_1} &< \frac{q_n/q_0}{\rho J_{n}} < 1+\epsilon_1,\ n \geq 1,\\ \frac{1}{1+\epsilon_1} &\leq \frac{g_n/g_0}{\rho (J_{n} + J_{n-1})} \leq 1+\epsilon_1,\ n \geq 2. \end{split}$$* *Proof.* The proof is for $q_n$ without loss of generality. Since $n^{1+\alpha} J_n \xrightarrow[n\rightarrow \infty]{} A$, there exists $N \in \mathbb{N}$ such that for any $n \geq N$, we have $|J_n - \frac{A}{n^{1+\alpha}}|<\frac{A}{2n^{1+\alpha}}$. Define $$\rho_* = \min\left(\frac{\epsilon_1}{2J},\min\limits_{2 \leq n \leq 2N}\left(\frac{\epsilon_1 J_n}{2(1+\epsilon_1)(n-1)J^2}\right), \frac{\epsilon_1}{3(1+2^{2+\alpha})(1+\epsilon_1)J}\right).$$ By [\[dnls2\]](#dnls2){reference-type="eqref" reference="dnls2"}, $$\frac{q_n}{q_0} \geq \gamma_n J_n > \frac{\rho J_n}{1+2\rho J} > \frac{\rho J_n}{1+\epsilon_1},$$ and hence the lower bound of [\[uniformbound\]](#uniformbound){reference-type="eqref" reference="uniformbound"}. The proof for the upper bound is by induction. For the base case, we have $$\frac{q_1}{\rho J_1 q_0} = \frac{1}{1+\rho(2J-J_2)} < 1+\epsilon_1.$$ Let $C = (1+\epsilon_1)\rho q_0$. For $2 \leq n \leq 2N$, $$q_n < \rho\left(J_n q_0 + C \sum_{m=1}^{n-1} (J_{n-m}+J_{n+m})J_m\right) \leq \rho \left( J_n q_0 + 2C(n-1)J^2\right) < C J_n,$$ since $\rho < \rho_*$. Let $n > 2N$. Suppose $q_m < C J_m$ for all $1 \leq m \leq n-1$. Observe that $$\begin{split} \sum_{m=1}^{n-1} J_{n-m}q_m &< C\left(\sum_{1 \leq m \leq \frac{n}{2}} J_{n-m}J_m + \sum_{\frac{n}{2} < m \leq n-1} J_{n-m}J_m\right)\\ &< \frac{3CA}{2} \left(\sum_{1 \leq m \leq \frac{n}{2}} \frac{J_m}{(n-m)^{1+\alpha}} + \sum_{\frac{n}{2}<m\leq n-1} \frac{J_{n-m}}{m^{1+\alpha}}\right)\\ &\leq 3 \cdot 2^{1+\alpha} CAn^{-(1+\alpha)}J < 3 \cdot 2^{2+\alpha} C J J_n. \end{split}$$ A similar computation yields $$\sum_{m=1}^{n-1} J_{n+m} q_m < 3CJ J_n,$$ and altogether, $$q_n < \rho \left(q_0 + 3(1+2^{2+\alpha})C J \right) J_n < C J_n.$$ This completes the induction, and the claim for $g_n$ follows similarly. ◻ **Corollary 3**. *Assuming the hypotheses of , $$\frac{q_n}{q_0} \sim \frac{A\rho}{n^{1+\alpha}},\hspace{20pt} \frac{q_{n+1}}{q_n} \sim \left(\frac{n}{n+1}\right)^{1+\alpha},\hspace{20pt} \frac{q_{2n}}{q_n} \sim 2^{-(1+\alpha)},$$ as $\rho \rightarrow 0$ and $n \rightarrow \infty$, and similarly for $\{g_n\}$.* As , it is shown that [\[dnls2\]](#dnls2){reference-type="eqref" reference="dnls2"}, [\[fdnls5\]](#fdnls5){reference-type="eqref" reference="fdnls5"} solve [\[generalmodel_stationary\]](#generalmodel_stationary){reference-type="eqref" reference="generalmodel_stationary"} in an asymptotic sense. **Proposition 9**. *For any $\epsilon,w,\alpha>0$, define $\{q_n\}$ by [\[dnls2\]](#dnls2){reference-type="eqref" reference="dnls2"}. Then $q_{n+1} = O(q_n)$ as $\rho \rightarrow 0$, and $$\label{fdnls_asymp} \lim_{\rho \rightarrow 0} \sup_{n \in \mathbb{Z}}\left| \frac{-wq_n}{\epsilon \mathscr{L}_\alpha q_n - q_n^3} - 1\right| = 0,$$ and similarly for $\{g_n\}$.* *Proof.* The big-O bound follows from . The proof is for $q_n$ and $n \geq 1$. By , $$-wq_n \sim -w \rho J_n q_0 = -w^{\frac{3}{2}}\rho J_n (1+2\rho J)^{\frac{1}{2}}.$$ The interaction term is given by $$\epsilon\sum_{m \neq n} J_{|n-m|}(q_n - q_m) = \epsilon J_{n}(q_n - q_0) + \epsilon\sum_{m \notin \{0,n\}} J_{|n-m|}(q_n - q_m) =: I + II,$$ and estimating the terms separately, we have $$I \sim -\epsilon J_n q_0 = - w^{\frac{3}{2}} \rho J_n (1+2\rho J)^{\frac{1}{2}},$$ and $$\label{asymp_estimate} II \sim \epsilon \rho q_0 \sum_{m \notin \{0,n\}} J_{|n-m|} (J_{|n|} - J_{|m|}) \leq 2 w^{\frac{3}{2}} \rho^2 (1+2\rho J)^{\frac{1}{2}}J^2.$$ It follows that $II$ is $O(\rho^2)$ and hence negligible. So is the nonlinear term, which is $O(w^{\frac{3}{2}} \rho^{3})$. The uniformity in $n$ follows as . The limit [\[fdnls_asymp\]](#fdnls_asymp){reference-type="eqref" reference="fdnls_asymp"} can be shown similarly for the case $n=0$. ◻ ## Peierls-Nabarro Barrier for fDNLS. In this part, we assume $J_n = n^{-(1+\alpha)}$. The PNB for fDNLS is fundamentally different from that of the nearest-neighbor interaction. The PNB can take both positive and negative values, and it can increase in $\epsilon$ or $w$ depending on $\alpha$; see . This suggests that the onsite solution need not always be the energy minimizer of the Hamiltonian. It is of interest to further investigate the role of non-locality in the framework of variational approach to fDNLS and the stability properties of onsite/offsite solutions. By [\[energy\]](#energy){reference-type="eqref" reference="energy"} and the symmetry properties of the onsite/offsite solutions, an analogue of is derived. The proof follows from and standard algebra, and thus is omitted. **Proposition 10**. *Let $q_n,g_n$ be given by [\[dnls2\]](#dnls2){reference-type="eqref" reference="dnls2"}, [\[fdnls5\]](#fdnls5){reference-type="eqref" reference="fdnls5"}, respectively. Then, $$\label{energy_fdnls} \begin{split} E_A &= \epsilon\sum_{n=1}^\infty\Biggl( \frac{|q_n - q_0|^2}{n^{1+\alpha}}+ \frac{1}{2} \sum_{\substack{m=1 \\ m \neq n}}^{\infty} \left(\frac{1}{|n-m|^{1+\alpha}} + \frac{1}{|n+m|^{1+\alpha}}\right)|q_n - q_m|^2\Biggl)-\left(\frac{1}{4} q_0^4 + \frac{1}{2} \sum_{n=1}^\infty q_n^4\right),\\ E_B &= \epsilon \sum_{n=2}^\infty \Biggl(\left(\frac{1}{n^{1+\alpha}}+\frac{1}{(n-1)^{1+\alpha}}\right)|g_n - g_0|^2\\ &\hspace{120pt}+ \frac{1}{2}\sum_{\substack{m=2 \\ m\neq n}}^{\infty}\left(\frac{1}{|n-m|^{1+\alpha}}+\frac{1}{|n+m-1|^{1+\alpha}}\right)|g_n - g_m|^2\Biggl) -\frac{1}{2} \sum_{n=1}^\infty g_n^4. \end{split}$$* The analysis in was simplified under the assumption $\rho = \frac{\epsilon}{w} \ll 1$. A similar analysis is further developed in the context of PNB using . By the conservation of particle number, assume $w_A = 2w_B$. {#fig_PNBfDNLS width="45%"} {#fig_PNBfDNLS width="45%"} **Corollary 4**. *As $w_A, w_B \rightarrow \infty$, $$\label{energy_fdnls2} \begin{split} E_A &\sim -\frac{w_A^2}{4} -(2\zeta(2+2\alpha) + \zeta(1+\alpha)^2)\epsilon^2,\\ E_B &\sim -\frac{w_B^2}{2} - \left(2 \sum_{n=2}^\infty \left(\frac{1}{n^{1+\alpha}} + \frac{1}{(n-1)^{1+\alpha}}\right)^2 + \frac{1}{2}(2\zeta(1+\alpha) - 1)^2\right)\epsilon^2. \end{split}$$ Therefore, $E_A(w_A) \sim 2 E_B(w_B)$ and $$\label{PNB_fdnls_wlarge} \Delta E_{AB} \sim -\frac{w_A^2}{8}.$$* **Corollary 5**. *As $\epsilon \rightarrow 0$, $$\label{energy_fdnls3} \begin{split} E_A &\sim -\frac{w_A^2}{4} + \left(-2\zeta(2+2\alpha) + \zeta(1+\alpha)^2\right)\epsilon^2,\\ E_B &\sim -\frac{w_B^2}{2} + \left(-2\sum_{n=2}^\infty \left(\frac{1}{n^{1+\alpha}} + \frac{1}{(n-1)^{1+\alpha}}\right)^2 + \frac{1}{2}(2\zeta(1+\alpha)-1)^2\right)\epsilon^2. \end{split}$$ Therefore, $E_A(w_A) \sim 2 E_B(w_B)$ and $$\label{PNB_fdnls_esmall} \Delta E_{AB} \sim -\frac{w_A^2}{8} + \left(2\sum_{n=2}^\infty \left(\frac{1}{n^{1+\alpha}} + \frac{1}{(n-1)^{1+\alpha}}\right)^2 - (\zeta(1+\alpha)-1)^2 - 2 \zeta(2+2\alpha) + \frac{1}{2}\right)\epsilon^2.$$* For $w_A>0$ not sufficiently large, $PNB(w_A)$ may not be well approximated by the quadratic term; in fact, $PNB(w_A)$ may be increasing for $w_A>0$ sufficiently small. For $\epsilon>0$ not sufficiently small, the behavior of PNB is generally not quadratic in $\epsilon$ since the higher order terms cannot be neglected. For $\epsilon \ll 1$, note that PNB may increase or decrease depending on the sign of the coefficient of $\epsilon^2$. **Remark 4**. *As $\alpha \rightarrow 0$, or more precisely, for $\alpha \ll \min(1,\frac{\epsilon}{w})$, observe that $\frac{1}{\alpha}<\zeta(1+\alpha)<1+\frac{1}{\alpha}$ yields $$N_A \sim q_0^2 \sim \frac{2\epsilon}{\alpha},\ N_B \sim 2 g_0^2 \sim \frac{4\epsilon}{\alpha}.$$ Since $N_B \sim 2 N_A$, the conservation of particle number as a localized wave travels along the lattice in the forms of onsite and offsite waves, if such solutions exist at all, does not hold. This suggests that the PNB for small $\alpha>0$ calculated from [\[energy_fdnls\]](#energy_fdnls){reference-type="eqref" reference="energy_fdnls"} is non-physical.* {#fig:mobility_figure width="75%"} {#fig:6 width="\\linewidth"} {#fig:6 width="\\linewidth"} {#fig:6 width="\\linewidth"} {#fig:6 width="\\linewidth"} In , the mobility/pinning of peak intensity is observed in the nonlocal dynamics given by $J_{n} = |n|^{-(1+\alpha)}$ for $n \neq 0$ with the initial condition as the onsite sequence defined in [\[dnls2\]](#dnls2){reference-type="eqref" reference="dnls2"}. As $\alpha \rightarrow 0$, the nonlocal coupling blows up as $\lim\limits_{\alpha \rightarrow 0+}\zeta(1+\alpha)=\infty$. Moreover the conservation of particle numbers between the onsite and offsite solutions fails in the sense described in , resulting in the erratic behavior illustrated in the top-left plot of . For $\alpha$ not too small, the intensity drifts and eventually pins at a lattice point. More precisely as $\alpha \to \infty$, the non-locality weakens, leading to a weaker drift (pinning) at earlier times. See [@jenkinson_weinstein_2017 Figure 1] for the drift and pinning under the DNLS dynamics for varying mesh grid sizes $h>0$ whereas plots the argmax of peak intensity for various $\alpha$. # Conclusion The study of existence, dynamics and interactions of coherent, localized structures in nonlinear lattices remains an active field given the wide range of applications. Here we presented work for a model where the coupling between units (oscillators, waveguides, resonators) of a one dimensional arrays is global, with strength decaying algebraically with respect to the index difference. Results include the characterization of modulational instability and emergence of nonlinear modes for large times, obtained by numerical simulations. By use of asymptotic techniques we show the existence of localized modes and to assess mobility, derived formulas for the Peierls-Nabarro barrier. In all instances we point to the behavior in terms of the coupling strength parameter $\alpha$. This parametric dependece of the dynamics can have potential applications. Future work will further explore these applications, extend work to consider interaction properties (collisions) between localized modes. Finally, there are two natural extensions of this work; first the study of two-dimensional lattices with similar coupling functionality and second is for the long wave limit, to identify the continuum limit. # Acknowledgements Both B.C. and A.M. acknowledge support from the NSF RTG award, DMS-1840260. # Appendix For our numerical simulations, a particular interaction kernel $J_n = |n|^{-(1+\alpha)}$ was used. Consider $(u_n)$ for $n = -N, \dots, N$. Then for the Dirichlet boundary condition, we have $$\label{fraclap_dirichet} \mathscr{L}_\alpha u_n = \sum_{-N \leq m \leq N, m \neq n} \frac{u_n - u_m}{|n-m|^{1+\alpha}},$$ where $u_m = 0$ for all $|m| > N$. For the periodic boundary condition, consider the modular arithmetics where the quotient space of $\mathbb{Z}$, with $I_N := \{-N,\dots,N-1\}$ as the fundamental cell, is considered, as $u_{-N} = u_{N}$. Given $m \in \mathbb{Z}$, let $m = 2Nq + r$ where $q \in \mathbb{Z},\ r \in I_N$, and assume $u_m = u_r$. Then for $n \in I_N$, $$\label{fraclap_periodic} \begin{split} \mathscr{L}_\alpha u_n &= \sum_{m \neq n} \frac{u_n - u_m}{|n-m|^{1+\alpha}}\\ &= 2\zeta(1+\alpha)\left(1-\frac{1}{(2N)^{1+\alpha}}\right)u_n - \sum \limits_{r\neq n} c_{nr}(N,\alpha)u_r, \end{split}$$ where $$\label{fraclap_periodic2} c_{nr} := \frac{1}{|n-r|^{1+\alpha}} + \frac{1}{(2N)^{1+\alpha}} \bigg\{ \zeta(1+\alpha, \frac{r-n}{2N}) + \zeta(1+\alpha, -\frac{r-n}{2N}) - |\frac{r-n}{2N}|^{-(1+\alpha)}(1+e^{-i(1+\alpha)\pi}) \bigg\},$$ and $\zeta(s) = \sum\limits_{k=1}^\infty \frac{1}{k^s}$ is the Riemann zeta function and $\zeta(s,a) = \sum\limits_{k=0}^\infty \frac{1}{(k+a)^s}$ is the Hurwitz zeta function. Lastly we provide a brief derivation of [\[fraclap_periodic2\]](#fraclap_periodic2){reference-type="eqref" reference="fraclap_periodic2"}. Using $m = 2Nq + r$ as above, $$\begin{split} \sum_{m \neq n} \frac{u_n - u_m}{|n-m|^{1+\alpha}} &= 2 \zeta(1+\alpha) u_n - \sum_{m \neq n} \frac{u_m}{|n-m|^{1+\alpha}}\\ &= 2 \zeta(1+\alpha) u_n - \sum_{r \in I_N \setminus \{n\}} \frac{u_r}{|n-r|^{1+\alpha}} - \sum_{r \in I_N} \sum_{q \in \mathbb{Z} \setminus \{0\}} \frac{u_r}{|q(2N) + r - n|^{1+\alpha}} \end{split}$$ The last sum simplifies to $2\zeta(1+\alpha) (2N)^{-(1+\alpha)}u_n$ when $r = n$. When $r \neq n$, $$\begin{split} \sum_{q \in \mathbb{Z} \setminus \{0\}} \frac{(2N)^{1+\alpha}u_r}{|q(2N) + r - n|^{1+\alpha}} &= \sum_{q=1}^\infty\frac{1}{(q+\frac{r-n}{N})^{1+\alpha}} + \sum_{q=1}^\infty\frac{1}{(q-\frac{r-n}{N})^{1+\alpha}}\\ &=\zeta(1+\alpha,\frac{r-n}{N}) + \zeta(1+\alpha,-\frac{r-n}{N}) - |\frac{r-n}{N}|^{-(1+\alpha)}\left(1+e^{-i(1+\alpha)\pi}\right). \end{split}$$ Rearranging terms, the expression for $\mathscr{L}_\alpha u_n$ is derived as a matrix multiplication with dense entries. [^1]: Corresponding author. Southern Methodist University, United States Military Academy, `[email protected]` [^2]: Southern Methodist University, `[email protected]` [^3]: Southern Methodist University, `[email protected]`

arxiv_math

--- abstract: | In this paper, we consider the travel time tomography problem for conformal metrics on a bounded domain, which seeks to determine the conformal factor of the metric from the lengths of geodesics joining boundary points. We establish forward and inverse stability estimates for simple conformal metrics under some a priori conditions. We then apply the stability estimates to show the consistency of a Bayesian statistical inversion technique for travel time tomography with discrete, noisy measurements. address: - Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106-3080, USA - Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106-3080, USA author: - Ashwin Tarikere - Hanming Zhou bibliography: - bibfile.bib title: Stability and statistical inversion of travel time tomography --- # Introduction Consider a smooth, bounded, and simply connected domain $\Omega\subseteq\mathbb R^m$, with $m\geq 2$. Given a Riemannian metric $g$ on $\overline{\Omega}$, we define the associated *boundary distance function* $\Gamma_g:\partial\Omega\times \partial\Omega\to [0,\infty)$ by $$\Gamma_g(\xi,\eta) = \inf \left\{ \int_\gamma \,d|g|:=\int_0^T |\dot\gamma(t)|_g\,dt \ : \ \gamma \in C^1([0,T],\overline{\Omega}), \ \gamma(0)=\xi,\ \gamma(T)=\eta \right\}$$ for all $\xi,\eta \in \partial\Omega$. In other words, $\Gamma_g(\xi,\eta)$ is the Riemannian distance (with respect to $g$) between the boundary points $\xi$ and $\eta$. We consider the following inverse problem: *Can we recover the metric $g$ in the interior of the domain from the boundary distance function $\Gamma_g$?* This inverse problem, called the *boundary rigidity problem* in mathematics literature, arose in geophysics in an attempt to determine the inner structure of the earth, such as the sound speed or index of refraction, from measurements of travel times of seismic waves on the earth's surface. This is called the *inverse kinematic problem* or the *travel time tomography problem* in seismology [@He1905; @WZ1907]. The boundary rigidity problem is not solvable in general. Consider, for example, a unit disk with a metric whose magnitude is large (and therefore, geodesic speed is low) near the center of the disk. In such cases, it is possible that all distance minimizing geodesics connecting boundary points avoid the large metric region, and therefore one can not expect to recover the metric in this region from the boundary distance function. In view of this restriction, one needs to impose additional geometric conditions on the metric to be reconstructed. One such condition is *simplicity*. A metric $g$ on $\overline{\Omega}$ is said to be simple if the boundary $\partial\Omega$ is strictly convex w.r.t. to $g$ and any two points on $\overline{\Omega}$ can be joined by a unique distance minimizing geodesic. Michel conjectured that simple metrics are boundary distance rigid [@Mi81], and this has been proved in dimension two [@PU05]. In dimensions $\geq 3$, this is known for generic simple metrics [@SU05]. When caustics appear, a completely new approach was established in [@SUV16; @SUV21] for the boundary rigidity problem in dimensions $\geq 3$, assuming a convex foliation condition. Boundary rigidity problems for more general dynamical systems can be found in [@DPSU07; @AZ15; @Zhou18; @PUZ19; @LOY16; @YUZ21; @Plamen22]. We also refer to [@Croke04; @SUVZ19] for summaries of recent developments on the boundary rigidity problem. The boundary rigidity problem for general Riemannian metrics has a natural gauge: isometries of $(\overline{\Omega},g)$ that preserve $\partial\Omega$ will also preserve the boundary distance function. In this paper, we restrict our attention to the problem of determining metrics from a fixed conformal class. Let $\bar g$ be a fixed "background\" metric on $\overline{\Omega}$ which is simple and has $C^3$ regularity. For any positive function $n \in C^3(\overline{\Omega})$, define $$g_n:=n^2\bar g,$$ which is a new Riemannian metric on $\overline{\Omega}$ that is conformal to $\bar g$. Our goal is to recover the parameter $n$ from the boundary distance function of $g_n$. In this problem, the gauge of isometries does not appear, and one expects to be able to uniquely determine the conformal factor $n$ from $\Gamma_{g_n}$. It is known that simple metrics from the same conformal class are boundary rigid for all $m\geq 2$ [@Mu75a; @Mu75b; @MR78]. To be precise, if $n_1,n_2 \in C^3(\overline{\Omega})$ are such that $g_{n_1},g_{n_2}$ are both simple metrics on $\overline{\Omega}$, then $\Gamma_{g_{n_1}}=\Gamma_{g_{n_2}}$ if and only if $n_1=n_2$. To simplify notation, we will henceforth denote $\Gamma_{g_n}$ by simply $\Gamma_n$. ## Stability estimates for the deterministic inverse problem The uniqueness aspect of the boundary rigidity problem for conformal simple metrics has been quite well understood through the aforementioned studies. The first topic of this paper is the *stability* of the boundary rigidity problem, i.e., quantitative lower bounds on the change in $\Gamma_n$ corresponding to a change in the parameter $n$. Stability is important in practice, as we hope the inversion method for travel time tomography will be stable under perturbations of the data, e.g., by noise. Conditional stability estimates for simple metrics can be found in [@Wa99; @SU05; @SUV16], where the metrics are assumed *a priori* to be close to a given one. When considering a fixed conformal class, various stability estimates without the closeness assumption have been established in [@Mu75b; @Mu81; @Be79]. In [@Mu75b] the following stability result has been proved for the 2D boundary rigidity problem with the Euclidean background metric: $$\label{2D nonlinear stability} \|n_1-n_2\|_{L^2(\Omega)}\leq \frac{1}{\sqrt{2\pi}}\|d_\xi (\Gamma_{n_1}-\Gamma_{n_2})(\xi,\eta)\|_{L^2(\partial\Omega\times\partial\Omega)}.$$ Here, $d_\xi$ is the exterior derivative operator with respect to $\xi$ and the $L^2$ norms are taken with respect to the standard Euclidean metric. Notice that since the boundary distance function is symmetric, this estimate essentially says that the $L^2$-norm of $n_1-n_2$ can be controlled by the $H^1$-norm of $\Gamma_{n_1}-\Gamma_{n_2}$. For dimensions $\geq 3$, there are generalizations [@Be79; @Mu81] of [\[2D nonlinear stability\]](#2D nonlinear stability){reference-type="eqref" reference="2D nonlinear stability"} with more complicated expressions (see also Theorem [Theorem 6](#beylkin){reference-type="ref" reference="beylkin"}). However, the estimates of [@Be79; @Mu81] are not in standard Sobolev or Hölder norms, which makes them inconvenient for applications. In this paper, we establish stability estimates similar to [\[2D nonlinear stability\]](#2D nonlinear stability){reference-type="eqref" reference="2D nonlinear stability"} for all dimensions $\geq 2$, without any *a priori* closeness assumptions on $n_1,n_2$. Before giving the statement of our results, we need to define some function spaces for the conformal parameter $n$. **Definition 1**. Let $\Omega_0$ be a smooth, relatively compact subdomain of $\Omega$, and let $\lambda,\Lambda, \ell, L$ be real numbers such that $$0< \lambda< 1 < \Lambda, \qquad 0<\ell<L.$$ We define $\mathcal{N}_{\lambda,\Lambda,\ell,L}(\Omega_0)$ to be the set of all functions $n \in C^3(\overline{\Omega})$ that satisfy the following conditions: (i) The metric $g_n = n^2\bar{g}$ is a simple metric on $\overline{\Omega}$. (ii) $\lambda< n(x) < \Lambda$ for all $x \in \overline{\Omega}$ and $n \equiv 1$ on $\overline{\Omega}\setminus \Omega_0$. (iii) Let $\exp_n(x,w)$ denote the exponential map with respect to $g_n$ based at $x \in \overline{\Omega}$ and acting on $w \in T_x\overline{\Omega}$. Then the derivative of $\exp_n(x,\cdot)$ satisfies $$\label{operatornorm0} \ell|v|_{\bar{g}} < |D_w \exp_n(x,w)(v)|_{\bar{g}} < L|v|_{\bar{g}}$$ for all $x \in \overline{\Omega}$, $w\in \operatorname{dom}(\exp_n(x,\cdot))$, and $v\in T_{w}T_x\overline{\Omega}\cong T_x\overline{\Omega}$. We also let $$\mathcal{N}_{\lambda, \ell}(\Omega_0) := \bigcup_{\Lambda>1, \, L>0}\mathcal{N}_{\lambda,\Lambda,\ell,L}(\Omega_0).$$ The class of metrics associated with these function spaces includes any metric with non-positive sectional curvature that is conformal to $\bar{g}$ and equal to $\bar{g}$ in a neighborhood of $\partial\Omega$ . Indeed, suppose $g_n = n^2\bar{g}$ is such a metric. Then $(\overline{\Omega},g_n)$ is free of conjugate points by the curvature assumption, and $\partial\Omega$ remains strictly convex with respect to $g_n$ since $g_n \equiv \bar{g}$ near $\partial\Omega$. Therefore, $g_n$ is a simple metric. Moreover, it follows from the Rauch Comparison Theorem that its exponential map $\exp_n$ satisfies [\[operatornorm0\]](#operatornorm0){reference-type="eqref" reference="operatornorm0"} for sufficiently large $L$ and any $\ell<1$ (see, e.g., [@CE08 Corollary 1.35]). *Remark 1* (Notation). Let $(\mathcal{M}_1,g_1)$ and $(\mathcal{M}_2,g_2)$ be Riemannian manifolds, and let $F:\mathcal{M}_1\to \mathcal{M}_2$ be a $C^1$ map. The derivative of $F$ at $x\in \mathcal{M}_1$ is a linear map from $T_x\mathcal{M}_1 \to T_{F(x)}\mathcal{M}_2$. We define the *operator norm* of $DF$ at $x$ as $$\|DF(x)\|_{op} := \sup \left\{ |DF(x)(v)|_{g_2} \ : \ v \in T_x\mathcal{M}_1, \ |v|_{g_1} =1\right\}.$$ Using this notation, [\[operatornorm0\]](#operatornorm0){reference-type="eqref" reference="operatornorm0"} can be rewritten as $$\label{operatornorm} \ell < \|D_w \exp_n(x,w)\|_{op} < L.$$ *Remark 2*. Let $\delta>0$ be the distance (w.r.t. to $\bar{g}$) between $\partial\Omega$ and $\overline{\Omega}_0$, and let $\xi,\eta \in \partial\Omega$ be any pair of boundary points such that $\operatorname{dist}_{\bar{g}}(\xi,\eta)<\delta$. For any $n \in \mathcal{N}_{\lambda,\ell}(\Omega_0)$, $g_{n}$ coincides with $\bar{g}$ on $\overline{\Omega}\setminus \Omega_0$, and consequently, we have $\Gamma_n(\xi,\eta)= \operatorname{dist}_{\bar{g}}(\xi,\eta)$. In particular, $\Gamma_{n_1}(\xi,\eta)=\Gamma_{n_2}(\xi,\eta)$ for all $n_1,n_2 \in \mathcal{N}_{\lambda,\ell}(\Omega_0)$. We are now ready to state our result on stability estimates for the boundary rigidity problem. **Theorem 2**. *Let $\Omega,\Omega_0,\bar{g}$ be as before, and let $\lambda,\ell$ be real numbers such that $$0 < \lambda< 1, \qquad 0 < \ell.$$ Then there exists a constant $C_1(\Omega,\Omega_0,\bar{g},\ell)>0$ such that for all $n_1,n_2 \in \mathcal{N}_{\lambda,\ell}(\Omega_0)$, $$\|n_1-n_2\|_{L^2(\Omega)} \leq C_1\lambda^{2-m}\|d_{\xi}(\Gamma_{n_1}-\Gamma_{n_2})(\xi,\eta)\|_{L^2(\partial\Omega\times \partial\Omega)}.$$* Here, the $L^2$ norms are taken with respect to the background metric $\bar{g}$, and $d_\xi$ represents the exterior derivative operator with respect to $\xi$. We will apply the above stability estimate to study a statistical inversion technique for travel time tomography. For this purpose, we also need the following continuity (or "forward stability\") estimate of $\Gamma_n$. To the best of our knowledge, no such continuity estimate has been proved before. **Theorem 3**. *Let $\Omega,\Omega_0,\bar{g}$ be as before, and let $\lambda,\Lambda,\ell,L$ be real numbers such that $$0 < \lambda< 1 < \Lambda, \qquad 0 < \ell < L.$$ Then there exists a constant $C_2(\Omega,\Omega_0,\bar{g},\ell,L)>0$ such that for all $n_1,n_2 \in \mathcal{N}_{\lambda,\Lambda,\ell,L}(\Omega_0)$, $$\|\Gamma_{n_1}-\Gamma_{n_2}\|_{L^2(\partial\Omega\times \partial\Omega)} \leq C_2 \frac{\Lambda^{m/2}}{\lambda}\|n_1-n_2\|_{L^2(\Omega)}.$$* ## The statistical inverse problem The boundary rigidity problem is nonlinear, and geodesics are curved in general, so it is hard to derive explicit inversion formulas. Some reconstruction algorithms and numerical implementations based on theoretical analyses can be found in [@CQUZ07; @CQUZ08; @ACU19]. Typically, inversion methods in travel time tomography take an optimization approach with appropriate regularization. This is a deterministic approach which seeks to minimize some mismatch functional that quantifies the difference between the observations and the forecasts (synthetic data). However, this approach generally does not work well for non-convex problems. Moreover, various approximations in numerical methods can introduce systematic (random) error to the reconstruction procedure. In this paper, we apply the above stability estimates (Theorems [Theorem 2](#inverse){reference-type="ref" reference="inverse"} and [Theorem 3](#forward){reference-type="ref" reference="forward"}) to study a Bayesian inversion technique for the travel time tomography problem. The Bayesian inversion technique provides a reasonable solution for ill-posed inverse problems when the number of available observations is limited, which is a common scenario in practice. Applications of Bayesian inversion to seismology can be found in [@MWBG12; @TGMS13], which are based on the general paradigm of infinite dimensional Bayesian inverse problems developed by Stuart [@Stuart10]. However, most studies in the literature are concerned with waveform inversion, which is more PDE-based. On the other hand, there are very few results on statistical guarantees for the Bayesian approach to seismic inverse problems. These motivate us to apply Stuart's Bayesian inversion framework to produce a rigorous statistical analysis of the problem of recovering the wave speed from the (noisy) travel time measurements. For statistical inversion, it is convenient to rewrite the conformal factor $n$ using an exponential parameter: For any $\beta \geq 3$, let $C_0^\beta(\Omega_0)$ denote the closure in the Hölder space $C^{\lfloor \beta\rfloor, \beta - \lfloor \beta \rfloor}(\overline{\Omega}_0)$ of the subspace of all smooth functions compactly supported in $\Omega_0$. Given any function $c\in C_0^3(\Omega_0)$, we define the corresponding conformal factor $n_c$ by $$\label{ncdef} n_c(x) = \begin{cases} e^{c(x)} & \textrm{if } x \in \Omega_0, \\ 1 & \textrm{if } x \in \overline{\Omega}\setminus \Omega_0. \end{cases}$$ It is easy to see that $n_c$ is a positive $C^3$ function on $\overline{\Omega}$. To simplify notation, we will denote the corresponding boundary distance function $\Gamma_{n_c}$ by simply $\Gamma_c$. Our goal is to reconstruct the exponential parameter $c$ from error-prone measurements of $\Gamma_c$ on finitely many pairs of boundary points $(X_i,Y_i)$, $i = 1, \ldots, N$. Following the general paradigm of Bayesian inverse problems, we assume that $c$ arises from a prior probability distribution $\Pi$ on $C^3_0(\Omega_0)$. We will construct $\Pi$ so that it is supported in a subset of $C_0^3(\Omega_0)$ of the following form: **Definition 4**. Let $\ell,M>0$ and $\beta \geq 3$. We define $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ as the set of all functions $c \in C_0^\beta(\Omega_0)$ that satisfy the following conditions: (i) The metric $g_{n_c}=n_c^{2}\bar{g}$ is a simple metric on $\overline{\Omega}$. (ii) The derivative of $\exp_{n_c}(x,\cdot)$ satisfies $$\|D_w \exp_{n_c}(x,w)\|_{op} > \ell$$ for all $x\in \overline{\Omega}$ and $w\in \operatorname{dom}(\exp_{n_c}(x,\cdot))$. (iii) $\|c\|_{C^{\lfloor \beta\rfloor, \beta - \lfloor \beta \rfloor}(\overline{\Omega}_0)} < M$. We will show in Section 2 that if $c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$, the corresponding conformal parameter $n_c \in \mathcal{N}_{\lambda,\Lambda,\ell,L}(\Omega_0)$ for appropriate choices of $\lambda,\Lambda$ and $L$. The precise construction of $\Pi$ is described in Section 3. *Remark 3* (Notation). Henceforth, we will denote $C^{\lfloor \beta\rfloor, \beta - \lfloor \beta \rfloor}$ by simply $C^\beta$. *Remark 4*. It is known that small perturbations of simple metrics are again simple. Therefore, $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ is an open subset of $C^\beta_0(\Omega_0)$. The pairs of boundary points $(X_i,Y_i)$ between which the distance measurements are to be made are chosen according to the rule $$(X_i,Y_i) \stackrel{\textrm{i.i.d.}}{\sim} \mu,$$ where $\mu$ is the uniform probability measure on $\partial\Omega\times \partial\Omega$ induced by the background metric $\bar g$. The actual distance measurements between these points are assumed to be of the form $$\Gamma_i = e^{\epsilon_i}\Gamma_c(X_i,Y_i),$$ where $\epsilon_i$ are i.i.d. $N(0, \sigma^2)$ normal random variables ($\sigma >0$ is fixed) that are also independent of $(X_j,Y_j)_{j=1}^N$. We will assume for simplicity that $\sigma = 1$. Define $$Z_c = \log \Gamma_c,$$ and for $i= 1, \ldots, N$, $$\begin{aligned} Z_i &= \log \Gamma_i\\ &= Z_c(X_i,Y_i) +\epsilon_i.\end{aligned}$$ All of our measurements can be summarized using the data vector $$\label{obs} \mathcal{D}_N = (X_i,Y_i,Z_i)_{i=1}^N \in (\partial\Omega\times \partial\Omega\times \mathbb R)^N.$$ For convenience, let us define $\mathcal{X}= \partial\Omega\times \partial\Omega\times \mathbb R$. Next, let $P^N_c$ denote the probability law of $\mathcal{D}_N|c$. It is easy to see that $P^N_c = \times_{i=1}^N P^{(i)}_c$, where for each $i \in \{1, \ldots , N\}$, $P^{(i)}_c$ is equal to the probability law of $(X_i,Y_i,Z_i)$. More explicitly, for each $i\in \{1,\ldots , N\}$, $$dP^{(i)}_c(x,y,z) = p_cd\mu(x,y)dz,$$ where $$p_c(x,y,z) = \frac{1}{\sqrt{2\pi}}\exp\left\{-\frac{1}{2}\left(z-Z_c(x,y)\right)^2\right\}.$$ We denote the posterior distribution of $c|\mathcal{D}_N$ by $\Pi(\cdot |\mathcal{D}_N)$. By Corollary [Corollary 12](#forward-c){reference-type="ref" reference="forward-c"}, the map $(c,(x,y,z))\mapsto p_c(x,y,z)$ is jointly Borel-measurable from $C^3_0(\Omega_0)\times \mathcal{X}$ to $\mathbb R$. So it follows from standard arguments (see [@GvdV17 p. 7] ) that the posterior distribution is well-defined and takes the form $$\Pi(A|\mathcal{D}_N) = \frac{\int_A \prod_{i=1}^N p_c(X_i,Y_i,Z_i)d\Pi(c)}{\int \prod_{i=1}^N p_c(X_i,Y_i,Z_i) d\Pi(c)}$$ for any Borel set $A \subseteq C^3_0(\Omega_0)$. Our posterior estimator for $c$ will be the posterior mean $$\label{postmean} \overline{c}_N = \mathbb{E}^{\Pi}[c|\mathcal{D}_N].$$ **Theorem 5**. *Suppose that the true parameter $c_0$ is smooth and compactly supported in $\Omega_0$, and is such that $g_{n_{c_0}}$ is a simple metric on $\overline{\Omega}$. Then there is a well defined prior distribution $\Pi$ on $C^3_0(\Omega_0)$ such that the posterior mean $\overline{c}_N$ satisfies $$\|\overline{c}_N - c_0\|_{L^2(\Omega)} \to 0$$ in $P^N_{c_0}$- probability, as $N \to \infty$.* A more precise version of this result is stated in Theorem [Theorem 14](#main){reference-type="ref" reference="main"} in Section [3](#stats){reference-type="ref" reference="stats"}, which in fact requires significantly weaker regularity assumptions on $c_0$. It also specifies an explicit $N^{-\omega}$ rate of convergence, where $\omega$ is a positive constant that can be made arbitrarily close to $1/4$. To prove Theorem [Theorem 5](#mainth){reference-type="ref" reference="mainth"}, we apply the analytic techniques developed in recent consistency studies of statistical inversion of the geodesic X-ray transform [@MNP19] and related non-linear problem arising in polarimetric neutron tomography [@MNP21a; @MNP21b]. The forward and inverse stability estimates for the measurement operators (like the ones in Theorems [Theorem 2](#inverse){reference-type="ref" reference="inverse"} and [Theorem 3](#forward){reference-type="ref" reference="forward"}) play a key role in the arguments of these references. The analysis of theoretical guarantees for statistical inverse problems is currently a very active topic. Recent progress for various linear and non-linear inverse problems include [@DaS16; @DuS16; @AN19; @MNP19; @Nickl20; @MNP21a; @MNP21b; @NP21; @BN21; @Bohr22]. See also the recent lecture notes [@Nickl22]. The paper is structured as follows. In Section 2, we establish the forward and inverse stability estimates for the boundary distance function. Section 3 is devoted to proving the statistical consistency of Bayesian inversion for the boundary rigidity problem. HZ is partly supported by the NSF grant DMS-2109116. # Forward and Inverse continuity estimates In order to prove the statistical consistency of the proposed Bayesian estimator, we need to establish quantitative upper and lower bounds on the magnitude of change in the boundary distance function $\Gamma_n$ corresponding to a change in the conformal parameter $n$ of the metric. This is the content of Theorems [Theorem 2](#inverse){reference-type="ref" reference="inverse"} and [Theorem 3](#forward){reference-type="ref" reference="forward"}, which we will prove in this section. We will also use these estimates to establish similar bounds for the map $c \mapsto Z_c = \log \Gamma_c$, when $c$ belongs to the parameter space $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ defined in Definition [Definition 4](#CMb){reference-type="ref" reference="CMb"}. ## Stability estimates We begin with the proof of Theorem [Theorem 2](#inverse){reference-type="ref" reference="inverse"}. As we noted in the introduction, such an estimate has already been proved for dimension $m=2$ by Mukhometov in [@Mu75b]. For general $m\geq 2$, we have the following result by Beylkin [@Be79]. Also see [@Mu81 Lemma 4]. **Theorem 6** ([@Be79]). *Let $n_1,n_2 \in C^3(\overline{\Omega})$ be such that $g_{n_1},g_{n_2}$ are simple metrics on $\overline{\Omega}$. Then $$\label{hds} \begin{split} \int_{\Omega} (n_1-n_2) & (n_1^{m-1}-n_2^{m-1})\,d\textrm{Vol}_{\bar{g}}\\ & \leq C_m \int_{\partial\Omega_{\xi}\times\partial\Omega_{\eta}} \sum_{a+b=m-2} d_\xi (\Gamma_{n_1}-\Gamma_{n_2})\wedge d_\eta (\Gamma_{n_1}-\Gamma_{n_2})\wedge (d_\xi d_\eta \Gamma_{n_1})^a \wedge (d_\xi d_\eta \Gamma_{n_2})^b\, , \end{split}$$ where $d\textrm{Vol}_{\bar{g}}$ is the Riemannian volume form induced by $\bar{g}$, and $d_\xi$ and $d_\eta$ represent the exterior derivative operators on $\partial\Omega$ with respect to $\xi$ and $\eta$ respectively. Given local coordinates $(\xi^1, \ldots , \xi^{m-1})$ for $\xi$ and $(\eta^1, \ldots , \eta^{m-1})$ for $\eta$, we have $d_\xi = d\xi^i \frac{\partial}{\partial\xi^i}$, $d_\eta = d\eta^j\frac{\partial}{\partial\eta^j}$, and $d_\xi d_\eta=d\xi^i\wedge d\eta^j \frac{\partial^2}{\partial\xi^i \partial\eta^j}$. The constant $$C_m=\frac{(-1)^{\frac{(m-1)(m-2)}{2}}\Gamma(m/2)}{2\pi^{m/2}(m-1)!}$$ depends only on the dimension $m$.* We will show that when $n_1,n_2 \in \mathcal{N}_{\lambda,\ell}(\Omega_0)$, the inequality [\[hds\]](#hds){reference-type="eqref" reference="hds"} leads to the desired stability estimate. **Lemma 7**. *Let $n \in \mathcal{N}_{\lambda,\ell}(\Omega_0)$. Then the corresponding boundary distance function $\Gamma_n$ satisfies $$|d_{\xi}\Gamma_n(\xi,\eta)|_{\bar{g}} \leq 1, \quad |d_{\eta}\Gamma_n(\xi,\eta)|_{\bar{g}} \leq 1,$$ and $$|\nabla^{\xi}\nabla^{\eta}\Gamma_n(\xi,\eta)|_{\bar{g}} \leq \frac{(1+\ell^{-1})}{\lambda}\operatorname{dist}_{\bar{g}}(\xi,\eta)^{-1}$$ for all $\xi,\eta \in \partial\Omega$ with $\xi \neq \eta$. Here, $\nabla^\xi, \nabla^\eta$ denote the covariant derivative operators with respect to $\xi$ and $\eta$ respectively, and $\operatorname{dist}_{\bar{g}}(\xi,\eta)$ is the distance from $\xi$ to $\eta$ with respect to the metric $\bar{g}$.* *Proof.* Given $\xi,\eta \in \partial\Omega$ with $\xi \neq \eta$, let $v(\xi,\eta)$ denote the unit vector (with respect to $g_n$) at $\eta$ tangent to the geodesic from $\xi$ to $\eta$. It follows from the First Variation Formula that the gradient (with respect to $g_n$) of $\Gamma_n(\xi,\cdot)$ is given by $$\label{gradeta} \operatorname{grad}_{\eta}\Gamma_n(\xi,\eta) = \Pi_{\eta} v(\xi,\eta),$$ where $\Pi_{\eta} : T_\eta \overline{\Omega}\to T_\eta \partial\Omega$ is the orthogonal projection map onto the tangent space of the boundary. Since $g_n = \bar{g}$ on $\partial\Omega$, it follows immediately that $$|d_{\eta}\Gamma_n(\xi,\eta)|_{\bar{g}} = |\operatorname{grad}_{\eta}\Gamma_n(\xi,\eta)|_{g_n} = \left|\Pi_\eta v(\xi,\eta)\right|_{g_n} \leq |v(\xi,\eta)|_{g_n}=1.$$ Similar arguments show that $|d_{\xi}\Gamma_n(\xi,\eta)|_{\bar{g}}\leq 1$ as well. Next, let $(\xi^1, \ldots , \xi^{m-1})$ and $(\eta^1, \ldots , \eta^{m-1})$ be local coordinates for $\partial\Omega$ around $\xi$ and $\eta$ respectively. We can extend these coordinate charts to boundary normal coordinates $(\xi^1, \ldots , \xi^m)$ and $(\eta^1, \ldots , \eta^m)$ by taking $\xi^m$ and $\eta^m$ to be the corresponding distance functions from the boundary. With respect to these coordinates, we may rewrite [\[gradeta\]](#gradeta){reference-type="eqref" reference="gradeta"} as $$\label{gradetac} \operatorname{grad}_\eta \Gamma_n(\xi,\eta) = \sum_{j=1}^{m-1} v^j(\xi,\eta)\frac{\partial}{\partial\eta^j}.$$ We can extend both sides of this equality to $(1,0)$-tensor fields on $\partial\Omega_\xi \times \partial\Omega_\eta$, while maintaining the equality. Taking covariant derivatives of both sides with respect to $\xi$, we get $$\label{dxigradetac} \nabla^\xi \operatorname{grad}_\eta \Gamma_n(\xi,\eta) = \sum_{i,j=1}^{m-1}\frac{\partial v^j}{\partial\xi^i}(\xi,\eta) \frac{\partial}{\partial\eta^j}\otimes d\xi^i.$$ Here, we have used the fact that the product connection on $\partial\Omega_{\xi}\times \partial\Omega_{\eta}$ satisfies $\nabla_{\partial_{\xi_i}}\partial_{\eta_j}=0$ for all $i,j$. Recall that $g_n$ is a simple metric, and its exponential map $\exp_n(x,\cdot)$ at any $x \in \overline{\Omega}$ is a diffeomorphism onto $\overline{\Omega}$. Let $w(x,\cdot):\overline{\Omega}\to T_x\overline{\Omega}$ denote its inverse map. Since $\|D_v \exp_n(x,v)\|_{op} > \ell$ for all $v$ in the domain of $\exp_n(x,\cdot)$, we have $$\label{dwnorm} \|D_y w(x,y)\|_{op} < \ell^{-1} \qquad \textrm{for all } y \in \overline{\Omega}.$$ Now observe that we have the identity $$v(\xi,\eta) = -\frac{w(\eta, \xi)}{\Gamma_n(\xi,\eta)}.$$ So by [\[gradetac\]](#gradetac){reference-type="eqref" reference="gradetac"} and [\[dxigradetac\]](#dxigradetac){reference-type="eqref" reference="dxigradetac"}, $$\begin{aligned} \nabla^\xi \operatorname{grad}_{\eta}\Gamma_n(\xi,\eta) &=-\sum_{i,j=1}^{m-1}\left\{\frac{1}{\Gamma_n(\xi,\eta)}\frac{\partial w^j (\eta, \xi)}{\partial\xi^i}-\frac{w^j(\eta,\xi)}{\Gamma_n(\xi,\eta)^2}\frac{\partial\Gamma_n(\xi,\eta)}{\partial\xi^i}\right\} \frac{\partial}{\partial\eta^j}\otimes d\xi^i \nonumber \\ &= -\frac{1}{\Gamma_n(\xi,\eta)}\left\{\sum_{i,j=1}^{m-1}\frac{\partial w^j (\eta, \xi)}{\partial\xi^i} \frac{\partial}{\partial\eta^j} \otimes d\xi^i\right\} + \frac{1}{\Gamma_n(\xi,\eta)}v(\xi,\eta)\otimes d_\xi \Gamma_n(\xi,\eta). \label{coordfree} \end{aligned}$$ Observe that $\sum_{i,j=1}^{m-1}\frac{\partial w^j (\eta, \xi)}{\partial\xi^i} \frac{\partial}{\partial\eta^j} \otimes d\xi^i$ is precisely the tensor form of the linear map $$\Pi_\eta \circ D_y w(\eta, y)\big|_{y=\xi}\circ \Pi_\xi,$$ where $\Pi_\xi$ and $\Pi_\eta$ are, as before, orthogonal projections from $T_\xi \overline{\Omega}\to T_\xi \partial\Omega$ and $T_\eta \overline{\Omega}\to T_\eta \partial\Omega$ respectively. Therefore, $$\left|\sum_{i,j=1}^{m-1}\frac{\partial w^j (\eta, \xi)}{\partial\xi^i} \frac{\partial}{\partial\eta^j} \otimes d\xi^i\right|_{\bar{g}} \leq \left\|D_y w(\eta, y)\big|_{y=\xi}\right\|_{op} < \ell^{-1}.$$ Combining this with [\[coordfree\]](#coordfree){reference-type="eqref" reference="coordfree"}, we get $$\begin{aligned} |\nabla^\xi d_\eta \Gamma_n(\xi,\eta)|_{\bar{g}} &= |\nabla^\xi \operatorname{grad}_\eta \Gamma_n(\xi,\eta)|_{\bar{g}} \\ &\leq \frac{\ell^{-1}}{\Gamma_n(\xi,\eta)} + \frac{|v(\xi,\eta)|_{\bar{g}}|d_\xi \Gamma_n(\xi,\eta)|_{\bar{g}}}{\Gamma_n(\xi,\eta)} \\ &\leq \frac{(1+\ell^{-1})}{\Gamma_n(\xi,\eta)}. \end{aligned}$$ Finally, applying the simple estimate $$\operatorname{dist}_{\bar{g}}(\xi,\eta) \leq \frac{1}{\lambda}\Gamma_n(\xi,\eta),$$ we get $$|\nabla^\xi \nabla^\eta\Gamma_n(\xi,\eta)|_{\bar{g}} = |\nabla^\xi d_\eta\Gamma_n(\xi,\eta)|_{\bar{g}} \leq \frac{(1+\ell^{-1})}{\lambda}\operatorname{dist}_{\bar{g}}(\xi,\eta)^{-1}.$$ This completes the proof. ◻ With these estimates in hand, we're now ready to prove Theorem [Theorem 2](#inverse){reference-type="ref" reference="inverse"}. *Proof of Theorem [Theorem 2](#inverse){reference-type="ref" reference="inverse"}.* Consider the inequality [\[hds\]](#hds){reference-type="eqref" reference="hds"} from Theorem [Theorem 6](#beylkin){reference-type="ref" reference="beylkin"}. For $n_1,n_2 \in \mathcal{N}_{\lambda,\ell}(\Omega_0)$, the left hand side becomes $$\label{lowboundn} \int_{\Omega} (n_1-n_2)^2(n_1^{m-2}+n_1^{m-3}n_2+\cdots +n_2^{m-2})d \textrm{Vol}_{\bar{g}} \geq (m-1)\lambda^{m-2}\|n_1-n_2\|_{L^2(\Omega)}^2.$$ Now consider the right hand side of [\[hds\]](#hds){reference-type="eqref" reference="hds"}. By Lemma [Lemma 7](#derivative estimates){reference-type="ref" reference="derivative estimates"}, $$|d_\xi d_\eta \Gamma_n|_{\bar{g}} = \left|\textrm{Alt}\left(\nabla^\xi \nabla^\eta \Gamma_n\right)\right|_{\bar{g}} \leq \frac{(1+\ell^{-1})}{\lambda}\operatorname{dist}_{\bar{g}}(\xi,\eta)^{-1}.$$ Therefore, the right hand side of [\[hds\]](#hds){reference-type="eqref" reference="hds"} is bounded above by $$\begin{aligned} & \phantom{\leq }|C_m|\int_{\partial\Omega\times \partial\Omega} |d_{\xi}(\Gamma_{n_1}-\Gamma_{n_2})|_{\bar{g}}|d_\eta(\Gamma_{n_1}-\Gamma_{n_2})|_{\bar{g}}\sum_{a+b=m-2}|d_{\xi}d_{\eta}\Gamma_{n_1}|_{\bar{g}}^a|d_{\xi}d_{\eta}\Gamma_{n_2}|_{\bar{g}}^b\, d\sigma_{\bar{g}} \\ &\leq (m-1)|C_m|\frac{(1+\ell^{-1})^{m-2}}{\lambda^{m-2}} \int_{\partial\Omega\times \partial\Omega} |d_{\xi}(\Gamma_{n_1}-\Gamma_{n_2})|_{\bar{g}}|d_\eta (\Gamma_{n_1}-\Gamma_{n_2})|_{\bar{g}}|\operatorname{dist}_{\bar{g}}(\xi,\eta)|^{2-m}\, d\sigma_{\bar{g}}, \end{aligned}$$ where $d\sigma_{\bar{g}}$ is the surface measure on $\partial\Omega\times \partial\Omega$ induced by $\bar{g}$. Observe that by Remark [Remark 2](#d){reference-type="ref" reference="d"}, we have $(\Gamma_{n_1}-\Gamma_{n_2})(\xi,\eta) =0$ for all $\xi,\eta \in \partial\Omega$ with $\operatorname{dist}_{\bar{g}}(\xi,\eta)<\delta$. Therefore, the above expression is further bounded above by $$\begin{aligned} & \phantom{\leq } (m-1)|C_m|\frac{(1+\ell^{-1})^{m-2}}{\lambda^{m-2}}\delta^{2-m}\int_{\partial\Omega\times \partial\Omega} |d_{\xi}(\Gamma_{n_1}-\Gamma_{n_2})|_{\bar{g}}|d_\eta (\Gamma_{n_1}-\Gamma_{n_2})|_{\bar{g}}|d\sigma_{\bar{g}}. \\ &\lesssim_{m,\delta, \ell}\lambda^{2-m}\left( \|d_{\xi}(\Gamma_{n_1}-\Gamma_{n_2})\|^2_{L^2(\partial\Omega\times \partial\Omega)} + \|d_{\eta}(\Gamma_{n_1}-\Gamma_{n_2})\|^2_{L^2(\partial\Omega\times \partial\Omega)}\right)\\ & \lesssim_{m,\delta, \ell} \lambda^{2-m} \|d_{\xi}(\Gamma_{n_1}-\Gamma_{n_2})\|^2_{L^2(\partial\Omega\times \partial\Omega)} \end{aligned}$$ since $\|d_\xi(\Gamma_{n_1}-\Gamma_{n_2})\|_{L^2} = \|d_\eta(\Gamma_{n_1}-\Gamma_{n_2})\|_{L^2}$ by symmetry. Combining this with [\[lowboundn\]](#lowboundn){reference-type="eqref" reference="lowboundn"}, we get $$\|n_1-n_2\|^2_{L^2(\Omega)} \lesssim_{m,\delta,\ell} \lambda^{2(2-m)}\|d_{\xi}(\Gamma_{n_1}-\Gamma_{n_2})\|^2_{L^2(\partial\Omega\times \partial\Omega)}$$ and the theorem follows. ◻ Recall that we parametrized the conformal parameter $n$ of the metric $g_n$ by a function $c$ belonging to the parameter space $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$, as defined in [\[ncdef\]](#ncdef){reference-type="eqref" reference="ncdef"}. We assumed that our input data consists of finitely many measurements of the function $Z_c = \log \Gamma_{c}$. In the following corollary, we translate Theorem [Theorem 2](#inverse){reference-type="ref" reference="inverse"} into stability estimates for the map $c \mapsto Z_c$ using simple Lipschitz estimates for the exponential function: For all $x,y \in [M_1,M_2]$, $$\label{lipexp} e^{M_1}|x-y| \leq |e^x-e^y| \leq e^{M_2}|x-y|.$$ This immediately implies that for all $c_1,c_2\in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$, $$\label{link-L2} e^{-M}\|c_1-c_2\|_{L^2(\Omega_0)} \leq \|n_{c_1}-n_{c_2}\|_{L^2(\Omega)} \leq e^M\|c_1-c_2\|_{L^2(\Omega_0)}.$$ **Corollary 8**. *For any $M>0$, there exists a constant $C_1' = C_1'(\Omega,\Omega_0,\bar{g},\ell,M)>0$ such that $$\|c_1-c_2\|_{L^2(\Omega_0)} \leq C_1'\|Z_{c_1}-Z_{c_2}\|_{H^1(\partial\Omega\times \partial\Omega)}$$ for all $c_1,c_2 \in \mathcal{C}^{3}_{\ell,M}(\Omega_0)$.* *Proof.* Let $c_1,c_2 \in \mathcal{C}^{3}_{\ell,M}(\Omega_0)$. Then $n_{c_1}, n_{c_2} \in \mathcal{N}_{\lambda,\ell}(\Omega_0)$ for $\lambda= e^{-M}$. So it follows from Theorem [Theorem 2](#inverse){reference-type="ref" reference="inverse"} that $$\label{c2.3temp} \|n_{c_1}-n_{c_2}\|_{L^2(\Omega)}\leq C_1e^{(m-2)M}\|d_\xi(\Gamma_{c_1}-\Gamma_{c_2})\|_{L^2(\partial\Omega\times \partial\Omega)}.$$ By [\[link-L2\]](#link-L2){reference-type="eqref" reference="link-L2"}, the left hand side of the above equation is bounded below by $e^{-M}\|c_1-c_2\|_{L^2(\Omega_0)}$. Now, rewrite $d_{\xi}(\Gamma_{c_1}-\Gamma_{c_2})$ as $$\begin{split} d_{\xi}(\Gamma_{c_1}-\Gamma_{c_2}) & = d_\xi(e^{Z_{c_1}}-e^{Z_{c_2}})\\ & = e^{Z_{c_1}}d_{\xi}Z_{c_1}-e^{Z_{c_2}}d_{\xi}Z_{c_2} \\ & = e^{Z_{c_1}}d_{\xi}(Z_{c_1}-Z_{c_2})+(e^{Z_{c_1}}-e^{Z_{c_2}})d_\xi Z_{c_2}. \end{split}$$ It follows from Remark [Remark 2](#d){reference-type="ref" reference="d"} that if $(\xi,\eta) \in \operatorname{supp}(\Gamma_{c_1}-\Gamma_{c_2})$, we have $\operatorname{dist}_{\bar{g}}(\xi,\eta)\geq \delta$, and consequently, $$e^{-M}\delta \leq \Gamma_{c_j}(\xi,\eta) \leq e^M\operatorname{diam}_{\bar{g}}(\Omega), \qquad j=1,2.$$ Therefore, by applying [\[lipexp\]](#lipexp){reference-type="eqref" reference="lipexp"} along with the fact that $|d_\xi\Gamma_{c_j}|_{\bar{g}}\leq 1$ by Lemma [Lemma 7](#derivative estimates){reference-type="ref" reference="derivative estimates"}, we get $$\begin{split} |d_\xi(\Gamma_{c_1}-\Gamma_{c_2})|_{\bar{g}} & \leq |\Gamma_{c_1}||d_\xi(Z_{c_1}-Z_{c_2})|_{\bar{g}}+|\Gamma_{c_1}-\Gamma_{c_2}||d_\xi\Gamma_{c_2}|_{\bar{g}}/|\Gamma_{c_2}|\\ &\leq e^M\operatorname{diam}_{\bar{g}}(\Omega)|d_\xi(Z_{c_1}-Z_{c_2})|_{\bar{g}}+\frac{|e^{Z_{c_1}}-e^{Z_{c_2}}|}{e^{-M}\delta} \\ &\leq e^M\operatorname{diam}_{\bar{g}}(\Omega)|d_\xi(Z_{c_1}-Z_{c_2})|_{\bar{g}}+\frac{e^M\operatorname{diam}_{\bar{g}}(\Omega)}{e^{-M}\delta}|Z_{c_1}-Z_{c_2}|, \end{split}$$ where $\operatorname{diam}_{\bar{g}}(\Omega)$ denotes the diameter of $\Omega$ with respect to the metric $\bar{g}$. This further implies $$\|d_{\xi}(\Gamma_{c_1}-\Gamma_{c_2})\|_{L^2(\partial\Omega\times \partial\Omega)} \lesssim_{\Omega,\bar{g}, \delta, \ell, M} \|Z_{c_1}-Z_{c_2}\|_{H^1(\partial\Omega\times \partial\Omega)}.$$ Combining this with [\[link-L2\]](#link-L2){reference-type="eqref" reference="link-L2"} and [\[c2.3temp\]](#c2.3temp){reference-type="eqref" reference="c2.3temp"}, we get $$\|c_1-c_2\|_{L^2(\Omega_0)} \lesssim_{\Omega,\bar{g}, \delta, \ell, M} \|Z_{c_1}-Z_{c_2}\|_{H^1(\partial\Omega\times \partial\Omega)}.$$ This completes the proof. ◻ ## Forward continuity estimates We now move on to the proof of Theorem [Theorem 3](#forward){reference-type="ref" reference="forward"}. The key idea is to use *upper bounds* on $\|D_v \exp_{n_j}(x,v)\|_{op}$ to control $\|\Gamma_{n_1}-\Gamma_{n_2}\|_{L^2}$ with respect to $\|n_1-n_2\|_{L^2}$. We begin by introducing some notation. Let $S\overline{\Omega}$ denote the unit sphere bundle on $\overline{\Omega}$, that is, $$S\overline{\Omega}= \{(x,v) \in T\overline{\Omega}\ : \ |v|_{\bar{g}}=1 \}.$$ The boundary of $S\overline{\Omega}$ consists of unit tangent vectors at $\partial\Omega$. Specifically, $$\partial S\overline{\Omega}= \{(x,v) \in S\overline{\Omega}\ : \ x \in \partial\Omega\}.$$ Let $\nu$ denote the inward unit normal vector field along $\partial\Omega$ with respect to the metric $\bar{g}$. We define the bundles of *inward pointing* and *outward pointing* unit tangent vectors on $\partial\Omega$ as follows: $$\begin{aligned} \partial_+S\overline{\Omega}&:= \left\{ (\xi,v) \in \partial S\overline{\Omega}\ : \ \langle v, \nu_{\xi}\rangle_{\bar{g}} \geq 0 \right\}, \quad \textrm{and } \\ \partial_{-}S\overline{\Omega}&:= \left\{ (\xi,v) \in \partial S\overline{\Omega}\ : \ \ \langle v, \nu_{\xi}\rangle_{\bar{g}} \leq 0 \right\}.\end{aligned}$$ We also set $$\partial_0S\overline{\Omega}:= \partial_{+}S\overline{\Omega}\cap \partial_{-}S\overline{\Omega}.$$ This coincides with $S\partial\Omega$, the unit sphere bundle on $\partial\Omega$. Next, let $n \in N_{\lambda,\ell}(\Omega_0)$. For $(\xi,v) \in \partial_{+}S\overline{\Omega}$, we let $\gamma_n(\xi,v,t) = \exp_n(\xi,tv)$ denote the unit speed geodesic (with respect to $g_n$) starting at $\xi$ with initial direction $v$ at time $t=0$. We define $\tau_n(\xi,v)$ to be the time at which $\gamma_n(\xi,v,\cdot)$ exits $\overline{\Omega}$. It is known (see [@GeomIP]) that for simple manifolds, $\tau_n$ is a $C^1$ function of $\partial_{+}S\overline{\Omega}$, and $\tau_n(\xi,v)=0$ if and only if $v \in S_\xi \partial\Omega$. We also define $\eta_n(\xi,v)$ and $u_n(\xi,v)$ as the point and direction at which $\gamma_n(\xi,v,\cdot)$ exits $\overline{\Omega}$. In other words, $$\begin{aligned} \eta_n(\xi,v) &:= \gamma_n(\xi,v,\tau_n(\xi,v)), \quad \textrm{and} \\ u_n(\xi,v) &:= \dot{\gamma}_{n}(\xi,v,\tau_n(\xi,v)). \end{aligned}$$ **Lemma 9**. *Let $n \in \mathcal{N}_{\lambda,\Lambda,\ell,L}(\Omega_0)$. Then for all $(\xi,v)\in \partial_+S\overline{\Omega}$, $$\|D_v\tau_n(\xi,v)\|_{op} \leq L\frac{\tau_n(\xi,v)}{\langle \nu, u\rangle_{\bar{g}}} \leq \frac{L\Lambda\operatorname{diam}_{\bar{g}}(\Omega)}{\langle\nu,u\rangle_{\bar{g}}},$$ where $\nu = \nu_{\eta_n(\xi,v)}$ and $u=u_n(\xi,v)$.* *Proof.* Let $\rho \in C^1(\overline{\Omega})$ be such that $\rho^{-1}(0) = \partial\Omega$ and $\rho(x) = \operatorname{dist}_{\bar{g}}(x,\partial\Omega)$ for $x$ near $\partial\Omega$. Consider the function $$f(t,v) = \rho(\exp_n(\xi,tv)).$$ Observe that $$\frac{\partial f}{\partial t}\Big|_{t=\tau_n(\xi,v)} = \left\langle (\operatorname{grad}\rho)_{\eta_n(\xi,v)}, u_n(\xi,v)\right\rangle_{\bar{g}} = \langle \nu, u\rangle_{\bar{g}}.$$ On the other hand, $$\begin{aligned} D_vf(t,v) &= D\rho_{\exp_n(\xi,tv)}\circ \left(tD_w\exp_n(\xi,w)\big|_{w=tv}\right) \\ \Rightarrow D_vf\big|_{(\tau_n(\xi,v),v)} &= \tau_n(\xi,v)\Pi^{\nu}\circ D_w\exp_n(\xi,w)\big|_{w = \tau_n(\xi,v)v}, \end{aligned}$$ where $\Pi^{\nu}$ is the linear map given by $$\Pi^{\nu}(w) = \langle \nu, w\rangle_{\bar{g}} \qquad \textrm{for all } w \in T_{\eta_n(\xi,v)}\overline{\Omega}.$$ Now differentiating the identity $f(\tau_n(\xi,v),v)=0$ with respect to $v$, we get $$\begin{split} 0 &= \frac{\partial f}{\partial t}\Big|_{(\tau_n(\xi,v),v)}D_v\tau_n(\xi,v)+D_vf\big|_{(\tau_n(\xi,v),v)} \\ &= \langle\nu,u\rangle_{\bar{g}} D_v\tau_n(\xi,v)+\tau_n(\xi,v)\Pi^\nu \circ D_w\exp_n(\xi,w)\big|_{w=\tau_n(\xi,v)v}. \end{split}$$ Therefore, $$\begin{aligned} D_v\tau_n(\xi,v) &= -\frac{\tau_n(\xi,v)}{\langle\nu,u\rangle_{\bar{g}}}\Pi^\nu \circ D_w\exp_n(\xi, w)\big|_{w=\tau_n(\xi,v)v} \\ \Rightarrow \|D_v\tau_n(\xi,v)\|_{op} &\leq \frac{\tau_n(\xi,v)}{\langle\nu,u\rangle_{\bar{g}}}\left\|D_w\exp_n(\xi,w)\big|_{w=\tau_n(\xi,v)v}\right\|_{op} \\ &\leq L\left[\frac{\tau_n(\xi,v)}{\langle\nu,u\rangle_{\bar{g}}}\right], \end{aligned}$$ as required. Now the lemma follows by observing that $$\tau_{n}(\xi,v) \leq \operatorname{diam}_{g_n}(\Omega) \leq \Lambda\operatorname{diam}_{\bar{g}}(\Omega)$$ for all $(\xi,v)\in \partial_+S\overline{\Omega}$. ◻ We are now ready to prove Theorem [Theorem 3](#forward){reference-type="ref" reference="forward"}. Recall that the notation $\int_\gamma f d|g|$ denotes the integral of a function $f$ along the curve $\gamma$ with respect to the arc-length metric induced by $g$. *Proof of Theorem [Theorem 3](#forward){reference-type="ref" reference="forward"}.* Fix $\xi \in \partial\Omega$, and define the sets $$\begin{aligned} B_1(\xi) &:= \{\eta \in \partial\Omega\ : \ \Gamma_{n_1}(\xi,\eta) \leq \Gamma_{n_2}(\xi,\eta)\},\\ B_2(\xi) &:= \{\eta \in \partial\Omega\ : \ \Gamma_{n_2}(\xi,\eta) \leq \Gamma_{n_1}(\xi,\eta)\}. \end{aligned}$$Suppose $\eta \in B_1(\xi)$, and let $\gamma_1(\xi,\eta)$ denote the unit speed geodesic with respect to $g_{n_1}$ from $\xi$ to $\eta$. Clearly, $\Gamma_{n_1}(\xi,\eta) = \int_{\gamma_1(\xi,\eta)}n_1d|\bar{g}|$, whereas $\Gamma_{n_2}(\xi,\eta) \leq \int_{\gamma_1(\xi,\eta)}n_2 d|\bar{g}|$. So we have $$(\Gamma_{n_2}-\Gamma_{n_1})(\xi,\eta) \leq \int_{\gamma_1(\xi,\eta)}(n_2-n_1)d|\bar{g}| = \int_{\gamma_1(\xi,\eta)}\frac{(n_2-n_1)}{n_1}d|g_{n_1}|.$$ This implies $$\begin{aligned} (\Gamma_{n_2}-\Gamma_{n_1})^2(\xi,\eta) &\leq \Gamma_{n_1}(\xi,\eta)\int_{\gamma_1(\xi,\eta)}\frac{(n_2-n_1)^2}{n_1^2} d|g_{n_1}| \quad \textrm{(by Cauchy-Schwarz)}\\ &= \Gamma_{n_1}(\xi,\eta)\int_0^{\Gamma_{n_1}(\xi,\eta)}\frac{(n_2-n_1)^2}{n_1^2}(\gamma_1(\xi,\eta,t))dt \\ &\leq \frac{\Gamma_{n_1}(\xi,\eta)}{\lambda^2}\int_0^{\Gamma_{n_1}(\xi,\eta)}(n_2-n_1)^2(\exp_{n_1}(\xi,tv_{n_1}(\xi,\eta)))dt, \end{aligned}$$ where $v_{n_1}(\xi,\eta) = \dot{\gamma}_{n_1}(\xi,\eta,0)$, that is, the unit tangent vector at $\xi$ that points towards $\eta$. This implies $$\begin{aligned} \int_{B_1(\xi)}(\Gamma_{n_2}-\Gamma_{n_1})^2(\xi,\eta)d\eta &\leq \frac{\Lambda \operatorname{diam}_{\bar{g}}(\Omega)}{\lambda^2} \int_{\partial\Omega}\int_0^{\Gamma_{n_1}(\xi,\eta)}(n_2-n_1)^2(\exp_{n_1}(\xi,tv_{n_1}(\xi,\eta)))dtd\eta \nonumber\\ &= \frac{\Lambda\operatorname{diam}_{\bar{g}}(\Omega)}{\lambda^2} \int_{\partial_{+}S_\xi\overline{\Omega}}\int_0^{\tau_{n_1}(\xi,v)}(n_2-n_1)^2(\exp_{n_1}(\xi,tv))|\det[D_v\eta_{n_1}(\xi,v)]dt dv. \label{changeofvar} \end{aligned}$$ by the change of variables formula. (Here, $d\eta$ is the surface measure on $\eta \in \partial\Omega$ with respect to $\bar{g}$.) We now find an upper bound for $|\det[D_v\eta_{n_1}]|$ on the support of the integrand. Recall that by definition, $$\eta_{n_1}(\xi,v) = \exp_{n_1}(\xi, \tau_{n_1}(\xi,v)v).$$ With the canonical identification of $T_v S_\xi\overline{\Omega}$ with a subspace of $T_\xi \overline{\Omega}$, we get $$\begin{aligned} D_v\eta_{n_1}(\xi,v) &= D_w\exp_{n_1}(\xi,w)\big|_{w =\tau_{n_1}(\xi,v)v}\circ D_v(\tau_{n_1}(\xi,v)v) \\ &= D_w \exp_{n_1}(\xi,w)\big|_{w=\tau_{n_1}(\xi,v)v} \circ \big( \tau_{n_1}(\xi,v)\text{Id}+v\otimes D_v\tau_{n_1}(\xi,v)\big). \end{aligned}$$ Here, $v\otimes D_v\tau_{n_1}(\xi,v)$ should be interpreted as the map $$w \in T_vS_\xi \overline{\Omega}\subseteq T_\xi\overline{\Omega}\qquad \mapsto \qquad [D_v\tau_{n_1}|_{(\xi,v)}(w)]v \in T_\xi \overline{\Omega}.$$ So we have $$\begin{aligned} \|D_v\eta_{n_1}(\xi,v)\|_{op} &\leq \left\|D_w\exp_{n_1}(\xi,w)\big|_{w=\tau_{n_1}(\xi,v)v}\right\|_{op}|\big( \tau_{n_1}(\xi,v) +\|D_v\tau_{n_1}(\xi,v)\|_{op}\big) \\ &\leq L\left(\Lambda\operatorname{diam}_{\bar{g}}(\Omega) +\frac{L\Lambda\operatorname{diam}_{\bar{g}}(\Omega)}{\left\langle \nu(\eta_{n_1}(\xi,v)),u_{n_1}(\xi,v)\right\rangle_{\bar{g}}}\right) \end{aligned}$$ by Lemma [Lemma 9](#derivative of tau){reference-type="ref" reference="derivative of tau"}. Now since $\Omega_0$ is a relatively compact subset of $\Omega$, there exists an $\varepsilon \in (0,1)$ such that if $\langle \nu(\eta_{n_1}(\xi,v)),u_{n_1}(\xi,v)\rangle_{\bar{g}} <\varepsilon$, the geodesic $\gamma_{n_1}(\xi,v,\cdot)$ lies entirely within $\overline{\Omega}\setminus \Omega_0$, and therefore, $$(n_2-n_1)^2(\exp_{n_1}(\xi,tv)) = 0 \qquad \textrm{for all } t \in [0,\tau_{n_1}(\xi,v)].$$ Therefore, on the support of the integrand in the right hand side of [\[changeofvar\]](#changeofvar){reference-type="eqref" reference="changeofvar"}, we have the bounds $$\|D_v\eta_{n_1}(\xi,v)\|_{op} \leq L\left( \Lambda\operatorname{diam}_{\bar{g}}(\Omega) +\frac{L\Lambda\operatorname{diam}_{\bar{g}}(\Omega)}{\varepsilon}\right) \lesssim_{\Omega,\Omega_0,\bar{g},L} \Lambda,$$ and consequently $$|\det[D_v(\eta_{n_1}(\xi,v))]| \lesssim_{\Omega,\Omega_0,\bar{g},L} \Lambda^{m-1}.$$ Applying this bound to the right hand side of [\[changeofvar\]](#changeofvar){reference-type="eqref" reference="changeofvar"}, we get $$\begin{aligned} \int_{B_1(\xi)}(\Gamma_{n_1}-\Gamma_{n_2})^2(\xi,\eta)d\eta &\lesssim\frac{\Lambda^m}{\lambda^2}\int_{\partial_{+}S_{\xi}\overline{\Omega}}\int_0^{\tau_{n_1}(\xi,v)}(n_2-n_1)^2(\exp_{n_1}(\xi,tv))dtdv \\ &\sim \frac{\Lambda^m}{\lambda^2}\int_{\operatorname{dom}(\exp_{n_1}(\xi, \cdot))} \frac{(n_2-n_1)^2(\exp_{n_1}(\xi,w))}{|w|_{\bar{g}}^{m-1}}dw \end{aligned}$$ Again by Remark [Remark 2](#d){reference-type="ref" reference="d"}, we have $(n_2-n_1)^2(\exp_{n_1}(\xi,w))=0$ for all $w \in \operatorname{dom}(\exp_{n_1}(\xi, \cdot))$ with $|w|_{\bar{g}}\leq \delta$. Therefore, we get $$\int_{B_1(\xi)} (\Gamma_{n_1}-\Gamma_{n_2})^2(\xi,\eta)d\eta \lesssim\frac{\Lambda^m }{\lambda^2 \delta^{m-1}}\int_{\operatorname{dom}(\exp_{n_1}(\xi, \cdot))}(n_2-n_1)^2(\exp_{n_1}(\xi,w)) dw.$$ We now make the change of variable $x = \exp_{n_1}(\xi, w)$. The assumption that $\|D_w \exp_{n_1}(\xi,w)\|_{op} > \ell$ implies that the inverse $w_{n_1}(\xi,\cdot)$ of $\exp_{n_1}(\xi,\cdot)$ satisfies $\|D_x w_{n_1}(\xi,x)\|_{op} < \ell^{-1}$, and consequently, $$|\det(D_x w_{n_1}(\xi,x))| < \ell^{-m}.$$ Therefore, $$\begin{aligned} \int_{B_1(\xi)} (\Gamma_{n_1}-\Gamma_{n_2})^2(\xi,\eta)d\eta &\lesssim\frac{\Lambda^m}{\lambda^2} \int_{\Omega}(n_2-n_1)^2(x)|\det(D_x w_{n_1}(\xi,x))|d\operatorname{Vol}_{\bar{g}}(x) \\ & \lesssim\frac{\Lambda^m}{\lambda^2\ell^m}\int_{\Omega}(n_2-n_1)^2(x)d\operatorname{Vol}_{\bar{g}}(x). \end{aligned}$$ By analogous arguments, we also have $$\int_{B_2(\xi)}(\Gamma_{n_1}-\Gamma_{n_2})^2(\xi,\eta)d\eta \lesssim\frac{\Lambda^m}{\lambda^2 \ell^m}\int_{\Omega} (n_2-n_1)^2(x)d\operatorname{Vol}_{\bar{g}}(x).$$ Adding the last two inequalities, we get $$\begin{aligned} \int_{\partial\Omega} (\Gamma_{n_1}-\Gamma_{n_2})^2(\xi,\eta) d\eta &\lesssim\frac{\Lambda^m}{\lambda^2 \ell^m}\|n_1-n_2\|^2_{L^2(\Omega)} \\ \Rightarrow \int_{\partial\Omega}\int_{\partial\Omega} (\Gamma_{n_1}-\Gamma_{n_2})^2(\xi,\eta)d\eta d\xi &\lesssim\frac{\Lambda^m}{\lambda^2 \ell^m}\|n_1-n_2\|^2_{L^2(\Omega)} \\ \Rightarrow \|\Gamma_{n_1}-\Gamma_{n_2}\|_{L^2(\partial\Omega\times \partial\Omega)} &\lesssim_{\Omega,\Omega_0,\bar{g},\ell,L} \frac{\Lambda^{m/2}}{\lambda}\|n_1-n_2\|_{L^2(\Omega)}. \end{aligned}$$ This completes the proof. ◻ Next, we derive the analogous continuity estimate for the map $c \mapsto Z_c$. The key step is to show that for any $M>0$, the operator norm of the derivative of $\exp_{n_c}(x,v)$ is uniformly bounded for all $c \in \mathcal{C}^{3}_{\ell,M}(\Omega_0)$ and $(x,v) \in \operatorname{dom}(\exp_{n_c})$. We begin with a simple lemma. **Lemma 10**. *Let $(\mathcal{M},g)$ be a Riemannian manifold whose curvature tensor $R$ satisfies $$\|R\| = \sup \left\{ |R(u,v)w|_g : u,v,w \in S\mathcal{M}\right\} < \infty.$$ Then any Jacobi field $J$ along a unit speed geodesic $\gamma:[0,T]\to \mathcal{M}$ satisfies the norm bounds $$|J(t)|_g^2+|\dot{J}(t)|_g^2 \leq e^{(1+\|R\|)t}\left(|J(0)|_g^2+|\dot{J}(0)|_g^2\right) \qquad \textrm{for all }t \in [0,T].$$* *Proof.* Set $f(t) = |J(t)|_g^2+|\dot{J}(t)|_g^2$. Since $J$ is a Jacobi field, it satisfies the equation $$\ddot{J}(t)+R(J(t),\dot{\gamma}(t))\dot{\gamma}(t) = 0.$$ Therefore, $$\begin{aligned} f'(t) &= 2\langle J(t),\dot{J}(t)\rangle_g+2\langle\dot{J}(t),\ddot{J}(t)\rangle_g \\ &= 2\langle J,\dot{J}\rangle_g+2\langle\dot{J}, -R(J,\dot{\gamma})\dot{\gamma}\rangle_g \\ &\leq 2|J|_g|\dot{J}|_g +2|\dot{J}|_g\|R\||J|_g|\dot{\gamma}|_g^2 \\ &\leq (1+\|R\|)f(t). \end{aligned}$$ So it follows that $$f(t) \leq e^{(1+\|\mathbb R\|)t}f(0) \qquad \textrm{for all }t \in [0,T].$$ ◻ Next, let us recall the definition of the canonical metric on the tangent bundle of a Riemannian manifold, also called the Sasaki metric. Let $(\mathcal{M},g)$ be a Riemannian manifold, $(x,w)\in T\mathcal{M}$, and $V_1,V_2 \in T_{(x,w)}T\mathcal{M}$. Then we may choose curves $\alpha_j(s)=(\sigma_j(s),v_j(s))$ in $T\mathcal{M}$, defined on $(-\varepsilon,\varepsilon)$, such that $$\alpha_j(0) = (x,w), \qquad \dot{\alpha}_j(0)= V_j, \qquad \textrm{for }j=1,2.$$ The inner product of $V_1,V_2$ with respect to the Sasaki metric is defined to be $$\langle V_1, V_2\rangle_{g} := \langle v_1(0),v_2(0)\rangle_{g}+ \langle\dot{v}_1(0),\dot{v}_2(0)\rangle_{g},$$ where $\dot{v}_1(s), \dot{v}_2(s)$ are the covariant derivatives of $v_1(s),v_2(s)$ along the curves $\sigma_1(s),\sigma_2(s)$ respectively. Note that we are using the same notation for the Sasaki metric as for the original metric $g$. Now, for any $C^1$ map $F:T\mathcal{M}\to \mathcal{M}$, the operator norm of the total derivative of $F$ at $(x,w)\in T\mathcal{M}$ is given by $$\|DF(x,w)\|_{op} := \sup \{ |DF(x,w)(V)|_{g} \ : \ V \in T_{(x,w)}T\mathcal{M}, \, |V|_{g}=1\}.$$ We will show that if $c\in \mathcal{C}^{3}_{\ell,M}(\Omega_0)$, the total derivative of $\exp_{n_c}$ is bounded above in the operator norm. **Proposition 11**. *For any $M>0$, there exists $L=L(M)>0$ such that for all $c\in \mathcal{C}^{3}_{\ell,M}(\Omega_0)$, the total derivative of the exponential map of $g_{n_c}$ satisfies $$\|D \exp_{n_c}(x,w)\|_{op} <L$$ for all $x \in \overline\Omega$ and $w \in \operatorname{dom}(\exp_{n_c}(x,\cdot))$. In particular, $n_c \in \mathcal{N}_{\lambda,\Lambda,\ell,L}(\Omega_0)$.* *Proof.* Suppose $c \in \mathcal{C}^{3}_{\ell,M}(\Omega_0)$. Fix $(x,w)\in \operatorname{dom}(\exp_{n_c})$, and let $V \in T_{(x,w)}T\overline{\Omega}$. It suffices to show that $$|D\exp_{n_c}(x,w)(V)|_{\bar{g}} < L|V|_{\bar{g}}.$$ Choose a curve $\alpha(s) = (\sigma(s),v(s))$ in $T\overline{\Omega}$, defined on $(-\varepsilon,\varepsilon)$, such that $\alpha(0)=(x,w)$ and $\dot{\alpha}(0)=V$. Consider the family of geodesics $\Phi:(-\varepsilon,\varepsilon)\times[0,1] \to \overline{\Omega}$ defined by $$\Phi(s,t) = \exp_{n_c}(\sigma(s),tv(s)).$$ The variation field of this family of geodesics is $$J(t) := \partial_s\exp_{n_c}(\sigma(s),tv(s))\big|_{s=0},$$ which is a Jacobi field along $\gamma(t) := \Phi(0,t)$. Observe that $$J(1) = \partial_s \exp_{n_c}(\sigma(s), v(s))\big|_{s=0} = D\exp_{n_c}(x,w)(V),$$ which is precisely the quantity whose norm we want to estimate. Let $R$ be the Riemann curvature tensor of $(\overline{\Omega},g_{n_c})$, and let $R^i_{jkl}$ denote its tensor coefficients with respect to a fixed global coordinate chart on $\overline{\Omega}$. Then we have $$R^i_{jkl} = \partial_k\Gamma^i_{lj}-\partial_l\Gamma^i_{kj}+\Gamma^i_{km}\Gamma^m_{lj}-\Gamma^i_{lm}\Gamma^m_{kj},$$ where $$\Gamma^l_{jk} = \frac{1}{2}n_c^{-2}\bar{g}^{lm}\left(\partial_j(n_c^2\bar{g}_{km})+\partial_k(n_c^2\bar{g}_{jm})-\partial_m(n_c^2\bar{g}_{jk})\right).$$ This implies that for any $x\in \overline{\Omega}$, $$\max_{ijkl}|R^i_{jkl}(x)| \lesssim_{\bar{g}} 1+ n_c(x)^{-2}\|n_c\|^2_{C^2} \lesssim e^{4M}(1+M)^4.$$ Therefore, for any $x\in \overline{\Omega}$ and unit tangent vectors $u,v,w \in S_x\Omega$, $$\begin{aligned} |R(u,v)w|_{g_c} &\lesssim n_c(x)\left(\max_{ijkl}|R^i_{jkl}(x)u^jv^kw^l|\right) \lesssim e^{5M}(1+M)^4 \\ \Rightarrow \|R\| &\leq Ce^{5M}(1+M)^4 \end{aligned}$$ for some $C>0$. Taking $L^2 > \exp(1+C'e^{5M}(1+M)^4)$ and applying Lemma [Lemma 10](#curvature Jacobi){reference-type="ref" reference="curvature Jacobi"}, we get $$\begin{aligned} |D\exp_c(x,w)(V)|_{g_c}^2 = |J(1)|_{g_{n_c}}^2 & < L^2\left(|J(0)|_{g_{n_c}}^2+|\dot{J}(0)|_{g_{n_c}}^2\right)\\ &= L^2\left( |\dot{\sigma}(0)|^2 +\left|\dot{v}(0)\right|^2\right) = L^2|V|_{\bar{g}}^2. \end{aligned}$$ This completes the proof. ◻ **Corollary 12**. *There exists a constant $C_2' = C_2'(\Omega,\Omega_0,\bar{g},\ell,M)>0$ such that for all $c_1,c_2 \in \mathcal{C}^{3}_{\ell,M}(\Omega_0)$, $$\|Z_{c_1}-Z_{c_2}\|_{L^2(\partial\Omega\times \partial\Omega)} \leq C_2'\|c_1-c_2\|_{L^2(\Omega_0)}.$$* *Proof.* We know from Theorem [Theorem 3](#forward){reference-type="ref" reference="forward"}, Proposition [Proposition 11](#ncbounded){reference-type="ref" reference="ncbounded"}, and equation [\[link-L2\]](#link-L2){reference-type="eqref" reference="link-L2"} that $$\|\Gamma_{c_1}-\Gamma_{c_2}\|_{L^2(\partial\Omega \times \partial\Omega)} \lesssim_{\Omega,\Omega_0,\bar{g},\ell,M}\|c_1-c_2\|_{L^2(\Omega_0)}.$$ Now consider $$\|\Gamma_{c_1}-\Gamma_{c_2}\|_{L^2(\partial\Omega \times \partial\Omega)}^2 = \int_{\partial\Omega \times \partial\Omega}\left|e^{Z_{c_1}}-e^{Z_{c_2}}\right|^2\ d\xi d\eta.$$ Recall that there exists $\delta>0$ such that $Z_{c_1}(\xi,\eta)=Z_{c_2}(\xi,\eta)$ whenever $\operatorname{dist}_{\bar{g}}(\xi,\eta)<\delta$. On the set $\{\operatorname{dist}_{\bar{g}}(\xi,\eta)\geq \delta\}$, $$\begin{aligned} e^{-M}\delta &\leq \Gamma_{c_j}(\xi,\eta) \leq e^M\operatorname{diam}_{\bar{g}}(\Omega) \nonumber \\ \Rightarrow -M+\log\delta &\leq Z_{c_j}(\xi,\eta) \leq M+ \log|\operatorname{diam}_{\bar{g}}(\Omega)|. \label{zcbound} \end{aligned}$$ So by [\[lipexp\]](#lipexp){reference-type="eqref" reference="lipexp"}, $$|e^{Z_{c_1}(\xi,\eta)}-e^{Z_{c_2}(\xi,\eta)}| \geq e^{-M}\delta|Z_{c_1}(\xi,\eta)- Z_{c_2}(\xi,\eta)|$$ for all $(\xi,\eta) \in \partial\Omega\times \partial\Omega$. Consequently, $$\|\Gamma_{c_1}-\Gamma_{c_2}\|^2_{L^2(\partial\Omega \times \partial\Omega)} = \int \left|e^{Z_{c_1}}-e^{Z_{c_2}}\right|^2\ d\xi d\eta \geq e^{-2M}\delta^2\int |Z_{c_1}-Z_{c_2}|^2\ d\xi d\eta.$$ So we conclude that $$\|Z_{c_1}-Z_{c_2}\|_{L^2} \lesssim\|\Gamma_{c_1}-\Gamma_{c_2}\|_{L^2} \lesssim\|c_1-c_2\|_{L^2}.$$ ◻ We conclude this section with a technical result that will be necessary for the proof of Theorem [Theorem 20](#postcontr){reference-type="ref" reference="postcontr"} in Section 3. **Theorem 13**. *Given $M>0$, there exists a constant $C_3' = C_3'(\Omega,\Omega_0, \bar{g},\ell,M)>0$ such that for all $c_1,c_2 \in \mathcal{C}^{3}_{\ell,M}(\Omega_0)$, $$\|Z_{c_1}-Z_{c_2}\|_{H^2(\partial\Omega\times \partial\Omega)} \leq C_3'.$$* *Proof.* We know from Theorem [Theorem 3](#forward){reference-type="ref" reference="forward"} that $$\|Z_{c_1}-Z_{c_2}\|_{L^2} \lesssim\|c_1-c_2\|_{L^2} \lesssim 2M.$$ Next, let $\xi,\eta \in \partial\Omega$. It follows from Remark [Remark 2](#d){reference-type="ref" reference="d"} that if $\operatorname{dist}_{\bar{g}}(\xi,\eta)<\delta$, then $Z_{c_1}-Z_{c_2}$ and all its derivatives are identically $0$ in a neighborhood of $(\xi,\eta)$. On the other hand, if $\operatorname{dist}_{\bar{g}}(\xi,\eta)>\delta$, Lemma [Lemma 7](#derivative estimates){reference-type="ref" reference="derivative estimates"} implies $$|d_\xi(Z_{c_1}-Z_{c_2})(\xi,\eta)|_{\bar{g}} \leq \frac{|d_\xi \Gamma_{c_1}(\xi,\eta)|_{\bar{g}}}{\Gamma_{c_1}(\xi,\eta)} +\frac{|d_\xi \Gamma_{c_2}(\xi,\eta)|_{\bar{g}}}{\Gamma_{c_2}(\xi,\eta)} \lesssim\frac{e^M}{\delta}.$$ This shows that $\|d_\xi(Z_{c_1}-Z_{c_2})\|_{L^2}$ is uniformly bounded for $c_1,c_2 \in \mathcal{C}^{3}_{\ell,M}(\Omega_0)$. By symmetry, $\|d_\eta(Z_{c_1}-Z_{c_2})\|_{L^2}$ is also uniformly bounded. So it only remains to consider the Hessian tensor of $Z_{c_1}-Z_{c_2}$. Let $\nabla$ denote the Levi-Civita connection on $\partial\Omega_{\xi}\times \partial\Omega_{\eta}$, and let $\pi^\xi:\partial\Omega_{\xi}\times \partial\Omega_{\eta} \to \partial\Omega_{\xi}$ and $\pi^\eta:\partial\Omega_{\xi}\times \partial\Omega_{\eta} \to \partial\Omega_{\eta}$ denote the canonical projection maps. We may decompose $\nabla$ as $\nabla^\xi +\nabla^{\eta}$, where $\nabla^{\xi}$ and $\nabla^{\eta}$ are the covariant derivative operations with respect to $\xi$ and $\eta$ respectively. More precisely, given any tensor field $F$ on $\partial\Omega_\xi \times \partial\Omega_\eta$, and any tangent vector $v\in T(\partial\Omega_\xi \times \partial\Omega_\eta)$, we have $$\nabla^{\xi}_vF = \nabla_{(\pi^\xi)_*v_\xi}F, \qquad \nabla^{\eta}_vF = \nabla_{(\pi^\eta)_*v_\eta}F,$$ where $(v_\xi,v_\eta)$ is the image of $v$ under the canonical isomorphism from $T(\partial\Omega_\xi\times\partial\Omega_\eta)$ to $(T\partial\Omega_\xi)\times (T\partial\Omega_\eta)$. Correspondingly, the Hessian operator on $\partial\Omega_\xi \times \partial\Omega_\eta$ can be decomposed as $$\begin{aligned} \operatorname{Hess}= \nabla^2 &= (\nabla^\xi+\nabla^\eta)(\nabla^\xi+\nabla^\eta) \\ &= \nabla^\xi \nabla^\xi +\nabla^\xi\nabla^\eta + \nabla^\eta\nabla^\xi +\nabla^\eta\nabla^\eta \\ &= \operatorname{Hess}_\xi +\nabla^\xi\nabla^\eta+ \nabla^\eta\nabla^\xi +\operatorname{Hess}_\eta, \end{aligned}$$ where $\operatorname{Hess}_\xi$ and $\operatorname{Hess}_\eta$ are the Hessian operators with respect to $\xi$ and $\eta$ respectively. Now let $\xi,\eta \in \partial\Omega$ be such that $\operatorname{dist}_{\bar{g}}(\xi,\eta)>\delta$. Then for $j=1,2$, $$\begin{aligned} \nabla^\xi\nabla^\eta Z_{c_j}(\xi,\eta) &= \nabla^\xi\nabla^\eta \log\Gamma_{c_j}(\xi,\eta) \\ &= \left(\frac{\nabla^\xi\nabla^\eta\Gamma_{c_j}}{\Gamma_{c_j}} - \frac{d_\xi\Gamma_{c_j} \otimes d_\eta \Gamma_{c_j}}{\Gamma_{c_j}^2}\right)(\xi,\eta). \end{aligned}$$ By Lemma [Lemma 7](#derivative estimates){reference-type="ref" reference="derivative estimates"}, this implies $$\begin{aligned} |\nabla^\xi\nabla^\eta Z_{c_j}(\xi,\eta)|_{\bar{g}} &\leq \frac{|\nabla^\xi\nabla^\eta\Gamma_{c_j}(\xi,\eta)|_{\bar{g}}}{\Gamma_{c_j}(\xi,\eta)} + \frac{|d_\xi \Gamma_{c_j}(\xi,\eta)|_{\bar{g}}|d_\eta \Gamma_{c_j}(\xi,\eta)|_{\bar{g}}}{\Gamma_{c_j}^2(\xi,\eta)} \\ &\lesssim\frac{1+\ell^{-1}}{\lambda\delta^2} + \frac{1}{\delta^2}. \end{aligned}$$ This implies that $\|\nabla^\xi\nabla^\eta (Z_{c_1}-Z_{c_2})\|_{L^2}$ is uniformly bounded as well. Finally, consider the fact [@V11] that $$\operatorname{Hess}_\xi \Gamma_{c_j}(\xi,\eta) = (D_w\exp_{c_j}(\xi,w(\xi,\eta)))^{-1}(D_\xi \exp_{c_j}(\xi, w(\xi,\eta))),$$ where $w(\xi,\cdot)$ is the inverse of $\exp_{c_j}(\xi,\cdot)$ as in Lemma [Lemma 7](#derivative estimates){reference-type="ref" reference="derivative estimates"}. Therefore, by Proposition [Proposition 11](#ncbounded){reference-type="ref" reference="ncbounded"}, $$|\operatorname{Hess}_\xi \Gamma_{c_j}(\xi,\eta)|_{\bar{g}} \lesssim\ell^{-1}L(M).$$ Writing $Z_{c_j}=\log \Gamma_{c_j}$, we get $$\begin{aligned} \operatorname{Hess}_\xi Z_{c_j}(\xi,\eta) &= \operatorname{Hess}_{\xi} \log \Gamma_{c_j}(\xi,\eta) \\ &= \left(\frac{\operatorname{Hess}_\xi \Gamma_{c_j}}{\Gamma_{c_j}} - \frac{d_\xi \Gamma_{c_j}\otimes d_\xi \Gamma_{c_j}}{\Gamma_{c_j}^2}\right)(\xi,\eta), \end{aligned}$$ which implies $$\begin{aligned} |\operatorname{Hess}_\xi Z_{c_j}(\xi,\eta)|_{\bar{g}} &\leq \frac{|\operatorname{Hess}_\xi \Gamma_{c_j}(\xi,\eta)|_{\bar{g}}}{\Gamma_{c_j}(\xi,\eta)} + \frac{|d_\xi\Gamma_{c_j}(\xi,\eta)|_{\bar{g}}^2}{\Gamma_{c_j}^2(\xi,eta)} \\ &\lesssim\frac{\ell^{-1}L}{\lambda \delta^2} +\frac{1}{\delta^2}. \end{aligned}$$ So we conclude that $\|\operatorname{Hess}_\xi(Z_{c_1}-Z_{c_2})\|_{L^2}$, and by similar arguments, $\|\operatorname{Hess}_\eta(Z_{c_1}-Z_{c_2})\|_{L^2}$, are both uniformly bounded on $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ as well. This proves the result. ◻ # Statistical Inversion through the Bayesian framework {#stats} As discussed in the Introduction, we will be using the posterior mean of $c$ given finitely many measurements $\mathcal{D}_N = (X_i,Y_i,Z_i)_{i=1}^N$, as an estimator for the true metric parameter $c_0$. Let us begin by describing the prior distribution $\Pi$ for $c \in C^3_0(\Omega_0)$. We will assume that $\Pi$ arises from a centered Gaussian probability distribution $\widetilde{\Pi}$ on the Banach space $C(\overline{\Omega}_0)$ that satisfies the following conditions. *Condition 1*. Let $\beta\geq 3$ and $\alpha > \beta +\frac{m}{2}$. We assume that $\widetilde{\Pi}$ is a centered Gaussian Borel probability measure on $C(\overline{\Omega}_0)$ that is supported in a separable subspace of $C^\beta_0(\Omega_0)$. Moreover, its *Reproducing Kernel Hilbert space (RKHS)* $(\mathcal{H},\|\cdot\|_{\mathcal{H}})$ must be continuously embedded in the Sobolev space $H^\alpha(\Omega_0)$. We refer the reader to [@GvdV17 Chapter 11] or [@GN15 Sections 2.1 and 2.6] for basic facts about Gaussian probability measures and their Reproducing Kernel Hilbert Spaces. We now define the prior $\Pi$ to be the restriction of $\widetilde{\Pi}$ to $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ in the sense that $$\label{Pi} \Pi(A) = \frac{\widetilde{\Pi}\left(A\cap \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)\right)}{\widetilde{\Pi}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0))}$$ for all Borel sets $A \subseteq C^3_0(\Omega_0)$. We will see in Lemma [Lemma 18](#smallball){reference-type="ref" reference="smallball"} that $C^\beta$-balls have positive $\widetilde{\Pi}$-measure. This together with the fact that $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ is an open subset of $C_0^\beta(\Omega_0)$ (c.f. Remark [Remark 4](#open){reference-type="ref" reference="open"}) implies that $\widetilde{\Pi}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)) >0$. Therefore, [\[Pi\]](#Pi){reference-type="eqref" reference="Pi"} yields a well-defined probability distribution on $C^3_0(\Omega_0)$. **Theorem 14**. *Let $\Pi$ be a prior distribution on $C^3_0(\Omega_0)$ defined by [\[Pi\]](#Pi){reference-type="eqref" reference="Pi"}. Assume that the true parameter $c_0 \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)\cap \mathcal{H}$, and let $\overline{c}_N$ be the mean [\[postmean\]](#postmean){reference-type="eqref" reference="postmean"} of the posterior distribution $\Pi(\cdot |\mathcal{D}_N)$ arising from observations [\[obs\]](#obs){reference-type="eqref" reference="obs"}. Then there exists $\omega\in(0,1/4)$ such that $$P^N_{c_0}\left( \|\overline{c}_N-c_0\|_{L^2(\Omega_0)} > N^{-\omega}\right) \to 0 \qquad \textrm{as } N \to \infty.$$ Moreover, $\omega$ can be made arbitrarily close to $1/4$ for $\beta$ large enough.* *Remark 5*. The assumption that $c_0\in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)\cap\mathcal{H}$ is weaker than in Theorem [Theorem 5](#mainth){reference-type="ref" reference="mainth"}, where we assumed that $c_0$ is smooth, compactly supported in $\Omega_0$, and that $g_{n_{c_0}}$ is simple. Indeed, if $g_{n_{c_0}}$ is a smooth simple metric, $c_0$ necessarily belongs to $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ for appropriate values of $\ell,M$, and any $\beta$. Moreover, given any $c_0\in H^\alpha_0(\Omega_0)$, it is possible to choose $\widetilde{\Pi}$ so that its RKHS $\mathcal{H}$ contains $c_0$. Indeed, let $(f(x): x \in \Omega_0)$ be the so-called *Matérn-Whittle process of regularity $\alpha$* (see [@GvdV17 Example 11.8]), whose corresponding RKHS is $H^\alpha(\Omega_0)$. It follows from Lemma I.4 in [@GvdV17] that the sample paths of this process belong almost surely to $C^\beta(\overline{\Omega}_0)$. Now choose a cut-off function $\varphi \in C^\infty(\overline{\Omega}_0)$ such that $\varphi >0$ on $\Omega_0$, $\varphi$ and all its partial derivatives vanish on $\partial\Omega_0$, and $\varphi^{-1}c_0 \in H^\alpha(\Omega_0)$. Define $\widetilde{\Pi}$ to be the probability law of $(\varphi(x)f(x): x \in \Omega_0)$. Then $\mathcal{H}= \left\{\varphi f : f \in H^\alpha(\Omega_0) \right\}$, which contains $c_0$. Therefore, Theorem [Theorem 14](#main){reference-type="ref" reference="main"} is a more general and precise version of Theorem [Theorem 5](#mainth){reference-type="ref" reference="mainth"}. ## A General Contraction Theorem {#gencontr} Our proof of Theorem [Theorem 14](#main){reference-type="ref" reference="main"} will follow the same general strategy as in [@MNP21a], with some modifications necessitated by the fact that our prior $\Pi$ is not in itself a Gaussian probability measure, but rather the restriction of such a measure to $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$. We begin with a general posterior contraction result (Theorem [Theorem 15](#g-contr){reference-type="ref" reference="g-contr"}). This is a simplified version of [@MNP21a Theorem 5.13], which suffices for us since our prior $\Pi$ independent of $N$. Before stating the result, we need to introduce some notation. Recall that for $c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$, we defined $p_c$ as the probability density function $$p_c(x,y,z) = \frac{1}{\sqrt{2\pi}}\exp\left\{-\frac{1}{2}(z-Z_c(x,y))^2\right\} \qquad \textrm{for all } (x,y,z)\in \mathcal{X},$$ where $\mathcal{X}= \partial\Omega\times \partial\Omega\times \mathbb R$. Given $c_1,c_2 \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$, let $$h(c_1,c_2) := \left(\int_{\mathcal{X}}(\sqrt{p_{c_1}}-\sqrt{p_{c_2}})^2 d\mu(x,y)\, dz\right)^{1/2}$$ denote the Hellinger distance between $p_{c_1}$ and $p_{c_2}$, $$K(c_1,c_2) := \mathbb{E}_{c_1}\left[\log\left(\frac{p_{c_1}}{p_{c_2}}\right)\right] = \int_{\mathcal{X}}\log\left(\frac{p_{c_1}}{p_{c_2}}\right)p_{c_1}d\mu(x,y)\, dz$$ the Kullback-Leibler divergence, and $$V(c_1,c_2) := \mathbb{E}_{c_1}\left[\log\left(\frac{p_{c_1}}{p_{c_2}}\right)\right]^2.$$ Also, for any $F \subseteq \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ and $\delta >0$, we let $\mathcal{N}(F, h, \delta)$ denote the minimum number of $h$-balls of radius $\delta$ needed to cover $F$. **Theorem 15**. *Let $\widehat{\Pi}$ be a Borel probability measure on $C_0^3(\Omega_0)$ supported on $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$. Let $c_0 \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ be fixed, and let $\delta_N$ be a sequence of positive numbers such that $\delta_N \to 0$ and $\sqrt{N}\delta_N \to \infty$ as $N \to \infty$. Assume that the following two conditions hold:* (1) *There exists $C>0$ such that for all $N \in \mathbb{N}$, $$\label{pmass} \widehat{\Pi}\left(\left\{c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0): K(c,c_0) \leq \delta_N^2, V(c,c_0) \leq \delta_N^2 \right\}\right) \geq e^{-CN\delta_N^2}.$$* (2) *There exists $\widetilde{C}>0$ such that $$\label{pcomp} \log \mathcal{N}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0), h, \delta_N) \leq \widetilde{C}N\delta_N^2.$$* *Now suppose that we make i.i.d. observations $\mathcal{D}_N = (X_i,Y_i,Z_i)_{i=1}^N \sim P^N_{c_0}$. Then for some $k>0$ large enough, we have $$P^N_{c_0}\left( \widehat{\Pi}\left( \left\{c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0): h(c,c_0) \leq k\delta_N\right\}|\mathcal{D}_N\right) \leq 1-e^{-(C+3)N\delta_N^2}\right) \to 0$$ as $N \to \infty$.* *Proof.* Define $$\label{bn} B_N = \left\{c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0): K(c,c_0) \leq \delta_N^2, V(c,c_0) \leq \delta_N^2 \right\}, \qquad N \in \mathbb{N}.$$ By condition (1) and [@GN15 Lemma 7.3.2], we have that for any $\zeta >0$ and any probability measure $\widetilde{m}$ on $B_N$, $$P^N_{c_0}\left( \int_{B_N} \prod_{i=1}^N \frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\widetilde{m}(c) \leq e^{-(1+\zeta)N\delta_N^2}\right) \leq \frac{1}{\zeta^2 N\delta_N^2}.$$ In particular, choosing $\zeta =1$ and taking $\widetilde{m}$ to be the restriction of $\widehat{\Pi}$ to $B_N$ followed by normalization, we get that $$P^N_{c_0}\left( \int_{B_N} \prod_{i=1}^N \frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\widehat{\Pi}(c) \leq \widehat{\Pi}(B_N) e^{-2N\delta_N^2}\right) \leq \frac{1}{N\delta_N^2} \xrightarrow{N \to \infty} 0.$$ Set $$A_N = \left\{\int_{B_N} \prod_{i=1}^N \frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\widehat{\Pi}(c) \geq e^{-(2+C)N\delta_N^2}\right\},$$ where $C$ is as in condition (1). It is clear that $A_N \supseteq \left\{ \int_{B_N} \prod_{i=1}^N\frac{p_c}{p_{c_0}}d\widehat{\Pi}(c) \geq \widehat{\Pi}(B_N)e^{-2N\delta_N^2}\right\}$, and therefore, $P^N_{c_0}(A_N) \to 1$ as $N \to \infty$. Next, we consider condition (2). Let $k > k' > 0$ be numbers to be determined later. Fix $N$ and define the function $N(\varepsilon) = e^{\widetilde{C}N\delta_N^2}$ for all $\varepsilon> \varepsilon_0 = k'\delta_N$. It follows from condition (2) that for any $\varepsilon> \varepsilon_0$, $$\mathcal{N}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0), h, \varepsilon/4) \leq \mathcal{N}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0), h, k'\delta_N/4) \leq e^{\widetilde{C}N\delta_N^2} = N(\varepsilon).$$ Therefore, by [@GN15 Theorem 7.1.4], there exist test functions $\Psi_N = \Psi_N(\mathcal{D}_N)$ such that for some $K >0$, $$P^N_{c_0}[\Psi_N = 1] \leq \frac{N(\varepsilon)}{K}e^{-KN\varepsilon^2} \quad ; \quad \sup_{c: h(c,c_0) > \varepsilon}\mathbb{E}^N_c[1-\Psi_N] \leq e^{-KN\varepsilon^2}.$$ Now let $l > \widetilde{C}$ be arbitrary. Setting $k = \sqrt{l / K}$ and $\varepsilon= k\delta_N$, we can see that this implies $$\label{tests} P^N_{c_0}[\Psi_N = 1] \to 0 \, \textrm{ as } N \to \infty \quad ; \quad \sup_{c: h(c,c_0)>k\delta_N}\mathbb{E}^N_c[1-\Psi_N] \leq e^{-l N\delta_N^2}.$$ Now define $$F_N = \{ c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0): h(c,c_0) \leq k\delta_N\}$$ which is the event whose probability we want to bound. Then by [\[tests\]](#tests){reference-type="eqref" reference="tests"}, $$\begin{aligned} & \ P^N_{c_0}\left( \widehat{\Pi}(F_N^c|\mathcal{D}_N) \geq e^{-(C+3)N\delta_N^2}\right) \\ = & \ P^N_{c_0}\left( \frac{\int_{F_N^c}\prod_{i=1}^N\frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\widehat{\Pi}(c)}{\int \prod_{i=1}^N \frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\widehat{\Pi}(c)} \geq e^{-(C+3)N\delta_N^2}, \ \Psi_N =0, \ A_N\right) +o(1) \\ \leq & \ P^N_{c_0}\left((1-\Psi_N)\int_{F_N^c}\prod_{i=1}^N\frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\widehat{\Pi}(c) \geq e^{-(2C+5)N\delta_N^2}\right) +o(1). \end{aligned}$$ Now by Markov's inequality, this is further bounded above by $$\begin{aligned} & \ \mathbb{E}^N_{c_0}\left[(1-\Psi_N)\int_{F_N^c}\prod_{i=1}^N \frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\widehat{\Pi}(c)\right]e^{(2C+5)N\delta_N^2} +o(1) \\ = & \ \left[\int_{F_N^c} \mathbb{E}^N_{c_0}\left[(1-\Psi_N)\prod_{i=1}^N\frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)\right]d\widehat{\Pi}(c)\right]e^{(2C+5)N\delta_N^2} +o(1) \quad \textrm{(by Fubini's Theorem)} \\ = & \ \left[ \int_{c: h(c,c_0) > k\delta_N} \mathbb{E}^N_c[(1-\Psi_N)]d\widehat{\Pi}(c)\right]e^{(2C+5)N\delta_N^2} +o(1) \\ \leq & \ e^{(2C+5-l)N\delta_N^2} +o(1). \end{aligned}$$ Now choosing $l > 2C+5$, the Theorem follows. ◻ ## Properties of the Prior {#priorprop} In this section, we will verify the assumptions of Theorem [Theorem 15](#g-contr){reference-type="ref" reference="g-contr"} when $\widehat{\Pi}=\Pi$. The key ingredient in the arguments is the forward continuity estimate from Corollary [Corollary 12](#forward-c){reference-type="ref" reference="forward-c"}. We begin by observing that the Hellinger distance between $c_1, c_2 \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ is equivalent to the $L^2(\partial\Omega\times \partial\Omega)$ distance between $Z_{c_1}$ and $Z_{c_2}$. **Lemma 16**. *There exists $\kappa=\kappa(\Omega,\bar{g},M)>0$ such that for all $c_1,c_2 \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$, $$\kappa\|Z_{c_1}-Z_{c_2}\|_{L^2}^2 \leq h^2(c_1,c_2) \leq \frac{1}{4\operatorname{Vol}_{\bar{g}}(\partial\Omega)^2}\|Z_{c_1}-Z_{c_2}\|_{L^2}^2.$$* *Proof.* Consider the "Hellinger affinity" function $$\rho(c_1,c_2) = \int_{\mathcal{X}}\sqrt{p_{c_1}p_{c_2}}d\mu = 1-\frac{1}{2}h^2(c_1,c_2).$$ We have $$\begin{aligned} \rho(c_1,c_2) =& \ \frac{1}{\sqrt{2\pi}}\int_{\mathcal{X}}\exp\left\{-\frac{1}{4}((z-Z_{c_1}(x,y))^2+(z-Z_{c_2}(x,y))^2)\right\} d\mu(x,y)\, dz \nonumber \\ =& \ \frac{1}{\operatorname{Vol}_{\bar{g}}(\partial\Omega\times \partial\Omega)}\int_{\partial\Omega\times \partial\Omega} \exp\left\{-\frac{1}{4}(Z_{c_1}(x,y)^2+Z_{c_2}(x,y)^2)\right\} \nonumber\\ & \times \left[\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp\left\{-\frac{1}{2}\left(z-\frac{Z_{c_1}+Z_{c_2}}{2}\right)^2\right\}dz\right]\exp\left\{\frac{1}{8}(Z_{c_1}+Z_{c_2})^2\right\}dx\, dy \nonumber \\ =& \ \frac{1}{\operatorname{Vol}_{\bar{g}}(\partial\Omega)^2}\int_{\partial\Omega\times \partial\Omega}\exp\left\{-\frac{1}{8}(Z_{c_1}(x,y)-Z_{c_2}(x,y))^2\right\}dx\, dy \label{rhoform}. \end{aligned}$$ Now applying the simple estimate $e^{-t} \geq 1-t$ for all $t \geq 0$, we get $$\begin{aligned} \rho(c_1,c_2) &\geq \frac{1}{\operatorname{Vol}_{\bar{g}}(\partial\Omega)^2}\int_{\partial\Omega\times \partial\Omega}\left[1-\frac{1}{8}(Z_{c_1}-Z_{c_2})^2\right] dx \, dy \\ &= 1-\frac{1}{8\operatorname{Vol}_{\bar{g}}(\partial\Omega)^2}\|Z_{c_1}-Z_{c_2}\|_{L^2}^2. \end{aligned}$$ Consequently, $$h^2(c_1,c_2) = 2(1-\rho(c_1,c_2)) \leq \frac{1}{4\operatorname{Vol}_{\bar{g}}(\partial\Omega)^2}\|Z_{c_1}-Z_{c_2}\|_{L^2}^2.$$ Next, we use the fact $Z_{c_1},Z_{c_2}$ satisfy the uniform bounds [\[zcbound\]](#zcbound){reference-type="eqref" reference="zcbound"} on the support of $Z_{c_1}-Z_{c_2}$. Consequently, for all $x,y\in \partial\Omega$, we have $$|Z_{c_1}(x,y)-Z_{c_2}(x,y)|\leq \Delta, \label{zcdiff}$$ where $\Delta = 2M+\log \operatorname{diam}_{\bar{g}}(\Omega)-\log\delta$. Set $T=\Delta^2/8$ and observe that for all $t \in [0,T]$, $$e^{-t} \leq 1- \left(\frac{1-e^{-T}}{T}\right)t$$ by the convexity of $t \mapsto e^{-t}$. Therefore, for $\kappa = \frac{1-e^{-T}}{4T}$, we have $$\exp\left\{-\frac{1}{8}(Z_{c_1}(x,y)-Z_{c_2}(x,y))^2\right\} \leq 1-\frac{\kappa}{2}|Z_{c_1}(x,y)-Z_{c_2}(x,y)|^2$$ for all $(x,y) \in \partial\Omega\times \partial\Omega$. Integrating both sides of this inequality with respect to $d\mu(x,y)$ and applying [\[rhoform\]](#rhoform){reference-type="eqref" reference="rhoform"}, we get $$\begin{aligned} \rho(c_1,c_2) &\leq 1- \frac{\kappa}{2}\|Z_{c_1}-Z_{c_2}\|_{L^2}^2 \\ \Rightarrow \ h^2(c_1,c_2) & \geq \kappa\|Z_{c_1}-Z_{c_2}\|_{L^2}^2. \end{aligned}$$ This completes the proof. ◻ Now let us verify Condition (1) of Theorem [Theorem 15](#g-contr){reference-type="ref" reference="g-contr"} for $\Pi$. **Lemma 17**. *For $c_0 \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ and $t>0$, define $$\mathcal{B}_N(t) = \{c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0): \|c-c_0\|_{C^\beta} \leq \delta_N/t \},$$ and let $B_N, \Pi,$ and $\delta_N$ be as in Theorem [Theorem 15](#g-contr){reference-type="ref" reference="g-contr"}. Then for some $t>0$ large enough, $\mathcal{B}_N(t) \subset B_N$ for all $N \in \mathbb{N}$. In particular, $$\Pi(B_N) \geq \Pi(\mathcal{B}_N(t)).$$* *Proof.* We need to verify that if $t$ is large enough, then for any $c\in \mathcal{B}_N(t)$, we have $K(c,c_0) \leq \delta_N^2$ and $V(c,c_0) \leq \delta_N^2$. Consider a random observation $(X,Y,Z)$, where $(X,Y)$ is a pair of boundary points chosen with respect to the uniform probability measure $\mu$, and $Z= Z_{c_0}(X,Y)+\epsilon$, with $\epsilon \sim N(0,1)$ independent of $(X,Y)$. Observe that for any $c \in \mathcal{B}_N(t)$, $$\begin{aligned} \log \frac{p_{c_0}}{p_{c}}(X,Y,Z) &= -\frac{1}{2}[(Z-Z_{c_0}(X,Y))^2-(Z-Z_c(X,Y))^2] \nonumber\\ &= \frac{1}{2}(Z_c(X,Y)-Z_{c_0}(X,Y))^2 -\epsilon(Z_c(X,Y)-Z_{c_0}(X,Y)). \label{pcpform} \end{aligned}$$ Since $\mathbb{E}[\epsilon|X,Y]=0$, we have $$\begin{aligned} K(c,c_0) &= \mathbb{E}_{c_0}\left[\log\frac{p_{c_0}}{p_c}(X,Y,Z)\right] \nonumber \\ &= \mathbb{E}^{\mu}\left[\frac{1}{2}(Z_c(X,Y)-Z_{c_0}(X,Y))^2\right] \label{kform}\\ &= \frac{1}{2\operatorname{Vol}_{\bar{g}}(\partial\Omega\times \partial\Omega)}\int_{\partial\Omega\times \partial\Omega}(Z_c(x,y)-Z_{c_0}(x,y))^2 \, dx \, dy \nonumber\\ &= \frac{1}{2\operatorname{Vol}_{\bar{g}}(\partial\Omega)^2}\|Z_c-Z_{c_0}\|_{L^2}^2 \nonumber\\ &\lesssim\|c-c_0\|^2_{L^2} \qquad (\textrm{by Corollary \ref{forward-c}}) \nonumber\\ &\lesssim\frac{\delta_N^2}{t^2}.\label{kdelta} \end{aligned}$$ So it follows that if $t$ is large enough, $K(c,c_0)\leq \delta_N^2$ for all $c \in \mathcal{B}_N(t)$. Next, consider $$\begin{aligned} V(c,c_0) &= \mathbb{E}_{c_0}\left[\log \frac{p_{c_0}}{p_{c}}(X,Y,Z)\right]^2 \\ &\leq 2\mathbb{E}^\mu\left[\frac{1}{2}(Z_{c}-Z_{c_0})^2\right]^2 +2\mathbb{E}^\mu\left[(Z_c-Z_{c_0})^2\mathbb{E}_\epsilon[\epsilon^2]\right] \qquad \textrm{(by \eqref{pcpform})} \\ &= \frac{1}{2}\int_{\partial\Omega\times \partial\Omega}|Z_c-Z_{c_0}|^4d\mu(x,y) + 2\mathbb{E}^{\mu}[Z_c-Z_{c_0}]^2 \qquad \textrm{(since $\mathbb{E}[\epsilon^2]=1$)}\\ &\leq \frac{\|Z_c-Z_{c_0}\|^2_{L^\infty}}{2\operatorname{Vol}_{\bar{g}}(\partial\Omega)^2}\|Z_c-Z_{c_0}\|^2_{L^2}+4K(c,c_0) \end{aligned}$$ by [\[kform\]](#kform){reference-type="eqref" reference="kform"}. It follows from [\[zcdiff\]](#zcdiff){reference-type="eqref" reference="zcdiff"} that $\|Z_c-Z_{c_0}\|_{L^\infty}< \Delta$, where $\Delta>0$ depends only on $\Omega,\bar{g},\delta$. Consequently, $$\begin{aligned} V(c,c_0) &\lesssim\|Z_c-Z_{c_0}\|^2_{L^2} +K(c,c_0) \\ &\lesssim C_2'^2\|c-c_0\|^2_{L^2} +K(c,c_0) \qquad (\textrm{by Corollary \ref{forward-c}}) \\ &\lesssim\|c-c_0\|^2_{C^\beta} + \frac{\delta_N^2}{t^2} \qquad \textrm{(by \eqref{kdelta})} \\ &\lesssim\frac{\delta_N^2}{t^2}. \end{aligned}$$ This shows that for $t>0$ large enough, we also get $V(c,c_0)\leq \delta_N^2$ for all $c \in \mathcal{B}_N(t)$. ◻ Next, we will establish a lower bound for $\Pi(\mathcal{B}_N(t))$, which will follow from estimates of $\widetilde{\Pi}$- measures of sets of the form $\{c : \|c\|_{C^\beta}\leq \varepsilon\}$ when $\varepsilon>0$ is small. To this end, it is convenient to work with Hölder-Zygmund spaces $C^s_*(\Omega_0)$, with $s>0$ (see [@Triebel] for a detailed treatment). If $s$ is not an integer, $C^s_*(\Omega_0)$ is simply the Hölder space $C^s(\overline{\Omega}_0)$. On the other hand, if $s$ is a positive integer, $C^s_*(\Omega_0)$ is a larger space than $C^s(\overline{\Omega}_0)$, and is defined by the norm $$\|f\|_{C^s_*(\Omega_0)} = \sum_{|a|\leq s-1}\sup_{x \in \Omega_0}|\partial^a f(x)| + \sum_{|a|= s-1}\sup_{x\in \Omega_0, \, h\neq 0}\frac{|\partial^af(x+h)+\partial^af(x-h)-2f\partial^a(x)|}{|h|}.$$ In either case, it is easy to see that $\|f\|_{C^s_*}\leq \|f\|_{C^s}$ for all $f \in C^s(\overline{\Omega}_0)$. It turns out that $C^s_*(\Omega_0)$ coincides with the Besov space $B^s_{\infty,\infty}(\Omega_0)$, which allows us to use various embedding and approximation results from Besov space theory. Before proceeding, let us fix $\nu>0$ such that $$\label{nu} \nu > \max\left\{\frac{2m}{2(\alpha-\beta)-m},\frac{m}{\beta}\right\}, \qquad \textrm{and define} \qquad \delta_N = N^{-1/(2+\nu)}.$$ It is easy to verify that $\delta_N \to 0$ and $\sqrt{N}\delta_N = N^{\tfrac{\nu}{2(2+\nu)}} \to \infty$ as $N \to \infty$. **Lemma 18**. *Let $c_0 \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)\cap \mathcal{H}$, and define $\delta_N$ as in [\[nu\]](#nu){reference-type="eqref" reference="nu"}. Then for $t>0$ large enough, there exists $C' = C'(\Omega,\Omega_0, \bar{g},\alpha,\beta,\ell,M,c_0,t)>0$ such that for all $N \in \mathbb{N}$, $$\Pi(\mathcal{B}_N(t)) \geq \exp\{-C'N\delta_N^2\}.$$ In particular, there exists $C = C(\Omega,\Omega_0,\bar{g}, \alpha, \beta, \ell, M, c_0)>0$ such that for all $N \in \mathbb{N}$, $$\Pi(B_N) \geq \exp\{-CN\delta_N^2\}.$$* *Proof.* The sets $\{b \in C^3_0(\Omega_0): \|b\|_{C^\beta}\leq \delta\}$ for $\delta>0$ are convex and symmetric. Hence by [@GN15 Corollary 2.6.18], $$\widetilde{\Pi}(\|c-c_0\|_{C^\beta} \leq \delta_N/t) \geq e^{-\|c_0\|_{\mathcal{H}}^2/2}\widetilde{\Pi}(\|c\|_{C^\beta} \leq \delta_N/t).$$ Moreover, since $c_0 \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$, which is open with respect to the $C^\beta$ metric, we have for all sufficiently large $t >0$, $$\Pi(\mathcal{B}_N(t)) = \Pi(\|c-c_0\|_{C^\beta}\leq \delta_N/t) = \frac{ \widetilde{\Pi}(\|c-c_0\|_{C^\beta}\leq \delta_N/t)}{\widetilde{\Pi}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0))},$$ and therefore, $$\Pi(\mathcal{B}_N(t)) \geq e^{-\|c_0\|_{\mathcal{H}}^2/2}\frac{\widetilde{\Pi}(\|c\|_{C^\beta}\leq \delta_N/t)}{\widetilde{\Pi}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0))}. \label{pibn}$$ Next, choose a real number $\gamma$ such that $$\beta < \gamma < \alpha-\frac{m}{2}, \qquad \nu > \frac{2m}{2(\alpha-\gamma)-m}. \label{gammadef}$$ Alternatively, if $\beta$ is not an integer, we can simply set $\gamma=\beta$. In either case, we have $\|f\|_{C^\beta}\leq \|f\|_{C^\gamma_*}$ for all $f \in C^\gamma_*(\Omega_0)$. Now recall our assumption that the RKHS $\mathcal{H}$ of $\widetilde{\Pi}$ is continuously embedded into $H^\alpha(\Omega_0)$. We know from [@ET92 Theorem 3.1.2] that the unit ball $U$ of this space satisfies $$\log \mathcal{N}(U, \|\cdot\|_{C^\gamma_*}, \varepsilon) \leq \left(\frac{A}{\varepsilon}\right)^{\tfrac{m}{(\alpha-\gamma)}}$$ for some fixed $A >0$ and all $\varepsilon>0$ small enough. Therefore, by [@LL99 Theorem 1.2], there exists $D >0$ such that for all $\varepsilon>0$ small enough, $$\widetilde{\Pi}(\|c\|_{C^\beta}\leq \varepsilon) \geq \widetilde{\Pi}(\|c\|_{C^\gamma_*}\leq \varepsilon) \geq \exp\left\{-D\varepsilon^{-\tfrac{2m}{2(\alpha-\gamma)-m}}\right\}.$$ Consequently, [\[pibn\]](#pibn){reference-type="eqref" reference="pibn"} implies that for $t>0$ large enough, $$\begin{aligned} \Pi(\mathcal{B}_N(t)) &\geq \frac{1}{\widetilde{\Pi}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0))}\exp\left\{-\frac{\|c_0\|^2_{\mathcal{H}}}{2}-Dt^{\tfrac{2m}{2(\alpha-\gamma)-m}}\delta_N^{-\tfrac{2m}{2(\alpha-\gamma)-m}}\right\} \\ &> \frac{1}{\widetilde{\Pi}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0))}\exp\left\{-\frac{\|c_0\|^2_{\mathcal{H}}}{2}-Dt^{\tfrac{2m}{2(\alpha-\gamma)-m}}\delta_N^{-\nu}\right\} \qquad \textrm{(by \eqref{nu} and \eqref{gammadef})}\\ &= \frac{1}{\widetilde{\Pi}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0))}\exp\left\{-\frac{\|c_0\|^2_{\mathcal{H}}}{2}-Dt^{\tfrac{2m}{2(\alpha-\gamma)-m}}N\delta_N^2\right\} \\ &\geq \exp\{-C'N\delta_N^2\} \end{aligned}$$ for $C'= \log\left(\widetilde{\Pi}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0))\right) + \frac{\|c_0\|^2_{\mathcal{H}}}{2}+Dt^{\tfrac{2m}{2(\alpha-\gamma)-m}}$. It now follows from Lemma [Lemma 17](#smallball0){reference-type="ref" reference="smallball0"} that for $t>0$ sufficiently large, there exists $C>0$ such that $\Pi(B_N) \geq \exp\{-CN\delta_N^2\}$. This completes the proof. ◻ Thus, we have verified Condition (1) of Theorem [Theorem 15](#g-contr){reference-type="ref" reference="g-contr"}. The next Lemma verifies Condition (2). **Lemma 19**. *There exists $\widetilde{C} = \widetilde{C}(\Omega,\Omega_0,\bar{g},\beta,\ell)>0$ such that $$\log \mathcal{N}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0), h, \delta_N) \leq \widetilde{C}N\delta_N^2.$$* *Proof.* In order to construct a covering of $\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$, it suffices to construct such a covering of the $C^\beta_*(\Omega_0)$ - ball of radius $M$ centered at $0$. Therefore, if $U_\beta$ denotes the unit ball of $C^\beta_*(\Omega_0)$, $$\log \mathcal{N}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0), \|\cdot \|_{L^2}, \delta_N) \leq \log \mathcal{N}(MU_\beta, \|\cdot\|_{L^2}, \delta_N).$$ Now applying [@ET92 Theorem 3.1.2] to the inclusion $C^\beta_*(\Omega_0) \hookrightarrow L^2(\Omega_0)$, we have $$\log \mathcal{N}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0), \|\cdot \|_{L^2}, \delta_N) \leq \left(\frac{A'}{\delta_N}\right)^{\frac{m}{\beta}}$$ for some $A' >0$. Since $\nu > m/\beta$, we get $$\log \mathcal{N}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0), \|\cdot\|_{L^2},\delta_N) \leq b\delta_N^{-\nu} = b N\delta_N^2,$$ where $b>0$. Now, Lemma [Lemma 16](#hellinger-l2){reference-type="ref" reference="hellinger-l2"} and Corollary [Corollary 12](#forward-c){reference-type="ref" reference="forward-c"} imply that an $L^2$ ball of radius $\delta_N$ centered at any $c\in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0)$ is contained in the Hellinger ball of radius $\frac{C_2'}{2\operatorname{Vol}_{\bar{g}}(\partial\Omega)}\delta_N$ centered at the same point. Therefore, by suitably rescaling the constant $b$ to $\widetilde{C}(\Omega,\Omega_0,\bar{g},\beta,\ell,M)>0$, we get the desired complexity bound $$\log \mathcal{N}(\mathcal{C}^{\beta}_{\ell,M}(\Omega_0), h, \delta_N) \leq \widetilde{C}N\delta_N^2.$$ ◻ ## Posterior Convergence In this section, we will combine the results of Sections [3.1](#gencontr){reference-type="ref" reference="gencontr"} and [3.2](#priorprop){reference-type="ref" reference="priorprop"} to prove Theorem [Theorem 14](#main){reference-type="ref" reference="main"}. **Theorem 20**. *Let $\Pi,\alpha, \beta, M, c_0$ be as in Theorem [Theorem 14](#main){reference-type="ref" reference="main"}, $\nu,\delta_N$ as in [\[nu\]](#nu){reference-type="eqref" reference="nu"}, and $C>0$ as in Lemma [Lemma 18](#smallball){reference-type="ref" reference="smallball"}. Then for $k'>0$ large enough, we have $$\label{post1} P^N_{c_0}\left( \Pi(\{c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0): \|Z_c-Z_{c_0}\|_{L^2} \leq k'\delta_N\}|\mathcal{D}_N) \geq 1-e^{-(C+3)N\delta_N^2}\right) \to 1$$ as $N \to \infty$. Moreover, for all $k''>0$ large enough, $$\label{post2} P^N_{c_0}\left(\Pi(\{c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0): \|c-c_0\|_{L^2} \geq k''\delta_N^{1/2} \}|\mathcal{D}_N)\geq e^{-(C+3)N\delta_N^2}\right) \to 0$$ as $N \to \infty$.* *Proof.* Combining Lemmas [Lemma 18](#smallball){reference-type="ref" reference="smallball"} and [Lemma 19](#complexity){reference-type="ref" reference="complexity"} with Theorem [Theorem 15](#g-contr){reference-type="ref" reference="g-contr"}, we get [\[post1\]](#post1){reference-type="eqref" reference="post1"} for all sufficiently large $k'>0$. To get [\[post2\]](#post2){reference-type="eqref" reference="post2"}, consider the event $$E_N = \{c \in \mathcal{C}^{\beta}_{\ell,M}(\Omega_0): \|Z_c-Z_{c_0}\|_{L^2} \leq k'\delta_N \}.$$ By Corollary [Corollary 8](#inverse-c){reference-type="ref" reference="inverse-c"}, for any $c \in E_N$, $$\begin{aligned} \|c-c_0\|_{L^2} &\leq & C_1'\|Z_c-Z_{c_0}\|_{H^1} \\ &\leq& C_1'\|Z_c-Z_{c_0}\|_{L^2}^{1/2}\|Z_c-Z_{c_0}\|_{H^{2}}^{1/2} \end{aligned}$$ by the standard interpolation result for Sobolev spaces. Therefore, by Theorem [Theorem 13](#h2bounds){reference-type="ref" reference="h2bounds"}, $$\|c-c_0\|_{L^2} \leq C_1'(C_3')^{1/2}(k'\delta_N)^{1/2}$$ Taking $k'' > C_1'(k'C_3')^{1/2}$, we conclude that $$\|c-c_0\|_{L^2} \leq k''\delta_N^{1/2}.$$ Combining this with [\[post1\]](#post1){reference-type="eqref" reference="post1"} gives us [\[post2\]](#post2){reference-type="eqref" reference="post2"}. ◻ The final step in the proof of Theorem [Theorem 14](#main){reference-type="ref" reference="main"} is to prove that the posterior contraction rate in the above Theorem carries over to the posterior mean $\overline{c}_N = \mathbb{E}^\Pi[c|\mathcal{D}_N]$ as well. Let $$0 < \omega < \frac{1}{2(2+\nu)}.$$ We note that $\omega$ can be made arbitrarily close to $1/4$ by choosing $\alpha, \beta$ appropriately. Indeed, if $\alpha$ and $\beta$ are sufficiently large, [\[nu\]](#nu){reference-type="eqref" reference="nu"} allows $\nu$ to be arbitrarily close to $0$. Correspondingly, $\omega$ can be made arbitrarily close to $1/4$. Next, define $$\omega_N := k''\delta_N^{1/2} = k''N^{-\frac{1}{2(2+\nu)}} = o(N^{-\omega})$$ where $k''>0$ is as in Theorem [Theorem 20](#postcontr){reference-type="ref" reference="postcontr"}. *Proof of Theorem [Theorem 14](#main){reference-type="ref" reference="main"}.* Observe that $$\begin{aligned} \|\overline{c}_N-c_0\|_{L^2} &=& \left\|\mathbb{E}^{\Pi}[c|\mathcal{D}_N]-c_0\right\|_{L^2} \\ &\leq & \mathbb{E}^{\Pi}\left[\|c-c_0\|_{L^2}|\mathcal{D}_N\right] \quad \textrm{(by Jensen's inequality)} \\ &\leq & \omega_N +\mathbb{E}^{\Pi}\left[\|c-c_0\|_{L^2}\mathds{1}_{\{\|c-c_0\|_{L^2}\geq \omega_N\}}\big|\mathcal{D}_N\right] \\ &\leq & \omega_N +\mathbb{E}^{\Pi}\left[\|c-c_0\|_{L^2}^2|\mathcal{D}_N\right]^{1/2}\left[\Pi(\|c-c_0\|_{L^2}\geq \omega_N|\mathcal{D}_N)\right]^{1/2} \end{aligned}$$ by Cauchy-Schwarz inequality. Now it suffices to show that the second summand on the right hand side is stochastically $O(\omega_N)$ as $N \to \infty$. Arguing as in the proof of Theorem [Theorem 15](#g-contr){reference-type="ref" reference="g-contr"} and applying Lemma [Lemma 18](#smallball){reference-type="ref" reference="smallball"}, we get that the events $$A_N' = \left\{ \int_{\mathcal{C}^{\beta}_{\ell,M}(\Omega_0)}\prod_{i=1}^N\frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\Pi(c) \geq e^{-(2+C)N\delta_N^2}\right\}$$ satisfy $P^N_{c_0}(A_N')\to 1$ as $N \to \infty$. Here, $C$ is as in Lemma [Lemma 18](#smallball){reference-type="ref" reference="smallball"}. Now, Theorem [Theorem 20](#postcontr){reference-type="ref" reference="postcontr"} implies $$\begin{aligned} & P^N_{c_0}\left(\mathbb{E}^{\Pi}\left[\|c-c_0\|_{L^2}^2|\mathcal{D}_N\right]\times \Pi(\|c-c_0\|_{L^2}\geq \omega_N|\mathcal{D}_N) > \omega_N^2\right) \\ & \leq P^N_{c_0}\left( \mathbb{E}^{\Pi}\left[\|c-c_0\|_{L^2}^2|\mathcal{D}_N\right]e^{-(C+3)N\delta_N^2}>\omega_N^2\right) +o(1), \end{aligned}$$ which is bounded above by $$\begin{aligned} & P^N_{c_0}\left( e^{-(C+3)N\delta_N^2}\mathbb{E}^{\Pi}\left[\|c-c_0\|_{L^2}^2|\mathcal{D}_N\right] > \omega_N^2 , A_N'\right) + o(1) \nonumber \\ &= P^N_{c_0}\left(e^{-(C+3)N\delta_N^2}\frac{\int \|c-c_0\|_{L^2}^2\prod_{i=1}^N \frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\Pi(c)}{\int \prod_{i=1}^N \frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\Pi(c)} > \omega_N^2, A_N'\right) +o(1) \nonumber \\ &\leq P^N_{c_0}\left( \int \|c-c_0\|_{L^2}^2 \prod_{i=1}^N\frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\Pi(c) > \omega_N^2e^{N\delta_N^2}\right) +o(1) \label{inter} \end{aligned}$$ using the fact that $\int \prod_{i=1}^N\frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\Pi(c) \geq e^{-(C+2)N\delta_N^2}$ on $A_N'$. Next, using Markov's inequality, [\[inter\]](#inter){reference-type="eqref" reference="inter"} can be further bounded above by $$\begin{aligned} &\leq e^{-N\delta_N^2}\omega_N^{-2}\mathbb{E}^N_{c_0}\left[\int \|c-c_0\|_{L^2}^2\prod_{i=1}^N \frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)d\Pi(c)\right] +o(1) \\ &= e^{-N\delta_N^2}\omega_N^{-2}\int \|c-c_0\|_{L^2}^2 \mathbb{E}_{c_0}^N\left[\prod_1^N\frac{p_c}{p_{c_0}}(X_i,Y_i,Z_i)\right]d\Pi(c) +o(1) \quad \textrm{(by Fubini's Theorem)} \\ &\leq e^{-N\delta_N^2}\omega_N^{-2}\int \|c-c_0\|_{L^2}^2d\Pi(c) + o(1) \quad \left(\textrm{since } \mathbb{E}^N_{c_0}\left[\prod_1^N\frac{p_c}{p_{c_0}}\right]=1\right) \\ &\lesssim e^{-N\delta_N^2}\omega_N^{-2} +o(1) \lesssim e^{-N\delta_N^2}N^{2\omega} +o(1) \to 0 \textrm{ as } N \to \infty \end{aligned}$$ This completes the proof. ◻

arxiv_math

--- abstract: | This study analyzes the derivative-free loss method to solve a certain class of elliptic PDEs using neural networks. The derivative-free loss method uses the Feynman-Kac formulation, incorporating stochastic walkers and their corresponding average values. We investigate the effect of the time interval related to the Feynman-Kac formulation and the walker size in the context of computational efficiency, trainability, and sampling errors. Our analysis shows that the training loss bias is proportional to the time interval and the spatial gradient of the neural network while inversely proportional to the walker size. We also show that the time interval must be sufficiently long to train the network. These analytic results tell that we can choose the walker size as small as possible based on the optimal lower bound of the time interval. We also provide numerical tests supporting our analysis. author: - "Jihun Han[^1] and Yoonsang Lee[^2]" bibliography: - dflm_analysis.bib title: An analysis of the derivative-free loss method for solving PDEs --- **MSC codes.** 65N15, 65N75, 65C05, 60G46 # Introduction The neural network is well known for its flexibility to represent complicated functions in a high-dimensional space [@cybenko; @hornik]. In recent years, this strong property of the neural network has naturally led to representing the solution of partial differential equations (PDEs). Physics-informed neural network [@PINN] and Deep Galerkin [@DGM] use the strong form of the PDE to define the training loss, while the Deep Ritz [@DRM] method uses a weak (or variational) formulation of PDEs to train the network. Also, a class of methods uses a stochastic representation of PDEs to train the neural network [@BSDE; @DFLM]. All these methods have shown successful results in a wide range of problems in science and engineering, particularly for high-dimensional problems where the standard numerical PDE methods have limitations [@BSDE; @DGM; @PINNreview]. The goal of the current study is an analysis of the derivative-free loss method (DFLM; [@DFLM]). DFLM employs a stochastic representation of the solution for a certain class of PDEs, averaging stochastic samples as a generalized Feynman-Kac formulation. The loss formulation of DFLM directly guides a neural network to learn the point-to-neighborhood relationships of the solution. DFLM adopts bootstrapping in the context of reinforcement learning, where the neural network's target function is computed based on its current state through the point-to-neighborhood relation. This iterative process incrementally refines the neural network toward solving the PDE. {#fig:sampling_diagram width="70%"} It is shown in [@DFLM] that the derivative-free formulation offers advantages in handling singularities that arise due to the geometric characteristics of the domain, particularly in cases involving sharp boundaries. Also, using the intrinsic averaging character in representing the solution, DFLM has been successfully applied to the homogenization problem [@DFLMHomo] and a nonlinear flow problem [@DFLMflow]. As in the other methods to solve PDEs numerically, there are collocation points in DFLM to impose constraints for the solution to satisfy. Additionally, there are $N_s$ walkers centered at each collocation point, which is related to the expected value calculation of the Feynman-Kac formula. By solving a certain stochastic process related to the given PDE operator for some time $\Delta t$, DFLM uses an expected value related to the walkers to represent the PDE solution at each collocation point. Note that $\Delta t$ is not the time step to solve a time-dependent problem; we consider only elliptic problems in the current study. Figure [1](#fig:sampling_diagram){reference-type="ref" reference="fig:sampling_diagram"} shows schematics of DFLM for the two parameters $N_s$ and $\Delta t$ of various values. As $N_s$ increases (that is, we have more walkers to explore a neighborhood), we have more sample values related to the expectation. In contrast, a large $\Delta t$ increases the spread of the walkers around each collocation point. A large $N_s$ will decrease the sampling error while a long $\Delta t$ will cover a wide range of neighborhoods. However, a large $N_s$ and a long $\Delta t$ will increase the overall computational cost; a large $N_s$ requires more walkers to run (or more samples to draw using $\tilde{q}$-martingale in Section [2](#sec:DFLM){reference-type="ref" reference="sec:DFLM"}), and $\Delta t$ requires a more extended time period to run the walkers. Our analysis (Theorem [Theorem 1](#thm:lossbias){reference-type="ref" reference="thm:lossbias"} in Section [3](#sec:analysis){reference-type="ref" reference="sec:analysis"}) shows that the empirical training loss has a bias bounded by $\frac{\Delta t}{N_s}$. This result implies that we can use a small $N_s$ for computational efficiency while keeping $\Delta t$ small accordingly. In this study, we also show (Theorem [Theorem 2](#thm:pointwise_learning){reference-type="ref" reference="thm:pointwise_learning"} in Section [3](#sec:analysis){reference-type="ref" reference="sec:analysis"}) that the time interval $\Delta t$ must be sufficiently large to guarantee the training of the PDE solution. When $\Delta t$ is too small, the walkers do not move enough to explore neighborhoods and thus cannot see the local variations of the solution. The overall message of our analysis for DFLM is that there exists an optimal lower bound for the time interval $\Delta t$, and we can choose the number of walkers $N_s$ (at a collocation point) as small as possible based on the optimal $\Delta t$. The current paper is structured as follows. In Section [2](#sec:DFLM){reference-type="ref" reference="sec:DFLM"}, we review DFLM for an elliptic problem, explaining all parameters related to our analysis. Section [3](#sec:analysis){reference-type="ref" reference="sec:analysis"} has all our analytic results for DFLM, including the upper bound of the training loss bias and the training issue using a small $\Delta t$ for the stochastic walkers. We provide numerical examples in Section [4](#sec:experiment){reference-type="ref" reference="sec:experiment"}, supporting our analysis. This paper concludes in Section [5](#sec:discussion){reference-type="ref" reference="sec:discussion"} with discussions about limitations and future directions. # Derivative-Free Neural Network Training Method {#sec:DFLM} This section reviews the derivative-free loss method (DFLM) for solving a certain type of elliptic PDEs [@DFLM]. DFLM tackles the PDEs through the underlying stochastic representation in the spirit of the Feynman-Kac formula that describes how the solution at a point interrelates to its neighborhood. DFLM guides a neural network directly to satisfy the interrelation across the entire domain, leading to the solution of the PDE. This intercorrelation characteristic makes DFLM different from other methods that use pointwise residuals of the strong form PDE, such as PINN [@PINN], in which a neural network implicitly learns the interrelation among different points. Moreover, DFLM gradually and alternately updates a neural network and corresponding target values toward the PDE solution in the same manner as bootstrapping in the context of reinforcement learning, which differs from supervised learning methods that optimize neural network parameters within a fixed topology defined by loss functions. In this paper, we consider DFLM for the following type of elliptic PDEs of an unknown function $u(\bm{x}) \in \mathbb{R}$: $$\label{eq:Quasi-linear elliptic} \mathcal{N}[u](\bm{x}):= \frac{1}{2}\Delta u(\bm{x}) + \bm{V}\cdot \nabla_{\bm{x}} u(\bm{x}) - G = 0, ~\textrm{in}~ \Omega \subset \mathbb{R}^k.$$ Here $\bm{V}=\bm{V}(\bm{x},u(\bm{x})) \in \mathbb{R}^k$ is the advection velocity and $G=G(\bm{x}, u(\bm{x})) \in \mathbb{R}$ is the force term, both of which can depend on $u$. From the standard application of It$\hat{\text{o}}$'s lemma (e.g., in [@karatzas]), we have the stochastic representation of the solution of Eq. [\[eq:Quasi-linear elliptic\]](#eq:Quasi-linear elliptic){reference-type="eqref" reference="eq:Quasi-linear elliptic"} through the following equivalence; - $u:\Omega \rightarrow \mathbb{R}$ is a solution of Eq. [\[eq:Quasi-linear elliptic\]](#eq:Quasi-linear elliptic){reference-type="eqref" reference="eq:Quasi-linear elliptic"}. - the stochastic process $q(t;u, \bm{x}, \{\bm{X}_s\}_{0\leq s\leq t}) \in \mathbb{R}$ defined as $$\label{eq:q_martingale_stochastic_process} q(t;u, \bm{x},\{\bm{X}_s\}_{0\leq s\leq t}) := u(\bm{X}_t) - \int_0^t G(\bm{X}_s,u(\bm{X}_s))ds,\\$$ where $\bm{X}_t \in \mathbb{R}^k$ is a stochastic process of the following SDE $$\label{eq:q_stochastic_walkers} d\bm{X}_t = \bm{V}(\bm{X}_t, u(\bm{X}_t))dt + d\bm{B}_t,\quad \bm{B}_t:\mbox{ standard Brownian motion in }\mathbb{R}^k,$$ satisfies the martingale property $$\begin{aligned} \label{eq:q-martingale} \begin{split} u(\bm{x})&=q(0;u,\bm{x}, \bm{X}_0)=\mathbb{E}\left[q\left(t;u,\bm{x}, \{\bm{X}_s\}_{0\leq s\leq t} \right) | \bm{X}_0=\bm{x}\right] \\ &= \mathbb{E}\left[u(\bm{X}_t) - \int_{0}^{t}G(\bm{X}_s, u(\bm{X}_s)) ds \middle |\bm{X}_0=\bm{x} \right],~\forall\bm{x}\in \Omega, \forall t>0. \end{split}\end{aligned}$$ Regarding to the definition of stochastic process $q(t;u, \bm{x},\{\bm{X}_s\}_{0\leq s\leq t})$, the infinitesimal drift $d(\cdot)$ of the stochastic process $u(\bm{X}_t)$ is connected to the differential operator $\mathcal{N}[u]$ as $$\label{eq:u_x_ito} d(u(\bm{X}_t)) = \left(\mathcal{N}[u](\bm{X}_t) + G(\bm{X}_t, u(\bm{X}_t)\right) dt + \nabla u (\bm{X}_t) \cdot d\bm{B}_t.$$ The martingale property, Eq. [\[eq:q-martingale\]](#eq:q-martingale){reference-type="eqref" reference="eq:q-martingale"}, shows that the solution at a point $\bm{x}$, $u(\bm{X})$ can be represented through its neighborhood statistics observed by the stochastic process $\bm{X}_t$ starting at the point $\bm{x}$ during the time period $[0, t]$. We note that the representation holds for an arbitrary time $t>0$ and any stopping time $\tau$ by the optional stopping theorem [@karatzas]. In particular, the exit time from the domain as the stopping time, $\tau=\inf \{s: \bm{X}_s \notin \Omega \}$, induces the well-known Feynman-Kac formula for the PDE [@oksendal]. Note that other methods are based on the classical Monte-Carlo of the Feynman-Kac formula [@classicFeynman1; @classicFeynman2; @classicFeynman3; @classicFeynman4]. Such methods estimate the solution of a PDE at an individual point independently with the realizations of the stochastic processes $\bm{X}_t$ until it exits from the given domain. DFLM, on the other hand, approximates the PDE solution over the domain at once through a neural network $u(\bm{x};\bm{\theta})$, which is trained to satisfy the martingale property Eq. [\[eq:q-martingale\]](#eq:q-martingale){reference-type="eqref" reference="eq:q-martingale"}. In particular, DFLM considers a short period, say $\Delta t$, rather than waiting for whole complete trajectories until $\bm{X}_t$ is out of the domain. This character of DFLM allows a neural network to learn more frequently for a short time period. DFLM constructs the loss function for training a neural network as $$\begin{aligned} \label{eq:q_loss_continuous} \mathcal{L}^{\Omega}(\bm{\theta})&=\mathbb{E}_{\bm{x}\sim \Omega}\left[\left|u(\bm{x};\bm{\theta})-\mathbb{E}\left[q\left(\Delta t;u(\cdot;\bm{\theta}),\bm{x}, \{\bm{X}_s\}_{0\leq s\leq \Delta t}\right) | \bm{X}_0=\bm{x}\right] \right|^2\right]\\ &=\mathbb{E}_{\bm{x}\sim \Omega} \left[\left|u(\bm{x};\bm{\theta}) - \mathbb{E}_{\{\bm{X}_s\}_{0\leq s\leq \Delta t}}\left[u(\bm{X}_t;\bm{\theta}) - \int_{0}^{\Delta t}G(\bm{X}_s, u(\bm{X}_s;\bm{\theta})) ds \middle |\bm{X}_0=\bm{x} \right] \right|^2 \right]\end{aligned}$$ where the outer expection is over the sample collocation point $\bm{x}$ in the domain $\Omega$ and the inner expectation is over the stochastic path $\bm{X}_t$ starting at $\bm{X}_0=\bm{x}$ during $[0,\Delta t]$. In the presence of the drive term $\bm{V}$ that can depend on $u$, the statistics of $\bm{V}$ will be nontrivial and thus a numerical approximation to $\bm{X}_t$ must be calculated by solving Eq. [\[eq:q_stochastic_walkers\]](#eq:q_stochastic_walkers){reference-type="eqref" reference="eq:q_stochastic_walkers"}. As an alternative to avoid the calculation of the solution to Eq. [\[eq:q_stochastic_walkers\]](#eq:q_stochastic_walkers){reference-type="eqref" reference="eq:q_stochastic_walkers"}, another martingale process $\tilde{q}(t;u,\bm{x},\{\bm{B}_s\}_{0\leq s\leq t})$ based on the standard Brownian motion $\bm{B}_t$ is proposed as $$\begin{aligned} \label{eq:q_tilde_martingale_process} \tilde{q}(t;u, \bm{x}&,\{\bm{B}_s\}_{0\leq s\leq t}) := \left(u(\bm{B}_t) - \int_0^t G(\bm{B}_s,u(\bm{B}_s))ds\right) \mathcal{D}(\bm{V},u,t),\\ &\text{where}{\quad}\mathcal{D}(\bm{V},u,t)=\exp \biggl(\int^t_0\bm{V}(\bm{B}_s, u(\bm{B}_s))\cdot d\bm{B}_s-\frac{1}{2}\int^{t}_{0}|\bm{V}(\bm{B}_s, u(\bm{B}_s))|^2 ds \biggr). \nonumber \end{aligned}$$ Here, $\tilde{q}$-process is of form replacing the $\bm{X}_t$ to $\bm{B}_t$ in $q$-process with additional exponential factor $\mathcal{D}(\bm{V}, u,t)$ compensating the removal of the drift effect in $\bm{X}_t$ [@karatzas; @oksendal]. Using the alternative $\tilde{q}$-martingale allows the standard Brownian walkers to explore the domain regardless of the form of the given PDE, which can be drawn from the standard Gaussian distribution without solving SDEs. The alternative loss function corresponding to $\tilde{q}$-martingale is $$\mathcal{L}^{\Omega}(\bm{\theta})=\mathbb{E}_{\bm{x}\sim \Omega}\left[\left|u(\bm{x};\bm{\theta})-\mathbb{E}\left[\tilde{q}(t;u(\cdot;\bm{\theta}),\bm{x}, \{\bm{B}_s\}_{0\leq s\leq \Delta t}) | \bm{B}_0=\bm{x}\right] \right|^2\right].$$ In the standard DFLM [@DFLM], for the Dirichlet boundary condition, $u(\bm{x})=g(\bm{x})$ on $\partial \Omega$, we consider $\bm{X}_t$ to be absorbed to the boundary $\partial \Omega$ at the exit position and the value of the neural network is replaced by the given boundary value at the exit position, which makes the information propagate from the boundary into the domain's interior. In this study, to enhance the constraint on the boundary, we add the following boundary loss term $$\mathcal{L}^{\partial \Omega}(\bm{\theta}) = \mathbb{E}_{\bm{x}\sim \partial \Omega}\left[|u(\bm{x};\bm{\theta})-g(\bm{x})|^2 \right]$$ in the total loss $${\mathcal{L}}(\bm{\theta})={\mathcal{L}}^{\Omega}(\bm{\theta}) + {\mathcal{L}}^{\partial\Omega}(\bm{\theta}).$$ The loss function ${\mathcal{L}}(\bm{\theta})$ is optimized by a stochastic gradient descent method, and, in particular, the bootstrapping approach is used as the target of the neural network (i.e., the expectation component of $q$- or $\tilde{q}$-process) is pre-evaluated using the current state of neural network parameters $\bm{\theta}$. The $n$-th iteration step for updating the parameters $\bm{\theta}_n$ is $$\label{eq:sgd_update} \bm{\theta}_n = \bm{\theta}_{n-1} - \alpha \nabla\widetilde{\mathcal{L}}_n(\bm{\theta}_{n-1}), {\quad}\text{where}{\quad} \widetilde{\mathcal{L}}_n(\bm{\theta})=\widetilde{\mathcal{L}}_n^{\Omega}(\bm{\theta}) + \widetilde{\mathcal{L}}^{\partial\Omega}(\bm{\theta}).$$ Here, the term $\widetilde{\mathcal{L}}_n^{\Omega}(\bm{\theta})$ is the empirical interior loss function using $N_r$ sample collocation points $\{\bm{x}_i\}_{i=1}^{N_r}$ in the interior of the domain $\Omega$, and $N_s$ stochastic walkers $\left\{\bm{X}_s^{(i,j)};s \in [0,\Delta t], \bm{X}_0=\bm{x}_i\right \}_{j=1}^{N_s}$ at each sample collocation point $\bm{x}_i$, $$\begin{aligned} \label{eq:q_loss_discrete} \widetilde{\mathcal{L}}^{\Omega}_{n}(\bm{\theta}) :&= \widetilde{\mathbb{E}}_{\bm{x}\sim \Omega}\left[\left|u(\bm{x};\bm{\theta})-\widetilde{\mathbb{E}}\left[q\left(\Delta t;u(\cdot;\bm{\theta}_{n-1}),\bm{x}, \{\bm{X}_s\}_{0\leq s\leq \Delta t}\right) | \bm{X}_0=\bm{x}\right] \right|^2\right] \\ &= \frac{1}{N_r} \sum \limits_{i=1}^{N_r}\left| u(\bm{x}_i;\bm{\theta})-\frac{1}{N_s}\sum\limits_{j=1}^{N_s}\left\{ u\left(\bm{X}_{\Delta t}^{(i,j)};\bm{\theta}_{n-1} \right)-\int_0^{\Delta t}G\left(\bm{X}_s^{(i,j)}, u\left(\bm{X}_{\Delta t}^{(i,j)};\bm{\theta}_{n-1} \right)\right) ds \right\} \right |^2.\end{aligned}$$ The other term $\widetilde{\mathcal{L}}^{\partial \Omega}(\bm{\theta})$ is the empirical boundary loss function using $N_b$ random boundary collocation points $\{\bm{x}_l\}_{l=1}^{N_b}$ on $\partial \Omega$ $$\widetilde{\mathcal{L}}^{\partial \Omega}(\bm{\theta}) := \widetilde{\mathbb{E}}_{\bm{x}\sim \partial \Omega}\left[|u(\bm{x};\bm{\theta})-g(\bm{x})|^2 \right] = \frac{1}{N_b}\sum \limits_{l=1}^{N_b} \left| u(\bm{x}_l;\bm{\theta}) - g(\bm{x}_l)\right|^2.$$ The random interior and boundary collocation points $\{\bm{x}_i\}_{i=1}^{N_r}$ and $\{\bm{x}_l\}_{l=1}^{N_b}$ can follow a distribution whose support covers the domain $\Omega$ and the boundary $\partial \Omega$, respectively. The learning rate $\alpha$ could be tuned at each step and the gradient descent step can be optimized by considering the previous steps, such as Adam optimization [@adam]. # Analysis {#sec:analysis} DFLM utilizes stochastic walkers within each local neighborhood of collocation points over the domain. Our primary interest is to comprehend these stochastic walkers' influence on the training of a neural network. The information integrated into the neural network during each iteration is affected by two key parameters: i) the time interval $\Delta t$ and ii) the number of stochastic walkers $N_s$. Both parameters determine how far (by $\Delta t$) and densely (by $N_s$) the walkers collect the information around the corresponding collocation points (see Fig. [1](#fig:sampling_diagram){reference-type="ref" reference="fig:sampling_diagram"}). We show in this section that the network's training to approximate a PDE solution is hindered by the bias of the empirical loss function, which depends on the two parameters. Moreover, we show that DFLM requires suitably spacious neighborhood observations through the stochastic walkers to approximate the solution effectively. This implies that a lower bound $\Delta t^{\ast}$ exists for the time interval $\Delta t$. ## Bias in the empirical martingale loss function **Theorem 1**. *The empirical loss $\widetilde{\mathcal{L}}^{\Omega}(\bm{\theta})$ in Eq. [\[eq:q_loss_discrete\]](#eq:q_loss_discrete){reference-type="eqref" reference="eq:q_loss_discrete"} is a biased estimator of the exact martingale loss $\mathcal{L}^{\Omega}(\bm{\theta})$ in Eq. [\[eq:q_loss_continuous\]](#eq:q_loss_continuous){reference-type="eqref" reference="eq:q_loss_continuous"}. Moreover, when $u(\cdot;\bm{\theta})$ has a small PDE residual $\mathcal{N}[u(\cdot;\bm{\theta})]$ in Eq. [\[eq:Quasi-linear elliptic\]](#eq:Quasi-linear elliptic){reference-type="eqref" reference="eq:Quasi-linear elliptic"}, the bias is proprotional to $\Delta t$ and the $\mathcal{L}^2$-norm of $\nabla_{\bm{x}}u(\cdot;\bm{\theta})$ with respect to the sampling measure $\bm{x}\sim \Omega$, while the bias is inversely proportional to $N_s$ $$\normalfont{\mbox{Bias}}_{\mathcal{L}^{\Omega}}\left[\widetilde{\mathcal{L}}^{\Omega} \right]{\quad}\propto{\quad} \frac{\Delta t}{N_s}\mathbb{E}_{\bm{x}\sim \Omega}\left[|\nabla_{\bm{x}} u(\bm{x};\bm{\theta})|^2\right].$$* *Proof.* For notational simplicity, for a fixed $\Delta t$, we use $y_{\bm{x}}$ for the target random variable $$y_{\bm{x}}:=q\left(\Delta t;u(\cdot;\bm{\theta}),\bm{x}, \{\bm{X}_s\}_{0\leq s\leq \Delta t}\right)$$ where the subscript $\bm{x}$ describes for the intial value of the stochastic process $\bm{X}_0=\bm{x}$. We also denote the unbiased sample mean statistic for the target as $\overline{y_{\bm{x}}}$ $$\overline{y_{\bm{x}}}:=\widetilde{\mathbb{E}}\left[q\left(\Delta t;u(\cdot;\bm{\theta}_{n-1}),\bm{x}, \{\bm{X}_s\}_{0\leq s\leq \Delta t}\right) | \bm{X}_0=\bm{x}\right].$$ We denote the sampling measure of $\bm{x}$ as $\mathbb{P}(\bm{x})$ and the distribution of $\overline{y_{\bm{x}}}$ conditioned on $\bm{x}$ as $\mathbb{P}_{\bm{\theta}}(\overline{y_{\bm{x}}}|\bm{x})$ where the subcript $\bm{\theta}$ corresponds to the dependency of the distribution on the neural network's state. We now take the expectation of the empirical loss with respect to $\bm{x}$, which yields $$\begin{aligned} \mathbb{E}\left[\widetilde{\mathcal{L}}^{\Omega}(\bm{\theta})\right] &= \int_{\bm{x}}\left(\int_{\overline{y_{\bm{x}}}}(u(\bm{x};\bm{\theta})-\overline{y_{\bm{x}}})^2 \mathbb{P}_{\bm{\theta}}(\overline{y_{\bm{x}}}|\bm{x}) d\overline{y_{\bm{x}}} \right)\mathbb{P}(\bm{x})d\bm{x} \\ &=\int_{\bm{x}}u^2(\bm{x};\bm{\theta})\mathbb{P}(\bm{x})d\bm{x}-\int_{\bm{x}}2u(\bm{x};\bm{\theta})\mathbb{E}_{\mathbb{P}_{\bm{\theta}}(\cdot|\bm{x})}[\overline{y_{\bm{x}}}]\mathbb{P}(\bm{x})d\bm{x} + \int_{\bm{x}}\mathbb{E}_{\mathbb{P}_{\bm{\theta}}(\cdot|\bm{x})}\left[\overline{y_{\bm{x}}}^2 \right]\mathbb{P}(\bm{x})d\bm{x} \\ &= \int_{\bm{x}}\left(u(\bm{x};\bm{\theta})-\mathbb{E}_{\mathbb{P}_{\bm{\theta}}(\cdot|\bm{x})}[\overline{y_{\bm{x}}}]\right)^2\mathbb{P}(\bm{x})d\bm{x} + \int_{\bm{x}}\left(\mathbb{E}_{\mathbb{P}_{\bm{\theta}}(\cdot|\bm{x})}\left[\overline{y_{\bm{x}}}^2\right]-\mathbb{E}_{\mathbb{P}_{\bm{\theta}}(\cdot|\bm{x})}[\overline{y_{\bm{x}}}]^2 \right) \mathbb{P}(\bm{x})d\bm{x} \\ &= \mathbb{E}_{\bm{x}\sim\mathbb{P}}\left[\left(u(\bm{x};\bm{\theta})-\mathbb{E}_{\mathbb{P}_{\bm{\theta}}(\cdot|\bm{x})}[\overline{y_{\bm{x}}}]\right)^2 \right] + \mathbb{E}_{\bm{x}\sim\mathbb{P}}\left[ \mathbb{V}_{\mathbb{P}_{\bm{\theta}}(\cdot|\bm{x})}(\overline{y_{\bm{x}}})\right]\\ &= \mathcal{L}^{\Omega}(\bm{\theta}) + \mathbb{E}_{\bm{x}\sim\mathbb{P}}\left[ \mathbb{V}_{\mathbb{P}_{\bm{\theta}}(\cdot|\bm{x})}(\overline{y_{\bm{x}}})\right],\end{aligned}$$ which implies that the empirical loss $\widetilde{\mathcal{L}}^{\Omega}(\bm{\theta})$ estimates the exact martingale loss $\mathcal{L}^{\Omega}(\bm{\theta})$ with the bias $\mathbb{E}_{\bm{x}\sim\mathbb{P}}\left[ \mathbb{V}_{\mathbb{P}_{\bm{\theta}}(\cdot|\bm{x})}(\overline{y_{\bm{x}}})\right]$. When the PDE residual of $u(\cdot;\bm{\theta})$, $\mathcal{N}[u(\cdot;\bm{\theta})]$, is sufficiently small, the stochastic process $q(\Delta t;$ $u(\cdot;\bm{\theta}),\bm{x}, \{\bm{X}_s\}_{0\leq s\leq \Delta t})$ corresponding to $y_{\bm{x}}$ can be approximated as follows $$\begin{aligned} d(q\left(\Delta t;u(\cdot;\bm{\theta}),\bm{x}, \{\bm{X}_s\}_{0\leq s\leq \Delta t}\right)) &= (\mathcal{N}[u](\bm{X}_t)) dt + \nabla_{\bm{x}} u (\bm{X}_t;\bm{\theta}) \cdot d\bm{B}_t \\ &\simeq \nabla_{\bm{x}} u (\bm{X}_t;\bm{\theta}) \cdot d\bm{B}_t.\end{aligned}$$ The variance of $y_{\bm{x}}$ is $$\begin{aligned} \mathbb{V}[y_{\bm{x}}] &= \mathbb{V}\left[\int_0^{\Delta t} \nabla_{\bm{x}} u(\bm{X}_s;\bm{\theta})\cdot d\bm{B}_s \middle | \bm{X}_0=\bm{x} \right] \\ &= \mathbb{E}\left[\left(\int_0^{\Delta t} \nabla_{\bm{x}} u(\bm{X}_s;\bm{\theta})\cdot d\bm{B}_s \right)^2 \middle | \bm{X}_0=\bm{x} \right] \\ &= \mathbb{E}\left[\int_0^{\Delta t} |\nabla_{\bm{x}} u(\bm{X}_s;\bm{\theta})|^2 ds \middle | \bm{X}_0=\bm{x} \right] {\quad}{\quad}~(\because \text{It$\hat{\text{o}}$ isometry}) \\ & \simeq \left |\nabla_{\bm{x}} u (\bm{x};\bm{\theta}) \right|^2\Delta t \hspace{5mm}( \Delta t \ll 1)\end{aligned}$$ Therefore the variance of the sample mean $\overline{y_{\bm{x}}}$ of $y_{\bm{x}}$ is approximated as $$\label{eq:each_target_sample_mean_variance} \mathbb{V}[\overline{y_{\bm{x}}}] = \frac{1}{N_s}\mathbb{V}[y_{\bm{x}}] \simeq \frac{\left |\nabla_{\bm{x}} u (\bm{x};\bm{\theta}) \right|^2\Delta t}{N_s}.$$ By taking the expectation over the sampling measure, the bias of the empirical loss $\widetilde{\mathcal{L}}^{\Omega}(\bm{\theta})$ is $$\normalfont{\mbox{Bias}}_{\mathcal{L}^{\Omega}}\left[\widetilde{\mathcal{L}}^{\Omega} \right]{\quad} \simeq {\quad}\frac{\Delta t}{N_s} \mathbb{E}_{\bm{x}\sim \Omega}\left[| \nabla_{\bm{x}} u(\bm{x};\bm{\theta}) |^2 \right].$$ ◻ Theorem [Theorem 1](#thm:lossbias){reference-type="ref" reference="thm:lossbias"} shows that the neural network optimization using the empirical loss is inappropriately regularized to the variance associated with target samples. This could impede the convergence of the neural network to satisfy the martingale property. Moreover, as the neural network approaches the solution, the target variance at each point primarily depends on the local topology of the neural network approximation; the larger the gradient's magnitude is, the higher the variance of the target sample. The two key parameters determine the scale of this variance. When the time interval $\Delta t$ extends, each stochastic walker can explore a broader neighborhood and collect more information regarding the solution over a larger measure. However, the expanded observation can be counterproductive for training as the expansion increases the bias unless sufficiently large enough stochastic walkers (i.e., large $N_s$) are engaged in the target computation and thus mitigate the bias. Conversely, when we reduce the time interval $\Delta t$, achieving a comparable bias level requires a relatively smaller size stochastic walkers (i.e., small $N_s$) compared to a larger time interval setting. However, the reduction in the time interval entails observing a smaller neighborhood, leading to a decrease in the collected information. From the variance estimation of the mean statistic in Eq. [\[eq:each_target_sample_mean_variance\]](#eq:each_target_sample_mean_variance){reference-type="eqref" reference="eq:each_target_sample_mean_variance"}, we quantify the uncertainty of the empirical target value at each point $\bm{x}$ through the Chebyshev's inequality as $$\forall \epsilon >0, \hspace{3mm} \mathbb{P}\left(\left|\overline{y_{\bm{x}}}- \mathbb{E}\left[\overline{y_{\bm{x}}} \right] \right| > \epsilon \right) \lesssim \frac{|\nabla_{\bm{x}} u(\bm{x};\bm{\theta})|^2\Delta t}{\epsilon^2 N_s}.$$ **Corollary 1**. *The empirical loss $\widetilde{\mathcal{L}}^{\Omega}(\bm{\theta})$ in Eq. [\[eq:q_loss_discrete\]](#eq:q_loss_discrete){reference-type="eqref" reference="eq:q_loss_discrete"} is an asymptotically unbiased estimator of the exact martingale loss $\mathcal{L}^{\Omega}(\bm{\theta})$ in Eq. [\[eq:q_loss_continuous\]](#eq:q_loss_continuous){reference-type="eqref" reference="eq:q_loss_continuous"} with respect to both the time interval (i.e., $\Delta t \rightarrow 0$) and the number of stochastic walkers (i.e., $N_s \rightarrow \infty$).* ## Analysis for the time interval of the Feynman-Kac formulation We now focus on the trainability issue for a small $\Delta t$. For a $k$-dimensional vector $\bm{\mu}\in \mathbb{R}^{k}$ and a positive value $\sigma^2 \in \mathbb{R}^{+}$, we detnote $f_{\bm{\mu}, \sigma^2}$ as the probability density function (PDF) of multivariate normal distribution with mean $\bm{\mu}$ and covariance matrix $\sigma^2 \mathbb{I}$ where $\mathbb{I}$ is the identity matrix in $\mathbb{R}^{k\times k}$. Hereafter, for notational simplicity, we suppress the dependence of the advection and force terms on $u(\bm{x})$, that is, $\bm{V}(\bm{x})=\bm{V}(\bm{x},u(\bm{x}))$ and $G(\bm{x})=G(\bm{x},u(\bm{x}))$. We first need the following lemma to analyze the trainability issue concerning $\Delta t$. **Lemma 1**. *The numerical target evaluation using $\tilde{q}$-martingale in Eq. [\[eq:q_tilde_martingale_process\]](#eq:q_tilde_martingale_process){reference-type="eqref" reference="eq:q_tilde_martingale_process"} for a small time interval $\Delta t$ is decomposed into a convolution with a normal distribution and the force effect $$\label{eq:qtilde_target_eval} \mathbb{E}[\tilde{q}(\Delta t; u, \bm{x},\bm{B}_{\Delta t})]=(u * f_{-\bm{V}(\bm{x})\Delta t, \Delta t})(\bm{x}) -G(\bm{x})\Delta t.$$* *Proof.* For a small time interval $\Delta t$, we consider the approximation of the stochastic integrals associated with the $\tilde{q}$-martingale as follows: $$\mathbb{E}[\tilde{q}(\Delta t; u, \bm{x},\bm{B}_{\Delta t})]= \mathbb{E}\left[\left(u(\bm{B}_{\Delta t}) -G(\bm{x})\Delta t\right)\exp\left(\bm{V}(\bm{x}))\cdot \Delta\bm{B}_{\Delta t} -\frac{1}{2}|\bm{V}(\bm{x}))|^2\Delta t \right)\middle |\bm{B}_0=\bm{x} \right].$$ Since the standard Brownian motion follows $\bm{B}_{\Delta t} \sim \mathcal{N}(\bm{x}, \Delta t\mathbb{I})$, $$\begin{aligned} \mathbb{E}[\tilde{q}(\Delta t; u, \bm{x},\bm{B}_{\Delta t})]&=\int_{\bm{y}\in\mathbb{R}^{d}}(u(\bm{y})-G(\bm{x})\Delta t)\exp\left(\bm{V}(\bm{x})\cdot (\bm{y}-\bm{x})-\frac{1}{2}|\bm{V}(\bm{x})|^2\Delta t \right)f_{\bm{x},\Delta t}(\bm{y})\bm{d}\bm{y} \\ &=\int_{\bm{z}\in\mathbb{R}^{d}}(u(\bm{x}-\bm{z})-G(\bm{x})\Delta t)\exp\left(-\bm{V}(\bm{x})\cdot \bm{z}-\frac{1}{2}|\bm{V}(\bm{x})|^2\Delta t \right)f_{\bm{0},\Delta t}(\bm{z})\bm{d}\bm{z} \\ &= \int_{\bm{z}\in\mathbb{R}^{d}}(u(\bm{x}-\bm{z})-G(\bm{x})\Delta t)f_{-\bm{V}(\bm{x})\Delta t, \Delta t}(\bm{z})\bm{d}\bm{z} \\ &= \int_{\bm{z}\in\mathbb{R}^{d}}u(\bm{x}-\bm{z})f_{-\bm{V}(\bm{x})\Delta t, \Delta t}(\bm{z})\bm{d}\bm{z} -G(\bm{x})\Delta t.\end{aligned}$$ where the third equality holds by the algebraic property $$-\bm{V}(\bm{x})\cdot \bm{z}-\frac{1}{2}|\bm{V}(\bm{x})|^2\Delta t -\frac{1}{2\Delta t} \bm{z}\cdot \bm{z}=-\frac{1}{2\Delta t}(\bm{z}+\bm{V}(\bm{x})\Delta t)\cdot (\bm{z}+\bm{V}(\bm{x})\Delta t)$$ in the exponent. ◻ We note that for a nontrivial advection field $\bm{V}(\bm{x})$, the convolution is inhomogenous over the domain as, for each $\bm{x}$, it takes account for the neighborhood shifted toward the advection vector $-\bm{V}(\bm{x})$ for a small time interval $\Delta t$ using the normal density function $\mathcal{N}(-\bm{V}(\bm{x})\Delta t, \Delta t\mathbb{I})$. Instead of using the standard Brownian walkers $\bm{B}_t$, the stochastic process $\bm{X}_t$ in Eq. [\[eq:q_stochastic_walkers\]](#eq:q_stochastic_walkers){reference-type="eqref" reference="eq:q_stochastic_walkers"} directly reflects the shift toward the field direction in the random sampling. **Remark 1**. *The numerical target evalution using $q$-martingale in Eq. [\[eq:q_stochastic_walkers\]](#eq:q_stochastic_walkers){reference-type="eqref" reference="eq:q_stochastic_walkers"} and Eq. [\[eq:q_martingale_stochastic_process\]](#eq:q_martingale_stochastic_process){reference-type="eqref" reference="eq:q_martingale_stochastic_process"} has the same representation as Eq. [\[eq:qtilde_target_eval\]](#eq:qtilde_target_eval){reference-type="eqref" reference="eq:qtilde_target_eval"}.* The bootstrapping approach in DFLM demonstrated in Eq. [\[eq:sgd_update\]](#eq:sgd_update){reference-type="eqref" reference="eq:sgd_update"} and Eq. [\[eq:q_loss_discrete\]](#eq:q_loss_discrete){reference-type="eqref" reference="eq:q_loss_discrete"} is to update the neural network *toward* the pre-evaluated target value using the current state of the neural network. That is, $$u(\bm{x};\bm{\theta}_{n}) \leftarrow (u(\cdot;\bm{\theta}_{n-1}) * f_{-\bm{V}(\bm{x})\Delta t, \Delta t})(\bm{x}) -G(\bm{x})\Delta t.$$ To achieve the pre-evaluated target, it may require multiple gradient descent steps in Eq. [\[eq:sgd_update\]](#eq:sgd_update){reference-type="eqref" reference="eq:sgd_update"}. For instance, when updating $\bm{\theta}_n$ from $\bm{\theta}_{n-1}$, $M$ number of additional gradient descent steps could be considered as $$\bm{\theta}_{n-1}^{(m+1)}=\bm{\theta}_{n-1}^{(m)}-\alpha \nabla \widetilde{\mathcal{L}}_n\left(\bm{\theta}_{n-1}^{(m)}\right),~m=0,1,\cdots,M-1, {\quad} \bm{\theta}_{n-1}^{(0)}=\bm{\theta}_{n-1},~\bm{\theta}_{n-1}^{(M)} = \bm{\theta}_n.$$ In the subsequent analysis, we assume that the update $u(\cdot;\bm{\theta}_{n+1})$ is equal to the target value function evaluated using $u(\cdot;\bm{\theta}_n)$. Formally, we define the target operator $T_n:\mathcal{L}^2(\Omega) \rightarrow \mathcal{L}^2(\Omega)$ at the $n$-th iteration as $$T_n :\mathcal{L}^2(\Omega) \rightarrow \mathcal{L}^2(\Omega), ~~~ (T_nu)(\bm{x})=(u * f_{-\bm{V}(\bm{x})\Delta t, \Delta t})(\bm{x}) -G(\bm{x})\Delta t$$ under the regularity assumptions $\bm{V}, G \in \mathcal{L}^2(\Omega)$. When $\bm{V}$ and $G$ are independent from $u$, $T_{n+1}=T_n, \forall n \in \mathbb{N}_0$. The learning of DFLM is understood as the recursion of the operator $T_n$ with an initial function $u_0=u(\cdot;\bm{\theta}_0)$ as $$u(\cdot;\bm{\theta}_{n+1}) = u_{n+1} =T_nu_{n} = T_nu(\cdot;\bm{\theta}_n), \forall n \in \mathbb{N}_0.$$ **Example 1**. *For the Laplace equation $\Delta u = 0$ in $\Omega$, the training in DFLM is the recursion of the convolution of the normal density function $f_{\bm{0}, \Delta t}$ as $$u_{n+1} = u_n * f_{\bm{0}, \Delta t}, \forall n \in \mathbb{N}_0.$$ When the time interval $\Delta t$ is large, the convolution considers a broader neighborhood around each collocation point $\bm{x}$, given that the density function exhibits a long tail. Conversely, for a smaller time interval $\Delta t$, the convolution considers a more localized neighborhood. When $\Delta t \rightarrow 0$, $f_{\bm{0}, \Delta t}(\bm{x}) \rightarrow \delta(\bm{x})$ in the sense of distribution, in which $u_{n+1}(\bm{x}) = (u_n * \delta(\bm{x}))(\bm{x})=u_n(\bm{x})$,$\forall \bm{x} \in \Omega$, $\forall n \in \mathbb{N}_0$. In this case, the training process does not advance while staying at the initial function $u_0$.* The above example shows that the training procedure depends on the choice of the time interval $\Delta t$. Opting for an excessively small time interval $\Delta t$ can result in the target function being too proximate to the current function, potentially leading to slow or hindered training progress. We aim to quantify how much training can be done at each iteration depending on the time interval $\Delta t$. For this goal, we need a lemma for the normal distribution. **Lemma 2**. *For a $k$-dimensional normal random variable $\bm{w}=(w_1,w_2,\cdots, w_k)$ with mean $\bm{\mu}$ and variance $\sigma^2\mathbb{I}, \sigma\in\mathbb{R}$, the expectation of the absolute value of $\bm{w}$ is bounded as $$\mathbb{E}[|\bm{w}|] \leq C_1\sigma \exp\left(-\frac{|\bm{\mu}|^2}{2\sigma^2} \right) + C_2|\bm{\mu}|$$ where the constant $C_1=k\sqrt{\frac{2}{\pi}}$ and $C_2 = k$ are independent of $\bm{\mu}$ and $\sigma$.* *Proof.* Let $f_{\mu_i, \sigma^2}$ be the density of the univariate normal with mean $\mu_i$ and variance $\sigma^2$. Also, let $\Phi$ be the cumulative distribution function (CDF) of the standard normal distribution. Since $w_i$, $i=1,2,\cdots,k$, are pairwise independent, the density function of $\bm{w}$ is equal to $\prod \limits_{i=1}^{k} f_{\mu_i, \sigma^2}(w_i)$. Thus, we have $$\begin{aligned} \mathbb{E}[|\bm{w}|]&=\int_{\mathbb{R}^d}|\bm{w}|\prod \limits_{i=1}^{k} f_{\mu_i, \sigma^2}(w_i)d\bm{w} \\ &\leq \int_{\mathbb{R}^d} \sum \limits_{j=1}^{d}|w_j|\prod \limits_{i=1}^{k} f_{\mu_i, \sigma^2}(w_i)d\bm{w} \\ &= \sum \limits_{j=1}^{d}\int_{\mathbb{R}^d} |w_j|\prod \limits_{i=1}^{k} f_{\mu_i, \sigma^2}(w_i)d\bm{w} \\ &= \sum \limits_{j=1}^{k}\int_{\mathbb{R}^d} |w_j| f_{\mu_j, \sigma^2}(w_j) \prod \limits_{i=1, i\neq j}^{k} f_{\mu_i, \sigma^2}(w_i)d\bm{w} \\ &= \sum \limits_{j=1}^{k}\int_{\mathbb{R}} |w_j| f_{\mu_j, \sigma^2}(w_j)dw_j \\ &= \sum \limits_{j=1}^{k} \sqrt{\frac{2}{\pi}} \sigma\exp\left(-\frac{\mu_j^2}{2\sigma^2}\right)+\mu_j\left[1-2\Phi\left(-\frac{\mu_j}{\sigma}\right) \right] \\ &\leq k\sqrt{\frac{2}{\pi}} \sigma\exp\left(-\frac{|\bm{\mu}|^2}{2\sigma^2}\right)+k|\bm{\mu}|,\end{aligned}$$ where the second equality from the last holds as the sum of the expectations of the folded normal distributions. ◻ Now, we are ready to prove the following theorem. **Theorem 2**. *Let $\{u_n\}_{n=0}^{\infty}=\{u(\cdot;\bm{\theta}_n)\}_{n=0}^{\infty}$ be the sequence of the states of a neural network in the training procedure of DFLM starting from $u_0=u(;\bm{\theta}_0)$. We assume that $u_n \in C^{1}\left(\overline{\Omega}\right)$, $\forall n \in \mathbb{N}_0$. Then, the learning amount at each iteration measured by the pointwise difference in the consecutive states is approximated as $$|u_{n+1}(\bm{x})-u_n(\bm{x})| \leq \left|\nabla_{\bm{x}} u_n(\bm{x})\right| \left(C_1\sqrt{\Delta t} + C_2|\bm{V}(\bm{x})|\Delta t \right) + |G(\bm{x})|\Delta t,$$ with constants $C_1=k\sqrt{\frac{2}{\pi}}$ and $C_2 = k$.* *Proof.* $$\begin{aligned} |u_{n+1}(\bm{x})-u_n(\bm{x})| &= |(T_nu_n)(\bm{x})-u_n(\bm{x})|\\ &= \int_{\bm{z}\in\mathbb{R}^{d}}|u_n(\bm{x}-\bm{z})-u_n(\bm{x})|f_{-\bm{V}(\bm{x})\Delta t, \Delta t}(\bm{z})\bm{d}\bm{z} +|G(\bm{x})|\Delta t \\ &= \int_{\bm{z}\in\mathbb{R}^{d}}\left|\nabla_{\bm{x}} u_n(\bm{x})\cdot \bm{x}^{\prime}(\bm{z})\right|f_{-\bm{V}(\bm{x})\Delta t, \Delta t}(\bm{z})\bm{d}\bm{z} +|G(\bm{x})|\Delta t, ~~~ \left|\bm{x}^{\prime}(\bm{z}) \right| \leq |\bm{z}| \end{aligned}$$ $$\begin{aligned} \hspace{31mm}&\leq \int_{\bm{z}\in\mathbb{R}^{d}}\left|\nabla_{\bm{x}} u_n(\bm{x})\right| |\bm{z}| f_{-\bm{V}(\bm{x})\Delta t, \Delta t}(\bm{z})\bm{d}\bm{z} +|G(\bm{x})|\Delta t \\ & = \left|\nabla_{\bm{x}} u_n(\bm{x})\right| \left(\int_{\bm{z}\in\mathbb{R}^{d}} |\bm{z}| f_{-\bm{V}(\bm{x})\Delta t, \Delta t}(\bm{z})\bm{d}\bm{z}\right) +|G(\bm{x})|\Delta t \\ & \leq \left|\nabla_{\bm{x}} u_n(\bm{x})\right| \left(C_1\sqrt{\Delta t}\exp\left(-\frac{|\bm{V}(\bm{x})|^2}{2} \Delta t \right) + C_2|\bm{V}(\bm{x})|\Delta t \right) + |G(\bm{x})|\Delta t\\ & \leq \left|\nabla_{\bm{x}} u_n(\bm{x})\right| \left(C_1\sqrt{\Delta t} + C_2|\bm{V}(\bm{x})|\Delta t \right) + |G(\bm{x})|\Delta t,\end{aligned}$$ where the third equality holds by the taylor expansion of $u_n$ at $\bm{x}$ and the second inequality from the last holds by Lemma [Lemma 2](#lemma:expactation_abs_normal){reference-type="ref" reference="lemma:expactation_abs_normal"}. ◻ Theorem [Theorem 2](#thm:pointwise_learning){reference-type="ref" reference="thm:pointwise_learning"} states that for each point $\bm{x}$, the learning from the convolution is proportional to i) the magnitude of the gradient, ii) $\mathcal{O}(\sqrt{\Delta t})$ from the shape of the normal distribution and iii) $\mathcal{O}(\Delta t)$ in the advection. Also, the learning from the forcing term is of order $\mathcal{O}(\Delta t)$. **Corollary 2**. *Let $\{u_n\}_{n=0}^{\infty}=\{u(\cdot;\bm{\theta}_n)\}_{n=0}^{\infty}$ be the sequence of the states of a neural network in the training procedure of DFLM starting from $u_0=u(\cdot;\bm{\theta}_0)$. We assume that $u_n \in C^{1}\left(\overline{\Omega}\right)$, $\forall n \in \mathbb{N}_0$ and $\bm{V}, G \in C(\overline{\Omega})$. Then $$\|u_{n+1}-u_n\|_2 \leq (C_1\sqrt{\Delta t}+ C_2\|\bm{V}\|_{\infty}\Delta t) \|\nabla_{\bm{x}} u_n\|_2 + \Delta t\|G\|_2.$$ In particular, if $\|\nabla u_n\|_2$ is uniformly bounded, $\|u_{n+1}-u_n\|_2 \rightarrow 0$ in order $\mathcal{O}(\sqrt{\Delta t})$ as $\Delta t \rightarrow 0$.* The analysis highlights the importance of selecting an appropriate $\Delta t$ to allow stochastic walkers to explore a broader neighborhood effectively, promoting suitable training progress. As demonstrated in Theorem [Theorem 2](#thm:pointwise_learning){reference-type="ref" reference="thm:pointwise_learning"} and Corollary [Corollary 2](#cor:uniform_learning){reference-type="ref" reference="cor:uniform_learning"}, the choice of $\Delta t$ is contingent not only on the advection field $\bm{V}$ and the force $G$ but also the network's topology, which can be associated with the PDE solution. However, it is essential to note that increasing $\Delta t$ also amplifies the bias in the loss function, as discussed in Theorem [Theorem 1](#thm:lossbias){reference-type="ref" reference="thm:lossbias"}, potentially impeding the training of the neural network. One effective strategy to mitigate the increased bias impact during training is to employ a larger number of stochastic walkers (i.e., increasing $N_s$). # Numerical Experiments {#sec:experiment} In this section, we provide numerical examples to validate the analysis of DFLM associated with the time interval $\Delta t$ and the number of stochastic walkers $N_s$. We choose other parameters (such as $N_r$ and $N_b$) to minimize the errors from those parameters while focusing on variations in the error induced by $\Delta t$ and $N_s$. In particular, we solve the Poisson problem in 2D. In addition to being one of the standard problems for numerical PDE methods, the Poisson problem enables the $q$- and $\tilde{q}$-martingales to be identical as there is no advection term. Thus, in solving the stochastic walkers, we can quickly draw samples from the normal distribution without solving the stochastic differential equation, which is computationally efficient. We also use the homogeneous Dirichlet boundary condition so that the boundary treatment has a minimal impact on the performance. ## Experiment setup {#subsec:experimentsetup} We solve the Poisson equation in the unit square $\Omega=(-0.5,0.5)^2$ with the homogeneous Dirichlet boundary condition $$\begin{aligned} \label{eq:Poisson} \begin{split} \Delta u &= f \quad\textrm{in}\quad \Omega \\ u &= 0 \quad\textrm{on}\quad \partial \Omega. \end{split}\end{aligned}$$ We choose the force term $f$ to be $-(2m\pi)^2\sin(2m\pi x_1)\sin(2m\pi x_2)$ so that the exact solution is $$\label{eq:expsoln} u(\bm{x})=\sin(2m\pi x_1)\sin(2m\pi x_2),\quad m \in \mathbb{N}.$$ The empirical loss function at the $n$-th iteration step for updating the neural network $\bm{\theta}_n$ is $$\widetilde{\mathcal{L}}^{\Omega}_{n}(\bm{\theta}) = \frac{1}{N_r} \sum \limits_{i=1}^{N_r}\left| u(\bm{x}_i;\bm{\theta})-\frac{1}{N_s}\sum\limits_{j=1}^{N_s}\left\{ u\left(\bm{B}_{\Delta t}^{(i,j)};\bm{\theta}_{n-1} \right)-\int_0^{\Delta t}\frac{1}{2}f\left(\bm{B}_s^{(i,j)} \right) ds \right\} \right |^2.$$ Here we use $N_r$ random sample collocation points $\{\bm{x}_i: 1\leq i \leq N_r\}$ and $N_s$ Brownian walkers $\left\{\bm{B}^{(i,j)}_s:\bm{B}^{(i,j)}_{0}=\bm{x}_i, 1\leq i \leq N_r, 1\leq j \leq N_s\right\}$ for each $\bm{x}_i$. To minimize the error in calculating the term related to $f$ and handling the boundary treatment, we use a small time step $\delta t\leq \Delta t$ to evolve the discrete Brownian motion by the Euler-Maruyama method $$\bm{B}_{m \delta t} = \bm{B}_{(m-1) \delta t} + \sqrt{\delta t} \bm{Z}, {\quad} \bm{Z} \sim \mathcal{N}(0, \mathbb{I}_2), ~ m \in \mathbb{N}.$$ We note again that we can quickly draw samples from $\sqrt{\delta t}\bm{Z}$ and add to $\bm{B}_{(m-1)\delta t}$, which can be computed efficiently. Using these Brownian paths, the stochastic integral during the time period $[0, \Delta t]$, $\Delta t = M \delta t$, $M \in \mathbb{N}$, is estimated as $$\int_0^{\Delta t}\frac{1}{2}f\left(\bm{B}_s^{(i,j)} \right) ds \simeq \sum \limits_{m=0}^{M-1}\frac{1}{2}f\left(\bm{B}^{(i,j)}_{m\delta t}\right)\delta t.$$ We impose the homogeneous Dirichlet boundary condition on the stochastic process $\bm{B}_t$ by allowing it to be absorbed to the boundary $\partial \Omega$ at the exit position. We estimate the exit position and the time by linear approximation within a short time period. When the simulation of a Brownian motion comes across the boundary between the time $m\delta t$ and $(m+1)\delta t$, we estimate the exit information $t_{\text{exit}}$, $t_{\text{exit}}\in [m\delta t, (m+1)\delta t]$ and $\bm{B}_{t_{\text{exit}}}$ by the intersection of line segment between $\bm{B}_{m \delta t}$ and $\bm{B}_{(m+1)\delta t}$ and the boundary $\partial \Omega$. Once a walker is absorbed, the value of the neural network at the exit position $u(\bm{B}_{t_{\text{exit}}};\bm{\theta})$ in the target computation is set to $0$, the homogeneous boundary value, and the time step $\delta t$ in the integral approximation is replaced by $(t_{\text{exit}}-m\delta t)$. To enhance the information on the boundary, we also impose the boundary loss term $$\widetilde{\mathcal{L}}^{\partial \Omega}(\bm{\theta}) = \frac{1}{N_b}\sum \limits_{l=1}^{N_b} \left| u(\bm{x}_l;\bm{\theta}) - g(\bm{x}_l)\right|^2$$ using $N_b$ boundary random collocation points. {#fig:lossNs width="100%"} To investigate the dependency of training trajectory on the choice of the time interval $\Delta t$ and the number of stochastic walkers $N_s$, we solve the problem with various combinations of these two parameters while keeping the other parameters fixed. Regarding the network structure, we use a standard multilayer perceptron (MLP) with three hidden layers, each comprising $200$ neurons and employing the ReLU activation function. The neural network is trained using the Adam optimizer [@adam] with learning parameters $\beta_1 = 0.99$ and $\beta_2 = 0.99$. At each iteration, we randomly sample $N_r=2000$ interior and $N_b=400$ boundary points from the uniform distribution. We consider ten different time intervals $\Delta t= 2^{p}$, $p=0,1,2,\cdots, 9$ and six different stochastic walker sizes $N_s=1,4,10,40,100,400$, resulting in a total of sixty combinations of $(\Delta t, N_s)$. In considering the randomness of the training procedure, we run ten independent trials with extensive $1.5 \times 10^5$ iterations for each parameter pair, which guarantees the convergence of the training loss. ## Experiment results {#subsec:experimentresult} We use the average interior training loss out of 10 trials to measure the training loss bias in Theorem [Theorem 1](#thm:lossbias){reference-type="ref" reference="thm:lossbias"}, which we call 'training loss.' We also consider two problems with different wavenumber $m=1$ and $3$ for Eq. [\[eq:expsoln\]](#eq:expsoln){reference-type="eqref" reference="eq:expsoln"}. We consider two solutions with $m=1$ and $m=2$ to check the contribution from the different magnitude $\ell_2$ norms of the gradient. Figure [2](#fig:lossNs){reference-type="ref" reference="fig:lossNs"} shows the log-log plot of the training loss after convergence as a function of the walker size $N_s$ (horizontal axis) with various $\Delta t$ (different line types). Figure [2](#fig:lossNs){reference-type="ref" reference="fig:lossNs"} (a) and (b) are the cases with $m=1$ and $3$, respectively, and we can see that the training loss has a larger value for the more complicated case $m=3$ (about three times larger than the case of $m=1$), which the gradient magnitudes can explain for $m=1$ and $m=3$. As the analysis in the previous section predicts, the training loss decreases as the walker size increases for both cases, which aligns with the reference line of $\frac{1}{N_s}$ (dash-dot). {#fig:lossDt width="100%"} {#fig:errDt width="100%"} In comparison between different line types, we can also check that the training loss decreases as the time interval $\Delta t$ decreases. The (linear) dependence of the training loss on $\Delta t$ is more explicit in Figure [3](#fig:lossDt){reference-type="ref" reference="fig:lossDt"}. Figure [3](#fig:lossDt){reference-type="ref" reference="fig:lossDt"} shows the training loss as a function of $\Delta t$ (horizontal axis) with various $N_s$ (different line types). As in the previous figure, Figure [3](#fig:lossDt){reference-type="ref" reference="fig:lossDt"} (a) and (b) show the results for the solution with $m=1$ and $3$, respectively. Compared to the reference line of $\Delta t$ (dash-dot), all training losses show a linear increase as $\Delta t$ increases. We now check the test error after the training loss converges. In particular, we use the relative $\mathcal{L}^2$ error as a performance measure, calculated using the $1001\times 1001$ uniform grid. We want to note that a small training loss does not always imply a small test error. In DFLM, if $\Delta t$ is sufficiently small, the left- and right-hand sides of Eq. [\[eq:q-martingale\]](#eq:q-martingale){reference-type="eqref" reference="eq:q-martingale"} get close enough that the training loss can be small for an arbitrary initial guess, which fails to learn the PDE solution. For the case of $N_s=1$ and $400$, Figure [4](#fig:errDt){reference-type="ref" reference="fig:errDt"} shows the relative $\mathcal{L}^2$ test error as $\Delta t$ increases for the solution Eq. [\[eq:expsoln\]](#eq:expsoln){reference-type="eqref" reference="eq:expsoln"} with $m=1$. {#fig:errDtNs width="100%"} {#fig:errDtscale1n3 width=".8\\textwidth"} When $\Delta t$ is small ($\leq 5\times 10^{-3}$), on the other hand, we can check that the training loss bias cannot explain the performance anymore. The test error increases as $\Delta t$ decreases regardless of the size of $N_s$. In other words, the result shows that the time interval $\Delta t$ must be sufficiently large for the network to learn the PDE solution by minimizing the training loss. When the training loss bias makes a non-negligible contribution with $N_s=1$, the test error can obtain the minimal value with an optimal $\Delta t\approx 5\times 10^{-3}$. If the bias contribution is small with $N_s=400$, the test error will be minimal with $\Delta t\geq 5\times 10^{-3}$. The increasing test error for decreasing $\Delta t$ is similar for other values of $N_s$ and solution types. Figure [5](#fig:errDtNs){reference-type="ref" reference="fig:errDtNs"} shows the test relative error as a function of $\Delta t$ for the two solutions with $m=1$ (a) and $3$ (b) for various values of $N_s$. As $\Delta t$ decreases, the test error increases. Also, as $N_s$ decreases, which increases the sampling error in calculating the expectation, the test error increases for both solutions. From Figure [6](#fig:errDtscale1n3){reference-type="ref" reference="fig:errDtscale1n3"}, which shows the test error of both solutions with $N_s=400$ and various $\Delta t$ values, we can also see that the optimal $\Delta t$ is related to the local variations of the solution. First, as we mentioned before, the solution with $m=3$ has a larger error as its gradient $\ell_2$ norm is larger than that of $m=1$, related to the loss bound and training update. Also, we use the same network structure, and all other parameters are equal for both solutions. Thus, it is natural to expect a much larger test error in the more complicated solution with $m=3$ than in the case of $m=1$. In comparison between the two solutions, we find that the test error of the more oscillatory solution ($m=3$) stabilizes much faster for a small $\Delta t$. That is, even using a small $\Delta t$, which yields a small neighborhood to explore and average, the more oscillatory solution case can see more variations than the simple solution case. Thus, the training loss can lead to a trainable result. Quantitatively, the optimal time intervals between the two solutions differ by a factor of about ten (that is, the more oscillatory solution can use $\Delta t$ ten times smaller than the one of the simple solution). This difference can be explained by the fact that the variance of the stochastic walkers is proportional to $\Delta t$, or the standard deviation is proportional to $\sqrt{\Delta t}$. As the more oscillatory solution has a wavenumber three times larger than the simple case, we can see that nine times shorter $\Delta t$ will cover the same variations as in the simple case, which matches the numerical result. # Discussions and conclusions {#sec:discussion} The derivative-free loss method (DFLM) uses the Feynman-Kac formulation to train a neural network to satisfy a PDE in the form of [\[eq:Quasi-linear elliptic\]](#eq:Quasi-linear elliptic){reference-type="eqref" reference="eq:Quasi-linear elliptic"}. In this study, we analyze the training loss bias and show that the training loss is asymptotically unbiased as $N_s$, the number of walkers at a collocation point, increases. The bias is also proportional to the time interval $\Delta t$, the time period for stochastic walkers to run for the expectation. We also showed that $\Delta t$ must be sufficiently large to guarantee a significant change in the training loss to make a meaningful update to find the PDE solution. Our numerical results also show that there exists a lower bound for the time interval $\Delta t$, which can depend on the local variations of the solution. Regarding computational efficiency, the analysis tells us that DFLM requires finding the optimal lower bound for the time interval and then choosing $N_s$ as small as possible based on the time interval. Although we suspect that the lower bound for $\Delta t$ depends on the characteristics of the PDE solution, such as local variations, we do not have a quantitative method to specify the lower bound at the moment. It is natural to extend to current work to develop an approach to find the optimal lower bound of the time interval. We also plan to work on the optimal lower bound for multiscale solutions of different wave components. As the lower bound of the time interval shrinks as the solution has more highly oscillatory behaviors, it is natural to speculate how the lower bound behaves for such multiscale solutions. Along this line, we consider adaptive or hierarchical time stepping following the hierarchical learning approach [@HiPINN]. The hierarchical time stepping can incorporate different time intervals for various scale components of the solution and use a hierarchical training procedure to expedite the learning process, which we leave as future work. # Acknowledgments {#acknowledgments .unnumbered} This work was supported by ONR MURI N00014-20-1-2595. [^1]: jihun.han\@dartmouth.edu [^2]: yoonsang.lee\@dartmouth.edu

arxiv_math

--- abstract: | In this short note, we prove the existence of solutions to a Monge-Ampère equation of entire type derived by a weighted version of the classical Minkowski problem. [^1] author: - Jacopo Ulivelli bibliography: - ref.bib title: Entire Monge-Ampère equations and weighted Minkowski problems --- # Introduction Consider a Monge-Ampère equation of the form $$\label{MAequation} c_u\phi(Du(x),u^*(Du(x)))\det D^2 u(x)=f(x), \quad x \in \mathbb{R}^n.$$ Here, $c_u>0$ is a constant depending on the solution, $u^*$ is the *Fenchel-Legendre transform* of $u$ $$u^*(x)=\sup_{y \in \mathbb{R}^n}\{ x\cdot y-u(y)\},$$ $f \in L^1(\mathbb{R}^n)$ is non-negative, and $\phi:\mathbb{R}^{n+1}\to \mathbb{R}$ is a non-negative, continuous, and even function. With more generality, we can work with a Borel measure $\rho$ on the right-hand side of [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"}. To do so, we focus on weak solutions of [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"}. That is, convex functions $u$ such that $$\label{eq:weak_r_curv} \rho(B)=\int_{\partial u(B)}c_u\phi((x,u^*(x)))\,dx,$$ where $\partial u$ is the *subgradient* of $u$. The reader can find details about the theory of convex functions in [@Rockafellar1970]. Clearly, [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"} is recovered when $\rho$ has continuous density $f$ with respect to the Lebesgue measure. Following the notation of Bakelman [@Bakelman], we denote the right-hand side of [\[eq:weak_r\_curv\]](#eq:weak_r_curv){reference-type="eqref" reference="eq:weak_r_curv"} as $\omega(B,u,c_u\phi)$. By the same procedure as in [@Bakelman Section 9.6] it is easy to check that this is a Borel measure. There, [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"} was studied when $\phi$ depends only on $Du$. Other instances of the problem have been studied by Chow and Wang [@Wang_Minkowski; @EntireSolutions] and Bielawski [@Bielawski]. For our strategy we make use of a recent result from Kryvonos and Langharst [@DylLyu] (see Theorem [\[thm:general_Minkowski\]](#thm:general_Minkowski){reference-type="ref" reference="thm:general_Minkowski"} below) on the existence of convex compact sets with prescribed weighted surface-area measure. The method we propose allows a quick translation of results on convex bodies to results on convex functions, following a procedure similar to the one employed by Knoerr and the author in [@FromConvToFunc2023]. Moreover, the specific structure of [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"} allows weaker assumptions than the ones usually required in Monge-Ampère equations depending on the solution and its gradient (see, for example, [@Bakelman Section 18]). Let $\mu$ be a Borel measure on $\mathbb{R}^{n+1}$ with continuous density $\phi$ such that $$\label{eq:cond_weigh_mink} \lim_{r\to \infty} \frac{\mu(rB^2_{n+1})^{\frac{\beta}{n+1}}}{r}=0 \text{ and } \lim_{r\to 0^+} \frac{\mu(rB^2_{n+1})^{\frac{\beta}{n+1}}}{r}=+\infty$$ for some $\beta>0$, where $B^2_{n+1}$ is the Euclidean unit ball in $\mathbb{R}^{n+1}$. These hypotheses are precisely those prescribed by Kryvonos and Langharst in [@DylLyu]. Our main result reads as follows. **Theorem 1**. *Consider a Borel measure $\rho$ on $\mathbb{R}^n$ that is not concentrated on an affine hyperplane. Consider, moreover, a continuous even function $\phi:\mathbb{R}^{n+1}\to [0,\infty)$. Then, if $\rho$ has finite first moment, i.e. $$\int_{\mathbb{R}^n}|x|\,d\rho(x)<+\infty,$$ and the measure $\mu$ with density $\phi$ with respect to the Lebesgue measure satisfies [\[eq:cond_weigh_mink\]](#eq:cond_weigh_mink){reference-type="eqref" reference="eq:cond_weigh_mink"}, there exist a constant $c_u>0$ and a convex function $u$ such that $$\label{eq:weak_Ma} \omega(B, u,c_u\phi)=\rho(B)$$ for every Borel set $B \subset \mathbb{R}^n$.* We remark that the constant $c_u$ is not necessarily unique, nor are the solutions of [\[eq:weak_Ma\]](#eq:weak_Ma){reference-type="eqref" reference="eq:weak_Ma"}. For a further discussion see Section [3](#remarks){reference-type="ref" reference="remarks"}. The role of such constant is due to the high non-homogeneity of the problem and is clarified later with Theorem [\[thm:general_Minkowski\]](#thm:general_Minkowski){reference-type="ref" reference="thm:general_Minkowski"}. Notice, moreover, that $\phi$ is required to be even as a function on $\mathbb{R}^{n+1}$, which is weaker than asking for symmetry on the first $n$ components. In particular, the solutions found in Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} are not necessarily symmetric. Furthermore, we provide the following expected regularity result. **Theorem 2**. *In the hypotheses of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}, suppose moreover that $\rho$ has continuous density $f$ with respect to the Lebesgue measure. If $f$ and $\phi$ are such that there exists $c>0$ such that $f,\phi>c$ and of class $C^{k,\alpha}$ for some $k \geq 0$ and $\alpha>0$, then any convex weak solution of [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"} is of class $C^{k+2,\alpha}$.* # Proofs In the following we consider the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$ with the Euclidean norm $|\cdot|$ and the usual scalar product $x \cdot y$ for $x,y \in \mathbb{R}^{n+1}$. Let $K$ be a compact convex subset of $\mathbb{R}^{n+1}$. For background material on convex geometry, see Schneider's monograph [@schneider_2013]. For every $\xi$ in the unit sphere $\mathbb{S}^{n}$ of $\mathbb{R}^{n+1}$ consider the set $\tau_K(\xi)$ of all the points on the boundary of $K$, denoted $\partial K$, such that there is a tangent hyperplane at $x$ with outer normal $\xi$ for every $x \in \tau_K(\xi)$. The map $\tau_K$ is known as *reverse spherical image*. If $\mu$ is a Borel measure on $\mathbb{R}^{n+1}$ with continuous density $\phi$ with respect to the Lebesgue measure, we define on $\mathbb{S}^{n+1}$ the $\mu$-surface-area measure of $K$ by $$\label{eq:weighted_measures} S_K^\mu(B)\coloneq\int_{\tau_K(B)}\phi(y)\, d \mathcal{H}^n(y)$$ for every Borel set $B \subset \mathbb{S}^n$, where $\mathcal{H}^n$ is the $n$-dimensional Hausdorff measure restricted to $\partial K$. When $\mu$ is the Lebesgue measure, the apex $\mu$ is omitted and $S_K$ is the classical surface-area measure (see [@schneider_2013 Section 4]). Existence theorems for compact convex sets with given $\mu$-surface-area measure have been studied, for example, by Livschytz [@LivMi] and Kryvonos and Langharst [@DylLyu]. The particular case where $\mu$ is the Gaussian measure has been covered in depth by Huang, Xi, and Zhao [@Zhao_Gauss]. Recovering a convex set by its surface-area measure and generalizations of this procedure are known as Minkowski problems. We recommend, for example, [@schneider_2013 Section 8] for an introduction. Our main instrument is the following result. **Theorem 3**. *[@DylLyu Theorem 1.2][\[thm:general_Minkowski\]]{#thm:general_Minkowski label="thm:general_Minkowski"} Let $\mu$ be an even Borel measure on $\mathbb{R}^{n+1}$ satisfying [\[eq:cond_weigh_mink\]](#eq:cond_weigh_mink){reference-type="eqref" reference="eq:cond_weigh_mink"}. Suppose $\rho$ is a finite, even Borel measure on $\mathbb{S}^n$ that is not concentrated in any great subsphere. Then, there exists a centrally symmetric convex compact set $K \subset \mathbb{R}^{n+1}$ such that $$d\rho(\xi)=c_{\mu,K}dS^\mu_K(\xi), \quad c_{\mu,K}\coloneq \mu(K)^{\frac{\beta}{n+1}-1}.$$* The constant $c_{\mu,K}$ appearing in the statement of Theorem [\[thm:general_Minkowski\]](#thm:general_Minkowski){reference-type="ref" reference="thm:general_Minkowski"} plays the same role as $c_u$, the one appearing in Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}. Terms of this kind are often involved in non-homogeneous Minkowski problems. However, if the measure $\mu$ is homogeneous of some positive degree, the constant can be chosen equal to one. See [@LivMi] and the discussion in [@DylLyu] after Theorem 1.2 for more details on this topic. We are now ready to prove our main result. *Proof of Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"}.* For a fixed $v \in \mathbb{R}^{n+1}$, we consider $\mathbb{R}^n \subset \mathbb{R}^{n+1}$ as the hyperplane orthogonal to $v$, which will be the domain of [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"}. Once we have a measure $\rho$ on $\mathbb{R}^n$, we can lift it to $\mathbb{S}^n_{-}\coloneq \{\xi \in \mathbb{S}^n: \xi \cdot v<0 \}$ through the diffeomorphism $$\begin{aligned} L\colon \mathbb{R}^n &\to \mathbb{S}^n_{-}\\ x &\mapsto \frac{(x,-1)}{\sqrt{1+|x|^2}}. \end{aligned}$$ Notice that $L$ brings affine hyperplanes to the intersection of a great subsphere with $\mathbb{S}^n_{-}$. Consider a finite Borel measure $\rho$ on $\mathbb{R}^n$ with finite first moment, and define the measure $$\rho'(B)=\int_{B}\sqrt{1+|x|^2}\,d\rho(x)$$ for every Borel set $B \subset \mathbb{R}^n$. Notice that $\mathrm{supp}\ (\rho)=\mathrm{supp}\ (\rho')$. Now, we lift $\rho'$ through $L$ as a measure on $\mathbb{S}^n_{-}$ considering its push-forward $(L)_{\sharp}\rho'$ which, for every Borel set $\omega \subset \mathbb{S}^n_{-}$, is defined as $$(L)_{\sharp}\rho'(\omega)=\rho'(L^{-1}(\omega)).$$ Then, we extend it trivially on the rest of $\mathbb{S}^n$ and consider the symmetrized measure $\bar{\rho}=(L)_{\sharp}\rho'+(L)_{\sharp}\rho'\circ R$, where $R(\xi)=-\xi$. It is easy to check that it is not concentrated on a great subsphere and is symmetric. By Theorem [\[thm:general_Minkowski\]](#thm:general_Minkowski){reference-type="ref" reference="thm:general_Minkowski"} there exists a centrally symmetric convex compact set $K\subset \mathbb{R}^{n+1}$ and a positive constant $c_{\mu,K}$ such that $\bar{\rho}=c_{\mu,K}S_K^\mu$. This body is the starting point to construct our solution. The rest of the proof will follow the diagram below. $$\label{diagram} \begin{tikzcd} \partial {K}_{-} \arrow[swap]{d}{\pi} & \arrow{l}{\tau_K} \mathbb{S}^n_{-} \\ \mathrm{dom}(w) \arrow{r}{\partial w} & \mathbb{R}^n \ar{u}{L} \end{tikzcd}$$ Here, $\pi: \mathbb{R}^{n+1}\to \mathbb{R}^n$ is the orthogonal projection along $v$, $\partial K_{-}$ is the closure of $\tau_K(\mathbb{S}^n_{-})$, and $w$ is the convex function on $\mathbb{R}^n$ defined as $w(x)\coloneq \inf\{t: x+tv \in K\}$. We claim that $u=w^*$ satisfies [\[eq:weak_Ma\]](#eq:weak_Ma){reference-type="eqref" reference="eq:weak_Ma"}. Notice that since $\mathrm{dom}(w)\coloneq \{p \in \mathbb{R}^n \colon w(p)<+\infty \}$ is compact, $u(x)<+\infty$ for every $x \in \mathbb{R}^n$ and thus $\partial u(x) \neq \emptyset$ for every $x \in \mathbb{R}^n$. Since $\partial u(x) \in \mathrm{dom}(w)$ for every $x \in \mathbb{R}^n$, we have that $u$ is Lipschitz. Moreover, notice that $K$ is centrally symmetric with non-empty interior. In particular, $\mathrm{dom}(w)$ contains an open neighborhood of the origin, and thus $u$ is coercive. First, consider $x \in \mathbb{R}^n$. By definition, $\partial u(x)=\{p \in \mathbb{R}^n\colon x \in \partial w(p)\}$. Then, by construction $x \in \partial w(p)$ if and only if $$L(x)=\frac{(x,-1)}{\sqrt{1+|x|^2}} \in \tau_K^{-1}\circ \pi^{-1}(p).$$ In particular, even if $\partial u(x)$ is not a singleton, it is brought back to a singleton by $\tau_K^{-1}$ (which is just the Gauss map) and thus, for every $x \in \mathbb{R}^n$ $$x=L^{-1}\circ \tau_K^{-1} \circ \pi^{-1}\circ \partial u(x).$$ To conclude, we now check that $u$ is a weak solution of [\[eq:weak_Ma\]](#eq:weak_Ma){reference-type="eqref" reference="eq:weak_Ma"}, where $c_u=c_{\mu,K}$. Consider a Borel set $B \subset \mathbb{R}^n$. We have the changes of variables $$\begin{aligned} \omega(B,u,c_u\phi)&=\int_{\partial u(B)}c_u\phi((x,w(x)))\, dx=\int_{\pi^{-1}\circ \partial u(B)} c_u\frac{\phi(y)}{\sqrt{1+|Dw(\pi(y))|^2}}\, d\mathcal{H}^n(y) \\ &=\int_{\tau_K^{-1}\circ \pi^{-1}\circ \partial u(B)}|\xi \cdot v| c_{\mu,K }\, dS_K^\mu(\xi)=\int_{L^{-1}\circ \tau_K^{-1}\circ \pi^{-1}\circ \partial u(B)} \frac{1}{\sqrt{1+|z|^2}} \, d\rho'(z)\\ &=\int_{L^{-1}\circ \tau_K^{-1}\circ \pi^{-1}\circ \partial u(B)} \, d\rho(z)=\rho(B), \end{aligned}$$ concluding the proof. Here, $1/\sqrt{1+|Dw(\pi(y))|^2}$ is the approximate Jacobian of $\pi$ and $Dw$ is defined almost everywhere on $\mathrm{dom}(w)$ since $w$ is convex. Moreover, we used the fact that the unit normal vector at $(x,w(x))$ is $$\frac{(D w(x),-1)}{\sqrt{1+|D w(x)|^2}}$$ whenever $D w$ is defined, and thus, for $z \in \mathbb{R}^n$ and $x \in \partial u(z)$, $$\frac{1}{\sqrt{1+|D w(\pi((x,w(x))))|^2}}=|L(z)\cdot v|=\frac{1}{\sqrt{1+|z|^2}}.$$ ◻ Suppose now that $\rho$ has continuous density $f$, and that there exists $c>0$ such that $c<f,\phi$. The proof of Theorem [Theorem 2](#thm:regularity){reference-type="ref" reference="thm:regularity"} can be considered classic and follows, for example, the same steps of [@Bielawski Theorem 0.2]. We include it for the convenience of the reader. For a lighter notation, since we suppose that a convex solution exists, the constant $c_u$ is absorbed in $\phi$. *Proof of Theorem [Theorem 2](#thm:regularity){reference-type="ref" reference="thm:regularity"}.* As remarked in the previous proof, $u$ is finite and coercive. Therefore, the level sets $\Omega_t\coloneq\{x \in \mathbb{R}^n \colon u(x)<t\}$ are compact for every $t\in \mathbb{R}$. Moreover since $u$ is Lipschitz and $f,\phi$ are strictly positive and continuous, by [\[eq:weak_Ma\]](#eq:weak_Ma){reference-type="eqref" reference="eq:weak_Ma"} there exist $0<c_1<c_2$ such that $$\label{eq:Ma_ineq} c_1 |B| \leq |\partial u(B)|\leq c_2 |B|$$ for every Borel set $B \subset \Omega_t$ once $t$ is fixed. Now, thanks to [@Caffa_adv Corollary 2], $u$ is strictly convex in $\Omega_t$, and since $t$ is arbitrary, $u$ is strictly convex everywhere. In [@Caffa_comm] it was proved that if $u$ satisfies [\[eq:Ma_ineq\]](#eq:Ma_ineq){reference-type="eqref" reference="eq:Ma_ineq"} and is strictly convex, then $u \in C^{1,\beta}(\Omega_t)$, where $\beta$ depends on $t$. Suppose now that $\phi,f$ are of class $C^{0,\alpha}$. First, rewrite [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"} as $$\det D^2u=\frac{f}{\phi((Du,u^*(Du)))}=g,$$ where $g$ is locally of class $C^{0,\gamma}$ for some $\gamma>0$ since both $Du$ and $u^*$ are Hölder continuous. By [@Caff_ann Theorem 1.2] applied to $u-t$ on $\Omega_t$, we infer $u \in C^{2,\gamma}(\Omega_t)$, and therefore $u\in C^2(\mathbb{R}^n)$. Thus, $g$ is $C^{0,\alpha}$ everywhere, and a reproduction of the argument above proves that $u$ is $C^{2,\alpha}(\mathbb{R}^n)$, settling the case $k=0$. The extension to $k\geq 1$ is achieved through a classic induction argument. ◻ # Final Remarks {#remarks} The proof of Theorem [Theorem 2](#thm:regularity){reference-type="ref" reference="thm:regularity"} shows that if $\rho$ has continuous density $f$ and both $f$ and $\phi$ are strictly positive, then a solution of [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"} is differentiable, and thus it can be rewritten as $$\phi(Du(x),x\cdot Du(x)-u(x))\det D^2 u(x)=f(x),$$ weakening considerably the assumptions that usually are required for the existence of solutions to problems depending explicitly on $u$ (see, for example, [@Bakelman (18.4),(18.5)]). In particular, no regularity or monotonicity is required for $\phi$. Notice, moreover, that Theorem [Theorem 2](#thm:regularity){reference-type="ref" reference="thm:regularity"} provides a regularity result for Theorem [\[thm:general_Minkowski\]](#thm:general_Minkowski){reference-type="ref" reference="thm:general_Minkowski"} in case the measure on the sphere and the ambient measure have Hölder-continuous densities. Concerning the non-uniqueness of the solution and the constant $c_u$, we now adapt some remarks by Huang, Xi, and Zhao [@Zhao_Gauss], who studied the instance of the Gaussian measure $\gamma$, that is, $\phi((x_1,\dots,x_{n+1}))=e^{-\sum_{i=1}^{n+1}x_i^{2}}/\sqrt{2\pi}^{n+1}$. A first source of non-uniqueness concerns the exponent $\beta$ in the condition [\[eq:cond_weigh_mink\]](#eq:cond_weigh_mink){reference-type="eqref" reference="eq:cond_weigh_mink"}. **Theorem 4**. *[@Zhao_Gauss Theorem 1.2][\[thm:Zhao_Gauss\]]{#thm:Zhao_Gauss label="thm:Zhao_Gauss"} Suppose $\rho$ is a finite even Borel measure on $\mathbb{S}^{n}$ not concentrated in any closed hemisphere. Then for each $0<\beta<\frac{1}{n+1}$, there exists a centrally symmetric convex compact set $K \subset \mathbb{R}^{n+1}$ that $$d\rho(\xi)=c_{\gamma,K}dS_{K}^{\gamma}(\xi)$$ where $$c_{\gamma,K}=\gamma(K)^{\frac{\beta}{n+1}-1}.$$* Thus, for every admissible $\beta$ one can find a potentially different solution. In Theorem 1.4 of the same work, uniqueness is retrieved under further conditions. To give a specific example in our setting, for $a>0$ consider in [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"} the particular $f(x)=\frac{a}{\sqrt{1+|x|^2}}$. Calculations show that this is equivalent to prescribing a constant measure in Theorem [\[thm:Zhao_Gauss\]](#thm:Zhao_Gauss){reference-type="ref" reference="thm:Zhao_Gauss"}. Then, as noted in [@Zhao_Gauss], since the Gaussian surface area measure of a sphere $B_r \subset \mathbb{R}^{n+1}$ of radius $r>0$ is $dS^{\gamma}_{B_r}(\xi)=\frac{e^{-\frac{r^2}{2}}}{\sqrt{2\pi}^{n+1}}r^n$, the equation $$a=dS^{\gamma}_{B_r}(\xi)$$ has two different solutions for $a$ sufficiently small. Denote the radii of these solutions as $r_1$ and $r_2$ respectively. Then, we have that $u_i(x)\coloneq r_i \sqrt{1+|x|^2}$ are both solutions in [\[MAequation\]](#MAequation){reference-type="eqref" reference="MAequation"} with $c_u=a^{-1}$. We conclude by noting that a further source of non-uniqueness is hidden in an inheritance vice of the proof. Indeed, when extending $(L)_{\sharp}\rho'$ to the whole sphere, we could have added another even measure concentrated in $\mathbb{S}^n \cap v^{\perp}$, which would have given a different body as result. A possible fix for this issue would be prescribing an asymptotic cone, which is the usual practice for this kind of problem (see, for example, [@Bakelman], or the more recent [@Schneider_cones]). We are not aware of how to implement this method in our setting but, if this issue were fixed and the body $K$ found in Theorem [Theorem 1](#thm:main){reference-type="ref" reference="thm:main"} was uniformly convex, under suitable regularity (granted by Theorem [Theorem 2](#thm:regularity){reference-type="ref" reference="thm:regularity"}) uniqueness would be achieved following [@Gilbarg_Trudinger Section 17.8]. Jacopo Ulivelli\ Dipartimento di Matematica\ Sapienza, University of Rome\ Piazzale Aldo Moro 5\ 00185 Rome, Italy\ e-mail: jacopo.ulivelli\@uniroma1.it [^1]: MSC 2020 Classification: 52A20 ,35J96, 26B25.\ Keywords: Convex function, Monge-Ampère equation, Minkowski problem.

arxiv_math

--- abstract: | In this paper we first prove a number of important inequalities with explicit constants in the setting of the Heisenberg group $\mathbb{H}^{n}$. This includes the fractional and integer Sobolev, Gagliardo-Nirenberg, (weighted) Hardy-Sobolev, Nash inequalities, and their logarithmic versions. In the case of the first order Sobolev inequality, our constant recovers the sharp constant of Jerison and Lee. Remarkably, we also establish the analogue of the Gross inequality with a semi-probability measure on $\mathbb{H}^{n}$ that allows- as it happens in the Euclidean setting- an extension to infinite dimensions, and particularly can be regarded as an inequality on the infinite dimensional $\mathbb{H}^{\infty}$. Finally, we prove the so called generalised Poincaré inequality on $\mathbb{H}^{n}$ both with respect to the aforementioned semi-probability measure and the Haar measure on $\mathbb{H}^{n}$, also with explicit constants. address: - " Marianna Chatzakou: Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Belgium *E-mail address* marianna.chatzakou\\@ugent.be " - " Aidyn Kassymov: Institute of Mathematics and Mathematical Modeling Shevchenko str. Almaty Kazakhstan *E-mail address* kassymov\\@math.kz" - " Michael Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Belgium and School of Mathematical Sciences Queen Mary University of London United Kingdom *E-mail address* michael.ruzhansky\\@ugent.be " author: - Marianna Chatzakou - Aidyn Kassymov - Michael Ruzhansky title: Logarithmic Sobolev, Hardy and Poincaré inequalities on the Heisenberg group --- [^1] # Introduction and main results One of the aims of this work is to obtain the fractional, logarithmic and in some cases weighted versions of a family of well-studied inequalities such as the Sobolev, Hardy-Sobolev, Hardy, and Gagliardo-Nirenberg inequalities in the setting of the Heisenberg group $\mathbb{H}^n$. We put an emphasis on establishing there inequalities with explicit constants. It is true that even though a lot of work has been done in order to prove and extend these inequalities in the Euclidean setting, regarding their consideration in the sub-Riemannian setting there is still a lot to be done. Exceptional to this, we shall refer to the important works of Frank and Lieb [@FL12] and Roncal and Thangavelu [@RT16] where the authors prove several sharp inequalities in the setting of the Heisenberg group $\mathbb{H}^{n}$. Additionally, in the last section we prove the so-called generalised Poincaré inequality (see [@Bec98]) in the regarded setting, with respect to two different and essential for our considerations measures on $\mathbb{H}^{n}$. The versions of the Poincaré inequalities proven are both global; integration is regarded on the whole of $\mathbb{H}^{n}$, and local; integration is regarded on balls in $\mathbb{H}^{n}$. One of the main contributions of the current work is that the involved constants in the aforementioned inequalities are explicit. In this respect, the current work can be served as a special case of the works [@CKR21c; @CKR21b; @CKR21a] in the sense that in these works the authors established similar type inequalities for general or graded/stratified Lie groups including just representations of the appearing constants, but also an extension of the latter works since the current work contains several Poincaré types inequalities. The importance of knowing explicitly the appearing constants is self-evident in view of the applications of these inequalities in PDEs, variational calculus, differential geometry, and other branches of mathematics as discussed in more details in the sequel. Additionally to that, having control over the constants is necessary in view of the possibly of extending the consideration of specific inequalities to infinite-dimensional objects like, in this case, the infinite dimensional Heisenberg groups $\mathbb{H}^{\infty}$, see e.g. the works [@DM08] and [@BGM13] on the setting $\mathbb{H}^{\infty}$, when equipped with a suitable probability measure. Details of this consideration are given below in [\[iii\]](#iii){reference-type="ref" reference="iii"}. On the other hand, it is known, see e.g [@BZ05], [@ABD07], [@DV12], that there is a strong link between the logarithmic-Sobolev inequalities and the Poincaré inequalities. Here we establish several versions of the Poincaré inequalities in the setting of the Heisenberg group $\mathbb{H}^{n}$, where either the Haar measure is involved, or a semi-probability measure that reads as the natural extension of the Gaussian measure on $\mathbb{R}^n$. This appears in the analogue of the logarithmic Sobolev inequality in the setting of $\mathbb{H}^{n}$ that is proven in the current work. Hence the stimulus behind this work is mainly based on the following elements: 1. [\[a\]]{#a label="a"} Logarithmic, or more generally coercive inequalities, are strongly linked other inequalities and properties of the related semigroups; see e.g. [@AB00; @GZ03; @BH99; @Car91; @HZ10]; 2. [\[b\]]{#b label="b"} Particularly, the log-Sobolev inequality has links to measuring uncertainty, c.f. [@Bec95; @Bec00], and to blow-up results on certain PDEs even in the quite general setting of homogeneous Lie groups, c.f. [@KRS20]; Nash inequality considered in the setting of Markov semigroups can be used to proving estimates for the related heat kernel; Hardy inequality can be used to proving stability of the relativistic matter [@FLS08]; and Gagliardo-Nirenberg inequalities (also known as "interpolation theorem" in nonlinear problems) can be used to proving estimates for the study of nonlinear evolution equations, c.f. [@FNQ18]; 3. [\[iii\]]{#iii label="iii"} While [\[a\]](#a){reference-type="ref" reference="a"} and [\[b\]](#b){reference-type="ref" reference="b"} explain the general motivation behind our analysis, here we obtain explicit expression of the appearing constants, and having good control of these constants allows one to pass to infinite dimensions. In particular, this means that we can still consider the Gross-type "semi-Gaussian" inequality with the Gaussian measure on the first stratum of the group, see Theorem [Theorem 8](#semi-g){reference-type="ref" reference="semi-g"}, on an object like $\mathbb{H}^{\infty}$ with the infinite-dimensional first stratum and a $1$-dimensional center. This relies on the appearing normalisation constant $\gamma$, see Remark [Observation 9](#mainrem){reference-type="ref" reference="mainrem"}, that essentially allows one to interpret the measure as the probability measure on the whole $\mathbb{H}^{\infty}$ as it happens in the Euclidean setting. Hence it becomes consequential to study properties of the analogues of the Ornstein--Uhlenbeck operator on $\mathbb{H}^{\infty}$, see e.g. [@Pic10]. The last realisation unlocks new investigation quests, eventually including the direction of stochastic analysis on $\mathbb{H}^{\infty}$. 4. [\[iv\]]{#iv label="iv"} Sobolev and Poincaré iequalities are closely related [@LR04]. The Poincaré inequality is the main protagonist in the domain of coercive inequalities and isoperimetry; see [@BZ05] for a general overview. In the case of the global Poincaré inequalities with respect to a probability measure very little is known even in the simple setting of the Heisenberg group $\mathbb{H}^{n}$, see e.g. [@CFZ21], [@DZ22], [@HZ10]. On the other hand, Poincaré inequalities on a bounded domain in the setting of nilpotent Lie groups are known only with respect to the horizontal gradient on the group, see [@Jer86]. Next, we give a short presentation of the main results of this work, sometimes combined with some additional comments about related works and approaches. Before doing so, let us note that in the sequel $\mathbb{H}^{n}$ stands for the Heisenberg group, and $Q=2n+2$ stands for the homogeneous dimension of $\mathbb{H}^{n}$. We also assume that for the parameter $s$ we have $s \in (0,1]$.[^2] - **The (fractional) (log-)Sobolev inequality on $\mathbb{H}^{n}$**: The works [@Bec12; @KRS20; @KS20] contain functional inequalities, in some cases of Sobolev type, that involve powers of the sub-Laplacian. Our version of the fractional Sobolev inequality as appears in Theorem [Theorem 2](#thm1){reference-type="ref" reference="thm1"} involves either the modified fractional operator $\mathcal{L}_s$ given by ([\[Ls.Thang\]](#Ls.Thang){reference-type="ref" reference="Ls.Thang"}) (introduced by Roncal and Thangavelu in [@RT16] that in the case where $s=1$ boils down to the sub-Laplacian on $\mathbb{H}^{n}$) or fractional powers of the sub-Laplacian $\mathcal{L}^{s}$ and the first reads as follows: $$\label{into1} \|f\|^{2}_{L^{\frac{2Q}{Q-2s}}(\mathbb{H}^{n})}\leq C_{B,s}\langle \mathcal{L}_{s}f,f\rangle_{L^{2}(\mathbb{H}^{n})},$$ where $C_{B,s}$ is given in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}). The fractional Sobolev inequality that involves $\mathcal{L}^{s}$ is the inequality ([\[2b\]](#2b){reference-type="ref" reference="2b"}) in Theorem [Theorem 2](#thm1){reference-type="ref" reference="thm1"}. Inequality ([\[into1\]](#into1){reference-type="ref" reference="into1"}) essentially extends the sharp inequality of Jerison and Lee in [@JL88] in the sense that when $s=1$, i.e. when $\mathcal{L}_s$ is simply the sub-Laplacian on $\mathbb{H}^{n}$, inequality ([\[into1\]](#into1){reference-type="ref" reference="into1"}) coincides with the latter inequality, see also Remark [Remark 3](#remJL){reference-type="ref" reference="remJL"}. It is important to note that when combining the latter with the logarithmic Hölder inequality as in Lemma [Lemma 5](#holder){reference-type="ref" reference="holder"}, then the resulting inequality gives as a special case for $s=1$ the following so called Log-Sobolev inequality on $\mathbb{H}^{n}$: $$\label{LogSobolevint.intro} \int_{\mathbb{H}^{n}}\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx \leq \frac{Q}{2}\log\left(C_{B,1}\|\nabla_{\mathbb{H}^{n}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}\right),$$ where the constant $C_{B,1}$ is given explicitly. A generalisation of the log-Sobolev inequality is its weighted version and is shown in Theorem [Theorem 19](#wwsobingrthm){reference-type="ref" reference="wwsobingrthm"}. - **The (log-)fractional Gagliardo-Nirenberg inequality on $\mathbb{H}^{n}$**: As one sees in the seminal work of Brezis and Mironescu [@BM18] on the history of Gagliardo-Nirenberg inequality, not much attention was paid until recently to the determination of the appearing constant even in the simple case of $\mathbb{R}^n$. In 2020 in the work [@RTY20] the authors prove the Gagliardo-Nirenberg inequality on graded groups with a constant that cannot (apart from the trivial cases of $\mathbb{R}^n$) be explicitly computed. Here, in Theorem [\[GN4\]](#GN4){reference-type="ref" reference="GN4"}, in the case of $\mathbb{H}^{n}$ we obtain: $$\label{into.GN} \int_{\mathbb{H}^{n}}|u(x)|^{q}dx\leq C_{GN,s_1,s_2}\langle\mathcal{L}_{s_{1}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{Q(q-2)-2s_{2}q}{4(s_{1}-s_{2})}}\langle\mathcal{L}_{s_{2}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{s_{1}(q-2)-Q(q-2)}{4(s_{1}-s_{2})}},$$ where $C_{GN,s_1,s_2}=C_{B,s_1}^{\frac{qa}{2}} C_{B,s_2}^{\frac{q(1-a)}{2}}$ for $C_{B,s_i}$, $i=1,2$, explicitly given in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}), whenever $1>s_1>s_2 \geq 0$, and $\frac{2Q}{Q-2s_2}\leq q \leq \frac{2Q}{Q-2s_1}$. In principle, the Gagliardo-Nirenberg inequality on $\mathbb{H}^{n}$ in its general form is not sharp. However, in the case where $s_1=1$ and $s_2=0$ inequality ([\[into.GN\]](#into.GN){reference-type="ref" reference="into.GN"}) is sharp, see Remark [Remark 15](#rem.GN){reference-type="ref" reference="rem.GN"}. Logarithmic versions of the general form of ([\[into.GN\]](#into.GN){reference-type="ref" reference="into.GN"}) are given in Theorem [Theorem 16](#thmlogGN){reference-type="ref" reference="thmlogGN"}. - **(log-)fractional Hardy--Sobolev inequality on $\mathbb{H}^{n}$:** The (log-)Sobolev type inequalities with weights are sometimes called (log-)Hardy inequalities. The fractional Hardy inequality on $\mathbb{H}^{n}$ is shown in [@RT16] by Roncal and Thangavelu. Here in Theorem [Theorem 18](#FHS){reference-type="ref" reference="FHS"} we prove the log-Hardy-Sobolev inequality on $\mathbb{H}^{n}$, which is for $s \in (0,1)$, $2^{*}_{\beta}=\frac{2(Q-\beta)}{Q-2s}$, and $0< \beta <2s$ the following inequality $$\begin{split} &\int_{\mathbb{H}^{n}}\frac{|x|^{-\frac{2\beta}{2^{*}_{\beta}}}|u(x)|^{2}}{\||\cdot|^{-\frac{2\beta}{2^{*}_{\beta}}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|x|^{-\frac{2\beta}{2^{*}_{\beta}}}|u(x)|^{2}}{\||\cdot|^{-\frac{2\beta}{2^{*}_{\beta}}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx\\& \leq \frac{Q-\beta}{2s-\beta}\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx\log\left(\left(C^{\frac{\beta}{2s}}_{BH,s}(C_{B,s}\|U_{s}\|_{\text{op}})^{\frac{n+1}{n+1-s}\frac{2s-\beta}{2s}}\right)^{\frac{2}{2^{*}_{\beta}}}\frac{\langle \mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}}{\||\cdot|^{-\frac{2\beta}{2^{*}_{\beta}}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right)\,, \end{split}$$ where $\|U_{s}\|_{\text{op}}$ stands for the norm of the operator $U_s$ defined in ([\[Us\]](#Us){reference-type="ref" reference="Us"}). The latter inequality, see the discussion given after Theorem [Theorem 18](#FHS){reference-type="ref" reference="FHS"}, gives under suitable choices of $\beta$ both the Hardy and the Sobolev inequalities. The fractional log-Hardy inequality on $\mathbb{H}^{n}$ is given in Theorem [Theorem 21](#THM:weHarnon){reference-type="ref" reference="THM:weHarnon"}. - **The Nash inequality and its application to heat equation on $\mathbb{H}^{n}$**: Nash inequality was shown by Nash [@Nas58] in the Euclidean setting. The best-appearing constant was determined by Carlen and Loss in [@CL] a few years later. In their monograph Varopoulos, Salof-Coste and Coulhon [@VSCC93] show Nash inequality in the form $$\label{intro.VSDD}\|u\|^{2+\frac{4}{Q}}_{L^2(\mathbb{H}^{n})}\leq C \|u\|^{\frac{4}{Q}}_{L^1(\mathbb{H}^{n})}\|\nabla_{\mathbb{H}^{n}}u\|^{2}_{L^2(\mathbb{H}^{n})}\,,$$ for some $C>0$. Here we find an explicit formula for $C$ in ([\[intro.VSDD\]](#intro.VSDD){reference-type="ref" reference="intro.VSDD"}) which, as it happens on $\mathbb{R}^n$ depends only on the dimension $n$ of the first stratum, and consequently only on $Q$, and generalises ([\[intro.VSDD\]](#intro.VSDD){reference-type="ref" reference="intro.VSDD"}) in the following sense: $$\label{intro.Nash} \|u\|_{L^2(\mathbb{H}^{n})}^{2+\frac{4s}{Q}} \leq C_{B,s}\|U\|_{\rm op} \|u\|_{L^1(\mathbb{H}^{n})}^{\frac{4s}{Q}}\langle\mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}\,,$$ see Theorem [Theorem 23](#THM:Nash){reference-type="ref" reference="THM:Nash"}. The modified fractional version of Nash inequality is given in ([\[Nash.g\]](#Nash.g){reference-type="ref" reference="Nash.g"}). As before, the inequality ([\[intro.Nash\]](#intro.Nash){reference-type="ref" reference="intro.Nash"}) for $s=1$, boils down to the inequality ([\[intro.VSDD\]](#intro.VSDD){reference-type="ref" reference="intro.VSDD"}) with an explicit constant. An application of Nash inequality for the time decay of the solution to the heat equation with respect to the sub-Laplacian $\mathcal{L}$ on $\mathbb{H}^{n}$ is given in Corollary [Corollary 25](#cor:par){reference-type="ref" reference="cor:par"}. - **The Gross-type log-Sobolev inequality on $\mathbb{H}^{n}$**: Gross in [@Gro75] proved the logarithmic Sobolev inequality on $\mathbb{R}^n$ with respect to the Gaussian measure $d\mu(x)=(2\pi)^{-\frac{n}{2}}e^{-\frac{|x|^2}{2}}dx$ of the form: $$\label{EQ:Gross} \int_{\mathbb{R}^{n}}|u(x)|^{2}\log \left(\frac{|u(x)|}{\|u\|_{L^{2}(\mathbb{R}^{n},\mu)}}\right)d\mu(x) \leq \int_{\mathbb{R}^{n}}|\nabla u(x)|^{2}d\mu(x)\,,$$ with the idea of extending it to infinite dimensions. Indeed, the appearing normalisation constant $(2\pi)^{-n}$ allows for such a consideration. Gross's inequality is linked to ultracontractivity and hypercontractivity properties of the corresponding Markovian semigroups, see e.g. [@Ad79; @Tos97], uncertainty principles [@Bec95] and Poincaré inequalities [@Bec89]. In Theorem [Theorem 8](#semi-g){reference-type="ref" reference="semi-g"} we prove the following analogue of Gross's inequality on $\mathbb{H}^{n}$ which we shall call the "semi-Gaussian" log-Sobolev (or Gross type) inequality on $\mathbb{H}^{n}$ since the appearing measure is Gaussian on the first stratum. The latter inequality is of the form $$\label{intro:semi-g} \int_{\mathbb{H}^{n}}|g|^2\log|g|\,d\mu \leq \int_{\mathbb{H}^{n}} |\nabla_{\mathbb{H}^{n}}g|^2\,d\mu\,,$$ for any $g$ such that $\|g\|_{L^2(\mathbb{H}^{n},\mu)}=1,$ where $\mu=\mu_1 \otimes \mu_2$, and $\mu_1$ is the Gaussian measure on $\mathbb{R}^{2n}$ given by $d\mu_1=\gamma e^{-\frac{|\xi|^2}{2}}d\xi$, for $\xi\in \mathbb{R}^{2n}$, and $|\xi|$ being the Euclidean norm of $\xi$, where $$\gamma:=n!\left(\frac{(n+1)e^{\frac{2n}{n+1}{-1}}}{2\pi n^{2}}\right)^{n+1}\,,$$ where $\mu_2$ is the Lebesgue measure $d\tau$ on $\mathbb{R}$, where we have denoted by $x=(\xi,\tau) \in \mathbb{R}^{2n} \times \mathbb{R}$ an element in $\mathbb{H}^{n}$. As it happens on $\mathbb{R}^n$, the latter inequality is actually equivalent to ([\[LogSobolevint.intro\]](#LogSobolevint.intro){reference-type="ref" reference="LogSobolevint.intro"}); see Theorem [Theorem 11](#equiv.thm){reference-type="ref" reference="equiv.thm"}. Let us underline that $\gamma$ for large $n$ (and so also for large $Q$) is uniformly bounded, see Observation [Observation 9](#mainrem){reference-type="ref" reference="mainrem"}, and most importantly, in this limiting case, the appearing probability measure on the first stratum of $\mathbb{H}^{n}$ becomes identical to the Gaussian measure on $\mathbb{R}^{2n}$. Hence, the idea of passing to infinite dimensions is justified in the case of $\mathbb{H}^{n}$ by this expression. - **Poincaré inequalities on $\mathbb{H}^{n}$**: Regarding Poincaré inequalities on the whole of $\mathbb{H}^{n}$, in Theorem [Theorem 26](#thm.poi.H){reference-type="ref" reference="thm.poi.H"}, we prove the generalised Poincaré inequality, see e.g. [@Bec89], with respect to the semi-Gaussian measure $\mu$ as in ([\[intro:semi-g\]](#intro:semi-g){reference-type="ref" reference="intro:semi-g"}). The latter reads as follows $$\int_{\mathbb{H}^{n}}|g|^2\,d\mu-\left(\int_{\mathbb{H}^{n}}|g|^p\,d\mu \right)^{\frac{2}{p}}\leq \frac{2(2-p)}{p} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}g|^2\,d\mu\,,$$ for $p<2$. Later on, in Corollary [Corollary 31](#cor.jer){reference-type="ref" reference="cor.jer"}, we deal with the Poincaré inequality on bounded domains in $\mathbb{H}^{n}$, and in particular we give an upper bound when $p=2$ for the term $$\int_{B_r(y)}|g-g_{B_r(y)}|^p\,dx$$ appearing in Jerison's Poincaré inequality [@Jer86] in terms of the infimum over $s\in (0,1]$ of the quantity $$C_{B,s}\frac{Q}{2s} \langle \mathcal{L}_sg,g \rangle_{L^2(B_r(y))}\,,$$ where $B_r(y)$ is the ball of radius $r$ with center $y$ with respect to the homogeneous quasi-norm on $\mathbb{H}^{n}$ which satisfies $\text{Vol}(B_r(y))\leq 1$, and where $C_{B,s}$ depends only on $s$ and is given in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}). Actually, Bakry-Emery conditions give the conditions on (the full) probability measures under which logarithmic inequalities are satisfied; see [@GZ03]. Such an approach is not part of the current work and will be addressed in the sequel. In this spirit let us mention an important result by Hebisch and Zegarlinski in [@HZ10] that in our case can be viewed as follows: there is no probability measure of the form $e^{-c\rho^2}dx$ with $\rho$ being a smooth homogeneous quasi-norm on $\mathbb{H}^{n}$ that gives rise to a Gross-inequality of the form ([\[intro:semi-g\]](#intro:semi-g){reference-type="ref" reference="intro:semi-g"}). There will be a subsequent work that initiates from the current one, as mentioned earlier in [\[iii\]](#iii){reference-type="ref" reference="iii"}, with the study of the properties of the corresponding Ornstein--Uhlenbeck operator on $\mathbb{H}^{n}$ and on $\mathbb{H}^{\infty}$. Recall that the Gross inequality ([\[EQ:Gross\]](#EQ:Gross){reference-type="ref" reference="EQ:Gross"}) on $\mathbb{R}^n$ is corresponding essentially to a reformulation of the hypercontractivity of the associated Markovian semigroup; see e.g. [@BE85]. # Preliminaries In this section, we briefly recall definitions and main properties of the Heisenberg group $\mathbb{H}^{n}$ as well as some notions and ideas from [@RT16] and [@FL12] that are useful for the subsequent analysis. Let us note that the comprehensive study on stratified groups, that include $\mathbb{H}^{n}$ as a special case, has been initiated in the works of Folland and Stein [@FS]. However, in our exposition below we chose to follow a more recent presentation of the open access book [@FR16]. The Heisenberg group $\mathbb{H}^{n}$ can be viewed via the identification of manifolds $\mathbb{H}^{n}:=(\mathbb{C}^{n}\times \mathbb{R}, \circ),\,n\in\mathbb N$, under the following group law $$\begin{split} (\xi,\tau)\circ(\tilde{\xi},\tilde{\tau})=\left(\xi+\tilde{\xi},\tau+\tilde{\tau}+\frac{1}{2}\text{Im}(\xi\cdot\tilde{\xi})\right), \end{split}$$ where $\xi,\tilde{\xi}\in \mathbb{C}^{n}$ and $\tau, \tilde{\tau}\in \mathbb{R}$. The family of dilations for $z=(\xi,\tau)\in\mathbb{H}^{n}$ has the following form $$\delta_{\lambda}(z):=(\lambda \xi, \lambda^{2}\tau),\,\,\,\,\, \lambda>0,\,\,\,\,\xi=\xi'+\text{i}\overline{\xi},$$ yielding also the homogeneous dimension $Q$ of $\mathbb{H}^{n}$ which is computed as $Q=2n+2$. By definition, the topological dimension is $2n+1$. The Kaplan norm on $\mathbb{H}^{n}$ is for $z=(\xi,\tau)\in \mathbb{H}^{n}$ given by $$|z|=\left(|\xi|^{4}+16\tau^{2}\right)^{\frac{1}{4}}\,.$$ The Lie algebra $\mathfrak{h}$ of the left-invariant vector fields on the Heisenberg group $\mathbb{H}^{n}$ is spanned by the vector fields $$X_{i}=\frac{\partial}{\partial\xi'_{i}}+\frac{1}{2}\overline{\xi}_{i}\frac{\partial}{\partial \tau}, \,\,\, Y_{i}=\frac{\partial}{\partial \overline{\xi}_{i}}-\frac{1}{2}\xi'_{i}\frac{\partial}{\partial \tau},\,\,\xi_{i}=\xi'_{i}+\text{i}\overline{\xi}_{i},$$ for $i=1,\ldots,n,$ having as the (only) non-zero commutator the vector field $$T=[X_{i},Y_{i}]=-\partial_{\tau},\,\,\,\,\,i=1,\ldots,n.$$ By the general theory, the horizontal gradient on $\mathbb{H}^{n}$ is then given by $$\nabla_{\mathbb{H}^{n}}:=(X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n})\,,$$ while the positive sub-Laplacian on $\mathbb{H}^{n}$ is of the form $$\mathcal{L}:=-\nabla_{\mathbb{H}^{n}}^{*}\cdot\nabla_{\mathbb{H}^{n}}=-\sum_{i=1}^{n}\left(X_{i}^{2}+Y_{i}^{2}\right).$$ Each $\lambda \in \mathbb{R}^{*}=\mathbb{R}\setminus \{0\}$ gives rise to an irreducible unitary representation $\pi_{\lambda}$ of $\mathbb{H}^{n}$ that is realised on $L^{2}(\mathbb{R}^{n})$ via the action $$\pi_{\lambda} (\xi,\tau)\varphi(t)=e^{\text{i}\lambda\tau}e^{\text{i}(\xi'\cdot t+\frac{1}{2}\xi'\cdot \overline{\xi})}\varphi(t+\overline{\xi}),$$ where $\xi=\xi'+\text{i}\overline{\xi}$. The group Fourier transform $\widehat{f}(\lambda)$ of a function $f \in L^1(\mathbb{H}^{n})$ is the operator-valued function defined, for each $\lambda \in \mathbb{R}^{*}$, by $$\widehat{f}(\lambda):=\pi_{\lambda}(f)=\int_{\mathbb{H}^{n}}f(\xi,\tau)\pi_{\lambda}(\xi,\tau)^{*}d\xi d\tau.$$ The Parseval identity on $\mathbb{H}^{n}$ reads as follows $$\label{parseval} \int_{\mathbb{H}^{n}}f(x)\overline{g(x)}dx=\int_{\mathbb{H}^{n}}f(\xi,\tau)\overline{g(\xi,\tau)}d\xi d\tau=\frac{2^{n-1}}{\pi^{n+1}}\int_{\mathbb{R}^{*}}\text{tr}(\widehat{f}(\lambda)\widehat{g}(\lambda)^{*})|\lambda|^{n}d\lambda,$$ where $\text{tr}(\widehat{f}(\lambda)\widehat{g}(\lambda)^{*})$ stands for the trace of the operator $\widehat{f}(\lambda)\widehat{g}(\lambda)^{*}$. By taking $f=g$ in ([\[parseval\]](#parseval){reference-type="ref" reference="parseval"}), the latter boils down to the Plancherel identity on $\mathbb{H}^{n}$, i.e, we have $$\int_{\mathbb{H}^{n}}|f(x)|^{2}dx=\int_{\mathbb{H}^{n}}|f(\xi,\tau)|^{2}d\xi d\tau=\frac{2^{n-1}}{\pi^{n+1}}\int_{\mathbb{R}^{*}}\|\widehat{f}(\lambda)\|^{2}_{\text{HS}}|\lambda|^{n}d\lambda,$$ where $\|\hat{f}(\lambda)\|_{\text{HS}}=\text{tr}(\widehat{f}(\lambda)\widehat{f}(\lambda)^{*})^{\frac{1}{2}}$ is the Hilbert-Schmidt norm of $\widehat{f}(\lambda)$. Recall that that group Fourier transform of the sub-Laplacian $\mathcal{L}$ on $\mathbb{H}^{n}$ is the harmonic oscillator on $\mathbb{R}^n$; that is we have $$\pi_{\lambda}(\mathcal{L})=\sigma_{\mathcal{L}}(\lambda)=|\lambda|\left(-\sum_{k=1}^{n}\partial^{2}_{w_{j}}+\sum_{k=1}^{n}w_{j}^{2}\right)\,,\quad w \in \mathbb{R}^n\,.$$ Next, we recall the spectral decomposition of the fractional powers of the sub-Laplacian as in [@RT16]. To this end, let us introduce the following notation: we shall denote by $f^{\lambda}$ the inverse Fourier transform of $f$ in the central variable $\tau$, that is $$f^{\lambda}(\xi):=\int_{-\infty}^{+\infty}f(\xi,\tau)e^{\text{i}\lambda \tau}d\tau\,.$$ The following functions on $\mathbb{R}^n$ are called in [@RT16] the scaled Laguerre functions of type $(n - 1)$: $$\varphi_{k}^{\lambda}(\xi)=L^{n-1}_{k}\left(\frac{|\lambda||\xi|^{2}}{2}\right)e^{-\frac{|\lambda||\xi|^{2}}{4}},$$ where $L_{k}^{n-1}$ is the Laguerre polynomial of type $(n-1)$. Using the above notation, the spectral decomposition of $\mathcal{L}$ was described in [@RT16] to be given by $$\mathcal{L}u(\xi,\tau)=(2\pi)^{n-1}\int_{-\infty}^{+\infty}\left(\sum_{k=0}^{\infty}(2k+n)|\lambda|f^{\lambda}\ast_{\lambda}\varphi_{k}^{\lambda}(\xi)\right)e^{-i\lambda\tau}|\lambda|^{n}d\lambda,$$ where $*_{\lambda}$ is called the $\lambda$-twisted convolution and is defined via $f^{\lambda} \ast_{\lambda} g^{\lambda}:=(f \ast g)^{\lambda}$. Hence, under the above notation, a natural way to extend the latter to define fractional powers of the sub-Laplacian is via the following formula: $$\mathcal{L}^{s}u(\xi,\tau)=(2\pi)^{n-1}\int_{-\infty}^{+\infty}\left(\sum_{k=0}^{\infty}((2k+n)|\lambda|)^{s}f^{\lambda}\ast_{\lambda}\varphi_{k}^{\lambda}(\xi)\right)e^{-i\lambda\tau}|\lambda|^{n}d\lambda.$$ By $\mathcal{L}_{s}$, $s\in (0,1)$, Roncal and Thangavelu in [@RT16] denoted the operator that they call the modified fractional power (of $\mathcal{L}$): $$\label{Ls.Thang} \mathcal{L}_{s}:=|2T|^{s}\frac{\Gamma\left(\frac{\mathcal{L}}{2|T|}+\frac{1+s}{2}\right)}{\Gamma\left(\frac{\mathcal{L}}{2|T|}+\frac{1-s}{2}\right)},$$ where $T=-\frac{\partial}{\partial \tau}$. The authors also observe that $\mathcal{L}_{s}$ corresponds to the following spectral multiplier: $$(2|\lambda|)^{s}\frac{\Gamma\left(\frac{2k+n}{2}+\frac{1+s}{2}\right)}{\Gamma\left(\frac{2k+n}{2}+\frac{1-s}{2}\right)},\,\,\,\,k\in\mathbb{N},$$ with the group Fourier transform of the operator $\mathcal{L}_{s}$ denoted by $$\sigma_{\mathcal{L}_{s}}(\lambda)=\pi_{\lambda}(\mathcal{L}_{s}).$$ In [@CH89] the authors found the fundamental solution of $\mathcal{L}_{s}$. In particular, we have $$(\mathcal{L}_{s}^{-1}\delta_{0})(x)=a_{s}|x|^{-Q+2s},\,\,\,\,\,a_{s}=\frac{2^{n+1-3s}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\pi^{n+1}\Gamma(s)},$$ where $\delta_{0}$ denotes a Dirac delta at the point $0$. It is easy to see that $\mathcal{L}_{1}=\mathcal{L}$, hence by the above the explicit representation of the fundamental solution of $\mathcal{L}$ is given by $\frac{2^{n-2}\Gamma^{2}\left(\frac{n}{2}\right)}{\pi^{n+1}}|x|^{-Q+2}$; see also Folland and Stein [@FS74]. Since for our analysis, we are referring to several results from the work [@FL12] of Frank and Lieb, let us see how the above operators become under the group law on $\mathbb{H}^{n}$ that was adopted there. We have in [@FL12], $$\begin{split} (\xi,\tau) \circ_{1} (\tilde{\xi},\tilde{\tau})=(\xi+\tilde{\xi},\tau+\tilde{\tau}+2\text{Im}(\xi\cdot\tilde{\xi})),\,\,(\xi,\tau)\in\mathbb{C}^{n}\times\mathbb{R}. \end{split}$$ Consequently, the vector fields (with $\mathbb{C}^{n}\ni\xi=\xi'+\text{i}\overline{\xi}$) take the form $$\widetilde{X}_{i}=\frac{\partial}{\partial\xi'_{i}}+2\overline{\xi}_{i}\frac{\partial}{\partial \tau}, \,\,\, \widetilde{Y}_{i}=\frac{\partial}{\partial\overline{\xi}_{i}}-2\xi'_{i}\frac{\partial}{\partial \tau},\,\,\,\,i=1,\ldots,n,$$ while the Kaplan distance of $x$, in this case denoted by $|x|_1$, takes the following form $$\label{fln} |x|^{4}_{1}:=|\xi|^{4}+\tau^{2},\,\,\,\,\,\,x=(\xi,\tau)\in\mathbb{H}^{n}.$$ Hence, for the sub-Laplacian, now denoted by $\tilde{\mathcal{L}}$, we have $$\tilde{\mathcal{L}}=-\frac{1}{4}\sum_{i=1}^{n}\left(\tilde{X}_{i}^{2}+\tilde{Y}_{i}^{2}\right)\,.$$ The analogue of the modified fractional operator $\mathcal{L}_{s}$ with respect to the sub-Laplacian is given by $$\widetilde{\mathcal{L}}_{s}:=|2\widetilde{T}|^{s}\frac{\Gamma\left(\frac{\widetilde{\mathcal{L}}}{|2\widetilde{T}|}+\frac{1+s}{2}\right)}{\Gamma\left(\frac{\widetilde{\mathcal{L}}}{|2\widetilde{T}|}+\frac{1-s}{2}\right)},$$ where $\widetilde{T}=\frac{\partial}{\partial \tau}$. As in [@C82] the representation of the fundamental solution in this case is given by $$\label{bs} \tilde{\mathcal{L}}_{s}^{-1}\delta_{0}=b_{s}|x|_{1}^{2s-Q},\,\,\,\,\,b_{s}=\frac{2^{n-1-s}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\pi^{n+1}\Gamma(s)},$$ where $\delta_{0}$ is the Dirac delta at the point $0.$ To show the relation between $\mathcal{L}_{s}$ and $\tilde{\mathcal{L}_{s}}$, let us take $u(\xi,\tau)=v(2\xi,\tau)$. Then we have $$\tilde{\mathcal{L}}u(\xi,\tau)=(\mathcal{L}v)(2\xi,\tau)\,,$$ and consequently also $$\tilde{\mathcal{L}_{s}}u(\xi,\tau)=(\mathcal{L}_{s}v)(2\xi,\tau)\,,$$ where the last equality implies that $$\label{dil} \langle\tilde{\mathcal{L}_{s}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}=2^{-2n}\langle\mathcal{L}_{s}v,v\rangle_{L^{2}(\mathbb{H}^{n})}.$$ The auxiliary operator $U_s$, $s \in (0,1)$, was introduced in [@RT16] as being the following composition of fractional sub-Laplacians resulting in a bounded operator $$\label{Us} U_{s}=\mathcal{L}_{s}(\mathcal{L}^{s})^{-1}\,.$$ Its norm was computed there as $$\label{Us.op.norm} \|U_{s}\|_{\rm op}=\sup_{k\geq0}\left(\frac{2k+n}{2}\right)^{-s}\frac{\Gamma\left(\frac{2k+n}{2}+\frac{1+s}{2}\right)}{\Gamma\left(\frac{2k+n}{2}+\frac{1-s}{2}\right)}.$$ Giving an upper bound for $\|U_{s}\|_{\text{op}}$ that is convenient for our purposes and more explicit, and actually improves the one given in Subsection 5.3 in [@RT16], requires the Wendel inequality for the ratio of Gamma functions, see [@W48 Formula (7)], which reads as follows $$\frac{\Gamma(x+s)}{\Gamma(x)}\leq x^{s},\,\,\,\,\,\,x>0\,\,\,\,\text{and}\,\,\,\,\,s\in(0,1).$$ Choosing $x=\frac{2k+n+1-s}{2}$ and $s\in(0,1)$, we get $$\begin{split} \left(\frac{2k+n}{2}\right)^{-s}\frac{\Gamma\left(\frac{2k+n}{2}+\frac{1+s}{2}\right)}{\Gamma\left(\frac{2k+n}{2}+\frac{1-s}{2}\right)}&=\left(\frac{2k+n}{2}\right)^{-s}\frac{\Gamma\left(\frac{2k+n+1-s}{2}+s\right)}{\Gamma\left(\frac{2k+n+1-s}{2}\right)}\\& \leq\left(\frac{2k+n}{2}\right)^{-s}\left(\frac{2k+n+1-s}{2}\right)^{s}\\& =\left(1+\frac{1-s}{2k+n}\right)^{s}\\& \leq \left(\frac{n+1-s}{n}\right)^{s}. \end{split}$$ A combination of the latter computations together with ([\[Us.op.norm\]](#Us.op.norm){reference-type="ref" reference="Us.op.norm"}) yields $$\label{est1} \|U_{s}\|_{\text{op}}\leq \left(\frac{n+1-s}{n}\right)^{s}.$$ In [@RT16] the authors also considered the auxiliary bounded operator $V_{s}=\mathcal{L}^{-1}_{1-s}\mathcal{L}\mathcal{L}^{-s}$, for $s \in (0,1)$. By the properties of the gamma function, one has the following estimate for the norm of $V_s$: $$\label{est2} \|V_{s}\|_{\text{op}}:=\|\mathcal{L}^{-1}_{1-s}\mathcal{L}\mathcal{L}^{-s}\|_{\text{op}}\leq \frac{n+2-s}{n+s}\,,$$ see [@RT16 Subsection 5.3]. We conclude this section with the next result which is a diagonalised version of the Hardy-Littlewood-Sobolev inequality with the best constant on $\mathbb{H}^{n}$ as shown in [@FL12]. **Theorem 1** (The Hardy-Littlewood-Sobolev inequality). *Let $0 < \lambda < Q$ and $p := \frac{2Q}{2Q-\lambda}$. Then for any $f,g \in L^p(\mathbb{H}^{n})$ we have $$\begin{split} \int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}}\frac{g(y)\overline{f(x)}}{|y^{-1}\circ_{1}x|_{1}^{\lambda}}dxdy&\leq C_{\text{HLS},\lambda}\|f\|_{L^{p}(\mathbb{H}^{n})}\|g\|_{L^{p}(\mathbb{H}^{n})}, \end{split}$$ where $|\cdot|_1$ is the Kaplan norm given in ([\[fln\]](#fln){reference-type="ref" reference="fln"}), and the best constant $C_{\textit{HLS},\lambda}$ is given by $$C_{\text{HLS},\lambda}=\left(\frac{\pi^{n+1}}{2^{n-1}n!}\right)^{\frac{\lambda}{Q}}\frac{n!\Gamma\left(\frac{Q-\lambda}{2}\right)}{\Gamma^{2}\left(\frac{2Q-\lambda}{4}\right)}.$$* # Fractional (logarithmic) Sobolev inequalities on $\mathbb{H}^{n}$ {#sec:LS} In this section, we show the fractional and the modified fractional (with respect to the operator $\mathcal{L}_{s}$ as in ([\[Ls.Thang\]](#Ls.Thang){reference-type="ref" reference="Ls.Thang"})) Sobolev inequalities on $\mathbb{H}^{n}$. We note that the first order Sobolev inequality on $\mathbb{H}^{n}$ with best constant was obtained by Jerison and Lee in [@JL88], and here we extend their results for the modified fractional sub-Laplacian and for fractional powers of it. The log-Sobolev inequalities are also obtained, as well as the "horizontal" log-Sobolev inequality that eventually leads to the analogue of the Gross inequality on $\mathbb{H}^{n}$. **Theorem 2** (Fractional Sobolev inequality). *Let $s\in (0,1]$. The (modified) fractional Sobolev inequality on $\mathbb{H}^{n}$ is given by $$\label{1b} \|f\|^{2}_{L^{\frac{2Q}{Q-2s}}}\leq C_{B,s}\langle \mathcal{L}_{s}f,f\rangle_{L^{2}(\mathbb{H}^{n})},$$ where $$\label{bestfrac} C_{B,s}=\frac{2^{-2s}\pi^{-s}(n!)^{\frac{s}{n+1}}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\Gamma^{2}\left(\frac{Q+2s}{4}\right)}.$$ Alternatively, inequality ([\[1b\]](#1b){reference-type="ref" reference="1b"}) can be expressed in terms of fractional powers of $\mathcal{L}$ as $$\label{2b} \begin{split} \|f\|^{2}_{L^{\frac{2Q}{Q-2s}}}\leq C_{B,s}\|U_{s}\|_{\rm op} \langle \mathcal{L}^{s}f,f\rangle_{L^{2}(\mathbb{H}^{n})}, \end{split}$$ where $\|U_{s}\|_{\rm op}$ has been estimated in ([\[est1\]](#est1){reference-type="ref" reference="est1"}).* *Proof.* By using Parsevals's theorem and [@GK Corollary 4.1], we have $$\begin{split} |\langle u,g\rangle_{L^{2}(\mathbb{H}^{n})}|^{2}&=\left|\int_{\mathbb{R}\setminus \{0\}}\text{Tr}\left(\pi_{\lambda}(u)\pi_{\lambda}(g)^{*}\right)d\mu(\lambda)\right|^{2}\\& =\left|\int_{\mathbb{R}\setminus \{0\}}\text{Tr}\left(\sigma_{\tilde{\mathcal{L}_{s}}}^{\frac{1}{2}}(\lambda)\pi_{\lambda}(u)\pi_{\lambda}(g)^{*}\sigma_{\tilde{\mathcal{L}_{s}}}^{-\frac{1}{2}}(\lambda)\right)d\mu(\lambda)\right|^{2}\\& \leq \left(\int_{\mathbb{R}\setminus \{0\}}\|\sigma_{\tilde{\mathcal{L}_{s}}}^{\frac{1}{2}}(\lambda)\pi_{\lambda}(u)\|^{2}_{\text{HS}}d\mu(\lambda)\right)\left(\int_{\mathbb{R}\setminus \{0\}}\|\sigma_{\tilde{\mathcal{L}_{s}}}^{-\frac{1}{2}}(\lambda)\pi_{\lambda}(g)\|^{2}_{\text{HS}}d\mu(\lambda)\right)\\& =\|\tilde{\mathcal{L}_{s}}^{\frac{1}{2}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}\|\tilde{\mathcal{L}_{s}}^{-\frac{1}{2}}g\|^{2}_{L^{2}(\mathbb{H}^{n})}\\& =\langle \tilde{\mathcal{L}_{s}} u,u\rangle_{L^{2}(\mathbb{H}^{n})}\langle g,\tilde{\mathcal{L}_{s}}^{-1}g\rangle_{L^{2}(\mathbb{H}^{n})}, \end{split}$$ where $d\mu=\frac{2^{n-1}}{\pi^{n+1}}|\lambda|^{n}d\lambda$. Using the Hardy-Littlewood-Sobolev inequality as in Theorem [Theorem 1](#H.liitle){reference-type="ref" reference="H.liitle"} we obtain, with $p=\frac{2Q}{Q+2s}$, $$\begin{split} \langle g,\tilde{\mathcal{L}_{s}}^{-1}g\rangle_{L^{2}(\mathbb{H}^{n})}&=b_{s}\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}}\frac{g(y)\overline{g(x)}}{|y^{-1}\circ_{1}x|_{1}^{Q-2s}}dxdy\\& \leq b_{s}C_{\text{HLS},Q-2s}\|g\|^{2}_{L^{p}(\mathbb{H}^{n})}\\& =\frac{2^{n-1-s}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\pi^{n+1}\Gamma(s)}\left(\frac{\pi^{n+1}}{2^{n-1}n!}\right)^{\frac{Q-2s}{Q}}\frac{n!\Gamma\left(s\right)}{\Gamma^{2}\left(\frac{Q+2s}{4}\right)}\|g\|^{2}_{L^{p}(\mathbb{H}^{n})}\\& =\frac{2^{-\frac{2s}{n+1}}\pi^{-s}(n!)^{\frac{s}{n+1}}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\Gamma^{2}\left(\frac{Q+2s}{4}\right)}\|g\|^{2}_{L^{p}(\mathbb{H}^{n})}, \end{split}$$ where $p=\frac{2Q}{Q+2s}$, while the constant $b_{s}$ and the distance function $|\cdot|_{1}$ are defined in ([\[bs\]](#bs){reference-type="ref" reference="bs"}) and ([\[fln\]](#fln){reference-type="ref" reference="fln"}), respectively. Hence, by the above we have $$\label{2sc} \begin{split} \|u\|^{2q}_{L^{q}(\mathbb{H}^{n})}&= |\langle u,|u|^{q}u^{-1}\rangle_{L^{2}(\mathbb{H}^{n})}|^{2}\\& \leq \langle \tilde{\mathcal{L}_{s}} u,u\rangle_{L^{2}(\mathbb{H}^{n})}\langle |u|^{q}u^{-1},\tilde{\mathcal{L}_{s}}^{-1}|u|^{q}u^{-1}\rangle_{L^{2}(\mathbb{H}^{n})}\\& \leq \frac{2^{-\frac{2s}{n+1}}\pi^{-s}(n!)^{\frac{s}{n+1}}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\Gamma^{2}\left(\frac{Q+2s}{4}\right)}\langle \tilde{\mathcal{L}_{s}} u,u\rangle_{L^{2}(\mathbb{H}^{n})} \|u\|^{2q-2}_{L^{q}(\mathbb{H}^{n})}, \end{split}$$ for $q=\frac{p}{p-1}=\frac{2Q}{Q-2s}$. We note that $|u|^q u^{-1}$ is well defined in view of $||u|^q u^{-1}|\leq |u|^{q-1}$, and $q>1$. The latter can be rewritten as $$\|u\|^{2}_{L^{\frac{2Q}{Q-2s}(\mathbb{H}^{n})}}\leq \frac{2^{-\frac{2s}{n+1}}\pi^{-s}(n!)^{\frac{s}{n+1}}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\Gamma^{2}\left(\frac{Q+2s}{4}\right)}\langle \tilde{\mathcal{L}_{s}} u,u\rangle_{L^{2}(\mathbb{H}^{n})}.$$ Now by ([\[dil\]](#dil){reference-type="ref" reference="dil"}) and substituting $u(\xi,\tau_{1})=f(2\xi,\tau_{1})$ we get, noting $\frac{2Q}{Q-2s}=\frac{2n+2}{n+1-s}$, $$\label{new} \begin{split} \left(\int_{\mathbb{H}^{n}}2^{-2n}|f(\xi_{1},\tau_{1})|^{\frac{2n+2}{n+1-s}}d\xi_{1} d\tau_{1}\right)^{\frac{n+1-s}{n+1}}&=\left(\int_{\mathbb{H}^{n}}|f(2\xi,\tau_{1})|^{\frac{2n+2}{n+1-s}}d\xi d\tau_{1}\right)^{\frac{n+1-s}{n+1}}\\& =\left(\int_{\mathbb{H}^{n}}|u(\xi,\tau_{1})|^{\frac{2n+2}{n+1-s}}d\xi d\tau_{1}\right)^{\frac{n+1-s}{n+1}}=\\& \leq\frac{2^{-\frac{2s}{n+1}}\pi^{-s}(n!)^{\frac{s}{n+1}}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\Gamma^{2}\left(\frac{Q+2s}{4}\right)}\langle \tilde{\mathcal{L}_{s}} u,u\rangle_{L^{2}(\mathbb{H}^{n})}\\& =\frac{2^{-\frac{2s}{n+1}-2n}\pi^{-s}(n!)^{\frac{s}{n+1}}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\Gamma^{2}\left(\frac{Q+2s}{4}\right)}\langle \mathcal{L}_{s}f,f\rangle_{L^{2}(\mathbb{H}^{n})}\,. \end{split}$$ We note that $2^{-\frac{2s}{n+1}-2n}2^{2n\frac{n+1-s}{n+1}}=2^{-2s}$. Hence ([\[new\]](#new){reference-type="ref" reference="new"}) implies $$\begin{aligned} \left(\int_{\mathbb{H}^{n}}|f(\xi_{1},\tau_{1})|^{\frac{2n+2}{n+1-s}}d\xi_{1} d\tau_{1}\right)^{\frac{n+1-s}{n+1}} &\leq & 2^{-2s} \frac{\pi^{-s}(n!)^{\frac{s}{n+1}}\Gamma^{2}\left(\frac{Q-2s}{4}\right)}{\Gamma^{2}\left(\frac{Q+2s}{4}\right)} \langle \mathcal{L}_{s}f,f\rangle_{L^{2}(\mathbb{H}^{n})}\,,\end{aligned}$$ the last can be rewritten as $$\|f\|^{2}_{L^{\frac{2Q}{Q-2s}}}\leq C_{B,s}\langle \mathcal{L}_{s}f,f\rangle_{L^{2}(\mathbb{H}^{n})}\,,$$ with $C_{B,s}$ as in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}). Equivalently, using the fact that $U_s=\mathcal{L}_{s}(\mathcal{L}^{s})^{-1}$, the latter can be expressed as in ([\[2b\]](#2b){reference-type="ref" reference="2b"}), and the proof is complete. ◻ **Remark 3**. We note that when $s=1$ in Theorem [Theorem 2](#thm1){reference-type="ref" reference="thm1"}, the appearing constant in the inequality ([\[1b\]](#1b){reference-type="ref" reference="1b"}) coincides with the one in the Sobolev inequality with respect to the sub-Laplacian involving the vector fields $\tilde{X}_{i}$ and $\tilde{Y}_{i},$ for $i=1,\ldots,n$, that was shown by Jerison and Lee in [@JL88]. Additionally, it is clear from the proof of Theorem [Theorem 2](#thm1){reference-type="ref" reference="thm1"} and the sharpness of the Sobolev inequality involving the modified sub-Laplacian in [@JL88] that the Sobolev inequality involving the sub-Laplacian $\mathcal{L}^{s}$ is also sharp in general, and for the appearing constant $C_{B,s}$ for $s=1$, one has $$\label{bestintsob} C_{B,1}=\frac{(n!)^{\frac{1}{n+1}}}{\pi n^{2}}\,,$$ since $$\frac{\Gamma\left(\frac{Q-2}{4} \right)}{\Gamma\left(\frac{Q+2}{4} \right)}=\frac{\Gamma\left(\frac{Q-2}{4} \right)}{\Gamma\left(\frac{Q-2}{4}+1 \right)}=\frac{4}{Q-2}=\frac{2}{n}\,,$$ where we have used the property that for $x>0$ we have $\Gamma(x+1)=x \Gamma(x)$. **Remark 4**. In [@RT16 p. 151] the authors deduce a weaker form of Hardy inequality which is essentially the (modified) fractional inequality on $\mathbb{H}^{n}$ and conclude that the resulting inequality is not the one obtained by Frank and Lieb in [@FL12 p.353]. However, comparing the latter inequality with the one in [@RT16] for $s=1$, one can observe that there is the extra constant $2^{-\frac{1}{n+1}}$ on the left-hand side of the inequality in [@RT16]. The authors of [@RT16] have confirmed to us that this factor arose from a confusion related to the use of two different sub-Laplacian in these two papers, and should not be there. The arguments in the proof of Theorem [Theorem 2](#thm1){reference-type="ref" reference="thm1"} take this fact into account, and yield the correct constant in these inequalities. To obtain the logarithmic analogues of the (fractional) Sobolev inequalities as in Theorem [Theorem 2](#thm1){reference-type="ref" reference="thm1"}, the latter must be combined with the logarithmic Hölder's inequality on general measure spaces. The following lemma states the desired inequality that was first shown in [@CKR21c]; see also [@KRS20 Lemma 3.2] for a more general version of it. **Lemma 5** (Logarithmic Hölder inequality). *Let $\mathbb{X}$ be a measure space and let $u\in L^{p}(\mathbb{X})\cap L^{q}(\mathbb{X})\setminus\{0\}$, where $1<p<q< \infty.$ Then we have $$\label{holdernn} \int_{\mathbb{X}}\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{X})}}\log\left(\frac{|u|^{p}}{\|u\|^{p}_{L^{p}(\mathbb{X})}}\right)dx\leq \frac{q}{q-p}\log\left(\frac{\|u\|^{p}_{L^{q}(\mathbb{X})}}{\|u\|^{p}_{L^{p}(\mathbb{X})}}\right).$$* A combination of Lemma [Lemma 5](#holder){reference-type="ref" reference="holder"} together with Theorem [Theorem 2](#thm1){reference-type="ref" reference="thm1"} yields the logarithmic version of the Sobolev inequality involving both the modified fractional sub-Laplacian $\mathcal{L}_{s}$ and powers of the sub-Laplacian $\mathcal{L}^{s}$. **Theorem 6** (Logarithmic Sobolev inequality). *Let $s \in (0,1]$. The (modified) fractional logarithmic Sobolev inequality on $\mathbb{H}^{n}$ is as follows $$\label{LogSobolev1} \int_{\mathbb{H}^{n}}\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx \leq \frac{Q}{2s}\log\left(C_{B,s}\frac{\langle \mathcal{L}_{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right),$$ where $C_{B,s}$ is given in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}). Additionally, the logarithmic Sobolev inequality on $\mathbb{H}^{n}$ in terms of fractional powers of $\mathcal{L}$ reads as follows $$\label{LogSobolev2} \int_{\mathbb{H}^{n}}\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx \leq \frac{Q}{2s}\log\left(C_{B,s}\|U_{s}\|_{\rm op}\frac{\langle \mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right),$$ where $\|U_{s}\|_{\rm op}$ has been estimated in ([\[est1\]](#est1){reference-type="ref" reference="est1"}).* *Proof.* From the logarithmic Hölder inequality ([\[holdernn\]](#holdernn){reference-type="ref" reference="holdernn"}) we have $$\int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|u(x)|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right)dx\leq \frac{q}{q-2}\log\left(\frac{\|u\|^{2}_{L^{q}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right).$$ For $p=2<q=p^{*}=\frac{2Q}{Q-2s}$ by ([\[1b\]](#1b){reference-type="ref" reference="1b"}) we have $$\begin{aligned} \label{log.sob.} \int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|u(x)|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx & \leq & \frac{q}{q-2}\log\left(\frac{\|u\|^{2}_{L^{q}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right)\nonumber\\ & \leq & \frac{q}{q-2} \log\left(C_{B,s}\frac{\langle \mathcal{L}_{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right)\nonumber\\ & = & \frac{Q}{2s} \log\left(C_{B,s}\frac{\langle \mathcal{L}_{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right),\end{aligned}$$ where $C_{B,s}$ is defined in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}), and this shows ([\[LogSobolev1\]](#LogSobolev1){reference-type="ref" reference="LogSobolev1"}). In the same way, we get ([\[LogSobolev2\]](#LogSobolev2){reference-type="ref" reference="LogSobolev2"}), and the proof of the theorem is complete. ◻ As the next result shows one can rewrite the fractional logarithmic Sobolev inequality ([\[LogSobolev1\]](#LogSobolev1){reference-type="ref" reference="LogSobolev1"}) for $s=1$ using the horizontal gradient on $\mathbb{H}^{n}$. The proof of Corollary [Corollary 7](#cor.s=1){reference-type="ref" reference="cor.s=1"} is immediate since $\mathcal{L}=-\nabla_{\mathbb{H}^{n}}^{*}\nabla_{\mathbb{H}^{n}}$. **Corollary 7**. *We have $$\label{LogSobolevint} \int_{\mathbb{H}^{n}}\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx \leq \frac{Q}{2}\log\left(C_{B,1}\frac{\|\nabla_{\mathbb{H}^{n}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right)\,.$$ Additionally, whenever $u$ is such that $\|u\|_{L^{2}(\mathbb{H}^{n})}=1$, we have $$\label{u=1} \int_{\mathbb{H}^{n}}|u|^{2}\log\left(|u|\right) dx \leq \frac{Q}{4}\log\left(C_{B,1}\|\nabla_{\mathbb{H}^{n}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}\right),$$ where $C_{B,1}$ is given in ([\[bestintsob\]](#bestintsob){reference-type="ref" reference="bestintsob"}).* # Gross inequalities In this section, we prove that the analogue of the Gross inequality holds true in the $\mathbb{H}^{n}$-setting. Even more, in Theorem [Theorem 11](#equiv.thm){reference-type="ref" reference="equiv.thm"} we show that the log-Sobolev inequality involving the horizontal gradient, the so-called "horizontal" log-Sobolev inequality, ([\[LogSobolev1\]](#LogSobolev1){reference-type="ref" reference="LogSobolev1"}) is equivalent to the latter, as it happens in the Euclidean setting, see [@Bec98]. The idea of passing the analogue of the Gross inequality on $\mathbb{H}^{n}$ to an infinite dimensional object like $\mathbb{H}^{\infty}$ is discussed in Observation [Observation 9](#mainrem){reference-type="ref" reference="mainrem"} where we see that in the limiting case, i.e., when the dimension of the first stratum of $\mathbb{H}^{n}$ becomes very big, the probability measure that is considered on it, becomes exactly the Gaussian measure on $\mathbb{R}^{2n}$. **Theorem 8** (Semi-Gaussian log-Sobolev inequality on $\mathbb{H}^{n}$). *The following "semi-Gaussian" log-Sobolev inequality is satisfied $$\label{gaus.log.sob} \int_{\mathbb{H}^{n}}|g|^2\log|g|\,d\mu \leq \int_{\mathbb{H}^{n}} |\nabla_{\mathbb{H}^{n}}g|^2\,d\mu\,,$$ for any $g$ such that $\|g\|_{L^2(\mathbb{H}^{n},\mu)}=1,$ where $\mu=\mu_1 \otimes \mu_2$, and $\mu_1$ is the Gaussian measure on $\mathbb{R}^{2n}$ given by $d\mu_1=\gamma e^{-\frac{|\xi|^2}{2}}d\xi$, for $\xi\in \mathbb{R}^{2n}$, with $|\xi|$ being the Euclidean norm of $\xi$, where $$\label{k} \gamma=n!\left(\frac{n+1}{2\pi n^{2}}\right)^{n+1}e^{n-1}\,,$$ and $d\mu_2=d\tau$ is the Lebesgue measure on $\mathbb{R}$ with respect to the variable $\tau$, for $x=(\xi,\tau) \in \mathbb{H}^{n}$.* As mentioned already in the introduction the "semi-Gaussian" inequality comes along with the idea of extending it to the infinite-dimensional Heisenberg group $\mathbb{H}^{\infty}$. The following observation argues in this direction. Very interestingly it shows that in the limiting case when $n \rightarrow \infty$ the probability measure in the case of $\mathbb{H}^{n}$ behaves exactly like the Gaussian measure on $\mathbb{R}^{2n}$. This means that the analogue of the Gross inequality in the case of $\mathbb{H}^{\infty}$ is the exact extension of the Gross inequality on $\mathbb{H}^{n}$. **Observation 9**. *First, observe that for large $n\gg 1$ (that is $Q\gg1$), we have that the constant $\gamma$ has the following form $$\gamma_{n \gg 1}\simeq(2\pi)^{-n-\frac{1}{2}}n^{-\frac{1}{2}}=\frac{1}{(2\pi)^n}\frac{1}{\sqrt{2\pi n}}\,.$$ Indeed by using Stirling's approximation formula (see e.g. [@Rom18]), i.e., that for large $n$ we have $n! \sim \sqrt{2 \pi n} \left( \frac{n}{e}\right)^{n}$, one can compute $$\begin{aligned} \gamma & = & n!\left(\frac{n+1}{2\pi n^{2}}\right)^{n+1}e^{n-1}\\ & = & n! e^{n-1}(2\pi n)^{-n-1}\left(1+\frac{1}{n} \right)^{n+1}\\ & \simeq & \sqrt{2\pi n}\left( \frac{n}{e}\right)^n e^{n-1}(2\pi n)^{-n-1} e \\ & = & (2\pi)^{-n-\frac{1}{2}}n^{-\frac{1}{2}}\,,\end{aligned}$$ where we have used the fact that $\lim\limits_{n \rightarrow \infty} \left(1+\frac{1}{n} \right)^{n+1}=e$.* *Now, since the normalisation constant $\gamma$ corresponds to the probability measure of the first stratum of topological dimension $2n$, each coordinate $\xi_i$, $i=1,\cdots, 2n$, in it should correspond to a probability measure $d\mu_{1,j}$, $j=1,\cdots, 2n$, with normalisation constant $\gamma_{n \gg 1, j}=\left(\gamma_{n \gg 1}\right)^{\frac{1}{2n}}$, so that $$d\mu_1=d\mu_{1,1} \times \cdots \times d\mu_{1,2n}:= \left(\gamma_{n \gg 1, 1} e^{-\frac{\xi_{1}^{2}}{2}}d\xi_1\right)\times \cdots \times \left(\gamma_{n \gg 1, 2n} e^{-\frac{\xi_{2n}^{2}}{2}}d\xi_{2n}\right)\,.$$ Hence, in order to be able to extend ([\[gaus.log.sob\]](#gaus.log.sob){reference-type="ref" reference="gaus.log.sob"}) to $\mathbb{H}^{\infty}$ with infinite dimensional first stratum we must have $\lim\limits_{n \rightarrow \infty} \left(\gamma_{n \gg 1}\right)^{\frac{1}{2n}}=c$, for some $c>0$. Indeed we have $$\begin{aligned} \label{ninfinity} \lim_{n \rightarrow \infty} \left(\gamma_{n \gg 1}\right)^{\frac{1}{2n}} & = & \lim_{n \rightarrow \infty} \left((2\pi)^{-n-\frac{1}{2}}n^{-\frac{1}{2}}\right)^{\frac{1}{2n}} \nonumber \\ & = & (2\pi)^{-\frac{1}{2}} \lim_{n \rightarrow \infty} (2\pi)^{-\frac{1}{4n}}n^{-\frac{1}{4n}} \nonumber\\ & = & (2\pi)^{-\frac{1}{2}} \lim_{n \rightarrow \infty}n^{-\frac{1}{4n}}= (2\pi)^{-\frac{1}{2}}\,, \end{aligned}$$ since $\lim\limits_{n \rightarrow \infty}(2\pi)^{-\frac{1}{4n}}=1$ and $$\lim_{n \rightarrow \infty}n^{-\frac{1}{4n}}=\lim_{n \rightarrow \infty}e^{\log n^{-\frac{1}{4n}}}= \lim_{n \rightarrow \infty}e^{-\frac{1}{4n}\log n}=1\,.$$ Equality ([\[ninfinity\]](#ninfinity){reference-type="ref" reference="ninfinity"}) shows not only that $\lim\limits_{n \rightarrow \infty} \left(\gamma_{n \gg 1}\right)^{\frac{1}{2n}}=\frac{1}{\sqrt{2\pi}}$, as required, but also it shows that the probability measure on the first stratum of $\mathbb{H}^{n}$ behaves in the limiting case when $n \rightarrow \infty$ exactly like $\mathbb{R}^{2n}$. Recall, see [@Gro92], that the normalisation constant in the classical Gross inequality is $(2\pi)^{-\frac{1}{2n}}$ on the whole $\mathbb{R}^n$, meaning that the probability measure corresponding to each of the $n$ coordinates is $(2\pi)^{-\frac{1}{2}}$ with the measure $\prod\limits_{j=1}^{n}\left(\frac{1}{\sqrt{2\pi}}e^{x_{j}^{2}}dx_{j}\right)$; exactly as it happens in the case of $\mathbb{H}^{n}$ when $n \rightarrow \infty$.* *Proof of Theorem [Theorem 8](#semi-g){reference-type="ref" reference="semi-g"}.* Assume that $g\in C^{\infty}_{0}(\mathbb{H}^{n})$ is a compactly supported function that satisfies $\|g\|_{L^2(\mathbb{H}^{n},\mu)}=1,$ where $\mu$ is the measure given in the hypothesis. Let us define $f(x)$ by $$f(x)=\gamma^{\frac{1}{2}}e^{-\frac{|\xi|^2}{4}}g(x),\,\,\,\,x=(\xi,\tau)\in\mathbb{H}^{n},$$ where $\gamma$ is as in ([\[k\]](#k){reference-type="ref" reference="k"}) and $|\xi|^{2}=\sum\limits_{k=1}\limits^{2n}\xi^{2}_{k}$ for $\xi\in \mathbb{R}^{2n}$. Then clearly $f\in C^{\infty}_{0}(\mathbb{H}^{n})$, while it is easy to check that $\|f\|_{L^2(\mathbb{H}^{n})}=1$. Indeed we have $$\label{norms1} 1=\|g\|_{L^2(\mathbb{H}^{n},\mu)}^2=\int_{\mathbb{H}^{n}}\gamma^{-1} e^{\frac{|\xi|^2}{2}}|f(x)|^2\,d\mu=\int_{\mathbb{H}^{n}}|f(x)|^2\,dx\,.$$ An application of the log-Sobolev inequality ([\[LogSobolevint\]](#LogSobolevint){reference-type="ref" reference="LogSobolevint"}) yields $$\begin{aligned} \label{thm.eq.glogg} \int_{\mathbb{H}^{n}}|g(x)|^2\log |g(x)|\,d\mu & = & \int_{\mathbb{H}^{n}} |f(x)|^2 \log |\gamma^{-\frac{1}{2}}e^{\frac{|\xi|^2}{4}}f(x)|\,dx \nonumber\\ & \leq & \frac{Q}{4}\log \left(C_{B,1} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx\right) \nonumber \\ & & +\log (\gamma^{-\frac{1}{2}})+\int_{\mathbb{H}^{n}}\frac{|\xi|^2}{4} |f(x)|^2\,dx\,.\end{aligned}$$ For $\xi\in \mathbb{R}^{2n}$, it is worth recalling that $$X_{i}=\frac{\partial}{\partial\xi_{i}}+\frac{1}{2}\xi_{n+i}\frac{\partial}{\partial \tau}, \,\,\, Y_{i}=\frac{\partial}{\partial \xi_{n+i}}-\frac{1}{2}\xi_{i}\frac{\partial}{\partial \tau}\,, \quad i=1,\ldots,n\,.$$ For each $i=1,\ldots,n$, we have $$\begin{aligned} \label{Xg} |X_ig(x)|^2 &=& \gamma^{-1} e^{\frac{|\xi|^2}{2}} \left|X_if(x)+\frac{\xi_{i}}{2}f(x) \right|^{2}\nonumber\\ &=& \gamma^{-1} e^{\frac{|\xi|^2}{2}} \left(|X_if(x)|^2+\frac{(\xi_{i})^2}{4}|f(x)|^2+{\rm Re} \overline{(X_if(x))}\xi_{i}f(x) \right)\,,\end{aligned}$$ and $$\begin{aligned} \label{Yg} |Y_ig(x)|^2 &=& \gamma^{-1} e^{\frac{|\xi|^2}{2}} \left|Y_if(x)+\frac{\xi_{n+i}}{2}f(x) \right|^{2}\nonumber\\ &=& \gamma^{-1} e^{\frac{|\xi|^2}{2}} \left(|Y_if(x)|^2+\frac{(\xi_{n+i})^2}{4}|f(x)|^2+{\rm Re} \overline{(Y_if(x))}\xi_{n+i}f(x) \right)\,.\end{aligned}$$ Moreover, notice that for each $\xi_{i}$, $i=1,\ldots,2n$, we have $$\begin{aligned} {\rm Re}\int_{\mathbb{H}^{n}}\overline{(\partial_{\xi_{i}}f(x))}\xi_{i}f(x)\,dx & =& -{\rm Re}\int_{\mathbb{H}^{n}}(\partial_{\xi_{i}}f(x))\xi_{i}\overline{f(x)}\,dx-\int_{\mathbb{H}^{n}}|f(x)|^2\,dx\\ &=& -{\rm Re}\int_{\mathbb{H}^{n}}\overline{(\partial_{\xi_{i}}f(x))}\xi_{i}f(x)\,dx-1\,.\end{aligned}$$ The above computations imply that for each $i=1,\ldots,2n$ we have $$\label{int.parts1} {\rm Re}\int_{\mathbb{H}^{n}}\overline{(\partial_{\xi_{i}}f(x))}\xi_{i}f(x)\,d\xi d\tau=-\frac{1}{2}\,.$$ To explicitly compute $\int_{\mathbb{H}^{n}}{\rm Re}\overline{(X_if(x))}\xi_{i}f(x)\,dx$ it remains to compute the term $$\frac{1}{2}{\rm Re} \int_{\mathbb{H}^{n}}\xi_{n+i}\overline{(\partial_{\tau}f(x))}\xi_{i}f(x)dx\,\,\,,i=1,\ldots,n.$$ To this end we have $$\begin{aligned} \frac{1}{2}{\rm Re} \int_{\mathbb{H}^{n}}\xi_{n+i}\overline{(\partial_{\tau}f(x))}\xi_{i}f(x)dx& = & - \frac{1}{2}{\rm Re}\int_{\mathbb{H}^{n}}\partial_{\tau}((\xi_{n+i}f(x)\xi_{i})\overline{f(x)}\,dx\\ &=&- \frac{1}{2}{\rm Re}\int_{\mathbb{H}^{n}}\xi_{n+i}\overline{(\partial_{\tau}f(x))}\xi_{i}f(x)\,dx\,,\end{aligned}$$ implying that $$\label{int.parts2} {\rm Re} \int_{\mathbb{H}^{n}}\xi_{n+i}\overline{(\partial_{\tau}f(x))}\xi_{i}f(x)\,dx=0,\,\,\,i=1,\ldots,n.$$ A similar argument applies to $Y_i$. Summarising, by ([\[int.parts1\]](#int.parts1){reference-type="ref" reference="int.parts1"}) and ([\[int.parts2\]](#int.parts2){reference-type="ref" reference="int.parts2"}) we get $$\int_{\mathbb{H}^{n}}{\rm Re}\overline{(X_if(x))}\xi_{i}f(x)\,dx=-\frac{1}{2}\,,$$ and similarly $$\int_{\mathbb{H}^{n}}{\rm Re}\overline{(Y_if(x))}\xi_{n+i}f(x)\,dx=-\frac{1}{2}\,,$$ for each $i=1,\ldots,n$. By the above and expression ([\[Xg\]](#Xg){reference-type="ref" reference="Xg"}) we get $$\label{thm.eq.nablag} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}g(x)|^2\,d\mu=\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx+\int_{\mathbb{H}^{n}}\frac{|\xi|^2}{4}|f(x)|^2\,dx-n\,.$$ Now, taking into account ([\[thm.eq.glogg\]](#thm.eq.glogg){reference-type="ref" reference="thm.eq.glogg"}) and ([\[thm.eq.nablag\]](#thm.eq.nablag){reference-type="ref" reference="thm.eq.nablag"}), we see that showing ([\[gaus.log.sob\]](#gaus.log.sob){reference-type="ref" reference="gaus.log.sob"}) reduces to proving that $$\frac{Q}{4}\log \left(C_{B,1} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx\right)+\log (\gamma^{-1/2}) \leq \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx-n\,,$$ or $$\frac{Q}{4}\log \left( C_{B,1} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx \right)+\log (\gamma^{-\frac{1}{2}}) + \log e^{n}\leq \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx\,.$$ The latter can be rewritten as $$\frac{Q}{4} \log \left(C_{B,1} \gamma^{-\frac{2}{Q}} e^{\frac{4n}{Q}} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx \right)\leq \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx\,,$$ or as $$\log \left(C_{B,1} \gamma^{-\frac{2}{Q}} e^{\frac{4n}{Q}} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx \right)\leq \frac{4}{Q} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx\,.$$ By the above and the property that for all $r>0$ we have $\log r \leq r-1$, we conclude that showing ([\[gaus.log.sob\]](#gaus.log.sob){reference-type="ref" reference="gaus.log.sob"}) amounts to proving the following $$\label{s-gaus.f} {e^{-1}}C_{B,1}\gamma^{-\frac{2}{Q}}e^{\frac{4n}{Q}} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx \leq \frac{4}{Q} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx\,.$$ Indeed, observe that ([\[s-gaus.f\]](#s-gaus.f){reference-type="ref" reference="s-gaus.f"}) holds true even as an equality for the choice of $\gamma$ as in ([\[k\]](#k){reference-type="ref" reference="k"}). Hence, we have shown the desired inequality, i.e., that $$\begin{split} &\log \left(C_{B,1} \gamma^{-\frac{2}{Q}} e^{\frac{4n}{Q}} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx \right)\\& = \log \left(e^{-1} \frac{(n!)^{\frac{1}{n+1}}}{\pi n^{2}}\gamma^{-\frac{1}{n+1}}e^{\frac{2n}{n+1}} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx\right)+1\\& \leq e^{-1} \frac{(n!)^{\frac{1}{n+1}}}{\pi n^{2}}\gamma^{-\frac{1}{n+1}}e^{\frac{2n}{n+1}} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx\\& = \frac{4}{Q} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx, \end{split}$$ and so we have shown ([\[gaus.log.sob\]](#gaus.log.sob){reference-type="ref" reference="gaus.log.sob"}) for $g \in C_{0}^{\infty}(\mathbb{H}^{n})$. The proof is now complete by the density of $C_{0}^{\infty}(\mathbb{H}^{n})$ in $L^2(\mathbb{H}^{n}, \mu)$. ◻ **Remark 10**. As mentioned earlier, the logarithmic Sobolev inequality with respect to a probability measure is a key inequality in the field of infinite dimensional analysis, see e.g. [@BZ05] and references therein. The associated terminology suggests the following expression for the Gaussian log-Sobolev inequality ([\[gaus.log.sob\]](#gaus.log.sob){reference-type="ref" reference="gaus.log.sob"}) $$\label{ent.gross} \frac{1}{2}\text{Ent}(|g|^2)\leq \int_{\mathbb{H}^{n}} |\nabla_{\mathbb{H}^{n}}g|^2\,d\mu\,, \quad \text{for}\quad \|g\|_{L^2(\mathbb{H}^{n},d\mu)}=1\,,$$ where the entropy functional $\text{Ent}(g)$ for a measurable function $g$ is defined as $$\text{Ent}(g):=\int g \log g \,d\mu-\int g \,d\mu \log \int g\,d\mu\,.$$ It is then clear that for $g$ such that $\|g\|_{L^2(\mathbb{H}^{n},d\mu)}=1$, we have $$\text{Ent}(|g|^2)=2 \int_{\mathbb{H}^{n}}|g|^2\log|g|\,d\mu\,,$$ implying that the inequalities ([\[ent.gross\]](#ent.gross){reference-type="ref" reference="ent.gross"}) and ([\[gaus.log.sob\]](#gaus.log.sob){reference-type="ref" reference="gaus.log.sob"}) are euivalent. **Theorem 11**. *Let $d\mu$ be the semi-Gaussian measure given in the hypothesis of Theorem [Theorem 8](#semi-g){reference-type="ref" reference="semi-g"}. The following statements are both true and imply each other:* 1. *[\[11i\]]{#11i label="11i"} for $g$ such that $\|g\|_{L^2(\mathbb{H}^{n},\mu)}=1,$ we have $$\label{thm.eq.itmq} \int_{\mathbb{H}^{n}}|g|^2 \log |g|\,d\mu \leq \int_{\mathbb{H}^{n}} |\nabla_{\mathbb{H}^{n}}g|^2\,d\mu\,;$$* 2. *[\[2i\]]{#2i label="2i"} for $f$ such that $\|f\|_{L^2(\mathbb{H}^{n})}=1,$ we have $$\label{thm.eq.itm2}\int_{\mathbb{H}^{n}}|f|^{2}\log |f|\,dx \leq \frac{Q}{4}\log \left(C_{B,1} \int_{\mathbb{H}^{n}} |\nabla_{\mathbb{H}^{n}}f|^2\,dx \right)\,,$$ where $C_{B,1}$ is as in ([\[bestintsob\]](#bestintsob){reference-type="ref" reference="bestintsob"}).* *Proof.* The implication (ii) $\Rightarrow$ (i) is exactly the proof of Theorem [Theorem 8](#semi-g){reference-type="ref" reference="semi-g"}. Hence we will show that (i) $\Rightarrow$ (ii). For $\gamma$ as in ([\[k\]](#k){reference-type="ref" reference="k"}) and for $f$ as in (i), we define $g$ by $$g(x)=\gamma^{-\frac{1}{2}}e^{\frac{|\xi|^2}{4}}f(x)\,.$$ It is then clear that $\|g\|_{L^2(\mathbb{H}^{n},\mu)}=1$, see ([\[norms1\]](#norms1){reference-type="ref" reference="norms1"}). We also have $$\int_{\mathbb{H}^{n}}|g(x)|^2\log |g(x)|\,d\mu = \int_{\mathbb{H}^{n}} |f(x)|^2 \log |\gamma^{-\frac{1}{2}}e^{\frac{|\xi|^2}{4}}f(x)|\,dx\,.$$ Hence by the assumptions in (ii) and equality ([\[thm.eq.nablag\]](#thm.eq.nablag){reference-type="ref" reference="thm.eq.nablag"}) we get $$\label{before.eps} \log (\gamma^{-\frac{1}{2}}e^{n})+\int_{\mathbb{H}^{n}}|f(x)|^2\log |f(x)|\,dx \leq \int_{\mathbb{H}^{n}} |\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx\,,$$ for any function $f$ such that $\|f\|_{L^2(\mathbb{H}^{n})}=1$. Now, if for $\epsilon>0$ the mapping $\delta_\epsilon: \mathbb{H}^{n}\rightarrow \mathbb{H}^{n}$ denotes the dilations on $\mathbb{H}^{n}$, i.e., for $x=(\xi, \tau) \in \mathbb{H}^{n}$ we have $\delta_\epsilon(\xi,\tau)=(\epsilon \xi, \epsilon^2 \tau)$, then for $f_\epsilon(x):=\epsilon^{\frac{Q}{2}}f(\delta_{\epsilon}(x))$ we still have $\|f_{\epsilon}\|_{L^2(\mathbb{H}^{n})}=1$. Now, since $$\int_{\mathbb{H}^{n}}|f_\epsilon|^2 \log|f_\epsilon|\,dx=\int_{\mathbb{H}^{n}}|f|^2 \log\left|\epsilon^{\frac{Q}{2}}f \right|\,dx\,,$$ and $$\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f_\epsilon|^2\,dx=\epsilon^2 \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f|^2\,dx\,,$$ an application of ([\[before.eps\]](#before.eps){reference-type="ref" reference="before.eps"}) to the function $f_\epsilon$ yields $$\label{after.eps} \begin{split} \int_{\mathbb{H}^{n}}&|f|^2\log |f|\,dx \leq \epsilon^2 \int_{\mathbb{H}^{n}} |\nabla_{\mathbb{H}^{n}}f|^2\,dx-\frac{Q}{2}\log \epsilon-\log (\gamma^{-\frac{1}{2}}e^{n})\\& =\epsilon^2 \int_{\mathbb{H}^{n}} |\nabla_{\mathbb{H}^{n}}f|^2\,dx-\frac{Q}{2}\log \epsilon-\log \left(\left[n!e^{n-1}\left(\frac{(n+1)}{2\pi n^{2}}\right)^{n+1}\right]^{-\frac{1}{2}}e^{n} \right)\\& =\epsilon^2 \int_{\mathbb{H}^{n}} |\nabla_{\mathbb{H}^{n}}f|^2\,dx-\frac{Q}{2}\log \epsilon+\frac{n+1}{2} \log \left(\frac{(n+1)(n!)^{\frac{1}{n+1}}}{2\pi n^{2}e} \right) \end{split}$$ for all $\epsilon>0$, where in the last inequality we have plugged in the expression for $\gamma$. Finally, minimizing the right-hand side of ([\[after.eps\]](#after.eps){reference-type="ref" reference="after.eps"}) over $\epsilon$ gives the desired inequality. Indeed for $\epsilon=\sqrt{\frac{Q}{4 \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx}}$ we have $$\begin{aligned} \int_{\mathbb{H}^{n}}|f|^2\log |f|\,dx & \leq & \frac{Q}{4}-\frac{Q}{2}\log \left( \sqrt{\frac{Q}{4 \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f(x)|^2\,dx}}\right)\\ &+&\frac{n+1}{2} \log \left(\frac{(n+1)(n!)^{\frac{1}{n+1}}}{2\pi n^{2}e} \right)\\ & = & \frac{n+1}{2}+\frac{n+1}{2} \log \left(4 \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f|^2\,dx \right)- \frac{n+1}{2}\log 2(n+1)\\ &+&\frac{n+1}{2} \log \left(\frac{(n+1)(n!)^{\frac{1}{n+1}}}{2\pi n^{2}e} \right)\\ & = & \frac{n+1}{2} \log \left(\frac{(n!)^{\frac{1}{n+1}}}{\pi n^{2}} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f|^2\,dx \right)\,\\ &=&\frac{Q}{4}\log \left(C_{B,1} \int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}f|^2\,dx \right)\,, \end{aligned}$$ and this shows that (i) implies (ii). The proof of Theorem [Theorem 11](#equiv.thm){reference-type="ref" reference="equiv.thm"} is now complete. ◻ # (logarithmic) Gagliardo-Nirenberg inequalities on $\mathbb{H}^{n}$ In this section, we prove the Gagliardo-Nirenberg inequality on $\mathbb{H}^{n}$, as well the "the derivates-type Gagliardo-Nirenberg inequality" on $\mathbb{H}^{n}$. As earlier in Section [3](#sec:LS){reference-type="ref" reference="sec:LS"}, we prove the aforementioned inequalities also with respect to the modified fractional sub-Laplacian $\mathcal{L}_{s}$ on $\mathbb{H}^{n}$. Let us first recall the next auxiliary result as in [@CKR21c]. **Lemma 12** (An interpolation inequality for general measure spaces). *Let $\mathbb{X}$ be any measure space, and let $1<\theta_1\leq q\leq \theta_2\leq\infty$. Then for any $u\in L^{\theta_1}(\mathbb{X})\cap L^{\theta_2}(\mathbb{X})$ with $$\frac{1}{q}=\frac{a}{\theta_1}+\frac{1-a}{\theta_2}\,,$$ for some $a \in [0,1]$, we have $$\label{holder.gen} \|u\|_{L^{q}(\mathbb{X})}\leq \|u\|^{a}_{L^{\theta_1}(\mathbb{X})}\|u\|^{1-a}_{L^{\theta_2}(\mathbb{X})}.$$* **Theorem 13** (Gagliardo-Nirenberg inequality). *Let $s\in(0,1]$ and $Q>2s$ satisfy the relation $\frac{1}{q}=a\left(\frac{1}{2}-\frac{s}{Q}\right)+\frac{1-a}{\sigma}$, for some $\sigma\geq q\geq \frac{2Q}{Q-2s}$ and $a \in (0,1]$. Then the Gagliardo-Nirenberg inequality on $\mathbb{H}^{n}$ with respect to the (modified) sub-Laplacian is given by $$\label{gnin1} \int_{\mathbb{H}^{n}}|u(x)|^{q}dx \leq C^{\frac{aq}{2}}_{B,s}\langle \mathcal{L}_{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{aq}{2}}\|u\|^{(1-a)q}_{L^{\sigma}(\mathbb{H}^{n})}\,,$$ where $C_{B,s}$ is as in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}). Moreover, we can rewrite the inequality ([\[gnin1\]](#gnin1){reference-type="ref" reference="gnin1"}) with respect to fractional powers of the sub-Laplacian $\mathcal{L}_{s}$ as follows $$\label{gnin2} \int_{\mathbb{H}^{n}}|u(x)|^{q}dx \leq C^{\frac{aq}{2}}_{B,s}\|U_{s}\|^{\frac{aq}{2}}_{\rm op}\langle \mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{aq}{2}}\|u\|^{(1-a)q}_{L^{\sigma}(\mathbb{H}^{n})},$$ where the operator norm $\|U_{s}\|_{\rm op}$ has been estimated in ([\[est1\]](#est1){reference-type="ref" reference="est1"}).* *Proof.* By using the Hölder inequality ([\[holder.gen\]](#holder.gen){reference-type="ref" reference="holder.gen"}), and the fractional Sobolev inequality as in ([\[1b\]](#1b){reference-type="ref" reference="1b"}), we have $$\begin{split} \int_{\mathbb{H}^{n}}|u(x)|^{q}dx&= \int_{\mathbb{H}^{n}}|u(x)|^{aq}|u(x)|^{(1-a)q}dx\\& \leq\|u\|^{aq}_{L^{2^{*}}(\mathbb{H}^{n})}\|u\|^{(1-a)q}_{L^{\sigma}(\mathbb{H}^{n})} \\& \leq C^{\frac{aq}{2}}_{B,s}\langle \mathcal{L}_{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{aq}{2}}\|u\|^{(1-a)q}_{L^{\sigma}(\mathbb{H}^{n})}\,, \end{split}$$ and we have proved inequality ([\[gnin1\]](#gnin1){reference-type="ref" reference="gnin1"}). Inequality ([\[gnin2\]](#gnin2){reference-type="ref" reference="gnin2"}) follows now by ([\[gnin1\]](#gnin1){reference-type="ref" reference="gnin1"}) and the definition of the operator $U_s$ as in ([\[Us\]](#Us){reference-type="ref" reference="Us"}). ◻ One can repeat the main arguments and show the following generalisation of ([\[gnin1\]](#gnin1){reference-type="ref" reference="gnin1"}). **Theorem 14** (Two derivative-type Gagliardo-Nirenberg inequality). *Let $1\geq s_{1}\geq s_{2}\geq0$ be such that $\frac{2Q}{Q-2s_{2}}\leq q\leq \frac{2Q}{Q-2s_{1}}$. Then, the Gagliardo-Nirenberg inequality with two derivatives on $\mathbb{H}^{n}$ is as follows $$\label{GN4} \int_{\mathbb{H}^{n}}|u(x)|^{q}dx\leq C_{GN,s_1,s_2}\langle\mathcal{L}_{s_{1}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{Q(q-2)-2s_{2}q}{4(s_{1}-s_{2})}}\langle\mathcal{L}_{s_{2}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{s_{1}(q-2)-Q(q-2)}{4(s_{1}-s_{2})}},$$ where $C_{GN,s_1,s_2}=C_{B,s_1}^{\frac{qa}{2}} C_{B,s_2}^{\frac{q(1-a)}{2}}$ for $a=\frac{Q(q-2)-2s_{2}q}{2(s_{1}-s_{2})q}$, where $C_{B,s_i}$, $i=1,2$, is given in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}). Additionally, the modified version of it is given by $$\label{GN5} \begin{split} \int_{\mathbb{H}^{n}}|u(x)|^{q}dx&\leq C_{GN,s_1,s_2}\|U_{s_1}\|^{\frac{Q(q-2)-2s_{2}q}{4(s_{1}-s_{2})}}_{\rm op}\|U_{s_2}\|^{\frac{s_{1}(q-2)-Q(q-2)}{4(s_{1}-s_{2})}}_{\rm op}\\& \times\langle\mathcal{L}^{s_{1}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{Q(q-2)-2s_{2}q}{4(s_{1}-s_{2})}}\langle\mathcal{L}^{s_{2}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{s_{1}(q-2)-Q(q-2)}{4(s_{1}-s_{2})}}\,, \end{split}$$ where the norms $\|U_{s_i}\|_{\rm op}$ for $i=1,2$ are given in ([\[est1\]](#est1){reference-type="ref" reference="est1"}).* *Proof.* First, let us show that $a=\frac{Q(q-2)-2s_{2}q}{2(s_{1}-s_{2})q}\in [0,1]$. To this end, we define the function $$A(q)=\frac{Q(q-2)-2s_{2}q}{2(s_{1}-s_{2})q}=\frac{Q-2s_2}{2(s_1-s_2)}-\frac{Q}{(s_1-s_2)}\frac{1}{q}\,,$$ for $\frac{2Q}{Q-2s_{2}}\leq q\leq \frac{2Q}{Q-2s_{1}}$ as in the hypothesis. It is then easy to see that since $A'(q)=\frac{Q}{s_1-s_2}\frac{1}{q^2} >0$, we have that $$0=A\left( \frac{2Q}{Q-2s_{2}}\right) \leq A(q) \leq A\left( \frac{2Q}{Q-2s_{1}}\right)=1\,.$$ and this proves our claim. By the Hölder inequality ([\[holder.gen\]](#holder.gen){reference-type="ref" reference="holder.gen"}) with $\theta_{1}=\frac{2Q}{Q-2s_{1}}$, $\theta_{2}=\frac{2Q}{Q-2s_{2}}$ and $a=\frac{Q(q-2)-2s_{2}q}{2(s_{1}-s_{2})q}\in [0,1]$, and Sobolev inequality ([\[1b\]](#1b){reference-type="ref" reference="1b"}) we get $$\begin{split} \int_{\mathbb{H}^{n}}|u(x)|^{q}dx&=\int_{\mathbb{H}^{n}}|u(x)|^{aq}|u(x)|^{(1-a)q}dx\\& \leq \left(\int_{\mathbb{H}^{n}}|u(x)|^{\theta_{1}}dx\right)^{\frac{qa}{\theta_{1}}}\left(\int_{\mathbb{H}^{n}}|u(x)|^{\theta_{2}}dx\right)^{\frac{q(1-a)}{\theta_{2}}}\\& \leq C_{B,s_1}^{\frac{qa}{2}}\langle\mathcal{L}_{s_{1}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{qa}{2}}C_{B,s_2}^{\frac{q(1-a)}{2}}\langle\mathcal{L}_{s_{2}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{q(1-a)}{2}}\\& =C_{GN,s_1,s_2}\langle\mathcal{L}_{s_{1}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{Q(q-2)-2s_{2}q}{4(s_{1}-s_{2})}}\langle\mathcal{L}_{s_{2}}u,u\rangle_{L^{2}(\mathbb{H}^{n})}^{\frac{s_{1}(q-2)-Q(q-2)}{4(s_{1}-s_{2})}}\,, \end{split}$$ where we have defined $C_{GN,s_1,s_2}:=C_{B,s_1}^{\frac{qa}{2}} C_{B,s_2}^{\frac{q(1-a)}{2}}$ and we have shown ([\[GN4\]](#GN4){reference-type="ref" reference="GN4"}). Inequality ([\[GN5\]](#GN5){reference-type="ref" reference="GN5"}) follows. ◻ **Remark 15** (Sharp version of Gagliardo-Nirenberg inequalities on $\mathbb{H}^{n}$). The appearing constants in the Gagliardo-Nirenberg inequalities on $\mathbb{H}^{n}$ may be in general not sharp. However, in view of Remark 6.2 in [@RTY20] that relates the best constant in the Sobolev inequality with the one in the Gagliardo-Nirenberg inequality, the Gagliardo-Nirenberg inequality ([\[GN4\]](#GN4){reference-type="ref" reference="GN4"}) when $s_1=1$, $s_2=0$ and $2\leq q \leq \frac{2Q}{Q-2}$ is sharp and the best constant $C_{GN,1,0}$ is given by $$C_{GN,1,0}=\left[C_{B,1}\frac{2q}{2q-Q(q-2)}\left( \frac{Q(q-2)}{2q-Q(q-2)}\right) \right]^{\frac{q}{2}}\,,$$ where $C_{B,1}$ is the best constant in the Sobolev inequality on $\mathbb{H}^{n}$ given by ([\[bestintsob\]](#bestintsob){reference-type="ref" reference="bestintsob"}). The logarithmic version of the Galirado-Nirenberg inequalities ([\[gnin1\]](#gnin1){reference-type="ref" reference="gnin1"}) and ([\[gnin2\]](#gnin2){reference-type="ref" reference="gnin2"}) are given in the next theorem. Following similar arguments, one can get the logarithmic versions of the inequalities ([\[GN4\]](#GN4){reference-type="ref" reference="GN4"}) and ([\[GN5\]](#GN5){reference-type="ref" reference="GN5"}). The latter is a simple exercise and its details are left for the interested reader. **Theorem 16** (Logarithmic Gagliardo-Nirenberg inequality). *Let $s\in(0,1]$ and $Q>2s$ satisfy the relation $\frac{1}{q}=a\left(\frac{1}{2}-\frac{s}{Q}\right)+\frac{1-a}{\sigma}$, for some $\sigma\geq q\geq \frac{2Q}{Q-2s}$ and $a \in (0,1]$. Then the logarithmic version of the Gagliardo-Nirenberg inequality with respect to the (modified) fractional sub-Laplacian is as follows $$\label{LogGN1} \int_{\mathbb{H}^{n}}\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx \leq \frac{q}{q-2}\log\left(C_{B,s}^{a}\frac{\langle \mathcal{L}_{s}u,u\rangle^{a}_{L^{2}(\mathbb{H}^{n})}\|u\|^{2(1-a)}_{L^{\sigma}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right)\,.$$ Alternatively, inequality ([\[LogGN1\]](#LogGN1){reference-type="ref" reference="LogGN1"}) can be expressed with respect to fractional powers of the sub-Laplacian via $$\label{LogGN2} \int_{\mathbb{H}^{n}}\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|u|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx \leq \frac{q}{q-2}\log\left(C_{B,s}^{a}\|U_{s}\|^{a}_{\rm{op}}\frac{\langle \mathcal{L}^{s}u,u\rangle^{a}_{L^{2}(\mathbb{H}^{n})}\|u\|^{2(1-a)}_{L^{\sigma}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right),$$ where the estimate for the norm $\|U_{s}\|_{\text{op}}$ is given by ([\[est1\]](#est1){reference-type="ref" reference="est1"}).* *Proof of Theorem [Theorem 16](#thmlogGN){reference-type="ref" reference="thmlogGN"}.* From the logarithmic Hölder inequality ([\[holdernn\]](#holdernn){reference-type="ref" reference="holdernn"}) we have $$\begin{split} \int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|u(x)|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right)dx&\leq \frac{q}{q-2}\log\left(\frac{\|u\|^{2}_{L^{q}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right). \end{split}$$ Hence by using ([\[gnin1\]](#gnin1){reference-type="ref" reference="gnin1"}), we have $$\begin{split} \int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}&\log\left(\frac{|u(x)|^{2}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx \leq \frac{q}{q-2}\log\left(\frac{\|u\|^{2}_{L^{q}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right)\\& \leq \frac{q}{q-2}\log\left(C_{B,s}^{a}\frac{\langle \mathcal{L}_{s}u,u\rangle^{a}_{L^{2}(\mathbb{H}^{n})}\|u\|^{2(1-a)}_{L^{\tau}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right). \end{split}$$[\[log.sob.123\]]{#log.sob.123 label="log.sob.123"} Inequality ([\[LogGN2\]](#LogGN2){reference-type="ref" reference="LogGN2"}) then follows, and the proof of the theorem is complete. ◻ # Hardy-Sobolev inequalities on $\mathbb{H}^{n}$ and applications The main results in this section are the weighted log-Sobolev inequality on $\mathbb{H}^{n}$ and the fractional log-Hardy inequality on $\mathbb{H}^{n}$. The next result that was proved in [@RT16] is the fractional Hardy inequality on $\mathbb{H}^{n}$ with the best constant and is used for the proof of the aforesaid inequalities. **Theorem 17** (Fractional Hardy inequality). *Let $s\in(0,1]$. We have the following inequality on $\mathbb{H}^{n}$ $$\label{eq.Hardy} \int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{|x|^{2s}}dx\leq C_{BH,s}\langle\mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})},$$ where $|\cdot|$ denotes the Kaplan norm, and the sharp constant $C_{BH,s}$ is given by $$\label{besthar} C_{BH,s}:=\|V_{s}\|_{\rm{op}}\frac{\Gamma(1-s)\Gamma^{2}\left(\frac{n}{2}\right)}{2^{2n+3s}\Gamma^{2}\left(\frac{n+s}{2}\right)}$$ where estimate for the norm of $V_{s}$ is given in ([\[est2\]](#est2){reference-type="ref" reference="est2"}).* With the use of Theorem [Theorem 17](#thm.Hardy){reference-type="ref" reference="thm.Hardy"} we prove the following generalised version of it which includes it as a special case. **Theorem 18** (Fractional Hardy-Sobolev inequality). *Let $s\in(0,1]$ and let $0\leq\beta\leq 2s$. Then we have* *$$\label{eq.Hardy-S} \left(\int_{\mathbb{H}^{n}}\frac{|u(x)|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{2}{2^{*}_{\beta}}}\leq \left(C^{\frac{\beta}{2s}}_{BH,s}\left(C_{B,s}\|U_{s}\|_{\rm{op}}\right)^{\frac{n+1}{n+1-s}\frac{2s-\beta}{2s}}\right)^{\frac{2}{2^{*}_{\beta}}} \langle\mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})},$$ where $2^{*}_{\beta}=\frac{2(Q-\beta)}{Q-2s}$ and $C_{B,s}$ and $C_{BH,s}$ are given by ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}) and ([\[besthar\]](#besthar){reference-type="ref" reference="besthar"}), respectively.* Indeed observe that the fractional Hardy-Sobolev inequality ([\[eq.Hardy-S\]](#eq.Hardy-S){reference-type="ref" reference="eq.Hardy-S"}) gives the fractional Hardy inequality ([\[eq.Hardy\]](#eq.Hardy){reference-type="ref" reference="eq.Hardy"}) when $\beta=2s$, and the fractional Sobolev ([\[1b\]](#1b){reference-type="ref" reference="1b"}) inequality when $\beta=0$. *Proof of Theorem [Theorem 18](#FHS){reference-type="ref" reference="FHS"}.* For $\beta=0$ and $\beta=2s$, we already have the fractional Sobolev and Hardy inequalities in ([\[1b\]](#1b){reference-type="ref" reference="1b"}) and ([\[eq.Hardy\]](#eq.Hardy){reference-type="ref" reference="eq.Hardy"}), respectively. Now for $0<\beta<2s$, using Hölder's inequality with $\frac{\beta}{2s}+\frac{2s-\beta}{2s}=1$, we have $$\begin{split} \int_{\mathbb{H}^{n}}\frac{|u(x)|^{2^{*}_{\beta}}}{|x|^{\beta}}dx&=\int_{\mathbb{H}^{n}}\frac{|u(x)|^{\frac{\beta}{s}}|u(x)|^{2^{*}_{\beta}-\frac{\beta}{s}}}{|x|^{\beta}}dx\\& \leq \left(\int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{|x|^{2s}}dx\right)^{\frac{\beta}{2s}}\left(\int_{\mathbb{H}^{n}}|u(x)|^{\frac{2Q}{Q-2s}}dx\right)^{\frac{2s-\beta}{2s}}\\& \stackrel{(\ref{1b}), (\ref{eq.Hardy})}\leq C^{\frac{\beta}{2s}}_{BH,s}\left(C_{B,s}\|U_{s}\|_{\text{op}}\right)^{\frac{n+1}{n+1-s}\frac{2s-\beta}{2s}} \langle\mathcal{L}^{s}u,u\rangle^{\frac{2^{*}_{\beta}}{2}}_{L^{2}(\mathbb{H}^{n})}, \end{split}$$ completing the proof. ◻ Next, we prove a weighted version of the log-Sobolev inequality in $\mathbb{H}^{n}$ that involves fractional powers of the sub-Laplacian $\mathcal{L}^{s}$. Following similar arguments, the analogous result can be shown for the log-Sobolev inequality involving the modified sub-Laplacian $\mathcal{L}_{s}$. **Theorem 19**. *Let $s\in(0,1]$ and let $0\leq \beta < 2s$. Then we have the following weighted log-Sobolev inequality of fractional order on $\mathbb{H}^{n}$* *$$\label{LogwSobolev2} \begin{split} &\int_{\mathbb{H}^{n}}\frac{|x|^{-\frac{2\beta}{2^{*}_{\beta}}}|u(x)|^{2}}{\||\cdot|^{-\frac{\beta}{2^{*}_{\beta}}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|x|^{-\frac{2\beta}{2^{*}_{\beta}}}|u(x)|^{2}}{\||\cdot|^{-\frac{\beta}{2^{*}_{\beta}}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx\\& \leq \frac{Q-\beta}{2s-\beta}\log\left(\left(C^{\frac{\beta}{2s}}_{BH,s}(C_{B,s}\|U_{s}\|_{\rm{op}})^{\frac{n+1}{n+1-s}\frac{2s-\beta}{2s}}\right)^{\frac{2}{2^{*}_{\beta}}}\frac{\langle \mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}}{\||\cdot|^{-\frac{\beta}{2^{*}_{\beta}}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right), \end{split}$$ where $2^{*}_{\beta}=\frac{2(Q-\beta)}{Q-2s}$.* *Proof.* Let $q=\frac{2(Q-\beta)}{Q-2s}$. Using the fact that $0\leq \beta <2s$, we have $$q-2=\frac{2(Q-\beta)}{Q-2s}-2=\frac{2(Q-\beta-Q+2s)}{Q-2s}=\frac{2(2s-\beta)}{Q-2s}> 0.$$ The latter implies that $q> 2=p$, and we compute $$\label{vychetstploghsgr} 0<\frac{q}{q-2}=\frac{\frac{2(Q-\beta)}{Q-2s}}{\frac{2(Q-\beta)}{Q-2s}-2} =\frac{\frac{Q-\beta}{Q-2s}}{\frac{(Q-\beta)}{Q-2s}-1} =\frac{Q-\beta}{2s-\beta}.$$ By using the logarithmic Hölder's inequality as in Lemma [Lemma 5](#holder){reference-type="ref" reference="holder"}, and Theorem [Theorem 18](#FHS){reference-type="ref" reference="FHS"}, we have $$\begin{split} \int_{\mathbb{H}^{n}}&\frac{|x|^{-\frac{2\beta}{2^{*}_{\beta}}}|u(x)|^{2}}{\||\cdot|^{-\frac{\beta}{2^{*}_{\beta}}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|x|^{-\frac{2\beta}{2^{*}_{\beta}}}|u(x)|^{2}}{\||\cdot|^{-\frac{\beta}{2^{*}_{\beta}}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx\leq \frac{q}{q-2}\log\left(\frac{\||\cdot|^{-\frac{\beta}{2^{*}_{\beta}}} u\|^{2}_{L^{2^{*}_{\beta}}(\mathbb{H}^{n})}}{\||\cdot|^{-\frac{\beta}{2^{*}_{\beta}}} u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right)\\& \leq \frac{Q-\beta}{2s-\beta}\log\left(\left(C^{\frac{\beta}{2s}}_{BH,s}(C_{B,s}\|U_{s}\|_{\text{op}})^{\frac{n+1}{n+1-s}\frac{2s-\beta}{2s}}\right)^{\frac{2}{2^{*}_{\beta}}}\frac{\langle \mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}}{\||\cdot|^{-\frac{\beta}{2^{*}_{\beta}}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right), \end{split}$$ completing the proof. ◻ **Remark 20**. We note that inequality ([\[LogwSobolev2\]](#LogwSobolev2){reference-type="ref" reference="LogwSobolev2"}) for $\beta=0$ gives the logarithmic Sobolev inequality ([\[LogSobolev2\]](#LogSobolev2){reference-type="ref" reference="LogSobolev2"}). However, the logarithmic Hardy inequality cannot be deduced from ([\[LogwSobolev2\]](#LogwSobolev2){reference-type="ref" reference="LogwSobolev2"}) since then we would need to consider the forbidden, by the logarithmic Hölder inequality ([\[holdernn\]](#holdernn){reference-type="ref" reference="holdernn"}), case where $p=q$. In the next theorem, we develop a different line of arguments to prove the logarithmic Hardy inequality on $\mathbb{H}^{n}$. In the Euclidean setting, the logarithmic first order ($s=1$) Hardy inequality was obtained in [@DDFT10]. **Theorem 21**. *Let $s\in(0,1]$, be such that $0\leq \beta<2s<Q.$ Then, the log-Hardy inequality on $\mathbb{H}^{n}$ reads as follows* *$$\begin{split} \int_{\mathbb{H}^{n}}&\frac{\frac{|u(x)|^{2}}{|x|^{2s-\beta}}}{\left\|\frac{u}{|\cdot|^{\frac{2s-\beta}{2}}}\right\|^{2}_{L^{2}(\mathbb{H}^{n})}}\log\left(\frac{|x|^{(Q-2s)(1-\frac{\beta}{2s-\beta})}|u(x)|^{2}}{\left\|\frac{u}{|\cdot|^{\frac{2s-\beta}{2}}}\right\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right) dx\\& \leq \frac{Q-\beta}{2s-\beta}\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx\log\left(C_{Lw,s}\frac{\langle\mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}}{\left\|\frac{u}{|\cdot|^{\frac{2s-\beta}{2}}}\right\|^{2}_{L^{2}(\mathbb{H}^{n})}}\right), \end{split}$$ where we have set $$C_{Lw,s}=\left(C^{\frac{\beta}{2s}}_{BH,s}\left(C_{B,s}\|U_{s}\|_{\text{op}}\right)^{\frac{n+1}{n+1-s}\frac{2s-\beta}{2s}}\right)^{\frac{2}{2^{*}_{\beta}}},\,\,\,2^{*}_{\beta}=\frac{2(Q-\beta)}{Q-2s}.$$ In particular, for all $u$ such that $\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx=1$, we have $$\label{weloghar1non} \int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{|x|^{2s-\beta}}\log\left(|x|^{(Q-2s)(1-\frac{\beta}{2s-\beta})}|u(x)|^{2}\right) dx \leq \frac{Q-\beta}{2s-\beta}\log\left(C_{Lw,s}\langle\mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}\right).$$* *Proof.* Let us choose $q\in(2,2_{\beta}^{*})$, where $2^{*}_{\beta}=\frac{2(Q-\beta)}{Q-2s}$. If we set $\theta=\frac{2(2^{*}_{\beta}-q)}{2^{*}_{\beta}-2}$, then we have $\theta\in(0,2)$. Let us make some preparatory computations: $$\begin{gathered} \label{wevyrtheta} \theta=\frac{2(2^{*}_{\beta}-q)}{2^{*}_{\beta}-2}=\frac{2\left(\frac{2(Q-\beta)}{Q-2s}-q\right)}{\frac{2(Q-\beta)}{Q-2s}-2} \\ =\frac{\frac{2}{Q-2s}\left(2(Q-\beta)-q(Q-2s)\right)}{\frac{1}{Q-2s}(2(Q-\beta)-2Q+4s)}=\frac{2(Q-\beta)}{2s-\beta}-\frac{(Q-2s)q}{2s-\beta},\end{gathered}$$ $$\label{wep(qth)} 2\frac{q-\theta}{2-\theta}=2\frac{q-2\frac{2^{*}_{\beta}-q}{2^{*}_{\beta}-2}}{2-2\frac{2^{*}_{\beta}-q}{2^{*}_{\beta}-2}}=\frac{q(2^{*}_{\beta}-2)-2(2^{*}_{\beta}-q)}{2^{*}_{\beta}-2-2^{*}_{\beta}+q}=\frac{2^{*}_{\beta}(q-2)}{q-2}=2^{*}_{\beta},$$ $$\label{weptp} \frac{2-\theta}{2}=1-\frac{\theta}{2}=\frac{q-2}{2^{*}_{\beta}-2},$$ and $$\label{hbeta} -\frac{\beta \theta}{2}=\beta\frac{2-\theta}{2}-\beta.$$ By using Hölder's inequality ([\[holder.gen\]](#holder.gen){reference-type="ref" reference="holder.gen"}) with $\frac{\theta}{2}+\frac{2-\theta}{2}=1$ and the above calculations we get, $$\begin{split} \int_{\mathbb{H}^{n}}\frac{|u(x)|^{q}}{|x|^{Q-\beta-\frac{q}{2}(Q-2s)}}dx &=\int_{\mathbb{H}^{n}}\frac{|u(x)|^{\theta}}{|x|^{s\theta }}\frac{|u(x)|^{q-\theta}}{|x|^{Q-\beta-\frac{q}{2}(Q-2s)-\theta s}}dx\\& \stackrel{(\ref{wevyrtheta})}=\int_{\mathbb{H}^{n}}\frac{|u(x)|^{\theta}}{|x|^{\theta s}}\frac{|u(x)|^{q-\theta}}{|x|^{-\frac{\theta \beta}{2}}}dx\\& \stackrel{(\ref{hbeta})}= \int_{\mathbb{H}^{n}}\frac{|u(x)|^{\theta}}{|x|^{\theta s}}\frac{|u(x)|^{q-\theta}}{|x|^{\beta\frac{2-\theta}{2}-\beta}}dx\\& =\int_{\mathbb{H}^{n}}\frac{|u(x)|^{\theta}}{|x|^{\theta s-\beta}}\frac{|u(x)|^{q-\theta}}{|x|^{\beta\frac{2-\theta}{2}}}dx\\& \leq \left(\int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{|x|^{2s-\frac{2\beta }{\theta}}}dx\right)^{\frac{\theta}{2}}\left(\int_{\mathbb{H}^{n}}\frac{|u(x)|^{\frac{2(q-\theta)}{2-\theta}}}{|x|^{\beta}}dx\right)^{\frac{2-\theta}{2}}\\& \stackrel{(\ref{wep(qth)}),(\ref{weptp})}=\left(\int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-q}}}dx\right)^{\frac{2^{*}_{\beta}-q}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u(x)|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{q-2}{2^{*}_{\beta}-2}}, \end{split}$$ also, in the last line we used $\theta=\frac{2(2^{*}_{\beta}-q)}{2^{*}_{\beta}-2}.$ If for $r>0$ we set $q=2+r$ then the above inequalities can be summarised as follows: $$\begin{gathered} \label{wesrnon} \int_{\mathbb{H}^{n}}\frac{|u(x)|^{r+2}}{|x|^{Q-\beta-\frac{(r+2)}{2}(Q-2s)}}dx \\ \leq \left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}dx\right)^{\frac{2^{*}_{\beta}-(r+2)}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}.\end{gathered}$$ On the other hand, when $r \rightarrow 0$, inequality ([\[wesrnon\]](#wesrnon){reference-type="ref" reference="wesrnon"}) is the equality $$\label{webrnon} \begin{split} \int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{Q-\beta-\frac{2}{2}(Q-2s)}}dx& =\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-2}}}dx\right)^{\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{2-2}{2^{*}_{\beta}-2}}. \end{split}$$ Hence using ([\[wesrnon\]](#wesrnon){reference-type="ref" reference="wesrnon"}) and ([\[webrnon\]](#webrnon){reference-type="ref" reference="webrnon"}) we get $$\label{welogosn2non} \begin{split} &\lim_{r\rightarrow 0}\frac{1}{r}\int_{\mathbb{H}^{n}}\left(\frac{|u|^{r+2}}{|x|^{Q-\beta-\frac{(r+2)}{2}(Q-2s)}}-\frac{|u|^{2}}{|x|^{Q-\beta-\frac{2}{2}(Q-2s)}}\right)dx\\& \leq \lim_{r\rightarrow 0}\frac{1}{r}\Biggl{[}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}dx\right)^{\frac{2^{*}_{\beta}-(r+2)}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}\\& -\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-2}}}dx\right)^{\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{2-2}{2^{*}_{\beta}-2}}\Biggl{]}. \end{split}$$ Let us now compute the left-hand side of the inequality ([\[welogosn2non\]](#welogosn2non){reference-type="ref" reference="welogosn2non"}) with the use of the l'Hôpital rule in the variable $r$: $$\label{otkrlevchnon} \begin{split} \lim_{r\rightarrow 0}\frac{1}{r}\left(\frac{|u|^{r+2}}{|x|^{Q-\beta-\frac{(r+2)}{2}(Q-2s)}}-\frac{|u|^{2}}{|x|^{Q-\beta-\frac{2}{2}(Q-2s)}}\right)&=\lim_{r\rightarrow 0}\frac{d}{dr}\frac{|u|^{r+2}}{|x|^{Q-\beta-\frac{(r+2)}{2}(Q-2s)}}\\& =\frac{|u|^{2}\log(|u|^{2}|x|^{Q-2s})}{2|x|^{2s-\beta}}. \end{split}$$ For the right-hand side of ([\[welogosn2non\]](#welogosn2non){reference-type="ref" reference="welogosn2non"}), we define the auxiliary function $z(r):=(f(r))^{g(r)}$, where $$f(r)=\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}dx,$$ and $$g(r)=\frac{2^{*}_{\beta}-(r+2)}{2^{*}_{\beta}-2}.$$ The derivatives of the functions $f,g$ with respect to the variable $r$ are as follows $$\label{derfrnon} \begin{split} \frac{df(r)}{dr}&=\frac{d}{dr}\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}dx\\& =\frac{\beta(2^{*}_{\beta}-2)}{(2^{*}_{\beta}-(r+2))^{2}}\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}\log(|x|)dx, \end{split}$$ and $$\label{dergrnon} \frac{dg(r)}{dr}=\frac{d}{dr}\frac{2^{*}_{\beta}-(r+2)}{2^{*}_{\beta}-2}=-\frac{1}{2^{*}_{\beta}-2}.$$ Hence the derivative of $z(r)$ is given by $$\label{compzrnon} \begin{split} \frac{dz(r)}{dr}&=(f(r))^{g(r)}\left(\frac{dg(r)}{dr}\log(f(r))+\frac{g(r)\frac{df(r)}{dr}}{f(r)}\right)\\& \stackrel{(\ref{derfrnon}),(\ref{dergrnon})}=z(r)\Biggl{(}-\frac{1}{2^{*}_{\beta}-2}\log\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}dx\right)\\& +\frac{\beta}{2^{*}_{\beta}-(r+2)}\frac{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}\log(|x|)dx}{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}dx}\Biggl{)}. \end{split}$$ We also have $$\label{compharsobnon} \begin{split} \frac{d}{dr}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}&= \frac{1}{2^{*}_{\beta}-2}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}\\& \times\log\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right). \end{split}$$ Using the above derivatives and l'Hôpital's rule, we have $$\begin{split} &\lim_{r\rightarrow 0}\frac{1}{r}\Biggl{[}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}dx\right)^{\frac{2^{*}_{\beta}-(r+2)}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}\\& -\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-2}}}dx\right)^{\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{p-p}{2^{*}_{\beta}-p}}\Biggl{]}\\& =\lim_{r\rightarrow 0}\frac{d}{dr}\left[\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}dx\right)^{\frac{2^{*}_{\beta}-(r+2)}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}\right] \end{split}$$ $$\begin{split} & =\lim_{r\rightarrow 0}\frac{d}{dr}\left(z(r)\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}\right)\\& =\lim_{r\rightarrow 0}\left[\frac{dz(r)}{dr}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}+z(r)\frac{d}{dr}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}\right]\\& \stackrel{(\ref{compzrnon}),(\ref{compharsobnon})}=\frac{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}{2^{*}_{\beta}-2}\Biggl{[}-\log\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx\right)\\& +\beta\frac{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}\log(|x|)dx}{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}+\log\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)\Biggl{]}\\& =\frac{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}{2^{*}_{\beta}-2}\Biggl{[}\beta\frac{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}\log(|x|)dx}{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}+\log\frac{\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)}{\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx\right)}\Biggl{]}\\& =\frac{I_{1}}{2^{*}_{\beta}-2}\left(\beta\frac{I_{3}}{I_{1}}+\log\frac{I_{2}}{I_{1}}\right), \end{split}$$ where $I_1$, $I_2$ and $I_3$ stand for the next integrals $$I_{1}=\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx,$$ $$I_{2}=\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx,$$ and $$I_{3}=\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}\log(|x|)dx.$$ Therefore, we also have $$\begin{split} &\lim_{r\rightarrow 0}\frac{1}{r}\Biggl{[}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-(r+2)}}}dx\right)^{\frac{2^{*}_{\beta}-(r+2)}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{(r+2)-2}{2^{*}_{\beta}-2}}\\& -\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-2}}}dx\right)^{\frac{2^{*}_{\beta}-2}{2^{*}_{\beta}-2}}\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{2-2}{2^{*}_{\beta}-2}}\Biggl{]} \end{split}$$ $$\label{otkrpravchnon} \begin{split} &=\frac{I_{1}}{2^{*}_{\beta}-2}\left(\beta\frac{I_{3}}{I_{1}}+\log\frac{I_{2}}{I_{1}}\right)\\& =\frac{\beta}{2^{*}_{\beta}-2}I_{3}+\frac{I_{1}}{2^{*}_{\beta}-2}\frac{22^{*}_{\beta}}{22^{*}_{\beta}}\log\frac{I_{2}}{I_{1}}\\& =\frac{\beta}{2^{*}_{\beta}-2}I_{3}+\frac{I_{1}2^{*}_{\beta}}{2(2^{*}_{\beta}-2)}\log\frac{I^{\frac{2}{2^{*}_{\beta}}}_{2}}{I_{1}^{1-1+\frac{2}{2^{*}_{\beta}}}}\\& =\frac{\beta}{2^{*}_{\beta}-2}I_{3}+\frac{I_{1}2^{*}_{\beta}}{2(2^{*}_{\beta}-2)}\log\frac{I^{\frac{2}{2^{*}_{\beta}}}_{2}}{I_{1}}-\frac{I_{1}2^{*}_{\beta}}{2(2^{*}_{\beta}-2)}\log I_{1}^{-1+\frac{2}{2^{*}_{\beta}}}\\& =\frac{\beta}{2^{*}_{\beta}-2}I_{3}+\frac{I_{1}2^{*}_{\beta}}{2(2^{*}_{\beta}-2)}\log\frac{I^{\frac{2}{2^{*}_{\beta}}}_{2}}{I_{1}}+\frac{I_{1}}{2}\log I_{1}. \end{split}$$ Using the expression ([\[otkrlevchnon\]](#otkrlevchnon){reference-type="ref" reference="otkrlevchnon"}) for the left-hand side of ([\[welogosn2non\]](#welogosn2non){reference-type="ref" reference="welogosn2non"}) and the expression ([\[otkrpravchnon\]](#otkrpravchnon){reference-type="ref" reference="otkrpravchnon"}) for the right-hand side of it we get $$\label{predposnon} \begin{split} \int_{\mathbb{H}^{n}}\frac{|u|^{2}\log(|u|^{2}|x|^{Q-2s})}{|x|^{2s-\beta}}&dx\leq \frac{2\beta }{2^{*}_{\beta}-2}I_{3}+\frac{I_{1}2^{*}_{\beta}}{2^{*}_{\beta}-2}\log\frac{I^{\frac{2}{2^{*}_{\beta}}}_{2}}{I_{1}}+I_{1}\log I_{1}\\& =\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}\log\left(|x|^{\frac{\beta(Q-2s)}{2s-\beta}}\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|y|^{2s-\beta}}dy\right)dx\\& +\frac{(Q-\beta)\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}{2s-\beta}\log \left(\frac{\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{2}{2^{*}_{\beta}}}}{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}\right)\,, \end{split}$$ since $\frac{2^{*}_{\beta}}{2(2^{*}_{\beta}-2)}=\frac{Q-\beta}{2s-\beta}$, and so $$\frac{I_{1}2^{*}_{\beta}}{2(2^{*}_{\beta}-2)}\log\frac{I^{\frac{2}{2^{*}_{\beta}}}_{2}}{I_{1}}=\frac{(Q-\beta)}{2s-\beta}\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx \log \left(\frac{\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{2}{2^{*}_{\beta}}}}{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}\right)\,,$$ while also, since $\frac{\beta }{2^{*}_{\beta}-2}= \frac{\beta(Q-2s)}{2s-\beta}$, it is easy to check that $$\frac{2\beta }{2^{*}_{\beta}-2}I_{3}+I_{1}\log I_{1}=\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}\log\left(|x|^{\frac{\beta(Q-2s)}{2s-\beta}}\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx\right)dx\,.$$ From Theorem [Theorem 18](#FHS){reference-type="ref" reference="FHS"}, if $0\leq \beta<2s<Q$, we have the Hardy-Sobolev inequality with $2^{*}_{\beta}=\frac{2(Q-\beta)}{Q-2s}$ in the following form: $$\left\|\frac{u}{|x|^{\frac{\beta}{2^{*}_{\beta}}}}\right\|_{L^{2^{*}_{\beta}}(\mathbb{H}^{n})}\leq C^{\frac{1}{2}}_{Lw,s}\langle\mathcal{L}^{s}u,u\rangle^{\frac{1}{2}}_{L^{2}(\mathbb{H}^{n})}\,,$$ with $C_{Lw,s}=\left(C^{\frac{\beta}{2s}}_{BH,s}\left(C_{B,s}\|U_{s}\|_{\text{op}}\right)^{\frac{n+1}{n+1-s}\frac{2s-\beta}{2s}}\right)^{\frac{2}{2^{*}_{\beta}}}$. Finally, by using this in ([\[predposnon\]](#predposnon){reference-type="ref" reference="predposnon"}), we arrive at $$\begin{split} &\int_{\mathbb{H}^{n}}\frac{\frac{|u|^{2}}{|x|^{2s-\beta}}}{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}\log\left(\frac{|u|^{2}|x|^{(Q-2s)(1-\frac{\beta}{2s-\beta})}}{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}\right)dx \\ &\leq\frac{Q-\beta}{2s-\beta} \int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx\log\frac{\left(\int_{\mathbb{H}^{n}}\frac{|u|^{2^{*}_{\beta}}}{|x|^{\beta}}dx\right)^{\frac{2}{2^{*}_{\beta}}}}{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}\\& \leq\frac{Q-\beta}{2s-\beta}\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx\log\left(C_{Lw,s}\frac{\langle\mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}}{\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{2s-\beta}}dx}\right), \end{split}$$ where for the first inequality we have subtracted the the first term of the right-hand side of ([\[predposnon\]](#predposnon){reference-type="ref" reference="predposnon"}) from the left-hand side of ([\[predposnon\]](#predposnon){reference-type="ref" reference="predposnon"}), and the proof is complete. ◻ **Remark 22**. By taking $\beta=s$ in ([\[weloghar1non\]](#weloghar1non){reference-type="ref" reference="weloghar1non"}), we obtain an interesting inequality $$\label{weloghar1non-nw} \int_{\mathbb{H}^{n}}\frac{|u(x)|^{2}}{|x|^{s}}\log\left(|u(x)|\right) dx \leq \frac{Q-s}{2s}\log\left(C_{Lw,s}\langle\mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}\right),$$ for all $u$ such that $\int_{\mathbb{H}^{n}}\frac{|u|^{2}}{|x|^{s}}dx=1$. Inequality ([\[weloghar1non-nw\]](#weloghar1non-nw){reference-type="ref" reference="weloghar1non-nw"}) is a combination of radial and logarithmic weights in the usual Hardy inequality. # Nash inequality on $\mathbb{H}^{n}$ and applications On the Euclidean space it is proved, see [@Bec98], that the Nash inequality is equivalent to the $L^2$-log-Sobolev inequality. In this section we show that the $L^2$-log-Sobolev inequality on $\mathbb{H}^{n}$ implies the Nash inequality on $\mathbb{H}^{n}$ with an explicit constant. **Theorem 23**. *Let $0< s\leq 1.$ Then, Nash's inequality on $\mathbb{H}^{n}$ with respect to the (modified) fractional sub-Laplacian reads as follows $$\label{Nash.g} \|u\|_{L^2(\mathbb{H}^{n})}^{2+\frac{4s}{Q}} \leq C_{B,s} \|u\|_{L^1(\mathbb{H}^{n})}^{\frac{4s}{Q}}\langle\mathcal{L}_{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}\,.$$ Alternatively, Nash's inequality on $\mathbb{H}^{n}$ with respect to fractional powers of the sub-Laplacian is given by $$\label{Nash.g1} \|u\|_{L^2(\mathbb{H}^{n})}^{2+\frac{4s}{Q}} \leq C_{B,s}\|U\|_{\text{op}} \|u\|_{L^1(\mathbb{H}^{n})}^{\frac{4s}{Q}}\langle\mathcal{L}^{s}u,u\rangle_{L^{2}(\mathbb{H}^{n})}\,,$$ where $C_{B,s}$ is defined in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}).* *Proof.* Using Jensen's inequality for the function $\theta(u)=\log \frac{1}{u}$ and for the probability measure $|u|^2 \|u\|^{-2}_{L^2(\mathbb{H}^{n})}\,dx$ we get $$\begin{aligned} \label{Nash1} \log \left(\frac{\|u\|^{2}_{L^2(\mathbb{H}^{n})}}{\|u\|_{L^1(\mathbb{H}^{n})}} \right) & = & \theta \left( \frac{\|u\|_{L^1(\mathbb{H}^{n})}}{\|u\|^{2}_{L^2(\mathbb{H}^{n})}}\right)=\theta \left( \int_{\mathbb{H}^{n}}\frac{1}{|u|}|u|^2 \|u\|^{-2}_{L^2(\mathbb{H}^{n})}\,dx\right)\nonumber\\ & \leq & \int_{\mathbb{H}^{n}} \theta \left( \frac{1}{|u|}\right) |u|^2 \|u\|^{-2}_{L^2(\mathbb{H}^{n})}\,dx=\int_{\mathbb{H}^{n}} \frac{|u|^2}{\|u\|^{2}_{L^2(\mathbb{H}^{n})}}\log |u| \,dx\,. \end{aligned}$$ An application of the log-Sobolev inequality as in Theorem [Theorem 6](#thm:Logsob){reference-type="ref" reference="thm:Logsob"} yields $$\label{Nash2} \int_{\mathbb{H}^{n}} \frac{|u|^2}{\|u\|^{2}_{L^2(\mathbb{H}^{n})}}\log |u| \,dx \leq \log \|u\|_{L^2(\mathbb{H}^{n})}+\frac{Q}{4s} \log \left( C_{B,s} \frac{\langle\mathcal{L}_{s}u, u\rangle_{L^{2}(\mathbb{H}^{n})}}{\|u\|^{2}_{L^2(\mathbb{H}^{n})}}\right)\,.$$ Now using the properties of the logarithm, summing up inequalities ([\[Nash1\]](#Nash1){reference-type="ref" reference="Nash1"}) and ([\[Nash2\]](#Nash2){reference-type="ref" reference="Nash2"}), and rearranging the terms of this sum, we get $$\left(1+\frac{Q}{2s} \right) \log \|u\|_{L^2(\mathbb{H}^{n})} \leq \frac{Q}{4s} \log \left(C_{B,s} \|u\|_{L^1(\mathbb{H}^{n})}^{\frac{4s}{Q}}\langle\mathcal{L}_{s}u, u\rangle_{L^{2}(\mathbb{H}^{n})} \right)\,,$$ which is equivalent to ([\[Nash.g\]](#Nash.g){reference-type="ref" reference="Nash.g"}). Now, ([\[Nash.g\]](#Nash.g){reference-type="ref" reference="Nash.g"}) implies, as before, ([\[Nash.g1\]](#Nash.g1){reference-type="ref" reference="Nash.g1"}) and the proof is complete. ◻ **Corollary 24**. *For $C_{B,1}$ as in ([\[bestintsob\]](#bestintsob){reference-type="ref" reference="bestintsob"}) we get the following standard form of Nash's inequality on $\mathbb{H}^{n}$ $$\label{Nash.hor} \|u\|_{L^2(\mathbb{H}^{n})}^{2+\frac{4}{Q}} \leq C_{B,1} \|u\|_{L^1(\mathbb{H}^{n})}^{\frac{4}{Q}}\|\nabla_{\mathbb{H}^{n}}u\|^{2}_{L^{2}(\mathbb{H}^{n})}\,.$$* *Proof.* The proof follows immediately from ([\[Nash.g\]](#Nash.g){reference-type="ref" reference="Nash.g"}) for $s=1$ and the equality $$\langle\mathcal{L}u,u\rangle^\frac{1}{2}_{L^{2}(\mathbb{H}^{n})}=\|\mathcal{L}^{1/2} u\|_{L^2(\mathbb{H}^{n})}=\||\nabla_{\mathbb{H}^{n}}| u\|_{L^2(\mathbb{H}^{n})}=\|\nabla_{\mathbb{H}^{n}}u\|_{L^2(\mathbb{H}^{n})}\,.$$ The proof is complete. ◻ Standard applications of Nash's inequality relate to Markovian semigroups, see e.g. [@VSCC93 Section II.5]. Below we give a short application of Nash's inequality to estimate the time-decay rate for the solution to the heat equation on $\mathbb{H}^{n}$ with respect to $\mathcal{L}$. **Corollary 25**. *Let $f_0 \geq 0$ be such that $f_0\in L^1(\mathbb{H}^{n})\cap L^2(\mathbb{H}^{n})$. Then the solution $f$ to the linear heat equation on $\mathbb{H}^{n}$ $$\label{heat.eq} \partial_t f+\mathcal{L} f=0,\quad f(0,x)=f_0(x),$$ satisfies the following time-decay estimate for all $t \geq 0$ $$\label{heatest} \|f(t,\cdot)\|_{L^2(\mathbb{H}^{n})}\leq \left( \|f_0\|_{L^2(\mathbb{H}^{n})}^{-\frac{4}{Q}} +\frac{4}{QC_{B,1}}\|f_0\|_{L^1(\mathbb{H}^{n})}^{-\frac{4}{Q}}t\right)^{-\frac{Q}{4}}\,,$$ where $C_{B,1}$ is given in ([\[bestintsob\]](#bestintsob){reference-type="ref" reference="bestintsob"}).* *Proof.* Since $f_0\geq 0$, the positivity of the heat kernel (see e.g. [@VSCC93 p. 48]) $h_t$ implies that $f(t,x)\geq 0$, as well. Now, the mass conservation implies the equality of the $L^1$-norms $$\label{L1norm} \int_{\mathbb{H}^{n}} f(t,x) dx=\int_{\mathbb{H}^{n}}\int_{\mathbb{H}^{n}} h_t(x y^{-1})f_0(y) dydx=\int_{\mathbb{H}^{n}} f_0(y) dy,$$ where we applied Fubini's theorem and the fact that $\|h_t\|_{L^1(\mathbb{H}^{n})}=1.$ Now, multiplying by $f$ the heat equation ([\[heat.eq\]](#heat.eq){reference-type="ref" reference="heat.eq"}), and integrating over $\mathbb{H}^{n}$ we get $$\langle \partial_t f(t,\cdot), f(t,\cdot) \rangle_{L^2(\mathbb{H}^{n})}+\langle \mathcal{L} f(t,\cdot), f(t,\cdot) \rangle_{L^2(\mathbb{H}^{n})}=0\,,$$ or, equivalently, $$\frac{d}{dt}\|f(t,\cdot)\|^{2}_{L^2(\mathbb{H}^{n})}=-2\|\nabla_{\mathbb{H}^{n}} f(t,\cdot)\|^{2}_{L^2(\mathbb{H}^{n})}\,.$$ If we denote by $y(t)$ the norm $y(t):=\|f(t,\cdot)\|^{2}_{L^2(\mathbb{H}^{n})}$, by ([\[Nash.hor\]](#Nash.hor){reference-type="ref" reference="Nash.hor"}) we get $$\begin{split} y'(t)&=\frac{d}{dt}\|f(t,\cdot)\|^{2}_{L^2(\mathbb{H}^{n})}\\& =-2\|\nabla_{\mathbb{H}^{n}} f(t,\cdot)\|^{2}_{L^2(\mathbb{H}^{n})}\\& \stackrel{(\ref{Nash.hor})}\leq -2C^{-1}_{B,1}\|f(t,\cdot)\|^{2+\frac{4}{Q}}_{L^{2}(\mathbb{H}^{n})}\|f(t,\cdot)\|^{\frac{4}{Q}}_{L^{1}(\mathbb{H}^{n})}\\& \stackrel{(\ref{L1norm})}=-2C^{-1}_{B,1}y^{1+\frac{2}{Q}}(t)\|f_{0}\|^{\frac{4}{Q}}_{L^{1}(\mathbb{H}^{n})}, \end{split}$$ which, after integration on $t \geq 0$, gives the estimate $$\|f(t,\cdot)\|_{L^2(\mathbb{H}^{n})}\leq \left(\|f_0\|_{L^2(\mathbb{H}^{n})}^{-\frac{4}{Q}}+\frac{4}{QC_{B,1}}\|f_0\|_{L^1(\mathbb{H}^{n})}^{-\frac{Q}{4}}t \right)^{-\frac{Q}{4}}\,,$$ and this finishes the proof. ◻ # Poincaré inequality on $\mathbb{H}^{n}$ In the section, we prove several versions of the Poincaré inequality on $\mathbb{H}^{n}$. In 1989 Beckner [@Bec89] proved the following generalised Poincaré inequality on $\mathbb{R}^n$: $$\label{Poi.bec} \frac{1}{2-p}\left[\int_{\mathbb{R}^{n}}f^2\,d\mu-\left(\int_{\mathbb{R}^{n}}|f|^p\,d\mu \right)^{\frac{2}{p}} \right]\leq \int_{\mathbb{R}^{n}}|\nabla f|^2\,d\mu\,, \quad p \geq 1\,,$$ for $d\mu$ the Gaussian measure on $\mathbb{R}^n$. Recall the classical Poincaré inequality on $\mathbb{R}^{n}$: $$\label{Poi} \int_{\mathbb{R}^{n}}|f-\mu(f)|^{q}\,d\mu \leq c_o \mu(|\nabla f|^q)\,, \quad q \geq 1\,,$$ where we have denoted $\mu(f):=\int_{\mathbb{R}^{n}}f\,d\mu$. It is easy to check that the left-hand side of the generalised Poincaré inequality ([\[Poi.bec\]](#Poi.bec){reference-type="ref" reference="Poi.bec"}) for $p=1$ equals the left-hand side of the classical Poincaré inequality ([\[Poi\]](#Poi){reference-type="ref" reference="Poi"}) for $q=2$. This is true in more generality; indeed if $\nu$ is a probability measure on any measure space $\mathbb{X}$, then we have $$\int_{\mathbb{X}}f^2\,d\nu-\left(\int_{\mathbb{X}}f\,dx\right)^2=\int_{\mathbb{X}}(f-\nu(f))^2\,d\nu\,.$$ **Theorem 26**. *Let $\mu$ be the semi-Gaussian measure as in Theorem [Theorem 8](#semi-g){reference-type="ref" reference="semi-g"}. Then, for $p\leq 2$, the following generalised Poincaré inequality with respect to $\mu$ holds true on $\mathbb{H}^{n}$: $$\label{Poin.H} \int_{\mathbb{H}^{n}}|g|^2\,d\mu-\left( \int_{\mathbb{H}^{n}}|g|^{p}\,d\mu\right)^{\frac{2}{p}} \leq \frac{2(2-p)}{p}\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}g|^2\,d\mu\,.$$ In particular, for $p=1$ we have $$\int_{\mathbb{H}^{n}}|g|^2\,d\mu -\left(\int_{\mathbb{H}^{n}}|g|\,d\mu \right)^2 \leq 2\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}g|^2\,d\mu\,.$$* *Proof.* For some suitable $g$ we consider the function $b=b(q):=q \log \left( \int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,d\mu \right)$ for $q> 1$. We denote by $I$ the integral $$I(q)=\int_{\mathbb{H}^{n}} |g|^{\frac{2}{q}}\,d\mu\,,$$ and hence, we rewrite $b(q)$ as $$b(q)=q\log I(q).$$ Let us compute derivatives of $I(q)$. Then, we have $$I'(q)=\int_{\mathbb{H}^{n}}\log |g||g|^{\frac{2}{q}}\left(-\frac{2}{q^2}\right)\,d\mu,$$ and $$I''(q)=\int_{\mathbb{H}^{n}}\left((\log|g|)^{2}|g|^{\frac{2}{q}}\left(-\frac{2}{q^2}\right)^{2}+\log |g||g|^{\frac{2}{q}}\left(\frac{4}{q^3}\right)\right)\,d\mu.$$ Then, by using the last facts, we have $$b'(q)=\log I(q)+\frac{q I'(q)}{I(q)}\,,$$ and $$\begin{split} b''(q)&=\frac{I(q)(2I'(q)+qI''(q))-q(I'(q))^{2}}{I^{2}(q)}\\& =\frac{q}{I^{2}(q)}\Biggl{[}\left(\int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}d\mu\right)\left(\int_{\mathbb{H}^{n}}(\log|g|)^{2}|g|^{\frac{2}{q}}\left(-\frac{2}{q^2}\right)^{2}d\mu\right)\\& -\left(\int_{\mathbb{H}^{n}}\log |g||g|^{\frac{2}{q}}\left(-\frac{2}{q^2}\right)\,d\mu\right)^{2}\Biggl{]}. \end{split}$$ It is then clear that $b$ is a convex function since by the Cauchy--Schwarz inequality we have $$\left(\int_{\mathbb{H}^{n}}(\log |g|)^2 |g|^{\frac{2}{q}}\left(-\frac{2}{q^2}\right)^2\,d\mu \right)\left( \int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,d\mu\right) \geq \left(\int_{\mathbb{H}^{n}}\log |g||g|^{\frac{2}{q}}\left(-\frac{2}{q^2}\right)\,d\mu \right)^2\,,$$ which implies that $b'' \geq 0$. The convexity of $b$ implies in turn the convexity of the function $q \mapsto e^{b(q)}$. Now let us recall the following characterisation of the convexity of functions of one variable: the function $f$ is convex if and only if the function $$R(x,y)=\frac{f(y)-f(x)}{y-x}$$ is monotonically non-decreasing in $x$ for fixed $y$, and vice versa. Hence by the latter characterisation we have that the function $$\phi(q):= \frac{e^{b(1)}-e^{b(q)}}{1-q}\,,$$ is monotonically non-increasing for $q \geq 1$. Consequently, we get $$\label{phi.decreasing} \phi(q)\leq \lim_{q \rightarrow 1}\phi(q)\,.$$ We note that $$\label{phi(q)} \phi(q)=\frac{\int_{\mathbb{H}^{n}}|g|^2\,d\mu-\left( \int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,d\mu\right)^q}{1-q}\,.$$ For the computation of the limit $\lim\limits_{q \rightarrow 1}\phi(q)$ we will use the l'Hôpital rule. We have $$\begin{aligned} \frac{d}{dq}\left(\int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,d\mu \right)^q & = & \left(\int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,d\mu \right)^q \frac{d}{dq} \left[q \log \left(\int_{\mathbb{H}^{n}}|g|^{2/q}\,d\mu \right) \right] \\ & = & \left(\int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,d\mu \right)^q \left[\log \left(\int_{\mathbb{H}^{n}}|g|^{2/q}\,d\mu \right)-\frac{2}{q^2} \log \int_{\mathbb{H}^{n}}|g|\,d\mu \right], \end{aligned}$$ which implies $$\begin{aligned} \lim_{q \rightarrow 1}\frac{d}{dq}\left(\int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,d\mu \right)^q & = & 2 \int_{\mathbb{H}^{n}}\ |g|^2 \log|g|\,d\mu \end{aligned}$$ and so $\lim\limits_{q \rightarrow 1}\phi(q)=2 \int_{\mathbb{H}^{n}}\ |g|^2 \log|g|\,d\mu$. Combining this together with ([\[phi.decreasing\]](#phi.decreasing){reference-type="ref" reference="phi.decreasing"}) and ([\[phi(q)\]](#phi(q)){reference-type="ref" reference="phi(q)"}) we get $$\int_{\mathbb{H}^{n}}|g|^2\,d\mu-\left( \int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,d\mu\right)^q \leq 2(q-1) \int_{\mathbb{H}^{n}}|g|^2 \log |g|\,d\mu\,,$$ or after replacing $g$ by $\frac{g}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}}$ $$\label{before.P} \begin{split} &\int_{\mathbb{H}^{n}}\frac{|g|^2}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}^2}\,d\mu-\left(\int_{\mathbb{H}^{n}}\left(\frac{|g|}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}}\right)^{\frac{2}{q}}\,d\mu\right)^{q} \\&\leq 2 (q-1)\int_{\mathbb{H}^{n}}\frac{|g|^2}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}^2} \log \left(\frac{|g|}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}} \right)\,d\mu\,. \end{split}$$ On the other hand, the semi-Gaussian inequality ([\[gaus.log.sob\]](#gaus.log.sob){reference-type="ref" reference="gaus.log.sob"}) gives $$\label{semi=g-rescaled} \int_{\mathbb{H}^{n}}\frac{|g|^2}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}^2}\log \frac{|g|}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}}\,d\mu \leq \frac{1}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}^2}\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}g|^2\,d\mu\,.$$ Finally, a combination of ([\[before.P\]](#before.P){reference-type="ref" reference="before.P"}) together with ([\[semi=g-rescaled\]](#semi=g-rescaled){reference-type="ref" reference="semi=g-rescaled"}) gives $$\label{D+semig} \begin{split} & \int_{\mathbb{H}^{n}}\frac{|g|^2}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}^2}\,d\mu-\left(\int_{\mathbb{H}^{n}}\left(\frac{|g|}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}}\right)^{\frac{2}{q}}\,d\mu\right)^q\\& \leq 2 (q-1) \frac{1}{\|g\|_{L^{2}(\mathbb{H}^{n},\mu)}^2}\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}g|^2\,d\mu\,, \end{split}$$ or after simplification $$\label{forq} \int_{\mathbb{H}^{n}}|g|^2d\mu-\left(\int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\right)^q\leq 2 (q-1)\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}g|^2\,d\mu \,,$$ and the proof is complete if one sets $p=\frac{2}{q}$ in ([\[forq\]](#forq){reference-type="ref" reference="forq"}). ◻ **Theorem 27**. *For $q > 1$ the following Poincaré type inequality with respect to the Haar measure on $\mathbb{H}^{n}$ holds true $$\label{Poi.haar.a} \int_{\mathbb{H}^{n}}\frac{|g|^2}{\|g\|_{L^2(\mathbb{H}^{n})}^2}\,dx-\left( \int_{\mathbb{H}^{n}}\left(\frac{|g|}{\|g\|_{L^2(\mathbb{H}^{n})}} \right)^{\frac{2}{q}}\,dx\right)^q \leq \frac{Q(q-1)}{2} \log \left( C_{B,1} \frac{\|\nabla_{\mathbb{H}^{n}}g\|_{L^2(\mathbb{H}^{n})}^{2}}{\|g\|^{2}_{L^2(\mathbb{H}^{n})}}\right)\,,$$ where $C_{B,1}$ is given in ([\[bestintsob\]](#bestintsob){reference-type="ref" reference="bestintsob"}). Additionally, for $s \in (0,1]$ and for $q \geq 1$, we have the following Poincaré type (modified) fractional inequality $$\label{Poi.haar.b} \int_{\mathbb{H}^{n}}\frac{|g|^2}{\|g\|_{L^2(\mathbb{H}^{n})}^2}\,dx-\left( \int_{\mathbb{H}^{n}}\left(\frac{|g|}{\|g\|_{L^2(\mathbb{H}^{n})}} \right)^{\frac{2}{q}}\,dx\right)^q \leq \frac{Q(q-1)}{2s} \log \left( C_{B,s} \frac{\langle \mathcal{L}_sg,g \rangle_{L^2(\mathbb{H}^{n})}}{\|g\|_{L^2(\mathbb{H}^{n})}^{2}}\right)\,,$$ where the constant $C_{B,s}$ is as in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}), as well as the following Poincaré type inequality that involves fractional powers of the sub-Laplacian $\mathcal{L}$ $$\label{Poi.haar.c} \int_{\mathbb{H}^{n}}\frac{|g|^2}{\|g\|_{L^2(\mathbb{H}^{n})}^2}\,dx-\left( \int_{\mathbb{H}^{n}}\left(\frac{|g|}{\|g\|_{L^2(\mathbb{H}^{n})}} \right)^{\frac{2}{q}}\,dx\right)^q \leq \frac{Q(q-1)}{2s} \log \left( C_{B,s}\|U_s\|_{\text{op}} \frac{\langle \mathcal{L}^s g,g \rangle_{L^2(\mathbb{H}^{n})}}{\|g\|_{L^2(\mathbb{H}^{n})}^{2}}\right)\,,$$where the operator norm $\|U_s\|_{\text{op}}$ has been estimated in ([\[est1\]](#est1){reference-type="ref" reference="est1"}).* *Proof.* Recall that in the proof of Theorem [Theorem 26](#thm.poi.H){reference-type="ref" reference="thm.poi.H"} the inequality ([\[before.P\]](#before.P){reference-type="ref" reference="before.P"}) is obtained independently of the choice of the considered measure. Hence, with respect to the Haar measure on $\mathbb{H}^{n}$ we have $$\int_{\mathbb{H}^{n}}|g|^2\,dx-\left( \int_{\mathbb{H}^{n}}|g|^{\frac{2}{q} }\,dx\right)^q \leq 2(q-1) \int_{\mathbb{H}^{n}}|g|^2 \log |g|\,dx\,,$$ or $$\label{Poin.haar.1} \int_{\mathbb{H}^{n}}\frac{|g|^2}{\|g\|_{L^2(\mathbb{H}^{n})}^2}\,dx-\left(\int_{\mathbb{H}^{n}}\left(\frac{|g|}{\|g\|_{L^2(\mathbb{H}^{n})}}\right)^{\frac{2}{q}}\,dx\right)^q\leq 2 (q-1)\int_{\mathbb{H}^{n}}\frac{|g|^2}{\|g\|_{L^2(\mathbb{H}^{n})}^2} \log \left(\frac{|g|}{\|g\|_{L^2(\mathbb{H}^{n})}} \right)\,dx\,.$$ A combination of ([\[Poin.haar.1\]](#Poin.haar.1){reference-type="ref" reference="Poin.haar.1"}) together with the horizontal log-Sobolev inequality ([\[LogSobolevint\]](#LogSobolevint){reference-type="ref" reference="LogSobolevint"}) proves ([\[Poi.haar.a\]](#Poi.haar.a){reference-type="ref" reference="Poi.haar.a"}). For the proof of ([\[Poi.haar.b\]](#Poi.haar.b){reference-type="ref" reference="Poi.haar.b"}) we combine inequality ([\[Poin.haar.1\]](#Poin.haar.1){reference-type="ref" reference="Poin.haar.1"}) and ([\[LogSobolev1\]](#LogSobolev1){reference-type="ref" reference="LogSobolev1"}), while the proof of ([\[Poi.haar.c\]](#Poi.haar.c){reference-type="ref" reference="Poi.haar.c"}) is a combination of ([\[Poin.haar.1\]](#Poin.haar.1){reference-type="ref" reference="Poin.haar.1"}) and ([\[LogSobolev2\]](#LogSobolev2){reference-type="ref" reference="LogSobolev2"}). This concludes the proof of the theorem. ◻ The following result is an immediate consequence of Theorem [Theorem 27](#thm.poi.haar){reference-type="ref" reference="thm.poi.haar"} since for all $x>0$ we have $\log x \leq x-1$. **Corollary 28**. *For $q > 1$ the following Poincaré type inequality with respect to the Haar measure on $\mathbb{H}^{n}$ holds true $$\label{Poi.haar.a3} \int_{\mathbb{H}^{n}}|g|^2\,dx-\left( \int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,dx\right)^q \leq C_{B,1}\frac{Q(q-1)}{2}\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}g|^2\,dx\,,$$ where $C_{B,1}$ is given in ([\[bestintsob\]](#bestintsob){reference-type="ref" reference="bestintsob"}). Additionally, for $s \in (0,1]$ and for $q \geq 1$, we have the following Poincaré type (modified) fractional inequality $$\label{Poi.haar.b3} \int_{\mathbb{H}^{n}}|g|^2\,dx-\left( \int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,dx\right)^q \leq C_{B,s}\frac{Q(q-1)}{2s} \langle \mathcal{L}_sg,g \rangle_{L^2(\mathbb{H}^{n})}\,,$$ where the constant $C_{B,s}$ is as in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}), as well as the following Poincaré type inequality that involves fractional powers of the sub-Laplacian $\mathcal{L}$ $$\label{Poi.haar.c3} \int_{\mathbb{H}^{n}}|g|^2\,dx-\left( \int_{\mathbb{H}^{n}}|g|^{\frac{2}{q}}\,dx\right)^q\leq C_{B,s} \|U_s\|_{\text{op}} \frac{Q(q-1)}{2s} \langle \mathcal{L}^s g,g \rangle_{L^2(\mathbb{H}^{n})}\,,$$where the operator norm $\|U_s\|_{\text{op}}$ has been estimated in ([\[est1\]](#est1){reference-type="ref" reference="est1"}).* **Remark 29**. Let us point out that the inequalities ([\[Poi.haar.a3\]](#Poi.haar.a3){reference-type="ref" reference="Poi.haar.a3"}), ([\[Poi.haar.b3\]](#Poi.haar.b3){reference-type="ref" reference="Poi.haar.b3"}) and ([\[Poi.haar.c3\]](#Poi.haar.c3){reference-type="ref" reference="Poi.haar.c3"}) hold true also when the integration is taken on a ball $B_r(y)$ centered at $y$ with respect to a homogeneous norm on $\mathbb{H}^{n}$. Indeed, can repeat the previous argument for the function $b=b(q):= q \log \left(\int_{B_r(y)}|g|^{\frac{2}{q}} \,dx\right)$, for $q >1$, and get $$\int_{B_r(y)}|g|^2\,dx-\left( \int_{B_r(y)}|g|^{\frac{2}{q}}\,dx\right)^q \leq 2(q-1) \int_{B_r(y)}|g|^2 \log |g|\,dx\,.$$ The latter inequality for $\frac{g}{\|g\|_{L^2(B_r(y))}}$, combined with the the horizontal log-Sobolev inequality ([\[LogSobolevint\]](#LogSobolevint){reference-type="ref" reference="LogSobolevint"}) that holds true on any open set in $\mathbb{H}^{n}$, and the properties of the logarithm give $$\label{Jer.comp.1} \int_{B_r(y)}|g|^2\,dx-\left( \int_{B_r(y)}|g|^{\frac{2}{q}}\,dx\right)^q \leq C_{B,1} \frac{Q(q-1)}{2} \int_{B_r(y)} |\nabla_{\mathbb{H}^{n}}g|^2\,dx\,.$$ Moreover, since the inequalities ([\[LogSobolev1\]](#LogSobolev1){reference-type="ref" reference="LogSobolev1"}) and ([\[LogSobolev2\]](#LogSobolev2){reference-type="ref" reference="LogSobolev2"}) also hold true on any open set in $\mathbb{H}^{n}$, we also have $$\label{Jer.comp.2} \int_{B_r(y)}|g|^2\,dx-\left( \int_{B_r(y)}|g|^{\frac{2}{q}}\,dx\right)^q \leq C_{B,s}\frac{Q(q-1)}{2s} \langle \mathcal{L}_sg,g \rangle_{L^2(B_r(y))}\,,$$ and $$\label{Jer.comp.3} \int_{B_r(y)}|g|^2\,dx-\left( \int_{B_r(y)}|g|^{\frac{2}{q}}\,dx\right)^q \leq C_{B,s} \|U_s\|_{\text{op}} \frac{Q(q-1)}{2s} \langle \mathcal{L}^s g,g \rangle_{L^2(B_r(y))}\,.$$ Note that all the above inequalities are proving an upper bound for the quantity $$\int_{B_r(y)}|g|^2\,dx-\left( \int_{B_r(y)}|g|^{\frac{2}{q}}\,dx\right)^q\,,$$ which can be compared with the quantity that appears on the left hand side of the famous Poincaré inequality on the balls due to Jerison. Let us recall the latter result: In 1986 Jerison [@Jer86] in his seminal paper proved the following Poincaré inequality on any nilpotent Lie group with respect to the Haar measure. In particular for $B_r(y)$ the ball with respect to the Carnot-Carathéodory distance centered at $y$ and of radius $r$, Jerison showed the following: **Theorem 30**. *For any $p \in [1,\infty)$, there exists a constant $P_{0}(r)=P_{0}(r,p)$ such that for all $f \in C^{\infty}(B_r(x))$ $$\label{jer} \int_{B_{r}(y)} |f(x)-f_{B_{r}(y)}|^{p}\,dx \leq P_{0}(r) \int_{B_{r}(y)} |\nabla_{\mathbb G}f(x)|^p\,dx\, ,$$ where $f_{B_{r}(y)}:= \frac{1}{|B_{r}(y)|}\int_{B_{r}(y)}f(x)\,dx$.* In the next theorem, we generalise Jerison's result in the case of $\mathbb{H}^{n}$ and when $p=2$, by allowing the ball $B_r(y)$ to be regarded with respect to any homogeneous quasi-norm on $\mathbb{H}^{n}$ and the right-hand side of ([\[jer\]](#jer){reference-type="ref" reference="jer"}) to be bounded by the infimum of certain quantities that involve the (modified) fractional sub-Laplacian $\mathcal{L}$. **Corollary 31**. *Let $B_r(y)$ be the ball of radius $r$ centered at $y \in \mathbb{H}^{n}$ and with respect to some homogeneous quasi-norm on $\mathbb{H}^{n}$ such that $\text{Vol}(B_r(y)) \leq 1$. Then, we have $$\label{upper.b.j} \int_{B_{r}(y)} |g(x)-g_{B_{r}(y)}|^{2}\,dx \leq M$$ where $$M=\inf_{s \in(0,1]}C_{B,s}\frac{Q}{2s} \langle \mathcal{L}_sg,g \rangle_{L^2(B_r(y)}\,,$$ where the constant $C_{B,s}$ is given in ([\[bestfrac\]](#bestfrac){reference-type="ref" reference="bestfrac"}), respectively. In particular, we have $$\int_{B_{r}(y)} |g(x)-g_{B_{r}(y)}|^{2}\,dx \leq \frac{(n!)^{\frac{1}{n+1}}}{\pi n^{2}}\frac{Q}{2}\int_{B_r(y)}|\nabla_{\mathbb{H}^{n}} g(x)|^2\,dx\,,$$ since $C_{B,1}=\frac{(n!)^{\frac{1}{n+1}}}{\pi n^{2}}$.* **Remark 32**. Note that the constant $C_{B,s}$ depends only on the dimension of $\mathbb{H}^{n}$; i.e., it can be regarded as fixed. Hence the constant that appears in the upper bound for the quantity ([\[upper.b.j\]](#upper.b.j){reference-type="ref" reference="upper.b.j"}) is independent of any involved parameters. *Proof of Corollary [Corollary 31](#cor.jer){reference-type="ref" reference="cor.jer"}.* Observe that the integral $I=\int_{B_{r}(y)} |g-g_{B_{r}(y)}|^{2}\,dx$ can be rewritten as $$\begin{aligned} \label{thmJer.b} I & = & \int_{B_r(y)} \left[g^2-\frac{2g}{\text{Vol}(B_r(y))} \int_{B_r(y)}g\,dx+\frac{1}{\text{Vol}^{2}(B_r(y))}\left( \int_{B_r(y)}g\,dx \right)^2\right]\,dx \nonumber\\ & = & \int_{B_r(y)} g^2\,dx-\frac{2}{\text{Vol}(B_r(y))} \left( \int_{B_r(y)}gdx\right)^2+ \frac{1}{\text{Vol}(B_r(y))} \left( \int_{B_r(y)}g\,dx\right)^2 \nonumber\\ & = & \left[\int_{B_r(y)} g^2\,dx-\frac{1}{\text{Vol}(B_r(y))}\left( \int_{B_r(y)}g\,dx\right)^2\right]\,. \end{aligned}$$ Now, since $\text{Vol}(B_r(y)) \leq 1$, we get $$\int_{B_{r}(y)} |g(x)-g_{B_{r}(y)}|^{2}\,dx\leq \int_{B_r(y)}|g|^2\,dx-\left( \int_{B_r(y)}|g|\,dx\right)^2\,.$$ Combining the latter with ([\[Jer.comp.2\]](#Jer.comp.2){reference-type="ref" reference="Jer.comp.2"}) for $q=2$ gives the desired result, and the proof is complete. **Remark 33**. 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[^1]: The authors are supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). Marianna Chatzakou is a postdoctoral fellow of the Research Foundation -- Flanders (FWO) under the postdoctoral grant No 12B1223N. Michael Ruzhansky is also supported by EPSRC grants EP/R003025/2 and EP/V005529/1, and Aidyn Kassymov by the MESRK grant AP14869275.\ *Keywords:* log-Sobolev inequality; log-Gagliardo-Nirenberg inequality; Nash inequality; Gross inequality; Poincaré inequality; Heisenberg group; fractional operator. [^2]: While the definition of the modified sub-Laplacian $\mathcal{L}_s$ is given for $s\in (0,1)$, here we allow the parameter $s$ to also take the extreme value $1$ since in this case $\mathcal{L}^s$ boils down to the sub-Laplacian itself, and this consideration makes sense in our setting.

arxiv_math

--- abstract: | Weinstein trisection is a trisection of a symplectic 4-manifold whose 1-handlebodies are the Weinstein domain for the symplectic structure induced from an ambient manifold. Lambert-Cole, Meier, and Starkston showed that every closed symplectic 4-manifold admits a Weinstein trisection. In this paper, we construct a Weinstein trisection of $\Sigma_g\times \Sigma_h$. As a consequence of this construction, we construct a little explicit Weinstein trisection of $S^2\times S^2$. address: Mathematical science center for co-creative society, Tohoku University, Aoba-6-3 Aramaki, Aoba Ward, Sendai, Miyagi 980-0845 author: - Masaki Ogawa title: Weinstein trisections of trivial surface bundles. --- # introduction A trisection, introduced in [@GK] has been studied by many authors. One of the kind of the field in trisection theory is a Weinstein trisection. Lambert-Cole reproved the Thom conjecture by using trisection of $\mathbb{C}P^2$ and inequality about an invariant induced by Khovanov homology. In the proof, he used a contact structure induced in the spine of trisection. Recently, Lambert-Cole, Meier, and Starkston introduce a trisection adapted to a symplectic 4-manifold called a Weinstein trisection[@LMS]. They also showed that every closed symplectic 4-manifold admits a Weinstein trisection. Weinstein trisection sometimes gives us a tool to study symplectic closed 4-manifolds and symplectic surfaces in them [@L1; @L2]. There are some examples of Weinstein trisection. For example, Weinstein trisection of $\mathbb{C}P^2$ and $\mathbb{C}P^2\# \overline{\mathbb{C}P^2}$ is described in [@LMS]. On the other hand, given a finite group $G$, there is a symplectic closed 4-manifold with fundamental group isomorphic $G$. This means that there are so many symplectic closed 4-manifolds. Therefore, it is very important to make an example of Weinstein trisection as the first step to studying symplectic closed 4-manifolds with Weinstein trisection. The main result of this paper is the following: **Theorem 1**. *$\Sigma_g\times \Sigma_h$ admits a genus $(2g+1)(2h+1)+1$ Weinstein trisection for some symplectic structure.* This implies that the trisection genus of $\Sigma_g\times \Sigma_h$ equals the Weinstein trisection genus of its. **Corollary 2**. *The trisection genus of $\Sigma_g\times \Sigma_h$ equals the Weinstein trisection genus of its.* This corollary immediately follows from the result in [@W2]. It states that the trisection genus of $\Sigma_g\times \Sigma_h$ is $(2g+1)(2h+1)+1$. This paper is organized as follows: In Section 2, we review definitions of trisection and Weinstein trisection. After that, we introduce a trisection of $\Sigma_g\times \Sigma_h$ which is constructed in [@W2] in Section 3. Then, we show this trisection can be seen as a Weinstein trisection in Section 4. # preliminalies In this section, we set up the notion and objects we use in this paper. Trisection of 4-manifolds is a decomposition of a four-manifold. **Definition 3**. Let $g,k_1,k_2$ and $k_3$ be non-negative integers with $\max\{k_1,k_2,k_3\}\leq g$. A $(g; k_1, k_2, k_3)$-*trisection* of a closed 4-manifold $X$ is a decomposition $X=X_1\cup X_2\cup X_3$ such that for $i,j\in\{1,2,3\}$, - $X_i\cong \natural^{k_i}(S^1\times B^3)$, - $H_{ij}=X_i\cap X_j\cong \natural^g (S^1\times B^2)$ if $i\neq j$, and - $\Sigma=X_1\cap X_2\cap X_3 \cong \#^g (S^1\times S^1)$. Symplectic 4-manifolds is a 4-manifold with non-degenerate closed 2-form $\omega$. We denote it by $(X, \omega)$. In the field of symplectic topology, symplectic manifolds are considered the same if they a symplectomorphic. **Definition 4**. Let $(X, \omega)$ be a symplectic manifold and $\varphi$ a diffeomorphism between itself. We say $\varphi$ is a symplectomorphism if it preserves the symplectic form $\omega$ (i.e. $\varphi^\ast \omega=\omega$). Since $\omega$ is a non-degenerate, we can obtain a volume form $\Omega=\omega\wedge\cdots \wedge \omega$ of $X$ after taking a wedge product n times. Hence in dimension two, the volume form of it is also a symplectic form so, volume-preserving diffeomorphism is the same as a symplectomorphism. Lambert-Cole, Meier, and Starkston define a trisection of a symplectic closed 4-manifold adopted to the symplectic structure. The Weinstein domain, first introduced in [@W], is a symplectic manifold with a contact boundary. More precisely, let $(X, \omega)$ be a compact symplectic manifold with boundaries. Then, $(X, \omega)$ is called a Weinstein domain if there exists a Morse function $f$ on $X$ and a gradient-like vector field $X_f$ of $f$ such that $X_f$ is Liouville vector field (i.e. $d(\iota_{X_f}\omega)=\omega$). **Definition 5**. Let $(X, \omega)$ be a symplectic closed 4-manifold and $X=X_1\cup X_2\cup X_3$ a trisection of $X$. We say $X=X_1\cup X_2\cup X_3$ is a Weinstein trisection if there is a Morse function $f_i: X \rightarrow \mathbb{R}$ and gradient-like vector field $X_f$ of $f$ such that $(X_i, \omega|_{X_i}, f_i, X_f)$ is a Weinstein domain, Lambert-Cole, Meier, and Starkston showed that any symplectic closed 4-manifold admits a Weinstein trisection by using a branched covering [@LMS]. Weinstein domains induce a contact structure in their boundaries by a 1-form $\iota_{X_{f}} \omega$. Also, $H_{ij}$ has contact structures induced by a Weinstein structure of $X_i$ and $X_j$ [@L2]. And they give the following question. **Question 6**. *Is a trisection genus equal to the Weinstein trisection genus?* In this paper, we answer this question for the $\Sigma_g\times \Sigma_h$. The known trisections of symplectic four-dimensional manifolds are known as Weinstein trisections. To positively affirm that there are no trisections that are not Weinstein trisections, we still have too few examples. # trisection of trivial surface bundle In this section we review the construction of a trisection of $\Sigma_g\times \Sigma_h$ in [@W2]. Let $X=\Sigma_g\times \Sigma_h$. First, we consider the decomposition of $\Sigma_g$. We can decompose any closed surface into three disks. **Lemma 7** (Lemma 3.1 in [@W2]). *$\Sigma_g$ admits a decomposition $\Sigma_g=B_1\cup B_2\cup B_3$ satisfies the following:* 1. *Each $B_i$ are disks,* 2. *$b_{ij}=B_i\cap B_j$ is $2g+1$ disjoint arcs.* 3. *$B_1\cap B_2\cap B_3$ is $4g+2$ disjoint vertices.* If $g=1$, this decomposition is illustrated in Figure [1](#surfdeco){reference-type="ref" reference="surfdeco"}. To obtain the case of a higher genus, remove a neighborhood of a vertex and glue a copy of it to itself along their boundary. {reference-type="ref" reference="decolem1"}.](surfdeco.pdf){#surfdeco} We take mutually disjoint disks $N_i$ in $\Sigma_h$ for $i=1, 2, 3$. Then we define the 1-handlebodies of a trisection as follows: $$X_i = (B_i \times (\Sigma_h - Int(N_i ; \Sigma_h)) \cup (B_{i+1}\times N_{i+1}).$$ To check $X_i$ is a 1-handlebody, we show that $(B_i \times (\Sigma_h - Int(N_i ; \Sigma_h))$ is a 1-handlebody. Since $(\Sigma_h - Int(N_i ; \Sigma_h)$ is a punctured genus $h$ surface, this can be represented by a disk and $2h$ 1-handles attached to it. Since $B_i$ is a disk, $(B_i \times (\Sigma_h - Int(N_i ; \Sigma_h))$ is a 1-handlebody diffeomorphic to $\natural^{2h} S^1\times B_3$. The intersection $(B_i \times (\Sigma_h - Int(N_i ; \Sigma_h))$ and $(B_{i+1}\times N_{i+1})$ is a $(B_i\cap B_{i+1})\times N_i$. This is a disjoint $2g+1$ 3-balls in their boundaries. Since $(B_{i+1}\times N_{i+1})$ is a 4-ball, $X_i$ is a 1-handlebody of genus $2g+2h$. In [@W2], Williams showed that $X=X_1\cup X_2\cup X_2$ is a trisection. **Theorem 8** (Theorem 3.3 in [@W2]). *$X=X_1\cup X_2\cup X_2$ is a genus $(2g+1)(2h+1)+1$ trisection.* The direction of this paper involves examining the above trisection in more detail to demonstrate how it will become a Weinstein trisection. To show this, we consider the Weinstein structure of $\Sigma_g - int (B_i)$ and $\Sigma_h - int (N_i)$. This is constructed in Section 5. It is easy to show that each of $N_i$ is mutually ambient isotopic since they are disjoint disks. Also, we can show that each of $B_i$ is mutually ambient isotopic. Actually, in Figure [1](#surfdeco){reference-type="ref" reference="surfdeco"}, $B_i$ is sent to $B_{i+1}$ by ambient isotopy along the diagonal from bottom-left to top-right. We can extend this ambient isotopy to an arbitrary genus since we can construct a decomposition of $\Sigma_g=B_1\cup B_2\cup B_2$ by connecting summing a torus by taking the regular neighborhood of a vertex depicted in Figure [1](#surfdeco){reference-type="ref" reference="surfdeco"}. We consider this feature with a symplectic structure in the next subsection. ## symplectomorphisms compatible to the trisection. In this subsection, we see the self-symplectomorphisms of symplectic surfaces. First of all, we provide a definition of symplectic and Hamiltonian isotopy. Let $(X, \omega)$ be a closed symplectic manifold and $H: X\to \mathbb{R}$ a smooth function on $X$. Then there is a unique vector field $X_H$ such that $$\iota_{X_H}\omega = \omega(X_H, \cdot \ )=dH$$ since $\omega$ is non-degenerate. We call $X_H$ a *Hamiltonian vector field* associated to the *Hamlltonian function* $H$. **Definition 9**. Let $\varphi_t$ be an ambient isotopy on $X$. We say $\varphi_t$ is *symplectic isotopy* if $\varphi_t$ is symplectomorphism for every $t\in [0, 1]$. A symplectic isotopy is called a *Hamiltonian isotopy* if $\iota_{X_t}\omega$ is exact 1-form (i.e. $X_t$ is Hamiltonian vector field for every $t$) where $X_t$ is a vecotor field such that $$\frac{d}{dt}\varphi_t = X_t\circ \varphi_t.$$ By definition, a Hamiltonian isotopy is a symplectic isotopy. In Section 5, we use the following lemmas to construct Weinstein structures of $\Sigma_g - int (B_i)$ and $\Sigma_h - int (N_i)$. To show Lemma [Lemma 11](#lemma1){reference-type="ref" reference="lemma1"}, we use the following lemma. **Lemma 10** (cf. [@MS], P.113). *Let $(\Sigma_g, \omega)$ be a symplectic closed surface and $B(r)^2$ a standard disk with radius $r$ for some $r\in \mathbb{R}$. $$\begin{array}{rccc} [0, 1]\ \times&B(r)^2 &\longrightarrow& \Sigma_g \\ \rotatebox{90}{$\in$}&& & \rotatebox{90}{$\in$} \\ (t, z)& &\longmapsto & \psi_t(z) \end{array}$$ be a smooth map such that $\psi_t: B(r)^2\rightarrow \Sigma_g$ is a symplectic embedding for every $t$. Then there exists a Hamiltonian isotopy $$\begin{array}{rccc} [0, 1]\times& \Sigma_g & \longrightarrow & \Sigma_g \\ \rotatebox{90}{$\in$}& & & \rotatebox{90}{$\in$} \\ (t, p) & &\longmapsto & \phi_t(p) \end{array}$$ such that $$\phi_0=id,\ \phi_t\circ\psi_0=\psi_t$$ for all $t$.* **Lemma 11**. *Let $\Sigma_g=B_1\cup B_2\cup B_3$ be a decomposition as in Lemma [Lemma 7](#decolem1){reference-type="ref" reference="decolem1"} and $\omega$ a symplectic form of $\Sigma_g$. Suppose that each area of $B_i$ is identical for $i=1, 2, 3$ with respect to $\omega$. Then there exists a symplectomorphism $\varphi_i$ such that $\varphi_i(B_i)=B_{i+1}$.* *Proof.* Let $\psi_t: \Sigma_g\rightarrow \Sigma_g$ be a ambient isotopy, such that $$\psi_1(B_i)= B_{i+1}.$$ Let $p_i$ be a point in the interior of $B_i$. Then, $\psi(p_i)$ is a point in the interior of $\psi(B_i)$. Then we can construct a symplectic embedding $\varphi'_t (B(r_i(t)))\rightarrow \Sigma_g$ so that $$\varphi'_t(B(r_i(t)))=\psi_t(B_i)$$ where $r_i(t)\in \mathbb{R}$ is a smooth function on $[0, 1]$ for $i=1, 2, 3$. Then, we perturb $\varphi'$ so that $r_i(t)=r_j$ for $i\neq j$ and any $t\in [0, 1]$, we obtain the follwoing family of symplectic embeddings $\varphi'_t$ such that $$\varphi'_t: B(r)\rightarrow \Sigma_g$$ $$\varphi'_0(B(r))=B_i, \varphi'_1(B_i)=B_{i+1}.$$ By Lemma [Lemma 10](#hamiso){reference-type="ref" reference="hamiso"}, we obtain a Hamiltonian isotopy $\varphi_t: \Sigma_g\rightarrow \Sigma_g$ such that $$\varphi_0=id,\ \varphi_t\circ \varphi'_0=\varphi'_t.$$ Then $\varphi_1$ is a symplectomorphism we want. ◻ From the result below, We can also assume that the $N_i$ in $\Sigma_h$ are sent to each other by symplectomorphism. **Lemma 12** (Theorem A in [@B]). *Let $(\Sigma_h, \omega)$ be a symplectic closed surface and $\{p_1, \ldots, p_n\}$ and $\{q_1, \ldots, q_n\}$ are two sets of distinct points of $\Sigma_h$. Then there is a symplectomorphism $\varphi$ such that $\varphi(p_i)=q_i$ for $i=1, \ldots, n$ that is isotopic to identity by an isotopy which preserves the structure and leaves fixed every point of $\Sigma_h$ outside a compact set of arbitrarily small volume.* # Stein and Weinstein structure and Riemann surface. ## Stein structure of a Riemann surface We will review the concepts of Stein domains and that of Riemann surfaces. A Riemann surface is a 2-manifold with a complex structure $J$. A Stein manifold is a complex manifold that is embedded in $\mathbb{C}^N$, where $N$ is a natural number. Grauert provided a characterization of Stein manifolds based on the function they admit (refer to [@GR]). This function is known as a strictly plurisubharmonic or $J$-convex function. In the context of a smooth function $f: M\rightarrow \mathbb{R}$, we say that it is exhausting if it is both proper and bounded from below. **Definition 13**. Let $(M, J)$ be a complex manifold and $f:M\to \mathbb{R}$ a function. $f$ is called a *plurisubharmonic or $J$-convex* function if the 2-form $$\omega_f:=-d(df\circ J)=-dd^{\mathbb{C}} f$$ satisfies $$\omega_f(v, Jv)>0$$ for every non-zero tangent vector $v\in TM$. It is well-known that a Riemann surface is a Stein manifold if and only if it is a non-compact. Hence, $\Sigma^n_g$ becomes a Stein domain where $\Sigma^n_g$ is genus $g$ closed surface removed $n$ disks. Hence we obtain the following: **Lemma 14**. *Let $\Sigma_g$ and $\Sigma_h$ be surfaces with genera $g$ and $h$ respectively, and $B_i$ and $N_i$ for $i=1, 2, 3$ are disks in $\Sigma_g$ and $\Sigma_h$ respectively that are defined in Section 3. Then, $\Sigma_h-int (N_i)$ and $\Sigma_g- B_{i+2}$ will be Stein domains for $i=1, 2, 3$.* ## Weinstein structure of Riemann surface Liouville vector field $\xi$ on symplectic manifold $(X, \omega)$ is a vector field such that $d (i_X \omega)=\omega$. If the Lie derivative of $\omega$ along $X$ is $\omega$, then $X$ is a Liouville vector field. This follows from $\omega$ is closed. Liouville domain is the symplectic manifold with some compatible vector fields with contact boundaries. **Definition 15**. Let $(W, \omega)$ be a compact symplectic manifold with no empty boundary. $(W, \omega)$ is called a *Liouville domain* if there is a Liouville vector field defined globally and it is transversally out of the boundary. We denote it by $(W, \omega, X)$ If the Liouville domain has a \"compatible\" Morse function, it is called a Weinstein domain. **Definition 16**. Let $(W, \omega, X)$ be a Liouville domain. $(W, \omega, X)$ is called a *Weinstein domain* if there exists a Morse function $f$ it is locally constant in $\partial W$ and $X$ is gradient-like for $f$. For a given symplectic manifold, it is difficult to determine whether it admits a Weinstein structure or not. Also, generally, it is difficult to construct a Weinstein structure but we sometimes construct it from a Stein structure. We say exhausting J-convex function $f$ is *completely exhausting* if its gradient vector fields $\nabla_{f}f$ is complete where $\nabla_{f}f$ is a gradient respect to a Riemann metric $\omega_{f}(\cdot, J\cdot)$ for $\omega_f=-dd^{\mathbb{C}} f$. The following is a well-known theorem: **Theorem 17** ([@EG]). *Let $(V, J)$ be a Stein manifold and $f: V\rightarrow \mathbb{R}$ a completely exhausting $J$-convex Morse function. Then, $$(\omega_{f}:=-dd^{\mathbb{C}}f, X_{f}:=\nabla_{f}f, f)$$ is a Weinstein structure on $V$.* Hence, we can construct a Weinstein structure of $\Sigma_h-int (N_i)$ from a Morse $J$-convex function $f_i$ for $i=1, 2, 3$. # Proof of the main theorem Let us consider a 1-handlebody $$X_i=(\Sigma_h - int(N_i))\times B_i\cup N_{i+1}\times B_{i+1}$$ that constructs a trisection prescribed in Section 3. We show that each of $X_i$ admits a Weinstein structure with respect to a symplectic structure defined by a product structure of $\Sigma_g\times \Sigma_h$. Let $g_1: \Sigma_g-B_3\rightarrow \mathbb{R}$ be a $J$-convex function such that $B_1$ contains only one critical point and its index is $0$. Then we define the symplectic structure $\omega_g$ on $\Sigma_g$ so that $$\omega_g |_{\Sigma_g-B_3}=- dd^{\mathbb{C}} g_1.$$ By Lemma [Lemma 11](#lemma1){reference-type="ref" reference="lemma1"}, there is a symplectomorphism $\varphi$ such that $\varphi(B_i)=B_{i+1}$. Then we define the $J$-convex function $g_2$ and $g_3$ as follows: $$g_2=g_1\circ \varphi^{-1} |_{\Sigma_g - B_1}: \Sigma_g - B_1 \rightarrow \mathbb{R},$$ $$g_3=g_2\circ \varphi^{-1}i |_{\Sigma_g - B_2}: \Sigma_g - B_2 \rightarrow \mathbb{R}.$$ Then we can define the Weinstein structures on $\Sigma_g - B_{i+2}$ for $i=1, 2, 3$ by Theorem [Theorem 17](#StoW){reference-type="ref" reference="StoW"}. Let $N_1$ be a sufficiently small disk in $\Sigma_h$ and $f_1: \Sigma_h - N_1\rightarrow \mathbb{R}$ a $J$-convex function. Also, we suppose that $N_2$ is a sufficiently small regular neighborhood of index $0$ critical point of $f_1$ and $N_3$ a disk in the interior of $\Sigma_h - (N_1\cup N_2)$. Then we define the symplectic structure $\omega_h$ of $\Sigma_h$ so that $$\omega_h|_{ \Sigma_h - N_1}= - dd^{\mathbb{C}} f_1.$$ By Lemma [Lemma 12](#hlem){reference-type="ref" reference="hlem"}, we can obtain the symplectomorphism $\varphi_2: (\Sigma, \omega_h)\rightarrow (\Sigma_h, \omega_h)$ such that $$\varphi(N_1)=N_2, \ \varphi(N_2)=N_3.$$ Then, we obtain the $J$ convex function $f_2: \Sigma - N_2\rightarrow \mathbb{R}$ as follows: $$f_2=f_1\circ \varphi_2^{-1}|_{\Sigma_h - N_2}: \Sigma_h - N_2 \rightarrow \mathbb{R}.$$ Also, we can obtain a symplectomorphism $\varphi_3: (\Sigma, \omega_h)\rightarrow (\Sigma_h, \omega_h)$ such that $$\varphi(N_2)=N_3, \ \varphi(N_3)=N_1.$$ by Lemma [Lemma 12](#hlem){reference-type="ref" reference="hlem"}. Then, we can define the $J$-convex function $f_3: \Sigma - N_3\rightarrow \mathbb{R}$ as follows: $$f_3=f_2\circ \varphi_3^{-1}|_{\Sigma_h - N_3}: \Sigma_h - N_3 \rightarrow \mathbb{R}.$$ Finally, we can construct a Weinstein structure on $\Sigma_h - N_i$ for $i=1, 2, 3$ by Theorem [Theorem 17](#StoW){reference-type="ref" reference="StoW"}. *Proof of Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"}.* Let $(\Sigma_h \times \Sigma_g, \omega)$ be a trivial bundle over surface with symplectic structure $\omega=pr_1^{\ast} \omega_h+pr_2^{\ast}\omega_g$ that is defined above. Now, the function $f_i+g_i$ is a $J$-convex Morse function of $(\Sigma_h- N_i)\times (\Sigma_g - B_{i+2})$. So, we have to show that the gradient vector field of this Morse function is outward and it is a Liouville vector field on $X_i$ with respect to a symplectic structure $\omega$. Let $\xi_i$ be the gradient vector field of $f_i+g_i$. First, we show that $\xi_i$ is a Liouville vector field on $X_i$ with respect to $\omega$. Now, $f_i$ and $g_i$ are $J$-convex Morse function, and $\omega_h|\Sigma_h- N_i$ and $\omega_g|\Sigma_g - B_{i+2}$ are symplectic structure defined by $-dd^{\mathbb{C}}f_i$ and $-dd^{\mathbb{C}}g_i$ respectively. Hence, by Theorem [Theorem 17](#StoW){reference-type="ref" reference="StoW"}, $\xi_i$ gives a Liouville vector fields on $(\Sigma_h- N_i)\times (\Sigma_g - B_{i+2})$, particularly on $X_i$ for $\omega$. Next, we show that $\xi_i$ is transverse outwardly on the boudary of $X_i$. we recall that $$X_i=(\Sigma_h - int(N_i))\times B_i\cup N_{i+1}\times B_{i+1}.$$ Since $(\Sigma_h - int(N_i))\times int(B_i)$ and $N_{i+1}\times B_{i+1}$ intersects their boundaries, $\partial X_i$ can be desicribed as following: $$\begin{aligned} \partial X_i =&( (\partial(\Sigma_h- int(N_i))\times B_i)\cup ((\Sigma_h- int(N_i)\times \partial B_i) )) \\ &\cup ((\partial N_{i+1}\times B_{i+1} )\cup (N_{i+1}\cup \partial B_{i+1})) \\ & - ( ((\Sigma_h - int(N_i))\times int(B_i))\cap (N_{i+1}\times B_{i+1} )) \end{aligned}$$ We note that $((\Sigma_h - int(N_i))\times int(B_i))\cap (N_{i+1}\times B_{i+1} )= N_{i+1}\times (B_i\cap B_{i+1})$. Then, we will see whether $\xi_i$ is outward for each region. We note that $B_i$ is a neighborhood of an index $0$ critical point of $g_i$ by definition of $g_i$. Then, the restriction of $\xi_i$ to $B_i$ is transverse outwardly on its boundary. Since the restriction of $\xi_i$ to $B_i$ and $\Sigma_h-int (N_i)$ is transverse outwardly on its boundary, $\xi_i$ transverse outwardly on $(\partial(\Sigma_h- int(N_i))\times B_i)$ and $((\Sigma_h- int(N_i)\times \partial B_i)$. We note that $N_{i+1}$ is a neighborhood of an index $0$ critical point of $f_i$ by definition of $f_i$. $\xi_i$ transverse outwardly on $\partial (N_{i+1}\times B_{i+1})$ except $N_{i+1}\times (B_{i} \cap \partial B_{i+1})$ since the restriction of $\xi$ to $B_{i+1}$ transeverse outwardly on $\partial B_{i+1}$ except $B_{i}\cap B_{i+1}$. But we see that $$((\Sigma_h - int(N_i))\times int(B_i))\cap ( N_{i+1}\times B_{i+1}) = N_{i+1}\times (B_i\cap B_{i+1}).$$ Hence the region $N_{i+1}\times (B_i\cap \partial B_{i+1})$ is not included a boundary of $X_i$. Hence $\xi_i$ transverse outwardly on $\partial X_i$ entirely. ◻ # The case where $g=h=0$. In this section, we give an example of the Weinstein trisection of $S^2\times S^2$. It is a trivial $S^2$ bundle over $S^2$. We denote it $S_1\times S_2$. Also, $S_2$ has a decomposition described above. More precisely, we assume that $S_2$ is decomposed into three disks $B_1$, $B_2$, and $B_3$ as in Figure [1](#surfdeco){reference-type="ref" reference="surfdeco"}. We note that the Weinstein trisection of $S^2\times S^2$ is constructed in former articles [@LMS]. But in this section, we will describe it slightly more explicitly. We construct a Weinstein trisection of $S^2\times S^2$ as the following steps: 1. For a given symplectic structure on $S_1 - N_i$, we define a Morse function $f_i$ so that $N_{i+1}$ is a neighborhood of index $0$ critical point of $f_i$, it does not contain the other critical point and whose gradient flow is a Liouville vector field of the symplectic structure. Furthermore, each of $N_i$ is mutually symplectically isotopic to each other. 2. For a given symplectic structure on $S_2 - B_i$, we define a Morse function $g_i$ so that $B_i$ is a neighborhood of index $0$ critical point of $g$ and does not contain the other critical points of $g$, $B_{i+1}$ is a neighborhood of index $2$ critical point of $g$ and does not contain the other critical points of $g$ and whose gradient vector field is a Liouville vector field for the symplectic structure. Furthermore, each of $B_i$ is mutually symplectically isotopic to each other. 3. $f+g$ is a Morse function on $X_i$ and a gradient vector field of it is a Liouville vector field for the products of the symplectic structure of $S_1$ and $S_2$ and $(X_i, \omega, g+f, grad(g+f))$ will be a Weinstein domain. To begin the steps above, we review the Kähler structure on $S^2$. First of all, we review the Fubini-study form of $\mathbb{C}P^1$. Let $(z_1, z_2)$ be a homogenious coordinate of $\mathbb{C}P^1$. Then we can take a chart of $\mathbb{C}P^1$ by taking $\phi_i:\mathbb{C}P^1\to \mathbb{C}$ for $z_j\neq 0$ $$\phi_i(z_1, z_2)=\frac{z_i}{z_j} \ (i\neq j).$$ We denote this chart by $\{U_i, \phi_i\}_{i=1, 2}$. Then we define the Fubini-Study form by $$\omega_{FS} = 2 \frac{\partial^2 \log(1+|z|^2 )}{\partial z\partial \bar{z}} dz \wedge d\bar{z}$$ where $z=z_1/z_2$. It is known that this form gives us the Kähler form on $U_1$, and in this case, $f= \log(1+|z|^2)$ gives a Kähler potential. The critical point of $f$ is only $z=0$. This means that $z_1=0$ since $z_2\neq 0$. Also, we note that this is a $J$-holomorphic function on $U_1$. Hence, $U_1$ is a Stein manifold with $J$-holomorphic function $f$. Next, we review the identification between $S^2$ and $\mathbb{C}P^1$. We denote the polar coordinate of the unit sphere by the following (See Figure [2](#stproj){reference-type="ref" reference="stproj"}): $$\begin{aligned} x=&\sin \theta_1 \cos\theta_2, \\ y=&sin\theta_1 \sin \theta_2, \\ z=&\cos \theta_1.\end{aligned}$$ {#stproj} Then we can represent a point on $S^2$ by $(x, y, z)\in S^2\subset \mathbb{R}^3$. We denote the stereographic projection $\phi_N: S^2\backslash p_N\to \mathbb{R}^2$ where $p_N=(0, 0, 1)$. Then we can define the diffeomorphism $\Phi : \mathbb{C}P^1\to S^2$ by $$\Phi(z_1, z_2)= \begin{cases}{} \phi_N(z)& (z_2\neq 0)\\ p_N& (z_2=0) \end{cases}$$ where $z=z_1/z_2$. We sometimes use this diffeomorphism to define the function on $S^2$ below. We identify $f\circ \Phi^{-1}$ and ${\Phi^{-1}}^\ast \omega_{FS}$ by $f$ and $\omega_{FS}$ respectively. ## Step 1: Weinstein structure on $S_1- N_i$ We define the region $N_1$, $N_2$ and $N_3$ on $S_1$. Let $\varphi$ be a $4/3 \pi$-rotation along a $y$-axis. Then we denote $P_N=p_1$, $\varphi(P_N)=p_2$ and $\varphi^2(P_N)=p_3$. Also, we denote $P_S=q_1$, $\varphi(P_S)=q_2$ and $\varphi^2(P_S)=q_3$. Then we define $N_1$ as a neighborhood of $p_1$ so that it contains $q_3$ and does not contain $p_i$ and $q_j$ for $i=2, 3, j=1, 2$. $N_2$ and $N_3$ are defined by $\varphi(N_1)$ and $\varphi^2(N_3)$ respectively (See Figure [3](#S_1){reference-type="ref" reference="S_1"}). {#S_1} We take rotation $\varphi$ along a $y-axis$ such that $\varphi(N_1)\cap N_1$ is an empty set. Also, the we define $\varphi(N_1)=N_2$ and $\varphi(N_2)=N_3$. Since a rotation is volume-preserving, this is a sympelctomorphism that sends $N_i$ to each other. Then $f$ define the $J$-convex function on $S^2-N_1$ respect to $\omega_{FS}$. By using Theorem [Theorem 17](#StoW){reference-type="ref" reference="StoW"}, we can obtain the Weinstein structure on $S_1- N_1$. By permuting $N_i$ by rotation, we can also obtain the Weinstein structure on $S_1 - N_i$ for $i=2, 3$. ## Step 2: Weinstein structure on $S_2 - B_{i+2}$ We define the region $B_i$ in $S_2$ for $i=1, 2, 3$ as in Lemma [1](#surfdeco){reference-type="ref" reference="surfdeco"}. In this case, the intersection $B_i\cap B_j$ for $i\neq j$ is an arc. We define the region $B_i$ as follows: $$B_1=\left\{(\sin \theta_1 \cos\theta_2, \sin\theta_1 \sin \theta_2, \cos \theta_1)\in S^2\mid \frac{4}{12}\pi \leq \theta_2 \leq \frac{12}{12}\pi \right \},$$ $$B_2 = \left\{(\sin \theta_1 \cos\theta_2, \sin\theta_1 \sin \theta_2, \cos \theta_1)\in S^2 \mid \frac{12}{12}\pi \leq \theta_2 \leq \frac{20}{12}\pi \right \},$$ $$B_2 = \left\{(\sin \theta_1 \cos\theta_2, \sin\theta_1 \sin \theta_2, \cos \theta_1)\in S^2 \mid \frac{20}{12}\pi \leq \theta_2 \leq \frac{24}{12}\pi \right \}.$$ We can see each region as in Figure [4](#Bregion){reference-type="ref" reference="Bregion"}. {#Bregion} It is easy to see that each of $B_i$ is mutually permuted by rotating along a $y$-axis. Also, we can define the $J$-holomorphic function $g_1$ respect to $\omega_{FS}$ on $S_2 - int (B_3)$ from the function $f$ defined above. Also, composing a $g_1$ and rotating, we can also obtain the $J$-convex function on $S_2 - int (B_i)$ for $i=1, 2$. We will check that the critical point of $g_1$ is included in $B_2$. $g_1$ is a restriction of $f$ to $S^2- int(B_3)$ it was defined before the steps. Its critical point is $z_1=0$. This is corresponds to a point $p_S=(0, 0, -1)$ by the identification map $\Phi$. Also, this critical point has an index $0$ since by Poincare-Hopf theorem, $f$ is a Morse function on disk and it has only one critical point. Hence we can assume $B_2$ is a neighborhood of index $0$ critical point of $g_1$. ## Step 3: $(X_i, \omega\mid_{X_i}, (g+f)\mid_{X_i}, grad(g+f))$ is a Weinstein domain. We remain the following: First, we have to check the gradient flow of $g_i + f_i$ is the Liouville vector field. Next, we have to check it is outward. By Theorem [Theorem 17](#StoW){reference-type="ref" reference="StoW"}, we can see that the gradient-like flow of $f_i$ and $g_i$ are Liouville vector fields with respect to $\omega_{FS}$. Hence we can show that the gradient-like flow of $f_i + g_i$ is a Liouville vector field with respect to $\omega_{FS}+\omega_{FS}=2\omega_{FS}$. Finally, we will check it is outward. Following the construction of trisection above, $X_i$ is a set as follows: $$X_i = ((S_1 - N_i)\times B_{i} )\cup (N_{i+1}\times B_{i+1})$$ We show the case where $i=1$. We note that each of $(S_1 - N_1) \times B_1$ and $N_{2}\times B_{2}$ are diffeomorphic to a 4-ball since $S_1 - N_1$ is a disk. We defined the vector field on $S_{1} - N_1$ and $B_1\cup B_2$ by the gradient-like vector field of $f_1$ and $g_1$ respectively. They are outward on each of $S_{1} - N_1$ and $B_1\cup B_2$ respectively. We denote an arc $B_1\cap B_2$ $\alpha$. Then the intersection of two 4-ball $(S_1 - N_1) \times B_1$ and $N_{2}\times B_{2}$ is a 3-ball $N_2\times \alpha$. We shall consider the boundary of $X_i$. $\partial X_i$ is a union of $\{(S_1 - N_1) \times B_1 \} - (N_2\times \alpha)$ and $N_{2}\times B_{2} - N_2\times \alpha$. Now, the gradient-like vector field of $f_1 + g_1$ on $(S_1 - N_1) \times B_1$ outward since $(S_1 - N_1)$ and $B_1$ are neiborhood of index $0$ critical point of $f_1$ and $g_1$ respectively. Also, the gradient-like vector field of $f_1 + g_1$ on $N_{2}\times B_{2}$ is outward except $N_2\times \alpha$ since we can assume that $N_2$ is a neighborhood of index $0$ critical point of $f_1$ and the gradient-like vector field of $g_1$ is outward on $B_1\cup B_2$. This is enough to show that $f_1 + g_1$ is outward on $X_1$. # Acknowledgement The author thanks Takahiro Oba for a very meaningful discussion and suggestion about research in symplectic topology, also giving comments on a draft of this paper. 99 William M. Boothby, Transitivity of the automorphisms of certain geometric structures, Trans. Am. Math. Soc. **137** (1969), 93-100. Y. Eliashberg and M. Gromov, Convex Symplectic Manifolds, Proceedings of Symposia in Pure Mathematics, **52**, Part 2, 135-162 (1991). D. Gay and R. Kirby. Trisecting 4-manifolds. Geom. Topol. **20** (2016), no. 6, 3097-3132. H. Grauert and R. Remmert, Theory of Stein Spaces, Grundlehren der Mathematischen Wissenschaften **236**, Springer-Verlag, Berlin (1979). Lambert-Cole. Bridge trisections in $\mathbb{C}P^2$ and the Thom conjecture. Geom. Topol, **24(3)** (2020), 1571-1614. Lambert-Cole. Symplectic surfaces and bridge position. Geometriae Dedicata, **217(1)** (2023) p.8. P. Lambert-Cole, J.Meier, L.Starkston Symplectic 4‐manifolds admit Weinstein trisections. Journal of Topology. 2021 Jun;14(2):641-73. D. McDuff and D. Salamon, Introduction to symplectic topology. Oxford University Press, (2017). A. Weinstein. Contact surgery and symplectic handlebodies. Hokkaido Mathematical Journal, **20(2)** (1991), pp.241-251. M. Williams. Trisections of flat surface bundles over surfaces Doctoral Thesis, The University of Nebraska-Lincoln (2020).

arxiv_math

--- abstract: | We show that the local-global divisibility in commutative algebraic groups defined over number fields can be tested on sets of primes of arbitrary small density, i.e. stable and persistent sets. We also give a new description of the cohomological group giving an obstruction to the problem. In addition, we show new examples of stable sets. address: - Alexander Ivanov, Ruhr-Universität Bochum, Fakultät für Mathematik, IB 2/153, Universitätsstrasse 150, D-44780 Bochum - Laura Paladino, Università della Calabria, Ponte Bucci, Cubo 30B, 87075, Rende (CS), Italy author: - Alexander B. Ivanov - Laura Paladino bibliography: - bib_ADLV.bib title: Testing local-global divisibility at a small set of primes --- # Introduction Let $k$ be a number field. Let $A$ be a connected commutative group scheme of finite type over $k$. Dvornicich and Zannier investigated a local-global principle for divisibility of rational points on $A$: **Problem 1** (Local-global divisibility problem [@DvornicichZ_01]). *Let $r$ be a positive integer and let $P \in A(k)$. Suppose that for all but finitely many primes ${\mathfrak p}$ of $k$, we have $rD_{\mathfrak p}= P$ for some $D_{{\mathfrak p}} \in A(k_{{\mathfrak p}})$, where $k_{{\mathfrak p}}$ denotes the completion of $k$ at ${\mathfrak p}$. Can one conclude that there exists $D \in A(k)$ with $rD = P$?* Of course, one may assume that $r$ is a power of a rational prime without loosing any generality. The solution to Problem [Problem 1](#problem:DZ){reference-type="ref" reference="problem:DZ"} and to variants of it are known in many cases, in particular, for tori [@DvornicichZ_01; @Illengo_08] and for elliptic curves [@PaladinoRV_12; @PaladinoRV_14; @Cre]. In elliptic curves the answer is affirmative for every power $r=p^n$, with $p>B(d)$, where $d:=[k:{\mathbb Q}]$ and $$B(d):=\left\{\begin{array}{ll} 3 & if \hspace{0.1cm} k={\mathbb Q};\\ (3^{\frac{d}{2}}+1)^2 & if \hspace{0.1cm} d>1.\\ \end{array} \right.$$ Observe that for $k\neq {\mathbb Q}$ the bound $B(d)=(3^{\frac{d}{2}}+1)^2$ is the one appearing in [@Oesterle_], giving an effective version of Merel's Theorem on torsion points of an elliptic curve [@Merel_96], and in particular $E(k)[p] = 0$ for all primes $p>B(d)$. It is shown in [@DvornicichP_21 §5] that for a *fixed* elliptic curve $E$, a *fixed* number field $k$ and a *fixed* power $r$, Problem [Problem 1](#problem:DZ){reference-type="ref" reference="problem:DZ"} admits an explicit and effective solution, that is there is an effectively computable constant $C(k,E,r)>0$ such that to deduce global divisibility, it suffices to test the local one for all primes ${\mathfrak p}$ of $k$ with norm $N{\mathfrak p}<C(k,E,r)$. In this note we ask, whether in Problem [Problem 1](#problem:DZ){reference-type="ref" reference="problem:DZ"} one can considerably shrink the set of primes where local divisibility is tested, simultaneously for *all* $A$, *all* $k$ and *all* $r$. It turns out that this strengthened version of the problem admits a solution. More precisely, we show that certain sets of primes with arbitrary small Dirichlet density suffice, cf. Theorem [Proposition 8](#thm:p_stable_sets_test_p_divisibility){reference-type="ref" reference="thm:p_stable_sets_test_p_divisibility"}. For clarity, let us state our main result in the case of elliptic curves, where the original Problem [Problem 1](#problem:DZ){reference-type="ref" reference="problem:DZ"} is quite well-understood. **Theorem 1** (see Corollary [Corollary 10](#cor:for_absolutely_stable_S){reference-type="ref" reference="cor:for_absolutely_stable_S"} for most general statement). *For any $\varepsilon > 0$ there exists a set $S$ of primes of ${\mathbb Q}$, such that for all number fields $k$, all elliptic curves $E/k$ and all primes $p$ one has $0 < \delta_k(S) < \varepsilon$ and the following hold:* - *If a point $P \in E(k)$ is locally divisible by $p$ at any ${\mathfrak p}\in S_k$, then it is globally divisible by $p$ in $E(k)$.* - *Suppose that $p > B(d)$. Then for all $n \geq 1$, if a point $P \in E(k)$ is locally divisible by $p^n$ at any ${\mathfrak p}\in S$, then it is globally divisible by $p^n$ in $E(k)$.* Moreover, sets $S$ as in Theorem [Theorem 1](#thm:small_test_set){reference-type="ref" reference="thm:small_test_set"} exist in abundance, and examples of them can be given explicitly, cf. §[3](#sec:stable_set){reference-type="ref" reference="sec:stable_set"}. It is well known that an obstruction to the validity of Problem [Problem 1](#problem:DZ){reference-type="ref" reference="problem:DZ"} is given by the first local cohomology group (see [@DvornicichZ_01],[@DvornicichZ_07] and Equation [\[eq:cohomology_with_local_conditions\]](#eq:cohomology_with_local_conditions){reference-type="eqref" reference="eq:cohomology_with_local_conditions"} below). Such a group is isomorphic to some modified Tate-Shafarevich groups defined by sets $S$ of primes of density one (see for instance [@Creutz_12; @DP_22]). If one shrinks the set $S$, the Tate-Shafarevich group can *a priori* become bigger. Our main point is that if we shrink $S$ is an appropriate way, then the Tate-Shafarevich group will stay small. This is based on the properties of the so-called *stable* and *persistent* sets of primes studied in [@Ivanov_stable_sets]. In §[3](#sec:stable_set){reference-type="ref" reference="sec:stable_set"} we also give new examples of such sets with particularly nice properties. At the end of the paper we state a generalization of a classical question posed by Cassels about the $p$-divisibility of elements of the Tate-Shafarevich group and we discuss the implications of our results for such a question. ## Notation {#notation .unnumbered} We denote by $p$ a rational prime. We fix once and for all an algebraic closure $\overline {\mathbb Q}$ of ${\mathbb Q}$, and consider all algebraic extensions of ${\mathbb Q}$ as subfields of $\overline {\mathbb Q}$. Unless stated otherwise, $k$ always denote a number field of degree $d = [K:{\mathbb Q}]$. If $\ell/k$ is a Galois extension, we denote by $\mathop{\mathrm{Gal}}_{\ell/k}$ its Galois group. We denote by $\Sigma_k$ the set of primes of $k$. If $S \subseteq \Sigma_k$ and $\ell/k$ is a finite extension, we write $S_\ell$ for the preimage of $S$ under the natural map $\Sigma_\ell \rightarrow \Sigma_k$. We denote by $\delta_k(S) \in [0,1]$ the Dirichlet density of a set $S \subseteq \Sigma_k$, whenever it exists. Whenever we write "density" below, we mean "Dirichlet density". By $A$ we denote a commutative algebraic group defined over $k$ and by $K_n$ the extension of $k$ trivializing $A[p^n]$, i.e. the $p^n$-division field of $A$ over $k$, where $p$ is a fixed prime and $n$ is a positive integer. ## Acknowledgements {#acknowledgements .unnumbered} The first author was supported by a Heisenberg grant of the DFG (grant nr. IV 177/3-1), based at the Universities of Bonn and Bochum. Also he was supported by the Leibniz Universität Hannover. The second author is a member of INdAM-GNSAGA. # Testing local-global divisibility at a stable set ## Review of stable sets We recall the following definition. **Definition 2** ([@Ivanov_stable_sets], §2). Let ${\mathscr L}/k$ be an algebraic extension, $S \subseteq \Sigma_k$ and $\lambda > 1$. - A finite subextension ${\mathscr L}/K/k$ is called *$\lambda$-stabilizing* (resp. *persisting*) for $S$, if there exists a subset $T \subseteq S$ and some $a \in (0,1]$ such that for all finite subextensions ${\mathscr L}/L/K$ one has $\lambda a > \delta_L(T_L) \geq a$ (resp. $\delta_L(T_L) = \delta_K(T_K) > 0$). - $S$ is *$\lambda$-stable* (resp. *persisting*) for ${\mathscr L}/k$, if it has a $\lambda$-stabilizing (resp. persisting) subextension of ${\mathscr L}/k$. There are many natural examples of stable and persistent sets, cf. [@Ivanov_stable_sets §3]. For example, if $\ell/k$ is a finite Galois extension, then for $\sigma \in \mathop{\mathrm{Gal}}_{\ell/k}$ the set $$\label{eq:chebotarev_set} P_{\ell/k}(\sigma) = \{ {\mathfrak p}\in \Sigma_k \colon {\mathfrak p}\text{ is unramified in $\ell/k$ and } \mathop{\mathrm{Frob}}_{{\mathfrak p}} = C(\sigma,\mathop{\mathrm{Gal}}_{l/k}) \},$$ where $C(\sigma,G_{\ell/k})$ denotes the conjugacy class of $\sigma$, is persistent for any algebraic extension ${\mathscr L}/k$ satisfying $C(\sigma,G_{\ell/k}) \cap \mathop{\mathrm{Gal}}_{\ell/ {\mathscr L}\cap \ell} \neq \varnothing$, with persisting field ${\mathscr L}\cap \ell$. In §[3](#sec:stable_set){reference-type="ref" reference="sec:stable_set"} we give new interesting examples of persistent sets. Stable sets generalize sets of density one in the following sense. Let $\ell/k$ be a finite extension. If $S$ is a set of primes of $k$ with density one, then any element of $\mathop{\mathrm{Gal}}_{\ell/k}$ is a Frobenius at $S$, and consequently any cyclic subgroup is a decomposition subgroup of a prime in $S$. Weakening the assumption on $S$ to be $p$-stable for $\ell/k$ with stabilizing field $k$, destroys the claim about *elements*, but the claim about *cyclic $p$-subgroups* (that is, of $p$-power order and not just of order $p$) remains true: **Lemma 3** ([@Ivanov_stable_sets], Lemma 4.4). *Let $\ell/k$ be a finite Galois extension, $S$ a set of primes of $k$ and $p$ a rational prime such that $S$ is $p$-stable for $\ell/k$ with $p$-stabilizing field $k$. Then any cyclic $p$-subgroup of $\mathop{\mathrm{Gal}}_{\ell/k}$ is the decomposition subgroup of a prime in $S$.* ## $p$-stable sets detect local-global divisibility by $p^n$ Recall the definition of the cohomology group satisfying the *local conditions* [@DvornicichZ_01]. Let $\Gamma$ be a finite group and $M$ a discrete $\Gamma$-module. Then $$\label{eq:cohomology_with_local_conditions} H^1_{\rm loc}(\Gamma,M) := \ker\left( H^1(\Gamma,M) \rightarrow \prod_{C\subseteq \Gamma} H^1(C,M) \right),$$ where the product is taken over all cyclic subgroups $C \subseteq \Gamma$, and the map is the product of restriction maps. **Lemma 4**. *Let $\Gamma,M$ be as above, and suppose that $M$ is $p$-primary for a rational prime $p$. Then $H^1_{\rm loc}(\Gamma,M) = \ker\left( H^1(\Gamma,M) \rightarrow \prod_{C\subseteq \Gamma} H^1(C,M) \right)$, where the product is taken over all cyclic $p$-subgroups $C \subseteq \Gamma$.* *Proof.* This follows from the fact that for any finite group $H$ and any $p$-primary module $M$, $H^1(H,M) \rightarrow H^1(H_p,M)$ is injective, where $H_p$ is a $p$-Sylow subgroup of $H$ (see [@NSW (1.6.10)]). ◻ Let $K_n^{\rm ab}(p)/k$ be the maximal abelian pro-$p$ extension of $K_n$. The following generalization of [@DvornicichZ_01 Prop. 2.1] shows that it suffices to test local-global divisibility by $p^n$ at a $p$-stable set of primes: **Proposition 5**. *Let $p$ be a rational prime and $n \geq 1$ an integer. Let $A/k$ be a commutative algebraic group. Let $S$ be a set of primes of $k$, which is $p$-stable for $K_n^{\rm ab}(p)/k$ with $p$-stabilizing field $k$. Assume that $H^1_{\rm loc}(\mathop{\mathrm{Gal}}_{K_n/k}, A[p^n]) = 0$. Then the following holds. Let $P \in A(k)$, such that for all ${\mathfrak p}\in S$, there is some $Q_{\mathfrak p}\in A(k_{\mathfrak p})$ with $P_{{\mathfrak p}} = p^n Q_{{\mathfrak p}}$. Then there is some $Q \in A(k)$ with $P = p^n Q$.* *Proof.* Let $D \in A(\overline {\mathbb Q})$ be a point with $p^n D = P$ and let $\ell=k(D)$ be the corresponding extension of $k$. Put $F = K_n\cdot \ell$. Then $F/k$ is Galois and $\ell/k$ is cyclic of $p$-power degree, so in particular $F \subseteq K_n(p)^{\rm ab}$. One can define a $1$-cocycle $c \colon \mathop{\mathrm{Gal}}_{F/k} \rightarrow A[p^n]$, by $c(\sigma):=\sigma(D)-D$, for all $\sigma\in \mathop{\mathrm{Gal}}_{F/k}$. Its image $[c] \in H^1(\mathop{\mathrm{Gal}}_{F/k}, A[p^n])$ is zero if and only if $P = p^n D'$ for some $D' \in A(k)$, see [@DvornicichZ_01 p. 320]. Moreover, as by assumpion $P$ is locally $p^n$-divisible at any ${\mathfrak p}\in S$, the same argument with cocycles show that the restriction of $[c]$ to $H^1(C,A[p^n])$ is zero, where $C \subseteq \mathop{\mathrm{Gal}}_{F/k}$ is the decomposition subgroup of any prime in $S$. Now, as $F \subseteq K_n(p)^{{\rm ab}}$, by Lemma [Lemma 3](#lm:main_lemma_about_stable_sets){reference-type="ref" reference="lm:main_lemma_about_stable_sets"} the set of decomposition subgroups in $\mathop{\mathrm{Gal}}_{F/k}$ at primes in $S$ contains the set of all cyclic $p$-subgroups of $\mathop{\mathrm{Gal}}_{F/k}$, and so by Lemma [Lemma 4](#lm:local_conditions_only_psubgroups){reference-type="ref" reference="lm:local_conditions_only_psubgroups"} we deduce $[c] \in H^1_{\rm loc}(\mathop{\mathrm{Gal}}_{F/k},A[p^n])$. To finish the proof of Proposition [Proposition 5](#prop:p_stable_sets_test_p_divisibility){reference-type="ref" reference="prop:p_stable_sets_test_p_divisibility"} it thus remains to show that the restriction via $\mathop{\mathrm{Gal}}_{F/k} \twoheadrightarrow \mathop{\mathrm{Gal}}_{K_n/k}$ induces an isomorphism $H^1_{\rm loc}(\mathop{\mathrm{Gal}}_{K_n/k},A[p^n]) \stackrel{\sim}{ \rightarrow } H^1_{\rm loc}(\mathop{\mathrm{Gal}}_{F/k},A[p^n])$. But this is done in the proof of [@DvornicichZ_01 Prop. 2.1]. ◻ **Corollary 6**. *Let $p$ be a rational prime and $n \geq 1$ an integer. Let $A/k$ be a commutative algebraic group. Let $S$ be a set of primes of $k$, which is $p$-stable for $K_n^{\rm ab}(p)/k$ with $p$-stabilizing field $k$. Assume that $P \in A(k)$, such that for all ${\mathfrak p}\in S$, there is some $Q_{\mathfrak p}\in A(k_{\mathfrak p})$ with $P_{{\mathfrak p}} = p^n Q_{{\mathfrak p}}$. Then $D\in A(K_n)$, for all $D$ such that $P=p^nD$.* *Proof.* One can apply the same argument used in the proof of [@DvornicichZ_01 Corollary 2.3], by substituting the set of density one with the set $S$. ◻ We denote by $G_k$ the absolute Galois group $\mathop{\mathrm{Gal}}_{\bar{{\mathbb Q}}/k}$ and by $G_{k_{{\mathfrak p}}}$ the absolute Galois group $\mathop{\mathrm{Gal}}_{\bar{k}_{{\mathfrak p}}/k_{{\mathfrak p}}}$, where $\bar{k}_{{\mathfrak p}}$ is an algebraic closure of $k_{\mathfrak p}$. Let $S$ be a subset of primes of $k$ which is $p$-stable at $K_n^{\rm ab}(p)/k$ with $p$-stabilizing field $k$. We define a modified Tate-Shafarevich group related to $S$ and we will prove that it is isomorphic to $H^1_{{\rm loc}}(\mathop{\mathrm{Gal}}_{K_n/k},A[p^n])$. **Definition 7**. Let $A$ be a comutative algebraic group defined over a number field $k$ and let $S$ be a set of primes of $k$, unramified in $K_n$, which is $p$-stable at $K_n^{\rm ab}(p)/k$ with $p$-stabilizing field $k$. We denote by $\Sha_{S}(k,A[p^n])$ the subgroup of $H^1(G_k,A[p^n])$ formed by the classes of the cocycles vanishing in $k_{{\mathfrak p}}$, for all ${\mathfrak p}\in S$, i.e.  $$\label{Sha} \Sha_{S}(k,A[p^n]):=\bigcap_{{\mathfrak p}\in S} \ker ( H^1(G_k,A[p^n])\xrightarrow{\makebox[1cm]{{\small $res_{\mathfrak p}$}}} H^1(G_{k_{{\mathfrak p}}},A[p^n]))$$ Notice that in Equation [\[Sha\]](#Sha){reference-type="eqref" reference="Sha"} by replacing $S$ with the set of primes of $k$ one gets the definition of the classical Tate-Shafarevich group $\Sha(k,A[p^n])$. We are going to prove that $H^1_{\rm loc}(G, A[p^n])\simeq \Sha_{S}(k, A[p^n])$. **Proposition 8**. *Let $p$ be a rational prime and $n \geq 1$ an integer. Let $A/k$ be a commutative algebraic group. Let $S$ be a set of primes of $k$, unramified in $K_n$, which is $p$-stable for $K_n^{\rm ab}(p)/k$ with $p$-stabilizing field $k$. Then $H^1_{\rm loc}(G, A[p^n]) \simeq \Sha_{S}(k,A[p^n])$.* *Proof.* Let $S_{K_n}$ denote the set of primes $w_{{\mathfrak p}}$ of $K_n$ extending the primes ${\mathfrak p}$ in $S$ and by $K_{n,w_{{\mathfrak p}}}$ we denote the completion of $K_n$ at the place $w_{{\mathfrak p}}$. Let $G_{{\mathfrak p}}:= \mathop{\mathrm{Gal}}_{K_{n,w_{{\mathfrak p}}}/k_{{\mathfrak p}}}$ and consider the following diagram given by the inflation restrictions exact sequence $$\begin{array}{ccccccc} 0 & \xrightarrow{\hspace{0.4cm}} & H^1(G,A[p^n]) & \xrightarrow{\hspace{0.1cm}inf\hspace{0.1cm}} & H^1(G_k,A[p^n]) & \xrightarrow{\hspace{0.1cm}res\hspace{0.1cm}} & H^1(G_{K_n},A[p^n]) \\ & &\hspace{0.8cm} \Big\downarrow {\prod res{_{\mathfrak p}}} & &\hspace{0.8cm}\Big\downarrow {\prod res{_{\mathfrak p}}} & &\hspace{0.8cm}\Big\downarrow {\prod res_{w_{\mathfrak p}}} \\ 0 & \xrightarrow{\hspace{0.4cm}} &\prod_{{\mathfrak p}\in S} H^1(G{_{\mathfrak p}},A[p^n]) & \xrightarrow{\hspace{0.1cm}inf\hspace{0.1cm}} & \prod_{{\mathfrak p}\in S}H^1(G_{k{_{\mathfrak p}}},A[p^n]) & \xrightarrow{\hspace{0.1cm}res\hspace{0.1cm}} &\prod_{{w_{\mathfrak p}}\in S_{K_n}} H^1(G_{K_{n,w_{\mathfrak p}}},A[p^n]) \\ \end{array}$$ The kernel of the vertical map on the left is $H^1_{{\rm loc}}(G,A[p^n])$ by assumption on $S$ and by Lemma [Lemma 3](#lm:main_lemma_about_stable_sets){reference-type="ref" reference="lm:main_lemma_about_stable_sets"} and the kernel of the central vertical map is $\Sha_{S}(k,A[p^n])$. The vertical map on the right is injective because of $G_{K_n}$ acting trivially on $A[p^n]$ and because of $G_{K_{n,w}}$ varying over all cyclic subgroups of $G_{K_n}$ as $w$ varies in $S_{K_n}$ by Lemma [Lemma 3](#lm:main_lemma_about_stable_sets){reference-type="ref" reference="lm:main_lemma_about_stable_sets"}. Therefore $H^1_{{\rm loc}}(G,A[p^n])\simeq \Sha_{S}(k,A[p^n])$. ◻ At cost of slightly strengthening the assumption on $S$, we can eliminate the dependence on the particular algebraic group $A$ in Proposition [Proposition 5](#prop:p_stable_sets_test_p_divisibility){reference-type="ref" reference="prop:p_stable_sets_test_p_divisibility"}. For an integer $M > 0$, consider the class ${\mathfrak C}_{\leq M}(k)$ of all commutative algebraic groups over $k$, such that $\dim_{{\mathbb F}_p}\#A[p](\overline {\mathbb Q}) \leq M$. Moreover, for a set ${\mathbb P}\subseteq \Sigma_{\mathbb Q}$ of rational primes, let $k({\mathbb P})$ be the compositum of all finite extensions of $k$ whose orders are products of elements of ${\mathbb P}$. **Corollary 9**. *Let $p$ be a rational prime and let $M>0$. Let $S$ be a set of primes of $k$. Assume that $S$ is $p$-stable with $p$-stabilizing field $k$ for the extension $k({\mathbb P}(M))/k$, where ${\mathbb P}(M)$ is the set of all prime divisors of $\#\mathop{\mathrm{GL}}_{M'}({\mathbb F}_p)$ for all $M' \leq M$. Then for any $A \in {\mathfrak C}_{\leq M}(k)$, such that $H^1_{\rm loc}(\mathop{\mathrm{Gal}}_{K_n/k}, A[p^n]) = 0$, the following holds.* *Let $P \in A(k)$, such that for all ${\mathfrak p}\in S$, there is some $Q_{\mathfrak p}\in A(k_{\mathfrak p})$ with $P_{{\mathfrak p}} = p^n Q_{{\mathfrak p}}$. Then there is some $Q \in A(k)$ with $P = p^n Q$.* For example, if one restricts further to the class of elliptic curves one can take $M = 2$ and ${\mathbb P}$ the set of prime divisors of $p,p\pm 1$. For abelian varieties of dimension $\leq g$, one can take $M=2g$. *Proof of Corollary [Corollary 9](#cor:p_stable_sets_test_p_divisibility){reference-type="ref" reference="cor:p_stable_sets_test_p_divisibility"}.* We have to show that for a particular $A \in {\mathfrak C}_M(k)$, the assumptions of Proposition [Proposition 5](#prop:p_stable_sets_test_p_divisibility){reference-type="ref" reference="prop:p_stable_sets_test_p_divisibility"} hold, i.e., that $K_n(p)^{{\rm ab}} \subseteq k({\mathbb P}(M))$. As $p \in {\mathbb P}(M)$, it suffices to show that $K_n \subseteq k({\mathbb P}(M))$. Also as $K_n/K_1$ is a $p$-extension, it suffices to show that $K_1/k \subseteq k({\mathbb P}(M))$. But $\mathop{\mathrm{Gal}}_{K_1/k} \subseteq \mathop{\mathrm{GL}}(A[p])$, which is a subgroup of $\mathop{\mathrm{GL}}_{M'}({\mathbb F}_p)$ for some $M'\leq M$. ◻ Strengthening assumptions on $S$ even further, Corollary [Corollary 9](#cor:p_stable_sets_test_p_divisibility){reference-type="ref" reference="cor:p_stable_sets_test_p_divisibility"} immediately gives the following: **Corollary 10**. *Fix a prime $p$. Let $S$ be a set of primes of $k$, which is persistent for $\overline {\mathbb Q}/k$ with persisting field $k$. For any commutative algebraic group $A/k$ and any $n > 0$, if $H^1_{\rm loc}(\mathop{\mathrm{Gal}}_{K_n/k},A[p^n]) = 0$, the following holds.* *Let $P \in A(k)$, such that for all ${\mathfrak p}\in S$, there is some $Q_{\mathfrak p}\in A(k_{\mathfrak p})$ with $P_{{\mathfrak p}} = p^n Q_{{\mathfrak p}}$. Then there is some $Q \in A(k)$ with $P = p^n Q$.* We show in Proposition [Proposition 11](#prop:absolutely_persistent_set){reference-type="ref" reference="prop:absolutely_persistent_set"} below that sets $S$ satisfying the requirements of the corollary exist in abundance. Using this along with existing results on the original form of Problem [Problem 1](#problem:DZ){reference-type="ref" reference="problem:DZ"}, Corollary [Corollary 10](#cor:for_absolutely_stable_S){reference-type="ref" reference="cor:for_absolutely_stable_S"} specializes to the case of elliptic curves: *Proof of Theorem [Theorem 1](#thm:small_test_set){reference-type="ref" reference="thm:small_test_set"}.* Pick a set $S \subseteq \Sigma_{\mathbb Q}$ as constructed in Proposition [Proposition 11](#prop:absolutely_persistent_set){reference-type="ref" reference="prop:absolutely_persistent_set"}. Let $E/k$ be an elliptic curve. By Corollary [Corollary 10](#cor:for_absolutely_stable_S){reference-type="ref" reference="cor:for_absolutely_stable_S"} it suffices to show that $H^1_{\rm loc}(\mathop{\mathrm{Gal}}_{K_p/k},E[p]) = 0$ for all $p$, resp. that $H^1_{\rm loc}(\mathop{\mathrm{Gal}}_{K{p^n}/k},E[p^n]) = 0$ for all $p>B(d)$ and all $n > 1$. The first case is easy, as was observed in [@DvornicichZ_01 beginning of §3]: $\mathop{\mathrm{Gal}}_{K_p/k}$ is then a subgroup of $\mathop{\mathrm{GL}}_2({\mathbb F}_p)$, which implies that the $p$-Sylow subgroup of $\mathop{\mathrm{Gal}}_{K_p/k}$ is cyclic and so $H^1_{\rm loc}(\mathop{\mathrm{Gal}}_{K_p/k},E[p]) = 0$. In the second case we may apply [@PaladinoRV_12 Theorem 1' (on p. 8)] (see also Corollary 2 of *loc. cit.*) and [@PaladinoRV_14]. ◻ # New examples of stable sets {#sec:stable_set} It is easy to give examples of stable sets $S$ with arbitrary small density in the whole tower ${\mathscr L}/k$, when ${\mathscr L}$ is some reasonably small subextension of $\overline{\mathbb Q}/k$. However, those examples will often not be stable for other towers ${\mathscr L}'/k$. Consider, for example, $S = P_{\ell/k}(\sigma)$ as in [\[eq:chebotarev_set\]](#eq:chebotarev_set){reference-type="eqref" reference="eq:chebotarev_set"}. If $\sigma = 1$, then $S$ will be stable --even persistent-- for any extension ${\mathscr L}/k$, but if ${\mathscr L}\supseteq \ell$, the persisting field is $\ell$ and $\delta_\ell(S_\ell) = 1$, that is $S_\ell$ eventually becomes "big". On the other hand, if $\sigma \neq 1$, then $\delta_{\ell'}(S_{\ell'}) = 0$ for any finite $\ell'/\ell$, and hence $S$ is not stable for ${\mathscr L}/k$ whenever ${\mathscr L}\supseteq \ell$. With this in mind, we now produce now many examples of sets persistent for $\overline{\mathbb Q}/k$ with persisting field $k$ and arbitrary small positive density. **Proposition 11**. *For any number field $k$ and any $\varepsilon > 0$, there exists a set $S$ of primes of $k$ satisfying $$0 < \delta_k(S) = \delta_\ell(S_\ell) < \varepsilon$$ for all finite extensions $\ell/k$. In particular, $S$ is persistent for $\overline{\mathbb Q}/k$ with persisting field $k$.* *Proof.* Let $p$ be a prime such that $\frac{1}{p-1} < \varepsilon$. Let $k_\infty/k$ be a ${\mathbb Z}_p$-extension (e.g. the cyclotomic one) with Galois group identified with ${\mathbb Z}_p$. For $n\geq 1$, choose $a_n \in {\mathbb F}_p^\times$ and consider the set $$A = \bigcup_{n\geq 1} \left(a_n p^{n-1} + p^n {\mathbb Z}_p\right) \subseteq {\mathbb Z}_p.$$ We put $$S = P_{k_\infty/k}(A),$$ the set of all primes unramified in $k_\infty/k$, whose Frobenius lies in $A$. Clearly, $A$ is open in ${\mathbb Z}_p$ and one computes $\overline A = A \cup \{0\}$. Equip ${\mathbb Z}_p$ with the invariant Haar measure $\mu$ normalized such that $\mu({\mathbb Z}_p) = 1$. Then the boundary $\overline A {\,\smallsetminus\,}A^\circ = \{0\}$ of $A$ has measure $0$, and the infinite Chebotarev theorem [@Serre_abelian I.2.2 Corollary 2b)] (which we may apply as $k_\infty/k$ is ramified at most in the finitely many primes above $p$ and $\infty$) then shows that $S$ has a density and that it is equal to $$\delta_k(S) = \mu(A) = \sum_{n\geq 1}^\infty p^{-n} = \frac{1}{p-1}.$$ Now, let $\ell/k$ be a finite extension. Then there is some $m \geq 0$ such that $\ell \cap k_\infty = k_{m} := (k_\infty)^{p^{m}{\mathbb Z}_p}$. Let $\ell_\infty = k_\infty . \ell$. Via $\mathop{\mathrm{Gal}}_{\ell_\infty/\ell} \cong \mathop{\mathrm{Gal}}_{k_\infty/k_{m}}$, we may identify $\mathop{\mathrm{Gal}}_{\ell_\infty/\ell}$ with $p^{m}{\mathbb Z}_p \subseteq {\mathbb Z}_p$. Let ${\rm Spl}_{\ell/k}$ denotes the set of primes of $\ell$, which are split (and unramified) over $k$. Then $$\label{eq:formula_for_pullback_of_primes} S_\ell \cap {\rm Spl}_{\ell/k} = P_{\ell_\infty/\ell}(A \cap p^{m}{\mathbb Z}_p),$$ where we ignore the finitely many primes of $\ell$ which ramify in $\ell_\infty/k$. Now, $\Sigma_\ell {\,\smallsetminus\,}{\rm Spl}_{\ell/k}$ consists of primes of $\ell$, which are not split over ${\mathbb Q}$, so it has density $0$. In particular, $\delta_\ell(S_\ell)$ exists if and only if $\delta_\ell(S_\ell \cap {\rm Spl}_{\ell/k})$ exists, in which case both agree. On the other hand, the argument using infinite Chebotarev applied above to compute $\delta_k(S)$ applies also to $P_{\ell_\infty/\ell}(A \cap p^{m}{\mathbb Z}_p)$, giving $\delta_\ell(P_{\ell_\infty/\ell}(A \cap p^{m}{\mathbb Z}_p)) = \frac{1}{p-1}$. Combining the two computations, we get $\delta_\ell(S_\ell) = \delta_\ell(P_{\ell_\infty/\ell}(A \cap p^{m}{\mathbb Z}_p)) = \frac{1}{p-1}$, finishing the proof. ◻ **Remark 12**. One has to be careful in the above proof, as the Dirichlet density does not satisfy $\sigma$-additivity: suppose that $T_n \subseteq \Sigma_k$ ($n \geq 1$) is a collection of mutually disjoint subsets, such that $\delta_k(T_n)$ exists. Let $T = \bigcup_n T_n$. Then it might happen that $\delta_k(T)$ does not exist, and even if it exists, it might happen that $\delta_k(T) \neq \sum_{n\geq 1} \delta_k(T_n)$ (it is enough to consider singletons $T_n = \{{\mathfrak p}_n\}$ for any $n$). However, by the argument in the proof of Proposition [Proposition 11](#prop:absolutely_persistent_set){reference-type="ref" reference="prop:absolutely_persistent_set"}, the density of the set $S$, which is in fact a disjoint union of infinitely many Chebotarev sets, exists and is equal to the sum of densities of these Chebotarev sets. # On a question posed by Cassels In addition Problem [Problem 1](#problem:DZ){reference-type="ref" reference="problem:DZ"} is strongly related to the following question stated by Cassels in 1962 that remained open for 50 years (see [@DP_22] for further details). **Cassels' question 1**. *Let $k$ be a number field and $E$ be an elliptic curve defined over $k$. Are the elements of $\Sha(k,E)$ infinitely divisible by a prime $p$ when considered as elements of the Weil-Châtelet group $H^1(G_k,A)$ of all classes of principal homogeneous spaces for $E$ defined over $k$?* An affirmative answer for $p>B(d)$ is implied by [@Cre2 Theorem 3] and the results in [@PaladinoRV_12; @PaladinoRV_14]. Since 1972 this question was considered in abelian varieties of every dimension by various authors [@Bas; @Cre2; @CS1]. In particular, Creutz showed sufficient and necessary conditions to get an affirmative answer and the existence of counterexamples for every $p$ in infinitely many abelian varieties [@Cre2]. [@Cre]. Moreover, Çiperiani and Stix also showed sufficient conditions to get an affirmative answer [@CS1]. In the spirit of this article, we can pose the following more general question. **Problem 2**. *Let $k$ be a number field and $A$ an abelian variety defined over $k$. Let $S$ be an infinite set of places of $k$ and let $$\Sha_{S}(k,A):=\bigcap_{{\mathfrak p}\in S} \ker ( H^1(G_k,A[p^n])\xrightarrow{\makebox[1cm]{{\small $res_{\mathfrak p}$}}} H^1(G_{k_{{\mathfrak p}}},A[p^n])).$$ Are the elements of $\Sha_{S}(k,A)$ infinitely divisible by a prime $p$ when considered as elements of the Weil-Châtelet group $H^1(G_k,A)$?* As a consequence of Proposition [Proposition 8](#thm:p_stable_sets_test_p_divisibility){reference-type="ref" reference="thm:p_stable_sets_test_p_divisibility"}, in the case of elliptic curves curves Problem [Problem 2](#prob2){reference-type="ref" reference="prob2"} has an affirmative answer for all $p>B(d)$, for every sets $S$ which is $p$-stable for $\cup_{n\in {\mathbb N}} K_n^{ab}(p)/k$ with $p$-stabilizing field $k$. **Corollary 13**. *Let $k$ be a number field and $E$ an elliptic curve over $k$. Let $S$ be a set of places of $k$ which is $p$-stable for $\cup_{n\in {\mathbb N}} K_n^{ab}(p)/k$ with stabilizing field $k$. Then the elements of $\Sha_{S}(k,E)$ are infinitely divisible by every $p>B(d)$ when considered as elements of $H^1(G_k,E)$.* *Proof.* By the results in [@PaladinoRV_12] and [@PaladinoRV_14], we have $H^1_{{\rm loc}}(G,E[p^n])=0$, for all $p>B(d)$ and all $n\geq 1$. By Proposition [Proposition 8](#thm:p_stable_sets_test_p_divisibility){reference-type="ref" reference="thm:p_stable_sets_test_p_divisibility"}, we have $\Sha_{S}(k,E[p^n])=0$, for all $p>B(d)$ and all $n\geq 1$. The conclusion is then implied by [@Cre2 Theorem 3]. ◻ Observe that by Proposition [Proposition 11](#prop:absolutely_persistent_set){reference-type="ref" reference="prop:absolutely_persistent_set"} there are many sets $S$ of primes satisfying the assumtpions of Corollary [Corollary 13](#cor_Cass){reference-type="ref" reference="cor_Cass"}.

arxiv_math

--- abstract: | We consider inverse problems for a Westervelt equation with a strong damping and a time-dependent potential $q$. We first prove that all boundary measurements, including the initial data, final data, and the lateral boundary measurements, uniquely determine $q$ and the nonlinear coefficient $\beta$. The proof is based on complex geometric optics construction and the approach proposed by Isakov. Further, by considering fundamental solutions supported in a half-space constructed by Hörmander, we prove that with vanishing initial conditions the Dirichlet-to-Neumann map determines $q$ and $\beta$. address: - Department of Mathematics, University of California, Irvine, CA 92697, USA - Department of Mathematics, University of Washington, Seattle, WA 98195, USA author: - Li Li - Yang Zhang bibliography: - local_ma.bib title: Inverse problems for a quasilinear strongly damped wave equation arising in nonlinear acoustics --- # Introduction The Westervelt equation is a classical model in the field of nonlinear acoustics. Within a thermoviscous medium, in some situations the Westervelt equation is accompanied by a strong damping term, as exemplified in [@kaltenbacher2018fundamental]. In this work, we consider inverse problems of recovering a potential and a nonlinear coefficient for the strongly damped Westervelt equation, from two types of boundary measurements. More explicitly, for $n \geq 2$, let $T > 0$ be fixed and $\Omega \subset \mathbb{R}^n$ be a bounded set with a smooth boundary $\partial \Omega$. Consider the following quasilinear strongly damped wave equation $$\label{eq_nl} \begin{aligned} (\partial^2_t - \Delta - \partial_t\Delta + q(t,x))u - \beta(t,x) \partial_t^2 (u^2) &= 0, & \ & \mbox{on } (0, T) \times \Omega,\\ u(t,x) &= h, & \ &\mbox{for } x\in \partial \Omega,\\ u = g_0, \ \partial_t u &= g_1, & \ &\mbox{for } t=0,\\ \end{aligned}$$ where $q$ is the potential, $\beta$ is the nonlinearity coefficient for the medium, and $\Delta$ is the Laplacian operator with respect to $x$. This initial-boundary value problem is locally well-posed for sufficiently small data $(g_0, g_1, h)$ satisfying the compatibility condition, see Proposition [Proposition 1](#pp_nl){reference-type="ref" reference="pp_nl"}. ## Main results {#main-results .unnumbered} We introduce the following notations. For fixed large $m$, let $\mathcal{G}^m$ be the space of all $$(g_0, g_1, h) \in H^{2m+1}(\Omega) \times H^{2m-1}(\Omega) \times C^{2m + 2}([0,T] \times \partial \Omega)$$satisfying the $m$th-order compatibility condition defined in ([\[def_comp_nl\]](#def_comp_nl){reference-type="ref" reference="def_comp_nl"}). Then there exists a unique solution $u$ to ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) for sufficiently small $(g_0, g_1, h) \in \mathcal{G}^m$. First, we consider the recovery from the so-called all boundary measurements. We define $${L}_{q, \beta}: (g_0, g_1, h) \rightarrow (u(T), \partial_t u(T), \partial_\nu u|_{\partial \Omega}),$$ for small initial-boundary data $(g_0, g_1, h) \in \mathcal{G}^m$. We prove that the following result. **Theorem 1**. *Let $n \geq 2$, $m \geq 2(\lfloor n/2 \rfloor +1)$, and $T > 0$ be fixed. Let $\Omega \subset \mathbb{R}^n$ be a bounded open set with smooth boundary. For $k = 1,2$, suppose $q^{(k)}, \beta^{(k)} \in C^\infty([0,T] \times \bar{\Omega})$ with $\partial_t^j \beta(0,x) = 0$ for $x \in \partial \Omega$ and $j = 0, \ldots, m$. Suppose $$L_{{q^{(1)}}, \beta^{(1)}}(g_0, g_1, h) = L_{{q^{(2)}}, \beta^{(2)}}(g_0, g_1, h)$$ for small $(g_0, g_1, h) \in \mathcal{G}^m$, then $${q^{(1)}}= {q^{(2)}}, \qquad \beta^{(1)}= \beta^{(2)}, \qquad \text{for } (t,x)\in [0,T] \times \bar{\Omega}.$$* To prove this result, we construct a exponentially growing geometric optics solution (i.e., the phase function has complex frequency) and use regular fundamental solutions constructed in [@H2005analysis Chapter 10] to estimate the remainder term. This approach is first introduced in [@isakov1991completeness], which provides an alternative proof of the unique determination theorem for the operator $\Delta+ q$ in [@sylvester1987global] and can be applied to other differential operators with constant coefficients. This idea is also used in [@krupchyk2010borg] to prove a multidimensional Borg-Levinson theorem for elliptic operators of higher order with constant coefficients. With the remainder term relatively small, for our model, the recovery of $q$ comes from the linear problem and that of $\beta$ comes from the second-order linearization. In many applications the initial data vanish, and in some cases measuring the final data is impossible. To address this, in the second part of this paper, we consider the recovery of the potential $q$ and the nonlinear coefficient $\beta$ from lateral boundary measurements. More precisely, we define $$\mathcal{H}^m= \{h \in C^{2m+2}([0,T]\times \partial \Omega): \ \partial_t^j h(0,x) = 0, \text{ for } x \in \partial \Omega \text{ and } j = 0, \ldots, m\}.$$ When $g_0 = g_1 = 0$, the boundary value problem ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) has a unique solution for small $h \in \mathcal{H}^m$, as a special example of Proposition [Proposition 1](#pp_nl){reference-type="ref" reference="pp_nl"}. We consider the Dirichlet-to-Neumann map $$\Lambda_{q, \beta}: h \rightarrow \partial_\nu u|_{\partial \Omega},$$ where $u$ satisfies ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) with $g_0 = g_1 =0$. We prove the following result. **Theorem 2**. *Let $n \geq 2$, $m \geq 2(\lfloor n/2 \rfloor +1)$, and $T > 0$ be fixed. Let $\Omega \subset \mathbb{R}^n$ be a bounded and convex open set with smooth boundary. For $k = 1,2$, suppose $q^{(k)}, \beta^{(k)} \in C^\infty([0,T] \times \bar{\Omega})$. Suppose $$\Lambda_{{q^{(1)}}, \beta^{(1)}}(h) = \Lambda_{{q^{(2)}}, \beta^{(2)}}(h)$$ for small $h \in \mathcal{H}^m$, then $${q^{(1)}}= {q^{(2)}}, \qquad \beta^{(1)}= \beta^{(2)}, \qquad \text{for } (t,x)\in [0,T] \times \bar{\Omega}.$$* To prove this theorem, we construct a complex geometric optics solution and would like to follow the same idea that we use earlier. To construct a remainder term with vanishing initial conditions, we consider the characteristic Cauchy problems studied in [@H2005analysis Chap 12]. We observe that for the differential operator involved in the equation for the remainder term, see ([\[eq_wR_new\]](#eq_wR_new){reference-type="ref" reference="eq_wR_new"}), the boundary of the half-space $H = \{t \geq \varepsilon\}$ is characteristic, where $\varepsilon>0$ is small. Then there is no uniqueness for the initial value problem and the aim is to find a solution with support in $H$ (equivalently, a fundamental solution with support in $H$). We describe under which condition such fundamental solution almost exists and particularly with a desired estimate, see Proposition [Proposition 9](#pp_12813){reference-type="ref" reference="pp_12813"} and [Proposition 10](#pp_Eestimate_new){reference-type="ref" reference="pp_Eestimate_new"}. Then we verify for our model, this condition is satisfied and therefore the remainder term is relatively small. Then the recovery of $q$ and $\beta$ follows from a similar argument as before. We note that in these two theorems above, the assumption that $q$ and $\beta$ are smooth is just for convenience. One may consider non-smooth functions but with certain regularity, which mainly depends on the assumptions for the local well-posedness. Inverse problems for Westervelt equations has been considered in [@acosta2022nonlinear; @eptaminitakis2022weakly], for a general nonlinearity in [@uhlmann2023inverse], and with various damping effects in [@kaltenbacher2021identification; @kaltenbacher2023simultaneous; @kaltenbacher2022inverse; @kaltenbacher2023nonlinearity; @zhang2023nonlinear; @li2023inverse]. Most recently in [@fu2023inverse], the recovery of a time-dependent nonlinearity for the Jordan-Moore-Gibson-Thompson equation is considered, from all boundary measurements. The model considered in [@fu2023inverse] has the principal part given by $\partial_t^3 - b \partial_t\Delta$, for a constant $b>0$. In this paper, the principal part of our model is given by the strong damping $\partial_t\Delta$, which is not hypoelliptic or of principal type. The idea of using multi-fold linearization and nonlinear interaction of solutions to solve inverse problems can be found in [@kurylev2018inverse] for a semilinear wave equation, see also [@lassas2018inverse]. In [@acosta2022nonlinear], the authors use Gaussian beams and the second-order linearization to recover the nonlinear coefficient for a Westervelt equation without damping. For inverse problems for time-dependent nonlinear Schrödinger equations, in [@lassas2022inverse] the authors use geometric optics construction and the second-order linearization to recover a potential and the nonlinear coefficient; later in [@lai2023partial], the stability and partial data problem are considered. The multi-fold linearization method has also been used to solve inverse problems for semilinear elliptic and fractional equations (see [@krupchyk2020remark; @lassas2020partial; @li2023elas] for instance). The rest of this paper is organized as follows. In Section [2](#sec_well){reference-type="ref" reference="sec_well"}, we consider the initial-boundary value problem in ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) and prove the local well-posedness. In Section [3](#sec_prelim){reference-type="ref" reference="sec_prelim"}, we briefly recall some results in [@H2005analysis Chap 10, 12] and we prove Proposition [Proposition 4](#pp_E0){reference-type="ref" reference="pp_E0"}, [Proposition 9](#pp_12813){reference-type="ref" reference="pp_12813"}, [Proposition 10](#pp_Eestimate_new){reference-type="ref" reference="pp_Eestimate_new"}, which provides estimates for regular fundamental solutions and fundamental solutions with support in a half space. Note that these results may be used for other differential operators with constant coefficients. In Section [4](#sec_goc){reference-type="ref" reference="sec_goc"}, we construct geometric optic solutions using a complex linear phase function. In Section [5](#sec_allbd){reference-type="ref" reference="sec_allbd"}, we prove Theorem [Theorem 1](#thm1){reference-type="ref" reference="thm1"} using regular fundamental solutions. In Section [6](#sec_gocnew){reference-type="ref" reference="sec_gocnew"}, we prove Theorem [Theorem 2](#thm2){reference-type="ref" reference="thm2"} using fundamental solutions with support in a half space. In both sections, we recover $q$ from the first-order linearization of the measurements and we recover $\beta$ from the second-order linearization. In Appendix [7](#sec_apdx){reference-type="ref" reference="sec_apdx"}, we prove the local well-posedness for the linearized problem, which is used in the proof of Section [2](#sec_well){reference-type="ref" reference="sec_well"}. **Acknowledgements.** L.L. and Y.Z. would like to thank Professor Katya Krupchyk and Professor Gunther Uhlmann for helpful discussions. # The local well-posedness {#sec_well} In this section, we prove the initial-boundary value problem ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) is locally well-posed. The local and global well-posedness for the strongly damped Westervelt equation without the potential $q(t,x)$ is proved in [@kaltenbacher2011well], where properties of the semigroup generated by strongly damped wave equations are used. Here we follow the ideas in [@dafermos1985energy] for our model. For $r \in \mathbb{R}$, we use $\lfloor r \rfloor$ to denote the largest integer that is no greater than $r$ and we use $\lceil r \rceil$ to denote the largest integer that is no less than $r$. For convenience, we define $$\begin{aligned} \label{eq_kn} k_n= \lfloor n/2 \rfloor +1.\end{aligned}$$ The aim of this section is to establish local well-posedness for the following initial-boundary value problem $$\label{eq_nonlinear_bvp} \begin{aligned} \partial^2_tu - \Delta u - \partial_t\Delta u + q(t,x)u - \beta(t,x) \partial_t^2( u^2) &= 0, & \ & \mbox{on } (0,T)\times \Omega,\\ u &= h, & \ &\mbox{for } x\in \partial \Omega,\\ u = g_0(x), \quad \partial_t u &= g_1(x) , & \ &\mbox{for } t=0. \end{aligned}$$ Note that the initial-boundary data $g_0,g_1, h$ must be compatible in some sense. For this purpose, we recursively define $$\begin{aligned} \label{def_gl_nl} g_{l+1} \coloneqq (1-2\beta(0)g_0)^{-1} \tilde{G}_l(0,x,g_0, \ldots, g_l),\end{aligned}$$ where we write $$\begin{aligned} \label{def_tG} &\tilde{G}_l(t, x, u, \partial_tu, \ldots, \partial_t^l u)\\ %= - %\sum_{j=2}^{l}\Cjl\dt^{l+1-j} %&(1-2\beta u) \dt^j u %+ \Laplace \dt^{l-1} u %+ \Laplace \dt^l u\\ %&- \sum_{j=0}^{l-1}\Cjl\dt^{l-1-j}q \dt^j u =& %G_l(t,x, u, \dt u, \ldots, \dt^{l-1}u; g_0, \ldots, g_l) -\sum_{j=2}^{l}C_{l,j}\partial_t^{l+1-j} (1-2\beta u) \partial_t^j u + \Delta\partial_t^{l-1} u + \Delta\partial_t^{l} u %- \sum_{j=0}^{l-1}\Cjl\dt^{l-1-j}q(t) g_j - \partial_t^{l-1}(qu) -2 \partial_t^{l-1}(\beta (\partial_tu)^2). \nonumber %- 2\sum_{j=0}^{l-1}\Cjl\dt^{l-1-j}\beta \dt^j ((\dt u)^2).\end{aligned}$$ When $l=1$ we do not have the first term on the right-hand side. We say $(g_0, g_1, h)$ satisfies the *$m$th-order compatible condition*, if $$\begin{aligned} \label{def_comp_nl} g_{l} = \partial_t^{l} h(0), \qquad \text{for }l = 0, \ldots, m.\end{aligned}$$ Recall $\mathcal{G}^m$ is the space of all $(g_0, g_1, h) \in H^{2m+1}(\Omega) \times H^{2m-1}(\Omega) \times C^{2m + 2}([0,T] \times \partial \Omega)$ satisfying the $m$th-order compatibility condition ([\[def_comp_nl\]](#def_comp_nl){reference-type="ref" reference="def_comp_nl"}). We define $$\|(g_0, g_1, h)\|_{\mathcal{G}^m} = \|h\|_{C^{2m + 2}([0,T] \times {\partial \Omega})} + \|g_0\|_{H^{2m+1}}+ \|g_1\|_{H^{2m-1}}.$$ Then we prove the following proposition. **Proposition 1**. *For fixed $m \geq 2k_n$ and $T>0$, let $q, \beta \in C^{\infty}([0,T] \times \bar{\Omega})$. Suppose $(g_0, g_1, h) \in \mathcal{G}^m$ with $\|(g_0, g_1, h)\|_{\mathcal{G}^m} \leq \rho_h$. Then for sufficiently small $\rho_h >0$, the initial-boundary value problem ([\[eq_nonlinear_bvp\]](#eq_nonlinear_bvp){reference-type="ref" reference="eq_nonlinear_bvp"}) has a unique solution $$u \in \bigcap_{k=0}^{m} C^{m-k}([0,T]; H^{2k+1}(\Omega)),$$ such that for some constant $C>0$, we have $$\begin{aligned} \label{est_nl} \sum_{k=0}^m \sup_{s \in [0, T]} \|\partial_t^{m-k} u(s)\|^2_{{H^{2k+1}}} %\|u\|^2_{\zm} \leq C \|(g_0, g_1, h)\|_{\mathcal{G}^m}. \end{aligned}$$* *Proof.* Note that for $h \in C^{2m + 2}((0,T) \times \partial \Omega)$, there exists a function $u_h \in C^{2m+1}([0, T] \times \bar{\Omega})$ such that $$u_h|_{\partial \Omega} = h \quad \text{with} \quad \| u_h (t)\|_{C^{2m+2}([0,T] \times \bar{\Omega})} \leq C \|h (t)\|_{C^{2m + 2}([0,T] \times \partial \Omega)}.$$ In particular, for $\rho > 0$ to be specified later, we choose $\rho_h$ small enough such that $$\|u_h\|_{C^{2m + 2}([0,T] \times \bar{\Omega})} + \|g_0\|_{H^{2m+1}}+ \|g_1\|_{H^{2m-1}}\leq \rho.$$ Now we consider $w = u - u_h$ and we rewrite the nonlinear term as $$\beta\partial^2_t(u^2) %= \beta\ddt((w + u_h)^2) = 2\beta(w + u_h)\partial^2_tw + 2 \beta( w + u_h)\partial_t^2 u_h + 2 \beta (\partial_t(w + u_h))^2.$$ Then $w$ solves the initial value problem $$\label{eq_w} \begin{aligned} (1-2\beta(w + u_h))\partial^2_tw - \Delta w - \partial_t\Delta w + q(t,x)w &= F(u_h, w), & \ & \mbox{on } (0,T)\times \Omega,\\ w(t,x) &= 0, & \ &\mbox{for } x\in \partial \Omega,\\ w = \tilde{g}_0(x), \quad \partial_t u &= \tilde{g}_1(x), & \ &\mbox{for } t=0, \end{aligned}$$ where we write $$\begin{aligned} &\tilde{g}_0(x) = g_0(x) - u_h(0), \qquad \tilde{g}_1(x) = g_1(x)- \partial_tu_h(0),\\ &F(u_h, w) = -Pu_h + 2\beta (\partial_t(w + u_h))^2 + 2 \beta( w + u_h)\partial_t^2 u_h. \end{aligned}$$ Further, we define $\tilde{g}_l = g_l - \partial_t^l u_h(0)$, for $l = 2, \ldots, m$. With $(g_0, g_1, h)\in \mathcal{G}^m$, we have $$\tilde{g}_l \in H_0^1(\Omega) \cap H^{2(m-l)+1}(\Omega).$$ In the following, we choose fixed $r_0 > 0$ to satisfy Proposition [Proposition 15](#pp_energy){reference-type="ref" reference="pp_energy"}. For $r < r_0/2$ to be specified later, we define the set ${\widetilde{Z}^m}(r, T)$, which containing all the functions $w$ such that $$\begin{aligned} &w \in \bigcap_{k=0}^{m} C^{m-k}([0,T]; H^{2k+1}(\Omega)), \quad \text{with } \|w\|_{Z^m}\leq r \text{ and } \partial_t^l w(0) = \tilde{g}_l, \text{ for } l = 0, \ldots, m. \end{aligned}$$ We consider the linearized problem, i.e., for fixed $v \in {\widetilde{Z}^m}(r, T)$, we consider $$\label{eq_linear_w} \begin{aligned} (1-2\beta (v + u_h)) \partial^2_tw - \Delta w - \partial_t\Delta w + q(t,x)w &= F(u_h, v), & \ & \mbox{on } (0,T)\times \Omega,\\ w(t,x) &= 0, & \ &\mbox{for } x\in \partial \Omega,\\ w = \tilde{g}_0(x), \quad \partial_t w & = \tilde{g}_1(x), & \ &\mbox{for } t=0. \end{aligned}$$ Let ${J}$ be the operator that maps $v$ to the solution $w$. Then with $u_h \in C^{2m+2}([0, T] \times \bar{\Omega})$ and $v \in {Z^m}(r,T)$, we have $$\begin{aligned} % &\tg_l = g_l - \dt^l u_h(0) \in H^{2(m-l)+1}(\Omega) \cap {\teal H_0^1(\Omega)},\\ % %\qquad \tg_1 \in H^{2m-1}(\Omega)\cap {\teal H_0^1(\Omega)},\\ &\|F\|_{Z^{m-1}}\leq C(\|u_h\|_{C^{2m+2}([0, T] \times \bar{\Omega})} + (\|v\|_{Z^{m-1}}+ \|u_h\|_{C^{2m+2}([0, T] \times \bar{\Omega})})^2), \end{aligned}$$ where we use [@Uhlmann2021a Claim 1]. According to Lemma [Lemma 4](#lm_compt){reference-type="ref" reference="lm_compt"} in Appendix [7](#sec_apdx){reference-type="ref" reference="sec_apdx"}, the data $\tilde{g}_0, \tilde{g}_1, F$ satisfy the $m$th-order compatibility condition ([\[def_comp_linear\]](#def_comp_linear){reference-type="ref" reference="def_comp_linear"}) for the linearized equation ([\[eq_linear_w\]](#eq_linear_w){reference-type="ref" reference="eq_linear_w"}). Then by Proposition [Proposition 15](#pp_energy){reference-type="ref" reference="pp_energy"}, there exists a unique solution $$w \in \bigcap_{k=0}^{m} C^{m-k}([0,T]; H^{2k+1}(\Omega))$$ such that $$\begin{aligned} \label{eq_fandu} \| w \|_{Z^m} % \leq C(\|F\|_\zmm + \|g_0\|_{H^{2m+1}}+ \|g_1\|_{H^{2m-1}}) \nonumber\\ \leq C (\|u_h\|_{C^{2m+2}([0, T] \times \bar{\Omega})} & + \|g_0\|_{H^{2m+1}}+ \|g_1\|_{H^{2m-1}}\\ & +(\|v\|_{Z^{m-1}}+ \|u_h\|_{C^{2m+2}([0, T] \times \bar{\Omega})})^2) ,\nonumber \end{aligned}$$ for a new constant $C$ independent of $r$ and $\rho$. For $r < \min\{r_0/2, 1, {1}/{(5C)}\}$ and $\rho = r^2$, the above inequality implies $$\| w \|_{Z^m} \leq C(\rho + (r + \rho)^2) \leq 5C r^2 \leq r.$$ Thus ${J}$ maps ${\widetilde{Z}^m}(r, T)$ to itself. In the following, we prove ${J}$ is a contraction for sufficiently small $r$. Indeed, for $w_j = {J}(v_j)$ with $v_j \in {\widetilde{Z}^m}(r, T)$, we write $w_0 = u_2 - u_1$ and it satisfies $$\begin{aligned} &(1 - 2\beta (v_1 + u_h)) \partial^2_tw_0 - \Delta w_0- \partial_t(\Delta) w_0 + q(t,x) w_0\\ =& 2\beta(v_2 - v_1) \partial^2_tw_2 + 2 \beta (\partial_tv_2 - \partial_tv_1)(\partial_tv_2 + \partial_tv_1 + 2\partial_tu_h) + 2\beta(v_2 - v_1) \partial_t^2 u_h. \end{aligned}$$ We denote the right-hand side by ${I}$ and compute $$\begin{aligned} \|{I}\|_{Z^{m-1}}&\leq C' (\|w_2\|_{Z^m}+ \|v_1\|_{Z^m}+ \| v_2\|_{Z^m}+ \|u_h\|_{C^{2m+2}([0, T] \times \bar{\Omega})})\|v_1 - v_2\|_{Z^m}, \end{aligned}$$ for some positive constant $C'$ independent of $r$ and $\rho$, again by [@Uhlmann2021a Claim 1]. Note that previously we choose $r$ small enough such that $\rho = r^2 \leq r$ and $\|u_j\|_{Z^m}\leq r$, for $j =1,2$. Then we have $$\|{I}\|_{Z^{m-1}}\leq 4C'r \|v_1 - v_2\|_{Z^m}.$$ By Proposition [Proposition 15](#pp_energy){reference-type="ref" reference="pp_energy"}, one obtains $$\begin{aligned} &\| w_2 - w_1 \|_{Z^m}\leq C \|{I}\|_{Z^{m-1}} \leq 4CC'r \|v_1 - v_2\|_{Z^m}. \end{aligned}$$ If we additionally assume $r < 1/(4CC')$, then $$\|{J}(v_2 -v_1)\|_{Z^m}< \|v_2 - v_1\|_{Z^m},$$ which proves that ${J}$ is a contraction. Thus, there exists a unique solution $\widetilde{w}$ as the fixed point of $J$ in ${\widetilde{Z}^m}(r, T)$, which is the solution to the nonlinear problem ([\[eq_w\]](#eq_w){reference-type="ref" reference="eq_w"}). Moreover, if we choose $r$ small enough such that $C r \leq 1/4$ in ([\[eq_fandu\]](#eq_fandu){reference-type="ref" reference="eq_fandu"}), then we have $$\|\tilde{w}\|_{Z^m}\leq 2C (\|u_h\|_{C^{2m+2}([0, T] \times \bar{\Omega})} + \|g_0\|_{H^{2m+1}}+ \|g_1\|_{H^{2m-1}}).$$ This implies there exists a unique solution $\tilde{u}$ to the initial-boundary value problem ([\[eq_nonlinear_bvp\]](#eq_nonlinear_bvp){reference-type="ref" reference="eq_nonlinear_bvp"}) in ${\widetilde{Z}^m}$ satisfying $$\|\tilde{u}\|_{Z^m}\leq C (\|u_h\|_{C^{2m+2}([0, T] \times \bar{\Omega})} + \|g_0\|_{H^{2m+1}}+ \|g_1\|_{H^{2m-1}}),$$ where $C$ is a new constant depending on $T$, $\beta$, $q$, and the domain $\Omega$. ◻ # Preliminaries {#sec_prelim} In this section, for convenience, we use $x$ to denote a point in $\mathbb{R}^n$ and $P(D)$ to denote a differential operator, where $D = -{i}\partial_x$. Let $\alpha = (\alpha_1, \ldots, \alpha_n)$ be a multi-index and we write $\partial^\alpha = \partial_1^{\alpha_1} \ldots\partial_n^{\alpha_n}. %\frac{\partial^{\alpha_1}}{\partial x_1} \ldots \frac{\partial^{\alpha_n}}{\partial x_n}.$ To apply these results to our model, we actually abuse the notation by considering $(t,x) \in \mathbb{R} \times \mathbb{R}^n$ and a differential operator $P(D) = P(D_t, D_x)$. ## The spaces $B_{p, k}$ Let $H^s(\mathbb{R}^n)$ be the Sobolev space. To describe the regularity of fundamental solutions, we introduce spaces of distributions which generalize the $H^s$ spaces, for more details see [@H2005analysis Section 10.1]. We briefly recall some definitions and results in the following. Let $k$ be a positive function in $\mathbb{R}^n$. We say $k$ is a **temperate weight function** if there exist positive constants $C$ and $N$ such that $$k(\xi + \eta) \leq (1+ C|\xi|)^N k(\eta), \quad \xi, \eta \in \mathbb{R}^n.$$ For example, the function $k_s(\xi) = (1+ |\xi|^2)^{{s}/{2}}$ is a temperate weight that corresponds to $H^s(\mathbb{R}^n)$. Another example is the function $\widetilde{P}$ define by $$\widetilde{P}(\xi)^2 = \sum_{|\alpha| \geq 0 } |P^{(\alpha)}(\xi)|^2, \quad \xi \in \mathbb{R}^n$$ where $P(\xi)$ is a polynomial and we write $P^{(\alpha)}(\xi) \coloneqq \partial^\alpha P(\xi)$. One can show by Taylor expansion that there is a positive constant $C$ only depending on $\deg P$ and $n$ such that $$\begin{aligned} \label{eq_tP} \widetilde{P}(\xi + \eta) \leq C(1+ |\xi|)^m \widetilde{P}(\eta).\end{aligned}$$ Thus, it is a temperate weight function. **Definition 1** ([@H2005analysis Definition 10.1.6]). For a temperate weight function $k(\xi)$ and $1 \leq p \leq \infty$, we denote by $B_{p, k}(\mathbb{R}^n)$ the set of all distributions $u \in \mathscr{S}'$ such that $\hat{u}$ is a function and $$\begin{aligned} \label{def_Bpk} \|u \|_{p,k} = ((2\pi)^{-n} \int |k(\xi)\hat{u}(\xi)|^p \mathop{}\!\mathrm{d}\xi)^{1/p} \leq \infty.\end{aligned}$$ When $p = \infty$, we define $\|u \|_{\infty,k} = \mathrm{ess.\ sup} |k(\xi)\hat{u}(\xi)|$. One can show $B_{p, k}(\mathbb{R}^n)$ is a Banach space with the norm ([\[def_Bpk\]](#def_Bpk){reference-type="ref" reference="def_Bpk"}). Moreover, we have $\mathscr{S}(\mathbb{R}^n) \subset B_{p, k}(\mathbb{R}^n) \subset \mathscr{S}'(\mathbb{R}^n),$ with $C_0^\infty(\mathbb{R}^n)$ dense in $B_{p, k}(\mathbb{R}^n)$ for $p < \infty$. When $p = 2$, one has $$\|u \|_{2,\widetilde{P}} = (\sum_{|\alpha| \geq 0} \| P^{(\alpha)}(D) u\|_{L^2(\mathbb{R}^n)}^2)^{1/2} .$$ In particular, we have $\|u \|_{2,1} = \|u\|_{{L^2}(\mathbb{R}^n)}$. Additionally, we have the following properties. **Proposition 2** ([@H2005analysis Theorem 10.1.12]). *If $u_1 \in B_{p, k_1}(\mathbb{R}^n) \cap \mathcal{E}'(\mathbb{R}^n)$ and $u_2 \in B_{\infty, k_2}(\mathbb{R}^n)$, it follows that $u_1 \ast u_2 \in B_{p, k_1k_2}(\mathbb{R}^n)$ and we have the estimate $$\begin{aligned} \| u_1 \ast u_2 \|_{p, k_1k_2} \leq \|u_1\|_{p, k_1} \|u_2\|_{\infty, k_2}.\end{aligned}$$* **Proposition 3** ([@H2005analysis Theorem 10.1.15]). *If $u \in B_{p, k}(\mathbb{R}^n)$ and $\psi \in \mathscr{S}(\mathbb{R}^n)$, it follows that $\psi u \in B_{p, k}(\mathbb{R}^n)$ with $$\begin{aligned} \| \psi u \|_{p, k} \leq \|\psi\|_{1, M_k} \|u\|_{p, k}, \end{aligned}$$ where $M_k(\xi) = \sup_{\eta\in \mathbb{R}^n} \frac{k(\xi + \eta)}{k(\eta)}$ is a temperate weight function induced by $k$.* Next, let $X$ be an open subset of $\mathbb{R}^n$ and we define $B_{p, k}^\mathrm{loc}(X)$ as local spaces containing $u \in \mathcal{D}'(X)$ such that for every $x_0 \in X$, there exists $\phi \in C_0^\infty(X)$ with $\phi(x_0) \neq 0$ and $\phi u \in B_{p, k}(\mathbb{R}^n)$. With this notation, we have the following result for fundamental solutions to differential operators with constant coefficients. **Proposition 4** ([@H2005analysis Theorem 10.2.1]). *Let $P(D)$ be a partial differential operator with constant coefficients which is not equal to $0$. Then there is a fundamental solution $E_0 \in B_{\infty, \widetilde{P}}^\mathrm{loc}(\mathbb{R}^n)$ such that $PE_0 = \delta$, where $\delta$ is the Dirac measure at $0$. In particular, we have $$\begin{aligned} \label{eq_E0norm} \|E_0/(\cosh|x|)\|_{\infty, \widetilde{P}} \leq C,\end{aligned}$$ for a positive constant $C$ only depending on $\deg P$ and $n$.* Furthermore, we prove the following proposition, by modifying [@isakov1991completeness Theorem 1.2] based on [@H2005analysis Theorem 10.3.7]. **Proposition 5**. *Let $X$ be a bounded open set in $\mathbb{R}^n$ with smooth boundary. There exists a bounded linear operator $E$ such that $$PE f = f, \quad \text{ for any } f \in H^s(X), \text{ where } s\geq0 \text{ is an integer,}$$ and for any differential operator $Q$ with constant coefficients we have $$\begin{aligned} \label{eq_Eestimate} \|Q(D) Ef\|_{H^s(X)} \leq C \sup_{\xi \in \mathbb{R}^n} \frac{\widetilde{Q}(\xi)}{\widetilde{P}(\xi)} \|f\|_{H^s(X)},\end{aligned}$$ where $C$ depends only on $\deg P$, $n$, and $X$.* *Proof.* When $s \geq 0$ is an integer, recall $H^s(X)$ consist of all $f \in L^2(X)$ such that $\partial^\alpha u \in L^2(X)$, for any multi-index $\alpha = (\alpha_1, \ldots, \alpha_n)$ with $|\alpha| \leq s$. Moreover, for $f \in H^s(X)$, there exists $f_0 \in H^s(\mathbb{R}^n)$ such that $\|f_0\|_{H^s(\mathbb{R}^n)} \leq C \|f\|_{H^s(X)}$ with $f_0 = f$ in $X$. Let $Ef$ be the restriction of $u_0 = E_0 \ast f_0$ to $X$, where $E_0$ is a fundamental solution given by Proposition [Proposition 4](#pp_E0){reference-type="ref" reference="pp_E0"}. It follows that $PEf = f$. To prove that $Q(D)E$ is bounded, we consider a smooth cutoff function $\psi \in C_0^\infty(X)$ with $\psi = 1$ in a neighborhood of the closure of $X- X = \{x-y: x, y \in X\}$. We set $F_0 = \psi E_0$ and $g(x) = \cosh(x)$. Then by Proposition [Proposition 3](#pp_psiu){reference-type="ref" reference="pp_psiu"}, we have $F_0 \in B_{\infty, \widetilde{P}}(\mathbb{R}^n)$ with $$\begin{aligned} \|F\|_{\infty, \widetilde{P}} = \|(\psi g) (E_0/g)\|_{\infty, \widetilde{P}} \leq \|\psi g\|_{1, M_{\widetilde{P}}} \|E_0/g\|_{\infty, \widetilde{P}}.\end{aligned}$$ Here by ([\[eq_tP\]](#eq_tP){reference-type="ref" reference="eq_tP"}) there exists a constant $C> 0$ only depending on $\deg P$ and $n$ such that $$M_{\widetilde{P}}(\xi) = \sup_{\eta\in \mathbb{R}^n} \frac{\widetilde{P}(\xi + \eta)}{\widetilde{P}(\eta)} \leq C(1+ |\xi|)^m,$$ and therefore $\|\psi g\|_{1, M_{\widetilde{P}}}$ is bounded. Combining it with ([\[eq_E0norm\]](#eq_E0norm){reference-type="ref" reference="eq_E0norm"}), we have $\|F\|_{\infty, \widetilde{P}} \leq C$, for a new constant $C$ only depending on $\psi$, $\deg P$, $n$, $X$. Hence, for any multi-index $\alpha$ with $|\alpha| \leq s$, we have $$\begin{aligned} \| \partial^\alpha (Q(D) Ef)\|_{L^2(X)} = \| Q(D) \partial^\alpha (Ef)\|_{L^2(X)} \leq \| Q(D) \partial^\alpha (F_0 \ast f_0)\|_{2, 1}.\end{aligned}$$ Note $F_0$ has compact support and therefore $Q(D)\partial^\alpha (F_0 \ast f_0) = (Q(D)F_0) \ast \partial^\alpha f_0$. This implies $$\begin{aligned} \| \partial^\alpha (Q(D) Ef)\|_{L^2(X)} &\leq \| Q(D) (F_0 \ast \partial^\alpha f_0)\|_{2, 1} \leq \|\partial^\alpha f_0\|_{2,1} \|Q(D) F_0\|_{\infty, 1}\\ &\leq C \|f\|_{H^s(X)} \|Q(D) F_0\|_{\infty, 1} \leq C \|f\|_{H^s(X)} \| F_0\|_{\infty, \widetilde{P}}\sup_{\xi \in \mathbb{R}^n} \frac{\widetilde{Q}(\xi)}{\widetilde{P}(\xi)},\end{aligned}$$ where we use Proposition [Proposition 2](#pp_convl){reference-type="ref" reference="pp_convl"} and the last inequality comes from the estimate $$|\widehat{Q(D)F_0}(\xi)| = |Q(\xi) \widehat{F}_0(\xi)| \leq \sup_{\xi \in \mathbb{R}^n} \frac{\widetilde{Q}(\xi)}{\widetilde{P}(\xi)} | \widetilde{P}(\xi)\widehat{F}_0(\xi)|.$$ Thus, we have the desired estimate. ◻ ## The characteristic Cauchy problem {#subsec_cauchy} In Section [6](#sec_gocnew){reference-type="ref" reference="sec_gocnew"}, we would like to show the reminder term is relatively small with respect to a large parameter, and moreover it is supported in a half space $\{t \geq \varepsilon\}$, for some small number $\varepsilon> 0$. This enables us to obtain solutions satisfying the zero initial conditions. For this purpose, we recall some notations and results in [@H2005analysis Section 12.8]. In the following, we use the notation $\check{u}(x) = u(-x)$. By [@Hoermander2003 Theorem 7.1.10], for any $u \in \mathscr{S}'(\mathbb{R}^n)$, one has $$% \settoheight{\dhatheight}{\ensuremath{\hat{u}}}% \addtolength{\dhatheight}{-0.35ex}% \hat{\vphantom{\rule{1pt}{\dhatheight}}% \smash{\hat{u}}} = (2\pi)^n \check{u},$$ where $\hat{u} = \int e^{-i x\cdot \xi} f(x) \mathop{}\!\mathrm{d}x$ is the Fourier transform. **Proposition 6** ([@H2005analysis Theorem 10.1.14]). *If $L$ is a continuous linear form on $B_{p,k}(\mathbb{R}^n)$ for $p < \infty$, then we have $$L(u) = \check{v}(u), \qquad u \in \mathscr{S}(\mathbb{R}^n),$$ for some $v \in B_{p', 1/k}(\mathbb{R}^n)$ with ${1}/{p} + {1}/{p'} = 1$. The norm of this linear form is $\|v\|_{p', 1/k}$. Hence $B_{p', 1/k}(\mathbb{R}^n)$ is the dual space of $B_{p,k}$ and the canonical bilinear form in $B_{p,k}(\mathbb{R}^n) \times B_{p', 1/k}(\mathbb{R}^n)$ is the continuous extension of the bilinear form $\check{v}(u)$, where $v \in B_{p', 1/k}(\mathbb{R}^n)$ and $u \in \mathscr{S}(\mathbb{R}^n)$.* Recall $P(D)$ is a partial differential operator with constant coefficients in $\mathbb{R}^n$. Let $N = (0, \ldots,0, 1) \in \mathbb{R}^n$ and for convenience we consider the half space $$H = \{x: x_n \geq 0\} = \{x: x \cdot N \geq 0\}.$$ We study the Cauchy problem for $P(D)$ in $H$ with vanishing data on the boundary $\partial H$. We suppose $\partial H$ is characteristic with respect to $P(D)$, i.e., $P_m(N) = 0$, where $P_m$ is the principal part. By [@Hoermander2003 Theorem 8.6.7], there is no uniqueness for the Cauchy problem $(P+q) u = f$ with $f$ supported in $H$, unless growth conditions are imposed. Our aim is to find a solution supported in $H$ with the desired estimate, following the proof of Hörmander. First, we recall the following proposition. **Proposition 7** ([@H2005analysis Theorem 12.8.1]). *Let $P(D)$, $N$, and $H$ be defined as above. Then the following statements are equivalent.* - *$P(D)$ has a fundamental solution with support in $H$.* - *[\[condition_b\]]{#condition_b label="condition_b"} There exist constant $A_1$ and $A_2$ with $A_1 > 0$ such that for every solution $\sigma(\zeta)$ of $P(\zeta + \sigma N) = 0$ which is analytic and single-valued in a ball $B$ with real center and radius $A_1$, we have $$\sup_B \mathrm{Im} \sigma(\zeta) \geq A_2.$$* - *If $1 \leq p < \infty$ and $k$ is a temperate weight function, the equation $P(D)u = f$ has a solution $u \in B_{p, k}^\mathrm{loc}(\mathbb{R}^n)$ with support in $H$, for every $f \in B_{p, k}^\mathrm{loc}(\mathbb{R}^n)$ with support in $H$.* We emphasize that condition ([b](#condition_b)) is invariant under translation, i.e., it holds for $P(D)$ exactly when it holds for $P(D + \zeta^0)$, where $\zeta^0 \in \mathbb{C}^n$. However, the constant $A_1, A_2$ are changed by $\zeta^0$ correspondingly. Next, we define the following quotient norm $$\|u\|_{p,k}^- = \inf_{v}\|v\|_{p,k}, \quad \text{where } u, v \in \mathscr{S}(\mathbb{R}^n) \text{ with $u = v$ in $-H$},$$ where we define $-H = \{x: -x \in H \}$. Note this norm allows us to focus on the restrictions of $u$ to $-H$. We have the following proposition. **Proposition 8** ([@H2005analysis Theorem 12.8.12]). *Assume $P$ satisfies condition ([b](#condition_b)) in Proposition [Proposition 7](#pp_1281){reference-type="ref" reference="pp_1281"} with $A_2 = A_1 +1$. Then one can find constants $\kappa$ and $C$ for arbitrary $p \in [1, \infty]$ and any temperate weight function $k$ such that $$\| v\|_{p,k}^- \leq C e^{\kappa M} \|P(D) v\|_{p, k/\widetilde{P}}^-,$$ when $v \in C_0^\infty(\mathbb{R}^n)$ and we assume $\mathop{\mathrm{supp}}v \cap (-H) \subset \{(x',x_n): |x'| \leq M\}$.* According to the remarks in the proof of [@H2005analysis Theorem 12.8.17], this constant $C e^{\kappa M}$ only depends on $A_1, A_2$, the dimension $n$, and $\deg P$. Indeed, [@H2005analysis Theorem 12.8.17] is proved based on Proposition 12.8.4, which depends on Lemma 12.8.5 to 12.8.10 there. In particular, the set $F_m(Z)$ defined in [@H2005analysis Lemma 12.8.7] and the estimate there are independent of the choice of the polynomial $R$. Thus, one can show a local existence result using estimates from Proposition [Proposition 8](#pp_12812){reference-type="ref" reference="pp_12812"}, based on the proof of [@H2005analysis Theorem 12.8.3]. **Proposition 9**. *Let $1 < p \leq \infty$, $k$ be a temperate weight function on $\mathbb{R}^n$, and $X \subset \mathbb{R}^n$ be a bounded open set. Moreover, suppose $X \subset \{(x', x_n): |x'| \leq M\}$ for some constant $M$. Assume $P(D)$ satisfies condition ([b](#condition_b)) with $A_1 = A_2 +1$. If $f \in B_{p, k}(\mathbb{R}^n) \cap \mathcal{E}'(\mathbb{R}^n)$ and $\mathop{\mathrm{supp}}f \subset H$, then one can find $u \in B_{p, k \widetilde{P}}(\mathbb{R}^n)$ such that $\mathop{\mathrm{supp}}u \subset H$ and $P(D) u = f$ in $X$ with $$\| u\|_{p,k\widetilde{P}} \leq C \|f\|_{p, k},$$ where $C$ depends on $n$, $X$, $A_1, A_2$, and $\deg P$.* *Proof.* Recall we write $\check{u} = u(-x)$ and let $^t \! P$ be the formal adjoint of $P$. Note that we have $P(D) u = f$ in $X$ if and only if $^t \! P(D) \check{u} = \check{f}$ in $-X$, which means $$\check{u} (P(D) v) = \check{f}(v), \qquad \text{for any } v \in C_0^\infty(-X).$$ By Proposition [Proposition 6](#pp_10114){reference-type="ref" reference="pp_10114"}, with $f \in B_{p, k}(\mathbb{R}^n)$, it defines a continuous linear form $\check{f}(w)$ for $w \in \mathscr{S}(\mathbb{R}^n)$. This linear form can be extended to $B_{p', 1/k}(\mathbb{R}^n)$, when $p'$ satisfies ${1}/{p}+ {1}/{p'} = 1$. The norm of this linear form is given by $\|f\|_{p,k}$. Then we have $$|\check{f}(v)| \leq \|f\|_{p,k} \|v\|_{p', 1/k}, \qquad v \in B_{p', 1/k}(\mathbb{R}^n). %\leq C\|f\|_{p,k} \|P(D)v\|^-_{p', 1/{k\tP}}$$ We emphasize that the notations $p$ and $p'$ are switched, compared to Proposition [Proposition 6](#pp_10114){reference-type="ref" reference="pp_10114"}. Thus, we assume $p > 1$ for convenience. Note that $-X \subset \{(x', x_n): |x'| \leq M\}$ as well. According to Proposition [Proposition 8](#pp_12812){reference-type="ref" reference="pp_12812"}, there exist constants $\kappa$ and $C'$ depending on $A_1, A_2$, the dimension $n$, and $\deg P$, such that $$\|f\|_{p,k} \|v\|_{p', 1/k} \leq C' e^{\kappa M}\|f\|_{p,k} \|P(D)v\|^-_{p', \widetilde{P}/{k}},$$ for any $v \in C_0^\infty(-X)$. This implies that we can regard $\check{u}: P(D)v \rightarrow \check{f}(v)$ as a continuous linear form on $P(D)(C_0^\infty(-X))$. By Hahn-Banach theorem, there exists a continuous linear form $\check{U}$ on $C_0^\infty(\mathbb{R}^n)$, such that $$|\check{U}(w)| \leq C \|w\|^-_{p', \widetilde{P}/{k}}, \qquad \text{for any } w \in C_0^\infty(\mathbb{R}^n),$$ where we write $C = C' e^{\kappa M} \|f\|_{p,k}$. We must have $\check{U}$ supported in $-H$, which implies $U$ is supported in $H$. Again by Proposition [Proposition 6](#pp_10114){reference-type="ref" reference="pp_10114"}, we have $U \in B_{p, k\widetilde{P}}(\mathbb{R}^n)$ with $$\| u\|_{p,k\widetilde{P}} \leq C \|f\|_{p, k}.$$ ◻ **Corollary 1**. *Let $X, P(D)$ be as in Proposition [Proposition 9](#pp_12813){reference-type="ref" reference="pp_12813"}. In particular, there exists a solution $E_1\in B_{\infty,\widetilde{P}}(\mathbb{R}^n)$ with $\mathop{\mathrm{supp}}E_1\subset H$ such that $P(D) E_1= \delta$ in $X$ and $$\| E_1\|_{\infty,\widetilde{P}} \leq C,$$ where $C$ depends on $n$, $X$, $A_1, A_2$, and $\deg P$.* *Proof.* Note that $\delta \in B_{\infty,1}(\mathbb{R}^n) \cap \mathcal{E}'(\mathbb{R}^n)$ with $\mathop{\mathrm{supp}}\delta \subset H$ and $\|\delta\|_{\infty, 1} = 1$. By Proposition [Proposition 9](#pp_12813){reference-type="ref" reference="pp_12813"}, we have the desired estimate for $E$. ◻ We emphasize such $E_1$ is almost a fundamental solution, if we choose $X$ large enough. More precisely, we prove the following proposition. **Proposition 10**. *Let $X, P(D)$ be as in Proposition [Proposition 9](#pp_12813){reference-type="ref" reference="pp_12813"}. Let $\widetilde{X}$ be a small open neighborhood of $X$ and $X_0$ be a small open neighborhood of $\widetilde{X}-\widetilde{X}$. Then there exists a bounded linear operator $E_H$ such that $$PE_Hf = f \quad \text{in } X, \quad \text{ for any } f \in H^{s}(\widetilde{X}) \text{ and } \mathop{\mathrm{supp}}f \subset H ,$$ with $\mathop{\mathrm{supp}}E_Hf \subset H$ and $$\begin{aligned} \label{eq_Eestimate_H} \| E_Hf\|_{H^s(\widetilde{X})} \leq C \sup_{\xi \in \mathbb{R}^n} \frac{1}{\widetilde{P}(\xi)} \|f\|_{H^s(\widetilde{X})}, \end{aligned}$$ where $C$ depends on $X_0$, $A_1, A_2$, the dimension $n$, and $\deg P$.* *Proof.* We are motivated by the proof of [@H2005analysis Theorem 12.8.14]. Note $X_0$ as a small neighborhood of $\widetilde{X}- \widetilde{X}$ is also bounded. By Corollary [Corollary 1](#crl_fundamental){reference-type="ref" reference="crl_fundamental"}, there exists $E_1$ with $\mathop{\mathrm{supp}}E_1\subset H$ such that $P(D) E_1= \delta$ in $X_0$. Thus, we set $g = P(D) E_1- \delta$ and $g$ vanishes in $X_0$. Now let $\psi \in C_0^\infty(\widetilde{X})$ be a smooth cutoff function with $\psi = 1$ in the closure of $X$. Since $f \in H^s(\widetilde{X})$ with $\mathop{\mathrm{supp}}f \subset H$, then $\tilde{f}= \psi f$ is supported in $H$ satisfying $$\|\tilde{f}\|_{H^s(\mathbb{R}^n)} \leq C \|f\|_{H^s(\widetilde{X})}, \qquad \tilde{f}= f \text{ in } X.$$ Then we set $u = E_1\ast \tilde{f}$ to have $$P(D)u = (P(D)E_1) \ast \tilde{f}= (\delta + g) \ast \tilde{f}= \tilde{f}+ g \ast \tilde{f}.$$ Recall $g$ is supported in $\mathbb{R}^n\setminus X_0$ and $\tilde{f}$ is supported in $\widetilde{X}$. Since $(\mathbb{R}^n\setminus X_0) \cap (\widetilde{X}- \widetilde{X}) = \emptyset$, we have $$g \ast \tilde{f}(x) = \int g(x-y)\tilde{f}(y) \mathop{}\!\mathrm{d}y= 0, \qquad \text{for any } x \in \widetilde{X}.$$ This implies $P(D) u = f$ in $X$. Since $\mathop{\mathrm{supp}}E_1\subset H$ and $\mathop{\mathrm{supp}}\tilde{f}\subset H$, we have $\mathop{\mathrm{supp}}u \subset H$. Moreover, we estimate $$\begin{aligned} \|u \|_{p, k} = \|E_1\ast \tilde{f}\|_{p,k}\leq \|E_1\|_{\infty, 1}\|\tilde{f}\|_{p,k} \leq \sup_{\xi \in \mathbb{R}^n} \frac{1}{\widetilde{P}(\xi)} \|E_1\|_{\infty, \widetilde{P}}\|\tilde{f}\|_{p,k}\leq C\sup_{\xi \in \mathbb{R}^n} \frac{1}{\widetilde{P}(\xi)} \|\tilde{f}\|_{p,k},\end{aligned}$$ where we use Proposition [Proposition 2](#pp_convl){reference-type="ref" reference="pp_convl"} and the second inequality comes from the estimate $$|\widehat{E_1}(\xi)| \leq \sup_{\xi \in \mathbb{R}^n} \frac{1}{\widetilde{P}(\xi)} |\widetilde{P}(\xi) \widehat{E_1}(\xi)|.$$ Here the constant $C$ depends on the dimension $n$, $X_0$, $A_1, A_2$, and $\deg P$. In particular, we can choose $p = 2$ and $k = k_s = {(1 + |\xi|^2)^{s/2}}$ to have $$\| u\|_{H^s(\mathbb{R}^n)} = \|{u}\|_{2,k_s} \leq C \sup_{\xi \in \mathbb{R}^n} \frac{1}{\widetilde{P}(\xi)} \|\tilde{f}\|_{2,k_s} \leq C \sup_{\xi \in \mathbb{R}^n} \frac{1}{\widetilde{P}(\xi)} \|f\|_{H^s(\widetilde{X})}.$$ Thus, we define $E_Hf = \psi u$ to have the desired result. ◻ # Geometric optics constructions {#sec_goc} In this section, we would like to construct an approximate solution to the linear problem $$\begin{aligned} \label{eq_linear1} \begin{aligned} (\partial^2_t - \Delta - \partial_t\Delta + q(t,x))u &= 0, & \ & \mbox{on } (0, T) \times \Omega, \\ %{u = \partial_t u} &= 0, & \ &\mbox{for } t=0. \end{aligned}\end{aligned}$$ For a large parameter $\rho$ and fixed $\omega \in S^{n-1}$, we consider an asymptotic solution of the form $$\begin{aligned} \label{eq_vN} v_{N}(t,x) = e^{i\varphi(t,x,\rho, \omega)} a(t,x, \rho, \omega), % = e^{i\varphi(t,x,\rho, \omega)} \sum_{j=1}^N\frac{a_j(t,x, \omega)}{h(\rho)^j},\end{aligned}$$ where $\varphi$ is the phase function and $a$ is the amplitude satisfying $$a(t,x, \rho, \omega) = \sum_{j=0}^N{\rho^{-j}}{a_j(t,x, \omega)}.$$ Plugging this form into the linear problem, we have $$\begin{aligned} &(\partial^2_t - \Delta - \partial_t\Delta + q(t,x))(e^{i\varphi} a ) \\ = &e^{i\varphi}( -(\partial_t \varphi)^2 a + i \partial_t^2 \varphi a + 2i \partial_t \varphi \partial_t a + \partial_t^2 a -(-(\nabla \varphi \cdot \nabla \varphi) a + i \Delta \varphi a + 2 i\nabla \varphi \cdot \nabla a + \Delta a)\\ &-(-i \partial_t \varphi(\nabla \varphi \cdot \nabla \varphi) a - 2(\partial_t \nabla \varphi \cdot \nabla \varphi)a - (\nabla \varphi \cdot \nabla \varphi) \partial_t a - \partial_t \varphi \Delta \varphi a + i \partial_t \Delta \varphi a + i\Delta \varphi \partial_t a \\ &- 2 \partial_t \varphi \nabla \varphi \cdot \nabla a + 2i (\partial_t \nabla \varphi) \cdot \nabla a + 2i \nabla \varphi \cdot \nabla \partial_t a + i \partial_t \varphi \Delta a + \partial_t \Delta a ) + q a ).\end{aligned}$$ We choose the Eikonal equation $$-(\partial_t \varphi)^2 +i \partial_t \varphi(\nabla \varphi \cdot \nabla \varphi) = 0,$$ which is satisfied by a linear phase function $$\varphi(t,x, \rho, \omega) = -i\rho^2 t + i\rho(x \cdot \omega).$$ This phase function has complex frequency and gives us exponentially growing solutions and we have $$\begin{aligned} &(\partial^2_t - \Delta - \partial_t\Delta + q(t,x))(e^{\rho^2 t-\rho(x \cdot \omega)} a ) % = &e^{i\varphi}( % 2 \rho^2 \partial_t a % + \partial_t^2 a % -(\rho^2 a - 2\rho \omega \cdot \nabla a + \Delta a)\\ % &- (\rho^2 \partial_t a % + 2 \rho^2 \rho \omega \cdot \nabla a % - 2 \rho \omega \cdot \nabla \partial_t a % + \rho^2 \Delta a + \partial_t \Delta a % ) + qa % )\\ % = &e^{i\varphi}( % 2 \rho^3 \omega \cdot \nabla a % + \rho^2 (\partial_t a - \Delta a - a) % + 2 \rho( \omega \cdot \nabla a + \omega \cdot \nabla \partial_t a) % + (P+q) a)\\ = e^{\rho^2 t-\rho(x \cdot \omega)}( \rho^3 T_\omega a + \rho^2 T_1a + \rho T_2a + (P+q) a),\end{aligned}$$ where we introduce the following operators $$\begin{aligned} T_\omega= 2 \omega \cdot \nabla, \quad T_1= \partial_t - \Delta - 1, \quad T_2= 2 ( \omega \cdot \nabla + \omega \cdot \nabla \partial_t), \quad P = (\partial^2_t - \Delta - \partial_t\Delta ).\end{aligned}$$ It follows that the amplitude satisfies $$\begin{aligned} & T_\omega a_0(t, x, \omega) = 0, \label{transport_0}\\ &T_\omega a_{1}(t, x, \omega) = -T_1a_0, \label{transport_1}\\ &T_\omega a_{2}(t, x, \omega) = -T_1a_1 - T_2a_0, \label{transport_2}\\ &T_\omega a_{j}(t, x, \omega) = -T_1a_{j-1} - T_2a_{j-2} - (P+q) a_{j-3}, \quad \text{ for } k \geq 3. \label{transport_j}\end{aligned}$$ To find the amplitude $a(t, x, \rho, \omega)$, one solves $a_0$ from ([\[transport_0\]](#transport_0){reference-type="ref" reference="transport_0"}) first and then plug it in ([\[transport_1\]](#transport_1){reference-type="ref" reference="transport_1"}) to solve $a_1$, and then plug $a_0, a_1$ in ([\[transport_2\]](#transport_2){reference-type="ref" reference="transport_2"}) to solve $a_2$, and next solve $a_j$ from ([\[transport_j\]](#transport_j){reference-type="ref" reference="transport_j"}) for $j \geq 3$. One possible solution to ([\[transport_0\]](#transport_0){reference-type="ref" reference="transport_0"}) is $$\begin{aligned} \label{eq_a0} a_0(t, x,\omega) = \phi(t) \prod_{j = 2}^{n} \chi(\omega_j \cdot (x - y_0)),\end{aligned}$$ where $\phi(t) \in C_0^\infty(\mathbb{R})$ is supported in $(0, T)$ and $\chi \in C_0^\infty(\mathbb{R})$ is supported in $(-\epsilon, \epsilon)$ with $\chi(s) = 1$ near $0$, for some $\epsilon> 0$. In addition, here we pick a fixed point $y_0 \in \partial \Omega$ and suppose $\omega_1, \ldots, \omega_{n}$ form an orthogonal basis in $\mathbb{R}^n$ with $\omega_1 = \omega$. Note that for each fixed $t$, the leading term $a_0(t,x, \omega)$ is a smooth function supported in a small $\epsilon$-neighborhood of the ray $x = s \omega + y_0$. This amplitude is used in [@lassas2022inverse] for a time-dependent Schrödinger equation and related inverse problems. Then we define the set $$\Sigma({y_0, \omega}) = \{y \in \mathbb{R}^n: \omega \cdot (y-y_0) = 0\}.$$ For any $x \in \mathbb{R}^n$, there exists a unique $y \in \Sigma({y_0, \omega})$ such that $x = s \omega + y$ for some $s \in \mathbb{R}$. With vanishing initial conditions on $\Sigma({y_0, \omega})$, we have $$\begin{aligned} & a_{1}(t, s\omega + y, \omega) = -\frac{1}{2}\int_0^s T_1a_0(s' \omega + y) \mathop{}\!\mathrm{d}s',\\ &a_{2}(t, s\omega + y, \omega) =-\frac{1}{2} \int_0^s (T_1a_1+ T_2a_0)(s' \omega + y) \mathop{}\!\mathrm{d}s',\\ &a_{j}(t, s\omega + y, \omega) = -\frac{1}{2}\int_0^s (T_1a_{j-1} + T_2a_{j-2} + (P+q) a_{j-3})(s' \omega + y) \mathop{}\!\mathrm{d}s', \quad \text{ for } k \geq 3.\end{aligned}$$ Note that $a_0(t, x, \omega)$ is supported with respect to $t$ in $(0, T)$ and $T_\omega$ is independent of $t$. This implies $a_j(t, x)$ as well as $v_N(t, x)$ have the same support with respect to $t$. In addition, the construction above implies that $$\begin{aligned} \label{eq_errorterm} (P+ q) v_{N}&= -\rho^{-N+2}e^{i\varphi} ((T_1 a_N + T_2 a_{N-1} + (P+q) a_{N-2}) \\ &\quad \quad \quad - \rho^{-1} (T_2 a_N + (P+q) a_{N-1}) - \rho^{-2}(P+q) a_N) \coloneqq \rho^{-N+2} e^{i\varphi} F_N \nonumber,\end{aligned}$$ for $N \geq 2$. When $N=0$, only $a_N$ is involved and when $N = 1$, only $a_N, a_{N-1}$ are involved. Note that for $\rho \gg 1$, we have $\|F_N \|_{H^s((0, T) \times \Omega)} \leq C_{F_N}, %(\|a_N \|_{H^{s+3}((0, T) \times \Omega)} + \|a_{N-1} \|_{H^{s+3}((0, T) \times \Omega)} + \|a_{N-2} \|_{H^{s+3}((0, T) \times \Omega)} ).$ where $C_{F_N}$ depends on $q, n, N, \Omega$ and the choice of $\phi, \chi$. ## The backward problem {#subsec_backward_goc} Next, we consider the backward strongly damped wave equation $(\partial^2_t - \Delta + \partial_t\Delta + q(t,x))w = 0$. Similarly, for a large parameter $\rho$ and fixed $\varpi \in S^{n-1}$, we construct an approximate solution by considering $$\begin{aligned} \label{eq_wN} w_N(t,x) = e^{i\psi(t,x,\rho, \varpi)} b(t,x, \rho, \varpi), % = e^{i\varphi(t,x,\rho, \varpi)} \sum_{j=1}^N\frac{a_j(t,x, \varpi)}{h(\rho)^j},\end{aligned}$$ where $\psi$ is a linear phase function given by $$\psi(t,x,\rho, \varpi) = i\rho^2 t + i\rho (x \cdot \varpi)$$ and $b(t,x, \rho, \varpi) = \sum_{j=0}^N \rho^{-j}{b_j(t,x, \varpi)}$ is smooth amplitude. In this case, we compute $$\begin{aligned} &(\partial^2_t - \Delta + \partial_t\Delta + q(t,x))(e^{-\rho^2 t -\rho(x \cdot \varpi)} b ) \\ % = &e^{i\psi}( % 2 \rho^2 \partial_t b % + \partial_t^2 b % -(\rho^2 b - 2\rho \varpi \cdot \nabla b + \Delta b)\\ % &+ (-\rho^2 \partial_t b % - 2 \rho^2 \rho \varpi \cdot \nabla b % - 2 \rho \varpi \cdot \nabla \partial_t b % - \rho^2 \Delta b + \partial_t \Delta b % ) + qb % )\\ % = &e^{i\psi}( % -2 \rho^3 \varpi \cdot \nabla b % + \rho^2 (\partial_t b - \Delta b - b) % +2 \rho( \varpi \cdot \nabla b + \varpi \cdot \nabla \partial_t b) % + (P+q) b)\\ = &e^{e^{-\rho^2 t -\rho(x \cdot \varpi)} b}( -\rho^3 T_\omega b + \rho^2 \widetilde{T}_1b + \rho \widetilde{T}_2b + (P+q) b),\end{aligned}$$ where $T_\omega$ is as before and we define $$\widetilde{T}_1= -\partial_t + \Delta - 1, \qquad \widetilde{T}_2= 2 ( \omega \cdot \nabla - \omega \cdot \nabla \partial_t).$$ Then $b_0$ solves the same transport equation ([\[transport_0\]](#transport_0){reference-type="ref" reference="transport_0"}) and therefore can be given by ([\[eq_a0\]](#eq_a0){reference-type="ref" reference="eq_a0"}). Other terms $b_1, \ldots, b_N$ can be constructed in a similar way. # The recovery from all boundary measurements {#sec_allbd} In this section we would like to use the geometric optics solution constructed above to solve the inverse problem. For this purpose, we need to first prove it is indeed an approximate solution to the linear problem, in the sense that its remainder term is relatively small, for sufficiently large $\rho$. ## The remainder term Suppose $v_N$ is constructed as in Section [4](#sec_goc){reference-type="ref" reference="sec_goc"}. In this part, we construct a remainder term $r_N$ such that $$(\partial^2_t - \Delta - \partial_t\Delta + q(t,x))(v_N + e^{i\varphi} r_N) = 0$$ in $(0, T) \times \Omega$, with $r_N$ relatively small. Recall we write $P = \partial^2_t - \Delta - \partial_t\Delta$. Its symbol is given by $$P(\tau, \xi) = -\tau^2 + \xi \cdot \xi + i\tau \xi \cdot \xi, \qquad \text{for }(\tau, \xi) \in \mathbb{R} \times \mathbb{R}^n.$$ For a multi-index $\alpha = (\alpha_0, \alpha_1, \ldots, \alpha_n)$, recall we write $$P^{(\alpha)}(\tau, \xi) = \partial^\alpha P(\tau, \xi), \qquad \widetilde{P}(\tau, \xi) = (\sum_\alpha |P^{(\alpha)}(\tau, \xi) |^2)^{1/2}.$$ Then a direct computation shows that $$\begin{aligned} \label{eq_tPgeq} \widetilde{P}^2(\tau, \xi) \geq %|-\tau^2 + \xi \cdot \xi + i\tau \xi \cdot \xi|^2 + %|-2 \tau + i \xi \cdot \xi|^2 + |\partial_\xi P(\tau, \xi)|^2 \geq |2 \xi (1 + i\tau)|^2.\end{aligned}$$ Now we prove the following proposition about the remainder term. **Proposition 11**. *For fixed $N \geq 0$, a large parameter $\rho$, and $\omega \in S^{n-1}$, let $v _N(t,x)$ be the approximate solution constructed in ([\[eq_vN\]](#eq_vN){reference-type="ref" reference="eq_vN"}), with the phase function $\varphi = -i \rho^2 t + i\rho(x \cdot \omega)$ and the smooth amplitude $a = \sum_{j=0}^N\rho^{-j}{a_j(t,x, \omega)}$, where $a_0$ is given by ([\[eq_a0\]](#eq_a0){reference-type="ref" reference="eq_a0"}) and $a_j$ satisfies ([\[transport_1\]](#transport_1){reference-type="ref" reference="transport_1"}, [\[transport_2\]](#transport_2){reference-type="ref" reference="transport_2"}, [\[transport_j\]](#transport_j){reference-type="ref" reference="transport_j"}) for $j=1, \ldots, N$. Then there exists a solution $r_N$ to the equation $$\begin{aligned} \label{eq_rN} (P+q)(e^{i \varphi} r_N) = -(P+q) v_N \qquad \text{in }(0, T) \times \Omega \end{aligned}$$ such that for $s \geq 0$, we have $$\|r_N\|_{H^s((0,T)\times \Omega)} \leq {C_N}{\rho^{-N-1}}, %\|F_N\|_{H^s((0,T)\times \Omega)},$$ where $C_N$ depends on $s, n, N, q, \Omega$ and the choice of $\phi, \chi$.* *Proof.* We are inspired by [@isakov1991completeness Theorem 1.3]. Let $\zeta^o= (-i \rho^2,i\rho \omega)$ and we have $$%P(e^{i \varphi}) = -\rho^2, \qquad P^{(\alpha)} (D) e^{i \varphi}= P^{(\alpha)}(\zeta^o) e^{i \varphi}.$$ Then we compute $$(P+q)(e^{i \varphi } r_N) = e^{i \varphi} (\sum_\alpha \frac{1}{\alpha!}P^{(\alpha)}(\zeta^o) D^\alpha r_N + q r_N) = e^{i \varphi} (P(D + \zeta^o) r_N + q r_N).$$ For convenience, we write $P_o(D) = P(D + \zeta^o)$. Equation ([\[eq_rN\]](#eq_rN){reference-type="ref" reference="eq_rN"}) can be written as $$\begin{aligned} \label{eq_wR} P_o(D) r_N = -(\rho^{-N+2} F_N + q r_N) \end{aligned}$$ where we write $(P+q) v_N = \rho^{-N+2} e^{i\varphi} F_N$ in ([\[eq_errorterm\]](#eq_errorterm){reference-type="ref" reference="eq_errorterm"}). Now let $E$ be the bounded linear operator from Proposition [Proposition 5](#pp_Eestimate){reference-type="ref" reference="pp_Eestimate"} for $P_o(D)$, $X = (0, T) \times \Omega$, and $Q(D) = 1$. For $r \in H^s(X)$, we consider the map $\mathcal{J}(r) = -E(\rho^{-N+2} F_N + q r)$. Note that by ([\[eq_Eestimate\]](#eq_Eestimate){reference-type="ref" reference="eq_Eestimate"}), one has $$\|\mathcal{J}(r)\|_{H^s(X)} \leq C\sup_{\xi \in \mathbb{R}^n} \frac{1}{\widetilde{P_o}(\xi)}(\rho^{-N+2}\|F_N\|_{H^s(X)} + \|q\|_{C^{\lceil s \rceil}([0,T] \times \bar{\Omega})} \|r\|_{H^s(X)}).$$ By ([\[eq_tPgeq\]](#eq_tPgeq){reference-type="ref" reference="eq_tPgeq"}), for $(\tau, \xi) \in \mathbb{R} \times \mathbb{R}^n$, we have $$\begin{aligned} \widetilde{P}_o^2(\zeta) = \widetilde{P}^2(\zeta + \zeta^o) &\geq |2(\xi + i\rho \omega)(1 + i(\tau - i \rho^2)|^2 \\ &= 4|\xi + i\rho \omega|^2 |(1 + \rho^2) + i\tau |^2\\ & =4(|\xi|^2 + \rho^2)((1+ \rho^2)^2 + \tau^2) \geq 4 \rho^6. \end{aligned}$$ It follows that $$\begin{aligned} \label{eq_Jw} \|\mathcal{J}(r)\|_{H^s(X)} \leq C(\rho^{-N-1}\|F_N\|_{H^s(X)} + \rho^{-3}\|q\|_{C^{\lceil s \rceil}([0,T] \times \bar{\Omega})} \|r\|_{H^s(X)}). \end{aligned}$$ We choose $\rho$ large enough such that $\rho > \max\{2, C\|q\|_{C^{\lceil s \rceil}([0,T] \times \bar{\Omega})}, CC_{F_N}\}$. This implies $$\|\mathcal{J}(r)\|_{H^s(X)} \leq (1+ \|r\|_{H^s(X)}))/2 \leq 1,$$ when $\|r\|_{H^s(X)} \leq 1$. Thus, the operator $\mathcal{J}$ maps the ball $B_1 = \{r: \|r\|_{H^s(X)} \leq 1\}$ to itself. Next, we prove $\mathcal{J}$ is contraction when $\rho$ is sufficiently large. Indeed, for $r_1, r_2 \in B_1$, we compute $$\begin{aligned} \|\mathcal{J}(r_2) - \mathcal{J}(r_1)\|_{H^s(X)} = & \|E(q(r_2 - r_1))\|_{H^s(X)} \\ \leq & \rho^{-3} C \|q\|_{{C^{\lceil s \rceil}([0,T] \times \bar{\Omega})}} \|r_2 - r_1\|_{H^s(X)} < \frac{1}{2}\|r_2 - r_1\|_{H^s(X)}. \end{aligned}$$ Then by Banach's contraction theorem, there is a function $r_N$ in $B_1$ satisfying $r_N = \mathcal{J}(r_N)$, which solves ([\[eq_wR\]](#eq_wR){reference-type="ref" reference="eq_wR"}). Note that $r_N$ satisfies the inequality ([\[eq_Jw\]](#eq_Jw){reference-type="ref" reference="eq_Jw"}), and therefore we have $$\frac{1}{2}\|r_N\|_{H^s(X)} \leq \|\mathcal{J}(r_N)\|_{H^s(X)} - \rho^{-1}\|r_N\|_{H^s(X)} \leq C \rho^{-N-1}\|F_N\|_{H^s(X)}.$$ ◻ **Remark 1**. One can consider other linear phase functions and the corresponding geometric optics construction, for example, the exponentially decaying solution given by $\varphi = i \rho t + \rho (\omega \cdot x)$. We emphasize that we still have the control of $\|r_N\|_{H^s((0, T) \times \Omega)}$, but we do not have the vanishing initial conditions, as the support of $E_0$ is unknown. ## The backward problem {#subsec_backward_rm} Consider the geometric optics solution constructed for the backward problem in Section [4.1](#subsec_backward_goc){reference-type="ref" reference="subsec_backward_goc"}. To construct the remainder term, we write $^t \! P= \partial^2_t - \Delta + \partial_t\Delta$. Note it has the symbol $$^t \! P(\tau, \xi) = -\tau^2 + \xi \cdot \xi - i\tau \xi \cdot \xi, \qquad \text{for }(\tau, \xi) \in \mathbb{R} \times \mathbb{R}^n.$$ We have $$\begin{aligned} %\label{eq_tPgeq} \widetilde{{^t \! P}}^2(\tau, \xi) \geq %|-\tau^2 + \xi \cdot \xi + i\tau \xi \cdot \xi|^2 + %|-2 \tau - i \xi \cdot \xi|^2 + |2 \xi (1 - i\tau)|^2.\end{aligned}$$ Following the idea of Proposition [Proposition 11](#pp_remainder){reference-type="ref" reference="pp_remainder"}, we prove the following corollary. **Corollary 2**. *For fixed $N \geq 0$, a large parameter $\rho$, and $\varpi \in \mathbb{S}^{n-1}$, let $w _N(t,x)$ be the approximate solution in ([\[eq_wN\]](#eq_wN){reference-type="ref" reference="eq_wN"}), with the phase function $\psi = i \rho^2 t + i\rho(x \cdot \varpi)$ and the smooth amplitude $b = \sum_{j=0}^N\rho^{-j}{b_j(t,x, \varpi)}$ constructed above, where $b_0$ is given by ([\[eq_a0\]](#eq_a0){reference-type="ref" reference="eq_a0"}). Then there exists a solution $d_N$ to the equation $$\begin{aligned} \label{eq_rN_pstar} (P^*+q)(e^{i \psi} d_N) = -(P^*+q) w_N \qquad \text{in }(0,T)\times \Omega \end{aligned}$$ such that $$\|d_N\|_{H^s((0,T)\times \Omega)} \leq {C_N}{\rho^{-N-1}}, %\|F_N\|_{H^s((0,T)\times \Omega)},$$ where $C_N$ depends on $s, n, N, q, \Omega$ and the choice of $\phi, \chi$.* *Proof.* Let $\zeta^o= (i \rho^2, i\rho \varpi)$. It suffices to verify $\widetilde{{^t \! P}}^2(\zeta+ \zeta^o) \geq C \rho^6$. Indeed, we have $$\begin{aligned} \widetilde{{^t \! P}}^2(\zeta + \zeta^o) &\geq |2(\xi + i\rho \varpi)(1 - i(\tau + i \rho^2)|^2 = 4|\xi + i\rho \varpi|^2 |(1 - \rho^2) - i\tau |^2\\ & =4(|\xi|^2 + \rho^2)((1- \rho^2)^2 + \tau^2) \geq 4 \rho^2(\rho^4 - 2 \rho^2 + 1) \geq 2 \rho^6, \end{aligned}$$ for sufficiently large $\rho$. ◻ ## Recovering $q$ We consider the recovery of $q$ from the linear problem, which is related to the first-order linearization of $L_{q, \beta}$. More explicitly, consider the initial-boundary value problem $$\label{eq_linear} \begin{aligned} (\partial^2_t - \Delta - \partial_t\Delta + q(t,x))u &= 0, & \ & \mbox{on } (0, T) \times \Omega,\\ u(t,x) &= h, & \ &\mbox{for } x\in \partial \Omega,\\ u = g_0, \ \partial_t u &= g_1, & \ &\mbox{for } t=0\\ \end{aligned}$$ Let $(g_0, g_1, h) \in \mathcal{G}^m(\rho, \rho_h)$. Then this linear problem has a unique solution $u$ satisfying ([\[est_nl\]](#est_nl){reference-type="ref" reference="est_nl"}), if the data $(g_0, g_1, h)$ satisfies the $m$th-order compatibility condition ([\[def_comp_nl\]](#def_comp_nl){reference-type="ref" reference="def_comp_nl"}) with $\beta \equiv 0$, by Proposition [Proposition 1](#pp_nl){reference-type="ref" reference="pp_nl"}. We emphasize without the nonlinear term, we do not need to assume the data are sufficiently small. Thus, for such $(g_0, g_1, h)$, we define the map $${L}^\mathrm{lin}_{q}: (g_0, g_1, h) \rightarrow (u(T), \partial_t u(T), \partial_\nu u|_{\partial \Omega}).$$ First, we prove the following proposition, which shows all boundary data for the linear problem determines the potential $q$. **Proposition 12**. *Let $q^{(j)} \in {C^\infty([0,T]\times \bar{\Omega})}$ be two potentials, $j = 1,2$. For some $\rho, \rho_h>0$, suppose $${L}^\mathrm{lin}_{q^{(1)}}(g_0,g_1,h) = {L}^\mathrm{lin}_{q^{(2)}}(g_0,g_1,h)$$ for any $(g_0,g_1,h)\in \mathcal{G}^m(\rho, \rho_h)$ satisfying the $m$th-order compatibility condition ([\[def_comp_nl\]](#def_comp_nl){reference-type="ref" reference="def_comp_nl"}), where $\beta \equiv 0$. Then we have ${q^{(1)}}= {q^{(2)}}$ in $[0,T]\times \bar{\Omega}$.* *Proof.* We are inspired by [@isakov1991completeness Theorem 3.3]. For large $\rho$ and fixed $\omega \in S^{n-1}$, let $u^{(1)}= e^{i\varphi}(a_0 + r_0)$ be constructed as in Proposition [Proposition 11](#pp_remainder){reference-type="ref" reference="pp_remainder"}, which solves $$(P + {q^{(1)}}) u^{(1)}= 0$$ with the phase function $\varphi = -i\rho^2 t + i\rho(x \cdot \omega)$. Recall in ([\[eq_a0\]](#eq_a0){reference-type="ref" reference="eq_a0"}), we choose $$a_0(t, x,\omega) = \phi(t) \prod_{j = 2}^{n} \chi(\omega_j \cdot (x - y_0)) \coloneqq \phi(t)\tilde{a}_0(x, \omega).$$ where $\phi \in C_0^\infty((0, T))$ and $\chi \in C_0^\infty(\mathbb{R})$ is supported in $(-\epsilon, \epsilon)$ with $\chi = 1$ near $0$. Note that $\tilde{a}_0(x, \omega)$ is a smooth function supported in a small $\epsilon$-neighborhood of the ray $\gamma(s) = s \omega + y_0$, for a fixed point $y_0 \in \partial \Omega$. Recall the remainder term satisfies $\|r_0\|_{H^s((0,T) \times \Omega)} \leq C\rho^{-1}$, where we choose $s$ large enough. This implies $u^{(1)}\in H^{s}((0,T) \times \Omega)$. Then we set $$g_0 = u^{(1)}(0),\qquad g_1 = \partial_t u^{(1)}(0),\qquad h = u^{(1)}|_{\partial \Omega},$$ which belongs to $H^{2m+1}(\Omega)\times H^{2m-1}(\Omega) \times C^{2m+2}(\partial \Omega)$ and satisfies the compatibility condition ([\[def_comp_nl\]](#def_comp_nl){reference-type="ref" reference="def_comp_nl"}) with $\beta \equiv 0$. Thus, there exists a unique $u^{(2)}$ solving the linear problem ([\[eq_linear\]](#eq_linear){reference-type="ref" reference="eq_linear"}) with the potential ${q^{(2)}}$ and the initial-boundary data $(g_0, g_1, h)$. We set $u = u^{(2)}- u^{(1)}$ and $q = {q^{(2)}}- {q^{(1)}}$, which satisfies $$\begin{aligned} (P+{q^{(2)}}) u &= qu^{(1)}, & \ & \mbox{on } (0, T) \times \Omega,\\ u(t,x) &= 0, & \ &\mbox{for } x\in \partial \Omega,\\ u = 0, \ \partial_t u &= 0, & \ &\mbox{for } t=0.\\ \end{aligned}$$ Moreover, with the assumption that ${L}^\mathrm{lin}_{q^{(1)}}(g_0, g_1, h) = {L}^\mathrm{lin}_{q^{(2)}}(g_0, g_1, h)$, we have $$\partial_\nu u|_\Omega = 0, \qquad u(T,x) = \partial_tu(T,x) = 0.$$ Now let $w = e^{i\psi}(b_0 + d_0)$ be as in Corollary [Corollary 2](#cr_remainder){reference-type="ref" reference="cr_remainder"}, which solves the backward problem $$(^t \! P+ {q^{(2)}}) w = 0$$ with the phase function $\psi = i\rho^2 t + i\rho(x \cdot \varpi)$. Here we set $\varpi = -\omega$. We choose $$b_0(t, x,\varphi) = \phi(t)\prod_{j = 2}^{n} \chi(\omega_j \cdot (x - y)) \coloneqq \phi(t)\tilde{b}_0(x, \omega).$$ as a smooth function supported in a small $\epsilon$-neighborhood of the same ray $\gamma(s)$, with the same $\phi$. In this case, the remainder term satisfies $\|d_0\|_{H^s((0,T) \times \Omega)} \leq C \rho^{-1}$, for sufficiently large $s$. We multiply $w$ with the equation above and integrate over $t,x$ to have $$\begin{aligned} 0 = \int_0^T \int_\Omega (P+{q^{(2)}}) uw - qu^{(1)}w \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = \int_0^T \int_\Omega u (^t \! P+{q^{(2)}}) w - qu^{(1)}w \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t,\end{aligned}$$ which implies $$\int_0^T \int_\Omega - qu^{(1)}w \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = 0.$$ Then we plug in the expansion of $u^{(1)}, w$ to have $$\begin{aligned} \mathcal{I}_q = \int_0^T \int_\Omega q a_0 b_0 \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = -\int_0^T \int_\Omega q ( r_0 b_0 + a_0 d_0) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t \leq C \rho^{-1}. %(\|q\|_{C^\qCm([0,T]\times \bar{\Omega})}). %(\|a_0\|_{C([0,T]\times \bar{\Omega})} + \|b_0\|_{C([0,T]\times \bar{\Omega})} ).\end{aligned}$$ When $\rho \rightarrow \infty$, this implies $\mathcal{I}_q = 0$. Since $\phi \in C_0^\infty((0,T))$ is arbitrary, we must have $$\int_\Omega q(t,x) \tilde{a}_0 \tilde{b}_0\mathop{}\!\mathrm{d}x = 0, \qquad t \in [0,T].$$ Note that $\tilde{a}_0$ and $\tilde{b}_0$ are both supported in a small $\epsilon$-neighborhood of the ray $\gamma(s) = s \omega + y$. Let $\epsilon\rightarrow 0$ and we can extract the line integral $$X q(\gamma) = \int q(t, \gamma(s)) \mathop{}\!\mathrm{d}s = 0, \qquad t \in [0,T].$$ Since $\omega \in S^{n-1}$ and $y_0\in \partial \Omega$ are arbitrary, the X-ray transform of $q$ over all rays vanishes. With the assumption on $\Omega$, we have $q = {q^{(2)}}- {q^{(1)}}= 0$. Indeed, the X-ray transform is injective on $L^1(\mathbb{R}^n)$ by the Fourier Slice Theorem, for example, see [@book_SU Chapter 2]. One can extend $q \in C^\infty([0,T] \times \bar{\Omega})$ to $L^1(\mathbb{R}^n)$ by setting it equal to zero in $\mathbb{R}^n\setminus \bar{\Omega}$, as $\Omega$ is bounded. ◻ Moreover, when we have $\partial_t^j \beta(0,x) = 0$ for $x \in \Omega$ and $j = 0, \ldots, m$, the data $(g_0, g_1, h)$ with the $m$th-order compatibility for the linear problem also satisfies the $m$th-order compatibility for the nonlinear problem, see ([\[def_comp_nl\]](#def_comp_nl){reference-type="ref" reference="def_comp_nl"}). In this case, we can say $$\begin{aligned} \label{eq_Lin} {L}^\mathrm{lin}_{q}(g_0, g_1, h) = %\lim_{\ep \rightarrow 0} \partial_\varepsilon{L}_{q, \beta}(\varepsilon g_0, \varepsilon g_1, \varepsilon h)|_{\varepsilon=0},\end{aligned}$$ for small $(g_0, g_1, h) \in \mathcal{G}^m$. Therefore, Proposition [Proposition 12](#pp_recoverq){reference-type="ref" reference="pp_recoverq"} implies the first-order linearization of $L_{q,\beta}$ determines $q$, under this assumption on $\beta$. ## The second-order Linearization {#subsec_2nd} In this subsection, let $\varepsilon_1, \varepsilon_2 > 0$ be small parameters and let $u_{\varepsilon_1,\varepsilon_2}$ be the solution to the nonlinear problem ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) with data $$\begin{aligned} \label{def_2data} (g_0, g_1, h) = \varepsilon_1 (g_{0,1}, g_{1,1}, h_1) + \varepsilon_2 (g_{0,2}, g_{1,2}, h_2).\end{aligned}$$ We consider the second-order linearization $$U_2 = \partial_{\varepsilon_1} \partial_{\varepsilon_2} u_{\varepsilon_1, \varepsilon_2}|_{\varepsilon_1 = \varepsilon_2 = 0}.$$ Let $u_j$ be solutions to the linear problem $$\begin{aligned} \label{LS} \begin{aligned} (P+q) u_j(t,x) &= 0, & \ & \mbox{on } (0, T) \times \Omega,\\ u_j(t,x) &= f_j, & \ &\mbox{for } x\in \partial \Omega,\\ u_j = g_{0,j},\ \partial_tu_j &= g_{1,j}, & \ &\mbox{for } t=0.\\ \end{aligned}\end{aligned}$$ Then $U_2$ solves $$\begin{aligned} \begin{aligned} (P+q) U_2 &= \beta(t,x) \partial_t^2 (u_1u_2), & \ & \mbox{on } (0, T) \times \Omega,\\ U_2(t,x) &= 0, & \ &\mbox{for } x\in \partial \Omega,\\ U_2 = \partial_t U_2 &= 0, & \ &\mbox{for } t=0.\\ \end{aligned}\end{aligned}$$ Now for any $u_0$ solving $(^t \! P+ q) u_0 = 0$, we integrate by parts to have $$\begin{aligned} \label{eq_Ibeta} \int_0^T \int_\Omega \beta(t,x) \partial_t^2(u_1 u_2) u_0 \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t %= \int_0^T \int_\Omega (P+q) U_2 u_0 \diff x \diff t\\ = %\int_0^T \int_V U_2 (P^*+q) u_0 \diff x \diff t \int_\Omega \partial_tU_2(T) &u_0(T) - U_2(T)(\partial_tu_0(T) - \Delta u_0(T)) \mathop{}\!\mathrm{d}x\\ &- \int_0^T (\partial_\nu U_2 u_0)|_{\partial \Omega} + (\partial_\nu \partial_tU_2 u_0)|_{\partial \Omega} \mathop{}\!\mathrm{d}t. \nonumber\end{aligned}$$ ## Recovering $\beta$. {#subsec_beta} Now we would like to recover the nonlinear coefficient from all boundary measurements. Suppose we have two coefficients $\beta^{(1)}, \beta^{(2)}\in C^{\infty}([0,T] \times \bar{\Omega})$. Let $u^{(k)}_{\varepsilon_1,\varepsilon_2}$ be the solutions to the nonlinear problem ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) with $\beta^{(k)}$ and data $(g_0, g_1, h)$ given in ([\[def_2data\]](#def_2data){reference-type="ref" reference="def_2data"}), for $k=1,2$. We consider the corresponding source-to-solution maps ${L}_{q^{(k)}, \beta^{(k)}}$ and suppose $${L}_{{q^{(1)}}, \beta^{(1)}}(g_0, g_1, h) = {L}_{{q^{(2)}}, \beta^{(2)}}(g_0, g_1, h),$$ for any small and compatible data $(g_0, g_1, h)$. First, according to ([\[eq_Lin\]](#eq_Lin){reference-type="ref" reference="eq_Lin"}) and Proposition [Proposition 12](#pp_recoverq){reference-type="ref" reference="pp_recoverq"}, we have ${q^{(1)}}= {q^{(2)}}$. Next, we would like to construct approximate solutions $u_j$ to the linear problems using Section [4](#sec_goc){reference-type="ref" reference="sec_goc"}, which are corresponding to the boundary data $(g_{0,j}, g_{1,j}, f_j)$, for $j = 1,2,0$. For this purpose, let $\omega_1, \omega_2, \ldots, \omega_{n}$ be an orthonormal basis for $\mathbb{R}^n$. Let $\varpi = - (\omega_1 + \omega_2)/\sqrt{2} \in S^{n-1}$ and we choose $\varpi_2, \ldots \varpi_n$ such that they form an orthonormal basis with $\varpi$. For a fixed point $x_0 \in \Omega$, we choose $y_1, y_2, y_0\in \partial \Omega$ such that the three rays $$\gamma_j(s) = s \omega_j + y_j, %\quad x = s \omega_2 + y_2, \quad x = s \omega_0 + y_0 \ \text{ for } j = 1,2, \qquad \text{and } \gamma_0(s) = s \varpi + y_0,$$ intersect at $x_0$. We construct approximate solutions $$\begin{aligned} u_{j} &= e^{\rho^2t - \rho (w_j \cdot x)} (a_{0,j} + r_{0,j}) ,\quad j = 1,2, % v_{1,0} &= e^{+2\rho^2t + i\sqrt{2}\rho (w_0 \cdot x)} a_{0,0}(t,x)\end{aligned}$$ to $(P + {q^{(1)}}) u_j =(P + {q^{(2)}}) u_j = 0$, according to Proposition [Proposition 11](#pp_remainder){reference-type="ref" reference="pp_remainder"}. Here the amplitudes given by $$\begin{aligned} a_{0,1}(t,x,\omega_j) = \phi(t) \prod_{i \neq 1} \chi(\omega_i \cdot (x - y_1)) \coloneqq \phi(t)\tilde{a}_{0,1}(x, \omega),\\ a_{0,2}(t,x,\omega_j) = \phi(t) \prod_{i \neq 2} \chi(\omega_i \cdot (x - y_2)) \coloneqq \phi(t)\tilde{a}_{0,2}(x, \omega),\end{aligned}$$ are smooth functions supported in a small $\epsilon$-neighborhood of the ray $\gamma_1(s)$ and $\gamma_2(s)$ respectively. Moreover, the remainder term is given by $\|r_{0,j}\|_{H^s((0,T) \times \Omega)} \leq C\rho^{-1}$, for $j = 1,2$, where $s$ is chosen to be large enough. Then we set $$(g_{0,j}, g_{1,j}, h_j) = (u_j(0), \partial_tu_j(0), u_j|_{\partial \Omega}), \quad j =1,2,$$ which satisfy the compatibility condition ([\[def_comp_nl\]](#def_comp_nl){reference-type="ref" reference="def_comp_nl"}), under the assumption $\partial_t^j \beta(0,x) = 0$ for $x \in \Omega$ and $j = 0, \ldots, m$. For large $s$, we have $(g_0, g_1, h) \in \mathcal{G}^m$. Then with small $\varepsilon_1, \varepsilon_2$, the initial-boundary data $(g_0, g_1, h)$ given by ([\[def_2data\]](#def_2data){reference-type="ref" reference="def_2data"}) allows a unique solution to the nonlinear problem. Thus, let $u^{(k)}_{\varepsilon_1,\varepsilon_2}$ be the solutions to the nonlinear problem ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) with $\beta^{(k)}$ and ([\[def_2data\]](#def_2data){reference-type="ref" reference="def_2data"}). Note the second-order linearization $U_2^{(k)}$ satisfies $$U_2^{(1)} (T) = U_2^{(2)} (T), \quad \partial_tU_2^{(1)}(T) = \partial_tU_2^{(2)}(T), \quad \partial_\nu U_2^{(1)}|_{\partial \Omega} = \partial_\nu U_2^{(2)}|_{\partial \Omega},$$ since ${L}_{{q^{(1)}}, \beta^{(1)}}(g_0, g_1, h) = {L}_{{q^{(2)}}, \beta^{(2)}}(g_0, g_1, h)$. Next, we construct approximate solutions $u_0$ to the backward problem $(^t \! P+ {q^{(1)}}) u_0 =(^t \! P+ {q^{(2)}}) u_0 = 0$ by $$\begin{aligned} u_{0} &= e^{-2\rho^2t - \sqrt{2}\rho (\varpi \cdot x)} (b_{0} + d_{0}),\end{aligned}$$ according to Corollary [Corollary 2](#cr_remainder){reference-type="ref" reference="cr_remainder"}. Here the amplitudes given by $$\begin{aligned} b_{0}(t,x,\varpi) = \phi(t) \prod_{i \neq 1} \chi(\varpi_i \cdot (x - y_0)) \coloneqq \phi(t)\tilde{b}_{0}(x, \varpi),\end{aligned}$$ is a smooth function supported in a small $\epsilon$-neighborhood of the ray $\gamma_0(s)$. In this case, the remainder term is given by $\|d_{0}\|_{H^s((0,T) \times \Omega)} \leq C\rho^{-1}$, where $s$ is large enough. Now we plug in $u_1,u_2,u_0, U_2^{(1)}, U_2^{(2)}$ in ([\[eq_Ibeta\]](#eq_Ibeta){reference-type="ref" reference="eq_Ibeta"}) to have $$\begin{aligned} \int_0^T \int_\Omega (\beta^{(1)}- \beta^{(2)}) \partial^2_t(u_1 u_2) u_0 \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = 0. % \mI_\beta = \int_0^T \int_\Omega (\betaone- \betatwo) \ddt (e^{-2 \rho^2 t + i\rho (\omega_1 + \omega_2) \cdot x} (a_{0,1} &+ \rho^{-1} a_{1,1} + r_{1,1})(a_{0,2} + \rho^{-1} a_{1,2} + r_{1,2})) \times\\ % &e^{2\rho^2t + i\sqrt{2}\rho (w_0 \cdot x)} (b_{0} + \rho^{-1} b_{1} + d_{1}) % \diff x \diff t = 0.\end{aligned}$$ We compute $$\begin{aligned} \partial^2_t(u_1 u_2) &= \partial^2_t(e^{2 \rho^2 t - \rho (\omega_1 + \omega_2) \cdot x} (a_{0,1} +r_{0,1})(a_{0,2} + r_{0,2})) = e^{2 \rho^2 t -\rho (\omega_1 + \omega_2) \cdot x} (4 \rho^4 \phi^2(t) \tilde{a}_{0,1}\tilde{a}_{0,2} + R_1),\end{aligned}$$ where we write $$\begin{aligned} R_1 &= 4 \rho^4 (r_{0,1}(a_{0,2} + r_{0,2}) + (a_{0,1} + r_{0,1}) r_{0,2})\\ &-2 \rho^2 \partial_t((a_{0,1} + r_{0,1})(a_{0,2} + r_{0,2})) + \partial^2_t((a_{0,1} + r_{0,1})(a_{0,2} + r_{0,2})).\end{aligned}$$ A straightforward computation shows that $\|R_1\|_{H^{s-2}((0,T)\times \Omega)} \leq C \rho^3$. Then we have $$\begin{aligned} 0 &= \int_0^T \int_\Omega (\beta^{(1)}- \beta^{(2)}) (4 \rho^4 \phi^2(t) \tilde{a}_{0,1}\tilde{a}_{0,2} + R_1) (b_{0} + d_{0}) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t \\ & = \int_0^T \int_\Omega (\beta^{(1)}- \beta^{(2)}) (4 \rho^4 \phi^3(t) \tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0}+ R_2) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t\end{aligned}$$ where we write $$R_2 = R_1 (b_{0} + d_{0}) + (4 \rho^4 \phi^2(t) \tilde{a}_{0,1}\tilde{a}_{0,2} + R_1) d_{0}.$$ Similarly, we have $\|R_2\|_{H^{s-2}((0,T)\times \Omega)} \leq C \rho^3$. It follows that $$\begin{aligned} \int_0^T \int_\Omega (\beta^{(1)}- \beta^{(2)}) \phi^3(t) \tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0} \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = 0.\end{aligned}$$ Since $\phi \in C_0^\infty((0,T))$ is arbitrary, this implies $$\int_\Omega (\beta^{(1)}(t,x)- \beta^{(2)}(t,x)) \tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0} \mathop{}\!\mathrm{d}x = 0, \qquad t \in [0,T].$$ Recall the rays $\gamma_1, \gamma_2, \gamma_0$ intersect at the fixed point $x_0 \in \Omega$. Then we can assume $\tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0}$ is supported in a small neighborhood $N_\epsilon$ of $x_0$, which is contained in a ball of radius $\epsilon$. By shrinking the support of $\chi$, we have $$\beta^{(1)}(t,x_0)- \beta^{(2)}(t,x_0) = \lim_{\epsilon\rightarrow 0} \frac{1}{|N_\epsilon|}\int_\Omega (\beta^{(1)}(t,x)- \beta^{(2)}(t,x)) \tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0} \mathop{}\!\mathrm{d}x = 0.$$ Since $x_0 \in \Omega$ is arbitrary, we must have $\beta^{(1)}= \beta^{(2)}$ in $[0,T] \times \bar{\Omega}$. # Recovery from the DN map {#sec_gocnew} In this section, we would like to improve the results in Section [5](#sec_allbd){reference-type="ref" reference="sec_allbd"} using fundamental solutions constructed for the characteristic Cauchy problem, see Section [3.2](#subsec_cauchy){reference-type="ref" reference="subsec_cauchy"}. We prove the recovery of $q, \beta$ from the lateral boundary measurements. ## The reminder term {#subsec_reminder_new} We reconsider the approximate solution constructed for the linear problem. The aim is to construct a reminder term $r_N$ satisfying $$(\partial^2_t - \Delta - \partial_t\Delta + q(t,x))(v + e^{i\varphi} r_N) = 0 \qquad \text{in }(0, T) \times \Omega$$ with $r_N$ relatively small, as well as, with the zero initial conditions. Recall we write $P = \partial^2_t - \Delta - \partial_t\Delta$. Now we prove the following proposition about the reminder term. **Proposition 13**. *For $N \geq 0$, a large parameter $\rho$, and $\omega \in \mathbb{S}^{n-1}$, let $v _N(t,x)$ be the approximate solution constructed in ([\[eq_vN\]](#eq_vN){reference-type="ref" reference="eq_vN"}), with the phase function $\varphi = -i \rho^2 t + i\rho(x \cdot \omega)$ and the smooth amplitude $a = \sum_{j=0}^N\rho^{-j}{a_j(t,x, \omega)}$, where $a_0$ is given by ([\[eq_a0\]](#eq_a0){reference-type="ref" reference="eq_a0"}) and satisfies ([\[transport_1\]](#transport_1){reference-type="ref" reference="transport_1"}, [\[transport_2\]](#transport_2){reference-type="ref" reference="transport_2"}, [\[transport_j\]](#transport_j){reference-type="ref" reference="transport_j"}) for $j = 1, \ldots, N$. Then there exists a solution $r_N$ to the equation $$\begin{aligned} \label{eq_rN_new} (P+q)(e^{i \varphi} r_N) = -(P+q) v_N \qquad \text{in }(0, T) \times \Omega \end{aligned}$$ with $r_N(0,x) = \partial_t r_N(0, x) = 0$, such that for $s \geq 0$, we have $$\|r_N\|_{H^s((0,T)\times \Omega)} \leq {C_N}{\rho^{-N-1}}, %\|F_N\|_{H^s((0,T)\times \Omega)},$$ where $C_N$ depends on $q, n, \Omega$ and the choice of $\phi, \chi$.* *Proof.* We would like to show the reminder term $r_N$ is relatively small for large $\rho$ and additionally it is supported in $t \geq \varepsilon$ for some small number $\varepsilon> 0$. This implies it satisfies vanishing initial conditions at $t = 0$. For this purpose, we recall some conclusions in Section [3.2](#subsec_cauchy){reference-type="ref" reference="subsec_cauchy"} and modify the notations there for our model. In the following, let $$\begin{aligned} N = (1, 0, \ldots, 0) \in \mathbb{R} \times \mathbb{R}^n, \qquad H = \{(t,x): (t,x) \cdot N \geq \varepsilon\}. \end{aligned}$$ Note that here we translate the half space by a parameter $\varepsilon$ independent of $\rho$ and it does not affect the estimate in Proposition [Proposition 10](#pp_Eestimate_new){reference-type="ref" reference="pp_Eestimate_new"}. As before, we rewrite ([\[eq_rN_new\]](#eq_rN_new){reference-type="ref" reference="eq_rN_new"}) using $\zeta^o= (-i\rho, i\rho \omega)$ to have $$(P+q)(e^{i \varphi } r_N) %= e^{i \varphi} (\sum_\alpha %\frac{1}{\alpha!}P^{(\alpha)}(\zetaz) D^\alpha r_0 + q r_0) = e^{i \varphi} (P(D + \zeta^o) r_N + q r_N) = e^{i \varphi} (P_or_N + q r_N).$$ Equation ([\[eq_rN_new\]](#eq_rN_new){reference-type="ref" reference="eq_rN_new"}) can be written as $$\begin{aligned} \label{eq_wR_new} P_o(D) r_N = -(\rho^{-N+2} F_N + q r_N) \end{aligned}$$ where we write $(P+q) v_N = \rho^{-N+2} e^{i\varphi} F_N$ in ([\[eq_errorterm\]](#eq_errorterm){reference-type="ref" reference="eq_errorterm"}). Note $F_N$ is supported in $(0, T) \time \Omega$ with respect to $t$. Then there exists $\varepsilon> 0$ such that its support in $t$ is contained in the set of $(\varepsilon, T)$. Thus, we have $\mathop{\mathrm{supp}}F_N \subset H$. Note for $P_o(D)$, the boundary $\partial H$ is characteristic, since its principal symbol satisfies $P_{o,3}(N) = 0$. There is no uniqueness for the initial value problem $P_o(D) u = f$ with $f$ supported in $H$. As mentioned before, we would like to find a solution supported in $H$ with the desired estimate, using Proposition [Proposition 10](#pp_Eestimate_new){reference-type="ref" reference="pp_Eestimate_new"}. For this purpose, we prove in Lemma [Lemma 1](#lm_A1A2){reference-type="ref" reference="lm_A1A2"} that for our operator $P_o$, there exist $A_1, A_2$ independent of $\rho$ such that condition ([b](#condition_b)) is satisfied. With this lemma, we set $A_1 = 1$ and $A_2 = 0$. Thus, we have $P_o(D)$ satisfies ([b](#condition_b)) with $A_1 = A_2 + 1$, which is independent of $\rho$. Now let $X = (0, T) \times \Omega$ and $\widetilde{X}$ be a small open neighborhood of $X$. Let $X_0$ be a small open neighborhood of $\widetilde{X}- \widetilde{X}$, which is bounded in $\mathbb{R}^n$. Then we choose $E_H$ as the bounded linear operator in Proposition [Proposition 10](#pp_Eestimate_new){reference-type="ref" reference="pp_Eestimate_new"} for $P_o(D)$ with $X, \widetilde{X}, X_0$. For every $r \in H^s(\widetilde{X})$ with $\mathop{\mathrm{supp}}r \subset H$, we consider the map $\mathcal{J}(r) = -E_H(\rho^{-N+2}F_N + q r)$, which has support in $H$ and satisfies $$P_o(D)\mathcal{J}(r) = - (\rho^{-N+2}F_N + q r) \quad \text{in } X,$$ with the estimate $$\| \mathcal{J}(r)\|_{H^s(\widetilde{X})} \leq C \sup_{\zeta \in \mathbb{R}^{n+1}} \frac{1}{\widetilde{P}_o(\zeta)} \|\rho^{-N+2}F_N + q r\|_{H^s(\widetilde{X})},$$ where $C$ is independent of $\rho$. On the other hand, we compute $$\begin{aligned} \widetilde{P}_o^2(\zeta) = \widetilde{P}^2(\zeta + \zeta^o) \geq |2(1 + i(\tau - i\rho^2))(\xi + i\rho \eta)|^2 = 4((1 + \rho^2)^2 + \tau^2)(|\xi|^2 + \rho^2) \geq 4 \rho^6. \end{aligned}$$ It follows that for any $r \in H^s(\widetilde{X})$ supported in $H$, we have $$\begin{aligned} \label{eq_Jw_new} \| \mathcal{J}(r)\|_{H^s(\widetilde{X})} \leq {C}({\rho}^{-N-1} \|F_N\|_{H^s(\widetilde{X})} + \rho^{-3}\|q\|_{C^{\lceil s \rceil}([0,T] \times \bar{\Omega})} \|r\|_{H^s(\widetilde{X})}). \end{aligned}$$ where we extend $q\in C^{\infty}(\bar{X})$ to a function in $C^{\infty}(\widetilde{X})$. This motivates us to define $$H^{s,H}(\widetilde{X}) = \{u: u \in H^s(\widetilde{X}) \text{ with $\mathop{\mathrm{supp}}u \subset H$}\}.$$ If we choose $\rho$ large enough such that $\rho > \max\{2C\|q\|_{C^{\lceil s \rceil}([0,T] \times \bar{\Omega})} , 2CC_{F_0}\}$, then we have $$\|\mathcal{J}(r)\|_{H^s(\widetilde{X})} \leq (1+ \|r\|_{H^s(\widetilde{X})})/2 \leq 1,$$ for any $\|r\|_{H^s(\widetilde{X})} \leq 1$. Thus, the operator $\mathcal{J}$ maps the ball $B_1 = \{r \in H^{s,H}(\widetilde{X}): \|r\|_{H^{s}(\widetilde{X})} \leq 1\}$ to itself. Next, we prove $\mathcal{J}$ is a contraction for large $\rho$. Indeed, for $r_1, r_2 \in H^s(\widetilde{X})$, we compute $$\begin{aligned} \|\mathcal{J}(r_2) - \mathcal{J}(r_1)\|_{H^s(\widetilde{X})} = & \|E_H(q(r_2 - r_1))\|_{H^s(\widetilde{X})} \\ \leq & \rho^{-3} C \|q\|_{C^{\lceil s \rceil}([0,T] \times \bar{\Omega})} \|r_2 - r_1\|_{H^s(\widetilde{X})} < \frac{1}{2}\|r_2 - r_1\|_{H^s(\widetilde{X})}. \end{aligned}$$ Then by Banach's contraction theorem, there is a function $r_N$ in $B_1$ satisfying $r_N = \mathcal{J}(r_N)$, which solves ([\[eq_rN_new\]](#eq_rN_new){reference-type="ref" reference="eq_rN_new"}). Note that $r_N$ satisfies the inequality ([\[eq_Jw_new\]](#eq_Jw_new){reference-type="ref" reference="eq_Jw_new"}), and therefore we have $$\frac{1}{2}\|r_N\|_{H^s(\widetilde{X})} = \|\mathcal{J}(r_N)\|_{H^s(\widetilde{X})} - \frac{1}{2}\|r_N\|_{H^s(\widetilde{X})} \leq C \rho^{-N-1}\|F_N\|_{H^s(\widetilde{X})}.$$ In particular, this implies $\|r_N\|_{H^s(X)} \leq C_N \rho^{-N-1}\|F_N\|_{H^s(\widetilde{X})},$ for some constant $C_N$ independent of $\rho$. ◻ **Lemma 1**. *Let $P$, $\zeta_0$, $N$ be defined as above. Suppose $\sigma(\zeta)$ is a solution to $P_o(\zeta + \sigma N)$, where $\zeta = (\tau, \xi) \in \mathbb{C} \times \mathbb{C}^n$. If $\sigma(\zeta)$ is analytic and single-valued in a ball $B \subset \mathbb{C} \times \mathbb{C}^n$ with real center and radius $A_1 = 1$, then we have $$\sup_{\zeta \in B} \mathrm{Im} (\sigma(\zeta)) \geq 0.$$* *Proof.* First, we would like to solve $\tau(\xi)$ from $P(\tau, \xi) = 0$ for $\xi \in \mathbb{C}^n$. Indeed, the equation $-\tau^2 + i \tau \xi \cdot \xi + \xi \cdot \xi = 0$ have two complex roots $$\begin{aligned} \tau_\pm(\xi) %= \frac{1}{2}(i\xi \cdot \xi \pm \sqrt{-(\xi \cdot \xi)^2 + 4 \xi \cdot \xi}). = \frac{1}{2}(iR(\xi) \pm \sqrt{-R^2(\xi) + 4 R (\xi)}), \qquad \text{where }R(\xi) = \xi \cdot \xi. % \qquad % \tau_2 = \frac{1}{2}(i\xi \cdot \xi ) - \sqrt{-(\xi \cdot \xi)^2 + 4 \xi \cdot \xi}. \end{aligned}$$ Recall $\zeta^o= (-i \rho^2, i \rho \omega)$. Then the roots for $P_o(\zeta + \sigma N) = P(\zeta + \zeta^0 + \sigma N) = 0$ are given by $$\begin{aligned} \sigma_\pm(\tau, \xi) &= \frac{1}{2}(iR(\xi + i \rho \omega) \pm \sqrt{\triangle(\xi + i \rho \omega)}) + i\rho^2 - \tau. \end{aligned}$$ For a ball $B$ with real center and radius $1$, we can choose a point $(\tau, \xi)\in \mathbb{R} \times \mathbb{R}^n$ in $B$ such that $(\tau, \xi) = (\tau, r \theta)$, where $r \geq 1$ and $\theta \in \mathbb{S}^{n-1}$. First we compute $$R(\xi + i \rho \omega) = (r^2 - \rho^2) + i 2 r \rho (\omega \cdot \theta) \coloneqq c + id,$$ and $$-R^2(\xi) + 4 R (\xi) = (d^2 - c^2 + 4c) + i 2d(2-c) \coloneqq C + iD,$$ where $c, d, C, D \in \mathbb{R}$. Suppose $C + iD = (a + ib)^2$ for some real numbers $a, b$. Then a straightforward computation shows that $$b^2 = \frac{1}{2}(-C + \sqrt{C^2 + D^2}).$$ Note that $$\begin{aligned} C^2 + D^2 &= (d^2 - c^2 + 4c)^2 + 4d^2(2-c)^2 \\ & = ((d^2 +4) - (c-2)^2)^2 + 4(d^2+4)(c-2)^2 -16(c-2)^2 \\ & \leq ((d^2 +4) + (c-2)^2)^2. \end{aligned}$$ This implies that $$b^2 \leq \frac{1}{2}(-(d^2 - c^2 + 4c) +(d^2 +4) + (c-2)^2) = (c-2)^2.$$ Then we have $$(c + 2\rho^2)^2 - b^2 \geq (r^2 + \rho^2)^2 - (r^2 - \rho^2 -2)^2 = 4(\rho^2 + 1)(r^2 -1) \geq 0.$$ Note $c +2 \rho^2 = r^2 + \rho^2 \geq 0$. Thus, the imaginary part of the roots satisfies $$\mathrm{Im}(\sigma_\pm(0, r\theta)) = \frac{1}{2}(c + 2\rho^2 \pm b) \geq 0.$$ This proves for any ball $B$ with real center and radius $A_1$, we have $$\sup_{\zeta \in B} \mathrm{Im} (\sigma(\zeta)) \geq A_2 = 0,$$ if $\sigma(\zeta)$ is a real-valued and analytic root. ◻ ## The backward problem {#subsec_backward_new} Next, we consider the backward problem $$\label{eq_linearback} \begin{aligned} (\partial^2_t - \Delta + \partial_t\Delta + q(t,x))w &= 0, & \ & \mbox{on } (0, T) \times \Omega,\\ w(t,x) &= f, & \ &\mbox{for } x\in \partial \Omega,\\ w = 0, \ \partial_t w &= 0, & \ &\mbox{for } t=T.\\ \end{aligned}$$ As in Section [4.1](#subsec_backward_goc){reference-type="ref" reference="subsec_backward_goc"} , for a large parameter $\rho$ and fixed $\varpi \in \mathbb{S}^{n-1}$, we construct its approximate solution $$\begin{aligned} %\label{eq_wN_new} w(t,x) = e^{i\psi(t, x,\rho, \varpi)} b(t,x,\rho, \varpi).\end{aligned}$$ For the reminder term, we follow the same idea of Proposition [Proposition 13](#pp_reminder_new){reference-type="ref" reference="pp_reminder_new"}. More explicitly, here we consider $$\begin{aligned} \widetilde{N} = (-1, 0, \ldots, 0) \in \mathbb{R} \times \mathbb{R}^n, \qquad \widetilde{H} = \{(t,x): (t,x) \cdot N \geq -T + \varepsilon\}, \qquad \widetilde{\zeta^o} = (i\rho^2, i \rho \varpi).\end{aligned}$$ Recall we write $^t \! P= \partial^2_t - \Delta + \partial_t\Delta$ and $^t \! P_o = ^t \! P(D + \widetilde{\zeta^o})$. We would like to verify the condition ([b](#condition_b)) for $^t \! P_o$. We have following lemma, as an analog to Lemma [Lemma 1](#lm_A1A2){reference-type="ref" reference="lm_A1A2"}. **Lemma 2**. *Let $^t \! P$, $\widetilde{\zeta^o}$, $\widetilde{N}$ be defined as above. Let ${\sigma}(\zeta)$ be the solution to $^t \! P_o(\zeta + \sigma \widetilde{N})$, where $\zeta = (\tau, \xi) \in \mathbb{C} \times \mathbb{C}^n$. If ${\sigma}(\zeta)$ is analytic and single-valued in a ball $B \subset \mathbb{C} \times \mathbb{C}^n$ with real center and radius $A_1 = 1$, then we have $$\sup_{\zeta \in B} \mathrm{Im} (\sigma(\zeta)) \geq 0.$$* *Proof.* In this case, we solve ${\tau}(\xi)$ from $^t \! P(\tau, \xi) = 0$ for $\xi \in \mathbb{C}^n$. Indeed, the equation $-{\tau}^2 - i {\tau} \xi \cdot \xi + \xi \cdot \xi = 0$ have two complex roots $$\begin{aligned} \tau_\pm(\xi) %= \frac{1}{2}(i\xi \cdot \xi \pm \sqrt{-(\xi \cdot \xi)^2 + 4 \xi \cdot \xi}). = \frac{1}{2}(- iR(\xi) \pm \sqrt{-R^2(\xi) + 4 R (\xi)}), \qquad \text{where }R(\xi) = \xi \cdot \xi. % \qquad % \tau_2 = \frac{1}{2}(i\xi \cdot \xi ) - \sqrt{-(\xi \cdot \xi)^2 + 4 \xi \cdot \xi}. \end{aligned}$$ Then the roots for $^t \! P_o(\zeta + \sigma N) = ^t \! P(\zeta + \widetilde{\zeta^o} + \sigma \widetilde{N}) = 0$ are given by $$\begin{aligned} \sigma_\pm(\tau, \xi) &= \tau + i\rho^2- {\tau}_\pm(\xi + i\rho \varpi)\\ &= \frac{1}{2}(iR(\xi + i \rho \varpi) \mp \sqrt{-R^2(\xi + i \rho \varpi) + 4 R(\xi + i \rho \varpi)}) + i\rho^2 + \tau. \end{aligned}$$ For a ball $B$ with real center and radius $1$, we can choose a point $(\tau, \xi)\in \mathbb{R} \times \mathbb{R}^n$ in $B$ such that $(\tau, \xi) = (\tau, r \theta)$, where $r \geq 1$ and $\theta \in \mathbb{S}^{n-1}$. The proof of Lemma [Lemma 1](#lm_A1A2){reference-type="ref" reference="lm_A1A2"} shows that the imaginary part of the roots satisfies $$\mathrm{Im}({\sigma}_\pm(0, r\theta)) \geq 0.$$ This proves for any ball $B$ with real center and radius $A_1$, we have $$\sup_{\zeta \in B} \mathrm{Im} ({\sigma}(\zeta)) \geq A_2 = 0,$$ if ${\sigma}(\zeta)$ is a real-valued and analytic root. ◻ **Corollary 3**. *For $N \geq 0$, a large parameter $\rho$, and $\varpi \in \mathbb{S}^{n-1}$, let $w(t,x)$ be the approximate solution constructed in ([\[eq_vN\]](#eq_vN){reference-type="ref" reference="eq_vN"}), with the phase function $\psi = i\rho^2 t + i\rho(x \cdot \varpi)$ and the smooth amplitude $b = \sum_{j=0}^N\rho^{-j}{b_j(t,x, \varpi)}$ constructed in Section [4.1](#subsec_backward_goc){reference-type="ref" reference="subsec_backward_goc"}. Then there exists a solution $d_N$ to the equation $$\begin{aligned} (^t \! P+q)(e^{i \varphi} d_N) = -(^t \! P+q) w\qquad \text{in }(0, T) \times \Omega \end{aligned}$$ with $d_N(T,x) = \partial_t d_N(T, x) = 0$, such that for $s \geq 0$, we have $$\|d_N\|_{H^s((0,T)\times \Omega)} \leq {C_N}{\rho^{-N-1}},$$ where $C_N$ depends on $q, n, \Omega$ and the choice of $\phi, \chi$.* ## Recovering $q$ In this part, we consider the recovery of $q$ from the DN map for the linear problem, which is related to the first-order linearization of $\Lambda_{q, \beta}$. More explicitly, consider the boundary value problem $$\label{eq_linear_f_new} \begin{aligned} (\partial^2_t - \Delta - \partial_t\Delta + q(t,x))u &= 0, & \ & \mbox{on } (0, T) \times \Omega,\\ u(t,x) &= h, & \ &\mbox{for } x\in \partial \Omega,\\ u = 0, \ \partial_t u &= 0, & \ &\mbox{for } t=0.\\ \end{aligned}$$ Let $h \in C^{2m+2}([0,T] \times \partial \Omega)$. Then this linear problem has a unique solution $u$ satisfying ([\[est_nl\]](#est_nl){reference-type="ref" reference="est_nl"}), if the data $(0, 0, h)$ satisfies the $m$th-order compatibility condition ([\[def_comp_nl\]](#def_comp_nl){reference-type="ref" reference="def_comp_nl"}) with $\beta \equiv 0$, by Proposition [Proposition 1](#pp_nl){reference-type="ref" reference="pp_nl"}. This is equivalent to require $\partial_t^j h(0) = 0$ for $j = 0, \ldots, m$. Thus, for $h \in \mathcal{H}^m$, we define the linear DN map $$\Lambda^\mathrm{lin}_{q}: h \rightarrow \partial_\nu u|_{\partial \Omega}.$$ First, we prove the following proposition, which shows the DN map for the linear problem determines the potential $q$. **Proposition 14**. *Let $q^{(j)} \in {C^\infty([0,T]\times \bar{\Omega})}$ be two potentials, $j = 1,2$. For some $\rho, \rho_h>0$, suppose $$\Lambda^\mathrm{lin}_{q^{(1)}}(h) = \Lambda^\mathrm{lin}_{q^{(2)}}(h)$$ for $h \in \mathcal{H}^m$. Then we have ${q^{(1)}}= {q^{(2)}}$ in $[0,T]\times \bar{\Omega}$.* *Proof.* We follow exactly the same idea as before. For large $\rho$ and fixed $\omega \in S^{n-1}$, let $u^{(1)}= e^{i\varphi}(a_0 + r_0)$ be constructed as in Proposition [Proposition 13](#pp_reminder_new){reference-type="ref" reference="pp_reminder_new"}, which solves $$(P + {q^{(1)}}) u^{(1)}= 0, \qquad u^{(1)}(0) = \partial_tu^{(1)}(0) = 0,$$ with the phase function $\varphi = -i\rho^2 t + i\rho(x \cdot \omega)$. Recall in ([\[eq_a0\]](#eq_a0){reference-type="ref" reference="eq_a0"}), we choose $$a_0(t, x,\omega) = \phi(t) \prod_{j = 2}^{n} \chi(\omega_j \cdot (x - y_0)) \coloneqq \phi(t)\tilde{a}_0(x, \omega).$$ where $\phi \in C_0^\infty((0, T))$ and $\chi \in C_0^\infty(\mathbb{R})$ is supported in $(-\epsilon, \epsilon)$ with $\chi = 1$ near $0$. Note that $\tilde{a}_0(x, \omega)$ is a smooth function supported in a small $\epsilon$-neighborhood of the ray $\gamma(s) = s \omega + y_0$, for a fixed point $y_0 \in \partial \Omega$. Recall the reminder term satisfies $\|r_0\|_{H^s((0,T) \times \Omega)} \leq C\rho^{-1}$, where we choose $s$ large enough. This implies $u^{(1)}\in H^{s}((0,T) \times \Omega)$. Then we set $$h = u^{(1)}|_{\partial \Omega},$$ which belongs to $C^{2m+{2}}([0,T] \times \partial \Omega)$, for large enough $s$. In particular, we have $\partial_t^j h(0) = 0$ for $j = 0, \ldots, m$, as in Proposition [Proposition 13](#pp_reminder_new){reference-type="ref" reference="pp_reminder_new"} we construct $a_0, r_0$ supported in $(\varepsilon, T)$ for some $\varepsilon> 0$. Thus, there exists a unique $u^{(2)}$ solving the linear problem ([\[eq_linear\]](#eq_linear){reference-type="ref" reference="eq_linear"}) with the potential ${q^{(2)}}$, the Dirichlet data $h$, and vanishing initial conditions. We set $u = u^{(2)}- u^{(1)}$ and $q = {q^{(2)}}- {q^{(1)}}$, which satisfies $$\begin{aligned} (P+{q^{(2)}}) u &= qu^{(1)}, & \ & \mbox{on } (0, T) \times \Omega,\\ u(t,x) &= 0, & \ &\mbox{for } x\in \partial \Omega,\\ u = 0, \ \partial_t u &= 0, & \ &\mbox{for } t=0.\\ \end{aligned}$$ Moreover, with the assumption that $\Lambda^\mathrm{lin}_{q^{(1)}}(h) = \Lambda^\mathrm{lin}_{q^{(2)}}(h)$, we have $$\partial_\nu u|_\Omega = 0.$$ Now let $w = e^{i\psi}(b_0 + d_0)$ be as in Corollary [Corollary 3](#cr_reminder_new){reference-type="ref" reference="cr_reminder_new"}, which solves the backward problem ([\[eq_linearback\]](#eq_linearback){reference-type="ref" reference="eq_linearback"}), with the phase function $\psi = i\rho^2 t + i\rho(x \cdot \varpi)$. Here we set $\varpi = -\omega$. We choose $$b_0(t, x,\varphi) = \phi(t)\prod_{j = 2}^{n} \chi(\omega_j \cdot (x - y)) \coloneqq \phi(t)\tilde{b}_0(x, \omega).$$ as a smooth function supported in a small $\epsilon$-neighborhood of the same ray $\gamma(s)$, with the same $\phi$. In this case, note that $w(T,x) = \partial_tw(T,x) = 0$ and the reminder term satisfies $\|d_0\|_{H^s((0,T) \times \Omega)} \leq C \rho^{-1}$. We multiply $w$ with the equation above and integrate over $t,x$ to have $$\begin{aligned} 0 = \int_0^T \int_\Omega (P+{q^{(2)}}) uw - qu^{(1)}w \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = \int_0^T \int_\Omega u (^t \! P+{q^{(2)}}) w - qu^{(1)}w \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t, \end{aligned}$$ which implies $$\int_0^T \int_\Omega - qu^{(1)}w \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = 0.$$ The following proof is just the same as that of Proposition [Proposition 12](#pp_recoverq){reference-type="ref" reference="pp_recoverq"}. For completeness, we repeat it here. We plug in the expansion of $u^{(1)}, w$ to have $$\begin{aligned} \mathcal{I}_q = \int_0^T \int_\Omega q a_0 b_0 \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = -\int_0^T \int_\Omega q ( r_0 b_0 + a_0 d_0) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t \leq C \rho^{-1}. %(\|q\|_{C^\qCm([0,T]\times \bar{\Omega})}). %(\|a_0\|_{C([0,T]\times \bar{\Omega})} + \|b_0\|_{C([0,T]\times \bar{\Omega})} ). \end{aligned}$$ When $\rho \rightarrow \infty$, this implies $\mathcal{I}_q = 0$. Since $\phi \in C_0^\infty((0,T))$ is arbitrary, we must have $$\int_\Omega q(t,x) \tilde{a}_0 \tilde{b}_0\mathop{}\!\mathrm{d}x = 0, \qquad t \in [0,T].$$ Note that $\tilde{a}_0$ and $\tilde{b}_0$ are both supported in a small $\epsilon$-neighborhood of the ray $\gamma(s) = s \omega + y$. Let $\epsilon\rightarrow 0$ and we can extract the line integral $$X q(\gamma) = \int q(t, \gamma(s)) \mathop{}\!\mathrm{d}s = 0, \qquad t \in [0,T].$$ Since $\omega \in S^{n-1}$ and $y_0\in \partial \Omega$ are arbitrary, the X-ray transform of $q$ over all rays vanishes. With the assumption on $\Omega$, we have $q = {q^{(2)}}- {q^{(1)}}= 0$. Indeed, the X-ray transform is injective on $L^1(\mathbb{R}^n)$ by the Fourier Slice Theorem, for example, see [@book_SU Chapter 2]. One can extend $q \in C^\infty([0,T] \times \bar{\Omega})$ to $L^1(\mathbb{R}^n)$ by setting it equal to zero in $\mathbb{R}^n\setminus \bar{\Omega}$, as $\Omega$ is bounded. ◻ In this case, compatibility condition of the Dirichlet data $h$ for the linear problem coincides with the compatibility condition for the nonlinear one. Indeed, both of them requires $\partial_t^j h(0) = 0, j = 0, \ldots, m$. In this case, we can say $$\begin{aligned} \label{eq_Lam} \Lambda^\mathrm{lin}_{q}(g_0, g_1, h) = %\lim_{\ep \rightarrow 0} \partial_\varepsilon\Lambda_{q, \beta}(\varepsilon g_0, \varepsilon g_1, \varepsilon h)|_{\varepsilon=0},\end{aligned}$$ for small $h \in \mathcal{H}^m$. Therefore, Proposition [Proposition 12](#pp_recoverq){reference-type="ref" reference="pp_recoverq"} implies the first-order linearization of $L_{q,\beta}$ determines $q$. ## The second-order Linearization {#subsec_2nd_new} In this subsection, let $\varepsilon_1, \varepsilon_2 > 0$ be small parameters and let $u_{\varepsilon_1,\varepsilon_2}$ be the solution to the nonlinear problem ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) with data $$\begin{aligned} \label{def_2data_new} g_0 = g_1 = 0, \qquad h = \varepsilon_1 h_1 + \varepsilon_2 h_2.\end{aligned}$$ We consider the second-order linearization $$U_2 = \partial_{\varepsilon_1} \partial_{\varepsilon_2} u_{\varepsilon_1, \varepsilon_2}|_{\varepsilon_1 = \varepsilon_2 = 0}.$$ Now we have $$\partial_\nu U_2|_{\partial \Omega} = \partial_{\varepsilon_1} \partial_{\varepsilon_2}\Lambda_{q, \beta}(h)|_{\varepsilon_1 = \varepsilon_2 = 0}.$$ Let $u_j$ be solutions to the linear problem $$\begin{aligned} \label{LS_new} \begin{aligned} (P+q) u_j(t,x) &= 0, & \ & \mbox{on } (0, T) \times \Omega,\\ u_j(t,x) &= h_j, & \ &\mbox{for } x\in \partial \Omega,\\ u_j = 0,\ \partial_tv_j &= 0, & \ &\mbox{for } t=0.\\ \end{aligned}\end{aligned}$$ Then $U_2$ solves $$\begin{aligned} \begin{aligned} (P+q) U_2 &= \beta(t,x) \partial_t^2 (u_1u_2), & \ & \mbox{on } (0, T) \times \Omega,\\ U_2(t,x) &= 0, & \ &\mbox{for } x\in \partial \Omega,\\ U_2 = \partial_t U_2 &= 0, & \ &\mbox{for } t=0.\\ \end{aligned}\end{aligned}$$ Now for any $u_0$ solving ([\[eq_linearback\]](#eq_linearback){reference-type="ref" reference="eq_linearback"}) with the Dirichlet data $h_0$, we integrate by parts to have $$\begin{aligned} \label{eq_Ibeta_new} &\int_0^T \int_\Omega \beta(t,x) \partial_t^2(u_1 u_2) u_0 \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t\\ = & \int_0^T \int_\Omega (P+q) U_2 u_0 \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = - \int_0^T \partial_\nu U_2|_{\partial \Omega} h_0 + \partial_t\partial_\nu U_2 |_{\partial \Omega} h_0 \mathop{}\!\mathrm{d}t \nonumber \\ = &- \int_0^T \partial_{\varepsilon_1} \partial_{\varepsilon_2}\Lambda_{q, \beta}(f)|_{\varepsilon_1 = \varepsilon_2 = 0} h_0 + \partial_t\partial_{\varepsilon_1} \partial_{\varepsilon_2}\Lambda_{q, \beta}(f)|_{\varepsilon_1 = \varepsilon_2 = 0} h_0 \mathop{}\!\mathrm{d}t . \nonumber\end{aligned}$$ ## Recovering $\beta$. {#recovering-beta.} Now we would like to recover the nonlinear coefficient from the DN map. Suppose we have two coefficients $\beta^{(1)}, \beta^{(2)}\in C^\infty([0,T] \times \bar{\Omega})$. Let $u^{(k)}_{\varepsilon_1,\varepsilon_2}$ be the solutions to the nonlinear problem ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) with $\beta^{(k)}$, the vanishing initial conditions, and the Dirichlet data given in ([\[def_2data_new\]](#def_2data_new){reference-type="ref" reference="def_2data_new"}), for $k=1,2$. We consider the corresponding DN maps $\Lambda_{q^{(k)}, \beta^{(k)}}$ and suppose $$\Lambda_{{q^{(1)}}, \beta^{(1)}}(h) = \Lambda_{{q^{(2)}}, \beta^{(2)}}(h),$$ for any small and compatible Dirichlet data $h$. First, according to ([\[eq_Lam\]](#eq_Lam){reference-type="ref" reference="eq_Lam"}) and Proposition [Proposition 14](#pp_recoverq_new){reference-type="ref" reference="pp_recoverq_new"} we have ${q^{(1)}}= {q^{(2)}}$. Next, we would like to construct approximate solutions $u_j$ to linear problems using Section [4](#sec_goc){reference-type="ref" reference="sec_goc"} and [6.1](#subsec_reminder_new){reference-type="ref" reference="subsec_reminder_new"}, which are corresponding to the Dirichlet data $h_j$, for $j = 1,2,0$. As before, by ([\[eq_Ibeta_new\]](#eq_Ibeta_new){reference-type="ref" reference="eq_Ibeta_new"}), we have $$\begin{aligned} \label{eq_Ibeta_zero} \int_0^T \int_\Omega (\beta^{(1)}- \beta^{(2)}) \partial^2_t(u_1 u_2) u_0 \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = 0.\end{aligned}$$ Then we follow exactly the same idea as in the proof of Section [5.5](#subsec_beta){reference-type="ref" reference="subsec_beta"}. But for completeness, we repeat it here. As before, let $\omega_1, \omega_2, \ldots, \omega_{n}$ be an orthonormal basis for $\mathbb{R}^n$. Let $\varpi = - (\omega_1 + \omega_2)/\sqrt{2} \in S^{n-1}$ and we choose $\varpi_2, \ldots \varpi_n$ such that they form an orthonormal basis with $\varpi$. For a fixed point $x_0 \in \Omega$, we choose $y_1, y_2, y_0\in \partial \Omega$ such that the three rays $$\gamma_j(s) = s \omega_j + y_j, %\quad x = s \omega_2 + y_2, \quad x = s \omega_0 + y_0 \ \text{ for } j = 1,2, \qquad \text{and } \gamma_0(s) = s \varpi + y_0,$$ intersect at $x_0$. We construct approximate solutions $$\begin{aligned} u_{j} &= e^{\rho^2t - \rho (w_j \cdot x)} (a_{0,j} + r_{0,j}) ,\quad j = 1,2, % v_{1,0} &= e^{+2\rho^2t + i\sqrt{2}\rho (w_0 \cdot x)} a_{0,0}(t,x)\end{aligned}$$ to $(P + {q^{(1)}}) u_j =(P + {q^{(2)}}) u_j = 0$, according to Proposition [Proposition 13](#pp_reminder_new){reference-type="ref" reference="pp_reminder_new"}. Here the amplitudes given by $$\begin{aligned} a_{0,1}(t,x,\omega_j) = \phi(t) \prod_{i \neq 1} \chi(\omega_i \cdot (x - y_1)) \coloneqq \phi(t)\tilde{a}_{0,1}(x, \omega),\\ a_{0,2}(t,x,\omega_j) = \phi(t) \prod_{i \neq 2} \chi(\omega_i \cdot (x - y_2)) \coloneqq \phi(t)\tilde{a}_{0,2}(x, \omega),\end{aligned}$$ are smooth function supported in a small $\epsilon$-neighborhood of the ray $\gamma_1(s)$ and $\gamma_2(s)$ respectively. Moreover, the reminder term is given by $\|r_{0,j}\|_{H^s((0,T) \times \Omega)} \leq C\rho^{-1}$, for $j = 1,2$, where $s$ is chosen to be large enough. Then we set $$h_j = u_j|_{\partial \Omega}, \quad j =1,2,$$ which satisfy the compatibility condition ([\[def_comp_nl\]](#def_comp_nl){reference-type="ref" reference="def_comp_nl"}), by our construction in Proposition [Proposition 13](#pp_reminder_new){reference-type="ref" reference="pp_reminder_new"}. Then with small $\varepsilon_1, \varepsilon_2$, the vanishing initial conditions and the Dirichlet data $h$ given by ([\[def_2data_new\]](#def_2data_new){reference-type="ref" reference="def_2data_new"}) allows a unique solution to the nonlinear problem. Thus, let $u^{(k)}_{\varepsilon_1,\varepsilon 2}$ be the solutions to the nonlinear problem ([\[eq_nl\]](#eq_nl){reference-type="ref" reference="eq_nl"}) with $\beta^{(k)}$ and ([\[def_2data_new\]](#def_2data_new){reference-type="ref" reference="def_2data_new"}). Note the second-order linearization $U_2^{(k)}$ satisfies $$\partial_\nu U_2^{(1)}|_{\partial \Omega} = \partial_\nu U_2^{(2)}|_{\partial \Omega},$$ since $\Lambda_{{q^{(1)}}, \beta^{(1)}}( h) = \Lambda_{{q^{(2)}}, \beta^{(2)}}( h)$. Then we construction approximate solutions $u_0$ to the backward problem ([\[eq_linearback\]](#eq_linearback){reference-type="ref" reference="eq_linearback"}) by $$\begin{aligned} u_{0} &= e^{-2\rho^2t - \sqrt{2}\rho (\varpi \cdot x)} (b_{0} + d_{0}),\end{aligned}$$ according to Corollary [Corollary 3](#cr_reminder_new){reference-type="ref" reference="cr_reminder_new"}. Here the amplitudes given by $$\begin{aligned} b_{0}(t,x,\varpi) = \phi(t) \prod_{i \neq 1} \chi(\varpi_i \cdot (x - y_0)) \coloneqq \phi(t)\tilde{b}_{0}(x, \varpi),\end{aligned}$$ is a smooth function supported in a small $\epsilon$-neighborhood of the ray $\gamma_0(s)$. In this case, the reminder term is given by $\|d_{0}\|_{H^s((0,T) \times \Omega)} \leq C\rho^{-1}$. Now we plug in $u_1,u_2,u_0, U_2^{(1)}, U_2^{(2)}$ in ([\[eq_Ibeta_new\]](#eq_Ibeta_new){reference-type="ref" reference="eq_Ibeta_new"}) to have $$\begin{aligned} \int_0^T \int_\Omega (\beta^{(1)}- \beta^{(2)}) \partial^2_t(u_1 u_2) u_0 \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = 0.\end{aligned}$$ As before, we have $$\begin{aligned} \partial^2_t(u_1 u_2) & = e^{2 \rho^2 t -\rho (\omega_1 + \omega_2) \cdot x} (4 \rho^4 \phi^2(t) \tilde{a}_{0,1}\tilde{a}_{0,2} + R_1),\end{aligned}$$ where we write $$\begin{aligned} R_1 &= 4 \rho^4 (r_{0,1}(a_{0,2} + r_{0,2}) + (a_{0,1} + r_{0,1}) r_{0,2})\\ &-2 \rho^2 \partial_t((a_{0,1} + r_{0,1})(a_{0,2} + r_{0,2})) + \partial^2_t((a_{0,1} + r_{0,1})(a_{0,2} + r_{0,2})).\end{aligned}$$ A straightforward computation shows that $\|R_1\|_{H^{s-2}((0,T)\times \Omega)} \leq C \rho^3$. Then we have $$\begin{aligned} 0 & = \int_0^T \int_\Omega (\beta^{(1)}- \beta^{(2)}) (4 \rho^4 \phi^3(t) \tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0}+ R_2) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t\end{aligned}$$ where we write $$R_2 = R_1 (b_{0} + d_{0}) + (4 \rho^4 \phi^2(t) \tilde{a}_{0,1}\tilde{a}_{0,2} + R_1) d_{0}.$$ Similarly, we have $\|R_2\|_{H^{s-2}((0,T)\times \Omega)} \leq C \rho^3$. It follows that $$\begin{aligned} \int_0^T \int_\Omega (\beta^{(1)}- \beta^{(2)}) \phi^3(t) \tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0} \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}t = 0.\end{aligned}$$ Since $\phi \in C_0^\infty((0,T))$ is arbitrary, this implies $$\int_\Omega (\beta^{(1)}(t,x)- \beta^{(2)}(t,x)) \tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0} \mathop{}\!\mathrm{d}x = 0, \qquad t \in [0,T].$$ Recall the rays $\gamma_1, \gamma_2, \gamma_0$ intersect at the fixed point $x_0 \in \Omega$. Then we can assume $\tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0}$ is supported in a small neighborhood $N_\epsilon$ of $x_0$, which is contained in a ball of radius $\epsilon$. By shrinking the support of $\chi$, we have $$\beta^{(1)}(t,x_0)- \beta^{(2)}(t,x_0) = \lim_{\epsilon\rightarrow 0} \frac{1}{|N_\epsilon|}\int_\Omega (\beta^{(1)}(t,x)- \beta^{(2)}(t,x)) \tilde{a}_{0,1}\tilde{a}_{0,2} \tilde{b}_{0} \mathop{}\!\mathrm{d}x = 0.$$ Since $x_0 \in \Omega$ is arbitrary, we must have $\beta^{(1)}= \beta^{(2)}$ in $[0,T] \times \bar{\Omega}$. # Appendix {#sec_apdx} ## Preliminaries {#preliminaries} Suppose $X$ is a Banach space and we denote by $H^k([0,T]; X)$ the set of all measurable functions $u: [0,T] \rightarrow X$ with $$\|u\|_{H^k([0,T]; X)} = (\sum_{j = 0}^k \int_0^T \| \partial_t^j u(t)\|^2_X \mathop{}\!\mathrm{d}t)^{\frac{1}{2}} < +\infty.$$ We have the following Sobolev embedding result, according to [@evans2022partial Section 5.9.2]. **Lemma 3**. *Let $\Omega \subset \mathbb{R}^n$ be a bounded open set with smooth boundary. For integers $l \geq 0$ and $k \geq k_n$, suppose $u \in H^{l+1}([0,T]; H^k ({\Omega}))$. Then we have $$u \in C^{l}([0,T]; H^k(\Omega)) \subset C^{l}([0,T]; C^{k-k_n}(\bar{\Omega}))$$ with the estimates $$\sum_{j = 0}^{l} \sup_{t \in [0, T]}\| \partial_t^{j} u(t) \|_{C^{k-k_n}(\bar{\Omega})} \leq C\sum_{j = 0}^{l} \sup_{t \in [0, T]}\| \partial_t^{j} u(t) \|_{H^k} \leq C'\| u\|_{H^{l+1}([0,T]; H^k (\Omega))}. %C\sum_{j = 0}^k \int_0^T (\| \dt^j u(t)\|^2_l) \diff t,$$* *Proof.* By [@evans2022partial Section 5.9, Theorem 2], we have $$\sup_{t \in [0, T]}\| \partial_t^{j} u(t) \|_{k} \leq \| u\|_{H^{l+1}([0,T]; H^k (\Omega))},$$ for any $j = 0, 1, \ldots, l$. Then by the Sobolev embedding inequality, with $\partial_t^{j} u(t) \in H^k(\Omega)$, we have $$\| \partial_t^{j} u(t) \|_{C^{k-k_n}(\bar{\Omega})} \leq C \| \partial_t^j u(t)\|^2_{H^k},$$ which proves the desired result. ◻ ## The linear problem with variable coefficients {#sec_linear} Let $\Omega \subset \mathbb{R}^n$ be a bounded open set with smooth boundary, for $n \geq 2$. In this section, we consider the following linear problem $$\begin{aligned} \label{eq_linear_f} \begin{aligned} (1 - 2\beta v) \partial^2_tu - \Delta u - \partial_t\Delta u -q(t,x)u &= f(t,x), & \ & \mbox{on } (0,T)\times \Omega,\\ u(t,x) &= 0, & \ &\mbox{for } x\in \partial \Omega,\\ %{u = \partial_t u} &= 0, & \ &\mbox{for } t=0. u = g_0, \quad & \partial_t u = g_1 , & \ &\mbox{for } t=0, \end{aligned}\end{aligned}$$ where $\beta, q \in C^\infty([0,T]\times \bar{\Omega})$, $f$ is the source, and $v$ is a function to be specified later. For the purpose of the nonlinear problem, we consider the space ${Z^m}(R,T)$, the set containing all functions $u$ such that $$u \in \bigcap_{k=0}^{m} C^{m-k}([0,T]; H^{2k+1}(\Omega)), \quad \|u\|^2_{{Z^m}} = \sum_{k=0}^m \sup_{s \in [0, T]} \|\partial_t^{m-k} v(s)\|^2_{{H^{2k+1}}} \leq R^2,$$ where $R> 0$ is a parameter to be chosen later. To deal with the initial value problem with a source, we consider $$\begin{aligned} \label{def_gf} g_0 \in H_0^1(\Omega) \cap H^{2m+1}(\Omega), \qquad g_1 \in H_0^1(\Omega) \cap H^{2m-1}(\Omega),\qquad f \in {Z^{m-1}}(R, T),\end{aligned}$$ for some $R > 0$. For convenience, we write $C_{l,j}= \binom{l-1}{j}$. For $l = 1, \ldots, m-1$, suppose $1- 2\beta(0)v(0) \neq 0$. We recursively define $$\begin{aligned} \label{def_gl} g_{l+1} \coloneqq (1- 2\beta(0)v(0))^{-1}(\partial_t^{l-1} f(0) + G_l(0, x, v(0), \partial_tv(0), \ldots, \partial_t^{l-1}v(0)); g_0, \ldots, g_l),\end{aligned}$$ where we introduce the notation $$\begin{aligned} \label{def_Gl} %G_l(u, \dt u, \ldots, \dt^l u; v, \dt v, \ldots, \dt^{l-1}v) &G_l(t,x, v, \partial_tv, \ldots, \partial_t^{l-1}v; g_0, \ldots, g_l)\\ %&G_l(t,x, v, \dt v, \ldots, \dt^{l-1}v; g_0, \ldots, g_l)\\ =&-\sum_{j=2}^{l}C_{l,j}\partial_t^{l+1-j} (1-2\beta(t) v(t)) g_j + \Delta g_{l-1} + \Delta g_l - \sum_{j=0}^{l-1}C_{l,j}\partial_t^{l-1-j}q(t) g_j,\nonumber\end{aligned}$$ We say $g_0, g_1, f$ satisfy the *$m$th-order compatibility condition*, if $$\begin{aligned} \label{def_comp_linear} g_{l+1} \in H_0^1(\Omega) \cap H^{2(m-l) - 1}(\Omega), \qquad \text{for }l = 1, \ldots, m-1.\end{aligned}$$ when $a(0) \neq 0$ and $\partial_t^{l-1} a(0) \in C(\bar{\Omega})$ is valid, for $l = 1, \ldots, m$. For convenience, we write $$\begin{aligned} \label{def_Ngf} N_{fg}= \|f\|_{Z^{m-1}}+ \sum_{l=0}^m \|g_l\|_{H^{2(m-l)+1}}.\end{aligned}$$ **Proposition 15**. *Let $m \geq 2k_n$ and $T > 0$ be fixed. Let $q, \beta \in C^{{\color{teal}Q}}([0,T]\times \bar{\Omega})$. Suppose $g_0, g_1$ and $f$ satisfy ([\[def_gf\]](#def_gf){reference-type="ref" reference="def_gf"}) and the $m$th-order compatibility condition defined as above. Then there exists $r_0> 0$ depending on $\beta$ and the domain $\Omega$, such that for any $v \in {Z^m}(r_0, T)$, the linear problem ([\[eq_linear_f\]](#eq_linear_f){reference-type="ref" reference="eq_linear_f"}) has a unique solution $$\begin{aligned} u \in \bigcap_{k=0}^{m} C^{m-k}([0,T]; H^{2k+1}(\Omega)),\end{aligned}$$ satisfying $$\begin{aligned} \label{energy} \|u\|^2_{{Z^m}} \leq C (\|f\|_{Z^{m-1}}+ \|g_0\|_{H^{2m+1}} + \|g_1\|_{H^{2m-1}}),\end{aligned}$$ where $C$ depends on $m$, $T$, $\beta$, $q$, and the domain $\Omega$.* *Proof.* First, we would like to prove there exists a solution $u$ such that $$\begin{aligned} \label{eq_Hm} u \in \bigcap_{k=0}^{m+1} H^{m+1-k}([0,T]; H^{2k+1}(\Omega)), \quad \text{with } \sum_{k=0}^{m+1} \int_0^T \|\partial_t^{m+1-k} v(s)\|^2_{{H^{2k+1}}} \mathop{}\!\mathrm{d}s \leq C N_{fg}. \end{aligned}$$ Then by Lemma [Lemma 3](#lm_emb1){reference-type="ref" reference="lm_emb1"} and an estimate for $N_{fg}$, we have the desired result. In the following, recall we write $a(t) = 1 - 2\beta v$ and $C$ denotes a generic positive constant that depends on $m$, $T$, $\beta$, $q$, $r_0$, and the domain $\Omega$. With $\beta \in C^\infty([0,T]\times \bar{\Omega})$, we choose $r_0$ small enough such that $a$ satisfies $$\frac{1}{2}\leq a(t, x) \leq \frac{3}{2}, \quad \text{for any } (t,x) \in [0, T] \times \Omega.$$ Moreover, by Lemma [Lemma 3](#lm_emb1){reference-type="ref" reference="lm_emb1"}, we have $$\begin{aligned} \label{eq_at} %\dt^l a \in C([0,T]; C^{2(m-l)-\kn -1}(\Omega)), \qquad \text{ for any } 0 \leq l \leq m - \Ckn.\\ \partial_t^{l} a \in C([0,T]; C^{k}(\bar{\Omega})), \quad \text{ if } 2l+k \leq 2m-k_n+ 1. \end{aligned}$$ We use the Galerkin approximation method and construct a sequence of approximate solutions $u_i(t)$ to the equation above. Let $0= \lambda_0< \lambda_1\leq \lambda_2<\cdots$ be the eigenvalues $-\Delta$ with vanishing Dirichlet boundary condition, listed according to their multiplicity, with the corresponding orthonormal eigenfunctions $\phi_k$ in $L^2(\Omega)$, where $k\in \mathbb{N}$. We consider approximate solutions $u_i(t)$ given by $u_i(t)= \sum_{k=1}^i u_{i,k}(t) \phi_k$, which satisfies $$\begin{aligned} \label{eq_dtl} &\langle a(t) \partial_t^{l+1} u_i, w \rangle+ \langle\sum_{j=2}^{l} C_{l,j}\partial_t^{l+1-j} a(t) \partial_t^{j} u_i, w \rangle- \langle\partial_t^{l-1}\Delta u_i, w \rangle - \langle\partial_t^l \Delta u_i, w \rangle\\ &\qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad +\langle\sum_{j=0}^{l-1}C_{l,j}\partial_t^{l-1-j}q \partial_t^j u_i, w \rangle = \langle\partial_t^{l-1}f(t,x), w \rangle, \nonumber \end{aligned}$$ for any $t \in [0, T]$ and any $w$ in the space spanned by $\phi_1, \ldots, \phi_i$. To get this, we formally differentiate the original equation $l-1$ times with respect to $t$ and the initial conditions are $$\partial_t^j u_i(0) = g_j, \quad j= 0, 1, \ldots, l,$$ where $g_j$ is defined in ([\[def_gl\]](#def_gl){reference-type="ref" reference="def_gl"}). Note when $l = 1$, we do not have the second term. There exists a solution $u_{i,k}(t)$ to the ODE obtained from the equation above. We derive prior energy estimates for $u_i(t)$ in the following. **Step 1.** We set $w = \partial_t^l u_i$ and we integrate it with respect to $t$. We estimate each term below. From the first term, we have $$\begin{aligned} \int_0^t \langle a(s) \partial_t^{l+1} u_i(s), \partial_t^l u_i(s) \rangle\mathop{}\!\mathrm{d}s &= \frac{1}{2} \langle a(s) \partial_t^l u_i(s), \partial_t^l u_i(s) \rangle|_0^t -\frac{1}{2} \int_0^t \langle\partial_ta(s) \partial_t^l u_i(s), \partial_t^l u_i(s) \rangle\mathop{}\!\mathrm{d}s,\\ & \geq \frac{1}{4} \|\partial_t^l u_i(t)\|_{L^2}^2 - \frac{3}{4} \|\partial_t^l u_i(0)\|_{L^2}^2 - C \int_0^t \|\partial_t^l u_i(s)\|_{L^2}^2 \mathop{}\!\mathrm{d}s. \end{aligned}$$ Recall we have the compatible initial conditions $\partial_t^l u_i(0) = g_l \in H^1_0(\Omega)$. Next, we estimate $$\begin{aligned} -\int_0^t \langle\partial_t^{l-1} \Delta u_i(s) , \partial_t^l u_i(s) \rangle\mathop{}\!\mathrm{d}s &= \frac{1}{2}\|\partial_t^{l-1} \nabla u_i(t)\|_{L^2}^2 - \frac{1}{2}\|\partial_t^{l-1} \nabla u_i(0)\|_{L^2}^2,\\ -\int_0^t \langle\partial_t^l \Delta u_i(s) , \partial_t^l u_i(s) \rangle\mathop{}\!\mathrm{d}s &= \int_0^t \|\partial_t^{l} \nabla u_i(s) \|_{L^2}^2 \mathop{}\!\mathrm{d}s,\\ -\int_0^t \langle\sum_{j=0}^{l-1}C_{l,j}\partial_t^{l-1-j}q \partial_t^j u_i, \partial_t^l u_i(s) \rangle & \leq %C_m \|q\|_{C^m([0,T] \times \bar{\Omega})} C\sum_{j=0}^{l} \int_0^t \|\partial_t^j u(s)\|_{L^2}\mathop{}\!\mathrm{d}s, \end{aligned}$$ and $$\begin{aligned} \int_0^t \langle\partial_t^{l-1} f(s) , \partial_t^l u_i(s) \rangle\mathop{}\!\mathrm{d}s \leq \int_0^t \|\partial_t^{l-1} f(s)\|_{L^2}^2 + \|\partial_t^{l} u_i(s)\|_{L^2}^2 \mathop{}\!\mathrm{d}s. \end{aligned}$$ To get an estimate for the second term of ([\[eq_dtl\]](#eq_dtl){reference-type="ref" reference="eq_dtl"}), we consider two different cases, when $l \leq L$ and $l \geq L+1$, where we define $$\begin{aligned} \label{eq_L} L = m-\lceil (k_n-1)/2\rceil+ 1,\end{aligned}$$ with $k_n$ defined in ([\[eq_kn\]](#eq_kn){reference-type="ref" reference="eq_kn"}). In the first case, we have $l+1-j \leq m-\lceil (k_n-1)/2\rceil$ for any $j \geq 2$, which implies $\partial_t^{l+1-j}a \in C([0,T]; C(\bar{\Omega}))$ by ([\[eq_at\]](#eq_at){reference-type="ref" reference="eq_at"}). Then we have $$\begin{aligned} \label{eq_modify1} &\sum_{j=2}^{l} \int_0^t \langle C_{l,j}\partial_t^{l+1-j} a(s) \partial_t^{j} u_i(s) , \partial_t^l u_i(s) \rangle\mathop{}\!\mathrm{d}s \leq C\sum_{j=2}^{l} \int_0^t \| \partial_t^{j} u_i(s) \|_{L^2}^2 \mathop{}\!\mathrm{d}s,\end{aligned}$$ for $l = 1, \ldots, L$. It follows that $$\begin{aligned} \frac{1}{4} \|\partial_t^l u_i(t)\|_{L^2}^2 + \frac{1}{2}\|\partial_t^{l-1} \nabla u_i\|_{L^2}^2 + \int_0^t \|\partial_t^{l} \nabla u_i(s) \|_{L^2}^2 \mathop{}\!\mathrm{d}s \leq N_{fg} + C \sum_{j=2}^{l} \int_0^t \| \partial_t^{j} u_i\|_{L^2}^2 \mathop{}\!\mathrm{d}s. \end{aligned}$$ We summarize over $l=1, \ldots, L$ and use the Grönwall inequality to have $$\begin{aligned} %\label{eq_dtu} \sum_{l=1}^{L} \|\partial_t^l u(t)\|_{L^2}^2 + \|\partial_t^{l-1} \nabla u_i\|_{L^2}^2 + \int_0^t \|\partial_t^{l} \nabla u_i(s) \|_{L^2}^2 \mathop{}\!\mathrm{d}s \leq CN_{fg}. %+ \sum_{l=0}^L \|g_l\|_{H^{1}}). %\leq C \|f\|_\zmm. \end{aligned}$$ In particular, using the Poincaré inequality, we have $$\begin{aligned} \label{eq_dtlu} \sum_{l=0}^{L} \int_0^t \|\partial_t^l u(s)\|_{L^2}^2 + \|\partial_t^{l} \nabla u_i(s) \|_{L^2}^2 \mathop{}\!\mathrm{d}s \leq C N_{fg}. \end{aligned}$$ When $l \geq L+1$, we need a new estimate for ([\[eq_modify1\]](#eq_modify1){reference-type="ref" reference="eq_modify1"}) using higher order regularity. **Step 2.** We would like to derive higher-order regularity estimates. More explicitly, for $l = 1, \ldots, L$, we rewrite ([\[eq_dtl\]](#eq_dtl){reference-type="ref" reference="eq_dtl"}) as $$\begin{aligned} \label{eq_dtl_new} \langle\partial_t^{l-1}(-\Delta) u_i, w \rangle + \langle\partial_t^l(-\Delta) u_i, w \rangle = -\langle& a(t) \partial_t^{l+1} u_i, w \rangle - \langle\sum_{j=2}^{l} C_{l,j}\partial_t^{l+1-j} a(t) \partial_t^{j} u_i, w \rangle\\ &- \langle\sum_{j=0}^{l-1}C_{l,j}\partial_t^{l-1-j}q \partial_t^j u_i, w \rangle + \langle\partial_t^{l-1}f(t,x), w \rangle,\nonumber \end{aligned}$$ where we set $w = \partial_t^{l} (-\Delta)^{2k} u_i$, for non-negative integer $k$ satisfying ${k+l \leq L}$. By ([\[eq_at\]](#eq_at){reference-type="ref" reference="eq_at"}), it follows that $$\begin{aligned} \label{eq_a_ljk} {\partial_t^{l-1}a \in C([0, T]; C^{2k}(\bar{\Omega})).} \end{aligned}$$ We have $$\begin{aligned} \langle\partial_t^{l-1}(-\Delta) u_i, \partial_t^{l} (-\Delta)^{2k} u_i\rangle &= \frac{1}{2} (\partial_t\| \partial_t^{l-1} (-\Delta)^{k} \nabla u_i\|_{L^2}^2), % = \| \dt^{l-1} \un \|_{k}^2, \\ \langle\partial_t^l (-\Delta) u , \partial_t^{l} (-\Delta)^{2k} u_i\rangle &= \| \partial_t^{l} (-\Delta)^{k} \nabla u_i\|_{L^2}^2, % = \| \dt^{l-1} \un\|_{k + \beta}^2, \end{aligned}$$ where we integrate by parts and using the property that $((-\Delta)^j u_i) |_{\partial \Omega} = \sum_{k=1}^i \lambda_k^j \phi_k|_{\partial \Omega} = 0$. Recall the initial condition $\partial_t^{l-1} u(0) = g_{l-1} \in H^{2(m-l)+3}(\Omega)$. Moreover, we have $$\begin{aligned} \label{eq_modify3} &-\langle a \partial_t^{l+1} u_i, \partial_t^{l} (-\Delta)^{2k} u_i\rangle\\ =& \langle\nabla (-\Delta)^{k-1}(a \partial_t^{l+1} u_i), \partial_t^{l} \nabla(-\Delta)^{k} u_i\rangle\nonumber \\ \leq &C \|\partial_t^{{l+1}} u_i\|_{H^{2k-1}}^2 + \frac{1}{4}\|\partial_t^{{l}} u_i\|_{H^{2k+1}}^2. \nonumber \end{aligned}$$ In addition, using ([\[eq_a\_ljk\]](#eq_a_ljk){reference-type="ref" reference="eq_a_ljk"}), we have $$\begin{aligned} \label{eq_modify4} & - \sum_{j=2}^{l} \langle C_{l,j}\partial_t^{l+1-j} a \partial_t^{j} u_i, \partial_t^{l} (-\Delta)^{2k} u_i\rangle\\ = & - \langle\sum_{j=2}^{l} C_{l,j}\nabla (-\Delta)^{k-1} (\partial_t^{l+1-j} a(t) \partial_t^{j} u_i) , \partial_t^{l} \nabla (-\Delta)^k u_i\rangle\nonumber \\ \leq & C\sum_{j=2}^{l} \| \partial_t^{j} u_i\|_{H^{2k-1}}^2 + \frac{1}{4}\|\partial_t^{l} u_i\|_{H^{2k+1}}^2\nonumber, \end{aligned}$$ and $$\begin{aligned} \langle\partial_t^{l-1} f, \partial_t^{l} (-\Delta)^{2k} u_i\rangle \leq & 4\| \partial_t^{l-1} (-\Delta)^{k-1}\nabla f\|_{L^2}^2 + \frac{1}{4}\| \partial_t^{l} (-\Delta)^{k} \nabla u_i\|_{L^2}^2\\ \leq & 4\| \partial_t^{l-1} f\|_{H^{2k-1}}^2 +\frac{1}{4}\| \partial_t^{l} u_i\|_{H^{2k+1}}^2. \end{aligned}$$ Combining the inequalities above, we have $$\begin{aligned} \label{eq_induction_even} \frac{1}{4}\int_0^t \| \partial_t^{l} u_i(s)\|_{H^{2k+1}}^2 \mathop{}\!\mathrm{d}s \leq C (\sum_{j=2}^{l+1} \int_0^t \| \partial_t^{j} u_i(s) \|_{H^{2k-1}}^2 \mathop{}\!\mathrm{d}s + N_{fg}), \end{aligned}$$ for any non-negative integer $l + k \leq L$. Further, we would like to use an inductive procedure to show $$\begin{aligned} \label{rm_Hmk} \sum_{l=0}^{L-k}\int_0^t \| \partial_t^{l} u_i(s)\|_{H^{2k+1}}^2 \mathop{}\!\mathrm{d}s \leq C N_{fg}, \quad \text{for any } k = 0, \ldots, L. \end{aligned}$$ Indeed, this is true for $k=0$, since in ([\[eq_dtlu\]](#eq_dtlu){reference-type="ref" reference="eq_dtlu"}) we have the estimates for $\|\partial_t^{l} u_i\|_{H^1}$, when $l = 0, \ldots, L$. Let $k' \geq 1$ and assume ([\[rm_Hmk\]](#rm_Hmk){reference-type="ref" reference="rm_Hmk"}) holds for $k \leq k'$. We would like to prove it is also true for $k = k'+1$. Using ([\[eq_induction_even\]](#eq_induction_even){reference-type="ref" reference="eq_induction_even"}), we have $$\begin{aligned} \int_0^t \| \partial_t^{l} u_i(s)\|_{H^{2k'+3}}^2 \mathop{}\!\mathrm{d}s \leq C (\sum_{j=2}^{l+1} \int_0^t \| \partial_t^{j} u_i(s) \|_{H^{2k' + 1}}^2 \mathop{}\!\mathrm{d}s + N_{fg}) \end{aligned}$$ if $l \leq L - k'-1$. By the assumption, this is bounded by $CN_{fg}$, since $j \leq L-k'$ when $j \leq l+1$. Thus, we have the desired estimate for $k'+1$. By induction, we prove ([\[rm_Hmk\]](#rm_Hmk){reference-type="ref" reference="rm_Hmk"}). **Step 3.** We would like to finish Step 1 and prove ([\[eq_dtlu\]](#eq_dtlu){reference-type="ref" reference="eq_dtlu"}) for $L +1 \leq l \leq m+1$. In this case, we rewrite ([\[eq_modify1\]](#eq_modify1){reference-type="ref" reference="eq_modify1"}) as $$\begin{aligned} &\sum_{j=2}^{l} \int_0^t \langle C_{l,j}\partial_t^{l+1-j} a(s) \partial_t^{j} u_i(s) , \partial_t^l u_i(s) \rangle\mathop{}\!\mathrm{d}s\\ =&\int_0^t \sum_{j=l-L+2}^{l} \langle C_{l,j}\partial_t^{l+1-j} a(s) \partial_t^{j} u_i(s) , \partial_t^l u_i(s) \rangle + \sum_{j=2}^{l-L+1} \langle C_{l,j}\partial_t^{l+1-j} a(s) \partial_t^{j} u_i(s) , \partial_t^l u_i(s) \rangle\mathop{}\!\mathrm{d}s. \end{aligned}$$ When $j \geq l-L+2$, by ([\[eq_at\]](#eq_at){reference-type="ref" reference="eq_at"}) and ([\[eq_L\]](#eq_L){reference-type="ref" reference="eq_L"}) we have $\partial_t^{l+1-j} a \in C([0,T]; C(\bar{\Omega}))$. In this case, one can estimate the first term as before. When $j \leq l-L+1$, we have $$j \leq (m+1)-L+1 = \lceil (k_n+1)/2 \rceil \leq L - 1,$$ since $m \geq 2 k_n$. By Lemma [Lemma 3](#lm_emb1){reference-type="ref" reference="lm_emb1"}, this implies $$\begin{aligned} \int_0^t \|\partial_t^{l+1-j} a(s) \partial_t^{j} u_i(s) \|_{L^2}\mathop{}\!\mathrm{d}s &\leq \|\partial_t^j u_i\|_{ C([0,T]; C(\Omega))} \|\partial_t^{l+1-j} a\|_{L^2}\\ &\leq C\|\partial_t^j u_i\|_{H^1([0,T]; H^{2k+1}(\Omega))},\end{aligned}$$ where we set $k = \lceil (k_n-1)/2 \rceil$. With $j \leq l-L+1$, note that $$j + k \leq k_n+ 1 \leq L,$$ where the last inequality holds as $m \geq 2k_n$. By ([\[rm_Hmk\]](#rm_Hmk){reference-type="ref" reference="rm_Hmk"}), we have $\|\partial_t^j u_i\|_{H^1([0,T]; H^{2k+1}(\Omega))} \leq CN_{fg}$, which implies $$\begin{aligned} &\sum_{j=2}^{l} \int_0^t \langle C_{l,j}\partial_t^{l+1-j} a(s) \partial_t^{j} u_i(s) , \partial_t^l u_i(s) \rangle\mathop{}\!\mathrm{d}s \leq C(\sum_{j=l-L+2}^{l} \int_0^t \| \partial_t^{j} u_i(s) \|_{L^2}\mathop{}\!\mathrm{d}s + N_{fg}).\end{aligned}$$ The rest of terms have the same estimates as before. This proves a complete version of ([\[eq_dtlu\]](#eq_dtlu){reference-type="ref" reference="eq_dtlu"}), i.e., $$\begin{aligned} \label{eq_dtlu_new} \sum_{l=0}^{m+1} \int_0^t \|\partial_t^{l} u_i\|_{L^2}^2 \mathop{}\!\mathrm{d}s + \int_0^t \|\partial_t^{l} \nabla u_i\|_{L^2}^2 \mathop{}\!\mathrm{d}s \leq C N_{fg}.\end{aligned}$$ **Step 4.** Thus, from ([\[eq_dtlu_new\]](#eq_dtlu_new){reference-type="ref" reference="eq_dtlu_new"}), we can conclude that $\{u_i\}_{n=1}^\infty$ is bounded in $H^{m}([0,T]; H^1_0(\Omega))$, with desired estimates. We may extract a subsequence which converges weakly to some $$u \in H^{m+1}([0,T]; H^1_0(\Omega)).$$ By a standard argument, we can show $u$ is a weak solution to ([\[eq_linear\]](#eq_linear){reference-type="ref" reference="eq_linear"}). Last, we would like to show such $u$ is in $H^{m+1-k}([0,T]; H^{2k+1}(\Omega))$ with the desired estimate ([\[eq_Hm\]](#eq_Hm){reference-type="ref" reference="eq_Hm"}) for $k = 0, \ldots, m$ by an inductive procedure. Following the same proof of ([\[eq_dtlu_new\]](#eq_dtlu_new){reference-type="ref" reference="eq_dtlu_new"}) and passing to limits as $i \rightarrow +\infty$, this statement is true for $k = 0$. Now we prove by induction. Assume that $u \in H^{m+1-k}([0,T]; H^{2k+1}({\Omega}))$ satisfies $$\begin{aligned} \label{assump_Hmk} \sum_{j=0}^{m+1-k} \int_{0}^t \|\partial_t^{l} u(s)\|^2_{H^{2k+1}} \mathop{}\!\mathrm{d}s \leq CN_{fg}, \end{aligned}$$ for any $k \leq k' -1$, where $1 \leq k' \leq m+1$ is an integer. In the following, we would like to prove $u \in H^{m-k'}([0,T]; H^{2k'+1}(\Omega))$ with the desired estimate. It suffices to show that $$\int_{0}^t \|\partial_t^{l} u(s)\|^2_{H^{2k'+1}} \mathop{}\!\mathrm{d}s \leq CN_{fg},$$ for any $l +k' \leq m+1$. We prove this for $u_i(t)$ and then pass to limits as $i \rightarrow +\infty$ as before. Indeed, we consider ([\[eq_dtl_new\]](#eq_dtl_new){reference-type="ref" reference="eq_dtl_new"}) and set $w = \partial_t^{l} (-\Delta)^{2k'} u_i$. Previously, we have shown this conclusion for $k'+ l \leq L$, see ([\[eq_induction_even\]](#eq_induction_even){reference-type="ref" reference="eq_induction_even"}). Now we consider the case when $l + k' \leq m+1$ but $l + k' \geq L+1$. We follow the same estimates as before, except for ([\[eq_modify3\]](#eq_modify3){reference-type="ref" reference="eq_modify3"}) and ([\[eq_modify4\]](#eq_modify4){reference-type="ref" reference="eq_modify4"}). Recall for ([\[eq_modify3\]](#eq_modify3){reference-type="ref" reference="eq_modify3"}) and ([\[eq_modify4\]](#eq_modify4){reference-type="ref" reference="eq_modify4"}), actually we need to derive an estimate for the following terms $$\begin{aligned} %\label{eq_modify0} %&\int_0^t \|\nabla (-\Laplace)^{k-1}(a(t) \dt^{l+1} \un(t)\|_\ltwo \diff t,\quad &\sum_{j = 2}^{l+1} \int_0^t \|\nabla (-\Delta)^{k'-1}(\partial_t^{l+1-j} a(s) \partial_t^{j} u(s)) \|_{L^2}^2 \mathop{}\!\mathrm{d}s. \end{aligned}$$ Let $k_1, k_2$ be non-negative integers satisfying $k_1 + k_2 = 2k'-1$. According to ([\[eq_at\]](#eq_at){reference-type="ref" reference="eq_at"}), we have $\partial_t^{l+1-j} a \in C([0,T];C^{k_1}(\bar{\Omega}))$ if $$\begin{aligned} %\label{eq_k1} 2(l+1-j) + k_1 \leq 2m-k_n+1 \quad \Rightarrow \quad 2l-2j + k_1 \leq 2m - k_n-1. \end{aligned}$$ On the other hand, we have $\partial_t^{j} u \in C([0,T];C^{k_2}(\bar{\Omega}))$ if $$\begin{aligned} %\label{eq_k2} 2j + k_2 \leq 2m-k_n+1. \end{aligned}$$ Note that $$(2l-2j + k_1) + (2j + k_2) = 2l+2k'-1 \leq 2m + 1 \leq 4m -2k_n$$ is always true, as $m \geq 2 k_n$. This implies we have either $\partial_t^{l+1-j} a \in C([0,T];C^{k_1}(\bar{\Omega}))$ or $\partial_t^{j} u \in C([0,T];C^{k_2}(\bar{\Omega}))$. In the first case, i.e., $2l-2j + k_1 \leq 2m - k_n-1$, we derive the estimate as before. Otherwise, suppose $2l-2j + k_1 \geq 2m - k_n$. On the one hand, we need $\partial_t^j u \in H^1([0,t]; H^{k_2 + k_n}(\Omega))$. On the other hand, note that we have $$2j + k_2 \leq (2l+2k'-1) - (2m - k_n) \leq k_n+ 1,$$ which implies $$\begin{aligned} (j+1) + \lceil{(k_2+k_n-1)}/{2}\rceil &\leq (j+1) + (k_2+k_n-1)/{2} + {1}/{2}\\ & = (2j + k_2)/2 + k_n/2 + 1 < k_n+ 2. \end{aligned}$$ To use ([\[rm_Hmk\]](#rm_Hmk){reference-type="ref" reference="rm_Hmk"}), recall $m \geq 2 k_n$ such that $k_n+ 1 \leq L$. Hence, we use Lemma [Lemma 3](#lm_emb1){reference-type="ref" reference="lm_emb1"} and ([\[rm_Hmk\]](#rm_Hmk){reference-type="ref" reference="rm_Hmk"}) to have $$\|\partial_t^j u \|_{C([0,T];C^{k_2}(\bar{\Omega}))} \leq C\|\partial_t^j u \|_{H^1([0,T];H^{k_2 + k_n}(\Omega))} \leq CN_{fg}.$$ To conclude, we have $$\begin{aligned} \int_0^t \| \partial_t^{l} u(s)\|_{H^{2k'+1}}^2 \mathop{}\!\mathrm{d}s \leq C (\sum_{j=2}^{l+1} \int_0^t \| \partial_t^{j} u_i(s) \|_{H^{2k'-1}}^2 \mathop{}\!\mathrm{d}s + N_{fg}), \end{aligned}$$ for any non-negative integers $l+k' \leq m+1$. By induction, we have ([\[eq_Hm\]](#eq_Hm){reference-type="ref" reference="eq_Hm"}). Note with $v \in {Z^m}(r_0,T)$ and $f \in {Z^{m-1}}(R,T)$, a similar inductive procure shows from ([\[def_gl\]](#def_gl){reference-type="ref" reference="def_gl"}) we have $$\|g_{l+1}\|_{H^{2(m-l-1)+1}} \leq C(\|f\|_{Z^{m-1}}+ \|g_0\|_{H^{2m+1}} + \|g_1\|_{H^{2m-1}}), \quad l = 1, \ldots, m-1,$$ for a constant $C$ depends on $m$, $\beta$, $q$, and the domain $\Omega$. Then by Lemma [Lemma 3](#lm_emb1){reference-type="ref" reference="lm_emb1"}, we prove the result. ◻ For the nonlinear problem, we need the following lemma. **Lemma 4**. *For $v \in {\widetilde{Z}^m}(r,T)$, the initial conditions $\tilde{g}_0, \tilde{g}_1$ and the source $F(u_h, v)$ satisfy the $m$th-order compatibility condition ([\[def_comp_linear\]](#def_comp_linear){reference-type="ref" reference="def_comp_linear"}) for the linearized equation ([\[eq_linear_w\]](#eq_linear_w){reference-type="ref" reference="eq_linear_w"}).* *Proof.* Indeed, we need to check whether $$\begin{aligned} \hat{g}_{l+1} \coloneqq &(1-2\beta(0)(v(0)+u_h(0)))^{-1}(\partial_t^{l-1} F(0) \\ &+ G_l(0, x,v(0)+u_h(0), \partial_t(v(0)+u_h(0)), \ldots, \partial_t^{l-1}(v(0)+u_h(0))); \hat{g}_0, \ldots, \hat{g}_l) % -\sum_{j=2}^{l}\Cjl\dt^{l+1-j} a(0) g_j % &+ \Laplace g_{l-1}\\ % &+ \Laplace g_l % - \sum_{j=0}^{l-1}\Cjl\dt^{l-1-j}q(0) g_j), \nonumber \end{aligned}$$ belongs to $H_0^1(\Omega) \cap H^{2(m-l)-1}(\Omega)$ or not, where we have $\hat{g}_0 = \tilde{g}_0$, $\hat{g}_1 = \tilde{g}_1$, and $G_l$ defined in ([\[def_Gl\]](#def_Gl){reference-type="ref" reference="def_Gl"}). Note with the initial conditions $\partial_t^l v(0) = \tilde{g}_l$, one has $\partial_t^l (v(0) + u_h(0)) =g_l$, for $l = 0, \ldots, m$. This implies $$\begin{aligned} \hat{g}_{l+1} = (1-2\beta(0)g_0)^{-1}(\partial_t^{l-1} F(0) + G_l(0, x, g_0, g_1, \ldots, g_{l-1}; \hat{g}_0, \ldots, \hat{g}_l) % -\sum_{j=2}^{l}\Cjl\dt^{l+1-j} a(0) g_j % &+ \Laplace g_{l-1}\\ % &+ \Laplace g_l % - \sum_{j=0}^{l-1}\Cjl\dt^{l-1-j}q(0) g_j), \nonumber \end{aligned}$$ In the following, we would like to show $\hat{g}_{l+1}= \tilde{g}_{l+1}$ also holds for $l = 1, \ldots, m-1$, by an inductive procedure. Assume this is true for any $l \leq L-1$, where $L \geq 1$. We check the case $l = L$. We compute $$\begin{aligned} \partial_t^{l-1}F(u_h, v) = &-(1-2\beta(v+u_h))\partial_t^{l+1} u_h + 2 \partial_t^{l-1}(\beta (\partial_t(v+u_h))^2)\\ &+ G_l(t,x, v+u_h, \partial_t(v+u_h), \ldots, \partial_t^{l-1}(v+u_h); h(0), \ldots, \partial_t^l h(0)). \end{aligned}$$ Note that $G_l$ is linear in the second group of arguments. It follows that $$\hat{g}_{l+1} =- \partial_t^{l+1}h(0) + (1-2\beta(0)g_0)^{-1}( \tilde{G}(0,x,g_0, \ldots, g_l)),$$ where we compare ([\[def_tG\]](#def_tG){reference-type="ref" reference="def_tG"}) and ([\[def_Gl\]](#def_Gl){reference-type="ref" reference="def_Gl"}). Then by definition we have $\hat{g}_{l+1} = \tilde{g}_{l+1}$ is true for $l = L$. By induction, this is true for $l = 1, \ldots, m-1$. As $\tilde{g}_l \in H_0^1(\Omega) \cap H^{2(m-l)+1}(\Omega)$, we have the desired result. ◻

arxiv_math

# Introduction {#s:Intro} We present extensions of some celebrated models of random integer partitions, to the case when such partitions are decorated by a subordinate specification, for simplicity described as a categorically distributed coloring. The *fil rouge* of our presentation is an algebraic approach to the count of integer partitions, which we draw from well-known connections among the *Dirichlet distribution*, *Ewens sampling formula* (ESF), *Hoppe's urn model*, the *Chinese restaurant process* (CRP), etc. Our starting point is the observation that univariate moments of the Dirichlet distribution are the generating functions of the (standard, 'monocromatic') ESF (cf. [\[eq:DirichletEwens\]](#eq:DirichletEwens){reference-type="eqref" reference="eq:DirichletEwens"} below). Here, our goal is to describe the relation between *multi*variate moments of the Dirichlet distribution and a '*poly*chromatic' ESF on colored partitions. A systematic treatment of the arising 'colored partition structure', including a representation theorem in the sense of Kingman [@Kin78], will be the subject of future work. Denote by $\Gamma$ the *Euler Gamma function*, by $\left\langle\alpha\right\rangle_{k}\mathop{\mathrm{\coloneqq}}\Gamma(\alpha+k)/\Gamma(\alpha)$ the *Pochhammer symbol* of $\alpha>0$, and by $\mathrm{B}(x_1,\dotsc,x_k)\mathop{\mathrm{\coloneqq}}\Gamma(x_1)\cdots \Gamma(x_k)/\Gamma(x_1+\cdots + x_k)$ the *multivariate Euler Beta function*. For $k\geq 1$ further let $\Delta^{k-1}$ be the *standard simplex* [\[eq:Simplex\]](#eq:Simplex){reference-type="eqref" reference="eq:Simplex"}. For $\boldsymbol\alpha\in {\mathbb R}_+^k$, the *Dirichlet distribution* $D_{\boldsymbol\alpha}$ is the probability measure with density $$\label{eq:DirichletDistribution} \frac{\mathop{\mathrm{\mathbf 1}}_{\Delta^{k-1}}(x_1,\dotsc, x_k)}{\mathrm{B}(\alpha_1,\dotsc,\alpha_k)}\, x_1^{\alpha_1-1}\cdots x_k^{\alpha_k-1}$$ w.r.t. the standard Lebesgue measure on the hyperplane of equation $x_1+\cdots +x_k=1$. *Moments of Dirichlet measures*. To find useful representations for the moments of $D_{\boldsymbol\alpha}$ is a difficult problem, of which we present a brief historical account in §[3.1](#ss:MomentsOverview){reference-type="ref" reference="ss:MomentsOverview"}. As a first main result, we provide a simple, elementary, closed formula for all multivariate moments of $D_{\boldsymbol\alpha}$. Precisely, fix integers $q\in {\mathbb N}_1$ and $\mbfn\mathop{\mathrm{\coloneqq}}\left(n_1,\dotsc, n_q\right)\in{\mathbb N}_1^q$, and let $\msZ_\mbfn$ be the *pattern inventory* [\[eq:PatternInventory\]](#eq:PatternInventory){reference-type="eqref" reference="eq:PatternInventory"} of $\mbfn$, also see [\[eq:t:Zn:0\]](#eq:t:Zn:0){reference-type="eqref" reference="eq:t:Zn:0"}. **Theorem 1 1** (see Thm. [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"}). *For every $\mbfs_1,\dotsc,\mbfs_q\in{\mathbb C}^k$ and $\boldsymbol\alpha\in{\mathbb R}^k_+$, $$\label{eq:t:IntroMain:0} \int_{\Delta^{k-1}} \prod_{j=1}^q (\mbfs_j\cdot\mbfy)^{n_j} \mathop{}\!\mathrm{d}D_{\boldsymbol\alpha}(\mbfy)= \frac{n_1! \cdots n_q!}{\left\langle\alpha_1+\cdots +\alpha_k\right\rangle_{n_1+\cdots+n_q}}\, \msZ_\mbfn[\mbfs_1,\dotsc,\mbfs_q;\boldsymbol\alpha]\,\mathrm{.}$$* By 'simple' we mean that our formula is not further simplifiable in terms of actions of the symmetric groups $\mfS_{n_1},\dotsc, \mfS_{n_q}$, by 'elementary' that it is expressed only in terms of elementary functions, and by 'closed' that it is both non-recursive and non-iterative. *Ewens Sampling Formula*. For a permutation $\pi$ in the symmetric group $\mfS_n$, denote by $r\mathop{\mathrm{\coloneqq}}r(\pi)$ the total number of its cycles (including fixed points). Let $\theta>0$ and recall that a probability distribution on $\mfS_n$ is *$\theta$-biased* if its value on each $\pi$ is proportional to $\theta^r$. The *Ewens Sampling Formula* (ESF) with parameter $\theta$ is the probability distribution $$E_{\theta}(\boldsymbol\lambda)\mathop{\mathrm{\coloneqq}}\frac{n!}{\left\langle\theta\right\rangle_{n}} \prod_{i=1}^n \frac{\theta^{\lambda_i}}{i^{\lambda_i}\lambda_i!} \,\mathrm{,}\;\,\qquad \boldsymbol\lambda\mathop{\mathrm{\coloneqq}}\left(\lambda_1,\dotsc, \lambda_n\right)\,\mathrm{,}\;\,$$ on the set of integer partitions $\boldsymbol\lambda$ of $n$, i.e. satisfying $\sum_{i} i \lambda_i=n$. It is the probability that a $\theta$-biased permutation has given cycle structure $\boldsymbol\lambda$, i.e. with $\lambda_1$ fixed points, $\lambda_2$ transpositions, $\lambda_3$ $3$-cycles, etc. In particular, the distribution $E_{1}$ describes the frequency of a permutation in $\mfS_n$ with a given cycle structure. We refer the reader to the recent surveys [@Cra16; @Tav21] and references therein for a complete account of the history and importance of the ESF throughout mathematics and beyond. *A Polychromatic ESF*. The proof of Theorem [Theorem 1 1](#t:IntroMain){reference-type="ref" reference="t:IntroMain"} will partly consist in counting the cardinality of the orbits of a certain group action with homogeneous space the symmetric group $\mfS_{n_1+\cdots + n_q}$. As a byproduct we derive a *polychromatic ESF* which we now describe. For positive integers $q$ and $\mbfn\mathop{\mathrm{\coloneqq}}\left(n_1,\dotsc, n_q\right)$ we set $n\mathop{\mathrm{\coloneqq}}n_1+\cdots + n_q$ and consider the set $[n]\mathop{\mathrm{\coloneqq}}\left\{1,\dotsc, n\right\}$. We interpret $[q]\mathop{\mathrm{\coloneqq}}\left\{1,\dotsc, q\right\}$ as a set of colors ---or, more generally, of categories--- and assign color $c_1$ to $n_1$ elements of $[n]$, color $c_2$ to $n_2$ elements, and so on, in a fixed deterministic way. Taking into account the coloring of the elements in $[n]$, one may ask for the following refinement of the standard ESF. **Question 1**. *What is the probability that a $\theta$-biased random permutation $\pi\in\mfS_n$, has a given cycle structure and each orbit of $\pi$ has a given number of elements of color $c_j$, $j\in [q]$?* In order to answer Question [Question 1](#quest:MultiEwens){reference-type="ref" reference="quest:MultiEwens"}, it is convenient to encode both the cycle structure of $\pi$ and the number of $c_j$-colored elements in each cycle (orbit) of $\pi$ into a multiset, namely a *$q$-colored partition* which we now describe; also see Dfn. [Definition 2](#d:MultiYoung){reference-type="ref" reference="d:MultiYoung"} below. Suppose that $\pi=\kappa_1\cdots\kappa_r$ is a permutation with cycles $\kappa_i$, including (!) fixed points. To each cycle $\kappa = (y_1 \cdots y_m)$ of $\pi$ we associate its *color count*, i.e. the vector $\mbfa=(a_1,\dotsc, a_q)$ where $a_j$ is the number of elements of color $c_j$ in $\left\{y_1, …, y_m\right\}\subset [n]$. The colored partition associated to $\pi$ is the function $A$ assigning to each fixed $\mbfa$ the number of cycles $\kappa$ of $\pi$ with color count $\mbfa$. We say that $\pi$ *has (cycle structure and) coloring $A$*. As it turns out, the number of permutations with given coloring $A$ is the multinomial coefficient [\[eq:MultinomialMultiYoung\]](#eq:MultinomialMultiYoung){reference-type="eqref" reference="eq:MultinomialMultiYoung"} of $A$. Now, let $\theta>0$ be a rate parameter, and $\mbfp\in\Delta^{q-1}$ be the parameter of a categorical distribution on $[q]$. We define a probability measure $E_{\theta,\mbfp}^n$ (Dfn. [Definition 25](#d:PolyESF){reference-type="ref" reference="d:PolyESF"}) on the set of all $q$-colored partitions of $n$, the properties of which we collect hereafter. **Theorem 2 1** (Polychromatic ESF). *For every $\theta>0$ and every $\mbfp\in\Delta^{q-1}$,* 1. *when $q=1$, hence $\mbfp=p=1$, $E_{\theta,1}^n$ is the Ewens distribution $E_{\theta}$ on partitions of $n$;* 2. *(Prop. [Proposition 28](#p:PolyEwens){reference-type="ref" reference="p:PolyEwens"}) conditioning $E_{\theta,\mbfp}^n$ on a $q$-colored partition $A$ coloring $\mbfn$ gives the probability that a $\theta$-biased random permutation $\pi$ has cycle structure and coloring $A$; (This answers Question [Question 1](#quest:MultiEwens){reference-type="ref" reference="quest:MultiEwens"}.)* 3. *(Prop. [Proposition 31](#p:Hoppe){reference-type="ref" reference="p:Hoppe"}) $E_{\theta,\mbfp}^n$ is the marginal distribution at time $n$ of the *polychromatic Hoppe urn model* described in §[4.1](#ss:Hoppe){reference-type="ref" reference="ss:Hoppe"} and of the extension of the CRP described below;* 4. *(Thm. [Theorem 34](#t:Consistency){reference-type="ref" reference="t:Consistency"}) the family $E_{\theta,\mbfp}^n$, $n\in{\mathbb N}_1$, is *consistent* in a suitable sense extending the notion of Kingman's consistency [@Kin78] to $q$-colored partitions.* The ESF appears in connection with a variety of models. In order to illustrate the analogies between $E_{\theta,\mbfp}^n$ and $E_{\theta}$, let us briefly discuss two of them: Ewens' original allelic partition, and the CRP. In §[4.1](#ss:Hoppe){reference-type="ref" reference="ss:Hoppe"} we present in full detail the polychromatic analogue to Hoppe's urn model [@Hop84]. *The ESF in population genetics*. In the seminal work [@Ewe72], W.J. Ewens introduced the formula later named after him, and showed that $E_{\theta}$ is the joint probability distribution of the number of selectively neutral alleles $A_i^{\scriptscriptstyle{(n)}}$ represented $i$ times in a sample of $n$ genes taken from a large $(\gg n)$ population, viz. $$\mbfP[A^{\scriptscriptstyle{(n)}}_1=\lambda_1,\dotsc, A^{\scriptscriptstyle{(n)}}_n=\lambda_n]= E_{\theta}(\boldsymbol\lambda) \,\mathrm{,}\;\,$$ where the parameter $\theta>0$ defines the rate $\tfrac{\theta}{\theta+n}$ at which novel alleles appear. The polychromatic analogue $E_{\theta,\mbfp}^n$ to the ESF is the distribution of the very same model, when alleles are additionally marked by a 'color' in $[q]$. Such a marking describes any of $q$ (hereditary or non-hereditary) features specific to a given allele and which are not reflected by the sequence of its base pairs. This includes, for instance, *in situ* epigenetic modifications such as DNA-methylation. *Tourists at the Chinese restaurant*. It would not be difficult to introduce polychromatic generalizations to many well-known problems and constructions in the theory, such as the *Spaghetti Loop distribution*, or the *Feller coupling*. For the sake of brevity, we only discuss the *Chinese restaurant process* (CRP). In [@Ald85], D.J. Aldous introduced[^1] the CRP as a sequential description of the sampling of random partitions distributed according to the Poisson--Dirichlet distribution. The process (and many of its variations) has proven a very successful tool in the study of random partitions/permutations. Let us briefly discuss a variation[^2] of the CRP well-suited to describe our colored partitions. As usual, *\[customers\] $1,2,\dotsc, n$ arrive sequentially at an initially empty restaurant with a large number of large \[circular\] tables. \[Customer\] $j$ either sits at the same table as \[customer\] $i$, with probability $1/(j-1+\theta)$ for each $i<j$, or else sits at an empty table, with probability $\theta/(j-1+\theta)$*. [@Ald85 (11.19), p. 91]. Additionally however, each customer randomly chooses to order from one out of the $q$ proposed menus, independently of the other customers and according to a fixed categorical distribution with parameter $\mbfp$. The colored partition 'people at each table ordering from each menu' is distributed according to $E_{\theta,\mbfp}^n$. *Plan of the work*. In §[2.1](#ss:Patterns){reference-type="ref" reference="ss:Patterns"} we introduce some necessary notation and define the pattern inventory $\msZ_\mbfn$ in the right-hand side of [\[eq:t:IntroMain:0\]](#eq:t:IntroMain:0){reference-type="eqref" reference="eq:t:IntroMain:0"}. In §[2.2](#ss:Groups){reference-type="ref" reference="ss:Groups"} we show that $\msZ_\mbfn$ coincides with a 'refined' cycle index polynomial $Z_\mbfn$ of a certain group action, counting $q$-colored partitions coloring $\mbfn$. We then move to prove Theorem [Theorem 1 1](#t:IntroMain){reference-type="ref" reference="t:IntroMain"} (§[3.4](#ss:MomentsProof){reference-type="ref" reference="ss:MomentsProof"}) together with an overview of previously known results (§[3.1](#ss:MomentsOverview){reference-type="ref" reference="ss:MomentsOverview"}), some corollaries (§[3.2](#ss:Corollaries){reference-type="ref" reference="ss:Corollaries"}), and applications to other measures (§[3.3](#ss:DirichletGamma){reference-type="ref" reference="ss:DirichletGamma"}). Finally, we study the polychromatic ESF by means of a polychromatic Hoppe urn model (§[4.1](#ss:Hoppe){reference-type="ref" reference="ss:Hoppe"}) and discuss its consistency in the sense of Kingman (§[4.2](#ss:Consistency){reference-type="ref" reference="ss:Consistency"}). # Counting pattern inventories {#s:Patterns} For $n\in{\mathbb N}_1$ let $[n]\mathop{\mathrm{\coloneqq}}\left\{1,\dotsc,n \right\}$, and $\mfS_n$ be the symmetric group of degree $n$, naturally acting on $[n]$ by permutation of its elements. *Multisets*. Given a set $S$, an *$S$-multiset* is any map $m\colon S\to {\mathbb N}_0$. We denote by $\mathsf{supp}\,{m}$ the *support* of $m$. The *cardinality* ${\mathsf{card}(m)}$ of $m$ is the sum $\sum_{s\in S}m(s)$ of all its values. Given a map $f\colon S\to T$, the *push-forward via $f$* of an $S$-multiset $m$ is the $T$-multiset $$\label{eq:MultiPush} f_*m \mathop{\mathrm{\coloneqq}}\sum_{s\in\mathsf{supp}\,{m}} m(s) \mathop{\mathrm{\mathbf 1}}_{f(s)} \,\mathrm{.}$$ *Vectors*. Whenever no confusion may arise, we do not distinguish between row vectors and column vectors. When relevant, we write $\mbfx^{\scriptscriptstyle{(n)}}$ to indicate that $\mbfx\in{\mathbb R}^n$ or, more generally, that $\mbfx$ has $n$ entries. Let $\mbfe_i^{\scriptscriptstyle{(n)}}$ be the $i^\textrm{th}$ vector of the canonical basis of ${\mathbb R}^n$, and set ${\mathbf 1}^{\scriptscriptstyle{(n)}}\mathop{\mathrm{\coloneqq}}\left(1\right)_{i\in [n]}$ and analogously for $\mathop{\mathrm{{\mathbf 0}}}^{{\scriptscriptstyle{(n)}}}$. For vectors $\mbfx,\mbfy\in {\mathbb R}^n$ and $\pi\in\mfS_n$, write $$\begin{aligned} \mbfx\cdot\mbfy\mathop{\mathrm{\coloneqq}}&\ x_1y_1+\cdots + x_ny_n\,\mathrm{,}\;\,& \mbfx\diamond\mbfy\mathop{\mathrm{\coloneqq}}& \left(x_1y_1,\dotsc, x_ny_n\right)\,\mathrm{,}\;\, \\ \mbfx^{\diamond n}\mathop{\mathrm{\coloneqq}}& \left(x_1^n,\dotsc, x_n^n\right) \,\mathrm{,}\;\, & {{}\mbfx}_\bullet\mathop{\mathrm{\coloneqq}}&\ \mbfx\cdot{\mathbf 1} \,\mathrm{.}\end{aligned}$$ For any $f\colon{\mathbb C}\to{\mathbb C}$ further write $f(\mbfx)\mathop{\mathrm{\coloneqq}}f(x_1)\cdots f(x_n)$. *Matrices*. For a matrix $\mbfM\mathop{\mathrm{\coloneqq}}\big[m_{i,j}\big]_{i\in[a], j\in [b]}\in {\mathbb R}^{a\times b}$ ($a$ rows, $b$ columns) set $$\begin{aligned} \mbfM_i\mathop{\mathrm{\coloneqq}}&\ \mbfe_i^{\scriptscriptstyle{(a)}} \mbfM \,\mathrm{,}\;\,& \mathsf{row}(\mbfM)\mathop{\mathrm{\coloneqq}}&\ \mbfM {\mathbf 1}^{{\scriptscriptstyle{(b)}}}\in {\mathbb R}^a\,\mathrm{,}\;\, \\ \mbfM^j\mathop{\mathrm{\coloneqq}}&\ \mbfM\, \mbfe_j^{{\scriptscriptstyle{(b)}}} \,\mathrm{,}\;\,& \mathsf{col}(\mbfM)\mathop{\mathrm{\coloneqq}}&\ {\mathbf 1}^{\scriptscriptstyle{(a)}}\mbfM\in {\mathbb R}^b \,\mathrm{.}\end{aligned}$$ In words: $\mbfM_i$ is the $i^\textrm{th}$ row of $\mbfM$ and $\mbfM^j$ is the $j^\textrm{th}$ column of $\mbfM$, while $\mathsf{row}(\mbfM)$ is the vector of the rows' lengths of $\mbfM$ and $\mathsf{col}(\mbfM)$ is the vector of the columns' lengths of $\mbfM$. For matrices $\mbfS,\mbfM\in{\mathbb R}^{a\times b}$ set $\mbfS^\mbfM\mathop{\mathrm{\coloneqq}}\prod_{i,j}^{a,b} s_{i,j}^{m_{i,j}}$ and $\mbfM!\mathop{\mathrm{\coloneqq}}\prod_{i,j}^{a,b} m_{i,j}!$. ## Colored partitions, permutations, patterns {#ss:Patterns} Throughout, let $q,r\in{\mathbb N}_1$ and let $\mbfn\mathop{\mathrm{\coloneqq}}\left(n_1,\dotsc,n_q\right)\in{\mathbb N}_0^q$. **Definition 1**. A *$q$-coloring* of $[{{}\mbfn}_\bullet]$ is any function assigning to each element of $[{{}\mbfn}_\bullet]$ a color in $[q]$. An *$\mbfn$-coloring* is a $q$-coloring $\mfc$ of $[{{}\mbfn}_\bullet]$ with $\mfc^{-1}(j)=n_j$ for each $j\in [q]$. If not otherwise stated, everywhere in the following we fix an $\mbfn$-coloring $\mfc_\mbfn$. Many assertions will depend on $\mfc_\mbfn$, but they do so only via the choice of $\mbfn$. ### Colored partitions Set ${\mathbb N}_*^q\mathop{\mathrm{\coloneqq}}{\mathbb N}_0^q\setminus \left\{\mathop{\mathrm{{\mathbf 0}}}^{\scriptscriptstyle{(q)}}\right\}$. For an ${\mathbb N}^q_*$-multiset $A$, we define the quantity $$\mathsf{col}(A)\mathop{\mathrm{\coloneqq}}\sum_{\mbfa\in \mathsf{supp}A} A(\mbfa)\, \mbfa \quad \in \overline{\mathbb N}_0^q \,\mathrm{.}$$ **Definition 2** (Colored partitions). A *$q$-colored partition of $\mbfn$* is any ${\mathbb N}_*^q$-multiset $A$ satisfying $\mathsf{col}(A)= \mbfn$, in which case we write  $A\vDash\mbfn$. We denote by $\mcA_\mbfn$ the family of all $q$-colored partitions of $\mbfn$. The *shape* of a $q$-colored partition $A$ is the integer partition $\mathsf{shape}(A)\vdash {{}\mbfn}_\bullet$ given by $$\mathsf{shape}(A)_i\mathop{\mathrm{\coloneqq}}\sum_{\mbfa\in \mathsf{supp}A\, :\, {{}\mbfa}_\bullet=i} A(\mbfa) \,\mathrm{,}\;\,\qquad i\in [{{}\mbfn}_\bullet]\,\mathrm{.}$$ **Definition 3**. The *multinomial coefficient* of a $q$-colored partition $A$ of $\mbfn$ is $$\begin{aligned} \label{eq:MultinomialMultiYoung} M_{\mbfn}(A)\mathop{\mathrm{\coloneqq}}&\ \mbfn! \prod_{\mbfa\in \mathsf{supp}A} \frac{\binom{{{}\mbfa}_\bullet}{\mbfa}^{A(\mbfa)}}{{{}\mbfa}_\bullet^{A(\mbfa)} A(\mbfa)!}\,\mathrm{.}\end{aligned}$$ *Remark 4* ($q=1$). When $q=1$, every $1$-colored partition is identical to its shape. Letting $n\mathop{\mathrm{\coloneqq}}n_1$, hence $\mbfn=(n)$, the multinomial coefficient of $A\in\mcA_\mbfn$ reduces to the *multinomial coefficient of the second kind* of the integer partition $\boldsymbol\lambda\mathop{\mathrm{\coloneqq}}\mathsf{shape}(A)\vdash n$, viz. $$\begin{aligned} \label{eq:MultiSecond} M_{\mbfn}(A)=M_2(\boldsymbol\lambda)\mathop{\mathrm{\coloneqq}}n! \prod_{i\in \mathsf{supp}A}\frac{1}{i^{\lambda_i} \lambda_i!}\,\mathrm{,}\;\,\qquad \lambda_i\mathop{\mathrm{\coloneqq}}A(i) \,\mathrm{.}\end{aligned}$$ ### Pattern inventory Let $G< \mfS_n$ be a permutation group of degree $n$. The *cycle index polynomial* $Z^G$ of $G$ is $$Z^G(\mbft)\mathop{\mathrm{\coloneqq}}\frac{1}{\left\lvert G\right\rvert}\sum_{\pi\in G}\mbft^{\boldsymbol\lambda(\pi)} \,\mathrm{,}\;\,\qquad \mbft=\left(t_1,\dotsc, t_n\right) \,\mathrm{,}\;\,$$ where $\boldsymbol\lambda(\pi)\vdash n$ accounts for the number of cycles in $\pi$ of given length, i.e. $\lambda_1(\pi)$ is the number of fixed points of $\pi$, $\lambda_2(\pi)$ the number of $2$-cycles in $\pi$, and so on. We denote by $Z_n\mathop{\mathrm{\coloneqq}}Z^{\mfS_n}$ the cycle index polynomial of $\mfS_n$. It is not difficult to show that (cf. [\[eq:MultiSecond\]](#eq:MultiSecond){reference-type="eqref" reference="eq:MultiSecond"}) $$\label{eq:CycleIndex} Z_n(\mbft)=\frac{1}{n!} \sum_{\boldsymbol\lambda\vdash n} M_2(\boldsymbol\lambda)\, \mbft^{\boldsymbol\lambda}\,\mathrm{,}\;\,\qquad \mbft=\left(t_1,\dotsc, t_n\right) \,\mathrm{.}$$ *Pattern inventory*. We represent a permutation $\pi$ in its cycle notation, viz. $$\label{eq:StdRepPi} \pi=(y_{1,1}y_{1,2}\cdots)(y_{2,1}y_{2,2}\cdots)\cdots (y_{r,1}y_{r,2}\cdots) \,\mathrm{.}$$ Let $\mbfS\mathop{\mathrm{\coloneqq}}\left(\mbfs_1,\dotsc,\mbfs_q\right)$ be a $k\times q$-matrix of dummy variables. We denote by $\mbfS^1=\mbfs_1,\dotsc,\mbfS^q=\mbfs_q$ the columns of $\mbfS$ and by $\mbfS_1,\dotsc,\mbfS_k$ the rows of $\mbfS$. Further let $\boldsymbol\alpha\in{\mathbb R}^k$. The following definition is inspired by Pólya Enumeration Theory. **Definition 5** (Pattern inventory). The $\mbfn$-*pattern* of a permutation $\pi$ is $$w_\mbfn[\mbfS;\boldsymbol\alpha](\pi)\mathop{\mathrm{\coloneqq}}\prod_i^r \big({\mbfs_{\mfc_\mbfn(y_{i,1})}\diamond\mbfs_{\mfc_\mbfn(y_{i,2})} \diamond\cdots}\big)\cdot \boldsymbol\alpha\,\mathrm{.}$$ The *pattern inventory of $\mbfn$* is the polynomial $$\label{eq:PatternInventory} \msZ_\mbfn[\mbfS;\boldsymbol\alpha]\mathop{\mathrm{\coloneqq}}\frac{1}{\mbfn!}\sum_{\pi\in\mfS_{{{}\mbfn}_\bullet}} w_\mbfn[\mbfS;\boldsymbol\alpha](\pi) \,\mathrm{.}$$ Up to a different normalization, $\msZ_\mbfn$ is a refinement of the cycle index polynomial of $\mfS_{{{}\mbfn}_\bullet}$, in the sense that each monomial in $\msZ_\mbfn$ depends not only on the cycle structure of a permutation, but also on its coloring. In order to simplify the expression of $\msZ_\mbfn$, let $$\begin{aligned} \label{eq:ZDefinition} Z_\mbfn(\mbft)\mathop{\mathrm{\coloneqq}}&\ \frac{1}{\mbfn!} \sum_{A\vDash \mbfn} M_{\mbfn}(A)\, \prod_{\mbfa\in \mathsf{supp}A} t_\mbfa^{A(\mbfa)}\,\mathrm{,}\;\,\qquad \mbft\mathop{\mathrm{\coloneqq}}\left(t_\mbfa\right)_{\mbfa\leq_\diamond\mbfn} \,\mathrm{.}\end{aligned}$$ Finally, for every $\mbfa\leq_\diamond\mbfn$ set $$\begin{aligned} \label{eq:YDefinition} \omega_\mbfa[\mbfS;\boldsymbol\alpha]\mathop{\mathrm{\coloneqq}}\big({\mbfs_1^{\diamond a_1}\diamond\cdots \diamond\mbfs_q^{\diamond a_q}}\big)\cdot\boldsymbol\alpha\,\mathrm{,}\;\,\quad \text{and} \quad \Omega_\mbfn[\mbfS;\boldsymbol\alpha]\mathop{\mathrm{\coloneqq}}&\ \left(\omega_\mbfa[\mbfS;\boldsymbol\alpha]\right)_{\mbfa\leq_\diamond\mbfn} \,\mathrm{.}\end{aligned}$$ In Theorem [Theorem 14](#t:Zn){reference-type="ref" reference="t:Zn"} below, we will prove that $$\begin{aligned} \label{eq:t:Zn:0} \msZ_\mbfn[\mbfS;\boldsymbol\alpha]=Z_\mbfn(\Omega_\mbfn[\mbfS;\boldsymbol\alpha]) \,\mathrm{.}\end{aligned}$$ *Remark 6* ($q=1$). When $q=1$, the polynomial $Z_\mbfn$ in [\[eq:ZDefinition\]](#eq:ZDefinition){reference-type="eqref" reference="eq:ZDefinition"} reduces to $Z_n$ in [\[eq:CycleIndex\]](#eq:CycleIndex){reference-type="eqref" reference="eq:CycleIndex"}. ## Group actions {#ss:Groups} In order to prove [\[eq:t:Zn:0\]](#eq:t:Zn:0){reference-type="eqref" reference="eq:t:Zn:0"}, we identify the algebraic meaning of $\msZ_\mbfn$ in terms of the action of a certain group of permutations. ### Some bijections of the symmetric group Let $G$ be any finite group. For $h\in G$ we denote by $\tau_h\colon G\to G$ the conjugation map $\tau_h\colon g\mapsto hgh^{-1}$. For each $\pi$ in $\mfS_n$ and $i,j\in[n]$ we write $$i \mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}} j \quad \text{if} \quad j=\pi^p(i) \quad \text{for some } p\in\mathop{\mathrm{{\mathbb Z}}}\,\mathrm{,}\;\,$$ i.e., if $i,j\in [n]$ belong to the same orbit (cycle) of $\pi$. We note that $\mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}}$ is an equivalence relation on $[n]$, and that $$\label{eq:ConjugateRel} i \mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}} j \iff \sigma(i) \mathrel{\underset{\tau_\sigma(\pi)}{\scalebox{2}[1]{$\sim$}}} \sigma(j) \,\mathrm{,}\;\,\qquad i,j\in [n]\,\mathrm{,}\;\,\quad \pi,\sigma\in\mfS_n\,\mathrm{.}$$ Let $(B_n,\circ)$ be the group of bijections of $\mfS_n$ leaving conjugacy classes invariant. That is, $g\in B_n$ if and only if $g(\pi)$ has the same cycle structure as $\pi$ for every $\pi\in\mfS_n$. We have $B_n\cong \bigtimes_{\boldsymbol\lambda\vdash n} \mfS_{M_2(\boldsymbol\lambda)}$ in a natural way. We denote by $H_n$ the subset of all $h\in B_n$ such that $$\begin{aligned} \label{eq:Group} i \mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}} j \implies i \mathrel{\underset{h(\pi)}{\scalebox{2}[1]{$\sim$}}} j \,\mathrm{,}\;\,\qquad i,j\in [n]\,\mathrm{,}\;\,\quad \pi\in\mfS_n \,\mathrm{.}\end{aligned}$$ Let $h_1,h_2\in H_n$. Consecutive applications of [\[eq:Group\]](#eq:Group){reference-type="eqref" reference="eq:Group"} show that $$i \mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}} j \implies i \mathrel{\underset{h_1(\pi)}{\scalebox{1.5}[1]{$\sim$}}} j \implies i \mathrel{\underset{(h_2\circ h_1)(\pi)}{\scalebox{2.5}[1]{$\sim$}}} j\,\mathrm{,}\;\,\qquad i,j\in [n]\,\mathrm{,}\;\,\pi\in\mfS_n \,\mathrm{.}$$ Thus, $H_n$ is closed under $\circ$ and therefore it is a subgroup of (the *finite* group) $B_n$. In order to exemplify some elements of $H_n$, for $\pi,\sigma\in\mfS_n$, set $$\begin{aligned} \label{eq:DefHsigma0} f_{\sigma,\pi}(i)\mathop{\mathrm{\coloneqq}}\sigma^{p_{\sigma,\pi}(i)}(i)\,\mathrm{,}\;\,\quad p_{\sigma,\pi}(i)\mathop{\mathrm{\coloneqq}}\min\left\{p\geq 1: \sigma^p(i) \mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}} i\right\}\,\mathrm{,}\;\,\quad i\in [n]\,\mathrm{.}%\end{aligned}$$ For each $\pi\in\mfS_n$ it is readily verified that $f_{\sigma,\pi}\in\mfS_n$ is a bijection and $f_{\sigma,\pi}^{-1}=f_{\sigma^{-1},\pi}$. *Example 7*. For $\sigma\in\mfS_n$ define $h_\sigma$ by $h_\sigma\colon \pi \mapsto \tau_{f_{\sigma,\pi}}(\pi)$. Then, $h_\sigma\in H_n$ and $h_\sigma^{-1}=h_{\sigma^{-1}}$. *Proof.* We verify that $h_\sigma\in H_n$. Firstly, since $f_{\sigma,\pi}^{-1}=f_{\sigma^{-1},\pi}$, we have that $h_\sigma$ is a bijection with inverse $h_{\sigma^{-1}}$. By definition, $h_\sigma(\pi)$ is conjugate to $\pi$ for every $\pi\in\mfS_n$. Thus, $h_\sigma\in B_n$ and it suffices to show [\[eq:Group\]](#eq:Group){reference-type="eqref" reference="eq:Group"}. Furthermore, $f_{\sigma,\pi}(i) \mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}} i$ for every $i$ by the definition [\[eq:DefHsigma0\]](#eq:DefHsigma0){reference-type="eqref" reference="eq:DefHsigma0"} of $f_{\sigma,\pi}$. Thus, $\pi$ and $h_\sigma(\pi)$ define the same equivalence relation, viz. $$\label{eq:p:Induced:1} i \mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}} j \iff i \mathrel{\underset{h_\sigma(\pi)}{\scalebox{2.5}[1]{$\sim$}}} j \,\mathrm{,}\;\,\quad i,j\in [n] \,\mathrm{,}\;\,\qquad \pi\in\mfS_n\,\mathrm{,}\;\,$$ which implies [\[eq:Group\]](#eq:Group){reference-type="eqref" reference="eq:Group"}. ◻ *Remark 8*. For $n\geq 4$, the map $\sigma\mapsto h_\sigma$ is *not* a group homomorphism. For instance, choose $\sigma_1\mathop{\mathrm{\coloneqq}}(24)$, $\sigma_2\mathop{\mathrm{\coloneqq}}(34)$, and $\pi=(123)$, and notice that $% (h_{\sigma_1}\circ h_{\sigma_2})(\pi)=\pi\neq (132)= h_{\sigma_1\sigma_2}(\pi)%$. *Remark 9*. [\[i:r:FixTransposition:1\]]{#i:r:FixTransposition:1 label="i:r:FixTransposition:1"} For every $n$, every $h\in H_n$ fixes transpositions, hence $H_1,H_2$ are the trivial group, and $H_3=\mfS_2$ is the group with non-trivial element exchanging the $3$-cycles in $\mfS_3$. For $n\geq 3$, the group $H_n$ is not a subgroup of the automorphism group of $\mfS_n$. Indeed, elements of $h$ are in general *not* group homomorphisms. For example, let $h\in H_n$ be the map exchanging $(123)$ and $(132)$ and fixing all other permutations. Assuming $h$ is a homomorphism we would have $$(12)=(123)(23)=h\big({(132)}\big) h\big({(23)}\big)= h\big({(132)(23)}\big)=h\big({(13)}\big) \,\mathrm{,}\;\,$$ which contradicts [\[i:r:FixTransposition:1\]](#i:r:FixTransposition:1){reference-type="ref" reference="i:r:FixTransposition:1"}. $H_n$ is not normal in $B_n$ for $n\geq 4$. Indeed, let $f\in B_n$ be the map exchanging $(123)$ with $(124)$ and fixing all other permutations, and $h\mathop{\mathrm{\coloneqq}}h_{(123)}$. Then, $$\big({\tau_f(h)}\big)\big({(123)}\big)=(214)\,\mathrm{,}\;\,$$ which violates [\[eq:Group\]](#eq:Group){reference-type="eqref" reference="eq:Group"} with $\tau_f(h)$ in place of $h$. Whereas $H_n$ is not a normal subgroup of $B_n$, we still have the following. **Lemma 10**. *Let $\sigma\in\mfS_n$. Then, $$\label{eq:Automorph} \varphi_\sigma\mathop{\mathrm{\coloneqq}}\tau_{\tau_\sigma}\colon h\longmapsto \tau_\sigma \circ h \circ \tau_\sigma^{-1}$$ is an automorphism of $H_n$, and $\varphi_{\,\cdot\,}\colon \mfS_n \to \Aut(H_n)$ is a group homomorphism.* **Proof.* Since $\tau_\sigma$ leaves conjugacy classes in $\mfS_n$ invariant, we have $\tau_\sigma\in B_n$. Thus $\varphi_\sigma$ is an inner automorphism of $B_n$. Furthermore, since for every group $G$ the map $\tau^G\colon g\mapsto \tau_g$ is a group homomorphism $G\to \Aut(G)$, the map $\varphi_{\,\cdot\,}=\tau^{B_n}\circ\tau^{\mfS_n}$ is a group homomorphism as well. Thus, it suffices to show that $\varphi_\sigma(H_n)\subset H_n$ for every $\sigma\in\mfS_n$. To this end, it suffices to verify [\[eq:Group\]](#eq:Group){reference-type="eqref" reference="eq:Group"} with $\varphi_\sigma(h)$ in place of $h$. Indeed, respectively by [\[eq:ConjugateRel\]](#eq:ConjugateRel){reference-type="eqref" reference="eq:ConjugateRel"} with $\sigma^{-1}$ in place of $\sigma$, by [\[eq:Group\]](#eq:Group){reference-type="eqref" reference="eq:Group"}, and by [\[eq:ConjugateRel\]](#eq:ConjugateRel){reference-type="eqref" reference="eq:ConjugateRel"}, $$\begin{aligned} i \mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}} j \implies&\ \sigma^{-1}(i) \mathrel{\underset{\tau_{\sigma^{-1}}(\pi)}{\scalebox{2}[1]{$\sim$}}} \sigma^{-1}(j) \implies \sigma^{-1}(i) \mathrel{\underset{(h\circ \tau_{\sigma^{-1}})(\pi)}{\scalebox{2}[1]{$\sim$}}} \sigma^{-1}(j) \\ \implies&\ \sigma\sigma^{-1}(i) \mathrel{\underset{(\tau_\sigma\circ h \circ \tau_{\sigma^{-1}})(\pi)}{\scalebox{2.5}[1]{$\sim$}}} \sigma\sigma^{-1}(j)\end{aligned}$$ and the conclusion follows since $\tau_\sigma^{-1}=\tau_{\sigma^{-1}}$. ◻* ### Semi-direct product and group action Fix an $\mbfn$-coloring $\mfc_\mbfn$. All results in the following hold for every such coloring. Proposition [Proposition 13](#p:Quotient){reference-type="ref" reference="p:Quotient"} below will provide an algebraic interpretation of the multinomial coefficient $M_{\mbfn}$ in [\[eq:MultinomialMultiYoung\]](#eq:MultinomialMultiYoung){reference-type="eqref" reference="eq:MultinomialMultiYoung"} by means of the surjective map $\Pi=\Pi_{\mfc_\mbfn}\colon \mfS_{{{}\mbfn}_\bullet} \to \mcA_\mbfn$ which we now define. Firstly, to every *cycle* $\kappa= (y_1y_2 \cdots)$ we associate a vector $\boldsymbol\varepsilon(\kappa)$ in ${\mathbb N}_0^q$ by $$\label{eq:Epsilon} \boldsymbol\varepsilon(\kappa)_j \mathop{\mathrm{\coloneqq}}\left\lvert\left\{h: \mfc_\mbfn(y_h)=j\right\}\right\rvert\,\mathrm{,}\;\,\qquad j\in [q] \,\mathrm{.}$$ For $\pi = \kappa_1 \cdots \kappa_r \in \mfS_{{{}\mbfn}_\bullet}$, with cycles $\kappa_1,\dotsc, \kappa_r$ (including fixed points), we then set $$\label{eq:Pi} \Pi\colon \pi \longmapsto \sum_{i=1}^r\mathop{\mathrm{\mathbf 1}}_{\boldsymbol\varepsilon(\kappa_i)} \,\mathrm{.}$$ *Semi-direct product*. In the following, we regard $$\mfS_\mbfn\mathop{\mathrm{\coloneqq}}\mfS_{\mfc_\mbfn^{-1}(1)}\times \cdots \times \mfS_{\mfc_\mbfn^{-1}(q)}\cong \mfS_{n_1}\times \cdots \times \mfS_{n_q}$$ as a subgroup of $\mfS_{{{}\mbfn}_\bullet}$. **Definition 11**. Let $(G_\mbfn,\star)\mathop{\mathrm{\coloneqq}}H_{{{}\mbfn}_\bullet} \rtimes \mfS_\mbfn$ be the semi-direct product induced by the group homomorphism $\varphi_{\,\cdot\,}$ defined by [\[eq:Automorph\]](#eq:Automorph){reference-type="eqref" reference="eq:Automorph"}, that is $$(h_1,\sigma_1) \star (h_2,\sigma_2)\mathop{\mathrm{\coloneqq}}\ (h_1\circ \varphi_{\sigma_1}(h_2),\sigma_1\sigma_2) \,\mathrm{.}$$ **Lemma 12**. *The function $\,\raisebox{\depth}{\scalebox{1}[-1]{$\circlearrowleft$}}\,\colon G_\mbfn\times \mfS_{{{}\mbfn}_\bullet} \to \mfS_{{{}\mbfn}_\bullet}$ given by $$\begin{aligned} \label{eq:l:Action:0} \,\raisebox{\depth}{\scalebox{1}[-1]{$\circlearrowleft$}}\,\colon \big({(h,\sigma), \pi}\big)\longmapsto (h,\sigma).\pi\mathop{\mathrm{\coloneqq}}\, (h\circ \tau_\sigma)(\pi)\end{aligned}$$ defines a group action of $G_\mbfn$ on $\mfS_{{{}\mbfn}_\bullet}$, faithful if ${{}\mbfn}_\bullet\geq 3$.* **Proof.* In order to show that $\,\raisebox{\depth}{\scalebox{1}[-1]{$\circlearrowleft$}}\,$ is a group action it suffices to verify that $$\begin{aligned} \big({(h_1,\sigma_1)\star (h_2,\sigma_2)}\big).\pi=&\ \big({h_1\circ \varphi_{\sigma_1}(h_2)}\big)(\sigma_1\sigma_2\pi\sigma_2^{-1}\sigma_1^{-1}) \\ =&\ h_1\big({\sigma_1 h_2(\sigma_2\pi\sigma_2^{-1})\sigma_1^{-1}}\big) \\ =&\ (h_1,\sigma_1).(h_2,\sigma_2).\pi\,\mathrm{.}\end{aligned}$$* *In order to show faithfulness, it suffices to prove that $(h,\sigma)=(\id,e)$ whenever $$\label{eq:l:Action:1} (h,\sigma).\pi=\pi\,\mathrm{,}\;\,\qquad \pi\in\mfS_{{{}\mbfn}_\bullet}\,\mathrm{.}$$* *If $\sigma=e$, since $B_{{{}\mbfn}_\bullet}$ (hence $H_{{{}\mbfn}_\bullet}$) acts faithfully on $\mfS_{{{}\mbfn}_\bullet}$, [\[eq:l:Action:1\]](#eq:l:Action:1){reference-type="eqref" reference="eq:l:Action:1"} implies $h=\id$. If $\sigma\neq e$, since ${{}\mbfn}_\bullet\geq 3$, there exist mutually different $i,j,k\in[n]$ with $\sigma(i)=j$. Choosing $\pi\mathop{\mathrm{\coloneqq}}(ik)$, $$(h,\sigma).\pi=h\big({(\sigma(i),\sigma(k))}\big)=h\big({(j, \sigma(k))}\big)=(j, \sigma(k))\neq \pi\,\mathrm{,}\;\,$$ where the last equality follows again from Remark [Remark 9](#r:FixTransposition){reference-type="ref" reference="r:FixTransposition"}. ◻* **Proposition 13**. *The orbit space $\mfS_{{{}\mbfn}_\bullet}/G$ is (parametrized by) the set $\mcA_\mbfn$ of all $q$-colored partitions, and $\left\lvert G.\pi\right\rvert=M_{\mbfn}(\Pi(\pi))$ for every $\pi\in\mfS_{{{}\mbfn}_\bullet}$.* **Proof.* For every $\pi, \pi' \in \mfS_{{{}\mbfn}_\bullet}$, let us prove that $\Pi(\pi)=\Pi(\pi')$ if and only if $\pi \in G.\pi'$. Let $\pi = \kappa_1 \cdots \kappa_r$ and $\pi' = \kappa_1' \cdots \kappa_{r'}'$ be cycle decompositions. If $\Pi(\pi) = \Pi(\pi')$, then $r = r'$ and, up to reordering the cycles, we may assume without loss of generality that $\Pi(\kappa_i) = \Pi(\kappa_i')$ for every $i$. Therefore, there exists $\sigma \in \mfS_\mbfn$ such that for every $i$ the cycles $\kappa_i$ and $\sigma \kappa_i' \sigma^{-1}$ transitively permute the same set of numbers. Equivalently, $$i \mathrel{\underset{\pi}{\scalebox{1.5}[1]{$\sim$}}} j \iff i \mathrel{\underset{\tau_\sigma(\pi')}{\scalebox{2}[1]{$\sim$}}} j \,\mathrm{,}\;\,\qquad i,j\in [{{}\mbfn}_\bullet] \,\mathrm{.}$$ Hence, the map $h \in B_n$ that swaps $\pi$ and $\tau_\sigma(\pi')$, and fixes every other element of $\mfS_{{{}\mbfn}_\bullet}$ is in $H_{{{}\mbfn}_\bullet}$. We can thus write $\pi = (h,\sigma).\pi'$. Conversely, if $\pi = (h,\sigma).\pi'$ holds for some $h$ and $\sigma$, then we can rearrange the cycle decompositions $\pi = \kappa_1 \cdots \kappa_r$ and $\tau_\sigma(\pi') = \tau_\sigma(\kappa'_1) \cdots \tau_\sigma(\kappa'_r)$ in such a way that $\kappa_i$ and $\sigma \kappa_i' \sigma^{-1}$ transitively permute the same set of numbers for every $i$. Therefore, $\Pi(\kappa_i) = \Pi(\sigma \kappa_i' \sigma^{-1})$. Furthermore, since $\sigma \in \mfS_\mbfn$, we have $\Pi(\sigma \kappa_i' \sigma^{-1})=\Pi(\kappa_i')$, whence $\Pi(\kappa_i) =\Pi(\kappa_i')$ as desired.* **Cardinality of the orbits*. Let $A\in\mcA_\mbfn$. We aim to show that $\left\lvert\Pi^{-1}(A)\right\rvert = M_{\mbfn}(A)$. In order to do so, it is convenient to introduce some new sets and maps, as schematized in Figure [\[fig:diagram\]](#fig:diagram){reference-type="ref" reference="fig:diagram"} below, and compute the cardinality of their fibers.* 1. *Firstly, given a vector $\mbfc\mathop{\mathrm{\coloneqq}}\left(c_1,c_2,\dotsc\right)$ with entries in $[q]$ and arbitrary (possibly zero) length, we consider the ${\mathbb N}_0^q$-valued map $\boldsymbol\varepsilon$ defined by $$\label{eq:BoldEps} \boldsymbol\varepsilon(\mbfc)_j \mathop{\mathrm{\coloneqq}}\left\lvert\left\{h: c_h=j\right\}\right\rvert\,\mathrm{,}\;\,\qquad j\in [q] \,\mathrm{.}$$* 2. *We denote by $\#\mbfM$ the number of rows of a matrix $\mbfM$. The map $\#$ is naturally extended to matrix-valued functions by post-composition.* 3. *Let $\mcY$ be the space of all matrix-valued functions $Y$ on ${\mathbb N}^q_*$ satisfying, for all $\mbfa\in {\mathbb N}^q_*$, $$Y(\mbfa)_i\in \boldsymbol\varepsilon^{-1}(\mbfa)\,\mathrm{,}\;\,\quad i\in [\# Y(\mbfa)]\,\mathrm{,}\;\,\qquad \text{and} \qquad \#\circ Y \in \mcA_\mbfn \,\mathrm{.}$$ We explicitly allow for $Y(\mbfa)$ to possibly be the empty matrix for some $\mbfa\in{\mathbb N}^q_*$.* 4. *Denote by $\mfc_\mbfn^\diamond$ the entry-by-entry extension of $\mfc_\mbfn$ to vectors and matrices. We define $\mcX$ as the set of all matrix-valued functions $X$ on ${\mathbb N}_*^q$, $$X(\mbfa) = % \begin{bmatrix}y_{\mbfa,1,1}& y_{\mbfa,1,2} &\dotsc\\ y_{\mbfa,2,1}& y_{\mbfa,2,2} & \dotsc\\ \vdots & \vdots & \ddots \end{bmatrix} \,\mathrm{,}\;\,$$ satisfying, for all $\mbfa\in{\mathbb N}^q_*$, $$\begin{gathered} X(\mbfa)_i \in (\boldsymbol\varepsilon\circ \mfc_\mbfn^\diamond)^{-1} (\mbfa)\,\mathrm{,}\;\,\quad i\in [\#X(\mbfa)] \,\mathrm{,}\;\, \\ \left\{y_{\mbfa,i,j}\right\}_{\mbfa,i,j} = [{{}\mbfn}_\bullet]\,\mathrm{,}\;\,\qquad \text{and} \qquad y_{\mbfa,i,j} \neq y_{\mbfa',i',j'}\,\mathrm{,}\;\,\quad \left(\mbfa,i,j\right) \neq \left(\mbfa',i',j'\right) \,\mathrm{.}\end{gathered}$$* 5. *Denote by $\mcZ$ the family of set-valued functions of the form $$\label{eq:p:Quotient:1bis} Z \colon \mbfa \longmapsto \big\{\left(y_{\mbfa,1,1}, y_{\mbfa,1,2},\dotsc\right),\left(y_{\mbfa,2,1},y_{\mbfa,2,2},\dotsc\right),\dotsc \big\}%$$ additionally so that $$\left(\mbfa\longmapsto \begin{bmatrix}y_{\mbfa,1,1}& y_{\mbfa,1,2} &\dotsc\\ y_{\mbfa,2,1}& y_{\mbfa,2,2} & \dotsc\\ \vdots & \vdots & \ddots \end{bmatrix}\right)\in\mcX \,\mathrm{.}$$* 6. *Finally let $f_1\colon\mcX\to\mcZ$ and $f_2\colon\mcZ\to \mfS_{{{}\mbfn}_\bullet}$ be maps *forgetting* part of the structure: $$f_1(X)(\mbfa) \mathop{\mathrm{\coloneqq}}\left\{X(\mbfa)_i\right\}_{i\in [\#X(\mbfa)]} \,\mathrm{,}\;\,\qquad \mbfa \in {\mathbb N}_*^q\,\mathrm{,}\;\,$$ and, using the notation of [\[eq:p:Quotient:1bis\]](#eq:p:Quotient:1bis){reference-type="eqref" reference="eq:p:Quotient:1bis"}, $$f_2 \colon Z \longmapsto \pi \mathop{\mathrm{\coloneqq}}\prod_{\substack{\mbfa \in {\mathbb N}_*^q \\ Z(\mbfa)\neq \mathop{\mathrm{\varnothing}}}} \left(y_{\mbfa,1,1}\ y_{\mbfa,1,2}\ \cdots\right)\left(y_{\mbfa,2,1}\ y_{\mbfa,2,2}\ \cdots\right)\cdots \in\mfS_{{{}\mbfn}_\bullet}\,\mathrm{.}$$* *It is a tedious verification that the diagram in Figure [\[fig:diagram\]](#fig:diagram){reference-type="ref" reference="fig:diagram"} commutes.* *Now, let $\pi = (y_{1,1} y_{1,2} \cdots)\cdots (y_{r,1} y_{r,2} \cdots) \in \Pi^{-1}(A)$ and define $\mbfa_i\leq_\diamond\mbfn$ by $$\Pi\big({(y_{i,1} y_{i,2} \cdots)}\big) = \mathop{\mathrm{\mathbf 1}}_{\mbfa_i} \,\mathrm{,}\;\,\qquad i \in [r] \,\mathrm{.}$$ The fiber $f_2^{-1}(\pi)$ consists of all the (distinct) set-valued functions $$Z_{k_1,\dotsc,k_r} \colon \mbfa \longmapsto \big\{{\big(\pi^{k_i}(y_{i,1}), \pi^{k_i}(y_{i,2}),\dotsc\big)}\big\}_{i \colon \mbfa = \mbfa_i} \,\mathrm{,}\;\,\qquad k_1 \in [{{}\mbfa_1}_\bullet],\dotsc,k_r \in [{{}\mbfa_r}_\bullet] \,\mathrm{,}\;\,$$ and has therefore cardinality $\left\lvert f_2^{-1}(\pi)\right\rvert = {{}\mbfa_1}_\bullet \cdots {{}\mbfa_r}_\bullet= \prod_{\mbfa\in \mathsf{supp}A} {{}\mbfa}_\bullet^{A(\mbfa)}$. As for the fibers of $f_1$, given $Z \in( \Pi\circ f_2)^{-1}(A)$ and $X\in f_1^{-1}(Z)$, every element of $f_1^{-1}(Z)$ is induced by a permutation-valued function $\varsigma$ on ${\mathbb N}_*^q$ such that $$\varsigma \colon \mbfa \longmapsto \varsigma_\mbfa \in \mfS_{A(\mbfa)} \,\mathrm{,}\;\,\qquad \mbfa \in {\mathbb N}_*^q\,\mathrm{,}\;\,$$ via the formula $$X_\varsigma \colon \mbfa \longmapsto P_{\varsigma_\mbfa} X(\mbfa) \,\mathrm{.}$$ where $P_{\varsigma_\mbfa}$ is the permutation matrix induced by $\varsigma_\mbfa$. It follows that $\left\lvert f_1^{-1}(Z)\right\rvert=\prod_{\mbfa \in \mathsf{supp}A} A(\mbfa)!$. It is easy to see that the fibers of $\mfc_\mbfn^\diamond\colon \mcX \to \mcY$ all have cardinality $\mbfn!$. Lastly, the computation of the cardinality of the fibers of $\# \colon \mcY \to \mcA_\mbfn$ can be performed $\mbfa$ by $\mbfa$ and, thanks to the properties of the multinomial coefficient, $$\left\lvert\#^{-1}(A)\right\rvert=\prod_{\mbfa\in\mathsf{supp}A} \left\lvert\boldsymbol\varepsilon^{-1}(\mbfa)\right\rvert^{A(\mbfa)} = \prod_{\mbfa\in\mathsf{supp}A} \binom{{{}\mbfa}_\bullet}{\mbfa}^{A(\mbfa)} \,\mathrm{.}$$ In conclusion, $$\begin{aligned} \mbfn! \prod_{\mbfa\in\mathsf{supp}A} \binom{{{}\mbfa}_\bullet}{\mbfa}^{A(\mbfa)} =&\ \left\lvert(\# \circ \mfc_\mbfn^\diamond)^{-1}(A)\right\rvert = \sum_{\pi \in \Pi^{-1}(A)} \left\lvert(f_2\circ f_1)^{-1}(\pi)\right\rvert \\ =&\ \left\lvert\Pi^{-1}(A)\right\rvert\prod_{\mbfa\in \mathsf{supp}A} {{}\mbfa}_\bullet^{A(\mbfa)} A(\mbfa)! \,\mathrm{,}\;\,\end{aligned}$$ which yields the desired identity. ◻* We conclude this section with the proof of [\[eq:t:Zn:0\]](#eq:t:Zn:0){reference-type="eqref" reference="eq:t:Zn:0"}. **Theorem 14**. *The polynomial $Z_\mbfn$ in [\[eq:ZDefinition\]](#eq:ZDefinition){reference-type="eqref" reference="eq:ZDefinition"} is the orbit generating function of the action [\[eq:l:Action:0\]](#eq:l:Action:0){reference-type="eqref" reference="eq:l:Action:0"}. Furthermore, [\[eq:t:Zn:0\]](#eq:t:Zn:0){reference-type="eqref" reference="eq:t:Zn:0"} holds.* **Proof.* It suffices to collect all terms in $\msZ_\mbfn$ with the same monomials. By Proposition [Proposition 13](#p:Quotient){reference-type="ref" reference="p:Quotient"}, for each $\pi\in\mfS_{{{}\mbfn}_\bullet}$ there are exactly $\left\lvert G.\pi\right\rvert=M_{\mbfn}(\Pi(\pi))$ monomials indexed by $A=\Pi(\pi)$, and the conclusion follows using that $w_\mbfn[\mbfS;\boldsymbol\alpha](\pi)= \prod_{\mbfa\in\mathsf{supp}A}\omega_\mbfa[\mbfS;\boldsymbol\alpha]^{A(\mbfa)}$. ◻* ### Necklaces Theorem [Theorem 14](#t:Zn){reference-type="ref" reference="t:Zn"} provides an algebraic interpretation for [\[eq:t:Zn:0\]](#eq:t:Zn:0){reference-type="eqref" reference="eq:t:Zn:0"}. Let us now give a combinatorial interpretation of the same formula, i.e. of the multinomial coefficient $M_{\mbfn}$, in terms of necklaces, which will in turn provide a connection to ESF via the extension of the CRP discussed in §[1](#s:Intro){reference-type="ref" reference="s:Intro"}. On the one hand, waiters in our busy restaurant take care to remember, for every table, which clients order from each menu. The arrangement of the customers around the table is important in serving them efficiently. All the information the waiters need about the customers' arrangement is thus contained in a $q$-colored necklace. On the other hand, chefs in the restaurant only care about how many customers at each table order from each menu, so that customers at the same table may be served at the same time. All the information the chefs need about the customers' arrangement is thus contained in a $q$-colored partition. Let us now count $q$-colored partitions by collecting together $q$-colored necklaces with the same occurrences of each color. For integer $q\in{\mathbb N}_1$ denote by $[q]^*$ the free monoid generated by $[q]$. Elements of $[q]^*$ are called ($q$-)*words*. Two words $u,v$ are *conjugate* if there exist words $s,t$ so that $u=st$ and $v=ts$. Two conjugate words are cyclic shifts of one another. Thus, conjugacy is an equivalence relation on words. Its equivalence classes are called ($q$-)*necklaces*. Let $\nu=\llbracket w\rrbracket_{}$ be a necklace and $w=c_1c_2\cdots c_\ell$ be any of its representatives. The *length* $\ell_\nu$ of $\nu$ is the total number $\ell$ of characters in $w$. The *period* $p_\nu$ of $\nu$ is the minimal integer $p\geq 1$ with $c_i=c_{i+p-1 \pmod \ell +1}$ for every $i\in [\ell]$. Clearly, $p_\nu$ divides $\ell_\nu$. 1. Let $w=c_1c_2\cdots\in [q]^*$. Consistently with [\[eq:BoldEps\]](#eq:BoldEps){reference-type="eqref" reference="eq:BoldEps"}, we denote by $\boldsymbol\varepsilon(w)\in {\mathbb N}_0^q$ the vector of occurrences of its characters, viz. $$\boldsymbol\varepsilon(w)_j\mathop{\mathrm{\coloneqq}}\left\lvert\left\{h: c_h=j\right\}\right\rvert \,\mathrm{.}$$ It is readily seen that $\boldsymbol\varepsilon$ descends to a (non-relabeled) map on necklaces. 2. Let $\mcN_\mbfn$ be the family of all multisets $N$ of $q$-necklaces satisfying $\boldsymbol\varepsilon_*N \in \mcA_\mbfn$, cf. [\[eq:MultiPush\]](#eq:MultiPush){reference-type="eqref" reference="eq:MultiPush"}. 3. Define a map $\mfc_\mbfn^\diamond$ on $\mfS_{{{}\mbfn}_\bullet}$ in the following way. For a cyclic permutation $\kappa=\left(y_1y_2\cdots\right)$ let $\nu$ be the necklace $\llbracket\mfc_\mbfn(y_1)\ \mfc_\mbfn(y_2)\ \cdots\rrbracket_{}$ and set $\mfc_\mbfn^\diamond(\kappa)\mathop{\mathrm{\coloneqq}}\mathop{\mathrm{\mathbf 1}}_\nu$. Extend $\mfc_\mbfn^\diamond$ by $$\mfc_\mbfn^\diamond\colon \pi\mathop{\mathrm{\coloneqq}}\kappa_1\cdots\kappa_r \longmapsto \sum_{i=1}^r \mfc_\mbfn^\diamond(\kappa_i) \,\mathrm{.}$$ 4. It is readily verified that $\Pi=\boldsymbol\varepsilon_*\circ\mfc_\mbfn^\diamond\colon\mfS_{{{}\mbfn}_\bullet}\to \mcA_\mbfn$ factors over $\mcN_\mbfn$. **Proposition 15**. *It holds that $$\left\lvert(\mfc_\mbfn^\diamond)^{-1}(N)\right\rvert=\mbfn! \prod_{\nu\in\mathsf{supp}N} \frac{p_\nu/\ell_\nu}{N(\nu)!} \qquad \text{and} \qquad M_{\mbfn}(A)=\mbfn!\sum_{\substack{N \in\mcN_\mbfn\\ \boldsymbol\varepsilon_*N=A}} \prod_{\nu\in\mathsf{supp}N} \frac{p_\nu/\ell_\nu}{N(\nu)!} \,\mathrm{.}$$* **Proof.* We provide a sketch of the proof, the details being similar to Proposition [Proposition 13](#p:Quotient){reference-type="ref" reference="p:Quotient"}.* 1. *A word in $[{{}\mbfn}_\bullet]^*$ is *simple* if each of its characters appears exactly once. Two words in $[{{}\mbfn}_\bullet]^*$ are *disjoint* if they share no common character. We denote by $\ell_w$ the length of $w\in [{{}\mbfn}_\bullet]^*$. Further set $$\begin{aligned} \mcX\mathop{\mathrm{\coloneqq}}&\ \left\{\left(w_1,\dotsc, w_r\right) : r\in{\mathbb N}_1\,\mathrm{,}\;\,w_i\in [{{}\mbfn}_\bullet]^*\,\mathrm{,}\;\,w_i \text{ simple, pairwise disjoint} \,\mathrm{,}\;\,\ \sum_{i=1}^r \ell_{w_i}={{}\mbfn}_\bullet\right\}\,\mathrm{,}\;\, \\ \mcZ\mathop{\mathrm{\coloneqq}}&\ \big\{\left\{w_1,\dotsc, w_r\right\} : \left(w_1,\dotsc, w_r\right)\in\mcX\big\}\,\mathrm{,}\;\,\end{aligned}$$ and note that, since the necklace $\llbracket w\rrbracket_{}$ of a simple word $w\in [{{}\mbfn}_\bullet]^*$ is a cycle in $\mfS_{{{}\mbfn}_\bullet}$, then $$\mfS_{{{}\mbfn}_\bullet}= \big\{\llbracket w_1\rrbracket_{}\cdots \llbracket w_r\rrbracket_{} : \left\{w_1,\dotsc, w_r\right\}\in\mcZ\big\} \,\mathrm{.}$$* 2. *Let $\mfc_\mbfn^*\colon [{{}\mbfn}_\bullet]^*\to [q]^*$ be defined by $% \mfc_\mbfn^*\colon w\mathop{\mathrm{\coloneqq}}y_1\cdots y_\ell \longmapsto \mfc_\mbfn(y_1)\cdots \mfc_\mbfn(y_\ell)$, and denote again by $\mfc_\mbfn^*$ its component-wise extension to $\mcX$.* 3. *Set $\mcV\mathop{\mathrm{\coloneqq}}\mfc_\mbfn^*(\mcX)$, denote again by $\llbracket{\,\cdot\,}\rrbracket_{}$ the component-wise extension to $\mcV$ of the quotient map $\llbracket{\,\cdot\,}\rrbracket_{}$ from $[q]^*$ to necklaces, and set $% \mcU\mathop{\mathrm{\coloneqq}}\left\{{\big(\llbracket v_1\rrbracket_{},\dotsc, \llbracket v_r\rrbracket_{}\big)}: \left(v_1,\dotsc, v_r\right)\in\mcV \right\} %$.* 4. *Define a map $\boldsymbol\nu$ on $\mcU$ by $% \boldsymbol\nu\colon {\big(\llbracket v_1\rrbracket_{}, \dotsc, \llbracket v_r\rrbracket_{}\big)} \longmapsto \sum_{i=1}^r \mathop{\mathrm{\mathbf 1}}_{\llbracket v_i\rrbracket_{}} %$.* 5. *Finally, define maps $f\colon\mcX\to\mcZ$ and $\llbracket{\,\cdot\,}\rrbracket_{}^*\colon\mcZ \to\mfS_{{{}\mbfn}_\bullet}$ by $$f\colon \left(w_1,\dotsc, w_r\right) \longmapsto \left\{w_1,\dotsc, w_r\right\}\,\mathrm{,}\;\,\qquad \llbracket{\,\cdot\,}\rrbracket_{}^*\colon \left\{w_1,\dotsc, w_r\right\}\longmapsto \llbracket w_1\rrbracket_{}\cdots \llbracket w_r\rrbracket_{} \,\mathrm{.}$$* *It is a tedious verification that the diagram in Figure [\[fig:diagram2\]](#fig:diagram2){reference-type="ref" reference="fig:diagram2"} commutes, and a simple computation of the cardinality of the fibers of the maps involved yields the conclusion. ◻* # Multivariate moments For $k\geq 1$ let $\Delta^{k-1}$ be the *standard simplex* $$\label{eq:Simplex} \Delta^{k-1}\mathop{\mathrm{\coloneqq}}\big\{\mbfx\in {\mathbb R}^k : x_i\geq 0\,\mathrm{,}\;\,x_1+\cdots+x_k=1\big\} \,\mathrm{,}\;\,$$ and recall the definition [\[eq:DirichletDistribution\]](#eq:DirichletDistribution){reference-type="eqref" reference="eq:DirichletDistribution"} of the Dirichlet distribution $D_{\boldsymbol\alpha}$. Our main result in this section is a formula for the multivariate moments of $D_{\boldsymbol\alpha}$. **Theorem 16** (Multivariate moments of $D_{\boldsymbol\alpha}$). *The following identity holds $$\label{eq:t:Moments:0} \mu_\mbfn[\mbfS;\boldsymbol\alpha]\mathop{\mathrm{\coloneqq}}\int_{\Delta^{k-1}} \prod_j^q (\mbfs_j\cdot\mbfx)^{n_j} \mathop{}\!\mathrm{d}D_{\boldsymbol\alpha}(\mbfx)= \frac{\mbfn!}{\left\langle{{}\boldsymbol\alpha}_\bullet\right\rangle_{{{}\mbfn}_\bullet}}\, Z_\mbfn\big({\Omega_\mbfn[\mbfS;\boldsymbol\alpha]}\big)\eqqcolon\zeta_\mbfn[\mbfS;\boldsymbol\alpha] \,\mathrm{.}$$* In order to put Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"} into context, we briefly survey previously known results on moments of Dirichlet and related measures. ## Overview on Dirichlet measures {#ss:MomentsOverview} Moment and Laplace/Fourier-transform methods for $D_{\boldsymbol\alpha}$ and its infinite-dimensional counterpart, the Dirichlet--Ferguson measure $\mcD_\alpha$ [@Fer73] over a measure space $(X,\alpha)$ are notoriously difficult, as we briefly summarize below. *Transforms*. It is well-known that the Fourier transform $\widehat{D_{\boldsymbol\alpha}}$ of the Dirichlet distribution $D_{\boldsymbol\alpha}$ may be expressed in terms of ${}_k\Phi_2$, the $k$-variate *confluent hypergeometric Lauricella function of type $D$* [@Lau1893; @Ext76]. The power-series representation of ${}_k\Phi_2$ is *inconvenient for numerical calculations when $k>2$* [@Phi88 p. 4]. Instead, the complex-contour-integral representation [@Erd40 Eqn. (7)] is preferred, but its treatment remains quite involved, see e.g. [@RegGugDiN02]. In particular, differentiating $\widehat{D_{\boldsymbol\alpha}}$ in this form does not provide any useful representation for the moments of the measure. For decades the Fourier transform $\widehat{\mcD_\alpha}$ of $\mcD_\alpha$ was widely considered intractable [@JiaDicKuo04], which led to the introduction of other characterizing transforms, such as the *Markov--Krein transform* [@KerTsi01] and the *$c$-transform* [@JiaDicKuo04]. These methods too are unsatisfactory, since there is no counterpart for such transforms of foundational results available for the Fourier transform, such as, for instance, Bochner--Minlos--Sazonov's (BMS) or Lévy's Continuity Theorem. The Fourier transform $\widehat{\mcD_\alpha}$ was eventually computed in [@LzDS19a] by methods in combinatorics and representation theory. *Moments*. Multivariate moments of $D_{\boldsymbol\alpha}$ are virtually well-known in the form [\[eq:l:Moments:0\]](#eq:l:Moments:0){reference-type="eqref" reference="eq:l:Moments:0"} below, which may be regarded as an extension of the ESF. Whereas easily computable for small $k$, this form is unsuitable for letting $k\to\infty$ and thus provides no insight on multivariate moments of $\mcD_\alpha$. Partially in order to overcome this issue, other expressions have been considered: the univariate moment $\mcD_\alpha(f^n)$ has appeared in [@Reg98] in terms of incomplete Bell polynomials, solely in the case $X\Subset {\mathbb R}$ and $f=\id_{\mathbb R}$; more general univariate moments for $D_{\boldsymbol\alpha}$ have implicitly appeared in [@LetPic18 proof of Prop. 3.3] in iterative form; univariate moments for both $D_{\boldsymbol\alpha}$ and $\mcD_\alpha$ have appeared in full generality in [@LzDS19a] in terms of the cycle index polynomials $Z_n$, which allowed the aforementioned computation of $\widehat{\mcD_\alpha}$. As for multivariate moments, they have appeared: in [@KerTsi01 Prop. 7.4], in terms of summations over constrained permutations, only in the case ${{}\boldsymbol\alpha}_\bullet=1$; in [@EthKur94 Eqn. (4.20)], [@Fen10 Lem. 5.2], and [@LzDSLyt17 Cor. 3.5], in terms of summations over constrained *set* partitions. *Other measures*. The measure $\mcD_\alpha$ is the *simplicial part* of other known measures on the space $\msM^+$ of non-negative Borel measures on $X$. Among them are: the law $\mcG_\alpha$ of the *$\gamma$-point process* [@KonDaSStrUs98] with intensity $\alpha$, and A.M. Vershik's *multiplicative infinite-dimensional Lebesgue measure* $\mcL_\alpha$ [@Ver07; @Ver08] with intensity $\alpha$. Together with $\mcD_\alpha$, these measures have a wide range of applications, from the theory of point processes and of measure-valued Markov diffusions, see [@LzDS17+ §1] and references therein, to the representation theory of infinite-dimensional Lie groups of currents/multipliers, see [@TsiVerYor01], or [@LzDS19a §1] for a unified treatment. In §[3.3](#ss:DirichletGamma){reference-type="ref" reference="ss:DirichletGamma"} we give moment formulas for $\mcD_\alpha$ and $\mcG_\alpha$ analog to the one in Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"}. *Relations to the ESF*. One relation between the Dirichlet distribution and the ESF is made apparent by the expression of the generating function of $E_{\theta}$ in the dummy variables $\mbft\mathop{\mathrm{\coloneqq}}\left(t_1,\dotsc, t_n\right)$ in terms of the cycle index polynomial $Z_n$ [\[eq:CycleIndex\]](#eq:CycleIndex){reference-type="eqref" reference="eq:CycleIndex"} of $\mfS_n$, viz. $$\label{eq:DirichletEwens} \sum_{\boldsymbol\lambda\vdash n} E_{\theta}(\boldsymbol\lambda) \, \mbft^{\boldsymbol\lambda} = \frac{n!}{\left\langle\theta\right\rangle_{n}} Z_n[\theta\, \mbft]\,\mathrm{,}\;\,\qquad \mbft^{\boldsymbol\lambda} \mathop{\mathrm{\coloneqq}}t_1^{\lambda_1} \cdots t_n^{\lambda_n} \,\mathrm{.}$$ ## Some corollaries {#ss:Corollaries} Let us collect some corollaries and special cases of Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"}. **Corollary 17**. *Let $P_\pi$ be the permutation matrix of a permutation $\pi\in\mfS_q$. Then, $$Z_\mbfn[\boldsymbol\alpha;\mbfS]=Z_{P_\pi\mbfn}[\boldsymbol\alpha; \mbfS P_\pi] \,\mathrm{,}\;\,\qquad \mbfS\in{\mathbb R}^{k\times q}\,\mathrm{.}$$* **Corollary 18**. *For every $\mbfn\in{\mathbb N}^q_*$ we have $$\sum_{\substack{A \vDash \mbfn\\ \mathsf{shape}(A)=\boldsymbol\lambda}} M_{\mbfn}(A)= M_2(\boldsymbol\lambda) \,\mathrm{.}$$* **Proof.* In [\[eq:t:Moments:0\]](#eq:t:Moments:0){reference-type="eqref" reference="eq:t:Moments:0"}, choose $\mbfs_1=\dotsc=\mbfs_q\eqqcolon\mbfs$ and $\boldsymbol\alpha$ with ${{}\boldsymbol\alpha}_\bullet=1$, and set $n\mathop{\mathrm{\coloneqq}}{{}\mbfn}_\bullet$. Then, the left-hand side of [\[eq:t:Moments:0\]](#eq:t:Moments:0){reference-type="eqref" reference="eq:t:Moments:0"} becomes the $n^\textrm{th}$-moment of the linear functional $\mbfx\mapsto\mbfs\cdot \mbfx$ of $D_{\boldsymbol\alpha}$ and is thus equal to $Z_{n}[\mbfs^{\diamond 1}\cdot \boldsymbol\alpha,\dotsc, \mbfs^{\diamond n}\cdot\boldsymbol\alpha]$ by [@LzDS19a Thm. 3.2]. As for the right-hand side, for the above choice of the $\mbfs_i$'s the monomials $\omega_\mbfa[\mbfS;\boldsymbol\alpha]$ satisfy $\omega_{\mbfa}[\mbfS;\boldsymbol\alpha]=\omega_{\mbfa'}[\mbfS;\boldsymbol\alpha]$ whenever ${{}\mbfa}_\bullet={{}\mbfa'}_\bullet$. Collecting terms in the right-hand side and equating the coefficients of the corresponding monomials on both sides yields the assertion. ◻* **Corollary 19**. *The following identity holds $$\label{eq:c:SumRows:0} \sum_{\substack{\mbfn\in {\mathbb N}_0^q\\ {{}\mbfn}_\bullet=n}} Z_\mbfn\big({\Omega_\mbfn[\mbfS;\boldsymbol\alpha]}\big)= Z_n[\mathsf{row}(\mbfS) \cdot \boldsymbol\alpha, \mathsf{row}(\mbfS)^{\diamond 2} \cdot \boldsymbol\alpha,\dotsc, \mathsf{row}(\mbfS)^{\diamond n} \cdot\boldsymbol\alpha] \,\mathrm{.}$$* **Proof.* Set $$\begin{aligned} \Phi[\boldsymbol\alpha; \mbfS]\mathop{\mathrm{\coloneqq}}\sum_{\mbfM\in {\mathbb N}_0^{k\times q}} \frac{\left\langle\boldsymbol\alpha\right\rangle_{\mathsf{row}(\mbfM)}}{\left\langle{{}\boldsymbol\alpha}_\bullet\right\rangle_{{{}\mathsf{col}(\mbfM)}_\bullet}} \frac{\mbfS^\mbfM}{\mbfM!} \,\mathrm{.}\end{aligned}$$ By definition of $\Phi[\boldsymbol\alpha;\mbfS]$ and Lemma [Lemma 23](#l:AuxiliaryNu){reference-type="ref" reference="l:AuxiliaryNu"} below, and by [@LzDS19a Eqn. (2.9)], for every $t\in{\mathbb R}$, $$\Phi[\boldsymbol\alpha;t\,\mbfS]= \int_{\Delta^{k-1}} e^{t (\mbfs_1+\cdots+\mbfs_q)\cdot\mbfx} \mathop{}\!\mathrm{d}D_{\boldsymbol\alpha}(\mbfx) \eqqcolon\widehat{D_{\boldsymbol\alpha}}\big({\mathsf{row}(\mbfS)}\big)={}_k\Phi_2[\boldsymbol\alpha;{{}\boldsymbol\alpha}_\bullet;t\, \mathsf{row}(\mbfS)] \,\mathrm{.}$$ Expanding the left-hand side as a series in $n\in {\mathbb N}_0$, each summand is the left-hand side of [\[eq:c:SumRows:0\]](#eq:c:SumRows:0){reference-type="eqref" reference="eq:c:SumRows:0"} by Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"}. Expanding the right-hand side as series in $n\in {\mathbb N}_0$, each summand is the right-hand side of [\[eq:c:SumRows:0\]](#eq:c:SumRows:0){reference-type="eqref" reference="eq:c:SumRows:0"} by [@LzDS19a Prop. 3.5]. Since, for *same* $n$, the summands in each of these expansions are polynomials of *same* degree equal to $n$ in the variables $\mbfS$, we may equate the summands one by one, which yields [\[eq:c:SumRows:0\]](#eq:c:SumRows:0){reference-type="eqref" reference="eq:c:SumRows:0"}. ◻* ## Dirichlet--Ferguson and Gamma measures {#ss:DirichletGamma} Let $X$ be a second countable locally compact Hausdorff space, and $\msP$ be the space of all Borel probability measures on $X$, endowed with the Borel $\sigma$-algebra of the narrow topology. For any finite Borel measure $\eta$ on $X$ and any bounded Borel $f\colon X\to{\mathbb R}$ we set $\eta f\mathop{\mathrm{\coloneqq}}\int f\mathop{}\!\mathrm{d}\eta$. *Dirichlet--Ferguson measures*. For $\beta>0$ and $\sigma\in\msP$, let $\alpha\mathop{\mathrm{\coloneqq}}\beta\sigma$ be the finite Borel measure on $X$ with total mass $\beta$ and *shape* (also: *simplicial part*) $\sigma$. The *Dirichlet--Ferguson measure $\mcD_{\alpha}$ with intensity* (*measure*) $\alpha$ is the unique Borel probability measure on $\msP$ with Fourier transform [@LzDS19a Thm. 3.10] $$\begin{aligned} \widehat{\mcD_\alpha}(f)\mathop{\mathrm{\coloneqq}}\int_\msP e^{\mathbbm{i} \, \eta f} \mathop{}\!\mathrm{d}\mcD_\alpha(f)= \sum_{n=0}^\infty \frac{\mathbbm{i}^n}{\left\langle\beta\right\rangle_{n}} Z_n\big({\alpha f, \alpha f^2, \dotsc, \alpha f^n}\big) \,\mathrm{,}\;\,\qquad f\in\mcC_b \,\mathrm{.}\end{aligned}$$ For continuous bounded $f_1,\dotsc, f_q\colon X\to{\mathbb R}$, set $$\begin{aligned} \Omega_\mbfn[f_1,\dotsc, f_q;\alpha]\mathop{\mathrm{\coloneqq}}&\ \left(\alpha\big({f_1^{h_1}\cdots f_q^{h_q}}\big)\right)_{\mbfh\leq_\diamond\mbfn} \,\mathrm{.}\end{aligned}$$ By a straightforward adaptation of the proof for the univariate case [@LzDS19a Thm. 3.10], as a corollary of Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"} we obtain an explicit expression for the moments of $\mcD_\alpha$. **Corollary 20** (Multivariate moments of $\mcD_\alpha$). *We have $$\int_{\msP} \prod_j^q (\eta f_j)^{n_j}\mathop{}\!\mathrm{d}\mcD_\alpha(\eta) = \frac{\mbfn!}{\left\langle\beta\right\rangle_{{{}\mbfn}_\bullet}}Z_\mbfn\big({\Omega_\mbfn[f_1,\dotsc, f_q;\alpha]}\big) \,\mathrm{.}$$* We recover Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"} by choosing a Borel partition $\left(X_i\right)_i^k$ of $X$ with $\alpha_i\mathop{\mathrm{\coloneqq}}\alpha X_i$, and simple functions $f_1,\dotsc, f_q$, constantly equal to, respectively, $s_{1,i},\dotsc, s_{q,i}$ on each set $X_i$ for each $i\in [k]$. *Gamma measures*. Let $\mcG_\alpha$ be the law of the Gamma point process with intensity $\alpha$, e.g. [@KonDaSStrUs98]. **Corollary 21** (Multivariate moments of $\mcG_\alpha$). *We have $$\int_{\msM_b^+} \prod_j^q (\eta f_j)^{n_j}\mathop{}\!\mathrm{d}\mcG_\alpha(\eta) = \mbfn! \, Z_\mbfn\big({\Omega_\mbfn[f_1,\dotsc, f_q;\alpha]}\big) \,\mathrm{.}$$* *Remark 22*. Alternative expressions for the multivariate moments of the Gamma measure may be obtained by differentiating its characteristic functional (e.g. [@LzDSLyt17 p. 5]). Such expressions are however not informative on their algebraic and combinatorial meaning in connection with $Z_\mbfn$, as they rather rely on the multivariate multi-factor Leibniz rule. A similar approach does not apply to the Dirichlet--Ferguson measure, due to the convoluted form of its characteristic functional. ## Proof of Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"} {#ss:MomentsProof} **Lemma 23**. *The following identity holds $$\label{eq:l:Moments:0} \mu_\mbfn[\mbfS;\boldsymbol\alpha] = \frac{\mbfn!}{\left\langle{{}\boldsymbol\alpha}_\bullet\right\rangle_{{{}\mbfn}_\bullet}} \sum_{\substack{\mbfM\in {\mathbb N}_0^{k\times q}\\ \mathsf{col}(\mbfM)=\mbfn}} \left\langle\boldsymbol\alpha\right\rangle_{\mathsf{row}(\mbfM)} \frac{\mbfS^\mbfM}{\mbfM!}\eqqcolon\nu_\mbfn[\mbfS;\boldsymbol\alpha] \,\mathrm{.}$$* *Proof.* By the Multinomial Theorem and by properties of the Dirichlet distribution $$\begin{aligned} \mu_\mbfn[\mbfS;\boldsymbol\alpha]% =&\ \frac{1}{\mathrm{B}[\boldsymbol\alpha]}\int_{\Delta^{k-1}} \left(\prod_j^q \sum_{\substack{\mbfm\in {\mathbb N}_0^k\\ {{}\mbfm}_\bullet=n_j}}\binom{n_j}{\mbfm}\, \mbfs_j^\mbfm\, \mbfx^\mbfm\right) \mbfx^{\boldsymbol\alpha-{\mathbf 1}}\mathop{}\!\mathrm{d}\mbfx \\ =&\ \frac{1}{\mathrm{B}[\boldsymbol\alpha]}\int_{\Delta^{k-1}} \left(\sum_{\substack{\mbfm_1,\dotsc, \mbfm_q\in {\mathbb N}_0^k\\ {{}\mbfm_1}_\bullet=n_1,\dotsc, {{}\mbfm_q}_\bullet=n_q}} \prod_j^q \binom{n_j}{\mbfm_j}\, \mbfs_j^{\mbfm_j} \mbfx^{\mbfm_j}\right)\mbfx^{\boldsymbol\alpha-{\mathbf 1}}\mathop{}\!\mathrm{d}\mbfx \\ =&\ \sum_{\substack{\mbfm_1,\dotsc, \mbfm_q\in {\mathbb N}_0^k\\ {{}\mbfm_1}_\bullet=n_1,\dotsc, {{}\mbfm_q}_\bullet=n_q}} \frac{1}{\mathrm{B}[\boldsymbol\alpha]} \left(\prod_j^q \binom{n_j}{\mbfm_j}\, \mbfs_j^{\mbfm_j} \right) \int_{\Delta^{k-1}} \mbfx^{\mbfm_1+\cdots+\mbfm_q+\boldsymbol\alpha-{\mathbf 1}} \mathop{}\!\mathrm{d}\mbfx \\ =&\ \sum_{\substack{\mbfm_1,\dotsc, \mbfm_q\in {\mathbb N}_0^k\\ {{}\mbfm_1}_\bullet=n_1,\dotsc, {{}\mbfm_q}_\bullet=n_q}} \frac{\mathrm{B}[\mbfm_1+\cdots+\mbfm_q+\boldsymbol\alpha]}{\mathrm{B}[\boldsymbol\alpha]} \prod_j^q \binom{n_j}{\mbfm_j}\,\mbfs_j^{\mbfm_j} \,\mathrm{.}\end{aligned}$$ Reindexing the summation over $\mbfM=(\mbfm_1, \dotsc, \mbfm_q)\in{\mathbb R}^{k\times q}$, we conclude that $$\begin{aligned} \mu_\mbfn[\mbfS;\boldsymbol\alpha]=&\ \frac{\mbfn!}{\left\langle{{}\boldsymbol\alpha}_\bullet\right\rangle_{{{}\mbfn}_\bullet}}\sum_{\substack{\mbfM\in {\mathbb N}_0^{q\times k}\\ \mathsf{col}(\mbfM)=\mbfn}} \left\langle\boldsymbol\alpha\right\rangle_{\mathsf{row}(\mbfM)} \frac{\mbfS^\mbfM}{\mbfM!} \,\mathrm{.}\qedhere\end{aligned}$$ ◻ Let us recall the following fact, e.g. [@Tau63 Eqn. (8), p. 1060], or [@Car63 Eqn. (10), p. 39]. **Lemma 24**. *For every $k\in {\mathbb N}$, every $\mbfv\in {\mathbb N}_0^k$, and every integer $0\leq m \leq {{}\mbfv}_\bullet$, $$\label{eq:l:Multinomial:0} \binom{{{}\mbfv}_\bullet}{\mbfv}=\sum_{\substack{\mbfw\in {\mathbb N}_0^k\\ \mbfw\leq_\diamond\mbfv\\ {{}\mbfw}_\bullet=m}} \binom{{{}\mbfw}_\bullet}{\mbfw} \binom{{{}\mbfv}_\bullet-{{}\mbfw}_\bullet}{\mbfv-\mbfw} \,\mathrm{.}$$* *Proof of Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"}.* Set $$\tilde \nu_\mbfn[\mbfS;\boldsymbol\alpha]\mathop{\mathrm{\coloneqq}}\frac{\left\langle\boldsymbol\alpha\right\rangle_{{{}\mbfn}_\bullet}}{\mbfn!} \nu_\mbfn[\mbfS;\boldsymbol\alpha]\qquad \text{and} \qquad \tilde \zeta_\mbfn[\mbfS;\boldsymbol\alpha]\mathop{\mathrm{\coloneqq}}\frac{\left\langle\boldsymbol\alpha\right\rangle_{{{}\mbfn}_\bullet}}{\mbfn!} \zeta_\mbfn[\mbfS;\boldsymbol\alpha] \,\mathrm{.}$$ We show that $\tilde\nu_\mbfn[\mbfS;\boldsymbol\alpha]=\tilde\zeta_\mbfn[\mbfS;\boldsymbol\alpha]$ and conclude the assertion by Lemma [Lemma 23](#l:AuxiliaryNu){reference-type="ref" reference="l:AuxiliaryNu"}. *Step 1*. We claim that $$\begin{aligned} \label{eq:t:Moments:1} \tilde\nu_{\mbfn-\mbfe_j}[\mbfS;\boldsymbol\alpha+\mbfe_\ell] = \sum_{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn} \mbfS_\ell^{\mbfh-\mbfe_j} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!} \, \tilde\nu_{\mbfn-\mbfh}[\mbfS;\boldsymbol\alpha] % \,\mathrm{,}\;\,\end{aligned}$$ where, conventionally, $$\label{eq:t:Moments:2} \tilde\nu_\mbfn[\mbfS;\boldsymbol\alpha]=0 \quad \text{whenever} \quad \mbfn\not\geq^\diamond\mathop{\mathrm{{\mathbf 0}}}_q\,\mathrm{.}$$ We argue by induction on ${{}\mbfn}_\bullet$ with trivial (i.e. $1=1$) base step for ${{}\mbfn}_\bullet=1$. *Inductive step.* Let $\partial_a^b\mathop{\mathrm{\coloneqq}}\partial_{s_a^b}$, set $\mbfE_a^b\mathop{\mathrm{\coloneqq}}[\delta_{ai}\delta_{bj}]_i^j\in \left\{0,1\right\}^{k\times q}$, and note that $$\begin{aligned} \nonumber \partial_a^b \tilde\nu_\mbfn[\mbfS;\boldsymbol\alpha]=&\ \sum_{\substack{\mbfM\in {\mathbb N}_0^{k\times q} \\ \mathsf{col}(\mbfM)=\mbfn}} \left\langle\boldsymbol\alpha\right\rangle_{\mathsf{row}(\mbfM)} \partial_a^b \frac{\mbfS^\mbfM}{\mbfM!}= \sum_{\substack{\mbfE_a^b\leq_\diamond\mbfM\in {\mathbb N}_0^{k\times q} \\ \mathsf{col}(\mbfM)=\mbfn}} \alpha_a \left\langle\boldsymbol\alpha+\mbfe_a\right\rangle_{\mathsf{row}(\mbfM-\mbfE_a^b)} \frac{\mbfS^{\mbfM-\mbfE_a^b}}{(\mbfM-\mbfE_a^b)!} \\ \nonumber =&\ \sum_{\substack{\mbfM\in {\mathbb N}_0^{k\times q} \\ \mathsf{col}(\mbfM)=\mbfn-\mbfe_b}} \alpha_a \left\langle\boldsymbol\alpha+\mbfe_a\right\rangle_{\mathsf{row}(\mbfM)} \frac{\mbfS^{\mbfM}}{\mbfM!} \\ \label{eq:t:Moments:3} =&\ \alpha_a \tilde\nu_{\mbfn-\mbfe_b}[\mbfS;\boldsymbol\alpha+\mbfe_a] \,\mathrm{.}\end{aligned}$$ Applying the inductive hypothesis to $\mbfn-\mbfe_b$ with $\boldsymbol\alpha+\mbfe_a$ in place of $\boldsymbol\alpha$, we have $$\begin{aligned} \nonumber \alpha_a \tilde\nu_{\mbfn-\mbfe_j-\mbfe_b}[\mbfS;\boldsymbol\alpha+\mbfe_\ell+\mbfe_a]=&\ \alpha_a \sum_{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn-\mbfe_b} \mbfS_\ell^{\mbfh-\mbfe_j} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!} \, \tilde\nu_{\mbfn-\mbfh-\mbfe_b}[\mbfS;\boldsymbol\alpha+\mbfe_a] \\ \label{eq:t:Moments:3.5} =&\ \alpha_a \sum_{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn} \mbfS_\ell^{\mbfh-\mbfe_j} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!}\, \tilde\nu_{\mbfn-\mbfh-\mbfe_b}[\mbfS;\boldsymbol\alpha+\mbfe_a]\,\mathrm{,}\;\,\end{aligned}$$ where the last equality holds by [\[eq:t:Moments:2\]](#eq:t:Moments:2){reference-type="eqref" reference="eq:t:Moments:2"}. Now, let $\ell\neq a$. Applying [\[eq:t:Moments:3\]](#eq:t:Moments:3){reference-type="eqref" reference="eq:t:Moments:3"} with $\mbfn-\mbfe_j$ in place of $\mbfn$ and $\boldsymbol\alpha+\mbfe_\ell$ in place of $\boldsymbol\alpha$, $$\begin{aligned} \label{eq:t:Moments:3.7} \partial_a^b \tilde\nu_{\mbfn-\mbfe_j}[\mbfS;\boldsymbol\alpha+\mbfe_\ell] = \alpha_a \tilde\nu_{\mbfn-\mbfe_j-\mbfe_b}[\mbfS;\boldsymbol\alpha+\mbfe_\ell+\mbfe_a] \,\mathrm{.}\end{aligned}$$ Since $\ell\neq a$, applying [\[eq:t:Moments:3\]](#eq:t:Moments:3){reference-type="eqref" reference="eq:t:Moments:3"} with $\mbfn-\mbfh$ in place of $\mbfn$ for every $\mbfh \leq_\diamond\mbfn$ yields $$\label{eq:t:Moments:3.8} \begin{aligned} \partial_a^b& \left(\sum_{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn} \mbfS_\ell^{\mbfh-\mbfe_j} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!} \tilde\nu_{\mbfn-\mbfh}[\mbfS;\boldsymbol\alpha]\right)= \\ &\ =\alpha_a \sum_{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn} \mbfS_\ell^{\mbfh-\mbfe_j} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!}\, \tilde\nu_{\mbfn-\mbfh-\mbfe_b}[\mbfS;\boldsymbol\alpha+\mbfe_a] \,\mathrm{.} \end{aligned}$$ Combining [\[eq:t:Moments:3.7\]](#eq:t:Moments:3.7){reference-type="eqref" reference="eq:t:Moments:3.7"}, [\[eq:t:Moments:3.8\]](#eq:t:Moments:3.8){reference-type="eqref" reference="eq:t:Moments:3.8"}, and [\[eq:t:Moments:3.5\]](#eq:t:Moments:3.5){reference-type="eqref" reference="eq:t:Moments:3.5"} yields $$\begin{aligned} \partial_a^b \left(\tilde\nu_{\mbfn-\mbfe_j}[\mbfS;\boldsymbol\alpha+\mbfe_\ell] - \sum_{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn} \mbfS_\ell^{\mbfh-\mbfe_j} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!} \tilde\nu_{\mbfn-\mbfh}[\mbfS;\boldsymbol\alpha]\right)=0 \,\mathrm{.}\end{aligned}$$ By arbitrariness of $a,b$ we conclude that the bracketed quantity is a polynomial in the sole variables $s_\ell^1,\dotsc, s_\ell^q$. As a consequence, every monomial not in the sole variables $s_\ell^1,\dotsc, s_\ell^q$ cancels out by arbitrariness of the variables $\mbfS$. Thus, $$\begin{aligned} \tilde\nu_{\mbfn-\mbfe_j}&[\mbfS;\boldsymbol\alpha+\mbfe_\ell] - \sum_{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn} \mbfS_\ell^{\mbfh-\mbfe_j} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!} \tilde\nu_{\mbfn-\mbfh}[\mbfS;\boldsymbol\alpha] \\ &= \left\langle\alpha_\ell+1\right\rangle_{{{}\mbfn}_\bullet-1}\frac{\mbfS_\ell^{\mbfn-\mbfe_j}}{(\mbfn-\mbfe_j)!}-\sum_{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn} \mbfS_\ell^{\mbfh-\mbfe_j} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!} \left\langle\alpha_\ell\right\rangle_{{{}\mbfn}_\bullet-{{}\mbfh}_\bullet}\frac{\mbfS_\ell^{\mbfn-\mbfh}}{(\mbfn-\mbfh)!} \,\mathrm{.}\end{aligned}$$ The latter quantity is proved to vanish as soon as $$\begin{aligned} \frac{\left\langle\alpha_\ell+1\right\rangle_{{{}\mbfn}_\bullet-1}}{(\mbfn-\mbfe_j)!}=\sum_{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!} \frac{\left\langle\alpha_\ell\right\rangle_{{{}\mbfn}_\bullet-{{}\mbfh}_\bullet}}{(\mbfn-\mbfh)!}\,\mathrm{,}\;\,\end{aligned}$$ or, equivalently, by the Chu--Vandermonde identity, $$\begin{aligned} \frac{({{}\mbfn}_\bullet-1)!}{(\mbfn-\mbfe_j)!}\sum_{i=1}^{{{}\mbfn}_\bullet} \frac{\left\langle\alpha_\ell\right\rangle_{{{}\mbfn}_\bullet-i}}{({{}\mbfn}_\bullet-i)!} =\sum_{i=1}^{{{}\mbfn}_\bullet}\sum_{\substack{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn\\ {{}\mbfh}_\bullet=i}}\frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!} \frac{\left\langle\alpha_\ell\right\rangle_{{{}\mbfn}_\bullet-i}}{(\mbfn-\mbfh)!}\,\mathrm{.}\end{aligned}$$ The latter is implied by the equality of each of the summands, viz. $$\begin{aligned} \frac{({{}\mbfn}_\bullet-1)!}{(\mbfn-\mbfe_j)!}\frac{1}{({{}\mbfn}_\bullet-i)!} =\sum_{\substack{{\mbfe_j\leq_\diamond\mbfh\leq_\diamond\mbfn}\\ {{}\mbfh}_\bullet=i}} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_j)!} \frac{1}{(\mbfn-\mbfh)!}\,\mathrm{,}\;\,\end{aligned}$$ which is in turn a consequence of Lemma [Lemma 24](#l:Auxiliary){reference-type="ref" reference="l:Auxiliary"}, after relabeling $\mbfn$ as $\mbfn-\mbfe_j$. *Step 2*. We now verify that $\tilde\nu_\mbfn=\tilde\zeta_\mbfn$. We argue by strong induction on ${{}\mbfn}_\bullet$ with trivial (i.e. $1=1$) base step ${{}\mbfn}_\bullet=0$. *Inductive step.* Assume for every $\boldsymbol\alpha\in{\mathbb R}^k_+$ that $\tilde\nu_{\mbfn-\mbfh}[\mbfS;\boldsymbol\alpha]=\tilde\zeta_{\mbfn-\mbfh}[\mbfS;\boldsymbol\alpha]$ for every $\mbfh\leq_\diamond\mbfn$ with $\mbfh\neq \mathop{\mathrm{{\mathbf 0}}}$. Now, $$\begin{aligned} \nonumber \mbfn! \, \partial_p^b \tilde\zeta_\mbfn[\mbfS;\boldsymbol\alpha]=&\ \sum_{A\vDash \mbfn} M_{\mbfn}(A)\, \partial_p^b \underbrace{\prod_{\mbfa\in\mathsf{supp}A} \left( \big({\mbfs_1^{\diamond a_1}\diamond\cdots \diamond\mbfs_q^{\diamond a_q}}\big)\cdot\boldsymbol\alpha\right)^{A(\mbfa)}}_{\eqqcolon J(\mbfS;\boldsymbol\alpha;\mbfA)} \\ \nonumber =&\ \sum_{A\vDash \mbfn} M_{\mbfn}(A)\, \sum_{\substack{\mbfa\in\mathsf{supp}A\\ \mbfa\geq_\diamond\mbfe_b}} \frac{A(\mbfa)\, a_b\, \alpha_p \,\mbfS_p^{\mbfa-\mbfe_b}}{ \big({\mbfs_1^{\diamond a_1}\diamond\cdots \diamond\mbfs_q^{\diamond a_q}}\big)\cdot\boldsymbol\alpha} \, J(\mbfS;\boldsymbol\alpha;A) \\ \label{eq:t:Moments:4} =&\ \alpha_p\sum_{\mbfe_b\leq_\diamond\mbfh\leq_\diamond\mbfn} \sum_{A\vDash \mbfn} M_{\mbfn}(A) \frac{A(\mbfh)\, h_b \,\mbfS_p^{\mbfh-\mbfe_b}}{ \big({\mbfs_1^{\diamond h_1}\diamond\cdots \diamond\mbfs_q^{\diamond h_q}}\big)\cdot\boldsymbol\alpha} \, J(\mbfS;\boldsymbol\alpha;A) \,\mathrm{.}\end{aligned}$$ For each $\mbfh\leq_\diamond\mbfn$ and each $A\vDash\mbfn$ with $A\geq \mathop{\mathrm{\mathbf 1}}_\mbfh$, set $C\mathop{\mathrm{\coloneqq}}A-\mathop{\mathrm{\mathbf 1}}_\mbfh$. Note that $$\begin{aligned} \nonumber M_{\mbfn}(A)=&\ \mbfn! \prod_{\mbfa\in \mathsf{supp}A} \frac{\binom{{{}\mbfa}_\bullet}{\mbfa}^{A(\mbfa)}}{{{}\mbfa}_\bullet^{A(\mbfa)}A(\mbfa)!} = \frac{\mbfn!\, {{}\mbfh}_\bullet!}{{{}\mbfh}_\bullet\, \mbfh!\, A(\mbfh)}\prod_{\mbfa\in \mathsf{supp}C} \frac{\binom{{{}\mbfa}_\bullet}{\mbfa}^{C(\mbfa)}}{{{}\mbfa}_\bullet^{C(\mbfa)}C(\mbfa)!} \\ \label{eq:t:Moments:5} =&\ \frac{M_{\mbfn}(C)}{A(\mbfh)} \frac{({{}\mbfh}_\bullet-1)!}{\mbfh!} \frac{\mbfn!}{(\mbfn-\mbfh)!} \intertext{and} \label{eq:t:Moments:6} J(\mbfS;\boldsymbol\alpha;A)=&\ J(\mbfS;\boldsymbol\alpha;C) \, \big({\mbfs_1^{\diamond h_1}\diamond\cdots \diamond\mbfs_q^{\diamond h_q}}\big)\cdot\boldsymbol\alpha\,\mathrm{.}\end{aligned}$$ Substituting [\[eq:t:Moments:5\]](#eq:t:Moments:5){reference-type="eqref" reference="eq:t:Moments:5"} and [\[eq:t:Moments:6\]](#eq:t:Moments:6){reference-type="eqref" reference="eq:t:Moments:6"} in [\[eq:t:Moments:4\]](#eq:t:Moments:4){reference-type="eqref" reference="eq:t:Moments:4"} above, and simplifying $\mbfn!$, $$\begin{aligned} \partial_p^b\, \tilde\zeta_\mbfn[\mbfS;\boldsymbol\alpha]=&\ \alpha_p \sum_{\mbfe_b\leq_\diamond\mbfh\leq_\diamond\mbfn} \sum_{C\vDash \mbfn-\mbfh} \frac{M_{\mbfn}(C)}{(\mbfn-\mbfh)!} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_b)!}\ \mbfS_p^{\mbfh-\mbfe_b} \,J(\mbfS;\boldsymbol\alpha;C) \\ =&\ \alpha_p \sum_{\mbfe_b\leq_\diamond\mbfh\leq_\diamond\mbfn} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_b)!}\ \mbfS_p^{\mbfh-\mbfe_b} \,\tilde\zeta_{\mbfn-\mbfh}[\mbfS;\boldsymbol\alpha] \,\mathrm{.}\end{aligned}$$ Combining the inductive hypothesis with [\[eq:t:Moments:1\]](#eq:t:Moments:1){reference-type="eqref" reference="eq:t:Moments:1"} and [\[eq:t:Moments:3\]](#eq:t:Moments:3){reference-type="eqref" reference="eq:t:Moments:3"} with $a=\ell$ and $b=j$, $$\begin{aligned} \partial_p^b\, \tilde\zeta_\mbfn[\mbfS;\boldsymbol\alpha]=&\ \alpha_p \sum_{\mbfe_b\leq_\diamond\mbfh\leq_\diamond\mbfn} \frac{({{}\mbfh}_\bullet-1)!}{(\mbfh-\mbfe_b)!}\ \mbfS_p^{\mbfh-\mbfe_b}\, \tilde\nu_{\mbfn-\mbfh}[\mbfS;\boldsymbol\alpha] = \partial_p^b\, \tilde\nu_\mbfn[\mbfS;\boldsymbol\alpha] \,\mathrm{.}\end{aligned}$$ By arbitrariness of $p$ and $b$ we conclude that $\tilde\zeta_\mbfn[\mbfS;\boldsymbol\alpha]-\tilde\nu_\mbfn[\mbfS;\boldsymbol\alpha]$ is constant as a function of $\mbfS$, hence vanishing by choosing $\mbfS=\mathop{\mathrm{{\mathbf 0}}}$. ◻ # A Polychromatic ESF {#s:Polychromatic} Let $r$ be the number of cycles of a random permutation $\pi\in\mfS_{{{}\mbfn}_\bullet}$. Assume that $\pi$ is chosen with a probability proportional to $\theta^r$ for some $\theta>0$. Then, the probability that $\pi$ has cycle structure $\boldsymbol\lambda\vdash n$ is precisely the Ewens distribution $E_{\theta}(\boldsymbol\lambda)$. We provide a generalization of this statement to the case of colored permutations, with coloring and cycle structure indexed by a $q$-colored partition. Let $$\begin{aligned} \label{eq:MultiSets} \mcA_n\mathop{\mathrm{\coloneqq}}\bigcup_{\mbfn\in {\mathbb N}^q_*: {{}\mbfn}_\bullet=n} \mcA_\mbfn\end{aligned}$$ be the family of all multisets $A$ on ${\mathbb N}^q_*$ with $\mathsf{shape}(A)\vdash n$. **Definition 25** (Polychromatic ESF). Fix $n,q\in{\mathbb N}_1$, $\theta>0$, and $\mbfp\in \Delta^{q-1}$. The *polychromatic ESF* $E_{\theta,\mbfp}^n$ is the probability distribution on $\mcA_n$ given by $$\label{d:eq:PolyESF} \begin{aligned} E_{\theta,\mbfp}^n(A)\mathop{\mathrm{\coloneqq}}% \frac{n!}{\left\langle\theta\right\rangle_{n}}\, \theta^{{\mathsf{card}(A)}}\, \frac{\mbfp^{\mathsf{col}(A)}}{\mathsf{col}(A)!}M_{\mathsf{col}(A)}(A) \end{aligned} \,\mathrm{,}\;\,\qquad A\in \mcA_n \,\mathrm{.}$$ *Proof.* Let us verify that $E_{\theta,\mbfp}^n$ is indeed a probability distribution on $\mcA_n$. For fixed $k>n$ set $\mbfs_j\mathop{\mathrm{\coloneqq}}p_j {\mathbf 1}^{\scriptscriptstyle{(k)}}$, $j\in [q]$, and $\boldsymbol\alpha\mathop{\mathrm{\coloneqq}}(\theta/k){\mathbf 1}^{\scriptscriptstyle{(k)}}$. Respectively by: the Multinomial Theorem, Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"}, and the definition [\[eq:ZDefinition\]](#eq:ZDefinition){reference-type="eqref" reference="eq:ZDefinition"} of $Z_\mbfn$, $$\begin{aligned} 1=& \sum_{\mbfn\in {\mathbb N}_0^q : {{}\mbfn}_\bullet=n} \binom{n}{\mbfn}\mbfp^\mbfn = \sum_{\mbfn\in {\mathbb N}_0^q : {{}\mbfn}_\bullet=n} \binom{n}{\mbfn} \int_{\Delta^{k-1}} \prod_j^q (\mbfs_j\cdot \mbfx)^{n_j} \mathop{}\!\mathrm{d}D_{\boldsymbol\alpha}(\mbfx) \\ =& \sum_{\mbfn\in {\mathbb N}_0^q : {{}\mbfn}_\bullet=n} \binom{n}{\mbfn} \frac{\mbfn!}{\left\langle\theta\right\rangle_{n}}\frac{1}{\mbfn!} \sum_{A\vDash\mbfn} M_{\mbfn}(A) \prod_{\mbfa\in\mathsf{supp}A} (\theta\, \mbfp^\mbfa)^{A(\mbfa)} \\ =& \sum_{A\in\mcA_n} \frac{\theta^{{\mathsf{card}(A)}}}{\left\langle\theta\right\rangle_{n}}\binom{n}{\mathsf{col}(A)} \mbfp^{\mathsf{col}(A)} M_{\mathsf{col}(A)}(A) \,\mathrm{.}\qedhere\end{aligned}$$ ◻ *Remark 26* ($q=1$). When $q=1$, we have $\mbfp=p=1$ and $\mathsf{col}(A)=n$ for every $A\in\mcA_n$, thus [\[d:eq:PolyESF\]](#d:eq:PolyESF){reference-type="eqref" reference="d:eq:PolyESF"} reduces to the standard ESF by Remark [Remark 4](#r:UnidimensionalFirst){reference-type="ref" reference="r:UnidimensionalFirst"}. **Lemma 27** (Conditioning). *Fix $\mbfn\in {\mathbb N}_0^q$ with ${{}\mbfn}_\bullet=n$. Then, the conditional probability $E_{\theta,\mbfp}^n\left[{\,\cdot\,}| \mathsf{col}({\,\cdot\,})=\mbfn\right]$ satisfies $$\label{l:Conditioning:0} E_{\theta,\mbfp}^n\left[{\,\cdot\,}| \mathsf{col}({\,\cdot\,})=\mbfn\right]= \frac{\theta^{{\mathsf{card}(A)}}}{\left\langle\theta\right\rangle_{n}} M_{\mbfn}(A)\,\mathrm{,}\;\,\qquad A\in\mcA_\mbfn \,\mathrm{.}$$* **Proof.* For fixed $k>n$ set $\mbfs_1=\cdots= \mbfs_q\mathop{\mathrm{\coloneqq}}{\mathbf 1}^{\scriptscriptstyle{(k)}}$, and $\boldsymbol\alpha\mathop{\mathrm{\coloneqq}}(\theta/k){\mathbf 1}^{\scriptscriptstyle{(k)}}$. By Theorem [Theorem 16](#t:Moments){reference-type="ref" reference="t:Moments"} and by the definition [\[eq:ZDefinition\]](#eq:ZDefinition){reference-type="eqref" reference="eq:ZDefinition"} of $Z_\mbfn$, $$\begin{aligned} 1=\int_{\Delta^{k-1}} \prod_j^q(\mbfs_j\cdot \mbfx)^{n_j} \mathop{}\!\mathrm{d}D_{\boldsymbol\alpha}= \frac{\mbfn!}{\left\langle\theta\right\rangle_{n}} \frac{1}{\mbfn!} \sum_{A\vDash\mbfn} M_{\mbfn}(A) \prod_{\mbfa\in\mathsf{supp}A} \theta^{A(\mbfa)}\,\mathrm{,}\;\,\end{aligned}$$ hence $$\label{eq:l:Conditioning:1} \sum_{A\vDash\mbfn} \theta^{{\mathsf{card}(A)}}M_{\mbfn}(A)=\left\langle\theta\right\rangle_{n}\,\mathrm{.}$$* *Now, $$\label{eq:l:Conditioning:2} E_{\theta,\mbfp}^n\left[A | \mathsf{col}(A)=\mbfn\right] = \frac{E_{\theta,\mbfp}^n(A)}{E_{\theta,\mbfp}^\mbfn\left[\mathsf{col}({\,\cdot\,})=\mbfn\right]} \quad \text{if} \quad \mathsf{col}(A)=\mbfn$$ and $0$ otherwise. Furthermore, $$\label{eq:l:Conditioning:3} \begin{aligned} E_{\theta,\mbfp}^n\left[\mathsf{col}({\,\cdot\,})=\mbfn\right] =& \sum_{A\vDash\mbfn} E_{\theta,\mbfp}^\mbfn(A) = \sum_{A\vDash\mbfn} \frac{n!}{\left\langle\theta\right\rangle_{n}}\, \theta^{{\mathsf{card}(A)}}\, \frac{\mbfp^{\mathsf{col}(A)}}{\mathsf{col}(A)!}M_{\mathsf{col}(A)}(A) \\ =& \frac{n!}{\left\langle\theta\right\rangle_{n}} \frac{\mbfp^\mbfn}{\mbfn!} \sum_{A\vDash \mbfn} \theta^{{\mathsf{card}(A)}}M_{\mathsf{col}(A)}(A)= n! \frac{\mbfp^\mbfn}{\mbfn!} \end{aligned}$$ by [\[eq:l:Conditioning:1\]](#eq:l:Conditioning:1){reference-type="eqref" reference="eq:l:Conditioning:1"}. Combining [\[eq:l:Conditioning:2\]](#eq:l:Conditioning:2){reference-type="eqref" reference="eq:l:Conditioning:2"}, [\[eq:l:Conditioning:3\]](#eq:l:Conditioning:3){reference-type="eqref" reference="eq:l:Conditioning:3"}, and [\[d:eq:PolyESF\]](#d:eq:PolyESF){reference-type="eqref" reference="d:eq:PolyESF"} thus yields $$E_{\theta,\mbfp}^n\left[A | \mathsf{col}(A)=\mbfn\right]= \frac{\theta^{{\mathsf{card}(A)}}}{\left\langle\theta\right\rangle_{n}} M_{\mbfn}(A) \quad \text{if} \quad \mathsf{col}(A)=\mbfn$$ and $0$ otherwise. ◻* Since $E_{\theta,\mbfp}^n\left[{\,\cdot\,}| \mathsf{col}({\,\cdot\,})=\mbfn\right]$ does not depend on $\mbfp$, let us set $$E_{\theta}^\mbfn \mathop{\mathrm{\coloneqq}}E_{\theta,\mbfp}^n\left[{\,\cdot\,}| \mathsf{col}({\,\cdot\,})=\mbfn\right] \quad \text{on} \quad \mcA_\mbfn\,\mathrm{.}$$ In analogy with the standard ESF, the conditional probability $E_{\theta}^\mbfn$ counts $\theta$-biased $q$-colored permutations, as we now show. **Proposition 28**. *Fix $\theta>0$ and let $\pi\in\mfS_{{{}\mbfn}_\bullet}$ be a $\theta$-biased random permutation. Then, $$\label{eq:PolyEwens} \mbfP\big[\Pi(\pi)=A\big]= E_{\theta}^\mbfn(A)\,\mathrm{,}\;\,\qquad A\in\mcA_\mbfn\,\mathrm{.}$$* **Proof.* Let $r$ be the number of cycles of $\pi$ including fixed points. Since $\pi$ is $\theta$-biased and applying Proposition [Proposition 13](#p:Quotient){reference-type="ref" reference="p:Quotient"}, we have $$\begin{aligned} \mbfP\big[\Pi(\pi)=A\big]= C_\theta\, \theta^r \left\lvert\Pi^{-1}(A)\right\rvert = C_\theta\, \theta^r M_{\mathsf{col}(A)}(A) \,\mathrm{.}\end{aligned}$$ The conclusion follows since $E_{\theta}^\mbfn$ is a probability measure by Lemma [Lemma 27](#l:Conditioning){reference-type="ref" reference="l:Conditioning"}. ◻* *Remark 29*. We can rephrase Proposition [Proposition 28](#p:PolyEwens){reference-type="ref" reference="p:PolyEwens"} by saying that $E_{\theta}^\mbfn$ is the push-forward via $\Pi$ of the law $\mbfP$ of a $\theta$-biased random permutation in $\mfS_{{{}\mbfn}_\bullet}$. Furthermore, as a consequence of Lemma [Lemma 27](#l:Conditioning){reference-type="ref" reference="l:Conditioning"} and Corollary [Corollary 18](#c:Multi){reference-type="ref" reference="c:Multi"}, we see that $$E_{\theta}(\boldsymbol\lambda)=\sum_{\substack{A\vDash\mbfn\\ \mathsf{shape}{A}=\boldsymbol\lambda}} E_{\theta}^\mbfn(A) \,\mathrm{,}\;\,\qquad \boldsymbol\lambda\vdash n\,\mathrm{.}$$ That is, $E_{\theta}$ is the push-forward of $E_{\theta}^\mbfn$ via the function $\mathsf{shape}$. In this sense, the newly defined measure $E_{\theta}^\mbfn$ can be seen as 'intermediate' between $\mbfP$ and $E_{\theta}$. Finally, let us collect here the main properties of $E_{\theta,\mbfp}^n$ with respect to manipulations of $\mbfp$. For each set partition $\mbfL\mathop{\mathrm{\coloneqq}}\left\{L_1,\dotsc, L_r\right\}\vdash [q]$ denote by $s_\mbfL\colon [q]\to [r]$ the $\mbfL$-*degeneracy map* defined by $s_{\mbfL}^{-1}(k)=L_k$ for $k\in [r]$. Further let $\mbfS_\mbfL\in\left\{0,1\right\}^{r\times q}$ be the matrix $[\mbfS_\mbfL]_i^j\mathop{\mathrm{\coloneqq}}\mathop{\mathrm{\mathbf 1}}_{j\in s_\mbfL^{-1}(i)}$ and note that $\mbfS_\mbfL\colon {\mathbb N}^q_*\to {\mathbb N}^r_*$ and $\mbfS_\mbfL\colon \Delta^{q-1}\to\Delta^{r-1}$. Arguing similarly as in the proof of Definition [Definition 25](#d:PolyESF){reference-type="ref" reference="d:PolyESF"}, choosing $\mbfs_j=\mbfs_{j'}$ in [\[eq:t:Moments:0\]](#eq:t:Moments:0){reference-type="eqref" reference="eq:t:Moments:0"} whenever $j,j'\in L_i$ for some $i$, we have the following. **Proposition 30** (Aggregation). *Let $n,q\in{\mathbb N}_1$, $\theta>0$, and $\mbfp\in\Delta^{q-1}$. Then, cf. [\[eq:MultiPush\]](#eq:MultiPush){reference-type="eqref" reference="eq:MultiPush"}, $${(\mbfS_\mbfL)_*}_\sharp E_{\theta,\mbfp}^n= E_{\theta,\mbfS_\mbfL\mbfp}^n \,\mathrm{,}\;\,\qquad \mbfL\vdash [q] \,\mathrm{.}$$* ## A Hoppe-type urn model {#ss:Hoppe} In [@Hop84], F. M. Hoppe showed that the ESF $E_{\theta}$ is the marginal distribution of a discrete-time Markov process $\left(\Pi_t\right)_t$ of integer partitions $\Pi_t\vdash t$ obtained from the sampling process $\left(X_t\right)_t$ of what is now known as *Hoppe's urn model*. We adapt his construction to a similar urn model, resulting in a Markov process with values in the space of colored integer partitions and with marginal distribution $E_{\theta,\mbfp}^t$ at time $t$. Denote by $\Cat_\mbfp$ the categorical distribution on $[q]$ with parameters $\mbfp\in\Delta^{q-1}$. Consider a process $Y_\circ\mathop{\mathrm{\coloneqq}}\left(Y_t\right)_t$ generated by sampling from an urn containing one cube and various numbers of labelled colored balls. At time $0$, the urn contains only the cube. At every (integer) time $t$, the labels are consecutive and ranging in ${\mathbb N}_1$, while the colors range in $[q]$. The cube has mass $\theta$ and every ball has mass $1$. At time $t$, an object in the urn is selected at random with a probability proportional to its mass. If it is a ball, it is returned together with one additional ball of the same label and of a color chosen according to $\Cat_\mbfp$ independently of the label. If it is the cube, it is returned together with a ball with the smallest label previously not present in the urn and of a color chosen according to $\Cat_\mbfp$. We define random variables $r_t\in {\mathbb N}_1$ and $Y_t\in {\mathbb N}_1\times [q]$ as the number of distinct labels (i.e. the maximal label) present in the urn, and the label and color of the additional ball returned after the $t^\text{th}$ drawing. Observe that, for every $T\in {\mathbb N}_1$, the process $Y_\circ$ defines a random $q$-colored partition $\msA_T$ by letting $$\begin{gathered} \label{eq:YtoA} \mbfa_T(i)\mathop{\mathrm{\coloneqq}}\left(a_{T,1}(i),\dotsc, a_{T,q}(i)\right)\,\mathrm{,}\;\,a_{T,j}(i)\mathop{\mathrm{\coloneqq}}\left\lvert\left\{t\in [T]: Y_t=(i,j)\right\}\right\rvert\,\mathrm{,}\;\, \qquad \msA_T\mathop{\mathrm{\coloneqq}}\sum_i^{r_T}\mathop{\mathrm{\mathbf 1}}_{\mbfa_T(i)} \,\mathrm{.}\end{gathered}$$ As a consequence, in the notation of [@Hop84], the first component $Y_{t,1}$ of $Y_t$ satisfies $Y_{t,1}=X_t$, while $\mathsf{shape}(\msA_T)$ coincides with $\Pi_T$. We call the Markov process $Y_\circ$ the *polychromatic Hoppe urn* (PHU), and the process $\msA_\circ\mathop{\mathrm{\coloneqq}}\left(\msA_T\right)_T$ the *PHU-partition process*. **Proposition 31**. *$\msA_\circ$ is a Markov process with marginal distribution $$\label{eq:p:Hoppe:0} \mbfP[\msA_T=A]=E_{\theta,\mbfp}^T(A)\,\mathrm{,}\;\,\qquad A\in\mcA_T \,\mathrm{.}$$* *Proof.* The Markov property is trivially satisfied. With the notation of [\[eq:YtoA\]](#eq:YtoA){reference-type="eqref" reference="eq:YtoA"}, the random variables $\left({{}\mbfa_T(i)}_\bullet\right)_i$ are $\left(Y_{t,1}\right)_{t \leq T}$-measurable. In order to compute the marginal distribution at time $T$, fix $A\in\mcA_T$, and set $\boldsymbol\lambda\mathop{\mathrm{\coloneqq}}\mathsf{shape}(A)$ and $r\mathop{\mathrm{\coloneqq}}{{}\boldsymbol\lambda}_\bullet$. We introduce two families of functions: $$\begin{gathered} \mcF \coloneqq \big\{ \mbff : [r] \to \mathsf{supp}(A) \, \colon \, \left\lvert\mbff^{-1}(\mbfa)\right\rvert=A(\mbfa)\,\mathrm{,}\;\,\quad \mbfa \in \mathsf{supp}(A) \big\} \,\mathrm{,}\;\, \\ \mcG \coloneqq \big\{ g = {{}({\,\cdot\,})}_\bullet \circ \mbff = {{}\mbff({\,\cdot\,})}_\bullet\, \colon \, \mbff \in \mcF\big\} \,\mathrm{.}\end{gathered}$$ Since the colors $Y_{t,2}$ are chosen independently of one another and of the labels $Y_{t,1}$, $$\begin{aligned} \mbfP&\big[\msA_T = A \big| \left(Y_{t,1}\right)_{t\leq T}\big]= \\ &= \sum_{\mbff \in \mcF} \mbfP\big[\mbff({\,\cdot\,}) = \mbfa_T({\,\cdot\,}) \big| \left(Y_{t,1}\right)_{t\leq T}\big] = \sum_{\mbff \in \mcF} \prod_{i=1}^r \mbfP\big[\mbff(i) = \mbfa_T(i) \big | \left(Y_{t,1}\right)_{t\leq T}\big] \\ &= \sum_{\mbff \in \mcF} \prod_{i=1}^r \binom{{{}\mbff(i)}_\bullet}{\mbff(i)} \mbfp^{\mbff(i)} \mathop{\mathrm{\mathbf 1}}_{\left\{{{}\mbff(i)}_\bullet = {{}\mbfa_T(i)}_\bullet\right\}} =\sum_{\mbff \in \mcF} \mbfp^{\mathsf{col}(A)} \mathop{\mathrm{\mathbf 1}}_{\left\{{{}\mbff({\,\cdot\,})}_\bullet = {{}\mbfa_T({\,\cdot\,})}_\bullet\right\}} \prod_{\mbfa \in \mathsf{supp}(A)} \binom{{{}\mbfa}_\bullet}{\mbfa}^{A(\mbfa)} \\ &= \left\lvert\left\{\mbff \in \mcF \, \colon \, {{}\mbff({\,\cdot\,})}_\bullet = {{}\mbfa_T({\,\cdot\,})}_\bullet \right\}\right\rvert \mbfp^{\mathsf{col}(A)} \prod_{\mbfa \in \mathsf{supp}(A)} \binom{{{}\mbfa}_\bullet}{\mbfa}^{A(\mbfa)} \,\mathrm{.}\end{aligned}$$ It can be easily checked that for every $g \in \mcG$ the following identities hold: $$\begin{aligned} \left\lvert\left\{\mbff \in \mcF: g = {{}({\,\cdot\,})}_\bullet \circ \mbff\right\}\right\rvert = \prod_i \binom{\lambda_i}{\left(A(\mbfa)\right)_{\mbfa \in \mathsf{supp}(A)\, : \, {{}\mbfa}_\bullet=i}} = \frac{\boldsymbol\lambda!}{\displaystyle\prod_{\mbfa \in \mathsf{supp}(A)} A(\mbfa)!} \,\mathrm{.}\end{aligned}$$ Thus, $$\begin{aligned} \nonumber \mbfP\big[\msA_T = A \big | \left(Y_{t,1}\right)_{t\leq T}\big] &= \left\lvert\left\{g \in \mcG : g({\,\cdot\,}) = {{}\mbfa_T({\,\cdot\,})}_\bullet\right\}\right\rvert \boldsymbol\lambda!\, \mbfp^{\mathsf{col}(A)} \prod_{\mbfa \in \mathsf{supp}(A)} \frac{\binom{{{}\mbfa}_\bullet}{\mbfa}^{A(\mbfa)}}{A(\mbfa)!} \\ \label{eq:p:Hoppe:05} &=\mathop{\mathrm{\mathbf 1}}_{\left\{\mathsf{shape}(\msA_T)=\boldsymbol\lambda\right\}}\boldsymbol\lambda!\, \mbfp^{\mathsf{col}(A)} \prod_{\mbfa \in \mathsf{supp}(A)} \frac{\binom{{{}\mbfa}_\bullet}{\mbfa}^{A(\mbfa)}}{A(\mbfa)!} \,\mathrm{.}\end{aligned}$$ Taking the expectation over $\left(Y_t\right)_{t\leq T}$ on both sides of [\[eq:p:Hoppe:05\]](#eq:p:Hoppe:05){reference-type="eqref" reference="eq:p:Hoppe:05"}, we infer that $$\label{eq:p:Hoppe:1} \mbfP[\msA_T = A] = \mbfP[\mathsf{shape}(\msA_T)=\boldsymbol\lambda] \, \boldsymbol\lambda!\, \mbfp^{\mathsf{col}(A)} \prod_{\mbfa \in \mathsf{supp}(A)} \frac{\binom{{{}\mbfa}_\bullet}{\mbfa}^{A(\mbfa)}}{A(\mbfa)!} \,\mathrm{.}$$ By the formula for the marginal distribution of Hoppe's urn model, [@Hop84 Eqn. (1)], $$\label{eq:p:Hoppe:2} \mbfP[\mathsf{shape}(\msA_T)=\boldsymbol\lambda]=\frac{T!}{\left\langle\theta\right\rangle_{T}}\prod_{i=1}^T \frac{\theta^{\lambda_i}}{i^{\lambda_i}\lambda_i!}= \frac{T!}{\boldsymbol\lambda!} \frac{\theta^r}{\left\langle\theta\right\rangle_{T}} \prod_{\mbfa\in\mathsf{supp}A} \frac{1}{{{}\mbfa}_\bullet^{A(\mbfa)}} \,\mathrm{.}$$ Combining [\[eq:p:Hoppe:1\]](#eq:p:Hoppe:1){reference-type="eqref" reference="eq:p:Hoppe:1"} and [\[eq:p:Hoppe:2\]](#eq:p:Hoppe:2){reference-type="eqref" reference="eq:p:Hoppe:2"}, the identity [\[eq:p:Hoppe:0\]](#eq:p:Hoppe:0){reference-type="eqref" reference="eq:p:Hoppe:0"} follows. ◻ ## Consistency {#ss:Consistency} In [@Kin78b; @Kin78], J.F.C. Kingman introduced a celebrated notion of *consistency* for stochastic processes on partitions, and showed that a sequence of random partitions $\left(\boldsymbol\lambda_n\right)_n$ with $\boldsymbol\lambda_n\vdash n$ distributed according to $E_{\theta}$, satisfies this notion. Precisely, if $n$ objects are partitioned into classes with sizes given by $\boldsymbol\lambda_n$, and one object is deleted uniformly at random, independently of $\boldsymbol\lambda_n$, the partition of the $n-1$ remaining objects has class sizes distributed as $\boldsymbol\lambda_{n-1}$, cf. e.g. [@Pit95 p. 146]. In this section, we show that the polychromatic ESF satisfies a similar consistency property. Denote by $\mcA\mathop{\mathrm{\coloneqq}}\bigcup_n \mcA_n$ the family of all finite multisets on ${\mathbb N}^q_*$, and set $$A_{\setminus \mbfa,j}\mathop{\mathrm{\coloneqq}}\begin{cases} A-\mathop{\mathrm{\mathbf 1}}_\mbfa & \text{if } \mbfa=\mbfe_j \,\mathrm{,}\;\, \\ A-\mathop{\mathrm{\mathbf 1}}_\mbfa +\mathop{\mathrm{\mathbf 1}}_{\mbfa-\mbfe_j} & \text{otherwise} \end{cases}\,\mathrm{,}\;\,\qquad \mbfa\in \mathsf{supp}A\,\mathrm{,}\;\,j\in [q] \,\mathrm{.}$$ Following [@Kin78], we define a system $S=S_{nm}$, $n\in {\mathbb N}_1$, $m\leq n$, of probability kernels on $\mcA$. Firstly, set [\[s:Kernels\]]{#s:Kernels label="s:Kernels"} $$\begin{aligned} S(A,B)\mathop{\mathrm{\coloneqq}}&\ \mathop{\mathrm{\mathbf 1}}_{A=B}\,\mathrm{,}\;\,& A,B&\in\mcA_n \,\mathrm{,}\;\, \\ S(A,B)\mathop{\mathrm{\coloneqq}}&\ \begin{cases} \frac{a_j A(\mbfa)}{n} & \text{if } B=A_{\setminus \mbfa,j} % \,\mathrm{,}\;\, \\ 0 & \text{otherwise} \end{cases} \,\mathrm{,}\;\,& A\in\mcA_n\,\mathrm{,}\;\,& B\in\mcA_{n-1}\,\mathrm{,}\;\,\end{aligned}$$ and note that $S(A,{\,\cdot\,})$ is a probability on $\mcA_{n-1}$ for every $A\in\mcA_n$. Secondly, let $S$ be the unique system of kernels extending [\[s:Kernels\]](#s:Kernels){reference-type="eqref" reference="s:Kernels"} and satisfying the cocycle relation $$\label{eq:Cocycle} S(A,C)= \sum_{B\in\mcA_m} S(A,B)\, S(B,C)\,\mathrm{,}\;\,\qquad A\in\mcA_n\,\mathrm{,}\;\,C\in\mcA_\ell\,\mathrm{,}\;\,\quad \ell < m < n\,\mathrm{.}$$ Note that $S_{nm}(A,{\,\cdot\,})$ is a probability on $\mcA_m$ for every $m$ and every $A\in\mcA_n$, since it is so for $m=n-1$ as noted above, and in light of [\[eq:Cocycle\]](#eq:Cocycle){reference-type="eqref" reference="eq:Cocycle"}. *Remark 32*. Analogously to the case of usual integer partitions, the system $S$ may be interpreted as the selection of a random sampling (uniform, without replacement) of $m$ elements from a given $q$-colored partition $A\in\mcA_n$, resulting in the $q$-colored partition $B\in\mcA_m$. The cocycle relation [\[eq:Cocycle\]](#eq:Cocycle){reference-type="eqref" reference="eq:Cocycle"} is then a consequence of the consistency of random sub-sampling. Let us now turn to probability measures on $\mcA$. For $n\in{\mathbb N}_1$ let $\msP(\mcA_n)$ be the set of all probability measures on $\mcA_n$. Define a system $\sigma$ of maps $\sigma_{nm}\colon \msP(\mcA_n)\to\msP(\mcA_m)$ by $$\big({\sigma_{nm} \mbfP}\big)(B) \longmapsto \mbfP[S({\,\cdot\,},B)] \,\mathrm{,}\;\,$$ and note that $\sigma$ satisfies the cocycle relation $$\label{eq:CocycleSigma} \sigma_{n\ell}=\sigma_{m\ell}\circ\sigma_{nm} \,\mathrm{,}\;\,\qquad \ell<m<n\,\mathrm{.}$$ **Definition 33** (Consistency). We say that a family $\left(\mbfP_n\right)_n$ of probability measures $\mbfP_n$ on $\mcA_n$ is *consistent* (w.r.t. the system $\sigma$) if $\mbfP_m=\sigma_{nm}\mbfP_n$ for every $m\leq n$. **Theorem 34**. *For every $\theta>0$ and $\mbfp\in\Delta^{q-1}$ the family ${\big(E_{\theta,\mbfp}^n\big)}_n$ is consistent.* **Proof.* In light of [\[eq:CocycleSigma\]](#eq:CocycleSigma){reference-type="eqref" reference="eq:CocycleSigma"}, it suffices to verify that $\sigma_{nm} E_{\theta,\mbfp}^n= E_{\theta,\mbfp}^{m}$ for $m= n-1$ and for every $n$. To this end, let $\mbfQ$ be the law of the PHU partition $\msA_\circ$ on its path space. By Bayes formula, and Proposition [Proposition 31](#p:Hoppe){reference-type="ref" reference="p:Hoppe"}, $$\begin{aligned} \nonumber \mbfQ[\msA_{n-1}=B\mid \msA_n=A] =&\ \frac{\mbfQ[\msA_n=A\mid \msA_{n-1}=B]\, \mbfQ[\msA_{n-1}=B]}{\mbfQ[\msA_n=A]} \\ \label{eq:t:Consistency:2} =&\ \frac{\mbfQ[\msA_n=A\mid \msA_{n-1}=B]\, E_{\theta,\mbfp}^{n-1}(B)}{E_{\theta,\mbfp}^n(A)} \,\mathrm{.}\end{aligned}$$ Furthermore, it follows from the definition of $\msA_\bullet$ that $$\label{eq:t:Consistency:3} \begin{aligned} \mbfQ[\msA_n=A\mid \msA_{n-1}=B]=& \sum_{\mbfa\in\mathsf{supp}A} \sum_{\substack{j\in[q]:\\ \mbfe_j\leq_\diamond\mbfa, \mbfe_j\neq \mbfa}} \mathop{\mathrm{\mathbf 1}}_{A=B+\mathop{\mathrm{\mathbf 1}}_\mbfa-\mathop{\mathrm{\mathbf 1}}_{\mbfa-\mbfe_j}} \frac{{{}\mbfa}_\bullet-1}{\theta+n-1} B(\mbfa-\mbfe_j) \, p_j \\ &+\sum_{j=1}^q \mathop{\mathrm{\mathbf 1}}_{A=B+\mathop{\mathrm{\mathbf 1}}_{\mbfe_j}} \frac{\theta}{\theta+n-1}\, p_j \,\mathrm{.} \end{aligned}$$* *On the other hand, by definition [\[d:eq:PolyESF\]](#d:eq:PolyESF){reference-type="eqref" reference="d:eq:PolyESF"} of $E_{\theta,\mbfp}^n$, $$\begin{aligned} \label{eq:t:Consistency:4} \frac{E_{\theta,\mbfp}^{n-1}(B)}{E_{\theta,\mbfp}^n(A)}=\begin{cases} \displaystyle\frac{\theta+n-1}{np_j} \frac{a_j}{{{}\mbfa}_\bullet-1}\frac{A(\mbfa)}{A(\mbfa-\mbfe_j)+1} & \text{if } A=B+\mathop{\mathrm{\mathbf 1}}_\mbfa-\mathop{\mathrm{\mathbf 1}}_{\mbfe_j} \\ \displaystyle\frac{\theta+n-1}{\theta np_j} A(\mbfe_j) &\text{if } A=B+\mathop{\mathrm{\mathbf 1}}_{\mbfe_j} \end{cases}\end{aligned}$$* *Combining [\[eq:t:Consistency:2\]](#eq:t:Consistency:2){reference-type="eqref" reference="eq:t:Consistency:2"}--[\[eq:t:Consistency:4\]](#eq:t:Consistency:4){reference-type="eqref" reference="eq:t:Consistency:4"}, we thus have $$\begin{aligned} \mbfQ[\msA_{n-1}=B\mid \msA_n=A] =& \sum_{\mbfa\in\mathsf{supp}A} \sum_{j\in[q]:\mbfe_j<\mbfa} \mathop{\mathrm{\mathbf 1}}_{A=B+\mathop{\mathrm{\mathbf 1}}_\mbfa-\mathop{\mathrm{\mathbf 1}}_{\mbfa-\mbfe_j}} \frac{a_j A(\mbfa)}{n} +\sum_{j=1}^q \mathop{\mathrm{\mathbf 1}}_{A=B+\mathop{\mathrm{\mathbf 1}}_{\mbfe_j}} \frac{A(\mbfe_j)}{n} \\ =&\sum_{\mbfa\in\mathsf{supp}A}\sum_j^q \mathop{\mathrm{\mathbf 1}}_{A_{\setminus \mbfa,j}=B} \frac{a_j A(\mbfa)}{n} =S_{n\ n-1}(A,B) \,\mathrm{.}\end{aligned}$$* *Finally, respectively by: the definition of $\sigma$, the previous equality and Proposition [Proposition 31](#p:Hoppe){reference-type="ref" reference="p:Hoppe"}, the law of total probability, and again Proposition [Proposition 31](#p:Hoppe){reference-type="ref" reference="p:Hoppe"}, $$\begin{aligned} \big({\sigma_{n\ n-1}E_{\theta,\mbfp}^n}\big)(B)=& \sum_{A\in\mcA_n} S_{n\ n-1}(A,B)\, E_{\theta,\mbfp}^n(A) =\sum_{A\in\mcA_n} \mbfQ[\msA_{n-1}=B\mid \msA_n=A] \, \mbfQ[\msA_n=A] \\ =&\ \mbfQ[\msA_{n-1}=B] = E_{\theta,\mbfp}^{n-1}(B) \,\mathrm{.}\qedhere\end{aligned}$$ ◻* [^1]: In fact, Aldous credits the introduction of the CRP to J. Pitman, who in turn acknowledges the contribution of L. Dubins, see e.g. the attribution to Dubins and Pitman in [@Tav21 §4.1]. [^2]: A variation which may already be familiar to those non-Chinese speaking customers ordering blindly from the "secret menu".

arxiv_math

--- abstract: | The first goal of this article is to give a complete classification (up to Real biholomorphisms) of Real primary Hopf surfaces $(H,s)$, and, for any such pair, to describe in detail the following naturally associated objects : the group $\mathrm{Aut}_h(H,s)$ of Real automorphisms, the Real Picard group $(\mathrm{Pic}(H),\hat s^*)$, and the Picard group of Real holomorphic line bundles $\mathrm{Pic}_{\mathbb{R}}(H)$. Our second goal: the classification of Real primary Hopf surfaces up to equivariant diffeomorphisms, which will allow us to describe explicitly in each case the real locus $H(\mathbb{R})=H^s$ and the quotient $H/\langle s\rangle$. address: Aix Marseille Univ, CNRS, I2M, Marseille, France. author: - Zahraa KHALED title: Real structures on primary Hopf surfaces --- # Acknowledgements {#acknowledgements .unnumbered} I am indebted to my PhD advisor Andrei Teleman for suggesting me the problems treated in this article and for many helpful ideas. I also thank Karl Oeljeklaus and Anne Pichon for their advices and their interest in my results. # Introduction {#intro} Let $X$ be a complex manifold, and let $J$ be the (integrable) almost holomorphic structure on its underlying differentiable manifold $X$ defining its complex structure. We will denote by $\bar X$ the complex manifold defined by $-J$. Note that the data of an anti-holomorphic isomorphism $X\to X$ is equivalent to the data of a biholomorphism $X\to \bar X$. A Real structure (in the sense of Atiyah) on $X$ is an anti-holomorphic involution of $X$ [@At], [@S]. A Real complex manifold is a pair $(X,s)$ consisting of a complex manifold and a Real structure on it. The real locus of a Real complex manifold $(X,s)$ is just the fixed point locus $X^s$ (also denoted $X({\mathbb R})$ if $s$ has been fixed) of its Real structure. Let $(X,s)$, $(Y,\sigma)$ be Real complex manifolds. A biholomorphism $f:X\to Y$ is called Real (or compatible with the Real structures) if $\sigma\circ f=f\circ s$. The fundamental problem of the theory is the classification of Real complex manifolds up to Real biholomorphisms. The group of real biholomorphisms of a Real complex manifold $(X,s)$ is the subgroup $$\mathrm {Aut}(X,s)\coloneq \{f:X\to X|\ f \hbox{ biholomorphism, }f\circ s=s\circ f\}$$ of the biholomorphism group $\mathrm {Aut}_h(X)$. Let $(M,s)$ be a differentiable manifold endowed with an involution $s$ and $E$ be a complex vector bundle on $M$. We recall [@At] that a **Definition 1**. *A Real structure on $E$ is a fiberwise anti-linear $s$-covering isomorphic involution $\varphi:E\to E$. A Real bundle on $(M,s)$ is pair $(E,\phi)$ consisting of a complex bundle $E$ on $M$ and a Real structure $\phi$ on $E$.* Let $(X,s)$ be a Real complex manifold. **Definition 2**. *A Real holomorphic bundle on $X$ is a pair $(E,\phi)$, where $E$ is a holomorphic bundle on $X$ and $\phi$ an anti-holomorphic Real structure on $E$.* Let $E$ be a holomorphic bundle of rank $r$ on $X$. The pull-back $s^*(\bar E)$ has a natural structure of a holomorphic bundle on $X$ (see for instance [@OT section 1.2]): it is just the pull-back of $\bar E$, regarded as a holomorphic bundle on $\bar X$, via the holomorphic map $s:X\to \bar X$. The map $[E]\mapsto [s^*(\bar E)]$ defines a natural involution on the set of isomorphism classes of holomorphic bundles on $X$. In particular, for $r=1$, we obtain an involution $$\bar s^*:\mathrm {Pic}(X)\to \mathrm {Pic}(X),\ \bar s^*([L])\coloneq [s^*(\bar L)]$$ on the Picard group of $X$; this involution is an anti-holomorphic group isomorphism, so $(\mathrm {Pic}(X),\bar s^*)$ becomes a Real complex Lie group. The definitions above allow us to associate to a compact, connected Real complex manifold two natural invariants constructed using holomorphic line bundles: - The group $\mathrm {Pic}_{\mathbb R}(X)$ of isomorphism classes of Real holomorphic line bundles on $X$. - The Real complex Lie group $(\mathrm {Pic}(X),\bar s^*)$. Note that one has an obvious comparison real Lie group morphism $$\mathrm {Pic}_{\mathbb R}(X)\to \mathrm {Pic}(X)({\mathbb R}),\ [L,\phi]\mapsto [L],$$ which is always injective and is an isomorphism when $X({\mathbb R})\ne\emptyset$.\ The goals of this article are: 1. [\[G1\]]{#G1 label="G1"} To give complete classification of Real primary Hopf surfaces (up to Real biholomorphisms) with an explicit description of the set of isomorphism classes. 2. To describe explicitly, for any Real primary Hopf surface $(H,s)$, the following naturally associated objects: 1. its automorphism group $\mathrm {Aut}(H,s)\subset \mathrm {Aut}_h(H)$. 2. its Real Picard group $(\mathrm {Pic}(H),\bar s^*)$ of holomorphic line bundles, and its Picard group $\mathrm {Pic}_{\mathbb R}(H)$ of Real holomorphic line bundles. 3. To classify differential-topologically the Real primary Hopf surfaces, and, for any Real primary Hopf surface $(H,s)$, to describe explicitly the fixed point (real) locus $H^s$ and the quotient $H/\langle s\rangle$. For (G[\[G1\]](#G1){reference-type="ref" reference="G1"}), recall first that [@BHPV]: **Definition 3**. *A primary Hopf surface is a compact complex surface $H$ whose universal covering is biholomorphic to $W\coloneq {\mathbb C}^2\setminus\{0\}$, and whose fundamental group is isomorphic to ${\mathbb Z}$.* From this definition, it follows that any primary Hopf surface is biholomorphic to a quotient of the form $$H_f={{\hbox{}^{\displaystyle{W}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f\rangle}}} \ ,$$ where $\langle f\rangle$ is the cyclic group generated by a biholomorphism $f\in \mathrm {Aut}_h(W).$ By a fundamental theorem of Kodaira [@Ko1], it follows that any primary Hopf surface is biholomorphic to ${W}/{\langle f\rangle}$ where $f$ is a biholomorphism of the form $$f(z,w)=(\alpha z+\lambda w^n,\beta w)$$ where $$0<\vert\alpha\vert \leq\vert\beta\vert<1 , \ n\in {\mathbb N},\ \lambda(\alpha-\beta^n)=0.$$ If the coefficients of $f$ are real, the standard conjugation $c:W\to W$ will obviously descend to a Real structure on $H_f$. We will see that there exists interesting classes of Real primary Hopf surfaces which are not of this type. Moreover, there exists Real primary Hopf surfaces defined by holomorphic contractions $f$ whose coefficients are not real. Note first that Kodaira's theorem does not give a precise classification of primary Hopf surfaces, because it is not clear under which conditions the surfaces associated with two 4-tuples $(\alpha,\beta,\lambda,n)$, $(\alpha',\beta',\lambda',n')$ as above are biholomorphic. Following [@We] we introduce five classes of holomorphic contractions: $$\label{WehlerCLasses} \begin{split} {IV}\coloneq& \left\{f:W\to W|\ f\begin{pmatrix} z\\w \end{pmatrix} =\begin{pmatrix}\alpha z\\ \alpha w\end{pmatrix}\vline\ 0< |\alpha| <1 \right\}, \\ {III}\coloneq& \left\{f:W\to W|\ f\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix}\delta^r z\\ \delta w\end{pmatrix}\vline\ \ r\in {\mathbb N}_{\geq 2},\ 0<|\delta|<1 \right\}, \\ {II}_a\coloneq &\left\{f:W\to W|\ f\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix}\delta^r z+w^r\\ \delta w\end{pmatrix}\vline\ r\in {\mathbb N}_{\geq 2},\ 0<|\delta|<1 \right\}, \\ % {II}_b\coloneq &\left\{f:W\to W|\ f\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix}\alpha z+w\\ \alpha w\end{pmatrix}\vline\ \ 0<|\alpha|<1 \right\}, \\ % {II}_c\coloneq &\left\{f:W\to W|\ f\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix}\alpha z\\ \delta w\end{pmatrix}\vline\ \begin{array}{c} 0<|\alpha|<1\\ 0< |\delta|<1\end{array},\ \alpha\ne \delta^r\, \forall r\in{\mathbb N}\right\}. \end{split}$$ The map $$\label{maptoisotypes} IV\cup III\cup II_a\cup II_b\cup II_c\ni f\mapsto [H_f]$$ which assigns to a holomorphic contraction $f$ the biholomorphism class of the corresponding Hopf surface $H_f$ is surjective, but *not* injective. Indeed, the contractions $f$, $f'\in II_c$ associated with the pairs $(\alpha,\delta)$, $(\alpha',\delta')=(\delta,\alpha)$ are biholomorphic. Note that this exception to injectivity is not mentioned in [@We]. In fact in [@We] the class $II_c$ is defined by imposing the additional condition $|\alpha|<|\delta|$. Unfortunately with this restrictive definition of $II_c$ *one loses the surjectivity of the map* ([\[maptoisotypes\]](#maptoisotypes){reference-type="ref" reference="maptoisotypes"}), because biholomorphism types of Hopf surfaces associated with pairs $(\alpha,\delta)$ satisfying $$\label{subclIIc} 0<|\alpha|=|\delta|<1,\ \alpha\ne \delta^r\, \forall r\in{\mathbb N}$$ will not belong to its image. This remark is important for our purposes because precisely in the subclass of $II_c$ defined by ([\[subclIIc\]](#subclIIc){reference-type="ref" reference="subclIIc"}) -- the subclass which is omitted in [@We] -- we will find contractions $f$ for which $H_f$ admits Real structures although the coefficients of $f$ are *not* real. Our first step in the classification of Real structures on primary Hopf surfaces is to divide them in two classes: a Real structure $\phi$ on $H_f$ will be called *even* (*odd*) if it admits a lift $\hat \phi:W\to W$ with $\phi^2= \mathrm{id}_W$ (respectively $\phi^2=f$). The even (odd) Real structures are classified by Theorem [Theorem 16](#ClassEven){reference-type="ref" reference="ClassEven"} (respectively Theorem [Theorem 25](#ClassOdd){reference-type="ref" reference="ClassOdd"}). Concerning (G2) we will give explicit descriptions of the automorphism group $\mathrm {Aut}(X,s)$ of all Real primary Hopf surface. For instance, when $f\in IV$ with negative coefficient $\alpha$ we obtain $\mathrm {Aut}(H_f,{\mathfrak s}_f)\simeq\mathrm {Spin}^c(3)$, where ${\mathfrak s}_f$ denotes the canonical odd Real structure on $H_f$ (see Corollary [Corollary 30](#Spinc(3)){reference-type="ref" reference="Spinc(3)"}). Our results concerning the Real complex group $(\mathrm {Pic}(X),\bar s^*)$ and the group $\mathrm {Pic}_{\mathbb R}(X)$ of a Real primary Hopf surface $(X,s)$ are (see Proposition [Proposition 34](#Pic(X)R){reference-type="ref" reference="Pic(X)R"}): 1. $(\mathrm {Pic}(X),\bar s^*)$ is always isomorphic to $({\mathbb C}^*,\bar{\ } )$. 2. The canonical monomorphism $\mathrm {Pic}_{\mathbb R}(X)\to \mathrm {Pic}(X)({\mathbb R})={\mathbb R}^*$ is an isomorphism if $(X,s)$ is even and identifies $\mathrm {Pic}_{\mathbb R}(X)$ with ${\mathbb R}_{>0}$ if $(X,s)$ odd. Our results for the goal (G3) give a complete differential topological classification of Real primary Hopf surfaces (see Theorems [Theorem 35](#ClassDiffEven){reference-type="ref" reference="ClassDiffEven"}, [Theorem 45](#ClassDiffOdd){reference-type="ref" reference="ClassDiffOdd"} and Remark [Remark 46](#mu-mu0){reference-type="ref" reference="mu-mu0"}). The final result is: - Any even Real primary Hopf surface is equivariantly diffeomorphic to either $$\big(S^1\times S^3, (\zeta,(u,v))\mapsto (\zeta,(\bar u,\bar v))\big),$$ or $$\big(S^1\times S^3, (\zeta,(u,v))\mapsto (\zeta,(\bar u,\zeta\bar v))\big).$$ - Any odd Real primary Hopf surface is equivariantly diffeomorphic to $$\big (S^1\times S^3, (\zeta,Z)\mapsto (-\zeta,Z)\big).$$ Taking into account the results of section [5.3.1](#RealLocusSection){reference-type="ref" reference="RealLocusSection"}, this shows that the equivariant differential topological type of a Real primary Hopf surface is determined by the type (even or odd) and the orientability of the real locus. The main idea in the proof of this classification result is: for a contraction $f\in IV\cup III\cup II_a\cup II_b\cup II_c$ with Real coefficients and positive diagonal coefficients, we construct 1-parameter group of diffeomorphisms $(f^t)_{t\in{\mathbb R}}$ of $W$ acting freely on $W$ such that $f=f^1$. Moreover we also construct a compact 3-dimensional submanifold $\Sigma\subset W$ which is transversal to the orbits of this group and can be identified to $S^3$ via a diffeomeorphism which commutes with the conjugation and the involutions $(z,w)\mapsto (\pm z,\pm w)$. Finally will show that: - The real locus $X^s$ of an even Real primary Hopf surface $(X,s)$ is either a torus, or a Klein bottle, whereas the real locus of an odd Real primary Hopf surface is always empty. - The quotient $X/\langle s\rangle$ associated with a Real primary Hopf surface $(X,s)$ is always homeomorphic to $S^1\times S^3$, and we describe the position of the fixed point locus $X^s$ in this quotient. Note that, by the equivariant slice theorem, for any Real complex surface $(X,s)$, the quotient $X/\langle s\rangle$ is a topological 4-manifold. # Holomorphic and anti-holomorphic automorphisms {#sect1} A fundamental role in this article will be played by the results of Wehler on the classification of primary Hopf surfaces and their automorphism group. In this section we review these results and we continue with the classification of the anti-holomorphic automorphisms of primary Hopf surfaces. ## Wehler's classification of primary Hopf surfaces {#WehlClass} A precise classification of primary Hopf surfaces -- with explicit descriptions of the automorphism groups -- has been given by Wehler [@We]. His result can be formulated as follows: **Theorem 4**. *Consider the sets ${IV}$, ${III}$, ${II_a}$, ${II_b}$, ${II_c}\subset\mathrm {Aut}_h(W)$ defined in ([\[WehlerCLasses\]](#WehlerCLasses){reference-type="ref" reference="WehlerCLasses"}).* 1. *For every primary Hopf surface $H$ there exists $f\in {IV}\cup {III}\cup {II_a}\cup {II_b}\cup {II_c}$ such that $H\simeq H_f$.* 2. *[\[no-inj\]]{#no-inj label="no-inj"} For $f$, $f'\in {IV}\cup {III}\cup {II_a}\cup {II_b}\cup {II_c}$ we have $H_f\simeq H_{f'}$ if and only if either $f=f'$, or $f$ and $f'$ belong to $II_c$, and the corresponding coefficients $\alpha$, $\delta$, $\alpha'$, $\delta'$ satisfy $\alpha'=\delta$, $\delta'=\alpha$.* 3. *For any $f$ the group $\mathrm {Aut}_h(W)^f$ of holomorphic automorphisms of $W$ commuting with $f$ is given by the table below:* *$$\begin{array} {|c|c|} \hline & \\ [-0.8em] \hbox{The class of } f & \mathrm {Aut}_h(W)^f \\ [0.1em] \hline & \\ [-0.8em] IV & \mathrm {GL}(2,{\mathbb C}) \\ [0.2em] \hline & \\ [-0.8em] III & \left\{\begin{pmatrix} z\\w \end{pmatrix}\mapsto\ \begin{pmatrix}az+bw^r\\dw\end{pmatrix}\vline\ a\in{\mathbb C}^*,\ d\in{\mathbb C}^*, b\in {\mathbb C}\right\} \\ [0.8em] \hline & \\ [-0.8em] II_a &\left\{\begin{pmatrix} z\\w \end{pmatrix}\mapsto\ \begin{pmatrix}a^rz+bw^r\\aw\end{pmatrix}\vline\ a\in {\mathbb C}^*, b\in {\mathbb C}\right\} \\ [0.8em] \hline & \\ [-0.8em] II_b & \left\{\begin{pmatrix} z\\w \end{pmatrix}\mapsto\ \begin{pmatrix}az+bw\\aw\end{pmatrix}\vline\ a\in {\mathbb C}^*, b\in {\mathbb C}\right\} \\ [0.8em] \hline & \\ [-0.8em] II_c & \left\{\begin{pmatrix} z\\w \end{pmatrix}\mapsto\ \begin{pmatrix}az\\dw\end{pmatrix}\vline\ a\in{\mathbb C}^*,\ d\in{\mathbb C}^*\right\} \\ [0.8em] \hline \end{array}$$* 4. *In each case the cyclic group $\langle f\rangle$ is a central subgroup of $\mathrm {Aut}_h(W)^f$, and the automorphism group $\mathrm {Aut}_h(H_f)$ is identified with $\mathrm {Aut}_h(W)^f/\langle f\rangle$.* Therefore the name of the class gives the dimension of the automorphism group of the corresponding surface. **Remark 5**. *In [@We] the case $|\alpha|=|\delta|$ is omitted in the definition of $II_c$, so the exception ([\[no-inj\]](#no-inj){reference-type="ref" reference="no-inj"}) to the injectivity of the map $${IV}\cup {III}\cup {II_a}\cup {II_b}\cup {II_c}\ni f\mapsto [H_f]$$ is not mentioned either.* ## Anti-holomorphic automorphisms of primary Hopf surfaces We start with a general remark about topological automorphisms of Hopf surfaces: **Proposition 6**. *Let $H=H_f= {W}/{\langle f \rangle}$ be a primary Hopf surface and let $\pi:W\to\ H$ be the canonical map. Let $\sigma:H\to\ H$ be a homeomorphism. Then* 1. *There exists a homeomorphism $\hat{\sigma}:W\to\ W$ such that $\pi \circ \hat{\sigma}=\sigma \circ \pi$.* 2. *For any such homeomorphism $\hat{\sigma}$ we have $\hat{\sigma}\circ\ f \circ\ \hat{\sigma}^{-1}\in\{f,f^{-1}\}$.* *Proof.* $$\begin{tikzcd}[row sep=large, column sep=large] W\ar[d,"\pi"']\ar[dr, "\sigma\circ \pi" description]\ar[r, dashed, "\hat\sigma"] & W\ar[d,"\pi"]\\ H\ar[r,"\sigma"] & H \end{tikzcd}$$ \(1\) The composition $\sigma\circ \pi$ remains a covering and, since $W$ is simply connected, the uniqueness theorem of the universal covering, guarantees the existence of a homeomorphism $\hat{\sigma}:W\to\ W$ verifying the equality $$\pi\circ\hat{\sigma} = \sigma \circ \pi.$$ (2) The group $$\mathrm {Aut}_{H}(W)=\{g:W\to\ W|\ g \hbox{ homeomorphism},\ \pi\circ\ g=\pi\}$$ of topological automorphisms of the universal covering $\pi$ (of deck transformations) coincides with the cyclic group $\langle f\rangle$. On the other hand the map $$g\mapsto \hat{\sigma}\circ g \circ \hat{\sigma}^{-1}$$ is a group automorphism of $\mathrm {Aut}_{H}(W)$, so it coincides either with $\mathrm{id}_{\mathrm {Aut}_{H}(W)}$ or with the automorphism $g\mapsto g^{-1}$. Replacing $g$ by $f$ we obtain $\hat{\sigma}\circ f \circ \hat{\sigma}^{-1}\in\{f, f^{-1}\}$ as claimed. ◻ In the case when $\sigma:H \to\ H$ is holomorphic or anti-holomorphic we have a more precise result: **Proposition 7**. *Let $\sigma:H \to\ H$ be a holomorphic (anti-holomorphic) automorphism of $H=H_f$. Then* 1. *There exists a a holomorphic (anti-holomorphic) automorphism $\hat \sigma$ of $W$ such that $\pi\circ\hat \sigma=\sigma\circ\pi$.* 2. *For any such automorphism $\hat{\sigma}$ we have $\hat{\sigma}\circ\ f \circ\ \hat{\sigma}^{-1}=f$.* *Proof.* Since $\pi$ is locally biholomorphic, it follows that $\hat{\sigma}$ is holomorphic (anti-holomorphic) if $\sigma$ is holomorphic (anti-holomorphic). By Hartogs theorem (applied to $\hat{\sigma}$ or to its composition with the conjugation automorphism) it follows that $\hat{\sigma}$ extends to a holomorphic (anti-holomorphic) automorphism $\tilde\sigma$ of ${\mathbb C}^2$ with $\tilde\sigma(0)=0$. We can suppose that $f\in {IV}\cup {III}\cup {II_a}\cup {II_b}\cup {II_c}$. Any such $f$ is a holomorphic contraction. It follows that for any $w_0\in W$ we have $$\lim_{n\to\infty} f^n(w_0)=0,\ \lim_{n\to\infty} f^{-n}(w_0)=\infty$$ in the end compactification $W\cup\{0,\infty\}$ of $W$. Since $\hat \sigma$ extends to a homeomorphism $\tilde\sigma$ of ${\mathbb C}^2$ with $\tilde\sigma(0)=0$, it follows that the permutation $\mathrm{end}(\hat\sigma)$ induced by $\hat \sigma$ on the set of ends $\{0,\infty\}$ is $\mathrm{id}_{\{0,\infty\}}$. Therefore $$\begin{split} \lim_{n\to \infty} (\hat{\sigma}\circ\ f \circ\ \hat{\sigma}^{-1})^n(w_0)&=\lim_{n\to \infty} (\hat{\sigma}\circ\ f^n \circ\ \hat{\sigma}^{-1})(w_0)=\mathrm{end}(\hat\sigma)(\lim_{n\to \infty}f^n(\hat{\sigma}^{-1}(w_0)))\\ &=\mathrm{end}(\hat\sigma)(0)=0, \end{split}$$ whereas $\lim_{n\to \infty} f^{-1}(w_0)=\infty$. Therefore the case $\hat{\sigma}\circ\ f \circ\ \hat{\sigma}^{-1}=f^{-1}$ is ruled out. ◻ We introduce the subclass $II'_c$ of $II_c$ defined by $$II'_c\coloneq \left\{f:W\to W|\ f\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix}\alpha z\\ \bar \alpha w\end{pmatrix}\vline\ 0<|\alpha|<1,\ \alpha\not\in{\mathbb R}\right\}.$$ **Proposition 8**. *Let $f\in {IV}\cup {III}\cup {II_a}\cup {II_b}\cup {II_c}$. The following conditions are equivalent:* 1. *The primary Hopf surface $H_f$ admits anti-holomorphic automorphisms.* 2. *Either the coefficients of $f$ are real, or $f\in II'_c$.* *Proof.* The $H_f$ admits an anti-holomorphic automorphism if and only if $H_f$ is biholomorphic to $\bar H_f$. On the other hand the conjugation automorphism $c:W\to W$ induces an anti-holomorphic isomorphism $s:H_f\to H_{{\mathfrak f}}$, where ${\mathfrak f}\coloneq c\circ f\circ c^{-1}$. Therefore $\bar H_f\simeq H_{{\mathfrak f}}$, so $H_f$ admits an anti-holomorphic automorphism if and only if $H_f\simeq H_{{\mathfrak f}}$. Now note that ${\mathfrak f}$ is obtained from $f$ by conjugating the coefficients of the polynomial expression which defines $f$. On the other hand Werner's classes are conjugation invariant, in particular ${\mathfrak f}$ also belongs to ${IV}\cup {III}\cup {II_a}\cup {II_b}\cup {II_c}$. By the classification Theorem [Theorem 4](#Wehl){reference-type="ref" reference="Wehl"}, it follows that $H_f\simeq H_{{\mathfrak f}}$ if and only if either the coefficients of $f$ and ${\mathfrak f}$ coincide (in other words the coefficients of $f$ are real), or $f$ and ${\mathfrak f}$ belong to $II_c$ and the coefficients $\bar\alpha$, $\bar\delta$ of ${\mathfrak f}$ are obtained from the the coefficients $\alpha$, $\delta$ of $f$ by changing the order. The latter condition is equivalent to $f\in II'_c$. ◻ **Remark 9**. *A direct proof of Proposition [Proposition 8](#AH-autom){reference-type="ref" reference="AH-autom"} can be obtained using Proposition [Proposition 7](#PrHolAnti){reference-type="ref" reference="PrHolAnti"} and the Taylor expansion of the anti-holomorphic automorphism $\tilde \sigma$ of ${\mathbb C}^2$ obtained by applying Hartogs Theorem to the lift $\hat\sigma$ of an anti-holomorphic automorphism $\sigma$.* For a primary Hopf surface $H$ we denote by $\mathrm{Ah}(H)$ the set of anti-holomorphic automorphisms. If $H=H_f$ with $f\in {IV}\cup {III}\cup {II_a}\cup {II_b}\cup {II_c}$ this set can be obtained explicitly using the idea of Remark [Remark 9](#new-direct-proof){reference-type="ref" reference="new-direct-proof"}. An anti-holomorphic automorphism $\sigma\in \mathrm{Ah}(H)$ has a lift $\hat \sigma\in \mathrm{Ah}(W)$, which extends to an anti-holomorphic automorphism $\tilde\sigma\in \mathrm{Ah}({\mathbb C}^2)$ with $\tilde\sigma (0)=0$. Denoting by $\tilde f\in\mathrm {Aut}_h({\mathbb C}^2)$ the extension of $f$, we see that the condition $\hat \sigma\circ f= f\circ \hat \sigma$ is equivalent to $\tilde \sigma\circ\tilde f=\tilde f\circ \tilde \sigma$, which can be interpreted in terms of the Taylor expansion $$\tilde \sigma(z,w):=\left(\sum_{p,q\in {\mathbb N}}a_{pq}\bar z^p\bar w^q,\sum_{p,q\in {\mathbb N}}b_{pq}\bar z^p\bar w^q\right)$$ of $\tilde \sigma$ at 0. Using this method we obtain easily **Proposition 10**. *Let $f\in {IV}\cup {III}\cup {II_a}\cup {II_b}\cup {II_c}$ with real coefficients. The set $$\mathrm{Ah}(W)^f\coloneq \{u\in \mathrm{Ah}(W)|\ u\circ f=f\circ u\}$$ is given by the table: $$\begin{array}{|c|c|} \hline &\\ [-0.8em] \hbox{The class of }f & \mathrm{Ah}(W)^f \\ [0.2em] \hline &\\ [-0.8em] IV &\left\{\begin{pmatrix}z\\w\end{pmatrix}\mapsto A\begin{pmatrix}\bar z\\\bar w\end{pmatrix}\vline\ A\in\mathrm {GL}(2,{\mathbb C})\right\}\\ [0.8em] \hline &\\ [-0.8em] III &\left\{\begin{pmatrix}z\\w\end{pmatrix}\mapsto \begin{pmatrix}a\bar z+b\bar w^r\\d\bar w\end{pmatrix}\vline\ a\in{\mathbb C}^*,\ d\in{\mathbb C}^*,\ b\in {\mathbb C}\right\}\\ [0.8em] \hline &\\ [-0.8em] II_a &\left\{\begin{pmatrix}z\\w\end{pmatrix}\mapsto \begin{pmatrix}a^r\bar z+b\bar w^r\\a \bar w\end{pmatrix}\vline\ a\in{\mathbb C}^*,\ b\in {\mathbb C}\right\}\\ [0.8em] \hline &\\ [-0.8em] II_b &\left\{\begin{pmatrix}z\\w\end{pmatrix}\mapsto \begin{pmatrix}a \bar z+b\bar w \\a \bar w\end{pmatrix}\vline\ a\in{\mathbb C}^*,\ b\in {\mathbb C}\right\}\\ [0.8em] \hline &\\ [-0.8em] II_c\setminus II'_c &\left\{\begin{pmatrix}z\\w\end{pmatrix}\mapsto \begin{pmatrix}a \bar z \\ d \bar w\end{pmatrix}\vline\ a\in{\mathbb C}^*,\ d\in{\mathbb C}^*\right\}\\ [0.8em] \hline &\\ [-0.8em] II'_c &\left\{\begin{pmatrix}z\\w\end{pmatrix}\mapsto \begin{pmatrix}a \bar w \\ d \bar z\end{pmatrix}\vline\ a\in{\mathbb C}^*,\ d\in{\mathbb C}^*\right\}\\ [0.8em] \hline \end{array}\ .$$* *In each case the cyclic group $\langle f\rangle$ acts freely on the set $\mathrm{Ah}(W)^f$, and $\mathrm{Ah}(H_f)$ is identified with the quotient set $\mathrm{Ah}(W)^f/\langle f\rangle$.* # The classification of Real primary Hopf surfaces {#sect2} ## Even Real structures, odd Real structures on primary Hopf surfaces We start by the following simple result: **Proposition 11**. *Let $H=H_f$ be a primary Hopf surface. Let $\sigma:H\to\ H$ be an anti-holomorphic involution of $H$ and $\hat{\sigma}:W\to\ W$ be a lift of $\sigma$ (see Proposition [Proposition 7](#PrHolAnti){reference-type="ref" reference="PrHolAnti"}). Then* 1. *There exists $n\in {\mathbb Z}$ such that $\hat{\sigma}^2=f^n$.* 2. *The parity of $n$ in the previous formula is well defined (depends only on $\sigma$, not on the lift $\hat{\sigma}$).* *Proof.* Indeed, since $\sigma^2= \mathrm{id}_{H_f}$ it follows that $\hat{\sigma}^2\in\mathrm {Aut}_H(W)=\langle f\rangle$. This proves the first claim. Let now $\hat\sigma$, $\hat\sigma'$ two lifts of $\sigma$ and $n$, $n'\in{\mathbb Z}$ be the associated integers. We have $\hat\sigma'\circ \hat\sigma^{-1}\in\mathrm {Aut}_H(W)$, so there exists $k\in{\mathbb N}$ such that $\hat\sigma'=\hat\sigma\circ f^k$. Since $\hat\sigma$ commutes with $f$, we get $\hat\sigma'^2=\hat\sigma^2\circ f^{2k}$, so $n'=n+2k$. ◻ Taking into account this proposition, it's natural to define: **Definition 12**. *A Real structure $\sigma:H \to\ H$ on $H$ is said to be:* 1. *even, if one of the following equivalent conditions is verified:* 1. *For any lift $\hat{\sigma}:W\to\ W$ of $\sigma$, $\hat{\sigma}^2$ coincides with an even power of $f$.* 2. *There exists a lift $\hat{\sigma}:W\to\ W$ of $\sigma$ such that $\hat{\sigma}^2= \mathrm{id}_W$.* 2. *odd, if one of the following equivalent conditions is verified:* 1. *For any lift $\hat{\sigma}:W\to\ W$ of $\sigma$, $\hat{\sigma}^2$ coincides with an odd power of $f$.* 2. *There exists a lift $\hat{\sigma}:W\to\ W$ of $\sigma$ such that $\hat{\sigma}^2=f$.* *Example 1*. Let $f\in IV\cup III\cup II_a\cup II_b\cup II_c$ be with real coefficients. The standard complex conjugation $c:W\to W$ induces a Real structure $s_f$ on $H_f$, which is obviously even. The Real structure will be called *the standard Real structure* of $H_f$. Let now $f\in II'_c$ be given by $f\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix} \alpha z\\\bar \alpha w \end{pmatrix}$, where $0<|\alpha|<1$ and $\alpha\not\in{\mathbb R}$. The anti-holomorphic automorphism of $c':W\to W$ defined by $$c'\begin{pmatrix} z\\w \end{pmatrix}= \begin{pmatrix} \bar w\\\bar z \end{pmatrix}$$ commutes with $f$ and is involutive, so it defines an even Real structure $s_f$ on $H_f$, which will also be called *the standard Real structure* on $H_f$. ## The classification of even Real structures on primary Hopf surfaces Let $E$ be a complex vector space of dimension $n$. Recall that a real structure on $E$ is an anti-linear involution $a:E\to E$. Recall that any two real structures $a$, $b:E\to E$ on $E$ are equivalent, i.e. there exists $l\in\mathrm {GL}(E)$ such that $a=l\circ b\circ l^{-1}$. Indeed, putting $$E^a_\pm\coloneq \ker( a\mp \mathrm{id}_E), \ E^b_\pm\coloneq \ker( a\mp \mathrm{id}_E)$$ we have real direct sum decompositions $$E=E^a_+\oplus E^a_-,\ E=E^b_+\oplus E^b_-$$ with $E^a_-=i E^a_+$, $E^b_-=i E^b_+$. Choose an ${\mathbb R}$-linear isomorphism $h:E^b_+\to E^a_+$ and note that $l:E\to E$ defined by $$l(x+iy)\coloneq h(x)+i h(y) \hbox{ for any } x,\ y\in E^b_+$$ is a ${\mathbb C}$-linear isomorphism satisfying $a\circ l=l\circ b$. In particular, in the special case when $E={\mathbb C}^n$ and $b$ is the standard conjugation $c:{\mathbb C}^n\to{\mathbb C}^n$, we obtain **Remark 13**. *Let $a:{\mathbb C}^n\to{\mathbb C}^n$ be an anti-linear involution on ${\mathbb C}^n$. Then there exists a ${\mathbb C}$-linear automorphism $l:{\mathbb C}^n\to {\mathbb C}^n$ such that $a=l\circ c\circ l^{-1}$.* *In other words any anti-linear Real structure on ${\mathbb C}^n$ is $\mathrm {GL}(n,{\mathbb C})$-conjugate to the standard conjugation.* The classification of even Real structures on primary Hopf surfaces will follow from the following proposition: **Proposition 14**. *Let $f\in IV\cup III\cup II_a\cup II_b\cup II_c$ be with real coefficients and let $\phi$ be a Real structure on $W$ such that $\phi\circ f=f\circ \phi$. Then there exists $\psi\in \mathrm {Aut}_h(W)^f$ such that $\phi=\psi\circ c\circ \psi^{-1}$.* *Proof.* 1. $f\in IV$. In this case, using Proposition [Proposition 10](#ahAuto){reference-type="ref" reference="ahAuto"} we see that $\phi$ has the form $$\phi\begin{pmatrix} z\\ w \end{pmatrix}=\begin{pmatrix} a & b\\ c & d \end{pmatrix}\begin{pmatrix} \bar z\\ \bar w \end{pmatrix}$$ with $\begin{pmatrix} a & b\\ c & d \end{pmatrix}\in\mathrm {GL}(2,{\mathbb C})$. The obvious extension $\tilde\phi:{\mathbb C}^2\to{\mathbb C}^2$ of $\phi$ to ${\mathbb C}^2$ is an anti-linear Real structure on ${\mathbb C}^2$, so Remark [Remark 13](#linearRealRem){reference-type="ref" reference="linearRealRem"} applies and gives a ${\mathbb C}$-linear isomorphism $l:{\mathbb C}^2\to{\mathbb C}^2$ such that $\tilde \phi=l\circ c\circ l^{-1}$. Denoting by $\psi:W\to W$ the automorphism induced by $l$ (which obviously commutes with $f$), we get $\phi=\psi\circ c\circ \psi^{-1}$ as claimed. 2. $f\in III$. In this case Proposition [Proposition 10](#ahAuto){reference-type="ref" reference="ahAuto"} shows that $\phi$ is given by $$\phi\begin{pmatrix}z\\w\end{pmatrix}= \begin{pmatrix}a\bar z+b\bar w^r\\d\bar w\end{pmatrix},$$ where $a$, $d\in{\mathbb C}^*$, $b\in {\mathbb C}$. We have $$\phi^2\begin{pmatrix}z\\w\end{pmatrix}=\begin{pmatrix}\vert a\vert^2z+(a\bar b+b\bar d^r)w^r\\\vert d\vert^2w\end{pmatrix}.$$ The condition $\phi^2= \mathrm{id}$ becomes $$\label{phi2=idIII} \left\{\begin{array}{ccc} \vert a\vert^2&=&1 \\ a\bar b+b\bar d^r&=&0 \\ \vert d\vert^2&=&1 \end{array}\right..$$ Let now $\psi\in\mathrm {Aut}_h(W)^f$. By Wehler's Theorem [Theorem 4](#Wehl){reference-type="ref" reference="Wehl"} we have $$\psi\begin{pmatrix}z\\w\end{pmatrix}=\begin{pmatrix} Az+Bw^r\\ Dw\end{pmatrix}$$ with $A$, $D\in{\mathbb C}^*$, $B\in{\mathbb C}$. The condition $$\phi=\psi\circ\ c \circ\ \psi^{-1}$$ is equivalent to the system $$\label{eqIII} \begin{cases} a=A\bar A^{-1} \\ b=B\bar D^{-r}-A\bar A^{-1} \bar B\bar D^{-r} \\ d=D\bar D^{-1}. \end{cases}$$ Since $|a|=|d|=1$ we can write $a=e^{i\theta}$, $d=e^{i\delta}$ with $\theta$, $\delta\in{\mathbb R}$. Put $A\coloneq e^{i\frac{\theta}{2}}$, $D\coloneq e^{i\frac{\delta}{2}}$. With this choice we have $A\bar A^{-1}=A^2=a$, $D\bar D^{-1}=D^2=d$, so the first and the third equations in ([\[eqIII\]](#eqIII){reference-type="ref" reference="eqIII"}) are satisfied. We are seeking $B\in{\mathbb C}$ such that the second equation is also satisfied. Consider the ${\mathbb R}$-linear map $\lambda:{\mathbb C}\to{\mathbb C}$ defined by $$\lambda(z)=u z-v\bar z,$$ where $u\coloneq \bar D^{-r}$, $v\coloneq A\bar A^{-1} \bar D^{-r}= A^2\bar D^{-r}$. This map is not surjective. Using $|u|=|v|=1$ it follows easily that its image is the real line $$\label{imlambda} \mathrm{im}(\lambda)=\{\zeta\in{\mathbb C}|\ u^{-1}\zeta+v \bar \zeta=0\}\subset{\mathbb C}.$$ Now note that $b\in\mathrm{im}(\lambda)$, because $$\hspace*{6mm}u^{-1}b+ v\bar b=\bar D^{r}b+A^2\bar D^{-r}\bar b=\bar D^{-r}(\bar D^{2r}b+ A^2\bar b)=\bar D^{-r}(\bar d^rb+a\bar b)=0$$ by the second equation in ([\[phi2=idIII\]](#phi2=idIII){reference-type="ref" reference="phi2=idIII"}). Therefore, there exists $B\in{\mathbb C}$ such that $$b=\lambda(B)=\bar D^{-r} B-A\bar A^{-1}\bar D^{-r} \bar B,$$ which proves that, with this choice, the second equation in ([\[eqIII\]](#eqIII){reference-type="ref" reference="eqIII"}) is satisfied, too. 3. $f\in II_a$. In this case Proposition [Proposition 10](#ahAuto){reference-type="ref" reference="ahAuto"} shows that $\phi$ has the form $$\phi\begin{pmatrix} z\\w\end{pmatrix}=\begin{pmatrix}a^r\bar z+b\bar w^r\\a\bar w\end{pmatrix}$$ with $a\in{\mathbb C}^*$ and $b\in {\mathbb C}$. The condition $\phi^2= \mathrm{id}_W$ is equivalent to the system $$\label{phi2=idIIa} \left\{\begin{array}{ccc} \vert a\vert^2&=&1 \\ a^r\bar b+b\bar a^r&=&0 \end{array}\right..$$ By Wehler's theorem, an automorphism $\psi\in \mathrm {Aut}_h(W)^f$ has the form $$\psi\begin{pmatrix}z\\w\end{pmatrix}=\begin{pmatrix}A^rz+Bw^r\\Aw\end{pmatrix}$$ with $A\in{\mathbb C}^*$, $B\in {\mathbb C}$. The condition $$\phi=\psi\circ\ c \circ\ \psi^{-1}$$ is equivalent to the system $$\label{eqIIa} \begin{cases} a=A\bar A^{-1} \\ b=B\bar A^{-r}-A^{r} \bar B\bar A^{-2r}. \end{cases}$$ Since $|a| ^2=1$ we can write $a=e^{i\theta}$ with $\theta\in{\mathbb R}$. Putting $A=e^{\frac{i\theta}{2}}$ we have $A\bar A^{-1}=A^2=a$. As in the previous case consider the ${\mathbb R}$-linear map $\lambda:{\mathbb C}\to {\mathbb C}$ $$\lambda(z)=uz-v\bar z,$$ where this time we choose $u\coloneq A^r$ and $v\coloneq A^{3r}$. We have again $|u|=|v|=1$, so the image of $\lambda$ is again given by ([\[imlambda\]](#imlambda){reference-type="ref" reference="imlambda"}). Note that $b\in\mathrm{im}(\lambda)$ because $$u^{-1}b+ v \bar b=A^{-r} b+A^{3r} \bar b=A^r(A^{2r}\bar b+ A^{-2r} b)=A^r(a^r \bar b+ b\bar a^r)=0$$ by the second equation in ([\[phi2=idIIa\]](#phi2=idIIa){reference-type="ref" reference="phi2=idIIa"}). Therefore there exists $B\in{\mathbb C}$ such that $$b=\lambda(B)= A^r B- A^{3r} \bar B=B\bar A^{-r}- A^r \bar B\bar A^{-2r},$$ which shows that $(A,B)\in{\mathbb C}^*\times{\mathbb C}$ is a solution of the system ([\[eqIIa\]](#eqIIa){reference-type="ref" reference="eqIIa"}). 4. $f\in II_b$. In this case Proposition [Proposition 10](#ahAuto){reference-type="ref" reference="ahAuto"} shows that $\phi$ has the form $$\phi\begin{pmatrix} z\\w \end{pmatrix} =\begin{pmatrix} a\bar z+b\bar w\\a\bar w\end{pmatrix}$$ where $a\in {\mathbb C}^*$, $b\in{\mathbb C}$. The condition $\phi^2= \mathrm{id}_W$ is equivalent to $$\label{phi2=idIIb} \left\{\begin{array}{ccc} \vert a\vert^2&=&1 \\ a\bar b+b\bar a&=&0 \end{array}\right..$$ By Wehler's theorem, an automorphism $\psi\in \mathrm {Aut}_h(W)^f$ has the form $$\psi\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix} Az+Bw\\Aw \end{pmatrix}$$ where $A\in {\mathbb C}^*$, $B\in{\mathbb C}$. We use the same arguments as in the case $II_a$ but taking $r=1$ in all formulae used in this case. 5. $f\in II_c$. In this case Proposition [Proposition 10](#ahAuto){reference-type="ref" reference="ahAuto"} shows that $\phi$ has the form $$\phi\begin{pmatrix} z\\w \end{pmatrix} =\begin{pmatrix} a\bar z \\ b\bar w\end{pmatrix}$$ where $a$, $b\in {\mathbb C}^*$. The condition $\phi^2= \mathrm{id}_W$ is equivalent to $$\label{phi2=idIIc} \left\{\begin{array}{ccc} | a|^2&=&1 \\ |b|^2&=&1 \end{array}\right..$$ By Wehler's theorem, in this case an automorphism $\psi\in \mathrm {Aut}_h(W)^f$ has the form $$\psi\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix} Az \\Bw \end{pmatrix}$$ where $A$, $B\in {\mathbb C}^*$, and the condition $$\phi=\psi\circ\ c \circ\ \psi^{-1}$$ is equivalent to the system $$\label{eqIIc} \begin{cases} a=A\bar A^{-1} \\ b=B\bar B^{-1}. \end{cases}$$ It suffices to put $A\coloneq e^{i\frac{\theta}{2}}$, $B\coloneq e^{i\frac{\beta}{2}}$, where $a=e^{i \theta}$, $b=e^{i \beta}$.  ◻ Using the notation $c'$ introduced in Example [Example 1](#ExStandard){reference-type="ref" reference="ExStandard"} we can state the following analogue of Proposition [Proposition 14](#PropEven){reference-type="ref" reference="PropEven"} for the the subclass $II'_c$: **Proposition 15**. *Let $f\in II'_c$ be given by $f\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix} \alpha z\\\bar \alpha w \end{pmatrix}$, where $0<|\alpha|<1$ and $\alpha\not\in{\mathbb R}$. Let $\phi$ be a Real structure on $W$ such that $\phi\circ f=f\circ \phi$. There exists $\psi\in\mathrm {Aut}_h(W)^f$ such that $\phi=\psi\circ c'\circ \psi^{-1}$.* *Proof.* Using Proposition [Proposition 10](#ahAuto){reference-type="ref" reference="ahAuto"} we see that $\phi$ has the form $\phi\begin{pmatrix} z\\ w \end{pmatrix}=\begin{pmatrix} a \bar w\\ d \bar z \end{pmatrix}$, and the condition $\phi^2= \mathrm{id}_W$ becomes $\bar a= d^{-1}$. It suffices to note that $$\begin{pmatrix} z\\ w \end{pmatrix}\mapsto \begin{pmatrix} a z\\ w \end{pmatrix}$$ defines an element $\psi\in \mathrm {Aut}_h(W)^f$ and that $\phi=\psi\circ c'\circ \psi^{-1}$. ◻ Using Propositions [Proposition 14](#PropEven){reference-type="ref" reference="PropEven"}, [Proposition 15](#PropEvenII'c){reference-type="ref" reference="PropEvenII'c"} we obtain the following classification theorem for even Real structures on primary Hopf surfaces: **Theorem 16**. *Let $f\in IV\cup III\cup II_a\cup II_b\cup II_c$ be either with real coefficients or element of the subclass $II'_c$. Let $\sigma:H_f\to H_f$ be an even Real structure on $H_f$. There exists a holomorphic automorphism $g\in \mathrm {Aut}_h(H_f)$ such that $\sigma=g\circ s_f\circ g^{-1}$, where $s_f$ is the standard Real structure on $H_f$.* In other words, any *even* Real structure on $H_f$ is equivalent to its standard Real structure $s_f$. *Proof.* Since $\sigma$ is even, there exists a lift $\hat \sigma:W\to W$ which is an involution, hence a Real structure on $W$. By Propositions [Proposition 14](#PropEven){reference-type="ref" reference="PropEven"}, [Proposition 15](#PropEvenII'c){reference-type="ref" reference="PropEvenII'c"} there exists $\psi\in\mathrm {Aut}_h(W)^f$ such that $\hat\sigma=\psi\circ c\circ \psi^{-1}$, respectively $\hat\sigma=\psi\circ c'\circ \psi^{-1}$. Denoting by $g\in\mathrm {Aut}_h(H_f)$ the automorphism induced by $\psi$, we obtain $\sigma= g\circ s_f\circ g^{-1}$. ◻ Note that Proposition [Proposition 11](#liftsOfsigma){reference-type="ref" reference="liftsOfsigma"} and Definition [Definition 12](#even-odd-def){reference-type="ref" reference="even-odd-def"} generalize in a natural way to primary Hopf $n$-folds for any $n\geq 2$. Moreover, for a primary Hopf $n$-fold $H_f$ defined by a holomorphic contraction $f$ given by a polynomial formula with real coefficients, we can define the standard Real structure $s_f$ of $H_f$ as in Example [Example 1](#ExStandard){reference-type="ref" reference="ExStandard"}. Using Remark [Remark 13](#linearRealRem){reference-type="ref" reference="linearRealRem"} we obtain in the same way the classification of even Real structures on a primary Hopf $n$-fold $$H_{f_\alpha}\coloneq {\hbox{}^{\displaystyle{{\mathbb C}^n\setminus\{0\}}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f_\alpha\rangle}},$$ where $0<|\alpha|<1$ and $f_\alpha(z)=\alpha z$. **Remark 17**. *The primary Hopf $n$-fold $H_{f_\alpha}$ admits an even Real structure if and only if $\alpha\in{\mathbb R}$. If this is the case, any even Real structure on $H_{f_\alpha}$ is equivalent to its standard Real structure $s_{f_\alpha}$.* ## The classification of odd Real structures on primary Hopf surfaces The classification of odd Real structures is more difficult. We consider first the case when the diagonal coefficients of $f$ (denoted by $\alpha$, $\delta$, $\delta^r$ in Theorem [Theorem 4](#Wehl){reference-type="ref" reference="Wehl"}) are positive. We will start with the following remark which can be proved easily by direct computations: **Remark 18**. *Let $f\in IV \cup III\cup II_a\cup II_b\cup II_c$ be with real coefficients and positive diagonal coefficients, and let $k\in{\mathbb N}^*$. The automorphism $f^{\frac{1}{k}}\in\mathrm {Aut}_h(W)$ defined in the table below has the properties:* 1. *$f^{\frac{1}{k}}$ is a polynomial holomorphic contraction with real coefficients.* 2. *$f^{\frac{1}{k}}$ is a root of order $k$ of $f$.* 3. *[\[Auf-k-root\]]{#Auf-k-root label="Auf-k-root"} $\mathrm {Aut}_h(W)^{f^{\frac{1}{k}}}=\mathrm {Aut}_h(W)^{f}$, $\mathrm{Ah}(W)^{f^{\frac{1}{k}}}=\mathrm{Ah}(W)^{f}$.* *$$\begin{array} {|c|c|c|} \hline &&\\ [-1em] \hbox{The class of } f & f\begin{pmatrix} z\\w \end{pmatrix}& f^{\frac{1}{k}}\begin{pmatrix} z\\w \end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-1em] IV & \begin{pmatrix}\alpha z\\ \alpha w\end{pmatrix} & \begin{pmatrix}\alpha^{\frac{1}{k}} z\\ \alpha^{\frac{1}{k}} w\end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-1em] III & \begin{pmatrix}\delta^r z\\ \delta w\end{pmatrix} & \begin{pmatrix} \delta^{\frac{r}{k}}z\\ \delta^{\frac{1}{k}}w \end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-1em] II_a & \begin{pmatrix}\delta^r z+w^r\\ \delta w\end{pmatrix} & \begin{pmatrix} \delta^{\frac{r}{k}}z+\frac{1}{k}\delta^{r\big(\frac{1-k}{k}\big)}w^r \\ \delta^{\frac{1}{k}}w \end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-0.7em] II_b & \begin{pmatrix}\alpha z+w\\ \alpha w\end{pmatrix} & \begin{pmatrix} \alpha^{\frac{1}{k}}z+\frac{1}{k}\alpha^{\frac{1-k}{k}}w \\ \alpha^{\frac{1}{k}}w \end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-1em] II_c &\begin{pmatrix}\alpha z\\ \delta w\end{pmatrix} & \begin{pmatrix} \alpha^{\frac{1}{k}}z \\ \delta^{\frac{1}{k}}w \end{pmatrix} \\ [0.8em] \hline \end{array}$$* **Proposition 19**. *Let $f\in IV \cup III\cup II_a\cup II_b\cup II_c$ be with real coefficients and positive diagonal coefficients, and let $f^{\frac{1}{2}}$ be the square root of $f$ given by Remark [Remark 18](#kroot){reference-type="ref" reference="kroot"}. Let $c:W\to W$ be the standard conjugation. Then:* 1. *[\[ccircf1/2\]]{#ccircf1/2 label="ccircf1/2"} The composition $c\circ f^{\frac{1}{2}}$ belongs to $\mathrm{Ah}(W)^f$ and satisfies $(c\circ f^{\frac{1}{2}})^2=f$.* 2. *For any $\phi\in \mathrm{Ah}(W)^f$ with $\phi^2=f$ there exists $\psi\in\mathrm {Aut}_h(W)^f$ such that $$\phi=\psi\circ (c\circ f^{\frac{1}{2}})\circ \psi^{-1}.$$* In other words, under the assumption of the proposition, any odd Real structure on $H_f$ is equivalent to the odd real structure $\sigma_f$ induced by $c\circ f^{\frac{1}{2}}$. *Proof.* (1) This follows taking into account that, since $f^{\frac{1}{2}}$ has real coefficients, it commutes with $c$.\ (2) Put $\phi'=f^{-\frac{1}{2}}\circ\phi$, and note that $\phi$ belongs to $\mathrm{Ah}(W)^f$ and (since $\phi$ commutes with $f^{-\frac{1}{2}}$ by Remark [Remark 18](#kroot){reference-type="ref" reference="kroot"} ([\[Auf-k-root\]](#Auf-k-root){reference-type="ref" reference="Auf-k-root"})) satisfies $\phi'^2= \mathrm{id}_W$. Therefore $\phi'$ is a Real structure on $W$ which commutes with $f$. By Proposition [Proposition 14](#PropEven){reference-type="ref" reference="PropEven"}, there exists $\psi\in \mathrm {Aut}_h(W)^f$ such that $\phi'=\psi\circ c\circ \psi^{-1}$. Therefore $$f^{-\frac{1}{2}}\circ\phi=\psi\circ c\circ \psi^{-1}.$$ By Remark [Remark 18](#kroot){reference-type="ref" reference="kroot"} ([\[Auf-k-root\]](#Auf-k-root){reference-type="ref" reference="Auf-k-root"}) again $\psi$ commutes with $f^{\frac{1}{2}}$, so we obtain $$\phi= \psi\circ (c\circ f^{\frac{1}{2}})\circ \psi^{-1},$$ as claimed. ◻ The proposition below shows that if $H_f$ admits an odd Real structure and $$f\in III\cup II_a\cup II_b\cup II_c,$$ has real coefficients, then its diagonal coefficients are always positive, so Proposition [Proposition 19](#class-odd-pos-coeff){reference-type="ref" reference="class-odd-pos-coeff"} classifies odd Real structure on all primary Hopf surfaces except those of class $II'_c$ and those of class $IV$ with negative parameter $\alpha$. **Proposition 20**. *Let $f\in III\cup II_a\cup II_b\cup II_c$ be with real coefficients. The following conditions are equivalent:* 1. *There exists $\phi\in \mathrm{Ah}(W)^f$ such that $\phi^2=f$.* 2. *The diagonal coefficients of $f$ are positive.* *Proof.* $(1)\Rightarrow(2)$: Suppose $f\in III$, so $f\begin{pmatrix} z\\ w\end{pmatrix}=\begin{pmatrix}\delta^r z\\ \delta w\end{pmatrix}$ with $0<|\delta|<1$. An element $\phi\in \mathrm{Ah}(W)^f$ has the form $$\phi\begin{pmatrix} z\\ w\end{pmatrix}= \begin{pmatrix}a\bar z+b\bar w^r\\d\bar w\end{pmatrix}.$$ The condition $\phi^2=f$ becomes $$\left\{\begin{array}{ccc} \vert a\vert^2&=&\delta^r \\ a\bar b+b\bar d^r&=&0 \\ \vert d\vert^2&=&\delta \end{array}\right.,$$ which obviously implies $\delta>0$. The other cases are treated in a similar way.\ $(2)\Rightarrow(1)$: This follows from Proposition [Proposition 19](#class-odd-pos-coeff){reference-type="ref" reference="class-odd-pos-coeff"} ([\[ccircf1/2\]](#ccircf1/2){reference-type="ref" reference="ccircf1/2"}). ◻ The analogue of Proposition [Proposition 19](#class-odd-pos-coeff){reference-type="ref" reference="class-odd-pos-coeff"} for $f\in II'_c$ follows easily by direct computation: **Proposition 21**. *Let $f\in II'_c$ be given by $f\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix} \alpha z\\\bar \alpha w \end{pmatrix}$, where $0<|\alpha|<1$ and $\alpha\not\in{\mathbb R}$. Let $\alpha^{\frac{1}{2}}$ be a square root of $\alpha$, $\bar \alpha^{\frac{1}{2}}$ be its conjugate, and let $f^{\frac{1}{2}}$ be the square root of $f$ defined by $f^{\frac{1}{2}}\begin{pmatrix} z\\w \end{pmatrix}=\begin{pmatrix} \alpha^{\frac{1}{2}} z\\ \bar \alpha^{\frac{1}{2}} w \end{pmatrix}$. Then* 1. *$c'$ commutes with $f^{\frac{1}{2}}$.* 2. *$c'\circ f^{\frac{1}{2}}\in \mathrm{Ah}(W)^f$ satisfies $(c'\circ f^{\frac{1}{2}})^2=f$.* 3. *For any $\phi\in \mathrm{Ah}(W)^f$ with $\phi^2=f$ there exists $\psi\in\mathrm {Aut}_h(W)^f$ such that $\phi=\psi\circ(c'\circ f^{\frac{1}{2}})\circ\psi^{-1}$.* In other words, under the assumption of the proposition, for $f\in II'_c$, any odd Real structure on $H_f$ is equivalent to the odd real structure $\sigma_f$ induced by $c'\circ f^{\frac{1}{2}}$.\ For the classification of odd Real structures on class IV primary Hopf surfaces with negative parameter $\alpha$ we will need a simple remark concerning the classification of quaternionic structures on a finite dimensional real vector space. Let $F$ be a real vector space of dimension $4k$. Recall that a left ${\mathbb H}$-vector space structure on $F$ is equivalent to the data of a pair $(I,J)\in\mathrm {End}(F)\times\mathrm {End}(F)$ such that $I^2=J^2=- \mathrm{id}_F$ and $I\circ J=-J\circ I$. In the presence of such a pair $(I,J)$, we put $K\coloneq I\circ J$, and we define a left quaternonic vector space structure on $E$ (extending its real space structure) by mapping the quaternonic units $i$, $j$, $k\in{\mathbb H}$ to $I$, $J$, $K$ respectively. Since two left ${\mathbb H}$-vector spaces of the same dimension are isomorphic, it follows that any two left ${\mathbb H}$-vector space structures $(I,J)$, $(I',J')$ on $F$ are always equivalent, i.e. there exists an automorphism $l\in\mathrm {GL}(F)$ such that $l\circ I=I'\circ l$, $l\circ J=J'\circ l$. In the special case $I=I'$ we obtain $l\circ I=I\circ l$, i.e. $l$ is linear with respect to the complex structure defined by $I$, and $J'=l\circ J\circ l^{-1}$. This shows that **Remark 22**. *Let $E$ be a complex vector space of dimension $2k$ and let $J$, $J'$ be anti-linear isomorphisms such that $J^2=J'^2=- \mathrm{id}_E$. Then there exists an automorphism $l\in\mathrm {GL}(E)$ such that $J'=l\circ J\circ l^{-1}$.* Consider the anti-linear map $J:{\mathbb C}^{2k} \to {\mathbb C}^{2k}$ defined by $$J\begin{pmatrix} z\\w\end{pmatrix} =\begin{pmatrix}-\bar w\\ \bar z\end{pmatrix}=\begin{pmatrix} 0 & - \mathrm{id}_{{\mathbb C}^k} \\ \mathrm{id}_{{\mathbb C}^k}& 0 \end{pmatrix}\begin{pmatrix} \overline z\\ \overline w \end{pmatrix}$$ for pairs $(z,w)\in {\mathbb C}^k\times{\mathbb C}^k={\mathbb C}^{2k}$. Note that $J^2=- \mathrm{id}_{{\mathbb C}^{2k}}$. By Remark [Remark 22](#RemQuat){reference-type="ref" reference="RemQuat"} we obtain the following analogue of Remark [Remark 13](#linearRealRem){reference-type="ref" reference="linearRealRem"}: **Remark 23**. *Let $a:{\mathbb C}^{2k}\to{\mathbb C}^{2k}$ ba an anti-linear isomorphism satisfying $a^2=- \mathrm{id}_{{\mathbb C}^2}$. Then there exists $l\in\mathrm {GL}(2k,{\mathbb C})$ such that $a=l\circ J\circ l^{-1}$, i.e. $a$ is $\mathrm {GL}(2k,{\mathbb C})$-conjugate to $J$.* With this preparation we can prove: **Proposition 24**. *Let $f:W\to\ W\in IV$ be given by $f\begin{pmatrix}z\\w\end{pmatrix}=\begin{pmatrix}\alpha z\\\alpha w\end{pmatrix}$, where $\alpha\in(-1,0)$. Let $\alpha^{\frac{1}{2}}$ be a square root of $\alpha$, and let $f^{\frac{1}{2}}$ be the square root of $f$ defined by $f^{\frac{1}{2}}\begin{pmatrix}z\\w\end{pmatrix}=\begin{pmatrix}\alpha^{\frac{1}{2}} z\\\alpha^{\frac{1}{2}} w\end{pmatrix}$. Then* 1. *$J$ anti-commutes with $f^{\frac{1}{2}}$.* 2. *$J\circ f^{\frac{1}{2}}\in \mathrm{Ah}(W)^f$ and satisfies $(J\circ f^{\frac{1}{2}})^2=f$.* 3. *For any $\phi\in \mathrm{Ah}(W)^f$ with $\phi^2=f$ there exists $\psi\in \mathrm {Aut}_h(W)$ such that $\phi=\psi\circ(J\circ f^{\frac{1}{2}})\circ\psi^{-1}$.* Therefore, under the assumptions of the proposition, any odd Real structure $\sigma$ on $H_f$ is equivalent to ${\mathfrak s}_f$, where ${\mathfrak s}_f$ is the odd Real structure induced by $J\circ f^{\frac{1}{2}}$. *Proof.* (1) It suffices to note that $f^{\frac{1}{2}}=\alpha^{\frac{1}{2}} \mathrm{id}_W$ with $\alpha^{\frac{1}{2}}$ pure imaginary and to recall that $J$ is anti-linear.\ (2) Follows from (1) taking into account that $J^2=- \mathrm{id}_{{\mathbb C}^2}$.\ (3) Let $\phi\in \mathrm{Ah}(W)^f$ with $\phi^2=f$. The extension $\tilde \phi$ of $\phi$ to ${\mathbb C}^2$ gives an anti-linear isomorphism $\tilde\phi:{\mathbb C}^2\to {\mathbb C}^2$. Since $\tilde\phi$ anti-commutes with $f^{\pm\frac{1}{2}}$, the condition $\phi^2=f$ is equivalent to $(\tilde\phi\circ f^{-\frac{1}{2}})^2=- \mathrm{id}_{{\mathbb C}^2}$. By Remark [Remark 23](#ClassQuatStr){reference-type="ref" reference="ClassQuatStr"}, there exists $l\in\mathrm {GL}(2,{\mathbb C})$ such that $$\tilde\phi\circ f^{-\frac{1}{2}}=l\circ J\circ l^{-1}.$$ Denoting by $\psi$ the automorphism of $W$ induced by $l$, which obviously commutes with $f^{\frac{1}{2}}$, we obtain $\phi=\psi\circ (J\circ f^{\frac{1}{2}})\circ \psi^{-1}$, as claimed. ◻ Using Propositions [Proposition 19](#class-odd-pos-coeff){reference-type="ref" reference="class-odd-pos-coeff"}, [Proposition 24](#oddIV-neg-coeff){reference-type="ref" reference="oddIV-neg-coeff"} we obtain the following classification theorem for odd Real structures on primary Hopf surfaces: **Theorem 25**. 1. *Let $f\in III\cup II_a\cup II_b\cup II_c$.* 1. *The following conditions are equivalent:* 1. *$H_f$ admits an odd Real structure.* 2. *$f$ either has real coefficients and positive diagonal coefficients, or belongs to $II'_c$.* 2. *If one of these equivalent conditions is satisfied, any odd Real structure on $H_f$ is equivalent to the Real structure $\sigma_f$ defined above.* 2. *Let $f\in IV$ be given by $f(z,w)=(\alpha z,\alpha w)$ where $0<|\alpha|<1$.* 1. *The following conditions are equivalent:* 1. *$H_f$ admits an odd Real structure.* 2. *$\alpha\in{\mathbb R}$.* 2. *If $\alpha\in (0,1)$, any odd Real structure on $H_f$ is equivalent to $\sigma_f$. If $\alpha\in (-1,0)$, any odd Real structure on $H_f$ is equivalent to ${\mathfrak s}_f$.* Consider again the primary Hopf $n$-fold $H_{f_\alpha}\coloneq {{\mathbb C}^{n}\setminus\{0\}}/{\langle f_\alpha\rangle}$. Using Remark [Remark 23](#ClassQuatStr){reference-type="ref" reference="ClassQuatStr"} and defining the odd Real structures $\sigma_{f_\alpha}$ (for $\alpha \in(0,1)$), ${\mathfrak s}_{f_\alpha}$ (for $\alpha\in (-1,0)$ and $n$ even) as above, we obtain in a similar way the classification of odd Real structures on $H_{f_\alpha}$: **Remark 26**. *Let $H_{f_\alpha}$ be the primary Hopf $n$-fold associated with the holomorphic contraction $f_\alpha$, where $0<|\alpha|<1$.* 1. *Suppose $n$ is odd. $H_{f_\alpha}$ admits an odd Real structure if and only if $\alpha\in(0,1)$. If this is the case, any odd Real structure on $H_{f_\alpha}$ is equivalent to $\sigma_{f_\alpha}$.* 2. *Suppose $n$ is even. $H_{f_\alpha}$ admits an odd Real structure if and only if $\alpha\in{\mathbb R}$. If $\alpha\in (0,1)$ any odd Real structure on $H_{f_\alpha}$ is equivalent to $\sigma_{f_\alpha}$. If $\alpha\in (-1,0)$ any odd Real structure on $H_{f_\alpha}$ is equivalent to ${\mathfrak s}_{f_\alpha}$.* # The automorphism group and the Real Picard group ## The automorphism group For the automorphism group of an even Real primary Hopf surface $(H_f,s_f)$ we have the following result which follows easily from Theorem [Theorem 4](#Wehl){reference-type="ref" reference="Wehl"}: **Theorem 27**. *Let $f\in {IV}\cup {III}\cup {II_a}\cup {II_b}\cup {II_c}$.* 1. *Suppose is with real coefficients. The group $\mathrm {Aut}_h(W)^{f,c}$ of holomorphic automorphisms of $W$ commuting with $f$ and the standard conjugation $c$ is given by the table below:* *$$\begin{array} {|c|c|} \hline & \\ [-0.8em] \hbox{The class of } f & \mathrm {Aut}_h(W)^{f,c} \\ [0.1em] \hline & \\ [-0.8em] IV & \mathrm {GL}(2,{\mathbb R}) \\ [0.2em] \hline & \\ [-0.8em] III & \left\{\begin{pmatrix} z\\w \end{pmatrix}\mapsto\ \begin{pmatrix}az+bw^r\\dw\end{pmatrix}\vline\ a\in{\mathbb R}^*,\ d\in{\mathbb R}^*, b\in {\mathbb R}\right\} \\ [0.8em] \hline & \\ [-0.8em] II_a &\left\{\begin{pmatrix} z\\w \end{pmatrix}\mapsto\ \begin{pmatrix}a^rz+bw^r\\aw\end{pmatrix}\vline\ a\in {\mathbb R}^*, b\in {\mathbb R}\right\} \\ [0.8em] \hline & \\ [-0.8em] II_b & \left\{\begin{pmatrix} z\\w \end{pmatrix}\mapsto\ \begin{pmatrix}az+bw\\aw\end{pmatrix}\vline\ a\in {\mathbb R}^*, b\in {\mathbb R}\right\} \\ [0.8em] \hline & \\ [-0.8em] II_c & \left\{\begin{pmatrix} z\\w \end{pmatrix}\mapsto\ \begin{pmatrix}az\\dw\end{pmatrix}\vline\ a\in{\mathbb R}^*,\ d\in{\mathbb R}^*\right\} \\ [0.8em] \hline \end{array}$$* 2. *Suppose $f\in II'_c$. Then $$\mathrm {Aut}_h(W)^{f,c'}=\left\{\begin{pmatrix} z\\w \end{pmatrix}\mapsto\ \begin{pmatrix}az\\ \bar a w\end{pmatrix}\vline\ a\in{\mathbb C}^*\right\}.$$* 3. *In each case the cyclic group $\langle f\rangle$ is a central subgroup of $\mathrm {Aut}_h(W)^{f,c}$, respectively $\mathrm {Aut}_h(W)^{f,c'}$, and the automorphism group $\mathrm {Aut}_h(H_f,s_f)$ is identified with the quotient $\mathrm {Aut}_h(W)^{f,c}/\langle f\rangle$, respectively $\mathrm {Aut}_h(W)^{f,c'}/\langle f\rangle$.* *Proof.* The claims (1), (2) follow directly from Theorem [Theorem 4](#Wehl){reference-type="ref" reference="Wehl"}. For (3) it suffices to prove that a holomorphic automorphism $\varphi\in\mathrm {Aut}_h(H_f)$ induced by $\hat \varphi\in \mathrm {Aut}_h(W)^f$ commutes with $s_f$ if and only if $\hat\varphi$ commutes with $c$, respectively $c'$. In other words we have to prove that $c\circ \hat \varphi= \hat \varphi\circ c\circ f^k$, respectively $c'\circ \hat \varphi= \hat \varphi\circ c'\circ f^k$ (with $k\in{\mathbb Z})$, then $k=0$. This follows by elementary computations. ◻ Using Theorem [Theorem 27](#AutEven){reference-type="ref" reference="AutEven"} we can describe the automorphism groups of even Real primary Hopf surfaces in terms of (semi-direct products of) classical groups. For instance, for $f\in IV$, we obtain $\mathrm {Aut}(H_f,s_f)=\mathrm {GL}(2,{\mathbb R})/\langle \alpha I_2\rangle$. For $f\in III$ the group $\mathrm {Aut}_h(W)^{f,c}$ coincides with the group $$\big\{ g_{a,d,b}|\ (a,d,b)\in{\mathbb R}^*\times{\mathbb R}^*\times{\mathbb R}\big\},\hbox{ where } g_{a,d,b}\begin{pmatrix}z\\w\end{pmatrix}=\begin{pmatrix}az+abw^r\\dw\end{pmatrix}.$$ This group can be identified with the semi-direct product $({\mathbb R}^*\times{\mathbb R}^*)\ltimes_{\rho_r} {\mathbb R}$ associated with the morphism $\rho_r:({\mathbb R}^*\times{\mathbb R}^*)\to \mathrm {GL}(1,{\mathbb R})$ given by $$\rho_r(a,d)(b)=ad^{-r}b.$$ Since $\rho_r(\delta^r,\delta)= \mathrm{id}_{\mathbb R}$, it follows that $\rho_r$ descends to a morphism $$\hat \rho_r: {\hbox{}^{\displaystyle{{\mathbb R}^*\times{\mathbb R}^*}}}\!\big/\!\hbox{}_{ \displaystyle{\langle (\delta^r,\delta)\rangle}}\to \mathrm {GL}(1,{\mathbb R}).$$ With this remark we obtain: **Corollary 28**. *Let $f\in III$ with real coefficients $\delta^r$, $\delta$. The automorphism group $\mathrm {Aut}(H_f,s_f)$ can be identified with the semi-direct product $$\left[{\hbox{}^{\displaystyle{{\mathbb R}^*\times{\mathbb R}^*}}}\!\big/\!\hbox{}_{ \displaystyle{\langle (\delta^r,\delta)\rangle}}\right]\ltimes_{\hat \rho_r} {\mathbb R}.$$* Similar descriptions are obtained for $f\in II_a$ and $f\in II_b$. For $f\in II_c$ with real coefficients we have obviously $\mathrm {Aut}(H_f,s_f)=({\mathbb R}^*\times{\mathbb R}^*)/\langle(\alpha,\delta)\rangle$, and for $f\in II'_c$ we obtain $\mathrm {Aut}(H_f,s_f)={\mathbb C}^*/\langle \alpha\rangle$, which is a 1-dimensional complex torus. The automorphism group of the odd Real Hopf surfaces is given by the following **Theorem 29**. *Let $f\in IV\cup III\cup II_a\cup II_b\cup II_c$ .* 1. *Suppose $f$ has real coefficients and positive diagonal coefficients. Then $$\mathrm {Aut}(H_f,\sigma_f)=\mathrm {Aut}(H_f,s_f).$$* 2. *Suppose $f\in II'_c$. Then again $$\mathrm {Aut}(H_f,\sigma_f)=\mathrm {Aut}(H_f,s_f).$$* 3. *[\[neg-alpha\]]{#neg-alpha label="neg-alpha"} Suppose that $f\in IV$ with negative diagonal coefficient $\alpha$. Then $$\mathrm {Aut}(H_f,{\mathfrak s}_f)={\hbox{}^{\displaystyle{\left\{\begin{pmatrix} a &-\bar b\\ b & \bar a \end{pmatrix}\vline\ (a, b)\in {\mathbb C}^2\setminus\{0\} \right\} }}}\!\big/\!\hbox{}_{ \displaystyle{\langle \alpha I_2\rangle}}.$$* *Proof.* Use similar arguments, based on elementary computations, as in the proof of Theorem [Theorem 27](#AutEven){reference-type="ref" reference="AutEven"}. ◻ The group $\left\{\begin{pmatrix} a &-\bar b\\ b & \bar a \end{pmatrix}\vline\ (a, b)\in {\mathbb C}^2\setminus\{0\} \right\}$ can be identified with the subgroup ${\mathbb R}^*_+\mathrm {SU}(2)$ of $\mathrm {GL}(2,{\mathbb C})$. This subgroup is isomorphic with ${\mathbb H}^*$ via the map: $$z+jw\mapsto \begin{pmatrix} z &-\bar w\\ w & \bar z \end{pmatrix}.$$ Therefore in case ([\[neg-alpha\]](#neg-alpha){reference-type="ref" reference="neg-alpha"}) we have $$\label{NewQuot} \mathrm {Aut}(H_f,{\mathfrak s}_f)\simeq {\hbox{}^{\displaystyle{{\mathbb R}^*_+\mathrm {SU}(2)}}}\!\big/\!\hbox{}_{ \displaystyle{\langle \alpha I_2\rangle}}.$$ The right hand quotient in ([\[NewQuot\]](#NewQuot){reference-type="ref" reference="NewQuot"}) can be written as the quotient of ${{\mathbb R}^*_+ \mathrm {SU}(2)}/{\langle \alpha^2 I_2\rangle}$ by the order 2 group $\langle \alpha I_2\rangle/\langle \alpha^2 I_2\rangle$. Via the Lie group isomorphism $$\Phi: {\mathbb R}^*_+\mathrm {SU}(2)\to {\mathbb R}^*_+\times\mathrm {SU}(2), \ \Phi(A)=( \det(A)^{\frac{1}{2}},\det(A)^{-\frac{1}{2}}A)$$ the matrices $\alpha I_2$, $\alpha^2I_2$ are mapped to $(|\alpha|,-I_2)$, $(|\alpha|^2,I_2)$ respectively. Therefore $\Phi$ induces an isomorphism $$\phi:{\hbox{}^{\displaystyle{{\mathbb R}^*_+ \mathrm {SU}(2)}}}\!\big/\!\hbox{}_{ \displaystyle{\langle \alpha^2 I_2\rangle}}\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\simeq\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;} {\hbox{}^{\displaystyle{{\mathbb R}^*_+ \times \mathrm {SU}(2)}}}\!\big/\!\hbox{}_{ \displaystyle{\langle (\alpha^2, I_2)\rangle}}={\hbox{}^{\displaystyle{{\mathbb R}^*_+}}}\!\big/\!\hbox{}_{ \displaystyle{\langle \alpha^2\rangle}}\times \mathrm {SU}(2),$$ and the image of $\alpha I_2$ in the right hand group is $([|\alpha|],-I_2)$. Identifying ${{\mathbb R}^*_+}/{\langle \alpha^2\rangle}$ with $S^1$ via the isomorphisms $[\rho]\mapsto e^{\pi i\frac{\ln(\rho)}{\ln|\alpha|}}$, and noting that the image of $[|\alpha|]$ in $S^1$ via this identification is $-1$, we obtain an isomorphism $${\hbox{}^{\displaystyle{{\mathbb R}^*_+\mathrm {SU}(2)}}}\!\big/\!\hbox{}_{ \displaystyle{\langle \alpha I_2\rangle}}\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\simeq\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;} {\hbox{}^{\displaystyle{S^1\times \mathrm {SU}(2)}}}\!\big/\!\hbox{}_{ \displaystyle{\langle (-1,-I_2)\rangle}}.$$ Therefore: **Corollary 30**. *Let $\alpha\in (-1,0)$ and $f\coloneq\alpha I_2$. The automorphism group of the odd Real Hopf surface $(H_f,{\mathfrak s}_f)$ is naturally isomorphic to the group $$\mathrm {Spin}^c(3)=S^1\times_{{\mathbb Z}_2} \mathrm {Spin}(3)=S^1\times_{{\mathbb Z}_2} \mathrm {SU}(2).$$* ## The Real Picard group of a Real primary Hopf surface For a class VII surface $X$ the canonical Lie group morphism $$\label{isoPic} \mathrm{Hom}(\pi_1(X,x_0),{\mathbb C}^*)=\mathrm{Hom}(H_1(X,{\mathbb Z}),{\mathbb C}^*)\to \mathrm {Pic}(X)$$ is injective and its image is the subgroup $\mathrm {Pic}^T(X)$ of isomorphism classes of holomorphic line bundles with torsion Chern class (see [@Te Remark 3.2.3]). For a primary Hopf surface $X$ we have $H^2(X,{\mathbb Z})=\{0\}$, so $\mathrm {Pic}^0(X)=\mathrm {Pic}^T(X)=\mathrm {Pic}(X)$, so ([\[isoPic\]](#isoPic){reference-type="ref" reference="isoPic"}) is an isomorphism. Since $\pi_1(X,x_0)\simeq {\mathbb Z}$, we obtain an isomorphism $$\lambda:{\mathbb C}^*=\mathrm{Hom}(\pi_1(X,x_0),{\mathbb C}^*)\to\mathrm {Pic}(X)$$ which can be obtained explicitly as follows. Suppose $X=H_f$ for a holomorphic contraction $f\in\mathrm {Aut}_h(W)$ (see section [2.1](#WehlClass){reference-type="ref" reference="WehlClass"}). For $\zeta\in{\mathbb C}^*$ put $$L_\zeta\coloneq {\hbox{}^{\displaystyle{W\times{\mathbb C}}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f_\zeta\rangle}},$$ where $f_\zeta: W\times{\mathbb C}\to W\times{\mathbb C}$ is the fiberwise linear automorphism $$f_\zeta (x,z)\coloneq (f(x),\zeta z).$$ Endowed with the obvious surjective submersion $L_\zeta\to H_f$ given by $$[(x,z)]_{\langle f_\zeta\rangle}\mapsto [x]_{\langle f\rangle}\,,$$ $L_\zeta$ is naturally a holomorphic line bundle on $H_f$. Recall (see for instance [@Te Section 2.2] that **Remark 31**. *The map $\lambda:{\mathbb C}^*\to \mathrm {Pic}(H_f)$ defined by $\lambda(\zeta)=[L_\zeta]$ is a Lie group isomorphism.* The following proposition shows that, for any $f\in IV\cup III\cup II_a\cup II_b\cup II_c$ and any Real structure $s$ on $f$, the anti-holomorphic involutive isomorphism $$\bar s^*: \mathrm {Pic}(H_f)\to \mathrm {Pic}(H_f)$$ induced by $s$ (see section [1](#intro){reference-type="ref" reference="intro"}) is given by the same formula. Recall that, by Proposition [Proposition 8](#AH-autom){reference-type="ref" reference="AH-autom"}, if $H_f$ admits a Real structure, then $f$ has real coefficients. **Proposition 32**. *Let $H$ be a primary Hopf surface, and let $s\in \mathrm{Ah}(H)$ (not necessarily involutive). Then for any $\zeta\in {\mathbb C}^*$ we have $$\bar s^*([L_\zeta])\simeq [L_{\bar \zeta}].$$* *Proof.* By Proposition [Proposition 7](#PrHolAnti){reference-type="ref" reference="PrHolAnti"} we know that $s$ is induced by an anti-holomorphic automorphism $\hat s\in \mathrm{Ah}(W)^f$. The commutative diagram $$\begin{tikzcd}[row sep=large, column sep=large ] W\times{\mathbb C}\ar[r, " (\hat s{,}\bar{\ })"] \ar[d, "f_{\bar\zeta}=(f{,}\bar\zeta\cdot)"']& W\times {\mathbb C}\ar[d, "f_{\zeta}=(f{,} \zeta\cdot)"] \\ W\times{\mathbb C}\ar[r, " (\hat s{,}\bar{\ })"]& W\times {\mathbb C} \end{tikzcd}$$ shows that the map $(\hat s{,}\bar{\ })$ descends to a well defined, fiberwise anti-linear, anti-holomorphic, $s$-lifting isomorphism $L_{\bar \zeta}\to L_{\zeta}$. The same map can be regarded as a fiberwise linear, holomorphic, $s$-lifting isomorphism $L_{\bar \zeta}\to \bar L_{\zeta}$, where this time $s$ has been regarded as a holomorphic map $H_f\to \bar H_f$. Therefore $L_{\bar \zeta}\simeq s^*(\bar L_\zeta)=\bar s^*(L_\zeta)$. ◻ Proposition [Proposition 32](#sbar*){reference-type="ref" reference="sbar*"} shows in particular that, for any Real primary Hopf surface $(H,s)$, the associated Real structure $\bar s^*$ on $\mathrm {Pic}(H)$ is always given, via the isomorphism $\lambda$, by the standard conjugation ${\mathbb C}^*\to{\mathbb C}^*$.\ If follows that if a line bundle $L_\zeta$ on a Real Hopf surface $(H,s)$ admits an anti-holomorphic Real structure $\phi$, then $\zeta\in{\mathbb R}^*$. Let $\zeta\in{\mathbb R}^*$. The proof of Proposition [Proposition 32](#sbar*){reference-type="ref" reference="sbar*"} shows that the map $(\hat s, \bar{\ })$ descends to a fiberwise anti-linear, anti-holomorphic, $s$-lifting isomorphism $\phi_0:L_\zeta\to L_\zeta$. Any fiberwise anti-linear, anti-holomorphic, $s$-lifting isomorphism $\phi:L_\zeta\to L_\zeta$ has the form $\phi=\nu \phi_0$ for a constant $\nu \in{\mathbb C}^*$. Indeed, the composition $\phi\circ \phi_0^{-1}$ is a holomorphic $\mathrm{id}_H$-covering automorphism of $L$, so $\phi\circ \phi_0^{-1}=\nu \mathrm{id}_{L_ \zeta}$ with $\nu \in{\mathbb C}^*$. We have $$\label{phi2onL} \phi^2=(\nu \phi_0)\circ (\nu \phi_0)=|\nu |^2\phi_0^2.$$ On the other hand $\phi_0^2$ is induced by the map $W\times{\mathbb C}\to{\mathbb C}$ given by $$\label{phi02onL} W\ni (x,z)\mapsto (\hat s^2(x),z).$$ - If $s$ is even, we can choose $\hat s$ such that $\hat s^2= \mathrm{id}_W$; in this case $\phi_0$ is already an anti-holomorphic Real structure on $L_ \zeta$ and formula ([\[phi2onL\]](#phi2onL){reference-type="ref" reference="phi2onL"}) shows that the set of all anti-holomorphic Real structures on $L_ \zeta$ is $S^1\phi_0$. - If $\phi$ is odd, we can choose $\hat s$ such that $\hat s^2=f$; in this case formula ([\[phi02onL\]](#phi02onL){reference-type="ref" reference="phi02onL"}) shows that $$\phi^2_0([x,z])=[f(x),z]=[x,\zeta^{-1}z],$$ so $\phi^2_0=\zeta^{-1} \mathrm{id}_{L_\zeta}$. Taking into account ([\[phi2onL\]](#phi2onL){reference-type="ref" reference="phi2onL"}) it follows that $\phi=\nu \phi_0$ is involutive if and only if $\zeta=|\nu|^2$. Therefore, in this case, $L_\zeta$ admits anti-holomorphic Real structures if and only if $\zeta>0$, and if this is the case, the set of anti-holomorphic Real structures on $L_\zeta$ is $S^1\sqrt{\zeta}\phi_0$. We have proved: **Proposition 33**. *Let $(H_f,s)$ be a Real primary Hopf surface, let $\hat s\in \mathrm{Ah}(W)^f$ be a lift of $s$ with $\hat s^2\in \{ \mathrm{id}_W,f\}$ and let $\zeta\in {\mathbb R}^*$.* 1. *If $\hat s^2= \mathrm{id}_W$ (i.e. if $s$ is even) then the set of anti-holomorphic Real structures on $L_\zeta$ is $S^1\phi_0$.* 2. *If $\hat s^2=f$ (i.e. if $s$ is odd) then $L_ \zeta$ admits anti-holomorphic Real structures if and only if $\zeta>0$, and, if this the case, the set of anti-holomorphic Real structures on $L_\zeta$ is $S^1\sqrt{\zeta}\phi_0$.* In all cases the group ${\mathbb C}^* \mathrm{id}_{L_\zeta}$ of holomorphic automorphisms acts on the set of anti-holomorphic Real structures on $L_\zeta$ by conjugation, and the explicit formula for this action is $$(z \mathrm{id}) \circ \phi \circ (z \mathrm{id})^{-1}=z\bar z^{-1}\phi=({z}{|z|^{-1}})^2\phi.$$ Since the map ${\mathbb C}^*\to S^1$ given by $z\mapsto ({z}{|z|^{-1}})^2$ is obviously surjective, it follows that any $\psi\in S^1\phi$ is isomorphic (equivalent) to $\phi$. This shows that, under the assumptions of Proposition [Proposition 33](#SetAHRealStrL){reference-type="ref" reference="SetAHRealStrL"}, all anti-holomorphic Real structures on $L_\zeta$ are isomorphic to either $\phi_0$ (if $\hat s^2= \mathrm{id}_W$), or to $\sqrt{\zeta}\phi_0$ (if $\hat s^2=f$ and $\zeta>0$). In conclusion, we obtain: **Proposition 34**. *Let $(H,s)$ be a Real primary Hopf surface.* 1. *Via the isomorphism ${\mathbb C}^*\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\lambda\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;}\mathrm {Pic}(H)$, the Real structure $\mathrm {Pic}(H)\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\bar s^*\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;}\mathrm {Pic}(H)$ induced by $s$ coincides with the standard conjugation.* 2. *The map $[L_\zeta,\phi]\mapsto \zeta$ defines* 1. *An isomorphism $\mathrm {Pic}_{\mathbb R}(H)\to {\mathbb R}^*$ if $s$ is an even Real structure.* 2. *An isomorphism $\mathrm {Pic}_{\mathbb R}(H)\to {\mathbb R}_{>0}$ if $s$ is an odd Real structure.* The second statement shows that, for odd Real primary Hopf surfaces, the obvious group morphism $\mathrm {Pic}_{\mathbb R}(H)\to \mathrm {Pic}(H)({\mathbb R})$ is not surjective; the classes $[L_\zeta]$ with $\zeta\in{\mathbb R}_{<0}$ do not correspond to Real holomorphic line bundles in the sense of Definition [Definition 2](#RealBdlsHol){reference-type="ref" reference="RealBdlsHol"}, although they are fixed points of $\bar s^*$. # The differential-topological classification ## The differential-topological classification of even Real Hopf surfaces On the product $S^1\times S^3$ we consider the following involutions $$\label{model-involutions} \begin{split} \tau(\zeta,(u,v))&\coloneq(\zeta,(\bar u,\bar v))\\ \tau'(\zeta,(u,v))&\coloneq(\zeta,(\bar u,\zeta\bar v)). \end{split}$$ Our goal is the following classification result: **Theorem 35**. *Any even Real primary Hopf surface $(H,\sigma)$ is diffeomorphic, as a ${\mathbb Z}_2$-manifold, to either $(S^1\times S^3,\tau)$ or $(S^1\times S^3,\tau')$ .* We will also need the following involutions on $S^1\times S^3$: $$j'(\zeta, (u,v))\coloneq(-\zeta,(u,-v)), \ j''(\zeta, (u,v))\coloneq(-\zeta,(-u,-v)).$$ The order 2 groups $\langle j'\rangle$, $\langle j''\rangle$ act freely and properly discontinuously on $S^1\times S^3$, so the we obtain double covers $$q':S^1\times S^3\to Q'\coloneq{\hbox{}^{\displaystyle{S^1\times S^3}}}\!\big/\!\hbox{}_{ \displaystyle{\langle j'\rangle}}, \ q'':S^1\times S^3\to Q''\coloneq{\hbox{}^{\displaystyle{S^1\times S^3}}}\!\big/\!\hbox{}_{ \displaystyle{\langle j''\rangle}}.$$ Note that $\tau$ commutes with $j'$ and $j''$, so we obtain induced involutions $$\theta': Q'\to Q',\ \theta'': Q''\to Q''$$ induced by $\tau$. We will need the following notation: **Definition 36**. *For $\zeta\in S^1$ we denote by $R_\zeta\in \mathrm {SO}(2)\subset\mathrm {GL}(2,{\mathbb C})$ the $2\times 2$ matrix which corresponds to $\zeta$ via the standard isomorphism $S^1\to\mathrm {SO}(2)$.* **Lemma 37**. *Consider the maps $a': S^1\times S^3\to S^1\times S^3$, $a'': S^1\times S^3\to S^1\times S^3$ defined by $$a'(\zeta,(u,v))\coloneq(\zeta^2, (u,\zeta v),\ a''(\zeta,(u,v))\coloneq(\zeta^2,R_\zeta(u,v)),$$* 1. *$a'$ is $\langle j'\rangle$-invariant and induces a diffeomorphism of ${\mathbb Z}_2$-spaces $$\hat a': (Q',\theta')\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\simeq\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;} (S^1\times S^3,\tau').$$* 2. *$a''$ is $\langle j''\rangle$-invariant and induces a diffeomorphism of ${\mathbb Z}_2$-spaces $$\hat a'': (Q'',\theta'')\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\simeq\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;} (S^1\times S^3,\tau).$$* *Proof.* It's easy to see that $a'$ ($a''$) is $\langle j'\rangle$-invariant (respectively $\langle j''\rangle$-invariant) and that the induced maps $\hat a':Q'\to S^1\times S^3$, $\hat a'':Q''\to S^1\times S^3$ are bijective and verify $\tau'\circ \hat a'=\hat a'\circ \theta'$, $\tau\circ \hat a''=\hat a''\circ \theta''$. On the other hand $\hat a'$, $\hat a''$ are local diffeomorphisms because $a'$, $a''$ have this property. Therefore $\hat a'$, $\hat a''$ are diffeomorphisms. ◻ The idea of the proof of Theorem [Theorem 35](#ClassDiffEven){reference-type="ref" reference="ClassDiffEven"} is the following: using our classification Theorem [Theorem 16](#ClassEven){reference-type="ref" reference="ClassEven"} we may suppose that $(H,\sigma)=(H_f,s_f)$, where $f\in IV\cup III\cup {II}_a\cup {II}_b\cup II_c$ is either with real coefficients, or $f\in II'_c$ and $s_f$ is the canonical Real structure on $H_f$. We will show (see Propositions [Proposition 38](#SignsCoeff){reference-type="ref" reference="SignsCoeff"}, [Proposition 44](#ClassDiffII'c){reference-type="ref" reference="ClassDiffII'c"} proved below) that $(H_f,s_f)$ is equivariantly diffeomorphic to $(S^1\times S^3,\tau)$, to $(Q',\theta')$ or to $(Q'',\theta'')$. Theorem [Theorem 35](#ClassDiffEven){reference-type="ref" reference="ClassDiffEven"} will then follow by Lemma [Lemma 37](#Q'Q''){reference-type="ref" reference="Q'Q''"}. **Proposition 38**. *Let $f\in IV\cup III\cup {II}_a\cup {II}_b\cup II_c$ be with real coefficients.* 1. *If the diagonal coefficients of $f$ are positive, then $(H_f,s_f)$ is equivariantly diffeomorphic to $(S^1\times S^3,\tau)$.* 2. *If a single diagonal coefficients of $f$ is negative, then $(H_f,s_f)$ is equivariantly diffeomorphic to $(Q',\theta')$.* 3. *If both diagonal coefficients of $f$ are negative, then $(H_f,s_f)$ is equivariantly diffeomorphic to $(Q'',\theta'')$.* The proof of Proposition [Proposition 38](#SignsCoeff){reference-type="ref" reference="SignsCoeff"} requires a preparation. Let $f\in II_a$, $$f\begin{pmatrix}z\\ w\end{pmatrix}=\begin{pmatrix}\delta^r z+w^r\\ \delta w\end{pmatrix}.$$ Suppose $\delta\in{\mathbb R}$. The second power $f^2$ of $f$ given by $$f^2\begin{pmatrix}z\\ w\end{pmatrix}=\begin{pmatrix} \delta^{2r}z+ 2\delta^{r}w^r\\ \delta^2 w\end{pmatrix}$$ has always positive diagonal coefficients, but, unfortunately $f^2\not\in II_a$. Similar remark for $f\in II_b$. Therefore we will need the slightly larger classes $\widetilde{II}_a\supset II_a$, $\widetilde{II}_b\supset II_b$ defined by $$\widetilde{II}_a\coloneq\left\{ f:W\to W\vert \ f\begin{pmatrix}z\\ w\end{pmatrix}=\begin{pmatrix}\delta^r z+cw^r\\ \delta w\end{pmatrix}, \ 0<|\delta|<1, \ c\in {\mathbb C}\right\}\ \ $$ $$\widetilde{II}_b\coloneq\left\{ f:W\to W\vert \ f\begin{pmatrix}z\\ w\end{pmatrix}=\begin{pmatrix}\alpha z+cw\\ \alpha w\end{pmatrix}, \ 0<|\delta|<1, \ c\in {\mathbb C}\right\}.$$ We begin with the following remark which shows that, any contraction $f\in IV\cup III\cup II_a\cup II_b\cup II_c$ with real coefficients and positive diagonal coefficients can be identified with the term $f^1$ of an explicit smooth 1-parameter group $(f^t)_{t\in{\mathbb R}}$ of holomorphic automorphisms of $W$. More precisely: **Remark 39**. *Let $f\in IV\cup III\cup \widetilde{II}_a\cup \widetilde{II}_b\cup II_c$ with real coefficients and positive diagonal coefficients. For $t\in{\mathbb R}$ we define $f^t\in\mathrm {Aut}_h(W)$ by the formula specified in the third column of the following table: $$\begin{array} {|c|c|c|} \hline &&\\ [-1em] \hbox{The class of } f & f\begin{pmatrix} z\\w \end{pmatrix}& f^{t}\begin{pmatrix} z\\w \end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-1em] IV & \begin{pmatrix}\alpha z\\ \alpha w\end{pmatrix} & \begin{pmatrix}\alpha^{t} z\\ \alpha^{t} w\end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-1em] III & \begin{pmatrix}\delta^r z\\ \delta w\end{pmatrix} & \begin{pmatrix} \delta^{rt}z\\ \delta^{t}w \end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-1em] \widetilde{II}_a & \begin{pmatrix}\delta^r z+cw^r\\ \delta w\end{pmatrix} & \begin{pmatrix} \delta^{rt}z+ct\delta^{r(t-1)}w^r \\ \delta^{t}w \end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-0.7em] \widetilde{II}_b & \begin{pmatrix}\alpha z+cw\\ \alpha w\end{pmatrix} & \begin{pmatrix} \alpha^{t}z+ct\alpha^{t-1}w \\ \alpha^{t}w \end{pmatrix} \\ && \\ [-1em] \hline &&\\ [-1em] II_c &\begin{pmatrix}\alpha z\\ \delta w\end{pmatrix} & \begin{pmatrix} \alpha^{t}z \\ \delta^{t}w \end{pmatrix} \\ [0.8em] \hline \end{array}$$* *Then* 1. *The family $(f^t)_{t\in{\mathbb R}}$ a 1-parameter group of automorphisms of $W$, i.e. the map ${\mathbb R}\ni t\mapsto f^t\in\mathrm {Aut}_h(W)$ is a group morphism.* 2. *$f^1=f$.* *Proof.* This follows by elementary computations. ◻ For a map $\eta:W\to {\mathbb R}$ and $Z=(z,w)\in W$ we define $\eta_Z:{\mathbb R}\to {\mathbb R}$ by $$\label{etaZ} \eta_Z(t)\coloneq\eta (f^t(Z)).$$ Since $(f^t)_{t\in{\mathbb R}}$ is a 1-parameter group of diffeomorphisms we have the identity $$\label{eta'Z} \eta'_Z(t)= \eta'_{f^t(Z)}(0).$$ **Remark 40**. *Let $B\in (0,\infty)$ and $q\in {\mathbb N}^*$. Let $\eta_{B}^{q}:W\to (0,\infty)$ be the map defined by $$\eta_{B}^{q}(z,w)=|z|^{2}+B|w|^{2q}.$$* 1. *$\eta_{B}^{q}$ is a submersion, in particular the fiber $\Sigma_{B}^{q}\coloneq(\eta_{B}^{q})^{-1}(1)$ is a smooth hypersurface of $W$.* 2. *The restriction $n_B^q:\Sigma_B^q\to S^3$ of the normalization map $N:W\to S^3$, $N(Z)\coloneq\frac{1}{\|Z\|} Z$ to $\Sigma_{B}^{q}$ is a diffeomorphism which commmutes with the involutions $(z,w)\mapsto (z,-w)$, $(z,w)\mapsto (-z,-w)$, $(z,w)\mapsto (\bar z,\bar w)$.* *Proof.* The first claim follows by elementary computation. For the second, note first that for any $Z\in W$ the half-line ${\mathbb R}^*_+ Z$ intersects $\Sigma_{B}^{q}$ in a unique point, which will be denoted ${\mathfrak N}_{B}^{q}(Z)$. Using the implicit function theorem it follows easily that the obtained map ${\mathfrak N}_{B}^{q}:W\to \Sigma_{B}^{q}$ is smooth, so the restriction ${\mathfrak n}^q_B\coloneq{\mathfrak N}_{B}^{q}|_{S^3}:S^3\to \Sigma_{B}^{q}$ will also be smooth. It suffices to note that $${\mathfrak n}_{B}^{q} \circ n_{B}^{q}= \mathrm{id}_{\Sigma_{B}^{q}},\ n_{B}^{q}\circ {\mathfrak n}_{B}^{q}= \mathrm{id}_{S^3}.$$ ◻ **Lemma 41**. *Let $f\in IV\cup III\cup \widetilde{II}_a\cup \widetilde{II}_b\cup II_c$ with real coefficients and positive diagonal coefficients. Let $\eta:W\to (0,+\infty)$ and $C<0$ be given by the following table: $$\begin{array} {|c|c|c|c|} \hline &&\\ [-1em] \hbox{The class of } f & f\begin{pmatrix} z\\w \end{pmatrix}& \eta & C \\ && \\ [-1em] \hline &&\\ [-1em] IV & \begin{pmatrix}\alpha z\\ \alpha w\end{pmatrix} & \eta^1_1 & 2\ln(\alpha) \\ && \\ [-1em] \hline &&\\ [-1em] III & \begin{pmatrix}\delta^r z\\ \delta w\end{pmatrix} & \eta^1_1 & 2\ln(\delta)\\ && \\ [-1em] \hline &&\\ [-1em] \widetilde{II}_a & \begin{pmatrix}\delta^r z+cw^r\\ \delta w\end{pmatrix} & \eta^{r}_{B} \hbox{ with} \ B\geq c^2 \frac{1}{ r^2\delta^{2r}\ln(\delta)^2} & r\ln(\delta) \\ && \\ [-1em] \hline &&\\ [-0.7em] \widetilde{II}_b & \begin{pmatrix}\alpha z+cw\\ \alpha w\end{pmatrix} & \eta^{1}_{B} \hbox{ with} \ B\geq c^2\frac{1}{\alpha^2\ln(\alpha)^2} & \ln(\alpha)\\ && \\ [-1em] \hline &&\\ [-1em] II_c &\begin{pmatrix}\alpha z\\ \delta w\end{pmatrix} & \eta^1_1 & 2\max(\ln(\alpha),\ln(\delta))\\ [0.8em] \hline \end{array}$$* 1. *In each case and for any $Z\in W$, the map $\eta_Z$ satisfies the differential inequality $$\label{DiffIneq} \eta_Z'(t)\leq C \eta_Z(t).$$ In particular $\eta_Z$ is strictly decreasing and $$\label{limits} \lim_{t\to \infty} \eta_{Z}(t)=0,\ \lim_{t\to -\infty} \eta_{Z}(t)=+\infty.$$* 2. *Put $\Sigma\coloneq\eta^{-1}(1)$. The map $F:{\mathbb R}\times \Sigma \to W$ defined by $F(t,Z)=f^t(Z)$ is a diffeomorphism.* 3. *Endowing $W$ with the conjugation $c$ and ${\mathbb R}\times \Sigma$ with the involution $\mathrm{id}\times c_\Sigma$ (where $c_\Sigma$ denotes the involution induced by $c$ on $\Sigma$), $F$ is equivariant.* *Proof.* (1) The proof of ([\[DiffIneq\]](#DiffIneq){reference-type="ref" reference="DiffIneq"}) is based on formula ([\[eta\'Z\]](#eta'Z){reference-type="ref" reference="eta'Z"}). For $f\in IV\cup III\cup II_c$ we have $\eta(z,w)=|z|^2+|w|^2$ and the computation of $\eta_Z'(0)$ is very easy. For $f\in \widetilde{II}_a$ one obtains for any $\varepsilon>0$: $$\begin{split} \eta'_Z(0)&=2r\ln(\delta)\big(|z|^2+ \frac{\delta^{-r}}{r\ln(\delta)}c\Re(z{\bar w^r})+ B|w|^{2r}) \\ &\leq 2r\ln(\delta)\left( \bigg( 1+ \frac{\delta^{-r}\varepsilon}{2r \ln(\delta)}\bigg)|z|^2+\bigg(B+ \frac{\delta^{-r}c^2}{2r \ln(\delta)\varepsilon}\bigg)|w^r|^2 \right). \end{split}$$ We first choose $\varepsilon=\delta^r {r|\ln(\delta)|}$ and we note that, for $B\geq \frac{c^2}{ r^2\delta^{2r}\ln(\delta)^2}$ we have $$B+ \frac{\delta^{-r}c^2}{2r \ln(\delta)\varepsilon}\geq \frac{1}{2}B.$$ The case $f\in II_b$ is similar. The formulae ([\[limits\]](#limits){reference-type="ref" reference="limits"}) follow from ([\[DiffIneq\]](#DiffIneq){reference-type="ref" reference="DiffIneq"}) by integrating the inequality $\frac{\eta'}{\eta}\leq C$.\ (2) For the injectivity of $F$ let $(t,Z)$, $(t',Z')\in {\mathbb R}\times\Sigma$ such that $F(t,Z)=F(t',Z')$. This implies $f^{t-t'}(Z')=Z$. Applying $\eta$ on both sides we obtain $\eta_{Z'}(t-t')=1=\eta_{Z'}(0)$. Since $\eta_{Z'}$ is strictly decreasing, this implies $t=t'$, so we also have $Z'=Z$. For the surjectivity, its suffices to note that for any $Z\in W$ there exists $t\in{\mathbb R}$ such that $f^{-t}(Z)\in\Sigma$, which is equivalent to $\eta(f^{-t}(Z))=1$, i.e. $\eta_Z(-t)=1$. But ([\[limits\]](#limits){reference-type="ref" reference="limits"}) shows that $\eta_Z({\mathbb R})=(0,+\infty)$. $F$ is obviously differentiable. To see that it is a diffeomorphism it suffices to prove that $F$ is a local diffeomorphism, i.e. that the differential $d_{(t,Z)} F$ is invertible for any $(t,Z)\in {\mathbb R}\times\Sigma$. For $t\in {\mathbb R}$ denote by $\tau_t:{\mathbb R}\to {\mathbb R}$ the translation by $t$, i.e. $\tau_t(s)=s+t$. Note that $$F\circ (\tau_t, \mathrm{id}_\Sigma)=f^t\circ F.$$ This implies $$F_{* (s+t,Z)}=F_{*(\tau_t, \mathrm{id}_\Sigma)(s,Z)}\circ (\tau_t, \mathrm{id}_\Sigma)_{*(s,Z)}=f^t_{*F(s,Z)}\circ F_{*(s,Z)}.$$ For $s=0$ we get $$F_{*(t,Z)} =f^t_{*Z}\circ F_{*(0,Z)}.$$ Since $f^t$ is a diffeomorphism, $f^t_{*Z}$ is a linear isomorphism, so it suffices to prove that $F_{*(0,Z)}$ is a linear isomorphism. Taking into account the dimensions, it suffices to prove that $\ker(F_{*(0,Z)})=0$. But $$\label{F*(0,Z)} F_{*(0,Z)}(h,v)=\frac{\partial F}{\partial t}(0,Z) h+ v, \ \forall (h,v)\in{\mathbb R}\times T_Z(\Sigma).$$ Let $(h,v)\in{\mathbb R}\times T_Z(\Sigma)$ such that $F_{*(0,Z)}(h,v)=0$. It follows $$0=d\eta(F_{*(0,Z)}(h,v))=h\, d\eta(\frac{\partial F}{\partial t}(0,Z))+ d\eta(v).$$ Since $\eta|_{\Sigma}\equiv 1$ and $v\in T_Z(\Sigma)$ we have $d\eta(v)=0$. On the other hand $$d\eta(\frac{\partial F}{\partial t}(0,Z))=\frac{d}{dt}|_{t=0}\, \eta (F(t,Z))=\frac{d}{dt}|_{t=0}\, \eta_Z (t),$$ which is negative by ([\[DiffIneq\]](#DiffIneq){reference-type="ref" reference="DiffIneq"}). Therefore $F_{*(0,Z)}(h,v)=0$ implies $h=0$. Coming back to ([\[F\*(0,Z)\]](#F*(0,Z)){reference-type="ref" reference="F*(0,Z)"}) we obtain $v=0$.\ (3) Since $f^t$ has real coefficients, for any $t$, we have $$F(t,\bar Z)=f^t(\bar Z)=\overline{f^t(Z)}=\overline{F(t, Z)}.$$ ◻ *Proof.* (of Proposition [Proposition 38](#SignsCoeff){reference-type="ref" reference="SignsCoeff"}) \(1\) By Lemma [Lemma 41](#ft-F){reference-type="ref" reference="ft-F"} the map $F:{\mathbb R}\times \Sigma \to W$ defined by $F(t,Z)=f^t(Z)$ is a diffeomorphism. $F$ induces a diffeomorphism $$\tilde F:({\mathbb R}/{\mathbb Z})\times \Sigma\to {\hbox{}^{\displaystyle{W}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f\rangle}} = H_f$$ given by $\tilde F([t]_{{\mathbb Z}},Z)=[f^t(Z)]_{\langle f \rangle}$. Let $e:{\mathbb R}/{\mathbb Z}\to S^1$ be the standard diffeomorphism and $n:\Sigma\to S^3$ be the given by Remark [Remark 40](#eta-n){reference-type="ref" reference="eta-n"}. We obtain the diffeomorphisms $$S^1\times S^3\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\; e^{-1}\times n^{-1}\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;} ({\mathbb R}/{\mathbb Z})\times \Sigma\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\tilde F\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;} H_f$$ which (by Remark [Remark 40](#eta-n){reference-type="ref" reference="eta-n"} (2) and Lemma [Lemma 41](#ft-F){reference-type="ref" reference="ft-F"} (3)) are equivariant with respect to the involutions $\tau$ on $S^1\times S^3$, $\mathrm{id}\times c_\Sigma$ and $s_f$ on $H_f$. \(2\) Note first that, under our assumption, we have either $f\in III\cup II_a$ with $\delta<0$ and $r$ even (in which cases the second diagonal coefficient of $f$ is negative), or $f\in II_c$ with $\alpha\delta<0$. In the latter case we may suppose $\delta<0$ (see Theorem [Theorem 4](#Wehl){reference-type="ref" reference="Wehl"}(2)). Therefore we may always assume that the second diagonal coefficient of $f$ is negative. We apply Lemma [Lemma 41](#ft-F){reference-type="ref" reference="ft-F"} to $g=f^2$ which has real coefficients and positive diagonal coefficients. Note that for $f\in II_a$ with diagonal coefficients $\delta^r$, $\delta$ we have $f^2\in \widetilde{II}_a$ with diagonal coefficients $(\delta^2)^r$, $\delta^2$ and non-diagonal coefficient $c=2\delta^r$. The diffeomorphism $G:{\mathbb R}\times \Sigma \to W$, $G(t,Z)=g^t(Z)$ given by Lemma [Lemma 41](#ft-F){reference-type="ref" reference="ft-F"} applied to $g$ induces a diffeomorphism $\tilde G:({\mathbb R}/{\mathbb Z})\times \Sigma\to H_g=H_{f^2}$ (as above) which is equivariant with respect to the involutions $\mathrm{id}\times c_\Sigma$ and $s_g$. Our primary Hopf surface $H_f=W/\langle f\rangle$ is identified with the quotient of $H_g$ by the involution $\hat f$ induced by $f$ on $H_g$, which is given explicitly by $$\hat f([Z]_{\langle g\rangle})=[f(Z)]_{\langle g\rangle}$$ and whose fixed point locus is empty. Let ${\mathfrak J}':{\mathbb R}\times\Sigma\to {\mathbb R}\times\Sigma$ be the diffeomorphism $${\mathfrak J}'(t,(u,v))\coloneq\big(t+\frac{1}{2},(u,-v)\big),$$ and let ${\mathfrak j}':({\mathbb R}/{\mathbb Z})\times\Sigma\to ({\mathbb R}/{\mathbb Z})\times\Sigma$ be the *involution* induced by ${\mathfrak J}'$. Direct computations give $$G\circ{\mathfrak J}'=f\circ G,$$ which obviously implies $$\tilde G\circ{\mathfrak j}'=\hat f\circ \tilde G.$$ Therefore $\tilde G$ induces a diffeomorphism $$\hat G: {\mathfrak Q}'\coloneq{\hbox{}^{\displaystyle{{\mathbb R}/{\mathbb Z}\times\Sigma}}}\!\big/\!\hbox{}_{ \displaystyle{\langle {\mathfrak j}'\rangle }}\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\simeq\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;} {\hbox{}^{\displaystyle{H_g}}}\!\big/\!\hbox{}_{ \displaystyle{\langle \hat f\rangle}}=H_f$$ (between the indicated free quotients) which is equivariant with respect to the following involutions: $s_f$ on $H_f$ and the involution ${\mathfrak t}'$ induced by $\mathrm{id}\times c_\Sigma$ on ${\mathfrak Q}'$. It suffices to note that the diffeomorphism $e\times n:{\mathbb R}/{\mathbb Z}\times\Sigma\to S^1\times S^3$ induces a diffeomorphism ${\mathfrak Q}'\to Q'$, which, by Remark [Remark 40](#eta-n){reference-type="ref" reference="eta-n"} (2) is equivariant with respect to the involutions ${\mathfrak t}'$, $\theta'$.\ (3) We use similar arguments noting that in this case $$G\circ{\mathfrak J}''=f\circ G,$$ where ${\mathfrak J}'':{\mathbb R}\times\Sigma\to {\mathbb R}\times\Sigma$ is given by ${\mathfrak J}''(t,(u,v))\coloneq\big(t+\frac{1}{2},(-u,-v)\big)$. Denoting by ${\mathfrak Q}''$ the quotient of ${\mathbb R}/{\mathbb Z}\times\Sigma$ by the involution ${\mathfrak j}''$ induced by ${\mathfrak J}''$, we obtain a diffeomorphism ${\mathfrak Q}''\to Q''$ induced again by $e\times n$ which is equivariant with respect to the involutions ${\mathfrak t}''$ (defined similarly) and $\theta''$. ◻ Let $f\in II'_c$ be of the form $$f\begin{pmatrix}z\\w\end{pmatrix}=\begin{pmatrix}\alpha z\\\bar \alpha w\end{pmatrix}$$ with $|\alpha|<1$, $\alpha\in{\mathbb C}\setminus{\mathbb R}$. Consider the Real structure $c'$ on $W$ given in Example [Example 1](#ExStandard){reference-type="ref" reference="ExStandard"}: $$c'\begin{pmatrix}z\\w\end{pmatrix}=\begin{pmatrix}\bar w\\ \bar z\end{pmatrix}.$$ **Lemma 42**. *Let $\tau:(0,+\infty)\to {\mathbb R}$ be a ${\mathcal C}^\infty$ map. The map $$\Psi_\tau:W\to W,\ \Psi_\tau(Z)\coloneq R_{e^{i\tau(\|Z\|)}} Z$$ is a diffeomorphism.* *Proof.* It suffices to note that $\Psi_\tau$ is obviously differentiable and that, for $\tau$, $\theta\in {\mathcal C}^\infty((0,+\infty), {\mathbb R})$, we have $\Psi_\tau\circ \Psi_\theta=\Psi_{\tau+\theta}$. It follows that $\Psi_\tau\circ \Psi_{-\tau}=\Psi_{-\tau}\circ \Psi_\tau= \mathrm{id}_W$, in particular $\Psi_\tau$ is bijective and its inverse is $\Psi_{-\tau}$, which is also differentiable. ◻ **Lemma 43**. *Let $\theta\in{\mathbb R}$ be such that by $\frac{1}{|\alpha|}\alpha=e^{i\theta}$ and $\tau\in {\mathcal C}^\infty((0,+\infty), {\mathbb R})$ given by $\tau(t)=\frac{\theta}{\ln|\alpha|}\ln(t)$. Let $L\coloneq\begin{pmatrix}1&i\\ 1 & - i\end{pmatrix}\in\mathrm {GL}(2,{\mathbb C})$, $l:W\to W$ the associated diffeomorphism, and ${\mathfrak l}\coloneq l\circ\Psi_\tau$. Then* 1. *We have $$l^{-1}\circ c'\circ l=c,\ l^{-1}\circ f\circ l=|\alpha| R_{\frac{1}{|\alpha|}\alpha}.$$* 2. *We have $${\mathfrak l}^{-1}\circ f\circ {\mathfrak l}=f_{|\alpha|},\ {\mathfrak l}^{-1}\circ c'\circ {\mathfrak l}=c.$$ Therefore ${\mathfrak l}$ induces an equivariant diffeomorphism $(H_{f_{|\alpha|}},s_{f_{|\alpha|}})\stackrel{\hat{\mathfrak l}}{\to}(H_f,s_f)$.* *Proof.* Direct computations ◻ Taking into account that $f_{|\alpha|}$ belongs to the class $IV$ and has positive diagonal coefficients, we obtain by Proposition [Proposition 38](#SignsCoeff){reference-type="ref" reference="SignsCoeff"}(1): **Proposition 44**. *Any even Real Hopf surface $(H_f,s_f)$ with $f\in II'_c$ is equivariantly diffeomorphic to $(S^1\times S^3,\tau)$.* ## The differential-topological classification of odd Real Hopf surfaces The goal of this section is the following classification theorem **Theorem 45**. *Every odd Real primary Hopf surface is equivariantly diffeomorphic to $(S^1\times S^3,\mu)$, where $\mu$ is the involution $$\mu(\zeta, Z)\coloneq(-\zeta, \bar Z).$$* *Proof.* By the classification Theorem [Theorem 25](#ClassOdd){reference-type="ref" reference="ClassOdd"} we know that any odd Real primary Hopf surface is (equivariantly biholomorphically) isomorphic to one of the following: 1. $(H_f,\sigma_f)$, where $f\in IV\cup III\cup II_a\cup II_b\cup II_c$ has real coefficients and postive diagonal coefficients and $\sigma_f$ is induced by $c\circ f^{\frac{1}{2}}$, where $f^{\frac{1}{2}}$ is defined in Remark [Remark 18](#kroot){reference-type="ref" reference="kroot"}. 2. $(H_f,\sigma_f)$, where $f\in II'_c$ and $\sigma_f$ is induced by $c'\circ f^{\frac{1}{2}}$, where $f^{\frac{1}{2}}$ is defined in Proposition [Proposition 21](#class-odd-II'c){reference-type="ref" reference="class-odd-II'c"}. 3. $(H_f,{\mathfrak s}_f)$, where $f\in IV$ has negative diagonal coefficient and ${\mathfrak s}$ is induced by $J\circ f^{\frac{1}{2}}$, where $f^{\frac{1}{2}}$ is defined in Proposition [Proposition 24](#oddIV-neg-coeff){reference-type="ref" reference="oddIV-neg-coeff"}. \(1\) In the first case note that the square root we use the diffeomorphism $$\tilde F:({\mathbb R}/{\mathbb Z})\times \Sigma\to {\hbox{}^{\displaystyle{W}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f\rangle}} = H_f$$ as in the proof of Proposition [Proposition 38](#SignsCoeff){reference-type="ref" reference="SignsCoeff"}, and we note that the involution $\sigma_f$ on $H_f$ corresponds via $\tilde F$ to the involution $$([t],Z)\mapsto ([t+\frac{1}{2}],\bar Z)$$ on $({\mathbb R}/{\mathbb Z})\times \Sigma$, which corresponds to the involution $\mu$ on $S^1\times S^3$ via $e\times n$.\ (2) Lemma [Lemma 43](#II'c-lg){reference-type="ref" reference="II'c-lg"} gives an equivariant diffeomorphism $\hat{\mathfrak l}: H_{f_{|\alpha|}} \to H_f$ induced by ${\mathfrak l}:W\to W$. A direct computation gives $${\mathfrak l}\circ (c\circ f_{|\alpha|}^{\frac{1}{2}})\circ{\mathfrak l}^{-1}=c'\circ f^{\frac{1}{2}},$$ which implies $\hat{\mathfrak l}\circ \sigma_{f_{|\alpha|}}\circ\hat{\mathfrak l}^{-1}=\sigma_f$. Therefore $(H_f,\sigma_f)$ is equivariantly diffeomorphic to $(H_{f_{|\alpha|}}, \sigma_{f_{|\alpha|}})$, which belongs to the class considered above.\ (3) Let $f=f_\alpha$ with $\alpha\in (-1,0)$. As in the even case, put $g\coloneq f^2=f_{\alpha^2}$, note that by Lemma [Lemma 41](#ft-F){reference-type="ref" reference="ft-F"}, the hypersurface $\Sigma$ associated with $g$ coincides with $S^3$, and consider the diffeomorphism $\tilde G:({\mathbb R}/{\mathbb Z})\times S^3\to H_g$ induced by $G:{\mathbb R}\times S^3\to W$. Let ${\mathfrak m}:{\mathbb R}\times S^3\to {\mathbb R}\times S^3$ be the diffeomorphism defined by $${\mathfrak m}(t,Z)\coloneq\big(t+\frac{1}{4},JiZ\big).$$ The induced map $\tilde {\mathfrak m}: ({\mathbb R}/{\mathbb Z})\times S^3\to ({\mathbb R}/{\mathbb Z})\times S^3$ is a diffeomorphism of order 4 of $({\mathbb R}/{\mathbb Z})\times S^3$ whose second power $\tilde{\mathfrak m}^2$ is given by the formula: $$\tilde {\mathfrak m}^2([t],Z)=([t+\frac{1}{2}],-Z).$$ Direct computations give $$\label{mJf1/2} G\circ {\mathfrak m}=(J\circ f^{\frac{1}{2}})\circ G.$$ Recall that our primary Hopf surface $H_f=W/\langle f\rangle$ is identified with the (free) quotient of $H_g$ by the involution $\hat f$ induced by $f$ on $H_g$, and, via $\tilde G$, $\hat f$ corresponds to the involution $${\mathfrak j}'': ({\mathbb R}/{\mathbb Z})\times S^3\to ({\mathbb R}/{\mathbb Z})\times S^3,\ {\mathfrak j}''([t],Z)=([t+\frac{1}{2}],-Z).$$ Therefore $\tilde G$ induces a diffeomorphism $$\hat G: {\mathfrak Q}''\coloneq{\hbox{}^{\displaystyle{({\mathbb R}/{\mathbb Z})\times S^3}}}\!\big/\!\hbox{}_{ \displaystyle{\langle{\mathfrak j}''\rangle}}\to H_f.$$ Formula ([\[mJf1/2\]](#mJf1/2){reference-type="ref" reference="mJf1/2"}) shows that, via $\hat G$, the involution ${\mathfrak s}_f$ on $H_f$ corresponds to the involution $\hat {\mathfrak m}$ induced by $\tilde {\mathfrak m}$ on ${\mathfrak Q}''$. Identifying ${\mathbb R}/{\mathbb Z}$ with $S^1$ and ${\mathfrak Q}''$ with $Q''$ in the usual way, we see that the involution $\hat m$ on $Q''$ which corresponds to $\hat {\mathfrak m}$ is given by $$\hat m([\zeta,Z])=[e^{i\frac{\pi}{2}}\zeta, JiZ].$$ Via the diffeomorphism $\hat a''$ given by Lemma [Lemma 37](#Q'Q''){reference-type="ref" reference="Q'Q''"} the involution on $S^1\times S^3$ which corresponds to $\hat m$ is $$\mu':S^1\times S^3 \to S^1\times S^3,\ \mu'(\zeta, Z)=(-\zeta, i\bar Z).$$ It suffices to note that $(\zeta,Z)\mapsto (\zeta,e^{-i\frac{\pi}{4}}Z)$ defines an equivariant diffeomorphism $(S^1\times S^3,\mu')\to (S^1\times S^3,\mu)$. ◻ **Remark 46**. *Let $\mu_0:S^1\times S^3\to S^1\times S^3$ be the involution $$\mu_0(\zeta,Z)=(-\zeta,Z).$$ Let $\Phi$ be the ${\mathbb R}$-linear orientation preserving isometry $$(z,w)\mapsto (\Re(z)+i\Re(w),\Im(z)+i\Im(w))$$ and $\phi:S^3\to S^3$ the induced diffeomorphism of $S^3$. Let $\psi:S^1\times S^3\to S^1\times S^3$ be the diffeomorphism $(\zeta,(m,n))\mapsto (\zeta,(m,\zeta n))$. The composition $\psi\circ( \mathrm{id}\times\phi)$ is an equivariant diffeomorphism $$(S^1\times S^3,\mu)\to (S^1\times S^3,\mu_0).$$ Therefore, in the classification Theorem [Theorem 45](#ClassDiffOdd){reference-type="ref" reference="ClassDiffOdd"}, we may replace $\mu$ by $\mu_0$.* *Proof.* It suffices to note that, putting $\mu'(\zeta, (z,w) )\coloneq(-\zeta,(z,-w))$, we have $$( \mathrm{id}\times \phi)\circ \mu=\mu'\circ ( \mathrm{id}\times \phi)$$ and that $\mu_0\circ \psi=\psi\circ\mu'$. ◻ ## The Real locus $H^s$ and the quotient $H/\langle s\rangle$ ### The Real locus {#RealLocusSection} Note first that the fixed point locus $M^\sigma$ of any involution $\sigma$ of a differentiable manifold $M$ is a submanifold of $M$. This follows by the equivariant slice theorem ([@TTD Theorem 5.6]), which shows that an point $x\in X^\sigma$ has a $\sigma$-invariant open neighborhood which is equivariantly diffeomorphic to $(T_xX,\sigma_{*,x})$. We have proved that any even Real structure on a primary Hopf surface $H_f$ with $f\in IV\cup III\cup II_a\cup II_b\cup II_c$ is equivalent to the standard Real structure $s_f$ (which is induced by the standard conjugation $c$ when $f$ is with real coefficients, and by $c'$ when $f\in II'_c$). Therefore, for describing the real locus of an arbitrary even Real primary Hopf surface, it suffices to consider only this standard Real structure. The fixed point locus $W^c$ (respectively $W^{c'}$) is $$W^c={\mathbb R}^2\setminus\{0\},\ W^{c'}=\{(z,\bar z)|\ z\in {\mathbb C}^*\}.$$ Let $f^{c}$, respectively $f^{c'}$ be the contraction induced by $f$ on $W^c$ (respectively $W^{c'}$). The canonical maps $${\hbox{}^{\displaystyle{W^c}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f^{c}\rangle }}\to \left[{\hbox{}^{\displaystyle{W}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f\rangle}}\right]^{s_f}=H_f^{s_f},\ {\hbox{}^{\displaystyle{W^{c'}}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f^{c'}\rangle }}\to \left[{\hbox{}^{\displaystyle{W}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f\rangle}}\right]^{s_f}=H_f^{s_f}$$ are obviously diffeomorphisms. **Proposition 47**. *The fixed point locus $H_f^{s_f}$ of a standard even Real primary Hopf surface is diffeomorphic either to the torus $T^2$ or to a Klein bottle according to the sign of the determinant of the real part of $f$. In particular, if $f\in II'_c$, then $H_f^{s_f}$ is a torus.* *Proof.* By the classification Theorem [Theorem 35](#ClassDiffEven){reference-type="ref" reference="ClassDiffEven"} we have only two equivariant diffeomorphism classes of even Real primary Hopf surface. The proof of this theorem shows any even Real primary Hopf surface is equivariantly diffeomorphic to either $(H_g, s_g)$ or $(H_h, s_h)$, where $g$, $h:W\to W$ are the contractions defined by $$g(z,w)=\big(\frac{1}{2}z, \frac{1}{2}w),\ h(z,w)=\big(\frac{1}{2}z, -\frac{1}{2}w).$$ Moreover, this proof also shows that if $f\in II'_c$, then $(H_f, s_f)$ is equivariantly diffeomorphic to $(H_g, s_g)$. It is easy to see that the quotient $W^c/\langle g^c \rangle$ is a torus, whereas the quotient $W^c/\langle h^c\rangle$ is a Klein bottle. ◻ Note that, for $f\in II'_c$, the projections $(z,\bar z)\mapsto z$, $(z,\bar z)\mapsto \bar z$ induce identifications $${\hbox{}^{\displaystyle{W^{c'}}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f^{c'}\rangle}}\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\simeq\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;} {\hbox{}^{\displaystyle{{\mathbb C}^*}}}\!\big/\!\hbox{}_{ \displaystyle{\langle \alpha\rangle}},\ {\hbox{}^{\displaystyle{W^{c'}}}}\!\big/\!\hbox{}_{ \displaystyle{\langle f^{c'}\rangle}}\mathop{\vbox{\ialign{ ##\crcr ${\scriptstyle\hfil\;\;\simeq\;\;\hfil}$\crcr \noalign{\kern 1pt\nointerlineskip} \rightarrowfill\crcr}}\;} {\hbox{}^{\displaystyle{{\mathbb C}^*}}}\!\big/\!\hbox{}_{ \displaystyle{\langle \bar \alpha\rangle}},$$ where $\alpha$, $\bar\alpha$ are the coefficients of $f$. This shows that, in this case, the Real locus $H_f^{s_f}$ comes with a canonical (non-oriented) conformal structure, which is conformally isomorphic to the elliptic curve $E_\alpha\coloneq{{\mathbb C}^*}/{\langle \alpha\rangle}$. In other words: **Remark 48**. *When $f$ belongs to the subclass $II'_c$, the real locus $H_f^{s_f}$ is a 2-torus which comes with a natural (non-oriented) conformal structure.* For the odd Real structures we have: **Remark 49**. *The Real locus of any odd Real primary Hopf surface is empty.* *Proof.* Let $(H_f,\sigma)$ be an odd Real Hopf surface and $\hat\sigma\in Aut_h(W)$ be a lift of $\sigma$ to $W$ such that $\hat\sigma^2=f$. Suppose that $x=[(z,w)]\in H_f$ is a fixed point of $\sigma$. Therefore there exists $k\in {\mathbb Z}$ such that $\hat\sigma(z,w)=f^k(z,w)$. Therefore $\hat\sigma^{2k-1}(z,w)=(z,w)$, which implies $\hat\sigma^{2(2k-1)}(z,w)=\hat\sigma^{2k-1}(z,w)=(z,w)$. We obtain $f^{2k-1}(z,w)=(z,w)$. Since $\langle f\rangle$ acts freely on $W$, it follows $2k-1=0$ (contradiction). ◻ ### The quotient of a Real Hopf surface by its involution Note first that the quotient $X/\langle \sigma\rangle$ associated with *any* Real complex surface $(X,\sigma)$ is a topological 4 manifold. This follows using 1. The equivariant slice theorem quoted above. 2. The classification of anti-linear involutions on a complex vector space (see for instance Remark [Remark 13](#linearRealRem){reference-type="ref" reference="linearRealRem"} in this article). 3. The homeomorphism ${\mathbb R}^2/\langle - \mathrm{id}_{{\mathbb R}^2}\rangle\simeq{\mathbb R}^2$ induced (for instance) by the $- \mathrm{id}_{{\mathbb R}^2}$-invariant map $$\beta:{\mathbb R}^2\to{\mathbb R}^2,\ \beta(\rho\cos(\theta),\rho\sin(\theta))=(\rho^2\cos(2\theta),\rho^2\sin(2\theta)).$$ Using a complet coordinate $\zeta$ on ${\mathbb R}^2$, this map is given by $\zeta\mapsto \zeta^2$. Taking into account this remark we will describe the quotient associated with a Real primary Hopf surface as a topological 4-manifold. We have seen that any even Real primary Hopf surface is isomorphic to $(H_f,s_f)$, where either $f\in IV\cup II\cup II_a\cup II_b\cup II_c$ with real coefficients, or $f\in II'_c$. The involution $s_f$ is induced by the *anti-linear* involution $c$, respectively $c'$ on $W$. The quotient $$Q_f\coloneq{\hbox{}^{\displaystyle{H_f}}}\!\big/\!\hbox{}_{ \displaystyle{\langle s_f\rangle }}$$ can be identified with the quotient ${\mathfrak W}\coloneq W/\langle c \rangle$, respectively ${\mathfrak W}'\coloneq W/\langle c' \rangle$ by the contraction ${\mathfrak f}$ induced by $f$ on ${\mathfrak W}$, respectively ${\mathfrak W}'$. We obtain a decomposition ${\mathbb C}^2=V_+\oplus V_-$ of ${\mathbb C}^2$ as direct sum of $c$-invariant (respectively $c'$-invariant) 2-dimensional real linear subspaces $V_\pm$ such that $c|_{V_\pm }=\pm \mathrm{id}_{V_\pm}$ (respectively $c'|_{V_\pm }=\pm \mathrm{id}_{V_\pm}$). Therefore the quotient ${\mathbb C}^2/\langle c\rangle$ (${\mathbb C}^2/\langle c'\rangle$) can be identified with the quotient $${\mathbb V}_+\times {\hbox{}^{\displaystyle{V_-}}}\!\big/\!\hbox{}_{ \displaystyle{\langle- \mathrm{id}_{V_-}\rangle}}\simeq {\mathbb R}^4$$ Here we have used linear isomorphisms $V_\pm\simeq{\mathbb R}^2$ and the homeomorphism $${\mathbb R}^2/\langle- \mathrm{id}_{{\mathbb R}^2}\rangle\to {\mathbb R}^2$$ induced by $\beta$. The image of $W/\langle c\rangle$ ($W/\langle c'\rangle$) via this homeomorphism is ${\mathbb R}^4\setminus\{0\}$ and the image of the fixed point locus $W^c$ ($W^{c'}$) in ${\mathfrak W}$ (${\mathfrak W}'$) is $({\mathbb R}^2\setminus\{0\})\times\{0\}$. Using the classification Theorem [Theorem 35](#ClassDiffEven){reference-type="ref" reference="ClassDiffEven"} we see that in all cases the contraction ${\mathcal F}$ induced by $f$ on $W/\langle c\rangle$ ($W/\langle c'\rangle$) is orientation preserving. Therefore we obtain the following result, which describes the quotient $Q_f$ associated with an even Real primary Hopf surface, as well as the image of the fixed point locus $H_f^{s_f}$ in this quotient: **Proposition 50**. *With the notations above we have* 1. *The quotient $Q_f\coloneq{H_f}/{\langle s_f\rangle}$ can be identified topologically with the quotient $${\mathcal Q}_f={\hbox{}^{\displaystyle{{\mathbb R}^4\setminus\{0\}}}}\!\big/\!\hbox{}_{ \displaystyle{\langle {\mathcal F}\rangle }}$$ of ${\mathbb R}^4\setminus\{0\}$ by the cyclic group generated by an orientation preserving contraction ${\mathcal F}$. Therefore, in all cases $Q_f$ is homeomorphic to $S^1\times S^3$.* 2. *${\mathcal F}$ leaves invariant the punctured plane $({\mathbb R}^2\setminus\{0\})\times\{0\}$ and, via the above identification, the fixed point locus $H_f^{s_f}$ corresponds to the quotient of ${\mathbb R}^2\setminus\{0\}$ by the contraction ${\mathcal F}_0$ induced by ${\mathcal F}$.* 3. *${\mathcal F}_0$ is orientation preserving if and only if the diagonal coefficients of $f$ have the same sign. If this is the case ${\mathbb R}^2\setminus\{0\}/\langle {\mathcal F}_0\rangle$ is a 2-torus. If the diagonal coefficients of $f$ have opposite signs, ${\mathbb R}^2\setminus\{0\}/\langle {\mathcal F}_0\rangle$ is a Klein bottle.* Taking into account the classification Theorem [Theorem 45](#ClassDiffOdd){reference-type="ref" reference="ClassDiffOdd"} and Remark [Remark 46](#mu-mu0){reference-type="ref" reference="mu-mu0"} we obtain the following simple description of the quotient $H/\langle\sigma\rangle$ of any odd Real primary Hopf surface $(H,\sigma)$: **Proposition 51**. *Let $(H,\sigma)$ be an odd Real primary Hopf surface. The quotient $H/\langle\sigma\rangle$ can be identified with $S^1\times S^3$ and the canonical projection $$H\to H/\langle\sigma\rangle$$ is a double cover whose (non-trivial) deck transformation is an anti-holomorphic involution.* AAAa M. Atiyah, *K-theory*, W.A. Benjamin, New York (1967). W. Barth, K. Hulek, Ch. Peters, A. Van de Ven: *Compact Complex Surfaces*, Springer (2004). B. Gross and J. Harris, Real algebraic curves, Ann. scient. Éc. Norm. Sup., $4^e$ série 14, 157--182 (1981). K. Kodaira, On the structure of compact complex analytic surfaces. Americain Journal of Mathematics, The Johns Hopkins University Press, 88 (3), 682-721 (1966). Ch. Okonek, A. Teleman, Abelian Yang-Mills Theory on Real Tori and Theta Divisors of Klein Surfaces, Commun. Math. Phys. 323, 813--858 (2013). R. Silhol: Real Algebraic Surfaces, Lecture Notes in Math. 1392, Springer-Verlag, Berlin (1989). A. Teleman, Non-Kählerian compact complex surfaces, In: Angella, D., Arosio, L., Di Nezza, E. (eds) Complex Non-Kähler Geometry. Lecture Notes in Mathematics, vol 2246 (2019), Springer, Cham. Tammo tom Dieck, Transformation Groups, de Gruyter, Berlin-New York (1987). J. Wehler, Versal deformation of Hopf Surfaces, Journal für die reine und angewandte Mathematik, Vol. 328, 22-32 (1981).

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--- abstract: | We construct and analyze a multiscale finite element method for an elliptic distributed optimal control problem with pointwise control constraints, where the state equation has rough coefficients. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for smooth problems and present corroborating numerical results. address: - Department of Mathematics and Center for Computation & Technology, Louisiana State University, Baton Rouge, LA 70803, USA - Institute of Mathematics, Universtät Augsburg, 86159 Augsburg, Germany - Department of Mathematics and Center for Computation & Technology, Louisiana State University, Baton Rouge, LA 70803, USA author: - Susanne C. Brenner - José C. Garay - Li-yeng Sung date: September 11, 2023 title: | A Multiscale Finite Element Method for an Elliptic Distributed Optimal Control Problem\ with Rough Coefficients and Control Constraints --- [^1] # Introduction {#sec:Introduction} Let $\Omega\subset \mathbb{R}^d$ ($d=1,2,3$) be a polytopal domain, $y_d\in{L_2(\Omega)}$ and $\gamma\leq 1$ be a positive constant. The model optimal control problem (cf. [@Lions:1971:OC; @Troltzsch:2010:OC]) is to find $$\label{eq:OCP} (\bar y,\bar u)=\mathop{\rm argmin}_{(y,u)\in \mathbb{K}} J(y,u),$$ where the cost function $J:{H^1_0(\Omega)}\times{L_2(\Omega)}\longrightarrow [0,\infty)$ is defined by $$\label{eq:JDef} J(y,u)=\frac12\big(\|y-y_d\|_{L_2(\Omega)}^2+\gamma \|u\|_{L_2(\Omega)}^2\big),$$ the closed convex subset $\mathbb{K}$ of ${H^1_0(\Omega)}\times{L_2(\Omega)}$ is defined by the conditions $$\begin{aligned} {3} a(y,z)&=\int_\Omega uz\,dx &\qquad&\forall\,z\in {H^1_0(\Omega)},\label{eq:PDEConstraint}\\ \phi_1\leq &\, u\leq\phi_2&\qquad&\text{a.e. in $\Omega$}, \label{eq:ControlConstraints}\end{aligned}$$ and the bilinear form $a(\cdot,\cdot)$ is given by $$\label{eq:aDef} a(y,z)=\int_\Omega\mathcal{A}\nabla y\cdot\nabla z\,dx.$$ We assume that the components of the symmetric positive definite matrix $\mathcal{A}$ belong to $L_\infty(\Omega)$, and that there exist positive constants $\alpha$ and $\beta$ such that $$\label{eq:Ellipticity} \text{the eigenvalues of $\mathcal{A}$ are bounded below (resp., above) by $\alpha$ (resp., $\beta$).}$$ For the constraint functions $\phi_1$ and $\phi_2$, we assume $$\label{eq:ConstraintRegularity} \text{$\phi_1$ and $\phi_2$ belong to $H^1(\Omega)$},$$ and $$\label{eq:ConstraintInequality} \text{$\phi_1\leq\phi_2\;$ a.e. on $\Omega$.}$$ **Remark 1**. *Throughout this paper we follow the standard notation for differential operators, functions spaces and norms that can be found for example in [@Ciarlet:1978:FEM; @ADAMS:2003:Sobolev; @BScott:2008:FEM].* **Remark 2**. *The rough coefficients in the title of the paper refer to the fact that [\[eq:Ellipticity\]](#eq:Ellipticity){reference-type="eqref" reference="eq:Ellipticity"} is the only assumption on the matrix $\mathcal{A}$. Under this assumption we have the relation $$\label{eq:SpaceComparison} \alpha|v|_{H^1(\Omega)}^2\leq \|v\|_a^2={a(v,v)}\leq \beta|v|_{H^1(\Omega)}^2 \qquad\forall\,v\in H^1(\Omega)$$ and nothing more. In particular, we do not assume the solution $y$ of [\[eq:PDEConstraint\]](#eq:PDEConstraint){reference-type="eqref" reference="eq:PDEConstraint"} belongs to $H^{1+s}(\Omega)$ for some positive $s$.* It is well-known that standard finite element methods for elliptic boundary value problems with rough coefficients can converge arbitrarily slowly (cf. [@BO:2000:Bad]). This is of course also the case for the optimal control problem defined by [\[eq:OCP\]](#eq:OCP){reference-type="eqref" reference="eq:OCP"}--[\[eq:aDef\]](#eq:aDef){reference-type="eqref" reference="eq:aDef"}. Our goal is to design a multiscale finite element method whose performance is in some sense similar to that of the standard finite element methods for smooth problems. The literature on the numerical solution of this optimal control problem is relatively small. For problems with scale separations and periodic structures, the method in [@LCY:2013:MS] is based on an asymptotic expansion of the solution, the method in [@CHLY:2015:MS] is based on the multiscale finite element space in [@CH:2003:Oscillating], and the method in [@GYWLY:2018:HMMOCP] is based on the heterogeneous multiscale method in [@EE:2003:HMSFEM]. For problems that do not assume scale separations or periodic structures, a numerical method based on the multiscale finite element space in [@CEL:2018:Multiscale] was investigated in [@AC:2022:DD26], and a numerical method based on a generalization of the multiscale finite element space in [@OZB:2014:Homogenization] has just appeared in [@CLZZ:2023:MultiscaleCC]. Our method is based on the local orthogonal decomposition (LOD) methodology (cf. [@MP:2021:LOD]) which, like the methods in [@AC:2022:DD26; @CLZZ:2023:MultiscaleCC], does not require scale separations or periodic structures in the coefficient matrix $\mathcal{A}(x)$. A variant of the LOD method for elliptic optimal control problems with rough coefficients but without control constraints can also be found in [@BGS:2022:LOD_OCP]. The rest of the paper is organized as follows. The properties of the continuous problem are recalled in Section [2](#sec:Continuous){reference-type="ref" reference="sec:Continuous"} and a discretization of the optimal control problem is analyzed in Section [3](#sec:FEMs){reference-type="ref" reference="sec:FEMs"}, where we present error estimates that are convenient for the error analysis of multiscale finite element methods. The construction and analysis of our multiscale finite element method are presented in Section [4](#sec:DDLOD){reference-type="ref" reference="sec:DDLOD"}, followed by numerical results in Section [5](#sec:Numerics){reference-type="ref" reference="sec:Numerics"}. We end with some concluding remarks in Section [6](#sec:Conclusions){reference-type="ref" reference="sec:Conclusions"}. # The Continuous Problem {#sec:Continuous} In this section we recall some well-known facts about the optimal control problem that can be found for example in [@Lions:1971:OC; @Troltzsch:2010:OC]. Since $\mathbb{K}$ is nonempty under [\[eq:ConstraintInequality\]](#eq:ConstraintInequality){reference-type="eqref" reference="eq:ConstraintInequality"} and $J$ is strictly convex and coercive, the minimization problem defined by [\[eq:OCP\]](#eq:OCP){reference-type="eqref" reference="eq:OCP"}--[\[eq:aDef\]](#eq:aDef){reference-type="eqref" reference="eq:aDef"} has a unique solution characterized by the first order optimality condition (cf. [@KS:1980:VarInequalities; @ET:1999:Convex]) $$\label{eq:OptimalityCondition1} \int_\Omega(\bar y-y_d)(y-\bar y)dx+\gamma\int_\Omega\bar u(u-\bar u)dx\geq0 \qquad\forall\,(y,u)\in\mathbb{K}.$$ Let the adjoint state $\bar p\in{H^1_0(\Omega)}$ be defined by $$\label{eq:AdjointState} a(q,\bar p)=\int_\Omega(\bar y-y_d)q\,dx\qquad\forall\,q\in {H^1_0(\Omega)}.$$ One can use [\[eq:PDEConstraint\]](#eq:PDEConstraint){reference-type="eqref" reference="eq:PDEConstraint"} and [\[eq:AdjointState\]](#eq:AdjointState){reference-type="eqref" reference="eq:AdjointState"} to write $$\label{eq:ASRelation} \int_\Omega(\bar y-y_d)y\,dx= a(y,\bar p)=\int_\Omega u\bar p\,dx \qquad\forall\,(y,u)\in\mathbb{K},$$ and then [\[eq:OptimalityCondition1\]](#eq:OptimalityCondition1){reference-type="eqref" reference="eq:OptimalityCondition1"} is equivalent to the inequality $$\label{eq:OptimalityCondition2} \int_\Omega(\bar p+\gamma \bar u)(u-\bar u)dx\geq0 \qquad \forall\,u\in K,$$ where $$K=\{u\in {L_2(\Omega)}:\,\phi_1\leq u\leq\phi_2\quad\text{a.e. in $\Omega$}\}.$$ The inequality [\[eq:OptimalityCondition2\]](#eq:OptimalityCondition2){reference-type="eqref" reference="eq:OptimalityCondition2"} is equivalent to the statement that $\bar u$ is the ${L_2(\Omega)}$-orthogonal projection of $-\gamma^{-1}\bar p$ on the closed convex subset $K$, i.e., $$\label{eq:Truncation} \bar u=\max(\phi_1,\min(\phi_2,-\gamma^{-1}\bar p)),$$ which, in view of [\[eq:ConstraintRegularity\]](#eq:ConstraintRegularity){reference-type="eqref" reference="eq:ConstraintRegularity"}, implies in particular that (cf. [@GT:2001:EllipticPDE Lemma 7.6]) $$\bar u\in H^1(\Omega).$$ For the analysis of problems with rough coefficients, it is desirable to keep track of the dependence of $|\bar u|_{H^1(\Omega)}$ on $\alpha$ and $\beta$. This can be achieved by using the estimate $$\|\bar y-y_d\|_{L_2(\Omega)}^2\leq 2 J(y_*,u_*)$$ that holds for any convenient choice of $(y_*,u_*)\in \mathbb{K}$. One can then bound $|\bar p|_{H^1(\Omega)}$ through [\[eq:AdjointState\]](#eq:AdjointState){reference-type="eqref" reference="eq:AdjointState"} and then obtain an estimate of $|\bar u|_{H^1(\Omega)}$ through [\[eq:Truncation\]](#eq:Truncation){reference-type="eqref" reference="eq:Truncation"}. For example, under the additional assumption $\phi_1\leq 0\leq \phi_2$ almost everywhere in $\Omega$, we can take $(y_*,u_*)=(0,0)$ to obtain a simple bound $$\label{eq:AprioriBound1} \|\bar y-y_d\|_{L_2(\Omega)}^2\leq 2J(0,0)=\|y_d\|_{L_2(\Omega)}^2.$$ It then follows from [\[eq:SpaceComparison\]](#eq:SpaceComparison){reference-type="eqref" reference="eq:SpaceComparison"}, [\[eq:AdjointState\]](#eq:AdjointState){reference-type="eqref" reference="eq:AdjointState"} and [\[eq:AprioriBound1\]](#eq:AprioriBound1){reference-type="eqref" reference="eq:AprioriBound1"} that $$\alpha|\bar p|_{H^1(\Omega)}^2\leq a(\bar p,\bar p) =\int_\Omega(\bar y-y_d)\bar p\,dx \leq \| y_d\|_{L_2(\Omega)}\|\bar p\|_{L_2(\Omega)},$$ which implies $$\label{eq:AprioriBound2} |\bar p|_{H^1(\Omega)}\leq (\mathrm{C}_{\mathrm{PF}}/\alpha)\|y_d\|_{L_2(\Omega)}$$ through the Poincaré-Friedrichs inequality $$\label{eq:PF} \|v\|_{L_2(\Omega)}\leq \mathrm{C}_{\mathrm{PF}}|v|_{H^1(\Omega)}\qquad\forall\,v\in {H^1_0(\Omega)}.$$ Putting [\[eq:Truncation\]](#eq:Truncation){reference-type="eqref" reference="eq:Truncation"} and [\[eq:AprioriBound2\]](#eq:AprioriBound2){reference-type="eqref" reference="eq:AprioriBound2"} together, we arrive at the bound $$|\bar u|_{H^1(\Omega)}\leq \max\big(|\phi_1|_{H^1(\Omega)},|\phi_2|_{H^1(\Omega)}, \gamma^{-1}({\mathrm{C}_{\mathrm{PF}}}/\alpha)\|y_d\|_{L_2(\Omega)}\big).$$ **Remark 3**. *Under the general assumption [\[eq:ConstraintInequality\]](#eq:ConstraintInequality){reference-type="eqref" reference="eq:ConstraintInequality"}, we can take $u_*=(\phi_1+\phi_2)/2$ and obtain a (more complicated) upper bound for $|u|_{H^1(\Omega)}$ that depends only on $\|\phi_1\|_{H^1(\Omega)}$, $\|\phi_2\|_{H^1(\Omega)}$, $\|y_d\|_{L_2(\Omega)}$, $\gamma^{-1}$ and $\alpha^{-1}$.* Next we define $$\label{eq:lambda} \lambda=\bar p+\gamma \bar u$$ and obtain through [\[eq:Truncation\]](#eq:Truncation){reference-type="eqref" reference="eq:Truncation"} the decomposition $$\label{eq:LambdaDecomposition} \lambda=\lambda_1+\lambda_2,$$ where $$\lambda_1=\max(\bar p+\gamma\phi_1,0)\in H^1(\Omega) \quad\text{and} \quad \lambda_2=\min(\bar p+\gamma\phi_2,0)\in H^1(\Omega)$$ satisfy [\[subeq:lambda\]]{#subeq:lambda label="subeq:lambda"} $$\begin{aligned} \lambda_1&\geq0 ,\label{eq:lambda1Sign}\\ \lambda_1(\phi_1-\bar u)&=0, \label{eq:lambda1Complementarity}\\ \nabla\lambda_1&=\begin{cases} \nabla\bar p+\gamma\nabla \phi_1&\quad\text{in $\mathfrak{A}_1$}\\[2pt] 0 &\quad\text{in $\Omega\setminus\mathfrak{A}_1$} \end{cases},\label{eq:lambda1Gradient}\\ \lambda_2&\leq0,\label{eq:lambda2Sign}\\ \lambda_2(\phi_2-\bar u)&=0, \label{eq:lambda2Complementarity}\\ \nabla\lambda_2&=\begin{cases} \nabla\bar p+\gamma\nabla \phi_2&\quad\text{in $\mathfrak{A}_2$}\\[2pt] 0 &\quad\text{in $\Omega\setminus\mathfrak{A}_2$} \end{cases}.\label{eq:lambda2Gradient}\end{aligned}$$ Here the active set $\mathfrak{A}_j$ is the closure in $\Omega$ of the set of the Lebesgue points where $\bar u-\phi_j=0$. # A Discretization of the Optimal Control Problem {#sec:FEMs} Let $\mathcal{T}_\rho$ be a simplicial/quadrilateral triangulation of $\Omega$ with mesh size $\rho$ and $W_\rho\subset {L_2(\Omega)}$ be the space of piecewise constant functions with respect to $\mathcal{T}_\rho$. The optimal control $\bar u$ will be approximated by functions in $W_\rho$, while the approximation of $\bar y$ comes from a subspace $V_*$ of ${H^1_0(\Omega)}$. **Remark 4**. *By allowing $V_*$ to be an arbitrary subspace of $H^1_0(\Omega)$, the analysis developed below can be applied to standard finite element methods and multiscale finite element methods.* The discrete problem is to find $$\label{eq:DOCP} (\bar y_{*,\rho},\bar u_{*,\rho})=\mathop{\rm argmin}_{(y_*,u_\rho)\in \mathbb{K}_{*,\rho}} J(y_*,u_\rho),$$ where the closed convex subset $\mathbb{K}_{*,\rho}$ of $V_*\times W_\rho$ is defined by the following conditions: $$\begin{aligned} {3} a(y_*,z_*)&=\int_\Omega u_\rho z_* dx&\qquad&\forall\,z_*\in V_*, \label{eq:DPDE}\\ Q_\rho \phi_1\leq\,&u_\rho\leq Q_\rho\phi_2&\qquad&\text{a.e. in $\Omega$}, \label{eq:DCC}\end{aligned}$$ and $Q_\rho$ is the orthogonal projection from ${L_2(\Omega)}$ onto $W_\rho$. We have a standard interpolation error estimate (cf. [@Ciarlet:1978:FEM; @BScott:2008:FEM]) $$\label{eq:QEstimate} \|\zeta-Q_\rho\zeta\|_{L_2(\Omega)}\leq C_\maltese\rho|\zeta|_{H^1(\Omega)},$$ where the positive constant $C_\maltese$ only depends on the shape regularity of $\mathcal{T}_\rho$. Since $Q_\rho u$ satisfies [\[eq:DCC\]](#eq:DCC){reference-type="eqref" reference="eq:DCC"} for any $u$ that satisfies [\[eq:ControlConstraints\]](#eq:ControlConstraints){reference-type="eqref" reference="eq:ControlConstraints"}, the set $\mathbb{K}_{*,\rho}$ is nonempty and the discrete convex minimization problem has a unique solution characterized by the first order optimality condition $$\label{eq:DOC} \int_\Omega(\bar y_{*,\rho}-y_d)(y_*-\bar y_{*,\rho})dx+ \gamma\int_\Omega\bar u_{*,\rho}(u_\rho-\bar u_{*,\rho})dx\geq0 \qquad \forall\, (y_*,u_\rho)\in \mathbb{K}_{*,\rho}.$$ The error analysis for $(\bar y_{*,\rho},\bar u_{*,\rho})$ was carried out in the pioneering work [@Falk:1973:Control] on finite element methods for elliptic optimal control problems. Here we present a self-contained treatment that is suitable for the analysis of the multiscale finite element method in Section [4](#sec:DDLOD){reference-type="ref" reference="sec:DDLOD"}. The following lemma is useful for the error analysis. **Lemma 5**. *Let $g\in{L_2(\Omega)}$ and $w_*\in V_*$ satisfy $$a(w_*,v_*)=\int_\Omega g v_* dx\qquad\forall\,v_*\in V_*.$$ Then we have $$\begin{aligned} \|w_*\|_{L_2(\Omega)}&\leq (\mathrm{C}_{\mathrm{PF}}^2/\alpha)\|g\|_{L_2(\Omega)},\label{eq:EasyEstimate}\\ \|w_*\|_a&\leq (\mathrm{C}_{\mathrm{PF}}/\sqrt\alpha)\|g\|_{L_2(\Omega)}.\label{eq:EasyEnergyEstimate}\end{aligned}$$* *Proof.* It follows from [\[eq:SpaceComparison\]](#eq:SpaceComparison){reference-type="eqref" reference="eq:SpaceComparison"}, [\[eq:PF\]](#eq:PF){reference-type="eqref" reference="eq:PF"} and the Cauchy-Schwarz inequality that $$\begin{aligned} \|w_*\|_{L_2(\Omega)}^2\leq \mathrm{C}_{\mathrm{PF}}^2|w_*|_{H^1(\Omega)}^2 &\leq (\mathrm{C}_{\mathrm{PF}}^2/\alpha)a(w_*,w_*)\\ &= (\mathrm{C}_{\mathrm{PF}}^2/\alpha)\int_\Omega gw_*dx\leq (\mathrm{C}_{\mathrm{PF}}^2/\alpha)\|g\|_{L_2(\Omega)}\|w_*\|_{L_2(\Omega)},\end{aligned}$$ which implies [\[eq:EasyEstimate\]](#eq:EasyEstimate){reference-type="eqref" reference="eq:EasyEstimate"}. The estimate [\[eq:EasyEnergyEstimate\]](#eq:EasyEnergyEstimate){reference-type="eqref" reference="eq:EasyEnergyEstimate"} also follows from [\[eq:SpaceComparison\]](#eq:SpaceComparison){reference-type="eqref" reference="eq:SpaceComparison"}, [\[eq:PF\]](#eq:PF){reference-type="eqref" reference="eq:PF"} and the Cauchy-Schwarz inequality: $$\|w_*\|_a^2=a(w_*,w_*)=\int_\Omega gw_*dx\leq \|g\|_{L_2(\Omega)}\|w_*\|_{L_2(\Omega)}\leq \|g\|_{L_2(\Omega)}(\mathrm{C}_{\mathrm{PF}}/\sqrt\alpha)\|w_*\|_a.$$ ◻ We will include the approximation of $\bar p$ by $\bar p_{*,\rho}$ in the error analysis, where $\bar p_{*,\rho}\in V_*$ is defined by $$\label{eq:bps} a(q_*,\bar p_{*,\rho})=\int_\Omega(\bar y_{*,\rho}-y_d)q_* dx\qquad\forall\,q_*\in V_*.$$ **Theorem 6**. *There exists a positive constant $C_\dag$, depending only on $\|y_d\|_{L_2(\Omega)}$, $\|\phi_1\|_{H^1(\Omega)}$, $\|\phi_2\|_{H^1(\Omega)}$, $\gamma^{-1}$, $\alpha^{-1}$ and the shape regularity of $\mathcal{T}_\rho$, such that $$\label{eq:AbstractErrorEstimate} \|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}+\|\bar p-\bar p_{*,\rho}\|_{L_2(\Omega)} \leq C_\dag(\|\bar y-\dot y_{*}\|_{L_2(\Omega)}+ \|\bar p-\dot p_{*}\|_{L_2(\Omega)}+\rho),$$ where $\dot y_{*},\dot p_{*}\in V_*$ are defined by $$\begin{aligned} {3} a(\dot y_{*},z_*)&=\int_\Omega\bar u z_* dx&\qquad&\forall\,z_*\in V_*,\label{eq:dys}\\ a(q_*,\dot p_{*})&=\int_\Omega(\bar y-y_d)q_* dx&\qquad&\forall\,q_*\in V_*.\label{eq:dps}\end{aligned}$$* *Proof.* First we note the following analog of [\[eq:ASRelation\]](#eq:ASRelation){reference-type="eqref" reference="eq:ASRelation"}: $$\label{eq:DiscreteASRelation} \int_\Omega(\bar y-y_d)y_*dx= a(y_*,\dot p_{*})=\int_\Omega u_\rho\dot p_{*}dx \qquad\forall\,(y_*,u_\rho)\in\mathbb{K}_{*,\rho}$$ by [\[eq:DPDE\]](#eq:DPDE){reference-type="eqref" reference="eq:DPDE"} and [\[eq:dps\]](#eq:dps){reference-type="eqref" reference="eq:dps"}. Let $(\tilde{y}_*,\tilde{u}_\rho)\in\mathbb{K}_{*,\rho}$ be defined by $$\label{eq:tur} \tilde{u}_\rho=Q_\rho\bar u$$ and $$\label{eq:tys} a(\tilde{y}_*,z_*)=\int_\Omega\tilde{u}_\rho z_*\,dx \qquad\forall\,z_*\in V_*.$$ It follows from [\[eq:QEstimate\]](#eq:QEstimate){reference-type="eqref" reference="eq:QEstimate"} and [\[eq:tur\]](#eq:tur){reference-type="eqref" reference="eq:tur"} that $$\label{eq:turEstimate} \|\bar u-\tilde{u}_\rho\|_{L_2(\Omega)}\leq C_\maltese\rho|\bar u|_{H^1(\Omega)}.$$ We have $$\begin{aligned} \label{eq:SError1} &\|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}^2+\gamma\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}^2\notag\\ &\hspace{30pt}=\int_\Omega(\bar y-\bar y_{*,\rho})(\bar y-\tilde{y}_*)dx+ \gamma\int_\Omega(\bar u-\bar u_{*,\rho})(\bar u-\tilde{u}_\rho)dx\\ &\hspace{60pt}+ \int_\Omega(\bar y-\bar y_{*,\rho})(\tilde{y}_*-\bar y_{*,\rho})dx+ \gamma\int_\Omega(\bar u-\bar u_{*,\rho})(\tilde{u}_\rho-\bar u_{*,\rho})dx,\notag\end{aligned}$$ and, in view of [\[eq:lambda\]](#eq:lambda){reference-type="eqref" reference="eq:lambda"}, [\[eq:DOC\]](#eq:DOC){reference-type="eqref" reference="eq:DOC"} and [\[eq:DiscreteASRelation\]](#eq:DiscreteASRelation){reference-type="eqref" reference="eq:DiscreteASRelation"}, $$\begin{aligned} \label{eq:SError2} &\int_\Omega(\bar y-\bar y_{*,\rho})(\tilde{y}_*-\bar y_{*,\rho})dx+ \gamma\int_\Omega(\bar u-\bar u_{*,\rho})(\tilde{u}_\rho-\bar u_{*,\rho})dx\notag\\ &\hspace{30pt}=\int_\Omega\bar y(\tilde{y}_*-\bar y_{*,\rho})dx+ \gamma\int_\Omega\bar u(\tilde{u}_\rho-\bar u_{*,\rho})dx\notag\\ &\hspace{60pt}-\int_\Omega\bar y_{*,\rho}(\tilde{y}_*-\bar y_{*,\rho})dx- \gamma\int_\Omega\bar u_{*,\rho}(\tilde{u}_\rho-\bar u_{*,\rho})dx\\ &\hspace{30pt}\leq\int_\Omega(\bar y-y_d)(\tilde{y}_*-\bar y_{*,\rho})dx +\gamma\int_\Omega\bar u(\tilde{u}_\rho-\bar u_{*,\rho})dx\notag\\ &\hspace{30pt}=\int_\Omega(\dot p_{*}+\gamma\bar u)(\tilde{u}_\rho-\bar u_{*,\rho})dx\notag\\ &\hspace{30pt}=\int_\Omega\lambda(\tilde{u}_\rho-\bar u_{*,\rho})dx +\int_\Omega(\dot p_{*}-\bar p)(\tilde{u}_\rho-\bar u_{*,\rho})dx.\notag\end{aligned}$$ We can bound the first integral on the right-hand side of [\[eq:SError2\]](#eq:SError2){reference-type="eqref" reference="eq:SError2"} by [\[eq:ConstraintRegularity\]](#eq:ConstraintRegularity){reference-type="eqref" reference="eq:ConstraintRegularity"}, Remark [Remark 3](#rem:GeneralConstraints){reference-type="ref" reference="rem:GeneralConstraints"}, [\[eq:LambdaDecomposition\]](#eq:LambdaDecomposition){reference-type="eqref" reference="eq:LambdaDecomposition"}, [\[subeq:lambda\]](#subeq:lambda){reference-type="eqref" reference="subeq:lambda"}, [\[eq:DCC\]](#eq:DCC){reference-type="eqref" reference="eq:DCC"}, [\[eq:QEstimate\]](#eq:QEstimate){reference-type="eqref" reference="eq:QEstimate"} and [\[eq:turEstimate\]](#eq:turEstimate){reference-type="eqref" reference="eq:turEstimate"}: $$\begin{aligned} \label{eq:SError3} \int_\Omega\lambda(\tilde{u}_\rho-\bar u_{*,\rho})dx&=\int_\Omega\lambda_1(\tilde{u}_\rho-\bar u_{*,\rho})dx +\int_\Omega\lambda_2(\tilde{u}_\rho-\bar u_{*,\rho})dx\notag\\ &=\int_\Omega\lambda_1(Q_\rho\bar u-\bar u)dx +\int_\Omega\lambda_2(Q_\rho\bar u-\bar u)dx\notag\\ &\hspace{40pt}+\int_\Omega\lambda_1(\bar u-\phi_1)dx +\int_\Omega\lambda_2(\bar u-\phi_2)dx\notag\\ &\hspace{60pt}+\int_\Omega\lambda_1(\phi_1-Q_\rho\phi_1)dx +\int_\Omega\lambda_2(\phi_2-Q_\rho\phi_2)dx\notag\\ &\hspace{80pt}+\int_\Omega\lambda_1(Q_\rho\phi_1-\bar u_{*,\rho})dx+ \int_\Omega\lambda_2(Q_\rho\phi_2-\bar u_{*,\rho})dx\notag\\ &\leq \int_\Omega\lambda_1 (Q_\rho\bar u-\bar u)dx+ \int_\Omega\lambda_2(Q_\rho\bar u-\bar u)dx\\ &\hspace{40pt}+\int_\Omega\lambda_1(\phi_1-Q_\rho\phi_1)dx+ \int_\Omega\lambda_2(\phi_2-Q_\rho\phi_2)dx\notag\\ &=\int_\Omega(\lambda_1-Q_\rho\lambda_1) (Q_\rho\bar u-\bar u)dx+ \int_\Omega(\lambda_2-Q_\rho\lambda_2)(Q_\rho\bar u-\bar u)dx\notag\\ &\hspace{40pt}+\int_\Omega(\lambda_1-Q_\rho\lambda_1)(\phi_1-Q_\rho\phi_1)dx+ \int_\Omega(\lambda_2-Q_\rho\lambda_2)(\phi_2-Q_\rho\phi_2)dx\notag\\ &\leq C_1\rho^2.\notag\end{aligned}$$ For the second integral on the right-hand side of [\[eq:SError2\]](#eq:SError2){reference-type="eqref" reference="eq:SError2"}, we have $$\begin{aligned} \label{eq:SError4} \int_\Omega(\dot p_{*}-\bar p)(\tilde{u}_\rho-\bar u_{*,\rho})dx& \leq \|\bar p-\dot p_{*}\|_{L_2(\Omega)}(\|\tilde{u}_\rho-\bar u\|_{L_2(\Omega)}+\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)})\\ &\leq \|\bar p-\dot p_{*}\|_{L_2(\Omega)}\big(C_\maltese\rho|\bar u|_{H^1(\Omega)} +\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}\big)\notag\end{aligned}$$ by [\[eq:turEstimate\]](#eq:turEstimate){reference-type="eqref" reference="eq:turEstimate"}. It follows from Remark [Remark 3](#rem:GeneralConstraints){reference-type="ref" reference="rem:GeneralConstraints"}, [\[eq:turEstimate\]](#eq:turEstimate){reference-type="eqref" reference="eq:turEstimate"}, and [\[eq:SError1\]](#eq:SError1){reference-type="eqref" reference="eq:SError1"}--[\[eq:SError4\]](#eq:SError4){reference-type="eqref" reference="eq:SError4"} that $$\begin{aligned} &\|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}^2+\gamma\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}^2\\ &\hspace{70pt}\leq \|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}\|\bar y-\tilde{y}_*\|_{L_2(\Omega)} +\gamma\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}C_\maltese\rho|\bar u|_{H^1(\Omega)}\\ &\hspace{120pt}+C_1\rho^2+C_2 \rho\|\bar p-\dot p_{*}\|_{L_2(\Omega)} +\|\bar p-\dot p_{*}\|_{L_2(\Omega)}\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)},\end{aligned}$$ which together with the inequality of arithmetic and geometric means implies $$\label{eq:SError5} \|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}\leq C_3\big(\|\bar y-\tilde{y}_*\|_{L_2(\Omega)} +\|\bar p-\dot p_{*}\|_{L_2(\Omega)}+\rho).$$ On the other hand, we have $$a(\dot y_{*}-\tilde{y}_*,z_*)=\int_\Omega(\bar u-\tilde{u}_\rho)z_*dx\qquad\forall\,z_h\in V_*,$$ by [\[eq:dys\]](#eq:dys){reference-type="eqref" reference="eq:dys"} and [\[eq:tys\]](#eq:tys){reference-type="eqref" reference="eq:tys"}, and hence $$\label{eq:SError6} \|\dot y_{*}-\tilde{y}_*\|_{L_2(\Omega)}\leq (\mathrm{C}_{\mathrm{PF}}^2/\alpha)\|\bar u-\tilde{u}_\rho\|_{L_2(\Omega)} \leq (\mathrm{C}_{\mathrm{PF}}^2/\alpha)C_\maltese\rho|\bar u|_{H^1(\Omega)}$$ by [\[eq:QEstimate\]](#eq:QEstimate){reference-type="eqref" reference="eq:QEstimate"} and Lemma [Lemma 5](#lem:EasyEstimate){reference-type="ref" reference="lem:EasyEstimate"}. Putting [\[eq:SError5\]](#eq:SError5){reference-type="eqref" reference="eq:SError5"} and [\[eq:SError6\]](#eq:SError6){reference-type="eqref" reference="eq:SError6"} together, we arrive at the estimate $$\label{eq:SError7} \|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}\leq C_4\big(\|\bar y-\dot y_{*}\|_{L_2(\Omega)} +\|\bar p-\dot p_{*}\|_{L_2(\Omega)}+\rho).$$ For the estimate of $\bar p-\bar p_{*,\rho}$, we begin with $$\label{eq:SError8} \|\bar p-\bar p_{*,\rho}\|_{L_2(\Omega)}\leq \|\bar p-\dot p_{*}\|_{L_2(\Omega)}+\|\dot p_{*}-\bar p_{*,\rho}\|_{L_2(\Omega)}$$ and note that $$\label{eq:SError10} a(q_*,\dot p_{*}-\bar p_{*,\rho})=\int_\Omega(\bar y-\bar y_{*,\rho})q_*dx \qquad\forall\,q_*\in V_*$$ by [\[eq:bps\]](#eq:bps){reference-type="eqref" reference="eq:bps"} and [\[eq:dps\]](#eq:dps){reference-type="eqref" reference="eq:dps"}, which implies $$\label{eq:SError9} \|\dot p_{*}-\bar p_{*,\rho}\|_{L_2(\Omega)}\leq (\mathrm{C}_{\mathrm{PF}}^2/\alpha)\|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}$$ through Lemma [Lemma 5](#lem:EasyEstimate){reference-type="ref" reference="lem:EasyEstimate"}. The estimate [\[eq:AbstractErrorEstimate\]](#eq:AbstractErrorEstimate){reference-type="eqref" reference="eq:AbstractErrorEstimate"} follows from [\[eq:SError7\]](#eq:SError7){reference-type="eqref" reference="eq:SError7"}--[\[eq:SError9\]](#eq:SError9){reference-type="eqref" reference="eq:SError9"}. ◻ The following result shows that the estimate [\[eq:AbstractErrorEstimate\]](#eq:AbstractErrorEstimate){reference-type="eqref" reference="eq:AbstractErrorEstimate"} is a tight estimate. **Theorem 7**. *There exists a positive constant $C_\ddag$, depending only on $\alpha^{-1}$, such that $$\label{eq:ReverseErrorEstiamte} \|\bar y-\dot y_{*}\|_{L_2(\Omega)}+\|\bar p-\dot p_{*}\|_{L_2(\Omega)}\leq C_\ddag\big(\|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)} +\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}+\|\bar p-\bar p_{*,\rho}\|_{L_2(\Omega)}\big),$$ where $\dot y_{*}$ $($resp., $\dot p_{*})$ is defined by [\[eq:dys\]](#eq:dys){reference-type="eqref" reference="eq:dys"} $($resp., $\eqref{eq:dps})$.* *Proof.* We have $$\label{eq:REstimate1} \|\bar y-\dot y_{*}\|_{L_2(\Omega)}\leq \|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}+\|\bar y_{*,\rho}-\dot y_{*}\|_{L_2(\Omega)},$$ and $$a(\bar y_{*,\rho}-\dot y_{*},z_*)=\int_\Omega(\bar u_{*,\rho}-\bar u)z_* dx\qquad\forall\,z_*\in V_*$$ by [\[eq:DPDE\]](#eq:DPDE){reference-type="eqref" reference="eq:DPDE"} and [\[eq:dys\]](#eq:dys){reference-type="eqref" reference="eq:dys"}, which implies $$\label{eq:REstiamte2} \|\bar y_{*,\rho}-\dot y_{*}\|_{L_2(\Omega)}\leq (\mathrm{C}_{\mathrm{PF}}^2/\alpha)\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}$$ by Lemma [Lemma 5](#lem:EasyEstimate){reference-type="ref" reference="lem:EasyEstimate"}. Similarly we have $$\begin{aligned} \label{eq:REstimate3} \|\bar p-\dot p_{*}\|_{L_2(\Omega)}&\leq \|\bar p-\bar p_{*,\rho}\|_{L_2(\Omega)}+\|\bar p_{*,\rho}-\dot p_{*}\|_{L_2(\Omega)}\\ &\leq \|\bar p-\bar p_{*,\rho}\|_{L_2(\Omega)}+(\mathrm{C}_{\mathrm{PF}}^2/\alpha)\|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}\notag \end{aligned}$$ by [\[eq:SError9\]](#eq:SError9){reference-type="eqref" reference="eq:SError9"}. The estimate [\[eq:ReverseErrorEstiamte\]](#eq:ReverseErrorEstiamte){reference-type="eqref" reference="eq:ReverseErrorEstiamte"} follows from [\[eq:REstimate1\]](#eq:REstimate1){reference-type="eqref" reference="eq:REstimate1"}--[\[eq:REstimate3\]](#eq:REstimate3){reference-type="eqref" reference="eq:REstimate3"}. ◻ It is straightforward to derive error estimates in the energy norm from the estimate [\[eq:AbstractErrorEstimate\]](#eq:AbstractErrorEstimate){reference-type="eqref" reference="eq:AbstractErrorEstimate"}. **Theorem 8**. *There exists a positive constant $C_\S$, depending only on $\|y_d\|_{L_2(\Omega)}$, $\|\phi_1\|_{H^1(\Omega)}$, $\|\phi_2\|_{H^1(\Omega)}$, $\gamma^{-1}$, $\alpha^{-1}$ and the shape regularity of $\mathcal{T}_\rho$, such that $$\label{eq:AbastractEnergyError} \|\bar y-\bar y_{*,\rho}\|_a+\|\bar p-\bar p_{*,\rho}\|_a\leq C_\S \big(\|\bar y-\dot y_{*}\|_a +\|\bar p-\dot p_{*}\|_a+\rho\big),$$ where $\dot y_{*}, \dot p_{*}\in V_*$ are defined in [\[eq:dys\]](#eq:dys){reference-type="eqref" reference="eq:dys"} and [\[eq:dps\]](#eq:dps){reference-type="eqref" reference="eq:dps"}.* *Proof.* We have $$\label{eq:AEnergy1} \|\bar y-\bar y_{*,\rho}\|_a+\|\bar p-\bar p_{*,\rho}\|_a\leq \|\bar y-\dot y_{*}\|_a +\|\bar p-\dot p_{*}\|_a+\|\dot y_{*}-\bar y_{*,\rho}\|_a +\|\dot p_{*}-\bar p_{*,\rho}\|_a.$$ It follows from [\[eq:DPDE\]](#eq:DPDE){reference-type="eqref" reference="eq:DPDE"} and [\[eq:dys\]](#eq:dys){reference-type="eqref" reference="eq:dys"} that $$a(\dot y_{*}-\bar y_{*,\rho},z_*)=\int_\Omega(\bar u-\bar u_{*,\rho})z_*dx\qquad\forall\,z_*\in V_*,$$ and hence $$\label{eq:AEnergy2} \|\dot y_{*}-\bar y_{*,\rho}\|_a\leq (\mathrm{C}_{\mathrm{PF}}/\sqrt\alpha)\|\bar u-\bar u_{*,\rho}\|_{L_2(\Omega)}$$ by Lemma [Lemma 5](#lem:EasyEstimate){reference-type="ref" reference="lem:EasyEstimate"}. Similarly the relation [\[eq:SError10\]](#eq:SError10){reference-type="eqref" reference="eq:SError10"} and Lemma [Lemma 5](#lem:EasyEstimate){reference-type="ref" reference="lem:EasyEstimate"} imply $$\label{eq:AEnergy3} \|\dot p_{*}-\bar p_{*,\rho}\|_a\leq (\mathrm{C}_{\mathrm{PF}}/\sqrt\alpha)\|\bar y-\bar y_{*,\rho}\|_{L_2(\Omega)}.$$ The estimate [\[eq:AbastractEnergyError\]](#eq:AbastractEnergyError){reference-type="eqref" reference="eq:AbastractEnergyError"} is obtained by combining [\[eq:AbstractErrorEstimate\]](#eq:AbstractErrorEstimate){reference-type="eqref" reference="eq:AbstractErrorEstimate"}, [\[eq:AEnergy1\]](#eq:AEnergy1){reference-type="eqref" reference="eq:AEnergy1"}--[\[eq:AEnergy3\]](#eq:AEnergy3){reference-type="eqref" reference="eq:AEnergy3"} and the relation $$\|\bar y-\dot y_{*}\|_{L_2(\Omega)}+\|\bar p-\dot p_{*}\|_{L_2(\Omega)}\leq(\mathrm{C}_{\mathrm{PF}}/\sqrt\alpha)\big( \|\bar y-\dot y_{*}\|_a+\|\bar p-\dot p_{*}\|_a\big)$$ that follows from [\[eq:SpaceComparison\]](#eq:SpaceComparison){reference-type="eqref" reference="eq:SpaceComparison"} and [\[eq:PF\]](#eq:PF){reference-type="eqref" reference="eq:PF"}. ◻ **Remark 9**. *Note that [\[eq:AdjointState\]](#eq:AdjointState){reference-type="eqref" reference="eq:AdjointState"} and [\[eq:dps\]](#eq:dps){reference-type="eqref" reference="eq:dps"} imply $\dot p_{*}\in V_*$ is the projection of $\bar p$ with respect to the bilinear form $a(\cdot,\cdot)$. Therefore we have $$\|\bar p-\dot p_{*}\|_a=\inf_{q_*\in V_*}\|\bar p-q_*\|_a.$$ Similarly we have $$\|\bar y-\dot y_{*}\|_a=\inf_{z_*\in V_*}\|\bar y-z_*\|_a$$ by [\[eq:PDEConstraint\]](#eq:PDEConstraint){reference-type="eqref" reference="eq:PDEConstraint"} and [\[eq:dys\]](#eq:dys){reference-type="eqref" reference="eq:dys"}.* Let $V_*=V_h$ be the $P_1/Q_1$ finite element space associated with a simplicial/quadrilateral triangulation $\mathcal{T}_h$ of $\Omega$ with mesh size $h$ and let $(\bar y_{*,\rho},\bar u_{*,\rho},\bar p_{*,\rho})$ be written as $(\bar y_{h,\rho},\bar u_{h,\rho},\bar p_{h,\rho})$. The estimate [\[eq:AbstractErrorEstimate\]](#eq:AbstractErrorEstimate){reference-type="eqref" reference="eq:AbstractErrorEstimate"} becomes $$\label{eq:StandardErrorEstimate} \|\bar y-\bar y_{h,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{h,\rho}\|_{L_2(\Omega)}+\|\bar p-\bar p_{h,\rho}\|_{L_2(\Omega)} \leq C_\dag(\|\bar y-\dot y_h\|_{L_2(\Omega)}+ \|\bar p-\dot p_h\|_{L_2(\Omega)}+\rho),$$ where $\dot y_h,\dot p_h\in V_h$ are defined by $$\begin{aligned} {3} a(\dot y_h,z_h)&=\int_\Omega\bar u z_h dx&\qquad&\forall\,z_h\in V_h,\label{eq:dyh}\\ a(q_h,\dot p_h)&=\int_\Omega(\bar y-y_d)q_h dx&\qquad&\forall\,q_h\in V_h,\label{eq:dph}\end{aligned}$$ and the estimate [\[eq:AbastractEnergyError\]](#eq:AbastractEnergyError){reference-type="eqref" reference="eq:AbastractEnergyError"} becomes $$\label{eq:StandardEnergyError} \|\bar y-\bar y_{h,\rho}\|_a+\|\bar p-\bar p_{h,\rho}\|_a\leq C_\S\big(\|\bar y-\dot y_h\|_a+\|\bar p-\dot p_h\|_a+\rho\big).$$ In the case where $\mathcal{A}$ is the identity matrix and $\Omega$ is convex, we have $\bar y,\bar p \in H^2(\Omega)$ by the elliptic regularity theory for polygonal domains (cf. [@Grisvard:1985:EPN; @Dauge:1988:EBV; @MR:2010:Polyhedral]). It follows from Remark [Remark 9](#rem:GalerkinProjection){reference-type="ref" reference="rem:GalerkinProjection"}, [\[eq:StandardErrorEstimate\]](#eq:StandardErrorEstimate){reference-type="eqref" reference="eq:StandardErrorEstimate"} and a standard duality argument (cf. [@Ciarlet:1978:FEM; @BScott:2008:FEM]) that $$\label{eq:Smooth} \|\bar y-\bar y_{h,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{h,\rho}\|_{L_2(\Omega)}+\|\bar p-\bar p_{h,\rho}\|_{L_2(\Omega)}\leq C(h^2+\rho).$$ In this case the estimate [\[eq:StandardEnergyError\]](#eq:StandardEnergyError){reference-type="eqref" reference="eq:StandardEnergyError"} yields $$\label{eq:SmoothEnergy} |\bar y-\bar y_{h,\rho}|_{H^1(\Omega)}+|\bar p-\bar p_{h,\rho}|_{H^1(\Omega)}\leq C(h+\rho).$$ In the case of rough coefficients, we can derive from [\[eq:PF\]](#eq:PF){reference-type="eqref" reference="eq:PF"}, [\[eq:AbstractErrorEstimate\]](#eq:AbstractErrorEstimate){reference-type="eqref" reference="eq:AbstractErrorEstimate"}, [\[eq:AbastractEnergyError\]](#eq:AbastractEnergyError){reference-type="eqref" reference="eq:AbastractEnergyError"} and Remark [Remark 9](#rem:GalerkinProjection){reference-type="ref" reference="rem:GalerkinProjection"} that $$\begin{aligned} &\|\bar y-\bar y_{h,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{h,\rho}\|_{L_2(\Omega)}+\|\bar p-\bar p_{h,\rho}\|_{L_2(\Omega)} +\|\bar y-\bar y_{h,\rho}\|_a+\|\bar p-\bar p_{h,\rho}\|_a\\ &\hspace{80pt}\leq C\Big(\inf_{z_h\in V_h}\|\bar y-z_h\|_a +\inf_{q_h\in V_h}\|\bar p-q_h\|_a+\rho\Big),\end{aligned}$$ which implies $$\lim_{h,\rho\downarrow0}\big(\|\bar y-\bar y_{h,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{h,\rho}\|_{L_2(\Omega)} +\|\bar p-\bar p_{h,\rho}\|_{L_2(\Omega)} +\|\bar y-\bar y_{h,\rho}\|_a+\|\bar p-\bar p_{h,\rho}\|_a\big)=0.$$ However the convergence with respect to $h$ can be very slow. Therefore a satisfactory approximate solution of the optimal control problem obtained by standard finite element methods will require a very fine mesh $\mathcal{T}_h$. Below we will show that it is possible to recover on coarse meshes a performance similar to [\[eq:Smooth\]](#eq:Smooth){reference-type="eqref" reference="eq:Smooth"} and [\[eq:SmoothEnergy\]](#eq:SmoothEnergy){reference-type="eqref" reference="eq:SmoothEnergy"} for rough coefficients and general $\Omega$ provided that one takes a multiscale finite element space to be $V_*$. # A DD-LOD Multiscale Finite Element Method {#sec:DDLOD} First we recall the construction of the multiscale finite element space from [@BGS:2021:LOD]. It begins with a simplicial/quadrilateral triangulation $\mathcal{T}_H$ of $\Omega$, and a refinement $\mathcal{T}_h$ ($h\ll H$) of $\mathcal{T}_H$. The $P_1/Q_1$ finite element subspace of $H^1_0(\Omega)$ associated with $\mathcal{T}_H$ (resp., $\mathcal{T}_h$) is denoted by $V_H$ (resp., $V_h$). The first step is to construct a projection operator $\Pi_{\scriptscriptstyle H}:H^1_0(\Omega)\longrightarrow V_H$ such that $$\frac{1}{H}\|v-\Pi_{\scriptscriptstyle H}v\|_{L_2(\Omega)}+|\Pi_{\scriptscriptstyle H}v|_{H^1(\Omega)}\leq C_\flat|v|_{H^1(\Omega)} \qquad \forall\,v\in H^1_0(\Omega).$$ **Remark 10**. *The operator $\Pi_{\scriptscriptstyle H}$ in [@BGS:2021:LOD] is constructed by taking the averages of local $L_2$ projections. There are other constructions that are adapted to the coefficient matrix $\mathcal{A}(x)$ (cf. [@PS:2016:Contrast; @HM:2017:Contrast]).* Let $K_h^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}=\{v\in V_h:\Pi_{\scriptscriptstyle H}v=0\}$ be the kernel of $\Pi_{\scriptscriptstyle H}$ in $V_h$ and the correction operator $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_h:V_h\longrightarrow K_h^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}$ be the projection operator with respect to $a(\cdot,\cdot)$, i.e., $$%\label{eq:Correction} a(\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_hv,w)=a(v,w) \qquad\forall\,w\in K_h^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}.$$ The multiscale finite element space ${V}_H^{\raise 2pt\hbox{$\scriptstyle\rm {ms},h$}}\subset V_h$ is the orthogonal complement of $K_h^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}$ with respect to $a(\cdot,\cdot)$. Let $\phi_1,\ldots,\phi_m$ be the standard nodal basis functions of $V_H$ associated with the interior vertices $p_1,\ldots,p_m$ of $\mathcal{T}_H$. Then ${V}_H^{\raise 2pt\hbox{$\scriptstyle\rm {ms},h$}}$ is spanned by $\phi_1-\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_h\phi_1,\ldots,\phi_m-\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_h\phi_m$. The performance of the finite element method based on ${V}_H^{\raise 2pt\hbox{$\scriptstyle\rm {ms},h$}}$ for the problem $$\label{eq:RBVP} a(u,v)=\int_\Omega fv\,dx \qquad\forall\,v\in {H^1_0(\Omega)}$$ with rough coefficients is similar to the performance of $V_H$ for problems with smooth coefficients on convex domains (cf. [@MP:2014:LOD; @MP:2021:LOD]). However, the construction of ${V}_H^{\raise 2pt\hbox{$\scriptstyle\rm {ms},h$}}$ requires solving $m$ problems on the fine mesh $\mathcal{T}_h$, which is expensive. The localized orthogonal decomposition (LOD) method is based on replacing the correction $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_h\phi_i$ by a correction computed in a subdomain consisting of a certain number of layers of elements from $\mathcal{T}_H$ around $p_i$. It significantly reduces the computational cost and at the same time it preserves the good approximation property of ${V}_H^{\raise 2pt\hbox{$\scriptstyle\rm {ms},h$}}$ because the function $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_h\phi_i$ decays exponentially away from $p_i$ (cf. [@MP:2014:LOD; @MP:2021:LOD; @AHP:2021:Acta]). The multiscale finite element method from [@BGS:2021:LOD] is a variant of the LOD method which is based on the ideas in [@KPY:2018:LOD]. It computes an approximate solution $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_i$ of the corrector equation $$a(\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_h\phi_i,w)=a(\phi_i,w) \qquad \forall\,w \in K_h^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}$$ by applying $k$ iterations of a preconditioned conjugate gradient (PCG) method with initial guess $0$. The theory of PCG (cf. [@Saad:2003:IM]) implies that the convergence of $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_i$ to $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_h\phi_i$ in $\|\cdot\|_a$ is approximately $q^k$, where $q\in (0,1)$ depends on the condition number of the preconditioned operator. The key is to use an additive Schwarz domain decomposition preconditioner (cf. [@TW:2005:DD]) where the subdomains are small patches $\omega_i$ around $p_i$ so that $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_i$ is supported on a subdomain obtained by adding approximately $2k$ layers of elements from $\mathcal{T}_H$ around $\omega_i$, i.e., $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_i$ is also a localized correction of $\phi_i$. The computation of $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_i$ only involves solving local small problems and $\|\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_h\phi_i-\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_i\|_a=O(H)$ provided $k$ is proportional to $|\ln H|$. The multiscale finite element space $V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}\subset V_h$ is spanned by $\phi_1-\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_1,\ldots,\phi_m-\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_m$. We will refer to it as the DD-LOD multiscale finite element space. The corresponding finite element method for [\[eq:RBVP\]](#eq:RBVP){reference-type="eqref" reference="eq:RBVP"} can be viewed as a reduced order method, where the functions $\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_1,\ldots,\mathcal{C}^{\raise 1pt\hbox{$\scriptstyle\Pi_{\scriptscriptstyle H}$}}_{h,k}\phi_m$ are computed off-line. The on-line computation only involves solving an $m\times m$ system. The following is the main result from [@BGS:2021:LOD] whose derivation only involves basic results from finite element methods, domain decomposition methods and numerical linear algebra. **Lemma 11**. *Let $f\in{L_2(\Omega)}$, $y_h\in V_h$ and ${y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\in V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}$ such that $$\begin{aligned} {3} a(y_h,z_h)&=\int_\Omega fz_h dx&\qquad&\forall\,z_h\in V_h, \\ a({y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{z}_{\scriptscriptstyle H,k}^{\mathrm{ms},h})&=\int_\Omega f{z}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}dx &\qquad&\forall\,{z}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\in V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}.\end{aligned}$$ There exists a positive constant $C_\sharp$ depending on the shape regularity of $\mathcal{T}_H$ but independent of $\alpha$, $\beta$, $h$ and $H$, such that $$\begin{aligned} \|y_h-{y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_a&\leq (C_\sharp/\sqrt{\alpha})H\|f\|_{L_2(\Omega)},\\ \|y_h-{y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_{L_2(\Omega)}&\leq (C_\sharp/\sqrt{\alpha})^2H^2\|f\|_{L_2(\Omega)},\end{aligned}$$ provided $k=\lceil -j \ln H\rceil$ for a sufficiently large $j$.* **Remark 12**. *The magnitude of $j$ depends on the condition number of the preconditioned operator in the PCG algorithm.* The DD-LOD finite element method for [\[eq:OCP\]](#eq:OCP){reference-type="eqref" reference="eq:OCP"}--[\[eq:aDef\]](#eq:aDef){reference-type="eqref" reference="eq:aDef"} is defined by [\[eq:DOCP\]](#eq:DOCP){reference-type="eqref" reference="eq:DOCP"}--[\[eq:DCC\]](#eq:DCC){reference-type="eqref" reference="eq:DCC"}, where $V_*=V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}$ and its solution is denoted by $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},\bar u_\rho)$. We also include the approximation of $\bar p$ by ${\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$ in the error analysis of the multiscale finite element method, where ${\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\in V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}$ is defined by $$\label{eq:bMSLp} a({q}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h})=\int_\Omega({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}-y_d){q}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\,dx\qquad\forall\,{q}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\in V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}.$$ **Remark 13**. *Strictly speaking ${\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$ and ${\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$ also depend on $\rho$ and $\bar u_\rho$ also depends on $h$, $H$ and $k$. These dependencies are suppressed for the sake of readability.* **Theorem 14**. *There exists a positive constant $C_\natural$, depending only on $\|y_d\|_{L_2(\Omega)}$, $\|\phi_1\|_{H^1(\Omega)}$, $\|\phi_2\|_{H^1(\Omega)}$, $\gamma^{-1}$, $\alpha^{-1}$ and the shape regularities of $\mathcal{T}_H$ and $\mathcal{T}_\rho$, such that $$\begin{aligned} \label{eq:DDLODError} &\|\bar y-{\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_{L_2(\Omega)}+\|\bar u-\bar u_\rho\|_{L_2(\Omega)}+\|\bar p-{\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_{L_2(\Omega)}\\ &\hspace{80pt}\leq C_\natural\big(\|\bar y-\bar y_{h,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{h,\rho}\|_{L_2(\Omega)}+\|\bar p-\bar p_{h,\rho}\|_{L_2(\Omega)} + H^2 + \rho \big),\notag\end{aligned}$$ where $(\bar y_{h,\rho},\bar u_{h,\rho},\bar p_{h,\rho})$ is the approximation of $(\bar y,\bar u,\bar p)$ obtained by using the standard finite element space $V_h\times W_\rho$ in the discretization defined by [\[eq:DOCP\]](#eq:DOCP){reference-type="eqref" reference="eq:DOCP"}--[\[eq:DCC\]](#eq:DCC){reference-type="eqref" reference="eq:DCC"}.* *Proof.* We apply Theorem [Theorem 6](#thm:AbstractErrorEstimate){reference-type="ref" reference="thm:AbstractErrorEstimate"} (with $V_*=V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}$) to obtain $$\begin{aligned} \label{eq:Analog1} &\|\bar y-{\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_{L_2(\Omega)}+\|\bar u-\bar u_\rho\|_{L_2(\Omega)}+\|\bar p-{\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_{L_2(\Omega)}\\ &\hspace{50pt}\leq C_\dag\big(\|\bar y-{\dot y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_{L_2(\Omega)}+\|\bar p-{\dot p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_{L_2(\Omega)}+ \rho\big),\notag\end{aligned}$$ where ${\dot y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\in V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}$ (resp., ${\dot p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\in V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}$) is the analog of $\dot y_{*}$ in [\[eq:dys\]](#eq:dys){reference-type="eqref" reference="eq:dys"} (resp., $\dot p_{*}$ in [\[eq:dps\]](#eq:dps){reference-type="eqref" reference="eq:dps"}), i.e., ${\dot y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$ is defined by $$\label{eq:dMSLy} a({\dot y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{z}_{\scriptscriptstyle H,k}^{\mathrm{ms},h})=\int_\Omega\bar u {z}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}dx\qquad\forall\,{z}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\in V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}},$$ and ${\dot p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$ is defined by $$\label{eq:dMSLp} a({q}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\dot p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h})=\int_\Omega(\bar y-y_d){q}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}dx\qquad\forall\,{q}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\in V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}.$$ Let $\dot y_h\in V_h$ $($resp., $\dot p_h\in V_h)$ be defined by [\[eq:dyh\]](#eq:dyh){reference-type="eqref" reference="eq:dyh"} (resp., [\[eq:dph\]](#eq:dph){reference-type="eqref" reference="eq:dph"}). According to Theorem [Theorem 7](#thm:ReverseErrorEstimate){reference-type="ref" reference="thm:ReverseErrorEstimate"}, we have $$\label{eq:LODDDEst0} \|\bar y-\dot y_h\|_{L_2(\Omega)}+\|\bar p-\dot p_h\|_{L_2(\Omega)} \leq C_\ddag\big(\|\bar y-\bar y_{h,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{h,\rho}\|_{L_2(\Omega)}+ \|\bar p-\bar p_{h,\rho}\|_{L_2(\Omega)}\big).$$ On the other hand, in view of Lemma [Lemma 11](#lem:ETNA){reference-type="ref" reference="lem:ETNA"}, we have $$\label{eq:LODDDEst1} \|\dot y_h-{\dot y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_{L_2(\Omega)}\leq (C_\sharp/\sqrt{\alpha})^2H^2\|\bar u\|_{L_2(\Omega)}$$ by [\[eq:dyh\]](#eq:dyh){reference-type="eqref" reference="eq:dyh"} and [\[eq:dMSLy\]](#eq:dMSLy){reference-type="eqref" reference="eq:dMSLy"}, and $$\label{eq:LODDDEst2} \|\dot p_h-{\dot p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_{L_2(\Omega)}\leq (C_\sharp/\sqrt{\alpha})^2H^2\|\bar y-y_d\|_{L_2(\Omega)}$$ by [\[eq:dph\]](#eq:dph){reference-type="eqref" reference="eq:dph"} and [\[eq:dMSLp\]](#eq:dMSLp){reference-type="eqref" reference="eq:dMSLp"}. The estimate [\[eq:DDLODError\]](#eq:DDLODError){reference-type="eqref" reference="eq:DDLODError"} follows from [\[eq:Analog1\]](#eq:Analog1){reference-type="eqref" reference="eq:Analog1"}, [\[eq:LODDDEst0\]](#eq:LODDDEst0){reference-type="eqref" reference="eq:LODDDEst0"}--[\[eq:LODDDEst2\]](#eq:LODDDEst2){reference-type="eqref" reference="eq:LODDDEst2"} and the triangle inequality. ◻ **Remark 15**. *The estimate [\[eq:DDLODError\]](#eq:DDLODError){reference-type="eqref" reference="eq:DDLODError"} indicates that up to an $O(H^2+\rho)$ error the approximation of $(\bar y,\bar u,\bar p)$ by $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},\bar u_\rho,{\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h})$ is as good as the approximation by $(\bar y_{h,\rho},\bar u_{h,\rho},\bar p_{h,\rho})$. On the other hand, by comparing [\[eq:Smooth\]](#eq:Smooth){reference-type="eqref" reference="eq:Smooth"} and [\[eq:DDLODError\]](#eq:DDLODError){reference-type="eqref" reference="eq:DDLODError"}, we can also say that, up to the fine scale error, the performance of the multiscale finite element method on coarse meshes (with respect to the ${L_2(\Omega)}$ norm) is similar to the performance of standard finite element methods for problems with smooth coefficients on convex domains.* We also have error estimates in the energy norm. **Theorem 16**. *There exists a positive constant $C_\diamond$, depending only on $\|y_d\|_{L_2(\Omega)}$, $\|\phi_1\|_{H^1(\Omega)}$, $\|\phi_2\|_{H^1(\Omega)}$, $\gamma^{-1}$, $\alpha^{-1}$ and the shape regularities of $\mathcal{T}_H$ and $\mathcal{T}_\rho$, such that $$\label{eq:EnergyErrors} \|\bar y-{\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_a+\|\bar p-{\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_a\leq C_\diamond \big(\|\bar y-\bar y_{h,\rho}\|_a+\|\bar p-\bar p_{h,\rho}\|_a + H + \rho\big),$$ where $(\bar y_{h,\rho},\bar p_{h,\rho})$ is the approximation of $(\bar y,\bar p)$ obtained by using the standard finite element space $V_h\times W_\rho$ in the discretization defined by [\[eq:DOCP\]](#eq:DOCP){reference-type="eqref" reference="eq:DOCP"}--[\[eq:DCC\]](#eq:DCC){reference-type="eqref" reference="eq:DCC"}.* *Proof.* It follows from Theorem [Theorem 8](#thm:AbstractEnergyError){reference-type="ref" reference="thm:AbstractEnergyError"} that $$\label{eq:Energy1} \|\bar y-{\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_a+\|\bar p-{\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_a \leq C_\S\big(\|\bar y-{\dot y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_a+\|\bar p-{\dot p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_a+\rho\big),$$ where ${\dot y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\dot p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\in V_{H,k}^{\hspace{1pt}\lower 3pt\hbox{$\scriptstyle\rm {ms},h$}}$ are defined by [\[eq:dMSLy\]](#eq:dMSLy){reference-type="eqref" reference="eq:dMSLy"} and [\[eq:dMSLp\]](#eq:dMSLp){reference-type="eqref" reference="eq:dMSLp"}. Let $\dot y_h\in V_h$ $($resp., $\dot p_h\in V_h)$ be defined by [\[eq:dyh\]](#eq:dyh){reference-type="eqref" reference="eq:dyh"} (resp., [\[eq:dph\]](#eq:dph){reference-type="eqref" reference="eq:dph"}). In view of Lemma [Lemma 11](#lem:ETNA){reference-type="ref" reference="lem:ETNA"}, we have $$\label{eq:Energy2} \|\dot y_h-{\dot y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_a\leq \big(C_\sharp/\sqrt{\alpha})H\|\bar u\|_{L_2(\Omega)}$$ by [\[eq:dyh\]](#eq:dyh){reference-type="eqref" reference="eq:dyh"} and [\[eq:dMSLy\]](#eq:dMSLy){reference-type="eqref" reference="eq:dMSLy"}, and also $$\label{eq:Energy3} \|\dot p_h-{\dot p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}\|_a \leq C_\sharp/\sqrt{\alpha})H\|\bar y-y_d\|_{L_2(\Omega)}$$ by [\[eq:dph\]](#eq:dph){reference-type="eqref" reference="eq:dph"} and [\[eq:dMSLp\]](#eq:dMSLp){reference-type="eqref" reference="eq:dMSLp"}. Finally we note that $$\begin{aligned} \label{eq:Energy4} \|\bar y-\dot y_h\|_a\leq \|\bar y-\bar y_{h,\rho}\|_a \quad\text{and}\quad \|\bar p-\dot p_h\|_a\leq \|\bar p-\bar p_{h,\rho}\|_a\end{aligned}$$ by Remark [Remark 9](#rem:GalerkinProjection){reference-type="ref" reference="rem:GalerkinProjection"}. The estimate [\[eq:EnergyErrors\]](#eq:EnergyErrors){reference-type="eqref" reference="eq:EnergyErrors"} follows from [\[eq:Energy1\]](#eq:Energy1){reference-type="eqref" reference="eq:Energy1"}--[\[eq:Energy4\]](#eq:Energy4){reference-type="eqref" reference="eq:Energy4"} and the triangle inequality. ◻ **Remark 17**. *The estimate [\[eq:EnergyErrors\]](#eq:EnergyErrors){reference-type="eqref" reference="eq:EnergyErrors"} indicates that, up to an $O(H+\rho)$ error, the approximation of $(\bar y,\bar p)$ by $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar p}_{\scriptscriptstyle H,k}^{\mathrm{ms},h})$ in the energy norm is as good as the fine scale approximation by $(\bar y_{h,\rho},\bar p_{h,\rho})$. By comparing [\[eq:SmoothEnergy\]](#eq:SmoothEnergy){reference-type="eqref" reference="eq:SmoothEnergy"} with [\[eq:EnergyErrors\]](#eq:EnergyErrors){reference-type="eqref" reference="eq:EnergyErrors"}, we can also say that up to the fine scale error the performance of the multiscale finite element method (with respect to the energy norm) on coarse meshes is similar to the performance of standard finite element methods for problems with smooth coefficients on convex domains.* # Numerical Results {#sec:Numerics} In this section we report the numerical results of two examples, one with highly heterogeneous coefficients and one with highly oscillatory coefficients. The domain is the unit square $\Omega=(0,1)\times(0,1)$ for both examples, and we use the $Q_1$ element on uniform rectangular meshes. The regularization parameter $\gamma$ is taken to be $1$. The objective function in our computations is given by $$\label{eq:newJ} \tilde J(y,u)=\frac12\big(\|y\|_{L_2(\Omega)}^2+\gamma\| u\|_{L_2(\Omega)}^2\big)-\int_\Omega yy_ddx$$ that differs from $J(y,u)$ by the constant $\|y_d\|_{L_2(\Omega)}^2/2$. The fine scale solution $(\bar y_h,\bar u_h)$ (where $\mathcal{T}_\rho=\mathcal{T}_h$) is computed by using the primal-dual interior point method in the PETSc/TAO library with 20 processors on the SuperMIC supercomputer at Louisiana State University. Each compute node is equipped with two 2.8GHz 10-Core Ivy Bridge-EP E5-2680 Xeon 64-bit Processors, two Intel Xeon Phi 7120P Coprocessors, 64GB DDR3 1866MHz Ram, 500GB HD, 56 Gigabit/sec Infiniband network interface, and 1 Gigabit Ethernet network interface. The DD-LOD solution $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ (with $\mathcal{T}_\rho=\mathcal{T}_H$) is computed by using the quadprog algorithm in MATLAB on a Lenovo Thinkpad X1 Carbon laptop with a 12th Gen Intel(R) Core(TM) i7-1260P processor, 4.70 GHz Max Turbo Frequency, an 18MB Intel(R) Smart Cache and 32 GB of RAM. **Example 18** (Highly Heterogeneous Coefficients). *The coefficient matrix for this example is given by $$\mathcal{A}=\begin{bmatrix} \mathcal{A}_{11}& {\bf 0}\\ {\bf 0} &\mathcal{A}_{22} \end{bmatrix},$$ where $\mathcal{A}_{11}$ and $\mathcal{A}_{22}$ are piecewise constant matrices with respect to a $40\times 40$ uniform rectangular subdivision of $\Omega$. The values of $\mathcal{A}_{11}$ and $\mathcal{A}_{22}$ on each square of the subdivision are randomly generated and range between 1 and 1350 (cf. Figure [2](#fig:Heterogeneous){reference-type="ref" reference="fig:Heterogeneous"}).* *{#fig:Heterogeneous width=".35\\linewidth"} {#fig:Heterogeneous width=".35\\linewidth"}* *We choose $y_d=1$ and the control constraints are given by $\phi_1(x)=0.0002x_1-0.0001$ and $\phi_2(x)=0.0002x_2+0.0001$ (cf. Figure [3](#fig:HHCC){reference-type="ref" reference="fig:HHCC"}).* {reference-type="ref" reference="example:HH"}.](Control-bounds-AL.png){#fig:HHCC width="0.3\\linewidth"} *We take $h=1/320$ for the fine scale solution $(\bar y_h,\bar u_h)$. In the first set of experiments we take $H=1/10, 1/20,1/40,1/80$ for the DD-LOD solution $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ with $\mathcal{T}_\rho=\mathcal{T}_H$. The number of iterations $k$ used in the solution of the corrector equation equals $\lceil-3\ln H\rceil$ for $H=1/10$, $1/20$ and $1/40$, and equals $\lceil-6\ln H\rceil$ for $H=1/80$. The relative errors for the approximation of the standard finite element solution $(\bar y_h,\bar u_h)$ by the multiscale finite element solution $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ are presented in Figure [6](#fig:HHControlStateErrors){reference-type="ref" reference="fig:HHControlStateErrors"}.* {reference-type="ref" reference="example:HH"} with $H=1/10,1/20,1/40,1/80$.](Control-L2Error-hh-AL-319.png){#fig:HHControlStateErrors width="1\\linewidth"} *(a)* {reference-type="ref" reference="example:HH"} with $H=1/10,1/20,1/40,1/80$.](L2Error-OptState-hh-AL-319.png){#fig:HHControlStateErrors width="1\\linewidth"} *(b)* {reference-type="ref" reference="example:HH"} with $H=1/10,1/20,1/40,1/80$.](Energy_error_optState_hh.png){#fig:HHControlStateErrors width="1\\linewidth"} *(c)* *The $O(H)$ convergence of ${\bar u}_{\scriptscriptstyle H}$ predicted by Theorem [Theorem 14](#thm:DDLODError){reference-type="ref" reference="thm:DDLODError"} is observed. The convergence of ${\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$ in the $L_2$ norm is $O(H^2)$, which is better than the $O(H)$ convergence predicted by Theorem [Theorem 14](#thm:DDLODError){reference-type="ref" reference="thm:DDLODError"}. It should be noted that the error estimate in [\[eq:DDLODError\]](#eq:DDLODError){reference-type="eqref" reference="eq:DDLODError"} concerns the approximation of $(\bar y,\bar u)$ by $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$, and the results reported in Figure [6](#fig:HHControlStateErrors){reference-type="ref" reference="fig:HHControlStateErrors"} measure the approximation of $(\bar y_h,\bar u_h)$ by $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$. The convergence of ${\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$ in the energy norm is $O(H)$, which agrees with Theorem [Theorem 16](#thm:EnergyErrors){reference-type="ref" reference="thm:EnergyErrors"}.* *For this example, the value of the modified cost function $\tilde J$ in [\[eq:newJ\]](#eq:newJ){reference-type="eqref" reference="eq:newJ"} is $-3.60479\times 10^{-8}$ for the fine scale standard finite element solution $(\bar y_h,\bar u_h)$. The values of $\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ are displayed in Table [1](#table:HHMinimum){reference-type="ref" reference="table:HHMinimum"}. The order of convergence of $\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ is roughly $O(H^2)$, which is consistent with Theorem [Theorem 14](#thm:DDLODError){reference-type="ref" reference="thm:DDLODError"}.* ---------- ------------------------------------------------------------------------------------------------- *$H$* *$\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$* *$1/10$* *$-3.55321\times10^{-8}$* *$1/20$* *$-3.59107\times10^{-8}$* *$1/40$* *$-3.60102\times 10^{-8}$* *$1/80$* *$-3.60372\times 10^{-8}$* ---------- ------------------------------------------------------------------------------------------------- : *Values of $\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ for Example [Example 18](#example:HH){reference-type="ref" reference="example:HH"}.* *We compare the graphs of $\bar y_h$ and ${\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$ (with $H=1/20$) in Figure [8](#fig:HHStateComparison){reference-type="ref" reference="fig:HHStateComparison"}, and the graphs of $\bar u_h$ and ${\bar u}_{\scriptscriptstyle H}$ (with $H=1/20$) in Figure [10](#fig:HHControlComparison){reference-type="ref" reference="fig:HHControlComparison"}.* *{reference-type="ref" reference="example:HH"}.](OptState-STD-319-hh-AL.png "fig:"){#fig:HHStateComparison width="0.3\\linewidth"} {reference-type="ref" reference="example:HH"}.](OptState-LOD-19-319-hh-AL.png "fig:"){#fig:HHStateComparison width="0.3\\linewidth"}* *{reference-type="ref" reference="example:HH"}.](OptControl-STD-319-hh-AL.png "fig:"){#fig:HHControlComparison width="0.3\\linewidth"} {reference-type="ref" reference="example:HH"}.](OptControl-LOD-19-319-hh-AL.png "fig:"){#fig:HHControlComparison width="0.3\\linewidth"}* *The active sets for $\bar u_h$ and ${\bar u}_{\scriptscriptstyle H}$ (with $H=1/20$) are depicted in Figure [12](#fig:HHActiveSet1){reference-type="ref" reference="fig:HHActiveSet1"} and Figure [14](#fig:HHActiveSet2){reference-type="ref" reference="fig:HHActiveSet2"}.* *{reference-type="ref" reference="example:HH"}: $\bar u_h$ (left) and ${\bar u}_{\scriptscriptstyle H}$ (right, $H=1/20$). ](Active-set-part1-af-19-319-hh.png "fig:"){#fig:HHActiveSet1 width="0.3\\linewidth"} {reference-type="ref" reference="example:HH"}: $\bar u_h$ (left) and ${\bar u}_{\scriptscriptstyle H}$ (right, $H=1/20$). ](Active-set-part1-af-LOD-19-319-hh.png "fig:"){#fig:HHActiveSet1 width="0.3\\linewidth"}* *{reference-type="ref" reference="example:HH"}: $\bar u_h$ (left) and ${\bar u}_{\scriptscriptstyle H}$ (right, $H=1/20$).](Active-set-part2-af-19-319-hh.png "fig:"){#fig:HHActiveSet2 width="0.3\\linewidth"} {reference-type="ref" reference="example:HH"}: $\bar u_h$ (left) and ${\bar u}_{\scriptscriptstyle H}$ (right, $H=1/20$).](Active-set-part2-af-LOD-19-319-hh.png "fig:"){#fig:HHActiveSet2 width="0.3\\linewidth"}* *The computation of the fine scale standard finite element solution $(\bar y_h,\bar u_h)$ of the discrete optimization problem takes $3.90\times 10^{+1}$ seconds by using the PETSc/TAO library with 20 processors. The computational time (in seconds) for $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ using MATLAB on a laptop are presented in Table [2](#table:HHTimes){reference-type="ref" reference="table:HHTimes"} for $H=1/10,1/20,1/40$.* *$H$* *Time* ---------- ------------------------ *$1/10$* *$1.26\times 10^{-2}$* *$1/20$* *$1.74\times 10^{-1}$* *$1/40$* *$1.04\times 10^{+1}$* : *Computational time in seconds for $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ (Example [Example 18](#example:HH){reference-type="ref" reference="example:HH"}).* *For $H=1/20$, the DD-LOD solution $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ yields a reasonable approximation of $(\bar y_h,\bar u_h)$ (cf. Figures [8](#fig:HHStateComparison){reference-type="ref" reference="fig:HHStateComparison"}--[14](#fig:HHActiveSet2){reference-type="ref" reference="fig:HHActiveSet2"}) and its computation is more than 100 times faster than the computation of $(\bar y_h,\bar u_h)$.* *In the second set of experiments we take $H=1/20$ and $\rho=1/40, 1/80, 1/160$ for the DD-LOD solution $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},\bar u_{\rho})$. In view of Theorem [Theorem 14](#thm:DDLODError){reference-type="ref" reference="thm:DDLODError"} and Theorem [Theorem 16](#thm:EnergyErrors){reference-type="ref" reference="thm:EnergyErrors"}, we expect these approximate solutions will improve over the approximate solution $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ with $H=1/20$ and $\mathcal{T}_\rho=\mathcal{T}_H$ obtained in the first set of experiments. This is confirmed by comparing the values of the cost function $\tilde J$ in Table [3](#table:HHrhoMinimum){reference-type="ref" reference="table:HHrhoMinimum"} with the value $\tilde J(\bar y_h,\bar u_h)=-3.60479\times 10^{-8}$ for the fine scale solution. The number of significant digits increases from $2$ to $4$ as $\rho$ decreases from $1/20$ to $1/160$.* ----------- ------------------------------------------------------------------------------- *$\rho$* *$\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},\bar u_{\rho})$* *$1/20$* *$-3.59107\times10^{-8}$* *$1/40$* *$-3.60090\times 10^{-8}$* *$1/80$* *$-3.60357\times 10^{-8}$* *$1/160$* *$-3.60431\times 10^{-8}$* ----------- ------------------------------------------------------------------------------- : *Values of $\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},\bar u_\rho)$ with $H=1/20$ and various $\rho$ for Example [Example 18](#example:HH){reference-type="ref" reference="example:HH"}.* *We can also visualize the improvement due to a smaller $\rho$ by comparing the graph of the fine scale solution $\bar u_h$ for the optimal control and the graph of the DD-LOD solution $\bar u_\rho$ for the optimal control (with $H=1/20$ and $\rho=1/160$) in Figure [16](#fig:HHrhoControlComparison){reference-type="ref" reference="fig:HHrhoControlComparison"}. They are hardly distinguishable, which is not the case for the graphs in Figure [10](#fig:HHControlComparison){reference-type="ref" reference="fig:HHControlComparison"}.* *{reference-type="ref" reference="example:HH"}.](OptControl-STD-319-hh-AL.png "fig:"){#fig:HHrhoControlComparison width="0.3\\linewidth"} {reference-type="ref" reference="example:HH"}.](Control_LOD_sol_19_160_hh.png "fig:"){#fig:HHrhoControlComparison width="0.3\\linewidth"}* *This is also true for the active sets, where the ones for the fine scale solution $\bar u_h$ and the ones for the DD-LOD solution $\bar u_\rho$ (with $H=1/20$ and $\rho=1/160$) are almost identical in Figure [18](#fig:HHrhoActiveSet1){reference-type="ref" reference="fig:HHrhoActiveSet1"} and Figure [20](#fig:HHrhoActiveSet2){reference-type="ref" reference="fig:HHrhoActiveSet2"}.* *{reference-type="ref" reference="example:HH"}: $\bar u_h$ (left) and $\bar u_\rho$ (right, $H=1/20$ and $\rho=1/160$). ](Active-set-part1-af-19-319-hh.png "fig:"){#fig:HHrhoActiveSet1 width="0.3\\linewidth"} {reference-type="ref" reference="example:HH"}: $\bar u_h$ (left) and $\bar u_\rho$ (right, $H=1/20$ and $\rho=1/160$). ](Active_set_phi1_19_160_hh.png "fig:"){#fig:HHrhoActiveSet1 width="0.3\\linewidth"}* *{reference-type="ref" reference="example:HH"}: $\bar u_h$ (left) and $\bar u_\rho$ (right, $H=1/20$ and $\rho=1/160$).](Active-set-part2-af-19-319-hh.png "fig:"){#fig:HHrhoActiveSet2 width="0.3\\linewidth"} {reference-type="ref" reference="example:HH"}: $\bar u_h$ (left) and $\bar u_\rho$ (right, $H=1/20$ and $\rho=1/160$).](Active_set_phi2_19_160_hh.png "fig:"){#fig:HHrhoActiveSet2 width="0.3\\linewidth"}* **Example 19** (Highly Oscillatory Coefficients). *The coefficient matrix for this example is given by $$\mathcal{A}=\begin{bmatrix} c(x)& {\bf 0}\\ {\bf 0} & c(x) \end{bmatrix},$$ where $$c(x)=\frac{2+1.8\sin\left(\frac{2\pi x_1}{\epsilon}\right)} {2+1.8\sin\left(\frac{2\pi x_2}{\epsilon}\right)}+ \frac{2+\sin\left(\frac{2\pi x_2}{\epsilon}\right)} {2+1.8\sin\left(\frac{2\pi x_1}{\epsilon}\right)}$$ with $\epsilon=0.025$. This choice of coefficients originates from the pioneering work [@HW:1999:MS] in numerical homogenization.* *We choose $y_d=-1$ and the control constraints are given by $\phi_1(x)=-0.01x_1-0.005$ and $\phi_2(x)=0.0007x_2-0.005$ (cf. Figure [21](#fig:HOCC){reference-type="ref" reference="fig:HOCC"}).* {reference-type="ref" reference="example:HO"}. ](Control-bounds-AL-osc.png){#fig:HOCC width="0.4\\linewidth"} *We take $h=1/320$ for the fine scale solution $(\bar y_h,\bar u_h)$. In the first set of experiments we compute the DD-LOD solution $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ for $H=1/10, 1/20,1/40,1/80$ (with $\mathcal{T}_\rho=\mathcal{T}_H$). The number of iterations $k$ used in the solution of the corrector equation equals $\lceil-3\ln H\rceil$ for all $H$. The relative errors for the approximation of the fine scale standard finite element solution $(\bar y_h,\bar u_h)$ by the multiscale finite element solution $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ are presented in Figure [24](#fig:HOControlStateErrors){reference-type="ref" reference="fig:HOControlStateErrors"}. The $O(H)$ convergence is observed for both ${\bar u}_{\scriptscriptstyle H}$ and ${\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$, which agrees with Theorem [Theorem 14](#thm:DDLODError){reference-type="ref" reference="thm:DDLODError"} and Theorem [Theorem 16](#thm:EnergyErrors){reference-type="ref" reference="thm:EnergyErrors"}.* {reference-type="ref" reference="example:HO"} with $H=1/10,1/20,1/20,1/80$.](Control-L2Error-osc-AL-319.png){#fig:HOControlStateErrors width="\\linewidth"} *(a)* {reference-type="ref" reference="example:HO"} with $H=1/10,1/20,1/20,1/80$.](L2Error-OptState-osc-AL-319.png){#fig:HOControlStateErrors width="\\linewidth"} *(b)* {reference-type="ref" reference="example:HO"} with $H=1/10,1/20,1/20,1/80$.](Energy_error_optState_osc.png){#fig:HOControlStateErrors width="\\linewidth"} *(c)* *For this example, the value of the modified cost function $\tilde J$ in [\[eq:newJ\]](#eq:newJ){reference-type="eqref" reference="eq:newJ"} is $-8.29631\times 10^{-5}$ for the fine scale standard finite element solution $(\bar y_h,\bar u_h)$. The values of $\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ are displayed in Table [4](#table:HOMinimum){reference-type="ref" reference="table:HOMinimum"}. The $O(H^2)$ convergence of $\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ also agrees with Theorem [Theorem 14](#thm:DDLODError){reference-type="ref" reference="thm:DDLODError"}.* ---------- ------------------------------------------------------------------------------------------------- *$H$* *$\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$* *$1/10$* *$-8.22171\times10^{-5}$* *$1/20$* *$-8.28313\times10^{-5}$* *$1/40$* *$-8.29343\times10^{-5}$* *$1/80$* *$-8.29550\times10^{-5}$* ---------- ------------------------------------------------------------------------------------------------- : *Values of $\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ for Example [Example 19](#example:HO){reference-type="ref" reference="example:HO"}.* *We compare the graphs of $\bar y_h$ and ${\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h}$ (with $H=1/20$) in Figure [26](#fig:HOStateComparison){reference-type="ref" reference="fig:HOStateComparison"}, and the graphs of $\bar u_h$ and ${\bar u}_{\scriptscriptstyle H}$ (with $H=1/20$) in Figure [28](#fig:HOControlComparison){reference-type="ref" reference="fig:HOControlComparison"}. The active sets for $\bar u_h$ and ${\bar u}_{\scriptscriptstyle H}$ (with $H=1/20$) are depicted in Figure [30](#fig:HOActiveSet1){reference-type="ref" reference="fig:HOActiveSet1"} and Figure [32](#fig:HOActiveSet2){reference-type="ref" reference="fig:HOActiveSet2"}.* *{reference-type="ref" reference="example:HO"}.](OptState-STD-319-osc-AL.png "fig:"){#fig:HOStateComparison width="0.3\\linewidth"} {reference-type="ref" reference="example:HO"}.](OptState-LOD-19-319-osc-AL.png "fig:"){#fig:HOStateComparison width="0.3\\linewidth"}* *{reference-type="ref" reference="example:HO"}.](OptControl-STD-319-osc-AL.png "fig:"){#fig:HOControlComparison width="0.3\\linewidth"} {reference-type="ref" reference="example:HO"}.](OptControl-LOD-19-319-osc-AL.png "fig:"){#fig:HOControlComparison width="0.3\\linewidth"}* *{reference-type="ref" reference="example:HO"}: $\bar u_h$ (left) and ${\bar u}_{\scriptscriptstyle H}$ (right, $H=1/20$).](Active-set-part1-af-19-319-osc.png "fig:"){#fig:HOActiveSet1 width="0.3\\linewidth"} {reference-type="ref" reference="example:HO"}: $\bar u_h$ (left) and ${\bar u}_{\scriptscriptstyle H}$ (right, $H=1/20$).](Active-set-part1-af-LOD-19-319-osc.png "fig:"){#fig:HOActiveSet1 width="0.3\\linewidth"}* *{reference-type="ref" reference="example:HO"}: $\bar u_h$ (left) and ${\bar u}_{\scriptscriptstyle H}$ (right, $H=1/20$).](Active-set-part2-af-19-319-osc.png "fig:"){#fig:HOActiveSet2 width="0.3\\linewidth"} {reference-type="ref" reference="example:HO"}: $\bar u_h$ (left) and ${\bar u}_{\scriptscriptstyle H}$ (right, $H=1/20$).](Active-set-part2-af-LOD-19-319-osc.png "fig:"){#fig:HOActiveSet2 width="0.3\\linewidth"}* *The computation of the fine scale standard finite element solution $(\bar y_h,\bar u_h)$ of the discrete optimization problem takes $4.36\times 10^{+1}$ seconds by using the PETSc/TAO library with 20 processors. The computational time (in seconds) for $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ using MATLAB on a laptop are presented in Table [5](#table:HOTimes){reference-type="ref" reference="table:HOTimes"} for $H=1/10,1/20,1/40$. For $H=1/20$, the DD-LOD solution $({y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ is a reasonable approximation of $(\bar y_h,\bar u_h)$ (cf. Figures [26](#fig:HOStateComparison){reference-type="ref" reference="fig:HOStateComparison"}--[32](#fig:HOActiveSet2){reference-type="ref" reference="fig:HOActiveSet2"}) and its computation is more than 200 times faster than the computation of $(\bar y_h,\bar u_h)$.* *$H$* *Time* ---------- ------------------------ *$1/10$* *$1.71\times 10^{-2}$* *$1/20$* *$1.27\times 10^{-1}$* *$1/40$* *$1.40\times 10^{+1}$* : *Computational time in seconds for $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},{\bar u}_{\scriptscriptstyle H})$ for Example [Example 19](#example:HO){reference-type="ref" reference="example:HO"}.* *In the second set of experiments we take $H=1/20$ and test the improved approximation by the DD-LOD solution $({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},\bar u_\rho)$ for $\rho=1/40,1/80/1/160$ that is predicted by the estimates in Theorem [Theorem 14](#thm:DDLODError){reference-type="ref" reference="thm:DDLODError"} and Theorem [Theorem 16](#thm:EnergyErrors){reference-type="ref" reference="thm:EnergyErrors"}. This improvement can be observed by comparing the values of the cost function $\tilde J$ in Table [6](#table:HOrhoMinimum){reference-type="ref" reference="table:HOrhoMinimum"} with the value $\tilde J(\bar y_h,\bar u_h)=-8.29631\times 10^{-5}$ for the fine scale solution. The number of significant digits improves from 2 to $3$ as $\rho$ decreases from $1/20$ to $1/160$.* ----------- ----------------------------------------------------------------------------- *$\rho$* *$\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},\bar u_\rho)$* *$1/20$* *$-8.28313\times10^{-5}$* *$1/40$* *$-8.29252\times10^{-5}$* *$1/80$* *$-8.29448\times10^{-5}$* *$1/160$* *$-8.29510\times 10^{-5}$* ----------- ----------------------------------------------------------------------------- : *Values of $\tilde J({\bar y}_{\scriptscriptstyle H,k}^{\mathrm{ms},h},\bar u_\rho)$ with $H=1/20$ and various $\rho$ for Example [Example 19](#example:HO){reference-type="ref" reference="example:HO"}.* *The improvement can also be visualized through a comparison of the graphs of the fine scale solution $\bar u_h$ and the DD-LOD solution $\bar u_\rho$ (with $H=1/20$ and $\rho=1/160$) in Figure [34](#fig:HOrhoControlComparison){reference-type="ref" reference="fig:HOrhoControlComparison"}. They are almost identical, which is not the case in Figure  [28](#fig:HOControlComparison){reference-type="ref" reference="fig:HOControlComparison"}.* *{reference-type="ref" reference="example:HO"}.](OptControl-STD-319-osc-AL.png "fig:"){#fig:HOrhoControlComparison width="0.3\\linewidth"} {reference-type="ref" reference="example:HO"}.](Control_LOD_sol_19_160_osc.png "fig:"){#fig:HOrhoControlComparison width="0.3\\linewidth"}* *We can also observe the improvement due to smaller $\rho$ by comparing the active sets depicted in Figure [36](#fig:HOrhoActiveSet1){reference-type="ref" reference="fig:HOrhoActiveSet1"} and Figure [38](#fig:HOrhoActiveSet2){reference-type="ref" reference="fig:HOrhoActiveSet2"}. These sets are almost identical, which is not the case in Figure [30](#fig:HOActiveSet1){reference-type="ref" reference="fig:HOActiveSet1"} and Figure [32](#fig:HOActiveSet2){reference-type="ref" reference="fig:HOActiveSet2"}.* *{reference-type="ref" reference="example:HO"}: $\bar u_h$ (left) and $\bar u_\rho$ (right, $H=1/20$ and $\rho=1/160$).](Active-set-part1-af-19-319-osc.png "fig:"){#fig:HOrhoActiveSet1 width="0.3\\linewidth"} {reference-type="ref" reference="example:HO"}: $\bar u_h$ (left) and $\bar u_\rho$ (right, $H=1/20$ and $\rho=1/160$).](Active_set_phi1_19_160_osc.png "fig:"){#fig:HOrhoActiveSet1 width="0.3\\linewidth"}* *{reference-type="ref" reference="example:HO"}: $\bar u_h$ (left) and $\bar u_\rho$ (right, $H=1/20$ and $\rho=1/160$).](Active-set-part2-af-19-319-osc.png "fig:"){#fig:HOrhoActiveSet2 width="0.3\\linewidth"} {reference-type="ref" reference="example:HO"}: $\bar u_h$ (left) and $\bar u_\rho$ (right, $H=1/20$ and $\rho=1/160$).](Active_set_phi2_19_160_osc.png "fig:"){#fig:HOrhoActiveSet2 width="0.3\\linewidth"}* # Concluding Remarks {#sec:Conclusions} We have constructed and analyzed a multiscale finite element method for the optimal control problem defined by [\[eq:OCP\]](#eq:OCP){reference-type="eqref" reference="eq:OCP"}--[\[eq:aDef\]](#eq:aDef){reference-type="eqref" reference="eq:aDef"}. We showed that the approximate solution obtained by the DD-LOD finite element method on the coarse mesh $\mathcal{T}_H$ is, up to an $O(H^2+\rho)$ term for the $L_2$ error and an $O(H+\rho)$ term for the energy error, as good as the approximate solution obtained by a standard finite element method on a fine mesh $\mathcal{T}_h$. Alternatively we can say that up to the fine scale error the performance of the DD-LOD method is as good as standard finite element methods for smooth problems. The DD-LOD multiscale finite element method is one of the simplest multiscale finite element methods in terms of construction and analysis. There is inherent parallelism in the construction of the DD-LOD finite element space that comes from domain decomposition so that it can readily benefit from high performance computing (cf. [@BGS:2022:LOD_OCP]), and its analysis only requires basic knowledge in finite element methods, domain decomposition methods and numerical linear algebra. After a multiscale basis has been computed off-line, the on-line solution with the coarse scale DD-LOD finite element method is fast. The multiscale finite element method is particularly useful for applications where the optimal control problem has to be solved repeatedly for different $y_d$, $\phi_1$ and $\phi_2$. We note that the error estimates in Theorem [Theorem 6](#thm:AbstractErrorEstimate){reference-type="ref" reference="thm:AbstractErrorEstimate"} and Theorem [Theorem 8](#thm:AbstractEnergyError){reference-type="ref" reference="thm:AbstractEnergyError"} are applicable to any subspace $V_*$ of $H^1_0(\Omega)$. The key is to have good error estimates for the Galerkin solution of [\[eq:RBVP\]](#eq:RBVP){reference-type="eqref" reference="eq:RBVP"}. In particular, we can take $V_*$ to be the LOD multiscale finite element spaces in [@PS:2016:Contrast; @HM:2017:Contrast; @HP:2013:LOD] and arrive at similar results. Note that the LOD methods in [@PS:2016:Contrast; @HM:2017:Contrast] are suitable for problems with high contrast. We can also take $V_*$ to be the multiscale finite element space $V_h$ from [@HW:1999:MS; @HWC:1999:Multiscale; @EH:2009:MSFEM] for problems with highly oscillatory and periodic coefficients (such as the problem in Example [Example 19](#example:HO){reference-type="ref" reference="example:HO"}), where $h$ stands for the coarse mesh size. The corresponding $L_2$ error estimate then takes the form $$\|\bar y-\bar y_{h,\rho}\|_{L_2(\Omega)}+\|\bar u-\bar u_{h,\rho}\|_{L_2(\Omega)}+\|\bar p-\bar p_{h,\rho}\|_{L_2(\Omega)} \leq C\Big(h^2+\epsilon+\frac{\epsilon}{h}+\rho\Big),$$ where $\epsilon \,(<h)$ is the parameter for the small scale, and the positive constant $C$ only depends on $\|y_d\|_{L_2(\Omega)}$, $\|\phi_1\|_{H^1(\Omega)}$, $\|\phi_2\|_{H^1(\Omega)}$, $\gamma^{-1}$, $\alpha^{-1}$ and the shape regularities of $\mathcal{T}_h$ and $\mathcal{T}_\rho$. Similarly, the multiscale finite element methods in [@AC:2022:DD26; @CLZZ:2023:MultiscaleCC] can also be analyzed by Theorem [Theorem 6](#thm:AbstractErrorEstimate){reference-type="ref" reference="thm:AbstractErrorEstimate"}, Theorem [Theorem 8](#thm:AbstractEnergyError){reference-type="ref" reference="thm:AbstractEnergyError"} and the estimates in [@OZB:2014:Homogenization; @CEL:2018:Multiscale]. # Acknowledgements {#acknowledgements .unnumbered} Portions of this research were conducted with high performance computing resources provided by Louisiana State University (http://www.hpc.lsu.edu). # Funding {#funding .unnumbered} This work was supported in part by the National Science Foundation under Grant No. DMS-19-13035 and Grant No. DMS-22-08404. # Data Availability {#data-availability .unnumbered} The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. 10 R.A. Adams and J.J.F. Fournier. . 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[^1]: This work was supported in part by the National Science Foundation under Grant No. DMS-19-13035 and Grant No. DMS-22-08404.

arxiv_math

--- abstract: | Consider $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ as real $C_0$-semigroups generated by closed and symmetric sesquilinear forms on a standard form of a von Neumann algebra. We provide a characterisation for the domination of the semigroup $(T_t)_{t\ge 0}$ by $(S_t)_{t\ge 0}$, which means that $-S_t v\le T_t u\le S_t v$ holds for all $t\ge 0$ and all real $u$ and $v$ that satisfy $-v\le u\le v$. This characterisation extends the Ouhabaz characterisation for semigroup domination to the non-commutative $L^2$ spaces. Additionally, we present a simpler characterisation when both semigroups are positive as well as consider the setting in which $(T_t)_{t\ge 0}$ need not be real. address: - Sahiba Arora, Technische Universität Dresden, Institut für Analysis, Fakultät für Mathematik , 01062 Dresden, Germany - Ralph Chill, Technische Universität Dresden, Institut für Analysis, Fakultät für Mathematik , 01062 Dresden, Germany - Sachi Srivastava, Department of Mathematics, University of Delhi, South Campus, New Delhi-21, India author: - Sahiba Arora - Ralph Chill - Sachi Srivastava bibliography: - ralph.bib title: Domination of semigroups on standard forms of von Neumann algebras --- [^1] # Introduction Domination of $C_0$-semigroups was first investigated by Simon [@Si77] inspired by the concept of domination of operators that occurred in the form of Kato's inequality [@Ka72]. Since then, the notion of domination has garnered considerable attention in the theory of $C_0$-semigroups. Whereas characterisations in terms of the generators were known much earlier (see, for instance, [@HeScUh77; @Si79], and [@Na86 Section C-II-4]), a characterisation in terms of the generating forms was given by Ouhabaz in [@Ou96] and by Barthélemy in [@By96] and was generalised to semigroups acting on two different $L^2$-spaces in [@MaVoVo05]. These results were recently considered within an abstract framework on more general ordered Hilbert spaces in [@LeScWi20]. In this article, we generalise Ouhabaz's result to the non-commutative setting. Following Cipriani [@Ci97; @Ci08] -- who generalised the Beurling-Deny criterion to the non-commutative setting (see [@Ci08 Theorem 2.53] or [@Ci97 Proposition 4.5 and Theorem 4.7]) -- we consider our state space to be a standard form of a von Neumann algebra. This has the advantage that for real elements, there is a concept of positive and negative parts (recalled below). It turns out that when generalising the known domination results from the commutative setting to the non-commutative setting one has to avoid the modulus as much as possible, both in the definition of domination and in the proofs. This is due to the well-known fact that in the non-commutative setting, the modulus no longer satisfies the triangle inequality so some convex structure is lost. Even though we define the modulus below, we only make use of it in the commutative setting. In the remainder of this section, we fix our notations about sesquilinear forms, briefly recall the notion of standard forms, and introduce the concept of generalised ideals. Subsequently, in Section [2](#sec:domination-general){reference-type="ref" reference="sec:domination-general"}, we state and prove our main result about the domination of real semigroups. Then, in Section [3](#sec:domination-both-positive){reference-type="ref" reference="sec:domination-both-positive"}, we restrict ourselves to the particular case when both semigroups are positive. Finally, in Section [4](#sec:domination-not-real){reference-type="ref" reference="sec:domination-not-real"}, we drop the assumption for the dominated semigroup to be real. ## Sesquilinear forms on Hilbert spaces and their associated semigroups {#sesquilinear-forms-on-hilbert-spaces-and-their-associated-semigroups .unnumbered} Throughout the article, we consider $C_0$-semigroups on complex Hilbert spaces, the generators of which arise from sesquilinear forms. Let $\mathcal{H}$ be a complex Hilbert space, let $\operatorname{dom}\left(\mathfrak a\right)\subseteq\mathcal{H}$ be a subspace and let $\mathfrak a: \operatorname{dom}\left(\mathfrak a\right) \times \operatorname{dom}\left(\mathfrak a\right) \to \mathbb{C}$ be a sesquilinear form. The subspace $\operatorname{dom}\left(\mathfrak a\right)$ is called the *domain* of the form $\mathfrak a$. We say that $\mathfrak a$ is - *densely defined* if $\operatorname{dom}\left(\mathfrak a\right)$ is a dense subspace of $\mathcal{H}$, - *symmetric* if $\mathfrak a(u,v) = \overline{\mathfrak a(v,u)}$ for every $u$, $v\in\operatorname{dom}\left(\mathfrak a\right)$, - *accretive* if $\mathop{\mathrm{Re}}\mathfrak a(u,u) \geq 0$ for every $u\in\operatorname{dom}\left(\mathfrak a\right)$, and - *closed* if there exists $\omega\in\mathbb{R}$ such that $\| u\|_{\operatorname{dom}\left(\mathfrak a\right)}^2 := \mathop{\mathrm{Re}}\mathfrak a(u,u) + \omega \| u\|_{\mathcal{H}}^2$ defines a complete norm on $\operatorname{dom}\left(\mathfrak a\right)$. We recall that every closed and symmetric form is continuous. In particular, if the sesquilinear form $\mathfrak a$ is densely defined, closed, and symmetric, then the operator $A$ on $\mathcal{H}$ given by $$\begin{aligned} \operatorname{dom}\left(A\right) & := \{ u\in \operatorname{dom}\left(\mathfrak a\right) : \exists f\in\mathcal{H}\ \forall y\in\operatorname{dom}\left(\mathfrak a\right) : \mathfrak a(u,v) = \left\langle f\, ,\, v\right\rangle \} , \\ Au & := f ,\end{aligned}$$ is self-adjoint, bounded from below, and therefore the negative generator of an analytic $C_0$-semigroup of self-adjoint operators. In short, we simply say that the sesquilinear form generates the semigroup. If the form $\mathfrak a$ is, in addition, accretive, then the semigroup is contractive. As we are interested in the domination of semigroups, and since domination is invariant under simultaneous scaling of both semigroups, we restrict ourselves to accretive forms. Throughout, we deliberately identify symmetric sesquilinear forms with their quadratic counterparts $\mathfrak a: \operatorname{dom}\left(\mathfrak a\right) \to \mathbb{R}$, $\mathfrak a(u) := \mathfrak a(u,u)$, since one can reconstruct the symmetric sesquilinear form from the quadratic form via the polarisation identity. As is customary in the literature, we denote the sesquilinear form and the quadratic form by the same letter. We further identify the quadratic form (and in turn the sesquilinear form) with its extension $\mathfrak a: \mathcal{H}\to \mathbb{R}\cup \{ +\infty\}$, where $$\mathfrak a(u) := \begin{cases} \mathfrak a(u) & \text{if } u\in\operatorname{dom}\left(\mathfrak a\right) , \\ +\infty & \text{if } u\in\mathcal{H}\setminus \operatorname{dom}\left(\mathfrak a\right) . \end{cases}$$ This extended quadratic form renders some statements easier to formulate. Note that if the sesquilinear form is symmetric and accretive, then the quadratic form (extended quadratic form) takes values in $[0,\infty )$ (respectively, $[0,\infty ]$). Frequently, we require the following theorem about the invariance of closed, convex sets in Hilbert spaces. The equivalence of (i)-(iv) is well-known (see, for example, [@Ou04 Theorems 2.2 and 2.3]) and the equivalences of (i), (v), and (vi) follow from [@By96 Théorème 1.9]. **Theorem 1**. *Let $(R_t)_{t\geq 0}$ be a $C_0$-semigroup on a Hilbert space $\mathcal{H}$ which is generated by a densely defined, closed, symmetric, and accretive sesquilinear form $\mathfrak c: \operatorname{dom}\left(\mathfrak a\right) \times\operatorname{dom}\left(\mathfrak c\right) \to\mathbb{C}$. Let $C \subseteq\mathcal{H}$ be a closed, convex subset and let $P$ be the orthogonal projection onto $C$. The following are equivalent:* 1. *$R_tC\subseteq C$ for every $t\geq 0$.* 2. *$\mathfrak c(Pu) \leq \mathfrak c(u)$ for every $u\in\mathcal{H}$.* 3. *$P(\operatorname{dom}\left(\mathfrak c\right))\subseteq \operatorname{dom}\left(\mathfrak c\right)$ and $\mathop{\mathrm{Re}}\mathfrak c( u, u-Pu )\ge 0$ for all $u\in\operatorname{dom}\left(\mathfrak c\right)$.* 4. *$P(\operatorname{dom}\left(\mathfrak c\right))\subseteq \operatorname{dom}\left(\mathfrak c\right)$ and $\mathop{\mathrm{Re}}\mathfrak c( Pu, u-Pu )\ge 0$ for all $u\in\operatorname{dom}\left(\mathfrak c\right)$.* *If $C_0\subseteq\mathcal{H}$ is another closed, convex subset such that $C\subseteq C_0$ and $R_tC_0\subseteq C_0$ for every $t\geq 0$, then (i)-(iv) are also equivalent to:* 1. *$\mathfrak c(Pu) \leq \mathfrak c(u)$ for every $u\in C_0$.* 2. *$P(\operatorname{dom}\left(\mathfrak c\right) \cap C_0)\subseteq \operatorname{dom}\left(\mathfrak c\right)$ and $\mathop{\mathrm{Re}}\mathfrak c( u, u-Pu )\ge 0$ for all $u\in\operatorname{dom}\left(\mathfrak c\right)\cap C_0$.* ## Standard forms of von Neumann algebras {#standard-forms-of-von-neumann-algebras .unnumbered} Let $\mathcal{H}$ be a complex Hilbert space, and let $\mathcal{H}_+\subseteq\mathcal{H}$ be a positive cone, that is, for all $u,v\in\mathcal{H}_+$ and all $\lambda\geq 0$, we have $u+\lambda v\in\mathcal{H}_+$ and $\mathcal{H}_+ \cap (-\mathcal{H}_+) = \{ 0\}$. Elements of $\mathcal{H}_+$ are called the *positive* elements of $\mathcal{H}$. Throughout, the cone $\mathcal{H}_+$ is *self-polar*, that is, $$\mathcal{H}_+ = \{ u\in \mathcal{H}: \left\langle u\, ,\, v\right\rangle \ge 0 \text{ for all }v\in \mathcal{H}_+\} .$$ Self-polarity of the cone ensures the following decomposition [@Ci08 Proposition 2.3]: $\mathcal{H}$ is the complexification of the real Hilbert space $$\mathcal{H}^J := \{ u \in \mathcal{H}: \left\langle u\, ,\, v\right\rangle \in \mathbb{R}\text{ for all }v\in \mathcal{H}_+\} , % = \calH_+ - \calH_+ ,$$ that is, $\mathcal{H}= \mathcal{H}^J + i \mathcal{H}^J.$ Elements of $\mathcal{H}^J$ are called *real*. For a subspace $\mathcal{U}$ of the Hilbert space $\mathcal{H}$, we use the notation $\mathcal{U}^J:= \mathcal{U}\cap \mathcal{H}^J$ for simplicity. A self-polar cone $\mathcal{H}_+$ turns $\mathcal{H}^J$ into a real ordered space with order denoted by $\leq$, where for $u,v \in \mathcal{H}^J,$ we have $u \le v$ if $v - u \in \mathcal{H}_+$. It also gives rise to an anti-unitary involution $$J: \mathcal{H}\to \mathcal{H}, \qquad (u+iv)\mapsto (u-iv) \text{ for } u,v \in \mathcal{H}^J.$$ Let $u_+ :=\operatorname{Proj}(u,\mathcal{H}_+)$ (respectively, $u_- := \operatorname{Proj}(-u,\mathcal{H}_+)$) denote the orthogonal projection of a vector $u\in\mathcal{H}^J$ (respectively, $-u$) onto the closed convex set $\mathcal{H}_+$. For every $u\in \mathcal{H}^J$, we have $u_+$, $u_-\in \mathcal{H}_+$ and $u=u_+ - u_-$ (Jordan decomposition) with $\left\langle u_+\, ,\, u_-\right\rangle=0$. This decomposition is unique. Similarly, define $$u \vee v := \operatorname{Proj}(u,v+\mathcal{H}_+) \quad \text{and} \quad u \wedge v:= \operatorname{Proj}(u,v-\mathcal{H}_+)$$ for $u,v\in \mathcal{H}^J$. This notation is reminiscent of the vector lattice case and is justified by [@Ci97 Lemma 4.4(i)], which says that if $\sup(u,v)$ exists (to be understood in the order sense) then $\sup(u,v)=u \vee v$ and if $\inf(u,v)$ exists (in the order sense), then $\inf(u,v)=u\wedge v$. For this reason, the vectors $u_+$ and $u_-$ are respectively called the *positive* and *negative* parts of $u$. In fact, we even know from [@Ci97 Lemma 4.4(iii)] that $$u \vee v= u+ (v-u)_+ = v+ (u-v)_+$$ In particular, $u \vee v$ is, at least, an upper bound of both $u$ and $v$. Analogously, $u \wedge v$ is a lower bound of both $u$ and $v$. Moreover, just like in the vector lattice case, the above projections satisfy the standard properties. For instance, $u\vee 0=u_+$, $u\wedge 0=-u_-$, and $u \vee v+ u\wedge v=u+v$ for all $u,v\in \mathcal{H}^J$. A proof of these identities can be found in [@Ci97 Lemma 4.4]. Our arguments freely make use of the properties listed in [@Ci97 Lemma 4.4] without citing. Throughout, we let $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ be a *standard form*, that is, a quadruple consisting of a von Neumann algebra $\mathfrak M$ acting on a complex Hilbert space $\mathcal{H}$ (in particular, $\mathfrak M\subseteq \mathcal{L}(\mathcal{H}))$ endowed with a self-polar cone $\mathcal{H}_+$ and an anti-linear involution $J : \mathcal{H}\to \mathcal{H}$ such that (a) $J\mathfrak MJ=\mathfrak M'$, where $\mathfrak M'$ is the commutant of $\mathfrak M$ in $\mathcal{L}(\mathcal{H})$, (b) $JaJ= a^*$ for all $a \in \mathfrak M\cap \mathfrak M'$, (c) $J u=u$ for all $u \in \mathcal{H}_+$, and (d) $aJaJ(\mathcal{H}_+)\subseteq \mathcal{H}_+$ for all $a\in \mathfrak M$. For every von Neumann algebra, there is a standard form, and if $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ and $(\hat{\mathfrak M}, \hat{\mathcal{H}}, \hat{\mathcal{H}}_+,\hat{J})$ are two standard forms such that the von Neumann algebras $\mathfrak M$ and $\hat{\mathfrak M}$ are isomorphic via an isomorphism, say $\Phi$, then there exists a unitary operator $U: \mathcal{H}\to \hat{\mathcal{H}}$ such that $\Phi (\mathfrak M) = U\mathfrak MU^{-1}$, $\hat{J} = UJU^{-1}$, and $U (\mathcal{H}_+) = \hat{\mathcal{H}}_+$ (see [@Ta03II Theorem IX.1.4] or [@Ci08 Proposition 2.16]). In this sense, the standard form is uniquely determined by the von Neumann algebra and is usually denoted by $(\mathfrak M, L^2(\mathfrak M), L_+^2(\mathfrak M), J)$. This notation is reminiscent of the case when the von Neumann algebra $\mathfrak M$ has a semi-finite, normal, and faithful trace $\tau$ because then $L^2 (\mathfrak M)$ is the associated non-commutative $L^2$-space which is, by definition, the completion of the space $\{ a\in\mathfrak M: \tau (aa^*) <\infty \}$ with respect to the inner product $\left\langle a\, ,\, b\right\rangle := \tau (ab^*)$. An operator $T: \mathcal{H}\to \mathcal{H}$ is said to be *real* if $T (\mathcal{H}^J) \subseteq \mathcal{H}^J$ and *positive* if $T (\mathcal{H}_{+}) \subseteq \mathcal{H}_{+}$. Similarly, a semigroup $(T_t)_{t\geq 0}$ is *real* (respectively, *positive*) if the operators $T_t$ are real (respectively, positive) for every $t\geq 0$. Lastly, if $(T_t)_{t\geq 0}$ is a real $C_0$-semigroup generated by a sesquilinear form $\mathfrak a$, then $\operatorname{dom}\left(\mathfrak a\right)^J$ is generating for $\operatorname{dom}\left(\mathfrak a\right)$ and $\mathfrak a(u,v)\in \mathbb{R}$ whenever $u,v\in \operatorname{dom}\left(\mathfrak a\right)^J$. Indeed, this follows Theorem [Theorem 1](#thm:barthelemy){reference-type="ref" reference="thm:barthelemy"}; the proof actually follows exactly as the proof of [@Ou04 Proposition 2.5] taking $C=\mathcal{H}^J$. ## Generalised ideals {#generalised-ideals .unnumbered} Let $(\Omega,\mu)$ be a $\sigma$-finite measure space and $\mathcal{H}:=L^2(\Omega,\mu)$. Let $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ be real and self-adjoint $C_0$-semigroups on the Hilbert space $\mathcal{H}$ generated by densely defined, closed, and symmetric sesquilinear forms $\mathfrak a: \operatorname{dom}\left(\mathfrak a\right) \times \operatorname{dom}\left(\mathfrak a\right) \to \mathbb{C}$ and $\mathfrak b: \operatorname{dom}\left(\mathfrak b\right) \times \operatorname{dom}\left(\mathfrak b\right) \to \mathbb{C}$ respectively. If $(S_t)_{t\ge 0}$ dominates $(T_t)_{t\ge 0}$, that is, $$\left\lvert T_t u \right\rvert\le S_t\left\lvert u \right\rvert$$ for all $u\in \mathcal{H}$, then $\operatorname{dom}\left(\mathfrak b\right)$ is a sublattice of $\mathcal{H}$ [@Ou04 Proposition 2.20] and $\operatorname{dom}\left(\mathfrak a\right)$ is a so-called *generalised ideal* of $\operatorname{dom}\left(\mathfrak b\right)$; see [@MaVoVo05 Theorem 4.1] or [@Ou04 Theorem 2.21]. We point out that generalised ideals were simply called *ideals* in [@Ou04]. However, we refrain from using that terminology as, in general, it does not coincide with the notion of a *lattice ideal*. In the non-commutative setting, when $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ is a standard form of a von Neumann algebra $\mathfrak M$, and when $\mathcal{U}$ and $\mathcal{V}$ are two subspaces of the Hilbert space $\mathcal{H}$, then we will say that the real space $\mathcal{U}^J$ is a *generalised ideal* of the real space $\mathcal{V}^J$ if both the conditions $$\label{cond:one-generalised ideal} (u-v)_+ - (u+v)_- \in \mathcal{U}$$ and $$\label{cond:two-generalised ideal} (u-v)_+ + (u+v)_- \in \mathcal{V}$$ hold whenever $u\in \mathcal{U}^J$ and $v\in\mathcal{V}^J$. Note that if $\mathcal{U}^J$ is a generalised ideal of $\mathcal{V}^J$, then condition [\[cond:two-generalised ideal\]](#cond:two-generalised ideal){reference-type="eqref" reference="cond:two-generalised ideal"} above implies that $$\label{eq:ideal-modulus} u\in \mathcal{U}^J \Rightarrow \left\lvert u \right\rvert := u_+ + u_- \in \mathcal{V}.$$ A rewriting of the terms in [\[cond:one-generalised ideal\]](#cond:one-generalised ideal){reference-type="eqref" reference="cond:one-generalised ideal"} and [\[cond:two-generalised ideal\]](#cond:two-generalised ideal){reference-type="eqref" reference="cond:two-generalised ideal"} yields the following alternate definition of generalised ideals. **Proposition 2**. *Let $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ be a standard form of the von Neumann algebra $\mathfrak M$ and let $\mathcal{U}$ and $\mathcal{V}$ be two subspaces of $\mathcal{H}$.* *Then $\mathcal{U}^J$ is a generalised ideal of $\mathcal{V}^J$ if and only if $$(u+v)_+ - (v-u)_+ \in \mathcal{U}\quad \text{ and} \quad (u+v)_+ + (v-u)_+ \in \mathcal{V}$$ whenever $u\in \mathcal{U}^J$ and $v\in \mathcal{V}^J$.* *Proof.* For $u$, $v\in \mathcal{H}^J$, we observe that $$\label{eq:ideal-equivalence-1} \begin{aligned} (u+v)_+ - (v-u)_+ &= u + v +(u+v)_- -\big(v-u + (u-v)_+\big)\\ &= 2u - \big((u-v)_+ - (u+v)_-\big) \end{aligned}$$ and $$\label{eq:ideal-equivalence-2} \begin{aligned} (u+v)_+ + (v-u)_+ &= u + v +(u+v)_- +\big(v-u + (u-v)_+\big)\\ &= 2v + \big((u-v)_+ + (u+v)_-\big); \end{aligned}$$ from which the assertion is immediate. ◻ We end this section by showing that in the commutative setting, if $\mathcal{V}$ is a sublattice, then $\mathcal{U}^J$ is a generalised ideal of $\mathcal{V}^J$ in the sense of our definition if and only if $\mathcal{U}^J$ is a generalised ideal of $\mathcal{V}^J$ in the sense of [@MaVoVo05 Definition 3.3]. Keep in mind that this is only a statement about the real parts of the spaces $\mathcal{U}$ and $\mathcal{V}$, and not about the full spaces themselves. **Proposition 3**. *Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, let $\mathcal{H}:=L^2(\Omega,\mu)$, and let $\mathcal{U}$ and $\mathcal{V}$ be two subspaces of $\mathcal{H}$.* *If $\mathcal{V}$ is a sublattice, then $\mathcal{U}^J$ is a generalised ideal of $\mathcal{V}^J$ in the sense of our definition if and only if* 1. *whenever $u\in \mathcal{U}^J$, then $\left\lvert u \right\rvert\in \mathcal{V}$, and* 2. *if $u \in \mathcal{U}^J, v\in \mathcal{V}^J$, and $\left\lvert v \right\rvert\le \left\lvert u \right\rvert$, then $v\mathop{\mathrm{sgn}}u \in \mathcal{U}$,* *that is, $\mathcal{U}^J$ is a generalised ideal of $\mathcal{V}^J$ in the sense of [@MaVoVo05 Definition 3.3].* *Proof.* For real $u,v\in \mathcal{H}$, we set $$\tilde u:= \frac{1}{2} (u+v)_+ -\frac{1}{2} (v-u)_+ \quad \text{and}\quad \tilde v=\frac{1}{2} (u+v)_+ +\frac{1}{2} (v-u)_+.$$ The commutative setting allows us to rewrite $$\begin{aligned} \big( \tilde u , \tilde v \big) &= \begin{cases} (u,v),\quad &\text{if }\left\lvert u \right\rvert\le v\\ (0,0),\quad &\text{if }\left\lvert u \right\rvert\le -v\\ \frac{1}{2}(u+v,u+v),\quad &\text{if }\left\lvert v \right\rvert\le u\\ \frac{1}{2}(u-v,v-u),\quad &\text{if }\left\lvert v \right\rvert\le -u \end{cases} \\ &= \frac{1}{2}\big((\left\lvert u \right\rvert+ \left\lvert u \right\rvert\wedge v)_+\mathop{\mathrm{sgn}}u, (v+ \left\lvert u \right\rvert\vee v)_+ \big); \end{aligned}$$ which is exactly the projection $P$ onto the set $$\{(a,b)\in \mathcal{H}\times \mathcal{H}: \left\lvert a \right\rvert\le b\}$$ for real $u,v\in \mathcal{H}$. By assumption, $\mathcal{V}$ is a sublattice of $\mathcal{H}$. Therefore, by [@MaVoVo05 Propositon 3.5], the projection $P$ leaves $\mathcal{U}^J\times \mathcal{V}^J$ invariant if and only if both (a) and (b) hold. With the aid of Proposition [Proposition 2](#prop:prop-ideal-equivalence){reference-type="ref" reference="prop:prop-ideal-equivalence"}, it follows that $\mathcal{U}^J$ is a generalised ideal of $\mathcal{V}^J$ if and only if both (a) and (b) are true. ◻ # Domination of a real semigroup by a positive semigroup {#sec:domination-general} Let $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ be a standard form of the von Neumann algebra $\mathfrak M$ and let $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ be real $C_0$-semigroups on the Hilbert space $\mathcal{H}$. We shall say that the semigroup $(T_t)_{t\ge 0}$ is *dominated* by $(S_t)_{t\ge 0}$ if $$\left( -v \le u \le v \right) \Rightarrow \left(- S_t v \le T_t u\le S_t v \right)$$ for all $u,v\in \mathcal{H}^J$ and all $t\ge 0$. In the commutative case, that is, if $\mathcal{H}$ is a Hilbert lattice, each $u\in \mathcal{H}^J$ satisfies $\left\lvert u \right\rvert=\sup(-u,u)$. Therefore, in this case, the above definition of domination is equivalent to the usual definition of domination [@Ou04 Section 2.3], which is, $$\left\lvert T_t u \right\rvert\le S_t \left\lvert u \right\rvert \quad (u\in \mathcal{H}, t\ge 0).$$ In this section, we characterise the domination of $(T_t)_{t\ge 0}$ by $(S_t)_{t\ge 0}$ generated by sesquilinear forms $\mathfrak a$ and $\mathfrak b$ in terms of $\mathfrak a$ and $\mathfrak b$. Recall that we deliberately identify symmetric sesquilinear forms with extended quadratic forms. **Theorem 4**. *Let $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ be a standard form of the von Neumann algebra $\mathfrak M$ and let $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ be real and self-adjoint $C_0$-semigroups on the Hilbert space $\mathcal{H}$ generated by densely defined, closed, symmetric, and accretive sesquilinear forms $\mathfrak a: \operatorname{dom}\left(\mathfrak a\right) \times \operatorname{dom}\left(\mathfrak a\right) \to \mathbb{C}$ and $\mathfrak b: \operatorname{dom}\left(\mathfrak b\right) \times \operatorname{dom}\left(\mathfrak b\right) \to \mathbb{C}$ respectively.* *Assume that the semigroup $(S_t)_{t\ge 0}$ is positive. Then the following conditions are equivalent.* 1. *The semigroup $(T_t)_{t\ge 0}$ is dominated by $(S_t)_{t\ge 0}$.* 2. *The inequality $\mathfrak a(u-\hat u) + \mathfrak b(v+\hat v)\le \mathfrak a(u)+\mathfrak b(v)$ holds for all $u,v\in \mathcal{H}^J$.* 3. *The inequality $\mathfrak a(\tilde u) + \mathfrak b(\tilde v)\le \mathfrak a(u)+\mathfrak b(v)$ holds for all $u,v\in \mathcal{H}^J$.* 4. *The subspace $\operatorname{dom}\left(\mathfrak a\right)^J$ is a generalised ideal of $\operatorname{dom}\left(\mathfrak b\right)^J$ and the inequality $\mathfrak a(u, \hat u) \ge \mathfrak b(v, \hat v)$ holds for all $(u,v)\in \operatorname{dom}\left(\mathfrak a\right)^J\times\operatorname{dom}\left(\mathfrak b\right)^J$.* 5. *The subspace $\operatorname{dom}\left(\mathfrak a\right)^J$ is a generalised ideal of $\operatorname{dom}\left(\mathfrak b\right)^J$ and the inequality $\mathfrak a(u,\tilde u) +\mathfrak b(v,\tilde v) \le \mathfrak a(u)+\mathfrak b(v)$ holds for all $(u,v)\in \operatorname{dom}\left(\mathfrak a\right)^J\times\operatorname{dom}\left(\mathfrak b\right)^J$.* 6. *The subspace $\operatorname{dom}\left(\mathfrak a\right)^J$ is a generalised ideal of $\operatorname{dom}\left(\mathfrak b\right)^J$ and the inequality $\mathfrak a(\hat u)+\mathfrak b(\hat v) \le \mathfrak a(u, \hat u) - \mathfrak b(v, \hat v)$ holds for all $(u,v)\in \operatorname{dom}\left(\mathfrak a\right)^J\times\operatorname{dom}\left(\mathfrak b\right)^J$.* 7. *The subspace $\operatorname{dom}\left(\mathfrak a\right)^J$ is a generalised ideal of $\operatorname{dom}\left(\mathfrak b\right)^J$ and the inequality $\mathfrak a(\tilde u)+\mathfrak b(\tilde u)\le \mathfrak a(u,\tilde u)+\mathfrak b(v,\tilde v)$ holds for all $(u,v)\in \operatorname{dom}\left(\mathfrak a\right)^J\times\operatorname{dom}\left(\mathfrak b\right)^J$.* *Here, we have used the notations, $$\hat u:= \frac{1}{2} (u-v)_+ -\frac{1}{2} (u+v)_- \quad \text{and}\quad \hat v:=\frac{1}{2} (u-v)_+ +\frac{1}{2} (u+v)_-$$ and $$\tilde u:= \frac{1}{2} (u+v)_+ -\frac{1}{2} (v-u)_+ \quad \text{and}\quad \tilde v:=\frac{1}{2} (u+v)_+ +\frac{1}{2} (v-u)_+$$ for $u,v\in\mathcal{H}^J$.* We point out that if $(u,v)\in \operatorname{dom}\left(\mathfrak a\right)^J\times\operatorname{dom}\left(\mathfrak b\right)^J$, then the inequality in condition (iii) of Theorem [Theorem 4](#thm:general){reference-type="ref" reference="thm:general"} implies that $(\tilde u,\tilde v)\in \operatorname{dom}\left(\mathfrak a\right)\times\operatorname{dom}\left(\mathfrak b\right)$. In particular, $\operatorname{dom}\left(\mathfrak a\right)^J$ is a generalised ideal of $\operatorname{dom}\left(\mathfrak b\right)^J$ by Proposition [Proposition 2](#prop:prop-ideal-equivalence){reference-type="ref" reference="prop:prop-ideal-equivalence"}. The same conclusion holds for the inequality in condition (ii). *Proof of Theorem [Theorem 4](#thm:general){reference-type="ref" reference="thm:general"}.* We start by considering the closed convex set $$C := \{ (a,b)\in \mathcal{H}\times \mathcal{H}: -b\le a\le b\}$$ and the product semigroup on $\mathcal{H}\times \mathcal{H}$ given by $$R_t:= \begin{pmatrix} T_t & 0 \\ 0 & S_t \end{pmatrix} \qquad (t\ge 0).$$ Since the semigroups $(R_t)_{t\ge 0}$ is real, the domination condition (i) is equivalent to the closed convex set $C$ being invariant under the semigroup $(R_t)_{t\ge 0}$, that is, $R_t C \subseteq C$ for all $t\ge 0$. This allows us to make use of Theorem [Theorem 1](#thm:barthelemy){reference-type="ref" reference="thm:barthelemy"}. For this purpose, we show that the orthogonal projection $P$ of $\mathcal{H}\times \mathcal{H}$ onto $C$ satisfies $$\label{eq:projection-general} P(u,v)=(u-\hat u, v +\hat v)= (\tilde u,\tilde v)\quad (u,v\in \mathcal{H}^J).$$ Actually, the second equality is a direct consequence of [\[eq:ideal-equivalence-1\]](#eq:ideal-equivalence-1){reference-type="eqref" reference="eq:ideal-equivalence-1"} and [\[eq:ideal-equivalence-2\]](#eq:ideal-equivalence-2){reference-type="eqref" reference="eq:ideal-equivalence-2"}. In order to show the first equality in [\[eq:projection-general\]](#eq:projection-general){reference-type="eqref" reference="eq:projection-general"}, we make use of the following equivalence: $(u',v')=P(u,v)$ if and only if $$\label{eq:projection-criterion} (u',v')\in C \text{ and } \mathop{\mathrm{Re}}\left\langle(u,v)-(u',v')\, ,\, (a,b)-(u',v')\right\rangle\le 0 \text{ for all }(a,b)\in C.$$ So, fix $u,v\in \mathcal{H}^J$. The inequalities $$\begin{aligned} v+\hat v-(u-\hat u) = v - u + (u-v)_+ = (u-v)_- \ge 0 \end{aligned}$$ and $$\begin{aligned} v+\hat v+(u-\hat u) = u + v + (u+v)_- = (u+v)_+ \ge 0 \end{aligned}$$ tell us that $(u-\hat u,v+\hat v)\in C$. In addition, for $(a,b)\in C$, we have $$\begin{aligned} 4A &:= 4\left\langle(u,v)-(u-\hat u,v+\hat v)\, ,\, (a,b)-(u-\hat u,v+\hat v)\right\rangle\\ & = 2\left\langle\big((u-v)_+ -(u+v)_-, - (u-v)_+ -(u+v)_- \big) \, ,\, (a+\hat u,b-\hat v) \right\rangle\\ & = 2\left\langle(u-v)_+\, ,\, a-b\right\rangle -2\left\langle(u+v)_-\, ,\, a+b\right\rangle\\ &\qquad\qquad+\left\langle(u-v)_+\, ,\, (u-v)_-\right\rangle +\left\langle(u+v)_-\, ,\, (u+v)_+\right\rangle \end{aligned}$$ Using orthogonality of the positive and the negative part, we deduce that $$2A = \left\langle(u-v)_+\, ,\, a-b\right\rangle -\left\langle(u+v)_-\, ,\, a+b\right\rangle \le 0.$$ We have thus verified that both the conditions in [\[eq:projection-criterion\]](#eq:projection-criterion){reference-type="eqref" reference="eq:projection-criterion"} are fulfilled. Consequently, the first equality in [\[eq:projection-general\]](#eq:projection-general){reference-type="eqref" reference="eq:projection-general"} is true. Next, note that the semigroup $(R_t)_{t\ge 0}$ on $\mathcal{H}\times \mathcal{H}$ is generated by the form $\mathfrak c$ with domain $\operatorname{dom}\left(\mathfrak c\right)=\operatorname{dom}\left(\mathfrak a\right)\times \operatorname{dom}\left(\mathfrak b\right)$ and $$\mathfrak c( (u_0,v_0), (u_1, v_1) ) = \mathfrak a(u_0, u_1)+ \mathfrak b(v_0,v_1).$$ As the semigroup is real and the form is symmetric, so the semigroup $(R_t)_{t\ge 0}$ leaves $C$ invariant if and only if $\mathfrak c(P(u,v))\le \mathfrak c( (u,v))$ for all $u,v\in \mathcal{H}^J$ (Theorem [Theorem 1](#thm:barthelemy){reference-type="ref" reference="thm:barthelemy"}). The equivalences "(i) $\Leftrightarrow$ (ii)" and "(i) $\Leftrightarrow$ (iii)" can now be deduced by substituting [\[eq:projection-general\]](#eq:projection-general){reference-type="eqref" reference="eq:projection-general"}. Next, recalling that the semigroup $(R_t)_{t\ge 0}$ is real, it follows from [\[eq:projection-general\]](#eq:projection-general){reference-type="eqref" reference="eq:projection-general"} that $P$ leaves the subspace $\operatorname{dom}\left(\mathfrak c\right)$ invariant if and only if $$(\hat u,\hat v) \in \operatorname{dom}\left(\mathfrak a\right)\times \operatorname{dom}\left(\mathfrak b\right) \quad \text{for all }(u,v)\in \operatorname{dom}\left(\mathfrak a\right)^J\times\operatorname{dom}\left(\mathfrak b\right)^J$$ or equivalently $$(\tilde u,\tilde v) \in \operatorname{dom}\left(\mathfrak a\right)\times\operatorname{dom}\left(\mathfrak b\right) \quad \text{for all }(u,v)\in \operatorname{dom}\left(\mathfrak a\right)^J\times\operatorname{dom}\left(\mathfrak b\right)^J.$$ In other words, $P$ leaves $\operatorname{dom}\left(\mathfrak c\right)$ invariant if and only if $\operatorname{dom}\left(\mathfrak a\right)^J$ is a generalised ideal of $\operatorname{dom}\left(\mathfrak b\right)^J$; see Proposition [Proposition 2](#prop:prop-ideal-equivalence){reference-type="ref" reference="prop:prop-ideal-equivalence"}. Now, fix $(u,v)\in \operatorname{dom}\left(\mathfrak a\right)^J\times\operatorname{dom}\left(\mathfrak b\right)^J$. Once again employing [\[eq:projection-general\]](#eq:projection-general){reference-type="eqref" reference="eq:projection-general"}, we obtain that $$\begin{aligned} \mathfrak c( (u,v), (u,v)-P(u,v) ) & = \mathfrak a(u, \hat u) - \mathfrak b(v, \hat v)\\ & = \mathfrak a(u)- \mathfrak a(u,\tilde u) +\mathfrak b(v)- \mathfrak b(v,\tilde v) \end{aligned}$$ and $$\begin{aligned} \mathfrak c( P(u,v), (u,v)-P(u,v) ) & = \mathfrak a(u,\hat u)-\mathfrak a(\hat u) - \mathfrak b(v,\hat v)-\mathfrak b(\hat v)\\ & = \mathfrak a(u,\tilde u)-\mathfrak a(\tilde u)+\mathfrak b(v,\tilde v)-\mathfrak b(\tilde v). \end{aligned}$$ In particular, $$\begin{aligned} \mathfrak c( (u,v), (u,v)-P(u,v) )\ge 0 &\Leftrightarrow \mathfrak a(u, \hat u) \ge \mathfrak b(v, \hat v) \\ & \Leftrightarrow \mathfrak a(u,\tilde u) +\mathfrak b(v,\tilde v) \le \mathfrak a(u)+\mathfrak b(v) \end{aligned}$$ and $$\begin{aligned} \mathfrak c( (Pu,v), (u,v)-P(u,v) )\ge 0 &\Leftrightarrow \mathfrak a(\hat u)+\mathfrak b(\hat v) \le \mathfrak a(u, \hat u) - \mathfrak b(v, \hat v)\\ & \Leftrightarrow \mathfrak a(\tilde u)+\mathfrak b(\tilde u)\le \mathfrak a(u,\tilde u)+\mathfrak b(v,\tilde v). \end{aligned}$$ We are now in a position to once again employ the characterisation of semigroup-invariance of a closed convex set Theorem [Theorem 1](#thm:barthelemy){reference-type="ref" reference="thm:barthelemy"} which gives the equivalence of (i) with each of (iv)-(vii). ◻ Note that the results in [@LeScWi20] are actually considered in the general setting of ordered Hilbert spaces with self-polar cones as well. However, their definition of domination of semigroups requires the existence of *absolute pairings* (see, for instance, [@LeScWi20 Definition 1.24]) which cannot be guaranteed in the non-commutative setting. In particular, the results of [@LeScWi20] are inapplicable in our setting. In fact, the implication "(iii) $\Rightarrow$ (i)" in [@LeScWi20 Theorem 3.5] is only true for *isotone projection cones*, that is, in the Hilbert lattice setting; see [@LeScWi20 Theorem 1.15]. # Domination of positive semigroups {#sec:domination-both-positive} Let $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ be a standard form of the von Neumann algebra $\mathfrak M$ and let $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ be self-adjoint $C_0$-semigroups on the Hilbert space $\mathcal{H}$ generated by densely defined, closed, symmetric, and accretive sesquilinear forms $\mathfrak a: \operatorname{dom}\left(\mathfrak a\right) \times \operatorname{dom}\left(\mathfrak a\right) \to \mathbb{C}$ and $\mathfrak b: \operatorname{dom}\left(\mathfrak b\right) \times \operatorname{dom}\left(\mathfrak b\right) \to \mathbb{C}$ respectively. In this section, we show that the situation of Theorem [Theorem 4](#thm:general){reference-type="ref" reference="thm:general"} becomes simpler if both the semigroups $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ are positive. In particular, our characterisation is the same in the commutative case [@Ou04 Theorem 2.24 and Proposition 2.23]. It is easy to see that due to the positivity of both semigroups, the definition -- stated in Section [2](#sec:domination-general){reference-type="ref" reference="sec:domination-general"} -- of domination of $(T_t)_{t\ge 0}$ by $(S_t)_{t\ge 0}$ simplifies to $$T_t u\le S_t u \quad (u \in \mathcal{H}_+, t\ge 0).$$ **Theorem 5**. *Let $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ be a standard form of the von Neumann algebra $\mathfrak M$ and let $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ be real and self-adjoint $C_0$-semigroups on the Hilbert space $\mathcal{H}$ generated by densely defined, closed, symmetric, and accretive sesquilinear forms $\mathfrak a: \operatorname{dom}\left(\mathfrak a\right) \times \operatorname{dom}\left(\mathfrak a\right) \to \mathbb{C}$ and $\mathfrak b: \operatorname{dom}\left(\mathfrak b\right) \times \operatorname{dom}\left(\mathfrak b\right) \to \mathbb{C}$ respectively.* *If $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ are both positive, then the following conditions are equivalent.* 1. *The semigroup $(T_t)_{t\ge 0}$ is dominated by $(S_t)_{t\ge 0}$, that is, $T_t u\le S_t u$ for all $u\in \mathcal{H}_+$ and all $t\ge 0$.* 2. *The inequality $\mathfrak a\left(\frac{u + (u \wedge v)}{2}\right) + \mathfrak b\left(\frac{v + (u \vee v)}{2}\right) \le \mathfrak a(u)+\mathfrak b(v)$ is true for all $u,v\in \mathcal{H}_+$.* 3. *Each of the following conditions is satisfied:* 1. *The inclusion $\operatorname{dom}\left(\mathfrak a\right)\subseteq \operatorname{dom}\left(\mathfrak b\right)$ holds.* 2. *If $u\in \operatorname{dom}\left(\mathfrak a\right)$ and $v\in \operatorname{dom}\left(\mathfrak b\right)$ such that $0\le v\le u$, then $v \in \operatorname{dom}\left(\mathfrak a\right)$.* 3. *For each $0\le u,v\in \operatorname{dom}\left(\mathfrak a\right)$, we have $\mathfrak a(u,v)\ge \mathfrak b(u,v)$.* *Proof.* First of all, consider the closed convex set $$C:= \{ (a,b)\in \mathcal{H}\times \mathcal{H}: 0\le a\le b\}$$ and the product semigroup on $\mathcal{H}\times \mathcal{H}$ given by $$R_t:= \begin{pmatrix} T_t & 0 \\ 0 & S_t \end{pmatrix} \qquad (t\ge 0).$$ The semigroup $(R_t)_{t\ge 0}$ is generated by the form $\mathfrak c$ with $\operatorname{dom}\left(\mathfrak c\right)=\operatorname{dom}\left(\mathfrak a\right)\times \operatorname{dom}\left(\mathfrak b\right)$ and $$\mathfrak c( (u_0,v_0), (u_1, v_1) ) = \mathfrak a(u_0, u_1)+ \mathfrak b(v_0,v_1).$$ Because the semigroups are positive, the domination of $(T_t)_{t\ge 0}$ by $(S_t)_{t\ge 0}$ is equivalent to the closed convex set $C$ being invariant under the semigroup $(R_t)_{t\ge 0}$. As in the proof of Theorem [Theorem 4](#thm:general){reference-type="ref" reference="thm:general"}, we compute the orthogonal projection $P$ of $\mathcal{H}\times \mathcal{H}$ onto $C$. We claim $$\label{eq:projection-positive} P(u,v) = \frac12\big(u + (u \wedge v), v+ (u \vee v)\big) \quad \text{whenever }u,v\ge 0.$$ To prove the claim, let $u,v\in \mathcal{H}_+$. We of course have $0\le u + (u \wedge v)\le v+ (u \vee v)$. Moreover, for each $(a,b)\in C$, we have $$\begin{aligned} A&:= \left\langle(u,v)-\frac 12\left(u + (u \wedge v), v+ (u \vee v)\right)\, ,\, (a,b)-\frac 12\left(u + (u \wedge v), v+ (u \vee v)\right)\right\rangle \\ &= \left\langle\frac{u-(u\wedge v)}{2}\, ,\, a-\frac{u + (u \wedge v)}{2}\right\rangle- \left\langle\frac{(u\vee v)-v}{2}\, ,\, b-\frac{v + (u \vee v)}{2}\right\rangle\\ &= \left\langle\frac{u-(u\wedge v)}{2}\, ,\, a-\frac{u + (u \wedge v)}{2}\right\rangle- \left\langle\frac{u-(u\wedge v)}{2}\, ,\, b-\frac{v + (u \vee v)}{2}\right\rangle. \end{aligned}$$ On further simplification, we get $$\begin{aligned} A &= \left\langle\frac{u-(u\wedge v)}{2}\, ,\, a-b+\frac{ (u \vee v) - u + v - (u \wedge v) }{2}\right\rangle\\ &= \left\langle\frac{u-(u\wedge v)}{2}\, ,\, a-b+(u \vee v) - u\right\rangle\\ %&= \frac{1}{2}\duality{(v-u)_-}{a-b+(v-u)_+}\\ &= \frac{1}{2}\left\langle(v-u)_-\, ,\, a-b\right\rangle+\frac12 \left\langle(v-u)_-\, ,\, (v-u)_+\right\rangle\\ &= \frac{1}{2}\left\langle(v-u)_-\, ,\, a-b\right\rangle \end{aligned}$$ where the last equality is obtained using the orthogonality of the positive and negative parts. As $(a,b)\in C$, it follows that $A\le 0$. As a consequence, the expression [\[eq:projection-positive\]](#eq:projection-positive){reference-type="eqref" reference="eq:projection-positive"} follows due to the criterion [\[eq:projection-criterion\]](#eq:projection-criterion){reference-type="eqref" reference="eq:projection-criterion"}; here we have implicitly used that the cone is self-polar. "(i) $\Leftrightarrow$ (ii)": As the forms are symmetric, so taking $C_0=\mathcal{H}_+\times \mathcal{H}_+$ in Theorem [Theorem 1](#thm:barthelemy){reference-type="ref" reference="thm:barthelemy"}, we obtain $(R_t)_{t\ge 0}$ leaves $C$ invariant if and only if $\mathfrak c(P(u,v))\le \mathfrak c( (u,v))$ for all $u,v\in \mathcal{H}_+$. The equivalence can now be obtained at once from [\[eq:projection-positive\]](#eq:projection-positive){reference-type="eqref" reference="eq:projection-positive"}. "(i) $\Rightarrow$ (iii)": Suppose that $(S_t)_{t\ge 0}$ dominates $(T_t)_{t\ge 0}$. This yields, as noted above, the invariance of the closed convex set $C$ under the semigroup $(R_t)_{t\ge 0}$. In particular, $P(\operatorname{dom}\left(\mathfrak a\right)\times \operatorname{dom}\left(\mathfrak b\right))\subseteq \operatorname{dom}\left(\mathfrak a\right)\times \operatorname{dom}\left(\mathfrak b\right)$ by Theorem [Theorem 1](#thm:barthelemy){reference-type="ref" reference="thm:barthelemy"}. So if $u,v$ are given as in (b), then using [\[eq:projection-positive\]](#eq:projection-positive){reference-type="eqref" reference="eq:projection-positive"}, we get $$\frac12(u+v,u+v)= P(u,v) \in \operatorname{dom}\left(\mathfrak a\right)\times \operatorname{dom}\left(\mathfrak b\right).$$ In particular, $v\in \operatorname{dom}\left(\mathfrak a\right)$, as $\operatorname{dom}\left(\mathfrak a\right)$ is a subspace. Moreover, using [\[eq:projection-positive\]](#eq:projection-positive){reference-type="eqref" reference="eq:projection-positive"}, we get $$\frac12(u,u)=P(u,0)\in \operatorname{dom}\left(\mathfrak a\right)\times \operatorname{dom}\left(\mathfrak b\right),$$ for all $0\le u \in \operatorname{dom}\left(\mathfrak a\right)$. Whence, $\operatorname{dom}\left(\mathfrak a\right) \cap \mathcal{H}_+ \subseteq \operatorname{dom}\left(\mathfrak b\right)$. Now, let $u\in \operatorname{dom}\left(\mathfrak a\right)$. Since $(T_t)_{t\ge 0}$ is positive, the Beurling-Deny criterion [@Ci08 Theorem 2.53] gives that $u_+, u_-\in \operatorname{dom}\left(\mathfrak a\right)\cap \mathcal{H}_+ \subseteq \operatorname{dom}\left(\mathfrak b\right)$, and consequently $u=u_+-u_-\in \operatorname{dom}\left(\mathfrak b\right)$. In fact, positivity of $(T_t)_{t\geq 0}$ even implies that it is real and so, $\operatorname{dom}\left(\mathfrak a\right)^J$ is generating for $\operatorname{dom}\left(\mathfrak a\right)$. We infer that $\operatorname{dom}\left(\mathfrak a\right)\subseteq \operatorname{dom}\left(\mathfrak b\right)$. We are left to prove (c). For this purpose, we define the bounded, symmetric, and accretive sesquilinear forms $\mathfrak a^t$ as $$\mathfrak a^t(\mathord{\,\cdot\,},\mathord{\,\cdot\,}):= \frac{1}{t}\left\langle(I-T_t)\mathord{\,\cdot\,}\, ,\, \mathord{\,\cdot\,}\right\rangle$$ and analogously the forms $\mathfrak b^t$ for $t>0$. These forms satisfy $$\mathfrak a^t(u,v)-\mathfrak b^t(u,v)= \frac{1}{t} \left\langle(S_t-T_t)u\, ,\, v\right\rangle \ge 0$$ for each $0\le u,v\in \operatorname{dom}\left(\mathfrak a\right)\subseteq \operatorname{dom}\left(\mathfrak b\right)$ and all $t>0$. Letting $t\downarrow 0$ in the above inequality yields (c); here we have used that $\lim_{t\to 0}\mathfrak a^t(\mathord{\,\cdot\,}, \mathord{\,\cdot\,})= \mathfrak a(\mathord{\,\cdot\,},\mathord{\,\cdot\,})$ and similarly for $\mathfrak b^t$ (see, for instance, [@Ou04 Lemma 1.56]). "(iii) $\Rightarrow$ (i)": Let $(u,v)\in \operatorname{dom}\left(\mathfrak a\right)\times\operatorname{dom}\left(\mathfrak b\right)$ with $u,v\ge 0$. By (a), we have $u,v\in \operatorname{dom}\left(\mathfrak b\right)$. This allows us to infer from the positivity of the semigroup $(S_t)_{t\ge 0}$ that $u \vee v, u \wedge v \in \operatorname{dom}\left(\mathfrak b\right)$ (see [@Ci97 Proposition 4.5 and Theorem 4.7]). On the other hand, $0 \le u \wedge v \le u$, so we can employ (b) to get $u\wedge v\in \operatorname{dom}\left(\mathfrak a\right)$. Wherefore, the equality [\[eq:projection-positive\]](#eq:projection-positive){reference-type="eqref" reference="eq:projection-positive"} yields $P(u,v)\in \operatorname{dom}\left(\mathfrak a\right)\times \operatorname{dom}\left(\mathfrak b\right)$. A direct application of [\[eq:projection-positive\]](#eq:projection-positive){reference-type="eqref" reference="eq:projection-positive"} and (c) gives that for every $(u,v)\in \operatorname{dom}\left(\mathfrak a\right)\times \operatorname{dom}\left(\mathfrak b\right)$ with $u,v\ge 0$, $$\begin{aligned} \mathfrak c( (u,v), (u,v)-P(u,v) ) &= \mathfrak a\left(u, \frac{u-(u\wedge v)}{2}\right)+\mathfrak b\left(v,\frac{v-(u\vee v)}{2}\right)\\ &\ge \mathfrak b\left(u, \frac{u-(u\wedge v)}{2}\right)+\mathfrak b\left(v,\frac{v-(u\vee v)}{2}\right)\\ &= \mathfrak d( (u,v), (u,v)-P(u,v) ); \end{aligned}$$ where $\mathfrak d$ is the form with domain $\operatorname{dom}\left(\mathfrak b\right)\times \operatorname{dom}\left(\mathfrak b\right)$ and $$\mathfrak d( (u_0,v_0), (u_1, v_1) ) := \mathfrak b(u_0, u_1)+ \mathfrak b(v_0,v_1).$$ Of course, the semigroup generated by $\mathfrak d$ on $\mathcal{H}\times \mathcal{H}$ is given by $$U_t:= \begin{pmatrix} S_t & 0 \\ 0 & S_t \end{pmatrix} \qquad (t\ge 0).$$ Positivity of $(S _t)_{t\ge 0}$ implies that $U_tC\subseteq C$ for $t\ge 0$. Therefore, the characterisation of invariance of closed convex sets Theorem [Theorem 1](#thm:barthelemy){reference-type="ref" reference="thm:barthelemy"} yields $$\mathfrak d( (u,v), (u,v)-P(u,v) )\ge 0.$$ Substituting above, we get $\mathfrak c( (u,v), (u,v)-P(u,v) )\ge 0$. Finally, let $C_0$ be the closed convex set $\mathcal{H}_+\times \mathcal{H}_+$. We have prove that the inclusion $P(\operatorname{dom}\left(\mathfrak c\right)\cap C_0)\subseteq \operatorname{dom}\left(\mathfrak c\right)\cap C_0$ and the inequality $\mathop{\mathrm{Re}}\mathfrak c( (u,v), (u,v)-P(u,v) )\ge 0$ are satisfied for all $(u,v)\in\operatorname{dom}\left(\mathfrak c\right) \cap C_0$. Another application of Theorem [Theorem 1](#thm:barthelemy){reference-type="ref" reference="thm:barthelemy"} gives that $R_tC\subseteq C$ for all $t\ge 0$, which proves (i). ◻ **Example 6**. Let $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ be a standard form of the von Neumann algebra $\mathfrak M$. Given a self-adjoint operator $(\mathcal b, \operatorname{dom}\left(\mathcal b\right))$ on $\mathcal{H}$ affiliated to $\mathfrak M$, we let $d_{\mathcal b}$ be the unbounded derivation $$\operatorname{dom}\left(d_{\mathcal b} \right) : = \operatorname{dom}\left(\mathcal b\right) \cap J \operatorname{dom}\left(\mathcal b\right),\qquad d_{\mathcal b} : = i [ a - JaJ];$$ see [@Ci97 Section 5] for details. It was shown in [@Ci97 Proposition 5.4] that the form $$\operatorname{dom}\left(\mathfrak b\right)=\operatorname{dom}\left(d_{\mathcal b}\right), \qquad \mathfrak b(u):= \left\lVert d_{\mathcal b}u \right\rVert^2$$ is closable and its closure $\overline{\mathfrak b}$ generates a positive semigroup $(S_t)_{t\ge 0}$. Therefore, if $M:\mathcal{H}\to \mathcal{H}$ is a positive bounded operator, then the closure of the form $$\operatorname{dom}\left(\mathfrak a\right):=\operatorname{dom}\left(\mathfrak b\right), \qquad \mathfrak a(u):= \mathfrak b(u) + \left\langle Mu\, ,\, u\right\rangle$$ also generates a positive semigroup, say $(T_t)_{t\ge 0}$. Clearly, Theorem [Theorem 5](#thm:both-positive){reference-type="ref" reference="thm:both-positive"} implies that the semigroup $(S_t)_{t\ge 0}$ dominates $(T_t)_{t\ge 0}$. # Domination of a semigroup by a positive semigroup {#sec:domination-not-real} Every element $u$ of a Hilbert lattice $\mathcal{H}$ satisfies $$\left\lvert u \right\rvert= \sup\{\mathop{\mathrm{Re}}(e^{i\theta}u): \theta \in [0,2\pi]\}.$$ This allows us to generalise the definition of domination of semigroups to the case when the dominated semigroup is not necessarily real and, in turn, generalise Theorem [Theorem 4](#thm:general){reference-type="ref" reference="thm:general"}. Let $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ be a standard form of the von Neumann algebra $\mathfrak M$ and let $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ be (not necessarily real) self-adjoint $C_0$-semigroups on the Hilbert space $\mathcal{H}$ generated by closed quadratic forms $\mathfrak a: \mathcal{H}\to [0,\infty]$ and $\mathfrak b: \mathcal{H}\to [0,\infty]$ respectively. We say that the semigroup $(T_t)_{t\ge 0}$ is *dominated* by $(S_t)_{t\ge 0}$ if $$\left( \mathop{\mathrm{Re}}(e^{i\theta} u) \le v \text{ for all }\theta \in [0,2\pi] \right) \Rightarrow \left(\mathop{\mathrm{Re}}(e^{i\theta}T_t u)\le S_t v \text{ for all }\theta \in [0,2\pi]\right)$$ for all $u,v \in \mathcal{H}$ and all $t\ge 0$. In this case, of course, $(S_t)_{t\ge 0}$ is automatically positive and, in particular, real. In particular, the above domination is equivalent to the following condition: the semigroup $(R_t)_{t\ge 0}$ given by $$R_t:= \begin{pmatrix} T_t & 0 \\ 0 & S_t \end{pmatrix} \qquad (t\ge 0)$$ leaves the closed convex sets $$C_{\theta}:= \{ (a,b) \in \mathcal{H}\times \mathcal{H}: \mathop{\mathrm{Re}}(e^{i \theta} a)\le b\}$$ invariant for each $\theta \in [0,2\pi]$. As in the preceding sections, one can show that the orthogonal projections $P_{\theta}$ onto the set $C_{\theta}$ satisfy $$P_{\theta} (u,v) = \left(u-\frac{1}{2} \left(\mathop{\mathrm{Re}}(e^{i\theta} u)-v\right)^+, v+\frac{1}{2} \left(\mathop{\mathrm{Re}}(e^{i\theta} u)-v\right)^+\right)$$ for each $(u,v)\in \mathcal{H}\times \mathcal{H}^J$ and all $\theta \in [0,2\pi]$. Now, we may proceed exactly as in Section [2](#sec:domination-general){reference-type="ref" reference="sec:domination-general"} (employing Theorem [Theorem 1](#thm:barthelemy){reference-type="ref" reference="thm:barthelemy"}) in order to obtain the following characterisation: **Theorem 7**. *Let $(\mathfrak M, \mathcal{H}, \mathcal{H}_+,J)$ be a standard form of the von Neumann algebra $\mathfrak M$ and let $(T_t)_{t\ge 0}$ and $(S_t)_{t\ge 0}$ be self-adjoint $C_0$-semigroups on the Hilbert space $\mathcal{H}$ generated by closed quadratic forms $\mathfrak a: \mathcal{H}\to [0,\infty]$ and $\mathfrak b: \mathcal{H}\to [0,\infty]$ respectively.* *If the semigroup $(S_t)_{t\ge 0}$ is real, then the following are equivalent.* 1. *The semigroup $(T_t)_{t\ge 0}$ is dominated by $(S_t)_{t\ge 0}$, that is, $$\left( \mathop{\mathrm{Re}}(e^{i\theta} u) \le v \text{ for all }\theta \in [0,2\pi] \right) \Rightarrow \left(\mathop{\mathrm{Re}}(e^{i\theta}T_t u)\le S_t v \text{ for all }\theta \in [0,2\pi]\right)$$ for all $u,v \in \mathcal{H}$ and all $t\ge 0$.* 2. *The inequality $$\mathfrak a\left( u - \frac{1}{2}\left(\mathop{\mathrm{Re}}(e^{i\theta} u-v\right)^+\right) +\mathfrak b\left(v+ \frac{1}{2}\left(\mathop{\mathrm{Re}}(e^{i\theta} u-v\right)^+\right) \le \mathfrak a(u)+\mathfrak b(v)$$ holds for all $(u,v)\in \mathcal{H}\times \mathcal{H}^J$ and for all $\theta \in [0,2\pi]$.* [^1]: This project is supported by the VAJRA scheme VJR/2018/000127, of the Science and Engineering Research Board, Department of Science and Technology, Govt. of India. The second and third authors gratefully acknowledge the support from VAJRA

arxiv_math

--- abstract: | Most of the literature on causality considers the structural framework of Pearl and the potential-outcome framework of Neyman and Rubin to be formally equivalent, and therefore interchangeably uses the do-notation and the potential-outcome subscript notation to write counterfactual outcomes. In this paper, we superimpose the two causal model to prove that structural counterfactual outcomes and potential outcomes do not coincide in general---not even in law. More precisely, we express the law of the potential outcomes in terms of the latent structural causal model under the fundamental assumptions of causal inference. This enables us to precisely identify when counterfactual inference is or is not equivalent between approaches, and to clarify the meaning of each kind of counterfactuals. author: - "Lucas De Lara[^1]" bibliography: - references.bib title: The difference between structural counterfactuals and potential outcomes --- **Keywords:** causality, structural causal models, potential outcomes # Introduction Understanding causation between phenomena rather than mere association is a fundamental scientific challenge. Over the last decades, two mathematical frameworks using a terminology based on random variables have become the gold standards to address this problem. On the one hand, the notorious *structural account* of [@pearl2009causality] rests on the knowledge of a *structural causal model* (SCM) which specifies all cause-effect equations between observed random variables (often depicted by a graph). The interest of such equations comes from the possibility of carrying out *do-interventions*: forcing a variable to take a given value while keeping the rest of the mechanism untouched. More concretely, let $T$ and $Y$ be observed variables of the model such that we would like to understand the downstream effect of $T$ onto $Y$. Replacing the formula generating $T$ by $T = t$ for a given possible value $t \in \mathcal T$ and propagating this change through the other equations defines the altered variable $Y_{T=t}$, representing $Y$ *had $T$ been equal to $t$*. On the other hand, the widely-used *potential-outcome account* of [@rubin1974estimating] mathematically formalizes causal inference in clinical trials. Letting $T$ denote a *treatment status* (e.g., taking a drug or not) and $Y$ an *outcome* of interest (e.g., recovering or not), this framework postulates the existence of *potential outcomes* $(Y_t)_{t \in \mathcal T}$ representing what the outcome would be *were $T$ equal to $t$* for any $t \in \mathcal T$. The *fundamental problem of causal inference* [@holland1986statistics] refers to the fact that in practice we cannot observe simultaneously all the potential outcomes, rendering unidentifiable the causal effect of $T$ onto $Y$. Nevertheless, causal inference can still be achieved thanks to a mix of untestable assumptions and statistical tools: adjusting on a set of available covariates $X$ containing all possible *confounders* between the treatment and the potential outcomes notoriously permits to identify the law of counterfactual outcomes. Each of these causal theories enables to carry out *counterfactual reasoning*, that is answering contrary-to-fact questions such as : by applying do-interventions on an SCM, one can compute the outcomes $(Y_{T=t})_{t \in \mathcal T}$ for all possible treatment statuses; using the Neyman-Rubin causal model, one can infer the law of the potential outcomes $(Y_t)_{t \in \mathcal T}$. Both approaches involve variables describing counterfactual outcomes, more precisely outcomes *had the variable $T$ taken a certain value*. This naturally raises the question: are these outcome variables equal (almost surely or in law) across frameworks? We believe the literature on causal inference to be strongly misleading on this matter. A plethora of scientific books and survey papers interchangeably use Pearl's do-notation and the potential-outcome subscript notation to write outcomes after interventions, suggesting that the corresponding definitions of counterfactuals are identical and differ only from theirs perspectives [@barocas-hardt-narayanan; @colnet2020causal; @imbens2020potential; @makhlouf2020survey; @neal2020introduction]. To justify this, they often refer to Pearl, who argued that .[^2] However, to our knowledge, works on equivalences between the two causal frameworks focus on translating conditional-independence restrictions into graphical assumptions instead of actually proving whether counterfactual outcomes are *algebraically* equal across models, or implicitly address specific cases. Notably, both [@pearl2009causality Chapter 7] and [@richardson2013single]---acclaimed references unifying both causal frameworks---consider *ex nihilo* the algebraic equivalence between the two notations. In this paper, we prove that structural counterfactual outcomes and potential outcomes do not coincide in general---not even in law---and discuss the implications of this result. More specifically, the rest of the article is organized as follows. In Section [2](#sec:prelim){reference-type="ref" reference="sec:prelim"}, we introduce the basic knowledge on structural causal models and the potential-outcome framework. In Section [3](#sec:main){reference-type="ref" reference="sec:main"}, we superimpose the two causal frameworks to derive a mathematical analysis of their similarities and differences. As our main result, we express under classical assumptions the law of the potential outcomes $(Y_t)_{t \in \mathcal T}$ using the causal equations of the SCM. This critically implies that---in contrary to what the mainstream literature often suggests---the two definitions of counterfactual outcomes are not always equal in law. In Section [4](#sec:consequences){reference-type="ref" reference="sec:consequences"}, we precisely analyze when counterfactual inference in one framework does (not) yield the same conclusions as in the other, and clarify how such results relate to the formal equivalence between causal frameworks accepted by the causal-inference community. This contribution notably highlights that $(Y_t)_{t \in \mathcal T}$ represents *ceteris paribus* counterfactuals under the standard causal-inference setting while $(Y_{t=t})_{t \in \mathcal T}$ always represents *mutatis mutandis* counterfactuals, which has concrete consequences on the computation of causal effects. In doing this work, we aim at clarifying the role of each framework in the past, current, and future causal-inference research. # Preliminaries {#sec:prelim} This section provides the necessary background on structural causal models and potential outcomes. It is meant to keep the paper self-contained and can be skipped by a reader familiar with these frameworks. Let us fix some notations before proceeding. Throughout, we consider a probability space $(\Omega, \Sigma, \mathbb P)$ with $\Omega$ a sample space, $\Sigma$ a $\sigma$-algebra, and $\mathbb P: \Sigma \to [0,1]$ a probability measure. We write $\mathcal{L}(W)$ and $\mathbb E[W]$ for respectively the law and expectation under $\mathbb P$ of a random variable or random vector $W$. Two variables $W_1$ and $W_2$ are *$\mathbb P$-almost surely equal*, denoted by $W_1 \stackrel{\mathbb{P}-a.s.}{=} W_2$, if $\mathbb P(W_1=W_2)=1$; they are *equal in law* (under $\mathbb P$), denoted by $W_1 \stackrel{\mathcal{L}}{=} W_2$, if $\mathcal{L}(W_1) = \mathcal{L}(W_2)$. Additionally, $W_1 \protect\mathpalette{\protect\independenT}{\perp}W_2$ means that $W_1$ and $W_2$ are independent (under $\mathbb P$). Besides, for any tuple $w := (w_i)_{i \in \mathcal I}$ indexed by a finite index set $\mathcal I$ and any subset $I \subseteq \mathcal I$ we write $w_I := (w_i)_{i \in I}$. Similarly, we define $\mathcal{W}_I := \prod_{i \in I} \mathcal{W}_i$ for any collection of spaces $(\mathcal{W}_i)_{i \in \mathcal I}$. ## Pearl's causal framework Pearl's causal modeling mathematically formalizes associations that standard probability calculus cannot describe through the notions of structural causal models and do-interventions [@pearl2009causality]. This section recalls the basics on this topic, borrowing the introduction proposed by [@bongers2021foundations]. ### Structural causal models A *structural causal model* (SCM) represents the causal relationships between the studied variables. It is the cornerstone of Pearl's causal framework. **Definition 1** (Structural causal model). *Let $\mathcal I$ and $\mathcal J$ be two disjoint finite index sets, and write $\mathcal{V} := \prod_{i \in \mathcal I} \mathcal{V}_i \subseteq \mathbb{R}^{|\mathcal I|}$, $\mathcal{U} := \prod_{i \in \mathcal J} \mathcal{U}_i \subseteq \mathbb{R}^{|\mathcal J|}$ for two measurable product spaces. A *structural causal model* $\mathcal{M}$ is a couple $\langle U, G \rangle$ where:* 1. *$U : \Omega \to \mathcal{U}$ is a vector of mutually independent random variables, sometimes called the *random noise*;* 2. *$G = \{G_i\}_{i \in \mathcal I}$ is a collection of measurable $\mathbb{R}$-valued functions, where for every $i \in \mathcal I$ there exist two subsets of indices $\operatorname{Endo}(i) \subseteq \mathcal I$ and $\operatorname{Exo}(i) \subseteq \mathcal J$, respectively called the *endogenous* and *exogenous parents* of $i$, such that $G_i$ is from $\mathcal{V}_{\operatorname{Endo}(i)} \times \mathcal{U}_{\operatorname{Exo}(i)}$ to $\mathcal{V}_i$.[^3]* *A random vector $V : \Omega \to \mathcal{V}$ is a solution of $\mathcal{M}$ if for every $i \in \mathcal I$, $$\label{eq:causal_eq} V_i \stackrel{\mathbb P-a.s.}{=} G_i(V_{{\text{Endo}}(i)},U_{\text{Exo}(i)}).$$ The collection of equations defined by [\[eq:causal_eq\]](#eq:causal_eq){reference-type="eqref" reference="eq:causal_eq"} and characterized by $G$ and $U$ are called the *structural equations*.* Such a model explains how some *endogenous* variables $V$, representing observed data, are generated from *exogenous* variables $U$, describing background factors. The structural equations quantify the causal dependencies between all these variables and are frequently illustrated by the directed graph with nodes $\mathcal I\cup \mathcal J$, and such that a directed edge points from node $k$ to node $l$ if and only if $k \in \text{Endo}(l) \cup \text{Exo}(l)$ (we say in this case that $k$ is a parent of $l$). For convenience, we make the common assumption that the studied models are *acyclic*, which means that their associated graphs do not contain any cycles. **Assumption 2** (Acyclicity). *The structural causal model $\mathcal{M}$ induces a directed *acyclic* graph.* Not only acyclicity simplifies the interpretation of causal dependencies, but it entails *unique solvability* of the SCM: according to [@bongers2021foundations Proposition 3.4], Equation $\eqref{eq:causal_eq}$ admits a unique solution up to $\mathbb P$-negligible sets. We will abusively refer to such a solution as *the* solution of the SCM. The purpose of causal structures is to capture the assumption that features are not independently manipulable. As we detail next, they enable to understand the downstream effect of fixing some variables to certain values onto nonintervened variables. ### Do-intervention A *do-intervention* is an operation forcing a set of endogenous variables to take predefined values while keeping all the rest of the causal mechanism equal. **Definition 3** (Do-intervention). *Let $\mathcal{M} = \langle U, G \rangle$ be an SCM, $I \subseteq \mathcal I$ a subset of endogenous variables, and $v_I \in \mathcal{V}_I$ a value. The action $\operatorname{do}(I,v_I)$ defines the modified model $\mathcal{M}_{\operatorname{do}(I,v_I)} = \langle U, \tilde{G} \rangle$ where $\tilde{G}$ is given by $$\tilde{G_i} := \begin{cases} v_i \text{ if } i \in I,\\ G_i \text{ if } i \in \mathcal I\setminus I. \end{cases}$$* Do-interventions preserve acyclicity, and therefore unique solvability. As a consequence, if $V$ is the solution of an acyclic $\mathcal M$, one can define (up to $\mathbb P$-negligible sets) its post-intervention counterpart $V_{\operatorname{do}(I,v_I)}$ solution to $\mathcal M_{\operatorname{do}(I,v_I)}$. It describes an alternative world where every $V_i$ for $i \in I$ is set to value $v_i$. In the sequel, we simply write $\operatorname{do}(V_I = v_I)$ for the operation $\operatorname{do}(I,v_I)$, and use the subscript $V_I=v_I$ to indicate results of this operation. Crucially, intervening does not amount to conditioning in general, that is $\mathcal{L}(V \mid V_I=v_I) \neq \mathcal{L}(V_{V_I=v_I})$. The next proposition provides a general expression of the solution before and after intervention, and will play a key role throughout this paper. **Proposition 4** (Do-calculus on variables). *Let $\mathcal{M} = \langle U, G \rangle$ be an SCM satisfying Assumption [Assumption 2](#hyp:acyclic){reference-type="ref" reference="hyp:acyclic"} with solution $V$, and consider a partition $\{I, J\}$ of $\mathcal I$. There exists a deterministic measurable function $F_J$ such that $$V_J \stackrel{\mathbb{P}-a.s.}{=} F_J(V_{\operatorname{Endo}(J) \setminus J}, U_{\operatorname{Exo}(J)}).$$ Moreover, for any intervention $\operatorname{do}(V_I = v_I)$ the solution $\tilde{V}$ of $\mathcal{M}_{V_I = v_I}$ verifies $$\begin{aligned} \tilde{V_J} & \stackrel{\mathbb{P}-a.s.}{=} F_J(v_{\operatorname{Endo}(J) \setminus J}, U_{\operatorname{Exo}(J)}),\\ \tilde{V_I} & \stackrel{\mathbb{P}-a.s.}{=} v_I. \end{aligned}$$* Importantly, this is the same deterministic function $F_J$ that generates $V_J$ and its post-intervention counterpart $\tilde{V_J}$, the only change being the assignment $V_I = v_I$. Slightly abusing notations, we will sometimes artificially extend the input variables of $F_J$ to write $V_J \stackrel{\mathbb{P}-a.s.}{=} F_J(V_I,U_{\operatorname{Exo}(J)})$ and $\tilde{V_J} \stackrel{\mathbb{P}-a.s.}{=} F_J(v_I,U_{\operatorname{Exo}(J)})$. ### Counterfactual inference Do-calculus provides a natural framework to address counterfactual queries. Let for instance $V := (T,X,Y)$ be the solution to an acyclical SCM $\mathcal M:= \langle U, G \rangle$. We aim at answering the counterfactual question: *had $T$ been equal to $t$, what would have been the value of $Y$ for a unit factually described by $X=x$?* [@pearl2009causality] answers this question using the so-called *three-step procedure*: 1. **Abduction:** Deduce the posterior distribution of $U$ given the reference $\{X=x\}$; 2. **Action:** Carry out do-calculus on $\mathcal M$ to obtain the intervened causal mechanism $G_{T=t}$ of $\mathcal M_{T=t}$; 3. **Prediction:** Pass the posterior distribution $\mathcal{L}(U \mid X=x)$ through $G_{T=t}$ to generate in particular the distribution $\mathcal{L}(Y_{T=t} \mid X = x)$ of counterfactual outcomes. More generally, an SCM enables one to compute counterfactual distributions for any choices of events of reference, variables to alter by do-intervention, and outcomes of interest. ## Neyman-Rubin causal framework The potential-outcome framework, also known as *Neyman-Rubin causal modeling* [@rubin1974estimating], was designed to understand the causal effect of a treatment onto an outcome of interest, for instance when one aims at assessing the contribution of a drug to recovering from some disease in clinical trials. In this section, we introduce this widely-used framework in the specific case of a binary treatment. ### Potential outcomes Let $T : \Omega \to \{0,1\}$ represent a binary *treatment status*, typically such that $T=0$ indicates the absence of treatment and $T=1$ indicates a treatment. More generally, it can encode any distinction between some groups (e.g., men and women). Assuming *no interference between units*, this framework postulates two *potential outcomes* $Y_0 : \Omega \to \mathbb{R}$ and $Y_1 : \Omega \to \mathbb{R}$, one for each treatment status. These potential outcomes as well as the treatment may depend on some covariates $X : \Omega \to \mathbb{R}^d$ (such as the patient's weight, height, or historical data in clinical trials). Critically, we cannot observe simultaneously $Y_0$ and $Y_1$ for a single unit $\omega$: a problem referred as the *fundamental problem of causal inference* [@holland1986statistics]. We only have access to the realized *outcome variable* $Y : \Omega \to \mathbb{R}$ which is supposed to be *consistent* with $(Y_0,Y_1)$, that is satisfying $Y = (1-T) \cdot Y_0 + T \cdot Y_1$. Concretely, if $T(\omega)=1$ for some $\omega \in \Omega$, then $Y(\omega)=Y_1(\omega)$, and $Y_0(\omega)$ becomes unidentifiable by mere observations. In this case, $Y_1(\omega)$ is called the *factual* outcome while $Y_0(\omega)$ is called the *counterfactual* outcome. Understanding the causal relationship between the treatment and the outcome in this framework consists in answering counterfactual questions such as . In practice, this amounts to estimating the discrepancy between $\mathcal{L}(Y_1)$ and $\mathcal{L}(Y_0)$, or $\mathcal{L}(Y_1 \mid X=x)$ and $\mathcal{L}(Y_0 \mid X=x)$. People commonly focus on computing the *average treatment effect* $\mathbb E[Y_1-Y_0]$ or the *conditional average treatment effect* $\mathbb E[Y_1-Y_0 \mid X=x]$. The main challenge lies in the fact that *association is not causation* in general. In particular, the observable quantity $\mathbb E[Y \mid T=t]$ does not necessarily coincide with the unobservable quantity $\mathbb E[Y_t]$ for $t \in \{0,1\}$. Typically, if some medical treatment is more likely to be taken by weaker patients, we may observe a lower rate of recovery among the treated group compared to the nontreated group due to the health condition even though the medicine does increase recovery all other things being kept equal: we would observe $\mathbb E[Y \mid T=1] < \mathbb E[Y \mid T=0]$ while $\mathbb E[Y_1] > \mathbb E[Y_0]$ (a phenomenon referred as *Simpson's paradox* [@blyth1972simpson]). In this case, the health condition is called a *confounder*: a variable associated with both the distribution of the treatment and the outcome. However, causal inference from observational data is still possible, as explained next. ### Estimation of causal effects {#sec:causal_estimation} A treatment effect is *identifiable* if it can be expressed with observational quantities only, that is in terms of $X$, $T$ and $Y$. Identifiability requires two fundamental assumptions. The first one goes by many names through the literature: *conditional ignorability*, *conditional exchangeability*, *conditional exogeneity*, and *conditional unconfoundedness* (among others). Originally formulated by [@rosenbaum1983central], it states that the potential outcomes are independent of the treatment conditional to the covariates, that is $(Y_0,Y_1) \protect\mathpalette{\protect\independenT}{\perp}T \mid X$. Said differently, it prevents the existence of unmodeled confounders between the treatment and the potential outcomes. Note that this assumption is untestable, as it would require to observe simultaneously the two potential outcomes. The second key hypothesis is *positivity*, which ensures that all individuals can be exposed to both treatment statuses, that is $0 < \mathbb P(T=1 \mid X) < 1$. It readily follows from positivity and conditional ignorability that $\mathcal{L}(Y \mid X,T=t)$ is well defined and coincides with $\mathcal{L}(Y_t \mid X)$ for $t \in \{0,1\}$, meaning that observable outcomes have a causal interpretation. Several statistical methods coexist to estimate the (conditional) average causal effect, all building upon this implication (see for instance [@imbens2004nonparametric; @yao2021survey]). We do not detail them for concision and clarity since it is not the topic of this paper. We only point out that, similarly to structural causal models, the potential-outcome framework enables one to carry out counterfactual inference. # Main result {#sec:main} In this section, we show that a Neyman-Rubin causal model and a structural causal model compatible with a same distribution of observations produce counterfactual outcomes that are generally not almost-surely equal, nor equal in law. ## Setting Let $N, d, p \geq 1$ be integers, and define three random variables $T : \Omega \to \mathcal T:= \{0,1,\ldots,N\}$, $X : \Omega \to \mathbb{R}^d$, and $Y : \Omega \to \mathbb{R}^p$. In order to study the consistency of counterfactual statements between the Neyman-Rubin causal framework and Pearl's causal framework, we consider a superimposed construction where the observations described by $(T,X,Y)$ are concurrently governed by a potential-outcome model and a structural causal model. On the one hand, we assume that $Y$ is the outcome of interest, $T$ the treatment status, and $X$ some covariates in a potential-outcome framework. This amounts to postulating $N$ random vectors $(Y_t)_{t \in \mathcal T}$ satisfying the *consistency rule*: $$Y \stackrel{\mathbb{P}-a.s.}{=} \sum_{t \in \mathcal T} \mathbf{1}_{\{T=t\}} Y_t.$$ Note that we address a more general framework than in Section [2](#sec:prelim){reference-type="ref" reference="sec:prelim"}, considering a nonbinary treatment and a multivariate outcome. In this setting, the two fundamental assumptions for causal inference can be written as follows. **Assumption 5** (Positivity). *$$0 < \mathbb P(T=t \mid X) < 1,\ \text{for all}\ t \in \mathcal T.$$* **Assumption 6** (Conditional ignorability). *$$(Y_t)_{t \in \mathcal T} \protect\mathpalette{\protect\independenT}{\perp}T \mid X.$$* On the other hand, we assume that these variables are generated by a latent, unknown structural causal model: the random vector $V := (T,X,Y)$ is the solution to an acyclical SCM $\mathcal{M} = \langle U, G \rangle$ where $U_T$, $U_X$ and $U_Y$ denote the exogenous parents of respectively $T$, $X$, and $Y$. Moreover, we suppose that $\mathcal M$ satisfies: **Assumption 7** (Outcome). *$$Y_{\operatorname{Endo}(T)} = Y_{\operatorname{Endo}(X)} = \emptyset \ \text{and}\ U_Y \protect\mathpalette{\protect\independenT}{\perp}(U_T,U_X).$$* The first item of Assumption [Assumption 7](#hyp:outcome){reference-type="ref" reference="hyp:outcome"} is a graphical condition that formally defines the variable $Y$ as the *outcome*; it changes in response to $X$ and $T$ but not the contrary. Through Proposition [Proposition 4](#prop:docalculus){reference-type="ref" reference="prop:docalculus"}, it permits to write $$\begin{aligned} T & \stackrel{\mathbb{P}-a.s.}{=} F_T(X,U_T),\\ X & \stackrel{\mathbb{P}-a.s.}{=} F_X(T,U_X),\\ Y & \stackrel{\mathbb{P}-a.s.}{=} F_Y(T,X,U_Y),\end{aligned}$$ where $F_X, F_T$ and $F_Y$ are deterministic measurable functions derived from $G$. The artificial cycle in these formulas (i.e., $X$ and $T$ are both functions of each other) merely serves to consider all configurations of causal links between $T$ and $X$ (see Figure [\[fig:decisive\]](#fig:decisive){reference-type="ref" reference="fig:decisive"}); strictly, $\mathcal{M}$ satisfies Assumption [Assumption 2](#hyp:acyclic){reference-type="ref" reference="hyp:acyclic"}. The following lemma clarifies the role of second item in Assumption [Assumption 7](#hyp:outcome){reference-type="ref" reference="hyp:outcome"}. **Lemma 8** (Random noise). *Let $(T,X,Y)$ be the solution of an SCM $\mathcal M$ satisfying Assumptions [Assumption 2](#hyp:acyclic){reference-type="ref" reference="hyp:acyclic"} and [Assumption 7](#hyp:outcome){reference-type="ref" reference="hyp:outcome"} where $U_T,U_X$ and $U_Y$ denote the exogenous parents of respectively $T,X$ and $Y$. Then $U_Y \protect\mathpalette{\protect\independenT}{\perp}(T,X)$.* It guarantees that all potential confounders between $T$ and $Y$---except $T$ itself---are included in $X$. All in all, Assumption [Assumption 7](#hyp:outcome){reference-type="ref" reference="hyp:outcome"} simply means through Lemma [Lemma 8](#lm:noise){reference-type="ref" reference="lm:noise"} that the randomness of the outcome $Y \stackrel{\mathbb{P}-a.s.}{=} F_Y(T,X,U_Y)$ can be divided into three sources: the direct effect of the treatment status $T$, the direct effect of the covariates $X$, and any other possible effects $U_Y$ independent to $T$ and $X$. To conclude this setup, recall that Proposition [Proposition 4](#prop:docalculus){reference-type="ref" reference="prop:docalculus"} also enables to define for every $t \in \mathcal T$ the post-intervention outcome under $\operatorname{do}(T=t)$ as $$Y_{T=t} \stackrel{\mathbb{P}-a.s.}{=} F_Y(t,X_{T=t},U_Y),$$ where the altered covariates are $X_{T=t} \stackrel{\mathbb{P}-a.s.}{=} F_X(t,U_X)$. We have now set the stage to present our main results. ## Three levels of difference We argue that counterfactual outcomes are not equivalent between the two models at three levels: at the definition level, at the variable level, at the distributional level. ### The definition level Before addressing mathematical equalities, we would like to underline a more conceptual distinction: the two considered causal models differ fundamentally in their constructions of counterfactual outcomes. As noted by [@pearl2010brief], the potential outcomes $(Y_t)_{t \in \mathcal T}$ are of the Neyman-Rubin causal model, not related to any formal of measurable quantities, while the intervened outcomes $(Y_{T=t})_{t \in \mathcal T}$ are of the structural causal model by application of do-calculus. Said differently, the firsts are inputs *defining* the causal model, whereas the seconds are post-intervention outputs *defined by* the causal model. However, Pearl uses the same notation for both constructions, suggesting a mathematical equality. Are they truly equivalent in the sense that $Y_t \stackrel{\mathbb{P}-a.s.}{=} Y_{T=t}$, or at least $Y_t \stackrel{\mathcal{L}}{=} Y_{T=t}$? This is what we address next. ### The variable level The aforementioned input/output difference is in fact more than just conceptual. Because an input can be arbitrarily chosen whereas an output is a necessary consequence, it feels that we could easily find settings where they are not equal. Of course, potential outcomes are not completely arbitrary: they must follow the consistency rule, that is $Y \stackrel{\mathbb{P}-a.s.}{=} \sum_{t \in \mathcal T} \mathbf{1}_{\{T=t\}} Y_t$. But this property does not fully characterize the outcomes in the sense that there is no unique choice of $(Y_t)_{t \in \mathcal T}$ satisfying the consistency rule. More precisely, while necessarily $Y_t = Y$ on $\{T=t\}$ for $t \in \mathcal T$, there is no restriction on $Y_t$ over $\Omega \setminus \{T=t\}$; it could take any value over it without violating the consistency rule. Consequently, it is mathematically impossible to relate $Y_t$---well-identifiable on the event $\{T=t\}$ only---to $Y_{T=t}$---defined (almost) everywhere through the structural causal model $\mathcal{M}$. Without further assumptions, we only have identification of the *observed outcomes*, namely $Y_t = Y_{T=t} = Y$ on $\{T=t\}$, as a direct consequence of the proposition below. **Proposition 9** (Consistency rule for structural counterfactuals). *Let $(T,X,Y)$ be the solution of an SCM $\mathcal M$ satisfying Assumptions [Assumption 2](#hyp:acyclic){reference-type="ref" reference="hyp:acyclic"}.[^4] Then, $(Y_{T=t})_{t \in \mathcal T}$ verifies the consistency rule, $$Y \stackrel{\mathbb{P}-a.s.}{=} \sum_{t \in \mathcal T} \mathbf{1}_{\{T=t\}} Y_{T=t}.$$* As such structural counterfactuals can be defined as potential outcomes, since the only requirement to be admissible potential outcomes is to follow the consistency rule. However, this does not signify that any pair or each causal model compatible with a same dataset produces potential outcomes and structural counterfactuals that coincide. In the next examples, we exhibit structural causal models and potential outcomes such that counterfactual outcomes are not almost-surely equal between frameworks. **Example 1** (Nonequality almost-surely). *Consider any acyclic structural causal model with solution $(T,X,Y)$ such that $Y$ is unidimensional and $0 < \mathbb P(T=t)$ for all $t \in \mathcal T$. We can construct via do-calculus the structural counterfactual outcomes $(Y_{T=t})_{t \in \mathcal T}$. Now, define the potential outcomes $(Y_t)_{t \in \mathcal T}$ as follows. For any $t \in \mathcal T$, $$Y_t = \mathbf{1}_{\{T=t\}} Y_{T=t} + \mathbf{1}_{\{T \neq t\}} (Y_{T=t} + 1).$$ The tuple $(Y_t)_{t \in \mathcal T}$ satisfies the consistency rule according to Proposition [Proposition 9](#prop:consistency_do){reference-type="ref" reference="prop:consistency_do"}, but is clearly not almost-surely equal to $(Y_{T=t})_{t \in \mathcal T}$.* Note that this example focuses on the most general setting, where the potential outcomes verify only consistency. The next example proves that counterfactual outcomes are not necessarily equal when the fundamental assumptions of causal inference hold. **Example 2** (Nonequality almost-surely under causal-inference assumptions). *Consider the following structural causal model: $$\begin{aligned} T & \stackrel{\mathbb{P}-a.s.}{=} U_T,\\ X & \stackrel{\mathbb{P}-a.s.}{=} U_X,\\ Y & \stackrel{\mathbb{P}-a.s.}{=} T + U_Y,\end{aligned}$$ where $U_T$ follows a Bernoulli distribution with parameter $1/2$, $U_X$ is any random variable, and $U_Y$ follows a centered Gaussian distribution with unit variance, such that $U_T,U_X$ and $U_Y$ are mutually independent. According to the rules of do-calculus, there exists a set $\Omega^*$ satisfying $\mathbb P(\Omega^*)=1$ such that for any $\omega \in \Omega^*$, $Y_{T=0}(\omega) = U_Y(\omega)$ and $Y_{T=1}(\omega) = 1+U_Y(\omega)$.* *Next, set $U'_Y := -U_Y$ and define the potential outcomes as follows: $$\begin{aligned} Y_0 &:= (1-T) \cdot U_Y + T \cdot U'_Y = (1-T) \cdot U_Y - T \cdot U_Y,\\ Y_1 &:= (1-T) \cdot (1+U'_Y) + T \cdot (1+U_Y) = (1-T) \cdot (1-U_Y) + T \cdot (1+U_Y).\end{aligned}$$ They clearly satisfy the consistency rule. Let us show that Assumptions [Assumption 5](#hyp:positivity){reference-type="ref" reference="hyp:positivity"} and [Assumption 6](#hyp:ignorability){reference-type="ref" reference="hyp:ignorability"} hold. Firstly, since $T \protect\mathpalette{\protect\independenT}{\perp}X$, we have $\mathbb P(T=1 \mid X) = \mathbb P(T=1) = 1/2$ which entails positivity. Secondly, $\mathcal{L}((Y_0,Y_1) \mid X=x, T=1) = \mathcal{L}((Y_0,Y_1) \mid T=1)$ because $U_Y \protect\mathpalette{\protect\independenT}{\perp}(T,X)$. Additionally, $\mathcal{L}((Y_0,Y_1) \mid T=1) = \mathcal{L}((-U_Y,1+U_Y)) = \mathcal{L}((U_Y, 1-U_Y))$ since $U_Y \stackrel{\mathcal{L}}{=} -U_Y$, and $\mathcal{L}((U_Y, 1-U_Y)) = \mathcal{L}((Y_0,Y_1) \mid T=0) = \mathcal{L}((Y_0,Y_1) \mid X=x, T=0)$ since $U_Y \protect\mathpalette{\protect\independenT}{\perp}(T,X)$. Therefore, $\mathcal{L}((Y_0,Y_1) \mid X=x, T=1) = \mathcal{L}((Y_0,Y_1) \mid X=x, T=0)$, meaning that conditional ignorability holds.* *We now turn to proving that $(Y_0,Y_1)$ is not almost-surely equal to $(Y_{T=0},Y_{T=1})$. On the event $\Omega^* \cap \{T=0\}$, we have $Y_{T=1} = 1+U_Y$ while $Y_1 = 1 + U'_Y = 1 - U_Y$. From $\mathbb P(U_Y=0) = 0$ and $\mathbb P(\Omega^* \cap \{T=0\}) = 1/2$ it follows that $\mathbb P(Y_{T=1} \neq Y_1)>0$, meaning that the two counterfactual outcomes are not almost surely equal. We can prove a similar result for $Y_{T=0}$ and $Y_0$.* *Remark also that $Y_t \stackrel{\mathcal{L}}{=} Y_{T=t}$ for every $t \in \{0, 1\}$, since $U_Y \stackrel{\mathcal{L}}{=} U'_Y$. In this scenario, counterfactual outcomes are not almost surely equal but equal in law.* Nevertheless, one could argue that this level of difference is not significant, since people doing counterfactual inference do not work with the variables themselves but their laws. They typically aim at computing quantities such as $\mathbb E[Y_1 - Y_0]$ or $\mathcal{L}(Y_{T=1} \mid X=x, T=0)$, which only depend on the (conditional) distributions of counterfactual outcomes. Therefore, as long as these distributions are equal across causal models, the two approaches will produce the same answers to counterfactual questions, even if the variables are not almost-surely equal. Note that there is equality in law in Example [Example 2](#ex:nas_cia){reference-type="ref" reference="ex:nas_cia"} (which verifies positivity and conditional ignorability), but not necessarily in Example [Example 1](#ex:nas){reference-type="ref" reference="ex:nas"}. In what follows, we show that equality in law does not always hold under the fundamental assumptions of causal inference. ### The distributional level We now address the most critical level: the one that concerns counterfactual inference. All answers to counterfactual questions in the structural framework and in the potential-outcome framework are respectively characterized by the joint probability distributions $\mathcal{L}((T,X,(Y_{T=t})_{t \in \mathcal T}))$ and $\mathcal{L}((T,X,(Y_t)_{t \in \mathcal T}))$. Should these distributions be equal, counterfactual reasoning would be equivalent between models. To prove that such an equality does not hold in general, we provide a *structural* identification of the law of potential outcomes. Counterfactual inference in the potential-outcome framework can be achieved when both positivity and conditional ignorability hold. Although these two fundamental assumptions of causal inference cannot fully solve the identification of the potential outcomes, as they do not constrain the variables almost surely, they permit to completely identify the law of the potential outcomes in terms of observable quantities: they entail that $\mathcal{L}(Y_t \mid X=x) = \mathcal{L}(Y \mid X=x, T=t)$ for any $t \in \mathcal T$. As our main mathematical result, we propose a different kind of identification under the same assumptions. The theorem below identifies the law of the potential outcomes through the latent SCM $\mathcal{M}$, thereby enabling us to compare $(Y_t)_{t \in \mathcal T}$ with $(Y_{T=t})_{t \in \mathcal T}$. **Theorem 10** (Structural law of potential outcomes). *Let $(Y_t)_{t \in \mathcal T}$ be random variables such that $$Y \stackrel{\mathbb{P}-a.s.}{=} \sum_{t \in \mathcal T} \mathbf{1}_{\{T=t\}} Y_t,$$ and suppose that $T$, $X$, and $(Y_t)_{t \in \mathcal T}$ verify Assumption [Assumption 5](#hyp:positivity){reference-type="ref" reference="hyp:positivity"} along with Assumption [Assumption 6](#hyp:ignorability){reference-type="ref" reference="hyp:ignorability"}. Additionally, assume that $V := (T,X,Y)$ is the solution to an SCM $\mathcal{M} = \langle U, G \rangle$ satisfying Assumptions [Assumption 2](#hyp:acyclic){reference-type="ref" reference="hyp:acyclic"} and [Assumption 7](#hyp:outcome){reference-type="ref" reference="hyp:outcome"} where $U_Y$ denotes the exogenous parents of $Y$. This notably entails that there exists a deterministic function $F_Y$ such that $Y \stackrel{\mathbb{P}-a.s.}{=} F_Y(T,X,U_Y)$ where $U_Y \protect\mathpalette{\protect\independenT}{\perp}(T,X)$. Then, $$(T,X,(Y_t)_{t \in \mathcal T}) \stackrel{\mathcal{L}}{=} (T,X,(F_Y(t,X,U_Y))_{t \in \mathcal T}),$$* This means in particular that under the assumptions of Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"}, we concurrently have $$\begin{aligned} &(Y_t)_{t \in \mathcal T} \stackrel{\mathcal{L}}{=} (F_Y(t,X,U_Y))_{t \in \mathcal T},\\ &(Y_{T=t})_{t \in \mathcal T} \stackrel{\mathbb{P}-a.s.}{=} (F_Y(t,X_{T=t},U_Y))_{t \in \mathcal T}.\end{aligned}$$ Therefore, $(Y_t)_{t \in \mathcal T}$ and $(Y_{T=t})_{t \in \mathcal T}$ are not necessarily equal in law since $\mathcal{L}(X) \neq \mathcal{L}(X_{T=t})$ in general. This difference also occurs at the individual level, that is conditional to $X$: $\mathcal{L}(Y_t \mid X=x) \neq \mathcal{L}(Y_{T=t} \mid X=x)$, since $\mathcal{L}(X_{T=t} \mid X=x) \neq \delta_x$ in general (here $\delta_x$ denotes the Dirac measure at $x$). As a consequence, in contrast to what many papers suggest, *the potential-outcome subscript notation and the do notation are not equivalent for counterfactual inference.* We provide a concrete example in the next section. Nevertheless, one can derive potential-outcome counterfactuals from the latent SCM by intervening on *both* $T$ and $X$ instead of $T$ only. According to the rules of do-calculus, $Y_{T=t,X=x} \stackrel{\mathbb{P}-a.s.}{=} F_Y(t,x,U_Y)$ whose law coincides with $\mathcal{L}(Y_t \mid X=x, T=t)$ under the assumptions of Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"}. We use this result in Remark [Remark 11](#rem:effect){reference-type="ref" reference="rem:effect"}. # Discussion {#sec:consequences} In this section, we discuss the consequences of the three levels of difference explained in Section [3](#sec:main){reference-type="ref" reference="sec:main"}. Firstly, Section [4.1](#sec:nonequi){reference-type="ref" reference="sec:nonequi"} specifies when counterfactual inference is equivalent or not between approaches. Secondly, Section [4.2](#sec:notation){reference-type="ref" reference="sec:notation"} focuses on the importance of using distinct notations for potential outcomes and structural counterfactuals. Finally, Section [4.3](#sec:choice){reference-type="ref" reference="sec:choice"} clarifies the apparent dissonance between our main results and the formal equivalence between frameworks mentioned in the literature. ## When potential outcomes and structural counterfactuals are (not) equivalent {#sec:nonequi} In this subsection, we study the assumptions and implications of Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"} to specify when counterfactual inference in one framework gives (or not) the same results as in the other. ### Nature of the treatment Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"} implies under the fundamental assumptions for causal inference that the law of potential outcomes coincides with the one of structural outcomes if: (1) $X$ is not altered by do-interventions on $T$, or (2) $Y$ is not impacted by $X$ (as in Example [Example 2](#ex:nas_cia){reference-type="ref" reference="ex:nas_cia"}). Let us focus on scenario (1), which is the most relevant in causal inference. The covariates are not altered by the treatment if $T$ is not a parent of $X$ in $\mathcal{M}$, as illustrated in Figure [\[fig:exogenous_covariates\]](#fig:exogenous_covariates){reference-type="ref" reference="fig:exogenous_covariates"}. Notably, this configuration encompasses various typical causal-inference scenarios: in clinical trials, the covariates $X$ may influence the treatment allocation $T$ but never the contrary. Therefore, both the Neyman-Rubin causal model and Pearl's causal model produce the same counterfactuals in common situations. However, it is also mathematically possible for $T$ to be a parent of several (or even all) covariates in $\mathcal{M}$. Figures [\[fig:nonexogenous_covariates\]](#fig:nonexogenous_covariates){reference-type="ref" reference="fig:nonexogenous_covariates"} and [\[fig:exogenous_treatment\]](#fig:exogenous_treatment){reference-type="ref" reference="fig:exogenous_treatment"} illustrate the possible causal graphs. In these situations, Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"} does not guarantee equality in law between potential outcomes and structural counterfactuals. In fact, it follows from the expressions of $\mathcal{L}((Y_t)_{t \in \mathcal T})$ and $\mathcal{L}((Y_{T=t})_{t \in \mathcal T})$ that equality in law will generally not hold in such configurations. Consequently, confusion between the two causal approaches can lead to misleading results: the Neyman-Rubin causal model considers counterfactual outcomes at fixed $X$, whereas Pearl's causal model alters the covariates into $X_{T=t}$. These cases are empirically relevant, since people also rely on causal inference outside the scope of clinical trials, in settings where the treatment cannot be completely manipulated and thereby impacts the covariates. For example, $T$ drives $X$ but not the contrary (as in Figure [\[fig:exogenous_treatment\]](#fig:exogenous_treatment){reference-type="ref" reference="fig:exogenous_treatment"}) in emblematic causal problems such as the Berkeley's admission paradox where $T$ represents the sex and $X$ the course choice [@bickel1975sex]. This is more generally true in the whole causal-fairness literature, where the variable to alter typically encodes the sex, the race, or the age of individuals (see for instance [@kusner2017counterfactual; @barocas-hardt-narayanan; @nilforoshan2022causal] for machine-learning-related research). Section [4.1.2](#sec:illustration){reference-type="ref" reference="sec:illustration"} below exemplifies all these points by studying an SCM corresponding to Figure [\[fig:exogenous_treatment\]](#fig:exogenous_treatment){reference-type="ref" reference="fig:exogenous_treatment"}. All in all, under the fundamental assumptions of causal inference, equivalence of counterfactuals across causal frameworks depends on the relationships between the treatment and the covariates, as described in Figure [\[fig:decisive\]](#fig:decisive){reference-type="ref" reference="fig:decisive"}. What distinguishes the different configurations is the nature of the so-called treatment. In particular, if the treatment can be assigned *a posteriori* to units (as in Figure [\[fig:exogenous_covariates\]](#fig:exogenous_covariates){reference-type="ref" reference="fig:exogenous_covariates"}), then the two notions of counterfactuals coincide. If the treatment is an intrinsic feature of units (as in Figure [\[fig:exogenous_treatment\]](#fig:exogenous_treatment){reference-type="ref" reference="fig:exogenous_treatment"}), such as individuals' race or sex, then structural counterfactuals and potential-outcomes counterfactuals are generally not equal. ### Illustration: an immutable treatment and two different kinds of counterfactuals {#sec:illustration} Counterfactual reasoning can be defined as thinking about outcomes in hypothetical worlds where some circumstances changes from what factually happened while others are kept equal. Crucially, there is not a single way of reasoning counterfactually. Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"} clearly shows that the potential-outcome framework compares worlds sharing the same observed features $X$ but differing in $T$, while the structural account compares worlds sharing the same exogenous parents $U_X$ but differing in $T$. Said differently, potential-outcome counterfactuals are *ceteris paribus* counterfactuals (i.e., all other things being kept equal) with respect to the covariates, whereas structural counterfactuals are *mutatis mutandis* counterfactuals (i.e., after changing what must be changed) with respect to the covariates. We emphasize that both definitions are perfectly legitimate, but convey distinct meanings and thereby correspond to different causal effects. Therefore, *they should not be employed for the same purpose*. Let us illustrate their implications on a concrete case. The following fairness-inspired example generalizes and circumstantiates the discussion from [@kusner2017counterfactual Appendix S1]. The treatment status $T$ indicates the gender, $T=0$ standing for women and $T=1$ standing for men; the covariate $X$ quantifies the level of work experience, a higher score encoding a richer experience; the outcome $Y$ evaluates a candidate's application for some position, a better score giving a higher probability of acceptance. Suppose that these three variables are ruled by the following SCM fitting Figure [\[fig:exogenous_treatment\]](#fig:exogenous_treatment){reference-type="ref" reference="fig:exogenous_treatment"}: $$\begin{aligned} T & \stackrel{\mathbb{P}-a.s.}{=} U_T,\\ X & \stackrel{\mathbb{P}-a.s.}{=} \alpha T + U_X,\\ Y & \stackrel{\mathbb{P}-a.s.}{=} X + \beta T + U_Y,\end{aligned}$$ where $\alpha$ and $\beta$ are deterministic parameters quantifying the causal influence of $T$ onto respectively $X$ and $Y$, and $U_X$ represents the hidden merit or effort of an individual. Typically, a positive parameter $\alpha$ describes the societal inequalities leading women to have a lower level of work experience than men with equal merit $U_X$. Moreover, we suppose that Assumption [Assumption 5](#hyp:positivity){reference-type="ref" reference="hyp:positivity"} is true, and that $U_Y \protect\mathpalette{\protect\independenT}{\perp}(U_T,U_X)$ so that Assumption [Assumption 7](#hyp:outcome){reference-type="ref" reference="hyp:outcome"} holds. Finally, we set two potential outcomes $(Y_0,Y_1)$ verifying the consistency rule and Assumption [Assumption 6](#hyp:ignorability){reference-type="ref" reference="hyp:ignorability"}. We consider the problem of assessing the causal effect of $T$ onto $Y$ conditional to $X=x$. In the potential-outcome approach this amounts to computing the conditional average treatment effect: $$\begin{aligned} \operatorname{CATE}(x) &:= \mathbb E[Y_1-Y_0 \mid X=x]\\ &= \mathbb E[(X + \beta + U_Y) - (X + U_Y) \mid X=x]\\ &= \beta,\end{aligned}$$ where we used Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"}. Observe that this first quantity measures only the *direct effect* of the treatment: it completely ignores the dependence of $Y$ on $T$ through $X$, as it involves only $\beta$. This is due to the fact that the $\operatorname{CATE}$ keeps the covariate $X$ fixed, comparing two *distinct* individuals with identical profiles but different genders. In contrast, Pearl's approach assesses the following structural counterfactual effect: $$\begin{aligned} \operatorname{SCE}(x) &:= \mathbb E[Y_{T=1}-Y_{T=0} \mid X=x]\\ &= \mathbb E[(X_{T=1} + \beta + U_Y) - (X_{T=0} + U_Y) \mid X=x]\\ &= \mathbb E[X_{T=1} - X_{T=0} \mid X=x] + \beta\\ &= \mathbb E[(\alpha + U_X) - U_X \mid X=x] + \beta\\ &= \alpha + \beta.\end{aligned}$$ Remark that this second quantity measures the *total effect* of the treatment: it takes into account the whole path of influence of $T$ onto $Y$, involving both $\alpha$ and $\beta$. This comes from the fact that the $\operatorname{SCE}$ fixes the random seed $U$ and not the covariates, comparing a *same* individual in two alternative realities where the gender is switched. Most importantly, $\operatorname{CATE} \neq \operatorname{SCE}$ if $\alpha \neq 0$, and consequently $\mathcal{L}((T,X,Y_{T=0},Y_{T=1})) \neq \mathcal{L}((T,X,Y_0,Y_1))$. From a fairness perspective, the $\operatorname{CATE}$ says that if $\beta = 0$, that is if $T$ is not a *direct* cause of $Y$, then the application process if fair; whether it is unfair towards men or women when $\beta \neq 0$ depends on the sign of $\beta$. In contrast, the $\operatorname{SCE}$ says that if $\beta = -\alpha$, that is if the decision rule $Y$ compensates the discrepancy of work experiences $X$ across genders $T$, then the application process is fair. Each analysis points out a different notion of fairness: considering the $\operatorname{SCE}$ as a fairness criterion suggests that recruiters should correct societal inequalities by preferring women with potentially lower work experience but higher merit whereas relying on the $\operatorname{CATE}$ suggests it is only explicitly including the gender in the decision-rule pipeline that is unfair. Critically, if $\alpha \neq 0$, practitioners mixing potential outcomes with structural counterfactuals could reach contradictory conclusions on fairness. **Remark 11** (Computing direct effects from an SCM). *One can still compute the $\operatorname{CATE}$, that is the *direct* effect, using do-interventions on the SCM. Under the same assumptions as above, $$\begin{aligned} \mathbb E[Y_{T=1,X=x} - Y_{T=0,X=x}] &= \mathbb E[(x + \beta + U_Y) - (x + U_Y)],\\ &= \beta,\\ &= \operatorname{CATE}(x).\end{aligned}$$ This is due to the fact that, more generally, $\mathcal{L}(Y_{T=t,X=x}) = \mathcal{L}(Y_t \mid X=x)$ under the fundamental assumptions of causal inference. In contrast, one cannot always compute the $\operatorname{SCE}$, that is the *indirect* effect from potential outcomes. Besides, we point out that even though the $\operatorname{CATE}$ can be theoretically derived from both a potential-outcome model and an SCM, the practical methods to estimate them from data differ between approaches. In the Neyman-Rubin causal framework, one only needs to estimate $\mathbb E[Y \mid X=x, T=t]$ (as it equals $\mathbb E[Y_t \mid X=x]$); in Pearl's causal framework, one needs to learn the full SCM beforehand (which is a notoriously difficult task), and then to apply the three-step procedure with $\operatorname{do}(T=t, X=x)$.* To sum-up, each approach has a different signification, and therefore corresponds to a specific way of reasoning counterfactually. This signifies that the difference between frameworks does not amount to practical considerations only. Analysts and researchers should also justify the chosen model depending on the kind of causal effects they want to compute. ### Nonequivalence in applicability Up until now, we have discussed the differences between frameworks from a theoretical viewpoint, not focusing on the practical aspects. Summing-up: to apply Pearl's causal framework, one must postulate a plausible structural causal model or infer it from data to then carry out do-calculus; to apply the Neyman-Rubin causal framework, one must find a set of covariates believed to satisfy conditional ignorability and then use statistical methods (e.g., matching, stratification, re-weighting, regression) to estimate observable quantities with causal meanings. As such, the two approaches are different in *how* they are applied; but there also exists a critical difference in *when* they can be applied due to the positivity assumption. Let us detail this point, as it also concerns a distinction between counterfactuals across frameworks. Causal inference in the potential-outcome framework requires two fundamental assumptions: conditional ignorability (Assumption [Assumption 6](#hyp:ignorability){reference-type="ref" reference="hyp:ignorability"}) and positivity (Assumption [Assumption 5](#hyp:positivity){reference-type="ref" reference="hyp:positivity"}). While the second is testable in contrast to the first, it raises practical issues. It basically states that the distributions $\mathcal{L}(X \mid T=t)$ for $t \in \mathcal T$ share the same support, which is violated as soon as the groups represented by $T$ bear unique properties. Consider for example that we study individuals where $T$ encodes their genders, and that the covariates $X$ specify their jobs (among other attributes). Positivity would not hold if gender-locked positions existed, and consequently we could not identify the counterfactual outcome *had she been a man* of every woman occupying a women-only job. In contrast, a structural causal model always allows such a computation. Therefore, there exist problems where the two causal models cannot be simultaneously applied to carry out counterfactual inference in practice. Not only the two kinds of counterfactuals should not be used for the same purpose (as they provably have different significations), but they cannot always be used for the same tasks. ## On the importance of notations {#sec:notation} All in all, it is inappropriate to exchange the do-notation and the potential-outcome notation, since they refer to variables with different distributions in common scenarios, such as almost every problem tackled by the causal-fairness literature. Going further, we argue that distinct notations should always be used---even when the laws are identical---for two reasons: a practical reason and a conceptual one. First, papers bridging the two causal approaches with a common notation often address favorable settings where the treatment is controllable, as in [@colnet2020causal Section 5], which has no consequence on the consistency of counterfactual reasoning. While this may seem harmless, this could create confusion since the authors never explicitly state that they focus on controllable treatments, and even less justify why the laws of counterfactuals are equals. Therefore, people could wrongly believe that this equivalence holds in general. Notably, [@makhlouf2020survey] study the application of both the structural account of counterfactuals and potential outcomes in fairness settings where the treatment encodes sex or race, and suggest that the appropriate choice of framework is mostly a matter of practical considerations. This could lead to contradictory results, as previously exemplified. Second, a notation is the essence of mathematical object. Therefore, a same notation can be used for two objects just in case: (1) the objects have the same definition, (2) the objects are mathematically equal. The first two levels of difference we studied in Section [3](#sec:main){reference-type="ref" reference="sec:main"} show that none of these conditions hold in general for structural counterfactuals and potential outcomes, even when their laws are equal (recall Example [Example 2](#ex:nas_cia){reference-type="ref" reference="ex:nas_cia"}); the third level shows that none of these conditions hold in general for law-dependent quantities. ## On the formal equivalence between frameworks {#sec:choice} Before concluding, we emphasize that our work does not contradict the formal equivalence between causal frameworks addressed notably by [@pearl2009causality] and [@richardson2013single]. We think, however, that some people possibly drew incorrect conclusions from this equivalence, in particular the systematic identification of $\mathcal{L}((T,X,(Y_t)_{t \in \mathcal T}))$ to $\mathcal{L}((T,X,(Y_{T=t})_{t \in \mathcal T}))$. In what follows, we try to explain where the confusion comes from. ### Treating potential outcomes as structural counterfactuals is a choice Let us start with a crucial reminder: in all generality, potential outcomes are merely variables $(Y_t)_{t \in \mathcal T}$ satisfying the consistency rule for a treatment of interest $T$ and an outcome $Y$. In particular, even though the potential-outcome framework is impractical without the two fundamental assumptions of causal inference, one can perfectly consider outcomes that do not satisfy conditional ignorability in theory.[^5] In this sense, structural counterfactuals can always be defined as potential outcomes according to Proposition [Proposition 9](#prop:consistency_do){reference-type="ref" reference="prop:consistency_do"} (which only requires acyclicity). Interestingly, if one chooses to define potential outcomes as structural counterfactuals, then the Neyman-Rubin causal model and Pearl's causal model become two different languages to talk about the same objects. In the Neyman-Rubin causal model, assumptions for causal inference are generally framed as conditional-independence restrictions (e.g., conditional ignorability); in Pearl's causal framework, assumptions on causal relationships are generally framed in terms of graphical conditions (e.g., backdoor criterion). Both [@pearl2009causality Chapter 7] and [@richardson2013single] focus on unifying these two mathematical languages by providing rules for translating assumptions and theorems from one viewpoint to the other. This ensures what people often refer as the *logical* or *formal equivalence* between frameworks. It notably allows analysts to work symbiotically with both models, as long as equivalent assumptions are made across them. However, we underline that defining potential outcomes as structural counterfactuals is a *choice*---it does not rest on any proof. When Pearl writes $Y_t := Y_{T=t}$ in [@pearl2009causality Equation 3.51], claiming that the operation $\operatorname{do}(T=t)$ on the SCM gives a physical meaning to the vague of the potential outcome, this is nothing more than an arbitrary choice. Naively looking at the definitions of the Neyman-Rubin causal framework and of Pearl's causal framework, there is nothing that mathematically constrains potential outcomes to coincide with the structural counterfactuals of a given SCM (as demonstrated in Examples [Example 1](#ex:nas){reference-type="ref" reference="ex:nas"} and [Example 2](#ex:nas_cia){reference-type="ref" reference="ex:nas_cia"}, and Section [4.1.2](#sec:illustration){reference-type="ref" reference="sec:illustration"}).[^6] What [@pearl2009causality Chapter 7] and [@richardson2013single] prove is not the algebraic equality between potential outcomes and structural counterfactual. Instead, they show that *if* potential outcomes are chosen to be structural counterfactuals, then one can translate the assumptions made on potential outcomes into assumptions on the underlying causal graph and *vice versa*. However, these references did not make their definition choices clear enough. As a consequence, it seems that the formal equivalence between viewpoints has sometimes been understood as the systematic interchangeability between $(Y_t)_{t \in \mathcal T}$ and $(Y_{T=t})_{t \in \mathcal T}$, or between $\mathcal{L}((T,X,(Y_t)_{t \in \mathcal T}))$ and $\mathcal{L}((T,X,(Y_{T=t})_{t \in \mathcal T}))$. ### The meaning of treating potential outcomes as distinct to structural counterfactuals If we consider the two models to be different causal mechanisms rather than just different perspectives, then we can make nonequivalent assumptions on potential outcomes and structural counterfactuals. Supposing distinct axioms across models means giving distinct interpretations to their respective counterfactuals, which mathematically translates into $\mathcal{L}((T,X,(Y_t)_{t \in \mathcal T})) \neq \mathcal{L}((T,X,(Y_{T=t})_{t \in \mathcal T}))$. Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"} and its implications precisely illustrate this aspect: assuming the conditional ignorability of the potential outcomes but not making an equivalent hypothesis on the SCM defining the structural counterfactuals entails that counterfactuals do not have the same law in general hence not the same interpretation. As illustrated in Section [4.1.2](#sec:illustration){reference-type="ref" reference="sec:illustration"}, the conditional-ignorability assumption defines potential outcomes as counterfactuals switching the treatment but keeping all other variables equal, whereas Pearl's do-intervention on the treatment defines structural counterfactuals altering the remaining variables. In a scenario where the treatment is controllable, these definitions happen to coincide. As such, the interest of defining potential outcomes as distinct to structural counterfactuals is notably to estimate the *ceteris paribus* causal effects of immutable treatments by leveraging statistical methods on observable distributions rather than by learning a fully specified plausible SCM and to then apply the three-step procedure with $\operatorname{do}(T=t, X=x)$ (as mentioned in Remark [Remark 11](#rem:effect){reference-type="ref" reference="rem:effect"}). An important corollary of Section [4.1](#sec:nonequi){reference-type="ref" reference="sec:nonequi"} is that one *cannot always* make an equivalent assumption to conditional ignorability on the structural counterfactuals. From the SCM perspective, conditional ignorability is simply $(Y_{T=t})_{t \in \mathcal T} \protect\mathpalette{\protect\independenT}{\perp}T \mid X$, which under Assumption [Assumption 7](#hyp:outcome){reference-type="ref" reference="hyp:outcome"} occurs when the treatment is fully controllable (that is fitting Figure [\[fig:exogenous_covariates\]](#fig:exogenous_covariates){reference-type="ref" reference="fig:exogenous_covariates"}). Therefore, in the many aforementioned settings where the treatment is factually not controllable (that is fitting Figure [\[fig:exogenous_treatment\]](#fig:exogenous_treatment){reference-type="ref" reference="fig:exogenous_treatment"}) potential outcomes verifying conditional ignorability must be defined as distinct to the *true* structural counterfactuals.[^7] Critically, this means that even though it is generally implicitly done, *not* defining potential outcomes as structural counterfactuals is actually something common in the scientific literature. Notably, [@li2017discrimination; @glymour2017evaluating; @khademi2019fairness; @khademi2020algorithmic; @makhlouf2020survey; @qureshi2020causal] rely on (or refer to) the potential-outcome framework equipped the fundamental assumption of causal inference to understand the influence of a noncontrollable treatment like sex, race, and biological factors.[^8] As explained, $\mathcal{L}((T,X,(Y_t)_{t \in \mathcal T}))$ generally differs from $\mathcal{L}((T,X,(Y_{T=t})_{t \in \mathcal T}))$ in such studies. Therefore, if we consider this corpus of the causal-inference literature to be admissible, then we must accept that unifying potential outcomes and structural counterfactuals is not an obligation. ### What is the best paradigm? The results from Section [3](#sec:main){reference-type="ref" reference="sec:main"} show that leveraging a structural causal model as the axiomatic characterization of potential outcomes is a choice. This outlines two paradigms for defining and applying potential outcomes: in synergy with the latent SCM generating the observations; as hypothetical outcomes verifying conditional ignorability. Settling the debate on whether there is a most relevant or inappropriate choice lies outside the scope of this paper. On the one hand, Pearl has advocated for long to *always* use the potential-outcome framework in symbiosis with an SCM, as the latter enables to formulate its conditional-independence conditions deemed nebulous in the intelligible language of causal graphs. On the other hand, deviating from this rule to work with SCM-free potential outcomes allows to compute the direct causal effects of noncontrollable treatments through statistical methods, as done in the aforementioned studies. Each approach can be legitimate; *what crucially matters is having a clear understanding of the produced counterfactuals*. Modeling potential outcomes through an SCM formally defines their as the operation $\operatorname{do}(T=t)$ which changes the remaining variables accordingly. Dressing potential outcomes with conditional ignorability defines their as switching $T$ into $t$ while keeping the remaining variables unchanged. We hope this discussion to make clear the (sometimes implicit) signification of potential outcomes in the causal-inference literature. To summarize, we think that the scope and implications of the formal equivalence between causal frameworks can be misleading. In particular, it does not mean that $\mathcal{L}((T,X,(Y_t)_{t \in \mathcal T})) = \mathcal{L}((T,X,(Y_{T=t})_{t \in \mathcal T}))$ in general; it means that such an equality in law holds *if equivalent assumptions are made on the counterfactual outcomes across models*. However, using the potential-outcome framework to estimate direct causal effects requires assuming conditional ignorability, which cannot always be assumed on the structural counterfactuals from the *true* latent SCM due to the possibly immutable nature of the treatment. This why we recommend to present the Neyman-Rubin causal framework and Pearl's causal framework as distinct ways of reasoning counterfactually, that coincide under specific assumptions and choices. # Conclusion In this paper, we superimposed Pearl's causal framework and the Neyman-Rubin causal framework without presuppositions to show that structural counterfactual outcomes and potential outcomes do not coincide at three levels: they are defined differently, they are not equal in general, they do not have the same law in general. To prove the third level of difference, we expressed the law of potential outcomes in terms of the latent structural causal model under classical causal-inference assumptions. On the basis of this result, we gave a detailed interpretation of counterfactuals in each causal framework, specifying when they entailed different conclusions. More specifically, counterfactual inference with potential outcomes under conditional ignorability yields *ceteris paribus* counterfactuals, whereas counterfactual inference with a do-intervention on a structural causal model yields *mutatis mutandis* counterfactuals. If the cause of interest is immutable, these constructions are generally not equivalent. For these reasons, we call the community to not interchangeably use the do-notation and the potential-outcome notation, unless the justification is explicitly made. We emphasize that this work is not an argument in favor of using one causal model rather than the other, or against the formal equivalence between frameworks. It is meant to shed light on the different mathematical choices that analysts can make when working with counterfactual outcomes, and to precise their implications in order to prevent incorrect or ambiguous conclusions in causal studies. In doing this paper, we hope to clarify the similarities and differences between the two major causal approaches. # Proofs **of Proposition [Proposition 4](#prop:docalculus){reference-type="ref" reference="prop:docalculus"}**  Since $\mathcal{M}$ is acyclical, there exists a topological ordering on the indices in $\mathcal I$, and therefore on the subset $J$. This means in particular that there exist some $j \in J$ such that $G_j$ takes only variables in $V_I$ as endogenous inputs. Starting from these indices, and recursively substituting along the topological ordering produces a measurable $F_J$ such that $$V_J \stackrel{\mathbb{P}-a.s.}{=} F_J(V_{\operatorname{Endo}(J) \setminus J}, U_{\operatorname{Exo}(J)}).$$ Note that $\operatorname{Endo}(J) \setminus J \subseteq I$. Carrying out the same substitution on the intervened model $\mathcal{M}_{V_I = v_I}$ with solution $\tilde{V}$ gives $$\tilde{V}_J \stackrel{\mathbb{P}-a.s.}{=} F_J(v_{\operatorname{Endo}(J) \setminus J}, U_{\operatorname{Exo}(J)}),$$ while by definition $\tilde{V}_I \stackrel{\mathbb{P}-a.s.}{=} v_I$. ------------------------------------------------------------------------ \ **of Lemma [Lemma 8](#lm:noise){reference-type="ref" reference="lm:noise"}**  By assumption, the random vector $V := (T,X,Y)$ is the solution to an acylical structural causal model where $U_T$, $U_X$ and $U_Y$ denote the exogenous parents of respectively $T$, $X$ and $Y$. Since additionally $Y_{\operatorname{Endo}(T)} = Y_{\operatorname{Endo}(X)} = \emptyset$, Proposition [Proposition 4](#prop:docalculus){reference-type="ref" reference="prop:docalculus"} ensures the existence of a measurable function $F_{T,X}$ such that $(T,X) \stackrel{\mathbb{P}-a.s.}{=} F_{T,X}(U_T,U_X).$ Therefore, if $U_Y \protect\mathpalette{\protect\independenT}{\perp}(U_T,U_X)$, then $U_Y \protect\mathpalette{\protect\independenT}{\perp}(T,X)$. ------------------------------------------------------------------------ \ **of Proposition [Proposition 9](#prop:consistency_do){reference-type="ref" reference="prop:consistency_do"}**  Let $t \in \mathcal T$. By assumption, the random vector $V := (T,X,Y)$ is the solution to an acylical structural causal model. We write $U_X$ and $U_Y$ the exogenous parents of respectively $X$ and $Y$. Therefore, by partitioning $V$ into $T$ and $(X,Y)$, Proposition [Proposition 4](#prop:docalculus){reference-type="ref" reference="prop:docalculus"} guarantees the existence of a measurable function $F_{X,Y}$ such that $$\begin{aligned} (X,Y) & \stackrel{\mathbb{P}-a.s.}{=} F_{X,Y}(T,U_X,U_Y)\\ (X_{T=t},Y_{T=t}) & \stackrel{\mathbb{P}-a.s.}{=} F_{X,Y}(t,U_X,U_Y).\end{aligned}$$ Therefore, selecting the coordinates corresponding to $Y$ furnishes a measurable function $\tilde{F}_Y$ such that $$\begin{aligned} Y & \stackrel{\mathbb{P}-a.s.}{=} \tilde{F}_Y(T,U_X,U_Y)\\ Y_{T=t} & \stackrel{\mathbb{P}-a.s.}{=} \tilde{F}_Y(t,U_X,U_Y).\end{aligned}$$ These identities hold on a measurable set $\Omega^* \subseteq \Omega$ such that $\mathbb P(\Omega^*) = 1$. To conclude, simply observe that for any $\omega \in \Omega^*$ such that $T(\omega)=t$, we have $$Y(\omega) = \tilde{F}_Y(t,U_X(\omega),U_Y(\omega)) = Y_{T=t}(\omega).$$ ------------------------------------------------------------------------ \ **of Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"}**  Let us compute the conditional joint distribution $\mathcal{L}((Y_{t''})_{t'' \in \mathcal T} \mid X=x, T=t)$ which is well-defined for all $x \in X(\Omega)$ and $t \in \mathcal T$ by positivity. The consistency rule entails that $$\mathcal{L}((Y_{t''})_{t'' \in \mathcal T} \mid X=x, T=t) = \mathcal{L}((Y_0,\ldots,Y_{t-1},Y,Y_{t+1},\ldots,Y_N) \mid X=x, T=t).$$ Moreover, according to $\mathcal{M}$ the observed outcome can be written as $Y \stackrel{\mathbb{P}-a.s.}{=} F_Y(T,X,U_Y)$, leading to $$\mathcal{L}((Y_{t''})_{t'' \in \mathcal T} \mid X=x, T=t) = \mathcal{L}((Y_0,\ldots,Y_{t-1},F_Y(t,x,U_Y),Y_{t+1},\ldots,Y_N) \mid X=x, T=t).$$ Next, recall that Assumption [Assumption 7](#hyp:outcome){reference-type="ref" reference="hyp:outcome"} entails through Lemma [Lemma 8](#lm:noise){reference-type="ref" reference="lm:noise"} that $U_Y \protect\mathpalette{\protect\independenT}{\perp}(T,X)$. Therefore, it follows from $(Y_{t''})_{t'' \in \mathcal T} \protect\mathpalette{\protect\independenT}{\perp}T \mid X$ that for any $t' \neq t$ the above equality is equivalent to $$\mathcal{L}((Y_{t''})_{t'' \in \mathcal T} \mid X=x) = \mathcal{L}((Y_0,\ldots,Y_{t-1},F_Y(t,x,U_Y),Y_{t+1},\ldots,Y_N) \mid X=x, T=t').$$ Then, using once the again the consistency rule we obtain $$\mathcal{L}((Y_{t''})_{t'' \in \mathcal T} \mid X=x) = \mathcal{L}((Y_0,\ldots,F_Y(t,x,U_Y),\ldots,Y_{t'-1},Y,Y_{t'+1},\ldots) \mid X=x, T=t'),$$ and the expression of $Y$ through $F_Y$ yields $$\mathcal{L}((Y_{t''})_{t'' \in \mathcal T} \mid X=x) = \mathcal{L}((Y_0,\ldots,F_Y(t,x,U_Y),\ldots,F_Y(t',x,U_Y),\ldots) \mid X=x, T=t').$$ We repeat this step by conditioning on all possible values of $T$ to finally obtain $$\mathcal{L}((Y_{t''})_{t'' \in \mathcal T} \mid X=x) = \mathcal{L}((F_Y(t'',x,U_Y)_{t'' \in \mathcal T}) \mid X=x) = \mathcal{L}((F_Y(t'',x,U_Y)_{t'' \in \mathcal T})).$$ It then follows from $U_Y \protect\mathpalette{\protect\independenT}{\perp}(T,X)$ that for any $t \in \mathcal T$, $$\mathcal{L}((t,x,(Y_{t''})_{t'' \in \mathcal T}) \mid X=x, T=t) = \mathcal{L}((t,x,(F_Y(t'',x,U_Y)_{t'' \in \mathcal T})) \mid X=x, T=t).$$ Therefore, marginalizing on $(T,X)$ yields $$\mathcal{L}((T,X,(Y_t)_{t \in \mathcal T})) = \mathcal{L}((T,X,(F_Y(t,X,U_Y))_{t \in \mathcal T})).$$ ------------------------------------------------------------------------ \ [^1]: E-mail: lucas.de_lara\@math.univ-toulouse.fr [^2]: <http://causality.cs.ucla.edu/blog/index.php/2012/12/03/judea-pearl-on-potential-outcomes/> [^3]: *This definition tolerates that distinct endogenous variables share the same exogenous parents, that is $\operatorname{Exo}(i) \cap \operatorname{Exo}(i') \neq \emptyset$ for some $i \neq i'$. Therefore, the $(U_{\operatorname{Exo}(i)})_{i \in \mathcal I}$ are not necessarily mutually independent.* [^4]: *We point out that Assumption [Assumption 7](#hyp:outcome){reference-type="ref" reference="hyp:outcome"} is not required.* [^5]: We dot not focus on positivity since it does not constrain the potential outcomes. [^6]: From a more philosophical angle, [@markus2021causal Section 2.1] made a similar remark to argue that the two frameworks were *weakly* equivalent rather than *strongly* equivalent. [^7]: More generally, the logical equivalence between framework ensures that there always exists an SCM that induces the law of given potential outcomes, but does not guarantee that this SCM correctly describe the observations $(T,X,Y)$. The discussion that followed Theorem [Theorem 10](#thm:identification){reference-type="ref" reference="thm:identification"} is based on the principle that the considered SCM is the true one, not necessarily one that fits the potential-outcome assumptions. [^8]: Other articles, for instance [@ridgeway2006assessing] and [@gaebler2022causal], explicitly focus on sex and race as *perceived* by a decider. Such a perception could depend on the covariates, hence not be immutable.

arxiv_math

--- abstract: | A conjecture attributed to Smith states that every pair of longest cycles in a $k$-connected graph intersect each other in at least $k$ vertices. In this paper, we show that every pair of longest cycles in a $k$-connected graph on $n$ vertices intersect each other in at least $\min\{n,8k-n-16\}$ vertices, which confirms Smith's conjecture when $k\geq (n+16)/7$. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either $k \leq 6$ or $k \geq (n+9)/7$. address: - | Departamento de Ciencia de la Computación\ Universidad de Ingeniería y Tecnología (UTEC), Perú - Pontificia Universidad Católica del Perú, Sección Matemáticas, PUCP, Av. Universitaria 1801, San Miguel, Lima 32, Perú author: - Juan Gutiérrez - Christian Valqui bibliography: - bibliografia.bib title: On two conjectures about the intersection of longest paths and cycles. --- # Introduction and Preliminaries It is known that every pair of longest cycles (paths) in a 2-connected (connected) graph intersect each other in at least two vertices (one vertex). A conjecture proposed by Grötschel, and attributed to Scott Smith [@Grotschel84 Conjecture 5.2], states that, in a $k$-connected graph, with $k\geq 2$, every pair of longest cycles intersect each other in at least $k$ vertices. Years later, Hippchen [@Hippchen08 Conjecture 2.2.4] conjectured that, for $k$-connected graphs, every pair of longest paths intersect each other in at least $k$ vertices. Smith's conjecture has been verified up to $k=7$ [@Grotschel84], and, for a general $k$, it was proved that every pair of longest cycles intersect in at least $ck^{3/5}$ vertices, for a constant $c \thickapprox 0.2615$ [@Chen98]. For Hippchen's conjecture, the case $k=3$ was proved by Hippchen himself [@Hippchen08 Lemma 2.2.3]. Later, the first author show it for $k=4$, and recently, Cho *et al.*[@Cho2022] show it for $k=5$. As we can observe in the literature, efforts on both conjectures has been made independently. The first main contribution of this paper is relate these conjectures. We will show that if Smith's conjecture is valid for a fixed $k$, then Hippchen's conjecture is valid for $k-1$. This relationship implies that efforts can be concentrated on Smith's conjecture. So, as Smith's conjecture is known to be valid for $k\leq 7$, an easy corollary is that Hippchen's conjecture is true for $k\leq 6$. (For an independent proof of this fact, see [@Gutierrez2021-bitraceable]). The second main contribution of this paper is an improvement of a result on Hippchen's conjecture when $k=\Omega(n)$, where $n$ is the number of vertices of the graph. In [@Gutierrez2021-Pro], the first author showed that Hippchen's conjecture is valid when $k \geq (n-2)/3$. This result was improved by Cho *et al.*[@Cho2022]. They showed that Hippchen's conjecture is valid when $k \geq (n+2)/5$. In this paper, we improve this result, by showing that Hippchen's conjecture is valid when $k \geq (n+9)/7$. In fact, we show the stronger statement that every pair of longest paths in a $k$-connected graph intersect in at least $\min\{k,8k-n-9\}$ vertices. Moreover, this result is a consequence of an analog result on cycles: we show that every pair of longest cycles intersect in at least $\min\{k,8k-n-16\}$ vertices. In this paper all graphs are simple (without loops or parallel edges) and the notation and terminology are standard. We also consider simple paths and cycles, that is, repetitions of vertices or edges is not allowed. The **length** of a path $P$ (cycle $C$) is the number of edges it has, and it is denoted by $|P|$ ($|C|$). A **longest path** (cycle) in a graph is a path (cycle) with maximum length among all paths (cycles). Given a path $P$ and two vertices $x$ and $y$ in $P$, we denote by $P[x,y]$ the subpath of $P$ with extremes $x$ and $y$. Given two set of vertices $S$ and $T$ in a graph $G$, an $\bm{S}$-$\bm{T}$ **path** is a path with one end in $S$, the other end in $T$, and whose internal vertices are neither in $S$ nor $T$. If $S=\{v\}$, we also say that an $S$-$T$ path is a $v$-$T$ path. When we refer to the intersection of two paths or cycles in a graph, we mean vertex-intersection, that is, the set of vertices they share. Two paths are **internally disjoint** if they have no internal vertices in common. A graph $G$ is $\bm{k}$**-connected** if, for any two distinct vertices $u$ and $v$ in $G$, there exists a set of $k$ $u$-$v$ internally disjoint paths. It is easy to see that for a $k$-connected graph on $n$ vertices, we have $k\leq n-1$. A set $S \subseteq V(G)$ of a connected graph $G$ is a **separator** if $G-S$ has more than one component. It is known that a graph is $k$-connected if and only if every separator in the graph has size at least $k$. # Two families of conjectures on longest paths and cycles For every $k\geq 1$ and every $r \leq k$, we consider the following families of conjectures. **Conjecture 1** (Conjecture P$\mathbf{(k,r)}$). *In any $k$-connected graph, every pair of longest paths intersect in at least $r$ vertices.* **Conjecture 2** (Conjecture C$\mathbf{(k,r)}$). *In any $k$-connected graph, every pair of longest cycles intersect in at least $r$ vertices.* Note that Conjecture P$\mathbf{(k,k)}$ is Hippchen's Conjecture [@Hippchen08], and Conjecture C$\mathbf{(k,k)}$ is Smith's Conjecture [@Grotschel84 Conjecture 5.2]. The first main purpose of this paper is to relate Conjecture P$\mathbf{(k+1,r+1)}$ and Conjecture C$\mathbf{(k,r)}$, as in the next Theorem. **Theorem 3**. *For every $k\geq 1$ and every $r\leq k$, Conjecture $C(k+1,r+1)$ implies Conjecture $P(k,r)$.* *Proof.* Let $G$ be a $k$-connected graph on $n$ vertices. Let $P$ and $Q$ be two longest paths in $G$. We construct a new graph, $G'$, by adding a new vertex $s$ to $G$ and joining it to every vertex of $G$. We will show that $G'$ is $(k+1)$-connected. Suppose by contradiction that it is not the case. Then $G'$ has a separator $X'$ with size at most $k$. As $s$ is universal in $G'$, we must have $s\in X'$. Indeed, otherwise, any pair of vertices of $G'$ different from $s$ are joined by $s$. As $X'$ is a separator of $G'$, there exists two distinct vertices $u$ and $v$ in $G'$ such that there is no $uv$-path in $G'-X'$. As $G$ is a subgraph of $G'$, there is also no $uv$-path in $G-X'$, which implies that $X' \cap V(G)$ is a separator in $G$. As $s \in X'$, $|X' \cap V(G)| \leq k-1$, which contradicts the fact that $G$ is $k$-connected. Let $P'$ and $Q'$ be the cycles in $G'$ with vertex set $P \cup \{s\}$ and $Q \cup \{s\}$, respectively. If Conjecture $C(k+1,r+1)$ is true, then $P'$ and $Q'$ intersect in at least $r+1$ vertices, thus $P$ and $Q$ intersect in at least $r$ vertices. ◻ Conjecture P$\mathbf{(k,k)}$ has been proved for $k=3$ [@Hippchen08], $k=4$ [@Gutierrez2021] and $k=5$ [@Cho2022]. Conjecture C$\mathbf{(k,k)}$ has been proved for $k \leq 7$, but with some conditions. For the sake of completeness, we state exactly what has been proved for this conjecture in the next three propositions. **Proposition 4** ([@Gutierrez2021 Theorem 1.2]). *Every pair of longest cycles in a 2-connected graph intersect in at least two vertices.* **Proposition 5** ([@Grotschel84 Theorem 1.2]). *Let $k \in \{3,\ldots,5\}$ and let $G$ be a 2-connected graph with at least $k+1$ vertices. Let $C$ and $D$ be two longest cycles in $G$. Then $V(C) \cap V(D)$ separates $G$.* **Proposition 6** ([@Steward1995 Theorem 1.2]). *Let $k \in \{6,7\}$ and let $G$ be a graph with circumference at least $k+1$. Let $C$ and $D$ be two distinct longest cycles in $G$. Then $V(C) \cap V(D)$ separates $G$.* **Theorem 7** ([@Grotschel84; @Gutierrez2021; @Steward1995]). *For $2 \leq k\leq 7$, Conjecture $C(k,k)$ is true.* *Proof.* When $k=2$, the proof follows by Proposition [Proposition 4](#prop:2conn2vertices){reference-type="ref" reference="prop:2conn2vertices"}. So let us assume that $k \geq 3$. Let $G$ be a $k$-connected graph with two longest cycles $C$ and $D$. As $G$ is $k$-connected, $\delta(G) \geq k$, which implies $|V(G)| \geq |V(C)| \geq k+1$. If $C=D$, then $|V(C)| \cap |V(D)| \geq k$, so let us assume that $C \neq D$. Then, by Propositions [Proposition 5](#prop:k3a5){reference-type="ref" reference="prop:k3a5"} and [Proposition 6](#prop:k6a7){reference-type="ref" reference="prop:k6a7"}, $V(C) \cap V(D)$ separates $G$. As $G$ is $k$-connected, $|V(C) \cap V(D)|\geq k$ and the proof follows. ◻ By Theorems [Theorem 3](#thm:CimpliesP){reference-type="ref" reference="thm:CimpliesP"} and [Theorem 7](#thm:Ckk2a7){reference-type="ref" reference="thm:Ckk2a7"}, we have the next result. **Theorem 8**. *For $1 \leq k\leq 6$, Conjecture $P(k,k)$ is true. That is, in every $k$-connected graph with $k\leq 6$, every pair of longest paths intersect in at least $k$ vertices.* For an arbitrary $k$, we have the next result due to Chen *et al.*[@Chen98]. **Theorem 9** ([@Chen98 Theorem 2]). *For any $k \geq 2$, Conjecture $C(k,ck^{3/5})$ is true, where ${c=1/(\sqrt[3]{256}+3)^{3/5}}$.* By Theorems [Theorem 3](#thm:CimpliesP){reference-type="ref" reference="thm:CimpliesP"} and [Theorem 9](#thm:Chen){reference-type="ref" reference="thm:Chen"}, we have the next result. **Theorem 10**. *For any $k$, Conjecture $P(k,c(k+1)^{3/5}-1)$ is true, where $c=1/(\sqrt[3]{256}+3)^{3/5}$.* Cho *et al.*[@Cho2022] showed that $P(k,k)$ is true when $k \geq \frac{n+2}{5}$, where $n$ is the number of vertices of the graph [^1]. In fact they proved the following stronger statement. **Theorem 11** ([@Cho2022 Theorem 1.4]). *For any $k \geq 2$, Conjecture $P(k,\min\{k,\ceil{\frac{8k-n-4}{3}}\})$ is true.* No similar result for $C(k,r)$ has been given in the literature. In Section [3](#section:Ckr){reference-type="ref" reference="section:Ckr"} we show that for any $k \geq 2$, Conjecture $C(k,\min\{k,8k-n-16\})$ is true (Theorem [Theorem 16](#thm:Ck8k-n-16){reference-type="ref" reference="thm:Ck8k-n-16"}). This will imply the following two results. **Theorem 12**. *For any $k \geq 2$, Conjecture $P(k,\min\{k,8k-n-9\})$ is true.* **Corollary 13**. *For any $k \geq \frac{n+16}{7}$, Conjecture $C(k,k)$ is true. For any $k \geq \frac{n+9}{7}$, Conjecture $P(k,k)$ is true.* # A new result on C(k,r) {#section:Ckr} Our proof rely in two well-known facts, that we state in the following propositions. The first proposition is also known as Fan lemma. **Proposition 14** ([@BondyM08 Proposition 9.5]). *Let $G$ be a $k$-connected graph. Let ${v \in V(G)}$ and $S \subseteq V(G) \setminus \{v\}$. If $|S|\geq k$, then there exists a set of $k$ $v$-$S$ internally disjoint paths. Moreover, every two paths in this set have $\{v\}$ as their intersection.* **Proposition 15** ([@Dirac52 Theorems 3 and 4]). *If $G$ is a 2-connected graph on $n$ vertices with minimum degree $k$, then $G$ has a longest cycle of length at least $\min\{2k,n\}$.* We are now ready to show the main result of this section. **Theorem 16**. *For any $k\geq 2$, Conjecture $C(k,\min\{k,8k-n-16\})$ is true.* *Proof.* Let $G$ be a $k$-connected graph on $n$ vertices. Let $C$ and $D$ be two longest cycles in $G$. Let $L=|C|$ and $X=V(C) \cap V(D)$. Let $R$ be a longest path in a component of $G-C$ or $G-D$ with length at most two. Furthermore, if $R \in G-C$ ($G-D$), then we choose $R$ as a path that maximizes $|E(R) \cap E(D)|$ ($|E(R) \cap E(C)|$). Without loss of generality, let us suppose that $R \in G-C$, and let $p$ and $r$ be the ends of $R$. By Proposition [Proposition 14](#prop:fan-lemma){reference-type="ref" reference="prop:fan-lemma"}, as $|V(C)| \geq k$, there exists a set, of $k$ $p$-$V(C)$ internally disjoint paths that end at different vertices of $C$. As at most $|R|$ of these paths includes a vertex in $V(R) \setminus \{p\}$, there is a set, say $\mathcal{A}$, of at least $(k-|R|)$ $p$-$V(C)$ internally disjoint paths that end at different vertices of $C$ and not contain any vertex in $V(R) \setminus \{p\}$. Analogously, there is a set, say $\mathcal{B}$, of least $(k-|R|)$ $r$-$V(C)$ internally disjoint paths that end at different vertices of $D$ and not contain any vertex of in $V(R) \setminus \{r\}$. Let $F$ be the set of ends of $\mathcal{A} \cup \mathcal{B}$ in $C$. We construct and auxiliary weighted graph $C^{*}$ as follows. The vertex set of $C^{*}$ is $F$. Two vertices $u$ and $v$ are adjacent in $C^{*}$ if $C[u,v]$ does not contain any other vertex from $F$ besides $u$ and $v$. The weight of such an edge, denote it by $w(uv)$, is $|C[u,v]|$. It is clear that $L=w(C^{*})$. Given a vertex $u \in V(C^{*})$, we say that $u$ is $A$-*colored* ($B$-*colored*) if $u$ is an end of a path in $\mathcal{A}$ ($\mathcal{B}$). If $u$ is $A$-colored and $B$-colored, we say that $u$ is *bicolored*. Moreover, we say that an edge $uv \in E(C^{*})$ is *bicolored* if some of $u$ or $v$ is bicolored. Let $F'$ be the set of bicolored vertices, and let $C'$ be the set of bicolored edges. The following claim is clear from the definition of $F'$ and $C'$. **Claim 17**. *$|C'| \geq |F'|$.* Let $F''=F \setminus F'$ and $C''=E(C^*) \setminus C'$. **Claim 18**. *Let $uv \in E(C^{*})$. If $uv \in C''$ then $w(uv) \geq 2$. If $uv \in C'$ then $w(uv) \geq 2+|R|$.* *Proof.* First suppose that $uv \in C''$. Then, neither of $u$ or $v$ is bicolored. Hence, without loss of generality, either $u$ and $v$ are $A$-colored or $u$ is $A$-colored and $v$ is $B$-colored. For the first case, let $P_u$ and $P_v$ be the corresponding paths in $\mathcal{A}$. As $C-C[u,v]+P_u+P_v$ is also a cycle, we must have $|C[u,v]| \geq 2$. For the second case, let $P_u$ and $Q_v$ be the corresponding paths in $\mathcal{A}$ and $\mathcal{B}$, respectively. If $P_u$ and $Q_v$ are disjoint, then $C-C[u,v]+P_u+R+Q_v$ is also a cycle, so we must have $|C[u,v]| \geq 2+|R| \geq 2$. Otherwise, let $\tilde{Q}_v$ be the shortest subpath of $Q_v$ with $v$ as one of its ends, and the other end, say $x$, in $P_u$. As $u$ is not $B$-colored, then $x \neq u$. Let $\tilde{P}_u$ be the subpath of $P_u$ with ends $u$ and $x$. As $C-C[u,v]+\tilde{P}_u + \tilde{Q}_v$ is also a cycle, we must have $|C[u,v]| \geq 2$. Now suppose that $uv \in E(C')$, and, without loss of generality, that $u$ is $A$-colored and $v$ is bicolored. Let $P_u,P_v$ and $Q_v$ be the corresponding paths in $\mathcal{A}$, $\mathcal{A}$ and $\mathcal{B}$, respectively. If $P_u$ and $Q_v$ are disjoint, then $C-C[u,v]+P_u+R+Q_v$ is also a cycle, so we must have $|C[u,v]| \geq 2+|R|$. Otherwise, let $\tilde{Q}_v$ be the shortest subpath of $Q_v$ with $r$ as one of its ends, and the other end, say $x$ in $P_u \cup P_v$. If $x \in P_u$, let $\tilde{P}_u$ be the subpath of $P_u$ with ends $u$ and $x$. As $C-C[u,v]+\tilde{P}_u + \tilde{Q}_v+R +P_v$ is also a cycle, we must have $|C[u,v]| \geq 3+|R|$. Otherwise, let $\tilde{P}_v$ be the subpath of $P_v$ with ends $v$ and $x$. As $C-C[u,v]+\tilde{P}_v + \tilde{Q}_v+R +P_u$ is also a cycle, we must have $|C[u,v]| \geq 3+|R|$. ◻ **Claim 19**. *$|E(C^*)|=|\mathcal{A}|+|\mathcal{B}|-|F'|$.* *Proof.* Note that $|E(C^*)|=|F|$. Let $F_a$ be the set of $A$-colored vertices, and let $F_b$ be the set of $B$-colored vertices. Then $|F|=|F_a| + |F_b| -|F_a \cap F_b|=|\mathcal{A}|+|\mathcal{B}|-|F'|$. ◻ **Claim 20**. *$L \geq 4(k-|R|)+|R||C'|-2|F'|$.* *Proof.* By Claim [Claim 18](#claim:wuvgeq){reference-type="ref" reference="claim:wuvgeq"}, $w(C')\geq (2+|R|)|C'|$ and $w(C'') \geq 2|C''|$. So $$\begin{aligned} L &=&w(C')+w(C'') \\ &\geq& (2+|R|)|C'|+2|C''| \\ &=&2(|C'|+|C''|)+|R||C'|\\ &=&2|E(C^*)|+|R||C'|\\ &=&2|\mathcal{A}|+2|\mathcal{B}|-2|F'|+|R||C'|\\ &\geq& 4(k-|R|)+|R||C'|-2|F'|,\end{aligned}$$ as we want. ◻ By Claim [Claim 20](#claim:Lgeq4(k-2)){reference-type="ref" reference="claim:Lgeq4(k-2)"}, if $|R|=2$, then $L \geq 4(k-2)$. Now, by Proposition [Claim 20](#claim:Lgeq4(k-2)){reference-type="ref" reference="claim:Lgeq4(k-2)"}, $|X|\geq 2L-n$. Thus, $|X| \geq 8k-n-16$, as we want. Hence, from now on, we may assume that $|R|\leq 1$. This condition will also imply that any component of $D-C$ has at most two vertices. Thus, every component of $D-C$ is either an edge or a vertex, and $$\label{eq:3XgeqL} L=|V(C) \cap V(D)|+ |V(D) \setminus V(C)| \leq |X|+2|X| = 3|X|.$$ Suppose for a moment that every component in $D-C$ is a vertex. By a similar reasoning to the previous paragraph, $L\leq 2|X|$. By Proposition [Proposition 15](#prop:dirac){reference-type="ref" reference="prop:dirac"}, $L \geq \min\{2k,n\}$. If $\min\{2k,n\}=n$, then $L=n$ and $G$ is hamiltonian, so $|X|=n=k$ and we are done; otherwise, $|X|\geq L/2 \geq k$ and we are also done. Hence, there exists an edge in $D-C$ and, by the choose of $R$, $R$ is an edge in $D-C$. From now on, if $uv \in E(C^{*})$ has a vertex in $X$ we say that $uv$ is *covered*. **Claim 21**. *Let $u \in V(C^{*})$. If $u$ is bicolored, then it is also covered. Moreover, if $v$ is a neighbor of $u$ in $C^{*}$, then $w(uv) \geq 3$.* *Proof.* Let $P_u$ and $Q_u$ be the corresponding paths in $\mathcal{A}$, and $\mathcal{B}$, respectively. If $|P_u|>1$, then there exists a path of length two in $G-C$, a contradiction to the choise of $R$. Hence $|P_u|=1$, and, analogously, $|Q_u|=1$. Suppose by contradiction that $u$ is not covered. In that case, $D-R+P_u+Q_u$ is a cycle longer than $L$, a contradiction. This proves the first part of the claim. For the second part, suppose, without loss of generality, that $v$ is $A$-colored, and let $P_v \in \mathcal{A}$ the corresponding path. As before, we can show that $|P_v|=1$. Hence, as $C-C[u,v]+Q_u+R+P_v$ is a cycle, we must have $|C[u,v]|\geq 3$. ◻ Suppose for a moment that $|F'|=0$. In that case, $|C'|=0$, so, by Claim [Claim 20](#claim:Lgeq4(k-2)){reference-type="ref" reference="claim:Lgeq4(k-2)"}, $L \geq 4(k-1)$. If $k\leq 3$, then the proof follows by Theorem [Theorem 7](#thm:Ckk2a7){reference-type="ref" reference="thm:Ckk2a7"}. Hence, we may assume that $k\geq 4$, which implies that $L \geq 3k$, and, as $3X\geq L$, the proof follows. Thus, from now on, we may assume that $|F'|>0$. **Claim 22**. *If $|C'| = |F'|$ then $|X| \geq k$.* *Proof.* Suppose that $|C'|=|F'|$. In that case, as $|F'|>0$, every vertex of $F$ is bicolored, so $F=F'$. As $|R|=1$, we have $|\mathcal{A}|\in \{k-1,k\}$. If $|\mathcal{A}|=k$ then, as every vertex of $F$ is bicolored, we have $|F|=|\mathcal{A}|=k$. By Claim [Claim 21](#claim:bicolimplcovmoreover){reference-type="ref" reference="claim:bicolimplcovmoreover"}, $|X| \geq |F|=k$ and the proof follows. Hence, from now on, we may assume that $|\mathcal{A}|=k-1$. Then, there exists a $p$-$V(C)$ path, say $\tilde{P}$, that contains $r$ and does not end in any vertex of $F$. Let $w$ be the other end of $\tilde{P}$ and let $\tilde{P}_{qw}$ be the subpath of $\tilde{P}$ with ends $r$ and $w$. Let $u$ and $v$ be the vertices of $F$ such that $r\in C[u,v]$. Let $P_u$ and $Q_v$ be the corresponding paths in $\mathcal{A}$ and $\mathcal{B}$, respectively. Then, as $C-C[u,w]+P_u+R+ \tilde{P}_{qw}$ is a path, $|C[u,w]|\geq 3$ So $|C[u,v]| \geq 4$ and there exists a path in $C-D$ with size at least 2, a contradiction to the choice of $R$. ◻ By Claim [Claim 21](#claim:bicolimplcovmoreover){reference-type="ref" reference="claim:bicolimplcovmoreover"}, $|X| \geq |C'|$ and $w(C')\geq 3|C'|$. By Claim [Claim 18](#claim:wuvgeq){reference-type="ref" reference="claim:wuvgeq"}, $w(C'') \geq 2|C''|$. Also, we may assume that $|F'| \leq |C'|-1$ by Claim [Claim 22](#claim:C'>F'){reference-type="ref" reference="claim:C'>F'"}. So, as $|R|=1$, by Claim [Claim 20](#claim:Lgeq4(k-2)){reference-type="ref" reference="claim:Lgeq4(k-2)"} we have that $$\begin{aligned} L &\geq& 4(k-1)+|C'|-2|F'| \\ &\geq& 4k-4+2-|C'| \\ &\geq& 4k-4+2-|X|.\end{aligned}$$ As $3|X| \geq L$ by [\[eq:3XgeqL\]](#eq:3XgeqL){reference-type="eqref" reference="eq:3XgeqL"}, we have $|X| \geq k-1/2$, which finishes the proof. ◻ # Concluding remarks {#section:conclusion} In this paper, we show that every pair of longest cycles in a $k$-connected graph intersect each other in at least $\min\{k,8k-n-16\}$ vertices, and that every pair of longest paths intersect each other in at least $\min\{k,8k-n-9\}$ vertices. A direct corollary of these results is that, if $k \geq (n+16)/7$, then every pair of longest cycles intersect in at least $k$ vertices and, if $k \geq (n+9)/7$, then every pair of longest paths intersect in at least $k$ vertices. In [@Gutierrez2021-Pro], a set of conjectures, called CHC($r$), affirm that if $k \geq n/r$, then every pair of longest paths intersect in at least $k$ vertices. In this paper, we progress towards the case $r=7$. We believe the techniques presented here can be used for proving the cases when $r>7$. [^1]: If we try to be complete formal, we should add the number of vertices as a third parameter to our familes of conjectures, but we will prefer to not complicate the notation.

arxiv_math

--- abstract: | Our work is motivated by the connection established between Lorentzian polynomials and the Dressian in the seminal work of Brändén and Huh on Lorentzian polynomials. We analyze this relation for the class of positroids, and are able to show that in this case, we can relate a multiaffine homogenous stable polynomial to it. Additionally, we also highlight that a conjecture for matroids posed by Brändén and Huh is true when considered over the class of Rayleigh matroids which strictly contain the class of positroids. We collect these findings along with other results for further exploration. address: author: - Ayush Kumar Tewari bibliography: - biblio.bib title: Positroids, Dressian and Stable polynomials --- # Introduction The combinatorics associated with electrical networks carries a lot of really intricate details related to matroids, an initial documentation of which can be found in [@recski1989matroid]. In subsequent studies [@choe2004homogeneous] the relation between matroids and the geometry of polynomials is described. In recent work [@branden2020lorentzian] with the introduction of the notion of *Lorentzian polynomials* it has been demonstrated that highly intrinsic properties concerning matroids can be obtained via findings that are based on the geometry of polynomials. For example, the well-known Mason conjecture for matroids can be resolved by methods developed in [@branden2020lorentzian; @huh2021correlation]. These results are also extended to show negative dependence properties for the $q$-state Potts model partition function [@branden2018hodge]. In [@choe2004homogeneous] the authors are interested in studying the *half-plane property* for multivariate polynomials. For matroids, this is equivalent to the case when the basis generating polynomial is *stable*. The authors in [@choe2004homogeneous] also discuss the various examples of matroids that satisfy the half-plane matroids: uniform matroids, sixth-root of unity matroids, and all matroids of rank and corank two satisfy the half-plane property. Moreover, in recent work [@kummer2022matroids] the authors provide a complete classification of which matroids on at most $8$ elements have the half-plane property. These investigations have also now forayed into asking slightly refined formulations, for example, the *Rayleigh* property for polynomials which captures negative dependence correlation between elements of the bases in a matroid. In [@choe2006rayleigh] the authors introduce the notion of *Rayleigh* matroid, which is the class of matroids whose basis generating polynomials is Rayleigh. They also provide examples of matroids which are Rayleigh: regular matroids and all matroids of rank three are Rayleigh. In this paper, our emphasis is on the class of *positroids*, which in essence captures positivity in the class of matroids. In one of the first results, we build on the work done in [@purbhoo2018total] in which they show that a certain multi affine homogenous polynomial over $\mathbb{C}$ which *represents* a point in the Grassmannian is stable if and only if this point represents a nonegative point which corresponds to a positroid. We extend this result over real and real closed fields, and use this to refine the characterization of points in the Dressian of a positroid in the form of this result **Theorem 1**. *Let $V$ represent a nonegative point in $Gr(k,n)$ and let $\mathcal{M}$ be a positroid of rank $k$ on $[n]$ elements with bases $\mathcal{B}$ that corresponds to $V$. Let $p_{B}$ represent the nonzero *Plücker coordinates* of $V$, which also satisfy the quadratic Plücker relations. Then for $\mu \in \text{Trop}\>Gr(\mathcal{M}) \subseteq Dr(\mathcal{M})$, the multi affine homogenous stable polynomial $f$ that *represents* $V$, $$f = \sum_{B \in \mathcal{B}} p_{B} \> x^{B}$$ in $\mathbb{K}[x_{1} , \hdots , x_{n}]$ is the one whose tropicalization is $\mu$.* This result is sort of intermediate between the weak half-plane property and strong half-plane property; in the sense that in the case of the strong plane property, the polynomial being considered is fixed and it is the basis generating polynomial of the matroid, whereas, in the case of the weak half-plane property, it guarantees only the existence of a stable polynomial whose support is the bases of the matroid. Additionally, we also share an observation to highlight that Conjecture 3.12 in [@branden2020lorentzian] is true when considered over the class of Rayleigh matroids [@choe2006rayleigh]. This follows by the definition used in [@choe2006rayleigh], and as this class of matroids strictly contains the class of positroids [@marcott2016positroids], we refurbish this result in the following form, **Corollary 2**. *The following conditions are equivalent for any non-empty $J \subseteq \{0, 1\}^{n}$ :* 1. *$J$ is the set of bases of a Rayleigh matroid on $[n]$.* 2. *The generating function $f_{J}$ is a homogeneous $1-$Rayleigh polynomial.* *Essentially, Conjecture 3.12 [@branden2020lorentzian] is true for the class of Rayleigh matroids.* We conclude with an extension of results relating Lorentzian polynomials and Dressians to the case of Flag Dressians and we also list some problems which we aim to work on in the future. # Preliminaries We try to introduce the basic notions in this section which we use in our subsequent sections. We refer the reader to [@oxley] for details on matroid theory. A *matroid* $\mathcal{M}$ of rank $k$ on the set $E$ is a nonempty collection $\mathcal{B} \subseteq \binom{E}{k}$ of $k$-element subsets of $E$, called *bases* of $\mathcal{M}$, that satisfies the basis exchange axiom: For any $I , J \in \mathcal{B}$ and $a \in I$, there exists $b \in J$ such that $I \setminus \{ a \} \cup \{ b \} \in \mathcal{B}$. A matroid is called *representable* if it can be represented by columns of a matrix over some field $\mathbb{K}$. We index the columns of a $k \times n$ matrix by the set $[n]$. The generating function of a matroid $\mathcal{M}$ with bases $\mathcal{B}$ (also referred to as the basis enumeration polynomials in [@marcott2016positroids]) is defined as the polynomial $$\label{eq:gen_func_matroid} M(x) = \sum_{B \in \mathcal{B}} x^{B}$$ A slightly general notion of *generating function* of a subset $J \subseteq \mathbb{N}^{n}$, is defined as follows $$f_{J} = \sum_{\alpha \in J} \frac{x^{\alpha}}{\alpha!} \quad \text{where} \quad \alpha! = \prod_{i=1}^{n} \alpha_{i}!$$ and we see that these two notions agree when $J \subseteq \{0, 1\}^{n}$ . The *Grassmannian* $Gr(k,n)$ is the parameterization of the family of all $k$-dimensional subspaces of $n$-dimensional vector space in $\mathbb{K}^{n}$. It enjoys a smooth projective variety structure, corresponding to the vanishing set of the *Plücker ideal*. An element $V$ in the Grassmannian $Gr(k,n)$ is understood as a collection of $n$ vectors $v_{1}, \hdots, v_{n} \in \mathbb{K}^{k}$ spanning the space $\mathbb{K}^{k}$ modulo the simultaneous action of $GL(k,n)$. Let $A$ be a $k \times n$-matrix consisting of column vectors $v_1, v_2, \ldots, v_n$. We call $A$, a *representative* of $V$. This defines a matroid $\mathcal{M}_V$ whose bases are the $k$-subsets $I \subseteq [n]$ such that $\text{det}(A[I]) \neq 0$. It is important to note that this is independent of the choice of $A$, and only depends on $V$. Here, $\text{det}(A[I])$ denotes the determinant of the $k \times k$ submatrix of $A$ with the column set $I \in \binom{[n]}{k}$. A *positroid* $P$ of rank $k$ is a matroid that can be represented by a $k \times n$-matrix $A$ such that the maximal minor $p_I$ is non-negative for each $I \in \binom{[n]}{k}$. We now introduce the definition of *Lorentzian* polynomials [@branden2020lorentzian]. We mostly rely on [@branden2020lorentzian] and [@dcamurota] for our definitions. Let $n$ and $d$ be nonnegative integers, and set $[n] = \{1, \hdots , n\}$ and $H^{d}_{n}$ to be the set of degree $d$ homogeneous polynomials in $\mathbb{R}[w_{1} , \hdots , w_{n}]$. Let $f$ be a polynomial in $\mathbb{R}[w_{1} , \cdots , w_{n}]$, $$f(x) = \sum_{\alpha \in \mathbb{N}^{n}} \frac{c_{\alpha}}{\alpha!} x^{\alpha}$$ The *support* of the polynomial $f$ is the subset of $\mathbb{N}^{n}$ n defined by $$\text{supp}(f) = \{\alpha \in \mathbb{N}^{n} \>\> | \>\> c_{\alpha} \neq 0 \}$$ Let $P^{d}_{n} \subset H^{d}_{n}$ be the open subset of polynomials all of whose coefficients are positive. The *Hessian* of $f \in \mathbb{R}[w_{1}, \cdots , w_{n}]$ is the symmetric matrix $$\mathcal{H}_{f}(w) = (\partial_{i} \partial_{j}f )^{n}_{i,j=1}$$ where $\partial$ denotes the partial derivative $\frac{\partial}{\partial w_{i}}$. **Definition 3**. Set $L^{0}_{n} = P^{0}_{n}, L^{1}_{n} = P^{1}_{n}$ , and $$L^{2}_{n} = \{ f \in P^{2}_{n} \> \> | \> \> \mathcal{H}_{f} \> \> \text{is nonsingular and has exactly one positive eigenvalue} \}$$ For $d\geq 2$, $L^{d}_{n}$ is defined recursively as follows $$L^{d}_{n} = \{ f \in P^{d}_{n} \> \> | \> \> \partial_{i}f \in L^{d-1}_{n} \> \> \text{for all} \> \> i \in [n] \}$$ The polynomials in $L^{d}_{n}$ are called *strictly Lorentzian*, and the limits of strictly Lorentzian polynomials are called *Lorentzian*. $L^{d}_{n}$ can also be identified with the set of $n \times n$ symmetric matrices with positive entries that have the *Lorentzian signature* $(+, -, . . . , -)$. **Definition 4**. A subset $J \subset \mathbb{N}^{n}$ to be $M$-convex if it satisfies any one of the following equivalent conditions: - For any $\alpha, \beta \in J$ and any index $i$ satisfying $\alpha_{i} > \beta_{i}$, there is an index $j$ satisfying $$\alpha_{j} < \beta_{j} \quad \text{and} \quad \alpha - e_{i} + e_{j} \in J$$ - For any $\alpha, \beta \in J$ and any index $i$ satisfying $\alpha_{i} > \beta_{i}$, there is an index $j$ satisfying $$\alpha_{j} < \beta_{j} \quad \text{and} \quad \alpha - e_{i} + e_{j} \in J \quad \text{and} \quad \beta - e_{i} + e_{j} \in J$$ Let $\mu$ be a function from $\mathbb{N}^{n}$ to $\mathbb{R} \cup \{\infty\}$. The effective domain of $\mu$ is, by definition, $$\text{dom}(\mu) = \{ \alpha \in \mathbb{N}^{n} \>\> | \>\> \mu(\alpha) < \infty \}$$ **Definition 5**. A function $\mu$ from $\mathbb{N}^{n}$ to $\mathbb{R} \cup \{\infty\}$ is said to be $M$-convex if it satisfies the *symmetric exchange property*: - For any $\alpha, \beta \in \text{dom}(\mu)$ and any $i$ satisfying $\alpha_{i} > \beta_{i}$ , there is $j$ satisfying $$\alpha_{j} < \beta_{j} \quad \text{and} \quad \mu(\alpha) + \mu(\beta) \geq \mu(\alpha - e_{i} + e_{j} ) + \mu(\beta - e_{j} + e_{i} ).$$ The effective domain of an $M$-convex function on $\mathbb{N}^{n}$ is an $M$-convex subset of $\mathbb{N}^{n}$. When the effective domain of a function $\mu$ is is $M$-convex, the symmetric exchange property for $\mu$ is equivalent to the following *local exchange property*, - For any $\alpha, \beta \in \text{dom}(\mu)$ with $|\alpha - \beta|_{1} = 4$, there exist $i$ and $j$ satisfying $$\alpha_{i} > \beta_{i} , \alpha_{j} < \beta_{j} \quad \text{and} \quad \mu(\alpha) + \mu(\beta) \geq \mu(\alpha - e_{i} + e_{j} ) + \mu(\beta - e_{j} + e_{i} ).$$ A function $\mu : \mathbb{N}^{n} \rightarrow \mathbb{R} \cup \{\infty\}$ is said to be $M$-concave if $-\mu$ is $M$-convex. The effective domain of an M-concave function $\mu$ is, $$\text{dom}(\mu) = \{ \alpha \in \mathbb{N}^{n} \>\> | \>\> \mu(\alpha) > -\infty \}$$ A *valuated matroid* on $[n]$ is an $M$-concave function on $\mathbb{N}^{n}$ whose effective domain is a nonempty subset of $\{0, 1\}^{n}$ . The effective domain of a valuated matroid $\mu$ on $[n]$ is the set of bases of a matroid on $[n]$, the *underlying matroid* of $\mu$. Let $M^{d}_{n}(\mathbb{K})$ denote the set of all all degree $d$ homogeneous polynomials in $K_{\geq 0}[w_{1} , \cdots, w_{n}]$ whose support is $M$-convex. Although, the standard definition of Lorentzian polynomials is over $\mathbb{R}$ in [@branden2020lorentzian] itself the authors provide a definition over any field $\mathbb{K}$ as follows, **Definition 6**. Set $L^{0}_{n}(\mathbb{K}) = M^{0}_{n}(\mathbb{K}), L^{1}_{n}(\mathbb{K}) = M^{1}_{n}(\mathbb{K})$, and $$L^{2}_{n}(\mathbb{K}) = \{ f_{t} \in M^{2}_{n}(\mathbb{K}) \>\> | \>\> \text{The Hessian of} \>\> f_{t} \>\> \text{has at most one eigenvalue in} \>\> \mathbb{K}_{\geq 0} \}$$ For $d \geq 3$, we define $L^{d}_{n}(\mathbb{K})$ as follows $$L^{d}_{n}(\mathbb{K}) = \{ f_{t} \in M^{d}_{n}(\mathbb{K}) \>\> | \>\> \partial^{\alpha}f_{t} \in L^{2}_{n}(\mathbb{K}) \>\> \text{for all} \>\> \alpha \in \Delta^{d-2}_{n} \}$$ where $\Delta^{d-2}_{n}$ represents the discrete $(d-2)-$th simplex. We now discuss the background of the definition of stable polynomials and how they are related to the study of Lorentzian polynomials. In earlier works, the definition of stable polynomials is referred to as the *half plane property* [@choe2004homogeneous]. Let $\mathcal{H} = \{ x \in \mathbb{C} \>\> | \>\> \text{Re} x > 0 \}$ denote the open upper half plane. **Definition 7**. A polynomial $f$ in $\mathbb{R}[w_{1} , \cdots , w_{n}]$ is called *stable* if $f$ is nonvanishing on $\mathcal{H}^{n}$ or identically zero. In this case, $f$ is also said to satisfy the *half plane property*. *Remark 8*. In [@choe2004homogeneous], the half-plane property is also defined for a matroid. $\mathcal{M}$ is said to satisfy the *half plane property* if the basis generating polynomial of $\mathcal{M}$ satisfies the half-plane property. Additionally, $\mathcal{M}$ is said to satisfy the *weak half plane* property if there exists a polynomial that satisfies the half-plane property and whose support is the basis of $\mathcal{M}$ [@choe2004homogeneous]. Let $S^{d}_{n}$ be the set of degree $d$ homogeneous stable polynomials in $n$ variables with nonnegative coefficients and by Hurwitz Theorem $S^{d}_{n} \subset H^{d}_{n}$. We refer the reader to [@wagner2011multivariate] to explore the rich theory of stable polynomials. Any polynomial $f \in S^{d}_{n}$ is the limit of *strictly stable polynomials* [@nuij1968note]. Lorentzian polynomials are a generalization of stable polynomials, i.e., all stable polynomials are Lorentzian, moreover, any homogeneous stable polynomial is a constant multiple of a Lorentzian polynomial [@branden2020lorentzian]. In [@choe2004homogeneous] the authors envisaged that for large classes of matroids, the *basis generating polynomials* are stable. Subsequent studies also concentrate on certain relaxations of this class, the most prominent of them being *Rayleigh* polynomials, **Definition 9**. Let $c$ be a fixed positive real number, and let $f$ be a polynomial in $\mathbb{R}[w_{1} , \cdots , w_{n}]$. $f$ is called $c$-*Rayleigh* if $f$ has nonnegative coefficients and $$\partial^{\alpha}f(x) \> \partial^{\alpha+e_{i}+e_{j}}f(x) \leq c \> \partial^{\alpha+e_{i}}f(x) \> \partial^{\alpha+e_{j}}f(x) \quad \text{for all} \>\> i,j \in [n], \alpha \in \mathbb{N}^{n}, x \in \mathbb{R}^{n}_{\geq 0}$$ When $f$ is multi-affine, that is, when $f$ has degree at most one in each variable, the $c$-Rayleigh condition for $f$ is equivalent to $$f(x) \partial^{i} \partial^{j}f(x) \leq c \>\> \partial^{i}f(x) \partial^{j}f(x) \>\> \text{for all distinct} \>\> i,j \in \mathbb{N}^{n}, \>\> \text{and} \>\> x \in \mathbb{R}^{n}_{\geq 0}$$ A multi-affine polynomial $f$ is said to be *strongly Rayleigh* if $$f(x) \partial^{i} \partial^{j}f(x) \leq c \>\> \partial^{i}f(x) \partial^{j}f(x) \>\> \text{for all distinct} \>\> i,j \in \mathbb{N}^{n}, \>\> \text{and} \>\> x \in \mathbb{R}^{n}$$ A multi-affine polynomial is stable if and only if it is strongly Rayleigh [@branden2007polynomials Theorem 5.6]. Based on the definition of Rayleigh polynomials, the notion of *Rayleigh* matroids [@choe2006rayleigh] was introduced, where a matroid $\mathcal{M}$ is a Rayleigh matroid if its basis generating polynomial is Rayleigh. Similarly, there also exists the class of *strongly Rayleigh matroid*, where a matroid is a *strongly Rayleigh matroid* if its basis generating polynomial is strongly Rayleigh. It is straightforward to see from [@branden2007polynomials Theorem 5.6] that all strongly Rayleigh matroids satisfy the half-plane property. We now define notions related to tropical geometry that appear in our discussion and we refer the reader to [@maclagan2021introduction] for further details. Tropical geometry is the study of vanishing sets of polynomials defined over the *tropical* semiring $\mathbb{T} = \{\mathbb{R} \cup \{-\infty\}, \text{max} = \oplus, + = \odot\}$. A *tropical* polynomial is a polynomial defined over $\mathbb{T}$ with the binary operations replaced by $\oplus$ and $\odot$. $\text{Trop}(f)$ denotes the *tropical hypersurface* associated with $f$ which is the collection of all points where the maxima is achieved at least twice. The *tropical Grassmannian* $\text{TropGr}(k,n)$ is the intersection of the tropical hypersurfaces $\text{Trop}(f)$, where $f$ ranges over all elements of the *Plücker ideal* $\mathcal{I}_{k,n}$ which is generated by the *quadratic Plücker relations*, and therefore it is also a tropical variety [@maclagan2021introduction]. The *Dressian* $Dr(k,n)$ is the intersection of the tropical hypersurfaces $\text{Trop}(f)$, where $f$ ranges over all three-term Plücker relations, that generate the *Plücker ideal* and hence it possesses the structure of a tropical prevariety [@maclagan2021introduction]. The underlying matroid for the definitions of the tropical Grassmannian and Dressian is the *uniform matroid* $\mathcal{U}_{k,n}$. However, the notion of Dressian has been extended to arbitrary matroids with the idea of the *local Dressian*. The *local Dressian* $Dr(M)$ is defined as the tropical pre-variety given by the set of quadrics obtained from the three-term Plücker relations by setting the variables $p_{B}$ to zero, where $B$ is not a basis of $M$ [@olarte2019local]. *Remark 10*. We acknowledge that in some instances, all Plücker relations (not necessarily only the 3-term Plücker relations) are considered for the definition of the Dressian. However, by the work in [@baker2019matroids] we know that these two notions coincide over the tropical hyperfield and hence our definition would remain consistent for our discussion. # Dressian, Stable and Lorentzian Polynomials Since the introduction of the notion of a Dressian, it has been noted that the points residing in the Dressian enjoy multiple avatars in which they can be treated and satisfy multiple properties. The subject for initial findings was the Dressian $Dr(k,n)$ with the underlying matroid being the uniform matroid $\mathcal{U}_{k,n}$. We collect the various previously known [@maclagan2021introduction; @speyer2008tropical; @branden2020lorentzian; @tewari2022generalized] equivalent notions for a point residing in the Dressian $Dr(k,n)$ in the form of Theorem [Theorem 11](#thm:Dressian){reference-type="ref" reference="thm:Dressian"}. **Theorem 11**. *Let $\mu$ be a point in $Dr(k,n)$. Then the following are equivalent* 1. *$\mu$ is a valuated matroid with the underlying matroid $\mathcal{U}_{k,n}$.* 2. *$\mu$ defines a $M-convex$ function on the matroid $\mathcal{U}_{k,n}$.* 3. *$\mu$ satisfies the tropical three-term Plücker relations.* 4. *$\mu$ as a weight vector induces a regular matroidal subdivision of $\Delta(k,n)$.* 5. *$\mu = \text{trop}(f_{t})$, where $f_{t}$ is a Lorentzial polynomial defined on the bases set $\mathcal{B}$ of $\mathcal{U}_{k,n}$.* 6. *$\mu$ defines a metric for a generalised metric tree arrangement.* These equivalences can also be extended to the local Dressian $Dr(\mathcal{M})$ [@olarte2019local], which can be defined for an arbitrary matroid $\mathcal{M}$, and it generalizes the notion of Dressian $Dr(k,n)$, and we list the corresponding equivalences for a point residing in the local Dressian in Theorem [Theorem 12](#thm:Dressian_local){reference-type="ref" reference="thm:Dressian_local"}, **Theorem 12**. *Let $\mu$ be a point in $Dr(\mathcal{M})$. Then the following are equivalent* 1. *$\mu$ is a valuated matroid with the underlying matroid $\mathcal{M}$.* 2. *$\mu$ defines a $M-convex$ function on the matroid $\mathcal{M}$.* 3. *$\mu$ satisfies the tropical three-term Plücker relations.* 4. *$\mu$ as a weight vector induces a regular matroidal subdivision of $\mathcal{P}_{M}$.* 5. *$\mu = \text{trop}(f_{t})$, where $f_{t}$ is a Lorentzial polynomial defined on the bases set $\mathcal{B}$ of $\mathcal{M}$.* The categorization given in (v) is one of the most recent ones, proven in [@branden2020lorentzian Theorem 3.20]. Our aim now is to possibly refine the characterization of the Dressian, concerning Lorentzian polynomials. We know that the class of Loentzian polynomials contains the class of stable polynomials [@branden2020lorentzian]. Of interest in this regard is the work in [@purbhoo2018total], in which the author relates the notions of stability and total nonnegativity. We first briefly describe the setup used in [@purbhoo2018total] to make ideas clearer. Consider the matrix $M \in \text{Mat(k,n)}$ of rank $k$ over the field of complex number. The column space of the matrix provides us a k-dimensional subspace $V$ which lies in the $Gr(k,n)$. Let $M[I]$ denote the $k \times k$ submatrix of $M$ with row set $I \in \binom{n}{k}$. The *Plücker coordinates* of $V$ are the maximal minors $\Bigl\{ \text{det}(M [I]) : I \in \binom{n}{k} \Bigr\}$. The homogenous multi-affine polynomial, $$\label{eq:ply_nonnegative} f = \sum_{I \in \binom{n}{k}} \text{det}(M [I]) \> x^{I}$$ where $x^{I} = \prod_{i \in I} x_{i}$, is said to *represent* $V$ in $Gr(k,n)$ [@purbhoo2018total]. Additionally, a necessary and sufficient condition for a polynomial to represent $V$ is that the coefficients satisfy the quadratic *Plücker* relations, which are the defining equations for the $Gr(k,n)$, and provide it the structure of a projective variety. We recall the following result from [@purbhoo2018total] concerning the multi affine polynomial in Equation [\[eq:ply_nonnegative\]](#eq:ply_nonnegative){reference-type="ref" reference="eq:ply_nonnegative"} **Theorem 13** (Theorem 1.1 [@purbhoo2018total]). *Let $f \in \mathbb{C}[x]$ be a multi-affine homogenous polynomial of degree $k$ that represents a point $V$ in $Gr(k,n)$. Then, $f$ is stable if and only if $V$ is totally nonnegative.* We state the above result over the field of real numbers, **Corollary 14**. *Let $f \in \mathbb{R}[x]$ be a multi-affine homogenous polynomial of degree $k$ that represents a point $V$ in $Gr(k,n)$. Then, $f$ is stable if and only if $V$ is totally nonnegative. Moreover, if $\mathbb{K}$ is a real closed field and $f \in \mathbb{K}[x]$ be a multi-affine homogenous polynomial of degree $k$ that represents a point $V$ in $Gr(k,n)$ then, $f$ is stable if and only if $V$ is totally nonnegative.* *Proof.* With the well-known "phase theorem" [@choe2004homogeneous Theorem 6.1] we know that every homogeneous stable polynomial has non-negative real coefficients up to scalar multiples. Hence, the stability of the polynomial in Theorem [Theorem 13](#thm:complex_stable){reference-type="ref" reference="thm:complex_stable"} over $\mathbb{C}$ is equivalent to stability of over $\mathbb{R}$ [@purbhooemail]. Since, the field of real numbers also allows quantifier elimination, therefore by invoking the Tarski principle as done in [@branden2020lorentzian] , we can extend Corollary [Corollary 14](#cor:real_stable){reference-type="ref" reference="cor:real_stable"} over real closed fields as well. ◻ We also know that $V \in Gr(k,n)$ determines a representable matroid of rank $k$ on the set $[n]$, by taking the bases to be the indices of the nonzero Plücker coordinates. If $V$ is totally nonnegative, this matroid is a positroid [@purbhoo2018total]. With this observation, we want to consider a restriction of a result on matroids [@branden2020lorentzian Theorem 3.20] to the class of positroids. We first recall this result, **Theorem 15** (Theorem 3.20 [@branden2020lorentzian]). *The following conditions are equivalent for any function $\mu : \Delta(k,n) \rightarrow \mathbb{Q} \cup \{ \infty \}$* 1. *the function $\mu$ is M-convex;* 2. *there is a Lorentzian polynomial in $\mathbb{K}[w_{1} , \hdots , w_{n}]$ whose tropicalization is $\mu$.* We also elaborate on the background setup for this statement used in [@branden2020lorentzian]. In this statement, the field $\mathbb{K}$ is the real convergent Puiseux series $$\mathbb{K} = \bigcup_{k \geq 1} \mathbb{R}((t^{1/k}))_{\text{conv}}$$ which is a *real closed field* and its corresponding algebraic closure is the field $$\overline{\mathbb{K}} = \bigcup_{k \geq 1} \mathbb{C}((t^{1/k}))_{\text{conv}}$$ The definition of Lorentzian polynomials over $\mathbb{K}$ are also provided in [@branden2020lorentzian Definition 3.17]. We propose the following refinement of Theorem [Theorem 15](#thm:lorent){reference-type="ref" reference="thm:lorent"}, when we restrict ourselves to the class of positroids, in which case we see that we obtain a relation to homogenous stable polynomials which are a subclass of Lorentzian polynomials. **Theorem 16**. *Let $V$ represent a nonnegative point in $Gr(k,n)$ and let $\mathcal{M}$ be a positroid of rank $k$ on $[n]$ elements with bases $\mathcal{B}$ that corresponds to $V$. Let $p_{B}$ represent the nonzero *Plücker coordinates* of $V$, which also satisfy the quadratic Plücker relations. Then for $\mu \in \text{Trop}\>Gr(\mathcal{M}) \subseteq Dr(\mathcal{M})$, the multi affine homogenous stable polynomial $f$ that *represents* $V$, $$f = \sum_{B \in \mathcal{B}} p_{B} \> x^{B}$$ in $\mathbb{K}[x_{1} , \hdots , x_{n}]$ is the one whose tropicalization is $\mu$.* *Proof.* We consider the homogenous multi-affine polynomial, $$f = \sum_{B \in \mathcal{B}} p_{B} \> x^{B}$$ where $x^{B} = \prod_{b \in B} x_{b}$, which *represents* $V$. By Corollary [Corollary 14](#cor:real_stable){reference-type="ref" reference="cor:real_stable"}, this polynomial is stable over $\mathbb{K}$ with coefficients satisfying the quadratic Plücker relations. We also realize that the tropicalization $\text{trop}(f)$ is defined by the tropicalization of the coefficients which are just Plücker coordinates. However, these tropical Plücker relations are the relations that define a point in the tropical Grassmannian $\text{TropGr}(k,n)$. Therefore, we realize that $\text{trop}(f)$ has coefficients that satisfy the tropical Plücker relations, which defines a point $\mu \in \text{TropGr}(k,n) \subseteq Dr(\mathcal{M})$. ◻ We also note that this relation between stable polynomials and positroids does not extend to the class of matroids, for example in [@branden2007polynomials] it is shown that if we consider the matroid $\mathcal{M}$ as the Fano plane then there is no stable polynomial whose support is $B$. However, the existence of a homogenous multi affine stable polynomial over the support of a matroid is known for matroids representable over $\mathbb{C}$ [@choe2004homogeneous Corollary 8.2]. We want to highlight that Theorem [Theorem 16](#thm:positroid_stable){reference-type="ref" reference="thm:positroid_stable"} emphasizes the fact that for the class of positroids, we can actually provide the exact form of the homogenous multi affine stable polynomial whose support is the basis of the positroid and whose tropicalization resides as a point in the Dressian. Another result from [@branden2020lorentzian] that we are interested in studying is the conjecture concerning the generating function of matroids, which the authors state as follows **Conjecture 17** (Conjecture 3.12 [@branden2020lorentzian]). *The following conditions are equivalent for any non-empty $J \subseteq \{0, 1\}^{n}$ :* 1. *$J$ is the set of bases of a matroid on $[n]$.* 2. *The generating function $f_{J}$ is a homogeneous $\frac{8}{7}$ -Rayleigh polynomial.* The constant $\frac{8}{7}$ appearing in this conjecture is strict in the sense that for any positive real number $c < \frac{8}{7}$, there is a matroid whose basis generating function is not $c$-Rayleigh [@huh2021correlation]. We now highlight that this conjecture is true for the class of Rayleigh matroids which strictly contain the class of positroids [@marcott2016positroids] and moreover, a stronger statement is true for them in the form of Theorem [Corollary 18](#thm:positroid_rayleigh){reference-type="ref" reference="thm:positroid_rayleigh"}. **Corollary 18**. *The following conditions are equivalent for any non-empty $J \subseteq \{0, 1\}^{n}$ :* 1. *$J$ is the set of bases of a Rayleigh matroid on $[n]$.* 2. *The generating function $f_{J}$ is a homogeneous $1-$Rayleigh polynomial.* *Essentially, Conjecture [Conjecture 17](#conj:lorent_conj){reference-type="ref" reference="conj:lorent_conj"} is true for the class of Rayleigh matroids.* *Proof.* For the implication (i) $\implies$ (ii) by the definition of Rayleigh matroids in [@choe2006rayleigh], the basis generating function of a Rayleigh matroid is 1-Rayleigh. Also, if $f$ is a 1-Rayleigh polynomial, then $f$ is $c$-Rayleigh polynomial for all $c \geq 1$. Additionally, for the opposite implication i.e., (ii) $\implies$ (i) is straightforward in this case by [@branden2020lorentzian Theorem 3.10], which implies that $J$ is the basis of a matroid, which in turn is Rayleigh by our assumption in this implication. Therefore, this helps us to see that Conjecture [Conjecture 17](#conj:lorent_conj){reference-type="ref" reference="conj:lorent_conj"} is true for the class of Rayleigh matroids. ◻ An interesting observation is that the statement of Corollary [Corollary 18](#thm:positroid_rayleigh){reference-type="ref" reference="thm:positroid_rayleigh"} when considered over the class of positroids, fails to hold true which is confirmed from the example of the *Vamos* matroid, which is not representable over any field and hence is not a positroid, even though it is 1-Rayleigh. This also explains the strict inclusion of positroids in the class of Rayleigh matroids. # Future Work Motivated by Theorem [Theorem 11](#thm:Dressian){reference-type="ref" reference="thm:Dressian"}, we now consider the case of the positive Dressian $Dr^{+}(k,n)$, which is based on the definition of the positive part of a tropical variety introduced in [@speyer2005tropical]. A very significant result pertaining to the positive Dressian is the following [@speyer2021positive], **Theorem 19** (Theorem 3.9 [@speyer2021positive]). *The positive tropical Grassmannian $\text{Trop}^{+} Gr(k,n)$ equals the positive Dressian $Dr^{+}(k,n)$.* A version of this theorem in terms of the local Dressian is proven in [@arkani2021positive] **Theorem 20** (Theorem 9.2 [@arkani2021positive]). *Let $\mathcal{M}$ be a positroid. Then the positive Dressian $Dr^{+}(\mathcal{M})$ equals the positive tropical positroidal cell $\text{Trop}^{+} \Pi_{\mathcal{M}}$.* Theorem [\[thm:Dress_pos\]](#thm:Dress_pos){reference-type="ref" reference="thm:Dress_pos"} collects all previously known equivalent notions [@speyer2021positive; @m=2amplut; @arkani2021positive] of a point residing in the positive Dressian and captures the essence of Theorem [Theorem 11](#thm:Dressian){reference-type="ref" reference="thm:Dressian"}, **Theorem 21**. *Let $\mu$ be a point in $Dr^{+}(d,n)$. Then the following are equivalent* 1. *$\mu$ is a realizable valuated matroid with the underlying positroid $\mathcal{U}_{d,n}$.* 2. *$\mu$ defines a realizable $M-convex$ function.* 3. *$\mu$ satisfies the positive tropical three-term Plücker relations.* 4. *$\mu$ as a weight vector induces a regular positroidal subdivision of $\Delta(d,n)$.* 5. *$\mu = \text{trop}(f_{t})$, where $f_{t}$ is a Lorentzial polynomial defined on the bases set $\mathcal{B}$ of the positroid $\mathcal{U}_{d,n}$.* We now present an extension of this theorem to the case of an arbitrary positroid in Theorem [Theorem 22](#thm:pos_Dress_equiv){reference-type="ref" reference="thm:pos_Dress_equiv"}, which mostly builds on previously known results in [@arkani2021positive; @speyer2021positive]. **Theorem 22**. *Let $\mathcal{M}$ be a positroid and $\mu$ be a point in $Dr^{+}(\mathcal{M})$. Then the following are equivalent* 1. *$\mu$ is a realizable valuated matroid with the underlying positroid $\mathcal{M}$.* 2. *$\mu$ defines a $M-convex$ function on the positroid $\mathcal{M}$.* 3. *$\mu$ satisfies the positive tropical three-term Plücker relations.* 4. *$\mu$ as a weight vector induces a regular positroidal subdivision of $\mathcal{P}_{\mathcal{M}}$.* 5. *$\mu = \text{trop}(f_{t})$, where $f_{t}$ is a Lorentzial polynomial defined on the bases set $\mathcal{B}$ of the positroid $\mathcal{M}$.* *Proof.* The equivalence between (i) $\iff$ (ii) is by definition. The equivalences for (ii) $\iff$ (iii) are established in [@arkani2021positive Proposition 8.2]. We concentrate on proving the equivalence of (i). We first concentrate on proving the equivalence (i) $\iff$ (iii). We already know that a point $\mu \in Dr^{+}(\mathcal{M}) \implies \mu \in Dr(\mathcal{M})$, hence the implication that $\mu$ defines a valuated matroid follows from the inclusion in the Dressian $Dr(\mathcal{M})$, the only nontrivial part being the realizability of $\mu$. The stratification of the tropical positive Grasmmanian for a positroid is discussed in [@arkani2021positive] and it is defined in terms of positroidal cells as $$\label{eq:dress_real} \text{Trop}^{+} \Pi_{\mathcal{M}} = \text{val}(\Tilde{\Pi}_{\mathcal{M}}(\mathcal{R}_{\geq 0})) = Dr^{+}(\mathcal{M})$$ where $\mathcal{R}_{\geq 0}$ represents the positive part of the generalized Puiseux series and $val()$ represents the *valuaton* map. The equality in [\[eq:dress_real\]](#eq:dress_real){reference-type="ref" reference="eq:dress_real"} implies that all positive Plücker vectors are realizable therefore by [@arkani2021positive Theorem 9.2], [@brandt2021tropical Remark 3.1.7] implies that the valuated matroid $\mu$ on the positroid $\mathcal{M}$ is realizable. So (i) $\iff$ (iii). The implication in (v) is straightforward with the inclusion of $\mu \in Dr(\mathcal{M})$ restricted to a positroid. ◻ We also look at a counterpart of the Dressian, defined in the case of flag matroids, the *Flag Dressian* introduced in [@brandt2021tropical] $FlDr(M)$, where $M := (M_{1}, \hdots M_{n})$ is a flag matroid, and it is the tropical variety cut out by the incidence-Plücker relations and the Grassmann-Plücker relations. A classification result similar to Theorem [Theorem 11](#thm:Dressian){reference-type="ref" reference="thm:Dressian"} is proven in [@brandt2021tropical] **Theorem 23** (Theorem A [@brandt2021tropical]). *Let $\mu = (\mu_{1} , \hdots, \mu_{k})$ be a sequence of valuated matroids such that its sequence of underlying matroids $M = (M_{1}, \hdots, M_{k})$ is a flag matroid. Then the following are equivalent:* 1. *$\mu$ is a point on $FlDr(M)$, i.e. it satisfies tropical incidence-Plücker relations and Grassmann-Plücker relations,* 2. *$\mu$ is a valuated flag matroid with underlying flag matroid $M$.* 3. *$\mu$ induces a subdivision of the base polytope of $M$ into base polytopes of flag matroids,* 4. *the projective tropical linear spaces $\overline{trop(\mu_{i})}$ form a flag $\overline{trop(\mu_{1})} \subseteq \hdots \subseteq \overline{trop(\mu_{k})}$* Our first move concerning the Flag Dressian is establishing the equivalence between points in the Flag Dressian and a sequence of Lorentzian polynomials. **Theorem 24**. *Let $\mu = (\mu_{1}, \hdots, \mu_{k})$ be a sequence of valuated matroids such that its sequence of underlying matroids $M = (M_{1}, \hdots, M_{k})$ is a flag matroid. Then the following are equivalent* 1. *$\mu$ is a point on $FlDr(M)$,* 2. *there exist a sequence of Lorentzian polynomials $f_{1}, \hdots f_{k}$ such that $\mu$ = $(\text{trop}(f_{1}) = \mu_{1}, \hdots, \text{trop}(f_{k})= \mu_{k})$, where $(\mu_{1}, \hdots \mu_{k})$ forms a valuated flag matroid and $f_{i}$ is a Lorentzial polynomial defined on the bases set $B_{i}$ of $M_{i}$.* *Proof.* We first consider the implication (i) $\implies$ (ii). Each valuated matroid $\mu_{i}$ corresponds to a M-convex function on $M_{i}$ by definition, and by [@branden2020lorentzian Theorem 3.20] this further corresponds to a Lorentzian polynomial $f_{i}$ such that $\text{trop}(f_{i}) = \mu_{i}$. Therefore, for $\mu \in FlDr(M)$, there exists a sequence of Lorentzian polynomials $f_{1}, \hdots f_{k}$, such that $\mu$ = $(\text{trop}(f_{1}) = \mu_{1}, \hdots, \text{trop}(f_{k}) = \mu_{k})$, where $f_{i}$ is a Lorentzial polynomial defined on the bases set $B_{i}$ of $M_{i}$. We now try to prove the implication (ii) $\implies$ (i). Given a sequence of Lorentzian polynomials such that $\mu$ = $(\text{trop}(f_{1}) = \mu_{1}, \hdots, \text{trop}(f_{k})= \mu_{k})$ is a valuated flag matroid on the underlying matroid $M = (M_{1}, \hdots , M_{k})$, which is equivalent to saying that $\mu \in FlDr(M)$ by [Theorem 23](#thm:FlagDressian_equ){reference-type="ref" reference="thm:FlagDressian_equ"}. ◻ We acknowledge that stipulating that a sequence of flag matroids form a valuated flag matroid is a very strong condition and it might be worthwhile to explore if a refinement of the statement above is possible with weaker conditions. There has been recent work on extending the equivalent conditions for points residing in the Flag Dressian $FlDr(M)$ to the case of the positive Flag Dressian $FlDr^{+}(M)$, specifically in the case of the *complete tropical flag variety* [@boretsky2022polyhedral; @10.1093/imrn/rnac349 Theorem A] and we wish to explore the relation between the positive Flag Dressian and Lorentzian polynomials in this case in future work. Another important aspect closely related to stable polynomials that we want to highlight is their connection with *hyperbolic polynomials* and *positively hyperbolic* varieties, studied in detail in [@borcea2010multivariate] and [@rincon2021positively] respectively. We first recall the definitions of a *hyperbolic polynomial*[@borcea2010multivariate; @amini2018non] and *positively hyperbolic* variety [@rincon2021positively]. **Definition 25**. A homogeneous polynomial $h(x) \in \mathbb{R}[x_{1} , \cdots , x_{n}]$ is hyperbolic with respect to a vector $e \in \mathbb{R}^{n}$ if $h(e) \neq 0$, and if for all $x \in \mathbb{R}^{n}$ the univariate polynomial $t \rightarrow h(te - x)$ has only real zeros. If h is a hyperbolic polynomial of degree d, then we may write $$h(te - x) = h(e) \prod_{j=1}^{d} (t- \lambda_{j}(x))$$ where $$\lambda_{\text{max}}(x) = \lambda_{1}(x) \geq \lambda_{2}(x) \geq \cdots \geq \lambda_{d}(x) = \lambda_{\text{min}}(x)$$ are called the *eigenvalues* of $x$ (with respect to $e$). The *hyperbolicity cone* of $h$ with respect to $e$ is the set $\Lambda_{+}(h, e) = \{ x \in \mathbb{R}^{n} : \lambda_{\text{min}}(x) \geq 0 \}$. We now relate the connection between stable and hyperbolic polynomials [@amini2018non Lemma 2.7], **Lemma 26**. *Let $P \in \mathbb{R}[x_{1} , \cdots , x_{n}]$ be a homogeneous polynomial. Then $P$ is stable if and only if $P$ is hyperbolic with $\mathbb{R}^{n}_{+} \subseteq \Lambda_{+}$.* We wish to explore these connections between hyperbolic polynomials and stable polynomials in the context of the Dressian in future work. **Definition 27**. Let $X \subset \mathbb{C}^{n}$ be a variety which is equidimensional of codimension $c \leq n-1$. X is called *positively hyperbolic* if for every linear subspace $L$ in the positive Grassmannian $Gr^{+}(c, n)$ and every $x \in X$, the imaginary part Im(x) does not belong to $L \setminus \{0\}$. A projective variety in $\mathbb{P}^{n-1}$ is positively hyperbolic if its affine cone in $\mathbb{C}^{n}$ is. A straightforward conclusion from [@rincon2021positively Proposition 2.2] and [@purbhoo2018total Theorem 1.1] is that the variety defined by the homogenous multi-affine polynomial in Equation [\[eq:ply_nonnegative\]](#eq:ply_nonnegative){reference-type="ref" reference="eq:ply_nonnegative"}, which *represents* $V$ in $Gr(k,n)$, is positively hyperbolic if $V$ corresponds to a positroid. Another aspect of our discussion is the classification of various classes of matroids on whether they are Rayleigh, strongly Rayleigh, or if they satisfy the half-plane property. Before moving on, we also briefly discuss the notion of *balanced matroids* [@feder1992balanced] and how they are related to our discussion. **Definition 28**. Let $\mathcal{M}$ be a matroid on the ground set $E$ with basis $\mathcal{B}$. Let $\text{Pr}(e)$ denote the probability of an element $e$ being present in basis element $B$, which is chosen uniformly at random. The matroid $\mathcal{M}(E, \mathcal{B})$ is said to satisfy the *negatively correlated* property if $$\text{Pr}(ef) \leq \text{Pr}(e)\text{Pr}(f)$$ for all pair of distinct $e,f \in E$. **Definition 29**. A matroid $\mathcal{M}$ is said to be *balanced* if all its minors including itself satisfy the negative correlation property. It is clear from the definition that Rayleigh matroids are balanced. In [@choe2006rayleigh] the authors also provide an explicit example of a transversal matroid of rank 4 [@choe2004homogeneous Proposition 5.9] (also updated in [@huh2021correlation]), which is not balanced and hence is also not Rayleigh. We recall the example here. Let $\mathcal{L}$ be a matroid on the ground set $E = \{1,2, \cdots, 10, e, f\}$ and $\mathcal{B}$ denote its bases, where the bases are the four transversals to the sets $\{1, 2, 3, 4, f\} , \{5, 6, 7, f\}, \{8, 9, 10, f\}$ and $\{1, 2, 3, 5, 6, 8, 9, e, f\}$ Equivalently the Rayleigh condition can be restated in terms of probability distribution in the following way [@huh2021correlation] for $B \in \mathcal{B}$ chosen uniformly at random, $$\text{Pr}(i \in B, j \in B) \>\> \text{Pr}( i \not \in B, j \not \in B) \leq \text{Pr}(i \in B, j \not \in B) \>\> \text{Pr}(j \in B, i \not \in B)$$ But, $$\text{Pr}(i \in B, j \in B) \>\> \text{Pr}( i \not \in B, j \not \in B) = 0.04355$$ and $$\text{Pr}(i \in B, j \not \in B) \>\> \text{Pr}(j \in B, i \not \in B) = 0.04298$$ which shows that $L$ is not Rayleigh. However, we want to point out the nuance here that this still does not provide any counterexample to Conjecture [Conjecture 17](#conj:lorent_conj){reference-type="ref" reference="conj:lorent_conj"}, as we see clearly that $L$ is still $\frac{8}{7}$- Rayleigh. Therefore, one of the problems that we want to work on is **Problem 30**. Verify Conjecture [Conjecture 17](#conj:lorent_conj){reference-type="ref" reference="conj:lorent_conj"} for the class of transversal matroids. In the class of transversal matroids, it is already known that lattice path matroids are Rayleigh [@cohen2015lattice]. Hence, we initially aim to consider families of transversal matroids that are not necessarily lattice path matroids, and try to verify if Conjecture [Conjecture 17](#conj:lorent_conj){reference-type="ref" reference="conj:lorent_conj"} holds true for them. Also as previously mentioned the authors in [@choe2004homogeneous] suggested that it might be the case that a large subclass of transversal matroids satisfy the half-plane property. However, since all matroids that satisfy the half-plane property are Rayleigh and now we know that the class of transversal matroids is not Rayleigh, therefore it also does not satisfy the half-plane property. But we wish to modify the problem in the following way, **Problem 31**. Find the largest class of transversal matroids for which it satisfies the half-plane property. Pertaining to positroids, we wish to work on the following problem **Problem 32**. Are positroids strongly Rayleigh? With the complete classification of matroids on at most eight elements in [@kummer2022matroids], one is tempted to search for a counterexample for Problem [Problem 32](#prob:pos_strong_ray){reference-type="ref" reference="prob:pos_strong_ray"} by looking for a positroid in the list of 22 matroids in [@kummer2022matroids Theorem 5.13] which do not satisfy the half-plane property, and hence are not strongly Rayleigh. We did verify that none of these 22 matroids are positroids and hence no counterexample comes from this list. However, we want to highlight that this does not guarantee that a counterexample of Problem [Problem 32](#prob:pos_strong_ray){reference-type="ref" reference="prob:pos_strong_ray"} does not exist in the form of a matroid on eight elements, because the property of being a positroid is not invariant under matroid isomorphisms, and the classification in [@kummer2022matroids] relies on a listing of matroids up to isomorphisms.

arxiv_math

--- abstract: | Derivative-free optimization (DFO) consists in finding the best value of an objective function without relying on derivatives. To tackle such problems, one may build approximate derivatives, using for instance finite-difference estimates. One may also design algorithmic strategies that perform space exploration and seek improvement over the current point. The first type of strategy often provides good performance on smooth problems but at the expense of more function evaluations. The second type is cheaper and typically handles non-smoothness or noise in the objective better. Recently, full-low evaluation methods have been proposed as a hybrid class of DFO algorithms that combine both strategies, respectively denoted as Full-Eval and Low-Eval. In the unconstrained case, these methods showed promising numerical performance. In this paper, we extend the full-low evaluation framework to bound and linearly constrained derivative-free optimization. We derive convergence results for an instance of this framework, that combines finite-difference quasi-Newton steps with probabilistic direct-search steps. The former are projected onto the feasible set, while the latter are defined within tangent cones identified by nearby active constraints. We illustrate the practical performance of our instance on standard linearly constrained problems, that we adapt to introduce noisy evaluations as well as non-smoothness. In all cases, our method performs favorably compared to algorithms that rely solely on Full-eval or Low-eval iterations. author: - "C. W. Royer [^1]" - "O. Sohab[^2]" - "L. N. Vicente[^3]" bibliography: - fle_const.bib title: Full-Low Evaluation Methods For Bound and Linearly Constrained Derivative-Free Optimization --- # Introduction {#sec:intro} Derivative-Free Optimization (DFO) methods are particularly useful for targeting complex optimization problems, where the objective function is computed via numerical simulations, so that derivatives are unavailable for algorithmic purposes. Derivative-free optimization now encompasses a wide range of algorithms, and has found applications in numerous fields of engineering and applied science [@CAudet_WHare_2017; @ARConn_KScheinberg_LNVicente_2009b; @ALCustodio_KScheinberg_LNVicente_2017; @JLarson_MMenickelly_SWild_2019; @LMRios_NVSahinidis_2013]. In these settings, evaluating the objective function represents the main computational bottleneck, that must be accounted for while designing DFO algorithms. Besides, simulation codes often enforce hard constraints on their parameters, typically under the form of bounds or linear relationships, that must be satisfied for the simulation to terminate and for function information to be obtained. Such constraints must also be handled by DFO schemes. Direct-search methods [@TGKolda_RMLewis_VTorczon_2003] are a common choice of derivative-free algorithms due to their ease of implementation. These iterative methods sample new function evaluations along suitably chosen directions at every iteration, in order to find a point at which the function value decreases. Direct-search schemes have been endowed with theoretical guarantees even in presence of non-smooth objectives, while being intrinsically robust to the presence of noise in the function evaluations [@CAudet_WHare_2017]. In presence of linear constraints, direct-search methods generally use directions that conform to the geometry of the feasible set, thereby ensuring feasible descent without relying on derivative information [@kolda2007stationarity]. Recent results have proposed probabilistic variants of direct search, in which the directions are only guaranteed to be feasible descent with a given probability [@gratton2019direct]. A probabilistic direct-search iteration can be performed using a significantly smaller number of directions (and, thus, of function evaluations) than its deterministic counterpart. An alternative to direct-search techniques consists in building an approximate derivative from function evaluations, which then enables the calculation of steps similar to those in the derivative-based setting. Model-based derivative-free techniques obey this logic, and rely on trust-region globalization arguments from nonlinear optimization to guarantee convergence of the methods [@ARConn_KScheinberg_LNVicente_2009b]. As a result, bounds and linear constraints are typically handled in a similar fashion than in the derivative-based setting [@ARConn_NIMGould_PhLToint_2000], even though ad hoc strategies have also been considered [@SGratton_PhLToint_ATroltzsch_2011]. Another widely common approach consists in using finite differences to approximate derivatives, so as to leverage existing algorithms from the derivative-based literature [@JNocedal_SJWright_2006 Chapter 8]. In particular, recent advances in applying quasi-Newton updates using finite-difference gradients have demonstrated good numerical performance, especially in a smooth setting [@ASBerahas_RHByrd_JNocedal_2019; @ASBerahas_OSohab_LNVicente_2022; @HJMShi_MQXuan_FOztoprak_JNocedal_2023]. This performance is mitigated by the inherent cost of finite-difference estimates, that scales at least linearly with the dimension, and thus may be expensive to perform in a simulation-based environment. The full-low evaluation (`Full-Low Evaluation`) framework was recently proposed as a principled way of combining derivative-free steps with different costs and properties [@ASBerahas_OSohab_LNVicente_2022]. In the unconstrained setting, it was proposed to instantiate this framework using a BFGS step computed through finite differences as well as a probabilistic direct-search iteration. This hybrid approach was shown to outperform the individual strategies while being competitive with an established solver on smooth and piecewise smooth problems. In this paper, we propose an extension of the `Full-Low Evaluation` framework that handles bounds and linear constraints by producing feasible iterates and feasible steps. We analyze an instance of this approach that combines projected steps built on finite-difference gradient estimates with direct-search steps based on probabilistic feasible descent. The former (`Full-Eval` step) is considered expensive in terms of evaluations but provides good performance and convergence results in the presence of a smooth objective. The latter (`Low-Eval` step) is cheaper in terms of evaluations, while being more robust to noise or non-smoothness in the objective. Similarly to the unconstrained setting, a switching condition determines the nature of the step taken at each iteration. The rest of this paper is organized as follows. Section [2](#sec:pbtan){reference-type="ref" reference="sec:pbtan"} states our problem of interest, as well as the key geometrical concepts used to design our algorithm. Section [3](#sec:flec){reference-type="ref" reference="sec:flec"} describes our generic `Full-Low Evaluation` framework, as well as the two subroutines that define our instance of interest. Section [4](#sec:cv){reference-type="ref" reference="sec:cv"} provides convergence results for both smooth and non-smooth objectives. Section [5](#sec:numsetup){reference-type="ref" reference="sec:numsetup"} details our implementation and our experimental setup, while the output of our tests is analyzed in Section [6](#sec:numres){reference-type="ref" reference="sec:numres"}. Final remarks are given in Section [7](#sec:conc){reference-type="ref" reference="sec:conc"}. A list of our test problems is provided in Appendix [8](#sec:pblist){reference-type="ref" reference="sec:pblist"}. # Linearly constrained optimization and tangent cones {#sec:pbtan} The purpose of this section is twofold. First, we describe our problem of interest as well as associated optimality measures in Section [2.1](#subsec:optimeas){reference-type="ref" reference="subsec:optimeas"}. These concepts will serve as a basis for the `Full-Eval` part of our algorithm. Secondly, we discuss the notion of tangent cones and its connection to feasibility in Section [2.2](#subsec:tangcone){reference-type="ref" reference="subsec:tangcone"}. Those definitions will be instrumental in designing our `Low-Eval` step based on direct search. ## Problem and optimality measure {#subsec:optimeas} In this paper, we are interested in solving linearly constrained problems of the form $$\label{eq:prob} \begin{split} \displaystyle \min_{x \in \mathbb{R}^n}&~f(x) \\ \text{s.t.} &~Ax = b\\ &\ell \leq A_\mathcal{I} x \leq u, \end{split}$$ where $f: \mathbb{R}^n \to \mathbb{R}$, $A \in \mathbb{R}^{m \times n}$, $A_\mathcal{I} \in \mathbb{R}^{m_\mathcal{I} \times n}$, $b \in \mathbb{R}^m$ and $(\ell,u) \in \bar{\mathbb{R}}^{m_\mathcal{I}} \times \bar{\mathbb{R}}^{m_\mathcal{I}}$ where $\bar{\mathbb{R}} = \mathbb{R}\cup \{-\infty, \infty\}$ with $\ell < u$. To encompass bound constrained problems into the general formulation [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"}, we consider the possibility that the matrix $A$ is empty, in which case $m$ is equal to zero. When $m > 0$, the matrix $A$ is assumed to have full row rank, and we let $W\in \mathbb{R}^{n \times (n-m)}$ be an orthonormal basis for the null space of $A$. When $m=0$, we consider $W$ to be the identity matrix in $\mathbb{R}^n$. Finally, we denote the set of feasible points by $\mathcal{F}$. Assuming that the function $f$ is continuously differentiable, it is possible to define a criticality measure for problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} that characterizes first-order stationary point. We focus on the quantity $q(\cdot)$ defined by $$\label{eq:qmeas} \forall x \in \mathcal{F}, \quad q(x) \; := \; \left\| P_{\mathcal{F}}\left[x -\nabla f(x)\right]-x \right\|.$$ In derivative-free optimization, the metric [\[eq:qmeas\]](#eq:qmeas){reference-type="eqref" reference="eq:qmeas"} has been employed for analyzing the convergence of algorithms designed for the linearly constrained setting [@RMLewis_VTorczon_1999; @RMLewis_VTorczon_2000]. Although more recent approaches have focused on another metric bearing a close connection with the direct-search stepsize [@gratton2019direct; @TGKolda_RMLewis_VTorczon_2003], the measure [\[eq:qmeas\]](#eq:qmeas){reference-type="eqref" reference="eq:qmeas"} is quite common in projected gradient techniques [@DPBertsekas_2016]. Since our theory relies on that of projected gradient techniques in the smooth setting, we naturally focus on the measure [\[eq:qmeas\]](#eq:qmeas){reference-type="eqref" reference="eq:qmeas"}. When the smoothness of the function $f$ is not guaranteed but $f$ is locally Lipschitz continuous, necessary optimality condition for problem [\[eq:prob\]](#eq:prob){reference-type="eqref" reference="eq:prob"} can be formulated based on the Clarke-Jahn generalized directional derivative of $f$ [@jahn1994introduction]. For a given point $x \in \mathcal{F}$ and a feasible direction $d$, the Clarke-Jahn generalized directional derivative is defined as $$\label{eq:clarke} f^\circ (x;d) := \limsup_{\begin{array}{c} y \to x, y \in \mathcal{F} \\ t \downarrow 0, y + td \in \mathcal{F} \end{array} } \frac{f(y+td) - f(y)}{t}.$$ Any $x^* \in \mathcal{F}$ such that $f^\circ (x^*;d) \geq 0$ for any feasible direction $d$ is called a Clarke-Jahn stationary point. Note that such a condition was recently used in the context of non-smooth optimization with linear constraints [@beck2020convergence]. In the linearly constrained case, the set of feasible directions at $x^*$ coincides with the tangent cone $T(x^*)$. ## Approximate Tangent Cones {#subsec:tangcone} Tangent cones are key concepts to characterize feasibility and optimality in constrained optimization [@JNocedal_SJWright_2006]. Although approximate tangent cones have been less studied, they have proven quite useful in the context of derivative-free optimization, as they characterize directions that are feasible for a given step size, by accounting for constraints that are either active or approximately active [@TGKolda_RMLewis_VTorczon_2003]. We recall below the key definitions related to approximate tangent cones, by following the presentation in Gratton et al. [@gratton2019direct]. For convenience, we will define approximate tangent cones based on a parameterization of the feasible set. More precisely, we fix a reference vector $\bar{x} \in \mathbb{R}^{n-m}$ such that $A \bar{x} = b$. Then, any feasible point $x \in \mathcal{F}$ can be written as $x=W \tilde{x}+\bar{x}$, where $\tilde{x} \in \mathbb{R}^{n-m}$ is such that $$\ell - A_{\mathcal{I}} \bar{x} \; \leq \; A_{\mathcal{I}} W \tilde{x} \; \le \; u - A_{\mathcal{I}} \bar{x} .$$ where $\bar{x}$ is fixed such that $A \bar{x} = b$ and $W \Tilde{x} = x - \bar{x}$. Using this decomposition, we define the *approximate active inequality constraints* at $x = W \tilde{x}+\bar{x} \in \mathcal{F}$ according to a step size $\xi>0$ as $$\label{eq:activecons} \Bigg\{ \begin{aligned} I_u(x,\xi) &:=& \left\{ i :~ |u_i -[A_\mathcal{I} \bar{x}]_i - [A_\mathcal{I} W \tilde{x}]_i | \leq \xi \left\lVert W^\top A_\mathcal{I}^\top e_i\right\rVert \right\} \\ I_\ell(x,\xi) &:=& \left\{ i :~ |\ell_i -[A_\mathcal{I} \bar{x}]_i - [A_\mathcal{I} W \tilde{x}]_i | \leq \xi \left\lVert W^\top A_\mathcal{I}^\top e_i\right\rVert \right\}, \end{aligned}$$ where $e_1, \dots, e_{m_\mathcal{I}}$ denote the coordinate vectors in $\mathbb{R}^{m_\mathcal{I}}$. Those indices in turn define the *approximate normal cone* associated with $(x,\xi)$ as $$\label{eq:normalcone} N(x, \xi) \; := \; \text{Cone}\left( \{W^\top A^\top_\mathcal{I} e_i \}_{i \in I_u(x, \xi)} \} \cup \{-W^\top A^\top_\mathcal{I} e_i \}_{i \in I_l(x, \xi)} \} \right).$$ Rather than using directions from the approximate normal cone to compute steps, we rely on the polar of this cone, called the *approximate tangent cone* and defined by $$\label{eq:tangentcone} T(x,\xi) \; := \; \left\{ v \in \mathbb{R}^n\ | v^\mathrm{T}u \le 0\ \forall u \in N(x,\xi) \right\}.$$ An important property of the approximate tangent cone is that it approximates the feasible region around $x$, and that moving along all its directions for a distance of $\xi$ from $x$ does not break feasibility [@gratton2019direct]. Lemma [Lemma 1](#lem:feas){reference-type="ref" reference="lem:feas"} below provides a formal description of this property. **Lemma 1** ([@gratton2019direct], Lemma 2.1). *Let $x \in \mathcal{F}$ and $\xi > 0$. Then, for any vector $\tilde{d} \in T(x,\xi)$ such that $\|\tilde{d}\| \leq \xi$, we have $x + W \tilde{d} \in \mathcal{F}$.* Direct-search techniques rely on approximate tangent cones to define new feasible points in a way that guarantees convergence to first-order stationarity [@kolda2007stationarity]. # Full-low evaluation framework with linear constraints {#sec:flec} In this section, we describe our main algorithmic framework, that belongs to the class of `Full-Low Evaluation` algorithms. The main idea behind this technique is the combination of two categories of steps. On the one hand, `Full-Eval` steps, that are produced at a significant cost in terms of function evaluations, are used to yield good performance of the method especially in the presence of smoothness. On the other hand, `Low-Eval` steps are cheaper to compute because they require less evaluations, and are often designed to handle the presence of noise and/or non-smoothness in the objective function. We first present our general algorithm that combines both types of steps, then dedicate a section to each category. The general mechanism of the `Full-Low Evaluation` approach is described in Algorithm [\[alg:flalg\]](#alg:flalg){reference-type="ref" reference="alg:flalg"}. `Full-Eval` steps are consecutively taken until a certain condition is triggered, after which one switches to `Low-Eval` iterations. The number of consecutive unsuccessful `Low-Eval` iterations is a user-defined parameter and can be selected as a function of the last successful `Full-Eval` iteration. **Initialization:** Choose an initial iterate $x_0 \in \mathcal{F}$. Set iteration $t_0 = \texttt{Full-Eval}{}$. **For** $k=0,1,2,\dots$ **If** $t_k = \texttt{Full-Eval}{}$, attempt to compute a $\texttt{Full-Eval}{}$ step. **If** success, update $x_{k+1}$ and set $t_{k+1} = \texttt{Full-Eval}{}$. **Else**, $x_{k+1}=x_k$ and $t_{k+1} = \texttt{Low-Eval}{}$. **If** $t_k = \texttt{Low-Eval}{}$, compute a $\texttt{Low-Eval}{}$ step. Update $x_{k+1}$. Decide on $t_{k+1} \in \{ \texttt{Low-Eval}{}, \texttt{Full-Eval}{} \}$. Apart from requiring feasibility of the initial point, note that Algorithm [\[alg:flalg\]](#alg:flalg){reference-type="ref" reference="alg:flalg"} is identical to that of Berahas et al. [@ASBerahas_OSohab_LNVicente_2022], and that linear constraints are assumed to be handled upon computation of a `Low-Eval` or a `Full-Eval` step. In the next sections, we detail our choices for computing those steps. ## Full-eval step based on projections {#subsec:fulleval} `Full-Eval` steps can be implemented by building a model of the objective function around the current point and minimizing it to define the next point. A popular approach that lies within the derivative-free paradigm consists in computing a finite-difference gradient approximation to define a search direction, as well as a stepsize computed via line search based on this approximation [@ASBerahas_OSohab_LNVicente_2022]. We extend here this approach to the linearly constrained setting by considering projections onto the feasible set, a popular technique for dealing with linear constraints [@DPBertsekas_2016]. If the $k$-th iteration of Algorithm [\[alg:flalg\]](#alg:flalg){reference-type="ref" reference="alg:flalg"} is a `Full-Eval` step, we define a search direction $p_k$ based on an approximate gradient $g_k$ computed through finite differences. We then compute candidate steps by considering the feasible direction $$\label{eq:projpt} \bar{x}_k \; = \; P_{\mathcal{F}}\left[x_k-p_k\right],$$ and performing a line search along the direction $\bar{x}_k-x_k$. More precisely, we seek the largest value $\beta \in (0,\bar{\beta}]$ such that $$\label{eq:sdccons} f\left (x_k + \beta (\bar{x}_k-x_k)\right) \; \le \; f(x_k) + c \beta g_k^\top (\bar{x}_k-x_k).$$ where $c \in (0,1)$. We will show in Section [4](#sec:cv){reference-type="ref" reference="sec:cv"} that condition [\[eq:sdccons\]](#eq:sdccons){reference-type="eqref" reference="eq:sdccons"} is satisfied for a sufficiently small value of $\beta$. Algorithm [\[alg:fevconsalg\]](#alg:fevconsalg){reference-type="ref" reference="alg:fevconsalg"} describes the calculation of a `Full-Eval` step for the $k$-th iteration of Algorithm [\[alg:flalg\]](#alg:flalg){reference-type="ref" reference="alg:flalg"}. Similarly to the unconstrained case [@ASBerahas_OSohab_LNVicente_2022], we introduce a switching condition[^4] that controls the norm of the `Full-Eval` step. A value $\beta$ is accepted if it satisfies the decrease condition [\[eq:sdccons\]](#eq:sdccons){reference-type="eqref" reference="eq:sdccons"} and $$\label{eq:key_const} \beta \; \geq \; \gamma \alpha_k,$$ where $\gamma>0$ is independent of $k$. Condition [\[eq:key_const\]](#eq:key_const){reference-type="eqref" reference="eq:key_const"} guarantees that $\beta$ does not go below a certain function of $\alpha_k$, which is the stepsize used for computing `Low-Eval` steps (see Section [3.2](#subsec:loweval){reference-type="ref" reference="subsec:loweval"}). When both [\[eq:sdccons\]](#eq:sdccons){reference-type="eqref" reference="eq:sdccons"} and [\[eq:key_const\]](#eq:key_const){reference-type="eqref" reference="eq:key_const"} are satisfied, we set $\beta_k=\beta$ and define the new iterate as $x_k+\beta\left(\bar{x}_k-x_k\right)$. On the other hand, if condition [\[eq:key_const\]](#eq:key_const){reference-type="eqref" reference="eq:key_const"} is violated, the `Full-Eval` step is skipped. **Input**: Iterate $x_k$. Backtracking parameters $\bar{\beta}>0$, $\tau \in (0,1)$, and $\xi_{k}$. **Output**: $\mathop{\mathrm{i-type}}(k+1)$, $x_{k+1}$, and $\alpha_{k+1}$. Compute the gradient approximation $g_k$ as well as a search direction $p_k$. Compute $\bar{x}_k$ according to [\[eq:projpt\]](#eq:projpt){reference-type="eqref" reference="eq:projpt"}. Backtracking line-search: Set $\beta=\bar{\beta}$. **If** ([\[eq:key_const\]](#eq:key_const){reference-type="ref" reference="eq:key_const"}) is false, **go to line 6**. **While True** **if** [\[eq:sdccons\]](#eq:sdccons){reference-type="eqref" reference="eq:sdccons"} is true or [\[eq:key_const\]](#eq:key_const){reference-type="eqref" reference="eq:key_const"} is false, **break**. Set $\beta = \tau \beta$. **If** ([\[eq:key_const\]](#eq:key_const){reference-type="ref" reference="eq:key_const"}) is true, set $\beta_k = \beta$, $x_{k+1}=x_k+\beta(\bar{x}_k-x_k)$, and $t_{k+1} = \texttt{Full-Eval}{}$. **Else**, set $x_{k+1}=x_k$ and $t_{k+1} = \texttt{Low-Eval}{}$. (The $\texttt{Low-Eval}{}$ parameter $\alpha_k$ remains unchanged, $\alpha_{k+1} = \alpha_k$; see Algorithm [\[alg:pdsf\]](#alg:pdsf){reference-type="ref" reference="alg:pdsf"}.) ## Low-eval step based on feasible descent cones {#subsec:loweval} $\texttt{Low-Eval}{}$ steps are based on the low evaluation paradigm of probabilistic direct search. This approach can be extended to the linearly constrained case as described in Algorithm [\[alg:pdsf\]](#alg:pdsf){reference-type="ref" reference="alg:pdsf"}. We suppose that a feasible initial point is provided by the user. At every iteration, the algorithm uses a finite number of polling directions to seek a new feasible iterate $x^+$ that reduces the objective function value by a sufficient amount $$\label{eq:ds-sdc} f(x^+) \; \leq \; f(x) - \rho(\alpha),$$ where $\rho$ is a forcing function classically employed in direct-search methods. The characteristics of $\rho$ are specified in Section [4.2](#sec:convergence_nonsmooth_const){reference-type="ref" reference="sec:convergence_nonsmooth_const"}. **Input**: Iterate $x_k \in \mathcal{F}$ and stepsize $\alpha_k$. Direct-search parameters $\lambda \geq 1$ and $\theta \in (0,1)$. **Output**: $\mathop{\mathrm{i-type}}(k+1)$, $x_{k+1}$, and $\alpha_{k+1}$. Generate a finite set $D_k$ of non-zero polling directions. **If** a feasible poll point $x_k + \alpha_k d_k$ is found such that [\[eq:ds-sdc\]](#eq:ds-sdc){reference-type="eqref" reference="eq:ds-sdc"} is true for some $d_k\in D_k$, set $x_{k+1} = x_k + \alpha_k d_k$ and $\alpha_{k+1} = \lambda \alpha_k$. **Else**, set $x_{k+1} = x_k$ and $\alpha_{k+1} = \theta \alpha_k$. Decide **if** $t_{k+1}=\texttt{Low-Eval}$ **or if** $t_{k+1}=\texttt{Full-Eval}$. As shown by Lemma [Lemma 1](#lem:feas){reference-type="ref" reference="lem:feas"}, choosing directions of the form $W \tilde{d}$, where $\tilde{d} \in T(x_k,\alpha_k)$ with a stepsize less or equal than $\alpha_k$ ensures the feasibility of the `Low-Eval` step [@gratton2019direct]. # Convergence Analysis {#sec:cv} ## Rate of convergence in the smooth case {#subsec:cvsmooth} In this section, we analyze the behavior of the class of `Full-Low Evaluation` methods in the smooth case. We show that if the `Full-Eval` step generates an infinite sequence of iterates, then the norm of $q(x_k)$ converges to zero with a rate of $1/ \sqrt{k}$. We now introduce the assumptions needed for the analysis, starting with standard boundedness and smoothness requirements. **Assumption 1**. *The objective function $f$ is bounded below by $f_\mathrm{low}\in \mathbb{R}$, i.e., $f(x) \ge f_\mathrm{low}$ for all $x \in \mathbb{R}^n$.* **Assumption 2**. *The function $f$ is continuously differentiable and its gradient $\nabla f$ is Lipschitz continuous with constant $L>0$.* The next assumptions are related to our approximate gradient and stationary measure. For any iterate $x_k$ in $\mathcal{F}$ computed by Algorithm [\[alg:flalg\]](#alg:flalg){reference-type="ref" reference="alg:flalg"}, we define $$\label{eq:qfunctionsxk} q_k = P_{\mathcal{F}}\left[x_k-\nabla f(x_k)\right]-x_k, \quad q_k^g = P_{\mathcal{F}}\left[x_k-g_k\right]-x_k \quad \mbox{and} \quad q_k^p = \bar{x}_k - x_k = P_{\mathcal{F}}\left[x_k+p_k\right]-x_k.$$ In oue algorithm, we rely on directions defined using $g_k$. Those should be close to the negative of that approximate gradient, in the sense of Assumption [Assumption 3](#ass:pkgk){reference-type="ref" reference="ass:pkgk"} below. **Assumption 3**. *For every iteration $k$, $$\frac{(-g_k)^\top q^p_k}{\|q^g_k\|\|q^p_k\|} \; \ge \; \kappa \quad \mbox{and} \quad u_p \|q_k^p\| \; \le \; \|q_k^g\| \; \le \; U_p \|q_k^p\|.$$ with $u_p>0$, $U_p > 0$ and $\kappa \in (0, 1]$.* When $p_k = -g_k$, Assumption [Assumption 3](#ass:pkgk){reference-type="ref" reference="ass:pkgk"} holds with $\kappa = u_p = U_p = 1$. Indeed, the first inequality can be proved using the property of the projection [@DPBertsekas_2016 Proposition 1.1.4(b)] that implies that $$(x_k - g_k-\bar{x}_k)^\top (x-\bar{x}_k) \; \le \; 0 \qquad \forall x \in \mathcal{F}.$$ Moreover, using $x=x_k$ in the previous inequality as well as $\bar{x}_k - x_k = q^g_k = q^p_k$ gives $$g_k^\top (\bar{x}_k-x_k) \le -\|\bar{x}_k-x_k\|^2 \quad \Rightarrow \quad - g_k^\top q_k^p \ge \|q^g_k\|\|q^p_k\|,$$ Finally, in order to relate the control the discrepancy between the true criticality measure and its approximation using $g_k$, we require the following assumption. **Assumption 4**. *The approximate gradient $g_k$ computed at $x_k$ satisfies $$\label{eq:accgk} \| \nabla f(x_k) - g_k \| \; \le \; u_g \|q_k^g\|,$$ where $u_g \in (0,\kappa(1-c))$ is independent of $k$.* This condition generalizes that in full-low methods for unconstrained optimization [@ASBerahas_OSohab_LNVicente_2022 Assumption 3.2], albeit with a restriction on the constant $u_g$ that becomes unnecessary in the unconstrained setting. Nevertheless, condition [\[eq:accgk\]](#eq:accgk){reference-type="eqref" reference="eq:accgk"} can be guaranteed in a finite number of steps when the gradient is estimated using finite differences as shown in Section [4.3](#subsec:cric){reference-type="ref" reference="subsec:cric"}. We now start our analysis by establishing a lower bound on the stepsize $\beta_k$. **Lemma 2**. *Let Assumptions [Assumption 2](#ass:lip){reference-type="ref" reference="ass:lip"}, [Assumption 3](#ass:pkgk){reference-type="ref" reference="ass:pkgk"}, and [Assumption 4](#ass:accgk){reference-type="ref" reference="ass:accgk"} hold. If the $k$-th iteration is a successful $\texttt{Full-Eval}{}$ iteration, then $$\label{eq:boundbetaSF} \beta_k \; \ge \; \beta_{\min} \; := \; \min\left\{ \bar{\beta}, \frac{2\tau(\kappa(1-c) -u_g )u_p}{L}\right\}.$$* **Proof.** If $\beta=\bar{\beta}$ satisfies the decrease condition [\[eq:sdccons\]](#eq:sdccons){reference-type="eqref" reference="eq:sdccons"}, then [\[eq:boundbetaSF\]](#eq:boundbetaSF){reference-type="eqref" reference="eq:boundbetaSF"} holds trivially. We thus suppose in the rest of the proof that there exists $\beta$ such that [\[eq:boundbetaSF\]](#eq:boundbetaSF){reference-type="eqref" reference="eq:boundbetaSF"} does not hold, i.e., that $$\label{eq:notsuffdecbndbetaSF} c \beta g_k^\top (\bar{x}_k-x_k) \; \le \; f(x_k+\beta(\bar{x}_k-x_k))-f(x_k).$$ Using a Taylor expansion of $f$ around $x_k$ on the right-hand side of [\[eq:notsuffdecbndbetaSF\]](#eq:notsuffdecbndbetaSF){reference-type="eqref" reference="eq:notsuffdecbndbetaSF"}, we obtain the following inequalities $$\begin{aligned} c\beta g_k^\top (\bar{x}_k-x_k) &\le &\beta\nabla f(x_k)^\top (\bar{x}_k-x_k) + \frac{L}{2}\beta^2 \|\bar{x}_k-x_k\|^2 \nonumber \\ c \beta g_k^\top (\bar{x}_k-x_k) &\le &\beta g_k^\top (\bar{x}_k-x_k) + \beta [\nabla f(x_k)-g_k]^\top (\bar{x}_k-x_k) \nonumber \\ & &+ \frac{L}{2}\beta^2\|\bar{x}_k-x_k\|^2 \nonumber \\ 0 &\le &(1-c)\beta g_k^\top (\bar{x}_k-x_k) + \beta [\nabla f(x_k)-g_k]^\top (\bar{x}_k-x_k) \nonumber \\ & &+ \frac{L}{2}\beta^2\|\bar{x}_k-x_k\|^2. \label{eq:taylorbetaSF} \end{aligned}$$ Using Assumption [Assumption 3](#ass:pkgk){reference-type="ref" reference="ass:pkgk"}, we have $$(g_k)^\top q_k^p \; \le \; - \kappa \|q_k^g\| \|q_k^p\| \quad \Leftrightarrow \quad g_k^\top (\bar{x}_k-x_k) \; \le \; - \kappa \|q_k^g\| \|\bar{x}_k-x_k\|,$$ hence $$\label{eq:taylorbetaSF1} (1-c)\beta g_k^\top (\bar{x}_k-x_k) \; \le \; -(1-c) \kappa \beta \|q_k^g\| \|\bar{x}_k-x_k\|.$$ We now turn to the second term in the right-hand side of [\[eq:taylorbetaSF\]](#eq:taylorbetaSF){reference-type="eqref" reference="eq:taylorbetaSF"}. Using Cauchy-Schwarz inequality together with Assumption [Assumption 4](#ass:accgk){reference-type="ref" reference="ass:accgk"}, we obtain $$\begin{aligned} ^\top (\bar{x}_k-x_k) &\le &\|\nabla f(x_k)-g_k\| \|\bar{x}_k-x_k\| \\ &\le &u_g \|q^g_k\| \|\bar{x}_k-x_k\|. \end{aligned}$$ Overall, we thus obtain that $$\label{eq:taylorbetaSF2} \beta [\nabla f(x_k)-g_k]^\top (\bar{x}_k-x_k) \; \le \; u_g \beta \|q^g_k\| \|\bar{x}_k-x_k\|.$$ Putting [\[eq:taylorbetaSF1\]](#eq:taylorbetaSF1){reference-type="eqref" reference="eq:taylorbetaSF1"} and [\[eq:taylorbetaSF2\]](#eq:taylorbetaSF2){reference-type="eqref" reference="eq:taylorbetaSF2"} into [\[eq:taylorbetaSF\]](#eq:taylorbetaSF){reference-type="eqref" reference="eq:taylorbetaSF"}, we obtain $$\begin{aligned} 0 &\le &-(1-c) \kappa \beta \|q_k^g\| \|\bar{x}_k-x_k\| + u_g \beta \|q_k^g\| \|\bar{x}_k-x_k\| \\ & &+ \frac{L}{2}\beta^2 \|\bar{x}_k-x_k\|^2 \\ 0 &\le &-(\kappa(1-c) -u_g)\beta \|q_k^g\| \|\bar{x}_k-x_k\| + \frac{L}{2}\beta^2 \|\bar{x}_k-x_k\|^2. \end{aligned}$$ Using $\kappa(1-c) -u_g \geq 0$ from Assumption [Assumption 4](#ass:accgk){reference-type="ref" reference="ass:accgk"} together with Assumption [Assumption 3](#ass:pkgk){reference-type="ref" reference="ass:pkgk"}, we show $$0 \; \leq \; -(\kappa(1-c) -u_g) u_q \beta \| \bar{x}_k-x_k\|^2 + \frac{L}{2}\beta^2 \|\bar{x}_k-x_k\|^2$$ The latter inequality only holds as long as $$\beta \; \ge \; \frac{2(\kappa(1-c) - u_g)u_p}{L},$$ hence the line-search procedure must terminate with $$\beta \; \ge \; \frac{2\tau(\kappa(1-c) - u_g )u_p}{L}.$$ Combining this result with the case $\beta=\bar{\beta}$ finally gives the desired result. $\Box$ We can now establish the main result of the smooth case. **Theorem 1**. *Let Assumptions [Assumption 1](#ass:flow){reference-type="ref" reference="ass:flow"}--[Assumption 3](#ass:pkgk){reference-type="ref" reference="ass:pkgk"} hold. For any $K \ge 1$, $$\label{eq:cvsmooth} \min_{k=0,\dots,K-1} \|\mathcal{P}_{\mathcal{F}}\left[x_k-\nabla f(x_k)\right] -x_k\| \le \frac{(u_g+1) \sqrt{U_p}}{\sqrt{\kappa \beta_{\min}}} \sqrt{\frac{f(x_0)-f_{\mathrm{low}}}{c}} \frac{1}{\sqrt{n_{SF}^K}},$$ where $n_{SF}^K$ is the number of successful `Full-Eval` iterations up to iteration $K$.* **Proof.** We denote by $\mathcal{I}_{SF}^K$ the set of indices corresponding to successful `Full-Eval` iterations. Let $k \in \mathcal{I}_{SF}^K$. By definition of such an iteration, the sufficient decrease condition [\[eq:sdccons\]](#eq:sdccons){reference-type="eqref" reference="eq:sdccons"} is satisfied for $x_{k+1}=\bar{x}_k(\beta_k)$, where $\beta_k$ satisfies [\[eq:boundbetaSF\]](#eq:boundbetaSF){reference-type="eqref" reference="eq:boundbetaSF"}. Moreover, as shown in the proof of Lemma [Lemma 2](#le:boundbetaSF){reference-type="ref" reference="le:boundbetaSF"}, we have $$g_k^\top (\bar{x}_k-x_k) \; \le \; - \kappa \|q_k^g\| \|\bar{x}_k-x_k\|.$$ Overall, we obtain $$\begin{aligned} f(x_k) - f(x_{k+1}) &\ge &-c \beta_k g_k^\top (\bar{x}_k-x_k) \\ &\ge &c \kappa \beta_k \|q_k^g\| \|\bar{x}_k-x_k\| \\ &\ge &\frac{c \kappa \beta_{\min}}{U_p} \|q_k^g\|^2. \end{aligned}$$ Meanwhile, using Assumption [Assumption 4](#ass:accgk){reference-type="ref" reference="ass:accgk"} gives $$\|q_k\| \; \le \; \|q_k-q_k^g\| + \|q_k^g\| \le (u_g+1)\|q_k^g\|.$$ Therefore, the decrease achieved at iteration $k$ satisfies $$\label{eq:decreaseSF} f(x_k)-f(x_{k+1}) \; \ge \; \frac{c\kappa \beta_{\min}}{U_p(u_g+1)^2}\|q_k\|^2.$$ We now consider the changes in function values across all iterations in $\{0,\dots,K-1\}$. Since the iterate does not change on unsuccessful iterations and the function value decreases on successful `Low-Eval` iterations, we have $f(x_k)-f(x_{k+1}) \ge 0$ for all $k \le K-1$. Combining this observation with Assumption [Assumption 1](#ass:flow){reference-type="ref" reference="ass:flow"} and [\[eq:decreaseSF\]](#eq:decreaseSF){reference-type="eqref" reference="eq:decreaseSF"} leads to $$\begin{aligned} f(x_0)-f_{\mathrm{low}} &\ge &f(x_0)-f(x_K) \\ &= &\sum_{k=0}^{K-1} f(x_k)-f(x_{k+1}) \\ &\ge &\sum_{k \in \mathcal{I}_{SF}^K} f(x_k)-f(x_{k+1}) \\ &\ge &\frac{c \kappa \beta_{\min}}{U_p(u_g+1)^2} \sum_{k \in \mathcal{I}_{SF}^K} \|q_k\|^2 \\ &\ge &\frac{c \kappa \beta_{\min}}{U_p(u_g+1)^2} n_{SF}^k \left(\min_{0 \le k \le K-1} \|q_k\|\right)^2. \end{aligned}$$ Re-arranging the terms and using the formula for $q_k$ leads to the desired conclusion. $\Box$ The rate [\[eq:cvsmooth\]](#eq:cvsmooth){reference-type="eqref" reference="eq:cvsmooth"} matches existing result for the unconstrained case [@ASBerahas_OSohab_LNVicente_2022]. ## Convergence in the non-smooth case {#sec:convergence_nonsmooth_const} When the smoothness of the function $f$ is not guaranteed, we rely on the properties of the `Low-Eval` steps, and in particular on the sufficient decrease guarantees certified by the forcing function. To this end, we make the following assumption. **Assumption 5**. *The function $\rho$ is positive, non-decreasing, and satisfies $\lim_{\alpha \rightarrow 0^+} \rho(\alpha)/\alpha = 0$.* As in the unconstrained setting [@ASBerahas_OSohab_LNVicente_2022], we require the following assumption on the failure of `Full-Eval` iterations. **Assumption 6**. *There exists $\epsilon_g>0$ such that for any $k \in I_{SF}$, where $I_{SF}$ denotes the set of successful `Full-Eval` iterations, $\|q_k^g\|>\epsilon_g$.* However, we still rely in the analysis on the switching condition ([\[eq:key_const\]](#eq:key_const){reference-type="ref" reference="eq:key_const"}), along with the assumption that the `Low-Eval` iterations generate an infinite subsequence of iterates to prove that the direct-search parameter $\alpha_k$ goes to zero. This result requires the forcing function to satisfy Assumption [Assumption 5](#ass:rho){reference-type="ref" reference="ass:rho"} used in the unconstrained regime. **Lemma 3**. *Let Assumption [Assumption 5](#ass:rho){reference-type="ref" reference="ass:rho"} hold. Assume that the sequence of iterates $\{ x_k \}$ is bounded. Then, there exists a point $x_*$ and a subsequence $\mathcal{K} \subset \mathcal{I}_{UL}$ of unsuccessful `Low-Eval` iterates for which $$\begin{array}{ccc} \displaystyle\lim_{k \in \mathcal{K}} x_k \; = \; x_* & \text{ and } & \displaystyle \lim_{k \in \mathcal{K}} \alpha_k \; = \; 0. \end{array} \label{eq:xstarconst}$$* **Proof.** First, suppose that the set $\mathcal{I}_{SF} \cup \mathcal{I}_{UF} \cup \mathcal{I}_{SL}$ is of infinite cardinality, where $\mathcal{I}_{UF}$ and $\mathcal{I}_{SL}$ are the sets of unsuccessful `Full-Eval` and successful `Low-Eval` iterations, respectively. Note that this set represents all iterations $k$ for which $\alpha_k$ does not decrease. For all successful `Full-Eval` iterations $k \in \mathcal{I}_{SF}$, recall from [\[eq:decreaseSF\]](#eq:decreaseSF){reference-type="eqref" reference="eq:decreaseSF"} that $$f(x_k) - f(x_{k+1}) \ge c \beta_k - g_k^\mathrm{T}(\bar{x}_k - x_k) \ge \frac{c \kappa \beta_{k}}{U_p} \|q_k^g\|^2.$$ From the fact that $\beta_k \geq \gamma \alpha_k$, we get $$\label{eq:dec1const} f(x_k)-f(x_{k+1}) \ge \frac{c \kappa \gamma }{U_p} \alpha_k \|q_k^g\|^2 \ge \frac{c \kappa \gamma \epsilon_g^2}{U_p} \alpha_k.$$ Meanwhile, successful `Low-Eval` iterations $k \in \mathcal{I}_{SL}$ achieve sufficient decrease, $$\label{eq:dec2const} f(x_k) - f(x_{k+1}) \ge \rho(\alpha_k).$$ Note that in `Full-Eval` unsuccessful iterations $k \in \mathcal{I}_{UF}$ neither $x_k$ nor $\alpha_k$ changes. Hence, given that for unsuccessful `Low-Eval` iterations ($\mathcal{I}_{UL}$) the function does not decrease, we can sum from $0$ to $k \in \mathcal{I}_{SF} \cup \mathcal{I}_{UF} \cup \mathcal{I}_{SL}$ the inequalities ([\[eq:dec1const\]](#eq:dec1const){reference-type="ref" reference="eq:dec1const"}) and ([\[eq:dec2const\]](#eq:dec2const){reference-type="ref" reference="eq:dec2const"}) to obtain $$\begin{aligned} f(x_0) - f(x_{k+1}) &\ge &\sum_{k \in \mathcal{I}_{SF}} (f(x_k)-f(x_{k+1})) + \sum_{k \in \mathcal{I}_{SL}} (f(x_k)-f(x_{k+1})) \\ &\ge & \frac{c \kappa \gamma \epsilon_g^2}{U_p} \sum_{k \in \mathcal{I}_{SF}} \alpha_k + \sum_{k \in \mathcal{I}_{SL}}\rho(\alpha_k).\end{aligned}$$ By the boundedness (from below) of $f$, we conclude that the series are summable, which implies that $\lim_{k \in \mathcal{I}_{SF}} \alpha_k = 0$ and $\lim_{k \in \mathcal{I}_{SL}} \rho(\alpha_k) = 0$. Since $\alpha$ remains unchanged during unsuccessful `Full-Eval` steps, and under Assumption [Assumption 5](#ass:rho){reference-type="ref" reference="ass:rho"}, it follows that $\lim_{k \in \mathcal{I}_{SF} \cup \mathcal{I}_{UF} \cup \mathcal{I}_{SL}} \alpha_k = 0$. It remains to consider the iterations in $\mathcal{I}_{UL}$. For each $k \in \mathcal{I}_{UL}$ corresponding to an unsuccessful `Low-Eval` iteration, consider the previous iteration $k'=k'(k) \in \mathcal{I}_{SL} \cup \mathcal{I}_{UF} \cup \mathcal{I}_{SL}$ with $k' < k$ ($k'$ could be zero). The direct-search stepsize can then be written as $\alpha_k = \theta^{k - k'} \alpha_{k'}$. Since $k' \to \infty$ and $\tau\in (0,1)$, one obtains $\alpha_k \to 0$ for all $k \in \mathcal{I}_{UL}$. We have thus proved that $\alpha_k$ goes to zero for all $k$. Since $\alpha_k$ is only decreased in unsuccessful `Low-Eval` iterations, there must be an infinite subsequence of those. From the boundedness of the sequence of iterates, one can extract a subsequence $\mathcal{K}$ of that subsequence satisfying the statement of the lemma. $\Box$ As in the unconstrained case, convergence results are established using the notion of generalized Clarke-Jahn derivative [@FHClarke_1990] at $x$ along a direction $d$. In Theorem [Theorem 2](#th:Clarke-stat){reference-type="ref" reference="th:Clarke-stat"}, we show that there exists a limit point which is Clarke-Jahn stationary, provided the so-called refining directions are dense in the tangent cone. **Theorem 2**. *Let Assumption [Assumption 5](#ass:rho){reference-type="ref" reference="ass:rho"} hold. Assume that the sequence of iterates $\{ x_k \}$ is bounded. Let the function $f$ be Lipschitz continuous around the point $x_*$ defined in Lemma [\[eq:xstarconst\]](#eq:xstarconst){reference-type="ref" reference="eq:xstarconst"}. Let the set of limit points of $$\label{eq: ref-dirconst} \left\{ \frac{d_k}{\left\lVert d_k\right\rVert}, \; d_k \in D_k, k \in \mathcal{K} \right\}$$ be dense in the tangent cone $T(x_*)$, where $\mathcal{K} \subset \mathcal{I}_{UL}$ is given in Lemma [Lemma 3](#the:nsc){reference-type="ref" reference="the:nsc"}.* *Then, $x_*$ is a Clarke-Jahn stationary point, i.e., $f^{\circ}(x_*;d) \geq 0$ for all normalized $d$ in $T(x^*)$.* **Proof.** The proof follows standard arguments in [@CAudet_JEDennis_2002; @CAudet_JEDennis_2006; @LNVicente_ALCustodio_2012]. Let $\bar{d}$ be a limit point of ([\[eq: ref-dirconst\]](#eq: ref-dirconst){reference-type="ref" reference="eq: ref-dirconst"}), identified for a certain subsequence $\mathcal{L} \subseteq \mathcal{K}$. Then, from the basic properties of the generalized Clarke-Jahn derivative, and $k \in \mathcal{L}$, $$\begin{aligned} f^{\circ}(x_*;\bar{d}) & = \limsup_{\small \begin{array}{c} x_k \to x_*, x_k \in \mathcal{F} \\ \alpha_k \downarrow 0, x_k + \alpha_k \bar{d} \in \mathcal{F} \end{array}} \frac{f(x_k + \alpha_k \bar{d}) - f(x_k)}{\alpha_k}\\ & \geq \limsup_{\small \begin{array}{c} x_k \to x_*, x_k \in \mathcal{F} \\ \alpha_k \downarrow 0, x_k + \alpha_k d_k \in \mathcal{F} \end{array}} \left\{ \frac{f(x_k + \alpha_k d_k) - f(x_k)}{\alpha_k} - L_f^* \| d_k - \bar{d} \| \right\} \\ & = \limsup_{\small \begin{array}{c} x_k \to x_*, x_k \in \mathcal{F} \\ \alpha_k \downarrow 0, x_k + \alpha_k d_k \in \mathcal{F} \end{array}} \left\{ \frac{f(x_k + \alpha_k d_k) - f(x_k)}{\alpha_k} + \frac{\rho(\alpha_k)}{\alpha_k} \right\},\end{aligned}$$ where $L_f^*$ is the Lipschitz constant of $f$ around $x_*$. Since $k \in \mathcal{L}$ are unsuccessful `Low-Eval` iterations, it follows that $f(x_k + \alpha_k d_k) > f(x_k) - \rho(\alpha_k)$ which implies that $$\limsup_{\small \begin{array}{c} x_k \to x_*, x_k \in \mathcal{F} \\ \alpha_k \downarrow 0, x_k + \alpha_k d_k \in \mathcal{F} \end{array}} \frac{f(x_k + \alpha_k d_k) - f(x_k) + \rho(\alpha_k)}{\alpha_k} \; \geq \; 0.$$ From this and Assumption [Assumption 5](#ass:rho){reference-type="ref" reference="ass:rho"}, we obtain $f^{\circ}(x_*;\bar{d}) \geq 0$. Given the continuity of $f^{\circ}(x_*;\cdot)$, one has for any $d \in T(x_*)$ such that $\|d\|= 1$, $f^{\circ}(x_*;d) = \lim_{\bar{d} \to d} f^{\circ}(x_*;\bar{d}) \geq 0$. $\Box$ ## More on the smooth case (use of finite difference gradients) {#subsec:cric} Let us return to the smooth case to clarify the imposition of Assumption [Assumption 4](#ass:accgk){reference-type="ref" reference="ass:accgk"}. Such an assumption is related to the satisfaction of the so-called criticality step in DFO trust-region methods [@ARConn_KScheinberg_LNVicente_2009a; @ARConn_KScheinberg_LNVicente_2009b] based on fully linear models. In the context of Algorithm [\[alg:fevconsalg\]](#alg:fevconsalg){reference-type="ref" reference="alg:fevconsalg"}, those models correspond to an approximate gradient $g_k$ built from finite differences. The $i$-th component of the forward FD approximation of the gradient at $x_k$ is defined as $$\label{eq:FD} [\nabla_{h_k} f(x_k)]_i \; = \; \frac{f(x_k + h_k e_i) - f(x_k)}{h_k}, \quad i=1,\ldots,n,$$ where $h_k$ is the finite difference parameter and $e_i \in \mathbb{R}^n$ is the $i$-th canonical vector. Computing such a gradient approximation costs $n$ function evaluations per iteration, and it is implicitly assumed that such evaluations can be made. By using a Taylor expansion, the error in the finite-differences gradient (in the smooth and noiseless setting) can be shown [@ARConn_KScheinberg_LNVicente_2009b] to satisfy $$\label{eq:FDbound} \|\nabla f(x_k) - \nabla_{h_k} f(x_k) \| \; \leq \; \frac{1}{2} \sqrt{n} \, L \, h_k.$$ It becomes then clear that one way to ensure Assumption [Assumption 4](#ass:accgk){reference-type="ref" reference="ass:accgk"} in practice, when $g_k = \nabla_{h_k} f(x_k)$, is to enforce $h_k \leq u_g' \| q_k^{h_k} \|$, where $q_k^{h_k} = P_{\mathcal{F}}\left[x_k- \nabla_{h_k} f(x_k)\right]-x_k$ for some $u_g'>0$, in which case $u_g = \frac{1}{2} \sqrt{n} \, L \, u_g'$. Enforcing such a condition is expensive but can be rigorously done through a criticality-step type argument (see Algorithm [\[alg:cric\]](#alg:cric){reference-type="ref" reference="alg:cric"}). **Input:** $h_k$, ${ q_k^{h_k}}^{(0)} = q_k^{h_k}$, and $\omega \in (0,1)$. Let $j=0$. **Output**: $q_k^{h_k} = {q_k^{h_k}}^{(j)}$ and $h_k$. **While** $h_k > u_g' \left\lVert {q_k^{h_k}}^{(j)}\right\rVert$ **Do** Set $j = j + 1$ and $h_k = \omega^j u_g' \left\lVert{q_k^{h_k}}^{(0)}\right\rVert$. Compute $\nabla_{h_k} f(x_k)$ using ([\[eq:FD\]](#eq:FD){reference-type="ref" reference="eq:FD"}) and set ${q_k^{h_k}}^{(j)} = P_{\mathcal{F}}\left[x_k- \nabla_{h_k} f(x_k)\right]-x_k$ [\[alg:cric\]]{#alg:cric label="alg:cric"} Proposition [Proposition 1](#prop:crit){reference-type="ref" reference="prop:crit"} shows that Algorithm [\[alg:cric\]](#alg:cric){reference-type="ref" reference="alg:cric"} terminates in a finite number of steps. **Proposition 1**. *Let Assumption [Assumption 2](#ass:lip){reference-type="ref" reference="ass:lip"} hold. If $\|q_k\|>0$, then Algorithm [\[alg:cric\]](#alg:cric){reference-type="ref" reference="alg:cric"} terminates in finitely many iterations by computing $h_k$ such that the condition $h_k \leq u_g' \| q_k^{h_k}\|$ is satisfied.* **Proof.** Let us suppose that the algorithm loops infinitely. Then, for all $j \geq 1$, using Step 2 and the satisfaction of the while--condition in Step 1, $$\label{eq:crit1} \| {q_k^{h_k}}^{(j)} \|\; \leq \; \omega^j \| {q_k^{h_k}}^{(0)} \|.$$ On the other hand, for all $j \geq 1$, the FD bound ([\[eq:FDbound\]](#eq:FDbound){reference-type="ref" reference="eq:FDbound"}), followed by Step 2, gives us $$\label{eq:crit2} \| \nabla f(x_k) - \nabla_{h_k} f(x_k)^{(j)} \| \; \leq \; \frac{1}{2} \sqrt{n} L \, \omega^j u_g' \| {q_k^{h_k}}^{(0)} \|.$$ Hence, using ([\[eq:crit1\]](#eq:crit1){reference-type="ref" reference="eq:crit1"})--([\[eq:crit2\]](#eq:crit2){reference-type="ref" reference="eq:crit2"}), we have $$\begin{aligned} \| q_k \| \le \| q_k - {q_k^{h_k}}^{(j)} \| + \| {q_k^{h_k}}^{(j)} \| &\le &\| \nabla f(x_k) - \nabla_{h_k} f(x_k)^{(j)} \| + \| {q_k^{h_k}}^{(j)} \| \\ &\le &\| \nabla f(x_k) - \nabla_{h_k} f(x_k)^{(j)} \| + \omega^j \| {q_k^{h_k}}^{(0)} \| \\ &\le &\left( \frac{\sqrt{n} L u_g'}{2}+ 1\right) \omega^j \| {q_k^{h_k}}^{(0)} \|,\end{aligned}$$ where the second inequality on the first line comes from the \[non-expansiveness of orthogonal projection. By taking limits (and noting that $\omega \in (0,1)$), we conclude that $q_k = 0$, which yields a contradiction. $\Box$ # Numerical Setup {#sec:numsetup} In this section, we will first present our implementation choices for the `Full-Low Evaluation` linearly constrained method. The complete MATLAB implementation is available on GitHub[^5]. The repository includes all the necessary algorithms and testing scripts. The numerical environment of our experiments is also introduced (other methods/solvers tested, test problems chosen, and performance profiles). The tests were run using MATLAB R2019b on an Asus Zenbook with 16GB of RAM and an Intel Core i7-8565U processor running at 1.80GHz. ## Practical `Full-Eval` implementation In this section, we present a detailed discussion of the implementation of the $\texttt{Full-Low Evaluation}{}$ algorithm in the linearly constrained case. Building upon the principles used in the unconstrained case, we introduce a direction $p_k$ that leverages second-order information for faster convergence. Specifically, we define $p_k = W H_k W^\top g_k$, where $H_k$ represents an approximation of the inverse Hessian using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton update [@CGBroyden_1970; @RFletcher_1970; @DGoldfarb_1970; @DFShanno_1970], as described in Algorithm [\[alg:BFGS-FD\]](#alg:BFGS-FD){reference-type="ref" reference="alg:BFGS-FD"}. Here, $W\in \mathbb{R}^{n \times (n-m)}$ denotes an orthonormal basis for the null space of matrix $A$. Notably, due to the positive definiteness of $H_k$, it follows that $W H_k W^\top$ is also positive definite. Using $W H_k W^\top g_k$ instead of $H_k g_k$ offers two significant advantages. Firstly, the resulting value of $x_k - p_k$ automatically satisfies the equality constraints, since $$A(x_k - W H_k W^\top g_k) \; = \; b - A W (H_kW^\top g_k) \;= \;b.$$ Secondly, using this direction allows us to compute $W^\top g_k$ rather than directly calculating $g_k$, thus reducing the computational cost of finite differences from $n$ to $n-m$ function evaluations. Indeed, the forward finite-differences approximation can be reduced to the null space of the linear equality constraints: $$\label{eq:FDw} [W^\top g_k]_i \; = \; \frac{f(x_k + h_k w_i) - f(x_k)}{h_k}, \quad \text{for} \quad i=1,\ldots,n-m,$$ where $h_k$ is the finite difference parameter, and $w_i \in \mathbb{R}^n$ is the $i$-th column vector of $W$. In the numerical experiments, the parameter $h_k$ is set to the square root of Matlab's machine precision. Our `Full-Eval` line-search iteration is described in Algorithm [\[alg:BFGS-FD\]](#alg:BFGS-FD){reference-type="ref" reference="alg:BFGS-FD"}, which includes BFGS updates for the inverse Hessian approximation $H_k$ using ([\[eq:BFGS\]](#eq:BFGS){reference-type="ref" reference="eq:BFGS"}). Here, $j_k$ refers to the previous `Full-Eval` iteration, and $s_k$ and $y_k$ are given in ([\[eq:s-and-y\]](#eq:s-and-y){reference-type="ref" reference="eq:s-and-y"}). Notably, in the non-convex case, the inner product $s_k^\top y_k$ cannot be ensured to be positive. To maintain the positive definiteness of the matrix $H_k$, we skip the BFGS update if $s_k^\top y_k \geq \epsilon_{c} \|s_k\| \|y_k\|$ is not satisfied, with $\epsilon_{c} \in (0,1)$ being independent of $k$. In our implementation, we use $\epsilon_{c}=10^{-10}$. The line search follows the backtracking scheme described in Algorithm [\[alg:fevconsalg\]](#alg:fevconsalg){reference-type="ref" reference="alg:fevconsalg"}, using standard values $\bar{\beta} = 1$ and $\tau=0.5$. A key feature of our `Full-Low Evaluation` methodology that led to rigorous results (see the proof of Lemma [Lemma 3](#the:nsc){reference-type="ref" reference="the:nsc"}) is to stop the line search once condition ([\[eq:key_const\]](#eq:key_const){reference-type="ref" reference="eq:key_const"}) is violated. In our implementation, we use: $$\label{eq:forcing-imp} \gamma \; = \; 1, \quad \rho(\alpha_k) \; = \; \min(\gamma_1, \gamma_2 \alpha_k^2), \quad \text{with} \quad \gamma_1\;=\;\gamma_2\;=\;10^{-5}.$$ For $k=0$, we perform a backtracking line search using $p_0=-W W^\top g_0$ (and update $t_1$ and $x_1$) as in Algorithm [\[alg:fevconsalg\]](#alg:fevconsalg){reference-type="ref" reference="alg:fevconsalg"} (with constants as in Algorithm [\[alg:BFGS-FD\]](#alg:BFGS-FD){reference-type="ref" reference="alg:BFGS-FD"}). The initialization of $H_0$ is done as follows: If $t_1=\texttt{Full-Eval}{}$, then we set $H_0 = (y_0^\top s_0)/(y_0^\top y_0) I$, in an attempt to make the size of $H_0$ similar to that of $\nabla^2 f(x_0)^{-1}$ [@JNocedal_SJWright_2006]. However, if $t_1=\texttt{Low-Eval}{}$, we set $H_0=I$. **Input**: Iterate $x_k$ with $k \geq 1$. Information $(x_{j_k},g_{j_k},H_{j_k})$ from the previous `Full-Eval` iteration $j_k$ (if $k>0$). Backtracking parameters $\bar{\beta}>0$ and $\tau \in (0,1)$. Other parameters $\epsilon_{c},\gamma,\gamma_1>0$. **Output**: $t_{k+1}$ and $(x_{k+1},H_k,g_k)$. Return the number $nb_k$ of backtrack attempts. Compute the FD gradient $W^\top g_k = W^\top \nabla_{h_k} f(x_k)$ using ([\[eq:FDw\]](#eq:FDw){reference-type="ref" reference="eq:FDw"}). Set $$\label{eq:s-and-y} s_k = x_{k} - x_{j_k} \quad \mbox{and} \quad y_{k} = g_k - g_{j_k}.$$ $s_k^\top y_k \geq \epsilon_{c} \|s_k\| \|y_k\|$, set $$\label{eq:BFGS} H_{k} \; = \; \left (I - \frac{s_k y_k^\top}{y_k^\top s_k}\right) H_{j_k} \left(I - \frac{y_k s_k^\top }{y_k^\top s_k}\right) + \frac{s_k s_k^\top }{y_k^\top s_k}.$$ , set $H_k = H_{j_k}$. Compute the direction $- W H_k W^\top g_k$. Perform a backtracking line-search and update $t_{k+1}$ and $x_{k+1}$ as in Algorithm [\[alg:fevconsalg\]](#alg:fevconsalg){reference-type="ref" reference="alg:fevconsalg"}. ## `Low-Eval` implementation {#subsec:setuplow} We now elaborate on our implementation of Algorithm [\[alg:pdsf\]](#alg:pdsf){reference-type="ref" reference="alg:pdsf"}, and more precisely on the calculation of the polling sets. Our algorithm uses positive generators of the approximate tangent cones described in Section [2.2](#subsec:tangcone){reference-type="ref" reference="subsec:tangcone"}. By describing an approximate tangent cone as a conic hull of a finite set of vectors, we can then use those vectors as (feasible) directions. The problem of finding such positive generators from a description of the cone through linear inequalities has attracted significant research in computational geometry, and is sometimes referred to as the representation conversion problem [@matheiss1980survey]. Recent advances in linearly constrained optimization have featured off-the-shelf softwares to compute those generators [@beck2020convergence]. We follow here a popular approach in the direct-search community [@lewisetal2007], that splits the problem of computing positive generators in two cases. In the first case, we are able to leverage the description of the approximate normal cone through positive generators given by [\[eq:normalcone\]](#eq:normalcone){reference-type="eqref" reference="eq:normalcone"} to directly define that of the approximate tangent cone. In the second case, we compute positive generators for a subset of the cone, and positive generators of the tangent cone are then obtained by considering the union of all these sets for all possible subsets of columns that yield a full row rank matrix [@CJPrice_IDCoope_2003]. One drawback of this strategy is that it leads to combinatorial explosion in the subsets of columns that must be considered and the number of positive generators that are obtained. For this reason, several implementations [@TGKolda_RMLewis_VTorczon_2003; @lewisetal2007] have relied on the double description method from computational geometry [@KFukuda_AProdon_1996]. This technique can significantly reduce the number of generators that are used to describe the approximate tangent cone, in the minority of cases where it is needed on standard test problems [@lewisetal2007]. Our implementation is that of a probabilistic variant of the aforementioned approach proposed by Gratton et al. [@gratton2019direct], in which the approximate tangent cone is decomposed into a subspace part and a pointed cone part (i.e. a cone that does not contain a straight line). Given a set of generators for the approximate tangent cone, we can then replace the subset related to the subspace by a direction drawn uniformly at random within that subspace and its negative, while we can randomly sample a fraction of the other generators corresponding to the pointed cone part. Such an approach reduces the number of polling directions even further, while being endowed with almost-sure convergence guarantees [@gratton2019direct Proposition 7.1]. Our implementation follows that of the `dspfd` MATLAB code [@gratton2019direct], that uses its own implementation of the double description method. ## Classes of problems tested and profiles used Evaluating optimization methods crucially involves assessing their performance across diverse scenarios. In pursuit of this, we perform experiments on smooth, noisy, and non-smooth problems. For each category, the test set is classified into three distinct classes, namely bound constrained problems, general linearly constrained problems, and problems with at least one linear inequality constraint. Detailed dimensions and inequality counts for each problem are provided in the Appendix for reference. For smooth bound constrained problems, we selected $41$ instances from the CUTEst library. The dimensions of these instances range from $2$ to $20$, and the number of bounds varies between $1$ and $40$. The relevant details are summarized in Table [3](#tab:bound){reference-type="ref" reference="tab:bound"}. In the context of smooth general linearly constrained problems, we consider a comprehensive set of $76$ CUTEst problems. Each of these problems involves at least one linear constraint, which is not a bound on the variable. The dimensions vary from $2$ to $24$, and in cases where linear inequalities are present, their count ranges from $1$ to $2000$. A detailed overview of these general constrained problems can be found in Tables [5](#tab:linequ){reference-type="ref" reference="tab:linequ"} to [7](#tab:linineequ){reference-type="ref" reference="tab:linineequ"}. To investigate the behavior of the optimization solvers on noisy functions, we conduct experiments using perturbed versions of the aforementioned problems. Following the approach of [@JJMore_SMWild_2009], the perturbed functions are formulated as $f(x) = \phi(x)(1 + \xi (x))$, where $\phi$ represents the original smooth function. In this case, $\xi(x)$ is a realization of a uniform random variable $U(-\epsilon_f, \epsilon_f)$. These noisy functions provide valuable insights into the robustness of optimization algorithms in practical scenarios. For the non-smooth problems, we transformed the smooth problem set by incorporating $\ell_1$ penalties into the objective function. This conversion involved relocating some of the constraints to the objective function. To create such non-smooth constrained problems, we considered problems with both bounds and general linear constraints, and moved either the general linear constraints or the bounds into the objective function. As a result of this transformation, we generated a total of 52 bound constrained problems and 107 problems with general linear constraints, out of which 52 included at least one inequality constraint. Comprehensive details about these problems are presented in Tables [10](#tab:nonsmoothbound){reference-type="ref" reference="tab:nonsmoothbound"} to [16](#tab:nonsmoothlininequ){reference-type="ref" reference="tab:nonsmoothlininequ"}. In generating general linearly constrained optimization problems, we adopt a method where we penalize only the first portion of the bound constraints in certain cases. This prevents the outcome from being dominated solely by linear equality or inequality constraints. We denote this category as \"$1/2 B$\" in the tables for ease of reference. As an illustrative example, let us consider the transformation of problem LSQFIT. The original problem is formulated as follows: $$\begin{aligned} \min_{x,y} \quad & \sum_{i=1}^5 (a_ix+y - b_i)^2\\ \textrm{s.t.} \quad & x + y \leq 0.85\\ & x \geq 0, \end{aligned}$$ where $a = [0.1, 0.3, 0.5, 0.7, 0.9]$ and $b =[0.25, 0.3, 0.625, 0.701, 1.0]$. After the transformation, the problem becomes: $$\begin{aligned} \min_{x,y} \quad & \sum_{i=1}^5 (a_ix+y - b_i)^2 + \lambda |x + y - 0.85|\\ \textrm{s.t.} \quad & x \geq 0, \end{aligned}$$ where $\lambda$ represents the penalty parameter. By incorporating the $\ell_1$ penalty term, we effectively convert the original constrained problem into a non-smooth bound constrained problem, enabling the exploration of optimization algorithms in scenarios involving non-smooth objective functions and bound constraints. Having outlined the landscape of our test problems, we now shift our focus to the metric employed for comparative evaluation. Here, we introduce performance profiles as a tool to gauge optimization solvers' effectiveness. As outlined in [@EDDolan_JJMore_2002], these profiles provide a mean of assessing the performance of a designated set of solvers $\mathcal{S}$ across a given set of problems $\mathcal{P}$. They are a visual tool where the highest curve corresponds to the solver with the best overall performance. Let $t_{p,s}>0$ be a performance measure of the solver $s \in \mathcal{S}$ on the problem $p \in \mathcal{P}$, which in our case was set to the number of function evaluations. The curve for a solver $s$ is defined as the fraction of problems where the performance ratio is at most $\alpha$, $$\rho_s(\alpha) \; = \; \frac{1}{|\mathcal{P}|} \text{size} \left\{ p \in \mathcal{P} : r_{p,s} \leq \alpha \right\},$$ where the performance ratio $r_{p,s}$ is defined as $$r_{p,s} \; = \; \frac{t_{p,s}}{\min \{ t_{p,s} : s \in \mathcal{S}\}}.$$ The convention $r_{p,s} = +\infty$ is used when a solver $s$ fails to satisfy the convergence test for problem $p$. The convergence test used is $$\label{eq:convprof} f(x_0) - f(x) \; \geq \; (1 - \tau) (f(x_0) - f_L),$$ where $\tau > 0$ is a tolerance, $x_0$ is the starting point for the problem, and $f_L$ is computed for each problem $p \in \mathcal{P}$ as the smallest value of $f$ obtained by any solver within a given number of function evaluations. Solvers with the highest values of $\rho_s(1)$ are the most efficient, and those with the highest values of $\rho_s(\alpha)$, for large $\alpha$, are the most robust. # Numerical Results {#sec:numres} ## Smooth problems ### Bound constrained problems {#bound-constrained-problems .unnumbered} Analyzing Figure [2](#fig:boundsmooth){reference-type="ref" reference="fig:boundsmooth"}, one observes that `Full-Low Evaluation` (red curve) demonstrates the best performance in terms of efficiency (as indicated by the highest curve at a ratio of 1). It is closely followed by pure `Full-Eval` (magenta curve), with `NOMAD` ranking third. However, when considering robustness, `NOMAD` (black curve) outperforms the others, while our method ranks second, making it the best overall. -25ex -30ex {#fig:boundsmooth} 5ex {#fig:boundsmooth} -25ex [\[fig:boundsmooth\]]{#fig:boundsmooth label="fig:boundsmooth"} ### Linearly constrained problems {#linearly-constrained-problems .unnumbered} On general linear equality problems, Figure [4](#fig:gensmooth){reference-type="ref" reference="fig:gensmooth"} illustrates that our method outperforms the three solvers in terms of both efficiency and robustness. Pure `Full-Eval` comes second in terms of efficiency, while pure `Low-Eval` performs exceptionally well in terms of robustness and ranks second for that metric. On the other hand, `NOMAD` exhibits lower performance due to its limited handling of linear equality constraints, which are present in some of the problems. Figure [6](#fig:linsmooth){reference-type="ref" reference="fig:linsmooth"} provides a more specific comparison of the four solvers on the subset of problems that contain at least one inequality constraint. Even in this context, `Full-Low Evaluation` demonstrates the best performance, followed by `Low-Eval` and then `Full-Eval`. It is worth noting that `NOMAD` shows improved performance compared to the previous experiment given the lack of equality constraints. -25ex -30ex {#fig:gensmooth} 5ex {#fig:gensmooth} -25ex [\[fig:gensmooth\]]{#fig:gensmooth label="fig:gensmooth"} -25ex -30ex {#fig:linsmooth} 5ex {#fig:linsmooth} -25ex [\[fig:linsmooth\]]{#fig:linsmooth label="fig:linsmooth"} ## Non-smooth problems ### Bound constrained problems {#bound-constrained-problems-1 .unnumbered} Figure [8](#fig:boundnonsmooth){reference-type="ref" reference="fig:boundnonsmooth"} displays the results obtained from testing non-smooth bound constrained problems. `Full-Low Evaluation` is here the most efficient solver, while `Full-Eval` takes the second spot for both low and high accuracy. Meanwhile, `NOMAD` showcases the best robustness. This observed ranking of solvers in the bound constrained setting remains consistent even with the introduction of the non-smooth regularization. -25ex -30ex {#fig:boundnonsmooth} 5ex {#fig:boundnonsmooth} -25ex [\[fig:boundnonsmooth\]]{#fig:boundnonsmooth label="fig:boundnonsmooth"} ### Linearly constrained problems {#linearly-constrained-problems-1 .unnumbered} In Figure [10](#fig:gennonsmooth){reference-type="ref" reference="fig:gennonsmooth"}, we present the results on general non-smooth problems. One can see that the `Full-Low Evaluation` curve is above all, followed by `Full-Eval` and `Low-Eval`, then `NOMAD`. As with the smooth case, employing `Full-Low Evaluation` yields better results than using individual steps alone, providing further confirmation of the effectiveness of our approach. Furthermore, even within this context, `NOMAD` faces challenges posed by equality constraints. However, upon their removal as shown in Figure [18](#fig:linnoisy){reference-type="ref" reference="fig:linnoisy"}, `NOMAD` demonstrates improved robustness compared to `Full-Eval`. -25ex -30ex {#fig:gennonsmooth} 5ex {#fig:gennonsmooth} -25ex [\[fig:gennonsmooth\]]{#fig:gennonsmooth label="fig:gennonsmooth"} -25ex -30ex {#fig:comp4-determult} 5ex {#fig:comp4-determult} -25ex [\[fig:comp4-determult\]]{#fig:comp4-determult label="fig:comp4-determult"} ## Noisy functions ### Bound constrained problems {#bound-constrained-problems-2 .unnumbered} In this context, `NOMAD` demonstrates the performance in terms of efficiency and robustness. Referring to Figure [14](#fig:boundnoisy){reference-type="ref" reference="fig:boundnoisy"}, we can see that the curve corresponding to `Full-Low Evaluation` is between the `Full-Eval` and `Low-Eval` curves. Such results are conform to observations made in the unconstrained case [@ASBerahas_OSohab_LNVicente_2022], especially for low accuracy. This correspondence arises from `Full-Eval` performing poorly when $h$ is equal to the square root of machine precision. Note that `Full-Low Evaluation` is able to outperform `Low-Eval` for high accuracy in term of robustness. -25ex -30ex {#fig:boundnoisy} 5ex {#fig:boundnoisy} -25ex [\[fig:boundnoisy\]]{#fig:boundnoisy label="fig:boundnoisy"} ### Linearly constrained problems {#linearly-constrained-problems-2 .unnumbered} When tested on general linear equality constrained problems, pure `Low-Eval` (probabilistic direct search) stands out as the most efficient solver, followed by `Full-Low Evaluation` which demonstrates superior robustness. Conversely as observed in Figure [16](#fig:gennoisy){reference-type="ref" reference="fig:gennoisy"}, `NOMAD` experiences a performance decline similar to observations in both smooth and non-smooth cases. Figure [18](#fig:linnoisy){reference-type="ref" reference="fig:linnoisy"} sheds light on problems featuring linear inequalities. Notably, in this context, `NOMAD`'s performance stands on par with `Full-Eval`, and it even surpasses it, especially under conditions demanding higher accuracy. -25ex -30ex {#fig:gennoisy} 5ex {#fig:gennoisy} -25ex [\[fig:gennoisy\]]{#fig:gennoisy label="fig:gennoisy"} -25ex -30ex {#fig:linnoisy} 5ex {#fig:linnoisy} -25ex [\[fig:linnoisy\]]{#fig:linnoisy label="fig:linnoisy"} # Conclusions {#sec:conc} We have proposed an instance of the `Full-Low Evaluation` framework tailored to the presence of bound and linear constraints, by combining projected BFGS steps with probabilistic direct-search steps within approximate tangent cones. The result method is equipped with similar guarantees than in the unconstrained case. In addition, its performance has been validated in linearly constrained problems with smooth, non-smooth, and noisy objectives. Those experiments overall suggest that our algorithm is able to get the best of both worlds, and improve over existing algorithms that do not combine `Full-Eval` and `Low-Eval` steps. Other variants of the `Full-Low Evaluation` framework may be able to improve on our current implementation. In particular, one could rely on trust-region steps as `Full-Eval`, while one- or two-point feedback feasible approaches that have been proposed more generally in the convexly constrained setting. In fact, extending the `Full-Low Evaluation` framework to nonlinear, convex constraints is a natural continuation of our work, which may benefit from existing results in feedback methods as well as mature theory regarding projected gradient techniques. # Acknowledgments {#acknowledgments .unnumbered} This work is partially supported by the U.S. Air Force Office of Scientific Research (AFOSR) award FA9550-23-1-0217, and by Agence Nationale de la Recherche through program ANR-19-P3IA-0001 (PRAIRIE 3IA Institute). # List of Problems {#sec:pblist} Name Size Bounds ---------- ------ -------- chenhark 10 10 explin 12 24 harkerp2 10 10 hatfldb 4 5 hs3 2 1 hs4 2 2 maxlika 8 16 ncvxbqp1 10 20 oslbqp 8 11 pspdoc 4 1 weeds 3 4 camel6 2 4 eg1 3 4 cvxbqp1 10 20 : Bound constrained problems. Name Size Bounds ---------- ------ -------- explin2 12 24 hart6 6 12 himmelp1 2 4 hs2 2 1 hs3mod 2 1 hs5 2 4 mccormck 10 20 ncvxbqp2 10 20 palmer1a 6 2 palmer4a 6 2 qrtquad 12 12 simbqp 2 2 yfit 3 1 expquad 12 24 : Bound constrained problems. Name Size Bounds ---------- ------ -------- hatflda 4 4 hs1 2 1 hs38 4 8 hs45 5 10 hs110 10 20 logros 2 2 mdhole 2 1 ncvxbqp3 10 20 palmer2b 4 2 palmer5b 9 2 probpenl 10 20 s368 8 16 sineali 20 40 : Bound constrained problems. Name Size Bounds LE ---------- ------ -------- ---- aug2d  24 0 9 bt3 5 0 3 hs28 3 0 1 hs49 5 0 2 hs51 5 0 3 cvxqp2 10 20 2 fccu 19 19 8 hs041 4 8 1 hs54 6 12 1 hs62 3 6 1 ncvxqp1 10 20 5 ncvxqp3 10 20 5 ncvxqp5 10 20 2 fits 10 10 6 portfl2 12 24 1 portfl4 12 24 1 reading2 9 14 4 sosqp2 20 40 11 : Linear equality constrained problems. Name Size Bounds LE ---------- ------ -------- ---- genhs28 10 0 8 hs9 2 0 1 hs48 5 0 2 hs50 5 0 3 hs52 5 0 3 cvxqp1 10 20 5 degenlpa 20 40 15 hong 4 8 1 hs53 5 10 3 hs55 6 8 6 hs112 10 10 3 ncvxqp2 10 20 5 ncvxqp4 10 20 2 ncvxqp6 10 20 2 portfl1 12 24 1 portfl3 12 24 1 portfl6 12 24 1 sosqp1 20 40 11 : Linear equality constrained problems. Name Size Bounds LE LI ---------- ------ -------- ---- ------ avgasa 8 16 0 10 biggsc4 4 8 0 7 dualc2 7 14 1 228 expfitb 5 0 0 102 hatfldh 4 8 0 7 hs118 15 30 0 17 hs21mod 7 8 0 1 hs268 5 0 0 5 hs35mod 3 4 0 1 hs36 3 6 0 1 hs44 4 4 0 6 hs76 4 4 0 3 hs86 5 5 0 10 lsqfit 2 1 0 1 oet3 4 0 0 1002 simpllpa 2 2 0 2 sipow1 2 0 0 2000 sipow2 2 0 0 2000 sipow3 4 0 0 2000 stancmin 3 3 0 2 : Linear inequality constrained problems. Name Size Bounds LE LI ---------- ------ -------- ---- ------ tfi2 3 0 0 101 avgasb 8 16 0 10 dualc1 9 18 1 214 dualc5 8 16 1 277 expfita 5 0 0 22 expfitc 5 0 0 502 hs105 8 16 0 1 hs21 2 4 0 1 hs24 2 2 0 3 hs35 3 3 0 1 hs37 3 6 0 2 hs44new 4 4 0 6 hubfit 2 1 0 1 oet1 3 0 0 1002 pentagon 6 0 0 15 simpllpb 2 2 0 3 sipow1m 2 0 0 2000 sipow2m 2 0 0 2000 sipow4 4 0 0 2000 zecevic2 2 4 0 2 : Linear inequality constrained problems. Name Pen. Const ---------- ------------ avgasa LI biggsc4 LI dualc2 LE & LI hatfldh LI hs118 LI hs21mod LI hs35mod LI hs36 LI hs44 LI hs76 LI hs86 LI lsqfit LI simpllpa LI stancmin LI avgasb LI dualc1 LE & LI dualc5 LE & LI : Non-smooth bound constrained problems. Name Pen. Const ---------- ------------ hs105 LI hs21 LI hs24 LI hs35 LI hs37 LI hs44new LI hubfit LI simpllpb LI zecevic2 LI cvxqp2 LE fccu LE hs41 LE hs54 LE hs62 LE ncvxqp1 LE ncvxqp3 LE ncvxqp5 LE : Non-smooth bound constrained problems. Name Pen. Const ---------- ------------ odfits LE portfl2 LE portfl4 LE reading2 LE sosqp2 LE cvxqp1 LE degenlpa LE hong LE hs53 LE hs55 LE hs112 LE ncvxqp2 LE ncvxqp4 LE ncvxqp6 LE portfl1 LE portfl3 LE portfl6 LE sosqp1 LE : Non-smooth bound constrained problems. Name Pen. Const --------- ------------ dualc2 LI dualc1 LI dualc5 LI cvxqp2 B cvxqp2 1/2 B fccu B fccu 1/2 B hs41 B hs41 1/2 B hs54 B hs54 1/2 B hs62 B hs62 1/2 B ncvxqp1 B ncvxqp1 1/2 B ncvxqp3 B ncvxqp3 1/2 B ncvxqp5 B ncvxqp5 1/2 B : Non-smooth linear equality constrained problems. Name Pen. Const ---------- ------------ odfits B odfits 1/2 B portfl2 B portfl2 1/2 B portfl4 B portfl4 1/2 B reading2 B reading2 1/2 B sosqp2 B sosqp2 1/2 B cvxqp1 B cvxqp1 1/2 B degenlpa B degenlpa 1/2 B hong B hong 1/2 B hs53 B hs53 1/2 B : Non-smooth linear equality constrained problems. Name Pen. Const --------- ------------ hs55 B hs55 1/2 B hs112 B hs112 1/2 B ncvxqp2 B ncvxqp2 1/2 B ncvxqp4 B ncvxqp4 1/2 B ncvxqp6 B ncvxqp6 1/2 B portfl1 B portfl1 1/2 B portfl3 B portfl3 1/2 B portfl6 B portfl6 1/2 B sosqp1 B sosqp1 1/2 B : Non-smooth linear equality constrained problems. Name Pen. Const --------- ------------ avgasa B avgasa 1/2 B biggsc4 B biggsc4 1/2 B dualc2 B dualc2 LE hatfldh B hatfldh 1/2 B hs118 B hs118 1/2 B hs21mod B hs21mod 1/2 B hs35mod B hs35mod 1/2 B hs36 B hs36 1/2 B hs44 B : Non-smooth linear inequality constrained problems. Name Pen. Const ---------- ------------ hs44 1/2 B hs76 B hs76 1/2 B hs86 B hs86 1/2 B lsqfit B lsqfit 1/2 B simpllpa B simpllpa LE stancmin B stancmin 1/2 B avgasb B avgasb 1/2 B dualc1 B dualc1 LE dualc5 B dualc5 LE hs105 B : Non-smooth linear inequality constrained problems. Name Pen. Const ---------- ------------ hs105 1/2 B hs21 B hs21 1/2 B hs24 B hs24 1/2 B hs35 B hs35 1/2 B hs37 B hs37 1/2 B hs44new B hs44new 1/2 B hubfit B hubfit 1/2 B simpllpb B simpllpb 1/2 B zecevic2 B zecevic2 1/2 B : Non-smooth linear inequality constrained problems. [^1]: LAMSADE, CNRS, Université Paris Dauphine-PSL, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France (`[email protected]`). [^2]: Department of Industrial and Systems Engineering, Lehigh University, 200 West Packer Avenue, Bethlehem, PA 18015-1582, USA (`[email protected]`). [^3]: Department of Industrial and Systems Engineering, Lehigh University, 200 West Packer Avenue, Bethlehem, PA 18015-1582, USA (`[email protected]`). [^4]: In the unconstrained case [@ASBerahas_OSohab_LNVicente_2022], we have proposed the switching condition $\beta \geq \gamma \rho(\alpha_k)$. Both work for the convergence theory, in the sense of Lemma [Lemma 3](#the:nsc){reference-type="ref" reference="the:nsc"}. [^5]: <https://github.com/sohaboumaima/FLE>

arxiv_math

--- abstract: | We consider the deconstruction/reconstruction of extensions in varieties of algebras which are modules expanded by multilinear operators. The parametrization of extensions determined by abelian ideals with unary actions agrees with the previous development of extensions realizing affine datum in arbitrary varieties of universal algebras. We establish a Well's type theorem which, for a fixed affine ideal, characterizes those ideal-preserving derivations of a group-trivial extension as a Lie algebra extension of the compatible pairs of derivations of the datum algebras associated to the ideal by the cohomological derivations of the datum. For these varieties, we establish a low-dimensional Hochschild-Serre exact sequence associated to an arbitrary extension equipped with an additional affine action. address: School of Mathematics, Sichuan University, Chengdu,610065, Sichuan, PRC author: - Alexander Wires title: Extensions of Multilinear Module Expansions --- # Introduction In this manuscript, we examine the parameters for characterizing arbitrary extensions in varieties of $R$-modules expanded by multilinear operations. These algebras can be seen as a special case of Higgin's [@higgins] groups with multiple operators formalisms but only slightly broader than Kurosh's formalism of $\Omega$-algebras since we consider modules rather than vector spaces. We describe a low-dimensional cohomology which is replete with the expected machinery of 2-cocycles, 2-coboundaries, derivations and stabilizing automorphisms in order for second-cohomology to classify extensions; in addition, we define a notion of principal derivation connected to special stabilizing automorphisms which determines a first cohomology which is not always defined for cohomologies of nonassociative or general structures. Those extensions realizing affine datum as defined in Wires [@wiresI] can be characterized by datum with abelian ideals and unary actions only. The 2-cocycle identity for a subvariety is formalized by a satisfaction relation associated to a multisorted signature formed by the action terms and factor sets; in this way, the variety membership (or equational theory) of extensions is incorporated into cohomology as a parameter. This affords a Galois connection between the lattice of subvarieties and second-cohomology. The notion of a derivation (as opposed to cohomological derivation of datum) makes sense for algebras in these varieties and so we prove a Well's-type theorem characterizing kernel-preserving derivations of a group-trivial extension realizing affine datum by a Lie algebra extension of derivations. A similar result for automorphisms is established in Wires [@wiresIII Thm 1.4]. Since we are able to define a first cohomology group for any variety of multilinear module expansions, we establish a low-dimension Hochschild-Serre (or inflation/deflation) exact sequence associated to a general extension equipped with an additional affine action. The topic and literature on the cohomology or parametrization of extensions in various varieties of modules expanded by bilinear operations is surely too vast to give adequate justice here so those references cited reflect only the author's very limited knowledge - mostly to more recent papers. The work parametrizing nonabelian extensions in the present manuscript recovers in a uniform manner the cohomological classification of extensions previously developed for rings (Everett [@rings]), associative algebras (Hochschild [@hoch], Agore and Militaru [@agore]), Lie algebras (Inassaridze, Khmaladze and Ladra [@lie]), Liebniz algebras (Casas, Khmaladze and Ladra [@leibniz]), dendriform and bilinear Rota-Baxter algebras (Das and Rathee [@das]), Lie-Yamaguti algebras ( Yamaguti [@yamaguti]) or conformal algebras (Bakalov, Kac and Voronov[@conform1], Hou and Zhao [@conform2], Smith [@smith1]) to give just a few well-studied examples. Here we are able to generalize to any equational class of $R$-modules expanded by any set of multilinear operations. Our Theorem [Theorem 45](#thm:HSexact){reference-type="ref" reference="thm:HSexact"} extends to general extensions in our particular varieties the Hochschild-Serre exact sequence established in Wires [@wiresII] for central extensions with an idempotent element in varieties with a difference term. Two applications for the Hochschild-Serre exact sequence (as outlined for groups in Karpilovsky [@karpil]) is in the development of the Schur multiplier and its characterization by relative commutators in free presentation, and the connection with perfect algebras (which for groups can be read in Milnor [@milnor]). For varieties of multilinear expansions of $R$-modules, the analogous results are already recovered by specialization of the more general results in [@wiresII]; for examples of Schur-type results in recent work on some varieties of interest which would fall under the general umbrella of [@wiresII], we may look at Batten [@batten1] for Lie algebras, Elyse [@elyse1] and Mainellis [@mainellis1] for Leibniz algebras and Mainellis [@mainellis2; @mainellis3] for diassociative algebras. The inspiration for Theorem [Theorem 39](#thm:derivations){reference-type="ref" reference="thm:derivations"} on kernel-preserving derivations in arbitrary varieties of multilinear module expansions can be found in Hou and Zhao [@conform2] where a similar result is established for abelian extensions of associative conformal algebras. The principal motivation for the present manuscript derives from the work in [@wiresI; @wiresII] for extensions realizing affine datum in arbitrary varieties of universal algebras. We would like to consider the parametrization of general or "nonabelian" extensions of universal algebras and develop proper interpretations for any possible higher cohomologies with general and affine datum. Varieties of $R$-modules expanded by multilinear operations form a particularly well-behaved class of Mal'cev varieties in which the generators of the commutator relations can be given a manageable form; therefore, these classes of algebras are a good place to start. The abelian group reduct and multilinearity of the higher arity operations appears to simplify structure and calculations but still allows for nonabelian extensions. These varieties may serve as a test case or set of examples for continuing the work from [@wiresI; @wiresII]. In light of Theorem [Theorem 45](#thm:HSexact){reference-type="ref" reference="thm:HSexact"} and [@wiresII], we think there is a good argument to be made that a fair amount of the compiled theory and results presented in Karpilovsky's book [@karpil] can be successfully developed for these varieties. We do not consider higher cohomologies at all in this manuscript. Since we consider nonabelian extensions in arbitrary varieties (so any equational axiomatization), we find it difficult to fit the present development into the chain complex framework of modern cohomology inspired by algebraic topology, even in the affine datum case. In our view, the 2-cocycle condition for groups which is determined by a coboundary map in a chain complex, in the case of general varieties is explicitly tied to the equational theory of extensions; that is, 2-cocycles are interpretations of certain multisorted signatures satisfying some equations in that mixed signature. Our personal intuition is that it is be possible to give a more explicit and concrete interpretation for higher cohomologies analogous to that given by Holt [@holt] for groups. Such an explicit description would involve equations among different levels of action terms which may by easier to directly modify in order to describe the "correct" multisorted structures for nonabelian cohomologies. In any case, all this deserves its own more focused future effort. We leave these questions for Section [6](#section:6){reference-type="ref" reference="section:6"}. # Preliminaries {#section:2} Let $\mathcal V$ be a variety of $R$-modules for a fixed ring $R$ expanded by multilinear operations indexed by the set $F$; formally, the signature $\tau = \{+,-1,0, r : r \in R \} \mathrel{\cup} F$ is the union of the signature $\{+, -, 0, r : r \in R \}$ of an $R$-module with $F$. Note the ring element $r \in R$ in the signature is interpreted in a module as the unary operation which is scalar multiplication by $r$. An algebra $M \in \mathcal V$ can be written in a slightly incorrect notation as $M = \left\langle _{R}M,F^{M} \right\rangle$ where $_{R} M$ is an $R$-module and $F^{M} = \{f^{M}: f \in F \}$ is the part of the signature which names the multilinear operations; that is, for $f \in F$ with arity $\mathop{\mathrm{ar}}f=n$, we have for each coordinate $1 \leq i \leq n$ $$f(x_{1},\ldots,r \cdot y + z,\ldots,x_{n}) = r \cdot f(x_{1},\ldots,y,\ldots,x_{n}) + f(x_{1},\ldots,z,\ldots,x_{n})$$ for $r \in R$, $x_{i},y,z \in M$. If we let $_{R} \mathcal M_{F}$ denote the largest variety of $R$-modules expanded by multilinear operation named by $F$, then $\mathcal V \, \leq \, _{R}\mathcal M_{F}$. Let $\mathop{\mathrm{Id}}\mathcal V$ denote the set of identities of the variety $\mathcal V$. Congruences $\alpha \in \mathop{\mathrm{Con}}M$ are in bijective correspondence with *ideals* $I \triangleleft M$ which are submodules of $M$ that are *absorbing* for each of the multilinear operations; that is, for each $f \in F$ with $n=\mathop{\mathrm{ar}}f$, $f(x_{1},\ldots,x_{n}) \in I$ whenever some $x_{i} \in I$. For a congruence $\alpha \in \mathop{\mathrm{Con}}M$, the corresponding ideal $I_{\alpha}$ is the $\alpha$-class which contains $0 \in M$. For an ideal $I \triangleleft M$, the corresponding congruence is given by $\alpha_{I} = \{(a,b) \in M^{2}: a-b \in I \} \in \mathop{\mathrm{Con}}M$. Given a function $f: X \rightarrow A$ and $\vec{x} \in X^{n}$ we write $f(\vec{x}):= (f(x_{1}),\ldots,f(x_{n}))$ for the tuple which is the evaluation of the function at each coordinate. We will have to write formulas which sum the values of higher arity functions on substitutions over different subsets of coordinates between fixed tuples. It will be convenient to introduce notational conventions for this. For $0 < n \in \mathds{N}$, we write $[n] = \{1,\ldots,n\}$ for the initial segment of positive integers and $[n]^{\ast} = \{ s \subseteq \{ 1,\ldots,n \}: 0 < |s| < n \}$ for the non-empty proper subsets of $[n]$. Given two tuples $\vec{q},\vec{p} \in Q^{n}$ and $s \in [n]^{\ast}$, we form the tuple $\vec{q}_{s} \left[ \vec{p} \right] \in Q^{n}$ $$\vec{q}_{s} [\vec{p}] (i) = \begin{cases} \vec{p}(i) & , i \in s \\ \vec{q}(i) & , i \not\in s; \end{cases}$$ that is, $\vec{q}_{s} [\vec{p}]$ is formed by substituting the values of the tuple $\vec{p}$ into the tuple $\vec{q}$ at those coordinates specified by $s$. Then for a function $f: Q^{n} \rightarrow M$, we can write the evaluation of the function on the constructed tuple in two ways as $f(\vec{q})_{s}[\vec{p}] = f \left( \vec{q}_{s} [\vec{p}] \right)$. Both ways will be convenient if we have to make additional substitutions of the coordinates over overlapping subsets of coordinates. For $\vec{x} \in Q^{n}$ and $s \in [n]^{\ast}$, $\vec{x}|_{s} \in Q^{s}$ is the tuple $\vec{x}$ restricted to the coordinates $s$. For an abelian group $A$, suppose we have an operation $f: A^{n} \rightarrow A$ and a map $l: Q \rightarrow A$. Given $s \in [n]^{\ast}$, this defines a function $a(f,s) : Q^n \times A^{n} \rightarrow A$ by the rule $a(f,s)(\vec{q},\vec{a}) := f(l(\vec{q}))_{s}[\vec{a}]$; certainly, the function is independent of the coordinates of $\vec{q}$ in $s$ and independent of the coordinates of $\vec{a}$ outside of $s$. Then by fixing the tuples $\vec{q} \in Q^{n}, \vec{a} \in A^{n}$, we can sum $$\begin{aligned} \label{eqn:1} \sum_{s \in [n]^{\ast}} a(f,s)(\vec{q},\vec{a})\end{aligned}$$ the values of the function on substitutions over all nonempty proper subsets of the coordinates. Of course, what really matters are the value of the functions over the possibly dependent coordinates so that we could naturally consider the function $a'(f,s): Q^{[n] - s} \times A^{s} \rightarrow A$ where $a'(f,s)(\vec{q}|_{[n]-s},\vec{a}|_{s}) = a(f,s)(\vec{q},\vec{a})$. Since we will later need to consider additional substitutions in our fomulas, we have found the form in Eq.([\[eqn:1\]](#eqn:1){reference-type="ref" reference="eqn:1"}) more convenient. An exception is sometimes made for unary subsets $s=\{i\} \in [n]^{\ast}$; in this case, we may sometimes write $a(f,i)(q_{1},\ldots,q_{i-1},a_{i},q_{i+1},\ldots,q_{n})$ rather than $a(f,i)(\vec{q},\vec{a})$ when the choice of $\vec{q} \in Q^{n},\vec{a} \in A^{n}$ is understood. In writing $a(f,s)$ we decided to take the viewpoint of substituting $\vec{a}$ into the $s$-coordinates of $f(l(\vec{q}))$ for the following reason: if $f$ is multilinear over a module structure on $A$, then for each $\vec{q} \in Q^{n}$, $a(f,s)(\vec{q},-)$ is multilinear over the restriction to the $s$-coordinates over $A$. # Machinery {#section:3} In this section, we develop the machinery around the first and second-cohomologies. **Definition 1**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations. A pair of algebras $(Q,I)$ with $Q,I \in \mathcal V$ is called *datum* in $\mathcal V$. **Definition 2**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations and $Q,I \in \mathcal V$. An *action* $Q \ast I$ is a sequence of operations $\ast = \{a(f,s): f \in F, s \in [\mathop{\mathrm{ar}}f]^{\ast} \}$ where $a(f,s):Q^{\mathop{\mathrm{ar}}f} \times I^{\mathop{\mathrm{ar}}f} \rightarrow I$. **Definition 3**. Let $\tau$ be a signature for a variety $\mathcal V$ of $R$-modules expanded by multilinear operations $F$ and $(Q,I)$ be datum in $\mathcal V$. A 2-*cocycle* for the datum $(Q,I)$ is a sequence $T = \{ T_{+}, T_{r}, T_{f}, a(f,s) : r \in R, f \in F, s \in [\mathop{\mathrm{ar}}f]^{\ast} \}$ where 1. $T_{+} : Q^{2} \rightarrow I$ such that $T_{+}(x,0)=T_{+}(0,x)=0$; 2. $T_{r} : Q \rightarrow I$ for each $r \in R$ such that $T_{r}(0)=0$; 3. $T_{f} : Q^{\mathop{\mathrm{ar}}f} \rightarrow I$ for $f \in F$ such that $T_{f}(\vec{x})=0$ whenever some $x_{i}=0$; 4. $\{ a(f,s): f \in F, s \in [\mathop{\mathrm{ar}}f]^{\ast} \}$ is an action $Q \ast I$ such that for each $\vec{q} \in Q^{\mathop{\mathrm{ar}}f}$, $a(f,s)(\vec{q},\cdot)|_{s} \in \mathop{\mathrm{Hom}}_{R}(\otimes^{s} \, _{R}I ,\, _{R}I)$ and $a(f,s)(\vec{q},\cdot) \equiv 0$ for any $\vec{q} \in Q^{\mathop{\mathrm{ar}}f}$ with $\vec{q}(i) = 0$ for some $i \not\in s$. The symbols $\{ a(f,s): f \in F, s \in [\mathop{\mathrm{ar}}f]^{\ast} \}$ in a 2-cocycle will be called the *action terms* of the 2-cocycle and the symbols $T_{+}, T_{r}$ and $T_{f}$ the *factor-sets*. A 2-cocycle $T = \{ T_{+}, T_{r}, T_{f}, a(f,s) : r \in R, f \in F, s \in [\mathop{\mathrm{ar}}f]^{\ast} \}$ is a multisorted signature in two diffferent sorts; for example, one sort for the domains of the symbols $T_{+}$, $T_{r}$ and $T_{f}$ and another sort for their codomains, but the symbols $a(f,s)$ use both sorts for the domain. Note that the datum $(Q,I)$ consists of two actual algebras so we have interpretations of the signature $\tau^{I}$ and $\tau^{Q}$ giving the module and multilinear operations for the algebras $I$ and $Q$. A 2-cocycle $T$ for datum $(Q,I)$ is then an interpretation of the multisorted signature $T$ in the multisorted structure $\left\langle I \cup Q, \tau^{I}, \tau^{Q}, T \right\rangle$ which satisfies the properties specified in (T1) - (T4). The structure $\left\langle I \cup Q, \tau^{I}, \tau^{Q}, T \right\rangle$ has $\tau_{I} \cup \tau_{Q} \cup T$ as an appropriate multisorted signature where we have written $\tau_{I}$ for the operation symbols for the sort which have the algebra $I$ as our intended interpretation. In case where the full signature $\tau$ is not stated explicitly but we have named the set of multilinear operations by $F$, we may also informally write the intended structure as $\left\langle _{R} I \cup \, _{R} Q, F^{I}, F^{Q}, T \right\rangle$ where we are using the informal notation $_{R} I \mathrel{\cup} \, _{R} Q$ to denote the different $R$-module structures over the two sorts of $I$ and $Q$. **Definition 4**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $\mathcal F$. Let $T$ be a 2-cocycle for the signature of $\mathcal V$ and $(Q,I)$ be datum in $\mathcal V$. The algebra $I \rtimes_{T} Q$ is defined with universe $I \times Q$ and operations 1. $\left\langle \, a,x \, \right\rangle + \left\langle \, b,y \, \right\rangle := \left\langle \, a+b + T_{+}(x,y), x+y \, \right\rangle$; 2. $r \cdot \left\langle \, a,x \, \right\rangle := \left\langle \, r \cdot a + T_{r}(x), r \cdot x \, \right\rangle$ for $r \in R$; 3. $F_f \left(\left\langle \, a_1, x_1 \, \right\rangle, \ldots, \left\langle \, a_n, x_n \, \right\rangle \right) := \left\langle \, f(\vec{a}) + \sum_{s \in [n]^{\ast}} a(f,s)(\vec{x},\vec{a}) + T_{f}(\vec{x}), f(\vec{x}) \, \right\rangle$ for $f \in F$ with $n = \mathop{\mathrm{ar}}f$. Note there is an embedding $i: I \rightarrow I \rtimes_{T} Q$ given by $i(a):= \left\langle a,0 \right\rangle$ and the second-projection $p_{2}: I \rtimes_{T} Q \rightarrow Q$ is an extension over $Q$ with $i(I) = I_{\ker p_2}$. We would like to place a condition on the 2-cocycle $T$ so that $I \rtimes_{T} Q \in \mathcal V$ whenever $(Q,I)$ is datum in $\mathcal V$. This is done by writing a representation of the terms in $I \rtimes_{T} Q$. **Lemma 5**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$. Let $T$ be a 2-cocycle for the datum $(Q,I)$. For each term $t(\vec{x})$ in the signature $\tau$ of $\mathcal V$, there are terms $t^{\ast,T}, t^{\partial,T}$ in the multisorted signature $\tau_{I} \cup \tau_{Q} \cup T$ such that $I \rtimes_{T} Q \in \mathcal V$ if and only if $Q,I \in \mathcal V$ and $\left\langle I \cup Q, \tau^{I}, \tau^{Q}, T \right\rangle \vDash t^{\ast,T} + t^{\partial,T} = s^{\ast,T} + s^{\partial,T}$ for all $t=s \in \mathrm{Id} \mathcal V$. *Proof.* By the recursive generation of terms, it can be shown that for each term $t(x_{1},\ldots,x_{n})$ in the signature $\tau$ of $\mathcal V$ there are terms $t^{\ast,T}$ and $t^{\partial,T}$ in the multisorted signature $\tau_{I} \cup \tau_{Q} \cup T$ such that $$\begin{aligned} \label{eqn:55} F_{t}(\vec{a},\vec{q}) = \left\langle \, t^{I}(\vec{a}) + t^{\ast,T}(\vec{a},\vec{q}) + t^{\partial,T}(\vec{a},\vec{q}), t^{Q}(\vec{q}) \, \right\rangle\end{aligned}$$ computed in the algebra $I \rtimes_{T} Q$ where we have written $\left( \vec{a}, \vec{q} \right) = \left( \left\langle \, a_{1}, q_{1} \, \right\rangle, \ldots, \left\langle \, a_{n}, q_{n} \, \right\rangle \right)$. The term $t^{\ast,T}$ is recursively constructed from only the module operations and the actions terms and $t^{\partial,T}$ is recursively constructed from the module operations and the operations in $T$ and must contain a factor-set in its composition tree. Informally, we use the module operations and multilinearity to expand $F_{t}(\vec{a},\vec{q})$ into a sum of various composed operations: all the summands that utilize a factor-set are summed together to form $t^{\partial,T}(\vec{a},\vec{q})$, all the parts that utlize action terms but no factor-sets are summed together to form $t^{\ast,T}(\vec{a},\vec{q})$, and the remaining parts must be just the interpretation of the term $t$ in the algebra $I$. Assume $I \rtimes_{T} Q \in \mathcal V$. Since $Q$ is a quotient determined by ideal $i(I)$, we have $I,Q \in \mathcal V$. Then for $t=s \in \mathrm{Id} \mathcal V$ we have $t^{I} = s^{I}$ and $F_{t} = F_{s}$. Then Eqn ([\[eqn:55\]](#eqn:55){reference-type="ref" reference="eqn:55"}) implies $t^{\ast,T} + t^{\partial,T} = s^{\ast,T} + s^{\partial,T}$. Conversely, $I,Q \in \mathcal V$ implies $t^{I}=s^{I}$ and $t^{Q}=s^{Q}$ for $t=s \in \mathrm{Id} \mathcal V$. Then substituting this and $\left\langle I \cup Q, \tau^{I}, \tau^{Q}, T \right\rangle \vDash t^{\ast,T} + t^{\partial,T} = s^{\ast,T} + s^{\partial,T}$ into Eqn ([\[eqn:55\]](#eqn:55){reference-type="ref" reference="eqn:55"}) yields $F_{t}=F_{s}$; thus, $I \rtimes_{T} Q \in \mathcal V$. ◻ **Definition 6**. Let $\tau$ be the signature of a variety $\mathcal V$ of $R$-modules expanded by multilinear operations and $(Q,I)$ datum in the same signature as $\mathcal V$. A 2-cocycle $T$ for the datum $(Q,I)$ is $\mathcal V$-*compatible* if $$\left\langle I \cup Q, \tau^{I}, \tau^{Q}, T \right\rangle \vDash t^{\ast,T} + t^{\partial,T} = s^{\ast,T} + s^{\partial,T} \quad \text{for all} \quad t=s \in \mathrm{Id} \mathcal V.$$ Given an equation $t=s$ appropriate for a variety of multilinear expansions of $R$-modules, we refer to the equation $t^{\ast,T} + t^{\partial,T} = s^{\ast,T} + s^{\partial,T}$ in Definition [Definition 6](#def:3){reference-type="ref" reference="def:3"} as the *general 2-cocycle identity* for $T$ of the equation $t=s$. We may refer to the equation $t^{\ast,T} = s^{\ast,T}$ as the *action identity* for compatibility with $t=s$ and the equation $t^{\partial,T} = s^{\partial,T}$ as the *strict 2-cocycle identity* for $T$ of the equation $t=s$. With the help of Lemma [Lemma 5](#lem:semident){reference-type="ref" reference="lem:semident"}, we shall later see that for extensions realizing affine datum that includes a fixed $\mathcal V$-compatible action as part of the datum, then it is the strict 2-cocycle identity that determines membership of an extension in a variety. Given an action $Q \ast I$, we can form a 2-cocycle $T^{\ast}$ for the datum $(Q,I)$ where the action terms of $T^{\ast}$ are the same as the operations in $\ast$ and the factor-sets are exactly zero. **Definition 7**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations and $(Q,I)$ datum in the same signature as $\mathcal V$. We say an action $Q \ast I$ is $\mathcal V$-*compatible* if the 2-cocycle $T^{\ast}$ is $\mathcal V$-compatible. Since the factor-sets of $T^{\ast}$ are all zero, we must have $t^{\partial,T^{\ast}}=0$ for all terms $t$; therefore, an action $Q \ast I$ is $\mathcal V$-compatible if and only if $t^{\ast,T^{\ast}} = s^{\ast,T^{\ast}}$ is true for all $t=s \in \mathrm{Id} \mathcal V$ if and only if $I \rtimes_{T^{\ast}} Q \in \mathcal V$. We will also allow ourselves the notational convenience to write $I \rtimes_{\ast} Q = I \rtimes_{T^{\ast}} Q$. **Example 8**. We illustrate by writing the 2-cocycle identities for $f \in F$ to be multilinear. Assume $n=\mathop{\mathrm{ar}}f$ and $r\in R$. The equations for linearity in the i-th coordinate are $$\begin{aligned} \label{eqn:3} f(x_{1},\ldots,rx_{i},\ldots,x_{n}) = r \cdot f(x_{1},\ldots,x_{n})\end{aligned}$$ and $$\begin{aligned} \label{eqn:4} f(x_{1},\ldots,x_{i}+y_{i},\ldots,x_{n}) = f(x_{1},\ldots,x_{i},\ldots,x_{n}) + f(x_{1},\ldots,y_{i},\ldots,x_{n}).\end{aligned}$$ To compute the corresponding general 2-cocycle identities of $T$ for datum $(Q,I)$ in $\mathcal V$, we evaluate the corresponding left and right-hand terms in $I \rtimes_{T} Q$. Since $I$ satisfies the identities in $\mathcal V$, we set the first-coordinates equal and delete the identity in the operations of $I$. For Eqn ([\[eqn:3\]](#eqn:3){reference-type="ref" reference="eqn:3"}), calculating the operations $$\begin{aligned} F_{f} \left(\left\langle a_{1},x_{1} \right\rangle, \ldots, r \cdot \left\langle a_{i},x_{i} \right\rangle , \ldots, \left\langle a_{n},x_{n} \right\rangle \right)\end{aligned}$$ and $$\begin{aligned} r \cdot F_{f} \left(\left\langle a_{1},x_{1} \right\rangle, \ldots, \left\langle a_{i},x_{i} \right\rangle , \ldots, \left\langle a_{n},x_{n} \right\rangle \right)\end{aligned}$$ yields the 2-cocycle identity $$\begin{aligned} f^{I}\left(a_{1},\ldots,T_{r}(x_{i}),\ldots,a_{n} \right) &+ \sum_{s \in [n]^{\ast}} a(f,s)\left( (x_{1},\ldots, r \cdot x_{i},\ldots,x_{n}), (a_{1},\ldots,r \cdot a_{i} + T_{r}(x_{i}),\ldots,a_{n}) \right) \\ + \ T_{f}(x_{1},\ldots, r \cdot x_{i},\ldots,x_{n}) \ &= \ \sum_{s \in [n]^{\ast}} r \cdot a(f,s)(\vec{x},\vec{a}) + r \cdot T_{f}(\vec{x}) + T_{r}(f^{Q}(\vec{x}))\end{aligned}$$ for respecting the $r$-action in the i-th coordinate; of course, by cancellation and multilinearity of the action terms this is equivalent to the identity $$\begin{aligned} f^{I} &\left(a_{1},\ldots,T_{r}(x_{i}),\ldots,a_{n} \right) + \sum_{i \not\in s \in [n]^{\ast}} a(f,s)\left( (x_{1},\ldots, r \cdot x_{i},\ldots,x_{n}), \vec{a} \right) \\ &+ \sum_{i \in s \in [n]^{\ast}} a(f,s)\left( \vec{x}, (a_{1},\ldots,T_{r}(x_{i}),\ldots,a_{n}) \right) + T_{f}(x_{1},\ldots, r \cdot x_{i},\ldots,x_{n}) \\ &= \sum_{i \not\in s \in [n]^{\ast}} r \cdot a(f,s)(\vec{x},\vec{a}) + r \cdot T_{f}(\vec{x}) + T_{r}(f^{Q}(\vec{x})).\end{aligned}$$ In the case $f$ is binary, the identity for the $1^{\mathrm{st}}$-coordinate simplifies to $$\begin{aligned} f^{I}(T_{r}(x_1),x_{2}) &+ a(f,2)(r \cdot x_{1}, a_{2}) + a(f,1)(T_{r}(x_{1}),x_{2}) + T_{f}(r \cdot x_{1},x_{2}) \\ &= r \cdot a(f,2)(x_{1},a_{2}) + r \cdot T_{r}(x_{1},x_{2}) + T_{r}(f^{Q}(x_{1},x_{2})).\end{aligned}$$ For Eqn ([\[eqn:4\]](#eqn:4){reference-type="ref" reference="eqn:4"}), calculating for the operations $$\begin{aligned} F_{f} \left(\left\langle a_{1},x_{1} \right\rangle, \ldots, \left\langle a_{i},x_{i} \right\rangle + \left\langle b_{i},y_{i} \right\rangle, \ldots, \left\langle a_{n},x_{n} \right\rangle \right)\end{aligned}$$ and $$\begin{aligned} F_{f} \left(\left\langle a_{1},x_{1} \right\rangle, \ldots, \left\langle a_{i},x_{i} \right\rangle , \ldots, \left\langle a_{n},x_{n} \right\rangle \right) + F_{f} \left(\left\langle a_{1},x_{1} \right\rangle, \ldots, \left\langle b_{i},y_{i} \right\rangle, \ldots, \left\langle a_{n},x_{n} \right\rangle \right)\end{aligned}$$ yields the 2-cocycle identity $$\begin{aligned} f^{I}(a_{1},\ldots,T_{+}(x_{i},y_{i}),\ldots,a_{n}) &+ \sum_{i \not\in s \in [n]^{\ast}} a(f,s)\left( (x_{1},\ldots,x_{i}+y_{i},\ldots,x_{n}), \vec{a} \right) \\ &+ \sum_{i \in s \in [n]^{\ast}} a(f,s)\left( \vec{x}, (a_{1},\ldots,T_{+}(x_{i},y_{i}),\ldots,a_{n}) \right) + T_{f}(x_{1},\ldots,x_{i}+y_{i},\ldots,x_{n}) \\ &= \sum_{i \not\in s \in [n]^{\ast}} a(f,s)\left( \vec{x}, \vec{a} \right) + a(f,s)\left( (x_{1},\ldots,y_{i},\ldots,x_{n}), \vec{a} \right) \\ &+ T_{f}(\vec{x}) + T_{f}(x_{1},\ldots,y_{i},\ldots,x_{n}) + T_{+}(f^{Q}(\vec{x}),f^{Q}(x_{1},\ldots,y_{i},\ldots,x_{n}))\end{aligned}$$ for additivity of $f \in F$ in the i-th coordinate. In the case $f$ is binary, the identity for additivity in the $1^{\mathrm{st}}$-coordinate simplifies to $$\begin{aligned} f^{I}(T_{+}(x_{1},y_{1}),a_{2}) &+ a(f,1)(T_{+}(x_{1},y_{1}),x_{2}) + a(f,2)(x_{1} + y_{1}, a_{2}) + T_{f}(x_{1}+y_{1},x_{2}) \\ &= a(f,2)(x_1,a_{2}) + a(f,2)(y_{1},a_{2}) + T_{f}(x_{1},x_{2}) + T_{f}(y_{1},x_{2}) + T_{+}(f^{Q}(x_{1},x_{2}),f^{Q}(y_{1},x_{2})).\end{aligned}$$ In the case of a single binary operation $F = \{ f \}$, it may be more convenient to use the notations $a \circ y := a(f,1)(a,y)$ and $x \ast b := a(f,2)(x,b)$ for the associated action terms. **Example 9**. A *general Rota-Baxter algebra of weight* $\lambda$ is of the form $\left\langle _{R} M, \cdot, P \right\rangle$ where $\cdot$ is a bilinear operation over the $R$-module $M_{R}$ and $P$ is a linear transformation satisfying the identity $$\begin{aligned} \label{eqn:6} P(x) \cdot P(y) = P(P(x) \cdot y) + P(x \cdot P(y)) + \lambda P(x \cdot y).\end{aligned}$$ If $Q,I$ are Rota-Baxter algebras over the same ring $R$, then a 2-cocycle for the datum $(Q,I)$ is of the form $T=\{T_{+}, T_{r}, T_{\cdot}, \ast, \circ : r \in R \}$ where we are using our convention to write actions associated to bilinear operations. By evaluating $$\begin{aligned} F_{P}\left( \left\langle a, x \right\rangle \right) \cdot F_{P}\left( \left\langle b, y \right\rangle \right) = F_{P} \left( F_{P}\left( \left\langle a, x \right\rangle\right) \cdot \left\langle b, y \right\rangle \right) + F_{P} \left( \left\langle a, x \right\rangle \cdot F_{P}\left( \left\langle b, y \right\rangle\right) \right) + \lambda F_{P} \left( \left\langle a, x \right\rangle \cdot \left\langle b, y \right\rangle \right)\end{aligned}$$ and deleting identity Eq ([\[eqn:6\]](#eqn:6){reference-type="ref" reference="eqn:6"}) calculated in $I$-operations from the first coordinate, we arrive at the general 2-cocycle identity for $T$ $$\begin{aligned} P(x) \ast P(b) &+ P(a) \circ P(y) + P(a) \cdot T_{P}(y) + T_{P}(x) \cdot P(b) + T_{P}(x) \cdot T_{P}(y) + P(x) \ast T_{P}(y) + T_{P}(x) \ast P(y) \\ &+ T_{\cdot}(P(x),P(y)) \ = \ P(P(x) \ast b) + P(P(a) \circ y) + P(x \ast P(b)) + P(a \circ P(y)) \\ &+ \lambda P(a \circ y) + \lambda P(x \ast b) + P(T_{P}(x) \cdot b) + P(T_{P}(x) \circ y) + P(T_{\cdot}(P(x),y)) + T_{P}(P(x) \cdot y) \\ &+ P(a \cdot T_{P}(y)) + P(x \ast T_{P}(y)) + P(T_{\cdot}(x,P(y))) + T_{P}(x \cdot P(y)) + \lambda P(T_{\cdot}(x,y)) + \lambda T_{P}(x \cdot y) \\ &+ T_{\lambda}(P(x \cdot y)) + T_{+}(P(P(x) \cdot y),P(x \cdot P(y))) + T_{+}( P(P(x) \cdot y), P(x \cdot P(y)) ) + \lambda P(x \cdot y)).\end{aligned}$$ If we write $s(x,y) := P(x) \cdot P(y)$ and $t(x,y):= P(P(x) \cdot y) + P(x \cdot P(y)) + \lambda \cdot P(x \cdot y)$, then we see that $$\begin{aligned} s^{\ast,T}(a,b,x,y) &= P(x) \ast P(b) + P(a) \circ P(y) \\ t^{\ast,T}(a,b,x,y) &= P(P(x) \ast b) + P(P(a) \circ y) + P(x \ast P(b)) + P(a \circ P(y)) + \lambda P(a \circ y) + \lambda P(x \ast b)\end{aligned}$$ and so the action identity for compatibility with the Rota-Baxter identity is $s^{\ast,T}(a,b,x,y) = t^{\ast,T}(a,b,x,y)$; that is, $$\begin{aligned} P(x) \ast P(b) + P(a) \circ P(y) &= P(P(x) \ast b) + P(P(a) \circ y) + P(x \ast P(b)) + P(a \circ P(y)) \\ &+ \ \lambda P(a \circ y) + \lambda P(x \ast b).\end{aligned}$$ **Example 10**. The identity for a bilinear operation $[-,-]: M^{2} \rightarrow M$ to be *left-Leibniz* is $$[x,[y,z]] = [[x,y],z] + [y,[x,z]].$$ Then enforcing equality of $$F \left( \left\langle a,x \right\rangle, F \left( \left\langle b,y \right\rangle, \left\langle c,z \right\rangle \right) \right) = F \left( F \left( \left\langle a,x \right\rangle, \left\langle b,y \right\rangle \right) , \left\langle c,z \right\rangle \right) + F \left( \left\langle b,y \right\rangle, F \left( \left\langle a,x \right\rangle, \left\langle c,z \right\rangle \right) \right)$$ and calculating the first coordinate we see that the action identity for compatibility with the left-Leibniz identity is $$\begin{aligned} + [a, y \ast z] &+ a \circ [y,z] + x \ast [b,c] + x \ast (b \circ z) + x \ast (y \ast c) \\ &= [x \ast b,c] + [a \circ y,c] + [x,y] \ast c + [a,b] \circ z +(x \ast b) \circ z + (a \circ y) \circ z + [b,a \circ z] \\ &+ \ [b,x \ast c] + b \circ [x,z] + y \ast [a,c] + y \ast (a \circ z) + y \ast (x \ast c) \end{aligned}$$ and the strict 2-cocycle identity for $T$ of a left-Leibniz operation is $$\begin{aligned} + x \ast T(y,z) + T(x,[y,z]) &= [T(x,y),c] + T(x,y) \circ z + T([x,y],z) + [b,T(y,z)] + y \ast T(x,z) \\ & \quad + \ T(y,[x,z]) + T([[x,y],z],[y,[x,z]]).\end{aligned}$$ **Definition 11**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$. The set of $\mathcal V$-compatible 2-cocycles of datum $(Q,I)$ is denoted by $Z^{2}_{\mathcal V}(Q,I)$. **Definition 12**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$. Let $(Q,I)$ be datum in the signature of $\mathcal V$ and $T$ a 2-cocycle for the datum. An algebra $M$ *realizes* the 2-cocycle $T$ if there is an extension $\pi: M \rightarrow Q$ with an isomorphism $i: I \rightarrow \ker \pi$ and a lifting $l: Q \rightarrow \ker \pi$ such that 1. $i \circ T_{+}(x,y) = l(x) + l(y) - l(x+y)$; 2. $i \circ T_{r}(x,y) = r \cdot l(x) - l(r\cdot x)$; 3. $i \circ T_{f}(\vec{x}) = f^{M}(l(\vec{x})) - l(f^{Q}(\vec{x}))$; 4. $i \circ a(f,s)(\vec{x},\vec{a}) = f^{M}(l(\vec{x}))_{s}[\vec{a}]$ for each $f \in F$. **Lemma 13**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations and $(Q,I)$ datum in the signature of $\mathcal V$. If $T$ is a 2-cocycle of the datum, then $I \rtimes_{T} Q$ realizes the 2-cocycle $T$. *Proof.* Fix the lifting $l: Q \rightarrow I \times Q$ defined by $l(x) = \left\langle 0, x \right\rangle$. We calculate for the operations. For the module operations, we have $$\begin{aligned} l(x) + l(y) - l(x+y) &= \left\langle \, 0, x \, \right\rangle + \left\langle \, 0, y \, \right\rangle - \left\langle \, 0, x + y \, \right\rangle \\ &= \left\langle \, T_{+}(x,y), x + y \, \right\rangle + \left\langle \, - T_{+}(x+y,-x-y), -x - y \, \right\rangle \\ &= \left\langle \, T_{+}(x,y), 0 \, \right\rangle\end{aligned}$$ and $$\begin{aligned} r \cdot l(x) - l(r\cdot x) = r \cdot \left\langle \, 0, x \, \right\rangle - \left\langle \, 0, r \cdot x \, \right\rangle &= \left\langle \, T_{r}(x), r \cdot x \, \right\rangle + \left\langle \, - T_{+}(r \cdot x, - r \cdot x ), - r \cdot x \, \right\rangle \\ &= \left\langle \, T_{r}(x), 0 \, \right\rangle.\end{aligned}$$ For $f \in F$ with $n = \mathop{\mathrm{ar}}f$, we have $$\begin{aligned} F_{f} \left( l(\vec{x}) \right) - l(f^{Q}(\vec{x})) &= \left\langle \, f^{I}(\vec{0}) + \sum_{s \in [n]^{\ast}} a(f,s)(\vec{x},\vec{0}) + T_{f}(\vec{x}) , f^{Q}(\vec{x}) \, \right\rangle - \left\langle 0, f^{Q}(\vec{x}) \right\rangle \\ &= \left\langle \, T_{f}(\vec{x}) , f^{Q}(\vec{x}) \, \right\rangle + \left\langle -T_{+}(f^{Q}(\vec{x}),-f^{Q}(\vec{x})), -f^{Q}(\vec{x}) \right\rangle \\ &= \left\langle T_{f}(\vec{x}) , 0 \right\rangle\end{aligned}$$ and for $\emptyset \neq s \subseteq [n]$, $$\begin{aligned} F_{f}\left( l(\vec{x}) \right)_{s} [i(\vec{a})] = \left\langle \, f^{I}(\vec{0})_{s}[\vec{a}] + \sum_{r \in [n]^{s}} a(f,r)\left( \vec{x}_{s}[\vec{0}], \vec{0}_{s}[\vec{a}] \right) + T_{f}(\vec{x})_{s}[\vec{0}] , \, f^{Q}(\vec{x})_{s}[\vec{0}] \right\rangle &= \left\langle \, a(f,s)(\vec{x},\vec{a}) , 0 \, \right\rangle \end{aligned}$$ where we have used the (T4) property of the action terms to cancel in the sum those terms for $r \neq s$. ◻ **Theorem 14**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations. Let $(Q,I)$ be datum in $\mathcal V$ and $\pi: M \rightarrow Q$ an extension with $\ker \pi = I$. Then $M \in \mathcal V$ if and only if $\pi : M \rightarrow Q$ realizes a $\mathcal V$-compatible 2-cocycle $T$ of the datum such that $M \approx I \rtimes_{T} Q$. *Proof.* Let $M \in \mathcal V$ and $\pi: M \rightarrow Q$ a surjective homomorphism with $\ker \pi = I$. Choose a lifting $l : Q \rightarrow M$ of $\pi$ and define a 2-cocycle $T$ by the rules 1. $T_{+}(x,y) := l(x) + l(y) - l(x+y)$; 2. $T_{r}(x,y) := r \cdot l(x) - l(r\cdot x)$; 3. $T_{f}(\vec{x}) := f^{M}(l(\vec{x})) - l(f^{Q}(\vec{x}))$; 4. $i \circ a(f,s)(\vec{x},\vec{a}) = f^{M}(l(\vec{x}))_{s}[\vec{a}]$ for each $f \in F$. It is easy to see that properties (T1)-(T4) hold so that $T$ is a 2-cocycle for datum $(Q,I)$. Define $\psi: M \rightarrow I \rtimes_{T} Q$ by $\psi(a):= \left\langle a - l \circ \pi(a),\pi(a) \right\rangle$. Since $l$ is a lifting of $\pi$, every $a \in M$ is uniquely represented as $a = \left( a - l \circ \pi(a) \right) + l \circ \pi(a)$ with $a - l \circ \pi(a) \in I$; consequently, $\psi$ is bijective. We verify $\psi$ is a homomorphism. Take $f \in F$ with $n = \mathop{\mathrm{ar}}f$. For $\vec{m} \in M^{n}$, write $x_{i}=\pi(m_{i})$ and $a_{i}= m_{i} - l (x_{i})$ so that we have $m_{i} = a_{i} + l(x_{i})$ with $a_{i} \in I, x_{i} \in Q$; thus, $\psi(m_{i}) = \left\langle a_{i}, x_{i} \right\rangle$. Then using multilinearity of $f$ we have $$\begin{aligned} \psi \left(f^{M}(\vec{m}) \right) &= \left\langle f^{M}(\vec{m}) - l \circ \pi (f^{M}(\vec{m})) , \pi \circ f^{M}(\vec{m}) \right\rangle \\ &= \left\langle f^{M}(a_{1} + l(x_{1}),\ldots,a_{n} + l(x_{n})) - l(f^{Q}(\vec{x})) , f^{Q}(\vec{x}) \right\rangle \\ &= \left\langle f^{M}(\vec{a}) + f^{M}(l(\vec{x})) + \sum_{s \in [n]^{\ast}} f^{M}(l(\vec{x}))_{s}[\vec{a}] - l(f^{Q}(\vec{x})) , f^{Q}(\vec{x}) \right\rangle \\ &= \left\langle f^{I}(\vec{a}) + \sum_{s \in [n]^{\ast}} a(f,s)(\vec{x},\vec{a}) + T_{f}(\vec{x}) , f^{Q}(\vec{x}) \right\rangle \\ &= F_{f}\left( \left\langle a_{1}, x_{1} \right\rangle, \ldots, \left\langle a_{n}, x_{n} \right\rangle \right) = F_{f} \left( \psi(\vec{m}) \right).\end{aligned}$$ We also see that $$\begin{aligned} \psi( r \cdot m_{1} + m_{2} ) &= \left\langle r \cdot a_{1} + a_{2} + r\cdot l(x_{1}) + l(x_{2}) - l(r \cdot x_{1} + x_{2}) , r \cdot x_{1} + x_{2} \right\rangle \\ &= \left\langle r \cdot a_{1} + a_{2} + r\cdot l(x_{1}) - l(r \cdot x_{1}) + l(r \cdot x_{1}) + l(x_{2}) - l(r \cdot x_{1} + x_{2}) , r \cdot x_{1} + x_{2} \right\rangle \\ &= \left\langle r \cdot a_{1} + T_{r}(x_{1}) + a_{2} + T_{+}(r \cdot x_{1},x_{2}) , r \cdot x_{1} + x_{2} \right\rangle \\ &= \left\langle r \cdot a_{1} + T_{r}(x_{1}), r \cdot x_{1} \right\rangle + \left\langle a_{2}, x_{2} \right\rangle\\ &= r \cdot \left\langle a_{1}, x_{1} \right\rangle + \left\langle a_{2}, x_{2} \right\rangle = r \cdot \psi(m_{1}) + \psi(m_{2});\end{aligned}$$ altogether, $M \approx I \rtimes_{T} Q$. Since $M \in \mathcal V$, Lemma [Lemma 5](#lem:semident){reference-type="ref" reference="lem:semident"} implies $T$ is $\mathcal V$-compatible. Conversely, assume $T$ is a $\mathcal V$-compatible 2-cocycle of the datum $(Q,I)$. Then since we assumed $Q,I \in \mathcal V$, Lemma [Lemma 5](#lem:semident){reference-type="ref" reference="lem:semident"} implies $M \approx I \rtimes_{T} Q \in \mathcal V$. ◻ Lemma [Lemma 5](#lem:semident){reference-type="ref" reference="lem:semident"} and Theorem [Theorem 14](#thm:multirep){reference-type="ref" reference="thm:multirep"} guarantee that the algebras $I \rtimes_{T} Q$ for $\mathcal V$-compatible 2-cocycles $T$ for datum $(Q,I)$ provide up to isomorphism all extensions of $Q$ by $I$; however, the identification in Theorem [Theorem 14](#thm:multirep){reference-type="ref" reference="thm:multirep"} depends on the choice of a lifting. Choosing different liftings lead to a combinatorial notion of equivalence of extensions via the 2-cocycles. **Definition 15**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$ and $(Q,I)$ datum in $\mathcal V$. A 2-*coboundary* for the datum the $(Q,I)$ is a sequence $G = \{G_{+}, G_{r}, G_{f}, g(f,s): r \in R, f \in F, s \in [\mathop{\mathrm{ar}}f]^{\ast} \}$ such that there exists an action $Q \ast I$ and a function $h : Q \rightarrow I$ with $h(0)=0$ which satisfies 1. $G_{+}(x,y) = h(x) + h(y) - h(x+y)$; 2. $G_{r}(x) = r \cdot h(x) - h(r \cdot x)$; 3. $G_{f}(\vec{x}) = \sum_{s \in [\mathop{\mathrm{ar}}f]^{\ast}} (-1)^{1+|s|} a(f,s)(\vec{x},h(\vec{x})) + (-1)^{1+\mathop{\mathrm{ar}}f}f^{I}(h(\vec{x})) - h(f^{Q}(\vec{x}))$; 4. $g(f,s)(\vec{x},\vec{a}) = \sum_{s \subsetneq r \subseteq [\mathop{\mathrm{ar}}f]^{\ast}} (-1)^{1 + |r| - |s|} a(f,r)(\vec{x},h(\vec{x}))_{s}[ \vec{a} ] + (-1)^{1 + |\mathop{\mathrm{ar}}f| - |s|}f^{I}\left( h(\vec{x}) \right)_{s}[ \vec{a} ]$. The map $h:Q \rightarrow I$ in Definition [Definition 15](#def:30){reference-type="ref" reference="def:30"} is said to *determine* or *witness* the 2-coboundary. The *null* 2-coboundary has all entries zero. **Definition 16**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$. The set of 2-coboundaries of the datum $(Q,I)$ is denoted by $B^{2}(Q,I)$. Note the notation for the set of 2-coboundaries has no index for the variety. We shall show in the comments after Theorem [Theorem 20](#thm:10){reference-type="ref" reference="thm:10"} that this is because $B^{2}(Q,I) \subseteq Z^{2}_{\mathcal V}(Q,I)$ for any variety $\mathcal V$ containing the datum. **Definition 17**. The 2-cocycles $T$ and $T'$ for datum $(Q,I)$ are *equivalent*, written $T \sim T'$, if $T - T' \in B^{2}(Q,I)$. This is indeed an equivalence relation on 2-cocycles but it will be easier to see this by relating it to an equivalence on extensions given in Definition [Definition 19](#def:5){reference-type="ref" reference="def:5"}. Let us first observe that the combinatorial definition of a 2-coboundary is derived by choosing different liftings for an extension. **Lemma 18**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$ and $\pi : M \rightarrow Q$ an extension with $\ker \pi = I$. If $l,l': Q \rightarrow I$ are liftings of $\pi$ which respectively define 2-cocycles $T$ and $T'$ by $(R1) - (R4)$, then $T \sim T'$. *Proof.* Define $h = l - l': Q \rightarrow I$. We will show $h$ determines the 2-coboundary $T-T'$. We see that $$\begin{aligned} T_{+}(x,y) = l(x) + l(y) - l(x+y) &= l'(x) + h(x) + l'(y) + h(y) - \left( l'(x+y) + h(x+y) \right) \\ &= h(x) + h(y) - h(x+y) + T'_{+}(x,y)\end{aligned}$$ and $$T_{r}(x) = r \cdot l(x) - l(r \cdot x) = r \cdot (l'(x) + h(x)) - \left( l'(r \cdot x) + h(r \cdot x) \right) = r \cdot h(x) + h(r \cdot x) + T'_{r}(x)$$ which covers (B1) and (B2) in Definition [Definition 15](#def:30){reference-type="ref" reference="def:30"}. Now fix $f \in F$ with $n=\mathop{\mathrm{ar}}f$. By realization, we see that $$\begin{aligned} T_{f}(\vec{x}) &= f^{M} \left( l'(\vec{x}) \right) + h(\vec{x})) - l(f^{Q}(\vec{x})) \\ &= \sum_{r \in [n]^{\ast}} f^{M}\left( l'(\vec{x}) \right)_{r} [h(\vec{x})] + f^{I}(h(\vec{x})) - h(f^{Q}(\vec{x})) + f^{M} \left( l'(\vec{x}) \right) - l'(f^{Q}(\vec{x})) \\ &= \sum_{r \in [n]^{\ast}} f^{M}\left( l(\vec{x}) - h(\vec{x}) \right)_{r} [h(\vec{x})] + f^{I}(h(\vec{x})) - h(f^{Q}(\vec{x})) + T'_{f}(\vec{x}) \\ &= \sum_{r \in [n]^{\ast}} \sum_{r \subseteq s \subseteq [n]} (-1)^{|s|-|r|} f^{M}\left( l(\vec{x}) \right)_{s}[h(\vec{x})] + f^{I}(h(\vec{x})) - h(f^{Q}(\vec{x})) + T'_{f}(\vec{x}) \\ &= \sum_{s \in [n]^{\ast}} \left( \sum_{\emptyset \neq r \subseteq s } (-1)^{|s| - |r|} \right) f^{M} \left( l(\vec{x}) \right)_{s}[h(\vec{x})] + \left(1 + \sum_{r \in [n]^{\ast}} (-1)^{n - |r|} \right) f^{I} \left( h(\vec{x}) \right) - h(f^{Q}(\vec{x})) + T'_{f}(\vec{x}) \\ &= \sum_{s \in [n]^{\ast}} (-1)^{1+|s|}f^{M} \left( l(\vec{x}) \right)_{s}[h(\vec{x})] + (-1)^{1+n}f^{I} \left( h(\vec{x}) \right) - h(f^{Q}(\vec{x})) + T'_{f}(\vec{x}) \\ &= \sum_{s \in [n]^{\ast}} (-1)^{1+|s|} a(f,s)(\vec{x},h(\vec{x})) + (-1)^{1+n}f^{I}(h(\vec{x})) - h(f^{Q}(\vec{x})) + T'_{f}(\vec{x})\end{aligned}$$ which covers (B3). For an action term $a(f,s)$ with $s \in [n]^{\ast}$, we use realization and the previous calculation applied to the action term as a function of $Q^{n-s}$ to get $$\begin{aligned} a(f,s)(\vec{x},\vec{a}) &= f(l(\vec{x}))_{s}[\vec{a}] \\ &= \sum_{t \in ([n]-s)^{\ast}} (-1)^{1+|t|} a(f,t)\left( \vec{x},h(\vec{x}) \right)_{s}[\vec{a}] + (-1)^{1+|n|-|s|} f^{I}\left( h(\vec{x})\right)_{s}[\vec{a}] + f(l'(\vec{x}))_{s}[\vec{a}] \\ &= \sum_{s \subseteq r \subseteq [n]^{\ast}} (-1)^{1 + |r| - |s|} a(f,r)(\vec{x},h(\vec{x}))_{s}[ \vec{a} ] + (-1)^{1 + |n| - |s|}f^{I}\left( h(\vec{x}) \right)_{s}[ \vec{a} ] + a'(f,s)(\vec{x},\vec{a})\end{aligned}$$ using $r = t \cup s$ in the last line. ◻ There is another notion of equivalence defined on the extensions themselves. **Definition 19**. Two extensions $\pi : M \rightarrow Q$ and $\pi': M' \rightarrow Q$ with $\ker \pi = I = \ker \pi'$ are *equivalent* if there is an isomorphism $\gamma: M \rightarrow M'$ such that $\gamma|_{I} = \mathop{\mathrm{id}}_{I}$ and $\pi' \circ \gamma = \pi$. The following theorem shows that equivalence on extensions is the same as the combinatorial equivalence on the associated 2-cocycles. **Theorem 20**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$ and $(Q,I)$ datum in $\mathcal V$. Two extensions of $Q$ with kernel $I$ in the variety $\mathcal V$ are equivalent if and only they realize equivalent $\mathcal V$-compatible 2-cocycles. *Proof.* Let $\pi: M \rightarrow Q$ and $\pi': M' \rightarrow Q$ be extensions of $Q$ by $I$. Assume $\pi$ and $\pi'$ are equivalent. Let $\gamma: M \rightarrow M'$ be an isomorphism such that $\gamma|_{I} = \mathop{\mathrm{id}}_{I}$ and $\pi' \circ \gamma = \pi$. Let $l:Q \rightarrow M$ be a lifting of $\pi$ and $T$ be the $\mathcal V$-compatible 2-cocycle defined by $l$ according to (R1)-(R4). Then $\gamma \circ l: Q \rightarrow M'$ is a lifting of $\pi'$. Let $S$ be the 2-cocycle defined by the lifting $\gamma \circ l$. If we apply $\gamma$ to (R1)-(R4) defining $T$, then $\gamma|_{I} = \mathop{\mathrm{id}}_{I}$ implies that $T=S$; thus, $T$ is a 2-cocycle realized by the extension $\pi'$. So if $l': Q \rightarrow M'$ is another lifting which defines a 2-cocycle $T'$ realized by $\pi'$, Lemma [Lemma 18](#lem:2coboundmulti){reference-type="ref" reference="lem:2coboundmulti"} yields $T \sim T'$. Conversely, assume $T$ is a 2-cocycle realized by $\pi$, $T'$ is a 2-cocycle realized by $\pi'$ and $T \sim T'$. Then there is a map $h: Q \rightarrow I$ which determines the 2-coboundary $T'-T$ by (B1)-(B4). According to Theorem [Theorem 14](#thm:multirep){reference-type="ref" reference="thm:multirep"}, we have the isomorphisms $\phi_{1}: M \rightarrow I \rtimes_{T} Q$ and $\phi_{2} : M' \rightarrow I \rtimes_{T'} Q$. We then define $\gamma': I \rtimes_{T} Q \rightarrow I \rtimes_{T'} Q$ by $\gamma' \left\langle a,x \right\rangle : = \left\langle a - h(x), x \right\rangle$ which is clearly bijective. To see that $\gamma'$ is a homomorphism, we calculate for the module operations $$\begin{aligned} \gamma'\left\langle a,x \right\rangle + \gamma'\left\langle b,y \right\rangle = \left\langle a + b - h(x) - h(y) + T'_{+}(x,y) , x + y \right\rangle &= \left\langle a + b + T_{+}(x,y) - h(x+y), x + y \right\rangle\\ &= \gamma' \left\langle a + b + T_{+}(x,y), x+y \right\rangle \\ &= \gamma' \left( \left\langle a,x \right\rangle + \left\langle b,y \right\rangle\right) \end{aligned}$$ and $$\begin{aligned} r \cdot \gamma'\left\langle a,x \right\rangle = \left\langle r \cdot a - r \cdot h(x) + T'_{r}(x), r \cdot x \right\rangle &= \left\langle r \cdot a + T_{r}(x) - h(r \cdot x), r \cdot x\right\rangle \\ &= \gamma'\left\langle r \cdot a + T_{r}(x) ,r \cdot x \right\rangle \\ &= \gamma'\left( r \cdot \left\langle a, x\right\rangle \right).\end{aligned}$$ Now take $f \in F$ with $n=\mathop{\mathrm{ar}}f$, $\vec{a} \in I^{n}$, $\vec{x} \in Q^{n}$ and calculate $$\begin{aligned} F_{f} &\left( \gamma'\left\langle a_1,x_1 \right\rangle, \ldots, \gamma'\left\langle a_n,x_n \right\rangle \right) \\ &= F_{f} \left( \left\langle a_{1} - h(x_{1}),x_{1} \right\rangle,\ldots,\left\langle a_{n} - h(x_{n}),x_{n} \right\rangle \right) \\ &= \left\langle f^{I} \left(\vec{a} - h(\vec{x}) \right) + \sum_{s \in [n]^{\ast}} a'(f,s)\left(\vec{x}, \vec{a} - h(\vec{x}) \right) + T'_{f}(\vec{x}) , f^{Q}(\vec{x}) \right\rangle \\ &= \Bigg< f^{I}(\vec{a}) + \sum_{ \emptyset \neq t \subseteq [n]} (-1)^{|t|} f^{I} \left( \vec{a} \right)_{t}[h(\vec{x})] + \sum_{s \in [n]^{\ast}} \sum_{r \subseteq s} (-1)^{|r|} a'(f,s)\left(\vec{x}, \vec{a} \right)_{r}[h(\vec{x})] + T'_{f}(\vec{x}), f^{Q}(\vec{x}) \Bigg> \\ \\ &= \Bigg< f^{I}(\vec{a}) + \sum_{u \in [n]^{\ast}} (-1)^{n - |u|} f^{I} \left(h(\vec{x}) \right)_{u}[\vec{a}] + (-1)^{n} f^{I} \left( h(\vec{x}) \right) \\ &\quad + \sum_{u \in [n]^{\ast}} \sum_{u \subseteq v \in [n]^{\ast}} (-1)^{|v| - |u|} a'(f,v)(\vec{x},h(\vec{x}))_{u}[\vec{a}] + T'_{f}(\vec{x}) , f^{Q}(\vec{x}) \Bigg> \\ &= \left\langle f^{I}(\vec{a}) + \sum_{s \in [n]^{\ast}} a(f,s)(\vec{x},\vec{a}) + T_{f}(\vec{x}) - h \left( f^{Q}(\vec{x}) \right), f^{Q}(\vec{x}) \right\rangle \\ &= \gamma' \left\langle f^{I} \left( \vec{a} \right) + \sum_{s \in [n]^{\ast}} a(f,s)(\vec{x},\vec{a}) + T_{f}(\vec{x}), f^{Q}(\vec{x}) \right\rangle \\ &= \gamma' \left( F_{f}\left( \left\langle a_{1},x_{1} \right\rangle, \ldots, \left\langle a_{1},x_{1} \right\rangle \right) \right).\end{aligned}$$ It is immediate that $\gamma'|_{I \times 0} = \mathop{\mathrm{id}}_{I \times 0}$ and $p_{2} \circ \gamma' = p_{2}$. The required isomorphism is then $\gamma = \phi_{2}^{-1} \circ \gamma' \circ \phi_{1}$. ◻ Let $G$ be a 2-coboundary for $(Q,I)$ witnessed by $h: Q \rightarrow I$. It is straightforward to verify that $G$ satisfies properties (T1) - (T4), and so it is a 2-cocycle of $(Q,I)$. Since $G \sim 0$, we see by Theorem [Theorem 20](#thm:10){reference-type="ref" reference="thm:10"} that $I \rtimes_{G} Q$ and $I \times Q$ are equivalent extensions, and so isomorphic. If $I,Q \in \mathcal V$, then $I \rtimes_{G} Q \approx I \times Q \in \mathcal V$ and so by Theorem [Theorem 14](#thm:multirep){reference-type="ref" reference="thm:multirep"}, $G$ is $\mathcal V$-compatible. This shows $B^{2}(Q,I) \subseteq Z^{2}_{\mathcal V}(Q,I)$ for any variety $\mathcal V$ containing the datum algebras. This also shows that if $T$ is $\mathcal V$-compatible and $T \sim T'$, then $T'$ is also $\mathcal V$-compatible; therefore, the equivalence on 2-cocycles respects the compatibility with equational theories. The set of 2-cocycles equivalent to $T$ is denoted by $[T]$. **Definition 21**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$. The $2^{\mathrm{nd}}$-cohomology of the datum $(Q,I)$ in the variety $\mathcal V$ is the set $H^{2}_{\mathcal V}(Q,I)$ of $\mathcal V$-compatible 2-cocycles modulo equivalence. **Proposition 22**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$ and $M \in \mathcal V$. The following are equivalent: 1. there is a retraction $r: M \rightarrow M$; 2. there is a surjective homomorphism $\pi: M \rightarrow Q$ and homomorphism $l: Q \rightarrow M$ such that $\pi \circ l = \mathop{\mathrm{id}}_{Q}$; 3. there is an ideal $I \triangleleft M$ and subalgebra $Q \leq M$ such that $I \cap Q = 0$ and $M = I + Q$; 4. $M \approx I \rtimes_{T^{\ast}} Q$ for a $\mathcal V$-compatible action $Q \ast I$. *Proof.* The equivalence of (1)-(3) is standard. We show $(2)$ and (4) are equivalent. Suppose $\phi: M \rightarrow I \rtimes_{T^{\ast}} Q$ witnesses an isomorphism. Define $l:Q \rightarrow I \times Q$ by $l(x):= \left\langle 0, x \right\rangle$; clearly, $l$ is a right-inverse map for the second-projection $p_{2}: I \rtimes_{T^{\ast}} Q \rightarrow Q$ which is always a homomorphism. Since $T^{\ast}$ is a 2-cocycle for $(Q,I)$ which satisfies properties (T1) -(T4) and all the non-action terms are zero, we see that $l$ is a homomorphism. Conversely, assume $(2)$ holds. Then we define the 2-cocycle $T$ with the homomorphism $l: Q \rightarrow M$ in the same manner as Theorem [Theorem 14](#thm:multirep){reference-type="ref" reference="thm:multirep"}. Since $l$ is a homomorphism we see that $T_{+} \equiv T_{r} \equiv T_{f} \equiv 0$ for all $r \in R$ and $f \in F$; thus $T=T^{\ast}$ for the action terms defined by $l$ and $M \approx I \rtimes_{T^{\ast}} Q$. ◻ An algebra $M$ satisfying condition (2) in Proposition [Proposition 22](#prop:1){reference-type="ref" reference="prop:1"} is called a *semidirect product* of $Q$ by $I$. It is then tempting to consider a general extension of datum $(Q,I)$ as a semidirect product translated by the non-action terms of the 2-cocycle. **Theorem 23**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$ and $(Q,I)$ datum in $\mathcal V$. The set of equivalence classes of extensions of $Q$ by $I$ in the variety $\mathcal V$ is in bijective correspondence with $H^{2}_{\mathcal V}(Q,I)$. The equivalence class of a semidirect product corresponds to an equivalence class of a 2-cocycle with only action terms The following is the more general version of the result in [@wiresI] for our varieties of interest. **Proposition 24**. Let $(Q,I)$ be datum in the signature appropriate for multilinear $R$-module expansions. The class of varieties in the same signature which contain the datum forms a complete lattice $\mathcal L(Q,I)$. The maps $\Psi(\mathcal V) := H^{2}_{\mathcal V}(Q,I)$ and $\Theta (E) := \mathcal V \left( \{ I \rtimes_{T} Q : [T] \in E \} \right)$ define a Galois connection $(\Theta,\Psi)$ between the ordered sets $\mathop{\mathrm{Sub}}H^{2}_{_{R} \mathcal M_{F}}(Q,I)$ and $\mathcal L(Q,I)$. *Proof.* It is easy to see that $\Psi$ is monotone on varieties and so yields the inequality $\Psi ( \mathcal U_{1} \wedge \mathcal U_{1} ) \leq \Psi (\mathcal U_{1}) \wedge \Psi(\mathcal U_{2})$. Now take $[T] \in H^{2}_{\mathcal U_{1}}(Q,I) \cap H^{2}_{\mathcal U_{2}}(Q,I)$; thus, $T$ is both $\mathcal U_{1}$ and $\mathcal U_{2}$-compatible. Then from Theorem [Theorem 14](#thm:multirep){reference-type="ref" reference="thm:multirep"} and the class operator we see that $I \rtimes_{T} Q \in \mathop{\mathrm{Mod}}\left( \mathop{\mathrm{Id}}\mathcal U_{1} \cup \mathop{\mathrm{Id}}\mathcal U_{2} \right) = \mathcal U_{1} \wedge \mathcal U_{2}$. This implies $[T] \in H^{2}_{\Psi (\mathcal U_{1}) \wedge \Psi(\mathcal U_{2})}(Q,I)$ and so $\Psi$ preserves the meet. From the definitions of the maps, it is not difficult to see that $\Theta \circ \Psi \geq \mathop{\mathrm{id}}_{\mathcal L(Q,I)}$ and so $\Theta$ is the lower-adjoint for $\Psi$. ◻ We can restrict the notion of equivalent extensions to the automorphisms of a fixed extension. An automorphism $\gamma \in \mathop{\mathrm{Aut}}M$ *stabilizes* the extension $\pi: M \rightarrow Q$ where $\ker \pi = I$ if $\gamma|_{I} = \mathop{\mathrm{id}}_{I}$ and $\pi \circ \gamma = \pi$. The set of stabilizing automorphisms of the extension $\pi$ is denoted by $\mathrm{Stab}(\pi)$ and is a subgroup of $\mathop{\mathrm{Aut}}M$. Since the second-projection is the canonical map for extensions $I \rtimes_{T} Q$, we may also write $\mathrm{Stab}( I \rtimes_{T} Q )$ to denote its group of stabilizing automorphisms. As we saw in the definition of 2-coboundaries, we may sometimes need to invoke an action as part of the definitions which leads to situations where an action is part of the notion of datum. This will become more apparent when we discuss extensions with abelian kernels and the set of extensions realizing particular fixed action terms. For algebras $Q$ and $I$ with an available action $Q \ast I$, the triple $(Q,I,\ast)$ may also be referred to as datum. **Definition 25**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$, $Q, I \in \mathcal V$ and $Q \ast I$ an action. A 1-cocycle or *derivation* of the datum $(Q,I,\ast)$ is a map $h: Q \rightarrow I$ which determines the null 2-coboundary. Let $\mathrm{Der}(Q,I,\ast)$ denote the set of derivations of the datum $(Q,I,\ast)$. It is possible to see directly from (B1)-(B4) that the set of derivations is an abelian group under the addition induced by $I$ in the codomain; however, this can more easily seen by making the following connection between derivations and stabilizing automorphisms. **Theorem 26**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$ and $(Q,I)$ datum in $\mathcal V$. The set of 1-cocycles is an abelian group and for any extension $\pi : M \rightarrow Q$ with $\ker \pi = I$, $\mathrm{Stab}(\pi) \approx \mathrm{Der}(Q,I,\ast)$ where the $Q \ast I$ is the associated action of the extension; in particular, $\mathrm{Stab}(\pi)$ is an abelian group. *Proof.* Note stabilizing automorphisms are just the restriction to a fixed extension of the isomorphisms which witness equivalence and derivations determine the 2-coboundaries which witness reflexivity $T \sim T$. Then the proofs of Theorem [Theorem 20](#thm:10){reference-type="ref" reference="thm:10"} and Lemma [Lemma 18](#lem:2coboundmulti){reference-type="ref" reference="lem:2coboundmulti"} show that $\gamma: I \rtimes_{T} Q \rightarrow I \rtimes_{T} Q$ is a stabilizing automorphism if and only if it is of the form $\gamma(a,x) = \left\langle a - d(x), x \right\rangle$ for a derivation $d: Q \rightarrow I$. We define a map $\Psi: \mathrm{Stab}(\pi) \approx \mathrm{Der}(Q,I,\ast)$ by $\Psi(\gamma) := d_{\gamma}$ where $\gamma(a,x) = \left\langle a - d_{\gamma}(x), x \right\rangle$. It follows that $\Psi$ is bijective by directly verifying $d_{\gamma_{d}} = d$ and $\gamma_{d_{\gamma}} = \gamma$ from the definitions. For two stabilizing automorphisms $\gamma$ and $\sigma$, the evaluation $\gamma \circ \sigma \left\langle a,x \right\rangle = \gamma \left\langle a - d_{\sigma}(x), x \right\rangle = \left\langle a - d_{\sigma}(x) - d_{\gamma}(x), x \right\rangle = \sigma \circ \gamma \left\langle a,x \right\rangle$ implies $\Psi(\gamma \circ \sigma) = \Psi(\gamma) + \Psi(\sigma)=\Psi(\sigma \circ \gamma)$ by bijectivity of $\Psi$; therefore, $\Psi$ is an isomorphism. ◻ We essentially follow [@wiresI] in the definition of principal derivations and $1^{\mathrm{st}}$-cohomology but with a modification to accommodate the fact that congruences are determine by ideals. The motivation come from the fact that under the analogous isomorphism of Theorem [Theorem 26](#thm:stabderiv){reference-type="ref" reference="thm:stabderiv"} for groups, principal derivations correspond to the stabilizing automorphisms which are inner automorphisms of the semidirect product of the datum. Given an ideal $I \triangleleft M$, polynomials $p$ and $q$ are $I$-twins if there is a term $t(x,\vec{y})$ and $\vec{c},\vec{d} \in I^{k}$ such that $p(x)=t(x,\vec{c})$ and $q(x)=t(x,\vec{d})$. The set of $I$-twins of the identity is denoted by $\mathrm{Tw}_{I} \, M$; that is, $p \in \mathrm{Tw}_{I} \, M$ if there is a term $t(x,\vec{y})$ and $\vec{c},\vec{d} \in I^{k}$ such that $p(x)=t(x,\vec{c})$ and $x=t(x,\vec{d})$. Note $\mathrm{Tw}_{I} \, M$ is closed under composition. If $p,q \in \mathrm{Tw}_{I} \, M$, then there are terms $t(x, \vec{y}), s(x,\vec{z})$ and tuples $\vec{c},\vec{d} \in I^{k}$, $\vec{a}, \vec{b}\in I^{m}$ such that $p(x) = t(x,\vec{c}), x=t(x,\vec{d})$ and $q(x)=s(x,\vec{a}), x = s(x,\vec{b})$. Then the term $r(x,\vec{y},\vec{z}) := t(s(x,\vec{z}),\vec{y})$ with the tuples $(\vec{c},\vec{a})$ and $(\vec{d},\vec{b})$ shows $p \circ q \in \mathrm{Tw}_{I} \, M$. We then consider the subset $\mathrm{Tw}_{I,F} \, M = \{ p \in \mathrm{Tw}_{I} \, M : \exists a \in M, p(a)=a \}$ of $I$-twins of the identity which have a fixed point. The *principal stabilizing automorphisms* of an extension $\pi: M \rightarrow Q$ realizing datum $(Q,I)$ are defined as $$\mathrm{PStab}(\pi) := \mathrm{Tw}_{I,F} \, M \cap \mathrm{Stab}(\pi).$$ In general, $\mathrm{Tw}_{I,F} \, M$ is not closed under composition, but $\mathrm{Tw}_{I,F} \, M \cap \mathrm{Stab}(\pi)$ will be because for the algebras under consideration stabilizing automorphisms restrict to the identity on the ideal of the extension. **Lemma 27**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations, $Q,I \in \mathcal V$ and $Q \ast I$ an action. Then $\mathrm{PStab}(I \rtimes_{T^{\ast}} Q ) \leq \mathrm{Stab}(I \rtimes_{T^{\ast}} Q )$. *Proof.* We need only show closure under inverses. We make the identification of $I$ with $I \times \{ 0 \}$. Let $\gamma \in \mathrm{PStab}(I \rtimes_{T^{\ast}} Q )$ witnessed by the term $t(x,\vec{y})$ and tuples $\vec{c},\vec{d}$; that is, $$\left\langle a - d_{\gamma}(x), x \right\rangle = p \left( a, x \right) = F_{t} \left( \left\langle a, x \right\rangle, \vec{c} \right) \quad \quad \text{and} \quad \quad \left\langle a, x \right\rangle = F_{t} \left( \left\langle a, x \right\rangle, \vec{d} \right).$$ Consider the term $r(x,\vec{y},\vec{z}) := t(x,\vec{z}) - t(x,\vec{y}) + x$. Since in the semidirect product, the non-action terms in the 2-cocycle are all trivial, we calculate $$\begin{aligned} F_{r} \left( \left\langle a, x \right\rangle, \vec{c}, \vec{d} \right) &= \left\langle a + d(x), x \right\rangle = p^{-1}(a,x)\\ F_{r} \left( \left\langle a, x \right\rangle, \vec{d}, \vec{d} \right) &= \left\langle a, x \right\rangle \end{aligned}$$ so that $p^{-1} \in \mathrm{PStab}(I \rtimes_{T^{\ast}} Q )$. ◻ The set of *principal derivations* of the datum $(Q,I,\ast)$ is then defined as $$\mathrm{PDer}(Q,I,\ast) := \{ \, d(x): \gamma(a,x) = \left\langle a - d(x), x \right\rangle, \gamma \in \mathrm{PStab}(I \rtimes_{T^{\ast}} Q ) \}.$$ The principal derivations are those derivations which under the isomorphism of Theorem [Theorem 26](#thm:stabderiv){reference-type="ref" reference="thm:stabderiv"} correspond to the principal stabilizing automorphisms of the semidirect product $I \rtimes_{T^{\ast}} Q$. **Definition 28**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations, $Q,I \in \mathcal V$ and $Q \ast I$ an action. The $1^{\mathrm{st}}$-cohomology of the datum $(Q,I,\ast)$ is $$H^{1}(Q,I,\ast) := \mathrm{Der}(Q,I,\ast)/\mathrm{PDer}(Q,I,\ast)$$ and the $1^{\mathrm{st}}$-cohomology of $(Q,I)$ is $H^{1}(Q,I) := \bigcup_{\ast} H^{1}(Q,I,\ast)$ where the union is over possible actions. Since the notion of derivation depends on the choice of action, we can restrict to a particular variety $\mathcal V$ by taking the union of $1^{\mathrm{st}}$-cohomology only over those $\mathcal V$-compatible actions $$H^{1}_{\mathcal V}(Q,I) = \bigcup \{ H^{1}(Q,I,\ast) : \ast \text{ is } \mathcal V\text{-compatible} \}.$$ We say a 2-cocycle $T$ for datum $(Q,I)$ is *linear* if the action terms are unary in $I$. Let us first observe that extensions with kernels which are abelian congruences are characterized by abelian ideals with linear 2-cocycles. This will then give a characterization of solvable algebras in the varieties of interest. **Theorem 29**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations. A cohomology class $[T] \in H^{2}_{\mathcal V}(Q,I)$ represents an extension $\pi: M \rightarrow Q$ in which $I = \ker \pi \triangleleft M$ determines an abelian congruence if and only if $I$ is an abelian algebra and $T$ is linear. *Proof.* The 2-cocycle $T$ determines the extensions $M = I \rtimes_{T} Q \rightarrow Q$. We identify $I$ with its copy in $I \rtimes_{T} Q$. Let $\alpha$ be the congruence determined by $I$. First, assume $\alpha$ is an abelian congruence. By realization, the actions terms $a(f,s)$ are given by the multilinear operation symbols $a(f,s)(\vec{x},\vec{a}) = f(l(\vec{x}))_{s}[\vec{a}]$ for a fixed lifting $l:Q \rightarrow M$. Take multilinear $f \in F$ with $n \geq 2$. For any $\vec{z} \in M^{n-2}$, $a,b \in I$ we have $f(a,0,\vec{z})=0=f(0,0,\vec{z}) \Longrightarrow f(a,b,\vec{z}) = f(0,b,\vec{z}) = 0$ since $\alpha$ is abelian; therefore, $a(f,s) = 0$ for $s=\{1,2\}$. A similar argument shows $a(f,s) = 0$ for all $f \in F$ with $\mathop{\mathrm{ar}}f \geq 2$ and $|s|>1$; thus, $a(f,i)$ are the only possible nontrivial action terms. The same argument shows the mutlilinear operations $f \in F$ on $I$ are all trivial. This implies the ideal $I \triangleleft M$ carries only a module structure which is abelian in the TC-commutator . Now, assume $I$ is abelian and $T$ is linear. We directly show the congruence $\alpha$ is abelian in the TC-commutator. Take a term $t$ and tuples $(\vec{a},\vec{x}), (\vec{b},\vec{x}) \in (I \times Q)^{k}$, $(\vec{c},\vec{z}), (\vec{d},\vec{z}) \in (I \times Q)^{m}$. Note we are using the convention $(\vec{a},\vec{x}) = \big( \left\langle a_{1},x_{1} \right\rangle, \ldots, \left\langle a_{k},x_{k} \right\rangle \big)$ to write the tuples. Assume $$\begin{aligned} \label{eqn:51} F_{t} \big( (\vec{a},\vec{x}),(\vec{c},\vec{z}) \big) = F_{t} \big( (\vec{a},\vec{x}),(\vec{d},\vec{z}) \big).\end{aligned}$$ The following claim on the representation of terms in $I \rtimes_{T} Q$ can be established by induction on their generation. The useful facts are that the action terms are unary and so linear in $I$, and the multilinear operations in $I$ are trivial. **Claim 30**. Let $(Q,I,\ast)$ be affine datum in a variety $\mathcal V$ of modules expanded by multilinear operations. If $t(\vec{x})$ is a term in the signature and $T$ a $\mathcal V$-compatible 2-cocycle of the datum, then there exists operations $s$ and $s^{\ast}$ where $s$ and $s^{\ast}$ are iterations of action terms such that the interpretation of the term $t$ in the algebra $I \rtimes_{T} Q$ is given by $$\begin{aligned} \label{eqn:52} F_{t} \big( (\vec{a},\vec{x}),(\vec{c},\vec{z}) \big) = \left\langle \, s((\vec{x},\vec{z}),\vec{a}) + s^{\ast}((\vec{x},\vec{z}), \vec{c}) + t^{\partial,T}(\vec{x},\vec{z}), t^{Q}(\vec{x},\vec{z}) \,\right\rangle\end{aligned}$$ for $(\vec{a},\vec{x}) \in (I \times Q)^{k}$, $(\vec{c},\vec{z}) \in (I \times Q)^{m}$. Now apply the representation in Eq.([\[eqn:52\]](#eqn:52){reference-type="ref" reference="eqn:52"}) to both terms in Eq.([\[eqn:51\]](#eqn:51){reference-type="ref" reference="eqn:51"}) to conclude $s^{\ast}((\vec{x},\vec{z}),\vec{c}) = s^{\ast}((\vec{x},\vec{z}),\vec{d})$. To this equality we can add the operations $s((\vec{x},\vec{z}),\vec{b})$ and $t^{\partial,T}(\vec{x},\vec{z})$ to both sides and again use Eq.([\[eqn:52\]](#eqn:52){reference-type="ref" reference="eqn:52"}) to conclude $F_{t} \big( (\vec{b},\vec{x}),(\vec{c},\vec{z}) \big) = F_{t} \big( (\vec{b},\vec{x}),(\vec{d},\vec{z}) \big)$. This shows $\alpha$ satisfies the term-condition and so is an abelian congruence. ◻ For an ideal $I \triangleleft M$, the derived series is defined by $[I]^{0} = I$ and $[I]^{n+1} = [[I]^{n},[I]^{n}]$. The ideal $I$ is *n-step solvable* if $[I]^{n}=0$ and the algebra $M$ is *n-step solvable* if $[M]^{n}=0$. **Proposition 31**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations. Then $M \in \mathcal V$ is a n-step solvable algebra if and only if $M$ can be represented as a right-associated product $$M \approx Q_{n} \rtimes_{T_{n-1}} Q_{n-1}\rtimes_{T_{n-2}} \cdots \rtimes_{T_{1}} Q_{1}$$ where each $Q_{i} \in \mathcal V$ is abelian and $T_i$ is linear. *Proof.* Assume $M \in \mathcal V$ is n-step solvable. Set $Q_{k} = [M]^{k-1}/[M]^{k}$ for $1 \leq k \leq n$. By Theorem [Theorem 29](#thm:multiabel){reference-type="ref" reference="thm:multiabel"} and Theorem [Theorem 14](#thm:multirep){reference-type="ref" reference="thm:multirep"}, we have $M/[M]^{k+1} \approx [M]^{k}/[M]^{k+1} \rtimes_{T_{k}} M/[M]^{k}$ where $[M]^{k}/[M]^{k+1}$ is abelian and each $T_k$ is linear. The right-associated product follows recursively. Assume now we have the right-associated representation $$M \approx Q_{n} \rtimes_{T_{n-1}} Q_{n-1}\rtimes_{T_{n-2}} \cdots \rtimes_{T_{1}} Q_{1}$$ for abelian algebras $Q_i \in \mathcal V$ and compatible linear 2-cocycles $T_i$. Set $A_{i} := Q_{i} \rtimes_{T_{n-1}} Q{n-1}\rtimes_{T_{n-2}} \cdots \rtimes_{T_{1}} Q_{1}$. By Theorem [Theorem 29](#thm:multiabel){reference-type="ref" reference="thm:multiabel"}, the second-projection $p_{2}: A_{i} = Q_{i} \rtimes_{T_{i-1}} A_{i-1} \rightarrow A_{i-1}$ realizes an extension in which $Q_{i} = \ker p_{2}$ determines an abelian congruence. We can apply the homomorphism property of the commutator to see that $$[A_{2},A_{2}] \vee Q_{2}/Q_{2} = [A_{2} \vee Q_{2}/Q_{2}, A_{2} \vee Q_{2}/Q_{2} ] = [Q_{1},Q_{1}] = 0$$ since $Q_{1}$ is abelian; thus, $[A_{2},A_{2}] \leq Q_{2}$ and so $[A_{2}]^{2} \leq [Q_{2},Q_{2}]=0$. Assume we have that $A_{k}$ is k-step solvable. Then $$\begin{aligned} ^{k} \vee Q_{k+1}/Q_{k+1} &= \left[ [A_{k+1}]^{k-1} \vee Q_{k+1}/Q_{k+1}, [A_{k+1}]^{k-1} \vee Q_{k+1}/Q_{k+1} \right] \\ &= \left[ [A_{k+1}]^{k-2} \vee Q_{k+1}/Q_{k+1} \right]^{2} \\ &= \vdots \\ &= [A_{k+1} \vee Q_{k+1}/Q_{k+1}]^{k} = \left[ A_{k} \right]^{k} = 0 \end{aligned}$$ which implies $[A_{k+1},A_{k+1}]^{k} \leq Q^{k+1}$ and so $[A_{k+1}]^{k+1} \leq [Q_{k+1},Q_{k+1}] = 0$; therefore, $A_{k+1}$ is k+1-step solvable. The proof is completed by induction since $A_{n} \approx M$. ◻ We can loosely paraphrase Proposition [Proposition 31](#prop:solvable){reference-type="ref" reference="prop:solvable"} as stating that solvable algebras are iterated linear translations of semidirect products of abelian algebras; analogously, Proposition [Proposition 35](#prop:nilpotent){reference-type="ref" reference="prop:nilpotent"} will state that nilpotent algebras are iterated linear translations of direct products of abelian algebras. We say $(Q,I,\ast)$ is *affine datum* if $I$ is an abelian algebra and if the terms in the action $Q \ast I$ are all unary in $I$. If $T$ is a linear 2-cocycle for datum $(Q,I)$ in which $I$ is an abelian algebra and $T \sim T'$, then $T'$ has the same action terms as $T$; to see this, for any 2-coboundary which witnesses $T-T' \in B^{2}(Q,I)$, the operations in (B4) must be zero because the higher-arity action terms and multilinear operation in $I$ are both trivial. This means equivalence respects the class of extensions which realize a fixed affine datum. There is an addition on equivalence classes of 2-cocycles $[T] + [T']:=[T + T']$ by the addition induced by $I$ on the non-action terms; that is, $$(T + T')_{+} := T_{+} + T_{+}' \quad \quad (T + T')_{r} := T_{r} + T_{r}' \quad \quad (T + T')_{f} := T_{f} + T_{f}'$$ and the action terms of $T+T'$ are exactly the same as $T$ and $T'$. Since the action terms are unary and the multilinear operations in $I$ are trivial, it is easy to see from (B1)-(B3) that the addition on equivalence classes is a well-defined abelian group operation. We say $(Q,I,\ast)$ is datum in $\mathcal V$ if $Q,I \in \mathcal V$ and $Q \ast I$ is a $\mathcal V$-compatible action. We say $T$ is a 2-cocycle for $(Q,I,\ast)$ if it is a 2-cocycle for $(Q,I)$ and the action-terms of $T$ are given by the action $Q \ast I$. The $2^{\mathrm{nd}}$-cohomology for the datum $(Q,I,\ast)$ in the variety $\mathcal V$ is the set $H^{2}_{\mathcal V}(Q,I,\ast)$ of equivalence classes of strictly $\mathcal V$-compatible 2-cocycles for the datum $(Q,I,\ast$). **Theorem 32**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$ and $(Q,I,\ast)$ affine datum in $\mathcal V$. The set of equivalence classes of extensions which realize the datum $(Q,I,\ast)$ in the variety $\mathcal V$ is in bijective correspondence with the abelian group $H^{2}_{\mathcal V}(Q,I,\ast)$. The equivalence class of the semidirect product corresponds to the equivalence class of the trivial 2-cocycle. *Proof.* Since we are given that the action $Q \ast I$ is $\mathcal V$-compatible, it follows from Lemma [Lemma 5](#lem:semident){reference-type="ref" reference="lem:semident"} and Theorem [Theorem 14](#thm:multirep){reference-type="ref" reference="thm:multirep"} that for any 2-cocycle $T$ for $(Q,I,\ast)$, $I \rtimes_{T} Q \in \mathcal V$ if and only if $T$ is strictly $\mathcal V$-compatible. Then Theorem [Theorem 20](#thm:10){reference-type="ref" reference="thm:10"} guarantees the set $H^{2}_{\mathcal V}(Q,I,\ast)$ parametrizes the equivalence classes of extensions in $\mathcal V$ which realize $(Q,I,\ast)$. Since the action terms are linear and the multilinear operations are trivial in $I$, it follows by the same derivation as in Lemma [Lemma 5](#lem:semident){reference-type="ref" reference="lem:semident"} that for any term $t$ in the signature of $\mathcal V$ we have $t^{\partial,T+T'} = t^{\partial,T} + t^{\partial,T'}$. Because the action terms of $T + T'$ are the same as $T$ and $T'$, it follows that $T + T'$ is again strictly $\mathcal V$-compatible whenever both $T$ and $T'$ are; altogether, the abelian group addition on 2-cocycles for affine datum respects equivalence classes and compatibility. ◻ **Remark 33**. In the case of affine datum $(Q,I,\ast)$, there is slight modification to the representation of terms from Lemma [Lemma 5](#lem:semident){reference-type="ref" reference="lem:semident"}. Let $M = I \rtimes_{T} Q$ realize the datum. In the evaluation $F_{t}(\vec{m})$ of the term $t(\vec{x})$ in the algebra $I \rtimes_{T} Q$, the operation $t^{\partial,T}$ gathers together all instances of the non-action parts of the 2-cocycle $T_{+}$, $T_{r}$ and $T_{f}$; for general datum, it involves evaluation of the action terms and all the operations of the algebra $I$ and so $t^{\partial,T}$ may depend on both the $I$ and $Q$ coordinates of $\vec{m}$. In the case of affine datum, the multilinear operations of $I$ are trivial and all the action terms are unary in $I$. Since the non-action parts of the 2-cocycle must appear in the operations of $t^{\partial,T}$, it cannot depend on the $I$-coordinates. Then for $m_i = \left\langle b_{i}, x_{i} \right\rangle \in I \times Q$ we can write $$F^{M}_{t}(\vec{m}) = \left\langle t^{\ast,T}(\vec{b},\vec{x}) + t^{\partial,T}(\vec{x}), t^{Q}(\vec{x}) \right\rangle.$$ Take an extension $\pi: M \rightarrow Q$ realizing affine datum $(Q,I,\ast)$ and determined by the 2-cocycle $T$. Then Theorem [Theorem 20](#thm:10){reference-type="ref" reference="thm:10"} yields the isomorphism $\psi: M \ni a \longmapsto \left\langle a - l \circ \pi (a), \pi(a) \right\rangle \in I \rtimes_{T} Q$ where in the algebra $I \rtimes_{T} Q$ the multilinear operations are computed by $$\begin{aligned} \label{eqn:20} F_{f}\left( \vec{a}, \vec{x} \right) = \left\langle \sum_{i \in [\mathop{\mathrm{ar}}f]} a(f,i)(\vec{x},\vec{a}) + T_{f}(\vec{x}), f^{Q}(\vec{x}) \right\rangle .\end{aligned}$$ This agrees with the more general notion of extensions realizing affine datum developed for arbitrary varieties of universal algebras in [@wiresI]. Since varieties of multilinear module expansions form a special subclass of varieties with a difference term, the next two results follow directly by comparison of Eq [\[eqn:20\]](#eqn:20){reference-type="eqref" reference="eqn:20"} with the characterization of central extensions and nilpotent algebras from [@wiresI]; however, we will argue separately. A 2-cocycle for datum $(Q,I)$ is *action-trivial* if the action terms are all zero. **Theorem 34**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations. A class $[T] \in H^{2}_{\mathcal V}(Q,I)$ represents an extension $\pi: M \rightarrow Q$ in which $I = \ker \pi \triangleleft M$ determines a central congruence if and only if $I$ is an abelian algebra and $T$ is action-trivial. *Proof.* Assume the congruence $\alpha_{I}$ determined by the ideal $I$ is central; in particular, $I$ is an abelian algebra and the action terms are unary in $I$ according to Theorem [Theorem 29](#thm:multiabel){reference-type="ref" reference="thm:multiabel"}. Take $f \in F$, $i \in [f]$ and $\vec{x} \in Q^{\mathop{\mathrm{ar}}f}$, $\vec{a} \in I^{\mathop{\mathrm{ar}}f}$. Since $\mathop{\mathrm{ar}}f \geq 2$, choose $i \neq j \in [ \mathop{\mathrm{ar}}f ]$ and $\vec{z} \in Q^{\mathop{\mathrm{ar}}f}$ such that $z_{j} = 0$. Then applying the term-condition we see that $$a(f,i)(\vec{z},\vec{0}) = 0 = a(f,i)(\vec{x},\vec{0}) \quad \Longrightarrow \quad 0 = a(f,i)(\vec{z},\vec{a}) = a(f,i)(\vec{x},\vec{a})$$ since $[I,M]=0$; thus, $a(f,i) \equiv 0$ and so $T$ is action-trivial. Now assume $I$ is an abelian algebra and $T$ is action-trivial. Then for a term $t$, the interpretation in $I \rtimes_{T} Q$ is given by $F_{t}\left( \vec{a}, \vec{x} \right) = \left\langle t^{I}(\vec{a}) + t^{\partial,T}(\vec{a},\vec{x}), t^{Q}(\vec{x}) \right\rangle$ where no action-terms appear in $t^{\partial,T}$. We can follow the argument in Theorem [Theorem 29](#thm:multiabel){reference-type="ref" reference="thm:multiabel"} and verify the term-condition to show $\alpha_{I}$ is a central congruence. ◻ **Proposition 35**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations. Then $M \in \mathcal V$ is a n-step nilpotent algebra if and only if $M$ can be represented as a right-associated product $$M \approx Q_{n} \rtimes_{T_{n-1}} Q_{n-1}\rtimes_{T_{n-2}} \cdots \rtimes_{T_{1}} Q_{1}$$ where each $Q_{i} \in \mathcal V$ is abelian and $T_i$ is action-trivial. *Proof.* This is precisely the argument in Proposition [Proposition 31](#prop:solvable){reference-type="ref" reference="prop:solvable"} where we use the the characterization of central extensions in Theorem [Theorem 34](#thm:multicentral){reference-type="ref" reference="thm:multicentral"} instead of Theorem [Theorem 29](#thm:multiabel){reference-type="ref" reference="thm:multiabel"}. ◻ We denote by $(Q,I,0)$ datum in which the action is trivial; that is, $a(f,s) \equiv 0$ for all $f \in F$, $s \in [\mathop{\mathrm{ar}}f]^{\ast}$. **Theorem 36**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$ and $(Q,I)$ datum in $\mathcal V$ with $I$ an abelian algebra. The set of equivalence classes of central extensions which realize the datum $(Q,I)$ in the variety $\mathcal V$ is in bijective correspondence with the abelian group $H^{2}_{\mathcal V}(Q,I,0)$. The equivalence class of the direct product $I \times Q$ corresponds to the equivalence class of the trivial 2-cocycle. In light of the last theorem, when $I$ is an abelian algebra we may refer to $(Q,I,0)$ as *central datum*. The next lemma gives a description of $1^{\mathrm{st}}$-cohomology for central datum. **Lemma 37**. Let $\mathcal V$ be a variety of $R$-modules expanded by multilinear operations $F$ and $Q,I \in \mathcal V$ with $I$ an abelian algebra. Then $H^{1}(Q,I,0) = \mathrm{Der}(Q,I,0) = \{ h \in \mathop{\mathrm{Hom}}_{R}(Q,I): h([Q,Q]) = 0 \}$. *Proof.* By Proposition [Theorem 34](#thm:multicentral){reference-type="ref" reference="thm:multicentral"}, the copy of $I$ in an extension realizing the datum determines a central congruence. Let $\gamma$ be a principal stabilizing automorphism of $I \times Q$. Then there is term $t(x,\vec{y})$ and $\vec{c},\vec{d} \in (I \times \{0\})^{k}$ such that $\gamma(x)=t(x,\vec{c})$ and $x=t(x,\vec{d})$. Since $\gamma$ is a stabilizing automorphism, it restricts to the identity on $I \times \{ 0 \}$ and so it has fixed-points, say $a \in I \times \{ 0 \}$. Then for any $x \in I \times Q$, the matrix $$\begin{bmatrix} a & \gamma(x) \\ a & x \end{bmatrix} = \begin{bmatrix} \gamma(a) & \gamma(x) \\ a & x \end{bmatrix} = \begin{bmatrix} t(a,\vec{c}) & t(x,\vec{c}) \\ t(a,\vec{d}) & t(x,\vec{d}) \end{bmatrix} \in M(\alpha_{I \times \{0 \}},1)$$ implies $(\gamma(x),x) \in [1, \alpha_{I \times \{0\}}] = 0$; thus, $0 = \mathrm{PStab}(I \times Q) \approx \mathrm{PDer}(Q,I,0)$ and so $H^{1}(Q,I,0) = \mathrm{Der}(Q,I,0)$. Since the action terms are all trivial, it follows from (B1) - (B3) that $h \in \mathrm{Der}(Q,I,0)$ if and only if $h$ is an $R$-module homomorphism from $Q$ to $I$ such that $h(f^{Q}(\vec{x})) = 0$ for all $f \in F$ and $\vec{x} \in Q^{\mathop{\mathrm{ar}}f}$; that is, $\mathrm{Der}(Q,I,0) = \{ h \in \mathop{\mathrm{Hom}}_{R}(Q,I): h([Q,Q]) = 0 \}$. ◻ It is possible to give a more concrete form to principal derivations. **Remark 38**. Let $(Q,I,\ast)$ be datum and $\gamma$ a nontrivial principal stabilizing automorphism. There is a term $t(x,\vec{y})$ and $\vec{c},\vec{d} \in I^{n}$ such that $$\begin{aligned} \left\langle a - d_{\gamma}(x), x \right\rangle = \gamma(a,x) &= F_{t} \left( \left\langle a,x \right\rangle, \left\langle c_{1},0 \right\rangle, \ldots, \left\langle c_{n},0 \right\rangle \right) = \left\langle t^{I}(a,\vec{c}) + t^{\ast,T}((a,\vec{c}),(x,\vec{0})) , t^{Q}(x,\vec{0})\right\rangle \\ \left\langle a, x \right\rangle &= F_{t} \left( \left\langle a,x \right\rangle, \left\langle d_{1},0 \right\rangle, \ldots, \left\langle d_{n},0 \right\rangle\right) = \left\langle t^{I}(a,\vec{d}) + t^{\ast,T}((a,\vec{d}),(x,\vec{0})) , t^{Q}(x,\vec{0})\right\rangle\end{aligned}$$ where we have used the representation from Lemma [Lemma 5](#lem:semident){reference-type="ref" reference="lem:semident"} in the semidirect product $I \rtimes_{T} Q$. Taking $x=0$ and using $d_{\gamma}(0)=0$, we conclude that $t(a,\vec{c}) = t(a,\vec{d})=a$. Then from the above we see that $$- d_{\gamma}(x) = t^{\ast,T}((0,\vec{c}),(x,\vec{0})) \quad \quad \text{ and } \quad \quad 0 = t^{\ast,T}((0,\vec{d}),(x,\vec{0}))$$ since the derivation $d_{\gamma}$ is independent of $a$. Note that $t^{\ast,T}$ is a sum of iterated action terms and so cannot depend on $0 \in Q$. In this way, we can just write $d_{\gamma}(x) = s^{\ast,T}(\vec{c},x)$ which is a sum of iterated action terms which depend only on $x \in Q$ and evaluated on the constants $\vec{c} \in I^{n}$. Then $\left\langle d_{\gamma}(x), 0 \right\rangle = F_{s} \left(\left\langle 0,x \right\rangle, \left\langle c_{1},0 \right\rangle ,\ldots, \left\langle c_{n}, \right\rangle \right)$. # Ideal-Preserving Derivations {#section:4} For the varieties under consideration, the notion of a derivation (as different from a cohomological derivation) is meaningful. For affine datum, we prove a Well's type theorem which characterizes the Lie algebra of derivations which preserve a fixed ideal; more formally, for $I \triangleleft M$ which realizes a group-trivial extension, the Lie algebra of derivations of $M$ which fix $I$ is a Lie algebra extension of compatible pairs of derivations of $I$ and $M/I$ by the cohomological derivations of the datum $\mathrm{Der}(Q,I,\ast)$. This is very similar to the theorem of Well's for automorphisms fixing a normal subgroup and generalizes the analogous result for associative conformal algebras in [@conform2]. **Theorem 39**. Let $\mathcal V$ be a variety of modules expanded by multilinear operations and $M \in \mathcal V$ Let $\pi: M \rightarrow Q$ be a group-trivial extension realizing affine datum $(Q,I,\ast)$. If $T$ is the 2-cocycle associated to the extension, then $$0 \longrightarrow \mathrm{Der}(Q,I,\ast) \longrightarrow \mathrm{Der}_{I} M \stackrel{\psi}{\longrightarrow} c(I,Q,\ast) \stackrel{W_{T}}{\longrightarrow} H^{2}_{_{R} \mathcal M_{F}}(Q,I)$$ is an exact sequence of Lie algebras. We begin by developing the items in the statement of the theorem. Let $T$ be a 2-cocycle which determines the extension $\pi: M \rightarrow Q$. The extension is *group-trivial* if $T \sim T'$ with $T_{+}'=0$; consequently, the abelian group reduct of a group-trivial extension factors as a direct product of the datum groups. **Definition 40**. Let $\mathcal V$ be a variety of multilinear expansions of $R$-modules and $M \in \mathcal V$. A map $h: M \rightarrow M$ is a *derivation* if it is an $R$-module homomorphism such that $$\begin{aligned} h \left( f^{M}(x_{1},\ldots,x_{n}) \right) &= f^{M}(h(x_{1}),x_{2},\ldots,x_{n}) + f^{M}(x_{1},h(x_{2}),\ldots,x_{n}) + \cdots + f^{M}(x_{1},x_{2},\ldots, h(x_{n})) \\ &= \sum_{i=1}^{n} f^{M}(\vec{x})_{i}[h(\vec{x})] \end{aligned}$$ for all multilinear operations $f \in F$ with $n= \mathop{\mathrm{ar}}f$. The set of derivations of $M$ is denoted by $\mathrm{Der} \, M$ and is an $R$-module under the module operations induced by $M$; consequently, it is a Lie algebra over $R$ by the standard bracket $[\alpha,\beta] := \alpha \circ \beta - \beta \circ \alpha$ through composition. Given $(\alpha,\beta) \in \mathrm{Der} \, I \times \mathrm{Der} \, Q$, we define a map $\phi_{(\alpha,\beta)} : I \times Q \rightarrow I \times Q$ by the rule $\phi_{(\alpha,\beta)} \left\langle a,x \right\rangle:= \left\langle \alpha(a),\beta (x) \right\rangle$. Let $T$ be a linear group-trivial 2-cocycle. We can query what conditions must be true in order for $\phi_{(\alpha,\beta)}$ to be a derivation of the semidirect product $I \rtimes_{T} Q$. **Lemma 41**. Let $\mathcal V$ be a variety of multilinear expansions of $R$-modules which contains datum $(Q,I)$. Let $T$ be group-trivial linear 2-cocycle and $(\alpha,\beta) \in \mathrm{Der} \, I \times \mathrm{Der} \, Q$. Then $\phi_{(\alpha,\beta)} \in \mathrm{Der} \, \left( I \rtimes_{T} Q \right)$ if and only if 1. $\alpha \circ T_{r}(x) = T_{r}(\beta(x))$ for all $r \in R$; 2. for each $f \in F$ with $n= \mathop{\mathrm{ar}}f$ and $k \in [n]$, $$\begin{aligned} \alpha \circ a(f,k)(\vec{x},\vec{a}) = \sum_{i \in [n]} a(f,k)\left( (x_{1},\ldots,\beta(x_{i}),\ldots,x_{n}), (a_{1},\ldots,\alpha(a_{i}),\ldots,a_{n}) \right);\end{aligned}$$ 3. for each $f \in F$ with $n= \mathop{\mathrm{ar}}f$, $\alpha \circ T_{f}(\vec{x}) = \sum_{i \in [n]} T_{f}(x_{1},\ldots,\beta(x_{i}),\ldots,x_{n})$. *Proof.* Since $T$ is group-trivial and linear, the action terms are unary in $I$ and $T_{+} = 0$. It is then easy to see that $\phi_{(\alpha,\beta)}$ is an abelian group homomorphism. We first see how to derive conditions (C1)-(C3). In order for $\phi_{(\alpha,\beta)}$ to a be module homomorphism we must have equality between $$\begin{aligned} \phi_{(\alpha,\beta)} \left( r \cdot \left\langle a,x\right\rangle \right) = \phi_{(\alpha,\beta)} \left( \left\langle ra + T_{r}(x), rx \right\rangle\right) = \left\langle \alpha(rx) + \alpha \circ T_{r}(x), \beta(rx) \right\rangle\end{aligned}$$ and $$\begin{aligned} r \cdot \phi_{(\alpha,\beta)} \left(\left\langle a, x \right\rangle \right) = r \cdot \left\langle \alpha(a), \beta(x) \right\rangle = \left\langle r \alpha(x) + T_{r}(\beta(x)), r\beta(x) \right\rangle.\end{aligned}$$ Since $\alpha$ and $\beta$ are module homomorphisms, we must have $\alpha \circ T_{r}(x) = T_{r}(\beta(x))$ for all $r \in R$. Calculating for multilinear $f \in F$ with $n=\mathop{\mathrm{ar}}f$, $$\begin{aligned} \phi_{(\alpha,\beta)} \circ F_{f}\left( \left\langle a_{1}, x_{1} \right\rangle,\ldots, \left\langle a_{1}, x_{1} \right\rangle \right) &= \phi_{(\alpha,\beta)} \circ \left\langle f^{I}(\vec{a}) + \sum_{i \in [n]} a(f,i)(\vec{x},\vec{a}) + T_{f}(\vec{x}) , f^{Q}(\vec{x}) \right\rangle \\ &= \left\langle \alpha \circ f^{I}(\vec{a}) + \sum_{i \in [n]} \alpha \circ a(f,i)(\vec{x},\vec{a}) + \alpha \circ T_{f}(\vec{x}), \beta \circ f^{Q}(\vec{x}) \right\rangle \end{aligned}$$ and $$\begin{aligned} \sum_{i \in [n]} &F_{f} \left( \left\langle a_{1},x_{1} \right\rangle,\ldots,\left\langle \alpha(a_{i}),\beta(x_{i}) \right\rangle,\ldots,\left\langle a_{n},x_{n} \right\rangle \right) \\ &= \sum_{i \in [n]} \left\langle f^{I}(\vec{a})_{i}[\alpha(\vec{a})] + \sum_{j \in [n]} a(f,j) \left( (\vec{x})_{i} [\beta(\vec{x})], (\vec{a})_{i} [\alpha(\vec{a})] \right) + T_{f}(\vec{x})_{i} [\beta(\vec{x})], f^{Q}(\vec{x})_{i} [\beta(\vec{x})] \right\rangle \\ &= \left\langle \sum_{i \in [n]} f^{I}(\vec{a})_{i}[\alpha(\vec{a})] + \sum_{i \in [n]} \sum_{j \in [n]} a(f,j) \left( (\vec{x})_{i} [\beta(\vec{x})], (\vec{a})_{i} [\alpha(\vec{a})] \right) + \sum_{i \in [n]} T_{f}(\vec{x})_{i} [\beta(\vec{x})], \sum_{i \in [n]} f^{Q}(\vec{x})_{i} [\beta(\vec{x})] \right\rangle \end{aligned}$$ using the fact that $T_{+}=0$. Let us assume $\phi_{(\alpha,\beta)}$ is a derivation which produces the equality $$\phi_{(\alpha,\beta)} \circ F_{f}\left( \left\langle a_{1}, x_{1} \right\rangle,\ldots, \left\langle a_{1}, x_{1} \right\rangle \right) = \sum_{i \in [n]} F_{f} \left( \left\langle a_{1},x_{1} \right\rangle,\ldots,\left\langle \alpha(a_{i}),\beta(x_{i}) \right\rangle,\ldots,\left\langle a_{n},x_{n} \right\rangle \right).$$ Since $\alpha$ and $\beta$ are derivations, the above calculations yield $$\begin{aligned} \label{eqn:48} \sum_{i \in [n]} \alpha \circ a(f,i)(\vec{x},\vec{a}) + \alpha \circ T_{f}(\vec{x}) = \sum_{i \in [n]} \sum_{j \in [n]} a(f,j) \left( (\vec{x})_{i} [\beta(\vec{x})], (\vec{a})_{i} [\alpha(\vec{a})] \right) + \sum_{i \in [n]} T_{f}(\vec{x})_{i} [\beta(\vec{x})].\end{aligned}$$ Suppose for each $k \in [n]$ we set $x_{k}=0$. Then $a(f,i)(\vec{x},\vec{a}) = 0$ for $i \neq k$ and $T_{f}(\vec{x}) = 0$. We also see that each $T_{f}(\vec{x})_{i} [\beta(\vec{x})] = 0$ since either $\beta(x_{k})=0$ or $x_{k}=0$ appears in the evaluated input of $T_{f}(\vec{x})_{i} [\beta(\vec{x})]$. Similarly, for $j \neq k$ either $x_{k}=0$ is an evaluated input in the action when $i \neq k$ or $\beta(x_{k})=0$ is an evaluated input in the action when $i=k$; thus, $a(f,j) \left( (\vec{x})_{i} [\beta(\vec{x})], (\vec{a})_{i} [\alpha(\vec{a})] \right) = 0$ for $j \neq k$. Then Eqn.([\[eqn:48\]](#eqn:48){reference-type="ref" reference="eqn:48"}) implies $$\alpha \circ a(f,k)(\vec{x},\vec{a}) = \sum_{i \in [n]} a(f,k) \left( (\vec{x})_{i} [\beta(\vec{x})], (\vec{a})_{i} [\alpha(\vec{a})] \right)$$ which confirms condition (C2). Subtracting this from Eqn.([\[eqn:48\]](#eqn:48){reference-type="ref" reference="eqn:48"}) yields $$\alpha \circ T_{f}(\vec{x}) = \sum_{i \in [n]} T_{f}(\vec{x})_{i} [\beta(\vec{x})]$$ which confirms condition (C3). From the above, it is straightforward to confirm that conditions (C1)-(C3) guarantee $\phi_{(\alpha,\beta)}$ is a derivation of $I \rtimes_{T} Q$. ◻ Since we are interested in reconstructing the derivations of an extension which realizes affine datum, note condition (C2) in Lemma [Lemma 41](#lem:compderiv){reference-type="ref" reference="lem:compderiv"} only depends on the datum since the action terms are incorporated into the datum. **Definition 42**. Let $\mathcal V$ be a variety of multilinear expansions of $R$-modules containing affine datum $(Q,I,\ast)$. The set of *compatible derivations* $c(I,Q,\ast)$ consists of all pairs $(\alpha,\beta) \in \mathrm{Der} \, I \times \mathrm{Der} \, Q$ which satisfy for each $f \in F$ with $n= \mathop{\mathrm{ar}}f$ and $k \in [n]$, $$\begin{aligned} \alpha \circ a(f,k)(\vec{x},\vec{a}) = \sum_{i \in [n]} a(f,k)\left( (x_{1},\ldots,\beta(x_{i}),\ldots,x_{n}), (a_{1},\ldots,\alpha(a_{i}),\ldots,a_{n}) \right);\end{aligned}$$ Given $(\alpha,\beta), (\sigma,\kappa) \in c(I,Q,\ast)$, we can take $T$ to be trivial except for the action terms and observe that Lemma [Lemma 41](#lem:compderiv){reference-type="ref" reference="lem:compderiv"} shows $\phi_{(\alpha,\beta)}$ and $\phi_{(\sigma,\kappa)}$ are both derivations of $I \times_{T} Q$. Then the standard bracket $[\phi_{(\alpha,\beta)},\phi_{(\sigma,\kappa)}]$ will again be a derivation of $I \rtimes_{T} Q$ which must satisfy (C2) by Lemma [Lemma 41](#lem:compderiv){reference-type="ref" reference="lem:compderiv"}. This shows $c(I,Q,\ast)$ is a Lie subalgebra of $\mathrm{Der} \, I \times \mathrm{Der} \, Q$. Fix an extension $\pi: M \rightarrow Q$ with $I=\ker \pi$. For any lifting $l:Q \rightarrow M$ of $\pi$, we have $$\begin{aligned} \label{eqn:47} \pi \circ l = \mathop{\mathrm{id}}_{Q} \quad \quad \quad \text{and} \quad \quad \quad a - l \circ \pi(a) \in I \quad \quad (a \in M).\end{aligned}$$ For any $\phi \in \mathrm{Der}_{I} \, M$, define $\phi_{l} := \pi \circ \phi \circ l$. If $l': Q \rightarrow M$ is another lifting for $\pi$, then $l(x) - l'(x) \in I$ for all $x \in Q$ which implies $\phi \circ l(x) - \phi \circ l'(x) \in I$ because $\phi \in \mathrm{Der}_{I} \, M$. It follows that $\phi_{l} = \phi_{l'}$ and so the value of $\phi_{l}$ is independent of the lifting. We can also see that $\phi_{l}$ is a derivation of $Q$. For a multilinear operation $f \in F$ with $n = \mathop{\mathrm{ar}}f$, if we substitute $f^{M}(l(\vec{x}))$ for $a$ in Eq. [\[eqn:47\]](#eqn:47){reference-type="ref" reference="eqn:47"} and apply $\phi$, then $I \ni \phi \left( f^{M}(l(\vec{x})) \right) - \phi \left( l(f^{Q}(\vec{x})) \right)$. Then applying $\pi$ we see that $$\begin{aligned} 0 = \pi \circ \phi \left( f^{M}(l(\vec{x})) \right) - \phi_{l}(f^{Q}(\vec{x})) &= \pi \left( \sum_{i \in [n]} f^{M}(l(\vec{x}))_{i} [ \phi(l(\vec{x})) ] \right) - \phi_{l}(f^{Q}(\vec{x})) \\ &= \sum_{i \in [n]} f^{Q}(\vec{x})_{i}[\phi_{l}(\vec{x})] - \phi_{l}(f^{Q}(\vec{x})).\end{aligned}$$ A similar argument shows $\phi_{l}$ is an $R$-module homomorphism; therefore, $\phi_{l} \in \mathrm{Der} \, Q$. If $\phi \in \mathrm{Der}_{I} \, M$, then clearly the restriction $\phi|_{I} \in \mathrm{Der} \, I$. **Lemma 43**. Let $\mathcal V$ be a variety of multilinear expansions of $R$-modules and $M \in \mathcal V$. If $\pi: M \rightarrow Q$ is an extension realizing affine datum $(Q,I,\ast)$ and $l: Q \rightarrow M$ is a lifting of $\pi$, then $\psi(\phi) := (\phi|_{I},\phi_l)$ defines a Lie algebra homomorphism $\psi: \mathrm{Der}_{I} \, M \rightarrow c(I,Q,\ast)$. *Proof.* We first show $(\phi|_{I},\phi_l)$ is a compatible pair for $\phi \in \mathrm{Der}_{I} \, M$. Since $(Q,I,\ast)$ is affine datum, the action terms $a(f,s)(\vec{x},\vec{a})=0$ for $|s|>1$. If we substitute $\phi \circ l(x_i)$ into a in Eq.([\[eqn:47\]](#eqn:47){reference-type="ref" reference="eqn:47"}), we see that $I \ni b_{i} = \phi \circ l(x_i) - l \circ \pi \circ \phi \circ l(x_i) = \phi \circ l(x_i) - l \circ \phi_{l}(x_i)$ from some $b_i \in I$. Since the extension $M$ realizes the datum, we have for multilinear $f \in F$ with $n = \mathop{\mathrm{ar}}f$, $\vec{x} \in Q^{n}$, $\vec{a} \in I^{n}$ and $k \neq i$, $$\begin{aligned} f^{M} &(l(x_{1}),\ldots,a_{k},\ldots,(x_{n}))_{i}[\phi \circ l(x_{i})] \\ &= f^{M}(l(x_{1}),\ldots,a_{k},\ldots,(x_{n}))_{i}[b_{i} + l \circ \phi_{l}(x_{i})] \\ &= f^{M}(l(x_{1}),\ldots,a_{k},\ldots,(x_{n}))_{i}[b_{i}] + f^{M}(l(x_{1}),\ldots,a_{k},\ldots,(x_{n}))_{i}[l \circ \phi_{l}(x_{i})] \\ &= f^{M}(l(x_{1}),\ldots,a_{k},\ldots,(x_{n}))_{i}[l \circ \phi_{l}(x_{i})] \\ &= a(f,k) \left( (x_{1},\ldots, \phi_{l}(x_{i}),\ldots,x_{n}), \vec{a} \right) \\ &= a(f,k) \left( (x_{1},\ldots, \phi_{l}(x_{i}),\ldots,x_{n}), (a_{1},\ldots,\phi(a_{i}),\ldots,a_{n}) \right).\end{aligned}$$ Then using that $\phi$ is a derivation we have $$\begin{aligned} \phi \circ a(f,k)(\vec{x},\vec{a}) &= \phi \circ f^{M} \left( l(x_{1}),\ldots,a_{k},\ldots,l(x_{n}) \right) \\ &= f^{M}\left( \phi \circ l(x_1),\ldots,a_{k},\ldots,x_{n} \right) + \cdots + f^{M}\left( x_1,\ldots,a_{k},\ldots,\phi \circ l(x_{n}) \right) \\ &= f^{M}(l(x_{1}),\ldots,\phi(a_{k}),\ldots,l(x_{n})) + \sum_{i \neq k} f^{M} (l(x_{1}),\ldots,a_{k},\ldots,(x_{n}))_{i}[\phi \circ l(x_{i})] \\ &= \sum_{i \in [n]} a(f,k)\left( (x_{1},\ldots,\phi_{l}(x_{i}),\ldots,x_{n}), (a_{1},\ldots,\phi(a_{i}),\ldots,a_{n}) \right)\end{aligned}$$ which shows $\psi(\phi) = (\phi|_{I},\phi_l) \in c(I,Q)$. Take $\phi,\sigma \in \mathrm{Der}_{I} \, M$. From Eq.([\[eqn:47\]](#eqn:47){reference-type="ref" reference="eqn:47"}) we have $\sigma \circ l(x) - l \circ \pi (\sigma \circ l(x)) \in I$ and so $\phi \circ \sigma \circ l(x) - \phi \circ l \circ \pi (\sigma \circ l(x)) \in I$ since $\phi(I) \subseteq I$. Then applying the canonical epimorphism $\pi$ we have $(\phi \circ \sigma)_{l}(x) = \phi_{l} \circ \sigma_{l}(x)$. Then by direct calculation $$\begin{aligned} _{l}(x) = \pi \circ [\phi,\sigma] \circ l (x) = \pi \circ (\phi \circ \sigma) \circ l(x) - \pi \circ (\sigma \circ \phi) \circ l(x) &= \phi_{l} \circ \sigma_{l}(x) - \sigma_{l} \circ \phi_{l} (x) = [\phi_{l},\sigma_{l}](x).\end{aligned}$$ Clearly, restriction to $I$ respects the bracket. Since the bracket in $c(I,Q,\ast)$ is the restriction of the bracket in $\mathrm{Der} \, I \times \mathrm{Der} \, Q$, we have shown $\psi$ is a Lie algebra homomorphism. ◻ For each $(\alpha,\beta) \in c(I,Q,\ast)$ and function $f: Q^{n} \rightarrow I$, define $f^{(\alpha,\beta)}$ by $$\begin{aligned} f^{(\alpha,\beta)}(\vec{x}):= \alpha \circ f(\vec{x}) - \sum_{i\in [n]} f(x_{1},\ldots,x_{i-1},\beta(x_{i}),x_{i+1},\ldots,x_{n}).\end{aligned}$$ If $T$ is a 2-cocycle for affine datum $(Q,I,\ast)$, then define $T^{(\alpha,\beta)} := \{T^{(\alpha,\beta)}_{+}, T^{(\alpha,\beta)}_{r},T^{(\alpha,\beta)}_{f} : r \in R, f \in F \}$. It may not be the case that $T^{(\alpha,\beta)}$ is $\mathcal V$-compatible when $T$ is, but we shall see that it is $_{R} \mathcal M_{F}$-compatible. Let $H^{2}_{\mathcal V}(Q,I,\ast)^{\mathrm{gr}}$ denote the subset of group-trivial 2-cocycles and note it is a subgroup of $2^{\mathrm{nd}}$-cohomology for affine datum. **Lemma 44**. Let $\mathcal V$ be a variety of multilinear expansions of $R$-modules and $(Q,I,\ast)$ affine datum in $\mathcal V$. Then $W((\alpha,\beta),[T]) := [T^{(\alpha,\beta)}]$ defines a Lie algebra representation $$W: c(I,Q,\ast) \times H^{2}_{\mathcal V}(Q,I,\ast)^{\mathrm{gr}} \rightarrow H^{2}_{_{R} \mathcal M_{F}}(Q,I,\ast)$$ of $c(I,Q,\ast)$ on $H^{2}_{_{R} \mathcal M_{F}}(Q,I,\ast)$. *Proof.* We must first show $W$ is well-defined on cohomology classes. This is done by showing the action on a group-trivial 2-coboundary is again a 2-coboundary. Suppose $G$ is a 2-coboundary for the datum; therefore, there is $h: Q \rightarrow I$ which satisfies 1. $0 = h(x) + h(y) - h(x + y)$; 2. $G_{r}(x) = r \cdot h(x) - h(r \cdot x)$ for all $r \in R$; 3. for all $f \in F$ with $n = \mathop{\mathrm{ar}}f$, $G_{f}(\vec{x}) = \sum_{i \in n} a(f,i)(\vec{x},h(\vec{x})) - h(f^{Q}(\vec{x}))$. Take $(\alpha,\beta) \in c(I, Q, \ast)$. For the ring action we have $$\begin{aligned} G_{r}(x)^{(\alpha,\beta)} = \alpha \circ G_{r}(x) - G_{r}(\beta(x)) &= \alpha \circ \left( r \cdot h(x) - h(r \cdot x) \right) - \left( r \cdot h(\beta(x)) - h(r \cdot \beta(x)) \right) \\ &= r \cdot \left( \alpha \circ h - h \circ \beta \right)(x) - \left( \alpha \circ h - h \circ \beta \right) ( r \cdot x) \\ &= r \cdot s(x) - s(r \cdot x) \end{aligned}$$ where we set $s:= (\alpha \circ h - h \circ \beta) : Q \rightarrow I$. Note we explicitly used that $\alpha$ and $\beta$ are module homomorphisms and $h$ is a group homomorphism by (1). In the next calculation, we use the fact that the action terms are linear are unary in $I$. For multilinear $f \in F$ with $n = \mathop{\mathrm{ar}}f$, we have $$\begin{aligned} &G_{f}^{(\alpha,\beta)}(\vec{x}) \\ &= \alpha \circ G_{f}(\vec{x}) - \sum_{i \in [n]} G_{f}(x_{1},\ldots,\beta(x_{i}),\ldots,x_{n}) \\ &= \sum_{k \in [n]} \alpha \circ a(f,k) \left( \vec{x}, h(\vec{x}) \right) - \alpha \circ h(f^{Q}(\vec{x})) \\ &\quad - \sum_{i \in [n]} \Big( \sum_{k \in [n]} a(f,k)\left( (x_{1},\ldots,\beta(x_{i}),\ldots,x_{n}) , (h(x_{1}),\ldots, h \circ \beta(x_{i}),\ldots,h(x_{n})) \right) - h(f^{Q}(\vec{x}))_{i} [\beta(\vec{x})] \Big) \\ &= \sum_{k \in [n]} \sum_{i \in [n]} a(f,k) \big( (x_{1},\ldots,\beta(x_{i}),\ldots,x_{n}), (h(x_{1}),\ldots,\alpha \circ h(x_{i}),\ldots,h(x_{n})) \big) - \alpha \circ h(f^{Q}(\vec{x})) \\ &\quad - \sum_{i \in [n]} \sum_{k \in [n]} a(f,k)\big( (x_{1},\ldots,\beta(x_{i}),\ldots,x_{n}) , (h(x_{1}),\ldots, h \circ \beta(x_{i}),\ldots,h(x_{n})) \big) + \sum_{i \in [n]} h(f^{Q}(\vec{x}))_{i} [\beta(\vec{x})] \\ &= \sum_{k \in [n]} a(f,k) \big( (x_{1},\ldots,\beta(x_{k}),\ldots,x_{n}) , (h(x_{1}),\ldots, \alpha \circ h(x_{k}) - h \circ \beta(x_{k}),\ldots,h(x_{n})) \big) \\ &\quad - \alpha \circ h(f^{Q}(\vec{x})) + h \circ \beta \left( f^{Q}(\vec{x}) \right) \\ &= \sum_{k \in [n]} a(f,k)(\vec{x},s(\vec{x})) - s \left( f^{Q}(\vec{x}) \right).\end{aligned}$$ This shows $G^{\alpha,\beta}$ is a 2-coboundary witnessed by $s:Q \rightarrow I$. Next, we show for fixed $[T] \in H^{2}_{\mathcal V}(Q,I,\ast)^{\mathrm{gr}}$ that $W(-,[T]): c(I,Q,\ast) \rightarrow H^{2}_{_{R} \mathcal M_{F}}(Q,I,\ast)$ respects the bracket. Since $[T]$ is group-trivial, then we can observe that $T$ is linear with respect to equivalence. For simplicity, we may assume $T_{+} = 0$. Then we may take an extension $M$ and lifting $l:Q \rightarrow M$ which is a group homomorphism and for this lifting the 2-cocycle is realized by definitions (R1)-(R4); in particular, for multilinear operation $f \in F$ we have $T_{f}(\vec{x}) = f^{M}(l(\vec{x})) - l(f^{Q}(\vec{x}))$. Then $$\begin{aligned} T_{f}(x_{1},\ldots,x_{i-1},q + p,x_{i+1},\ldots,x_{n}) &= f^{M}(l(\vec{x}))_{i}[l(q + p)] - l \big( f^{Q}(\vec{x})_{i}[q+p] \big) \\ &= f^{M}(l(\vec{x}))_{i}[l(q) + l(p)] - l \big( f^{Q}(\vec{x})_{i}[q] + f^{Q}(\vec{x})_{i}[p] \big) \\ &= f^{M}(l(\vec{x}))_{i}[l(q)] + f^{M}(l(\vec{x}))_{i}[l(p)] - l \big( f^{Q}(\vec{x})_{i}[q] \big) - l \big( f^{Q}(\vec{x})_{i}[p] \big) \\ &= T_{f}(x_{1},\ldots,x_{i-1},q,x_{i+1},\ldots,x_{n}) + T_{f}(x_{1},\ldots,x_{i-1},p,x_{i+1},\ldots,x_{n}).\end{aligned}$$ Now take $(\alpha,\beta),(\sigma,\kappa) \in c(I,Q,\ast)$ and note $[(\alpha,\beta),(\sigma,\kappa)] = ([\alpha,\sigma],[\beta,\kappa])$. Then $$\begin{aligned} (T^{(\sigma,\kappa)})^{(\alpha,\beta)}(\vec{x}) &- (T^{(\alpha,\beta)})^{(\sigma,\kappa)}(\vec{x}) \\ &= \Big( \sigma \circ T(\vec{x}) - \sum_{i \in [n]} T(\vec{x})_{i}[\kappa(\vec{x})] \Big)^{(\alpha,\beta)} - \Big( \alpha \circ T(\vec{x}) - \sum_{i \in [n]} T(\vec{x})_{i}[\beta(\vec{x})] \Big)^{(\sigma,\kappa)}\\ &= \alpha \circ \big( \sigma \circ T(\vec{x}) - \sum_{i \in [n]} T(\vec{x})_{i}[\kappa(\vec{x})] \big) - \sum_{j \in [n]} \big( \sigma \circ T(\vec{x}) - \sum_{i \in [n]} T(\vec{x})_{i}[\kappa(\vec{x})] \big)_{j}[\beta(\vec{x})] \\ &\quad - \sigma \circ \big( \alpha \circ T(\vec{x}) - \sum_{i \in [n]} T(\vec{x})_{i}[\beta(\vec{x})] \big) + \sum_{j \in [n]} \big( \alpha \circ T(\vec{x}) - \sum_{i \in [n]} T(\vec{x})_{i}[\beta(\vec{x})] \big)_{j}[\kappa(\vec{x})] \\ &= [\alpha,\sigma] \circ T(\vec{x}) + \sum_{j \in [n] } \sum_{i \in [n]} \big( T(\vec{x})_{i} [\kappa(\vec{x})] \big)_{j} [\beta(\vec{x})] - \sum_{j \in [n] } \sum_{i \in [n]} \big( T(\vec{x})_{i} [\beta(\vec{x})] \big)_{j} [\kappa(\vec{x})] \\ &= [\alpha,\sigma] \circ T(\vec{x}) + \sum_{i \in n} T(\vec{x})_{i} \left[ \kappa \circ \beta (\vec{x}) \right] - \sum_{i \in n} T(\vec{x})_{i} \left[ \beta \circ \kappa(\vec{x}) \right] \\ &= [\alpha,\sigma] \circ T(\vec{x}) - \sum_{i \in n} T(\vec{x})_{i} \left[ [\beta,\kappa](\vec{x}) \right] = T^{([\alpha,\sigma],[\beta,\kappa])}(\vec{x}).\end{aligned}$$ It follows that $W \big( [(\alpha,\beta),(\sigma,\kappa)],[T] \big) = \big[ W \big( \alpha,\beta),[T] \big), W \big( (\alpha,\beta),[T] \big) \big]$. Finally, we show $T^{(\alpha,\beta)}$ is $_{R} \mathcal M_{F}$-compatible. Since the datum $(Q,I,\ast)$ is affine, the equations in Example [Example 8](#ex:multiequations){reference-type="ref" reference="ex:multiequations"} to enforce $_{R} \mathcal M_{F}$-compatibility for a group trivial 2-cocycle $T$ of the datum simplifies to just stating that $T_{f}$ is a multilinear operation for each $f \in F$. It then follows that if we take $\mathcal V$-compatible group-trivial $[T] \in H^{2}_{\mathcal V}(Q,I,\ast)^{\mathrm{gr}}$ and since $\mathcal V \leq \, _{R} \mathcal M_{F}$, then $T_{f}$ is multilinear for each $f \in F$. Since both $\alpha$ and $\beta$ are module homomorphisms, the compositions $\alpha \circ T(\vec{a})$ and $T_{f}(\vec{x})_{i} [\beta(\vec{x})]$ are multilinear. Since the cohomology classes for affine datum forms an abelian group, $[T^{(\alpha,\beta)}] \in H^{2}_{_{R} \mathcal M_{F}}(Q,I,\ast)$. ◻ *Proof.* (of Theorem [Theorem 39](#thm:derivations){reference-type="ref" reference="thm:derivations"}) According to Theorem [Theorem 14](#thm:multirep){reference-type="ref" reference="thm:multirep"}, we may assume the extension is given by $M = I \rtimes_{T} Q$ and $\pi: I \rtimes_{T} Q \rightarrow Q$ is given by second-projection. Fix the lifting $l: Q \rightarrow I \rtimes_{T} Q$ defined by $l(x) = \left\langle 0, x \right\rangle$. We may identify $I$ with it's copy $I \times \{0\}$ in $I \rtimes_{T} Q$. Since the extension realizes a group-trivial 2-cocycle, we may assume $T_{+}=0$ and the lifting $l$ is a group homomorphism. First, we show $\ker \psi \approx \mathrm{Der}(Q,I,\ast)$. Assume $\psi(\phi) = (\phi|_{I},\phi_{l}) = (0,0)$. Note $0 = \phi_{l} = \pi \circ \phi \circ l(x)$ implies $\phi \circ l(x) \in I$ for all $x \in Q$. Then $\phi(a,0) = (0,0)$ implies $\phi(a,x) = \phi(a,0) + \phi(0,x) = \phi(0,x)$. We see that $F_{f}\left(l(\vec{x})\right) = \left\langle T_{f}(\vec{x}), f^{Q}(\vec{x}) \right\rangle = \left\langle T_{f}(\vec{x}), 0 \right\rangle + \left\langle 0, f^{Q}(\vec{x}) \right\rangle$. Then since $\phi$ is a derivation, we have $$\begin{aligned} \phi \circ l (f^{Q}(\vec{x})) = \phi(0,f^{Q}(\vec{x})) = \phi \left( F_{f}\left(l(\vec{x})\right) - \left\langle T_{f}(\vec{x}), 0 \right\rangle \right) &= \phi \left( F_{f}\left(l(\vec{x})\right) \right) \\ &= \sum_{i \in [n]} F_{f} \left( l(\vec{x}) \right)_{i} [\phi(l(\vec{x}))] \\ &= \left\langle \, \sum_{i \in [n]} a(f,i)\left( \vec{x}, \phi \circ l(\vec{x}) \right) , 0 \, \right\rangle\end{aligned}$$ by realization; thus, $\phi \circ l \in \mathrm{Der} (Q,I,\ast)$. Define $\Delta: \ker \psi \rightarrow \mathrm{Der} (Q,I,\ast)$ by $\Delta(\phi):= \phi \circ l$. We show $\Delta$ is a Lie algebra isomorphism. It is clearly seen to respect the standard bracket operation. For injectivity, assume $\phi \circ l = \Delta(\phi)=0$. Since $\phi \in \ker \psi$, we see that $\phi(a,x) = \phi(a,0) + \phi(0,x) = \phi \circ l(x) = 0$; thus, $\phi = 0$. For surjectivity, take $\sigma \in \mathrm{Der} (Q,I,\ast)$. Define $\gamma(a,x) := \left\langle \sigma(x), 0 \right\rangle$. Then $\gamma(a,0)= \left\langle 0,0 \right\rangle$ and $\pi \circ \gamma \circ l(x) = \pi(\sigma(x),0) = 0$ which implies $\gamma \in \ker \psi$. We also easily see that $\Delta(\gamma) = \sigma$ by the identification $I$ with $I \times \{0\}$. The last step is to show $\gamma$ is a derivation. It is easy to see that $\gamma$ is a module homomorphism. For $f \in F$ with $n = \mathop{\mathrm{ar}}f$, we calculate using that $\sigma \in \mathrm{Der} (Q,I,\ast)$, $$\begin{aligned} \gamma \circ F_{f} \left( \left\langle a_1,x_1 \right\rangle,\ldots, \left\langle a_n,x_n \right\rangle \right) = \gamma \left( \, \sum_{i \in [n]} a(f,i)(\vec{x},\vec{a}) + T_{f}(\vec{x}), f^{Q}(\vec{x}) \, \right) &= \left\langle \, \sigma \circ f^{Q}(\vec{x}), 0 \, \right\rangle \\ &= \left\langle \, \sum_{i \in [n]} a(f,i)(\vec{x},\sigma(\vec{x})), 0 \, \right\rangle \\ &= \sum_{i \in [n]} F_{f} \left( \vec{x} \right)_{i} [\sigma(\vec{x})]\end{aligned}$$ by realization. We have finished showing $\ker \psi \approx \mathrm{Der}(Q,I,\ast)$. Next we show $\mathop{\mathrm{im}}\psi \subseteq \ker W_{T}$. Take $\phi \in \mathrm{Der}_{I} M$. From Eq.([\[eqn:47\]](#eqn:47){reference-type="ref" reference="eqn:47"}), we have $\phi \circ l(x) - l \circ \pi \circ \phi \circ l(x) \in I$ and so the map $h:Q \rightarrow I$ defined by $h(x):= \phi \circ l(x) - l \circ \phi_{l}(x)$ is the failure of the lifting $l$ to commute with the two derivations. Fix $f \in F$ with $n = \mathop{\mathrm{ar}}f$. Using the fact that both $\phi$ and $\phi_{l}$ are derivations and $l$ is group homomorphism we have $$\begin{aligned} \phi \circ F_{f}\left( l(\vec{x}) \right) = \sum_{i \in [n]} F_{f} \left( l(\vec{x}) \right)_{i} [\phi(l(\vec{x}))] &= \sum_{i \in [n]} F_{f} \left( l(\vec{x}) \right)_{i} [ (h + l \circ \phi_{l})(\vec{x}) ] \\ &= \sum_{i \in [n]} F_{f} \left( l(\vec{x}) \right)_{i} [ h(\vec{x}) ] + \sum_{i \in [n]} F_{f} \left( l(\vec{x}) \right)_{i} [ l \circ \phi_{l} (\vec{x}) ] \\ &= \sum_{i \in [n]} a(f,i)\left( \vec{x}, h(\vec{x}) \right) + \sum_{i \in [n]} F_{f} \left( l(\vec{x}) \right)_{i} [ l \circ \phi_{l} (\vec{x}) ].\end{aligned}$$ We also can calculate $$\begin{aligned} \phi \circ F_{f}\left( l(\vec{x}) \right) &= \phi \left( l \left( f^{Q}(\vec{x}) \right) + T_{f}(\vec{x}) \right) \\ &= \phi \circ l \left( f^{Q}(\vec{x}) \right) + \phi \circ T_{f}(\vec{x}) \\ &= h(f^{Q}(\vec{x})) + l \circ \phi_{l}\left( f^{Q}(\vec{x}) \right) + \phi \circ T_{f}(\vec{x}) \\ &= h(f^{Q}(\vec{x})) + \sum_{i \in [n]} l \circ f^{Q}(\vec{x})_{i} [\phi_{l}(\vec{x})] + \phi \circ T_{f}(\vec{x}). \end{aligned}$$ Rewriting the above two equations yields $$\begin{aligned} \sum_{i \in [n]} a(f,i)\left( \vec{x}, h(\vec{x}) \right) - h(f^{Q}(\vec{x})) &= \sum_{i \in [n]} l \circ f^{Q}(\vec{x})_{i} [\phi_{l}(\vec{x})] + \phi \circ T_{f}(\vec{x}) - \sum_{i \in [n]} F_{f} \left( l(\vec{x}) \right)_{i} [ l \circ \phi_{l} (\vec{x}) ] \\ &= \phi \circ T_{f}(\vec{x}) - \sum_{i \in [n]} \Big( F_{f} \left( l(\vec{x}) \right)_{i} [ l \circ \phi_{l} (\vec{x}) ] - l \circ f^{Q}(\vec{x})_{i} [\phi_{l}(\vec{x})] \Big) \\ &= \phi \circ T_{f}(\vec{x}) - \sum_{i \in [n]} T_{f}(\vec{x})_{i} [l \circ \phi_{l}(\vec{x})] \\ &= T^{(\phi|_{I},\phi_{l})}_{f}.\end{aligned}$$ This shows $W(\psi(\phi),[T]) = W((\phi|_{I},\phi_{l}), [T]) = 0$ witnessed by $h$. We now complete the proof of exactness. Assume $(\alpha,\beta) \in c(I,Q,\ast)$ such that $[T^{(\alpha,\beta)}]=0$. There exists $h: Q \rightarrow I$ such that - $0 = h(x) + h(y) - h(x+y)$; - for $r \in R$, $\alpha \circ T_{r}(x) - T_{r}(\beta(x)) = r \cdot h(x) - h(r \cdot x)$; - for $f \in F$ with $n = \mathop{\mathrm{ar}}f$, $$\alpha \circ T_{f}(\vec{x}) - \sum_{i \in [n]} T_{f}(\vec{x})_{i} [\beta(\vec{x})] = \sum_{i \in [n]} a(f,i)(\vec{x},h(\vec{x})) - h(f^{Q}(\vec{x})).$$ Define $\phi: I \times Q \rightarrow I \times Q$ by $\phi (a,x):= \left\langle \alpha(a) + h(x), \beta(x) \right\rangle$. It is easy to see that $\phi (I) \subseteq I$ and $\phi|_{I} = \alpha$, and $\pi \circ \phi \circ l(x) = \pi \circ \phi(0,x)= \pi(h(x),\beta(x)) = \beta(x)$ and so $\phi_{l} = \beta$; thus, $\psi(\phi) = (\alpha,\beta)$. We show $\phi \in \mathrm{Der}_{I} M$. To see that $\phi$ is a module homomorphism, we calculate $$\begin{aligned} \phi \big( r \cdot \left\langle a, x \right\rangle + \left\langle b, y \right\rangle \big) &= \phi \big( \left\langle \, r \cdot a + b + T_{r}(x), r \cdot x + y \, \right\rangle \big) \\ &= \left\langle \, \alpha(r \cdot a) + \alpha(b) + \alpha \circ T_{r}(x) + h(r \cdot x + y), \beta(r \cdot x + y) \, \right\rangle \\ &= \left\langle \, r \cdot \alpha(a) + \alpha(b) + \alpha \circ T_{r}(x) + h(r \cdot x) + h(y), r \cdot \beta(x) + \beta(y) \, \right\rangle \\ &= \left\langle \, r \cdot \alpha(a) + \alpha(b) + T_{r}(\beta(x)) + r \cdot h(x) + h(y), r \cdot \beta(x) + \beta(y) \, \right\rangle \\ &= \left\langle \, r \cdot \alpha(a) + T_{r}(\beta(x)) + r \cdot h(x) , r \cdot \beta(x) \, \right\rangle + \left\langle \, \alpha(b) + h(y) , \beta(y) \, \right\rangle \\ &= r \cdot \left\langle \alpha(a) + h(x) , \beta(x) \right\rangle + \phi(b,y) \\ &= r \cdot \phi\left(a,x \right) + \phi(b,y)\end{aligned}$$ Now for for $f \in F$ with $n = \mathop{\mathrm{ar}}f$, we have $$\begin{aligned} \sum_{i \in [n]} &F_{f} \Big( \left\langle a_{1}, x_{1} \right\rangle, \ldots, \phi \big( \left\langle a_{i}, x_{i} \right\rangle \big) , \ldots, \left\langle a_{n}, x_{n} \right\rangle \Big) \\ &= \sum_{i \in [n]} F_{f} \Big( \left\langle a_{1}, x_{1} \right\rangle, \ldots, \left\langle \alpha(a_{i}) + h(x_{i}), \beta(x_{i}) \right\rangle , \ldots, \left\langle a_{n}, x_{n} \right\rangle \Big) \\ &= \sum_{i \in [n]} \left\langle \ \sum_{k \in [n]} a(f,k) \big( \vec{x}_{i} [\beta(\vec{x})], (a_{1},\ldots,\alpha(a_{i}) + h(x_{i}),\ldots,a_{n}) \big) + T_{f}(\vec{x})_{i} [\beta(\vec{x})] , f^{Q}(\vec{x})_{i} [\beta(\vec{x})] \ \right\rangle \\ &= \Bigg< \ \sum_{i \in [n]} \sum_{k \in [n]} a(f,k) \big( \vec{x}_{i} [\beta(\vec{x})], \vec{a}_{i} [\alpha(\vec{a})] \big) + \sum_{i \in [n]} a(f,i) \big( \vec{x}_{i} [\beta(\vec{x})], \vec{a}_{i} [ h(\vec{x}) ] \big) \\ &\quad + \sum_{i \in [n]} T_{f}( \vec{x} )_{i} [\beta(\vec{x})] , \sum_{i \in [n]} f^{Q} (\vec{x})_{i} [\beta(\vec{x})] \ \Bigg> \\ &= \Bigg< \ \sum_{k \in [n]} \alpha \circ a(f,k)(\vec{x},\vec{a}) + \sum_{i \in [n]} a(f,i) \big( \vec{x}, h(\vec{x}) \big) + \sum_{i \in [n]} T_{f}( \vec{x} )_{i} [\beta(\vec{x})] , \beta \left( f^{Q}(\vec{x}) \right) \ \Bigg> \\ &= \Bigg< \ \sum_{k \in [n]} \alpha \circ a(f,k)(\vec{x},\vec{a}) + \alpha \circ T_{f}(\vec{x}) + h \left( f^{Q}(\vec{x}) \right) , \beta \left( f^{Q}(\vec{x}) \right) \ \Bigg> \\ \\ &= \phi \left( \, \sum_{k \in [n]} a(f,k)(\vec{x}) + T_{f}(\vec{x}) , f^{Q}(\vec{x}) \, \right) = \phi \circ F_{f} \left( \left\langle a_{1}, x_{1} \right\rangle, \ldots, \left\langle a_{n}, x_{n} \right\rangle \right) \end{aligned}$$ Altogether, we have shown $\phi$ is a derivation and so $(\alpha,\beta) \in\mathop{\mathrm{im}}\psi$. ◻ # Low-dimensional Hochschild-Serre sequence {#section:5} In this section, we establish a Hochschild-Serre exact sequence [@hochserre] for the first and second cohomology groups associated to affine datum and a general extension in a variety of modules expanded by multilinear operations. The inflation, restriction and transgression maps will be defined the same way as in the group case; however, there will be two main alterations in the development. For each of the cohomology groups, rather than just the cohomology associated to the quotient algebra, we must restrict to a certain null submodule of the given affine datum. The second alteration is the introduction of the $\square$-condition of an action which will restrict the domain of the transfer map on $1^{\mathrm{st}}$-cohomology. When restricted to the null submodule, the $\square$-condition is greatly simplified. **Theorem 45**. Let $\mathcal V$ be a variety of modules expanded by multilinear operations and $M \in \mathcal V$ which is an extension $0 \rightarrow I \rightarrow M \rightarrow Q \rightarrow 0$. Let $(M,A,\ast)$ be affine datum in $\mathcal V$. There is an exact sequence $$0 \longrightarrow H^{1}(Q,A^{I},\hat{\ast}) \stackrel{\sigma}{\longrightarrow} H^{1}(M,A^{I},\ast) \stackrel{r}{\longrightarrow} H^{1}(I,A^{I},\ast)^{\square} \stackrel{\delta_{T}}{\longrightarrow} H^{2}_{\mathcal V}(Q,A^{I},\hat{\ast}) \stackrel{\sigma}{\longrightarrow} H^{2}_{\mathcal V}(M,A^{I},\ast).$$ We begin by developing the terms in the statement of the theorem. Fix an extension $0 \rightarrow I \rightarrow M \rightarrow Q \rightarrow 0$ and affine datum $(M,A,\ast)$ in $\mathcal V$. By Theorem [Theorem 29](#thm:multiabel){reference-type="ref" reference="thm:multiabel"}, the multilinear operations in $A$ are all trivial and the operations in the action $M \ast A$ are all unary in $A$ and given by the sequence $\{a(f,i): f \in F, i \in [\mathop{\mathrm{ar}}f] \}$. The extension $\pi: M \rightarrow Q$ induces an action $Q \star I$ with operations given by $\{ b(f,s): f \in F, s \in [\mathop{\mathrm{ar}}f]^{\ast} \}$. Let $T$ be the 2-cocycle determined by the extension so that $\pi: M \rightarrow Q$ is equivalent to $p_{2}: I \rtimes_{T} Q \rightarrow Q$. For simplicity, we make the identification $M=I \rtimes_{T} Q \stackrel{\pi}{\rightarrow} Q$. Let $\mathcal S^{\ast}$ be the set of operations which are formed by composing the action terms of $M \ast A$ with each other; that is, $\mathcal S^{\ast}$ is the smallest set of operations such that each $a(f,i) \in \mathcal S^{\ast}$ for $f \in F$, $i \in [\mathop{\mathrm{ar}}f]$ and whenever $t(x,\vec{y}) \in \mathcal S^{\ast}$, we have $a(f,i)(z_{1},\ldots,z_{i-1},t(x,\vec{y}),z_{i+1},\ldots,z_{\mathop{\mathrm{ar}}f}) \in \mathcal S^{\ast}$ for $f \in F$, $i \in [\mathop{\mathrm{ar}}f]$. For the ideal $I \triangleleft M$, the *null* submodule of $A$ determined by $I$ is $$\begin{aligned} \label{eqn:induceaction} A^{I} = \{ a \in A : t(a,\vec{m})=0 \text{ for all } t \in \mathcal S^{\ast} \text{ with some } m_{j} \in B \}.\end{aligned}$$ We see that $A^{I}$ is indeed a submodule since the action terms are unary in $A$. Let us first observe for the 2-cocycle $T$ that for the group part we have $T_{+}(0,x) = T_{+}(x,0) = 0$ which implies we can write $\left\langle b, x \right\rangle = \left\langle b, 0 \right\rangle + \left\langle 0, x \right\rangle$ in $M = I \rtimes_{T} Q$. Note by realization in $A \rtimes_{\ast} M$, for $f \in F$ with $\mathop{\mathrm{ar}}f = n$, if we take $a \in A^{I}$ and write $m_i = \left\langle b_{i}, x_{i} \right\rangle \in I \times Q$, then $$\begin{aligned} \left\langle \ a(f,i)(m_{1},\ldots,a_{i},\ldots,m_{n}), 0 \ \right\rangle &= F_{f} \big( \left\langle 0, m_{1} \right\rangle, \ldots, \left\langle a, 0 \right\rangle, \ldots, \left\langle 0, m_{n} \right\rangle \big) \\ &= F_{f} \big( \left\langle 0, \left\langle b_{1}, 0 \right\rangle + \left\langle 0, x_{1} \right\rangle \right\rangle, \ldots, \left\langle a, 0 \right\rangle, \ldots, \left\langle 0, m_{n} \right\rangle \big) \\ &= F_{f} \big( \left\langle 0, \left\langle b_{1}, 0 \right\rangle \right\rangle + \left\langle 0, \left\langle 0, x_{1} \right\rangle \right\rangle, \ldots,\left\langle a, 0 \right\rangle, \ldots, \left\langle 0, m_{n} \right\rangle \big) \\ &= F_{f} \big( \left\langle 0, \left\langle b_{1}, 0 \right\rangle \right\rangle, \ldots,\left\langle a, 0 \right\rangle, \ldots, \left\langle 0, m_{n} \right\rangle \big) \\ &\quad \quad + \ \ F_{f} \big( \left\langle 0, \left\langle 0, x_{1} \right\rangle \right\rangle, \ldots,\left\langle a, 0 \right\rangle, \ldots, \left\langle 0, m_{n} \right\rangle \big) \\ &= F_{f} \big( \left\langle 0, \left\langle 0, x_{1} \right\rangle \right\rangle, \ldots, \left\langle a, 0 \right\rangle, \ldots, \left\langle 0, m_{n} \right\rangle \big) \\ &\vdots \\ &= F_{f} \big( \left\langle 0, \left\langle 0, x_{1} \right\rangle \right\rangle, \ldots,\left\langle a, 0 \right\rangle, \ldots, \left\langle 0, \left\langle 0, x_{1} \right\rangle \right\rangle\big)\end{aligned}$$ which establishes that $$\begin{aligned} \label{eqn:welldef} a(f,i) \left( m_{1},\ldots,a,\ldots,m_{n} \right) = a(f,i)\left( \left\langle 0,x_{1} \right\rangle,\ldots,a,\ldots,\left\langle 0,x_{n} \right\rangle \right).\end{aligned}$$ Inductively on the composition of action terms we see that for $t(x,\vec{y}) \in \mathcal S^{\ast}$, $a \in A^{B}$ and $m_i = \left\langle b_{i}, x_{i} \right\rangle \in I \times Q$ that $$\begin{aligned} \label{eqn:welldef2} t(a,\vec{m}) &= t(a,\left\langle 0, \vec{x}\right\rangle) &\quad \quad \quad \quad \quad \quad &(t(x,\vec{y}) \in \mathcal S^{\ast}).\end{aligned}$$ For the ideal $I \triangleleft M$, we define an action $Q \hat{\ast} A^{I}$ such that $(Q,A^{I},\hat{\ast})$ is affine datum in $\mathcal V$. For each $f \in F$ and $i \in [ \mathop{\mathrm{ar}}f]$, define $$\begin{aligned} \label{eqn:actioninflated} \hat{a}(f,i)(x_{1},\ldots,x_{i-1},a,x_{i+1},\ldots,x_{\mathop{\mathrm{ar}}f}) := a(f,i) \left( \left\langle 0,x_{1} \right\rangle, \ldots, \left\langle 0,x_{i-1} \right\rangle, a, \left\langle 0, x_{i+1} \right\rangle, \ldots, \left\langle 0,x_{\mathop{\mathrm{ar}}f} \right\rangle \right)\end{aligned}$$ for any $a \in A^{I}, x_{1},\ldots,x_{i-1},x_{i+1},\ldots,x_{\mathop{\mathrm{ar}}f} \in Q$. By Eq [\[eqn:welldef2\]](#eqn:welldef2){reference-type="eqref" reference="eqn:welldef2"} and because the definition in Eq [\[eqn:induceaction\]](#eqn:induceaction){reference-type="eqref" reference="eqn:induceaction"} is taken over all terms in $\mathcal S^{\ast}$, we see that the action is well-defined and closed on $A^{I}$. We can succinctly write the definition relating the two actions as $$\begin{aligned} \label{eqn:definfl} \hat{a}(f,i) \left( \pi(\vec{m}), \vec{a} \right) = a(f,i) \left( \vec{m}, \vec{a} \right)\end{aligned}$$ for $f \in F$, $\vec{m} \in M^{\mathop{\mathrm{ar}}f}$. If we note that for terms $s$ in the signature of the variety $\mathcal V$, the action terms $s^{\ast,T^{\ast}}$ from Lemma [Lemma 5](#lem:semident){reference-type="ref" reference="lem:semident"} are sums of iterated action terms from $\mathcal S^{\ast}$, then it is easy to that $Q \hat{\ast} A^{I}$ is also a $\mathcal V$-compatible action. The *inflation* maps $$\begin{aligned} \sigma &: H^{1}( Q,A^{I},\hat{\ast} ) \rightarrow H^{1}( M,A,\ast ) \quad \quad \quad \quad \sigma : H^{2}_{\mathcal V}( Q,A^{I},\hat{\ast} ) \rightarrow H^{2}_{\mathcal V}( M,A,\ast )\end{aligned}$$ are defined by precomposition for derivations $\sigma([d]) := [d \circ \pi]$ and for 2-cocycles $\sigma([T]) := [T \circ \pi]$ where $(T \circ \pi)_{+}(x,y) := T_{+}(\pi(x),\pi(y))$, $(T \circ \pi)_{+}(x) := T_{r}(\pi(x))$ and $(T \circ \pi)_{f}(x_{1},\ldots,x_{\mathop{\mathrm{ar}}f}) := T_{f}(\pi(x_{1}),\ldots,\pi(x_{\mathop{\mathrm{ar}}f}))$ for $r \in R$, $f \in F$ with action terms given by $M \ast A$. Let us show inflation is well-defined on cohomology; that is, precomposition maps coboundaries to coboundaries. First, suppose $G$ is a 2-coboundary for $(Q,A^{I},\hat{\ast})$ witnessed by $h: Q \rightarrow A^{I}$. Using (B1)-(B3) for trivial multilinear operations and unary actions, we see that for $z,w \in M, \vec{z} \in M^{\mathop{\mathrm{ar}}f}$ and $x = \pi(w), y = \pi(z), \vec{x} = \pi(\vec{z})$ we have $$\begin{aligned} G_{+}(\pi(w),\pi(z)) = G_{+}(x,y) = h(x) + h(y) - h(x+y) &= h(\pi (w)) + h (\pi (z)) - h(\pi (x)+ \pi(y)) \\ &= h \circ \pi (w) + h \circ \pi (z) - h \circ \pi (x+y),\end{aligned}$$ $$\begin{aligned} G_{r}(\pi(w)) = G_{r}(x) = r \cdot h(x) - h(r \cdot x) = r \cdot h(\pi(w)) - h(r \cdot \pi(w)) &= r \cdot h \circ \pi(w) - h \circ \pi (r \cdot w)\end{aligned}$$ and by the definition in Eq [\[eqn:definfl\]](#eqn:definfl){reference-type="eqref" reference="eqn:definfl"} $$\begin{aligned} G_{f}(\pi(\vec{z})) = G_{f}(\vec{x}) = \sum_{i \in [\mathop{\mathrm{ar}}f]^{\ast}} \hat{a}(f,i)(\vec{x},h(\vec{x})) - h(f^{Q}(\vec{x})) &= \sum_{i \in [\mathop{\mathrm{ar}}f]^{\ast}} \hat{a}(f,i)(\pi(\vec{z}),h(\pi(\vec{z}))) - h(f^{Q}(\pi(\vec{z}))) \\ &= \sum_{i \in [\mathop{\mathrm{ar}}f]^{\ast}} a(f,i)(\vec{z},h \circ \pi(\vec{z})) - h \circ \pi (f^{M}(\vec{z})).\end{aligned}$$ This shows $G \circ \pi$ is a 2-coboundary $(Q,A,\ast)$ witnessed by $h \circ \pi$. Observe that since $Q \hat{\ast} A^{I}$ is $\mathcal V$-compatible, we see that $[T] \in H^{2}_{\mathcal V}(Q,A^{I},\hat{\ast})$ if and only if $T$ is strictly $\mathcal V$-compatible. Then for the same reason as in the calculation verifying the (B3) property for the 2-coboundary above, $T \circ \pi$ is $\mathcal V$-compatible using the action $M \ast A$; that is, $[T \circ \pi] \in H^{2}_{\mathcal V}(M,A,\ast)$. We now consider $1^{\mathrm{st}}$-cohomology. If we take a derivation $d: Q \rightarrow A^{I}$ of the action $\hat{\ast}$, then we can verify for $m_{i} = \left\langle b_i, x_{i} \right\rangle$ and $f \in F$ with $n = \mathop{\mathrm{ar}}f$ that $$\begin{aligned} d \circ \pi \left( f^{M}(\vec{m}) \right) = d \left( f^{Q}(\vec{x}) \right) = \sum_{i \in [n]} \hat{a}(f,i) \left( \vec{x}, d(\vec{x}) \right) = \sum_{i \in [n]} a(f,i) \left( \vec{m}, d \circ \pi (\vec{m}) \right),\end{aligned}$$ and so $d \circ \pi$ is a derivation of $(M,A,\ast)$. Now let $\gamma$ be a principal stabilizing automorphism of the datum $(Q,A^{I},\hat{\ast})$; that is, there is a term $t(x,\vec{y})$ and elements $\vec{c},\vec{d} \in (A^{I})^{n}$ such that in $A^{I} \rtimes_{\hat{\ast}} Q$ $$\begin{aligned} \left\langle a - d_{\gamma}(x), x \right\rangle = \gamma(a,x) = F_{t}\left( \left\langle a, x \right\rangle, \left\langle c_{1}, 0 \right\rangle, \ldots, \left\langle c_{n}, 0 \right\rangle \right) &= \left\langle a + t^{\hat{\ast},T^{\hat{\ast}}}((0,\vec{c}),(x,\vec{0})) , x \right\rangle \\ \left\langle a, x \right\rangle = F_{t}\left( \left\langle a, x \right\rangle, \left\langle d_{1}, 0 \right\rangle, \ldots, \left\langle d_{n}, 0 \right\rangle \right) &= \left\langle a + t^{\hat{\ast},T^{\hat{\ast}}}((0,\vec{d}),(x,\vec{0})) , x \right\rangle\\\end{aligned}$$ where $a=t(a,\vec{c})=t(a,\vec{d})$ and $d_{\gamma}$ is a derivation with $d_{\gamma}(x) = t^{\hat{\ast},T^{\hat{\ast}}}((0,\vec{c}),(x,\vec{0}))$ (see Remark [Remark 38](#rem:principal){reference-type="ref" reference="rem:principal"}). Then in the semidirect product $A \rtimes_{\ast} M$ we have $$\begin{aligned} F_{t} &\left( \left\langle a, \left\langle b,x \right\rangle \right\rangle, \left\langle c_{1}, \left\langle 0,0\right\rangle \right\rangle, \ldots, \left\langle c_{n}, \left\langle 0,0\right\rangle \right\rangle \right) - F_{t}\left( \left\langle a, \left\langle b,x \right\rangle \right\rangle, \left\langle d_{1}, \left\langle 0,0\right\rangle \right\rangle, \ldots, \left\langle d_{n}, \left\langle 0,0\right\rangle \right\rangle \right) + \left\langle a, \left\langle b,x \right\rangle \right\rangle \\ &= \left\langle a + t^{\ast,T^{\ast}} \left( (0,\vec{c}),(b,x),(0,0),\ldots,(0,0) \right) , t^{M}((b,x),(0,0),\ldots,(0,0)) \right\rangle \\ &- \left\langle a + t^{\ast,T^{\ast}} \left( (0,\vec{d}),(b,x),(0,0),\ldots,(0,0) \right) , t^{M}((b,x),(0,0),\ldots,(0,0)) \right\rangle + \left\langle a, \left\langle b,x \right\rangle \right\rangle \\ &= \left\langle a + t^{\hat{\ast},T^{\hat{\ast}}} \left( (0,\vec{c}),(x,\vec{0}) \right) , x \right\rangle - \left\langle a + t^{\hat{\ast},T^{\hat{\ast}}} \left( (0,\vec{d}),(x,\vec{0}) \right) , x \right\rangle + \left\langle a, \left\langle b,x \right\rangle \right\rangle \\ &= \left\langle a - d(x), x \right\rangle - \left\langle a, x \right\rangle + \left\langle a, \left\langle b,x \right\rangle \right\rangle \\ &= \left\langle a - d \circ \pi (b,x), \left\langle b,x \right\rangle \right\rangle .\end{aligned}$$ Now that we have shown inflation is well-defined on cohomology classes, it is easy to see that it is a homomorphism. There is a *restriction* map $$\begin{aligned} r: H^{1}( M,A,\ast ) \rightarrow H^{1}( I,A,\ast )\end{aligned}$$ defined by $r([d]):= [d|_{I}]$; that is, we take the cohomology class of the derivation $d$ restricted to $I$. Note that in the polynomial which defines a principal stabilizing automorphism in $A \rtimes_{T^{\ast}} M$, the constants come from the isomorphic copy of $A$; thus, the restriction of the automorphism to $A \rtimes_{T^{\ast}} I \leq A \rtimes_{T^{\ast}} M$ is still a principal stabilizing automorphism according to definition. It is easy to see that the restriction map is a homomorphism. We are interested in a property of the derivations which are in the image of the restriction map. Fix a derivation $d: M \rightarrow A$, $f \in F$ with $n=\mathop{\mathrm{ar}}f$ and for simplicity, let us consider an action term for the coordinates $s = \{i, i+1,\ldots,n\}$. Then $$\begin{aligned} d|_{I} \big( b(f,s)(x_{1},\ldots,x_{i-1},b_{i},\ldots,b_{n}) \big) &= d \left( f^{M}( \left\langle 0, x_{1} \right\rangle,\ldots,\left\langle 0, x_{i-1} \right\rangle, \left\langle b_{i}, 0 \right\rangle,\ldots, \left\langle b_{n}, 0 \right\rangle ) \right) \\ &= \sum_{k=1}^{i-1} a(f,k)( \left\langle 0, x_{1} \right\rangle,\ldots, d(0,x_{k}) ,\ldots, \left\langle 0, x_{i-1} \right\rangle, \left\langle b_{i}, 0 \right\rangle,\ldots, \left\langle b_{n}, 0 \right\rangle ) \\ &+ \quad \sum_{k=i}^{n} a(f,k)( \left\langle 0, x_{1} \right\rangle, \ldots, \left\langle 0, x_{i-1} \right\rangle, \left\langle b_{i}, 0 \right\rangle,\ldots,d(b_{k},0), \ldots, \left\langle b_{n}, 0 \right\rangle ) \\ &= \sum_{k=1}^{i-1} a(f,k)( \left\langle 0, x_{1} \right\rangle,\ldots, h(x_{k}) ,\ldots, \left\langle 0, x_{i-1} \right\rangle, \left\langle b_{i}, 0 \right\rangle,\ldots, \left\langle b_{n}, 0 \right\rangle ) \\ &+ \quad \sum_{k=i}^{n} a(f,k)( \left\langle 0, x_{1} \right\rangle, \ldots, \left\langle 0, x_{i-1} \right\rangle, \left\langle b_{i}, 0 \right\rangle,\ldots,d|_{I}(b_{k}), \ldots, \left\langle b_{n}, 0 \right\rangle ) \end{aligned}$$ where we have written $h(x):=d(0,x) : Q \rightarrow A$. This determines a condition on derivations which relates the given affine action $M \ast A$ and the action $Q \star I$ induced from the extension $\pi: M \rightarrow Q$. A derivation $d \in \mathrm{Der}(I,A,\ast)$ satisfies the *$\square$-condition* if there exists a map $h: Q \rightarrow A$ such that $$\begin{aligned} \label{eqn:square} d \left( b(f,s)(\vec{x}, \vec{b} ) \right) = \sum_{i \not\in s} a(f,i)\left( \left\langle \vec{0}, \vec{x} \right\rangle_{s} \left\langle \vec{b}, \vec{0} \right\rangle , h(\vec{x}) \right) \ + \ \sum_{i \in s} a(f,i)\left( \left\langle \vec{0}, \vec{x} \right\rangle_{s} \left\langle \vec{b}, \vec{0} \right\rangle , d(\vec{b}) \right)\end{aligned}$$ for all $f \in F$, $s \in [\mathop{\mathrm{ar}}f]^{\ast}$. Note we are using the convention to write the tuple $\left\langle \vec{b}, \vec{x} \right\rangle = \left( \left\langle b_{1}, x_{1} \right\rangle,\ldots, \left\langle b_{n}, x_{n} \right\rangle \right)$ and $\left\langle \vec{0}, \vec{x} \right\rangle_{s} \left\langle \vec{b}, \vec{0} \right\rangle$ is the tuple formed by substituting the s-coordinates of $\left\langle \vec{b}, \vec{0} \right\rangle$ into the s-coordinates of $\left\langle \vec{0}, \vec{x} \right\rangle$. Since derivations are linear module transformation and the right-hand side of the definition in Eq [\[eqn:square\]](#eqn:square){reference-type="eqref" reference="eqn:square"} is in terms of the unary actions, the set of derivations which satisfy the $\square$-condition forms a subgroup. We then define $$H^{1}(I,A,\ast)^{\square} = \{ [d] \in H^{1}(I,A,\ast): d \text{ satisfies the } \square-\text{condition} \}$$ which forms a subgroup of $1^{\mathrm{st}}$-cohomology. The above calculation can be extended to show the restriction map $r: H^{1}( M,A,\ast ) \rightarrow H^{1}( I,A,\ast )^{\square}$ is well-defined. We now consider our analogue of the trangression map which takes $1^{\mathrm{st}}$-cohomology to $2^{\mathrm{nd}}$-cohomology. In order for the codomain of the transgression to be contained in the set of $\mathcal V$-compatible 2-cocycles, we must enforce the $\square$-condition on derivations of the datum $(I,A^{I},\ast)$; that is, on the "coefficients" $A^{I}$ and not just $A$. Let us consider why this is so. Take $[d] \in H^{1}(I,A^{I},\ast)^{\square}$ where $d$ satisfies the $\square$-condition; thus, there is a map $h: Q \rightarrow A^{I}$ which satisfies Eq [\[eqn:square\]](#eqn:square){reference-type="eqref" reference="eqn:square"}. Consider $|s| > 1$. Then for any $i \not\in s$, $a(f,i)\left( \left\langle \vec{0}, \vec{x} \right\rangle_{s} \left\langle \vec{b}, \vec{0} \right\rangle , h(\vec{x}) \right) = 0$ since both $\left\langle b_{j}, 0 \right\rangle$ for $j \in s$ and $h(x_{i})$ appear in the coordinates. For any $i \in s$, there is $i \neq j \in$ such that both $\left\langle b_{j}, 0 \right\rangle$ and $d(b_{i}) \in A^{I}$ appear in the coordinate; thus, we again see that $a(f,i)\left( \left\langle \vec{0}, \vec{x} \right\rangle_{s} \left\langle \vec{b}, \vec{0} \right\rangle , d(\vec{b}) \right) = 0$. Altogether, we conclude that $$\begin{aligned} \label{eqn:square1} d \big( b(f,s)(\vec{x},\vec{b}) \big) = 0 \quad \quad \text{for} \quad \quad |s| > 1.\end{aligned}$$ Now fix $s=\{k\}$. Then for the same reason we have $a(f,i)\left( \left\langle \vec{0}, \vec{x} \right\rangle_{s} \left\langle \vec{b}, \vec{0} \right\rangle , h(\vec{x}) \right) = 0$ for $i \neq k$. Then in this case we have $$\begin{aligned} \label{eqn:square2} d \big( b(f,i)(\vec{x},\vec{b}) \big) = a(f,i)( \left\langle 0,x_{1} \right\rangle, \ldots, d(b_{i}), \ldots, \left\langle 0,x_{n} \right\rangle ) = \hat{a}(f,i)(\vec{x},d(\vec{b})).\end{aligned}$$ So for reference later, we conclude that a derivation $d$ of the datum $(I,A^{I},\ast)$ which satisfies the $\square$-condition implies $$\begin{aligned} \label{eqn:squarenull} d \big( b(f,i)(\vec{x},\vec{b}) \big) = \hat{a}(f,i)(\vec{x},d(\vec{b})) \quad \quad \quad \text{ and } \quad \quad \quad d \big( b(f,s)(\vec{x},\vec{b}) \big) = 0 \quad \quad \text{for} \quad \quad |s| > 1.\end{aligned}$$ We can very roughly summarize Eq [\[eqn:square1\]](#eqn:square1){reference-type="eqref" reference="eqn:square1"} and Eq [\[eqn:square2\]](#eqn:square2){reference-type="eqref" reference="eqn:square2"} by stating that the $\square$-condition on the datum $(I,A^{I},\ast)$ implies the relation $$\begin{aligned} \label{eqn:dersquare} d(Q \mathrel{\star} I) = Q \mathrel{\hat{\ast}} d(I).\end{aligned}$$ We can also observe for the semidirect product $A^{I} \rtimes_{\ast} I = A^{I} \times I$ since the action of $I$ on $A^{I}$ is null. Then by Remark [Remark 38](#rem:principal){reference-type="ref" reference="rem:principal"} we see that the principal derivations of the datum $(I,A^{I},\ast)$ must be trivial; therefore, $H^{1}(I,A^{I},\ast) = \mathrm{Der}(I,A^{I},\ast)$. The *transgression* map $$\begin{aligned} \partial : H^{1}(I,A^{I},\ast)^{\square} \times H^{2}_{\mathcal V}(Q,I) \rightarrow H^{2}_{\mathcal V}(Q,A^{I},\hat{\ast})\end{aligned}$$ is defined by $\delta ([d],[T]):= [d \circ T]$ where $$(d \circ T)_{+} = d \circ T_{+}, \quad \quad (d \circ T)_{r} = d \circ T_{r}, \quad \quad (d \circ T)_{f} = d \circ T_{f}$$ and the action terms are exactly the action terms of $Q \hat{\ast} A^{I}$. In order to show that it is well-defined on cohomology classes, let us first take a derivation $d:I \rightarrow A^{I}$. Then observe that for any $f \in F$ with $n=\mathop{\mathrm{ar}}f$, $$\begin{aligned} \label{eqn:nullder} d ( f^{I}(\vec{b}) ) = \sum_{i \in [n]} a(f,i) \left( \left\langle \vec{0}, \vec{b} \right\rangle, d(\vec{b}) \right) = 0\end{aligned}$$ since $d(b_i) \in A^{I}$. This fact, together with Eq [\[eqn:square1\]](#eqn:square1){reference-type="eqref" reference="eqn:square1"} and Eq [\[eqn:square2\]](#eqn:square2){reference-type="eqref" reference="eqn:square2"}, implies that if we take $[d] \in H^{1}(I,A^{I},\ast)^{\square}$ and $T \sim T'$ witnessed by $h: Q \rightarrow I$, then $d \circ T \sim d \circ T'$ witnessed by $d \circ h: Q \rightarrow A^{I}$; therefore, the transgression does not depend on our choice of representative 2-cocycle for the extension $\pi: M \rightarrow Q$. The final step is to verify that $[d \circ T] \in H^{2}_{V}(Q, A^{I},\hat{\ast})$ for any $d \in H^{1}(I,A^{I},\ast)^{\square}$. The relation Eq [\[eqn:dersquare\]](#eqn:dersquare){reference-type="eqref" reference="eqn:dersquare"} expresses the central reason why $d \circ T$ should be a $\mathcal V$-compatible 2-cocycle for datum $(Q, A^{I},\hat{\ast})$, but a more formal argument is based on the representation of terms evaluated in the algebra $A^{I} \rtimes_{d \circ T} Q$. Let $p_{1}: A^{I} \rtimes_{d \circ T} Q \rightarrow A^{I}$ denote the first-projection set map. Then using Eq [\[eqn:square1\]](#eqn:square1){reference-type="eqref" reference="eqn:square1"}, Eq [\[eqn:square2\]](#eqn:square2){reference-type="eqref" reference="eqn:square2"} and Eq [\[eqn:nullder\]](#eqn:nullder){reference-type="eqref" reference="eqn:nullder"}, one can show inductively on the generation of terms that for any term $t(\vec{x})$ in the language of $\mathcal V$, we have the evaluation $$\begin{aligned} \label{eqn:100} F_{t} \left( \left\langle d \circ p_{1}(\vec{m}), \pi(\vec{m}) \right\rangle \right) = \left\langle d \circ p_{1} \left( F_{t}^{M}(\vec{m}) \right) , t^{Q}(\pi(\vec{m})) \right\rangle \quad \quad \quad \quad \quad \left( \vec{m} \in (A^{I} \times Q )^{\mathop{\mathrm{ar}}t} \right)\end{aligned}$$ computed in the algebra $A^{I} \rtimes_{d \circ T} Q$. Since we have already seen that the action $Q \hat{\ast} A^{I}$ is $\mathcal V$-compatible, the 2-cocycle $d \circ T$ is $\mathcal V$-compatible if and only if it is strictly $\mathcal V$-compatible. Take $t=s \in \mathrm{Id} \, \mathcal V$. Since $M \in \mathcal V$, we have $F^{M}_{t}(\vec{m}) = F^{M}_{s}(\vec{m})$. Then using Remark [Remark 33](#remark:affineident){reference-type="ref" reference="remark:affineident"} for the evaluation of terms with affine datum, Eq [\[eqn:100\]](#eqn:100){reference-type="eqref" reference="eqn:100"} yields $$\begin{aligned} t^{\hat{\ast},d \circ T}( d \circ p_{1}(\vec{m}), \pi(\vec{m}) ) + t^{\partial,d \circ T}(\pi(\vec{m})) &= d \circ p_{1} \left( F_{t}^{M}(\vec{m}) \right) \\ &= d \circ p_{1} \left( F_{s}^{M}(\vec{m}) \right) \\ &= s^{\hat{\ast},d \circ T}( d \circ p_{1}(\vec{m}), \pi(\vec{m}) ) + s^{\partial,d \circ T}(\pi(\vec{m}))\end{aligned}$$ Since the action terms of $d \circ T$ are given by the $\mathcal V$-compatible action $Q \hat{\ast} A^{I}$, we conclude from the above that $t^{\partial,d \circ T}(\pi(\vec{m})) = s^{\partial,d \circ T}(\pi(\vec{m}))$; therefore, $d \circ T$ is strictly $\mathcal V$-compatible. *Proof.* (of Theorem [Theorem 45](#thm:HSexact){reference-type="ref" reference="thm:HSexact"}) We show exactness at each of the groups. $H^{1}(Q,A^{I},\hat{\ast})$ : Take $[d] \in H^{1}(Q,A^{I},\hat{\ast})$ such that $\sigma([d]) = [d \circ \pi] = 0$. Then there is a term $t(x,\vec{y})$ and $\vec{c},\vec{d} \in (A^{I})^{n}$ such that $$\begin{aligned} \left\langle a - d \circ \pi (b,x), \left\langle b,x \right\rangle \right\rangle &= F_{t} \left( \left\langle a, \left\langle b,x \right\rangle \right\rangle, \left\langle c_{1} , \left\langle 0, 0 \right\rangle \right\rangle, \ldots, \left\langle c_{n}, \left\langle 0, 0 \right\rangle \right\rangle \right) \\ &= \left\langle t^{A^{I}}(a,\vec{c}) + t^{\ast,T^{\ast}}((a,\vec{c}), \left\langle b,x \right\rangle) , t^{M}(\left\langle b, x \right\rangle,\left\langle 0, 0 \right\rangle, \ldots, \left\langle 0, 0 \right\rangle ) \right\rangle \\ \left\langle a, \left\langle b,x \right\rangle \right\rangle &= F_{t} \left( \left\langle a, \left\langle b,x \right\rangle \right\rangle, \left\langle d_{1} , \left\langle 0, 0 \right\rangle \right\rangle, \ldots, \left\langle d_{n}, \left\langle 0, 0 \right\rangle \right\rangle \right)\end{aligned}$$ in the semidirect product $A^{I} \rtimes_{\ast} M$. Since $a \in A^{I}$ we can rewrite in terms of the induced action $Q \hat{\ast} A^{I}$ $$\begin{aligned} a - d(x) = a - d \circ \pi (0,x) &= t^{A^{I}}(a,\vec{c}) + t^{\ast,T^{\ast}}((a,\vec{c}), \left\langle 0,x \right\rangle) = t^{A^{I}}(a,\vec{c}) + t^{\hat{\ast},T^{\hat{\ast}}}((a,\vec{c}), x).\end{aligned}$$ Then $$\begin{aligned} F_{t}\left( \left\langle a, x \right\rangle,\ldots, \left\langle c_{1}, 0 \right\rangle, \ldots, \left\langle c_{n}, 0 \right\rangle \right) &= \left\langle t^{A^{I}}(a,\vec{c}) +t^{\hat{\ast},T^{\hat{\ast}}}((a,\vec{c}), x), t^{Q}(x,0,\ldots,0) \right\rangle = \left\langle a - d(x), x \right\rangle. \end{aligned}$$ A similar calculation applies to the tuple $\vec{d} \in (A^{I})^{n}$ to show $d$ is a principal derivation of the datum $(Q,A^{I},\hat{\ast})$. $H^{1}(M,A^{I},\ast)$ : It is clear that $r \circ \sigma = 0$. Take $[d] \in H^{1}(M,A^{I},\ast)$ such that $r(d) = d|_{I} \equiv 0$. Note in $M=I \times_{T} Q$ we can write $d(b,x) = d \big( \left\langle b, 0 \right\rangle + \left\langle 0, x \right\rangle \big) = d(b,0) + d(0,q) = d(0,q)$. Define $\gamma: Q \rightarrow A^{I}$ by $\gamma(x):= d(0,q)$. Then $d = \gamma \circ \pi$ shows $\sigma([\gamma]) = [d]$. That $\gamma$ is a derivation of the datum $(Q,A^{I},\hat{\ast})$ follows from that fact that $d$ is independent of the first-coordinate. We have shown $\mathop{\mathrm{im}}\sigma = \ker r$. $H^{1}(I,A^{I},\ast)^{\square}$ : Take $[d] \in H^{1}(M,A^{I},\ast)$. We want to show $\delta_{T} \circ r ([d]) = [d|_{I} \circ T] = 0$ in $H^{2}_{\mathcal V}(Q,A^{I},\hat{\ast})$. Define $\psi: Q \rightarrow A^{I}$ by $\psi(x):= d(0,x)$. Then we see that for $f \in F$ with $n = ar f$, $$\begin{aligned} \sum_{i \in [n]} \hat{a}(f,i)(x_{1},\ldots,\psi(x_{i}),\ldots,x_{n}) &= \sum_{i \in [n]} a(f,i)(\left\langle 0 ,x_{1} \right\rangle,\ldots,d0,(x_{i}),\ldots,\left\langle 0 ,x_{n} \right\rangle) \\ &= d\left( f^{M}\left( \left\langle 0 ,x_{1} \right\rangle,\ldots, \left\langle 0 ,x_{n} \right\rangle \right) \right) \\ &= d \left( T_{f}(\vec{x}), f^{Q}(\vec{x}) \right) \\ &= d \left( T_{f}(\vec{x}), 0 \right) + d\left(0, f^{Q}(\vec{x}) \right) = d|_{I} \circ T_{f}(\vec{x}) + \psi \left( f^{Q}(\vec{x}) \right).\end{aligned}$$ Similarly, we have $$\begin{aligned} d|_{I} \circ T_{+}(x,y) + \psi(x+y) = d \left( T_{+}(x,y), x+y \right) = d\left( \left\langle 0, x \right\rangle + \left\langle 0, y \right\rangle \right) = d(0,x) + d(y) = \psi(x) + \psi(y)\end{aligned}$$ and $$\begin{aligned} d|_{I} \circ T_{r}(x) + \psi(r \cdot x) = d \left( T_{r}(x), r \cdot x \right) = d \left( r \cdot \left\langle 0, x \right\rangle \right) = r \cdot d(0,x) = r \cdot \psi(x).\end{aligned}$$ Altogether, we have shown $\psi$ witnesses that $d|_{I} \circ T$ is a 2-coboundary of the datum $(Q,A^{I},\hat{\ast})$. Now, take $d \in H^{1}(I,A^{I},\ast)^{\square}$ such that $[d \circ T]=0$. Then there is a map $h: Q \rightarrow A^{I}$ such that 1. $d \circ T_{+}(x,y) = h(x) + h(y) - h(x+y)$; 2. $d \circ T_{r}(x) = r \cdot h(x) - h(r \cdot x)$; 3. $d \circ T_{f}(\vec{x}) = \sum_{i \in [n]} \hat{a}(f,i)(x_{1},\ldots,h(x_{i}),\ldots,x_{n}) - h(f^{Q}(\vec{x}))$ $(f \in F, n = \mathop{\mathrm{ar}}f)$. Define $\gamma: M \rightarrow A^{I}$ by $\gamma(b,x) = d(b) + h(x)$. Since we see that $\gamma|_{I} = d$, we need to show $\gamma$ is a derivation. This can be done by using the $\square$-condition and (3) above for $f \in F$ with $n = \mathop{\mathrm{ar}}f$: calculating, we find $$\begin{aligned} \gamma \circ F_{f}^{M} \big( \left\langle b_{1}, x_{1} \right\rangle, \ldots, \left\langle b_{n}, x_{n} \right\rangle \big) &= \gamma \left( f^{I}(\vec{b}) + \sum_{s \in [n]^{\ast}} b(f,s)(\vec{x},\vec{b}) + T_{f}(\vec{x}), f^{Q}(\vec{x}) \right) \\ &= d \left(f^{I}(\vec{b}) \right) + d \left( \sum_{s \in [n]^{\ast}} b(f,s)(\vec{x},\vec{b}) \right) + d \circ T_{f}(\vec{x}) + h \left( f^{Q}(\vec{x}) \right) \\ &= \sum_{i \in [n]} \hat{a}(f,i)(x_{1},\ldots,d(b_{i}),\ldots,x_{n}) + d \circ T_{f}(\vec{x}) + h \left( f^{Q}(\vec{x}) \right) \\ &= \sum_{i \in [n]} a(f,i)( \left\langle b_{1}, x_{1} \right\rangle, \ldots,d(b_{i}),\ldots, \left\langle b_{n}, x_{n} \right\rangle ) \\ &\quad + \sum_{i \in [n]} \hat{a}(f,i)(x_{1},\ldots,h(x_{i}),\ldots,x_{n}) \\ &= \sum_{i \in [n]} a(f,i)( \left\langle b_{1}, x_{1} \right\rangle, \ldots,d(b_{i}),\ldots, \left\langle b_{n}, x_{n} \right\rangle ) \\ &\quad + \sum_{i \in [n]} a(f,i)(\left\langle b_{1}, x_{1} \right\rangle, \ldots, h(x_{i}), \ldots, \left\langle b_{1}, x_{1} \right\rangle ) \\ &= \sum_{i \in [n]} a(f,i)( \left\langle b_{1}, x_{1} \right\rangle, \ldots,\gamma(b_{i},x_{i}),\ldots, \left\langle b_{n}, x_{n} \right\rangle ).\end{aligned}$$ Similarly, by using (1) and (2) above we find that $$\begin{aligned} \gamma \left( \left\langle b, x \right\rangle + \left\langle c, y \right\rangle \right) = \gamma \left( \left\langle b + c + T_{+}(x,y), x + y \right\rangle \right) &= d(b) + d(c) + d \circ T_{+}(x,y) + h(x + y) \\ &= d(b) + d(c) + h(x) + h(y) \\ &= \gamma(b,x) + \gamma(c,y)\end{aligned}$$ and $$\begin{aligned} \gamma \left( r \cdot \left\langle b, x \right\rangle \right) = \gamma \left( r \cdot b + T_{r}(x), r \cdot x \right) = d(r \cdot b) + d \circ T_{r}(x) + h \left( r \cdot x \right) = r \cdot d(b) + r \cdot h(x) = r \cdot \gamma(b,x). \\\end{aligned}$$ We have shown $\mathop{\mathrm{im}}r = \ker \delta_{T}$. $H^{2}_{\mathcal V}(Q,A^{I},\hat{\ast})$ : We first show $\sigma \circ \delta_{T} = 0$. Fix $d \in H^{1}(I,A^{I},\ast)^{\square}$ and note $\sigma \circ \delta_{T}(d) = [d \circ T \circ \pi]$. Define $s: M \rightarrow I$ by $s(m): l \circ m - m$ where $l: Q \rightarrow M$ is the lifting $l(x) = \left\langle 0, x \right\rangle \in I \rtimes_{T} Q = M$. If we write $m = \left\langle b, x \right\rangle$, then $s(m) = \left\langle 0, x \right\rangle - \left\langle b, x \right\rangle = - \left\langle b, 0 \right\rangle = \left\langle -b, 0 \right\rangle$. Then by realization of the 2-cocycle, we have $$\begin{aligned} d \circ T_{+}(\pi(m_{1}),\pi(m_{2})) &= d \Big( l \circ \pi(m_{1}) + l \circ \pi(m_{1}) - l \circ \pi(m_{1} + m_{2}) \Big) \\ &= d \Big( l \circ \pi(m_{1}) + l \circ \pi(m_{1}) - (m_{1} + m_{2}) \Big) + d \Big( (m_{1} + m_{2}) - l \circ \pi(m_{1} + m_{2}) \Big) \\ &= d \circ s(m_{1}) + d \circ s(m_{2}) - d \circ s( m_{1} + m_{2} ) \end{aligned}$$ and $$\begin{aligned} d \circ T_{r}(\pi(m)) = d \Big( r \cdot ( l \circ \pi(m)) - l \circ \pi(r \cdot m) \Big) &= d \Big( r \cdot ( l \circ \pi(m)) - r \cdot m \Big) + d \Big( r \cdot m - l \circ \pi(r \cdot m) \Big) \\ &= r \cdot (d \circ s (m)) - d \circ s (r \cdot m).\end{aligned}$$ For a multilinear operation $f \in F$ with $n =\mathop{\mathrm{ar}}f$, we similarly observe that $$\begin{aligned} d \circ T_{f}(\pi(\vec{m})) &= d \big( f^{M}(l \circ \pi(\vec{m})) - l \circ \pi (f^{Q}(\vec{m})) \big) \\ &= d \big( f^{M}(l \circ \pi(\vec{m})) - f^{M}(\vec{m}) \big) - d \circ s ( f^{M}(\vec{m})) \\ &= d \left( - f^{I}(p_{1}(\vec{m})) - \sum_{s \in [n]^{\ast}} b(f,s)(\pi(\vec{m}), p_{1}(\vec{m}) ) \right) - d \circ s ( f^{M}(\vec{m})) \\ &= \sum_{i \in [n]} \hat{a}(f,i)( \pi(m_{1}), \ldots, d(-p_{1}(m_{i})), \ldots, \pi(m_{n}) ) - d \circ s ( f^{M}(\vec{m})) \\ &= \sum_{i \in [n]} \hat{a}(f,i)( \pi(m_{1}), \ldots, d \circ s (m_{i}), \ldots, \pi(m_{n}) ) - d \circ s ( f^{M}(\vec{m}))\end{aligned}$$ Altogether, we have shown $d \circ s: M \rightarrow A^{I}$ witnesses that $d \circ T \circ \pi$ is a 2-coboundary for the datum $(M,A^{I},\ast)$. Now fix $[S] \in H^{2}_{\mathcal V}(Q,A^{I},\hat{\ast})$ such that $\sigma([S]) = [S \circ \pi] = 0$. By definition, there exists $h:M \rightarrow A^{I}$ such that 1. $S_{+}(\pi(m_{1}),\pi(m_{2})) = h(m_{1}) + h(m_{2}) - h(m_{1} + m_{2})$; 2. $S_{r}(\pi(m)) = r \cdot h(m) - h(r \cdot m)$; 3. $S_{f}(\pi(\vec{m})) = \sum_{i \in [n]} a(f,i) ( m_{1},\ldots,h(m_{i}),\ldots,m_{n} ) - h(f^{M}(\vec{m}))$ $( f \in F, n = \mathop{\mathrm{ar}}f)$. We can also assume that $0=S_{+}(x,0) = S(0,x) = S_{r}(0) = S_{f}(\vec{x})$ whenever some $x_{i}=0$. If we evaluate on the elements $m_{i} = \left\langle b_{i}, 0 \right\rangle$, we see that (1) - (3) above implies the restriction $h|_{I}: I \rightarrow A^{I}$ is a derivation since the left-hand side are all zero. We wish to show $h|_{I}$ satisfies the $\square$-condition according to Eq [\[eqn:squarenull\]](#eqn:squarenull){reference-type="eqref" reference="eqn:squarenull"}. This follows by (3) since for $f \in F$ with $n = \mathop{\mathrm{ar}}f$, we have $$\begin{aligned} h \left( b(f,i)(x_{1},\ldots,b_{i},\ldots,x_{n}), 0 \right) &= h \left( f^{M} \left( \left\langle 0, x_{1} \right\rangle, \ldots, \left\langle b_{i}, 0 \right\rangle, \ldots, \left\langle 0, x_{n} \right\rangle \right) \right) \\ &= \sum_{k \in [n]} a(f,i) ( \left\langle 0, x_{1} \right\rangle, \ldots, h(0,x_{k}) ,\ldots , \left\langle b_{i}, 0 \right\rangle ,\ldots, \left\langle 0, x_{n} \right\rangle ) \\ &\quad - S_{f}(x_{1},\ldots,0,\ldots,x_{n})\\ &= a(f,i) ( \left\langle 0, x_{1} \right\rangle, \ldots, h( b_{i}, 0), \ldots, \left\langle 0, x_{n} \right\rangle ) \\ &= \hat{a}(f,i)(x_{1},\ldots,h|_{I}(b_{i}),\ldots,x_{n})\end{aligned}$$ since $\mathop{\mathrm{im}}h \subseteq A^{I}$. The same reason also shows $h \left( b(f,i)(x_{1},\ldots,b_{i},\ldots,x_{n}), 0 \right) = 0$. Let us observe from (1) above that $0 = S_{+}(0,x) = S_{+}(\pi(b,0),\pi(0,x)) = h(b,0) + h(0,x) - h ( \left\langle b, 0 \right\rangle + \left\langle 0, x \right\rangle ) = h(b,0) + h(0,x) - h(a,x)$; thus, we can write $h(a,x) = h(a,0) + h(0,x)$. If we define $\gamma: Q \rightarrow A^{I}$ by $\gamma(x):=h(0,x)$, then we see that for $f \in F$ with $n = \mathop{\mathrm{ar}}f$, $$\begin{aligned} h|_{I} \circ T_{f}(\vec{x}) = h(T_{f}(\vec{x}),0) &= h(T_{f}(\vec{x}),f^{Q}(\vec{x}) ) - h(0,f^{Q}(\vec{x})) \\ &= h \left( f^{M} \left( \left\langle 0, x_{1} \right\rangle, \ldots, \left\langle 0, x_{n} \right\rangle \right) \right) - h(0,f^{Q}(\vec{x})) \\ &= \sum_{i \in [n]} \hat{a}(f,i)(x_{1},\ldots,h(x_{i}),\ldots,x_{n} ) - S_{f}(\vec{x}) - \gamma \left( f^{Q}(\vec{x}) \right).\end{aligned}$$ A similar calculation for $T_{+}$ and $T_{r}$ contributes to show $[S] \in \mathop{\mathrm{im}}\delta_{T}$; altogether, $\mathop{\mathrm{im}}\delta_{T} = \ker \sigma$. The demonstration of the theorem is complete. ◻ # Discussion {#section:6} It should be possible in both the affine and nonabelian cases to give a more concrete development of higher cohomologies by focusing on the intended interpretations. **Problem 46**. Let $\mathcal V$ be a variety of multilinear expansions of $R$-modules and $(Q,I)$ be datum in $\mathcal V$. À la Holt [@holt], is it possible to construct higher cohomologies for general datum $(Q,I)$ in $\mathcal V$ which are characterized by equivalence classes of certain "decorated" exact sequences ? Can these higher cohomologies be realized by classes of models of multisorted signatures extending 2-cocycles ? For our varieties of interests, is there an analogue of the description for groups of nonabelian $2^{\mathrm{nd}}$-cohomology by weak 2-functors between certain 2-categories? Is this instructive for determining what higher cohomology should be characterizing ? **Problem 47**. For varieties of multilinear expansions of $R$-modules, is it possible to describe the general cohomologies in the framework of weak functors and n-categories ? In particular, the suggestion here is that for these varieties there may be three equivalent descriptions of cohomology given be models of certain multisorted expansions of 2-cocycles together with their equational theories, isomorphism classes of certain decorated exact sequences and weak natural isomorphism classes of weak functors between certain n-categories. **Acknowledgments 48**. The research in this manuscript was supported by NSF China Grant \#12071374. 15 A.L. Agore, G. Militaru: *Hochschild products and global non-abelian cohomology for algebras. Applications*. J. Pure Appl. Algebra **221**:366-392, (2017) B. Bakalov, V.G. Kac, A. Voronov: *Cohomology of conformal algebras*. Comm. Math. Phys. **200**:561-589, (1999) P. Batten: *Covers and multipliers of Lie algebras* Dissertation, North Carolina State University (1993) J.M. Casas, E. Khmaladze, M. Ladra: *Low-dimensional on-abelian Leibniz cohomology*. Forum Math., **25**:443-469, (2013) A. Das, N. Rathee: *A unified extension theory of Rota-Baxter algebras, dendriform algebras, and a fundamental sequence of Wells*. arXiv:2206.09019 \[math.RA\] (2022) R. Elyse: *Multipliers and covers of Liebniz algebras*, Dissertation, North Carolina State University (2019) C.J. Everett Jr.: *An extension theory for rings*, Amer. J. Math. **64**:363-370, (1942) P.J. Higgins: *Groups with multiple operators*, Proceedings of the London Math. Soc. **s3-6**(3):366-416, (1956) G. Hochschild: *Cohomology and representations of associative algebras*, Duke Math. J., **14**:921-948, (1947) G. Hochschilde, J.P. Serre: *Cohomology of group extensions*, Transactions of the American Mathematical Society, **74**, 110--135 (1953) D. Holt: *An Interpretation of the Cohomology Groups $H^{n}(G,M)$*. Journal of Algebra, **60**:307-320, (1979) B. Hou, J. Zhao: *Crossed Modules, non-abelian extensions of associative conformal algebras and Wells exact sequences*, arXiv:2211.10842 \[math.QA\], (2022) G. Karpilovsky: *The Schur Multiplier*, London Mathematical Society monographs; new ser. 2, Oxford University Press, Oxford (1987) N. Inassaridze, E. Khmaladze, M. Ladra: *Non-abelian cohomology and extensions of Lie algebras*, J. Lie Theory, **18**:413-432, (2008) E. Mainellis: *Multipliers and unicentral Leibniz algebras*. arXiv:2106.13294 \[math.RA\] (2021) E. Mainellis: *Multipliers and unicentral diassociative algebras*. arXiv:2107.06947 \[math.RA\] (2021) E. Mainellis: *Multipliers and covers of perfect diassociative algebras*. arXiv:2201.07148 \[math.RA\] (2022) J. Milnor: *Introduction to Algebraic K-theory*. Annals of Math. Studies, vol.72, Princeton Univ.Press, Princeton (1971) J. D. H. Smith: *Conformal algebras, vertex algebras, and the logic of locality*. Math. Slovaca **66**(2):407-420 (2016) C. Wells: *Automorphisms group of extensions*. Transactions of the American Mathematical Society **155**(1):189-194 (1971) A. Wires: *Extensions realizing affine datum : low-dimensional cohomology*. (manuscript) A. Wires: *Extensions realizing affine datum : central extensions*. (manuscript) A. Wires: *Extensions realizing affine datum : the Well's derivation*. (manuscript) K. Yamaguti: *On cohmology groups of general Lie triple systems*. Kumamoto J. Sci. A **8**:135-146 (1967/1969)

arxiv_math

--- abstract: | In the past decades, model averaging (MA) has attracted much attention as it has emerged as an alternative tool to the model selection (MS) statistical approach. Hansen \[*Econometrica* **75** (2007) 1175--1189\] introduced a Mallows model averaging (MMA) method with model weights selected by minimizing a Mallows' $C_p$ criterion. The main theoretical justification for MMA is an asymptotic optimality (AOP), which states that the risk/loss of the resulting MA estimator is asymptotically equivalent to that of the best but infeasible averaged model. MMA's AOP is proved in the literature by either constraining weights in a special discrete weight set or limiting the number of candidate models. In this work, it is first shown that under these restrictions, however, the optimal risk of MA becomes an unreachable target, and MMA may converge more slowly than MS. In this background, a foundational issue that has not been addressed is: When a suitably large set of candidate models is considered, and the model weights are not harmfully constrained, can the MMA estimator perform asymptotically as well as the optimal convex combination of the candidate models? We answer this question in a nested model setting commonly adopted in the area of MA. We provide finite sample inequalities for the risk of MMA and show that without unnatural restrictions on the candidate models, MMA's AOP holds in a general continuous weight set under certain mild conditions. Several specific methods for constructing the candidate model sets are proposed. Implications on minimax adaptivity are given as well. The results from simulations back up our theoretical findings. author: - Jingfu Peng - Yang Li - Yuhong Yang bibliography: - mybibfile.bib title: "On optimality of Mallows model averaging[^1]" --- **Keywords: Model averaging, model selection, asymptotic optimality, minimax adaptivity.** # Introduction {#sec:introduction} In statistical modeling, multiple candidate models are usually considered to explore the data. Model selection (MS) guides us in search for the best model among candidates based on a traditional selection criterion, such as AIC [@Akaike1973], $C_p$ [@Mallows1973], and BIC [@Schwarz1978], the use of cross-validation [@Allen1974; @Stone1974], and solving a penalized regression problem, such as Lasso [@Tibshirani1996], adaptive Lasso [@zou2006adaptive], SCAD [@Fan2001], and MCP [@Zhang2010Nearly] (see [@Ding2018] for a recent review). The key theoretical properties of these methods, namely consistency in selection, asymptotic efficiency, and minimax-rate optimality, have been well established in the literature. Once a final model is selected, all subsequent estimation, prediction, and inference are typically based on the selected model as if it were given in advance. However, it has been increasingly recognized that choosing just one model inherently ignores possibly high uncertainty in the selection process [@Chatfield1995model; @Draper1995; @Yuan2005]. Model averaging (MA), on the other hand, provides an alternative to reduce the variability in MS while offering a possibility of reducing modeling bias by averaging over the candidate models properly. MA has a rich heritage in Bayesian statistics, see, e.g., [@Draper1995], [@George1997], and [@Hoeting1999] for more details and references therein. From a frequentist perspective, several attractive strategies have been proposed to combine models, including boosting [@FREUND1995256], bagging [@Breiman1996b], random forest [@Amit1997], information criterion weighting [@Buckland1997; @Hjort2003], progressive mixture [@Yang2000Mixing; @Catoni2004; @Juditsky2008], exponentially weighted aggregation [@George1986; @Leung2006; @Dalalyan2012], Q-aggregation [@Dai2012; @Rigollet2012kl; @Lecue2014], to name a few (see Section [12](#sec:a:related){reference-type="ref" reference="sec:a:related"} of the appendix for other related works). In particular, by minimizing some specific performance measures, a growing MA literature develops methods to pursue the optimal convex combination of the candidate models based on the same data. To the best of our knowledge, this problem was first considered by [@Blaker1999adaptive] in a two candidate model setting, and studied by @Hansen2007 in a general context, who proposed a Mallows model averaging (MMA) method to select weights for averaging across nested linear models by minimizing the Mallows' $C_p$ criterion [@Mallows1973]. Adopting other performance measures like cross-validation error and Kullback-Leibler divergence, the MMA-type strategies have been developed explicitly for other or more general frameworks, such as heteroskedastic error regression model [@Hansen2012; @Liu2013], time-series error models [@HANSEN2008342; @ZHANG201382; @Cheng2015], high-dimensional regression model [@Ando2014; @Ando2017; @Zhang2020], generalized linear model [@Ando2017; @Zhang2016jlm], quantile regression model [@Lu2015], varying-coefficient model [@zhu2019mallows], semiparametric model [@FANG2022219], general supervised learning framework [@WOLPERT1992241; @Breiman1996a; @vanderLaanPolleyHubbard+2007], among many useful others. Given the increasing and potential wide applications of the MMA-type methods, an essential question arising from an estimation perspective is how good this popular class of methods for constructing an MA estimator is. This paper focuses on MMA introduced by [@Hansen2007] and revisits its optimality. Note that the MMA criterion is an unbiased estimate of the squared risk of the MA estimator plus a constant, and the resulting MMA estimator targets the minimization of the squared risk/loss of MA. The optimality of MMA has certainly been studied from an asymptotic viewpoint in the MA literature. An asymptotic optimality (AOP) theory states that a good MA estimator can be asymptotically equivalent to the optimal convex combination of the given candidates in terms of the statistical risk/loss. There are two major approaches to establishing the MMA's AOP. @Hansen2007 first proved it when the weight vectors are contained in a special discrete set. His results require that the candidates are nested and do not impose any additional assumption on the number of candidate models. Since the discrete weight set is quite restrictive, @Wan2010 made an important contribution by considering direct minimization of the MMA criterion over the continuous weight set with possibly non-nested models. Their paper justifies the MMA's AOP but requires a restriction on the candidate model set. Similar assumptions also arise in a number of subsequent papers, see [@Ando2014; @Ando2017; @Zhang2020; @Zhang2021]. In summarizing the literature in relation to the real goal of AOP, while the aforementioned theoretical advancements are novel and valuable, the consequences of the restrictions imposed on weight/candidate models are still unclear. Consider a typical nested model framework with the $m$-th candidate model containing the first $m$ regressors. For @Hansen2007's approach, a sensible choice for the candidate model set is to include $M_n \geq m_n^*$ nested models, where $m_n^*$ is the size of the optimal single model. We show in Section [3.1](#subsec:review_hansen){reference-type="ref" reference="subsec:review_hansen"} that when $m_n^*$ is not too small relative to the sample size $n$ (e.g., $m_n^*$ grows at order $n^{\alpha}$ for some $0<\alpha<1$), the best possible MA risk in the discrete weight set is suboptimal. For the approach in [@Wan2010], as shown in Section [3.2](#subsec:review_wan){reference-type="ref" reference="subsec:review_wan"}, the required restriction on the candidate models is so strong that the optimal single model $m_n^*$ is excluded, and the MMA criterion can only combine a set of underperforming models. Note that the MMA-type literature often motivates their approaches to overcome the problems of MS and hence perform better. However, the MA estimator based on such candidate model sets actually converges more slowly than MS. In this background, a critical issue that has not been addressed in the existing literature is: When the weight vector is allowed for the full potential of MA, and the number of candidate models is not harmfully constrained, can the MMA estimator perform asymptotically as well as the infeasible optimal averaged model? Inspired by the previous work of @Hansen2007 and @Wan2010, this paper answers the aforementioned foundational question on MMA in the context of linear regression with nested models. We derive non-asymptotic risk bounds for MMA when the random errors follow the sub-Gaussian assumption, which show that the squared risk of the MMA estimator is bounded above by the optimal MA risk plus a couple of additional terms associated with the estimation errors of the weights and the variance of the error term, respectively. Based on these risk bounds, there are mainly three implications. First, when the convergence rate of the optimal MA risk is not too fast (e.g., the optimal MA risk converges slower than $(\log n)^3/n$), the MMA estimator asymptotically attains the optimal risk among all averaged models without any unnatural restrictions on the weight set or the candidate model set. Second, instead of incorporating all nested models, the full advantage of MA can still be realized by grouping regressors properly or removing inferior models at the outset prior to implementing MMA. Third, the resulting MMA estimator exhibits optimal minimax adaptivity over some general coefficient classes, such as ellipsoids and hyperrectangles. The results from our finite sample simulations support these findings. The rest of the paper is organized as follows. In Section [2](#sec:setup){reference-type="ref" reference="sec:setup"}, we set up the regression framework and give the MMA estimators. In Section [3](#sec:review){reference-type="ref" reference="sec:review"}, we theoretically investigate the consequences of using a discrete weight set or restricting the candidate model set. We then in Section [4](#sec:main_results){reference-type="ref" reference="sec:main_results"} develop non-asymptotic risk bounds for MMA. Consequently, the MMA's AOP theory is obtained. Section [5](#sec:reduced){reference-type="ref" reference="sec:reduced"} suggests two strategies for constructing the candidate model set. Section [6](#sec:minimax){reference-type="ref" reference="sec:minimax"} shows minimax adaptivity of MMA. Section [7](#sec:simulation){reference-type="ref" reference="sec:simulation"} presents the results of simulation experiments. Concluding remarks are given in Section [8](#sec:conclusion){reference-type="ref" reference="sec:conclusion"}. The proofs, additional simulation results, and discussions on the other related works can be found in the Appendix. # Problem setup {#sec:setup} ## Setup and notation {#sec:setup:1} Consider the linear regression model $$\label{eq:model} y_i=f_i+\epsilon_i=\sum_{j=1}^{p_n}\beta_jx_{ij}+\epsilon_i,\quad i=1,\ldots,n,$$ where $\epsilon_1, \ldots, \epsilon_n$ are i.i.d. sub-Gaussian random variables with $\mathbb{E}{\epsilon_i}=0$ and $\mathbb{E}{\epsilon_i^2}=\sigma^2$, and $\mathbf{x}_j = (x_{1j}, \ldots , x_{nj})^{\top}$, $j=1,\ldots,p_n$ are nonstochastic regressor vectors. Defining the response vector $\mathbf{y}=(y_1,\ldots,y_n)^{\top}$, the regression mean vector $\mathbf{f}=(f_1,\ldots,f_n)^{\top}$, the coefficient vector $\boldsymbol{\beta}=\left(\beta_1,\ldots,\beta_{p_n}\right)^{\top}$, the regressor matrix $\mathbf{X}=\left[\mathbf{x}_1,\ldots, \mathbf{x}_{p_n}\right]\in \mathbb{R}^{n \times p_n}$, and the noise vector $\boldsymbol{\epsilon}=(\epsilon_1,\ldots,\epsilon_n)^{\top}$, we can write ([\[eq:model\]](#eq:model){reference-type="ref" reference="eq:model"}) in matrix form $$\label{eq:model_matrix} \mathbf{y}= \mathbf{f}+ \boldsymbol{\epsilon}= \mathbf{X}\boldsymbol{\beta}+ \boldsymbol{\epsilon}.$$ For the sake of simplicity, we assume $p_n \leq n$ and $\mathbf{X}$ has full column rank. To estimate the true regression mean vector $\mathbf{f}$, $M_n$ strictly nested linear models are considered as candidates. The $m$-th candidate model includes the first $k_m$ regressors, where $1\leq k_1<k_2<\cdots<k_{M_n}\leq p_n$. The information about the sizes of candidate models is stored in a set $\mathcal{M}=\left\{k_1,\ldots,k_{M_n} \right\}$, and then $M_n=|\mathcal{M}|$, where $|\mathcal{S}|$ denotes the cardinality of a set $\mathcal{S}$ throughout this paper. Let $\mathbf{X}_{k_m}=\left[\mathbf{x}_1,\ldots,\mathbf{x}_{k_m}\right]$ be the design matrix of the $m$-th candidate model, which estimates $\mathbf{f}$ by the least squares method $\widehat{\mathbf{f}}_{k_m}=\mathbf{X}_{k_m}(\mathbf{X}_{k_m}^{\top}\mathbf{X}_{k_m})^{-1}\mathbf{X}_{k_m}^{\top}\mathbf{y}\triangleq \mathbf{P}_{k_m}\mathbf{y}$. Let $\mathbf{w}=(w_1,\ldots,w_{M_n})^{\top}$ denote a weight vector in the unit simplex of $\mathbb{R}^{M_n}$: $$\label{eq:weight_general} \mathcal{W}_{M_n}=\left\{\mathbf{w}\in[0,1]^{M_n}:\sum_{m=1}^{M_n}w_m=1\right\}.$$ Given the candidate model set $\mathcal{M}$, the MA estimator of $\mathbf{f}$ is $\widehat{\mathbf{f}}_{\mathbf{w}| \mathcal{M}}=\sum_{m=1}^{M_n}w_m\widehat{\mathbf{f}}_{k_m}$, where the subscript $\mathbf{w}| \mathcal{M}$ is to emphasize the dependence of the MA estimator on the candidate model set $\mathcal{M}$. For the theoretical work, we consider the normalized squared $\ell_2$ loss $L_n(\widehat{\mathbf{f}},\mathbf{f})= n^{-1}\|\widehat{\mathbf{f}}-\mathbf{f}\|^2$ and its corresponding risk $R_n(\widehat{\mathbf{f}},\mathbf{f})=\mathbb{E}L_n(\widehat{\mathbf{f}},\mathbf{f})$ as measures of the performance of an estimator $\widehat{\mathbf{f}}$, where $\|\cdot\|$ refers to the Euclidean norm. For abbreviation, let $L_n(m,\mathbf{f})$, $R_n(m,\mathbf{f})$, $L_n(\mathbf{w}| \mathcal{M},\mathbf{f})$ and $R_n(\mathbf{w}| \mathcal{M},\mathbf{f})$ stand for $L_n(\widehat{\mathbf{f}}_m,\mathbf{f})$, $R_n(\widehat{\mathbf{f}}_m,\mathbf{f})$, $L_n(\widehat{\mathbf{f}}_{\mathbf{w}| \mathcal{M}},\mathbf{f})$ and $R_n(\widehat{\mathbf{f}}_{\mathbf{w}| \mathcal{M}},\mathbf{f})$ respectively. We denote $m_n^*=\arg\min_{m \in \{1,\ldots,p_n \}}R_n(m,\mathbf{f})$ the size of the optimal single model, $m^*|\mathcal{M}=\arg\min_{m \in \mathcal{M}}R_n(m,\mathbf{f})$ the size of the optimal candidate model in $\mathcal{M}$, and $\mathbf{w}^*|\mathcal{M}=\arg\min_{\mathbf{w} \in \mathcal{W}_{M_n}}R_n(\mathbf{w}| \mathcal{M},\mathbf{f})$ the optimal weight vector based on the candidate model set $\mathcal{M}$ and the general continuous weight set $\mathcal{W}_{M_n}$. The quantities $m_n^*$, $m^*|\mathcal{M}$, and $\mathbf{w}^*|\mathcal{M}$ are all infeasible in practice since they depend on the unknown parameters $\mathbf{f}$ and $\sigma^2$. In this paper, we estimate the weights by minimizing the MMA criterion proposed by [@Hansen2007] $$\label{eq:criterion} C_n(\mathbf{w}|\mathcal{M},\mathbf{y})=\frac{1}{n}\|\mathbf{y}-\widehat{\mathbf{f}}_{\mathbf{w}| \mathcal{M}} \|^2+\frac{2\widehat{\sigma}^2}{n}\mathbf{k}^{\top}\mathbf{w},$$ that is, $\widehat{\mathbf{w}}|\mathcal{M}=\arg\min_{\mathbf{w} \in \mathcal{W}_{M_n}}C_n(\mathbf{w}|\mathcal{M},\mathbf{y})$, where $\widehat{\sigma}^2$ is an estimator of $\sigma^2$, and $\mathbf{k}=(k_1,\ldots,k_{M_n})^{\top}$ is the vector of the sizes of the candidate models in $\mathcal{M}$. The resulting MMA estimator of $\mathbf{f}$ is $$\label{eq:MMA_estimator} \widehat{\mathbf{f}}_{\widehat{\mathbf{w}} | \mathcal{M}} =\sum_{m=1}^{M_n}\widehat{w}_m\widehat{\mathbf{f}}_{k_m}.$$ Note that when $\sigma^2$ is known, $\widehat{\mathbf{w}}|\mathcal{M}$ is chosen based on the minimization of an unbiased estimate for $R_n(\mathbf{w}|\mathcal{M},\mathbf{f})$ plus a constant, since $\mathbb{E} C_n(\mathbf{w}|\mathcal{M},\mathbf{y})=R_n(\mathbf{w}|\mathcal{M},\mathbf{f})+\sigma^2$. Let $\mathbb{E}L_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$ and $\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$ denote the risk functions of the resulting MMA estimator, which take the randomness of $\widehat{\mathbf{w}}$ into account. But the former is a little different from the latter since in the latter function, $\widehat{\mathbf{w}}$ is directly plugged in the expression of $R_n(\mathbf{w}|\mathcal{M},\mathbf{f})$. Let $Q_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$ denote any one of two quantities: $\mathbb{E}L_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$ and $\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$. From now on, we will use the symbols $\lesssim$, $\gtrsim$, and $\asymp$ for comparison of positive sequences, where $a_n\lesssim b_n$ means $a_n=O(b_n)$, $a_n\gtrsim b_n$ means $b_n=O(a_n)$, and $a_n\asymp b_n$ means both $a_n\lesssim b_n$ and $a_n\gtrsim b_n$. Also, $a_n\sim b_n$ means that $a_n/b_n\rightarrow 1$ as $n\rightarrow \infty$. Let $\lfloor a \rfloor$ and $\lceil a \rceil$ return the floor and the ceiling of $a$ respectively. For any two real numbers $a$ and $b$, we use notation $a \wedge b = \min(a,b)$ and $a \vee b = \max(a,b)$. ## Definitions of optimality {#sec:setup2} We first give some notations that will play a key role in our theoretical analysis. Let $\mathbf{P}_{j}\triangleq\mathbf{X}_{j}(\mathbf{X}_{j}^{\top}\mathbf{X}_{j})^{-1}\mathbf{X}_{j}^{\top}$ be the projection matrix on the column space of the first $j$ columns of the full design matrix $\mathbf{X}$. As pointed out by [@Xu2022From], the successive subtraction of $\mathbf{P}_{j}, j=1,\ldots,p_n$ yields $p_n$ mutually orthogonal matrixes $\mathbf{D}_{j}\triangleq\mathbf{P}_{j}-\mathbf{P}_{j-1}=\boldsymbol{\phi}_j\boldsymbol{\phi}_j^{\top}$, $j=1,\ldots,p_n$, where $\mathbf{P}_{0}=\boldsymbol{0}_{n \times n}$ and $\boldsymbol{\phi}_j\in \mathbb{R}^{n}$ is an eigenvector of $\mathbf{D}_{j}$ satisfying $\|\boldsymbol{\phi}_j\|=1$. Obviously, $\{\boldsymbol{\phi}_1,\ldots,\boldsymbol{\phi}_{p_n} \}$ forms an orthonormal basis for the column space of $\mathbf{X}$. Let us denote the *transformed coefficients* $\boldsymbol{\theta}=(\theta_1,\ldots,\theta_{p_n})^{\top}$ of $\mathbf{f}$ by $$\label{eq:trans_para} \theta_j=\theta_j(\mathbf{f})=\frac{\boldsymbol{\phi}_j^{\top}\mathbf{f}}{\sqrt{n}},\quad j=1,\ldots,p_n.$$ When the columns of $\mathbf{X}$ are mutually orthogonal with $\ell_2$ norm $n$, we see that the transformed coefficient $\theta_j$ coincides with the regression coefficient $\beta_j$. Otherwise, $\theta_j$ depends additionally on the dependence between the covariates. There are two important approaches to defining the optimality of MMA: AOP within a given class of averaged estimators and minimax adaptivity within given classes of true regression mean vectors. **Definition 1**. *Given a candidate model set $\mathcal{M}$ and a weight set $\mathcal{W}$, an MA estimator $\widehat{\mathbf{f}}_{\widetilde{\mathbf{w}} | \mathcal{M}}$ with $\widetilde{\mathbf{w}}$ trained on the data is said to be asymptotically optimal (AOP) if it satisfies $$\label{eq:opt1} Q_n\left(\widetilde{\mathbf{w}}|\mathcal{M},\mathbf{f}\right)=[1+o(1)]\min_{\mathbf{w} \in \mathcal{W}}R_n\left(\mathbf{w}|\mathcal{M},\mathbf{f}\right)$$ as $n\to \infty$.* The existing literature showed that the AOP property can be obtained for the MMA estimator with certain restrictions on the weight set $\mathcal{W}$ or the candidate model set $\mathcal{M}$. Specifically, @Hansen2007 proved the MMA's AOP by minimizing the criterion over a special discrete set $$\label{eq:condition_discrete} \mathcal{W}_{|\mathcal{M}|}(N)=\left\{\sum_{m=1}^{|\mathcal{M}|}w_m=1, w_m\in \left\{0,\frac{1}{N},\frac{2}{N},\ldots,1\right\}\right\},$$ in which $N$ is a fixed positive integer [see also @HANSEN2008342; @Hansen2012]. @Hansen2007's approach does not impose any additional restriction on the candidate model set $\mathcal{M}$ in the nested model setting. With a different technique, @Wan2010 established the MMA's AOP in the continuous set $\mathcal{W}_{|\mathcal{M}|}$ defined in ([\[eq:weight_general\]](#eq:weight_general){reference-type="ref" reference="eq:weight_general"}) but with a condition on $\mathcal{M}$, that is $$\label{eq:condition_wan} \frac{|\mathcal{M}|\sum_{m=1}^{|\mathcal{M}|}R_n\left(\mathbf{w}_m^0|\mathcal{M},\mathbf{f}\right)}{nR_n^2\left(\mathbf{w}^*|\mathcal{M},\mathbf{f}\right)}\to 0,$$ where $\mathbf{w}_m^0$ is a $|\mathcal{M}| \times 1$ vector in which the $m$-th element is one and the others are zeros. In this paper, we refer the AOP theories in [@Hansen2007] and [@Wan2010] as the *restricted AOP* since these results do not allow all the possible convex combinations of the candidate models, hence may lead to a suboptimal convergence rate (see Section [3](#sec:review){reference-type="ref" reference="sec:review"} for the detailed discussion). Let $\mathcal{M}_a=\{1, 2, \ldots, p_n \}$ denote the candidate model set with all the nested models. Note that the relation $R_n\left(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f}\right)\leq R_n\left(\mathbf{w}^*|\mathcal{M},\mathbf{f}\right)$ holds for any $\mathcal{M}\subseteq \mathcal{M}_a$. Thus, $R_n\left(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f}\right)$ can be seen as the full potential of MA under the nested model setting we consider. Therefore, in contrast to the restricted AOP, a more natural definition of the optimality of MA is the *full AOP*. **Definition 2**. *An MA estimator $\widehat{\mathbf{f}}_{\widetilde{\mathbf{w}} | \mathcal{M}}$ with $\widetilde{\mathbf{w}}$ trained on the data is said to achieve the full AOP if it satisfies $$\label{eq:fullopt} Q_n\left(\widetilde{\mathbf{w}}|\mathcal{M},\mathbf{f}\right)=[1+o(1)]R_n\left(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f}\right)$$ as $n\to \infty$.* Then two important questions arise: Q1. : Does the MMA estimator ([\[eq:MMA_estimator\]](#eq:MMA_estimator){reference-type="ref" reference="eq:MMA_estimator"}) obtain the full AOP by combining candidates in $\mathcal{M}_a$ and minimizing the criterion ([\[eq:criterion\]](#eq:criterion){reference-type="ref" reference="eq:criterion"}) over $\mathcal{W}_{|\mathcal{M}_a|}$ directly? Q2. : Can we reduce the candidate model set $\widehat{\mathcal{M}} \subset \mathcal{M}_a$ yet it still satisfies the full AOP property $$\label{eq:opt3} \mathbb{E}Q_n(\widehat{\mathbf{w}}|\widehat{\mathcal{M}},\mathbf{f})=[1+o(1)] R_n\left(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f}\right)?$$ The second question is particularly interesting from an application perspective. Another approach to defining the optimality of MA is the minimax adaptivity. Suppose the transformed coefficients $\boldsymbol{\theta}$ defined in ([\[eq:trans_para\]](#eq:trans_para){reference-type="ref" reference="eq:trans_para"}) belongs to the parameter space $\Theta \subseteq \mathbb{R}^{p_n}$, and the corresponding mean vector space of $\mathbf{f}$ is defined by $\mathcal{F}_{\Theta}=\{\mathbf{f}=\sum_{j=1}^{p_n}\theta_j\boldsymbol{\phi}_j:\boldsymbol{\theta}\in\Theta\}$. Define the minimax risk $R_{M}(\mathcal{F}_{\Theta})=\inf_{\widehat{\mathbf{f}}}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta}}R_n(\widehat{\mathbf{f}},\mathbf{f}),$ where the infimum is over all estimator $\widehat{\mathbf{f}}$. In addition, define the minimax risk of the linear-combined estimators $R_L(\mathcal{F}_{\Theta})=\inf_{\mathbf{w}}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f}),$ where $\inf_{\mathbf{w}}$ denote the infimum over all $\mathbf{w}\in\mathbb{R}^{p_n}$, and the subscript $L$ here is to emphasize that $\widehat{\mathbf{f}}$ is restricted to the class of all the linear combinations of the models in $\mathcal{M}_a$. **Definition 3**. *An estimator $\widetilde{\mathbf{f}}$ is called *adaptive in the exact minimax sense* on the family of the mean vector spaces $\boldsymbol{\mathcal{F}}=\{\mathcal{F}_{\Theta}:\Theta\in \boldsymbol{\mathcal{A}}\}$ if $$\label{eq:minimax_all} \sup_{\mathbf{f}\in \mathcal{F}_{\Theta}}R_n(\widetilde{\mathbf{f}},\mathbf{f})=[1+o(1)]R_M(\mathcal{F}_{\Theta})$$ holds for every $\mathcal{F}_{\Theta}\in \boldsymbol{\mathcal{F}}$. An MA estimator $\widehat{\mathbf{f}}_{\widetilde{\mathbf{w}}|\mathcal{M}_a}$ with $\widetilde{\mathbf{w}}$ estimated on data is called *adaptive in the exact linear-combined minimax sense* on the family of classes $\boldsymbol{\mathcal{F}}$ if $$\label{eq:minimax_ma} \sup_{\mathbf{f}\in \mathcal{F}_{\Theta}}R_n(\widetilde{\mathbf{w}}|\mathcal{M}_a,\mathbf{f})=[1+o(1)]R_L(\mathcal{F}_{\Theta})$$ holds for every $\mathcal{F}_{\Theta}\in \boldsymbol{\mathcal{F}}$.* Q3. : Is the MMA estimator adaptive in the exact minimax sense or adaptive in the exact linear-combined minimax sense on some general families of coefficient classes $\Theta$, such as the families of Sobolev ellipsoids and hyperrectangles? The answers to questions Q1--Q3 may provide a previously unavailable insight on the theoretical foundation of MMA. # Revisiting the existing AOP theories on MMA {#sec:review} The main purpose of this section is to investigate the consequences of using the discrete weight set [@Hansen2007] or restricting the candidate model set [@Wan2010] in the restricted-AOP theory. ## Discrete weight set {#subsec:review_hansen} Recall that @Hansen2007 established the MMA's AOP when the weights are contained in the discrete weight set ([\[eq:condition_discrete\]](#eq:condition_discrete){reference-type="ref" reference="eq:condition_discrete"}) but without imposing any additional restriction on $\mathcal{M}$. For simplicity, we consider a set of successive candidate models $\mathcal{M}_{s}=\{1,2, \ldots, M_n\}$, which has usually been adopted to implement the MMA-type methods [@Hansen2007; @Zhang2016jlm; @Zhang2020]. And let $\mathbf{w}_N^*|\mathcal{M}_s=\arg\min_{\mathbf{w}\in \mathcal{W}_{|\mathcal{M}_s|}(N)}R_n(\mathbf{w}| \mathcal{M}_s,\mathbf{f})$ denote the optimal discrete weight vector in $\mathcal{W}_{|\mathcal{M}_s|}(N)$. We first focus on the magnitude of the risk increment $R_n\left(\mathbf{w}_N^*|\mathcal{M}_s,\mathbf{f}\right)-R_n\left(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}\right)$. Certain assumptions on the nature of the regression mean vector are made to evaluate this risk increment in a feasible way. **Assumption 1**. *The regression mean vector $\mathbf{f}$ satisfies $\lim\sup_n n^{-1}\|\mathbf{f}\|^2 < \infty$.* **Assumption 2**. *The transformed coefficients ([\[eq:trans_para\]](#eq:trans_para){reference-type="ref" reference="eq:trans_para"}) are ordered, which means $\{|\theta_j|,j\geq 1\}$ is a non-increasing positive sequence.* Assumption [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"} is a standard assumption for regression estimation problems. Assumption [Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"} can give us many conveniences in characterizing the unknown optimal weights. When the columns of $\mathbf{X}$ are mutually orthogonal, we see that $\theta_j$ is proportional to $\beta_j$. In this case, Assumption [Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"} ensures that the regressors are ordered from most important to least important. The idea of ordering regressors to prepare candidate models has been commonly adopted in the implementation of MA; for example, see [@Hansen2007; @Ando2017; @Zhang2016jlm; @Zhang2020]. Under Assumptions [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"}--[Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"}, we further provide two different conditions on the transformed coefficients $\theta_j,j=1,\ldots,p_n$. **Condition 1**. *(Slowly decaying coefficients) There exist constants $k>1$ and $0<\delta\leq\nu<1$ with $k\nu^2<1$ such that $\delta\leq|\theta_{\lfloor kl \rfloor}/\theta_l|\leq\nu$ when $l$ is large enough.* **Condition 2**. *(Fast decaying coefficients) For every constant $k>1$, $\lim\limits_{l \to \infty}|\theta_{\lfloor kl \rfloor}/\theta_l|= 0$.* Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"} contains the case $\theta_j=j^{-\alpha_1}$ for $\alpha_1 > 1/2$, which serves as the principal case in the MA literature [@Hansen2007]. In contrast, the coefficients satisfying Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"} decay much faster. An example is the exponentially decaying coefficients $\theta_j=\exp(-j^{\alpha_2})$ for some $\alpha_2>0$. **Proposition 1**. *Suppose Assumptions [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"}--[Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"} hold. When both Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"} and $M_n\gtrsim m_n^*$ are satisfied, we have $$R_n\left(\mathbf{w}_N^*|\mathcal{M}_s,\mathbf{f}\right)-R_n\left(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}\right) \asymp R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}).$$ When either Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"} or $M_n=o(m_n^*)$ holds, we have $$R_n\left(\mathbf{w}_N^*|\mathcal{M}_s,\mathbf{f}\right)-R_n\left(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}\right)=o\left[ R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f})\right].$$* Proposition [Proposition 1](#lemma:delta_1){reference-type="ref" reference="lemma:delta_1"} theoretically clarifies the effects of weight discretization and $M_n$ on the optimal MA risk. When $\theta_l$ decays slowly and $M_n$ is large, the difference $R_n\left(\mathbf{w}_N^*|\mathcal{M}_s,\mathbf{f}\right)-R_n\left(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}\right)$ is of the same order as the risk $R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f})$. In this case, weight discretization increases the optimal risk in the general continuous weight set $\mathcal{W}_{|\mathcal{M}_s|}$ by a significant fraction. However, when $\theta_s$ decays fast or $M_n$ is small relative to the size of the optimal model, the discrete weight set asymptotically does not influence the optimal risk of MA. This proposition implies that in some important scenarios, such as $p_n=n$ and $\theta_j=j^{-\alpha_1},\alpha_1 > 1/2$, where the optimal single model $m_n^*$ grows at order $n^{1/(2\alpha_1)}$, it is impossible to achieve the full potential of MA by minimizing the MMA criterion in a discrete weight set with any fixed $N$. On the other hand, MS can be viewed as MA in the discrete set $\mathcal{W}_{|\mathcal{M}_s|}(1)$. Recall that $m_n^*$ denotes the optimal single model among all candidate models and $m^*|\mathcal{M}_s$ stands for the optimal model in $\mathcal{M}_s$. Thus we have $R_n\left(m^*|\mathcal{M}_s,\mathbf{f}\right)\geq R_n\left(\mathbf{w}_N^*|\mathcal{M}_s,\mathbf{f}\right)$. A natural question to ask is whether $R_n\left(\mathbf{w}_N^*|\mathcal{M}_s,\mathbf{f}\right)$ has a substantial improvement over $R_n\left(m^*|\mathcal{M}_s,\mathbf{f}\right)$ when $N\geq 2$. **Proposition 2**. *Suppose Assumptions [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"}--[Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"} hold. Under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"} and $M_n\gtrsim m_n^*$, define $$\kappa\triangleq\log_k\left( \frac{m_n^*}{M_n}\vee 1 \right),$$ where $k$ is the constant given in Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}. If $N>(1+\delta^{2\kappa+2})/(2\delta^{2\kappa+2})$, we have $$R_n\left( m^*|\mathcal{M}_{s},\mathbf{f}\right) - R_n\left( \mathbf{w}_N^*|\mathcal{M}_{s}, \mathbf{f}\right) \asymp R_n\left( m^*|\mathcal{M}_{s},\mathbf{f}\right).$$ Under Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"} or $M_n=o(m_n^*)$, for any $N$, we have $$R_n\left( m^*|\mathcal{M}_{s},\mathbf{f}\right) - R_n\left( \mathbf{w}_N^*|\mathcal{M}_{s}, \mathbf{f}\right)=o\left[ R_n\left( m^*|\mathcal{M}_{s},\mathbf{f}\right)\right].$$* When $\theta_l$ decays slowly and $M_n$ is large, the optimal model size $m^*|\mathcal{M}_{s}$ is not very small relative to the sample size $n$. In this case, the MS uncertainty is relatively high, and MA under the discrete weight set still reduces the risk of MS substantially, although it does not provide the full potential of MA. For example, when $\theta_j=j^{-\alpha_1},\alpha_1>1/2$, Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"} is satisfied for any $k>1$ and $\delta=k^{-\alpha_1}$. Then, for a large candidate model set with $M_n\geq m_n^*$, the condition for improving over MS is $$N>\frac{1+\delta^{2}}{2\delta^{2}}=\frac{1+k^{2\alpha_1}}{2}.$$ Due to the arbitrariness of $k$, it suffices to require $N\geq 2$. ## Restriction of the candidate model set {#subsec:review_wan} Directly minimizing the MMA criterion over the continuous weight set $\mathcal{W}_{|\mathcal{M}|}$ was considered by [@Wan2010]. But they impose another restriction ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) on $\mathcal{M}$. As will be seen, ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) is a rather strong condition that can lead to exclusion of some important models. In this subsection, we continue to focus on a nested framework with successive candidates $\mathcal{M}_s=\{ 1, 2, \ldots, M_n \}$. **Example 1** (Polynomially decaying coefficients). *Consider $\theta_j=j^{-\alpha_1}, \alpha_1>1/2$, and assume $M_n=o(p_n)$. Condition ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) is equivalent to the restriction on the rate of increase of the number of candidate models in $\mathcal{M}_s$ $$\label{eq:M_n_order1} M_n=\left\{\begin{array}{ll} o(n^{\frac{1}{2\alpha_1+1}} ) &\quad 1/2<\alpha_1<1, \\ o(n^{\frac{1}{4\alpha_1-1}} ) &\quad \alpha_1 \geq 1. \end{array}\right.$$ Therefore we need $M_n=c_n(m_n^*)^{2\alpha_1/(2\alpha_1+1)}$ with $c_n \to 0$ as $n \to \infty$, where $m_n^*\sim(n/\sigma^2)^{1/(2\alpha_1)}$. In this case, the optimal rate of convergence of MS is $R_n\left(m_n^*,\mathbf{f}\right)\asymp n^{-1+1/(2\alpha_1)}$. But the rate of convergence of MA based $\mathcal{M}_s$ is $M_n^{-2\alpha_1+1}$, which converges no faster than $n^{-(2\alpha_1-1)/(2\alpha_1+1)}$ and thus much slower than MS. For a specific example, if $\alpha_1=1$, the MMA converges slower than $n^{-1/3}$ in contrast to the rate $n^{-1/2}$ for MS.* **Example 2** (Exponentially decaying coefficients). *Now the transformed coefficients decay fast: $\theta_j=\exp(-cj^{\alpha_2})$, $\alpha_2>0$. A sufficient condition for ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) is $M_n<\left(1/2 \right)^{1/\alpha_2}m_n^*$, where $m_n^{\ast}\sim [\log(n/\sigma^2)^{1/(2c)}]^{1/\alpha_2}$. In this case, MA based on $\mathcal{M}_s$ converges at the rate of $M_n^{1-\alpha_2}/n^{1/2}$, which is still slower than the optimal MS rate $m_n^*/n$.* In both representative examples, an undesired consequence of reducing the candidate model set to $\mathcal{M}_s$ with ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) is that the optimal single model $m_n^*$ is excluded, and the resulting MA estimators converge more slowly than MS. In more general cases of coefficients, the implications of the condition ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) on $M_n$ and $R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f})$ are summarized in the following proposition. **Proposition 3**. *Suppose Assumptions [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"}--[Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"} are satisfied. Under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}, a necessary condition of ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) is $M_n=o\left(m_n^* \right)$. In such a case, we have $$\label{eq:prop3} R_n(m_n^*,\mathbf{f})=o[R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f})].$$ Under Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"}, for ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) to hold, it is also necessary to require $M_n\leq \lfloor C m_n^* \rfloor$ with a constant $0<C<1$. In this case, ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) still leads to the relation ([\[eq:prop3\]](#eq:prop3){reference-type="ref" reference="eq:prop3"}).* Proposition [Proposition 3](#lemma:delta_2){reference-type="ref" reference="lemma:delta_2"} confirms that the widely used condition ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) excludes even the optimal single model $m_n^*$. When $\theta_l$ decays slowly, MA based on the restrictive candidate model set has a significant disadvantage compared to MS in terms of rate of convergence, which is against the motivation of MA. When $\theta_l$ decays fast, MS uncertainty is relatively low, and MA generally does not have any real benefit compared to MS. The restricted MMA with ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}), however is actually worse. Comparing the two restricted-AOP theories given by [@Hansen2007] and [@Wan2010], it seems that MA with the discrete weight set is safer since it always leads to the optimal MS rate when $M_n\gtrsim m_n^*$, while MA based on the restrictive candidate set does not. Nevertheless, both theories have the same drawback of not achieving the MA's full potential. Therefore, the first question we raised remains largely unanswered. The next section sheds some new light on this matter. Both non-asymptotic and asymptotic results will be given. **Remark 1**. *Note that a recent work of [@Zhang2021] proved the MMA's AOP under a milder and more interpretable assumption $$\label{eq:condition_zhang} \frac{|\mathcal{M}|^2}{nR_n\left(\mathbf{w}^*|\mathcal{M},\mathbf{f}\right)}\to 0$$ than ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}). Following the proof in Proposition [Proposition 3](#lemma:delta_2){reference-type="ref" reference="lemma:delta_2"}, we can see that ([\[eq:condition_zhang\]](#eq:condition_zhang){reference-type="ref" reference="eq:condition_zhang"}) still fails to include $m_n^*$ and thus suffers the same consequence ([\[eq:prop3\]](#eq:prop3){reference-type="ref" reference="eq:prop3"}).* # Main results {#sec:main_results} ## A risk bound {#sec:main_results:non-asymptotic} We start with non-asymptotic results. Recall that $Q_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$ is any one of two quantities: $\mathbb{E}L_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$ and $\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$. Given a general nested candidate model set $\mathcal{M}=\{k_1,k_2,\ldots,k_{M_n} \}$, define $$\label{eq:psi_M} \psi(\mathcal{M}) = \left(1+ \sum_{j=1}^{M_n-1}\frac{k_{j+1} - k_{j}}{k_j}\right)(1+\log M_n)^2.$$ Then we have the following upper bound on $Q_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$. **Theorem 1**. *Suppose that Assumption [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"} holds, then we have $$\label{eq:risk_bound_general} \begin{split} Q_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})\leq R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})&+\frac{C\sigma^2}{n}\psi(\mathcal{M})+\frac{C\sigma}{\sqrt{n}}[\psi(\mathcal{M})]^{\frac{1}{2}}[R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}\\ &+C\rho\left(n, \mathcal{M},\mathbf{f},\widehat{\sigma}^2,\sigma^2\right),\\ \end{split}$$ where $C$ is some universal constant, and $\rho\left(n, \mathcal{M},\mathbf{f},\widehat{\sigma}^2,\sigma^2\right)$ is the estimation error related to $\widehat{\sigma}^2$, which is defined by $$\begin{split} \rho\left(n, \mathcal{M},\mathbf{f},\widehat{\sigma}^2,\sigma^2\right) & = \frac{k_{M_n}}{n\sigma^2}\mathbb{E}\left( \widehat{\sigma}^2-\sigma^2 \right)^2+ \left[ \frac{k_{M_n}}{n\sigma^2}\mathbb{E}\left( \widehat{\sigma}^2-\sigma^2 \right)^2 \right]^{\frac{1}{2}}[R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}. \end{split}$$* The risk bound ([\[eq:risk_bound_general\]](#eq:risk_bound_general){reference-type="ref" reference="eq:risk_bound_general"}) is valid for any sample size and does not rely on Assumption [Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"} that the transformed coefficients are ordered. Note that the risk of the MMA estimator $Q_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$ is bounded by the infeasible optimal MA risk $R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})$ plus three additional terms. The first two terms are related to the candidate model set $\mathcal{M}$. The third term $\rho\left(n, \mathcal{M},\mathbf{f},\widehat{\sigma}^2,\sigma^2\right)$ is mainly about the estimation error of $\widehat{\sigma}^2$. As the risk bound suggests, the variance estimation may also have a significant effect on the performance of MMA. When a poor estimator of $\sigma^2$ with non-converging squared risk is considered, the upper bound in ([\[eq:risk_bound_general\]](#eq:risk_bound_general){reference-type="ref" reference="eq:risk_bound_general"}) becomes non-converging if the largest size $k_{M_n}$ is of order $n$. In contrast, when $\mathbb{E}\left( \widehat{\sigma}^2-\sigma^2 \right)^2$ converges at the parametric rate $1/n$, the term $\rho\left(n, \mathcal{M},\mathbf{f},\widehat{\sigma}^2,\sigma^2\right)$ does not affect the rate of convergence of the upper bound. ## Estimation of $\sigma^2$ {#sec:estimator_sigma} Here we present two variance estimators that prove useful under different situations. Consider a model-based estimator from the least squares theory $$\label{eq:sigma_estimator_1} \widehat{\sigma}_{m_n}^2=\frac{1}{n-m_n}\| \mathbf{y}- \widehat{\mathbf{f}}_{m_n} \|^2,$$ where $\widehat{\mathbf{f}}_{m_n}=\mathbf{X}_{m_n}(\mathbf{X}_{m_n}^{\top}\mathbf{X}_{m_n})^{-1}\mathbf{X}_{m_n}^{\top}\mathbf{y}$ is the least squares estimator involving the first $m_n$ regressors. With an elementary calculation, we have $$\label{eq:variance_risk} \mathbb{E}(\widehat{\sigma}_{m_n}^2-\sigma^2)^2\lesssim \frac{1}{n-m_n}\vee\frac{n\|\boldsymbol{\theta}_{-m_n} \|^2}{(n-m_n)^2} \vee \frac{n^2 \|\boldsymbol{\theta}_{-m_n} \|^4}{(n-m_n)^2},$$ where $\boldsymbol{\theta}_{-m_n}=(\theta_{m_n+1},\ldots, \theta_{p_n})^{\top}$. When $n-p_n \asymp n$, the variance estimator $\widehat{\sigma}_{p_n}^2$ based on the largest candidate model converges at the parametric rate $1/n$. When $p_n = n$, the estimation error of $\widehat{\sigma}_{m_n}^2$ with $m_n = \lfloor kn \rfloor$ ($0<k<1$) is not slower than $(1/n)\vee \|\boldsymbol{\theta}_{-m_n} \|^4$. As will be seen in the next subsection, $\widehat{\sigma}_{m_n}^2$ may be sufficient for the AOP of MMA (e.g., in the examples of polynomially and exponentially decaying coefficients), even if it does not converge at the parametric rate in some cases. Moreover, when $p_n=n$, the first difference variance estimator proposed by [@rice1984bandwidth] can also be used. For the one-dimensional nonparametric regression $y_i=f(u_i)+\epsilon_i$, where the model ([\[eq:model\]](#eq:model){reference-type="ref" reference="eq:model"}) is a linear approximation for $f$, consider $$\widehat{\sigma}_{D}^2=\frac{1}{2(n-1)}\sum_{i=2}^{n}\left[y_{(i+1)}-y_{(i)}\right]^2,$$ where $y_{(i)}$ denotes the observed response at the $i$-th smallest $u$ value. Under a mild smoothness assumption on $f$, $\widehat{\sigma}_{D}^2$ has the property $\mathbb{E}(\widehat{\sigma}_D^2 -\sigma^2)^2 \sim cn^{-1}\mbox{\mbox{Var}}(\epsilon^2)$. This estimator extends to design points in a multidimensional case [@Munk2005]. ## AOP {#subsec:aop} With a suitable estimator $\widehat{\sigma}^2$, the AOP of MMA is readily available as shown in the following theorem. **Theorem 2**. *Suppose Assumption [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"} holds. As $n \to \infty$, if $$\label{eq:variance_rate} k_{M_n}\mathbb{E}\left( \widehat{\sigma}^2-\sigma^2 \right)^2=o\left[nR_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})\right]$$ and $$\label{eq:minimum_marisk_rate} \psi(\mathcal{M})=o\left[nR_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})\right],$$ then $\widehat{\mathbf{w}}|\mathcal{M}$ is AOP in the sense that ([\[eq:opt1\]](#eq:opt1){reference-type="ref" reference="eq:opt1"}) holds for the continuous weight set $\mathcal{W}_{|\mathcal{M}|}$.* *In particular, using the estimator ([\[eq:sigma_estimator_1\]](#eq:sigma_estimator_1){reference-type="ref" reference="eq:sigma_estimator_1"}) with $m_n = \lfloor kn \rfloor \wedge p_n$ ($0<k<1$), if Assumptions [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"}--[Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"}, $$\label{eq:variance_rate_2} (1/n)\vee \|\boldsymbol{\theta}_{-m_n} \|^4=o\left(\frac{m_n^*}{n}\right),$$ and $$\label{eq:minimum_marisk_rate_2} (\log p_n)^3=o\left(m_n^*\right)$$ hold, then $\widehat{\mathbf{w}}|\mathcal{M}_a$ achieves the full AOP in terms of ([\[eq:fullopt\]](#eq:fullopt){reference-type="ref" reference="eq:fullopt"}).* Theorem [Theorem 2](#theorem:aop){reference-type="ref" reference="theorem:aop"} establishes the MMA's AOP for the general nested model set $\mathcal{M}$ and weight set $\mathcal{W}_{|\mathcal{M}|}$ with variance estimation. Compared with the restricted-AOP theory in [@Hansen2007], our result does not restrict the model weights to the discrete set $\mathcal{W}_{|\mathcal{M}|}(N)$. As demonstrated in Proposition [Proposition 1](#lemma:delta_1){reference-type="ref" reference="lemma:delta_1"}, relaxing the model weights from $\mathcal{W}_{|\mathcal{M}|}(N)$ to $\mathcal{W}_{|\mathcal{M}|}$ improves (substantially in various situations) the optimal MA risk. Second, the condition ([\[eq:minimum_marisk_rate\]](#eq:minimum_marisk_rate){reference-type="ref" reference="eq:minimum_marisk_rate"}) in Theorem [Theorem 2](#theorem:aop){reference-type="ref" reference="theorem:aop"} significantly improves the condition ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) in [@Wan2010] by allowing more helpful candidate models to be combined. In fact, Theorem [Theorem 2](#theorem:aop){reference-type="ref" reference="theorem:aop"} permits the use of the largest candidate model set $\mathcal{M}_a$, which answers the question Q1 raised in Section [2.2](#sec:setup2){reference-type="ref" reference="sec:setup2"} that MMA can achieve the full AOP by combining all models in $\mathcal{M}_a$ without additional restriction on the weight set. The conditions ([\[eq:variance_rate_2\]](#eq:variance_rate_2){reference-type="ref" reference="eq:variance_rate_2"})--([\[eq:minimum_marisk_rate_2\]](#eq:minimum_marisk_rate_2){reference-type="ref" reference="eq:minimum_marisk_rate_2"}) are two specific forms of ([\[eq:variance_rate\]](#eq:variance_rate){reference-type="ref" reference="eq:variance_rate"})--([\[eq:minimum_marisk_rate\]](#eq:minimum_marisk_rate){reference-type="ref" reference="eq:minimum_marisk_rate"}) when $\widehat{\sigma}^2=\widehat{\sigma}_{m_n}^2$ and $\mathcal{M}=\mathcal{M}_a$. Note that a prerequisite for ([\[eq:variance_rate_2\]](#eq:variance_rate_2){reference-type="ref" reference="eq:variance_rate_2"})--([\[eq:minimum_marisk_rate_2\]](#eq:minimum_marisk_rate_2){reference-type="ref" reference="eq:minimum_marisk_rate_2"}) is $$\label{eq:prerequisite} m_n^*\to \infty,$$ which is required in [@Hansen2007] for MA and [@Li1987] for MS. This condition means that there are no candidate models with fixed dimensions for which the approximation error is zero. When $p_n=n$ and $m_n = \lfloor kn \rfloor$, $0<k<1$, the condition ([\[eq:variance_rate_2\]](#eq:variance_rate_2){reference-type="ref" reference="eq:variance_rate_2"}) is satisfied in Examples 1--2, and ([\[eq:minimum_marisk_rate_2\]](#eq:minimum_marisk_rate_2){reference-type="ref" reference="eq:minimum_marisk_rate_2"}) may impose an additional condition on the situation to apply the largest candidate model set $\mathcal{M}_a$, as seen below. **Example 3** (continued). *The transformed coefficients are $\theta_j=j^{-\alpha_1}$, $\alpha_1>1/2$. In this case, we have $m_n^*\asymp n^{1/(2\alpha_1)}$. When $m_n = \lfloor kn \rfloor$, $0<k<1$, we obtain $\|\boldsymbol{\theta}_{-m_n} \|^4 =O( 1/n^{4\alpha_1-2})$ and $[(1/n)\vee (1/n^{4\alpha_1-2})]=o(m_n^*/n)$, which implies ([\[eq:variance_rate_2\]](#eq:variance_rate_2){reference-type="ref" reference="eq:variance_rate_2"}). And note that $(\log n)^3/n^{1/(2\alpha_1)}\to 0$. Thus the condition ([\[eq:minimum_marisk_rate_2\]](#eq:minimum_marisk_rate_2){reference-type="ref" reference="eq:minimum_marisk_rate_2"}) is also satisfied.* **Example 4** (continued). *When the coefficients decay as $\theta_j=\exp(-j^{\alpha_2})$, $\alpha_2>0$, we have $m_n^*\asymp (\log n)^{1/\alpha_2}$. In this case, the condition ([\[eq:variance_rate_2\]](#eq:variance_rate_2){reference-type="ref" reference="eq:variance_rate_2"}) is satisfied by observing $\|\boldsymbol{\theta}_{-m_n} \|^4 =O[\exp(-2m_n^{\alpha_2})]=o(1/n)$ when $m_n = \lfloor kn \rfloor$, $0<k<1$. Furthermore, if $0<\alpha_2<1/3$, the condition ([\[eq:minimum_marisk_rate_2\]](#eq:minimum_marisk_rate_2){reference-type="ref" reference="eq:minimum_marisk_rate_2"}) is also satisfied due to $(\log n)^3/(\log n)^{1/\alpha_2}\to 0$.* Based on the above analysis, we observe that with the optimal single model $m_n^*$ being included in $\mathcal{M}_a$ in both examples, the full AOP is achieved for the MMA estimator based on $\mathcal{M}_a$ when the coefficients do not decay too fast, which much strengthens the restricted-AOP theories established by [@Hansen2007] and [@Wan2010]. # Construction of candidate model set {#sec:reduced} This section proposes two types of reduced candidate model sets, on which the MMA estimators achieve the full AOP on broader parameter regions than that based on the largest candidate set $\mathcal{M}_a$. ## Candidate model set with grouped regressors {#subsec:grouped} Instead of combining the candidate models with successively increasing sizes, we consider a smaller set $\mathcal{M}_g=\{k_1,k_2,\ldots,k_{M_n}\}$, where the size of each candidate model is group-wise added. Define $k_0=0$. **Theorem 3**. *Let $k_{M_n}=p_n$ and $$\max_{1\leq j \leq M_n-1}\frac{k_{j+1}-k_j}{k_j-k_{j-1}}\leq 1+\zeta_n,$$ where $\zeta_n \geq 0$. Suppose Assumptions [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"}--[Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"}, $k_1=o(m_n^*)$, and $\zeta_n =o(1)$ hold, and the conditions ([\[eq:variance_rate\]](#eq:variance_rate){reference-type="ref" reference="eq:variance_rate"})--([\[eq:minimum_marisk_rate\]](#eq:minimum_marisk_rate){reference-type="ref" reference="eq:minimum_marisk_rate"}) are satisfied for $\mathcal{M}_g$, then we have $$Q_n\left(\widehat{\mathbf{w}}|\mathcal{M}_g,\mathbf{f}\right)=[1+o(1)]R_n\left(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f}\right).$$* Theorem [Theorem 3](#cor:grouped){reference-type="ref" reference="cor:grouped"} indicates that rather than combining all nested models, the MMA estimator based on $\mathcal{M}_g$ still achieves the optimal risk $R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})$ asymptotically when the sizes of the candidate models are appropriately selected. The similar strategies of constructing group-wise estimators have been widely used in various nonparametric estimation problems [@cavalier2001penalized; @Cavalier2002Sharp; @Rigollet2006blockwise]; See Section 6 of [@Dalalyan2012] for its application in the context of model aggregation. The conditions in Theorem [Theorem 3](#cor:grouped){reference-type="ref" reference="cor:grouped"} are quite mild. First, notice that the condition ([\[eq:variance_rate\]](#eq:variance_rate){reference-type="ref" reference="eq:variance_rate"}) is satisfied for $\mathcal{M}_g$ with the variance estimators given in Section [4.2](#sec:estimator_sigma){reference-type="ref" reference="sec:estimator_sigma"}. Next, we provide some specific choices of $k_1$ and $\zeta_n$ that satisfy the remaining assumptions in Theorem [Theorem 3](#cor:grouped){reference-type="ref" reference="cor:grouped"}. To keep in line with the analysis in Section [4.3](#subsec:aop){reference-type="ref" reference="subsec:aop"}, we here focus on the case $p_n = n$. ### Equal size groups Consider $\mathcal{M}_{g1}$ with $\zeta_n = 0$, $k_1 = \lceil(\log n)^{t}\rceil$, $k_m = mk_1$ for $m=2,\ldots,M_n-1$, and $k_{M_n}=p_n$, where $0<t<3$ and $M_n= \arg\min_{m \in \mathbb{N}} k_m \geq p_n$. We have $$\psi(\mathcal{M}_{g1}) \asymp (\log M_n)^3 \asymp (\log n - t \log\log n)^3.$$ Now we verify the conditions $k_1=o(m_n^*)$ and ([\[eq:minimum_marisk_rate\]](#eq:minimum_marisk_rate){reference-type="ref" reference="eq:minimum_marisk_rate"}) in the following examples. **Example 5** (continued). *Note that $k_1/n^{1/(2\alpha_1)} \to 0$ and $\psi(\mathcal{M}_{g1})/n^{1/(2\alpha_1)} \to 0$. Thus the MMA estimator based on $\mathcal{M}_{g1}$ still attains the full AOP in the case of polynomially decaying coefficients.* **Example 6** (continued). *Since $k_1/(\log n)^{1/\alpha_2} \to 0$ and $\psi(\mathcal{M}_{g1})/(\log n)^{1/\alpha_2} \to 0$ when $0<\alpha_2\leq1/3$. Therefore, $\mathcal{M}_{g1}$ improves $\mathcal{M}_{a}$ a little by achieving the full AOP of MMA when $\theta_j=\exp(-j^{\alpha_2})$, $0<\alpha_2\leq1/3$.* ### Increasing size groups In this subsection, we construct the groups in the same spirit as the weakly geometrically increasing blocks in [@cavalier2001penalized]. For two constants $t_1>0$ and $t_2>0$, define $\zeta_n = t_1/(\log n)^{t_2}$. Consider $\mathcal{M}_{g2}$ with $k_1 = \lceil \zeta_{n}^{-1} \rceil$, $k_m =k_{m-1} +\lfloor k_1(1+\zeta_{n})^{m-1}\rfloor$ for $m=2,\ldots,M_n-1$, and $k_{M_n}=p_n$, where $$M_n = \arg\min_{m \in \mathbb{N}} \left(k_1 + \sum_{j=2}^{m}\lfloor k_1(1+\zeta_{n})^{j-1}\rfloor\right) \geq p_n.$$ When $p_n= n$, the result in [@cavalier2001penalized] shows that $M_n \lesssim (\log n)^{t_2+1}$. Thus we have $$\psi(\mathcal{M}_{g2}) \asymp \zeta_n M_n (\log M_n)^2 \lesssim (\log n) (\log\log n)^2.$$ **Example 7** (continued). *Since $k_1/n^{1/(2\alpha_1)} \to 0$ and $\psi(\mathcal{M}_{g2})/n^{1/(2\alpha_1)} \to 0$, the MMA estimator based on $\mathcal{M}_{g2}$ attains the same full-AOP property as those based on $\mathcal{M}_{a}$ and $\mathcal{M}_{g1}$.* **Example 8** (continued). *Set $t_2 = 1$. When $\theta_j=\exp(-j^{\alpha_2})$, $0<\alpha_2< 1$, and $m_n^* \asymp (\log n)^{1/\alpha_2}$, note that $k_1/m_n^* \to 0$ and $\psi(\mathcal{M}_{g2})/m_n^* \to 0$. Thus the MMA estimator with $\mathcal{M}_{g2}$ achieves the full AOP on a broader parameter region compared to those based on $\mathcal{M}_{a}$ and $\mathcal{M}_{g1}$.* ## Candidate model set based on MS {#subsec:minimum} Another approach is to combine a smaller number of candidate models with the size centering on $m_n^*$. Since $m_n^*$ is unknown in practice, we estimate it by some MS method and then consider the candidate model set $\widehat{\mathcal{M}}_{MS}=\widehat{\mathcal{M}}_{MS}(k_l, k_u)=\{ \widehat{l}_n,\ldots,\widehat{m}_n,\ldots,\widehat{u}_n \}$, where $\widehat{l}_n=1\vee\left\lfloor k_l^{-1}\widehat{m}_n\right\rfloor$, $\widehat{u}_n= p_n\wedge \left\lfloor k_u\widehat{m}_n\right\rfloor$, $k_l>1$, and $k_u>1$. To get asymptotic properties of $\widehat{\mathcal{M}}_{MS}$, we need another assumption on transformed coefficients, which is naturally satisfied for both polynomially and exponentially decaying coefficients. **Assumption 3**. *The transformed coefficients satisfy $\lim_{k \to \infty}\left|\theta_{\lfloor kl\rfloor}/\theta_l\right| \to 0$ for any $l \in \mathbb{N}$.* Define $c_1$ and $c_2$ two constants with $0<c_1<1<c_2$. Let $F_n$ denote the event $\left\lfloor c_1m_n^*\right\rfloor\leq\widehat{m}_n\leq \left\lfloor c_2m_n^*\right\rfloor$ and $\bar{F}_n$ be its complement. ### Increasing $k_l$ and $k_u$ Consider a candidate model set $\widehat{\mathcal{M}}_{MS1}=\widehat{\mathcal{M}}_{MS}$ with $k_l \to \infty$ and $k_u \to \infty$. **Theorem 4**. *Suppose that Assumptions [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"}--[Assumption 3](#asmp:regressor_order2){reference-type="ref" reference="asmp:regressor_order2"} hold. If the condition ([\[eq:variance_rate\]](#eq:variance_rate){reference-type="ref" reference="eq:variance_rate"}) is satisfied for $\mathcal{M}_a$, $$\label{eq:Ephi} \mathbb{E}\psi(\widehat{\mathcal{M}}_{MS1}) = o(m_n^*),$$ and the event $F_n$ satisfies $$\label{eq:pG} \mathbb{P}(\bar{F}_n)= o\left(\frac{m_n^*}{n}\right),$$ then the equation ([\[eq:opt3\]](#eq:opt3){reference-type="ref" reference="eq:opt3"}) holds for $\widehat{\mathcal{M}}_{MS1}$.* Theorem [Theorem 4](#cor:minimum_1){reference-type="ref" reference="cor:minimum_1"} states that MMA achieves the full AOP in terms of ([\[eq:opt3\]](#eq:opt3){reference-type="ref" reference="eq:opt3"}) with the estimated candidate model set $\widehat{\mathcal{M}}_{MS1}$ under certain regularity conditions. Observe that the condition ([\[eq:Ephi\]](#eq:Ephi){reference-type="ref" reference="eq:Ephi"}) is quite mild. Based on the definition of ([\[eq:psi_M\]](#eq:psi_M){reference-type="ref" reference="eq:psi_M"}), we have $$\begin{split} \mathbb{E}\psi(\widehat{\mathcal{M}}_{MS1})& \asymp \mathbb{E}\log(k_lk_u)\left\{\log\left[(k_u - k_l^{-1})\widehat{m}_n\right]\right\}^2\\ &\leq \log(k_lk_u)\left\{\log\left[(k_u - k_l^{-1})\mathbb{E}\widehat{m}_n\right]\right\}^2\\ & \lesssim (\log k_l + \log k_u) \left[ \log(k_u - k_l^{-1}) + \log m_n^* \right]^2, \end{split}$$ where the first inequality follows from Jensen's inequality, and the second inequality is due to $$\mathbb{E}\widehat{m}_n=\mathbb{E}(\widehat{m}_n1_{F_n})+\mathbb{E}(\widehat{m}_n1_{\bar{F}_n})\lesssim c_2m_n^*+n\cdot\frac{m_n^*}{n}\lesssim m_n^*.$$ If we set $k_l=k_u=\log n$, then a sufficient condition for ([\[eq:Ephi\]](#eq:Ephi){reference-type="ref" reference="eq:Ephi"}) is $(\log\log n)[\log\log n + \log m_n^*]^2 = o\left(m_n^*\right)$, which holds in Examples 1--2. Then we will see that the condition ([\[eq:pG\]](#eq:pG){reference-type="ref" reference="eq:pG"}) is satisfied when Mallows' $C_p$ MS criterion [@Mallows1973] is adopted. Suppose $\sigma^2$ is known, from [@Kneip1994], we obtain $$\label{eq:kneip} \mathbb{P}\left(\left| R_n(\widehat{m}_n,\mathbf{f})-R_n(m_n^*,\mathbf{f}) \right|> n^{-1}[x^2\vee x(m_n^*)^{1/2}] \right)\leq C_1\exp(-C_2x)\quad \mbox{for}\,x\geq 0,$$ where $C_1$ and $C_2$ are two constants that depend only on $\sigma^2$. Combining ([\[eq:kneip\]](#eq:kneip){reference-type="ref" reference="eq:kneip"}) with the fact $\varpi_n\triangleq[R_n(c_1m_n^*,\mathbf{f})-R_n(m_n^*,\mathbf{f})]\wedge[R_n(c_2m_n^*,\mathbf{f})-R_n(m_n^*,\mathbf{f})]\gtrsim m_n^*/n$ under Conditions [Condition 1](#condition1){reference-type="ref" reference="condition1"}--[Condition 2](#condition2){reference-type="ref" reference="condition2"}, we see $$\label{eq:relate} \begin{split} \mathbb{P}\left(\bar{F}_n\right)& \leq \mathbb{P}\left(\left|R_n(\widehat{m}_n,\mathbf{f})-R_n(m_n^*,\mathbf{f})\right|>\varpi_n\right)\lesssim \exp{\left[-C(m_n^*)^{\frac{1}{2}}\right]}, \end{split}$$ where $C$ is a fixed constant. To connect ([\[eq:relate\]](#eq:relate){reference-type="ref" reference="eq:relate"}) with the condition ([\[eq:pG\]](#eq:pG){reference-type="ref" reference="eq:pG"}), consider the following two examples. **Example 9** (continued). *When $\theta_j=j^{-\alpha_1}$, $\alpha_1>1/2$, and $m_n^* \asymp n^{1/(2\alpha_1)}$, we have $\exp{[-C(m_n^*)^{1/2}]}=o\left(m_n^*/n\right)$ for any fixed $C$, which meets the condition ([\[eq:pG\]](#eq:pG){reference-type="ref" reference="eq:pG"}).* **Example 10** (continued). *When $\theta_j=\exp(-j^{\alpha_2})$, $0<\alpha_2< 1/2$, and $m_n^* \asymp (\log n)^{1/\alpha_2}$, note that $\exp{[-C(m_n^*)^{1/2}]} = 1/[n^{C(\log n)^{1/(2\alpha_2)-1}}] = o(m_n^*/n)$ for any constant $C$. It also verifies ([\[eq:pG\]](#eq:pG){reference-type="ref" reference="eq:pG"}).* The above analysis implies that the MMA estimator with $\widehat{\mathcal{M}}_{MS1}$ retains the full AOP as that based on $\mathcal{M}_a$ when the transformed coefficients $\boldsymbol{\theta}$ decay slowly. It also expands the region for the full AOP of $\mathcal{M}_a$ when the coefficients decay fast. ### Bounded $k_l$ and $k_u$ Let $\widehat{\mathcal{M}}_{MS2}=\widehat{\mathcal{M}}_{MS}$ with $k_l\vee k_u$ being upper bounded by some positive constant $C$. **Theorem 5**. *Suppose that Assumptions [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"}--[Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"} hold. Under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}, if there exists a constant $0<C_1<1$ such that $\mathbb{P}(F_n)\geq C_1$, then we have $$\label{eq:corpart1} \mathbb{E}R_n(\widehat{\mathbf{w}}|\widehat{\mathcal{M}}_{MS2},\mathbf{f})-R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f}) \gtrsim R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f}).$$* *Under Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"}, if $m_n^*\to \infty$, $\mathbb{E}R_n(\widehat{m}_n,\mathbf{f})/R_n(m_n^*,\mathbf{f}) \to 1$, and there exists a constant $C_2\geq 1$ such that $\widehat{u}_n-\widehat{l}_n \leq C_2$ almost surely, then we get ([\[eq:opt3\]](#eq:opt3){reference-type="ref" reference="eq:opt3"}) for $\widehat{\mathcal{M}}_{MS2}$.* This theorem states that when the coefficients decay slowly, such as in the case $\theta_j=j^{-\alpha_1},\alpha_1>1/2$, the MMA estimator based on a restricted $\widehat{\mathcal{M}}_{MS2}$ cannot achieve the full potential of MA. However, when the coefficients decay fast, reducing the number of candidate models around $\widehat{m}_n$ to a constant level is beneficial for MMA. Indeed, Theorem [Theorem 5](#cor:minimum_2){reference-type="ref" reference="cor:minimum_2"} states that MMA based on $\widehat{\mathcal{M}}_{MS2}$ with bounded $\widehat{u}_n-\widehat{l}_n$ achieves the optimal MA risk when $\theta_j=\exp(-j^{\alpha_2})$, $0<\alpha_2<\infty$. Nevertheless, requiring $k_l$ and $k_u$ to increase to $\infty$ is still necessary for the full AOP in the case of polynomially decaying coefficients. Table [\[tab:method\]](#tab:method){reference-type="ref" reference="tab:method"} summarizes the available MMA strategies discussed in Sections [3](#sec:review){reference-type="ref" reference="sec:review"}--[5](#sec:reduced){reference-type="ref" reference="sec:reduced"}. We emphasize that the parameter regions given in the last two columns are the known sufficient conditions for the full AOP of MMA. Whether these methods achieve the full AOP in larger regions remains open. More comparisons are available through simulations in the Appendix. [\[tab:method\]]{#tab:method label="tab:method"} # Minimax adaptivity {#sec:minimax} To the best of our knowledge, minimax properties have not been established on MMA, although some minimax results have been obtained for very different MA methods [see, e.g., @Yang2001; @Yang2004; @Leung2006; @Dalalyan2012; @Bellec2018]. The purpose of this section is to fill in this gap for MMA. For simplicity, in this section we assume $p_n=n$ and $\epsilon_1, \ldots, \epsilon_n$ are i.i.d. $N(0,\sigma^2)$. We investigate the exact minimax adaptivity (defined in Definition [Definition 3](#def:minimax){reference-type="ref" reference="def:minimax"}) of the MMA estimator based on $\mathcal{M}_a=\{1,\ldots,n \}$ when the transformed coefficient $\boldsymbol{\theta}$ belongs to two types of classes, respectively. The first class is the ellipsoid $$\label{eq:ellipsoid} \Theta(\alpha, R)=\left\{ \boldsymbol{\theta}\in \mathbb{R}^{n}:\sum_{j=1}^{n}j^{2\alpha}\theta_j^2\leq R \right\},$$ where $\alpha>0$ and $R>0$. Let $\mathcal{F}_{\Theta(\alpha, R)}=\{\mathbf{f}=\sum_{j=1}^{n}\theta_j\boldsymbol{\phi}_j:\boldsymbol{\theta}\in\Theta(\alpha, R)\}$ denote the class of regression mean vector associated with $\Theta(\alpha, R)$. Another is the hyperrectangle $$\label{eq:hyperrectangle} \Theta^{H}(c,q)=\left\{\boldsymbol{\theta}\in \mathbb{R}^{n}: |\theta_j|\leq cj^{-q},j=1,\ldots,n \right\},$$ where $c>0$ and $q>1/2$. And let $\mathcal{F}_{\Theta^{H}(c,q)}$ be the corresponding mean vector class of $\Theta^{H}(c,q)$. **Theorem 6**. *Suppose $\widehat{\sigma}_D^2$ or $\widehat{\sigma}_{m_n}^2$ with $m_n = \lfloor kn \rfloor$ $(0<k<1)$ is adopted. Then the MMA estimator $\widehat{\mathbf{f}}_{\widehat{\mathbf{w}}|\mathcal{M}_a}$ is adaptive in the exact minimax sense on the family of the ellipsoids $\boldsymbol{\mathcal{F}}=\left\{\mathcal{F}_{\Theta(\alpha, R)},\alpha>0,R>0\right\}$, and it is adaptive in the exact linear-combined minimax sense on the family of the hyperrectangles $\boldsymbol{\mathcal{F}}^{H}=\left\{\mathcal{F}_{\Theta^{H}(c,q)},c>0,q>1/2\right\}$.* This theorem answers the question Q3 that the MMA estimator is minimax optimal in the sense of Definition [Definition 3](#def:minimax){reference-type="ref" reference="def:minimax"} with the estimated $\sigma^2$. The detailed definitions of the variance estimators $\widehat{\sigma}_D^2$ and $\widehat{\sigma}_{m_n}^2$are given in Section [4.2](#sec:estimator_sigma){reference-type="ref" reference="sec:estimator_sigma"}. Note that $\widehat{\mathbf{f}}_{\widehat{\mathbf{w}}|\mathcal{M}_a}$ is a linear combination of candidate estimators in $\mathcal{M}_a$; thus, $\widehat{\mathbf{f}}_{\widehat{\mathbf{w}}|\mathcal{M}_a}$ is also adaptive in the exact linear-combined minimax sense on the family of the ellipsoids. However, based on Theorem 5 of [@Donoho1990], we deduce that the MMA estimator $\widehat{\mathbf{f}}_{\widehat{\mathbf{w}}|\mathcal{M}_a}$ is not adaptive in the exact minimax sense on the family of the hyperrectangles due to $R_L[\mathcal{F}_{\Theta^{H}(c,q)}]/R_M[\mathcal{F}_{\Theta^{H}(c,q)}]\to \rho, 1<\rho<\infty$. But it is still seen that $\widehat{\mathbf{f}}_{\widehat{\mathbf{w}}|\mathcal{M}_a}$ achieves minimax-rate optimality over all the estimators. # Simulation studies {#sec:simulation} Although the discrete weight set restriction ([\[eq:condition_discrete\]](#eq:condition_discrete){reference-type="ref" reference="eq:condition_discrete"}) and the candidate model set restriction ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) have been commonly used to develop the theoretical properties of MMA, they have rarely been examined numerically. This subsection examines the MMA estimators with these two restrictions relative to the unrestricted MMA. The data is simulated from the linear regression model ([\[eq:model\]](#eq:model){reference-type="ref" reference="eq:model"}), where $p_n=\lfloor 2n/3 \rfloor$, $x_{1i} = 1$, the remaining $x_{ji}$ are independently generated from $N(0,1)$, and the random error terms $\epsilon_i$ are i.i.d. from $N(0,\sigma^2)$ and are independent of ${x_{ji}}'$s. We consider two cases of the regression coefficients: - *Case 1* (Polynomially decaying coefficients). Here, $\beta_j=j^{-\alpha_1}$ and $\alpha_1$ is varied from $0.5$ to $1.5$. - *Case 2* (Exponentially decaying coefficients). Here, $\beta_j=\exp(-j^{\alpha_2})$ and $\alpha_2$ is varied from $0.25$ to $1.25$. The signal-to-noise ratio, which is defined by $\sum_{j=2}^{p_n}\beta_j^2/\sigma^2$, is set to be one via the parameter $\sigma^2$. And the sample size $n$ increases from $30$ to $1000$. The candidate models used to implement MA are nested and estimated by least squares. To highlight the issue of the weight/candidate model restriction, we assume that $\sigma^2$ is known for all methods. Let $\mathbf{f}=(f_1,\ldots,f_n)^{\top}$ denote the mean vector of the true regression function. The accuracy of an estimation procedure is evaluated in terms of the squared $\ell_2$ loss $n^{-1}\|\mathbf{f}- \widehat{\mathbf{f}} \|^2$, where $\widehat{\mathbf{f}}=(\widehat{f}_1,\ldots,\widehat{f}_n)^{\top}$ is the estimated mean vector. We replicate the data generation process $R=1000$ times to approximate the risks of the competing methods. The restricted-AOP MMA estimators considered are WR with $N=2$ (WR1), WR with $N=5$ (WR2), MR with $M_n = 2\vee\lfloor(m_n^*)^{1/2}\rfloor$ (MR1), and MR with $M_n = 2\vee\lfloor m_n^*/2\rfloor$ (MR2). Detailed definitions of these methods are given in Section [3](#sec:review){reference-type="ref" reference="sec:review"} and Table [\[tab:method\]](#tab:method){reference-type="ref" reference="tab:method"}. In each replication, we normalize the squared $\ell_2$ loss of these four methods by dividing the $\ell_2$ loss of the MMA estimator based on $\mathcal{M}_a$ and $\mathcal{W}_{p_n}$ (representing a full-AOP MMA method). [\[tab:simu2\]]{#tab:simu2 label="tab:simu2"} From Table [\[tab:simu2\]](#tab:simu2){reference-type="ref" reference="tab:simu2"}, the relative risks of the methods WR1 and WR2 are significantly larger than 1 in Case 1, which implies that using the discrete weight sets increases the risk of the full-AOP MMA by a sizable edge. This result is consistent with Proposition [Proposition 1](#lemma:delta_1){reference-type="ref" reference="lemma:delta_1"}. In Case 2, however, when $\alpha_2=1.25$ and $n=1000$, the relative risks of WR1 and WR2 are 0.898 (0.030) and 1.011 (0.023), respectively, which shows that WR methods perform better than and comparably to the MMA based on $\mathcal{W}_{p_n}$. This phenomenon is not surprising. Although Proposition [Proposition 1](#lemma:delta_1){reference-type="ref" reference="lemma:delta_1"} states that MA with the discrete weight restriction has an asymptotically equivalent oracle risk to that under the continuous weight set in Case 2, the latter actually pays a higher price to pursue the oracle MA risk when $n$ is finite, and the trade-off favors simplicity in this special case. We find that the MR1 and MR2 methods mostly have much larger relative risks than the WR methods in both cases. Moreover, their relative risks become increasingly greater as the sample size increases from 30 to 1000. These findings support our theoretical understandings in Section [3.2](#subsec:review_wan){reference-type="ref" reference="subsec:review_wan"}. Another interesting observation is about the result when $\alpha_2= 0.25$ in Case 2. Although the data is generated from a true regression model with exponentially decaying coefficients, this setting is more like a polynomially decaying case in the finite sample situation. Indeed, when $\alpha_1= 0.75$ and $n=1000$, we have $m_n^* \approx 75$. While in Case 2 with $\alpha_2= 0.25$ and $n=1000$, $m_n^*$ is around $77$, which does not exhibit the significant difference as in Case 1. Thus it is not surprising that the numerical performance of the competing methods in Case 2 ($\alpha_2= 0.25$) is similar to that in Case 1. More discussions related to this phenomenon can be found in [@Liu2011PI] and [@ZHANG201595]. In Section [11](#sec:a:simu){reference-type="ref" reference="sec:a:simu"} of the Appendix, we provide more simulation results to assess the full-AOP theory in Section [4](#sec:main_results){reference-type="ref" reference="sec:main_results"} and to compare the different candidate model sets given in Section [5](#sec:reduced){reference-type="ref" reference="sec:reduced"}. Overall, these results support our full-AOP theory on MMA and present evidence favoring the use of the candidate model sets with reduced sizes. # Discussion {#sec:conclusion} This paper focuses on the problem of combining a set of nested linear models by minimizing an MMA criterion. As a background, we first revisited two well-known AOP theories of MMA, which are based on the weight set restriction [@Hansen2007] and the candidate model set restriction [@Wan2010], respectively. We found that under these restrictions, MMA may not achieve its full potential, and it can perform much worse than MS. In this paper, inspired by the pioneering work of [@Hansen2007], [@Wan2010], and [@Zhang2021], we have addressed three key questions about the optimality of MMA: Can MMA achieve the performance of the optimal convex combination of all the nested models (i.e., the full-AOP property)? How to construct the candidate model set optimally? Is MMA adaptive in an exact minimax sense for some nonparametric classes? Correspondingly, our main contribution is threefold. First, a non-asymptotic risk bound of MMA is obtained under the sub-Gaussian assumption, which shows that when the optimal MA risk does not converge too fast, the full AOP can be achieved by minimizing the MMA criterion over the largest candidate model set. Second, two types of reduced candidate model sets are proposed, on which the full-AOP property of MMA can be realized and further improved in some aspects. Third, the MMA estimator is shown to be adaptive in the exact minimax sense over the family of ellipsoids. It is also proved to be adaptive in the exact linear-combined minimax sense on the family of hyperrectangles. To the best of our knowledge, it was previously unknown if MMA has any minimax property. In closing, we provide several directions for future research. The focus of this paper has been on a linear regression setup with nested models. It is of great interest to extend the theoretical framework to combining ordered linear smoothers [@Chernousova2013; @Bellec2020] and other non-nested models [@Wan2010; @Zhang2021]. Another extension, motivated by an observation from Table [\[tab:method\]](#tab:method){reference-type="ref" reference="tab:method"}, is to develop an MA method that can achieve the full AOP on the whole parameter region, if possible. Based on the works of [@ZHANG201595] and [@QIAN2022193], we conjecture that a universally full AOP may be established by properly using cross-validation or hypothesis testing. We leave these for future work. # Appendix {#appendix .unnumbered} Section [9](#sec:a:proof){reference-type="ref" reference="sec:a:proof"} contains the proofs of all the theorems, corollaries, and propositions in this paper. Section [10](#sec:a:aop_loss){reference-type="ref" reference="sec:a:aop_loss"} proves that MMA is asymptotically optimal (AOP) in terms of statistical loss. Section [11](#sec:a:simu){reference-type="ref" reference="sec:a:simu"} provides additional simulation results. And other related works are discussed in Section [12](#sec:a:related){reference-type="ref" reference="sec:a:related"}. # Proofs {#sec:a:proof} ## Notations In this appendix, we will use the symbols defined in Section [2.1](#sec:setup:1){reference-type="ref" reference="sec:setup:1"} of the main text. In addition, for any $n\times n$ real matrix $\mathbf{A}$, let $\|\mathbf{A}\|_2$ and $\|\mathbf{A}\|_{\mathrm{F}}$ denote the operator norm and the Frobenius norm of $\mathbf{A}$, respectively. ## Preliminaries {#sec:preliminaries} Define $\mathbf{P}_j=\mathbf{X}_j(\mathbf{X}_j^{\top}\mathbf{X}_j)^{-1}\mathbf{X}_j^{\top}$ the projection matrix based on the first $j$ columns of $\mathbf{X}$. Let $\mathbf{D}_j=\mathbf{P}_j - \mathbf{P}_{j-1},j=1,\ldots,p_n$, where $\mathbf{P}_0=\boldsymbol{0}_{n\times n}$. Note that $\mathbf{D}_j$ is a projection matrix, and $\mathbf{D}_j,j=1,\ldots,p_n$ are mutually orthogonal, i.e., $\mathbf{D}_j\mathbf{D}_{j'}=\mathbf{D}_{j'}\mathbf{D}_j=\mathbf{D}_j\delta_{jj'}$, where $\delta_{jj'}$ is the Kronecker delta. Using eigendecomposition, we have $\mathbf{D}_j=\boldsymbol{\phi}_j\boldsymbol{\phi}_j^{\top}$, where $\boldsymbol{\phi}_j\in \mathbb{R}^{n}$ satisfying $\|\boldsymbol{\phi}_j\|=1$. Due to the orthogonality of $\mathbf{D}_j,j=1,\ldots,p_n$, we see that $\{\boldsymbol{\phi}_1,\ldots,\boldsymbol{\phi}_{p_n} \}$ forms an orthonormal basis for the column space of $\mathbf{X}$. Thus, we can represent the model ([\[eq:model_matrix\]](#eq:model_matrix){reference-type="ref" reference="eq:model_matrix"}) as an equivalent sequence model $$\label{eq:sequence} \widehat{\theta}_{j} = \theta_j + e_j, \quad j=1,\ldots,p_n,$$ where $\widehat{\theta}_{j} =\boldsymbol{\phi}_j^{\top}\mathbf{y}/\sqrt{n}$, $\theta_j=\boldsymbol{\phi}_j^{\top}\mathbf{f}/\sqrt{n}$, and $e_j=\boldsymbol{\phi}_j^{\top}\boldsymbol{\epsilon}/\sqrt{n}$. Assume $\epsilon_1, \ldots, \epsilon_n$ are i.i.d. $\eta$-sub-Gaussian random variables. Note that $e_j,j=1,\ldots,p_n$ are $(\eta/\sqrt{n})$-sub-Gaussian variables, which satisfy $\mathbb{E}e_j=0$, $\mathbb{E}e_j^2=\sigma^2/n$, and $\mathbb{E}e_je_{j'}=0$ when $j \neq j'$. Based on the sequence model ([\[eq:sequence\]](#eq:sequence){reference-type="ref" reference="eq:sequence"}), the least squares estimator $\widehat{\mathbf{f}}_{m}$ has the following equivalent form $$\widehat{\mathbf{f}}_{m}=\mathbf{P}_m\mathbf{y}= \sum_{j=1}^{m}\mathbf{D}_j\mathbf{y}=\sum_{j=1}^{m}\boldsymbol{\phi}_j\boldsymbol{\phi}_j^{\top}\mathbf{y}=\sqrt{n}\sum_{j=1}^{m}\boldsymbol{\phi}_j\widehat{\theta}_{j}.$$ The $\ell_2$ risk of $\widehat{\mathbf{f}}_m$ is $$\label{eq:risk_m} \begin{split} R_n(m,\mathbf{f}) & =\frac{1}{n}\mathbb{E}\left\|\widehat{\mathbf{f}}_m-\mathbf{f}\right\|^2=\frac{1}{n}\mathbb{E}\left\|\sum_{j=1}^{m}\mathbf{D}_j\mathbf{y}-\sum_{j=1}^{p_n}\mathbf{D}_j\mathbf{f}\right\|^2\\ &=\mathbb{E} \left\| \sum_{j=1}^{m}\boldsymbol{\phi}_j\widehat{\theta}_{j} - \sum_{j=1}^{p_n}\boldsymbol{\phi}_j\theta_{j} \right\|^2 =\mathbb{E}\left\|\sum_{j=1}^{m}\boldsymbol{\phi}_je_j-\sum_{j=m+1}^{p_n}\boldsymbol{\phi}_j\theta_j \right\|^2\\ &=\frac{m\sigma^2}{n}+\sum_{j=m+1}^{p_n}\theta_j^2, \end{split}$$ where the last equality is due to the orthogonality of $\{\boldsymbol{\phi}_1,\ldots,\boldsymbol{\phi}_{p_n} \}$ and $\mathbb{E}e_j^2=\sigma^2/n$. Define $k_0=0$. The MA estimator based on $\mathcal{M}=\left\{k_1,k_2,\ldots, k_{M_n} \right\}$ is $$\label{eq:fw} \begin{split} \widehat{\mathbf{f}}_{\mathbf{w}|\mathcal{M}} & =\sum_{m=1}^{M_n}w_m\widehat{\mathbf{f}}_{k_m}=\sum_{m=1}^{M_n}w_m\left(\sqrt{n}\sum_{j=1}^{k_m}\boldsymbol{\phi}_j\widehat{\theta}_{j}\right) \\ & =\sqrt{n}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\boldsymbol{\phi}_l\widehat{\theta}_l, \end{split}$$ where $\gamma_j=\sum_{m=j}^{M_n}w_m$ is the cumulative weight. A similar calculation to ([\[eq:risk_m\]](#eq:risk_m){reference-type="ref" reference="eq:risk_m"}) yields the $\ell_2$ loss of $\widehat{\mathbf{f}}_{\mathbf{w}|\mathcal{M}}$ $$\label{eq:lossw} \begin{split} & L_n(\mathbf{w}|\mathcal{M},\mathbf{f})=\frac{1}{n}\left\|\widehat{\mathbf{f}}_{\mathbf{w}|\mathcal{M}}-\mathbf{f}\right\|^2 \\ & =\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\left( \gamma_j\widehat{\theta}_l-\theta_l \right)^2+\sum_{j=k_{M_n}+1}^{p_n}\theta_j^2 \end{split}$$ and the corresponding MA risk $$\label{eq:riskw} \begin{split} & R_n(\mathbf{w}|\mathcal{M},\mathbf{f})=\mathbb{E}L_n(\mathbf{w}|\mathcal{M},\mathbf{f}) \\ & =\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\left[ \left(1-\gamma_j \right)^2\theta_l^2+\frac{\sigma^2}{n}\gamma_j^2 \right]+\sum_{j=k_{M_n}+1}^{p_n}\theta_j^2. \end{split}$$ Furthermore, the MMA criterion ([\[eq:criterion\]](#eq:criterion){reference-type="ref" reference="eq:criterion"}) can also be rewritten as $$\label{eq:cr} \begin{split} &C_{n}(\mathbf{w}|\mathcal{M},\mathbf{y}) =\frac{1}{n}\left\|\mathbf{y}-\widehat{\mathbf{f}}_{\mathbf{w}| \mathcal{M}} \right\|^2+\frac{2\widehat{\sigma}^2}{n}\mathbf{k}^{\top}\mathbf{w}\\ & = \frac{1}{n}\left\|\mathbf{y}-\sqrt{n}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\boldsymbol{\phi}_l\widehat{\theta}_l \right\|^2+\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}2\gamma_j\frac{\widehat{\sigma}^2}{n}\\ & =\frac{1}{n}\left\|\mathbf{y}\right\|^2 - \frac{2}{\sqrt{n}}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\mathbf{y}^{\top}\boldsymbol{\phi}_l\widehat{\theta}_l + \sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j^2\widehat{\theta}_l^2 +\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}2\gamma_j\frac{\widehat{\sigma}^2}{n}\\ & =\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\left[ \gamma_j^2\widehat{\theta}_l^2+2\gamma_j\left(\frac{\widehat{\sigma}^2}{n}-\widehat{\theta}_l^2\right) \right]+\frac{1}{n}\sum_{i=1}^{n}y_i^2, \end{split}$$ where the last equality follows from $\mathbf{y}^{\top}\boldsymbol{\phi}_l/\sqrt{n}=\widehat{\theta}_l$. ## Technical lemmas We state or prove several preliminary lemmas used to prove the propositions in Section [3](#sec:review){reference-type="ref" reference="sec:review"} and the main results in Section [4](#sec:main_results){reference-type="ref" reference="sec:main_results"}. Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"} compares the optimal risks of MS and MA based on the successive candidate model set $\mathcal{M}_s=\{1,2,\ldots,M_n \}$. Define $m_n^*=\arg\min_{m \in \{1,\ldots,p_n \}}R_n(m,\mathbf{f})$ the size of the optimal single model, $m^*|\mathcal{M}_s=\arg\min_{m \in \mathcal{M}_s}R_n(m,\mathbf{f})$ the size of the optimal candidate model in $\mathcal{M}_s$, and $\mathbf{w}^*|\mathcal{M}_s=\arg\min_{\mathbf{w} \in \mathcal{W}_{M_n}}R_n(\mathbf{w}| \mathcal{M}_s,\mathbf{f})$ the optimal weight vector based on the candidate model set $\mathcal{M}_s$. **Lemma 1**. *Suppose that Assumptions [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"}--[Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"} hold. For the set of successive candidate models $\mathcal{M}_{s}=\{1,2, \ldots, M_n\}$, we always have $$\label{eq:lemma:peng:1} R_n(m^*|\mathcal{M}_{s},\mathbf{f}) \asymp R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}).$$* *For a large set $\mathcal{M}_{s}$ with $M_n\gtrsim m_n^*$, we have $$\label{eq:lemma:peng:2} R_n(m^*|\mathcal{M}_{s},\mathbf{f}) \asymp R_n(\mathbf{w}^*|\mathcal{M}_{s},\mathbf{f}) \asymp R_n(m_n^*,\mathbf{f}) \asymp \frac{m_n^*}{n}.$$ Under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}, we get $$\label{eq:lemma:peng:3} R_n(m^*|\mathcal{M}_{s},\mathbf{f}) - R_n(\mathbf{w}^*|\mathcal{M}_{s},\mathbf{f}) \asymp R_n(m^*|\mathcal{M}_{s},\mathbf{f}).$$ Under Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"}, we get $$\label{eq:lemma:peng:4} R_n(m^*|\mathcal{M}_{s},\mathbf{f}) - R_n(\mathbf{w}^*|\mathcal{M}_{s},\mathbf{f}) = o\left[R_n(m^*|\mathcal{M}_{s},\mathbf{f})\right].$$* *For a small set $\mathcal{M}_{s}=\{1,2, \ldots, M_n\}$ with $M_n=o(m_n^*)$, we have $$\label{eq:lemma:peng:5} R_n(m^*|\mathcal{M}_{s},\mathbf{f}) - R_n(\mathbf{w}^*|\mathcal{M}_{s},\mathbf{f}) = o\left[R_n(m^*|\mathcal{M}_{s},\mathbf{f})\right].$$* *Proof.* Note that Assumption [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"} is equivalent to $$\label{eq:sum_theta} \frac{1}{n}\left\|\mathbf{f}\right\|^2 = \frac{1}{n}\left\|\sum_{j=1}^{p_n}\mathbf{D}_j\mathbf{f}\right\|^2= \frac{1}{n}\left\|\sqrt{n}\sum_{j=1}^{p_n}\boldsymbol{\phi}_j\theta_j \right\|^2 = \sum_{j=1}^{p_n}\theta_j^2<\infty.$$ This coincides with Assumption 1 in [@Peng2021]. Thus, Theorems 1--2 of [@Peng2021] and Theorems 1--4 of [@Xu2022From] imply the results of this lemma. ◻ **Lemma 2**. *Let $\{\xi(t), t \in \mathcal{T}\}$ be a stochastic process with $\mathbb{E}\xi(t) = 0$ and finite variance $\mathbb{E}[\xi(t)]^2 = \sigma^2(t)$ for all $t \in \mathcal{T}$, where $\mathcal{T}$ is a finite index set. Suppose that there exist $\lambda>0$ and $\varphi(\lambda)<\infty$ such that $$\label{eq:lemma_con} \max_{t \in \mathcal{T}} \mathbb{E}\exp(\lambda| \xi(t) |)\leq \varphi(\lambda).$$ Then for all $r\geq 1$, there exists a constant $C$ depending on $\lambda$ and $r$ such that $$\left(\mathbb{E} \max_{t \in \mathcal{T}}| \xi(t) |^r\right)^{\frac{1}{r}} \leq C (\log |\mathcal{T}|+1).$$* *Proof.* The proof of this lemma is motivated by Lemma 1 in [@Golubev2010]. Notice that for $r \geq 1$, the function $F(x)=\log^r[x+\exp(r-1)]$ is concave on $(0,\infty)$ since $$F''(x) = \frac{r\log^{r-2}[x+\exp(r-1)]}{[x+\exp(r-1)]^2}\left\{r-1-\log\left[ x+\exp(r-1)\right] \right\}\leq 0.$$ Using Jensen's inequality, we have $$\begin{split} \left[\mathbb{E} \max_{t \in \mathcal{T}}| \xi(t) |^r\right]^{\frac{1}{r}} &=\frac{1}{\lambda}\left\{ \mathbb{E} \left[\max_{t \in \mathcal{T}}|\lambda \xi(t) |\right]^r \right\}^{\frac{1}{r}}= \frac{1}{\lambda}\left\{\mathbb{E}\log^r \left[\exp\left( \max_{t \in \mathcal{T}}|\lambda \xi(t) | \right)\right]\right\}^{\frac{1}{r}} \\ & \leq \frac{1}{\lambda}\left\{\mathbb{E}\log^r \left[\exp\left( \max_{t \in \mathcal{T}}|\lambda \xi(t) | \right)+\exp(r-1)\right]\right\}^{\frac{1}{r}} \\ &\leq \frac{1}{\lambda}\log \left[\mathbb{E}\exp\left(\max_{t \in \mathcal{T}}\lambda| \xi(t) |\right)+ \exp(r-1) \right]\\ &\leq \frac{1}{\lambda}\log \left[\sum_{t\in\mathcal{T}}\mathbb{E}\exp(\lambda| \xi(t) |)+ \exp(r-1) \right]\\ &\leq \frac{\log \left[ \varphi(\lambda)|\mathcal{T}| + \exp(r-1) \right]}{\lambda} \leq C(\log |\mathcal{T}|+1), \end{split}$$ which proves the lemma. ◻ ## Proof of Proposition [Proposition 1](#lemma:delta_1){reference-type="ref" reference="lemma:delta_1"} {#proof-of-proposition-lemmadelta_1} From Assumption [Assumption 2](#asmp:regressor_order){reference-type="ref" reference="asmp:regressor_order"} and ([\[eq:risk_m\]](#eq:risk_m){reference-type="ref" reference="eq:risk_m"}), we see that the optimal single model $m_n^{\ast}$ satisfies $$\label{eq:mnstar} \theta_{m_n^{\ast}}^2>\frac{\sigma^2}{n}\geq \theta_{m_n^{\ast}+1}^2.$$ Hence the optimal MS risk is $$\label{eq:optimal_ms_risk} R_n(m_n^{\ast},\mathbf{f})=\frac{m_n^{\ast}\sigma^2}{n}+\sum_{j=m_n^{\ast}+1}^{p_n}\theta_j^2.$$ Using ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}), we get the MA risk $$\label{eq:riskw_large} \begin{split} R_n\left(\mathbf{w}|\mathcal{M}_s,\mathbf{f}\right)&=\sum_{j=1}^{M_n}\left( \frac{\sigma^2}{n} + \theta_j^2 \right) \left( \gamma_j - \frac{\theta_j^2}{\theta_j^2+\frac{\sigma^2}{n}} \right)^2\\ &+\sum_{j=1}^{M_n}\frac{\theta_j^2\sigma^2}{n\theta_j^2+\sigma^2}+\sum_{j=M_n+1}^{p_n}\theta_j^2. \end{split}$$ The infeasible optimal weights $\mathbf{w}^*|\mathcal{M}_s=(w_1^*,\ldots,w_{M_n}^*)^{\top}$ can be obtained by setting $$\label{eq:gamma_star} \gamma_1^*=1,\,\gamma_j^*=\frac{\theta_j^2}{\theta_j^2+\frac{\sigma^2}{n}},\, j=2,\ldots,M_n,$$ where $\gamma_j^*=\sum_{m=j}^{M_n}w_m^*$. Hence the optimal MA risk based on $\mathcal{M}_s$ is $$R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f})=\frac{\sigma^2}{n}+\sum_{j=2}^{M_n}\frac{\theta_j^2\sigma^2}{n\theta_j^2+\sigma^2}+\sum_{j=M_n+1}^{p_n}\theta_j^2.$$ We first prove the results when Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"} and $M_n\gtrsim m_n^*$ hold. Let $G:\mathbb{N}\rightarrow \mathbb{N}$ by $$G(x)=\arg\min_{t \in \mathbb{N}}\left( \lfloor kt \rfloor \geq x \right),$$ where $k$ is the constant given in Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"} and $\mathbb{N}$ is the set of natural numbers. Define a sequence of functions $G_d(x)$ indexed by integer $d$ $$\label{eq:Gfunction} G_d(x) =\left\{\begin{array}{ll} x &\quad d = 0, \\ \left(G \circ G_{d-1} \right)(x) &\quad d \geq 1, \end{array}\right.$$ where the notation $(f\circ g)(x)$ means the composition of functions $f(g(x))$. Given a fixed $N$, define $d_1^*=\arg\min_{d \in \mathbb{N}}\nu^{2d} \leq 1/(N-1)$ and $i_n^*=M_n \wedge G_{d_1^*+1}(m_n^*)$, where $0<\nu<1$ is the constant defined in Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}. Since $M_n\gtrsim m_n^*$ and $d_1^*$ is a fixed integer, we see $i_n^* \asymp m_n^*$. We have $$\label{eq:a135} \begin{split} \frac{\theta_{m_n^*}^2}{\theta_{i_n^*}^2} & \leq\frac{\theta_{m_n^*}^2}{\theta_{G_1(m_n^*)}^2}\times\frac{\theta_{G_1(m_n^*)}^2}{\theta_{G_2(m_n^*)}^2}\times\cdots\times\frac{\theta_{G_{d_1^*}(m_n^*)}^2}{\theta_{G_{d_1^*+1}(m_n^*)}^2}\times\frac{\theta_{G_{d_1^*+1}(m_n^*)}^2}{\theta_{i_n^*}^2} \\ &\leq \nu^{2d_1^*+2}\leq\frac{\nu^2}{N-1}, \end{split}$$ where the second inequality follows from Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"} and $\theta_{i_n^*}^2\geq \theta_{G_{d_1^*+1}(m_n^*)}^2$, and the last inequality is due to the definition of $d_1^*$. Therefore $$\label{eq:dis_weight_1} \gamma_{i_n^*}^{*}- \frac{N-1}{N}\geq \frac{\theta_{i_n^*}^2}{\theta_{i_n^*}^2+\theta_{m_n^*}^2}- \frac{N-1}{N} \geq \frac{N-1}{N-1+\nu^2}- \frac{N-1}{N}\triangleq C_1>0,$$ where the first inequality is due to ([\[eq:mnstar\]](#eq:mnstar){reference-type="ref" reference="eq:mnstar"}) and ([\[eq:gamma_star\]](#eq:gamma_star){reference-type="ref" reference="eq:gamma_star"}), and the second inequality is due to ([\[eq:a135\]](#eq:a135){reference-type="ref" reference="eq:a135"}). Define another model index $j_n^*=G_{1}(i_n^*)$. Note that $$\begin{split} \frac{\theta_{m_n^*+1}^2}{\theta_{j_n^*}^2} & =\frac{\theta_{m_n^*+1}^2}{\theta_{G_1(m_n^*+1)}^2}\times\frac{\theta_{G_1(m_n^*+1)}^2}{\theta_{G_2(m_n^*+1)}^2}\times\cdots\times\frac{\theta_{G_{d_1^*+1}(m_n^*+1)}^2}{\theta_{i_n^*}^2}\times\frac{\theta_{i_n^*}^2}{\theta_{G_{1}(i_n^*)}^2} \\ & \geq \delta^{2d_1^*+4}\frac{\theta_{G_{d_1^*+1}(m_n^*+1)}^2}{\theta_{i_n^*}^2}, \end{split}$$ where $0<\delta<1$ is the constant defined in Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}. Since $i_n^*=M_n \wedge G_{d_1^*+1}(m_n^*)$ and $M_n\gtrsim m_n^*$, there must exist a constant $0<c\leq 1$ such that $$\frac{\theta_{G_{d_1^*+1}(m_n^*+1)}^2}{\theta_{i_n^*}^2}>c$$ under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}. We thus have $$\label{eq:dis_weight_2} 1-\gamma_{j_n^*}^{*}\geq 1- \frac{\theta_{j_n^*}^2}{\theta_{j_n^*}^2+\theta_{m_n^*+1}^2}\geq 1- \frac{1}{1+ c\delta^{2d_1^*+4}}\triangleq C_2>0.$$ Let $\mathbf{w}_N^*|\mathcal{M}_s=\arg\min_{\mathbf{w}\in \mathcal{W}_{|\mathcal{M}_s|}(N)}R_n(\mathbf{w}| \mathcal{M}_s,\mathbf{f})$ denote the optimal discrete weight vector in $\mathcal{W}_{|\mathcal{M}_s|}(N)$. Note that restricting $\mathbf{w}_N|\mathcal{M}_{s}=(w_{N,1},\ldots,w_{N,M_n})^{\top} \in \mathcal{W}_{|\mathcal{M}_s|}(N)$ is equivalent to restricting $\boldsymbol{\gamma}_N|\mathcal{M}_s=(\gamma_{N,1},\ldots,\gamma_{N,M_n})^{\top} \in \Gamma_{|\mathcal{M}_s|}(N)=\{\gamma_{N,j}=t_j/N:N=t_1\geq t_2\geq\cdots\geq t_{M_n}\geq0, t_j \in \mathbb{N}\cup \{0\} \}$, where $\gamma_{N,j}=\sum_{m=j}^{M_n}w_{N,m}$. Based on ([\[eq:dis_weight_1\]](#eq:dis_weight_1){reference-type="ref" reference="eq:dis_weight_1"}) and ([\[eq:dis_weight_2\]](#eq:dis_weight_2){reference-type="ref" reference="eq:dis_weight_2"}), when $j_n^*<j\leq i_n^*$, we see that the optimal cumulative weights satisfy $$\frac{N-1}{N}+C_1\leq \gamma_{j}^{*}\leq 1-C_2.$$ However, the optimal discrete cumulative weight $\gamma_{N,j}^*=\sum_{m=j}^{M_n}w_{N,m}^*$ is either 1 or $(N-1)/N$ when $j_n^*<j\leq i_n^*$. Combining ([\[eq:riskw_large\]](#eq:riskw_large){reference-type="ref" reference="eq:riskw_large"}) with ([\[eq:dis_weight_1\]](#eq:dis_weight_1){reference-type="ref" reference="eq:dis_weight_1"}) and ([\[eq:dis_weight_2\]](#eq:dis_weight_2){reference-type="ref" reference="eq:dis_weight_2"}), we see at once that $$\label{eq:risk_difference_bound} \begin{split} &R_n\left(\mathbf{w}_N^*|\mathcal{M}_{s},\mathbf{f}\right)-R_n\left(\mathbf{w}^*|\mathcal{M}_{s},\mathbf{f}\right)\\ &=\sum_{j=1}^{M_n}\left( \frac{\sigma^2}{n} + \theta_j^2 \right) \left( \gamma_{N,j}^* - \frac{\theta_j^2}{\theta_j^2+\frac{\sigma^2}{n}} \right)^2-\sum_{j=1}^{M_n}\left( \frac{\sigma^2}{n} + \theta_j^2 \right) \left( \gamma_{j}^* - \frac{\theta_j^2}{\theta_j^2+\frac{\sigma^2}{n}} \right)^2\\ & \geq \sum_{j=2}^{M_n}\left( \frac{\sigma^2}{n} + \theta_j^2 \right) \left( \gamma_{N,j}^*- \gamma_{j}^* \right)^2\geq \sum_{j=j_n^*+1}^{i_n^*}\frac{\sigma^2}{n}(C_1^2\wedge C_2^2) \\ &= \frac{(C_1^2\wedge C_2^2)(i_n^*-j_n^*)\sigma^2}{n}\asymp \frac{m_n^*}{n} \asymp R_n(\mathbf{w}^*|\mathcal{M}_{s},\mathbf{f}), \end{split}$$ where the constants $C_1$ and $C_2$ are defined in ([\[eq:dis_weight_1\]](#eq:dis_weight_1){reference-type="ref" reference="eq:dis_weight_1"}) and ([\[eq:dis_weight_2\]](#eq:dis_weight_2){reference-type="ref" reference="eq:dis_weight_2"}) respectively, and the last approximation follows from Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"}. Due to $$\label{eq:relation3} R_n\left(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}\right)\leq R_n\left(\mathbf{w}_N^*|\mathcal{M}_s,\mathbf{f}\right)\leq R_n\left(m^*|\mathcal{M}_s,\mathbf{f}\right),$$ the proof of the results under Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"} or $M_n=o(m_n^*)$ is a direct application of Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"}. This completes the proof. ## Proof of Proposition [Proposition 2](#lemma:delta_11){reference-type="ref" reference="lemma:delta_11"} {#proof-of-proposition-lemmadelta_11} We first prove the claim under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"} and $M_n\gtrsim m_n^*$. Recall that $\mathbf{w}_N^*|\mathcal{M}_s=\arg\min_{\mathbf{w}\in \mathcal{W}_{|\mathcal{M}_s|}(N)}R_n(\mathbf{w}| \mathcal{M}_s,\mathbf{f})$ denotes the optimal discrete weight vector in $\mathcal{W}_{|\mathcal{M}_s|}(N)$, and $m^*|\mathcal{M}_s=\arg\min_{m \in \mathcal{M}_s}R_n(m,\mathbf{f})$ is the size of the optimal candidate model in $\mathcal{M}_s$. Since MS can be seen as the MA on the discrete weight set with $N=1$, we have $R_n\left( m^*|\mathcal{M}_{s},\mathbf{f}\right)=R_n\left( \mathbf{w}_1^*|\mathcal{M}_{s}, \mathbf{f}\right)$, where $\mathbf{w}_1^*|\mathcal{M}_{s}=(w_{1,1}^*,\ldots,w_{1,M_n}^*)^{\top}$ and the optimal discrete cumulative weights for MS is $\gamma_{1,j}^*=\sum_{m=j}^{M_n}w_{1,m}^*$. From ([\[eq:risk_m\]](#eq:risk_m){reference-type="ref" reference="eq:risk_m"}) and ([\[eq:riskw_large\]](#eq:riskw_large){reference-type="ref" reference="eq:riskw_large"}), we have $$\label{eq:gamma1j} \gamma_{1,j}^*=\left\{\begin{array}{ll} 1 &\quad 1\leq j \leq (m_n^*\wedge M_n), \\ 0 &\quad (m_n^*\wedge M_n)<j\leq M_n. \end{array}\right.$$ From ([\[eq:riskw_large\]](#eq:riskw_large){reference-type="ref" reference="eq:riskw_large"}), we see that the risk difference between MS and MA is $$\label{eq:p111} \begin{split} &R_n\left( m^*|\mathcal{M}_{s},\mathbf{f}\right) - R_n\left( \mathbf{w}_N^*|\mathcal{M}_{s}, \mathbf{f}\right)\\ &=R_n\left( \mathbf{w}_1^*|\mathcal{M}_{s}, \mathbf{f}\right) -R_n\left( \mathbf{w}_N^*|\mathcal{M}_{s}, \mathbf{f}\right) \\ & =\sum_{j=1}^{M_n}\left( \frac{\sigma^2}{n} + \theta_j^2 \right) \left( \gamma_{1,j}^* - \frac{\theta_j^2}{\theta_j^2+\frac{\sigma^2}{n}} \right)^2 - \sum_{j=1}^{M_n}\left( \frac{\sigma^2}{n} + \theta_j^2 \right) \left( \gamma_{N,j}^* - \frac{\theta_j^2}{\theta_j^2+\frac{\sigma^2}{n}} \right)^2 \\ &\geq \sum_{j=1}^{M_n\wedge m_n^*}\left( \frac{\sigma^2}{n} + \theta_j^2 \right)\left( 1 - \frac{\theta_j^2}{\theta_j^2+\frac{\sigma^2}{n}} \right)^2- \frac{1}{4N^2}\sum_{j=1}^{M_n\wedge m_n^*}\left( \frac{\sigma^2}{n} + \theta_j^2 \right), \end{split}$$ where the inequality is due to ([\[eq:gamma1j\]](#eq:gamma1j){reference-type="ref" reference="eq:gamma1j"}) and the fact $$\left|\gamma_{N,j}^* - \frac{\theta_j^2}{\theta_j^2+\frac{\sigma^2}{n}} \right| \leq \frac{1}{2N}.$$ Define $$d_2^*=\left\{\begin{array}{ll} \arg\min_{d \in \mathbb{N}}\{G_d(m_n^*+1)<M_n\} &\quad M_n<m_n^*, \\ 0 &\quad M_n\geq m_n^*. \end{array}\right.$$ where the function $G_d$ is given by ([\[eq:Gfunction\]](#eq:Gfunction){reference-type="ref" reference="eq:Gfunction"}). It is easy to check that $$\label{eq:check_d2star} d_2^* \sim \log_k\left( \frac{m_n^*}{M_n}\vee 1 \right),$$ where $k>1$ is the constant given in Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}. When $N>(1+\delta^{2d_2^*+2})/(2\delta^{2d_2^*+2})$, there must exist a positive constant $\tau$ that satisfies $\delta^{2d_2^*+2}\geq (1+\tau)/[2N-(1+\tau)]$, where $0<\delta<1$ is the constant given in Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}. Then we define $$\label{eq:d3_star} d_3^*=\mathop{\mathrm{argmax}}_{d\in \mathbb{N}\cup\{0\}}\delta^{2d+2d_2^*+2}\geq \frac{1+\tau}{2N-(1+\tau)}$$ and the model index $j_n^*=G_1(M_n) \wedge G_{d_3^*}\left(m_n^*+1 \right)$. When $j_n^*=G_1(M_n)$, we have $$\label{eq:what12} \begin{split} \frac{\theta_{m_n^*+1}^2}{\theta_{j_n^*}^2} & =\frac{\theta_{m_n^*+1}^2}{\theta_{G_1(m_n^*+1)}^2}\times\frac{\theta_{G_1(m_n^*+1)}^2}{\theta_{G_2(m_n^*+1)}^2}\times\cdots\times\frac{\theta_{G_{d_2^*}(m_n^*+1)}^2}{\theta_{G_1(M_n)}^2} \\ & \geq \delta^{2d_2^*} \frac{\theta_{G_{d_2^*}(m_n^*+1)}^2}{\theta_{G_1(M_n)}^2}\geq \delta^{2d_2^*} \frac{\theta_{G_{d_2^*}(m_n^*+1)}^2}{\theta_{G_{d_2^*+1}(m_n^*+1)}^2}\\ &\geq \delta^{2d_2^*+2}, \end{split}$$ where the first inequality follows Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}, and the second inequality is due to $G_{d_2^*}(m_n^*+1)<M_n$ and $G_{d_2^*+1}(m_n^*+1)<G_1(M_n)$. When $j_n^*=G_{d_3^*}\left(m_n^*+1 \right)$, we have $$\label{eq:what123} \begin{split} \frac{\theta_{m_n^*+1}^2}{\theta_{j_n^*}^2} & =\frac{\theta_{m_n^*+1}^2}{\theta_{G_1(m_n^*+1)}^2}\times\frac{\theta_{G_1(m_n^*+1)}^2}{\theta_{G_2(m_n^*+1)}^2}\times\cdots\times\frac{\theta_{G_{d_3^*-1}(m_n^*+1)}^2}{\theta_{G_{d_3^*}(m_n^*+1)}^2} \geq \delta^{2d_3^*}. \end{split}$$ Combining ([\[eq:what12\]](#eq:what12){reference-type="ref" reference="eq:what12"}) with ([\[eq:what123\]](#eq:what123){reference-type="ref" reference="eq:what123"}), we have $$\begin{split} \frac{\theta_{m_n^*+1}^2}{\theta_{j_n^*}^2} & \geq \delta^{2d_2^*+2}\wedge \delta^{2d_3^*}= \delta^{(2d_2^*+2)\vee(2d_3^*)} \\ & \geq \delta^{2d_2^*+2d_3^*+2} \geq \frac{1+\tau}{2N-(1+\tau)}, \end{split}$$ where the second inequality is due to $0<\delta<1$, and the last inequality is due to the definition ([\[eq:d3_star\]](#eq:d3_star){reference-type="ref" reference="eq:d3_star"}). Thus when $j\geq j_n^*$, we have $$\label{eq:p112} \begin{split} 1 - \frac{\theta_j^2}{\theta_j^2+\frac{\sigma^2}{n}}&\geq 1 - \frac{1}{1+\frac{\theta_{m_n^*+1}^2}{\theta_j^2}} \geq 1-\frac{1}{1+\frac{\theta_{m_n^*+1}^2}{\theta_{j_n^*}^2}}\\ &\geq 1 - \frac{1}{1+\frac{1+\tau}{2N-(1+\tau)}}= \frac{1+\tau}{2N}. \end{split}$$ Substituting ([\[eq:p112\]](#eq:p112){reference-type="ref" reference="eq:p112"}) into ([\[eq:p111\]](#eq:p111){reference-type="ref" reference="eq:p111"}) gives the desired claim $$\begin{split} &R_n\left( m^*|\mathcal{M}_{s},\mathbf{f}\right) - R_n\left( \mathbf{w}_N^*|\mathcal{M}_{s}, \mathbf{f}\right)\\ &\geq \sum_{j=1}^{M_n\wedge m_n^*}\left( \frac{\sigma^2}{n} + \theta_j^2 \right) \left[\frac{(1+\tau)^2}{4N^2}-\frac{1}{4N^2}\right] \\ & \geq \frac{(\tau^2+2\tau)(M_n\wedge m_n^*-j_n^*)\sigma^2}{4N^2 n} \asymp \frac{m_n^*}{n}\asymp R_n(m^*|\mathcal{M}_{s},\mathbf{f}). \end{split}$$ The proof of the result under Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"} or the condition $M_n=o(m_n^*)$ is straightforward based on Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"} and ([\[eq:relation3\]](#eq:relation3){reference-type="ref" reference="eq:relation3"}). This completes the proof of this proposition. ## Proof of Proposition [Proposition 3](#lemma:delta_2){reference-type="ref" reference="lemma:delta_2"} {#proof-of-proposition-lemmadelta_2} Under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}, in a manner of proof by contradiction, we first check that a necessary condition for ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) is $M_n=o(m_n^*)$. Suppose $M_n\geq m_n^*$, it is already seen from [@Peng2021] that $$R_n\left(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}\right) \asymp R_n(m_n^*,\mathbf{f}) \asymp \frac{m_n^*}{n}$$ under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}. We thus obtain $$\label{eq:condition_wan2} \begin{split} & \frac{|\mathcal{M}_s|\sum_{m=1}^{|\mathcal{M}_s|}R_n\left(w_m^0|\mathcal{M}_s,\mathbf{f}\right)}{nR_n^2\left(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}\right)} \gtrsim \frac{M_n^2m_n^*/n}{(m_n^*)^2/n}\\ &=\frac{M_n^2}{m_n^*}\geq M_n \geq m_n^* \to \infty, \end{split}$$ which contradicts the assumption ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}). Suppose $M_n < m_n^*$ but $M_n \asymp m_n^*$, there must exist a constant $C>1$ and a positive integer $K$ such that for any $n>K$, we have $m_n^*<CM_n$. In this case, the main task is to show the risk of the optimal single model in $\mathcal{M}_s$ and the risk of the optimal averaged model based on $\mathcal{M}_s$ both have the order $m_n^*/n$. Note first that the optimal single model in $\mathcal{M}_s$ needs to include $M_n$ terms, which has the risk $$R_n\left( m^*|\mathcal{M}_s,\mathbf{f}\right) = \frac{M_n\sigma^2}{n} + \sum_{j=M_n+1}^{m_n^*}\theta_j^2+\sum_{j=m_n^*+1}^{p_n}\theta_j^2.$$ As there must exist an index $d_4^*$ such that $G_{d_4^*}(m_n^*+1)\leq m_n^*/C<M_n$, it follows that the second term in $R_n\left( m^*|\mathcal{M}_s,\mathbf{f}\right)$ is bounded by $$\begin{split} \sum_{j=M_n+1}^{m_n^*}\theta_j^2 & \leq \left(m_n^*-M_n\right)\theta_{G_{d_4^*}(m_n^*+1)}^2\leq \frac{\left(m_n^*-M_n\right)\theta_{m_n^*+1}^2}{\delta^{2d_4^*}} \\ & \leq \frac{\left(m_n^*-M_n\right)\sigma^2}{n\delta^{2d_4^*}}\lesssim \frac{m_n^*}{n}, \end{split}$$ where the second inequality follows from Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"} and the third inequality follows from ([\[eq:mnstar\]](#eq:mnstar){reference-type="ref" reference="eq:mnstar"}). Since the order of the last term in $R_n\left( m^*|\mathcal{M}_s,\mathbf{f}\right)$ is also no bigger than $m_n^*/n$ [@Peng2021], we thus get $R_n\left( m^*|\mathcal{M}_s,\mathbf{f}\right)\asymp m_n^*/n$. Furthermore, it is easy to check that $$\begin{split} & R_n\left( m^*|\mathcal{M}_s,\mathbf{f}\right) \geq R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}) \\ & \geq R_n(\mathbf{w}^*|\mathcal{M}_l,\mathbf{f}) \asymp R_n(m_n^*,\mathbf{f}) \asymp \frac{m_n^*}{n}, \end{split}$$ where $\mathcal{M}_l$ is a large candidate model set which includes $m_n^*$, and the last two approximations are due to [@Peng2021]. It follows immediately that $R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}) \asymp m_n^*/n$. In the same manner of ([\[eq:condition_wan2\]](#eq:condition_wan2){reference-type="ref" reference="eq:condition_wan2"}), we also obtain a contradiction of assumption ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) when $M_n < m_n^*$ and $M_n \asymp m_n^*$. Thus, under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}, a necessary condition for ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) is $M_n=o(m_n^*)$. Define $d_5^*=\arg\max_{d\in \mathbb{N}}\{G_{d}(m_n^*)\geq M_n\}$. Since $M_n=o(m_n^*)$, we have $d_5^*\to \infty$ as $n\to \infty$. Then the MA risk is lower bounded by $$\label{eq:dndkna} \begin{split} &R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}) \geq \sum_{j=M_n+1}^{m_n^*}\theta_j^2\\ &= \sum_{j=G_1(m_n^*)+1}^{m_n^*}\theta_j^2 + \sum_{j=G_2(m_n^*)+1}^{G_1(m_n^*)}\theta_j^2 + \cdots + \sum_{j=G_{d_5^*}(m_n^*)+1}^{G_{d_5^*-1}(m_n^*)}\theta_j^2 \\ &\geq \theta_{m_n^*}^2[m_n^*-G_1(m_n^*)]+ \theta_{G_1(m_n^*)}^2[G_1(m_n^*)-G_2(m_n^*)]\\ &\qquad\quad\quad\quad+\cdots+\theta_{G_{d_5^*-1}(m_n^*)}^2[G_{d_5^*-1}(m_n^*)-G_{d_5^*}(m_n^*)] \\ &\geq \frac{\sigma^2}{n}\left(m_n^*-\frac{m_n^*}{k}\right)+\frac{\sigma^2}{n\nu^2}\left(\frac{m_n^*}{k}-\frac{m_n^*}{k^2}\right)+\cdots+\frac{\sigma^2}{n\nu^{2(d_5^*-1)}}\left(\frac{m_n^*}{k^{d_5^*-1}}-\frac{m_n^*}{k^{d_5^*}}\right)\\ & \geq \frac{m_n^*\sigma^2}{n}\left(1-\frac{1}{k}\right)\sum_{l=0}^{d_5^*-1}\frac{1}{(k\nu^{2})^l}, \end{split}$$ where the first inequality follows from ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}), and the third inequality is due to ([\[eq:mnstar\]](#eq:mnstar){reference-type="ref" reference="eq:mnstar"}) and Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}. Since $d_5^*\to \infty$ and $k\nu^{2}<1$, we thus get $$\sum_{l=0}^{d_5^*-1}\frac{1}{(k\nu^{2})^l}\to \infty.$$ Due to $R_n(m_n^*,\mathbf{f})\asymp m_n^*/n$, from ([\[eq:dndkna\]](#eq:dndkna){reference-type="ref" reference="eq:dndkna"}) we conclude $R_n(m_n^*,\mathbf{f})=o[R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f})]$. When Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"} holds, using the proof by contradiction again, we see that a necessary condition for ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) is $M_n \leq \lfloor C m_n^* \rfloor$ with a constant $0<C<1$. Note that $\lfloor C m_n^* \rfloor \leq \lfloor (C+1) m_n^*/2 \rfloor\leq m_n^*$. Then the MA risk is lower bounded by $$\begin{split} & R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f})\geq \sum_{j=M_n+1}^{\lfloor (C+1) m_n^*/2 \rfloor}\theta_j^2 \\ & \geq ( \lfloor (C+1) m_n^*/2 \rfloor - M_n)\theta_{\lfloor (C+1) m_n^*/2 \rfloor}^2. \end{split}$$ Under Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"}, we have $\theta_{\lfloor (C+1) m_n^*/2 \rfloor}^2/\theta_{m_n^*}^2\to \infty$ and $\theta_{m_n^*}^2 \asymp 1/n$. Thus we get $R_n(m_n^*,\mathbf{f})=o[R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f})]$, which proves the proposition. ## Proof of the results in the two examples Based on the risk of MS ([\[eq:risk_m\]](#eq:risk_m){reference-type="ref" reference="eq:risk_m"}), we have $$\label{eq:add1} \begin{split} &\sum_{m=1}^{|\mathcal{M}_s|}R_n\left(\mathbf{w}_m^0|\mathcal{M}_s,\mathbf{f}\right)= \sum_{m=1}^{M_n}R_n\left(m,\mathbf{f}\right)\\ &= \sum_{j=1}^{M_n}\frac{j}{n}\sigma^2+\sum_{j=2}^{p_n}\theta_j^2+\cdots+\sum_{j=M_n+1}^{p_n}\theta_j^2 \\ & = \sum_{j=1}^{M_n}\frac{j}{n}\sigma^2 + \sum_{j=2}^{M_n}(j-1)\theta_j^2+M_n\sum_{j=M_n+1}^{p_n}\theta_j^2. \end{split}$$ When $\theta_j=j^{-\alpha_1}, \alpha_1>1/2$, approximating the sums in ([\[eq:add1\]](#eq:add1){reference-type="ref" reference="eq:add1"}) by integrals, we obtain that the numerator of ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) has the order $$M_n\sum_{m=1}^{M_n}R_n\left(m,\mathbf{f}\right) \asymp\left\{\begin{array}{ll} M_n^{-2\alpha_1+3} &\quad 1/2<\alpha_1<1, \\ M_n\log{M_n} &\quad \alpha_1=1,\\ M_n &\quad \alpha_1>1. \end{array}\right.$$ We now turn to evaluate the order of the denominator of ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}). Define $g(x)=\int_{0}^{\frac{1}{1+x^{2\alpha_1}}}t^{1-\frac{1}{2\alpha_1}}(1-t)^{\frac{1}{2\alpha_1}}dt$ and $g'(x)=-\frac{2\alpha_1}{1+x^{2\alpha_1}}$. Based on the proof of Example 1 in [@Peng2021], we have $$\label{eq:add2} \begin{split} R_n(\mathbf{w}^*|\mathcal{M}_s, \mathbf{f})& \asymp n^{-1+1/(2\alpha_1)}\left[g(0) - g\left(\frac{M_n}{m_n^*}\right) \right]+M_n^{-2\alpha_1+1} \\ &\asymp n^{-1+1/(2\alpha_1)} \left[-g'(0) \left(\frac{M_n}{m_n^*}\right) \right]+M_n^{-2\alpha_1+1}\\ &\asymp \frac{M_n}{n} + M_n^{-2\alpha_1+1} \asymp M_n^{-2\alpha_1+1}, \end{split}$$ where the second approximation follows from Taylor's expansion, the third approximation follows from $m_n^*\asymp n^{1/(2\alpha_1)}$, and the last approximation follows from the fact $M_n=o(m_n^*)$ and $m_n^*\asymp n^{1/(2\alpha_1)}$. Combining ([\[eq:add1\]](#eq:add1){reference-type="ref" reference="eq:add1"}) with ([\[eq:add2\]](#eq:add2){reference-type="ref" reference="eq:add2"}) gives ([\[eq:M_n\_order1\]](#eq:M_n_order1){reference-type="ref" reference="eq:M_n_order1"}). When $\theta_j=\exp(-j^{\alpha_2})$, $\alpha_2>0$, in the same manner, we can see that the numerator of ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) has the order $M_n$. Define $\mbox{Ga}(x;a)=\int_{t=x}^{\infty}t^{a-1}\exp(-t)dt$ for $x>0$. Based on the proof of Example 2 in [@Peng2021], we have $$\begin{split} & R_n(\mathbf{w}^*|\mathcal{M}_s,\mathbf{f}) \asymp \frac{M_n}{n}+\mbox{Ga}\left(2M_n^{\alpha_2};\frac{1}{\alpha_2}\right) \\ & \asymp \frac{M_n}{n} +(2M_n^{\alpha_2})^{\frac{1}{\alpha_2}-1}\exp(-2M_n^{\alpha_2}), \end{split}$$ where the second approximation is based on the asymptotic expansion of the incomplete gamma-function. Thus ([\[eq:condition_wan\]](#eq:condition_wan){reference-type="ref" reference="eq:condition_wan"}) is reduced to $M_n<\left(1/2 \right)^{1/\alpha_2}m_n^*$, where $m_n^*=[(1/2)\log(n/\sigma^2)]^{1/\alpha_2}$. This completes the proof. ## Proof of Theorem [Theorem 1](#theorem:main){reference-type="ref" reference="theorem:main"} {#sec:Proof_main} Recall that $\widehat{\theta}_{l} =\boldsymbol{\phi}_l^{\top}\mathbf{y}/\sqrt{n}$, $\theta_l=\boldsymbol{\phi}_l^{\top}\mathbf{f}/\sqrt{n}$, and $e_l=\boldsymbol{\phi}_l^{\top}\boldsymbol{\epsilon}/\sqrt{n}$, $l=1,\ldots,p_n$. Define $z_l=\sqrt{n}e_l/\sigma,l=1,\ldots,k_{M_n}$, $\widehat{\gamma}_j=\sum_{m=j}^{M_n}\widehat{w}_m$, $\gamma_j^*=\sum_{m=j}^{M_n}w_m^*$, $j=1,\ldots,M_n$, where $\widehat{w}_m$ and $w_m^*$ are $m$-th elements of $\widehat{\mathbf{w}}|\mathcal{M}$ and $\mathbf{w}^*|\mathcal{M}$, respectively. Based on ([\[eq:lossw\]](#eq:lossw){reference-type="ref" reference="eq:lossw"}) and ([\[eq:cr\]](#eq:cr){reference-type="ref" reference="eq:cr"}), we have $$\label{eq:addd1} \begin{split} & L_n(\mathbf{w}|\mathcal{M},\mathbf{f})- C_{n}(\mathbf{w}|\mathcal{M},\mathbf{y}) \\ & = 2\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\left[\gamma_j\widehat{\theta}_l(\widehat{\theta}_l-\theta_l) \right]-2\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\frac{\widehat{\sigma}^2}{n}+\sum_{j=1}^{p_n}\theta_j^2-\frac{1}{n}\sum_{i=1}^{n}y_i^2\\ & = 2\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( e_l^2-\frac{\sigma^2}{n}+\theta_le_l \right)+2\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( \frac{\sigma^2}{n} - \frac{\widehat{\sigma}^2}{n} \right)\\ & \quad+ \frac{1}{n}\sum_{j=1}^{p_n} \left(\boldsymbol{\phi}_j^{\top}\mathbf{f}\right)^2 - \frac{1}{n}\|\mathbf{f}\|^2 - \frac{1}{n}\mathbf{f}^{\top}\boldsymbol{\epsilon}- \frac{1}{n}\|\boldsymbol{\epsilon}\|^2\\ & = 2\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( e_l^2-\frac{\sigma^2}{n}+\theta_le_l \right)+2\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( \frac{\sigma^2}{n} - \frac{\widehat{\sigma}^2}{n} \right)\\ & \quad- \frac{1}{n}\mathbf{f}^{\top}\boldsymbol{\epsilon}- \frac{1}{n}\|\boldsymbol{\epsilon}\|^2, \end{split}$$ where the second equality follows from $\widehat{\theta}_{l} = \theta_l + e_l$ and $\theta_j=\boldsymbol{\phi}_j^{\top}\mathbf{f}/\sqrt{n}$, and the last step follows from $\|\mathbf{f}\|^2=\sum_{j=1}^{p_n} \left(\boldsymbol{\phi}_j^{\top}\mathbf{f}\right)^2$. In addition, for any non-random $\mathbf{w}|\mathcal{M}$, we have $$\label{eq:addd2} \begin{split} & \mathbb{E}C_{n}(\mathbf{w}|\mathcal{M},\mathbf{y})-R_n(\mathbf{w}|\mathcal{M},\mathbf{f}) \\ & =\mathbb{E}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\left[(\gamma_j^2-2\gamma_j)(\widehat{\theta}_l^2 - \theta_l^2) + 2\gamma_j \frac{\widehat{\sigma}^2}{n} - \gamma_j^2\frac{\sigma^2}{n} \right]\\ &\quad + \frac{1}{n}\mathbb{E}\sum_{i=1}^{n}y_i^2 - \sum_{j=1}^{p_n}\theta_j^2\\ & = 2\mathbb{E}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( \frac{\widehat{\sigma}^2}{n} - \frac{\sigma^2}{n}\right)+ \frac{1}{n}\mathbb{E}\left(\|\mathbf{f}\|^2+2\mathbf{f}^{\top}\boldsymbol{\epsilon}+ \|\boldsymbol{\epsilon}\|^2 \right) - \sum_{j=1}^{p_n}\theta_j^2\\ & = 2\mathbb{E}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( \frac{\widehat{\sigma}^2}{n} - \frac{\sigma^2}{n}\right) + \sigma^2, \end{split}$$ where the first equality follows from ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}) and ([\[eq:cr\]](#eq:cr){reference-type="ref" reference="eq:cr"}), the second equality follows from $\mathbb{E}{\widehat{\theta}_l^2}= \theta_l^2+\sigma^2/n$, and the last equality is due to $\|\mathbf{f}\|^2=\sum_{j=1}^{p_n} \left(\boldsymbol{\phi}_j^{\top}\mathbf{f}\right)^2= n\sum_{j=1}^{p_n}\theta_j^2$. Combining ([\[eq:addd1\]](#eq:addd1){reference-type="ref" reference="eq:addd1"}) with ([\[eq:addd2\]](#eq:addd2){reference-type="ref" reference="eq:addd2"}), we have $$\label{eq:difference1} \begin{split} \mathbb{E}L_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})&= \mathbb{E}C_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{y})-\sigma^2 +2\mathbb{E}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j\left( e_l^2-\frac{\sigma^2}{n}+\theta_le_l \right)\\ &+2\mathbb{E}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j\left( \frac{\sigma^2}{n} - \frac{\widehat{\sigma}^2}{n} \right)\\ \leq R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})&+\frac{2\sigma^2}{n}\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j\left( z_l^2-1\right) \right| +\frac{2\sigma}{\sqrt{n}}\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}(1-\widehat{\gamma}_j)\theta_lz_l\right|\\ &+2\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j\left(\frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right)\right|+2\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j^*\left(\frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right)\right|, \end{split}$$ where the inequality in ([\[eq:difference1\]](#eq:difference1){reference-type="ref" reference="eq:difference1"}) follows from $C_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{y}) \leq C_n(\mathbf{w}^*|\mathcal{M},\mathbf{y})$ and the absolute value inequalities, and $z_l=\sqrt{n}e_l/\sigma$, $l=1,\ldots,k_{M_n}$. From ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}) with ([\[eq:cr\]](#eq:cr){reference-type="ref" reference="eq:cr"}), in the same manner we can see that $$\label{eq:addd3} \begin{split} & R_n(\mathbf{w}|\mathcal{M},\mathbf{f})- C_{n}(\mathbf{w}|\mathcal{M},\mathbf{y}) \\ & = \sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j} \left[ (\gamma_j^2-2\gamma_j)(\theta_l^2 - \widehat{\theta}_l^2) +\gamma_j^2\frac{\sigma^2}{n} -2\gamma_j\frac{\widehat{\sigma}^2}{n} \right]+\sum_{j=1}^{p_n}\theta_j^2-\frac{1}{n}\sum_{i=1}^{n}y_i^2\\ & = \sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}(\gamma_j^2-2\gamma_j)\left(\frac{\sigma^2}{n}-e_l^2-2\theta_le_l\right)+2 \sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left(\frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right)\\ & \quad- \frac{1}{n}\mathbf{f}^{\top}\boldsymbol{\epsilon}- \frac{1}{n}\|\boldsymbol{\epsilon}\|^2, \end{split}$$ where the second equality follows from $\widehat{\theta}_l^2=\theta_l^2 + 2\theta_le_l +e_l^2$. Combining ([\[eq:addd3\]](#eq:addd3){reference-type="ref" reference="eq:addd3"}) with ([\[eq:addd2\]](#eq:addd2){reference-type="ref" reference="eq:addd2"}), we have $$\label{eq:difference2} \begin{split} \mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f}) &=\mathbb{E}C_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{y})-\sigma^2+\mathbb{E}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}(\widehat{\gamma}_j^2-2\widehat{\gamma}_j)\left(\frac{\sigma^2}{n}-e_l^2-2\theta_le_l\right)\\ &+2 \mathbb{E}\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j\left(\frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right) \\ \leq R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})&+\frac{\sigma^2}{n}\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j^2\left( z_l^2-1\right) \right| +\frac{2\sigma^2}{n}\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j\left( z_l^2-1\right) \right| \\ &+\frac{2\sigma}{\sqrt{n}}\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}(1-\widehat{\gamma}_j)^2\theta_lz_l\right|+2\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j\left(\frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right)\right|\\ &+2\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j^*\left(\frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right)\right|. \end{split}$$ The main idea of the proof is to take the upper bounds of the terms in ([\[eq:difference1\]](#eq:difference1){reference-type="ref" reference="eq:difference1"}) and ([\[eq:difference2\]](#eq:difference2){reference-type="ref" reference="eq:difference2"}). We first bound $\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j( z_l^2-1) \right|$. Define $k_0=0$, $\widehat{\gamma}_{M_n+1}=0$, and a random variable $\kappa_1=\max_{1\leq j \leq M_n}\{| \sum_{l=1}^{k_j}(z_l^2-1) |k_j^{-1/2}\}$. Note that $$\label{eq:F1} \begin{split} &\sum_{j=1}^{M_n}\frac{\left( k_j^{\frac{1}{2}} -k_{j-1}^{\frac{1}{2}}\right)^2}{k_j-k_{j-1}} = 1+ \sum_{j=2}^{M_n}\left(\frac{ k_j^{\frac{1}{2}} -k_{j-1}^{\frac{1}{2}}}{k_j-k_{j-1}}\right)^2(k_j-k_{j-1})\\ & \leq 1+ \sum_{j=2}^{M_n}\frac{k_{j} - k_{j-1}}{4k_{j-1}} =1+ \sum_{j=1}^{M_n-1}\frac{k_{j+1} - k_{j}}{4k_{j}}, \end{split}$$ where the inequality is due to the concavity of the function $h_1(x)=x^{1/2}$. Using summation by parts, we can rewrite the first term as $$\label{eq:step21} \begin{split} &\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j( z_l^2-1) \right|\\ &= \mathbb{E}\left|\sum_{j=1}^{M_n}(\widehat{\gamma}_j-\widehat{\gamma}_{j+1})\sum_{l=1}^{k_j}(z_l^2-1)\right|\\ &\leq \mathbb{E}\left\{\kappa_1\sum_{j=1}^{M_n}(\widehat{\gamma}_j-\widehat{\gamma}_{j+1})k_j^{\frac{1}{2}}\right\}\\ &=\mathbb{E}\left\{ \kappa_1\sum_{j=1}^{M_n}\widehat{\gamma}_j\left( k_j^{\frac{1}{2}}-k_{j-1}^{\frac{1}{2}}\right)\right\}\\ &\leq\mathbb{E}\left\{ \kappa_1 \left[\sum_{j=1}^{M_n}\widehat{\gamma}_j^2\left(k_j-k_{j-1} \right)\right]^{\frac{1}{2}} \left[\sum_{j=1}^{M_n}\frac{\left( k_j^{\frac{1}{2}} -k_{j-1}^{\frac{1}{2}}\right)^2}{k_j-k_{j-1}}\right]^{\frac{1}{2}}\right\}\\ &\leq \frac{C\sqrt{n}}{\sigma}\left(\mathbb{E}\kappa_1^2\right)^{\frac{1}{2}}\left[\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})\right]^{\frac{1}{2}}\left(1+ \sum_{j=1}^{M_n-1}\frac{k_{j+1} - k_{j}}{4k_{j}}\right)^{\frac{1}{2}},\\ \end{split}$$ where the first inequality follows from the definition of $\kappa_1$, the second inequality follows from the Cauchy-Schwarz inequality, and the third inequality follows from the Cauchy-Schwarz inequality, ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}), and ([\[eq:F1\]](#eq:F1){reference-type="ref" reference="eq:F1"}). The task is now to construct an upper bound for $\left(\mathbb{E}\kappa_1^2\right)^{1/2}$ by Lemma [Lemma 2](#lemma){reference-type="ref" reference="lemma"}. It remains to check ([\[eq:lemma_con\]](#eq:lemma_con){reference-type="ref" reference="eq:lemma_con"}) for the stochastic process $\xi_1(t)=\sum_{l=1}^{k_t}(z_l^2-1) k_t^{-1/2}$. Recall that $z_l=\sqrt{n}e_l/\sigma=\boldsymbol{\phi}_l^{\top}\boldsymbol{\epsilon}/\sigma$. Define an $n \times n$ matrix $$\mathbf{A}\triangleq \frac{\sum_{l=1}^{k_t}\boldsymbol{\phi}_l \boldsymbol{\phi}_l^{\top}}{\sigma^2\sqrt{k_t}}.$$ Then we can write $\xi_1(t)$ as $$\begin{split} \xi_1(t) & = \boldsymbol{\epsilon}^{\top}\left(\frac{\sum_{l=1}^{k_t}\boldsymbol{\phi}_l \boldsymbol{\phi}_l^{\top}}{\sigma^2\sqrt{k_t}} \right)\boldsymbol{\epsilon}- \sqrt{k_t} = \boldsymbol{\epsilon}^{\top} \mathbf{A}\boldsymbol{\epsilon}- \mathbb{E}\boldsymbol{\epsilon}^{\top} \mathbf{A}\boldsymbol{\epsilon}. \end{split}$$ Using Hansen-Wright inequality for sub-Gaussian random variables (Theorem 1.1 of [@Rudelson2013]), we know that there exists a positive absolute constant $c$ such that for any $x\geq 0$, $$\label{eq:HW_ineq} \begin{split} \mathbb{P}\left( \left|\xi_1(t)\right| >x \right)& = \mathbb{P}\left( \left|\boldsymbol{\epsilon}^{\top} \mathbf{A}\boldsymbol{\epsilon}- \mathbb{E}\boldsymbol{\epsilon}^{\top} \mathbf{A}\boldsymbol{\epsilon}\right|>x \right)\\ &\leq 2\exp\left[ -c\min\left(\frac{x}{\eta^2\|\mathbf{A}\|_2} , \frac{x^2}{\eta^4\|\mathbf{A}\|_{\mathrm{F}}^2} \right) \right] \\ &\leq 2\exp\left[ -c\min\left(x , x^2 \right) \right], \end{split}$$ where the second inequality follows from $\|\mathbf{A}\|_2 = 1/(\sigma^2\sqrt{k_t})\leq 1/\sigma^2$ and $\|\mathbf{A}\|_{\mathrm{F}}^2=\operatorname{tr}(\mathbf{A}^{\top}\mathbf{A})=1/\sigma^4$. The inequality ([\[eq:HW_ineq\]](#eq:HW_ineq){reference-type="ref" reference="eq:HW_ineq"}) also implies that $$\mathbb{P}\left( \left|\xi_1(t)\right|>\frac{\log x}{\lambda} \right)\leq \left\{\begin{array}{ll} 2x^{-\frac{c}{\lambda^2}\log x} &\quad 0\leq x<\exp(\lambda), \\ 2x^{-\frac{c}{\lambda}} &\quad x\geq \exp(\lambda),\\ \end{array}\right.$$ where $\lambda>0$. Thus we have $$\label{eq:expect_xi} \begin{split} \mathbb{E}\exp(\lambda| \xi_1(t) |) & = \int_{0}^{\infty}\mathbb{P}(\exp(\lambda| \xi_1(t) |)>x)dx = \int_{0}^{\infty}\mathbb{P}\left(| \xi_1(t) |>\frac{\log x}{\lambda}\right)dx \\ & \leq 2\int_{0}^{\exp(\lambda)}x^{-\frac{c}{\lambda^2}\log x}dx+2\int_{\exp(\lambda)}^{\infty}x^{-\frac{c}{\lambda}}dx. \end{split}$$ When $0<\lambda<c$, the first term of ([\[eq:expect_xi\]](#eq:expect_xi){reference-type="ref" reference="eq:expect_xi"}) is upper bounded by $$\label{eq:expect_xi_1} \begin{split} 2\int_{0}^{\exp(\lambda)}x^{-\frac{c}{\lambda^2}\log x}dx & = \frac{2\lambda^2}{c}\int_{-\frac{c}{\lambda}}^{\infty}\exp\left[ -\frac{\lambda^2(u^2+u)}{c} \right]du\\ & \leq \frac{2\lambda^2}{c}\exp\left(\frac{\lambda^2}{4c} \right)\sqrt{\frac{\pi c}{\lambda^2}}\\ &\leq 2\exp\left(\frac{c}{4}\right)\sqrt{\pi c}<\infty. \end{split}$$ And the second term of ([\[eq:expect_xi\]](#eq:expect_xi){reference-type="ref" reference="eq:expect_xi"}) is $$\label{eq:expect_xi_2} 2\int_{\exp(\lambda)}^{\infty}x^{-\frac{c}{\lambda}}dx = \frac{2}{\frac{c}{\lambda}-1}\exp(-c+\lambda)< \infty.$$ Combining ([\[eq:expect_xi_1\]](#eq:expect_xi_1){reference-type="ref" reference="eq:expect_xi_1"})--([\[eq:expect_xi_2\]](#eq:expect_xi_2){reference-type="ref" reference="eq:expect_xi_2"}) with ([\[eq:expect_xi\]](#eq:expect_xi){reference-type="ref" reference="eq:expect_xi"}), we see that when $0<\lambda<c$, $\mathbb{E}\exp(\lambda| \xi_1(t) |)$ is uniformly upper bounded for any $t=1,\ldots,M_n$, which meets the condition ([\[eq:lemma_con\]](#eq:lemma_con){reference-type="ref" reference="eq:lemma_con"}) of Lemma [Lemma 2](#lemma){reference-type="ref" reference="lemma"}. Thus we have $\left(\mathbb{E}\kappa_1^2\right)^{1/2} \leq C (1+\log M_n)$, and the term $\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j( z_l^2-1) \right|$ is upper bounded by $$\label{eq:step211} \mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j( z_l^2-1) \right| \leq \frac{C\sqrt{n}}{\sigma}[\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}\left[\psi(\mathcal{M})\right]^{\frac{1}{2}},$$ where $\psi(\mathcal{M})$ is defined in ([\[eq:psi_M\]](#eq:psi_M){reference-type="ref" reference="eq:psi_M"}). We now turn to find the upper bound of $\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}(1-\widehat{\gamma}_j)\theta_lz_l\right|$. Define $S_t=\sum_{l=k_t+1}^{k_{M_n}}\theta_l^2$ and a random variable $\kappa_2=\max_{1\leq t \leq M_n}\left\{\left| \sum_{l=k_t+1}^{k_{M_n}}\theta_lz_l \right|S_t^{-1/2}\right\}$. Note that $$\label{eq:func2} \begin{split} &\sum_{j=1}^{M_n}\frac{\left[(S_{j-1}+1)^{\frac{1}{2}}-(S_{j}+1)^{\frac{1}{2}}\right]^2}{S_{j-1} - S_{j}}\\ &=\sum_{j=1}^{M_n}\left[\frac{(S_{j-1}+1)^{\frac{1}{2}}-(S_{j}+1)^{\frac{1}{2}}}{S_{j-1} - S_{j}}\right]^2(S_{j-1} - S_{j})\\ & \leq \frac{1}{4}\sum_{j=1}^{M_n}\left( S_{j-1} - S_{j} \right) < \infty, \end{split}$$ where the inequality follows from $h_2(x)=(x+1)^{1/2}$ and $h_2'(x)=(1/2)(x+1)^{-1/2}\leq 1/2$ when $x\geq 0$, and the second inequality is due to ([\[eq:sum_theta\]](#eq:sum_theta){reference-type="ref" reference="eq:sum_theta"}). Using summation by parts again, we see that $$\label{eq:important2} \begin{split} &\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}(1-\widehat{\gamma}_j)\theta_lz_l\right|\\ &= \mathbb{E}\left|\sum_{j=2}^{M_n}(\widehat{\gamma}_{j-1}-\widehat{\gamma}_j)\sum_{l=k_{j-1}+1}^{k_{M_n}}\theta_lz_l\right|\\ &\leq \mathbb{E}\left\{\kappa_2 \sum_{j=2}^{M_n}(\widehat{\gamma}_{j-1}-\widehat{\gamma}_j)(S_{j-1}+1)^{\frac{1}{2}}(S_{j-1})^{\frac{1}{2}}(S_{j-1}+1)^{-\frac{1}{2}}\right\}\\ &\leq \mathbb{E}\left\{\kappa_2 \sum_{j=2}^{M_n}(\widehat{\gamma}_{j-1}-\widehat{\gamma}_j)(S_{j-1}+1)^{\frac{1}{2}}\right\}\\ &= \mathbb{E}\left\{\kappa_2 \sum_{j=1}^{M_n}(1-\widehat{\gamma}_j)\left[(S_{j-1}+1)^{\frac{1}{2}}-(S_{j}+1)^{\frac{1}{2}}\right]\right\}\\ & \leq \mathbb{E}\left\{\kappa_2 \left[\sum_{j=1}^{M_n}(1-\widehat{\gamma}_j)^2(S_{j-1} - S_{j} )\right]^{\frac{1}{2}}\left[\sum_{j=1}^{M_n}\frac{\left[(S_{j-1}+1)^{\frac{1}{2}}-(S_{j}+1)^{\frac{1}{2}}\right]^2}{S_{j-1} - S_{j}}\right]^{\frac{1}{2}}\right\}\\ & \leq C (\mathbb{E} \kappa_2^2)^{\frac{1}{2}}[\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}, \end{split}$$ where the first inequality is due to the definition of $\kappa_2$, the third inequality follows from the Cauchy-Schwarz inequality, and the last inequality is due to the Cauchy-Schwarz inequality, ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}), and ([\[eq:func2\]](#eq:func2){reference-type="ref" reference="eq:func2"}). Now we construct upper bound for $(\mathbb{E} \kappa_2^2)^{1/2}$ by Lemma [Lemma 2](#lemma){reference-type="ref" reference="lemma"}. Consider the stochastic process $\xi_2(t) = ( \sum_{l=k_t+1}^{k_{M_n}}\theta_lz_l )S_t^{-1/2}$. Recall that $z_l=\boldsymbol{\phi}_l^{\top}\boldsymbol{\epsilon}/\sigma$. Define an $n$-dimensional vector $$\mathbf{a}\triangleq \frac{1}{\sigma S_t^{\frac{1}{2}}}\left(\boldsymbol{\phi}_{k_t+1},\ldots, \boldsymbol{\phi}_{k_{M_n}} \right)\begin{pmatrix} \theta_{k_t+1} \\ \vdots \\ \theta_{k_{M_n}} \end{pmatrix}.$$ We write $\xi_2(t)$ as $$\xi_2(t)= \frac{1}{\sigma S_t^{\frac{1}{2}}}\left(\theta_{k_t+1},\ldots,\theta_{k_{M_n}} \right)\begin{pmatrix} \boldsymbol{\phi}_{k_t+1}^{\top} \\ \vdots \\ \boldsymbol{\phi}_{k_{M_n}}^{\top} \end{pmatrix}\boldsymbol{\epsilon}= \mathbf{a}^{\top}\boldsymbol{\epsilon}.$$ Since the elements of $\boldsymbol{\epsilon}$ are i.i.d. $\eta$-sub-Gaussian variables, from Theorem 2.6 in [@Wainwright2019], we have for any $\lambda\in \mathbb{R}$, $$\mathbb{E}\exp[\lambda\xi_2(t)]=\mathbb{E}\exp(\lambda\mathbf{a}^{\top}\boldsymbol{\epsilon})\leq \exp\left(\frac{\lambda^2\eta^2\|\mathbf{a}\|^2}{2}\right) = \exp\left(\frac{\lambda^2\eta^2}{2\sigma^2}\right),$$ where the last equality is due to $\|\mathbf{a}\|^2=1/\sigma^2$. This leads to $$\begin{split} & \mathbb{E}\exp(\lambda| \xi_2(t) |) \leq \mathbb{E}\exp[\lambda \xi_2(t) ] + \mathbb{E}\exp[-\lambda \xi_2(t) ] = 2\exp\left(\frac{\lambda^2\eta^2}{2\sigma^2}\right)< \infty,\\ \end{split}$$ which verifies the condition ([\[eq:lemma_con\]](#eq:lemma_con){reference-type="ref" reference="eq:lemma_con"}) of Lemma [Lemma 2](#lemma){reference-type="ref" reference="lemma"}. Thus combining Lemma [Lemma 2](#lemma){reference-type="ref" reference="lemma"} with ([\[eq:important2\]](#eq:important2){reference-type="ref" reference="eq:important2"}), we have the second term $$\label{eq:important211} \mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}(1-\widehat{\gamma}_j)\theta_lz_l\right| \leq C [\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})]^{\frac{1}{2}} (1+\log M_n).$$ Based on the same reasoning adopted in ([\[eq:step211\]](#eq:step211){reference-type="ref" reference="eq:step211"}) and ([\[eq:important211\]](#eq:important211){reference-type="ref" reference="eq:important211"}), and the fact that $0\leq \widehat{\gamma}_j\leq 1,j=1,\ldots,M_n$, we can also prove that $$\label{eq:important3} \begin{split} \mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j^2( z_l^2-1) \right|&\leq \frac{C\sqrt{n}}{\sigma}[\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}\left[\psi(\mathcal{M})\right]^{\frac{1}{2}} \end{split}$$ and $$\label{eq:important4} \begin{split} \mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}(1-\widehat{\gamma}_j)^2\theta_lz_l\right| \leq C [\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})]^{\frac{1}{2}} (1+\log M_n). \end{split}$$ Using the Cauchy-Schwarz inequality and ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}), we observe that $$\label{eq:important5} \begin{split} &\mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j\left(\frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right)\right| \\ &\leq \mathbb{E}\left\{\left(\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\widehat{\gamma}_j^2\frac{\sigma^2}{n}\right)^{\frac{1}{2}}\left[\frac{nk_{M_n}}{\sigma^2}\left( \frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right)^2\right]^{\frac{1}{2}} \right\} \\ &\leq [\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}\left[\frac{k_{M_n}}{n\sigma^2}\mathbb{E}\left( \sigma^2-\widehat{\sigma}^2 \right)^2\right]^{\frac{1}{2}}, \end{split}$$ and $$\label{eq:important6} \begin{split} \mathbb{E}\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j^*\left(\frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right)\right| & \leq [R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}\left[\frac{k_{M_n}}{n\sigma^2}\mathbb{E}\left( \sigma^2-\widehat{\sigma}^2 \right)^2\right]^{\frac{1}{2}}. \end{split}$$ Substituting ([\[eq:step211\]](#eq:step211){reference-type="ref" reference="eq:step211"}), ([\[eq:important2\]](#eq:important2){reference-type="ref" reference="eq:important2"}), and ([\[eq:important3\]](#eq:important3){reference-type="ref" reference="eq:important3"})--([\[eq:important6\]](#eq:important6){reference-type="ref" reference="eq:important6"}) into ([\[eq:difference1\]](#eq:difference1){reference-type="ref" reference="eq:difference1"}) and ([\[eq:difference2\]](#eq:difference2){reference-type="ref" reference="eq:difference2"}) yields $$\label{eq:inequality1} \begin{split} Q_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})&\leq R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f}) +\frac{C\sigma}{\sqrt{n}}[\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}\left[\psi(\mathcal{M})\right]^{\frac{1}{2}}\\ &+\left[\frac{k_{M_n}}{n\sigma^2}\mathbb{E}\left( \sigma^2-\widehat{\sigma}^2 \right)^2\right]^{\frac{1}{2}}\left[[\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}+[R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}\right]. \end{split}$$ In particular, when $Q_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$ represents $\mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$, ([\[eq:inequality1\]](#eq:inequality1){reference-type="ref" reference="eq:inequality1"}) also implies that $$\label{eq:inequality2} \begin{split} \mathbb{E}R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})&\leq 2\left\{R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})+\left[\frac{k_{M_n}}{n\sigma^2}\mathbb{E}\left( \sigma^2-\widehat{\sigma}^2 \right)^2\right]^{\frac{1}{2}}[R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})]^{\frac{1}{2}}\right\}\\ &+\left\{\frac{C\sigma}{\sqrt{n}}\left[\psi(\mathcal{M})\right]^{\frac{1}{2}}+ \left[\frac{2k_{M_n}}{n\sigma^2}\mathbb{E}\left( \sigma^2-\widehat{\sigma}^2 \right)^2\right]^{\frac{1}{2}}\right\}^2. \end{split}$$ Therefore, after inserting ([\[eq:inequality2\]](#eq:inequality2){reference-type="ref" reference="eq:inequality2"}) into the right side of ([\[eq:inequality1\]](#eq:inequality1){reference-type="ref" reference="eq:inequality1"}) and some additional algebra, we see that ([\[eq:risk_bound_general\]](#eq:risk_bound_general){reference-type="ref" reference="eq:risk_bound_general"}) holds. ## Proof of ([\[eq:variance_risk\]](#eq:variance_risk){reference-type="ref" reference="eq:variance_risk"}) {#sec:proof:eq:variance_risk} For completeness, we provide a brief proof for ([\[eq:variance_risk\]](#eq:variance_risk){reference-type="ref" reference="eq:variance_risk"}). We first decompose $\mathbb{E}(\widehat{\sigma}_{m_n}^2-\sigma^2)^2$ as the variance term and the bias term $$\label{eq:variance_1} \mathbb{E}(\widehat{\sigma}_{m_n}^2-\sigma^2)^2=\mathbb{E}(\widehat{\sigma}_{m_n}^2-\mathbb{E}\widehat{\sigma}_{m_n}^2)^2 + (\mathbb{E}\widehat{\sigma}_{m_n}^2-\sigma^2)^2.$$ Note that $$\label{eq:variance_2} \begin{split} \widehat{\sigma}_{m_n}^2 & =\frac{1}{n-m_n}\left\| \mathbf{y}- \widehat{\mathbf{f}}_{m_n} \right\|^2 \\ & =\frac{n\| \boldsymbol{\theta}_{-m_n} \|^2}{n-m_n}+\frac{\boldsymbol{\epsilon}^{\top}(\mathbf{I}-\mathbf{P}_{m_n})\boldsymbol{\epsilon}}{n-m_n}+\frac{2\mathbf{f}^{\top}(\mathbf{P}_{p_n}-\mathbf{P}_{m_n})\boldsymbol{\epsilon}}{n-m_n}, \end{split}$$ where $\boldsymbol{\theta}_{-m_n}=(\theta_{m_n+1},\ldots, \theta_{p_n})^{\top}$. Thus, the bias term of ([\[eq:variance_1\]](#eq:variance_1){reference-type="ref" reference="eq:variance_1"}) equals to $$\label{eq:variance_3} (\mathbb{E}\widehat{\sigma}_{m_n}^2-\sigma^2)^2 = \left( \frac{n\| \boldsymbol{\theta}_{-m_n} \|^2}{n-m_n} + \sigma^2- \sigma^2 \right)^2 = \frac{n^2 \|\boldsymbol{\theta}_{-m_n} \|^4}{(n-m_n)^2}.$$ We proceed to construct an upper bound for the variance term $\mathbb{E}(\widehat{\sigma}_{m_n}^2-\mathbb{E}\widehat{\sigma}_{m_n}^2)^2$. According to Theorem 1.1 of [@Rudelson2013], we have $$\label{eq:HW_ineq_2} \begin{split} &\mathbb{P}\left( \left|\frac{\boldsymbol{\epsilon}^{\top}(\mathbf{I}-\mathbf{P}_{m_n})\boldsymbol{\epsilon}}{n-m_n} - \mathbb{E}\frac{\boldsymbol{\epsilon}^{\top}(\mathbf{I}-\mathbf{P}_{m_n})\boldsymbol{\epsilon}}{n-m_n}\right|>x \right)\\ &\leq 2\exp\left[ -c(n-m_n)(x \wedge x^2 ) \right]. \end{split}$$ And due to the sub-Gaussian property of $\boldsymbol{\epsilon}$, we have $$\label{eq:v_o_1} \mathbb{P}\left( \left| \frac{2\mathbf{f}^{\top}(\mathbf{P}_{p_n}-\mathbf{P}_{m_n})\boldsymbol{\epsilon}}{n-m_n} \right|>x \right)\leq 2\exp\left[ -\frac{c(n-m_n)^2x^2}{n\| \boldsymbol{\theta}_{-m_n} \|^2} \right].$$ Combining ([\[eq:HW_ineq_2\]](#eq:HW_ineq_2){reference-type="ref" reference="eq:HW_ineq_2"})--([\[eq:v_o\_1\]](#eq:v_o_1){reference-type="ref" reference="eq:v_o_1"}) with ([\[eq:variance_2\]](#eq:variance_2){reference-type="ref" reference="eq:variance_2"}) yields $$\label{eq:v_o_2} \begin{split} & \mathbb{P}\left( |\widehat{\sigma}_{m_n}^2-\mathbb{E}\widehat{\sigma}_{m_n}^2|>x \right) \\ & \leq 4\exp\left\{ -c\min\left[(n-m_n)x , \frac{(n-m_n)^2x^2}{(n-m_n)\vee(n\| \boldsymbol{\theta}_{-m_n} \|^2)} \right] \right\}. \end{split}$$ By integrating the tail probability, we have $$\label{eq:variance_term} \begin{split} \mathbb{E}(\widehat{\sigma}_{m_n}^2-\mathbb{E}\widehat{\sigma}_{m_n}^2)^2 & = \int_{0}^{\infty}\mathbb{P}\left( |\widehat{\sigma}_{m_n}^2-\mathbb{E}\widehat{\sigma}_{m_n}^2|>\sqrt{x} \right)dx \\ &\lesssim \frac{1}{n-m_n} \vee \frac{n\| \boldsymbol{\theta}_{-m_n} \|^2}{(n-m_n)^2}.\\ \end{split}$$ Combining ([\[eq:variance_3\]](#eq:variance_3){reference-type="ref" reference="eq:variance_3"}) with ([\[eq:variance_term\]](#eq:variance_term){reference-type="ref" reference="eq:variance_term"}) gives ([\[eq:variance_risk\]](#eq:variance_risk){reference-type="ref" reference="eq:variance_risk"}). ## Proof of Theorem [Theorem 2](#theorem:aop){reference-type="ref" reference="theorem:aop"} {#sec:proof:theorem:aop} The proof of this theorem is straightforward in view of Theorem [Theorem 1](#theorem:main){reference-type="ref" reference="theorem:main"}, ([\[eq:variance_risk\]](#eq:variance_risk){reference-type="ref" reference="eq:variance_risk"}), and Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"}. ## Proof of Theorem [Theorem 3](#cor:grouped){reference-type="ref" reference="cor:grouped"} {#proof-of-theorem-corgrouped} The proof of this theorem follows from the techniques in [@cavalier2001penalized]. We first show that $$\label{eq:ppc} R_n(\mathbf{w}^*|\mathcal{M}_g,\mathbf{f})\leq (1+\zeta_n)R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})+\frac{k_1\sigma^2}{n}.$$ Define an $M_n$-dimensional weight vector $\bar{\mathbf{w}}=(\bar{w}_1,\ldots,\bar{w}_{M_n})^{\top}$, where $\bar{w}_m=\sum_{j=k_{m-1}+1}^{k_m}w_j^*$, $\bar{\gamma}_m=\sum_{j=m}^{M_n}\bar{w}_m$, and $w_j^*$ is the $j$-th element of $\mathbf{w}^*|\mathcal{M}_a$. According to ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}), we have $$\label{eq:a38} R_n(\bar{\mathbf{w}}|\mathcal{M}_g,\mathbf{f})\leq \sum_{j=1}^{p_n}(1-\gamma_j^*)^2\theta_j^2+\frac{\sigma^2}{n}\sum_{j=1}^{M_n}(k_j-k_{j-1})\bar{\gamma}_j^2,$$ where the inequality follows the fact that $\bar{\gamma}_m\geq \gamma_j^*$ for any $k_{m-1}+1\leq j \leq k_m$. Note that $$\label{eq:a39} \begin{split} \sum_{j=1}^{M_n}(k_j-k_{j-1})\bar{\gamma}_j^2& \leq k_1+(1+\zeta_n)\sum_{j=2}^{M_n}(k_{j-1}-k_{j-2})\bar{\gamma}_j^2\\ & \leq k_1+(1+\zeta_n)\sum_{j=1}^{p_n}(\gamma_j^*)^2, \end{split}$$ where the second inequality is due to $\bar{\gamma}_m\leq \gamma_j^*$ when $k_{m-2}+1\leq j \leq k_{m-1}$. Substituting ([\[eq:a39\]](#eq:a39){reference-type="ref" reference="eq:a39"}) into ([\[eq:a38\]](#eq:a38){reference-type="ref" reference="eq:a38"}), we obtain ([\[eq:ppc\]](#eq:ppc){reference-type="ref" reference="eq:ppc"}). Then provided $k_1=o(m_n^*)$ and $\zeta_n =o(1)$, we have $R_n(\mathbf{w}^*|\mathcal{M}_g,\mathbf{f})\sim R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})$. The proof is completed using the AOP theory of MMA given in Theorem [Theorem 2](#theorem:aop){reference-type="ref" reference="theorem:aop"}. ## Proof of Theorem [Theorem 4](#cor:minimum_1){reference-type="ref" reference="cor:minimum_1"} {#proof-of-theorem-corminimum_1} Define the random variable $\Delta_{n1}=R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS1},\mathbf{f})-R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})$, which measures the risk increment of using the reduced candidate model set $\widehat{\mathcal{M}}_{MS1}$. In view of the risk bound ([\[eq:risk_bound_general\]](#eq:risk_bound_general){reference-type="ref" reference="eq:risk_bound_general"}), it suffices to prove $$\label{eq:411} \frac{\mathbb{E}\Delta_{n1}}{R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})}=\frac{\mathbb{E}(\Delta_{n1}1_{\bar{F}_n})+\mathbb{E}(\Delta_{n1}1_{F_n})}{R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})} \to 0$$ and $$\label{eq:condition_psi} \frac{\mathbb{E}\psi(\widehat{\mathcal{M}}_{MS1})}{nR_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})}\to 0.$$ The condition ([\[eq:condition_psi\]](#eq:condition_psi){reference-type="ref" reference="eq:condition_psi"}) is satisfied due to ([\[eq:Ephi\]](#eq:Ephi){reference-type="ref" reference="eq:Ephi"}) and Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"}. Then our main task is to prove ([\[eq:411\]](#eq:411){reference-type="ref" reference="eq:411"}). We have the first part of ([\[eq:411\]](#eq:411){reference-type="ref" reference="eq:411"}) $$\label{eq:first_part_411} \frac{\mathbb{E}(\Delta_{n1}1_{\bar{F}_n})}{R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})}\lesssim \frac{\mathbb{P}(\bar{F}_n)}{m_n^*/n}\to 0,$$ where the inequality is due to Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"} and $$\begin{split} \Delta_{n1} & \leq R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS1},\mathbf{f})\leq \max_{\mathcal{M}\subseteq\{1,\ldots,p_n\}}R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f}) \\ & \leq \max_{m\in\{1,\ldots,p_n\}}R_n(m,\mathbf{f})< C, \end{split}$$ and the approximation is due to the assumption ([\[eq:pG\]](#eq:pG){reference-type="ref" reference="eq:pG"}). Now we turn to prove the second part of ([\[eq:411\]](#eq:411){reference-type="ref" reference="eq:411"}). From ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}), we have $$R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})=\frac{\sigma^2}{n}+\sum_{j=2}^{p_n}\frac{\theta_j^2\sigma^2}{n\theta_j^2+\sigma^2}.$$ Since $R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS1},\mathbf{f})$ is defined by directly plugging $\widehat{\mathcal{M}}_{MS1}$ into the expression of $R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})$, we have $$R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS1},\mathbf{f})=\frac{\widehat{l}_n\sigma^2}{n}+\sum_{j=\widehat{l}_n+1}^{\widehat{u}_n}\frac{\theta_j^2\sigma^2}{n\theta_j^2+\sigma^2}+\sum_{j=\widehat{u}_n+1}^{p_n}\theta_j^2.$$ When $F_n$ holds, $\Delta_{n1}$ is upper bounded by $$\begin{split} \Delta_{n1}&=R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS1},\mathbf{f})-R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})\\ &=\sum_{j=2}^{\widehat{l}_n}\left( \frac{\sigma^2}{n} - \frac{\sigma^2}{n+\frac{\sigma^2}{\theta_j^2}} \right) + \sum_{j=\widehat{u}_n+1}^{p_n}\frac{\theta_j^2}{1+\frac{\sigma^2}{n\theta_j^2}}\\ &\leq \frac{\widehat{l}_n}{n}\sigma^2+\sum_{j=\widehat{u}_n+1}^{p_n}\frac{\theta_j^2}{1+\frac{\theta_{m_n^*+1}^2}{\theta_j^2}}\\ &\leq \frac{\widehat{l}_n}{n}\sigma^2+\sum_{j=m_n^*+1}^{p_n}\frac{\theta_j^2}{1+\frac{\theta_{m_n^*+1}^2}{\theta_{\widehat{u}_n}^2}}\\ &\leq \frac{c_2m_n^*}{nk_l }+ \sum_{j=m_n^*+1}^{p_n}\frac{\theta_j^2}{1+\frac{\theta_{m_n^*+1}^2}{\theta_{\lfloor c_1m_n^*k_u\rfloor}^2}}, \end{split}$$ where the first inequality follows from ([\[eq:mnstar\]](#eq:mnstar){reference-type="ref" reference="eq:mnstar"}), and the last step is due to the definitions of $\widehat{l}_n$, $\widehat{u}_n$, and the event $F_n$. From this, we see that when $k_l\to \infty$ and $k_u \to \infty$ $$\begin{split} \mathbb{E}(\Delta_{n1}1_{F_n}) & \leq \frac{c_2m_n^*}{nk_l }+ \sum_{j=m_n^*+1}^{p_n}\frac{\theta_j^2}{1+\frac{\theta_{m_n^*+1}^2}{\theta_{\lfloor c_1m_n^*k_u\rfloor}^2}} =o\left(\frac{m_n^*}{n} \right)+o\left(\sum_{j=m_n^*+1}^{p_n}\theta_j^2 \right)\\ &=o\left[R_n(m_n^*,\mathbf{f}) \right]=o\left[R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f}) \right], \end{split}$$ where the first equality is due to Assumption [Assumption 3](#asmp:regressor_order2){reference-type="ref" reference="asmp:regressor_order2"}, and the second equality follows from ([\[eq:optimal_ms_risk\]](#eq:optimal_ms_risk){reference-type="ref" reference="eq:optimal_ms_risk"}), and the last equality is due to Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"}. Thus, we have proved the theorem. ## Proof of Theorem [Theorem 5](#cor:minimum_2){reference-type="ref" reference="cor:minimum_2"} {#proof-of-theorem-corminimum_2} Define the random variable $\Delta_{n2}=R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS2},\mathbf{f})-R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})$. Let us first prove the results under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}. It is evident that $$\label{eq:60} \mathbb{E}R_n(\widehat{\mathbf{w}}|\widehat{\mathcal{M}}_{MS2},\mathbf{f})-R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})\geq \mathbb{E}\Delta_{n2}\geq \mathbb{E}(\Delta_{n2}1_{F_n}),$$ where $R_n(\widehat{\mathbf{w}}|\widehat{\mathcal{M}}_{MS2},\mathbf{f})$ is defined by plugging $\widehat{\mathcal{M}}_{MS2}$ into the expression of $R_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f})$. When the event $F_n$ holds, we have $$\label{eq:61} \begin{split} \Delta_{n2} \geq \sum_{j=2}^{\lfloor c_1k_l^{-1}m_n^*\rfloor}\left( \frac{\sigma^2}{n} - \frac{\sigma^2}{n+\frac{\sigma^2}{\theta_j^2}} \right) + \sum_{j=\lfloor c_2k_um_n^*\rfloor+1}^{p_n}\frac{\theta_j^2}{1+\frac{\sigma^2}{n\theta_j^2}}. \end{split}$$ Recall the function $G_d$ defined in ([\[eq:Gfunction\]](#eq:Gfunction){reference-type="ref" reference="eq:Gfunction"}). Under Condition [Condition 1](#condition1){reference-type="ref" reference="condition1"}, there must exist two integers $d_3^*$ and $t_n^*=G_{d_3^*}(m_n^*+1)$ such that $\theta_{m_n^*+1}^2/\theta_{t_n^*}^2 \geq \delta^{2d_3^*}$ and $\lfloor c_1k_l^{-1}m_n^*\rfloor-t_n^*\asymp m_n^*$ when $k_l$ is bounded. Hence the first term on the right side of ([\[eq:61\]](#eq:61){reference-type="ref" reference="eq:61"}) can be lower bounded by $$\begin{split} &\sum_{j=2}^{\lfloor c_1k_l^{-1}m_n^*\rfloor}\left( \frac{\sigma^2}{n} - \frac{\sigma^2}{n+\frac{\sigma^2}{\theta_j^2}} \right)\\ & =\sum_{j=2}^{\lfloor c_1k_l^{-1}m_n^*\rfloor}\frac{\sigma^2}{n}-\sum_{j=2}^{t_n^*}\frac{\sigma^2}{n+\frac{\sigma^2}{\theta_j^2}}-\sum_{j=t_n^*+1}^{\lfloor c_1k_l^{-1}m_n^*\rfloor}\frac{\sigma^2}{n+\frac{\sigma^2}{\theta_j^2}}\\ &\geq \frac{(\lfloor c_1k_l^{-1}m_n^*\rfloor-t_n^*)\sigma^2}{n}-\frac{(\lfloor c_1k_l^{-1}m_n^*\rfloor-t_n^*)\sigma^2}{n(1+\delta^{2d_3^*})}\\ &\asymp \frac{m_n^*}{n}. \end{split}$$ Similarly, when $k_u$ is bounded, the second term in ([\[eq:61\]](#eq:61){reference-type="ref" reference="eq:61"}) has a lower bound with the order $m_n^*/n$. Combining this with ([\[eq:60\]](#eq:60){reference-type="ref" reference="eq:60"}), we have $$\mathbb{E}(\Delta_{n2}1_{F_n}) \gtrsim \frac{m_n^*}{n}\mathbb{P}(F_n) \gtrsim \frac{m_n^*}{n} \sim R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f}),$$ where the second inequality is due to the condition $\mathbb{P}(F_n)>C_1$, and last approximation follows from Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"}. Under Condition [Condition 2](#condition2){reference-type="ref" reference="condition2"}, it is easy to see $$\label{eq:show} \mathbb{E}R_n(\widehat{\mathbf{w}}|\widehat{\mathcal{M}}_{MS2},\mathbf{f}) = [1+o(1)]\mathbb{E}R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS2},\mathbf{f}).$$ Indeed, based on the risk bound ([\[eq:risk_bound_general\]](#eq:risk_bound_general){reference-type="ref" reference="eq:risk_bound_general"}), we only need to show that $\mathbb{E}\psi(\widehat{\mathcal{M}}_{MS2})=o(m_n^*)$. Note that $$\mathbb{E}\psi(\widehat{\mathcal{M}}_{MS2}) \asymp \mathbb{E}\log(k_lk_u)[\log(\widehat{u}_n - \widehat{l}_n)]^2 \leq C = o(m_n^*),$$ where the inequality is due to $\widehat{u}_n-\widehat{l}_n$ is bounded almost surely. Thus ([\[eq:show\]](#eq:show){reference-type="ref" reference="eq:show"}) is proved. Then define a candidate model set that contains a single model $\widehat{\mathcal{M}}_{MS3}=\{ \widehat{m}_n \}$. We see that $$\label{eq:final1} \begin{split} &\mathbb{E}R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS2},\mathbf{f}) \leq \mathbb{E}R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS3},\mathbf{f})\\ &= \mathbb{E}R_n(\widehat{m}_n,\mathbf{f})\sim R_n(m_n^*,\mathbf{f}) \sim R_n(\mathbf{w}^*|\mathcal{M}_{a},\mathbf{f}), \end{split}$$ where the last approximation follows from Lemma [Lemma 1](#lemma:peng){reference-type="ref" reference="lemma:peng"}. On the other hand, we have $$\label{eq:final2} R_n(\mathbf{w}^*|\mathcal{M}_{a},\mathbf{f}) \leq \mathbb{E}R_n(\mathbf{w}^*|\widehat{\mathcal{M}}_{MS2},\mathbf{f}).$$ By combining ([\[eq:final1\]](#eq:final1){reference-type="ref" reference="eq:final1"})--([\[eq:final2\]](#eq:final2){reference-type="ref" reference="eq:final2"}) with ([\[eq:show\]](#eq:show){reference-type="ref" reference="eq:show"}), we obtain the desired conclusion. ## Proof of Theorem [Theorem 6](#tho:minimax){reference-type="ref" reference="tho:minimax"} {#proof-of-theorem-thominimax} We first give some well-established minimax results. According to ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}), we have $$\label{eq:risk_minimax_proof} R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})=\sum_{j=1}^{n}\left[(1-\gamma_j)^2\theta_j^2+\frac{\sigma^2\gamma_j^2}{n}\right],$$ where $\gamma_j=\sum_{m=j}^{n}w_m$. Note that the MA risk ([\[eq:risk_minimax_proof\]](#eq:risk_minimax_proof){reference-type="ref" reference="eq:risk_minimax_proof"}) coincides with the risk of the linear estimator $\widehat{\boldsymbol{\theta}}(\boldsymbol{\gamma})=(\gamma_1\widehat{\theta}_1,\ldots,\gamma_n\widehat{\theta}_n)^{\top}$ in the Gaussian sequence model ([\[eq:sequence\]](#eq:sequence){reference-type="ref" reference="eq:sequence"}), i.e., $R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})=\mathbb{E}\|\widehat{\boldsymbol{\theta}}(\boldsymbol{\gamma}) -\boldsymbol{\theta}\|^2$. For the Gaussian sequence model, @Pinsker1980 obtained an exact evaluation for the linear minimax risk over the ellipsoid $\Theta(\alpha, R)$ and showed that the optimal minimax risk is asymptotically equivalent to the optimal linear minimax risk. @Pinsker1980's results yield the minimax risk and the linear-combined minimax risk of MA $$\label{eq:pinsker} R_M\left[\mathcal{F}_{\Theta(\alpha, R)}\right]\sim R_L\left[\mathcal{F}_{\Theta(\alpha, R)}\right] \sim C_1\left(\frac{\sigma^2}{n} \right)^{\frac{2\alpha}{2\alpha+1}},$$ where $C_1$ is the Pinsker constant which only depends on $\alpha$ and $R$. Define $x_{+}=\max(x, 0)$. The minimax optimal weights are given by $\widetilde{w}_j^*=\widetilde{\gamma}_j^*-\widetilde{\gamma}^*_{j-1}, j=1\ldots,n$, where $$\label{eq:pinsker2} \widetilde{\gamma}_j^*=\left[1-C_2\left(\frac{\sigma^2}{n} \right)^{\frac{\alpha}{2\alpha+1}} j^{\alpha}\right]_+,$$ and $C_2$ a constant that depends on $\alpha$ and $R$. Since $\widetilde{\gamma}_1^* \to 1$ and $\widetilde{\gamma}_j^*\geq \widetilde{\gamma}_{j+1}^*$, we see that $(\widetilde{w}_1^*,\ldots,\widetilde{w}_n^*)$ approximately lies in the unit simplex $\mathcal{W}_n$. Then, taking the upper bound on both sides of ([\[eq:risk_bound_general\]](#eq:risk_bound_general){reference-type="ref" reference="eq:risk_bound_general"}) with respect to $\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}$ gives $$\label{eq:mini1} \begin{split} &\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\widehat{\mathbf{w}}|\mathcal{M}_a,\mathbf{f}) \leq \sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})+\frac{C\sigma^2}{n}\psi(\mathcal{M}_a)\\ & +\frac{C\sigma}{\sqrt{n}}\left[\psi(\mathcal{M}_a)\right]^{\frac{1}{2}}\left[\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f})\right]^{\frac{1}{2}}+ C\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}\rho\left(n, \mathcal{M}_a,\mathbf{f},\widehat{\sigma}^2,\sigma^2\right).\\ \end{split}$$ The first term on the right side of ([\[eq:mini1\]](#eq:mini1){reference-type="ref" reference="eq:mini1"}) is upper bounded by $$\label{eq:mini1_1} \begin{split} &\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}^*|\mathcal{M}_a,\mathbf{f}) \leq \inf_{\mathbf{w}\in \mathcal{W}_n}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f}) \\ & =R_L\left[\mathcal{F}_{\Theta(\alpha, R)}\right]+\inf_{\mathbf{w}\in \mathcal{W}_n}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})-\inf_{\mathbf{w}}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f}), \end{split}$$ where the first inequality is due to the definition of $\mathbf{w}^*|\mathcal{M}_a$, and the second equality is due to the definition of $R_L\left[\mathcal{F}_{\Theta(\alpha, R)}\right]$. The last term on the right side of ([\[eq:mini1\]](#eq:mini1){reference-type="ref" reference="eq:mini1"}) is upper bounded by $$\label{eq:mini1_2} \begin{split} &\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}\rho\left(n, \mathcal{M}_a,\mathbf{f},\widehat{\sigma}^2,\sigma^2\right) \leq \sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}\left[\frac{1}{\sigma^2}\mathbb{E}\left( \widehat{\sigma}^2-\sigma^2 \right)^2\right]\\ & + \left\{\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}\left[\frac{1}{\sigma^2}\mathbb{E}\left( \widehat{\sigma}^2-\sigma^2 \right)^2\right]\right\}^{\frac{1}{2}}\left\{ R_L\left[\mathcal{F}_{\Theta(\alpha, R)}\right]+\inf_{\mathbf{w}\in \mathcal{W}_n}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})\right.\\ &\left.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\inf_{\mathbf{w}}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f}) \right\}^{\frac{1}{2}}. \end{split}$$ Thus, it remains to prove $$\label{eq:remain0} \inf_{\mathbf{w}\in \mathcal{W}_n}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})-\inf_{\mathbf{w}}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})= o\left(R_L\left[\mathcal{F}_{\Theta(\alpha, R)}\right]\right),$$ $$\label{eq:remain1} \psi(\mathcal{M}_a) = o\left(R_L\left[\mathcal{F}_{\Theta(\alpha, R)}\right]\right),$$ and $$\label{eq:remain2} \sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}\mathbb{E}\left( \widehat{\sigma}^2-\sigma^2 \right)^2 = o\left(R_L\left[\mathcal{F}_{\Theta(\alpha, R)}\right]\right)$$ for all $\alpha>0$ and $R>0$. For ([\[eq:remain0\]](#eq:remain0){reference-type="ref" reference="eq:remain0"}), using the arguments in Chapter 3 of [@tsybakov2008introduction], we have $$\begin{split} &\inf_{\mathbf{w}\in \mathcal{W}_n}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})-\inf_{\mathbf{w}}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})\\ &\asymp \frac{1-\widetilde{\gamma}_1^*}{n} = o\left(R_L\left[\mathcal{F}_{\Theta(\alpha, R)}\right]\right), \end{split}$$ where the last equality is due to ([\[eq:pinsker\]](#eq:pinsker){reference-type="ref" reference="eq:pinsker"})--([\[eq:pinsker2\]](#eq:pinsker2){reference-type="ref" reference="eq:pinsker2"}). The condition ([\[eq:remain1\]](#eq:remain1){reference-type="ref" reference="eq:remain1"}) can be easily proved for all $\alpha>0$ and $R>0$ by noticing $\psi(\mathcal{M}_a)\asymp(\log n)^3$ and ([\[eq:pinsker\]](#eq:pinsker){reference-type="ref" reference="eq:pinsker"}). The condition ([\[eq:remain2\]](#eq:remain2){reference-type="ref" reference="eq:remain2"}) is satisfied when the estimator $\widehat{\sigma}^2_{D}$ with the parametric rate $1/n$ is adopted. When $\widehat{\sigma}^2 = \widehat{\sigma}_{m_n}^2$ with $m_n = \lfloor kn \rfloor$ ($0<k<1$), we have $$\begin{split} &\mathbb{E}(\widehat{\sigma}_{m_n}^2-\sigma^2)^2 \lesssim n^{-1}\vee \left(\sum_{j=\lfloor kn \rfloor+1}^{n}\theta_j^2\right)^2\\ &\leq n^{-1}\vee \left[(kn)^{-2\alpha}\sum_{j=\lfloor kn \rfloor+1}^{n}j^{2\alpha}\theta_j^2\right]^2 \lesssim n^{-1} \vee n^{-4\alpha}, \end{split}$$ where the first inequality follows from ([\[eq:variance_risk\]](#eq:variance_risk){reference-type="ref" reference="eq:variance_risk"}), and the third inequality follows from ([\[eq:ellipsoid\]](#eq:ellipsoid){reference-type="ref" reference="eq:ellipsoid"}). Thus, we obtain $$\sup_{\mathbf{f}\in \mathcal{F}_{\Theta(\alpha, R)}}\mathbb{E}\left( \widehat{\sigma}^2-\sigma^2 \right)^2\lesssim n^{-1} \vee n^{-4\alpha}=o\left(R_L\left[\mathcal{F}_{\Theta(\alpha, R)}\right]\right)$$ for all $\alpha>0$ and $R>0$. Combining ([\[eq:mini1\]](#eq:mini1){reference-type="ref" reference="eq:mini1"})--([\[eq:remain2\]](#eq:remain2){reference-type="ref" reference="eq:remain2"}), we have proved the exact linear-combined minimax adaptivity of MMA on the family of ellipsoids. According to ([\[eq:pinsker\]](#eq:pinsker){reference-type="ref" reference="eq:pinsker"}), MMA also achieves the exact minimax adaptivity on the family of ellipsoids. The linear-combined minimax risk over the hyperrectangle is $$\label{eq:minin_risk_hyper} \begin{split} & R_L\left[\mathcal{F}_{\Theta^H(c,q)}\right]=\inf_{\mathbf{w}}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta^H(c,q)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f}) \\ & =\sum_{j=1}^{n}\frac{c^2j^{-2q}\sigma^2}{nc^2j^{-2q}+\sigma^2}\asymp n^{-1+\frac{1}{2q}}, \end{split}$$ where the second equality is due to ([\[eq:hyperrectangle\]](#eq:hyperrectangle){reference-type="ref" reference="eq:hyperrectangle"}) and ([\[eq:risk_minimax_proof\]](#eq:risk_minimax_proof){reference-type="ref" reference="eq:risk_minimax_proof"}), and the last approximation can be obtained based on the similar technique in the proof of Theorem 1 of [@Peng2021]. Likewise, by taking the upper bound on both sides of ([\[eq:risk_bound_general\]](#eq:risk_bound_general){reference-type="ref" reference="eq:risk_bound_general"}) over $\mathbf{f}\in \mathcal{F}_{\Theta^H(c,q)}$, we see that the results can be proved if we show $$\label{eq:remain10} \inf_{\mathbf{w}\in \mathcal{W}_n}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta^H(c,q)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})-\inf_{\mathbf{w}}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta^H(c,q)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})= o\left(R_L\left[\mathcal{F}_{\Theta^H(c,q)}\right]\right),$$ $$\label{eq:remain11} \psi(\mathcal{M}_a) = o\left(R_L\left[\mathcal{F}_{\Theta^H(c,q)}\right]\right),$$ and $$\label{eq:remain12} \sup_{\mathbf{f}\in \mathcal{F}_{\Theta^H(c,q)}}\mathbb{E}\left( \widehat{\sigma}^2-\sigma^2 \right)^2 = o\left(R_L\left[\mathcal{F}_{\Theta^H(c,q)}\right]\right)$$ for all $c>0$ and $q>1/2$. Note that $$\begin{split} & \inf_{\mathbf{w}\in \mathcal{W}_n}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta^H(c,q)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f})-\inf_{\mathbf{w}}\sup_{\mathbf{f}\in \mathcal{F}_{\Theta^H(c,q)}}R_n(\mathbf{w}|\mathcal{M}_a,\mathbf{f}) \\ & =\frac{\sigma^4}{n^2c^2+n\sigma^2}=o\left(R_L\left[\mathcal{F}_{\Theta^H(c,q)}\right]\right), \end{split}$$ which implies ([\[eq:remain10\]](#eq:remain10){reference-type="ref" reference="eq:remain10"}). The equation ([\[eq:remain11\]](#eq:remain11){reference-type="ref" reference="eq:remain11"}) holds for all $c>0$ and $q>1/2$ since $\psi(\mathcal{M}_a)\asymp(\log n)^3$ and ([\[eq:minin_risk_hyper\]](#eq:minin_risk_hyper){reference-type="ref" reference="eq:minin_risk_hyper"}). The condition ([\[eq:remain12\]](#eq:remain12){reference-type="ref" reference="eq:remain12"}) is naturally satisfied for the estimator $\widehat{\sigma}^2_{D}$. When $\widehat{\sigma}^2 = \widehat{\sigma}_{m_n}^2$ with $m_n = \lfloor kn \rfloor$ ($0<k<1$) is adopted, we have $$\begin{split} \mathbb{E}(\widehat{\sigma}_{m_n}^2-\sigma^2)^2 & \lesssim n^{-1}\vee \left(\sum_{j=\lfloor kn \rfloor+1}^{n}\theta_j^2\right)^2\leq n^{-1}\vee \left(c^2\sum_{j=\lfloor kn \rfloor+1}^{n}j^{-2q}\right)^2 \\ & \lesssim n^{-1} \vee n^{-2q+1}=o\left(n^{-1+\frac{1}{2q}}\right) \end{split}$$ for all $q>1/2$, which implies ([\[eq:remain12\]](#eq:remain12){reference-type="ref" reference="eq:remain12"}). Thus, we see that the MMA estimator is adaptive in the exact linear-combined minimax sense on the family of hyperrectangles. # AOP in terms of the squared loss {#sec:a:aop_loss} Theorems [Theorem 1](#theorem:main){reference-type="ref" reference="theorem:main"}--[Theorem 2](#theorem:aop){reference-type="ref" reference="theorem:aop"} in the main text focus on the squared risk of the MMA estimator. Note that the definitions of AOP in terms of statistical loss have also been commonly adopted in MS [@Stone1984; @Li1987; @Shao1997] and MA literature [@Hansen2007; @Wan2010]. The following corollary shows that under the same assumptions in Theorem [Theorem 2](#theorem:aop){reference-type="ref" reference="theorem:aop"}, MMA is optimal in the sense that its squared loss asymptotically converges to that of the oracle MA estimator in probability. **Corollary 1**. *Suppose Assumption [Assumption 1](#asmp:square_summable){reference-type="ref" reference="asmp:square_summable"} holds. As $n \to \infty$, if the conditions ([\[eq:variance_rate\]](#eq:variance_rate){reference-type="ref" reference="eq:variance_rate"})--([\[eq:minimum_marisk_rate\]](#eq:minimum_marisk_rate){reference-type="ref" reference="eq:minimum_marisk_rate"}) are satisfied, then we have $$\frac{L_n(\widehat{\mathbf{w}} | \mathcal{M},\mathbf{f})}{\inf_{\mathbf{w}\in\mathcal{W}_{M_n}}L_n(\mathbf{w}| \mathcal{M},\mathbf{f})}\to_p 1,$$ where $\to_p$ means convergence in probability.* *Proof.* From ([\[eq:addd1\]](#eq:addd1){reference-type="ref" reference="eq:addd1"}), the MMA criterion can be decomposed as $$\label{eq:addd10} \begin{split} & C_{n}(\mathbf{w}|\mathcal{M},\mathbf{y})= L_n(\mathbf{w}|\mathcal{M},\mathbf{f}) - 2\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( e_l^2-\frac{\sigma^2}{n}\right) \\ & -2\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\theta_le_l-2\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( \frac{\sigma^2}{n} - \frac{\widehat{\sigma}^2}{n} \right)\\ & + \frac{1}{n}\mathbf{f}^{\top}\boldsymbol{\epsilon}+ \frac{1}{n}\|\boldsymbol{\epsilon}\|^2. \end{split}$$ Following the technique in [@Li1987], it is sufficient to verify $$\label{eq:wan_1} \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( e_l^2-\frac{\sigma^2}{n}\right)\right| }{R_n(\mathbf{w}|\mathcal{M},\mathbf{f})}\to_p 0,$$ $$\label{eq:wan_2} \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\theta_le_l \right| }{R_n(\mathbf{w}|\mathcal{M},\mathbf{f})}\to_p 0,$$ $$\label{eq:wan_4} \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( \frac{\sigma^2}{n} - \frac{\widehat{\sigma}^2}{n} \right) \right| }{R_n(\mathbf{w}|\mathcal{M},\mathbf{f})}\to_p 0,$$ and $$\label{eq:wan_3} \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\left| \frac{L_n(\mathbf{w}|\mathcal{M},\mathbf{f})}{R_n(\mathbf{w}|\mathcal{M},\mathbf{f})}-1 \right|\to_p 0.$$ In particular, ([\[eq:wan_3\]](#eq:wan_3){reference-type="ref" reference="eq:wan_3"}) is equivalent to $$\label{eq:wan_31} \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j^2\left( e_l^2-\frac{\sigma^2}{n}\right)\right| }{R_n(\mathbf{w}|\mathcal{M},\mathbf{f})}\to_p 0$$ and $$\label{eq:wan_32} \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j^2\theta_le_l \right| }{R_n(\mathbf{w}|\mathcal{M},\mathbf{f})}\to_p 0.$$ As an example, we prove ([\[eq:wan_1\]](#eq:wan_1){reference-type="ref" reference="eq:wan_1"}) and ([\[eq:wan_4\]](#eq:wan_4){reference-type="ref" reference="eq:wan_4"}). Recall that $z_l=\sqrt{n}e_l/\sigma$, $l=1,\ldots,k_{M_n}$. For any $\delta>0$, we have $$\label{eq:aop_loss_prove} \begin{split} \mathbb{P}&\left\{ \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( e_l^2-\frac{\sigma^2}{n}\right)\right| }{R_n(\mathbf{w}|\mathcal{M},\mathbf{f})}>\delta \right\} \\ =\mathbb{P}&\left\{ \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\sigma^2\gamma_j\left( z_l^2-1\right) \right| }{nR_n(\mathbf{w}|\mathcal{M},\mathbf{f})}>\delta \right\} \\ \leq \mathbb{P}&\left\{ \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\kappa_1 \left[\sum_{j=1}^{M_n}\sigma^4\gamma_j^2\left(k_j-k_{j-1} \right)\right]^{\frac{1}{2}} \left[\sum_{j=1}^{M_n}\frac{\left( k_j^{\frac{1}{2}} -k_{j-1}^{\frac{1}{2}}\right)^2}{k_j-k_{j-1}}\right]^{\frac{1}{2}} }{\left[nR_n(\mathbf{w}|\mathcal{M},\mathbf{f})\right]^{\frac{1}{2}}\left[nR_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})\right]^{\frac{1}{2}} }>\delta \right\}\\ \leq \mathbb{P}&\left\{\kappa_1\sigma\left[\sum_{j=1}^{M_n}\frac{\left( k_j^{\frac{1}{2}} -k_{j-1}^{\frac{1}{2}}\right)^2}{k_j-k_{j-1}}\right]^{\frac{1}{2}}>\delta\left[nR_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})\right]^{\frac{1}{2}} \right\}\\ \leq(&\mathbb{E}\kappa_1^2)\delta^{-2}\left[nR_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})\right]^{-1} \sigma^{2}\left[\sum_{j=1}^{M_n}\frac{\left( k_j^{\frac{1}{2}} -k_{j-1}^{\frac{1}{2}}\right)^2}{k_j-k_{j-1}}\right]\\ &\!\!\!\!\!\!\!\!\!\!\!\leq \frac{C\psi(\mathcal{M})}{nR_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})}\to 0, \end{split}$$ where the first inequality follows from ([\[eq:step21\]](#eq:step21){reference-type="ref" reference="eq:step21"}), the second inequality follows from ([\[eq:riskw\]](#eq:riskw){reference-type="ref" reference="eq:riskw"}), the third inequality is due to Markov's inequality, and the last inequality follows from the upper bound on $\mathbb{E}\kappa_1^2$ and the definition of $\psi(\mathcal{M})$. For any $\delta>0$, we have $$\label{eq:aop_loss_prove2} \begin{split} \mathbb{P}&\left\{ \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left|\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j\left( \frac{\sigma^2}{n} - \frac{\widehat{\sigma}^2}{n} \right) \right| }{R_n(\mathbf{w}|\mathcal{M},\mathbf{f})}>\delta \right\} \\ \leq\mathbb{P}&\left\{ \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left(\sum_{j=1}^{M_n}\sum_{l=k_{j-1}+1}^{k_j}\gamma_j^2\frac{\sigma^2}{n}\right)^{\frac{1}{2}}\left[\frac{nk_{M_n}}{\sigma^2}\left( \frac{\sigma^2}{n}-\frac{\widehat{\sigma}^2}{n}\right)^2\right]^{\frac{1}{2}} }{R_n(\mathbf{w}|\mathcal{M},\mathbf{f})}>\delta \right\} \\ \leq\mathbb{P}&\left\{ \sup_{\mathbf{w}\in \mathcal{W}_{M_n}}\frac{\left[\frac{k_{M_n}}{n\sigma^2}\left( \sigma^2-\widehat{\sigma}^2\right)^2\right]^{\frac{1}{2}} }{\left[R_n(\mathbf{w}|\mathcal{M},\mathbf{f})\right]^{\frac{1}{2}}}>\delta \right\} \\ \leq\mathbb{P}&\left\{ \left[\frac{k_{M_n}}{n\sigma^2}\left( \sigma^2-\widehat{\sigma}^2\right)^2\right]^{\frac{1}{2}} >\delta\left[R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})\right]^{\frac{1}{2}} \right\} \\ &\!\!\!\!\!\!\!\!\!\!\!\!\leq \frac{\mathbb{E}\left[\frac{k_{M_n}}{\sigma^2}\left( \sigma^2-\widehat{\sigma}^2\right)^2\right]}{\delta^2 n R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})}\to 0. \end{split}$$ The remaining equations in ([\[eq:wan_1\]](#eq:wan_1){reference-type="ref" reference="eq:wan_1"})--([\[eq:wan_31\]](#eq:wan_31){reference-type="ref" reference="eq:wan_31"}) can also be proved using similar techniques in Section [9.8](#sec:Proof_main){reference-type="ref" reference="sec:Proof_main"} and ([\[eq:aop_loss_prove\]](#eq:aop_loss_prove){reference-type="ref" reference="eq:aop_loss_prove"}). Thus we skip the similar materials here. ◻ # Additional Numerical Results {#sec:a:simu} ## Assessing the full AOP of MMA {#sec:simulation:subsec1} To illustrate the full-AOP theory in Section [4](#sec:main_results){reference-type="ref" reference="sec:main_results"}, we focus on the MMA estimator based on the largest candidate model set $\mathcal{M}_a$ as a representative. Let $\mathbf{f}^{(r)}$ and $\widehat{\mathbf{f}}^{(r)}$ denote the true mean vector and the estimated mean vector in the $r$-th replicate, respectively. We plot the risk ratio $$\label{eq:risk_ratio} \text{Ratio} = \frac{R^{-1}\sum_{r=1}^{R}\|\mathbf{f}^{(r)} - \widehat{\mathbf{f}}^{(r)}_{\widehat{\mathbf{w}}|\mathcal{M}_a} \|^2}{R^{-1}\sum_{r=1}^{R}\min_{\mathbf{w}\in\mathcal{W}_{p_n}}\|\mathbf{f}^{(r)} - \widehat{\mathbf{f}}^{(r)}_{\mathbf{w}|\mathcal{M}_a} \|^2}$$ as a function of $n$, where $\widehat{\mathbf{f}}^{(r)}_{\widehat{\mathbf{w}}|\mathcal{M}_a}$ is the MMA estimator in the $r$-th replicate. The optimizations involved in ([\[eq:risk_ratio\]](#eq:risk_ratio){reference-type="ref" reference="eq:risk_ratio"}) can be efficiently performed by quadratic programming. For example, `quadprog` package in R language is applicable. The simulation results are displayed in Figure [1](#fig:aop){reference-type="ref" reference="fig:aop"}. {#fig:aop width="5in"} As shown in the left panel of Figure [1](#fig:aop){reference-type="ref" reference="fig:aop"}, the curves decrease gradually and tend to $1$ as the sample size $n$ increases. This feature confirms our theoretical understanding that MMA attains the full AOP without restricting the weight or candidate model set when coefficients decay at a polynomial rate. Another observation is that when the sample size $n$ is fixed, the risk ratio increases as $\alpha_1$ increases from $0.51$ to $1.5$. This phenomenon implies that it is more difficult for MMA to achieve the full AOP when coefficients decay fast, which is expected. The simulation results in Case 2 also seem to support our AOP theory in Section [4](#sec:main_results){reference-type="ref" reference="sec:main_results"}, which claims that the MMA estimator based on $\mathcal{M}_a$ achieves the full AOP when $1<\alpha_2<1/3$. Indeed, as observed in the right panel of the figure, the curve with $\alpha_2< 1/3$ still shows an apparent downward trend. However, the curves with large $\alpha_2$ exhibit quite different patterns. It seems that the risk ratio experiences a two-phase process, a sharp increase when $n\leq 300$ followed by a slight decrease when $n$ approaches $1000$. Due to the limit of computing power, it is not easy to check by simulation whether these curves will finally tend to $1$ when $n$ is sufficiently large. ## Comparing different choices of candidate model set {#sec:simulation:subsec3} The primary purpose of this subsection is to compare several full-AOP MMA strategies, which are based on different candidate model sets as summarized in Table [\[tab:method\]](#tab:method){reference-type="ref" reference="tab:method"}. The competing methods include M-G1 with $k_1 = \lceil \log n \rceil$ and $\zeta_n=0$, M-G2 with $k_1 = \lceil \log n \rceil$ and $\zeta_n=1/\log n$, M-MS1 with $k_l=k_u=\log n$, and M-MS2 with $\widehat{l}_n = 1\vee (\widehat{m}_n-5)$ and $\widehat{u}_n = p_n \wedge (\widehat{m}_n+5)$, where $\widehat{m}_n$ in M-MS1 and M-MS2 is selected by Mallows' $C_p$ criterion. To show the differences between the competing methods, we divide the $\ell_2$ loss of these four methods by the $\ell_2$ loss of the full-AOP MMA based on $\mathcal{M}_a$. The simulation results are presented in Figure [\[fig:risk\]](#fig:risk){reference-type="ref" reference="fig:risk"}. As can be seen from Figure [\[fig:risk\]](#fig:risk){reference-type="ref" reference="fig:risk"} (a), the relative risks of the methods M-G1, M-G2, and M-MS1 are near 1. This feature corroborates the findings in Theorems [Theorem 3](#cor:grouped){reference-type="ref" reference="cor:grouped"}--[Theorem 4](#cor:minimum_1){reference-type="ref" reference="cor:minimum_1"} that the full AOP is still realized based on these properly constructed candidate model sets. Figure [\[fig:risk\]](#fig:risk){reference-type="ref" reference="fig:risk"} (a) also illustrates the consequence of over-reducing the number of candidate models. The M-MS2 method, which combines at most 11 models around $\widehat{m}_n$, exhibits much higher relative risks than 1 when the coefficients decay slowly. This observation accords with our statement in Theorem [Theorem 5](#cor:minimum_2){reference-type="ref" reference="cor:minimum_2"} that M-MS2 cannot achieve the full potential of MA in Case 1. From Figure [\[fig:risk\]](#fig:risk){reference-type="ref" reference="fig:risk"} (b), we observe that the methods M-G1, M-G2, and M-MS1 perform slightly better than the MMA estimator based on $\mathcal{M}_a$ when $\alpha_2=0.45$ and $0.75$. In addition, the methods M-MS1 and M-MS2 show an obvious advantage when $\alpha_2 = 1.25$. These results further support our understanding in Section [5](#sec:reduced){reference-type="ref" reference="sec:reduced"} that contracting the candidate model set provides certain benefits for MMA when coefficients decay fast. Interestingly, when coefficients decay extremely fast ($\alpha_2 = 1.25$), the curves of the methods M-G1 and M-G2 show an upward trend with some fluctuations. A sensible explanation is that the M-G methods exclude the best candidate model in this case. Note that their smallest candidate model has size $k_1= \lceil \log n \rceil$, while the optimal single model, in this case, is $m_n^* \asymp (\log n)^{4/5}$. Therefore, excluding the best candidate models from below can be harmful as well due to unnecessarily large variances in the models. This is in contrast to the situation of excluding the best models from above, as done in the MR methods, which induces unnecessarily large biases in the candidate models. We also notice that the results with $\alpha_1=1.5$ in Case 1 show more similar patterns to those in Case 2, while the relative risk curves with $\alpha_2=0.25$ in Case 2 are more like those in Case 1. Indeed, this phenomenon is caused by the same reason stated at the end of Section [\[sec:simulation:subsec2\]](#sec:simulation:subsec2){reference-type="ref" reference="sec:simulation:subsec2"}. See [@Liu2011PI] and [@ZHANG201595] for more related theoretical and numerical discussions. # Discussions on other related works {#sec:a:related} ## Aggregation It is worth mentioning that our work relates to a vast literature on aggregation procedures, which were first studied by [@Yang2000Mixing; @Yang2001; @Yang2004], [@Nemirovski2000; @juditsky2000functional], and [@Catoni2004], respectively. The optimal rates of aggregation have been established by [@Tsybakov2003; @Wang2014] and various rate-optimal procedures have been proposed with different weight constraints [see, e.g., @Tsybakov2003; @Yang2004; @Bunea2007; @Lounici2007; @Rigollet2011; @Dalalyan2012Mirror; @Lecue2013; @Wang2014]. A significant difference between the traditional aggregation procedures and the MMA-type methods is that the formers often focus on the step of combining models, namely, *pure aggregation*, wherein one has already obtained the candidate estimates based on previous studies, or from data splitting [see, e.g., @Yang2001; @Lecue2007; @Rigollet2007]. When candidate models and aggregation are trained on the same sample, some substantial progress has also been made in the aggregation literature. The exponential weighting (EW) methods in [@Leung2006; @Alquier2011; @Rigollet2011; @Dalalyan2012] and the Q-aggregation in [@Dai2014; @Bellec2018] are suitable for combining least squares or affine estimators from the same data. In particular, the EW method described in @Dalalyan2012 can be applied for convex aggregation of a list of affine estimators. Note that the EW method can be formulated as the entropy-penalized empirical risk minimization problem $$\label{eq:ds1} \widehat{\pi}_{EW}=\arg\inf_{\pi}\left\{ \int_{\mathcal{W}_{M_n}} C_n(\mathbf{w})\pi(d\mathbf{w})+\frac{\lambda}{n}D_{\mathrm{KL}}(\pi||\pi_0) \right\},$$ where $\pi$ is a probability measure on $\mathcal{W}_{M_n}$, $C_n(\mathbf{w})$ is the MMA criterion ([\[eq:criterion\]](#eq:criterion){reference-type="ref" reference="eq:criterion"}), $\lambda$ is a temperature parameter, $\pi_0$ is a given prior, and $D_{\mathrm{KL}}$ stands for the Kullback-Leibler divergence. The final EW estimator is $$\label{eq:ds2} \widehat{\mathbf{f}}_{EW}=\int_{\mathcal{W}_{M_n}}\widehat{\mathbf{f}}_{\mathbf{w}|\mathcal{M}}\widehat{\pi}_{EW}(d\mathbf{w}).$$ When $\pi_0$ is the uniform distribution on $\mathcal{W}_{M_n}$ and $\lambda\geq8\sigma^2$, Proposition 2 of @Dalalyan2012 implies that $$\label{eq:dsbound} \mathbb{E}L_n(\widehat{\mathbf{f}}_{EW},\mathbf{f}) \leq R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f}) + \frac{CM_n\log(n)}{n}.$$ When $M_n$ is large, [@Dalalyan2012] suggest a heavy tailed prior $\pi_0$ which favors sparse weight vectors. Their Proposition 3 shows that with a properly defined $\pi_0$, $$\label{eq:dsbound2} \mathbb{E}L_n(\widehat{\mathbf{f}}_{EW},\mathbf{f}) \leq R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f}) + \frac{C\log(nM_n)}{n}.$$ First, notice that the EW estimator ([\[eq:ds2\]](#eq:ds2){reference-type="ref" reference="eq:ds2"}) coincides with the MMA estimator ([\[eq:MMA_estimator\]](#eq:MMA_estimator){reference-type="ref" reference="eq:MMA_estimator"}) when $\lambda=0$ but differs from ([\[eq:MMA_estimator\]](#eq:MMA_estimator){reference-type="ref" reference="eq:MMA_estimator"}) when $\lambda>0$. The risk bounds ([\[eq:dsbound\]](#eq:dsbound){reference-type="ref" reference="eq:dsbound"}) and ([\[eq:dsbound2\]](#eq:dsbound2){reference-type="ref" reference="eq:dsbound2"}), which are obtained under the condition $\lambda\geq8\sigma^2$, are not applicable for the understanding of the MMA method as intended in this paper. Second, the core proof technique in [@Dalalyan2012] is based on Stein's lemma [@Stein1981Estimation], which requires $\epsilon$ to follow a Gaussian distribution and the error variance is estimated based on independent data, which is typically unavailable. In contrast, our MMA approach can handle the sub-Gaussian errors with $\sigma^2$ being estimated based on the same data. It is worthy mentioning the risk bounds ([\[eq:dsbound\]](#eq:dsbound){reference-type="ref" reference="eq:dsbound"})--([\[eq:dsbound2\]](#eq:dsbound2){reference-type="ref" reference="eq:dsbound2"}) also target the optimal MA risk $R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})$ as the MMA approach does. They can justify the full AOP of the EW method when the priors are properly selected. Proposition 7.2 of @Bellec2018 gives a risk bound for MMA when $\epsilon$ is normally distributed and $\sigma^2$ is known. Integrating the tail probability of their equation (7.4) yields $$\label{eq:bellec} \mathbb{E}L_n(\widehat{\mathbf{w}}|\mathcal{M},\mathbf{f}) \leq R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f}) + \frac{C\log M_n}{n} + \left(\frac{C\log M_n}{n}\right)^{\frac{1}{2}}.$$ The bound ([\[eq:bellec\]](#eq:bellec){reference-type="ref" reference="eq:bellec"}) cannot achieve MMA's AOP unless the optimal MA risk $R_n(\mathbf{w}^*|\mathcal{M},\mathbf{f})$ converges slower than $(\log M_n/n)^{1/2}$. Note that the framework in [@Bellec2018] allows one to combine a set of affine estimators, which may be applicable to some other MA problems. However, in our MMA context, Theorem [Theorem 1](#theorem:main){reference-type="ref" reference="theorem:main"} substantially improves ([\[eq:bellec\]](#eq:bellec){reference-type="ref" reference="eq:bellec"}) for AOP under much milder conditions. Our examination of MMA is in the nested model framework, which serves as a representative setup in the MS and MA literature [see, e.g., @Polyak1991; @Li1987; @Hansen2007]. Nested models can be seen as a special case of the ordered linear smoother [@Kneip1994]. Aggregation of ordered linear smoothers has been studied in [@Chernousova2013] and [@Bellec2020]. However, their risk bounds are in terms of the best model instead of their optimal combination. As shown in [@Peng2021], the optimal MS risk can be substantially reduced by MA under certain conditions. ## Minimax adaptivity {#minimax-adaptivity} The minimax statement in Definition [Definition 3](#def:minimax){reference-type="ref" reference="def:minimax"} is known as the exact minimax adaptivity, which was first introduced by [@efroimovich1984learning] in the Gaussian white noise model and was further investigated for various estimators in other specific problems [see, e.g., @Donoho1995; @Efromovich1996; @Nemirovski2000; @Yang2000Mixing; @Cavalier2002Sharp; @Dalalyan2012; @Bellec2018]. Our setup focuses on the minimax adaptivity on the spaces of the transformed parameters $\boldsymbol{\theta}$ rather than the spaces of the original regression coefficient $\boldsymbol{\beta}$. Similar setup was adopted by [@Dalalyan2012] based on a discrete-cosine transformation of $\mathbf{f}$. Another goal considered in the literature is the minimax-rate adaptation, which is less demanding but more tangible with much wider applicability. Some MS and MA schemes have been considered to construct the minimax-rate optimal estimators that require almost no assumption on the behaviors of the candidate models. For example, see [@Barron1999], [@juditsky2000functional], and [@YANG2000135; @Yang2000PATTERN; @Yang1998ms] for early representative work. In this paper, we show that the MMA estimator is adaptive in the exact minimax sense over the family of ellipsoids and hyperrectangles. Some other approaches, such as the blockwise constant (BC) rules [@efroimovich1984learning; @Efromovich1996; @Donoho1995; @Nemirovski2000; @cavalier2001penalized; @Cavalier2002Sharp], have also been used to derive the exact minimax adaptive estimators on various classes. There are two notable differences between the BC rule and the MMA method. First, the adaptivity of the BC rule can be obtained only when the orders of some hyperparameters, such as the lengths of blocks, are set correctly, while there are no parameters needed to be determined prior to implementing MMA. Second, the BC rule requires $\sigma^2$ to be known, while the MMA method can accommodate the setting with unknown $\sigma^2$, which is more applicable in regression problems. The effects of the variance estimation on MMA are seen in the risk bound ([\[eq:risk_bound_general\]](#eq:risk_bound_general){reference-type="ref" reference="eq:risk_bound_general"}). It is worth noting that the exact minimax adaptivity property over the family of ellipsoids can also be obtained by aggregation methods in [@Dalalyan2012] and [@Bellec2018], in which the candidate models are constructed from the Pinsker filters and the variance $\sigma^2$ is assumed to be known or estimated from an independent sample. [^1]: The comments from the Annals of Statistics are greatly appreciated. The simulation part of this paper was supported by Public Computing Cloud, Renmin University of China.

arxiv_math

--- title: Roundoff error analysis of the double exponential formula-based method for the matrix sign function --- # Introduction Let $A\in\mathbb{R}^{n\times n}$ be a real matrix and $A=XJX^{-1}$ be its Jordan decomposition. Here, $X\in\mathbb{R}^{n\times n}$ is a nonsingular matrix and $J=J_{n_1}(\lambda_1)\oplus\cdots\oplus J_{n_r}(\lambda_r)$, where $J_{n_i}(\lambda_i)$ ($i=1, \ldots r$) is a Jordan cell of size $n_i\times n_i$ with eigenvalue $\lambda_i$ and $n_1+\cdots+n_r=n$. We assume that $A$ has no purely imaginary eigenvalues. Also, let $I_n$ be the identity matrix of order $n$ and ${\rm sign}(z)$, where $z\in\mathbb{C}$, be the (scalar) sign function defined by $${\rm sign}(z)=\begin{cases} 1, & {\rm Re}(z)>0, \\ -1, & {\rm Re}(z)<0. \end{cases}$$ Then, the *matrix sign function* [@Higham08] is defined by $${\rm sign}(A)\equiv X({\rm sign}(\lambda_1)I_{n_1}\oplus\cdots\oplus{\rm sign}(\lambda_r) I_{n_r})X^{-1}. \label{eq:definition}$$ Note that the matrix sign function is undefined when $A$ has purely imaginary eigenvalues. The matrix sign function has applications to the solution of the Sylvester equation and computation of eigendecomposition [@Nakatsukasa16]. Several approaches have been proposed for computing the matrix sign function, including Schur's method, Newton's method and rational function approximation [@Higham08]. Recently, Nakaya and Tanaka proposed a new method for computing the matrix sign function using numerical quadrature [@Nakaya21]. The idea is to expresses ${\rm sign}(A)$ using an integral representation: $${\rm sign}(A)=\frac{2}{\pi}\int_0^{\infty}(t^2 I+A^2)^{-1}A\,dt \label{eq:integral_representation}$$ and compute it numerically with the double-exponential (DE) formula [@Takahashi74]. In the following, We refer to this method as the *DE-based method* for the matrix sign function. Since the computation of $(t^2 I+A^2)^{-1}$ can be done for each sample point independently, the method has large-grain parallelism and is well-suited for modern high performance computers. Detailed analysis of the discretization and truncation errors of the DE-based method, assuming exact arithmetic, is given in [@Nakaya21]. The reader is also referred to [@Miyashita22] for an analysis in the case where $A$ is diagonalizable. According to our numerical experiments, the method works well when $A$ is a well conditioned and close-to-normal matrix and delivers results with accuracy comparable to that of Schur's method. However, we observed that as $A$ becomes ill-conditioned or deviates from normality (which means that $X$ becomes ill-conditioned), the numerical error of the method increases rapidly. This problem is not fixed even if we make the step size sufficiently small or make the truncation points sufficiently distant from the origin. Hence, we suppose that this degradation of accuracy originates not from discretization or truncation errors but from rounding errors. In this paper, we present a roundoff error analysis of the DE-based method for the matrix sign function to find the cause of this accuracy degradation and look for a direction for possible improvement. For simplicity, we focus on the case where $A$ is diagonalizable and consider two main sources of rounding errors, namely, the error arising in the computation of $(t^2 I+A^2)^{-1}A$ at each sample point and the error that occurs when summing up the contributions from the sample points. The rest of this paper is structured as follows. In Section 2, we review the DE-based method for the matrix sign function. Roundoff error analysis of this method is given in Section 3. Numerical results that illustrate the validity of the derived error bound are presented in Section 4. Finally, Section 5 provides some conclusion. # DE-based method for the matrix sign function In the DE-based method, we approximate the integral [\[eq:integral_representation\]](#eq:integral_representation){reference-type="eqref" reference="eq:integral_representation"} using the DE formula. By letting $Y(t)\equiv (t^2I+A^2)^{-1}A$ and $\phi(x)=\exp(\frac{\pi}{2}\sinh x)$, we have $$\begin{aligned} {\rm sign}(A) &= \frac{2}{\pi}\int_0^{\infty}Y(t)dt = \frac{2}{\pi}\int_{-\infty}^{\infty}Y(\phi(x))\phi'(x)dx \nonumber \\ &\simeq \frac{2}{\pi}h\sum_{k=N^-}^{N^+}Y(\phi(kh))\phi'(kh),\end{aligned}$$ where $h$ is the step size and $-N^->0$ and $N^+>0$ are the number of sample points in the negative and positive part of the $x$-axis, respectively. It is shown that by choosing $N^-=N^+=N$ and $h=\log(8dN)/N$, where $d$ is some constant depending on $A$, the discretization and truncation errors of the DE-based method decrease exponentially with $N$ [@Nakaya21]. In this method, the most computationally intensive part is the calculation of $Y(\phi(kh))$ for $k=N^-, \ldots, N^+$. Since this can be done for each $k$ independently, the method has large-grain parallelism. # Roundoff error analysis We present a roundoff error analysis of the DE-based method. We denote a quantity computed in floating-point arithmetic by $fl(\cdot)$ or by a symbol with a hat. ${\bf u}$ denotes the unit roundoff and $\gamma_m\equiv m{\bf u}/(1-m{\bf u})$. For a matrix $A=(a_{ij})$, $|A|$ means a matrix whose elements are $|a_{ij}|$. For matrices $A$ and $B$ of the same dimension, $A\le B$ means componentwise inequality. $\|\cdot\|_2$ and $\|\cdot\|_F$ denote the 2-norm and the Frobenius norm, respectively. The condition number of $A$ is denoted by $\kappa_2(A)$. ## Sources of roundoff errors To simplify the analysis, we assume that scalar functions such as $\phi(x)$ and $\phi'(x)=\frac{\pi}{2}\exp(\frac{\pi}{2}\sinh x)\cosh x$ can be computed without errors. We also assume that multiplications by scalars such as $\phi'(x)$ and $\frac{2}{\pi}h$ can be done without errors. Thus, we consider the following two sources of roundoff errors. - Errors in the computation of $Y(t)$: Let $\hat{Y}(t)\equiv fl(Y(t))$ and $\tilde{E}_1(t)\equiv \hat{Y}(t)-Y(t)$, where $t=\phi(kh)$ and $N^-\le k\le N^+$. The weighted sum of these errors, $$E_1\equiv \frac{2}{\pi}h\sum_{k=N^-}^{N^+}|\tilde{E}_1(\phi(kh))|\phi'(kh), \label{eq:E1}$$ contributes to the total roundoff error. - Errors in the summation. Strictly speaking, the summand is $fl(Y(\phi(kh))\phi'(kh))$, but we substitute it with its exact counterpart, $Y(\phi(kh))\phi'(kh)$, for simplicity. This can be justified because their difference is $O({\bf u})$, as will be shown later (see [\[eq:hatXerror\]](#eq:hatXerror){reference-type="eqref" reference="eq:hatXerror"}), and therefore causes only $O({\bf u}^2)$ difference in the value of $E_2$. Thus, the error is defined as $$\begin{aligned} E_2 &\equiv fl\left(\sum_{k=N^-}^{N^+} Y(\phi(kh))\phi'(kh)\right) \nonumber \\ & \quad\quad - \sum_{k=N^-}^{N^+} Y(\phi(kh))\phi'(kh). \label{eq:E2}\end{aligned}$$ In the following, we evaluate $E_1$ and $E_2$ separately. ## Evaluation of $\tilde{E}_1(t)=\hat{Y}(t)-Y(t)$ Let $B\equiv t^2 I+A^2$ and denote the $j$th column of $A$, $Y(t)$ and $\hat{Y}(t)$ by ${\bf a}_j$, ${\bf y}_j$ and $\hat{\bf y}_j$, respectively. Then, ${\bf y}_j$ is computed as the solution of the linear simultaneous equations $B{\bf y}_j={\bf a}_j$. Thus, $\hat{\bf y}_j-{\bf y}_j$ consists of two parts, the error in the formation of $B$ and that in the solution of the linear simultaneous equations. Denote the former error by $\Delta B'$. Then, if we consider only the error in the computation of $A^2$ and ignore the error arising from the addition of $t^2 I$, we have $|\Delta B'|\le\gamma_n|A|^2$ from the result of the standard error analysis [@Higham02 §3.5] and therefore $$\|\Delta B^{\prime}\|_F \le \gamma_n\|A\|_F^2. \label{eq:deltaBbound}$$ Now, $\hat{\bf y}_j$ is obtained by solving the linear simultaneous equation with the coefficient matrix $\tilde{B}\equiv B+\Delta B'$ using Gaussian elimination with partial pivoting in floating-point arithmetic. In that case, it is well known that $\hat{\bf y}_j$ satisfies the following equation [@Higham02 Theorem 9.4]: $$(\tilde{B}+\Delta B_j^{\prime\prime})\hat{\bf y}_j={\bf a}_j, \quad |\Delta B_j^{\prime\prime}|\le \gamma_{3n}|\hat{L}|\,|\hat{U}|, \label{eq:Higham9-4}$$ where $\hat{L}$ and $\hat{U}$ are computed LU factors of $\tilde{B}$ and $\Delta B_j^{\prime\prime}$ is the backward error in the solution of the linear simultaneous equation. Now, we evaluate $\|\Delta B_j^{\prime\prime}\|_F$. First, since $|\hat{l}_{ij}|\le 1$, we have $\|\hat{L}\|_F\le n$. Next, let the coefficient matrix in the $k$th step of the Gaussian elimination be $\tilde{B}^{(k)}=(\tilde{b}_{ij}^{(k)})$ and define the *growth factor* as $$\hat{\rho}_n = \frac{\max_{i,j,k}|\tilde{b}_{i,j}^{(k)}|}{\max_{i,j}|\tilde{b}_{i,j}|}.$$ Then, $$|\hat{u}_{i,j}|=|\tilde{b}_{i,j}^{(i)}| \le \hat{\rho}_n\max_{i',j'}|\tilde{b}_{i',j'}| \le \hat{\rho}_n\|\tilde{B}\|_2$$ and we have $\|\hat{U}\|_F\le n\hat{\rho}_n\|\tilde{B}\|_2$. From these results, it follows that $$\begin{aligned} \|\Delta B_j^{\prime\prime}\|_F &\le n^2\gamma_{3n}\hat{\rho}_n\|\tilde{B}\|_2 \nonumber \\ &\le n^2\gamma_{3n}\hat{\rho}_n(\|B\|_2+\gamma_n\|A\|_F^2) \nonumber \\ &\simeq n^2\gamma_{3n}\hat{\rho}_n\|t^2I + A^2\|_2. \label{eq:DeltaAbound}\end{aligned}$$ It is known that the growth factor $\hat{\rho}_n$ is almost independent of $n$ and is typically around 10 [@Higham02 §9.4]. Combining [\[eq:deltaBbound\]](#eq:deltaBbound){reference-type="eqref" reference="eq:deltaBbound"}, [\[eq:Higham9-4\]](#eq:Higham9-4){reference-type="eqref" reference="eq:Higham9-4"} and [\[eq:DeltaAbound\]](#eq:DeltaAbound){reference-type="eqref" reference="eq:DeltaAbound"}, we know that $\hat{\bf y}_j$ satisfies $$\begin{aligned} & (B+\Delta B_j)\hat{\bf y}_j={\bf a}_j, \\ & \|\Delta B_j\|_F \le \gamma_n\|A\|_F^2+n^2\gamma_{3n}\hat{\rho}_n\|t^2I + A^2\|_2.\end{aligned}$$ Assuming that $\|B^{-1}\|_2\|\Delta B_j\|_2\le 1/2$ and applying the perturbation theory for linear simultaneous equations [@Higham02 Theorem 7.2] gives $$\begin{aligned} \|\hat{\bf y}_j-{\bf y}_j\| &\le \frac{\|B^{-1}\|_2\|\Delta B_j\|_2}{1-\|B^{-1}\|_2\|\Delta B_j\|_2}\,\|{\bf y}_j\| \nonumber \\ &\le 2\|B^{-1}\|_2\|\Delta B_j\|_2\|{\bf y}_j\| \nonumber \\ &\le 2\|(t^2I+A^2)^{-1}\|_2 \nonumber \\ & \quad \times(\gamma_n\|A\|_F^2+n^2\gamma_{3n}\hat{\rho}_n\|t^2I + A^2\|_2)\|{\bf y}_j\| . \nonumber %\label{eq:yjerror}\end{aligned}$$ From this, it is immediate to show that $$\begin{aligned} & \|\tilde{E}_1(t)\|_F=\|\hat{Y}(t)-Y(t)\|_F \nonumber \\ &\le 2\|(t^2I+A^2)^{-1}\|_2(\gamma_n\|A\|_F^2+n^2\gamma_{3n}\hat{\rho}_n\|t^2I + A^2\|_2) \nonumber \\ & \quad \times\|(t^2I+A^2)^{-1}A\|_F. \label{eq:hatXerror}\end{aligned}$$ ## Evaluation of $E_1$ Now we evaluate $E_1$ using [\[eq:E1\]](#eq:E1){reference-type="eqref" reference="eq:E1"} and [\[eq:hatXerror\]](#eq:hatXerror){reference-type="eqref" reference="eq:hatXerror"}. From the assumption of diagonalizability, $A$ can be written as $A=X\Lambda X^{-1}$, where $\Lambda={\rm diag}(\lambda_1,\ldots,\lambda_n)$. Hence, $$\begin{aligned} \|A\|_F^2 &= \|X\Lambda X^{-1}\|_F^2 \nonumber \\ &\le (\|X\|_2\|\Lambda\|_F\|X^{-1}\|_2)^2 = (\kappa_2(X))^2\|\Lambda\|_F^2, \label{eq:AF2}\end{aligned}$$ where we used $\|AB\|_F \le \|A\|_2\|B\|_F$. Similarly, we have $$\begin{aligned} \|t^2 I + A^2\|_2 &\le \kappa_2(X)\|t^2I+\Lambda^2\|_2, \label{eq:YF1} \\ \|(t^2 I + A^2)^{-1}\|_2 &\le \kappa_2(X)\|(t^2I+\Lambda^2)^{-1}\|_2, \label{eq:YF2} \\ \|(t^2I+A^2)^{-1}A\|_F &\le \kappa_2(X)\|(t^2I+\Lambda^2)^{-1}\Lambda\|_F. \label{eq:YF3}\end{aligned}$$ Substituting [\[eq:AF2\]](#eq:AF2){reference-type="eqref" reference="eq:AF2"} through [\[eq:YF3\]](#eq:YF3){reference-type="eqref" reference="eq:YF3"} into [\[eq:hatXerror\]](#eq:hatXerror){reference-type="eqref" reference="eq:hatXerror"} gives $$\begin{aligned} & \|\tilde{E}_1(t)\|_F \nonumber \\ &\le 2\left\{\gamma_n(\kappa_2(X))^4\|\Lambda\|_F^2+n^2\gamma_{3n}\hat{\rho}_n(\kappa_2(X))^3\|t^2I+\Lambda^2\|_2\right\} \nonumber \\ & \quad \times\|(t^2I+\Lambda^2)^{-1}\|_2 \|(t^2I+\Lambda^2)^{-1}\Lambda\|_F. \label{eq:Yterror}\end{aligned}$$ From [\[eq:E1\]](#eq:E1){reference-type="eqref" reference="eq:E1"} and [\[eq:Yterror\]](#eq:Yterror){reference-type="eqref" reference="eq:Yterror"}, the contribution to the total roundoff error can be computed as $$\begin{aligned} & \|E_1\|_F = \frac{2}{\pi}h\sum_{k=N^-}^{N^+}\|\tilde{E}_1(\phi(kh))\|_F\phi'(kh) \nonumber \\ &\simeq \frac{2}{\pi}\int_{-\infty}^{\infty}\|\tilde{E}_1(\phi(x))\|_F\phi'(x)\,dx = \frac{2}{\pi}\int_0^{\infty}\|\tilde{E}_1(t)\|_F\,dt \nonumber \\ &\le \frac{4}{\pi}\int_0^{\infty}\left\{\gamma_n(\kappa_2(X))^4\|\Lambda\|_F^2 \right.\nonumber \\ &\quad\quad\quad\quad\quad \left.+n^2\gamma_{3n}\hat{\rho}_n(\kappa_2(X))^3\|t^2I+\Lambda^2\|_2\right\} \nonumber \\ &\quad\quad\quad\quad \times \|(t^2I+\Lambda^2)^{-1}\|_2 \|(t^2I+\Lambda^2)^{-1}\Lambda\|_F\,dt. \label{eq:Yphibound}\end{aligned}$$ Let us write the integrand of the last integral as $e_{n,X,\Lambda}(t)$. To evaluate the integral, we let $t_1=\sqrt{2}\|\Lambda\|_F$ and divide the integration interval into two parts, $[0,t_1]$ and $[t_1,\infty)$. When $t\ge t_1$, we have $$\begin{aligned} \|\Lambda\|_F^2 &\le \frac{t^2}{2}, \label{eq:LFbound} \\ \|t^2I+\Lambda^2\|_2 &= \max_{1\le j\le n}|t^2+\lambda_j^2| \le \max_{1\le j\le n}(t^2+|\lambda_j|^2) \nonumber \\ &\le t^2 + \frac{t^2}{2} = \frac{3}{2}t^2, \\ \|(t^2I+\Lambda^2)^{-1}\|_2 &= \max_{1\le j\le n}\frac{1}{|t^2+\lambda_j^2|} \le \max_{1\le j\le n}\frac{1}{t^2-|\lambda_j|^2} \nonumber \\ &\le \frac{1}{t^2-\frac{1}{2}t^2} =\frac{2}{t^2}, \\ \|(t^2I+\Lambda^2)^{-1}\Lambda\|_F &= \sqrt{\sum_{j=1}^n\frac{|\lambda_j|^2}{|t^2+\lambda_j^2|^2}} \le \frac{2}{t^2}\sqrt{\sum_{j=1}^n|\lambda_j|^2} \nonumber \\ &= \frac{2}{t^2}\|\Lambda\|_F. \label{eq:integrandbound3}\end{aligned}$$ Now, let $$c_{n,X,\Lambda}=\gamma_n(\kappa_2(X))^4+3n^2\gamma_{3n}\hat{\rho}_n(\kappa_2(X))^3.$$ Then, we have from [\[eq:LFbound\]](#eq:LFbound){reference-type="eqref" reference="eq:LFbound"} through [\[eq:integrandbound3\]](#eq:integrandbound3){reference-type="eqref" reference="eq:integrandbound3"}, $$\begin{aligned} 0 \le \int_{t_1}^{\infty}e_{n,X,\Lambda}(t)\,dt &\le c_{n,X,\Lambda}\int_{t_1}^{\infty}\frac{t^2}{2}\cdot\frac{2}{t^2}\cdot\frac{2}{t^2}\|\Lambda\|_F\,dt \nonumber \\ &= c_{n,X,\Lambda}\|\Lambda\|_F\int_{t_1}^{\infty}\frac{2}{t^2}\,dt \nonumber \\ &= c_{n,X,\Lambda}\|\Lambda\|_F\cdot\frac{\sqrt{2}}{\|\Lambda\|_F} \nonumber \\ &= \sqrt{2}\,c_{n,X,\Lambda}. \label{eq:integralbound3b}\end{aligned}$$ On the other hand, when $0\le t\le t_1$, $$\begin{aligned} \|t^2I+\Lambda^2\|_2 &\le \max_{1\le j\le n}(t^2+|\lambda_j|^2) \le 3\|\Lambda\|_F^2, \\ \|(t^2I+\Lambda^2)^{-1}\|_2 &= \max_{1\le j\le n}\frac{1}{|t^2+\lambda_j^2|}, \\ \|(t^2I+\Lambda^2)^{-1}\Lambda\|_F &\le \|\Lambda\|_F\|(t^2I+\Lambda^2)^{-1}\|_2 \nonumber \\ &\le \|\Lambda\|_F\max_{1\le j\le n}\frac{1}{|t^2+\lambda_j^2|}. \label{eq:integrandbound6}\end{aligned}$$ Hence, $$\begin{aligned} 0 &\le \int_0^{t_1}e_{n,X,\Lambda}(t)\,dt \le c_{n,X,\Lambda}\|\Lambda\|_F^3\int_0^{t_1}\max_{1\le j\le n}\frac{dt}{|t^2+\lambda_j^2|^2} \nonumber \\ &\quad\le c_{n,X,\Lambda}\|\Lambda\|_F^3\sum_{j=1}^n\int_0^{t_1}\frac{dt}{|t^2+\lambda_j^2|^2} \nonumber \\ &\quad\le \frac{1}{2}c_{n,X,\Lambda}\|\Lambda\|_F^3\sum_{j=1}^n\int_{-\infty}^{\infty}\frac{dt}{|t^2+\lambda_j^2|^2}.\end{aligned}$$ Noting that ${\rm Re}(\lambda_j)\ne 0$, we can evaluate this integral as $$\int_{-\infty}^{\infty}\frac{dt}{|t^2+\lambda_j^2|^2} = \frac{\pi}{2|\lambda_j|^2|{\rm Re}(\lambda_j)|}. \label{eq:quadraticintegral}$$ See Lemma [Lemma 1](#Lemma1){reference-type="ref" reference="Lemma1"} in the Appendix. Thus, it follows that $$\begin{aligned} 0&\le \int_0^{t_1}e_{n,X,\Lambda}(t)\,dt \le \frac{\pi}{4}c_{n,X,\Lambda}\|\Lambda\|_F^3\sum_{j=1}^n \frac{\pi}{|\lambda_j|^2|{\rm Re}(\lambda_j)|} \nonumber \\ &= \frac{\pi}{4}c_{n,X,\Lambda}\|\Lambda\|_F^3\|\Lambda^{-1}|{\rm Re}(\Lambda)|^{-\frac{1}{2}}\|_F^2. \label{eq:quadraticintegral2}\end{aligned}$$ Combining [\[eq:Yphibound\]](#eq:Yphibound){reference-type="eqref" reference="eq:Yphibound"}, [\[eq:integralbound3b\]](#eq:integralbound3b){reference-type="eqref" reference="eq:integralbound3b"} and [\[eq:quadraticintegral2\]](#eq:quadraticintegral2){reference-type="eqref" reference="eq:quadraticintegral2"} gives $$\begin{aligned} \|E_1\|_F &\le c_{n,X,\Lambda}\left(\frac{4\sqrt{2}}{\pi}+\|\Lambda\|_F^3\|\Lambda^{-1}|{\rm Re}(\Lambda)|^{-\frac{1}{2}}\|_F^2\right) \nonumber \\ &=\left(\gamma_n(\kappa_2(X))^4+3n^2\gamma_{3n}\hat{\rho}_n(\kappa_2(X))^3\right) \nonumber \\&\quad \times\left(\frac{4\sqrt{2}}{\pi}+\|\Lambda\|_F^3\|\Lambda^{-1}|{\rm Re}(\Lambda)|^{-\frac{1}{2}}\|_F^2\right). \label{TermwiseError}\end{aligned}$$ ## Evaluation of $E_2$ Next, we evaluate $E_2$, the roundoff error in the summation. Let us consider the sum of $m$ matrices, $S_m=\sum_{i=1}^m T_i$, and denote the computed result by $\hat{S}_m$. Then, from the formula of roundoff error bound for scalar summation [@Higham02 Problem 4.3], $|\hat{S}_m-S_m|$ can be bounded as follows [@Yamamoto22]. $$|\hat{S}_m-S_m| \le \gamma_{m-1}(|T_1|+|T_2|)+\sum_{i=3}^m\gamma_{m-i+1}|T_i|.$$ Taking the Frobenius norm of both sides and replacing $\gamma_{m-i+1}$ with $\gamma_{m-1}$ for simplicity gives $$\|\hat{S}_m-S_m\|_F \le \gamma_{m-1}\sum_{i=1}^m\|T_i\|_F.$$ By applying this result to [\[eq:E2\]](#eq:E2){reference-type="eqref" reference="eq:E2"} and writing $M=N_+-N_-$, we have $$\begin{aligned} \|E_2\|_F &=\frac{2}{\pi}h\gamma_{M}\sum_{k=N^-}^{N^+}\|Y(\phi(kh))\|_F\phi'(kh) \nonumber \\ &\simeq \frac{2}{\pi}\gamma_{M}\int_{-\infty}^{\infty}\|Y(\phi(x))\|_F\phi'(x)\,dx \nonumber \\ &= \frac{2}{\pi}\gamma_{M}\int_0^{\infty}\|Y(t)\|_F\,dt \nonumber \\ &= \frac{2}{\pi}\gamma_{M}\int_0^{\infty}\|(t^2I+A^2)^{-1}A\|_F\,dt \nonumber \\ &\le \frac{2}{\pi}\gamma_{M}\kappa_2(X)\int_0^{\infty}\|(t^2I+\Lambda^2)^{-1}\Lambda\|_F\,dt \nonumber \\ &= \frac{2}{\pi}\gamma_{M}\kappa_2(X)\int_0^{\infty}\sqrt{\sum_{j=1}^n\frac{|\lambda_j|^2}{|t^2+\lambda_j^2|^2}}\,dt \nonumber \\ &\le \frac{2}{\pi}\gamma_{M}\kappa_2(X)\int_0^{\infty}\sum_{j=1}^n\frac{|\lambda_j|}{|t^2+\lambda_j^2|}\,dt \nonumber \\ &= \frac{2}{\pi}\gamma_{M}\kappa_2(X)\sum_{j=1}^n|\lambda_j|\int_0^{\infty}\frac{1}{|t^2+\lambda_j^2|}\,dt. \label{eq:Yphibound2}\end{aligned}$$ Now, we explain how to evaluate the integral in the last expression. Let $\lambda_j=\mu_j+{\rm i}\nu_j$, where ${\rm i}=\sqrt{-1}$. Then, the denominator of the integrand can be written as the square root of a real quartic polynomial in $t$. Hence, as is well known, the integral can be transformed into the complete elliptic integral of the first kind: $$K(k)=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}. \label{eq:elliptic2}$$ Details of this transformation are given in Lemma [Lemma 2](#Lemma2){reference-type="ref" reference="Lemma2"}. Further, by employing an upper bound on $K(k)$, which is provided in Lemma [Lemma 3](#Lemma3){reference-type="ref" reference="Lemma3"}, we can obtain an upper bound on this integral. The evaluation proceeds as follows. $$\begin{aligned} & \int_0^{\infty}\frac{1}{|t^2+\lambda_j^2|}\,dt \nonumber \\ &= \int_0^{\infty}\frac{1}{\sqrt{t^4+2(\mu_j^2-\nu_j^2)t^2+(\mu_j^2+\nu_j^2)^2}}\,dt \nonumber \\ &= \frac{1}{\sqrt{\mu_j^2+\nu_j^2}}K\left(\frac{|\nu_j|}{\sqrt{\mu_j^2+\nu_j^2}}\right) \nonumber \\ &\le \frac{\pi}{2}\cdot\frac{1}{\sqrt{\mu_j^2+\nu_j^2}}\left\{1-\frac{1}{\pi}\log\left(\frac{\mu_j^2}{\mu_j^2+\nu_j^2}\right)\right\} \nonumber \\ &= \frac{\pi}{2}\cdot\frac{1}{|\lambda_j|}\left\{1-\frac{1}{\pi}\log\left(\frac{({\rm Re}(\lambda_j))^2}{|\lambda_j|^2}\right)\right\},\end{aligned}$$ where we used Lemma [Lemma 2](#Lemma2){reference-type="ref" reference="Lemma2"} in the second equality and Lemma [Lemma 3](#Lemma3){reference-type="ref" reference="Lemma3"} in the first inequality. Inserting this into Eq. [\[eq:Yphibound2\]](#eq:Yphibound2){reference-type="eqref" reference="eq:Yphibound2"} gives $$\|E_2\|_F \le \gamma_{M}\kappa_2(X)\left\{n-\frac{2}{\pi}\sum_{j=1}^n\log\left(\frac{|{\rm Re}(\lambda_j)|}{|\lambda_j|}\right)\right\}. \label{eq:SumError}$$ ## Total roundoff error The total roundoff error is given as the sum of $E_1$ and $E_2$. From Eqs. [\[TermwiseError\]](#TermwiseError){reference-type="eqref" reference="TermwiseError"} and [\[eq:SumError\]](#eq:SumError){reference-type="eqref" reference="eq:SumError"}, we see that the bound on $\|E_1\|_F$ is cubic in $n$ and quartic in $\kappa_2(X)$, while that of $\|E_2\|_F$ is linear in both $n$ and $\kappa_2(X)$. Also, whereas both bounds show singularity when ${\rm Re}(\lambda_j)$ approaches zero, the singularity in the former is $O(1/{\rm Re}(\lambda_j))$, while that of the latter is logarithmic. From these facts, we can say that $E_1$ is dominant in the roundoff error. # Numerical results We performed numerical experiments to check the validity of our error bound. In the experiments, we used a PC with the AMD Ryzen 7 3700X Processor (8-Core, 3.59GHz). Our program was written in Python. To compute the matrix products and inverses, we used NumPy. All the computations were performed in double precision arithmetic. In the experiments, we computed ${\rm sign}(A)$ with the DE-based method, compared the result with that computed by Eq. [\[eq:definition\]](#eq:definition){reference-type="eqref" reference="eq:definition"}, and evaluated their difference $E$. In the DE-based method, $h$ was chosen sufficiently small and $N^+$ and $N^-$ were chosen sufficiently large so that the discretization and truncation errors can be neglected. The test matrices were constructed as $A=X\Lambda X^{\top}$, where $X$ is a nonsingular matrix with a specified condition number $\kappa_2(X)$ and $\Lambda$ is a real random diagonal matrix with a specified condition number $\kappa_2(\Lambda)$. To control the condition number of $X$, we generated $X$ as $X=QDQ^{\top}$, where $Q$ is a random orthogonal matrix and $D$ is a real random diagonal matrix with the condition number $\kappa_2(X)$. The matrix size $n$ was fixed to 100. In the first experiment, we fixed $\Lambda$ and varied $\kappa_2(X)$ from $10^1$ to $10^6$. The results are shown in Fig. [1](#fig:fig1){reference-type="ref" reference="fig:fig1"}(a). Here, the horizontal axis and the vertical axis denote $\kappa_2(X)$ and $\|E\|_F$, respectively, and log-log plot is used. By regression, the slope is estimated to be 3.886. This is in accordance with our theoretical result, which shows that the dominant component of the roundoff error, $\|E_1\|_F$, is bounded by a quantity proportional to $(\kappa_2(X))^4$. In the second experiment, we fixed $X$ and varied $\kappa_2(\Lambda)$ from $10^0$ to $10^6$. The result is shown in Fig. [1](#fig:fig1){reference-type="ref" reference="fig:fig1"}(b). In this case, the slope is estimated to be 1.957. This is smaller than expected from our bound [\[TermwiseError\]](#TermwiseError){reference-type="eqref" reference="TermwiseError"}, which suggests that $\|E_1\|_F$ grows cubically with the condition number of $\Lambda$ when all eigenvalues of $A$ are real. Thus, our bound seems to be somewhat overestimate with respect to $\Lambda$, although it still is a valid upper bound. {#fig:fig1 height="1.4in"} # Conclusion In this paper, we presented a roundoff error analysis of the DE-based method for the matrix sign function. Our upper bound on the roundoff error is in good accordance with the actual error and partially explains why the accuracy of the method deteriorates when the input matrix is ill-conditioned or highly nonnormal. One possible remedy would be to expand $(t^2 I+A^2)^{-1}$ into partial fractions, by allowing complex arithmetic. This will reduce the dependency of Eq. [\[eq:AF2\]](#eq:AF2){reference-type="eqref" reference="eq:AF2"} on $\kappa_2(X)$ from quadratic to linear, improving the overall dependency of the total error to $(\kappa_2(X))^3$. This will be a topic in our future study. # Acknowledgment {#acknowledgment .unnumbered} This study is supported by JSPS KAKENHI Grant Numbers 19KK0255 and 22KK19772. 99 N. J. Higham: *Functions of Matrices: Theory and Computation*, SIAM, Philadelphia, 2008. Y. Nakatsukasa and N. J. Higham: Stable and efficient spectral divide and conquer algorithms for the symmetric eigenvalue decomposition and the SVD, *SIAM J. Sci. Comput.*, Vol. 35, No. 3, pp. (2013). T. Nakaya and K. Tanaka: Numerical computation of matrix sign function by double exponential formula, *Transactions of JSIAM*, Vol. 31, No. 3, pp. 105--132 (2021) (in Japanese). H. Takahasi and M. Mori: Double exponential formulas for numerical integration, *Publ. RIMS Kyoto Univ.*, Vol. 9, pp. 721--741 (1974). T. Miyashita and Y. Yamamoto: Improvement and application of the double exponential formula-based computation method for the matrix sign function, *Proceedings of the 2022 Annual Meeting of JSIAM*, 2022 (in Japanese). N. J. Higham: *Accuracy and Stability of Numerical Algorithms*, 2nd Ed., SIAM, Philadelphia, 2002. Y. Yamamoto, S. Kudo and T. Hoshi: Error analysis of the truncated Taylor series expansion method for computing matrix exponential, *JSIAM Letters*, Vol. 14, pp. 147--150 (2022). https://mamekebi-science.com/math/integral/biquadratic-elliptic-20221222/ S. András and A. Baricz: Bounds for complete elliptic integrals of the first kind, *Expositiones Mathematicae*, Vol. 28, pp. 357--364 (2010). S. Koumandos: Remarks on a paper by Chao-Ping Chen and Feng Qi, *Proc. Amer. Math. Soc.*, Vol. 134, No. 5, pp. 1365--1367 (2005). # Appendix A {#appendix-a .unnumbered} **Lemma 1**. *Let $\lambda_j\in\mathbb{C}$ and ${\rm Re}(\lambda_j)\ne 0$. Then, Eq. [\[eq:quadraticintegral\]](#eq:quadraticintegral){reference-type="eqref" reference="eq:quadraticintegral"} holds.* *Proof.* Let $\lambda_j=\mu_j+{\rm i}\nu_j$, where ${\rm i}=\sqrt{-1}$. Then, the integrand can be rewritten as follows. $$\begin{aligned} &\frac{1}{|t^2+\lambda_j^2|^2} \nonumber \\ &= \frac{1}{(t^2+\mu_j^2-\nu_j^2)^2+4\mu_j^2\nu_j^2} \nonumber \\ &= \frac{1}{(t^2+\mu_j^2+\nu_j^2)^2-(2\nu_j t)^2} \nonumber \\ &= \frac{1}{(t+\nu_j)^2+\mu_j^2}\cdot\frac{1}{(t-\nu_j)^2+\mu_j^2} \nonumber \\ &= \frac{1}{4\nu_j(\mu_j^2+\nu_j^2)}\left\{\frac{t+2\nu_j}{(t+\nu_j)^2+\mu_j^2}+\frac{-t+2\nu_j}{(t-\nu_j)^2+\mu_j^2}\right\} \nonumber \\ &= \frac{1}{8\nu_j(\mu_j^2+\nu_j^2)}\left\{\frac{2(t+\nu_j)}{(t+\nu_j)^2+\mu_j^2}-\frac{2(t-\nu_j)}{(t-\nu_j)^2+\mu_j^2}\right\} \nonumber \\ & \quad + \frac{1}{4(\mu_j^2+\nu_j^2)}\left\{\frac{1}{(t+\nu_j)^2+\mu_j^2}+\frac{1}{(t-\nu_j)^2+\mu_j^2}\right\}. \label{eq:lambdaint}\end{aligned}$$ Now, let us choose $L\in\mathbb{R}$ so that $L>\max_{1\le j\le n}|\nu_j|$. Then, if $\nu_j\ge 0$, we have $$\begin{aligned} \int_{-L}^L\frac{2(t+\nu_j)}{(t+\nu_j)^2+\mu_j^2}\,dt &= \int_{-L+\nu_j}^{L+\nu_j}\frac{2t'}{{t'}^2+\mu_j^2}\,dt' \nonumber \\ &= \int_{L-\nu_j}^{L+\nu_j}\frac{2t'}{{t'}^2+\mu_j^2}\,dt' \nonumber \\ &= \log\frac{(L+\nu_j)^2+\mu_j^2}{(L-\nu_j)^2+\mu_j^2} \nonumber \\ &\rightarrow 0 \quad (L\rightarrow\infty),\end{aligned}$$ where the second inequality is due to the fact that the integrand is an odd function in $t$. The same result holds also when $\nu_j<0$ and also for the integral of $2(t-\nu_j)/\{(t-\nu_j)^2+\mu_j^2\}$. Thus, when the integration interval is $(-\infty, \infty)$, the contribution from the second line from the last of [\[eq:lambdaint\]](#eq:lambdaint){reference-type="eqref" reference="eq:lambdaint"} is zero. On the other hand, by letting $t=\mp\nu_j+|\mu_j|\tan\theta$, the contribution from the last line of [\[eq:lambdaint\]](#eq:lambdaint){reference-type="eqref" reference="eq:lambdaint"} can be computed as follows. $$\begin{aligned} & \frac{1}{4(\mu_j^2+\nu_j^2)}\left\{\int_{-\infty}^{\infty}\frac{dt}{(t+\nu_j)^2+\mu_j^2} + \int_{-\infty}^{\infty}\frac{dt}{(t-\nu_j)^2+\mu_j^2}\right\} \nonumber \\ &= \frac{1}{4(\mu_j^2+\nu_j^2)}\cdot\frac{2\pi}{|\mu_j|}=\frac{\pi}{2|\lambda_j|^2|{\rm Re}(\lambda_j)|}.\end{aligned}$$ Hence, the lemma is proved. ◻ **Lemma 2**. *Let $a,c\in\mathbb{R}$ satisfy $a>0$ and $|c|\le a^2$. Then, the following equality holds. $$\int_0^{\infty}\frac{dx}{\sqrt{x^4+2cx^2+a^4}}=\frac{1}{a}K\left(\frac{\sqrt{a^2-c}}{a\sqrt{2}}\right), \label{eq:elliptic1}$$ where $K(k)$ is the complete elliptic integral of the first kind defined by $$K(k)=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}. \label{eq:elliptic2b}$$* *Proof.* Following [@biquadratic], we compute the right-hand side of [\[eq:elliptic1\]](#eq:elliptic1){reference-type="eqref" reference="eq:elliptic1"} and show that it is equal to the left-hand side. Let $$\cos\theta=\frac{x^2-a^2}{x^2+a^2}.$$ Then, the right-hand side is a monotonically increasing function of $x$ in $[0,\infty)$ and $0 \le x <\infty$ corresponds to $\pi \ge \theta > 0$. Since $$-\sin\theta\,d\theta = \frac{4a^2 x}{(x^2+a^2)^2}\,dx,$$ we have $$\begin{aligned} d\theta&=-\frac{1}{\sin\theta}\cdot\frac{4a^2 x}{(x^2+a^2)^2}\,dx \nonumber \\ &= -\frac{1}{\sqrt{1-\left(\frac{x^2-a^2}{x^2+a^2}\right)^2}}\cdot\frac{4a^2 x}{(x^2+a^2)^2}\,dx \nonumber \\ &= -\frac{2a}{x^2+a^2}\,dx. \label{eq:dtheta}\end{aligned}$$ Also, note that $$\sin^2\theta=1-\left(\frac{x^2-a^2}{x^2+a^2}\right)^2=\frac{4a^2 x^2}{(x^2+a^2)^2}. \label{eq:sinsqtheta}$$ Substituting [\[eq:dtheta\]](#eq:dtheta){reference-type="eqref" reference="eq:dtheta"} and [\[eq:sinsqtheta\]](#eq:sinsqtheta){reference-type="eqref" reference="eq:sinsqtheta"} into the right-hand side of [\[eq:elliptic1\]](#eq:elliptic1){reference-type="eqref" reference="eq:elliptic1"} gives $$\begin{aligned} & K\left(\frac{\sqrt{a^2-c}}{a\sqrt{2}}\right) \nonumber \\ &= \int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-\frac{a^2-c}{2a^2}\sin^2\theta}} \nonumber \\ &= \frac{1}{2}\int_0^{\pi}\frac{d\theta}{\sqrt{1-\frac{a^2-c}{2a^2}\sin^2\theta}} \nonumber \\ &= \frac{1}{2}\int_0^{\infty}\frac{1}{\sqrt{1-\frac{a^2-c}{2a^2}\cdot\frac{4a^2 x^2}{(x^2+a^2)^2}}}\cdot\frac{2a}{x^2+a^2}\,dx \nonumber \\ &= a\int_0^{\infty}\frac{dx}{\sqrt{x^4+2cx^2+a^4}}.\end{aligned}$$ Hence, the lemma is proved. ◻ **Lemma 3**. *For $k\in(-1,1)$, the following inequality holds. $$K(k) \le \frac{\pi}{2}\left\{1-\frac{1}{\pi}\log(1-k^2)\right\}.$$* *Proof.* This inequality can be shown as a special case of the results given in [@Andras10]. For real numbers $a,b,c$ and $k$ satisfying $c\ne 0, -1, -2, \ldots$ and $k\in(-1,1)$, the Gauss hypergeometric function is defined by $$_2 F_1(a,b,c,k) = \sum_{n\ge 0}\frac{(a)_n(b)_n}{(c)_n}\cdot\frac{k^n}{n!},$$ where $(a)_n=\Gamma(a+n)/\Gamma(a)=a(a+1)\cdots(a+n-1)$. It is known that $K(k)$ can be expressed with $_2 F_1(a,b,c,k)$ as follows. $$K(k)=\frac{\pi}{2}\cdot\,_2F_1\left(\frac{1}{2},\frac{1}{2},1,k^2\right).$$ Now, let us consider a generalized complete elliptic integral of the first kind defined by $$K_a(k)=\frac{\pi}{2}\cdot\,_2F_1\left(a,1-a,1,k^2\right).$$ Koumandos [@Koumandos05] showed that $K_a(k)$ has the following lower and upper bounds when $a,k\in(0,1)$. $$\begin{aligned} & \frac{\pi}{2}\left\{1-\frac{\sin(\pi a)}{\pi}\left(\frac{1}{k^2}\log(1-k^2)+1\right)\right\} \nonumber \\ &< K_a(k) < \frac{\pi}{2}\left\{1-\frac{\sin(\pi a)}{\pi}\log(1-k^2)\right\}.\end{aligned}$$ By setting $a=1/2$, we immediately obtain the following bound on $K(k)$ that is valid when $k\in(0,1)$. $$K(k) < \frac{\pi}{2}\left\{1-\frac{1}{\pi}\log(1-k^2)\right\}.$$ Noting that $K(0)=\frac{\pi}{2}$ and replacing $<$ with $\le$, we obtain an inequality that is valid also for $k=0$. Hence, the lemma is proved. ◻

arxiv_math

--- abstract: | This is a brief introduction to link homology theories that categorify Reshetikhin--Turaev $\mathsf{SL}(N)$-quantum link invariants. A recently discovered surprising connection between finite state automata and Boolean TQFTs in dimension one is explained as a warm-up. address: - Department of Mathematics, United States Naval Academy, Annapolis, MD 21402, USA - Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA - Department of Mathematics, Columbia University, New York, NY 10027, USA author: - Mee Seong Im - Mikhail Khovanov bibliography: - z_brauer-group.bib date: September 17, 2023 title: "From finite state automata to tangle cobordisms: a TQFT journey from one to four dimensions" --- # Introduction {#section:intro} This note is based on the talk that one of us (M.K.) gave at the First International Congress of Basic Science which was held at the Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (BIMSA) in July 2023. We briefly discuss topological quantum field theories (TQFTs) and explain a recent surprising observation [@GIKKL23] that Boolean-valued one-dimensional TQFTs with defects correspond to nondeterministic finite state automata. The precise statement is given in Theorem [Theorem 2](#thm_automata_TQFT){reference-type="ref" reference="thm_automata_TQFT"} below and in [@GIKKL23]. We then review various approaches to categorification of Reshetikhin--Turaev link invariants for the fundamental representations of quantum $\mathsf{SL}(N)$ emphasizing the original approach via matrix factorizations [@KR04; @KR05; @Yon11; @WuEquiv12] and Robert--Wagner foam evaluation approach [@RW20]. Reshetikhin--Turaev link invariants [@RT90; @Resh91] are part of the Chern--Simons three-dimensional TQFT discovered by Witten and Reshetikhin--Turaev [@Witten89; @RT91]. Categorification of Reshetikhin--Turaev link invariants can be viewed as a 4-dimensional TQFT restricted to links in $\mathbb R^3$ and to link cobordisms. Boolean one-dimensional TQFTs and finite state automata connection is explained first, as a warm-up to discussing these sophisticated TQFTs in dimensions three and four. This is a relatively short paper which reviews Khovanov--Rozansky link homology in its second part. We provide key references but can not give a fully comprehensive coverage of the substantial amount of literature on this and closely related subjects due to the immense and rapid development over the last twenty-five years. **Acknowledgments:** The authors are grateful to Tsinghua University and Yau Mathematical Sciences Center for their hospitality. M.K. would like to acknowledge support from Beijing Institute of Mathematical Sciences and Applications (BIMSA) and the opportunity to give a plenary lecture at the first International Congress of Basic Science (ICBS). The authors would like to thank Joshua Sussan for valuable feedback on the paper. M.K. was partially supported by NSF grant DMS-2204033 and Simons Collaboration Award 994328 while working on this paper. # TQFTs, Boolean TQFTs and automata {#sec_boolean} In this section, we survey the correspondence between automata and one-dimensional Boolean TQFTs explained in [@GIKKL23]. ## TQFTs An $n$-dimensional TQFT (topological quantum field theory) over a field $\mathbf{k}$ is a tensor functor from the category of $n$-dimensional cobordisms to the category of vector spaces $$\mathcal{F}\ : \ \mathsf{Cob}_n \longrightarrow\mathbf{k}\mathsf{-vect}.$$ The category $\mathsf{Cob}_n$ has oriented $(n-1)$-manifolds as objects and $n$-dimensional cobordisms between them as morphisms. This category is symmetric tensor, and the functor $\mathcal{F}$ must respect this structure so that, in particular, $$\mathcal{F}(K_1\sqcup K_2) \cong \mathcal{F}(K_1)\otimes \mathcal{F}(K_2)$$ for $(n-1)$-manifolds $K_1$ and $K_2$. ## One-dimensional TQFTs A TQFT $\mathcal{F}$ in one dimension assigns a vector space $V$ to a point with positive orientation, a space $W$ to a point with negative orientation, which we can write as $$\mathcal{F}(+)= V, \hspace{1cm} \mathcal{F}(-)=W,$$ and maps $\mathbf{k}\longrightarrow V\otimes W$ and $W\otimes V\longrightarrow\mathbf{k}$ to the *cup*, respectively, $\emph{cap}$, cobordisms shown in the top row of Figure [\[figure-0.1\]](#figure-0.1){reference-type="ref" reference="figure-0.1"}. Transposition cobordisms, see Figure [\[figure-0.1\]](#figure-0.1){reference-type="ref" reference="figure-0.1"} top right, are mapped to transposition maps between products of $V$ and $W$, such as $V\otimes W\longrightarrow W\otimes V$, $v\otimes w\mapsto w\otimes v$. Isotopy relations shown in the second row of that figure imply that vector space $V$ is finite-dimensional and vector space $W\cong V^{\ast}$ can be taken to be the dual of $V$, so that cup and cap cobordisms are sent by $\mathcal{F}$ to the *evaluation* and *coevaluation* maps: $$\mathsf{coev}: \mathbf{k}\longrightarrow V \otimes V^{\ast}, \ \ \ \ \mathsf{ev}: V^{\ast}\otimes V \longrightarrow\mathbf{k}$$ given in a basis $\{v_1,\ldots, v_n\}$ of $V$ and the dual basis $\{v^1,\ldots, v^n\}$ of $V^{\ast}$ by $$\begin{aligned} \mathsf{coev}(1) & = & \sum_{i=1}^n v_i\otimes v^i, \\ \mathsf{ev}(v^i\otimes v_j) & = & \delta_{i,j}, \ \ \mathsf{ev}(v^{\ast}\otimes v)=v^{\ast}(v), \ \ v^{\ast}\in V^{\ast}, \ v \in V. \end{aligned}$$ Composing with the transposition morphism in Figure [\[figure-0.1\]](#figure-0.1){reference-type="ref" reference="figure-0.1"} in top right gives the cup and cap morphisms for the opposite orientation of the arc. The TQFT $\mathcal{F}$ evaluated on a circle gives a linear map $\mathbf{k}\longrightarrow\mathbf{k}$ which is multiplication by $\dim V$, and that is the only invariant of a one-dimensional TQFT. Two such TQFTs $$\mathcal{F}\ : \ \mathsf{Cob}_1 \longrightarrow\mathbf{k}\mathsf{-vect}, \hspace{1cm} \mathcal{F}(+)=V, \hspace{1cm} \mathcal{F}(-)=V^{\ast}$$ are isomorphic if and only if the corresponding vector spaces are isomorphic, that is, if they have the same dimension (evaluate the same on the circle). To spice up this simple classification, let us add $0$-dimensional defects, which are points on one-manifolds, labelled by elements of a finite set $\Sigma$, and also allow one-manifolds to end in the middle of a cobordism, see Figure [\[figure-A1\]](#figure-A1){reference-type="ref" reference="figure-A1"}. In this way, the boundary of a one-manifold cobordism splits into the *outer boundary* and *inner boundary*. Connected components of a morphism in $\mathsf{Cob}_{\Sigma,1}$ can be classified into four types: - arcs with two outer endpoints, - *half-intervals*, which are arcs with one outer and one inner endpoint, - *floating intervals*, which are arcs with two inner endpoints, - circles. The morphism in Figure [\[figure-A1\]](#figure-A1){reference-type="ref" reference="figure-A1"} consists of two arcs, three half-intervals, one floating interval and one circle. Each connected component of a morphism may be decorated by $\Sigma$-labelled dots. This bigger category $\mathsf{Cob}_{\Sigma,1}$ is also a symmetric tensor category, containing $\mathsf{Cob}_1$ as a subcategory. How can one classify TQFTs (tensor functors) $$\mathcal{F}\ : \ \mathsf{Cob}_{\Sigma,1} \longrightarrow\mathbf{k}\mathsf{-vect} \ ?$$ Necessarily $\mathcal{F}(+)\cong V$ and $\mathcal{F}(-)\cong V^{\ast}$ for some finite-dimensional vector space $V$, with cup and cap maps given by evaluation and coevaluation. Applying $\mathcal{F}$ to a dot labelled $a$ on an upward-oriented strand gives a morphism $\mathcal{F}(+)\longrightarrow\mathcal{F}(+)$, that is, a linear map $m_a:V\longrightarrow V$, see Figure [\[figure-3\]](#figure-3){reference-type="ref" reference="figure-3"}. Applying $\mathcal{F}$ to a dot labelled $a$ on a downward-oriented strand then necessarily gives the dual linear map $m_a^{\ast}:V^{\ast}\longrightarrow V^{\ast}$ (to check the duality property, move a dot across a local maximum or minimum of a cobordism shown in Figure [\[figure-0.1\]](#figure-0.1){reference-type="ref" reference="figure-0.1"} top left). A collection of labelled dots on a strand can be encoded by a word $\omega=a_1\cdots a_n$, with the corresponding map on $V$ given by the product, see Figure [\[figure-3\]](#figure-3){reference-type="ref" reference="figure-3"}. A *half-interval* (an interval with one outer and one inner endpoint) oriented upward with the outer endpoint at the top, upon applying $\mathcal{F}$, gives a linear map $\mathbf{k}\cong \mathcal{F}(\varnothing)\longrightarrow\mathcal{F}(+)\cong V$, that is, a vector $v_0\in V$, see Figure [\[figure-2\]](#figure-2){reference-type="ref" reference="figure-2"}. For the other half-interval with a $+$ endpoint, also see Figure [\[figure-2\]](#figure-2){reference-type="ref" reference="figure-2"}, applying $\mathcal{F}$ results in a morphism $V\longrightarrow\mathbf{k}$, described by a covector $v^{\ast}\in V^{\ast}$. No relations on $m_a$'s, $v_0$ and $v^{\ast}$ are imposed. Thus, a TQFT for the category $\mathsf{Cob}_{\Sigma,1}$ is given by a finite-dimensional vector space $V$, a collection of linear maps $m_a:V\longrightarrow V$, for $a\in \Sigma$, a vector $v_0$ and a covector $v^{\ast}$. Two such theories are isomorphic if there is an isomorphism between the corresponding vector spaces that intertwines maps $m_a$, for all $a$, vectors $v_0$ and covectors $v^{\ast}$ for the two theories. Topological quantum field theory $\mathcal{F}$ has a large number of invariants, which are values of $\mathcal{F}$ on: - A circle carrying a word $\omega=a_1\cdots a_n$, up to cyclic permutation. The functor $\mathcal{F}$ evaluates this decorated circle to the trace ${\sf tr}_V(m_{a_n}\cdots m_{a_1})\in \mathbf{k}$. - A floating interval carrying a word $\omega =a_1\cdots a_n$. The functor $\mathcal{F}$ evaluates this interval to $v^{\ast}(m_{a_1}\cdots m_{a_n}v_0)\in \mathbf{k}$. These evaluations are depicted in Figures [\[figure-4\]](#figure-4){reference-type="ref" reference="figure-4"} and [\[figure-5\]](#figure-5){reference-type="ref" reference="figure-5"}. For instance, for an interval, start with $v_0\in V$ at the endpoint where the orientation looks into the interval, then apply the map $m_{a_n}$ to $v_0$ and so on until the opposite end of the interval is reached. Then we apply the covector $v^{\ast}$ to the product. ## Automata and regular languages {#subsection:automata_regular_lang} By an *alphabet* or *set of letters* we mean a finite set $\Sigma$. A *language* $L$ is any subset of the free monoid $\Sigma^{\ast}$ on $\Sigma$. A (nondeterministic) finite state automaton $(Q)$ over alphabet $\Sigma$ consists of a finite set of states $Q$, a *transition function* $$\label{eq_delta} \delta: \Sigma\times Q\longrightarrow\mathcal{P}(Q),$$ where $\mathcal{P}(Q)$ is the powerset of $Q$, a subset $Q_{\mathsf{in}}\subset Q$ of *initial* states and a subset $Q_{\mathsf{t}}\subset Q$ of *accepting* or *terminal* states. To an automaton there is associated a graph $\Gamma_{(Q)}$ with $Q$ as the set of vertices, an oriented edge $q\longrightarrow q'$ labelled $a\in \Sigma$ if and only if $q'\in \delta(a,q)\subset Q$ and two subsets $Q_{\mathsf{in}}, Q_{\mathsf{t}}$ of distinguished vertices. Vice versa, such a data of a decorated oriented graph determines an automaton. A word $\omega=a_1\cdots a_n\in \Sigma^{\ast}$ is *accepted* by the automaton $(Q)$ if there exists an initial state $q_{\mathsf{in}}\in Q_{\mathsf{in}}$, an accepting state $q_{\mathsf{t}}\in Q_{\mathsf{t}}$ and an oriented path from $q_{\mathsf{in}}$ to $q_{\mathsf{t}}$ where consecutive labels of oriented edges are $a_1,\ldots, a_n$, see Figure [\[fig_A1\]](#fig_A1){reference-type="ref" reference="fig_A1"}. The set of words $\omega\in \Sigma^{\ast}$ accepted by $(Q)$ is called *the language of automaton $(Q)$*. We denote this language by $L_{(Q)}$. A language $L\subset \Sigma^{\ast}$ is called *regular* if it is the language of some automaton. One can check that a language is regular if and only if it can be described by a regular expression. **Example 1**. Consider the language $L$ for the alphabet $\Sigma=\{a,b\}$ given by the regular expression $L=(a+b)^{\ast}b(a+b)$. This expression describes all words which have $b$ as the second from the last letter. An example of a nondeterministic automaton for $L$ is shown in Figure [\[figure-X1\]](#figure-X1){reference-type="ref" reference="figure-X1"}. ## Boolean TQFT from an automaton {#subsection:boolean_TQFT} In the definition of a TQFT, we can replace field $\mathbf{k}$ by any commutative semiring. Commutativity is needed for the following reason: floating components of a cobordism are evaluated to elements of a ground ring or semiring. These components can change relative position, that is, float up or down past each other, which correspond to requiring that the ground (semi)ring is *commutative*. By analogy with the category $\mathsf{Cob}_{\Sigma,1}$ and a TQFT $\mathcal{F}$ on it we can imagine encoding words $\omega$ in some language $L$ by placing their labels next to dots along a one-manifold. A floating interval that carries a word $\omega$ can evaluate to two values, depending on whether or not $\omega$ is in the language $L$. We make these values $0,1$, with $\omega$ evaluating to $1$ if it belongs to a language $L$ and to $0$ otherwise. It is then natural to replace a field $\mathbf{k}$ by the Boolean semiring $\mathbb{B}=\{0,1 : 1+1=1\}$ which consists of these two elements, with the additional rule that $1+1=1$. In particular, $\mathbb{B}$ is not a ring, but a commutative semiring. We specialize to the commutative semiring $\mathbb{B}$ and fix an automaton $(Q)$ as above. Replace a $\mathbf{k}$-vector space $V\cong \mathbf{k}^n$ by a free module $\mathbb{B}Q$ over $\mathbb{B}$ with a set of generators $Q$. Elements of $\mathbb{B}Q$ are formal finite sums of distinct elements of $Q$ or, equivalently, subsets of $Q$, so we can identify $\mathbb{B}Q\cong \mathcal{P}(Q)$ with the set of subsets of $Q$. The zero element of $\mathbb{B}Q$ corresponds to the empty subset of $Q$. Note that $q+q=q$ for $q\in Q$ since $1+1=1$. We view $\mathbb{B}Q$ as a free $\mathbb{B}$-module with a basis $Q$. It has $2^{|Q|}$ elements. Unlike the case of a field, where the group $\mathsf{GL}(V)$ acts freely and transitively on the set of bases of $V$, the only basis of $\mathbb{B}Q$, up to changing the order of elements, is $Q$. To the automaton $(Q)$ we associate a Boolean-valued TQFT $$\mathcal{F}\ : \ \mathsf{Cob}_{\Sigma,1} \longrightarrow\mathbb{B}\mathsf{-fmod}$$ taking values in the category of free $\mathbb{B}$-modules by assigning $\mathbb{B}Q$ to the $+$ point and its dual $\mathbb{B}Q^*$ to the $-$ point: $$\begin{aligned} &\mathcal{F}(+) = \mathbb{B}Q, \hspace{3.0cm} \mathcal{F}(-) = \mathbb{B}Q^{\ast}, \\ &\mathsf{ev} : \ \mathbb{B}Q^{\ast} \otimes \mathbb{B}Q\longrightarrow\mathbb{B}, \hspace{1.35cm} \mathsf{ev}(q^{\ast}\otimes q') = \delta_{q,q'} \\ &\mathsf{coev}: \ \mathbb{B}\longrightarrow\mathbb{B}Q \otimes \mathbb{B}Q^{\ast}, \hspace{1cm} \mathsf{coev}(1) = \sum_{q\in Q}q\otimes q^{\ast}. \end{aligned}$$ Here $Q^{\ast}$ is a copy of the set $Q$ with elements labelled $q^{\ast}$, over $q\in Q$, with the evaluation and coevaluation maps $\mathsf{ev}$ and $\mathsf{coev}$ given by the above formulas. For each $a\in \Sigma$ the transition map $\delta_a : Q\longrightarrow\mathcal{P}(Q)$ extends to a $\mathbb{B}$-linear endomorphism $\delta_a$ of the free module $\mathbb{B}Q$ that takes $q$ to $\delta(a,q)$, using the identification $\mathcal{P}(Q)\cong \mathbb{B}Q$, see above. Note that $\delta_a$ denotes the transition map while $\delta_{q,q'}$ stands for the delta function on a pair of states. For example, in the basis $\{q_1,q_2,q_3\}$ of $\mathbb{B}Q$ in Example [Example 1](#ex_language){reference-type="ref" reference="ex_language"} the two maps for $a,b\in \Sigma$ are given by $\mathbb{B}$-valued matrices $$\delta_a = \begin{pmatrix} 1 & 0 & 1\\ 0 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix}, \hspace{1cm} \delta_b = \begin{pmatrix} 0 & 0 & 0\\ 1 & 1 & 1\\ 0 & 1 & 0 \end{pmatrix},$$ see Figure [\[figure-X1\]](#figure-X1){reference-type="ref" reference="figure-X1"}. To an upward-oriented arc cobordism from $+$ to $+$ with an $a$-dot, $a\in \Sigma$, we associate the endomorphism $\delta_a$ of $\mathbb{B}Q$, see Figure [\[figure-X2\]](#figure-X2){reference-type="ref" reference="figure-X2"}. To a downward-oriented arc, which is a cobordism from $-$ to $-$, we associate the dual map $\delta_a^{\ast}:\mathbb{B}Q^{\ast}\longrightarrow\mathbb{B}Q^{\ast}$ given by the transposed matrix to that of $\delta_a$, see Figure [\[figure-X2\]](#figure-X2){reference-type="ref" reference="figure-X2"}. To upward-oriented half-intervals (cobordisms from the empty 0-manifold $\varnothing_0$ to $+$ and from $+$ to $\varnothing_0$) associate maps between free $\mathbb{B}$-modules $\mathbb{B}$ and $\mathbb{B}Q$ given by $$\begin{aligned} & \mathbb{B}\longrightarrow\mathbb{B}Q, \ \ 1\longmapsto \displaystyle{\sum_{q\in Q_{\mathsf{in}}}} q, \\ & \mathbb{B}Q\longrightarrow\mathbb{B}, \ \ q\longmapsto \begin{cases} 1 & \mathsf{if }\ q\in Q_{\mathsf{t}}, \\ 0 & \mathsf{if }\ q\notin Q_{\mathsf{t}}, \end{cases}\end{aligned}$$ see Figure [\[figure-1-1\]](#figure-1-1){reference-type="ref" reference="figure-1-1"}, and likewise for downward-oriented half-intervals. These maps are determined by subsets $Q_{\mathsf{in}}$ of initial and $Q_{\mathsf{t}}$ of accepting states. Alternatively the image of $1$ under the first map can be written as $Q_{\mathsf{in}}$, using identification $\mathbb{B}Q\cong \mathcal{P}(Q)$, and the second map can be denoted $Q_{\mathsf{t}}^{\ast}$. Given a floating interval with the word $\omega=a_1\cdots a_n$ written on it so that $a_1$ is at the tail and $a_n$ at the head, the interval evaluates to $1\in \mathbb{B}$ if and only if $\omega$ is in the language $L_{(Q)}$ defined by the automaton $(Q)$, see Figure [\[figure-5a\]](#figure-5a){reference-type="ref" reference="figure-5a"}. If $\omega$ is not in $L_{(Q)}$ the interval evaluates to $0$. The evaluation can also be written as $Q_{\mathsf{t}}^{\ast}(\delta_{a_n}\cdots \delta_{a_1}Q_{\mathsf{in}})$, that is, evaluating the product $\delta_{a_n}\cdots \delta_{a_1}Q_{\mathsf{in}}\in \mathbb{B}Q$ on the functional $Q_{\mathsf{t}}^{\ast}$. A circle with a circular word $\omega=a_1\cdots a_n$ placed on it evaluates to the trace of operator $\delta_{\omega}:=\delta_{a_n}\cdots \delta_{a_1}$ on $\mathbb{B}Q$, see Figure [\[figure-4a\]](#figure-4a){reference-type="ref" reference="figure-4a"}. Equivalently, it evaluates to $1$ if and only if there is a state $q\in Q$ and a sequence of arrows labelled $a_1,a_2,\ldots, a_n$ terminating back in $q$ (*i.e.*, and oriented cycle with word $\omega$ in the graph of the automaton). We state a main result in [@GIKKL23]: **Theorem 2** (Gustafson--Im--Kaldawy--Khovanov--Lihn). *A nondeterministic finite state automaton $(Q)$ on alphabet $\Sigma$ defines a one-dimensional Boolean-valued TQFT $$\label{eq_thm_TQFT} \mathcal{F}\ : \ \mathsf{Cob}_{\Sigma,1} \longrightarrow\mathbb{B}\mathsf{-fmod}$$ with $\mathcal{F}(+)=\mathbb{B}Q$, transition function of $(Q)$ encoding TQFT maps for $\Sigma$-labelled defects on strands, and sets of initial and accepting states encoding the maps for undecorated half-intervals. This correspondence gives a bijection between isomorphism classes of Boolean-valued TQFTs for $\mathsf{Cob}_{\Sigma,1}$ with $\mathcal{F}(+)$ a free $\mathbb{B}$-module and isomorphism classes of nondeterministic finite state automata on alphabet $\Sigma$.* $\mathbb{B}\mathsf{-fmod}$ in [\[eq_thm_TQFT\]](#eq_thm_TQFT){reference-type="eqref" reference="eq_thm_TQFT"} denotes the category of free $\mathbb{B}$-semimodules and semimodule maps. Regular language $L_{(Q)}$ of the automaton $(Q)$ describes evaluation of floating intervals decorated by words in $\Sigma^{\ast}$ (with $\omega$-decorated interval evaluating to $1$ if and only if $\omega \in L_{(Q)}$), while evaluation of circles with defects in the TQFT $\mathcal{F}$ is determined by oriented cycles in the graph of $(Q)$. The table in Figure [\[mfig_022\]](#mfig_022){reference-type="ref" reference="mfig_022"} summarizes the correspondence between generating morphisms in $\mathsf{Cob}_{\Sigma,1}$ and structural parts of an automaton (the set of states, transition maps, and sets of initial and accepting states). Consider a one-dimensional TQFT $$\label{eq_thm_TQFT_2} \mathcal{F}\ : \ \mathsf{Cob}_{\Sigma,1} \longrightarrow\mathbb{B}\mathsf{-mod}$$ valued, more generally, in the category of all $\mathbb{B}$-modules rather than free ones. Then necessarily $P=\mathcal{F}(+)$ is a finitely-generated semimodule which is *projective* in the sense of being a retract of a free module [@GIKKL23; @IK-top-automata]. Namely, there are semimodule maps $$\label{eq_retract} P \stackrel{\iota}{\longrightarrow} \mathbb{B}^n \stackrel{p}{\longrightarrow} P, \hspace{1cm} p\circ\iota=\mathsf{id}_P$$ for some $n$. Note that $P$ is a direct summand of $\mathbb{B}^n$ only if $P$ is free, otherwise it is just a retract. Then $P^{\ast}\cong \mathcal{F}(-)$ is finitely-generated projective as well, with the retract maps $p^{\ast},\iota^{\ast}$ given by dualizing those for $P$. Finitely-generated projective $\mathbb{B}$-modules $P$ are described by finite distributive lattices, *i.e.*, see [@IK-top-automata]: it is a theorem that goes back at least to Birkhoff that any such $P$ is isomorphic to the $\mathbb{B}$-semimodule $\mathcal{U}(X)$ of open sets in a finite topological space $X$, with the empty set $\varnothing$ the $0$ element of the semimodule and addition $U_1+U_2:=U_1 \cup U_2$ given by the union of sets. The structure of a TQFT $\mathcal{F}$ in this case is given by - a collection of endomorphisms $\delta_a:\mathcal{U}(X)\longrightarrow\mathcal{U}(X)$ for $a\in \Sigma$, taking open sets to open sets, $\varnothing$ to $\varnothing$, and preserving unions of sets, - initial element $Q_{\mathsf{in}}\in \mathcal{U}(X)$ and a terminal map $Q_{\mathsf{t}}:\mathcal{U}(X)\longrightarrow\mathbb{B}$ taking $\varnothing$ to $0$ and intertwining union of sets with addition in $\mathbb{B}$. Such structures are called *quasi-automata* in [@GIKKL23]. It is an interesting question whether they can be of use in computer science. ## Extending to arbitrary commutative semirings It is straightforward to extend the above TQFT construction from vector spaces over a field $\mathbf{k}$ and $\mathbb{B}$-semimodules to semimodules over a commutative semiring $R$. A commutative semiring $R$ is an abelian group under addition, a commutative monoid under multiplication, and distributivity property holds, $a(b+c)=ab+ac$. Semiring $R$ has the zero element $0$ and the unit element $1$. Subtraction operation $a-b$ is usually not available in semirings. It is straightforward to define the notion of a module $M$ over $R$ (alternatively called a *semimodule*) and introduce the category $R\mathsf{-dmod}$ of $R$-modules. In the definition of a TQFT over $R$ the tensor product of vector spaces is replaced by the tensor product of $R$-modules, and vector space $V=\mathcal{F}(+)$ is replaced by an $R$-module $P=\mathcal{F}(+)$. To have cup and cap morphisms subject to the isotopy relations above requires $P$ to be a projective $R$-module of finite rank, see [@GIKKL23], for instance. This observation quickly leads to the following in [@GIKKL23]: **Theorem 3** (Gustafson--Im--Kaldawy--Khovanov--Lihn). *One-dimensional TQFTs $$\mathcal{F}:\mathsf{Cob}_1\longrightarrow R\mathsf{-mod}$$ over a commutative semiring $R$ correspond to finitely-generated projective $R$-modules. One-dimensional TQFTs with defects, that is, tensor functors $$\label{TQFT_S_R} \mathcal{F}\ : \ \mathsf{Cob}_{\Sigma,1}\longrightarrow R\mathsf{-mod}$$ are in a correspondence with finitely-generated projective $R$-modules $P$ equipped with endomorphisms $m_a:P\longrightarrow P$ for $a\in \Sigma$, an element $v_0\in P$ and a covector $v^{\ast}\in \mathsf{Hom}_R(P,R)$. This correspondence is a bijection between isomorphism classes of TQFTs and isomorphism classes of data $(P,\{m_a\}_{a\in\Sigma},v_0,v^{\ast})$.* Here, an $R$-semimodule $P$ is defined to be finitely-generated projective if it is a retract of free semimodule $R^n$ for some $n$: $$\label{eq_retract_2} P \stackrel{\iota}{\longrightarrow} R^n \stackrel{p}{\longrightarrow} P, \hspace{1cm} p\circ\iota=\mathsf{id}_P .$$ It may be interesting to look at examples of such TQFTs when $R$ is, for instance, the tropical semiring, see [@JK_tropical14; @IJK_tropical_poly16] where projective modules over the tropical semiring are studied. Another question is whether the notion of a *finite state machine*, which extends the notion of a finite state automaton, has the TQFT counterpart. The authors are not aware of any studies or results on Boolean TQFTs (and, generally, TQFTs over commutative semirings that are not rings) in dimension two and higher. The above correspondence between finite state automata and one-dimensional Boolean TQFTs with defects, observed in [@GIKKL23], see also [@IK-top-automata], remains a curiosity, for now. Two-dimensional TQFTs for oriented surfaces, without defects and over a field $\mathbf{k}$, are classified by commutative Frobenius $\mathbf{k}$-algebras, and one open problem is to find a supply of commutative Frobenius $\mathbf{k}$-semialgebras when $\mathbf{k}$ is the Boolean semiring $\mathbb{B}$ or the tropical semiring. # Reshetikhin--Turaev $\mathsf{SL}(N)$-invariants and their categorification Quantum link polynomials were discovered by Vaughan Jones [@Jones85; @Jones97] (the Jones polynomial), Louis Kauffman [@Kauffman90] (the Kauffman polynomial and bracket), J. Hoste, A. Ocneanu, K. Millet, P. Freyd, W.B.R. Lickorish, D. Yetter [@FYHLMO85], J. Przytycki and P. Traczyk [@Przytycki88] (the HOMFLYPT polynomial) and others (Alexander polynomial, discovered several decades prior, was an outlier). N. Reshetikhin and V. Turaev [@RT90] put these polynomials into the framework of quantum groups and their representations (V. Drinfeld [@Drinfeld86; @Drinfeld87], M. Jimbo [@Jimbo85]), see also earlier preprints of N. Reshetikhin [@Resh87; @Resh87_II] and many other references. Furthermore, while at generic values of the parameter $q$ of the quantum group $U_q(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$, its representation theory produces link invariants, root of unity values give rise to Witten--Reshetikhin--Turaev [@Witten89; @RT91; @Resh91] and related invariants of 3-manifolds, see [@MR1191386; @Kuperberg96; @KL01], which can also be thought of as three-dimensional TQFTs. Quantum group $U_q(\mathfrak{g})$ is a Hopf algebra deformation of the universal enveloping algebra $U(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$. For generic $q$, representation theory of $U_q(\mathfrak{g})$ gives rise to Reshetikhin--Turaev invariants $P_{\mathfrak{g}}(L)\in q^{\ell}\mathbb Z[q,q^{-1}]$ of knots and links $L$ in $\mathbb{R}^3$, where $\ell$ is a rational number that depends on $\mathfrak{g}$ and the linking number of $L$, see [@Le00]. A key property of $U_q(\mathfrak{g})$ is that it is *quasitriangular*. To define the Reshetikhin--Turaev invariant of a link, its connected components need to be labelled by irreducible representations of $U_q(\mathfrak{g})$. The latter are parametrized by positive integral weights $\lambda\in \Lambda^+$ of $\mathfrak{g}$. Quantum invariants of 3-manifolds (Witten--Reshetikhin--Turaev invariants) are given by an appropriate sum of these invariants when $q$ is a root of unity [@RT91; @KM91]. A 3-manifold is given by surgery on a link. ## Tangles and Reshetikhin--Turaev invariants Reshetikhin--Turaev invariants [@RT90] are defined for tangles, which are links with boundary (more carefully, the invariant is usually defined for framed links and tangles). Tangles constitute a braided monoidal category $\mathsf{Tan}$. Composition of tangles is given by concatenation, while the tensor product on morphisms is given by placing tangles in parallel. Objects in the category of tangles are finite sequences of signs, which are orientations of a tangle at its endpoints. The Reshetikhin--Turaev invariant $f(T)$ of a tangle $T$ is an intertwiner (homomorphism of representations) between tensor products of representations, read off from the endpoints of a tangle. If one picks a representation $V$ of the quantum group $U_q(\mathfrak{g})$, the invariant is an intertwiner between tensor products of $V$ and its dual $V^{\ast}$, according to the orientations of the endpoints, see an example in Figure [\[mfig_002\]](#mfig_002){reference-type="ref" reference="mfig_002"}. To define the Reshetikhin--Turaev invariant in full generality, one first modifies the category $\mathsf{Tan}$ by labeling the components of a tangle by positive integral weights $\lambda\in \Lambda^+$. The Reshetikhin--Turaev invariant for a labelled tangle is built from intertwiners between tensor products of irreducible representations $V_{\lambda}$ of $U_q(\mathfrak{g})$. A link $L$ is a tangle with the empty boundary. Reshetikhin--Turaev invariant $f(L): \mathbb C\longrightarrow\mathbb C$ is then a scalar, depending on $q$, see Figure [\[mfig_003\]](#mfig_003){reference-type="ref" reference="mfig_003"}, and $f(L)\in \mathbb Z[q^{1/D},q^{-1/D}]$, where $D$ divides the determinant of the Cartan matrix of $\mathfrak{g}$, see [@Le00]. This integrality is a special property of Reshetikhin--Turaev invariants. ## Crane--Frenkel conjecture Around 1994, Igor B. Frenkel (the graduate advisor of the second author) and Louis Crane proposed in [@CF94]: **Conjecture 4** (Crane--Frenkel). There exists a categorification of the quantum group $U_q(\mathfrak{sl}_2)$ at roots of unity giving rise to a 4D TQFT. One motivation for the conjecture was that Floer homology was already known at the time. It has the Casson invariant as its Euler characteristic. Floer homology can be viewed as a 4-dimensional TQFT, defined for at least some 3-manifolds and 4-cobordisms. It was natural to wonder whether other quantum invariants of 3-manifolds can be realized as Euler characteristics. Another motivation came from geometric representation theory, with the discovery of the Kazhdan--Lusztig basis in the Hecke algebra in [@KL79] and its geometric interpretation via sheaves on flag varieties. This was followed by the Beilinson--Lusztig--MacPherson's [@BLM90] geometric interpretation of $V^{\otimes k}$, for $V$ a fundamental representation of $U_q(\mathfrak{sl}_n)$, via sheaves on partial flag varieties, with generators $E_i,F_i$ of the quantum group acting by correspondences. Lusztig's geometric realization [@Lusztig91] of Kashiwara--Lusztig bases [@Kash94] of $U_q(\mathfrak{g})$ and bases of its irreducible representations and Lusztig's discovery of his bases of idempotented quantum groups [@Lusztig10]. I. Frenkel's insight, beyond the above conjecture, was that positivity and integrality structure of these bases should be used to systematically lift Hecke algebra and quantum group elements to functors acting on categories that replace representations of quantum groups. On the TQFT level this should correspond to lifting quantum invariants one dimension up, from 3D to 4D (categorification). While Conjecture [Conjecture 4](#conjecture:Crane-Frenkel){reference-type="ref" reference="conjecture:Crane-Frenkel"} is still open, significant progress in the past thirty years has been made on: - Link homology theories, which are four-dimensional counterparts of Reshetikhin--Turaev quantum link invariants, - Categorification of quantum groups at generic $q$ (A. Lauda, A. Lauda--Khovanov, R. Rouquier and further foundational work by many researchers), - Categorification at prime roots of unity (Y. Qi, J. Sussan, B. Elias, Khovanov). ## Semisimple versus triangulated Reshetikhin--Turaev link invariants are governed by *semisimple* categories of representations of quantum groups $U_q(\mathfrak{g})$. In one dimension up (4D), categories cannot be semisimple. They are also unlikely to be abelian. In four dimensions, an *extended* TQFT for link cobordisms requires one to assign: - a category $\mathcal{C}_n$ to a plane with $n$ points (ignoring orientations, for simplicity), - a functor $F(T):\mathcal{C}_n\longrightarrow\mathcal{C}_m$ between these categories to a tangle $T$ with $n$ bottom and $m$ top boundary points, and - a natural transformation $F(T_1)\Rightarrow F(T_2)$ between these functors to a tangle cobordism between $T_1$ and $T_2$. This assignment should assemble into a 2-functor $$\mathcal{F}\ : \ \mathsf{Tan}_2 \longrightarrow\mathcal{NT}$$ from the 2-category of tangle cobordisms to the 2-category $\mathcal{NT}$ of natural transformations (between functors between appropriate categories). Usually one wants to convert topological structures into something algebraic, so functors and natural transformations must be additive and defined over a field or a commutative ring. In particular, the braid group $\mathsf{B}_n$ on $n$ strands (the mapping class group of a plane with $n$ points) needs to act on $\mathcal{C}_n$, the category assigned to the plane with $n$ points. An action of a group $G$ on a semisimple category essentially just permutes its simple objects, thus reduces to a homomorphism $G\longrightarrow S_n$ into the symmetric group. Homomorphisms from braid groups $\mathsf{B}_n$ to symmetric groups $S_n$ are unlikely to be part of a sophisticated structure that carries interesting information about four-dimensional topology. The same argument applies to full four-dimensional TQFTs, replacing braid groups by mapping class groups of closed surfaces. In some cases one can expect that endomorphism rings $\mathsf{End}(L_i)$ of simple objects are not the ground field $\mathbf{k}$ but a field or a division ring $D$ over $\mathbf{k}$, in which case homomorphisms of $\mathsf{B}_n$ into $\mathsf{Aut}_{\mathbf{k}}(D)$ or into cross-products of its direct products with the symmetric group may be available, but the crux of this informal argument (even just a gut feeling) that such homomorphisms cannot be upgraded to a sophisticated 4D TQFT remains. This informal argument about unsuitability of semisimple categories and the need for triangulated categories in four-dimensional TQFTs can be found in the old paper of one of us [@Kho02 Section 6.5] which further argues that interesting four-dimensional TQFTs are unlikely to assign abelian categories to surfaces. For a much more recent and precise work we refer to Reutter [@Reu20] and [@RSP22]. Replacing semisimple or abelian categories by triangulated categories removes the obstacle which is the lack of interesting categorical actions. Consider a ring $A$, take its category of modules $A\mathsf{-mod}$ and form the category $\mathcal{H}(A\mathsf{-mod})$ of finite length complexes of $A$-modules up to chain homotopies: $$\xymatrix{ \ldots \ar[r] & M^i \ar[r]^{d^i} & M^{i+1} \ar[r]^{d^{i+1}} & M^{i+2} \ar[r]^{d^{i+2}} & \ldots, & \hspace{-0.5cm} \mbox{where } M^i\in A\mathsf{-mod}, \ \ d\circ d=0. }$$ One can come up with a specific finite-dimensional ring $A$ (interestingly, graded $A$-modules are essentially the same as double complexes) and a *faithful* action of the braid group on the category $\mathcal{H}(A\mathsf{-mod})$, see [@KS02], also [@ST01] in the context of algebraic geometry. Passing to the Grothendieck group of $\mathcal{H}(A\mathsf{-mod})$ recovers the Burau representation of the braid group (or the permutation representation, ignoring the extra grading. An additional grading on the category of modules turns the Grothendieck group into a $\mathbb Z[q,q^{-1}]$-module). Faithfulness holds in either case. There are many ways to construct algebras with a faithful braid group action on their homotopy categories but the above example seems minimal, in a sense. One can, for instance, ask for a finite-dimensional algebra $A_n$ over a field $\mathbf{k}$ with a faithful action of the braid group $\mathsf{B}_n$ on $n$ strands on its homotopy category. In the example in [@KS02] $\dim_{\mathbf{k}}(A_n)=4n-6$, and we do not know examples of algebras of dimension less than $4n-6$ for $n\ge 3$ with a faithful action of $\mathsf{B}_n$ on their homotopy categories of complexes. Graded $A_n$ and $A$-modules correspond to bicomplexes, and one is working in the homotopy category of complexes over them. The latter category can be replaced by the stable category of tricomplexes, with a faithful braid group action on it [@KQ20]. It is an open question whether this step into stable categories and tricomplexes can be extended beyond categorified Burau representation, to categorification of other braid group and quantum group representations. ## The HOMFLYPT polynomial The Reshetikhin--Turaev invariant for the quantum group $U_q(\mathfrak{sl}(N))$ of $\mathfrak{sl}(N)$ and its fundamental $N$-dimensional representation $V$ is determined by the skein relation and normalization on the unknot: Sometimes the normalization on the unknot is taken to be $1$, since otherwise the invariant of any nonempty link is divisible by $[N]$. The disjoint union with the unknot multiplies either invariant by $[N]$ and the above normalization is natural from the categorification viewpoint. The skein relation above is due to the space of intertwiners (homomorphisms of quantum group representations) $V^{\otimes 2}\longrightarrow V^{\otimes 2}$ being two-dimensional, so that any three maps are related by a linear equation. The HOMFLYPT polynomial [@FYHLMO85; @Przytycki88] is a 2-variable invariant $P(L)\in \mathbb Z[a^{\pm 1},b^{\pm 1}]$ given by replacing the coefficients in the above skein relation by $a=q^N,b = q-q^{-1}$. One-variable specializations $P_N(L)$ of $P(L)$ have representation-theoretic interpretation via the quantum group $U_q(\mathfrak{sl}(N))$, as briefly explained earlier. Replacing $N$ by $-N$ and $q$ by $q^{-1}$ preserves the invariant, so one can restrict to $N\ge 0$. We record special cases: - $N=0$: $P_0(L)$ is the Alexander polynomial. - $N=1$: $P_1(L)$ is a trivial invariant. - $N=2$: $P_2(L)$ is the Jones polynomial. - $N=3$: $P_3(L)$ is the Kuperberg $\mathfrak{sl}(3)$ quantum invariant [@Kup96]. ## MOY graphs and their invariants The above specializations $P_N(L)$ take values in the ring $\mathbb Z[q,q^{-1}]$ of Laurent polynomials with *integer* coefficients. It is possible to reduce links to linear combinations of planar objects (webs or graphs) on which the invariant takes values in $\mathbb Z_+[q,q^{-1}]$, the semiring of Laurent polynomials with *non-negative integer* coefficients. Decompose $V^{\otimes 2}\cong \Lambda^2_q(V)\oplus S^2_q(V)$ into the sum of two irreducible representations---the second quantum exterior and symmetric powers of $V$---and consider projection and inclusion operators $$\xymatrix{ V^{\otimes 2} \ar[r]^p & \Lambda^2_q(V) \ar[r]^{\iota} & V^{\otimes 2} }$$ scaled so that $p\circ\iota = (q+q^{-1}) \mathsf{id}$ is the identity map of $\Lambda^2_q(V)$ times $q+q^{-1}$, see Figure [\[mfig_005\]](#mfig_005){reference-type="ref" reference="mfig_005"}. From these two basic pieces one can assemble oriented planar graphs with thin and thin edges and all vertices of valency three, as in Figure [\[mfig_005\]](#mfig_005){reference-type="ref" reference="mfig_005"} left and center, with one thick and two thin edges at each vertex. Vertices correspond to the intertwiners $p$ and $\iota$. These are the simplest instances of Murakami--Ohtsuki--Yamada (MOY) graphs. We think of thin edges as carrying label one (for $V$) and thick edges as carrying label two (for $\Lambda_q^2 V$). Such a graph with boundary defines an intertwiner between tensor products of $V$, $\Lambda_q^2 V$ and their duals. A closed MOY graph $\Gamma$ defines an endomorphism of the trivial representation of $U_q(\mathfrak{sl}(N))$, thus a function $P_N(\Gamma)\in \mathbb C(q)$. The invariant $P_N(\Gamma)$ is both integral and positive, see [@MOY98]: **Theorem 5** (Murakami--Ohtsuki--Yamada). *For any planar MOY graph $\Gamma$ as above, its invariant $P_N(\Gamma)\in \mathbb Z_+[q,q^{-1}]$.* Integrality and positivity properties of $P_N(\Gamma)$ are key to its categorification and categorification of the corresponding link invariants $P_N(L)$. Let us first observe that the link invariant $P_N(L)$ reduces to the invariants $P_N(\Gamma)$ of MOY planar graphs via skein relations: implying the HOMFLYPT relation More generally, one can allow lines of thickness $a$, $1\le a\le N-1$, which correspond to fundamental representations $\Lambda^a_q V$ of the quantum group $U_q(\mathfrak{sl}(N))$ and $(a,b,a+b)$-vertices, see the top row of Figure [\[mfig_007\]](#mfig_007){reference-type="ref" reference="mfig_007"}, which denote the scaled projection and inclusion between the corresponding representations $$\xymatrix{ \Lambda^a_q V \otimes \Lambda^b_q V \ar[r]^{\hspace{0.35cm} p_{a,b}} & \Lambda^{a+b}_q V, } \qquad \quad \xymatrix{ \Lambda^{a+b}_q V \ar[r]^{\iota_{a,b}\hspace{0.50cm}} & \Lambda^a_q V \otimes \Lambda^b_q V. }$$ normalized so that the composition $$p_{a,b}\circ \iota_{a,b} = \begin{bmatrix} a+b \\ b \\ \end{bmatrix} \,\mathsf{id}= \frac{[a+b]!}{[a]![b]!}\cdot\mathsf{id}, \qquad [a]! :=\prod_{c=1}^a[c], \qquad [c]=\frac{q^c-q^{-c}}{q-q^{-1}}$$ is the identity endomorphism of $\Lambda_q^{a+b}V$ scaled by the $q$-binomial coefficient. It is also natural to allow thickness $N$ as well and replace $\mathfrak{sl}(N)$ by the Lie algebra $\mathfrak{gl}(N)$. The exterior power $\Lambda^N V$ is a trivial representation of $\mathfrak{sl}(N)$ but not of $\mathfrak{gl}(N)$, and likewise for their quantum deformations, and vertices $(a,N-a)$ are now allowed, with lines of thickness $a$ and $N-a$ merging into a line of thickness $N$. Murakami--Ohtsuki--Yamada planar graphs or webs $\Gamma$ with edges of arbitrary thickness from $1$ to $N$ have vertices $(a,b)$ as above ($a+b\le N$). The quantum invariant $P_N(\Gamma)$ of an MOY graph with edges of arbitrary thickness takes values in $\mathbb Z_+[q,q^{-1}]$, and Theorem [Theorem 5](#thm_MOY){reference-type="ref" reference="thm_MOY"} holds for such graphs as well. Consider a link $L$ with components labelled by numbers in $\{1,2,\dots, N\}$ and pick a planar projection. The analogue of the skein relations above $(I)$ and $(II)$ are the relations in Figures [\[mfig_023\]](#mfig_023){reference-type="ref" reference="mfig_023"}, [\[mfig_024\]](#mfig_024){reference-type="ref" reference="mfig_024"}, see for instance [@Wu12]. These skein relations allow to define $P_N(L)\in \mathbb Z[q,q^{-1}]$ for links with components labelled by numbers from $1$ to $N$. ## Matrix factorizations and link homology **Idea:** To realize $P_N(L)$ as the Euler characteristic of a bigraded homology theory $H_N(L)$ of links, first build homology $H_N(\Gamma)$ for planar graphs $\Gamma$ as *singly-graded* vector spaces. There must be an equality of Laurent polynomials $$P_N(\Gamma) \ = \ \mathsf{gdim} H_N(\Gamma),$$ where for a $\mathbb Z$-graded vector space $V=\oplus_i V_i$, the graded dimension $\mathsf{gdim} V := \sum_i \dim(V_i)q^i.$ Then lift skein relations above to the long exact sequence in Figure [\[mfig_009\]](#mfig_009){reference-type="ref" reference="mfig_009"}. Commutative ground ring $R$ of the theory may be different from a field, in which case one expects $H_N(\Gamma)$ to be a free graded $R$-module of graded rank $P_N(\Gamma)$. This idea was successfully realized by L. Rozansky and one of us back in 2004 in [@KR04]. To define homology groups (or state spaces) $H_N(\Gamma)$ of planar graphs $\Gamma$ *matrix factorizations* were used. Start with a polynomial ring $S=\mathbf{k}[x_1,\ldots, x_k]$ in several variables over a field $\mathbf{k}$, where $\mathsf{char}(\mathbf{k})=0$. A polynomial $\omega\in S$ is called a *potential* if the ideal $(\partial \omega/\partial x_1,\dots, \partial \omega/\partial x_k)$ has finite codimension in $S$. Informally, this means that $\omega$ is sufficiently generic. In this case the quotient algebra $J_{\omega}:=S/(\partial \omega/\partial x_1,\dots, \partial \omega/\partial x_k)$ is known to be Frobenius, via the Grothendieck residue construction. The quotient algebra is called *the Milnor algebra* of $\omega$. For a potential $\omega\in S$, consider 2-periodic generalized complexes $M$ of free $S$-modules and maps between them such that $d^2(m) = \omega m$. Modulo homotopies, these constitute a triangulated category of matrix factorizations $MF_{\omega}$. When matrix factorizations $M,M'$ have finite ranks (meaning $M^0,M'^0$ are finite rank free $S$-modules) hom spaces $\mathsf{Hom}(M,M')$ in the category $MF_{\omega}$ are finite-dimensional $\mathbf{k}$-vector spaces, due to multiplications by $\partial \omega/\partial x_i$ being homotopic to $0$. The Milnor algebra $J_{\omega}$ then acts on $\mathsf{Hom}(M,M')$, implying finite-dimensionality of hom spaces. To a potential $\omega$ there is assigned a two-dimensional TQFT with corners, built out of categories of matrix factorizations for potentials that are signed sums of copies of $\omega$ over several sets of variables. This construction goes back to Kapustin--Li [@KL03], see [@CM16] for a more recent treatment, while the one-variable case is thoroughly explained in [@KR04]. Without going into full details and specializing to $\mathbf{k}=\mathbb Q$ and $\omega=x^{N+1}$, to an arc (viewed as a one-manifold with boundary) one assigns a factorization $L$ for $\omega = x_1^{N+1}-x_2^{N+1}$, with $S=\mathbb Q[x_1,x_2]$: Term $x_1-x_2$ ensures that $x_1,x_2$ act the same up to homotopy, when viewed as endomorphisms of factorization $L$ in the homotopy category. One thinks of $L$ as the identity factorization, implementing the identity functor on the category of matrix factorizations. Namely, tensoring $L$, say over the subring $\mathbb Q[x_1]$ with a factorization $M$, in variables $x_1$ and some other variables (not $x_2$, say $x_3,x_4$) for the potential $-x_1^{N+1}+\omega'(x_3,x_4)$ results in the factorization $M':=L\otimes_{\mathbb Q[x_1]}M$ over $\mathbb Q[x_2,x_3,x_4]$, and one can check that upon substitution $x_2\mapsto x_1$ factorizations $M,M'$ are naturally isomorphic in the homotopy category. Factorization $M'$ is of infinite rank over $\mathbb Q[x_2,x_3,x_4]$, but upon removing contractible summands it becomes finite rank and isomorphic to $M$. Closing up the arc into a circle and equating $x_1=x_2$, see Figure [\[mfig_012\]](#mfig_012){reference-type="ref" reference="mfig_012"} on the right, gives a complex $$\xymatrix{ \mathbb Q[x_1] \ar[rr]^{(N+1)x^N} & & \mathbb Q[x_1] \ar[rr]^0 & & \mathbb Q[x_1], }$$ where the ring is reduced to $\mathbb Q[x_1,x_2]/(x_1-x_2)\cong \mathbb Q[x_1]$. This a general feature of building topological theories from matrix factorizations: closing up a factorization with boundary points and equating the variables, as long as the potentials at the endpoints match, results in a two-periodic complex, with $d^2=0$, rather than just a factorization, due to the cancellation of terms in $\omega$ upon identifying the variables. For the circle as above, the homology group of the complex is $\mathbb Q[x]/(x^N)\cong H^{\ast}(\mathbb{CP}^{N-1})$ (the other homology group is $0$). Algebra $\mathbb Q[x]/(x^N)$ is commutative Frobenius and gives rise to a 2D TQFT once a nondegenerate trace on it is chosen. Building an extended TQFT from matrix factorizations is discussed in [@KR04], also see [@CM16] and references there. In the above example one can work with graded polynomial rings with $\deg(x_j)=2$ and factorizations $\sum_j \pm x_j^{N+1}$. Then the graded degree of the vector space of the circle is $$\mathsf{gdeg} (H_N(\mathsf{circle}))= 1+q^2+\ldots + q^{2(N-1)}=q^{N-1}[N]=q^{N-1}P_N(\mathsf{unknot}).$$ This equality points to a possible match between the 2D TQFT for the one-variable potential $x^{N+1}$ and homology of links and planar graphs, for the simplest possible link and MOY graph, which is the circle in the plane (of thickness 1). Up to a power of $q$, the quantum invariant of the unknot equals the graded dimension of the commutative Frobenius algebra that the matrix factorization TQFT for the potential $\omega=x^{N+1}$ assigns to the circle. That algebra is also isomorphic to the cohomology ring of $\mathbb{CP}^{N-1}$. To move beyond circles, recall that, for link diagrams with components colored by $V$, the corresponding MOY graphs have edges of thickness one and two only, with any thickness two edge having two thin "in" edges and two thin "out" edges, see Figure [\[mfig_013\]](#mfig_013){reference-type="ref" reference="mfig_013"} on the left. Matrix factorization associated to a neighbourhood of a double edge should have the potential $$\label{eq_omega}\omega = x_1^{N+1}+x_2^{N+1}-x_3^{N+1}-x_4^{N+1},$$ see Figure [\[mfig_013\]](#mfig_013){reference-type="ref" reference="mfig_013"} on the left. The four variables $x_1,x_2,x_3,x_4$ are assigned to the endpoints of the diagram. The term $x_i^{N+1}$ enters $\omega$ with the plus sign, respectively the minus sign, if the orientation at that point is out, respectively into, the diagram. Potential $\omega$ is the difference of two terms, and the first term $x_1^{N+1}+x_2^{N+1}$ can be written as a polynomial in the elementary symmetric functions: $x_1^{N+1}+x_2^{N+1}= g(x_1+x_2,x_1x_2).$ To write the identity factorization for the 2-variable polynomial $g$ we decompose the difference of two $g$'s for two different sets of variables: $$\begin{aligned} g(y_1,y_2)-g(z_1,z_2)& = & (g(y_1,y_2)-g(z_1,y_2))+(g(z_1,y_2)-g(z_1,z_2)) \\ & = & (y_1-z_1)u_1(\underline{y},\underline{z}) + (y_2-z_2)u_2(\underline{y},\underline{z})\end{aligned}$$ for some polynomials $u_1,u_2$ in the four variables $y_1,y_2,z_1,z_2$. Tensor the two factorizations $$\xymatrix{ (S \ar[r]^{u_1} & S \ar[r]^{y_1 - z_1} & S) } \quad \mbox{ and } \quad \xymatrix{ (S \ar[r]^{u_2} & S \ar[r]^{y_2 - z_2} & S) }$$ to get a factorization associated to a double edge $j$ of $\Gamma$: $$\xymatrix{ M_j \ := \ (S \ar[r]^{\hspace{0.6cm} u_1} & S \ar[r]^{y_1 - z_1 \hspace{0.60cm}} & S) \otimes_S (S \ar[r]^{\hspace{0.50cm} u_2} & S \ar[r]^{y_2-z_2 \hspace{0.1cm}} & S). }$$ Notice that $M_j$ has the potential given by [\[eq_omega\]](#eq_omega){reference-type="eqref" reference="eq_omega"}. For a general planar MOY graph $\Gamma$ with edges of thickness one and two, place marks on thin edges, with at least one mark on each edge. Denote by $I$ the set of marks and consider the ring $\mathbb Q[x_i]_{i\in I}$. To a thick edge there is assigned a factorization with the potential $x_{i_1}^{N+1}+x_{i_2}^{N+1}-x_{i_3}^{N+1}-x_{i_4}^{N+1}$, to a thin arc -- an identity factorization as described earlier, see Figure [\[mfig_014\]](#mfig_014){reference-type="ref" reference="mfig_014"} on the left. In the Figure [\[mfig_014\]](#mfig_014){reference-type="ref" reference="mfig_014"} example, the factorization for the thin arc shown has potential $x_{i_5}^{N+1}-x_{i_1}^{N+1}$. Given a graph $\Gamma$ and a set of markings $I$ of thin edges as described, tensor together factorizations $M_j$ over all double edges $j$ and factorizations for thin arcs (if some thin edge carries more than one mark) to get the product factorization $M_{\Gamma}$ . Then the differential $D= \displaystyle{\sum_j} d_j$ in $M_{\Gamma}$ satisfies $D^2 = \displaystyle{\sum_j} d_j^2 = \displaystyle{\sum_i} (x_i^{N+1}-x_i^{N+1}) = 0$. Define $H_N(\Gamma) := H_N(M_{\Gamma},D)$. This is our homology (or state space) of the graph $\Gamma$. Placing extra marks on any thin edges of $\Gamma$ does not change $H_N(\Gamma)$. From [@KR04], we cite the following: **Theorem 6** (Khovanov--Rozansky). *$H_N(\Gamma)$ lives is a single $\mathbb Z/2$ degree and $\mathsf{gdim}(H_N(\Gamma))= P_N(\Gamma).$* To extend the homology to links, consider resolutions $\Gamma_0,\Gamma_1$ of a crossing in Figure [\[mfig_015\]](#mfig_015){reference-type="ref" reference="mfig_015"}. There are homomorphisms of factorizations $$\chi_0: M_{\Gamma_0}\longrightarrow M_{\Gamma_1}, \qquad \qquad \chi_1: M_{\Gamma_1}\longrightarrow M_{\Gamma_0},$$ and to positive and negative crossings one assigns two-step complexes of factorizations with the differential given by $\chi_0,\chi_1$, respectively, see Figure [\[mfig_015\]](#mfig_015){reference-type="ref" reference="mfig_015"} on the right. Homology of a link diagram $D$ is given by tensoring these two-step complexes over all crossings of $D$, computing the homology of all terms (for the inner differential in factorizations), and then taking the homology again for the differential build from maps $\chi_0,\chi_1$ over all crossings. From [@KR04], we also recall: **Theorem 7** (Khovanov--Rozansky). *The resulting homology does not depend on the choice of a link diagram $D$ of an oriented link $L$ and can be denoted $H_N(L)$.* 1. *$H_N(L)$ is bigraded and its Euler characteristic $$\chi_N(L) = \displaystyle{\sum_{i,j}} (-1)^i q^j \dim H^{i,j}_N(L)=P_N(L)$$ is the Reshetikhin--Turaev link invariant for the fundamental $\mathsf{SL}(N)$ representation.* 2. *$H_N(L)$ is functorial under link cobordisms.* Homology theory $H_N$ of links and link cobordisms can be viewed as a categorification of quantum invariant $P_N$, the latter a one-variable specialization of the HOMFLYPT polynomial. The functoriality under link cobordisms, shown in [@KR04], is up to overall scaling by elements of $\mathbb Q^{\ast}$. The theory extends to tangles and tangle cobordisms as well. Y. Yonezawa [@Yon11] and H. Wu [@Wu14] extended the homology from *thin* MOY graphs to arbitrary MOY graphs, with general $(a,b,a+b)$ trivalent vertices, and from a categorification of the link invariant $P_N$ to categorification of Reshetikhin--Turaev invariants where components of a link are colored by arbitrary quantum exterior powers $\Lambda^a_q V$ of the fundamental representation $V\cong \mathbb C^N$, $1\le a\le N$. In their construction the homology of the unknot colored $a$ is the cohomology of the Grassmannian of $a$-planes in $\mathbb C^N$: This cohomology ring is a commutative Frobenius algebra and gives rise to a 2D TQFT. Standard merge, split, birth and death cobordisms between unnested circles in the plane and relations on compositions of these cobordisms show that in a functorial link homology theory homology of the unknot is a commutative Frobenius algebra, and for the theory above that algebra is the cohomology ring of the complex Grassmannian. This theory admits a generalization where the potential $x^{N+1}$ is replaced by the potential $\omega = x^{N+1} + a_1 X^N + \ldots + a_N x$, where $a_i$ is a formal variable of degree $2i$. The potential is then homogeneous and defined over the ground ring $R_N=\mathbb Q[a_1,\dots, a_N]$. The 2D TQFT for matrix factorizations and link homology construction can be generalized to the ground ring $R_N$. In the resulting homology theory $\widetilde{H}_N$ the ring $R_N$, which is the homology of the empty link, can be interpreted as the $U(N)$-equivariant cohomology of a point $p$: $$\widetilde{H}_N(\varnothing) \ = \ R_N \ \cong \ \mathsf{H}^{\ast}_{U(N)}(p,\mathbb Q).$$ Homology of an $a$-labelled unknot is then the $U(N)$-equivariant cohomology of the complex Grassmannian: We refer to papers [@Gornik04; @Kra10; @WuEquiv12; @Wu_equivar15] for the construction of equivariant link homology, see also [@LL16; @Ras15; @MY19] for some applications. Deforming the potential proved to be a prolific idea, also leading to a categorification of the bigraded HOMFLYPT polynomial [@KR05] rather than of its singly-graded specializations $P_N$. This triply-graded link homology theory is closely related to the category of Soergel bimodules [@Kho_triply07], which is a fundamental object at the center of modern geometric representation theory [@EMTW20]. We refer to [@GSV05; @Witt18; @Aga22] and many other papers for physical interpretations of $\mathsf{SL}(N)$ link homology. Bigraded and triply-graded categorifications of the HOMFLYPT polynomial and its specializations, including for torus and algebraic links, relate to deep structures in representation theory and geometry [@GORS14; @GHM21]. A refinement of bigraded and triply-graded homology groups, now known as *$y$-ified homology*, was discovered by J. Batson and C. Seed [@BS_spectral15] in the $N=2$ case, S. Cautis and J. Kamnitzer [@CK_coherent17] in the geometric language for any $N$ and E. Gorsky and M. Hogancamp for the HOMFLYPT homology [@GH_yification22], also see [@CLS_Rickard20] for a generalization and an approach via the KLR categories and a related paper [@HRW21]. This homology exhibits additional symmetries and relates link homology to Hilbert schemes. Sophisticated relations between flavors of link homology and equivariant matrix factorizations have been investigated by A. Oblomkov and L. Rozansky in [@OR20] and follow-up papers, see [@Oblomkov19] for a review. ## Model scenario for categorification of Reshetikhin--Turaev link invariants {#subsubsection:model_scenario_RT_inv} Reshetikhin--Turaev link invariant functor [@RT90] assigns the tensor product $V_{\underline{\lambda}}=V_{\lambda_1}\otimes \dots \otimes V_{\lambda_n}$ of irreducible $U_q(\mathfrak{g})$ representations to a plane with $n$ points marked by positive integral weights $\lambda_1,\dots, \lambda_n$, where $\underline{\lambda}:=(\lambda_1,\dots, \lambda_n)$, and taking orientations into account. The invariant of a decorated tangle $T$ with $\underline{\lambda}$ and $\underline{\mu}$ endpoint label sequences at the bottom and top planes is an intertwiner $$f(T) \ : \ V_{\underline{\lambda}}\longrightarrow V_{\underline{\mu}}.$$ Let us explain the best possible scenario for categorification of Reshetikhin--Turaev link invariants. Upon a categorification, the tensor product $V_{\underline{\lambda}}$ of representations should be replaced by a triangulated category $\mathcal{C}_{\underline{\lambda}}$. The Grothendieck group $K_0(\mathcal{C}_{\underline{\lambda}})$ of that category should be related to $V_{\underline{\lambda}}$, for instance, be a $\mathbb Z[q,q^{-1}]$-lattice in $V_{\underline{\lambda}}$, so that $$K_0(\mathcal{C}_{\underline{\lambda}}) \otimes_{\mathbb Z[q,q^{-1}]} \mathbb C(q) \ \cong \ V_{\underline{\lambda}}$$ (assuming $U_q(\mathfrak{g})$ is defined over the field of rational functions $\mathbb Q(q)$). To a collection of $n$ points on the plane labelled by $\underline{\lambda}$ assign the category $\mathcal{C}_{\underline{\lambda}}$ as above. To a tangle $T$ there should be assigned an exact functor $\mathcal{F}(T):\mathcal{C}_{\underline{\lambda}}\longrightarrow\mathcal{C}_{\underline{\mu}}$, see Figure [\[mfig_020\]](#mfig_020){reference-type="ref" reference="mfig_020"} on the left. On the Grothendieck group level the functor $\mathcal{F}(T)$ should give the Reshetikhin--Turaev tangle invariant, $[\mathcal{F}(T)]=f(T)$. This equality can be written as a commutative diagram $$\xymatrix{ K_0(\mathcal{C}_{\underline{\mu}}) \ar@{^{(}->}[rr]^{}& & V_{\underline{\mu}} \\ & & \\ K_0(\mathcal{C}_{\underline{\lambda}}) \ar@{^{(}->}[rr]^{} \ar[uu]^{[\mathcal{F}(T)]} & & V_{\underline{\lambda}} \ar[uu]_{f(T).} \\ }$$ Horizontal inclusions are those of $\mathbb Z[q,q^{-1}]$-modules and become isomorphisms upon tensoring with $\mathbb Q(q)$. Alternatively, $K_0(\mathcal{C}_{\underline{\lambda}}) \otimes_{\mathbb Z[q,q^{-1}]} \mathbb C(q)$ could be a subspace of $V_{\underline{\lambda}}$, such as the subspace $\mathsf{Inv}_{U_q(\mathfrak{g})}(V_{\underline{\lambda}})$ of quantum group invariants in the tensor product. For composable tangles $T',T$ there should be fixed an isomorphism of functors $F(T'\circ T)\cong F(T')\circ F(T)$. A tangle is an oriented decorated one-manifold properly embedded in $\mathbb R^2\times [0,1]$. A tangle cobordism between oriented tangles $T_0,T_1$ is a surface $S$ with boundary and corners properly embedded in $\mathbb R^2\times [0,1]^2$ such that its boundary is the union of $T_0,T_1$ on the intersections $S\cap (\mathbb R^2\times[0,1]\times \{i\})$, $i=0,1$, and the union of product tangles (finite sets of points times $[0,1]$) on the intersections $\mathbb R^2\times \{i\}\times [0,1]$, $i=0,1$. A tangle cobordism is depicted rather schematically in Figure [\[mfig_020\]](#mfig_020){reference-type="ref" reference="mfig_020"}, in the middle. Since tangles are additionally decorated by positive integral weights $\lambda$, connected components of $S$ must be decorated by integral weights as well, with boundary decorations induced by those of $S$. To a tangle cobordism $S$ between tangles $T_0,T_1$ there should be assigned a natural transformation of functors $F(S):\mathcal{F}(T_0)\longrightarrow \mathcal{F}(T_1)$. As one varies over all collections of labelled points in the plane, functors $\mathcal{F}(T)$ and natural transformations $\mathcal{F}(S)$, these should fit into a *2-functor* $\mathcal{F}$ from the *2-category of tangle cobordisms* to the 2-category of natural transformations between exact functors. (Tangle cobordisms are oriented and decorated by positive integral weights of $\mathfrak{g}$.) Having such a 2-functor is the most functorial scenario for a link homology theory. In the matrix factorization approach to link homology, described earlier, the link homology does extend to a 2-functor from the category of tangle cobordisms to the category of bigraded vector spaces (or bigraded $R_N$-modules, for the equivariant theory). The Lie algebra $\mathfrak{g}=\mathfrak{sl}(N)$ and all components are colored by $V$. For $n$ points on the plane one forms the category $MF_{\omega}$ with the potential $\omega = \displaystyle{\sum_{i=1}^n} \pm x_i^{N+1}$, where the signs depend on orientations of points. Then one forgets that $MF_{\omega}$ is a triangulated category, views it only as an additive category, and forms the homotopy category of complexes $\mathcal{C}_n=\mathcal{H}(MF_{\omega})$ over it. To a tangle $T$ there is assigned a complex of matrix factorizations and to a tangle cobordism -- a homomorphism of complexes. The entire construction results in a 2-functor as above: Objects of the category $\mathsf{CMF}$ are potentials $\omega$ in finite sets of variables $\underline{x}$, morphisms from $\omega_1(\underline{x})$ to $\omega_2(\underline{y})$ are complexes of matrix factorizations with potential $\omega_2(\underline{y})-\omega_1(\underline{x})$, and two-morphisms are homomorphisms of complexes of factorizations modulo chain homotopies. The 2-functor $\mathcal{F}$ above is $\mathbb Q^{\ast}$-projective, with the map associated to a link cobordism well-defined up to scaling by elements of $\mathbb Q^{\ast}$. Extending T. Sano's approach to strict cobordism invariance [@Sano20] from $N=2$ to any $N$ should be one way to resolve the $\mathbb Q^\ast$-indeterminacy. To get a homomorphism from the Grothendieck group of the category $\mathcal{C}_n$ to the tensor product $V_{\underline{\lambda}}$, more precisely, to its subspace of invariants $\mathsf{Inv}(V_{\underline{\lambda}})$, one should restrict to the subcategory generated by matrix factorizations associated to planar graphs with a given boundary. Versions of $\mathsf{SL}(N)$ homology and the 2-functor $\mathcal{F}$ can also be recovered from - Parabolic-singular blocks of highest weight categories for $\mathfrak{sl}(k)$, where $k$ depends on $\underline{\lambda}$, see [@Sussan07; @MS09], and, in the $\mathfrak{sl}(2)$ case, [@BFK99; @Str_Duke_05; @FKS_quantum06; @Str_perverse09; @BS11; @CK14]. - Fukaya--Floer categories of quiver varieties for $\mathsf{SL}(k)$, $k$ a function of weights $\underline{\lambda}$ ($N=2$ case [@SS06; @Reza09; @AS19]) and arbitrary $N$ case restricted to braid and braid closures [@Man07], see also [@WW15]. - Derived categories of coherent sheaves on the convolution varieties of affine Grassmannians; $N=2$ case: [@CK_Duke08] and arbitrary $N$: [@CK_slm08], and also on quiver varieties [@AN23]. For a recent physics-motivated approach, see M. Aganagic [@Aga22; @Agan21]. - Various categories of importance in representation theory, see [@Web17] and related papers. The principle here is that in each of these cases there is (or expected to exist) a categorical action of the 2-category of tangle cobordisms $\mathsf{Tan}_2$ on the corresponding derived or homotopy categories, giving a 2-functor similar to $(I)$ and $(II)$ in Figure [\[mfig_010\]](#mfig_010){reference-type="ref" reference="mfig_010"}. Tangles act by functors on the categories and tangle cobordisms act by natural transformations. Often, the corresponding categories for various examples are equivalent [@MW_Howe18], sometimes with minor modifications. One expects the equivalences to respect the 2-functor actions. Motivation to relate quiver varieties (Kronheimer--Nakajima varieties) and link homology came from observing that (a) homology groups of $\mathsf{SL}(k)$ quiver varieties carry an action of that Lie algebra and can be identified with weight spaces of $\mathsf{SL}(k)$ representations [@Naka94], (b) due to the level-rank duality, those weight spaces can be identified with the invariants of tensor products of exterior powers $\Lambda^a V$ of the fundamental $\mathfrak{sl}(N)$ representation $V$, where $k$ depends on the sequence $(a_1,\dots, a_n)$ of these parameters in the tensor product. One then expects that replacing homology by the derived category of coherent sheaves will result in a categorical action of the Lie algebra and a commuting categorical action of tangles and tangle cobordisms. This program was realized in [@CK_slm08] restricting to tensor products of $V$ and $V^{\ast}$ and using related varieties which are suitable convolution varieties of the affine Grassmannian and should contain quiver varieties as open subvarities. *Extended TQFTs and biadjoint functors:* When one has an extended TQFT in dimension $n$, to an $(n-2)$-manifold $M$ assign a category $\mathcal{F}(M)$, to an $(n-1)$-cobordism $K$ a functor $\mathcal{F}(K):\mathcal{F}(\partial_0 K)\longrightarrow\mathcal{F}(\partial_1 K)$, to an $n$-cobordism $L$ with corners a natural transformation $\mathcal{F}(\partial_0 L)\longrightarrow\mathcal{F}(\partial_1 L)$. For each $(n-1)$-cobordism $K$ there is a dual cobordism $K^{\ast}$ from $\partial_1 K$ to $\partial_0 K$ (reflect $K$). There are four canonical $n$-cobordisms between compositions $K K^{\ast}$, $K^{\ast}K$ and identity cobordisms $\mathsf{id}_{\partial_1 K}$ and $\mathsf{id}_{\partial_0 K}$. Applying $\mathcal{F}$ to them tells us that functors $\mathcal{F}(K)$ and $\mathcal{F}(K^{\ast})$ are biadjoint (both left and right adjoint). Examples of biadjoint functors appear in algebraic geometry (Fourier--Mukai kernels between Calabi--Yau manifolds) and in symplectic topology (convolutions with Lagrangians in Fukaya--Floer categories). In particular, quiver varieties are Calabi--Yau (around their compact part) and their derived categories of coherent sheaves admit plenty of biadjoint functors. In three dimensions, the model example of a TQFT is the Witten--Reshetikhin--Turaev theory, for which the categories assigned to $3-2=1$-dimensional manifolds are semisimple. Any linear functor between these categories has a biadjoint functor. In higher dimensions, for an extended TQFT, one expects non-semisimple, likely triangulated, categories associated to $n-2$-dimensional manifolds. For an exact functor between triangulated categories having a biadjoint is a strong condition. It is then natural to pay special attention to Calabi--Yau varieties (and their derived categories of coherent sheaves), Fukaya--Floer categories, and suitable categories of representations built out of symmetric Frobenius algebras, for these types of categories provide large supply of biadjoint functors that can be constructed naturally [@Kho02]. Reshetikhin--Turaev $\mathsf{SL}(N)$ (or $\mathsf{GL}(N)$) invariants for fundamental representations $\Lambda^a_q V$ are distinguished by the existence of *positive integral* diagrammatical calculus of intertwiners (*MOY calculus*). It guides the categorification of Reshetikhin--Turaev invariants. Positivity and integrality property is missing already for other representations of quantum $\mathsf{SL}(2)$, since $q$-spin networks are not positive and have denominators [@KL94]. This creates serious problems trying to extend the above approaches beyond $(\mathsf{SL}(N),\Lambda_q^{\ast}V)$ case. In the remarkable work [@Web17], Webster categorified Reshetikhin--Turaev invariants for any simple Lie algebra $\mathfrak{g}$ and any labelling of a link's components by irreducible $\mathfrak{g}$-modules. Categorification of quantum groups [@KL01; @rouquier2008] was one of the motivations for Webster's construction. It is likely that for $\mathfrak{sl}(N)$ and coloring by minuscule representations his construction gives homology isomorphic to those coming from matrix factorizations (and those from Robert--Wagner foam evaluation). Beyond the minuscule representation case, Webster homology and a number of other known link homology theories do not fully extend or not known to extend to link cobordisms. For a subset of these theories [@FSS_fractional12; @Rozan_Jones_Wenzl14; @CK_Jones_Wenzl12; @Web17; @GH_yification22; @SS_cat22; @OR19], the reason is the following. Take a functorial link homology theory (and subject to TQFT assumption) and consider standardly embedded cobordisms in $\mathbb R^3\times [0,1]$ between unlinks (disjoint unions of unknots). Restricting a link homology theory to these cobordisms results in a 2D TQFT. Consequently, homology $A$ of the unknot must be a commutative Frobenius algebra over the homology $R$ of the empty link. In particular, if $R$ is a field, $A$ must be finite dimensional over $R$, and this property fails in the above examples. Other theories [@KR05; @Caut17; @QRS18; @RW_sym20; @OR_glmk22] have finite-dimensional homology groups over $R$ (upon normalization, if needed), but their construction is only available for braid closures, making it hard to extend them to link cobordism. For an early categorification of the colored Jones polynomial [@Kho_colored_Jones05] functoriality is not known either, see also [@BW_Rasmussen08]. It is an important open problem to (in the first case) modify these link homology theories, including the Webster homology, to make them functorial under link cobordisms and (in the second) find whether they can redefined in a more functorial way, for all link diagrams and extending to link cobordisms or even to tangle cobordisms. ## Foam evaluation and link homology L.-H. Robert and E. Wagner [@RW20] found a purely combinatorial approach to $\mathsf{SL}(N)$ link homology (again, for minuscule representations $\Lambda^a_q V$). Their approach is based on foam evaluations. Foams are cobordisms between planar graphs $\Gamma$ as above and implicitly appear in most approaches to $\mathsf{SL}(N)$ link homology. Maps in the long exact sequences in Figure [\[mfig_009\]](#mfig_009){reference-type="ref" reference="mfig_009"} between the two planar graphs should be induced by foam cobordisms shown in Figure [\[mfig_010\]](#mfig_010){reference-type="ref" reference="mfig_010"} between graphs $\Gamma_0,\Gamma_1$, also see Figure [\[mfig_016\]](#mfig_016){reference-type="ref" reference="mfig_016"}. To build a combinatorial theory, Robert and Wagner construct a subtle evaluation of closed foams to symmetric functions $R=\mathbb Z[x_1,\ldots, x_N]^{S_N}$ in $N$ variables. Let us briefly review foams and Robert--Wagner foam evaluation. $\mathsf{GL}(N)$ foams can have facets of thickness $a\in \{1,\ldots, N\}$, seams where facets of thickness $a,b,a+b$ come together and vertices with 6 adjoint facets of thickness $a,b,c,a+b,b+c,a+b+c$, see Figure [\[mfig_021\]](#mfig_021){reference-type="ref" reference="mfig_021"}. At first, consider *closed foams* embedded in $\mathbb R^3$, that is, foams without boundary, while later we will need foam with boundary, viewed as cobordisms in $\mathbb R^2\times [0,1]$ between MOY graphs. A *coloring* of foam $F$ is a map $c: \mathsf{facets} \longrightarrow\mathcal{P}(N)$, where $|c(f)|$ is the thickness of a facet $f$, with the flow condition at each seam, $c(f_3)=c(f_1)\sqcup c(f_2)$, see Figures [\[mfig_000\]](#mfig_000){reference-type="ref" reference="mfig_000"}, [\[mfig_017\]](#mfig_017){reference-type="ref" reference="mfig_017"}. In particular, $c(f_1),c(f_2)$ are disjoint sets. Here $\mathcal{P}(N)$ is the set of subsets of $\{1,\ldots, N\}$, so that a coloring maps facets to subsets of the set of colors from $1$ to $N$. A facet of thickness $a$ is mapped to a subset of cardinality $a$; subsets for $a$, $b$ thickness facets that meet along a seam and become an $a+b$ facet are disjoint and their union is the subset for the $a+b$ facet. Define the *bicolored surface* $F_{ij}(c)$, $1\le i<j\le N$ as the union of facets that contain exactly one color from $\{i,j\}$. For a closed foam $F$, its bicolored surface is a closed compact surface without boundary embedded in $\mathbb R^3$. Figures [\[mfig_000\]](#mfig_000){reference-type="ref" reference="mfig_000"}, [\[mfig_017\]](#mfig_017){reference-type="ref" reference="mfig_017"} on the right explain why $F_{ij}(c)$ has no singularities along seams of the foam, always containing either none or two facets along a seam. A similar computation shows that $F_{ij}(c)$ has no singularities at vertices of the foam. Thus, $F_{ij}(c)$ is an orientable surface and the union of its connected components, each of Euler characteristic $2-2g$, for a component of genus $g\ge 0$. Robert--Wagner evaluation $\langle F,c\rangle$ of a closed foam $F$ and its coloring $c$ has the form $$\langle F,c\rangle = \pm \prod_{i<j} (x_i-x_j)^{-\chi(F_{ij}(c))/2}\prod (\mathsf{facet\ decoration\ contributions}).$$ We refer to [@RW20] and a review in [@KK_deformation_RW20] for the subtle formula for the minus sign, facet decorations and their contributions. Facets are decorated by dots (observables) labelled by homogeneous symmetric functions $f$ in $a$ variables, for a facet of thickness $a$. A coloring tells one which $a$ variables out of $N$ to select from $x_1,\dots, x_N$ to turn $f$ into a function in $x$'s. In general $\langle F,c\rangle$ has denominators $x_i-x_j$ but the sum $$\langle F \rangle \ = \ \sum_c \langle F,c\rangle \in R$$ is a symmetric polynomial in $x_1,\ldots, x_N$. With that sophisticated yet beautiful foam evaluation $\langle F \rangle$ at hand, Robert and Wagner build the state space (or homology) for a planar MOY graph $\Gamma$ using the universal construction. Universal construction of topological theories [@BHMV95; @Kh20_univ_const_two; @KQR; @IK_22_linear; @IK-top-automata] begins with an evaluation of closed objects (closed foams, in our case) and builds state spaces for generic cross-sections of these objects. A generic cross-section $\Gamma$ of a foam in $\mathbb R^3$ by a plane $\mathbb R^2$ is a planar MOY graph $\Gamma$. Fix $\Gamma$ and consider the free $R$-module $\mathsf{Fr}(\Gamma)$ with a basis $[F]$ of all foams $F$ with $\partial F = \Gamma$, see Figure [\[mfig_X1\]](#mfig_X1){reference-type="ref" reference="mfig_X1"} on the left. Notice that we no longer restrict to closed foams, instead looking at foams $F$ in $\mathbb R^2\times (-\infty,0]$ with $\Gamma$ as the boundary. Define an $R$-bilinear form $$(\:\:,\:\:):\mathsf{Fr}(\Gamma)\times \mathsf{Fr}(\Gamma) \longrightarrow R$$ by $([F],[F_1])= \ensuremath{\left\langle \overline{F}F_1\right\rangle}$ and extending via $R$-bilinearity. Here we glue $F$ and $F_1$ along the common boundary $\Gamma$ into a closed foam $\overline{F}F_1$, where $\overline{F}$ denotes the reflection of $F$ in a horizontal plane. Define the state space $$H(\Gamma) \ := \ \mathsf{Fr}(\Gamma)/\ker((\:\:,\:\:)_\Gamma)$$ of an MOY graph $\Gamma$ as the quotient of the (large) free module $\mathsf{Fr}(\Gamma)$ by the kernel of the bilinear form $(\:\:,\:\:)_\Gamma$. This means that an $R$-linear combination of foams $\sum_i \lambda_i F_i$, with $\lambda_i\in R$ and $\partial F_i = \Gamma$ is $0$ in $H(\Gamma)$ if and only if for any foam $F$ with $\partial F=\Gamma$, we have $$\sum_i \lambda_i \langle \overline{F}F_i \rangle =0 \in R.$$ One thinks of $\sum_i \lambda_i F_i=0$ as a linear skein relation on foams with boundary $\Gamma$. It is easy to see that state spaces $H(\Gamma)$, over all $\Gamma$, form a functorial topological theory. Namely, given a foam $F\in \mathbb R^2\times [0,1]$ with top and bottom boundary, so that $\partial F= \partial_1 F \sqcup (- \partial_0 F),$ composition with $F$ induces a map from $\mathsf{Fr}(F_0)$ to $\mathsf{Fr}(F_1)$. The map takes the kernel of the bilinear form $(\:\:,\:\:)_{\Gamma_0}$ to the kernel of the bilinear form $(\:\:,\:\:)_{\Gamma_1}$. Consequently, there is an induced map on state spaces $$H(F) \ : \ H(\partial_0 F)\longrightarrow H(\partial_1 F) .$$ Varying over all foams $F$ with boundary, this results in a functor $H$ from the category of foams to the category of $R$-modules. For a general foam evaluation $\langle F\rangle$ such a functor is not interesting. In particular, one wants state spaces $H(\Gamma)$ to be sufficiently small, for instance, be finitely-generated $R$-modules. Even under such an assumption, functor $H$ will not be a TQFT in general, with the natural map $H(\Gamma_1)\otimes_R H(\Gamma_2) \longrightarrow H(\Gamma_1\sqcup \Gamma_2)$ not an isomorphism of $R$-modules. For their evaluation sketched above, Robert and Wagner in [@RW20] proved: **Theorem 8** (Robert--Wagner). *$H$ is a functorial TQFT from *Foams* to free graded $R$-modules, with graded ranks the Murakami--Ohtsuki--Yamada planar graph invariants: $$\mathsf{grank}_R(\Gamma) = P_N(\Gamma).$$* Methods of Yonezawa--Wu [@Yon11; @Wu12] for constructing homology from matrix factorizations for MOY graphs with edges of arbitrary thickness apply to Robert--Wagner state spaces. Robert--Wagner foam TQFT gives rise to a link homology theory which categorifies Reshetikhin--Turaev invariant for $\mathsf{SL}(N)$ and link components labelled by exterior powers of the fundamental representation [@ETW18]. Furthermore, in [@ETW18] this result is extended to: **Theorem 9** (M. Ehrig--D. Tubbenhauer--P. Wedrich). *Robert--Wagner link homology theory is functorial for link cobordisms.* Robert--Wagner's approach [@RW20; @ETW18] to categorification of Reshetikhin--Turaev $\mathsf{SL}(N)$ invariants is *complementary* to all the others, which require *specific categories* (highest weight categories, coherent sheaves or Fukaya--Floer categories on quiver varieties, particular representation theory categories). In the Robert--Wagner construction, categories appear at the last step only (when extending to graphs with boundary, tangles, and their cobordisms). It is a *categorification of the state sum approach to quantum invariants* and should also be relevant to some models of 3-dimensional statistical mechanics. Prior to Robert--Wagner's work, foams were heavily used in link homology and categorified quantum groups, see for example [@MSV_foam09; @Mack09; @QR_comb16; @Cap13; @RW_deformation16; @Wed19]. Foam evaluation should help to streamline and clarify a significant amount of prior work on the subject, including replacing the ground ring $\mathbb Q$ by $\mathbb Z$ or by the ring of symmetric functions $\mathbb Z[x_1,\ldots, x_N]^{S_N}$. # Interactions and applications ## Some interactions *Soergel category.* A version of Robert--Wagner evaluation [@RW_sym20] allows one to describe the category of Soergel bimodules [@EMTW20] for the symmetric group $S_N$, with homs given by foams in $\mathbb R^3$ between braid-like graphs in the plane modulo universal construction relations [@RW_sym20; @KRW_inprogress]. The Soergel category is central in geometric representation theory and categorifies the Hecke algebra of the symmetric group. Originally, the relation between foams and the Soergel category was established by D. Rose and P. Wedrich [@RW_deformation16] and P. Wedrich [@Wed19]. An earlier diagrammatic approach to the Soergel category by Elias--Khovanov [@EK10] is via a two-dimensional graphical calculus rather than foams in $\mathbb R^3$, with the missing dimension encoded by labels in the ordered set $\{1,\dots, N-1\}$. *Kronheimer--Mrowka theory.* Inspired by the Robert--Wagner foam evaluation, Robert and one of us [@KR21] related *unoriented* $\mathsf{SL}(3)$ foams and Kronheimer--Mrowka gauge $SO(3)$ theory for 3-orbifolds [@KM_Tait19], proving one of Kronheimer--Mrowka conjectures. Kronheimer--Mrowka theory, in a rather special case, assigns homology groups to planar trivalent graphs $\Gamma$, and similar groups can be recovered from an unoriented version of Robert--Wagner foam evaluation. (Kronheimer--Mrowka theory is much more general, and, in particular, assigns homology groups to spacial trivalent graphs, for which a combinatorial counterpart is unknown.) Kronheimer--Mrowka theory and its combinatorial counterpart in [@KR21] for planar trivalent graphs relate to the 4-Color Theorem, aiming to prove and rethink the 4-Color Theorem in a more conceptual way and via its relations to TQFTs, gauge theory, and low-dimensional topology. These two theories have been investigated by D. Boozer [@Booz19; @Booz23], who have shown, in particular, that they give rise to two non-isomorphic functors on foams in $\mathbb R^3$. *APS homology.* Foam evaluation allows for a natural extension of the Asaeda--Przytycki--Sikora [@APS_Kauffman04] annular $\mathsf{SL}(2)$ homology to the equivariant setting of a larger ground ring as well as its extension to annular $\mathsf{SL}(N)$ homology [@AkKh21; @Akh23]. ## $\mathsf{SL}(2)$ and $\mathsf{SL}(3)$ homology theories {#subsection:SL2_SL3} $\mathsf{SL}(2)$ homology (aka Khovanov homology) is noticeably simpler than $\mathsf{SL}(N)$ homology and was discovered first [@Kho_Duke00]. Foams are replaced by surfaces, and homology of links is built from a 2-dimensional TQFT where homology of the unknot has rank two over homology of the empty link. Khovanov homology categorifies the Jones polynomial. Odd Khovanov homology, discovered by P. Ozsvath, J. Rasmussen and Z. Szabo [@ORS13], is another bigraded categorification of the Jones polynomial. $\mathsf{SL}(3)$ homology, which categorifies the Kuperberg quantum $\mathfrak{sl}(3)$ invariant [@Kup96], is in-between $N=2$ and $N\ge 4$ cases complexity-wise [@Kho04; @MV_universalsl307; @MN_su308; @Clark_functoriality09]. $\mathsf{SL}(3)$ foams do not have vertices, and closed $\mathsf{SL}(3)$ foams can be evaluated via "localization along singular circles" and manipulation of foams with a single such circle. Single-circle foam evaluation is encoded in the cohomology groups of the flag variety of $\mathbb C^3$ (commutative algebra structure plus the trace map). ## Two applications to 4D topology **I.** *Rasmussen invariant.* The Rasmussen invariant $s(K)$ comes from an equivariant version of Khovanov homology ($N=2$) given by replacing the ground ring $\mathbb Z$ by $\mathbb Q[t]$ and the Frobenius algebra $A=\mathbb Z[X]/(X^2)$ of homology of the unknot by $A_t=\mathbb Q[t,X]/(X^2-t)$ over $\mathbb Q[t]$. Then $\mathsf{SL}(2)$-homology $H_t(K)$ of a knot $K$ becomes a $\mathbb Q[t]$-module, and it can be written as a sum of its $t$-torsion and a free summand, where the free summand is a free rank one $A_t$-module in homological degree $0$ and $q$-degree $s(K)-1$: $$H_t(K) \ \cong \ \mathsf{Tor}(H_t(K))\oplus A_t\{s(K)-1\}.$$ From [@Ras_slicegenus10], one obtains the following: **Theorem 10** (J. Rasmussen). *The value $s(K)$ is an invariant of knot concordance and gives a lower bound on the slice genus of $K$. This bound is explicitly computable and sharp on positive knots.* The Rasmussen invariant provides a combinatorial proof of Kronheimer--Mrowka--Milnor theorem (formerly Milnor's conjecture) that the slice genus of $(p,q)$-torus knot is $\frac{(p-1)(q-1)}{2}$. Replacing $\mathbb Q$ by $\mathbb{F}_p$ or extending to $\mathsf{SL}(N)$ homology, $N\ge 3$ and its deformations leads to families of Rasmussen-like concordance invariants [@MTV07; @MTV_rasmusseninv13; @Lobb_perturbation12; @Lewark_spectral14]. Some linear independence results on these invariants are known and there is even a postcard on the topic [@postcard_21], but the general theory of such concordance invariants remains a mystery. **II.** *Exotic surfaces in $\mathbb R^4$.* K. Haiden and I. Sundberg in [@HS_exotic21] give examples of surfaces $S_1,S_2\subset \mathbb{D}^4$ that bound a knot $K\subset \mathbb{S}^3$ such that $S_1,S_2$ are homeomorphic but not diffeomorphic rel boundary. Non-diffeomorphic property follows by checking that maps induced by $S_1,S_2$ on Khovanov homology $$H(S_i) \ : \ H(K) \longrightarrow H(\mathsf{empty} \ \mathsf{link})\cong\mathbb Z$$ are not equal. S. Akbulut showed in [@Akb91] that $S_1,S_2$ have this property by using Donaldson theory. Thirty years later, there is a combinatorial proof of his result. *Link homology relates smooth 4D topology and geometric representation theory in a highly intricate way. Braid groups and tangle categories act by exact functors on key categories in geometric representation theory. Tangle cobordisms act by natural transformations between these functors. These tangle cobordism invariants are subtle enough to distinguish between distinct smooth structures on properly embedded surfaces in the four-ball $\mathbb{D}^4$ with the same underlying topological structure.* Going beyond mere homology groups, we should mention the spectrification of Khovanov homology by R. Lipshitz and S. Sarkar [@LS_stable14; @LLS_Burnside20] and [@HKK16]; see N. Kitchloo [@Kitch_I19] for spectrifications of $\mathsf{SL}(N)$ homology. Heegaard--Floer theory [@OS06_lectures; @OS06_intro] and link Floer homology of Ozsvath--Szabo and Rasmussen [@Manol16; @OS_Floer18] play a fundamental role in modern low-dimensional topology and should relate to yet-unknown categorified quantum group $\mathsf{GL}(1|1)$ and its categorified representations, as studied by Y. Tian, A. Ellis--I. Petkova--V. Vertesi, A. Manion--R. Rouquier and others [@Tian_UT14; @EPV_tangle19; @MR_Hegaard20]. This informal write-up discusses only a fraction of the rich developments in the past twenty-five years that were inspired by the work [@CF94] of Igor Frenkel and Louis Crane and other related ideas of Igor Frenkel. We are delighted to celebrate Igor Frenkel's anniversary and wish him many more years of inspiring and exciting mathematical discoveries.

arxiv_math

--- abstract: | We investigate nonparametric drift estimation for multidimensional jump diffusions based on continuous observations. The results are derived under anisotropic smoothness assumptions and the estimators' performance is measured in terms of the $\sup$-norm loss. We present two different Nadaraya--Watson type estimators, which are both shown to achieve the classical nonparametric rate of convergence under varying assumptions on the jump measure. Fully data-driven versions of both estimators are also introduced and shown to attain the same rate of convergence. The results rely on novel uniform moment bounds for empirical processes associated to the investigated jump diffusion, which are of independent interest. author: - "Niklas Dexheimer[^1]" bibliography: - jumpdrift.bib title: Adaptive nonparametric drift estimation for multivariate jump diffusions under $\sup$-norm risk --- # Introduction {#sec: intro} Diffusion processes are one of the most fundamental classes of stochastic processes, with their applications ranging from physics and chemistry up to finance or life sciences. Thus a plethora of research has been devoted to their statistical analysis. However, much of this has been focussed on the case of reversible diffusions. [ It is known by Kolmogorov's characterisation, that a stochastic process $\mathbf{X}=(X_t)_{t\geq0}$ satisfying a stochastic differential equation of the form $$\mathop{}\!\mathrm{d} X_t= b(X_t)\mathop{}\!\mathrm{d} t+\mathop{}\!\mathrm{d} W_t,\quad X_0=\xi, t\geq0,$$ is reversible if, and only if, $b=-\nabla V$ for some suitable potential function $V\colon \mathbb{R}^d\to\mathbb{R}.$ Here $(W_t)_{t\geq0}$ is a $d$-dimensional standard Brownian motion and the random vector $\xi\in\mathbb{R}^d$ is assumed to be independent of $(W_t)_{t\geq0}$.]{style="color: dex"} As the generator of $\mathbf{X}$ is then self-adjoint, this allows the usage of functional inequalities, such as Poincaré or Nash inequalities, for the statistical analysis of $\mathbf{X}$ (see e.g. [@strauch18; @dalrei07; @str15]). Apart from the obvious downside of restricting the drift function to be of gradient form, this framework also does not permit the inclusion of jump structures, i.e. it does not allow the study of processes $\mathbf{X}=(X_t)_{t\geq0}$ of the form $$\label{eq: jumpdiff intro} X_t=\xi+\int_0^t b(X_s)\mathop{}\!\mathrm{d} s+\int_0^t\sigma(X_s)\mathop{}\!\mathrm{d} W_s+\int_0^t \int_{\mathbb{R}^d} \gamma(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s, \mathop{}\!\mathrm{d} z),\quad t\geq0.$$ Here $b\colon \mathbb{R}^d\to\mathbb{R}^d,\sigma\colon\mathbb{R}^d\to\mathbb{R}^{d\times d}, \gamma\colon\mathbb{R}^d\to\mathbb{R}^{d\times d},$ $(W_t)_{t\geq0}$ is as above and $\widetilde{N}$ is a compensated Poisson random measure on $[0,\infty)\times \mathbb{R}^d\backslash\{0\}$ with intensity measure $\pi(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z)=\mathop{}\!\mathrm{d} s\otimes \nu(\mathop{}\!\mathrm{d} z),$ where $\nu$ is a Lévy measure, and $\xi\in\mathbb{R}^d$ is assumed to be independent of $N$ and $(W_t)_{t\geq0}$. The focus of this work lies on nonparametric estimation of the drift function $b$ for jump diffusions of the form [\[eq: jumpdiff intro\]](#eq: jumpdiff intro){reference-type="eqref" reference="eq: jumpdiff intro"}, based on a continuous record of observations. We impose anisotropic Hölder smoothness assumptions and measure the estimation error in the $\sup$-norm, which compared to the $L^2$-risk is generally more demanding as it requires a precise knowledge about the process's concentration behaviour. In order to investigate the arising challenges through the diffusion's discontinuity, we firstly investigate the same Nadaraya--Watson type estimator as employed in the continuous setting in [@aeck22], and show that it attains the classical nonparametric rate of convergence as soon as $\nu$ admits an exponential moment. Building on this we also introduce a more delicate, truncated estimator, which achieves this result as soon as the Lévy-measure admits a third moment. Furthermore adaptive, i.e. fully data-driven, variants of both estimators are introduced and it is proved that they also achieve the classical nonparametric rate of convergence, without requiring any previous knowledge on the drift function. The essential ingredient for all of these results are novel uniform moment bounds for empirical processes associated with the investigated jump diffusion, which are proved through Talagrand's generic chaining device. As mentioned above, a similar framework has been investigated for nonreversible continuous diffusion processes in the recent reference [@aeck22]. Therein an adaptive estimator is presented, which attains the classical nonparametric rate of convergence, thus achieving minimax optimality. The results also rely on uniform moment bounds, which are proved through a similar technique as introduced in [@kindiff] for drift estimation for a class of hypoelliptic diffusions, so-called stochastic damping Hamiltonian systems. As this technique strongly relies on the continuity of the investigated processes, generalising this approach towards jump diffusions of the form [\[eq: jumpdiff intro\]](#eq: jumpdiff intro){reference-type="eqref" reference="eq: jumpdiff intro"} is nontrivial. For a more detailed discussion see Section [3](#sec: unif mom){reference-type="ref" reference="sec: unif mom"}. Apart from this there exist many other works focussed on nonparametric drift estimation for continuous diffusions (see e.g. [@str15; @dalrei07; @dala05]) or parametric drift estimation for jump diffusions (see e.g.[@gloteraos18; @onlinedrift; @dexheimer2022lasso; @maiou]), however the literature on nonparametric drift estimation for jump diffusions is rather scarce (for an overview see e.g. [@park21; @schmisser14]). In particular, to the best of our knowledge, no results exist in the multidimensional case or on the $\sup$-norm risk. The most closely related work to ours is given by [@schmisser14], where nonparametric drift estimation for jump diffusions based on discrete observations is investigated. The estimator is obtained by minimising a certain contrast function and also an adaptive version is presented. It is shown that the classical nonparametric rate of convergence is achieved by the estimators as soon as the distance between the observations is small enough. However, only the scalar setting is investigated and the aforementioned result on the rate of convergence is shown for the less involved empirical squared risk. Additionally, due to the different observation structure, the assumptions are more restrictive than in our work, as the reference requires symmetry of the Lévy-measure $\nu$ or the jump coefficient $\gamma$ to be constant, as well as a fourth moment of $\nu$. The structure of this paper is as follows. In Section [2](#sec: ass){reference-type="ref" reference="sec: ass"} we introduce the mathematical framework of this work including all major assumptions on the investigated jump diffusion $\mathbf{X}$ together with some notation. In Section [3](#sec: unif mom){reference-type="ref" reference="sec: unif mom"} we then prove the needed uniform moment bounds for empirical processes associated with $\mathbf{X}$ under differing assumptions on the Lévy measure $\nu$. In the following Section [4](#sec: drift est){reference-type="ref" reference="sec: drift est"} these results are then applied to bounding the $\sup$-norm risk of two variations of Nadaraya--Watson type estimator for the drift function $b$ under anisotropic Hölder smoothness assumptions on $b$. Section [5](#sec: adap){reference-type="ref" reference="sec: adap"} then contains adaptive extensions of the estimators introduced in the previous segment, which are shown to achieve the same rate of convergence, even if the drift's smoothness is unknown. Lastly, the results of this paper are discussed in Section [6](#sec: dis){reference-type="ref" reference="sec: dis"}. # Preliminaries {#sec: ass} [In this section we introduce the required assumptions for our statistical analysis.]{style="color: dex"} ## Assumptions Throughout the whole paper, we assume that the stochastic integral equation [\[eq: jumpdiff intro\]](#eq: jumpdiff intro){reference-type="eqref" reference="eq: jumpdiff intro"} admits a unique, non-explosive weak solution, which we denote by $\mathbf{X}=(X_t)_{t\geq0}$. [Additionally we assume that $(\mathbf{X},(\mathbb{P}^x)_{x\in\mathbb{R}^d}))$ is a Borel right Markov process.]{style="color: dex"} The following framework of assumptions will be referred to as ($\mathscr{A}$) for the rest of the paper. 1. [\[ass: inv\]]{#ass: inv label="ass: inv"} For any $t>0,$ exists a measurable function $p_t\colon\mathbb{R}^d\times \mathbb{R}^d\mapsto\mathbb{R}_+$, referred to as transition density, such that $$P_t(x,B)=\int_B p_t(x,y)\mathop{}\!\mathrm{d} y, \quad B\in \mathcal{B}(\mathbb{R}^d),x\in\mathbb{R}^d,$$ i.e. the marginal laws of $\mathbf{X}$ are absolutely continuous. Additionally we assume $\mathbf{X}$ to admit a unique absolutely continuous invariant distribution $\mu$ with density $\rho\colon \mathbb{R}^d\mapsto\mathbb{R}_+$, i.e. for any $t>0, B\in\mathcal{B}(\mathbb{R}^d),$ $$\mathbb{P}^\mu (X_t\in B)\coloneq \int P_t(x, B)\mu(\mathop{}\!\mathrm{d} x)=\int_{\mathbb{R}^d}\int_B p_t(x,y)\mathop{}\!\mathrm{d} y\rho(x)\mathop{}\!\mathrm{d} x=\int_B \rho(x)\mathop{}\!\mathrm{d} x=\mu(B). $$ 2. There exists a constant $c_0>0,$ such that the following transition density bound holds true: $$\forall t\in (0,1]: \quad \sup_{x,y \in \mathbb{R}^d} p_t(x,y) \leq c_0 t^{-d \slash 2}.$$ [\[ass: heat\]]{#ass: heat label="ass: heat"} 3. $\mathbf{X}$ started in the invariant measure $\mu$ is exponentially $\beta$-mixing, i.e. [\[cond2:mixing\]]{#cond2:mixing label="cond2:mixing"} there exist constants $c_\kappa,\kappa>0$ such that $$\int \Vert P_t(x,\cdot)-\mu(\cdot)\Vert_{\operatorname{TV}}\mu(\mathop{}\!\mathrm{d} x)\leq c_\kappa \exp(-\kappa t), \quad t\geq0,$$ where $\Vert \cdot\Vert_{\operatorname{TV}}$ denotes the total variation norm. [\[ass: mixing\]]{#ass: mixing label="ass: mixing"} These assumptions on the investigated process are of course rather abstract, but give an overview of which properties are actually needed for our analysis and also allow our statistical results to be adapted to new probabilistic results. In practice however, one may be more interested in precise conditions on the coefficients $b,\sigma,\gamma$ and the Lévy-measure $\nu,$ than the theoretical framework above. Therefore we introduce the following set of assumptions. 1. [\[Lipschitz assumptions\]]{#Lipschitz assumptions label="Lipschitz assumptions"}The functions $b,\gamma,\sigma$ are globally Lipschitz continuous, $b$ and $\gamma$ are bounded, and, for $\mathbb{I}_{d\times d}$ denoting the $d\times d$-identity matrix, there exists a constant $c\geq1$ such that $$\forall x\in\mathbb{R}^d\colon c^{-1}\mathbb{I}_{d\times d}\leq \sigma(x)\sigma^\top(x)\leq c\mathbb{I}_{d\times d},$$ where the ordering is in the sense of Loewner for positive semi-definite matrices. 2. [\[Kappa Assumptions\]]{#Kappa Assumptions label="Kappa Assumptions"}$\nu$ is absolutely continuous with respect to the Lebesgue measure and, for an $\alpha\in(0,2)$, $$(x,z) \mapsto \Vert \gamma(x)z\Vert^{d+\alpha}\nu(z)$$ is bounded and measurable, where, by abuse of notation, we denoted the density of $\nu$ also by $\nu$. Furthermore, if $\alpha=1$, $$\int_{r<\Vert\gamma(x) z\Vert\leq R} \gamma(x)z\,\nu(\mathrm{d}z)=0,\quad\text{ for any } 0<r<R<\infty,\ x\in\mathbb{R}^d.$$ 3. [\[Ergodicity Assumptions\]]{#Ergodicity Assumptions label="Ergodicity Assumptions"} There exist $\eta_0,\eta_1,\eta_2>0$ such that $$\langle x,b(x)\rangle\leq -\eta_0\Vert x\Vert, \quad \forall x:\Vert x\Vert\geq \eta_1,\quad\text{ and }\quad \int_{ \mathbb{R}^d}\Vert z\Vert^2 \mathrm{e}^{\eta_2\Vert z\Vert}\nu(\mathrm{d}z)<\infty.$$ The above assumptions are the same as investigated in Example 4.3 of [@dexheimeraihp], where it is also shown that $\mathscr{A}$ holds as soon as [\[Lipschitz assumptions\]](#Lipschitz assumptions){reference-type="ref" reference="Lipschitz assumptions"},[\[Kappa Assumptions\]](#Kappa Assumptions){reference-type="ref" reference="Kappa Assumptions"} and [\[Ergodicity Assumptions\]](#Ergodicity Assumptions){reference-type="ref" reference="Ergodicity Assumptions"} are fulfilled. ## Additional Assumptions and Notation {#subsec: not} Throughout the whole paper we let $X_0=\xi\sim\mu,$ i.e. $\mathbf{X}$ is stationary and denote $\mathbb{P}=\mathbb{P}^\mu,\mathbb{E}=\mathbb{E}^\mu$. For a function $f\in L^1(\mu)$ we denote $\mu(f)\coloneq\int f\mathop{}\!\mathrm{d} \mu$ and we introduce the following notation for the diffusion coefficient $a\colon \mathbb{R}^d\to\mathbb{R}^{d\times d}, x\mapsto\sigma(x) \sigma^\top(x)=a(x).$ Furthermore for $n\in\mathbb{N}$ we denote $[n]\coloneq\{1,\ldots,n\}$. Additionally, we assume the drift $b$ to be locally bounded, and $a,\gamma$ to be globally bounded. Moreover for $x\in\mathbb{R}^d,\varepsilon>0,$ we denote $B(x,\varepsilon)\coloneq\{y\in\mathbb{R}^d\colon \Vert x-y\Vert<\varepsilon\}$ and for a class of functions $\mathcal{G}$ mapping from $\mathbb{R}^d$ to $\mathbb{R}$ and a function $f\colon\mathbb{R}^d\to\mathbb{R}$ we set $\mathcal{G}f\coloneq \{fg\colon g\in\mathcal{G}\}.$ Lastly, we define the $\sup$-norm risk of an estimator $\widehat{f}$ of a function $f$ on a domain $D\subset \mathbb{R}^d$ as $$\mathcal{R}(\widehat{f},f;D)\coloneq \mathbb{E}[\Vert f-\widehat{f}\Vert^p_{L^\infty(D)} ]^{1/p}, \quad p\geq 1,$$ where $\Vert \cdot\Vert_{L^\infty(D)}$ denotes the restriction of the $\sup$-norm to $D$. In order to improve the paper's readability all proofs have been deferred to the appendix. Furthermore $c$ always denotes a strictly positive constant, whose value may change from line to line. In contrast, specific constants are denoted with an additional subscript. # Uniform moment bounds {#sec: unif mom} As we intend to bound the $\sup$-norm risk of a Nadaraya--Watson type drift estimator, we are in need of uniform moment bounds over certain classes of functions for empirical processes associated to the investigated jump diffusion $\mathbf{X}$. More precisely, for a bounded, measurable function $g$ on $\mathbb{R}^d$ and $j\in[d]$ we define $$\begin{aligned} \mathbb{I}^j_T(g)&\coloneq \frac{1}{\sqrt{T}}\int_0^T g(X_{s-}) \mathop{}\!\mathrm{d} X^j_s \\&= \mathbb{H}_T(gb^j)+\mathbb{M}^j_T(g)+\mathbb{J}^j_T(g),\end{aligned}$$ where $$\begin{aligned} \mathbb{H}_T(g)&= \frac{1}{\sqrt{T}}\int_0^T g(X_s) \mathop{}\!\mathrm{d} s,\notag \\ \mathbb{M}^j_T(g)&=\frac{1}{\sqrt{T}}\int_0^T g(X_s)\sum_{k=1}^{d}\sigma_{jk}(X_s) \mathop{}\!\mathrm{d} W^k_s,\label{def: emp proc} \\ \mathbb{J}^j_T(g)&=\frac{1}{\sqrt{T}}\int_0^T\int_{\mathbb{R}^d} g(X_{s-})\gamma^j(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z),\notag\end{aligned}$$ with $\gamma^j$ denoting the $j$-th row of $\gamma,$ and aim at finding a suitable uniform moment bound of the centred version of this functional over a countable class $\mathcal{G}$ of such functions, i.e. for $j\in[d],p\geq1,$ we want to bound $$\mathbb{E}\Big[\sup_{g\in\mathcal{G}}\Vert \mathbb{I}^j_T(g)-\sqrt{T}\mu(gb^j)\Vert^p\Big]^{1/p}.$$ By the Minkowski inequality it obviously suffices to find uniform moments bounds for the three summands defined in [\[def: emp proc\]](#def: emp proc){reference-type="eqref" reference="def: emp proc"}. A crucial ingredient for doing this will be the generalisation of Talagrand's generic chaining device contained in [@dirksen15], which entails the needed bounds as soon as the investigated stochastic process fulfills a Bernstein type concentration inequality. Regarding $\mathbb{H}_T$ we can employ the results of [@dexheimeraihp], where, as in [@viennet], Berbee's coupling method is applied together with the exponential $\beta$-mixing property to obtain the needed concentration behaviour through the standard Bernstein inequality for independent, identically distributed random variables. Using this result together with Bernstein's inequality for continuous martingales (see, e.g., p. 153 in [@revuzyor1999]) uniform moment bounds for functionals akin to $\mathbb{M}^j_T$ were shown in [@kindiff] and can be derived similarly in the setting of this paper. Before we present this result, recall that for $\varepsilon>0$ the covering number $\mathcal{N}(\varepsilon,\mathcal{G},d)$ of a set $\mathcal{G}$ with respect to a semi-metric $d$ denotes the smallest number of open balls of $d$-radius $\varepsilon$ needed to cover $\mathcal{G}$. Furthermore for two measurable, bounded functions $f,g\colon \mathbb{R}^d\to\mathbb{R}$ we define $$d_{\infty}(f,g)\coloneq \Vert f-g\Vert_\infty,\quad d^p_{L^p(\mu)}(f,g)\coloneq \mu(\vert f-g\vert^p),\quad p\geq1.$$ Our uniform moment bounds for $\mathbb{M}^j_T$ then read as follows. **Proposition 1**. *Grant assumptions [\[ass: inv\]](#ass: inv){reference-type="ref" reference="ass: inv"}, [\[ass: mixing\]](#ass: mixing){reference-type="ref" reference="ass: mixing"} and let $\mathcal{G}$ be a countable class of bounded, measurable functions mapping from $\mathbb{R}^d \to\mathbb{R}$. Then for large enough $T,$ it holds for any $p\in[1,\infty)$ and $j\in[d]$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}}\vert \mathbb{M}^j_T\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},24\sqrt{\Vert a_{jj}\Vert_{\infty}}((\kappa T)^{-1/4}d_\infty+(\kappa T)^{-1/8}d_{L^4(\mu)})) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},\sqrt{6\Vert a_{jj}\Vert_{\infty}} (128^{1/4}(\kappa T)^{-1/8}d_{L^4(\mu)}+d_{L^2(\mu)}))\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\&\quad+ 24p\sqrt{\Vert a_{jj}\Vert_\infty}\widetilde{c}_1\sup_{g\in\mathcal{G}}\mathopen{}\mathclose\bgroup\left((\kappa T)^{-1/4}\Vert g\Vert_\infty+(\kappa T)^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right) \\&\quad+\sqrt{6\Vert a_{jj}\Vert_\infty p}\widetilde{c}_2\sup_{g\in\mathcal{G}}\mathopen{}\mathclose\bgroup\left((128)^{1/4}(\kappa T)^{-1/8}\Vert g\Vert_{L^4(\mu)}+\Vert g\Vert_{L^2(\mu)}\aftergroup\egroup\right), \end{aligned}$$ where $$\widetilde{c}_1=4\mathrm{e}^{1/(2\mathrm{e})}(\sqrt{8\pi}\mathrm{e}^{1/(12p)})^{1/p}\mathrm{e}^{-1},\quad \widetilde{c}_2=4\mathrm{e}^{1/(2\mathrm{e})}(\sqrt{2\pi}\mathrm{e}^{1/(6p)})^{1/p}\mathrm{e}^{-1/2}.$$* For the analysis of the jump part $\mathbb{J}^j_T$ Bernstein's inequality for continuous martingales is naturally not available. For verifying the required concentration behaviour we therefore employ the exponential martingale inequality given in Theorem 2.2 of [@applebaum09paper], which adjusted to the setting of this paper gives for any $T,\alpha,\beta>0$ that $$\mathbb{P}\mathopen{}\mathclose\bgroup\left(\int_0^T\int \gamma(X_{s-})z\widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z)- \frac{1}{\alpha}\int_0^T\int \exp(\alpha \gamma(X_{s-}z))-1-\alpha \gamma(X_{s-})z\nu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} s>\beta \aftergroup\egroup\right)\leq \exp(-\alpha\beta). \label{eq: exp martingale ineq}$$ Clearly, the usefulness of this inequality strongly depends on the Lévy measure $\nu,$ which is why we introduce the following exponential moment assumption. 1. There exists a constant $c_{1,\nu}>0,$ such that[\[ass: exp mom\]]{#ass: exp mom label="ass: exp mom"} $$\int_{\Vert x\Vert\geq1} \exp(c_{1,\nu}\Vert x\Vert)\nu(\mathop{}\!\mathrm{d} x)<\infty.$$ Under the above assumption we are able to prove bounds similar to the ones for the continuous martingale part observed in Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"}. **Proposition 2**. *Let everything be given as in Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"} and assume [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"}. Then there exists a universal constant $C_1>0,$ such that for large enough $T,$ it holds for any $p\in[1,\infty)$ and $j\in[d]$ $$\begin{aligned} &\mathbb{E}\Big[\sup_{g\in\mathcal{G}}\vert \mathbb{J}^j_T(g)\vert^p\Big]^{1/p} \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},3T^{-1/8}d_{L^4(\mu)}+3(64c_1\kappa^{-1/2}+1)T^{-1/4}d_\infty) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},\sqrt{3}(1+c_1)d_{L^2(\mu)}+\sqrt{3072}c_1\kappa^{-1/4}T^{-1/8}d_{L^4(\mu)}) \aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+3p\widetilde{c}_1\sup_{g\in\mathcal{G}}\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+(64c_1\kappa^{-1/2}+1)T^{-1/4}\Vert g\Vert_\infty \aftergroup\egroup\right) \\&\quad+\sqrt{3p}\widetilde{c}_2\sup_{g\in\mathcal{G}}\mathopen{}\mathclose\bgroup\left((1+c_1)\Vert g\Vert_{L^2(\mu)}+32c_1\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right), \end{aligned}$$ where $c_1$ is defined in [\[eq: def c_1\]](#eq: def c_1){reference-type="eqref" reference="eq: def c_1"}.* One might be tempted to think that the exponential moment assumption [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"} is necessary for $\mathbf{X}$ to be exponentially $\beta$-mixing. However, in [@mas04] it was shown that under a certain drift condition Lévy-driven Ornstein--Uhlenbeck processes fulfill [\[ass: mixing\]](#ass: mixing){reference-type="ref" reference="ass: mixing"} as soon as the Lévy measure $\nu$ admits a polynomial moment. Thus, we also investigate the setting in which only the following assumption holds true. 1. The Lévy measure $\nu$ admits a third moment, i.e. [\[ass: th mom\]]{#ass: th mom label="ass: th mom"} $$\nu_3\coloneq \int \Vert x\Vert^3\nu(\mathop{}\!\mathrm{d} z)<\infty.$$ Additionally there exist $\gamma_{\min}>0,$ such that $\gamma_{\min}$ is a global lower bound for the smallest singular value of $\gamma$. The assumption on the smallest singular value of the jump coefficient $\gamma$ is merely a technical condition, which implies that any jump in the noise also leads to a jump of $\mathbf{X}$. While the stronger assumption [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"} allowed us to find uniform moment bounds for $\mathbb{I}^j_T,$ we can still show similar results under [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"} for a truncated version of $\mathbb{I}^j_T$. More precisely, for $T,\delta>0,j\in[d]$ we define $$X^{j,\delta}_T=X^j_T-\sum_{0\leq s\leq T} \Delta X^j_s\mathbf{1}_{(\delta,\infty)}(\Vert \Delta X_s\Vert),\quad \textrm{where}\quad \Delta X^j_s\coloneq X^j_{s}-X^j_{s-},$$ and $$\mathbb{I}^{j,\delta}_T(g)\coloneq \frac{1}{\sqrt{T}}\int_0^Tg(X_{s-})\mathop{}\!\mathrm{d} X^{j,\delta}_s.$$ Then the following decomposition for the centred moment of $\mathbb{I}^{j,\delta}_T$ holds true. **Lemma 3**. *Let $\mathcal{G}$ be as in Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"} and assume [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"}. Then it holds for any $j\in[d],T>0,p\geq1,\delta>0,$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}}\mathopen{}\mathclose\bgroup\left\vert\mathbb{I}^{j,\delta}_T(g)-\sqrt{T}\mu(gb)\aftergroup\egroup\right\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq \mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}} \vert \mathbb{H}_T(gb^j)-\sqrt{T}\mu(gb^j)\vert^p\aftergroup\egroup\right]^{1/p}+\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}}\vert \mathbb{M}^j_T(g)\vert^p\aftergroup\egroup\right]^{1/p}+\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}}\vert \mathbb{J}^{j,\delta,1}_T(g)\vert^p\aftergroup\egroup\right]^{1/p} \\ &\quad+\frac{\Vert \gamma\Vert_\infty\delta}{\gamma_{\min}}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}}\vert \mathbb{J}^{\delta,2}_T(g)\vert^p\aftergroup\egroup\right]^{1/p}+\frac{c_2}{\delta^2}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}} \vert \mathbb{H}_T(\vert g\vert)-\sqrt{T}\mu(\vert g\vert)\vert^p\aftergroup\egroup\right]^{1/p}+\frac{c_2\sqrt{T}}{\delta^2}\sup_{g\in\mathcal{G}}\mu(\vert g\vert), \end{aligned}$$ where $$\begin{aligned} \mathbb{J}^{j,\delta,1}_T(g)&\coloneq \frac{1}{\sqrt{T}}\int_0^T\int_{B(0,\delta/\gamma_{\min})} g(X_{s-})\gamma^j(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z), \\ \mathbb{J}^{\delta,2}_T(g)&\coloneq \frac{1}{\sqrt{T}}\int_0^T\int_{\delta/\Vert \gamma\Vert_\infty<\Vert z\Vert\leq \delta/\gamma_{\min}} \vert g(X_{s-})\vert \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z), \\ c_2&\coloneq \nu_3\Vert \gamma\Vert_\infty\mathopen{}\mathclose\bgroup\left(\frac{\Vert \gamma\Vert_\infty^3}{\gamma_{\min}}+\gamma_{\min}^2\aftergroup\egroup\right),\quad \textrm{with}\quad \nu_3=\int \Vert z\Vert ^3\nu(\mathop{}\!\mathrm{d} z). \end{aligned}$$* As we can again employ the results of [@dexheimeraihp] and Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"}, it now suffices to find suitable bounds for $\mathbb{J}^{j,\delta,1}_T,\mathbb{J}^{\delta,2}_T$ in order to obtain the desired uniform moment bounds for the truncated empirical process. These are given in the following Proposition. **Proposition 4**. *Let everything be given as in Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"}, assume [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"} and let $0<\delta\leq \alpha T^{1/4}$, for a fixed constant $\alpha>0$. Then for large enough $T$ it holds for any $j\in[d],p\geq 1,$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}}\vert \mathbb{J}^{j,\delta,1}_T\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G},3\mathopen{}\mathclose\bgroup\left(T^{-1/8}d_{L^4(\mu)}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)d_\infty\aftergroup\egroup\right)\aftergroup\egroup\right) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G},\sqrt{3}\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)d_{L^2(\mu)}+32\mathcal{E}_\alpha T^{-1/8}\kappa^{-1/4}d_{L^4(\mu)} \aftergroup\egroup\right)\aftergroup\egroup\right)\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+\sup_{g\in\mathcal{G}}\Bigg(3p\widetilde{c}_1\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)\Vert g\Vert_{\infty}\aftergroup\egroup\right) \\&\quad+\sqrt{3p}\widetilde{c}_2\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)\Vert g\Vert_{L^2(\mu)}+32\mathcal{E}_\alpha T^{-1/8}\kappa^{-1/4}\Vert g\Vert_{L^4(\mu)} \aftergroup\egroup\right) \Bigg), \end{aligned}$$ and $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}}\vert \mathbb{J}^{\delta,2}_T\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G},3\delta^{-1}\mathopen{}\mathclose\bgroup\left(T^{-1/8}d_{L^4(\mu)}+T^{-1/4}d_\infty\mathopen{}\mathclose\bgroup\left(1+32\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/2}\aftergroup\egroup\right) \aftergroup\egroup\right)\aftergroup\egroup\right) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G},\sqrt{3}\delta^{-1}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(1+\frac{\nu_2\Vert \gamma\Vert_\infty^2}{2}\exp(\alpha)\aftergroup\egroup\right)d_{L^2(\mu)}+16\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/4}T^{-1/8}d_{L^4(\mu)}\aftergroup\egroup\right)\aftergroup\egroup\right)\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+\delta^{-1}\sup_{g\in\mathcal{G}}\Bigg(3p\widetilde{c}_1\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\Vert g\Vert_\infty\mathopen{}\mathclose\bgroup\left(1+32\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/2}\aftergroup\egroup\right) \aftergroup\egroup\right) \\&\quad+\sqrt{3p}\widetilde{c}_2\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(1+\frac{\nu_2\Vert \gamma\Vert_\infty^2}{2}\exp(\alpha)\aftergroup\egroup\right)\Vert g\Vert_{L^2(\mu)}+16\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right) \Bigg), \end{aligned}$$ where $$\mathcal{E}_\alpha\coloneq\Vert \gamma\Vert_\infty^2\frac{\nu_2\exp(\alpha\Vert \gamma\Vert_\infty \sigma^{-1}_d)}{2}, \quad \nu_2\coloneq \int\Vert z\Vert^2\nu(\mathop{}\!\mathrm{d} z).$$* Comparable uniform moment bounds for jump processes as in Propositions [Proposition 2](#prop: jump moment bound){reference-type="ref" reference="prop: jump moment bound"} and [Proposition 4](#prop: trunc chain){reference-type="ref" reference="prop: trunc chain"} have been shown in [@wang19] and [@vdgeer95]. However, for this [@wang19] assumes the jumps to be bounded (see Theorem 1.2 in [@wang19]) and [@vdgeer95] assumes the jumps to be bounded from below (See Theorem 3.1 in [@vdgeer95]). To the best of our knowledge no results under an exponential moment assumption as in Proposition [Proposition 2](#prop: jump moment bound){reference-type="ref" reference="prop: jump moment bound"} were available so far. Lastly, denoting $\mathrm{supp}(\mathcal{G})\coloneq\bigcup_{g\in\mathcal{G}}\mathrm{supp}(g),$ one can actually achieve the same results as stated in this section under the less restrictive assumption, that $a$ and $\gamma$ are bounded on $\mathrm{supp}(\mathcal{G})$. However, as this does not offer much further insight, we stick to the global boundedness assumptions stated in Section [2.2](#subsec: not){reference-type="ref" reference="subsec: not"}. # Drift estimation {#sec: drift est} In this section we employ the powerful results of Section [3](#sec: unif mom){reference-type="ref" reference="sec: unif mom"} to investigate the statistical aim of this work, nonparametric drift estimation for jump diffusions. For the rest of this section $D$ will denote an open, bounded subset of $\mathbb{R}^d$. Furthermore we let $k\colon \mathbb{R}^d\to\mathbb{R}$ denote a symmetric, Lipschitz continuous kernel function with $\mathrm{supp}(k)= [-1/2,1/2]$ of order $\mathsf{k}\in\mathbb{N},$ i.e. $$\int k(x)\mathop{}\!\mathrm{d} x=1,\quad \textrm{and }\forall i\in[\mathsf{k}]: \int k(x)x^i\mathop{}\!\mathrm{d} x=0.$$ With this we define the following for any multi-index bandwidth $\boldsymbol{h}=\boldsymbol{h}(T)=(h_1(T),\ldots,h_d(T))\in(0,1)^d, x\in D,$ $$\begin{aligned} K(x)\coloneq \prod_{i=1}^d k(x_i),\quad K_{\boldsymbol{h}}(x)&\coloneq \prod_{i=1}^{d}h_i^{-1} k(x_i/h_i), \quad V(\boldsymbol{h})\coloneq \prod_{i=1}^d h_i.\end{aligned}$$ Furthermore for any $T>0$ we introduce the following class of multi-index bandwidths $$\mathsf{H}_T\coloneq \mathopen{}\mathclose\bgroup\left\{\boldsymbol{h}\in(0,1)^d\colon h_i\leq \log(1+T)^{-1}\, \forall i\in[d]\, \mathrm{and}\, V(\boldsymbol{h})\geq T^{-1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^dh_i^{-1}\aftergroup\egroup\right)^4 \aftergroup\egroup\right\},$$ and the anisotropic Hölder class of functions on $D$. **Definition 5**. Given $\boldsymbol{\beta}=(\beta_1,\ldots,\beta_d), \mathbf{\mathcal{L}}=(\mathcal{L}_1,\ldots,\mathcal{L}_d)\in(0,\infty)^d,$ the anisotropic Hölder class of functions $\mathcal{H}_D(\boldsymbol{\beta},\mathbf{\mathcal{L}})$ on $D$ is the set of all functions $f\colon D\to \mathbb{R}$ fulfilling for all $i\in[d],$ $$\begin{aligned} \Vert D^k_i f\Vert_\infty&\leq \mathcal{L}_i, \quad \forall k\in[\llfloor \beta_i\rrfloor], \\ \Vert D^{\llfloor \beta_i\rrfloor}_i f(\cdot +t\mathsf{e}_i)- D^{\llfloor \beta_i\rrfloor}_i f(\cdot)\Vert_\infty &\leq \mathcal{L}_i\vert t\vert^{\beta_i-\llfloor \beta_i\rrfloor}, \quad \forall t\in\mathbb{R}.\end{aligned}$$ Here $D^k_i f$ denotes the $k$-th order partial derivative of $f$ with respect to the $i$-th component, $\llfloor \beta\rrfloor$ the largest integer strictly smaller than $\beta$ and $\mathsf{e}_1,\ldots, \mathsf{e}_d$ the canonical basis of $\mathbb{R}^d$. Furthermore we let $$\bar{\boldsymbol{\beta}}\coloneq \frac{d}{\sum_{i=1}^{d}\beta_i^{-1}},$$ denote the harmonic mean of $\boldsymbol{\beta}$. In order to estimate the drift coefficient $b$ we firstly introduce the following two auxiliary estimators, defined for any $x\in D, \boldsymbol{h}\in(0,1)^d,j\in[d], \delta>0$ as $$\label{def: aux est} \bar{b}^{(j)}_{\boldsymbol{h},T}(x)\coloneq \frac 1 T \int_0^TK_{\boldsymbol{h}}(x-X_{s-})\mathop{}\!\mathrm{d} X^j_s,\quad \bar{b}^{(j),\delta}_{\boldsymbol{h},T}(x)\coloneq \frac 1 T \int_0^TK_{\boldsymbol{h}}(x-X_{s-})\mathop{}\!\mathrm{d} X^{j,\delta}_s.$$ Note that $\bar{b}^{(j)}_{\boldsymbol{h},T}$ is the same estimator as employed for continuous diffusions in [@aeck22], whereas $\bar{b}^{(j),\delta}_{\boldsymbol{h},T}$ has been adjusted to the given discountinuous setting. Recalling the classical decomposition of the $\sup$-norm risk into bias and stochastic error $$\mathcal{R}^{(p)}_\infty(b^j\rho,\bar{b}^{(j)}_{\boldsymbol{h},T};D)\leq \mathcal{B}_{b^j\rho}(\boldsymbol{h})+\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{x\in D} \vert \bar{b}^{(j)}_{\boldsymbol{h},T}-\mu(b^j\rho)\vert \aftergroup\egroup\right],$$ where the bias is given as $$\mathcal{B}_{b^j\rho}(\boldsymbol{h})\coloneq \sup_{x\in D}\vert (b^j\rho - (b^j\rho)\ast K_{\boldsymbol{h}})(x)\vert,  $$ we see that the uniform moment bounds of Section [3](#sec: unif mom){reference-type="ref" reference="sec: unif mom"} can be applied to the function class $$\mathcal{G}_{\boldsymbol{h}}\coloneq \mathopen{}\mathclose\bgroup\left\{\prod_{i=1}^d k((x_i-\cdot)/h_i)\colon x\in D\cap \mathbb{Q}^d\aftergroup\egroup\right\},$$ in order to bound the stochastic error. Indeed, this technique leads to the following result. **Theorem 6**. *Assume [($\mathscr{A}$)](#framework) and fix $\theta>0$. Then for any $j\in[d],$ the following holds true:* 1. *[\[thm: general rate exp\]]{#thm: general rate exp label="thm: general rate exp"} If [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"} is fulfilled, exists a constant $c>0$ depending on $\theta$, such that for large enough $T$ it holds for any $\boldsymbol{h}\in\mathsf{H}_T$ $$\forall p\in\Big[1,\theta\log\Big(\sum_{i=1}^d h_i^{-1}\Big)\Big]\colon\quad\mathcal{R}^{(p)}_\infty(b^j\rho,\bar{b}^{(j)}_{\boldsymbol{h},T};D)\leq \mathcal{B}_{b^j\rho}(\boldsymbol{h})+c(TV(\boldsymbol{h}))^{-1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2}.$$* 2. *[\[thm: rate trunc est\]]{#thm: rate trunc est label="thm: rate trunc est"} If [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"} is fulfilled and $\delta=\alpha T^{1/4}$ for fixed $\alpha>0,$ exists a constant $c>0$ depending on $\theta$ and $\alpha$, such that for large enough $T$ it holds for any $\boldsymbol{h}\in\mathsf{H}_T$ $$\forall p\in\Big[1,\theta\log\Big(\sum_{i=1}^d h_i^{-1}\Big)\Big]\colon\quad\mathcal{R}^{(p)}_\infty(b^j\rho,\bar{b}^{(j),\delta}_{\boldsymbol{h},T};D)\leq \mathcal{B}_{b^j\rho}(\boldsymbol{h})+c(TV(\boldsymbol{h}))^{-1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2}.$$* The estimators defined in [\[def: aux est\]](#def: aux est){reference-type="eqref" reference="def: aux est"} can be interpreted as estimators for $b\rho,$ thus natural modifications for directly estimating the drift $b$ are given by the following Nadaraya--Watson type estimators defined for $x\in D, j\in[d],\delta>0, \boldsymbol{h}^b,\boldsymbol{h}^\rho\in(0,1)^d,$ $$\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}(x)\coloneq \frac{ \bar{b}^{(j)}_{\boldsymbol{h}^b,T}(x)}{\vert \widehat{\rho}_{\boldsymbol{h}^\rho,T}(x)\vert +r(T)},\quad \widehat{b}^{(j),\delta}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}(x)\coloneq \frac{\bar{b}^{(j),\delta}_{\boldsymbol{h}^b,T}(x)}{\vert \widehat{\rho}_{\boldsymbol{h}^\rho,T}(x)\vert +r(T)},$$ where $$\label{def: dens est} \widehat{\rho}_{\boldsymbol{h}^\rho,T}(x)\coloneq \frac 1 T \int_0^T K_{\boldsymbol{h}^\rho}(x-X_s)\mathop{}\!\mathrm{d} s,$$ is a kernel density estimator, whose behaviour has been studied in [@dexheimeraihp] under [($\mathscr{A}$)](#framework), and $r(T)$ is a strictly positive function. Applying Theorem [Theorem 6](#thm: general rate){reference-type="ref" reference="thm: general rate"} with the rate-optimal choices of $\boldsymbol{h}^b,\boldsymbol{h}^\rho$ together with suitable $r(T)$ then yields that both estimators achieve the nonparametric rate of convergence for a weighted version of the $\sup$-norm risk (see Remark [Remark 8](#rem: weight){reference-type="ref" reference="rem: weight"}). **Corollary 7**. *Let $j\in[d],$ assume [($\mathscr{A}$)](#framework), $b^j\rho,\rho\in\mathcal{H}_D(\boldsymbol{\beta},\mathcal{L}),$ where $\boldsymbol{\beta},\mathcal{L}$ are given as in Definition [Definition 5](#def: hölder){reference-type="ref" reference="def: hölder"} with $\bar{\boldsymbol{\beta}}>d/2\lor (2\land d)$ and $\max_{i\in[d]}(\llfloor\beta_i\rrfloor)\leq \mathsf{k}$. Then choosing $$\begin{aligned} r(T)&= \Phi_{d,\boldsymbol{\beta}}(T)\exp(\sqrt{\log(T)}),\quad\textrm{where}\quad \Phi_{d,\boldsymbol{\beta}}\coloneq \begin{cases} \frac{\log(T)}{\sqrt{T}},& d\leq 2,\\ \mathopen{}\mathclose\bgroup\left(\frac{\log(T)}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{2\bar{\boldsymbol{\beta}}+d-2}}, & d\geq 3, \end{cases} \\ \boldsymbol{h}^b&=(h_1^b(T),\ldots,h^b_d(T)),\quad\textrm{where}\quad h_i^b(T)\sim\mathopen{}\mathclose\bgroup\left(\frac{\log(T)}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{(2\bar{\boldsymbol{\beta}}+d)\beta_i}}, \\ \boldsymbol{h}^\rho&=(h^\rho_1(T),\ldots,h^\rho_d(T)),\quad\textrm{where}\quad h^\rho_i(T)\sim\begin{cases} \frac{\log(T)^2}{\sqrt{T}},&d=1, \\ \mathopen{}\mathclose\bgroup\left(\frac{\log(T)^4}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{4\beta_i}}, &d=2, \\ \mathopen{}\mathclose\bgroup\left(\frac{\log(T)}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{(2\bar{\boldsymbol{\beta}}+d-2)\beta_i}}, &d\geq3, \end{cases}\end{aligned}$$ gives the following:* 1. *[\[cor: a\]]{#cor: a label="cor: a"}If [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"} is fulfilled, it holds $$\mathbb{E}\mathopen{}\mathclose\bgroup\left[\Vert(\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}-b^j)\rho\Vert_{L^\infty(D)} \aftergroup\egroup\right]\in\mathcal{O}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(\frac{\log(T)}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{2\bar{\boldsymbol{\beta}}+d}}\aftergroup\egroup\right).$$* 2. *[\[cor: b\]]{#cor: b label="cor: b"} If [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"} is fulfilled and $\delta=\alpha T^{1/4}$ for some fixed $\alpha>0,$ it holds $$\mathbb{E}\mathopen{}\mathclose\bgroup\left[\Vert(\widehat{b}^{(j),\delta}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}-b^j)\rho\Vert_{L^\infty(D)} \aftergroup\egroup\right]\in\mathcal{O}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(\frac{\log(T)}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{2\bar{\boldsymbol{\beta}}+d}}\aftergroup\egroup\right).$$* *Remark 8*. a) The requirement $\bar{\boldsymbol{\beta}}>d/2$ is frequent in the context of drift estimation for diffusion type processes, see, for example, Theorem 13 in [@str15] or Theorem 4.5 in [@kindiff] (note that the investigated process in this reference is $2d$ dimensional). It stems from the term involving $T^{-1/4}$ in Propositions [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"}, [Proposition 2](#prop: jump moment bound){reference-type="ref" reference="prop: jump moment bound"} and [Proposition 4](#prop: trunc chain){reference-type="ref" reference="prop: trunc chain"} dominating the uniform moment bounds if the bandwidths are too small, which is the case if $\bar{\boldsymbol{\beta}}\leq d/2$. The other condition $\bar{\boldsymbol{\beta}}>2\land d$ is needed for the analysis of the invariant density estimator $\widehat{\rho}$ and has been discussed in detail in Remark 4.3 of [@dexheimeraihp]. b) [The weighted version of the $\sup$-norm risk considered in Corollary [Corollary 7](#cor: rate){reference-type="ref" reference="cor: rate"} can be interpreted as the empirical counterpart to the traditional $\sup$-norm risk, similarly as the empirical $L^2$ risk investigated in [@schmisser14] compared to the standard $L^2$ risk. Clearly, if $\rho$ is bounded away from zero on the domain $D,$ the results of Corollary [Corollary 7](#cor: rate){reference-type="ref" reference="cor: rate"} also translate to the normal $\sup$-norm risk. Intuitively, the weight function $\rho$ is needed since $\rho$ being small on a subset of $D$ corresponds to the process $\mathbf{X}$ spending little time in that subset, and thus providing limited information about its drift there.]{style="color: dex"} c) [Although the estimator $\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}$ requires more restrictive assumptions than $\widehat{b}^{(j),\delta}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}$ for achieving the same rate of convergence, its investigation can also be justified apart from the theoretical comparison of the continuous and discontinuous setting. This is because if one wants to modify the given estimators for practical applications, in particular under discrete observations, the needed knowledge on $\mathbf{X}$'s jumps for the truncated estimator becomes challenging, whereas employing a discretised version of $\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}$ is relatively straightforward. Nevertheless, since $\widehat{b}^{(j),\delta}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}$ only requires insight on the jumps of magnitude larger than $\sim T^{1/4},$ their detection is possible for large enough $T$ through a similar truncation method as used in [@maiou]. ]{style="color: dex"} # Adaptive Estimation {#sec: adap} As the rate-optimal choices $\boldsymbol{h}^b$ and $\boldsymbol{h}^\rho$ in Corollary [Corollary 7](#cor: rate){reference-type="ref" reference="cor: rate"} obviously depend on the in general unknown smoothness indices, the question of finding an adaptive estimator suited for drift estimation without this knowledge naturally arises. For this we employ a variant of the Goldenshluger--Lepski procedure presented in [@goldenshluger2011; @lepski2013], which requires us to introduce the following setup. As in Section [4](#sec: drift est){reference-type="ref" reference="sec: drift est"} we fix a bounded open set $D\subset\mathbb{R}^d,$ on which we additionally assume that $\rho\geq \rho_{\min}$ holds for some a priori known constant $\rho_{\min}>0$. For $K$ given as in Section [4](#sec: drift est){reference-type="ref" reference="sec: drift est"} and any multi-bandwidths $\boldsymbol{h}=(h_1,\ldots,h_d),\boldsymbol{\eta}=(\eta_1,\ldots,\eta_d)\in(0,1)^d$ and $x=(x_1,\ldots,x_d)^\top\in\mathbb{R}^d$ we set $$\begin{aligned} \notag K_{\boldsymbol{h}}\star K_{\boldsymbol{\eta}}(x)&\coloneq \prod_{i=1}^d (h_i^{-1} k(x_i/h_i))\ast (\eta_i^{-1} k(x_i/\eta_i)) \\&= \prod_{i=1}^d \int_{\mathbb{R}} h_i^{-1} k((u-x_i)/h_i)\eta_i^{-1} k(u/\eta_i)\mathop{}\!\mathrm{d} u\notag \\ \notag &=\int_{\mathbb{R}^d} K_{\boldsymbol{h}}(u-x)K_{\boldsymbol{\eta}}(u)\mathop{}\!\mathrm{d} u,\end{aligned}$$ and for $(X^{j,\delta}_t)_{t\geq0},$ defined as in Section [4](#sec: drift est){reference-type="ref" reference="sec: drift est"} we let $$\begin{aligned} \bar{b}^{(j)}_{\boldsymbol{h},\boldsymbol{\eta},T}(x)&\coloneq \frac{1}{T}\int_0^T (K_{\boldsymbol{h}}\star K_{\boldsymbol{\eta}})(X_{s-}-x)\mathop{}\!\mathrm{d} X^j_s, \\ \bar{b}^{(j),\delta}_{\boldsymbol{h},\boldsymbol{\eta},T}(x)&\coloneq \frac{1}{T}\int_0^T (K_{\boldsymbol{h}}\star K_{\boldsymbol{\eta}})(X_{s-}-x)\mathop{}\!\mathrm{d} X^{j,\delta}_s.\end{aligned}$$ Additionally we introduce for some fixed, but arbitrary, $\iota>1,$ the set of candidate multi-bandwidths as $$\mathscr{H}_T\coloneq \mathopen{}\mathclose\bgroup\left\{\boldsymbol{h}=(h_1,\ldots,h_d)\in(0,\log(1+T)^{-1})^d\colon h_i=\iota^{-k_i}\,\mathrm{ with }\, k_i\in\mathbb{N}, \iota^{\sum_{i=1}^dk_i}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d \iota^{k_i}\aftergroup\egroup\right)^{4}\leq T^{1/2} \aftergroup\egroup\right\},$$ and for any $\boldsymbol{h}\in\mathscr{H}_T,\theta,\alpha>0,$ we denote $$\begin{aligned} A_{T,1}(\boldsymbol{h},\theta)&\coloneq2\mathrm{e}T^{-1/2} V(\boldsymbol{h})^{-1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2}\Vert K\Vert_\infty\Bigg(C_1\mathopen{}\mathclose\bgroup\left(21+29d\sqrt{\Vert a\Vert_{\infty}} +17c_1 \aftergroup\egroup\right) \\&\quad+\sqrt{\theta}\widetilde{c}_2\mathopen{}\mathclose\bgroup\left(3+4\sqrt{\Vert a_{jj}\Vert_\infty } +3c_1\aftergroup\egroup\right)\Bigg) \\ A_{T,2}(\boldsymbol{h},\theta,\alpha)&\coloneq 2\mathrm{e}T^{-1/2} V(\boldsymbol{h})^{-1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2}\Vert K\Vert_\infty\Bigg(29C_1d\sqrt{\Vert a\Vert_{\infty}} +4\sqrt{\theta\Vert a_{jj}\Vert_\infty }\widetilde{c}_2, \\ &\quad +(1+\mathcal{E}_\alpha)\mathopen{}\mathclose\bgroup\left(64C_1 \sqrt{d}+3\sqrt{\theta}\widetilde{c}_2\aftergroup\egroup\right) +\frac{\Vert \gamma\Vert_\infty}{\gamma_{\min}}\mathopen{}\mathclose\bgroup\left(1+\frac{\nu_2\Vert \gamma\Vert_\infty^2}{2}\exp(\alpha)\aftergroup\egroup\right)\mathopen{}\mathclose\bgroup\left(64\sqrt{d}C_1+3\sqrt{\theta}\widetilde{c}_2\aftergroup\egroup\right)\Bigg), \\ \Upsilon_{T,1}^j(\boldsymbol{h})&\coloneq \sup_{\eta\in \mathscr{H}_T}\mathopen{}\mathclose\bgroup\left(\Vert \bar{b}^{(j)}_{\boldsymbol{h},\boldsymbol{\eta},T}-\bar{b}^{(j)}_{\boldsymbol{\eta},T}\Vert_{D,\infty}-A_{T,1}(\boldsymbol{\eta},d) \aftergroup\egroup\right)_+, \\ \Upsilon_{T,2}^j(\boldsymbol{h})&\coloneq \sup_{\eta\in \mathscr{H}_T}\mathopen{}\mathclose\bgroup\left(\Vert \bar{b}^{(j),\delta}_{\boldsymbol{h},\boldsymbol{\eta},T}-\bar{b}^{(j),\delta}_{\boldsymbol{\eta},T}\Vert_{D,\infty}-A_{T,2}(\boldsymbol{\eta},d,\alpha) \aftergroup\egroup\right)_+,\end{aligned}$$ where $C_1$ is introduced in the proof of Proposition [Proposition 2](#prop: jump moment bound){reference-type="ref" reference="prop: jump moment bound"}, $c_1$ and $\widetilde{c}_2$ are given in Lemma [Lemma 13](#lemma: qv jump bound){reference-type="ref" reference="lemma: qv jump bound"}, respectively Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"}, $\gamma_{\min}$ is defined in Assumption [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"} and $\nu_2$ and $\mathcal{E}_\alpha$ are denoted in Proposition [Proposition 4](#prop: trunc chain){reference-type="ref" reference="prop: trunc chain"}. Choosing the bandwidths $\widehat{\boldsymbol{h}}^j_1,\widehat{\boldsymbol{h}}^j_2$ according to $$\begin{aligned} \Upsilon_{T,1}^j(\widehat{\boldsymbol{h}}^j_1)+A_{T,1}(\widehat{\boldsymbol{h}}^j_1,d)&=\inf_{\boldsymbol{h}\in\mathscr{H}_T}\mathopen{}\mathclose\bgroup\left(\Upsilon_{T,1}^j(\boldsymbol{h})+A_{T,1}(\boldsymbol{h},d) \aftergroup\egroup\right), \\ \Upsilon_{T,2}^j(\widehat{\boldsymbol{h}}^j_2)+A_{T,2}(\widehat{\boldsymbol{h}}^j_2,d,\alpha)&=\inf_{\boldsymbol{h}\in\mathscr{H}_T}\mathopen{}\mathclose\bgroup\left(\Upsilon_{T,2}^j(\boldsymbol{h})+A_{T,2}(\boldsymbol{h},d,\alpha) \aftergroup\egroup\right),\end{aligned}$$ then allows us to provide the following oracle inequalities over the class of candidate bandwidths $\mathscr{H}_T$ for the adaptive versions of the auxiliary estimators introduced in [\[def: aux est\]](#def: aux est){reference-type="eqref" reference="def: aux est"}. **Proposition 9**. *Assume [($\mathscr{A}$)](#framework). Then for any $j\in[d]$ the following holds true:* 1. *If [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"} is fulfilled then there exists a constant $c>0,$ such that for large enough $T$ $$\mathcal{R}^{(1)}_\infty(b^j\rho,\bar{b}^{(j)}_{\widehat{\boldsymbol{h}}^j_1,T};D)\leq c\mathopen{}\mathclose\bgroup\left(\inf_{\boldsymbol{h}\in\mathscr{H}_T}\mathopen{}\mathclose\bgroup\left(\mathcal{B}_{b^j\rho}(\boldsymbol{h})+(TV(\boldsymbol{h}))^{-1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^dh_i^{-1}\aftergroup\egroup\right)^{1/2}\aftergroup\egroup\right)+\log(T)^{d+1/2}T^{-1/2}\aftergroup\egroup\right).$$* 2. *If [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"} is fulfilled and $\delta=\alpha T^{1/4}$ then there exists a constant $c>0,$ such that for large enough $T$ $$\mathcal{R}^{(1)}_\infty(b^j\rho,\bar{b}^{(j),\delta}_{\widehat{\boldsymbol{h}}^j_2,T};D)\leq c\mathopen{}\mathclose\bgroup\left(\inf_{\boldsymbol{h}\in\mathscr{H}_T}\mathopen{}\mathclose\bgroup\left(\mathcal{B}_{b^j\rho}(\boldsymbol{h})+(TV(\boldsymbol{h}))^{-1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^dh_i^{-1}\aftergroup\egroup\right)^{1/2}\aftergroup\egroup\right)+\log(T)^{d+1/2}T^{-1/2}\aftergroup\egroup\right).$$* Similarly as in Section [4](#sec: drift est){reference-type="ref" reference="sec: drift est"}, we now introduce Nadaraya--Watson type estimators, more precisely for $x\in D, j\in[d],\delta>0$, we set $$\begin{aligned} \widehat{b}_{\textrm{adap},T}^{(j)}(x)&\coloneq \frac{\bar{b}^{(j)}_{\widehat{\boldsymbol{h}}^j_1,T}(x)}{ \widehat{\rho}_{\widehat{\boldsymbol{h}}_\rho,T}(x)\lor \rho_{\min}}, \quad \widehat{b}_{\textrm{adap},T}^{(j),\delta}(x)\coloneq \frac{\bar{b}^{(j),\delta}_{\widehat{\boldsymbol{h}}^j_2,T}(x)}{ \widehat{\rho}_{\widehat{\boldsymbol{h}}_\rho,T}(x)\lor \rho_{\min}}.\label{def: adap est}\end{aligned}$$ Here for a multi-bandwidth $\boldsymbol{h}$ the kernel density estimator $\widehat{\rho}_{\boldsymbol{h},T}$ is defined as in [\[def: dens est\]](#def: dens est){reference-type="eqref" reference="def: dens est"}, and $\widehat{\boldsymbol{h}}_\rho$ is assumed to be an adaptive bandwidth choice, such that $$\label{eq: adap dens} \mathcal{R}^{(1)}_\infty(\rho,\widehat{\rho}_{\widehat{\boldsymbol{h}}_\rho,T};D)\in \mathcal{O}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(\frac{\log(T)}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{2\bar{\boldsymbol{\beta}}+d}}\aftergroup\egroup\right),$$ if $\rho\in\mathcal{H}_D(\boldsymbol{\beta},\mathcal{L})$ with $\bar{\boldsymbol{\beta}}>d\land 2$ and [($\mathscr{A}$)](#framework) is fulfilled. This assumption is discussed in detail in Remark [Remark 11](#rem: adap){reference-type="ref" reference="rem: adap"}. As the optimal bandwidth choices in Corollary [Corollary 7](#cor: rate){reference-type="ref" reference="cor: rate"} are contained in $\mathscr{H}_T$ if $\bar{\boldsymbol{\beta}}>d/2,$ a straightforward application of the above oracle inequalities then yields the following result on the $\sup$-norm risk of the adaptive Nadaraya--Watson type estimators defined in [\[def: adap est\]](#def: adap est){reference-type="eqref" reference="def: adap est"}. **Theorem 10**. *Let $j\in[d],$ assume [($\mathscr{A}$)](#framework), $b^j\rho,\rho\in\mathcal{H}_D(\boldsymbol{\beta},\mathcal{L}),$ with $\bar{\boldsymbol{\beta}}>d/2\lor (2\land d)$ and $\max_{i\in[d]}(\llfloor\beta_i\rrfloor)\leq \mathsf{k}$. Then the following holds true:* a) *If [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"} is fulfilled it holds $$\notag \mathcal{R}^{(1)}_\infty(b^j,\widehat{b}^{(j)}_{\mathrm{adap},T};D)\in\mathcal{O}\mathopen{}\mathclose\bgroup\left( \mathopen{}\mathclose\bgroup\left(\frac{\log(T)}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{2\bar{\boldsymbol{\beta}}+d}}\aftergroup\egroup\right).$$* b) *If [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"} is fulfilled and $\delta=\alpha T^{1/4}$ it holds $$\notag \mathcal{R}^{(1)}_\infty(b^j,\widehat{b}^{(j),\delta}_{\mathrm{adap},T};D)\in\mathcal{O}\mathopen{}\mathclose\bgroup\left( \mathopen{}\mathclose\bgroup\left(\frac{\log(T)}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{2\bar{\boldsymbol{\beta}}+d}}\aftergroup\egroup\right).$$* *Remark 11*. In the isotropic setting, where $\rho$ is assumed to be in $\mathcal{H}_D(\boldsymbol{\beta},\mathcal{L})$ with $\beta_i=\beta_1$ for all $i\in[d],$ the assumption on the existence of an adaptive bandwidth choice $\widehat{\boldsymbol{h}}_\rho$ satisfying [\[eq: adap dens\]](#eq: adap dens){reference-type="eqref" reference="eq: adap dens"} can be justified through Theorem 4.2 in [@dexheimeraihp] as soon as [($\mathscr{A}$)](#framework) and $\beta_1>2$ are fulfilled and $d\geq 3$. The assumption $d\geq3$ in this context is no restriction as the optimal bandwidth choice $\boldsymbol{h}^\rho$ given in Corollary [Corollary 7](#cor: rate){reference-type="ref" reference="cor: rate"} is independent of $\boldsymbol{\beta}$ in the isotropic setting if $d<3$. In fact, this reference provides $\widehat{\boldsymbol{h}}_\rho,$ such that $$\mathcal{R}^{(1)}_\infty(\rho,\widehat{\rho}_{\widehat{\boldsymbol{h}}_\rho,T};D)\in\mathcal{O}\mathopen{}\mathclose\bgroup\left(\log(T)\Phi_{d,\boldsymbol{\beta}} \aftergroup\egroup\right),$$ where $\Phi_{d,\boldsymbol{\beta}}$ is defined in Corollary [Corollary 7](#cor: rate){reference-type="ref" reference="cor: rate"}, thus achieving better results than needed for our analysis. Using the uniform moment bounds of Theorem 3.1 in [@dexheimeraihp] and setting up a variant of the Goldenshluger--Lepski procedure analogous to the given section, it is straightforward to extend these results to the anisotropic case. Hence, the assumption on the existence of a suitable adaptive bandwidth choice for the kernel density estimator can also be justified in the general anisotropic setting. # Conclusion {#sec: dis} We have introduced two different adaptive Nadaraya--Watson type estimators for the drift of a jump diffusion of the form [\[eq: jumpdiff intro\]](#eq: jumpdiff intro){reference-type="eqref" reference="eq: jumpdiff intro"} and shown that they both achieve the classical nonparametric rate of convergence measured in the $\sup$-norm risk under anisotropic Hölder smoothness conditions and differing assumptions on the jump measure. As explained in Section [1](#sec: intro){reference-type="ref" reference="sec: intro"}, we investigate both estimators in detail for comparing the continuous with the discontinuous setting. We showed that under the exponential moment assumption [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"}, the same estimator as used in the continuous setting in [@aeck22] still achieves satisfying results and were able to generalise this to the less restrictive third moment assumption [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"} by introducing a truncated estimator. Crucial for the proofs were new uniform moment bounds for empirical processes related to jump diffusions, which are proven through the extension of Talagrand's generic chaining device in [@dirksen15], together with the exponential martingale inequality [\[eq: exp martingale ineq\]](#eq: exp martingale ineq){reference-type="eqref" reference="eq: exp martingale ineq"}. Since our results on the rate of convergence coincide with the benchmark case of continuous and reversible diffusion processes, in which the classical nonparametric rate of convergence is known to be minimax optimal (see [@str15]), our results can be regarded as optimal as well. # Auxilliary results Before we can start with the proofs of the main results of this paper we require some auxiliary results. Firstly, we introduce the following Bernstein type inequality for exponentially $\beta$-mixing Markov processes. **Lemma 12**. *Suppose that $\mathbf{X}$ is an exponentially $\beta$-mixing Markov process, and let $g$ be a bounded, measurable function satisfying $\mu(g)=0$. Then, for any $T,u>0$ and $m_T\in(0,T/4]$ exists $\tau \in [m_T,2m_T],$ such that $$\begin{aligned} &\mathbb{P}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{T}}\int_0^T g(X_s)\mathop{}\!\mathrm{d} s>32\sqrt{u}\mathopen{}\mathclose\bgroup\left(\sqrt{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)}+2\sqrt{u}\Vert g\Vert_\infty \frac{m_T}{\sqrt{T}} \aftergroup\egroup\right) \aftergroup\egroup\right) \\ &\leq 2\exp(-u)+\frac{T}{m_T}c_\kappa \exp(-\kappa m_T)\mathbf{1}_{(u,\infty)}\mathopen{}\mathclose\bgroup\left(\frac{T}{16m_T}\aftergroup\egroup\right) \end{aligned}$$* *Proof.* By Lemma 3.1 in [@dexheimer2020mixing] we have that for any $T,u>0$ and $m_T\in(0,T/4]$ there exists $\tau \in [m_T,2m_T],$ such that $$\begin{aligned} \mathbb{P}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{T}}\int_0^T g(X_s)\mathop{}\!\mathrm{d} s>u \aftergroup\egroup\right)&\leq 2\exp\mathopen{}\mathclose\bgroup\left(-\frac{u^2}{32\mathopen{}\mathclose\bgroup\left(\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)+2u\Vert g\Vert_\infty \frac{m_T}{\sqrt{T}}\aftergroup\egroup\right)} \aftergroup\egroup\right) \\ &\quad +\frac{T}{m_T}c_\kappa \exp(-\kappa m_T)\mathbf{1}_{(0,4\sqrt{T}\Vert g\Vert_\infty)}(u).\end{aligned}$$ Hence we get $$\begin{aligned} &\mathbb{P}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{T}}\int_0^T g(X_s)\mathop{}\!\mathrm{d} s>32\sqrt{u}\mathopen{}\mathclose\bgroup\left(\sqrt{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)}+2\sqrt{u}\Vert g\Vert_\infty \frac{m_T}{\sqrt{T}} \aftergroup\egroup\right) \aftergroup\egroup\right) \\ &\leq 2\exp\mathopen{}\mathclose\bgroup\left(-32u\frac{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)+4u\Vert g\Vert^2_\infty \frac{m^2_T}{T}+4\sqrt{u}\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)\Vert g\Vert_\infty \frac{m_T}{\sqrt{T}} }{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)\mathop{}\!\mathrm{d} s\aftergroup\egroup\right)+64\sqrt{u}\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)\mathop{}\!\mathrm{d} s\aftergroup\egroup\right)\Vert g\Vert_\infty \frac{m_T}{\sqrt{T}} +128u\Vert g\Vert^2_\infty \frac{m_T^2}{T}} \aftergroup\egroup\right) \\ &\quad +\frac{T}{m_T}c_\kappa \exp(-\kappa m_T)\mathbf{1}_{(0,4\sqrt{T}\Vert g\Vert_\infty)}\mathopen{}\mathclose\bgroup\left(32\sqrt{u}\mathopen{}\mathclose\bgroup\left(\sqrt{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)}+2\sqrt{u}\Vert g\Vert_\infty \frac{m_T}{\sqrt{T}} \aftergroup\egroup\right) \aftergroup\egroup\right) \\ &\leq 2\exp(-u)+\frac{T}{m_T}c_\kappa \exp(-\kappa m_T)\mathbf{1}_{(0,4\sqrt{T}\Vert g\Vert_\infty)}\mathopen{}\mathclose\bgroup\left(64u\Vert g\Vert_\infty \frac{m_T}{\sqrt{T}} \aftergroup\egroup\right) \\ &\leq 2\exp(-u)+\frac{T}{m_T}c_\kappa \exp(-\kappa m_T)\mathbf{1}_{(u,\infty)}\mathopen{}\mathclose\bgroup\left(\frac{T}{16m_T}\aftergroup\egroup\right).\end{aligned}$$ ◻ The two following lemmas will prove to be useful in combination with the exponential martingale inequality [\[eq: exp martingale ineq\]](#eq: exp martingale ineq){reference-type="ref" reference="eq: exp martingale ineq"}. **Lemma 13**. *Grant assumption [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"}. Then for any $0<y\leq c_{1,\nu}(2\Vert g\Vert_\infty \Vert \gamma\Vert_\infty)^{-1},j\in[d]$ it holds $$\int_0^T\int_{\mathbb{R}^d}\exp(yg(X_{s-})\gamma^j(X_{s-})z))-1-yg(X_{s-})\gamma^j(X_{s-})z)\nu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} s\leq c_1 y^2\int_0^Tg(X_s)^2\mathop{}\!\mathrm{d} s,$$ where $$\label{eq: def c_1} c_1\coloneq\Vert \gamma\Vert_\infty^2\mathopen{}\mathclose\bgroup\left( \frac{1}{2}\exp(c_{1,\nu}/2) \int_{B(0,1)} \Vert z\Vert^2\nu(\mathop{}\!\mathrm{d} z)+4 c_{1,\nu}^{-2}\int_{B(0,1)^\mathsf{C}}\exp(c_{1,\nu}\Vert z\Vert)\nu(\mathop{}\!\mathrm{d} z)\aftergroup\egroup\right).$$* *Proof.* Define $h\colon \mathbb{R}\to \mathbb{R}, h(x)=\exp(x)-1-x$. Then $h$ is monotonically increasing on $[0,\infty)$ and the classical inequality $\exp(x)\geq 1+x, x\in\mathbb{R}$ gives for any $x\in\mathbb{R}$ $$\begin{aligned} 0=\int_{-\vert x\vert}^{\vert x\vert} y\mathop{}\!\mathrm{d} y\leq \int_{-\vert x\vert}^{\vert x\vert} \exp(y)-1\mathop{}\!\mathrm{d} y=\exp(\vert x\vert)-\exp(-\vert x\vert)-2\vert x\vert.\end{aligned}$$ Rearranging this inequality yields $h(x)\leq h(\vert x\vert)$ for any $x\in\mathbb{R}$. By Taylor's theorem we then get for $c_\gamma\coloneq\Vert \gamma\Vert_\infty,$ $$\begin{aligned} \int_{\mathbb{R}^d}h(yg(X_{s-})\gamma^j(X_{s-})z))\nu(\mathop{}\!\mathrm{d} z) &\leq \int_{B(0,1)}h(yc_\gamma\vert g(X_{s-})\vert \Vert z\Vert)\nu(\mathop{}\!\mathrm{d} z)+\int_{B(0,1)^\mathsf{C}}h(yc_\gamma\vert g(X_{s-})\vert \Vert z\Vert)\nu(\mathop{}\!\mathrm{d} z) \\ &\leq \frac{1}{2}\exp(yc_\gamma\vert g(X_{s-})\vert)y^2c^2_\gamma\vert g(X_{s-})\vert^2 \int_{B(0,1)} \Vert z\Vert^2\nu(\mathop{}\!\mathrm{d} z)\\&\quad+\int_{B(0,1)^\mathsf{C}}\exp(yc_\gamma\vert g(X_{s-})\vert \Vert z\Vert)-1-yc_\gamma\vert g(X_{s-})\vert \Vert z\Vert\nu(\mathop{}\!\mathrm{d} z).\end{aligned}$$ Now by [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"} we get by Taylor's theorem and the assumption on $y$, $$\begin{aligned} &\int_{B(0,1)^\mathsf{C}}\exp(yc_\gamma\vert g(X_{s-})\vert \Vert z\Vert)-1-yc_\gamma\vert g(X_{s-})\vert \Vert z\Vert\nu(\mathop{}\!\mathrm{d} z) \\ &\leq \frac{1}{2}(yc_\gamma\vert g(X_{s-})\vert)^2\int_{B(0,1)^\mathsf{C}}\exp(yc_\gamma\vert g(X_{s-})\vert \Vert z\Vert) \Vert z\Vert^2\nu(\mathop{}\!\mathrm{d} z) \\ &\leq 4(yc_\gamma\vert g(X_{s-})\vert c_{1,\nu}^{-1})^2\int_{B(0,1)^\mathsf{C}}\exp(c_{1,\nu}\Vert z\Vert)\nu(\mathop{}\!\mathrm{d} z),\end{aligned}$$ where we used the inequality $\exp(x)\geq x^2/2,x>0$ in the last step. Hence if $2y c_\gamma\vert g(X_{s-})\vert\leq c_{1,\nu},$ it holds $$\begin{aligned} &\int_{\mathbb{R}^d}\exp(yg(X_{s-})\gamma^j(X_{s-})z))-1-yg(X_{s-})\gamma^j(X_{s-})z)\nu(\mathop{}\!\mathrm{d} z) \\ &\leq y^2 g(X_{s-})^2c_\gamma^2\mathopen{}\mathclose\bgroup\left( \frac{1}{2}\exp(c_{1,\nu}/2) \int_{B(0,1)} \Vert z\Vert^2\nu(\mathop{}\!\mathrm{d} z)+4 c_{1,\nu}^{-2}\int_{B(0,1)^\mathsf{C}}\exp(c_{1,\nu}\Vert z\Vert)\nu(\mathop{}\!\mathrm{d} z)\aftergroup\egroup\right),\end{aligned}$$ which concludes the proof. ◻ **Lemma 14**. *Grant assumption [\[ass: th mom\]](#ass: th mom){reference-type="ref" reference="ass: th mom"}. Then, for any $y,\delta>0$ it holds $$\begin{aligned} &\int_{B(0,\delta)}\exp(yg(X_{s-})\gamma^j(X_{s-})z))-1-yg(X_{s-})\gamma^j(X_{s-})z)\nu(\mathop{}\!\mathrm{d} z) \\ &\leq (y\Vert \gamma\Vert_\infty g(X_{s-}))^2\frac{\nu_2\exp(y\Vert \gamma\Vert_\infty\vert g(X_{s-})\vert \delta)}{2}.\end{aligned}$$* *Proof.* Arguing as in the proof of Lemma [Lemma 13](#lemma: qv jump bound){reference-type="ref" reference="lemma: qv jump bound"} Taylor's theorem gives for $c_\gamma=\Vert \gamma\Vert_\infty,$ $$\begin{aligned} &\int_{B(0,\delta)}\exp(yg(X_{s-})\gamma^j(X_{s-})z))-1-yg(X_{s-})\gamma^j(X_{s-})z)\nu(\mathop{}\!\mathrm{d} z) \\ &\leq \int_{B(0,\delta)}\exp(yc_\gamma\vert g(X_{s-})\vert \Vert z\Vert)-1-yc_\gamma\vert g(X_{s-})\vert \Vert z\Vert\nu(\mathop{}\!\mathrm{d} z) \\ &\leq (yc_\gamma\vert g(X_{s-})\vert)^2\frac{\exp(yc_\gamma\vert g(X_{s-})\vert \delta)}{2} \int_{B(0,\delta)} \Vert z\Vert^2\nu(\mathop{}\!\mathrm{d} z).\end{aligned}$$ ◻ # Proofs for Section [3](#sec: unif mom){reference-type="ref" reference="sec: unif mom"} {#proofs-for-section-sec-unif-mom} We start with the proof of Proposition [Proposition 2](#prop: jump moment bound){reference-type="ref" reference="prop: jump moment bound"} as it is more involved than the proof of Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"}. *Proof of Proposition [Proposition 2](#prop: jump moment bound){reference-type="ref" reference="prop: jump moment bound"}.* Firstly, Theorem 2.2 in [@applebaum09paper] gives for any $T,x,y>0$ $$\begin{aligned} \mathbb{P}\mathopen{}\mathclose\bgroup\left(\mathbb{J}^j_T(g)>\frac{x}{\sqrt{T}}+\frac{1}{\sqrt{T}y} \int_0^T\int_{\mathbb{R}^d}\exp(yg(X_{s-})\gamma^j(X_{s-})z))-1-yg(X_{s-})\gamma^j(X_{s-})z)\nu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)\leq \exp(-xy). \end{aligned}$$ Thus, applying Lemma [Lemma 13](#lemma: qv jump bound){reference-type="ref" reference="lemma: qv jump bound"} to $g$ and $-g$ we get for any $x,T>0$ and $0<y\Vert g\Vert_\infty \Vert\gamma^j\Vert_\infty\leq c_{1,\nu}/2$ $$\begin{aligned} \mathbb{P}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left\vert\mathbb{J}^j_T(g)\aftergroup\egroup\right\vert >\frac{x}{\sqrt{T}}+\frac{yc_1}{\sqrt{T}}\int_0^Tg(X_s)^2\mathop{}\!\mathrm{d} s \aftergroup\egroup\right) &\leq 2\exp(-xy), \end{aligned}$$ which implies for any $r>0,$ $$\begin{aligned} &\mathbb{P}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left\vert\mathbb{J}^j_T(g)\aftergroup\egroup\right\vert >\frac{x}{\sqrt{T}}+yc_1r+yc_1\sqrt{T}\mu(g^2)\aftergroup\egroup\right) \\ &\leq \mathbb{P}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left\vert\mathbb{J}^j_T(g)\aftergroup\egroup\right\vert >\frac{x}{\sqrt{T}}+yc_1r+yc_1\sqrt{T}\mu(g^2),\frac{1}{\sqrt{T}}\int_0^Tg(X_s)^2-\mu(g^2)\mathop{}\!\mathrm{d} s\leq r \aftergroup\egroup\right) \\&\quad +\mathbb{P}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{T}}\int_0^Tg(X_s)^2-\mu(g^2)\mathop{}\!\mathrm{d} s>r \aftergroup\egroup\right) \\ &\leq \mathbb{P}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left\vert\mathbb{J}^j_T(g)\aftergroup\egroup\right\vert >\frac{x}{\sqrt{T}}+\frac{yc_1}{\sqrt{T}}\int_0^Tg(X_s)^2\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)+\mathbb{P}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{T}}\int_0^Tg(X_s)^2-\mu(g^2)\mathop{}\!\mathrm{d} s>r \aftergroup\egroup\right) \\ &\leq 2\exp(-xy)+\mathbb{P}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{T}}\int_0^Tg(X_s)^2-\mu(g^2)\mathop{}\!\mathrm{d} s>r \aftergroup\egroup\right). \end{aligned}$$ Hence choosing $m_T=\sqrt{T}/(2\sqrt{\kappa})$ in Lemma [Lemma 12](#lemma: bernstein dirksen form){reference-type="ref" reference="lemma: bernstein dirksen form"} we have for large enough $T,$ $$\begin{aligned} &\mathbb{P}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left\vert\mathbb{J}^j_T(g)\aftergroup\egroup\right\vert >\frac{x}{\sqrt{T}}+yc_132\sqrt{u}\mathopen{}\mathclose\bgroup\left(\sqrt{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)^2\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)}+\kappa^{-1/2}\sqrt{u}\Vert g^2-\mu(g^2)\Vert_\infty \aftergroup\egroup\right) +yc_1\sqrt{T}\mu(g^2)\aftergroup\egroup\right)\notag \\ &\leq \notag 2\exp(-xy)+2\exp(-u)+2c_\kappa\sqrt{\kappa T} \exp(-\sqrt{\kappa T}/2)\mathbf{1}_{(u,\infty)}\mathopen{}\mathclose\bgroup\left(\frac{\sqrt{\kappa T}}{8}\aftergroup\egroup\right) \\ &\leq 2\exp(-xy)+4\exp(-u).\label{eq: jump chain 1} \end{aligned}$$ Now note, that for $u>0$ it holds for $x_u=u \mathopen{}\mathclose\bgroup\left(\sqrt{T\mu(g^2)/u}+T^{3/8}\Vert g\Vert_{L^4(\mu)}+T^{1/4}\Vert g\Vert_\infty \aftergroup\egroup\right), y_u=u/x_u$ $$\begin{aligned} &\frac{x_u}{\sqrt{T}}+y_uc_132\sqrt{u}\mathopen{}\mathclose\bgroup\left(\sqrt{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)^2\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)}+\kappa^{-1/2}\sqrt{u}\Vert g^2-\mu(g^2)\Vert_\infty \aftergroup\egroup\right) +y_uc_1\sqrt{T}\mu(g^2) \\ &\leq \sqrt{u}(1+c_1)\Vert g\Vert_{L^2(\mu)}+uT^{-1/8}\Vert g\Vert_{L^4(\mu)}+uT^{-1/4}\Vert g\Vert_\infty \\&\quad+32y_uc_1\sqrt{u}\mathopen{}\mathclose\bgroup\left(\sqrt{\tau\mu(g^4)}+\kappa^{-1/2}\sqrt{u}(\Vert g^2\Vert_\infty+\mu(g^2)) \aftergroup\egroup\right) \\ &\leq \sqrt{u}(1+c_1)\Vert g\Vert_{L^2(\mu)}+uT^{-1/8}\Vert g\Vert_{L^4(\mu)}+uT^{-1/4}\Vert g\Vert_\infty \\&\quad+32y_uc_1\sqrt{u}\mathopen{}\mathclose\bgroup\left((T/\kappa)^{1/4}\sqrt{\mu(g^4)}+2\kappa^{-1/2}\sqrt{u}\Vert g^2\Vert_\infty \aftergroup\egroup\right) \\ &\leq \sqrt{u}(1+c_1)\Vert g\Vert_{L^2(\mu)}+uT^{-1/8}\Vert g\Vert_{L^4(\mu)}+uT^{-1/4}\Vert g\Vert_\infty \\&\quad+32c_1\sqrt{u}\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}+64uc_1\kappa^{-1/2}T^{-1/4}\Vert g\Vert_\infty \\ &=u\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+(64c_1\kappa^{-1/2}+1)T^{-1/4}\Vert g\Vert_\infty \aftergroup\egroup\right)+\sqrt{u}\mathopen{}\mathclose\bgroup\left((1+c_1)\Vert g\Vert_{L^2(\mu)}+32c_1\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right) \end{aligned}$$ Since $2y_u\Vert g\Vert_\infty \Vert \gamma\Vert_\infty \leq c_{1,\nu}$ holds for large enough $T$ for any $u>0$, this implies for large enough values of $T$ together with [\[eq: jump chain 1\]](#eq: jump chain 1){reference-type="eqref" reference="eq: jump chain 1"} $$\begin{aligned} &\mathbb{P}\Bigg(\mathopen{}\mathclose\bgroup\left\vert\mathbb{J}^j_T(g)\aftergroup\egroup\right\vert > u\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+(64c_1\kappa^{-1/2}+1)T^{-1/4}\Vert g\Vert_\infty \aftergroup\egroup\right)\\&\hskip 2cm +\sqrt{u}\mathopen{}\mathclose\bgroup\left((1+c_1)\Vert g\Vert_{L^2(\mu)}+32c_1\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right)\Bigg) \\ &\leq 6\exp(-u). \end{aligned}$$ As $$\begin{aligned} \label{eq: conc jump} \notag &\mathbb{P}\Bigg(\mathopen{}\mathclose\bgroup\left\vert\mathbb{J}^j_T(g)\aftergroup\egroup\right\vert > 3u\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+(64c_1\kappa^{-1/2}+1)T^{-1/4}\Vert g\Vert_\infty \aftergroup\egroup\right)\\&\hskip 2cm +\sqrt{3u}\mathopen{}\mathclose\bgroup\left((1+c_1)\Vert g\Vert_{L^2(\mu)}+32c_1\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right)\Bigg) \\\notag &\leq 2\exp(-u), \end{aligned}$$ is trivially true for $u\leq \log(2),$ we then obtain for large enough $T,$ that [\[eq: conc jump\]](#eq: conc jump){reference-type="eqref" reference="eq: conc jump"} holds true for any $u>0$. Hence we can apply Theorem 3.5 of [@dirksen15], which yields for any $p\geq 1$ $$\begin{aligned} &\mathbb{E}\Big[\sup_{g\in\mathcal{G}}\vert \mathbb{J}^j_T(g)\vert^p\Big]^{1/p} \\ &\leq c\Bigg(\widetilde{c}_1\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},3T^{-1/8}d_{L^4(\mu)}+3(64c_1\kappa^{-1/2}+1)T^{-1/4}d_\infty) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\widetilde{c}_2\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},\sqrt{3}(1+c_1)d_{L^2(\mu)}+\sqrt{3072}c_1\kappa^{-1/4}T^{-1/8}d_{L^4(\mu)}) \aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) +2\sup_{g\in\mathcal{G}}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\vert \mathbb{J}^j_T(g)\vert^p \aftergroup\egroup\right]^{1/p}, \end{aligned}$$ where we bounded the $\gamma_\alpha$ functionals by the corresponding entropy integrals (see e.g. Equation (2.3) in [@dirksen15]). Noting that a slight adjustment of Lemma A.2 in [@dirksen15] implies for any $p\geq1,g\in\mathcal{G},$ together with [\[eq: conc jump\]](#eq: conc jump){reference-type="eqref" reference="eq: conc jump"} $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\vert \mathbb{J}^j_T(g)\vert^p \aftergroup\egroup\right]^{1/p} \\ &\leq 6p\mathrm{e}^{1/(2\mathrm{e})}(\sqrt{8\pi}\mathrm{e}^{1/(12p)})^{1/p}\mathrm{e}^{-1}\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+(64c_1\kappa^{-1/2}+1)T^{-1/4}\Vert g\Vert_\infty \aftergroup\egroup\right) \\&\quad+\sqrt{3p}2\mathrm{e}^{-1/2}(\sqrt{2\pi}\mathrm{e}^{1/(6p)})^{1/p}\mathrm{e}^{1/(2\mathrm{e})}\mathopen{}\mathclose\bgroup\left((1+c_1)\Vert g\Vert_{L^2(\mu)}+32c_1\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right), \end{aligned}$$ concludes the proof. ◻ *Proof of Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"}.* By Bernstein's inequality for continuous martingales (see e.g. p.153 in [@revuzyor1999]) and Lemma [Lemma 12](#lemma: bernstein dirksen form){reference-type="ref" reference="lemma: bernstein dirksen form"} we have that there exists $\tau\in [\sqrt{T}/(2\sqrt{\kappa}),\sqrt{T}/\sqrt{\kappa}]$ such that for any $x>0$ $$\begin{aligned} &\mathbb{P}\mathopen{}\mathclose\bgroup\left(\sqrt{T}\mathbb{M}^j_T(g)>\sqrt{T}x\aftergroup\egroup\right) \\ &\leq \mathbb{P}\Bigg(\sqrt{T}\mathbb{M}^j_T(g)>\sqrt{T}x, \langle\sqrt{T}\mathbb{M}^j_{\cdot}(g)\rangle_T\leq 32\sqrt{Tu}\mathopen{}\mathclose\bgroup\left(\sqrt{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g^2(X_s)a_{jj}(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)}+2\sqrt{u}\Vert g^2a_{jj}\Vert_\infty \frac{m_T}{\sqrt{T}} \aftergroup\egroup\right) \\&\quad+\sqrt{T}\mu(g^2a_{jj})\Bigg) +2\exp(-u)+\frac{T}{m_T}c_\kappa \exp(-\kappa m_T)\mathbf{1}_{(u,\infty)}\mathopen{}\mathclose\bgroup\left(\frac{T}{16m_T}\aftergroup\egroup\right) \\ &\leq 2\exp\mathopen{}\mathclose\bgroup\left(-\frac{Tx^2}{64\sqrt{Tu}\mathopen{}\mathclose\bgroup\left(\sqrt{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g^2(X_s)a_{jj}(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)}+\sqrt{u}\Vert g^2a_{jj}\Vert_\infty \frac{1}{\sqrt{\kappa}} \aftergroup\egroup\right)+2\sqrt{T}\mu(g^2a_{jj})}\aftergroup\egroup\right)+4\exp(-u) \\ &\leq 2\exp\mathopen{}\mathclose\bgroup\left(-\frac{\sqrt{T}x^2}{64\sqrt{u}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(\frac{T}{\kappa}\aftergroup\egroup\right)^{1/4}\sqrt{2\mu(g^4a^2_{jj})}+\sqrt{u}\Vert g^2a_{jj}\Vert_\infty \frac{1}{\sqrt{\kappa}} \aftergroup\egroup\right)+2\sqrt{T}\mu(g^2a_{jj})}\aftergroup\egroup\right)+4\exp(-u) \\ &\leq 2\exp\mathopen{}\mathclose\bgroup\left(-\frac{\sqrt{T}x^2}{64\Vert a_{jj}\Vert_\infty\sqrt{u}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(\frac{T}{\kappa}\aftergroup\egroup\right)^{1/4}\sqrt{2\mu(g^4)}+\sqrt{u}\Vert g^2\Vert_\infty \frac{1}{\sqrt{\kappa}} \aftergroup\egroup\right)+2\sqrt{T}\mu(g^2a_{jj})}\aftergroup\egroup\right)+4\exp(-u) \end{aligned}$$ where we choose $m_T=\sqrt{T}/(2\sqrt{\kappa})$ in Lemma [Lemma 12](#lemma: bernstein dirksen form){reference-type="ref" reference="lemma: bernstein dirksen form"} as in the derivation of [\[eq: jump chain 1\]](#eq: jump chain 1){reference-type="eqref" reference="eq: jump chain 1"}. Hence for any $u>0$ it holds $$\begin{aligned} &\mathbb{P}\mathopen{}\mathclose\bgroup\left(\mathbb{M}^j_T(g)>\sqrt{\Vert a_{jj}\Vert_\infty}\mathopen{}\mathclose\bgroup\left((512)^{1/4}(\kappa T)^{-1/8}(\sqrt{u}+u)\Vert g\Vert_{L^4(\mu)}+8u(\kappa T)^{-1/4}\Vert g\Vert_\infty+\sqrt{2u}\Vert g\Vert_{L^2(\mu)}\aftergroup\egroup\right)\aftergroup\egroup\right) \\ &\leq 6\exp(-u), \end{aligned}$$ and arguing as in the derivation of [\[eq: conc jump\]](#eq: conc jump){reference-type="eqref" reference="eq: conc jump"}, we obtain that for large enough $T$ it holds for any $u>0$ $$\begin{aligned} &\mathbb{P}\Bigg(\mathbb{M}^j_T(g)>\sqrt{\Vert a_{jj}\Vert_\infty}\Big(\sqrt{6u}\mathopen{}\mathclose\bgroup\left((128)^{1/4}(\kappa T)^{-1/8}\Vert g\Vert_{L^4(\mu)}+\Vert g\Vert_{L^2(\mu)}\aftergroup\egroup\right)\\&\quad\qquad\qquad+24u\mathopen{}\mathclose\bgroup\left((\kappa T)^{-1/4}\Vert g\Vert_\infty+(\kappa T)^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right)\Big)\Bigg) \\ &\leq 2\exp(-u). \end{aligned}$$ Applying Theorem 3.5 of [@dirksen15] now gives for any $p\geq 1$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}}\vert \mathbb{M}^j_T\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq c\Bigg(\widetilde{c}_1\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},24\sqrt{\Vert a_{jj}\Vert_{\infty}}((\kappa T)^{-1/4}d_\infty+(\kappa T)^{-1/8}d_{L^4(\mu)})) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\widetilde{c}_2\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},\sqrt{6\Vert a_{jj}\Vert_{\infty}} ((128)^{1/4}(\kappa T)^{-1/8}d_{L^4(\mu)}+d_{L^2(\mu))}\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) +2\sup_{g\in\mathcal{G}}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\vert \mathbb{M}^j_T(g)\vert^p \aftergroup\egroup\right]^{1/p}, \end{aligned}$$ where we argue analogously to the proof of Proposition [Proposition 2](#prop: jump moment bound){reference-type="ref" reference="prop: jump moment bound"}, which also gives for any $g\in\mathcal{G}$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\vert \mathbb{M}^j_T(g)\vert^p \aftergroup\egroup\right]^{1/p} \\ &\leq 12p\sqrt{\Vert a_{jj}\Vert_\infty}\widetilde{c}_1\mathopen{}\mathclose\bgroup\left((\kappa T)^{-1/4}\Vert g\Vert_\infty+(\kappa T)^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right) \\&\quad+\sqrt{\frac{3}{2}\Vert a_{jj}\Vert_\infty p}\widetilde{c}_2\mathopen{}\mathclose\bgroup\left((128)^{1/4}(\kappa T)^{-1/8}\Vert g\Vert_{L^4(\mu)}+\Vert g\Vert_{L^2(\mu)}\aftergroup\egroup\right), \end{aligned}$$ concluding the proof. ◻ *Proof of Lemma [Lemma 3](#prop: trunc decomp){reference-type="ref" reference="prop: trunc decomp"}.* By linearity it holds $$\begin{aligned} &\frac{1}{\sqrt{T}}\int_0^Tg(X_{s-})\mathop{}\!\mathrm{d} X^{j,\delta}_s \\ &=\mathbb{H}_T(gb^j)+\mathbb{M}^j_T(g)+\frac{1}{\sqrt{T}}\int_0^T\int_{B(0,\delta/\gamma_{\min})} g(X_{s-})\gamma^j(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\ &\quad+\frac{1}{\sqrt{T}}\int_0^T\int_{B(0,\delta/\gamma_{\min})^{^{\operatorname{c}}}} g(X_{s-})\gamma^j(X_{s-})z N(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\&\quad- \frac{1}{\sqrt{T}}\int_0^Tg(X_{s-})\int_{B(0,\delta/\gamma_{\min})^{^{\operatorname{c}}}} \gamma^j(X_{s-})z\nu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} s \\&\quad-\frac{1}{\sqrt{T}}\sum_{0\leq s\leq T} g(X_{s-})\Delta X^j_s\mathbf{1}_{(\delta,\infty)}(\Vert \Delta X_s\Vert) \\ &=\mathbb{H}_T(gb^j)+\mathbb{M}^j_T(g)+\frac{1}{\sqrt{T}}\int_0^T\int_{B(0,\delta/\gamma_{\min})} g(X_{s-})\gamma^j(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\ &\quad+\frac{1}{\sqrt{T}}\sum_{0\leq s\leq T} g(X_{s-})\Delta X^j_s \mathopen{}\mathclose\bgroup\left(\mathbf{1}_{(\delta/\gamma_{\min},\infty)}(\Vert \Delta N_s\Vert)-\mathbf{1}_{(\delta,\infty)}(\Vert \Delta X_s\Vert)\aftergroup\egroup\right) \\&\quad- \frac{1}{\sqrt{T}}\int_0^Tg(X_{s-})\int_{B(0,\delta/\gamma_{\min})^{^{\operatorname{c}}}} \gamma^j(X_{s-})z\nu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} s \\ &\leq \mathbb{H}_T(gb^j)+\mathbb{M}^j_T(g)+\frac{1}{\sqrt{T}}\int_0^T\int_{B(0,\delta/\gamma_{\min})} g(X_{s-})\gamma^j(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\ &\quad+\frac{1}{\sqrt{T}}\sum_{0\leq s\leq T} \vert g(X_{s-})\Delta X^j_s\vert \mathopen{}\mathclose\bgroup\left\vert\mathbf{1}_{(\delta/\gamma_{\min},\infty)}(\Vert \Delta N_s\Vert)-\mathbf{1}_{(\delta,\infty)}(\Vert \Delta X_s\Vert)\aftergroup\egroup\right\vert \\&\quad- \frac{1}{\sqrt{T}}\int_0^Tg(X_{s-})\int_{B(0,\delta/\gamma_{\min})^{^{\operatorname{c}}}} \gamma^j(X_{s-})z\nu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} s \\ &\leq \mathbb{H}_T(gb^j)+\mathbb{M}^j_T(g)+\frac{1}{\sqrt{T}}\int_0^T\int_{B(0,\delta/\gamma_{\min})} g(X_{s-})\gamma^j(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\ &\quad+\frac{1}{\sqrt{T}}\sum_{0\leq s\leq T} \vert g(X_{s-})\Delta X^j_s\vert \mathbf{1}_{(\delta/\Vert \gamma\Vert_\infty,\delta/\gamma_{\min}]}(\Vert \Delta N_s\Vert) \\&\quad- \frac{1}{\sqrt{T}}\int_0^Tg(X_{s-})\int_{B(0,\delta/\gamma_{\min})^{^{\operatorname{c}}}} \gamma^j(X_{s-})z\nu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} s \\ &\leq \mathbb{H}_T(gb^j)+\mathbb{M}^j_T(g)+\frac{1}{\sqrt{T}}\int_0^T\int_{B(0,\delta/\gamma_{\min})} g(X_{s-})\gamma^j(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\ &\quad+\frac{\Vert \gamma\Vert_\infty}{\gamma_{\min}\sqrt{T}}\delta\int_0^T\int_{\delta/\Vert \gamma\Vert_\infty<\Vert z\Vert\leq \delta/\gamma_{\min}} \vert g(X_{s-})\vert N(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\ &\quad+ \frac{\nu_3\Vert \gamma\Vert_\infty\sigma^2_d}{\delta^2\sqrt{T}}\int_0^T\vert g(X_{s-})\vert \mathop{}\!\mathrm{d} s \\ &\leq \mathbb{H}_T(gb^j)+\mathbb{M}^j_T(g)+\frac{1}{\sqrt{T}}\int_0^T\int_{B(0,\delta/\gamma_{\min})} g(X_{s-})\gamma^j(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\ &\quad+\frac{\Vert \gamma\Vert_\infty}{\gamma_{\min}\sqrt{T}}\delta\int_0^T\int_{\delta/\Vert \gamma\Vert_\infty<\Vert z\Vert\leq \delta/\gamma_{\min}} \vert g(X_{s-})\vert \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\ &\quad+\frac{\Vert \gamma\Vert_\infty}{\gamma_{\min}\sqrt{T}}\delta\int_0^T\vert g(X_{s-})\vert \nu((\delta/\Vert \gamma\Vert_\infty, \delta/\gamma_{\min}])\mathop{}\!\mathrm{d} s \\ &\quad+ \frac{\nu_3\Vert \gamma\Vert_\infty\sigma^2_d}{\delta^2\sqrt{T}}\int_0^T\vert g(X_{s-})\vert \mathop{}\!\mathrm{d} s \\ &\leq \mathbb{H}_T(gb^j)+\mathbb{M}^j_T(g)+\frac{1}{\sqrt{T}}\int_0^T\int_{B(0,\delta/\gamma_{\min})} g(X_{s-})\gamma^j(X_{s-})z \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z) \\ &\quad+\frac{\Vert \gamma\Vert_\infty}{\gamma_{\min}\sqrt{T}}\delta\int_0^T\int_{\delta/\Vert \gamma\Vert_\infty<\Vert z\Vert\leq \delta/\gamma_{\min}} \vert g(X_{s-})\vert \widetilde{N}(\mathop{}\!\mathrm{d} s,\mathop{}\!\mathrm{d} z)+\frac{\Vert \gamma\Vert_\infty^4\nu_3}{\gamma_{\min}\delta^2\sqrt{T}}\int_0^T\vert g(X_{s-})\vert \mathop{}\!\mathrm{d} s \\ &\quad+ \frac{\nu_3\Vert \gamma\Vert_\infty\sigma^2_d}{\delta^2\sqrt{T}}\int_0^T\vert g(X_{s-})\vert \mathop{}\!\mathrm{d} s. \end{aligned}$$ The assertion now follows by the Minkowski inequality. ◻ *Proof of Proposition [Proposition 4](#prop: trunc chain){reference-type="ref" reference="prop: trunc chain"}.* Theorem 2.2 in [@applebaum09paper] gives together with Lemma [Lemma 14](#lemma: qv jump bound comp supp){reference-type="ref" reference="lemma: qv jump bound comp supp"} for any $T,x,y>0$ $$\begin{aligned} &2\exp(-xy) \\ &\geq \mathbb{P}\Bigg(\vert \mathbb{J}^{j,\delta,1}_T(g)\vert>\frac{x}{\sqrt{T}}\\&\quad\qquad+\frac{1}{\sqrt{T}y} \int_0^T\int_{B(0,\delta/\gamma_{\min})}\exp(yg(X_{s-})\gamma^j(X_{s-})z))-1-yg(X_{s-})\gamma^j(X_{s-})z)\nu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} s \Bigg) \\ &\geq \mathbb{P}\mathopen{}\mathclose\bgroup\left(\vert\mathbb{J}^{j,\delta,1}_T(g)\vert>\frac{x}{\sqrt{T}}+\frac{y}{\sqrt{T}}\mathcal{E}(\delta,y)\int_0^Tg(X_{s-})^2\mathop{}\!\mathrm{d} s \aftergroup\egroup\right), \end{aligned}$$ where $$\mathcal{E}(\delta,y)\coloneq \Vert \gamma\Vert_\infty^2\frac{\nu_2\exp(y\Vert \gamma\Vert_\infty\Vert g\Vert_\infty \delta/\gamma_{\min})}{2}.$$ Hence, applying Lemma [Lemma 12](#lemma: bernstein dirksen form){reference-type="ref" reference="lemma: bernstein dirksen form"} with $m_T=\sqrt{T}/(2\sqrt{\kappa})$ gives for large enough $T$ $$\begin{aligned} &\mathbb{P}\mathopen{}\mathclose\bgroup\left(\vert\mathbb{J}^{j,\delta,1}_T(g)\vert>\frac{x}{\sqrt{T}}+y\mathcal{E}(\delta,y)32\sqrt{u}\mathopen{}\mathclose\bgroup\left((T/\kappa)^{1/4}\sqrt{\mu(g^4)}+2\sqrt{u/\kappa}\Vert g^2\Vert_\infty \aftergroup\egroup\right)+y\sqrt{T}\mathcal{E}(\delta,y)\mu(g^2) \aftergroup\egroup\right)\notag \\ &\leq \mathbb{P}\mathopen{}\mathclose\bgroup\left(\vert\mathbb{J}^{j,\delta,1}_T(g)\vert>\frac{x}{\sqrt{T}}+\frac{y}{\sqrt{T}}\mathcal{E}(\delta,y)\int_0^T g(X_s)^2\mathop{}\!\mathrm{d} s \aftergroup\egroup\right) \notag \\& \quad+\mathbb{P}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{T}}\int_0^T g(X_s)\mathop{}\!\mathrm{d} s>32\sqrt{u}\mathopen{}\mathclose\bgroup\left(\sqrt{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)}+\sqrt{u/\kappa}\Vert g\Vert_\infty\aftergroup\egroup\right) \aftergroup\egroup\right)\notag \\ &\leq2\exp(-xy)+ 2\exp(-u)+2\sqrt{\kappa T}c_\kappa \exp(-\sqrt{\kappa T}/2)\mathbf{1}_{(u,\infty)}\mathopen{}\mathclose\bgroup\left(\frac{\sqrt{\kappa T}}{8}\aftergroup\egroup\right)\notag \\ &\leq2\exp(-xy)+2\exp(-u)+2\exp(-\sqrt{\kappa T}/8)\mathbf{1}_{(u,\infty)}\mathopen{}\mathclose\bgroup\left(\frac{\kappa\sqrt{T}}{8}\aftergroup\egroup\right)\notag \\ &\leq2\exp(-xy)+4\exp(-u)\label{eq: trunc bern} \end{aligned}$$ where we again used that for any $\tau>0$ $$\sqrt{\mathrm{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^\tau g^2(X_s)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right)}\leq \sqrt{\tau\mu(g^4)}.$$ Now for $u>0,$ setting $x_u=u\mathopen{}\mathclose\bgroup\left(\sqrt{T\mu(g^2)/u} +T^{3/8}\Vert g\Vert_{L^4(\mu)}+T^{1/4}\Vert g\Vert_\infty\aftergroup\egroup\right),y_u=u/x_u$ gives $$\begin{aligned} &\frac{x_u}{\sqrt{T}}+y_u\mathcal{E}(\delta,y_u)32\sqrt{u}\mathopen{}\mathclose\bgroup\left((T/\kappa)^{1/4}\sqrt{\mu(g^4)}+2\sqrt{u/\kappa}\Vert g^2\Vert_\infty \aftergroup\egroup\right)+y_u\sqrt{T}\mathcal{E}(\delta,y_u)\mu(g^2) \\ &\leq \sqrt{u}\mathopen{}\mathclose\bgroup\left(\Vert g\Vert_{L^2(\mu)} +\sqrt{u}\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\Vert g\Vert_\infty\aftergroup\egroup\right)\aftergroup\egroup\right) \\&\quad+y_u\mathcal{E}(\delta,y_u)32\sqrt{u}\mathopen{}\mathclose\bgroup\left((T/\kappa)^{1/4}\sqrt{\mu(g^4)}+2\sqrt{u/\kappa}\Vert g^2\Vert_\infty \aftergroup\egroup\right)+y_u\sqrt{T}\mathcal{E}(\delta,y_u)\mu(g^2) \\ &\leq \sqrt{u}\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}(\delta,y_u))\Vert g\Vert_{L^2(\mu)}+32\mathcal{E}(\delta,y_u)T^{-1/8}\kappa^{-1/4}\Vert g\Vert_{L^4(\mu)} \aftergroup\egroup\right) \\&\quad+u\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}(\delta,y_u)\kappa^{-1/2} \aftergroup\egroup\right)\Vert g\Vert_\infty\aftergroup\egroup\right). \end{aligned}$$ Arguing as in the derivation of [\[eq: conc jump\]](#eq: conc jump){reference-type="eqref" reference="eq: conc jump"} this implies together with [\[eq: trunc bern\]](#eq: trunc bern){reference-type="eqref" reference="eq: trunc bern"} and $$\mathcal{E}(\delta,y_u)\leq \Vert \gamma\Vert_\infty^2\frac{\nu_2\exp(\alpha\Vert \gamma\Vert_\infty \sigma^{-1}_d)}{2}\eqcolon \mathcal{E}_\alpha,$$ for $\delta\leq \alpha T^{1/4},$ $$\begin{aligned} &\mathbb{P}\Big(\vert\mathbb{J}^{j,\delta,1}_T(g)\vert>\sqrt{3u}\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)\Vert g\Vert_{L^2(\mu)}+32\mathcal{E}_\alpha T^{-1/8}\kappa^{-1/4}\Vert g\Vert_{L^4(\mu)} \aftergroup\egroup\right) \\&\qquad+3u\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)\Vert g\Vert_\infty\aftergroup\egroup\right)\Big)\notag \\ &\leq2\exp(-u), \end{aligned}$$ which allows us to apply Theorem 3.5 and Lemma A.2 of [@dirksen15]. This gives for any $p\geq 1$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}}\vert \mathbb{J}^{j,\delta,1}_T\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G},3\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)\Vert g\Vert_\infty\aftergroup\egroup\right)\aftergroup\egroup\right) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G},\sqrt{3}\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)\Vert g\Vert_{L^2(\mu)}+32\mathcal{E}_\alpha T^{-1/8}\kappa^{-1/4}\Vert g\Vert_{L^4(\mu)} \aftergroup\egroup\right)\aftergroup\egroup\right)\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+2\sup_{g\in\mathcal{G}}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\vert \mathbb{J}^{j,\delta,1}_T(g)\vert^p \aftergroup\egroup\right]^{1/p} \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G},3\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)\Vert g\Vert_\infty\aftergroup\egroup\right)\aftergroup\egroup\right) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G},\sqrt{3}\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)\Vert g\Vert_{L^2(\mu)}+32\mathcal{E}_\alpha T^{-1/8}\kappa^{-1/4}\Vert g\Vert_{L^4(\mu)} \aftergroup\egroup\right)\aftergroup\egroup\right)\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+\sup_{g\in\mathcal{G}}\Bigg(3p\widetilde{c}_1\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)\Vert g\Vert_{\infty}\aftergroup\egroup\right) \\&\quad+\sqrt{3p}\widetilde{c}_2\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)\Vert g\Vert_{L^2(\mu)}+32\mathcal{E}_\alpha T^{-1/8}\kappa^{-1/4}\Vert g\Vert_{L^4(\mu)} \aftergroup\egroup\right) \Bigg), \end{aligned}$$ concluding the proof of the first assertion. For the proof of the second assertion, Theorem 2.2 in [@applebaum09paper] gives for any $T,x,y>0,$ together with Taylor's theorem $$\begin{aligned} &2\exp(-xy) \\ &\geq \mathbb{P}\mathopen{}\mathclose\bgroup\left(\vert \mathbb{J}^{\delta,2}_T(g)\vert>\frac{x}{\sqrt{T}}+\frac{1}{\sqrt{T}y} \int_0^T\int_{\delta/\Vert \gamma\Vert_\infty<\Vert z\Vert\leq \delta/\gamma_{\min}}\exp(yg(X_{s-}))-1-yg(X_{s-})\nu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} s \aftergroup\egroup\right) \\ &\geq \mathbb{P}\mathopen{}\mathclose\bgroup\left(\vert\mathbb{J}^{\delta,2}_T(g)\vert>\frac{x}{\sqrt{T}}+\frac{y}{2\sqrt{T}}\exp(y\Vert g\Vert_\infty)\nu(B(0,\delta/\Vert \gamma\Vert_\infty)^{^{\operatorname{c}}})\int_0^Tg(X_{s-})^2\mathop{}\!\mathrm{d} s \aftergroup\egroup\right) \\ &\geq \mathbb{P}\mathopen{}\mathclose\bgroup\left(\vert\mathbb{J}^{\delta,2}_T(g)\vert>\frac{x}{\sqrt{T}}+\frac{\nu_2\Vert \gamma\Vert_\infty^2 y}{2\delta^2\sqrt{T}}\exp(y\Vert g\Vert_\infty)\int_0^Tg(X_{s-})^2\mathop{}\!\mathrm{d} s \aftergroup\egroup\right). \end{aligned}$$ Hence arguing as above, Lemma [Lemma 12](#lemma: bernstein dirksen form){reference-type="ref" reference="lemma: bernstein dirksen form"} gives with $m_T=\sqrt{T}(2\sqrt{\kappa})$ that for large enough $T$ it holds for any $u>0$ $$\begin{aligned} &\mathbb{P}\Big(\vert\mathbb{J}^{\delta,2}_T(g)\vert>\frac{x}{\sqrt{T}}+16\frac{\nu_2\Vert \gamma\Vert_\infty^2 y}{\delta^2}\exp(y\Vert g\Vert_\infty)\sqrt{u}\mathopen{}\mathclose\bgroup\left((T/\kappa)^{1/4}\sqrt{\mu(g^4)}+2\sqrt{u/\kappa}\Vert g^2\Vert_\infty \aftergroup\egroup\right) \\&\qquad +\sqrt{T}\frac{\nu_2\Vert \gamma\Vert_\infty^2 y}{2\delta^2}\exp(y\Vert g\Vert_\infty)\mu(g^2) \Big) \\ &\leq2\exp(-xy)+ 4\exp(-u), \end{aligned}$$ and choosing $x_u=u\delta^{-1}\mathopen{}\mathclose\bgroup\left(\sqrt{T\mu(g^2)/u} +T^{3/8}\Vert g\Vert_{L^4(\mu)}+T^{1/4}\Vert g\Vert_\infty\aftergroup\egroup\right),y_u=u/x_u$ gives for large enough $T$ $$\begin{aligned} &\frac{x_u}{\sqrt{T}}+16\frac{\nu_2\Vert \gamma\Vert_\infty^2 y_u}{\delta^2}\exp(y_u\Vert g\Vert_\infty)\sqrt{u}\mathopen{}\mathclose\bgroup\left((T/\kappa)^{1/4}\sqrt{\mu(g^4)}+2\sqrt{u/\kappa}\Vert g^2\Vert_\infty \aftergroup\egroup\right) \\&\quad+\frac{\nu_2\Vert \gamma\Vert_\infty^2 y_u}{2\delta^2\sqrt{T}}\exp(y_u\Vert g\Vert_\infty)\mu(g^2) \\ &\leq\delta^{-1}\Vert g\Vert_{L^2(\mu)}\sqrt{u} +u\delta^{-1}\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\Vert g\Vert_\infty\aftergroup\egroup\right) \\&\quad+16\exp(\alpha)\frac{\nu_2\Vert \gamma\Vert_\infty^2 y_u}{\delta^2}\sqrt{u}\mathopen{}\mathclose\bgroup\left((T/\kappa)^{1/4}\sqrt{\mu(g^4)}+2\sqrt{u/\kappa}\Vert g^2\Vert_\infty \aftergroup\egroup\right) +\sqrt{T}\frac{\nu_2\Vert \gamma\Vert_\infty^2 y_u}{2\delta^2}\exp(\alpha)\mu(g^2) \\ &\leq\frac{\sqrt{u}}{\delta}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(1+\frac{\nu_2\Vert \gamma\Vert_\infty^2}{2}\exp(\alpha)\aftergroup\egroup\right)\Vert g\Vert_{L^2(\mu)}+16\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right) \\&\quad+\frac{u}{\delta}\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\Vert g\Vert_\infty\mathopen{}\mathclose\bgroup\left(1+32\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/2}\aftergroup\egroup\right) \aftergroup\egroup\right). \end{aligned}$$ Now following the same steps as in the proof of the first assertion concludes the proof. ◻ # Proofs for Section [4](#sec: drift est){reference-type="ref" reference="sec: drift est"}. {#proofs-for-section-sec-drift-est.} *Proof for Theorem [\[thm: general rate exp\]](#thm: general rate exp){reference-type="ref" reference="thm: general rate exp"}.* For notational convenience, we suppress the dependence on $j$ throughout this proof. Denoting $$d^2_{\mathbb{G},\tau}(f_1,f_2)=\Vert f_1-f_2\Vert_{\mathbb{G},\tau}^2= \operatorname{Var}\mathopen{}\mathclose\bgroup\left(\frac{1}{\sqrt{\tau}}\int_0^{\tau} (f_1-f_2)(X_s)\mathop{}\!\mathrm{d} s\aftergroup\egroup\right),\quad f_1,f_2\in L^2(\mu),$$ and $\Vert b\Vert_{D,\infty}=\sup_{x: d(x,D)\leq1}\vert b(x)\vert,$ Theorem 3.1 in [@dexheimeraihp] gives by choosing $m_T=(p/\kappa)\log(T),$ that there exists $\tau\in [m_t,2m_T],$ such that for large enough $T$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{H}_T(gb)-\sqrt{T}\mu(gb)\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq C_1\mathopen{}\mathclose\bgroup\left(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}}b,\frac{2p\log(T)}{\kappa\sqrt{T}}d_\infty\aftergroup\egroup\right)\mathop{}\!\mathrm{d} u\aftergroup\egroup\right)+\int_0^\infty \sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}}b,d_{\mathbb{G},\tau }\aftergroup\egroup\right)\aftergroup\egroup\right)} \aftergroup\egroup\right)\mathop{}\!\mathrm{d} u \\&\quad +4\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\mathopen{}\mathclose\bgroup\left(\widetilde{c}_1\frac{2p}{\kappa\sqrt{T}}\Vert gb\Vert_\infty +\widetilde{c}_2\sqrt{p}\Vert gb\Vert_{\mathbb{G},\tau}+\frac{1}{2\sqrt{T}}\Vert gb\Vert_\infty \aftergroup\egroup\right) \\ &\leq C_1\mathopen{}\mathclose\bgroup\left(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}}b,\frac{2p\log(T)}{\kappa\sqrt{T}}d_\infty\aftergroup\egroup\right) \aftergroup\egroup\right)\mathop{}\!\mathrm{d} u+\int_0^\infty \sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}}b,c V(\boldsymbol{h})\psi_d(V(\boldsymbol{h}))d_\infty\aftergroup\egroup\right)\aftergroup\egroup\right)} \mathop{}\!\mathrm{d} u \aftergroup\egroup\right) \\&\quad +4\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\mathopen{}\mathclose\bgroup\left(\widetilde{c}_1\frac{2p}{\kappa\sqrt{T}}\Vert gb\Vert_\infty +c\sqrt{ p}\Vert gb\Vert_\infty V(\boldsymbol{h})\psi_d(V(\boldsymbol{h}))+\frac{1}{2\sqrt{T}}\Vert gb\Vert_\infty \aftergroup\egroup\right) \\ &\leq c\Bigg(\frac{p\log(T)}{\sqrt{T}}\int_0^{ 2\Vert b\Vert_{D,\infty} \Vert K\Vert_\infty}\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}}b,d_\infty\aftergroup\egroup\right) \aftergroup\egroup\right)\mathop{}\!\mathrm{d} u\\&\quad+ V(\boldsymbol{h})\psi_d(V(\boldsymbol{h}))\int_0^{2\Vert b\Vert_{D,\infty} \Vert K\Vert_\infty} \sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}}b,d_\infty\aftergroup\egroup\right)\aftergroup\egroup\right)} \mathop{}\!\mathrm{d} u \Bigg) +c\sqrt{\log(T)} V(\boldsymbol{h})\psi_d(V(\boldsymbol{h})), \end{aligned}$$ where we used Proposition 2.5 in [@dexheimeraihp] for bounding the variance distance with $$\label{def: psi} \psi_d(x)\coloneq \begin{cases} \sqrt{1-\log(x)},& d\leq 2,\\ x^{1/d-1/2}, & d\geq 3. \end{cases}$$ For the bound of the variance distance in the case $d=1$ note that by slightly adapting the proof for $d=2$ in Proposition 2.5 of [@dexheimeraihp] the same result as in $d=2$ can be achieved. As the proof of Lemma D.2 in [@aeckdriftsupp] gives for any $u>0$ $$\begin{aligned} \label{eq: covering number bound} \mathcal{N}(u,\mathcal{G}_{\boldsymbol{h}}b,d_\infty)\leq \mathopen{}\mathclose\bgroup\left(\frac{2L\Vert b\Vert_{D,\infty}\operatorname{diam}(D)\sum_{i=1}^{d}h_i^{-1} }{u} \aftergroup\egroup\right)^d, \end{aligned}$$ where $L$ is the Lipschitz constant of $K,$ we get for large enough $T$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{H}_T(gb)-\sqrt{T}\mu(gb)\vert^p\aftergroup\egroup\right]^{1/p}\notag \\ &\leq c\Bigg(\frac{\log(T)}{\sqrt{T}} \log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1} \aftergroup\egroup\right)^2 + V(\boldsymbol{h})\psi_d(V(\boldsymbol{h}))\sqrt{\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1} \aftergroup\egroup\right) }+\sqrt{\log(T) }\notag V(\boldsymbol{h})\psi_d(V(\boldsymbol{h})) \Bigg) \\ &\leq V(\boldsymbol{h})^{1/2}, \label{eq: moment bound dens} \end{aligned}$$ where we used the inequality $$\label{eq: sqrt entropy ineq} \int_0^C\sqrt{\log(M/u)}\mathop{}\!\mathrm{d} u\leq 4 C\sqrt{\log(M/C)},\quad\mathrm{if}\quad \log(M/C)\geq 2$$ (see p. 592 of [@gine2015]). Furthermore, Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"} gives together with [\[eq: covering number bound\]](#eq: covering number bound){reference-type="eqref" reference="eq: covering number bound"} for large enough $T$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\vert \mathbb{M}_T(g)\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G}_{\boldsymbol{h}},24\sqrt{\Vert a\Vert_{\infty}}((\kappa T)^{-1/4}d_\infty+(\kappa T)^{-1/8}d_{L^4(\mu)})) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G}_{\boldsymbol{h}},\sqrt{6\Vert a\Vert_{\infty}} (128^{1/4}(\kappa T)^{-1/8}d_{L^4(\mu)}+d_{L^2(\mu)}))\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\&\quad + 24p\sqrt{\Vert a_{jj}\Vert_\infty}\widetilde{c}_1\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\mathopen{}\mathclose\bgroup\left((\kappa T)^{-1/4}\Vert g\Vert_\infty+(\kappa T)^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right) \\&\quad+\sqrt{6\Vert a_{jj}\Vert_\infty p}\widetilde{c}_2\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\mathopen{}\mathclose\bgroup\left((128)^{1/4}(\kappa T)^{-1/8}\Vert g\Vert_{L^4(\mu)}+\Vert g\Vert_{L^2(\mu)}\aftergroup\egroup\right), \\ &\leq C_1\Bigg(24\sqrt{\Vert a\Vert_{\infty}}((\kappa T)^{-1/4}+(\kappa T)^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4})\int_0^{2\Vert K\Vert_\infty} \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G}_{\boldsymbol{h}},d_\infty) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\sqrt{6\Vert a\Vert_{\infty}} (128^{1/4}(\kappa T)^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}+(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2})\int_0^{2\Vert K\Vert_\infty}\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G}_{\boldsymbol{h}},d_\infty)\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\&\quad + 24p\sqrt{\Vert a_{jj}\Vert_\infty}\widetilde{c}_1\Vert K\Vert_\infty\mathopen{}\mathclose\bgroup\left((\kappa T)^{-1/4}+(\kappa T)^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}\aftergroup\egroup\right) \\&\quad+\sqrt{6\Vert a_{jj}\Vert_\infty p}\widetilde{c}_2\Vert K\Vert_\infty\mathopen{}\mathclose\bgroup\left((128)^{1/4}(\kappa T)^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}+(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}\aftergroup\egroup\right) \\ &\leq C_1\Bigg(d\sqrt{\Vert a\Vert_{\infty}\Vert \rho\Vert_{\infty}V(\boldsymbol{h})}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{-1/2}\int_0^{2\Vert K\Vert_\infty} \log\mathopen{}\mathclose\bgroup\left(\frac{2L\sum_{i=1}^{d}h_i^{-1} \operatorname{diam}(D)}{u} \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +3\sqrt{d\Vert a\Vert_{\infty}\Vert \rho\Vert_{\infty}V(\boldsymbol{h})} \int_0^{2\Vert K\Vert_\infty}\sqrt{\log\mathopen{}\mathclose\bgroup\left(\frac{2L\sum_{i=1}^{d}h_i^{-1} \operatorname{diam}(D)}{u} \aftergroup\egroup\right)}\mathop{}\!\mathrm{d} u\Bigg) \\&\quad +4\sqrt{\theta\Vert a_{jj}\Vert_\infty\Vert \rho\Vert_{\infty} }\widetilde{c}_2\Vert K\Vert_\infty V(\boldsymbol{h})^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2} \\ &\leq C_1\Bigg(4d\Vert K\Vert_\infty\sqrt{\Vert a\Vert_{\infty}\Vert \rho\Vert_{\infty}V(\boldsymbol{h})}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2} +25\Vert K\Vert_\infty\sqrt{d\Vert a\Vert_{\infty}\Vert \rho\Vert_{\infty}V(\boldsymbol{h})}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1} \aftergroup\egroup\right)^{1/2}\Bigg) \\&\quad+4\sqrt{\theta\Vert a_{jj}\Vert_\infty\Vert \rho\Vert_{\infty} }\widetilde{c}_2\Vert K\Vert_\infty V(\boldsymbol{h})^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2} \\ &\leq 29C_1d\Vert K\Vert_\infty\sqrt{\Vert a\Vert_{\infty}\Vert \rho\Vert_{\infty}V(\boldsymbol{h})}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2} +4\sqrt{\theta\Vert a_{jj}\Vert_\infty\Vert \rho\Vert_{\infty} }\widetilde{c}_2\Vert K\Vert_\infty V(\boldsymbol{h})^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2}\stepcounter{equation}\tag{\theequation}\label{eq: moment bound cont} \end{aligned}$$ where we used $V(\boldsymbol{h})\geq T^{-1/2}\log(\sum_{i=1}^d h_i^{-1})^4$ and [\[eq: sqrt entropy ineq\]](#eq: sqrt entropy ineq){reference-type="eqref" reference="eq: sqrt entropy ineq"}. Turning our attention to the jump part, we obtain for large enough $T$ by Proposition [Proposition 2](#prop: jump moment bound){reference-type="ref" reference="prop: jump moment bound"} $$\begin{aligned} &\mathbb{E}\Big[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\vert \mathbb{J}^j_T(g)\vert^p\Big]^{1/p} \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},3T^{-1/8}d_{L^4(\mu)}+3(64c_1\kappa^{-1/2}+1)T^{-1/4}d_\infty) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},\sqrt{3}(1+c_1)d_{L^2(\mu)}+\sqrt{3072}c_1\kappa^{-1/4}T^{-1/8}d_{L^4(\mu)}) \aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+3p\widetilde{c}_1\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+(64c_1\kappa^{-1/2}+1)T^{-1/4}\Vert g\Vert_\infty \aftergroup\egroup\right) \\&\quad+\sqrt{3p}\widetilde{c}_2\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\mathopen{}\mathclose\bgroup\left((1+c_1)\Vert g\Vert_{L^2(\mu)}+32c_1\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right) \\ &\leq C_1\Bigg((3T^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}+3(64c_1\kappa^{-1/2}+1)T^{-1/4})\int_0^{2\Vert K\Vert_\infty} \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},d_\infty) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +(\sqrt{3}(1+c_1)(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}+\sqrt{3072}c_1\kappa^{-1/4}T^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4})\int_0^{2\Vert K\vert_\infty}\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},d_\infty \aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+3p\widetilde{c}_1\Vert K\Vert_\infty\mathopen{}\mathclose\bgroup\left(T^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}+(64c_1\kappa^{-1/2}+1)T^{-1/4} \aftergroup\egroup\right) \\&\quad+\sqrt{3p}\widetilde{c}_2\Vert K\Vert_\infty\mathopen{}\mathclose\bgroup\left((1+c_1)(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}+32c_1\kappa^{-1/4}T^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}\aftergroup\egroup\right) \\ &\leq C_1\Bigg((\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{-1/2}\int_0^{2\Vert K\Vert_\infty} \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},d_\infty) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +2(1+c_1)(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}\int_0^{2\Vert K\Vert_\infty}\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}(u,\mathcal{G},d_\infty \aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\&\quad+3\sqrt{\Vert \rho\Vert_{\infty}\theta}\widetilde{c}_2\Vert K\Vert_\infty(1+c_1)V(\boldsymbol{h})^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2} \\ &\leq C_1\Bigg(4\Vert K\Vert_\infty (\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2} +17\Vert K\Vert_\infty(1+c_1)(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2} \Bigg) \\&\quad+3\sqrt{\Vert \rho\Vert_{\infty}\theta}\widetilde{c}_2\Vert K\Vert_\infty(1+c_1)V(\boldsymbol{h})^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2} \\ &\leq C_1\Vert K\Vert_\infty (\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2} (4 +17(1+c_1) )\notag \\ &\quad +3\sqrt{\Vert \rho\Vert_{\infty}\theta}\widetilde{c}_2\Vert K\Vert_\infty(1+c_1)V(\boldsymbol{h})^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2}, \stepcounter{equation}\tag{\theequation}\label{eq: moment bound jump} \end{aligned}$$ where we argued analogously to the derivation of [\[eq: moment bound cont\]](#eq: moment bound cont){reference-type="eqref" reference="eq: moment bound cont"}. Combining [\[eq: moment bound dens\]](#eq: moment bound dens){reference-type="eqref" reference="eq: moment bound dens"}, [\[eq: moment bound cont\]](#eq: moment bound cont){reference-type="eqref" reference="eq: moment bound cont"} and [\[eq: moment bound jump\]](#eq: moment bound jump){reference-type="eqref" reference="eq: moment bound jump"} with the Minkowski inequality then shows $$\begin{aligned} \label{eq: moment adap} \mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{I}^j_T(g)-\sqrt{T}\mu(gb^j)\vert^p\aftergroup\egroup\right]^{1/p}\leq cV(\boldsymbol{h})^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2}, \end{aligned}$$ which completes the proof of the first assertion as $$\begin{aligned} \mathcal{R}^{(p)}_\infty(b^j\rho,\bar{b}^{(j)}_{\boldsymbol{h},T};D)& \leq \mathcal{B}_{b^j\rho}(\boldsymbol{h})+T^{-1/2}V(\boldsymbol{h})^{-1}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{I}^j_T(g)-\sqrt{T}\mu(gb^j)\vert^p\aftergroup\egroup\right]^{1/p}. \end{aligned}$$ ◻ *Proof of Theorem [\[thm: rate trunc est\]](#thm: rate trunc est){reference-type="ref" reference="thm: rate trunc est"}.* Firstly, it holds $$\begin{aligned} \mathcal{R}^{(p)}_\infty(b^j\rho,\bar{b}^{(j),\delta}_{\boldsymbol{h},T};D)& \leq \mathcal{B}_{b^j\rho}(\boldsymbol{h})+T^{-1/2}V(\boldsymbol{h})^{-1}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{I}^{j,\delta}_T(g)-\sqrt{T}\mu(gb^j)\vert^p\aftergroup\egroup\right]^{1/p}. \end{aligned}$$ and Lemma [Lemma 3](#prop: trunc decomp){reference-type="ref" reference="prop: trunc decomp"} gives $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{I}^{j,\delta}_T(g)-\sqrt{T}\mu(gb^j)\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq \mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{H}_T(gb^j)-\sqrt{T}\mu(gb^j)\vert^p\aftergroup\egroup\right]^{1/p}+\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\vert \mathbb{M}^j_T(g)\vert^p\aftergroup\egroup\right]^{1/p}+\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\vert \mathbb{J}^{j,\delta,1}_T(g)\vert^p\aftergroup\egroup\right]^{1/p} \\ &\quad+\frac{\Vert \gamma\Vert_\infty\delta}{\gamma_{\min}}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\vert \mathbb{J}^{\delta,2}_T(g)\vert^p\aftergroup\egroup\right]^{1/p}+\frac{c_2}{\delta^2}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{H}_T(\vert g\vert)-\sqrt{T}\mu(\vert g\vert)\vert^p\aftergroup\egroup\right]^{1/p}+\frac{c_2\sqrt{T}}{\delta^2}\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\mu(\vert g\vert). \end{aligned}$$ Now Proposition [Proposition 4](#prop: trunc chain){reference-type="ref" reference="prop: trunc chain"} yields for large enough $T$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\vert \mathbb{J}^{j,\delta,1}_T\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}},3\mathopen{}\mathclose\bgroup\left(T^{-1/8}d_{L^4(\mu)}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)d_\infty\aftergroup\egroup\right)\aftergroup\egroup\right) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}},\sqrt{3}\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)d_{L^2(\mu)}+32\mathcal{E}_\alpha T^{-1/8}\kappa^{-1/4}d_{L^4(\mu)} \aftergroup\egroup\right)\aftergroup\egroup\right)\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\Bigg(3p\widetilde{c}_1\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)\Vert g\Vert_{\infty}\aftergroup\egroup\right) \\&\quad+\sqrt{3p}\widetilde{c}_2\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)\Vert g\Vert_{L^2(\mu)}+32\mathcal{E}_\alpha T^{-1/8}\kappa^{-1/4}\Vert g\Vert_{L^4(\mu)} \aftergroup\egroup\right) \Bigg) \\ &\leq C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}},3d_\infty\mathopen{}\mathclose\bgroup\left(T^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)\aftergroup\egroup\right)\aftergroup\egroup\right) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}},\sqrt{3}d_\infty\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}+32\mathcal{E}_\alpha T^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}\kappa^{-1/4} \aftergroup\egroup\right)\aftergroup\egroup\right)\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\Bigg(3p\widetilde{c}_1\mathopen{}\mathclose\bgroup\left(T^{-1/8}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}\Vert g\Vert_\infty+T^{-1/4}\mathopen{}\mathclose\bgroup\left(1+64\mathcal{E}_\alpha\kappa^{-1/2} \aftergroup\egroup\right)\Vert g\Vert_{\infty}\aftergroup\egroup\right) \\&\quad+\sqrt{3p}\widetilde{c}_2\mathopen{}\mathclose\bgroup\left((1+\mathcal{E}_\alpha)(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}\Vert g\Vert_\infty+32\mathcal{E}_\alpha T^{-1/8}\kappa^{-1/4}(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/4}\Vert g\Vert_{\infty} \aftergroup\egroup\right) \Bigg) \\ &\leq C_1\Bigg(4(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}\log(\sum_{i=1}^d h_i^{-1})^{-1}\int_0^{2\Vert K\Vert_\infty} \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}},d_\infty\aftergroup\egroup\right)\aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +3(1+\mathcal{E}_\alpha)(\Vert \rho\Vert_{\infty}V(\boldsymbol{h}))^{1/2}\int_0^{2\Vert K\Vert_\infty}\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}},d_\infty\aftergroup\egroup\right)\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\&\quad+3\sqrt{\Vert \rho\Vert_{\infty}\theta\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)}\widetilde{c}_2(1+\mathcal{E}_\alpha)V(\boldsymbol{h})^{1/2}\Vert K\Vert_\infty \\ &\leq C_1\Bigg(5dV(\boldsymbol{h})^{1/2}\log(\sum_{i=1}^d h_i^{-1})^{-1}\int_0^{2\Vert K\Vert_\infty} \log\mathopen{}\mathclose\bgroup\left(\frac{\sum_{i=1}^{d}h_i^{-1} }{u}\aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +4\sqrt{d}(1+\mathcal{E}_\alpha)V(\boldsymbol{h})^{1/2}\int_0^{2\Vert K\Vert_\infty}\sqrt{\log\mathopen{}\mathclose\bgroup\left(\frac{\sum_{i=1}^{d}h_i^{-1}}{u}\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\&\quad+3\sqrt{\theta\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)}\widetilde{c}_2(1+\mathcal{E}_\alpha)V(\boldsymbol{h})^{1/2}\Vert K\Vert_\infty \\ &\leq V(\boldsymbol{h})^{1/2}\sqrt{\Vert \rho\Vert_{\infty}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1}\aftergroup\egroup\right) }\Vert K\Vert_\infty(1+\mathcal{E}_\alpha)\mathopen{}\mathclose\bgroup\left(64C_1 \sqrt{d}+3\sqrt{\theta}\widetilde{c}_2\aftergroup\egroup\right), \end{aligned}$$ where we argued analogously to the derivation of [\[eq: moment bound cont\]](#eq: moment bound cont){reference-type="eqref" reference="eq: moment bound cont"}. Similarly it also follows by Proposition [Proposition 4](#prop: trunc chain){reference-type="ref" reference="prop: trunc chain"} that for large enough $T$ $$\begin{aligned} &\delta\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\vert \mathbb{J}^{\delta,2}_T(g)\vert^p\aftergroup\egroup\right]^{1/p} \\ &\leq \delta C_1\Bigg(\int_0^\infty \log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G},3\delta^{-1}\mathopen{}\mathclose\bgroup\left(T^{-1/8}d_{L^4(\mu)}+T^{-1/4}d_\infty\mathopen{}\mathclose\bgroup\left(1+32\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/2}\aftergroup\egroup\right) \aftergroup\egroup\right)\aftergroup\egroup\right) \aftergroup\egroup\right) \mathop{}\!\mathrm{d} u \\ &\quad +\int_0^\infty\sqrt{\log\mathopen{}\mathclose\bgroup\left(\mathcal{N}\mathopen{}\mathclose\bgroup\left(u,\mathcal{G}_{\boldsymbol{h}},\sqrt{3}\delta^{-1}\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(1+\frac{\nu_2\Vert \gamma\Vert_\infty^2}{2}\exp(\alpha)\aftergroup\egroup\right)d_{L^2(\mu)}+16\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/4}T^{-1/8}d_{L^4(\mu)}\aftergroup\egroup\right)\aftergroup\egroup\right)\aftergroup\egroup\right) }\mathop{}\!\mathrm{d} u\Bigg) \\ &\quad+\sup_{g\in\mathcal{G}_{\boldsymbol{h}}}\Bigg(3p\widetilde{c}_1\mathopen{}\mathclose\bgroup\left(T^{-1/8}\Vert g\Vert_{L^4(\mu)}+T^{-1/4}\Vert g\Vert_\infty\mathopen{}\mathclose\bgroup\left(1+32\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/2}\aftergroup\egroup\right) \aftergroup\egroup\right) \\&\quad+\sqrt{3p}\widetilde{c}_2\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(1+\frac{\nu_2\Vert \gamma\Vert_\infty^2}{2}\exp(\alpha)\aftergroup\egroup\right)\Vert g\Vert_{L^2(\mu)}+16\exp(\alpha)\nu_2\Vert \gamma\Vert_\infty^2\kappa^{-1/4}T^{-1/8}\Vert g\Vert_{L^4(\mu)}\aftergroup\egroup\right) \Bigg) \\ &\leq V(\boldsymbol{h})^{1/2}\sqrt{\Vert \rho\Vert_{\infty}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1}\aftergroup\egroup\right) }\Vert K\Vert_\infty\mathopen{}\mathclose\bgroup\left(1+\frac{\nu_2\Vert \gamma\Vert_\infty^2}{2}\exp(\alpha)\aftergroup\egroup\right)\mathopen{}\mathclose\bgroup\left(64\sqrt{d}C_1+3\sqrt{\theta}\widetilde{c}_2\aftergroup\egroup\right). \end{aligned}$$ Arguing as in the proof of Theorem [\[thm: general rate exp\]](#thm: general rate exp){reference-type="ref" reference="thm: general rate exp"}, i.e. applying Proposition [Proposition 1](#prop: cont moment bound){reference-type="ref" reference="prop: cont moment bound"} and Theorem 3.1 in [@dexheimer2020mixing] now concludes the proof. ◻ *Proof of Corollary [Corollary 7](#cor: rate){reference-type="ref" reference="cor: rate"}.* As the proofs of [a)](#cor: a) and [b)](#cor: b) are analogous, we only give the proof for the first assertion. Arguing as in the derivation of [\[eq: moment bound dens\]](#eq: moment bound dens){reference-type="eqref" reference="eq: moment bound dens"} we obtain that for fixed $\alpha>0$ and large enough $T$ it holds for any $p\in[1,\alpha \log(T)]$ $$\begin{aligned} &\mathbb{E}[\Vert \widehat{\rho}_{\boldsymbol{h}^\rho,T}-\mathbb{E}[\widehat{\rho}_{\boldsymbol{h}^\rho,T}]\Vert_{L^\infty(D)}^p]^{1/p} \\ &\leq cT^{-1/2}V(\boldsymbol{h}^\rho)^{-1}\mathopen{}\mathclose\bgroup\left(\log(T)^3T^{-1/2}+\log(T)^{1/2}V(\boldsymbol{h}^\rho)\psi_d(V(\boldsymbol{h}^\rho))\aftergroup\egroup\right), \end{aligned}$$ where $\psi_d$ is defined in [\[def: psi\]](#def: psi){reference-type="eqref" reference="def: psi"}. Together with the standard bias bound under Hölder smoothness assumptions (see, e.g. Proposition 1 in [@comte2013]) and the specific choice of $\boldsymbol{h}^\rho$, this yields for large enough $T,$ that for any $p\in[1,\alpha\log(T)]$ $$\begin{aligned} &\mathbb{E}[\Vert \widehat{\rho}_{\boldsymbol{h}^\rho,T}-\rho\Vert_{L^\infty(D)}^p]^{1/p}\notag \\ &\leq c\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}(h_i^\rho)^{\beta_i}+T^{-1/2}V(\boldsymbol{h}^\rho)^{-1}\mathopen{}\mathclose\bgroup\left(\log(T)^3T^{-1/2}+\log(T)^{1/2}V(\boldsymbol{h}^\rho)\psi_d(V(\boldsymbol{h}^\rho))\aftergroup\egroup\right)\aftergroup\egroup\right)\notag \\ &\leq c\Phi_{d,\boldsymbol{\beta}}(T),\label{eq: cor 2} \end{aligned}$$ where we also used $\bar{\boldsymbol{\beta}}>d\land 2$. Thus for large enough $T,$ Markov's inequality entails for the event $B_T\coloneq\{\Vert \widehat{\rho}_{\boldsymbol{h}^\rho,T}-\rho\Vert_{L^\infty(D)}\leq r(T) \}$ that for any $p\in [1,\alpha\log(T)]$ $$\mathbb{P}(B_T^\mathsf{c})\leq c^p \exp(- p\sqrt{\log(T)}).$$ Hence we obtain if [\[ass: exp mom\]](#ass: exp mom){reference-type="ref" reference="ass: exp mom"} holds $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\Vert(\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}-b^j)\rho\Vert_{L^\infty(D)}\mathbf{1}_{B^\mathsf{c}_T} \aftergroup\egroup\right] \\ &\leq \mathbb{E}\mathopen{}\mathclose\bgroup\left[\Vert\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}\rho\Vert_{L^\infty(D)}\mathbf{1}_{B^\mathsf{c}_T} \aftergroup\egroup\right]+\mathbb{E}\mathopen{}\mathclose\bgroup\left[\Vert b^j\rho\Vert_{L^\infty(D)}\mathbf{1}_{B^\mathsf{c}_T} \aftergroup\egroup\right] \\ &\leq \mathbb{E}\mathopen{}\mathclose\bgroup\left[\Vert\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}\rho\Vert_{L^\infty(D)}^2 \aftergroup\egroup\right]^{1/2} c^{p/2}\exp(- p\sqrt{\log(T)}/2)+c^p \exp(- p\sqrt{\log(T)}) \\ &\leq r(T)^{-1} c^{p/2} \exp(- p\sqrt{\log(T)}/2)+c^p \exp(- p\sqrt{\log(T)}), \\ &\leq \sqrt{T} c^{p/2} \exp(- p\sqrt{\log(T)}/2)+c^p \exp(- p\sqrt{\log(T)}), \end{aligned}$$ where we used Theorem [\[thm: general rate exp\]](#thm: general rate exp){reference-type="ref" reference="thm: general rate exp"} in the second to last step, together with the aforementioned bias bound and that $\bar{\boldsymbol{\beta}}>d/2$ entails $\boldsymbol{h}^b\in\mathsf{H}_T$. Choosing $p=5\sqrt{\log(T)}$ then gives for large enough $T$ $$\label{eq: cor 1} \mathbb{E}\mathopen{}\mathclose\bgroup\left[\Vert(\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}-b^j)\rho\Vert_{L^\infty(D)}\mathbf{1}_{B^\mathsf{c}_T} \aftergroup\egroup\right] \leq cT^{-2}.$$ Furthermore the triangle inequality implies on the event $B_T$ that $\rho/(\vert\widehat{\rho}_{\boldsymbol{h}^\rho,T}\vert +r_T )\leq 1,$ which together with Theorem [\[thm: general rate exp\]](#thm: general rate exp){reference-type="ref" reference="thm: general rate exp"} gives by arguing as above $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\Vert(\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}-b^j)\rho\Vert_{L^\infty(D)}\mathbf{1}_{B_T} \aftergroup\egroup\right] \\ &\leq \mathbb{E}\mathopen{}\mathclose\bgroup\left[\mathopen{}\mathclose\bgroup\left\Vert\mathopen{}\mathclose\bgroup\left(\widehat{b}^{(j)}_{\boldsymbol{h}^b,\boldsymbol{h}^\rho,T}-\frac{b^j\rho}{\vert\widehat{\rho}_{\boldsymbol{h}^\rho,T}\vert+r(T) }\aftergroup\egroup\right)\rho\aftergroup\egroup\right\Vert_{L^\infty(D)}\mathbf{1}_{B_T} \aftergroup\egroup\right]+\mathbb{E}\mathopen{}\mathclose\bgroup\left[\mathopen{}\mathclose\bgroup\left\Vert\mathopen{}\mathclose\bgroup\left(\frac{b^j\rho}{\vert\widehat{\rho}_{\boldsymbol{h}^\rho,T}\vert+r(T) }-b^j\aftergroup\egroup\right)\rho\aftergroup\egroup\right\Vert_{L^\infty(D)}\mathbf{1}_{B_T} \aftergroup\egroup\right] \\ &\leq c\mathopen{}\mathclose\bgroup\left(\mathopen{}\mathclose\bgroup\left(\frac{\log(T)}{T}\aftergroup\egroup\right)^{\frac{\bar{\boldsymbol{\beta}}}{2\bar{\boldsymbol{\beta}}+d}}+r_T+\Phi_{d,\boldsymbol{\beta}}(T) \aftergroup\egroup\right), \end{aligned}$$ where we also used [\[eq: cor 2\]](#eq: cor 2){reference-type="eqref" reference="eq: cor 2"}. Combining this with [\[eq: cor 1\]](#eq: cor 1){reference-type="eqref" reference="eq: cor 1"} completes the proof. ◻ # Proofs for Section [5](#sec: adap){reference-type="ref" reference="sec: adap"} {#proofs-for-section-sec-adap} Our proofs for the adaptive estimators rely on the following concentration inequalities. **Lemma 15**. 1. *Let everything be given as in Theorem [\[thm: general rate exp\]](#thm: general rate exp){reference-type="ref" reference="thm: general rate exp"}. Then for large enough $T$ it holds for any $\boldsymbol{h}\in\mathscr{H}_T,j\in[d]$ $$\begin{aligned} \mathbb{P}\mathopen{}\mathclose\bgroup\left(\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{I}^j_T(g)-\sqrt{T}\mu(gb^j)\vert\geq T^{1/2}V(\boldsymbol{h})A_{T,1}(\boldsymbol{h},\theta) \aftergroup\egroup\right)&\leq \mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1} \aftergroup\egroup\right)^{-\theta}.\end{aligned}$$* 2. *Let everything be given as in Theorem [\[thm: rate trunc est\]](#thm: rate trunc est){reference-type="ref" reference="thm: rate trunc est"}. Then for large enough $T$ it holds for any $\boldsymbol{h}\in\mathscr{H}_T,j\in[d]$ $$\begin{aligned} \mathbb{P}\mathopen{}\mathclose\bgroup\left(\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{I}^{j,\delta}_T(g)-\sqrt{T}\mu(gb^j)\vert\geq T^{1/2}V(\boldsymbol{h})A_{T,2}(\boldsymbol{h},\theta,\alpha) \aftergroup\egroup\right)&\leq \mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1} \aftergroup\egroup\right)^{-\theta}.\end{aligned}$$* *Proof.* For fixed $\theta>0,$ set $p^*(\boldsymbol{h})=\theta \log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1} \aftergroup\egroup\right)$ for $\boldsymbol{h}\in\mathscr{H}_T$ and note that $\mathscr{H}_T\subset \mathsf{H}_T$. Inspecting the proof of Theorem [\[thm: general rate exp\]](#thm: general rate exp){reference-type="ref" reference="thm: general rate exp"} then gives that under the given assumptions for large enough $T$ it holds for any $\boldsymbol{h}\in\mathscr{H}_T$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{I}^j_T(g)-\sqrt{T}\mu(gb^j)\vert^{p^*(\boldsymbol{h})}\aftergroup\egroup\right]^{1/p^*(\boldsymbol{h})} \\ &\leq 2V(\boldsymbol{h})^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2}\Vert K\Vert_\infty\Bigg(C_1\mathopen{}\mathclose\bgroup\left(21+29d\sqrt{\Vert a\Vert_{\infty}} +17c_1 \aftergroup\egroup\right) \\&\quad+\sqrt{\theta}\widetilde{c}_2\mathopen{}\mathclose\bgroup\left(3+4\sqrt{\Vert a_{jj}\Vert_\infty } +3c_1\aftergroup\egroup\right)\Bigg) \\ &=\mathrm{e}^{-1}T^{1/2}V(\boldsymbol{h})A_{T,1}(\boldsymbol{h},\theta). \end{aligned}$$ Hence, Markov's inequality entails $$\begin{aligned} \mathbb{P}\mathopen{}\mathclose\bgroup\left(\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{I}^j_T(g)-\sqrt{T}\mu(gb^j)\vert\geq T^{1/2}V(\boldsymbol{h})A_{T,1}(\boldsymbol{h},\theta) \aftergroup\egroup\right)&\leq \exp(-p^*(\boldsymbol{h}))=\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1} \aftergroup\egroup\right)^{-\theta}. \end{aligned}$$ Similarly, the proof of Theorem [\[thm: rate trunc est\]](#thm: rate trunc est){reference-type="ref" reference="thm: rate trunc est"} gives that the imposed assumptions entail for large enough $T$ that for any $\boldsymbol{h}\in\mathscr{H}_T$ it holds $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\sup_{g\in\mathcal{G}_{\boldsymbol{h}}} \vert \mathbb{I}^{j,\delta}_T(g)-\sqrt{T}\mu(gb^j)\vert^{p^*(\boldsymbol{h})}\aftergroup\egroup\right]^{1/p^*(\boldsymbol{h})} \\ &\leq 2V(\boldsymbol{h})^{1/2}\log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^d h_i^{-1}\aftergroup\egroup\right)^{1/2}\Vert K\Vert_\infty\Bigg(29C_1d\sqrt{\Vert a\Vert_{\infty}} +4\sqrt{\theta\Vert a_{jj}\Vert_\infty }\widetilde{c}_2 \\ &\quad +(1+\mathcal{E}_\alpha)\mathopen{}\mathclose\bgroup\left(64C_1 \sqrt{d}+3\sqrt{\theta}\widetilde{c}_2\aftergroup\egroup\right) \\ &\quad+\frac{\Vert \gamma\Vert_\infty}{\gamma_{\min}}\mathopen{}\mathclose\bgroup\left(1+\frac{\nu_2\Vert \gamma\Vert_\infty^2}{2}\exp(\alpha)\aftergroup\egroup\right)\mathopen{}\mathclose\bgroup\left(64\sqrt{d}C_1+3\sqrt{\theta}\widetilde{c}_2\aftergroup\egroup\right)\Bigg) \\ &=\mathrm{e}^{-1}T^{1/2}V(\boldsymbol{h})A_{T,2}(\boldsymbol{h},\theta,\alpha), \end{aligned}$$ which concludes the proof through Markov's inequality. ◻ *Proof of Proposition [Proposition 9](#thm: adap){reference-type="ref" reference="thm: adap"}.* In the following we usually suppress the dependencies on $j\in[d]$. For any $\boldsymbol{h}\in\mathscr{H}_T$ define for $x\in\mathbb{R}^d$ $$\begin{aligned} s_{\boldsymbol{h},T}(x)\coloneq \mu(K_{\boldsymbol{h}}(x-\cdot)b^j(\cdot)),\quad \mathcal{S}_{\boldsymbol{h},T}(x)\coloneq \bar{b}_{\boldsymbol{h},T}(x)-s_{\boldsymbol{h},T}(x), \end{aligned}$$ and $$\begin{aligned} s_{\boldsymbol{h},\boldsymbol{\eta},T}(x)\coloneq \mu((K_{\boldsymbol{h}}\star K_{\boldsymbol{\eta}})(\cdot-x)b^j(\cdot)). \end{aligned}$$ Furthermore set $$\begin{aligned} \zeta_T&\coloneq \sup_{\boldsymbol{h}\in \mathcal{H}_T}\mathopen{}\mathclose\bgroup\left(\Vert\mathcal{S}^1_{\boldsymbol{h},T}\Vert_{\widetilde{D},\infty}-A_{T,1}(\boldsymbol{h},d) \aftergroup\egroup\right)_+, \end{aligned}$$ where $$\widetilde{D}\coloneq \{x\in\mathbb{R}^d\colon d(x,D)\leq 1 \}.$$ Now for any $\boldsymbol{h}\in\mathscr{H}_T$ it holds $$\begin{aligned} \notag \Vert \bar{b}_{\widehat{\boldsymbol{h}}_{1,T},T}-b^j\rho\Vert_{D,\infty}&\leq \Vert\bar{b}_{\widehat{\boldsymbol{h}}_{1,T},T}- \bar{b}_{\widehat{\boldsymbol{h}}_{1,T},\boldsymbol{h},T}\Vert_{D,\infty}+\Vert\bar{b}_{\widehat{\boldsymbol{h}}_{1,T},\boldsymbol{h},T}- \bar{b}_{\boldsymbol{h},T}\Vert_{D,\infty}+\Vert \bar{b}_{\boldsymbol{h},T}-b^j\rho\Vert_{D,\infty} \\\notag &\leq \Upsilon_{T,1}(\boldsymbol{h})+A_{T,1}(\boldsymbol{h},d)+\Upsilon_{T,1}(\widehat{\boldsymbol{h}}_{1,T})+A_{T,1}(\widehat{\boldsymbol{h}}_{1,T},d)+\Vert \bar{b}_{\boldsymbol{h},T}-b^j\rho\Vert_{D,\infty} \\\notag &\leq 2(\Upsilon_{T,1}(\boldsymbol{h})+A_{T,1}(\boldsymbol{h},d))+\Vert \bar{b}_{\boldsymbol{h},T}-b^j\rho\Vert_{D,\infty} \\\notag &\leq 2(\Upsilon_{T,1}(\boldsymbol{h})+A_{T,1}(\boldsymbol{h},d))+\Vert\mathcal{S}_{\boldsymbol{h},T}\Vert_{D,\infty}+\mathcal{B}_{b^j\rho}(\boldsymbol{h}) \\\label{eq: adap 1} &\leq 2(\Upsilon_{T,1}(\boldsymbol{h})+A_{T,1}(\boldsymbol{h},d))+\zeta_T+A_{T,1}(\boldsymbol{h},d)+\mathcal{B}_{b^j\rho}(\boldsymbol{h}), \end{aligned}$$ where we used $\bar{b}_{\widehat{\boldsymbol{h}}_{1,T},\boldsymbol{h},T}=\bar{b}_{\boldsymbol{h},\widehat{\boldsymbol{h}}_{1,T},T},$ together with the definition of $\widehat{\boldsymbol{h}}_{1,T}$. Furthermore Fubini's Theorem for stochastic integrals (see Theorem 65 in [@protter2004]) and the symmetry of $K$ give for any $\boldsymbol{h},\boldsymbol{\eta}\in\mathscr{H}_T$ $$\begin{aligned} &\Vert\bar{b}_{\boldsymbol{h},\boldsymbol{\eta},T}-s_{\boldsymbol{\eta},T} \Vert_{D,\infty} \\ &\leq \Vert\bar{b}_{\boldsymbol{h},\boldsymbol{\eta},T}-s_{\boldsymbol{h},\boldsymbol{\eta},T} \Vert_{D,\infty}+\Vert s_{\boldsymbol{\eta},T}-s_{\boldsymbol{h},\boldsymbol{\eta},T} \Vert_{D,\infty} \\ &=\sup_{x\in D}\mathopen{}\mathclose\bgroup\left\vert\int_{\mathbb{R}^d} K_{\boldsymbol{\eta}}(u)\frac{1}{T}\int_0^T K_{\boldsymbol{h}}(u-(X_{s-}-x))\mathop{}\!\mathrm{d} X^j_s\mathop{}\!\mathrm{d} u-\int_{\mathbb{R}^d}K_{\boldsymbol{\eta}}(u) \int_{\mathbb{R}^d} b^j(z)K_{\boldsymbol{h}}(u-(z-x))\mu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} u \aftergroup\egroup\right\vert \\&\quad +\sup_{x\in D}\mathopen{}\mathclose\bgroup\left\vert\int_{\mathbb{R}^d}K_{\boldsymbol{\eta}}(x-z) b^j(z)\mu(\mathop{}\!\mathrm{d} z)-\int_{\mathbb{R}^d}K_{\boldsymbol{\eta}}(u) \int_{\mathbb{R}^d} b^j(z)K_{\boldsymbol{h}}(u-(z-x))\mu(\mathop{}\!\mathrm{d} z)\mathop{}\!\mathrm{d} u \aftergroup\egroup\right\vert \\ &=\sup_{x\in D}\mathopen{}\mathclose\bgroup\left\vert\int_{\mathbb{R}^d} K_{\boldsymbol{\eta}}(u)\mathcal{S}^1_{\boldsymbol{h},T}(u+x)\mathop{}\!\mathrm{d} u \aftergroup\egroup\right\vert +\sup_{x\in D}\mathopen{}\mathclose\bgroup\left\vert\int_{\mathbb{R}^d}K_{\boldsymbol{\eta}}(u) ((b^j\rho)(u+x)- ((b^j\rho)\ast K_{\boldsymbol{h}})(u+x))\mathop{}\!\mathrm{d} u \aftergroup\egroup\right\vert \\ &=\Vert K\Vert_{L^1(\mathbb{R}^d)}(\Vert \mathcal{S}_{\boldsymbol{h},T}\Vert_{\widetilde{D},\infty}+\mathcal{B}_{b^j\rho}(\boldsymbol{h})), \end{aligned}$$ which implies for any $\boldsymbol{h}\in\mathscr{H}_T$ $$\begin{aligned} \Upsilon_{T,1}(\boldsymbol{h})&\leq \sup_{\eta\in \mathcal{H}_T}\mathopen{}\mathclose\bgroup\left(\Vert K\Vert_{L^1(\mathbb{R}^d)}(\Vert \mathcal{S}_{\boldsymbol{h},T}\Vert_{\widetilde{D},\infty}+\mathcal{B}_{b^j\rho}(\boldsymbol{h}))+\Vert \mathcal{S}_{\boldsymbol{\eta},T}\Vert_{D,\infty}-A_{T,1}(\boldsymbol{\eta},\ldots) \aftergroup\egroup\right)_+ \\ &\leq \Vert K\Vert_{L^1(\mathbb{R}^d)}(\zeta_T+A_{T,1}(\boldsymbol{h},d)+\mathcal{B}_{b^j\rho}(\boldsymbol{h}))+\zeta_T \\ &\leq (1\lor\Vert K\Vert_{L^1(\mathbb{R}^d)})(2\zeta_T+A_{T,1}(\boldsymbol{h},d)+\mathcal{B}_{b^j\rho}(\boldsymbol{h}) ). \end{aligned}$$ Combining this with [\[eq: adap 1\]](#eq: adap 1){reference-type="eqref" reference="eq: adap 1"} gives for any $\boldsymbol{h}\in\mathscr{H}_T$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\Vert \bar{b}_{\widehat{\boldsymbol{h}}_{1,T},T}-b^j\rho\Vert_\infty \aftergroup\egroup\right] \leq (1\lor\Vert K\Vert_{L^1(\mathbb{R}^d)})(3\mathcal{B}_{b^j\rho}(\boldsymbol{h}) +5A_{T,1}(\boldsymbol{h},d)+5\mathbb{E}[\zeta_T]). \end{aligned}$$ Thus it only remains to bound $\mathbb{E}[\zeta_T]$. For this note first that by Hölder's inequality, Lemma [Lemma 15](#lemma: adap){reference-type="ref" reference="lemma: adap"} and [\[eq: moment adap\]](#eq: moment adap){reference-type="eqref" reference="eq: moment adap"} there exists a constant $c>0$ such that for large enough $T$ it holds for any $\boldsymbol{h}\in\mathscr{H}_T,$ $$\begin{aligned} &\mathbb{E}\mathopen{}\mathclose\bgroup\left[\mathopen{}\mathclose\bgroup\left(\Vert\mathcal{S}_{\boldsymbol{h},T}\Vert_{\widetilde{D},\infty}-A_{T,1}(\boldsymbol{h},d) \aftergroup\egroup\right)_+\aftergroup\egroup\right] \\ &\leq \sqrt{\mathbb{E}\mathopen{}\mathclose\bgroup\left[\mathopen{}\mathclose\bgroup\left(\Vert\mathcal{S}_{\boldsymbol{h},T}\Vert_{\widetilde{D},\infty}-A_{T,1}(\boldsymbol{h},d) \aftergroup\egroup\right)^2\aftergroup\egroup\right]\mathbb{P}\mathopen{}\mathclose\bgroup\left(\Vert\mathcal{S}_{\boldsymbol{h},T}\Vert_\infty\geq A_{T,1}(\boldsymbol{h},d)\aftergroup\egroup\right)} \\ &\leq c T^{-1/2} V(\boldsymbol{h})^{-1/2} \log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1}\aftergroup\egroup\right)^{1/2}\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1} \aftergroup\egroup\right)^{-\theta/2} \\ &= c T^{-1/2} \log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1}\aftergroup\egroup\right)^{1/2}\mathopen{}\mathclose\bgroup\left(\frac{\mathopen{}\mathclose\bgroup\left(\prod_{i=1}^dh_i^{-1}\aftergroup\egroup\right)^{1/d}}{\sum_{i=1}^{d}h_i^{-1}} \aftergroup\egroup\right)^{d/2} \\ &\leq cd^{-d/2} T^{-1/2} \log\mathopen{}\mathclose\bgroup\left(\sum_{i=1}^{d}h_i^{-1}\aftergroup\egroup\right)^{1/2}, \end{aligned}$$ where we also used the AM--GM inequality in the last step. Hence, $$\begin{aligned} \mathbb{E}[\zeta_T]&\leq \sum_{\boldsymbol{h}\in\mathcal{H}_T}\mathbb{E}\mathopen{}\mathclose\bgroup\left[\mathopen{}\mathclose\bgroup\left(\Vert\mathcal{S}_{\boldsymbol{h},T}\Vert_{\widetilde{D},\infty}-A_{T,1}(\boldsymbol{h},d) \aftergroup\egroup\right)_+\aftergroup\egroup\right] \\ &\leq c\operatorname{card}(\mathcal{H}_T)\sqrt{\frac{\log(T)}{T}} \\ &\leq c \log(T)^d\sqrt{\frac{\log(T)}{T}}, \end{aligned}$$ which concludes the proof of the first assertion. The second assertion follows analogously from the results of Section [4](#sec: drift est){reference-type="ref" reference="sec: drift est"} and Lemma [Lemma 15](#lemma: adap){reference-type="ref" reference="lemma: adap"}. ◻ *Proof of Theorem [Theorem 10](#thm: adap final){reference-type="ref" reference="thm: adap final"}.* As the proofs of both assertions are almost identical, we omit the proof of the second statement. Fix $j\in[d]$. Noting that for any $\boldsymbol{h}\in\mathscr{H}_T,x\in D$ $$\begin{aligned} \vert b^j(x)-\widehat{b}^{(j)}_{\mathrm{adap},T}(x)\vert &\leq \rho_{\min}^{-1}\vert b^j(x)(\widehat{\rho}_{\widehat{\boldsymbol{h}}_\rho,T}(x)\lor \rho_{\min})-\bar{b}^{(j)}_{\widehat{\boldsymbol{h}}^j_1,T}(x) \vert \\ &\leq \rho_{\min}^{-1}\mathopen{}\mathclose\bgroup\left(\vert b^j(x)(\widehat{\rho}_{\widehat{\boldsymbol{h}}_\rho,T}(x)\lor \rho_{\min})-(b^j\rho)(x) \vert+\vert (b^j\rho)(x)-\bar{b}^{(j)}_{\widehat{\boldsymbol{h}}^j_1,T}(x) \vert\aftergroup\egroup\right) \\ &\leq \rho_{\min}^{-1}\mathopen{}\mathclose\bgroup\left(\Vert b^j\Vert_{L^\infty(D)}\vert \widehat{\rho}_{\widehat{\boldsymbol{h}}_\rho,T}(x)-\rho(x) \vert+\vert (b^j\rho)(x)-\bar{b}^{(j)}_{\widehat{\boldsymbol{h}}^j_1,T}(x) \vert\aftergroup\egroup\right), \end{aligned}$$ completes the proof by applying Proposition [Proposition 9](#thm: adap){reference-type="ref" reference="thm: adap"}, together with the classical anisotropic bias bound, since the optimal bandwidth choices in Corollary [Corollary 7](#cor: rate){reference-type="ref" reference="cor: rate"} are contained in $\mathscr{H}_T$ if $\bar{\boldsymbol{\beta}}>d/2$. ◻ [^1]: Aarhus University, Department of Mathematics, Ny Munkegade 118, 8000 Aarhus C, Denmark.Email: <[email protected]> The author gratefully acknowledges financial support of Sapere Aude: DFF-Starting Grant 0165-00061B "Learning diffusion dynamics and strategies for optimal control."

arxiv_math

--- abstract: | We prove novel asymptotic freeness results in tracial ultraproduct von Neumann algebras. In particular, we show that whenever $M = M_1 \ast M_2$ is a tracial free product von Neumann algebra and $u_1 \in \mathscr U(M_1)$, $u_2 \in \mathscr U(M_2)$ are Haar unitaries, the relative commutants $\{u_1\}' \cap M^{\mathcal U}$ and $\{u_2\}' \cap M^{\mathcal U}$ are freely independent in the ultraproduct $M^{\mathcal U}$. Our proof relies on Mei--Ricard's results [@MR16] regarding $\operatorname{L}^p$-boundedness (for all $1 < p < +\infty$) of certain Fourier multipliers in tracial amalgamated free products von Neumann algebras. We derive two applications. Firstly, we obtain a general absorption result in tracial amalgamated free products that recovers several previous maximal amenability/Gamma absorption results. Secondly, we prove a new lifting theorem which we combine with our asymptotic freeness results and Chifan--Ioana--Kunnawalkam Elayavalli's recent construction [@CIKE22] to provide the first example of a ${\rm II_1}$ factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras. address: - | École Normale Supérieure\ Département de Mathématiques et Applications\ Université Paris-Saclay\ 45 rue d'Ulm\ 75230 Paris Cedex 05\ FRANCE - | Department of Mathematics\ University of California San Diego\ 9500 Gilman Drive\ La Jolla\ CA 92093\ USA author: - Cyril Houdayer - Adrian Ioana title: Asymptotic freeness in tracial ultraproducts --- [^1] [^2] # Introduction In order to state our main results, we recall the following terminology regarding $n$-*independence* and *freeness*. ## Terminology {#terminology .unnumbered} Let $(M, \tau)$ be a tracial von Neumann algebra together with a von Neumann subalgebra $B \subset M$. We denote by $\operatorname{E}_B : M \to B$ the unique trace-preserving faithful normal conditional expectation and set $M \ominus B = \ker(\operatorname{E}_B)$. Let $n \geq 1$. Following Popa (see e.g. [@Po13a; @Po13b]), we say that two subsets $X, Y \subset M \ominus B$ are $n$-*independent* in $M$ with respect to $\operatorname{E}_{B}$ if $\operatorname{E}_B(x_1y_1\cdots x_ky_k)=0$, for every $1\leq k\leq n$, $x_1,\dots ,x_k\in X$ and $y_1,\dots,y_k\in Y$. We then say that two intermediate von Neumann subalgebras $B \subset M_1, M_2\subset M$ are $n$-*independent* in $M$ with respect to $\operatorname{E}_B$ if the sets $M_1 \ominus B$ and $M_2 \ominus B$ are $n$-independent with respect to $\operatorname{E}_B$. When $B = \mathbb{C}1$, two von Neumann subalgebras $M_1, M_2 \subset M$ are $1$-independent in $M$ with respect to $\tau$ if and only if $M_1$ and $M_2$ are $\tau$-orthogonal, i.e., $\tau(xy)=\tau(x)\tau(y)$, for every $x\in M_1$ and $y\in M_2$. Let $I$ be a nonempty index set. We say that a family $(X_i)_{i \in I}$ of subsets of $M \ominus B$ is *freely independent* in $M$ with respect to $\operatorname{E}_B$ if $\operatorname{E}_B(x_1 \cdots x_k)=0$ for every $k \geq 1$, $x_1 \in X_{\varepsilon_1}, \dots, x_k \in X_{\varepsilon_k}$ with $\varepsilon_1 \neq \cdots \neq \varepsilon_k$ in $I$. We say that a family $(M_i)_{i \in I}$ of intermediate von Neumann subalgebras $B \subset M_i \subset M$ is *freely independent* in $M$ with respect to $\operatorname{E}_B$ if the family of subsets $(M_i \ominus B)_{i \in I}$ is freely independent in $M$ with respect to $\operatorname{E}_B$. In this case, we denote by $\ast_{B, i \in I} (M_i, \tau_i) = \bigvee_{i \in I} M_i \subset M$ the tracial amalgamated free product von Neumann algebra where $\tau_i = \tau|_{M_i}$ for every $i \in I$. ## Main results {#main-results .unnumbered} Let $I$ be an at most countable index set such that $2 \leq | I | \leq +\infty$. Let $(M_i, \tau_i)_{i \in I}$ be a family of tracial von Neumann algebras with a common von Neumann subalgebra $(B, \tau_0)$ such that for every $i \in I$, we have $\tau_i|_B = \tau_0$. Denote by $(M, \tau) = \ast_{B, i \in I} (M_i, \tau_i)$ the tracial amalgamated free product von Neumann algebra. Let $\mathcal U$ be a nonprincipal ultrafilter on $\mathbb{N}$. Denote by $(M^{\mathcal U}, \tau^{\mathcal U})$ the tracial ultraproduct von Neumann algebra. Simply denote by $\operatorname{E}_B : M \to B$ (resp. $\operatorname{E}_{B^{\mathcal U}} : M^{\mathcal U} \to B^{\mathcal U}$) the unique trace-preserving faithful normal conditional expectation. Denote by $\mathscr W \subset M$ the linear span of $B$ and of all the reduced words in $M$ of the form $w = w_1 \cdots w_n$, with $n \geq 1$, $w_j \in M_{\varepsilon_j} \ominus B$ for every $j \in \{1, \dots, n\}$, and $\varepsilon_1, \dots, \varepsilon_n \in I$ such that $\varepsilon_1 \neq \cdots \neq \varepsilon_n$. For every $i \in I$, denote by $\mathscr W_i \subset M \ominus B$ the linear span of all the reduced words in $M$ whose first and last letter lie in $M_i \ominus B$. Moreover, denote by $P_{\mathscr W_i} : \operatorname{L}^2(M) \to \operatorname{L}^2(\mathscr W_i)$ the corresponding orthogonal projection. By construction, the family $(\mathscr W_i)_{i \in I}$ is freely independent in $M$ with respect to $\operatorname{E}_B$. Our first main result is an extension of this fact to the ultraproduct framework. **Theorem 1**. *Keep the same notation as above. For every $i \in I$, denote by $\mathbf X_i$ the set of all the elements $x = (x_n)^{\mathcal U} \in M^{\mathcal U} \ominus B^{\mathcal U}$ such that $\lim_{n \to \mathcal U} \|x_n - P_{\mathscr W_i}(x_n)\|_2 = 0$.* *Then the family $(\mathbf X_i)_{i \in I}$ is freely independent in $M^{\mathcal U}$ with respect to $\operatorname{E}_{B^{\mathcal U}}$.* The proof of Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"} relies on Mei--Ricard's results [@MR16] showing that the canonical projection $P_{\mathscr W_i} : \mathscr W \to \mathscr W_i$ extends to a completely bounded operator $P_{\mathscr W_i} : \operatorname{L}^p(M) \to \operatorname{L}^p(\mathscr W_i)$ for every $p \in (1, +\infty)$. In particular, we exploit $\operatorname{L}^p$-boundedness of $P_{\mathscr W_i} : \operatorname{L}^p(M) \to \operatorname{L}^p(\mathscr W_i)$ for every $2 \leq p < +\infty$. Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"} is a novel application of noncommutative $\operatorname{L}^p$-spaces to the structure theory of tracial von Neumann algebras. It follows from Popa's asymptotic orthogonality property [@Po83] (see Lemma [Lemma 5](#lem-commutant){reference-type="ref" reference="lem-commutant"} below) that for every $i \in I$ and every unitary $u \in \mathscr U(M_i^{\mathcal U})$ such that $\operatorname{E}_{B^{\mathcal U}}(u^k) = 0$ for every $k \in \mathbb{Z}\setminus \{0\}$, if $x \in \{u\}' \cap M^{\mathcal U}$ and $\operatorname{E}_{B^{\mathcal U}}(x) = 0$, then $x \in \mathbf X_i$. In particular, in the case $B = \mathbb{C}1$, Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"} implies the following **Theorem 2**. *Assume that $B= \mathbb{C}1$. For every $i\in I$, let $u_i \in \mathscr U(M_i^{\mathcal U})$ be a Haar unitary.* *Then the family $(\{u_i\}' \cap M^{\mathcal U})_{i\in I}$ is freely independent in $M^{\mathcal U}$ with respect to $\tau^{\mathcal U}$.* Assume that for every $i \in I$, $M_i$ is a diffuse abelian von Neumann algebra so that $M_i^{\mathcal U} \subset \{u_i\}' \cap M^{\mathcal U}$. Then Theorem [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"} can be regarded as a far-reaching generalization of the fact that the family $(M_i^{\mathcal U})_{i \in I}$ is freely independent in $M^{\mathcal U}$ with respect to $\tau^{\mathcal U}$. In the case $I = \{1, 2\}$, we also obtain the following variation of Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"}. **Theorem 3**. *Assume that $I = \{1, 2\}$. Keep the same notation as above. Let $\mathbf Y_1 \subset \mathbf X_1$ be a subset with the property that $a \mathbf Y_1 b \subset \mathbf X_1$ for all $a, b \in M_1$.* *Then the sets $\mathbf Y_1$ and $M \ominus M_1$ are freely independent in $M^{\mathcal U}$ with respect to $\operatorname{E}_{B^{\mathcal U}}$.* A typical example of a subset $\mathbf Y_1 \subset \mathbf X_1$ with the property that $a \mathbf Y_1 b \subset \mathbf X_1$ or all $a, b \in M_1$ is given by $\mathbf Y_1 = A' \cap (M^{\mathcal U} \ominus M_1^{\mathcal U})$, where $A \subset M_1$ is a von Neumann subalgebra such that $A \npreceq_{M_1} B$ (see Lemmas [Lemma 4](#lem-intertwining){reference-type="ref" reference="lem-intertwining"} and [Lemma 6](#lem-commutant-bis){reference-type="ref" reference="lem-commutant-bis"}). In the case $M = B \rtimes \mathbb{F}_n = (B \rtimes \mathbb{Z}) \ast_B \cdots \ast_B (B \rtimes \mathbb{Z}) = M_1 \ast_B \cdots \ast_B M_n$, Popa showed in [@Po83 Lemma 2.1] that for the canonical Haar unitary $u \in \operatorname{L}(\mathbb{Z}) \subset M_1$, the sets $\{u\}' \cap (M^{\mathcal U} \ominus M_1^{\mathcal U})$ and $M \ominus M_1$ are $2$-independent in $M^{\mathcal U}$ with respect to $\operatorname{E}_{B^{\mathcal U}}$ (see also [@HU15] for the free product case). Letting $\mathbf Y_1 = \{u\}' \cap (M^{\mathcal U} \ominus M_1^{\mathcal U})$, Theorem [Theorem 3](#thm-main-result-bis){reference-type="ref" reference="thm-main-result-bis"} can be regarded as a generalization and a strengthening of Popa's result. In the case $I = \{1, 2\}$ and $B = \mathbb{C}1$, we exploit Mei--Ricard's results [@MR16] to obtain the following indecomposability result in $M^{\mathcal U}$. **Theorem 4**. *Assume that $I=\{1,2\}$ and $B=\mathbb C1$. Let $u_1\in\mathscr U(M_1^{\mathcal U})$ be a Haar unitary and $u_2\in\mathscr U(M_2^{\mathcal U})$ such that $\tau^{\mathcal U}(u_2)=\tau^{\mathcal U}(u_2^2)=0$.* *Then there do not exist $v_1,v_2\in\mathscr U(M^{\mathcal U})$ such that $\tau^{\mathcal U}(v_1)=\tau^{\mathcal U}(v_1^2)=\tau^{\mathcal U}(v_2)=0$ and $[u_1,v_1]=[v_1,v_2]=[v_2,u_2]=0$.* Another way to formulate Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} is as follows. Let $u_1\in\mathscr U(M_1^{\mathcal U})$ be a Haar unitary, $u_2\in\mathscr U(M_2^{\mathcal U})$ such that $\tau^{\mathcal U}(u_2)=\tau^{\mathcal U}(u_2^2)=0$ and $v_1\in\mathscr U(M^{\mathcal U})$ such that $[u_1, v_1] = 0$ and $\tau^{\mathcal U}(v_1)=\tau^{\mathcal U}(v_1^2)= 0$. Then we have $\{v_1, u_2\}' \cap M^{\mathcal U} = \mathbb{C}1$. This generalizes the well-known fact (see e.g. [@Io12 Lemma 6.1]) that $\{u_1,u_2\}' \cap M^{\mathcal U} = \mathbb{C}1$. In Section [3](#section-main-results){reference-type="ref" reference="section-main-results"}, we generalize Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} to arbitrary tracial amalgamated free product von Neumann algebras (see Theorem [Theorem 9](#3infinity){reference-type="ref" reference="3infinity"}). Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} is new even in the case $M=\operatorname{L}(C_1*C_2)$, where $C_1, C_2$ are cyclic groups with $|C_1|>1$ and $|C_2|>2$. In this case, $M=M_1 \ast M_2$, where $M_1=\operatorname{L}(C_1), M_2=\operatorname{L}(C_2)$. Moreover, $M$ is an interpolated free group factor by [@Dy92 Corollary 5.3] and thus has positive $1$-bounded entropy, $h(M)>0$, in the sense of [@Ju05; @Ha15]. By [@Ha15 Corollary 4.8] (see also [@CIKE22 Facts 2.4 and 2.9]), if $u_1,u_2\in M$ are generating unitaries, then there are no Haar unitaries $v_1,v_2\in M^{\mathcal U}$ satisfying $[u_1,v_1]=[v_1,v_2]=[v_2,u_2]=0$. This fact was used in [@CIKE22] to construct two non-elementarily equivalent non-Gamma ${\rm II_1}$ factors. Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} considerably strengthens this fact when $C_1=\mathbb Z$, $u_1\in \mathscr U(M_1)$, $u_2\in \mathscr U(M_2)$. Unlike [@Ha15], we cannot say anything about arbitrary generating unitaries $u_1$ and $u_2$, that do not belong to $M_1$ and $M_2$, respectively. On the other hand, while the free entropy methods from [@Ha15] only rule out the existence of Haar unitaries $v_1,v_2 \in \mathscr U(M^{\mathcal U})$ satisfying $[u_1,v_1]=[v_1,v_2]=[v_2,u_2]=0$, Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} also excludes the existence of such finite order unitaries $v_1,v_2$ provided that $v_1,v_1^2,v_2$ have trace zero. Let $u_1\in\mathscr U(M_1^{\mathcal U})$ and $u_2\in\mathscr U(M_2^{\mathcal U})$ be as in Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"}, and assume that $u_2^m=1$, for some $m>2$. Then $\{u_2\}'\cap M^{\mathcal U}$ has finite index in $M^{\mathcal U}$ and therefore, unlike in Theorem [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"}, $\{u_1\}'\cap M^{\mathcal U}$ and $\{u_2\}'\cap M^{\mathcal U}$ are not freely independent in $M^{\mathcal U}$ with respect to $\tau^{\mathcal U}$. Instead, the proof of Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} relies on a subtler analysis of commuting unitaries belonging to $\{u_1\}'\cap M^{\mathcal U}$ and $\{u_2\}'\cap M^{\mathcal U}$. However, similarly to the proof of Theorem [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"}, we make crucial use of Mei--Ricard's results [@MR16]. We do not know if Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} holds if we remove the assumption that $\operatorname{E}_{B^{\mathcal U}}(u_2^2)=0$. However, a standard diagonal argument implies the existence of $N\in\mathbb N$ such that the assumption that $\operatorname{E}_{B^{\mathcal U}}(u_1^k)=0$, for every $k\in\mathbb Z\setminus\{0\}$, can be relaxed by assuming instead that $\operatorname{E}_{B^{\mathcal U}}(u_1^k)=0$, for every $k\in\mathbb Z\setminus\{0\}$ with $|k|\leq N$. ## Application to absorption in AFP von Neumann algebras {#application-to-absorption-in-afp-von-neumann-algebras .unnumbered} We use Theorem [Theorem 3](#thm-main-result-bis){reference-type="ref" reference="thm-main-result-bis"} to obtain a new absorption result for tracial amalgamated free product von Neumann algebras. **Theorem 5**. *Assume that $I = \{1, 2\}$. Keep the same notation as above and assume that $M$ is separable. Let $P \subset M$ be a von Neumann subalgebra such that $P \cap M_1 \npreceq_{M_1} B$ and $P' \cap M^{\mathcal U} \npreceq_{M^{\mathcal U}} B^{\mathcal U}$. Then we have $P \subset M_1$.* Theorem [Theorem 5](#cor-amenable-absorption){reference-type="ref" reference="cor-amenable-absorption"} vastly generalizes Popa's seminal result [@Po83] that the generator masa $\text{L}(\langle a\rangle)$ is maximal amenable inside the free group factor $\text{L}(\mathbb F_2)=\text{L}(\langle a,b\rangle)$. Specifically, it extends several maximal amenability/Gamma absorption results. Theorem [Theorem 5](#cor-amenable-absorption){reference-type="ref" reference="cor-amenable-absorption"} generalizes [@HU15 Theorem A] (see also [@Ho14 Theorem A]) to arbitrary tracial amalgamated free product von Neumann algebras. As we observe in Remark [Remark 13](#rem-amenable){reference-type="ref" reference="rem-amenable"}, if $P \subset M$ is an amenable von Neumann subalgebra such that $P \cap M_1 \npreceq_{M_1} B$, then we necessarily have $P' \cap M^{\mathcal U} \npreceq_{M^{\mathcal U}} B^{\mathcal U}$. Thus, Theorem [Theorem 5](#cor-amenable-absorption){reference-type="ref" reference="cor-amenable-absorption"} also yields a new proof of [@BH16 Main theorem] in the setting of tracial amalgamated free product von Neumann algebras. Let us point out that in the setting of tracial free products $M = M_1 \ast M_2$ of Connes-embeddable von Neumann algebras, the inclusion $M_1 \subset M$ satisfies a more general absorption property. Indeed, [@HJNS19 Theorem A] shows that if $P \subset M$ is a von Neumann subalgebra such that $P \cap M_1$ is diffuse and has $1$-bounded entropy zero, then $P \subset M_1$. In the case $M = \operatorname{L}(\mathbb{F}_n)$ is a free group factor, the aforementioned absorption property holds for *any* diffuse maximal amenable subalgebra $Q \subset M$, thanks to the recent resolution of the Peterson--Thom conjecture via random matrix theory [@BC22; @BC23] and $1$-bounded entropy [@Ha15] (see also [@HJKE23]). ## Application to continuous model theory of ${\rm II_1}$ factors {#application-to-continuous-model-theory-of-rm-ii_1-factors .unnumbered} We next present an application of Theorem [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"} to the continuous model theory of ${\rm II_1}$ factors. A main theme in this theory, initiated by Farah--Sherman--Hart in [@FHS11], is to determine whether two given ${\rm II_1}$ factors $M, N$ are elementarily equivalent. By the continuous version of the Keisler--Shelah theorem this amounts to $M$ and $N$ admitting isomorphic ultrapowers, $M^{\mathcal U}\cong N^{\mathcal V}$, for some ultrafilters $\mathcal U$ and $\mathcal V$ on arbitrary sets [@FHS11; @HI02]. It was shown in [@FHS11] that property Gamma and being McDuff are elementary properties, leading to three distinct elementary classes of ${\rm II_1}$ factors. A fourth such elementary class was then provided in [@GH16]. The problem of determining the number of elementary classes of ${\rm II_1}$ factors was solved in [@BCI15], where the continuum of non-isomorphic ${\rm II_1}$ factors constructed in [@Mc69] were shown to be pairwise non-elementarily equivalent. However, all the available techniques for distinguishing ${\rm II_1}$ factors up to elementary equivalence were based on central sequences. It thus remained open to construct any non-elementarily equivalent ${\rm II_1}$ factors which do not have any non-trivial central sequences, i.e., fail property Gamma. This problem was solved by Chifan--Ioana--Kunnawalkam Elayavalli in [@CIKE22] using a combination of techniques from Popa's deformation/rigidity theory and Voiculescu's free entropy theory. First, deformation/rigidity methods from [@IPP05] were used to construct a non-Gamma ${\rm II_1}$ factor $M$ via an iterative amalgamated free product construction. It was then shown that $M$ is not elementarily equivalent to any (necessarily non-Gamma) ${\rm II_1}$ factor $N$ having positive $1$-bounded entropy, $h(N)>0$, in the sense of Jung [@Ju05] and Hayes [@Ha15]. Examples of ${\rm II_1}$ factors $N$ with $h(N)>0$ include the interpolated free group factors $\operatorname{L}(\mathbb F_t)$, $1< t\leq\infty$, and, more generally, any tracial free product $N=N_1 \ast N_2$ of diffuse Connes-embeddable von Neumann algebras. For additional examples of such ${\rm II}_1$ factors, see [@CIKE22 Fact 2.7]. However, the methods from [@CIKE22] could not distinguish $M$ up to elementary equivalence from $N=N_1*N_2$, whenever $N_1$ or $N_2$ is a non-Connes-embeddable tracial von Neumann algebra (the existence of which has been announced in the preprint [@JNVWY20]). In particular, since it is unclear if $M$ is Connes-embeddable, it remained open whether $M$ is elementarily equivalent to $M*\operatorname{L}(\mathbb Z)$. Theorem [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"} allows us to settle this problem for a variant of the ${\rm II_1}$ factor constructed in [@CIKE22]: **Theorem 6**. *There exists a separable ${\rm II_1}$ factor $M$ which does not have property Gamma and that is not elementarily equivalent to $N=N_1*N_2$, for any diffuse tracial von Neumann algebras $(N_1,\tau_1)$ and $(N_2,\tau_2)$.* In particular, Theorem [Theorem 6](#non-ee){reference-type="ref" reference="non-ee"} provides the first example of a non-Gamma ${\rm II_1}$ factor $M$ which is not elementarily equivalent to $M*\operatorname{L}(\mathbb Z)$. The conclusion of Theorem [Theorem 6](#non-ee){reference-type="ref" reference="non-ee"} is verified by any ${\rm II_1}$ factor $M$ satisfying the following: **Theorem 7**. *There exists a separable ${\rm II_1}$ factor $M$ which does not have property Gamma and satisfies the following. For every countably cofinal ultrafilter $\mathcal U$ on a set $J$ and $u_1,u_2\in\mathscr U(M^{\mathcal U})$ such that $\{u_1\}^{\prime\prime}$ and $\{u_2\}^{\prime\prime}$ are $2$-independent in $M^{\mathcal U}$ with respect to $\tau^{\mathcal U}$, there exist Haar unitaries $v_1,v_2\in\mathscr U(M^{\mathcal U})$ such that $[u_1,v_1]=[u_2,v_2]=[v_1,v_2]=0$.* An ultrafilter $\mathcal U$ on a set $J$ is called *countably cofinal* if there exists a sequence $(A_n)_{n\in\mathbb N}$ in $\mathcal U$ with $\bigcap_{n\in\mathbb N}A_n=\emptyset$. Any free ultrafilter on $\mathbb N$ is countably cofinal. The proof of Theorem [Theorem 7](#inductive){reference-type="ref" reference="inductive"} uses the iterative amalgamated free product construction introduced in [@CIKE22]. In [@CIKE22 Theorem B], this construction was used to build a non-Gamma separable ${\rm II_1}$ factor $M$ with the following property: for any unitaries $u_1,u_2\in \mathscr U(M^{\mathcal U})$ such that $\{u_1\}^{\prime\prime}$ and $\{u_2\}^{\prime\prime}$ are orthogonal and $u_1^2=u_2^3=1$, there exist Haar unitaries $v_1,v_2\in\mathscr U(M^{\mathcal U})$ such that $[u_1,v_1]=[u_2,v_2]=[v_1,v_2]=0$. The proof of [@CIKE22 Theorem B] relies crucially on a lifting lemma showing that any unitaries $u_1,u_2\in \mathscr U(M^{\mathcal U})$ such that $\{u_1\}^{\prime\prime}$ and $\{u_2\}^{\prime\prime}$ are orthogonal and $u_1^2=u_2^3=1$ lift to unitaries in $M$ with the same properties. A key limitation in [@CIKE22] was the assumption that $u_1$ and $u_2$ have orders $2$ and $3$. We remove this limitation here by proving a general lifting result of independent interest (see Theorem [Theorem 14](#lifting){reference-type="ref" reference="lifting"}) which shows that any unitaries $u_1,u_2\in \mathscr U(M^{\mathcal U})$ such that $\{u_1\}^{\prime\prime}$ and $\{u_2\}^{\prime\prime}$ are $2$-independent admit lifts $u_1=(u_{1,n})^{\mathcal U}$ and $u_2=(u_{2,n})^{\mathcal U}$ with $\{u_{1,n}\}^{\prime\prime}$ and $\{u_{2,n}\}^{\prime\prime}$ orthogonal for every $n \in \mathbb{N}$. With this result in hand, adjusting the iterative construction from [@CIKE22] implies Theorem [Theorem 7](#inductive){reference-type="ref" reference="inductive"}. To explain how Theorem [Theorem 6](#non-ee){reference-type="ref" reference="non-ee"} follows by combining Theorem [Theorem 7](#inductive){reference-type="ref" reference="inductive"} and Theorem [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"}, let $M$ be a ${\rm II_1}$ factor as in Theorem [Theorem 7](#inductive){reference-type="ref" reference="inductive"}, $N=N_1 \ast N_2$ a free product of diffuse tracial von Neumann algebras and $u_1\in \mathscr U(N_1)$, $u_2\in \mathscr U(N_2)$ Haar unitaries. Since $\{u_1\}^{\prime\prime}$ and $\{u_2\}^{\prime\prime}$ are freely and thus $2$-independent, it follows that $M^{\mathcal U}\not\cong N^{\mathcal V}$, for any countably cofinal ultrafilter $\mathcal U$ and any ultrafilter $\mathcal V$. Indeed, Theorem [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"} implies that any Haar unitaries (more generally, any trace zero unitaries) $v_1,v_2\in\mathscr U(N^{\mathcal V})$ such that $[u_1,v_1]=[u_2,v_2]=0$ are freely independent and therefore do not commute. If $\mathcal U$ is an ultrafilter which is not countably cofinal, then we also have that $M^{\mathcal U}\not\cong N^{\mathcal V}$, for any ultrafilter $\mathcal V$. Otherwise, using [@BCI15 Lemma 2.3] we would get that $M^{\mathcal U}\cong M$, thus $N^{\mathcal V}\cong M$ is separable, hence $N^{\mathcal V}\cong N$ and therefore $M\cong N$. But then $M^{\mathcal W}\cong N^{\mathcal W}$, for any free ultrafilter $\mathcal W$ on $\mathbb N$. Since $\mathcal W$ is countably cofinal, this is a contradiction. Altogether, we conclude that $M^{\mathcal U}\not\cong N^{\mathcal V}$, for any ultrafilters $\mathcal U,\mathcal V$, and thus $M, N$ are not elementarily equivalent. ## Application to the orthogonalization problem {#application-to-the-orthogonalization-problem .unnumbered} We end the introduction with an application to the following orthogonalization problem: given a ${\rm II}_1$ factor $M$ and two subsets $X,Y\subset M\ominus\mathbb C1$, when can we find $u\in\mathscr U(M)$ such that $uXu^*$ and $Y$ are orthogonal? In the case $X=A\ominus\mathbb C1$ and $Y=B\ominus \mathbb C1$, for von Neumann subalgebras $A,B\subset M$, this and related independence problems have been studied extensively by Popa (see e.g. [@Po13a; @Po13b; @Po17]). When $X,Y\subset M\ominus\mathbb C1$ are finite, a standard averaging argument shows that we can find $u\in\mathscr U(M)$ such that $uXu^*$ and $Y$ are "almost orthogonal\": for every $\varepsilon>0$, there exists $u\in\mathscr U(M)$ such that $|\langle uxu^*,y\rangle|<\varepsilon$, for every $x\in X,y\in Y$. This implies the existence of $v\in\mathscr U(M^{\mathcal U})$, where $\mathcal U$ is a free ultrafilter on $\mathbb N$, such that $vXv^*$ and $Y$ are orthogonal. In this context, much more is true: by a result of Popa (see [@Po13a Corollary 0.2]), if $X,Y\subset M\ominus \mathbb C1$ are countable, then there exists $u\in\mathscr U(M^{\mathcal U})$ such that $uXu^*$ and $Y$ are freely independent. By combining this result with the proof of our lifting theorem (Theorem [Theorem 14](#lifting){reference-type="ref" reference="lifting"}) we settle affirmatively the above orthogonalization problem whenever $X,Y\subset M\ominus\mathbb C1$ are finite. **Theorem 8**. *Let $M$ be a ${\rm II}_1$ factor and $X,Y\subset M\ominus\mathbb C1$ be finite sets.* *Then there exists $u\in\mathscr U(M)$ such that $uXu^*$ and $Y$ are orthogonal.* ## Acknowledgements {#acknowledgements .unnumbered} This work was initiated when CH was visiting the University of California at San Diego (UCSD) in March 2023. He thanks the Department of Mathematics at UCSD for its kind hospitality. The authors thank Ben Hayes, Srivatsav Kunnawalkam Elayavalli and Sorin Popa for their useful comments. # Preliminaries ## Noncommutative $\operatorname{L}^p$-spaces Let $(M, \tau)$ be a tracial von Neumann algebra. For every $p \in [1, +\infty)$, we write $\operatorname{L}^p(M) = \operatorname{L}^p(M, \tau)$ for the completion of $M$ with respect to the norm $\|\cdot\|_p$ defined by $\|x\|_p = \tau(|x|^p)^{1/p}$ for every $x \in M$. More generally, given a subspace $\mathscr W\subset M$, we denote by $\operatorname{L}^p(\mathscr W)\subset\operatorname{L}^p(M)$ the closure of $\mathscr W$ with respect to $\|\cdot\|_p$. Then $\operatorname{L}^p(M)$ is the noncommutative $\operatorname{L}^p$-space associated with the tracial von Neumann algebra $M$. We simply write $\operatorname{L}^\infty(M) = M$. We will use the following generalized noncommutative Hölder inequality (see e.g. [@Ta03 Theorem IX.2.13]): for all $k \geq 2$, all $p_1, \dots, p_k, r \in [1, +\infty)$ such that $\frac1r = \sum_{j = 1}^k \frac{1}{p_j}$ and all $(x_j)_j \in \prod_{j = 1}^k \operatorname{L}^{p_j}(M)$, we have $x = x_1 \cdots x_k \in \operatorname{L}^r(M)$ and $\|x_1 \cdots x_k\|_r \leq \|x_1\|_{p_1} \cdots \|x_k\|_{p_k}$. For all $1 \leq p \leq q < +\infty$ and all $x \in M$, we have $\|x\|_1 \leq \|x\|_p \leq \|x\|_q \leq \|x\|_\infty$ and so we may regard $M \subset \operatorname{L}^q(M) \subset \operatorname{L}^p(M) \subset \operatorname{L}^1(M)$. ## Ultraproduct von Neumann algebras Let $\mathcal U$ be a nonprincipal ultrafilter on $\mathbb{N}$. Whenever $(M, \tau)$ is a tracial von Neumann algebra, we denote by $(M^{\mathcal U}, \tau^{\mathcal U})$ the tracial ultraproduct von Neumann algebra. We regard $\operatorname{L}^2(M^{\mathcal U}) \subset \operatorname{L}^2(M)^{\mathcal U}$ as a closed subspace and we denote by $e : \operatorname{L}^2(M)^{\mathcal U} \to \operatorname{L}^2(M^{\mathcal U})$ the corresponding orthogonal projection. Recall the following elementary yet useful facts. **Lemma 1**. *Keep the same notation as above. The following assertions hold:* - *Let $(\xi_n)_n$ be a $\|\cdot\|_2$-bounded sequence in $\operatorname{L}^2(M)$ and set $\xi = (\xi_n)^{\mathcal U} \in \operatorname{L}^2(M)^{\mathcal U}$. Then $\xi \in \operatorname{L}^2(M^{\mathcal U})$ if and only if for every $\varepsilon > 0$, there exists a $\|\cdot\|_\infty$-bounded sequence $(x_n)_n$ in $M$ such that $\lim_{n \to \mathcal U} \|\xi_n - x_n \|_2 \leq \varepsilon$.* - *Let $r > 2$. Then for every $\|\cdot\|_r$-bounded sequence $(\xi_n)_n$ in $\operatorname{L}^r(M)$, we have $\xi = (\xi_n)^{\mathcal U} \in \operatorname{L}^2(M^{\mathcal U})$.* - *Let $(\xi_n)_n$ be a $\|\cdot\|_2$-bounded sequence in $\operatorname{L}^2(M)$. Let $(x_n)_n$ and $(y_n)_n$ be $\|\cdot\|_\infty$-bounded sequences in $M$. Set $\xi = (\xi_n)^{\mathcal U} \in \operatorname{L}^2(M)^{\mathcal U}$, $x = (x_n)^{\mathcal U} \in M^{\mathcal U}$ and $y = (y_n)^{\mathcal U} \in M^{\mathcal U}$. If $\xi \in \operatorname{L}^2(M^{\mathcal U})$, then $(x_n \xi_n y_n)^{\mathcal U} = x \xi y \in \operatorname{L}^2(M^{\mathcal U})$.* *Proof.* $(\rm i)$ It is straightforward. $(\rm ii)$ Let $(\xi_n)_n$ be a $\|\cdot\|_r$-bounded sequence in $\operatorname{L}^r(M)$. There exists $\kappa > 0$ such that $\sup_{n \in \mathbb{N}} \tau(|\xi_n|^r) < \kappa$. Set $\xi = (\xi_n)^{\mathcal U} \in \operatorname{L}^2(M)^{\mathcal U}$. For every $n \in \mathbb{N}$, write $\xi_n = v_n |\xi_n|$ for the polar decomposition of $\xi_n \in \operatorname{L}^r(M)$. For every $n \in \mathbb{N}$ and every $t > 0$, define the spectral projection $p_{n, t} = \mathbf 1_{[0, t]}(|\xi_n|) \in M$. For every $n \in \mathbb{N}$ and every $t > 0$, we have $$\|\xi_n \, p_{n, t}^\perp\|_2^2 \leq \| |\xi_n| \, p_{n, t}^\perp \|_2^2 = \tau(|\xi_n|^2 \mathbf 1_{(t, +\infty)}(|\xi_n|)) \leq \frac{1}{t^{r - 2}} \tau(|\xi_n|^r \mathbf 1_{(t, +\infty)}(|\xi_n|)) \leq \frac{\kappa}{t^{r - 2}}.$$ Let $\varepsilon > 0$ and choose $t > 0$ large enough so that $\frac{\kappa}{t^{r - 2}} \leq \varepsilon^2$. For every $n \in \mathbb{N}$, set $x_n = \xi_n \, p_{n, t}\in M$ and observe that we have $\|\xi_n - x_n\|_2 = \|\xi_n \, p_{n, t}^\perp\|_2 \leq \varepsilon$. Since $\sup \left\{ \|x_n\|_\infty \mid n \in \mathbb{N}\right \} \leq t$, Item $(\rm i)$ implies that $\xi = (\xi_n)^{\mathcal U} \in \operatorname{L}^2(M^{\mathcal U})$. $(\rm iii)$ Assume that $\xi = (\xi_n)^{\mathcal U} \in \operatorname{L}^2(M^{\mathcal U})$. Choose $\kappa > 0$ large enough so that $\sup \left\{ \|x_n\|_\infty, \|y_n\|_\infty \mid n \in \mathbb{N}\right\} \leq \kappa$. Let $\varepsilon > 0$. By Item $(\rm i)$, there exists a $\|\cdot\|_\infty$-bounded sequence $(z_n)_n$ in $M$ such that $\lim_{n \to \mathcal U} \|\xi_n - z_n\|_2 \leq \varepsilon$. Set $z = (z_n)^{\mathcal U} \in M^{\mathcal U}$. Then $\|\xi - z\|_2 = \lim_{n \to \mathcal U} \|\xi_n - z_n\|_2 \leq \varepsilon$. Since $x z y = (x_n z_n y_n)^{\mathcal U} \in M^{\mathcal U}$, we have $$\begin{aligned} \|(x_n \xi_n y_n)^{\mathcal U} - x \xi y\|_2 &\leq \|(x_n \xi_n y_n)^{\mathcal U} - (x_n z_n y_n)^{\mathcal U} \|_2 + \| x z y - x \xi y\|_2 \\ &= \lim_{n \to \mathcal U} \|x_n (\xi_n - z_n) y_n\|_2 + \| x (z - \xi) y\|_2 \\ &\leq 2 \kappa^2 \|\xi - z\|_2 \leq 2 \kappa^2 \varepsilon.\end{aligned}$$ Since this holds for every $\varepsilon > 0$, it follows that $(x_n \xi_n y_n)^{\mathcal U} = x \xi y \in \operatorname{L}^2(M^{\mathcal U})$. ◻ We also record the following basic fact concerning tracial ultraproducts: **Lemma 2**. *Keep the same notation as above. Let $(x_n)_n$ be a $\|\cdot\|_\infty$-bounded sequence in $M$. Set $x=(x_n)^{\mathcal U}\in M^{\mathcal U}$. Then for every $p\in [1,+\infty)$ we have $\|x\|_p=\lim_{n\rightarrow\mathcal U}\|x_n\|_p$.* *Proof.* We may assume that $\sup \left\{\|x_n\|_\infty\mid n\in\mathbb N\right\}\leq 1$ and thus $\|x\|_\infty\leq 1$. Note that $|x|^2=x^*x=(x_n^*x_n)^{\mathcal U}=(|x_n|^2)^{\mathcal U}$. Then for every $k\in\mathbb N$, we have $|x|^{2k}=(|x_n|^{2k})^{\mathcal U}$ and thus $\tau^{\mathcal U}(|x|^{2k})=\lim_{n\rightarrow\mathcal U}\tau(|x_n|^{2k})$. Thus, if $P(t)\in\mathbb C[t]$ is a polynomial with complex coefficients and $Q(t)=P(t^2)$, then $\tau^{\mathcal U}(Q(|x|))=\lim_{n\rightarrow\mathcal U}\tau(Q(|x_n|))$. Since by the Stone--Weierstrass theorem $\{P(t^2)\mid P(t)\in\mathbb C[t]\}$ is dense in $\operatorname{C}([0,1])$ in the uniform norm, we get that $\tau^{\mathcal U}(f(|x|))=\lim_{n\rightarrow\mathcal U}\tau(f(|x_n|))$, for every $f\in \operatorname{C}([0,1])$. In particular, $\tau^{\mathcal U}(|x|^p)=\lim_{n\rightarrow\mathcal U}\tau(|x_n|^p)$, for every $p\in [1,+\infty)$, which implies the conclusion. ◻ ## Amalgamated free products {#subsection-amalgamated} Let $I$ be an at most countable index set such that $2 \leq | I | \leq +\infty$. Let $(M_i, \tau_i)_{i \in I}$ be a family of tracial von Neumann algebras with a common von Neumann subalgebra $(B, \tau_0)$ such that for every $i \in I$, we have $\tau_i|_B = \tau_0$. Denote by $(M, \tau) = \ast_{B, i \in I} (M_i, \tau_i)$ the tracial amalgamated free product von Neumann algebra. Simply denote by $\operatorname{E}_B : M \to B$ the unique trace-preserving faithful normal conditional expectation. Denote by $\mathscr W \subset M$ the linear span of $B$ and of all the reduced words in $M$ of the form $w = w_1 \cdots w_n$, with $n \geq 1$, $w_j \in M_{\varepsilon_j} \ominus B$ for every $j \in \{1, \dots, n\}$, and $\varepsilon_1, \dots, \varepsilon_n \in I$ such that $\varepsilon_1 \neq \cdots \neq \varepsilon_n$. For every subset $J \subset I$, denote by $\mathscr L_J \subset \mathscr W$ (resp. $\mathscr R_J \subset \mathscr W$) the linear span of all the reduced words whose first (resp. last) letter lies in $M_j \ominus B$ for some $j \in J$. For every $i \in I$, denote by $\mathscr W_i \subset \mathscr W$ the linear span of all the reduced words whose first and last letter lie in $M_i \ominus B$. We will use the following consequences of Mei--Ricard's results (see [@MR16 Theorem 3.5]). **Theorem 3** (Mei--Ricard [@MR16]). *Let $p \in (1, +\infty)$, $J \subset I$, and $i \in I$. The following assertions hold:* - *The projection map $P_{\mathscr L_J} : \mathscr W \to \mathscr L_J$ extends to a completely bounded operator $P_{\mathscr L_J} : \operatorname{L}^p(M) \to \operatorname{L}^p(\mathscr L_J)$.* - *The projection map $P_{\mathscr R_J} : \mathscr W \to \mathscr R_J$ extends to a completely bounded operator $P_{\mathscr R_J} : \operatorname{L}^p(M) \to \operatorname{L}^p(\mathscr R_J)$.* - *The projection map $P_{\mathscr W_i} : \mathscr W \to \mathscr W_i$ extends to a completely bounded operator $P_{\mathscr W_i} : \operatorname{L}^p(M) \to \operatorname{L}^p(\mathscr W_i)$.* *Proof.* We use the notation $H_\varepsilon$ of [@MR16 Section 3]. $(\rm i)$ For every $j \in J$, set $\varepsilon_j = - 1$ and for every $j \in I\setminus J$, set $\varepsilon_j = 1$. Then with $\varepsilon = (\varepsilon_i)_{i \in I}$, we have $P_{\mathscr L_J} = \frac{\operatorname{Id}- H_\varepsilon}{2}$. Therefore, [@MR16 Theorem 3.5] implies that $P_{\mathscr L_J} : \mathscr W \to \mathscr L_J$ extends to a completely bounded operator $P_{\mathscr L_J} : \operatorname{L}^p(M) \to \operatorname{L}^p(\mathscr L_J)$. $(\rm ii)$ The proof is completely analogous to Item $(\rm i)$. $(\rm iii)$ We have $P_{\mathscr W_i} = P_{\mathscr L_{i}} \circ P_{\mathscr R_{i}} = P_{\mathscr R_{i}} \circ P_{\mathscr L_{i}}$. Therefore, Items $(\rm i)$ and $(\rm ii)$ imply that $P_{\mathscr W_i} : \mathscr W \to \mathscr W_i$ extends to a completely bounded operator $P_{\mathscr W_i} : \operatorname{L}^p(M) \to \operatorname{L}^p(\mathscr W_i)$. ◻ Let $P_p$ be one of the operators from Theorem [Theorem 3](#thm-Mei-Ricard){reference-type="ref" reference="thm-Mei-Ricard"} (i.e., $P_{\mathscr L_J}, P_{\mathscr R_J}$, or $P_{\mathscr W_i}$). Then the operators $(P_p)_{p\in (1,+\infty)}$ are consistent with the inclusions $\operatorname{L}^q(M)\subset\operatorname{L}^p(M)$, in the sense that $P_q=P_p |_{\operatorname{L}^q(M)}$, for every $1<p\leq q<+\infty$. This is why we denote $P$ instead of $P_p$. ## Popa's intertwining theory We review Popa's criterion for intertwining von Neumann subalgebras [@Po01; @Po03]. Let $(M, \tau)$ be a tracial von Neumann algebra and $A\subset 1_A M 1_A$, $B \subset 1_B M 1_B$ be von Neumann subalgebras. By [@Po03 Corollary 2.3] and [@Po01 Theorem A.1] (see also [@Va06 Proposition C.1]), the following conditions are equivalent: - There exist $n \geq 1$, a projection $q \in \mathbf M_n(B)$, a nonzero partial isometry $v \in \mathbf M_{1, n}(1_A M)q$ and a unital normal $\ast$-homomorphism $\pi : A \to q\mathbf M_n(B)q$ such that $a v = v \pi(a)$ for all $a \in A$. - There exist projections $p \in A$ and $q \in B$, a nonzero partial isometry $v \in pMq$ and a unital normal $\ast$-homomorphism $\pi : pAp \to qBq$ such that $a v = v \pi(a)$ for all $a \in A$. - There is no net of unitaries $(w_k)_k$ in $A$ such that $$\forall x, y \in 1_A M 1_B, \quad \lim_k \|\operatorname{E}_B(x^* w_k y)\|_2 = 0.$$ If one of the previous equivalent conditions is satisfied, we say that $A$ *embeds into* $B$ *inside* $M$ and write $A \preceq_M B$. Following [@Jo82; @PP84], we say that an inclusion of tracial von Neumann algebras $P \subset M$ has *finite index* if $\operatorname{L}^2(M, \tau)$ has finite dimension as a right $P$-module. If $A_0 \subset A$ is a von Neumann subalgebra with finite index and if $A \preceq_M B$, then $A_0 \preceq_M B$ (see [@Va07 Lemma 3.9]). We record the following new criterion for intertwining von Neumann subalgebras. **Lemma 4**. *Let $(M,\tau)$ be a separable tracial von Neumann algebra and $A,B\subset M$ be von Neumann subalgebras such that $A\npreceq_MB$. Then there exists $u\in\mathscr U(A^{\mathcal U})$ such that $\operatorname{E}_{B^{\mathcal U}}(xu^my)=0$, for all $x,y\in M$ and all $m\in\mathbb Z\setminus\{0\}$.* *Proof.* To prove the lemma, it suffices to argue that for every finite subset $F\subset M$, $\varepsilon>0$ and $K\in\mathbb N$, we can find $u\in\mathscr U(A)$ such that $\|\operatorname{E}_B(xu^my^*)\|_2<\varepsilon$, for all $m\in\mathbb Z\setminus\{0\}$ with $|m|\leq K$. To this end, fix a finite subset $F\subset M$, $\varepsilon>0$ and $K\in\mathbb N$. For $u\in\mathscr U(M)$, set $\psi(u)=\sum_{m\in \mathbb Z\setminus\{0\},|m|\leq K}\sum_{x,y\in F}\|\operatorname{E}_B(xu^my^*)\|_2^2$. Let $v\in\mathscr U(M)$ with $\{v\}^{\prime\prime}\npreceq_MB$. For every $N\in\mathbb N$, set $$\varphi(v,N)=\frac{1}{N}\sum_{k=1}^N\sum_{x,y\in F}\|\operatorname{E}_B(xv^ky^*)\|_2^2.$$ We claim that $\lim_{N \to \infty}\varphi(v,N)=0$. Indeed, set $\xi=\sum_{x\in F}xe_Bx^*\in\langle M,B\rangle$, where $(\langle M, B\rangle, \operatorname{Tr})$ is Jones basic construction of $B\subset M$. Using that $\|\operatorname{E}_B(z)\|_2^2=\text{Tr}(ze_Bz^*e_B)$ for every $z\in M$, we obtain that $$\label{average} \forall N \in \mathbb{N}, \quad \varphi(v,N)=\operatorname{Tr}\left( \left(\frac{1}{N}\sum_{k=1}^Nv^k\xi {v^{-k}} \right)\xi \right).$$ By von Neumann's ergodic theorem, there exists $\eta\in\operatorname{L}^2(\langle M,B\rangle, \operatorname{Tr})$ such that $v\eta v^*=\eta$ and $\lim_{N \to \infty}\|\frac{1}{N}\sum_{k=1}^Nv^k\xi{v^{-k}}-\eta\|_{2, \operatorname{Tr}}=0$. Then $w\eta=\eta w$, for all $w\in\{v\}^{\prime\prime}$. Since $\{v\}^{\prime\prime}\npreceq_MB$, we obtain that $\eta=0$. In combination with [\[average\]](#average){reference-type="eqref" reference="average"}, this proves our claim that $\lim_{N\to\infty}\varphi(v,N)=0$. We are now ready to finish the proof. Since $A\npreceq_MB$, we can find a diffuse abelian von Neumann subalgebra $A_0\subset A$ such that $A_0\npreceq_MB$ (see [@BO08 Corollary F.14]). Let $v\in\mathscr U(A_0)$ be a Haar unitary with $\{v\}^{\prime\prime}=A_0$. If $m\in\mathbb Z\setminus\{0\}$, then $\{v^m\}^{\prime\prime}\subset A_0$ has finite index, and thus $\{v^m\}^{\prime\prime}\npreceq_MB$. The above claim gives that $\lim_{N \to \infty}\varphi(v^m,N)=0$, for all $m\in\mathbb Z\setminus\{0\}$. Thus, we can find $N\in\mathbb N$ such that $\sum_{m\in\mathbb Z\setminus\{0\},|m|\leq K}\varphi(v^m,N)<\varepsilon^2$. Since $\sum_{m\in\mathbb Z\setminus\{0\},|m|\leq K}\varphi(v^m,N)=\frac{1}{N}\sum_{k=1}^N\psi(v^k)$, we find $1\leq k\leq N$ such that $\psi(v^k)<\varepsilon^2$. Thus, $u=v^k$ satisfies the desired conclusion, which finishes the proof. ◻ # Proofs of Theorems [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"}, [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"}, [Theorem 3](#thm-main-result-bis){reference-type="ref" reference="thm-main-result-bis"}, [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} {#section-main-results} ## Popa's asymptotic orthogonality property Let $I$ be an at most countable index set such that $2 \leq | I | \leq +\infty$. Let $(M_i, \tau_i)_{i \in I}$ be a family of tracial von Neumann algebras with a common von Neumann subalgebra $(B, \tau_0)$ such that for every $i \in I$, we have $\tau_i|_B = \tau_0$. Denote by $(M, \tau) = \ast_{B, i \in I} (M_i, \tau_i)$ the tracial amalgamated free product von Neumann algebra. Simply denote by $\operatorname{E}_B : M \to B$ (resp. $\operatorname{E}_{B^{\mathcal U}} : M^{\mathcal U} \to B^{\mathcal U}$) the unique trace-preserving faithful normal conditional expectation. The following lemma is a generalization of Popa's asymptotic orthogonality property (see [@Po83 Lemma 2.1]) in the framework of tracial amalgamated free product von Neumann algebras. The key new feature of the proof is that we exploit Theorem [Theorem 3](#thm-Mei-Ricard){reference-type="ref" reference="thm-Mei-Ricard"} to work inside the Hilbert space $\operatorname{L}^2(M^{\mathcal U})$ instead of $\operatorname{L}^2(M)^{\mathcal U}$ as in Popa's proof. **Lemma 5**. *Let $i \in I$. Let $u \in \mathscr U(M_i^{\mathcal U})$ be a unitary such that $\operatorname{E}_{B^{\mathcal U}}(u^k) = 0$ for every $k \in \mathbb{Z}\setminus \{0\}$. For every $x = (x_n)^{\mathcal U} \in \{u\}' \cap M^{\mathcal U}$ such that $\operatorname{E}_{B^{\mathcal U}}(x) = 0$, we have $\lim_{n \to \mathcal U} \|x_n - P_{\mathscr W_i}(x_n)\|_2 = 0$.* *Proof.* Let $x = (x_n)^{\mathcal U} \in \{u\}' \cap M^{\mathcal U}$ such that $\operatorname{E}_{B^{\mathcal U}}(x) = 0$. Without loss of generality, we may assume that $\|x_n\|_\infty \leq 1$ for every $n \in \mathbb{N}$. To prove that $\lim_{n \to \mathcal U} \|x_n - P_{\mathscr W_i}(x_n)\|_2 = 0$, we show that $\lim_{n \to \mathcal U} \|P_{\mathscr L_{I\setminus \{i\}}}(x_n)\|_2 = \lim_{n \to \mathcal U} \|P_{\mathscr R_{I\setminus \{i\}}}(x_n)\|_2 = 0$. Since $\mathscr R_{I \setminus \{i\}} = J \mathscr L_{I \setminus \{i\}} J$, it suffices to prove that $\lim_{n \to \mathcal U} \|P_{\mathscr L_{I\setminus \{i\}}}(x_n)\|_2 = 0$. To simplify the notation, we set $P_i = P_{\mathscr L_{I\setminus \{i\}}}$. By Lemma [Lemma 1](#lem-ultraproduct){reference-type="ref" reference="lem-ultraproduct"}$(\rm ii)$ and Theorem [Theorem 3](#thm-Mei-Ricard){reference-type="ref" reference="thm-Mei-Ricard"}$(\rm i)$, we have $(P_i(x_n))^{\mathcal U} \in \operatorname{L}^2(M^{\mathcal U})$. Set $\mathscr H_i = \operatorname{L}^2(M^{\mathcal U}) \cap (\mathscr L_{I \setminus \{i\}})^{\mathcal U} \subset \operatorname{L}^2(M^{\mathcal U})$ and denote by $P_{\mathscr H_i} : \operatorname{L}^2(M^{\mathcal U}) \to \mathscr H_i$ the corresponding orthogonal projection. Then we have $P_{\mathscr H_i}(x) = (P_i(x_n))^{\mathcal U} \in \mathscr H_i$. For every $N \geq 1$, we have $$\begin{aligned} \label{eq-parallelogram} N \cdot \| P_{\mathscr H_i}(x)\|^2_2 &= \sum_{k =1}^N \|u^k P_{\mathscr H_i}(x) u^{-k}\|^2_2 \\ \nonumber &= \sum_{k =1}^N \|P_{u^{k}\mathscr H_i u^{-k}}(u^k x u^{-k})\|^2_2 \\ \nonumber &= \sum_{k =1}^N \|P_{ u^{k}\mathscr H_i u^{-k}}(x)\|^2_2.\end{aligned}$$ We claim that the Hilbert subspaces $( u^{k}\mathscr H_i u^{-k} )_{k \in \mathbb{Z}}$ are mutually orthogonal in $\operatorname{L}^2(M^{\mathcal U})$ i.e. for every $k \in \mathbb{Z}\setminus \{0\}$, $u^{k}\mathscr H_i u^{-k}$ and $\mathscr H_i$ are orthogonal in $\operatorname{L}^2(M^{\mathcal U})$. Indeed, for every $k \in \mathbb{Z}\setminus \{0\}$, since $\operatorname{E}_{B^{\mathcal U}}(u^k) = 0$, we may write $u^k = (u_{n, k})^{\mathcal U} \in M_i^{\mathcal U}$ where $(u_{n, k})_{n}$ is a $\|\cdot\|_\infty$-bounded sequence in $M_i \ominus B$. Let $\xi = (\xi_n)^{\mathcal U} \in \mathscr H_i$ and $\eta = (\eta_n)^{\mathcal U} \in \mathscr H_i$ where $(\xi_n)_n$ and $(\eta_n)_n$ are $\|\cdot\|_2$-bounded sequences in $\mathscr L_{I \setminus \{i\}}$. By construction, it is plain to see that for all $n \in \mathbb{N}$ and all $k \in \mathbb{Z}\setminus \{0\}$, the vectors $u_{n, k} \xi_n u_{n, k}^*$ and $\eta_n$ are orthogonal in $\operatorname{L}^2(M)$. By Lemma [Lemma 1](#lem-ultraproduct){reference-type="ref" reference="lem-ultraproduct"}$(\rm iii)$, since $\xi = (\xi_n)^{\mathcal U} \in \operatorname{L}^2(M^{\mathcal U})$, we have $$\langle u^k \xi u^{-k} , \eta \rangle = \langle (u_{n, k} \xi_n u_{n, k}^*)^{\mathcal U} , (\eta_n)^{\mathcal U} \rangle = \lim_{n \to \mathcal U} \langle u_{n, k} \xi_n u_{n, k}^* , \eta_n \rangle = 0.$$ This finishes the proof of the claim. From [\[eq-parallelogram\]](#eq-parallelogram){reference-type="eqref" reference="eq-parallelogram"}, since the projections $(P_{u^k \mathscr H_i u^{-k}})_{k \in \mathbb{Z}}$ are mutually orthogonal, we infer that $$\label{eq-conclusion} N \cdot \| P_{\mathscr H_i}(x)\|^2_2 = \sum_{k =1}^N \|P_{ u^{k}\mathscr H_i u^{-k}}(x)\|^2_2 \leq \|x\|_2^2.$$ Since [\[eq-conclusion\]](#eq-conclusion){reference-type="eqref" reference="eq-conclusion"} holds for every $N \geq 1$, we have $\lim_{n \to \mathcal U} \|P_i(x_n)\|_2 = \| P_{\mathscr H_i}(x) \|_2 = 0$. This finishes the proof of the lemma. ◻ Using a $2 \times 2$ matrix trick, we obtain the following extension of Lemma [Lemma 5](#lem-commutant){reference-type="ref" reference="lem-commutant"}. **Lemma 6**. *Let $i \in I$. Let $u \in \mathscr U(M_i^{\mathcal U})$ be a unitary such that $\operatorname{E}_{B^{\mathcal U}}(v u^k v^*) = 0$ for every $k \in \mathbb{Z}\setminus \{0\}$ and every $v \in \mathscr U(M_i)$. For every $x = (x_n)^{\mathcal U} \in \{u\}' \cap M^{\mathcal U}$ such that $\operatorname{E}_{M_i^{\mathcal U}}(x) = 0$ and every $y, z \in M_i$, we have $\lim_{n \to \mathcal U} \|y x_n z - P_{\mathscr W_i}(y x_n z)\|_2 = 0$.* *Proof.* Set $\mathscr B = \mathbf M_2(B)$, $\mathscr M_j = \mathbf M_2(M_j)$ for every $j \in I$ and $\mathscr M = \mathbf M_2(M)$ so that we have $\mathscr M = \ast_{\mathscr B, j \in I} \mathscr M_j$. Let $x = (x_n)^{\mathcal U} \in \{u\}' \cap M^{\mathcal U}$ be such that $\operatorname{E}_{M_i^{\mathcal U}}(x) = 0$. Since any element of $M_i$ is a linear combination of at most four unitaries of $M_i$, it suffices to prove that for every $v, w \in \mathscr U(M_i)$, we have $\lim_{n \to \mathcal U} \|v x_n w - P_{\mathscr W_i}(v x_n w)\|_2 = 0$. Let $v, w \in \mathscr U(M_i)$. Set $$U = \begin{pmatrix} v uv^* & 0 \\ 0 & w^* u w \end{pmatrix} \in \mathscr U(\mathscr M_i^{\mathcal U}) \quad \text{and} \quad X = \begin{pmatrix} 0 & v x w \\ 0 & 0 \end{pmatrix} \in \mathscr M^{\mathcal U} \ominus \mathscr M_i^{\mathcal U}.$$ By construction, we have $U X = X U$ and $\operatorname{E}_{\mathscr B^{\mathcal U}}(U^k) = 0$ for every $k \in \mathbb{Z}\setminus \{0\}$. We may now apply Lemma [Lemma 5](#lem-commutant){reference-type="ref" reference="lem-commutant"} to $X = (X_n)^{\mathcal U} \in \mathscr M^{\mathcal U} \ominus \mathscr B^{\mathcal U}$ and conclude that $$\lim_{n \to \mathcal U} \|v x_n w - P_{\mathscr W_i}(v x_n w)\|_2 = \lim_{n \to \mathcal U} \| (X_n)_{12} - P_{\mathscr W_i}((X_n)_{12})\|_2 = 0.$$ This finishes the proof of the lemma. ◻ ## Proofs of Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"}, Theorem [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"}, Theorem [Theorem 3](#thm-main-result-bis){reference-type="ref" reference="thm-main-result-bis"} {#proofs-of-theorem-thm-main-result-theorem-cor-rel-comm-theorem-thm-main-result-bis} *Proof of Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"}.* Keep the same notation as in the statement of Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"}. Let $k \geq 1$ and $\varepsilon_1, \dots, \varepsilon_k \in I$ be such that $\varepsilon_1 \neq \cdots \neq \varepsilon_k$. For every $1 \leq j \leq k$, let $x_j = (x_{j, n})^{\mathcal U} \in \mathbf X_{\varepsilon_j}$. We may assume that $\sup \left\{\|x_{j, n}\|_\infty \mid 1 \leq j \leq k, n \in \mathbb{N}\right\} \leq 1$. We show that $\operatorname{E}_{B^{\mathcal U}}(x_1 \cdots x_k) = 0$. For every $i \in I$ and every $p \in (1, +\infty)$, we simply denote by $P_i : \operatorname{L}^p(M) \to \operatorname{L}^p(\mathscr W_i)$ the completely bounded operator (see Theorem [Theorem 3](#thm-Mei-Ricard){reference-type="ref" reference="thm-Mei-Ricard"}$(\rm iii)$). For every $p \in (1, +\infty)$, choose $\kappa_p > 0$ large enough so that $$\sup \left\{\|P_i(x)\|_p \mid i \in \{\varepsilon_1, \dots, \varepsilon_k\}, x \in \operatorname{L}^p(M), \|x\|_p \leq 1 \right\} \leq \kappa_p.$$ Fix $1 < r < 2$ (e.g., $r = \frac32$). Let $p \in (1, +\infty)$ be such that $\frac1r = \frac12 + \frac{k - 1}{p}$. For every $1 \leq j \leq k$ and every $n \in \mathbb{N}$, write $x_{j, n} = P_{\varepsilon_j}(x_{j, n}) + (x_{j, n} - P_{\varepsilon_j}(x_{j, n}))$ and observe that - $P_{\varepsilon_j}(x_{j, n}) \in \operatorname{L}^2(\mathscr W_{\varepsilon_j})$ and $\lim_{n \to \mathcal U} \|x_{j, n} - P_{\varepsilon_j}(x_{j, n})\|_2 = 0$; - $P_{\varepsilon_j}(x_{j, n}) \in \operatorname{L}^p(\mathscr W_{\varepsilon_j})$ and $\max \left\{ \|P_{\varepsilon_j}(x_{j, n}) \|_{p}, \|x_{j, n} - P_{\varepsilon_j}(x_{j, n})\|_p \right\} \leq 1 + \kappa_p$. For every $n \in \mathbb{N}$, we may write $x_{1, n} \cdots x_{k, n} - P_{\varepsilon_1}(x_{1, n}) \cdots P_{\varepsilon_k}(x_{k, n})$ as a sum of $2^k - 1$ terms that are products of length $k$ for which at least one of the factors is of the form $x_{j, n} - P_{\varepsilon_j}(x_{j, n})$ for some $1 \leq j \leq k$. For every $n \in \mathbb{N}$, using the triangle inequality and the generalized noncommutative Hölder inequality, we obtain $$\|x_{1, n} \cdots x_{k, n} - P_{\varepsilon_1}(x_{1, n}) \cdots P_{\varepsilon_k}(x_{k, n})\|_r \leq (2^{k} - 1) (1 + \kappa_p)^{k - 1} \max_j \|x_{j, n} - P_{\varepsilon_j}(x_{j, n})\|_2.$$ This implies that $$\label{eq-1} \lim_{n \to \mathcal U} \|x_{1, n} \cdots x_{k, n} - P_{\varepsilon_1}(x_{1, n}) \cdots P_{\varepsilon_k}(x_{k, n})\|_r = 0.$$ Next, set $q = kr$ so that $\frac1r = \frac{k}{q}$. For every $1 \leq j \leq k$ and every $n \in \mathbb{N}$, since $P_{\varepsilon_j}(x_{j, n}) \in \operatorname{L}^q(\mathscr W_{\varepsilon_j})$, we may choose $w_{j, n} \in \mathscr W_{\varepsilon_j}$ such that $\|P_{\varepsilon_j}(x_{j, n}) - w_{j, n}\|_q \leq \frac{1}{n + 1}$. For every $1 \leq j \leq k$ and every $n \in \mathbb{N}$, write $P_{\varepsilon_j}(x_{j, n}) = w_{j, n} + (P_{\varepsilon_j}(x_{j, n}) - w_{j, n})$ and observe that - $\max \left\{ \|w_{j, n} \|_{q}, \|P_{\varepsilon_j}(x_{j, n}) - w_{j, n} \|_q \right\} \leq 1 + \kappa_q$. We may then write $P_{\varepsilon_1}(x_{1, n}) \cdots P_{\varepsilon_k}(x_{k, n}) - w_{1, n} \cdots w_{k, n}$ as a sum of $2^k - 1$ terms that are products of length $k$ for which at least one of the factors is of the form $P_{\varepsilon_j}(x_{j, n}) - w_{j, n}$ for some $1 \leq j \leq k$. For every $n\in \mathbb{N}$, using the triangle inequality and the generalized noncommutative Hölder inequality, we obtain $$\|P_{\varepsilon_1}(x_{1, n}) \cdots P_{\varepsilon_k}(x_{k, n}) - w_{1, n} \cdots w_{k, n}\|_r \leq (2^{k} - 1) (1 + \kappa_q)^{k - 1} \max_j \|P_{\varepsilon_j}(x_{j, n}) - w_{j, n}\|_q.$$ This implies that $$\label{eq-2} \lim_{n \to \mathcal U} \| P_{\varepsilon_1}(x_{1, n}) \cdots P_{\varepsilon_k}(x_{k, n}) - w_{1, n} \cdots w_{k, n}\|_r = 0.$$ By combining [\[eq-1\]](#eq-1){reference-type="eqref" reference="eq-1"} and [\[eq-2\]](#eq-2){reference-type="eqref" reference="eq-2"}, it follows that $$\label{eq-3} \lim_{n \to \mathcal U} \| x_{1, n} \cdots x_{k, n} - w_{1, n} \cdots w_{k, n}\|_r = 0.$$ In particular, since $\operatorname{E}_{B^{\mathcal U}} (x_1 \cdots x_k)=(\operatorname{E}_B (x_{1, n} \cdots x_{k, n}))^{\mathcal U}$, using Lemma [Lemma 2](#p-norm){reference-type="ref" reference="p-norm"} and the fact that $\operatorname{E}_B$ is $\|\cdot\|_r$-contractive, we have $$\|\operatorname{E}_{B^{\mathcal U}} (x_1 \cdots x_k) \|_r = \lim_{n \to \mathcal U} \|\operatorname{E}_B (x_{1, n} \cdots x_{k, n}) \|_r = \lim_{n \to \mathcal U} \|\operatorname{E}_B (w_{1, n} \cdots w_{k, n}) \|_r.$$ Since for every $1 \leq j \leq k$ and every $n \in \mathbb{N}$, we have $w_{j, n} \in \mathscr W_{\varepsilon_j}$ and since $\varepsilon_1 \neq \cdots \neq \varepsilon_k$, it follows that $\operatorname{E}_B(w_{1, n} \cdots w_{k, n}) = 0$. Thus, we obtain $\operatorname{E}_{B^{\mathcal U}}(x_1 \cdots x_k) =0$. ◻ **Remark 7**. We were informed by Sorin Popa that he and Stefaan Vaes had recently made the following observation. In the case $\operatorname{L}(\mathbb{F}_2) = A_1 \ast A_2$ is a free group factor with $A_1 \cong A_2 \cong \operatorname{L}(\mathbb{Z})$, they showed that $A_1' \cap \operatorname{L}(\mathbb{F}_2)^{\mathcal U}$ and $A_2$ are freely independent in $\operatorname{L}(\mathbb{F}_2)^{\mathcal U}$ with respect to $\tau^{\mathcal U}$. We obtain the following consequence of Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"} which implies Theorem [Theorem 2](#cor-rel-comm){reference-type="ref" reference="cor-rel-comm"}. **Theorem 8**. *Assume that $B = \mathbb{C}1$. For every $i \in I$, let $(A_{i, k})_{k \in \mathbb{N}}$ be a decreasing sequence of separable diffuse abelian von Neumann subalgebras of $M_i^{\mathcal U}$ such that $\bigcap_{k = 1}^\infty A_{i, k} = \mathbb{C}1$.* *Then for every $i \in I$, $\mathscr M_i = \bigvee_{k = 1}^\infty (A_{i, k}' \cap M^{\mathcal U}) \subset M^{\mathcal U}$ is a nonamenable irreducible subfactor with property Gamma. Moreover, the family $(\mathscr M_i)_{i \in I}$ is freely independent in $M^{\mathcal U}$ with respect to $\tau^{\mathcal U}$.* *Proof.* Let $i \in I$. For every $k \in \mathbb{N}$, since $A_{i, k}$ is separable, we have $(A_{i, k}' \cap M^{\mathcal U})' \cap M^{\mathcal U} = A_{i, k}$ by [@Po13a Theorem 2.1]. This further implies that $$\mathscr M_i' \cap M^{\mathcal U} = (\bigvee_{k = 1}^\infty A_{i, k}' \cap M^{\mathcal U})' \cap M^{\mathcal U} = \bigcap_{k = 1}^\infty (A_{i, k}' \cap M^{\mathcal U})' \cap M^{\mathcal U} = \bigcap_{k = 1}^\infty A_{i, k} = \mathbb{C}1.$$ Therefore, $\mathscr M_i \subset M^{\mathcal U}$ is a irreducible subfactor. Since $A_{i, 0}' \cap M^{\mathcal U}$ is nonamenable, $\mathscr M_i$ is nonamenable as well. Let $\mathscr V$ be another nonprincipal ultrafilter on $\mathbb{N}$. For every $k \in \mathbb{N}$ and every $\lambda \in (0, 1)$, choose a projection $p_{\lambda, k} \in A_{i, k}$ such that $\tau^{\mathcal U}(p_{\lambda, k}) = \lambda$. Then $p_{\lambda} = (p_{\lambda, k})^{\mathcal V} \in \mathscr M_i' \cap \mathscr M_i^{\mathcal V}$ is a projection such that $(\tau^{\mathcal U})^{\mathcal V}(p_\lambda) = \lambda$. Therefore, $\mathscr M_i' \cap \mathscr M_i^{\mathcal V}$ is a diffuse von Neumann algebra and so $\mathscr M_i$ has property Gamma. A combination of Lemma [Lemma 5](#lem-commutant){reference-type="ref" reference="lem-commutant"} and Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"} implies that all $k, \ell \in \mathbb{N}$, the family $(A_{i, k}' \cap M^{\mathcal U})_{i \in I}$ is freely independent in $M^{\mathcal U}$ with respect to $\tau^{\mathcal U}$. Since for every $i \in I$, the sequence of von Neumann subalgebras $(A_{i, k}' \cap M^{\mathcal U})_k$ is increasing, Kaplansky's density theorem further implies that the family $(\mathscr M_i)_{i \in I}$ is freely independent in $M^{\mathcal U}$ with respect to $\tau^{\mathcal U}$. ◻ *Proof of Theorem [Theorem 3](#thm-main-result-bis){reference-type="ref" reference="thm-main-result-bis"}.* Keep the same notation as in Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"}. Let $\mathbf Y_1 \subset \mathbf X_1$ be a subset with the property that $a \mathbf Y_1 b \subset \mathbf X_1$ for all $a, b \in M_1$. Denote by $M_1 \mathbf Y_1 M_1$ the linear span of all the elements of the form $a Y b$ for $a, b \in M_1$ and $Y \in \mathbf Y_1$. Then we have $M_1 \mathbf Y_1 M_1 \subset \mathbf X_1$. Likewise, denote by $M_1 \mathscr W_2 M_1$ the linear span of all the elements of the form $a w b$ for $a, b \in M_1$ and $w \in \mathscr W_2$. Observe that any word with letters alternating from $\mathbf Y_1$ and $M_1 \mathscr W_2 M_1$ can be written as a linear combination of words with letters alternating from $M_1 \mathbf Y_1 M_1 \cup (M_1 \ominus B)$ and $\mathscr W_2$. Since $M_1 \mathbf Y_1 M_1 \cup (M_1 \ominus B) \subset \mathbf X_1$ and $\mathscr W_2 \subset \mathbf X_2$, Theorem [Theorem 1](#thm-main-result){reference-type="ref" reference="thm-main-result"} implies that the sets $\mathbf Y_1$ and $M_1 \mathscr W_2 M_1$ are freely independent in $M^{\mathcal U}$ with respect to $\operatorname{E}_{B^{\mathcal U}}$. Using Kaplansky's density theorem, for any element $x \in M \ominus M_1$, there exists a $\|\cdot\|_\infty$-bounded sequence $(x_n)_n$ in $M_1 \mathscr W_2 M_1$ such that $x_n \to x$ for the strong operator topology. Combining this fact with the first paragraph of the proof, we infer that the sets $\mathbf Y_1$ and $M \ominus M_1$ are freely independent in $M^{\mathcal U}$ with respect to $\operatorname{E}_{B^{\mathcal U}}$. ◻ ## Proof of Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} {#proof-of-theorem-31infinity} This subsection is devoted to the proof of Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"}. Moreover, we generalize Theorem [Theorem 4](#31infinity){reference-type="ref" reference="31infinity"} to arbitrary tracial amalgamated free product von Neumann algebras. **Theorem 9**. *Assume that $I=\{1,2\}$. Let $u_1\in\mathscr U(M_1^{\mathcal U})$ and $u_2\in\mathscr U(M_2^{\mathcal U})$ be such that $\operatorname{E}_{B^{\mathcal U}}(u_1^k)=0$, for every $k\in\mathbb Z\setminus\{0\}$, and $\operatorname{E}_{B^{\mathcal U}}(u_2)=\operatorname{E}_{B^{\mathcal U}}(u_2^2)=0$.* *Then there do not exist unitaries $v_1,v_2\in\mathscr U(M^{\mathcal U})$ such that $[u_1,v_1]=[v_1,v_2]=[v_2,u_2]=0$ and $\operatorname{E}_{B^{\mathcal U}}(v_1)=\operatorname{E}_{B^{\mathcal U}}(v_1^2)=\operatorname{E}_{B^{\mathcal U}}(v_2)=0$.* The proof of Theorem [Theorem 9](#3infinity){reference-type="ref" reference="3infinity"} relies on two lemmas. Using the notation from Section [2.3](#subsection-amalgamated){reference-type="ref" reference="subsection-amalgamated"}, for every $i,j\in I$, we put $\mathscr W_{i,j}=\mathscr L_i\cap\mathscr R_j$ and $P_{i,j}=P_{\mathscr L_i}\circ P_{\mathscr R_j}$. By Theorem [Theorem 3](#thm-Mei-Ricard){reference-type="ref" reference="thm-Mei-Ricard"}, we have a completely bounded operator $P_{i,j}:\operatorname{L}^p(M)\to \operatorname{L}^p(\mathscr W_{i,j})$, for every $p\in (1,+\infty)$. **Lemma 10**. *Assume that $I=\{1,2\}$. Let $u \in \mathscr U(M_1^{\mathcal U})$ be such that $\operatorname{E}_{B^{\mathcal U}}(u^k) = 0$ for all $k \in \mathbb{Z}\setminus \{0\}$. Let $x = (x_n)^{\mathcal U} \in \mathscr U(\{u\}' \cap M^{\mathcal U})$ and $y=(y_n)^{\mathcal U}\in \{x\}'\cap M^{\mathcal U}$ with $\operatorname{E}_{B^{\mathcal U}}(x)=\operatorname{E}_{B^{\mathcal U}}(y)=0$. Then $\lim_{n\rightarrow\mathcal U}\|P_{2,2}(y_n)\|_2^2=\lim_{n\rightarrow\mathcal U}\langle x_nP_{2,2}(y_n), P_{1,1}(y_n)x_n\rangle$. Thus, $\lim_{n\rightarrow \mathcal U}\|P_{2,2}(y_n)\|_2\leq\lim_{n\rightarrow\mathcal U}\|P_{1,1}(y_n)\|_2$.* *Proof.* Since $\operatorname{E}_{B^{\mathcal U}}(y)=0$, after replacing $y_n$ with $y_n-\operatorname{E}_B(y_n)$, we may assume that $\operatorname{E}_B(y_n)=0$, for all $n\in\mathbb N$. We may assume that $x_n\in\mathscr U(M)$ and $\|y_n\|_\infty\leq 1$, for all $n\in\mathbb N$. For $p\in (1,+\infty)$, let $\kappa_p=\sup \left\{\|P_{i,j}(x)\|_p\mid i,j\in \{1,2\}, x\in \operatorname{L}^p(M),\|x\|_p\leq 1\right\}.$ For every $n\in\mathbb N$ and $i,j\in\{1,2\}$, put $y_n^{i,j}=P_{{i,j}}(y_n)$. Then $y_n=\sum_{i,j=1}^2y_n^{i,j}$ and $\|y_n^{i,j}\|_p\leq\kappa_p$, for every $n\in\mathbb N,i,j\in\{1,2\}$ and $p\in (1,+\infty)$. We claim that $$\label{perp}\text{$\lim_{n\rightarrow\mathcal U}\|\operatorname{E}_B(x_ny_n^{2,2}x_n^*{y_n^{i,j}}^*)\|_1=0$, for every $(i,j)\not=(1,1)$.}$$ Let $(i,j)\not=(1,1)$. By Lemma [Lemma 5](#lem-commutant){reference-type="ref" reference="lem-commutant"}, we have $\lim_{n\rightarrow\mathcal U}\|x_n-P_{1,1}(x_n)\|_2=0$. By using the noncommutative Hölder inequality and that $1=\frac{1}{2}+3\cdot \frac{1}{6}$, we get that $$\|x_ny_n^{2,2}x_n^*{y_n^{i,j}}^*-P_{1,1}(x_n)y_n^{2,2}P_{1,1}(x_n)^*{y_n^{i,j}}^*\|_1\leq (\kappa_6^2+\kappa_6^3)\|x_n-P_{1,1}(x_n)\|_2.$$ This implies that $$\label{estimate1} \lim_{n\rightarrow\mathcal U}\|x_ny_n^{2,2}x_n^*{y_n^{i,j}}^*-P_{1,1}(x_n)y_n^{2,2}P_{1,1}(x_n)^*{y_n^{i,j}}^*\|_1=0.$$ Next, let $v_n\in\mathscr W_{1,1}$ and $w_n^{i,j}\in\mathscr W_{i,j}$ such that $\|P_{1,1}(x_n)-v_n\|_4\leq\frac{1}{n}$ and $\|y_n^{i,j}-w_n^{i,j}\|_4\leq\frac{1}{n}$, for every $n\in\mathbb N$ and $i,j\in\{1,2\}$. Then $\|v_n\|_4,\|w_n^{i,j}\|_4\leq\kappa_4+\frac{1}{n}\leq\kappa_4+1$. Since $1=4\cdot\frac{1}{4}$, applying the noncommutative Hölder inequality again gives that $$\|P_{1,1}(x_n)y_n^{2,2}P_{1,1}(x_n)^*{y_n^{i,j}}^*-v_nw_n^{2,2}v_n^*{w_n^{i,j}}^*\|_1\leq \frac{15(\kappa_4+1)^3}{n}.$$ This implies that $$\label{estimate2} \lim_{n\rightarrow\mathcal U}\|P_{1,1}(x_n)y_n^{2,2}P_{1,1}(x_n)^*{y_n^{i,j}}^*-v_nw_n^{2,2}v_n^*{w_n^{i,j}}^*\|_1=0.$$ By combining [\[estimate1\]](#estimate1){reference-type="eqref" reference="estimate1"} and [\[estimate2\]](#estimate2){reference-type="eqref" reference="estimate2"}, it follows that $$\label{estimate3}\lim_{n\rightarrow\mathcal U}\|x_ny_n^{2,2}x_n^*{y_n^{i,j}}^*-v_nw_n^{2,2}v_n^*{w_n^{i,j}}^*\|_1=0.$$ Now, $v_nw_n^{2,2}v_n^*{w_n^{i,j}}^*\in\mathscr W_{1,1}\mathscr W_{2,2}\mathscr W_{1,1}\mathscr W_{i,j}^*\subset \mathscr W_{1,1}\mathscr W_{i,j}^*$. Since $(i,j)\not=(1,1)$, $\mathscr W_{1,1}$ and $\mathscr W_{i,j}$ are orthogonal (algebraic) $B$-bimodules. Thus, $\operatorname{E}_B(\mathscr W_{1,1}\mathscr W_{i,j}^*)=\{0\}$ and therefore $\operatorname{E}_B(v_nw_n^{2,2}v_n^*{w_n^{i,j}}^*)=0$, for every $n\in\mathbb N$. In combination with [\[estimate3\]](#estimate3){reference-type="eqref" reference="estimate3"}, this proves [\[perp\]](#perp){reference-type="eqref" reference="perp"}. Finally, for every $n\in\mathbb N$, we have that $\|y_n^{2,2}\|_2^2=\langle y_n^{2,2},y_n\rangle=\langle x_ny_n^{2,2},x_ny_n\rangle$. Since $\lim_{n\rightarrow\mathcal U}\|x_ny_n-y_nx_n\|_2=0$, we get that $$\lim_{n\rightarrow\mathcal U}\|y_n^{2,2}\|_2^2=\lim_{n\rightarrow \mathcal U}\langle x_ny_n^{2,2},y_nx_n\rangle=\lim_{n\rightarrow\mathcal U}(\sum_{i,j=1}^2\langle x_ny_n^{2,2},y_n^{i,j}x_n\rangle).$$ On the other hand, [\[perp\]](#perp){reference-type="eqref" reference="perp"} gives that $\lim_{n\rightarrow\mathcal U}\langle x_ny_n^{2,2},y_n^{i,j}x_n\rangle=0$ if $(i,j)\not=(1,1)$. Thus, we get $\lim_{n\rightarrow\mathcal U}\|y_n^{2,2}\|_2^2=\lim_{n\rightarrow\mathcal U}\langle x_ny_n^{2,2}, y_n^{1,1}x_n\rangle$, which proves the main assertion. Since $|\langle x_ny_n^{2,2}, y_n^{1,1}x_n\rangle|\leq \|y_n^{2,2}\|_2\|y_n^{1,1}\|_2$, we get that $\lim_{n\rightarrow\mathcal U}\|y_n^{2,2}\|_2\leq\lim_{n\rightarrow\mathcal U}\|y_n^{1,1}\|_2$. ◻ **Lemma 11**. *In the setting of Lemma [Lemma 10](#almost-ortho){reference-type="ref" reference="almost-ortho"}, assume additionally that $\operatorname{E}_{B^{\mathcal U}}(x^2)=0$ and $y\in\{v\}'\cap M^{\mathcal U}$, for some $v=(v_n)^{\mathcal U}\in\mathscr U(M_2^{\mathcal U})$.* - *If $\operatorname{E}_{B^{\mathcal U}}(v)=0$, then $\lim_{n\rightarrow\mathcal U}\|P_{1,1}(y_n)\|_2=\lim_{n\rightarrow\mathcal U}\|P_{2,2}(y_n)\|_2=0$.* - *If $\operatorname{E}_{B^{\mathcal U}}(v)=\operatorname{E}_{B^{\mathcal U}}(v^2)=0$, then $y=0$.* *Proof.* We keep the notation from the proof of Lemma [Lemma 10](#almost-ortho){reference-type="ref" reference="almost-ortho"}. $(\rm i)$ Assume that $\operatorname{E}_{B^{\mathcal U}}(v)=0$. Write $v=(v_n)^{\mathcal U}$, where $v_n\in M_2\ominus B$, for every $n\in\mathbb N$, and $\sup\left\{\|v_n\|_\infty\mid n\in\mathbb N\right\}<\infty$. We first claim that $$\label{y_11y_22} \lim_{n\rightarrow\mathcal U}\|x_ny_n^{2,2}-y_n^{1,1}x_n\|_2=0.$$ Since $vyv^*=y$, we get that $\lim_{n\rightarrow\mathcal U}\|v_n^*y_nv_n-y_n\|_2=0$. Thus, we derive that $$\lim_{n\rightarrow\mathcal U}\langle v_ny_n^{1,1}v_n^*,y_n\rangle=\lim_{n\rightarrow\mathcal U}\langle y_n^{1,1},y_n\rangle=\lim_{n\rightarrow\mathcal U}\|y_n^{1,1}\|_2^2.$$ Since $(M_2\ominus B)\mathscr W_{1,1}(M_2\ominus B)\subset\mathscr W_{2,2}$ we also get that $P_{i,j}(v_ny_n^{1,1}v_n^*)=0$, for every $(i,j)\not=(2,2)$ and $n\in\mathbb N$. This implies that $\langle v_ny_n^{1,1}v_n^*,y_n\rangle=\langle v_ny_n^{1,1}v_n^*, y_n^{2,2}\rangle$, for every $n\in\mathbb N$. Since $|\langle v_ny_n^{1,1}v_n^*, y_n^{2,2}\rangle\leq \|y_n^{1,1}\|_2\|y_n^{2,2}\|_2$, for every $n\in\mathbb N$, we conclude that $\lim_{n\rightarrow\mathcal U}\|y_n^{1,1}\|_2^2\leq\lim_{n\rightarrow\mathcal U}\|y_n^{1,1}\|_2\|y_n^{2,2}\|_2$ and thus $$\label{est1} \lim_{n\rightarrow\mathcal U}\|y_n^{1,1}\|_2\leq\lim_{n\rightarrow\mathcal U}\|y_n^{2,2}\|_2.$$ On the other hand, Lemma [Lemma 10](#almost-ortho){reference-type="ref" reference="almost-ortho"} implies that $$\label{est2} \lim_{n\rightarrow\mathcal U}\|y_n^{2,2}\|_2^2=\lim_{n\rightarrow\mathcal U}\langle x_ny_n^{2,2},y_n^{1,1}x_n\rangle \quad \text{and} \quad \lim_{n\rightarrow\mathcal U}\|y_n^{2,2}\|_2\leq\lim_{n\rightarrow\mathcal U}\|y_n^{1,1}\|_2.$$ It is now clear that [\[est1\]](#est1){reference-type="eqref" reference="est1"} and [\[est2\]](#est2){reference-type="eqref" reference="est2"} together imply [\[y_11y_22\]](#y_11y_22){reference-type="eqref" reference="y_11y_22"}. Since $y$ also commutes with $x^*=(x_n^*)$, applying [\[y_11y_22\]](#y_11y_22){reference-type="eqref" reference="y_11y_22"} to $x^*$ instead of $x$ gives that $\lim_{n\rightarrow\mathcal U}\|x_n^*y_n^{2,2}-y_n^{1,1}x_n^*\|_2=0$ and thus $\lim_{n\rightarrow\mathcal U}\|y_n^{2,2}x_n-x_ny_n^{1,1}\|_2=0$. In combination with [\[y_11y_22\]](#y_11y_22){reference-type="eqref" reference="y_11y_22"}, this implies that $$\label{est3} \lim_{n\rightarrow\mathcal U}\|x_n^2y_n^{2,2}-y_n^{2,2}x_n^2\|_2=0.$$ Since $y$ commutes with $x^2=(x_n^2)$ and $\operatorname{E}_{B^{\mathcal U}}(x^2)=0$, [\[perp\]](#perp){reference-type="eqref" reference="perp"} from the proof of Lemma [Lemma 10](#almost-ortho){reference-type="ref" reference="almost-ortho"} gives that $\lim_{n\rightarrow\mathcal U}\|\operatorname{E}_B(x_n^2y_n^{2,2}{x_n^2}^*{y_n^{2,2}}^*)\|_1=0$, thus $\lim_{n\rightarrow\mathcal U}\langle x_n^2y_n^{2,2},y_n^{2,2}x_n^2\rangle=0$. Together with [\[est3\]](#est3){reference-type="eqref" reference="est3"} we get that $\lim_{n\rightarrow\mathcal U}\|x_n^2y_n^{2,2}\|_2=0$. Since $x_n\in\mathscr U(M)$ we get that $\lim_{n\rightarrow\mathcal U}\|y_n^{2,2}\|_2=0$ and [\[est1\]](#est1){reference-type="eqref" reference="est1"} gives that $\lim_{n\rightarrow\mathcal U}\|y_n^{1,1}\|_2=0$, proving part $(\rm i)$. $(\rm ii)$ Assume that $\operatorname{E}_{B^{\mathcal U}}(v)=\operatorname{E}_{B^{\mathcal U}}(v^2)=0$. By $(\rm i)$, we have $\lim_{n\rightarrow\mathcal U}\|y_n-(y_{1,2}^n+y_{2,1}^n)\|_2=~0$. Since $vy=yv$, we have $\lim_{n\rightarrow\mathcal U}\|v_ny_n-y_n v_n\|_2=0$ and so $$\label{12_to_21} \lim_{n\rightarrow\mathcal U}\|v_ny_{1,2}^n+ v_ny_{2,1}^n - y_{1,2}^n v_n - y_{2,1}^n v_n\|_2=0.$$ For every $n \in \mathbb{N}$, we have $v_ny_{1,2}^n = P_{2, 2}(v_ny_{1,2}^n)$, $y_{2,1}^n v_n = P_{2, 2}(y_{2,1}^n v_n)$, $v_ny_{2,1}^n = P_{1,1}(v_ny_{2,1}^n) + P_{2,1}(v_ny_{2,1}^n)$ and $y_{1,2}^n v_n = P_{1, 1}(y_{1,2}^n v_n) + P_{1, 2}(y_{1,2}^n v_n)$. In combination with [\[12_to_21\]](#12_to_21){reference-type="eqref" reference="12_to_21"}, we obtain $$\label{12_to_21-upgraded} \lim_{n \to \mathcal U} \|v_ny_{2,1}^n - y_{1,2}^n v_n\|_2 = 0 \quad \text{and} \quad \lim_{n \to \mathcal U} \|v_ny_{2,1}^n - P_{1, 1}(v_n y_{2, 1}^n)\|_2 = 0.$$ For every $n \in \mathbb{N}$, set $\eta_n = P_{1,1}(v_ny_{2,1}^n) \in \operatorname{L}^2(M)$. Then [\[12_to_21-upgraded\]](#12_to_21-upgraded){reference-type="eqref" reference="12_to_21-upgraded"}, Theorem [Theorem 3](#thm-Mei-Ricard){reference-type="ref" reference="thm-Mei-Ricard"} and Lemma [Lemma 1](#lem-ultraproduct){reference-type="ref" reference="lem-ultraproduct"} together imply that $\eta = (\eta_n)^{\mathcal U} = (v_ny_{2,1}^n)^{\mathcal U} = (y_{1,2}^n v_n)^{\mathcal U} \in \operatorname{L}^2(M^{\mathcal U})$ and that $y = v^* \eta + \eta v^*$. Since $v^* y = y v^*$, we obtain $(v^*)^2 \eta = \eta (v^*)^2$ and so $v^2 \eta = \eta v^2$. Since $\operatorname{E}_{B^{\mathcal U}}(v^2) = 0$, we may write $v^2 = (w_n)^{\mathcal U}$ where $w_n \in M_2 \ominus B$ for every $n \in \mathbb{N}$. For every $n \in \mathbb{N}$, since $\eta_n = P_{1,1}(\eta_n)$, we have $w_n \eta_n = P_{2, 1}(w_n \eta_n) \perp P_{1, 2}(\eta_n w_n) = \eta_n w_n$. Then we obtain $v^2 \eta \perp \eta v^2$. Since $v^2 \eta = \eta v^2$, this further implies that $v^2\eta = 0$ and so $\eta = 0$. Thus, $y = 0$. ◻ *Proof of Theorem [Theorem 9](#3infinity){reference-type="ref" reference="3infinity"}.* Theorem [Theorem 9](#3infinity){reference-type="ref" reference="3infinity"} follows directly from part $({\rm ii})$ of Lemma [Lemma 11](#3-infinity){reference-type="ref" reference="3-infinity"}. ◻ # Proof of Theorem [Theorem 5](#cor-amenable-absorption){reference-type="ref" reference="cor-amenable-absorption"} {#proof-of-theorem-cor-amenable-absorption} *Proof of Theorem [Theorem 5](#cor-amenable-absorption){reference-type="ref" reference="cor-amenable-absorption"}.* Let $P \subset M$ be a von Neumann subalgebra such that $P \cap M_1 \npreceq_{M_1} B$ and $P' \cap M^{\mathcal U} \npreceq_{M^{\mathcal U}} B^{\mathcal U}$. Set $A = P \cap M_1$. By [@IPP05 Theorem 1.1], since $A \npreceq_{M_1} B$, we have $P' \cap M \subset A' \cap M \subset M_1$ and so $P' \cap M = P' \cap M_1$. The set of projections $p \in P' \cap M_1$ for which $Pp \subset pM_1p$ attains its maximum in a projection $z \in \mathscr Z(P' \cap M_1)$. It suffices to prove that $z = 1$. By contradiction, assume that $z \neq 1$. Set $q = z^\perp$ and $Q = P q$. **Claim 12**. We have $Q \preceq_{M} M_1$. *Proof of Claim [Claim 12](#claim-intertwining){reference-type="ref" reference="claim-intertwining"}.* By contradiction, assume that $Q \npreceq_{M} M_1$. Choose a sequence $(w_k)_k$ in $\mathscr U(Q)$ such that $\lim_k \|\operatorname{E}_{M_1}(x^* w_k y)\|_2 = 0$ for all $x, y \in qM$. Set $\mathscr Q = Q' \cap (q M q)^{\mathcal U} = q(P' \cap M^{\mathcal U})q$. We have $\mathscr Q \npreceq_{M^{\mathcal U}} B^{\mathcal U}$. Firstly, we show that $\mathscr Q \preceq_{M^{\mathcal U}} M_1^{\mathcal U}$. By contradiction, assume that $\mathscr Q \npreceq_{M^{\mathcal U}} M_1^{\mathcal U}$. Since $A \npreceq_{M_1} B$, by Lemma [Lemma 4](#lem-intertwining){reference-type="ref" reference="lem-intertwining"}, we may choose $u \in \mathscr U(A^{\mathcal U})$ such that $\operatorname{E}_{B^{\mathcal U}}(a u^m b) = 0$ for all $a, b \in M_1$ and all $m \in \mathbb{Z}\setminus \{0\}$. Since $M$ is separable, $\mathscr Q \npreceq_{M^{\mathcal U}} M_1^{\mathcal U}$, $\mathscr Q\subset A'\cap M$ and $u\in \mathscr U(A^{\mathcal U})$, by a standard diagonal argument, we can construct a unitary $v \in \mathscr U(\mathscr Q)$ such that $\operatorname{E}_{M_1^{\mathcal U}}(v) = 0$ and $vu=uv$. By Lemma [Lemma 6](#lem-commutant-bis){reference-type="ref" reference="lem-commutant-bis"}, the set $\mathbf Y_1 = \{u\}' \cap (M^{\mathcal U} \ominus M_1^{\mathcal U})$ satisfies $a \mathbf Y_1 b \subset \mathbf X_1$ for all $a, b \in M_1$. On the one hand, applying Theorem [Theorem 3](#thm-main-result-bis){reference-type="ref" reference="thm-main-result-bis"}, since $v \in \mathbf Y_1$, we have $$\forall k \in \mathbb{N}, \quad \operatorname{E}_{B^{\mathcal U}} \left(v (w_k - \operatorname{E}_{M_1}(w_k)) v^* (w_k - \operatorname{E}_{M_1}(w_k))^* \right) = 0.$$ On the other hand, for every $k \in \mathbb{N}$, we have $v w_k = w_k v$ and $\operatorname{E}_{M_1}(w_k) \to 0$ strongly as $k \to \infty$. Altogether, since $vv^* = v^*v = q = w_kw_k^* = w_k^*w_k$, this implies that $\operatorname{E}_{B^{\mathcal U}}(q) = 0$, a contradiction. Therefore, we have $\mathscr Q \preceq_{M^{\mathcal U}} M_1^{\mathcal U}$. Secondly, we derive a contradiction using the proof of [@Io12 Lemma 9.5]. By [@Io12 Lemma 9.5, Claim 1], there exist $\delta > 0$ and a nonempty finite subset $\mathscr F \subset qM$ such that $$\forall v \in \mathscr U(\mathscr Q), \quad \sum_{a, b \in \mathscr F} \|\operatorname{E}_{M_1^{\mathcal U}}(b^* v a)\|_2^2 \geq \delta.$$ Denote by $\mathbf M_1 \subset M_1^{\mathcal U}$ the set of all elements $x \in M_1^{\mathcal U}$ such that $\operatorname{E}_{B^{\mathcal U}}(d^* x c) = 0$ for all $c, d \in M_1$. Then denote by $\mathscr K \subset \operatorname{L}^2((qMq)^{\mathcal U})$ the $\|\cdot\|_2$-closure of the linear span of the set $\left\{a x b^* \mid a, b \in qM, x \in \mathbf M_1\right\}$ and by $e : \operatorname{L}^2((qMq)^{\mathcal U}) \to \mathscr K$ the corresponding orthogonal projection. Since $\mathscr Q \npreceq_{M^{\mathcal U}} B^{\mathcal U}$ and since $M$ is separable, by a standard diagonal argument, we can construct a unitary $v \in \mathscr U(\mathscr Q)$ such that $\operatorname{E}_{B^{\mathcal U}}(d^* v c) = 0$ for all $c, d \in qM$. Set $\xi = e(v) \in \mathscr K$ and $\eta = \sum_{a, b \in \mathscr F} b \operatorname{E}_{M_1^{\mathcal U}}(b^* v a) a^* \in (qMq)^{\mathcal U}$. Then for every $c,d\in M_1$ and $a,b\in\mathscr F$, we have $\operatorname{E}_{B^{\mathcal U}}(d^* \operatorname{E}_{M_1^{\mathcal U}}(b^* v a) c) = \operatorname{E}_{B^{\mathcal U}}(d^* b^* v a c)=0$. Thus $\eta \in \mathscr K$ and we have $$\langle \xi, \eta \rangle = \langle v, \eta \rangle = \sum_{a, b \in \mathscr F} \|\operatorname{E}_{M_1^{\mathcal U}}(b^* v a)\|_2^2 \geq \delta.$$ It follows that $\xi = e(v) \neq 0$. On the one hand, since $\mathscr K \subset \operatorname{L}^2((qMq)^{\mathcal U})$ is a $qMq$-$qMq$-bimodule and since $v \in \mathscr Q$, for every $k \in \mathbb{N}$, we have $w_k \xi w_k^*= w_k e(v) w_k^* = e(w_k v w_k^*) = e(v) = \xi$. On the other hand, following the proof of [@Io12 Lemma 9.5, Claim 2], we show that $\lim_k \langle w_k \xi w_k^*, \xi \rangle = 0$. This will give a contradiction. By linearity and density, it suffices to show that $\lim_k \langle w_k \, a_1 x_1 b_1^* \, w_k^*, a_2 x_2 b_2^*\rangle = 0$ for all $a_1, a_2, b_1, b_2 \in q M$ and all $x_1, x_2 \in \mathbf M_1$. So let us fix $a_1, a_2, b_1, b_2 \in q M$ and $x_1, x_2 \in \mathbf M_1$. We may further assume that $\max \left\{\|a_i\|_\infty, \|b_i\|_\infty, \|x_i\|_\infty \mid i \in \{1, 2\}\right\} \leq 1$. Then for every $k \in \mathbb{N}$, we have $$|\langle w_k \, a_1 x_1 b_1^* \, w_k^*, a_2 x_2 b_2^*\rangle| = |\tau^{\mathcal U}(x_2^* a_2^* w_k a_1 x_1 b_1^* w_k^* b_2)| \leq \|\operatorname{E}_{M_1^{\mathcal U}}( a_2^* w_k a_1 \, x_1 \, b_1^* w_k^* b_2)\|_2.$$ Using the amalgamated free product structure $M = M_1 \ast_B M_2$, the inclusion $M_1 \subset M$ is mixing relative to $B$. In particular, since $x_1 \in \mathbf M_1$, we have $\operatorname{E}_{M_1^{\mathcal U}}(c^* x_1 d) = \operatorname{E}_{M_1^{\mathcal U}}(c^* x_1)= \operatorname{E}_{M_1^{\mathcal U}}(x_1 d)=0$ for all $c, d \in M \ominus M_1$ (see e.g. the proof of [@CH08 Claim 2.5]). This implies $$\forall k \in \mathbb{N}, \quad \operatorname{E}_{M_1^{\mathcal U}}( a_2^* w_k a_1 \, x_1 \, b_1^* w_k^* b_2) = \operatorname{E}_{M_1}(a_2^* w_k a_1) \, x_1 \operatorname{E}_{M_1}(b_1^* w_k^* b_2).$$ Thus, we have $$\limsup_k |\langle w_k \, a_1 x_1 b_1^* \, w_k^*, a_2 x_2 b_2^* \rangle| \leq \limsup_k \| \operatorname{E}_{M_1}(a_2^* w_k a_1)\|_2 = 0.$$ This gives a contradiction and finishes the proof of Claim [Claim 12](#claim-intertwining){reference-type="ref" reference="claim-intertwining"}. ◻ Since $Q \preceq_{M} M_1$, there exist $n \geq 1$, a projection $r \in \mathbf M_n(M_1)$, a nonzero partial isometry $v = [v_1, \dots, v_n] \in \mathbf M_{1, n}(z^\perp M)r$ and a unital normal $\ast$-homomorphism $\pi : Q \to r\mathbf M_n(M_1)r$ such that $a v = v \pi(a)$ for all $a \in Q$. In particular, we have $A v_i \subset \sum_{j = 1}^n v_j M_1$ for every $i \in \{1, \dots, n\}$. By [@IPP05 Theorem 1.1], since $A \npreceq_{M_1} B$, we have $v_i \in M_1$ for every $i \in \{1, \dots, n\}$. It follows that $vv^* \in Q' \cap M_1$ and $Q vv^* \subset vv^* M_1 vv^*$. Thus, we obtain $P(z + vv^*) \subset (z + vv^*) M_1 (z + vv^*)$. This contradicts the maximality of the projection $z \in P' \cap M_1$. Therefore, we have $z = 1$ and so $P \subset M_1$. ◻ **Remark 13**. We make two observations. - If $A \subset M_1$ is a von Neumann subalgebra such that $A \npreceq_{M_1} B$, then we have $A \npreceq_M B$. Indeed, this follows from the amalgamated free product structure $M = M_1 \ast_B M_2$ and the fact that the inclusion $M_1 \subset M$ is mixing relative to $B$ (see the proof of Claim [Claim 12](#claim-intertwining){reference-type="ref" reference="claim-intertwining"}). - If $P \subset M$ is an amenable von Neumann subalgebra such that $P \npreceq_{M} B$, then we have $P' \cap M^{\mathcal U} \npreceq_{M^{\mathcal U}} B^{\mathcal U}$. Indeed, by contradiction, assume that $P' \cap M^{\mathcal U} \preceq_{M^{\mathcal U}} B^{\mathcal U}$. On the one hand, by [@Io12 Lemma 9.5, Claim 1], there exist $\delta > 0$ and a nonempty finite subset $\mathscr F \subset M$ such that $$\label{eq-embedding} \forall v \in \mathscr U(P' \cap M^{\mathcal U}), \quad \sum_{a, b \in \mathscr F} \|\operatorname{E}_{B^{\mathcal U}}(b^* v a)\|_2^2 \geq \delta.$$ On the other hand, since $P$ is amenable hence hyperfinite by Connes' fundamental result [@Co75], there exists an increasing sequence $(P_k)_k$ of finite dimensional von Neumann subalgebras of $P$ such that $(\bigcup_kP_k)^{\prime\prime}=P$ and $P_k' \cap P \subset P$ has finite index for every $k \in \mathbb{N}$ (see e.g. the proof of [@Ho12 Theorem 8.1]). Since $P \npreceq_{M} B$, it follows that $P_k'\cap P \npreceq_{M} B$ for every $k \in \mathbb{N}$. Since $M$ is separable, by a standard diagonal argument, we can construct a unitary $v \in \mathscr U(P' \cap M^{\mathcal U})$ such that $\operatorname{E}_{B^{\mathcal U}}(b^* v a) = 0$ for all $a, b \in M$. This contradicts [\[eq-embedding\]](#eq-embedding){reference-type="eqref" reference="eq-embedding"}. Therefore, we have $P' \cap M^{\mathcal U} \npreceq_{M^{\mathcal U}} B^{\mathcal U}$. # A lifting theorem and proofs of Theorems [Theorem 7](#inductive){reference-type="ref" reference="inductive"} and [Theorem 8](#orthogonal){reference-type="ref" reference="orthogonal"} {#a-lifting-theorem-and-proofs-of-theorems-inductive-and-orthogonal} ## A lifting theorem The goal of this subsection is to establish the following lifting theorem which will be needed in the proof of Theorem [Theorem 7](#inductive){reference-type="ref" reference="inductive"}. **Theorem 14**. *Let $\mathcal U$ be an ultrafilter on a set $K$ and $(M_k,\tau_k),k\in K$, be tracial von Neumann algebras. Let $A,B\subset\prod_{\mathcal U}M_k$ be separable abelian von Neumann subalgebras which are $2$-independent in $\prod_{\mathcal U}M_k$ with respect to $(\tau_k)^{\mathcal U}$. Then there exist orthogonal abelian von Neumann subalgebras $C_k,D_k\subset M_k$, for every $k\in K$, such that $A\subset\prod_{\mathcal U}C_k$ and $B\subset\prod_{\mathcal U}D_k$.* We do not know whether Theorem [Theorem 14](#lifting){reference-type="ref" reference="lifting"} still holds if we replace the assumption that $A$ and $B$ are $2$-independent with the weaker assumption that $A$ and $B$ are orthogonal. When $\dim(A)=2$ and $\dim(B)=3$, Theorem [Theorem 14](#lifting){reference-type="ref" reference="lifting"} follows from [@CIKE22 Lemma 3.1], which moreover only assumes that $A$ and $B$ are orthogonal. Theorem [Theorem 14](#lifting){reference-type="ref" reference="lifting"} is new in all other cases, including when $A$ and $B$ are finite dimensional and of dimension at least $3$. The proof of Theorem [Theorem 14](#lifting){reference-type="ref" reference="lifting"} relies on the following perturbation lemma. First, we need to introduce some additional terminology. Let $(M,\tau)$ be a tracial von Neumann algebra. We denote by $M_{\text {sa},1}$ the set of $x\in M$ such that $x=x^*$ and $\|x\|_\infty \leq 1$. Let $x=(x_1,\dots, x_m)\in M^m$ and $y=(y_1,\dots,y_n)\in M^n$, for some $m,n\in\mathbb N$. For $u\in\mathscr U(M)$, we write $uxu^*=(ux_1u^*, \dots, ux_mu^*)$. We define $$\begin{aligned} \delta(x,y) &=\min \left\{\|[x_i,y_j]\|_2\mid 1\leq i\leq m,1\leq j\leq n \right\}, \\ \varepsilon(x,y) &=\max \left\{|\tau(x_iy_j)|\mid 1\leq i\leq m,1\leq j\leq n \right\}, \\ \gamma(x,y) &=\max \left\{|\langle [x_i,y_j],[x_{i'},y_{j'}]\rangle|\mid 1\leq i,i'\leq m,1\leq j,j'\leq n, (i,j)\not=(i',j') \right\}.\end{aligned}$$ **Lemma 15**. *Let $(M,\tau)$ be a tracial von Neumann algebra, $x=(x_1,\dots, x_m)\in M_{\emph{sa},1}^m$ and $y=(y_1,\dots,y_n)\in M_{\emph{sa},1}^n$, for $m,n\in\mathbb N$. Set $\delta_0=\delta(x,y), \varepsilon_0=\varepsilon(x,y),\gamma_0=\gamma(x,y)$. Assume that $13mn\sqrt{\varepsilon_0} < \delta_0^2 - (mn-1)\gamma_0$. Then there exists $v\in\mathscr U(M)$ such that $$\|v-1\|_\infty \leq \frac{8mn\varepsilon_0}{\delta_0^2-(mn-1)\gamma_0} \leq\frac{8}{13}\sqrt{\varepsilon_0} \quad \text{and} \quad \varepsilon(vxv^*,y)=0.$$* Note that Lemma [Lemma 15](#perturbation){reference-type="ref" reference="perturbation"} is interesting even when $M$ is finite dimensional. To prove Lemma [Lemma 15](#perturbation){reference-type="ref" reference="perturbation"}, we will need two auxiliary lemmas. **Lemma 16**. *Let $(M,\tau)$ be a tracial von Neumann algebra, $\xi_1,\dots,\xi_p\in M_{\emph{sa},1}$ and $\alpha_1,\dots,\alpha_p\in\mathbb R$, for some $p\geq 2$. Let $\delta\in (0,1)$ and $\varepsilon\in (0,\frac{\delta^2}{p-1})$. Assume that $\|\xi_i\|_2 \geq\delta$, for every $1\leq i\leq p$, and $|\langle \xi_i,\xi_j\rangle|\leq\varepsilon$, for every $1\leq i<j\leq p$. Then there exists $h\in M$ such that $h=h^*, \|h\|_\infty\leq\frac{\sum_{j=1}^p|\alpha_j|}{\delta^2 -(p-1)\varepsilon}$ and $\tau(h\xi_i)=\alpha_i$, for every $1\leq i\leq p$.* *Proof.* First, we claim that $\xi_1,\dots,\xi_p$ are linearly independent. Otherwise, we can find $\beta_1,\dots,\beta_p\in\mathbb R$ such that $\beta_1\xi_1+\cdots+\beta_p\xi_p=0$ and $\max \left\{|\beta_i|\mid 1\leq i\leq p \right\}>0$. Let $1\leq j\leq p$ such that $|\beta_j|=\max \left\{|\beta_i|\mid 1\leq i\leq p \right\}$. Then $-\beta_j\xi_j=\sum_{i\not=j}\beta_i\xi_i$ and thus $|\beta_j|\, \|\xi_j\|_2^2 \leq\sum_{i\not=j}|\beta_i|\, |\langle\xi_i,\xi_j\rangle|\leq |\beta_j|\sum_{i\not=j}|\langle\xi_i,\xi_j\rangle|$. Since $\beta_j\not=0$, we derive that $\|\xi_j\|_2^2\leq\sum_{i\not=j}|\langle \xi_i,\xi_j\rangle|$, which implies that $\delta^2\leq (p-1)\varepsilon$, contradicting that $\delta^2 > (p-1)\varepsilon$. Since $\xi_1,\dots,\xi_p$ are linearly independent, it follows that we can find $\lambda_1,\dots,\lambda_p\in\mathbb R$ such that $h=\sum_{i=1}^p\lambda_i\xi_i$ satisfies $\tau(h\xi_j)=\langle h,\xi_j\rangle=\alpha_j$, for every $1\leq j\leq p$. Then $|\alpha_j|=|\sum_{i=1}^p\lambda_i\langle\xi_i,\xi_j\rangle|\geq |\lambda_j|\|\xi_j\|_2^2-\sum_{i\not=j}|\lambda_i||\langle\xi_i,\xi_j\rangle|$ and thus $$\label{lambda} \forall 1\leq j\leq p, \quad |\alpha_j|\geq \delta^2 |\lambda_j|-\varepsilon\sum_{i\not=j}|\lambda_i|.$$ Adding the inequalities in [\[lambda\]](#lambda){reference-type="eqref" reference="lambda"} for $1\leq j\leq p$ gives $\sum_{j=1}^p|\alpha_j|\geq (\delta^2 - (p-1)\varepsilon)\sum_{j=1}^p|\lambda_j|$. Thus, $\|h\|_\infty \leq\sum_{j=1}^p|\lambda_j|\leq \frac{\sum_{j=1}^p|\alpha_j|}{\delta^2-(p-1)\varepsilon}$. Since $h=h^*$, this finishes the proof. ◻ **Lemma 17**. *Let $(M,\tau)$ be a tracial von Neumann algebra, $x=(x_1,\dots, x_m)\in M_{\emph{sa},1}^m$ and $y=(y_1,\dots,y_n)\in M_{\emph{sa},1}^n$, for some $m,n\in\mathbb N$. Set $\delta=\delta(x,y), \varepsilon=\varepsilon(x,y),\gamma=\gamma(x,y)$. Assume that $2mn\varepsilon<\delta^2-(mn-1)\gamma$ and set $\lambda=\frac{2 mn\varepsilon}{\delta^2 -(mn-1)\gamma}<1$.* *Then there exists $u\in\mathscr U(M)$ such that* - *$\|u-1\|_\infty \leq 2\lambda$.* - *$\delta(uxu^*,y)\geq \delta-8\lambda$.* - *$\varepsilon(uxu^*,y)\leq 4\lambda^2$.* - *$\gamma(uxu^*,y)\leq\gamma+32\lambda$.* *Proof.* For every $1\leq i\leq m,1\leq j\leq n$, set $\xi_{i,j}=-\frac{{\rm i}}{2}[x_i,y_j]$. Then $\xi_{i,j}\in M_{\text{sa},1}$ and $\|\xi_{i,j}\|_2=\frac{\|[x_i,y_j]\|_2}{2}\geq \frac{\delta}{2}$, for every $1\leq i\leq m,1\leq j\leq n$. On the other hand, for every $(i,j)\not=(i',j')$, we have $|\langle\xi_{i,j},\xi_{i',j'}\rangle|=\frac{|\langle [x_i,y_j],[x_{i'},y_{j'}]\rangle|}{4}\leq\frac{\gamma}{4}$. By applying Lemma [Lemma 16](#linearindep){reference-type="ref" reference="linearindep"} to $\xi_{i,j}$ and $\alpha_{i,j}=\frac{\tau(x_iy_j)}{2}$, we may find $h\in M$ such that $h=h^*$, $$\label{h} \forall 1\leq i\leq m,1\leq j\leq m, \quad \tau(h\xi_{i,j})=\frac{\tau(x_iy_j)}{2},$$ and $$\label{h2} \|h\|_\infty\leq \frac{\sum_{i,j}\frac{|\tau(x_iy_j)|}{2}}{\frac{\delta^2}{4}-(mn-1)\frac{\gamma}{4}}\leq\frac{2mn\varepsilon}{\delta^2 - (mn-1)\gamma}=\lambda.$$ Define $u=\exp({\rm i} h)\in\mathscr U(M)$. We will prove that $u$ satisfies the conclusion. Since for every $x\in\mathbb R$, $|\exp({\rm i} x)-1|\leq 2|x|$ and $|\exp({\rm i} x)-(1+ {\rm i} x)|\leq x^2$, using [\[h2\]](#h2){reference-type="eqref" reference="h2"} we get that $$\label{u} \|u-1\|_\infty\leq 2\lambda \quad \text{and} \quad \|u-(1+ {\rm i} h)\|_\infty \leq \lambda^2.$$ Let $1\leq i\leq m$ and $1\leq j\leq n$. Then using [\[h2\]](#h2){reference-type="eqref" reference="h2"} and the second part of [\[u\]](#u){reference-type="eqref" reference="u"} we get that $\|ux_iu^*y_j-(1+ {\rm i} h)x_i(1+ {\rm i} h)^*y_j\|_\infty \leq \|u-(1+ {\rm i} h)\|_\infty (1+\|1+ {\rm i} h\|_\infty)\leq \lambda^2(2+\lambda)\leq 3\lambda^2$ and $\|(1+ {\rm i} h)x_i(1+ {\rm i} h)^*y_j-(x_iy_j+ {\rm i} (hx_iy_j-x_ihy_j))\|_\infty =\|hx_ihy_j\|_\infty \leq \lambda^2.$ Thus, we get $$\|ux_iu^*y_j-(x_iy_j+{\rm i} (hx_iy_j-x_ihy_j))\|_\infty \leq 4\lambda^2$$ and therefore $|\tau(ux_iu^*y_j)-\tau(x_iy_j+ {\rm i} (hx_iy_j-x_ihy_j))|\leq 4\lambda^2$. On the other hand, [\[h\]](#h){reference-type="eqref" reference="h"} gives $\tau(x_iy_j+ {\rm i} (hx_iy_j-x_ihy_j))=\tau(x_iy_j)+\tau({\rm i} h[x_i,y_j])=\tau(x_iy_j)-2\tau(h\xi_{i,j})=0$. Altogether, we get that $|\tau(ux_iu^*y_j)|\leq 4\lambda^2$. Thus, $\varepsilon(uxu^*,y)\leq 4\lambda^2$, which proves $(\rm iii)$. Next, $\|[ux_iu^*,y_j]-[x_i,y_j]\|_2\leq 2\|ux_iu^*-x_i\|_2\leq 4\|u-1\|_2\leq 8\lambda$, by the first part of [\[u\]](#u){reference-type="eqref" reference="u"}. Hence, $\|[ux_iu^*,y_j]\|_2\geq \|[x_i,y_j]\|_2-8\lambda\geq\delta-8\lambda$, for every $1\leq i\leq m$ and $1\leq j\leq n$. This implies that $\delta(uxu^*,y)\geq\delta-8\lambda$, which proves $(\rm ii)$. Finally, for every $(i,j),(i',j')$ we have $\|[ux_{i'}u^*,y_{j'}]\|_2\leq 2$, $\|[x_i,y_j]\|_2\leq 2$ and thus $$\begin{aligned} &|\langle [ux_iu^*,y_j],[ux_{i'}u^*,y_{j'}]\rangle-\langle [x_i,y_j],[x_{i'},y_{j'}]\rangle|\\ & \quad \leq 2\big(\|[ux_iu^*,y_j]-[x_i,y_j]\|_2+\|[ux_{i'}u^*,y_{j'}]-[x_{i'},y_{j'}]\|_2\big)\leq 32\lambda. \end{aligned}$$ Thus, $|\langle [ux_iu^*,y_j],[ux_{i'}u^*,y_{j'}]\rangle|\leq |\langle [x_i,y_j],[x_{i'},y_{j'}]\rangle|+32\lambda\leq \gamma+32\lambda$. This implies that $\gamma(uxu^*,y)\leq\gamma+32\lambda$, which proves $(\rm iv)$. Since $(\rm i)$ also holds by the first part of [\[u\]](#u){reference-type="eqref" reference="u"}, this finishes the proof. ◻ *Proof of Lemma [Lemma 15](#perturbation){reference-type="ref" reference="perturbation"}.* We will inductively construct sequences $(u_k)_{k\in\mathbb N}\subset\mathscr U(M)$ and $(\lambda_k)_{k\in\mathbb N}\subset (0,\infty)$ with the following properties: $\lambda_0=1$, $\lambda_1=\frac{2mn\varepsilon_0}{\delta_0^2-(mn-1)\gamma_0}$ and if we define $v_0=1$, $v_k=u_ku_{k-1}\cdots u_1\in\mathscr U(M)$, $\delta_k=\delta(v_kxv_k^*,y),\varepsilon_k=\varepsilon(v_kxv_k^*,y)$ and $\gamma_k=\gamma(v_kxv_k^*,y)$, for every $k\geq 0$, then for every $k\geq 1$ we have that (i) $\|u_k-1\|_\infty \leq 2\lambda_k$. (ii) $\delta_k\geq \delta_{k-1}-8\lambda_k$. (iii) $\varepsilon_k\leq 4\lambda_k^2$. (iv) $\gamma_k\leq\gamma_{k-1}+32\lambda_k$. (v) $\lambda_k\leq\frac{\lambda_{k-1}}{2}.$ Since $\varepsilon_0\leq 1$, we have that $4mn\varepsilon_0\leq 13mn\sqrt{\varepsilon_0} < \delta_0^2 - (mn-1)\gamma_0$. Thus, $\lambda_1<\frac{1}{2}$ and hence condition (v) holds for $k=1$. By applying Lemma [Lemma 17](#moveone){reference-type="ref" reference="moveone"}, we can find $u_1\in\mathscr U(M)$ such that conditions (i)-(iv) hold for $k=1$. Next, assume that we have constructed $u_1,\dots, u_l\in\mathscr U(M)$ and $\lambda_1,\dots,\lambda_l\in (0,\infty)$, for some $l\in\mathbb N$, such that conditions (i)-(v) are satisfied for $k=1,\dots,l$. Our goal is to construct $u_{l+1}$ and $\lambda_{l+1}$. Let $\lambda_{l+1}=\frac{2mn\varepsilon_l}{\delta_l^2-(mn-1)\gamma_l}$. We continue with the following claim. **Claim 18**. $\lambda_{l+1}\leq\frac{\lambda_l}{2}$. *Proof of Claim [Claim 18](#lambda0){reference-type="ref" reference="lambda0"}.* First, $(\rm ii)$ implies that $\delta_k^2 \geq (\delta_{k-1}-8\lambda_k)^2 \geq \delta_{k - 1}^2 - 32 \lambda_k$. Then combining (ii) and (iv) gives that $$\forall 1\leq k\leq l, \quad \delta_k^2-(mn-1)\gamma_k\geq (\delta_{k-1}^2-(mn-1)\gamma_{k - 1}) - 32mn\lambda_k$$ which implies that $\delta_l^2-(mn-1)\gamma_l\geq (\delta_0^2-(mn-1)\gamma_0)- 32mn(\sum_{k=1}^l\lambda_k).$ By using that (v) holds for $k=1,\dots,l$, we also get that $\sum_{k=1}^l\lambda_k\leq 2\lambda_1$. By combining the last two inequalities we get that $$\label{lambda1} \delta_l^2-(mn-1)\gamma_l\geq (\delta_0^2-(mn-1)\gamma_0) - 64mn\lambda_1.$$ Since $13mn\sqrt{\varepsilon_0} < \delta_0^2 - (mn-1)\gamma_0$, we get that $(\delta_0^2-(mn-1)\gamma_0)^2 > 169(mn)^2 \varepsilon_0$ and thus $$\label{lambda2} \delta_0^2-(mn-1)\gamma_0 > 80mn\lambda_1.$$ By combining [\[lambda1\]](#lambda1){reference-type="eqref" reference="lambda1"} and [\[lambda2\]](#lambda2){reference-type="eqref" reference="lambda2"} we derive that $$\label{lambda3} \delta_l^2-(mn-1)\gamma_l\geq 16mn\lambda_1.$$ Since (v) holds for every $k=1,\dots,l$, we get that $\lambda_l\leq\lambda_1$. Since $\varepsilon_l\leq 4\lambda_l^2$ by (iii), using [\[lambda3\]](#lambda3){reference-type="eqref" reference="lambda3"} we get that $$\lambda_{l+1}=\frac{2 mn\varepsilon_l}{\delta_l^2 -(mn-1)\gamma_l}\leq \frac{8mn\lambda_l^2}{\delta_l^2-(mn-1)\gamma_l}\leq\frac{16mn\lambda_1}{\delta_l^2-(mn-1)\gamma_l}\cdot\frac{\lambda_l}{2}\leq\frac{\lambda_l}{2}.$$ This finishes the proof of the claim. $\square$ By using (v) and Claim [Claim 18](#lambda0){reference-type="ref" reference="lambda0"} we get that $\lambda_{l+1}\leq\frac{1}{2^{l+1}}<1$. Thus, $2 mn\varepsilon_l < \delta_l^2 -(mn-1)\gamma_l$. We can therefore apply Lemma [Lemma 17](#moveone){reference-type="ref" reference="moveone"} to $v_lxv_l^*$ and $y$ to find $u_{l+1}\in\mathscr U(M)$ such that 1. $\|u_{l+1}-1\|_\infty\leq 2\lambda_{l+1}$. 2. $\delta_{l+1}=\delta(u_{l+1}(v_lxv_l^*)u_{l+1}^*),y)\geq \delta_{l}-8\lambda_{l+1}$. 3. $\varepsilon_{l+1}=\varepsilon(u_{l+1}(v_lxv_l^*)u_{l+1}^*,y)\leq 4\lambda_{l+1}^2$. 4. $\gamma_{l+1}=\gamma(u_{l+1}(v_lxv_l^*)u_{l+1}^*,y)\leq\gamma_l+32\lambda_{l+1}$. By induction, this finishes the construction of $(u_k)_{k\in\mathbb N}\subset\mathscr U(M)$ and $(\lambda_k)_{k\in\mathbb N}\subset (0,\infty)$. Finally, since $\lambda_0=1$, (v) implies that $\lambda_k\leq \frac{1}{2^k}$, for every $k\geq 0$. Using (i), we derive that $\|v_k-v_{k-1}\|_\infty=\|u_k-1\|_\infty\leq\frac{1}{2^{k-1}}$, for every $k\geq 1$. Thus, the sequence $(v_k)_{k\in\mathbb N}$ is Cauchy in $\|\cdot\|_\infty$ and so we can find $v\in\mathscr U(M)$ such that $\lim_{k\rightarrow\infty}\|v_k-v\|_\infty=0$. Using (iii), we get that $\varepsilon_k\leq 4\lambda_k^2\leq \frac{1}{4^{k-1}}$, for every $k\geq 1$. Thus, $\varepsilon(vxv^*,y)=\lim_{k\rightarrow\infty}\varepsilon_k=0$. Moreover, using (i) and (v) we get that $\|v_k-1\|_\infty \leq\sum_{l=1}^k\|u_l-1\|_\infty\leq\sum_{l=1}^k2\lambda_l\leq 4\lambda_1.$ Hence $\|v-1\|_\infty=\lim_{k\rightarrow\infty}\|v_k-1\|_\infty \leq 4\lambda_1= \frac{8mn\varepsilon_0}{\delta_0^2-(mn-1)\gamma_0}$. This finishes the proof. ◻ *Proof of Theorem [Theorem 14](#lifting){reference-type="ref" reference="lifting"}.* We may clearly assume that $\dim(A)\geq 2$ and $\dim(B)\geq 2$. Since $A$ and $B$ are separable, we can write $A=(\bigcup_{n\in\mathbb N}A_n)^{\prime\prime}$, $B=(\bigcup_{n\in\mathbb N}B_n)^{\prime\prime}$, where $A_n\subset A, B_n\subset B$ are finite dimensional von Neumann subalgebras such that $A_n\subset A_{n+1}$, $B_n\subset B_{n+1}$, $a_n:=\dim(A_n)\geq 2$ and $b_n:=\dim(B_n)\geq 2$, for every $n\in\mathbb N$. Fix $n\in\mathbb N$. Write $A_n=\bigoplus_{i=1}^{a_n}\mathbb Cp_{n,i}$ and $B_n=\bigoplus_{j=1}^{b_n}\mathbb Cq_{n,j}$, where $(p_{n,i})_{i=1}^{a_n}$ and $(q_{n,j})_{j=1}^{b_n}$ are partitions of unity into projections from $A$ and $B$, respectively. For every $1\leq i\leq a_n$ and $1\leq j\leq b_n$, represent $p_{n,i},q_{n,j}\in\prod_{\mathcal U}M_k$ as $p_{n,i}=(p_{n,i}^k)^{\mathcal U}$ and $q_{n,j}=(q_{n,j}^k)^{\mathcal U}$, where for every $k\in K$, $(p_{n,i}^k)_{i=1}^{a_n}$ and $(q_{n,j}^k)_{j=1}^{b_n}$ are partitions of unity into projections from $M_k$. Denote $A_n^k=\bigoplus_{i=1}^{a_n}\mathbb Cp_{n,i}^k$ and $B_n^k=\bigoplus_{j=1}^{b_n}\mathbb Cq_{n,j}^k$. Moreover, we can arrange that $A_n^k\subset A_{n+1}^k$ and $B_n^k\subset B_{n+1}^k$, for every $n\in\mathbb N$ and $k\in K$. If $(r_l)_{l=1}^m$ is a partition of unity into nonzero projections from a tracial von Neumann algebra $(N,\tau)$, then $\left\{\tau(r_{l+1}+\cdots+r_m)r_l-\tau(r_l)(r_{l+1}+\cdots+r_m)\mid 1\leq l\leq m-1 \right\}$ is an orthogonal basis for $C\ominus\mathbb C1$ contained in $C_{\text{sa},1}$, where $C=\bigoplus_{l=1}^m\mathbb Cr_l$. Using this observation, for every $1\leq i\leq a_n-1,1\leq j\leq b_n-1$ and $k\in K$, we define $$x_{n,i}=\tau(p_{n,i+1}+\cdots+p_{n,a_n})p_{n,i}-\tau(p_{n,i})(p_{n,i+1}+\cdots+p_{n,a_n}),$$ $$y_{n,j}=\tau(q_{n,j+1}+\cdots+q_{n,b_n})q_{n,j}-\tau(q_{n,j})(q_{n,j+1}+\cdots+q_{n,b_n}),$$ $$x_{n,i}^k=\tau(p_{n,i+1}^k+\cdots+p_{n,a_n}^k)p_{n,i}^k-\tau(p_{n,i}^k)(p_{n,i+1}^k+\cdots+p_{n,a_n}^k),$$ $$y_{n,j}^k=\tau(q_{n,j+1}^k+\cdots+q_{n,b_n}^k)q_{n,j}^k-\tau(q_{n,j}^k)(q_{n,j+1}^k+\cdots+q_{n,b_n}^k).$$ Set $x_n=(x_{n,i})_{i=1}^{a_n-1}\in A_n^{a_n-1}, y_n=(y_{n,j})_{j=1}^{b_n-1}\in B_n^{b_n-1}, x_n^k=(x_{n,i}^k)_{i=1}^{a_n-1}\in M_k^{a_n-1}$ and $y_n^k=(y_{n,j}^k)_{j=1}^{b_n-1}\in M_k^{b_n-1}$. Let $n\in\mathbb N$, $1\leq i,i'\leq a_n-1$ and $1\leq j,j'\leq b_n-1$ with $(i,j)\not=(i',j')$. Since $A_n$ and $B_n$ are $2$-independent, $x_{n,i}\not=0$ and $y_{n,j}\not=0$, we have that $\|[x_{n,i},y_{n,j}]\|_2= \sqrt{2}\|x_{n,i}\|_2\|y_{n,j}\|_2>0$ and $\tau(x_{n,i}y_{j,n})=0$. Moreover, $\langle [x_{n,i},y_{n,j}],[x_{n,i'},y_{n,j'}]\rangle=2\tau(x_{n,i}x_{n,i'})\tau(y_{n,j}y_{n,j'}).$ Since $(x_{n,i})_{i=1}^{a_n-1}$ and $(y_{n,j})_{j=1}^{b_n-1}$ are pairwise orthogonal, we get that $\langle [x_{n,i},y_{n,j}],[x_{n,i'},y_{n,j'}]\rangle=0$. Altogether, we derive that $\delta(x_n,y_n)>0$ and $\varepsilon(x_n,y_n)=\gamma(x_n,y_n)=0$. Thus, we get that $\lim_{k\rightarrow\mathcal U}\delta(x_n^k,y_n^k)=\delta(x_n,y_n)>0$, $\lim_{k\rightarrow\mathcal U}\varepsilon(x_n^k,y_n^k)=\varepsilon(x_n,y_n)=0$ and $\lim_{k\rightarrow\mathcal U}\gamma(x_n^k,y_n^k)=\gamma(x_n,y_n)=0$. By applying Lemma [Lemma 15](#perturbation){reference-type="ref" reference="perturbation"}, we find $v_n^k\in\mathscr U(M_k)$, for every $k\in K$, such that $\varepsilon(v_n^kx_n^k{v_n^k}^*,y_n^k)=0$, for every $k\in K$, and $\lim_{k\rightarrow\mathcal U}\|v_n^k-1\|_\infty=~0$. Since $x_n^k$ and $y_n^k$ are bases for $A_n^k$ and $B_n^k$, respectively, we get that $v_n^kA_n^k{v_n^k}^*$ and $B_n^k$ are orthogonal, for every $k\in K$. To complete the proof we consider two cases: **Case 1.** $\mathcal U$ is countably cofinal. In this case, we proceed as in the proof of [@BCI15 Lemma 2.2]. Since $\mathcal U$ is countably cofinal, there exists a decreasing sequence $\{S_n\}_{n\geq 2}$ of sets in $\mathcal U$ such that $\bigcap_{n\geq 2}S_n=\emptyset$. For $n\geq 2$, let $T_n=\{k\in K\mid \|v_m^k-1\|_\infty<\frac{1}{n}, \forall 1\leq m\leq n\}\in\mathcal U$ and set $K_n=S_n\cap T_n$. Then $\{K_n\}_{n\geq 2}$ is a decreasing sequence of sets in $\mathcal U$ such that $\bigcap_{n\geq 2}K_n=\emptyset$. Let $K_1=K\setminus K_2$. For every $k\in K$, let $n(k)$ be the smallest integer $n\geq 1$ such that $k\in K_n$. Then $n(k)$ is well-defined and $\lim_{k\rightarrow\mathcal U}n(k)=+\infty$. For $k\in K$, let $C_k=A_{n(k)}^k, D_k^0=B_{n(k)}^k$ and $v_k=v_{n(k)}^k$. If $n(k)\geq 2$, then as $k\in K_{n(k)}$ we have $\|v_k-1\|_\infty < \frac{1}{n(k)}$. Since $\lim_{k\rightarrow\mathcal U}n(k)=+\infty$, we get that $\lim_{k\rightarrow\mathcal U}\|v_k-1\|_\infty =0$. Let $n\in\mathbb N$. Since $\{k\in K\mid n(k)\geq n\}\in\mathcal U$ and the sequences $\{A_m^k\}_{m\in\mathbb N}$ and $\{B_m^k\}_{m\in\mathbb N}$ are increasing for every $k\in K$, we get that $\prod_{\mathcal U}A_n^k\subset\prod_{\mathcal U}C_k$ and $\prod_{\mathcal U}B_n^k\subset\prod_{\mathcal U}D_k^0$. Since $A_n\subset\prod_{\mathcal U}A_n^k$ and $B_n\subset\prod_{\mathcal U}B_n^k$, we conclude that $A_n\subset\prod_{\mathcal U}C_k$ and $B_n\subset\prod_{\mathcal U}D_k^0$. As this holds for every $n\in\mathbb N$, we get that $A\subset\prod_{\mathcal U}C_k$ and $B\subset\prod_{\mathcal U}D_k^0$. Finally, let $D_k=v_kD_k^0v_k^*$. Then $C_k=A_{n(k)}^k$ and $D_k=v_{n(k)}^kB_{n(k)}^kv_{n(k)}^*$ are orthogonal, for every $k\in K$. Since $\lim_{k\rightarrow\mathcal U}\|v_k-1\|_\infty=0$, we get that $\prod_{\mathcal U}D_k^0=\prod_{\mathcal U}D_k$ and $B\subset\prod_{\mathcal U}D_k$. This finishes the proof of Case 1. **Case 2.** $\mathcal U$ is not countably cofinal. Since $\mathcal U$ is not countably cofinal, $\{k'\in K\mid f(k')=\lim_{k\rightarrow\mathcal U}f(k)\}\in\mathcal U$, for every $f\in\ell^\infty(K)$ (see the proof of [@BCI15 Lemma 2.3 (2)]). If $n\in\mathbb N$, since $\lim_{k\rightarrow\mathcal U}\|v_n^k-1\|_\infty=0$, we get that $R_n:=\{k\in K\mid v_n^k=1\}\in\mathcal U$. Using again that $\mathcal U$ is not countably cofinal, we further deduce that $R:=\bigcap_{n\in\mathbb N}R_n=\{k\in K\mid v_n^k=1,\forall n\in\mathbb N\}\in\mathcal U$. If $k\in R$, then $v_n^k=1$, hence $A_n^k$ and $B_n^k$ are orthogonal, for every $n\in\mathbb N$. Since the sequences $\{A_n^k\}_{n\in\mathbb N}$ and $\{B_n^k\}_{n\in\mathbb N}$ are increasing, we get that $C_k=(\bigcup_{n\in\mathbb N}A_n^k)^{\prime\prime}$ and $D_k=(\bigcup_{n\in\mathbb N}B_n^k)^{\prime\prime}$ are orthogonal, for every $k\in R$. For $k\in K\setminus R$, let $C_k=D_k=\mathbb C1$. If $n\in\mathbb N$, then $A_n\subset\prod_{\mathcal U}A_n^k\subset\prod_{\mathcal U}C_k$ and $B_n\subset\prod_{\mathcal U}B_n^k\subset\prod_{\mathcal U}D_k$. As this holds for every $n\in\mathbb N$, we get that $A\subset\prod_{\mathcal U}C_k$ and $B\subset\prod_{\mathcal U}D_k$. This finishes the proof of Case 2 and of the theorem. ◻ ## Proof of Theorem [Theorem 7](#inductive){reference-type="ref" reference="inductive"} {#proof-of-theorem-inductive} In order to construct a ${\rm II_1}$ factor satisfying the hypothesis of Theorem [Theorem 7](#inductive){reference-type="ref" reference="inductive"}, we follow closely the construction from [@CIKE22 Definition 5.1]. This construction uses the following key result from [@CIKE22]. **Corollary 19** (Corollary 4.3 in [@CIKE22]). *Let $(M,\tau)$ be a tracial von Neumann algebra having no type I direct summand. Let $u_1,u_2\in\mathscr U(M)$ such that $\{u_1\}^{\prime\prime}\perp\{u_2\}^{\prime\prime}$.* *Then there exists a ${\rm II_1}$ factor $P=\Phi(M,u_1,u_2)^{\prime\prime}$ generated by a copy of $M$ and Haar unitaries $v_1,v_2\in \mathscr U(P)$ so that $[u_1,v_1]=[u_2,v_2]=[v_1,v_2]=0$. Moreover, if $Q\subset M$ is a von Neumann subalgebra such that $Q\npreceq_{M}\{u_i\}^{\prime\prime}$, for every $1\leq i\leq 2$, then $Q'\cap P\subset M$.* For a ${\rm II_1}$ factor $M$, we let $\mathscr W(M)$ be the set of pairs $(u_1,u_2)\in\mathscr U(M)\times\mathscr U(M)$ such that $\{u_1\}^{\prime\prime}$ and $\{u_2\}^{\prime\prime}$ are orthogonal. We endow $\mathscr U(M)\times\mathscr U(M)$ with the product $\|\cdot\|_2$-topology. We next repeat the construction from [@CIKE22 Definition 5.1] where we replace $\mathscr V(M)$ (the set of pairs $(u_1,u_2)\in\mathscr W(M)$ such that $u_1^2=u_2^3=1$) with $\mathscr W(M)$. **Definition 20**. Let $M_1$ be a ${\rm II_1}$ factor. We construct a ${\rm II_1}$ factor $M$ which contains $M_1$ and arises as the inductive limit of an increasing sequence $(M_n)_{n\in\mathbb N}$ of ${\rm II_1}$ factors. To this end, let $\sigma=(\sigma_1,\sigma_2):\mathbb N\rightarrow\mathbb N\times\mathbb N$ be a bijection such that $\sigma_1(n)\leq n$, for every $n\in\mathbb N$. Assume that $M_1,\ldots,M_n$ have been constructed, for some $n\in\mathbb N$. Let $\{(u_1^{n,k},u_2^{n,k})\}_{k\in\mathbb N}\subset\mathscr W(M_n)$ be a $\|\cdot\|_2$-dense sequence. Since $\sigma_1(n)\leq n$, we have $(u_1^{\sigma(n)},u_2^{\sigma(n)})\in\mathscr W(M_n)$ and we can define $M_{n+1}:=\Phi(M_n,u_1^{\sigma(n)},u_2^{\sigma(n)}).$ Then $M_n\subset M_{n+1}$ and $M_{n+1}$ is a ${\rm II_1}$ factor by Corollary [Corollary 19](#amalgam){reference-type="ref" reference="amalgam"}. Thus, $M:=({\bigcup_{n\in\mathbb N}M_n})^{\prime\prime}$ a ${\rm II_1}$ factor. **Proposition 21**. *Let $M$ be the ${\rm II_1}$ factor introduced in Definition [Definition 20](#M){reference-type="ref" reference="M"} and $\mathcal U$ be a countably cofinal ultrafilter on a set $I$. Let $u_1,u_2\in\mathscr U(M^{\mathcal U})$ such that $\{u_1\}^{\prime\prime}$ and $\{u_2\}^{\prime\prime}$ are $2$-independent.* *Then there exist Haar unitaries $v_1,v_2\in M^{\mathcal U}$ so that $[u_1,v_1]=[u_2,v_2]=[v_1,v_2]=0$.* Proposition [Proposition 21](#2unitaries){reference-type="ref" reference="2unitaries"} follows by repeating the argument used in the proof of [@CIKE22 Proposition 5.3], which we recall for the reader's convenience. *Proof.* Since $M=(\bigcup_{n\in\mathbb N}M_n)^{\prime\prime}$ and $\mathcal U$ is countably cofinal, by applying [@BCI15 Lemma 2.2] we can find ${(n_i)}_{i\in I}\subset \mathbb N$ such that $u_1,u_2\in\prod_{i\in\mathcal U}M_{n_i}$. Also, the proof of [@BCI15 Lemma 2.2] provides a function $f:I\rightarrow\mathbb N$ such that $\lim_{i\rightarrow\mathcal U}f(i)=+\infty$. Since $\{u_1\}^{\prime\prime}$ and $\{u_2\}^{\prime\prime}$ are $2$-independent, Theorem [Theorem 14](#lifting){reference-type="ref" reference="lifting"} provides orthogonal von Neumann subalgebras $C_i,D_i\subset M_{n_i}$, for every $i\in I$, such that $u_1\in\prod_{\mathcal U}C_i$ and $u_2\in\prod_{\mathcal U}D_i$. Thus, we can represent $u_1=(u_{1,i})^{\mathcal U}$ and $u_2=(u_{2,i})^{\mathcal U}$, where $u_{1,i}\in \mathscr U(C_i)$ and $u_{2,i}\in \mathscr U(D_i)$, for every $i\in I$. In particular, $\{u_{1,i}\}^{\prime\prime}$ and $\{u_{2,i}\}^{\prime\prime}$ are orthogonal, and thus $(u_{1,i},u_{2,i})\in\mathscr W(M_{n_i})$, for every $i\in I$. As the sequence $\{(u_1^{n_i,j},u_2^{n_i,j})\}_{j\in\mathbb N}$ is dense in $\mathscr W(M_{n_i})$, we can find $j_i\in\mathbb N$ such that $\|u_{1,i}-u_1^{n_i,j_i}\|_2+\|u_{2,i}-u_2^{n_i,j_i}\|_2\leq\frac{1}{f(i)}$, for every $i\in I$. For $i\in I$, let $l_i\in\mathbb N$ with $\sigma(l_i)=(n_i,j_i)$. Then $M_{\sigma(l_i)+1}=\Phi(M_{\sigma(l_i)},u_1^{n_i,j_i},u_2^{n_i,j_i})$. Corollary [Corollary 19](#amalgam){reference-type="ref" reference="amalgam"} gives Haar unitaries $v_{1,i},v_{2,i}\in\mathscr U(M_{\sigma(l_i)+1})\subset\mathscr U(M)$ with $[u_1^{n_i,j_i},v_{1,i}]=[u_2^{n_i,j_i},v_{2,i}]=[v_{1,i},v_{2,i}]=0$. Using that $\lim_{i\rightarrow\mathcal U}f(i)=+\infty$, we conclude that $v_1=(v_{1,i})^{\mathcal U}, v_2=(v_{2,i})^{\mathcal U} \in\mathscr U(M^{\mathcal U})$ are Haar unitaries such that $[u_1,v_1]=[u_2,v_2]=[v_1,v_2]=0$. ◻ To ensure that $M$ does not have property Gamma, it suffices to take $M_1$ to have property (T), as the next result from [@CIKE22] shows: **Proposition 22** (Proposition 5.4 in [@CIKE22]). *Assume that $M_1$ has property *(T)*. Then $M$ does not have property Gamma.* *Proof of Theorem [Theorem 7](#inductive){reference-type="ref" reference="inductive"}.* Let $M_1$ be a separable ${\rm II_1}$ factor with property (T), e.g., take $M_1=\operatorname{L}(\operatorname{PSL}_n(\mathbb Z))$, for $n\geq 3$. Let $M$ be constructed as in Definition [Definition 20](#M){reference-type="ref" reference="M"}. The conclusion follows from Propositions [Proposition 21](#2unitaries){reference-type="ref" reference="2unitaries"} and [Proposition 22](#propT){reference-type="ref" reference="propT"}. ◻ ## Proof of Theorem [Theorem 8](#orthogonal){reference-type="ref" reference="orthogonal"} {#proof-of-theorem-orthogonal} We may clearly assume that $z\not=0$ and $z\in M_{\text{sa},1}$, for every $z\in X\cup Y$. Further, we may assume that $X$ and $Y$ consist of pairwise orthogonal vectors. Enumerate $X=\{x_1,\dots,x_m\}$ and $Y=\{y_1,\dots,y_n\}$ and define $x=(x_1,\dots,x_m)\in M^m_{\text{sa},1}$ and $y=(y_1,\dots,y_n)\in M^n_{\text{sa},1}$. By [@Po13a Corollary 0.2] there exists $v\in\mathscr U(M^{\mathcal U})$ such that $vMv^*$ and $M$ are freely and hence $2$-independent. Then $\|[vx_iv^*,y_j]\|_2=\sqrt{2}\|x_i\|_2\|y_j\|_2>0$ and $\tau^{\mathcal U}(vx_iv^*y_j)=~0$, for every $1\leq i\leq m$, $1\leq j\leq n$. Moreover, for every $(i,j)\not=(i',j')$, we have $\langle [vx_iv^*,y_j], [vx_{i'}v^*,y_{j'}]\rangle=\tau(x_ix_{i'})\tau(y_jy_{j'})=0.$ Thus, we conclude that $\delta(vxv^*,y)>0$ and $\varepsilon(vxv^*,y)=\gamma(vxv^*,y)=0$. In particular, $$\label{condition}13mn\sqrt{\varepsilon(vxv^*,y)}<\delta(vxv^*,y)^2-(mn-1)\gamma(vxv^*,y).$$ Writing $v=(v_k)^{\mathcal U}$, where $v_k\in\mathscr U(M)$, for all $k\in\mathbb N$. Then $\lim_{k\rightarrow\mathcal U}\delta(v_kxv_k^*,y)=\delta(vxv^*,y)$, $\lim_{k\rightarrow\mathcal U}\varepsilon(v_kxv_k^*,y)=\varepsilon(vxv^*,y)$ and $\lim_{k\rightarrow\mathcal U}\gamma(v_kxv_k^*,y)=\gamma(vxv^*,y)$. Using [\[condition\]](#condition){reference-type="eqref" reference="condition"} gives $k\in\mathbb N$ such that $13mn\sqrt{\varepsilon(v_kxv_k^*,y)}<\delta(v_kxv_k^*,y)^2-(mn-1)\gamma(v_kxv_k^*,y)$. By applying Lemma [Lemma 15](#perturbation){reference-type="ref" reference="perturbation"}, we can find $w\in\mathscr U(M)$ such that $\varepsilon(w(v_kxv_k^*)w^*,y)=0$. Letting $u=wv_k\in\mathscr U(M)$, we get that $\varepsilon(uXu^*,Y)=0$, i.e., $uXu^*$ and $Y$ are orthogonal. $\square$ JNVWY20 S. Belinschi, M. Capitaine, *Strong convergence of tensor products of independent G.U.E. matrices.* `arXiv:2205.07695` C. Bordenave, B. Collins, *Norm of matrix-valued polynomials in random unitaries and permutations.* `arXiv:2304.05714` R. Boutonnet, I. Chifan, A. Ioana, *${\rm II_1}$ factors with nonisomorphic ultrapowers.* Duke Math. J. **166** (2017), 2023--2051. R. Boutonnet, C. Houdayer, *Amenable absorption in amalgamated free product von Neumann algebras.* Kyoto J. Math. **58** (2018), 583--593. N.P. Brown, N. Ozawa, *$\operatorname{C}^*$-algebras and finite-dimensional approximations.* Graduate Studies in Mathematics, **88**. American Mathematical Society, Providence, RI, 2008. xvi+509 pp. I. Chifan, C. Houdayer, *Bass--Serre rigidity results in von Neumann algebras.* Duke Math. J. **153** (2010), 23--54. I. Chifan, A. Ioana, S. Kunnawalkam Elayavalli, *An exotic ${\rm II_1}$ factor without property Gamma.* To appear in Geom. Funct. Anal. `arXiv:2209.10645` A. Connes, *Classification of injective factors. Cases ${\rm II_1}$, ${\rm II_\infty}$, ${\rm III_\lambda}$, $\lambda \neq 1$.* Ann. of Math. **74** (1976), 73--115. K. Dykema, *Free products of hyperfinite von Neumann algebras and free dimension.* Duke Math. J. **69** (1993), 97--119. I. Farah, B. Hart, D. Sherman, *Model theory of operator algebras ${\rm III}$: elementary equivalence and ${\rm II_1}$ factors.* Bull. Lond. Math. Soc. **46** (2014), 609--628. I. Goldbring, B. Hart, *On the theories of McDuff's ${\rm II_1}$ factors.* Int. Math. Res. Not. IMRN (2017), 5609--5628. B. Hayes, *$1$-bounded entropy and regularity problems in von Neumann algebras.* Int. Math. Res. Not. IMRN (2018), 57--137. B. Hayes, D. Jekel, S. Kunnawalkam Elayavalli, *Consequences of the random matrix solution to the Peterson--Thom conjecture.* `arXiv:2308.14109` B. Hayes, D. Jekel, B. Nelson, T. Sinclair, *A random matrix approach to absorption in free products.* Int. Math. Res. Not. IMRN (2021), 1919--1979. C.W. Henson, J. Iovino, *Ultraproducts in analysis.* London Math. Soc. Lecture Note Ser., **262** Cambridge University Press, Cambridge, 2002, 1--110. C. Houdayer, *Structure of ${\rm II_1}$ factors arising from free Bogoljubov actions of arbitrary groups.* Adv. Math. **260** (2014), 414--457. C. Houdayer, *Gamma stability in free product von Neumann algebras.* Comm. Math. Phys. **336** (2015), 831--851. C. Houdayer, Y. Ueda, *Asymptotic structure of free product von Neumann algebras.* Math. Proc. Cambridge Philos. Soc. **161** (2016), 489--516. A. Ioana, *Cartan subalgebras of amalgamated free product ${\rm II_1}$ factors.* With an appendix joint with Stefaan Vaes. Ann. Sci. École Norm. Sup. **48** (2015), 71--130. A. Ioana, J. Peterson, S. Popa, *Amalgamated free products of $w$-rigid factors and calculation of their symmetry groups.* Acta Math. **200** (2008), 85--153. Z. Ji, A. Natarajan, T. Vidick, J. Wright, H. Yuen, *MIP\*$=$RE.* `arXiv:2001.04383` V.F.R. Jones, *Index for subfactors.* Invent. Math. **72** (1983), 1--25. K. Jung, *Strongly $1$-bounded von Neumann algebras.* Geom. Funct. Anal. **17** (2007), 1180--1200. D. McDuff, *Uncountably many ${\rm II_1}$ factors.* Ann. of Math. **90** (1969), 372--377. T. Mei, É. Ricard, *Free Hilbert transforms.* Duke Math. J. **166** (2017), 2153--2182. M. Pimsner, S. Popa, *Entropy and index for subfactors.* Ann. Sci. École Norm. Sup. **19** (1986), 57--106. S. Popa, *Maximal injective subalgebras in factors associated with free groups.* Adv. Math. **50** (1983), 27--48. S. Popa, *On a class of type ${\rm II_1}$ factors with Betti numbers invariants.* Ann. of Math. **163** (2006), 809--899. S. Popa, *Strong rigidity of ${\rm II_1}$ factors arising from malleable actions of $w$-rigid groups ${\rm I}$.* Invent. Math. **165** (2006), 369--408. S. Popa, *Independence properties in subalgebras of ultraproduct ${\rm II_1}$ factors.* J. Funct. Anal. **266** (2014), 5818--5846. S. Popa, *A ${\rm II_1}$ factor approach to the Kadison--Singer problem.* Comm. Math. Phys. **332** (2014), 379--414. S. Popa, *Asymptotic orthogonalization of subalgebras in ${\rm II_1}$ factors.* Publ. Res. Inst. Math. Sci. **55** (2019), 795--809. M. Takesaki, *Theory of operator algebras. ${\rm II}$.* Encyclopaedia of Mathematical Sciences, **125**. Operator Algebras and Non-commutative Geometry, 6. Springer-Verlag, Berlin, 2003. xxii+518 pp. S. Vaes, *Rigidity results for Bernoulli actions and their von Neumann algebras (after S. Popa).* Séminaire Bourbaki, exposé 961. Astérisque **311** (2007), 237--294. S. Vaes, *Explicit computations of all finite index bimodules for a family of ${\rm II_1}$ factors.* Ann. Sci. École Norm. Sup. **41** (2008), 743--788. [^1]: CH is supported by Institut Universitaire de France [^2]: AI is supported by NSF DMS grants 1854074 and 2153805, and a Simons Fellowship

arxiv_math

--- abstract: | We show that the moduli spaces of bounded global $\mathcal{G}$-Shtukas with pairwise colliding legs admit $p$-adic uniformization isomorphisms by Rapoport-Zink spaces. Here $\mathcal{G}$ is a smooth affine group scheme with connected fibers and reductive generic fiber, i.e. we do not assume it to be parahoric, or even hyperspecial. Moreover, we deduce the Langlands-Rapoport Conjecture over function fields in the case of colliding legs using our uniformization theorem. address: - Uni Muenster, Germany - Columbia University, New York City, USA author: - Urs Hartl - Yujie Xu bibliography: - bibfile.bib title: Uniformizing the moduli stacks of global $G$-shtukas II --- # Introduction ## Main Results Shimura varieties play an important role in arithmetic geometry. Their structure, especially their reduction at bad primes, has been intensively studied. One way to investigate their reduction is through $p$-adic uniformization (see for example [@RZ]). In this paper, we consider the function field analogues of Shimura varieties and their $p$-adic uniformization. The first such analogues are the moduli spaces of Drinfeld modules [@Drinfeld-elliptic-modules]. In order to prove Langlands reciprocity [@Drinfeld-GL2; @Drinfeld-moduli-Fsheaves] for $\operatorname{GL}_2$ over the function field $Q$ of a smooth projective curve $X$ over a finite field $\mathbb{F}_q$, Drinfeld [@Drinfeld-moduli-Fsheaves] defined *global $\mathcal{G}$-shtukas* (which he called "$F$-sheaves") and constructed their moduli spaces. These were later generalized by Varshavsky [@Varshavsky04] and V. Lafforgue [@Lafforgue12] to the case of arbitrary constant split reductive groups $\mathcal{G}$, and by Ngô and Ngô Dac [@NgoNgo; @NgoDac13] to the case of certain non-constant groups $\mathcal{G}$. For general flat affine group schemes $\mathcal{G}$ of finite type over $X$, the moduli spaces of bounded global $\mathcal{G}$-shtukas with $n$ legs were constructed by Arasteh Rad and the first author [@AH_Unif] as separated Deligne-Mumford stacks locally of finite type over $\mathbb{F}_q$. More precisely, an (*iterated*) *global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}=(\underline x,\mathcal{E}^{(i)},\varphi^{(i)}\colon i=0,\ldots,n-1)$ with $n$ legs* over a base scheme $S$ consists of a tuple $\underline x=(x_1,\ldots,x_n)\in X^n(S)$ called *legs*, and $\mathcal{G}$-bundles $\mathcal{E}^{(i)}$ over $X_S:=X\times_{\mathbb{F}_q} S$ together with isomorphisms (called *modifications*) $\varphi^{(i-1)}\colon \mathcal{E}^{(i-1)}|_{X_S\smallsetminus\Gamma_{x_i}}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}^{(i)}|_{X_S\smallsetminus\Gamma_{x_i}}$ of $\mathcal{G}$-bundles outside the graph $\Gamma_{x_i}$ of $x_i$, with $\mathcal{E}^{(n)}:={}^{\tau\!}\mathcal{E}^{(0)}$, where $\tau:=\operatorname{id}_X \times \operatorname{Frob}_{q,S}$ (see Definition [Definition 1](#Def_Sht){reference-type="ref" reference="Def_Sht"} for the general setup). We visualize $\underline{\mathcal{E}}$ as $$\label{EqDef_Sht} \xymatrix @C+1pc { \underline{\mathcal{E}}:=\Bigl(\underline x,\, \mathcal{E}^{(0)} \ar@{-->}[r]^-{\varphi^{(0)}}_-{x_1} & \mathcal{E}^{(1)} \ar@{-->}[r]^-{\varphi^{(1)}}_-{x_2} & \ldots \ar@{-->}[r]^-{\varphi^{(n-2)}}_-{x_1} & \mathcal{E}^{(n-1)} \ar@{-->}[r]^-{\varphi^{(n-1)}}_-{x_n} & {}^{\tau\!}\mathcal{E}^{(0)}\Bigr). }$$ Drinfeld considered the case where $\mathcal{G}:=\operatorname{GL}_r$ and $n=2$ (see Example [Example 1](#ExDrinfeld){reference-type="ref" reference="ExDrinfeld"}). In addition to [@AH_Unif], various other special cases have been treated in literature: Hilbert-Blumenthal Shtukas by Stuhler [@Stuhler] for $\mathcal{G}=\operatorname{Res}_{X'|X}\operatorname{SL}_r$ where $X'$ is a smooth curve, finite flat over $X$ on which $\infty$ splits completely ("totally real case"); $\mathscr{D}$-elliptic sheaves by Laumon, Rapoport and Stuhler [@Laumon-Rapoport-Stuhler] where $\mathcal{G}$ is the unit group of (a maximal order in) a central division algebra over $Q$ (see Example [Example 1](#ExLRS){reference-type="ref" reference="ExLRS"}); generalizations of $\mathscr{D}$-elliptic sheaves by L. Lafforgue [@Lafforgue-Ramanujan], Lau [@Lau07], Ngô [@Ngo06] and Spiess [@Spiess10]; global $\mathcal{G}$-shtukas for $\mathcal{G}=\operatorname{PGL}_2$ in [@YunZhang; @YunZhang2], and for unitary groups $\mathcal{G}$ in [@FYZ; @FYZ2]. In all these cases, the modifications $\varphi^{(i)}$ were suitably bounded. In the current article, we study uniformization of the moduli stacks of $\mathcal{G}$-shtukas with *colliding legs*. We generalize the notion of boundedness from [@AH_Unif; @Bieker]. We define *bounds* $\mathcal{Z}$ in Section [2.6](#subsec:Bounds){reference-type="ref" reference="subsec:Bounds"} as closed subschemes of the Beilinson-Drinfeld Grassmannian $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ which are defined over their *reflex scheme*. The latter generalizes the notion of a reflex field, which is familiar from the theory of Shimura varieties. Here we allow more general reflex schemes than in [@AH_Unif; @Bieker]. In particular, the modifications $\varphi^{(i)}$ can be bounded by cocharacters of $\mathcal{G}$. But we also allow the bound to be defined in the fiber of $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ at a closed point of $X$ when the leg is constant at that point. This is needed to recover and generalize the bounds used in [@Stuhler; @Laumon-Rapoport-Stuhler]. We define the stack $\operatorname{Sht}_{\mathcal{G},X^n,I_\bullet}^\mathcal{Z}$ of global $\mathcal{G}$-shtukas with $n$ legs bounded by $\mathcal{Z}$. In Theorems [Theorem 1](#Thm_ShtBounded){reference-type="ref" reference="Thm_ShtBounded"} and [Theorem 1](#ThmLocMod){reference-type="ref" reference="ThmLocMod"}, we prove the following result (in more detail and generality). **Theorem 1**. *The stack $\operatorname{Sht}_{\mathcal{G},X^n,I_\bullet}^\mathcal{Z}$ is a Deligne-Mumford stack locally of finite type and separated over $X^n$. The bound $\mathcal{Z}$ is a local model for $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}^{\mathcal{Z}}$, i.e. both are isomorphic locally for the étale topology.* Our main result in this article is the uniformization of these stacks. In Section [1.2](#subsect:history){reference-type="ref" reference="subsect:history"}, we give a summary of the history of uniformization for Shimura varieties and moduli stacks of shtukas. For the latter, uniformization was proven in full generality for smooth affine group schemes $\mathcal{G}$ over $X$ with connected fibers and reductive generic fiber by Arasteh Rad and the first author [@AH_Unif] under the assumption that the legs $x_i$ stay *disjoint*. For *colliding legs*, uniformization of shtuka stacks is only known in the very special case of $\mathcal{G}=\operatorname{GL}_r$ or $\operatorname{Res}_{X'|X}\operatorname{SL}_r$ as above, with two legs, one fixed at a closed point $\infty\in X$ with basic bound and the other leg moving into $\infty$ bounded by the cocharacter $\mu=(d,0,\ldots,0)$ for a positive integer $d$ (see [@Drinfeld-elliptic-modules; @Drinfeld-commutative-subrings; @Stuhler; @Blum-Stuhler; @HartlAbSh]). In all of the above articles, the bound at $\infty$ was imposed by using chains of $\mathcal{G}$-bundles. Our results recover the case of two legs, one fixed at $\infty$ with minimal basic bound and the other leg moving into $\infty$ (see $\S$[2.8](#subsec:Chains){reference-type="ref" reference="subsec:Chains"}), but we generalize to arbitrary smooth, affine group schemes $\mathcal{G}$ over $X$ with connected fibers and reductive generic fiber $G$ and arbitrary cocharacter $\mu$ as bound. Our *first innovation* is to replace the chains from [@Drinfeld-elliptic-modules; @Drinfeld-commutative-subrings; @Stuhler; @Blum-Stuhler; @HartlAbSh] by a suitably chosen bound at $\infty$. Let $Q_\infty$ denote the completion of $Q$ at $\infty$, and let $\Breve{Q}_\infty$ be the completion of the maximal unramified extension of $Q_\infty$. Let $\mathcal{G}_\infty:=\mathcal{G}\times_X \operatorname{Spec}\mathcal{O}_\infty$ be the base change to the valuation ring $\mathcal{O}_\infty$ of $Q_\infty$. Take $\beta\in G(\Breve{Q}_\infty)$ such that $\beta \mathcal{G}_\infty\beta^{-1}=\mathcal{G}_\infty$. Thus in a sense, $\beta$ is as small as possible (see § [ 1](#Def_beta){reference-type="ref" reference="Def_beta"}). We then consider the stack $\operatorname{Sht}_{\mathcal{G},X\times\infty}$ of global $\mathcal{G}$-shtukas with two legs, of which one is allowed to vary over all of $X$ and the other is fixed at $\infty$. For a conjugacy class of cocharacters $\mu\colon\mathbb{G}_m \to G$ and a compact open subgroup $H\subset G(\mathbb{A}^\infty)$, we define the stack $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}$ of such global $\mathcal{G}$-shtukas with $H$-level structure, whose varying leg $x\colon S\to X$ factors through $\widehat{\infty}:=\operatorname{Spf}\mathcal{O}_\infty$ and whose modification $\varphi^{(0)}$ (respectively $\varphi^{(1)}$) at $x$ (respectively $\infty$) is bounded by $\mu$ (respectively by $\beta$) (see Definitions [Definition 1](#DefZmubeta){reference-type="ref" reference="DefZmubeta"} and [Definition 1](#Def_ShtBounded){reference-type="ref" reference="Def_ShtBounded"}). Our *second innovation* is to construct the $\beta^{-1}$-twisted global-local functor $L^+_{\infty,\mathcal{M}}$ in Definition [Definition 1](#DefGlobLocM){reference-type="ref" reference="DefGlobLocM"}, which relates such global $\mathcal{G}$-shtukas to local $\mathcal{M}$-shtukas (see §[2.9](#subsec-LocSht-and-RZ){reference-type="ref" reference="subsec-LocSht-and-RZ"}), where $\mathcal{M}$ is the inner form of $\mathcal{G}_\infty$ given by $\beta^{-1}$. The cocharacter $\mu$ gives rise to a cocharacter $\mu$ of $\mathcal{M}$ by which we can bound local $\mathcal{M}$-shtukas (see Definition [Definition 1](#DefLocShtBounded){reference-type="ref" reference="DefLocShtBounded"} for more details). Let $\mathcal{O}_\mu$ be the ring of integers in the reflex field extension of $Q_\infty$ of $\mu$ and let $\Breve{\mathcal{O}}_\mu$ be the completion of the maximal unramified extension of $\mathcal{O}_\mu$. For any global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}\in \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)$, the local $\mathcal{M}$-shtuka $L^+_{\infty,\mathcal{M}}(\underline{\mathcal{E}})$ is bounded by $\mu$ (Corollary [Corollary 1](#CorGlobalLocalFunctorWithChains){reference-type="ref" reference="CorGlobalLocalFunctorWithChains"}). Via this twisted global-local functor, we obtain a Serre-Tate Theorem for Shtukas. **Theorem 1**. *There is an equivalence between deformations of $\underline{\mathcal{E}}$ and deformations of $L^+_{\infty,\mathcal{M}}(\underline{\mathcal{E}})$.* Theorem [Theorem 1](#Serre-Tate-thm-intro){reference-type="ref" reference="Serre-Tate-thm-intro"} allows us to prove the following Theorem [Theorem 1](#Uniformization1Intro){reference-type="ref" reference="Uniformization1Intro"}. More precisely, we fix a global framing object $\mathcal{G}$-shtuka ${\underline{\mathbb{E}}}\in \operatorname{Sht}_{\mathcal{G},\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(\overline{\mathbb{F}}_\infty)$ over $\overline{\mathbb{F}}_\infty$. Let ${\underline{{\mathbb{L}}}}$ be its associated local $\mathcal{M}$-shtuka under the $\beta$-twisted global-to-local functor (see Definition [Definition 1](#DefGlobLocM){reference-type="ref" reference="DefGlobLocM"}). Let $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ be the *Rapoport-Zink space* of ${\underline{{\mathbb{L}}}}$ bounded by $\mu$. $$\begin{aligned} \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S):=\Bigl\{\,(\underline{\mathcal{L}},\hat{\delta}) & \text{ where $\underline{\mathcal{L}}$ is local $\mathcal{M}$-shtuka over $S$ bounded by $\mu$}, \nonumber \\ & \text{and }\hat{\delta}:\underline{\mathcal{L}}\to {\underline{{\mathbb{L}}}}_S \text{ is a quasi-isogeny}\,\Bigr\}. \end{aligned}$$ It was shown by Arasteh Rad and the first author in [@AH_Local Theorem 4.18] that the Rapoport-Zink space is representable by a formal scheme locally formally of finite type over $\operatorname{Spf}\Breve{\mathcal{O}}_\mu$. Finally, let $I_{\underline{\mathbb{E}}}(Q)$ be the quasi-isogeny group of ${\underline{\mathbb{E}}}$ (see Remark [Remark 1](#RemIsogGlobalSht){reference-type="ref" reference="RemIsogGlobalSht"}). With this setup, we generalize the uniformization theorem of [@Drinfeld-elliptic-modules; @Drinfeld-commutative-subrings; @Stuhler; @Blum-Stuhler; @HartlAbSh] to the following. **Theorem 1**. *(a) There is a canonical morphism $$\label{EqUnifIntro} \Theta_{{\underline{\mathbb{E}}}}\colon I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times G(\mathbb{A}^\infty)/H\bigr) \;\longrightarrow\; \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}\operatorname{Spf}\Breve{\mathcal{O}}_\mu$$ of formal algebraic Deligne-Mumford stacks over $\operatorname{Spf}\Breve{\mathcal{O}}_\mu$, which is an ind-proper and formally étale monomorphism.* *(b) Let $\mathcal{X}:=\mathcal{X}_{\underline{\mathbb{E}}}$ be the image of $\Theta_{{\underline{\mathbb{E}}}}$ as in Lemma [Lemma 1](#LemmaImageOfTheta){reference-type="ref" reference="LemmaImageOfTheta"}[\[LemmaImageOfTheta_B\]](#LemmaImageOfTheta_B){reference-type="ref" reference="LemmaImageOfTheta_B"}. It is the isogeny class of ${\underline{\mathbb{E}}}$. Let $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}{}_{/\mathcal{X}}$ be the formal completion of $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}\operatorname{Spf}\Breve{\mathcal{O}}_\mu$ along the set $\mathcal{X}_{\underline{\mathbb{E}}}$. Then $\Theta_{{\underline{\mathbb{E}}}}$ induces an isomorphism of locally noetherian, adic formal algebraic Deligne-Mumford stacks locally formally of finite type over $\operatorname{Spf}\Breve{\mathcal{O}}_\mu$ $$\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times G(\mathbb{A}^\infty)/H\bigr)\;\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\; \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}_{/\mathcal{X}}\,.$$* Our theorem recovers all known uniformization results for moduli spaces of shtukas with colliding legs. For the moduli space of $\mathscr{D}$-elliptic sheaves, we hereby prove the result expected by Laumon, Rapoport and Stuhler [@Laumon-Rapoport-Stuhler (14.19)]. We prove Theorem [Theorem 1](#Uniformization1Intro){reference-type="ref" reference="Uniformization1Intro"} slightly more generally in Theorems [Theorem 1](#Uniformization1){reference-type="ref" reference="Uniformization1"} and [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"} and discuss the compatibility of the morphisms with the actions through Hecke correspondences and other actions. Combining our techniques with the ones from [@AH_Unif], one can even extend our result to the case of several disjoint pairs of two colliding legs, such that in each pair one leg varies and the other leg is fixed at a place $\infty_i$ and bounded by some element $\beta_i$. Then the uniformizing space will be a product of Rapoport-Zink spaces for inner forms of $\mathcal{G}$; see Remark [Remark 1](#RemManyPairsOfLegs){reference-type="ref" reference="RemManyPairsOfLegs"}. As an application of our Theorem [Theorem 1](#Uniformization1Intro){reference-type="ref" reference="Uniformization1Intro"}, we prove the function field analogue of the Langlands-Rapoport conjecture for moduli spaces of global $\mathcal{G}$-Shtukas with colliding legs. The case of disjoint legs was solved by Arasteh Rad and the first author in [@AH_LRConj]. The Langlands-Rapoport conjecture was first proposed for Shimura varieties in [@Langlands-Jugendtraum], which describes some of the ideas flowing from Kronecker's *Jugendtraum*. The conjecture, following the works of Ihara [@Ihara1; @Ihara2-Vol1; @Ihara2-Vol2], is an essential part of the Langlands program [@Langlands2; @Langlands3; @Langlands4] to express the zeta function of a Shimura variety as a product of automorphic $L$-functions. This conjecture was made more precise by Kottwitz [@Kottwitz-lambda-adic] and then further refined by Langlands-Rapoport [@Langlands-Rapoport-gerbes] using the formalism of motives. In the Shimura varieties setting, there has been progress towards this conjecture in the case of abelian type Shimura varieties at hyperspecial level at $p$ [@Kisin-mod-p-points; @Kisin-Shin-Zhu]. In the function field setting, we consider the $Q$-linear semi-simple Tannakian category $\mathcal{M}ot_X^{\infty }$ of "$X$-motives" away from $\infty$ (see Definition [Definition 1](#defn-category-Xmotives){reference-type="ref" reference="defn-category-Xmotives"}), which generalizes Anderson's $t$-motives [@Anderson-tmotives]. It is equipped with a fiber functor $\underline{\omega}$ to vector spaces over $Q\otimes_{\mathbb{F}_q}\overline{\mathbb{F}}_q$, whose Tannakian fundamental group $\mathfrak{P}:=\mathrm{Aut}^{\otimes}(\underline{\omega}|\mathcal{M}ot_X^{\infty })$ is the *motivic groupoid*. One can alternatively view $\mathfrak{P}$ as a motivic Galois gerbe via $1\to \mathfrak{P}^{\Delta}\to\mathfrak{P}\to \operatorname{Gal}(Q\otimes_{\mathbb{F}_q}{\overline{\mathbb{F}}_q}/Q)\to 1$. Given any global $\mathcal{G}$-Shtuka $\underline{\mathcal{E}}\in\operatorname{Sht}_{\mathcal{G},\varnothing,\widehat{\infty}\times\infty}(\overline{\mathbb{F}}_\infty)$, one can associate a corresponding "$G$-motive", which is a tensor functor $h_{\underline{\mathcal{E}}}$ from $\operatorname{Rep}_QG$ to the category $\mathcal{M}ot_X^{\infty }$ of "$X$-motives"; equivalently, $h_{\underline{\mathcal{E}}}$ gives a homomorphism (of Galois gerbes) from $\mathfrak{P}$ to the the neutral Galois gerbe $\mathfrak{G}_G:=G(\Breve{Q})\rtimes \operatorname{Gal}(\Breve{Q}/Q)$ of $G$. To each such homomorphism $h$, one can attach a set $X^{\infty}(h)$, which corresponds to "prime-to-$\infty$" quasi-isogenies (i.e. they are isomorphisms at $\infty$), and a set $X_{\infty}(h)$, which corresponds to "at-$\infty$" quasi-isogenies (i.e. they are isomorphisms away from $\infty$). Let $I_h$ be the "isogeny group" of $h$ (see Remark [Remark 1](#RemIsogGlobalSht){reference-type="ref" reference="RemIsogGlobalSht"}). The Langlands-Rapoport conjecture gives a precise description of the action of $I_h(Q)$ on $X_{\infty}(h)\times X^{\infty}(h)$. **Theorem 1**. *The $\overline{\mathbb{F}}_{\infty}$-points of the Shtuka space $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}$ has the form predicted by the Langlands-Rapoport conjecture, i.e. $$\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(\overline{\mathbb{F}}_{\infty})=\coprod\limits_h I_h(Q)\backslash X_{\infty}(h)\times X^{\infty}(h)/H,$$ compatible with Hecke correspondences, Frobenius, and the action of the center.* Note that for the level structure at $\infty$, we take the group scheme $\mathcal{G}_{\infty}$ which is only assumed to be smooth, affine, with connected fibers and reductive generic fiber. In particular, our level structure at $\infty$ does not need to be parahoric (or even hyperspecial). ## Historical overview of uniformization {#subsect:history} For convenience of the reader, let us summarize the history of uniformization for Shimura varieties and shtuka stacks. **A. Uniformization varieties at infinity.** The history begins in the 19th century with (1) elliptic modular curves over the complex numbers $\mathbb{C}$, which can be written as quotients of Poincaré's upper halfplane by congruence subgroups. It was generalized by Baily, Borel , and Shimura who showed that (2) certain quotients of Hermitian symmetric domains by discrete arithmetic groups are algebraic varieties and defined over number fields. Deligne [@Deligne-varietes-de-Shimura] systematically developed the theory of these varieties, which today are called *Shimura varieties* and wrote them as a double quotient of a Hermitian symmetric domain times the adèle points of the corresponding reductive group. All the cases (1), (2) of uniformization are for number fields and over $\mathbb{C}$, i.e. "at infinity". Function fields first came into play with (3) Drinfeld modular varieties [@Drinfeld-elliptic-modules] which parameterize Drinfeld $A$-modules of rank $r$ where $A=\Gamma(X\smallsetminus\{\infty\},\mathcal{O}_X)$ for a fixed closed point $\infty\in X$. Drinfeld $A$-modules have one leg $x\colon S\to\operatorname{Spec}A\subset X$. Let $\mathbb{C}_\infty$ be the completion of an algebraic closure of $Q_\infty$. Then Drinfeld showed that the points of the Drinfeld modular varieties with values in $S=\operatorname{Spec}\mathbb{C}_\infty$ are the quotient of his $(r-1)$-dimensional upper halfspace $\Omega^r_{Q_\infty}$ by a congruence subgroup. Deligne [@Deligne-Husemoller] explained that this can be rewritten as a double coset $\operatorname{GL}_r(Q)\backslash\Omega^r_{Q_\infty}\times \operatorname{GL}_r(\mathbb{A}^\infty)/H$, where $\mathbb{A}^\infty$ are the adèles of $Q$ outside $\infty$ and $H\subset\operatorname{GL}_r(\mathbb{A}^\infty)$ is a compact open subgroup. Also the uniformization of the Drinfeld modular varieties is "at infinity", because $\infty$ is the place forbidden for the leg $x\colon S\to\operatorname{Spec}A = X\smallsetminus\{\infty\}$, which still moves "close" to $\infty$ on $S=\operatorname{Spec}\mathbb{C}_\infty$. **B. Uniformization away from infinity.** At the same time, uniformization at a place $p$ different from infinity arose in the work (4) of Čerednik [@Cerednik76], who proved that certain Shimura curves of EL-type have $p$-adic uniformization by Deligne's formal model $\widehat{\Omega}^2_{\mathbb{Q}_p}$ of Drinfeld's upper halfplane $\Omega^2_{\mathbb{Q}_p}$. Drinfeld [@Drinfeld-padic-covering] explained that for any $r$ the formal model $\widehat{\Omega}^r_{\mathbb{Q}_p}$ is a Rapoport-Zink space for an inner form of $\operatorname{GL}_r$, i.e. a moduli space for $p$-divisible groups with extra structure that are isogenous to a fixed supersingular $p$-divisible group. See Boutot--Carayol [@Boutot-Carayol] and Genestier [@Genestier-Asterisque] for a detailed account. This was vastly generalized (5) by Rapoport and Zink [@RZ] to (partial) $p$-adic uniformization of integral models of higher dimensional Shimura varieties by more general moduli spaces for $p$-divisible groups. These integral models have a morphism to $\operatorname{Spec}\mathbb{Z}_p$, which can be called the "leg" of the data parameterized by the integral model. This leg stays disjoint from $\infty$, which is a kind of "fixed second leg" for all Shimura varieties, see Remark [Remark 1](#RemSecondLegShiVar){reference-type="ref" reference="RemSecondLegShiVar"}. In contrast, for the uniformizations (1), (2), (3) mentioned in the previous paragraph the varying leg moved towards $\infty$. **C. Uniformization of moduli spaces of shtukas.** Generalizing the uniformization (3) of the Drinfeld modular varieties, Stuhler [@Stuhler] proved (6) uniformization at $\infty$ of his moduli spaces of Hilbert-Blumenthal shtukas. In [@Blum-Stuhler], Blum and Stuhler reinterpreted and reproved the uniformization (3) in terms of (7) Drinfeld's "elliptic sheaves" [@Drinfeld-commutative-subrings]. The latter was generalized by the first author in [@HartlAbSh] where (8) partial uniformization at $\infty$ of moduli stacks of "abelian $\tau$-sheaves" was proven. Laumon, Rapoport and Stuhler mention (9) the uniformization at $\infty$ of their moduli spaces of $\mathscr{D}$-elliptic sheaves in [@Stuhler p. 493] and [@Laumon-Rapoport-Stuhler (14.19)], but do not prove it. Uniformization (10) for these spaces at a place $v$ different from $\infty$, i.e. for the two legs $v$ and $\infty$ staying disjoint, was proven by Hausberger [@Hausberger]. When all the legs stay *disjoint*, the uniformization of moduli spaces of $\mathcal{G}$-shtukas with $n$ legs was proven in full generality (11) for smooth affine $\mathcal{G}$ with connected fibers and reductive generic fiber by Arasteh Rad and the first author [@AH_Unif]. In all these cases (7), (8), (9), (10), (11) the uniformizing spaces are Rapoport-Zink spaces for local shtukas as above. For *colliding legs* uniformization of shtuka stacks (3), (6), (7), (8) is only known in the very special case for $\mathcal{G}=\operatorname{GL}_r$ or $\operatorname{Res}_{X'|X}\operatorname{SL}_r$ and two legs, one fixed at $\infty$ with basic bound and the other leg moving into $\infty$. The condition on the legs is analogous to (1), (2), see Remark [Remark 1](#RemSecondLegShiVar){reference-type="ref" reference="RemSecondLegShiVar"}. In (6), (7), (8), (9) the boundedness condition at $\infty$ on the shtukas was imposed by using chains. We give a detailed explanation of this in Section [2.8](#subsec:Chains){reference-type="ref" reference="subsec:Chains"}. **Remark 1**. We want to explain, why we think that Shimura varieties have two legs, i.e. a hidden leg at infinity in addition to the obvious leg, which is the structure morphism of the Shimura variety over $\operatorname{Spec}\mathbb{Z}$. For simplicity, we restrict to the case of Shimura varieties of PEL-type parameterizing abelian varieties with extra structures. Consider an abelian variety $\mathcal{A}$ over a finite field $\mathbb{F}_q$ of characteristic $p$. The (varying) leg of $\mathcal{A}$ is the morphism $\operatorname{Spec}\mathbb{F}_q\to \operatorname{Spec}\mathbb{Z}$ given by the natural homomorphism $\mathbb{Z}\to \mathbb{Z}/(p)\to \mathbb{F}_q$. This leg and the hidden leg at infinity of $\mathcal{A}$ can be seen by looking at the absolute values of the $q$-Frobenius endomorphism $\pi\colon \mathcal{A}\to\mathcal{A}$ of $\mathcal{A}$. Since the endomorphism algebra $\operatorname{End}^\circ(\mathcal{A})$ of $\mathcal{A}$ is a finite dimensional $\mathbb{Q}$-algebra, $\pi$ is the root of its minimal polynomial $m_\pi\in\mathbb{Q}[T]$. Fix an absolute value $|\,.\,|$ on an algebraic closure $\mathbb{Q}^{\rm alg}$ of $\mathbb{Q}$ and a root $\alpha\in\mathbb{Q}^{\rm alg}$ of $m_\pi$. If $|\,.\,|$ extends an $\ell$-adic absolute value on $\mathbb{Q}$ for $\ell\ne p$, then $|\alpha|=1$. On the other hand, if $\ell=p$ then $|\alpha|$ can be different from $1$ depending on the slopes of (the $p$-divisible group of) $\mathcal{A}$. Responsible for both cases $\ell\ne p$ and $\ell=p$ is the leg at $p$ which implies that at $\ell\ne p$ all slopes are zero. Finally, if $|\,.\,|$ is obtained from the archimedean absolute value on $\mathbb{C}$ by an inclusion $Q^{\rm alg}\hookrightarrow\mathbb{C}$, we have $|\alpha|=q^{1/2} \ne 1$. This hints at the presence of another leg at infinity, where all slopes of the Frobenius endomorphism $\pi$ are equal, that is $\pi$ and $\mathcal{A}$ and its $p$-divisible group could be called "basic". The authors would like to thank Eva Viehmann for helpful conversations. U.H. acknowledges support of the DFG (German Research Foundation) in form of Project-ID 427320536 -- SFB 1442, and Germany's Excellence Strategy EXC 2044--390685587 "Mathematics Münster: Dynamics--Geometry--Structure". Y.X. was supported by the National Science Foundation under Award No. 2202677. # Preliminaries ## Notations {#subsec-notations} Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $X$ denote a smooth, projective, geometrically connected curve over $\mathbb{F}_q$. Let $Q:=\mathbb{F}_q(X)$ be its function field. Let $Q^{\rm alg}$ and $Q^{{\operatorname{sep}}}$ denote an algebraic and separable closure of $Q$, respectively. For an $\mathbb{F}_q$-scheme $S$ and an open or closed subscheme $U\subset X$, denote $U_S:=U\times_{\mathbb{F}_q}S$. For a morphism $x\colon S\to X$ of $\mathbb{F}_q$-schemes, we denote by $\Gamma_x\subset X_S$ the graph of $x$. Closed points of $X$ are also called *places* of $Q$ or of $X$. For a closed point $v\in X$, we denote by $\mathbb{F}_v$ its residue field, $\mathcal{O}_v:=\widehat{\mathcal{O}}_{X,v}$ its complete local ring, and $Q_v=\operatorname{Frac}(\mathcal{O}_v)$ the fraction field of $\mathcal{O}_v$. Let $\overline{\mathbb{F}}_v$ be a separable closure of $\mathbb{F}_v$. Let $\Breve{\mathcal{O}}_v$ and $\Breve{Q}_v$ denote the completions of maximal unramified extensions of $\mathcal{O}_v$ and $Q_v$, respectively. Upon fixing a uniformizer $z_v$ at $v$, one has canonical identifications $\mathcal{O}_v=\mathbb{F}_v{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_v {\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}, Q_v=\mathbb{F}_v{\mathchoice{\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\scriptsize\rm (\hspace{-0.15em}(}} {\mbox{\tiny\rm (\hspace{-0.15em}(}}}z_v {\mathchoice{\mbox{\rm )\hspace{-0.15em})}} {\mbox{\rm )\hspace{-0.15em})}} {\mbox{\scriptsize\rm )\hspace{-0.15em})}} {\mbox{\tiny\rm )\hspace{-0.15em})}}}, \Breve{\mathcal{O}}_v=\overline{\mathbb{F}}_v{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_v {\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$ and $\Breve{Q}_v=\overline{\mathbb{F}}_v{\mathchoice{\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\scriptsize\rm (\hspace{-0.15em}(}} {\mbox{\tiny\rm (\hspace{-0.15em}(}}}z_v {\mathchoice{\mbox{\rm )\hspace{-0.15em})}} {\mbox{\rm )\hspace{-0.15em})}} {\mbox{\scriptsize\rm )\hspace{-0.15em})}} {\mbox{\tiny\rm )\hspace{-0.15em})}}}$. Let $\mathcal{N}ilp_{\mathcal{O}_v}$ (resp. $\mathcal{N}ilp_{\Breve{\mathcal{O}}_v}$) denote the category of all $\mathcal{O}_v$-schemes (resp. $\Breve{\mathcal{O}}_v$-schemes) on which $z_v$ is locally nilpotent (in the structure sheaf). In some parts of this article we will consider a point $\infty\in X(\mathbb{F}_q)$, which is assumed to be $\mathbb{F}_q$-rational for simplicity. Then $\mathbb{F}_\infty=\mathbb{F}_q$, but we will still use the notation $\mathbb{F}_\infty$ to emphasize that it comes with the morphism $\operatorname{Spec}\mathbb{F}_\infty\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\{\infty\}\subset X$. Let $\mathcal{G}$ be a smooth affine group scheme over $X$ with connected fibers and reductive generic fiber $G:=\mathcal{G}\times_X \operatorname{Spec}Q$. Set $\mathcal{G}_v:=\mathcal{G}\times_X \operatorname{Spec}\mathcal{O}_v$ and $G_v:=\mathcal{G}\times_X \operatorname{Spec}Q_v$. As usual, for elements $g,h\in\mathcal{G}(S)$ for some $S\to X$ we write $\operatorname{int}_h\colon g\mapsto h\,g\,h^{-1}$ for the conjugation action ("interior automorphism"). For a sheaf $\mathscr{H}$ of groups (in the *fppf*-topology) on a scheme $Y$, an *$\mathscr{H}$-bundle* (also called a *right* *$\mathscr{H}$-torsor*) on $Y$ is a sheaf $\mathcal{E}$ for the *fppf*-topology on $Y$, together with a right action of the sheaf $\mathscr{H}$ such that $\mathcal{E}$ is isomorphic to $\mathscr{H}$ on an *fppf*-covering of $Y$. Here $\mathscr{H}$ is viewed as an $\mathscr{H}$-torsor via right multiplication. We denote by $\tau:=\operatorname{Frob}_{q,S}$ the absolute $q$-Frobenius of an $\mathbb{F}_q$-scheme $S$, which is the identity on the topological space, and the $q$-power map on the structure sheaf. For a place $v\in X$ we let $q_v:=\#\mathbb{F}_v=q^{[\mathbb{F}_v:\mathbb{F}_q]}$ and $\hat{\tau}_v:=\tau^{[\mathbb{F}_v:\mathbb{F}_q]}=\operatorname{Frob}_{q_v,S}$. For data defined over $S$ (e.g. $\mathcal{G}$-bundles $\mathcal{E}$ on $X_S$), we denote the pullback under $\tau$ by a left superscript $\tau$ (e.g. ${}^{\tau\!}\mathcal{E}$). We use a similar notation with $\hat{\tau}_v$ or more generally with $\tau^n$ for $n\in \mathbb{N}$. For a linear algebraic group $M$ over $Q_v$, the Frobenius $$\label{EqTau_G} \tau_{M}\colon L_vM(S)\to L_vM(S),$$ for an $\mathbb{F}_v$-scheme $S$, is defined by sending $g\colon S\to L_vM$ to $\tau_{M}(g):=g\circ\operatorname{Frob}_{q_v,S}$. In particular, this applies to $M=G_v$. ## Loop groups {#subsec:LoopGp} We will recall the definition of the (positive) loop groups $L_\Delta \mathcal{G}$ and $L^+_\Delta \mathcal{G}$ in the general setting for an effective relative Cartier divisor $\Delta\subset X_R$ over $\operatorname{Spec}R$. We then modify the notation in the important special cases discussed in Example [Example 1](#ExDivisors){reference-type="ref" reference="ExDivisors"}. Let $S=\operatorname{Spec}R$ be affine and write $X_R:=X_S$. Let $\Delta\subset X_R$ be an effective relative Cartier divisor over $S$, i.e. $\Delta$ is an effective Cartier divisor on $X_R$ and is finite flat over $S$. In particular, $\Delta$ is an affine scheme. Its ideal sheaf $\mathscr{I}_\Delta\subseteq\mathcal{O}_{X_R}$ is invertible. Thus Zariski-locally on $X_R$, the sheaf $\mathscr{I}_\Delta=z_\Delta\cdot \mathcal{O}_{X_R}$ is principal, for some $z_\Delta\in \mathcal{O}_{X_R}$. In particular, $\Delta=\operatorname{Spec}\mathcal{O}_{X_R}/\mathscr{I}_\Delta$ is locally of the form $\operatorname{Spec}\mathcal{O}_{X_R}/(z_\Delta)$. Let $\widehat{\Delta}$ be the formal completion of $X_R$ at $\Delta$. It is an affine formal scheme of the form $\operatorname{Spf}\widehat{\mathcal{O}}_{X_R,\Delta}$, where $\widehat{\mathcal{O}}_{X_R,\Delta}:=\underset{n}{\varprojlim}\,\mathcal{O}_{X_R}/\mathscr{I}_\Delta^n$. Looking at an open neighborhood where $\mathscr{I}_\Delta=z_\Delta\cdot \mathcal{O}_{X_R}$ is principal, we see that Zariski-locally on $\operatorname{Spec}R$, the formal scheme $\widehat{\Delta}$ is of the form $\operatorname{Spf}R{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_\Delta {\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$. **Definition 1**. Let $\Delta\subset X_R$ be an effective relative Cartier divisor over $S=\operatorname{Spec}R$. 1. The *positive loop group of $\mathcal{G}$ at $\Delta$* is the ${\it fpqc\/}$-sheaf of groups $L^+_\Delta\mathcal{G}$ over $\operatorname{Spec}R$ whose $R'$-points, for an $R$-algebra $R'$ are given by $$\begin{aligned} \begin{split} L^+_\Delta\mathcal{G}(R')&:=\mathcal{G}(\widehat{\mathcal{O}}_{X_{R'},\Delta'})=\operatorname{Hom}_X(\operatorname{Spf}\widehat{\mathcal{O}}_{X_{R'},\Delta'},\mathcal{G})=\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{O}_{\mathcal{G}},\widehat{\mathcal{O}}_{X_{R'},\Delta'})\\ &=\underset{n}{\varprojlim} \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{O}_{\mathcal{G}},\mathcal{O}_{X_{R'}}/\mathscr{I}_{\Delta'}^n)=\underset{n}{\varprojlim}\,\mathcal{G}(\operatorname{Spec}\mathcal{O}_{X_{R'}}/\mathscr{I}_{\Delta'}^n), \end{split}\end{aligned}$$ where $\Delta'\subset X_{R'}$ denotes the pullback of $\Delta$ to $X_{R'}$. 2. The *loop group of $\mathcal{G}$ at $\Delta$* is the ${\it fpqc\/}$-sheaf of groups $L_\Delta\mathcal{G}$ over $\operatorname{Spec}R$ whose $R'$-points, for an $R$-algebra $R'$, are given by $$\label{Defn-LDeltaG} L_\Delta\mathcal{G}(R'):=\mathcal{G}(\widehat{\mathcal{O}}_{X_{R'},\Delta'}[z_\Delta^{-1}]).$$ Zariski-locally on $R$ the group $L_\Delta^+\mathcal{G}(R')$ is of the form $\mathcal{G}(R'{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_\Delta {\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}})$ and the group $L_\Delta\mathcal{G}(R')$ is of the form $\mathcal{G}(R'{\mathchoice{\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\scriptsize\rm (\hspace{-0.15em}(}} {\mbox{\tiny\rm (\hspace{-0.15em}(}}}z_\Delta {\mathchoice{\mbox{\rm )\hspace{-0.15em})}} {\mbox{\rm )\hspace{-0.15em})}} {\mbox{\scriptsize\rm )\hspace{-0.15em})}} {\mbox{\tiny\rm )\hspace{-0.15em})}}})$ for $R'{\mathchoice{\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\scriptsize\rm (\hspace{-0.15em}(}} {\mbox{\tiny\rm (\hspace{-0.15em}(}}}z_\Delta {\mathchoice{\mbox{\rm )\hspace{-0.15em})}} {\mbox{\rm )\hspace{-0.15em})}} {\mbox{\scriptsize\rm )\hspace{-0.15em})}} {\mbox{\tiny\rm )\hspace{-0.15em})}}}:=R'{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_\Delta {\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}[z_\Delta^{-1}]$. Note that [\[Defn-LDeltaG\]](#Defn-LDeltaG){reference-type="eqref" reference="Defn-LDeltaG"} is independent of the choice of $z_\Delta$, because any other $z_\Delta$ is of the form $\widetilde{z}_\Delta=u\cdot z_\Delta$ for some $u\in \mathcal{O}_{X_R}^{\times}\subseteq\widehat{\mathcal{O}}_{X_R,\Delta}^\times$. Thus, the functor $L_\Delta^+\mathcal{G}$ is representable by an infinite-dimensional affine group scheme over $R$. Moreover, the functor $L_\Delta\mathcal{G}$ is representable by an ind-affine ind-scheme of ind-finite type over $R$ by [@Richarz16 Lemma 2.11]. Let $[\operatorname{Spec}R/L_\Delta^+\mathcal{G}]$ (resp. $[\operatorname{Spec}R/L_\Delta\mathcal{G}]$) denote the classifying space of $L_\Delta^+\mathcal{G}$-bundles (resp. $L_\Delta\mathcal{G}$-bundles). It is a stack fibered in groupoids over the category of $R$-schemes $S'$ whose category $[\operatorname{Spec}R/L_\Delta^+\mathcal{G}](S')$ (resp. $[\operatorname{Spec}R/L_\Delta\mathcal{G}](S')$) consists of all $L_\Delta^+\mathcal{G}$-bundles (resp. $L_\Delta\mathcal{G}$-bundles) on $S'$. The inclusion of sheaves $L_\Delta^+\mathcal{G}\subset L_\Delta\mathcal{G}$ gives rise to the natural 1-morphism $$\label{EqLoopTorsorDelta} L_\Delta\colon[\operatorname{Spec}R/L_\Delta^+\mathcal{G}]\longrightarrow [\operatorname{Spec}R/L_\Delta\mathcal{G}],\quad \mathcal{L}\longmapsto L_\Delta\mathcal{L}.$$ **Definition 1**. The (*local*) *affine flag variety* of $\mathcal{G}$ at a divisor $\Delta\subset X_R$ is the *fpqc*-sheaf $\mathcal{F}\!\ell_{\mathcal{G},\Delta}:=L_\Delta \mathcal{G}/L^+_\Delta \mathcal{G}$ on $\operatorname{Spec}R$. **Lemma 1**. *The affine flag variety $\mathcal{F}\!\ell_{\mathcal{G},\Delta}$ represents the functor on $R$-schemes that sends an $R$-scheme $S$ to the set of isomorphism classes of pairs $(\mathcal{L},\hat{\delta})$, where $\mathcal{L}$ is an $L^+_\Delta \mathcal{G}$-bundle over $S$ and $\hat{\delta}\colon L_\Delta \mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L_\Delta \mathcal{G})_S$ is an isomorphism of $L_\Delta \mathcal{G}$-bundles over $S$.* *Proof.* This was proven in [@Richarz16 Lemma 2.12] (or [@Pappas-Rapoport-twisted-loop-groups] in the special case where $\Delta= \{v\}\times_{\mathbb{F}_q} S \subset X_S$ for a closed point $v$ in $X$). ◻ **Example 1**. (a) For any $x\in X(R)$, the graph $\Delta:=\Gamma_x\subset X_R$ of $x$ is an effective relative Cartier divisor over $\operatorname{Spec}R$ by [@Neron-models-book § 8.2, Lemma 6]. In this case we write $\widehat{\Gamma}_x:=\widehat{\Delta}$ for the formal completion of $X_R$ along $\Gamma_x$. We also write $L^+_x\mathcal{G}:=L^+_\Delta\mathcal{G}$ and $L_x\mathcal{G}:=L_\Delta\mathcal{G}$ for the (positive) loop group of $\mathcal{G}$ at $x$, and $\mathcal{F}\!\ell_{\mathcal{G},x}:=\mathcal{F}\!\ell_{\mathcal{G},\Delta}$ for the affine flag variety, and $L_x$ for the functor from [\[EqLoopTorsorDelta\]](#EqLoopTorsorDelta){reference-type="eqref" reference="EqLoopTorsorDelta"}. \(b\) As a special case of (a) consider a closed point $v\in X$ and view it as a point $x:=v\in X(\mathbb{F}_v)$ for $R=\mathbb{F}_v$. Let $z_v$ be a uniformizing parameter at $v$. In this case we write $\mathcal{F}\!\ell_{\mathcal{G},v}:=\mathcal{F}\!\ell_{\mathcal{G},x}$ for the affine flag variety, $L_v$ for the functor from [\[EqLoopTorsorDelta\]](#EqLoopTorsorDelta){reference-type="eqref" reference="EqLoopTorsorDelta"}, and $L^+_v\mathcal{G}:=L^+_x\mathcal{G}$ and $L_v\mathcal{G}:=L_x\mathcal{G}$ for the (positive) loop group of $\mathcal{G}$ at $v$. They are ${\it fpqc\/}$-sheaves of groups on $\operatorname{Spec}\mathbb{F}_v$. Their $R'$-valued points for an $\mathbb{F}_v$-algebra $R'$ are $$L_v^+\mathcal{G}(R')=\mathcal{G}(R'{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_v{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}})=\mathcal{G}_v(R'{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_v{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}) \qquad \text{and} \qquad L_v\mathcal{G}(R')=\mathcal{G}(R'{\mathchoice{\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\scriptsize\rm (\hspace{-0.15em}(}} {\mbox{\tiny\rm (\hspace{-0.15em}(}}}z_v{\mathchoice{\mbox{\rm )\hspace{-0.15em})}} {\mbox{\rm )\hspace{-0.15em})}} {\mbox{\scriptsize\rm )\hspace{-0.15em})}} {\mbox{\tiny\rm )\hspace{-0.15em})}}})=\mathcal{G}_v(R'{\mathchoice{\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\scriptsize\rm (\hspace{-0.15em}(}} {\mbox{\tiny\rm (\hspace{-0.15em}(}}}z_v{\mathchoice{\mbox{\rm )\hspace{-0.15em})}} {\mbox{\rm )\hspace{-0.15em})}} {\mbox{\scriptsize\rm )\hspace{-0.15em})}} {\mbox{\tiny\rm )\hspace{-0.15em})}}}).$$ This definition extends to arbitrary smooth affine group schemes $\mathcal{M}$ over $\mathcal{O}_v$ instead of $\mathcal{G}_v$. The group $L_v^+\mathcal{M}$ is also called the *positive loop group associated with $\mathcal{M}$*, and $L_v\mathcal{M}$ is called the *loop group associated with $\mathcal{M}$*. The latter only depends on the generic fiber $\mathcal{M}\times_{\mathcal{O}_v} Q_v$. The fact that $L_v\mathcal{M}$ is represented by an ind-scheme was proven earlier in [@Pappas-Rapoport-twisted-loop-groups § 1.a], or when $\mathcal{M}$ is constant in [@Beilinson-Drinfeld §4.5], and [@Ngo-Polo], and [@Faltings03]. \(c\) More generally than (a) let $n\in\mathbb{N}_{>0}$ and let $\underline x=(x_1,\ldots,x_n)\in X^n(R)$. Then $\Delta:=\Gamma_{\underline x}:=\Gamma_{x_1}+\ldots + \Gamma_{x_n}$ is an effective relative Cartier divisor over $\operatorname{Spec}R$. In this case we write $\widehat{\Gamma}_{\underline x}:=\widehat{\Delta}$ for the formal completion of $X_R$ along $\Gamma_{\underline x}$. We also write $L^+_{\underline x}\mathcal{G}:=L^+_\Delta\mathcal{G}$ and $L_{\underline x}\mathcal{G}:=L_\Delta\mathcal{G}$ for the (positive) loop group of $\mathcal{G}$ at $\underline x$, and $\mathcal{F}\!\ell_{\mathcal{G},\underline x}:=\mathcal{F}\!\ell_{\mathcal{G},\Delta}$ for the affine flag variety, and $L_{\underline x}$ for the functor from [\[EqLoopTorsorDelta\]](#EqLoopTorsorDelta){reference-type="eqref" reference="EqLoopTorsorDelta"}. **Definition 1**. Let $n\in\mathbb{N}_{>0}$. The *global positive loop group* is defined as the ${\it fpqc\/}$-sheaf on $\operatorname{Spec}\mathbb{F}_q$ whose $R$-valued points for an $\mathbb{F}_q$-algebra $R$ are given by $$\label{EqGlobalPosLoopGp} \mathcal{L}^+_{X^n}\mathcal{G}(R):=\{(\underline x,g) \colon \underline x\in X^n(R),g\in L_{\underline x}^+\mathcal{G}(R)\}.$$ The *global loop group* is defined as the ${\it fpqc\/}$-sheaf on $\operatorname{Spec}\mathbb{F}_q$ whose $R$-valued points for an $\mathbb{F}_q$-algebra $R$ are given by $$\label{EqGlobalLoopGp} \mathcal{L}_{X^n}\mathcal{G}(R):=\{(\underline x,g) \colon \underline x\in X^n(R),g\in L_{\underline x}\mathcal{G}(R)\}.$$ For an $n$-tuple of non-negative integers $(c_i)_i$ we also consider the *truncated global positive loop group* defined as the ${\it fpqc\/}$-sheaf on $\operatorname{Spec}\mathbb{F}_q$ whose $R$-valued points for an $\mathbb{F}_q$-algebra $R$ are given by $$\label{EqGlobalTruncLoopGp} \mathcal{L}^{+,(c_i)_i}_{X^n}\mathcal{G}(R):=\{(\underline x,g) \colon \underline x\in X^n(R),g\in \mathcal{G}(\Delta)\},$$ for the divisor $\Delta:=\sum_i c_i\cdot \Gamma_{x_i}\subset X_R$ considered as a scheme over $X$. Clearly, $\mathcal{L}^+_{X^n}\mathcal{G}$ is a subsheaf of $\mathcal{L}_{X^n}\mathcal{G}$, and $\mathcal{L}^{+,(c_i)_i}_{X^n}\mathcal{G}$ is a quotient of $\mathcal{L}^+_{X^n}\mathcal{G}$. The projection onto $\underline x$ defines morphisms $\mathcal{L}^+_{X^n}\mathcal{G}\to X^n$ and $\mathcal{L}_{X^n}\mathcal{G}\to X^n$ and $\mathcal{L}^{+,(c_i)_i}_{X^n}\mathcal{G}\to X^n$. **Lemma 1**. 1. *[\[LemmaGlobalLoopIsqc_A\]]{#LemmaGlobalLoopIsqc_A label="LemmaGlobalLoopIsqc_A"} The global loop group $\mathcal{L}_{X^n}\mathcal{G}$ is representable by an ind-group scheme which is ind-affine over $X^n$.* 2. *[\[LemmaGlobalLoopIsqc_B\]]{#LemmaGlobalLoopIsqc_B label="LemmaGlobalLoopIsqc_B"} The global positive loop group $\mathcal{L}^+_{X^n}\mathcal{G}$ is representable by a quasi-compact, reduced, infinite dimensional group scheme, which is affine and flat over $X^n$ with geometrically connected fibers.* 3. *[\[LemmaGlobalLoopIsqc_C\]]{#LemmaGlobalLoopIsqc_C label="LemmaGlobalLoopIsqc_C"} The truncated global positive loop group $\mathcal{L}^{+,(c_i)_i}_{X^n}\mathcal{G}$ is representable by a smooth, affine group scheme over $X^n$ of relative dimension equal to $(\sum_i c_i)\cdot \dim G$ with geometrically connected fibers.* *Proof.* The statement is local on $X^n$. Thus for [\[LemmaGlobalLoopIsqc_A\]](#LemmaGlobalLoopIsqc_A){reference-type="ref" reference="LemmaGlobalLoopIsqc_A"} and [\[LemmaGlobalLoopIsqc_B\]](#LemmaGlobalLoopIsqc_B){reference-type="ref" reference="LemmaGlobalLoopIsqc_B"} we can work on an affine open subscheme $U\subset X^n$, and assume that the divisor $\Gamma_{\underline x}\subset X_U$ is principal and the zero locus of an element $z_{\underline x}\in \mathcal{O}_{X_U}$. Then $L^+_{\underline x}\mathcal{G}(R)=\mathcal{G}(R{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_{\underline x}{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}})$ and $L_{\underline x}\mathcal{G}(R)=\mathcal{G}(R{\mathchoice{\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\scriptsize\rm (\hspace{-0.15em}(}} {\mbox{\tiny\rm (\hspace{-0.15em}(}}}z_{\underline x}{\mathchoice{\mbox{\rm )\hspace{-0.15em})}} {\mbox{\rm )\hspace{-0.15em})}} {\mbox{\scriptsize\rm )\hspace{-0.15em})}} {\mbox{\tiny\rm )\hspace{-0.15em})}}})$. After this reformulation, [\[LemmaGlobalLoopIsqc_A\]](#LemmaGlobalLoopIsqc_A){reference-type="ref" reference="LemmaGlobalLoopIsqc_A"} was proven by Heinloth [@Heinloth Proposition 2] and [\[LemmaGlobalLoopIsqc_B\]](#LemmaGlobalLoopIsqc_B){reference-type="ref" reference="LemmaGlobalLoopIsqc_B"} was proven by Richarz [@Richarz16 Lemma 2.11]. Since $\mathcal{L}^+_{X^n}\mathcal{G}$ is affine over the quasi-compact $X^n$, also $\mathcal{L}^+_{X^n}\mathcal{G}$ is quasi-compact. [\[LemmaGlobalLoopIsqc_C\]](#LemmaGlobalLoopIsqc_C){reference-type="ref" reference="LemmaGlobalLoopIsqc_C"} We consider the universal situation over $X^n$ in which the section $x_i\colon X^n\to X$ is the projection onto the $i$-th component. Then the divisor $\Delta:=\sum_i c_i\cdot \Gamma_{x_i}$ is a closed subscheme in $X\times_{\mathbb{F}_q} X^n$. Let $\operatorname{pr}_1\colon\Delta \to X$ and $\operatorname{pr}_2\colon\Delta\to X^n$ be the projections. The morphism $\operatorname{pr}_2$ is finite and flat of degree $\sum_i c_i$. Then the group $\mathcal{L}^{+,(c_i)_i}_{X^n}\mathcal{G}$ over $X^n$ is the Weil restriction $\operatorname{Res}_{\operatorname{pr}_2}(\operatorname{pr}_1^*\mathcal{G})$ of $\operatorname{pr}_1^*\mathcal{G}$ under $\operatorname{pr}_2$. Our assertions now follow from [@CGP Propositions A.5.2 and A.5.9]. ◻ ## The stacks of $G$-bundles {#subsec:Bun} **Definition 1**. Let $\operatorname{Bun}_\mathcal{G}:=\operatorname{Bun}^X_\mathcal{G}$ denote the category fiberd in groupoids over the category of $\mathbb{F}_q$-schemes, which assigns to an $\mathbb{F}_q$-scheme $S$ the category whose objects $\operatorname{Bun}_\mathcal{G}(S)$ are $\mathcal{G}$-bundles over $X_S$ and morphisms are isomorphisms of $\mathcal{G}$-bundles. Let $D\subset X$ be a proper closed subscheme. A *$D$-level structure* on a $\mathcal{G}$-bundle $\mathcal{E}$ on $X_S$ is a trivialization $\psi\colon \mathcal{E}\times_{X_S}{D_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{G}\times_X D_S$ along $D_S$. Let $\operatorname{Bun}_{\mathcal{G},D}$ denote the stack classifying $\mathcal{G}$-bundles with $D$-level structures, i.e. for an $\mathbb{F}_q$-scheme $S$ the objects of the category $\operatorname{Bun}_{\mathcal{G},D}(S)$ are $$\operatorname{Bun}_{\mathcal{G},D}(S):=\left\lbrace (\mathcal{E},\psi)\colon \mathcal{E}\in \operatorname{Bun}_\mathcal{G}(S),\, \psi\colon \mathcal{E}\times_{X_S}{D_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{G}\times_X D_S \right\rbrace,$$ and the morphisms are those isomorphisms of $\mathcal{G}$-bundles that preserve the $D$-level structure $\psi$. The following theorem is well known, see for example [@Beh § 4.4], [@Heinloth Proposition 1] and [@AH_Unif Theorem 2.5]. **Theorem 1**. *The stack $\operatorname{Bun}_{\mathcal{G},D}$ is a smooth Artin stack, locally of finite type over $\mathbb{F}_q$. It admits a covering by connected open substacks (given by bounds on the Harder-Narasinham filtration) of finite type over $\mathbb{F}_q$.* ** 1**. Generalizing [@AH_Local § 5.1] we define the global-local-functor $L_\Delta$ for $\operatorname{Bun}_\mathcal{G}$. Let $S=\operatorname{Spec}R$ be affine and let $\Delta\subset X_R$ be an effective relative Cartier divisor. Let $[X/\mathcal{G}](X_R\smallsetminus\Delta)$ be the category of $\mathcal{G}$-bundles $\overset{\circ}{\mathcal{E}}$ on $X_R\smallsetminus\Delta$ and let $[X/\mathcal{G}](X_R\smallsetminus\Delta)^\mathrm{ext}$ be the full subcategory of $[X/\mathcal{G}](X_R\smallsetminus\Delta)$ consisting of those $\mathcal{G}$-bundles $\overset{\circ}{\mathcal{E}}$ over $X_R\smallsetminus\Delta$ that can be extended to some $\mathcal{G}$-bundle $\mathcal{E}$ over the whole relative curve $X_R$. The restriction functor $${}_\bullet\,|_{X_R\smallsetminus\Delta}\colon\operatorname{Bun}_\mathcal{G}(R)\longrightarrow [X/\mathcal{G}](X_R\smallsetminus\Delta)^\mathrm{ext},\quad \mathcal{E}\longmapsto \mathcal{E}|_{X_R\smallsetminus\Delta}$$ assigns to a $\mathcal{G}$-bundle $\mathcal{E}$ over $X_R$ the $\mathcal{G}$-bundle $\mathcal{E}|_{X_R\smallsetminus\Delta}:=\mathcal{E}\times_{X_R}(X_R\smallsetminus\Delta)$ over $X_R\smallsetminus\Delta$. For $\mathcal{E}\in\operatorname{Bun}_\mathcal{G}(R)$, we also consider its formal completion $\mathcal{E}|_{\widehat{\Delta}} :=\mathcal{E}\times_{X_R} \widehat{\Delta}$ along $\Delta$. By [@AH_Local Proposition 2.4], the formal completion $\mathcal{E}|_{\widehat{\Delta}}$ corresponds to an $L^+_\Delta\mathcal{G}$-bundle over $\operatorname{Spec}R$ which we denote as $L^+_\Delta(\mathcal{E})$. This gives a functor $$\label{EqL^+_v} L^+_\Delta\colon \operatorname{Bun}_\mathcal{G}(R)\longrightarrow [\operatorname{Spec}R/L^+_\Delta\mathcal{G}](R)\,,\quad\mathcal{E}\mapsto L^+_\Delta(\mathcal{E})\,.$$ Moreover, we have a functor $$\label{EqL_v} L_\Delta\colon [X/\mathcal{G}](X_R\smallsetminus\Delta)^\mathrm{ext}\longrightarrow [\operatorname{Spec}R/L_\Delta G](R)\,,\quad\overset{\circ}{\mathcal{E}}\mapsto L_\Delta(\overset{\circ}{\mathcal{E}}):=L_\Delta L^+_\Delta(\mathcal{E})$$ which sends the $\mathcal{G}$-bundle $\overset{\circ}{\mathcal{E}}$ over $X_R\smallsetminus\Delta$ (equipped with some extension $\mathcal{E}$ over $X_R$) to the $L_\Delta G$-bundle $L_\Delta(\overset{\circ}{\mathcal{E}})$ associated with $L^+_\Delta(\mathcal{E})$ under the functor $L_\Delta$ from [\[EqLoopTorsorDelta\]](#EqLoopTorsorDelta){reference-type="eqref" reference="EqLoopTorsorDelta"}. As in [@AH_Local § 5.1] one can show that $L_\Delta(\overset{\circ}{\mathcal{E}})$ is independent of the choice of the extension $\mathcal{E}$. The functors from [\[EqL\^+\_v\]](#EqL^+_v){reference-type="eqref" reference="EqL^+_v"} and [\[EqL_v\]](#EqL_v){reference-type="eqref" reference="EqL_v"} are called the *global-local-functors at $\Delta$*. **Example 1**. We continue with Example [Example 1](#ExDivisors){reference-type="ref" reference="ExDivisors"} and introduce the following notation. \(a\) For a point $x\in X(R)$ we write $L^+_x$ and $L_x$ for the global-local functors from [\[EqL\^+\_v\]](#EqL^+_v){reference-type="eqref" reference="EqL^+_v"} and [\[EqL_v\]](#EqL_v){reference-type="eqref" reference="EqL_v"}. \(b\) When $v\in X$ is a closed point we write $L^+_v$ and $L_v$ for the global-local functors from [\[EqL\^+\_v\]](#EqL^+_v){reference-type="eqref" reference="EqL^+_v"} and [\[EqL_v\]](#EqL_v){reference-type="eqref" reference="EqL_v"}. Later on we will apply these functors for different groups. For clarity, we will then include the group in the subscript and write $L^+_{v,\mathcal{G}}$ (resp. $L_{v,\mathcal{G}}$) instead. Let $\mathrm{Tri}_\mathcal{G}(X_R,\Delta)$ denote the category whose objects are triples $(\overset{\circ}{\mathcal{E}},\mathcal{L},\gamma)$, where $\overset{\circ}{\mathcal{E}}\in [X/\mathcal{G}](X_R\smallsetminus\Delta)^\mathrm{ext}$, and $\mathcal{L}\in [\operatorname{Spec}R/L^+_\Delta\mathcal{G}](R)$, and $\gamma\colon L_\Delta(\overset{\circ}{\mathcal{E}})\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_\Delta(\mathcal{L})$ is an isomorphism of $L_\Delta G$-bundles on $\operatorname{Spec}R$. We obtain a functor $$\begin{aligned} \label{BunG-to-triples} \begin{split} \operatorname{Bun}_\mathcal{G}(R)&\longrightarrow\mathrm{Tri}_\mathcal{G}(X_R,\Delta)\\ \mathcal{E}&\longmapsto \bigl(\mathcal{E}|_{X_R\smallsetminus\Delta},L^+_\Delta(\mathcal{E}),\gamma), \end{split}\end{aligned}$$ where $\gamma$ is the identity morphism of the $L_\Delta G$-bundle $L_\Delta(\mathcal{E}|_{X_R\smallsetminus\Delta}):=L_\Delta L^+_\Delta(\mathcal{E})$. The following lemma generalizes [@AH_Local Lemma 5.1]. **Lemma 1**. *The functor [\[BunG-to-triples\]](#BunG-to-triples){reference-type="eqref" reference="BunG-to-triples"} is an equivalence of categories $\operatorname{Bun}_\mathcal{G}(R)\cong \mathrm{Tri}_\mathcal{G}(X_R,\Delta)$.* *Proof.* This follows from the glueing lemma of Beauville and Lazlo [@Beauville-Laszlo] as in [@AH_Local Lemma 5.1]. ◻ ## The Hecke stack {#subsec:Hecke-stack} We recall the definition of the $\operatorname{Hecke}$ stack with $n$ legs from [@Lafforgue12 Definition 1.2]. Let $n\in\mathbb{N}_0$, let $I=\{1,\ldots,n\}$, and let $I_{\bullet}=(I_1,\ldots,I_k)$ be an ordered partition of $I$, i.e. $I=I_1\sqcup\ldots\sqcup I_k$. Let $D\subset X$ be a proper closed subset. **Definition 1**. The *Hecke stack* $\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}$ *with $n$ legs and partition $I_\bullet$* is the stack over $\mathbb{F}_q$, whose $S$-valued points, for an $\mathbb{F}_q$-scheme $S$, are tuples $\bigl(\underline x,\,(\mathcal{E}^{(j)},\psi^{(j)})_{j=0\ldots k},\,(\varphi^{(j-1)})_{j=1\ldots k}\,\bigr)$ where[^1] - $x_i \in (X\smallsetminus D)(S)$ for $i=1,\ldots, n$ are sections, called *legs*, and $\underline x:=(x_i)_{i=1\ldots n}\in (X\smallsetminus D)^n(S)$ - $(\mathcal{E}^{(j)},\psi^{(j)})$ for $j=0,\ldots,k$ are objects in $\operatorname{Bun}_{\mathcal{G},D}(S)$, and - the *modifications* $\varphi^{(j-1)}\colon \mathcal{E}^{(j-1)}|_{{X_S}\smallsetminus\cup_{i\in I_j}\Gamma_{x_i}}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}^{(j)}|_{{X_S}\smallsetminus\cup_{i\in I_j}\Gamma_{x_i}}$ for $j=1,\ldots,k$ are isomorphisms preserving the $D$-level structures, i.e. $\psi^{(j)}\circ\varphi^{(j-1)}|_{D_S}=\psi^{(j-1)}$. Morphisms $\bigl(\underline x,\,(\mathcal{E}^{(j)},\psi^{(j)})_{j=0\ldots k},\,(\varphi^{(j-1)})_{j=1\ldots k}\,\bigr)\to \bigl(\underline x,\,(\widetilde{\mathcal{E}}^{(j)},\widetilde{\psi}^{(j)})_{j=0\ldots k},\,(\widetilde{\varphi}^{(j-1)})_{j=1\ldots k}\,\bigr)$ are tuples of isomorphisms $f^{(j)}\colon (\mathcal{E}^{(j)},\psi^{(j)})\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\widetilde{\mathcal{E}}^{(j)},\widetilde{\psi}^{(j)})$ in $\operatorname{Bun}_{\mathcal{G},D}(S)$ for all $j$ which are compatible with the $\varphi^{(j-1)}$ and $\widetilde{\varphi}^{(j-1)}$. We can visualize the above data as $$\label{Hecke-GDXn-visual} \xymatrix @C+1pc { (\mathcal{E}^{(0)},\psi^{(0)}) \ar@{-->}[r]^-{\varphi^{(0)}}_-{x_i\colon i\in I_1} & (\mathcal{E}^{(1)},\psi^{(1)}) \ar@{-->}[r]^-{\varphi^{(1)}}_-{x_i\colon i\in I_2} & \ldots \ar@{-->}[r]^-{\varphi^{(k-1)}}_-{x_i\colon i\in I_k} & (\mathcal{E}^{(k)},\psi^{(k)}) \,. }$$ The projection map of [\[Hecke-GDXn-visual\]](#Hecke-GDXn-visual){reference-type="eqref" reference="Hecke-GDXn-visual"} onto $(x_i)_{i=1\ldots n}$ defines a morphism $$\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}\to (X\smallsetminus D)^I.$$ When $D=\varnothing$, we will drop it (and the $\psi^{(j)}$) from the notation. For $n=1$, the set $I=\{1\}$ only has the trivial partition $I_1:=I$. Thus we drop $I_\bullet$ from the notation and simply write $\operatorname{Hecke}_{\mathcal{G},D,X}$. **Remark 1**. Let $\widetilde{I}_\bullet=(\widetilde{I}_1,\ldots,\widetilde{I}_{\tilde k})$ be a partition of $I$ and $I_\bullet=(I_1,\ldots,I_k)$ a coarsening of $\widetilde{I}_\bullet$ obtained by uniting certain $\widetilde{I}_{\tilde j}$ with neighboring indices. More precisely, we require that there are integers $0=\ell_0< \ell_1< \ldots <\ell_k=\tilde k$ and $I_j=\bigcup_{\ell_{j-1}<\tilde{j} \le\ell_j} \widetilde{I}_{\tilde j}$. Then there is an $X^n$-morphism $$\begin{aligned} \label{EqHeckeChangeI} \operatorname{Hecke}_{\mathcal{G},D,X^n,\widetilde{I}_\bullet} & \longrightarrow \operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet} \\ \bigl(\underline x,\,(\widetilde{\mathcal{E}}^{(\tilde j)},\widetilde{\psi}^{(\tilde j)})_{\tilde j=0\ldots \tilde k},\,(\widetilde{\varphi}^{(\tilde j-1)})_{\tilde j=1\ldots \tilde k}\,\bigr) & \longmapsto \bigl(\underline x,\,(\mathcal{E}^{(j)},\psi^{(j)})_{j=0\ldots k},\,(\varphi^{(j-1)})_{j=1\ldots k}\,\bigr) \nonumber\end{aligned}$$ given by forgetting the $(\widetilde{\mathcal{E}}^{(\tilde{j})},\widetilde{\psi}^{(\tilde{j})})$ for $\tilde{j}\notin\{\ell_0,\ldots,\ell_k\}$, reindexing by $(\mathcal{E}^{(j)},\psi^{(j)}) :=(\widetilde{\mathcal{E}}^{(\ell_j)},\widetilde{\psi}^{(\ell_j)})$, and composing the corresponding $\widetilde{\varphi}^{(\tilde{j})}$ to $\varphi^{(j)}:=\widetilde{\varphi}^{(-1+\ell_{j+1})}\circ\ldots\circ\widetilde{\varphi}^{(\ell_j)}$. **Proposition 1**. *$\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}$ is an ind-Artin stack locally of ind-finite type over $X$. The morphism $\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}\to X^n\times_{\mathbb{F}_q} \operatorname{Bun}_{\mathcal{G},D}$ sending $\bigl(\underline x,\,(\mathcal{E}^{(j)},\psi^{(j)})_{j=0\ldots k},\,(\varphi^{(j-1)})_{j=1\ldots k}\,\bigr)$ to $\bigl(\underline x,(\mathcal{E}^{(k)},\psi^{(k)})\bigr)$ is relatively representable by a morphism of ind-schemes which is of ind-finite type and ind-quasi-projective. It is even ind-projective if and only if the group scheme $\mathcal{G}$ is parahoric as in [@Pappas-Rapoport-twisted-loop-groups Appendix, Definition 1].* *Proof.* This was proven in [@AH_Unif Propositions 3.9 and 3.12], where instead of the $\varphi^{(j)}$, their inverses are considered (and called $\tau_{k-j}$, while also the numbering of the $\mathcal{G}$-bundles is reversed). ◻ **Proposition 1**. *Let $U:=\{(x_i)_i\in X^n \colon x_i\neq x_j\text{ for }i\neq j\}\subseteq X^n$ be the complement of all diagonals. Let $I_\bullet$ and $\widetilde{I}_\bullet$ be partitions of $I$, such that $I_\bullet$ is a coarsening of $\widetilde{I}_\bullet$ as in Remark [Remark 1](#RemHeckeChangeI){reference-type="ref" reference="RemHeckeChangeI"}. Then over the open set $U\subset X^n$, the morphism $\operatorname{Hecke}_{\mathcal{G},D,X^n,\widetilde{I}_\bullet}\times_{X^n} U\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}\times_{X^n} U$ from [\[EqHeckeChangeI\]](#EqHeckeChangeI){reference-type="eqref" reference="EqHeckeChangeI"} is an isomorphism.* *Proof.* We define the divisors $\Delta_{\tilde j}:=\sum_{i\in \widetilde{I}_{\tilde j}}\Gamma_{x_i}$, which are pairwise disjoint over $U$. Then the inverse of the morphism [\[EqHeckeChangeI\]](#EqHeckeChangeI){reference-type="eqref" reference="EqHeckeChangeI"} is given by reconstructing the forgotten $\widetilde{\mathcal{E}}^{(\tilde j)}$ with $\ell_j<\tilde j<\ell_{j+1}$ from $(\widetilde{\mathcal{E}}^{(\ell_j)},\widetilde{\psi}^{(\ell_j)}):=(\mathcal{E}^{(j)}, \psi^{(j)})$. This is done via Lemma [Lemma 1](#LemmaBL){reference-type="ref" reference="LemmaBL"} by succesively glueing $\mathcal{E}^{(\tilde j-1)}$ with $L^+_{\Delta_{\tilde j}}(\mathcal{E}^{(j+1)})$ via the isomorphism $L_{\Delta_{\tilde j}}(\varphi^{(j)})$ to obtain $\widetilde{\mathcal{E}}^{(\tilde j)}$ and $\widetilde{\varphi}^{(\tilde j-1)}$. Finally we set $\widetilde{\psi}^{(\tilde j)}:=\widetilde{\psi}^{(\tilde j-1)}\circ(\widetilde{\varphi}^{(\tilde j-1)})^{-1}|_{D_S}$. ◻ In the rest of this subsection, we fix a point $\infty\in X(\mathbb{F}_q)$. We are particularly interested in the Hecke stack with two legs, one fixed at $\infty$. For simplicity, we slightly change the notation and use $'$ and $''$ instead of numbers for upper-scripts. **Definition 1**. Let $I=\{1,2\}$ and let $D\subset X\smallsetminus\{\infty\}$ be a proper closed subset. \(a\) For the finest partition $I_\bullet=(\{1\},\{2\})$, we define the *Hecke stack with two legs, one fixed at $\infty$* as $$\operatorname{Hecke}_{\mathcal{G},D,X\times\infty} :=\operatorname{Hecke}_{\mathcal{G},D,X^2,(\{1\},\{2\})}\times_{X^2}(X\times_{\mathbb{F}_q}\operatorname{Spec}\mathbb{F}_\infty).$$ Its $S$-valued points, for $S$ an $\mathbb{F}_q$-scheme, are tuples $$\bigl(x,\,(\mathcal{E},\psi),(\mathcal{E}',\psi'),(\mathcal{E}'',\psi''),\,\varphi,\varphi'\,\bigr) \\ $$ where - $x \in (X\smallsetminus D)(S)$ is a section, called a *leg*, - $(\mathcal{E},\psi),(\mathcal{E}',\psi'),(\mathcal{E}'',\psi'')$ are objects in $\operatorname{Bun}_{\mathcal{G},D}(S)$, and - the *modifications* $\varphi\colon \mathcal{E}|_{{X_S}\smallsetminus\Gamma_{x}}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}'|_{{X_S}\smallsetminus\Gamma_{x}}$ and $\varphi'\colon \mathcal{E}'|_{(X\smallsetminus\{\infty\})_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}''|_{(X\smallsetminus\{\infty\})_S}$ are isomorphisms preserving the $D$-level structures, i.e. $\psi'\circ\varphi|_{D_S}=\psi$ and $\psi''\circ\varphi'|_{D_S}=\psi'$. We can visualize the above data as $$\xymatrix @C+1pc { (\mathcal{E},\psi) \ar@{-->}[r]^{\varphi}_{x} & (\mathcal{E}',\psi') \ar@{-->}[r]^{\varphi'}_\infty & (\mathcal{E}'',\psi'') \,. }$$ \(b\) We also define the stack for the coarsest partition $I_1:=I=\{1,2\}$ $${{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times\infty} :=\operatorname{Hecke}_{\mathcal{G},D,X^2,(\{1,2\})}\times_{X^2}(X\times_{\mathbb{F}_q}\operatorname{Spec}\mathbb{F}_\infty).$$ classifying data $\bigl(x,\,(\mathcal{E},\psi),(\mathcal{E}'',\psi''),\,\varphi\colon(\mathcal{E},\psi)|_{X_S\smallsetminus(\Gamma_x\cup\infty)} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\mathcal{E}'',\psi'')|_{X_S\smallsetminus(\Gamma_x\cup\infty)} \,\bigr)$, visualized as $$\xymatrix @C+1pc { (\mathcal{E},\psi) \ar@{-->}[r]^{\varphi}_{x,\infty} & (\mathcal{E}'',\psi'') \,. }$$ When $D=\varnothing$, we will drop it and the $\psi,\psi',\psi''$ from the notation. The projection map onto the leg $x$ defines morphisms $\operatorname{Hecke}_{\mathcal{G},D,X\times\infty}\to {{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times\infty}\to X$. "Creating" an unnecessary additional leg at $\infty$ defines a morphism $$\begin{aligned} \label{EqHeckeCreateInfty} \operatorname{Hecke}_{\mathcal{G},D,X} & \longrightarrow {{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times\infty} \\ \bigl(x,\,(\mathcal{E},\psi) \mathrel{ \mathpalette{\da@xarrow{x}{\varphi}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } (\mathcal{E}',\psi')\bigr) & \longmapsto \bigl(x,\,(\mathcal{E},\psi) \mathrel{ \mathpalette{\da@xarrow{x,\infty}{\varphi}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } (\mathcal{E}',\psi')\bigr) \nonumber\end{aligned}$$ where we view the isomorphism $\varphi\colon (\mathcal{E},\psi)|_{X_S\smallsetminus\Gamma_x} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\mathcal{E}',\psi')|_{X_S\smallsetminus\Gamma_x}$ outside $\Gamma_x$ as an isomorphism $\varphi\colon (\mathcal{E},\psi)|_{X_S\smallsetminus(\Gamma_x\cup\infty)} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\mathcal{E}',\psi')|_{X_S\smallsetminus(\Gamma_x\cup\infty)}$ outside $\Gamma_x\cup (\infty\times_{\mathbb{F}_q}S)$. This is a morphism over $X\times_{\mathbb{F}_q} \operatorname{Bun}_{\mathcal{G},D}$ as in Proposition [Proposition 1](#PropHeckeArtin){reference-type="ref" reference="PropHeckeArtin"}. **Lemma 1**. *The morphism [\[EqHeckeCreateInfty\]](#EqHeckeCreateInfty){reference-type="eqref" reference="EqHeckeCreateInfty"} is a monomorphism, and hence schematic.* *Proof.* To prove that it is a monomorphism we must show that it is fully faithful as a functor. Let $(x,(\mathcal{E},\psi),(\mathcal{E}',\psi'),\varphi)$ and $(x,(\widetilde{\mathcal{E}},\widetilde{\psi}),(\widetilde{\mathcal{E}}',\widetilde{\psi}'),\widetilde{\varphi})$ be two objects of $\operatorname{Hecke}_{\mathcal{G},D,X}(S)$. Then the isomorphism between these two objects in $\operatorname{Hecke}_{\mathcal{G},D,X}(S)$ and also in ${{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times\infty}(S)$ consist of isomorphisms $f\colon (\mathcal{E},\psi)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\widetilde{\mathcal{E}},\widetilde{\psi})$ and $f'\colon (\mathcal{E}',\psi')\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\widetilde{\mathcal{E}}',\widetilde{\psi}')$ which are compatible with $\varphi$ and $\widetilde{\varphi}$. This proves that [\[EqHeckeCreateInfty\]](#EqHeckeCreateInfty){reference-type="eqref" reference="EqHeckeCreateInfty"} is a monomorphism. It is schematic by [@Laumon-Moret-Bailly Corollaire 8.1.3 and Théorème A.2]. ◻ ## The Beilinson-Drinfeld Grassmannian {#subsec:Gr} Let $n\in\mathbb{N}_0$, let $I=\{1,\ldots,n\}$, and let $I_\bullet=(I_1,\ldots,I_k)$ be a partition of $I$. **Definition 1**. The *Beilinson-Drinfeld Grassmannian* $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ is the ${\it fpqc\/}$-sheaf of sets on $\mathbb{F}_q$, whose $S$-points, for an $\mathbb{F}_q$-scheme $S$, are tuples $$\bigl(\underline x,\,(\mathcal{E}^{(j)})_{j=0\ldots k},\,(\varphi^{(j-1)})_{j=1\ldots k}\,\bigr)\in\operatorname{Hecke}_{\mathcal{G},\varnothing,X^n,I_\bullet}(S)$$ as in Definition [Definition 1](#DefHecke_nlegs){reference-type="ref" reference="DefHecke_nlegs"} together with: - a trivialization $\epsilon\colon \mathcal{E}^{(k)}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{G}\times_X X_S$. For $n=1$, the set $I=\{1\}$ only has the trivial partition $I_1:=I$. Thus we drop $I_\bullet$ from the notation and simply write $\operatorname{Gr}_{\mathcal{G},X}$. Thus $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ is the fiber product $$\label{EqGrFiberProd} \xymatrix @C+2pc { \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet} \ar[r] \ar[d] & \operatorname{Hecke}_{\mathcal{G},\varnothing,X^n,I_\bullet} \ar[d]^{\mathcal{E}^{(k)}} \\ \operatorname{Spec}\mathbb{F}_q \ar[r]^{\mathcal{G}} & \operatorname{Bun}_\mathcal{G} }$$ where the vertical map on the right was defined in Proposition [Proposition 1](#PropHeckeArtin){reference-type="ref" reference="PropHeckeArtin"} and the horizontal map on the bottom comes from the trivial $\mathcal{G}$-bundle $\mathcal{G}\in\operatorname{Bun}_\mathcal{G}(\mathbb{F}_q)$. **Proposition 1**. *$\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ is an ind-scheme of ind-finite type, which is ind-quasi-projective over $X^n$. It is even ind-projective if and only if the group scheme $\mathcal{G}$ is parahoric.* *Proof.* This was proven by Richarz [@Richarz16 Lemma 2.8(ii)] in case $n=1$. In general it follows from Proposition [Proposition 1](#PropHeckeArtin){reference-type="ref" reference="PropHeckeArtin"} by the base change diagram [\[EqGrFiberProd\]](#EqGrFiberProd){reference-type="eqref" reference="EqGrFiberProd"}. ◻ We will need the following alternative description from [@Lafforgue12 (0.10)] of the Beilinson-Drinfeld Grassmannian in terms of modifications of $L^+_{\underline x}\mathcal{G}$-bundles. **Proposition 1**. *Via the functor $L_{\underline x}$ from Example [Example 1](#ExDivisors){reference-type="ref" reference="ExDivisors"}(c), the Beilinson-Drinfeld Grassmannian $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ is isomorphic to the ${\it fpqc\/}$-sheaf on $\operatorname{Spec}\mathbb{F}_q$, whose $S$-valued points are tuples $$\label{EqPropGrGlobLoc} \xymatrix @C+2pc { \Bigl(\underline x,\,\mathcal{L}^{(0)} \ar@{-->}[r]^-{\widehat{\varphi}^{(0)}}_-{x_i\colon i\in I_1} & \mathcal{L}^{(1)} \ar@{-->}[r]^-{\widehat{\varphi}^{(1)}}_-{x_i\colon i\in I_2} & \ldots \ar@{-->}[r]^-{\widehat{\varphi}^{(k-1)}}_-{x_i\colon i\in I_k} & \mathcal{L}^{(k)} \ar[r]^-{\hat{\epsilon}}_-\sim & (L^+_{\underline x}\mathcal{G})_S\Bigr)},$$ where the $\mathcal{L}^{(j)}$ are $L^+_{\underline x}\mathcal{G}$-bundles on $S$, the $\widehat{\varphi}^{(j-1)}$ are isomorphism of $L_{\underline x}\mathcal{G}$-bundles that above $\Gamma_{\underline x}\smallsetminus \bigcup_{i\in I_j}\Gamma_{x_i}$ are even isomorphisms of $L^+_{\underline x}\mathcal{G}$-bundles, and $\hat{\epsilon}$ is an isomorphism of $L^+_{\underline x}\mathcal{G}$-bundles.* **Remark 1**. The condition on $\widehat{\varphi}^{(j-1)}$ means the following. Let $\operatorname{Spec}R\subset S$ be an affine open subset. For every $j=1,\ldots,k$ let $\mathscr{I}_j$ be the ideal sheaf of the effective relative Cartier divisor $\sum_{i\in I_j}\Gamma_{x_i}$. Then locally on $X_S$ the sheaf $\mathscr{I}_j$ is generated by an element $z_j\in\mathcal{O}_{X_S}$. For any $j$, we can define the partial loop group $L^j_{\underline x}\mathcal{G}$ Zariski locally on $\operatorname{Spec}R$ as the ${\it fpqc\/}$-sheaf of groups on $\operatorname{Spec}R$ given by $L^j_{\underline x}\mathcal{G}(R'):=\mathcal{G}(\widehat{\mathcal{O}}_{X_{R'},\Delta'}[z_j^{-1}])$, where $\Delta'$ denotes the pullback of $\Delta$ to $X_{R'}$. As in Definition [Definition 1](#DefLoopGpAtDelta){reference-type="ref" reference="DefLoopGpAtDelta"}, this is independent of the chosen generator $z_j$, and hence glues to an ${\it fpqc\/}$-sheaf of groups on $S$. The inclusion of sheaves $L^+_{\underline x}\mathcal{G}\subset L^j_{\underline x}\mathcal{G}$ induces a functor $L_j\colon [S / L^+_{\underline x}\mathcal{G}] \to [S / L^j_{\underline x}\mathcal{G}]$. By the condition on $\widehat{\varphi}^{(j-1)}$ we mean that $\widehat{\varphi}^{(j-1)}$ is an isomorphism of the induced $L^j_{\underline x}\mathcal{G}$-bundles $\widehat{\varphi}^{(j-1)}\colon L_j(\mathcal{L}^{(j-1)})\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_j(\mathcal{L}^{(j)})$. *Proof of Proposition [Proposition 1](#PropGrGlobLoc){reference-type="ref" reference="PropGrGlobLoc"}..* The functor $L^+_{\underline x}$ sends an object $\bigl(\underline x,\,(\mathcal{E}^{(j)})_{j=0\ldots k},\,(\varphi^{(j-1)})_{j=1\ldots k},\,\epsilon\,\bigr)\in\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}(S)$ to $$\xymatrix @C+2pc { \Bigl(\underline x,\, L^+_{\underline x}(\mathcal{E}^{(0)}) \ar@{-->}[r]^-{L_{\underline x}(\varphi^{(0)})}_-{x_i\colon i\in I_1} & L^+_{\underline x}(\mathcal{E}^{(1)}) \ar@{-->}[r]^-{L_{\underline x}(\varphi^{(1)})}_-{x_i\colon i\in I_2} & \ldots \ar@{-->}[r]^-{L_{\underline x}(\varphi^{(k-1)})}_-{x_i\colon i\in I_k} & L_{\underline x}(\mathcal{E}^{(k)}) \ar[r]^-{L^+_{\underline x}(\epsilon)}_-\sim & L^+_{\underline x}\mathcal{G}\Bigr)}$$ That it is an equivalence follows from Lemma [Lemma 1](#LemmaBL){reference-type="ref" reference="LemmaBL"}, by gluing $\mathcal{G}\times_X (X_S\smallsetminus \Gamma_{\underline x})$ with $\mathcal{L}^{(j)}$ via the isomorphism $\hat{\epsilon} \circ \widehat{\varphi}^{(k-1)}\circ \ldots \circ \widehat{\varphi}^{(j)}$ to obtain $\mathcal{E}^{(j)}$ and $\varphi^{(j)}$. ◻ **Definition 1**. The global loop group $\mathcal{L}^+_{X^n}\mathcal{G}$ from Definition [Definition 1](#DefGlobalLoopGp){reference-type="ref" reference="DefGlobalLoopGp"} acts on $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ by acting on $\hat{\epsilon}$. More precisely, the action is given by a morphism $$\mathcal{L}^+_{X^n}\mathcal{G}\times_{X^n} \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet} \longrightarrow \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$$ such that $(\underline x,g)\in \mathcal{L}^+_{X^n}\mathcal{G}(S)$ sends an object [\[EqPropGrGlobLoc\]](#EqPropGrGlobLoc){reference-type="eqref" reference="EqPropGrGlobLoc"} of $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}(S)$ to the same object but with $\hat{\epsilon}$ replaced by $g\cdot\hat{\epsilon}$. Let $n=1$ and recall from Definition [Definition 1](#DefAffineFlagVarAtx){reference-type="ref" reference="DefAffineFlagVarAtx"} and Example [Example 1](#ExDivisors){reference-type="ref" reference="ExDivisors"}(a) the local affine flag variety $\mathcal{F}\!\ell_{\mathcal{G},x}$. For the global loop groups from Definition [Definition 1](#DefGlobalLoopGp){reference-type="ref" reference="DefGlobalLoopGp"}, the set of $S$-valued points of the ${\it fppf\/}$-quotient $\mathcal{L}_X\mathcal{G}/\mathcal{L}^+_X\mathcal{G}$ equals $\{\,(x,\overline{g})\colon x\in X(S),\, \overline{g}\in \mathcal{F}\!\ell_{\mathcal{G},x}(S)\,\}$. Via the interpretation of $\mathcal{F}\!\ell_{\mathcal{G},x}$ from Lemma [Lemma 1](#LemmaAffineFlagVar){reference-type="ref" reference="LemmaAffineFlagVar"}, we consider the morphism $$\begin{aligned} \label{GrGX-map} \begin{split} \operatorname{Gr}_{\mathcal{G},X} \qquad & \longrightarrow\qquad \mathcal{L}_X\mathcal{G}/\mathcal{L}^+_X\mathcal{G}\,,\\ \bigl(x,\,\mathcal{E}^{(0)} \mathrel{ \mathpalette{\da@xarrow{x}{\varphi^{(0)}}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } \mathcal{E}^{(1)}\xrightarrow[\sim]{\enspace\epsilon\enspace} \mathcal{G}\times_X X_S\bigr) & \longmapsto \bigl(\,x, L^+_x(\mathcal{E}^{(0)}),\, L_x(\epsilon\circ\varphi^{(0)})\,\bigr)\,. \end{split}\end{aligned}$$ **Corollary 1**. *The map in [\[GrGX-map\]](#GrGX-map){reference-type="eqref" reference="GrGX-map"} is an isomorphism $\operatorname{Gr}_{\mathcal{G},X} \cong \mathcal{L}_X\mathcal{G}/\mathcal{L}^+_X\mathcal{G}$. In particular, for a fixed closed point $v\in X$, we obtain a canonical isomorphism of ind-schemes over $\operatorname{Spf}\mathcal{O}_v$ $$\label{EqCorGrGlobLoc} \operatorname{Gr}_{\mathcal{G},X}\times_X \operatorname{Spf}\mathcal{O}_v \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{F}\!\ell_{\mathcal{G},v}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{\mathbb{F}_v} \operatorname{Spf}\mathcal{O}_v$$ where $\mathcal{F}\!\ell_{\mathcal{G},v}$ is the local affine flag variety from Example [Example 1](#ExDivisors){reference-type="ref" reference="ExDivisors"}(b) and $\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{\mathbb{F}_v}$ denotes the fiber product of ind-schemes over $\mathbb{F}_v$.* *Proof.* The first assertion follows from Proposition [Proposition 1](#PropGrGlobLoc){reference-type="ref" reference="PropGrGlobLoc"}. To prove the second assertion, note that $\operatorname{Spf}\mathcal{O}_v= \ifthenelse{\equal{}{}}% falls Argument leer {\displaystyle \lim_{\longrightarrow}}% verwende niedrige Version {\displaystyle \lim_{\underset{}{\longrightarrow}}}% sonst: verwende Argument \operatorname{Spec}\mathcal{O}_v/(z_v^n)$ where $z_v$ is a uniformizing parameter at $v$. Thus it suffices to prove that [\[EqCorGrGlobLoc\]](#EqCorGrGlobLoc){reference-type="eqref" reference="EqCorGrGlobLoc"} is an isomorphism modulo $z_v^n$ for every $n$. To this end we must show for every ring $R$ and morphism $x\colon \operatorname{Spec}R\to\operatorname{Spec}\mathcal{O}_v/(z_v^n)$ that $\mathcal{F}\!\ell_{\mathcal{G},x}(R)=\mathcal{F}\!\ell_{\mathcal{G},v}(R)$. Let $\zeta:=x^*(z_v)\in R$. Then $z_x:=z_v-\zeta$ is a generator of the ideal sheaf $\mathscr{I}_{\Gamma_x}$ defining the graph $\Gamma_x\subset X_R$ of $x$. Since $\zeta^n=0$ in $R$, we have $\widehat{\Gamma}_x = \operatorname{Spf}R{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_x{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}= \operatorname{Spf}R{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_v{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$ and $R{\mathchoice{\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\scriptsize\rm (\hspace{-0.15em}(}} {\mbox{\tiny\rm (\hspace{-0.15em}(}}}z_x{\mathchoice{\mbox{\rm )\hspace{-0.15em})}} {\mbox{\rm )\hspace{-0.15em})}} {\mbox{\scriptsize\rm )\hspace{-0.15em})}} {\mbox{\tiny\rm )\hspace{-0.15em})}}}= R{\mathchoice{\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\rm (\hspace{-0.15em}(}} {\mbox{\scriptsize\rm (\hspace{-0.15em}(}} {\mbox{\tiny\rm (\hspace{-0.15em}(}}}z_v{\mathchoice{\mbox{\rm )\hspace{-0.15em})}} {\mbox{\rm )\hspace{-0.15em})}} {\mbox{\scriptsize\rm )\hspace{-0.15em})}} {\mbox{\tiny\rm )\hspace{-0.15em})}}}$. This proves the equality $\mathcal{F}\!\ell_{\mathcal{G},x}(R)=\mathcal{F}\!\ell_{\mathcal{G},v}(R)$. ◻ ** 1**. Let $n=1$. We recall the definition of the affine Schubert varieties from [@Richarz16; @Lafforgue12]. Let $\mu\colon \mathbb{G}_{m,Q^{\operatorname{sep}}}\to G_{Q^{\operatorname{sep}}}$ be a cocharacter of $G$, and let $K/Q$ be a finite separable extension over which $\mu$ is defined. Let $E_{\mu}$ be the reflex field of $\mu$, i.e. $E_\mu$ is the field of definition of the conjugacy class of $\mu$. It is a finite separable field extension of $Q$, contained in $K$. Let $\widetilde{X}_K$ and $\widetilde{X}_\mu$ be the normalizations of $X$ in $K$ and $E_\mu$. The field extensions correspond to finite, flat, surjective morphisms $\widetilde{X}_K \to \widetilde{X}_\mu \to X$. We consider the canonical leg $x\colon \operatorname{Spec}K \hookrightarrow\widetilde{X}_K \twoheadrightarrow X$. The completion $\widehat{\Gamma}_x$ of its graph $\Gamma_x$ is of the form $\widehat{\Gamma}_x=\operatorname{Spf}K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_x{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$. If $z\in Q$ is an element for which $Q$ is a separable extension of $\mathbb{F}_q(z)$ and $\zeta$ denotes the image of $z$ in $K$, then $\widehat{\Gamma}_x=\operatorname{Spf}K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_x{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}=\operatorname{Spf}K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z-\zeta{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$ by [@HartlJuschka Lemma 1.3]. There is a ring homomorphism $\mathbb{F}_q(z)\hookrightarrow K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z-\zeta{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$ sending $z$ to $z=\zeta+(z-\zeta)$, which modulo the maximal ideal $(z-\zeta)$ induces the inclusion $\mathbb{F}_q(z)\hookrightarrow K$ sending $z$ to $\zeta$. Since $K$ is finite separable over $\mathbb{F}_q(z)$ and $K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z-\zeta{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$ is henselian, there is a unique $\mathbb{F}_q(z)$-homomorphism $K\hookrightarrow K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z-\zeta{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$, which modulo the maximal ideal $(z-\zeta)$ induces the identity on $K$. Note that this $\mathbb{F}_q(z)$-homomorphism is *not* the isomorphism $K\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}K\cdot(z-\zeta)^0\subset K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z-\zeta{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$, because it sends $z$ to $z=\zeta+(z-\zeta)\not\in K\cdot(z-\zeta)^0$. This $\mathbb{F}_q(z)$-homomorphism defines the upper row in the following diagram $$\xymatrix { \widehat{\Gamma}_x = \operatorname{Spf}K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z-\zeta{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}\ar[r] & \operatorname{Spec}(K\otimes_{\mathbb{F}_q} K) \ar[r] \ar[d] & \operatorname{Spec}K \ar[r] \ar[d] & \operatorname{Spec}Q\ar[r] \ar[d] & X \ar[d] \\ & \operatorname{Spec}K \ar[r] & \operatorname{Spec}\mathbb{F}_q \ar@{=}[r] & \operatorname{Spec}\mathbb{F}_q \ar@{=}[r] & \operatorname{Spec}\mathbb{F}_q }$$ and realizes $\widehat{\Gamma}_x = \operatorname{Spf}K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z-\zeta{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$ as the formal completion of $\operatorname{Spec}(K\otimes_{\mathbb{F}_q}K)$ along the diagonal embedding of $\operatorname{Spec}K$. Pulling back the group scheme $\mathcal{G}$ from $X$ to $\operatorname{Spec}K$ under the upper row in the diagram, we see that the cocharacter $\mu\colon\mathbb{G}_{m,K}\to G_K$ is defined over $\widehat{\Gamma}_x$. We now define the $K$-valued point $(x,\bar{g})\in \operatorname{Gr}_{\mathcal{G},X}(K)=(\mathcal{L}_X\mathcal{G}/\mathcal{L}^+_X\mathcal{G})(K)$ given as in Corollary [Corollary 1](#CorGrGlobLoc){reference-type="ref" reference="CorGrGlobLoc"} by the canonical leg $x$ as above, and $$\bar{g} \;:=\; \mu(z_x)\cdot L^+_x\mathcal{G}(K) \; \in \; \mathcal{F}\!\ell_{\mathcal{G},x}(K).$$ It is independent of the choice of $z_x$, because any other $\tilde{z}_x$ differs from $z_x$ by a unit $u\in K{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z_x{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}^\times$, which is mapped under $\mu$ to $\mu(u)\in L^+_x\mathcal{G}(K)$; see [@Richarz16 Remark A.2]. **Definition 1**. The *affine Schubert variety $\operatorname{Gr}_{\mathcal{G},X}^{\leq \mu}$ of $\mu$ in the Beilinson-Drinfeld Grassmannian* $\operatorname{Gr}_{\mathcal{G},X}$ is defined as the scheme-theoretic image of the morphism $(\mathcal{L}^+_X\mathcal{G}\times_X \operatorname{Spec}K)\cdot (x,\bar{g})\to \operatorname{Gr}_{\mathcal{G},X}\times_X\widetilde{X}_K$. It is an $\mathcal{L}^+_X\mathcal{G}\times_X \widetilde{X}_K$-invariant, closed subscheme, which is quasi-compact, because $\mathcal{L}^+_X\mathcal{G}$ is a quasi-compact scheme by Lemma [Lemma 1](#LemmaGlobalLoopIsqc){reference-type="ref" reference="LemmaGlobalLoopIsqc"}[\[LemmaGlobalLoopIsqc_B\]](#LemmaGlobalLoopIsqc_B){reference-type="ref" reference="LemmaGlobalLoopIsqc_B"}. It only depends on the conjugacy class of $\mu$, because if $\tilde{\mu}=\operatorname{int}_h\circ \mu$ is conjugate to $\mu$ by an element $h\in G(K)$, then $(x,h)\in \mathcal{L}^+_X\mathcal{G}(K)$. Therefore, the affine Schubert variety descends to a closed subscheme $\operatorname{Gr}_{\mathcal{G},X}^{\leq \mu}$ of $\operatorname{Gr}_{\mathcal{G},X}\times_X \widetilde{X}_\mu$. **Proposition 1**. *Let $U:=\{(x_i)_i\in X^n \colon x_i\neq x_j\text{ for }i\neq j\}\subseteq X^n$ be the complement of all diagonals. Over the open set $U\subset X^n$, the Beilinson-Drinfeld Grassmannian is a product $$\begin{aligned} \label{product-Grassmannian} \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet} \times_{X^n} U & \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\bigl(\operatorname{Gr}_{\mathcal{G},X} \times_{\mathbb{F}_q} \ldots \times_{\mathbb{F}_q} \operatorname{Gr}_{\mathcal{G},X}\bigr) \times_{X^n} U\\ \bigl(\underline x,\,(\mathcal{E}^{(j)}),\,(\varphi^{(j-1)}),\epsilon\,\bigr)& \longmapsto \bigl(x_i,L_{x_i}^+\mathcal{E}^{(0)},L_{x_i}\mathcal{E}^{(0)}\xrightarrow{L_{x_i}(\epsilon\circ\varphi^{(k-1)}\circ\ldots\circ\varphi^{(0)})}L_{x_i}\mathcal{G}\bigr)_{i=1,\ldots,n}. \nonumber\end{aligned}$$* *Proof.* Since the graphs $\Gamma_{x_i}$ are pairwise disjoint, we have $L^+_{\underline x}(\mathcal{E}^{(0)}) = \bigl(L^+_{x_i}(\mathcal{E}^{(0)})\bigr)_i$. Using the isomorphism [\[EqHeckeChangeI\]](#EqHeckeChangeI){reference-type="eqref" reference="EqHeckeChangeI"} from Proposition [Proposition 1](#PropHeckeChangeI){reference-type="ref" reference="PropHeckeChangeI"} we may assume that $I_\bullet$ is the coarsest partition with $I_1=I$. Then the data $\mathcal{E}^{(0)}$ and $\varphi^{(0)}\colon \mathcal{E}^{(0)}|_{X_S\smallsetminus\Gamma_{\underline x}}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}^{(1)}|_{X_S\smallsetminus\Gamma_{\underline x}}$ can be recovered from $\mathcal{E}^{(1)}:= \mathcal{G}\times_X X_S$ by gluing $\mathcal{E}^{(1)}|_{X_S\smallsetminus \Gamma_{\underline x}}$ with $L^+_{\underline x}(\mathcal{E}^{(0)}) = \bigl(L^+_{x_i}(\mathcal{E}^{(0)})\bigr)_i$ via $L_{\underline x}(\epsilon\circ\varphi^{(0)})$ as in Lemma [Lemma 1](#LemmaBL){reference-type="ref" reference="LemmaBL"}. ◻ This proposition allows us to define bounds on the left-hand side of [\[product-Grassmannian\]](#product-Grassmannian){reference-type="eqref" reference="product-Grassmannian"} using the bounds on the right-hand side of [\[product-Grassmannian\]](#product-Grassmannian){reference-type="eqref" reference="product-Grassmannian"} componentwise, see Construction [Construction 1](#Def_Bound_individual){reference-type="ref" reference="Def_Bound_individual"}. **Definition 1**. As in the introduction we fix a point $\infty\in X(\mathbb{F}_q)$. \(a\) For $I:=\{1,2\}$ with the finest partition $I_\bullet=(\{1\},\{2\})$ we define the *Beilinson-Drinfeld Grassmannian with two legs, one fixed at $\infty$* as $$\operatorname{Gr}_{\mathcal{G},X\times\infty}:=\operatorname{Gr}_{\mathcal{G},X^2,(\{1\},\{2\})}\times_{X^2}(X\times_{\mathbb{F}_q}\operatorname{Spec}\mathbb{F}_\infty)$$ Its $S$-valued points, for an $\mathbb{F}_q$-scheme $S$, are tuples $$\bigl(x,\mathcal{E},\mathcal{E}',\mathcal{E}'',\,\varphi,\varphi'\,\bigr)\in\operatorname{Hecke}_{\mathcal{G},\varnothing,X\times\infty}(S)$$ as in Definition [Definition 1](#DefHecke2legs){reference-type="ref" reference="DefHecke2legs"}(a) together with - a trivialization $\epsilon\colon \mathcal{E}''\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{G}\times_X X_S$. \(b\) For the coarsest partition $I_1:=I=\{1,2\}$ we also define $${{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty} :=\operatorname{Gr}_{\mathcal{G},X^2,(\{1,2\})}\times_{X^2}(X\times_{\mathbb{F}_q}\operatorname{Spec}\mathbb{F}_\infty).$$ classifying data $\bigl(x,\,\mathcal{E},\mathcal{E}'',\,\varphi\colon \mathcal{E}|_{X_S\smallsetminus(\Gamma_x\cup\infty)} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}''|_{X_S\smallsetminus(\Gamma_x\cup\infty)} \,\bigr)$, visualized as $$\xymatrix @C+1pc { \mathcal{E}\ar@{-->}[r]^{\varphi}_{x,\infty} & \mathcal{E}'' \,. }$$ Again, the projection map onto the leg $x$ defines morphisms $\operatorname{Gr}_{\mathcal{G},X\times\infty}\to {{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty}\to X$. Recall the isomorphism $\operatorname{Gr}_{\mathcal{G},X}\times_X \operatorname{Spec}\mathbb{F}_\infty \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{F}\!\ell_{\mathcal{G},\infty}$ from [\[EqCorGrGlobLoc\]](#EqCorGrGlobLoc){reference-type="eqref" reference="EqCorGrGlobLoc"} in Corollary [Corollary 1](#CorGrGlobLoc){reference-type="ref" reference="CorGrGlobLoc"}. **Proposition 1**. *Above $X\smallsetminus\{\infty\}$ the base changes $\operatorname{Gr}_{\mathcal{G},X\times\infty} \times_X (X\smallsetminus\{\infty\})$ and ${{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty} \times_X (X\smallsetminus\{\infty\})$ are isomorphic to the product $\operatorname{Gr}_{\mathcal{G},X} \times_X (X\smallsetminus\{\infty\})\times_{\mathbb{F}_q} \mathcal{F}\!\ell_{\mathcal{G},\infty}$.* *Proof.* This follows from Proposition [Proposition 1](#PropGrProduct){reference-type="ref" reference="PropGrProduct"} and Corollary [Corollary 1](#CorGrGlobLoc){reference-type="ref" reference="CorGrGlobLoc"}. As in that proposition the isomorphism is given by $$\begin{aligned} \begin{split} \operatorname{Gr}_{\mathcal{G},X\times\infty}\times_X(X\smallsetminus \{\infty\})&\xlongrightarrow{\sim}\operatorname{Gr}_{\mathcal{G},X}\times_X (X\smallsetminus\{\infty\}) \times_{\mathbb{F}_q}\mathcal{F}\!\ell_{\mathcal{G},\infty}\\ (x,\mathcal{E}\underset{x}{\xrightarrow{\varphi}}\mathcal{E}'\underset{\infty}{\xrightarrow{\varphi'}}\mathcal{E}''\underset{\sim}{\xrightarrow{\epsilon}}\mathcal{G}\times_X X_S)&\longmapsto \bigl(x,L_x^+\mathcal{E}, L_x\mathcal{E}\xrightarrow{L_x(\epsilon\circ\varphi'\circ\varphi)}L_x\mathcal{G}\bigr), \bigl(L_\infty^+\mathcal{E}', L_\infty\mathcal{E}'\xrightarrow{L_\infty(\epsilon\circ\varphi')}L_\infty\mathcal{G}\bigr). \end{split}\end{aligned}$$ ◻ As in [\[EqHeckeCreateInfty\]](#EqHeckeCreateInfty){reference-type="eqref" reference="EqHeckeCreateInfty"}, "creating" an unnecessary additional leg at $\infty$ defines a morphism $$\begin{aligned} \label{EqGrCreateInfty} \operatorname{Gr}_{\mathcal{G},X} & \longrightarrow {{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty} \\ \bigl(x,\,\mathcal{E} \mathrel{ \mathpalette{\da@xarrow{x}{\varphi}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } \mathcal{E}' \xrightarrow[\sim]{\enspace\epsilon\enspace} \mathcal{G}\times_X X_S\bigr) & \longmapsto \bigl(x,\,\mathcal{E} \mathrel{ \mathpalette{\da@xarrow{x,\infty}{\varphi}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } \mathcal{E}'':=\mathcal{E}' \xrightarrow[\sim]{\enspace\epsilon\enspace} \mathcal{G}\times_X X_S\bigr). \nonumber\end{aligned}$$ **Lemma 1**. *The morphism [\[EqGrCreateInfty\]](#EqGrCreateInfty){reference-type="eqref" reference="EqGrCreateInfty"} is an ind-proper monomorphism of ind-schemes.* *Proof.* We already saw in Lemma [Lemma 1](#LemmaHeckeCreateInfty){reference-type="ref" reference="LemmaHeckeCreateInfty"} that it is a monomorphism. To prove that it is ind-proper, we may work on the ${\it fpqc\/}$-covering $(X\smallsetminus\{\infty\})\coprod \operatorname{Spec}\mathcal{O}_\infty$ of $X$. Above $X\smallsetminus\{\infty\}$ the map is an ind-closed immersion, because there $${{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty}\times_X (X\smallsetminus\{\infty\})\cong \operatorname{Gr}_{\mathcal{G},X}\times_X (X\smallsetminus\{\infty\}) \times_{\mathbb{F}_q} \mathcal{F}\!\ell_{\mathcal{G},\infty}$$ by Proposition [Proposition 1](#PropGrGeneric){reference-type="ref" reference="PropGrGeneric"}. The image of the morphism [\[EqGrCreateInfty\]](#EqGrCreateInfty){reference-type="eqref" reference="EqGrCreateInfty"} equals $\operatorname{Gr}_{\mathcal{G},X}\times_X (X\smallsetminus\{\infty\}) \times_{\mathbb{F}_q} 1\cdot L^+_\infty \mathcal{G}$, which is an ind-closed ind-subscheme. Over $\mathcal{O}_\infty$ the ind-scheme $\operatorname{Gr}_{\mathcal{G},X}\times_X \operatorname{Spec}\mathcal{O}_\infty$ is ind-proper by [@Richarz16 § 2.5], because we assumed that $\mathcal{G}_\infty$ is parahoric. Therefore also the morphism [\[EqGrCreateInfty\]](#EqGrCreateInfty){reference-type="eqref" reference="EqGrCreateInfty"} is ind-proper over $\mathcal{O}_\infty$. This proves the lemma. ◻ ## Bounds {#subsec:Bounds} We recall the definition of bounds from Bieker [@Bieker § 2] and generalize it slightly (to case [\[DefThreeTypes_C\]](#DefThreeTypes_C){reference-type="ref" reference="DefThreeTypes_C"} in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"} below). We need three cases in which we want to define bounds. **Definition 1**. For the *ground field* $F=Q=\mathbb{F}_q(X)$ or $F=Q_v$ or $F=\mathbb{F}_v$ at a place $v\in X$, we fix an algebraic closure $F^{{\operatorname{sep}}}$ of the ground field $F$ and consider finite separable field extensions $F\subset K$ in $F^{{\operatorname{sep}}}$. For $K$ we define $\widetilde{X}_K$ as follows: 1. [\[DefThreeTypes_A\]]{#DefThreeTypes_A label="DefThreeTypes_A"} *global type:* For $F=Q=\mathbb{F}_q(X)$ let $\widetilde{X}_K$ be the normalization of $X$ in $K$. It is a smooth projective curve over $\mathbb{F}_q$. Then we obtain a finite morphism $\widetilde{X}_K\to X$. 2. [\[DefThreeTypes_B\]]{#DefThreeTypes_B label="DefThreeTypes_B"} *local type:* For $F=Q_v$ at a place $v\in X$ let $\widetilde{X}_K:=\operatorname{Spec}\mathcal{O}_K$, where $\mathcal{O}_K$ is the valuation ring of $K$. Then we obtain a finite morphism $\widetilde{X}_K\to \operatorname{Spec}\mathcal{O}_v$. 3. [\[DefThreeTypes_C\]]{#DefThreeTypes_C label="DefThreeTypes_C"} *finite type:* For $F=\mathbb{F}_v$ at a place $v\in X$ let $\widetilde{X}_K:=\operatorname{Spec}K$. Then we obtain a finite morphism $\widetilde{X}_K\to \operatorname{Spec}\mathbb{F}_v$. **Remark 1**. Since we are mainly interested in bounds defined by cocharacters of the group $G$ and bounds of the finite type [\[DefThreeTypes_C\]](#DefThreeTypes_C){reference-type="ref" reference="DefThreeTypes_C"}, we only consider separable field extensions $K$ of $F$ in this article. We slightly generalize the definition of bounds from Bieker [@Bieker Definition 2.8]. **Definition 1**. 1. [\[DefBound_A\]]{#DefBound_A label="DefBound_A"} For $i=1,\ldots, n$ we fix one of the three types in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"} and let $K_i,K'_i$ be two fields of that type (for the same place $v$ in the local and finite case). We let $\widetilde{X}_{K_i}$ for $K_i$ and $\widetilde{X}_{K'_i}$ for $K'_i$ as in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"}. We write $\prod_i \widetilde{X}_{K_i} := \widetilde{X}_{K_1}\times_{\mathbb{F}_q}\ldots \times_{\mathbb{F}_q} \widetilde{X}_{K_n}$ and likewise $\prod_i \widetilde{X}_{K'_i}$. For any two quasi-compact closed subschemes $Z\subseteq \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \prod_i \widetilde{X}_{K_i}$ and $Z'\subseteq \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \prod_i \widetilde{X}_{K'_i}$, we say they are *equivalent* if for each $i$ there is a finite extension $K''_i$ of both $K_i,K'_i$ such that $$Z\times_{\prod_i \widetilde{X}_{K_i}} \prod_i \widetilde{X}_{K''_i}=Z'\times_{\prod_i \widetilde{X}_{K'_i}}\prod_i \widetilde{X}_{K''_i}$$ inside $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \prod_i \widetilde{X}_{K''_i}$. 2. [\[DefBound_B\]]{#DefBound_B label="DefBound_B"} Let $\mathcal{Z}$ be an equivalence class as in [\[DefBound_A\]](#DefBound_A){reference-type="ref" reference="DefBound_A"} and let $Z_{(K_i)_i}\subseteq \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \prod_i \widetilde{X}_{K_i}$ be a quasi-compact closed subscheme for some tuple $(K_i)_i$ representating $\mathcal{Z}$. For every $i$ we let $F_i$ be the ground field of the corresponding type as in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"}, and $\widetilde{K}_i$ be the Galois closure of $K_i/F_i$. We define $$\operatorname{Aut}_{\mathcal{Z}}({\textstyle\prod_i}\widetilde{K}_i):=\{(g_i)_i \in \prod_i \operatorname{Gal}(\widetilde{K}_i/F_i):(g_i)_i^*\mathcal{Z}=\mathcal{Z}\}.$$ The quotient $$\widetilde{X}_{\mathcal{Z}} := \bigl(\prod_i \widetilde{X}_{\widetilde{K}_i}\bigr) \big/ \operatorname{Aut}_{\mathcal{Z}}({\textstyle\prod_i}\widetilde{K}_i)$$ is called the *reflex scheme* of $\mathcal{Z}$. Since $\prod_i \operatorname{Gal}(\widetilde{K}_i/K_i)\subset \operatorname{Aut}_{\mathcal{Z}}({\textstyle\prod_i}\widetilde{K}_i)$, the reflex scheme is equiped with a finite and faithfully flat morphism $\prod_i \widetilde{X}_{K_i} \to \widetilde{X}_{\mathcal{Z}}$. 3. [\[DefBound_C\]]{#DefBound_C label="DefBound_C"} A *bound in $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$* is an equivalence class $\mathcal{Z}$ as in [\[DefBound_A\]](#DefBound_A){reference-type="ref" reference="DefBound_A"}, such that all its representatives $Z_{(K_i)_i}\subseteq \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \prod_i \widetilde{X}_{K_i}$ are stable under the left $\mathcal{L}^+_{X^n}\mathcal{G}\times_{X^n} \prod_i \widetilde{X}_{K_i}$-action on $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \prod_i \widetilde{X}_{K_i}$ from Definition [Definition 1](#DefLoopActionOnGr){reference-type="ref" reference="DefLoopActionOnGr"}, and such that $\mathcal{Z}$ has a representative over its reflex scheme $\widetilde{X}_{\mathcal{Z}}$. **Remark 1**. The quotient $\widetilde{X}_{\mathcal{Z}}$ can be complicated and does not have to be a product. However, when $\mathcal{Z}$ is the product $\prod_i\mathcal{Z}^{(i)}$ as in Construction [Construction 1](#Def_Bound_individual){reference-type="ref" reference="Def_Bound_individual"}, the reflex scheme $\widetilde{X}_{\mathcal{Z}}$ will be the product of the reflex schemes $\widetilde{X}_{\mathcal{Z}^{(i)}}$ of the individual factors $\mathcal{Z}^{(i)}$. **Lemma 1**. *For every bound $\mathcal{Z}$ there is a tuple of non-negative integers $(c_i)_i$ such that the action of $\mathcal{L}^+_{X^n}\mathcal{G}$ on all the representatives $Z$ of $\mathcal{Z}$ factors through the finite group scheme $\mathcal{L}^{+,(c_i)_i}_{X^n}\mathcal{G}$ from [\[EqGlobalTruncLoopGp\]](#EqGlobalTruncLoopGp){reference-type="eqref" reference="EqGlobalTruncLoopGp"}.* We generalize the definition of bounded Hecke data from [@Arasteh-Habibi] and [@Bieker Definition 3.3] as follows. **Definition 1**. Let $\mathcal{Z}$ be a bound in $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ as in Definition [Definition 1](#DefBound){reference-type="ref" reference="DefBound"} with reflex scheme $\widetilde{X}_{\mathcal{Z}}$. Let $$\underline{\mathcal{E}}=\Bigl(\underline x,(\mathcal{E}^{(0)},\psi^{(0)}) \mathrel{ \mathpalette{\da@xarrow{}{\varphi^{(0)}}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } (\mathcal{E}^{(1)},\psi^{(1)}) \mathrel{ \mathpalette{\da@xarrow{}{\varphi^{(1)}}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } \ldots \mathrel{ \mathpalette{\da@xarrow{}{\varphi^{(k-1)}}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } (\mathcal{E}^{(k)},\psi^{(k)})\Bigr)$$ in $(\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}\times_{X^n} \widetilde{X}_\mathcal{Z})(S)$ be a Hecke datum over an $\widetilde{X}_\mathcal{Z}$-scheme $S$. By definition of $L^+_{\underline x}\mathcal{G}$-bundles, there is an étale covering $S'\to S$ and a trivialization $\epsilon\colon L^+_{\underline x}(\mathcal{E}^{(k)})_{S'} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_{\underline x}\mathcal{G})_{S'}$. As in Proposition [Proposition 1](#PropGrGlobLoc){reference-type="ref" reference="PropGrGlobLoc"}, the tuple $$\Bigl(L^+_{\underline x}\mathcal{E}^{(0)} \mathrel{ \mathpalette{\da@xarrow{}{L_{\underline x}\varphi^{(0)}}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } \ldots \mathrel{ \mathpalette{\da@xarrow{}{L_{\underline x}\varphi^{(k-1)}}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } L^+_{\underline x}\mathcal{E}^{(k)} \xrightarrow[\sim]{\enspace\epsilon\enspace} L^+_{\underline x}\mathcal{G}\Bigr)\in (\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \widetilde{X}_\mathcal{Z})(S')$$ defines a morphism $S'\to \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \widetilde{X}_\mathcal{Z}$. We say that $\underline{\mathcal{E}}$ is *bounded by $\mathcal{Z}$* if this morphism factors through $Z\subset \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \widetilde{X}_\mathcal{Z}$ for the representative $Z$ of $\mathcal{Z}$ that is defined over the reflex scheme $\widetilde{X}_{\mathcal{Z}}$. By the invariance of $Z$ under the left multiplication by $L^+_{\underline x}\mathcal{G}$, the definition is independent of the choice of $\epsilon$ and $S'\to S$. We denote the stack of Hecke data bounded by $\mathcal{Z}$ by $\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}$. Then diagram [\[EqGrFiberProd\]](#EqGrFiberProd){reference-type="eqref" reference="EqGrFiberProd"} induces an isomorphism $\operatorname{Hecke}_{\mathcal{G},D,X^m,I_\bullet}^\mathcal{Z}\times_{\operatorname{Bun}_{\mathcal{G},D}} \operatorname{Spec}\mathbb{F}_q\cong Z\times_X (X\smallsetminus D)$ for the representative $Z$ of $\mathcal{Z}$ over $\widetilde{X}_\mathcal{Z}$. **Remark 1**. For $n=1$, let $\mathcal{Z}$ be a bound in $\operatorname{Gr}_{\mathcal{G},X}$ with reflex scheme $\widetilde{X}_\mathcal{Z}$. Let $S$ be a scheme over $\widetilde{X}_\mathcal{Z}$ and let $\varphi\colon \mathcal{E}|_{X_S\smallsetminus\Gamma_x} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}'|_{X_S\smallsetminus\Gamma_x}$ be an isomorphism of $\mathcal{G}$-bundles on $X_S$ outside the graph of a leg $x\in X(S)$. The definition allows to say that *$\varphi$ is bounded by $\mathcal{Z}$*, by viewing $(x,\mathcal{E},\mathcal{E}',\varphi)\in\operatorname{Hecke}_{\mathcal{G},\varnothing,X}$. **Theorem 1**. *The stack $\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}$ is a closed substack of $\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}\times_{X^n} \widetilde{X}_\mathcal{Z}$. Moreover, it is an Artin-stack locally of finite type over $(X\smallsetminus D)^n$.* *Proof.* For any test scheme $S$ over $\widetilde{X}_\mathcal{Z}$ and Hecke datum $\underline{\mathcal{E}}\in \operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}(S)$, we must show that in the cartesian diagram $$\xymatrix @C+2pc { \widetilde{S}\ar[r] \ar[d] & \operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}\ar[d] \\ S\ar[r]^-{\underline{\mathcal{E}}} & \operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}\times_{X^n} \widetilde{X}_\mathcal{Z} }$$ the map $\widetilde{S}\to S$ is a closed immersion of schemes. This can be tested after base change to an étale covering $S'\to S$, which we may choose as in Definition [Definition 1](#Def_HeckeBounded){reference-type="ref" reference="Def_HeckeBounded"}. Then the base change morphism $\widetilde{S}\times_S S' \to S'$ arises as the base change of $Z\hookrightarrow\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \widetilde{X}_\mathcal{Z}$ which is a closed immersion. ◻ Next we give some examples of bounds. **Definition 1** (Bound in $\operatorname{Gr}_{\mathcal{G},X}$ given by $\mu$). Let $n=1$ and let $\mu\colon \mathbb{G}_{m,Q^{\operatorname{sep}}}\to G_{Q^{\operatorname{sep}}}$ be a cocharacter of $G$. Let $E_{\mu}$ be the reflex field of $\mu$, and $\widetilde{X}_\mu$ the normalization of $X$ in $E_\mu$ as in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"}. The affine Schubert variety $Z^{\leq \mu}:=\operatorname{Gr}_{\mathcal{G},X}^{\leq \mu}$ of $\mu$ from Definition [Definition 1](#Def_SchubertVariety){reference-type="ref" reference="Def_SchubertVariety"} defines a bound $\mathcal{Z}^{\leq\mu}$, which has the representative $Z^{\leq \mu}$ over the reflex scheme $\widetilde{X}_\mu$. We say that a Hecke datum $\underline{\mathcal{E}}\in (\operatorname{Hecke}_{\mathcal{G},D,X}\times_{X} \widetilde{X}_\mu)(S)$ is *bounded by $\mu$* if it is bounded by $\mathcal{Z}^{\leq\mu}$. Since we will need it later, we introduce the following notation. We choose a point $\widetilde{\infty}$ of $\widetilde{X}_\mu$ that lies above the point $\infty\in X(\mathbb{F}_q)$ from the introduction. We write $\mathcal{O}_\mu$ for the complete local ring of $\widetilde{X}_\mu$ at $\widetilde{\infty}$ and $\kappa_\mu$ for its residue field. We let $\Breve{\mathcal{O}}_\mu$ be the completion of the maximal unramified ring extension of $\mathcal{O}_\mu$. Then $\mathcal{O}_\infty\subset \mathcal{O}_\mu$ and $\Breve{\mathcal{O}}_\infty\subset \Breve{\mathcal{O}}_\mu$ and $\mathbb{F}_\infty\subset \kappa_\mu$ are finite extensions. ** 1**. Let $\infty\in X(\mathbb{F}_q)$ and let $\beta\in L_\infty\mathcal{G}(\overline{\mathbb{F}}_q)=G(\Breve{Q}_\infty)$ with $\beta\cdot L^+_\infty\mathcal{G}\cdot\beta^{-1}=L^+_\infty\mathcal{G}$. Then there is a smallest finite field extension $\mathbb{F}_\beta$ of $\mathbb{F}_q$ with $\beta\in L_\infty\mathcal{G}(\mathbb{F}_\beta)$. For the Frobenius $\tau_{G_\infty}$ from [\[EqTau_G\]](#EqTau_G){reference-type="eqref" reference="EqTau_G"} we have $\tau_{G_\infty}^j(\beta) \cdot L^+_\infty\mathcal{G}= L^+_\infty\mathcal{G}\cdot \tau_{G_\infty}^j(\beta)$ for any $j\in\mathbb{Z}$, because $\mathcal{G}_\infty$ is defined over $\mathbb{F}_q$. This equality implies that $\beta$ is small in the sense, that its orbit in $\mathcal{F}\!\ell_{\mathcal{G},\infty}(\mathbb{F}_\beta)$ under the left action of $L^+_\infty\mathcal{G}$ is the single point $\beta\cdot L^+_\infty\mathcal{G}(\mathbb{F}_\beta)$; see Definition [Definition 1](#Def_Zbeta){reference-type="ref" reference="Def_Zbeta"}. That $\beta$ is small can also be interpreted in the following way. Assume that $\mathcal{G}_\infty$ is an Iwahori group scheme corresponding to a $\tau$-stable alcove $\mathfrak{a}$ in the extended Bruhat-Tits building $\mathcal{B}(G_\infty,\Breve{Q}_\infty)$ of $G_\infty$ over $\Breve{Q}_\infty$. The condition $\beta\cdot L^+_\infty\mathcal{G}\cdot\beta^{-1}=L^+_\infty\mathcal{G}$ says that $\mathfrak{a}$ is a fixed point of $\beta$ under the action of $G(\Breve{Q}_\infty)=L_\infty\mathcal{G}(\overline{\mathbb{F}}_q)$ on $\mathcal{B}(G_\infty,\Breve{Q}_\infty)$. Assume further, that $\beta$ also stabilizes an appartment containing $\mathfrak{a}$. If $A$ is the maximal split torus of $G(\Breve{Q}_\infty)$ corresponding to that appartment, then $\beta$ normalizes $A$. In this case, $\beta$ induces an element in the *Iwahori-Weyl group* $\widetilde{W}_\infty:=\widetilde{W}(G_\infty,A,\Breve{Q}_\infty)$ of $A$ over $\Breve{Q}_\infty$; see Richarz [@RicharzIWGp]. Then the condition $\beta\cdot L^+_\infty\mathcal{G}\cdot\beta^{-1}=L^+_\infty\mathcal{G}$ means that $\beta$ has *length zero* in $\widetilde{W}_\infty$. **Definition 1** (Bound in $\operatorname{Gr}_{\mathcal{G},X}\times_X \operatorname{Spec}\mathbb{F}_\beta$ given by $\beta$). Let $n=1$. Let $\beta\in L_\infty\mathcal{G}(\mathbb{F}_\beta)$ as in § [ 1](#Def_beta){reference-type="ref" reference="Def_beta"}. We define the bound $$\begin{aligned} \mathcal{Z}(\beta) & \; := \; (L^+_\infty\mathcal{G})_{\mathbb{F}_\beta} \cdot\beta\cdot (L^+_\infty\mathcal{G})_{\mathbb{F}_\beta}/(L^+_\infty\mathcal{G})_{\mathbb{F}_\beta} \;=\; \beta\cdot (L^+_\infty\mathcal{G})_{\mathbb{F}_\beta}/(L^+_\infty\mathcal{G})_{\mathbb{F}_\beta} \\ & \; \subset \; \mathcal{F}\!\ell_{\mathcal{G},\infty}\times_{\mathbb{F}_q} \operatorname{Spec}\mathbb{F}_\beta \; = \; (L_\infty\mathcal{G}/L^+_\infty\mathcal{G})_{\mathbb{F}_\beta}. \nonumber\end{aligned}$$ Its reflex scheme $\widetilde{X}_{\mathcal{Z}(\beta)}$ is $\operatorname{Spec}\mathbb{F}_\beta$. This is a field of type [\[DefThreeTypes_C\]](#DefThreeTypes_C){reference-type="ref" reference="DefThreeTypes_C"} in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"}. **Construction 1** (Bounds in $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ as products of bounds in $\operatorname{Gr}_{\mathcal{G},X}$). Let $(\mathcal{Z}^{(i)})_{i=1\ldots n}$ be an $n$-tuple of bounds in $\operatorname{Gr}_{\mathcal{G},X}$ with reflex schemes $\widetilde{X}_{\mathcal{Z}^{(i)}}$, and let $Z^{(i)}$ be the representative of $\mathcal{Z}^{(i)}$ over $\widetilde{X}_{\mathcal{Z}^{(i)}}$ for all $i=1,\ldots,n$. We allow each $\widetilde{X}_{\mathcal{Z}^{(i)}}$ to be of any of the three types in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"}. Let $U\subseteq X^n$ be the complement of all diagonals as in Proposition [Proposition 1](#PropGrProduct){reference-type="ref" reference="PropGrProduct"}. If $\bigl(\prod_i \widetilde{X}_{\mathcal{Z}^{(i)}}\bigr) \times_{X^n} U$ is non-empty, we define $Z$ to be the scheme-theoretic image of the composite morphism $$\xymatrix { \Bigl(\prod\limits_{i=1}^n Z^{(i)}\Bigr) \times_{X^n} U \ar@{^{ (}->}[dr] \ar@{^{ (}->}[r] & \Bigl(\bigl(\operatorname{Gr}_{\mathcal{G},X}\times_X \widetilde{X}_{\mathcal{Z}^{(1)}}\bigr) \times_{\mathbb{F}_q} \ldots \times_{\mathbb{F}_q} \bigl(\operatorname{Gr}_{\mathcal{G},X}\times_X \widetilde{X}_{\mathcal{Z}^{(n)}}\bigr)\Bigr) \times_{X^n} U \ar@{^{ (}->}[d] \\ & \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}\times_{X^n} \prod_i\widetilde{X}_{\mathcal{Z}^{(i)}}, }$$ where the right vertical map is given in Proposition [Proposition 1](#PropGrProduct){reference-type="ref" reference="PropGrProduct"}. This $Z$ defines a bound $\mathcal{Z}$ in $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$, which we denote $\prod_i\mathcal{Z}^{(i)}$ and call the *product bound* of the $\mathcal{Z}^{(i)}$. Its reflex scheme is $\prod_i\widetilde{X}_{\mathcal{Z}^{(i)}}$, because the group $\operatorname{Aut}_{\mathcal{Z}}({\textstyle\prod_i}\widetilde{K}_i)$ from Definition [Definition 1](#DefBound){reference-type="ref" reference="DefBound"}[\[DefBound_B\]](#DefBound_B){reference-type="ref" reference="DefBound_B"} equals $\prod_i \operatorname{Gal}(\widetilde{K}_i/E_i)$ for the "reflex fields" $E_i$ for which $\widetilde{X}_{\mathcal{Z}^{(i)}}=\widetilde{X}_{E_i}$. Note that $\bigl(\prod_i \widetilde{X}_{\mathcal{Z}^{(i)}}\bigr) \times_{X^n} U$ is non-empty unless two different $\widetilde{X}_{\mathcal{Z}^{(i)}}$ are of type [\[DefThreeTypes_C\]](#DefThreeTypes_C){reference-type="ref" reference="DefThreeTypes_C"} in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"} for the same place $v\in X$. **Definition 1**. Let $\beta\in L_\infty\mathcal{G}(\mathbb{F}_\beta)$ as in § [ 1](#Def_beta){reference-type="ref" reference="Def_beta"} and let $\mu\in X_*(T)$. We define the bounds $\mathcal{Z}(\mu,\beta)$ in $\operatorname{Gr}_{\mathcal{G},X\times\infty}$ and ${{}'\!\mathcal{Z}}(\mu,\beta)$ in ${{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty}$ as the product bound $\mathcal{Z}^{\leq\mu}\times \mathcal{Z}(\beta)$ from Construction [Construction 1](#Def_Bound_individual){reference-type="ref" reference="Def_Bound_individual"}, such that the bound at the moving leg $x$ is $\mathcal{Z}^{\leq\mu}$ and the bound at the fixed leg $\infty$ equals $\mathcal{Z}(\beta)$ from Definition [Definition 1](#Def_Zbeta){reference-type="ref" reference="Def_Zbeta"}. The reflex scheme of these bounds is the product $\widetilde{X}_{\mu,\beta}:=\widetilde{X}_\mu\times_{\mathbb{F}_q} \operatorname{Spec}\mathbb{F}_\beta$. Since we will need it later, we introduce the following notation. We choose a point $\widetilde{\infty}$ of $\widetilde{X}_{\mu,\beta}$ that lies above the point $\infty$. We write ${\mathcal{O}_{\mu,\beta}}$ for the complete local ring of $\widetilde{X}_{\mu,\beta}$ at $\widetilde{\infty}$ and ${\kappa_{\mu,\beta}}$ for its residue field. We let ${\Breve{\mathcal{O}}_{\mu,\beta}}$ be the completion of the maximal unramified ring extension of ${\mathcal{O}_{\mu,\beta}}$. It is equal to the ring $\Breve{\mathcal{O}}_\mu$ from Definition [Definition 1](#Def_BoundBy_mu){reference-type="ref" reference="Def_BoundBy_mu"} if the two points on $\widetilde{X}_\mu$ and $\widetilde{X}_{\mu,\beta}$ above $\infty\in X$ are chosen compatibly. Then $\mathcal{O}_\infty\subset {\mathcal{O}_{\mu,\beta}}$ and $\Breve{\mathcal{O}}_\infty\subset {\Breve{\mathcal{O}}_{\mu,\beta}}$ and $\mathbb{F}_\infty\subset {\kappa_{\mu,\beta}}$ are finite extensions. **Proposition 1**. *The relative dimension of $\mathcal{Z}(\mu,\beta)$ over $\widetilde{X}_{\mu,\beta}$ equals $\langle \mu,2\check{\rho}\rangle$, where $2\check{\rho}$ is the sum of all positive coroots of $G_{Q^{\rm alg}}$ with respect to some Borel subgroup for which $\mu$ is dominant.* *Proof.* This was proven in [@Ngo-Polo Lemma 2.2 and the remarks thereafter]. ◻ **Lemma 1**. *Let $\beta=1$, and hence $\mathbb{F}_\beta=\mathbb{F}_q$. Let $\mathcal{Z}$ be a bound in $\operatorname{Gr}_{\mathcal{G},X}$ with reflex scheme $\widetilde{X}_\mathcal{Z}$ of global type in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"}[\[DefThreeTypes_A\]](#DefThreeTypes_A){reference-type="ref" reference="DefThreeTypes_A"}, such that its representative $Z$ over $\widetilde{X}_\mathcal{Z}$ is the scheme theoretic closure of its restriction $Z\times_X (X\smallsetminus\{\infty\})$. Let $\mathcal{Z}\times\mathcal{Z}(1)$ be the product bound in ${{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty}$ from Construction [Construction 1](#Def_Bound_individual){reference-type="ref" reference="Def_Bound_individual"}.* 1. *[\[LemmaZmu1_A\]]{#LemmaZmu1_A label="LemmaZmu1_A"} Under the morphism $\operatorname{Gr}_{\mathcal{G},X} \to {{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty}$ from [\[EqGrCreateInfty\]](#EqGrCreateInfty){reference-type="eqref" reference="EqGrCreateInfty"}, the bound $\mathcal{Z}$ in $\operatorname{Gr}_{\mathcal{G},X}$ is mapped isomorphically to the bound $\mathcal{Z}\times\mathcal{Z}(1)$ in ${{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty}$.* 2. *[\[LemmaZmu1_B\]]{#LemmaZmu1_B label="LemmaZmu1_B"} The morphism [\[EqHeckeCreateInfty\]](#EqHeckeCreateInfty){reference-type="eqref" reference="EqHeckeCreateInfty"} restricts to an isomorphism $\operatorname{Hecke}_{\mathcal{G},D,X}^{\mathcal{Z}}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times \infty}^{\mathcal{Z}\times\mathcal{Z}(1)}$.* *Proof.* [\[LemmaZmu1_A\]](#LemmaZmu1_A){reference-type="ref" reference="LemmaZmu1_A"} By Lemma [Lemma 1](#LemmaGrCreateInfty){reference-type="ref" reference="LemmaGrCreateInfty"} the morphism $\mathcal{Z}\to{{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty}\times_X \widetilde{X}_\mathcal{Z}$ is a proper monomorphism, hence a closed immersion. We write $\widetilde{X}_\mathcal{Z}\smallsetminus\{\infty\}:=\widetilde{X}_\mathcal{Z}\times_X (X\smallsetminus\{\infty\})$. Identifying $\mathcal{Z}$ with its image, both $\mathcal{Z}$ and $\mathcal{Z}\times\mathcal{Z}(1)$ are defined as the scheme theoretic closure of their intersection with ${{}'\!\operatorname{Gr}}_{\mathcal{G},X\times\infty}\times_X (\widetilde{X}_\mathcal{Z}\smallsetminus\{\infty\})$. The latter is isomorphic to the product $\operatorname{Gr}_{\mathcal{G},X}\times_X (\widetilde{X}_\mathcal{Z}\smallsetminus\{\infty\}) \times_{\mathbb{F}_q} \mathcal{F}\!\ell_{\mathcal{G},\infty}$ by Proposition [Proposition 1](#PropGrGeneric){reference-type="ref" reference="PropGrGeneric"}, and the intersections of $\mathcal{Z}$ and $\mathcal{Z}\times\mathcal{Z}(1)$ with that product are equal to $Z\times_{\widetilde{X}_\mathcal{Z}} (\widetilde{X}_\mathcal{Z}\smallsetminus\{\infty\})\times_{\mathbb{F}_q} (1\cdot L^+_\infty\mathcal{G})$. This proves [\[LemmaZmu1_A\]](#LemmaZmu1_A){reference-type="ref" reference="LemmaZmu1_A"}. [\[LemmaZmu1_B\]](#LemmaZmu1_B){reference-type="ref" reference="LemmaZmu1_B"} We must show that the two projection morphisms $\operatorname{pr}_1$ and $\operatorname{pr}_2$ in $$\label{EqLemmaZmu1} \operatorname{Hecke}_{\mathcal{G},D,X}^{\mathcal{Z}} \; \xleftarrow{\;\operatorname{pr}_1\;} \; \operatorname{Hecke}_{\mathcal{G},D,X}^{\mathcal{Z}} \underset{\;{{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times \infty}}{\times} {{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times \infty}^{\mathcal{Z}\times\mathcal{Z}(1)} \; \xrightarrow{\;\operatorname{pr}_2\;} \; {{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times \infty}^{\mathcal{Z}\times\mathcal{Z}(1)}$$ are isomorphisms, where the fiber product is defined via the morphism [\[EqHeckeCreateInfty\]](#EqHeckeCreateInfty){reference-type="eqref" reference="EqHeckeCreateInfty"}. The projection $\operatorname{pr}_1$ (respectively $\operatorname{pr}_2$) is a closed immersion (respectively a monomorphism) by Theorem [Theorem 1](#Thm_HeckeBounded){reference-type="ref" reference="Thm_HeckeBounded"} (respectively Lemma [\[EqHeckeCreateInfty\]](#EqHeckeCreateInfty){reference-type="ref" reference="EqHeckeCreateInfty"}). In particular, both projections are schematic by [@Laumon-Moret-Bailly Corollaire 8.1.3 and Théorème A.2]. Let $(x,(\mathcal{E},\psi),(\mathcal{E}',\psi'),\varphi)\in \operatorname{Hecke}_{\mathcal{G},D,X}^{\mathcal{Z}}(S)$ be a $\mathcal{G}$-shtuka bounded by $\mathcal{Z}$ over a test scheme $S$. There is an étale covering $S'\to S$ over which a trivialization $\epsilon\colon L^+_\Delta(\mathcal{E}'_{S'})\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\Delta\mathcal{G})_{S'}$ exists, where $\Delta:=\Gamma_x + (\infty\times_{\mathbb{F}_q}S')$. We let $S' \hookleftarrow \operatorname{pr}_1^*S'=:\widetilde{S}'$ be the base change of the closed immersion $\operatorname{pr}_1$ under the morphism $S'\to \operatorname{Hecke}_{\mathcal{G},D,X}^{\mathcal{Z}}$. The morphism from $\widetilde{S}'$ to the fiber product in [\[EqLemmaZmu1\]](#EqLemmaZmu1){reference-type="eqref" reference="EqLemmaZmu1"} corresponds to the object $(x,(\mathcal{E},\psi),(\mathcal{E}',\psi'),\varphi)\in \operatorname{Hecke}_{\mathcal{G},D,X}^{\mathcal{Z}}(\widetilde{S}')$ and an object $(x,(\widetilde{\mathcal{E}},\widetilde{\psi}),(\widetilde{\mathcal{E}}',\widetilde{\psi}'),\widetilde{\varphi})\in {{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times \infty}^{\mathcal{Z}\times\mathcal{Z}(1)}(\widetilde{S}')$ together with an isomorphism $(f,f')\colon (x,(\widetilde{\mathcal{E}},\widetilde{\psi}),(\widetilde{\mathcal{E}}',\widetilde{\psi}'),\widetilde{\varphi})\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(x,(\mathcal{E},\psi),(\mathcal{E}',\psi'),\varphi)$ in ${{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times \infty}(\widetilde{S}')$. In particular, $f'\colon\widetilde{\mathcal{E}}'\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}'$ is an isomorphism of $\mathcal{G}$-bundles. The trivialization $\epsilon$ induces the trivialization $\epsilon\circ L^+_\Delta(f')\colon L^+_\Delta(\widetilde{\mathcal{E}}')\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\Delta\mathcal{G})_{\widetilde{S}'}$. Using these trivializations, the morphism $S'\hookleftarrow \widetilde{S}'$ is obtained as base change under $S'\to\mathcal{Z}$ from the projection $\operatorname{pr}_1$ in the following diagram $$\label{EqLemmaZmu2} \mathcal{Z}\; \xleftarrow[\sim]{\;\operatorname{pr}_1\;} \; \mathcal{Z}\underset{\;{{}'\!\operatorname{Gr}}_{\mathcal{G},X\times \infty}}{\times} (\mathcal{Z}\times\mathcal{Z}(1)) \; \xrightarrow[\sim]{\;\operatorname{pr}_2\;} \; (\mathcal{Z}\times\mathcal{Z}(1)).$$ That diagram is obtained from [\[EqLemmaZmu1\]](#EqLemmaZmu1){reference-type="eqref" reference="EqLemmaZmu1"} under base change via $\operatorname{Spec}\mathbb{F}_q\to\operatorname{Bun}_{\mathcal{G},D}$ as in [\[EqGrFiberProd\]](#EqGrFiberProd){reference-type="eqref" reference="EqGrFiberProd"}. In diagram [\[EqLemmaZmu2\]](#EqLemmaZmu2){reference-type="eqref" reference="EqLemmaZmu2"} the projections $\operatorname{pr}_1$ and $\operatorname{pr}_2$ are isomorphisms by [\[LemmaZmu1_A\]](#LemmaZmu1_A){reference-type="ref" reference="LemmaZmu1_A"}. This shows that $\operatorname{pr}_1$ in [\[EqLemmaZmu1\]](#EqLemmaZmu1){reference-type="eqref" reference="EqLemmaZmu1"} is an isomorphism. The analogous argument for the projection $\operatorname{pr}_2$ in [\[EqLemmaZmu1\]](#EqLemmaZmu1){reference-type="eqref" reference="EqLemmaZmu1"} starts with an object $(x,(\widetilde{\mathcal{E}},\widetilde{\psi}),(\widetilde{\mathcal{E}}',\widetilde{\psi}'),\widetilde{\varphi})\in {{}'\!\operatorname{Hecke}}_{\mathcal{G},D,X\times \infty}^{\mathcal{Z}\times\mathcal{Z}(1)}(S)$ and proves that $\operatorname{pr}_2$ is an isomorphism. ◻ ## Moduli spaces of shtukas {#Shtuka-subsection} In this section, we recall the preliminaries on shtukas. **Definition 1**. Let $D\subset X$ be a finite subscheme. The *stack of global $\mathcal{G}$-shtukas* $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}$ *with $n$ legs and $D$-level structure* is the stack fibered in groupoids over the category of $\mathbb{F}_q$-schemes, whose $S$-valued points, for an $\mathbb{F}_q$-scheme $S$, are tuples $$\bigl((x_i)_{i=1\ldots n},\,(\mathcal{E}^{(i)},\psi^{(i)})_{i=0\ldots n},\,(\varphi^{(i-1)})_{i=1\ldots n}\,\bigr)\in\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}(S)$$ as in Definition [Definition 1](#DefHecke_nlegs){reference-type="ref" reference="DefHecke_nlegs"} together with: - *shtuka condition*: an isomorphism $\varphi^{(k)}\colon \mathcal{E}^{(k)}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!}\mathcal{E}^{(0)}$ compatible with the $D$-level structure, i.e. ${}^{\tau\!}\psi^{(0)}\circ\varphi^{(k)}=\psi^{(k)}$. Here the superscript ${}^{\tau\!}$ refers to the pullback under the absolute $q$-Frobenius $\tau=\operatorname{Frob}_{q,S}$ of $S$ as defined in §[2.1](#subsec-notations){reference-type="ref" reference="subsec-notations"}. We usually drop $(\mathcal{E}^{(k)},\psi^{(k)})$ and $\varphi^{(k)}$ from the notation and simply identify $(\mathcal{E}^{(k)},\psi^{(k)})$ with ${}^{\tau\!}(\mathcal{E}^{(0)},\psi^{(0)})$. Then we can visualize the above data as $$\label{EqDef_Sht} \xymatrix @C+1pc { \underline{\mathcal{E}}:=\Bigl(\underline x,\, (\mathcal{E}^{(0)},\psi^{(0)}) \ar@{-->}[r]^-{\varphi^{(0)}}_-{x_i\colon i\in I_1} & (\mathcal{E}^{(1)},\psi^{(1)}) \ar@{-->}[r]^-{\varphi^{(1)}}_-{x_i\colon i\in I_2} & \ldots \ar@{-->}[r]^-{\varphi^{(k-1)}}_-{x_i\colon i\in I_k} & {}^{\tau\!}(\mathcal{E}^{(0)},\psi^{(0)})\Bigr) }$$ and call it a *global $\mathcal{G}$-shtuka with $D$-level structure over $S$*. When $D=\varnothing$, we will drop it (and the $\psi^{(i)}$) from the notation. **Definition 1**. Consider a scheme $S$ together with legs $x_i\colon S\to X\smallsetminus D$ for $i=1,\ldots,n$ and let $\underline{\mathcal{E}},\underline{\widetilde{\mathcal{E}}}\in\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}(S)$ be two global $\mathcal{G}$-shtukas over $S$ with the same legs $x_i$. A *quasi-isogeny* from $\underline{\mathcal{E}}$ to $\underline{\widetilde{\mathcal{E}}}$ is a tuple of isomorphisms $f^{(i)}\colon(\mathcal{E}^{(i)},\psi^{(i)})|_{X_S \smallsetminus N_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\widetilde{\mathcal{E}}^{(i)},\widetilde{\psi}^{(i)})|_{X_S \smallsetminus N_S}$ for $i=0,\ldots,n$ of $\mathcal{G}$-bundles with $D$-level structure satisfying $\widetilde{\varphi}^{(i)}\circ f^{(i)}=f^{(i+1)}\circ\varphi^{(i)}$ for $i=0,\ldots,n-1$ and $\widetilde{\varphi}^{(k)}\circ f^{(k)}={}^{\tau\!}f^{(0)}\circ\varphi^{(k)}$, where $N\subset X$ is some proper closed subscheme. We denote the *group of quasi-isogenies* from $\underline{\mathcal{E}}$ to itself by $\operatorname{QIsog}_S(\underline{\mathcal{E}})$. **Remark 1**. If $S=\operatorname{Spec}\overline{\mathbb{F}}_q$ then we write $I_{\underline{\mathcal{E}}}(Q):=\mathrm{QIsog}_{\overline{{\mathbb{F}}}_\infty}(\underline{\mathcal{E}})$. We do not need the following result in this article. However, it justifies the notation. There is a linear algebraic group $I_{\underline{\mathcal{E}}}$ over $Q$ such that $I_{\underline{\mathcal{E}}}(Q)=\mathrm{QIsog}_{\overline{{\mathbb{F}}}_\infty}(\underline{\mathcal{E}})$. This can be proven as in [@AH_CMotives; @AH_LRConj] by noting that $\underline{\mathcal{E}}$ gives rise to a tensor functor from $\operatorname{Rep}_QG$ to the category of $C$-motives (with $C=X$) as in [@AH_CMotives (4.3)]. Here $\operatorname{Rep}_QG$ is the neutral Tannakian category of algebraic representations of $G$ in finite dimensional $Q$-vector spaces. By Tannakian duality the $\mathcal{G}$-shtuka $\underline{\mathcal{E}}$ corresponds to a homomorphism $h=h_{\underline{\mathcal{E}}}$ from the Tannakian fundamental groupoid $\mathfrak{P}$ of the category of $C$-motives to the neutral groupoid of $G$. Then $I_h:=\operatorname{Aut}(h)$ is the linear algebraic group over $Q$ defined by $$I_h(R) :=\operatorname{Aut}(h)(R) := \bigl\{\, g\in G(Q^{\rm alg}\otimes_QR)\colon \operatorname{int}_g \circ h = h\,\bigr\}$$ for $Q$-algebras $R$. Indeed, $I_h$ and $I_{\underline{\mathcal{E}}}$ are equal by Lemma [Lemma 1](#lemma-ADLV-identied-Xphi){reference-type="ref" reference="lemma-ADLV-identied-Xphi"}. By Definition [Definition 1](#Def_Sht){reference-type="ref" reference="Def_Sht"}, the stack $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}$ is defined as the fiber product $$\label{Eq_DiagSht} \xymatrix @C+4pc { \operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet} \ar[r] \ar[d] & \operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet} \ar[d]^{(\mathcal{E}^{(0)}, \mathcal{E}^{(k)})} \\ \operatorname{Bun}_{\mathcal{G},D} \ar[r]^-{\operatorname{id}\times \operatorname{Frob}_q} & \operatorname{Bun}_{\mathcal{G},D} \times_{\mathbb{F}_q} \operatorname{Bun}_{\mathcal{G},D} }$$ There is a map $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}\to (X\smallsetminus D)^n$ given by $\underline{\mathcal{E}}\mapsto (x_i)_{i=1,\ldots,n}$. **Theorem 1**. 1. *[\[ThmSht_A\]]{#ThmSht_A label="ThmSht_A"} The stack $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}$ is an ind-Deligne-Mumford stack, locally of ind-finite type and ind-separated over $(X\smallsetminus D)^n$.* 2. *[\[ThmSht_B\]]{#ThmSht_B label="ThmSht_B"} If $D\subset D'\subset X$ are proper closed subschemes, then the natural morphism $\operatorname{Sht}_{\mathcal{G},D',X^n,I_\bullet} \to \operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}\times_{(X\smallsetminus D)^n} (X\smallsetminus D')^n$ is finite, étale, surjective and a torsor for the finite abstract group $\ker\bigl(\mathcal{G}(D')\to\mathcal{G}(D)\bigr)$.* *Proof.* This was proven in [@AH_Unif Theorem 3.15] building on earlier work for constant split $\mathcal{G}$ of Varshavsky [@Varshavsky04] and Lafforgue [@Lafforgue12]. ◻ The stacks $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}$ and the theorem generalize results on the moduli space of $F$-sheaves $FSh_{D,r}$ which were considered by Drinfeld [@Drinfeld-moduli-Fsheaves] and Lafforgue [@LafforgueL02] in their proof of the Langlands correspondence for $\mathcal{G}=\operatorname{GL}_2$ (resp. $\mathcal{G}=\operatorname{GL}_r$), and its generalization $FBun$ by Varshavsky [@Varshavsky04]. It likewise generalizes the moduli stacks $\mathcal{E}\ell\ell_{X,\mathscr{D},I}$ of Laumon, Rapoport and Stuhler [@Laumon-Rapoport-Stuhler], their generalizations by L. Lafforgue [@Lafforgue-Ramanujan], Lau [@Lau07], Ngô [@Ngo06] and Spieß [@Spiess10], the spaces $\text{Cht}_{\underline\lambda}$ of Ngô and Ngô Dac [@NgoNgo; @NgoDac13], and the spaces ${\rm AbSh}^{r,d}_H$ of the first author [@HartlAbSh]. **Corollary 1**. *Let $\underline{\mathcal{E}}$ and $\underline{\mathcal{E}}'$ be global $\mathcal{G}$-shtukas with the same legs in $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}(S)$ over an $\mathbb{F}_q$-scheme $S$. Then the sheaf of sets on $S_{\it fpqc\/}$ given by $\underline\operatorname{Isom}_S(\underline{\mathcal{E}},\underline{\mathcal{E}}')\colon T\mapsto\operatorname{Isom}_T(\underline{\mathcal{E}}_T,\underline{\mathcal{E}}'_T)$ is representable by a scheme, which is finite and unramified over $S$. In particular, the (abstract) group of automorphisms $\operatorname{Aut}_S(\underline{\mathcal{E}})$ of $\underline{\mathcal{E}}$ over $S$ is finite.* *Proof.* Since $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}$ is an ind-separated ind-Deligne-Mumford stack, its diagonal is unramified and proper. The base change of the diagonal under the morphism $(\underline{\mathcal{E}},\underline{\mathcal{E}}')\colon S \to \operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}\times_{\mathbb{F}_q} \operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}$ equals $\underline\operatorname{Isom}_S(\underline{\mathcal{E}},\underline{\mathcal{E}}')$, which is hence an algebraic space unramified and proper over $S$. In particular it is finite and affine over $S$ and hence a scheme; c.f. [@Laumon-Moret-Bailly Lemma 4.2]. ◻ **Definition 1**. Let $\mathcal{Z}$ be a bound in $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ and let $\widetilde{X}_\mathcal{Z}$ be its reflex scheme. We define the closed substack $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}^{\mathcal{Z}}$ of $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}\times_{X^n} \widetilde{X}_\mathcal{Z}$ as the base change of $\operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}^{\mathcal{Z}}$ under the morphism $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}\to \operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet}$ from [\[Eq_DiagSht\]](#Eq_DiagSht){reference-type="eqref" reference="Eq_DiagSht"}. In terms of the data from [\[EqDef_Sht\]](#EqDef_Sht){reference-type="eqref" reference="EqDef_Sht"} this means that the boundedness is tested after trivializing $L^+_{\underline x}({}^{\tau\!}\mathcal{E}^{(0)})$. We say that a global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}\in(\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}\times_{X^n} \widetilde{X}_\mathcal{Z})(S)$ is *bounded by $\mathcal{Z}$* if it belongs to $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}^{\mathcal{Z}}(S)$. **Theorem 1**. *The stack $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}$ is a Deligne-Mumford stack locally of finite type and separated over $(X\smallsetminus D)^n \times_{X^n} \widetilde{X}_\mathcal{Z}$.* In the following we recall that the Beilinson-Drinfeld Grassmannian (and a bound therein) is a local model for the moduli spaces of (bounded) shtukas. **Definition 1**. Let $D\subset X$ be a finite subscheme. We define $\widetilde{\operatorname{Sht}}_{\mathcal{G},D,X^n,I_\bullet}$ as the stack, whose $S$-valued points, for an $\mathbb{F}_q$-scheme $S$, are tuples $$\bigl(\underline x=(x_i)_{i=1\ldots n},\,(\mathcal{E}^{(i)},\psi^{(i)})_{i=0\ldots n},\,(\varphi^{(i-1)})_{i=1\ldots n},\varphi^{(k)}\,\bigr)\in\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}(S)$$ as in Definition [Definition 1](#Def_Sht){reference-type="ref" reference="Def_Sht"} together with: - a trivialization $\hat{\epsilon}\colon L_{\underline x}\mathcal{E}^{(k)} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_{\underline x}\mathcal{G})_S$. Note that $\mathcal{L}^+_{X^n}\mathcal{G}$ is the automorphism group of the trivial $L^+_{\underline x}\mathcal{G}$-bundle. If $\mathcal{Z}$ is a bound in $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ and $Z$ is its representative over the reflex scheme $\widetilde{X}_{\mathcal{Z}}$, we define $\widetilde{\operatorname{Sht}}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}:= \widetilde{\operatorname{Sht}}_{\mathcal{G},D,X^n,I_\bullet} \times_{\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}} \operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}$. Forgetting the trivialization $\hat{\epsilon}$ (respectively the isomorphism $\varphi^{(k)}\colon \mathcal{E}^{(k)} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!}\mathcal{E}^{(0)}$) defines the $X^n$-morphisms $\pi_1$ (respectively $\pi_2$) in the following diagrams $$\label{EqLocalModel} \xymatrix @C-2pc { & \widetilde{\operatorname{Sht}}_{\mathcal{G},D,X^n,I_\bullet} \ar[dl]_{\pi_1} \ar[dr]^{\pi_2} & & & & \widetilde{\operatorname{Sht}}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}\ar[dl]_{\pi_1} \ar[dr]^{\pi_2} \\ \operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet} \ar[dr]_{\overline{\pi}_2} & & \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet} \ar[dl]^{\overline{\pi}_1} & \quad\text{and}\quad & \operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}\ar[dr]_{\overline{\pi}_2} & & {\enspace Z \qquad} \ar[dl]^{\overline{\pi}_1} \\ & [\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet} / \mathcal{L}^+_{X^n}\mathcal{G}] & & & & {\quad [Z / \mathcal{L}^+_{X^n}\mathcal{G}] \quad} }$$ in which the bottom entries are the stack quotients modulo the $\mathcal{L}^+_{X^n}\mathcal{G}$-action. These diagrams are called the *local model diagram* for $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}$ (respectively for $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}$). **Theorem 1**. *In the diagrams [\[EqLocalModel\]](#EqLocalModel){reference-type="eqref" reference="EqLocalModel"} the $X^n$-morphisms $\pi_2$ and $\overline{\pi}_2$ are formally smooth and both diagrams are cartesian. The $X^n$-morphisms $\pi_1$ and $\overline{\pi}_1$ are $\mathcal{L}^+_{X^n}\mathcal{G}$-torsors and have sections étale locally on the target. For any such section $s$ of $\pi_1$, the composition $\pi_2\circ s$ is étale. In particular, $\operatorname{Gr}_{\mathcal{G},X^n,I_\bullet}$ is a *local model* for $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}$, in the sense that both are isomorphic locally for the étale topology. Likewise, $Z$ is a local model for $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}$.* *If $(c_i)_i$ is a tuple of integers as in Lemma [Lemma 1](#LemmaLoopActionOnGr){reference-type="ref" reference="LemmaLoopActionOnGr"} for which the $\mathcal{L}^+_{X^n}\mathcal{G}$-action on the representatives of the bound $\mathcal{Z}$ factors through $\mathcal{L}^{+,(c_i)_i}_{X^n}\mathcal{G}$, then the morphism $\overline{\pi}_2$ factors through the morphism $$\label{EqThmLocMod} \operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}^\mathcal{Z}\to [Z / \mathcal{L}^{+,(c_i)_i}_{X^n}\mathcal{G}].$$ The latter is smooth of relative dimension $(\sum_i c_i)\cdot\dim G$, which is also equal do the relative dimension of the group scheme $\mathcal{L}^{+,(c_i)_i}_{X^n}\mathcal{G}$ over $X^n$; see Lemma [Lemma 1](#LemmaGlobalLoopIsqc){reference-type="ref" reference="LemmaGlobalLoopIsqc"}[\[LemmaGlobalLoopIsqc_C\]](#LemmaGlobalLoopIsqc_C){reference-type="ref" reference="LemmaGlobalLoopIsqc_C"}.* *Proof.* This goes back to Varshavsky [@Varshavsky04 Theorem 2.20] for constant split reductive $\mathcal{G}$ and was reproven by Lafforgue [@Lafforgue12 Proposition 2.11] and generalized by Arasteh Rad and Habibi [@Arasteh-Habibi Theorem 3.2.1] to smooth affine group schemes $\mathcal{G}$ over $X$. The proof relies on the observation that in the diagramm [\[Eq_DiagSht\]](#Eq_DiagSht){reference-type="eqref" reference="Eq_DiagSht"} which defines $\operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet}$ $$\xymatrix @C+4pc { \operatorname{Sht}_{\mathcal{G},D,X^n,I_\bullet} \ar[r] \ar[d] & \operatorname{Hecke}_{\mathcal{G},D,X^n,I_\bullet} \ar[d]^{(\mathcal{E}^{(0)}, \mathcal{E}^{(k)})} & \operatorname{Gr}_{\mathcal{G},X^n,I_\bullet} \ar[l] \ar[d] \\ \operatorname{Bun}_{\mathcal{G},D} \ar[r]^-{\operatorname{id}\times \operatorname{Frob}_q} & \operatorname{Bun}_{\mathcal{G},D} \times_{\mathbb{F}_q} \operatorname{Bun}_{\mathcal{G},D} & \operatorname{Bun}_{\mathcal{G},D} \ar[l]_-{\operatorname{id}\times \mathcal{G}} }$$ both horizontal morphisms in the bottom row have the same differential $(\operatorname{id},0)$. ◻ **Definition 1**. Let $\infty\in X(\mathbb{F}_q)$ and let $D\subset X\smallsetminus\{\infty\}$ be a proper closed subscheme. The *stack of global $\mathcal{G}$-shtukas* $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}$ *with two legs, one fixed at $\infty$*, is the stack, whose $S$-valued points, for $S$ an $\mathbb{F}_q$-scheme, are tuples $\bigl(x,\,(\mathcal{E},\psi),(\mathcal{E}',\psi'),\,\varphi,\varphi'\,\bigr)$ where - $x \in (X\smallsetminus D)(S)$ is a section, called a *leg*, - $(\mathcal{E},\psi),(\mathcal{E}',\psi')$ are objects in $\operatorname{Bun}_{\mathcal{G},D}(S)$, and - $\varphi\colon \mathcal{E}|_{{X_S}\smallsetminus\Gamma_{x}}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}'|_{{X_S}\smallsetminus\Gamma_{x}}$ and $\varphi'\colon \mathcal{E}'|_{(X\smallsetminus\{\infty\})_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!}\mathcal{E}|_{(X\smallsetminus\{\infty\})_S}$ are isomorphisms preserving the $D$-level structures, i.e. $\psi'\circ\varphi|_{D_S}=\psi$ and ${}^{\tau\!}\psi\circ\varphi'|_{D_S}=\psi'$. We can visualize the above data as $$\xymatrix @C+1pc { (\mathcal{E},\psi) \ar@{-->}[r]^\varphi_{x} & (\mathcal{E}',\psi') \ar@{-->}[r]^{\varphi'}_\infty & {}^{\tau\!}(\mathcal{E},\psi) \,. }$$ When $D=\varnothing$, we will drop it from the notation. The projection map $$\bigl(x,\,(\mathcal{E},\psi),(\mathcal{E}',\psi'),\,\varphi,\varphi'\,\bigr)\mapsto x$$ defines a morphism $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}\to X\smallsetminus D$. For a bound $\mathcal{Z}$ in $\operatorname{Gr}_{\mathcal{G},X\times\infty}$, we also define the stack $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}}$ of global $\mathcal{G}$-shtukas $\underline{\mathcal{E}}\in\operatorname{Sht}_{\mathcal{G},D,X\times\infty}$ that are bounded by $\mathcal{Z}$ as in Definition [Definition 1](#Def_ShtBounded){reference-type="ref" reference="Def_ShtBounded"}. **Theorem 1**. *The stack $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}^\mathcal{Z}$ is a Deligne-Mumford stack locally of finite type and separated over $X\smallsetminus D$. Moreover, the stack $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}$ is an ind-Deligne-Mumford stack, locally of ind-finite type and ind-separated over $X\smallsetminus D$.* **Proposition 1**. *Let $\mathcal{Z}$ be a bound in $\operatorname{Gr}_{\mathcal{G},X\times\infty}$ and let $Z$ be its representative over the reflex scheme $\widetilde{X}_{\mu,\beta}$. Then $Z$ is a local model for $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}}$, i.e. both are isomorphic locally for the étale topology.* *Proof.* This follows from Theoreom [Theorem 1](#ThmLocMod){reference-type="ref" reference="ThmLocMod"}. ◻ ## Interpretation in terms of chains {#subsec:Chains} In this section let $\infty\in X(\mathbb{F}_q)$ and let $\beta\in L_\infty\mathcal{G}(\mathbb{F}_\beta)$ be the element from § [ 1](#Def_beta){reference-type="ref" reference="Def_beta"}. Let $\mu\in X_*(T)$ and let $\mathcal{Z}(\mu,\beta)$ be the bound in $\operatorname{Gr}_{\mathcal{G},X\times\infty}$ from Definition [Definition 1](#DefZmubeta){reference-type="ref" reference="DefZmubeta"}. In Corollary [Corollary 1](#CorShtWithChains){reference-type="ref" reference="CorShtWithChains"} we shall give an interpretation of $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}$ in terms of chains of $\mathcal{G}$-bundles. This relates our global $\mathcal{G}$-shtukas bounded by $\mathcal{Z}(\mu,\beta)$ to the objects defined by "sequences", namely the elliptic sheaves of Drinfeld [@Drinfeld-commutative-subrings; @Blum-Stuhler] (see Example [Example 1](#ExDrinfeld){reference-type="ref" reference="ExDrinfeld"}), the $\mathscr{D}$-elliptic sheaves of Laumon, Rapoport and Stuhler [@Laumon-Rapoport-Stuhler] (see Example [Example 1](#ExLRS){reference-type="ref" reference="ExLRS"}), and the abelian $\tau$-sheaves of the first author [@HartlAbSh] (see Example [\[ExAbelianSheaves\]](#ExAbelianSheaves){reference-type="ref" reference="ExAbelianSheaves"}). Recall the Frobenius map $\tau_{G_\infty}$ of the group $G_\infty$ over $Q_\infty$ from [\[EqTau_G\]](#EqTau_G){reference-type="eqref" reference="EqTau_G"}. **Definition 1**. Let $D\subset X\smallsetminus\{\infty\}$ be a closed subscheme. Let $\mathcal{CB}un_{\mathcal{G},D,\beta}$ be the stack over $\mathbb{F}_\beta$ classifying chains $$\label{chain-eqn} \ldots \mathrel{ \mathpalette{\da@xarrow{}{\Pi_{-1}}\mathchar"0\hexnumber@\symAMSa 4C {}{}{\,}}{}% } (\mathcal{E}_{-1},\psi_{-1}) \mathrel{ \mathpalette{\da@xarrow{}{\Pi_0}\mathchar"0\hexnumber@\symAMSa 4C {}{}{\,}}{}% } (\mathcal{E}_{0},\psi_{0}) \mathrel{ \mathpalette{\da@xarrow{}{\Pi_1}\mathchar"0\hexnumber@\symAMSa 4C {}{}{\,}}{}% } (\mathcal{E}_1,\psi_1) \mathrel{ \mathpalette{\da@xarrow{}{\Pi_2}\mathchar"0\hexnumber@\symAMSa 4C {}{}{\,}}{}% } \ldots$$ of $\mathcal{G}$-bundles with $D$-level structure over $S$ such that for all $i\in\mathbb{Z}$ the modifications are isomorphisms of $\mathcal{G}$-bundles $\Pi_i\colon (\mathcal{E}_i,\psi_i)|_{(X\smallsetminus\{\infty\})_S} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\mathcal{E}_{i-1},\psi_{i-1})|_{(X\smallsetminus\{\infty\})_S}$ bounded by $\mathcal{Z}(\tau_{G_\infty}^{i-1}(\beta)^{-1})$, which preserve the $D$-level structures. Morphisms $(f_i)_i\colon(\mathcal{E}_i,\psi_i,\Pi_i)_i\to (\mathcal{E}'_i,\psi'_i,\Pi'_i)_i$ in $\mathcal{CB}un_{\mathcal{G},D,\beta}$ are tuples of isomorphisms $f_i\colon(\mathcal{E}_i,\psi_i)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\mathcal{E}'_i,\psi'_i)$ in $\operatorname{Bun}_{\mathcal{G},D}(S)$ satisfying $\Pi'_i\circ f_i = f_{i-1}\circ \Pi_i$ for all $i$. **Proposition 1**. *For any fixed $j\in\mathbb{Z}$, the functor $$\mathfrak{pr}_j\colon \mathcal{CB}un_{\mathcal{G},D,\beta} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\operatorname{Bun}_{\mathcal{G},D}\times_{\mathbb{F}_q} \operatorname{Spec}\mathbb{F}_\beta \,,\quad (\mathcal{E}_i,\psi_i,\Pi_i) \longmapsto (\mathcal{E}_j,\psi_j)$$ is an isomorphism of stacks. In particular, $\mathcal{CB}un_{\mathcal{G},D,\beta}$ is a smooth Artin-stack locally of finite type over $\mathbb{F}_\beta$.* *Proof.* Since the $D$-level structure is outside $\infty$, and the modifications $\Pi_i$ take place at $\infty$, the $D$-level structure is preserved. We will ignore the level structure in the rest of the proof. In the following, we prove the result for $j=0$. The same proof holds for arbitrary $j\in\mathbb{Z}$. \(1\) Full faithfulness of the functor $\mathfrak{pr}_0$: Let $(\mathcal{E}_i,\Pi_i)_i$ and $(\mathcal{E}_i',\Pi_i')_i$ be two chains over $S$, and let $f_0:\mathcal{E}_0\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}_0'$ be an isomorphism in $\operatorname{Bun}_\mathcal{G}(S)$. For any $i\geq 0$ (and similarly for $i\leq 0$), the map $f_0$ induces an isomorphism outside $\infty$ $$\label{Eq2.3.17} \mathcal{E}_i|_{(X\smallsetminus\{\infty\})_S}\cong \mathcal{E}_0|_{(X\smallsetminus\{\infty\})_S}\xrightarrow[f_0]{\;\sim\,\,}\mathcal{E}_0'|_{(X\smallsetminus\{\infty\})_S}\cong \mathcal{E}_i'|_{(X\smallsetminus\{\infty\})_S}.$$ We apply the functor $L_\infty^+$ from Definition [Definition 1](#DefLoopGpAtDelta){reference-type="ref" reference="DefLoopGpAtDelta"} and Example [Example 1](#ExDivisors){reference-type="ref" reference="ExDivisors"}(b), and choose an étale covering $S'\to S$, over which trivializations exist that form the vertical maps in the following diagram $$\begin{tikzcd}[column sep=12ex] L_\infty(\mathcal{E}_i)_{S'}\arrow[]{r}{\Pi_1\circ\ldots\circ\Pi_i}[swap]{\sim}\arrow[]{d}{\alpha_i}[swap]{\sim} & L_\infty(\mathcal{E}_0)_{S'}\arrow[]{r}{L_\infty(f_0)}[swap]{\sim}\arrow[]{d}{\alpha_0}[swap]{\sim} & L_\infty(\mathcal{E}_0')_{S'}\arrow[]{r}{(\Pi_1'\circ\ldots\circ\Pi_i')^{-1}}[swap]{\sim}\arrow[]{d}{\alpha_0'}[swap]{\sim}&L_\infty(\mathcal{E}_i')_{S'}\arrow[]{d}{\alpha_i'}[swap]{\sim}\\ (L_\infty G)_{S'}\arrow[]{r}{\sim}&(L_\infty G)_{S'}\arrow[]{r}{\sim}&(L_\infty G)_{S'}\arrow[]{r}{\sim}&(L_\infty G)_{S'} \end{tikzcd}$$ Since $\tau_{G_\infty}^k(\beta)\cdot L^+_\infty\mathcal{G}(S') = L^+_\infty\mathcal{G}(S')\cdot\tau_{G_\infty}^k(\beta)$ by § [ 1](#Def_beta){reference-type="ref" reference="Def_beta"}, the three horizontal isomorphisms in the bottom row lie in $\tau_{G_\infty}^{0}(\beta)^{-1}\cdot\ldots\cdot \tau_{G_\infty}^{i-1}(\beta)^{-1} L^+_\infty\mathcal{G}(S')$ and $L^+_\infty\mathcal{G}(S')$ and $\tau_{G_\infty}^{i-1}(\beta)\cdot\ldots\cdot \tau_{G_\infty}^{0}(\beta) L^+_\infty\mathcal{G}(S')$, respectively. Therefore, their composition lies in $L^+_\infty\mathcal{G}(S')$ and yields an isomorphism $L_\infty^+(\mathcal{E}_i)_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_\infty^+(\mathcal{E}_i')_{S'}$, which descends to $S$ and glues with [\[Eq2.3.17\]](#Eq2.3.17){reference-type="eqref" reference="Eq2.3.17"} to an isomorphism $f_i\colon\mathcal{E}_i \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}'_i$ on all of $X_S$ by Lemma [Lemma 1](#LemmaBL){reference-type="ref" reference="LemmaBL"}. The $f_i$ define an isomorphism of chains $(f_i)_i\colon (\mathcal{E}_i,\Pi_i)_i\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\mathcal{E}'_i,\Pi'_i)_i$. \(2\) Essential surjectivity of the functor $\mathfrak{pr}_0$: Fix any $\mathcal{E}\in \operatorname{Bun}_\mathcal{G}(S)$. We want to construct a chain of the form [\[chain-eqn\]](#chain-eqn){reference-type="eqref" reference="chain-eqn"} with $\mathcal{E}_0=\mathcal{E}$. For every $i\in\mathbb{Z}_{\ge 0}$, we can construct $\mathcal{E}_{i+1}$ from $\mathcal{E}_i$ via Lemma [Lemma 1](#LemmaBL){reference-type="ref" reference="LemmaBL"} by glueing $\overset{\circ}{\mathcal{E}}_{i+1}:=\mathcal{E}_i|_{(X\smallsetminus\{\infty\})_S}$ with $\mathcal{L}_{i+1}$, where $\mathcal{L}_{i+1}$ is constructed from $L_\infty^+(\mathcal{E}_i)$ as follows: choose an étale covering $S'\to S$ and a trivialization $\alpha_i: L_\infty^+(\mathcal{E}_i)_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{G})_{S'}$ over $S'$. Let $h_i\in L^+_\infty\mathcal{G}(S'')$, for $S'':=S'\times_S S'$, be given by $h_i:=\operatorname{pr}_2^*\alpha_i\circ\operatorname{pr}_1^*\alpha_i^{-1}$, where $\operatorname{pr}_k:S''\to S'$ is the projection onto the $k$-th factor, for $k=1,2$. Let $(\mathcal{L}_{i+1})_{S'}:=(L^+_\infty\mathcal{G})_{S'}$, let $\alpha_{i+1}:=\operatorname{id}\colon (\mathcal{L}_{i+1})_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{G})_{S'}$, and let $$h_{i+1}:=\tau_{G_\infty}^{i}(\beta)\cdot h_i\cdot \tau_{G_\infty}^{i}(\beta)^{-1}\in \tau_{G_\infty}^{i}(\beta)\cdot L^+_\infty\mathcal{G}(S'')\cdot \tau_{G_\infty}^{i}(\beta)^{-1}=L^+_\infty\mathcal{G}(S'').$$ Then $h_{i+1}$ defines a descent datum on $(\mathcal{L}_{i+1})_{S'}$ and the latter descends to an $L^+_\infty\mathcal{G}$-bundle $\mathcal{L}_{i+1}$ on $S$. Moreover, $$\label{EqDefPi_i} \Pi_{i+1}:=\alpha_i^{-1}\circ\tau_{G_\infty}^{i}(\beta)^{-1}\circ\alpha_{i+1}: L_\infty(\mathcal{L}_{i+1})_{S'}=(L_\infty\mathcal{G})_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_\infty(\mathcal{E}_i)_{S'}$$ satisfies $\operatorname{pr}_2^*\Pi_{i+1}=\operatorname{pr}_1^*\alpha_i^{-1}\circ h_i^{-1}\tau_{G_\infty}^i(\beta)^{-1} h_{i+1}\circ\operatorname{pr}_1^*\alpha_{i+1}=\operatorname{pr}_1^*\Pi_i$ and descends to an isomorphism $$\label{PropBunWithChainsEqPi} \Pi_{i+1}: L_\infty(\mathcal{L}_{i+1}) \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_\infty(\mathcal{E}_i)$$ over $S$. By Lemma [Lemma 1](#LemmaBL){reference-type="ref" reference="LemmaBL"} we glue $\mathcal{E}_{i+1}$ from $\overset{\circ}{\mathcal{E}}_{i+1}:=\mathcal{E}_i|_{(X\smallsetminus\{\infty\})_S}$ and $\mathcal{L}_{i+1}$ via the isomorphism $$\Pi_{i+1}^{-1} \colon L_\infty(\overset{\circ}{\mathcal{E}}_{i+1})=L_\infty(\mathcal{E}_i|_{(X\smallsetminus\{\infty\})_S})=L_\infty(\mathcal{E}_i)\xrightarrow{\Pi_{i+1}^{-1}}L_\infty(\mathcal{L}_{i+1}).$$ The isomorphism $\Pi_{i+1}$ from [\[PropBunWithChainsEqPi\]](#PropBunWithChainsEqPi){reference-type="eqref" reference="PropBunWithChainsEqPi"} extends to the isomorphism $$\Pi_{i+1}=\operatorname{id}\colon \mathcal{E}_{i+1}|_{(X\smallsetminus\{\infty\})_S} = \overset{\circ}{\mathcal{E}}_{i+1} = \mathcal{E}_i|_{(X\smallsetminus\{\infty\})_S} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}_i|_{(X\smallsetminus\{\infty\})_S}.$$ We proceed analogously for all $i<0$. This construction defines a chain [\[chain-eqn\]](#chain-eqn){reference-type="eqref" reference="chain-eqn"} in $\mathcal{CB}un_{G,\beta}(S)$. ◻ Let $\mu\in X_*(T)$. Also the bound $\mathcal{Z}(\mu,\beta)$ from Definition [Definition 1](#DefZmubeta){reference-type="ref" reference="DefZmubeta"} can be interpreted in terms of chains. Let $\widetilde{X}_{\mu,\beta}$ be its reflex scheme. **Proposition 1**. *The bound $\mathcal{Z}(\mu,\beta)$ is represented by the closed subscheme of $\operatorname{Gr}_{\mathcal{G},X\times\infty}\times_X \widetilde{X}_{\mu,\beta}$ whose $S$-valued points are tuples $\bigl(x,\,(\mathcal{E}_i,\Pi_i,\mathcal{E}_i'',\Pi_i'',\varphi^\lhd_i)_{i\in\mathbb{Z}},\,\epsilon\bigr)$ where* - *$x \colon S\to \widetilde{X}_{\mu,\beta}\to X$ is a leg,* - *together with a commutative diagram, $$\label{EqPropBoundWithChains} \begin{tikzcd} \ldots & \mathcal{E}_{-1} \arrow[dashed]{l}[swap]{\Pi_{-1}} \arrow[dashed]{ld}[swap]{\varphi^\lhd_{-1}} & \mathcal{E}_0\arrow[dashed]{l}[swap]{\Pi_0}\arrow[dashed]{ld}[swap]{\varphi^\lhd_0} & \mathcal{E}_1\arrow[dashed]{l}[swap]{\Pi_1}\arrow[dashed]{ld}[swap]{\varphi^\lhd_1}& \arrow[dashed]{l}[swap]{\Pi_2} \arrow[dashed]{ld}[swap]{\varphi^\lhd_2}\ldots\\ \ldots & \mathcal{E}''_{-1}\arrow[dashed]{l}{\Pi''_{-1}}&\mathcal{E}''_0\arrow[dashed]{l}{\Pi''_0}&\mathcal{E}''_1\arrow[dashed]{l}{\Pi''_1} & \arrow[dashed]{l}{\Pi''_2}\ldots \end{tikzcd},$$ where the $\varphi^\lhd_i$ are isomorphisms outside the graph $\Gamma_x$ bounded by $\mu$ in the sense of Definition [Definition 1](#Def_BoundBy_mu){reference-type="ref" reference="Def_BoundBy_mu"} and Remark [Remark 1](#Rem_HeckeBounded){reference-type="ref" reference="Rem_HeckeBounded"}, and the $\Pi_i$ and $\Pi_i''$ are isomorphisms outside $\infty$ with $\Pi_i$ bounded by $\mathcal{Z}(\tau_{G_\infty}^{i-1}(\beta)^{-1})$ and $\Pi_i''$ bounded by $\mathcal{Z}(\tau_{G_\infty}^{i}(\beta)^{-1})$,* - *and $\epsilon\colon\mathcal{E}''_0\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{G}\times_XX_S$ is an isomorphism of $\mathcal{G}$-bundles.* *Proof.* Let $Z(\mu,\beta)$ be the representative over $\widetilde{X}_{\mu,\beta}$ of the bound $\mathcal{Z}(\mu,\beta)$. We will work over the universal base scheme $S:=Z(\mu,\beta)$. It is defined as the scheme theoretic closure in $\operatorname{Gr}_{\mathcal{G},X\times\infty} \times_X \widetilde{X}_{\mu,\beta}$ of its restriction $\widetilde{S}:=Z(\mu,\beta)\times_X (X\smallsetminus\{\infty\})$ outside $\infty$. Therefore, $\widetilde{S}\subset S$ is schematically dense by [@EGA1 I, Corollaire 9.5.11]. The restriction $\widetilde{S}$ classifies tuples $(x,\mathcal{E}_0,\mathcal{E}'_0,\mathcal{E}''_0,\varphi_0,\varphi'_0,\epsilon)\in\operatorname{Gr}_{\mathcal{G},X\times\infty}$ as in Definition [Definition 1](#DefGr2legs){reference-type="ref" reference="DefGr2legs"}, where $\varphi_0$ is bounded by $\mathcal{Z}^{\leq\mu}$ and $\varphi'_0$ is bounded by $\mathcal{Z}(\beta)$. Under the isomorphism $\mathfrak{pr}_0$ from Proposition [Proposition 1](#PropBunWithChains){reference-type="ref" reference="PropBunWithChains"}, we can uniquely extend $\mathcal{E}_0$, $\mathcal{E}'_0$, and $\mathcal{E}''_0$ to chains of $\mathcal{G}$-bundles to obtain the rows of the following commutative diagram $$\label{EqPropBoundWithChains2} \begin{tikzcd} \ldots & \mathcal{E}_{-1}\arrow[dotted]{d}{\varphi_{-1}} \arrow[dashed]{l}[swap]{\Pi_{-1}} & \mathcal{E}_0\arrow[dashed]{l}[swap]{\Pi_0}\arrow[dashed]{d}{\varphi_0}&\mathcal{E}_1\arrow[dashed]{l}[swap]{\Pi_1}\arrow[dotted]{d}{\varphi_1}&\arrow[dashed]{l}[swap]{\Pi_2}\ldots\\ \ldots & \mathcal{E}'_{-1}\arrow[dotted]{d}{\varphi'_{-1}}\arrow[dashed]{l}[swap]{\Pi'_{-1}} & \mathcal{E}'_0\arrow[dashed]{l}[swap]{\Pi'_0}\arrow[dashed]{d}{\varphi'_0} & \mathcal{E}'_1\arrow[dashed]{l}[swap]{\Pi'_1}\arrow[dotted]{d}{\varphi'_1}&\arrow[dashed]{l}[swap]{\Pi'_2}\ldots\\ \ldots & \mathcal{E}''_{-1}\arrow[dashed]{l}[swap]{\Pi''_{-1}}&\mathcal{E}''_0\arrow[dashed]{l}[swap]{\Pi''_0}&\mathcal{E}''_1\arrow[dashed]{l}[swap]{\Pi''_1}&\arrow[dashed]{l}[swap]{\Pi''_2} \ldots \end{tikzcd}$$ with $\Pi_i$ and $\Pi'_i$ bounded by $\mathcal{Z}(\tau_{G_\infty}^{i-1}(\beta)^{-1})$, and $\Pi''_i$ bounded by $\mathcal{Z}(\tau_{G_\infty}^{i}(\beta)^{-1})$. In this diagram the dotted maps $\varphi_i$ and $\varphi'_i$ for $i\ne 0$ are the induced isomorphisms outside $\Gamma_x\cup\{\infty\}$. We claim for all $i\in\mathbb{Z}$ 1. [\[ProofPropBoundWithChains_A\]]{#ProofPropBoundWithChains_A label="ProofPropBoundWithChains_A"} that $\Pi''_i\circ\varphi'_i\colon\mathcal{E}'_i|_{(X\smallsetminus\{\infty\})_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{E}''_{i-1}|_{(X\smallsetminus\{\infty\})_S}$ extends to an isomorphism over all of $X_S$, 2. [\[ProofPropBoundWithChains_B\]]{#ProofPropBoundWithChains_B label="ProofPropBoundWithChains_B"} and that $\varphi_i$, and hence also $\varphi^\lhd_i:=\Pi''_i\circ \varphi'_i \circ \varphi_i$ is bounded by $\mathcal{Z}^{\leq\mu}$. We prove the claim for a fixed $i>0$. For $i<0$ we can argue analogously. We consider the associated $L^+_\infty\mathcal{G}$-bundles $L_\infty^+(\mathcal{E}^\Box_j)$ for $\Box\in\{\varnothing,\,',\,''\}$. Over some étale covering $S'\to S$ we choose trivializations $$\alpha^\Box_j\colon L_\infty^+(\mathcal{E}^\Box_j)_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{G})_{S'}$$ for all $j\in\{0,\ldots,i\}$. Over $\widetilde{S}:=Z(\mu,\beta)\times_X (X\smallsetminus\{\infty\})$, the divisors $\Gamma_x$ and $\infty\times_{\mathbb{F}_q} \widetilde{S}$ are disjoint. There, the isomorphisms $\alpha^\Box_{j-1}\circ L_\infty(\Pi^\Box_j)\circ(\alpha^\Box_{j})^{-1}: (L_\infty G)_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L_\infty G)_{S'}$ for $\Box\in\{\varnothing,\,'\}$, respectively $\alpha''_{j-1}\circ L_\infty(\Pi''_j)\circ(\alpha''_{j})^{-1}$, respectively $\alpha''_{0}\circ L_\infty(\varphi'_0)\circ(\alpha'_{0})^{-1}$ are given by multiplication on the left with an element of $\tau_{G_\infty}^{j-1}(\beta)^{-1}\cdot L^+_\infty\mathcal{G}(S')$, respectively of $\tau_{G_\infty}^{j}(\beta)^{-1}\cdot L^+_\infty\mathcal{G}(S')$, respectively of $\beta\cdot L^+_\infty\mathcal{G}(S')$; see [\[EqDefPi_i\]](#EqDefPi_i){reference-type="eqref" reference="EqDefPi_i"} in the proof of Proposition [Proposition 1](#PropBunWithChains){reference-type="ref" reference="PropBunWithChains"}. We now use $\tau_{G_\infty}^j(\beta)^{-1}\cdot L^+_\infty\mathcal{G}(S')\cdot \tau_{G_\infty}^j(\beta) = L^+_\infty\mathcal{G}(S')$. To prove [\[ProofPropBoundWithChains_A\]](#ProofPropBoundWithChains_A){reference-type="ref" reference="ProofPropBoundWithChains_A"} we observe that over $\widetilde{S}$ $$\alpha''_{i-1}\circ L_\infty\bigl(\Pi''_i\circ\varphi'_i)\circ(\alpha'_{i})^{-1}\;=\;\alpha''_{i-1}\circ L_\infty\bigl((\Pi''_1\circ\ldots\circ\Pi''_{i-1})^{-1} \circ\varphi'_0 \circ (\Pi'_{1}\circ\ldots\circ\Pi'_i)\bigr)\circ(\alpha'_{i})^{-1}$$ is given by multiplication with an element of $$\tau_{G_\infty}^{i-1}(\beta)\cdot\ldots\cdot \tau_{G_\infty}^{1}(\beta)\cdot\beta\cdot \tau_{G_\infty}^{0}(\beta)^{-1} \cdot\ldots\cdot \tau_{G_\infty}^{i-1}(\beta)^{-1} \cdot L^+_\infty\mathcal{G}(S')=L^+_\infty\mathcal{G}(S'),$$ and hence is an isomorphism at $\infty$. As $\Pi''_i\circ\varphi'_i$ is also an isomorphism outside $\infty$, it is an isomorphism on all of $X_{\widetilde{S}}$. We consider the tuple $(\infty,\Pi''_i\circ\varphi'_i\colon \mathcal{E}'_i \mathrel{ \mathpalette{\da@xarrow{\infty}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } \mathcal{E}''_{i-1})$ as an object in $\operatorname{Hecke}_{\mathcal{G},\varnothing,X}(S)$. The condition that $\Pi''_i\circ\varphi'_i$ is an isomorphism over all of $X$ can be formulated as boundedness by the trivial bound $\mathcal{Z}(1):=1\cdot \mathcal{L}^+_X\mathcal{G}\subset \operatorname{Gr}_{\mathcal{G},X}$ given as in Corollary [Corollary 1](#CorGrGlobLoc){reference-type="ref" reference="CorGrGlobLoc"}. Since $\Pi''_i\circ\varphi'_i$ is an isomorphism on $X_{\widetilde{S}}$, the composition $\widetilde{S}\to S\to \operatorname{Hecke}_{\mathcal{G},\varnothing,X}$ factors through the closed substack $\operatorname{Hecke}_{\mathcal{G},\varnothing,X}^{\mathcal{Z}(1)}$ from Theorem [Theorem 1](#Thm_HeckeBounded){reference-type="ref" reference="Thm_HeckeBounded"}. Since $\widetilde{S}$ is schematically dense in $S$, already the morphism $S\to \operatorname{Hecke}_{\mathcal{G},\varnothing,X}$ factors through $\operatorname{Hecke}_{\mathcal{G},\varnothing,X}^{\mathcal{Z}(1)}$. We conclude that $\Pi''_i\circ\varphi'_i$ is an isomorphism on all of $X_S$. To prove [\[ProofPropBoundWithChains_B\]](#ProofPropBoundWithChains_B){reference-type="ref" reference="ProofPropBoundWithChains_B"} we continue to work over $\widetilde{S}$, where the morphism $L^+_\infty(\varphi_0)\colon L^+_\infty\mathcal{E}_0\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L^+_\infty\mathcal{E}'_0$ is an isomorphism. Then $$\alpha'_{i}\circ L_\infty\bigl(\varphi_i)\circ(\alpha_{i})^{-1}\;=\;\alpha'_{i}\circ L_\infty\bigl((\Pi'_{1}\circ\ldots\circ\Pi'_i)^{-1} \circ\varphi_0 \circ (\Pi_{1}\circ\ldots\circ\Pi_i)\bigr)\circ(\alpha_{i})^{-1}$$ is given by multiplication with an element of $$\tau_{G_\infty}^{i-1}(\beta)\cdot\ldots\cdot \tau_{G_\infty}^{0}(\beta)\cdot\tau_{G_\infty}^{0}(\beta)^{-1} \cdot\ldots\cdot \tau_{G_\infty}^{i-1}(\beta)^{-1} \cdot L^+_\infty\mathcal{G}(S')=L^+_\infty\mathcal{G}(S').$$ Therefore, over $\widetilde{S}$ the morphism $\varphi_i$ is an isomorphism at $\infty$. Since the $\Pi_i$ and $\Pi'_i$ are isomorphisms outside $\infty$, we conclude that with $\varphi_0$ also $\varphi_i$ and by [\[ProofPropBoundWithChains_A\]](#ProofPropBoundWithChains_A){reference-type="ref" reference="ProofPropBoundWithChains_A"} also $\varphi^\lhd_i$ are isomorphisms on $X_{\widetilde{S}}\smallsetminus\Gamma_x$ bounded by $\mathcal{Z}^{\leq\mu}$ at $\Gamma_x$. We now consider the tuple $(x,\varphi_i\colon \mathcal{E}_i \mathrel{ \mathpalette{\da@xarrow{x,\infty}{}{}\mathchar"0\hexnumber@\symAMSa 4B {\,}{}}{} } \mathcal{E}'_i)$ as an object in ${{}'\!\operatorname{Hecke}}_{\mathcal{G},\varnothing,X\times\infty}(S)$. Our considerations show that the composition $\widetilde{S}\to S\to {{}'\!\operatorname{Hecke}}_{\mathcal{G},\varnothing,X\times\infty}$ factors through the closed substack ${{}'\!\operatorname{Hecke}}_{\mathcal{G},\varnothing,X\times \infty}^{\mathcal{Z}(\mu,1)}$ from Theorem [Theorem 1](#Thm_HeckeBounded){reference-type="ref" reference="Thm_HeckeBounded"} for the bound $\mathcal{Z}(\mu,1)$. Since $\widetilde{S}$ is schematically dense in $S$, already the morphism $S\to {{}'\!\operatorname{Hecke}}_{\mathcal{G},\varnothing,X\times \infty}$ factors through ${{}'\!\operatorname{Hecke}}_{\mathcal{G},\varnothing,X\times \infty}^{\mathcal{Z}(\mu,1)}$. But the latter is isomorphic to $\operatorname{Hecke}_{\mathcal{G},\varnothing,X}^{\mathcal{Z}^{\leq\mu}}$ by Lemma [Lemma 1](#LemmaZmu1){reference-type="ref" reference="LemmaZmu1"}[\[LemmaZmu1_B\]](#LemmaZmu1_B){reference-type="ref" reference="LemmaZmu1_B"}. We conclude that $\varphi_i$ is bounded by $\mathcal{Z}^{\leq\mu}$. Thus diagram [\[EqPropBoundWithChains2\]](#EqPropBoundWithChains2){reference-type="eqref" reference="EqPropBoundWithChains2"} can be written in the form [\[EqPropBoundWithChains\]](#EqPropBoundWithChains){reference-type="eqref" reference="EqPropBoundWithChains"}, such that $\varphi^\lhd_i:=\Pi''_i\circ \varphi'_i\circ \varphi_i$ is bounded by $\mu$. ◻ **Corollary 1**. *The stack $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}$ is isomorphic to the stack over $\mathbb{F}_q$ whose points over $\mathbb{F}_q$-schemes $S$ are tuples $\bigl(x,(\mathcal{E}_i,\psi_i,\Pi_i,\varphi^\lhd_i)_{i\in\mathbb{Z}}\bigr)$ consisting of* - *one leg $x \colon S\to \widetilde{X}_{\mu,\beta}\to X$ which factors through $X\smallsetminus D$,* - *a commutative diagram $$\label{Eq_ShtukaDiagTildephi} \begin{tikzcd} \ldots & (\mathcal{E}_{-1},\psi_{-1})\arrow[dashed]{ld}[swap]{\varphi^\lhd_{-1}}\arrow[dashed]{l}[swap]{\Pi_{-1}}&(\mathcal{E}_0,\psi_0)\arrow[dashed]{l}[swap]{\Pi_0}\arrow[dashed]{ld}[swap]{\varphi^\lhd_0}&(\mathcal{E}_1,\psi_1)\arrow[dashed]{l}[swap]{\Pi_1}\arrow[dashed]{ld}[swap]{\varphi^\lhd_1}&\arrow[dashed]{l}[swap]{\Pi_2}\arrow[dashed]{ld}[swap]{\varphi^\lhd_2}\ldots\\ \ldots & {}^{\tau\!}(\mathcal{E}_{-1},\psi_{-1})\arrow[dashed]{l}{{}^{\tau\!}\Pi_{-1}}&{}^{\tau\!}(\mathcal{E}_0,\psi_0)\arrow[dashed]{l}{{}^{\tau\!}\Pi_0}&{}^{\tau\!}(\mathcal{E}_1,\psi_1)\arrow[dashed]{l}{{}^{\tau\!}\Pi_1}&\arrow[dashed]{l}{{}^{\tau\!}\Pi_2} \ldots \end{tikzcd}$$* *of $\mathcal{G}$-bundles with $D$-level structures $(\mathcal{E}_i,\psi_i)$ on $X_S$ where all the $\varphi^\lhd_i$ are isomorphisms outside the graph $\Gamma_x$ bounded by $\mu$, and the $\Pi_i$ are isomorphisms outside $\infty$ bounded by $\mathcal{Z}(\tau_{G_\infty}^{i-1}(\beta)^{-1})$.* *Proof.* Let $\underline{\mathcal{E}}=(x,(\mathcal{E}_0,\psi_0),(\mathcal{E}_0',\psi_0'),(\mathcal{E}_0'',\psi_0''),\varphi_0,\varphi_0',\varphi_0'')\in \operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)$ be a global $\mathcal{G}$-shtuka over $S$ bounded by $\mathcal{Z}(\mu,\beta)$, where $\varphi_0''\colon (\mathcal{E}_0'',\psi_0'') \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!}(\mathcal{E}_0,\psi_0)$ is the isomorphism, which we often supress from the notation. Under the isomorphism $\mathfrak{pr}_0$ from Proposition [Proposition 1](#PropBunWithChains){reference-type="ref" reference="PropBunWithChains"}, we can uniquely extend $(\mathcal{E}_0,\psi_0)$, $(\mathcal{E}'_0,\psi'_0)$, and $(\mathcal{E}''_0,\psi''_0)$ to chains of $\mathcal{G}$-bundles to obtain the rows of the following commutative diagram $$\label{EqCorShtWithChains} \begin{tikzcd} \ldots & (\mathcal{E}_{-1},\psi_{-1})\arrow[dotted]{d}{\varphi_{-1}}\arrow[dashed]{l}[swap]{\Pi_{-1}}&(\mathcal{E}_0,\psi_0)\arrow[dashed]{l}[swap]{\Pi_0}\arrow[dashed]{d}{\varphi_0}&(\mathcal{E}_1,\psi_1)\arrow[dashed]{l}[swap]{\Pi_1}\arrow[dotted]{d}{\varphi_1}&\arrow[dashed]{l}[swap]{\Pi_2}\ldots\\ \ldots & (\mathcal{E}'_{-1},\psi'_{-1})\arrow[dotted]{d}{\varphi'_{-1}}\arrow[dashed]{l}[swap]{\Pi'_{-1}}&(\mathcal{E}'_0,\psi'_0)\arrow[dashed]{l}[swap]{\Pi'_0}\arrow[dashed]{d}{\varphi'_0}&(\mathcal{E}'_1,\psi'_1)\arrow[dashed]{l}[swap]{\Pi'_1}\arrow[dotted]{d}{\varphi'_1}&\arrow[dashed]{l}[swap]{\Pi'_2}\ldots\\ \ldots & (\mathcal{E}''_{-1},\psi''_{-1})\arrow[dotted]{d}{\varphi''_{-1}} \arrow[dashed]{l}[swap]{\Pi''_{-1}}&(\mathcal{E}''_0,\psi''_0)\arrow[dashed]{l}[swap]{\Pi''_0}\arrow{d}{\varphi''_0}[swap]{\cong} &(\mathcal{E}''_1,\psi''_1)\arrow[dashed]{l}[swap]{\Pi''_1}\arrow[dotted]{d}{\varphi''_1}&\arrow[dashed]{l}[swap]{\Pi''_2}\ldots\\ \ldots & {}^{\tau\!}(\mathcal{E}_{-1},\psi_{-1}) \arrow[dashed]{l}[swap]{{}^{\tau\!}\Pi_{-1}}&{}^{\tau\!}(\mathcal{E}_0,\psi_0)\arrow[dashed]{l}[swap]{{}^{\tau\!}\Pi_0}&{}^{\tau\!}(\mathcal{E}_1,\psi_1)\arrow[dashed]{l}[swap]{{}^{\tau\!}\Pi_1}&\arrow[dashed]{l}[swap]{{}^{\tau\!}\Pi_2}\ldots \end{tikzcd}$$ with $\Pi_i$ and $\Pi'_i$ bounded by $\mathcal{Z}(\tau_{G_\infty}^{i-1}(\beta)^{-1})$, and $\Pi''_i$ bounded by $\mathcal{Z}(\tau_{G_\infty}^{i}(\beta)^{-1})$. In this diagram the dotted maps $\varphi_i$, $\varphi'_i$, and $\varphi''_i$ for $i\ne 0$ are the induced isomorphisms outside $\Gamma_x\cup\{\infty\}$. We first show that the isomorphism $\varphi_0''\colon(\mathcal{E}''_0,\psi''_0)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!}(\mathcal{E}_0,\psi_0)$ outside $\infty$ inductively induces isomorphisms $\varphi''_i:={}^{\tau\!}\Pi_{i}^{-1} \circ \varphi''_{i-1}\circ \Pi''_{i}\colon (\mathcal{E}''_i,\psi''_i)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!}(\mathcal{E}_i,\psi_i)$ on all of $X_S$ for all $i>0$ and similarly for $i<0$. After choosing trivializations of $L^+_\infty(\mathcal{E}_i)$ and $L^+_\infty(\mathcal{E}''_i)$ over an étale covering of $S$, the $\Pi_{i}$ are given by multiplication with $\tau_{G_\infty}^{i-1}(\beta)^{-1}$, and hence $\Pi''_{i}$ and ${}^{\tau\!}\Pi_{i}$ are given by multiplication with $\tau_{G_\infty}^{i}(\beta)^{-1}$. Therefore, the $\varphi''_i$ are given by multiplication with an element of $\tau_{G_\infty}^{i}(\beta)\cdot L^+_\infty\mathcal{G}\cdot \tau_{G_\infty}^{i}(\beta)^{-1}=L^+_\infty\mathcal{G}$ for $i>0$, respectively $\tau_{G_\infty}^{i+1}(\beta)^{-1}\cdot L^+_\infty\mathcal{G}\cdot \tau_{G_\infty}^{i+1}(\beta)=L^+_\infty\mathcal{G}$ for $i<0$. This proves that all $\varphi''_i$ are isomorphisms on all of $X_S$. Now we choose an étale covering $S'\to S$ and a trivialization $\epsilon\colon L^+_\Delta(\mathcal{E}''_0)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\Delta\mathcal{G})_{S'}$ for the divisor $\Delta:=\Gamma_x + (\infty\times_{\mathbb{F}_q}S')$. Then the boundedness of $\underline{\mathcal{E}}$ by $\mathcal{Z}(\mu,\beta)$ implies that we get a morphism $S'\to Z(\mu,\beta)$ for the representative $Z(\mu,\beta)$ of $\mathcal{Z}(\mu,\beta)$ over $\widetilde{X}_{\mu,\beta}$. By (the proof of) Proposition [Proposition 1](#PropBoundWithChains){reference-type="ref" reference="PropBoundWithChains"} the unique extension of $\underline{\mathcal{E}}$ to the diagram [\[EqCorShtWithChains\]](#EqCorShtWithChains){reference-type="eqref" reference="EqCorShtWithChains"} corresponds to the unique extension of the morphism $S'\to Z(\mu,\beta)$ to data over $S'$ as in diagram [\[EqPropBoundWithChains2\]](#EqPropBoundWithChains2){reference-type="eqref" reference="EqPropBoundWithChains2"}. There we saw for all $i\in\mathbb{Z}$, that $\Pi''_i\circ \varphi'_i$ is an isomorphism on all of $X$, and that $\Pi''_i\circ\varphi'_i\circ\varphi_i$ is an isomorphism outside $\Gamma_x$ bounded by $\mu$. We now take $\varphi^\lhd_i:=\varphi''_{i-1}\circ\Pi''_i\circ\varphi'_i\circ\varphi_i\colon (\mathcal{E}_i,\psi_i)|_{X_S\smallsetminus\Gamma_x} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!}(\mathcal{E}_{i-1},\psi_{i-1})|_{X_S\smallsetminus\Gamma_x}$, which is bounded by $\mu$. Descending back to $S$ finishes the proof. ◻ **Corollary 1**. *We use the interpretation of $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}$ in terms of chains as in Corollary [Corollary 1](#CorShtWithChains){reference-type="ref" reference="CorShtWithChains"}.* 1. *[\[CorIsog\[n\]\_A\]]{#CorIsog[n]_A label="CorIsog[n]_A"} There is an action of $\mathbb{Z}$ on the stack $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}$, which for an integer $n\in\mathbb{Z}$ is given by the index shift $$\begin{split} [n]\colon\qquad\quad \operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)} & \longrightarrow \operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)} \\ (x,\mathcal{E}_i,\psi_i,\Pi_i,\varphi^\lhd_i)_{i\in\mathbb{Z}} & \longmapsto (x,\mathcal{E}^{\mathrm{new}}_i,\psi^{\mathrm{new}}_i,\Pi^{\mathrm{new}}_i,\varphi^\lhd_i{}^{\mathrm{new}})_{i\in \mathbb{Z}} \end{split}$$ with $(\mathcal{E}^{\mathrm{new}}_i,\psi^{\mathrm{new}}_i,\Pi^{\mathrm{new}}_i,\varphi^\lhd_i{}^{\mathrm{new}}):=(\mathcal{E}_{i+n},\psi_{i+n},\Pi_{i+n},\varphi^\lhd_{i+n})$.* 2. *[\[CorIsog\[n\]\_B\]]{#CorIsog[n]_B label="CorIsog[n]_B"} For all integers $m,n$ and every global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}=(x,\mathcal{E}_i,\psi_i,\Pi_i,\varphi^\lhd_i)\in \operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)$ over a scheme $S$, there is a quasi-isogeny $$\Pi^n\colon [n+m](\underline{\mathcal{E}}) \to [m](\underline{\mathcal{E}}),$$ which is given by the isomorphism $\Pi_{i+1}\circ\ldots\circ \Pi_{i+n}\colon \mathcal{E}_{i+n}\to \mathcal{E}_i$ for $n\ge 0$ (respectively $\Pi_i^{-1}\circ\ldots\circ \Pi_{i+n+1}^{-1}\colon \mathcal{E}_{i+n}\to \mathcal{E}_i$ for $n<0$) on $(X\smallsetminus\{\infty\})_S$. 0◻* ## Local shtukas and Rapoport-Zink spaces {#subsec-LocSht-and-RZ} Let $v\in X$ be a place of $Q$ and let $q_v:=\#\mathbb{F}_v=q^{[\mathbb{F}_v:\mathbb{F}_q]}$. For a scheme $S\in\mathcal{N}ilp_{\mathcal{O}_v}$ let $\hat{\tau}_v:=\operatorname{Frob}_{q_v,S}$ be the $\mathbb{F}_v$-Frobenius of $S$ as defined in §[2.1](#subsec-notations){reference-type="ref" reference="subsec-notations"}. **Definition 1**. Let $S\in\mathcal{N}ilp_{\mathcal{O}_v}$. A *local $\mathcal{G}_v$-shtuka over $S\in \mathcal{N}ilp_{\mathcal{O}_v}$* is a pair $\underline{\mathcal{L}} = (\mathcal{L},\widehat{\varphi})$ consisting of an $L^+_v\mathcal{G}$-bundle $\mathcal{L}$ on $S$ and an isomorphism of the associated $L_v\mathcal{G}$-bundles $\widehat{\varphi}\colon L_v\mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\hat{\tau}_v\!} L_v\mathcal{L}$ from [\[EqLoopTorsorDelta\]](#EqLoopTorsorDelta){reference-type="eqref" reference="EqLoopTorsorDelta"} and Example [Example 1](#ExDivisors){reference-type="ref" reference="ExDivisors"}(b). We denote the stack fibered in groupoids over $\mathcal{N}ilp_{\mathcal{O}_v}$ which classifies local $\mathcal{G}_v$-shtukas by $\operatorname{LocSht}_{\mathcal{G}_v}$. **Remark 1**. Note that in the literature on local $\mathcal{G}$-shtukas usually an isomorphism ${}^{\hat{\tau}_v\!} L_v\mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_v\mathcal{L}$ is considered. Our $\widehat{\varphi}$ is the inverse of that isomorphism. **Remark 1**. By definition, an $L^+_v\mathcal{G}$-bundle $\mathcal{L}$ over a scheme $S$ can be trivialized over a suitable étale covering $S'\to S$ via an isomorphism $\hat\alpha\colon \mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_v\mathcal{G})_{S'}$. In this case we usually write $\underline{\mathcal{L}}\cong\bigl((L^+_v\mathcal{G})_{S'},b\bigr)$ meaning that the isomorphism ${}^{\hat{\tau}_v\!}\hat\alpha\circ\widehat{\varphi}\circ \hat\alpha^{-1}$ of the trivial $L_v\mathcal{G}$-bundle is given as multiplication on the left with $b\in L_v\mathcal{G}(S')$. Note that if $S$ is the spectrum of a strictly henselian local ring, then it has no non-trivial étale coverings and we can take $S'=S$. **Definition 1**. A *quasi-isogeny* $f\colon\underline{\mathcal{L}}\to\underline{\mathcal{L}}'$ between two local $\mathcal{G}_v$-shtukas $\underline{\mathcal{L}}=(\mathcal{L},\widehat{\varphi})$ and $\underline{\mathcal{L}}'=(\mathcal{L}' ,\widehat{\varphi}')$ over $S\in\mathcal{N}ilp_{\mathcal{O}_v}$ is an isomorphism of the associated $L_v\mathcal{G}$-bundles $f \colon L_v\mathcal{L}\to L_v\mathcal{L}'$ satisfying ${}^{\hat{\tau}_v\!}f\circ\widehat{\varphi}=\widehat{\varphi}'\circ f$. We denote by $\operatorname{QIsog}_S(\underline{\mathcal{L}},\underline{\mathcal{L}}')$ the set of quasi-isogenies between $\underline{\mathcal{L}}$ and $\underline{\mathcal{L}}'$ over $S$, and we write $\operatorname{QIsog}_S(\underline{\mathcal{L}}):=\operatorname{QIsog}_S(\underline{\mathcal{L}},\underline{\mathcal{L}})$ for the quasi-isogeny group of $\underline{\mathcal{L}}$. **Example 1**. Let $\underline{\mathcal{L}}=(\mathcal{L},\widehat{\varphi})$ be a local $\mathcal{G}$-shtuka over $S\in\mathcal{N}ilp_{\mathcal{O}_v}$. Then ${}^{\hat{\tau}_v\!}\underline{\mathcal{L}}=({}^{\hat{\tau}_v\!}\mathcal{L},{}^{\hat{\tau}_v\!}\widehat{\varphi})$ is a local $\mathcal{G}$-shtuka over $S$, too, and $f:=\widehat{\varphi}\colon L_v\mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_v{}^{\hat{\tau}_v\!}\mathcal{L}$ satisfies ${}^{\hat{\tau}_v\!}f \circ \widehat{\varphi}={}^{\hat{\tau}_v\!}\widehat{\varphi}\circ f$. Therefore, $\widehat{\varphi}\colon \underline{\mathcal{L}}\to {}^{\hat{\tau}_v\!}\underline{\mathcal{L}}$ is a quasi-isogeny, called the *$q_v$-Frobenius isogeny* of $\underline{\mathcal{L}}$. We recall the rigidity of quasi-isogenies of local $\mathcal{G}_v$-shtukas from [@AH_Local Proposition 2.11]. **Proposition 1** (Rigidity of quasi-isogenies for local $\mathcal{G}_v$-shtukas). *Let $S$ be a scheme in $\mathcal{N}ilp_{\mathcal{O}_v}$ and let $j \colon \bar{S}\to S$ be a closed immersion defined by a sheaf of ideals which is locally nilpotent. Let $\underline{\mathcal{L}}=(\mathcal{L},\widehat{\varphi})$ and $\underline{\mathcal{L}}'=(\mathcal{L}',\widehat{\varphi}')$ be two local $\mathcal{G}_v$-shtukas over $S$. Then $$\operatorname{QIsog}_S(\underline{\mathcal{L}}, \underline{\mathcal{L}}') \longrightarrow\operatorname{QIsog}_{\bar{S}}(j^*\underline{\mathcal{L}}, j^*\underline{\mathcal{L}}') ,\quad f \mapsto j^*f$$ is a bijection of sets.* *Proof.* Let $\mathcal{I}$ be the ideal sheaf defining $j\colon \bar{S}\hookrightarrow S$. Arguing by induction over $\mathcal{O}_S/\mathcal{I}^{q_v^n}$ it suffices to treat the case where $\mathcal{I}^{q_v}=(0)$. In this case the morphism $\hat\tau_v=\operatorname{Frob}_{q_v,S}$ factors as $S\xrightarrow{\;i}\bar{S}\xrightarrow{\;j}S$ where $i$ is the identity on the underlying topological space $|\bar{S}|=|S|$ and on the structure sheaf this factorization is given by $$\begin{aligned} \label{EqTauFactors} \textstyle\mathcal{O}_S \enspace \xrightarrow{\enspace j^\ast\;} & \mathcal{O}_{\bar{S}} & \textstyle\xrightarrow{\enspace i^\ast\;} \enspace \mathcal{O}_S\\ \textstyle b\quad \mapsto\;\: & b \operatorname{mod}\mathcal{I}& \textstyle\;\:\mapsto\quad b^{q_v}\,. \nonumber\end{aligned}$$ Therefore ${}^{\hat\tau_v\!}f=i^\ast(j^\ast f)$ for any $f\in\operatorname{QIsog}_S(\underline{\mathcal{L}},\underline{\mathcal{L}}')$. We obtain the diagram $$\label{EqRigidity} \xymatrix @C=5pc @R+0.5pc { L_\infty\mathcal{L}\ar[d]^\cong_{\textstyle\widehat{\varphi}} \ar[r]_\cong^{\textstyle f} & L_\infty\mathcal{L}' \ar[d]_\cong^{\textstyle\widehat{\varphi}'} \\ {}^{\hat\tau_v\!}L_\infty\mathcal{L}\ar[r]_\cong^{\textstyle i^\ast(j^\ast f)\enspace} & {}^{\hat\tau_v\!}L_\infty\mathcal{L}' }$$ from which the bijectivity is obvious. ◻ **Definition 1**. Consider the local type [\[DefThreeTypes_B\]](#DefThreeTypes_B){reference-type="ref" reference="DefThreeTypes_B"} at the place $v$ in Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"} and let $\mathcal{Z}$ be a bound in $\operatorname{Gr}_{\mathcal{G},X}\times_X \widetilde{X}_{\mathcal{Z}}$ with reflex scheme $\widetilde{X}_{\mathcal{Z}}$. Then $\widetilde{X}_{\mathcal{Z}}=\operatorname{Spec}\mathcal{O}_{\mathcal{Z}}$ for a finite ring extension $\mathcal{O}_{\mathcal{Z}}$ of $\mathcal{O}_v$. Let $Z$ be the representative of $\mathcal{Z}$ over $\mathcal{O}_\mathcal{Z}$. Recall from Corollary [Corollary 1](#CorGrGlobLoc){reference-type="ref" reference="CorGrGlobLoc"} that $\operatorname{Gr}_{\mathcal{G},X}\times_X \operatorname{Spf}\mathcal{O}_{\mathcal{Z}} \cong \mathcal{F}\!\ell_{\mathcal{G},v} \mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{\mathbb{F}_v} \operatorname{Spf}\mathcal{O}_{\mathcal{Z}}$. Let $S\in\mathcal{N}ilp_{\mathcal{O}_{\mathcal{Z}}}$. A local $\mathcal{G}_v$-shtuka $\underline{\mathcal{L}}:=(\mathcal{L},\widehat{\varphi})$ is *bounded by $\mathcal{Z}$* if for every (some) étale covering $S'\to S$ and every (some) trivialization $\hat\alpha\colon \mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_v\mathcal{G})_{S'}$ the element $b={}^{\hat{\tau}_v\!}\hat\alpha\circ\widehat{\varphi}\circ \hat\alpha^{-1}\in L_v\mathcal{G}(S')$ factors through $Z\subset \mathcal{F}\!\ell_{\mathcal{G},v}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{\mathbb{F}_v}\operatorname{Spf}\mathcal{O}_{\mathcal{Z}}$ when viewed as a morphism $b\colon S'\to \mathcal{F}\!\ell_{\mathcal{G},v}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{\mathbb{F}_v}\operatorname{Spf}\mathcal{O}_{\mathcal{Z}}$. Note that the equivalences of "every" and "some" was proven in [@AH_Local Remark 4.9]. We write $\operatorname{LocSht}_{\mathcal{G}_v}^{\mathcal{Z}}$ for the stack of local $\mathcal{G}_v$-shtukas bounded by $\mathcal{Z}$. When $\mathcal{Z}=\mathcal{Z}^{\leq\mu}\times_{\widetilde{X}_\mu} \operatorname{Spf}\mathcal{O}_\mu$ for the bound $\mathcal{Z}^{\leq\mu}$ from Definition [Definition 1](#Def_BoundBy_mu){reference-type="ref" reference="Def_BoundBy_mu"}, we write $\operatorname{LocSht}_{\mathcal{G}_v}^{\leq \mu}$. **Remark 1**. We continue with Remark [Remark 1](#Rem_LocShtInversePhi){reference-type="ref" reference="Rem_LocShtInversePhi"}. Let $\underline{\mathcal{L}}=(\mathcal{L},\widehat{\varphi})$ be a local $\mathcal{G}_v$-shtuka as in our Definition [Definition 1](#Def_LocSht){reference-type="ref" reference="Def_LocSht"}. In the literature on local $\mathcal{G}$-shtukas like [@AH_Local; @AH_Unif; @HV1; @HV2; @HartlViehmann3] where the Frobenius of a local $\mathcal{G}_v$-shtuka is an isomorphism ${}^{\hat{\tau}_v\!} L_v\mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_v\mathcal{L}$ one would have to consider $(\mathcal{L},\widehat{\varphi}^{-1})$ as the local $\mathcal{G}_v$-shtuka. In the language of that literature one says that "$(\mathcal{L},\widehat{\varphi}^{-1})$ is bounded by a bound $\widetilde{\mathcal{Z}}$" if $\widehat{\varphi}^{-1}$ is bounded by $\widetilde{\mathcal{Z}}$. This coincides with our definition that "$(\mathcal{L},\widehat{\varphi})$ is bounded by $\mathcal{Z}$" if one takes $\widetilde{\mathcal{Z}}=\mathcal{Z}^{-1}$ in the sense of [@HartlViehmann3 Remark 2.3 and Lemma 2.12]. **Definition 1**. Let $S\in\mathcal{N}ilp_{\Breve{\mathcal{O}}_\infty}$. The *global-local functor* $$\label{Eq_DefGlobLocG} L^+_{\infty,\mathcal{G}}\colon \operatorname{Sht}_{\mathcal{G},D,X\times\infty}(S) \longrightarrow\operatorname{LocSht}_{\mathcal{G}_\infty}(S)$$ is defined as follows. Let $\underline{\mathcal{E}}=\bigl(x,\,(\mathcal{E},\psi,)(\mathcal{E}',\psi'),\varphi,\varphi'\bigr)\in \operatorname{Sht}_{\mathcal{G},D,X\times\infty}(S)$ be a global $\mathcal{G}$-shtuka over $S$ as in Definition [Definition 1](#Def_Sht2legs){reference-type="ref" reference="Def_Sht2legs"}. Then $$\label{Eq_DefGlobLocG2} L^+_{\infty,\mathcal{G}}(\underline{\mathcal{E}}) := \bigl(L^+_{\infty,\mathcal{G}}(\mathcal{E}), L_{\infty,\mathcal{G}}(\varphi'\circ\varphi)\bigr).$$ **Remark 1**. Note that in the previous definition $$\varphi'\circ\varphi\colon \mathcal{E}|_{(X\smallsetminus\{\infty\})_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!}\mathcal{E}|_{(X\smallsetminus\{\infty\})_S}$$ is an isomorphism of $\mathcal{G}$-bundles, because $S\in\mathcal{N}ilp_{\Breve{\mathcal{O}}_\infty}$ implies that $X_S\smallsetminus\Gamma_x = (X\smallsetminus\{\infty\})_S$. So $L^+_{\infty,\mathcal{G}}(\underline{\mathcal{E}})$ recovers the combined modification of $\mathcal{E}$ at the two legs $x$ and $\infty$. ** 1**. Let $\beta\in L_\infty\mathcal{G}(\mathbb{F}_\beta)$ as in § [ 1](#Def_beta){reference-type="ref" reference="Def_beta"}. Recall the Frobenius $\tau_{G_\infty}\colon L_\infty\mathcal{G}(S)\to L_\infty\mathcal{G}(S)$ from [\[EqTau_G\]](#EqTau_G){reference-type="eqref" reference="EqTau_G"}. Let $M:=M_{\beta^{-1}}$ be the inner form over $Q_\infty$ of $G_\infty$ given by $\beta^{-1}$, i.e. there is an isomorphism of linear algebraic groups over $\Breve{Q}_\infty$ $$\iota\colon G_\infty \times_{Q_\infty} \Breve{Q}_\infty \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}M_{\beta^{-1}} \times_{Q_\infty} \Breve{Q}_\infty$$ with $Q_\infty$-structure on $M_{\beta^{-1}}$ given by the Frobenius $$\label{Eq_tau_M} \tau_M(g)=\iota\bigl(\beta^{-1} \cdot \tau_{G_\infty}(\iota^{-1}(g)) \cdot \beta\bigr)=\iota\bigl(\beta^{-1} \cdot {}^{\tau\!}\iota^{-1}(\tau_M(g)) \cdot \beta\bigr)$$ i.e. given by $\operatorname{id}=\iota\circ\operatorname{int}_{\beta^{-1}}\circ {}^{\tau\!} \iota^{-1}$ and ${}^{\tau\!} \iota=\iota\circ \operatorname{int}_{\beta^{-1}}$. Since $\beta^{-1}\cdot \mathcal{G}_\infty\cdot\beta = \mathcal{G}_\infty$, there exists a smooth affine group scheme $\mathcal{M}=\mathcal{M}_{\beta^{-1}}$ over $\mathcal{O}_\infty$ such that $\iota$ restricts to an isomorphism of algebraic groups over $\Breve{\mathcal{O}}_\infty$ $$\label{Eq_mcM} \iota\colon \mathcal{G}_\infty \times_{\mathcal{O}_\infty} \Breve{\mathcal{O}}_\infty \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{M}_{\beta^{-1}} \times_{\mathcal{O}_\infty} \Breve{\mathcal{O}}_\infty\,.$$ We use $\iota$ to identify $L^+_\infty\mathcal{G}(R)=\mathcal{G}_\infty(R{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z {\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}) =\mathcal{M}(R{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}z {\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}})=L^+_\infty\mathcal{M}(R)$ for $\Breve{\mathcal{O}}_\infty$-algebras $R$. The bound $\mathcal{Z}^{\leq\mu}\times_{\widetilde{X}_\mu} \operatorname{Spf}\mathcal{O}_\mu$ from Definition [Definition 1](#DefLocShtBounded){reference-type="ref" reference="DefLocShtBounded"} induces a bound in $\mathcal{F}\!\ell_{\mathcal{M},\infty}$ which has a representative over ${\mathcal{O}_{\mu,\beta}}$, and which we will use to bound local $\mathcal{M}$-shtukas in Definition [Definition 1](#DefRZforM){reference-type="ref" reference="DefRZforM"}. That local bound depends on the choice of the map $\operatorname{Spec}\mathcal{O}_{\mu,\beta}\to\widetilde{X}_{\mu,\beta}$. **Proposition 1**. *For any $S\in\mathcal{N}ilp_{\Breve{\mathcal{O}}_\infty}$ and any $\beta\in L_\infty\mathcal{G}(\mathbb{F}_\beta)$ as in § [ 1](#Def_beta){reference-type="ref" reference="Def_beta"}, there is an equivalence of categories given by left translation by $\beta^{-1}$ $$\label{EqEquivG-M-Shtukas} t_{\beta^{-1}}\colon \operatorname{LocSht}_{\mathcal{G}_\infty}(S) \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\operatorname{LocSht}_{\mathcal{M}_{\beta^{-1}}}(S),$$ such that the underlying $L^+_\infty\mathcal{G}$-bundle $\mathcal{L}$ of a local $\mathcal{G}_\infty$-shtuka $\underline{\mathcal{L}}=(\mathcal{L},\widehat{\varphi})$ coincides with the underlying $L^+_\infty\mathcal{M}_{\beta^{-1}}$-bundle of $t_{\beta^{-1}}(\underline{\mathcal{L}})$. In terms of trivialized local shtukas, the functor $t_{\beta^{-1}}$ is given by $$\bigl((L^+_\infty\mathcal{G})_S, b\bigr) \longmapsto \bigl((L^+_\infty\mathcal{M}_{\beta^{-1}})_S, \beta^{-1} b\bigr)$$ for $b\in L_\infty\mathcal{G}(S)=L_\infty\mathcal{M}_{\beta^{-1}}(S)$. The functor $t_{\beta^{-1}}$ also sends quasi-isogenies to quasi-isogenies.* *Proof.* Let $\underline{\mathcal{L}}:=(\mathcal{L},\widehat{\varphi})$ be a local $\mathcal{G}_\infty$-shtuka over $S$. Let $S'\to S$ be an étale covering over which a trivialization $\alpha\colon \underline{\mathcal{L}}_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\bigl((L^+_\infty\mathcal{G})_{S'}, b'\bigr)$ exists with $b':={}^{\tau\!}\alpha\circ\widehat{\varphi}\circ\alpha^{-1}\in L_\infty\mathcal{G}(S')$. Over $S''=S'\times_S S'$, we obtain the descent datum $h:=\operatorname{pr}_2^*\alpha\circ \operatorname{pr}_1^* \alpha^{-1}\in L^+_\infty\mathcal{G}(S'')$ for $\mathcal{L}$, where $\operatorname{pr}_j\colon S''\to S'$ is the projection onto the $j$-th factor for $j=1,2$. Then ${}^{\tau\!}\mathcal{L}$ is trivialized over $S'$ by ${}^{\tau\!}\alpha$ with descent datum $\tau_{G_\infty}(h)=\operatorname{pr}_2^*({}^{\tau\!}\alpha)\circ \operatorname{pr}_1^*({}^{\tau\!}\alpha)^{-1}\in L^+_\infty\mathcal{G}(S'')$. The fact that $\widehat{\varphi}$ is defined over $S$ is equivalent to $\operatorname{pr}_1^*\widehat{\varphi}=\operatorname{pr}_2^*\widehat{\varphi}$, which in turn is equivalent to the equation $$\label{EqPropGvsM_1} \tau_{G_\infty}(h)\cdot\operatorname{pr}_1^*b'=\operatorname{pr}_2^*({}^{\tau\!}\alpha)\circ \operatorname{pr}_1^*\widehat{\varphi}\circ \operatorname{pr}_1^*\alpha^{-1}=\operatorname{pr}_2^*({}^{\tau\!}\alpha)\circ \operatorname{pr}_2^*\widehat{\varphi}\circ \operatorname{pr}_1^*\alpha^{-1}=\operatorname{pr}_2^*(b') \cdot h.$$ We now view $\mathcal{L}$ as an $L^+_\infty\mathcal{M}$-bundle via the isomorphism $\iota$ from [\[Eq_mcM\]](#Eq_mcM){reference-type="eqref" reference="Eq_mcM"}. Over $S'$, we obtain the trivialization $\iota\circ\alpha\colon \mathcal{L}_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{M})_{S'}$, which gives rise to the descent datum $\iota(h)=\iota\circ\operatorname{pr}_2^*\alpha\circ \operatorname{pr}_1^* \alpha^{-1}\circ\iota^{-1}\in L^+_\infty\mathcal{M}(S'')$. However, ${}^{\tau\!}\mathcal{L}$ is now trivialized over $S'$ by ${}^{\tau\!}(\iota\circ\alpha)\colon {}^{\tau\!}\mathcal{L}_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{M})_{S'}$ and has the descent datum $\operatorname{pr}_2^*{}^{\tau\!}(\iota\circ\alpha)\circ \operatorname{pr}_1^*{}^{\tau\!}(\iota\circ\alpha)^{-1}={}^{\tau\!}\iota \circ \operatorname{pr}_2^*({}^{\tau\!}\alpha)\circ \operatorname{pr}_1^*({}^{\tau\!}\alpha)^{-1} \circ {}^{\tau\!}\iota^{-1}$ which sends the neutral element $1\in L^+_\infty\mathcal{M}(S'')$ to ${}^{\tau\!}\iota (\tau_{G_\infty}(h))=\iota\bigl(\beta^{-1}\cdot \tau_{G_\infty}(h) \cdot\beta\bigr)=\tau_M(\iota(h))$ by definition of $\tau_M$. We equip $\mathcal{L}_{S'}$ with the new Frobenius $\widehat{\varphi}':={}^{\tau\!}(\iota\circ\alpha)^{-1}\circ \iota(\beta^{-1} b')\circ (\iota\circ\alpha)\colon L_\infty\mathcal{L}_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!} L_\infty\mathcal{L}_{S'}$ such that $\iota\circ\alpha\colon (\mathcal{L}_{S'},\widehat{\varphi}')\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\bigl((L^+_\infty\mathcal{M})_{S'},\iota(\beta^{-1}b')\bigr)$ is an isomorphism of local $\mathcal{M}$-shtukas over $S'$. From [\[EqPropGvsM_1\]](#EqPropGvsM_1){reference-type="eqref" reference="EqPropGvsM_1"}, we obtain the middle equality in the equation $$\tau_M(\iota(h))\cdot \operatorname{pr}_1^*\iota(\beta^{-1}b') = \iota\bigl(\beta^{-1}\tau_{G_\infty}(h)\beta\cdot \beta^{-1}\cdot \operatorname{pr}_1^*b'\bigr) = \iota\bigl(\beta^{-1}\cdot \operatorname{pr}_2^*b'\cdot h\bigr) = \operatorname{pr}_2^*\iota(\beta^{-1}b')\cdot \iota(h),$$ where the first (resp. last) equality follows because $\iota$ commutes with $\operatorname{pr}_1^*$ (resp. $\operatorname{pr}_2^*$). Replacing $\tau_{G_\infty}$ by $\tau_M\circ\iota$ and $b'$ by $\iota(\beta^{-1}b')$ in [\[EqPropGvsM_1\]](#EqPropGvsM_1){reference-type="eqref" reference="EqPropGvsM_1"}, this implies $\operatorname{pr}_1^*\widehat{\varphi}'=\operatorname{pr}_2^*\widehat{\varphi}'$, and thus $\widehat{\varphi}'$ descends to an isomorphism $L_\infty\mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!} L_\infty\mathcal{L}$ over $S$. This defines the local $\mathcal{M}$-shtuka $t_{\beta^{-1}}(\underline{\mathcal{L}}):=(\mathcal{L},\widehat{\varphi}')$ over $S$ and the functor $t_{\beta^{-1}}$. To visualize the construction, we have the diagram $$\begin{tikzcd} (\mathcal{L},\widehat{\varphi})_{S'}=\underline{\mathcal{L}}_{S'}\arrow[]{r}{\alpha}[swap]{\sim}\arrow[mapsto]{d}[swap]{t_{\beta^{-1}}}& ((L^+_\infty\mathcal{G})_{S'},b')\arrow[]{r}{\iota}[swap]{\sim}&((L^+_\infty\mathcal{M}_{\beta^{-1}})_{S'},\iota(b'))\arrow[mapsto]{d}{t_{\beta^{-1}}}\\ (\mathcal{L},\widehat{\varphi}')_{S'}=t_{\beta^{-1}}(\underline{\mathcal{L}})_{S'}\arrow[]{rr}{\iota\circ\alpha}[swap]{\sim}&&((L^+_\infty\mathcal{M}_{\beta^{-1}})_{S'},\iota(\beta^{-1}\cdot b')) \end{tikzcd}$$ We must show that $t_{\beta^{-1}}$ is independent of the choices of $S'$ and $\alpha$. If we choose a different $\widetilde{S}'$, we may as well replace it with a common refinement with the previous $S'$ and assume $\widetilde{S}'=S'$. Then the new $\widetilde{\alpha}$ differs from the previous $\alpha$ by left multiplication with an element $g:=\widetilde{\alpha}\circ \alpha^{-1}\in L^+_\infty\mathcal{G}(S')$. This gives the new descent datum $\tilde h:=\operatorname{pr}_2^*\widetilde{\alpha} \circ \operatorname{pr}_1^*\widetilde{\alpha}^{-1}=\operatorname{pr}_2^*g\cdot h\cdot \operatorname{pr}_1^*g^{-1}$ of the $L^+_\infty\mathcal{G}$-bundle $\mathcal{L}$ and changes $b'$ to $\tilde b'=\tau_{G_\infty}(g)\cdot b' \cdot g^{-1}$. It also changes the trivialization of the $L^+_\infty\mathcal{M}$-bundle $\mathcal{L}$ to $\iota\circ\widetilde{\alpha}=\iota(g)\cdot \iota\circ\alpha$ and $\iota(\beta^{-1}b')$ to $\iota(\beta^{-1}\tilde b')=\iota(\beta^{-1}\tau_{G_\infty}(g)\beta\cdot\beta^{-1} b' g^{-1})=\tau_M(\iota(g))\cdot \iota(\beta^{-1} b')\cdot \iota(g)^{-1}$. Therefore, $t_{\beta^{-1}}(g):=\iota(g)\colon \bigl((L^+_\infty\mathcal{M})_{S'}, \iota(\beta^{-1} b')\bigr)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\bigl((L^+_\infty\mathcal{M})_{S'}, \iota(\beta^{-1}\tilde b')\bigr)$ is an isomorphism over $S'$. This shows that the change of the trivialization from $\alpha$ to $\widetilde{\alpha}$ does not change the $L^+_\infty\mathcal{M}$-bundle $\mathcal{L}$ and its new Frobenius $\widehat{\varphi}'$. Thus the functor $t_{\beta^{-1}}$ is well defined. Clearly $t_{\beta^{-1}}$ is an equivalence, as its inverse functor is given by $t_{\iota(\beta)}$. Let $f:\underline{\mathcal{L}}\to \underline{\widetilde{\mathcal{L}}}$ be a quasi-isogeny. We have trivializations over a suitable étale covering $S'$ of $S$ $$\alpha: \underline{\mathcal{L}}_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}((L^+_\infty\mathcal{G})_{S'},b')\quad\text{and}\quad \widetilde{\alpha}: \underline{\widetilde{\mathcal{L}}}_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}((L^+_\infty\mathcal{G})_{S'},\widetilde{b}').$$ Let $\widetilde{\alpha}\circ f\circ\alpha^{-1}=:g\in L_\infty\mathcal{G}(S')$. The same computations as in the preceeding paragraph with $g\in L_\infty\mathcal{G}(S')$ instead of $L^+_\infty\mathcal{G}(S')$ show that $t_{\beta^{-1}}(f)$ is a quasi-isogeny $t_{\beta^{-1}}(\underline{\mathcal{L}})\to t_{\beta^{-1}}(\underline{\widetilde{\mathcal{L}}})$. Thus the functor $t_{\beta^{-1}}$ is compatible with quasi-isogenies. ◻ Let $\mathcal{Z}(\mu,\beta)$ be the bound in $\operatorname{Gr}_{\mathcal{G},X\times\infty}$ from Definition [Definition 1](#DefZmubeta){reference-type="ref" reference="DefZmubeta"}. The following is a variant of the *global-local functor* from [\[EqL\^+\_v\]](#EqL^+_v){reference-type="eqref" reference="EqL^+_v"} and [\[EqL_v\]](#EqL_v){reference-type="eqref" reference="EqL_v"}. It is the appropriate global-local functor for global shtukas in $\operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}$ when the two legs collide. **Definition 1**. Let $S\in\mathcal{N}ilp_{\Breve{\mathcal{O}}_\infty}$. The *$\beta^{-1}$-twisted global-local functor* $$\label{Eq_DefGlobLocM} L^+_{\infty,\mathcal{M}_{\beta^{-1}}}:=t_{\beta^{-1}}\circ L^+_{\infty,\mathcal{G}}\colon \operatorname{Sht}_{\mathcal{G},D,X\times\infty}(S) \longrightarrow\operatorname{LocSht}_{\mathcal{M}_{\beta^{-1}}}(S)$$ is defined as follows. Let $\underline{\mathcal{E}}=\bigl(x,\,(\mathcal{E},\psi),(\mathcal{E}',\psi'),\,\varphi,\varphi'\,\bigr)\in \operatorname{Sht}_{\mathcal{G},D,X\times\infty}(S)$ be a global $\mathcal{G}$-shtuka over $S$ as in Definition [Definition 1](#Def_Sht2legs){reference-type="ref" reference="Def_Sht2legs"}. Then $$\label{Eq_DefGlobLocM2} L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}}) := t_{\beta^{-1}}\bigl(L^+_{\infty,\mathcal{G}}(\mathcal{E}), L_{\infty,\mathcal{G}}(\varphi'\circ\varphi)\bigr).$$ Note that when $\beta=1$, the $\beta^{-1}$-twisted global-local functor $L_{\infty,\mathcal{M}_{\beta^{-1}}}^+=L_{\infty,\mathcal{G}}^+$ recovers the (usual) global-local functor given in [\[EqL\^+\_v\]](#EqL^+_v){reference-type="eqref" reference="EqL^+_v"} and [\[EqL_v\]](#EqL_v){reference-type="eqref" reference="EqL_v"}. **Corollary 1**. *Let $\underline{\mathcal{E}}\in \operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)$ be given in terms of Corollary [Corollary 1](#CorShtWithChains){reference-type="ref" reference="CorShtWithChains"} as the tuple $\underline{\mathcal{E}}=\bigl(x,(\mathcal{E}_i,\psi_i,\Pi_i,\varphi^\lhd_i)_{i\in\mathbb{Z}}\bigr)$. Then $$\label{Eq_CorGlobalLocalFunctorWithChains} L^+_{\infty,\mathcal{M}_{\beta^{-1}}}({}^{\tau\!}\mathcal{E}_{-1})\cong {}^{\tau\!} L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\mathcal{E}_0)$$ and the $\beta^{-1}$-twisted global-local functor $L^+_{\infty,\mathcal{M}_{\beta^{-1}}}$ sends $\underline{\mathcal{E}}$ to $$\label{Eq_CorGlobalLocalFunctorWithChains2} L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})=\bigl(L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\mathcal{E}_0), L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\varphi^\lhd_0)\bigr).$$ In particular, the local $\mathcal{M}$-shtuka $L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})$ is bounded by $\mu$.* *Proof.* To prove [\[Eq_CorGlobalLocalFunctorWithChains\]](#Eq_CorGlobalLocalFunctorWithChains){reference-type="eqref" reference="Eq_CorGlobalLocalFunctorWithChains"}, let $S'\to S$ be an étale covering over which trivializations $\alpha_i\colon L^+_{\infty,\mathcal{G}}(\mathcal{E}_i)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{G})_{S'}$ for $i=-1,0$ exist. After base change under $\tau=\operatorname{Frob}_{q,S'}$ this yields trivializations ${}^{\tau\!}\alpha_i\colon L^+_{\infty,\mathcal{G}}({}^{\tau\!}\mathcal{E}_i)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{G})_{S'}$. We now view these $L^+_\infty\mathcal{G}$-bundles as $L^+_\infty\mathcal{M}$-bundles via the isomorphism $\iota$ from [\[Eq_mcM\]](#Eq_mcM){reference-type="eqref" reference="Eq_mcM"}. As such they are trivialized by $\iota\circ\alpha_i\colon L^+_{\infty,\mathcal{M}}(\mathcal{E}_i)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{M})_{S'}$ and $\iota\circ{}^{\tau\!}\alpha_i\colon L^+_{\infty,\mathcal{M}}({}^{\tau\!}\mathcal{E}_i)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{M})_{S'}$ and ${}^{\tau\!}(\iota\circ\alpha_i)\colon {}^{\tau\!}L^+_{\infty,\mathcal{M}}(\mathcal{E}_i)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(L^+_\infty\mathcal{M})_{S'}$. Over $S''=S'\times_S S'$ we obtain the descent data $h_i:=\operatorname{pr}_2^*\alpha_i \circ \operatorname{pr}_1^*\alpha_i^{-1}\in L^+_\infty\mathcal{G}(S'')$ for $L^+_{\infty,\mathcal{G}}(\mathcal{E}_i)$ and $\tau_{G_\infty}(h_i)=\operatorname{pr}_2^*({}^{\tau\!}\alpha_i) \circ \operatorname{pr}_1^*({}^{\tau\!}\alpha_i)^{-1}\in L^+_\infty\mathcal{G}(S'')$ for $L^+_{\infty,\mathcal{G}}({}^{\tau\!}\mathcal{E}_i)$, as well as $\iota(h_i)=\iota\circ\operatorname{pr}_2^*\alpha_i \circ \operatorname{pr}_1^*\alpha_i^{-1}\circ\iota^{-1}\in L^+_\mathcal{M}(S'')$ for $L^+_{\infty,\mathcal{M}}(\mathcal{E}_i)$ and $\iota(\tau_{G_\infty}(h_i))=\iota\circ\operatorname{pr}_2^*({}^{\tau\!}\alpha_i) \circ \operatorname{pr}_1^*({}^{\tau\!}\alpha_i)^{-1}\circ\iota^{-1}\in L^+_\infty\mathcal{M}(S'')$ for $L^+_{\infty,\mathcal{M}}({}^{\tau\!}\mathcal{E}_i)$ and $\tau_M(\iota(h_i))=\operatorname{pr}_2^*{}^{\tau\!}(\iota\circ\alpha_i) \circ \operatorname{pr}_1^*{}^{\tau\!}(\iota\circ\alpha_i)^{-1}\in L^+_\infty\mathcal{M}(S'')$ for ${}^{\tau\!}L^+_{\infty,\mathcal{M}}(\mathcal{E}_i)$; see the proof of Proposition [Proposition 1](#PropGvsM){reference-type="ref" reference="PropGvsM"}. Now by construction of $\mathcal{E}_{-1}$ from $\mathcal{E}_0$ in Proposition [Proposition 1](#PropBunWithChains){reference-type="ref" reference="PropBunWithChains"}, we have $h_{-1}=\tau_{G_\infty}^{-1}(\beta)^{-1}\cdot h_0\cdot \tau_{G_\infty}^{-1}(\beta)$. This implies $\tau_M(\iota(h_0))=\iota(\beta^{-1}\cdot\tau_{G_\infty}(h_0)\cdot\beta)=\iota(\tau_{G_\infty}(\tau_{G_\infty}^{-1}(\beta)^{-1}\cdot h_0\cdot \tau_{G_\infty}^{-1}(\beta)))=\iota(\tau_{G_\infty}(h_{-1}))$, and hence [\[Eq_CorGlobalLocalFunctorWithChains\]](#Eq_CorGlobalLocalFunctorWithChains){reference-type="eqref" reference="Eq_CorGlobalLocalFunctorWithChains"} follows. To prove [\[Eq_CorGlobalLocalFunctorWithChains2\]](#Eq_CorGlobalLocalFunctorWithChains2){reference-type="eqref" reference="Eq_CorGlobalLocalFunctorWithChains2"}, we recall from the construction of $t_{\beta^{-1}}$ in the proof of Proposition [Proposition 1](#PropGvsM){reference-type="ref" reference="PropGvsM"}, that over $S'$ the Frobenius of $L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})_{S'}$ is given by the left hand side in the following equation $$\iota\bigl(\beta^{-1}\cdot {}^{\tau\!}\alpha_0\circ L_{\infty,\mathcal{G}}(\varphi'_0\circ\varphi_0)\circ \alpha_0^{-1}\bigr)\;=\;\iota\bigl({}^{\tau\!}\alpha_1\circ L_{\infty,\mathcal{G}}({}^{\tau\!}\Pi_0\circ\varphi'_0\circ\varphi_0)\circ \alpha_0^{-1}\bigr).$$ This equation comes from the construction of $L_{\infty,\mathcal{G}}(\Pi_0):=\alpha_{-1}^{-1}\circ \tau_{G_\infty}^{-1}(\beta)^{-1}\circ\alpha_0$ in [\[EqDefPi_i\]](#EqDefPi_i){reference-type="eqref" reference="EqDefPi_i"} in Proposition [Proposition 1](#PropBunWithChains){reference-type="ref" reference="PropBunWithChains"}, which implies $\beta^{-1}\cdot {}^{\tau\!}\alpha_0={}^{\tau\!}\alpha_1\circ L_{\infty,\mathcal{G}}({}^{\tau\!}\Pi_0)$. Now the claim follows from the definition of $\varphi^\lhd_0:={}^{\tau\!}\Pi_0\circ\varphi'_0 \circ \varphi_0$ in the proof of Corollary [Corollary 1](#CorShtWithChains){reference-type="ref" reference="CorShtWithChains"}. ◻ The following is an analogue of [@HartlAbSh Proposition 8.1]. **Proposition 1**. *Let $S\in \mathcal{N}ilp_{\Breve{\mathcal{O}}_\infty}$.* 1. *[\[PropQIsogLocalGlobal_A\]]{#PropQIsogLocalGlobal_A label="PropQIsogLocalGlobal_A"} Given global $\mathcal{G}$-shtukas $\underline{\mathcal{E}},\underline{\widetilde{\mathcal{E}}}\in \operatorname{Sht}_{\mathcal{G},D,X\times\infty}(S)$. Any quasi-isogeny $\delta:\underline{\widetilde{\mathcal{E}}}\to \underline{\mathcal{E}}$ induces a quasi-isogeny $$\hat{\delta}:=L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta)\colon L_{\mathcal{M},\infty }(\underline{\widetilde{\mathcal{E}}})\to L_{\mathcal{M},\infty }(\underline{\mathcal{E}})$$ of the corresponding local $\mathcal{M}$-shtukas.* 2. *[\[PropQIsogLocalGlobal_B\]]{#PropQIsogLocalGlobal_B label="PropQIsogLocalGlobal_B"} Conversely, fix a global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}\in \operatorname{Sht}_{\mathcal{G},D,X\times\infty}(S)$. Any quasi-isogeny $\hat{\delta}\colon \underline{\widetilde{\mathcal{L}}}\to L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})$ of local $\mathcal{M}$-shtukas over $S$ comes from a unique (up to a canonical isomorphism) global $\mathcal{G}$-shtuka $\underline{\widetilde{\mathcal{E}}}$ and a quasi-isogeny $\delta: \underline{\widetilde{\mathcal{E}}}\to\underline{\mathcal{E}}$ which is an isomorphism outside $\infty$ such that $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\widetilde{\mathcal{E}}})=\underline{\widetilde{\mathcal{L}}}$ and $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta) = \hat{\delta}$. In this setting, we will write $\hat{\delta}^*\underline{\mathcal{E}}$ to denote $\underline{\widetilde{\mathcal{E}}}$.* *Proof.* [\[PropQIsogLocalGlobal_A\]](#PropQIsogLocalGlobal_A){reference-type="ref" reference="PropQIsogLocalGlobal_A"} A quasi-isogeny $\delta\colon \underline{\widetilde{\mathcal{E}}}=(x,(\widetilde{\mathcal{E}},\widetilde{\psi}),(\widetilde{\mathcal{E}}',\widetilde{\psi}'),\widetilde{\varphi},\widetilde{\varphi}')\to \underline{\mathcal{E}}=(x,(\mathcal{E},\psi),(\mathcal{E}',\psi'),\varphi,\varphi')$ is given by two isomorphisms $f\colon (\widetilde{\mathcal{E}},\widetilde{\psi})|_{X_S\smallsetminus N_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\mathcal{E},\psi)|_{X_S\smallsetminus N_S}$ and $f'\colon (\widetilde{\mathcal{E}}',\widetilde{\psi}')|_{X_S\smallsetminus N_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}(\mathcal{E}',\psi')|_{X_S\smallsetminus N_S}$ for some proper closed subscheme $N\subset X$, such that $f'\circ\widetilde{\varphi}=\varphi\circ f$ and ${}^{\tau\!} f\circ\widetilde{\varphi}'=\varphi'\circ f'$. Then $\hat{\delta}:= L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(f)$ satisfies $\tau_M(\hat{\delta})\cdot\bigl(\beta^{-1} \circ L^+_{\infty,\mathcal{G}}(\widetilde{\varphi}'\circ\widetilde{\varphi})\bigr)=\beta^{-1}\cdot\tau_{G_\infty}(\hat{\delta})\circ L^+_{\infty,\mathcal{G}}(\widetilde{\varphi}'\circ\widetilde{\varphi})=\bigl(\beta^{-1}\cdot L^+_{\infty,\mathcal{G}}(\varphi'\circ\varphi)\bigr)\circ \hat{\delta}$, and hence is a quasi-isogeny $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\widetilde{\mathcal{E}}})\to L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})$. (Here, strictly speaking, the equality $\tau_M(\hat{\delta})\cdot\beta^{-1}=\beta^{-1}\cdot\tau_{G_\infty}(\hat{\delta})$ only makes sense after trivializing the $L^+_\infty\mathcal{M}$-bundles $L^+_\infty(\widetilde{\mathcal{E}})$ and $L^+_\infty(\mathcal{E})$ over an étale covering $S'\to S$ and viewing $\hat{\delta}$ as an element of $L^+_\infty\mathcal{M}(S')$, as in the proof of Proposition [Proposition 1](#PropGvsM){reference-type="ref" reference="PropGvsM"}.) [\[PropQIsogLocalGlobal_B\]](#PropQIsogLocalGlobal_B){reference-type="ref" reference="PropQIsogLocalGlobal_B"} We construct $\underline{\widetilde{\mathcal{E}}}=(x,(\widetilde{\mathcal{E}},\widetilde{\psi}),(\widetilde{\mathcal{E}}',\widetilde{\psi}'),\widetilde{\varphi},\widetilde{\varphi}')$ with the Beauvill-Laszlo glueing in Lemma [Lemma 1](#LemmaBL){reference-type="ref" reference="LemmaBL"} as follows. $\widetilde{\mathcal{E}}$ is obtained by glueing $\mathcal{E}|_{(X\smallsetminus\{\infty\})_S}$ with $\widetilde{\mathcal{L}}$ via the isomorphism $$\hat{\delta}\colon L_\infty\widetilde{\mathcal{L}}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_\infty(L^+_\infty\mathcal{E})=L_\infty(\mathcal{E}|_{(X\smallsetminus\{\infty\})_S}).$$ Moreover, $\widetilde{\mathcal{E}}'$ is obtained from ${}^{\tau\!}\widetilde{\mathcal{E}}$ as in the proof of Proposition [Proposition 1](#PropBunWithChains){reference-type="ref" reference="PropBunWithChains"}, such that the isomorphism $\widetilde{\varphi}'\colon \widetilde{\mathcal{E}}'|_{(X\smallsetminus\{\infty\})_S} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!}\widetilde{\mathcal{E}}|_{(X\smallsetminus\{\infty\})_S}$ is bounded by $\mathcal{Z}(\beta)$, or equivalently by glueing $\mathcal{E}'|_{(X\smallsetminus\{\infty\})_S}$ with $\widetilde{\mathcal{L}}$ via the isomorphism $$\varphi\circ\beta^{-1}\cdot\hat{\delta}\colon L_\infty\widetilde{\mathcal{L}}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_\infty(L^+_\infty\mathcal{E}')=L_\infty(\mathcal{E}'|_{(X\smallsetminus\{\infty\})_S}).$$ Finally $\widetilde{\psi}=\psi$ and $\widetilde{\psi}'=\psi'$ and $\widetilde{\varphi}=\varphi$ and $\widetilde{\varphi}'=\varphi'$ and $\delta=(\operatorname{id}_{\mathcal{E}},\operatorname{id}_{\mathcal{E}'})$ as isomorphisms over $(X\smallsetminus\{\infty\})_S=X_S\smallsetminus\Gamma_x$. ◻ When $\beta=1$, Proposition [Proposition 1](#PropQIsogLocalGlobal){reference-type="ref" reference="PropQIsogLocalGlobal"} is analogous to the theory of abelian varieties and $p$-divisible groups. Next we come to the analogue of the Serre-Tate-Theorem. **Definition 1**. Let $S\in\mathcal{N}ilp_{\Breve{\mathcal{O}}_\infty}$ and let $j:\bar{S}\hookrightarrow S$ be a closed subscheme defined by a locally nilpotent sheaf $\mathcal{I}$ of ideals. Let $\underline{\bar{\mathcal{E}}}\in\operatorname{Sht}_{\mathcal{G},D,X\times\infty}(\bar{S})$ be a global $\mathcal{G}$-shtuka over $\bar{S}$. The *category $\operatorname{Defo}_S(\underline{\bar{\mathcal{E}}})$ of deformations of $\underline{\bar{\mathcal{E}}}$ to $S$* has as objects : all pairs $(\underline{\mathcal{E}}\,,\,\alpha\colon j^*\underline{\mathcal{E}}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\underline{\bar{\mathcal{E}}})$ where $\underline{\mathcal{E}}$ is a global $\mathcal{G}$-shtuka over $S$ and $\alpha$ an isomorphism of global $\mathcal{G}$-shtukas over $\bar{S}$, as morphisms : isomorphisms between the $\underline{\mathcal{E}}$'s that are compatible with the $\alpha$'s. If $\underline{\bar\mathcal{L}}:=L^+_{\infty,\mathcal{M}_{\beta^{-1}}} (\underline{\bar{\mathcal{E}}})$ is the associated local $\mathcal{M}_{\beta^{-1}}$-shtuka over $\bar{S}$, we similarly define the *category $\operatorname{Defo}_S(\underline{\bar\mathcal{L}})$ of deformations of $\underline{\bar\mathcal{L}}$ to $S$*. By rigidity of quasi-isogenies ([@AH_Local Prop 5.9] and Proposition [Proposition 1](#PropRigidityLocal){reference-type="ref" reference="PropRigidityLocal"}), all $\operatorname{Hom}$-sets in these categories contain at most one element. The following result generalizes [@HartlAbSh Thm 8.4]. It is the analogue of the classical Serre-Tate theorem for abelian varieties. **Proposition 1**. *In the situation of Definition [Definition 1](#DefDefo){reference-type="ref" reference="DefDefo"} the functor $$\operatorname{Defo}_S(\underline{\bar{\mathcal{E}}}) \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\operatorname{Defo}_S(\underline{\bar\mathcal{L}})\,\qquad (\underline{\mathcal{E}}, \alpha)\longmapsto\bigl(L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}}), L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\alpha)\bigr)$$ induced from $L^+_{\infty,\mathcal{M}_{\beta^{-1}}}$ is an equivalence.* *Proof.* We construct the inverse of the above functor. Write $x\colon S\to \operatorname{Spf}\Breve{\mathcal{O}}_\infty$ for the structure morphism of $S\in\mathcal{N}ilp_{\Breve{\mathcal{O}}_\infty}$, and let $\bar{x}:=x\circ j\colon \bar{S}\to \operatorname{Spf}\Breve{\mathcal{O}}_\infty$. Write $\underline{\bar{\mathcal{E}}}\in\operatorname{Sht}_{\mathcal{G},D,X\times\infty}(\bar{S})$ as $\underline{\bar{\mathcal{E}}}=(\bar{x},(\bar{\mathcal{E}},\bar{\psi}),(\bar{\mathcal{E}}',\bar{\psi}'),\bar{\varphi},\bar{\varphi}')$. It suffices to treat the case where $\mathcal{I}^q = (0)$. In this case the morphism $\tau=\operatorname{Frob}_{q,S}$ factors as $\tau=j\circ i$ as in [\[Eq_TauFactors\]](#Eq_TauFactors){reference-type="eqref" reference="Eq_TauFactors"}. Let $(\underline\mathcal{L},\hat{\alpha}\colon j^*\underline\mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\underline{\bar\mathcal{L}})$ be an object of $\operatorname{Defo}_S(\underline{\bar\mathcal{L}})$. Since $X_S\smallsetminus\Gamma_x= (X\smallsetminus\{\infty\})_S$, we can define the global $\mathcal{G}$-shtuka $\underline{\widetilde{\mathcal{E}}}:= [-1](x,i^*(\bar{\mathcal{E}},\bar{\psi}),i^*(\bar{\mathcal{E}}',\bar{\psi}'),i^*\bar{\varphi},i^*\bar{\varphi}')$ over $S$, where $[-1]$ denotes the index shift from Corollary [Corollary 1](#CorIsog[n]){reference-type="ref" reference="CorIsog[n]"}[\[CorIsog\[n\]\_A\]](#CorIsog[n]_A){reference-type="ref" reference="CorIsog[n]_A"}. It satisfies $[1](j^*\underline{\widetilde{\mathcal{E}}})= (\bar{x},({}^{\tau\!}\bar{\mathcal{E}},{}^{\tau\!}\bar{\psi}),({}^{\tau\!}\bar{\mathcal{E}}',{}^{\tau\!}\bar{\psi}'),{}^{\tau\!}\bar{\varphi},{}^{\tau\!}\bar{\varphi}')$. The isomorphisms $\bar{\varphi}'\circ\bar{\varphi}\colon (\bar{\mathcal{E}},\bar{\psi})|_{(X\smallsetminus\{\infty\})_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}({}^{\tau\!}\bar{\mathcal{E}},{}^{\tau\!}\bar{\psi})|_{(X\smallsetminus\{\infty\})_S}$ and ${}^{\tau\!}\bar{\varphi}\circ\bar{\varphi}'\colon (\bar{\mathcal{E}}',\bar{\psi}')|_{(X\smallsetminus\{\infty\})_S}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}({}^{\tau\!}\bar{\mathcal{E}}',{}^{\tau\!}\bar{\psi}')|_{(X\smallsetminus\{\infty\})_S}$ define a quasi-isogeny $\underline{\bar{\mathcal{E}}} \rightarrow [1](j^* \underline{\widetilde{\mathcal{E}}})$. We compose it with the quasi-isogeny $\Pi\colon [1](j^* \underline{\widetilde{\mathcal{E}}}) \to j^*\underline{\widetilde{\mathcal{E}}}$ from Corollary [Corollary 1](#CorIsog[n]){reference-type="ref" reference="CorIsog[n]"}[\[CorIsog\[n\]\_B\]](#CorIsog[n]_B){reference-type="ref" reference="CorIsog[n]_B"} to obtain a quasi-isogeny $\bar{\delta}\colon \underline{\bar{\mathcal{E}}} \rightarrow j^* \underline{\widetilde{\mathcal{E}}}$ which is an isomorphism outside $\infty$. We write $L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\widetilde{\mathcal{E}}})=\underline{\widetilde\mathcal{L}}$ and $\underline{\bar\mathcal{L}}=(\bar\mathcal{L},\overline{\widehat{\varphi}})$. By Corollary [Corollary 1](#CorGlobalLocalFunctorWithChains){reference-type="ref" reference="CorGlobalLocalFunctorWithChains"} we have $${}^{\tau\!}\underline{\bar{\mathcal{L}}}= L^+_{\infty,\mathcal{M}}({}^{\tau\!}\bar{\mathcal{E}}_{-1}) = L^+_{\infty,\mathcal{M}}([-1]{}^{\tau\!}\underline{\bar{\mathcal{E}}}) = L^+_{\infty,\mathcal{M}}(j^*\underline{\widetilde{\mathcal{E}}})= j^*\underline{\widetilde{\mathcal{L}}}.$$ We view ${\overline{\widehat{\varphi}}}\colon \underline{\bar\mathcal{L}} \to {}^{\tau\!}\underline{\bar\mathcal{L}}$ as a quasi-isogeny as in Example [Example 1](#ExFrobIsogLocSht){reference-type="ref" reference="ExFrobIsogLocSht"}, and compose it with $\hat{\alpha}$ to obtain the quasi-isogeny $\overline{\hat{\gamma}}:= \overline{\widehat{\varphi}} \circ \hat{\alpha} \colon j^*\underline\mathcal{L}\rightarrow {}^{\tau\!}\underline{\bar{\mathcal{L}}}=j^*\underline{\widetilde\mathcal{L}}$. By rigidity of quasi-isogenies (Proposition [Proposition 1](#PropRigidityLocal){reference-type="ref" reference="PropRigidityLocal"}) it lifts to a quasi-isogeny $\hat{\gamma}\colon \underline\mathcal{L}\rightarrow \underline{\widetilde\mathcal{L}}$ with $j^*\hat\gamma=\overline{\hat{\gamma}}$. As in Proposition [Proposition 1](#PropQIsogLocalGlobal){reference-type="ref" reference="PropQIsogLocalGlobal"}[\[PropQIsogLocalGlobal_B\]](#PropQIsogLocalGlobal_B){reference-type="ref" reference="PropQIsogLocalGlobal_B"}, we put $\underline{\mathcal{E}}:=\hat\gamma^*\underline{\widetilde{\mathcal{E}}}$ and recall that there is a quasi-isogeny $\gamma\colon \underline{\mathcal{E}} \rightarrow \underline{\widetilde{\mathcal{E}}}$ of global $\mathcal{G}$-shtukas, which is an isomorphism outside $\infty$, and satisfies $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})=\underline{\mathcal{L}}$ and $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\gamma)=\hat\gamma$. We may now define the functor $$Defo_S(\underline{\bar\mathcal{L}})\rightarrow Defo(\bar{\underline{\mathcal{E}}})$$ by sending $(\underline\mathcal{L},\hat{\alpha}\colon j^*\underline\mathcal{L}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\underline{\bar\mathcal{L}})$ to $(\underline{\mathcal{E}},\bar{\delta}^{-1}\circ j^*\gamma)$. The quasi-isogeny $\alpha:=\bar{\delta}^{-1}\circ j^\ast \gamma\colon j^*\underline{\mathcal{E}}\to \underline{\bar{\mathcal{E}}}$ is an isomorphism outside $\infty$ by construction, and also at $\infty$ because $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\alpha) = L_{\infty,\mathcal{M}_{\beta^{-1}}}(\Pi\circ\bar{\varphi}'\circ\bar{\varphi})^{-1}\circ\overline{\widehat{\varphi}} \circ \hat{\alpha} = \hat\alpha$ by Corollary [Corollary 1](#CorGlobalLocalFunctorWithChains){reference-type="ref" reference="CorGlobalLocalFunctorWithChains"}. It can easily be seen by the above construction that these functors are actually inverse to each other. ◻ Next we recall the definition of the Rapoport-Zink spaces for local shtukas bounded by a cocharacter $\mu\in X_*(T)$. Recall the bound $\mathcal{Z}^{\leq\mu}$ from Definition [Definition 1](#Def_BoundBy_mu){reference-type="ref" reference="Def_BoundBy_mu"} and the local bound $\mathcal{Z}^{\leq\mu}\times_{\widetilde{X}_\mu} \operatorname{Spec}\mathcal{O}_\mu$ from Definition [Definition 1](#DefLocShtBounded){reference-type="ref" reference="DefLocShtBounded"} used to bound local shtukas by $\mu$. That local bound depends on the choice of the map $\operatorname{Spec}\mathcal{O}_\mu\to\widetilde{X}_\mu$. **Definition 1**. Fix an element $b\in LM(\overline{\mathbb{F}}_\infty)=L_\infty G(\overline{\mathbb{F}}_\infty)$. Consider the local $\mathcal{M}$-shtuka ${\underline{{\mathbb{L}}}}=(L^+_\infty\mathcal{M}_{\overline{\mathbb{F}}_\infty},b)\in \operatorname{LocSht}_{\mathcal{M}}$ over $\overline{\mathbb{F}}_\infty$. \(1\) We define the *Rapoport-Zink space* $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ *(of the framing object $\underline{\mathbb{L}}$)* as the functor whose $S$-points, for $S\in\mathcal{N}ilp_{\Breve{\mathcal{O}}_\mu}$, are given by $$\begin{aligned} \label{Eq_DefRZforM} \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S):=\Bigl\{\,(\underline{\mathcal{L}},\hat{\delta}) & \text{ where $\underline{\mathcal{L}}$ is local $\mathcal{M}$-shtuka over $S$ bounded by $\mu$}, \nonumber \\ & \text{and }\hat{\delta}:\underline{\mathcal{L}}\to {\underline{{\mathbb{L}}}}_S \text{ is a quasi-isogeny}\,\Bigr\}. \end{aligned}$$ Here "bounded by $\mu$" means bounded by the bound $\mathcal{Z}^{\leq\mu}$ from Definition [Definition 1](#Def_BoundBy_mu){reference-type="ref" reference="Def_BoundBy_mu"}; compare § [ 1](#ParInnerFormM){reference-type="ref" reference="ParInnerFormM"}. \(2\) We define the *affine Deligne--Lusztig variety* $X_{\mathcal{M}}^{\leq\mu}(b)$ by $$X_{\mathcal{M}}^{\leq \mu}(b)(\overline{\mathbb{F}}_\infty):=\{\overline{g}\in (L_\infty\mathcal{M}/L^+_\infty\mathcal{M})(\overline{\mathbb{F}}_\infty)\colon \tau_M(g)^{-1}b g\text{ is bounded by }\mu\}.$$ **Remark 1**. In the literature on affine Deligne-Lusztig varieties for an element $\tilde b\in M(\Breve{Q}_\infty)=LM(\overline{\mathbb{F}}_q)$ one usually requires that $g^{-1} \tilde{b}\,\tau_M(g)$ is bounded. However the $\tilde{b}$ in that literature is equal to our $b^{-1}$ as explained in Remarks [Remark 1](#Rem_LocShtInversePhi){reference-type="ref" reference="Rem_LocShtInversePhi"} and [Remark 1](#Rem_LocShtInversePhi2){reference-type="ref" reference="Rem_LocShtInversePhi2"}. Note that $\tau_M(g)^{-1} b g$ is bounded by $\mu$ if and only if $g^{-1}b^{-1}\tau_M(g)$ is bounded by $-\mu$. So our Rapoport-Zink spaces and affine Deligne-Lusztig varieties are the same as the ones usually considered in the literature after changing $\mu$ and $b$ to $-\mu$ and $b^{-1}$. **Remark 1**. Using the description of $\mathcal{F}\!\ell_{\mathcal{M},\infty}=L_\infty\mathcal{M}/L^+_\infty\mathcal{M}$ from Lemma [Lemma 1](#LemmaAffineFlagVar){reference-type="ref" reference="LemmaAffineFlagVar"}, the space $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ can also be described as $$\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S):=\{\overline{g}\in (L_\infty\mathcal{M}/L^+_\infty\mathcal{M})(S)\colon \tau_M(g)^{-1}b g\text{ is bounded by }\mu\},$$ where $\overline{g}\in (L_\infty\mathcal{M}/L^+_\infty\mathcal{M})(S)$ is represented by $g\in L_\infty\mathcal{M}(S')$ for some étale covering $S'\to S$. In particular, $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ is an ind-closed ind-subscheme of $\mathcal{F}\!\ell_{\mathcal{M},\infty}$, compare Remark [Remark 1](#Rem_HeckeBounded){reference-type="ref" reference="Rem_HeckeBounded"} and Theorem [Theorem 1](#Thm_HeckeBounded){reference-type="ref" reference="Thm_HeckeBounded"}. We recall the following **Theorem 1**. *$\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ is representable by a formal scheme over $\operatorname{Spf}\Breve{\mathcal{O}}_\mu$, which is locally formally of finite type and separated. Its underlying reduced subscheme is precisely $X_{\mathcal{M}}^{\leq \mu}(b)$. In particular, $X_{\mathcal{M}}^{\leq \mu}(b)$ is a reduced scheme locally of finite type and separated over $\overline{\mathbb{F}}_\infty$.* *Proof.* In view of Remarks [Remark 1](#Rem_LocShtInversePhi2){reference-type="ref" reference="Rem_LocShtInversePhi2"} and [Remark 1](#Rem_LocShtInversePhi3){reference-type="ref" reference="Rem_LocShtInversePhi3"} this was proven in [@AH_Local Theorem 4.18]. ◻ **Remark 1**. The group $\operatorname{QIsog}_{{\overline\mathbb{F}_\infty}}({\underline{{\mathbb{L}}}})$ of quasi-isogenies of ${\underline{{\mathbb{L}}}}:=(L^+_\infty\mathcal{M}_{\overline{\mathbb{F}}_\infty},b)$ acts on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ via $j\cdot(\underline{\widehat{\mathcal{L}}},\hat\delta):=(\underline{\widehat{\mathcal{L}}},j\circ\hat\delta)$, for $j\in \operatorname{QIsog}_{{\overline\mathbb{F}_\infty}}({\underline{{\mathbb{L}}}})$. There is a connected algebraic group $J_{{\underline{{\mathbb{L}}}}}$ over $Q_\infty$ whose $R$-points, for a $Q_\infty$-algebra $R$, is the group $$\label{EqGroupJ} J_{{\underline{{\mathbb{L}}}}}(R):=\bigl\{\,j \in \mathcal{M}(R\otimes_{Q_\infty} \Breve{Q}_\infty)\colon \tau_M(j)^{-1}b j=b\,\bigr\}\,,$$ see [@AH_Local Remark 4.16]. In particular, $\operatorname{QIsog}_{{\overline\mathbb{F}_\infty}}({\underline{{\mathbb{L}}}})=J_{{\underline{{\mathbb{L}}}}}(Q_\infty)$. This group also acts on $X_{\mathcal{M}}^{\leq \mu}(b)$ via multiplication $j\colon g\mapsto j\cdot g$ for $j\in J_{{\underline{{\mathbb{L}}}}}(Q_\infty)$. ## Tate modules In this section, we consider the fiber products $$\begin{aligned} \operatorname{Sht}_{\mathcal{G},D,\widehat{\infty}\times\infty}&:=\operatorname{Sht}_{\mathcal{G},D,X\times\infty} \times_X \operatorname{Spf}\mathcal{O}_\infty \qquad\text{and} \label{EqShtHatInfty} \\ \operatorname{Sht}_{\mathcal{G},D,\infty\times\infty}&:=\operatorname{Sht}_{\mathcal{G},D,X\times\infty} \times_X \operatorname{Spec}\mathbb{F}_\infty \label{EqShtInfty} \end{aligned}$$ on which the moving leg $x\colon S\to X$ factors through $\widehat{\infty}:=\operatorname{Spf}\mathcal{O}_\infty$ or $\infty:=\operatorname{Spec}\mathbb{F}_\infty$, respectively. Recall that we have an additional fixed leg at $\infty$. This guarantees that we have a Tate module at every place $v\neq \infty$. Let $\mathbb{O}^\infty:=\mathbb{O}_{Q}^\infty:=\prod_{v\neq\infty}\mathcal{O}_v$ be the ring of integral adeles of $X$ (i.e. of the function field $Q=\mathbb{F}_q(X)$) outside $\infty$. Let $\mathbb{A}^\infty:=\mathbb{A}_{Q}^\infty:=\mathbb{O}^\infty \otimes_{\mathcal{O}_X} Q$ be the ring of adeles of $X$ outside $\infty$. The group $\mathcal{G}(\mathbb{O}^\infty)$ acts through Hecke correspondences on the tower $\{\operatorname{Sht}_{\mathcal{G},D,\widehat{\infty}\times\infty}\}_D$. We want to extend this to an action of $G(\mathbb{A}^\infty)$; see Definition [Definition 1](#Defn-Hecke-corr){reference-type="ref" reference="Defn-Hecke-corr"} below. For this purpose, we generalize the notion of level structures on global $\mathcal{G}$-shtukas in this subsection. Let $S$ be a connected scheme in $\mathcal{N}ilp_{\mathcal{O}_\infty}$. We fix a geometric base point $\bar{s}$ of $S$. Let $\operatorname{Rep}_{\mathbb{O}^\infty}\mathcal{G}$ be the category of pairs $(V,\rho)$, where $V$ is a finite free $\mathbb{O}^\infty$-module and $\rho\colon\mathcal{G}\times_X\operatorname{Spec}\mathbb{O}^\infty\to\operatorname{GL}_{\mathbb{O}^\infty}(V)$ is an $\mathbb{O}^\infty\,$-morphism of algebraic groups. Let $\mathrm{Funct}^\otimes(\operatorname{Rep}_{\mathbb{O}^\infty}\mathcal{G},\; \mathfrak{M} od_{\mathbb{O}^\infty [\pi_1^{\rm \acute{e}t\/}(S,\bar{s})]})$ denote the category of tensor functors from $\operatorname{Rep}_{\mathbb{O}^\infty} \mathcal{G}$ to the category $\mathfrak{M} od_{\mathbb{O}^\infty[\pi_1^{\rm \acute{e}t\/}(S,\bar{s})]}$ of $\mathbb{O}^\infty[\pi_1^{\rm \acute{e}t\/}(S,\bar{s})]$-modules. For a proper closed subscheme $D$ of $X\smallsetminus\{\infty\}$ the sheaf $\mathcal{O}_D$ is an $\mathbb{O}^\infty$-module and we can consider $\rho|_D\colon\mathcal{G}|_D:=\mathcal{G}\times_X D\to \operatorname{GL}_{\mathcal{O}_D}(V\otimes_{\mathbb{O}^\infty} \mathcal{O}_D)$. Let $\underline{\mathcal{E}}=(x,\mathcal{E},\mathcal{E}',\varphi,\varphi')\in\operatorname{Sht}_{\mathcal{G},\varnothing,\widehat{\infty}\times\infty}(S)$ be a global $\mathcal{G}$-shtuka over $S$. Fix a proper closed subscheme $D$ of $X\smallsetminus\{\infty\}$ and let $\underline{\mathcal{E}}|_{D_S}:=\underline{\mathcal{E}}\times_{X_S}D_S$ denote the pullback of $\underline{\mathcal{E}}$ to $D_S$. Also fix $(V,\rho)\in \operatorname{Rep}_{\mathbb{O}^\infty}\mathcal{G}$ and let $$\mathcal{F}:=\rho_*\mathcal{E}|_{D_S}:=(\rho|_D)_*(\mathcal{E}|_{D_S}):=(\mathcal{E}|_{D_S})\overset{\mathcal{G}|_D}{\times}(V\otimes_{\mathbb{O}^\infty} \mathcal{O}_D)$$ denote the pushout vector bundle on $D_S$ of rank equal to $\dim V$. Since $D_S$ is disjoint from the graphs of the legs $x$ and $\infty$, the maps $\varphi|_{D_S}$ and $\varphi'|_{D_S}$ of $\underline{\mathcal{E}}|_{D_S}$ are isomorphisms. We equip $\mathcal{F}$ with the Frobenius isomorphism $\varphi_\mathcal{F}:=(\rho|_D)_*(\varphi'\circ\varphi)|_{D_S}\colon \mathcal{F}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^{\tau\!} \mathcal{F}$. It is an isomorphism of $\mathcal{O}_{D_S}$-modules. For the fixed geometric base point $\bar s=\operatorname{Spec}k$ of $S$ let $\mathcal{F}_{\bar s}:= \mathcal{F}\otimes_{\mathcal{O}_S} k$. Let $$(\rho_*\underline{\mathcal{E}}|_{D_{\bar{s}}})^\varphi:=\{m\in \mathcal{F}_{\bar s}\colon \varphi_\mathcal{F}(m)={}^{\tau\!} m\}$$ be the $\varphi$-invariants. They form a free $\mathcal{O}_D$-module of rank equal to $\dim V$, equipped with a continuous action of the étale fundamental group $\pi_1^{\rm \acute{e}t\/}(S,\bar{s})$. This module $(\rho_*\underline{\mathcal{E}}|_{D_{\bar{s}}})^\varphi$ is independent of $\bar{s}$ up to a change of base point. Let $\check{\mathcal{T}}_{\mathcal{E}}\in \mathrm{Funct}^\otimes(\operatorname{Rep}_{\mathbb{O}^\infty}\mathcal{G},\; \mathfrak{M} od_{\mathbb{O}^\infty [\pi_1^{\rm \acute{e}t\/}(S,\bar{s})]})$ be the tensor functor defined as $$\check{\mathcal{T}}_{\underline{\mathcal{E}}}(\rho):=\underset{D\subset X\smallsetminus\infty}{\varprojlim}(\rho_*\underline\mathcal{E}|_{D_{\bar{s}}})^\varphi.$$ Likewise, $\check{\mathcal{V}}_{\mathcal{E}}\in \mathrm{Funct}^\otimes(\operatorname{Rep}_{\mathbb{O}^\infty}\mathcal{G},\; \mathfrak{M} od_{\mathbb{A}^\infty [\pi_1^{\rm \acute{e}t\/}(S,\bar{s})]})$ is the tensor functor defined as $$\check{\mathcal{V}}_{\underline{\mathcal{E}}}(\rho):=\mathbb{A}^\infty \otimes_{\mathbb{O}^\infty} \underset{D\subset X\smallsetminus\infty}{\varprojlim}(\rho_*\underline{\mathcal{E}}|_{D_{\bar{s}}})^\varphi .$$ **Definition 1**. We define the (*dual*)[^2] *Tate-module functor* as follows. $$\begin{aligned} \label{tatefunctor} \begin{split} \check{\mathcal{T}}_{-}\colon \operatorname{Sht}_{\mathcal{G},D,\widehat{\infty}\times\infty}(S)&\longrightarrow \mathrm{Funct}^\otimes (\operatorname{Rep}_{\mathbb{O}^\infty}\mathcal{G}\,,\,\mathfrak{M} od_{\mathbb{O}^\infty[\pi_1^{\rm \acute{e}t\/}(S,\bar{s})]})\,\\ \underline{\mathcal{E}} &\longmapsto \check{\mathcal{T}}_{\underline\mathcal{E}}, \end{split}\end{aligned}$$ We define the *(dual) rational Tate-module functor* as follows. $$\begin{split} \check{\mathcal{V}}_{-}\colon \operatorname{Sht}_{\mathcal{G},D,\widehat{\infty}\times\infty}(S) &\longrightarrow \mathrm{Funct}^\otimes (\operatorname{Rep}_{\mathbb{O}^\infty}\mathcal{G}\,,\,\mathfrak{M} od_{\mathbb{A}^\infty[\pi_1^{\rm \acute{e}t\/}(S,\bar{s})]})\,\\ \label{rationaltatefunctor} \underline{\mathcal{E}} &\longmapsto \check{\mathcal{V}}_{\underline\mathcal{E}}. \end{split}$$ The functor $\check{\mathcal{V}}_{-}$ moreover transforms quasi-isogenies $\delta:\underline{\mathcal{E}}\to\underline{\mathcal{E}}'$ of global $\mathcal{G}$-shtukas into isomorphisms $\check{\mathcal{V}}_{\delta}: \check{\mathcal{V}}_{\underline{\mathcal{E}}}\to\check{\mathcal{V}}_{\underline{\mathcal{E}}'}$ of their rational Tate modules. Let $\omega_{\mathbb{O}^\infty}\colon \operatorname{Rep}_{\mathbb{O}^\infty}\mathcal{G}\to \mathfrak{M} od_{\mathbb{O}^\infty}$ and $\omega:=\omega_{\mathbb{O}^\infty}\otimes_{\mathbb{O}^\infty}\mathbb{A}^\infty\colon \operatorname{Rep}_{\mathbb{O}^\infty}\mathcal{G}\to \mathfrak{M} od_{\mathbb{A}^\infty}$ denote the forgetful functors sending $(V,\rho)$ to $V$, and $V\otimes_{\mathbb{O}^\infty} \mathbb{A}^\infty$, respectively. For a global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}$ over $S$, consider the sets of isomorphisms of tensor functors $\operatorname{Isom}^{\otimes}(\omega_{\mathbb{O}^\infty},\check{\mathcal{T}}_{\underline{\mathcal{E}}})$ and $\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{\underline{\mathcal{E}}})$. These sets are non-empty by Lemma [Lemma 1](#LSisnonempty){reference-type="ref" reference="LSisnonempty"} below. Since $X\smallsetminus\{\infty\}$ is the spectrum of a Dedekind domain, by the generalized Tannakian formalism [@Wed Corollary 5.20], we have $G(\mathbb{A}^\infty)=\operatorname{Aut}^\otimes(\omega)$ and $\mathcal{G}(\mathbb{O}^\infty)=\operatorname{Aut}^\otimes(\omega_{\mathbb{O}^\infty})$. By the definition of $\check{\mathcal{T}}_{\underline{\mathcal{E}}}$ in [\[tatefunctor\]](#tatefunctor){reference-type="eqref" reference="tatefunctor"}, $\operatorname{Isom}^{\otimes}(\omega_{\mathbb{O}^\infty},\check{\mathcal{T}}_{\underline{\mathcal{E}}})$ admits an action of $\pi_1^{\rm \acute{e}t\/}(S,\bar{s})\times\mathcal{G}(\mathbb{O}^\infty)$ where $\mathcal{G}(\mathbb{O}^\infty)$ acts through $\omega_{\mathbb{O}^\infty}$ and $\pi_1^{\rm \acute{e}t\/}(S,\bar{s})$ acts through $\check{\mathcal{T}}_{\underline{\mathcal{E}}}$. Likewise, $\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{\underline{\mathcal{E}}})$ admits an action of $\pi_1^{\rm \acute{e}t\/}(S,\bar{s})\times G(\mathbb{A}^\infty)$. **Lemma 1**. *The sets $\operatorname{Isom}^{\otimes}(\omega_{\mathbb{O}^\infty},\check{\mathcal{T}}_{\underline{\mathcal{E}}})$ and $\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{\underline{\mathcal{E}}})$ are non-empty. Moreover, $\operatorname{Isom}^{\otimes}(\omega_{\mathbb{O}^\infty},\check{\mathcal{T}}_{\underline{\mathcal{E}}})$ (resp. $\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{\underline{\mathcal{E}}})$) is a principal homogeneous space under the group $\mathcal{G}(\mathbb{O}^\infty)$ (resp. $G(\mathbb{A}^\infty)$).* *Proof.* This is [@AH_Unif Lemma 6.2]. ◻ **Definition 1**. Let $m\in\mathbb{N}_0$. For every global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}:=(x,(\mathcal{E},\psi),(\mathcal{E}',\psi'),\varphi,\varphi')\in \operatorname{Sht}_{\mathcal{G},D,\infty\times\infty}(S)$, the *$q^m$-Frobenius quasi-isogeny* $\Phi_{\underline{\mathcal{E}}}^m\colon\underline{\mathcal{E}}\longrightarrow{}^{\tau^m\!} \underline{\mathcal{E}}$ is defined as the tuple $\Phi_{\underline{\mathcal{E}}}^m = (f,f')$ with $$\begin{aligned} {4} f & :={}^{\tau^{m-1}}(\varphi'\circ\varphi)\circ\ldots\circ{}^{\tau\!} (\varphi'\circ\varphi)\circ (\varphi'\circ\varphi) && \colon\;(\mathcal{E},\psi)|_{(X\smallsetminus\infty)_S} && \longrightarrow{}^{\tau^m\!}(\mathcal{E},\psi)|_{(X\smallsetminus\infty)_S} \\ f' & :={}^{\tau^m\!}\varphi\circ {}^{\tau^{m-1}}(\varphi'\circ\varphi)\circ\ldots\circ{}^{\tau\!} (\varphi'\circ\varphi)\circ \varphi' && \colon\;(\mathcal{E}',\psi')|_{(X\smallsetminus\infty)_S} && \longrightarrow{}^{\tau^m\!}(\mathcal{E}',\psi')|_{(X\smallsetminus\infty)_S}\end{aligned}$$ satisfying ${}^{\tau^m\!}\varphi\circ f = f'\circ \varphi$ and ${}^{\tau^m\!}\varphi'\circ f'={}^{\tau\!}f \circ \varphi'$; see Definition [Definition 1](#DefIsogGlobalSht){reference-type="ref" reference="DefIsogGlobalSht"}. Here we observe that the global $\mathcal{G}$-shtuka ${}^{\tau^m\!}\underline{\mathcal{E}}$ is obtained by pulling back $\underline{\mathcal{E}}$ under the absolute $q^m$-Frobenius $\tau^m=\operatorname{Frob}_{q^m,S}\colon S\to S$. The leg $x$ of $\underline{\mathcal{E}}$ satisfies $x\circ\operatorname{Frob}_{q^m,S}=\operatorname{Frob}_{q^m,\mathbb{F}_\infty}\circ x=x$, because $x\colon S\to X$ factors through $\{\infty\}=\operatorname{Spec}\mathbb{F}_\infty\in X$ and $\operatorname{Frob}_{q^m,\mathbb{F}_\infty}=\operatorname{id}_{\mathbb{F}_\infty}$ by our assumption $\mathbb{F}_\infty=\mathbb{F}_q$. So $\underline{\mathcal{E}}$ and ${}^{\tau^m\!}\underline{\mathcal{E}}$ have the leg. **Corollary 1**. *Keep the situation of Definition [Definition 1](#DeFfrobIsog){reference-type="ref" reference="DeFfrobIsog"}.* *If $\gamma\in\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{\underline{\mathcal{E}}})$ then we have an equality ${}^{\tau^m\!} (\gamma)=\check{\mathcal{V}}_{\Phi_{\underline{\mathcal{E}}}^m}\circ\gamma$ inside $\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{}^{\tau^m\!} \underline{\mathcal{E}}})$.* *Proof.* This follows by the same argument as in [@AH_Unif Corollary 6.3]. ◻ **Example 1**. For $m=1$ the composition of the $q$-Frobenius quasi-isogeny $\Phi_{\underline{\mathcal{E}}}\colon\underline{\mathcal{E}}\longrightarrow{}^{\tau\!} \underline{\mathcal{E}}$ with the isogeny $\Pi\colon \underline{\mathcal{E}}\longrightarrow[-1](\underline{\mathcal{E}})$ from Corollary [Corollary 1](#CorIsog[n]){reference-type="ref" reference="CorIsog[n]"}[\[CorIsog\[n\]\_B\]](#CorIsog[n]_B){reference-type="ref" reference="CorIsog[n]_B"} is given by the maps $\varphi^\lhd_i\colon \mathcal{E}_i \to {}^{\tau\!}\mathcal{E}_{i+1}$ from diagram [\[Eq_ShtukaDiagTildephi\]](#Eq_ShtukaDiagTildephi){reference-type="eqref" reference="Eq_ShtukaDiagTildephi"}, which are isomorphisms on $(X\smallsetminus\{\infty\})_S$. **Definition 1**. Let $S\in\mathcal{N}ilp_{\mathcal{O}_\infty}$ be a connected scheme. For a compact open subgroup $H\subseteq G(\mathbb{A}^\infty)$, we define a *rational $H$-level structure* $\bar\gamma$ on a global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}$ over $S$ as a $\pi_1^{\rm \acute{e}t\/}(S,\bar{s})$-invariant $H$-orbit $\bar\gamma=\gamma H$ in $\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{\underline{\mathcal{E}}})$. For a non-connected scheme $S$, we make a similar definition by choosing a base point on each connected component and a rational $H$-level structure on the restriction to each connected component separately. Let $\mathcal{Z}$ be a bound in $\operatorname{Gr}_{\mathcal{G},X\times\infty}\times_X \operatorname{Spf}\mathcal{O}_\infty$ which in terms of Definition [Definition 1](#DefThreeTypes){reference-type="ref" reference="DefThreeTypes"} is of local type [\[DefThreeTypes_B\]](#DefThreeTypes_B){reference-type="ref" reference="DefThreeTypes_B"} at the moving leg and of finite type [\[DefThreeTypes_C\]](#DefThreeTypes_C){reference-type="ref" reference="DefThreeTypes_C"} at the fixed leg $\infty$. Its reflex scheme is $\operatorname{Spec}\mathcal{O}_\mathcal{Z}$ for a finite ring extension $\mathcal{O}_\infty\subset \mathcal{O}_\mathcal{Z}$. We write $\kappa_\mathcal{Z}$ for the residue field of $\mathcal{O}_\mathcal{Z}$. We denote by $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}$ (respectively $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}}$) the category fibered in groupoids over $\mathcal{N}ilp_{\mathcal{O}_\infty}$ (respectively $\mathcal{N}ilp_{\mathcal{O}_\mathcal{Z}}$) whose $S$-valued points $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}(S)$ (respectively $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}}(S)$) is the category whose - objects are tuples $(\underline{\mathcal{E}},\gamma H)$ consisting of a global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}$ over $S$ (respectively, which is bounded by $\mathcal{Z}$) together with a rational $H$-level structure $\gamma H$; - morphisms are quasi-isogenies of global $\mathcal{G}$-shtukas that are isomorphisms above $\infty$ and are compatible with the $H$-level structures. We also set $$\begin{aligned} \operatorname{Sht}_{\mathcal{G},H,\infty\times\infty} & := \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}\times_{\mathcal{O}_\infty} \operatorname{Spec}\mathbb{F}_\infty \qquad \text{and} \\ \operatorname{Sht}_{\mathcal{G},H,\infty\times\infty}^{\mathcal{Z}} & := \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}} \times_{\mathcal{O}_\mathcal{Z}} \operatorname{Spec}\kappa_\mathcal{Z}\end{aligned}$$ This definition of level structures generalizes our initial Definition [Definition 1](#DefD-LevelStr){reference-type="ref" reference="DefD-LevelStr"} as follows. **Proposition 1**. *Let $D\subset X$ be a proper closed subscheme disjoint from $\infty$. Consider the compact open subgroup $H_D:=\ker\bigl(\mathcal{G}(\mathbb{O}^\infty)\to\mathcal{G}(\mathcal{O}_D)\bigr)\subset G(\mathbb{A}^\infty)$. There is a canonical isomorphism of stacks $$\operatorname{Sht}_{\mathcal{G},D,\widehat{\infty}\times\infty} \enspace \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\enspace\operatorname{Sht}_{\mathcal{G},H_D,\widehat{\infty}\times\infty}.$$ In particular, it induces an isomorphism of the moduli stacks of bounded $\mathcal{G}$-shtukas $$\operatorname{Sht}_{\mathcal{G},D,\widehat{\infty}\times\infty}^\mathcal{Z}\enspace \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\enspace\operatorname{Sht}_{\mathcal{G},H_D,\widehat{\infty}\times\infty}^\mathcal{Z}.$$* *Proof.* This follows as in [@AH_Unif Theorem 6.5]. ◻ **Remark 1**. (a) Let $(\underline{\mathcal{E}},\gamma H)\in\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}(S)$. Then every choice of a representative $\gamma\in\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{\underline{\mathcal{E}}})$ of the $H$-level structure $\gamma H$ induces a representation of the étale fundamental group given by $$\label{EqRepOfpi1} \rho_{\underline{\mathcal{E}},\gamma}\colon \pi_1^{\rm \acute{e}t\/}(S,\bar s)\;\longrightarrow\; H\,,\quad g\;\longmapsto\;\gamma^{-1}\circ g(\gamma)\;=:\;\rho_{\underline{\mathcal{E}},\gamma}(g)\,.$$ It is a group homomorphism because $$\rho_{\underline{\mathcal{E}},\gamma}(gg')=\gamma^{-1}\circ g(\gamma)\circ g\bigl(\gamma^{-1}\circ g'(\gamma)\bigr)=\rho_{\underline{\mathcal{E}},\gamma}(g)\cdot\rho_{\underline{\mathcal{E}},\gamma}(g'),$$ as $\gamma^{-1}\circ g'(\gamma)$ lies in $H$ on which $\pi_1^{\rm \acute{e}t\/}(S,\bar s)$ acts trivially. Replacing $\gamma$ by $\gamma h$, for $h\in H$, gives $\rho_{\underline{\mathcal{E}},\gamma}=\operatorname{int}_h\circ\rho_{\underline{\mathcal{E}},\gamma h}$. \(b\) For any compact open subgroup $H\subseteq G(\mathbb{A}^\infty)$ and any element $h\in G(\mathbb{A}^\infty)$, there is an isomorphism $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}\;\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\;\operatorname{Sht}_{\mathcal{G},h^{-1}Hh,\widehat{\infty}\times\infty}$ of stacks, given by $(\underline{\mathcal{E}},\gamma H)\mapsto\bigl(\underline{\mathcal{E}},\gamma h(h^{-1}Hh)\bigr)$. **Proposition 1**. 1. *[\[PropLSGGsht1_A\]]{#PropLSGGsht1_A label="PropLSGGsht1_A"} For any compact open subgroup $H\subseteq G(\mathbb{A}^\infty)$, the stack $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}$ is an ind-Deligne-Mumford stack, ind-separated and locally of ind-finite type over $\operatorname{Spf}\mathcal{O}_\infty$. If $\mathfrak{m}_\mathcal{Z}$ denotes the maximal ideal of $\mathcal{O}_\mathcal{Z}$, then for every $e\in\mathbb{N}$ the fiber product $$\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^\mathcal{Z}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{\mathcal{O}_\mathcal{Z}} \operatorname{Spec}\mathcal{O}_\mathcal{Z}/\mathfrak{m}_\mathcal{Z}^e$$ is an (algebraic) Deligne-Mumford stack separated and locally of finite type over $\operatorname{Spec}\mathcal{O}_\mathcal{Z}/\mathfrak{m}_\mathcal{Z}^e$, and $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^\mathcal{Z}$ is a locally noetherian, adic formal algebraic Deligne-Mumford stack, separated and locally of finite type over $\operatorname{Spf}\mathcal{O}_\mathcal{Z}$ in the sense of [@HartlAbSh Appendix A].* 2. *[\[PropLSGGsht1_B\]]{#PropLSGGsht1_B label="PropLSGGsht1_B"} If $\widetilde H\subset H\subseteq G(\mathbb{A}^\infty)$ are compact open subgroups then the forgetful morphism $$\label{forgetting-level-map} \operatorname{Sht}_{\mathcal{G},\widetilde H,\widehat{\infty}\times\infty}\;\longrightarrow\;\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty},\quad (\underline{\mathcal{E}},\gamma\widetilde H)\;\longmapsto\;(\underline{\mathcal{E}},\gamma H)$$ is finite étale and surjective. Moreover, the same is true for the stack $\operatorname{Sht}_{\mathcal{G},\widetilde H,\widehat{\infty}\times\infty}^\mathcal{Z}$.* 3. *[\[PropLSGGsht1_C\]]{#PropLSGGsht1_C label="PropLSGGsht1_C"} Furthermore, if $\widetilde H$ is a normal subgroup of $H$, then the group $H/\widetilde H$ acts on $\operatorname{Sht}_{\mathcal{G},\widetilde H,\widehat{\infty}\times\infty}$ from the right via $h\widetilde H\colon(\underline{\mathcal{E}},\gamma\widetilde H)\mapsto(\underline{\mathcal{E}},\gamma h\widetilde H)$ for $h\widetilde H\in H/\widetilde H$.* *Moreover, the stack $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}$ is canonically isomorphic to the stack quotient $\bigl[\operatorname{Sht}_{\mathcal{G},\widetilde H,\widehat{\infty}\times\infty}\big/(H/\widetilde H)\bigr]$ and $\operatorname{Sht}_{\mathcal{G},\widetilde H,\widehat{\infty}\times\infty}$ is a right $H \slash \widetilde H$-torsor over $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}$ under the forgetful morphism [\[forgetting-level-map\]](#forgetting-level-map){reference-type="eqref" reference="forgetting-level-map"}. The same is true for the bounded shtukas.* *Proof.* Assertions [\[PropLSGGsht1_B\]](#PropLSGGsht1_B){reference-type="ref" reference="PropLSGGsht1_B"} and [\[PropLSGGsht1_C\]](#PropLSGGsht1_C){reference-type="ref" reference="PropLSGGsht1_C"} are standard. See for example [@AH_Unif Theorem 6.7] for a detailed proof. [\[PropLSGGsht1_A\]](#PropLSGGsht1_A){reference-type="ref" reference="PropLSGGsht1_A"} The intersection $H_1:=H\cap\mathcal{G}(\mathbb{O}^\infty)$ has finite index in $H$, because it is open and $H$ is compact. Thus the intersection $H_2:=\bigcap_{h\in H/H_1}h H_1 h^{-1}\subset H_1\subset \mathcal{G}(\mathbb{O}^\infty)$ is compact open, normal in $H$, and of finite index in $\mathcal{G}(\mathbb{O}^\infty)$. There is a proper closed subscheme $D\subset X$ with $H_D\subset H_2$, and this is a normal subgroup because $H_D$ is normal in $\mathcal{G}(\mathbb{O}^\infty)$. Therefore, statement [\[PropLSGGsht1_A\]](#PropLSGGsht1_A){reference-type="ref" reference="PropLSGGsht1_A"} holds for $\operatorname{Sht}_{\mathcal{G},H_D,\widehat{\infty}\times\infty}$ by Theorems [Proposition 1](#H_DL-Str){reference-type="ref" reference="H_DL-Str"} and [Theorem 1](#Thm_Sht2legsBounded){reference-type="ref" reference="Thm_Sht2legsBounded"}, see the definition in [\[EqShtHatInfty\]](#EqShtHatInfty){reference-type="eqref" reference="EqShtHatInfty"}. Consequently also $\operatorname{Sht}_{\mathcal{G},H_2,\widehat{\infty}\times\infty}$ and $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}$ are ind-Deligne-Mumford stacks locally of ind-finite type over $\operatorname{Spf}\mathcal{O}_\infty$ by [\[PropLSGGsht1_C\]](#PropLSGGsht1_C){reference-type="ref" reference="PropLSGGsht1_C"} because they are obtained as stack quotients by finite groups. They are (ind-)separated over $\operatorname{Spf}\mathcal{O}_\infty$, because the forgetful morphisms in [\[PropLSGGsht1_C\]](#PropLSGGsht1_C){reference-type="ref" reference="PropLSGGsht1_C"} are finite surjective with (ind-)separated source. ◻ ## Examples and relation to the previous literature We explain how our Theorems [Theorem 1](#Uniformization1){reference-type="ref" reference="Uniformization1"} and [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"} generalize uniformization results in the literature. **Example 1**. Drinfeld [@Drinfeld-elliptic-modules] defined "elliptic modules" (which are today called *Drinfeld modules*), and constructed moduli spaces for them. In [@Drinfeld-commutative-subrings] he also defined the equivalent notion of *elliptic sheaves*; see also [@Blum-Stuhler Chapter 3]. An elliptic sheaf over an $\mathbb{F}_q$-scheme $S$ is by definition a global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}=\bigl(x,(\mathcal{E}_i,\psi_i,\Pi_i,\varphi^\lhd_i)_{i\in\mathbb{Z}}\bigr)$ as in Corollary [Corollary 1](#CorShtWithChains){reference-type="ref" reference="CorShtWithChains"} for $G=\operatorname{GL}_r$, $\mu=(0,\ldots,0,-1)$. The group scheme $\mathcal{G}\times_X (X\smallsetminus\{\infty\})$ equals $\operatorname{GL}_r$, but $\mathcal{G}_\infty$ is an Iwahori subgroup of $\operatorname{GL}_r$ (e.g. whose reduction is the Borel of lower triangular matrices), and $$\label{EqExDrinfeld} \beta = \left( \raisebox{4.6ex}{$ \xymatrix @C=0.2pc @R=0pc { 0 \ar@{.}[rrr]\ar@{.}[drdrdrdr] & & & 0 & z_\infty\\ 1\ar@{.}[drdrdr] & & & & 0 \ar@{.}[ddd]\\ 0 \ar@{.}[dd]\ar@{.}[drdr] & & & & \\ & & & & \\ 0 \ar@{.}[rr] & & 0 & 1 & 0\\ }$} \right),$$ where $z_\infty$ is a uniformizer at $\infty$. Its reflex ring ${\Breve{\mathcal{O}}_{\mu,\beta}}=\Breve{\mathcal{O}}_\infty$. The elliptic sheaf $\underline{\mathcal{E}}$ is written in [@Drinfeld-commutative-subrings; @Blum-Stuhler] in terms of chains of vector bundles $\mathcal{F}_i$ of rank $r$. These vector bundles are obtained from our right $\mathcal{G}$-bundles $\mathcal{E}_i$ as $\mathcal{F}_i:=\mathcal{E}_i\times^{\mathcal{G}} \mathcal{O}_{X_S}^r$ via the action of $\mathcal{G}$ on $\mathcal{O}_{X_S}^r$ by left multiplication. The boundedness of $\varphi^\lhd_i$ by $\mu=(0,\ldots,0,-1)$ and of $\Pi_i$ by $\tau_{G_\infty}^{i-1}(\beta)^{-1}=\beta^{-1}$ implies that $(\varphi^\lhd_i)^{-1}\colon {}^{\tau\!}\mathcal{F}_{i-1}\to\mathcal{F}_i$ and $\Pi_i^{-1}\colon \mathcal{F}_{i-1}\to\mathcal{F}_i$ are morphisms of vector bundles on $X_S$ whose cokernels are locally free of rank $1$ over $\mathcal{O}_S$ supported on $\infty$, and $\Gamma_x$, respectively. Moreover, for any $i$ the composition $\Pi_{i+r}^{-1}\circ\ldots\circ\Pi_{i+1}^{-1}\colon\mathcal{F}_i\to\mathcal{F}_{i+r}$ is bounded by $\beta^r= z_\infty\in Q_\infty^\times\subset \operatorname{GL}_r(Q_\infty)$. This is equivalent to the periodicity condition formulated in [@Blum-Stuhler ii) on page 146]. In this case $\mathcal{M}_{\beta^{-1}}$ is the group of units in the maximal order of the central division algebra over $Q_\infty$ of Hasse-invariant $1/r$. Drinfeld [@Drinfeld-commutative-subrings Proposition 3] proved that the category of elliptic sheaves of rank $r$ over $S$ is equivalent to the category of Drinfeld $A$-modules over $S$ of rank $r$, where $A=\Gamma(X\smallsetminus\{\infty\},\mathcal{O}_X)$. In [@Drinfeld-elliptic-modules] he showed that the moduli spaces $M^r_{A,D}:=\operatorname{Sht}_{\mathcal{G},D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}$ of Drinfeld $A$-modules of rank $r$ with $D$-level structure are uniformized over $\mathbb{C}_\infty$ by his $(r-1)$-dimensional upper halfspace $\Omega^r_{Q_\infty}$. This uniformization was worked out by Blum and Stuhler [@Blum-Stuhler Chapter 4] in terms of elliptic sheaves. Both uniformization results are implied by our Theorem [Theorem 1](#Uniformization1Intro){reference-type="ref" reference="Uniformization1Intro"} as follows. As the global framing object ${\underline{\mathbb{E}}}$ we take the global $\mathcal{G}$-shtuka over $\mathbb{F}_q$ given by $\mathcal{E}_0=\mathcal{G}$ and $\varphi^\lhd_0={}^{\tau\!}\Pi_0$. Its quasi-isogeny group $I_{\underline{\mathbb{E}}}$ equals $G$. Then ${\underline{{\mathbb{L}}}}=L^+_{\infty,\mathcal{M}}({\underline{\mathbb{E}}})=\bigl((L^+_\infty\mathcal{M}_{\beta^{-1}})_{\overline{\mathbb{F}}_q}, \beta^{-1})$, i.e. $b=\beta^{-1}$. In view of Remark [Remark 1](#Rem_LocShtInversePhi3){reference-type="ref" reference="Rem_LocShtInversePhi3"}, the Rapoport-Zink space $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq\mu}$ is the disjoint union indexed by $\mathbb{Z}$ of Deligne's formal model $\widehat{\Omega}^r_{Q_\infty}$ of $\Omega^r_{Q_\infty}$; see for example [@HartlViehmann3 Example 6.14]. By the "Lemma of the critical index" [@Blum-Stuhler Lemma 3.3.1], the isogeny class $\mathcal{X}_{\underline{\mathbb{E}}}$ of ${\underline{\mathbb{E}}}$ from Theorem [Theorem 1](#Uniformization1Intro){reference-type="ref" reference="Uniformization1Intro"} is the whole fiber $\operatorname{Sht}_{\mathcal{G},X\times\infty}^{\mathcal{Z}(\mu,\beta)}\times_X \operatorname{Spec}\overline{\mathbb{F}}_\infty$. This also follows from the fact that $B(M_{\beta^{-1}},-\mu)$ is a set of one element by [@KottwitzII § 6.11]. Thus this whole fiber is the *Newton stratum* of ${\underline{{\mathbb{L}}}}:=L^+_{\infty,\mathcal{M}}({\underline{\mathbb{E}}})$, i.e. for every global $\mathcal{G}$-shtukas $\underline{\mathcal{E}}$ in this fiber the associated local $\mathcal{M}$-shtuka $L^+_{\infty,\mathcal{M}}(\underline{\mathcal{E}})$ is isogenous to ${\underline{{\mathbb{L}}}}$. By [@HartlAbSh Proposition 11.4] this implies that already $\underline{\mathcal{E}}$ and ${\underline{\mathbb{E}}}$ are isogenous. As a consequence $\mathcal{X}_{\underline{\mathbb{E}}}=\operatorname{Sht}_{\mathcal{G},X\times\infty}^{\mathcal{Z}(\mu,\beta)}\times_X \operatorname{Spec}\overline{\mathbb{F}}_\infty$. The adelic uniformization isomorphism described in [@Deligne-Husemoller Theorem 5.6] follows from our Theorem [Theorem 1](#Uniformization1Intro){reference-type="ref" reference="Uniformization1Intro"} $$\label{Eq_ExLRS} \Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon \operatorname{GL}_r(Q)\backslash \widehat{\Omega}^r_{Q_\infty} \times G(\mathbb{A}^\infty)/H_D \; \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\; M^r_{A,D} \times_X \operatorname{Spf}\Breve{\mathcal{O}}_\infty\,.$$ Note that we can use $\widehat{\Omega}^r_{Q_\infty}$ instead of $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq\mu}=\coprod_\mathbb{Z}\widehat{\Omega}^r_{Q_\infty}$, because the existence of elements in $Q^\times$ with vanishing order $1$ at $\infty$ implies $Q^\times\backslash\mathbb{Z}\times (\mathbb{A}^\infty)^\times=Q^\times\backslash\{1\}\times(\mathbb{A}^\infty)^\times$. **Example 1**. As a variant of the previous example, Laumon, Rapoport and Stuhler [@Laumon-Rapoport-Stuhler] defined $\mathscr{D}$-elliptic sheaves, where $\mathscr{D}$ is a sheaf on $X$ of maximal orders in a central division algebra over $Q$ of dimension $d^2$. The point $\infty$ is chosen such that $\mathscr{D}\otimes_{\mathcal{O}_X} \mathcal{O}_\infty\cong M_d(\mathcal{O}_\infty)$. The group scheme $\mathcal{G}\times_X (X\smallsetminus\{\infty\})$ equals the group $\mathscr{D}^\times$ of units in $\mathscr{D}$, and $\mathcal{G}_\infty$ is an Iwahori subgroup of $\operatorname{GL}_r$ (e.g. whose reduction is the Borel of lower triangular matrices). We take $\mu=(0,\ldots,0,-1)$, and $\beta$ is given by the matrix from [\[EqExDrinfeld\]](#EqExDrinfeld){reference-type="eqref" reference="EqExDrinfeld"}. Its reflex ring ${\Breve{\mathcal{O}}_{\mu,\beta}}=\Breve{\mathcal{O}}_\infty$. Then by definition, a $\mathscr{D}$-elliptic sheaf over an $\mathbb{F}_q$-scheme $S$ is a global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}=\bigl(x,(\mathcal{E}_i,\psi_i,\Pi_i,\varphi^\lhd_i)_{i\in\mathbb{Z}}\bigr)$ as in Corollary [Corollary 1](#CorShtWithChains){reference-type="ref" reference="CorShtWithChains"}. It is written in [@Laumon-Rapoport-Stuhler Definition 2.2] in terms of right modules $\mathcal{F}_i$ over $\mathscr{D}\otimes_{\mathbb{F}_q} \mathcal{O}_S$ which are locally free of rank one. These modules are obtained from our right $\mathcal{G}$-bundles $\mathcal{E}_i$ as $\mathcal{F}_i:=\mathcal{E}_i\times^{\mathscr{D}^\times} \mathscr{D}$ via the action of $\mathcal{G}=\mathscr{D}^\times$ on $\mathscr{D}$ by left multiplication. As in the previous example, $\mathcal{M}_{\beta^{-1}}$ is the group of units in the maximal order of the central division algebra over $Q_\infty$ of Hasse-invariant $1/r$. The uniformization at $\infty$ of the moduli space $\mathcal{E}\ell\ell_{X,\mathscr{D},D}:=\operatorname{Sht}_{\mathcal{G},H_D,X\times\infty}^{\mathcal{Z}(\mu,\beta)}$ of $\mathscr{D}$-elliptic sheaves was mentioned by Laumon, Rapoport and Stuhler [@Laumon-Rapoport-Stuhler (14.19)], but not proven. We prove it in our Theorem [Theorem 1](#Uniformization1Intro){reference-type="ref" reference="Uniformization1Intro"}. As the global framing object ${\underline{\mathbb{E}}}$ we take the global $\mathcal{G}$-shtuka over $\mathbb{F}_q$ given by $\mathcal{E}_0=\mathcal{G}$ and $\varphi^\lhd_0={}^{\tau\!}\Pi_0$. Then ${\underline{{\mathbb{L}}}}=L^+_{\infty,\mathcal{M}}({\underline{\mathbb{E}}})=\bigl((L^+_\infty\mathcal{M}_{\beta^{-1}})_{\overline{\mathbb{F}}_q}, \beta^{-1})$, i.e. $b=\beta^{-1}$. Its quasi-isogeny group $I_{\underline{\mathbb{E}}}$ equals $G$. As in the previous example, the Rapoport-Zink space $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq\mu}$ is the disjoint union indexed by $\mathbb{Z}$ of Deligne's formal model $\widehat{\Omega}^d_{Q_\infty}$ of $\Omega^d_{Q_\infty}$, and the isogeny class $\mathcal{X}_{\underline{\mathbb{E}}}$ of ${\underline{\mathbb{E}}}$ from Theorem [Theorem 1](#Uniformization1Intro){reference-type="ref" reference="Uniformization1Intro"} is the whole fiber $\operatorname{Sht}_{\mathcal{G},X\times\infty}^{\mathcal{Z}(\mu,\beta)}\times_X \operatorname{Spec}\overline{\mathbb{F}}_\infty$. We obtain the uniformization isomorphism $$\label{Eq_ExLRS} \Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon G(Q)\backslash \widehat{\Omega}^d_{Q_\infty} \times G(\mathbb{A}^\infty)/H_D \; \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\; \mathcal{E}\ell\ell_{X,\mathscr{D},D} \times_X \operatorname{Spf}\Breve{\mathcal{O}}_\infty\,.$$ As in the previous example note that we can use $\widehat{\Omega}^r_{Q_\infty}$ instead of $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq\mu}=\coprod_\mathbb{Z}\widehat{\Omega}^r_{Q_\infty}$, because $Q^\times\backslash\mathbb{Z}\times (\mathbb{A}^\infty)^\times=Q^\times\backslash\{1\}\times(\mathbb{A}^\infty)^\times$. # Uniformization {#ChaptUnif} ## Source and target of the uniformization map {#subsec:SourceTarget} ** 1**. Throughout Chapter [3](#ChaptUnif){reference-type="ref" reference="ChaptUnif"} we fix a global framing object ${\underline{\mathbb{E}}}\in \operatorname{Sht}_{\mathcal{G},\varnothing,X\times\infty}^{\mathcal{Z}(\mu,\beta)}(\overline{\mathbb{F}}_\infty)$ and let ${\underline{{\mathbb{L}}}}:=({\mathbb{L}},\widehat{\varphi}_{{\mathbb{L}}}):=L^+_{\infty,\mathcal{M}_{\beta^{-1}}}({\underline{\mathbb{E}}})$ denote the associated local $\mathcal{M}$-shtuka over ${\overline\mathbb{F}_\infty}$, where $L^+_{\infty,\mathcal{M}_{\beta^{-1}}}$ is the $\beta^{-1}$-twisted global-local functor from Definition [Definition 1](#DefGlobLocM){reference-type="ref" reference="DefGlobLocM"}. We fix a trivialization ${\underline{{\mathbb{L}}}}\cong(L^+_\infty\mathcal{M}_{\overline{{\mathbb{F}}}_\infty},b)$ for $[b]\in B(M,\mu)$ by Proposition [Proposition 1](#PropBoundWithChains){reference-type="ref" reference="PropBoundWithChains"}. Recall from Definition [Definition 1](#DefIsogGlobalSht){reference-type="ref" reference="DefIsogGlobalSht"} and Remark [Remark 1](#RemIsogGlobalSht){reference-type="ref" reference="RemIsogGlobalSht"} the group $I_{{\underline{\mathbb{E}}}} (Q):=\mathrm{QIsog}_{\overline{\mathbb{F}}_\infty}({\underline{\mathbb{E}}})$ of quasi-isogenies of the framing object. **Proposition 1**. *Let $S$ be a connected non-empty scheme in $\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$. Then the natural group homomorphism $\operatorname{QIsog}_{\overline\mathbb{F}_\infty}({\underline{\mathbb{E}}})\to\operatorname{QIsog}_{S}({\underline{\mathbb{E}}}_S)$, $g\mapsto g \times_{{\overline\mathbb{F}_\infty}} S=:g_S$ is an isomorphism.* *Proof.* This follows as in [@AH_Unif Proposition 7.2]. ◻ We next describe the source of the morphism $\Theta_{{\underline{\mathbb{E}}}}$ from [\[EqUnifIntro\]](#EqUnifIntro){reference-type="eqref" reference="EqUnifIntro"}. Let $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ be the Rapoport-Zink space of ${\underline{{\mathbb{L}}}}$ with underlying topological space $X_{\mathcal{M}}^{\leq \mu}(b)$, see Theorem [Theorem 1](#ThmRRZSp){reference-type="ref" reference="ThmRRZSp"}. ** 1**. Recall that the group $J_{{\underline{{\mathbb{L}}}}}(Q_\infty)=\operatorname{QIsog}_{\overline\mathbb{F}_\infty}({\underline{{\mathbb{L}}}})$ of quasi-isogenies of ${\underline{{\mathbb{L}}}}$ over ${\overline\mathbb{F}_\infty}$ acts naturally on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ and on $X_{\mathcal{M}}^{\leq \mu}(b)$; see Remark [Remark 1](#RemMActsOnRZ){reference-type="ref" reference="RemMActsOnRZ"}. In particular, we see that the group $I_{{\underline{\mathbb{E}}}}(Q)$ acts on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ and $X_{\mathcal{M}}^{\leq \mu}(b)$ via the natural group homomorphism $$\label{EqI(Q)} L_{\infty,\mathcal{M}_{\beta^{-1}}}\colon I_{{\underline{\mathbb{E}}}}(Q) \;\longrightarrow\; J_{{\underline{{\mathbb{L}}}}}(Q_\infty) \,,$$ which sends a quasi-isogeny $\eta\in I_{{\underline{\mathbb{E}}}}(Q)$ of ${\underline{\mathbb{E}}}$ to the quasi-isogeny $L_{\mathcal{M}_{\beta^{-1}}}(\eta)$ of ${\underline{{\mathbb{L}}}}=L_{\mathcal{M}_{\beta^{-1}}}({\underline{\mathbb{E}}})$. The group $I_{{\underline{\mathbb{E}}}}(Q)$ also acts naturally on $\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}}$ and $\operatorname{Isom}^{\otimes}(\omega,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})$ by sending $\gamma\in\operatorname{Isom}^{\otimes}(\omega,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})$ to $\check{\mathcal{V}}_\eta\circ\gamma$ for $\eta\in I_{{\underline{\mathbb{E}}}}(Q)$; see Definition [Definition 1](#DefTateModule){reference-type="ref" reference="DefTateModule"} and Lemma [Lemma 1](#LSisnonempty){reference-type="ref" reference="LSisnonempty"}. Upon choosing an element $\gamma_0\in\operatorname{Isom}^{\otimes}(\omega_{\mathbb{O}^\infty} ,\check{{\mathcal{T}}}_{{\underline{\mathbb{E}}}})$, this defines an injective morphism $$\label{EqEpsilon} \zeta\colon I_{{\underline{\mathbb{E}}}}(Q) \;\longrightarrow\; \operatorname{Aut}^\otimes(\omega )\;\cong\;G(\mathbb{A}^\infty),\quad\eta\mapsto\gamma_0^{-1}\circ\check{{\mathcal{V}}}_\eta\circ\gamma_0\,.$$ The map $\zeta$ depends on $\gamma_0$ only up to $G(\mathbb{A}^\infty)$-conjugation. **Lemma 1**. *The group homomorphisms $L_{\infty,\mathcal{M}_{\beta^{-1}}}$ and $\zeta$ from [\[EqI(Q)\]](#EqI(Q)){reference-type="eqref" reference="EqI(Q)"} and [\[EqEpsilon\]](#EqEpsilon){reference-type="eqref" reference="EqEpsilon"} are injective.* *Proof.* Consider a faithful representation $\rho\colon\mathcal{G}\hookrightarrow\operatorname{GL}({\mathcal{V}})$ as in [@AH_Unif Proposition 2.2(a)] for a vector bundle $\mathcal{V}$ on $X$. Then $\rho_*\eta$ induces a quasi-isogeny of the vector bundle $M:=\rho_*{\mathbb{E}}$ on $X_S$ associated with $\rho_*{\underline{\mathbb{E}}}$. If $\eta$ lies in the kernel of $\zeta$ then its restriction to $X_{\overline\mathbb{F}_\infty}\smallsetminus\{\infty\}$ is the identity on $M$ because of [@AH_Local Proposition 3.4]. Therefore $\eta$ must be the identity. This proves that $\zeta$ is injective. On the other hand, $L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(M)$ is the completion of $M$ at the graph $\Gamma_{x}$ of $x\colon\operatorname{Spec}{\overline\mathbb{F}_\infty}\to X$. Since this completion functor is faithful, also $L_{\infty,\mathcal{M}_{\beta^{-1}}}\colon I_{{\underline{\mathbb{E}}}}(Q) \to J_{{\underline{{\mathbb{L}}}}}(Q_\infty)$ is injective. ◻ By Lemma [Lemma 1](#injective-L-and-zeta){reference-type="ref" reference="injective-L-and-zeta"}, we have the following injective morphism $$(L_{\infty,\mathcal{M}_{\beta^{-1}}},\zeta)\colon I_{{\underline{\mathbb{E}}}}(Q)\hookrightarrow J_{{\underline{{\mathbb{L}}}}}(Q_\infty)\times G(\mathbb{A}^\infty)$$ and we identify $I_{{\underline{\mathbb{E}}}}(Q)$ with its image. **Lemma 1**. *$I_{{\underline{\mathbb{E}}}}(Q)$ is a discrete subgroup of $J_{{\underline{{\mathbb{L}}}}}(Q_\infty)\times G(\mathbb{A}^\infty)$.* *Proof.* To show this, we take an open subgroup $U\subset\operatorname{Aut}_{\overline\mathbb{F}_\infty}({\underline{{\mathbb{L}}}})$ and consider the open subgroup $U\times\mathcal{G}(\mathbb{O}^\infty)\subset J_{{\underline{{\mathbb{L}}}}}(Q_\infty)\times G(\mathbb{A}^\infty)$. Since $\mathcal{G}(\mathbb{O}^\infty)=\gamma_0\operatorname{Aut}^{\otimes}(\check{{\mathcal{T}}}_{{\underline{\mathbb{E}}}})\gamma_0^{-1}$, the elements of $I_{{\underline{\mathbb{E}}}}(Q)\cap \bigl(U\times\mathcal{G}(\mathbb{O}^\infty)\bigr)$ give automorphisms of the global $\mathcal{G}$-shtuka ${\underline{\mathbb{E}}}$. Then the finiteness of $I_{{\underline{\mathbb{E}}}}(Q)\cap \bigl(U\times\mathcal{G}(\mathbb{O}^\infty)\bigr)$ follows from Corollary [Corollary 1](#CorAutFinite){reference-type="ref" reference="CorAutFinite"}. ◻ To state the properties of the source of $\Theta_{{\underline{\mathbb{E}}}}$, we say that a formal algebraic Deligne-Mumford stack $\mathcal{Y}$ over $\operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ (see [@HartlAbSh Definition A.5]) is *$\mathcal{I}$-adic* for a sheaf of ideals $\mathcal{I}\subset\mathcal{O}_\mathcal{Y}$, if for some (any) presentation $Y\to\mathcal{Y}$, the formal scheme $Y$ is $\mathcal{I}\mathcal{O}_Y$-adic, i.e. $\mathcal{I}^r\mathcal{O}_Y$ is an ideal of definition of $Y$ for all $r\in \mathbb{N}_{>0}$. We then call $\mathcal{I}$ an *ideal of definition of $\mathcal{Y}$*. We say that $\mathcal{Y}$ is *locally formally of finite type* over $\operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ if $\mathcal{Y}$ is locally noetherian, adic, and if the closed substack defined by the largest ideal of definition (see [@HartlAbSh A.7]) is an algebraic stack locally of finite type over $\operatorname{Spec}{\overline\mathbb{F}_\infty}$. Let $H\subset G(\mathbb{A}^\infty)$ be a compact open subgroup. Consider the countable double coset $$\label{double-coset-IF} I_{{\underline{\mathbb{E}}}}(Q) \backslash\operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\cong I_{{\underline{\mathbb{E}}}}(Q) \backslash G(\mathbb{A}^\infty)/H.$$ For $\bar\gamma:=\gamma H\in \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H$ the coset $h_\gamma H\subset G(\mathbb{A}^\infty)/H$ of $h_\gamma:=\gamma_0^{-1}\gamma \in G(\mathbb{A}^\infty)$ is well defined. We set $$\Gamma_{\bar\gamma}\;:=\; I_{{\underline{\mathbb{E}}}}(Q)\cap \bigl(J_{{\underline{{\mathbb{L}}}}}(Q_\infty)\times h_\gamma H h_\gamma^{-1}\bigr)\;\subset\; J_{{\underline{{\mathbb{L}}}}}(Q_\infty).$$ This is a discrete subgroup for the $\infty$-adic topology by Lemma [Lemma 1](#LemmaIisDiscrete){reference-type="ref" reference="LemmaIisDiscrete"}. Moreover, $\Gamma_{\overline{\gamma}}$ is separated in the profinite topology, i.e. for every $1\ne \eta \in \Gamma_{\bar\gamma}$, there is a normal subgroup of finite index in $\Gamma_{\bar\gamma}$ that does not contain $g$. Indeed, there is a normal subgroup $\widetilde H\subset H$ of finite index such that $h_\gamma\widetilde H h_\gamma^{-1}$ does not contain the element $\zeta(\eta)\ne1$. **Proposition 1**. *(a) The quotient of $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$ by the abstract group $I_{{\underline{\mathbb{E}}}}(Q)$ exists as a locally noetherian, adic, formal algebraic Deligne-Mumford stack locally formally of finite type over $\operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ and is given by the following disjoint union $$\label{EqSourceOfTheta} I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr) \enspace\cong\enspace \coprod_{\bar\gamma} \Gamma_{\bar\gamma}\big{\backslash}\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\,.$$ Here $\bar\gamma:=\gamma H\in\operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$ runs through a set of representatives for the double coset [\[double-coset-IF\]](#double-coset-IF){reference-type="eqref" reference="double-coset-IF"}.* *(b) The quotient morphisms $$\begin{aligned} \label{EqPropQuotientByIEtale} \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu} & \enspace \mbox{$\kern 2pt\to\kern-8pt\to\kern 2pt$}\enspace \Gamma_{\bar\gamma}\big{\backslash}\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu} \qquad \text{and} \nonumber \\ \bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr) & \enspace \mbox{$\kern 2pt\to\kern-8pt\to\kern 2pt$}\enspace I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr) \end{aligned}$$ are adic and étale.* *(c) In particular, the closed substack of [\[EqSourceOfTheta\]](#EqSourceOfTheta){reference-type="eqref" reference="EqSourceOfTheta"} defined by the largest ideal of definition is the Deligne-Mumford stack locally of finite type over $\operatorname{Spec}{\overline\mathbb{F}_\infty}$ given by $$\label{EqReducedSourceOfTheta} I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H \enspace\cong\enspace \coprod_{\bar\gamma} \Gamma_{\bar\gamma}\big{\backslash}X_{\mathcal{M}}^{\leq \mu}(b)\,.$$* **Remark 1**. Note that all unipotent subgroups of $J_{{\underline{{\mathbb{L}}}}}(Q_\infty)$ are torsion, so they might not act fixed-point-freely on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq\mu}$ and $X_{\mathcal{M}}^{\leq\mu}(b)$, thus the quotients [\[EqSourceOfTheta\]](#EqSourceOfTheta){reference-type="eqref" reference="EqSourceOfTheta"} and [\[EqReducedSourceOfTheta\]](#EqReducedSourceOfTheta){reference-type="eqref" reference="EqReducedSourceOfTheta"} may not be (formal) schemes. *Proof of Proposition [Proposition 1](#PropQuotientByI){reference-type="ref" reference="PropQuotientByI"}.* The quotients $\Gamma_{\bar\gamma}\big{\backslash}\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ and $I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr)$ are formal algebraic Deligne-Mumford stacks by [@AH_Local Proposition 4.27]. That they are Deligne-Mumford and the last assertion about [\[EqReducedSourceOfTheta\]](#EqReducedSourceOfTheta){reference-type="eqref" reference="EqReducedSourceOfTheta"} follow from the proof of [@AH_Local Proposition 4.27] where it is shown that [\[EqReducedSourceOfTheta\]](#EqReducedSourceOfTheta){reference-type="eqref" reference="EqReducedSourceOfTheta"} (respectively [\[EqSourceOfTheta\]](#EqSourceOfTheta){reference-type="eqref" reference="EqSourceOfTheta"}) are locally the stack quotient of a (formal) scheme by a finite group. ◻ ## Various actions on the source and target We keep the situation of Section [3.1](#subsec:SourceTarget){reference-type="ref" reference="subsec:SourceTarget"}. There is an action of $G(\mathbb{A}^\infty)$ by Hecke correspondences, which is explicitly given as follows. **Definition 1**. Let $H,H'\subset G(\mathbb{A}^\infty)$ be compact open subgroups and let $h\in G(\mathbb{A}^\infty)$. We define the Hecke correspondence $\pi(h)_{H'\!,H}$ by the two diagrams: the local case $$\label{EqHeckeSource} \xymatrix @C=-4pc { & \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/(hHh^{-1}\cap H') \ar[ld] \ar[rd]& \\ \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H& & \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H'\ar@{-->}[ll]_{\pi(h)_{H',H}}\\ }$$ On the $S$-valued points, it is given by $$\begin{tikzcd} & (\underline{\mathcal{L}},\hat{\delta}) \times \gamma(hHh^{-1}\cap H')\arrow[mapsto]{ld}{} \arrow[]{rd}{} \\ (\underline{\mathcal{L}},\hat{\delta}) \times \gamma hH & & (\underline{\mathcal{L}},\hat{\delta}) \times \gamma H' \arrow[dotted,mapsto]{ll}{} \end{tikzcd}$$ We define the Hecke correspondence in the global case by $$\label{EqHeckeTarget} \begin{tikzcd} & \operatorname{Sht}_{\mathcal{G},(hHh^{-1}\cap H'),\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)} \arrow[]{ld}{} \arrow[]{rd}{} & \\ \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)} & & \operatorname{Sht}_{\mathcal{G},H'\!,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)} \arrow[dotted]{ll}{\pi(h)_{H',H}} \end{tikzcd}$$ On the level of $S$-points, it is given by $$\begin{tikzcd} & (\underline{\mathcal{E}},\gamma(hHh^{-1}\cap H')) \arrow[mapsto]{ld}{} \arrow[mapsto]{rd}{} \\ (\underline{\mathcal{E}},\gamma hH) & & (\underline{\mathcal{E}},\gamma H') \arrow[dotted,mapsto]{ll}{} \end{tikzcd}$$ **Example 1**. A special case for $H'\subset H$ and $h=1$ are the forgetful morphisms $$\pi(1)_{H'\!,H}\colon\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H'\to\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$$ and $\pi(1)_{H'\!,H}\colon\operatorname{Sht}_{\mathcal{G},H'\!,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\to\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}$, which are finite étale and surjective; see Proposition [Proposition 1](#PropLSGGsht1){reference-type="ref" reference="PropLSGGsht1"}[\[PropLSGGsht1_B\]](#PropLSGGsht1_B){reference-type="ref" reference="PropLSGGsht1_B"}. ** 1**. There is a another group acting on the source and the target of the morphism $\Theta_{{\underline{\mathbb{E}}}}$ from [\[EqUnifIntro\]](#EqUnifIntro){reference-type="eqref" reference="EqUnifIntro"}, namely the group $Z(Q_\infty)$ where $Z\subset \mathcal{G}\times_X\operatorname{Spec}Q=G$ is the center. Since the center of $M$ and that of $G_\infty$ coincide, $Z(Q_\infty)$ lies in the center of $L_\infty\mathcal{M}(\overline{\mathbb{F}}_\infty)$. Recall the maps $\tau_{G_\infty}$ and $\tau_M$ from [\[Eq_tau_M\]](#Eq_tau_M){reference-type="eqref" reference="Eq_tau_M"}. Writing ${\underline{{\mathbb{L}}}}\cong\bigl((L^+_\infty\mathcal{M})_{\overline\mathbb{F}_\infty},b\bigr)$ with $b\in L_\infty\mathcal{M}({\overline\mathbb{F}_\infty})=L_\infty\mathcal{G}({\overline\mathbb{F}_\infty})$, there is an inclusion $$\begin{aligned} \label{EqActionCenter1} \begin{split} Z(Q_\infty) \enspace\lhook\joinrel\longrightarrow\enspace & J_{{\underline{{\mathbb{L}}}}}(Q_\infty)\;=\;\bigl\{\,j\in L_\infty\mathcal{M}({\overline\mathbb{F}_\infty})\colon \tau_M(j)\, b=b\,j\,\bigr\}\;=\;\operatorname{QIsog}_{\overline\mathbb{F}_\infty}({\underline{{\mathbb{L}}}})\,, \\ c \quad\enspace\longmapsto\enspace & \enspace\quad c_{\underline{{\mathbb{L}}}}\enspace\; := \quad c=\tau_{G_\infty}(c) = \beta^{-1}\tau_{G_\infty}(c) \beta = \tau_M(c) = b^{-1}\tau_M(c)b \end{split}\end{aligned}$$ through which $c\in Z(Q_\infty)$ acts on $(\underline{\mathcal{L}},\hat{\delta})\in\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ via $c\colon (\underline{\mathcal{L}},\hat{\delta})\mapsto (\underline{\mathcal{L}},c_{{\underline{{\mathbb{L}}}}}\circ\hat{\delta})$, where we denote the image of $c$ inside $\operatorname{QIsog}_{\overline{\mathbb{F}}_\infty}({\underline{{\mathbb{L}}}})$ by $c_{{\underline{{\mathbb{L}}}}}$ whenever confusion may arise. This action can also be described in a different way as follows. For any local $\mathcal{M}$-shtuka $\underline{\mathcal{L}}$ over a scheme $S\in\mathcal{N}ilp_{\breve\mathcal{O}_\infty}$, we claim that $c$ induces an element $c_{\underline{\mathcal{L}}}\in\operatorname{QIsog}_S(\underline{\mathcal{L}})$ of the quasi-isogeny group of $\underline{\mathcal{L}}$ such that $\hat{\delta}\circ c_{\underline{\mathcal{L}}}=c_{{\underline{{\mathbb{L}}}}}\circ\hat{\delta}$. Over an étale covering $S'\to S$, we can choose a trivialization $\alpha\colon\underline{\mathcal{L}}_{S'}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\bigl((L^+_\infty\mathcal{M})_{S'},b'\bigr)$ for $b'\in L_\infty\mathcal{M}(S')$. Then $c=\tau_M(c)$ implies $b'c= \tau_M(c)b'$, i.e. $\alpha^{-1}\circ c\circ\alpha\in\operatorname{QIsog}_{S'}(\underline{\mathcal{L}})$. Next we show that this quasi-isogeny descends to $S$. Let $\operatorname{pr}_1,\operatorname{pr}_2\colon S'':=S'\times_S S' \to S'$ be the projections onto the first and second factor. Set $h:=\operatorname{pr}_2^*\alpha\circ \operatorname{pr}_1^*\alpha^{-1}\in L^+_\infty\mathcal{M}(S'')$. Then $h \circ \operatorname{pr}_1^*c\circ h^{-1}=\operatorname{pr}_1^*c = \operatorname{pr}_2^*c$ implies $$\operatorname{pr}_1^*(\alpha^{-1}\circ c\circ\alpha)\;=\;\operatorname{pr}_2^*\alpha^{-1}\circ h \circ \operatorname{pr}_1^*c\circ h^{-1}\circ \operatorname{pr}_2^*\alpha\;=\;\operatorname{pr}_2^*(\alpha^{-1}\circ c\circ\alpha)\,.$$ Therefore, $\alpha^{-1}\circ c\circ\alpha$ descends to a quasi-isogeny of $\underline{\mathcal{L}}$ over $S$ which we denote by $c_{\underline{\mathcal{L}}}\in\operatorname{QIsog}_S(\underline{\mathcal{L}})$. If moreover we are given a quasi-isogeny $\hat{\delta}\colon\underline{\mathcal{L}}\to {\underline{{\mathbb{L}}}}_{S}$, i.e. if $(\underline{\mathcal{L}},\hat{\delta})\in\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S)$, then $\hat{\delta}$ is automatically compatible with the quasi-isogenies $c_{\underline{\mathcal{L}}}\in\operatorname{QIsog}_S(\underline{\mathcal{L}})$ and $c_{{\underline{{\mathbb{L}}}}}\in\operatorname{QIsog}_{\overline\mathbb{F}_\infty}({\underline{{\mathbb{L}}}})$, i.e. $\hat{\delta}\circ c_{\underline{\mathcal{L}}}=c_{{\underline{{\mathbb{L}}}}}\circ\hat{\delta}$. To see this: using the trivialization $\alpha$ over $S'$ from above, $\hat{\delta}$ corresponds to $g:=\hat{\delta}\circ\alpha^{-1}\in L_\infty\mathcal{G}(S')$. Thus $$\hat{\delta}\circ c_{\underline{\mathcal{L}}}\;:=\;\hat{\delta}\circ(\alpha^{-1}\circ c\circ\alpha)\;=\;g\circ c\circ\alpha\;=\;c\circ g\circ\alpha\;=\;c_{{\underline{{\mathbb{L}}}}}\circ\hat{\delta}\,.$$ This shows that the action of $c\in Z(Q_\infty)$ on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ is given as $$\label{EqActionCenter3} c\colon\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\;\longrightarrow\;\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\,,\quad (\underline{\mathcal{L}},\hat{\delta})\;\longmapsto\; (\underline{\mathcal{L}},\hat{\delta}\circ c_{\underline{\mathcal{L}}})\;=\;(\underline{\mathcal{L}},c_{{\underline{{\mathbb{L}}}}}\circ\hat{\delta})\,.$$ As such, it does not matter which quasi-isogeny group we realize $Z(Q_\infty)$ in. The action of $Z(Q_\infty)$ on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ commutes with the action of $\eta\in I_{{\underline{\mathbb{E}}}}(Q)$, as $\eta$ and $c$ act on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$ by $$(\underline{\mathcal{L}},\hat{\delta}) \times \gamma H\;\longmapsto\;(\underline{\mathcal{L}},L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)\circ\hat{\delta}\circ c) \times \check{{\mathcal{V}}}_\eta\gamma H\,.$$ On the other hand, $c\in Z(Q_\infty)$ also acts on the target $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}$ of the map $\Theta_{{\underline{\mathbb{E}}}}$ as follows. Let $(\underline{\mathcal{E}},\gamma H)$ be in $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)$ for some $S\in\mathcal{N}ilp_{\Breve{\mathcal{O}}_{\mu,\beta}}$. Consider the associated local $\mathcal{M}$-shtuka $\underline{\mathcal{L}}:=L^+_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})$. We have seen above that $c$ induces an element $c_{\underline{\mathcal{L}}}\in\operatorname{QIsog}_S(\underline{\mathcal{L}})$ of the quasi-isogeny group of $\underline{\mathcal{L}}$. By Proposition [Proposition 1](#PropQIsogLocalGlobal){reference-type="ref" reference="PropQIsogLocalGlobal"}[\[PropQIsogLocalGlobal_B\]](#PropQIsogLocalGlobal_B){reference-type="ref" reference="PropQIsogLocalGlobal_B"}, there is a uniquely determined global $\mathcal{G}$-shtuka $c^*\,\underline{\mathcal{E}}$ and a quasi-isogeny $c_{\underline{\mathcal{E}}}\colon c^*\,\underline{\mathcal{E}}\to\underline{\mathcal{E}}$, which is an isomorphism outside $\infty$ and satisfies $L_{\infty,\mathcal{M}_{\beta^{-1}}}(c_{\underline{\mathcal{E}}})=c_{\underline{\mathcal{L}}}$. We can now define the action of $c\in Z(Q_\infty)$ on $(\underline{\mathcal{E}},\gamma H)\in\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)$ as $$\label{EqActionCenter2} c\colon\;(\underline{\mathcal{E}},\gamma H)\;\longmapsto\;(c^*\,\underline{\mathcal{E}}\,,\,\check{\mathcal{V}}_{c_{\underline{\mathcal{E}}}}^{-1}\gamma H)\,.$$ ** 1**. The source and target of $\Theta_{{\underline{\mathbb{E}}}}$ carry a *Weil-descent datum* for the ring extension ${\mathcal{O}_{\mu,\beta}}\subset{\Breve{\mathcal{O}}_{\mu,\beta}}$, compare [@RZ Definition 3.45]. We explain what this means. Recall that ${\kappa_{\mu,\beta}}$ is the residue field of ${\mathcal{O}_{\mu,\beta}}$. Consider the ${\mathcal{O}_{\mu,\beta}}$-automorphism $\lambda$ of ${\Breve{\mathcal{O}}_{\mu,\beta}}$ inducing $$\label{EqWeilDescent_lambda} \lambda|_{\overline\mathbb{F}_\infty}\;:=\;\operatorname{Frob}_{\#{\kappa_{\mu,\beta}},{\overline\mathbb{F}_\infty}}\colon x\;\longmapsto\;x^{\#{\kappa_{\mu,\beta}}}\quad\text{for }x\in{\overline\mathbb{F}_\infty}$$ on the residue field $\overline{\mathbb{F}}_\infty$ of ${\Breve{\mathcal{O}}_{\mu,\beta}}$. For a scheme $(S,\theta)\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$, where $\theta\colon S\to\operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ denotes the structure morphism of the scheme $S$, we denote by $S_{[\lambda]}\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$ the pair ($S,\lambda\circ\theta)$. For a stack $\mathcal{H}$ over $\operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$, we define the stack ${}^\lambda\mathcal{H}$ by $${}^\lambda\mathcal{H}(S)\;:=\;\mathcal{H}(S_{[\lambda]})\,.$$ **Definition 1**. A *Weil-descent datum* on $\mathcal{H}$ is an isomorphism of stacks $\mathcal{H}\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}{}^\lambda\mathcal{H}$, i.e. an equivalence $\mathcal{H}(S)\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\mathcal{H}(S_{[\lambda]})$ for every $S\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$ compatible with morphisms in $\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$. Let $S\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$. Under the inclusion $\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}\hookrightarrow\mathcal{N}ilp_{{\mathcal{O}_{\mu,\beta}}}$, we have $S=S_{[\lambda]}$ in $\mathcal{N}ilp_{{\mathcal{O}_{\mu,\beta}}}$. Therefore, on $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$, the canonical Weil-descent datum is given by the identity $$\label{EqDescentDatumOnNablaH} \operatorname{id}\colon\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)\;\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\;\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S_{[\lambda]})\,,\quad(\underline{\mathcal{E}},\gamma H)\;\longmapsto\;(\underline{\mathcal{E}},\gamma H).$$ On $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$, we consider the Weil descent datum given by $$\begin{aligned} \label{EqDescentDatumSource} \begin{split} \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S)\;\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\; & \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S_{[\lambda]}),\\ (\underline{\mathcal{L}},\hat{\delta}\colon\underline{\mathcal{L}}\to{\underline{{\mathbb{L}}}}_{S})\;\longmapsto\; & (\underline{\mathcal{L}},\theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]})\circ\hat{\delta}\colon\underline{\mathcal{L}}\to{\underline{{\mathbb{L}}}}_{S_{[\lambda]}})\,, \end{split}\end{aligned}$$ where $\widehat{\varphi}_{{\mathbb{L}}}$ is the Frobenius of the local shtuka ${\underline{{\mathbb{L}}}}=({\underline{{\mathbb{L}}}},\widehat{\varphi}_{{\mathbb{L}}})$. Here we observe that ${\underline{{\mathbb{L}}}}_{S}:=\theta^*{\underline{{\mathbb{L}}}}$ and ${\underline{{\mathbb{L}}}}_{S_{[\lambda]}}:=(\lambda\circ\theta)^*{\underline{{\mathbb{L}}}}=\theta^*\lambda^*{\underline{{\mathbb{L}}}}=\theta^*({}^{\tau^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}}{\underline{{\mathbb{L}}}})$, and that $\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}\colon{\underline{{\mathbb{L}}}}\to{}^{\tau^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}}{\underline{{\mathbb{L}}}}$ is a quasi-isogeny. **Remark 1**. On on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ there is even a Weil descent datum for the ring extension $\mathcal{O}_\mu\subset \Breve{\mathcal{O}}_\mu$. It is defined analogously by replacing ${\kappa_{\mu,\beta}}$ in [\[EqWeilDescent_lambda\]](#EqWeilDescent_lambda){reference-type="eqref" reference="EqWeilDescent_lambda"} and [\[EqDescentDatumSource\]](#EqDescentDatumSource){reference-type="eqref" reference="EqDescentDatumSource"} by $\mathcal{O}_\mu$. We do not discuss the question whether this Weil descent datum on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ is effective. In the analogous situation for $p$-divisible groups, this is true and proven by Rapoport and Zink in [@RZ Theorem 3.49]. Their argument uses a morphism $\mathbb{G}_m\to\mathcal{G}$, which might not exist in our setup. Moreover, on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$ we consider the product of the Weil Descent datum [\[EqDescentDatumSource\]](#EqDescentDatumSource){reference-type="eqref" reference="EqDescentDatumSource"} with the identity on $\operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$. Let $\eta\in I_{{\underline{\mathbb{E}}}}(Q)$. Since $$\begin{aligned} \theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}\circ L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)) & =\theta^*\Bigl({}^{\tau^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}}(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta))\circ\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}\Bigr) \\ & =(\lambda\circ\theta)^*(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta))\circ\theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}),\end{aligned}$$ this product Weil descent datum commutes with the action of $I_{{\underline{\mathbb{E}}}}(Q)$ via the following diagram $$\xymatrix @C+2pc @R=1pc { (\underline{\mathcal{L}},\hat{\delta}) \ar@{|->}[r]^-{\textstyle\eta} \ar@{|->}[dd]^(0.45){\textstyle\text{Weil descent}} & (\underline{\mathcal{L}},\theta^*(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta))\circ\hat{\delta}) \ar@{|->}[dd]^(0.45){\textstyle\text{Weil descent}} \\ \\ (\underline{\mathcal{L}},\theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]})\circ\hat{\delta}) \ar@{|->}[rd]^-{\textstyle\eta} & (\underline{\mathcal{L}},\theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]})\circ\theta^*(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta))\circ\hat{\delta}) \ar@{=}[d]\\ & \;(\underline{\mathcal{L}},(\lambda\circ\theta)^*(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta))\circ\theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]})\circ\hat{\delta})\,. }$$ This defines a Weil descent datum on $\mathcal{Y}:=I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$ by $$\begin{aligned} \label{EqDescentDatumSourceModI} \mathcal{Y}(S) \;\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\; & {}^\lambda\mathcal{Y}(S)\;=\;\mathcal{Y}(S_{[\lambda]}) \\ (\underline{\mathcal{L}},\hat{\delta}, \gamma H)\;\longmapsto\; & (\underline{\mathcal{L}},\theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]})\circ\hat{\delta}, \gamma H)\,. \nonumber\end{aligned}$$ ** 1**. Now we define the Frobenius endomorphism on the source and target of $\Theta_{{\underline{\mathbb{E}}}}$. For every multiple $m\in\mathbb{N}_0$ of $[\mathcal{O}_\mu:\mathbb{F}_q]$, the special fiber $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\Breve{\mathcal{O}}_{\mu,\beta}}}\operatorname{Spec}{\overline\mathbb{F}_\infty}$ of $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ carries a Frobenius endomorphism structure $\Phi_m$ defined as follows. Consider the absolute $q^m$-Frobenius $\tau^m:=\operatorname{Frob}_{q^m,S}\colon S\to S$ on an ${\overline\mathbb{F}_\infty}$-scheme $S$. Consider a pair $(\underline{\mathcal{L}},\hat{\delta})\in\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S)$, which induces the left horizontal morphisms in the diagram $$\label{qm-Frob-diagram} \xymatrix @C+1pc { S \ar[d]_{\textstyle\operatorname{Frob}_{q^m,S}} \ar[rrrr]^-{\textstyle(\underline{\mathcal{L}},\hat{\delta})} \ar[drrrr]_(.35){\textstyle({}^{\tau^m\!}\underline{\mathcal{L}},{}^{\tau^m\!}\hat{\delta})\qquad} & & & & \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\Breve{\mathcal{O}}_{\mu,\beta}}}\operatorname{Spec}{\overline\mathbb{F}_\infty}\ar[d]_{\textstyle\operatorname{Frob}_{q^m}} \ar[r] & \operatorname{Spec}{\overline\mathbb{F}_\infty}\ar[d]_{\textstyle\operatorname{Frob}_{q^m,{\overline\mathbb{F}_\infty}}} \\ S \ar[rrrr]_-{\textstyle(\underline{\mathcal{L}},\hat{\delta})} & & & &\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\Breve{\mathcal{O}}_{\mu,\beta}}}\operatorname{Spec}{\overline\mathbb{F}_\infty}\ar[r] & \operatorname{Spec}{\overline\mathbb{F}_\infty} }$$ Let $\theta\colon S\to\operatorname{Spec}{\overline\mathbb{F}_\infty}$ be the structure morphism of $S$. Then the upper-left $S$ in diagram [\[qm-Frob-diagram\]](#qm-Frob-diagram){reference-type="eqref" reference="qm-Frob-diagram"}, viewed as a scheme over the lower-right $\operatorname{Spec}{\overline\mathbb{F}_\infty}$, has structure morphism $\operatorname{Frob}_{q^m,{\overline\mathbb{F}_\infty}}\circ\theta$. Thus in terms of §[ 1](#Weil-descent-paragraph){reference-type="ref" reference="Weil-descent-paragraph"}, taking $\lambda:=\lambda_m:=\operatorname{Frob}_{q^m,{\overline\mathbb{F}_\infty}}$, the upper-left $S$ becomes $S_{[\lambda_m]}$ over the lower-right $\operatorname{Spec}{\overline\mathbb{F}_\infty}$. Let $\mathcal{Z}(\mu):=\mathcal{Z}^{\leq\mu}$ be the bound from Definition [Definition 1](#Def_BoundBy_mu){reference-type="ref" reference="Def_BoundBy_mu"}, which we used as the bound on $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq\mu}$ from Definition [Definition 1](#DefRZforM){reference-type="ref" reference="DefRZforM"}. Its special fiber $\mathcal{Z}(\mu)_\infty:=\mathcal{Z}(\mu)\times_{\widetilde{X}_\mu} \operatorname{Spec}\kappa_\mu$ is defined over $\kappa_\mu$. Since $m$ is a multiple of $[\kappa_\mu\colon\mathbb{F}_q]$, the Frobenius $\widehat{\varphi}_{{}^{\tau^m\!}\mathcal{L}}={}^{\tau^m\!}\widehat{\varphi}_{\mathcal{L}}$ lies in ${}^{\tau^m\!}\mathcal{Z}(\mu)_\infty=\mathcal{Z}(\mu)_\infty$. Thus ${}^{\tau^m\!}\underline{\mathcal{L}}$ is also bounded by $\mathcal{Z}(\mu)$, and therefore the diagonal arrow $({}^{\tau^m\!}\underline{\mathcal{L}},{}^{\tau^m\!}\hat{\delta})$ lies in $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S_{[\lambda_m]})={}^{\lambda_m}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\bigr)(S)$. Via the Weil descent datum [\[EqDescentDatumSource\]](#EqDescentDatumSource){reference-type="eqref" reference="EqDescentDatumSource"}, $({}^{\tau^m\!}\underline{\mathcal{L}},{}^{\tau^m\!}\hat{\delta})$ is mapped to $({}^{\tau^m\!}\underline{\mathcal{L}}, \theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{\;-m})\circ{}^{\tau^m\!}\hat{\delta})$, which lies in $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S)$. We therefore define the $q^m$-Frobenius endomorphism of $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{R}\operatorname{Spec}{\overline\mathbb{F}_\infty}$ as $$\begin{aligned} \label{EqFrobOnRZ} \Phi_m\colon\;\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{R}\operatorname{Spec}{\overline\mathbb{F}_\infty}\;\longrightarrow\; & \bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{R}\operatorname{Spec}{\overline\mathbb{F}_\infty}\bigr) \\ (\underline{\mathcal{L}},\hat{\delta}) \;\longmapsto\; & ({}^{\tau^m\!}\underline{\mathcal{L}}, \widehat{\varphi}_{{\mathbb{L}}}^{\;-m}\circ{}^{\tau^m\!}\hat{\delta})\;=\;({}^{\tau^m\!}\underline{\mathcal{L}}, \hat{\delta}\circ\widehat{\varphi}_{\mathcal{L}}^{\;-m})\,. \nonumber\end{aligned}$$ The product of the $q^m$-Frobenius endomorphism $\Phi_m$ from [\[EqFrobOnRZ\]](#EqFrobOnRZ){reference-type="eqref" reference="EqFrobOnRZ"} with the identity on $\operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$ gives a $q^m$-Frobenius morphism (which we again denote by $\Phi_m$) $$\begin{aligned} \label{EqFrobOnSource} \Phi_m\colon\;\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{R}\operatorname{Spec}{\overline\mathbb{F}_\infty}\bigr) & \times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\;\longrightarrow\; \\ & \longrightarrow\; \bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{R}\operatorname{Spec}{\overline\mathbb{F}_\infty}\bigr)\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\,.\nonumber\end{aligned}$$ Since the composition of $\eta\in I_{{\underline{\mathbb{E}}}}(Q)$ and $\Phi_m$ is given by $$(\underline{\mathcal{L}},\hat{\delta}) \times \gamma H\;\longmapsto\;(\underline{\mathcal{L}},L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)\circ\hat{\delta}\circ\widehat{\varphi}_{\mathcal{L}}^{\;-m}) \times \check{{\mathcal{V}}}_\eta\gamma H,$$ it follows that $\Phi_m$ commutes with the action of $I_{{\underline{\mathbb{E}}}}(Q)$. This defines the $q^m$-Frobenius endomorphism $\Phi_m$ of the source $I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$ of $\Theta_{{\underline{\mathbb{E}}}}$ . Now we discuss the Frobenius endomorphisms on the target. Let $m\in\mathbb{N}_0$ be a multiple of $[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]$. Since $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\times_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spec}{\overline\mathbb{F}_\infty}=(\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\times_{{\mathcal{O}_{\mu,\beta}}} {\kappa_{\mu,\beta}})\times_{{\kappa_{\mu,\beta}}}\operatorname{Spec}{\overline\mathbb{F}_\infty}$, the map $\operatorname{id}\times\tau^m$ defines the relative $q^m$-Frobenius of the left-hand side, which is an endomorphism, because this stack arises by base change from $\operatorname{Spec}{\kappa_{\mu,\beta}}$. Explicitly, this $q^m$-Frobenius endomorphism is given by $$\begin{aligned} \label{EqFrobOnTarget} \Phi_m\colon \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spec}{\overline\mathbb{F}_\infty}\;\longrightarrow\; & \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spec}{\overline\mathbb{F}_\infty}\nonumber \\ (\underline{\mathcal{E}},\gamma H) \;\longmapsto\; & ({}^{\tau^m\!}\underline{\mathcal{E}},{}^{\tau^m\!}(\gamma)H)\,.\end{aligned}$$ We observe that ${}^{\tau^m\!}\underline{\mathcal{E}}$ is indeed bounded by $\mathcal{Z}(\mu,\beta)$: Since $m$ is a multiple of $[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]$ and the special fiber $\mathcal{Z}(\mu,\beta)_\infty:={\mathcal{Z}(\mu,\beta)}\times_{\widetilde{X}_{\mu,\beta}} \operatorname{Spec}{\kappa_{\mu,\beta}}$ of the bound ${\mathcal{Z}(\mu,\beta)}$ is defined over ${\kappa_{\mu,\beta}}$, it follows that ${}^{\tau^m\!}\underline{\mathcal{E}}$ is bounded by ${}^{\tau^m\!}\mathcal{Z}(\mu,\beta)_\infty=\mathcal{Z}(\mu,\beta)_\infty$. ## Statement of the uniformization theorem In this subsection, we define the uniformization morphism and state our main results, Theorems [Theorem 1](#Uniformization1){reference-type="ref" reference="Uniformization1"} and [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"}, which will be proven in Section [3.4](#subsec:ProofMainThms){reference-type="ref" reference="subsec:ProofMainThms"}. Recall from §[ 1](#SetupUniformization){reference-type="ref" reference="SetupUniformization"} that we fix a global $\mathcal{G}$-shtuka ${\underline{\mathbb{E}}}\in\operatorname{Sht}_{\mathcal{G},\varnothing,X\times\infty}^{\mathcal{Z}(\mu,\beta)}(\overline{\mathbb{F}}_\infty)$ and its associated local $\mathcal{M}$-shtuka ${\underline{{\mathbb{L}}}}:=L^+_{\infty,\mathcal{M}_{\beta^{-1}}}({\underline{\mathbb{E}}})$. We fix a trivialization ${\underline{{\mathbb{L}}}}\cong\left((L^+_\infty\mathcal{M})_{\overline{\mathbb{F}}_\infty},b\right)$. ** 1**. To define the uniformization map $\Theta_{{\underline{\mathbb{E}}}}$, let $$(\underline{\mathcal{L}},\hat{\delta}:\underline{\mathcal{L}}\to {\underline{{\mathbb{L}}}}_S)\in \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq\mu}(S), \quad\text{for }S\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}},$$ where $\underline{\mathcal{L}}\in \mathrm{LocSht}_{\mathcal{M}}^{\leq \mu}(S)$ is a local $\mathcal{M}$-shtuka bounded by $\mu$. By Proposition [Proposition 1](#PropQIsogLocalGlobal){reference-type="ref" reference="PropQIsogLocalGlobal"}[\[PropQIsogLocalGlobal_B\]](#PropQIsogLocalGlobal_B){reference-type="ref" reference="PropQIsogLocalGlobal_B"}, there is a global $\mathcal{G}$-shtuka $\hat{\delta}^*{\underline{\mathbb{E}}}$ in $\operatorname{Sht}_{\mathcal{G},\varnothing,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)$ and a quasi-isogeny $\delta\colon \hat{\delta}^*{\underline{\mathbb{E}}}\to {\underline{\mathbb{E}}}$ with $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta) = \hat{\delta}$. Since $\delta$ is defined over $S$, the isomorphism $\check{\mathcal{V}}_\delta$ is equivariant for the action of $\pi_1^{\rm \acute{e}t\/}(S,\bar s)$ which acts on $\check{\mathcal{V}}_{\underline{\mathbb{E}}}$ through the map $\pi_1^{\rm \acute{e}t\/}(S,\bar s)\to \pi_1^{\rm \acute{e}t\/}(\operatorname{Spec}\overline{\mathbb{F}}_q,\bar s)=(1)$, that is trivially. Let $H\subset G(\mathbb{A}^\infty)$ be an arbitrary compact open subgroup. In particular, the $H$-orbit $\check{\mathcal{V}}_\delta^{-1}\circ\gamma H$ of the tensor isomorphism $\check{\mathcal{V}}_\delta^{-1}\circ\gamma\colon\omega \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\check{\mathcal{V}}_{\hat{\delta}^*{\underline{\mathbb{E}}}}$ is invariant under $\pi_1^{\rm \acute{e}t\/}(S,\bar s)$. This defines the following morphism $$\label{EqTheta} \begin{split} \widetilde{\Theta}_{{\underline{\mathbb{E}}}}: \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H & \longrightarrow \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}\\ (\underline{\mathcal{L}},\hat{\delta},\gamma H)&\longmapsto (\hat{\delta}^*{\underline{\mathbb{E}}},\check{\mathcal{V}}_{\delta}^{-1}\circ \gamma H), \end{split}$$ which is obviously equivariant for the action of the center $Z(Q_\infty)$ given in [\[EqActionCenter3\]](#EqActionCenter3){reference-type="eqref" reference="EqActionCenter3"} and [\[EqActionCenter2\]](#EqActionCenter2){reference-type="eqref" reference="EqActionCenter2"}, and the action of $G(\mathbb{A}^\infty)$ through Hecke correspondences given in [\[EqHeckeSource\]](#EqHeckeSource){reference-type="eqref" reference="EqHeckeSource"} and [\[EqHeckeTarget\]](#EqHeckeTarget){reference-type="eqref" reference="EqHeckeTarget"}. **Theorem 1**. *Consider a compact open subgroup $H\subset G(\mathbb{A}^\infty)$. The morphism $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ from [\[EqTheta\]](#EqTheta){reference-type="eqref" reference="EqTheta"} is $I_{{\underline{\mathbb{E}}}}(Q)$-invariant, where $I_{{\underline{\mathbb{E}}}}(Q)$ acts trivially on the target and diagonally on the source as described in §[ 1](#Point7.6){reference-type="ref" reference="Point7.6"}. Furthermore, this morphism factors through a morphism $$\label{EqUnifMorph} \Theta_{{\underline{\mathbb{E}}}}\colon I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H \;\longrightarrow\; \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$$ of formal algebraic Deligne-Mumford stacks over $\operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ that is a monomorphism, i.e. the functor $\Theta_{{\underline{\mathbb{E}}}}$ is fully faithful, or equivalently its diagonal is an isomorphism. Both $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ and $\Theta_{{\underline{\mathbb{E}}}}$ are ind-proper and formally étale.* *Proof.* An elelemt $\eta\in I_{{\underline{\mathbb{E}}}}(Q)$ acts on the source of the morphism $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ by sending an $S$-valued point $(\underline{\mathcal{L}},\hat{\delta}) \times \gamma H$ to $(\underline{\mathcal{L}},L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)\hat{\delta})\times \check{\mathcal{V}}_\eta\circ\gamma H$. These two $S$-valued points are mapped under $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ to global $\mathcal{G}$-shtukas with $H$-level structure $(\underline{\mathcal{E}},\check{\mathcal{V}}_\delta^{-1}\gamma H)$ and $(\underline{\widetilde{\mathcal{E}}},\check{\mathcal{V}}_{\tilde\delta}^{-1}\check{\mathcal{V}}_\eta\gamma H)$ over $S$, where $\delta\colon\underline{\mathcal{E}}:=\hat{\delta}^*{\underline{\mathbb{E}}}\to{\underline{\mathbb{E}}}$ is the isogeny satisfying $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta)=\hat{\delta}$ and $\tilde\delta\colon\underline{\widetilde{\mathcal{E}}}:=(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)\hat{\delta})^*\,{\underline{\mathbb{E}}}\to{\underline{\mathbb{E}}}$ is the isogeny satisfying $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\tilde\delta)=L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)\hat{\delta}$. Since $\check{\mathcal{V}}_{\tilde\delta^{-1}\eta\delta}\circ\check{\mathcal{V}}_\delta^{-1}\gamma H=\check{\mathcal{V}}_{\tilde\delta}^{-1}\check{\mathcal{V}}_\eta\gamma H$, these two global $\mathcal{G}$-shtukas with $H$-level structure are isomorphic via the quasi-isogeny $\tilde\delta^{-1}\eta\delta\colon\underline{\mathcal{E}}\to\underline{\widetilde{\mathcal{E}}}$, which is an isomorphism at $\infty$, because $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\tilde\delta^{-1}\eta\delta)=(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)\hat{\delta})^{-1}\circ L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)\hat{\delta}=\operatorname{id}$. In other words, $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is invariant under the action of $I_{{\underline{\mathbb{E}}}}(Q)$ and factors through the morphism $\Theta_{{\underline{\mathbb{E}}}}$ from [\[EqUnifMorph\]](#EqUnifMorph){reference-type="eqref" reference="EqUnifMorph"} of formal algebraic Deligne-Mumford stacks. The remaining statements will be proven in Lemmas [Lemma 1](#LemmaThetaEtaleProper){reference-type="ref" reference="LemmaThetaEtaleProper"} through [Lemma 1](#LemmaThetaisadic){reference-type="ref" reference="LemmaThetaisadic"}. ◻ Recall that the scheme $X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$ is locally of finite type over ${\overline\mathbb{F}_\infty}$. Let $\{T_j\}$ be a set of representatives of $I_{{\underline{\mathbb{E}}}}(Q)$-orbits of its irreducible components. **Lemma 1**. 1. *[\[LemmaImageOfTheta_A\]]{#LemmaImageOfTheta_A label="LemmaImageOfTheta_A"} The image $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$ of $T_j$ under $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is a closed substack with the reduced substack structure, and each $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$ intersects only finitely many $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j')$ for $j'\neq j$.* 2. *[\[LemmaImageOfTheta_B\]]{#LemmaImageOfTheta_B label="LemmaImageOfTheta_B"} Let $\mathcal{X}_{\underline{\mathbb{E}}}$ be the union of all the $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$. Then $\mathcal{X}_{\underline{\mathbb{E}}}$ (with its reduced structure) is a separated Deligne-Mumford stack over $\overline{\mathbb{F}}_q$. Its underlying set $|\mathcal{X}_{\underline{\mathbb{E}}}|\subset\operatorname{Sht}_{\mathcal{G},H,\infty\times\infty}^{\mathcal{Z}(\mu,\beta)}\times_{{\kappa_{\mu,\beta}}} \operatorname{Spec}\overline{\mathbb{F}}_q$ is the isogeny class of ${\underline{\mathbb{E}}}$, i.e. the set of all $(\underline{\mathcal{E}},\gamma H)$ for which $\underline{\mathcal{E}}$ is isogenous to ${\underline{\mathbb{E}}}$.* *Moreover, the morphism $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ restricted to $X_{\mathcal{M}}^{\leq \mu}(b)$ factors through a map $\iota_{\mathcal{X}}:\mathcal{X}_{\underline{\mathbb{E}}}\to \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$.* *Proof.* [\[LemmaImageOfTheta_A\]](#LemmaImageOfTheta_A){reference-type="ref" reference="LemmaImageOfTheta_A"} Note that the $T_j$ correspond bijectively to the irreducible components of the Deligne-Mumford stack $I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$ from [\[EqReducedSourceOfTheta\]](#EqReducedSourceOfTheta){reference-type="eqref" reference="EqReducedSourceOfTheta"}, which is locally of finite type over $\operatorname{Spec}{\overline\mathbb{F}_\infty}$. Since $\Theta_{{\underline{\mathbb{E}}}}$ is a monomorphism by Theorem [Theorem 1](#Uniformization1){reference-type="ref" reference="Uniformization1"}, each $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$ intersects only finitely many $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_{j'})$ for $j'\neq j$. Since $T_j$ is quasi-compact by [@AH_Local Corollary 4.26] and $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is ind-proper by Theorem [Theorem 1](#Uniformization1){reference-type="ref" reference="Uniformization1"}, the restriction of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ to $T_j$ is proper. Thus $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$ are closed substacks. [\[LemmaImageOfTheta_B\]](#LemmaImageOfTheta_B){reference-type="ref" reference="LemmaImageOfTheta_B"} For a field $K$, every $K$-valued point $(\underline{\mathcal{E}},\gamma H)$ of $\mathcal{X}_{\underline{\mathbb{E}}}$ lies in the image of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$, and hence is of the form $\underline{\mathcal{E}}=\hat{\delta}^*{\underline{\mathbb{E}}}$ with an isogeny $\delta\colon\underline{\mathcal{E}}\to{\underline{\mathbb{E}}}$. This shows that $\mathcal{X}_{\underline{\mathbb{E}}}$ is contained in the (quasi-)isogeny class of ${\underline{\mathbb{E}}}$. Conversely, let $(\underline{\mathcal{E}},\gamma H)$ be a $K$-valued point of $\operatorname{Sht}_{\mathcal{G},H,\infty\times\infty}^{\mathcal{Z}(\mu,\beta)}$ in the isogeny class of ${\underline{\mathbb{E}}}$, and let $\delta\colon\underline{\mathcal{E}}\to{\underline{\mathbb{E}}}_K$ be a quasi-isogeny. Let $\underline{\mathcal{L}}:=L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})$ and $\hat{\delta}:=L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta)\colon\underline{\mathcal{L}}\to{\underline{{\mathbb{L}}}}_K$ and $(\check{\mathcal{V}}_{\delta}\circ\gamma) H\in\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H$. Then $(\underline{\mathcal{L}},\hat{\delta}, (\check{\mathcal{V}}_{\delta}\circ\gamma) H)$ is a $K$-valued point of the source of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ which is mapped under $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ to $(\underline{\mathcal{E}}',(\check{\mathcal{V}}_{\delta'}^{-1}\circ(\check{\mathcal{V}}_{\delta}\circ\gamma) H)$, where $\underline{\mathcal{E}}':=\hat{\delta}^*\,{\underline{\mathbb{E}}}$ and $\delta'\colon\underline{\mathcal{E}}'\to{\underline{\mathbb{E}}}$ is the isogeny with $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta')=\hat{\delta}$ which is an isomorphism outside $\infty$. The isogeny $\delta^{-1}\circ\delta'\colon\underline{\mathcal{E}}'\to\underline{\mathcal{E}}$ satisfies $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta^{-1}\delta')=\operatorname{id}$ and $\check{\mathcal{V}}_{\delta^{-1}\delta'}\circ \check{\mathcal{V}}_{\delta'}^{-1}\circ\check{\mathcal{V}}_{\delta}\circ\gamma H=\gamma H$, and so $(\underline{\mathcal{E}}',\check{\mathcal{V}}_{\delta'}^{-1}\circ\check{\mathcal{V}}_{\delta}\circ \gamma H)\cong(\underline{\mathcal{E}},\gamma H)$ in $\operatorname{Sht}_{\mathcal{G},H,\infty\times\infty}^{\mathcal{Z}(\mu,\beta)}(K)$. The point $(\underline{\mathcal{L}},\hat{\delta}, \gamma H)$ lies on an irreducible component of $X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H$ belonging to the $I_{{\underline{\mathbb{E}}}}(Q)$-orbit of some irreducible component $T_j$. By the $I_{{\underline{\mathbb{E}}}}(Q)$-equivariance of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ we can move the point $(\underline{\mathcal{L}},\hat{\delta}, \gamma H)$ to $T_j$ and then its image $(\underline{\mathcal{E}},\gamma H)$ under $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ lies in $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)\subset\mathcal{X}_{\underline{\mathbb{E}}}$ as desired. To prove that $\mathcal{X}_{\underline{\mathbb{E}}}$ is separated over $\operatorname{Spec}{\overline\mathbb{F}_\infty}$ we use the valuative criterion [@Laumon-Moret-Bailly Proposition 7.8]. Let $R$ be a valuation ring containing ${\overline\mathbb{F}_\infty}$ with fraction field $K$. Consider two morphisms $f_1,f_2\colon\operatorname{Spec}R\to \mathcal{X}_{\underline{\mathbb{E}}}$ whose restrictions $f_{i,K}$ to $K$ are isomorphic in $\mathcal{X}_{\underline{\mathbb{E}}}(K)$. We must show that $f_1\cong f_2$ in $\mathcal{X}_{\underline{\mathbb{E}}}(R)$. The $K$-valued point $f_{1,K}\cong f_{2,K}$ lies in $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)(K)\subset\mathcal{X}_{\underline{\mathbb{E}}}(K)$ for some $j$. Since $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$ is a closed substack of $\operatorname{Sht}_{\mathcal{G},H,\infty\times\infty}^{\mathcal{Z}(\mu,\beta)}$, the two morphisms $f_1,f_2$ factor through $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$. Since $\operatorname{Sht}_{\mathcal{G},H,\infty\times\infty}^{\mathcal{Z}(\mu,\beta)}$ is separated over ${\overline\mathbb{F}_\infty}$, also $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$ is separated over ${\overline\mathbb{F}_\infty}$, and so $f_1\cong f_2$ in $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)(R)$. Thus $\mathcal{X}_{\underline{\mathbb{E}}}$ is separated over ${\overline\mathbb{F}_\infty}$. ◻ Reasoning as in [@RZ 6.22], we may form the formal completion $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}_{/\mathcal{X}}$ of $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ along the set $\mathcal{X}:=\mathcal{X}_{\underline{\mathbb{E}}}$ From Lemma [Lemma 1](#LemmaImageOfTheta){reference-type="ref" reference="LemmaImageOfTheta"}[\[LemmaImageOfTheta_B\]](#LemmaImageOfTheta_B){reference-type="ref" reference="LemmaImageOfTheta_B"}. It is the category fiberd over $\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$ whose $S$-valued points give the full subcategory $$\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}_{/\mathcal{X}}(S):=\bigl\{\,f\colon S\to\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}, \enspace\text{such that } f|_{S_{\rm red}}\text{ factors through }\mathcal{X}_{\underline{\mathbb{E}}}\,\bigr\}\,,$$ where $S_{\rm red}$ is the underlying reduced closed subscheme of $S$. Note that it follows immediately that the natural morphism $$\label{EqShtCompletion} \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}_{/\mathcal{X}}\;\longrightarrow\;\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$$ is an ind-proper monomorphism and formally étale, because for an affine scheme $S=\operatorname{Spec}B\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$ and an ideal $I\subset B$ with $I^2=(0)$ one has $S_{\rm red}=(\operatorname{Spec}B/I)_{\rm red}$. **Theorem 1**. *Let $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}{}_{/\mathcal{X}}$ be the formal completion of $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ along the set $\mathcal{X}_{\underline{\mathbb{E}}}$ from Lemma [Lemma 1](#LemmaImageOfTheta){reference-type="ref" reference="LemmaImageOfTheta"}[\[LemmaImageOfTheta_B\]](#LemmaImageOfTheta_B){reference-type="ref" reference="LemmaImageOfTheta_B"}.* 1. *[\[Uniformization2_B\]]{#Uniformization2_B label="Uniformization2_B"} Then $\Theta_{{\underline{\mathbb{E}}}}$ induces an isomorphism of locally noetherian, adic formal algebraic Deligne-Mumford stacks locally formally of finite type over $\operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ $$\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr)\;\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\; \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}_{/\mathcal{X}}\,,$$ and in particular of the underlying Deligne-Mumford stacks $$\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\;\stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\; \mathcal{X}_{\underline{\mathbb{E}}}$$ which are locally of finite type and separated over $\operatorname{Spec}{\overline\mathbb{F}_\infty}$.* 2. *[\[Uniformization2_C\]]{#Uniformization2_C label="Uniformization2_C"} The morphisms $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$, $\Theta_{{\underline{\mathbb{E}}}}$ and $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ are compatible with the following actions on source and target: the action of $Z(Q_\infty)$ described in [ 1](#center-action-paragraph){reference-type="ref" reference="center-action-paragraph"}, the action of $G(\mathbb{A}^\infty)$ through Hecke-correspondences described in Definition [Definition 1](#Defn-Hecke-corr){reference-type="ref" reference="Defn-Hecke-corr"}, and the Weil descent data described in [ 1](#Weil-descent-paragraph){reference-type="ref" reference="Weil-descent-paragraph"}.* *For every multiple $m\in\mathbb{N}_0$ of $[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]$, the base changes of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$, $\Theta_{{\underline{\mathbb{E}}}}$ and $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ to $\operatorname{Spec}{\overline\mathbb{F}_\infty}$ are compatible with the $q^m$-Frobenius endomorphisms $\Phi_m$ from [\[EqFrobOnSource\]](#EqFrobOnSource){reference-type="eqref" reference="EqFrobOnSource"} and [\[EqFrobOnTarget\]](#EqFrobOnTarget){reference-type="eqref" reference="EqFrobOnTarget"}.* The proof will be given in the next section. **Remark 1**. Isogeny classes on $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ have the structure of quotients of affine Deligne--Lusztig varieties by some $Q$-rational group $I_{{\underline{\mathbb{E}}}}(Q)$. Theorem [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"} has consequences for point-counting in the Langlands-Rapoport conjectures. **Remark 1**. Conbining our techniques with the ones from [@AH_Unif Theorem 7.11], one can extend Theorems [Theorem 1](#Uniformization1){reference-type="ref" reference="Uniformization1"} and [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"} to the case of $n$ disjoint pairs $(x_i,\infty_i)_{i=1\ldots n}$ of two colliding legs, such that in each pair the leg $x_i$ varies and the other leg is fixed at a place $\infty_i$ and bounded by some element $\beta_i\in L_{\infty_i}\mathcal{G}(\overline{\mathbb{F}}_q)$ with $\beta_i\cdot L^+_{\infty_i}\mathcal{G}\cdot \beta_i^{-1} = L^+_{\infty_i}\mathcal{G}$ for all $i$. Here disjoint means that $\infty_i\ne\infty_j$ for $i\ne j$. For each $i$ one considers the $\beta_i^{-1}$-twisted global-local functor $L^+_{\infty_i,\mathcal{M}_i}$ from global $\mathcal{G}$-shtukas to local $\mathcal{M}_i$-shtukas, where $\mathcal{M}_i$ is the inner form of $\mathcal{G}_{\infty_i}$ given by $\beta_i^{-1}$. The target space of the uniformization will be the stack $\operatorname{Sht}_{\mathcal{G},H,(\widehat{\infty}_i\times\infty_i)_i}^{\mathcal{Z}((\mu_i,\beta_i)_i)}$ of global $\mathcal{G}$-shtukas with $n$ varying legs $x_i\colon S\to\operatorname{Spf}\mathcal{O}_{\infty_i}$ (respectively $n$ fixed legs $\infty_i$) at which the modification is bounded by a cocharacter $\mu_i$ (respectively by $\beta_i$), and with a $H$-level structure for a compact open subgroups $H\subset G(\mathbb{A}^{\underline{\infty}})$ for $\underline{\infty}=(\infty_1,\ldots,\infty_n)$. As a global framing object one fixes a global $\mathcal{G}$-shtuka ${\underline{\mathbb{E}}}\in \operatorname{Sht}_{\mathcal{G},\varnothing,(\widehat{\infty}_i\times\infty_i)_i}^{\mathcal{Z}((\mu_i,\beta_i)_i)}(\overline{\mathbb{F}}_q)$ over $\overline{\mathbb{F}}_q$. For every $i$ the associated local $\mathcal{M}_i$-shtuka is ${\underline{{\mathbb{L}}}}_i:=L^+_{\infty_i,\mathcal{M}_i}({\underline{\mathbb{E}}})\cong\bigl((L^+\mathcal{M}_i)_{\overline{\mathbb{F}}_q},b_i)$ with $b_i\in L_{\infty_i}\mathcal{M}_i(\overline{\mathbb{F}}_q)$. One obtains the Rapoport-Zink space $\mathcal{R}\mathcal{Z}_{\mathcal{M}_i,{\underline{{\mathbb{L}}}}_i}^{\leq \mu_i}$ over $\Breve{\mathcal{O}}_{\mu_i}=:\overline{\mathbb{F}}_q{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}\xi_i{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$ with the affine Deligne-Lusztig varietie $X_{\mathcal{M}_i}^{\leq \mu_i}(b_i)$ as its underlying topological space. The uniformization is then given by an isomorphism $$\label{EqRemManyPairsOfLegs} \Theta_{{\underline{\mathbb{E}}},\mathcal{X}} \colon I_{\underline{\mathbb{E}}}(Q) \big\backslash \bigl( \prod_{i=1}^n X_{\mathcal{M}_i}^{\leq \mu_i}(b_i) \times \operatorname{Isom}^\otimes(\omega,\check{\mathcal{V}}_{\underline{\mathbb{E}}})/H \bigr) \; \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\; \mathcal{X}_{\underline{\mathbb{E}}}$$ of Deligne-Mumford stacks locally of finite type and separated over $\overline{\mathbb{F}}_q$, obtained as the restriction of an isomorphism $$\Theta_{{\underline{\mathbb{E}}},\mathcal{X}} \colon I_{\underline{\mathbb{E}}}(Q) \big\backslash \bigl( \prod_{i=1}^n \mathcal{R}\mathcal{Z}_{\mathcal{M}_i,{\underline{{\mathbb{L}}}}_i}^{\leq \mu_i} \times \operatorname{Isom}^\otimes(\omega,\check{\mathcal{V}}_{\underline{\mathbb{E}}})/H \bigr) \; \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\; \operatorname{Sht}_{\mathcal{G},H,(\widehat{\infty}_i\times\infty_i)_i}^{\mathcal{Z}((\mu_i,\beta_i)_i)}{}_{/\mathcal{X}}$$ of locally noetherian, adic formal algebraic Deligne-Mumford stacks locally formally of finite type over $\Breve{\mathcal{O}}_{(\mu_i,\beta_i)_i}:=\overline{\mathbb{F}}_q{\mathchoice{\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\rm [\hspace{-0.15em}[}} {\mbox{\scriptsize\rm [\hspace{-0.15em}[}} {\mbox{\tiny\rm [\hspace{-0.15em}[}}}\xi_1,\ldots,\xi_n{\mathchoice{\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\rm ]\hspace{-0.15em}]}} {\mbox{\scriptsize\rm ]\hspace{-0.15em}]}} {\mbox{\tiny\rm ]\hspace{-0.15em}]}}}$, where the target is the formal completion of $\operatorname{Sht}_{\mathcal{G},H,(\widehat{\infty}_i\times\infty_i)_i}^{\mathcal{Z}((\mu_i,\beta_i)_i)} \times \operatorname{Spf}\Breve{\mathcal{O}}_{(\mu_i,\beta_i)_i}$ along the underlying set $\mathcal{X}_{\underline{\mathbb{E}}}$ which is the image of the morphism [\[EqRemManyPairsOfLegs\]](#EqRemManyPairsOfLegs){reference-type="eqref" reference="EqRemManyPairsOfLegs"} and the isogeny class of ${\underline{\mathbb{E}}}$. ## Proof of the Main Theorems {#subsec:ProofMainThms} **Lemma 1**. *The maps $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$, $\Theta_{{\underline{\mathbb{E}}}}$, and $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ are ind-proper and formally étale.* *Proof.* The claim that $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is formally étale follows from the rigidity of quasi-isogenies. We give more details. Let $S\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$, and let $(\underline{\mathcal{E}},\gamma H)\in \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)$ with $\gamma\in\operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{\underline{\mathcal{E}}})$. Let $S'$ be a closed subscheme of $S$ defined by a locally nilpotent sheaf of ideals. Suppose that the base change $(\underline{\mathcal{E}}_{S'},\gamma H)$ to $S'$ lies in the image of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$, i.e. there is a tuple $$\label{tuple-in-RZtimesIsom} (\underline{\mathcal{L}}',\hat{\delta}'\colon \underline{\mathcal{L}}'\to {\underline{{\mathbb{L}}}}_{S'},\gamma'H) \in \mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S')\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$$ such that $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(\underline{\mathcal{L}}',\hat{\delta}',\gamma'H)=(\underline{\mathcal{E}}_{S'},\gamma H)$ in $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S')$. The last equality means that there is a quasi-isogeny $\tilde{\delta}'\colon \underline{\mathcal{E}}_{S'} \to (\hat{\delta}')^*{\underline{\mathbb{E}}}_{S'}$ of global $\mathcal{G}$-shtukas over $S'$ that is an isomorphism over $\infty$ with $\check{\mathcal{V}}_{\tilde{\delta}'}^{-1}\circ\check{\mathcal{V}}_{\delta'}^{-1}\circ\gamma' H=\gamma H$ in $\operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{\underline{\mathcal{E}}})/H$, where $\delta'\colon(\hat{\delta}')^*{\underline{\mathbb{E}}}_{S'}\to {\underline{\mathbb{E}}}_{S'}$ is the quasi-isogeny from Proposition [Proposition 1](#PropQIsogLocalGlobal){reference-type="ref" reference="PropQIsogLocalGlobal"}[\[PropQIsogLocalGlobal_B\]](#PropQIsogLocalGlobal_B){reference-type="ref" reference="PropQIsogLocalGlobal_B"} with $L_{\infty,\mathcal{M}_{\beta^{-1}}}((\hat{\delta}')^*{\underline{\mathbb{E}}}_{S'})=\underline{\mathcal{L}}'$ and $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta')=\hat{\delta}'$. In particular, $$\label{loop-isom-equation} L_{\infty,\mathcal{M}_{\beta^{-1}}}(\tilde{\delta}')\colon L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})_{S'} \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}L_{\infty,\mathcal{M}_{\beta^{-1}}}\bigl((\hat{\delta}')^*{\underline{\mathbb{E}}}_{S'}\bigr) = \underline{\mathcal{L}}'$$ is an isomorphism of local $\mathcal{M}$-shtukas. Thus $(\underline{\mathcal{L}}',\hat{\delta}')$ equals $(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})_{S'},\hat{\delta}'\circ L_{\infty,\mathcal{M}_{\beta^{-1}}}(\tilde{\delta}'))$ in $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S')$. We may replace the former by the latter in [\[tuple-in-RZtimesIsom\]](#tuple-in-RZtimesIsom){reference-type="eqref" reference="tuple-in-RZtimesIsom"} and thus assume $L_{\infty,\mathcal{M}_{\beta^{-1}}}((\hat{\delta}')^*{\underline{\mathbb{E}}}_{S'})=L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})_{S'}$ and $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\tilde{\delta}')=\operatorname{id}_{L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})_{S'}}$ in [\[loop-isom-equation\]](#loop-isom-equation){reference-type="eqref" reference="loop-isom-equation"} Now the quasi-isogeny $\hat{\delta}'=\hat{\delta}'\circ L_{\infty,\mathcal{M}_{\beta^{-1}}}(\tilde{\delta}')\colon L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})_{S'}\to{\underline{{\mathbb{L}}}}_{S'}$ lifts uniquely to a quasi-isogeny $\hat{\delta}\colon L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})\to {\underline{{\mathbb{L}}}}_S$ over $S$ by the rigidity of quasi-isogenies for local $\mathcal{M}$-shtukas; see Proposition [Proposition 1](#PropRigidityLocal){reference-type="ref" reference="PropRigidityLocal"}. Therefore, $(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}}), \hat{\delta},\gamma' H)$ is an $S$-valued point of $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}(S)\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H$. Its image under $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is $(\hat{\delta}^*{\underline{\mathbb{E}}}_S,\check{\mathcal{V}}_\delta^{-1}\circ \gamma'H)$, where $\delta\colon\hat{\delta}^*{\underline{\mathbb{E}}}_S\to {\underline{\mathbb{E}}}_S$ is the quasi-isogeny from Proposition [Proposition 1](#PropQIsogLocalGlobal){reference-type="ref" reference="PropQIsogLocalGlobal"}[\[PropQIsogLocalGlobal_B\]](#PropQIsogLocalGlobal_B){reference-type="ref" reference="PropQIsogLocalGlobal_B"} with $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\hat{\delta}^*{\underline{\mathbb{E}}}_S)=L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})$ and $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta)=\hat{\delta}$. Since $(\hat{\delta}^*{\underline{\mathbb{E}}}_S)_{S'}=(\hat{\delta}')^*{\underline{\mathbb{E}}}_{S'}$, the quasi-isogeny $\tilde{\delta'}$ over $S'$ lifts uniquely to a quasi-isogeny $\tilde{\delta}\colon\underline{\mathcal{E}}\to \hat{\delta}^*{\underline{\mathbb{E}}}_S$ over $S$ by rigidity of quasi-isogenies for global $\mathcal{G}$-shtukas; see [@AH_Local Proposition 5.9]. It satisfies $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\tilde{\delta})=\operatorname{id}_{L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}})}$ by the uniqueness of the quasi-isogeny lifting $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\tilde{\delta}')=\operatorname{id}$ to $S$. This shows that $\tilde{\delta}$ is a quasi-isogeny which is an isomorphism over $\infty$ and identifies $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(L_{\infty,\mathcal{M}_{\beta^{-1}}}(\underline{\mathcal{E}}), \hat{\delta},\gamma'H):=(\hat{\delta}^*{\underline{\mathbb{E}}}_S,\check{\mathcal{V}}_\delta^{-1}\circ \gamma'H)$ with $(\underline{\mathcal{E}},\gamma H)$ in $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S')$. This finishes the proof that $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is formally étale. Since the quotient morphism $$\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr) \enspace \mbox{$\kern 2pt\to\kern-8pt\to\kern 2pt$}\enspace I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr)$$ is étale by Proposition [Proposition 1](#PropQuotientByI){reference-type="ref" reference="PropQuotientByI"}(b) and $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is formally étale, also $\Theta_{{\underline{\mathbb{E}}}}$ is formally étale. And since the morphism [\[EqShtCompletion\]](#EqShtCompletion){reference-type="eqref" reference="EqShtCompletion"} is formally étale, also $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ is formally étale. Since $\mathcal{M}$ is parahoric, $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}$ is ind-proper over ${\Breve{\mathcal{O}}_{\mu,\beta}}$ by Remark [Remark 1](#RemRZIndProper){reference-type="ref" reference="RemRZIndProper"}. Therefore, $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is ind-proper. The morphism $\Theta_{{\underline{\mathbb{E}}}}$ is ind-proper, because $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is ind-proper and $$\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr) \mbox{$\kern 2pt\to\kern-8pt\to\kern 2pt$}I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr)$$ is surjective. Finally $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ is ind-proper, because the morphism [\[EqShtCompletion\]](#EqShtCompletion){reference-type="eqref" reference="EqShtCompletion"} is ind-proper. ◻ **Lemma 1**. *We use the abbreviations $Y_1:=\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq\mu}\times \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H$ and $Y_2:=X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H$. Then for $j=1$ or $2$, the action of $I_{{\underline{\mathbb{E}}}}(Q)$ on $Y_j$ induces an isomorphism of stacks $$\label{EqDiagIsIsom} I_{{\underline{\mathbb{E}}}}(Q) \times Y_j \enspace := \enspace \coprod_{I_{{\underline{\mathbb{E}}}}(Q)} Y_j \enspace \stackrel{}{\mbox{\hspace{1mm}\raisebox{+1.4mm}{$\scriptstyle\sim$}\hspace{-4.2mm}$\longrightarrow$}}\enspace Y_j \underset{\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} {\Breve{\mathcal{O}}_{\mu,\beta}}}{\times} Y_j\,,$$ where the map to the first copy of $Y_j$ is the identity and the map to the second copy is given by the action of $I_{{\underline{\mathbb{E}}}}(Q)$ on $Y_j$. In particular, $\Theta_{{\underline{\mathbb{E}}}}$ is a monomorphism in the sense stated in Theorem [Theorem 1](#Uniformization1){reference-type="ref" reference="Uniformization1"}.* *Proof.* By [@stacks-project Tag 04Z7], the two definitions of a monomorphism given in Theorem [Theorem 1](#Uniformization1){reference-type="ref" reference="Uniformization1"} are equivalent. By the $I_{{\underline{\mathbb{E}}}}(Q)$-equivariance of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$, the morphism [\[EqDiagIsIsom\]](#EqDiagIsIsom){reference-type="eqref" reference="EqDiagIsIsom"} is well defined. To describe its inverse, consider a connected scheme $S\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$ and two $S$-valued points of $Y_j$ $$y:=\bigl((\underline{\mathcal{L}},\hat{\delta}),\gamma H \bigr)~~~~ \text{and}~~~~ y':=\bigl((\underline{\mathcal{L}}',\hat{\delta}'),\gamma'H \bigr).$$ Under $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$, they are mapped to global $\mathcal{G}$-shtukas $(\underline{\mathcal{E}},\check{\mathcal{V}}_\delta^{-1} \gamma H)$ and $(\underline{\mathcal{E}}',\check{\mathcal{V}}_{\delta'}^{-1} \gamma'H)$ with $H$-level structures, where $\delta\colon\underline{\mathcal{E}} \to {\underline{\mathbb{E}}}_S$ and $\delta'\colon \underline{\mathcal{E}}'\to {\underline{\mathbb{E}}}_S$ are the canonical quasi-isogenies which are isomorphisms outside $\infty$ with $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta)=\hat{\delta}$ and $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta')=\hat{\delta}'$. Suppose that $(\underline{\mathcal{E}},\check{\mathcal{V}}_\delta^{-1} \gamma H)$ and $(\underline{\mathcal{E}}',\check{\mathcal{V}}_{\delta'}^{-1} \gamma'H)$ are isomorphic in $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(S)$ via a quasi-isogeny $\psi\colon\underline{\mathcal{E}}\to \underline{\mathcal{E}}'$ which is an isomorphism above $\infty$ and compatible with the $H$-level structures, i.e. $\check{\mathcal{V}}_\psi\circ\check{\mathcal{V}}_\delta^{-1} \gamma H=\check{\mathcal{V}}_{\delta'}^{-1} \gamma'H$ (see Definition [Definition 1](#DefRatLevelStr){reference-type="ref" reference="DefRatLevelStr"}). Consider the quasi-isogeny $\eta:=\delta'\psi \delta^{-1}$ from ${\underline{\mathbb{E}}}_S$ to itself. By Proposition [Proposition 1](#Prop7.1){reference-type="ref" reference="Prop7.1"} we may view $\eta$ as an element of $I_{{\underline{\mathbb{E}}}}(Q)$. Consider the corresponding quasi-isogenies between the associated local $\mathcal{M}$-shtukas $$\xymatrix @C+2pc { \underline{\mathcal{L}} \ar[r]^{L_{\infty,\mathcal{M}_{\beta^{-1}}}(\psi)} \ar[d]_{\hat{\delta}} & \underline{\mathcal{L}}' \ar[d]_{\hat{\delta}'} \\ {\underline{{\mathbb{L}}}}_S \ar[r]^{L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)} & {\underline{{\mathbb{L}}}}_S }$$ Since $\psi\colon\underline{\mathcal{E}}\to \underline{\mathcal{E}}'$ is an isomorphism above $\infty$, the quasi-isogeny $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\psi)$ is an isomorphism. Therefore, $\eta\cdot(\underline{\mathcal{L}},\hat{\delta}):=(\underline{\mathcal{L}},L_{\infty,\mathcal{M}_{\beta^{-1}}}(\eta)\circ\hat{\delta})\cong(\underline{\mathcal{L}}',\hat{\delta}')$ in $\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq\mu}(S)$. Moreover, $\eta$ sends $\gamma H\in\operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H$ to $\check{\mathcal{V}}_\eta\circ\gamma H=\check{\mathcal{V}}_{\delta'}\circ\check{\mathcal{V}}_\psi\circ \check{\mathcal{V}}_\delta^{-1}\circ\gamma H=\gamma'H$. This proves that $\eta\cdot y=y'$, i.e. $(\eta,y)$ maps to $(y,y')$ under [\[EqDiagIsIsom\]](#EqDiagIsIsom){reference-type="eqref" reference="EqDiagIsIsom"}. Thus $\Theta_{{\underline{\mathbb{E}}}}$ is a monomorphism. ◻ Next we turn to the proof of Theorem [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"}[\[Uniformization2_B\]](#Uniformization2_B){reference-type="ref" reference="Uniformization2_B"}. To shorten notations, we write $$\begin{aligned} \mathcal{R}& := I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})/H\bigr) \qquad\text{and} \\ \mathcal{S}& := \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}_{/\mathcal{X}}\end{aligned}$$ for the source and target of the morphism $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$. These are locally noetherian, adic formal algebraic Deligne-Mumford stacks. For $\mathcal{R}$, this was proven in Proposition [Proposition 1](#PropQuotientByI){reference-type="ref" reference="PropQuotientByI"}, and for $\mathcal{S}$, this follows from Proposition [Proposition 1](#PropLSGGsht1){reference-type="ref" reference="PropLSGGsht1"} and [@HartlAbSh Proposition A.14]. Thus both $\mathcal{R}$ and $\mathcal{S}$ have unique largest ideals of definition $\mathcal{I}_{\mathcal{R}}\subset\mathcal{O}_\mathcal{R}$ and $\mathcal{I}_{\mathcal{S}}\subset\mathcal{O}_\mathcal{S}$ containing the maximal ideal $\breve{\mathfrak{m}}_{\mu,\beta}$ of ${\Breve{\mathcal{O}}_{\mu,\beta}}$. For positive integers $m$, the closed substacks $\mathcal{R}_m:=\operatorname{V}(\mathcal{I}_{\mathcal{R}}^m)\subset\mathcal{R}$ and $\mathcal{S}_m:=\operatorname{V}(\mathcal{I}_{\mathcal{S}}^m)\subset\mathcal{S}$ are algebraic by [@HartlAbSh Proposition A.8]. **Lemma 1**. *(a) $\mathcal{R}_m$ and $\mathcal{S}_m$ are Deligne-Mumford stacks locally of finite type over $\operatorname{Spec}{\Breve{\mathcal{O}}_{\mu,\beta}}/\breve{\mathfrak{m}}_{\mu,\beta}^m$. We have $\mathcal{R}=\varinjlim\limits_m\mathcal{R}_m$ and $\mathcal{S}=\varinjlim\limits_m\mathcal{S}_m$. Moreover, $\mathcal{S}_1=\mathcal{X}_{\underline{\mathbb{E}}}$ and $\mathcal{R}_1=I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})$.* *(b) The morphism $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ induces a morphism $\mathcal{R}_m\to\mathcal{S}_m$ for every $m$, which is locally of finite presentation as a morphism between Deligne-Mumford stacks locally of finite type over $\operatorname{Spec}{\Breve{\mathcal{O}}_{\mu,\beta}}/\breve{\mathfrak{m}}_{\mu,\beta}^m$.* *Proof.* For $\mathcal{R}_m$, this follows since $\mathcal{R}$ is a formal algebraic Deligne-Mumford stack locally formally of finite type over $\operatorname{Spec}{\Breve{\mathcal{O}}_{\mu,\beta}}$. For $\mathcal{S}_m$, it follows because $\mathcal{S}_m$ is a closed substack of the Deligne-Mumford stack $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\times_{{\Breve{\mathcal{O}}_{\mu,\beta}}}\operatorname{Spec}{\Breve{\mathcal{O}}_{\mu,\beta}}/\breve{\mathfrak{m}}_{\mu,\beta}^m$ which is locally of finite type over $\operatorname{Spec}{\Breve{\mathcal{O}}_{\mu,\beta}}/\breve{\mathfrak{m}}_{\mu,\beta}^m$ by Proposition [Proposition 1](#PropLSGGsht1){reference-type="ref" reference="PropLSGGsht1"}. By definitions of $\mathcal{I}_{\mathcal{S}}$ and $\mathcal{I}_{\mathcal{R}}$, for $m=1$, the stacks $\mathcal{S}_1=\mathcal{S}_{{\rm red}}$ and $\mathcal{R}_1=\mathcal{R}_{{\rm red}}$ are reduced. Recall that $\mathcal{X}_{\underline{\mathbb{E}}}$ is reduced by Lemma [Lemma 1](#LemmaImageOfTheta){reference-type="ref" reference="LemmaImageOfTheta"}[\[LemmaImageOfTheta_B\]](#LemmaImageOfTheta_B){reference-type="ref" reference="LemmaImageOfTheta_B"}, we have $\mathcal{X}_{\underline{\mathbb{E}}}=\mathcal{S}_1$. Moreover, it is clear that $\mathcal{R}_1=I_{{\underline{\mathbb{E}}}}(Q) \big{\backslash}X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega ,\check{{\mathcal{V}}}_{{\underline{\mathbb{E}}}})$. Moreover, $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ induces a morphism $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon \mathcal{R}_1 \to \mathcal{S}_1$, because if $\mathcal{P}\mbox{$\kern 2pt\to\kern-8pt\to\kern 2pt$}\mathcal{R}_1$ is a presentation, then $\mathcal{P}$ is a reduced scheme, and hence $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ induces a morphism $\mathcal{P}\to \mathcal{X}_{\underline{\mathbb{E}}}=\mathcal{S}_1$ which descends to a morphism $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon\mathcal{R}_1\to\mathcal{S}_1$. In particular, $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}^*(\mathcal{I}_{\mathcal{S}})\subset\mathcal{I}_{\mathcal{R}}$. ◻ **Lemma 1**. *For every $m$, the induced morphism $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon \mathcal{R}_m \to \mathcal{S}_m$ is quasi-compact and surjective.* *Proof.* The assertion only depends on the underlying topological spaces $|\mathcal{S}_m|=|\mathcal{S}_1|$ and $|\mathcal{R}_m|=|\mathcal{R}_1|$ (see [@Laumon-Moret-Bailly Chapitre 5]), so we may assume that $m=1$. The morphism $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon \mathcal{R}_1 \to \mathcal{S}_1$ is surjective by the definition of $\mathcal{X}_{\underline{\mathbb{E}}}$ as the image of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$. We show that the morphism is quasi-compact. Let $S$ be a quasi-compact scheme and let $f\colon S\to\mathcal{S}_1$ be a morphism. We must show that $S\times_{\mathcal{S}_1}\mathcal{R}_1$ is quasi-compact. Consider the topological space $|\mathcal{X}_{\underline{\mathbb{E}}}|$ underlying $\mathcal{X}_{\underline{\mathbb{E}}}$, and the set $\{T_j\}_{j\in N}$ of representatives of $I_{{\underline{\mathbb{E}}}}(Q)$-orbits of the irreducible components of the scheme $X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H$ from Lemma [Lemma 1](#LemmaImageOfTheta){reference-type="ref" reference="LemmaImageOfTheta"}. By Lemma [Lemma 1](#LemmaImageOfTheta){reference-type="ref" reference="LemmaImageOfTheta"}[\[LemmaImageOfTheta_A\]](#LemmaImageOfTheta_A){reference-type="ref" reference="LemmaImageOfTheta_A"}, every point $z\in|\mathcal{X}_{\underline{\mathbb{E}}}|$ lies only in finitely many $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$. Let $N(z)\subset N$ be the set of $j\in N(z)$ such that $z\in\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$. The open substack $$U_z\;:=\;\bigcup_{j\in N(z)}\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)\;\smallsetminus\bigcup_{j\notin N(z)}\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)\;\subset\;\mathcal{X}_{\underline{\mathbb{E}}}$$ contains $z$, and hence $\mathcal{X}_{\underline{\mathbb{E}}}$ is covered by the $U_z$ for all $z\in|\mathcal{X}_{\underline{\mathbb{E}}}|$. Let $s\in S$ be a point and set $z=f(s)\in|\mathcal{X}_{\underline{\mathbb{E}}}|$. The preimage $f^{-1}(U_{f(s)})$ of $U_{f(s)}$ in $S$ contains $s$. We choose an affine open neighborhood $S_s$ of $s$ in $S$ which is contained in $f^{-1}(U_{f(s)})$. Then the $S_s$ cover $S$, and since $S$ is quasi-compact, we have $S=S_{s_1}\cup\ldots\cup S_{s_r}$ for finitely many points $s_k\in S$, $k=1,\ldots,r$. Since $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}\colon T_j\to\mathcal{X}_{\underline{\mathbb{E}}}$ is proper by Lemma [Lemma 1](#LemmaThetaEtaleProper){reference-type="ref" reference="LemmaThetaEtaleProper"}, $S_{s_k} \times_{\mathcal{X}_{\underline{\mathbb{E}}}} T_j$ is proper over the affine scheme $S_{s_k}$. The scheme $S'$ defined as the finite disjoint union $$S'\;:=\;\coprod_{k=1}^r\;\coprod_{j\in N(f(s_k))} S_{s_k} \underset{f,\mathcal{X}_{\underline{\mathbb{E}}},\widetilde{\Theta}_{{\underline{\mathbb{E}}}}}{\times} T_j$$ is quasi-compact. Since every point $s\in S$ lies in one $S_{s_k}$, and then $f(s)\in U_{f(s_k)}\subset\bigcup_{j\in N(f(s_k))}\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(T_j)$ has a preimage in one of the $T_j$, we conclude that the projection $S'\to S$ is surjective. Therefore, the projection $S'\to \coprod\limits_{k,j} T_j$ defines the upper horizontal morphism in the following commutative diagram $$\xymatrix @C+1pc { S' \ar@{->>}[d] \ar[rr] & & X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H \ar[d]^{\textstyle\widetilde{\Theta}_{{\underline{\mathbb{E}}}}} \\ S \ar[r]^{\textstyle f} & \mathcal{S}_1\ar[r]^{\textstyle \iota_{\mathcal{X}}} & \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}\; }$$ $$\xymatrix { S' \ar@{->>}[d] \ar[rr] & & X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H \ar[d]{\widetilde{\Theta}_{{\underline{\mathbb{E}}}}}\ar@{-->}[ld] \\ S \ar[r]_-{f} & \mathcal{S}_1 \ar[r]_-{\iota_{\mathcal{X}}} & \operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}} }$$ where $\iota_{\mathcal{X}}$ is as defined in Lemma [Lemma 1](#LemmaImageOfTheta){reference-type="ref" reference="LemmaImageOfTheta"}[\[LemmaImageOfTheta_B\]](#LemmaImageOfTheta_B){reference-type="ref" reference="LemmaImageOfTheta_B"}, through which $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ factors. Then we can consider the surjective morphisms $$\xymatrix @C=+8pc @R+1pc { **{!L(0.45) !U(0.5)} \objectbox{\; S' \underset{\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}}{\times} X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H} \ar@{->>}[d] & I_{{\underline{\mathbb{E}}}}(Q) \times S' \ar[l]_-{\textstyle\sim} \ar@{-->}[d] \\ \quad\; S'\times_{\mathcal{S}_1}\mathcal{R}_1 \ar@{->>}[r] & S\times_{\mathcal{S}_1}\mathcal{R}_1 }$$ $$\begin{tikzcd} {S' \underset{\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}}{\times} X_{\mathcal{M}}^{\leq \mu}(b)\times \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H} \arrow[two heads]{d}{} & I_{{\underline{\mathbb{E}}}}(Q) \times S' \ar[l]_-{\textstyle\sim} \ar@{-->}[d] \\ \quad\; S'\times_{\mathcal{S}_1}\mathcal{R}_1 \ar@{->>}[r] & S\times_{\mathcal{S}_1}\mathcal{R}_1 \end{tikzcd}$$ in which the isomorphism in the upper row comes from Lemma [Lemma 1](#LemmaThetaismono){reference-type="ref" reference="LemmaThetaismono"}. The left downward map is defined by taking the identity on $S'$, multiplied by the surjective quotient map by $I_{{\underline{\mathbb{E}}}}(Q)$, and observing that $\operatorname{Id}_{S'}$ and the quotient map form a fiber product over $\mathcal{S}_1$. Hence the left downward map is surjective. By the $I_{{\underline{\mathbb{E}}}}(Q)$-equivariance of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$, the composite surjective map $I_{{\underline{\mathbb{E}}}}(Q)\times S'\twoheadrightarrow S\times_{\mathcal{S}_1}\mathcal{R}_1$ gives a surjective map $S'\twoheadrightarrow S\times_{\mathcal{S}_1}\mathcal{R}_1$, and hence $S\times_{\mathcal{S}_1}\mathcal{R}_1$ is quasi-compact by [@stacks-project Tag 04YC]. ◻ **Lemma 1**. *We have $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}^*(\mathcal{I}_{\mathcal{S}})=\mathcal{I}_{\mathcal{R}}$. In particular, the morphism $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}\colon\mathcal{R}\to \mathcal{S}$ of formal algebraic stacks is adic.* *Proof.* Let $\mathcal{P}' \twoheadrightarrow \mathcal{S}_1=\operatorname{V}(\mathcal{I}_{\mathcal{S}})$ be an atlas. By Lemmas [Lemma 1](#LemmaThetaEtaleProper){reference-type="ref" reference="LemmaThetaEtaleProper"}, [Lemma 1](#LemmaThetaismono){reference-type="ref" reference="LemmaThetaismono"} and [Lemma 1](#new-lemma){reference-type="ref" reference="new-lemma"}, we see that $\Theta_{\mathcal{X},m}\colon\mathcal{R}_m\times_{\mathcal{S}}\mathcal{P}'=\mathcal{R}_m\times_{\mathcal{S}_m}\mathcal{P}' \to \mathcal{P}'$ is a monomorphism locally of finite presentation satisfying the valuative criterion for properness. Since $\Theta_{\mathcal{X},m}$ is quasi-compact by Lemma [Lemma 1](#LemmaThetaRedQC){reference-type="ref" reference="LemmaThetaRedQC"}, it is a proper monomorphism, hence a closed immersion of schemes by [@Laumon-Moret-Bailly Corollaire A.2.2]. Since $\Theta_{\mathcal{X},m}$ is surjective by Lemma [Lemma 1](#LemmaThetaRedQC){reference-type="ref" reference="LemmaThetaRedQC"} and $\mathcal{P}'$ is reduced, it must be an isomorphism for all $m$. Therefore $\mathcal{R}_m\times_{\mathcal{S}} \mathcal{P}'=\mathcal{P}'=\mathcal{R}_1\times_{\mathcal{S}} \mathcal{P}'$, and thus $\mathcal{R}\times_{\mathcal{S}} \mathcal{P}'=\ifthenelse{\equal{}{}}% falls Argument leer {\displaystyle \lim_{\longrightarrow}}% verwende niedrige Version {\displaystyle \lim_{\underset{}{\longrightarrow}}}% sonst: verwende Argument \mathcal{R}_m\times_{\mathcal{S}} \mathcal{P}'=\mathcal{P}'=\mathcal{R}_1\times_{\mathcal{S}} \mathcal{P}'$. This shows that $\operatorname{V}(\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}^*\mathcal{I}_{\mathcal{S}})=\mathcal{R}\times_{\mathcal{S}}\mathcal{S}_1=\mathcal{R}_1\times_{\mathcal{S}}\mathcal{S}_1\subset\mathcal{R}_1=\operatorname{V}(\mathcal{I}_{\mathcal{R}})$ as closed substacks of $\mathcal{R}$. Therefore $\mathcal{I}_{\mathcal{R}}\subset\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}^*(\mathcal{I}_{\mathcal{S}})$ as the corresponding ideals. With the opposite inclusion established in Lemma [Lemma 1](#new-lemma){reference-type="ref" reference="new-lemma"}, we have $\mathcal{I}_{\mathcal{R}}=\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}^*(\mathcal{I}_{\mathcal{S}})$, and hence $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ is adic. ◻ *Proof of Theorem [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"}[\[Uniformization2_B\]](#Uniformization2_B){reference-type="ref" reference="Uniformization2_B"}.* By Lemma [Lemma 1](#LemmaThetaisadic){reference-type="ref" reference="LemmaThetaisadic"}, we have $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}^*(\mathcal{I}_{\mathcal{S}})=\mathcal{I}_{\mathcal{R}}$. Then $\mathcal{R}_m\to\mathcal{S}_m$ is obtained from $\mathcal{R}\to\mathcal{S}$ by base change via $\mathcal{S}_m\to \mathcal{S}$, and hence is a formally étale morphism locally of finite presentation of algebraic stacks by Lemma [Lemma 1](#LemmaThetaEtaleProper){reference-type="ref" reference="LemmaThetaEtaleProper"}. Since $\Theta_{{\underline{\mathbb{E}}}}$ is a monomorphism by Lemma [Lemma 1](#LemmaThetaismono){reference-type="ref" reference="LemmaThetaismono"} and [\[EqShtCompletion\]](#EqShtCompletion){reference-type="eqref" reference="EqShtCompletion"} is a monomorphism, thus $\mathcal{R}_m\to\mathcal{S}_m$ is a monomorphism. In particular, $\mathcal{R}_m\to\mathcal{S}_m$ is relatively representable by an étale monomorphism of schemes; see [@Laumon-Moret-Bailly Corollaire 8.1.3 and Théorème A.2]. In addition, $\mathcal{R}_m\to\mathcal{S}_m$ is surjective by Lemma [Lemma 1](#LemmaThetaRedQC){reference-type="ref" reference="LemmaThetaRedQC"}, hence an isomorphism by \[EGA IV$_4$, Théorème 17.9.1\] . As this holds for all $m$, we conclude that $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}:\mathcal{R}\to\mathcal{S}$ is an isomorphism of stacks. ◻ *Proof of Theorem [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"}[\[Uniformization2_C\]](#Uniformization2_C){reference-type="ref" reference="Uniformization2_C"}.* Recall from earlier that $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is equivariant for the action of the center $Z(Q_\infty)$ given in [\[EqActionCenter3\]](#EqActionCenter3){reference-type="eqref" reference="EqActionCenter3"} and [\[EqActionCenter2\]](#EqActionCenter2){reference-type="eqref" reference="EqActionCenter2"}, and the action of $G(\mathbb{A}^\infty)$ through Hecke correspondences given in [\[EqHeckeSource\]](#EqHeckeSource){reference-type="eqref" reference="EqHeckeSource"} and [\[EqHeckeTarget\]](#EqHeckeTarget){reference-type="eqref" reference="EqHeckeTarget"}. Thus it suffices to check that $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is compatible with Weil descent data and Frobenius endomorphism structure. First we show that the morphism $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ is compatible with the Weil descent data [\[EqDescentDatumOnNablaH\]](#EqDescentDatumOnNablaH){reference-type="eqref" reference="EqDescentDatumOnNablaH"} and [\[EqDescentDatumSource\]](#EqDescentDatumSource){reference-type="eqref" reference="EqDescentDatumSource"}. Let $(S,\theta)\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$ and $S_{[\lambda]}=(S,\lambda\circ\theta)\in\mathcal{N}ilp_{{\Breve{\mathcal{O}}_{\mu,\beta}}}$ where $\lambda$ is defined in [\[EqWeilDescent_lambda\]](#EqWeilDescent_lambda){reference-type="eqref" reference="EqWeilDescent_lambda"}. The $S$-valued point $(\underline{\mathcal{L}},\hat{\delta}, \gamma H)$, respectively the $S_{[\lambda]}$-valued point $(\underline{\mathcal{L}},\theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]})\circ\hat{\delta}, \gamma H)$, of the source of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ are sent to $(\underline{\mathcal{E}},\check{\mathcal{V}}_\delta^{-1}\theta^*(\gamma) H)$ and $(\underline{\mathcal{E}}',\check{\mathcal{V}}_{\delta'}^{-1}(\lambda\circ\theta)^*(\gamma) H)$, respectively, where $\underline{\mathcal{E}}:=\hat{\delta}^*{\underline{\mathbb{E}}}_S$ and $\underline{\mathcal{E}}':=(\theta^*\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}\circ\hat{\delta})^*{\underline{\mathbb{E}}}_{S_{[\lambda]}}$, and $\delta\colon\underline{\mathcal{E}}\to{\underline{\mathbb{E}}}_S:=\theta^*{\underline{\mathbb{E}}}$ and $\delta'\colon\underline{\mathcal{E}}'\to{\underline{\mathbb{E}}}_{S_{[\lambda]}}:=(\lambda\circ\theta)^*{\underline{\mathbb{E}}}=\theta^*\lambda^*{\underline{\mathbb{E}}}=\theta^*({}^{\tau^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}\!}{\underline{\mathbb{E}}})$ are the quasi-isogenies with $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta)=\hat{\delta}$ and $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta')=\theta^*(\widehat{\varphi}_{{\mathbb{L}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]})\circ\hat{\delta}$, which are isomorphisms outside $\infty$. Then $\psi:=\delta^{-1}\circ\theta^*(\Phi_{{\underline{\mathbb{E}}}}^{-[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]})\circ\delta'\colon\underline{\mathcal{E}}'\to\underline{\mathcal{E}}$ is a quasi-isogeny by Definition [Definition 1](#DeFfrobIsog){reference-type="ref" reference="DeFfrobIsog"}, which by Corollary [Corollary 1](#CorFrobIsog){reference-type="ref" reference="CorFrobIsog"} satisfies $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\psi)=\operatorname{id}_{\underline{\mathcal{L}}}$ and $\check{\mathcal{V}}_\psi\circ\check{\mathcal{V}}_{\delta'}^{-1}\circ\theta^*\lambda^*(\gamma) H=\check{\mathcal{V}}_\delta^{-1}\theta^*(\gamma) H$, because $\lambda^*(\gamma)={}^{\tau^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}\!}\gamma=\check{\mathcal{V}}_{\Phi_{{\underline{\mathbb{E}}}}^{[{\kappa_{\mu,\beta}}\colon\mathbb{F}_q]}}\circ\gamma$. Therefore, $\psi$ identifies $(\underline{\mathcal{E}}',\check{\mathcal{V}}_{\delta'}^{-1}(\lambda\circ\theta)^*(\gamma) H)$ with $(\underline{\mathcal{E}},\check{\mathcal{V}}_\delta^{-1}\theta^*(\gamma) H)$, and this establishes the compatibility of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ with the Weil descent data [\[EqDescentDatumOnNablaH\]](#EqDescentDatumOnNablaH){reference-type="eqref" reference="EqDescentDatumOnNablaH"} and [\[EqDescentDatumSource\]](#EqDescentDatumSource){reference-type="eqref" reference="EqDescentDatumSource"}. Since the Weil descent datum on the source of $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ commutes with the action of $I_{{\underline{\mathbb{E}}}}(Q)$, this also proves the compatibility of $\Theta_{{\underline{\mathbb{E}}}}$ with the Weil descent data [\[EqDescentDatumOnNablaH\]](#EqDescentDatumOnNablaH){reference-type="eqref" reference="EqDescentDatumOnNablaH"} and [\[EqDescentDatumSourceModI\]](#EqDescentDatumSourceModI){reference-type="eqref" reference="EqDescentDatumSourceModI"}. Finally, the target $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}_{/\mathcal{X}}$ carries the Weil descent datum [\[EqDescentDatumOnNablaH\]](#EqDescentDatumOnNablaH){reference-type="eqref" reference="EqDescentDatumOnNablaH"} induced from $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)} \mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$. To see this: if a morphism $S_{\rm red}\to\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)} \mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ given by $(\underline{\mathcal{E}},\gamma H)$ factors through $\mathcal{X}_{\underline{\mathbb{E}}}=\operatorname{im}(\widetilde{\Theta}_{{\underline{\mathbb{E}}}})$, then the morphism $(S_{[\lambda]})_{\rm red}\to\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)} \mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ given by $(\underline{\mathcal{E}},\gamma H)$ also factors through $\mathcal{X}_{\underline{\mathbb{E}}}=\operatorname{im}(\widetilde{\Theta}_{{\underline{\mathbb{E}}}})$, because $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ commutes with the Weil descent data. This shows that $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ is also compatible with the Weil descent data [\[EqDescentDatumOnNablaH\]](#EqDescentDatumOnNablaH){reference-type="eqref" reference="EqDescentDatumOnNablaH"} and [\[EqDescentDatumSourceModI\]](#EqDescentDatumSourceModI){reference-type="eqref" reference="EqDescentDatumSourceModI"}. We also prove that $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ commutes with the $q^m$-Frobenius endomorphisms $\Phi_m$ from [\[EqFrobOnSource\]](#EqFrobOnSource){reference-type="eqref" reference="EqFrobOnSource"} and [\[EqFrobOnTarget\]](#EqFrobOnTarget){reference-type="eqref" reference="EqFrobOnTarget"}. Let $y:=(\underline{\mathcal{L}},\hat{\delta}, \gamma H)$ be an $S$-valued point of $$\bigl(\mathcal{R}\mathcal{Z}_{\mathcal{M},{\underline{{\mathbb{L}}}}}^{\leq \mu}\mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\Breve{\mathcal{O}}_{\mu,\beta}}}\operatorname{Spec}{\overline\mathbb{F}_\infty}\bigr) \times \operatorname{Isom}^{\otimes}(\omega,\check{\mathcal{V}}_{{\underline{\mathbb{E}}}})/H.$$ The images of this point and of $\Phi_m(y)=({}^{\tau^m\!}\underline{\mathcal{L}}, \widehat{\varphi}_{{\mathbb{L}}}^{\;-m}\circ{}^{\tau^m\!}\hat{\delta},\gamma H)$ in $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}$ are given by $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(y)=(\underline{\mathcal{E}},\check{\mathcal{V}}_\delta^{-1}\gamma H)$ and $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}\circ\Phi_m(y)=(\underline{\mathcal{E}}',\check{\mathcal{V}}_{\delta'}^{-1}\gamma H)$, respectively, where $\underline{\mathcal{E}}:=\hat{\delta}^*{\underline{\mathbb{E}}}_S$ and $\underline{\mathcal{E}}':=(\widehat{\varphi}_{{\mathbb{L}}}^{\;-m}\circ {}^{\tau^m\!}\hat{\delta})^*{\underline{\mathbb{E}}}_S$, and $\delta\colon\underline{\mathcal{E}}\to{\underline{\mathbb{E}}}_S$ and $\delta'\colon\underline{\mathcal{E}}'\to{\underline{\mathbb{E}}}_S$ are the quasi-isogenies with $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta)=\hat{\delta}$ and $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\delta')=\widehat{\varphi}_{{\mathbb{L}}}^{\;-m}\circ{}^{\tau^m\!}\hat{\delta}$, which are isomorphisms outside $\infty$. We obtain the image $\Phi_m\circ\widetilde{\Theta}_{{\underline{\mathbb{E}}}}(y)=({}^{\tau^m\!}\underline{\mathcal{E}},{}^{\tau^m\!}(\check{\mathcal{V}}_\delta^{-1}\gamma) H)$, which comes with the quasi-isogeny ${}^{\tau^m\!}\delta\colon{}^{\tau^m\!}\underline{\mathcal{E}}\to{}^{\tau^m\!}{\underline{\mathbb{E}}}_S$. Then $\psi:={}^{\tau^m\!}\delta^{-1}\circ\Phi_{{\underline{\mathbb{E}}}}^{m}\circ\delta'\colon\underline{\mathcal{E}}'\to{}^{\tau^m\!}\underline{\mathcal{E}}$ is a quasi-isogeny by Definition [Definition 1](#DeFfrobIsog){reference-type="ref" reference="DeFfrobIsog"} which by Corollary [Corollary 1](#CorFrobIsog){reference-type="ref" reference="CorFrobIsog"} satisfies $L_{\infty,\mathcal{M}_{\beta^{-1}}}(\psi)=\operatorname{id}_{{}^{\tau^m\!}\underline{\mathcal{L}}}$ and $\check{\mathcal{V}}_\psi\circ\check{\mathcal{V}}_{\delta'}^{-1}\gamma H=\check{\mathcal{V}}_{{}^{\tau^m\!}\delta}^{-1}\circ\check{\mathcal{V}}_{\Phi_{{\underline{\mathbb{E}}}}^m}\circ\gamma H={}^{\tau^m\!}(\check{\mathcal{V}}_\delta^{-1}\gamma) H$, because $\check{\mathcal{V}}_{\Phi_{{\underline{\mathbb{E}}}}^{m}}\circ\gamma={}^{\tau^m\!}(\gamma)$. Therefore, $\psi$ identifies $(\underline{\mathcal{E}}',\check{\mathcal{V}}_{\delta'}^{-1}\gamma H)$ with $({}^{\tau^m\!}\underline{\mathcal{E}},{}^{\tau^m\!}(\check{\mathcal{V}}_\delta^{-1}\gamma) H)$, and this proves $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}\circ\Phi_m=\Phi_m\circ\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$. Since $\Phi_m$ commutes with the action of $I_{{\underline{\mathbb{E}}}}(Q)$, this also proves that $\Theta_{{\underline{\mathbb{E}}}}$ commutes with the $q^m$-Frobenius endomorphisms $\Phi_m$. Finally, the target $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}{}_{/\mathcal{X}}$ carries the $q^m$-Frobenius endomorphism $\Phi_{\underline{\mathcal{E}}}^m$ induced from [\[EqFrobOnTarget\]](#EqFrobOnTarget){reference-type="eqref" reference="EqFrobOnTarget"} on $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)} \mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$. The reason is that if a morphism $S_{\rm red}\to\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)} \mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ given by $(\underline{\mathcal{E}},\gamma H)$ factors through $\mathcal{X}_{\underline{\mathbb{E}}}=\operatorname{im}(\widetilde{\Theta}_{{\underline{\mathbb{E}}}})$, then the morphism $S_{\rm red}\to\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)} \mathop{\mathrm{\mathchoice {\widehat{\raisebox{0ex}[0ex]{$\displaystyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\textstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptstyle\times$}}} {\widehat{\raisebox{0ex}[0ex]{$\scriptscriptstyle\times$}}}}}_{{\mathcal{O}_{\mu,\beta}}} \operatorname{Spf}{\Breve{\mathcal{O}}_{\mu,\beta}}$ given by $({}^{\tau^m\!}\underline{\mathcal{E}},{}^{\tau^m\!}(\lambda)H)$ also factors through $\mathcal{X}_{\underline{\mathbb{E}}}=\operatorname{im}(\widetilde{\Theta}_{{\underline{\mathbb{E}}}})$, because $\widetilde{\Theta}_{{\underline{\mathbb{E}}}}$ commutes with the $\Phi_m$. This shows that also $\Theta_{{\underline{\mathbb{E}}},\mathcal{X}}$ is compatible with the $q^m$-Frobenius endomorphisms $\Phi_m$ from [\[EqFrobOnTarget\]](#EqFrobOnTarget){reference-type="eqref" reference="EqFrobOnTarget"} and [\[EqFrobOnSource\]](#EqFrobOnSource){reference-type="eqref" reference="EqFrobOnSource"}. This completes the proof of Theorem [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"}[\[Uniformization2_C\]](#Uniformization2_C){reference-type="ref" reference="Uniformization2_C"}. ◻ ## Application to the Langlands-Rapoport Conjecture We finish this section with an application of our main theorem [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"} to the Langlands-Rapoport Conjecture over function fields. Consider the following category $\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)$, which generalizes Anderson's $t$-motives [@Anderson-tmotives]. Write $\Breve{Q}:=Q\otimes_{\mathbb{F}_q}\overline{\mathbb{F}}_q$. **Definition 1**. The category of $X$-motives $\mathcal{M}ot_X^{\infty}$ is defined as $$\begin{split} \mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q):=\{\, & (\mathcal{F},\varphi)\colon \mathcal{F}\text{ is a vector bundle on $X_{\overline{\mathbb{F}}_q}$, and } \\ & \varphi: \mathcal{F}|_{(X\smallsetminus\{\infty\})_{\overline{\mathbb{F}}_q}}\xrightarrow{\sim}{}^{\tau\!}\mathcal{F}|_{(X\smallsetminus\{\infty\})_{\overline{\mathbb{F}}_q}} \text{ is an isomorphism of vector bundles} \}, \end{split}$$ with morphisms in this category given by $$\mathrm{Hom}_{\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)}((\mathcal{F},\varphi),(\widetilde{\mathcal{F}},\widetilde{\varphi})):=\{f\in \mathrm{Hom}_{\Breve{Q}}(\mathcal{F}\otimes_{\mathcal{O}_X}Q,\widetilde{\mathcal{F}}\otimes_{\mathcal{O}_X}Q) \colon \widetilde{\varphi}\circ f={}^{\tau}f\circ\varphi\}.$$ The *motivic Galois gerbe* is the Tannakian fundamental group $$\mathfrak{P}:=\operatorname{Aut}^{\otimes}(\underline{\omega}|\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q))$$ corresponding to the fiber functor $\underline{\omega}$. **Remark 1**. A $X$-motive $\underline{\mathcal{F}}=(\mathcal{F},\varphi)$ of rank $r$ is just the same as a global $\operatorname{GL}_r$-shtuka in $\operatorname{Sht}_{\operatorname{GL}_r,\varnothing,\infty\times\infty}(\overline{\mathbb{F}}_\infty)$ over $\overline{\mathbb{F}}_\infty$. But note that the category $\operatorname{Sht}_{\operatorname{GL}_r,\varnothing,\infty\times\infty}(\overline{\mathbb{F}}_\infty)$ is a groupoid with all its morphisms being isomorphisms, while $\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)$ is abelian and contains morphisms which may fail to be isomorphisms, and even morphisms between $X$-motives of different rank are allowed. In [@AH_CMotives Theorem 1.5], Arasteh Rad and the first author proved the following result. **Lemma 1**. *$\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)$ is a semi-simple $Q$-linear Tannakian category, with a canonical fiber functor $\underline{\omega}$ over $\Breve{Q}$ given by $\underline{\omega}: (\mathcal{F},\varphi)\mapsto \mathcal{F}\otimes_{\mathcal{O}_X}Q=\mathcal{F}\otimes_{\mathcal{O}_{X_{\overline{\mathbb{F}}_q}}}\Breve{Q}$. The motivic Galois gerbe is an extension $$1\to \mathfrak{P}^{\Delta}\to\mathfrak{P}\to \operatorname{Gal}(\Breve{Q}/Q)\to 1,$$ where the kernel group $\mathfrak{P}^{\Delta}$ is a pro-reductive group over $\Breve{Q}$.* Note that $\underline{\omega}$ is not the only fiber functor for $\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)$. **Definition 1**. For any closed point $v\in X\smallsetminus\{\infty\}$, there is a fiber functor given by the $v$-adic dual Tate module (also called $v$-adic étale cohomology) of $\underline{\mathcal{F}}$ $${\rm H}^1_{v,{\rm \acute{e}t\/}}:\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)\longrightarrow Q_v\text{-Vect},\qquad \underline{\mathcal{F}} \mapsto {\rm H}^1_{v,{\rm \acute{e}t\/}}(\underline{\mathcal{F}},\mathcal{O}_v):= \{m\in \mathcal{F}\otimes_{\mathcal{O}_{X_{\overline{\mathbb{F}}_q}}} \Breve{\mathcal{O}}_v\colon \varphi(m) = {}^{\tau\!}m\}\otimes_{\mathcal{O}_v} Q_v.$$ The fiber functor ${\rm H}^1_{v,{\rm \acute{e}t\/}}$ corresponds to a homomorphism of Galois gerbs $\xi_v\colon \mathfrak{H}_v\to\mathfrak{P}$, where $\mathfrak{H}_v=\operatorname{Gal}(\Breve{Q}/Q)$ is the trivial Galois gerbe, i.e. the Tannakian fundamental group of the category of $Q_v$-vector spaces. At $\infty$ there is a fiber functor given by the "crystalline cohomology" of $\underline{\mathcal{F}}$ $${\rm H}^1_{\infty,\rm crys}:\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)\longrightarrow \Breve{Q}_{\infty}\text{-Vect},\qquad \underline{\mathcal{F}} \mapsto \underline{\mathcal{F}} \otimes_{\mathcal{O}_{X_{\overline{\mathbb{F}}_q}}} \Breve{Q}_\infty\,.$$ The fiber functor ${\rm H}^1_{\infty,\rm crys}$ corresponds to a homomorphism of Galois gerbs $\xi_\infty\colon \mathfrak{H}_\infty\to\mathfrak{P}$, where $\mathfrak{H}_\infty$ is the "Dieudonne gerbe", which is the Tannakian fundamental group of the category of $F$-crystals for $Q_\infty$. **Definition 1**. For a smooth affine group scheme $\mathcal{G}$ with connected fibers over $X$ and reductive generic fiber $G$, a *$G$-motive* is a tensor functor $\underline{\mathcal{M}}_G:\operatorname{Rep}_Q(G)\to \mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)$. Equivalently, a $G$-motive is a homomorphism of Galois gerbes $h\colon \mathfrak{P}\to\mathfrak{G}_G$, where $\mathfrak{G}_G:=G(\Breve{Q})\rtimes \operatorname{Gal}(\Breve{Q}/Q)$ is the neutral Galois gerbe of $G$. We denote by $G\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)$ the category of $G$-motives. For any morphism $h: \mathfrak{P}\to \mathfrak{G}_G$ of Galois gerbes, we let $h_v:=h\circ\xi_v: \mathfrak{H}_v\to \mathfrak{G}_G$ and $h_\infty:=h\circ\xi_\infty: \mathfrak{H}_{\infty}\to \mathfrak{G}_G$ be the compositum. The morphism $h_\infty$ defines a "local $G$-isoshtuka" $\bigl((L_\infty\mathcal{M})_{\overline{\mathbb{F}}_q},b)\bigr)$ over $\overline{\mathbb{F}}_q$; see [@AH_LRConj]. We define $$X^{\infty}(h):=\{(g_v)_{v\neq \infty}\in G(\mathbb{A}^{\infty}):\operatorname{int}(g_v)\circ\xi_v=\xi_v \}.$$ $$X_{\infty}(h):=\{\overline{g}\in (L_\infty\mathcal{M}/L^+_\infty\mathcal{M})(\overline{\mathbb{F}}_\infty)\colon \tau_M(g)^{-1}b g\text{ is bounded by }\mu\}.$$ There is a functor $\operatorname{Sht}_{\mathcal{G},\infty\times\infty}(\overline{\mathbb{F}}_q)\to G\mathcal{M}ot_X^{\infty}(\overline{\mathbb{F}}_q)$ given by sending a global $\mathcal{G}$-shtuka $\underline{\mathcal{E}}=(\mathcal{E},\mathcal{E}',\varphi,\varphi')$ to the tensor functor $$\underline{\mathcal{M}}_G^{\underline{\mathcal{E}}}:(V,\rho)\mapsto \bigl(\mathcal{E}\overset{G}{\times}V,(\varphi'\circ\varphi)\times 1\bigr),$$ where $\rho:G\to \operatorname{GL}(V)$. We denote its associated homomorphism from Definition [Definition 1](#Def_GMotives){reference-type="ref" reference="Def_GMotives"} of Galois gerbes $\mathfrak{P}\to\mathfrak{G}_G$ by $h_{\underline{\mathcal{E}}}$. **Lemma 1**. *We have an equality of the isogeny groups $I_{\underline{\mathcal{E}}}=I_{h_{\underline{\mathcal{E}}}}$. We have $X_{\infty}(h_{\underline{\mathcal{E}}})=X_{\mathcal{G}}^{\leq\mu}(b)$ and $X^{\infty}(h_{\underline{\mathcal{E}}})=\operatorname{Isom}^{\otimes}(\omega,\breve{\mathcal{V}}_{\underline{\mathcal{E}}})$.* *Proof.* This follows directly from the definitions. ◻ **Corollary 1**. *The $\overline{\mathbb{F}}_{\infty}$-points of the Shtuka space $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}$ has the form predicted by the Langlands-Rapoport conjecture, i.e. $\operatorname{Sht}_{\mathcal{G},H,\widehat{\infty}\times\infty}^{\mathcal{Z}(\mu,\beta)}(\overline{\mathbb{F}}_{\infty})=\coprod\limits_hI_h(Q)\backslash X_{\infty}(h)\times X^{\infty}(h)/H$ compatible with Hecke correspondences, Frobenius, action of the center.* *Proof.* This follows by combing Lemma [Lemma 1](#lemma-ADLV-identied-Xphi){reference-type="ref" reference="lemma-ADLV-identied-Xphi"} and Theorem [Theorem 1](#Uniformization2){reference-type="ref" reference="Uniformization2"}. ◻ [^1]: Here we use superscript for indexing inside the Hecke stack and subscript for indexing inside $\mathcal{CB}un$; see Definition [Definition 1](#DefCBun){reference-type="ref" reference="DefCBun"}. Both super- and sub- scripts will be used in Section [2.8](#subsec:Chains){reference-type="ref" reference="subsec:Chains"}, where we explain that our $\mathcal{G}$-shtukas generalize the $\mathscr{D}$-elliptic sheaves from [@Laumon-Rapoport-Stuhler]. [^2]: In the case of abelian varieties $\mathcal{A}$, the Tate module $T(\mathcal{A})$ is the homology $\prod\limits_{\ell}H_{1,{\rm \acute{e}t\/}}(\mathcal{A},\mathbb{Z}_{\ell})$, while for $\mathcal{G}$-shtukas $\underline{\mathcal{E}}$, the Tate module $\check{\mathcal{T}}_{\underline{\mathcal{E}}}(\rho)$ is the cohomology $\prod\limits_{v\neq\infty}H^1_{{\rm \acute{e}t\/}}(\rho_*\underline{\mathcal{E}},\mathcal{O}_v)$, see [@HartlKim Definition 3.4.1], hence the "dual" terminology.

arxiv_math

--- abstract: | In this paper we study the mean-field limit of a system of point vortices for the lake equations. These equations model the evolution of the horizontal component of the velocity field of a fluid in a lake of non-constant depth, when its vertical component can be neglected. As for the axisymmetric Euler equations there are non-trivial self interactions of the vortices consisting in the leading order of a transport term along the level sets of the depth function. If the self-interactions are negligible, we show that the system of point vortices converges to the lake equations as the number of points becomes very large. If the self-interactions are of order one, we show that it converges to a forced lake equations and if the self-interactions are predominant, then up to time rescaling we show that it converges to a transport equation. The proof is based on a modulated energy approach introduced by Duerinckx and Serfaty in (Duke Math. J., 2020) that we adapt to deal with the heterogeneity of the lake kernel. address: Univ. Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France. author: - Matthieu Ménard title: Mean-field limit of point vortices for the lake equations. --- # Introduction ## Lake equations The purpose of this article is to investigate the mean-field limit of point vortices (which are dirac masses in the vorticity field of a fluid) in a lake of non-constant depth. Namely we want to establish the convergence of an empirical distribution of point vortices to a continuous density solving the lake equations, as the number of vortices becomes very large. These equations describe the evolution of the horizontal velocity field of an incompressible fluid in a lake, when: - The depth is small with respect to the lengthscale of horizontal variations of the fluid velocity. - The surface of the fluid is almost flat (small Froude number). - The vertical velocity is small with respect to the horizontal velocity. For a rigorous derivation of these equations from the shallow water system we refer to the work of Bresch, Gisclon and Lin in [@BreschGisclonLin]. A more general introduction to depth-averaged models can be found in [@Greenspan Chapter 5] and a discussion on the three upper hypothesis can be found in [@Richardson]. These equations are similar to the planar Euler equations, but they take into account the depth of the lake, given by a positive function $b$. If $b$ is constant, then one recovers the usual planar Euler equations. The well-posedness of the lake equations on bounded domains have been first investigated by Levermore, Oliver and Titi in [@LevermoreOliverTiti]. In this paper they studied an analogue of the Yudovich theorem for Euler equations (see [@Yudovich]). This result was extended later by Bresch and Métivier in [@BreschMetivier] to include the case where the depth function $b$ vanishes at the boundary and by Lacave, Nguyen and Pausader in [@LacaveNguyenPausader] to deal with the case of rough bottoms. The existence and uniqueness of global classical solutions have been established by Al Taki and Lacave in [@AlTakiLacave]. In this paper we will study the case of an infinite lake modeled by the whole plane $\mathbb{R}^2$. We are interested in the following vorticity form of the equations: $$\label{lake_equation_vorticity} \left\{ \begin{aligned} & \partial_t \omega + \operatorname{div}\left(\left(u-\alpha\frac{\nabla^\bot b}{b}\right)\omega\right) = 0 \\ & \operatorname{div}(bu) = 0 \\ & \operatorname{curl}(u) = \omega \end{aligned}\right.$$ where - $\bot$ denotes the rotation by $\displaystyle{\frac{\pi}{2}}$ (that is $(x_1,x_2)^\bot := (-x_2,x_1)$). - $\alpha \in [0,+\infty)$ is a forcing parameter. - $b : \mathbb{R}^2 \longrightarrow (0,+\infty)$ is the depth function satisfying Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"} below. - $u : [0,+\infty)\times \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ is the velocity field of the fluid. - $\omega : [0,+\infty)\times \mathbb{R}^2 \longrightarrow \mathbb{R}$ is the vorticity field of the fluid, defined by $$\omega = \operatorname{curl}(u) := \partial_1 u_2 - \partial_2 u_1.$$ The true lake equations have no forcing term ($\alpha=0$), but we will study this more general model as it could arise as a mean-field limit of point vortices (in the regime where the self-interaction of the vortices are not negligible). It is a particular case of a model studied by Duerinckx and Fischer (see [@DuerinckxFischer Equation (1.9)]). In this work the authors proved the global existence and uniqueness of weak solutions and the local well-posedness of strong solutions. We will consider the following definition of weak solutions: **Definition 1**. *Let $T > 0$ and $\omega_0 \in L^\infty(\mathbb{R}^2)$ with compact support. We say that $(\omega,u)$ is a weak solution of [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} on $[0,T]$ with initial condition $\omega_0$ if $\omega \in L^1([0,T],L^\infty(\mathbb{R}^2,\mathbb{R}^2)) \cap \mathcal{C}^0([0,T],L^\infty(\mathbb{R}^2)-w^\ast)$ with compact support in space for all $t \in [0,T]$, $u \in L^2_{\rm loc}([0,T]\times\mathbb{R}^2,\mathbb{R}^2)$, for almost every $t \in [0,T]$, $\operatorname{div}(bu) = 0$ and $\operatorname{curl}(u) = \omega$ distributionally and for all $\varphi$ smooth with compact support in $[0,T)\times\mathbb{R}^2$ and $t \in [0,T)$, $$\label{formulation_faible} \iint_{[0,t]\times\mathbb{R}^2} \partial_t \varphi\omega + \nabla \varphi\cdot\left(u-\alpha\frac{\nabla^\bot b}{b}\right)\omega = \int_{\mathbb{R}^2} \varphi(t)\omega(t)-\int_{\mathbb{R}^2} \varphi(0)\omega_0.$$* In the regime where the self-interaction of the point vortices is predominant, the system of point vortices will converge in an accelerated timescale to a transport equation along the level sets of the topography: $$\label{transport_equation} \partial_t \overline{\omega} - \operatorname{div}\left(\frac{\nabla^\bot b}{b}\overline{\omega}\right) = 0.$$ For this equation we will use the following definition of weak solutions: **Definition 2**. *Let $T > 0$ and $\overline{\omega}_0 \in L^\infty(\mathbb{R}^2)$ with compact support. We say that is a weak solution of [\[transport_equation\]](#transport_equation){reference-type="eqref" reference="transport_equation"} on $[0,T]$ with initial condition $\overline{\omega}_0$ if $\overline{\omega}\in L^1([0,T],L^\infty(\mathbb{R}^2,\mathbb{R}^2))\cap \mathcal{C}^0([0,T],L^\infty(\mathbb{R}^2)-w^\ast)$ with compact support in space for all $t \in [0,T]$ and for all $\varphi$ smooth with compact support in $[0,T)\times\mathbb{R}^2$ and $t \in [0,T)$, $$\label{formulation_faible_transport} \iint_{[0,t]\times\mathbb{R}^2} \partial_t \varphi\overline{\omega}- \nabla \varphi\cdot\frac{\nabla^\bot b}{b}\overline{\omega} = \int_{\mathbb{R}^2} \varphi(t)\overline{\omega}(t)-\int_{\mathbb{R}^2} \varphi(0)\overline{\omega}_0.$$* ## Point vortices for the lake equations The forced lake equations [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} have been derived as the mean-field limit of complex Ginzburg-Landau vortices with forcing and pinning effects by Duerinckx and Serfaty in [@DuerinckxSerfaty]. The dynamics of these vortices comes from the physics of supraconductors or superfluids and is very different from the dynamics of vortices in a lake. In this paper we are interested in deriving Equations [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} as the mean-field limit of a model introduced by Richardson in [@Richardson]. In that work he established by a formal computation the equation followed by the center of vorticity $q(t)$ of a small vortex of size $\varepsilon$ in a lake of depth $b$. To leading order in $\varepsilon$, this equation gives $$\label{equation_pvs_richardson} \dot q(t) \approx -\frac{\Gamma|\ln(\varepsilon)|}{4\pi}\frac{\nabla^\bot b(q(t))}{q(t)}$$ where $\Gamma$ is the intensity of vorticity (that is $\displaystyle{\Gamma =\int_{B(q(0),\varepsilon)} \omega}$). This means that to leading order in $\varepsilon$, a very small vortex follows the level lines of the topography without seeing the interaction with other vortices remaining far from him. The latter equation was rigourously justified by Dekeyser and Van Schaftingen in [@DekeyserVanSchaftingen] for the motion of a single vortex and this result was extended later to the case of a finite number of vortices by Hientzsch, Lacave and Miot in [@HientzschLacaveMiot]. We want to investigate the behavior of $N$ point vortices of intensity $N^{-1}$ as $N$ becomes large. We will see in Section [2](#section:2){reference-type="ref" reference="section:2"} that the elliptic problem $$\left\{ \begin{aligned} & \operatorname{div}(bu) = 0 \\ & \operatorname{curl}(u) = \omega \end{aligned}\right.$$ has a unique solution given by the kernel $$g_b(x,y) := \sqrt{b(x)b(y)}g(x-y) + S_b(x,y)$$ where $S_b$ is a function solving a certain elliptic equation (see Equation [\[definition_S\_b\]](#definition_S_b){reference-type="eqref" reference="definition_S_b"}) and $\displaystyle{g(x) := -\frac{1}{2\pi}\ln|x|}$ is the opposite of the Green kernel of the Laplacian on the plane $\mathbb{R}^2$. More precisely, we have $$u(x) = -\frac{1}{b(x)}\int_{\mathbb{R}^2} \nabla_x^\bot g_b(x,y)\omega(y)\,\mathrm dy.$$ Recall that a point vortex is asymptotically represented by a dirac mass of vorticity. Therefore using the kernel $\nabla_x^\bot g_b$ we can compute the velocity field generated by $N-1$ vortices $\delta_{q_j}$ of intensity $\displaystyle{\frac{1}{N}}$ on a vortex $\delta_{q_i}$: $$-\frac{1}{N}\underset{j\neq i}{\sum_{j=1}^N}\frac{1}{b(q_i)}\nabla^\bot_x g_b(q_i,q_j).$$ This term correspond to the term $u_{reg}$ given by Richardson in [@Richardson Equation (2.90)]. Combining this equation with the self-interaction term of [\[equation_pvs_richardson\]](#equation_pvs_richardson){reference-type="eqref" reference="equation_pvs_richardson"} we get the model of point vortices we will study in this paper: $$\label{equation_pv} \dot q_i = -\alpha_N\frac{\nabla^\bot b(q_i)}{b(q_i)} -\frac{1}{N}\underset{j\neq i}{\sum_{j=1}^N}\frac{1}{b(q_i)}\nabla^\bot_x g_b(q_i,q_j)$$ where we have denoted $$\alpha_N := \frac{|\ln(\varepsilon_N)|}{4\pi N}$$ where $\varepsilon_N$ is the size of the vortices. *Remark 3*. Up to now there is no mathematical justification of Equation [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"}: We do not even expect this equation to describe precisely the motion of a fixed number of small vortices as we have neglected all self-interaction terms of order smaller than $|\ln(\varepsilon)|$. However Theorem [Theorem 8](#MFL_theorem){reference-type="ref" reference="MFL_theorem"} will justify that this simplified model is statistically relevant when $N$ becomes very large. *Remark 4*. There are several works establishing approximate analytical trajectories of vortices in a lake for some specific depth profiles, and also other numerical and experimental results related to vortex dynamics in lakes. For more details we refer to the results of [@Richardson] and the associated bibliography. Two quantities will be of interest for the study of this system. The interaction energy $$E_N(t) := \frac{1}{N^2}\sum_{i=1}^N\underset{j \neq i}{\sum_{j=1}^N} g_b(q_i(t),q_j(t))$$ and the moment of inertia $$I_N(t) := \frac{1}{N}\sum_{i=1}^N|q_i(t)|^2.$$ One could prove that the total energy $$E_N^{tot} := E_N + \frac{\alpha_N}{N}\sum_{i=1}^N b(q_i)$$ is a conserved quantity for the point vortex system [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"} or that if $\omega$ is a solution of [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} with enough regularity and decay, the quantity $$\iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y)\omega(t,x)\omega(t,y)\,\mathrm dx \,\mathrm dy + \alpha \int_{\mathbb{R}^2} b(x)\omega(t,x) \,\mathrm dx$$ is conserved by the flow. The moment of inertia $I_N$ and the interaction energy $E_N$ are not conserved quantity but they are bounded in time, and this will be useful both for our mean-field limit result and for the well-posedness of System [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"} (see Section [3](#section:3){reference-type="ref" reference="section:3"}). If $\alpha_N \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} +\infty$ the self-interactions are predominant. In order to study this regime we will study an accelerated timescale as it was done in [@DekeyserVanSchaftingen] and [@HientzschLacaveMiot] to study the motion of a finite number of vortices. Therefore we define: $$\overline{q_i}(t) := q_i(\alpha_N^{-1}t).$$ This gives $$\label{equation_pv_rescaled} \dot{\overline{q_i}} = -\frac{\nabla^\bot b(\overline{q_i})}{b(\overline{q_i})} -\frac{1}{N\alpha_N}\underset{j\neq i}{\sum_{j=1}^N}\frac{1}{b(\overline{q_i})}\nabla^\bot_x g_b(\overline{q_i},\overline{q_j})$$ We also define the rescaled interaction energy $$\overline{E_N}(t) := E_N(\alpha_N^{-1}t)$$ and the rescaled moment of inertia $$\overline{I_N}(t) := I_N(\alpha_N^{-1}t).$$ ## Mean-field limits Mean-field limits consist in studying the convergence of a system of ordinary differential equations modeling the evolution of a finite number of particles $$\label{edo_generic_order_1} \dot x_i = \frac{1}{N}\sum_{i=1}^NK(x_i - x_j)$$ to a Euler-like equation modeling the evolution of a continuous density $\mu(t,x)$: $$\label{edp_generic_order_1} \partial_t \mu + \operatorname{div}((K*\mu)\mu) = 0$$ when the number of particles becomes large (here $K : \mathbb{R}^d \longrightarrow \mathbb{R}^d$ is an interaction kernel). For systems of order two we are interested in the convergence of a system of particles following Newton's second law $$\ddot x_i = \frac{1}{N}\sum_{i=1}^NK(x_i - x_j)$$ to a Vlasov-like equation modeling the evolution of a continuous density $f(t,x,v)$: $$\label{edp_generic_order_2} \left\{ \begin{aligned} & \partial_t f + \operatorname{div}_x(fv) + \operatorname{div}_v((K*\mu)f) = 0 \\ & \mu(t,x) = \int_{\mathbb{R}^d} f(t,x,v)\,\mathrm dv. \end{aligned}\right.$$ A mean-field limit result consists in proving that if at time zero, the empirical distribution of the particles $$\frac{1}{N}\sum_{i=1}^N \delta_{x_i(t)} \qquad \left(\text{respectively} \quad \frac{1}{N}\sum_{i=1}^N \delta_{(x_i(t),\dot x_i(t))}\right)$$ converges to the continuous density $\mu(t,x)$ solution of [\[edp_generic_order_1\]](#edp_generic_order_1){reference-type="eqref" reference="edp_generic_order_1"} (respectively $f(t,x,v)$ solution of [\[edp_generic_order_2\]](#edp_generic_order_2){reference-type="eqref" reference="edp_generic_order_2"}) then the convergence also holds for any finite time. When $K$ is Lipschitz the mean-field limit of the upper system was established by compactness arguments in [@BraunHepp; @NeuzertWick] or by optimal transport theory and Wasserstein distances by Dobrushin in [@Dobrushin]. If $K$ is singular there are numerous results establishing the mean-field limit of systems of order one: Schochet proved in [@Schochet1] the mean-field convergence of the point vortex system (that is $\displaystyle{K = \frac{1}{2\pi}\frac{x^\bot}{|x|^2}}$ in dimension 2) to a measure-valued solution of Euler equations up to a subsequence, using arguments previously developed in [@Delort] and [@Schochet2] to prove existence of such solutions. For sub-coulombic interactions, that is $|K(x)|, |x||\nabla K(x)| \leqslant C|x|^{-\alpha}$ with $0 < \alpha < d-1$, the mean-field limit of [\[edo_generic_order_1\]](#edo_generic_order_1){reference-type="eqref" reference="edo_generic_order_1"} was proved by Hauray in [@Hauray] assuming $\operatorname{div}(K) = 0$ and using a Dobruschin-type approach (following the idea of [@HaurayJabin2; @HaurayJabin1]). It was also used by Carillo, Choi and Hauray to study with the mean-field limit of some aggregation models in [@CarrilloChoiHauray]. In [@Duerinckx] Duerinckx gave another proof of the mean-field limit of several Riesz interaction gradient flows using a \"modulated energy\" that was introduced by Serfaty in [@Serfaty2017]. In [@Serfaty], Serfaty used this modulated energy approach to prove the mean-field convergence of such systems where $K$ was a kernel given by Coulomb, logarithmic or Riesz interaction, that is $K = \nabla g$ for $g(x) = |x|^{-s}$ with $\max(d-2,0) \leqslant s < d$ for $d \geqslant 1$ or $g(x) = -\ln|x|$ for $d = 1$ or $2$. For this purpose $K*\mu$ was supposed to be Lipschitz. Rosenzweig proved in [@Rosenzweig] the mean-field convergence of the point vortex system without assuming Lipschitz regularity for the limit velocity field, using the same energy as in [@Serfaty] with refined estimates. Remark that it ensures that the point vortex system converges to any Yudovich solutions of the Euler equations (see [@Yudovich]). This result was extended later for higher dimensional systems ($d \geqslant 3$) in [@Rosenzweig4] by the same author. In [@NguyenRosenzweigSerfaty] Nguyen, Rosenzweig and Serfaty extended the modulated energy approach to a more general class of potentials $g$ using the commutator structure of the equations. With a modulated energy approach, Bresch, Jabin and Wang defined a modulated entropy functionnal which allowed them to prove mean-field limit of interacting particles with noise in [@BreschJabinWang; @BreschJabinWang2; @BreschJabinWang3]. This method was used later to obtain uniform in time convergence for Riesz-type flows by Rosenzweig and Serfaty in [@RosenzweigSerfaty] and by Rosenzweig, Serfaty and Chodron de Courcel in [@ChodronDeCourcelRosenzweigSerfaty]. For systems of order two, the mean-field limit has been established for several singular kernels: In [@HaurayJabin2; @HaurayJabin1], Hauray and Jabin dealt with the case of some sub-coulombian interactions (or more precisely $|K(x)| \leqslant c|x|^{-s}$ with $0 < s < 1$) by using a Dobrushin-type approach. In [@JabinWang2; @JabinWang1], Jabin and Wang studied the case of bounded and $W^{-1,\infty}$ gradients. In [@BoersPickl; @HuangLiuPickl; @Lazarovici; @LazaroviciPickl] the same kind of results is proved with some cutoff of the interaction kernel. In the appendix of [@Serfaty], Duerinckx and Serfaty studied the case of particles interacting with a Coulomb or a Riesz interaction kernel to the Vlasov equation in the monokinetic regime, that is the pressureless Euler-Poisson equations. The same method have been used to study the mean-field limit of more general models coming from quantum physics, biology or fluid dynamics (see for example [@CarilloChoi; @M1preprint; @BenPorat]). In [@HanKwanIacobelli], Han-Kwan and Iacobelli proved the mean-field limit of particles following Newton's second law to the Euler equation in a quasineutral regime or in the gyrokinetic limit. This result was extended later by Rosenzweig in [@Rosenzweig2] to allow a larger choice of scaling between the number of particles and the coupling constant. Recently, Bresch, Jabin and Soler were able in [@BreschJabinSoler] to prove the mean-field limit derivation of the Vlasov-Fokker-Planck equation with the true Coulomb interactions using the BBGKY hierarchy and the diffusivity in the velocity variables to get estimates on the marginals. Numerous other mean-field limit results were proved for interacting particles with noise with regular or singular kernels. See for example [@BermanOnheim; @BolleyChafaiJoaquin; @CarilloFerreiraPrecioso; @FournierHaurayMischler; @JabinWang2; @JabinWang1; @Lacker; @LiLiuYu; @NguyenRosenzweigSerfaty; @Osada]. For a more complete bibliography on the mean-field limit of interacting particles with noise we refer to the bibliography of [@ChodronDeCourcelRosenzweigSerfaty]. For a general introduction to the subject of mean-field limits we refer to the reviews [@Golse; @Jabin]. ## Notations and assumptions ### Notations - For $u \in L^1_{\rm loc}(\mathbb{R}^2,\mathbb{R}^2)$, we denote $\operatorname{curl}(u) = \partial_1 u_2 - \partial_2 u_1$. - For $h \in \dot H^1(\mathbb{R}^2)$, we denote $$\label{definition_SE} [h,h]_{i,j} := 2\partial_i h \partial_j h - |\nabla h|^2\delta_{i,j}.$$ It is the stress-energy tensor used in [@Serfaty] to prove the mean-field limit of several singular ODE's. Remark that for $h$ smooth enough, we have $$\operatorname{div}[h,h] = 2\Delta h \nabla h.$$ - We denote $<\! x \!> = (1+|x|^2)^\frac{1}{2}$. - $g$ is the opposite of the Green function of the laplacian: $$g(x) := -\frac{1}{2\pi}\ln|x|.$$ - $|\cdot|_{\mathcal{C}^{0,s}}$ is the semi-norm associated to the Hölder space $\mathcal{C}^{0,s}$: $$|f|_{\mathcal{C}^{0,s}} = \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-y|^s}.$$ - When $1 \leqslant p \leqslant+\infty$, $p'$ denotes the dual exponent of $p$. - If $\nu$ is a probability measure on $\mathbb{R}^2$, we will denote $\nu^{\otimes 2} := \nu \otimes \nu$. - $C$ is a generic constant. We will denote $C_{A,B}$ when a constant depends on some quantities $A$ and $B$. - $\mathcal{P}(\mathbb{R}^2)$ is the space of probability measures on $\mathbb{R}^2$. - For $Q_N = (q_1,...,q_N) \in (\mathbb{R}^2)^N$ we denote $\displaystyle{I(Q_N) = \frac{1}{N}\sum_{i=1}^N |q_i|^2}$. ### Assumptions We will make the following assumption on the depth function $b$: **Assumption 5**. *We assume that $b$ is a smooth function, $\inf{b} > 0$, $\sup{b} < +\infty$ and that there exists $\gamma > 0$ such that $$(1+|x|)^{4+\gamma}(|\nabla b(x)| + |D^2 b(x)|) < +\infty.$$* We will consider regular solutions of [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} and [\[transport_equation\]](#transport_equation){reference-type="eqref" reference="transport_equation"} in the following sense: **Assumption 6**. *We say that a function $\omega(t,x)$ satisfies Assumption [Assumption 6](#assumption_omega){reference-type="ref" reference="assumption_omega"} if $\omega \in L^{\infty}([0,T],L^\infty(\mathbb{R}^2)\cap \mathcal{P}(\mathbb{R}^2))\cap \mathcal{C}^0([0,T],L^\infty(\mathbb{R}^2)-w^\ast)$, if there exists a compact $K$ such that for every $t \in [0,T]$, $\operatorname{supp}(\omega(t)) \subset K$ and if $\nabla G_b[\omega] \in L^\infty([0,T],W^{1,\infty})$ where $G_b$ is the operator defined by Equation [\[existence_noyau_green\]](#existence_noyau_green){reference-type="eqref" reference="existence_noyau_green"}.* *Remark 7*. A weak solution of [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} in the sense of Definition [Definition 1](#definition_weak_solution){reference-type="ref" reference="definition_weak_solution"} (or a weak solution of [\[transport_equation\]](#transport_equation){reference-type="eqref" reference="transport_equation"} in the sense of Definition [Definition 2](#definition_weak_solution_transport){reference-type="ref" reference="definition_weak_solution_transport"}) does not necessarily verify Assumption [Assumption 6](#assumption_omega){reference-type="ref" reference="assumption_omega"} because of the regularity we ask for the velocity field $\nabla G_b[\omega]$. This assumption will be crucial to apply Proposition [Proposition 39](#controle_terme_principal_gronwall){reference-type="ref" reference="controle_terme_principal_gronwall"} and prove the mean-field limit Theorem [Theorem 8](#MFL_theorem){reference-type="ref" reference="MFL_theorem"}. The existence and uniqueness of sufficiently regular solutions of [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} locally in time is ensured by [@DuerinckxFischer Theorem 2]. One could also prove that $\omega \in L^\infty([0,T],\mathcal{C}^{0,s})$ is sufficient to have $\nabla G_b[\omega] \in L^\infty([0,T],W^{1,\infty})$. ## Main result and plan of the paper The main result of this paper is the following theorem which gives the mean-field limit of the point vortex system [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"} and its rescaled version [\[equation_pv_rescaled\]](#equation_pv_rescaled){reference-type="eqref" reference="equation_pv_rescaled"} (we recall that the kernel $g_b$ is defined by [\[definition_g\_b\]](#definition_g_b){reference-type="eqref" reference="definition_g_b"}): **Theorem 8**. *Assume that $b$ satisfies Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}. We have mean-field convergence of the point-vortex system in the two following regimes:* 1. *Let $\omega$ be a solution of [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} with initial datum $\omega_0$ in the sense of Definition [Definition 1](#definition_weak_solution){reference-type="ref" reference="definition_weak_solution"}, satisfying Assumption [Assumption 6](#assumption_omega){reference-type="ref" reference="assumption_omega"} and $(q_1,...,q_N)$ be a solution of [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"}. Assume that:* - *$(I_N(0))_N$ is bounded.* - *$\displaystyle{\frac{1}{N}\sum_{i=1}^N \delta_{q_i^0}} \underset{N \rightarrow +\infty}{\xrightharpoonup{\; \; \ast \; \;}} \omega_0$ for the weak-$\ast$ topology of probability measures.* - *$\alpha_N \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} \alpha$.* - *$\displaystyle{\frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} g_b(q_i^0,q_j^0) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y)\omega_0(x)\omega_0(y)\,\mathrm dx \,\mathrm dy}$.* *Then for all $t \in [0,T]$, $\displaystyle{\frac{1}{N}\sum_{i=1}^N \delta_{q_i(t)}} \underset{N \rightarrow +\infty}{\xrightharpoonup{\; \; \ast \; \;}} \omega(t)$ for the weak-$\ast$ topology of probability measures and $$\frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} g_b(q_i(t),q_j(t)) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y)\omega(t,x)\omega(t,y)\,\mathrm dx \,\mathrm dy.$$* 2. *Let $\overline{\omega}$ be a solution of [\[transport_equation\]](#transport_equation){reference-type="eqref" reference="transport_equation"} with initial datum $\omega_0$ in the sense of Definition [Definition 2](#definition_weak_solution_transport){reference-type="ref" reference="definition_weak_solution_transport"}, satisfying Assumption [Assumption 6](#assumption_omega){reference-type="ref" reference="assumption_omega"} and $(\overline{q_1},...,\overline{q_N})$ be a solution of [\[equation_pv_rescaled\]](#equation_pv_rescaled){reference-type="eqref" reference="equation_pv_rescaled"}. Assume that:* - *$(\overline{I_N}(0))_N$ is bounded.* - *$\displaystyle{\frac{1}{N}\sum_{i=1}^N \delta_{q_i^0}} \underset{N \rightarrow +\infty}{\xrightharpoonup{\; \; \ast \; \;}} \omega_0$ for the weak-$\ast$ topology of probability measures.* - *$\alpha_N \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} +\infty$.* - *$\displaystyle{\frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} g_b(q_i^0,q_j^0) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y)\omega_0(x)\omega_0(y)\,\mathrm dx \,\mathrm dy}$.* *Then for all $t \in [0,T]$, $\displaystyle{\frac{1}{N}\sum_{i=1}^N \delta_{\overline{q_i}(t)}} \underset{N \rightarrow +\infty}{\xrightharpoonup{\; \; \ast \; \;}} \overline{\omega}(t)$ for the weak-$\ast$ topology of probability measures and $$\frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} g_b(\overline{q_i}(t),\overline{q_j}(t)) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y)\overline{\omega}(t,x)\overline{\omega}(t,y)\,\mathrm dx \,\mathrm dy.$$* Remark that in the case $\alpha_N \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} 0$ we recover the classical lake equations ([\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} with $\alpha = 0$). The boundedness of $(I_N(0))$ is a technical assumption made to ensure that not too much vorticity is going to infinity. This assumption was not necessary in the original papers of Duerinckx in [@Duerinckx] and of Serfaty in [@Serfaty] but we will need it to deal with the heterogeneity of the kernel $g_b$ (defined in [\[definition_g\_b\]](#definition_g_b){reference-type="eqref" reference="definition_g_b"}). The convergence of the interaction energy and the weak$-\ast$ convergence of $(\omega_N)$ to $\omega$ ensure the convergence of $(\omega_N)$ to $\omega$ in a stronger sense: We will prove in Corollary [Corollary 32](#corollary_weak_star_cv){reference-type="ref" reference="corollary_weak_star_cv"} that provided certain technical assumptions are satisfied, it is equivalent to the convergence to zero of a \"modulated energy\" functionnal. For an empirical measure of point vortices $(q_1,...,q_N)$ and a vorticity field $\omega \in L^\infty$ with compact support, this modulated energy is defined by: $$\begin{gathered} \label{definition_modulated_energy} \mathcal{F}_b(Q_N,\omega) := \\ \iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} g_b(x,y)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N \delta_{q_i} - \omega\right)(x)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N \delta_{q_i} - \omega\right)(y)\end{gathered}$$ where $$\Delta := \{(x,x) \; ; \; x \in \mathbb{R}^2\}.$$ We will use this energy to control the distance between solutions $\omega$ and $Q_N$ of [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} and [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"} or solutions $\overline{\omega}$ and $\overline{Q_N}$ of [\[transport_equation\]](#transport_equation){reference-type="eqref" reference="transport_equation"} and [\[equation_pv_rescaled\]](#equation_pv_rescaled){reference-type="eqref" reference="equation_pv_rescaled"} at any given time $t$: $$\label{definition_F_b_N} \mathcal{F}_{b,N}(t) := \mathcal{F}_{b}(Q_N(t),\omega(t))$$ and $$\label{definition_F_b_N_rescaled} \overline{\mathcal{F}}_{b,N}(t) := \mathcal{F}_{b}(\overline{Q_N}(t),\overline{\omega}(t)).$$ The proof of Theorem [Theorem 8](#MFL_theorem){reference-type="ref" reference="MFL_theorem"} relies on Grönwall-type estimates on these two quantities. The paper is organised as follows: - In Section [2](#section:2){reference-type="ref" reference="section:2"} we prove the well-posedness of the elliptic problem linking a velocity field satisfying $\operatorname{div}(bu) = 0$ and its vorticity, the existence of a Green kernel for this elliptic problem and we establish several regularity estimates. - In Section [3](#section:3){reference-type="ref" reference="section:3"} we prove that the point-vortex system is well-posed and give some estimates on the interaction energy and on the moment of inertia of the system that we will need in Section [7](#section:7){reference-type="ref" reference="section:7"}. - In Section [4](#section:4){reference-type="ref" reference="section:4"} we compute the time derivative of $\mathcal{F}_{b,N}$ and of $\overline{\mathcal{F}}_{b,N}$. - In Section [5](#section:5){reference-type="ref" reference="section:5"} we state several properties of the modulated energy. We prove that it controls the convergence in $H^s$ for $s < -1$ (see Corollary [Corollary 31](#corollary_coercivity){reference-type="ref" reference="corollary_coercivity"}) and that having the convergence of the modulated energy is equivalent to have weak-$\ast$ convergence of the point vortex system and convergence of its interaction energy (see Corollary [Corollary 32](#corollary_weak_star_cv){reference-type="ref" reference="corollary_weak_star_cv"}). - In Section [6](#section:6){reference-type="ref" reference="section:6"} we bound the main term appearing in the derivatives of the modulated energies. - In Section [7](#section:7){reference-type="ref" reference="section:7"} we use the results of the other sections to prove Theorem [Theorem 8](#MFL_theorem){reference-type="ref" reference="MFL_theorem"}. The modulated energy $\mathcal{F}_b$ is similar to the modulated energy defined in [@Serfaty Equation (1.16)] and the proofs of Sections [4](#section:4){reference-type="ref" reference="section:4"} to [7](#section:7){reference-type="ref" reference="section:7"} follow the same global ideas. The main difference between Theorem [Theorem 8](#MFL_theorem){reference-type="ref" reference="MFL_theorem"} and other mean-field limit results using modulated energies is that the kernel $g_b$ is not of the form $a(x,y) = a(x-y)$. Most of the difficulties adressed by this paper consist in dealing with the heterogeneity of the kernel $g_b$. ## Funding {#funding .unnumbered} This work is supported by the French National Research Agency in the framework of the project "SINGFLOWS" (ANR-18-CE40-0027-01). # Velocity reconstruction {#section:2} There exists a Biot-Savart type law to reconstruct a velocity field $u$ satisfying $\operatorname{div}(bu) = 0$ from its vorticity. In this section we prove several results concerning this reconstruction. In Subsection [2.1](#subsection:21){reference-type="ref" reference="subsection:21"} we prove that the elliptic equations linking $u$ with its vorticity are well-posed. In Subsection [2.2](#subsection:22){reference-type="ref" reference="subsection:22"} we prove some results related to the asymptotic behavior of the velocity field as $|x| \longrightarrow \infty$. In Subsection [2.3](#subsection:23){reference-type="ref" reference="subsection:23"}, we give an analogue of the Biot-Savart law for a velocity field satisfying System [\[elliptic_problem_velocity\]](#elliptic_problem_velocity){reference-type="eqref" reference="elliptic_problem_velocity"}. Finally, in Subsection [2.4](#subsection:24){reference-type="ref" reference="subsection:24"} we define some regularisations of the Coulomb kernel and of the dirac mass that we will need in Sections [5](#section:5){reference-type="ref" reference="section:5"} and [6](#section:6){reference-type="ref" reference="section:6"}. ## Well-posedness of the elliptic problem {#subsection:21} In this subsection we justify the well-posedness of the elliptic equations satisfied by the velocity field: $$\label{elliptic_problem_velocity} \left\{ \begin{aligned} & \operatorname{div}(bu) = 0 \\ & \operatorname{curl}(u) = \omega. \end{aligned}\right.$$ As we will write $\displaystyle{u = -\frac{1}{b}\nabla^\bot \psi}$ we will also consider the \"stream function\" formulation of the upper system: $$\label{elliptic_problem_stream} -\operatorname{div}\left(\frac{1}{b}\nabla \psi\right) = \omega.$$ For this purpose we will consider the following weighted Sobolev spaces: **Definition 9**. *For $1 < p < \infty$ we consider the Banach space $W^{2,p}_{-1}(\mathbb{R}^2)$ defined by $$W^{2,p}_{-1}(\mathbb{R}^2) := \{u \in \mathcal{D}'(\mathbb{R}^2) \; ; \; \forall \alpha \in \mathbb{N}^2, |\alpha| \leqslant 2, <\! \cdot \!>^{|\alpha|-1}D^\alpha u \in L^p(\mathbb{R}^2)\}$$ and equipped with its natural norm $$\left\lVert u\right\rVert_{W^{2,p}_{-1}} := \left(\sum_{|\alpha| \leqslant 2} \left\lVert<\! \cdot \!>^{|\alpha|-1}D^\alpha u\right\rVert^p_{L^p}\right)^\frac{1}{p}.$$* These weighted spaces were first introduced by Cantor in [@Cantor] and have been investigated to study elliptic equations on unbounded domains. For a more precise study of these spaces and further references we refer to [@LockhartMcOwen; @McOwen79; @McOwen80]. The following proposition is a straightforward consequence of [@LockhartMcOwen Theorem 2] (which is the combination of two theorems proved in [@McOwen79] and [@McOwen80]) and states that Equations [\[elliptic_problem_velocity\]](#elliptic_problem_velocity){reference-type="eqref" reference="elliptic_problem_velocity"} and [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} are well-posed. **Proposition 10**. *Let $2 < p < +\infty$, assume that $<\! \cdot \!>\omega \in L^p(\mathbb{R}^2)$, then there exists a unique solution $\psi$ of [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} in $W^{2,p}_{-1}(\mathbb{R}^2) \slash \mathbb{R}$. Morever if $u \in L^p(\mathbb{R}^2,\mathbb{R}^2)$ is a solution of [\[elliptic_problem_velocity\]](#elliptic_problem_velocity){reference-type="eqref" reference="elliptic_problem_velocity"} in the sense of distributions, then $$u = -\frac{1}{b}\nabla^\bot \psi.$$* *Proof.* We can rewrite Equation [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} as $$\begin{aligned} -\Delta \psi - b\nabla\left(\frac{1}{b}\right)\cdot\nabla \psi = b\omega.\end{aligned}$$ We have that: - $-\Delta$ is an elliptic operator with constant coefficients and homogeneous of degree $2$. - $\displaystyle{b\nabla\left(\frac{1}{b}\right) \in \mathcal{C}^0}$ and $$\underset{|x|\rightarrow+\infty}{\lim} \left|<\! x \!>^{2-1+0}b(x)\nabla\left(\frac{1}{b}\right) (x)\right| = 0$$ since $b$ satisfies Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}. - $<\! \cdot \!> b\omega \in L^p$. - $\displaystyle{-1\leqslant-\frac{2}{p}}$ and $\displaystyle{1-\frac{2}{p} \notin \mathbb{N}}$. Therefore by [@LockhartMcOwen Theorem 2], there exists a unique solution $\psi$ (up to a constant) of Equation [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} in $W^{2,p}_{-1}(\mathbb{R}^2)$. Now if $u \in L^p$ is a solution of [\[elliptic_problem_velocity\]](#elliptic_problem_velocity){reference-type="eqref" reference="elliptic_problem_velocity"}, then $$\begin{aligned} \left\lVert<\! \cdot \!>\operatorname{curl}(bu)\right\rVert_{L^p} &= \left\lVert<\! \cdot \!>b\omega\right\rVert_{L^p}+\left\lVert<\! \cdot \!>\nabla^\bot b\cdot u\right\rVert \\ &\leqslant C_b(\left\lVert<\! \cdot \!>\omega\right\rVert_{L^p}+\left\lVert u\right\rVert_{L^p})\end{aligned}$$ since $b$ satisfies Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}. Let us consider $\pi \in W^{2,p}_{-1}(\mathbb{R}^2)$ to be the unique solution (up to a constant) of $-\Delta \pi = \operatorname{curl}(bu)$ given by [@LockhartMcOwen Theorem 1]. Then $bu + \nabla^\bot \pi$ is a div-curl free vector field in $L^p$ so it is zero. Moreover, $$\begin{aligned} -\operatorname{div}\left(\frac{1}{b}\nabla \pi\right) &= -\operatorname{curl}\left(\frac{1}{b}\nabla^\bot \pi\right) = \operatorname{curl}(u) = \omega\end{aligned}$$ so $\nabla \pi = \nabla \psi$ by uniqueness of solutions of [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} in $W^{2,p}_{-1}(\mathbb{R}^2)/\mathbb{R}$. ◻ Now we state several estimates for solutions of Equation [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"}, proved by Duerinckx in [@DuerinckxFischer]: **Lemma 11**. *\[From [@DuerinckxFischer Lemma 2.6]\] Let $p>2$, $\omega$ be such that $<\! \cdot \!>\omega \in L^p(\mathbb{R}^2)$. If $\psi \in W^{2,p}_{-1}(\mathbb{R}^2)$ is the solution of [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} given by Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"}, then:* 1. *There exists $p_0 >2$ depending only on $b$ such that for all $2 < p \leqslant p_0$, $$\left\lVert\nabla \psi\right\rVert_{L^p} \leqslant C_p\left\lVert\omega\right\rVert_{L^\frac{2p}{p+2}}.$$* 2. *For all $0 < s < 1$, $$\begin{aligned} |\nabla \psi|_{\mathcal{C}^{0,s}} \leqslant C_s \left\lVert\omega\right\rVert_{L^\frac{2}{1-s}}.\end{aligned}$$* 3. *$\left\lVert\nabla \psi\right\rVert_{L^\infty} \leqslant C\left\lVert\omega\right\rVert_{L^1\cap L^\infty}$.* *Remark 12*. In [@DuerinckxFischer], this lemma was stated for any solution of [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} with decreasing gradient (which is the case for a solution given by Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"} since its gradient is in $W^{1,p}$) and for $\omega$ smooth with compact support but by density it can be extended to all $\omega$ such that $<\! \cdot \!>\omega \in L^p(\mathbb{R}^2)$ and such that the upper inequalities make sense. ## Asymptotic behavior of the velocity field {#subsection:22} The main result of this subsection is the following proposition giving the asymptotic behavior of a velocity field satisfying [\[elliptic_problem_velocity\]](#elliptic_problem_velocity){reference-type="eqref" reference="elliptic_problem_velocity"}. **Proposition 13**. *Let $\omega \in L^\infty$ with compact support and $\displaystyle{u = -\frac{1}{b}\nabla^\bot \psi}$ where $\psi$ is the solution of [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} given by Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"}. There exists $C > 0$ depending only on $b$ and $\omega$ such that for all $x \in \mathbb{R}^2\backslash\{0\}$,* *$$\label{asymptotic_bound_velocity_field} \left|u(x) - \frac{1}{2\pi}\left(\int_{\mathbb{R}^2} \omega\right) \frac{x^\bot}{|x|^2}\right| \leqslant\frac{C}{|x|^2}.$$ Moreover there exists $\delta \in (0,1)$ and $C$ such that $$\label{bound_stream_function} |\psi(x)| \leqslant C(1+|x|^\delta).$$* To prove this proposition we will need to use the following result about the asymptotic behavior of a velocity field given by the usual Biot-Savart law: **Lemma 14**. *Let us assume that $\mu$ is a measurable function such that $\mu \in L^1((1+|x|^2)\,\mathrm dx)$ and $|\cdot|^2\mu \in L^p$ for some $p > 2$. Then there exists $C,R >0$ depending only on $\mu$ such that for all $x \in \mathbb{R}^2\backslash\{0\}$,* *$$\left|\int_{\mathbb{R}^2}\frac{x-y}{|x-y|^2}\,\mathrm d\mu(y)-\left(\int_{\mathbb{R}^2} \,\mathrm d\mu(y)\right)\frac{x}{|x|^2}\right| \leqslant\frac{C}{|x|^2}.$$* *In particular if $\displaystyle{\int_{\mathbb{R}^2} \,\mathrm d\mu = 0}$, then* *$$\int_{\mathbb{R}^2}\frac{x-y}{|x-y|^2}\,\mathrm d\mu(y) = \underset{|x|\rightarrow+\infty}{\mathcal{O}}(|x|^{-2}).$$* This lemma is a classical result in fluid dynamics (see for example [@MajdaBertozzi Proposition 3.3]) that we will prove for the sake of completeness. *Proof.* If $x \neq 0$, we have $$\int_{\mathbb{R}^2} \mu(y)\left(\frac{x-y}{|x-y|^2}-\frac{x}{|x|^2}\right)\,\mathrm dy = \frac{1}{|x|^2}\int_{\mathbb{R}^2} \mu(y)\frac{|x|^2(x-y) - x|x-y|^2}{|x-y|^2}\,\mathrm dy.$$ Now remark that $$\begin{aligned} |x|^2&(x-y) - x|x-y|^2 \\ &= |x|^2(x-y) - (x-y)(|x|^2 + |y|^2 -2x\cdot y) -y|x-y|^2 \\ &= (x-y)(|y|^2 -2(x-y)\cdot y -2|y|^2) -y|x-y|^2 \\ &= -y|x-y|^2 - 2[(x-y)\cdot y](x-y) -|y|^2(x-y).\end{aligned}$$ Thus $$\begin{aligned} \left|\int_{\mathbb{R}^2} \mu(y)\left(\frac{x-y}{|x-y|^2}-\frac{x}{|x|^2}\right)\,\mathrm dy\right| \leqslant& \frac{C}{|x|^2}\bigg(\int_{\mathbb{R}^2} |y||\mu(y)|\,\mathrm dy \\ &+ \int_{\mathbb{R}^2} \frac{|y|^2|\mu(y)|}{|x-y|}\,\mathrm dy\bigg).\end{aligned}$$ Now we have that for any $p > 2$, $$\begin{aligned} \int_{\mathbb{R}^2} \frac{|y|^2|\mu(y)|}{|x-y|}\,\mathrm dy &\leqslant\left\lVert|\cdot|^2\mu\right\rVert^\frac{p-2}{2p-2}_{L^1}\left\lVert|\cdot|^2\mu\right\rVert_{L^p}^\frac{p}{2p-2}\end{aligned}$$ (see for example [@Iftimie Lemma 1]) and therefore we get the proof of Lemma [Lemma 14](#theorem_asymptotic_velocity_field_euler){reference-type="ref" reference="theorem_asymptotic_velocity_field_euler"}. ◻ With this result we can now study the asymptotic behavior of a velocity field satisfying System [\[elliptic_problem_velocity\]](#elliptic_problem_velocity){reference-type="eqref" reference="elliptic_problem_velocity"}: *Proof of Proposition [Proposition 13](#theorem_asymptotic_velocity_field_lake){reference-type="ref" reference="theorem_asymptotic_velocity_field_lake"}.* We write $$\label{expression_mu_bu} \mu := \operatorname{div}(u) = \operatorname{div}\left(\frac{1}{b}bu\right) = \nabla\left(\frac{1}{b}\right)\cdot bu = -\frac{\nabla b\cdot u}{b}.$$ By Helmholtz decomposition we can write $$\label{helmoltz_u} u = -\nabla g\ast\mu -\nabla^\bot g\ast\omega.$$ Let $2 < p < +\infty$, then by Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}, $$\begin{aligned} \int_{\mathbb{R}^2} (1+|y|^2) |\mu(y)|\,\mathrm dy &\leqslant C_b\int_{\mathbb{R}^2} \frac{1+|y|^2}{(1+|y|)^{4+\gamma}}|u(y)|\,\mathrm dy \\ &\leqslant C_b \left\lVert(1+|\cdot|)^{-(2+\gamma)}\right\rVert_{L^{p'}}\left\lVert u\right\rVert_{L^p} < +\infty\end{aligned}$$ and $$\begin{aligned} \int_{\mathbb{R}^2} |y|^{2p}|\mu(y)|^p\,\mathrm dy &\leqslant C_b\int_{\mathbb{R}^2} |y|^{2p}(1+|y|)^{-p(4+\gamma)}|u(y)|^p \,\mathrm dy \\ &\leqslant C_b \int_{\mathbb{R}^2} |u(y)|^p \,\mathrm dy < + \infty.\end{aligned}$$ If we apply Lemma [Lemma 14](#theorem_asymptotic_velocity_field_euler){reference-type="ref" reference="theorem_asymptotic_velocity_field_euler"} on each term of [\[helmoltz_u\]](#helmoltz_u){reference-type="eqref" reference="helmoltz_u"} we only need to show that $\displaystyle{\int \mu = 0}$ to obtain [\[asymptotic_bound_velocity_field\]](#asymptotic_bound_velocity_field){reference-type="eqref" reference="asymptotic_bound_velocity_field"}. We define $$b_\infty := \underset{|x| \rightarrow +\infty}{\lim} b(x).$$ Remark that the existence of this limit is guaranteed by Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}. Let us prove by induction that for any integer $n$, $$\label{hyp_recurrence_asymptotique} \sum_{k=0}^n \frac{\ln^k(b_\infty)}{k!}\int_{\mathbb{R}^2} \mu = \frac{1}{n!}\int_{\mathbb{R}^2} \ln^n(b)\mu.$$ If $n = 0$ then this equality reduces to $\displaystyle{\int \mu = \int \mu}$. Now let us assume that it holds for some $n \geqslant 0$. Using Equation [\[expression_mu_bu\]](#expression_mu_bu){reference-type="eqref" reference="expression_mu_bu"}, we get $$\ln^n(b)\mu = -\frac{1}{n+1}\nabla \ln^{n+1}(b) \cdot u.$$ Inserting Equation [\[helmoltz_u\]](#helmoltz_u){reference-type="eqref" reference="helmoltz_u"}, we get $$\ln^n(b)\mu = \frac{1}{n+1}\nabla \ln^{n+1}(b)\cdot(\nabla g\ast\mu + \nabla^\bot g\ast\omega).$$ Integrating over a ball of center $0$ and radius $R$ and integrating by parts we get $$\label{ipp_recurrence} \begin{aligned} \int_{B(0,R)} \ln^n(b)\mu =& \frac{1}{n+1}\bigg(\int_{\partial B(0,R)} \ln^{n+1}(b)(\nabla g\ast\mu + \nabla^\bot g\ast\omega)\cdot \,\mathrm d\vec{S} \\ &- \int_{B(0,R)}\ln^{n+1}(b)\operatorname{div}(\nabla g\ast\mu + \nabla^\bot g\ast\omega)\bigg) \\ =&\frac{1}{n+1}\bigg(\int_{\partial B(0,R)} \ln^{n+1}(b)(\nabla g\ast\mu + \nabla^\bot g\ast\omega)\cdot \,\mathrm d\vec{S} \\ &+ \int_{B(0,R)}\ln^{n+1}(b)\mu\bigg) \\ \end{aligned}$$ where $\,\mathrm d\vec{S}(x) = 2\pi x \,\mathrm d\sigma(x)$ and $\sigma$ is the uniform probability measure on $\partial B(0,R)$. Using Lemma [Lemma 14](#theorem_asymptotic_velocity_field_euler){reference-type="ref" reference="theorem_asymptotic_velocity_field_euler"}, we get that for $x \in \partial B(0,R)$, $$\begin{aligned} (\nabla g\ast\mu &+ \nabla^\bot g\ast\omega)(x)\cdot x \\ &= -\frac{1}{2\pi}\left(\left(\int_{\mathbb{R}^2} \mu\right)\frac{x}{|x|^2}+\left(\int_{\mathbb{R}^2} \omega\right)\frac{x^\bot}{|x|^2} + \mathcal{O}(R^{-2})\right)\cdot x \\ &= -\frac{1}{2\pi}\left(\int_{\mathbb{R}^2} \mu\right)+ \mathcal{O}(R^{-1}).\end{aligned}$$ Thus we get that $$\frac{1}{n+1}\int_{\partial B(0,R)} \ln^{n+1}(b)(\nabla g\ast\mu + \nabla^\bot g\ast\omega)\cdot \,\mathrm d\vec{S} \mathop{\longrightarrow}\limits_{R\rightarrow+\infty} -\frac{\ln^{n+1}(b_\infty)}{n+1}\int_{\mathbb{R}^2} \mu.$$ Combining the upper equality with Equations [\[hyp_recurrence_asymptotique\]](#hyp_recurrence_asymptotique){reference-type="eqref" reference="hyp_recurrence_asymptotique"} and [\[ipp_recurrence\]](#ipp_recurrence){reference-type="eqref" reference="ipp_recurrence"} we get that $$\sum_{k=0}^{n+1} \frac{\ln^k(b_\infty)}{k!}\int_{\mathbb{R}^2} \mu = \frac{1}{(n+1)!}\int_{\mathbb{R}^2} \ln^{n+1}(b)\mu$$ which ends the proof of Equality [\[hyp_recurrence_asymptotique\]](#hyp_recurrence_asymptotique){reference-type="eqref" reference="hyp_recurrence_asymptotique"}. Now if $n$ goes to infinity, this gives $$e^{\ln(b_\infty)}\int_{\mathbb{R}^2} \mu = 0$$ and thus $$\int_{\mathbb{R}^2} \mu = 0.$$ Now by Lemma [Lemma 11](#lemma_link_solutions_duerinckx){reference-type="ref" reference="lemma_link_solutions_duerinckx"} and Morrey's inequality (see for example [@Brezis Theorem 9.12]), for any $2 < p \leqslant p_0$, $$\begin{aligned} |\psi(x)| &\leqslant|\psi(x)- \psi(0)| + |\psi(0)| \\ &\leqslant C_p\left\lVert\nabla \psi\right\rVert_{L^p}|x|^{1-\frac{2}{p}} + |\psi(0)|.\end{aligned}$$ Taking $\displaystyle{\delta=1-\frac{2}{p}}$ we obtain [\[bound_stream_function\]](#bound_stream_function){reference-type="eqref" reference="bound_stream_function"}. ◻ ## Construction of the Green kernel {#subsection:23} The main result of this subsection is a Biot-Savart type law for the lake equations, given by Proposition [Proposition 16](#proposition_bs_law){reference-type="ref" reference="proposition_bs_law"}. Let us begin by giving the definition and some estimates on the function $S_b$ that appears in the definition of the kernel $g_b$ (see Equation [\[definition_g\_b\]](#definition_g_b){reference-type="eqref" reference="definition_g_b"}): **Lemma 15**. *For $y \in \mathbb{R}^2$, let $S_b(\cdot,y)$ be a solution of $$\label{definition_S_b} -\operatorname{div}\left(\frac{1}{b}\nabla S_b(\cdot,y)\right) = -g(\cdot-y)\sqrt{b(y)}\Delta\left(\frac{1}{\sqrt{b}}\right)$$ given by Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"} applied to $\displaystyle{\omega = -g(\cdot-y)\sqrt{b(y)}\Delta\left(\frac{1}{\sqrt{b}}\right)}$ and $\psi = S_b(\cdot,y)$. Then:* 1. *For any $y \in \mathbb{R}^2$ and $2 < p \leqslant+\infty$, $\nabla_x S_b(\cdot,y) \in L^p$ and $$\left\lVert\nabla_x S_b(\cdot,y)\right\rVert_{L^p} \leqslant C_{b,p}(1+|y|).$$* 2. *There exists $s_0 \in (0,1)$ such that for all $0 < s < s_0$, $$\begin{aligned} |\nabla_x S_b(x,\cdot)|_{\mathcal{C}^{0,s}(B(y,1))} &\leqslant C_{b,s}(1+|y|) \\ |\nabla_x S_b(\cdot,y)|_{\mathcal{C}^{0,s}(\mathbb{R}^2)} &\leqslant C_{b,s}(1+|y|).\end{aligned}$$* *Proof.* For any $p$ such that $1 \leqslant p < + \infty$, we have $$\left\lVert\sqrt{b(y)}<\! \cdot \!>\Delta\left(\frac{1}{\sqrt{b}}\right)g(\cdot-y)\right\rVert_{L^p} \leqslant\left\lVert b\right\rVert_{L^\infty}^\frac{1}{2}\left\lVert g(\cdot-y)<\! \cdot \!>\Delta\left(\frac{1}{\sqrt{b}}\right)\right\rVert_{L^p}$$ and $$\begin{aligned} &\left\lVert<\! \cdot \!> g(\cdot-y)\Delta\left(\frac{1}{\sqrt{b}}\right)\right\rVert^p_{L^p} \\ \leqslant& \int_{B(y,1)} <\! x \!>^p|g(x-y)|^p\left|\Delta\left(\frac{1}{\sqrt{b}}\right)(x)\right|^p \,\mathrm dx \\ &+ \int_{B(y,1)^c} <\! x \!>^p|g(x-y)|^p\left|\Delta\left(\frac{1}{\sqrt{b}}\right)(x)\right|^p \,\mathrm dx \\ \leqslant& C\left\lVert g\right\rVert^p_{L^p(B(0,1))}\left\lVert<\! \cdot \!> \Delta\left(\frac{1}{\sqrt{b}}\right)\right\rVert^p_{L^\infty} \\ &+ \int_{B(y,1)^c} (1+|x|^2)^\frac{p}{2}(|x|+|y|)^p\left|\Delta\left(\frac{1}{\sqrt{b}}\right)(x)\right|^p \,\mathrm dx.\end{aligned}$$ By Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}, we have that $$\begin{aligned} \int_{B(y,1)^c} (1+|x|^2)^\frac{p}{2}(|x|+|y|)^p&\left|\Delta\left(\frac{1}{\sqrt{b}}\right)(x)\right|^p \,\mathrm dx \\ &\leqslant\int_{\mathbb{R}^2} \frac{(1+|x|^2)^\frac{p}{2}(|x|+|y|)^p}{(1+|x|)^{(4+\gamma)p}}\,\mathrm dx \\ &\leqslant C_b(1+|y|)^p.\end{aligned}$$ Therefore we can apply Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"} to show that there exists a solution $S_b(\cdot,y)$ of [\[definition_S\_b\]](#definition_S_b){reference-type="eqref" reference="definition_S_b"} in $W^{2,p}_{-1}(\mathbb{R}^2)$, unique up to a constant. Since $<\! x \!> \geqslant 1$ we also have that $$\begin{aligned} \left\lVert\sqrt{b(y)}g(\cdot-y)\Delta\left(\frac{1}{\sqrt{b}}\right)\right\rVert_{L^p} \leqslant C_{b,p}(1+|y|).\end{aligned}$$ By Lemma [Lemma 11](#lemma_link_solutions_duerinckx){reference-type="ref" reference="lemma_link_solutions_duerinckx"}, there exists $p_0$ such that for any $2 < p \leqslant p_0$ and $0 < s < 1$: $$\begin{aligned} \left\lVert\nabla_x S_b(\cdot,y)\right\rVert_{L^p} &\leqslant\left\lVert\sqrt{b(y)}\Delta\left(\frac{1}{\sqrt{b}}\right)g(\cdot-y)\right\rVert_{L^\frac{2p}{p+2}} \\ &\leqslant C_{b,p}(1+|y|)\end{aligned}$$ and $$\begin{aligned} |\nabla_x S_b(\cdot,y)|_{\mathcal{C}^{0,s}} &\leqslant C_s \left\lVert\sqrt{b(y)}\Delta\left(\frac{1}{\sqrt{b}}\right)g(\cdot-y)\right\rVert_{L^{\frac{2}{1-s}}} \\ &\leqslant C_{b,s}(1+|y|)\end{aligned}$$ that is the second inequality of Claim (2). Using that $$\left\lVert\cdot\right\rVert_{L^\infty} \leqslant C(\left\lVert\cdot\right\rVert_{L^p} + |\cdot|_{\mathcal{C}^{0,s}})$$ (see for example the proof of Morrey's embedding theorem in [@Brezis Theorem 9.12]), we get the bound we want on $\nabla_x S_b$: $$\begin{aligned} \left\lVert\nabla_x S_b(\cdot,y)\right\rVert_{L^\infty} \leqslant C_b(1+|y|).\end{aligned}$$ If we interpolate the inequalities on $\left\lVert\nabla_x S_b(\cdot,y)\right\rVert_{L^\infty}$ and $\left\lVert\nabla_x S_b(\cdot,y)\right\rVert_{L^p}$ for $2 < p \leqslant p_0$ we find that for any $p > 2$, $$\begin{aligned} \left\lVert\nabla_x S_b(\cdot,y)\right\rVert_{L^p} \leqslant C_{b,p}(1+|y|).\end{aligned}$$ For the first inequality of Claim (2), let us consider $z$ such that $|z|$ is small and remark that $S_b(x,y+z) - S_b(x,y)$ solves $$\begin{aligned} \operatorname{div}&\left(\frac{1}{b}(\nabla_x S_b(\cdot,y+z) -\nabla_x S_b(\cdot,y))\right) \\ &= \left(\sqrt{b(y+z)}g(y+z-\cdot) - \sqrt{b(y)}g(y-\cdot)\right)\Delta\left(\frac{1}{\sqrt{b}}\right).\end{aligned}$$ Let us find a bound for the second member in $L^p$: $$\begin{aligned} \bigg(&\sqrt{b(y+z)}g(y+z-x) - \sqrt{b(y)}g(y-x)\bigg)\Delta\left(\frac{1}{\sqrt{b}}\right)(x) \\ =& (\sqrt{b(y+z)} - \sqrt{b(y)})g(y-x) \Delta\left(\frac{1}{\sqrt{b}}\right)(x) \\ &+ \sqrt{b(y+z)}(g(y+z-x) - g(y-x))\Delta\left(\frac{1}{\sqrt{b}}\right)(x).\end{aligned}$$ For the first term, $$\begin{aligned} \left|(\sqrt{b(y+z)} - \sqrt{b(y)})g(y-x) \Delta\left(\frac{1}{\sqrt{b}}\right)(x)\right| &\leqslant C_b|z|\left|g(y-x)\Delta\left(\frac{1}{\sqrt{b}}\right)\right|\end{aligned}$$ and we can bound its $L^p$ norms by $C_b(1+|y|)|z|$ as in the proof of Claim $(1)$. For the second term, $$\begin{aligned} \int_{\mathbb{R}^2}& \left|\sqrt{b(y+z)}(g(y+z-x) - g(y-x))\Delta\left(\frac{1}{\sqrt{b}}\right)(x)\right|^p \,\mathrm dx \\ \leqslant& C_b \int_{\mathbb{R}^2} \left|(g(x+z) - g(x))\Delta\left(\frac{1}{\sqrt{b}}\right)(y-x)\right|^p \,\mathrm dx \\ \leqslant& C_b\int_{B(0,|z|^\alpha)}|g(x+z)-g(x)|^p\,\mathrm dx \\ &+ C_b\int_{B(0,|z|^\alpha)^c}|g(x+z)-g(x)|^p\left|\Delta\left(\frac{1}{\sqrt{b}}\right)(y-x)\right|^p\,\mathrm dx\end{aligned}$$ for any $0 < \alpha < 1$. Now, if $|z|$ is small enough, $$\begin{aligned} \int_{B(0,|z|^\alpha)}|g(x+z)-g(x)|^p\,\mathrm dx &\leqslant C\int_{B(0,|z|^\alpha)}g(x+z)^p+g(x)^p\,\mathrm dx. \end{aligned}$$ Now we use a classical rearrangement procedure to bound $$\begin{aligned} \int_{B(0,|z|^\alpha)}g(x+z)^p &- \int_{B(0,|z|^\alpha)}g(x)^p\,\mathrm dx \\ =& \int_{B(z,|z|^\alpha)}g(x)^p - \int_{B(0,|z|^\alpha)}g(x)^p\,\mathrm dx \\ =& \int_{B(0,|z|^\alpha)}g(x)^p(\mathbf{1}_{B(z,|z|^\alpha)}(x) - 1)\,\mathrm dx \\ &+ \int_{B(0,|z|^\alpha)^c\cap B(z,|z|^\alpha)} g(x)^p \,\mathrm dx\end{aligned}$$ Now remark that for $x \in B(0,|z|^\alpha)$, $\displaystyle{g(x)^p \geqslant-\frac{1}{2\pi}\ln^p(|z|^\alpha)}$ and therefore $$\begin{aligned} \int_{B(0,|z|^\alpha)}&g(x)^p(\mathbf{1}_{B(z,|z|^\alpha)}(x) - 1)\,\mathrm dx \\ &\leqslant-\frac{1}{2\pi}\ln^p(|z|^\alpha)\int_{B(0,|z|^\alpha)}(\mathbf{1}_{B(z,|z|^\alpha)}(x) - 1)\,\mathrm dx \\ &\leqslant-\frac{1}{2\pi}\ln^p(|z|^\alpha)(|B(0,|z|^\alpha)\cap B(z,|z|^\alpha)| - |B(0,|z|^\alpha)|) \end{aligned}$$ and on $B(0,|z|^\alpha)^c$, $g(x) \leqslant-\frac{1}{2\pi}\ln(|z|^\alpha)$ so $$\begin{aligned} \int_{B(0,|z|^\alpha)^c\cap B(z,|z|^\alpha)} g(x)^p \,\mathrm dx \leqslant-\frac{1}{2\pi}\ln^p(|z|^\alpha)|B(0,|z|^\alpha)^c\cap B(z,|z|^\alpha)|.\end{aligned}$$ We get $$\begin{aligned} \int_{B(0,|z|^\alpha)}g(x+z)^p - \int_{B(0,|z|^\alpha)}g(x)^p\,\mathrm dx \leqslant 0\end{aligned}$$ and therefore $$\begin{aligned} \int_{B(0,|z|^\alpha)}|g(x+z)-g(x)|^p\,\mathrm dx &\leqslant 2\int_{B(0,|z|^\alpha)}g(x)^p\,\mathrm dx \\ &\leqslant C|z|^{2\alpha}\int_{B(0,1)}g(|z|^{\alpha}y)^p\,\mathrm dy \\ &\leqslant C|z|^{2\alpha}\int_{B(0,1)}(\alpha g(z) + g(y))^p \,\mathrm dy \\ &\leqslant C_b|z|^{2\alpha}g(z)^p.\end{aligned}$$ Now if $|z|$ is small enough, $$\begin{aligned} C_b&\int_{B(0,|z|^\alpha)^c}|g(x+z)-g(x)|^p\,\mathrm dx \left|\Delta\left(\frac{1}{\sqrt{b}}\right)(y-x)\right|^p \\ &\leqslant C_b\bigg(|z|\frac{C}{|z|^\alpha}\bigg)^p\int_{\mathbb{R}^2} \left|\Delta\left(\frac{1}{\sqrt{b}}\right)(y-x)\right|^p \,\mathrm dx \\ &\leqslant C_b|z|^{p(1-\alpha)}\end{aligned}$$ by Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}. Finally, using Lemma [Lemma 11](#lemma_link_solutions_duerinckx){reference-type="ref" reference="lemma_link_solutions_duerinckx"} as for the first claim, we get that for any $0 < \alpha < 1$ and some $p > 2$, $$\begin{aligned} |\nabla_x S_b(x,y+z) - \nabla_x S_b(x,y)| \leqslant C_b(1+|y|)|z| + C_b(|z|^{\frac{2\alpha}{p}}g(z) + |z|^{1-\alpha}).\end{aligned}$$ Dividing both sides by $|z|^s$ for $s$ small enough proves the first inequality of Claim $(2)$. ◻ With this lemma we are now able to construct the lake kernel. The construction is similar to the one established in [@DekeyserVanSchaftingen Proposition 3.1] for bounded domains. **Proposition 16**. *There exists a symmetric solution $S_b$ of Equation [\[definition_S\_b\]](#definition_S_b){reference-type="eqref" reference="definition_S_b"} such that $S_b(0,0) = 0$. We define $g_b$ as $$\label{definition_g_b} g_b(x,y) := \sqrt{b(x)b(y)}g(x-y) + S_b(x,y).$$ Let $\omega \in L^\infty$ with compact support. We define $$\label{existence_noyau_green} G_b[\omega](x) = \int_{\mathbb{R}^2} g_b(x,y)\,\mathrm d\omega(y).$$ Then $G_b[\omega]$ is a distributional solution of [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"}.* *Moreover for $2 < p <+\infty$, $G_b[\omega]$ is the unique solution (up to a constant) of [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} in $W^{2,p}_{-1}(\mathbb{R}^2)$ given by Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"}.* *Proof of Proposition [Proposition 16](#proposition_bs_law){reference-type="ref" reference="proposition_bs_law"}.* Let us first define $$g_b(x,y) := \sqrt{b(x)b(y)}g(x-y) + S_b(x,y)$$ where $S_b$ is a solution of Equation [\[definition_S\_b\]](#definition_S_b){reference-type="eqref" reference="definition_S_b"} given by Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"} (not necessarily symmetric). Then we have the following result: *Claim 17*. If $\varphi$ is smooth with compact support, then $$-\int_{\mathbb{R}^2} g_b(x,y)\operatorname{div}\left(\frac{1}{b}\nabla \varphi\right)(x)\,\mathrm dx = \varphi(y).$$ *Proof of the Claim.* We have $$\begin{aligned} -\int_{\mathbb{R}^2} &g_b(x,y)\operatorname{div}\left(\frac{1}{b}\nabla \varphi\right)(x)\,\mathrm dx \\ =& -\int_{\mathbb{R}^2} \sqrt{b(x)b(y)}g(x-y)\operatorname{div}\left(\frac{1}{b}\nabla \varphi\right)(x)\,\mathrm dx \\ &- \int_{\mathbb{R}^2} S_b(x,y)\operatorname{div}\left(\frac{1}{b}\nabla \varphi\right)(x)\,\mathrm dx \\ =:& T_1 + T_2.\end{aligned}$$ We have $$\begin{aligned} T_1 =& - \sqrt{b(y)}\int_{\mathbb{R}^2}\sqrt{b(x)}g(x-y)\operatorname{div}\left(\frac{1}{b}\nabla \varphi\right)(x)\,\mathrm dx \\ =& \sqrt{b(y)}\int_{\mathbb{R}^2} g(x-y)\frac{1}{2b(x)\sqrt{b(x)}}\nabla b(x)\cdot \nabla \varphi(x) \,\mathrm dx \\ &+ \sqrt{b(y)}\int_{\mathbb{R}^2}\frac{1}{\sqrt{b(x)}}\nabla g(x-y)\cdot \nabla \varphi(x) \,\mathrm dx \\ =:& L_1 + L_2.\end{aligned}$$ Integrating by parts in the first integral we get $$\begin{aligned} L_1 =&- \sqrt{b(y)}\int_{\mathbb{R}^2} \varphi(x) \frac{1}{2b(x)\sqrt{b(x)}}\nabla g(x-y)\cdot \nabla b(x) \,\mathrm dx \\ &- \sqrt{b(y)}\int_{\mathbb{R}^2} \varphi(x) g(x-y) \operatorname{div}\left(\frac{1}{2b\sqrt{b}}\nabla b\right)(x)\,\mathrm dx.\end{aligned}$$ For $L_2$, we use $$\nabla\left(\frac{1}{\sqrt{b}}\varphi\right) = \frac{1}{\sqrt{b}}\nabla \varphi- \varphi\frac{1}{2b\sqrt{b}}\nabla b$$ to get $$\begin{aligned} L_2 =& \sqrt{b(y)} \int_{\mathbb{R}^2} \varphi(x) \frac{1}{2b(x)\sqrt{b(x)}}\nabla b(x) \cdot \nabla g(x-y) \,\mathrm dx \\ &+ \sqrt{b(y)}\int_{\mathbb{R}^2} \nabla\left(\frac{1}{\sqrt{b(x)}}\varphi(x)\right)\cdot\nabla g(x-y)\,\mathrm dx \\ =& \sqrt{b(y)} \int_{\mathbb{R}^2} \varphi(x) \frac{1}{2b(x)\sqrt{b(x)}}\nabla b(x) \cdot \nabla g(x-y) \,\mathrm dx \\ &+ \varphi(y)\end{aligned}$$ since $-\Delta_x g(x-y) = \delta_y$ distributionally. Now let us compute $T_2$: $$\begin{aligned} T_2 &= -\int_{\mathbb{R}^2} S_b(x,y) \operatorname{div}\left(\frac{1}{b}\nabla \varphi\right)(x)\,\mathrm dx \\ &= -\int_{\mathbb{R}^2} \operatorname{div}\left(\frac{1}{b}\nabla_x S_b(\cdot,y)\right)(x)\varphi(x)\,\mathrm dx \\ &= -\sqrt{b(y)}\int_{\mathbb{R}^2} g(x-y)\Delta\left(\frac{1}{\sqrt{b}}\right)(x) \varphi(x) \,\mathrm dx\end{aligned}$$ where we used that $S_b$ is a solution of [\[definition_S\_b\]](#definition_S_b){reference-type="eqref" reference="definition_S_b"} in the last line. Now just remark that $$\Delta\left(\frac{1}{\sqrt{b}}\right) = -\operatorname{div}\left(\frac{1}{2b\sqrt{b}}\nabla b\right)$$ and thus adding $L_1$ and $L_2$ we get $$-\int_{\mathbb{R}^2} g_b(x,y)\operatorname{div}\left(\frac{1}{b}\nabla \varphi\right)(x)\,\mathrm dx = \varphi(y)$$ and we get the proof of Claim [Claim 17](#g_b_is_a_kernel_in_x){reference-type="ref" reference="g_b_is_a_kernel_in_x"}. ◻ Now let $\omega \in L^\infty(\mathbb{R}^2)$ with compact support. We have $$\begin{aligned} -\int_{\mathbb{R}^2}&\left(\int_{\mathbb{R}^2} g_b(x,y)\omega(y) \,\mathrm dy\right) \operatorname{div}\left(\frac{1}{b}\nabla\varphi\right)(x) \,\mathrm dx \\ &= -\int_{\mathbb{R}^2} \left(\int_{\mathbb{R}^2} g_b(x,y) \operatorname{div}\left(\frac{1}{b}\nabla\varphi\right)(x) \,\mathrm dx\right) \omega(y) \,\mathrm dy\\ &= \int_{\mathbb{R}^2} \varphi(y) \omega(y) \,\mathrm dy\end{aligned}$$ where we used Claim [Claim 17](#g_b_is_a_kernel_in_x){reference-type="ref" reference="g_b_is_a_kernel_in_x"} in the last equality. Therefore $G_b[\omega]$ is a distributional solution of [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"}. Now we prove that with this kernel we recover solutions in the sense of Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"}: *Claim 18*. Let $\omega \in L^\infty$ with compact support, then for all $p \in (2,+\infty)$, we have that $\nabla G_b[\omega] \in L^p$. Moreover if $\psi$ is the solution of [\[elliptic_problem_stream\]](#elliptic_problem_stream){reference-type="eqref" reference="elliptic_problem_stream"} given by Proposition [\[proposition_well_posedness_elliptic\]](#proposition_well_posedness_elliptic){reference-type="eqref" reference="proposition_well_posedness_elliptic"}, then $\psi = G_b[\omega]$ up to a constant. *Proof of the claim.* We have: $$\begin{aligned} \nabla G_b[\omega](x) =& \int_{\mathbb{R}^2} \frac{\nabla b(x)}{2\sqrt{b(x)}}\sqrt{b(y)}g(x-y)\omega(y)\,\mathrm dy\\ &+ \int_{\mathbb{R}^2} \sqrt{b(x)b(y)}\nabla g(x-y)\omega(y) \,\mathrm dy\\ &+ \int_{\mathbb{R}^2} \nabla_x S_b(x,y) \omega(y)\,\mathrm dy\\ =:& T_1 + T_2 + T_3.\end{aligned}$$ Now, $$\begin{aligned} |T_1| \leqslant& C_b|\nabla b(x)|\bigg(\int_{B(x,1)} |(\ln|x-y|)\omega(y)|\,\mathrm dy \\ &+ \int_{\operatorname{supp}(\omega)\backslash B(x,1)} (|x|+|y|)|\omega(y)|\,\mathrm dy \bigg) \\ \leqslant& C_{b}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}(1+|x|)^{-(3+\gamma)}\end{aligned}$$ by Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}. Hence $T_1 \in L^p$. For the second term, we have $$\begin{aligned} T_2 = \sqrt{b(x)}\nabla g\ast(\sqrt{b}\omega)\end{aligned}$$ and therefore $T_2 \in L^p$ by Hardy-Littlewood-Sobolev inequality (see for example [@BahouriCheminDanchin Theorem 1.7]). For the third term, $$\begin{aligned} |T_3| \leqslant\left(\int_{\mathbb{R}^2} |\omega|\right)\int_{\mathbb{R}^2} |\nabla_x S_b(x,y)|\frac{|\omega(y)|\,\mathrm dy}{\int |\omega|}\end{aligned}$$ and thus by Jensen inequality $$\begin{aligned} \left\lVert T_3\right\rVert^p_{L^p} \leqslant\left(\int_{\mathbb{R}^2} |\omega|\right)^{p-1} \iint_{\mathbb{R}^2\times\mathbb{R}^2} |\nabla_x S_b(x,y)|^p |\omega(y)| \,\mathrm dy\,\mathrm dx.\end{aligned}$$ We have that $$\left\lVert\nabla S_b(\cdot,y)\right\rVert_{L^p} \leqslant C_b(1+|y|)$$ by Claim (1) of Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"}. Therefore $$\left\lVert T_3\right\rVert^p_{L^p} \leqslant C_b \left(\int_{\mathbb{R}^2} |\omega|\right)^{p-1} \int_{\mathbb{R}^2}(1+|y|)^p|\omega(y)|\,\mathrm dy$$ and it follows that $\nabla G_b[\omega] \in L^p$. By Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"} we get that $G_b[\omega] = \psi$ up to a constant. ◻ We are only left to justify that there exists a symmetric solution of [\[definition_S\_b\]](#definition_S_b){reference-type="eqref" reference="definition_S_b"}. Consider $\omega_1,\omega_2$ two smooth functions with average zero, then by Claim [Claim 18](#link_solutions_elliptic_kernel){reference-type="ref" reference="link_solutions_elliptic_kernel"}, we have $$\begin{aligned} \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y) \omega_1(x) \omega_2(y) \,\mathrm dx \,\mathrm dy &= \int_{\mathbb{R}^2} (\psi_2(x) +C)\omega_1(x) \,\mathrm dx \\ &= - \int_{\mathbb{R}^2} \psi_2(x)\operatorname{div}\left(\frac{1}{b}\nabla \psi_1\right)(x)\,\mathrm dx\end{aligned}$$ where $\psi_i$ is the solution of $$\begin{aligned} -\operatorname{div}\left(\frac{1}{b}\nabla \psi_i\right) = \omega_i\end{aligned}$$ given by Proposition [Proposition 10](#proposition_well_posedness_elliptic){reference-type="ref" reference="proposition_well_posedness_elliptic"}. If $R > 0$, we have that $$\begin{aligned} - \int_{B(0,R)} \psi_2(x)\operatorname{div}\left(\frac{1}{b}\nabla \psi_1\right)(x)\,\mathrm dx =& - \int_{\partial B(0,R)}\frac{1}{b}\psi_2 \nabla \psi_1\cdot \,\mathrm d\vec{S} \\ &+ \int_{B(0,R)} \frac{1}{b}\nabla \psi_2 \cdot \nabla \psi_1.\end{aligned}$$ Using Proposition [Proposition 13](#theorem_asymptotic_velocity_field_lake){reference-type="ref" reference="theorem_asymptotic_velocity_field_lake"}, we obtain $$\left|\int_{\partial B(0,R)}\frac{1}{b}\psi_2 \nabla \psi_1\cdot \,\mathrm d\vec{S}\right| \leqslant 2\pi R\left\lVert b^{-1}\right\rVert_{L^\infty} C(1+R^\delta)\frac{C}{R^2} \mathop{\longrightarrow}\limits_{R\rightarrow+\infty} 0$$ and therefore $$\iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y) \omega_1(x) \omega_2(y) \,\mathrm dx \,\mathrm dy = \int_{\mathbb{R}^2} \frac{1}{b}\nabla \psi_2 \cdot \nabla \psi_1$$ which is a symmetric expression of $\psi_1$ and $\psi_2$. It follows that $$\begin{aligned} \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y) \omega_1(x) \omega_2(y) \,\mathrm dx \,\mathrm dy = \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(y,x) \omega_1(x) \omega_2(y) \,\mathrm dx \,\mathrm dy.\end{aligned}$$ Since $\sqrt{b(x)b(y)}g(x-y)$ is symmetric we get that $$\iint_{\mathbb{R}^2\times\mathbb{R}^2} S_b(x,y) \omega_1(x) \omega_2(y) \,\mathrm dx \,\mathrm dy = \iint_{\mathbb{R}^2\times\mathbb{R}^2} S_b(y,x) \omega_1(x) \omega_2(y) \,\mathrm dx \,\mathrm dy$$ for any $\omega_1,\omega_2$ smooth with compact suport and average zero. Let us define $$\begin{aligned} A(x,y) := S_b(x,y) - S_b(y,x).\end{aligned}$$ Now we fix $\chi, \omega_1, \omega_2$ smooth functions with compact support such that $\displaystyle{\int_{\mathbb{R}^2} \omega_2 = 0}$ and $\displaystyle{\int_{\mathbb{R}^2}\chi = 1}$. Remark that we no longer assume that $\displaystyle{\int_{\mathbb{R}^2} \omega_1 = 0}$. We define $$A_2(x) := \int_{\mathbb{R}^2} A(x,y) \omega_2(y) \,\mathrm dy.$$ We have $$\begin{aligned} \int_{\mathbb{R}^2} A_2 \omega_1 &= \int_{\mathbb{R}^2} A_2\left(\omega_1 - \left(\int_{\mathbb{R}^2} \omega_1\right)\chi\right) + \left(\int_{\mathbb{R}^2} \omega_1\right)\int_{\mathbb{R}^2} A_2 \chi \\ &= 0 + \left(\int_{\mathbb{R}^2} \omega_1\right) \int_{\mathbb{R}^2} A_2 \chi.\end{aligned}$$ Thus $A_2$ is constant so for every $x \in \mathbb{R}^2$, $$\begin{aligned} \int_{\mathbb{R}^2} \nabla_x A(x,y) \omega_2(y)\,\mathrm dy = 0\end{aligned}$$ for all $\omega_2$ with mean zero and therefore $\nabla_x A(x,y) = U(x)$. It follows that $A(x,y) = c(x) + d(y)$. Since $A(x,y) = - A(y,x)$, we have $d = -c$. Now let us set $\widetilde{S_b}(x,y) := S_b(x,y) + c(y)$. We have: $$\begin{aligned} \widetilde{S_b}(x,y) - \widetilde{S_b}(y,x) &= S_b(x,y) - S_b(y,x) + c(y) - c(x) \\ &= c(x) - c(y) + c(y) - c(x) \\ &= 0\end{aligned}$$ which proves that $\widetilde{S_b}$ a symmetric solution of [\[definition_S\_b\]](#definition_S_b){reference-type="eqref" reference="definition_S_b"}. Up to adding a constant we can also assume that $\widetilde{S_b}(0,0) = 0$. ◻ The symmetry of $S_b$ allows us to obtain more regularity estimates: **Lemma 19**. *Let $S_b$ be the symmetric solution of Equation [\[definition_S\_b\]](#definition_S_b){reference-type="eqref" reference="definition_S_b"} given by Proposition [Proposition 16](#proposition_bs_law){reference-type="ref" reference="proposition_bs_law"}, then* 1. *$S_b$ is smooth on $\mathbb{R}^2\times\mathbb{R}^2\backslash \{(x,x) \; ; \; x \in\mathbb{R}^2\}$.* 2. *$|S_b(x,y)| \leqslant C_b(1+|x|^2+|y|^2)$.* *Proof.* For $0 < r < R$, we define $C(y,r,R) := B(y,R)\backslash B(y,r)$. We have that $S_b(\cdot,y)$ is a solution of $$\left\{ \begin{aligned} & \operatorname{div}\left(\frac{1}{b}\nabla S_b(\cdot,y)\right) = g(\cdot-y)\sqrt{b(y)}\Delta \left(\frac{1}{\sqrt{b}}\right) \qquad \text{in} \; C(y,r,R) \\ & S_b(\cdot,y) = S_b(\cdot,y) \in \mathcal{C}^{0,s} \qquad \text{in} \; \partial C(y,r,R). \end{aligned}\right.$$ Thus by elliptic regularity (see for example [@GilbargTrudinger Theorem 6.13]) we obtain that $S_b(\cdot,y) \in \mathcal{C}^{2,s}(\mathring{C}(y,r,R))$ for all $y \in \mathbb{R}^2$ and $0 < r < R$. By symmetry we get that $S_b$ is $\mathcal{C}^{2,s}$ on $$\mathbb{R}^2\times\mathbb{R}^2\backslash \{(x,x) \; ; \; x \in\mathbb{R}^2\}.$$ We can iterate the argument by writing the elliptic system satisfied by the derivatives of $S_b$ to show that $S_b$ is smooth on $$\mathbb{R}^2\times\mathbb{R}^2\backslash \{(x,x) \; ; \; x \in\mathbb{R}^2\}.$$ The second claim is just a consequence of Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"}, since $$\begin{aligned} |S_b(x,y)| &\leqslant|S_b(0,0) - S_b(x,0)| + |S_b(x,0) - S_b(x,y)| \\ &\leqslant|S_b(0,0) - S_b(x,0)| + |S_b(0,x) - S_b(y,x)| \\ &\leqslant\left\lVert\nabla_x S_b(\cdot,0)\right\rVert_{L^\infty}|x| + \left\lVert\nabla_x S_b(\cdot,x)\right\rVert_{L^\infty}|y| \\ &\leqslant C_b|x| + C_b(1+|x|)|y| \\ &\leqslant C_b(1+|x|^2 + |y|^2).\end{aligned}$$ ◻ We finish this subsection by giving a straightforward consequence of Proposition [Proposition 13](#theorem_asymptotic_velocity_field_lake){reference-type="ref" reference="theorem_asymptotic_velocity_field_lake"} and [@DuerinckxFischer Lemma 2.7] which will be useful to deal with the regularisation of the dirac mass we will introduce in Subsection [2.4](#subsection:24){reference-type="ref" reference="subsection:24"} and use in Sections [5](#section:5){reference-type="ref" reference="section:5"} and [6](#section:6){reference-type="ref" reference="section:6"}. **Lemma 20**. *$\mu \mapsto \nabla G_b[\mu]$ extends into a bounded operator from $\dot H^ {-1}$ to $L^2$.* *Proof.* Let $\mu$ be a smooth function with compact support and average zero. By Proposition [Proposition 13](#theorem_asymptotic_velocity_field_lake){reference-type="ref" reference="theorem_asymptotic_velocity_field_lake"}, $\nabla G_b[\mu] \in L^2$ and therefore it follows by [@DuerinckxFischer Lemma 2.7] that $$\begin{aligned} \left\lVert\nabla G_b[\mu]\right\rVert_{L^2} \leqslant C_b \left\lVert\mu\right\rVert_{\dot H^{-1}}\end{aligned}$$ and the lemma follows from the density of smooth functions with compact support and average zero in $\dot H^{-1}$. ◻ ## Regularisations of the Coulomb kernel and the dirac mass {#subsection:24} To study our modulated energy we will need to have suitable regularisations of $g$ and of the dirac mass $\delta_y$. For that purpose, let us first define $g^{(\eta)}$ for any $0 < \eta < 1$ as $$\label{definition_g_eta} g^{(\eta)}(x) := \left\{ \begin{aligned} &-\frac{1}{2\pi}\ln(\eta) \qquad &\text{if} \; |x| \leqslant\eta \\ &g(x) \qquad &\text{if} \; |x| \geqslant\eta \end{aligned}\right.$$ and we define $\delta_{y}^{(\eta)}$ as the uniform probability measure on the circle $\partial B(y,\eta)$. We also define $$\label{definition_delta_tilde_q} \widetilde{\delta}_y^{(\eta)} := m_b(y,\eta)\frac{\,\mathrm d\delta_{y}^{(\eta)}}{\sqrt{b}}$$ where $$\label{definition_m_b} m_b(y,\eta) := \left(\int\frac{\,\mathrm d\delta_y^{(\eta)}}{\sqrt{b}}\right)^{-1}.$$ In the following proposition we state several properties related to these regularisations. **Proposition 21**. *For any $0 < \eta < 1$ and $y \in \mathbb{R}^2$, we have $$\label{egalite_convolution_g_eta} \int g(x-z) \,\mathrm d\delta_y^{(\eta)}(z) = g^{(\eta)}(x-y)$$ and $$\label{estimee_m_b} |m_b(y,\eta)-\sqrt{b(y)}| \leqslant C_b\eta.$$* *Proof.* By a change of variable we may assume that $y = 0$. The function $$f(x) := \int_{\partial B(0,\eta)} g(x-z) \,\mathrm d\delta_0^{(\eta)}(z)$$ is locally bounded and satisfies $\Delta f = -\delta_0^{(\eta)} = \Delta g^{(\eta)}$. Now if $|x| \geqslant\eta$, we have $$\begin{aligned} \int_{\partial B(0,\eta)} g(x-z)\,\mathrm d\delta_0^{(\eta)}(z) - g^{(\eta)}(x) &= \int_{\partial B(0,\eta)}(g(x-z)-g(x))\,\mathrm d\delta_0^{(\eta)}(z) \\ &= \int_{\partial B(0,\eta)}g\bigg(\frac{x}{|x|}-\frac{z}{|x|}\bigg)\,\mathrm d\delta_0^{(\eta)}(z) \\ &\mathop{\longrightarrow}\limits_{|x|\rightarrow\infty} \int_{\partial B(0,\eta)} -\frac {1}{2\pi}\ln(1) = 0\end{aligned}$$ by dominated convergence theorem. Thus $f - g^{(\eta)}$ is a harmonic bounded function so it is constant. Since $f(z) = g(\eta) = g^{(\eta)}(z)$ for any $z$ of norm $\eta$, we get that $f = g^{(\eta)}$. Let us now prove [\[estimee_m\_b\]](#estimee_m_b){reference-type="eqref" reference="estimee_m_b"}: $$m_b(y,\eta)-\sqrt{b(y)} = m_b(y,\eta)\sqrt{b(y)}\left(\frac{1}{\sqrt{b(y)}}-\int\frac{ \,\mathrm d\delta_y^{(\eta)}(z)}{\sqrt{b(z)}}\right)$$ and thus $$|m_b(y,\eta)-\sqrt{b(y)}|\leqslant C_b \eta$$ by Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}. ◻ # Point vortices {#section:3} To prove Theorem [Theorem 8](#MFL_theorem){reference-type="ref" reference="MFL_theorem"} we will need to control the evolution of the interaction energy and of the moment of inertia. We recall that the moment of inertia is not conserved for the lake equations, nor for the point vortex system. Due to the self-interactions, the interaction energy $E_N$ is also not conserved. The following proposition gives bounds on the interaction energy and on the moment of inertia and the global well-posedness of the lake point-vortex system [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"}. **Proposition 22**. *Let $T > 0$ and $(q_1^0,...,q_N^0)$ be such that $q_i^0 \neq q_j^0$ if $i \neq j$. There exists a unique smooth solution of [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"} on $[0,T]$. Moreover, we have the following estimates: $$\label{bound_E_N} |E_N(t)| \leqslant e^{C_b(1+\alpha_N)t}(|E_N(0)|+I_N(0)+1)$$ $$\label{bound_I_N} I_N(t) \leqslant e^{C_b(1+\alpha_N)t}(|E_N(0)|+I_N(0)+1).$$ We also have similar estimates for the rescaled moment of inertia and for the interaction energy: $$\label{bound_E_N_resc} |\overline{E_N}(t)| \leqslant e^{C_b(1+\alpha_N^{-1})t}(|\overline{E_N}(0)| + \overline{I_N}(0)+1)$$ $$\label{bound_I_N_resc} \overline{I_N}(t) \leqslant e^{C_b(1+\alpha_N^{-1})t}(|\overline{E_N}(0)| + \overline{I_N}(0)+1).$$* *Proof.* Since $b$ is regular (see Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}) and $S_b,g,\nabla g$ are regular outside of the diagonal (see Claim (1) of Lemma [Lemma 19](#estimees_S_b_symmetric){reference-type="ref" reference="estimees_S_b_symmetric"}), System [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"} is well-posed up to the first collision time by Cauchy-Lipschitz theorem. We will first prove the bounds on $E_N$ and $I_N$ and then deduce that there is no collision between the points (this is the classical strategy to prove that the Euler point vortex system is well-posed when all the vorticities are positive, as explained for example in [@MarchioroPulvirenti Chapter 4.2]). Let us assume that there is no collision up to some time $T^\ast \leqslant T$. We first compute the time derivative of $E_N$. Since $g_b$ is symmetric, we have $$\begin{aligned} \dot E_N =& \frac{1}{N^2}\sum_{i=1}^N\bigg(\underset{j\neq i}{\sum_{j=1}^N} \dot q_i \cdot \nabla_x g_b(q_i,q_j) + \dot q_j\nabla_y g_b(q_i,q_j)\bigg) \\ =& \frac{2}{N^2}\sum_{i=1}^N\bigg(-\alpha_N\frac{\nabla^\bot b(q_i)}{b(q_i)} -\frac{1}{Nb(q_i)}\underset{k\neq i}{\sum_{k=1}^N}\nabla_x^\bot g_b(q_i,q_k)\bigg) \cdot\underset{j\neq i}{\sum_{j=1}^N}\nabla_x g_b(q_i,q_j) \\ =& -\frac{2\alpha_N}{N^2}\sum_{i=1}^N\frac{\nabla^\bot b(q_i)}{b(q_i)}\cdot\bigg(\frac{\sqrt{b(q_i)}}{2\sqrt{b(q_j)}}g(q_i - q_j)\nabla b(q_j) \\ &+\sqrt{b(q_i)b(q_j)}\nabla g(q_i - q_j) + \nabla_x S_b(q_i,q_j)\bigg)\end{aligned}$$ and thus we get that $$\label{derivative_E_N} \dot E_N = -\frac{2\alpha_N}{N^2}\sum_{i=1}^N\frac{\nabla^\bot b(q_i)}{b(q_i)} \cdot \bigg(\underset{j\neq i}{\sum_{j=1}^N}\sqrt{b(q_i)b(q_j)}\nabla g(q_i - q_j) + \nabla_x S_b(q_i,q_j)\bigg).$$ Now let us bound the right-handside of the upper equality. Using Claim $(1)$ of Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"} and Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}, we have $$\begin{aligned} \left|\frac{\nabla^\bot b(q_i)}{b(q_i)}\cdot \nabla_x S_b(q_i,q_j)\right| &\leqslant C_b(1+|q_j|)\end{aligned}$$ and thus $$\label{bound_E_N_1} \left|\frac{2\alpha_N}{N^2}\sum_{i=1}^N\frac{\nabla^\bot b(q_i)}{b(q_i)} \cdot \underset{j\neq i}{\sum_{j=1}^N} \nabla_x S_b(q_i,q_j)\right| \leqslant C_b\alpha_N(1+I_N).$$ Now remark that $$\begin{aligned} \frac{2\alpha_N}{N^2}&\sum_{i=1}^N\frac{\nabla^\bot b(q_i)}{b(q_i)} \cdot \bigg(\underset{j\neq i}{\sum_{j=1}^N}\sqrt{b(q_i)b(q_j)}\nabla g(q_i - q_j)\bigg) \\ &= \frac{\alpha_N}{N^2}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N}\left(\sqrt{\frac{b(q_j)}{b(q_i)}}\nabla^\bot b(q_i) - \sqrt{\frac{b(q_i)}{b(q_j)}}\nabla^\bot b(q_j)\right)\cdot \nabla g(q_i - q_j).\end{aligned}$$ Moreover, $$\begin{aligned} \sqrt{\frac{b(q_j)}{b(q_i)}}\nabla^\bot b(q_i) - \sqrt{\frac{b(q_i)}{b(q_j)}}\nabla^\bot b(q_j) =& \sqrt{\frac{b(q_j)}{b(q_i)}}(\nabla^\bot b(q_i) - \nabla^\bot b(q_j)) \\ &+ \frac{b(q_j) - b(q_i)}{\sqrt{b(q_i)b(q_j)}}\nabla^\bot b(q_j)\end{aligned}$$ and thus using the Lipschitz regularity of $b$ and $\nabla b$ (see Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}) and $|\nabla g(q_i-q_j)| = C|q_i-q_j|^{-1}$ we get that $$\label{bound_E_N_2} \left|\frac{2\alpha_N}{N^2}\sum_{i=1}^N\frac{\nabla^\bot b(q_i)}{b(q_i)} \cdot \bigg[\underset{j\neq i}{\sum_{j=1}^N}\sqrt{b(q_i)b(q_j)}\nabla g(q_i - q_j)\bigg]\right| \leqslant C_b\alpha_N.$$ Combining inequalities [\[bound_E\_N_1\]](#bound_E_N_1){reference-type="eqref" reference="bound_E_N_1"} and [\[bound_E\_N_2\]](#bound_E_N_2){reference-type="eqref" reference="bound_E_N_2"} we get that $$\label{bound_E_N_dot} |\dot E_N| \leqslant C_b(1+I_N)\alpha_N.$$ Now we compute the time derivative of $I_N$: $$\begin{aligned} \dot I_N =& \frac{2}{N}\sum_{i=1}^N q_i \cdot \dot q_i \\ =& -\frac{2\alpha_N}{N}\sum_{i=1}^N q_i \cdot \frac{\nabla^\bot b(q_i)}{b(q_i)} \\ &-\frac{2}{N}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N}\frac{\sqrt{b(q_j)}}{2b(q_i)\sqrt{b(q_i)}}g(q_i-q_j)q_i \cdot\nabla^\bot b(q_i) \\ &-\frac{2}{N}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N}\frac{\sqrt{b(q_i)b(q_j)}}{b(q_i)}q_i \cdot\nabla^\bot g(q_i - q_j) \\ &-\frac{2}{N}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N} q_i \cdot\nabla_x^\bot S_b(q_i,q_j) \\ =:& 2(T_1 + T_2 + T_3 + T_4).\end{aligned}$$ Using Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"} we have $$\label{bound_IN1} |T_1| \leqslant C_b\alpha_N.$$ For the second term, using Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"} we have $$\begin{aligned} |T_2| &\leqslant\frac{C_b}{N^2}\sum_{i=1}^N \bigg(\underset{j\neq i}{\sum_{j=1}^N} |g(q_i - q_j)|\bigg) \\ &\leqslant\frac{C_b}{N^2}\sum_{i=1}^N\bigg(\underset{j\neq i}{\sum_{j=1}^N} g(q_i-q_j)\mathbf{1}_{|q_i-q_j| \leqslant 1}+|q_i|^2 + |q_j|^2\bigg)\\ &\leqslant C_b I_N + \frac{C_b}{N^2}\underset{|q_i - q_j| \leqslant 1}{\sum_{1 \leqslant i \neq j \leqslant N}}g(q_i-q_j).\end{aligned}$$ Now by Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}, we have that $$\begin{aligned} \frac{1}{N^2}\underset{|q_i - q_j| \leqslant 1}{\sum_{1 \leqslant i \neq j \leqslant N}}g(q_i-q_j) \leqslant& \frac{C_b}{N^2}\sum_{1 \leqslant i \neq j \leqslant N}\bigg(\sqrt{b(q_i)b(q_j)}g(q_i - q_j) \\ &+ S_b(q_i,q_j)\bigg) + \frac{C_b}{N^2}\underset{|q_i - q_j| \geqslant 1}{\sum_{1 \leqslant i \neq j \leqslant N}}|g(q_i-q_j)| \\ &+ \frac{C_b}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} |S_b(q_i,q_j)| \\ \leqslant& C_b\bigg(E_N + \frac{1}{N^2}\underset{|q_i - q_j| \geqslant 1}{\sum_{1 \leqslant i \neq j \leqslant N}}|g(q_i-q_j)| \\ &+ \frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} |S_b(q_i,q_j)|\bigg).\end{aligned}$$ Moreover, $$\begin{aligned} \frac{C_b}{N^2}\underset{|q_i - q_j| \geqslant 1}{\sum_{1 \leqslant i \neq j \leqslant N}}|g(q_i-q_j)| &\leqslant\frac{C_b}{N^2}\underset{|q_i - q_j| \geqslant 1}{\sum_{1 \leqslant i \neq j \leqslant N}}|q_i|^2 + |q_j|^2 \\ &\leqslant C_b I_N\end{aligned}$$ and using Claim $(2)$ of Lemma [Lemma 19](#estimees_S_b_symmetric){reference-type="ref" reference="estimees_S_b_symmetric"}, $$\begin{aligned} \frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} |S_b(q_i,q_j)| &\leqslant\frac{C_b}{N^2}\sum_{1 \leqslant i \neq j \leqslant N}(1+|q_i|^2+|q_j|^2) \leqslant C_b(1+I_N).\end{aligned}$$ Therefore $$\label{bound_IN2} |T_2| \leqslant C_b(1+|E_N|+I_N).$$ For the third term we write $$\begin{aligned} T_3 =& -\frac{1}{N^2}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N}\frac{\sqrt{b(q_j)}-\sqrt{b(q_i)}}{\sqrt{b(q_i)}}\nabla^\bot g(q_i - q_j) \cdot q_i \\ &-\frac{1}{2N^2}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N}\nabla^\bot g(q_i - q_j) \cdot (q_i-q_j) \\ =& -\frac{1}{N^2}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N}\frac{\sqrt{b(q_j)}-\sqrt{b(q_i)}}{\sqrt{b(q_i)}}\nabla^\bot g(q_i - q_j) \cdot q_i -0\end{aligned}$$ and thus using the Lipschitz regularity of $b$ (see Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}) we get $$\label{bound_IN3} |T_3| \leqslant C_b(1+I_N).$$ For the fourth term, using Claim (1) of Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"} we get $$\label{bound_IN4} \begin{aligned} |T_4| &= \left|-\frac{1}{N^2}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N} \frac{1}{b(q_i)}q_i\cdot\nabla_x^\bot S_b(q_i,q_j)\right| \\ &\leqslant C_b\frac{1}{N^2}\sum_{i=1}^N\sum_{j=1}^N |q_i|(1+|q_j|) \\ &\leqslant C_b(1+I_N). \end{aligned}$$ Combining with inequalities [\[bound_IN1\]](#bound_IN1){reference-type="eqref" reference="bound_IN1"}, [\[bound_IN2\]](#bound_IN2){reference-type="eqref" reference="bound_IN2"}, [\[bound_IN3\]](#bound_IN3){reference-type="eqref" reference="bound_IN3"} and [\[bound_IN4\]](#bound_IN4){reference-type="eqref" reference="bound_IN4"} we get that $$\label{bound_I_N_dot} |\dot I_N| \leqslant C_b(1+ \alpha_N + |I_N| + |E_N|).$$ Let us write $U_N := (E_N,I_N)$. By equations [\[bound_E\_N_dot\]](#bound_E_N_dot){reference-type="eqref" reference="bound_E_N_dot"} and [\[bound_I\_N_dot\]](#bound_I_N_dot){reference-type="eqref" reference="bound_I_N_dot"} we have $$|\dot U_N| \leqslant C_b(1+\alpha_N)(1+|U_N|)$$ therefore by Grönwall's lemma we have $$|U_N(t)| \leqslant e^{C_b(1+\alpha_N)t}(|U_N(0)|+1)-1$$ from which [\[bound_E\_N\]](#bound_E_N){reference-type="eqref" reference="bound_E_N"} and [\[bound_I\_N\]](#bound_I_N){reference-type="eqref" reference="bound_I_N"} follows. Let us use these bounds to prove that there is no collision (and it will follow that System [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"} is globally well-posed). If $i \neq j$, then $$\begin{aligned} g(|q_i - q_j|) \leqslant& C_b\bigg(E_N + \frac{1}{N^2}\sum_{1 \leqslant k \neq l \leqslant N}{\sum}|S_b(q_k,q_l)| \\ &-\frac{1}{N^2}\underset{(k,l) \neq (i,j)}{\sum_{1 \leqslant k \neq l \leqslant N}}g(q_k - q_l)\bigg) \\ \leqslant& C_b\bigg(E_N + \frac{1}{N^2}\sum_{1 \leqslant k \neq l \leqslant N}(1+|q_k|^2 + |q_l|^2)\bigg).\end{aligned}$$ where we used Claim $(2)$ of Lemma [Lemma 19](#estimees_S_b_symmetric){reference-type="ref" reference="estimees_S_b_symmetric"} and $\ln|x-y| \leqslant|x| + |y|$. Thus by inequalities [\[bound_E\_N\]](#bound_E_N){reference-type="eqref" reference="bound_E_N"} and [\[bound_I\_N\]](#bound_I_N){reference-type="eqref" reference="bound_I_N"} we get $$\begin{aligned} g(|q_i - q_j|) &\leqslant C_b(e^{C_b(1+\alpha_N)t}(|E_N(0)|+I_N(0)+1)+1)\end{aligned}$$ and therefore $$|q_i(t) - q_j(t)| \geqslant\exp\bigg(-2\pi C_b(e^{C_b(1+\alpha_N)t}(|E_N(0)|+I_N(0)+1)+1)\bigg) > 0.$$ It follows that there is no collision on $[0,T]$. The bounds on $\overline{E_N}$ and $\overline{I_N}$ follow directly from Inequalities [\[bound_E\_N\]](#bound_E_N){reference-type="eqref" reference="bound_E_N"} and [\[bound_I\_N\]](#bound_I_N){reference-type="eqref" reference="bound_I_N"} applied to $t = \alpha_N^{-1}\tau$. ◻ # Time derivatives of the modulated energies {#section:4} The time derivatives of $\mathcal{F}_{b,N}$ and of $\overline{\mathcal{F}}_{b,N}$, defined in [\[definition_F\_b_N\]](#definition_F_b_N){reference-type="eqref" reference="definition_F_b_N"} and [\[definition_F\_b_N\_rescaled\]](#definition_F_b_N_rescaled){reference-type="eqref" reference="definition_F_b_N_rescaled"}, are given by the two following propositions: **Proposition 23**. *Let $\omega$ be a weak solution of [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} in the sense of Definition [Definition 1](#definition_weak_solution){reference-type="ref" reference="definition_weak_solution"}, $(q_1,...,q_N)$ be solutions of [\[equation_pv\]](#equation_pv){reference-type="eqref" reference="equation_pv"}. We denote $$\omega_N = \frac{1}{N}\sum_{i=1}^N \delta_{q_i(t)}.$$* *Assume that $\omega$ satisfies Assumption [Assumption 6](#assumption_omega){reference-type="ref" reference="assumption_omega"}. Then $\mathcal{F}_{b,N}$ is Lipschitz and for almost every $t \in [0,T]$, $$\begin{aligned} &\frac{\,\mathrm d}{\,\mathrm dt}\mathcal{F}_{b,N}(t) = \\ &2\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} \left(u(t,x) - \alpha\frac{\nabla^\bot b(x)}{b(x)}\right)\cdot \nabla_x g_b(x,y)\,\mathrm d(\omega(t)-\omega_N(t))^{\otimes 2}(x,y) \\ &+ 2(\alpha_N - \alpha)\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} \frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y) \,\mathrm d\omega_N(t,x) \,\mathrm d(\omega(t) - \omega_N(t))(y).\end{aligned}$$* **Proposition 24**. *Let $(\overline{q_1},...,\overline{q_N})$ be solutions of [\[equation_pv_rescaled\]](#equation_pv_rescaled){reference-type="eqref" reference="equation_pv_rescaled"} and $\overline{\omega}$ be a solution of [\[transport_equation\]](#transport_equation){reference-type="eqref" reference="transport_equation"} in the sense of Definition [Definition 2](#definition_weak_solution_transport){reference-type="ref" reference="definition_weak_solution_transport"}.* *We denote $$\overline{\omega}_N = \frac{1}{N}\sum_{i=1}^N \delta_{\overline{q_i}(t)}.$$* *Assume that $\overline{\omega}$ satisfies Assumption [Assumption 6](#assumption_omega){reference-type="ref" reference="assumption_omega"}. Denote $v = \nabla G_b[\overline{\omega}]$. Then $\overline{\mathcal{F}}_{b,N}$ is Lipschitz and for almost every $t \in [0,T]$, we have $$\begin{gathered} \frac{\,\mathrm d}{\,\mathrm dt}\overline{\mathcal{F}}_{b,N}(t) = -2\iint_{(\mathbb{R}^2\times \mathbb{R}^2)\backslash \Delta} \frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y)\,\mathrm d(\overline{\omega}(t)-\overline{\omega}_N(t))^{\otimes 2}(x,y) \\ + \frac{2}{N^2\alpha_N} \sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N} \frac{v(t,\overline{q_i})}{b(\overline{q_i})}\cdot\nabla_x g_b(\overline{q_i},\overline{q_j}).\end{gathered}$$* *Proof of Proposition [Proposition 23](#time_derivative_F_N){reference-type="ref" reference="time_derivative_F_N"}.* We split $\mathcal{F}_{b,N}$ in three terms: $$\begin{aligned} \mathcal{F}_{b,N} =& \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y) \omega(t,x)\omega(t,y) \,\mathrm dx \,\mathrm dy \\ &-\frac{2}{N}\sum_{i=1}^N \int_{\mathbb{R}^2} g_b(x,q_i)\omega(t,x)\,\mathrm dx + E_N \\ =:& T_1 + T_2 + E_N.\end{aligned}$$ Let us compute the time derivative of $T_1$. For that purpose, we will need to regularize the kernel $g_b$. The regularisation we will use is given by the following Claim: *Claim 25*. There exists a familly of smooth functions $(g_b^{\eta})_{0 < \eta < 1}$ such that: - $|g_b^{\eta}(x,y)| \leqslant C_b(|g(x-y)|+1+|x|^2+|y|^2)$ - $|\nabla_x g_b^{\eta}(x,y)|,|\nabla_y g_b^{\eta}(x,y)| \leqslant C_b(|x-y|^{-1}+1+|x|+|y|)$. - For any $(x,y) \in (\mathbb{R}^2)^2$ such that $x \neq y$, $$\begin{aligned} g_b^{\eta}(x,y) &\mathop{\longrightarrow}\limits_{\eta\rightarrow 0} g_b(x,y) \\ \nabla_x g_b^{\eta}(x,y) &\mathop{\longrightarrow}\limits_{\eta\rightarrow 0} \nabla_x g_b(x,y) \\ \nabla_y g_b^{\eta}(x,y) &\mathop{\longrightarrow}\limits_{\eta\rightarrow 0} \nabla_y g_b(x,y).\end{aligned}$$ *Proof of the claim.* We define $$g_b^\eta(x,y) = \sqrt{b(x)b(y)}g^\eta(x-y) + S_b^\eta(x,y)$$ where $g^\eta$ is a smooth function satisfying: - $g^\eta(x) = g(x)$ for $|x| \geqslant\eta$, - $|g^\eta(x)| \leqslant|g(x)|$, - $|\nabla g^\eta(x)| \leqslant C|x|^{-1}$. that we can obtain by extending $\displaystyle{\ln\vert_{x\geqslant\eta}}$ in a smooth function on $\mathbb{R^+}$. We define $S_b^\eta := S_b\ast\chi_\eta$ where $\chi_\eta$ is a mollifier on $\mathbb{R}^4$. Since $S_b$ is locally Lipschitz (see Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"}), $S_b^\eta$ is smooth and we get from Claim (1) of Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"} and Claim (2) of Lemma [Lemma 19](#estimees_S_b_symmetric){reference-type="ref" reference="estimees_S_b_symmetric"} that - $|S_b^\eta(x,y)| \leqslant C_b(1+|x|^2+|y|^2)$, - $|\nabla_x S_b^\eta(x,y)|,|\nabla_y S_b^\eta(x,y)| \leqslant C_b(1+|x|+|y|)$. Since $S_b$ is locally Lipschitz, $S_b^\eta$ and $\nabla S_b^\eta$ converge locally uniformly to $S_b$ and $\nabla S_b$ (see for example [@Brezis Proposition 4.21]) and therefore we get the convergence of $g_b^\eta(x,y)$ and $\nabla g_b^\eta(x,y)$ to $g_b(x,y)$ and $\nabla g_b(x,y)$ for any $x \neq y$. ◻ With this regularisation we can compute the time derivative of $T_1$: *Claim 26*. $T_1 \in W^{1,\infty}([0,T])$ and for almost every $t \in [0,T]$, we have $$\frac{\,\mathrm dT_1}{\,\mathrm dt} = 2 \iint_{\mathbb{R}^2\times\mathbb{R}^2} \left(u(t,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right)\cdot\nabla_x g_b(x,y)\omega(t,x)\omega(t,y)\,\mathrm dx \,\mathrm dy.$$ *Proof of the claim.* For $0 \leqslant s,t \leqslant T$ and $0 < \eta < 1$ we have: $$T_1(t) - T_1(s) = \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y)(\omega(t,x)\omega(t,y) - \omega(s,x)\omega(s,y))\,\mathrm dx \,\mathrm dy.$$ Now for almost all $x$ and $y$ such that $x \neq y$, $$\begin{gathered} |g_b^{\eta}(x,y)||\omega(t,x)\omega(t,y) - \omega(s,x)\omega(s,y)| \\ \leqslant C_b(|g(x-y)|+1+|x|^2+|y|^2) |\omega(t,x)\omega(t,y) - \omega(s,x)\omega(s,y)|\end{gathered}$$ and $$\begin{gathered} \iint_{\mathbb{R}^2\times\mathbb{R}^2}(|g(x-y)|+1+|x|^2+|y|^2)\\ \times |\omega(t,x)\omega(t,y) - \omega(s,x)\omega(s,y)|\,\mathrm dx \,\mathrm dy < +\infty\end{gathered}$$ because $\omega \in L^\infty$ with compact support. Therefore by dominated convergence theorem we get that $$\label{limit_T_1_b_eta} T_1(t) - T_1(s) = \underset{\eta \rightarrow 0}{\lim}\iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b^\eta(x,y)(\omega(t,x)\omega(t,y) - \omega(s,x)\omega(s,y))\,\mathrm dx \,\mathrm dy.$$ Since $g_b^\eta$ is smooth and $\omega$ has compact support, we can use [\[formulation_faible\]](#formulation_faible){reference-type="eqref" reference="formulation_faible"} to get that $$\begin{gathered} \int_{\mathbb{R}^2} g_b^\eta(x,y)(\omega(t,y)-\omega(s,y))\,\mathrm dy = \\ \int_s^t \int_{\mathbb{R}^2} \nabla_y g_b^\eta(x,y)\cdot \left(u(\tau,y)-\alpha\frac{\nabla^\bot b(y)}{b(y)}\right)\omega(\tau,y) \,\mathrm dy \,\mathrm d\tau.\end{gathered}$$ Let us write $$\varphi(t,x) := \int_{\mathbb{R}^2} g_b^\eta(x,y)\omega(t,y)\,\mathrm dy.$$ Since $g_b^\eta$ is smooth we have that for any compact $K \subset \mathbb{R}^2$, $$\begin{gathered} (t,x) \mapsto \\ \int_{\mathbb{R}^2} \nabla_y g_b^\eta(x,y)\cdot\left(u(t,y)-\alpha\frac{\nabla^\bot b(y)}{b(y)}\right) \omega(t,y)\,\mathrm dy \in L^\infty([0,T],\mathcal{C}^\infty(K))\end{gathered}$$ and thus $\varphi\in W^{1,\infty}([0,T],\mathcal{C}^\infty(K))$ and for almost every $t \in [0,T]$, $$\partial_t \varphi(t,x) = \int_{\mathbb{R}^2} \nabla_y g_b^\eta(x,y)\left(u(\tau,y)-\alpha\frac{\nabla^\bot b(y)}{b(y)}\right) \omega(t,y)\,\mathrm dy.$$ Therefore we can use $\varphi$ as a test function in [\[formulation_faible\]](#formulation_faible){reference-type="eqref" reference="formulation_faible"} (remark that we defined [\[formulation_faible\]](#formulation_faible){reference-type="eqref" reference="formulation_faible"} for smooth functions only but by density we can extend it to functions which are only $W^{1,\infty}$ in time) and we get that $$\begin{aligned} &\iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b^\eta(x,y)(\omega(t,x)\omega(t,y) - \omega(s,x)\omega(s,y))\,\mathrm dx \,\mathrm dy \\ =& \int_s^t\iint_{\mathbb{R}^2\times\mathbb{R}^2} \nabla_y g_b^\eta(x,y)\cdot \left(u(\tau,y)-\alpha\frac{\nabla^\bot b(y)}{b(y)}\right) \omega(\tau,y)\omega(\tau,x)\,\mathrm dy \,\mathrm dx \,\mathrm d\tau \\ &+\int_s^t\iint_{\mathbb{R}^2\times\mathbb{R}^2} \nabla_x g_b^\eta(x,y)\cdot \left(u(\tau,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right)\omega(\tau,x)\omega(\tau,y)\,\mathrm dx \,\mathrm dy \,\mathrm d\tau.\end{aligned}$$ Now we have that for almost every $x$ and $y$ such that $x \neq y$ and almost every $\tau \in [0,T]$, $$\begin{gathered} |\nabla_x g_b^\eta(x,y)\cdot \left(u(\tau,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right) \omega(\tau,x)\omega(\tau,y)| \\ \leqslant C_b (|x-y|^{-1}+1+|x|^2+|y|^2)|\left|u(\tau,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right||\omega(\tau,y)||\omega(\tau,x)|\end{gathered}$$ and $$\begin{gathered} \int_s^t\iint_{\mathbb{R}^2\times\mathbb{R}^2} (|x-y|^{-1}+1+|x|^2+|y|^2)\\ \times\left|u(\tau,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right||\omega(\tau,y)||\omega(\tau,x)| \,\mathrm dx \,\mathrm dy < +\infty.\end{gathered}$$ Applying dominated convergence theorem we find that $$\begin{gathered} \int_s^t\iint_{\mathbb{R}^2\times\mathbb{R}^2} \nabla_x g_b^\eta(x,y)\cdot \left(u(\tau,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right)\omega(\tau,x)\omega(\tau,y)\,\mathrm dx \,\mathrm dy \,\mathrm d\tau \\ \mathop{\longrightarrow}\limits_{\eta\rightarrow 0} \int_s^t\iint_{\mathbb{R}^2\times\mathbb{R}^2} \nabla_x g_b(x,y)\cdot \left(u(\tau,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right)\omega(\tau,x)\omega(\tau,y)\,\mathrm dx \,\mathrm dy \,\mathrm d\tau. \end{gathered}$$ We can do the same for the first term to get that $$\begin{gathered} \int_s^t\iint_{\mathbb{R}^2\times\mathbb{R}^2} \nabla_y g_b^\eta(x,y)\cdot\left(u(\tau,y)-\alpha\frac{\nabla^\bot b(y)}{b(y)}\right) \omega(\tau,y)\omega(\tau,x)\,\mathrm dy \,\mathrm dx \,\mathrm d\tau \\ \mathop{\longrightarrow}\limits_{\eta\rightarrow 0} \int_s^t\iint_{\mathbb{R}^2\times\mathbb{R}^2} \nabla_y g_b(x,y)\cdot \left(u(\tau,y)-\alpha\frac{\nabla^\bot b(y)}{b(y)}\right) \omega(\tau,y)\omega(\tau,x)\,\mathrm dy \,\mathrm dx \,\mathrm d\tau.\end{gathered}$$ Using that $\nabla_y g_b(x,y) = \nabla_x g_b(y,x)$ and [\[limit_T\_1_b\_eta\]](#limit_T_1_b_eta){reference-type="eqref" reference="limit_T_1_b_eta"} we get that $T_1 \in W^{1,\infty}([0,T])$ and for almost every $t \in [0,T]$, we get Claim [Claim 26](#expression_derivative_T1){reference-type="ref" reference="expression_derivative_T1"}. ◻ We know by Equation [\[derivative_E\_N\]](#derivative_E_N){reference-type="ref" reference="derivative_E_N"} that $$\begin{aligned} \dot E_N &= -\frac{2\alpha_N}{N^2}\sum_{i=1}^N\frac{\nabla^\bot b(q_i)}{b(q_i)}\cdot \underset{j\neq i}{\sum_{j=1}^N} \nabla_x g_b(q_i,q_j)\end{aligned}$$ and therefore $$\label{expression_derivative_EN} \dot E_N = -2 \alpha_N\iint_{\mathbb{R}^2\times \mathbb{R}^2 \backslash \Delta} \frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y) \,\mathrm d\omega_N(x) \,\mathrm d\omega_N (y).$$ Now we compute the derivative of the second term: *Claim 27*. $T_2$ is Lipschitz and for almost every $t \in [0,T]$, we have $$\begin{aligned} \frac{\,\mathrm d}{\,\mathrm dt}T_2(t)& \\ =& -2\iint_{\mathbb{R}^2\times \mathbb{R}^2} \left(u(t,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right)\cdot \nabla_x g_b(x,y) \omega(t,x)\,\mathrm dx\,\mathrm d\omega_N(t,y) \\ &+ 2\alpha_N\iint_{\mathbb{R}^2\times \mathbb{R}^2} \frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y) \,\mathrm d\omega_N(x)\omega(t,y)\,\mathrm dy \\ &+ 2\iint_{(\mathbb{R}^2\times \mathbb{R}^2)\backslash \Delta} u(t,x)\cdot \nabla_x g_b(x,y)\,\mathrm d\omega_N(x) \,\mathrm d\omega_N(y).\end{aligned}$$ *Proof of the Claim.* If we use the regularisation $g_b^\eta$ we defined in Claim [Claim 25](#regularisation_g_b){reference-type="ref" reference="regularisation_g_b"}, Equation [\[formulation_faible\]](#formulation_faible){reference-type="eqref" reference="formulation_faible"} and if we let $\eta$ tends to zero as we did for the proof of Claim [Claim 26](#expression_derivative_T1){reference-type="ref" reference="expression_derivative_T1"}, we can show that $T_2$ is Lipschitz and that for almost every $t \in [0,T]$, we have $$\begin{aligned} \frac{\,\mathrm dT_2}{\,\mathrm dt} = T_{2,1} + T_{2,2}\end{aligned}$$ where $$\label{expression_derivative_T21} \begin{aligned} T_{2,1} &:= -\frac{2}{N}\sum_{i=1}^N \int_{\mathbb{R}^2} \left(u(t,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right)\cdot \nabla_x g_b(x,q_i)\omega(t,x)\,\mathrm dx \\ &= -2\iint_{\mathbb{R}^2\times \mathbb{R}^2} \left(u(t,x)-\alpha\frac{\nabla^\bot b(x)}{b(x)}\right)\cdot \nabla_x g_b(x,y) \omega(t,x)\,\mathrm dx\,\mathrm d\omega_N(t,y) \end{aligned}$$ and $$\begin{aligned} T_{2,2} :=&- \frac{2}{N} \sum_{i=1}^N\dot q_i \cdot \int_{\mathbb{R}^2} \nabla_y g_b(x,q_i) \omega(t,x)\,\mathrm dx \\ =& -\frac{2}{N} \sum_{i=1}^N\dot q_i \cdot \int_{\mathbb{R}^2} \nabla_x g_b(q_i,x)\omega(t,x)\,\mathrm dx \\ =& \bigg[\frac{2\alpha_N}{N} \sum_{i=1}^N \frac{\nabla^\bot b(q_i)}{b(q_i)}\cdot \int_{\mathbb{R}^2} \nabla_x g_b(q_i,x) \omega(t,x)\,\mathrm dx\bigg] \\ &+ \bigg[\frac{2}{N^2}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N}\frac{1}{b(q_i)}\nabla_x^\bot g_b(q_i,q_j)\cdot \int_{\mathbb{R}^2} \nabla_x g_b(q_i,x) \omega(t,x)\,\mathrm dx \bigg]\\ =:& T_{2,2,1} + T_{2,2,2}.\end{aligned}$$ Now we have $$\label{expression_derivative_T221} \begin{aligned} T_{2,2,1} &= \frac{2\alpha_N}{N} \sum_{i=1}^N \frac{\nabla^\bot b(q_i)}{b(q_i)}\cdot \int_{\mathbb{R}^2} \nabla_x g_b(q_i,x) \omega(t,x)\,\mathrm dx \\ &= 2\alpha_N\iint_{\mathbb{R}^2\times\mathbb{R}^2} \frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y) \,\mathrm d\omega_N(x)\omega(t,y)\,\mathrm dy \end{aligned}$$ and using $\displaystyle{\int_{\mathbb{R}^2} \nabla_x g_b(q_i,y)\omega(y)\,\mathrm dy = b(q_i)u^\bot(t,q_i)}$ (see Proposition [Proposition 16](#proposition_bs_law){reference-type="ref" reference="proposition_bs_law"}), we get $$\begin{aligned} T_{2,2,2} =&\frac{2}{N^2} \sum_{i=1}^N u(t,q_i)\cdot\underset{j\neq i}{\sum_{j=1}^N}\nabla_x^\bot g_b(q_i,q_j)\\ =& 2\iint_{(\mathbb{R}^2\times \mathbb{R}^2)\backslash \Delta} u(t,x)\cdot \nabla_x g_b(x,y)\,\mathrm d\omega_N(x) \,\mathrm d\omega_N(y).\end{aligned}$$ Combining the upper equality with [\[expression_derivative_T21\]](#expression_derivative_T21){reference-type="eqref" reference="expression_derivative_T21"} and[\[expression_derivative_T221\]](#expression_derivative_T221){reference-type="eqref" reference="expression_derivative_T221"} we get the proof of Claim [\[expression_derivative_T2\]](#expression_derivative_T2){reference-type="eqref" reference="expression_derivative_T2"}. ◻ Now remark that $$\begin{aligned} \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\omega_N(x) \,\mathrm d\omega(y) = \int_{\mathbb{R}^2} u \cdot bu^\bot \,\mathrm d\omega_N = 0.\end{aligned}$$ Thus combining Claim [Claim 26](#expression_derivative_T1){reference-type="ref" reference="expression_derivative_T1"}, Equation [\[expression_derivative_EN\]](#expression_derivative_EN){reference-type="eqref" reference="expression_derivative_EN"} and Claim [Claim 27](#expression_derivative_T2){reference-type="ref" reference="expression_derivative_T2"} we obtain Proposition [Proposition 23](#time_derivative_F_N){reference-type="ref" reference="time_derivative_F_N"}. ◻ We now compute the derivative of the rescaled modulated energy: *Proof of Proposition [Proposition 24](#time_derivative_F_N_resc){reference-type="ref" reference="time_derivative_F_N_resc"}.* We split $\overline{\mathcal{F}}_{b,N}$ in three terms: $$\begin{aligned} \overline{\mathcal{F}}_{b,N} =& \iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y) \overline{\omega}(t,x)\overline{\omega}(t,y) \,\mathrm dx \,\mathrm dy \\ &-\frac{2}{N}\sum_{i=1}^N \int_{\mathbb{R}^2} g_b(x,\overline{q_i})\overline{\omega}(t,x)\,\mathrm dx + \overline{E_N}\\ =:& T_1 + T_2 + \overline{E_N}.\end{aligned}$$ Let us compute the time derivative of the first term. Using the regularisation $g_b^\eta$ we defined in Claim [Claim 25](#regularisation_g_b){reference-type="ref" reference="regularisation_g_b"} and using [\[formulation_faible\]](#formulation_faible){reference-type="eqref" reference="formulation_faible"} and letting $\eta$ tends to zero as we did for the proof of Claim [Claim 26](#expression_derivative_T1){reference-type="ref" reference="expression_derivative_T1"}, one can show that $T_1$ is Lipschitz and that for almost every $t \in [0,T]$, we have $$\label{expression_derivative_resc_T1} \begin{aligned} \frac{\,\mathrm dT_1}{\,\mathrm dt} &= -2 \iint_{\mathbb{R}^2\times\mathbb{R}^2} \frac{\nabla^\bot b(x)}{b(x)}\cdot\nabla_x g_b(x,y)\overline{\omega}(t,x)\overline{\omega}(t,y)\,\mathrm dx \,\mathrm dy. \end{aligned}$$ For the derivative of $\overline{E_N}$ we rescale Equation [\[expression_derivative_EN\]](#expression_derivative_EN){reference-type="eqref" reference="expression_derivative_EN"} to get $$\label{expression_derivative_resc_EN} \frac{\,\mathrm d}{\,\mathrm dt}\overline{E_N} = -2\iint_{\mathbb{R}^2\times \mathbb{R}^2 \backslash \Delta}\frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y) \,\mathrm d\overline{\omega}_N(x) \,\mathrm d\overline{\omega}_N (y).$$ Now let us compute the derivative of the second term: *Claim 28*. $$\label{expression_derivative_resc_T2} \begin{aligned} \frac{\,\mathrm dT_2}{\,\mathrm dt} =& 2\iint_{\mathbb{R}^2\times\mathbb{R}^2} \frac{\nabla^\bot b(x)}{b(x)} \cdot \nabla_x g_b(x,y) \overline{\omega}(t,x)\,\mathrm dx\,\mathrm d\overline{\omega}_N(t,y) \\ &+ 2\iint_{\mathbb{R}^2\times\mathbb{R}^2} \frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y)\,\mathrm d\overline{\omega}_N(x)\,\mathrm d\overline{\omega}(y) \\ &+ \frac{2}{N^2\alpha_N} \sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N} \frac{v(t,\overline{q_i})}{b(\overline{q_i})}\cdot\nabla_x g_b(\overline{q_i},\overline{q_j}). \end{aligned}$$ *Proof of [\[expression_derivative_resc_T2\]](#expression_derivative_resc_T2){reference-type="eqref" reference="expression_derivative_resc_T2"}.* Using the regularisation $g_b^\eta$ we defined in Claim [Claim 25](#regularisation_g_b){reference-type="ref" reference="regularisation_g_b"} and using [\[formulation_faible\]](#formulation_faible){reference-type="eqref" reference="formulation_faible"} and letting $\eta$ tends to zero as we did for the proof of Claim [Claim 26](#expression_derivative_T1){reference-type="ref" reference="expression_derivative_T1"}, one can show that $T_2$ is Lipschitz and that for almost every $t \in [0,T]$, we have $$\begin{aligned} \frac{\,\mathrm dT_2}{\,\mathrm dt} = T_{2,1} + T_{2,2}\end{aligned}$$ where $$\begin{aligned} T_{2,1} &:= \frac{2}{N}\sum_{i=1}^N \int_{\mathbb{R}^2} \frac{\nabla^\bot b(x)}{b(x)} \cdot \nabla_x g_b(x,q_i)\overline{\omega}(t,x)\,\mathrm dx \\ &= 2\iint_{\mathbb{R}^2\times\mathbb{R}^2} \frac{\nabla^\bot b(x)}{b(x)} \cdot \nabla_x g_b(x,y) \overline{\omega}(t,x)\,\mathrm dx\,\mathrm d\overline{\omega}_N(t,y) \end{aligned}$$ and $$\begin{aligned} T_{2,2} :=&- \frac{2}{N} \sum_{i=1}^N \dot{\overline{q_i}} \cdot \int_{\mathbb{R}^2} \nabla_y g_b(x,q_i) \overline{\omega}(t,x)\,\mathrm dx \\ =&-\frac{2}{N} \sum_{i=1}^N\dot{\overline{q_i}} \cdot v(t,\overline{q_i}) \\ =& \frac{2}{N}\sum_{i=1}^N v(t,\overline{q_i})\cdot\bigg[\frac{\nabla^\bot b(\overline{q_i})}{b(\overline{q_i})} +\frac{1}{N\alpha_N}\underset{j\neq i}{\sum_{j=1}^N}\frac{1}{b(\overline{q_i})}\nabla_x g_b(\overline{q_i},\overline{q_j})\bigg] \\ =& 2\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} \frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y)\,\mathrm d\overline{\omega}_N(x)\,\mathrm d\overline{\omega}(y) \\ &+ \frac{2}{N^2\alpha_N} \sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N} \frac{v(t,\overline{q_i})}{b(\overline{q_i})}\cdot\nabla_x g_b(\overline{q_i},\overline{q_j})\end{aligned}$$ and thus we have [\[expression_derivative_resc_T2\]](#expression_derivative_resc_T2){reference-type="eqref" reference="expression_derivative_resc_T2"}. ◻ Combining Equations [\[expression_derivative_resc_T1\]](#expression_derivative_resc_T1){reference-type="eqref" reference="expression_derivative_resc_T1"}, [\[expression_derivative_resc_EN\]](#expression_derivative_resc_EN){reference-type="eqref" reference="expression_derivative_resc_EN"} and [\[expression_derivative_resc_T2\]](#expression_derivative_resc_T2){reference-type="eqref" reference="expression_derivative_resc_T2"} we get [\[time_derivative_F\_N_resc\]](#time_derivative_F_N_resc){reference-type="eqref" reference="time_derivative_F_N_resc"}. ◻ # Properties of the modulated energy {#section:5} For $\displaystyle{0 < \eta < 1}$, we denote $$H_{N,\eta} := G_b\left[\frac{1}{N}\sum_{i=1}^N \widetilde{\delta}_{q_i}^{(\eta)} - \omega\right].$$ If $b=1$ this quantity is the electric potential introduced by Serfaty in [@Serfaty Equation (3.12)] divided by $N$. **Proposition 29**. *Let $\omega \in \mathcal{P}(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$ with compact support and $q_1,...,q_N \in \mathbb{R}^2$ be such that $q_i \neq q_j$ if $i \neq j$. Then the following inequality holds: $$\begin{gathered} \int_{\mathbb{R}^2} \frac{1}{b}|\nabla H_{N,\eta}|^2 + \frac{C_b}{N^2}\sum_{1 \leqslant i \neq j \leqslant N}(g(q_i - q_j) - g^{(\eta)}(q_i-q_j)) \\ \leqslant\mathcal{F}_b(Q_N,\omega) + C_b\bigg(\frac{g(\eta)}{N} + I(Q_N)(\eta+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\eta)\eta\bigg)\end{gathered}$$ where $g^{(\eta)}$ is defined by [\[definition_g\_eta\]](#definition_g_eta){reference-type="eqref" reference="definition_g_eta"}.* From this proposition we see that even if it is not necessarily positive, the modulated energy is bounded from below by some negative power of $N$ (provided that $(I(Q_N))$ is bounded). We will also prove the three following corollaries: **Corollary 30**. *If $\omega$ and $Q_N$ satisfy the hypothesis of Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"}, then there exists $c > 0$ such that $$\begin{aligned} \frac{c}{N^2}|\{(q_i,q_j) ; |q_i - q_j| \leqslant\varepsilon\}| \leqslant& \mathcal{F}_b(Q_N,\omega) + C_b\bigg(\frac{g(\varepsilon)}{N} + I(Q_N)(\varepsilon+N^{-1}) \\ &+ \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\varepsilon)\varepsilon\bigg).\end{aligned}$$* **Corollary 31**. *Let $\alpha \in (0,1)$ and $\xi$ be a test function (for example smooth with compact support or in the Schwartz space), then if $\omega$ and $Q_N$ satisfy the hypothesis of Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"} we have* *$$\begin{aligned} \left|\int_{\mathbb{R}^2} \xi\left(\frac{1}{N}\sum_{i=1}^N \delta_{q_i} - \omega\right)\right| \leqslant& C_b|\xi|_{\mathcal{C}^{0,\alpha}}N^{-\alpha} + C_b \left(\int_{\mathbb{R}^2} \frac{1}{b} |\nabla \xi|^2\right)^\frac{1}{2}\bigg(\mathcal{F}_b(\omega,Q_N) \\ &+ \frac{\ln(N)}{N} + I(Q_N)N^{-1}\\ &+ \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}\frac{\ln(N)}{N}\bigg)^\frac{1}{2}.\end{aligned}$$ In particular, there exists $\beta > 0$ such that for all $s < -1$, $$\begin{aligned} \left\lVert\frac{1}{N}\sum_{i=1}^N \delta_{q_i} - \omega\right\rVert_{H^s} \leqslant& C_b((1+I(Q_N)+\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty})N^{-\beta} \\ &+ \mathcal{F}_b(\omega,Q_N)).\end{aligned}$$* **Corollary 32**. *If $\omega$ and $Q_N$ satisfy the hypothesis of Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"} and if $(I(Q_N))$ is bounded, then the two following assertions are equivalent:* 1. *$\mathcal{F}_b(\omega,Q_N) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} 0$.* 2. *$\displaystyle{\frac{1}{N}\sum_{i=1}^N \delta_{q_i}} \underset{N \rightarrow +\infty}{\xrightharpoonup{\; \; \ast \; \;}} \omega$ for the weak-$\ast$ topology of probability measures and $$\frac{1}{N^2} \sum_{1 \leqslant i \neq j \leqslant N} g_b(q_i,q_j) \longrightarrow \iint_{\mathbb{R}^2\times \mathbb{R}^2} g_b(x,y)\omega(x)\omega(y)\,\mathrm dx \,\mathrm dy.$$* Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"} and Corollaries [Corollary 30](#corollary_counting){reference-type="ref" reference="corollary_counting"}, [Corollary 31](#corollary_coercivity){reference-type="ref" reference="corollary_coercivity"} and [Corollary 32](#corollary_weak_star_cv){reference-type="ref" reference="corollary_weak_star_cv"} are analogues of other results obtained in [@Duerinckx; @NguyenRosenzweigSerfaty; @Serfaty]. Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"} is an equivalent of [@Serfaty Proposition 3.3] or [@NguyenRosenzweigSerfaty Proposition 2.2] and the proof will follow the same steps: regularise the modulated energy and control the remainders. Some terms are very similar to the ones obtained in the Coulomb case whereas other terms are specific to the lake kernel and will be handled using the estimates proved in Section [2](#section:2){reference-type="ref" reference="section:2"}. Corollary [Corollary 30](#corollary_counting){reference-type="ref" reference="corollary_counting"} is an equivalent of [@NguyenRosenzweigSerfaty Corollary 2.3] and Corollary [Corollary 31](#corollary_coercivity){reference-type="ref" reference="corollary_coercivity"} is an equivalent of [@Serfaty Proposition 3.6]. Both can be deduced from Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"} in the same way [@NguyenRosenzweigSerfaty Corollary 2.3] and [@Serfaty Proposition 3.6] are deduced from [@Serfaty Proposition 3.3] or [@NguyenRosenzweigSerfaty Proposition 2.2]. Corollary [Corollary 32](#corollary_weak_star_cv){reference-type="ref" reference="corollary_weak_star_cv"} is an equivalent of [@Duerinckx Lemma 2.6] and its proof proceeds in the same way. Due to the bound we assumed on the moment of inertia, tightness issues will be easier to handle. Let us begin by proving the main proposition of this section: *Proof of Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"}.* Let us regularise the modulated energy [\[definition_modulated_energy\]](#definition_modulated_energy){reference-type="eqref" reference="definition_modulated_energy"} using the regularisation of the dirac mass $\widetilde{\delta}$ defined in [\[definition_delta_tilde_q\]](#definition_delta_tilde_q){reference-type="eqref" reference="definition_delta_tilde_q"}. We have $$\begin{aligned} &\mathcal{F}_b(Q_N,\omega) = \\ &\iint_{\mathbb{R}^2\times \mathbb{R}^2} g_b(x,y)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N \widetilde{\delta}_{q_i}^{(\eta)} - \omega\right)(x)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N \widetilde{\delta}_{q_i}^{(\eta)} - \omega\right)(y) \\ &+\frac{1}{N^2}\underset{1 \leqslant i \neq j \leqslant N}{\sum}\iint_{\mathbb{R}^2\times \mathbb{R}^2} (\sqrt{b(q_i)b(q_j)}g(q_i - q_j) \\ &- \sqrt{b(x)b(y)}g(x-y))\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(x)\,\mathrm d\widetilde{\delta}_{q_j}^{(\eta)}(y) \\ &+ \frac{1}{N^2}\underset{1 \leqslant i \neq j \leqslant N}{\sum} \iint_{\mathbb{R}^2\times \mathbb{R}^2} (S_b(q_i,q_j) - S_b(x,y))\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(x)\,\mathrm d\widetilde{\delta}_{q_j}^{(\eta)}(y) \\ &- \frac{1}{N^2}\sum_{i=1}^N\iint_{\mathbb{R}^2\times \mathbb{R}^2}g_b(x,y)\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(x)\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(y) \\ &+ \frac{2}{N}\sum_{i=1}^N \iint_{\mathbb{R}^2\times \mathbb{R}^2} \big(\sqrt{b(x)b(y)}g(x-y) \\ &- \sqrt{b(x)b(q_i)}g(x-q_i)\big)\omega(x)\,\mathrm dx \,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(y) \\ &+ \frac{2}{N}\sum_{i=1}^N \iint_{\mathbb{R}^2\times \mathbb{R}^2} (S_b(x,y) - S_b(x,q_i))\omega(x)\,\mathrm dx \,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(y) \\ =:& T_1 + T_2 + T_3 + T_4 + T_5 + T_6.\end{aligned}$$ *Claim 33*. We have $$T_1 = \int_{\mathbb{R}^2} \frac{1}{b}|\nabla H_{N,\eta}|^2.$$ *Proof of the claim.* Let us first fix $\mu$ smooth with compact support and average zero and write $H_\mu = G_b[\mu]$. By Proposition [Proposition 16](#proposition_bs_law){reference-type="ref" reference="proposition_bs_law"}, we have $$\begin{aligned} \iint_{\mathbb{R}^2\times \mathbb{R}^2} g_b(x,y)\mu(x)\mu(y) \,\mathrm dx \,\mathrm dy &= \int_{\mathbb{R}^2} H_{\mu}(x) \mu(x)\,\mathrm dx \\ &= - \int_{\mathbb{R}^2} H_{\mu}(x) \operatorname{div}\left(\frac{1}{b}\nabla H_{\mu}\right)(x)\,\mathrm dx.\end{aligned}$$ Let $R >0$, then integrating by parts we get $$\begin{aligned} -\int_{B(0,R)} H_{\mu} \operatorname{div}\left(\frac{1}{b}\nabla H_{\mu}\right) = - \int_{\partial B(0,R)}\frac{1}{b}H_{\mu} \nabla H_{\mu}\cdot \,\mathrm d\vec{S} + \int_{B(0,R)} \frac{1}{b}|\nabla H_{\mu}|^2.\end{aligned}$$ Using Proposition [Proposition 13](#theorem_asymptotic_velocity_field_lake){reference-type="ref" reference="theorem_asymptotic_velocity_field_lake"} applied to $\omega = \mu$, $\displaystyle{u = -\frac{1}{b}\nabla^\bot H_\mu}$ and $\psi = H_\mu$, we have $$\left|\int_{\partial B(0,R)}\frac{1}{b}H_{\mu} \nabla H_{\mu}\cdot \,\mathrm d\vec{S}\right| \leqslant\frac{C}{R^2}(1+R^\delta)R \mathop{\longrightarrow}\limits_{R\rightarrow+\infty} 0$$ and therefore $$\iint_{\mathbb{R}^2\times \mathbb{R}^2} g_b(x,y) \mu(x) \mu(y) \,\mathrm dx \,\mathrm dy = \int_{\mathbb{R}^2} \frac{1}{b}|\nabla H_{\mu}|^2.$$ Now consider a sequence $(\mu_k)$ of smooth functions with compact support and average zero converging to $\displaystyle{m := \frac{1}{N}\sum_{i=1}^N \widetilde{\delta}_{q_i}^{(\eta)} - \omega}$ in $\dot H^{-1}$, then by Lemma [Lemma 20](#nabla_G_b_bounded_H_minus_one){reference-type="ref" reference="nabla_G_b_bounded_H_minus_one"}, $$\begin{aligned} \nabla H_{\mu_k} \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} \nabla H_{N,\eta} \; \text{in} \; L^2.\end{aligned}$$ and therefore $$\begin{aligned} \int_{\mathbb{R}^2} \frac{1}{b}|\nabla H_{\mu_k}|^2 \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} \int_{\mathbb{R}^2} \frac{1}{b}|\nabla H_{N,\eta}|^2\end{aligned}$$ and $$\begin{aligned} \bigg|\iint_{\mathbb{R}^2\times \mathbb{R}^2} &g_b(x,y)\mu_k(x) \mu_k(y)\,\mathrm dx \,\mathrm dy - \iint_{\mathbb{R}^2\times \mathbb{R}^2} g_b(x,y)\,\mathrm dm(x) \,\mathrm dm(y)\bigg| \\ &= \bigg|\int_{\mathbb{R}^2}G_b[\mu_k-m]\,\mathrm d\mu_k + \int_{\mathbb{R}^2}G_b[m]\,\mathrm d(\mu_k-m)\bigg| \\ &\leqslant C\left\lVert\nabla G_b[\mu_k - m]\right\rVert_{L^2}\left\lVert\mu_k\right\rVert_{\dot H^{-1}} + C\left\lVert\nabla G_b[m]\right\rVert_{L^2}\left\lVert\mu_k-m\right\rVert_{\dot H^{-1}} \\ &\leqslant C\left\lVert\mu_k-m\right\rVert_{\dot H^{-1}} \end{aligned}$$ by Lemma [Lemma 20](#nabla_G_b_bounded_H_minus_one){reference-type="ref" reference="nabla_G_b_bounded_H_minus_one"} so we get Claim [Claim 33](#coerc_T1){reference-type="ref" reference="coerc_T1"}. ◻ Now let us bound the fourth term: *Claim 34*. $$|T_4| \leqslant\frac{C_b}{N}(g(\eta)+I(Q_N)).$$ *Proof.* We write $$\begin{aligned} T_4 =& -\frac{1}{N^2}\sum_{i=1}^N\iint_{\mathbb{R}^2\times \mathbb{R}^2}\sqrt{b(x)b(y)}g(x-y)\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(x)\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(y) \\ &-\frac{1}{N^2}\sum_{i=1}^N\iint_{\mathbb{R}^2\times \mathbb{R}^2} S_b(x,y)\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(x)\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(y) \\ =:& T_{4,1} + T_{4,2}.\end{aligned}$$ Using the definition of $\widetilde{\delta}_q$ [\[definition_delta_tilde_q\]](#definition_delta_tilde_q){reference-type="eqref" reference="definition_delta_tilde_q"} and Equality [\[egalite_convolution_g\_eta\]](#egalite_convolution_g_eta){reference-type="eqref" reference="egalite_convolution_g_eta"} we get $$\begin{aligned} T_{4,1} &= -\frac{1}{N^2}\sum_{i=1}^N m_b(q_i,\eta)^2 \iint_{\mathbb{R}^2\times \mathbb{R}^2} g(x-y) \,\mathrm d\delta_{q_i}^{(\eta)}(x)\,\mathrm d\delta_{q_i}^{(\eta)}(y) \\ &= -\frac{1}{N^2}\sum_{i=1}^Nm_b(q_i,\eta)^2\int_{\mathbb{R}^2} g^{(\eta)}(x-q_i) \,\mathrm d\delta_{q_i}^{(\eta)}(x).\end{aligned}$$ Therefore, $$|T_{4,1}| \leqslant\frac{C_bg(\eta)}{N}.$$ Now by Claim $(2)$ of Lemma [Lemma 19](#estimees_S_b_symmetric){reference-type="ref" reference="estimees_S_b_symmetric"}, we have $$|T_{4,2}| \leqslant\frac{C_b}{N^2}\sum_{i=1}^N(1+|q_i|^2).$$ We get that $$|T_4| \leqslant\frac{C_b}{N}(1+I(Q_N) + g(\eta)) \leqslant\frac{C_b}{N}(g(\eta)+I(Q_N)).$$ ◻ Now we bound the third and the sixth term: *Claim 35*. $$|T_3| + |T_6| \leqslant C_b(\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)}+I(Q_N))\eta.$$ *Proof.* For $x \in \partial B(q_i,\eta), y \in \partial B(q_j,\eta)$, we use Claim $(1)$ of Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"} and the symmetry of $S_b$ to get $$\begin{aligned} |S_b(q_i,q_j) - S_b(x,y)| &\leqslant|S_b(q_i,q_j) - S_b(x,q_j)| + |S_b(x,q_j) - S_b(x,y)| \\ &\leqslant C_b(1+|q_j|)\eta + C_b(1+|q_i|)\eta \\ &\leqslant C_b(1+|q_i| + |q_j|)\eta.\end{aligned}$$ Thus we can bound the third term: $$\label{coerc_T3} |T_3| \leqslant C_b(1+I(Q_N))\eta.$$ The sixth term can be bounded in the same way: $$|T_6| \leqslant\frac{C_b}{N}\sum_{i=1}^N \iint_{\mathbb{R}^2\times \mathbb{R}^2} (1+|x|+|q_i|)\eta\omega(x) \,\mathrm dx \,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(y).$$ We get that $$\label{coerc_T6} |T_6| \leqslant C_b(\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)}+I(Q_N))\eta.$$ and combining [\[coerc_T3\]](#coerc_T3){reference-type="eqref" reference="coerc_T3"} with [\[coerc_T6\]](#coerc_T6){reference-type="eqref" reference="coerc_T6"} we get Claim [Claim 35](#coerc_T36){reference-type="ref" reference="coerc_T36"}. ◻ Now let us bound the fifth term: *Claim 36*. $$|T_5| \leqslant C_b\left\lVert\omega\right\rVert_{L^1\cap L^\infty}\eta g(\eta).$$ *Proof.* Using Proposition [Proposition 21](#proposition_regularisation){reference-type="ref" reference="proposition_regularisation"} we write $T_5$ as $$\begin{aligned} T_5 =& \frac{2}{N}\sum_{i=1}^N \int_{\mathbb{R}^2}(m_b(q_i,\eta)g^{(\eta)}(x-q_i)-\sqrt{b(q_i)}g(x-q_i))\sqrt{b(x)}\omega(x)\,\mathrm dx \\ =& \frac{2}{N}\sum_{i=1}^N (m_b(q_i,\eta) - \sqrt{b(q_i)})\int_{\mathbb{R}^2} g^{(\eta)}(x-q_i)\sqrt{b(x)}\omega(x)\,\mathrm dx \\ &+ \frac{2}{N}\sum_{i=1}^N \sqrt{b(q_i)}\int_{\mathbb{R}^2} (g^{(\eta)}(x-q_i) - g(x-q_i))\sqrt{b(x)}\omega(x)\,\mathrm dx.\end{aligned}$$ and thus by [\[estimee_m\_b\]](#estimee_m_b){reference-type="eqref" reference="estimee_m_b"} and since $|g^{(\eta)}(x-q_i)| \leqslant C(g(\eta) + |x| + |q_i|)$ we have $$\begin{aligned} |T_5| \leqslant& C_b\left\lVert\omega\right\rVert_{L^1}\eta g(\eta) + C_b\left\lVert\omega\right\rVert_{L^1(|x|\,\mathrm dx)}\eta + C_b\left\lVert\omega\right\rVert_{L^1}(1+I(Q_N))\eta \\ &+C_b\left\lVert\omega\right\rVert_{L^\infty}\int_{B(0,\eta)}|g^{(\eta)}(x) - g(x)|\,\mathrm dx.\end{aligned}$$ We get that $$|T_5| \leqslant C_b\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}\eta g(\eta) + (1+I(Q_N))\eta$$ since $\omega$ is a probability density. ◻ We are only remained to estimate from below the second term: *Claim 37*. $$T_2 \geqslant\frac{C_b}{N^2}\sum_{1 \leqslant i \neq j \leqslant N}(g(q_i - q_j) - g^{(\eta)}(q_i-q_j)) - C_b\eta g(\eta).$$ *Proof.* We also split $T_2$ in two terms: $$\begin{aligned} T_2 =& \frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} \sqrt{b(q_i)b(q_j)}g(q_i - q_j) \\ &- m_b(q_i,\eta) m_b(q_j,\eta) \iint_{\mathbb{R}^2\times\mathbb{R}^2} g(x-y) \,\mathrm d\delta_{q_i}^{(\eta)}(x)\,\mathrm d\delta_{q_j}^{(\eta)}(y) \\ =& \frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} \sqrt{b(q_i)b(q_j)}g(q_i - q_j) \\ &- m_b(q_i,\eta) m_b(q_j,\eta) \int_{\mathbb{R}^2} g^{(\eta)}(q_i-y) \,\mathrm d\delta_{q_j}^{(\eta)}(y) \\ =& \frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N}(\sqrt{b(q_i)b(q_j)} - m_b(q_i,\eta) m_b(q_j,\eta)) \\ &\times\int_{\mathbb{R}^2} g^{(\eta)}(q_i-y) \,\mathrm d\delta_{q_j}^{(\eta)}(y) \\ &+ \frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} \sqrt{b(q_i)b(q_j)}\left(g(q_i - q_j) -\int_{\mathbb{R}^2} g^{(\eta)}(q_i-y) \,\mathrm d\delta_{q_j}^{(\eta)}(y)\right) \\ =& T_{2,1} + T_{2,2}.\end{aligned}$$ Writing $$\begin{aligned} \sqrt{b(q_i)b(q_j)} - m_b(q_i,\eta) m_b(q_j,\eta) &= \sqrt{b(q_i)}(\sqrt{b(q_j)} - m_b(q_j,\eta)) \\ &+ m_b(q_j,\eta)(\sqrt{b(q_i)} - m_b(q_i,\eta))\end{aligned}$$ and using [\[estimee_m\_b\]](#estimee_m_b){reference-type="eqref" reference="estimee_m_b"} we get that $$\label{coerc_T21} |T_{2,1}| \leqslant C_b\eta g(\eta).$$ Now by [\[egalite_convolution_g\_eta\]](#egalite_convolution_g_eta){reference-type="eqref" reference="egalite_convolution_g_eta"}, $$\begin{aligned} g&(q_i - q_j) -\int_{\mathbb{R}^2} g^{(\eta)}(q_i-y) \,\mathrm d\delta_{q_j}^{(\eta)}(y) \\ &= g(q_i - q_j) - g^{(\eta)}(q_i-q_j) + \int_{\mathbb{R}^2} (g(q_i - y) - g^{(\eta)}(q_i-y))\,\mathrm d\delta_{q_j}^{(\eta)}(y) \\ &\geqslant g(q_i - q_j) - g^{(\eta)}(q_i-q_j) + 0\end{aligned}$$ and thus $$\label{coerc_T22} T_{2,2} \geqslant\frac{C_b}{N^2}\sum_{1 \leqslant i \neq j \leqslant N}(g(q_i - q_j) - g^{(\eta)}(q_i-q_j)).$$ We get Claim [Claim 37](#coerc_T2){reference-type="ref" reference="coerc_T2"} combining Equations [\[coerc_T21\]](#coerc_T21){reference-type="eqref" reference="coerc_T21"} with [\[coerc_T22\]](#coerc_T22){reference-type="eqref" reference="coerc_T22"}. ◻ Combining Claims [Claim 33](#coerc_T1){reference-type="ref" reference="coerc_T1"}, [Claim 34](#coerc_T4){reference-type="ref" reference="coerc_T4"}, [Claim 35](#coerc_T36){reference-type="ref" reference="coerc_T36"}, [Claim 36](#coerc_T5){reference-type="ref" reference="coerc_T5"} and [Claim 37](#coerc_T2){reference-type="ref" reference="coerc_T2"} we get the proof of Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"}. ◻ Now we prove the \"counting close particles\" Corollary: *Proof of Corollary [Corollary 30](#corollary_counting){reference-type="ref" reference="corollary_counting"}.* The proof is exactly the same as the proof of [@Rosenzweig Lemma 3.7]. If $|q_i - q_j| \leqslant\varepsilon$ then $$\begin{aligned} g(q_i - q_j) - g^{(2\varepsilon)}(q_i - q_j) &= -\frac{1}{2\pi}\ln|q_i - q_j| + \frac{1}{2\pi}\ln(2\varepsilon) \\ &\geqslant-\frac{1}{2\pi}\ln(\varepsilon) + \frac{1}{2\pi}\ln(2\varepsilon) = \frac{1}{2\pi}\ln(2) > 0.\end{aligned}$$ Thus, since $g - g^{(2\varepsilon)} \geqslant 0$, $$\begin{aligned} &\frac{1}{2\pi N^2}\ln(2)|\{(q_i,q_j) ; |q_i - q_j| \leqslant\varepsilon\}| \\ \leqslant& \frac{1}{N^2}\underset{|q_i - q_j| \leqslant\varepsilon}{\sum_{1 \leqslant i \neq j\leqslant N}} (g(q_i - q_j) - g^{(2\varepsilon)}(q_i - q_j)) \\ \leqslant& \frac{1}{N^2}\sum_{1 \leqslant i \neq j\leqslant N} (g(q_i - q_j) - g^{(2\varepsilon)}(q_i - q_j)) \\ \leqslant& \mathcal{F}_b(Q_N,\omega) \\ &+ C_b\bigg(\frac{g(\varepsilon)}{N} + I(Q_N)(\varepsilon+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\varepsilon)\varepsilon\bigg).\end{aligned}$$ where we used Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"} in the last inequality. ◻ Now we prove the coercivity result: *Proof of Corollary [Corollary 31](#corollary_coercivity){reference-type="ref" reference="corollary_coercivity"}.* We have $$\begin{aligned} \int_{\mathbb{R}^2} \xi\left(\frac{1}{N}\sum_{i=1}^N \delta_{q_i} - \omega\right) =& \frac{1}{N}\int_{\mathbb{R}^2} \xi \left(\sum_{i=1}^N\delta_{q_i} - \widetilde{\delta}_{q_i}^{(\eta)}\right) \\ &+ \int_{\mathbb{R}^2} \xi\left(\frac{1}{N}\sum_{i=1}^N \widetilde{\delta}_{q_i}^{(\eta)} - \omega\right) \\ =&: T_1 + T_2.\end{aligned}$$ Now, $$\begin{aligned} T_1 &= \frac{1}{N}\sum_{i=1}^N \xi(q_i) - m_b(q_i,\eta)\int_{\partial B(q_i,\eta)} \frac{\xi(x)}{\sqrt{b(x)}}\,\mathrm d\delta_{q_i}^{(\eta)}(x) \\ &= \frac{1}{N}\sum_{i=1}^N m_b(q_i,\eta)\int_{\partial B(q_i,\eta)} \frac{\xi(q_i) - \xi(x)}{\sqrt{b(x)}}\,\mathrm d\delta_{q_i}^{(\eta)}(x).\end{aligned}$$ Thus $$|T_1| \leqslant C_b|\xi|_{\mathcal{C}^{0,\alpha}}\eta^\alpha.$$ Using a sequence $(\mu_k)$ of smooth functions with compact support and average $0$ converging to $\displaystyle{\frac{1}{N}\sum_{i=1}^N \widetilde{\delta}_{q_i}^{(\eta)} - \omega}$ as we have done for Claim [Claim 33](#coerc_T1){reference-type="ref" reference="coerc_T1"} we can show that $$\begin{aligned} T_2 &= \int_{\mathbb{R}^2} \frac{1}{b} \nabla \xi \cdot \nabla H_{N,\eta} \end{aligned}$$ and therefore $$\begin{aligned} |T_2| \leqslant& \left(\int_{\mathbb{R}^2} \frac{1}{b} |\nabla \xi|^2\right)^\frac{1}{2}\left(\int_{\mathbb{R}^2} \frac{1}{b} |\nabla H_{N,\eta}|^2\right)^\frac{1}{2} \\ \leqslant& C_b \left(\int_{\mathbb{R}^2} \frac{1}{b} |\nabla \xi|^2\right)^\frac{1}{2}\bigg(\mathcal{F}_b(Q_N,\omega)+ \frac{g(\eta)}{N} \\ &+ I(Q_N)(\eta+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\eta)\eta\bigg)^\frac{1}{2}.\end{aligned}$$ by Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"}. We conclude by taking $\eta = N^{-1}$. The bound on $$\left\lVert\frac{1}{N}\sum_{i=1}^N \delta_{q_i} - \omega\right\rVert_{H^s}$$ follows from Sobolev embeddings. ◻ We finish this section by proving the weak-$\ast$ convergence result: *Proof of Corollary [Corollary 32](#corollary_weak_star_cv){reference-type="ref" reference="corollary_weak_star_cv"}.* Let us denote $\displaystyle{\omega_N = \frac{1}{N}\sum_{i=1}^N\delta_{q_i}}$ and prove that $(\omega_N)$ is a tight sequence of probability measures. Let $R > 1$, then $$\label{bound_tightness} \begin{aligned} |\{i \in [1,N]\; ; \; |q_i| \geqslant R \}|R^2 &\leqslant\underset{|q_i| \geqslant R}{\sum_{i=1}^N}|q_i|^2 \\ &\leqslant N I(Q_N). \end{aligned}$$ Dividing by $NR^2$ both sides of the inequality we get $$\int_{B(0,R)^c} \,\mathrm d\omega_N \leqslant I(Q_N)R^{-2}$$ and since $(I(Q_N))$ is bounded we get that $(\omega_N)$ is tight. We will now prove the following Claim: *Claim 38*. Assume that $(\omega_N)$ converges to $\omega$ for the weak-$\ast$ topology of probability measures and that $(I(Q_N))$ is bounded. Then $\mathcal{F}_b(Q_N,\omega) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} 0$ if and only if we have $$\frac{1}{N^2} \sum_{1 \leqslant i \neq j \leqslant N} g_b(q_i,q_j) \longrightarrow \iint_{\mathbb{R}^2\times \mathbb{R}^2} g_b(x,y)\omega(x)\omega(y)\,\mathrm dx \,\mathrm dy.$$ *Proof.* Let $\varepsilon> 0$. We write the modulated energy as the sum of three terms: $$\label{split_mod_energy_3} \begin{aligned} \mathcal{F}_b(Q_N,\omega) =& -\iint_{\mathbb{R}^2\times\mathbb{R}^2} g_b(x,y) \omega(x)\omega(y)\,\mathrm dx \,\mathrm dy \\ &+ \frac{1}{N^2}\sum_{1 \leqslant i \neq j \leqslant N} g_b(q_i,q_j) \\ &-2 \int_{\mathbb{R}^2} \psi(y)\,\mathrm d(\omega_N-\omega)(y) \end{aligned}$$ where $\psi = G_b[\omega]$. Let $R \geqslant 1$ be such that $\operatorname{supp}(\omega) \subset B(0,R)$. We have $$\begin{aligned} \int_{\mathbb{R}^2}\psi\,\mathrm d(\omega-\omega_N) = -\int_{B(0,R)^c}\psi\,\mathrm d\omega_N + \int_{B(0,R)}\psi\,\mathrm d(\omega-\omega_N).\end{aligned}$$ We bound the first term as we did to obtain [\[bound_tightness\]](#bound_tightness){reference-type="eqref" reference="bound_tightness"}: $$\begin{aligned} \left|\int_{B(0,R)^c}\psi\,\mathrm d\omega_N\right| &\leqslant\frac{1}{N}\underset{|q_i| \geqslant R}{\sum_{i=1}^N}|\psi(q_i)| \\ &\leqslant\frac{C_b}{N}\underset{|q_i| \geqslant R}{\sum_{i=1}^N}(1+|q_i|^\delta) \\ &\leqslant C_b(R^{-2}I(Q_N) + R^{2-\delta} I(Q_N))\end{aligned}$$ for some $0 < \delta < 1$ (by Proposition [Proposition 13](#theorem_asymptotic_velocity_field_lake){reference-type="ref" reference="theorem_asymptotic_velocity_field_lake"}). Therefore, $$\left|\int_{B(0,R)^c}\psi\,\mathrm d\omega_N\right| \leqslant\varepsilon$$ if $R$ is big enough. Now let $\chi_{R,\beta}$ be a smooth function such that $0 \leqslant\chi \leqslant 1$, $\chi_{R,\beta}(x) = 1$ if $|x| \leqslant R$ and $\chi_{R,\beta}(x) = 0$ if $|x| \geqslant R+\beta$. Then $$\begin{aligned} \int_{B(0,R)}\psi\,\mathrm d(\omega-\omega_N) = \int \chi_{R,\beta}\psi\,\mathrm d(\omega-\omega_N) - \int_{R \leqslant|x| \leqslant R+\beta} \chi_{R,\beta}\psi\,\mathrm d(\omega-\omega_N)\end{aligned}$$ Choosing $\beta$ small enough we have $$\begin{aligned} \left|\int_{R \leqslant|x| \leqslant R+\beta} \chi_{R,\beta}\psi\,\mathrm d(\omega-\omega_N)\right| \leqslant\varepsilon.\end{aligned}$$ Now $\psi$ is continuous (see Lemma [Lemma 11](#lemma_link_solutions_duerinckx){reference-type="ref" reference="lemma_link_solutions_duerinckx"}) so by weak-$\ast$ convergence of $(\omega_N)$ to $\omega$ we get that $$\int\psi\chi_{R,\beta}\,\mathrm d(\omega-\omega_N) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} 0$$ and therefore $$\underset{N\rightarrow +\infty}{\limsup}\left|\int_{\mathbb{R}^2}\psi\,\mathrm d(\omega-\omega_N)\right| \leqslant 2\varepsilon.$$ for all $\varepsilon> 0$, so we get $$\int_{\mathbb{R}^2}\psi\,\mathrm d(\omega-\omega_N) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} 0.$$ Using [\[split_mod_energy_3\]](#split_mod_energy_3){reference-type="eqref" reference="split_mod_energy_3"} we get that $\mathcal{F}_b(Q_N,\omega) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} 0$ if and only if we have $$\frac{1}{N^2} \sum_{1 \leqslant i \neq j \leqslant N} g_b(q_i,q_j) \longrightarrow \iint_{\mathbb{R}^2\times \mathbb{R}^2} g_b(x,y)\omega(x)\omega(y)\,\mathrm dx \,\mathrm dy.$$ ◻ It follows directly from the Claim that $(2) \implies (1)$. Now if we have $(1)$, using Corollary [Corollary 31](#corollary_coercivity){reference-type="ref" reference="corollary_coercivity"} we have convergence of $(\omega_N)$ to $\omega$ in any $H^{s}$ for any $s < -1$. It follows by Prokhorov's theorem $(\omega_N)$ converges to $\omega$ for the weak-$\ast$ topology of probability measures. By the Claim we also have convergence of the interaction energy and therefore $(1) \implies (2)$. ◻ # Proof of the main Proposition [Proposition 39](#controle_terme_principal_gronwall){reference-type="ref" reference="controle_terme_principal_gronwall"} {#section:6} Let us recall that for $q \in \mathbb{R}^2$, $Q_N = (q_1,...,q_N) \in (\mathbb{R}^2)^N$ and $\displaystyle{0 < \eta < 1}$, we have denoted $$I(Q_N) = \frac{1}{N}\sum_{i=1}^N |q_i|^2,$$ $$\widetilde{\delta}_q^{(\eta)} = m_b(q,\eta)\frac{\,\mathrm d\delta_{q}^{(\eta)}}{\sqrt{b}}$$ and $$m_b(q,\eta) = \left(\int_{\mathbb{R}^2}\frac{\,\mathrm d\delta_{q}^{(\eta)}}{\sqrt{b}}\right)^{-1}$$ where $\displaystyle{\delta_{q}^{(\eta)}}$ is the uniform probability measure on the circle $\partial B(q,\eta)$. In this Section, we prove the following result: **Proposition 39**. *Let $Q_N = (q_1,...,q_N) \in (\mathbb{R}^2)^N$ such that $q_i \neq q_j$ if $i \neq j$, $u \in W^{1,\infty}(\mathbb{R}^2,\mathbb{R}^2)$ and $\omega \in \mathcal{P}(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$ with compact support such that $\nabla G_b[\omega]$ is continuous and bounded. There exists $\beta \in (0,1)$ (independent of $\omega$, $u$ and $Q_N$) such that $$\begin{aligned} \bigg|\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} &u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N\delta_{q_i}-\omega\right)^{\otimes 2}(x,y)\bigg| \\ \leqslant& C_b\left\lVert u\right\rVert_{W^{1,\infty}}|\mathcal{F}_b(Q_N,\omega)| \\ &+ C_b(1+\left\lVert u\right\rVert_{W^{1,\infty}})\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}(1+I(Q_N))N^{-\beta}.\end{aligned}$$* This proposition is an equivalent of [@Serfaty Proposition 1.1] or [@NguyenRosenzweigSerfaty Proposition 4.1] and the proof will follow the same steps: regularise the dirac masses, use the structure of the lake kernel to bound the regular part and control the remainders. Some terms are very similar to the ones obtained in the Coulomb case and we will use both the properties of our regularisation (see Subsection [2.4](#subsection:24){reference-type="ref" reference="subsection:24"}) and some estimates proved in [@NguyenRosenzweigSerfaty] to bound them. As in the proof of Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"} some terms are specific to the lake kernel and we will use results of Section [2](#section:2){reference-type="ref" reference="section:2"} to bound them. *Proof.* Let us fix $\displaystyle{0 < \eta < \frac{1}{8}}$ and write $$\label{decomposition_main_proposition} \begin{aligned} &\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N\delta_{q_i}-\omega\right)^{\otimes 2}(x,y) \\ =& \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N\widetilde{\delta}^{(\eta)}_{q_i}-\omega\right)^{\otimes 2}(x,y) \\ &+\bigg(-\frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\bigg[\,\mathrm d\omega(x)\,\mathrm d(\delta_{q_i}-\widetilde{\delta}^{(\eta)}_{q_i})(y) \\ &+ \,\mathrm d(\delta_{q_i}-\widetilde{\delta}^{(\eta)}_{q_i})(x) \,\mathrm d\omega(y)\bigg]\bigg) \\ &+\bigg(\frac{1}{N^2}\underset{1 \leqslant i,j \leqslant N}{\sum}\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} u(x)\cdot \nabla_x g_b(x,y)\\ &[\,\mathrm d\delta_{q_i}(x) \,\mathrm d\delta_{q_j}(y) - \,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_j}(y)]\bigg) \\ =:& T_1 + T_2 + T_3. \end{aligned}$$ Let us bound the first term. As in Section [5](#section:5){reference-type="ref" reference="section:5"} we write $$H_{N,\eta} := G_b\left[\frac{1}{N}\sum_{i=1}^N\widetilde{\delta}^{(\eta)}_{q_i}-\omega\right].$$ We claim: *Claim 40*. $$\begin{aligned} T_1 =& -\int_{\mathbb{R}^2} u(x) \cdot \nabla H_{N,\eta}(x) \nabla\left(\frac{1}{b}\right)\cdot\nabla H_{N,\eta}(x)\,\mathrm dx \\ &+\int_{\mathbb{R}^2} \nabla\left(\frac{1}{2b}u\right):[H_{N,\eta},H_{N,\eta}] \end{aligned}$$ *Proof of the Claim.* This claim is similar to [@Serfaty Lemma 4.3] and we proceed the same way: Let us first fix $\mu$ smooth with compact support and average zero and write $H_\mu = G_b[\mu]$. Then $$\begin{aligned} \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)&\,\mathrm d\mu^{\otimes 2}(x,y) \\ =& - \int_{\mathbb{R}^2} u(x)\cdot \nabla H_\mu(x) \operatorname{div}\left(\frac{1}{b}\nabla H_\mu\right)(x)\,\mathrm dx \\ =& -\int_{\mathbb{R}^2} u(x) \cdot \nabla H_\mu(x) \nabla\left(\frac{1}{b}\right)\cdot\nabla H_\mu(x)\,\mathrm dx \\ &- \int_{\mathbb{R}^2}\frac{1}{b}u\cdot \nabla H_\mu\Delta H_\mu.\end{aligned}$$ For the second integral of the right handside we proceed as in [@Serfaty] and use the stress-energy tensor defined by [\[definition_SE\]](#definition_SE){reference-type="eqref" reference="definition_SE"} (for more details, see [@Serfaty Equality (1.25)] and the associated references): $$\int_{\mathbb{R}^2}\frac{1}{b}u\cdot \nabla H_\mu\Delta H_\mu = \int_{\mathbb{R}^2} \frac{1}{2b}u\cdot \operatorname{div}([H_\mu,H_\mu]).$$ Integrating over a ball of radius $R$ and integrating by parts we get $$\begin{aligned} \int_{B(0,R)}\frac{1}{2b}u\cdot \operatorname{div}([H_\mu,H_\mu]) =& \int_{\partial B(0,R)} \frac{1}{2b}[H_\mu,H_\mu]u\cdot \,\mathrm d\vec{S} \\ &- \int_{B(0,R)}\nabla\left(\frac{1}{2b}u\right):[H_\mu,H_\mu].\end{aligned}$$ Using Proposition [Proposition 13](#theorem_asymptotic_velocity_field_lake){reference-type="ref" reference="theorem_asymptotic_velocity_field_lake"} (applied to $\omega = \mu$ and $\psi = H_\mu$) we have $$\left|\int_{\partial B(0,R)} \frac{1}{2b}[H_\mu,H_\mu]u\cdot \,\mathrm d\vec{S}\right| \leqslant\frac{C_{b,\mu}\left\lVert u\right\rVert_{L^\infty}}{R^4}R.$$ Letting $R \longrightarrow \infty$ we obtain $$\begin{aligned} \int_{\mathbb{R}^2} \frac{1}{2b}u\cdot \operatorname{div}([H_{\mu},H_{\mu}]) &= -\int_{\mathbb{R}^2} \nabla\left(\frac{1}{2b}u\right):[H_{\mu},H_{\mu}].\end{aligned}$$ Now if $(\mu_k)$ is a sequence of smooth functions with compact support and average zero such that $$\mu_k - \bigg(\frac{1}{N}\sum_{i=1}^N\widetilde{\delta}^{(\eta)}_{q_i} - \omega\bigg) \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} 0 \; \text{in} \; \dot H^{-1}$$ then by Lemma [Lemma 20](#nabla_G_b_bounded_H_minus_one){reference-type="ref" reference="nabla_G_b_bounded_H_minus_one"} we have $$\nabla H_{\mu_k} \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} \nabla H_{N,\eta} \; \text{in} \; L^2$$ and therefore since $u \in W^{1,\infty}$ and since $[H_{\mu_k},H_{\mu_k}]$ (defined by Equation [\[definition_SE\]](#definition_SE){reference-type="eqref" reference="definition_SE"}) is quadratic in the derivatives of $H_{\mu_k}$ we get that $$-\int_{\mathbb{R}^2} u(x) \cdot \nabla H_{\mu_k}(x) \nabla\left(\frac{1}{b}\right)\cdot\nabla H_{\mu_k}(x)\,\mathrm dx +\int_{\mathbb{R}^2} \nabla\left(\frac{1}{2b}u\right):[H_{\mu_k},H_{\mu_k}]$$ converges to $$-\int_{\mathbb{R}^2} u(x) \cdot \nabla H_{N,\eta}(x) \nabla\left(\frac{1}{b}\right)\cdot\nabla H_{N,\eta}(x)\,\mathrm dx +\int_{\mathbb{R}^2} \nabla\left(\frac{1}{2b}u\right):[H_{N,\eta},H_{N,\eta}]$$ as $k \longrightarrow +\infty$. We are only left to justify that $$\begin{gathered} \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\mu_k^{\otimes 2}(x,y) \\ \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N\widetilde{\delta}^{(\eta)}_{q_i}-\omega\right)^{\otimes 2}(x,y).\end{gathered}$$ We define $$\begin{aligned} m &= \frac{1}{N}\sum_{i=1}^N\widetilde{\delta}^{(\eta)}_{q_i}.\end{aligned}$$ Let us consider a sequence $(\nu_k)$ of smooth probability densities with support included in a ball $B(0,R)$ independent of $k$ (containing $\operatorname{supp}(m)$), such that $(\nu_k - m)$ converges to zero in $\dot H^{-1}$ and for the weak-$\ast$ topology of probability measures. If we set $\mu_k = \nu_k - \omega$, then $$\mu_k - (m-\omega) \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} 0 \; \text{in} \; \dot H^{-1}.$$ Now we write $$\begin{aligned} &\iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\mu_k^{\otimes 2}(x,y) \\ &- \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d(m-\omega)^{\otimes 2}(x,y) \\ =& \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\omega(x)\,\mathrm dx \,\mathrm d(m-\nu_k)(y) \\ &+ \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\omega(y)\,\mathrm dy\,\mathrm d(m-\nu_k)(x) \\ &+ \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d(\nu_k\otimes\nu_k - m\otimes m)(x,y) \\ =:& I_1 + I_2 + I_3.\end{aligned}$$ We have $$\begin{aligned} |I_1| = \bigg|\int_{\mathbb{R}^2} u \cdot \nabla G_b[m-\nu_k]\omega \bigg| &\leqslant\left\lVert u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^2}\left\lVert\nabla G_b[m-\nu_k]\right\rVert_{L^2} \\ &\leqslant C\left\lVert u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^2}\left\lVert m-\nu_k\right\rVert_{\dot H^{-1}}\end{aligned}$$ by Lemma [Lemma 20](#nabla_G_b_bounded_H_minus_one){reference-type="ref" reference="nabla_G_b_bounded_H_minus_one"} and therefore $I_1 \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} 0$. Recall that $(m-\nu_k)$ converges to zero for the weak-$\ast$ topology of probability measures. Therefore $$\begin{aligned} I_2 = \int_{\mathbb{R}^2} u\cdot \nabla G_b[\omega] \,\mathrm d(m-\nu_k) \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} 0\end{aligned}$$ since $u$ and $\nabla G_b[\omega]$ are continuous and bounded by assumption. Now we want to show that $I_3$ converges to zero. Remark that writing $\mu_k = \nu_k - \omega$ and proving that $I_1$ and $I_2$ converge to zero allowed us to restrict ourself to study the convergence of $$\iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\nu_k(x)\,\mathrm d\nu_k(y)$$ for $\nu_k$ nonnegative (which will be crucial for using Delort's argument below). We use the definition of $g_b$ [\[definition_g\_b\]](#definition_g_b){reference-type="eqref" reference="definition_g_b"} to write $$\begin{aligned} I_3 =& \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla \sqrt{b}(x)\sqrt{b(y)}g(x-y)\,\mathrm d(\nu_k\otimes\nu_k - m\otimes m)(x,y) \\ &+ \iint_{\mathbb{R}^2\times\mathbb{R}^2} \sqrt{b(x)b(y)} u(x)\cdot \nabla g(x-y)\,\mathrm d(\nu_k\otimes\nu_k - m\otimes m)(x,y) \\ &+ \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot\nabla_x S_b(x,y) \,\mathrm d(\nu_k\otimes\nu_k - m\otimes m)(x,y) \\ =:& I_{3,1} + I_{3,2} + I_{3,3}.\end{aligned}$$ We write $$\label{decomposition_I_3_1} \begin{aligned} I_{3,1} =& \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla \sqrt{b}(x)\sqrt{b(y)}g(x-y) \,\mathrm d(\nu_k-m)(x)\,\mathrm d\nu_k(y) \\ &+ \iint_{\mathbb{R}^2\times\mathbb{R}^2} u(x)\cdot \nabla \sqrt{b}(x)\sqrt{b(y)}g(x-y) \,\mathrm dm(x) \,\mathrm d(\nu_k-m)(y) \\ =& \int_{\mathbb{R}^2} (u\cdot \nabla \sqrt{b})(g\ast [\sqrt{b}\nu_k])\,\mathrm d(\nu_k-m) \\ &+ \int_{\mathbb{R}^2} (u\cdot \nabla \sqrt{b})(g\ast[\sqrt{b}(\nu_k-m)])\,\mathrm dm. \end{aligned}$$ Recall that $B(0,R)$ is a ball containing the supports of $m$ and $\nu_k$. Consider a smooth probability density $\rho$ with support in $B(0,R)$. We define $$\begin{aligned} \chi_k &= \left(\int_{\mathbb{R}^2} \sqrt{b}\nu_k\right)\rho, \\ \chi_\infty &= \left(\int_{\mathbb{R}^2} \sqrt{b}\,\mathrm dm\right)\rho \end{aligned}$$ and write $$\label{decomposition_claim_cv_H_1_loc} \begin{aligned} \nabla g\ast(\sqrt{b}(\nu_k - m)) =& \nabla g\ast(\sqrt{b}\nu_k - \chi_k+\chi_\infty - \sqrt{b}m) \\ &+ \left(\int_{\mathbb{R}^2} \sqrt{b}\nu_k-\int_{\mathbb{R}^2} \sqrt{b}\,\mathrm dm\right)\nabla g\ast\rho. \end{aligned}$$ Now $$\begin{gathered} \left\lVert\nabla g\ast(\sqrt{b}\nu_k - \chi_k+\chi_\infty - \sqrt{b}m)\right\rVert^2_{L^2} \\ = C\int_{\mathbb{R}^2}\frac{1}{|\xi|^2}|\widehat{\sqrt{b}\nu_k}(\xi) - \widehat{\chi_k}(\xi)+\widehat{\chi_\infty}(\xi) - \widehat{\sqrt{b}m}(\xi)|^2\,\mathrm d\xi. \end{gathered}$$ Remark that $\alpha_k=\sqrt{b}\nu_k - \chi_k+\chi_\infty - \sqrt{b}m$ is a Radon measure with support included in $B(0,R)$ such that $\widehat{\alpha_k}(0) = 0$. Therefore $$\begin{aligned} \left|\int_{\mathbb{R}^2} e^{-ix\cdot\xi}\,\mathrm d\alpha_k(x)\right| &= \left|\int_{\mathbb{R}^2}(e^{-ix\cdot\xi}-1)\,\mathrm d\alpha_k(x)\right| \\ &=2\left|\int_{\mathbb{R}^2}\sin\left(\frac{x\cdot\xi}{2}\right)\,\mathrm d\alpha_k(x)\right| \\ &\leqslant CR|\xi|\int_{\mathbb{R}^2}\,\mathrm d|\alpha_k|(x) \\ &\leqslant C_{b,R}|\xi|.\end{aligned}$$ It follows that for $\varepsilon> 0$ $$\begin{aligned} \int_{|\xi| \leqslant\varepsilon}\frac{1}{|\xi|^2}|\widehat{\alpha_k}(\xi)|^2\,\mathrm d\xi \leqslant C_{b,R}\varepsilon^2.\end{aligned}$$ Moreover, $$\begin{aligned} \int_{|\xi| \geqslant\varepsilon}&\frac{1}{|\xi|^2}|\widehat{\sqrt{b}\nu_k}(\xi) - \widehat{\chi_k}(\xi)+\widehat{\chi_\infty}(\xi) - \widehat{\sqrt{b}m}(\xi)|^2\,\mathrm d\xi \\ &\leqslant C_\varepsilon\left(\int_{\mathbb{R}^2}|\widehat{\chi_k}(\xi)-\widehat{\chi_\infty}(\xi)|^2\,\mathrm d\xi + \int_{\mathbb{R}^2}\frac{1}{1+|\xi|^2}|\widehat{\sqrt{b}\nu_k}(\xi) - \widehat{\sqrt{b}m}(\xi)|^2\right) \\ &\leqslant C_\varepsilon\left(\left\lVert\chi_k - \chi_\infty\right\rVert_{L^2} + \left\lVert\sqrt{b}\nu_k-\sqrt{b}m\right\rVert_{H^{-1}}\right) \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} 0\end{aligned}$$ since $b$ is smooth. Therefore $$\begin{aligned} \underset{k\rightarrow+\infty}{\limsup}\left\lVert\nabla g\ast(\sqrt{b}\nu_k - \chi_k+\chi_\infty - \sqrt{b}m)\right\rVert^2_{L^2} \leqslant C_{b,R}\varepsilon^2\end{aligned}$$ for all $\varepsilon> 0$ so $$\label{cv_grad_L_2_claim_cv_H_1_loc} \nabla g\ast(\sqrt{b}\nu_k - \chi_k+\chi_\infty - \sqrt{b}m) \overset{L^2}{\mathop{\longrightarrow}\limits_{k\rightarrow+\infty}} 0.$$ By Hardy-Littlewood-Sobolev inequality (see for example [@BahouriCheminDanchin Theorem 1.7]), $\nabla g\ast\rho \in L^p$ for all $2 < p < +\infty$ so $$\begin{aligned} \left(\int_{\mathbb{R}^2} \sqrt{b}\nu_k-\int_{\mathbb{R}^2} \sqrt{b}m\right)\nabla g\ast\rho \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} 0 \; \text{in} \; L^2(B(0,R)).\end{aligned}$$ Combining the upper limit with [\[decomposition_claim_cv_H\_1_loc\]](#decomposition_claim_cv_H_1_loc){reference-type="eqref" reference="decomposition_claim_cv_H_1_loc"} and [\[cv_grad_L\_2_claim_cv_H\_1_loc\]](#cv_grad_L_2_claim_cv_H_1_loc){reference-type="eqref" reference="cv_grad_L_2_claim_cv_H_1_loc"} we get that $$\nabla g\ast(\sqrt{b}\nu_k) \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} \nabla g\ast(\sqrt{b}m) \; \text{in} \; L^2(B(0,R)).$$ Now, by convolution inequality, we have $$\label{convergence_g_ast_nu_L_2} \left\lVert g\ast[\sqrt{b}\nu_k]\right\rVert_{L^2(B(0,R))} \leqslant C_b\left\lVert g\right\rVert_{L^2(B(0,2R))}\left\lVert\nu_k\right\rVert_{L^1} \leqslant C_b \left\lVert g\right\rVert_{L^2(B(0,2R))}$$ so $(g\ast[\sqrt{b}\nu_k])$ is bounded in $H^1(B(0,R))$ which is compactly embedded in $L^2(B(0,R))$. Therefore by [\[cv_grad_L\_2_claim_cv_H\_1_loc\]](#cv_grad_L_2_claim_cv_H_1_loc){reference-type="eqref" reference="cv_grad_L_2_claim_cv_H_1_loc"}, up to extraction, $(g\ast[\sqrt{b}\nu_k])$ converges to $g\ast[\sqrt{b}m] + C$ where $C$ is a constant. If $x_0 \in B(0,R)$ is at a positive distance from the supports of $\nu_k$ and $m$ then $g(x_0-\cdot)$ is continuous on the supports of $\nu_k$ and $m$ and therefore $$g\ast[\sqrt{b}\nu_k](x_0) \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} g\ast[\sqrt{b}m](x_0)$$ by dominated convergence theorem. It follows that $C = 0$, thus $$g\ast[\sqrt{b}\nu_k] \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} g\ast[\sqrt{b}m] \; \text{in} \; H^1(B(0,R)).$$ We recall that since $b$ is smooth, $$\begin{aligned} \sqrt{b}\nu_k \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} \sqrt{b}m \; \text{in} \; H^{-1}.\end{aligned}$$ Moreover, $m \in H^{-1}$ with compact support and $u\cdot\nabla \sqrt{b} \in W^{1,\infty}$ so it follows by Decomposition [\[decomposition_I\_3_1\]](#decomposition_I_3_1){reference-type="ref" reference="decomposition_I_3_1"} that $$\begin{aligned} I_{3,1} \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} 0.\end{aligned}$$ Since $\nabla g$ is antisymmetric we can write $$\begin{aligned} I_{3,2} =& \frac{1}{2}\iint_{\mathbb{R}^2\times\mathbb{R}^2}H_u(x,y)\,\mathrm d(\sqrt{b}\nu_k)(x)\,\mathrm d(\sqrt{b}\nu_k)(y)\\ &- \frac{1}{2}\iint_{\mathbb{R}^2\times\mathbb{R}^2}H_u(x,y)\,\mathrm d(\sqrt{b}m)(x)\,\mathrm d(\sqrt{b}m)(y)\end{aligned}$$ where $$\begin{aligned} H_u(x,y) = \frac{1}{2}(u(x)-u(y))\cdot\nabla g(x-y).\end{aligned}$$ We recall that $(\sqrt{b}\nu_k)$ is a sequence of nonnegative functions with supports in $B(0,R)$ converging to $\sqrt{b}m$ in $H^{-1}$ and for the weak-$\ast$ topology of measures with finite mass. Moreover, since $u$ is Lipschitz, $H_u$ is continuous outside of the diagonal and bounded. Therefore we can use Delort's argument (see [@Delort Proposition 1.2.6] or [@Schochet2 Inequalities (3.4) and (3.5)]) to prove that $$\begin{aligned} I_{3,2} \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} 0.\end{aligned}$$ Finally we write $$\begin{aligned} I_{3,3} =& \int_{\mathbb{R}^2} u(x)\cdot\left(\int_{\mathbb{R}^2}\nabla_x S_b(x,y)\,\mathrm d\nu_k(y)\right) \,\mathrm d(\nu_k-m)(x)\\ &+\int_{\mathbb{R}^2}u(x)\cdot\left(\int_{\mathbb{R}^2} \nabla_x S_b(x,y) \,\mathrm dm(x)\right) \,\mathrm d(\nu_k-m)(y).\end{aligned}$$ By Proposition [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"} $u(x)\cdot\nabla_x S_b(x,y)$ is locally Hölder with respect to both variables and therefore since $\nu_k\otimes\nu_k - m\otimes m$ has compact support we have that $I_{3,3} \mathop{\longrightarrow}\limits_{k\rightarrow+\infty} 0$. ◻ It follows from Claim [Claim 40](#claim_delort_type_limit_lake){reference-type="ref" reference="claim_delort_type_limit_lake"} that $$\begin{aligned} |T_1| \leqslant C_b\left\lVert u\right\rVert_{W^{1,\infty}}\int_{\mathbb{R}^2} |\nabla H_{N,\eta}|^2.\end{aligned}$$ Hence by Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"} we get $$\label{gronwall_bound_T1} \begin{aligned} |T_1| \leqslant& C_b\left\lVert u\right\rVert_{W^{1,\infty}}\bigg(|\mathcal{F}_b(Q_N,\omega)| + \frac{g(\eta)}{N} + I(Q_N)(\eta+N^{-1}) \\&+ \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\eta)\eta\bigg). \end{aligned}$$ Now let us split $T_2$ in three parts: $$\begin{aligned} T_2 =& -\frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2} \big(u(x)\cdot \nabla_x g_b(x,y) \\ &+ u(y)\cdot \nabla_x g_b(y,x)\big)\omega(x)\,\mathrm dx\,\mathrm d(\delta_{q_i}-\widetilde{\delta}^{(\eta)}_{q_i})(y) \\ =& -\frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2}\big(u(x)\cdot \nabla \sqrt{b}(x) \sqrt{b(y)} \\ &+ u(y)\cdot \nabla \sqrt{b}(y) \sqrt{b(x)}\big)g(x-y)\omega(x)\,\mathrm dx\,\mathrm d(\delta_{q_i}-\widetilde{\delta}^{(\eta)}_{q_i})(y) \\ &- \frac{1}{N}\sum_{i=1}^N \iint_{\mathbb{R}^2\times\mathbb{R}^2} \sqrt{b(x)b(y)}(u(x) - u(y))\\ &\cdot \nabla g(x-y) \omega(x)\,\mathrm dx\,\mathrm d(\delta_{q_i}-\widetilde{\delta}^{(\eta)}_{q_i})(y) \\ &- \frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2}\big(u(x)\cdot \nabla_x S_b(x,y) \\ &+ u(y)\cdot \nabla_x S_b(y,x)\big) \omega(x)\,\mathrm dx\,\mathrm d(\delta_{q_i}-\widetilde{\delta}^{(\eta)}_{q_i})(y) \\ =:& -(T_{2,1} + T_{2,2} + T_{2,3}).\end{aligned}$$ We will bound the three terms separately: *Claim 41*. There exists $0 < s < 1$ such that $$|T_{2,1}| \leqslant C_b\left\lVert u\right\rVert_{W^{1,\infty}}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}(1+I(Q_N))\eta^s.$$ *Proof of the claim.* Since $\widetilde{\delta}^{(\eta)}_{q_i}$ is a probability measure, we can write $$\begin{aligned} T_{2,1} =& \frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2} \nabla \sqrt{b}(x)\cdot u(x) \omega(x)(\sqrt{b(q_i)}g(x-q_i) \\ &- \sqrt{b(y)}g(x-y))\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx \\ &+ \frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2} \sqrt{b(x)}\omega(x)(\nabla \sqrt{b}(q_i)\cdot u(q_i)g(x-q_i) \\ &- \nabla \sqrt{b}(y)\cdot u(y)g(x-y))\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx \\ =& \frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2} (\nabla \sqrt{b}\cdot u \omega)(x)(\sqrt{b(q_i)}-\sqrt{b(y)})g(x-y)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx\\ &+ \frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2} (\nabla \sqrt{b}\cdot u \omega)(x)\sqrt{b(q_i)}\\ &\times(g(x-q_i) - g(x-y))\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx\\ &+ \frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2} \sqrt{b(x)}\omega(x)(\nabla \sqrt{b}(q_i)\cdot u(q_i) \\ &- \nabla \sqrt{b}(y)\cdot u(y))g(x-y)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx \\ &+ \frac{1}{N}\sum_{i=1}^N\iint_{\mathbb{R}^2\times\mathbb{R}^2} \sqrt{b(x)}\omega(x)\nabla \sqrt{b}(q_i)\cdot u(q_i)\\ &\times(g(x-q_i) -g(x-y))\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y)\,\mathrm dx.\end{aligned}$$ For the first integral, we use the Lipschitz regularity of $\sqrt{b}$ to bound $$\begin{gathered} \bigg|\iint_{\mathbb{R}^2\times\mathbb{R}^2} (\nabla \sqrt{b}\cdot u \omega)(x) (\sqrt{b(q_i)}-\sqrt{b(y)})g(x-y)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx\bigg| \\ \leqslant C_b\eta \iint_{\mathbb{R}^2\times\mathbb{R}^2} |(\nabla \sqrt{b}\cdot u \omega)(x)g(x-y)|\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y)\,\mathrm dx.\end{gathered}$$ Moreover for $y \in \partial B(q_i,\eta)$, we have $$\begin{aligned} \int_{\mathbb{R}^2} &|(\nabla \sqrt{b}\cdot u \omega)(x)g(x-y)|\,\mathrm dx \\ \leqslant& \int_{B(y,1)} |(\nabla \sqrt{b}\cdot u \omega)(x)g(x-y)|\,\mathrm dx \\ &+ \int_{B(y,1)^c} |(\nabla \sqrt{b}\cdot u \omega)(x) g(x-y)|\,\mathrm dx \\ \leqslant& \left\lVert\nabla \sqrt{b}\cdot u \omega\right\rVert_{L^\infty}\left\lVert g\right\rVert_{L^1(B(0,1))} +\int_{B(y,1)^c} |(\nabla \sqrt{b}\cdot u \omega)(x)|(|x|+|y|)\,\mathrm dx \\ \leqslant& C_b\left\lVert u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^\infty}(1+|q_i|)\end{aligned}$$ since $b$ satisfies Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}. Therefore $$\begin{gathered} \bigg|\iint_{\mathbb{R}^2\times\mathbb{R}^2} (\nabla \sqrt{b}\cdot u \omega)(x) (\sqrt{b(q_i)}-\sqrt{b(y)})g(x-y)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx\bigg| \\ \leqslant C_b\left\lVert u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^\infty}(1+|q_i|)\eta.\end{gathered}$$ The third integral can be bounded in the same way: $$\begin{aligned} \bigg|\iint_{\mathbb{R}^2\times\mathbb{R}^2} &\sqrt{b(x)}\omega(x)(\nabla \sqrt{b}(q_i)\cdot u(q_i)- \nabla \sqrt{b}(y)\cdot u(y))g(x-y)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx\bigg| \\ \leqslant& C_b\left\lVert u\right\rVert_{W^{1,\infty}}\eta \iint_{\mathbb{R}^2\times\mathbb{R}^2} |\sqrt{b(x)}\omega(x)g(x-y)|\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx \\ \leqslant& C_b\left\lVert u\right\rVert_{W^{1,\infty}}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}(1+|q_i|)\eta.\end{aligned}$$ Summing over $N$ we get that both the first and the third line can be bounded by $$\label{first_bound_T21} C_b\left\lVert u\right\rVert_{W^{1,\infty}}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}(1+I(Q_N))\eta.$$ Now the second integral is equal to $$\begin{aligned} \frac{1}{N}\sum_{i=1}^N\sqrt{b(q_i)}\int_{\mathbb{R}^2}(g\ast(\nabla \sqrt{b}\cdot u \omega)(q_i) - g\ast(\nabla \sqrt{b}\cdot u \omega)(y))\,\mathrm d\delta_{q_i}^{(\eta)}(y) \\\end{aligned}$$ and thus by Morrey's inequality (see [@Brezis Theorem 9.12]) its absolute value can be bounded by $$C_{b,p}\eta^{1-\frac{2}{p}}\left\lVert\nabla g\ast(\nabla \sqrt{b}\cdot u \omega)\right\rVert_{L^p}$$ for any finite $p > 2$. The fourth integral can be bounded in the same way by $$C_{b,p}\eta^{1-\frac{2}{p}}\left\lVert\nabla g\ast(\sqrt{b}\omega)\right\rVert_{L^p}.$$ Using Hardy-Littlewood-Sobolev inequality (see for example [@BahouriCheminDanchin Theorem 1.7]) we have $$\label{second_bound_T21} C_b\eta^{1-\frac{2}{p}}\left\lVert\nabla g\ast(\sqrt{b}\omega)\right\rVert_{L^p} \leqslant C_b\eta^{1-\frac{2}{p}}\left\lVert\omega\right\rVert_{L^{\frac{2p}{p+2}}}.$$ Combining [\[first_bound_T21\]](#first_bound_T21){reference-type="eqref" reference="first_bound_T21"} and [\[second_bound_T21\]](#second_bound_T21){reference-type="eqref" reference="second_bound_T21"} we get that $$|T_{2,1}| \leqslant C_b\left\lVert u\right\rVert_{W^{1,\infty}}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}(1+I(Q_N))\eta^s$$ for some $0 < s < 1$. ◻ Now we bound $T_{2,2}$: *Claim 42*. $$|T_{2,2}| \leqslant C_b\left\lVert\nabla u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^1\cap L^\infty}\eta.$$ *Proof of the claim.* Let us recall that $$\widetilde{\delta}^{(\eta)}_{q} = m_b(q,\eta)\frac{\,\mathrm d\delta_{q}^{(\eta)}}{\sqrt{b}}$$ and thus $$\begin{aligned} &\iint_{\mathbb{R}^2\times\mathbb{R}^2} \sqrt{b(x)b(y)}(u(x) - u(y))\cdot \nabla g(x-y) \omega(x)\,\mathrm dx \,\mathrm d(\delta_{q_i}-\widetilde{\delta}^{(\eta)}_{q_i})(y) \\ =& m_b(q_i,\eta)\iint_{\mathbb{R}^2\times\mathbb{R}^2} \sqrt{b(x)}(u(x) - u(y))\cdot \nabla g(x-y) \omega(x)\,\mathrm dx \,\mathrm d(\delta_{q_i}-\delta_{q_i}^{(\eta)})(y) \\ &+ \left(1- \frac{m_b(q_i,\eta)}{\sqrt{b(q_i)}}\right)\int_{\mathbb{R}^2} \sqrt{b(x)b(q_i)}(u(x)-u(q_i))\cdot \nabla g(x-q_i)\omega(x)\,\mathrm dx.\end{aligned}$$ The first integral is exactly the term defined in [@NguyenRosenzweigSerfaty Equation (4.10)] with $s=0$ and $m=0$ (remark that we can choose $m=0$ since no extension procedure is needed for $s=0$ and $d=2$, for more details we refer to the introduction of [@NguyenRosenzweigSerfaty Section 4]). It can be bounded by the right hand side of [@NguyenRosenzweigSerfaty Equation (4.24)] : $$\begin{aligned} C\left\lVert\nabla u\right\rVert_{L^\infty}\left\lVert|\nabla|^{-1}(\sqrt{b}\omega)\right\rVert_{L^\infty}\eta &\leqslant C_b\left\lVert\nabla u\right\rVert_{L^\infty}\left\lVert|\nabla|^{-1}\omega\right\rVert_{L^\infty}\eta \\ &\leqslant C_b\left\lVert\nabla u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^1\cap L^\infty}\eta.\end{aligned}$$ A proof of the last inequality can be found for example in [@Iftimie Lemma 1]. Now by [\[estimee_m\_b\]](#estimee_m_b){reference-type="eqref" reference="estimee_m_b"} and the Lipschitz regularity of $u$ we can bound the second line by $$C_b\left\lVert\nabla u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^1}\eta.$$ Combining the two upper equations we get $$|T_{2,2}| \leqslant C_b\left\lVert\nabla u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^1\cap L^\infty}\eta.$$ ◻ *Claim 43*. $$|T_{2,3}| \leqslant C_{b,s}\left\lVert u\right\rVert_{W^{1,\infty}}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)}(1+I(Q_N))\eta^s.$$ *Proof of the claim.* We write $T_{2,3}$ as $$\begin{aligned} T_{2,3} =& \frac{1}{N}\sum_{i=1}^N\bigg(\iint_{\mathbb{R}^2\times\mathbb{R}^2} \omega(x)u(x)\cdot(\nabla_x S_b(x,q_i) - \nabla_x S_b(x,y))\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx \\ &+ \iint_{\mathbb{R}^2\times\mathbb{R}^2} \omega(x)(u(q_i) - u(y))\cdot \nabla_x S_b(q_i,x) \,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx \\ &+ \iint_{\mathbb{R}^2\times\mathbb{R}^2} \omega(x) u(y)\cdot(\nabla_x S_b(q_i,x) - \nabla_x S_b(y,x))\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \,\mathrm dx\bigg).\end{aligned}$$ Using Claims $(1)$ and $(2)$ of Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"}, we get that for some $0 < s < 1$, $$\begin{aligned} |T_{2,3}| \leqslant& \frac{1}{N}\sum_{i=1}^N\bigg(C_{b,s}\left\lVert u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^1}(1+|q_i|)\eta^s \\ &+ \left\lVert\nabla u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)}\eta \\ &+ \left\lVert u\right\rVert_{L^\infty}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)}\eta^s\bigg) \\ \leqslant& C_{b,s}\left\lVert u\right\rVert_{W^{1,\infty}}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)}(1+I(Q_N))\eta^s.\end{aligned}$$ ◻ Combining Claims [Claim 41](#bound_T21){reference-type="ref" reference="bound_T21"}, [Claim 42](#bound_T22){reference-type="ref" reference="bound_T22"} and [Claim 43](#bound_T23){reference-type="ref" reference="bound_T23"} we get that $$\label{gronwall_bound_T2} |T_2| \leqslant C_{b,s}\left\lVert u\right\rVert_{W^{1,\infty}}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}(1+I(Q_N))\eta^s.$$ Now let us write $T_3$ as $$\begin{aligned} T_3 =& \frac{1}{N^2}\sum_{1\leqslant i,j \leqslant N} \iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} u(x)\cdot \nabla_x g_b(x,y) \\ &(\,\mathrm d\delta_{q_i}(x) \,\mathrm d\delta_{q_j}(y) - \,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_j}(y)) \\ =&\frac{1}{N^2}\sum_{1\leqslant i,j \leqslant N} \iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} u(x)\cdot \nabla \sqrt{b}(x) \sqrt{b(y)}g(x-y) \\ &(\,\mathrm d\delta_{q_i}(x) \,\mathrm d\delta_{q_j}(y) - \,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_j}(y)) \\ &+ \frac{1}{N^2}\sum_{1\leqslant i,j \leqslant N} \iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta}\sqrt{b(x)b(y)}u(x)\cdot \nabla g(x-y) \\ &(\,\mathrm d\delta_{q_i}(x) \,\mathrm d\delta_{q_j}(y) - \,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_j}(y)) \\ &+ \frac{1}{N^2}\sum_{1\leqslant i,j \leqslant N} \iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta}u(x)\cdot \nabla_x S_b(x,y) \\ &(\,\mathrm d\delta_{q_i}(x) \,\mathrm d\delta_{q_j}(y) - \,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_j}(y)) \\ =:& T_{3,1} + T_{3,2} + T_{3,3}.\end{aligned}$$ We bound the first term: *Claim 44*. $$\begin{gathered} |T_{3,1}| \leqslant C_b\left\lVert u\right\rVert_{L^\infty}|\mathcal{F}_b(Q_N,\omega)| + C_b\left\lVert u\right\rVert_{W^{1,\infty}}\bigg(\frac{g(\eta)}{N} \\ + I(Q_N)(\eta+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\eta)\eta\bigg).\end{gathered}$$ *Proof of the claim.* We write $$\begin{aligned} T_{3,1} =& -\frac{1}{N^2} \sum_{i=1}^N \iint_{\mathbb{R}^2\times\mathbb{R}^2}u(x)\cdot \nabla \sqrt{b}(x) \sqrt{b(y)}g(x-y) \,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(y) \\ &+ \frac{1}{N^2}\sum_{1\leqslant i \neq j \leqslant N}\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta}u(x)\cdot \nabla \sqrt{b}(x) \sqrt{b(y)}g(x-y) \\ &(\,\mathrm d\delta_{q_i}(x) \,\mathrm d\delta_{q_j}(y) - \,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_j}(y)) \\ =:& T_{3,1,1} + T_{3,1,2}.\end{aligned}$$ By the definition of $\widetilde{\delta}_{q_i}^{(\eta)}$ [\[definition_delta_tilde_q\]](#definition_delta_tilde_q){reference-type="eqref" reference="definition_delta_tilde_q"} we have $$\begin{aligned} T_{3,1,1} =& -\frac{1}{N^2}\sum_{i=1}^N m_b(q_i,\eta)^2\iint_{\mathbb{R}^2\times\mathbb{R}^2}u(x)\cdot \frac{\nabla \sqrt{b}(x)}{\sqrt{b(x)}}g(x-y)\,\mathrm d\delta_{q_i}^{(\eta)}(x)\,\mathrm d\delta_{q_i}^{(\eta)}(y) \\ =& -\frac{1}{N^2}\sum_{i=1}^N m_b(q_i,\eta)^2 \int_{\mathbb{R}^2} \frac{u(x)\cdot\nabla \sqrt{b}(x)}{\sqrt{b}(x)}g^{(\eta)}(x-q_i)\,\mathrm d\delta_{q_i}^{(\eta)}(x)\end{aligned}$$ by Claim [\[egalite_convolution_g\_eta\]](#egalite_convolution_g_eta){reference-type="eqref" reference="egalite_convolution_g_eta"}. It follows by Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"} that $$\label{boundT311} |T_{3,1,1}| \leqslant\frac{C_b\left\lVert u\right\rVert_{L^\infty}g(\eta)}{N}.$$ Now we write $$\begin{aligned} T_{3,1,2} =& \frac{1}{N^2}\sum_{1\leqslant i \neq j \leqslant N}\bigg((u\cdot \nabla \sqrt{b})(q_i) \sqrt{b(q_j)}g(q_i-q_j) \\ &- \iint_{\mathbb{R}^2\times\mathbb{R}^2}(u\cdot \nabla \sqrt{b})(x)\sqrt{b(y)}g(x-y)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_j}(y) \bigg)\\ =& \frac{1}{N^2}\sum_{1\leqslant i \neq j \leqslant N}\bigg((u\cdot \nabla \sqrt{b})(q_i) \sqrt{b(q_j)}g(q_i-q_j) \\ &- m_b(q_j,\eta)\int_{\mathbb{R}^2}(u\cdot \nabla \sqrt{b})(x)g^{(\eta)}(x-q_j)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x)\bigg)\end{aligned}$$ by the definition of $\widetilde{\delta}_{q_i}^{(\eta)}$ [\[definition_delta_tilde_q\]](#definition_delta_tilde_q){reference-type="eqref" reference="definition_delta_tilde_q"} and Claim [\[egalite_convolution_g\_eta\]](#egalite_convolution_g_eta){reference-type="eqref" reference="egalite_convolution_g_eta"}. Now, $$\begin{aligned} T_{3,1,2} =& \frac{1}{N^2}\sum_{1\leqslant i \neq j \leqslant N}(u\cdot \nabla \sqrt{b})(q_i) \sqrt{b(q_j)}(g(q_i-q_j)-g^{(\eta)}(q_i-q_j)) \\ &+ \frac{1}{N^2}\sum_{1\leqslant i \neq j \leqslant N}(u\cdot \nabla \sqrt{b})(q_i) \sqrt{b(q_j)}\\ &\times\int_{\mathbb{R}^2}(g^{(\eta)}(q_i-q_j)-g^{(\eta)}(x-q_j))\,\mathrm d\delta^{(\eta)}_{q_i}(x) \\ &+ \frac{1}{N^2}\sum_{1\leqslant i \neq j \leqslant N}(u\cdot \nabla \sqrt{b})(q_i) \sqrt{b(q_j)}\int_{\mathbb{R}^2}g^{(\eta)}(x-q_j)\,\mathrm d(\delta^{(\eta)}_{q_i}-\widetilde{\delta}^{(\eta)}_{q_i})(x) \\ &+ \frac{1}{N^2}\sum_{1\leqslant i \neq j \leqslant N}\sqrt{b(q_j)}\\ &\times\int_{\mathbb{R}^2}((u\cdot \nabla \sqrt{b})(q_i) - (u\cdot \nabla \sqrt{b})(x))g^{(\eta)}(x-q_j)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x) \\ &+ \frac{1}{N^2}\sum_{1\leqslant i \neq j \leqslant N}(\sqrt{b(q_j)}-m_b(q_j,\eta))\\ &\times\int_{\mathbb{R}^2}(u\cdot \nabla \sqrt{b})(x)g^{(\eta)}(x-q_j)\,\mathrm d\widetilde{\delta}^{(\eta)}_{q_i}(x) \\ =&: S_1 + S_2 + S_3 + S_4 + S_5.\end{aligned}$$ Since $g - g^{(\eta)}$ is nonnegative we can bound $$\label{bound_S_1} \begin{aligned} |S_1| \leqslant& C_b\left\lVert u\right\rVert_{L^\infty}\frac{1}{N^2} \sum_{1\leqslant i \neq j \leqslant N}(g(q_i-q_j)-g^{(\eta)}(q_i-q_j)) \\ \leqslant& C_b\left\lVert u\right\rVert_{L^\infty}|\mathcal{F}_b(Q_N,\omega)| + C_b\left\lVert u\right\rVert_{L^\infty}\bigg(\frac{g(\eta)}{N} \\ &+ I(Q_N)(\eta+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\eta)\eta\bigg) \end{aligned}$$ by Proposition [Proposition 29](#proposition_pre_coerc){reference-type="ref" reference="proposition_pre_coerc"}. Now remark that if $|q_i - q_j| \geqslant 2\eta$ and $x \in \partial B(q_i,\eta)$, $$\begin{aligned} |q_j - x| \geqslant|q_i-q_j| - |q_i-x| \geqslant 2\eta - \eta \geqslant\eta\end{aligned}$$ and it follows by Claim [\[egalite_convolution_g\_eta\]](#egalite_convolution_g_eta){reference-type="eqref" reference="egalite_convolution_g_eta"} that $$\begin{aligned} \int_{\mathbb{R}^2}g^{(\eta)}(x-q_j)\,\mathrm d\delta^{(\eta)}_{q_i}(x) &= \int_{\mathbb{R}^2}g(x-q_j)\,\mathrm d\delta^{(\eta)}_{q_i}(x) \\ &= g^{(\eta)}(q_i-q_j).\end{aligned}$$ Hence we can write $$S_2 = \frac{1}{N^2}\underset{|q_i - q_j| \leqslant 2\eta}{\sum_{1\leqslant i \neq j \leqslant N}}(u\cdot \nabla \sqrt{b})(q_i) \sqrt{b(q_j)}\int_{\mathbb{R}^2}(g^{(\eta)}(q_i-q_j)-g^{(\eta)}(x-q_j))\,\mathrm d\delta^{(\eta)}_{q_i}(x).$$ Notice that if $|q_i - q_j| \leqslant 2\eta$ and $x \in \partial B(q_i,\eta)$, then $$|g^{(\eta)}(q_i-q_j)-g^{(\eta)}(x-q_j)| \leqslant\left\lVert\nabla g^{(\eta)}\right\rVert_{L^\infty}\eta = C\eta^{-1}\eta \leqslant C.$$ Therefore, $$\label{bound_S_2} \begin{aligned} |S_2| \leqslant& \frac{C_b \left\lVert u\right\rVert_{L^\infty}}{N^2} |\{(q_i,q_j) ; |q_i - q_j| \leqslant 2\eta\}| \\ \leqslant& C_b \left\lVert u\right\rVert_{L^\infty}|\mathcal{F}_b(Q_N,\omega)| + C_b \left\lVert u\right\rVert_{L^\infty}\bigg(\frac{g(2\eta)}{N} \\ &+ I(Q_N)(2\eta+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(2\eta)2\eta\bigg) \end{aligned}$$ by Corollary [Corollary 30](#corollary_counting){reference-type="ref" reference="corollary_counting"} applied to $\varepsilon= 2\eta$. By definition of $\widetilde{\delta}_{q_i}^{(\eta)}$ [\[definition_delta_tilde_q\]](#definition_delta_tilde_q){reference-type="eqref" reference="definition_delta_tilde_q"} we can write $$\begin{aligned} S_3 =& \frac{1}{N^2}\sum_{1\leqslant i \neq j \leqslant N}(u\cdot \nabla \sqrt{b})(q_i) \sqrt{b(q_j)} \\ &\times\int_{\mathbb{R}^2}g^{(\eta)}(x-q_j)\left(1-\frac{m_b(q_i,\eta)}{\sqrt{b(x)}}\right)\,\mathrm d\delta^{(\eta)}_{q_i}(x)\end{aligned}$$ and therefore $$|S_3| \leqslant\frac{C_b\left\lVert u\right\rVert_{L^\infty}g(\eta)}{N}\sum_{i=1}^N\int_{\mathbb{R}^2}\left|\frac{m_b(q_i,\eta)}{\sqrt{b(x)}}-1\right|\,\mathrm d\delta^{(\eta)}_{q_i}(x).$$ For $x \in \partial B(q_i,\eta)$, we have $$\begin{aligned} \left|\frac{m_b(q_i,\eta)}{\sqrt{b(x)}}-1\right| &\leqslant C_b\left|m_b(q_i,\eta)^{-1}-\frac{1}{\sqrt{b(x)}}\right| \\ &\leqslant C_b\left|\int_{\mathbb{R}^2}\frac{\,\mathrm d\delta_{q_i}^{(\eta)}(y)}{\sqrt{b(y)}}-\frac{1}{\sqrt{b(x)}}\right| \\ &\leqslant C_b\eta\end{aligned}$$ since $b$ is Lipschitz by Assumption [Assumption 5](#assumption_b){reference-type="ref" reference="assumption_b"}. It follows that $$\label{bound_S_3} |S_3| \leqslant C_b\left\lVert u\right\rVert_{L^\infty}g(\eta)\eta.$$ Now by regularity of $u$, $b$ and Proposition [Proposition 21](#proposition_regularisation){reference-type="ref" reference="proposition_regularisation"}, we have $$\begin{aligned} |S_4| + |S_5| \leqslant C_b\left\lVert u\right\rVert_{W^{1,\infty}}\eta g(\eta).\end{aligned}$$ Combining the upper inequality with [\[boundT311\]](#boundT311){reference-type="eqref" reference="boundT311"}, [\[bound_S\_1\]](#bound_S_1){reference-type="eqref" reference="bound_S_1"}, [\[bound_S\_2\]](#bound_S_2){reference-type="eqref" reference="bound_S_2"} and [\[bound_S\_3\]](#bound_S_3){reference-type="eqref" reference="bound_S_3"} we obtain Claim [Claim 44](#gronwall_bound_T31){reference-type="ref" reference="gronwall_bound_T31"}. ◻ For the third term we have the following bound: *Claim 45*. For $s$ small enough, we have $$|T_{3,3}| \leqslant C_{b,s}\left\lVert u\right\rVert_{W^{1,\infty}}(1+I(Q_N))\eta^s.$$ *Proof.* We write $$\begin{aligned} T_{3,3} =& \frac{1}{N^2}\sum_{1 \leqslant i,j \leqslant N} u(q_i)\cdot \nabla_x S_b(q_i,q_j) \\ &- \frac{1}{N^2}\sum_{1 \leqslant i,j \leqslant N}\iint_{\mathbb{R}^2\times\mathbb{R}^2}u(x) \cdot \nabla_x S_b(x,y)\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(x)\,\mathrm d\widetilde{\delta}_{q_j}^{(\eta)}(y) \\ =& \frac{1}{N^2}\sum_{1 \leqslant i,j \leqslant N} \iint_{\mathbb{R}^2\times\mathbb{R}^2} \bigg(u(q_i)\cdot \nabla_x S_b(q_i,q_j) - u(q_i)\cdot \nabla_x S_b(q_i,y) \\ &+ u(q_i)\cdot \nabla_x S_b(q_i,y) - u(q_i)\cdot \nabla_x S_b(x,y) \\ &+ u(q_i) \cdot \nabla_x S_b(x,y) - u(x) \cdot \nabla_x S_b(x,y)\bigg)\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(x)\,\mathrm d\widetilde{\delta}_{q_j}^{(\eta)}(y).\end{aligned}$$ Therefore, $$\begin{aligned} |T_{3,3}| \leqslant& \frac{1}{N^2}\sum_{1 \leqslant i,j \leqslant N} \left\lVert u\right\rVert_{L^\infty}|\nabla_x S_b(q_i,\cdot)|_{\mathcal{C}^{0,s}(B(q_j,1))}\eta^s \\ &+ \frac{1}{N^2}\sum_{1 \leqslant i,j \leqslant N} \left\lVert u\right\rVert_{L^\infty}\eta^s\int_{\mathbb{R}^2}|\nabla_x S_b(\cdot,y)|_{\mathcal{C}^{0,s}}\,\mathrm d\widetilde{\delta}_{q_j}^{(\eta)}(y) \\ &+ \frac{1}{N^2}\sum_{1 \leqslant i,j \leqslant N}\iint_{\mathbb{R}^2\times\mathbb{R}^2} \left\lVert u\right\rVert_{W^{1,\infty}}\eta|\nabla_x S_b(x,y)|\,\mathrm d\widetilde{\delta}_{q_i}^{(\eta)}(x)\,\mathrm d\widetilde{\delta}_{q_j}^{(\eta)}(y).\end{aligned}$$ By Proposition [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"}, for $s$ small enough we have $$\begin{aligned} |T_{3,3}| \leqslant& \frac{C_{b,s}}{N^2}\sum_{1 \leqslant i,j \leqslant N} \left\lVert u\right\rVert_{L^\infty}|(1+|q_j|)\eta^s \\ &+ \frac{C_{b,s}}{N^2}\sum_{1 \leqslant i,j \leqslant N} \left\lVert u\right\rVert_{L^\infty}\eta^s(1+|q_j|) \\ &+ \frac{C_b}{N^2}\sum_{1 \leqslant i,j \leqslant N} \left\lVert u\right\rVert_{W^{1,\infty}}\eta(1+|q_j|) \\ \leqslant& C_{b,s}\left\lVert u\right\rVert_{W^{1,\infty}}(1+I(Q_N))\eta^s.\end{aligned}$$ ◻ We are only remained to bound $T_{3,2}$: *Claim 46*. For $\varepsilon> 2\eta$ small enough, we have $$\begin{aligned} |T_{3,2}| \leqslant& \frac{C_b}{N}\left\lVert\nabla u\right\rVert_{L^\infty} + \frac{C_b\eta\left\lVert\nabla u\right\rVert_{L^\infty}}{\varepsilon} + C_b\left\lVert\nabla u\right\rVert_{L^\infty}\bigg(|\mathcal{F}_b(Q_N,\omega)| +\frac{g(\varepsilon)}{N} +\eta \\ &+ I(Q_N)(\varepsilon+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\varepsilon)\varepsilon\bigg). \end{aligned}$$ *Proof.* Let us denote $$k_u(x,y) = (u(x) - u(y))\cdot \nabla g(x-y)$$ and remark that $$\label{bound_ku} |k_u(x,y)| \leqslant C\left\lVert\nabla u\right\rVert_{L^\infty}.$$ Since $\nabla g$ is antisymmetric we can write $T_{3,2}$ as $$\begin{gathered} T_{3,2} = \\ \frac{1}{2N^2}\sum_{1\leqslant i,j \leqslant N} \iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta}\sqrt{b(x)b(y)}k_u(x,y)\,\mathrm d(\delta_{q_i} + \widetilde{\delta}^{(\eta)}_{q_i})(x)\,\mathrm d(\delta_{q_j} - \widetilde{\delta}^{(\eta)}_{q_j})(y).\end{gathered}$$ Using the definition of $\widetilde{\delta}^{(\eta)}_{q_i}$ [\[definition_delta_tilde_q\]](#definition_delta_tilde_q){reference-type="eqref" reference="definition_delta_tilde_q"} we can write $$\begin{aligned} \,\mathrm d(\delta_{q_i} + \widetilde{\delta}^{(\eta)}_{q_i})(x)&\,\mathrm d(\delta_{q_j} - \widetilde{\delta}^{(\eta)}_{q_j})(y) \\ =& \,\mathrm d\bigg(\delta_{q_i} + \frac{m_b(q_i,\eta)}{\sqrt{b}}\delta_{q_i}^{(\eta)}\bigg)(x)\,\mathrm d\bigg(\delta_{q_j} - \frac{m_b(q_j,\eta)}{\sqrt{b}}\delta_{q_j}^{(\eta)}\bigg)(y) \\ =& \frac{m_b(q_i,\eta)m_b(q_j,\eta)}{\sqrt{b(x)b(y)}}\,\mathrm d(\delta_{q_i} + \delta_{q_i}^{(\eta)})(x)\,\mathrm d(\delta_{q_j} - \delta_{q_j}^{(\eta)})(y) \\ &+ \left(1-\frac{m_b(q_i,\eta)m_b(q_j,\eta)}{\sqrt{b(q_i)b(q_j)}}\right)\,\mathrm d\delta_{q_i}(x)\,\mathrm d\delta_{q_j}(y) \\ &+ \frac{m_b(q_i,\eta)}{\sqrt{b(q_i)}}\left(1-\frac{m_b(q_j,\eta)}{\sqrt{b(q_j)}}\right)\,\mathrm d\delta_{q_i}^{(\eta)}(x)\,\mathrm d\delta_{q_j}(y) \\\\ &+ \frac{m_b(q_j,\eta)}{\sqrt{b(y)}}\left(\frac{m_b(q_i,\eta)}{\sqrt{b(q_i)}}-1\right)\,\mathrm d\delta_{q_i}(x)\,\mathrm d\delta_{q_j}^{(\eta)}(y).\end{aligned}$$ We will use some inequalities proved in [@NguyenRosenzweigSerfaty] and Corollary [Corollary 30](#corollary_counting){reference-type="ref" reference="corollary_counting"} to control the first line, but let us begin by controling the three last remainders. Using the bound [\[bound_ku\]](#bound_ku){reference-type="eqref" reference="bound_ku"} and [\[estimee_m\_b\]](#estimee_m_b){reference-type="eqref" reference="estimee_m_b"} we can bound $$\begin{aligned} T_{3,2,2} :=& \frac{1}{2N^2}\sum_{1\leqslant i,j \leqslant N}\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} \sqrt{b(x)b(y)}k_u(x,y) \\ &\bigg(\left(1-\frac{m_b(q_i,\eta)m_b(q_j,\eta)}{\sqrt{b(q_i)b(q_j)}}\right)\,\mathrm d\delta_{q_i}(x)\,\mathrm d\delta_{q_j}(y) \\ &+ \frac{m_b(q_i,\eta)}{\sqrt{b(q_i)}}\left(1-\frac{m_b(q_j,\eta)}{\sqrt{b(q_j)}}\right)\,\mathrm d\delta_{q_i}^{(\eta)}(x)\,\mathrm d\delta_{q_j}(y) \\ &+ \frac{m_b(q_j,\eta)}{\sqrt{b(y)}}\left(\frac{m_b(q_i,\eta)}{\sqrt{b(q_i)}}-1\right)\,\mathrm d\delta_{q_i}(x)\,\mathrm d\delta_{q_j}^{(\eta)}(x)\bigg).\end{aligned}$$ by $$\label{gronwall_bound_T322} |T_{3,2,2}| \leqslant C_b\left\lVert\nabla u\right\rVert_{L^\infty}\eta.$$ We are remained to bound $$\begin{aligned} T_{3,2,1} :=& \frac{1}{2N^2}\sum_{1\leqslant i,j \leqslant N} m_b(q_i,\eta)m_b(q_j,\eta) \\ &\times\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} k_u(x,y) \,\mathrm d(\delta_{q_i} + \delta_{q_i}^{(\eta)})(x)\,\mathrm d(\delta_{q_j} - \delta_{q_j}^{(\eta)}(y).\end{aligned}$$ Using decomposition (4.26) and inequalities (4.27), (4.28) and (4.31) of [@NguyenRosenzweigSerfaty] with $s=0$ and $m=0$ (remark that we can choose $m=0$ since no extension procedure is needed for $s=0$ and $d=2$, for more details we refer to the introduction of [@NguyenRosenzweigSerfaty Section 4]), we get that for any small parameter $\varepsilon> 2\eta$, $$|T_{3,2,1}| \leqslant\frac{C_b}{N}\left\lVert\nabla u\right\rVert_{L^\infty} + \frac{C_b\left\lVert\nabla u\right\rVert_{L^\infty}}{N^2}|\{(q_i,q_j) ; |q_i - q_j| \leqslant\varepsilon\}| + \frac{C\eta\left\lVert\nabla u\right\rVert_{L^\infty}}{\varepsilon}.$$ Using Corollary [Corollary 30](#corollary_counting){reference-type="ref" reference="corollary_counting"}, we get that $$\label{gronwall_bound_T321} \begin{aligned} T_{3,2,1} \leqslant& \frac{C_b}{N}\left\lVert\nabla u\right\rVert_{L^\infty} + \frac{C_b\eta\left\lVert\nabla u\right\rVert_{L^\infty}}{\varepsilon} + C_b\left\lVert\nabla u\right\rVert_{L^\infty}\bigg(|\mathcal{F}_b(Q_N,\omega)| \\ &+\frac{g(\varepsilon)}{N} + I(Q_N)(\varepsilon+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\varepsilon)\varepsilon\bigg). \end{aligned}$$ And we get Claim [Claim 46](#gronwall_bound_T32){reference-type="ref" reference="gronwall_bound_T32"} by combining [\[gronwall_bound_T321\]](#gronwall_bound_T321){reference-type="eqref" reference="gronwall_bound_T321"} with [\[gronwall_bound_T322\]](#gronwall_bound_T322){reference-type="eqref" reference="gronwall_bound_T322"}. ◻ We finish the proof of Proposition [Proposition 39](#controle_terme_principal_gronwall){reference-type="ref" reference="controle_terme_principal_gronwall"} using Decomposition [\[decomposition_main_proposition\]](#decomposition_main_proposition){reference-type="eqref" reference="decomposition_main_proposition"}, Inequalities [\[gronwall_bound_T1\]](#gronwall_bound_T1){reference-type="eqref" reference="gronwall_bound_T1"}, [\[gronwall_bound_T2\]](#gronwall_bound_T2){reference-type="eqref" reference="gronwall_bound_T2"} and Claims [Claim 44](#gronwall_bound_T31){reference-type="ref" reference="gronwall_bound_T31"}, [Claim 45](#gronwall_bound_T33){reference-type="ref" reference="gronwall_bound_T33"} and [Claim 46](#gronwall_bound_T32){reference-type="ref" reference="gronwall_bound_T32"}. That gives $$\begin{aligned} \bigg|\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} &u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N\delta_{q_i}-\omega\right)^{\otimes 2}(x,y)\bigg| \\ \leqslant& C_b\left\lVert u\right\rVert_{W^{1,\infty}}\bigg(|\mathcal{F}_b(Q_N,\omega)| + \frac{g(\eta)}{N} + I(Q_N)(\eta+N^{-1}) \\ &+ \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\eta)\eta\bigg)\\ &+ C_{b,s}\left\lVert u\right\rVert_{W^{1,\infty}}\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}(1+I(Q_N))\eta^s\\ &+ C_b\left\lVert u\right\rVert_{L^\infty}\mathcal{F}_b(Q_N,\omega) + C_b\left\lVert u\right\rVert_{W^{1,\infty}}\bigg(\frac{g(\eta)}{N} \\ &+ I(Q_N)(\eta+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\eta)\eta\bigg)\\ &+ C_{b,s}\left\lVert u\right\rVert_{W^{1,\infty}}(1+I(Q_N))\eta^s \\ &+\frac{C_b}{N}\left\lVert\nabla u\right\rVert_{L^\infty} + \frac{C_b\eta\left\lVert\nabla u\right\rVert_{L^\infty}}{\varepsilon} + C_b\left\lVert\nabla u\right\rVert_{L^\infty}\bigg(|\mathcal{F}_b(Q_N,\omega)| \\ &+\frac{g(\varepsilon)}{N} + \eta + I(Q_N)(\varepsilon+N^{-1}) + \left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}g(\varepsilon)\varepsilon\bigg).\end{aligned}$$ Choosing $\varepsilon= N^{-1}$ and $\eta = N^{-2}$, and since $\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}$ is bounded by below (because $\omega$ is a probability density) we get that $$\begin{aligned} \bigg|\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} &u(x)\cdot \nabla_x g_b(x,y)\,\mathrm d\left(\frac{1}{N}\sum_{i=1}^N\delta_{q_i}-\omega\right)^{\otimes 2}(x,y)\bigg| \\ \leqslant& C_b\left\lVert u\right\rVert_{W^{1,\infty}}|\mathcal{F}_b(Q_N,\omega)| \\ &+ C_b(1+\left\lVert u\right\rVert_{W^{1,\infty}})\left\lVert\omega\right\rVert_{L^1((1+|x|)\,\mathrm dx)\cap L^\infty}(1+I(Q_N))N^{-\beta}\end{aligned}$$ for some $0 < \beta < 1$. ◻ # Mean-field limit {#section:7} In this section we prove the mean-field limit Theorem [Theorem 8](#MFL_theorem){reference-type="ref" reference="MFL_theorem"}. For this purpose let us first prove the following estimates: **Theorem 47**. *If $\omega$ is a weak solution of [\[lake_equation_vorticity\]](#lake_equation_vorticity){reference-type="eqref" reference="lake_equation_vorticity"} with initial datum $\omega_0$ (in the sense of Definition [Definition 1](#definition_weak_solution){reference-type="ref" reference="definition_weak_solution"}) that satisfies Assumption [Assumption 6](#assumption_omega){reference-type="ref" reference="assumption_omega"} and if $I_N(0)$ is bounded, there exists a constant $$A := A\left(b,T,\left\lVert u\right\rVert_{L^\infty([0,T],W^{1,\infty})},\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)},\underset{N}{\sup} \; I_N(0)\right)$$ such that for every $t \in [0,T]$, $$\label{gronwall_bound_alpha_regime} |\mathcal{F}_{b,N}(t)| \leqslant A(|\mathcal{F}_{b,N}(0)| + (1+|E_N(0)|)(N^{-\beta} + |\alpha_N - \alpha|)).$$ If $\overline{\omega}$ is a weak solution of [\[transport_equation\]](#transport_equation){reference-type="eqref" reference="transport_equation"} with initial datum $\omega_0$ (in the sense of Definition [Definition 2](#definition_weak_solution_transport){reference-type="ref" reference="definition_weak_solution_transport"}) that satisfies Assumption [Assumption 6](#assumption_omega){reference-type="ref" reference="assumption_omega"} and if $\overline{I_N}(0)$ is bounded, there exists a constant $$\begin{aligned} B :=& B\bigg(b, T,\left\lVert\overline{\omega}\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)}, \\ &\left\lVert\nabla g*\overline{\omega}\right\rVert_{L^\infty([0,T],W^{1,\infty})},\underset{N}{\sup} \;\overline{I_N}(0)\bigg)\end{aligned}$$ such that for every $t \in [0,T]$, $$\label{gronwall_bound_rescaled_regime} |\overline{\mathcal{F}}_{b,N}(t)| \leqslant B(|\overline{\mathcal{F}}_{b,N}(0)| + (1+|\overline{E_N}(0)|)(N^{-\beta} + \alpha_N^{-1})).$$* *Proof.* By Proposition [Proposition 23](#time_derivative_F_N){reference-type="ref" reference="time_derivative_F_N"}, we have that for almost every $t \in [0,T]$, $$\begin{aligned} &\frac{\,\mathrm d}{\,\mathrm dt}\mathcal{F}_{b,N}(t) \\ =& 2\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} \left(u(t,x) - \alpha\frac{\nabla^\bot b(x)}{b(x)}\right)\cdot \nabla_x g_b(x,y)\,\mathrm d(\omega(t)-\omega_N(t))^{\otimes 2}(x,y) \\ &+ 2(\alpha_N - \alpha)\iint_{(\mathbb{R}^2\times\mathbb{R}^2)\backslash\Delta} \frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y) \,\mathrm d\omega_N(t,x) \,\mathrm d(\omega(t) - \omega_N(t))(y) \\ =:& L_1 + 2(\alpha_N - \alpha)L_2.\end{aligned}$$ Using Proposition [Proposition 39](#controle_terme_principal_gronwall){reference-type="ref" reference="controle_terme_principal_gronwall"}, we have $$\begin{aligned} |L_1| \leqslant& C_b\left\lVert u-\alpha\frac{\nabla^\bot b}{b}\right\rVert_{L^\infty([0,T],W^{1,\infty})}|\mathcal{F}_b(Q_N,\omega)| \\ &+ C_b\left(1+\left\lVert u-\alpha\frac{\nabla^\bot b}{b}\right\rVert_{L^\infty([0,T],W^{1,\infty})}\right)\\ &\times\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)} (1+I_N(t))N^{-\beta}.\end{aligned}$$ By Proposition [Proposition 22](#proposition_pv_well_posed){reference-type="ref" reference="proposition_pv_well_posed"}, we have $$\begin{aligned} I_N(t) &\leqslant C_{b,T}(1+|E_N(0)| + I_N(0))\end{aligned}$$ since $(\alpha_N)$ is bounded (here we consider the case $\alpha_N \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} \alpha$). Therefore, $$\label{gronwall_alpha_regime_L1} \begin{aligned} |L_1| \leqslant& C_b\big(1+\left\lVert u\right\rVert_{L^\infty([0,T],W^{1,\infty})}\big)|\mathcal{F}_b(Q_N,\omega)| + C_{b,T}\left(1+\left\lVert u\right\rVert_{L^\infty([0,T],W^{1,\infty})}\right)\\ &\times\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)} (1+I_N(0)+|E_N(0)|)N^{-\beta}. \end{aligned}$$ Now $$\begin{aligned} L_2 =& \frac{1}{N}\sum_{i=1}^N\frac{\nabla^\bot b(q_i)}{b(q_i)} \\ &\cdot\bigg[\int_{\mathbb{R}^2\backslash\{q_i\}} \sqrt{b(q_i)b(y)}\nabla g(q_i - y)\,\mathrm d\bigg(\omega(t)-\frac{1}{N}\sum_{j=1}^N\delta_{q_j(t)}\bigg) \\ &+ \int_{\mathbb{R}^2\backslash\{q_i\}} \nabla_x S_b(q_i,y)\,\mathrm d\bigg(\omega(t)-\frac{1}{N}\sum_{j=1}^N\delta_{q_j(t)}\bigg)\bigg] \\ =:& L_{2,1} + L_{2,2} + L_{2,3}\end{aligned}$$ with $$\label{gronwall_alpha_regime_L21} \begin{aligned} |L_{2,1}| &= \bigg|\frac{1}{N}\sum_{i=1}^N\frac{\nabla^\bot b(q_i)}{\sqrt{b(q_i)}}\cdot\int_{\mathbb{R}^2} \nabla g(q_i - y)\sqrt{b(y)}\omega(t,y)\,\mathrm dy\bigg| \\ &\leqslant C_b\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1\cap L^\infty)} \end{aligned}$$ (for the last inequality see for example [@Iftimie Lemma 1]). For the second term $$\begin{aligned} L_{2,2} &= -\frac{1}{N^2}\sum_{i=1}^N\underset{j \neq i}{\sum_{j=1}^N} \sqrt{b(q_i)b(q_i)}\frac{\nabla^\bot b(q_i)}{b(q_i)}\cdot \nabla g(q_i-q_j).\end{aligned}$$ We can bound it as in [\[bound_E\_N_2\]](#bound_E_N_2){reference-type="eqref" reference="bound_E_N_2"} to get $$\label{gronwall_alpha_regime_L22} |L_{2,2}| \leqslant C_b.$$ For the last term, we use Claim $(1)$ of Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"} to get $$\begin{aligned} |L_{2,3}| &= \bigg|\frac{1}{N}\sum_{i=1}^N\frac{\nabla^\bot b(q_i)}{b(q_i)}\cdot\int_{\mathbb{R}^2} \nabla_x S_b(q_i,y)\,\mathrm d\bigg(\omega(t)-\frac{1}{N}\underset{j\neq i}{\sum_{j=1}^N}\delta_{q_j(t)}\bigg)(y)\bigg| \\ &\leqslant C_b \int_{\mathbb{R}^2} (1+|y|)\,\mathrm d\bigg(\omega(t)+\frac{1}{N}\underset{j\neq i}{\sum_{j=1}^N}\delta_{q_j(t)}\bigg)(y) \\ &\leqslant C_b(\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx))} + I_N(t)) \\ &\leqslant C_{b,T}(\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx))} + 1+I_N(0) + |E_N(0)|)\end{aligned}$$ by Proposition [Proposition 22](#proposition_pv_well_posed){reference-type="ref" reference="proposition_pv_well_posed"}. Combining the upper inequality with [\[gronwall_alpha_regime_L1\]](#gronwall_alpha_regime_L1){reference-type="eqref" reference="gronwall_alpha_regime_L1"}, [\[gronwall_alpha_regime_L21\]](#gronwall_alpha_regime_L21){reference-type="eqref" reference="gronwall_alpha_regime_L21"} and [\[gronwall_alpha_regime_L22\]](#gronwall_alpha_regime_L22){reference-type="eqref" reference="gronwall_alpha_regime_L22"} we get that for almost every $t \in [0,T]$, $$\begin{aligned} &\left|\frac{\,\mathrm d}{\,\mathrm dt}\mathcal{F}_{b,N}(t)\right| \\ \leqslant& C_b(1+\left\lVert u\right\rVert_{L^\infty([0,T],W^{1,\infty})})|\mathcal{F}_{b,N}(t)| + C_b\left(1+\left\lVert u\right\rVert_{L^\infty([0,T],W^{1,\infty})}\right)\\ &\times\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)} (1+I_N(0)+|E_N(0)|)N^{-\beta} \\ &+ C_{b,T}|\alpha_N - \alpha|\bigg(\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)} +1+ I_N(0) + |E_N(0)|\bigg).\end{aligned}$$ Therefore there exists a constant $A$ depending only on the quantities $b$, $T$, $\left\lVert u\right\rVert_{L^\infty([0,T],W^{1,\infty})}$, $\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)}$ and $I_N(0)$ (which is uniformly bounded in $N$ by assumption) such that for almost every $t \in [0,T]$, $$\left|\frac{\,\mathrm d}{\,\mathrm dt}\mathcal{F}_{b,N}(t)\right| \leqslant A(|\mathcal{F}_{b,N}(t)|+(1+|E_N(0)|)(N^{-\beta}+|\alpha_N - \alpha|)).$$ By Grönwall's lemma (up to taking another constant $A$ depending on the same quantities) we get [\[gronwall_bound_alpha_regime\]](#gronwall_bound_alpha_regime){reference-type="eqref" reference="gronwall_bound_alpha_regime"}. Now let us study the rescaled regime where $\alpha_N \mathop{\longrightarrow}\limits_{N\rightarrow+\infty} +\infty$. By Proposition [Proposition 24](#time_derivative_F_N_resc){reference-type="ref" reference="time_derivative_F_N_resc"} we have $$\begin{aligned} \frac{\,\mathrm d}{\,\mathrm dt}\overline{\mathcal{F}}_{b,N}(t) =& -2\iint_{(\mathbb{R}^2\times \mathbb{R}^2)\backslash \Delta} \frac{\nabla^\bot b(x)}{b(x)}\cdot \nabla_x g_b(x,y)\,\mathrm d(\overline{\omega}(t)-\overline{\omega}_N(t))^{\otimes 2}(x,y) \\ &+ \frac{2}{N^2\alpha_N} \sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N} \frac{v(t,\overline{q_i})}{b(\overline{q_i})}\cdot\nabla_x g_b(\overline{q_i},\overline{q_j}) \\ =:& L_1 + L_2.\end{aligned}$$ The first term can be bounded by Proposition [Proposition 39](#controle_terme_principal_gronwall){reference-type="ref" reference="controle_terme_principal_gronwall"}: $$\label{gronwall_rescaled_regime_L1} \begin{aligned} |L_1| \leqslant& C_b\left\lVert\frac{\nabla b}{b}\right\rVert_{W^{1,\infty}}|\overline{\mathcal{F}}_{b,N}(t)| + C_b\bigg(1+\left\lVert\frac{\nabla b}{b}\right\rVert_{W^{1,\infty}}\bigg)\\ &\times \left\lVert\overline{\omega}\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)}(1+I(\overline{Q_N}))N^{-\beta} \\ \leqslant& C_b|\overline{\mathcal{F}}_{b,N}(t)| + C_{b,T}\left\lVert\overline{\omega}\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)}\\ &\times(1 +\overline{I_N}(0)+|\overline{E_N}(0)|)N^{-\beta} \end{aligned}$$ where we used Proposition [Proposition 22](#proposition_pv_well_posed){reference-type="ref" reference="proposition_pv_well_posed"} in the last inequality. We split the second line in three terms: $$\begin{aligned} L_2 =& \frac{2}{N^2\alpha_N} \sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N} \frac{v(t,\overline{q_i})}{b(\overline{q_i})}\cdot \frac{\nabla b(\overline{q_i})}{2\sqrt{b(\overline{q_i})}}\sqrt{b(\overline{q_j})}g(\overline{q_i} - \overline{q_j}) \\ &+ \frac{2}{N^2\alpha_N} \sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N} \frac{v(t,\overline{q_i})}{b(\overline{q_i})}\cdot\nabla g(\overline{q_i} - \overline{q_j})\sqrt{b(\overline{q_i})b(\overline{q_i})} \\ &+\frac{2}{N^2\alpha_N}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N} \frac{v(t,\overline{q_i})}{b(\overline{q_i})}\cdot \nabla_x S_b(\overline{q_i},\overline{q_j}) \\ =:& L_{2,1} + L_{2,2} + L_{2,3}.\end{aligned}$$ We can bound the first term by $$\begin{aligned} |L_{2,1}| \leqslant\frac{C_b}{N^2\alpha_N}\left\lVert v\right\rVert_{L^\infty}\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N}|g(\overline{q_i}-\overline{q_j})|\end{aligned}$$ and applying Lemma [Lemma 11](#lemma_link_solutions_duerinckx){reference-type="ref" reference="lemma_link_solutions_duerinckx"} we get $$\left\lVert v\right\rVert_{L^\infty} = \left\lVert\nabla G_b[\overline{\omega}]\right\rVert_{L^\infty} \leqslant C_b\left\lVert\overline{\omega}\right\rVert_{L^\infty([0,T],L^1\cap L^\infty)}.$$ We can bound $\displaystyle{\sum_{i=1}^N\underset{j\neq i}{\sum_{j=1}^N}|g(\overline{q_i}-\overline{q_j})|}$ as we did for Inequality [\[bound_IN2\]](#bound_IN2){reference-type="eqref" reference="bound_IN2"} to get $$|L_{2,1}| \leqslant C_b\left\lVert\overline{\omega}\right\rVert_{L^\infty([0,T],L^1\cap L^\infty)}(1+|\overline{E_N}|+\overline{I_N})\alpha_N^{-1}.$$ The second term $L_{2,2}$ can be bounded as in [\[bound_E\_N_2\]](#bound_E_N_2){reference-type="eqref" reference="bound_E_N_2"} to get $$|L_{2,2}| \leqslant C_b(1+\left\lVert v\right\rVert_{L^\infty([0,T],W^{1,\infty})})\alpha_N^{-1}$$ and the last term can be bounded directly using Claim (1) of Lemma [Lemma 15](#estimees_S_b_not_necessarily_symmetric){reference-type="ref" reference="estimees_S_b_not_necessarily_symmetric"}: $$|L_{2,3}| \leqslant C_b\left\lVert v\right\rVert_{L^\infty}(1+\overline{I_N})\alpha_N^{-1} \leqslant C_b\left\lVert\overline{\omega}\right\rVert_{L^\infty([0,T],L^1\cap L^\infty)}(1+\overline{I_N})\alpha_N^{-1}.$$ Combining these three inequalities with [\[gronwall_rescaled_regime_L1\]](#gronwall_rescaled_regime_L1){reference-type="eqref" reference="gronwall_rescaled_regime_L1"} and using Proposition [Proposition 22](#proposition_pv_well_posed){reference-type="ref" reference="proposition_pv_well_posed"} to bound $\overline{I_N}$ we get that for almost every $t \in [0,T]$, $$\begin{aligned} \left|\frac{\,\mathrm d}{\,\mathrm dt}\overline{\mathcal{F}}_{b,N}(t)\right| \leqslant& C_b|\overline{\mathcal{F}}_{b,N}(t)| + C_b(\left\lVert\overline{\omega}\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)}\\ &+\left\lVert v\right\rVert_{L^\infty([0,T],W^{1,\infty})})\\ &\times(1+|\overline{I_N}(0)|+|\overline{E_N}(0)|)(N^{-\beta} + \alpha_N^{-1}).\end{aligned}$$ And therefore there exists a constant $B$ depending only on the quantities $b$, $T$,$\left\lVert\omega\right\rVert_{L^\infty([0,T],L^1((1+|x|)\,\mathrm dx)\cap L^\infty)}$ and $\overline{I_N}(0)$ (which is uniformly bounded in $N$ by assumption) such that for almost every $t \in [0,T]$, $$\begin{aligned} \left|\frac{\,\mathrm d}{\,\mathrm dt}\overline{\mathcal{F}}_{b,N}(t)\right| \leqslant& B(|\overline{\mathcal{F}}_{b,N}(t)| + (1+|\overline{E_N}(0)|)(N^{-\beta} + \alpha_N^{-1})).\end{aligned}$$ By Grönwall's lemma (up to taking another constant $B$ depending on the same quantities) we get [\[gronwall_bound_rescaled_regime\]](#gronwall_bound_rescaled_regime){reference-type="eqref" reference="gronwall_bound_rescaled_regime"}. ◻ *Proof of Theorem [Theorem 8](#MFL_theorem){reference-type="ref" reference="MFL_theorem"}.* By Corollary [Corollary 32](#corollary_weak_star_cv){reference-type="ref" reference="corollary_weak_star_cv"}, weak-$\ast$ convergence and convergence of the interaction energy gives that $(\mathcal{F}_{b,N}(0))$ and $(\overline{\mathcal{F}_{b,N}}(0))$ converge to zero. Using convergence of the interaction energy we also get that $|E_N(0)|$ and $|\overline{E_N}(0)|$ are bounded. Thus by Inequalities [\[gronwall_bound_alpha_regime\]](#gronwall_bound_alpha_regime){reference-type="eqref" reference="gronwall_bound_alpha_regime"} and [\[gronwall_bound_rescaled_regime\]](#gronwall_bound_rescaled_regime){reference-type="eqref" reference="gronwall_bound_rescaled_regime"} we get that for any $t \in [0,T]$ $(\mathcal{F}_{b,N}(t))$ and $(\overline{\mathcal{F}_{b,N}}(t))$ converge to zero and the theorem follows by Corollary [Corollary 32](#corollary_weak_star_cv){reference-type="ref" reference="corollary_weak_star_cv"}. ◻ 10 B. Al Taki and C. Lacave. Degenerate lake equations: classical solutions and vanishing viscosity limit. , 36(1):653--678, 2023. H. Bahouri, J.-Y. Chemin, and R. Danchin. , volume 343. Berlin: Heidelberg, 2011. R. J. Berman and M. Önnheim. . , 51(1):159--196, 2019. N. Boers and P. Pickl. . , 164(1):1--16, 2016. 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arxiv_math

--- abstract: | One-dependent first passage percolation is a spreading process on a graph where the transmission time through each edge depends on the direct surroundings of the edge. In particular, the classical iid transmission time $L_{xy}$ is multiplied by $(W_xW_y)^\mu$, a polynomial of the expected degrees $W_x, W_y$ of the endpoints of the edge $xy$, which we call the penalty function. Beyond the Markov case, we also allow any distribution for $L_{xy}$ with regularly varying distribution near $0$. We then run this process on three spatial scale-free random graph models: finite and infinite Geometric Inhomogeneous Random Graphs, and Scale-Free Percolation. In these spatial models, the connection probability between two vertices depends on their spatial distance and on their expected degrees. We show that as the penalty-function, i.e., $\mu$ increases, the transmission time between two far away vertices sweeps through four universal phases: *explosive* (with tight transmission times), *polylogarithmic*, *polynomial* but strictly sublinear, and *linear* in the Euclidean distance. The strictly polynomial growth phase here is a new phenomenon that so far was extremely rare in spatial graph models. The four growth phases are highly robust in the model parameters and are not restricted to phase boundaries. Further, the transition points between the phases depend non-trivially on the main model parameters: the tail of the degree distribution, a long-range parameter governing the presence of long edges, and the behaviour of the distribution $L$ near $0$. In this paper we develop new methods to prove the upper bounds in all sub-explosive phases. Our companion paper complements these results by providing matching lower bounds in the polynomial and linear regimes. author: - "Júlia Komjáthy[^1], John Lapinskas[^2], Johannes Lengler[^3], Ulysse Schaller[^4]" bibliography: - references.bib title: Four universal growth regimes in degree-dependent first passage percolation on spatial random graphs I --- # Introduction {#sec:intro} First passage percolation (FPP) is a natural way to understand geodesics in random metric spaces. Starting from some initial vertex at time $0$, the process spreads through the underlying graph so that the transmission time between any two vertices $x,y$ is the minimum sum of edge transmission times over all paths between $x$ and $y$. In classical FPP, edge transmission times are independent and identically distributed random variables. In the recent paper [@komjathy2020stopping] we introduced one-dependent FPP, where edge transmission times depend on the edge's direct surroundings in the underlying graph. There, we determined the phase transition for explosion (i.e., reaching infinitely many vertices in finite time). In this paper we study the sub-explosive regime, when explosion does not occur. We show that the process exhibits rich behaviour with several growth phases and non-smooth phase transitions between them. This holds across a large class of scale-free spatial random graph models (namely Scale-Free Percolation, Hyperbolic Random Graphs, and infinite and finite Geometric Inhomogeneous Random Graphs [@deijfen2013scale; @bringmann2019geometric; @krioukov2010hyperbolic]), and across all Markovian and non-Markovian transmission time distributions with reasonable limiting behaviour at zero. 0.5em **Universality classes of transmission times.** In one-dependent FPP, we set the transmission time through the edge $e=xy$ between vertices $x,y$ as the product of an independent and identically distributed (iid) random factor $L_{xy}$ and a factor $(W_xW_y)^{\mu}$ for $\mu \geq 0$, where $W_x$ and $W_y$ are (up to constant factors) the expected degrees of $x$ and $y$ in the graph models under consideration. As $\mu$ increases and/or the parameters of the underlying graph change, we prove that the following four different phases occur for the transmission time between the vertex at $0$ and a far away vertex $x$: - it converges to a limiting distribution that is independent of $|x|$ (*explosive phase*); This was the main result of [@komjathy2020stopping]. The main result of this paper is to characterise the other phases by the growth of the transmission time between $0$ and $x$: - it grows at most *polylogarithmically* in $|x|$, without being explosive; - it grows *polynomially* with exponent $0<\eta_0 < 1$; - it grows *linearly* with exponent $\eta_0=1$. These phases are *highly robust* in the parameters, they are not restricted to phase boundaries in either $\mu$ or the other model parameters. Moreover, all four phases occur on a single underlying graph universally for any distribution of $L_{xy}$ as long as it is regularly varying at $0$ with non-zero exponent ($\beta$ in Table [1](#table:summary){reference-type="ref" reference="table:summary"}). This rich behaviour arises despite the doubly-logarithmic graph distances in the underlying spatial graph models. 0.5em **Phases of FPP in other models.**[\[paragraph:FPP-other-graphs\]]{#paragraph:FPP-other-graphs label="paragraph:FPP-other-graphs"} By contrast, in other models the behaviour of transmission times in classical FPP is less rich, and the strict polynomial *phase (iii) is absent* or restricted only to phase transition boundaries. Indeed, on sparse *non-spatial* graph models with finite-variance degrees, both Markovian and non-Markovian classical FPP universally show Malthusian (exponential) growth [@bhamidi2017universality]. Transmission times between two uniformly chosen vertices are then logarithmic in the graph size (phase (ii)). Sparse *spatial* graphs with finite-variance degrees (e.g. percolation, long-range percolation, random geometric graphs etc.) are typically restricted to linear graph distances/transmission times (phase (iv)) in the absence of long edges [@antal1996chemical; @penrose2003random; @cox1981some], or to polylogarithmic distances (phase (ii)) in the presence of long edges [@biskup2004scaling; @biskup2019sharp; @hao2021graph]. In both spatial and non-spatial graph models with infinite-variance degrees, classical FPP typically either explodes or exhibits a smooth transition between explosion and doubly-logarithmic transmission times (which match the graph distances) [@adriaans2018weighted; @jorritsma2020weighted; @van2017explosion]; in particular, there is no analogue of phases (ii)--(iv). While there is not much work on one-dependent FPP on non-spatial graphs, there are strong indications that the process either explodes [@slangen19], with the same criterion for explosion as for spatial graphs in [@komjathy2020stopping], or becomes Malthusian [@Fransson1720143], the latter implying logarithmic transmission times between two uniformly chosen vertices by the universality in [@bhamidi2017universality], so only phases (i) and (ii) can occur. The only graph model to exhibit a transition from a fast-growing phase (phase (ii)) to a slow-growing phase (phase (iv)) is long-range percolation, where the polynomial phase is restricted to the phase boundary in the long-range parameter $\alpha=2$ (unpublished work in [@ding2013distances]). Even in degenerate models (i.e., those where the underlying graph is complete), long-range first passage percolation [@chatterjee2016multiple] is the only other model where a similarly rich set of phases is known to occur. Thus one-dependent FPP is the first process that displays a full interpolation between the four phases on a *single non-degenerate graph model*. Moreover, the phase boundaries for one-dependent FPP depend non-trivially on the main model parameters: the degree power-law exponent $\tau$, the parameter $\alpha$ controlling the prevalence of long-range edges, and the behaviour of $L_{xy}$ near $0$ characterised by $\beta$, see Table [1](#table:summary){reference-type="ref" reference="table:summary"} for our results, Table [2](#table:phases-top-FPP){reference-type="ref" reference="table:phases-top-FPP"} for phases of growth in other models, and Section [1.3](#sec:discussion){reference-type="ref" reference="sec:discussion"} for more details. ------------------------------------------------------------------- ------------------------------------------------------------------------------------------------- -- $\tau\in(2,3)$ $\alpha \in(1,2)$ $\mu < \frac{3-\tau}{2\beta}$ $d_{\mathcal{C}}(0,x) = \Theta(1)$ $\mu > \frac{3-\tau}{2\beta}$ $d_{\mathcal{C}}(0,x) = O((\log|x|)^{\Delta_0+o(1)}), \Delta_0>1$ $\tau\in(2,3)$ $\alpha > 2$ $\mu < \frac{3-\tau}{2\beta}$ $d_{\mathcal{C}}(0,x) = \Theta(1)$ $\mu\in\big(\frac{3-\tau}{2\beta}, \frac{3-\tau}{\beta}\big)$ $d_{\mathcal{C}}(0,x) = O((\log|x|)^{\Delta_0+o(1)}), \Delta_0 >1$ $\mu\in\big(\frac{3-\tau}{\beta}, \frac{3-\tau}{\min\{\beta, d(\alpha-2)\}} + \frac{1}{d}\big)$ $d_{\mathcal{C}}(0,x) = |x|^{\eta_0\pm o(1)}, \eta_0<1$ $\mu > \frac{3-\tau}{\min\{\beta, d(\alpha-2)\}} + \frac{1}{d}$ $d_{\mathcal{C}}(0,x) = \Theta(|x|)$ ------------------------------------------------------------------- ------------------------------------------------------------------------------------------------- -- : Summary of our main results. In 1-FPP, edge transmission times are $L_{xy} (W_xW_y)^\mu$ where $W_x, W_y$ are constant multiples of the expected degrees of the vertices $x,y$, and $L_{xy}$ is iid with distribution function that varies regularly near $0$ with exponent $\beta\in(0,\infty]$. The degree distribution follows a power law with exponent $\tau\in(2,3)$: graph distances are doubly-logarithmic in the underlying graph. The transmission time $d_\mathcal{C}(0,x)$ between $0$ and a far away vertex $x$ sweeps through four different phases as the penalty exponent $\mu$ increases. For long-range parameter $\alpha\in(1,2)$, long edges between low-degree vertices maintain polylogarithmic transmission times (similar to long-range percolation), so increasing $\mu$ stops explosion but it has no further effect. When $\alpha>2$, these edges are sparser and a larger $\mu$ slows down 1-FPP, to polynomial but sublinear transmission times in an interval of length at least $1/d$ for $\mu$. Then, all long edges have polynomial transmission times in the distance they bridge. For even higher penalty exponent $\mu$ the behaviour becomes similar to FPP on the grid $\mathbb{Z}^d$. We give the growth exponents $\Delta_0$ and $\eta_0$ explicitly in [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"} and [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}. 0.5em **Precise behaviour in the four phases.** In this paper we prove the upper bounds on transmission times in the sub-explosive regime (phase (i) was previous work [@komjathy2020stopping]). In phase (ii), we show that the transmission time is at most $(\log|x|)^{\Delta_0 + o(1)}$ with an explicit $\Delta_0 >1$ which we conjecture to be tight. In phases (iii) and (iv), we show that the transmission time is precisely $|x|^{\eta_0 \pm o(1)}$, where we give $\eta_0<1$ explicitly for phase (iii) and $\eta_0 =1$ for phase (iv). The companion paper [@komjathy2022one2] contains the matching lower bounds for phases (ii)-(iv). For phase (iv), [@komjathy2022one2] also contains a more precise upper bound $\overline \kappa |x|$ for some constant $\overline \kappa$ (i.e., strictly linear distances) as long as the underlying spatial graph has dimension at least two. When the dimension is one, we prove here an upper bound of $|x|^{1+o(1)}$ in phase (iv). The lower bound $\underline \kappa |x|$ for some non-zero constant $\underline \kappa$ is contained in [@komjathy2022one2] for all dimensions. We develop new techniques that allow us to treat upper bounds for all three sub-explosive phases *simultaneously*, which we expect to be of independent interest. 0.5em **New methodology: moving to quenched to replace FKG-inequality.** In this paper we develop a general technique -- *pseudorandom nets combined with multi-round exposure* -- that *replaces the FKG-inequality* in problems concerning vertex and/or edge-weighted graph models where this inequality does not hold. Let us explain why the FKG-inequality fails in the context of 1-FPP. Typically, for upper bounds one constructs paths connecting $0$ and $x$ by revealing vertices and/or edges of the graph sequentially, which destroys the independence of edges. For graph distances, the FKG-inequality resolves this problem [@biskup2004scaling]. However, in 1-FPP, the existence of a long edge is positively correlated to its endpoints having large vertex weights, which is *negatively* correlated to its other outgoing edges having short transmission times. To overcome this issue, we move to the *quenched setting* where we reveal the realisation of the whole weighted vertex set -- say $(\mathcal{V}, \mathcal{W})=(V, w_V)$ -- and thus events concerning only edges become independent. Working with 'arbitrary' realisations is impossible. We thus develop a 'multi-scale control' on the realisation of the weighted vertex set, which we call *pseudorandom nets* or just nets. A net $\mathcal{N}$ is a subset of vertices, together with their weights, such that for every not-too-small radius $r$, every "reasonable" weight $w$, and every vertex $v\in\mathcal{N}$, the net has *constant density* in $B_r(v)\times [w, 2w]$, shorthand for vertices of weight in $[w,2w]$ within Euclidean distance $r$ of $v$: $$\label{eq:net-heuristics} \frac{|\mathcal{N}\cap B_r(v)\times[w, 2w]|}{\mathbb{E}\big[|\mathcal{V}\cap B_r(v)\times [w,2w]|\big]} \in \Big(\frac{1}{16}, 8\Big).$$ We prove via a *multi-scale analysis* that as $|x|\to \infty$, asymptotically almost every realisation of the weighted vertex set contains a net in a box containing $0$ and $x$, with total density at least $1/4$. Given a net $\mathcal{N}$, the FKG-inequality in FPP and 1-FPP is still not applicable to find edges, since we select vertices based on the existence of cheap edges. Such a selection introduces dependencies, for example if we check edges one-by-one until we find a cheap edge, then selecting the $i$th edge implies that the first $i-1$ edges were not cheap. In spatial settings, the combinatorial options might also vanish in some bad instances of adaptively chosen sequences. Standardly these issues are resolved by vertex-sprinkling, however, vertex sprinkling is impossible when the vertex set of the underlying graph is $\mathbb{Z}^d$. Instead, we define an edge-weighted *multi-round exposure* process based on a careful thinning of the edges, which replaces the FKG-inequality. We fix the realisation of the weighted vertex set $(V, w_V)$. Then we couple $R$ rounds of exposure of edges (with their transmission times) to $R$ different, conditionally independent copies $H_i$ of the graph on the same weighted vertex set, each with edge probability $1/R$ times the original edge-probabilities. In Section [3](#sec:exposure){reference-type="ref" reference="sec:exposure"} we design a coupling so that the probability that an *adaptively* chosen sequence of set of edges $E_1, \dots, E_R$ (with constraints on their transmission times) is present in the original graph satisfies $$\mathbb{P}\Big(\text{adaptively chosen } E_1, \dots, E_R \text{ present}\mid (V,w_V)\Big) \ge \prod_{i\le R} \mathbb{P}\Big(E_i \text{ is present in }H_i \mid (V, w_V)\Big).$$ **Robustness of our techniques.** The technique of nets combined with multi-round edge-exposure is robust, and will be applicable elsewhere, for questions concerning *first passage percolation, robustness to percolation (random deletion of edges), graph distances, SIR-type and other epidemic processes, rumour spreading*, etc. on a larger class of vertex-weighted graphs; including random geometric graphs, Boolean models with random radii, the age-dependent and the weight-dependent random connection model (mimicking spatial preferential attachment), scale-free Gilbert graph, and the models used here [@AieBonCooJanss08; @CooFriePral12; @gracar2019age; @gracar2022chemical; @GraHeyMonMor19; @gracar2022finiteness; @gracar2021percolation; @hirsch2017gilbertgraph; @JacMor15], and can also be extended to dynamical versions of the above graph models on fixed vertex sets. 0.5em **Budget travel plan with 3-edge bridge-paths.** Switching to the quenched setting allows to prove the upper bounds in all subexponential phases (ii)--(iv) all-at-once. Our construction of a connecting path overcomes the problem that a long edge with a short transmission time typically occurs on high-degree vertices and thus typical outgoing edges from the same vertices have too long transmission times. The main idea resembles a 'budget travel plan': when someone travels with a low budget, one takes the cheapest mode of transport to the airport within a $200$km radius that offers the cheapest flight landing within a $200$km radius of the destination, then takes the cheapest transport to the destination city. Formally, we put balls of radius $|x|^\gamma$ for some $\gamma \in (0,1)$ around $0$ and around $x$, and we find a cheap $3$-edge path ("bridge") $\pi_1 = y_0aby_x$ between these two balls *using only vertices in the pseudorandom net*. The net guarantees enough vertices in each vertex-weight range of interest. Typically, $a,b$ in $\pi_1$ are high-weight vertices connected by an atypically cheap edge, that simultaneously have an atypically cheap edge to low-weight vertices $y_0, y_x$, respectively. (Here we use the common terminology of fast transmission corresponding to 'cheap' cost.) Then we have replaced the task of connecting $0$ and $x$ by the two tasks of connecting $0$ with $y_0$ and $x$ with $y_x$, where the new 'gaps' $|0-y_0|$ and $|x-y_x|$ are much smaller than $|x|$. The *multi-round exposure* and the *pseudorandom net* on the fixed vertex set together guarantee that we can iterate this process without running out vertices in the relevant weight-ranges, and without accumulated correlations in the presence of edges along the iteration (e.g. out of $y_0, y_x$). Iteration yields a set of multi-scale bridge-paths, which we call after Biskup a *hierarchy* [@biskup2004scaling]. The construction in [@biskup2004scaling] also uses recursion, with one-edge bridges instead of three-edge bridges, and yields polylogarithmic graph distances in long range percolation. The techniques in [@biskup2004scaling] would not work for 1-FPP because we need to balance distances vs costs vs the penalisation on high-weight vertices in very different regimes, and at the same time deal with edge-costs dependencies. Those can only be dealt with in the quenched setting. The cost (transmission time) of the bridge-paths $\pi$ in 1-FPP are either polynomial in the distance they bridge or constant. When it is *polynomial* -- with optimal exponent $\eta_0$ -- we are in the *polynomial phase*. The cost of the first bridge $\pi_1$ then dominates the cost of the whole path, and we only carry out a constant number of iterations (irrespective of $|x|$). When bridge-paths with constant cost exist, we are in the *polylogarithmic phase*. Then, the cost of all bridges together are negligible compared to the cost of the polylogarithmic number of gaps that remain after the last iteration. Here, we iterate until we can connect the remaining gaps via essentially constant cost paths. Connecting the gaps is a non-trivial task itself since GIRG/SFP does not contain nearest-neighbour edges. Solutions for filling gaps in [@biskup2004scaling], developed for graph distances in long-range percolation, are based on the infinite component having positive local density with very small error probability. Such ideas fail for filling gaps in 1-FPP since a vertex being in a local giant component increases the chance of having high degree and thus high penalty for 1-FPP. Instead, we connect the gaps with a construction that we call *'weight-increasing paths'* that crucially use that the underlying graphs are scale-free. We give a more detailed discussion about the hierarchical construction at the beginning of Section [5](#sec:hierarchy){reference-type="ref" reference="sec:hierarchy"} and back-of-the-envelope calculations about how to obtain the precise growth exponents in phases (ii) and (iii) at the beginning of Section [5.1](#sec:choices){reference-type="ref" reference="sec:choices"} with proof sketches below Corollaries [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} and [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"}. 0.5em **Two papers, two techniques and optimality.** The 'budget travel plan' together with the renormalisation group argument in [@komjathy2022one2] reveals that the strategy of polynomial paths is essentially optimal: in this phase, all long edges have polynomial transmission time in the distance they bridge. Our techniques for the lower bounds are entirely different and deserve their own exposition, hence we present them in the companion paper [@komjathy2022one2]. ## Graph Models {#sec:graph_model} We consider simple and undirected graphs with vertex set $\mathcal{V}\subseteq \mathbb{R}^d$. We use standard graph notation along with other common terminology, see Section [1.3.1](#sec:notation){reference-type="ref" reference="sec:notation"}. We consider three random graph models: *Scale-Free Percolation* (SFP), *Infinite Geometric Inhomogeneous Random Graphs* (IGIRG)[^5], and (finite) *Geometric Inhomogeneous Random Graphs* (GIRG). The latter model contains *Hyperbolic Random Graphs* (HypRG) as special case, so our results extend to HypRG. The main difference between SFP and IGIRG is the vertex set $\mathcal{V}$. For SFP, we use $\mathcal{V}:= \mathbb Z^d$, with $d \in \mathbb{N}$. For IGIRG, a unit-intensity Poisson point process on $\mathbb R^d$ forms $\mathcal{V}$. **Definition 1** (SFP, IGIRG, GIRG). *Let $d\in \mathbb{N}$, $\tau >2$, $\alpha\in(1,\infty)$, and $\overline{c}>\underline{c}>0$. Let $\ell:[1,\infty)\rightarrow(0,\infty)$ be function that varies slowly at infinity (see Section [1.3.1](#sec:notation){reference-type="ref" reference="sec:notation"}), and let $h:\mathbb{R}^d\times[1,\infty)\times[1,\infty)\rightarrow[0,1]$ be a function satisfying $$\begin{aligned} \label{eq:connection_prob} \underline{c}\cdot\min\left\{1, \dfrac{w_1w_2}{|x|^d}\right\}^{\alpha} \le h(x,w_1,w_2)\le \overline{c}\cdot\min\left\{1, \dfrac{w_1w_2}{|x|^d}\right\}^{\alpha}. \end{aligned}$$ The vertex set and vertex-weights: For SFP, set $\mathcal{V}:= \mathbb Z^d$, for IGIRG, let $\mathcal{V}$ be given by a Poisson point process on $\mathbb R^d$ of intensity one.[^6] For each $v\in\mathcal{V}$, we draw a *weight* $W_v$ independently from a probability distribution on $[1, \infty)$ satisfying $$\label{eq:power_law} F_W(w)=\mathbb{P}( W\le w)= 1-\ell(w)/w^{\tau-1}.$$ We denote $\widetilde \mathcal{V}(G):=(\mathcal{V}, \mathcal{W})$ the vertex set $\mathcal{V}$ together with the random weight vector $\mathcal{W}_{\mathcal{V}}:=(W_v)_{v\in \mathcal{V}}$, and $(V,w_V):=(V, (w_v)_{v\in V})$ a realisation of $\widetilde \mathcal{V}:=\widetilde \mathcal{V}(G)$, where $\tilde v:=(v, w_v)$ stands for a single weighted vertex.* *The edge set: Conditioned on $\widetilde \mathcal{V}=(V, w_V)$, consider all unordered pairs $\mathcal{V}^{\scriptscriptstyle{(2)}}$ of $\mathcal{V}$. Then every pair $xy\in \mathcal{V}^{\scriptscriptstyle{(2)}}$ is present in $\mathcal{E}(G)$ independently with probability $h(x-y,w_x,w_y)$.* *Finally, a GIRG $G_n$ is obtained as the induced subgraph $G[Q_n]$ of an IGIRG $G$ by the set of vertices in the cube $Q_n$ of volume $n$ centred at $0$. We call $h$ the *connection probability*, $d$ the *dimension*, $\tau$ the *power-law exponent*, and $\alpha$ the *long-range parameter*.* The above definition essentially merges the Euclidean space and the vertex-weight space by considering vertices with weights as points in $\mathbb{R}^d \times [1, \infty)$, i.e., we think of each vertex as a pair $\tilde v=(v, w_v)$, where $v\in \mathbb{R}^d$ is its spatial location and $w_v$ is its weight. For finite GIRG models, we are interested in the behaviour as $n\to \infty$. Definition [Definition 1](#def:girg){reference-type="ref" reference="def:girg"} leads to a slightly less general model than those e.g. in [@bringmann2019geometric] and [@komjathy2020stopping]. There, the original definition had a different scaling of the geometric space vs connection probabilities. However, the resulting graphs are identical in distribution after rescaling, see [@komjathy2020stopping] for a comparison. Finally, [@bringmann2019geometric] considered the torus topology on the cube, identifying "left" and "right" boundaries, but this does not make a difference for our results. Next we define $1$-dependent FPP on these graphs. **Definition 2** (1-dependent first passage percolation (1-FPP)). *Consider a graph $G = (\mathcal{V}, \mathcal{E})$ where each vertex $v\in \mathcal{V}$ has an associated vertex-weight $W_v$. For every edge $xy\in \mathcal{E}$, draw an i.i.d. copy $L_{xy}$ of a random variable $L$, and set the *(transmission) cost* of an edge $xy$ as $$\label{eq:cost} \mathcal{C}(xy):=L_{xy}(W_xW_y)^{\mu},$$ for a fixed parameter $\mu>0$ called the *penalty strength*. The costs define a *cost distance* $d_{\mathcal{C}}(x,y)$ between any two vertices $x$ and $y$, which is the minimal total cost of any path between $x$ and $y$ (see Section [1.3.1](#sec:notation){reference-type="ref" reference="sec:notation"}). We call $d_{\mathcal{C}}$ the 1-dependent first passage percolation.* We usually assume that the cumulative distribution function (cdf) $F_L:[0,\infty)\rightarrow[0,1]$ of $L$ satisfies the following assumption, (with exceptions of this assumption explicitly mentioned): **Assumption 3**. There exist constants $t_0,\,c_1,\,c_2,\,\beta>0$ such that $$\begin{aligned} \label{eq:F_L-condition} c_1t^{\beta}\le F_L(t)\le c_2t^{\beta}\mbox{ for all }t\in[0,t_0].\end{aligned}$$ Without much effort, one can relax Assumption [Assumption 3](#assu:L){reference-type="ref" reference="assu:L"} to $\lim_{x\to0} \log F_L(x)/\log x=\beta$. We work with [\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} for the sake of readability. We discuss extensions to $\alpha = \infty$ and $\beta = \infty$ separately in Section [1.2.1](#sec:threshold){reference-type="ref" reference="sec:threshold"}. We call the set of parameters $\textnormal{\texttt{par}}\xspace:= \{d, \tau, \alpha, \mu, \beta, \underline{c}, \overline{c}, c_1, c_2, t_0\}$ the *model parameters*. We say that a variable is *large* (or *small*) relative to a collection of other variables when it is bounded below (or above) by some finite positive function of those variables and the model parameters. We restrict to $\tau \in (2,3)$, (explicitly stated in the theorems), which ensures that there is a unique infinite component (or linear-sized "giant" component for finite GIRG)[^7] and that graph distances between vertices $x,y$ in the infinite/giant component grow like $d_G(x,y) \sim 2\log \log |x-y|/|\log(\tau-2)|$ in all three models [@komjathy2020explosion; @bringmann2016average; @deijfen2013scale; @van2017explosion]. We consider $\mu$ as the easiest parameter to change: increasing $\mu$ means gradually slowing down the spreading process around high-degree vertices, which corresponds to adjusting behaviour of individuals with high number of contacts. Hence, we will phrase our results from this perspective. ## Results {#sec:results} In this paper, we focus on the sub-explosive parameter regime $$\label{eq:no_explosion} \mu>\frac{3-\tau}{2\beta}:=\mu_{\mathrm{expl}},$$ since for $\mu < \mu_{\mathrm{expl}}$ we have shown in previous work [@komjathy2020stopping] that the model is *explosive*: the cost-distance of two vertices $x,y$ converges in distribution to an almost surely finite variable as $|x-y| \to \infty$, conditioned on $x$ and $y$ being in the infinite component.[^8] In other words, [\[eq:no_explosion\]](#eq:no_explosion){reference-type="eqref" reference="eq:no_explosion"} restricts us to the non-explosive phase. The following two quantities will serve as the boundaries of the new phases: $$\begin{aligned} \label{eq:mu_pol_log} \mu_{\log}:=\frac{3-\tau}{\beta}, \quad \mu_{\mathrm{pol}}:=\frac1d+\frac{3-\tau}{\min\{\beta, d(\alpha-2)\}} = \max\big\{1/d+\mu_{\log}, \mu_{\mathrm{pol}, \alpha}\big\}, \end{aligned}$$ with $\mu_{\mathrm{pol},\alpha}:=\tfrac{1}{d}+\tfrac{3-\tau}{d(\alpha-2)}=\tfrac{\alpha-(\tau-1)}{d(\alpha-2)}$ and $\mu_{\mathrm{pol}, \beta}:=\tfrac{1}{d}+\tfrac{3-\tau}{\beta}$. We also define two *growth exponents*. If $\alpha\in(1,2)$ or $\mu\in(\mu_{\mathrm{expl}}, \mu_{\log})$, we define $$\begin{aligned} \label{eq:Delta_0} \Delta_0 := \Delta_0(\alpha, \beta,\mu, \tau) := \frac{1}{1-\log_2(\min\{\alpha, \tau-1+\mu\beta\})} =\min\{\Delta_\alpha, \Delta_\beta\}> 1,\end{aligned}$$ with $\Delta_\alpha=1/(1-\log_2\alpha)$ and $\Delta_\beta=1/(1-\log_2(\tau-1+\mu\beta)$. $\Delta_0>1$ follows since when $\alpha\in(1,2)$ then $\Delta_\alpha>1$, while when $\mu \in( \mu_{\mathrm{expl}}, \mu_{\log})$ then $\tau-1+\mu\beta >\tfrac{\tau+1}{2} >1$ and also $\tau-1+\mu\beta<2$, so $\log_2(\tau-1+\mu\beta)$ is positive but less than $1$. If both $\alpha>2$ and $\mu > \mu_{\log}$, we define $$\begin{aligned} \label{eq:eta_0} \eta_0 := \eta_0(\alpha,\beta,\mu,\tau) := \begin{cases} 1 & \mbox{ if $\mu>\mu_{\mathrm{pol}}$,}\\ \min\left\{d(\mu-\mu_{\log}), \mu/\mu_{\mathrm{pol},\alpha}\right\} & \mbox{ if $\mu\le\mu_{\mathrm{pol}}$,} \end{cases}\end{aligned}$$ and note that $\eta_0>0$ for all $\mu>\mu_{\log}$, and $\eta_0<1$ exactly when $\mu< \mu_{\mathrm{pol}}$ by [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}. We often write $\eta_\beta:=d(\mu-\mu_{\log})$ and $\eta_\alpha:=\mu/\mu_{\mathrm{pol},\alpha}$. The formulas can be naturally extended by taking limits and hold also when $\alpha = \infty$ or $\beta=\infty$, which we elaborate in Section [1.2.1](#sec:threshold){reference-type="ref" reference="sec:threshold"} below. We first formulate the main results for the infinite models IGIRG and SFP. We denote by $\mathcal{C}_{\infty}$ the unique infinite component of IGIRG/SFP. theoremPolylogRegime [\[thm:polylog_regime\]]{#thm:polylog_regime label="thm:polylog_regime"} Consider $1$-FPP in Definition [Definition 2](#def:1-FPP){reference-type="ref" reference="def:1-FPP"} on the graphs IGIRG or SFP of Definition [Definition 1](#def:girg){reference-type="ref" reference="def:girg"} satisfying the assumptions given in [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} with $\tau\in(2,3), \alpha>1, \mu>0$. When either $\alpha\in(1,2)$ or $\mu\in(\mu_{\mathrm{expl}},\mu_{\log})$ or both hold, then for any $\varepsilon>0$, $$\begin{aligned} \lim_{|x|\to \infty}\mathbb{P}\big( d_{\mathcal{C}}(0,x) \le (\log |x|)^{\Delta_0+\varepsilon} \mid 0, x \in \mathcal{C}_{\infty} \big) =1. \end{aligned}$$ The proof of Theorem [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"} is valid when $\mu < \mu_{\mathrm{expl}}$, however, then the model is explosive [@komjathy2020stopping Theorem 1.1], and the bound is not sharp. With the restriction $\mu>\mu_{\mathrm{expl}}$, we conjecture that Theorem [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"} is actually sharp, i.e., that a corresponding lower bound with exponent $\Delta_0 - \varepsilon$ also holds. See Section [1.3](#sec:discussion){reference-type="ref" reference="sec:discussion"} below for results on polylogarithmic lower bounds. The exponent $\Delta_0>1$ intuitively corresponds to stretched exponential ball-growth, where the number of vertices in cost-distance at most $r$ scales as $\exp(r^{1/\Delta_0})$. Trapman in [@trapman2010growth] showed that strictly exponential ball growth, i.e., $\Delta_0=1$ is possible for long-range percolation when $\alpha=1$ under additional constraints. This is consistent with our formula for $\Delta_0$, since $\Delta_0 \to 1$ as $\alpha\to 1$. We leave the lower bound in this phase for future work. Slightly related is the work [@hao2021graph] that treats polylogarithmic graph distances in the same model class but in a different parameter regime (finite variance degrees), however, the proof techniques there neither extend to FPP nor to infinite variance degree underlying graphs. **Remark 4**. The proof reveals two different types of paths with polylogarithmic cost-distances present in the graph. When $\alpha<2$, randomly occurring long edges on low-weight vertices cause the existence of paths of cost at most $(\log |x|)^{\Delta_\alpha+o(1)}$ with $\Delta_\alpha=1/(1-\log_2(\alpha))$. The closest long edge of order $|x|$ lands at distance $|x|^{\alpha/2}$ from $0$ and $x$ respectively, resulting in $\Delta_ \alpha$ after iteration. When $\mu<\mu_{\log}$, there are also paths using a cheap yet long edge (of order $|x|$) between two high-weight vertices (weight roughly $|x|^{d/2}$) that lie within distance $|x|^{(\tau-1+\mu\beta)/2+o(1)}$ from $0$ and $x$ respectively, and these cause the existence of paths of cost at most $(\log|x|)^{\Delta_\beta+\varepsilon}$ with $\Delta_\beta=1/(1-\log_2(\tau-1+\mu\beta))$. $\Delta_\beta$ is the outcome of an optimisation: we minimise the distance between the high-weight vertices to $0$ and $x$, while maintaining that an edge with constant cost exists between them. The minimal distance possible is of order $|x|^{(\tau-1+ \mu\beta)/2+o(1)}$: the tail exponent $\tau-1$ of the weight distribution [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}, and $\mu\beta$, the penalty exponent in [\[eq:cost\]](#eq:cost){reference-type="eqref" reference="eq:cost"} times the behaviour of the cdf of $L$ in [\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} both play a role. When we increase $\mu$ above $\mu_{\log}$ and $\alpha$ above $2$, we enter a new universality class and cost distances become polynomial: theoremPolynomialRegime [\[thm:polynomial_regime\]]{#thm:polynomial_regime label="thm:polynomial_regime"} Consider $1$-FPP in Definition [Definition 2](#def:1-FPP){reference-type="ref" reference="def:1-FPP"} on the graphs IGIRG or SFP of Definition [Definition 1](#def:girg){reference-type="ref" reference="def:girg"} satisfying the assumptions given in [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} with $\tau\in(2,3)$. When $\alpha>2$ and $\mu > \mu_{\log}$ both hold, then for any $\varepsilon>0$, $$\begin{aligned} \lim_{|x|\to \infty}\mathbb{P}\left(d_{\mathcal{C}}(0,x) \le|x|^{\eta_0+\varepsilon} \mid 0,x \in \mathcal{C}_{\infty}\right) =1. \end{aligned}$$ In the accompanying [@komjathy2022one2] we prove the corresponding lower bound, which implies: **Corollary 5** (Polynomial Regime). *Consider $1$-FPP in Definition [Definition 2](#def:1-FPP){reference-type="ref" reference="def:1-FPP"} on the graphs IGIRG or SFP satisfying the assumptions given in [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} with $\tau\in(2,3)$. When $\alpha>2$ and $\mu > \mu_{\mathrm{log}}$ both hold, then for any $\varepsilon>0$, $$\lim_{|x|\to \infty}\mathbb{P}\left( |x|^{\eta_0-\varepsilon}\le d_{\mathcal{C}}(0,x) \le|x|^{\eta_0+\varepsilon} \mid 0,x \in \mathcal{C}_{\infty}\right) =1.$$* Corollary [Corollary 5](#cor:polynomial_regime){reference-type="ref" reference="cor:polynomial_regime"} together with Theorem [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"} implies that a phase transition happens at $\mu_{\log}$ (and/or at $\alpha =2$) from at most polylogarithmic distances to polynomial distances. Moreover, when $\eta_0=1$ and the dimension $d\ge 2$, [@komjathy2022one2] also proves *strictly* linear cost-distances (both upper and lower bounds), which together with Theorem [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"} implies that there is another phase transition at $\mu_{\mathrm{pol}}$, from sublinear ($\eta_0 < 1$) to linear ($\eta_0=1$) cost-distances. See Table [1](#table:summary){reference-type="ref" reference="table:summary"} for a summary. We find it remarkable that 1-FPP shows polynomial distances with exponent *strictly less than one* in a spread-out parameter regime $\mu\in(\mu_{\log}, \mu_{\mathrm{pol}})$. This implies polynomial growth faster than the dimension for 1-FPP, which is rare in spatial models, see Section [1.3](#sec:discussion){reference-type="ref" reference="sec:discussion"}. **Remark 6**. The proof reveals two different types of paths with polynomial cost-distances present in the graph. When $\mu\le \mu_{\mathrm{pol,\alpha}}$, there are a few very long edges (of order $|x|$) with endpoints polynomially near $0$ and $x$, emanating from vertices with weight $|x|^{\eta_\alpha/(2\mu)}$ with $\eta_\alpha=\mu/\mu_{\mathrm{pol,\alpha}}$, and these results in paths with cost at most $|x|^{\eta_\alpha+o(1)}$ (the second term in [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}). Since there are only few such edges, the optimisation effect of choosing the one with smallest cost is negligible and $\beta$ does not enter the formula. Further, when $\mu\le \mu_{\mathrm{pol},\beta}$ in [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}, there are many long edges (of order $|x|$) with respective endpoints polynomially near $0$ and $x$ on vertices with weight roughly $|x|^{d/2}$, and when we optimise to choose the one with cheapest cost, the effect of $F_L$, i.e., $\beta$ in [\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"}, enters the formula, and we obtain a path with cost at most $|x|^{d(\mu-\mu_{\log})+o(1)}$, the first term in [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}. The proof of the lower bound in [@komjathy2022one2] shows that in this phase *all* long edges near $0,x$ have polynomial costs in the Euclidean distance they bridge, which explains the qualitative difference between 1-FPP and classical FPP. The next theorem describes in which sense the results stay valid for finite-sized models: theoremFiniteGraph [\[thm:finite_graph\]]{#thm:finite_graph label="thm:finite_graph"} Consider 1-FPP in Definition [Definition 2](#def:1-FPP){reference-type="ref" reference="def:1-FPP"} on the graph GIRG of Definition [Definition 1](#def:girg){reference-type="ref" reference="def:girg"} satisfying the assumptions given in [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} with $\tau\in(2,3), \alpha>1, \mu>0$. Let $\mathcal{C}_{\max}^{(n)}$ be the largest component in $Q_n$. Let $u_n,v_n$ be two vertices chosen uniformly at random from $\mathcal{V}\cap Q_n$. \(i\) When either $\alpha\in(1,2)$ or $\mu\in(\mu_{\mathrm{expl}},\mu_{\log})$ or both hold, then for any $\varepsilon>0$, $$\begin{aligned} \label{eq:finite-polylog} \lim_{n\to \infty}\mathbb{P}\left( d_{\mathcal{C}}(u_n,v_n) \le (\log |u_n-v_n|)^{\Delta_0+\varepsilon} \ \mid \ u_n,v_n \in \mathcal{C}_{\max}^{\scriptscriptstyle{(n)}} \right)=1. \end{aligned}$$ (ii) When $\alpha>2$ and $\mu>\mu_{\log}$ both hold, then for any $\varepsilon>0$, $$\begin{aligned} \label{eq:finite-polynomial} \lim_{n\to \infty}\mathbb{P}\left( d_{\mathcal{C}}(u_n,v_n) \le |u_n-v_n|^{\eta_0+\varepsilon} \ \mid \ u_n,v_n \in \mathcal{C}_{\max}^{\scriptscriptstyle{(n)}} \right)=1. \end{aligned}$$ The lower bound in Corollary [Corollary 5](#cor:polynomial_regime){reference-type="ref" reference="cor:polynomial_regime"} also transfers to finite GIRGs, since GIRG is defined as a subgraph of IGIRG. We refer to [@komjathy2022one2] for details. The proofs of Theorems [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"}, [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"}, and [\[thm:finite_graph\]](#thm:finite_graph){reference-type="ref" reference="thm:finite_graph"} also reveal that the paths realising the upper bounds deviate only sublinearly from the straight line between the two vertices, cf. Definition [Definition 37](#def:deviation){reference-type="ref" reference="def:deviation"} and Lemmas [Lemma 48](#lem:polylog-deviation){reference-type="ref" reference="lem:polylog-deviation"} and [Lemma 49](#lem:polynomial-deviation){reference-type="ref" reference="lem:polynomial-deviation"} for more details. ### Limit Cases and Extensions {#sec:threshold} Theorems [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"}--[\[thm:finite_graph\]](#thm:finite_graph){reference-type="ref" reference="thm:finite_graph"} can be extended to interesting cases that may informally be described as $\alpha = \infty$ or $\beta = \infty$. In the case $\alpha = \infty$, all connection probabilities are either constant or zero, and we replace the condition [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"} by $$\begin{aligned} \label{eq:alpha_infty} h(x,w_1,w_2) \begin{cases} \ = 0,\quad & \text{if } \tfrac{w_1w_2}{|x|^d} < c', \\ \ \ge \underline{c}\quad & \text{if }\tfrac{w_1w_2}{|x|^d} \ge c'', \end{cases}\end{aligned}$$ for some constants $\underline{c} \in(0,1]$ and $c'' \ge c' > 0$. For the sake of simplicity we will assume $c''=1$ in all our proofs, however the results still hold for general $c''$. Models satisfying [\[eq:alpha_infty\]](#eq:alpha_infty){reference-type="eqref" reference="eq:alpha_infty"} are called threshold (or zero temperature) models, and include *hyperbolic random graphs* [@krioukov2010hyperbolic] when the dimension is one. The correspondence between GIRGs and threshold hyperbolic random graphs was established in [@bringmann2019geometric Theorem 2.3]. For models where [\[eq:alpha_infty\]](#eq:alpha_infty){reference-type="eqref" reference="eq:alpha_infty"} holds, we extend the definitions [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}-[\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"} in the natural way to $\alpha=\infty$, since $\lim_{\alpha\to \infty}\mu_{\mathrm{pol},\alpha}=1/d$: $$\begin{aligned} \label{eq:alpha-infty-definitions} \mu_{\log}:=\frac{3-\tau}{\beta}, \quad \mu_{\mathrm{pol}}:= \frac{1}{d}+\frac{3-\tau}{\beta},\quad \eta_0 := \begin{cases} 1 & \mbox{ if $\mu>\mu_{\mathrm{pol}}$,}\\ d\cdot(\mu-\mu_{\log}) & \mbox{ if $\mu\le\mu_{\mathrm{pol}}$,} \end{cases}\end{aligned}$$ and, when $\mu\in(\mu_{\mathrm{expl}},\mu_{\log})$, $$\begin{aligned} \label{eq:alpha-infty-Delta_0} \Delta_0 := \frac{1}{1-\log_2(\tau-1+\mu\beta)} >0.\end{aligned}$$ The case $\beta = \infty$ captures when the cdf of the edge transmission variable $L$ in [\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} is flatter near $0$ than any polynomial, and we replace [\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} by the condition that $$\begin{aligned} \label{eq:beta_infty} \lim_{t\to 0} F_L(t)/t^{\beta} = 0 \mbox{ for all }0<\beta <\infty.\end{aligned}$$ In particular, this condition is satisfied if $F_L$ has no probability mass around zero, for example in the case $L \equiv 1$.[^9] When $\beta=\infty$, using that $\tau\in(2,3)$ we replace [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}-[\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"} naturally by $$\begin{aligned} \label{eq:beta-infty-definitions} \mu_{\mathrm{expl}} := \mu_{\log}:=0, \quad \mu_{\mathrm{pol}}:= \frac{\alpha-(\tau-1)}{d(\alpha-2)}, \quad \eta_0 := \begin{cases} 1 & \mbox{ if $\mu>\mu_{\mathrm{pol}}$,}\\ \mu/\mu_{\mathrm{pol}} & \mbox{ if $\mu\le\mu_{\mathrm{pol}}$,} \end{cases}\end{aligned}$$ and, when $\alpha\in(1,2)$, $$\begin{aligned} \label{eq:beta-infty-Delta_0} \Delta_0 := \frac{1}{1-\log_2(\alpha)} >0.\end{aligned}$$ Finally, when both $\alpha=\beta=\infty$ we replace [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"} and [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"} by $$\begin{aligned} \label{eq:alpha-beta-infty-definitions} \mu_{\mathrm{expl}} := \mu_{\log}:=0, \quad \mu_{\mathrm{pol}}:= \tfrac{1}{d}, \quad \eta_0 := \min\{1,d\mu\},\end{aligned}$$ and in that case we do not define $\Delta_0$, since the polylogarithmic case is vacuous when $\alpha=\beta=\infty$ (see Remark [Remark 44](#rem:alpha-beta-infty-polyllog){reference-type="ref" reference="rem:alpha-beta-infty-polyllog"}). Our main results still hold for these limit regimes. We remark that the corresponding lower bounds also hold [@komjathy2022one2 Theorem 1.10]. **Theorem 7** (Extension to threshold GIRGs and $\beta=\infty$). *  [\[thm:threshold_regimes\]]{#thm:threshold_regimes label="thm:threshold_regimes"}* (a) *Theorems [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"}, [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"} and [\[thm:finite_graph\]](#thm:finite_graph){reference-type="ref" reference="thm:finite_graph"} still hold for $\alpha=\infty$ if we replace definitions [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}-[\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"} by definitions [\[eq:alpha-infty-definitions\]](#eq:alpha-infty-definitions){reference-type="eqref" reference="eq:alpha-infty-definitions"}-[\[eq:alpha-infty-Delta_0\]](#eq:alpha-infty-Delta_0){reference-type="eqref" reference="eq:alpha-infty-Delta_0"}.* (b) *Theorems [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"}, [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"} and [\[thm:finite_graph\]](#thm:finite_graph){reference-type="ref" reference="thm:finite_graph"} still hold for $\beta=\infty$ if we replace definitions [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}-[\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"} by definitions [\[eq:beta-infty-definitions\]](#eq:beta-infty-definitions){reference-type="eqref" reference="eq:beta-infty-definitions"}-[\[eq:beta-infty-Delta_0\]](#eq:beta-infty-Delta_0){reference-type="eqref" reference="eq:beta-infty-Delta_0"}.* (c) *Theorems [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"} and [\[thm:finite_graph\]](#thm:finite_graph){reference-type="ref" reference="thm:finite_graph"} still hold for $\alpha=\beta=\infty$ if we replace definitions [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}-[\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"} by definition [\[eq:alpha-beta-infty-definitions\]](#eq:alpha-beta-infty-definitions){reference-type="eqref" reference="eq:alpha-beta-infty-definitions"}.* Theorem [\[thm:threshold_regimes\]](#thm:threshold_regimes){reference-type="ref" reference="thm:threshold_regimes"}(a) implies the analogous result for hyperbolic random graphs (HypRG) by setting $d=1$ in [\[eq:alpha-infty-definitions\]](#eq:alpha-infty-definitions){reference-type="eqref" reference="eq:alpha-infty-definitions"}, except for some minor caveats. In Definition [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}, the number of vertices in GIRG is Poisson distributed with mean $n$, while in the usual definition of HypRG [@krioukov2010hyperbolic; @gugelmann2012random] and GIRG [@bringmann2019geometric] the number of vertices is exactly $n$. In HypRG the vertex-weights have an $n$-dependent distribution converging to a limiting distribution [@komjathy2020explosion]. However, these differences may be overcome by coupling techniques presented in e.g. [@komjathy2020explosion]: a model with exactly $n$ vertices can be squeezed between two GIRGs with Poisson intensity $1-\sqrt{4\log n/n}$ and $1+\sqrt{4\log n/n}$, and one can couple $n$-dependent and limiting vertex-weights to each other, respectively, but we avoid spelling out the details and refer the reader to [@komjathy2020explosion Claims 3.2, 3.3]. ## Discussion {#sec:discussion} Here we discuss our results in context with related results about (inhomogeneous) first passage percolation and graph distances on spatial random graphs. **Long-range first passage percolation.** The work on long-range first passage percolation (LR-FPP) [@chatterjee2016multiple] is closest to our work. In that model, the underlying graph is the *complete graph* of $\mathbb{Z}^d$, and the edge transmission time on any edge $uv$ is exponentially distributed with mean $|u-v|^{d\alpha'+o(1)}$, so $\beta=1$, the process is Markovian, and the penalty depends on the Euclidean distance of $u$ and $v$. This choice eliminates the correlations coming from the presence/absence of underlying edges, and the growth is strictly governed by the long-range transmission times. As $\alpha'$ grows, [@chatterjee2016multiple] finds the same sub-explosive phases for transmission times in LR-FPP that we find for 1-FPP in Table [1](#table:summary){reference-type="ref" reference="table:summary"}. The main difference is that the explosive phase is absent in LR-FPP, and is replaced by a 'super-fast' phase there where transmission times are $0$ almost surely. Moreover, the behaviour on phase boundaries are different. Using the symmetries in their model, [@chatterjee2016multiple] proves that whenever transmission times in LR-FPP are strictly positive, then they must be at least logarithmic. In contrast, in general 1-FPP on IGIRG and SFP we also see doubly-logarithmic distances, for example for graph-distances ($L\equiv1, \mu=0$). We summarise the results on LR-FPP in Table [2](#table:phases-top-FPP){reference-type="ref" reference="table:phases-top-FPP"}. Nevertheless, in 1-FPP, the cost function $\mathcal{C}(xy)$ in [\[eq:cost\]](#eq:cost){reference-type="eqref" reference="eq:cost"} could also depend on $|x-y|$, i.e., take the form $L_{xy}(W_x W_y)^\mu|x-y|^\zeta$. The result of [@komjathy2020stopping] on explosion carries through to this case without much effort [@reubsaet2022], with the model being explosive if and only if $\mu+\zeta/d < (3-\tau)/(2\beta)=\mu_{\mathrm{expl}}$, with $\tau\in(2,3)$. Extrapolating our results to the spatial penalty, we conjecture that we see polylogarithmic distances when $\mu+\zeta/d\in (\mu_{\mathrm{expl}}, \mu_{\log})$ and strictly polynomial distances when $\mu+\zeta/d\in (\mu_{\log}, \mu_{\mathrm{pol}})$. **SFP/LRP** with **Graph-distance** **Growth** ---------------------------------------------- ----------------------------------------------------------------------------------------------- --------------- --------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------- $\tau\in(2,3)$ $\tfrac{(2\pm o(1))\log \log |x|}{|\log(\tau-2)|}$ doubly- [@deijfen2013scale; @van2017explosion] [@van2017explosion] logarithmic $\tau>3$ and $(\log |x|)^{\Delta\pm o(1)}$ poly- SFP: [@hao2021graph], SFP: [@hao2021graph] $\alpha \in (1,2)$ for some $\Delta>0$ logarithmic LRP: [@biskup2004scaling; @trapman2010growth] (partly open) LRP:[@biskup2004scaling; @biskup2019sharp; @trapman2010growth] $\tau>3$ and $|x|^{\eta\pm o(1)}$, polynomial SFP: open, SFP: open $\alpha=2$ for some $\eta<1$ LRP, $d=1$: [@ding2013distances] LRP, $d=1$: [@ding2013distances] $\tau>3$ and $\Theta(|x|)$ linear partly open$^\ddag$ [@antal1996chemical] [@berger2004lower; @deprez2015inhomogeneous] $\alpha>2$ **LRFPP** with **Cost-distance** **Growth** **Upper bound** **Lower bound** $\alpha' < 1$ $0$ instantaneous [@chatterjee2016multiple] $\alpha' \in(1, 2)$ $(\log |x|)^{\Delta_{\alpha'}\pm o(1)} for$ poly- [@chatterjee2016multiple] $\Delta_{\alpha'}=1/(1-\log_2\alpha')$ logarithmic $\alpha'\in (2,2+1/d)$ $|x|^{d (\alpha'-2)\pm o(1)}$ polynomial [@chatterjee2016multiple] $\alpha'>2+1/d$ $\Theta(|x|)$ linear [@chatterjee2016multiple] **IGIRG/SFP** **Cost-distance** **Growth** **Upper bound** **Lower bound** with $\tau\in(2,3)$ $\mu<\mu_{\mathrm{expl}}$ converges explosion [@komjathy2020stopping] [@komjathy2020stopping] in distribution $\mu\in(\mu_{\mathrm{expl}},\mu_{\log})$ $(\log |x|)^{\Delta_0+ o(1)}$, poly- Theorem [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"} open or $\alpha\in(1,2)$ $\Delta_0$ as in [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"} logarithmic   $\mu\in\big(\mu_{\log}, \mu_{\mathrm{pol}})$ $|x|^{\eta_0\pm o(1)}$, polynomial Theorem [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"} [@komjathy2022one2] and $\alpha>2$ $\eta_0$ as in [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"} $\mu>\mu_{\mathrm{pol}}$ $\Theta(|x|)$ linear [@komjathy2022one2], $d\ge 2$ [@komjathy2022one2] and $\alpha>2$ : Known results about the universality classes of graph-distances on long-range percolation **LRP**, scale-free percolation **SFP**, long-range first-passage percolation **LRFPP** and infinite geometric inhomogeneous random graphs **IGIRG**.\ $^\ddag$An upper bound is only known for high enough edge-density or all nearest-neighbour edges present. 0.5em **Qualitative difference between one-dependent FPP and graph distances.** Some phases of 1-FPP in Table [1](#table:summary){reference-type="ref" reference="table:summary"} are also phases for *graph-distances* in spatial models in general. However, while the polynomial phase is spread-out in 1-FPP, this phase is essentially absent for graph distances in models. Indeed, the polynomial phase occurs when long edges all have polynomial spreading times in the Euclidean distance they bridge, both in 1-FPP here and in LR-FPP in [@chatterjee2016multiple]. Thus, transmission times in 1-FPP are not equivalent to graph distances in any inhomogeneous percolation on the underlying graph. Table [2](#table:phases-top-FPP){reference-type="ref" reference="table:phases-top-FPP"} summarises known results on 1-FPP, LR-FPP, and graph distances in spatial graphs. Now we elaborate on each phase. 0.5em **The polylogarithmic phase.** Theorem [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"} proves polylogarithmic cost-distances in 1-FPP when $\tau\in(2,3)$, and either $\mu\in (\mu_{\mathrm{exp}}, \mu_{\mathrm{log}})$ or $\alpha\in(1,2)$. The results here, in [@komjathy2020stopping] and the accompanying [@komjathy2022one2] (Corollary [Corollary 5](#cor:polynomial_regime){reference-type="ref" reference="cor:polynomial_regime"}) together imply that $\mu_{\mathrm{exp}}$ and $\mu_{\mathrm{log}}$ are true phase-transition points, separating this phase from both the explosive and the polynomial phases. Even though we do not have a matching lower bound, we conjecture that this phase is truly polylogarithmic, and the exponent $\Delta_0$ in [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"} is sharp. The exponent $\Delta_0$ also depends on the product $\mu\beta$, which does not allow to match it easily to exponents for graph-distances: For long-range percolation, where each edge $(u,v)\in\mathbb{Z}^d\times \mathbb{Z}^d$ is present independently with probability $\Theta(|u-v|^{-d\alpha})$, Biskup and Lin [@biskup2019sharp] show that graph distances grow polylogarithmically with exponent $\Delta_{\alpha} = 1/(1-\log_2 \alpha)$ when $\alpha\in(1,2)$. This coincides with our upper bound in Theorem [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"} if $\alpha\le \tau-1+\mu\beta$. The same type of paths are used in both cases, passing through only low-degree vertices (and typical edge-costs on them for 1-FPP). For scale-free percolation, Hao and Heydenreich proved in [@hao2021graph Theorem 1.1] that graph distances are also polylogarithmic in scale-free percolation when $\alpha\in(1,2)$ and additionally $\tau>3$ (using our parametrisation of SFP). Their exponent $\Delta_{\mathrm{SFP}}$ agrees with the exponent $\Delta_{\alpha}$ if $\alpha<(\tau-1)/2$, but when $\tau-2 < \alpha$ they obtain $\Delta_{\mathrm{SFP}}\le 1/(1-\log_2(\tau-2))< \Delta_\alpha$, realised by paths on high-degree vertices, and they give a (non-matching) polylogarithmic lower bound. It remains open to determine the exact exponent for graph distances in SFP, and to give a lower bound for 1-FPP. The lower-bound methods in [@biskup2019sharp] or [@hao2021graph] do not transfer to 1-FPP. 0.5em **The linear phase.** Linear distances are common in supercritical spatial graph models with bounded edge-lengths. For example, Random Geometric Graphs exhibit linear distances [@penrose2003random], and so does supercritical percolation on grids of dimension at least $2$ [@antal1996chemical]. Assuming high enough edge-density, a renormalisation argument to percolation on $\mathbb{Z}^d$ gives that SFP and LRP for $\tau>3$ and $\alpha >2$ also have at most linear graph-distances for $d\geq 2$. The corresponding lower bound was shown by Berger for LRP [@berger2004lower] and by Deprez *et al.* for SFP [@deprez2015inhomogeneous]. Our lower bound for 1-FPP contains these as special cases, and holds universally for classical FPP for arbitrary non-zero edge-transmission times in this phase [@komjathy2022one2 Corollary 1.12]. 0.5em **The strictly polynomial phase.** The phase where intrinsic distances scale as $|x-y|^{\eta_0+o(1)}$ with $\eta_0<1$ (the result of Theorem [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"}) is quite rare in spatial settings and we only know two examples. One is in one-dimensional LRP at a boundary line in the parameter space, when $\alpha=2$, the unpublished [@ding2013distances]. The other is for long-range first passage percolation (LR-FPP) in [@chatterjee2016multiple], mentioned at the beginning of this section. There are some similarities to 1-FPP: LR-FPP is Markovian, i.e., $\beta=1$ in [\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"}, and has strictly polynomial growth when $\alpha'\in(2,2+1/d)$, see Table [2](#table:phases-top-FPP){reference-type="ref" reference="table:phases-top-FPP"}. Using exponential $L$ in 1-FPP, the length of the parameter interval $(\mu_{\log}, \mu_{\mathrm{pol}})$ with polynomial growth is also exactly $1/d$ for $\mu$ when $\alpha>2+1/d$, but it is longer when $\alpha<2+1/d$, which shows that the penalty $\alpha'$ of LR-FPP plays a slightly different role as the long-range parameter $\alpha$ in [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"} here. 0.5em **Gaps at approaching the phase boundaries.** Here we discuss what happens as the parameters $\tau, \alpha, \mu, \beta$ approach the phase boundaries of growth. Polylogarithmic distances with exponent $\Delta_0$ heuristically imply *stretched exponential ball-growth*, where the number of vertices within intrinsic distance $r$ scales as $\exp(r^{1/\Delta_0})$. Our upper bound exponent $\Delta_0=\min\{\Delta_\alpha, \Delta_\beta\}$ in [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"} approaches $1$ as $\alpha \downarrow 1$, and so does the exponent $\Delta_{\alpha}$ of LRP [@biskup2004scaling], which also partly governs SFP. This means that as $\alpha\downarrow 1$ we approach exponential growth. In LRP, strictly exponential ball growth only occurs when $\alpha=1$ and the connectivity function has a suitably chosen slowly varying correction term $\ell(\cdot)$, i.e., $p(x,y)= \ell(|x-y|)/|x-y|^{\alpha d}$, see Trapman [@trapman2010growth]. Strictly exponential growth is a natural barrier, since (age-dependent) branching processes with finite first moments exhibit at most exponential growth, and non-Markovian FPP can be dominated by such branching processes. Interestingly, when $\alpha>2$ and we approach the phase transition of explosion by letting $\mu\downarrow\mu_{\mathrm{expl}}=(3-\tau)/(2\beta)$, then $\Delta_0$ in [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"} does not converge to $1$, but to $1/(2-\log_2(\tau+1))=:\Delta_{\tau}$. So, for the whole range $\tau \in (2,3)$, $\Delta_{\tau}\ge 1/(2-\log_2(3)) > 2.4 > 1$. As $\tau \uparrow 3$, $\Delta_{\tau}$ approaches $\infty$, which is natural, since already graph-distances are linear when $\tau >3$ and $\alpha >2$ [@deprez2015inhomogeneous]. This leaves two possibilities: either our upper bound $\Delta_0$ is not sharp for $\alpha>2$; or the ball growth jumps directly from subexponential ($\Delta_0 >1$) into the explosive phase. If the latter should be the case, it would be interesting to understand better how such a jump could happen. Such jumps at phase boundaries may occur. This paper together with [@komjathy2022one2] proves a gap in the *polynomial regime* as $\tau$ crosses the threshold $3$. The limits of $\lim_{\tau\uparrow 3}\mu_{\textrm{pol}}=1/d$ and $\lim_{\tau\uparrow 3}\eta_0=\mu d$ exist and are in $(0,1)$ in [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"} and [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}. So if we fix some $\mu < 1/d$ and let $\tau \uparrow 3$, the cost-distances grow polynomially with exponents bounded away from one (e.g., they approach $1/2$ from below for $\mu = 1/(2d)$). But as soon as $\tau > 3$ is reached, the exponent "jumps" to $1$ and distances become strictly linear [@komjathy2022one2 Theorem 1.11]. So the parameter space is discontinuous in $\eta_0$ and $\mu_{\textrm{pol}}$ with respect to $\tau$. Some important questions are centred around such gaps. The *gap conjecture* in geometric group theory is about the ball growth of finitely generated groups: it states that there are no groups with growth between polynomial and stretched exponential of order $\exp(\Theta(\sqrt{n}))$ [@grigorchuk1990growth]. While the polynomial side is understood by Gromov's theorem [@gromov1981groups], the conjecture remains open. We find it intriguing to discover a deeper connection between phases of intrinsic growth in spatial random graphs ("stochastic lattices") and group theory ("deterministic lattices"). 0.5em **One-dependent processes in the applied context.** While our paper is theoretical, we mention some broader contexts where 1-FPP may turn out to be useful. In the spread of physical epidemics, while many diseases spread at an exponential rate, others spread at a polynomial rate, dominated by the local geometry. Examples of the latter include HIV/AIDS, Ebola, and foot-and-mouth disease, see the survey [@polyepidemicsurvey] on polynomial epidemic growth. This distinction is widely believed to be driven by network effects, but classical epidemic models can typically only model either exponential or polynomial growth, not both. Arguably, $1$-FPP provides a natural explanation, since in $1$-FPP the transition can be driven by changes only to the transmission dynamics, not to the underlying network. Further, one-dependent processes in general (beyond FPP) allow for more realistic modelling of real phenomena. In social networks, actual contacts do not scale linearly with the degree due to limited time or awareness [@feldman2017high]. 1-FPP type penalisation has been used to model the sublinear impact of superspreaders as a function of contacts [@giuraniuc2006criticality; @karsai2006nonequilibrium; @miritello2013time]. One-dependent processes such as 'topology biased' random walk has been used for improving estimates on graph-parameters [@bonaventura2014characteristic; @ding2018centrality; @pu2015epidemic]. 0.5em **Organisation.** We start by moving to the quenched setting. In Section [2](#sec:nets){reference-type="ref" reference="sec:nets"} we develop the pseudorandom nets, and in Section [3](#sec:exposure){reference-type="ref" reference="sec:exposure"} the multi-round exposure of edges, with the main result in Proposition [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}. In Section [4](#sec:lemmas){reference-type="ref" reference="sec:lemmas"} we collect some connectivity-estimates that serve as our building blocks, while in Section [5](#sec:hierarchy){reference-type="ref" reference="sec:hierarchy"} we carry out the 'budget travel plan' and build a hierarchical path that only uses vertices of the pseudorandom net and connects vertices $y_0^\star, y_x^\star$ very close to $0$ and $x$, respectively. In Section [6](#sec:endpoints){reference-type="ref" reference="sec:endpoints"} we connect $0,x$ to these vertices with low cost, which is a nontrivial task itself, and prove the main theorems. ### Notation {#sec:notation} Throughout, we consider simple and undirected graphs with vertex set $\mathcal{V}\subseteq \mathbb{R}^d$. For a graph $G=(\mathcal{V},\mathcal{E})$ and a set $A\subseteq \mathbb{R}^d$, $G[A]$ stands for the induced subgraph of $G$ with vertex set $\mathcal{V}\cap A$. For two vertices $x,y\in \mathcal{V}$, we denote the edge between them by $xy$, and for a set $V\subseteq \mathcal{V}$ we write $V^{(2)} := \{\{x,y\} : x,y\in V, x\neq y\}$ for the set of possible edges among vertices in $V$. For a path $\pi$, $\mathcal{E}(\pi)$ is the set of edges forming $\pi$, and $|\pi|$ is the number of edges of $\pi$. Generally the size of a discrete set $S$ is $|S|$, while of a set $A\subseteq \mathbb{R}^d$, $\mathrm{Vol}(A)$ is its Lebesgue measure. Given a cost function $\mathcal{C}: \mathcal{E}\to [0,\infty]$ on the edges, the cost of a set of edges $\mathcal{P}$ is $\mathcal{C}(\mathcal{P}):=\sum_{e\in\mathcal{E}(\mathcal{P})}\mathcal{C}(e)$. We define $\mathcal{C}(xx):=0$ for all $x\in \mathcal{V}$. We define the *cost-distance* between vertices $x$ and $y$ as $$\begin{aligned} \label{eq:cost_distance} d_{\mathcal{C}}(x,y):=\inf\{\mathcal{C}(\pi) : \pi \textnormal{ is a path from } x \textnormal{ to } y \textnormal{ in $G$}\}.\end{aligned}$$ We define the graph distance $d_G(x,y)$ similarly, where all edge-costs are set to $1$. We denote the Euclidean norm of $x \in \mathbb R^d$ by $|x|$, the Euclidean ball with radius $r\geq 0$ around $x$ by $B_r(x) := \{y\in \mathbb{R}^d : |x-y|\le r\}$, and the set of vertices in this ball by $\mathcal{B}_r(x):=\{y\in\mathcal{V}: |x-y|\le r\} = B_r(x) \cap \mathcal{V}$. (The minimal notation difference is intentional). The *graph-distance ball* and *cost-distance ball* (or *cost-ball* for short) around a vertex $x$ are the vertex sets $\mathcal{B}_r^G(x):=\{y\in\mathcal{V}: d_G(x,y)\le r\}$ and $\mathcal{B}_r^{\mathcal{C}}(x):=\{y\in\mathcal{V}: d_{\mathcal{C}}(x,y)\le r\}$, respectively. We set $B_r := B_r(0)$, and do similarly for $\mathcal{B}_r$, $\mathcal{B}_r^G$, $\mathcal{B}_r^{\mathcal{C}}$ if $0$ is a vertex. We define $\partial B_r(x) := B_r(x)\setminus \{y\in \mathbb{R}^d : |x-y| < r\}$, and use similar definitions for $\partial \mathcal{B}_r$, $\partial \mathcal{B}_r^G$ and $\partial \mathcal{B}_r^{\mathcal{C}}$. In particular, $\partial \mathcal{B}_1^G(v)$ is the set of neighbours of $v$. In Definition [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}, $\widetilde v = (v,w_v)$ stands for a weighted vertex. For some $A \subset \mathbb{R}^d$, we simply write $\widetilde{v} \in A$ to indicate that $(v, w_v) \in A \times [1, \infty)$, i.e., we implicitly project $\widetilde{v}$ onto Euclidean space. The set of *model parameters* are $\textnormal{\texttt{par}}\xspace:= \{d, \tau, \alpha, \mu, \beta, \underline{c}, \overline{c}, c_1, c_2, t_0\}$. For parameters $a,b >0$ (model parameters or otherwise), we use "for all $a{\,\gg_{\star}\,}b$" as shortcut for "$\forall b>0:\, \exists a_0 = a_0(b):\, \forall a\ge a_0$". We also say "$a {\,\gg_{\star}\,}b$" to mean that $a \ge a_0(b)$. We use $a {\,\ll_{\star}\,}b$ analogously, and may use more than two parameters. For example, "for $a{\,\gg_{\star}\,}b,c$" means "$\forall b,c>0:\, \exists a_0=a_0(b,c):\, \forall a \ge a_0$". A measurable function $\ell:(0,\infty) \to (0,\infty)$ is said to be *slowly varying* at infinity if $\lim_{x\to \infty} \ell(cx)/\ell(x) =1$ for all $c>0$. We denote by $\log$ the natural logarithm, by $\log_2$ the logarithm with base $2$, and by $\log^{*k}$ the $k$-fold iterated logarithm, e.g. $\log^{*3}x = \log\log\log x$. For $n \in \mathbb{N}$ we write $[n]:= \{1,\ldots,n\}$. # Moving to the quenched setting: pseudorandom nets {#sec:nets} In proving the upper bounds (Theorems [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"} and [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"}), we will construct cheap paths along the lines of the 'budget travel plan' in Section [1](#sec:intro){reference-type="ref" reference="sec:intro"}, which is an iterative scheme of finding long 3-edge bridge-paths to connect two far-away vertices. Since low-cost events in 1-FPP are not increasing, we develop a technique that replaces the FKG-inequality. Moving to the quenched setting, we will first expose all vertex positions and weights (above some threshold weight, in the case of IGIRG); then, low-cost edge existence events become independent. To be able to work with *fixed realisations* of the vertex set, we find (with high probability as $|x|\to\infty$) a *pseudorandom* subset of the vertices that behaves regularly enough inside a box around $0,x$, as in [\[eq:net-heuristics\]](#eq:net-heuristics){reference-type="eqref" reference="eq:net-heuristics"}, which we call a *net*. We formalise the notion of the nets now. For a set $A\subset \mathbb{R}^d$ we write $\mathrm{Vol}(A)$ for its Lebesgue measure (volume), while for a discrete set $\mathcal{A}\subseteq (0,\infty)$ we write $|\mathcal{A}|$ for the cardinality (size) of the set. Recall the slowly varying function $\ell(\cdot)$ from [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}. Recall that weighted vertices are of the form $\tilde v=(v, w_v)$. **Definition 8** (Net). *Let $G=(\mathcal{V},\mathcal{E})$ be an IGIRG or SFP in Definition [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}. Let $\mathcal{R}\subseteq (0,\infty)$ be a set with $|\mathcal{R}|<\infty$, let $w_0 \in [1,\infty)$, and let $f\colon \mathcal{R}\to [w_0,\infty)$. Let $Q \subseteq \mathbb{R}^d$ be Lebesgue measurable. An *$(\mathcal{R},w_0,f)$-net for $Q$* is a set $\mathcal{N}\subseteq \widetilde{\mathcal{V}} \cap Q\times[1,\infty)$ of size at least $\mathrm{Vol}(Q)/4$ such that for all $\tilde v \in \mathcal{N}$, $r \in \mathcal{R}$, and all $w \in [w_0,f(r)]$, $$\label{eq:net-defining-crit} \left| \big\{\widetilde u\in \mathcal{N}\cap B_r(v) \times [w/2, 2w] \big\}\right| \ge r^d\cdot \ell(w)w^{-(\tau-1)}/(2d)^{d+\tau+5}.$$* It may seem very strong that we require [\[eq:net-defining-crit\]](#eq:net-defining-crit){reference-type="eqref" reference="eq:net-defining-crit"} for infinitely many $w$. We discretise $[w_0, f(r)]$ into a finite set of subintervals $(I_j)_{j\le j_{\max}}$ in a smart way. Then we ensure that [\[eq:net-defining-crit\]](#eq:net-defining-crit){reference-type="eqref" reference="eq:net-defining-crit"} holds with $[w/2, 2w]$ replaced by $I_j$ on the left hand side and the $\ell(w)w^{-(\tau-1)}$ replaced by $\mathbb P(W\in I_j)$ on the right hand side, and then this will imply that [\[eq:net-defining-crit\]](#eq:net-defining-crit){reference-type="eqref" reference="eq:net-defining-crit"} also holds for all values $w\in [w_0, f(r)]$. Now we set $f(r)$ only a little less than the typical largest vertex weight in a ball of radius $r$ (roughly $r^{d/(\tau-1)}$), and $w_0$ is a large constant, starting with $w_0$: **Definition 9** (Strong net). *Fix $\ell(w)$ from [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}. We define $w_0$ to be the smallest integer in $[1,\infty)$ such that for all $w \ge w_0$ and all $t \in [1/2,2]$, $$\label{eq:nets-ell-bound-0} \ell(w)w^{-(\tau-1)} < 2^{-\tau-8} \quad \mbox{and}\quad 0.99 \le \ell(t w)/\ell(w) \le 1.01$$ both hold. For all $\delta>0$, and $R>0$ we define the function $$\label{eq:nets-f-def} f_{R,\delta}(r) = r^{\frac{d}{\tau-1}} \big(1 \wedge \inf\big\{\ell(x)\colon x \in [w_0,r^{d/(\tau-1)}]\big\}\big)^{\frac{1}{\tau-1}}\cdot \Big(\frac{1}{(2d)^{2\tau+d+8}\log(16 R/\delta)}\Big)^{\frac{1}{\tau-1}}.$$ On fixing $\delta>0$, $w_0$ satisfying [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"}, and $R:=|\mathcal{R}|$ in [\[eq:nets-f-def\]](#eq:nets-f-def){reference-type="eqref" reference="eq:nets-f-def"}, we call an $(\mathcal{R},w_0,f_{R,\delta})$-net in Definition [Definition 8](#def:net){reference-type="ref" reference="def:net"} for a set $Q \subseteq \mathbb{R}^d$ an *$(\delta, \mathcal{R})$-net for $Q$*.* Note that $w_0$ satisfying [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"} must exist since $\ell$ is a slowly-varying function, and so Potter's bound [@bingham1989regular] ensures the first inequality. In order to find an $(\delta,\mathcal{R})$-net, and specify the values $r\in \mathcal{R}$, for technical reasons it is convenient to assume that the values in $\mathcal{R}$ grow at least exponentially with base depending on $|\mathcal{R}|$. In the main proofs, we later choose $|\mathcal{R}|$ to be a either a constant or doubly logarithmic in the distance of the two vertices we want to connect via a cheap path. The specific condition is the following. **Definition 10**. *Fix $\delta \in(0,1)$. We say a set $\mathcal{R}\subseteq (0,\infty)$ is *$\delta$-well-spaced* if $|\mathcal{R}|=:R<\infty$ and the following hold, writing $\mathcal{R}= \{r_1,\ldots,r_R\}$ with $r_1 < \ldots < r_R$:* *$$\begin{aligned} r_1 &\ge 24d\big(\log(4R/\delta)\big)^{1/d} \vee w_0^{(\tau-1)/d} \vee \inf\{r \colon f_{R,\delta}(r) \ge w_0\}; \label{eq:nets-small-r}\\ \frac{r_i}{r_{i-1}} &\ge 6R^{1/d}\Big(\frac{\log(2R/\delta)}{\delta}\Big)^{1/d} \quad \forall i\in [2,R]. \label{eq:nets-radii} \end{aligned}$$* We now state the main result of the section. Heuristically, a box $Q$ contains an $(\delta,\mathcal{R})$-net with sufficiently high probability, and we can also condition on the presence of a few vertices in the net. The condition $t\le 1/\delta$ in the following is added to avoid a vacuous statement. propositionNetsExist[\[lem:nets-exist\]]{#lem:nets-exist label="lem:nets-exist"} Consider IGIRG or SFP with $\tau > 2$. Let $\delta \in (0,1/16), \xi > 0$, and $Q \subseteq \mathbb{R}^d$ be a cube of side length $\xi$. Let $R\in \mathbb{N}$ and $\mathcal{R}=\{r_1, \ldots, r_R\}$ with $0<r_1<\ldots<r_R$ be a $\delta$-well-spaced set such that $r_R = \xi\sqrt{d}$. Let $x_1,\ldots,x_t \in Q$ with $t \le \min\{1/\delta,(r_1/4\sqrt{d})^d\}$. Then $$\begin{aligned} \mathbb{P}&(\text{$Q$ contains an $(\delta,\mathcal{R})$-net $\mathcal{N}$})\ge 1-\delta/R\label{eq:netsexist-1};\\ \mathbb{P}&(\text{$Q$ contains an $(\delta,\mathcal{R})$-net $\mathcal{N}$}, x_1,\ldots,x_t \in \mathcal{N}\mid x_1,\ldots,x_t \in \mathcal{V}) \ge 1-t\delta. \label{eq:strong-netsexist} \end{aligned}$$ The rest of this section is devoted to proving Proposition [\[lem:nets-exist\]](#lem:nets-exist){reference-type="ref" reference="lem:nets-exist"}. All remaining definitions and lemmas are used only within this section. We now give the setting throughout this section. **Setting 11**. Consider IGIRG or SFP with $\tau > 2$ in Definition [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}. Let $\delta \in (0,1/16)$, $\xi > 0$, $Q \subseteq \mathbb{R}^d$ be a $d$-dimensional cube of side length $\xi$, and $\mathcal{R}= \{r_1,\ldots,r_R\} \subseteq (0,\infty)$ a $\delta$-well-spaced set with $r_1 < \ldots < r_R = \xi\sqrt{d}$. Shortly we shall carry out a multi-scale analysis. We partition $Q\times [w_0, f(r_R)]$ into hyper-rectangles. On the weight-coordinate, we cover the interval $[w_0,f(r_R)]$ of weights with a set of disjoint intervals $(I_j)_{j=1, \ldots, j_{\max}}$ so that the first interval is of length $w_0$, and each consecutive interval is twice as long as the previous one. On the space-marginal, we partition $Q$ into nested boxes $B$. The side lengths of these nested boxes will be close to $r_1, \ldots, r_R$, with some minor perturbation so that they can form a proper nested partition: we write $r_i' \approx r_i$ for the side length of the $i$'th level of boxes. A depiction and extended example can be found in Figure [1](#fig:i-good){reference-type="ref" reference="fig:i-good"} on page  below, after the formal definition. After fixing this partitioning of $Q\times [w_0,f(r_R)]$, we look at $\widetilde {\mathcal{V}}\cap (Q\times [w_0,f(r_R)])$. For each $i \in [R]$, we show that with probability close to $1$ there is a dense subset of "good" boxes $B$ of side length $r_i'$, in the sense that $B \times I_j$ contains the right number of vertices for all $I_j$ with $\max(I_j) \approx f(r_i)$. We choose $r_i'<r_i/\sqrt{d}$ to ensure that for all vertices $v$ in a box of side length $r_i'$, the entire box will be contained in $B_{r_i}(v)$ -- the ball of radius $r_i$ around $v$ -- this will allow us to take the net to be the set of all vertices which lie in good boxes of all side lengths $r_1',\ldots,r_R'$. We start with the space marginal and now formally define the nested boxes. **Definition 12**. *Given $\mathcal{R}=\{r_1, \ldots, r_R\}$, and $Q$ as in Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"}, an *$\mathcal{R}$-partition* of $Q$ is a collection of partitions $\widehat{\mathcal{P}}(\mathcal{R}):=\{\mathcal{P}_1,\ldots,\mathcal{P}_R\}$ of $Q$ into boxes with the following properties:* 1. *[\[item:P1\]]{#item:P1 label="item:P1"} For all $i \in [R]$, every box in $\mathcal{P}_i$ has the same side length $r_i'$ with $$\label{eq:ri-bound} r_i'\in [r_i/(2\sqrt{d}), r_i/\sqrt{d}].$$* 2. *[\[item:P2\]]{#item:P2 label="item:P2"} For all $i \in [R-1]$, every box in $\mathcal{P}_{i+1}$ is partitioned into exactly $(r_{i+1}')^d/(r_i')^d$ boxes in $\mathcal{P}_i$.* 3. *[\[item:P3\]]{#item:P3 label="item:P3"} We have $\mathcal{P}_R = \{Q\}$.* 4. *[\[item:P4\]]{#item:P4 label="item:P4"} For $x \in Q, i \!\in\! [R]$, write $B^i(x)$ for the box in $\mathcal{P}_i$ containing $x$. Then $B^i(x) \subseteq B_{r_i}(x)$.* Observe that *(P2)* ensures that the partition $\mathcal{P}_i$ is a refinement of the partition of $\mathcal{P}_{i+1}$, i.e., that every box in $\mathcal{P}_{i+1}$ can be partitioned exactly into sub-boxes in $\mathcal{P}_i$. Also, $r_R=\xi\sqrt{d}$ ensures that *(P1)* and *(P3)* can be simultaneously satisfied for $i=R$. **Claim 13**. *Suppose $\mathcal{R}$ and $Q$ satisfy Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"}. Then an $\mathcal{R}$-partition $\widehat{\mathcal{P}}(\mathcal{R})$ of $Q$ exists.* *Proof.* We prove that given $r_1, \ldots, r_R$, there exist side lengths $r_1',\ldots,r_R'$ that satisfy *(P[\[item:P1\]](#item:P1){reference-type="ref" reference="item:P1"}) -- (P[\[item:P4\]](#item:P4){reference-type="ref" reference="item:P4"})*, i.e., that $r_{i+1}'/r_i'$ is an integer, [\[eq:ri-bound\]](#eq:ri-bound){reference-type="eqref" reference="eq:ri-bound"} holds, and $r_R'=\xi$. Clearly [\[eq:ri-bound\]](#eq:ri-bound){reference-type="eqref" reference="eq:ri-bound"} implies that for any vertex $v$ in a box $B$ of side-length $r_i'$, $B\subseteq B_{r_i}(v)$, hence *(P[\[item:P4\]](#item:P4){reference-type="ref" reference="item:P4"})* follows directly once we satisfy [\[eq:ri-bound\]](#eq:ri-bound){reference-type="eqref" reference="eq:ri-bound"} and allocate box-boundaries uniquely. We proceed by induction on $i$, starting from $i=R$ and decreasing $i$. We take $r_{R}' := r_{R}/\sqrt{d}=\xi$, then [\[eq:ri-bound\]](#eq:ri-bound){reference-type="eqref" reference="eq:ri-bound"} is satisfied immediately and *(P[\[item:P2\]](#item:P2){reference-type="ref" reference="item:P2"})--(P[\[item:P3\]](#item:P3){reference-type="ref" reference="item:P3"})* are vacuous. Suppose we have found $r_i',\ldots,r_{R}'$ satisfying *(P[\[item:P1\]](#item:P1){reference-type="ref" reference="item:P1"})--(P[\[item:P3\]](#item:P3){reference-type="ref" reference="item:P3"})* for some $2 \le i \le R$. Let $$\label{eq:def-ri-prime} r_{i-1}' = \frac{r_i'}{\lceil \sqrt{d} r_{i}'/r_{i-1} \rceil}.$$ This choice of $r_{i-1}'$ divides $r_i'$, hence *(P2)* can be satisfied, and $r_{i-1}' \le r_i'/(\sqrt{d}r_i'/r_{i-1}) = r_{i-1}/\sqrt{d}$. Moreover, $$\label{eq:nets-partition-exists} r_{i-1}' \ge \frac{r_i'}{1 + \sqrt{d} r_i'/r_{i-1}} = \frac{r_{i-1}}{r_{i-1}/r_i'+\sqrt{d}}.$$ Since [\[eq:ri-bound\]](#eq:ri-bound){reference-type="eqref" reference="eq:ri-bound"} holds for $i$ (by the inductive assumption), $r_i' \ge r_i/2\sqrt{d}$. Since $R$ is well-spaced, $r_{i-1} \le r_i/2$ by [\[eq:nets-radii\]](#eq:nets-radii){reference-type="eqref" reference="eq:nets-radii"}; hence $r_{i-1}/r_i' \le \sqrt{d}$. It follows from [\[eq:nets-partition-exists\]](#eq:nets-partition-exists){reference-type="eqref" reference="eq:nets-partition-exists"} that $r_{i-1}' \ge r_{i-1}/2\sqrt{d}$, and so [\[eq:ri-bound\]](#eq:ri-bound){reference-type="eqref" reference="eq:ri-bound"} holds also for $i-1$ and the induction is advanced. Given these $r_1',\ldots,r_R'$, we find an $\mathcal{R}$-partition of $Q$ by taking $\mathcal{P}_R = \{Q\}$ and iteratively forming each layer $\mathcal{P}_{i-1}$ by taking the unique partition of each box in $\mathcal{P}_i$ into $(r_i')^d/(r_{i-1}')^d$ sub-boxes of side length $r_{i-1}'$. We first define each partition box to be of the form $\prod_{j=1}^d[a_j, b_j)$, this allocates each point except $d$ of the $d-1$-dim faces of $\partial Q$ uniquely. Finally, we allocate the points $x\in\partial Q$ in $\mathcal{P}_i$ to the box in $\mathcal{P}_i$ that contains $x$ in its closure, this box is unique except on $d-2$ dimensional faces. Here we again use half-open $d-1$-dim boxes to determine the $d-2$ dim boundaries, and so on until only the corner-points are left which we allocate arbitrarily (but consistently across different $i$). ◻ We continue with the weight-marginal and cover the interval $[w_0,f(r_i)]$ of weights with a collection of intervals, forming later the weight-coordinate of the hyper-rectangles: **Definition 14** (Base-$2$-cover). *Given a closed interval $J = [a,b] \subset \mathbb{R}_+$, let $j_{\max}:=\lfloor\log_2(b/a)\rfloor+1$, and define $I_j:=[2^{j-1} a, 2^{j}a)$ for $j\in[j_{\max}]$. Then $J\subseteq \bigcup_{j=1}^{j_{\max}}I_j$ and we call $I=\{I_j\}_{j\le j_{\max}}$ the *base-$2$-cover* of $[a,b]$. For each $x\in [a,b]$ we define $I(w):=\{I_j: w\in I_j\}$ to be the unique interval that contains $x$, and write $I(w)=[w_-, w_+)$ for its endpoints.* For each $w\!\in\![a,b]$, $w/2\le w_-$ and $w_+\le 2x$, so $I(w)\subseteq [w/2, 2w]$; by the definition of $I_{j_{\max}}$, we also have $b \in I_{j_{\max}}$. We define the hyperrectangle-covering of the box $Q$ including vertex-weights now. Recall vertex-weight distribution $W$ from [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}, and $f(r)$ from Def. [Definition 8](#def:net){reference-type="ref" reference="def:net"}. **Definition 15** (Hyperrectangles). *Consider Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"} and Definitions [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"}, [Definition 14](#def:base-2-cover){reference-type="ref" reference="def:base-2-cover"}. Let $\widehat {\mathcal{P}}(\mathcal{R}):=\{\mathcal{P}_1, \ldots, \mathcal{P}_R\}$ be an $\mathcal{R}$-partition of the cube $Q$ with $\mathcal{R}=\{r_1, \ldots, r_R\}$, $r_i'$ be the side-lengths in $\mathcal{P}_i$, and let $I=\{I_j\}_{j\le j_{\max}}$ be a base-$2$-cover of $[w_0, f(r_R)]$. Let $j_{\star}(i)$ be the index of the interval that contains $f(r_i)$, i.e., $$\label{eq:jstar} f(r_i) \in I(f(r_i))=:I_{j_\star(i)}.$$ Then we say that the collection $\mathcal{H}(\mathcal{R}):=\big\{ B_i\times I_j: B_i\in \mathcal{P}_i, 1 \le j \le j_{\star}(i) \big\}$ is a *hyperrectangle-cover of $Q\times [w_0, f(r_R)]$*. For all $i \in [R]$ and all $A \subset [w_0,f(r_R)]$, we define $$\label{eq:measure-hyperrectangle} \mu_i(A) :=(r_i')^d\cdot \mathbb P(W\in A).$$* When we cover with boxes in $\mathcal{P}_i$ on the spatial coordinate, the number $j_{\star}(i)$ of weight intervals in $\mathcal{H}(\mathcal{R})$ depends on $i$. In particular, for smaller side-length we do not include intervals of very large weights. This is because there are too few (or no) vertices of large weight in a typical box of small side-length, so we cannot control their number. We illustrate a hyperrectangle cover on Figure [1](#fig:i-good){reference-type="ref" reference="fig:i-good"} in dimension $1$. In GIRG, $\mu_i(A)$ is the expected number of vertices with weights in $A$ in any box in $\mathcal{P}_i$. In SFP, $\mu_i(A)$ is only roughly the expectation, since e.g. a box touching the boundary $\partial Q$ in $\mathcal{P}_i$ may not contain exactly $(r_i')^d$ vertices. By Definition [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"} *(P[\[item:P4\]](#item:P4){reference-type="ref" reference="item:P4"})*, all vertices in the hyperrectangle $B^i(v) \times I_j$ are within distance $r_i$ of $v$. Hence once we control the number of vertices in a dense set of hyperrectangles in all partitions $i\in[R]$, we can find a net. We now define a hyperrectangle being "good", with respect to a realisation of $\widetilde{\mathcal{V}}$. Recall that $I(w)$ denotes the interval $I_j$ that contains $w$ in Definition [Definition 14](#def:base-2-cover){reference-type="ref" reference="def:base-2-cover"}. **Definition 16**. *Consider the setting of Def. [Definition 15](#def:hyperrectangles){reference-type="ref" reference="def:hyperrectangles"}, and let a $\mathcal{H}(\mathcal{R})$ be a hyperrectangle-cover of $Q\times [w_0, f(r_R)]$. Consider a realisation of the weighted vertex set $\widetilde {\mathcal{V}} =\big((v, w_v)\big)_{v\in \mathcal{V}}$.* *We recursively define when we call a vertex $\widetilde v\in \widetilde{\mathcal{V}}$ and a box $B\in \widehat{\mathcal{P}}(\mathcal{R})$ good. Every vertex is $1$-good. For all $i \in [R]$, we say a vertex $\tilde v=(v, w_v)\in \widetilde{\mathcal{V}}$ is *$i$-good* if the boxes $B^1(v),\ldots,B^{i-1}(v)$ are all good (which we define next). Denote the set of $i$-good (weighted) vertices by $\widetilde{\mathcal{G}}_i:=\{\widetilde v \in {\widetilde \mathcal{V}} \ i\text{-good}\}$ and $\mathcal{G}_i:=\{v: \widetilde v \in \widetilde{\mathcal{G}}_i\}$. We say that a box $B \in \mathcal{P}_i$ is $i$-*good* or simply good if the following conditions all hold:* 1. *[\[item:B1\]]{#item:B1 label="item:B1"} Either $i=1$, or $B$ contains at least $1-\tfrac{2\delta}{R}\cdot (r_i'/r_{i-1}')^d$ many $i-1$-good sub-boxes $B'\in\mathcal{P}_{i-1}$.* 2. *[\[item:B2\]]{#item:B2 label="item:B2"} The total number of $i$-good vertices in $B$ satisfies $$\label{eq:i-good-total} |\mathcal{G}_i \cap B | \in \Big[\Big(\frac{1}{2} - \frac{2(i-1)\delta}{R}\Big)(r_i')^d, 2(r_i')^d\Big].$$* 3. *[\[item:B3\]]{#item:B3 label="item:B3"} For all $w \in [w_0,f(r_i)]$, the number of $i$-good vertices in $B$ with weight in $I(w)$ satisfies $$\label{eq:i-good-weight} |\widetilde{\mathcal{G}_i}\cap (B\times I(w))| \in \Big[\frac{1}{8} \Big(1 - \frac{2i\delta}{R}\Big)\mu_i(I(w)), 8\mu_i(I(w))\Big].$$* *Finally, we say that the realisation $\widetilde {\mathcal{V}}$ is *good* wrt the hyperrectangle-cover $\widetilde {\mathcal{P}}(\mathcal{R})$ if $Q$ is $R$-good.* The above definition is *not circular*; the definition of $i$-good vertices depends only on the definition of good boxes in $\mathcal{P}_{i-1}$, i.e., one level lower, and then the definition of a good box in $\mathcal{P}_i$ depends only on the number of $i$-good vertices in it (and their weights) and its number of good subboxes in $\mathcal{P}_{i-1}$. For $i=1$, the (longer) definition of $1$-good vertices is vacuous, so every vertex is indeed $1$-good which we emphasised in the definition. Further, $\mathcal{G}_R\subseteq \mathcal{G}_{R-1}\subseteq \ldots \subseteq \mathcal{G}_1=\mathcal{V}\cap Q$, since each $i$-good vertex is also $(i\!-\!1)$-good for all $i\le R$. See Figure [1](#fig:i-good){reference-type="ref" reference="fig:i-good"} for a graphical depiction of $i$-good vertices and boxes. {#fig:i-good width="70%"} Before we relate goodness to our overall goal of finding an $(\delta,\mathcal{R})$-net, we give some easy algebraic bounds which we need multiple times in the rest of the section. We defer the proof to the appendix. Recall that $I(w)=I_j$ iff $w\in I_j$ (cf. Def. [Definition 14](#def:base-2-cover){reference-type="ref" reference="def:base-2-cover"}). claimClaimPolylogRegime [\[claim:nets-mu-bound\]]{#claim:nets-mu-bound label="claim:nets-mu-bound"} Consider Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"} and Definitions [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"}, [Definition 15](#def:hyperrectangles){reference-type="ref" reference="def:hyperrectangles"}. Suppose $\widehat{\mathcal{P}}(\mathcal{R})= \{\mathcal{P}_1,\ldots,\mathcal{P}_R\}$ is an $\mathcal{R}$-partition of $Q$, and for all $i \in [R]$, let $r_i'$ be the side length of boxes in $\mathcal{P}_i$. Then for all $i \in [R]$ and all $w \in [w_0,f(r_R)]$, we have $$\begin{aligned} r_i^d\ell(w)w^{-(\tau-1)}/(2d)^{\tau+d+1} &\le \mu_i(I(w))= (r_i')^d \cdot \mathbb{P}(W\in I(w)) \le 2^\tau r_i^d\ell(w)w^{-(\tau-1)}, \label{eq:mui-bound}\\ r_i^d \ell(f(r_i)) f(r_i)^{-(\tau-1)} &\ge (2d)^{2\tau+d+8}\log(16R/\delta). \label{eq:z-bound} \end{aligned}$$ We now show that given that the box $Q$ is good with respect to a hyperrectangle cover, we can find an $(\delta,\mathcal{R})$-net for $Q$ (see Def. [Definition 8](#def:net){reference-type="ref" reference="def:net"}, [Definition 9](#def:net-constants){reference-type="ref" reference="def:net-constants"}). **Lemma 17**. *Consider Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"} and Definitions [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"}, [Definition 15](#def:hyperrectangles){reference-type="ref" reference="def:hyperrectangles"}, [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"}, i.e., consider a hyper-rectangle cover ${\mathcal{H}}(\mathcal{R})$ of $Q\times [w_0, f(r_R)]$. Consider a realisation of $\widetilde {\mathcal{V}}$ for which $Q$ is $R$-good. Then $\widetilde{\mathcal{G}}_R$, the set of all $R$-good vertices, forms an $(\delta,\mathcal{R})$-net for $Q$.* *Proof.* Suppose that $Q$ is $R$-good. The side length of $Q$ equals $r_R'$ by *(P3)* in Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"}, and $\delta\in(0,1/16)$ in Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"}, hence we may apply *(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* in Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"} for $i=R$ to get $$|\widetilde{\mathcal{G}}_R| \ge \Big(\frac{1}{2} - \frac{2(R-1)\delta}{R}\Big)\mathrm{Vol}(Q) \ge \Big(\frac{1}{2} - 2\delta\Big)\mathrm{Vol}(Q) > \mathrm{Vol}(Q)/4,$$ hence the cardinality assumption in Definition [Definition 8](#def:net){reference-type="ref" reference="def:net"} is satisfied for $\widetilde{\mathcal{G}}_R$. To show that $\widetilde{\mathcal{G}}_R$ satisfies Definitions [Definition 8](#def:net){reference-type="ref" reference="def:net"} with $f$ from Definition [Definition 9](#def:net-constants){reference-type="ref" reference="def:net-constants"}, we first show that for all $v\in \mathcal{G}_R$, $$\label{eq:good-also} \mathcal{G}_i \cap B^i(v)= \mathcal{G}_R \cap B^i(v).$$ Given an $i$-good vertex $u \in B^i(v)$ we show that $u$ is also $R$-good, i.e., that $B_j(u)$ is good for all $j \in [R]$. By the definition of $i$-goodness (Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"}), $B_j(u)$ is good for all $j \le i-1$, and $B_R(u) = Q$ is good by hypothesis. Consider now a $j \in [i+1, R-1]$. By Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"}, the partition $\mathcal{P}_i$ is a refinement of the partition $\mathcal{P}_j$, so $B^j(u) = B^j(v)$. Since $v$ is $R$-good, it follows that $B^j(u)$ is good. So, $B^j(u)$ is good for all $j \in [R]$, so $u$ is $R$-good, showing [\[eq:good-also\]](#eq:good-also){reference-type="eqref" reference="eq:good-also"}. Recall now that $B^i(v)$ is the box in $\mathcal{P}_i$ containing $v$, and $I(w)\subseteq [w/2, 2w]$ is the interval containing $w$ in the base-$2$-cover of $[w_0, f(r_i)]$. We now show that $$\begin{aligned} | \widetilde {\mathcal{G}}_R \cap (B_{r_i}(v) \times [w/2,2w])| \ge |\widetilde {\mathcal{G}}_R \cap (B^i(v)\times I(w))|=|\widetilde {\mathcal{G}}_i \cap (B^i(v)\times I(w))|.\end{aligned}$$ Indeed, $B^{i}(v)\subseteq B_{r_i}(v)$ by Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"} *(P3)* and $I(w)\subseteq [w/2, 2w]$ by Def. [Definition 14](#def:base-2-cover){reference-type="ref" reference="def:base-2-cover"}, and since all $i$-good vertices in $B^i(v)$ are also $R$-good by [\[eq:good-also\]](#eq:good-also){reference-type="eqref" reference="eq:good-also"}, the last inequality follows. Now we apply, on the rhs $|\widetilde{\mathcal{G}_i} \cap (B^i(v) \times I(w))|$ above, the lower bound from Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"} *(B[\[item:B3\]](#item:B3){reference-type="ref" reference="item:B3"})*, i.e., [\[eq:i-good-weight\]](#eq:i-good-weight){reference-type="eqref" reference="eq:i-good-weight"}, to obtain $$\begin{aligned} \big| \widetilde{\mathcal{G}}_R\cap (B^{i}(v)\times [w/2,2w])\big| &\ {\buildrel \eqref{eq:i-good-weight} \over \ge}\ \frac18\Big(1-\frac{2i\delta}{R}\Big) \mu_i(I(w)) \\&\ {\buildrel \eqref{eq:mui-bound} \over \ge} \frac{1}{8} \Big(1-\frac{2i\delta}{R}\Big)r_i^d\ell(w)w^{-(\tau-1)}/(2d)^{\tau+d+1}.\end{aligned}$$ Observing that $\delta<1/16$ and $i\le R$ ensures that the prefactor on the rhs of the last row is at least $1/8\cdot 1/2=1/2^4$, establishing [\[eq:net-defining-crit\]](#eq:net-defining-crit){reference-type="eqref" reference="eq:net-defining-crit"} for all $w\le f(r_i)$, as required. ◻ A lower bound on the probability that any given box in an $\mathcal{R}$-partition is good, together with Claim [Claim 13](#lem:nets-partition-exists){reference-type="ref" reference="lem:nets-partition-exists"} and Lemma [Lemma 17](#lem:good-implies-net){reference-type="ref" reference="lem:good-implies-net"}, will yield the proof of Proposition [\[lem:nets-exist\]](#lem:nets-exist){reference-type="ref" reference="lem:nets-exist"}. The bound is by induction on $i$ together with Chernoff bounds. Recall $I$ and $I(w)$ from Def. [Definition 14](#def:base-2-cover){reference-type="ref" reference="def:base-2-cover"}, applied to the interval $[w_0,f(r_R)]$ for $\mathcal{R}=\{r_1, \ldots, r_R\}$. Recall that [\[eq:i-good-weight\]](#eq:i-good-weight){reference-type="eqref" reference="eq:i-good-weight"} of Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"} *(B3)* is required only when $w\in [w_0, f(r_i)]$, and that $j_\star(i)$ in [\[eq:jstar\]](#eq:jstar){reference-type="eqref" reference="eq:jstar"} is the index of $I_j$ that contains $f(r_i)$. We now describe a gradual revealment of vertex-weights. **Definition 18**. *Consider Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"} and Definitions [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"}, [Definition 15](#def:hyperrectangles){reference-type="ref" reference="def:hyperrectangles"}. Suppose $\mathcal{H}(\mathcal{R})$ is a hyperrectangle-cover of $Q\times [w_0, f(r_R)]$. Let $i \in [R]$ and $B \in \mathcal{P}_i$, and let $\widetilde{\mathcal{V}}$ be a realisation of the weighted vertices in Definition [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}. We define $$\begin{aligned} \mathcal{F}_i(B):= \begin{cases} \mathcal{V}\cap B & \text{when }i=1 \text{ and } B\in \mathcal{P}_1,\\ \mathcal{F}_1(B)\cup \Big(\widetilde{\mathcal{V}} \cap (B\times \cup_{j\le j_\star(i-1)}I_j)\Big) & \text{when } i>1\text{ and } B\in \mathcal{P}_i.\label{eq:fib-2} \end{cases} \end{aligned}$$* $\mathcal{F}_1(B)$ reveals the number and positions of vertices in $B$, while $\mathcal{F}_i(B)$ reveals the precise weights *only* of vertices whose weight falls in the interval $\cup_{j\le j_\star(i-1)}I_j \supseteq [w_0, f(r_{i-1})]$. The index shift in [\[eq:fib-2\]](#eq:fib-2){reference-type="eqref" reference="eq:fib-2"}, and the fact that $\mathcal{R}$ is $\delta$-well-spaced, means that vertex weights between $w_0 2^{j_\star(i-1)}\approx f(r_{i-1})$ and $f(r_i)$ are not revealed in $\mathcal{F}_i(B)$. Also, vertex weights in $[1,w_0]$ are not revealed at all; since $w_0$ is large, $\mathbb{P}(W\le w_0)$ is large and most vertex weights will not be revealed by exposing $\mathcal{F}_i(B)$. The filtration generated by $\mathcal{F}_i(B)$ determines whether or not boxes in $\cup_{j\le i-1} \mathcal{P}_{j}$ are good, and whether or not a *vertex* is $i$-good (see Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"}). So, $\mathcal{F}_i(B)$ determines whether or not $B\in \mathcal{P}_i$ satisfies Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"} *(B[\[item:B1\]](#item:B1){reference-type="ref" reference="item:B1"}) -(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})*, but it leaves *(B[\[item:B3\]](#item:B3){reference-type="ref" reference="item:B3"})* undecided for weights slightly below $f(r_i)$. The next lemma treats *(B[\[item:B3\]](#item:B3){reference-type="ref" reference="item:B3"})*, with $\mathcal{F}_i(B)$ exposed. **Lemma 19**. *Consider Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"} and Definitions [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"}, [Definition 15](#def:hyperrectangles){reference-type="ref" reference="def:hyperrectangles"}. Let $\mathcal{H}(\mathcal{R})$ be a hyperrectangle-cover of the cube $Q$. Let $i \in [R]$ and let $B \in \mathcal{P}_i$. Let $F$ be a realisation of $\mathcal{F}_i(B)$ that satisfies Definition [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"} (B[\[item:B1\]](#item:B1){reference-type="ref" reference="item:B1"}) and (B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"}) for $B$. Then independently of other boxes in $\mathcal{P}_i$, uniformly for all such $F$, $$\label{eq:b-good-b3} \mathbb{P}(B\mbox{ is good}\mid \mathcal{F}_i(B) = F) \ge 1 - \delta/(2R).$$* *Proof.* Recall $I_j$ from Def. [Definition 14](#def:base-2-cover){reference-type="ref" reference="def:base-2-cover"}, and let $B\in \mathcal{P}_i$. Let $a(I_j) := (1-2i\delta/R)\mu_i(I_j)/8, b(I_j) := 8\mu_i(I_j)$ the required lower and upper bounds in [\[eq:i-good-weight\]](#eq:i-good-weight){reference-type="eqref" reference="eq:i-good-weight"}. Let $\Xi_j(B)=|\widetilde{\mathcal{G}_i}\cap (B\times I_j)|$; thus (B3) holds for $B\times I_j$ iff $\Xi_j(B) \in [a(I_j),b(I_j)]$. Since $B$ satisfies *(B1)--(B2)* on $F$, by a union bound, $$\label{eq:nets-exist-b3-sum} \mathbb{P}\big(B\mbox{ is good}\mid \mathcal{F}_i(B) = F\big) \ge 1 - \sum_{I_j: j\le j_\star(i)} \mathbb{P}\big(\Xi_j(B) \notin [a(I_j),b(I_j)] \mid \mathcal{F}_i(B) = F \big).$$ We proceed by bounding each term above. By the definition of $\mathcal{F}_i(B)$ in [\[eq:fib-2\]](#eq:fib-2){reference-type="eqref" reference="eq:fib-2"}, we already exposed $\Xi_j(B)$ when $i > 1$ and $j\le j_*(i-1)$; the latter is equivalent to $\min(I_j) \le f(r_{i-1})$. **Case 1: $\boldsymbol{i \!>\! 1}$ and $\boldsymbol{j\!\le\! j_\star(i\!-\!1)}$.** We first show that $\Xi_j \ge a(I_j)$ holds deterministically on $\{\mathcal{F}_i(B)=F\}$. The goodness of each sub-box $B'\in \mathcal{P}_{i-1}$ of $B\in \mathcal{P}_i$ *is revealed* by $\mathcal{F}_i(B)$. If $B'\in \mathcal{P}_{i-1}$ is a good box, all vertices in $\mathcal{G}_{i-1} \cap B'$ are also $i$-good by Definition [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"}. So $$\label{eq:nets-good-likely-1} \Xi_j(B) = |\widetilde{\mathcal{G}_i}\cap (B\times I_j)|\ge \sum_{B'\in \mathcal{P}_{i-1}\colon B' \text{ good}} |\widetilde{\mathcal{G}}_{i-1}\cap (B'\times I_j)|.$$ Since $j\le j_\star(i-1)$, $\min(I_j) \le f(r_{i-1})$, so we may apply [\[eq:i-good-weight\]](#eq:i-good-weight){reference-type="eqref" reference="eq:i-good-weight"} to the good subboxes: $$\label{eq:nets-good-likely-2} |\widetilde{\mathcal{G}}_{i-1}\cap (B'\times I_j)|\ge \Big(1-\frac{2(i-1)\delta}{R}\Big)\frac{\mu_{i-1}(I_j)}{8} = \Big(1-\frac{2(i-1)\delta}{R}\Big)\frac{\mu_i(I_j)}{8}\Big(\frac{r_{i-1}'}{r_i'}\Big)^d,$$ where $\mu_i(I_j)/\mu_{i-1}(I_j)=(r_i'/r_{i-1}')^d$ follows from [\[eq:measure-hyperrectangle\]](#eq:measure-hyperrectangle){reference-type="eqref" reference="eq:measure-hyperrectangle"}. By *(B[\[item:B1\]](#item:B1){reference-type="ref" reference="item:B1"})* holding on $F$, $B$ contains at least $(1-2\delta/R)(r_i'/r_{i-1}')^d$ good sub-boxes in $\mathcal{P}_{i-1}$. Combining that with [\[eq:nets-good-likely-1\]](#eq:nets-good-likely-1){reference-type="eqref" reference="eq:nets-good-likely-1"}--[\[eq:nets-good-likely-2\]](#eq:nets-good-likely-2){reference-type="eqref" reference="eq:nets-good-likely-2"} yields $$\begin{aligned} \Xi_j(B)&\ge \Big(1-\frac{2\delta}{R}\Big) \cdot \Big(1-\frac{2(i-1)\delta}{R}\Big) \frac{\mu_i(I_j)}{8} \ge \Big(1-\frac{2i\delta}{R}\Big) \frac{\mu_i(I_j)}{8} = a(I_j). \end{aligned}$$ We show $\Xi_j(B)=|\widetilde{\mathcal{G}_i}\cap (B\times I_j)| \le b(I_j)$ also holds a.s.. $B$ contains $(r_i'/r_{i-1}')^d$ sub-boxes in $\mathcal{P}_{i-1}$ by Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"} *(P[\[item:P2\]](#item:P2){reference-type="ref" reference="item:P2"})*. If a sub-box is bad, it contains no $i$-good vertices. If it is good, *(B[\[item:B3\]](#item:B3){reference-type="ref" reference="item:B3"})* holds and it contains at most $8\mu_i(I_j)(r_{i-1}'/r_i')^d$ $i$-good vertices with weights in $B\!\times\! I_j$. We obtain $\Xi_j(B) \le 8\mu_i(I_j) = b(I_j)$. So overall we have shown that $$\label{eq:nets-exist-b3-term-1} \mbox{if $i>1$ and $j \le j_\star(i-1)$:}\quad \mathbb{P}\big(\Xi_j(B) \notin [a(I_j),b(I_j)] \mid \mathcal{F}_i(B) = F\big) = 0.$$ **Case 2: $\boldsymbol{i > 1}$ and $\boldsymbol{j> j_\star(i-1)}$.** Define the set $$\label{eq:calS} \mathcal{S}=\widetilde{\mathcal{G}_i}\cap \Big(B\times \big([1,w_0) \cup \bigcup\nolimits_{j> j_\star(i-1)} I_j\big)\Big),$$ the $i$-good vertices in $B$ with weights *not revealed* by $\mathcal{F}_i(B)$. $\mathcal{F}_i(B)$ does reveal the positions of vertices in $B$, and all weighted vertices not in $\mathcal{S}$; thus $\mathcal{F}_i(B)$ reveals $|\mathcal{S}|$, and the position of vertices in $\mathcal{S}$. By Def. [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}, vertex weights are independent of vertex positions and of each other; hence for a vertex $v \in \widetilde{\mathcal{V}}$, conditioning on $\mathcal{F}_i(B)$ is equivalent to either exposing its weight (if $w_v\in \cup_{j\le j_\star(i-1)}I_j$) or conditioning on $w_v \notin \cup_{j\le j_\star(i-1)} I_j$ (otherwise). Since $|\mathcal{S}|$ is determined by $\mathcal{F}_i(B)$, $\Xi_j(B)$ is binomially distributed on $\mathcal{F}_i(B)$, when $j> j_\star(i-1)$, with parameters $$\label{eq:withinS-distribution} \Xi_j(B) \mid \mathcal{F}_i(B)\ {\buildrel d \over =} \ \mathrm{Bin}\Big(|\mathcal{S}|, \mathbb P(W\in I_j)/\mathbb P(W\notin \cup_{j\le j_\star(i-1)} I_j) \Big).$$ We next bound the conditional expectation of $\Xi_j(B)\mid \mathcal{F}_i(B)$, starting with the upper bound. Using the lower bound [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"} on $w_0$, the success probability of the binomial in [\[eq:withinS-distribution\]](#eq:withinS-distribution){reference-type="eqref" reference="eq:withinS-distribution"} is $$\begin{aligned} \label{eq:successprob} \frac{\mathbb{P}(W\in I_j)}{\mathbb P(W\notin \cup_{j\le j_\star(i-1)} I_j)} \le \frac{\mathbb{P}(W\in I_j)}{\mathbb{P}(W\in [1,w_0))} = \frac{\mathbb{P}(W\in I_j)}{1-\ell(w_0)w_0^{\tau-1}}\le 2\mathbb{P}(W\in I_j). \end{aligned}$$ Since Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"} *(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* holds for $B$ on $\mathcal{F}_i(B)=F$, by [\[eq:i-good-total\]](#eq:i-good-total){reference-type="eqref" reference="eq:i-good-total"} there are at most $2(r_i')^d$ $i$-good vertices in $B$, so $|\mathcal{S}| \le 2(r_i')^d$. Recalling the definition of $\mu_i(I_j)$ from [\[eq:measure-hyperrectangle\]](#eq:measure-hyperrectangle){reference-type="eqref" reference="eq:measure-hyperrectangle"}, we thus obtain $$\label{eq:nets-b3-bound-1} \mathbb{E}\big[\Xi_j(B) \mid \mathcal{F}_i(B) = F\big] \le 4(r_i')^d\mathbb{P}(W\in I_j) = 4\mu_i(I_j) = b(I_j)/2.$$ We next prove the corresponding lower bound. By Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"} *(P[\[item:P2\]](#item:P2){reference-type="ref" reference="item:P2"})*, $B$ contains $(r_i'/r_{i-1}')^d$ sub-boxes in $\mathcal{P}_{i-1}$. Clearly by [\[eq:calS\]](#eq:calS){reference-type="eqref" reference="eq:calS"}, $|\mathcal{S}|=|\mathcal{G}_i\cap B|-|\cup_{j\le j_\star(i-1)} \widetilde{\mathcal{G}_i}\cap (B\times I_j)|$, both terms revealed by $F$. We can bound $|\mathcal{G}_i\cap B|$ from below using [\[eq:i-good-total\]](#eq:i-good-total){reference-type="eqref" reference="eq:i-good-total"} in Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"} *(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})*. Since there are no $i$-good vertices in bad boxes $B'\in \mathcal{P}_{i-1}$, we can bound $|\cup_{j\le j_\star(i-1)} \widetilde{\mathcal{G}_i}\cap (B\times I_j)|$ from above by applying [\[eq:i-good-weight\]](#eq:i-good-weight){reference-type="eqref" reference="eq:i-good-weight"} to each good sub-box $B'\in \mathcal{P}_{i-1}$ of $B$, yielding $$\label{eq:nets-b3-bound-2} |\mathcal{S}|\ge \Big(\frac{1}{2}-\frac{2(i-1)\delta}{R}\Big)(r_i')^d - \Big(\frac{r_i'}{r_{i-1}'}\Big)^d\cdot\sum_{j \le j_\star(i-1)} 8\mu_{i-1}(I_j).$$ Using that $I_j=[2^{j-1}w_0, w_0)$, Claim [\[claim:nets-mu-bound\]](#claim:nets-mu-bound){reference-type="ref" reference="claim:nets-mu-bound"} with $w=2^{j-1}w_0$ yields $$\begin{aligned} \begin{split}\label{eq:geometric} \big(r_i'/r_{i-1}'\big)^d\sum_{j \le j_\star(i-1)} 8\mu_{i-1}(I_j)&\le 2^{\tau+3}(r_i')^d\sum_{j \le j_\star(i-1)} \ell(2^{j-1}w_0)(2^{j-1}w_0)^{-(\tau-1)}\\ &< 2^{\tau+3}(r_i')^d\sum_{j = 0}^\infty \ell(2^jw_0)(2^jw_0)^{-(\tau-1)}, \end{split} \end{aligned}$$ where we switched indices in the last row. By the lower bound [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"} on $w_0$ and since $\tau>2$, for all $j \ge 0$ we have $\ell(2^{j+1}w_0)(2^{j+1}w_0)^{-(\tau-1)} < \tfrac{2}{3}\ell(2^jw_0)(2^jw_0)^{-(\tau-1)}$, so the sum on the rhs is bounded above termwise by a geometric series. It follows from [\[eq:nets-b3-bound-2\]](#eq:nets-b3-bound-2){reference-type="eqref" reference="eq:nets-b3-bound-2"} that $$|\mathcal{S}| \ge \Big(\frac{1}{2} - \frac{2(i-1)\delta}{R} - 2^{\tau+5}\ell(w_0)w_0^{-(\tau-1)}\Big)(r_i')^d \ge (r_i')^d/4,$$ where we used in the last step that $i \le R$ and $\delta < 1/16$, and the lower bound [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"} on $w_0$. The success probability of the binomial in [\[eq:withinS-distribution\]](#eq:withinS-distribution){reference-type="eqref" reference="eq:withinS-distribution"} is at least $\mathbb{P}(W\in I_j)$. We defined $a(I_j)= (1-2i\delta/R)\mu_i(I_j)/8$ at the beginning of the proof, so $$\label{eq:nets-b3-bound-3} \mathbb{E}(\Xi_j(B) \mid \mathcal{F}_i(B) = F) \ge (r_i')^d\mathbb{P}(W\in I_j)/4 = \mu_i(I_j)/4 \ge 2a(I_j).$$ Combining [\[eq:nets-b3-bound-1\]](#eq:nets-b3-bound-1){reference-type="eqref" reference="eq:nets-b3-bound-1"} with [\[eq:nets-b3-bound-3\]](#eq:nets-b3-bound-3){reference-type="eqref" reference="eq:nets-b3-bound-3"} yields that $\mathbb E(\Xi_j(B)\mid \mathcal{F}_i(B)=F) \in [2a(I_j), b(I_j)/2]$, which allows us to bound $\mathbb{P}(\Xi_j(B) \notin [a(I_j),b(I_j)])$ with standard Chernoff bounds. By [\[eq:nets-b3-bound-1\]](#eq:nets-b3-bound-1){reference-type="eqref" reference="eq:nets-b3-bound-1"} and Theorem [Theorem 50](#thm:chernoff){reference-type="ref" reference="thm:chernoff"} applied with $\varepsilon=1/2$, we have shown that $$\label{eq:nets-exist-b3-term-2a} \begin{aligned} \mbox{for $i > 1, j > j_\star(i-1)$:}\quad \mathbb{P}\big(\Xi_j(B) \notin [a(I_j), b(I_j)] \mid \mathcal{F}(B) = F\big) &\le 2\exp(-a(I_j)/6) \\ & \le 2\exp(-\mu_i(I_j)/96). \end{aligned}$$ **Case 3: $\boldsymbol{i\!=\!1}$.** When $i=1$, we set $j_\star(i-1):=0$ naturally, since in $\mathcal{F}_1(B)$ we revealed the total number of vertices in $B\in \mathcal{P}_1$, which are all $1$-good by Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"}. Conditioned on $\mathcal{F}_1(B)=F$, [\[eq:i-good-total\]](#eq:i-good-total){reference-type="eqref" reference="eq:i-good-total"} in *(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* is satisfied and $(r_1')^d/4 \le |\mathcal{S}| \le 2(r_1')^d$ directly. The rest of our previous calculations from Case 2 with $j>j_\star(0)=0$ all carry through for estimating the lhs of [\[eq:i-good-weight\]](#eq:i-good-weight){reference-type="eqref" reference="eq:i-good-weight"} in *(B[\[item:B3\]](#item:B3){reference-type="ref" reference="item:B3"})*. We obtain that [\[eq:nets-exist-b3-term-2a\]](#eq:nets-exist-b3-term-2a){reference-type="eqref" reference="eq:nets-exist-b3-term-2a"} holds also for $i=1$ and all $j\ge j_\star(0)=0$. **Combining the cases:** By [\[eq:nets-exist-b3-term-1\]](#eq:nets-exist-b3-term-1){reference-type="eqref" reference="eq:nets-exist-b3-term-1"}, [\[eq:nets-exist-b3-term-2a\]](#eq:nets-exist-b3-term-2a){reference-type="eqref" reference="eq:nets-exist-b3-term-2a"}, and Case 3, for all $i$ and $j \le j_\star(i)$ the bound in [\[eq:nets-exist-b3-term-2a\]](#eq:nets-exist-b3-term-2a){reference-type="eqref" reference="eq:nets-exist-b3-term-2a"} holds. Combining that with [\[eq:nets-exist-b3-sum\]](#eq:nets-exist-b3-sum){reference-type="eqref" reference="eq:nets-exist-b3-sum"}, for all $B\in \mathcal{P}_i$ and $\mathcal{F}_i(B)=F$ satisfying *(B1),(B2)*, $$\begin{aligned} p_i:=&\ \mathbb{P}\big(B\in \mathcal{P}_i \mbox{ is $i$-good}\mid \mathcal{F}_i(B) = F\big) \ge 1 - 2\sum_{j \le j_\star(i)} \exp(-\mu_i(I_j)/96)\\ &\ge 1 - 2\sum_{j \le j_\star(i)} \exp\Big({-}(2d)^{-(\tau+d+8)}r_i^d\ell(2^{j-1}w_0 )(2^{j-1}w_0)^{-(\tau-1)}\Big), \end{aligned}$$ where the second line follows from $\mu_i(I_j)=\mu_i(I(2^{j-1}w_0))$ in Claim [\[claim:nets-mu-bound\]](#claim:nets-mu-bound){reference-type="ref" reference="claim:nets-mu-bound"}. By the lower bound [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"} on $w_0$ and the fact that $\tau>2$, for all $j\ge 1$ we have $\ell(2^{j-1}w_0)(2^{j-1}w_0)^{-(\tau-1)} \ge \tfrac{3}{2}\ell(2^{j}w_0)(2^{j}w_0)^{-(\tau-1)}$. Using the same geometric-sum argument as below [\[eq:geometric\]](#eq:geometric){reference-type="eqref" reference="eq:geometric"} except now from the reversed viewpoint, writing $z:=w_02^{j_\star(i)-1}$, the lower endpoint of $I_{j_\star(i)}$, we have $$\begin{aligned} \label{eq:nets-likely-pen} p_i &\ge 1 - 2\sum_{t=0}^\infty \exp\Big({-}\frac{1}{(2d)^{\tau+d+8}}r_i^d\Big(\frac{3}{2}\Big)^t\ell(z)z^{-(\tau-1)}\Big). \end{aligned}$$ Since $z$ is the lower endpoint of $I_{j_\star(i)}=I(f(r_i))$, we have $z \le f(r_i) \le 2z$. Hence by [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"} $$\ell(z)z^{-(\tau-1)} \ge 2^{-\tau}\ell(f(r_i))f(r_i)^{-(\tau-1)}.$$ Using this bound in [\[eq:nets-likely-pen\]](#eq:nets-likely-pen){reference-type="eqref" reference="eq:nets-likely-pen"} and combining it with [\[eq:z-bound\]](#eq:z-bound){reference-type="eqref" reference="eq:z-bound"} from Claim [\[claim:nets-mu-bound\]](#claim:nets-mu-bound){reference-type="ref" reference="claim:nets-mu-bound"}, we obtain that $$\begin{aligned} p_i \ge 1 - 2\sum_{t=0}^\infty \exp\Big({-}\Big(\frac{3}{2}\Big)^t\log(16R/\delta)\Big) \ge 1 - \frac{\delta}{8R} - 2\sum_{t=1}^\infty (\delta/16R)^t \ge 1 - \frac{\delta}{2R}, \end{aligned}$$ where we used that $(3/2)^t \ge t$ for all $t \ge 1$, and $\delta < 1/16$. Independence across boxes in the same $\mathcal{P}_i$ is immediate, since whether $B$ satisfies *(B[\[item:B3\]](#item:B3){reference-type="ref" reference="item:B3"})* conditioned on $\mathcal{F}_i(B)=F$ satisfying *(B[\[item:B1\]](#item:B1){reference-type="ref" reference="item:B1"}), (B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* only depends on vertices in $B$. ◻ The next lemma gets rid of the conditioning in Lemma [Lemma 19](#lem:nets-good-likely-b3){reference-type="ref" reference="lem:nets-good-likely-b3"} on the filtration: **Lemma 20**. *Consider Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"} and Definitions [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"}, [Definition 15](#def:hyperrectangles){reference-type="ref" reference="def:hyperrectangles"}. Let $\mathcal{H}(\mathcal{R})$ be a hyperrectangle-cover of the cube $Q$. Let $t \le (r_1/4\sqrt{d})^d$, and let $x_1,\ldots,x_t$ be a (possibly empty) sequence of points in $\mathbb{R}^d$. Then for each $B\in \cup_i\mathcal{P}_i$ $$\label{eq:nets-good-likely} \mathbb{P}\big(B \mbox{ is good}\mid x_1,\ldots,x_t \in \mathcal{V}\big) \ge 1 - \delta/R.$$* *Proof.* We prove the statement by induction on $i$, the base case being $i=1$. Consider a box $B \in \mathcal{P}_1$. By Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"}, *(B[\[item:B1\]](#item:B1){reference-type="ref" reference="item:B1"})* holds vacuously. We next consider *(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})*. Every vertex in $B$ is $1$-good, and so $|\mathcal{G}_1\cap B|=|\mathcal{V}\cap B|$. In SFP, $\mathcal{V}$ is deterministic and $|\mathcal{G}_1\cap B|\in[(r_1')^d/2, 2(r_1')^d]$ holds with certainty by Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"} *(P[\[item:P1\]](#item:P1){reference-type="ref" reference="item:P1"})*. In (I)GIRG, $|\mathcal{G}_1\cap B|$ is a Poisson variable with mean $(r_1')^d$, so by a standard Chernoff bound (Theorem [Theorem 50](#thm:chernoff){reference-type="ref" reference="thm:chernoff"} with $\varepsilon=1/2$), and using $r_i' \in [r_i/(2\sqrt{d}), r_i/\sqrt{d}]$ in Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"} *(P[\[item:P1\]](#item:P1){reference-type="ref" reference="item:P1"})*, and the bound [\[eq:nets-small-r\]](#eq:nets-small-r){reference-type="eqref" reference="eq:nets-small-r"} on $r_1$ in Def. [Definition 10](#def:well-spaced){reference-type="ref" reference="def:well-spaced"}, $$\begin{aligned} p_1':=\mathbb{P}\Big( |\mathcal{G}_1\cap B| \in[t+\tfrac12(r_1')^d, t+\tfrac32(r_1')^d]\mid x_1, \dots, x_t\in \mathcal{V}\Big) \ge 1 - 2\mathrm{e}^{-(r_1/24d)^d} \ge 1-\delta/(2R), \end{aligned}$$ so the lower bound in [\[eq:i-good-total\]](#eq:i-good-total){reference-type="eqref" reference="eq:i-good-total"} in Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"} *(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* is satisfied. Moreover, since $t \le (r_1/4\sqrt{d})^d$, by Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"} *(P[\[item:P1\]](#item:P1){reference-type="ref" reference="item:P1"})*, $\tfrac32(r_1')^d + t \le 2(r_1')^d$, the upper bound in Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"} *(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* also holds, so independently for all boxes $B\in \mathcal{P}_1$, and regardless on where $x_1, \ldots, x_t$ fall, $$\begin{aligned} \mathbb{P}\big(B \in \mathcal{P}_1 \mbox{ satisfies \emph{(B2)}}\mid x_1,\ldots,x_t \in \mathcal{V}\big) \ge p_1'\ge 1 - \delta/(2R). \end{aligned}$$ Lemma [Lemma 19](#lem:nets-good-likely-b3){reference-type="ref" reference="lem:nets-good-likely-b3"} ensures that *(B[\[item:B3\]](#item:B3){reference-type="ref" reference="item:B3"})* holds with probability at least $1-\delta/(2R)$ conditioned on any realisation where *(B[\[item:B1\]](#item:B1){reference-type="ref" reference="item:B1"}),(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* holds. A union bound proves [\[eq:nets-good-likely\]](#eq:nets-good-likely){reference-type="eqref" reference="eq:nets-good-likely"} for $B\in \mathcal{P}_1$. Now we advance the induction. Suppose that [\[eq:nets-good-likely\]](#eq:nets-good-likely){reference-type="eqref" reference="eq:nets-good-likely"} holds for each $B\in\cup_{j\le i-1}\mathcal{P}_i$, and let $B \in \mathcal{P}_i$. $B$ contains $(r_i'/r_{i-1}')^d$ sub-boxes in $\mathcal{P}_{i-1}$ (by Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"}), and by induction these sub-boxes of $B$ are good *independently* (regardless of the positions and weights of $x_1,\ldots,x_t \in\mathcal{V}$), so the number of bad sub-boxes of $B$ is binomial with mean at most $(r_i'/r_{i-1}')^d\delta/R$. Let $$\label{eq:aib} \mathcal{A}_{i}(B):=\Big\{\big|\{B'\in \mathcal{P}_{i-1}\colon B'\subseteq B,\ B' \mbox{ not $(i-1)$-good}\}\big| < 2(r_i'/r_{i-1}')^d\delta/R\Big\}.$$ Then, $\mathcal{A}_{i}(B)$ implies Def. [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"}*(B[\[item:B1\]](#item:B1){reference-type="ref" reference="item:B1"})*. A Chernoff bound (Theorem [Theorem 50](#thm:chernoff){reference-type="ref" reference="thm:chernoff"} with $\varepsilon=1$) yields that $$\mathbb{P}\Big(\neg \mathcal{A}_{i}(B) \mid x_1,\ldots,x_t\in\mathcal{V}\Big) \le \exp\Big({-}\frac{\delta}{3R}\cdot \Big(\frac{r_i'}{r_{i-1}'}\Big)^d \Big).$$ By Def. [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"} *(P[\[item:P1\]](#item:P1){reference-type="ref" reference="item:P1"})*, $(r_i'/r_{i-1}')^d \ge 2^{-d} (r_i/r_{i-1})^d$, so by Def. [Definition 10](#def:well-spaced){reference-type="ref" reference="def:well-spaced"} [\[eq:nets-radii\]](#eq:nets-radii){reference-type="eqref" reference="eq:nets-radii"}, regardless of the positions of $x_1,\ldots,x_t\in\mathcal{V}$, and *independently* across different boxes in $\mathcal{P}_i$: $$\begin{aligned} \label{eq:induction-error-1} \mathbb{P}\Big(\neg \mathcal{A}_{i}(B)\mid x_1,\ldots,x_t\in\mathcal{V}\Big) &\le \exp\Big({-}\frac{\delta}{3R} \cdot 3^d R \cdot \frac{\log (2R/\delta)}{\delta}\Big) \le \frac{\delta}{2R}. \end{aligned}$$ We now show that $\mathcal{A}_{i}(B)$ implies *(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* as well, by inductively applying [\[eq:i-good-total\]](#eq:i-good-total){reference-type="eqref" reference="eq:i-good-total"} to the good sub-boxes of $B$. Consider an $(i-1)$-good vertex $v$ in a good sub-box $B'\in \mathcal{P}_{i-1}$ of $B$. Since $v$ is $(i-1)$-good, $B^1(v),\ldots,B^{i-2}(v)$ must all be good; since $B^{i-1}(v)=B'$ is also good, it follows that $v$ is in fact $i$-good. Thus, for all good $B'\in \mathcal{P}_{i-1}:$ $\mathcal{G}_{i-1}\cap B'=\mathcal{G}_{i}\cap B'$ holds. Since $B'$ is $(i\!-\!1)$-good, it satisfies [\[eq:i-good-total\]](#eq:i-good-total){reference-type="eqref" reference="eq:i-good-total"}, and we obtain: $$|\mathcal{G}_{i}\cap B| \ge \sum_{\text{good } B'\subset B} |\mathcal{G}_{i-1}\cap B'|\ge \big|\{B' \in \mathcal{P}_{i-1}\colon B'\subset B,\ B'\mbox{ good}\}\big|\Big(\frac{1}{2} - \frac{2(i-2)\delta}{R}\Big)(r_{i-1}')^d.$$ On $\mathcal{A}_i(B)$ in [\[eq:aib\]](#eq:aib){reference-type="eqref" reference="eq:aib"}, there are $(1-2\delta/R)(r_i'/r_{i-1}')^d$ good sub-boxes of $B$, so a.s. on $\mathcal{A}_i(B)$ $$\begin{aligned} |\mathcal{G}_{i}\cap B| &\ge \Big(1-\frac{2\delta}{R}\Big)\Big(\frac{r_i'}{r_{i-1}'}\Big)^d \Big(\frac{1}{2} - \frac{2(i-2)\delta}{R}\Big)(r_{i-1}')^d \ge \Big(\frac{1}{2}-\frac{2(i-1)\delta}{R}\Big)(r_i')^d. \end{aligned}$$ Vertices in bad sub-boxes cannot be $i$-good by Definition [Definition 16](#def:nets-good-box){reference-type="ref" reference="def:nets-good-box"}. So [\[eq:i-good-total\]](#eq:i-good-total){reference-type="eqref" reference="eq:i-good-total"} similarly implies that on $\mathcal{A}_{i}(B)$ there are at most $2(r_i')^d$ $i$-good vertices in $B$, and thus $\mathcal{A}_i(B)$ in [\[eq:aib\]](#eq:aib){reference-type="eqref" reference="eq:aib"} implies both *(B[\[item:B1\]](#item:B1){reference-type="ref" reference="item:B1"})--(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* for $B$; it follows from [\[eq:induction-error-1\]](#eq:induction-error-1){reference-type="eqref" reference="eq:induction-error-1"} that $$\label{eqn:boxes-B1B2} \mathbb{P}\big(\mbox{\emph{(B\ref{item:B1})} and \emph{(B\ref{item:B2})} hold for }B \mid x_1,\ldots,x_t \in \mathcal{V}\big) \ge 1 - \delta/(2R).$$ By Lemma [Lemma 19](#lem:nets-good-likely-b3){reference-type="ref" reference="lem:nets-good-likely-b3"}, $B$ is good (i.e., *(B[\[item:B3\]](#item:B3){reference-type="ref" reference="item:B3"})* also holds) with probability at least $1-\delta/(2R)$ for all $\mathcal{F}_{i-1}(B)=F$ with *(B[\[item:B1\]](#item:B1){reference-type="ref" reference="item:B1"})--(B[\[item:B2\]](#item:B2){reference-type="ref" reference="item:B2"})* holding for $B$; these events and $x_1,\ldots,x_t \in \mathcal{V}$ are all determined by $\mathcal{F}_{i-1}(B)$. So, a union bound on [\[eq:b-good-b3\]](#eq:b-good-b3){reference-type="eqref" reference="eq:b-good-b3"} and [\[eqn:boxes-B1B2\]](#eqn:boxes-B1B2){reference-type="eqref" reference="eqn:boxes-B1B2"} yields that independently across boxes in $\mathcal{P}_i$, and regardless of the vertices $x_1,\ldots,x_t\in \mathcal{V}$, [\[eq:nets-good-likely\]](#eq:nets-good-likely){reference-type="eqref" reference="eq:nets-good-likely"} holds. This advances the induction and finishes the proof. ◻ *Proof of Proposition [\[lem:nets-exist\]](#lem:nets-exist){reference-type="ref" reference="lem:nets-exist"}.* Recall Setting [Setting 11](#set:R-of-section){reference-type="ref" reference="set:R-of-section"}, and consider an $\mathcal{R}$-partition $\mathcal{P}_1,\ldots,\mathcal{P}_R$ of $Q$, (which exists by Claim [Claim 13](#lem:nets-partition-exists){reference-type="ref" reference="lem:nets-partition-exists"}), and let $\mathcal{H}(\mathcal{R})$ be the associated hyperrectangle cover of $Q\times [w_0, f(r_R)]$. By Lemma [Lemma 20](#lem:nets-good-likely){reference-type="ref" reference="lem:nets-good-likely"}, conditioned on $x_1,\ldots,x_t \in \mathcal{V}$, each box $B \in \mathcal{P}_1 \cup \dots \cup \mathcal{P}_R$ is good with probability at least $1 - \delta/R$. Let $\mathcal{A}$ be the event that all boxes in $\{B^i(x_j)\colon i \in [R], j \in [t]\}$ are good; a union bound over $i=1, \ldots, R$ and $1,\ldots,t$ implies that $$\mathbb{P}(\mathcal{A}\mid x_1,\ldots,x_t\in\mathcal{V}) \ge 1-t\delta.$$ In particular, if $\mathcal{A}$ occurs then $B^R(x_1)=Q$ is also good, so by Lemma [Lemma 17](#lem:good-implies-net){reference-type="ref" reference="lem:good-implies-net"}, the set $\widetilde{\mathcal{G}}_R=:\mathcal{N}$ of all $R$-good vertices forms an $(\delta,\mathcal{R})$-net of $Q$ with $x_1,\ldots,x_t \in \mathcal{N}$ as required, showing [\[eq:strong-netsexist\]](#eq:strong-netsexist){reference-type="eqref" reference="eq:strong-netsexist"}. To obtain [\[eq:netsexist-1\]](#eq:netsexist-1){reference-type="eqref" reference="eq:netsexist-1"}, note that Lemma [Lemma 20](#lem:nets-good-likely){reference-type="ref" reference="lem:nets-good-likely"} implies that $Q\subset \mathcal{P}_R$ is good with probability at least $1-\delta/R$, and then again Lemma [Lemma 17](#lem:good-implies-net){reference-type="ref" reference="lem:good-implies-net"} finishes the proof. ◻ **Weak nets.** After having established pseudorandom nets in a general and strong form (possibly better for re-use), we now give a relaxed version that suffices here. The following version -- *weak nets* -- only retains three parameters: the box $Q$; an error parameter $\varepsilon>0$; and a lower bound $w_1$ on the weights considered for the net. Recall the definition of an $(\mathcal{R},w_0, f)$-net from Def. [Definition 8](#def:net){reference-type="ref" reference="def:net"} and the choice of $w_0$ and of the function $f(r)=f_{R, \delta}(r)$ from Def. [Definition 9](#def:net-constants){reference-type="ref" reference="def:net-constants"}, yielding an $(\delta,\mathcal{R})$-net. **Definition 21** (Weak net). *Let $Q \subseteq \mathbb{R}^d$ be a box of side length $\xi$, $\varepsilon>0$, and $w_1 \geq w_0$. A *weak $(\varepsilon,w_1)$-net* for $Q$ is a set $\mathcal{N}\subseteq \widetilde{\mathcal{V}} \cap Q\times[1,\infty)$ of size at least $\mathrm{Vol}(Q)/4$ such that for all $\widetilde v \in \mathcal{N}$, *all* $r \in [(\log\log\xi\sqrt{d})^{4/\varepsilon}, \xi\sqrt{d}]$ and all $w \in [w_1, r^{d/(\tau-1)-\varepsilon}]$: $$\label{eq:net-defining-crit-eps} |\mathcal{N}\cap (B_r(v)\times [w/2, 2w])| \ge r^{d(1-\varepsilon)}\cdot \ell(w)w^{-(\tau-1)}.$$* In a weak $(\varepsilon,w_1)$-net we allow an error of order $r^{-d\varepsilon}$ on the rhs of [\[eq:net-defining-crit-eps\]](#eq:net-defining-crit-eps){reference-type="eqref" reference="eq:net-defining-crit-eps"} compared to the constant factor in a $(\delta,\mathcal{R})$-net in [\[eq:net-defining-crit\]](#eq:net-defining-crit){reference-type="eqref" reference="eq:net-defining-crit"}, and here the smallest radius $r$ also grows with $\xi$. **Lemma 22**. *Consider IGIRG or SFP with $\tau\!>\!2$ in Def. [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}. Then for all $\varepsilon\in(0,1/2)$, and for all $\xi$ sufficiently large relative to $\varepsilon$, and $t \le \min\{1/\varepsilon,\log\log\xi\}$ the following holds. Consider a cube $Q \subseteq \mathbb{R}^d$ of side length $\xi$, and let $x_1,\ldots,x_t \in Q$, and $w_0$ from Def. [Definition 9](#def:net-constants){reference-type="ref" reference="def:net-constants"}, then $$\begin{aligned} \mathbb{P}(\text{$Q$ contains a weak $(\varepsilon,w_0)$-net $\mathcal{N}$, and } x_1,\ldots,x_t \in \mathcal{N}\mid x_1,\ldots,x_t \in \mathcal{V}) \ge 1-t\varepsilon. \end{aligned}$$* The condition $t\le 1/\varepsilon$ is there to avoid a vacuous statement, and below we set $t=2$. Note that the condition [\[eq:net-defining-crit-eps\]](#eq:net-defining-crit-eps){reference-type="eqref" reference="eq:net-defining-crit-eps"} never counts vertices of weight less than $w_1/2$. So, we can decide whether a weak $(\varepsilon,w_1)$-net $\mathcal{N}$ exists by uncovering only the set of vertices of weight at least $w_1/2$ (beyond $x_1,\ldots,x_t \in \mathcal{N}$). For IGIRG, this set is independent of the set of vertices of weight smaller than $w_1/2$. This is the main reason for introducing the parameter $w_1$. *Proof of Lemma [Lemma 22](#lem:weak-nets-exist){reference-type="ref" reference="lem:weak-nets-exist"}.* Let $\varepsilon\in(0,1/2)$, set $\eta := 1 - \varepsilon/2$, and define $\mathcal{R}= \{r_1,\ldots,r_R\}$ as $$\begin{aligned} R &:= 2 + \lfloor(\log\log\xi\sqrt{d} - \log^{*4}\xi\sqrt{d}-\log\tfrac{4}{\varepsilon})/\log(1/\eta)\rfloor, \label{eq:weak-net-choices-1}\\ r_i &:= (\xi\sqrt{d})^{\eta^{R-i}}, \quad\mbox{for} \quad i\in[R]. \label{eq:weak-net-choices-2} \end{aligned}$$ We next prove that $\mathcal{R}$ is $\varepsilon$-well-spaced (Def. [Definition 10](#def:well-spaced){reference-type="ref" reference="def:well-spaced"}). Let $1-a\in[0,1)$ be the fractional part of the expression inside the $\lfloor \cdot \rfloor$ in [\[eq:weak-net-choices-1\]](#eq:weak-net-choices-1){reference-type="eqref" reference="eq:weak-net-choices-1"}. Then using that $\lfloor x\rfloor=x-1+a$, $$\label{eq:weak-nets-exist-r1} r_1 = (\xi\sqrt{d})^{\eta^{R-1}} = (\xi\sqrt{d})^{\eta^{a}4\log^{*3}\xi\sqrt{d}/(\varepsilon\log\xi\sqrt{d})} = (\log\log\xi\sqrt{d})^{(1-\varepsilon/2)^{a}4/\varepsilon}.$$ Since $\xi$ is large relative to $\varepsilon$, Def. [Definition 10](#def:well-spaced){reference-type="ref" reference="def:well-spaced"} [\[eq:nets-small-r\]](#eq:nets-small-r){reference-type="eqref" reference="eq:nets-small-r"} holds for $\varepsilon=\delta$. For all $i \in [R]$, since $\eta=1-\varepsilon/2$: $$r_i/r_{i-1} = (\xi\sqrt{d})^{\eta^{R-i}(1-\eta)} \ge (\xi\sqrt{d})^{\eta^{R-1}\varepsilon/2} = r_1^{\varepsilon/2} = (\log\log\xi\sqrt{d})^{2(1-\varepsilon/2)^{a}}.$$ Since $a\le 1$, $2(1-\varepsilon/2)^{a}>1$, for all $\varepsilon<1/2$, so Def. [Definition 10](#def:well-spaced){reference-type="ref" reference="def:well-spaced"} [\[eq:nets-radii\]](#eq:nets-radii){reference-type="eqref" reference="eq:nets-radii"} holds even in $d=1$, and $\mathcal{R}$ is $\varepsilon$-well-spaced as claimed. Moreover, since $t \le \log\log\xi$ by hypothesis, by [\[eq:weak-nets-exist-r1\]](#eq:weak-nets-exist-r1){reference-type="eqref" reference="eq:weak-nets-exist-r1"} we have $t \le (r_1/4\sqrt{d})^d$. By Proposition [\[lem:nets-exist\]](#lem:nets-exist){reference-type="ref" reference="lem:nets-exist"}, with probability at least $1-t\varepsilon$, conditioned on $x_1,\ldots,x_t\in \mathcal{V}$, $Q$ contains a (strong) $(\varepsilon,\mathcal{R})$-net $\mathcal{N}$, using Def. [Definition 9](#def:net-constants){reference-type="ref" reference="def:net-constants"}. We now prove that a strong net is also a weak net. Let $r \in [(\log\log\xi\sqrt{d})^{4/\varepsilon}, \xi\sqrt{d}]$ and $w \in [w_0,r^{d/(\tau-1)-\varepsilon}]$ as in Def. [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"} (this interval is non-empty since $\tau\in(2,3), \varepsilon<1/2$), and consider a vertex $v \in \mathcal{N}$. By [\[eq:weak-nets-exist-r1\]](#eq:weak-nets-exist-r1){reference-type="eqref" reference="eq:weak-nets-exist-r1"} since $a\ge 0$, we have $r_1 \le r \le r_R$. Let $r_j$ be such that $r\in [r_j,r_{j+1})$; thus $r^{\eta} \le r_j \le r$. Since $\xi$ is large relative to $\varepsilon$, we have $w \le f_{R,\varepsilon}(r_j)$ (using Def. [Definition 9](#def:net-constants){reference-type="ref" reference="def:net-constants"}) for $f$); thus by the definition of a strong $(\varepsilon, \mathcal{R})$-net in Def. [Definition 8](#def:net){reference-type="ref" reference="def:net"}, $|\mathcal{N}\cap (B_{r_j}(v)\times[w/2, 2w])|\ge \ell(w)w^{-(\tau-1)}r_j^d/(2d)^{d+\tau+5}$. Since $\xi$ is large relative to $\varepsilon$ and $r \ge r_j \ge r^{\eta}$, the required inequality in Def. [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"} follows since $$|\mathcal{N}\cap (B_{r}(v)\times[w/2, 2w])| \ge \ell(w)w^{-(\tau-1)}r^{d\eta}/(2d)^{d+\tau+5} \ge \ell(w)w^{-(\tau-1)}r^{d(1-\varepsilon)}.$$ -1em ◻ # Multi-round exposure with dependent edge-costs {#sec:exposure} Now with the pseudorandom nets at hand, we may switch to the quenched setting and reveal the realisation of the vertex set $\widetilde\mathcal{V}=(V,w_V)$. We shall now reveal edges adaptively to construct a fast-transmission path between $0,x$, according to the 'budget travel plan' in Section [1](#sec:intro){reference-type="ref" reference="sec:intro"}, see Fig. [\[fig:intuition_hierarchy\]](#fig:intuition_hierarchy){reference-type="ref" reference="fig:intuition_hierarchy"}. In particular, we need to find low-cost edges in spatial regions which depend on the previous low-cost edges we have found. When studying graph distances in Biskup [@biskup2004scaling] this is not a major obstacle, but with the presence of edge-costs we run into conditioning issues. To overcome these, we develop a multiple-round exposure -- essentially an elaborate edge-sprinkling method on the quenched vertex set -- where in each round we reveal multiple edges, which we explain now heuristically. We construct *$1/R$-percolated SFP/IGIRGs* $G_1, \ldots, G_R$ on the fixed vertex set $(V,w_V)$, where in each $G_i$ the probability that an edge $uv$ exists is given by $\mathbb{P}(uv\in\mathcal{E}(G_i)) = h(u-v,w_u,w_v)/R$, where $h$ is the connection probability from [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"}. To each edge $uv$ in $G_i$ we also assign $L_{uv}^{(i)}$, i.i.d. from the r.v. $L$ determining the costs in [\[eq:cost\]](#eq:cost){reference-type="eqref" reference="eq:cost"}. For each pair of vertices $u,v\in (V, w_V)$ we also draw a r.v. $Z_{uv}$ uniformly in $[R]$. Then we construct a single weighted graph $G$ on $(V, w_V)$ from the collection $(G_1, G_2, \ldots, G_R)$ by setting $\mathcal{E}(G):=\{uv : uv \textnormal{ is present in } G_{Z_{uv}}\}$ with transmission costs $\mathcal{C}(uv) = L_{uv}^{(Z_{uv})}\cdot(w_uw_v)^{\mu}$. We show that this is an alternative construction of graphs in Def. [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}. Given $(V, w_V)$, we then use edges in $G_i\cap G$ in the $i$th round of edge-revealment, independent of edges in earlier rounds. In a classical random graph setting, multiple-round exposure means to couple the base graph model $\mathcal{G}$ to a suite of sparser but independent random subgraphs $\mathcal{G}_1,\ldots,\mathcal{G}_R \subseteq \mathcal{G}$, and taking the $i$th edge of a path from the $i$th "round of exposure" $\mathcal{G}_i$. In the edge-weighted setting we design a (slightly more restrictive) construction incorporating *independent* edge-cost variables on $\mathcal{G}_1,\ldots,\mathcal{G}_R \subseteq \mathcal{G}$ that we describe in Prop. [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"} after some preliminary definitions. Recall that for a set $V$, we write $V^{(2)} := \{\{x,y\}\colon x,y \in V, x\ne y\}$ for the set of *possible* edges of a graph with vertex set $V$. For future re-usability, we formulate Prop. [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"} in a general class of random graph models, as set out below. **Definition 23**. *A *conditionally-independent edge-weighted vertex-marked random graph model (CIRG model)* $\mathcal{G}$ consists of distributions for a random vertex set $\mathcal{V}$ which is a.s. countable, a random parameter set $\mathcal{W}$, a random edge set $\mathcal{E}\subseteq \mathcal{V}^{(2)}$, and random edge costs $\mathcal{C}(xy)$ for each possible edge $\{x,y\} \in \mathcal{V}^{(2)}$. Conditioned on any given realisation of $(\mathcal{V},\mathcal{W})=(V, w_V)$, all costs $\mathcal{C}(xy)$ and all events $xy \in \mathcal{E}$ are independent across $\{x,y\} \in \mathcal{V}^{(2)}$. If $G \sim \mathcal{G}$, we say that $G$ is a realisation of a *CIRG*, or simply a CIRG. We write $\{\mathcal{G}\mid V, w_V\}$ for the distribution of $G$ conditioned on $(\mathcal{V}, \mathcal{W})=(V, w_V)$.* First passage percolation (1-FPP) on GIRG and SFP in Def. [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}--[Definition 2](#def:1-FPP){reference-type="ref" reference="def:1-FPP"} are both CIRG models, with $\mathcal{V}$ either a PPP on $\mathbb{R}^d$ or $\mathbb{Z}^d$, and $\mathcal{W}$ the vertex-weights, and $\mathcal{C}(xy)$ in [\[eq:cost\]](#eq:cost){reference-type="eqref" reference="eq:cost"} the edge-weights. For other graph models $\mathcal{W}$ could contain extra randomness. We now define the analogue of a single "round of exposure" for a CIRG model: **Definition 24** ($\theta$-percolated CIRG). *Let $G \sim \mathcal{G}$ be a CIRG from Definition [Definition 23](#def:CIRG){reference-type="ref" reference="def:CIRG"}. Then for all $\theta\in(0,1)$ the *$\theta$-percolation* of $G$ is the subgraph $G^{\theta}$ of $G$ which includes each $e\in \mathcal{E}(G)$ independently with probability $\theta$, and we write $\mathcal{G}^{\theta}$ for its law. We call $\mathcal{G}^{\theta}$ the *$\theta$-percolation* of $\mathcal{G}$, and $\theta$ the *percolation probability*.* **Remark 25** ($\theta$-percolated CIRGs are CIRGs). An alternative construction of $\mathcal{G}^\theta$ is to sample the realisation of $(V,w_V)\sim (\mathcal{V}, \mathcal{W})$ first, and then sample each edge $e$ with probability $\theta\mathbb P(e\in \mathcal{E}(G)\mid V, w_V)$. So the CIRG model class is closed under $\theta$-percolation. We now set out a specific coupling between a base CIRG model and percolated CIRGs, that will serve as graphs forming the rounds of exposure. Recall that $[r]:=\{1,2, \ldots, r\}$. **Definition 26** (Exposure setting of $G$). *Let $\mathcal{G}$ be a CIRG model from Def. [Definition 23](#def:CIRG){reference-type="ref" reference="def:CIRG"}. Fix $r\!\in\!\mathbb{N}$ and $\theta_1,\ldots,\theta_r \in [0,1]$ satisfying $\sum_{i\in[r]}\theta_i\!=\! 1$. We define the *exposure setting* of $\mathcal{G}$ with *percolation probabilities* $\theta_1,\ldots,\theta_r$ as follows. Reveal the realisation of $(\mathcal{V}, \mathcal{W})=:(V, w_V)$, and let $(Z_{uv})_{uv\in V^{(2)}}$ be iid random variables with $\mathbb{P}(Z_{uv}=i)=\theta_i$ for all $i \in [r]$. Take $G_1^\star, \ldots, G_{r}^\star$ to be conditionally iid realisations of CIRGs, with shared distributions $G_i^\star \sim \{\mathcal{G}\mid V, w_V\}$, and respective edge costs $\mathcal{C}_i(e)$ for $e\in \mathcal{E}(G_i^\star)$, independent across $i\le r$. Let $G_i^{\theta_i}$ be the subgraph of $G_i^\star$ with edge set $\mathcal{E}(G_i^{\theta_i}):=\{e \in \mathcal{E}(G_i^\star)\colon Z_{e}=i\}$ and edge costs $\{\mathcal{C}_i(e): e\in \mathcal{E}(G_i^{\theta_i})\}$.* The following claim reconstructs $G$ from the percolated versions. **Claim 27** (Realisation of a CIRG in the exposure setting). *Let $\mathcal{G}$ be a CIRG model from Def. [Definition 23](#def:CIRG){reference-type="ref" reference="def:CIRG"}. Let $\theta_1,\ldots,\theta_r$ be the percolation probabilities, and consider $(G_i^{\theta_i})_{i\le r}$ in Definition [Definition 26](#def:exposure-setting){reference-type="ref" reference="def:exposure-setting"}. Then marginally, each $G_i^{\theta_i}$ is a $\theta_i$-percolated CIRG. Define now $G$ as the graph with vertex set and parameters $(V,w_V)$, and with edge set $\mathcal{E}(G):= \cup_{i\in[r]}\mathcal{E}(G_i^{\theta_i})$, and with edge costs $\{\mathcal{C}(e):=\mathcal{C}_{Z_e}(e): e\in \mathcal{E}(G)\}$. Then $G \sim \{\mathcal{G}\mid V, w_V\}$.* *Proof.* That $G_i^{\theta_i}$ is marginally a $\theta_i$-percolated CIRG, i.e., that it has law $\mathcal{G}^{\theta_i}$, is immediate since $\mathbb{P}(e\in \mathcal{E}(G_i^{\theta_i})\mid V, w_V)=\mathbb{P}(Z_e=i)\cdot \mathbb{P}(e\in G_i^\star\mid V, w_V)$, and now one can integrate over the realisations $(V, w_V)$. To see that $G$ has distribution $\{\mathcal{G}\mid V, w_V\}$ we argue as follows. Since $Z_{uv}$ takes a single value in $[r]$ each possible edge $e=uv$ appears in at most one of $G_1^{\theta_1}, \ldots, G_r^{\theta_r}$. Hence the union $\cup_{i\in[r]}\mathcal{E}(G_i^{\theta_i})=\mathcal{E}(G)$ is *disjoint*, and using that $G_1^\star, \ldots, G_r^\star$ all have law $\{\mathcal{G}\mid V, w_V\}$, $$\begin{aligned}\mathbb{P}\big(uv \in \mathcal{E}(G) \mid V, w_V\big) &= \sum_{i\in[r]} \mathbb{P}(Z_{uv}=i)\mathbb{P}\big(uv \in \mathcal{E}(G_i^\star) \mid V, w_V\big)\\ &= \sum_{i\in[r]} \theta_i\mathbb{P}\big(uv \in \mathcal{E}(G_1^\star)\mid V, w_V \big) = \mathbb{P}\big(uv \in \mathcal{E}(G_1^\star)\mid V, w_V\big), \end{aligned}$$ and $G_1^\star\sim \{\mathcal{G}\mid V, w_V\}$. Further, edges are present in $G$ independently (conditional on $(V,w_V)$) since the variables $Z_e$ and $\mathcal{E}(G_i^\star)$ are conditionally independent. ◻ For two collections of random variables we write $\mathcal{X}\in \sigma(\mathcal{X}')$, if all elements in $\mathcal{X}$ are measurable with respect to the $\sigma$-algebra generated by the variables in $\mathcal{F}'$, i.e., they are a deterministic function of elements in $\mathcal{X}'$. In the next definition we formalise multi-round exposure with edge-cost constraints, in the setting of CIRGs with $(\mathcal{V}, \mathcal{W})=(V, w_V)$ already exposed, which guarantees that edge presence and edge costs are independent by Def. [Definition 23](#def:CIRG){reference-type="ref" reference="def:CIRG"}. **Definition 28** (Iterative cost construction). *Let $G_1,\ldots,G_r$ be edge-weighted CIRGs on a common realisation of the vertex set and parameters $(\mathcal{V}, \mathcal{W})=(V,w_V)$. Write $\mathcal{C}_i$ for the cost function of $G_i$. An *iterative cost construction* $\mathrm{Iter}$ is a sequence of tuples $(G_1,\mathcal{E}_1,\mathcal{F}_1,\mathcal{U}_1),\ldots,$ $(G_r,\mathcal{E}_r,\mathcal{F}_r,\mathcal{U}_r)$ satisfying the following properties:* (i) *[\[item:iter1\]]{#item:iter1 label="item:iter1"} $\mathcal{F}_i$ is a finite collection of tuples ("allowed sets") of potential edges without repetition (i.e. of $\binom{V}{2}$), in $\sigma(V, w_V, \mathcal{E}_1,\ldots,\mathcal{E}_{i-1})$. In the $i$th round of revealment, we will reveal a tuple of edges $\mathcal{E}_i \in \mathcal{F}_i$ chosen among $\mathcal{F}_i$:* (ii) *[\[item:iter2\]]{#item:iter2 label="item:iter2"} Define the *round-$i$ marginal cost* of an edge by $$\begin{aligned} \label{eq:marginal-cost} \textnormal{mcost}_i(e) = \begin{cases} 0 & \mbox{ if $e$ appears in some tuple $\mathcal{E}_1,\ldots,\mathcal{E}_{i-1}$,}\\ \mathcal{C}_i(e) & \mbox{ otherwise.} \end{cases} \end{aligned}$$* (iii) *[\[item:iter3\]]{#item:iter3 label="item:iter3"} Either $\mathcal{E}_i=\mathtt{None}$ or $\mathcal{E}_i$ is a tuple of edges of $G_i$ together with their round-$i$ marginal cost of the form $\mathcal{E}_i=((e_1,\textnormal{mcost}_i(e_1)), \ldots (e_t,\textnormal{mcost}_i(e_t)))$ for some $t$.* (iv) *[\[item:iter4\]]{#item:iter4 label="item:iter4"} Each $\mathcal{U}_i = \mathcal{U}_i(V,w_V,\mathcal{E}_1,\ldots,\mathcal{E}_i)$ ("cost constraint") is a (list of) event(s) or constraint(s), measurable wrt $\sigma(V, w_V, \mathcal{E}_1,\ldots,\mathcal{E}_i)$.* (v) *[\[item:iter5\]]{#item:iter5 label="item:iter5"} $\mathcal{E}_i$ is chosen in the following way. We first fix a deterministic ordering of all tuples on $V^{\scriptscriptstyle{(2)}}$ that may appear in $\mathcal{F}_i$ (the *canonical ordering*). Then we define $\mathcal{E}_i$ to be the first element $(e_1,\ldots,e_t)$ in this ordering such that $e_j \in \mathcal{E}(G_i) \cup \mathcal{E}_1 \cup \dots \cup \mathcal{E}_{i-1}$ for all $j$ and $\mathcal{U}_i(V,w_V,\mathcal{E}_1,\ldots,\mathcal{E}_i)$ occurs. If no such set of edges exists, we set $\mathcal{E}_k = \textnormal{\texttt{None}}$ for all $k \ge i$.* *We call $r$ the number of *rounds*, and $G_i$ the *round-$i$ graph*. The construction is *successful* if $\mathcal{E}_i \ne \textnormal{\texttt{None}}$ for all $i \in [r]$. We define the event of seeing a given outcome up until round $i$ as: $$\label{eq:ait-event} \mathcal{A}_{\mathrm{Iter}}(V,w_V,E_1,\ldots,E_i) := \{(\mathcal{V},\mathcal{W}) = (V,w_V)\} \cap \bigcap_{j\in [i]} \{\mathcal{E}_j = E_j\};$$ we omit $V$ and $w_V$ from $\mathcal{A}_{\mathrm{Iter}}$ when they are clear from context.* In further sections, $\mathcal{U}_i$ represents the upper bounds we require on the round-$i$ marginal costs of the edges that we reveal in the $i$'th round, and $\mathcal{U}_i$ may depend on $(\mathrm{mcost}_j(\mathcal{E}_j))_{j\le i}$. The marginal cost [\[eq:marginal-cost\]](#eq:marginal-cost){reference-type="eqref" reference="eq:marginal-cost"} of an edge drops to zero when an edge has been already chosen in previous rounds, so bounding total marginal cost rather than total cost corresponds to counting the cost of each chosen edge exactly once; this is fine in our application, since if an edge appears twice in a walk then we can pass to a cheaper sub-path in which it appears at most once. We illustrate Definition [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"} on toy example. Take $G_1$ and $G_2$ to be two conditionally independent realisations of SFP in a box $B$ with the same $(V, w_V)$, equipped with edge-costs as in [\[eq:cost\]](#eq:cost){reference-type="eqref" reference="eq:cost"}. Take $\mathcal{F}_1$ to be the set of all possible pairs of vertex-disjoint triangles in $B$, i.e., $\mathcal{F}_1$ is the collection of all $6$ possible edges of the form $\{v_1v_2, v_2v_3, v_3v_1, v_4v_5, v_5v_6, v_6v_4\}$ with $v_1,\ldots,v_6$ all distinct. Take $\mathcal{U}_1$ to be the constraint that each of the two triangles individually has total cost at most $1$. The canonical ordering is arbitrary and typically not important. The first round reveals edge presence and costs in $G_1$: it reveals tuples in $\mathcal{F}_1$ sequentially in the canonical ordering until it finds $\mathcal{E}_1$, two (random) triangles $T_1$ and $T_2$ in $G_1$, or else outputs $\mathtt{None}$. Take $\mathcal{F}_2$ to be the collection of all possible paths of any length from $T_1$ to $T_2$, and take $\mathcal{U}_2$ to be the constraint that $\mathcal{E}_2\in \mathcal{F}_2$ has total marginal cost at most $1$. The second round reveals edge presence and costs in $G_2$: it reveals tuples in $\mathcal{F}_2$ sequentially. If successful, we have found two triangles of cost at most $1$ joined by a path $\pi_{T_1T_2}$ of marginal cost at most $1$. The path $\pi_{T_1T_2}$ may reuse an edge from $T_1$ or $T_2$, nevertheless, the total cost of the construction is still at most $3$ as each edge's cost only counts once in the construction. The next proposition replaces the FKG-inequality for iterative cost constructions. Below, edges in the $\theta_i$-percolated CIRGs $(H_i)_{i\le r}$ are conditionally *independent*, in contrast to Def. [Definition 26](#def:exposure-setting){reference-type="ref" reference="def:exposure-setting"} where edges in $(G_i^{\theta_i})_{i\le r}$ are dependent through $Z_e$. The proof is via coupling: when two graph-collections use the *same vertex set*, one can carry out the *same* iterative cost construction on them, using the same $(\mathcal{F}_i, \mathcal{U}_i)$ as long as tuples chosen in previous rounds agree. **Proposition 29** (Multi-round exposure). *Let $G\sim\{\mathcal{G}\mid V, w_V\}$ be a CIRG model with a fixed realisation $(V,w_V)$ of $(\mathcal{V},\mathcal{W})$. Let $\theta_1,\ldots,\theta_r \in [0,1]$ with $\sum_{i\in[r]}\theta_i = 1$, and let $H_1,\ldots,H_r$ be independent with distributions $\{\mathcal{G}^{\theta_i}\mid V, w_V\}$ for all $i\le r$ as in Definition [Definition 24](#def:percolated){reference-type="ref" reference="def:percolated"}. Consider an iterative construction $\mathrm{Iter}_G = (G,\mathcal{E}_1^G,\mathcal{F}_1,\mathcal{U}_1),\ldots,(G,\mathcal{E}_r^G,\mathcal{F}_r,\mathcal{U}_r)$ with all rounds using $G$, and let $\mathrm{Iter}_H = (H_1,\mathcal{E}_1^H,\mathcal{F}_1,\mathcal{U}_1),\ldots,(H_r,\mathcal{E}_r^H,\mathcal{F}_r,\mathcal{U}_r)$ be the iterative cost construction on $H_1,\ldots,H_r$ using the same $(\mathcal{F}_i,\mathcal{U}_i)_{i\le [r]}$. Then, taking the minimum over all possible realisations $(E_1,\ldots,E_{r-1})$ of $(\mathcal{E}_1^H, \ldots, \mathcal{E}_r^H)$ not containing $\mathtt{None}$, $$\begin{aligned} \begin{split}\label{eq:multi-round-exp-goal} &\mathbb{P}\big(\mathrm{Iter}_G \textnormal{ succeeds} \mid V,w_V\big) \\ &\qquad\qquad \ge \min_{E_1,\ldots,E_{r-1} \nsupseteq \mathtt{None}} \prod_{i\in[r]}\mathbb{P}\big(\mathcal{E}_i^H \ne \mathtt{None} \mid \mathcal{A}_{\mathrm{Iter}_H}(V,w_V,E_1,\ldots,E_{i-1})\big). \end{split} \end{aligned}$$* *Proof of Proposition [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}.* Let $G \sim \{\mathcal{G}\mid V, w_V\}$. By repeated conditioning, and taking the minimum over all possible successful realisations, $$\begin{aligned} \begin{split}\label{eq:multi-round-exp-0} &\mathbb{P}\big(\mathrm{Iter}_G \textnormal{ succeeds} \mid V,w_V\big) = \prod_{i \in [r]} \mathbb{P}\big(\mathcal{E}_i^G \ne \mathtt{None} \mid V,w_V, \mathcal{E}_1,\ldots,\mathcal{E}_{i-1} \nsupseteq \mathtt{None} \big)\\ &\qquad\qquad \ge \min_{E_1,\ldots,E_{r-1} \nsupseteq \mathtt{None}} \prod_{i\in[r]}\mathbb{P}\big(\mathcal{E}_i^G \ne \mathtt{None} \mid \mathcal{A}_{\mathrm{Iter}_G}( V, w_V, E_1,\ldots,E_{i-1})\big). \end{split} \end{aligned}$$ Claim [Claim 27](#claim:exposure-coupling){reference-type="ref" reference="claim:exposure-coupling"} ensures that using the edge-disjoint graphs $G_i^{\theta_i}, i\le r$, we can realise $G$ as $$\label{eq:G-construction} G = \cup_{i\le r} G_i^{\theta_i},$$ where marginally $G_i^{\theta_i}\,{\buildrel d \over =}\, H_i$, but $G_i^{\theta_i}$ are dependent through the variables $(Z_{uv})_{uv\in V^{(2)}}$. Let $\mathrm{Iter}_G^{-} = (G_1^{\theta_1},\mathcal{E}_1^-,\mathcal{F}_1,\mathcal{U}_1),\ldots,(G_r^{\theta_r},\mathcal{E}_r^-,\mathcal{F}_r,\mathcal{U}_r)$ be the iterative cost construction on $(G_1^{\theta_1},\ldots,G_r^{\theta_r})$ that uses the same possible tuples and constraints $(\mathcal{F}_i, \mathcal{U}_i)$ as $\mathrm{Iter}_G$, but differs in that $\mathcal{E}_i^{-}$ is chosen from $G_i^{\theta_i}$ in round $i$ (while $\mathcal{E}_i^G$ is chosen from $G$). By [\[eq:multi-round-exp-0\]](#eq:multi-round-exp-0){reference-type="eqref" reference="eq:multi-round-exp-0"} and [\[eq:G-construction\]](#eq:G-construction){reference-type="eqref" reference="eq:G-construction"}, $$\begin{aligned} \label{eq:multi-round-exp-1} \mathbb{P}\big(\mathrm{Iter}_G \textnormal{ succeeds} \mid (V,w_V)\big) \ge \min_{E_1,\ldots,E_{r-1} \nsupseteq \mathtt{None}} \prod_{i\in[r]}\mathbb{P}\big(\mathcal{E}_i^- \ne \mathtt{None} \mid \mathcal{A}_{\mathrm{Iter}_G^-}(E_1,\ldots,E_{i-1})\big). \end{aligned}$$ In order to prove [\[eq:multi-round-exp-goal\]](#eq:multi-round-exp-goal){reference-type="eqref" reference="eq:multi-round-exp-goal"} from [\[eq:multi-round-exp-1\]](#eq:multi-round-exp-1){reference-type="eqref" reference="eq:multi-round-exp-1"}, for each $i \in [r]$ and each possible outcome $E_1,\ldots,E_{i-1} \nsupseteq \mathtt{None}$, for each $i\le r$ we will couple $G_i^{\theta_i}$ conditioned on $\mathcal{A}_{\mathrm{Iter}_G^-}(E_1,\ldots,E_{i-1})$ to $H_i$ conditioned on $\mathcal{A}_{\mathrm{Iter}_H}(E_1,\ldots,E_{i-1})$; satisfying for all $e \in V^{(2)}$: $$\begin{aligned} \{e \in \mathcal{E}(H_i)\}\cap\{ e\in \mathcal{E}(G_i^{\theta_i})\}& \Longrightarrow \mathcal{C}_{H_i}(e)= \mathcal{C}_{i}(e), \label{eq:multi-round-exp-coupling-1a}\\ \{ e \in \mathcal{E}(H_i) \} &\subseteq \{e \in \mathcal{E}(G_i^{\theta_i}) \cup \mathcal{E}_1^- \cup \dots \cup \mathcal{E}_{i-1}^-\}. \label{eq:multi-round-exp-coupling-1b} \end{aligned}$$ In words, if an edge appears in both graphs $H_i$ and $G_i^{\theta_i}$ then its cost agrees, and if an edge is in $H_i$ then either $e$ has been chosen already in previous rounds or $e$ is also in $G_i^{\theta_i}$. Under the conditioning, $\mathcal{E}_j^H = \mathcal{E}_j^- =: E_j \ne \texttt{None}$ for all $j \le i-1$. Suppose $\mathcal{E}_i^H =: E_i \ne \texttt{None}$. Then by [\[eq:multi-round-exp-coupling-1b\]](#eq:multi-round-exp-coupling-1b){reference-type="eqref" reference="eq:multi-round-exp-coupling-1b"} every set $E \in E_i \subseteq \mathcal{E}(H_i) \cup E_1 \cup \dots \cup E_{i-1}$ is also contained in $\mathcal{E}(G_i^{\theta_i}) \cup E_1 \cup \dots \cup E_{i-1}$, and by [\[eq:marginal-cost\]](#eq:marginal-cost){reference-type="eqref" reference="eq:marginal-cost"} their round-$i$ marginal costs in $\mathrm{Iter}_G^-$ are equal to those in $\mathrm{Iter}_H$. Hence $E_i$ provides a valid choice for $\mathcal{E}_i^-$, and $\mathcal{E}_i^- \ne \texttt{None}$ holds also. Thus $$\label{eq:multi-round-exp-2} \mathbb{P}\big(\mathcal{E}_i^- \ne \mathtt{None} \mid \mathcal{A}_{\mathrm{Iter}_G^-}(E_1,\ldots,E_{i-1})\big) \ge \mathbb{P}\big(\mathcal{E}_i^H \ne \mathtt{None} \mid \mathcal{A}_{\mathrm{Iter}_H}(E_1,\ldots,E_{i-1})\big),$$ and the result follows from [\[eq:multi-round-exp-1\]](#eq:multi-round-exp-1){reference-type="eqref" reference="eq:multi-round-exp-1"}. Now we provide the coupling achieving [\[eq:multi-round-exp-coupling-1a\]](#eq:multi-round-exp-coupling-1a){reference-type="eqref" reference="eq:multi-round-exp-coupling-1a"}--[\[eq:multi-round-exp-coupling-1b\]](#eq:multi-round-exp-coupling-1b){reference-type="eqref" reference="eq:multi-round-exp-coupling-1b"}. Def. [Definition 26](#def:exposure-setting){reference-type="ref" reference="def:exposure-setting"} uses the independent graphs $G_i^\star\sim\{\mathcal{G}\mid V, w_V\}$, and obtains $G_i^{\theta_i}$ as a dependent thinning of $G_i^\star$ using $(Z_{uv})_{uv \in V^{(2)}}$ (independent across different $uv$). For each $uv\in V^{(2)}$ sample iid uniform $U_{uv}^{(i)}\sim U[0,1]$ and realise the presence of $uv$ in $H_i$ (resp. $G_i^{\theta_i}$) as $$\label{eq:partial-coupling} \mathbf{1}_{uv\in H_i}=\mathbf{1}_{uv\in G_i^\star}\mathbf{1}_{U_{uv}^{(i)}\le \theta_i}, \qquad \mathbf{1}_{uv\in G_i^{\theta_i}}=\mathbf{1}_{uv\in G_i^\star}\mathbf{1}_{Z_{uv}=i}.$$ Then $H_1,\ldots,H_r$ are *independent* $\theta_i$-percolations of $G_1^\star,\ldots,G_r^\star$ respectively, and $H_1,\ldots,H_r$ are themselves independent as required in the statement, since $G_1^\star,\ldots,G_r^\star$ are independent conditionally on $(V, w_V)$. Note that [\[eq:partial-coupling\]](#eq:partial-coupling){reference-type="eqref" reference="eq:partial-coupling"} is only a *partial coupling*, since we can still specify the joint distribution of $(U_{uv}^{(i)})_{i\le r}, Z_{uv}$. By this partial coupling, $H_i$ and $G_i^{\theta_i}$ are both subgraphs of $G_i^\star$, and hence if an edge $e$ is in both subgraphs, then $\mathcal{C}_{H_i}(e)=\mathcal{C}_i(e)$ (the cost of $e$ in $G_i^\star$), so [\[eq:multi-round-exp-coupling-1a\]](#eq:multi-round-exp-coupling-1a){reference-type="eqref" reference="eq:multi-round-exp-coupling-1a"} holds. We now make this into a full coupling to satisfy [\[eq:multi-round-exp-coupling-1b\]](#eq:multi-round-exp-coupling-1b){reference-type="eqref" reference="eq:multi-round-exp-coupling-1b"}. Fix $i$ and $E_1,\ldots,E_{i-1}$. For [\[eq:multi-round-exp-coupling-1b\]](#eq:multi-round-exp-coupling-1b){reference-type="eqref" reference="eq:multi-round-exp-coupling-1b"}, we first claim the following distributional identities hold: $$\begin{aligned} \mathcal{E}(H_i)\mid \mathcal{A}_{\mathrm{Iter}_H}(V, w_V, E_1,\ldots,E_{i-1})\ &{\buildrel d \over =}\ \mathcal{E}(H_i)\mid (V, w_V),\label{eq:dist-id1}\\ \mathcal{E}(G_i^\star)\mid \mathcal{A}_{\mathrm{Iter}_G^-}(V, w_V, E_1, \ldots, E_{i-1})\ &{\buildrel d \over = }\ \mathcal{E}(G_i^\star)\mid (V, w_V).\label{eq:dist-id2} \end{aligned}$$ Indeed, by Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"} and $\mathrm{Iter_H}$ in Prop. [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}, the event $\mathcal{A}_{\mathrm{Iter}_H}(V, w_V, E_1,\ldots,E_{i-1})$ is measurable wrt $\sigma(H_1,\ldots,H_{i-1} \mid V, w_V)$, and $H_1, \ldots, H_{i-1}$ are *independent* of $H_i$ conditioned on $(V, w_V)$, so [\[eq:dist-id1\]](#eq:dist-id1){reference-type="eqref" reference="eq:dist-id1"} holds. By [\[eq:partial-coupling\]](#eq:partial-coupling){reference-type="eqref" reference="eq:partial-coupling"} and by the conditional independence of $G_1^\star, \ldots, G_i^{\star}$, $\mathcal{A}_{\mathrm{Iter}_G^-}(V, w_V, E_1, \ldots, E_{i-1})$ is in $\sigma(G_1^{\theta_1}, \ldots, G_{i-1}^{\theta_{i-1}}\mid V, w_V) \subseteq \sigma(G_1^{\star}, \ldots, G_{i-1}^{\star}, (Z_e)_{e\in V^{(2)}} \mid V, w_V)$, so [\[eq:dist-id2\]](#eq:dist-id2){reference-type="eqref" reference="eq:dist-id2"} follows similarly. By Strassen's theorem, to prove that a coupling in [\[eq:multi-round-exp-coupling-1b\]](#eq:multi-round-exp-coupling-1b){reference-type="eqref" reference="eq:multi-round-exp-coupling-1b"} exists, it now suffices to prove that for all $k\ge 1$ and all $e_1,\ldots,e_k \in V^{(2)}\setminus (E_1 \cup \dots \cup E_{i-1})$, $$\label{eq:multi-round-exp-coupling-3} \begin{aligned} \mathbb{P}\big(e_1,\ldots,e_k \in \mathcal{E}(G_i^{\theta_i}) &\mid \mathcal{A}_{\mathrm{Iter}_{G^-}}(V, w_V, E_1,\ldots,E_{i-1})\big) \\ &\quad\ge \mathbb{P}\big(e_1,\ldots,e_k \in \mathcal{E}(H_i) \mid \mathcal{A}_{\mathrm{Iter}_{H}}(V, w_V, E_1,\ldots,E_{i-1})\big)\\ &\quad=\theta_i^k \cdot \mathbb{P}\big(e_1,\ldots,e_k \in \mathcal{E}(G_i^\star) \mid \mathcal{A}_{\mathrm{Iter}_{G}^-}(V, w_V, E_1,\ldots,E_{i-1})\big), \end{aligned}$$ where the last row follows from [\[eq:partial-coupling\]](#eq:partial-coupling){reference-type="eqref" reference="eq:partial-coupling"}--[\[eq:dist-id2\]](#eq:dist-id2){reference-type="eqref" reference="eq:dist-id2"}. Dividing both sides by the second factor of the rhs and applying [\[eq:partial-coupling\]](#eq:partial-coupling){reference-type="eqref" reference="eq:partial-coupling"} yields $$\begin{aligned} \mathbb{P}\big(Z_{e_1}=i,\ldots,Z_{e_k} = i \mid \{e_1,\ldots,e_k \in \mathcal{E}(G_i^\star)\} \cap \mathcal{A}_{\mathrm{Iter}_{G}^-}(V, w_V, E_1,\ldots,E_{i-1})\big) \ge \theta_i^k. \end{aligned}$$ Since $\sigma(G_i^\star \mid V,w_V)$ is independent of $\sigma(G_1^\star,\dots,G_{i-1}^\star,(Z_e)_{e \in V^{(2)}}\mid V,w_V)$, we see it suffices to prove that for all $k\ge 1$ and all $e_1,\dots,e_k \in V^{(2)} \setminus (E_1 \cup \dots \cup E_{i-1})$, $$\label{eq:multi-round-exp-coupling-4} \begin{aligned} \mathbb{P}\big(Z_{e_1}=i,\ldots,Z_{e_k} = i \mid \mathcal{A}_{\mathrm{Iter}_{G}^-}(V, w_V, E_1,\ldots,E_{i-1})\big) \ge \theta_i^k. \end{aligned}$$ We next express $\mathcal{A}_{\mathrm{Iter}_{G}^-}(V, w_V, E_1,\ldots,E_{i-1})$ in terms of simpler events, using Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}[\[item:iter5\]](#item:iter5){reference-type="eqref" reference="item:iter5"}. Let $\underline t_{j,s}$ be the $s$-th tuple in the canonical ordering of $\mathcal{F}_j(V,w_V,E_1,\ldots,E_{j-1})$, with $E_j=:\underline t_{j,s_j^\star}$ the $s^\star_j$th tuple. For each tuple $\underline t_{j,s}$ we collect the edges in $\{e_1,\dots, e_k\}\cap \underline t_{j,s}$, and define $$\label{eq:a-and-c} \begin{aligned} A_{j,s}'&:=\{\exists e \in \underline t_{j,s}\cap \{e_1, \dots, e_k\}: Z_{e}\neq j\},\\ B_{j,s}&:= \{\exists e\in \underline t_{j,s}, e\notin\{e_1, \dots e_k\}, Z_e\neq j\},\\ C_{j,s}&:= \big(\neg\,\mathcal{U}_j(V,w_V,E_1,\ldots,E_{j-1},\underline t_{j,s}) \cup \{\exists e\in \underline t_{j,s}: e \notin \mathcal{E}(G_j^\star)\} \big), \end{aligned}$$ with the idea that if $t_{j,s}\cap \{e_1, \dots, e_k\}=\emptyset$ then $A_{j,s}'=\Omega$. Let now $A=\{Z_{e_1}=i, \dots, Z_{e_k}=1 \}$, then clearly $A\subseteq A_{j,s}'$. Further, for $j\le i-1$, the edges and costs of $G_i^\star$ are (conditionally on $(V, w_V)$) independent from those of $G_1^\star,\ldots,G_{i-1}^\star$, so $C_{j,s}$ is independent of $A$. $B_{j,s}$ is also independent of $A$ since it concerns different edges than $\{e_1,\dots, e_k\}$. Then, by Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}[\[item:iter5\]](#item:iter5){reference-type="eqref" reference="item:iter5"}, $\underline t_{j,s}$ is not chosen as $E_j$ iff $A'_{j,s}\cup B_{j,s}\cup C_{j,s}$ holds, while $\underline t_{j,s_{j}^\star}=E_j$ is chosen in round $j$ if $\neg A_{j,s_j^\star}\cap \neg B_{j,s_j^\star}\cap \neg C_{j,s_j^\star}$ holds. Hence $$\mathcal{A}_{\mathrm{Iter}_{G}^-}(V, w_V, E_1,\ldots,E_{i-1}) = \bigcap_{j \le i-1} \Big(\neg A_{j,s_j^\star}\cap \neg B_{j,s_j^\star}\cap \neg C_{j,s_j^\star} \cap \bigcap_{s < s_j^\star} \big( A'_{j,s}\cup B_{j,s}\cup C_{j,s} \big)\Big).$$ Since $e_1,\dots,e_k \notin E_1,\dots,E_{i-1}$, the event $A_{j,s_j^\star}$ is also independent of $Z_{e_1},\dots,Z_{e_k}$, so the event $B:=\cap_{j\le i-1}(\neg A_{j,s_j^\star}\cap \neg B_{j,s_j^\star}\cap \neg C_{j,s_j^\star})$ above is independent of $Z_{e_1},\dots,Z_{e_k}$ and thus of $A$, and since $A\subseteq \cap_{j\le i-1}\cap_{s<s_j^\star}A_{j,s}'$, the lhs of  [\[eq:multi-round-exp-coupling-4\]](#eq:multi-round-exp-coupling-4){reference-type="eqref" reference="eq:multi-round-exp-coupling-4"} equals $$\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B\cap \cap_{s\le s_j^\star} (A_{j,s}'\cup B_{j,s}\cup C_{j,s}))} \ge \frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}= \mathbb{P}(A)=\theta_i^k,$$ since $Z_{e_\ell}$ are independent across $\ell\le k$. This yields the rhs of [\[eq:multi-round-exp-coupling-4\]](#eq:multi-round-exp-coupling-4){reference-type="eqref" reference="eq:multi-round-exp-coupling-4"}, finishing the proof. ◻ # Building blocks: finding cheap edges {#sec:lemmas} In this section, we return from CIRGs to GIRGs and state a few ancillary claims that we shall use to construct the different parts of the low-cost path between $0$ and $x$. We work in the quenched setting with the realisation of vertices and their weights $\widetilde{\mathcal{V}}=(V, w_V)$ *exposed*, taking the role of $(V,w_V)$ for CIRGs of Definition [Definition 23](#def:CIRG){reference-type="ref" reference="def:CIRG"}. All claims here concern $\theta$-percolated SFP/IGIRG as in Definition [Definition 24](#def:percolated){reference-type="ref" reference="def:percolated"}, so that we can later use them on the graphs $H_i$ of the multi-round exposure Proposition [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}. We first set out some common notation for Sections [4](#sec:lemmas){reference-type="ref" reference="sec:lemmas"} and [5](#sec:hierarchy){reference-type="ref" reference="sec:hierarchy"}. **Setting 30**. (The setting)[\[set:hierarchy-common\]]{#set:hierarchy-common label="set:hierarchy-common"} Consider $1$-FPP in Definition [Definition 2](#def:1-FPP){reference-type="ref" reference="def:1-FPP"} on the graphs IGIRG or SFP satisfying the assumptions given in [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} with $d\ge 1, \alpha \in (1,\infty], \tau\in(2,3)$. Let $\underline{c}$, $\overline{c}$, $h$, $L$, $c_1$, $c_2$, and $\beta$ be as in [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"}; we allow $\beta = \infty$ and $\alpha=\infty$. Fix a realisation $(V, w_V)$ of $\widetilde{\cal V}$, and let $G \sim \{\mathcal{G}\mid V, w_V\}$, and for a $\theta\in(0,1]$, let $G'$ be a $\theta$-percolation $G'$ of $G$. For brevity we write $\mathbb{P}( \cdot \mid V,w_V)$ for $\mathbb{P}( \cdot \mid \widetilde{\mathcal{V}}(G') = (V,w_V))$. Let $x \in V$, and let $Q$ be a cube of side length $\xi$ containing $0$ and $x$. Let $\delta \in (0,1)$, $w_0\ge 1$, and assume that $(V, w_V)$ is such that $Q$ contains a weak $(\delta/4,w_0)$-net $\mathcal{N}$ with $0,x \in \mathcal{N}$ given in Definition [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"}. Finally, let $\gamma \in(0,1)$. We now define a function of crucial importance for the optimisation of the exponents $\Delta_0, \eta_0$ in [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"} [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}. The first claim joins two Euclidean balls with a low-cost edge in the net with endpoints having specified weights. For all $\gamma > 0$ and all $\eta, z \ge 0$, we define $$\begin{aligned} \label{eq:Lambda-def} \Lambda(\eta, z):= 2d\gamma-\alpha(d-z)-z(\tau-1)+\big(0\wedge\beta(\eta-\mu z)\big).\end{aligned}$$ **Claim 31** (Single bridge-edge). *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}. Let $z \in [0,d]$ satisfy $2d\gamma > z(\tau-1)$. Let $c_H, \eta \ge 0$. Suppose that $0 < \delta{\,\ll_{\star}\,}\gamma,\eta, z, c_H,\textnormal{\texttt{par}}\xspace$, and that $D {\,\gg_{\star}\,}\eta, z,c_H,\delta,w_0$. Assume that $D^\gamma \in[ (\log\log\xi\sqrt{d})^{16/\delta}, \xi\sqrt{d} ]$ and that $x, y \in \mathcal{N}$ satisfy $|x-y| \le c_HD$, and let $\underline{w}\in[w_0 \vee 4(c_H+2)^d \vee 4000c_1^{-1/(\mu\beta)}, D^{\delta}]$ satisfy $F_L((\underline{w}/4000)^{\mu})\ge 1/2$. For $v \in \{x,y\}$, define $$\begin{aligned} \label{eq:calZ-def} \mathcal{Z}(v)=\mathcal{Z}_{\gamma,z,\underline{w}}(v):=\mathcal{N}\cap \big(B_{D^\gamma}(v)\times [\underline{w}D^{z/2}/2, 2\underline{w}D^{z/2}]\big). \end{aligned}$$ Again for each $v \in \{x,y\}$, let $Z_v \subseteq \mathcal{Z}(v)$ with $|Z_v| \ge |\mathcal{Z}(v)|/4$. Let $$\begin{aligned} \label{eq:N-gamma-eta-z} N_{\eta,\gamma,z, \underline{w}} (Z_x,Z_y) &:=\left\{ (a,b) \in Z_x\times Z_y, ab\in \mathcal{E}(G'), \mathcal{C}(ab) \le (\underline{w}/10)^{3\mu}D^\eta \right\}. \end{aligned}$$ Then, (and also for $\alpha=\infty$ and/or $\beta=\infty$ under the convention that $\infty \cdot 0 = 0$ in [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}), $$\label{eq:lem-cheap-bridge} \mathbb{P}\big(N_{\eta,\gamma,z, \underline{w}}(Z_x,Z_y) = \emptyset \mid V,w_V\big) \le \exp\Big({-}\theta \underline{w}^{-2(\tau-1)}D^{\Lambda(\eta,z)-2\gamma d\delta/3}\Big).$$* Since $D{\,\gg_{\star}\,}\delta$, we can always ensure that the interval for $\underline{w}$ is non-empty. *Proof.* We first bound the number of possible edges in $N_{\eta,\gamma,z,\underline{w}}(Z_x,Z_y)$ from below. In Def. [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"}, we will take $\varepsilon=\delta/4$, $w=\underline{w}D^{z/2}$ and $r=D^\gamma$. We have $\underline{w}D^{z/2} \ge \underline{w}\ge w_0$. Since $2d\gamma > z(\tau-1)$, we have $z/2 < d\gamma/(\tau-1)$, and $\underline{w}\le D^\delta$ by hypothesis; for sufficiently small $\delta$, it follows that $\underline{w}D^{z/2} < D^\delta \cdot D^{d\gamma/(\tau-1) - 2\delta} \le D^{d\gamma/(\tau-1) - \delta/4}$; thus the requirement on $\underline{w}$ of Def. [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"} is satisfied. Also $D^\gamma \in [(\log\log\xi\sqrt{d})^{16/\delta}, \xi\sqrt{d}]$ by hypothesis. Thus [\[eq:net-defining-crit-eps\]](#eq:net-defining-crit-eps){reference-type="eqref" reference="eq:net-defining-crit-eps"} gives for $v \in \{x,y\}$: $$|\mathcal{Z}(v)| \ge D^{\gamma d(1-\delta/4)} \ell(\underline{w}D^{z/2})\underline{w}^{-(\tau-1)}D^{-z(\tau-1)/2}.$$ Since $\underline{w}D^{z/2} \le D^{\delta+z/2}$ and $D{\,\gg_{\star}\,}\delta, z$, by Potter's bound $|\mathcal{Z}(v)| \ge \underline{w}^{-(\tau-1)}D^{d\gamma - z(\tau-1)/2 - 3\delta\gamma d/10}$, so $|Z_v| \ge |\mathcal{Z}(v)|/4 \ge \underline{w}^{-(\tau-1)}D^{d\gamma - z(\tau-1)/2 - 3\delta\gamma d/10}/4$ for $v \in \{x,y\}$. Accounting for the possibility of overlap between $Z_x$ and $Z_y$, we obtain $$\label{eq:cheap-bridge-N} \big|\{\{a,b\}: a\in Z_x, b\in Z_y|\}\big|\! \ge\! \frac{(|Z_x|\wedge|Z_y|)^2}{4} \ge \underline{w}^{-2(\tau-1)}D^{2d\gamma - z(\tau-1) - 3 \delta \gamma d/5}/64.$$ We now lower-bound the probability that $a\in Z_x, b\in Z_y$ forms a low-cost edge as in [\[eq:N-gamma-eta-z\]](#eq:N-gamma-eta-z){reference-type="eqref" reference="eq:N-gamma-eta-z"}. By hypothesis $|x-y| \le c_HD$, $a \in B_{D^\gamma}(x)$, $b \in B_{D^\gamma}(y)$, and $\gamma<1$, so $|a-b| \le c_HD + 2D^\gamma \le (c_H+2)D.$ Since $w_a, w_b \in [\underline{w}D^{z/2}/2, 2\underline{w}D^{z/2}]$ by [\[eq:calZ-def\]](#eq:calZ-def){reference-type="eqref" reference="eq:calZ-def"}, it follows from [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"} that $$\label{eq:bb-conn-prob} \mathbb{P}\big(ab \in \mathcal{E}(G') \mid V, w_V \big) \ge \theta\underline{c}\Big(1 \wedge \frac{\underline{w}^2D^z}{4(c_H+2)^dD^d} \Big)^\alpha \ge \theta\underline{c}\big(1 \wedge \underline{w}D^{z-d} \big)^\alpha,$$ where we used the assumption that $\underline w\ge 4(c_H+2)^d$ to simplify the formula. Since $z \in [0,d]$, and $\delta$ is small relative to $z$, if $z-d < 0$ then we may assume $z-d \le -2\delta$. Since $1 \le \underline{w}\le D^\delta$, the minimum in [\[eq:bb-conn-prob\]](#eq:bb-conn-prob){reference-type="eqref" reference="eq:bb-conn-prob"} is attained at $1$ only for $z=d$. So, for all $\{a,b\} \in Z_x\times Z_y$, $$\label{eq:bb-conn-prob-2} \mathbb{P}\big(ab \in E(G') \mid V,w_V\big)\ge \begin{cases} \theta \underline{c} \mathbbm{1}\{z = d\}& \mbox{if }\alpha=\infty,\\ \theta \underline{c} D^{\alpha(z-d)}& \mbox{otherwise.} \end{cases}$$ Since $w_a,w_b \le 2\underline{w}D^{z/2}$ by [\[eq:calZ-def\]](#eq:calZ-def){reference-type="eqref" reference="eq:calZ-def"}, $$\begin{aligned} \mathbb{P}\big(\mathcal{C}(ab)\le (\underline{w}/10)^{3\mu}D^\eta \mid ab\in\mathcal{E}(G'), V,w_V\big) &\ge \mathbb{P}\big((4\underline{w}^2D^z)^{\mu}L_{ab} \le (\underline{w}/10)^{3\mu}D^\eta\big) \\ &= F_L(4000^{-\mu}\underline{w}^{\mu} D^{\eta-\mu z}). \end{aligned}$$ If $\eta < \mu z$, then since $\delta{\,\ll_{\star}\,}z,\eta,\textnormal{\texttt{par}}\xspace$ we may assume that $\eta-\mu z \le -2\mu\delta$. Since $\underline{w}\le D^\delta$ and $D{\,\gg_{\star}\,}\delta$, it follows that $4000^{-\mu}\underline{w}^{\mu} D^{\eta-\mu z} \le D^{-\mu\delta} \le t_0$ and hence using Assumption [Assumption 3](#assu:L){reference-type="ref" reference="assu:L"} and the assumption $\underline{w}\ge 4000 c_1^{-1/(\mu\beta)}$ we get $F_L(4000^{-\mu}\underline{w}^{\mu} D^{\eta-\mu z}) \ge D^{\beta(\eta-\mu z)}$ after simplifications. If instead $\eta \ge \mu z$, then $F_L(4000^{-\mu}\underline{w}^{\mu} D^{\eta-\mu z}) \ge F_L(4000^{-\mu}\underline{w}^{\mu}) \ge 1/2$ by hypothesis. Summarising the two cases with indicators we arrive at $$\begin{aligned} \begin{split}\label{eq:bb-cost-prob} \mathbb{P}\big(\mathcal{C}(ab)\le (\underline{w}/10)^{3\mu}D^\eta \mid ab\in \mathcal{E}(G'), V,w_V\big) \ge \begin{cases} \mathbbm{1}\{\eta \ge \mu z\}/2 & \mbox{if }\beta=\infty,\\ D^{0 \wedge \beta(\eta -\mu z)}/2 & \mbox{otherwise.} \end{cases} \end{split} \end{aligned}$$ With the convention that $\infty \cdot 0 = 0$, the second row equals the first row in both [\[eq:bb-conn-prob-2\]](#eq:bb-conn-prob-2){reference-type="eqref" reference="eq:bb-conn-prob-2"} and [\[eq:bb-cost-prob\]](#eq:bb-cost-prob){reference-type="eqref" reference="eq:bb-cost-prob"}. Combining [\[eq:bb-conn-prob-2\]](#eq:bb-conn-prob-2){reference-type="eqref" reference="eq:bb-conn-prob-2"} and [\[eq:bb-cost-prob\]](#eq:bb-cost-prob){reference-type="eqref" reference="eq:bb-cost-prob"}, we obtain that for all $\{a,b\}\in Z_x\times Z_y$: $$\label{eq:cheap-bridge-p} \mathbb{P}\big(\{a,b\} \in N_{\eta,\gamma,z,\underline{w}}(x,y) \mid V,w_V\big) \ge \theta \underline{c} D^{\alpha(z-d) + (0 \wedge \beta(\eta - \mu z))}/2.$$ Given $V, w_V$, the possible edges $\{a,b\}$ lie in $N_{\eta,\gamma,z,\underline{w}}(x,y)$ independently. Hence by [\[eq:cheap-bridge-N\]](#eq:cheap-bridge-N){reference-type="eqref" reference="eq:cheap-bridge-N"} and [\[eq:cheap-bridge-p\]](#eq:cheap-bridge-p){reference-type="eqref" reference="eq:cheap-bridge-p"}, $|N_{\eta,\gamma,z}(x,y)|$ stochastically dominates a binomial random variable whose mean $m$ is the product of the two equations' right-hand sides. On bounding $\underline{c}/128 \ge D^{-\delta \gamma d/15}$, we obtain $$m \ge \theta \underline{w}^{-2(\tau-1)}D^{\Lambda(\eta,z)-2\delta\gamma d/3}.$$ Inequality [\[eq:lem-cheap-bridge\]](#eq:lem-cheap-bridge){reference-type="eqref" reference="eq:lem-cheap-bridge"} follows since this binomial variable is zero with probability at most $e^{-m}$. ◻ The next claim finds a low-cost edge from a fix vertex in $\mathcal{N}$ with weight roughly $M$ to some nearby vertex in $\mathcal{N}$ with weight roughly $K$. We will use this claim later in two different ways, either $K$ being much lower than $M$; or $K$ being somewhat larger than $M$. **Claim 32** (Single cheap edge nearby). *[\[lem:CETNV\]]{#lem:CETNV label="lem:CETNV"} Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}. Let $M > 1$, and let $x\in\mathcal{N}$ be a vertex with $w_x\in [\tfrac12M, 2M]$. Let $U,D > 0$ and $K>w_0$, and define the event $$\label{eq:akdu} \mathcal{A}_{K,D,U}(x):=\left\{ \exists y\in \mathcal{N}\cap (B_D(x)\times[\tfrac12K,2K]): xy\in \mathcal{E}(G'), \ \mathcal{C}(xy) \le U \right\}.$$ Suppose that $\delta {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, that $K,M,D {\,\gg_{\star}\,}\delta,w_0$, and that $$\begin{aligned} (\log\log\xi\sqrt{d})^{16/\delta} &\le (D \wedge (KM)^{1/d})/4^{1/d} \le \xi\sqrt{d},\label{eq:KM-xi}\\ K &\le D^{d/(\tau-1)-\delta} \wedge M^{1/(\tau-2+\delta\tau)}.\label{eq:K-D-M} \end{aligned}$$ Then if $\beta = \infty$ and $U(KM)^{-\mu}{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$, or alternatively if $\beta < \infty$, then $$\label{eq:cheap-edge-to-nice-vertex} \begin{aligned} \mathbb{P}\big(\mathcal{A}_{K,D,U}(x) &\mid V,w_V\big) \ge 1 - \exp\Big({-}\theta K^{-(\tau-1)}(D^d \wedge KM)^{1-\delta}(1 \wedge (U(KM)^{-\mu})^\beta \Big). \end{aligned}$$* *Proof.* Let $r = 4^{-1/d}(D \wedge (KM)^{1/d})$, and define $\mathcal{Z}'(x):=\mathcal{N}\cap (B_r(x)\times [\tfrac12K, 2K])$. We will first lower-bound $|\mathcal{Z}'(x)|$ by applying Definition [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"} with $\varepsilon=\delta/4$, $w=K$ and the same value of $r$. Observe that [\[eq:KM-xi\]](#eq:KM-xi){reference-type="eqref" reference="eq:KM-xi"} and the fact that $K \ge w_0$ imply all the requirements of Definition [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"} except $K \le r^{d/(\tau-1)-\delta/4}$, which we now prove. By [\[eq:K-D-M\]](#eq:K-D-M){reference-type="eqref" reference="eq:K-D-M"}, $M \ge K^{\tau-2+\tau\delta}$ and hence $$\begin{aligned} r^{d/(\tau-1)-\delta/4}&\ge (KM/4)^{1/(\tau-1) - \delta/(4d)} \ge (K^{\tau-1+\tau\delta})^{1/(\tau-1)-\delta/4} /4\\ &= K^{1 + \delta(\tau/(\tau-1)-(\tau-1)/4-\tau\delta/4)}/4. \end{aligned}$$ Since $\tau<3$ and $\delta{\,\ll_{\star}\,}\tau$, the coefficient of $\delta$ is positive in the exponent so the rhs is at least $K$, as required required by Def. [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"}. Applying [\[eq:net-defining-crit-eps\]](#eq:net-defining-crit-eps){reference-type="eqref" reference="eq:net-defining-crit-eps"} followed by Potter's bound (since $D,K{\,\gg_{\star}\,}\delta$) yields that $$\label{eq:cheap-edge-to-nice-vertex-N} |\mathcal{Z}'(x)| \ge \ell(K)K^{-(\tau-1)}r^{d(1-\delta/4)} \ge K^{-(\tau-1)}\big(D \wedge KM)^{1-\delta/2}.$$ We now lower-bound the probability that for a $y\in \mathcal{Z}'(x)$ the edge $xy$ is present and has cost at most $U$, satisfying the requirements of $\mathcal{A}_{K,D,U}(x)$. Let $y\in \mathcal{Z}'(x)$. Since $w_x\in[M/2, 2M]$ and $w_y\in[K/2, 2K]$, by [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"} and the definition of $r=D \wedge (KM/4)^{1/d}$ we have $$\label{eq:cheap-edge-to-nice-vertex-p1} \mathbb{P}\big(xy\in \mathcal{E}\mid V, w_V\big) \ge \theta\underline{c}(1 \wedge KM/(4r^d))^\alpha = \theta\underline{c},$$ since the minimum is at the first term; also for $\alpha=\infty$. Moreover, if $\beta < \infty$, we apply [\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"}; otherwise, since $U(KM)^{-\mu}$ is large, $F_L(U(4KM)^{-\mu}) \ge 1/2$, to estimate the cost $$\begin{aligned} \begin{split}\label{eq:cheap-edge-to-nice-vertex-p2} \mathbb{P}\big(\mathcal{C}(xy) \le U \mid xy\in \mathcal{E}(G'), V,w_V\big) &\ge \mathbb{P}\big((4KM)^{\mu}L_{xy} \le U\big) = F_L((4KM)^{-\mu}U) \\ &\ge C(1 \wedge (U(KM)^{-\mu})^\beta), \end{split} \end{aligned}$$ for an appropriate choice of $C>0$ depending only on [`par`]{.nodecor}. Combining [\[eq:cheap-edge-to-nice-vertex-p1\]](#eq:cheap-edge-to-nice-vertex-p1){reference-type="eqref" reference="eq:cheap-edge-to-nice-vertex-p1"} and [\[eq:cheap-edge-to-nice-vertex-p2\]](#eq:cheap-edge-to-nice-vertex-p2){reference-type="eqref" reference="eq:cheap-edge-to-nice-vertex-p2"}, we obtain for any $y\in \mathcal{Z}'(x)$: $$\label{eq:cheap-edge-to-nice-vertex-p} \mathbb{P}\big(xy \in \mathcal{E}(G') \mbox{ with }\mathcal{C}(xy) \le U\mid V,w_V\big) \ge \theta C\underline{c}(1 \wedge (U(KM)^{-\mu})^\beta).$$ Conditioned on $(V,w_V)$, edges are present independently, so the number of low-cost edges between $x$ and $\mathcal{Z}'(x)$ stochastically dominates a binomial random variable with parameters the rhs of [\[eq:cheap-edge-to-nice-vertex-N\]](#eq:cheap-edge-to-nice-vertex-N){reference-type="eqref" reference="eq:cheap-edge-to-nice-vertex-N"} and [\[eq:cheap-edge-to-nice-vertex-p\]](#eq:cheap-edge-to-nice-vertex-p){reference-type="eqref" reference="eq:cheap-edge-to-nice-vertex-p"}. The mean is $$\theta C\underline{c} K^{-(\tau-1)}(D^d\wedge KM)^{1-\delta/2} (1 \wedge (U(KM)^{-\mu})^\beta).$$ Since $K,M,D{\,\gg_{\star}\,}\delta$, we may swallow the constant factor $\theta C\underline{c}$ by increasing $\delta/2$ to $\delta$. The result follows since for a binomial variable $Z$, $\mathbb{P}(Z=0)\le \exp(-\mathbb E[Z])$. ◻ The third claim builds cheap *weight-increasing paths*, from a low-weight vertex in $\mathcal{N}$ to a high-weight vertex in $\mathcal{N}$. The proof is via repeated application of Claim [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"}. **Claim 33** (Weight-increasing paths). *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}. Let $M > 1$, and let $y_0$ be a vertex in $\mathcal{N}$ with weight in $[\tfrac12M, 2M]$. Let $K,D>1$, $U\ge K^{2\mu}$, and let $$\begin{aligned} \label{eq:cond-q} q := \left\lceil\frac{\log(\log K / \log M)}{\log(1/(\tau-2+2d\tau\delta))}\right\rceil. \end{aligned}$$ Let $\mathcal{A}_{\pi(y_0)}$ be the event that $G'$ contains a path $\pi = y_0y_1\ldots y_q$ contained in $\mathcal{N}\cap B_{qD}(y_0)$ such that $W_{y_q} \in [\tfrac12K,2K]$ and $\mathcal{C}(\pi) \le q U$. Suppose that $\delta{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, that $K,M,D{\,\gg_{\star}\,}\theta,\delta,w_0$, and that $M \le K \le D^{d/2}$, $D \le \xi\sqrt{d}$, and $(M/2)^{2/d} \ge (\log\log\xi\sqrt{d})^{16/\delta}$. Then if $\beta = \infty$ and $U(KM)^{-\mu} {\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$, or if $\beta < \infty$, then $$\begin{aligned} \label{eq:weight-increasing-path-error} \mathbb{P}\big(\mathcal{A}_{\pi(y_0)} \mid V,w_V\big) \ge 1-\exp({-}\theta M^\delta). \end{aligned}$$* *Proof.* We will build $\pi$ vertex-by-vertex by applying Claim [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"} $q$ times. We first define target weights. Let $M_0 := M$, and for all $i \in [q]$, let $$\begin{aligned} \label{eq:Mi} M_i := M^{1/(\tau-2+2d\tau\delta)^i} \wedge K. \end{aligned}$$ Since $\tau<3$ and $\delta$ is small, $\tau-2+2d\tau\delta < 1$; hence on substituting the definition of $q$ in [\[eq:cond-q\]](#eq:cond-q){reference-type="eqref" reference="eq:cond-q"} into [\[eq:Mi\]](#eq:Mi){reference-type="eqref" reference="eq:Mi"} and removing the ceiling, we obtain $$M^{1/(\tau-2+2d\tau\delta)^q} = \exp\Big(\log M\cdot \mathrm{e}^{{-}q\log(\tau-2+2d\tau\delta)}\Big) \ge \exp\Big(\log M\cdot \mathrm{e}^{\log\big(\frac{\log K}{\log M}\big)}\Big) = K,$$ and hence $M_q = K$. By a very similar argument, $M_{q-1} < K$. We now define $Y_0 = y_0$, and define an arbitrary ordering on $\mathcal{N}$. For all $i \in [q]$, we define $Y_i$ to be the first vertex in $\mathcal{N}$ in $B_D(Y_{i-1}) \times [\tfrac{1}{2}M_i, 2M_i]$ so that the edge $Y_{i-1}Y_i$ is present in $G'$ and has cost at most $U$. If no such vertex exists, we define $Y_j = \texttt{None}$ for all $j \ge i$. Let $\mathcal{A}_i$ be the event that $Y_0,\ldots,Y_{i} \ne \texttt{None}$. Then, if $\mathcal{A}_q$ occurs, the path $\pi=Y_0\ldots Y_q$ yields $V(\pi)\subseteq \mathcal{N}\cap B_{qD}(y_0)$ and $\mathcal{C}(\pi) \le qU$, and that $w_{Y_q} \in [\tfrac{1}{2}K,2K]$ since $M_q = K$. So, (and because $\mathcal{A}_{i-1}\subseteq \mathcal{A}_i$), $$\label{eq:cheap-increasing-path-step-1} \mathbb{P}\big(\mathcal{A}_{\pi(y_0)} \mid V,w_V\big) \ge \mathbb{P}\big(\mathcal{A}_q \mid V,w_V\big) = \prod_{i=1}^q \mathbb{P}\big(\mathcal{A}_i \mid \mathcal{A}_{i-1}, V,w_V\big).$$ We now simplify the conditioning in [\[eq:cheap-increasing-path-step-1\]](#eq:cheap-increasing-path-step-1){reference-type="eqref" reference="eq:cheap-increasing-path-step-1"}. For all $i \in [q]$, let $$\begin{aligned} \mathcal{F}_i &:= \mathcal{E}(G') \cap \big\{\{x,y\}\colon x,y\in\mathcal{N},\ w_x \in [\tfrac{1}{2} M_{i-1},2M_{i-1}],\ w_y \in [\tfrac{1}{2}M_i,2M_i]\big\},\label{eq:fi-111}\\ \mathcal{F}_{\le i} &:= (\mathcal{F}_1,\dots,\mathcal{F}_i),\qquad\qquad p_i := \mathbb{P}\big(\mathcal{A}_i \mid \mathcal{A}_{i-1},V,w_V\big).\nonumber \end{aligned}$$ Observe that $\mathcal{A}_1,\dots,\mathcal{A}_i$ and $Y_1,\dots,Y_i$ are deterministic functions of $\mathcal{F}_{\le i}$. Moreover, if $F$ is a possible realisation of $\mathcal{F}_{\le i-1}$ such that $\mathcal{A}_{i-1}(F)$ occurs, then conditioned on $\mathcal{F}_{\le i-1}=F$, $\mathcal{A}_i$ occurs iff $Y_i \ne \texttt{None}$. Thus, with $\mathcal{A}_{K,D,U}$ from [\[eq:akdu\]](#eq:akdu){reference-type="eqref" reference="eq:akdu"}, [\[eq:cheap-increasing-path-step-1\]](#eq:cheap-increasing-path-step-1){reference-type="eqref" reference="eq:cheap-increasing-path-step-1"} implies that for all $i \in [q]$, $$\begin{aligned} \nonumber p_i&\ge \min_{F\colon \mathcal{A}_{i-1}(F)\text{ occurs}} \mathbb{P}\big(Y_i \ne \texttt{None} \mid \mathcal{F}_{\le i-1} = F,\,V,w_V\big)\\\label{eq:cheap-increasing-path-step-1b} &= \min_{F\colon \mathcal{A}_{i-1}(F)\text{ occurs}} \mathbb{P}\big(\mathcal{A}_{M_i,D,U}(Y_{i-1}(F)) \text{ occurs}\mid \mathcal{F}_{\le i-1} = F,\,V,w_V\big). \end{aligned}$$ Now observe that since $\tau \in (2,3)$, $\delta$ is small, and $M{\,\gg_{\star}\,}\delta$, for all $i \in [q-2]$, we may assume $M_{i} = M_{i-1}^{1/(\tau-2+2d\tau\delta)^i} > 4M_{i-1}$, and moreover $M_q \ge M_{q-1}$. Therefore, the intervals $[\tfrac{1}{2}M_1,2M_2],\dots,[\tfrac{1}{2}M_{q-1},2M_q]$ are all disjoint except possibly for $[\tfrac{1}{2}M_{q-2},2M_{q-1}]$ and $[\tfrac{1}{2}M_{q-1},2M_q]$. It follows that the variables $\mathcal{F}_1,\ldots,\mathcal{F}_q$ are determined by disjoint sets of possible edges. For $\mathcal{F}_{q-1}$ and $\mathcal{F}_q$ the weights $w_{q-1}$ and $w_q$ still fall into disjoint intervals, so the edge sets in $\mathcal{F}_i$ in [\[eq:fi-111\]](#eq:fi-111){reference-type="eqref" reference="eq:fi-111"} are disjoint across $i$. So, $\mathcal{A}_{M_i,D,U}(Y_{i-1}(F))$ is independent of $\mathcal{F}_{\le i-1}$ in [\[eq:cheap-increasing-path-step-1b\]](#eq:cheap-increasing-path-step-1b){reference-type="eqref" reference="eq:cheap-increasing-path-step-1b"} (conditioned on $(V,w_V)$), and hence $$\label{eq:cheap-increasing-path-step-2} p_i \ge \min_{y \colon w_y \in [M_{i-1}/2,2M_{i-1}]} \mathbb{P}\big(\mathcal{A}_{M_i,D,U}(y) \mid V,w_V\big).$$ We now apply Claim [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"} on the rhs: take there $\delta_{\ref{claim:cheap-edge-to-nice-vertex}} := \delta$, $\theta_{\ref{lem:CETNV}} := \theta$, $M_{\ref{lem:CETNV}}:=M_{i-1}, K_{\ref{lem:CETNV}}:=M_i$, $D_{\ref{lem:CETNV}}:=D$ and $U_{\ref{lem:CETNV}}:=U$. Observe $K_{\ref{lem:CETNV}}, M_{\ref{lem:CETNV}}\ge M$; thus by hypothesis we have that $\delta_{\ref{lem:CETNV}}{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ is small and that $K_{\ref{lem:CETNV}},M_{\ref{lem:CETNV}},D_{\ref{lem:CETNV}}{\,\gg_{\star}\,}\delta, w_0$, as required by Claim [\[lem:CETNV\]](#lem:CETNV){reference-type="ref" reference="lem:CETNV"}. Next, we have $(K_{\ref{lem:CETNV}}M_{\ref{lem:CETNV}})^\mu \le K^{2\mu}$, so if $\beta=\infty$ it follows that $U_{\ref{lem:CETNV}}(K_{\ref{lem:CETNV}}M_{\ref{lem:CETNV}})^{-\mu} \ge UK^{-2\mu}$ is large as required, by the assumptions before [\[eq:cond-q\]](#eq:cond-q){reference-type="eqref" reference="eq:cond-q"}. Next, $(D \wedge (M_iM_{i-1})^{1/d})/4^d \le D \le \xi\sqrt{d}$ by hypothesis, and $(D \wedge (M_iM_{i-1})^{1/d})/4^d \ge (M_0/2)^{2/d} \ge (\log\log\xi\sqrt{d})^{16/\delta}$ by hypothesis, so [\[eq:KM-xi\]](#eq:KM-xi){reference-type="eqref" reference="eq:KM-xi"} holds. Next, $M_i \le M_q = K \le D^{d/2} \le D^{d/(\tau-1)-\delta}$ by hypothesis and because $\delta$ is small; and finally $M_i \le M_{i-1}^{1/(\tau-2+2d\tau\delta)} < M_{i-1}^{1/(\tau-2+\tau\delta)}$ by definition, so [\[eq:K-D-M\]](#eq:K-D-M){reference-type="eqref" reference="eq:K-D-M"} holds. Thus the conditions of Claim [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"} all hold, and applying [\[eq:cheap-edge-to-nice-vertex\]](#eq:cheap-edge-to-nice-vertex){reference-type="eqref" reference="eq:cheap-edge-to-nice-vertex"} to [\[eq:cheap-increasing-path-step-2\]](#eq:cheap-increasing-path-step-2){reference-type="eqref" reference="eq:cheap-increasing-path-step-2"} yields that for all $i \in [q]$: $$\begin{aligned} \label{eq:cheap-increasing-path-step-3} p_i \ge1-\exp\Big({-}\theta M_i^{-(\tau-1)}\Big[(D^d \wedge M_iM_{i-1})^{1-\delta} (1\wedge (U(M_iM_{i-1})^{-\mu})^\beta\Big]\Big) . \end{aligned}$$ Clearly $M_iM_{i-1} \le M_q^2 = K^2$; and since $K \le D^{d/2}$ and $U\ge K^{2\mu}$ by hypothesis, the first minimum is at $M_iM_{i-1}$, while the second minimum is taken at $1$ on the rhs. Hence $$p_i \ge 1-\exp\Big({-}\theta M_i^{-(\tau-2)-\delta}M_{i-1}^{1-\delta} \Big).$$ Since $M_i \le M_{i-1}^{1/(\tau-2+2d\tau\delta)}$ by [\[eq:Mi\]](#eq:Mi){reference-type="eqref" reference="eq:Mi"}, $\delta$ is small, and $\tau \in (2,3)$, after simplification the exponent of $M_{i-1}$ is at least $\delta (\tau+1-2d\tau\delta)/(\tau-2+2d\tau\delta)\ge 3\delta$, so $p_i \ge 1-\exp({-}\theta M_{i-1}^{3\delta})$. Using this bound in [\[eq:cheap-increasing-path-step-1\]](#eq:cheap-increasing-path-step-1){reference-type="eqref" reference="eq:cheap-increasing-path-step-1"}, we obtain that $$\label{eq:cheap-increasing-path-step-4} \mathbb{P}\big(\mathcal{A}_q \mid V,w_V\big) \ge \prod_{i=1}^q \big(1-\exp({-}\theta M_i^{3\delta})\big) \ge 1 - \sum_{i=1}^q \exp({-}\theta M_i^{3\delta}).$$ Recall that for all $2 \le i \le q-1$, $M_i = M_{i-1}^{1/(\tau-2+2d\tau\delta)}$, and so since $\delta$ is small and $\tau \in (2,3)$ we have $M_i \ge M_{i-1}^{1+\delta}$. Since $M_0 = M{\,\gg_{\star}\,}\delta,\theta$, we obtain $\exp({-}\theta M_i^{3\delta}) \le \tfrac{1}{2}\exp({-}\theta M_{i-1}^{3\delta})$. It follows from [\[eq:cheap-increasing-path-step-4\]](#eq:cheap-increasing-path-step-4){reference-type="eqref" reference="eq:cheap-increasing-path-step-4"} that $$\begin{aligned} \mathbb{P}\big(\mathcal{A}_q \mid V,w_V\big) &\ge 1 - \sum_{i=1}^{q-1}\exp({-}\theta M_i^{3\delta}) - \exp(-M_q^{3\delta}) \ge 1 - 3\exp({-}\theta M^{3\delta}) \ge 1 - \exp({-}\theta M^\delta) \end{aligned}$$ as required, where the last step holds since $M{\,\gg_{\star}\,}\delta,\theta$. ◻ The last claim allows us to find a common neighbour for two vertices with roughly the same weight if the distance between them is not too large wrt their weights. **Claim 34** (Common neighbour). *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}. Let $\delta {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, let $c_H>0$, and let $D\ge w_0^{2/d}$ with $D {\,\gg_{\star}\,}c_H,\delta$ and $D\in[(\log\log\xi\sqrt{d})^{16/\delta}, \xi\sqrt{d}]$. Let $x_0,x_1\in\mathcal{N}$ be vertices with $w_{x_0}, w_{x_1}\in[D^{d/2}, 4D^{d/2}]$ at distance $|x_0-x_1| \le c_H D$, and let $\mathcal{A}_{x_0\star x_1}$ be the event that $x_0$ and $x_1$ have a common neighbour in $G'$, $v \in \mathcal{N}\cap B_D(x_0)$ with $\mathcal{C}(x_0v) + \mathcal{C}(vx_1) \le D^{2\mu d}$. Then $$\label{eq:common-neighbor} \mathbb{P}\big(\mathcal{A}_{x_0\star x_1} \mid V,w_V\big) \ge 1 - \exp\Big({-}\theta^2D^{(3-\tau-2\delta)d/2}\Big).$$* *Proof.* We define a vertex $v\in\widetilde{\mathcal{V}}$ as *good* if $v \in \mathcal{N}\cap (B_{D}(x_0)\times [(c_H+1)^{d}D^{d/2}, 4(c_H+1)^{d}D^{d/2}])$; thus for $\mathcal{A}_{x_0\star x_1}$ to occur, it suffices that there is a good vertex $v$ such that $x_0vx_1$ is a path of cost at most $D^{2\mu d}$ in $G'$. We call this a *good path*. We first lower-bound the number of good vertices. By assumption, $2(c_H+1)^{d}D^{d/2} \ge D^{d/2}\ge w_0$, and since $\tau < 3$, $\delta$ is small and $D{\,\gg_{\star}\,}c_H,\delta$ we have $2(c_H+1)^{d}D^{d/2} \le D^{d/(\tau-1)-\delta/4}$. Since $\mathcal{N}$ is a weak $(\delta/4, w_0)$ net, by [\[eq:net-defining-crit\]](#eq:net-defining-crit){reference-type="eqref" reference="eq:net-defining-crit"}, $$\begin{aligned}\label{eq:common-neighbour-N} &\big|\mathcal{N}\cap (B_{D}(x_0)\times [(c_H+1)^{d}D^{d/2}, 4(c_H+1)^{d}D^{d/2}])\big| \\ &\qquad\ge D^{d(1-\delta/4)}\ell\big(2(c_H+1)^{d}D^{d/2}\big)\big(2(c_H+1)^{d}D^{d/2}\big)^{-(\tau-1)} \ge D^{(3-\tau-\delta)d/2}, \end{aligned}$$ where the last inequality follows by Potter's bound since $D{\,\gg_{\star}\,}c_H,\delta$. We now lower-bound the probability that for a good $v\in \mathcal{N}$, the edges $x_0v, vx_1$ are present and have cost at most $D^{3\mu d/2}$ in $G'$. Observe that $|x_1-v| \le |x_1-x_0| + |x_0-v| \le (c_H+1)D$. Thus, by [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"}, and since $G'$ is a $\theta$-percolation, $\mathbb{P}(x_1v\in \mathcal{E}(G')|V, w_V)\ge \theta\underline{c}\big[1 \wedge (c_H+1)^d(D^{d/2})^2/((c_H+1)D)^d\big]^\alpha = \theta\underline{c}$, also when $\alpha=\infty$. Further, conditioned on the existence of the edge $x_1v$, $$\begin{aligned} \mathbb{P}\big(\mathcal{C}(x_1v) \le D^{3\mu d/2} \mid x_1v\in\mathcal{E}(G'), V,w_V\big) &\ge \mathbb{P}\big((16(c_H+1)^{d}D^{d})^{\mu}L \le D^{3\mu d/2}\big) \\ &= F_L(16^{-\mu}(c_H+1)^{-\mu d}D^{\mu d/2}) \ge 1/2, \end{aligned}$$ where the last inequality holds (including when $\beta=\infty$) since $D{\,\gg_{\star}\,}c_H$. Combining the two bounds, for all good vertices $v\in \widetilde{\mathcal{V}}$, $$\mathbb{P}\big(x_1v \in \mathcal{E}(G'), \mathcal{C}(x_1v) \le D^{3\mu d/2} \mid V,w_V\big) \ge \theta\underline{c}/2.$$ Since $|x_0-v| \le D$, the same lower bounds hold for the edge $x_0v$. The two events are independent conditioned on $(V,w_V)$, and since $2D^{3\mu d/2} < D^{2\mu d}$, for all good vertices $v\in \widetilde{\mathcal{V}}$, $$\label{eq:common-neighbour-p} \begin{aligned} &\mathbb{P}\big(x_0v,x_1v\in \mathcal{E}(G'), \mathcal{C}(x_0vx_1) \le D^{2\mu d} \mid V,w_V\big) \ge \theta^2\underline{c}^2/4. \end{aligned}$$ Conditioned on $(V,w_V)$, the presence and cost of $x_0vx_1$ vs $x_0v'x_1$ are independent, so the number of good paths between $x_0$ and $x_1$ stochastically dominates a binomial random variable with parameters given by the rhs of [\[eq:common-neighbour-N\]](#eq:common-neighbour-N){reference-type="eqref" reference="eq:common-neighbour-N"} and that of [\[eq:common-neighbour-p\]](#eq:common-neighbour-p){reference-type="eqref" reference="eq:common-neighbour-p"}. For a binomial variable $Z$, $\mathbb{P}(Z\neq0) \ge 1-\exp(-\mathbb{E}[Z])$, and so we obtain [\[eq:common-neighbor\]](#eq:common-neighbor){reference-type="eqref" reference="eq:common-neighbor"} by absorbing the constant $\underline{c}^2/4$ by replacing $\delta$ with $2\delta$ in the exponent of $D$, using that $D{\,\gg_{\star}\,}\delta$. ◻ # Budget travel plan: hierarchical bridge-paths {#sec:hierarchy} In this section, we present the main construction for the upper bounds in Theorems [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"} and [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"}. This construction is a "hierarchy" of cheap bridging paths connecting $x$ and $y$ that we heuristically described in Section [1](#sec:intro){reference-type="ref" reference="sec:intro"} as the 'budget travel plan'. Here we elaborate more on the heuristics before diving into proofs. Let $U$ be either polynomial in $|x-y|$ (when proving Theorem [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"}) or sub-logarithmic in $|x-y|$ (when proving Theorem [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"}). We first find a 3-edge *bridging-path* $\pi_1=x'aby'$ of cost at most $U$ between two vertices $x'$ and $y'$ with weights $w_{x'}, w_{y'}\in [w_{H_1},4w_{H_1}]$, such that $|x-x'|$ and $|y-y'|$ are both at most $|x-y|^\gamma$ for some $\gamma \in (0,1)$, see Figure [\[fig:intuition_hierarchy\]](#fig:intuition_hierarchy){reference-type="ref" reference="fig:intuition_hierarchy"}(a). This reduces the original problem of connecting $x$ and $y$ to two instances of connecting two vertices at distance $|x-y|^{\gamma}$, at the additional cost of $U$. We then work recursively, applying the same procedure to find a bridging-path with endpoints near $x$ and $x'$ and another one with endpoints near $y'$ and $y$, with all four distances at most $|x-y|^{\gamma^2}$, and both bridging-paths having cost at most $U$, obtaining the second level of the hierarchy, see Figure [\[fig:intuition_hierarchy\]](#fig:intuition_hierarchy){reference-type="ref" reference="fig:intuition_hierarchy"}(b). The endpoints of the bridging paths always have weight in $[w_{H_1},4w_{H_1}]$, hence iteration is possible. By repeating the process $R$ times we obtain a "broken path" of bridging-paths of cost $U(1+2+\dots +2^R)$ and $2^R$ gaps of length $|x|^{\gamma^R}$ between the bridging-paths. We call this "broken path" a *hierarchy* after Biskup, who developed the one-edge bridge construction for graph distances in long range percolation in [@biskup2004scaling]. There are two reasons for having a *bridging-path* instead of a single bridge-edge. Firstly, a typical single bridge-edge $ab$ has very high weights $w_a, w_b$, and even the cheapest edge out of $a$ and $b$ has high cost, which would cause high costs when filling the gaps. So instead we find an a-typical path of the form $(x'aby')$, with all three edges of cost $U/5$, and $x',y'$ having low weight in $[w_{H_1}, 4 w_{H_1}]$, giving a bridging path of length three. Secondly, to fill the $2^R$ gaps after $R$ iterations whp, the failure probability of finding a connecting path has to be extremely low, $o(2^{-R})$. In most regimes this is impossible via short paths (e.g. length two) and low enough failure probability. Instead, we find weight increasing paths $\pi_{x'x''}$ and $\pi_{y'y''}$ (as in Claim [Claim 33](#claim:cheap-path-to-larger-weight){reference-type="ref" reference="claim:cheap-path-to-larger-weight"}) of cost at most $U/5$ from each vertex $x'$ and $y'$ of the bridge paths $(x'aby')$ to respective vertices $x'', y''$ still near $a$ and $b$ but with much higher weights in $[w_{H_2}, 4w_{H_2}]$. The concatenated paths $(\pi_{x''x'},x'aby',\pi_{y'y''})$ then themselves form a second hierarchy (now with bridging paths of more than $3$ edges). Connecting all the new $2^R$ gaps whp is possible via paths of length two and cost $U'$, which is polynomial in the distance $|x-y|^{\gamma^{R}}$, using Claim [Claim 34](#claim:common-neighbour){reference-type="ref" reference="claim:common-neighbour"}. In the *polylogarithmic case*, $U$ and $U'$ are sublogarithmic, and the factor $2^R$ is of order $(\log |x-y|)^{\Delta_0 + o(1)}$ and dominates the overall cost. The bottleneck in this regime is the number of gaps, whereas the bridge-paths have negligible costs. In the *polynomial case*, however, the cost of the first bridge $U=|x-y|^{\eta_0 + o(1)}$ dominates, and all other costs (even with the factors $2^i$) are negligible in comparison, causing the total cost to be polynomial in $|x-y|$. In both cases we could use and optimise level-dependent costs $U_i$, but that does not improve the statements of Theorems [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"}, [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"}. Let us now formally define the concept of a *hierarchy* including edge-costs. In the rest of the paper, the symbol $\sigma$ denotes an index $\sigma=\sigma_1\sigma_2\ldots\sigma_R\in\{0,1\}^R$ indicating the place of a vertex in the hierarchy. This can be viewed as the Ulam-Harris labelling of the leaves of a binary tree of depth $R$, e.g. $\sigma = 1001$ corresponds to the leaf that we reach by starting at the root and then moving to the right child, the left child twice, and the right child again. We denote the string formed by concatenating $\sigma'$ to the end of $\sigma$ by $\sigma \sigma'$. We "pad" strings of length less than $R$ by adding copies of their last digit or its complement via the $T$ and $T^c$ operations we now define (and discuss further below): **Definition 35** (Binary strings). *For $\sigma=\sigma_1\ldots \sigma_i\in\{0,1\}^i$ for some $i\ge 1$, we define $\sigma T:=\sigma_1\ldots \sigma_i\sigma_i\in \{0,1\}^{i+1}$, while $\sigma T_0:=\sigma$, and $\sigma T_k:=(\sigma T_{k-1})T$ for any $k\ge 2$. Let $0_i:=0T_{i-1}$ and $1_i:=1T_{i-1}$ be the strings consisting of $i$ copies of $0$ and $1$, respectively. Fix an integer $R \ge 1$. Define the *equivalence relation* $\sim_T$ on $\cup_{i=1}^R\{0,1\}^i$, where $\sigma\sim_T \sigma'$ if either $\sigma T_k=\sigma'$ or $\sigma'T_k=\sigma$ for some $k\ge 0$, with $\{\sigma\}$ be the equivalence class of $\sigma$. Let $$\begin{aligned} \Xi_{i}:=\{ \sigma \in \cup_{j=i}^R\{0,1\}^j: \sigma_{i-1} \neq \sigma_i, \sigma_{j}=\sigma_i \ \forall j\ge i\}, \qquad \Xi_0:=\{\emptyset\},\end{aligned}$$ with $\emptyset$ the empty string. We say that $\{\sigma\}$ *appears first on level $i$* if any (the shortest) representative of the class $\{\sigma\}$ is contained in $\Xi_i$.* *For $\sigma=\sigma_1\ldots \sigma_i\in\{0,1\}^i$ for some $i\ge 1$, we define $\sigma T^c:=\sigma_1\ldots \sigma_i(1-\sigma_i)\in \{0,1\}^{i+1}$. For $\sigma\in \Xi_i$, we say that $(\sigma T_{j-1})T^c\in \{0,1\}^{i+j}$ is the *level-$(i+j)$ sibling* of $\{\sigma\}$. We say that two strings in level $i$ are *newly appearing cousins* on level $i$ if they are of the forms $\sigma01$ and $\sigma 10$ for some $\sigma\in \{0,1\}^{i-2}$.* The inverse of the operator $T$ \"cuts off\" all but one of the identical last digits from a $\sigma \in \{0,1\}^R$, hence, each class $\{\sigma\}$ has exactly one representative in $\{0,1\}^R$, and the number of equivalence classes is $2^R$. For $i>1$, there are exactly $2^{i-1}$ equivalence classes that first appear on level $i$ (i.e. the shortest representative of the class is in $\Xi_i$), and (since $0, 1\in \Xi_1$) the total number of equivalence classes that appear until level $i$ is $2^i$. To show an example of the sibling relationship, e.g. $01111\sim 01$ belongs to $\Xi_2$, and the level-$3$ sibling of $\{01\}$ is $010$, and the level-$(2+j)$ sibling of $\{01\}$ is $01_j0$. Similarly, $010$ and $001$ are newly appearing level-$3$ cousins, and on level $i$, there are $2^{i-2}$ pairs of newly appearing cousins. The hierarchy embeds each equivalence class $\{\sigma\}\in\cup_{j=1}^{R}\{0,1\}^R$ into the (weighted) vertex set of $G$ so that all cousins are joined by low-cost "bridge" paths, all siblings are close in Euclidean space, $0^R=x$ and $1^R=y$ are the vertices we start with, and the weights of all other vertices in the embedding are constrained. The Euclidean distances between siblings/cousins will decay doubly-exponentially in $i$. We formalise the embedding in the following definition. **Definition 36** (Hierarchy). *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}. Let $y_0, y_1 \in\widetilde\mathcal{V}$, $U,\overline{w},c_H\ge1$, and $R\ge2$ be an integer. Consider a set of vertices $\{y_{\sigma}\}_{\sigma\in\{0,1\}^R}$, divided into *levels* $\mathcal{L}_i:=\{y_{\sigma} \colon \sigma\in\Xi_i\}$ for $i\in \{1,\ldots,R\}$, satisfying that $y_{\sigma}=y_{\sigma'}$ if $\sigma\sim_T \sigma'$. We say that $\{y_{\sigma}\}_{\sigma\in\{0,1\}^R}\subset \widetilde \mathcal{V}$ is a *$(\gamma, U, \overline{w}, c_H)$-hierarchy of depth $R$* with $\mathcal{L}_1=\{y_0, y_1\}$ if it satisfies the following properties:* 1. *[\[item:H1\]]{#item:H1 label="item:H1"} $W_{y_{\sigma}}\in [\overline{w}, 4\overline{w}]$ for all $\sigma\in\{0,1\}^R \setminus \Xi_1$.* 2. *[\[item:H2\]]{#item:H2 label="item:H2"} $|y_{\sigma0}-y_{\sigma1}|\le c_H|y_0-y_1|^{\gamma^i}$ for all $\sigma\in\{0,1\}^i$, $i=0,\ldots,R-1$.* 3. *[\[item:H3\]]{#item:H3 label="item:H3"} There is a set $\{P_{\sigma} : \sigma\in\{0,1\}^i, 0\le i \le R-2\}$ of paths in $G$ such that for all $0 \le i \le R-2$ and all $\sigma \in \{0,1\}^i$, $P_{\sigma}$ connects $y_{\sigma01}$ to $y_{\sigma10}$. Moreover, we can partition $\bigcup_{\sigma\in\{0,1\}^R} \mathcal{E}(P_\sigma)$ into sets $\{\mathcal{E}^-(P_\sigma)\colon\sigma\in\{0,1\}^R\}$ in such a way that for all $\sigma$, we have $\mathcal{E}^-(P_\sigma) \subseteq \mathcal{E}(P_\sigma)$ and $\mathcal{C}(\mathcal{E}^-(P_\sigma)) \le U$. These paths $P_{\sigma}$ are called *bridges*.* *Given a set $\mathcal{N}\subseteq \widetilde{\mathcal{V}}$, we say that a hierarchy $\{y_\sigma\}_{\sigma \in \{0,1\}^R}$ is *fully contained in $\mathcal{N}$* if both $\{y_\sigma\}_{\sigma \in \{0,1\}^R} \subseteq \mathcal{N}$, and every vertex on the paths $P_{\sigma}$ in (H[\[item:H3\]](#item:H3){reference-type="ref" reference="item:H3"}) lies in $\mathcal{N}$.* Condition *(H[\[item:H3\]](#item:H3){reference-type="ref" reference="item:H3"})* is slightly weaker than requiring each bridge to have cost at most $U$. We shall construct the hierarchy via an iterative construction in Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}, using one round to embed each level $\mathcal{L}_i$. We shall use Prop. [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"} to estimate the success probability of the whole construction, which requires that we use marginal costs, and this gives the definition of $\mathcal{E}^-(P_\sigma)$ in [\[eq:marginal-cost\]](#eq:marginal-cost){reference-type="eqref" reference="eq:marginal-cost"}. Using marginal costs causes no problem, since our goal is to find a path $\pi$ between $y_0$ and $y_1$ of low cost. A path $\pi$ uses every edge in it once, so all the bridges $P_\sigma$ together will contribute to the cost of $\pi$ at most $$\begin{aligned} \mathcal{C}(\pi) = \sum_{\sigma\in\{0,1\}^i, 0\le i \le R-2} \mathcal{C}(\mathcal{E}^-(P_\sigma)) \le (2^{R-1}-1) U.\end{aligned}$$ Later we also need that the the hierarchy stays close to the straight line segment between the starting vertices. To track this, we have the following definition: **Definition 37**. *Given $u,v \in \mathbb{R}^d$, let $S_{u,v}$ denote the line segment between $u,v$. For $x\in \mathbb{R}^d$ we define the deviation $\mathrm{dev}_{uv}(x):=\min\{|x-y|: y\in S_{u,v}\}$. Given a set of vertices $\mathcal{H}$ in $\mathbb{R}^d$, we define the *deviation of $\mathcal{H}$ from $S_{uv}$* as $\mathrm{dev}_{uv}(\mathcal{H}):=\max\{\mathrm{dev}_{uv}(x) \colon x\in \mathcal{H}\}$. Finally, for a path $\pi=(x_1\dots x_k)$, let the *deviation of $\pi$* be $\mathrm{dev}(\pi) := \max\{\mathrm{dev}_{x_1x_k}(x_i) \colon i \in [k]\}$, i.e., the deviation of its vertex set from the segment between the endpoints.* Next we describe the procedure used to find the hierarchy in $G$. We iteratively embed the levels $\Xi_i$ into the vertex set $\widetilde\mathcal{V}$. We first embed $\Xi_1$, by setting $0\mapsto y_0$ and $1\mapsto y_1$, i.e., $\mathcal{L}_1:=\{y_0, y_1\}$, the two given starting vertices. Observe that this embedding trivially satisfies condition *(H[\[item:H2\]](#item:H2){reference-type="ref" reference="item:H2"})* for $i=0$, (i.e., $\sigma=\emptyset$ in *(H[\[item:H2\]](#item:H2){reference-type="ref" reference="item:H2"})*) for all $c_H\ge1$. Conditions *(H[\[item:H1\]](#item:H1){reference-type="ref" reference="item:H1"})* and *(H[\[item:H3\]](#item:H3){reference-type="ref" reference="item:H3"})* do not concern $y_0$ and $y_1$. In round $i+1$ we then embed all $\sigma \in \Xi_{i+1}$. Given the embedding of $\cup_{j\le i}\Xi_j$ of vertices in level $\cup_{j\le i}\mathcal{L}_j$, we will embed $\sigma\in \Xi_{i+1}$ by finding $\{y_{\sigma}\}_{\sigma\in\Xi_{i+1}}=\mathcal{L}_{i+1}$ as follows. For each sibling pair $\sigma 0, \sigma 1 \in \{0,1\}^i$, by the equivalence relation $\sim_T$ in Def. [Definition 35](#def:strings){reference-type="ref" reference="def:strings"}, $y_{\sigma 00} = y_{\sigma 0}$ and $y_{\sigma 11} = y_{\sigma 1}$. We then search for a pair of vertices $a$ and $b$ close to $y_{\sigma 00}$ and $y_{\sigma 11}$ respectively, so that $ab$ is a low-cost edge (typically covering a large Euclidean distance), and both $a$ and $b$ have a low-cost edge to a nearby vertex with weight in $[\overline{w},4\overline{w}]$; we embed these latter two vertices as $y_{\sigma 01}$ and $y_{\sigma10}$. The path $(y_{\sigma01}aby_{\sigma10})$ then constitutes the bridge-path $P_{\sigma}$ required by *(H[\[item:H3\]](#item:H3){reference-type="ref" reference="item:H3"})*. See Figure [\[fig:hierarchy\]](#fig:hierarchy){reference-type="ref" reference="fig:hierarchy"} for a visual explanation. We formalise our goal for this iterative construction of bridges in the following definition and remark. **Definition 38** (Valid bridges). *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"} and the notion of bridges in Definition [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"}, and let $S$ be a set of edges of $G$. For any $D,U>0, w\ge 1$, we say that a path $P \subseteq \mathcal{N}$ with endpoints $y,y'$ is a *$(D,U,w)$-valid* bridge for $x_0$ and $x_1$ with respect to $S$ if:* *$$\begin{aligned} &w_y, w_{y'} \in [w,4w], \label{eq:bridge-condition-b1}\\ &|x_0-y|\le D, \quad \mbox{and}\quad |x_1-y'| \le D, \label{eq:bridge-condition-b2}\\ & \mathcal{C}(P\setminus S)\le U. \label{eq:bridge-condition-b3} \end{aligned}$$* **Observation 39**. *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}. Fix any ordering on $\{0,1\}^R$, and let $\{y_\sigma\}_{\sigma \in \{0,1\}^R} \subseteq \mathcal{N}$, $U > 0$, and $\overline{w}\ge 1$. For all $0 \le i \le R-2$ and $\sigma \in \{0,1\}^i$, set $y_{\sigma'} := y_\sigma$ whenever $\sigma'\sim_T\sigma$. For all $i \le R-2$, let $D_{i} =|y_0-y_1|^{\gamma^{i}}$. Suppose that for all $0 \le i \le R-2$ and all $\sigma \in \{0,1\}^i$, there exists a bridge $P_{\sigma}$ with endpoints $y_{\sigma01}$ and $y_{\sigma10}$ that is $(c_HD_{i+1}, U, \overline{w})$-valid for $y_{\sigma0},y_{\sigma1} \in \{0,1\}^{i+1}$ with respect to $S=\bigcup_{\sigma' < \sigma}\mathcal{E}(P_{\sigma'})$. Then $\{y_\sigma\}_{\sigma \in \{0,1\}^R}$ is a $(\gamma,U,\overline{w}, c_H)$-hierarchy of depth $R$ with first level $\{y_0,y_1\}$ (i.e., satisfying Definition [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"}).* *Proof.* Conditions *(H[\[item:H1\]](#item:H1){reference-type="ref" reference="item:H1"})* of Definition [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"} is immediate from the weight constraint [\[eq:bridge-condition-b1\]](#eq:bridge-condition-b1){reference-type="eqref" reference="eq:bridge-condition-b1"}. *(H[\[item:H2\]](#item:H2){reference-type="ref" reference="item:H2"})* holds for the following reason. For $\sigma0, \sigma1 \in \{0,1\}^{i+1}$, $P_\sigma$ being a $(c_HD_{i+1}, U, \overline{w})$ valid bridge for $y_{\sigma0}, y_{\sigma1}$ implies by [\[eq:bridge-condition-b2\]](#eq:bridge-condition-b2){reference-type="eqref" reference="eq:bridge-condition-b2"} and $y_{\sigma0}=y_{\sigma00}, y_{\sigma1}=y_{\sigma11}$ that both $|y_{\sigma00}- y_{\sigma01}|, |y_{\sigma10}- y_{\sigma11}|$ are at most $c_H|y_0-y_1|^{\gamma^{i+1}}$ for all $\sigma\in \{0,1\}^i$. Setting now either $\sigma':=\sigma0$ or $\sigma':=\sigma1$, this is equivalent to $|y_{\sigma'0}-y_{\sigma'1}|\le c_H|y_0-y_1|^{\gamma^{i+1}}$ for all $\sigma'\in \{0,1\}^{i+1}, i\ge 0$, and this exactly corresponds to *(H[\[item:H2\]](#item:H2){reference-type="ref" reference="item:H2"})*, since the inequality in *(H[\[item:H2\]](#item:H2){reference-type="ref" reference="item:H2"})* holds for $i=0$ trivially. Finally, condition *(H[\[item:H3\]](#item:H3){reference-type="ref" reference="item:H3"})* follows from [\[eq:bridge-condition-b3\]](#eq:bridge-condition-b3){reference-type="eqref" reference="eq:bridge-condition-b3"} by setting $\mathcal{E}^-(P_{\sigma}) := \mathcal{E}(P_\sigma) \setminus \bigcup_{\sigma' < \sigma}\mathcal{E}(P_{\sigma'})$. ◻ We now lower-bound the probability of finding a valid bridge between two fixed vertices. **Lemma 40** (3-edge bridges). *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}. Let $z \in [0,d]$ and let $c_H,\eta \ge 0$. Suppose that $\delta{\,\ll_{\star}\,}\gamma,\eta,z,c_H,\textnormal{\texttt{par}}\xspace$ and that $D{\,\gg_{\star}\,}\gamma,\eta,z,c_H,\delta,w_0$. Suppose further that $D^\gamma \in[4^{1/d}(\log\log\xi\sqrt{d})^{16/\delta}, \xi\sqrt{d} ]$ and that $\theta D^{\Lambda(\eta,z)-\sqrt{\delta}} > 1$. Suppose that $x_0, x_1 \in \mathcal{N}$ satisfy $|x_0-x_1| \le c_HD$, and let $\underline{w}{\,\gg_{\star}\,}\delta,w_0$ satisfy $\underline{w}\in[(\log\log\xi\sqrt{d})^{16d/\delta}, D^{\delta}]$. Let $\mathcal{A}(x_0,x_1)$ denote the event that $G'$ contains a bridge $P$ that is $(2D^\gamma,3\underline{w}^{3\mu}D^{\eta},\underline{w})$-valid for $x_0$ and $x_1$ with respect to $\emptyset$, and $\mathrm{dev}_{x_0x_1}(P)\le 2D^\gamma$. Finally, suppose that $$\begin{aligned} \label{eq:p(.)} p(D, \underline{w}, \theta, \eta, z):= \theta \underline{w}^{-(\tau-1)}\left(D^{d\gamma} \wedge \underline{w}^2 D^{z/2}\right)^{1-\delta}\left(1 \wedge \underline{w}^{\mu\beta}D^{\eta\beta-\mu\beta z/2}\right) \ge 20^{\tau+\mu\beta}. \end{aligned}$$ Then, with $\Lambda(\eta,z)$ from [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}, $$\begin{aligned} \label{eq:bridge-proba} \mathbb{P}\big(\mathcal{A}(x_0,x_1) \mid V,w_V\big) \ge 1 - 3\exp\left(-\big(\theta D^{\Lambda(\eta,z)-\sqrt{\delta}}\big)^{1/4}\right). \end{aligned}$$ With the convention that $\infty \cdot 0 = 0$ in [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}, the statement is also valid when $\alpha=\infty$ or $\beta=\infty$.* *Proof of Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"}.* First, when $\beta=\infty$ and $\eta < \mu z$ then $\Lambda(\eta,z)=-\infty$ in [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}, and the condition $\theta D^{\Lambda(\eta,z)-\sqrt{\delta}} > 1$ cannot hold. Hence, we can wlog assume that if $\beta=\infty$ then $\eta \ge \mu z$. We will first apply Claim [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"} to show that most vertices of weight roughly $\underline{w}D^{z/2}$ close to $x_0$ and $x_1$ are "good", i.e., they have a cheap edge to a vertex with weight in $[\underline{w},4\underline{w}]$. We will then apply Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"} to find a cheap edge between some pair of good vertices. Formally, let $I^+ = [5\underline{w}D^{z/2}, 20\underline{w}D^{z/2}]$ and $I^- = [\underline{w}, 4\underline{w}]$. Note that $I_+\cap I_-=\emptyset$ for all $z\in[0,d]$. As in Claim [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"}, for all $v \in \mathcal{N}$ let $\mathcal{A}_{2\underline{w},D^\gamma,\underline{w}^{3\mu}D^{\eta}}(v)=:\mathcal{A}(v)$ be the event that there is an edge of cost at most $\underline{w}^{3\mu}D^{\eta}$ in $G'$ from $v$ to a vertex $y \in \mathcal{N}\cap (B_{D^\gamma}(v) \times I^-)$. Let $$\begin{aligned} \label{eq:Zi-new} Z_i := \big\{v \in \mathcal{N}\cap (B_{D^\gamma}(x_i) \times I^+) \colon \mathcal{A}(v)\mbox{ occurs}\big\}, \qquad i \in \{0,1\}.\end{aligned}$$ As in [\[eq:N-gamma-eta-z\]](#eq:N-gamma-eta-z){reference-type="eqref" reference="eq:N-gamma-eta-z"} of Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"}, let $N_{\eta,\gamma,z,10\underline{w}}(Z_0,Z_1)$ be the set of all edges between $Z_0$ and $Z_1$ of cost at most $\underline{w}^{3\mu}D^\eta$ and then $I^+$ exactly corresponds to the weight interval $[5\underline{w}D^{z/2}, 20\underline{w}D^{z/2}]$ as required for $Z_0\subseteq \mathcal{Z}(x_0), Z_1\subseteq \mathcal{Z}(x_1)$ in [\[eq:calZ-def\]](#eq:calZ-def){reference-type="eqref" reference="eq:calZ-def"}. With $Z_i$ in [\[eq:Zi-new\]](#eq:Zi-new){reference-type="eqref" reference="eq:Zi-new"}, we now show that $$\label{eq:triple-bridge-main} \mathbb{P}\big(\mathcal{A}(x_0,x_1) \mid V,w_V\big) \ge \mathbb{P}\big(N_{\eta,\gamma,z,\underline{w}}(Z_0,Z_1) \ne \emptyset \mid V,w_V\big).$$ Indeed, suppose there exists $(a,b) \in N_{\eta,\gamma,z,10\underline{w}}(Z_0,Z_1)$. Since $a \in Z_0$, there exists $x \in \mathcal{N}\cap (B_{D^\gamma}(a)\times I^-)$ such that $(x,a)$ is an edge of cost at most $\underline{w}^{3\mu}D^\eta$. Likewise, since $b \in Z_1$, there exists $y \in \mathcal{N}\cap (B_{D^\gamma}(b) \times I^-)$ such that $(y,b)$ is an edge of cost at most $\underline{w}^{3\mu}D^\eta$. Since $a \in B_{D^\gamma}(x_0)$ and $b \in B_{D^\gamma}(x_1)$, by the triangle inequality, $x \in B_{2D^\gamma}(x_0)$ and $y \in B_{2D^\gamma}(x_1)$. Thus $xaby$ is a $(2D^\gamma, 3\underline{w}^{3\mu}D^\eta, \underline{w})$-valid bridge with $\mathrm{dev}_{x_0x_1}\le 2D^\gamma$, as required by $\mathcal{A}(x_0,x_1)$, showing [\[eq:triple-bridge-main\]](#eq:triple-bridge-main){reference-type="eqref" reference="eq:triple-bridge-main"}. Now, for each $i \in \{0,1\}$, using [\[eq:calZ-def\]](#eq:calZ-def){reference-type="eqref" reference="eq:calZ-def"}, we set $\mathcal{Z}(x_i) = \mathcal{N}\cap (B_{D^\gamma}(x_i) \times I^+)$. For [\[eq:lem-cheap-bridge\]](#eq:lem-cheap-bridge){reference-type="eqref" reference="eq:lem-cheap-bridge"} to hold we need that $|Z_i| \ge |\mathcal{Z}(x_i)|/4$. We prove this by showing that any given vertex in $v \in \mathcal{Z}(x_i)$ lies in $Z_i$ with probability at least $1/2$, by recalling that in [\[eq:Zi-new\]](#eq:Zi-new){reference-type="eqref" reference="eq:Zi-new"}, $\mathcal{A}(v)=\mathcal{A}_{2\underline{w},D^\gamma,\underline{w}^{3\mu}D^{\eta}}(v)=\mathcal{A}_{K,D,U}(v)$ in Claim [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"}. Hence we set $K = 2\underline{w}, M = 10\underline{w}D^{z/2}, U = \underline{w}^{3\mu}D^{\eta}$, $D_{\ref{claim:cheap-edge-to-nice-vertex}}=D^\gamma$, and all other variables to match their current values. We check the requirements of Claim [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"}: By hypothesis in the statement of Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"}, $\delta$ is small; $\underline{w},D{\,\gg_{\star}\,}\delta,w_0$, and $D{\,\gg_{\star}\,}\gamma$. Since $M,K \ge \underline{w}$, it follows that $M,K,D^\gamma {\,\gg_{\star}\,}\delta, w_0$, as required above [\[eq:KM-xi\]](#eq:KM-xi){reference-type="eqref" reference="eq:KM-xi"}. Condition [\[eq:KM-xi\]](#eq:KM-xi){reference-type="eqref" reference="eq:KM-xi"} itself holds since $(D^\gamma \wedge (20\underline{w}^2 D^{z/2})^{1/d}/4^{1/d} \ge D^\gamma/4^{1/d} \wedge \underline{w}^{1/d} \ge (\log\log\xi\sqrt{d})^{16/\delta}$ by hypothesis, and $(D^\gamma \wedge (20\underline{w}^2 D^{z/2})^{1/d}/4^{1/d} \le D^\gamma \le \xi\sqrt{d}$ by hypothesis. Condition [\[eq:K-D-M\]](#eq:K-D-M){reference-type="eqref" reference="eq:K-D-M"} holds since $K=2\underline{w}\le 2D^{\delta}$ by hypothesis, so since $\delta{\,\ll_{\star}\,}\gamma$ and $D{\,\gg_{\star}\,}\delta$, we have $2\underline{w}\le D^{\gamma(d/(\tau-1)-\delta)}$, and similarly since $\tau\in(2,3)$ and $\delta {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, $K=2\underline{w}\le (2\underline{w})^{1/(\tau-2+\tau\delta)} \le (10\underline{w}D^{z/2})^{1/(\tau-2+\delta\tau)}=M^{1/(\tau-2+\delta \tau)}$. Finally, if $\beta=\infty$, then below [\[eq:K-D-M\]](#eq:K-D-M){reference-type="eqref" reference="eq:K-D-M"} we need to check $U(KM)^{-\mu}{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$. Since wlog we assumed that $\eta \ge \mu z$, clearly $\eta \ge \mu z/2$. Therefore, $U(KM)^{-\mu}=\underline{w}^{3\mu}D^{\eta}(20\underline{w}^2 D^{z/2})^{-\mu} = 20^{-\mu}\underline{w}^{\mu}D^{\eta-\mu z/2} \ge (\underline{w}/20)^{\mu}$, which is large since $\underline{w}$ is large by hypothesis. Hence, all conditions of Claim [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"} are met and [\[eq:cheap-edge-to-nice-vertex\]](#eq:cheap-edge-to-nice-vertex){reference-type="eqref" reference="eq:cheap-edge-to-nice-vertex"} applies, and substituting $K = 2\underline{w}, M = 10\underline{w}D^{z/2}, U = \underline{w}^{3\mu}D^{\eta}$ there, the exponent on the rhs of [\[eq:cheap-edge-to-nice-vertex\]](#eq:cheap-edge-to-nice-vertex){reference-type="eqref" reference="eq:cheap-edge-to-nice-vertex"} in our setting becomes $$-2^{-(\tau-1)} \theta \underline{w}^{-(\tau-1)}\big(D^{d\gamma} \wedge 20\underline{w}^2D^{z/2}\big)^{1-\delta}\big(1 \wedge (\underline{w}/20)^{\mu\beta}D^{\eta\beta-\mu\beta z/2}\Big),$$ where we recognise that this matches $p(\cdot)$ from [\[eq:p(.)\]](#eq:p(.)){reference-type="eqref" reference="eq:p(.)"} up to a factor of at most $20^{1-(\tau+\mu\beta)}$. Since we assumed $p(\cdot)\ge 20^{\tau+\mu\beta}$ in [\[eq:p(.)\]](#eq:p(.)){reference-type="eqref" reference="eq:p(.)"}, for any vertex $v \in \mathcal{Z}(x_i)=\mathcal{N}\cap (B_{D^\gamma}(x_i) \times I^+)$, $$\label{eq:triple-bridge-binom-prob} \mathbb{P}\big(v \in Z_i \mid V,w_V\big) = \mathbb{P}(\mathcal{A}(v) \mid V, w_V) \ge 1 - e^{-20} > 1/2.$$ Since $I^+$ and $I^-$ are disjoint, the events $\mathcal{A}(v), \mathcal{A}(v')$ are functions of disjoint edge sets and are therefore mutually independent conditioned on $(V, w_V)$. Hence, for $i \in \{0,1\}$, $|Z_i|$ is dominated below by a binomial variable with mean $|\mathcal{Z}(x_i)|/2$. By the standard Chernoff bound (Theorem [Theorem 50](#thm:chernoff){reference-type="ref" reference="thm:chernoff"} with $\lambda=1/2$), $$\label{eq:triple-bridge-chernoff} \mathbb{P}\big(|Z_i| < |\mathcal{Z}(x_i)|/4 \mid V,w_V\big) \le e^{-|\mathcal{Z}(x_i)|/16}.$$ To bound $|\mathcal{Z}(x_i)|$ below in [\[eq:triple-bridge-chernoff\]](#eq:triple-bridge-chernoff){reference-type="eqref" reference="eq:triple-bridge-chernoff"}, we will use that $x_0,x_1 \in \mathcal{N}$ and $\mathcal{N}$ is a weak $(\delta/4,w_0)$-net as assumed in Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and apply [\[eq:net-defining-crit-eps\]](#eq:net-defining-crit-eps){reference-type="eqref" reference="eq:net-defining-crit-eps"}. We check if the conditions to apply [\[eq:net-defining-crit-eps\]](#eq:net-defining-crit-eps){reference-type="eqref" reference="eq:net-defining-crit-eps"} in Def. [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"} hold. Since $\mathcal{Z}(x_i)=\mathcal{N}\cap (B_{D^\gamma}(x_i) \times [5\underline{w}D^{z/2}, 20\underline{w}D^{z/2}])$, we set there $r=D^\gamma$ and $w=10\underline{w}D^{z/2}$, and we must bound $10\underline{w}D^{z/2}$ above and below. Recall that by hypothesis, $\theta D^{\Lambda(\eta,z)-\sqrt{\delta}} > 1$; this implies $\Lambda(\eta,z) > 0$ and hence $2d\gamma > z(\tau-1)$ using [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}. Since $\delta{\,\ll_{\star}\,}\gamma,z$, we may therefore assume $z/2 \le d\gamma/(\tau-1) - 2\delta$. Also, we assumed $\underline{w}\le D^\delta$, so $10\underline{w}D^{z/2} \le 10D^{d\gamma/(\tau-1)-\delta} \le (D^{\gamma})^{d/(\tau-1)-\delta/4}$, where the second inequality holds since $\gamma<1$ and $D{\,\gg_{\star}\,}\delta$. Moreover, since $\underline{w}{\,\gg_{\star}\,}w_0$, we have $10\underline{w}D^{z/2} \ge w_0$. Thus all conditions in Def. [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"} are met, and [\[eq:net-defining-crit-eps\]](#eq:net-defining-crit-eps){reference-type="eqref" reference="eq:net-defining-crit-eps"} here becomes $$|\mathcal{Z}(x_i)| \ge D^{d\gamma(1-\delta/4)}\ell(10\underline{w}D^{z/2})(10\underline{w}D^{z/2})^{-(\tau-1)}\ge D^{d\gamma(1-\delta/4) - (\tau-1+\delta/4)(\delta + z/2)},$$ where the second inequality holds by Potter's bound since $D^{\delta} \ge \underline{w}\gg \delta$. The exponent of $D$ on the right-hand side is $$d\gamma-(\tau-1)z/2-\delta(d\gamma/4+z/8+\tau-1+\delta/4) \ge d\gamma/2-(\tau-1)z/2 \ge \Lambda(\eta,z)/4,$$ where we used $\delta{\,\ll_{\star}\,}\gamma$ and then the formula of $\Lambda(\eta,z)$ in [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}. So, $|\mathcal{Z}(x_i)| \ge D^{\Lambda(\eta,z)/4}$ in [\[eq:triple-bridge-chernoff\]](#eq:triple-bridge-chernoff){reference-type="eqref" reference="eq:triple-bridge-chernoff"}, and since $D{\,\gg_{\star}\,}\delta$, $$\label{eq:triple-bridge-N-bound} \mathbb{P}\big(|Z_i| < |\mathcal{Z}(x_i)|/4 \mid V,w_V\big) \le \exp(-D^{\Lambda(\eta,z)/4}/16)\le \exp(-(\theta D^{\Lambda(\eta,z) - \sqrt{\delta}})^{1/4}).$$ Returning to the event $\mathcal{A}(x_0, x_1)$ in [\[eq:triple-bridge-main\]](#eq:triple-bridge-main){reference-type="eqref" reference="eq:triple-bridge-main"}, let $\mathcal{A}'$ be the event that $|Z_i| \ge |\mathcal{Z}(x_i)|/4$ for each $i \in \{0,1\}$, and suppose that $\mathcal{A}'$ occurs. We apply Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"}, conditioned on the values of $Z_0$ and $Z_1$, to lower-bound the rhs of [\[eq:triple-bridge-main\]](#eq:triple-bridge-main){reference-type="eqref" reference="eq:triple-bridge-main"}. In the statement of Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"}, we will take $x = x_0$, $y = x_1$, $Z_x = Z_0$, $Z_y = Z_1$, $\underline{w}_{\ref{claim:cheap-bridge}} = 10\underline{w}$, and all other variables to match their current values. The event $N_{\eta,\gamma,z,10\underline{w}}(Z_0,Z_1) \ne \emptyset$ of [\[eq:triple-bridge-main\]](#eq:triple-bridge-main){reference-type="eqref" reference="eq:triple-bridge-main"} requires a low-cost edge between the set $Z_0$ and $Z_1$, connecting vertices with weights in $I_+$. Given $(V, w_V)$, the existence of such an edge $(u,v)$ is independent of the events $\mathcal{A}(u), \mathcal{A}(v)$ since in $\mathcal{A}(\cdot)$ the other endpoint of the edge has weight $I_-$, and $I_+\cap I_-=\emptyset$. We now check the requirements of Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"}: it requires $z\in[0,d]$ that we assumed, and $2d\gamma > z(\tau-1)$. The latter holds since here we assume $\theta D^{\Lambda(\eta,z)-\sqrt{\delta}} > 1$ implying that $\Lambda(\eta,z) > 0$, so $2d\gamma > z(\tau-1)$ then follows from [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}. Second, here we assume $\underline{w}\ge (\log\log\xi\sqrt{d})^{16d/\delta} \ge (\log\log D^{\gamma})^{16d/\delta}$, and also $D{\,\gg_{\star}\,}\gamma,c_H,w_0$. So $\underline{w}\ge w_0 \vee 4(c_H+2)^d \vee 4000$ and $F_L((\underline{w}/4000)^{\mu})\ge 1/2$ as required above [\[eq:calZ-def\]](#eq:calZ-def){reference-type="eqref" reference="eq:calZ-def"}. The requirement on $D^\gamma$ here is more restrictive than in Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"}, so all requirements hold. Then, since here we have $10\underline{w}$, [\[eq:lem-cheap-bridge\]](#eq:lem-cheap-bridge){reference-type="eqref" reference="eq:lem-cheap-bridge"} turns into the following, which we then estimate by using that $\underline{w}\le D^\delta$, that $\delta{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ and that $D{\,\gg_{\star}\,}\delta$, $$\begin{aligned} \mathbb{P}\big(N_{\eta,\gamma,z,10\underline{w}}(Z_0,Z_1) &= \emptyset \mid \mathcal{A}',V,w_V\big) \le \exp\Big({-}\theta (10\underline{w})^{-2(\tau-1)}D^{\Lambda(\eta,z)-2\gamma d\delta/3}\Big) \\ &\le \exp\Big({-}\theta D^{\Lambda(\eta,z) - \sqrt{\delta}}\Big) \le\exp\Big({-}\big(\theta D^{\Lambda(\eta,z) - \sqrt{\delta}}\big)^{1/4}\Big),\end{aligned}$$ since we assumed $\theta D^{\Lambda(\eta,z)-\sqrt{\delta}} > 1$. Since $\mathcal{A}'=\{Z_0\ge |\mathcal{Z}(x_0)|/4, Z_1\ge |\mathcal{Z}(x_1)|/4\}$, combining this with [\[eq:triple-bridge-N-bound\]](#eq:triple-bridge-N-bound){reference-type="eqref" reference="eq:triple-bridge-N-bound"} and a union bound, the result in [\[eq:bridge-proba\]](#eq:bridge-proba){reference-type="eqref" reference="eq:bridge-proba"} follows. ◻ We now construct a hierarchy by repeatedly applying Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"} to find a set of valid bridges as in Observation [Observation 39](#obs:valid-bridges){reference-type="ref" reference="obs:valid-bridges"}, using an iterative construction (Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}) to mitigate independence issues. We now define the second function that will be crucial in determining the optimal exponents $\Delta_0, \eta_0$ in [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"}, [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}. For all $\eta > 0$, $z \ge 0$, and integers $R \ge 2$, we define $$\begin{aligned} \label{eq:Phi-def} \Phi(\eta,z) &:= \Big[d\gamma \wedge \frac{z}{2}\Big] + \Big[0 \wedge \beta\Big(\eta - \frac{\mu z}{2}\Big)\Big].\end{aligned}$$ Recall the $(\gamma, U, \overline{w}, c_H)$-hierarchy of depth $R$ from Def. [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"}, and $\Lambda(\eta,z)$ from [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}. **Lemma 41** (Hierarchy with low weights $\underline{w}$). *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and let $y_0,y_1\in\mathcal{N}$ with $|y_0-y_1|=\xi$. Let $z\in[0,d]$, $\eta \ge 0$, and let $0<\delta{\,\ll_{\star}\,}\gamma,\eta, z,\textnormal{\texttt{par}}\xspace$ be such that $\Lambda(\eta,z) \ge 2\sqrt{\delta}$ and either $z=0$ or $\Phi(\eta,z)\ge\sqrt{\delta}$. Let $\xi{\,\gg_{\star}\,}\gamma,\eta,z,\delta, w_0$. Let $R\ge2$ be an integer satisfying $$\begin{aligned} \xi^{\gamma^{R-1}} &\ge (\log\log\xi\sqrt{d})^{16d/\delta^2} \qquad \mbox{and}\qquad R/\theta \le (\log\log\xi)^{1/\sqrt{\delta}},\label{eq:R-condition-in-prop} \\ \text{and let }\qquad \underline{w}&:= \xi^{\gamma^{R-1}\delta}.\label{eq:ulw-in-prop} \end{aligned}$$ Let $\mathcal{X}_{low\textnormal{-}hierarchy}(R,\eta,y_0,y_1)$ be the event that $G'$ contains a $(\gamma,3\underline{w}^{3\mu}\xi^{\eta},\underline{w},2)$-hierarchy $\mathcal{H}_{low}$ of depth $R$ with first level $\mathcal{L}_1 = \{y_0,y_1\}$, fully contained in $\mathcal{N}$, with $\mathrm{dev}_{y_0y_1}(\mathcal{H}_{low})\le 4\xi^\gamma$. Then $$\begin{aligned} \label{eq:hierarchy-intermediate-prob} \mathbb{P}\big(\mathcal{X}_{low\textnormal{-}hierarchy}(R,\eta,y_0,y_1)\mid V,w_V\big) \ge 1 - \exp\big({-}(\log\log\xi)^{1/\sqrt{\delta}}\big)=:1-\mathrm{err}_{\xi,\delta}. \end{aligned}$$ With the convention that $\infty \cdot 0 = 0$, the statement is also valid when $\alpha=\infty$ or $\beta=\infty$.* *Proof.* To construct a $(\gamma,3\underline{w}^{3\mu}\xi^{\eta},\underline{w},2)$-hierarchy in $\mathcal{N}$, we will use an iterative cost construction of $R-1$ rounds from Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"} on $G'$. Recall from Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"} that $G'$ with given $V, w_V$ is a $\theta$-percolated CIRG. By Remark [Remark 25](#remark:cirg){reference-type="ref" reference="remark:cirg"}, $G'$ is a CIRG itself (also when $\theta=\theta_n$) with distribution $\{\mathcal{G}^{\theta}|V,w_V\}$. In the $i$'th round we will construct all bridges of the $i$'th level of the hierarchy at once, using Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"} $2^{i-1}$ times to find each bridge in the level. We will use Prop. [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"} to deal with conditioning between rounds, and union bounds to deal with conditioning within rounds. For $2 \le i \le R$, in the $i$'th round we we will set the constraint $\mathcal{U}_i$ so that the chosen set $\mathcal{E}_i$ in Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}[\[item:iter5\]](#item:iter5){reference-type="eqref" reference="item:iter5"} consists of a $(2\xi^{\gamma^{i-1}}, 3\underline{w}^{2\mu}\xi^{\eta}, \underline{w})$-valid bridge for $\tilde y_{\sigma 0}$ and $\tilde y_{\sigma1}$ for all $\sigma \in \{0,1\}^{i-2}$, where $\tilde y_{\sigma 0};$ and $\tilde y_{\sigma1}$ are (all) endpoints of bridges from the previous levels. In other words, $\mathcal{E}_i$ will contain all the necessary bridges at the $i$'th level of the hierarchy for $\mathcal{X}_{low\textnormal{-}hierarchy}(R,\eta,y_0,y_1):=\mathcal{X}_{low\textnormal{-}h}$. Since $\mathcal{L}_1=\{y_0, y_1\}$ contains no bridges yet, we denote by $(G',\mathcal{E}_2,\mathcal{F}_2,\mathcal{U}_2),\ldots,(G',\mathcal{E}_R,\mathcal{F}_R,\mathcal{U}_R)$ the consecutive rounds of the iterative construction. We next inductively define the valid edge tuples $\mathcal{F}_i$ in Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}[\[item:iter1\]](#item:iter1){reference-type="eqref" reference="item:iter1"} and the cost constraint event $\mathcal{U}_i$ in Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}[\[item:iter5\]](#item:iter5){reference-type="eqref" reference="item:iter5"}; and the vertices $\tilde y_{\sigma}$ for all $\sigma \in \{0,1\}^{i}$. Assume that $E_1, \dots, E_{i-1}$ is already given, i.e., we constructed $(\mathcal{L}_j)_{j\le i-1}$. For each $\sigma \in \{0,1\}^{i-2}$ consider the vertices $\tilde y_{\sigma0}, \tilde y_{\sigma1}\in \{0,1\}^{i-1}$. Set $D_{i}=\xi^{\gamma^i}$ as in Observation [Observation 39](#obs:valid-bridges){reference-type="ref" reference="obs:valid-bridges"}, and write $\mathcal{P}(\sigma)$ for the set of all possible paths (i.e., sequence of vertices) contained in $\mathcal{N}$ between all $y\in \mathcal{N}\cap (B_{2D_{i-1}}(\tilde y_{\sigma0})\times [\underline{w}, 4\underline{w}])$ and all $y'\in \mathcal{N}\cap (B_{2D_{i-1}}(\tilde y_{\sigma1})\times [\underline{w}, 4\underline{w}])$ (so that if $\widetilde P_{\sigma}\in\mathcal{P}(\sigma)$, then $\widetilde P_{\sigma}$ satisfies both [\[eq:bridge-condition-b1\]](#eq:bridge-condition-b1){reference-type="eqref" reference="eq:bridge-condition-b1"}, [\[eq:bridge-condition-b2\]](#eq:bridge-condition-b2){reference-type="eqref" reference="eq:bridge-condition-b2"} in Def. [Definition 38](#def:valid-bridges){reference-type="ref" reference="def:valid-bridges"}). Given $V, w_V, E_1, \dots, E_{i-1}$, define now a tuple $\underline t$ to be level-$i$ admissible if it contains exactly one such potential path from $\mathcal{P}(\sigma)$ for each $\sigma\in\{0,1\}^{i-2}$, and let $\mathcal{F}_i$ be the collection of all level-$i$ admissible tuples, with an arbitrary ordering. Let $\mathcal{U}_i$ be the event that each potential path in the chosen tuple is present in the graph under consideration, and has round-$i$ marginal cost at most $3\underline{w}^{3\mu}\xi^{\eta}$ in [\[eq:marginal-cost\]](#eq:marginal-cost){reference-type="eqref" reference="eq:marginal-cost"}. Following Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}[\[item:iter5\]](#item:iter5){reference-type="eqref" reference="item:iter5"} and Prop [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}, we set $\mathcal{E}_i^{G'}$ to be the first tuple of bridges (in the ordering of $\mathcal{F}_i$) for which $\mathcal{U}_i$ occurs, and we set $\mathcal{E}_i^{G'}$ to $\mathtt{None}$ if no such tuple exists. Moreover, we define $\tilde y_{\sigma00} := \tilde y_{\sigma0}$, $\tilde y_{\sigma11} := \tilde y_{\sigma1}$, and $\tilde y_{\sigma01}$ and $\tilde y_{\sigma10}$ to be the endpoints of the bridge $\widetilde P_\sigma$ in the chosen tuple $\mathcal{E}_i^{G'}$, or $\texttt{None}$ if $\mathcal{E}_i^{G'}=\texttt{None}$. This gives the iterative cost construction $I_{G'} = ((G',\mathcal{E}_i^{G'},\mathcal{F}_i,\mathcal{U}_i)\colon i \in \{2,\ldots,R\})$. Note that the criteria above for $\mathcal{P}(\sigma), \mathcal{F}_i, \mathcal{U}_i$ exactly matches Observation [Observation 39](#obs:valid-bridges){reference-type="ref" reference="obs:valid-bridges"} with $c_H\!=\!2$ and marginal cost of each bridge $\widetilde P_\sigma$ at most $U\!=\!3\underline{w}^{3\mu}\xi^{\eta}$, implying [\[eq:bridge-condition-b3\]](#eq:bridge-condition-b3){reference-type="eqref" reference="eq:bridge-condition-b3"}, i.e., each chosen bridge $\widetilde P_\sigma\in E_i$ is $(2\xi^{\gamma^{i-1}}, 3\underline{w}^{3\mu}\xi^{\eta}, \underline{w})$-valid for $y_{\sigma0},y_{\sigma1} \in \{0,1\}^{i-1}$ wrt the chosen edges in earlier rounds, i.e., wrt $S=\cup_{j\le i-1}E_j$. Since all vertices in $\{P_\sigma\}_\sigma$ are contained in a $2(\xi^\gamma+\xi^{\gamma^2}+\dots+\xi^{\gamma^{R-1}})\le 4\xi^\gamma$ ball around $y_0$ and $y_1$, respectively, the deviation requirement is also satisfied, and so by Obs. [Observation 39](#obs:valid-bridges){reference-type="ref" reference="obs:valid-bridges"}, if $I_G'$ succeeds then $\{\tilde y_\sigma\}_{ \sigma\in\{0,1\}^R}$ is a $(\gamma,3\underline{w}^{3\mu}\xi^{\eta},\underline{w},2)$-hierarchy as needed in $\mathcal{X}_{low\textnormal{-}h}$. Following Prop. [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}, let $r=R-1$ and $\theta_i\equiv 1/(R\!-\!1)$ there, and let $H_2, \ldots, H_{R}$ be independent $1/(R\!-\!1)$-percolations of $\{\mathcal{G}^{\theta} \mid V, w_W\}$, i.e., with distribution $\{\mathcal{G}^{\theta/(R-1)}\mid V, w_W\}$ from Def. [Definition 24](#def:percolated){reference-type="ref" reference="def:percolated"}. Then, the iterative cost construction $I_H = ((H_i,\mathcal{E}_i^H,\mathcal{F}_i,\mathcal{U}_i)\colon i \in \{2,\ldots,R\})$ on $H_2,\ldots,H_{R}$ in Prop. [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"} uses the same $\mathcal{F}_i, \mathcal{U}_i$ as $I_{G'}$. Let us denote by $\mathcal{A}_{I_H}(V, w_V, E_2,\ldots,E_{i-1})$ the event that $\mathcal{E}_j^H = E_{j}$ for $2\!\le\! j\!\le\! i-1$. So applying Prop. [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}, [\[eq:multi-round-exp-goal\]](#eq:multi-round-exp-goal){reference-type="eqref" reference="eq:multi-round-exp-goal"} turns into $$\begin{aligned}\label{eq:hierarchy-intermediate-weights-0} \mathbb{P}\big(\mathcal{X}_{low\textnormal{-}h} &\mid V,w_V\big)\ge \mathbb{P}\big(I_{G'} \textnormal{ succeeds} \mid V,w_V\big) \\ &\ge \min_{E_2,\ldots,E_{R} \ne \mathtt{None}} \prod_{i=2}^R\mathbb{P}\big(\mathcal{E}_i^H \ne \mathtt{None} \mid \mathcal{A}_{I_H}(V, w_V, E_2,\ldots,E_{i-1})\big)\\ &\ge1-\sum_{i=2}^R\max_{E_2,\ldots,E_{i-1} \ne \mathtt{None}}\mathbb{P}\big(\mathcal{E}_i^H = \mathtt{None} \mid \mathcal{A}_{I_H}(V, w_V, E_2,\ldots,E_{i-1})\big), \end{aligned}$$ by a union bound over all rounds. We now break the rhs of [\[eq:hierarchy-intermediate-weights-0\]](#eq:hierarchy-intermediate-weights-0){reference-type="eqref" reference="eq:hierarchy-intermediate-weights-0"} down into bridge existence events under simpler conditioning. For each $2 \le i \le R$ and $\sigma0, \sigma1\in \{0,1\}^{i-1}$, let $$\label{eq:Ai-event-def} \mathcal{A}_{i}(\tilde y_{\sigma0},\tilde y_{\sigma1}):=\{ \exists \widetilde P_\sigma \in E_i: (2\xi^{\gamma^{i-1}},3\underline{w}^{2\mu}\xi^{\gamma^{i-2}\eta}, \underline{w})\text{-valid for } \tilde y_{\sigma0}, \tilde y_{\sigma1} \text{ wrt } S=\emptyset\}.$$ This is a stronger condition than what is required for a $(\gamma, 3\underline{w}^{3\mu}\xi^\eta, \underline{w}, 2)$-hierarchy to exist in Obs. [Observation 39](#obs:valid-bridges){reference-type="ref" reference="obs:valid-bridges"}, since $\gamma^{i-2}\eta\le\eta$ and validity with respect to $\emptyset$ implies validity with respect to any set of edges. Conditioned on $\mathcal{A}_{I_H}(V, w_V, E_2,\ldots,E_{i-1})$ so that none of the $(E_j)_{j\le i-1}$ equals $\mathtt{None}$, the event $\mathcal{E}_{i}^H = \mathtt{None}$ occurs only if for some pair $\sigma0, \sigma1 \in \{0,1\}^{i-1}$, the event $\neg\mathcal{A}_{i}(\tilde y_{\sigma 0}, \tilde y_{\sigma1})$ occurs; hence by a union bound, [\[eq:hierarchy-intermediate-weights-0\]](#eq:hierarchy-intermediate-weights-0){reference-type="eqref" reference="eq:hierarchy-intermediate-weights-0"} implies $$\begin{aligned} \label{eq:hierarchy-intermediate-weights-2} \mathbb{P}\big(\mathcal{X}_{low\textnormal{-}h}\! \mid\! V,w_V\big) \ge 1\!-\!\sum_{i=2}^R 2^{i-2}\!\!\!\!\!\!\!\!\!\max_{\substack{\sigma \in \{0,1\}^{i-2}\\E_2,\ldots,E_{i-1} \ne \mathtt{None}}}\!\!\!\!\!\mathbb{P}\big(\neg\mathcal{A}_{i}(\tilde y_{\sigma 0},\tilde y_{\sigma 1})\! \mid\! \mathcal{A}_{I_H}(V, w_V, E_2,\ldots,E_{i-1})\big). \end{aligned}$$ Recall that conditioned on $(V,w_V)$, the graphs $H_2, \dots H_{R-1}$ are iid $\{\mathcal{G}^{\theta/(R-1)}\mid V, w_W\}$. So, the events in $\mathcal{A}_{I_H}(V, w_V, E_2,\ldots,E_{i-1})$ are contained in the $\sigma$-algebra generated by $H_2,\ldots,H_{i-1}$, i.e., independent of $H_i$ and thus of $\neg\mathcal{A}_{i}(\tilde y_{\sigma 0},\tilde y_{\sigma 1})$. Hence [\[eq:hierarchy-intermediate-weights-2\]](#eq:hierarchy-intermediate-weights-2){reference-type="eqref" reference="eq:hierarchy-intermediate-weights-2"} simplifies to $$\begin{aligned} \label{eq:hierarchy-intermediate-weights-3} \mathbb{P}\big(\mathcal{X}_{low\textnormal{-}h} \mid V,w_V\big) \ge 1-\sum_{i=2}^R 2^{i-2}\max_{\tilde y_{\sigma0},\tilde y_{\sigma1}\ne\mathtt{None}}\mathbb{P}\big(\neg\mathcal{A}_i(\tilde y_{\sigma 0},\tilde y_{\sigma 1}) \mid V,w_V\big), \end{aligned}$$ where the maximum is taken over all possible values of $(\tilde y_{\sigma0},\tilde y_{\sigma1})$ occurring in non-`None` $E_{i-1}$. Finally, we will upper-bound the probabilities on the rhs of [\[eq:hierarchy-intermediate-weights-3\]](#eq:hierarchy-intermediate-weights-3){reference-type="eqref" reference="eq:hierarchy-intermediate-weights-3"} using Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"}. Let $2 \le i \le R$, let $\sigma \in \{0,1\}^{i-2}$, and let $\tilde y_{\sigma0},\tilde y_{\sigma1}$ be a possible non-`None` realisation of the embedding. Recall $D_i=\xi^{\gamma^{i}}$. Then the event [\[eq:Ai-event-def\]](#eq:Ai-event-def){reference-type="eqref" reference="eq:Ai-event-def"} requires a $(D_{i-2}^\gamma, 3\underline{w}^{3\mu}D_{i-2}^\eta, \underline{w})$-valid bridge $P_\sigma$, which formally matches Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"} with $D:=D_{i-2}, \tilde y_{\sigma0}:=x_0, \tilde y_{\sigma1}:=x_1$ there and the graph $H_i\sim \{\mathcal{G}^{\theta/(R-1)}\mid V, w_V\}$ in place of $G'$ there, i.e., with $\theta_{\ref{lem:triple-edge-bridge}}:=\theta/(R-1)$. We check the conditions of Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"} in order of their appearance. $z\in[0,d], \eta,\delta>0$ and $\delta{\,\ll_{\star}\,}\gamma,\eta, z$ is assumed both here and there. The assumption $\xi{\,\gg_{\star}\,}\gamma,\eta, z, \delta, w_0$ here implies $D_{i-2}{\,\gg_{\star}\,}\gamma,\eta, z, \delta, w_0$ since by [\[eq:R-condition-in-prop\]](#eq:R-condition-in-prop){reference-type="eqref" reference="eq:R-condition-in-prop"} $D_{i-2}^\gamma \ge \xi^{\gamma^{R-1}} \ge (\log\log\xi\sqrt{d})^{16d/\delta^2}$. The latter also implies the requirement on $D^\gamma$ in Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"}. Similarly, the upper bound requirement holds since $D_{i-2}^{\gamma} \le \xi \le \xi\sqrt{d}$. We now check whether $\theta D^{\Lambda(\eta,z)-\sqrt{\delta}} > 1$ holds in Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"} for our choices. Since we assumed here $\Lambda(\eta,z) \ge 2\sqrt{\delta}$, and also [\[eq:R-condition-in-prop\]](#eq:R-condition-in-prop){reference-type="eqref" reference="eq:R-condition-in-prop"}, we estimate $$\label{eq:inter-weights-theta-R} \tfrac{\theta}{R-1}D_{i-2}^{\Lambda(\eta,z)-\sqrt{\delta}} \ge (\log\log\xi)^{-1/\sqrt{\delta}}\cdot D_{i-2}^{\sqrt{\delta}} \ge (\log\log\xi)^{15/\sqrt{\delta}} > 1.$$ Next we need to check whether $x_0=\tilde y_{\sigma0}, x_1=\tilde y_{\sigma1}$ satisfies $|x_0- x_1|\le c_HD_{i-2}$. This is true since $\tilde y_{\sigma0},\tilde y_{\sigma1}$ are possible non-`None` values coming from chosen tuples $E_1, \dots, E_{i-1}$; and by construction of $\mathcal{P}(\sigma)$ above, we required that $|\tilde y_{\sigma'00}-\tilde y_{\sigma'01}|, |\tilde y_{\sigma'10}-\tilde y_{\sigma'11}|\le 2D_{i-1}$ for all $\sigma'\in \{0,1\}^{i-2}$, which, when shifting indices yields exactly that $|\tilde y_{\sigma0}-\tilde y_{\sigma1}|\le 2D_{i-2}$ for all $\sigma\in\{0,1\}^{i-2}$. Next we check the criterion on $\underline{w}$ in Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"}. Here, $\underline{w}$ is defined in [\[eq:ulw-in-prop\]](#eq:ulw-in-prop){reference-type="eqref" reference="eq:ulw-in-prop"}, hence, using [\[eq:R-condition-in-prop\]](#eq:R-condition-in-prop){reference-type="eqref" reference="eq:R-condition-in-prop"}, $\underline{w}= \xi^{\gamma^{R-1}\delta} \ge (\log\log\xi\sqrt{d})^{16d/\delta}$ as required. This also implies $\underline{w}{\,\gg_{\star}\,}\delta, w_0$ since $\xi {\,\gg_{\star}\,}\delta, w_0$. Moreover, $\underline{w}= \xi^{\gamma^{R-1}\delta} \le D_{i-2}^{\delta}=\xi^{\gamma^{i-2}\delta}$ holds since $i-2\le R-2$ and $\gamma<1$. Next, we check [\[eq:p(.)\]](#eq:p(.)){reference-type="eqref" reference="eq:p(.)"}, which can be lower bounded by omitting the prefactor $\underline{w}^{\mu\beta}$ in the last factor (the minimum): $$\begin{aligned} \label{eq:p(.)-bound} p(D_{i-2},\underline{w},\tfrac{\theta}{R-1},\eta,z) \ge \tfrac{\theta}{R-1}\underline{w}^{-(\tau-1)} \left(D_{i-2}^{d\gamma } \wedge \underline{w}^2 D_{i-2}^{z/2}\right)^{1-\delta} D_{i-2}^{[0 \wedge \beta(\eta-\mu z/2)]}. \end{aligned}$$ We distinguish cases wrt $z$ to handle the minimum in the middle of the rhs. If $z=0$, then $\underline{w}^2D_i^{z/2} = \underline{w}^2 = \xi^{2\gamma^{R-1}\delta} \le \xi^{\gamma^{i-1}d} = D_{i-2}^{\gamma d}$, where the inequality holds because $i\le R$ and $\delta {\,\ll_{\star}\,}\gamma$. Moreover $0 \le \eta-\mu z/2$ in that case, so when $z=0$, equation [\[eq:p(.)-bound\]](#eq:p(.)-bound){reference-type="eqref" reference="eq:p(.)-bound"} becomes $$\begin{aligned} \label{eq:p-bound-z-0} p(D_{i-2},\underline{w},\tfrac{\theta}{R-1},\eta,z) \ge \tfrac{\theta}{R-1}\underline{w}^{2(1-\delta)-(\tau-1)} = \tfrac{\theta}{R-1}\underline{w}^{3-\tau-2\delta} \ge \tfrac{\theta}{R-1}\underline{w}^{\sqrt{\delta}}, \end{aligned}$$ where the last inequality holds because $\delta {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$. If, however, $z\neq 0$, then we assumed that $\Phi(\eta,z)\ge\sqrt{\delta}$ in [\[eq:Phi-def\]](#eq:Phi-def){reference-type="eqref" reference="eq:Phi-def"}. Using again $\underline{w}\ge1$, we lower bound [\[eq:p(.)-bound\]](#eq:p(.)-bound){reference-type="eqref" reference="eq:p(.)-bound"} in this case $$\begin{aligned} p(D_{i-2},\underline{w},\tfrac{\theta}{(R-1)},\eta,z) \ge \tfrac{\theta}{R-1}\underline{w}^{-(\tau-1)} D_{i-2}^{(1-\delta)[d\gamma \wedge z/2] + [0 \wedge \beta(\eta-\mu z/2)]}\ge \tfrac{\theta}{R-1} D_{i-2}^{\Phi(\eta,z)-\delta(\tau-1+d)}, \end{aligned}$$ where we used that $\underline{w}\le D_{i-2}^\delta$ implies $\underline{w}^{-(\tau-1)} \ge D_{i-2}^{-\delta(\tau-1)}$ and $d\gamma \wedge z/2 \le d$ (since $\gamma<1$) to obtain the last inequality. Since $\delta$ is small, $\Phi(\eta,z)\ge\sqrt{\delta}$, and $\underline{w}\le D_{i-2}^{\delta}$, this implies $$\begin{aligned} \label{eq:p-bound-z-not-0} p(D_{i-2},\underline{w},\tfrac{\theta}{R-1},\eta,z) \ge \tfrac{\theta}{R-1} D_{i-2}^{\delta} \ge \tfrac{\theta}{R-1}\underline{w}\ge \tfrac{\theta}{R-1}\underline{w}^{\sqrt{\delta}}, \end{aligned}$$ the same lower bound as in [\[eq:p-bound-z-0\]](#eq:p-bound-z-0){reference-type="eqref" reference="eq:p-bound-z-0"} for $z=0$. Thus for all $z\in[0,d]$, using [\[eq:R-condition-in-prop\]](#eq:R-condition-in-prop){reference-type="eqref" reference="eq:R-condition-in-prop"} and [\[eq:ulw-in-prop\]](#eq:ulw-in-prop){reference-type="eqref" reference="eq:ulw-in-prop"}, $$\begin{aligned} p(D_{i-2},\underline{w},\tfrac{\theta}{R-1},\eta,z) \ge \tfrac{\theta}{R-1}\underline{w}^{\sqrt{\delta}} \ge \tfrac{\theta}{R-1}(\log\log\xi)^{16/\sqrt{\delta}} \ge (\log\log\xi)^{15/\sqrt{\delta}} \ge 20^{\tau+\mu\beta}, \end{aligned}$$ where the last inequality holds because $\xi {\,\gg_{\star}\,}\delta,\textnormal{\texttt{par}}\xspace$. With this, all conditions of Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"} are satisfied, so combining [\[eq:bridge-proba\]](#eq:bridge-proba){reference-type="eqref" reference="eq:bridge-proba"} with [\[eq:hierarchy-intermediate-weights-3\]](#eq:hierarchy-intermediate-weights-3){reference-type="eqref" reference="eq:hierarchy-intermediate-weights-3"} and then using the lower bound in [\[eq:inter-weights-theta-R\]](#eq:inter-weights-theta-R){reference-type="eqref" reference="eq:inter-weights-theta-R"} yields $$\label{eq:hierarchy-intermediate-weights-4} \begin{aligned} \mathbb{P}\big(&\mathcal{X}_{low\textnormal{-}h} \mid V,w_V\big) \ge 1 - 3\sum_{i=2}^{R}2^{i-2}\exp\Big({-}\Big[\tfrac{\theta}{R-1} D_{i-2}^{\Lambda(\eta,z)-\sqrt{\delta}}\Big]^{1/4}\Big)\\ & \ge 1 - 3\sum_{i=2}^{R}2^{i-2}\cdot \exp\big({-}(\log\log\xi)^{3/\sqrt{\delta}}\big)\ge 1 - 2^{R+1}\exp\big({-}(\log\log\xi)^{3/\sqrt{\delta}}\big). \end{aligned}$$ Finally, in [\[eq:R-condition-in-prop\]](#eq:R-condition-in-prop){reference-type="eqref" reference="eq:R-condition-in-prop"} the estimate $R\le(\log\log\xi)^{1/\sqrt{\delta}}$ can be used to upper bound $2^{R+1}$, yielding the required inequality in [\[eq:hierarchy-intermediate-prob\]](#eq:hierarchy-intermediate-prob){reference-type="eqref" reference="eq:hierarchy-intermediate-prob"}. ◻ Lemma [Lemma 41](#lem:hierarchy-intermediate-weights){reference-type="ref" reference="lem:hierarchy-intermediate-weights"} constructed a hierarchy with bridge endpoints $\tilde y_\sigma$ of weight roughly $\underline{w}= \xi^{\gamma^{R-1}\delta}$. This weight is too low to connect the final gaps (siblings) in the hierarchy via short paths. The next lemma extends this hierarchy to a new one with endpoints $y_\sigma$ of weight roughly $\overline{w}:= \xi^{d\gamma^{R-1}/2}$, where connecting the gaps is possible. Recall the $(\gamma, U, \overline{w}, c_H)$-hierarchy of depth $R$ from Def. [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"}, $\Lambda(\eta,z), \Phi(\eta,z)$ from [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"} and [\[eq:Phi-def\]](#eq:Phi-def){reference-type="eqref" reference="eq:Phi-def"}, respectively. **Lemma 42** (Hierarchy with high weights $\overline{w}$). *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and let $y_0,y_1\in\mathcal{N}$ with $|y_0-y_1|=\xi$. Let $z\in[0,d], \eta \ge 0$, and let $0<\delta{\,\ll_{\star}\,}\gamma,\eta,z,\textnormal{\texttt{par}}\xspace$ be such that $\Lambda(\eta,z) \ge 2\sqrt{\delta}$ and either $z=0$ or $\Phi(\eta,z)\ge\sqrt{\delta}$. Let $\xi{\,\gg_{\star}\,}\gamma,\eta,z,\theta, \delta,w_0$. Let $R\ge2$ be an integer satisfying $\xi^{\gamma^{R-1}} \ge (\log\log\xi\sqrt{d})^{16d/\delta^2}$ and $R\le (\log\log\xi)^2$, and set $$\begin{aligned} \label{eq:c_H} \overline{w}:= \xi^{\gamma^{R-1}d/2}, \qquad c_H:=8\bigg(1+\left\lceil\frac{\log(d/\delta)}{\log(1/(\tau-2+2d\tau\delta))}\right\rceil\bigg). \end{aligned}$$ Let $\mathcal{X}_{high\textnormal{-}hierarchy}(R,\eta,y_0,y_1)$ be the event that $G'$ contains a $(\gamma,c_H\overline{w}^{4\mu}\xi^{\eta},\overline{w},c_H)$-hierarchy $\mathcal{H}_{high}$ of depth $R$ with first level $\mathcal{L}_1 = \{y_0,y_1\}$, fully contained in $\mathcal{N}$, and $\mathrm{dev}_{y_0y_1}(\mathcal{H}_{high})\le 2c_H\xi^\gamma$. Then $$\begin{aligned} \label{eq:hierarchy-final-prob} \mathbb{P}\big(\mathcal{X}_{high\textnormal{-}hierarchy}(R,\eta,y_0,y_1) \mid V,w_V\big) \ge 1 - \exp\big({-}(\log\log\xi)^{13}\big); \end{aligned}$$ under the convention that $\infty \cdot 0 = 0$, the statement is also valid when $\alpha=\infty$ or $\beta=\infty$.* *Proof.* As in Lemma [Lemma 41](#lem:hierarchy-intermediate-weights){reference-type="ref" reference="lem:hierarchy-intermediate-weights"}, let $\underline{w}:= \xi^{\gamma^{R-1}\delta}$. To construct a $(\gamma,c_H\overline{w}^{4\mu}\xi^{\eta},\overline{w},c_H)$-hierarchy in $\mathcal{N}$, we use two rounds of exposure. In the first round we apply Lemma [Lemma 41](#lem:hierarchy-intermediate-weights){reference-type="ref" reference="lem:hierarchy-intermediate-weights"} to get $\mathcal{H}_{low}=\{\tilde y_\sigma\}$, a $(\gamma,3\underline{w}^{3\mu}\xi^{\eta},\underline{w},2)$-hierarchy with failure probability $\mathrm{err}_{\xi,\delta}$ in [\[eq:hierarchy-intermediate-prob\]](#eq:hierarchy-intermediate-prob){reference-type="eqref" reference="eq:hierarchy-intermediate-prob"}. In the second round, we use weight-increasing paths from Claim [Claim 33](#claim:cheap-path-to-larger-weight){reference-type="ref" reference="claim:cheap-path-to-larger-weight"} to connect each $\tilde y_\sigma\in \mathcal{H}_{low}$ to a vertex $y_\sigma$ of weight in $[\overline{w},4\overline{w}]$, transforming $\mathcal{H}_{low}$ into $\mathcal{H}_{high}=\{y_\sigma\}$, a $(\gamma,c_H\overline{w}^{4\mu}\xi^{\eta},\overline{w},c_H)$-hierarchy. We now define an iterative cost construction on $G' \sim\{\mathcal{G}^{\theta}|V, w_V\}$. Let $\mathcal{F}_1$ be the collection of all tuples of possible edges that could form the bridges $(P_\sigma)_{\sigma\in \{0,1\}^R}$ of a $(\gamma,3\underline{w}^{3\mu}\xi^{\eta},\underline{w},2)$-hierarchy $\{\tilde y_{\sigma}\}_{\sigma\in\{0,1\}^R}$ fully contained in $\mathcal{N}$, and $\mathcal{U}_1$ be the event that the costs of all $P_\sigma$ satisfy *(H[\[item:H3\]](#item:H3){reference-type="ref" reference="item:H3"})* of Definition [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"} with $U=3\underline{w}^{3\mu}\xi^{\eta}$, so that $\mathcal{H}_{low}$ is indeed a valid $(\gamma,3\underline{w}^{3\mu}\xi^{\eta},\underline{w},2)$-hierarchy. Here, $\mathcal{U}_1$ is measurable wrt $\sigma(V, w_V, \mathcal{E}_1)$ as required by Definition [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"} ([\[item:iter4\]](#item:iter4){reference-type="ref" reference="item:iter4"}), and the round-$1$ marginal costs in [\[eq:marginal-cost\]](#eq:marginal-cost){reference-type="eqref" reference="eq:marginal-cost"} are equal to the true edge costs (in $G'$). For each $\sigma\in\{0,1\}^R$, let $\mathcal{P}(\sigma)$ be the set of all paths $\pi_{\tilde y_\sigma, y_\sigma}$ in $\mathcal{N}$ connecting $\tilde y_{\sigma}$ to any vertex $y_{\sigma}\in \mathcal{N}\cap (B_{(c_H-2)\xi^{\gamma^{R-1}}/2}(\tilde y_{\sigma}) \times [\overline{w}, 4\overline{w}])$. Given $(V, w_V, E_1)$, call a tuple $\underline t$ of edges *admissible* if it contains exactly one such potential path from $\mathcal{P}(\sigma)$ for each $\sigma\in\{0,1\}^{R}$, and let $\mathcal{F}_2$ be the collection of all admissible tuples, with an arbitrary ordering. Let $\mathcal{U}_2$ be the event that each potential path in the chosen tuple is present in the graph under consideration, and has round-$2$ marginal cost at most $(c_H-3)\overline{w}^{4\mu}\xi^{\eta}/2$ in [\[eq:marginal-cost\]](#eq:marginal-cost){reference-type="eqref" reference="eq:marginal-cost"}. Following Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}[\[item:iter5\]](#item:iter5){reference-type="eqref" reference="item:iter5"} and Prop [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}, for $i=1,2$, we set $\mathcal{E}_i^{G'}$ to be the first tuple (in the ordering of $\mathcal{F}_i$) for which $\mathcal{U}_i$ occurs, and we set $\mathcal{E}_i^{G'}$ to $\mathtt{None}$ if no such tuple exists. This defines an iterative cost construction on $G'$, $I_{G'} = ((G',\mathcal{E}_1^{G'},\mathcal{F}_1,\mathcal{U}_1), (G',\mathcal{E}_2^{G'},\mathcal{F}_2,\mathcal{U}_2))$. If $I_{G'}$ succeeds, then $\{y_\sigma\}_{ \sigma\in\{0,1\}^R}$ is a $(\gamma,c_H\overline{w}^{4\mu}\xi^{\eta},\overline{w},c_H)$-hierarchy. Indeed, condition *(H[\[item:H1\]](#item:H1){reference-type="ref" reference="item:H1"})* of Def. [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"} is satisfied by construction. By the triangle inequality, *(H[\[item:H2\]](#item:H2){reference-type="ref" reference="item:H2"})* is satisfied since for all $\sigma \in \{0,1\}^i$, $y_{\sigma1} \in B_{(c_H-2)\xi^{\gamma^i}/2}(\tilde y_{\sigma1})$ and $y_{\sigma0} \in B_{(c_H-2)\xi^{\gamma^i}/2}(\tilde y_{\sigma0})$ by construction, and $|\tilde y_{\sigma1} - \tilde y_{\sigma0}| \le 2\xi^{\gamma^i}$ by *(H[\[item:H2\]](#item:H2){reference-type="ref" reference="item:H2"})* since $\tilde y_\sigma$ forms a $(\gamma, 3\underline{w}^{3\mu}\xi^{\eta},\underline{w}, 2)$-hierarchy. This also implies that $\mathrm{dev}_{y_0y_1}(\mathcal{H}_{high})\le c_H(\xi^\gamma+\xi^{\gamma^2}+\dots+ \xi^{\gamma^{R-1}})\le 2c_H\xi^\gamma$, since $\xi{\,\gg_{\star}\,}\gamma$, as required. Finally, $\widetilde P_\sigma$ is a path between $\tilde y_{\sigma01}$ and $\tilde y_{\sigma10}$, so let $P_\sigma$ be the concatenated path $\pi_{y_{\sigma01},\tilde y_{\sigma01}}\widetilde P_\sigma \pi_{\tilde y_{\sigma10}, y_{\sigma10}}$. Then the total cost of $P_\sigma$ is $$\begin{aligned} \mathcal{C}(P_\sigma)&\le \mathcal{C}(\widetilde P_\sigma) + \mathrm{mcost}_2(\pi_{y_{\sigma01},\tilde y_{\sigma01}}) + \mathrm{mcost}_2(\pi_{\tilde y_{\sigma10},y_{\sigma10}}) \\&\le 3\underline{w}^{3\mu}\xi^{\eta}+2(c_H-3)\overline{w}^{4\mu}\xi^{\eta}/2 \le c_H \overline{w}^{4\mu} \xi^\eta,\end{aligned}$$ since $\underline{w}=\overline{w}^{2\delta/d}$, see [\[eq:ulw-in-prop\]](#eq:ulw-in-prop){reference-type="eqref" reference="eq:ulw-in-prop"} vs [\[eq:c_H\]](#eq:c_H){reference-type="eqref" reference="eq:c_H"}. Let $H_1$ and $H_2$ be independent $\theta/2$-percolations of $\mathcal{G}$, i.e., with distribution $\{\mathcal{G}^{\theta/2}\mid V, w_V\}$, and consider the corresponding iterative cost construction $I_H = ((H_i,\mathcal{E}_i^H,\mathcal{F}_i,\mathcal{U}_i)\colon i = 1,2)$ on $H_1,H_2$ as in Proposition [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}, and let $\mathcal{A}_{I_H}(V, w_V, E_1)$ be the event that the first round returns the edge set $\mathcal{E}_1^H = E_1$. Then Proposition [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"} followed by a union bound gives $$\begin{aligned} \begin{split}\label{eq:hierarchy-final-weights-multi} &\mathbb{P}\big(\mathcal{X}_{high\textnormal{-}hierarchy}(R,\eta,y_0,y_1) \mid V,w_V\big)\ge \mathbb{P}\big(I_{G'} \textnormal{ succeeds}\mid V,w_V\big) \\ &\quad\quad \ge \mathbb{P}\big(\mathcal{E}_1^H \neq \texttt{None}\mid V,w_V\big) \cdot \min_{E_1 \neq \mathtt{None}} \mathbb{P}\big(\mathcal{E}_2^H \neq \mathtt{None} \mid \mathcal{A}_{I_H}(V, w_V,E_1)\big) \\ &\quad\quad \ge 1-\mathbb{P}\big(\mathcal{E}_1^H = \texttt{None}\mid V,w_V\big)- \max_{E_1 \neq \mathtt{None}} \mathbb{P}\big(\mathcal{E}_2^H = \mathtt{None} \mid \mathcal{A}_{I_H}(V,w_V, E_1)\big). \end{split}\end{aligned}$$ The event $\mathcal{E}_1^H \ne \texttt{None}$ is equivalent that the graph $H_1$ contains a $(\gamma,3\underline{w}^{3\mu}\xi^{\eta},\underline{w},2)$-hierarchy $\mathcal{H}_{low}:=\{\tilde y_\sigma\}$ of depth $R$ fully contained in $\mathcal{N}$ with first level $\mathcal{L}_1 = \{y_0,y_1\}$. Since $H_1\sim \{\mathcal{G}^{\theta/2}\mid V, w_V\}$ is a CIRG, and since the conditions here are stronger than those in Lemma [Lemma 41](#lem:hierarchy-intermediate-weights){reference-type="ref" reference="lem:hierarchy-intermediate-weights"}, all requirements of Lemma [Lemma 41](#lem:hierarchy-intermediate-weights){reference-type="ref" reference="lem:hierarchy-intermediate-weights"} hold with $\theta$ replaced by $\theta/2$, so the first error term in [\[eq:hierarchy-final-weights-multi\]](#eq:hierarchy-final-weights-multi){reference-type="eqref" reference="eq:hierarchy-final-weights-multi"} is at most $\mathrm{err}_{\xi,\delta}$ in [\[eq:hierarchy-intermediate-prob\]](#eq:hierarchy-intermediate-prob){reference-type="eqref" reference="eq:hierarchy-intermediate-prob"}. It remains to upper-bound the second error term in [\[eq:hierarchy-final-weights-multi\]](#eq:hierarchy-final-weights-multi){reference-type="eqref" reference="eq:hierarchy-final-weights-multi"}. Let $$\begin{aligned} q^\star:= \left\lceil\frac{\log(d/\delta)}{\log(1/(\tau-2+2d\tau\delta))}\right\rceil,\label{eq:q-star}\end{aligned}$$ and for each $v\in\mathcal{N}$ let $\mathcal{A}_{\mathrm{path}}(v)$ be the event that $H_2$ contains a path $\pi_{v,v'}$ from $v$ to some vertex $v'\in\mathcal{N}\cap (B_{4q^\star \xi^{\gamma^{R-1}}}(v) \times [\overline{w},4\overline{w}])$ with cost $\mathcal{C}_2(\pi_{v,v'}) \le q^\star \overline{w}^{4\mu}\xi^{\eta}$. The value $c_H$ in [\[eq:c_H\]](#eq:c_H){reference-type="eqref" reference="eq:c_H"} is chosen so that $4q^\star \le (c_H-2)/2$ and $q^\star \le (c_H-3)/2$ both hold; thus conditioned on $\mathcal{A}_{I_H}(V,w_V, E_1)$, the event $\mathcal{E}_2^H=\texttt{None}$ occurs only if for some $\sigma\in\{0,1\}^R$ the event $\neg\mathcal{A}_{\mathrm{path}}(\tilde y_{\sigma})$ occurs. There are $2^R$ strings $\sigma\in\{0,1\}^R$, and all the events in $\mathcal{A}_{I_H}(V, w_V, E_1)$ are contained in the $\sigma$-algebra generated by $H_1$, which is independent of $H_2$ given $V, w_V$. So, by a union bound, $$\begin{aligned} \max_{E_1 \neq \mathtt{None}} \mathbb{P}\big(\mathcal{E}_2^H = \mathtt{None} \mid \mathcal{A}_{I_H}(V,w_V,E_1)\big) &\le 2^R\cdot\max_{\substack{\sigma \in \{0,1\}^{R}\\E_1 \neq \mathtt{None}}}\mathbb{P}\big(\neg\mathcal{A}_{\mathrm{path}}(\tilde y_{\sigma}) \mid \mathcal{A}_{I_H}(V,w_V,E_1)\big)\nonumber\\ &\le 2^R\cdot\max_{\tilde y_{\sigma}\neq \mathtt{None}}\mathbb{P}\big(\neg\mathcal{A}_{\mathrm{path}}(\tilde y_{\sigma})\mid V,w_V\big), \label{eq:hierarchy-final-weights-3}\end{aligned}$$ where the maximum is taken over all possible values of $\tilde y_{\sigma}$ in non-`None` $E_1$. To bound [\[eq:hierarchy-final-weights-3\]](#eq:hierarchy-final-weights-3){reference-type="eqref" reference="eq:hierarchy-final-weights-3"}, we apply Claim [Claim 33](#claim:cheap-path-to-larger-weight){reference-type="ref" reference="claim:cheap-path-to-larger-weight"} with $G'=H_2$, $\theta$ replaced by $\theta/2$, $K=2\overline{w}$, $M=2\underline{w}$, $D=4\xi^{\gamma^{R-1}}$, $U=\overline{w}^{4\mu}\xi^{\eta}$, $y_0=y_{\sigma}$, and all other variables taking their present values. Using $\underline{w},\overline{w}$ from [\[eq:ulw-in-prop\]](#eq:ulw-in-prop){reference-type="eqref" reference="eq:ulw-in-prop"} and [\[eq:c_H\]](#eq:c_H){reference-type="eqref" reference="eq:c_H"}, we compute $$\frac{\log K}{\log M} = \frac{\log (2\overline{w})}{\log (2\underline{w})}= \frac{\log 2 + \tfrac12\gamma^{R-1}d\log\xi}{\log 2 + \gamma^{R-1}\delta\log\xi}\le 1 + \frac{d}{2\delta} \le \frac{d}{\delta};$$ and therefore $q$ from [\[eq:cond-q\]](#eq:cond-q){reference-type="eqref" reference="eq:cond-q"} with these choices satisfies $$\begin{aligned} q &= \bigg\lceil\frac{\log\big(\log K/\log M\big)}{\log(1/(\tau-2+2d\tau\delta))} \bigg\rceil \le \bigg\lceil\frac{ \log(d/\delta)}{\log(1/(\tau-2+2d\tau\delta))} \bigg\rceil = q^\star. \end{aligned}$$ Hence the event $\mathcal{A}_{\pi(\tilde y_{\sigma})}$ in Claim [Claim 33](#claim:cheap-path-to-larger-weight){reference-type="ref" reference="claim:cheap-path-to-larger-weight"} is contained in $\mathcal{A}_{\mathrm{path}}(\tilde y_{\sigma})$. We now verify that the requirements of Claim [Claim 33](#claim:cheap-path-to-larger-weight){reference-type="ref" reference="claim:cheap-path-to-larger-weight"} hold in order of their appearance there. Whenever $E_1 \neq \mathtt{None}$, $\tilde y_{\sigma}$ lies in $\mathcal{N}$ with weight in $[\underline{w},4\underline{w}]=[M/2,2M]$ by construction, where $M=2\underline{w}>1$. Similarly, $K=2\overline{w}>1$ and $D=4\xi^{\gamma^{R-1}}>1$ by our choices. We check the requirement $U\ge K^{2\mu}$. By definition of $\overline{w}$ in [\[eq:c_H\]](#eq:c_H){reference-type="eqref" reference="eq:c_H"} and the choices $U=\overline{w}^{4\mu}\xi^{\eta}$ and $K=2\underline{w}$, we compute $$\begin{aligned} UK^{-2\mu} = \overline{w}^{4\mu}\xi^{\eta}(2\overline{w})^{-2\mu} = 2^{-2\mu} \overline{w}^{2\mu} \xi^{\eta},\end{aligned}$$ which is larger than $1$ (even if $\eta\!=\!0$) since $\mu>1$ and $\overline{w}\ge (\log\log\xi\sqrt{d})^{8d^2/\delta^2}$ and $\xi{\,\gg_{\star}\,}\delta$. Next, since $\delta{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ by hypothesis, $\overline{w}=\xi^{\gamma^{R-1}d/2} > \xi^{\gamma^{R-1}\delta}=\underline{w}$ and so $K \ge M$. Moreover, $K = 2\xi^{\gamma^{R-1}d/2} \le 4^{d/2} \xi^{\gamma^{R-1}d/2}=D^{d/2}$ also holds. Since $M= 2\underline{w}=2 \xi^{\gamma^{R-1}\delta} \ge 2(\log\log\xi\sqrt{d})^{16d/\delta}$, $\xi {\,\gg_{\star}\,}\theta,\delta,w_0$, and $M\le K\le D^{d/2}$, we have $K,M,D{\,\gg_{\star}\,}\theta,\delta,w_0$. Clearly $D = 4\xi^{\gamma^{R-1}} < \xi \le \xi\sqrt{d}$ since $\gamma<1$ and $\xi$ is large. Next, we check $(M/2)^{2/d} = (\xi^{\gamma^{R-1}\delta})^{2/d} > \xi^{\gamma^{R-1}\delta/d} \ge (\log\log\xi\sqrt{d})^{16/\delta}$ as required. Finally, if $\beta=\infty$ then we also need that $U(KM)^{-\mu}=\overline{w}^{4\mu}\xi^{\eta}(4\overline{w}\underline{w})^{-\mu}\ge 4^{-\mu}\overline{w}^{2\mu}\xi^{\eta}$ is sufficiently large. This holds even when $\eta\!=\!0$ since $\overline{w}{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$. Hence, all requirements of Claim [Claim 33](#claim:cheap-path-to-larger-weight){reference-type="ref" reference="claim:cheap-path-to-larger-weight"} are satisfied, and since $\theta$ changes to $\theta/2$ and $M=2\underline{w}=2\xi^{\gamma^{R-1}\delta}$ in [\[eq:weight-increasing-path-error\]](#eq:weight-increasing-path-error){reference-type="eqref" reference="eq:weight-increasing-path-error"}, [\[eq:hierarchy-final-weights-3\]](#eq:hierarchy-final-weights-3){reference-type="eqref" reference="eq:hierarchy-final-weights-3"} can be bounded as $$\begin{aligned} \begin{split}\label{eq:hierarchy-final-weights-4} \max_{E_1 \neq \mathtt{None}} \mathbb{P}\big(\mathcal{E}_2^H = \mathtt{None} &\mid \mathcal{A}_{I_H}(V, w_V, E_1)\big) \le 2^R\exp\Big({-}(\theta/2)2^{\delta}\xi^{\gamma^{R-1}\delta^2}\Big)\\ &\le 2^R\exp\Big({-}(\log\log\xi)^{15}\Big)\le \exp\Big({-}(\log\log\xi)^{14}\Big), \end{split}\end{aligned}$$ where we obtained the second row from the hypotheses $\xi^{\gamma^{R-1}\delta^2} \ge (\log\log\xi)^{16}$ and $\xi{\,\gg_{\star}\,}\theta$, and then from $2^R \le e^{R}$ and $R\le (\log \log \xi)^2$. Combining [\[eq:hierarchy-final-weights-4\]](#eq:hierarchy-final-weights-4){reference-type="eqref" reference="eq:hierarchy-final-weights-4"} with [\[eq:hierarchy-final-weights-multi\]](#eq:hierarchy-final-weights-multi){reference-type="eqref" reference="eq:hierarchy-final-weights-multi"} and recalling that the first error term there is at most $\exp(- (\log \log \xi)^{1/\sqrt{\delta}})$ finishes the proof of [\[eq:hierarchy-final-prob\]](#eq:hierarchy-final-prob){reference-type="eqref" reference="eq:hierarchy-final-prob"} since $\delta$ is small and $\xi$ is large. ◻ The hierarchy we constructed in Lemma [Lemma 42](#lem:hierarchy-final-weights){reference-type="ref" reference="lem:hierarchy-final-weights"} is a \"broken path\" formed by the bridge paths between the starting vertices $y_0, y_1\in \mathcal{N}$. The next proposition connects the endpoints of bridge-paths and constructs a connected path via common neighbours using Claim [Claim 34](#claim:common-neighbour){reference-type="ref" reference="claim:common-neighbour"}, but not yet between $y_0, y_1$, only between $y_{0_{R-1}1}$ and $y_{1_{R-1}0}$, the closest vertices to $y_0, y_1$ in the hierarchy constructed in Lemma [Lemma 42](#lem:hierarchy-final-weights){reference-type="ref" reference="lem:hierarchy-final-weights"}. Connecting $y_0$ to $y_{0_{R-1}1}$ and $y_1$ to $y_{1_{R-1}0}$ needs different techniques, since $y_0,y_1$ have typically lower weights than $\overline{w}$ in [\[eq:c_H\]](#eq:c_H){reference-type="eqref" reference="eq:c_H"}, see Section [6](#sec:endpoints){reference-type="ref" reference="sec:endpoints"}. **Proposition 43** (Path from hierarchy). *Consider Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and let $y_0,y_1\in\mathcal{N}$ with $|y_0-y_1|=\xi$. Let $z\in[0,d], \eta \ge 0$. Let $0<\delta {\,\ll_{\star}\,}\gamma,\eta,z, \textnormal{\texttt{par}}\xspace$ be such that $\Lambda(\eta,z) \ge 2\sqrt\delta$ and either $z=0$ or $\Phi(\eta,z)\ge\sqrt{\delta}$. Let $\xi{\,\gg_{\star}\,}\gamma,\eta,z,\theta,\delta,w_0$. Let $R\ge2$ be an integer satisfying $\xi^{\gamma^{R-1}} \ge (\log\log\xi\sqrt{d})^{16d/\delta^2}$ and $R \le (\log\log\xi)^2$, let $\overline{w}:= \xi^{\gamma^{R-1}d/2}$, and $c_H$ be as in [\[eq:c_H\]](#eq:c_H){reference-type="eqref" reference="eq:c_H"}. Let $\mathcal{X}_{high\textnormal{-}path}=\mathcal{X}_{high\textnormal{-}path}(R,\eta,y_0,y_1)$ be the event that $G'$ contains a path $\pi_{y_0^\star, y_1^\star}$ fully contained in $\mathcal{N}$ between some vertices $y^{\star}_0 \in \mathcal{N}\cap (B_{c_H\xi^{\gamma^{R-1}}}(y_0) \times [\overline{w}, 4\overline{w}])$ and $y^{\star}_1 \in \mathcal{N}\cap (B_{c_H\xi^{\gamma^{R-1}}}(y_1)\times [\overline{w}, 4\overline{w}])$ with cost $\mathcal{C}(\pi_{y_0^\star y_1^\star})\le c_H 2^R\overline{w}^{4\mu}\xi^{\eta}$ and $\mathrm{dev}_{y_0y_1}(\pi_{y_0^\star y_1^\star})\le 3c_H\xi^\gamma$. Then $$\begin{aligned} \label{eq:high-path} &\mathbb{P}\big(\mathcal{X}_{high\textnormal{-}path} \mid V,w_V\big) \ge 1 - 2\exp\big({-}(\log\log\xi)^{13}\big); \end{aligned}$$ under the convention that $\infty \cdot 0 = 0$, the statement is also valid when $\alpha=\infty$ or $\beta=\infty$.* *Proof.* To construct the path $\pi_{y_0^\star, y_1^\star}$, we again use two rounds of exposure. In the first round we apply Lemma [Lemma 42](#lem:hierarchy-final-weights){reference-type="ref" reference="lem:hierarchy-final-weights"} to get a $(\gamma,c_H\overline{w}^{4\mu}\xi^{\eta},\overline{w},c_H)$-hierarchy $\mathcal{H}_{high}:=\{y_\sigma\}$ of depth $R$ fully contained in $\mathcal{N}$ with first level $\{y_0, y_1\}$. In the second round, we use Claim [Claim 34](#claim:common-neighbour){reference-type="ref" reference="claim:common-neighbour"} to connect, via a common neighbour, each pair of level-$R$ siblings $y_{\sigma0}, y_{\sigma1}, \sigma \in \{0,1\}^{R-1}\setminus\{0_{R-1}, 1_{R-1}\}$. This yields a path between $y_{0_{R-1}1}=:y^{\star}_0$ and $y_{1_{R-1}0}=:y^{\star}_1$. We now define an iterative cost construction on $G'\sim \{\mathcal{G}^{\theta}|V,w_V\}$. Let $\mathcal{F}_1$ be the collection of all tuples of possible edges $e$ with $\mathrm{dev}_{y_0y_1}(e)\le 2c_H\xi^\gamma$ that could form the bridges of a $(\gamma,c_H\overline{w}^{4\mu}\xi^{\eta},\overline{w},c_H)$-hierarchy $\mathcal{H}_{high}=\{y_{\sigma}\}_{\sigma\in\{0,1\}^R}$ fully contained in $\mathcal{N}$ with first level $\{y_0, y_1\}$. Moreover, let $\mathcal{U}_1$ be the event that the costs of all $P_\sigma$ satisfy *(H[\[item:H3\]](#item:H3){reference-type="ref" reference="item:H3"})* of Definition [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"} with $U=c_H\overline{w}^{4\mu}\xi^{\eta}$, so that $\mathcal{H}_{high}$ is indeed a valid $(\gamma,c_H\overline{w}^{4\mu}\xi^{\eta},\overline{w},c_H)$-hierarchy with $\mathrm{dev}_{y_0y_1}(\mathcal{H}_{high})\le 2c_H\xi^\gamma$. For each $\sigma\in\{0,1\}^{R-1} \setminus \{0_{R-1}, 1_{R-1}\}$, define $\mathcal{J}(\sigma)$ to be the set of all potential paths $J_\sigma$ fully contained in $\mathcal{N}$ connecting $y_{\sigma0}$ and $y_{\sigma1}$ with $\mathrm{dev}_{y_0y_1}(J_\sigma)\le 3c_H\xi^\gamma$. Given $(V, w_V, E_1)$, call a tuple $\underline t$ of edges *admissible* if it contains exactly one such potential path from $\mathcal{J}(\sigma)$ for each $\sigma\in\{0,1\}^{R} \setminus \{0_{R-1}, 1_{R-1}\}$, and let $\mathcal{F}_2$ be the collection of all admissible tuples, with an arbitrary ordering. Let $\mathcal{U}_2$ be the event that each potential path in the chosen tuple is present in the graph under consideration, and has round-$2$ marginal cost $\textnormal{mcost}_2(J_\sigma) \le \overline{w}^{4\mu}$ in [\[eq:marginal-cost\]](#eq:marginal-cost){reference-type="eqref" reference="eq:marginal-cost"}. Following Def. [Definition 28](#def:iter-construct){reference-type="ref" reference="def:iter-construct"}[\[item:iter5\]](#item:iter5){reference-type="eqref" reference="item:iter5"} and Prop [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}, for $i=1,2$, we set $\mathcal{E}_i^{G'}$ to be the first tuple (in the ordering of $\mathcal{F}_i$) for which $\mathcal{U}_i$ occurs, and we set $\mathcal{E}_i^{G'}$ to $\mathtt{None}$ if no such tuple exists. This defines an iterative cost construction on $G'$, $I_{G'} = ((G',\mathcal{E}_1^{G'},\mathcal{F}_1,\mathcal{U}_1), (G',\mathcal{E}_2^{G'},\mathcal{F}_2,\mathcal{U}_2))$. If $I_{G'}$ succeeds, then there is a path $\pi_{y_0^\star,y_1^\star}\subseteq\mathcal{N}$ between $y_0^\star:=y_{0_{R-1}1}$ and $y_1^\star:=y_{1_{R-1}0}$ with $\mathcal{C}(\pi_{y_0^\star,y_1^\star}) \le c_H 2^R \overline{w}^{4\mu}\xi^{\eta}$ and $\mathrm{dev}_{y_0y_1}(\pi_{y_0^\star,y_1^\star})\le 3c_H\xi^\gamma$. Indeed, let us order the elements $y_{\sigma}$ of $\mathcal{H}_{high}$ lexicographically by their index $\sigma$, omitting $y_0$ and $y_1$, i.e. $$\begin{aligned} y_0^\star=y_{0_{R-1}1}, y_{0_{R-2}10}, y_{0_{R-2}11}, \ldots, y_{1_{R-2}00}, y_{1_{R-2}01}, y_{1_{R-1}0}=y_1^\star, \end{aligned}$$ and notice that $P_\sigma\in \mathcal{H}_{high}$ is a path between every consecutive pair of the form $y_{\sigma01}, y_{\sigma10}$ while $J_\sigma$ is a path between every consecutive pair of the form $y_{\sigma00}, y_{\sigma01}$ or $y_{\sigma{10}}, y_{\sigma11}$, so the concatenation forms a connected walk $\pi^+$. We then remove any cycles from $\pi^+$, passing to an arbitrary sub-path $\pi_{y_0^\star, y_1^\star}\in \mathcal{N}$. Since $\mathcal{H}_{high}$ is a $(\gamma,c_H\overline{w}^{4\mu}\xi^{\eta},\overline{w},c_H)$ hierarchy with first level $y_0, y_1$, by Definition [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"} *(H[\[item:H1\]](#item:H1){reference-type="ref" reference="item:H1"})* $w_{y_0^\star}, w_{y_1^\star} \in [\overline{w}, 4\overline{w}]$, and by *(H[\[item:H2\]](#item:H2){reference-type="ref" reference="item:H2"})*, the distances $|y_0-y_{0}^\star| \le c_H\xi^{\gamma^{R-1}}$ and $|y_1-y_{1}^\star| \le c_H\xi^{\gamma^{R-1}}$ both hold. Finally, since each edge of $\pi^+$ is contained in $\pi_{y_0^\star,y_1^\star}$ only once, its cost is at most $$\mathcal{C}(\pi_{y_0^\star,y_1^\star}) \le \sum_{e \in E(\pi^+)} \mathcal{C}(e) \le \sum_{\sigma\in\{0,1\}^{i}: 0\le i\le R-2} \mathcal{C}(E^-(P_\sigma)) + \sum_{\sigma\in\{0,1\}^{R-1} \setminus \{0_{R-1}, 1_{R-1}\}} \textnormal{mcost}_2(J_\sigma).$$ The cost of each $E^-(P_\sigma)$ is at most $c_H \overline{w}^{4\mu}\xi^{\eta}$ by $\mathcal{H}_{high}$ (see *(H[\[item:H3\]](#item:H3){reference-type="ref" reference="item:H3"})*), and $\textnormal{mcost}_2(J_{\sigma}) \le \overline{w}^{4\mu}$ by construction; since $c_H \ge 1$ it follows that $$\mathcal{C}(\pi_{y_0^\star,y_1^\star}) \le (2^{R-1}-1) c_H\overline{w}^{4\mu}\xi^{\eta} + (2^{R-1}-2)\overline{w}^{4\mu}< c_H2^R\overline{w}^{4\mu}\xi^{\eta},$$ as required by $\mathcal{X}_{high\textnormal{-}path}$. The deviation bound $3c_H\xi^\gamma$ also holds since it holds individually for all $J_\sigma$ and it holds for $\mathcal{H}_{high}$ already by Lemma [Lemma 42](#lem:hierarchy-final-weights){reference-type="ref" reference="lem:hierarchy-final-weights"}. Let $H_1,H_2$ again be independent $\{\mathcal{G}^{\theta/2}\mid V, w_V\}$, and let $I_H = ((H_i,\mathcal{E}_i^H,\mathcal{F}_i,\mathcal{U}_i)\colon i = 1,2)$ be the corresponding iterative cost construction on $H_1,H_2$ in Proposition [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"}, and let $\mathcal{A}_{I_H}(V, w_V, E_1)$ be the event that the first round returns the edge set $\mathcal{E}_1^H = E_1$. Then Proposition [Proposition 29](#prop:multi-round-exposure){reference-type="ref" reference="prop:multi-round-exposure"} followed by a union bound gives similarly to [\[eq:hierarchy-final-weights-multi\]](#eq:hierarchy-final-weights-multi){reference-type="eqref" reference="eq:hierarchy-final-weights-multi"} that $$\begin{aligned} \begin{split}\label{eq:path-from-hierarchy-multi} \mathbb{P}(\mathcal{X}_{high\textnormal{-}path} &\mid V,w_V) \ge \mathbb{P}\big(I_{G'} \textnormal{ succeeds} \mid V,w_V\big) \\ &\ge 1-\mathbb{P}\big(\mathcal{E}_1^H = \texttt{None}\mid V,w_V\big)\ - \max_{E_1 \neq \mathtt{None}} \mathbb{P}\big(\mathcal{E}_2^H = \mathtt{None} \mid \mathcal{A}_{I_H}(V,w_V,E_1)\big). \end{split} \end{aligned}$$ We bound both errors on the rhs. The event $\mathcal{E}_1^H \neq \texttt{None}$ can be bounded using Lemma [Lemma 42](#lem:hierarchy-final-weights){reference-type="ref" reference="lem:hierarchy-final-weights"} with $\theta$ replaced by $\theta/2$, since $H_1\sim \{\mathcal{G}^{\theta/2}\mid V, w_V\}$. All the requirements of Lemma [Lemma 42](#lem:hierarchy-final-weights){reference-type="ref" reference="lem:hierarchy-final-weights"} are fulfilled by hypothesis, so the first term on the rhs is at most $\exp\big({-}(\log\log\xi)^{13}\big)$ by [\[eq:hierarchy-final-prob\]](#eq:hierarchy-final-prob){reference-type="eqref" reference="eq:hierarchy-final-prob"}. It remains to upper-bound the second term in [\[eq:path-from-hierarchy-multi\]](#eq:path-from-hierarchy-multi){reference-type="eqref" reference="eq:path-from-hierarchy-multi"}. For each $x_0,x_1\in \mathcal{N}$, let $\widetilde \mathcal{A}_{x_0\star x_1}$ be the event that $H_2$ contains a two-edge path $x_0vx_1 \subseteq\mathcal{N}$ of cost at most $\overline{w}^{4\mu}$ with $|x_0-v| \le c_H\xi^\gamma$. If $\mathrm{dev}_{y_0y_1}(x_0)\le 2c_H\xi^\gamma$, this implies $\mathrm{dev}_{y_0y_1}(v)\le 3c_H\xi^\gamma$. Hence, conditioned on $\mathcal{A}_{I_H}(V,w_V,E_1)$, the event $\mathcal{E}_2^H=\texttt{None}$ occurs only if for some $\sigma\in\{0,1\}^{R-1} \setminus \{0_{R-1}, 1_{R-1}\}$ the event $\neg\widetilde \mathcal{A}_{y_{\sigma0}\star y_{\sigma1}}$ occurs. Since all the events in $\mathcal{A}_{I_H}(V, w_V, E_1)$ are contained in the $\sigma$-algebra generated by $H_1$, which is independent of $H_2$ given $V, w_V$, we get by a union bound that $$\begin{aligned} \begin{split}\label{eq:path-from-hierarchy-3} &\max_{E_1 \neq \mathtt{None}} \mathbb{P}\big(\mathcal{E}_2^H = \mathtt{None} \mid \mathcal{A}_{I_H}(V, w_V,E_1)\big) \\ &\qquad\qquad\le (2^{R-1}-2)\cdot\max_{\substack{\sigma \in \{0,1\}^{R-1} \setminus \{0_{R-1}, 1_{R-1}\}\\E_1 \neq \mathtt{None}}}\mathbb{P}\big(\neg\widetilde \mathcal{A}_{y_{\sigma0}\star y_{\sigma1}} \mid \mathcal{A}_{I_H}(V, w_V, E_1)\big) \\ &\qquad\qquad\le (2^{R-1}-2)\cdot \max_{y_{\sigma0}, y_{\sigma1}\neq \mathtt{None}}\mathbb{P}\big(\neg\widetilde \mathcal{A}_{y_{\sigma0}\star y_{\sigma1}} \mid V, w_V\big), \end{split} \end{aligned}$$ where the maximum is taken over all possible values of $y_{\sigma0},y_{\sigma1}$ occurring in non-`None` $E_1$. To bound [\[eq:path-from-hierarchy-3\]](#eq:path-from-hierarchy-3){reference-type="eqref" reference="eq:path-from-hierarchy-3"}, we observe $|y_{\sigma0}-y_{\sigma1}|\le c_H \xi^{\gamma^{R-1}}$ and $w_{y_{\sigma0}}, w_{y_{\sigma1}} \in [\overline{w},4\overline{w}]$ when $\sigma\in\{0,1\}^R\setminus \{0_{R-1}, 1_{R-1}\}$, by Def. [Definition 36](#def:hierarchy){reference-type="ref" reference="def:hierarchy"} *(H[\[item:H2\]](#item:H2){reference-type="ref" reference="item:H2"})* and *(H[\[item:H1\]](#item:H1){reference-type="ref" reference="item:H1"})*, since $\mathcal{H}_{high}$ is a $(\gamma,c_H\overline{w}^{4\mu}\xi^{\eta},\overline{w},c_H)$ hierarchy. Thus we apply Claim [Claim 34](#claim:common-neighbour){reference-type="ref" reference="claim:common-neighbour"} with $G'$ replaced by $H_2$, $\theta$ replaced by $\theta/2$, $D=\xi^{\gamma^{R-1}}$, $x_0=y_{\sigma0}$, $x_1=y_{\sigma1}$ and all other variables taking their present values. We verify that the requirements of Claim [Claim 34](#claim:common-neighbour){reference-type="ref" reference="claim:common-neighbour"} all hold in order of their appearance there. $\delta {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ by hypothesis and $D=\xi^{\gamma^{R-1}} \ge (\log\log\xi\sqrt{d})^{16d/\delta^2} \ge(\log\log\xi\sqrt{d})^{16/\delta}$ by assumption, so in particular $D{\,\gg_{\star}\,}c_H, \delta$ and $D \ge w_0^{2/d}$ since $\xi {\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace,\delta,w_0$. Also, clearly $\xi^{\gamma^{R-1}} \le \xi\sqrt{d}$. Next, we check the distance and weights of $y_{\sigma0},y_{\sigma1}$. Since $E_1\neq \texttt{None}$, they must lie in $\mathcal{N}$, and as argued already, satisfy $|y_{\sigma0}-y_{\sigma1}| \le c_H\xi^{\gamma^{R-1}}=c_HD$ and $w_{y_{\sigma0}}, w_{y_{\sigma1}} \in [\overline{w},4\overline{w}] = [D^{d/2}, 4D^{d/2}]$. Finally, the cost-bound in Claim [Claim 34](#claim:common-neighbour){reference-type="ref" reference="claim:common-neighbour"} is $D^{2\mu d}=\xi^{2\mu d \gamma^{R-1}}=\overline{w}^{4\mu}$ exactly as we require it here, and the vertex $v$ satisfies $|x_0-v|\le D = \xi^{\gamma^{R-1}}\le c_H\xi^\gamma$, also as required. Claim [Claim 34](#claim:common-neighbour){reference-type="ref" reference="claim:common-neighbour"} applies and [\[eq:path-from-hierarchy-3\]](#eq:path-from-hierarchy-3){reference-type="eqref" reference="eq:path-from-hierarchy-3"} can be bounded as $$\begin{aligned} \max_{E_1 \neq \mathtt{None}} \mathbb{P}\big(\mathcal{E}_2^H = \mathtt{None} \mid \mathcal{A}_{I_H}(V, w_V, E_1)\big) &\le 2^{R-1}\exp\Big({-}(\theta^2/4) \xi^{\gamma^{R-1}(3-\tau-2\delta)d/2}\Big) \\ &\le 2^{R-1}\exp\big({-}(\log\log\xi)^{15/\delta}\big), \end{aligned}$$ where for the second row we used that $\delta {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, so $(3-\tau-2\delta)d/2 \ge \delta$ and by hypothesis $\xi^{\gamma^{R-1}\delta } \ge (\log\log\xi)^{16/\delta}$ and $\xi {\,\gg_{\star}\,}\theta$. This, together with that the first error term in [\[eq:path-from-hierarchy-multi\]](#eq:path-from-hierarchy-multi){reference-type="eqref" reference="eq:path-from-hierarchy-multi"} was at most $\exp\big({-}(\log\log\xi)^{13}\big)$ concludes the proof of [\[eq:high-path\]](#eq:high-path){reference-type="eqref" reference="eq:high-path"}. ◻ ## Cost optimisation of the constructed paths {#sec:choices} In this section we apply Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} and optimise the cost of the path $\pi_{y_0^\star y_1^\star}$ constructed there, yielding either polylogarithmic (Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"}) or polynomial cost-distances (Corollary [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"}). The cost of $\pi_{y_0^\star y_1^\star}$ will dominate the cost of the eventual path between $0, x$. These corollaries are rather immediate: we choose appropriate values of $\gamma,\eta,z,R$, apply Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"}, and read off the cost of $\pi_{y_0^\star y_1^\star}$. There are four possible optimal choices of $\gamma,\eta, z,R$ depending on the model parameters, and verifying that the conditions of Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"} and Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} hold for these choices and calculating the resulting path's cost requires some work. Thus, we defer a formal proof of Corollaries [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} and [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"} to Appendix [8](#app:path-proofs){reference-type="ref" reference="app:path-proofs"}, and instead focus on why these four optimisers arise and what they mean on a qualitative level. Thus, in Prop. [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"}, disregarding constant factors, our goal is to minimise the cost bound $\mathcal{C}(\pi_{y_0^\star y_1^\star})\le 2^R\overline{w}^{4\mu}|x|^\eta = 2^R|x|^{2\gamma^{R-1}d\mu + \eta}$ by choosing $\gamma, \eta,z,R$ optimally. To put our choices into context, we first summarise the construction of the path in Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} (drawing on the proofs of Lemmas [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"}--[Lemma 42](#lem:hierarchy-final-weights){reference-type="ref" reference="lem:hierarchy-final-weights"}). We first embed a hierarchy $\{y_\sigma\}_{\sigma \in \{0,1\}^R}$ by embedding its bridge-paths. Each bridge of $\{y_\sigma\}_{\sigma \in \{0,1\}^R}$ from $y_{\sigma 01}$ to $y_{\sigma 10}$ contains a single long edge obtained via Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"} from a vertex near $y_{\sigma 01}$ to a vertex near $y_{\sigma 10}$ (see Lemma [Lemma 40](#lem:triple-edge-bridge){reference-type="ref" reference="lem:triple-edge-bridge"}), which is then extended to first a 3-edge then a multiple-edge bridge-path using Claims [Claim 32](#claim:cheap-edge-to-nice-vertex){reference-type="ref" reference="claim:cheap-edge-to-nice-vertex"} and [Claim 33](#claim:cheap-path-to-larger-weight){reference-type="ref" reference="claim:cheap-path-to-larger-weight"}. We informally refer to this long edge as a *bridging edge*. Having obtained the hierarchy, we then join pairs of level-$R$ siblings (which we informally call *gaps*) using Claim [Claim 34](#claim:common-neighbour){reference-type="ref" reference="claim:common-neighbour"}; combined with the bridge-paths, this forms $\pi_{y_0^\star y_1^\star}$. The roles of $\gamma, \eta,z, R$ in the construction are as follows: $R$ is the depth of the hierarchy, and hence controls the number $2^R$ of gaps, while $\gamma$ controls the Euclidean length of the gaps and hence also the cost of joining them, with the total cost of joining a single gap being roughly $\overline{w}^{4\mu} = |x|^{2d\mu\gamma^{R-1}}$. The exponent $\eta$ controls the cost of bridge-paths in the hierarchy. When $\eta > 0$ the total cost is dominated by the cost of the very first bridge-edge of length $\Theta(|x|)$ of cost $|x|^\eta$ and when $\eta = 0$ the total cost of all bridge-paths is negligible compared to the cost of joining gaps. Finally, $z$ controls the weights of the endpoints of bridging edges relative to the distance they span: when $z>0$, each endpoint of a round-$i$ bridging edge has weight roughly $(|x|^{d\gamma^{i-1}})^{z/2}$ (see Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"}); when $z=0$, these weights are subpolynomial in $|x|$ and the same across all rounds. The final cost of $\pi_{y_0^\star y_1^\star}$ is thus roughly $2^R|x|^{2\mu\gamma^{R-1}} + |x|^{\eta}$, which we have bounded above by roughly $2^R|x|^{2\mu\gamma^{R-1}}|x|^\eta$ in Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} for convenience. From the many constraints in Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"}, the following are relevant when optimising the cost of the path. The requirement $\Lambda(\eta,z)>0$ ensures that low-cost bridging edges exist (Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"}). The requirement that either $z=0$ or $\Phi(\eta, z) > 0$ ensures that among the many potential bridging edges a few can be extended to low-cost 3-edge bridge-paths in Lemma [Lemma 41](#lem:hierarchy-intermediate-weights){reference-type="ref" reference="lem:hierarchy-intermediate-weights"}. The requirement $\gamma<1$ ensures that boxes where we search for the bridging edge shrink in size, while $z \le d$ is a formal requirement for applying Claim [Claim 31](#claim:cheap-bridge){reference-type="ref" reference="claim:cheap-bridge"} to find bridging edges, which we tolerate because increasing $z$ above $d$ will never be optimal. Heuristically, the effect of increasing $z$ is to increase the probability that a given bridging edge exists at the price of increasing its expected cost; at $z=d$ the existence probability is already in the interval $[\underline{c}, \overline{c}]$ and cannot be increased further, (however the penalty would increase and the number of combinatorial option decrease by increasing $z$, which is never optimal). The other constraints of Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} and Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"} (such as $2d\gamma < \tau-1$ and $R \le (\log \log |x|)^2$) never turn out to be tight for optimal choices of $\eta, R,\gamma, z$. Recall $\mu_{\log}, \mu_{\mathrm{pol}}$ from [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}. corollaryComputationsPolyLog[\[cor:computations-polylog\]]{#cor:computations-polylog label="cor:computations-polylog"} Consider $1$-FPP in Definition [Definition 2](#def:1-FPP){reference-type="ref" reference="def:1-FPP"} on the graphs IGIRG or SFP satisfying the assumptions given in [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} with $d\ge 1, \alpha \in (1,\infty], \tau\in(2,3), \mu>0$. Let $\underline{c},\overline{c},h,L,c_1,c_2,\beta$ be as in [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"}, we allow $\beta = \infty$ and/or $\alpha=\infty$. Let $q,\varepsilon,\zeta\in(0,1)$, let $0<\delta{\,\ll_{\star}\,}\varepsilon,q,\textnormal{\texttt{par}}\xspace$, and let $w_0>1$. Fix a realisation $(V, w_V)$ of $\widetilde{\cal V}$. Let $x \in V$ with $|x|{\,\gg_{\star}\,}q, \delta,\varepsilon,\zeta, w_0, \textnormal{\texttt{par}}\xspace$. Let $Q$ be a cube of side length $|x|$ containing $0$ and $x$, and assume that $(V, w_V)$ is such that $Q$ contains a weak $(\delta/4,w_0)$-net $\mathcal{N}$ with $0,x \in \mathcal{N}$ given in Definition [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"}. Let $G \sim \{\mathcal{G}\mid V, w_V\}$. Let $\mathcal{X}_{\mathrm{polylog}}(0,x)$ be the event that $G$ contains a path $\pi$, fully contained in $\mathcal{N}$, with endpoints say $y_0^\star, y_x^\star$, with the following properties: $$\begin{aligned} &w_{y_0^\star}, w_{y_x^\star} \in [\overline{w},4\overline{w}], \quad \mbox{where} \quad \overline{w}\in [\log\log|x|, (\log|x|)^\varepsilon],\label{eq:weight-crit-cor1}\\ &y_0^\star \in B_{\overline{w}^{3/d}}(0) \qquad\qquad \mbox{and} \qquad y_x^\star \in B_{\overline{w}^{3/d}}(x) \label{eq:dist-crit-cor1},\\ &\mathcal{C}(\pi) \le (\log |x|)^{\Delta_0+\varepsilon}, \qquad \mbox{and} \qquad \mathrm{dev}_{0x}(\pi)\le \zeta |x|,\label{eq:cost-crit-cor1}\end{aligned}$$ where $\Delta_0$ is defined in  [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"}, [\[eq:alpha-infty-Delta_0\]](#eq:alpha-infty-Delta_0){reference-type="eqref" reference="eq:alpha-infty-Delta_0"} or [\[eq:beta-infty-Delta_0\]](#eq:beta-infty-Delta_0){reference-type="eqref" reference="eq:beta-infty-Delta_0"} depending on whether $\alpha,\beta<\infty$, $\alpha=\infty$ or $\beta=\infty$. If either $\alpha\in(1,2)$ or $\mu\in(\mu_{\mathrm{expl}},\mu_{\log})$ or both hold, then $\mathbb{P}(\mathcal{X}_{\mathrm{polylog}}(0,x) \mid V,w_V) \ge 1-q$. *Sketch of proof.* Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} covers the polylogarithmic regime, which corresponds to solutions where $\eta = 0$ is possible -- such solutions exists when either $\alpha\in(1,2)$ or $\mu<\mu_{\log}$. When $\eta=0$, the cost of the path $\pi_{y_0^\star,y_x^\star}$ is dominated by the cost $2^R|x|^{2\mu\gamma^{R-1}}$ of joining gaps. Given $\gamma$, this has minimum $2^{(1+o(1))R}$ when setting $R = (1-o(1))\log\log |x|/\log(1/\gamma)$. To minimise the cost further, we must therefore minimise $\gamma\in(0,1)$ subject to the constraints $z\in[0,d], \Lambda(0,z)>0$, and either $z=0$ or $\Phi(0,z)>0$. This problem turns out to have two potentially optimal solutions corresponding to two possible strategies for finding bridging edges, with the optimal choice depending on the values of $\alpha,\tau,\beta,\mu$. One possible solution -- which only exists when $\alpha\in(1,2)$ -- takes $\gamma = \alpha/2 + o(1)$ and $z=0$, so that bridging edges are unusually long-range edges between pairs of low-weight vertices, yielding total path cost $(\log |x|)^{\Delta_\alpha}$ with $\Delta_\alpha=1/(1-\log_2\alpha)$, see Claim [Claim 54](#claim:polylog-low-alpha){reference-type="ref" reference="claim:polylog-low-alpha"}. The other possible solution -- which only exists when $\mu<\mu_{\log}$ -- takes $\gamma = (\tau-1+\mu\beta)/2 + o(1)$ and $z=d$, so that bridging edges are unusually low-cost edges between pairs of high-weight vertices and the total path cost is $(\log x)^{\Delta_\beta}$ with $\Delta_\beta=1/(1-\log_2(\tau-1+\mu\beta))$, see Claim [Claim 55](#claim:polylog-low-mu){reference-type="ref" reference="claim:polylog-low-mu"}. The proof is in Appendix [8](#app:path-proofs){reference-type="ref" reference="app:path-proofs"}. ◻ **Remark 44**. If both $\alpha=\beta=\infty$, then the conditions of Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} cannot be satisfied. Indeed, when $\alpha=\infty$ then $\alpha\in (1,2)$ is not satisfied. Since $\mu_{\mathrm{expl}}= \mu_{\log} = 0$ by [\[eq:beta-infty-definitions\]](#eq:beta-infty-definitions){reference-type="eqref" reference="eq:beta-infty-definitions"} when $\alpha=\beta=\infty$, so $\mu\in(\mu_{\mathrm{expl}},\mu_{\log})$ can also not be satisfied. corollaryComputationsPolynomial[\[cor:computations-polynomial\]]{#cor:computations-polynomial label="cor:computations-polynomial"} Consider $1$-FPP in Definition [Definition 2](#def:1-FPP){reference-type="ref" reference="def:1-FPP"} on the graphs IGIRG or SFP satisfying the assumptions given in [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} with $d\ge 1, \alpha \in (1,\infty], \tau\in(2,3), \mu>0$. Let $\underline{c},\overline{c},h,L,c_1,c_2,\beta$ be as in [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"}, we allow $\beta = \infty$ and/or $\alpha=\infty$. Let $q,\varepsilon,\zeta\in(0,1)$, and let $0<\delta{\,\ll_{\star}\,}\varepsilon,q,\textnormal{\texttt{par}}\xspace$, and $w_0>1$. Fix a realisation $(V, w_V)$ of $\widetilde{\cal V}$. Let $x \in V$ with $|x|{\,\gg_{\star}\,}q,\delta,\varepsilon, \zeta,w_0,\textnormal{\texttt{par}}\xspace$. Let $Q$ be a cube of side length $|x|$ containing $0$ and $x$, and assume that $(V, w_V)$ is such that $Q$ contains a weak $(\delta/4,w_0)$-net $\mathcal{N}$ with $0,x \in \mathcal{N}$ given in Definition [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"}. Let $G \sim \{\mathcal{G}\mid V, w_V\}$. Let $\mathcal{X}_{\mathrm{pol}}(0,x)$ be the event that $G$ contains a path $\pi$, fully contained in $\mathcal{N}$, with endpoints say $y_0^\star, y_x^\star$, with the following properties: $$\begin{aligned} &w_{y_0^\star}, w_{y_x^\star} \in [\overline{w},4\overline{w}], \quad \mbox{where} \quad \overline{w}\in [\log\log|x|, |x|^\varepsilon],\label{eq:weight-crit-cor2}\\ &y_0^\star \in B_{\overline{w}^{3/d}}(0) \qquad\qquad \mbox{and} \qquad y_x^\star \in B_{\overline{w}^{3/d}}(x) \label{eq:dist-crit-cor2},\\ &\mathcal{C}(\pi) \le |x|^{\eta_0+\varepsilon}, \qquad \mbox{and} \qquad \mathrm{dev}_{0x}(\pi)\le \zeta |x|,\label{eq:cost-crit-cor2}\end{aligned}$$ where $\eta_0$ is defined in [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}, [\[eq:alpha-infty-definitions\]](#eq:alpha-infty-definitions){reference-type="eqref" reference="eq:alpha-infty-definitions"}, [\[eq:beta-infty-definitions\]](#eq:beta-infty-definitions){reference-type="eqref" reference="eq:beta-infty-definitions"}, or [\[eq:alpha-beta-infty-definitions\]](#eq:alpha-beta-infty-definitions){reference-type="eqref" reference="eq:alpha-beta-infty-definitions"} depending on $\alpha,\beta<\infty$, $\alpha=\infty$, $\beta=\infty$, or $\alpha=\beta=\infty$. If both $\alpha>2$ and $\mu \in(\mu_{\log}, \mu_{\mathrm{pol}}]$ hold then $\mathbb{P}(\mathcal{X}_{\mathrm{pol}}(0,x) \mid V,w_V) \ge 1-q$. *Sketch of proof.* Corollary [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"} covers the polynomial regime, which corresponds to solutions where only $\eta > 0$ is possible, i.e. when $\alpha>2$ and $\mu>\mu_{\log}$. Here, on taking $R$ to be a suitably large constant, the cost bound on the path $2^R|x|^{2\mu\gamma^{R-1}}|x|^\eta=|x|^{\eta+o(1)}$, which is roughly the cost of the very first bridging edge. Our goal is thus to minimise $\eta$ under the constraints that $\Lambda(\eta,z)>0$, $z\in[0,d]$, $\gamma\in(0,1)$, and either $z=0$ or $\Phi(\eta,z)>0$. [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"} and [\[eq:Phi-def\]](#eq:Phi-def){reference-type="eqref" reference="eq:Phi-def"} show that both $\Phi$ and $\Lambda$ are increasing functions of $\gamma$; thus we can take $\gamma = 1-o(1)$. As in the polylogarithmic regime, this minimisation problem has two potentially optimal solutions. One possible solution -- which exists when $\mu\le\mu_{\mathrm{pol},\beta}$ -- takes $z=d$ and gives $\eta = \mu d-(3-\tau)d/\beta + o(1)$, so that bridging edges are unusually low-cost edges between pairs of high-weight vertices. The total path cost is then $|x|^{\eta_\beta+o(1)}$ with $\eta_\beta=d(\mu-(3-\tau)/\beta)$ (see Claim [Claim 58](#claim:polynomial-small-mu){reference-type="ref" reference="claim:polynomial-small-mu"}). The other possible solution -- which exists when $\mu\le \mu_{\mathrm{pol},\alpha}$ -- takes $z$ to be as small as possible, so that bridging edges are unusually long-range edges between pairs of relatively low-weight vertices. However, when $\alpha>2$, there are no bridging-edges between constant weight vertices, and the minimal $z$ where bridging-edges appear is $z = d(\alpha-2)/(\alpha - (\tau - 1)) + o(1)=1/\mu_{\mathrm{pol},\alpha}+o(1)$, i.e., between vertices of weight $|x|^{1/(2\mu_{\mathrm{pol},\alpha})+o(1)}$. This gives cost-exponent $\eta_\alpha := \mu/\mu_{\mathrm{pol},\alpha}$ and total cost $|x|^{\mu/\mu_{\mathrm{pol},\alpha} + o(1)}$ (see Claim [Claim 59](#claim:polynomial-large-mu){reference-type="ref" reference="claim:polynomial-large-mu"}). Whenever a solution exists among the above two possibilities, it gives an exponent $\eta$ at most $1$. So, whenever $\mu\le \max\{\mu_{\mathrm{pol},\alpha}, \mu_{\mathrm{pol},\beta}\}$, we obtain the cost bound $|x|^{\min\{\eta_\beta, \eta_\alpha\}+o(1)}$, which gives the definition of $\eta_0$ in [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}. The proof is in Appendix [8](#app:path-proofs){reference-type="ref" reference="app:path-proofs"}. ◻ The goal of this section has been to prove Corollaries [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} and [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"}; now that this has been achieved, notation internal to this section will no longer be used. # Connecting the endpoints $0, x$ to the path {#sec:endpoints} The final step is to connect the initial vertices $0$ and $x$ to the respective endpoints $y_0^\star$ and $y_x^\star$ of the path constructed in Section [5](#sec:hierarchy){reference-type="ref" reference="sec:hierarchy"}. For $d\ge 2$, we consider the graph $G_M$ induced by the vertices of weight in $[M,2M]$ for some large constant $M$. By a result in our companion paper, this graph has an infinite component $\mathcal{C}_\infty^M$ where cost-distances scale linearly with the Euclidean distance. We then connect $0, y_0^\star$ to respective nearby vertices $u_0, u_0^\star\in G_M$, and then use that the cost-distance $d_{\mathcal{C}}(u_0, u_0^\star)=\Theta(|u_0-u_0^\star|$) within $G_M$. We then do the same for $y_x^\star$ and $x$. For $d=1$, we take a similar approach with larger value of $M$: here we need $M$ to depend on $|x|$ to guarantee that $G_M$ contains a large connected subgraph in the section between $0$ and $x$. The construction of the path $\pi_{y_0^\star, y_x^\star}$ already revealed information about the graph, in particular the vertices $y_0^\star, y_x^\star$ are the outcomes of a selection procedure that might influence the graph around them. Thus, we ensure that cost-distances are linear in $G_M$ simultaneously for all potential candidate vertices for $u_0$ and $u_0^\star$, see Lemma [Lemma 46](#lem:new-external){reference-type="ref" reference="lem:new-external"} below. When the graph is finite (e.g. GIRG $G_n$ in Def. [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}), we additionally use that (near)-shortest paths within $G_M$ have very small deviation from the straight line, so that when two vertices are not too close to the boundary of the box $Q_n$, the constructed path stays in $Q_n$. For this we use that the constructed paths have low deviation, see Def. [Definition 37](#def:deviation){reference-type="ref" reference="def:deviation"}. We define the setting of this section. As in Theorems [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"}--[\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"}, we aim to bound $d_\mathcal{C}(0,x)$. **Setting 45**. Consider $1$-FPP in Definition [Definition 2](#def:1-FPP){reference-type="ref" reference="def:1-FPP"} on the graphs IGIRG or SFP satisfying the assumptions given in [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"} with $d\ge 1, \alpha \in (1,\infty], \tau\in(2,3), \mu>0$. Let $\underline{c},\overline{c},h,L,c_1,c_2,\beta$ be as in [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"}--[\[eq:F_L-condition\]](#eq:F_L-condition){reference-type="eqref" reference="eq:F_L-condition"}, we allow $\beta = \infty$ and/or $\alpha=\infty$. Let $G\sim \mathcal{G}$, let $\mathcal{F}_{0,x} := \{0,x \in \mathcal{V}\}$, and let $\mathcal{C}_\infty$ be the (unique) infinite component of $G$. Let $q \in (0,1)$, and let $M {\,\gg_{\star}\,}q,\textnormal{\texttt{par}}\xspace$. Let $I_M := [M,2M]$, and let $G_M=(\mathcal{V}_M,\mathcal{E}_M)$ be the subgraph of $G$ with vertices $\mathcal{V}_M:=\{(v,w_v)\in\widetilde \mathcal{V}\colon w_v\in I_M\}$ and edges $\mathcal{E}_M:=\{uv \colon u,v\in \mathcal{V}_M,\, \mathcal{C}(uv)\le M^{3\mu}\}$. Let $\mathcal{C}_\infty^M$ be the infinite component of $G_M$ if it exists and is unique, and define $\mathcal{C}_\infty^M:=\texttt{None}$ otherwise. The next claim is a technical necessity to remove conditioning on membership of $\mathcal{C}_\infty$ later. claimtwoinCinfty[\[claim:two-in-Cinfty\]]{#claim:two-in-Cinfty label="claim:two-in-Cinfty"} Consider Setting [Setting 45](#set:joining-common){reference-type="ref" reference="set:joining-common"}. There exists $\rho > 0$ such that for all distinct $a,b \in \mathbb{R}^d$, we have $\mathbb{P}(a,b \in \mathcal{C}_\infty \mid a,b \in \mathcal{V}) \ge \rho$. *Proof.* For $d \ge 2$, this is [@komjathy2022one2 Claim 3.10]. We provide a proof for $d=1$ on page in Appendix [9](#app:1d-endpoints){reference-type="ref" reference="app:1d-endpoints"}. ◻ The next lemma, that we prove in the companion paper [@komjathy2022one2], is the main tool of the section. We first need some definitions. We use the notation $\pi_{a,b}$ for a path between vertices $a$ and $b$. Let $r, \kappa, \zeta, C>0$ and $z\in \mathbb{R}^d$. We say that a set of vertices $\mathcal{H}\subseteq \mathcal{V}_M$ is *$r$-strongly dense* around $z\in \mathbb{R}^d$ in $\mathcal{V}_M$ if the following event holds: $$\label{eq:a-dense} \mathcal{A}_\mathrm{dense}(\mathcal{H}, \mathcal{V}_M, r,z):=\Big\{\forall y \in B_r(z): \big|B_{r^{1/3}}(y) \cap \mathcal{H}\big| \ge |B_{r^{1/3}}(y) \cap \mathcal{V}_M|/2\Big\}.$$ We say that a set of vertices $\mathcal{H}\subseteq \mathcal{C}_\infty^M$ shows *$r$-strongly $\kappa$-linear* distances with deviation $\zeta$ in $\mathcal{C}_\infty^M$ around $z\in \mathbb{R}^d$ if the following event holds: $$\label{eq:a-linear} \begin{aligned} \mathcal{A}_\mathrm{linear}(\mathcal{H}, \mathcal{C}_\infty^M,r,\kappa,\zeta,C, z):=&\Big\{\forall a \in \mathcal{B}_r(z) \cap \mathcal{H},\ \forall b \in \mathcal{H}: \exists\mbox{ a path } \pi_{a,b} \subseteq \mathcal{C}_\infty^M \mbox{ with}\\ &\quad\mathcal{C}(\pi_{a,b})\le \kappa |a-b|+C,\ \mathrm{dev}(\pi_{a,b})\le \zeta |a-b| + C\Big\}. \end{aligned}$$ Finally, we say that a set $\mathcal{H}$ is $(r,C)$-near to $z\in \mathcal{V}$ if the following event holds: $$\begin{aligned}\label{eq:a-near} \mathcal{A}_{\mathrm{near}}(\mathcal{H}, r,C,z):=&\Big\{\exists \mbox{ a path } \pi_{z,a} \mbox{ to some } a\in B_r(z)\cap \mathcal{H}\mbox{ with}\\ &\qquad \qquad\qquad\qquad\mathcal{C}(\pi_{z,a})\le C, \ \mathrm{dev}(\pi_{z,a})\le C \Big\}. \end{aligned}$$ **Lemma 46**. *Consider Setting [Setting 45](#set:joining-common){reference-type="ref" reference="set:joining-common"} and assume $d\ge 2$. Let $M,r_1,r_2,C,\kappa > 0$ and $q,\zeta \in (0,1)$. Whenever $C {\,\gg_{\star}\,}r_2$ and $r_1,r_2{\,\gg_{\star}\,}M,\zeta,q,\textnormal{\texttt{par}}\xspace$, and $\kappa{\,\gg_{\star}\,}M$, then a.s. $\mathcal{C}_\infty^M \neq \normalfont{\texttt{None}}$ and there is an infinite-sized vertex set $\mathcal{H}_\infty\subseteq \mathcal{C}_\infty^M$ determined by $(\widetilde \mathcal{V}, \mathcal{E}(G_M))$ so that $G_M[\mathcal{H}_\infty]$ is connected, and for all $z\in \mathbb{R}^d$, $$\begin{aligned} &\mathbb{P}(\mathcal{A}_\mathrm{dense}(\mathcal{H}_\infty,\mathcal{V}_M,r_1,z))\ge 1-q/10, \qquad\mathbb{P}(\mathcal{A}_\mathrm{near}(\mathcal{H}_\infty, r_2,C,z)\mid z\in \mathcal{C}_\infty) \ge 1-q/10, \label{eq:dense-near}\\ &\mathbb{P}(\mathcal{A}_{\mathrm{linear}}(\mathcal{H}_\infty, \mathcal{C}_\infty^M, r_2,\kappa, \zeta,C,z))\ge1-q/10.\label{eq:linear-again} \end{aligned}$$ The statement remains valid conditioned on $\mathcal{F}_{y,z}=\{y,z \in \mathcal{V}\}$; moreover, the constraints on $C,r,M,\kappa$ are uniform over $\{\mathcal{F}_{y,z}: y,z\in \mathbb{R}^d\}$.* *Sketch of proof.* In [@komjathy2022one2], we show that $\mathcal{H}_\infty$ exists, and is infinite and connected in $G_M$ in Corollary 3.9(ii). The $r_2$-strong $\kappa$-linearity comes from Corollary 3.9(iv) in [@komjathy2022one2] applied with $r_{3.9}=r_2$ and $C_{3.9}=C$, and the $(r_2,C)$-near property comes from in [@komjathy2022one2 Claim 3.11]. Moreover, we can apply  [@komjathy2022one2 Corollary 3.9(iii)] with $r_{3.9}=r_1$ to get the $r_1$-dense property with $(\log r_1)^2$ instead of $r_1^{1/3}$ and arbitrary density $1-\varepsilon$ instead of $1/2$ in [\[eq:a-dense\]](#eq:a-dense){reference-type="eqref" reference="eq:a-dense"}. This is a strictly stronger statement since we can cover any ball of radius $r_1^{1/3}$ with balls of radius $(\log r_1)^2$, at the cost of increasing the fraction $\varepsilon$ of non-covered vertices by a $d$-dependent factor. ◻ For $d=1$, the graph $G_M$ does not have an infinite component for any $M$ and the proof techniques in Lemma [Lemma 46](#lem:new-external){reference-type="ref" reference="lem:new-external"} do not apply. Instead, on page  in Appendix [9](#app:1d-endpoints){reference-type="ref" reference="app:1d-endpoints"} we directly prove the following analogous statement for $G_M$ in a *finite* interval. lemmanewexternaloned[\[lem:new-external-1d\]]{#lem:new-external-1d label="lem:new-external-1d"} Consider Setting [Setting 45](#set:joining-common){reference-type="ref" reference="set:joining-common"} with $d = 1$. Let $q,\zeta \in (0,1)$, $r_M := e^{(\log M)^2}$, $\kappa_M := 2M^{3\mu+2}$ and $C_M := M^{2(\tau-1)+3\mu}$. Let $z \in \mathbb{R}^d$, and $\mathcal{H}_M := B_{2r_M}(z) \cap \mathcal{V}_M$. Then whenever $M {\,\gg_{\star}\,}q,\textnormal{\texttt{par}}\xspace$, $$\begin{aligned} &\mathbb{P}(\mathcal{A}_\mathrm{dense}(\mathcal{H}_M,\mathcal{V}_M,r_M,z)) = 1, \qquad\mathbb{P}(\mathcal{A}_\mathrm{near}(\mathcal{H}_M, C_M,C_M,z)\mid z\in \mathcal{C}_\infty) \ge 1-q/10, \label{eq:dense-near-1d}\\ &\mathbb{P}(\mathcal{A}_{\mathrm{linear}}(\mathcal{H}_M, \mathcal{H}_M, r_M,\kappa_M, 0,2\kappa_M,z))\ge 1-q/10.\label{eq:linear-again-1d} \end{aligned}$$ The statement remains valid conditioned on $\mathcal{F}_{y,z}=\{y,z \in \mathcal{V}\}$; moreover, the constraints on $r_M$ are uniform over $\{\mathcal{F}_{y,z}: y,z\in \mathbb{R}^d\}$. We use the next claim to connect the endpoints $y_0^\star, y_x^\star$ of the path $\pi_{y_0^\star, y_x^\star}$ in Corollaries [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} and [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"} to $\mathcal{H}_\infty \subseteq \mathcal{C}_\infty^M$ from Lemma [Lemma 46](#lem:new-external){reference-type="ref" reference="lem:new-external"} (when $d \ge 2$) and $\mathcal{H}_{M}$ from Lemma [\[lem:new-external-1d\]](#lem:new-external-1d){reference-type="ref" reference="lem:new-external-1d"} (when $d=1$). **Claim 47**. *Consider Setting [Setting 45](#set:joining-common){reference-type="ref" reference="set:joining-common"} and any $d\ge 1$. Let $w{\,\gg_{\star}\,}q,\textnormal{\texttt{par}}\xspace$ with $w \ge M^{8(\tau-1)}$, let $r := w^{3/d}$ and let $z \in \mathbb{R}^d$. Let $\mathcal{H}\subseteq \mathcal{V}_M$ be a random vertex set which depends only on $(V,w_V,\mathcal{E}_M)$ and which satisfies $\mathbb{P}(\mathcal{A}_{\mathrm{dense}}(\mathcal{H},\mathcal{V}_M,r,z) \mid \mathcal{F}_{0,x}) \ge 1-q/10$. Let $$\begin{aligned} \mathcal{A}_{\mathrm{down}}(w,z):=\Big\{ \forall y \in \widetilde \mathcal{V}\cap (B_{r}(z)\times[w, 4w]): \exists u \in \mathcal{H}\cap B_{r^{1/3}}(y),\ yu\in\mathcal{E},\ \mathcal{C}(yu) \le w^{2\mu} \Big\}. \end{aligned}$$ Then for all $z \in \mathbb{R}^d$, $\mathbb{P}(\mathcal{A}_{\mathrm{down}}(w,z)\mid\mathcal{F}_{0,x}) \ge 1 - q/3$.* Before the proof, we show that the conditions are also satisfied for $d=1$. Fixing $w$ and requiring $w^{3/d}=r$ gives an equation for $M$ in terms of $w$ in Lemma [\[lem:new-external-1d\]](#lem:new-external-1d){reference-type="ref" reference="lem:new-external-1d"}: $w^{3/d}=r_M=e^{(\log M)^2}$, which implies that $M=e^{\sqrt{\log(w)\cdot 3/d}}$. Then $M^{8(\tau-1)}$ is a subpolynomial function of $w$, and so the requirement $w\ge M^{8(\tau-1)}$ is satisfied for all sufficiently large $w{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$. *Proof of Claim [Claim 47](#claim:new-new-join-Cm){reference-type="ref" reference="claim:new-new-join-Cm"}.* Fix $z\in \mathbb{R}^d$ and let $\mathcal{A}_1:=\mathcal{A}_{\mathrm{dense}}(\mathcal{H}, \mathcal{V}_M, r,z)$ in [\[eq:a-dense\]](#eq:a-dense){reference-type="eqref" reference="eq:a-dense"}, so that $\mathbb{P}(\neg \mathcal{A}_1\mid \mathcal{F}_{0,x})\le q/10$ by hypothesis. Considering the definition of $\mathcal{A}_{\mathrm{dense}}$ in [\[eq:a-dense\]](#eq:a-dense){reference-type="eqref" reference="eq:a-dense"}, let $\mathcal{A}_2$ be the event that for all $y \in B_r(z)$, $|B_{r^{1/3}}(y)\cap \mathcal{V}_M| \ge r^{d/4}$. Choose fixed points $x_1,\dots,x_{\lceil r^d\rceil} \in B_r(z)$ such that $\{B_{r^{1/3}/2}(x_i)\colon i \le \lceil r^d\rceil\}$ covers $B_r(z)$. For all $y$, the ball $B_{r^{1/3}}(y)$ must contain at least one ball $B_{r^{1/3}/2}(x_i)$, so if in each ball $B_{r^{1/3}/2}(x_i)$ we find at least $r^{d/4}$ vertices from $\mathcal{H}\subseteq \mathcal{V}_M$ then the event $\mathcal{A}_2$ holds. Let $c_d$ denote the volume of a unit-radius $d$-dimensional ball. By [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}, in IGIRG $|B_{r^{1/3}/2}(x_i) \cap \mathcal{V}_M|$ is a Poisson variable with mean $$2^{-d}c_d r^{d/3} \Big(\frac{\ell(M)}{M^{\tau-1}} - \frac{\ell(2M)}{(2M)^{\tau-1}}\Big) \ge 2r^{d/3}M^{-3(\tau-1)/2} \ge 2r^{d/4}$$ (also conditioned on $\mathcal{F}_{0,x}$), where the first inequality holds because $M {\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$ and the second inequality holds since $r^{d/12}=w^{1/4} \ge M^{2(\tau-1)}$ and $M\ge1$. Similarly, for SFP it is a binomial variable with mean greater than $2r^{d/4}$; in either case, the Chernoff bound of Theorem [Theorem 50](#thm:chernoff){reference-type="ref" reference="thm:chernoff"} applies, and since $r{\,\gg_{\star}\,}q$ we have $$\label{eq:njcm-a2} \begin{aligned} \mathbb{P}(\mathcal{A}_2\mid\mathcal{F}_{0,x}) &= \mathbb{P}\Big(\forall y \in B_r(z): |B_{r^{1/3}}(y)\cap \mathcal{V}_M| \ge r^{d/4}\mid \mathcal{F}_{0,x}\Big) \\ &\ge \mathbb{P}(\forall i\colon |B_{r^{1/3}/2}(x_i) \cap \mathcal{V}_M| \ge r^{d/4}\mid \mathcal{F}_{0,x}) \ge 1-\lceil r^d\rceil \cdot e^{-r^{d/4}/4} \ge 1-q/30. \end{aligned}$$ Let $\mathcal{A}_3$ be the event that $B_r(z)$ contains at most $2(c_dr^d+2)$ vertices. In SFP, $\mathbb{P}(\mathcal{A}_3\mid\mathcal{F}_{0,x}) = 1$; in IGIRG, Theorem [Theorem 50](#thm:chernoff){reference-type="ref" reference="thm:chernoff"} applies. In both cases, using $r{\,\gg_{\star}\,}q, \textnormal{\texttt{par}}\xspace$, $$\label{eq:njcm-a3} \mathbb{P}(\mathcal{A}_3\mid \mathcal{F}_{0,x})=\mathbb{P}(|B_r(z) \cap \mathcal{V}|\le 2(c_dr^d+2)\mid \mathcal{F}_{0,x}) \ge 1-e^{-c_dr^d/3} \ge 1-q/30.$$ Since $\mathbb{P}(\neg \mathcal{A}_1\mid \mathcal{F}_{0,x}) \le q/10$, a union bound with [\[eq:njcm-a2\]](#eq:njcm-a2){reference-type="eqref" reference="eq:njcm-a2"} and [\[eq:njcm-a3\]](#eq:njcm-a3){reference-type="eqref" reference="eq:njcm-a3"} yields $\mathbb{P}(\mathcal{A}_1\cap \mathcal{A}_2\cap \mathcal{A}_3 \mid \mathcal{F}_{0,x}) \ge 1-q/6$. We abbreviate $\mathbb{P}(\cdot \mid V,w_V,E_M)$ when we condition on the event that $\widetilde\mathcal{V}= (V,w_V)$ and $\mathcal{E}_M = E_M$. The events $\mathcal{F}_{0,x},\mathcal{A}_2,\mathcal{A}_3$, and also the set $\mathcal{H}$ and thus $\mathcal{A}_1$ are all determined by $(V,w_V,E_M)$. Let us call the realisation $(V,w_V,E_M)$ *good* if the event $\mathcal{A}_1\cap\mathcal{A}_2\cap\mathcal{A}_3\cap\mathcal{F}_{0,x}$ holds. Then $$\label{eq:njcm-goal} \mathbb{P}(\mathcal{A}_{\mathrm{down}}(w,z)\mid\mathcal{F}_{0,x}) \ge 1 - q/6 - \max_{(V,w_V,E_M)\text{ good}}\mathbb{P}\big(\neg\mathcal{A}_{\mathrm{down}}(w,z)\mid V,w_V,E_M\big).$$ Fix a good realisation $(V,w_V,E_M)$. Following $\mathcal{A}_{\mathrm{down}}$, let $y_1,\dots,y_k$ be the (fixed) vertices in $B_r(z)$ with weights in $[w,4w]$, and for each $i \in [k]$ let $a_{1}^{\scriptscriptstyle{(i)}},\dots,a_{\ell_i}^{\scriptscriptstyle{(i)}}$ be the (fixed) vertices in $B_{r^{1/3}}(y_i) \cap \mathcal{H}$. Thus, by definition of $\mathcal{A}_{\mathrm{down}}$, $$\begin{aligned} \begin{split}\label{eq:njcm-all-edges} &\mathbb{P}\big(\neg\mathcal{A}_{\mathrm{down}}(w,z)\mid V,w_V,E_M\big) \\ &\qquad =\mathbb{P}\big(\exists i\in[k]\colon\forall j\colon y_ia_{j}^{\scriptscriptstyle{(i)}}\notin \mathcal{E}(G) \mbox{ or }\mathcal{C}(y_ia_{j}^{\scriptscriptstyle{(i)}})\! >\! w^{2\mu}\mid V,w_V,E_M\big). \end{split} \end{aligned}$$ Conditioned on $(V,w_V,E_M)$, the edges $y_ia_j^{\scriptscriptstyle{(i)}}$ are present independently since $w_{y_i} \ge w \ge M^{8(\tau-1)} > 2M$ since $\tau > 2$ and $M{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$. Since $w_{y_i}\in[w, 4w]$, $w_{a_j^{\scriptscriptstyle{(i)}}}$ $\in[M, 2M]$, and $|y_i-a_j^{\scriptscriptstyle{(i)}}|\le r^{1/3}=w^{1/d}$, we get using [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"} and [\[eq:cost\]](#eq:cost){reference-type="eqref" reference="eq:cost"} that for all $i$ and $j$, $$\begin{aligned} &\mathbb{P}(y_ia_{j}^{\scriptscriptstyle{(i)}} \notin\mathcal{E}(G) \mbox{ or }\mathcal{C}(y_ia_{j}^{\scriptscriptstyle{(i)}}) > w^{2\mu}\mid V,w_V,E_M) \le 1 - \underline{c}\big(1 \wedge \tfrac{wM}{w}\big)^\alpha + \mathbb{P}\big(L> \tfrac{w^{2\mu}}{(8wM)^{\mu}}\big) \le 1 - \tfrac{\underline{c}}{2}, \end{aligned}$$ where the last inequality holds because $w^{2\mu}/(8wM)^{\mu} \ge M^{8\mu(\tau-1)-\mu}/8^\mu$ tends to infinity with $M$ and $M{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$. This computation also holds for $\alpha=\infty$ or $\beta=\infty$. Conditioned on a good realisation $(V,w_V,E_M)$, $\mathcal{A}_1\cap \mathcal{A}_2\cap\mathcal{A}_3$ occurs, so for each $i$, the number of vertices $a_j^{\scriptscriptstyle{(i)}}$ is $\ell_i \ge r^{d/4}$, and the number of vertices $y_i$ is $k\le 2c_dr^d+4$. By independence across $j$, [\[eq:njcm-all-edges\]](#eq:njcm-all-edges){reference-type="eqref" reference="eq:njcm-all-edges"}, and a union bound, $$\begin{aligned} \mathbb{P}\big(\neg\mathcal{A}_{\mathrm{down}}(w,z) \mid V,w_V,E_M\big) &\le \sum_{i\le k}\mathbb{P}\big(\forall j\colon y_ia_{j}^{\scriptscriptstyle{(i)}} \notin\mathcal{E}\mbox{ or }\mathcal{C}( y_ia_{j}^{\scriptscriptstyle{(i)}}) > w^{2\mu}\mid V,w_V,E_M\big) \\ &\le \sum_{i\le k}(1-\underline{c}/2)^{\ell_i} \le (2c_dr^d+4)e^{-r^{d/4}\underline{c}/2} \le q/6, \end{aligned}$$ where the last inequality holds since $r=w^{3/d}{\,\gg_{\star}\,}q,\textnormal{\texttt{par}}\xspace$. The claim then follows by [\[eq:njcm-goal\]](#eq:njcm-goal){reference-type="eqref" reference="eq:njcm-goal"}. ◻ We are now ready to prove the main results. Recall $\mu_{\log}, \mu_{\mathrm{pol}}$ from [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}. The following two lemmas contain the statements for the infinite graph cases. **Lemma 48**. *Consider Setting [Setting 45](#set:joining-common){reference-type="ref" reference="set:joining-common"}. Suppose that either $\alpha\in(1,2)$ or $\mu\in(\mu_{\mathrm{expl}},\mu_{\log})$ or both hold, and let $\Delta_0$ be as defined in [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"}, [\[eq:alpha-infty-Delta_0\]](#eq:alpha-infty-Delta_0){reference-type="eqref" reference="eq:alpha-infty-Delta_0"}, or [\[eq:beta-infty-Delta_0\]](#eq:beta-infty-Delta_0){reference-type="eqref" reference="eq:beta-infty-Delta_0"}, depending on whether $\alpha,\beta<\infty$, $\alpha=\infty$, or $\beta=\infty$. For every $q, \varepsilon,\zeta >0$ there exists $C>0$ such that the following holds. For any $x \in \mathbb{R}^d$ let $\mathcal{A}_{\mathrm{polylog}}$ be the event that $G$ contains a path $\pi_{0,x}$, with endpoints $0$ and $x$, of cost $\mathcal{C}(\pi_{0,x})\le (\log |x|)^{\Delta_0+\varepsilon} + C$ and deviation $\mathrm{dev}(\pi_{0,x})\le \zeta|x| + C$. Then $\mathbb{P}(\mathcal{A}_{\mathrm{polylog}} \mid 0,x \in \mathcal{C}_\infty) \ge 1-q$.* *Proof.* We first prove the result for $d\ge 2$, then describe the necessary modifications for $d=1$. Let $\rho > 0$ be as in Claim [\[claim:two-in-Cinfty\]](#claim:two-in-Cinfty){reference-type="ref" reference="claim:two-in-Cinfty"} and let $\delta {\,\ll_{\star}\,}\varepsilon,q,\rho,\textnormal{\texttt{par}}\xspace$ and $w_0 >1$. We want to apply Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"}, which holds for sufficiently large $|x|$. Thus there exists $r_{\ref{cor:computations-polylog}} {\,\gg_{\star}\,}q,\delta,\varepsilon,\zeta,w_0,\textnormal{\texttt{par}}\xspace$ such that Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} is applicable whenever $|x| \ge r_{\ref{cor:computations-polylog}}$. We may also assume $r_{\ref{cor:computations-polylog}} {\,\gg_{\star}\,}\kappa$. To cover $|x| < r_{\ref{cor:computations-polylog}}$, for all $v\in \mathcal{C}_\infty \cap B_{r_{\ref{cor:computations-polylog}}}(0)$, pick the cheapest path $\pi_{0,v}$ from $0$ to $v$. Then $R_1 := \max\{\mathcal{C}(\pi_{0,v}): v\in \mathcal{C}_\infty \cap B_{r_{\ref{cor:computations-polylog}}}(0)\}$ and $R_2 := \max\{\mathrm{dev}(\pi_{0,v}): v\in \mathcal{C}_\infty \cap B_{r_{\ref{cor:computations-polylog}}}(0)\}$ are almost surely finite random variables, and since we may assume $C {\,\gg_{\star}\,}r_{\ref{cor:computations-polylog}},q,\textnormal{\texttt{par}}\xspace$, we have $\mathbb{P}(R_1,R_2 \le C \mid 0,x\in \mathcal{C}_\infty) \ge 1-q$, as required. So from now on we may assume $|x| \ge r_{\ref{cor:computations-polylog}}$. Let $\overline{w}$ be as in [\[eq:weight-crit-cor1\]](#eq:weight-crit-cor1){reference-type="eqref" reference="eq:weight-crit-cor1"} in Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} and let $r_1:=\overline{w}^{3/d}$. Let $M,r_2,\kappa > 0$ satisfy $\overline{w}, C {\,\gg_{\star}\,}r_2 {\,\gg_{\star}\,}M,\zeta$, $q, \rho, \varepsilon,\textnormal{\texttt{par}}\xspace$ and $C {\,\gg_{\star}\,}r_{\ref{cor:computations-polylog}} {\,\gg_{\star}\,}\kappa {\,\gg_{\star}\,}M$ as in Lemma [Lemma 46](#lem:new-external){reference-type="ref" reference="lem:new-external"}, and note that $|x|\ge r_{\ref{cor:computations-polylog}}$ implies $\overline{w},r_1 {\,\gg_{\star}\,}\kappa$. Let $Q$ be a cube of side length $|x|$ containing 0 and $x$, and let $\mathcal{A}_{\mathrm{net}}$ be the event that $Q$ contains a weak $(\delta/4, w_0)$-net (as in Definition [Definition 21](#def:weak-net){reference-type="ref" reference="def:weak-net"}) which contains $0$ and $x$. Apply Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} with $\varepsilon_{\ref{cor:computations-polylog}} = \varepsilon/2$ and $q_{\ref{cor:computations-polylog}}:=q\rho/5$ to obtain $\mathcal{X}_{\mathrm{polylog}}(0,x)$. Then consider the intersection of the following events from Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"}, Lemma [Lemma 46](#lem:new-external){reference-type="ref" reference="lem:new-external"} (defined in [\[eq:a-linear\]](#eq:a-linear){reference-type="eqref" reference="eq:a-linear"}, [\[eq:a-near\]](#eq:a-near){reference-type="eqref" reference="eq:a-near"}) and Claim [Claim 47](#claim:new-new-join-Cm){reference-type="ref" reference="claim:new-new-join-Cm"}: $$\begin{aligned} \begin{split}\label{eq:big-intersection} &\mathcal{A}:=\mathcal{A}_{\mathrm{net}}\cap \mathcal{X}_{\mathrm{polylog}}(0,x)\\ &\quad\cap \bigcap_{v\in\{0,x\}}\Big(\mathcal{A}_\mathrm{near}(\mathcal{H}_\infty, r_2,C/4,v)\cap \mathcal{A}_\mathrm{linear}(\mathcal{H}_\infty, \mathcal{C}_{\infty}^M, r_2,\kappa,\zeta,C/4,v)\cap \mathcal{A}_{\mathrm{down}}(\overline{w},v)\Big). \end{split} \end{aligned}$$ When $\mathcal{A}$ occurs, $\mathcal{X}_{\mathrm{polylog}}(0,x)$ gives a path between endpoints $y_0^\star, y_x^\star$ with weights in $[\overline{w}, 4\overline{w}]$ and within distance $r_1=\overline{w}^{3/d}$ from $0,x$, with cost $\mathcal{C}(\pi_{y_0^\star, y_x^\star})\le (\log |x|)^{\Delta_0+\varepsilon/2}$ and deviation $\mathrm{dev}_{0x}(\pi_{y_0^\star, y_x^\star}) \le \zeta |x|$, respectively, see [\[eq:weight-crit-cor1\]](#eq:weight-crit-cor1){reference-type="eqref" reference="eq:weight-crit-cor1"}--[\[eq:cost-crit-cor1\]](#eq:cost-crit-cor1){reference-type="eqref" reference="eq:cost-crit-cor1"}. Then, the events $\mathcal{A}_{\mathrm{down}}(\overline{w},0), \mathcal{A}_{\mathrm{down}}(\overline{w},x)$ from Claim [Claim 47](#claim:new-new-join-Cm){reference-type="ref" reference="claim:new-new-join-Cm"} applied respectively to $y_0^\star, y_x^\star$ give us two paths $\pi_{y_0^\star, u_0^\star}$ and $\pi_{y_x^\star, u_x^\star}$ with $u_0^\star, u_x^\star\in \mathcal{H}_\infty$ and within respective distance $\overline{w}^{1/d}$ from $y_0^\star, y_x^\star$, and cost at most $\overline w^{2\mu}$. Further, the events $\mathcal{A}_{\mathrm{near}}(\mathcal{H}_\infty, r_2,C/4,0)$ and $\mathcal{A}_{\mathrm{near}}(\mathcal{H}_\infty, r_2,C/4,x)$ in [\[eq:a-near\]](#eq:a-near){reference-type="eqref" reference="eq:a-near"} also give us two paths $\pi_{0, u_0}$ and $\pi_{x, u_x}$, with respective endpoints $u_0, u_x\in \mathcal{H}_\infty$ within distance $r_2 \le \overline{w}^{3/d}$ from $0,x$ respectively, and cost at most $C/4$ each. Finally, since $u_0, u_0^\star, u_x, u_x^\star\in \mathcal{H}_\infty$, and $u_0, u_x$ is within distance $r_2$ from $0,x$, respectively, the events $\mathcal{A}_\mathrm{linear}(\mathcal{H}_\infty, \mathcal{C}_{\infty}^M, r_2,\kappa,\delta,C/4,v), v\in\{0,x\}$ in [\[eq:a-linear\]](#eq:a-linear){reference-type="eqref" reference="eq:a-linear"} ensure that there exist paths $\pi_{u_0, u_0^\star}$ and $\pi_{u_x, u_x^\star}$ in $G$ that have cost at most $\kappa|u_v-u_v^\star|+C/4 \le \kappa3\overline{w}^{3/d}+C/4$ since $|u_v-u_v^\star|\le 3r_1 = 3\overline{w}^{3/d}$ and deviation at most $\zeta|u_v-u_v^\star|+C/4\le \zeta 3\overline{w}^{3/d}+C/4$. The concatenated path is $\pi_{0,x}:=\pi_{0,u_0} \pi_{u_0,u_0^\star} \pi_{u_0^\star, y_0^\star} \pi_{y_0^\star, y_x^\star} \pi_{y_x^\star, u_x^\star} \pi_{u_x^\star, u_x} \pi_{u_x, x}$. Then, since $\overline{w}\le (\log |x|)^{\varepsilon/2}$ in [\[eq:weight-crit-cor1\]](#eq:weight-crit-cor1){reference-type="eqref" reference="eq:weight-crit-cor1"}, we can estimate the cost, and using that the vertices of the paths $\pi_{0,u_0},\pi_{u_0,u_0^\star}, \pi_{y_0^\star, y_x^\star}, \pi_{y_x^\star, u_x^\star},\pi_{u_x^\star, u_x}, \pi_{u_x, x}$ are all within distance $3r_1=3\overline{w}^{3/d}$ from $0$ and $x$ respectively, we can bound cost and deviation as $$\label{eq:cost-calculation-long} \begin{aligned} \mathcal{C}(\pi_{0,x})&\le 2\cdot C/4+ 2\overline{w}^{2\mu}+ 2(\kappa3\overline{w}^{3/d}+C/4) + (\log |x|)^{\Delta_0+\varepsilon/2}\le (\log |x|)^{\Delta_0+\varepsilon} +C,\\ \mathrm{dev}(\pi_{0,x})&\le \max\big\{\mathrm{dev}_{0,x}(\pi_{y_0^\star, y_x^\star}), 2\overline{w}^{3/d}+\zeta3\overline{w}^{3/d}+C/4\big\}\le \zeta |x|+C, \end{aligned}$$ using $|x| \ge r_{\ref{cor:computations-polylog}} {\,\gg_{\star}\,}\varepsilon, \kappa, \textnormal{\texttt{par}}\xspace$. Thus $\mathcal{A}\subseteq\mathcal{A}_{\mathrm{polylog}}$. A union bound on the complement of the events in [\[eq:big-intersection\]](#eq:big-intersection){reference-type="eqref" reference="eq:big-intersection"} from Lemma [Lemma 22](#lem:weak-nets-exist){reference-type="ref" reference="lem:weak-nets-exist"} with $t=2$ and $\varepsilon_{\ref{lem:weak-nets-exist}}:=\delta/4$ for $\mathcal{A}_{\mathrm{net}}$, Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"}, Lemma [Lemma 46](#lem:new-external){reference-type="ref" reference="lem:new-external"} with $q_{\ref{lem:new-external}}:=q\rho$, and Claim [Claim 47](#claim:new-new-join-Cm){reference-type="ref" reference="claim:new-new-join-Cm"} with $q_{\ref{claim:new-new-join-Cm}}:=q\rho$ gives $$\begin{aligned} \mathbb{P}(\neg\mathcal{A}_{\mathrm{polylog}}\mid 0,x\in\mathcal{C}_\infty) &\le \mathbb{P}(\neg\mathcal{A}\mid 0,x\in\mathcal{C}_\infty) =\frac{\mathbb{P}(\neg\mathcal{A}\cap \{0,x\in\mathcal{C}_\infty\}\mid 0,x\in \mathcal{V})}{\mathbb{P}(0,x\in\mathcal{C}_\infty\mid 0,x\in\mathcal{V})}\\ &\le \frac{q\rho}{\mathbb{P}(0,x\in\mathcal{C}_\infty\mid 0,x\in\mathcal{V})}.\end{aligned}$$ The result therefore follows from Claim [\[claim:two-in-Cinfty\]](#claim:two-in-Cinfty){reference-type="ref" reference="claim:two-in-Cinfty"}. When $d=1$, we construct $\pi_{0,x}$ in exactly the same way as below [\[eq:big-intersection\]](#eq:big-intersection){reference-type="eqref" reference="eq:big-intersection"}, using Lemma [\[lem:new-external-1d\]](#lem:new-external-1d){reference-type="ref" reference="lem:new-external-1d"} with $M_{\ref{lem:new-external-1d}} := \exp(\sqrt{(3/d)\log\overline{w}})$ (so that $r_{M_{\ref{lem:new-external-1d}}} = \overline{w}^{3/d}$), in place of Lemma [Lemma 46](#lem:new-external){reference-type="ref" reference="lem:new-external"}. We may assume $|x| {\,\gg_{\star}\,}\varepsilon,\zeta$ as for $d\ge 2$. Note that $M_{\ref{lem:new-external-1d}} \le \exp(\sqrt{\log\log|x|})$ since $\overline{w}\le (\log|x|)^{\varepsilon}$ and $\varepsilon{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, and in particular we may assume $\overline{w}\ge M_{\ref{lem:new-external-1d}}^{8(\tau-1)}$ as in Claim [Claim 47](#claim:new-new-join-Cm){reference-type="ref" reference="claim:new-new-join-Cm"} since $\overline{w}{\,\gg_{\star}\,}\varepsilon, \textnormal{\texttt{par}}\xspace$, see also the computation below Claim [Claim 47](#claim:new-new-join-Cm){reference-type="ref" reference="claim:new-new-join-Cm"}. Using $|x|{\,\gg_{\star}\,}\varepsilon,\zeta$, this implies the costs of all our subpaths counted in [\[eq:cost-calculation-long\]](#eq:cost-calculation-long){reference-type="eqref" reference="eq:cost-calculation-long"} except $\pi_{y_0^\star,y_x^\star}$ are negligible compared to the $(\log |x|)^{\Delta_0+\varepsilon/2}$ cost of $\pi_{y_0^\star,y_x^\star}$, as in the $d \ge 2$ case, and likewise that the deviation of these subpaths from the line segment $S_{0x}$ is negligible compared to $\zeta |x|$. The cost and deviation of $\pi_{y_0^\star,y_x^\star}$ are bounded using Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} exactly as in the $d \ge 2$ case in [\[eq:cost-calculation-long\]](#eq:cost-calculation-long){reference-type="eqref" reference="eq:cost-calculation-long"}. ◻ **Lemma 49**. *Consider Setting [Setting 45](#set:joining-common){reference-type="ref" reference="set:joining-common"}. Suppose that $\alpha>2$ and $\mu>\mu_{\log}$, and let $\eta_0$ be as defined in [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}, [\[eq:alpha-infty-definitions\]](#eq:alpha-infty-definitions){reference-type="eqref" reference="eq:alpha-infty-definitions"}, [\[eq:beta-infty-definitions\]](#eq:beta-infty-definitions){reference-type="eqref" reference="eq:beta-infty-definitions"}, or [\[eq:alpha-beta-infty-definitions\]](#eq:alpha-beta-infty-definitions){reference-type="eqref" reference="eq:alpha-beta-infty-definitions"}, depending on $\alpha,\beta<\infty$, $\beta <\alpha=\infty$, $\alpha <\beta=\infty$, or $\alpha=\beta =\infty$. For every $q, \varepsilon,\delta >0$ there is $C>0$ such that the following holds. For any $x \in \mathbb{R}^d$ let $\mathcal{A}_{\mathrm{pol}}$ be the event that $G$ contains a path $\pi_{0,x}$, with endpoints $0$ and $x$, of cost $\mathcal{C}(\pi_{0,x})\le |x|^{\eta_0+\varepsilon} + C$ and deviation $\mathrm{dev}(\pi_{0,x})\le \zeta|x| + C$. Then $\mathbb{P}(\mathcal{A}_{\mathrm{pol}} \mid 0,x \in \mathcal{C}_\infty) \ge 1-q$.* *Proof.* The proof when $\mu\in(\mu_{\log}, \mu_{\mathrm{pol}}]$ is identical to the proof of Lemma [Lemma 48](#lem:polylog-deviation){reference-type="ref" reference="lem:polylog-deviation"}, except that the event $\mathcal{X}_{\mathrm{polylog}}(0,x)$ from Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} is replaced by $\mathcal{X}_{\mathrm{pol}}(0,x)$ from Corollary [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"}. When $\mu>\mu_{\mathrm{pol}}$ and $d=1$, we have $\eta_0=1$ and we can prove that the cost distance is at most $|x|^{1+\varepsilon}$ by using Lemma [\[lem:new-external-1d\]](#lem:new-external-1d){reference-type="ref" reference="lem:new-external-1d"} directly as follows. We set $r_M=|x|$, which gives, using $r_M=\exp((\log M)^2)$, the value $M=\exp(\sqrt{\log |x|})$, which is slowly varying in $|x|$. Lemma [\[lem:new-external-1d\]](#lem:new-external-1d){reference-type="ref" reference="lem:new-external-1d"} defines $\mathcal{H}_M:=B_{2r_M}\cap \mathcal{V}_M$, and with $C_M=M^{2(\tau-1)+3\mu}$ and $\kappa_M=2M^{3\mu+2}$, it states that $$\begin{aligned} \mathbb{P}\Big(\mathcal{A}_\mathrm{near}(\mathcal{H}_M, C_M,C_M,0) &\cap \mathcal{A}_\mathrm{near}(\mathcal{H}_M, C_M,C_M,x)\\ &\cap \mathcal{A}_{\mathrm{linear}}(\mathcal{H}_M, \mathcal{H}_M, r_M,\kappa_M, 0,2\kappa_M,0) \mid 0,x \in \mathcal{C}_\infty\Big)\ge 1-3q/10.\end{aligned}$$ The first two events $\mathcal{A}_\mathrm{near}(\mathcal{H}_M, C_M,C_M,z)$ with $z\in\{0,x\}$ guarantee that we find two paths $\pi_{0,y^\star_0}$ and $\pi_{x, y_x^\star}$ with cost at most $C_M$ from $0$ and $x$ to respective vertices $y_0^\star, y_x^\star\in \mathcal{V}_M$, that are fully contained in $B_{C_M}(0)$ and $B_{C_M}(x)$, respectively. Here, $C_M=M^{2(\tau-1)+3\mu}\le |x|^{\varepsilon/2}$ for sufficiently large $|x|$. Then, since $y_0^\star$ is within distance $C_M<r_M$ from $0$, the third event $\mathcal{A}_{\mathrm{linear}}$ guarantees a path between $y_0^\star$ and every vertex in $\mathcal{H}_M=\mathcal{V}_M\cap B_{2|x|}(0)$ with $\kappa_M$-linear cost, in particular there is such a path $\pi_{y_0^\star, y_x^\star}$ between $y_0^\star$ and $y_x^\star$. Let $\pi_{0,x}:=\pi_{0,y_0^\star}\pi_{y_0^\star, y_x^\star}\pi_{y_x^\star, x}$ be the concatenation of these paths. Since the distance $|y_0^\star- y_x^\star|\le 2C_M+|x|$, the cost and deviation of this path is, using that $\kappa_M=2M^{3\mu+2}\le |x|^{\varepsilon/2}$ for $|x|$ large, $$\begin{aligned} \mathcal{C}(\pi_{0,x})&=\mathcal{C}(\pi_{0,y_0^\star})+\mathcal{C}(\pi_{y_x^\star, x})+ \mathcal{C}(\pi_{y_0^\star, y_x^\star}) \le 2C_M + \kappa_M|y_0^\star-y_x^\star| +2\kappa_M\\ &\le 2|x|^{\varepsilon/2} + |x|^{\varepsilon/2} (|x|+2|x|^{\varepsilon/2}) +2 |x|^{\varepsilon/2}\le |x|^{1+\varepsilon}, \\ \mathrm{dev}(\pi_{0,x})&\le \max\{\mathrm{dev}_{0,x}(\pi_{0,y_0^\star}), \mathrm{dev}_{0,x}(\pi_{y_0^\star, y_x^\star}), \mathrm{dev}_{0,x}(\pi_{y_x^\star},x)\} \\ &\le \max\{C_M , 0|y_0^\star-y_x^\star|+2\kappa_M\} \le |x|^\varepsilon,\end{aligned}$$ for $|x|$ large enough. For small $|x|$ we can absorb the costs and deviation in the constant $C$. This proves the lemma when $d=1$ and $\mu>\mu_{\mathrm{pol}}$ with $\eta_0=1$. When $\mu>\mu_{\mathrm{pol}}$ and $d\ge 2$, a straightforward adaptation of the above proof for $d=1$ could in principle also be used in higher dimensions. However, with some more effort one can get rid of the extra $+\varepsilon$ in the exponent, and prove fully linear cost distances. We prove this stronger version (without the $|x|^\varepsilon$ factor) in [@komjathy2022one2 Theorems 1.8, 1.10]. ◻ ## Proof of the Main Theorems {#sec:main_proof} The proofs of Theorems [\[thm:polylog_regime\]](#thm:polylog_regime){reference-type="ref" reference="thm:polylog_regime"} and [\[thm:polynomial_regime\]](#thm:polynomial_regime){reference-type="ref" reference="thm:polynomial_regime"} follow directly from Lemmas [Lemma 48](#lem:polylog-deviation){reference-type="ref" reference="lem:polylog-deviation"} and [Lemma 49](#lem:polynomial-deviation){reference-type="ref" reference="lem:polynomial-deviation"}, respectively, and so do their extensions to $\alpha=\infty$ and/or $\beta=\infty$ in Theorem [\[thm:threshold_regimes\]](#thm:threshold_regimes){reference-type="ref" reference="thm:threshold_regimes"}. It remains to prove Theorem [\[thm:finite_graph\]](#thm:finite_graph){reference-type="ref" reference="thm:finite_graph"}, including its extension to $\alpha=\infty$ and/or $\beta=\infty$. *Proof of Theorem [\[thm:finite_graph\]](#thm:finite_graph){reference-type="ref" reference="thm:finite_graph"}.* Following Def. [Definition 1](#def:girg){reference-type="ref" reference="def:girg"}, let $G_n$ be a finite GIRG obtained by intersecting an IGIRG $G = (\mathcal{V},\mathcal{E})$ with a finite cube $Q_n$ of volume $n$, and let $u_n,v_n$ be two vertices chosen uniformly at random from $\mathcal{V}\cap Q_n$. For the polylogarithmic case we must prove [\[eq:finite-polylog\]](#eq:finite-polylog){reference-type="eqref" reference="eq:finite-polylog"}. For this, first we prove the slightly different statement that for two uniformly random *positions* $x_n,y_n\in Q_n$, $$\begin{aligned} \label{eq:replace-giant-with-infinite2} \lim_{n\to\infty} \mathbb{P}\big( d_{\mathcal{C}}^{G_n}(x_n,y_n) > (\log |x_n-y_n|)^{\Delta_0+\varepsilon} \mid x_n, y_n \in \mathcal{C}_\infty \big) =0.\end{aligned}$$ Compared to [\[eq:finite-polylog\]](#eq:finite-polylog){reference-type="eqref" reference="eq:finite-polylog"}, there are two differences. First, $\mathcal{C}_\infty$ replaces $\mathcal{C}^{\scriptscriptstyle{(n)}}_{\max}$ in the conditioning. By [@komjathy2020explosion Theorem 3.11] there is a constant $\rho >0$ such that a.a.s. $|\mathcal{C}^{\scriptscriptstyle{(n)}}_{\max}| \ge \rho |\mathcal{V}\cap Q_n| \ge \rho|\mathcal{C}_\infty \cap Q_n|$, and on the other hand $\lim_{n\rightarrow\infty} \mathbb{P}\big(\mathcal{C}_{\max}^{(n)} \subseteq \mathcal{C}_{\infty}\big)=1$ since $\mathcal{C}_\infty$ is unique. Hence, the probability of the conditions $\mathbb{P}(u_n,v_n\in \mathcal{C}^{\scriptscriptstyle{(n)}}_{\max})$ and $\mathbb{P}(u_n,v_n \in \mathcal{C}_\infty)$ differ by at most a constant factor, which means that [\[eq:finite-polylog\]](#eq:finite-polylog){reference-type="eqref" reference="eq:finite-polylog"} is equivalent to conditioning on $\{u_n,v_n \in \mathcal{C}_\infty\}$. Secondly, in [\[eq:finite-polylog\]](#eq:finite-polylog){reference-type="eqref" reference="eq:finite-polylog"} we draw two random vertices $u_n,v_n$ from $\mathcal{V}\cap Q_n$, while in [\[eq:replace-giant-with-infinite2\]](#eq:replace-giant-with-infinite2){reference-type="eqref" reference="eq:replace-giant-with-infinite2"} we draw two random positions $x_n,y_n$ and condition on those being in the vertex set. This changes the number of vertices in $Q_n$ from $\textrm{Poisson}(n)$ to $\textrm{Poisson}(n)\!+\!2$. The total variation distance of these two distributions is vanishing as $n\to\infty$, so this difference can also be ignored, and proving [\[eq:replace-giant-with-infinite2\]](#eq:replace-giant-with-infinite2){reference-type="eqref" reference="eq:replace-giant-with-infinite2"} implies [\[eq:finite-polylog\]](#eq:finite-polylog){reference-type="eqref" reference="eq:finite-polylog"}. To prove [\[eq:replace-giant-with-infinite2\]](#eq:replace-giant-with-infinite2){reference-type="eqref" reference="eq:replace-giant-with-infinite2"}, let $C>0$ be the constant from Lemma [Lemma 48](#lem:polylog-deviation){reference-type="ref" reference="lem:polylog-deviation"}, let $0<\zeta{\,\ll_{\star}\,}q,\textnormal{\texttt{par}}\xspace$ and consider the event $\mathcal{A}_{\text{pos}}(x_n, y_n)$ that $|x_n-y_n| \ge \log n$ and that $x_n,y_n$ have distance at least $2\sqrt{d}\zeta n^{1/d}$ from the boundary of $Q_n$, a box of side-length $n^{1/d}$. Then, since $\zeta{\,\ll_{\star}\,}q, \textnormal{\texttt{par}}\xspace$, $$\begin{aligned} \label{eq:proof-of-main-theorems1} \mathbb{P}(\mathcal{A}_{\text{pos}}(x_n, y_n)) \ge 1-q/2.\end{aligned}$$ Consider now any given realisation $x_n,y_n\in Q_n$ of the random positions that satisfy $\mathcal{A}_{\text{pos}}(x_n, y_n)$. By Lemma [Lemma 48](#lem:polylog-deviation){reference-type="ref" reference="lem:polylog-deviation"} applied with $\varepsilon_{\ref{lem:polylog-deviation}}:=\varepsilon/2, q_{\ref{lem:polylog-deviation}}:=q/2$, conditional on $x_n,y_n\in \mathcal{C}_\infty$ there is a path $\pi_{x_n, y_n}$ from $x_n$ to $y_n$ with $\mathrm{dev}(\pi_{x_n,y_n})\le \zeta|x_n-y_n| + C \le 2\sqrt{d}\zeta n^{1/d}$ and cost at most $\mathcal{C}(\pi) \le (\log|x_n-y_n|)^{\Delta_0+\varepsilon/2}+C$ with probability at least $1-q/2$. Since $\mathcal{A}_{\text{pos}}(x_n, y_n)$ holds, the deviation bound of $\pi_{x_n, y_n}$ ensures that the path $\pi_{x_n,y_n}$ lies fully within $Q_n$ and thus in $G_n$. Moreover, since $|x_n-y_n|\ge \log n$ and $n$ is sufficiently large, $\mathcal{C}(\pi) \le (\log|x_n-y_n|)^{\Delta_0+\varepsilon/2}+C \le (\log|x_n-y_n|)^{\Delta_0+\varepsilon}$. Hence, for all $n$ large enough, whenever $x_n,y_n$ satisfies $\mathcal{A}_{\text{pos}}(x_n, y_n)$, $$\begin{aligned} \label{eq:proof-of-main-theorems2} \mathbb{P}\big( d_{\mathcal{C}}^{G_n}(x_n,y_n) \le (\log |x_n-y_n|)^{\Delta_0+\varepsilon} \mid x_n, y_n \in \mathcal{C}_\infty\big) \ge 1-q/2.\end{aligned}$$ Since $q$ was arbitrary, together with [\[eq:proof-of-main-theorems1\]](#eq:proof-of-main-theorems1){reference-type="eqref" reference="eq:proof-of-main-theorems1"}, this proves [\[eq:replace-giant-with-infinite2\]](#eq:replace-giant-with-infinite2){reference-type="eqref" reference="eq:replace-giant-with-infinite2"} and concludes the proof for the polylogarithmic case of Theorem [\[thm:finite_graph\]](#thm:finite_graph){reference-type="ref" reference="thm:finite_graph"} (including the extensions for $\alpha=\infty$, and/or $\beta =\infty$). The polynomial case is identical except that we use Lemma [Lemma 49](#lem:polynomial-deviation){reference-type="ref" reference="lem:polynomial-deviation"} instead of Lemma [Lemma 48](#lem:polylog-deviation){reference-type="ref" reference="lem:polylog-deviation"}. ◻ # Chernoff bound and proofs relating to nets **Theorem 50**. *(**Chernoff bounds**) Let $X_1,\ldots X_k$ be independent Bernoulli distributed random variables, and define $X:=\sum_{i=1}^k X_i$ and $m:=\mathbb{E}[X]$. Then, for all $\lambda\in(0,1]$ and all $t\ge 2em$, $$\begin{aligned} \mathbb{P}(X\leqslant (1-\lambda)m)\leqslant e^{-m\lambda^2/2},\quad\quad \mathbb{P}(X\geqslant (1+\lambda)m)\leqslant e^{-m\lambda^2/3}, \quad\quad \mathbb{P}(X\ge t) \le 2^{-t}.\end{aligned}$$ The same bounds hold when $X$ is instead a Poisson variable with mean $m$.* We restate the following claim before proving it: *Proof of Claim [\[claim:nets-mu-bound\]](#claim:nets-mu-bound){reference-type="ref" reference="claim:nets-mu-bound"}.* We start by showing [\[eq:mui-bound\]](#eq:mui-bound){reference-type="eqref" reference="eq:mui-bound"}. We write $I(w) = [w_-, w_+)$. The definition of a base-$2$-cover (Definition [Definition 14](#def:base-2-cover){reference-type="ref" reference="def:base-2-cover"}) ensures that $w_-,w_+ \in [w/2,2w]$ and $w_+/w_-=2$. Thus by the lower bound [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"} on $w_0$, for all $w\in[w_0, f(r_R)]$, $$\begin{aligned} \mathbb{P}(W \in I(w)) &= \ell(w_-)w_-^{-(\tau-1)} - \ell(w_+)w_+^{-(\tau-1)}\\ &\ge \ell(w)\Big(\frac{99}{100}w_-^{-(\tau-1)} - \frac{101}{100}w_+^{-(\tau-1)}\Big)\\ &= \ell(w)w_-^{-(\tau-1)}\Big(\frac{99}{100} - \frac{101}{100}\cdot 2^{-(\tau-1)}\Big). \end{aligned}$$ Since $\tau > 2$ and $w_- \le w$, it follows that $$\mu_i(I(w)) \ge (r_i')^d\ell(w) w^{-(\tau-1)}/2^{\tau+1}.$$ The required lower bound on $\mu_i(I(w))$ then follows by the lower bound (P1) in Definition [Definition 12](#def:nets-partition){reference-type="ref" reference="def:nets-partition"} on $r_i'$. By a very similar argument to the lower bound, [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"} also implies that $$\begin{aligned} \mathbb{P}(W \in I(w)) \le \frac{101}{100}\ell(w)w_-^{-(\tau-1)} \le 2^\tau\ell(w)w^{-(\tau-1)}, \end{aligned}$$ so the required upper bound on $\mu_i(I(w))$ likewise follows by the upper bound (P1) on $r_i'$. It remains to show [\[eq:z-bound\]](#eq:z-bound){reference-type="eqref" reference="eq:z-bound"}. First we show that $f(r_R) \ge f(r_{R-1}) \ge \ldots \ge f(r_1) \ge w_0$. Indeed, let $j \in [R]$. By the definition of $f$ in [\[eq:nets-f-def\]](#eq:nets-f-def){reference-type="eqref" reference="eq:nets-f-def"}, $$\begin{aligned} \label{eq:f-ratio} \frac{f(r_j)}{f(r_{j-1})} = \left(\frac{r_j}{r_{j-1}}\right)^{d/(\tau-1)} \cdot \left(\frac{1 \wedge \inf\big\{\ell(x)\colon x \in [w_0,r_j^{d/(\tau-1)}]\big\}}{1 \wedge \inf\big\{\ell(x)\colon x \in [w_0,r_{j-1}^{d/(\tau-1)}]\big\}}\right)^{1/(\tau-1)}. \end{aligned}$$ To bound the second factor, we will first observe that since $r_{j-1} \le r_j$, we have $\inf\{\ell(x)\colon x \in [w_0,r_j^{d/(\tau-1)}]\} \le \inf\{\ell(x)\colon x \in [w_0,r_{j-1}^{d/(\tau-1)}]\}$. By considering the two possible values of $1 \wedge \inf\{\ell(x)\colon x \in [w_0,r_j^{d/(\tau-1)}]\}$ separately, it follows that we can drop the minimum with $1$ in the ratio: $$\begin{aligned} \begin{split}\label{eq:f-ratio-inter} &\left(\frac{1 \wedge \inf\big\{\ell(x)\colon x \in [w_0,r_j^{d/(\tau-1)}]\big\}}{1 \wedge \inf\big\{\ell(x)\colon x \in [w_0,r_{j-1}^{d/(\tau-1)}]\big\}}\right)^{1/(\tau-1)} \\ &\qquad\ge \left(\frac{\inf\big\{\ell(x)\colon x \in [w_0,r_j^{d/(\tau-1)}]\big\}}{\inf\big\{\ell(x)\colon x \in [w_0,r_{j-1}^{d/(\tau-1)}]\big\}}\right)^{1/(\tau-1)}. \end{split} \end{aligned}$$ We now bound $\ell(x)$ on $[w_0, r_{j-1}^{d/(\tau-1)}]$ (in the denominator) by repeatedly applying [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"}. We write $\lg(\cdot)=\log_2(\cdot)$. Then, we have $r_j^{d/(\tau-1)} = 2^{\lg ((r_j/r_{j-1} )^{d/(\tau-1)})} r_{j-1}^{d/(\tau-1)}$, so we iterate the bound in [\[eq:nets-ell-bound-0\]](#eq:nets-ell-bound-0){reference-type="eqref" reference="eq:nets-ell-bound-0"} roughly $\lg ((r_j/r_{j-1} )^{d/(\tau-1)})$ many times to obtain that for all $x \in [r_{j-1}^{d/(\tau-1)}, r_j^{d/(\tau-1)}]$ we have $$\ell(x) \ge \Big(\frac{99}{100}\Big)^{1 + \lg ((r_j/r_{j-1} )^{d/(\tau-1)})} \ell(r_{j-1}).$$ Returning to [\[eq:f-ratio-inter\]](#eq:f-ratio-inter){reference-type="eqref" reference="eq:f-ratio-inter"}, the ratio of the two infima in the smaller interval $[w_0,r_{j-1}^{d/(\tau-1)}]$ is one. And since $r_j \ge 2r_{j-1}$ by (R2) of Definition [Definition 10](#def:well-spaced){reference-type="ref" reference="def:well-spaced"} (using $\delta <1/16 < 1$), it follows that $$\begin{aligned} \left(\frac{\inf\big\{\ell(x)\colon x \in [w_0,r_j^{d/(\tau-1)}]\big\}}{\inf\big\{\ell(x)\colon x \in [w_0,r_{j-1}^{d/(\tau-1)}]\big\}}\right)^{1/(\tau-1)} &\ge \Big(\frac{99}{100}\Big)^{1 + \lg ((r_j/r_{j-1} )^{d/(\tau-1)}) } \\ &\ge \Big(\frac{1}{2}\Big)^{\frac{1}{\tau-1}\lg ((r_j/r_{j-1} )^{d/(\tau-1)}) }\\ & = \Big(\frac{r_{j-1}}{r_j}\Big)^{d/(\tau-1)^2} . \end{aligned}$$ Combining this with [\[eq:f-ratio\]](#eq:f-ratio){reference-type="eqref" reference="eq:f-ratio"}, [\[eq:f-ratio-inter\]](#eq:f-ratio-inter){reference-type="eqref" reference="eq:f-ratio-inter"}, and the fact that $\tau > 2$, we obtain $$\frac{f(r_j)}{f(r_{j-1})} \ge \Big(\frac{r_j}{r_{j-1}}\Big)^{\frac{d}{\tau-1}(1-1/(\tau-1))} \ge 1.$$ Hence $f(r_R) \ge f(r_{R-1}) \ge \ldots \ge f(r_1)$, as claimed. It is now relatively easy to prove the desired lower bound. From the definition of $f$ in [\[eq:nets-f-def\]](#eq:nets-f-def){reference-type="eqref" reference="eq:nets-f-def"}, we have $f(r_i) \le r_i^{d/(\tau-1)}$, and [\[eq:nets-small-r\]](#eq:nets-small-r){reference-type="eqref" reference="eq:nets-small-r"} in the definition of well-spacedness ensures that $f(r_i)\ge w_0$, so $$\begin{aligned} \ell(f(r_i))f(r_i)^{-(\tau-1)} & = \frac{\ell(f(r_i))}{1 \wedge \inf\{\ell(x)\colon x \in [w_0,r_i^{d/(\tau-1)}]\}} \cdot \frac{(2d)^{2\tau+d+8}\log(16R/\delta)}{r_i^d} \\ &\ge \frac{(2d)^{2\tau+d+8}\log(16R/\delta)}{r_i^d}. \end{aligned}$$ Multiplying by $r_i^d$ finishes the statement of [\[eq:z-bound\]](#eq:z-bound){reference-type="eqref" reference="eq:z-bound"}. ◻ # Proof of Corollaries [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} and [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"} {#app:path-proofs} Both corollaries follow from Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} with suitably-chosen parameters. Throughout, we use the convention that $\infty\cdot 0 = 0$. In Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"}, the values of $(\gamma,z,\eta,R)$ are not set yet (and they are not part of the model parameters $\textnormal{\texttt{par}}\xspace$). We introduce constraints on these parameters in Definition [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} ("$(K,A)$-validity"), then we show in Lemma [Lemma 52](#lem:valid-works){reference-type="ref" reference="lem:valid-works"} that a $(K,A)$-valid assignment of values in Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} yields a path between $0$ and $x$ of cost $K$ with a multiplicative "error" of at most $A$. Recall $\Lambda$, $\Phi$ and $\overline{w}$ from [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}, [\[eq:Phi-def\]](#eq:Phi-def){reference-type="eqref" reference="eq:Phi-def"} and [\[eq:c_H\]](#eq:c_H){reference-type="eqref" reference="eq:c_H"}, and that $\xi$ is the side-length of the box $Q$ in which the net exist. **Definition 51** (Valid parameter choices). *The *reduced Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}* is Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, except without $\gamma$ being defined. Consider the reduced Setting  [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and let $K,A > 0$. A setting of parameters $(\gamma,z,\eta,R)$ is *$(K,A)$-valid* for $\xi$ if the following conditions all hold for $\xi{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$, writing $\overline{w}:= \xi^{\gamma^{R-1}d/2}$: $$\begin{aligned} \gamma=\gamma(\textnormal{\texttt{par}}\xspace)& \in (0,1),\quad z=z(\textnormal{\texttt{par}}\xspace) \in [0,d], \quad \eta=\eta(\textnormal{\texttt{par}}\xspace) \in [0,\infty), \label{eq:valid-trivial}\\ R&=R(\textnormal{\texttt{par}}\xspace, \xi) \in [2, (\log\log\xi)^2/4]\cap \mathbb{N}, \mbox{ with }\label{eq:valid-R}\\ \overline{w}^2& \in [e^{(\log^{*3}\xi)^2}, A/\log\log\xi],\label{eq:valid-rounds}\\ 2^R\overline{w}^{4\mu}\xi^{\eta} & \le KA/\log\log \xi, \label{eq:valid-low-cost}\\ \Lambda(\eta,z) &> 0 \qquad \mbox{and}\qquad \mbox{either } z=0 \mbox{ or } \Phi(\eta,z) > 0.\label{eq:valid-lambda-phi} \end{aligned}$$* **Lemma 52**. *Consider the reduced Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}. Let $q,\zeta > 0$, let $0<\delta{\,\ll_{\star}\,}q,\textnormal{\texttt{par}}\xspace$, and suppose that $\xi{\,\gg_{\star}\,}\delta, q, w_0,\zeta,\textnormal{\texttt{par}}\xspace$. Let $K,A > 0$, and suppose that $(\gamma,z,\eta,R)$ is $(K,A)$-valid for $\xi$. Let $\mathcal{X}_{(K,A)}$ be the event that there is a path $\pi_{y_0^\star, y_x^\star}$ in $G'$ with endpoints $y_0^\star$ and $y_x^\star$ satisfying $$\begin{aligned} \label{eq:valid-works-weights} &w_{y_0^\star}, w_{y_x^\star}\in[\overline{w}, 4\overline{w}],\\\label{eq:valid-works-positions} &y_0^\star \in B_{\overline{w}^{3/d}}(0), \quad y_x^\star \in B_{\overline{w}^{3/d}}(x),\\\label{eq:valid-works-costs} &\mathcal{C}(\pi_{y_0^\star, y_x^\star})\le KA \mbox{ and }\mathrm{dev}_{0x}(\pi) \le \zeta |x|. \end{aligned}$$ Then $\mathbb{P}(\mathcal{X}_{(K,A)} \mid V,w_V) \ge 1-q$.* *Proof.* Let $y_0 := 0$, let $y_1 := x$, let $\xi := |x|$, and let $\theta := 1$. We first verify that the conditions of Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} hold. Since $\delta{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, by [\[eq:valid-trivial\]](#eq:valid-trivial){reference-type="eqref" reference="eq:valid-trivial"} we may also assume $\delta {\,\ll_{\star}\,}\gamma, z, \eta$ as required by Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"}. Combined with [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"} this implies that $\Lambda(\eta,z) \ge 2\sqrt\delta$ as required, and that either $z=0$ or $\Phi(\eta,z) \ge \sqrt\delta$ as required by Prop. [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"}. Since $\xi {\,\gg_{\star}\,}\delta,\textnormal{\texttt{par}}\xspace$, the inequalities $\xi {\,\gg_{\star}\,}\gamma, z, \eta, \delta,w_0$ and $\xi^{\gamma^{R-1}} \ge (\log\log\xi\sqrt d)^{16d/\delta^2}$ by [\[eq:valid-rounds\]](#eq:valid-rounds){reference-type="eqref" reference="eq:valid-rounds"} and since $\overline{w}:= \xi^{\gamma^{R-1}d/2}$, which is also required. Finally, $R \in [2, (\log\log\xi)^2]$ by [\[eq:valid-R\]](#eq:valid-R){reference-type="eqref" reference="eq:valid-R"} and $\gamma \in (0,1)$, $z \in [0,d]$ and $\eta \ge 0$ by [\[eq:valid-trivial\]](#eq:valid-trivial){reference-type="eqref" reference="eq:valid-trivial"}. Suppose that the event $\mathcal{X}_{\mathrm{high\textnormal{-}path}}$ of Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} occurs, and let $\pi$ be a path as in the definition of $\mathcal{X}_{\mathrm{high\textnormal{-}path}}$. Then $\pi$ satisfies [\[eq:valid-works-weights\]](#eq:valid-works-weights){reference-type="eqref" reference="eq:valid-works-weights"} immediately, because $\mathcal{X}_{\mathrm{high\textnormal{-}path}}$ requires that the end-vertices of the path $\pi$ have weights in $[\overline{w}, 4\overline{w}]$. The event also requires that the end-vertices are within distance $c_H\xi^{\gamma^{R-1}}$ from $0,x$ respectively. Since $\xi{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace, \delta$, $c_H\xi^{\gamma^{R-1}} \le \xi^{\gamma^{R-1}3/2} = \overline{w}^{3/d}$, and so $\pi$ satisfies [\[eq:valid-works-positions\]](#eq:valid-works-positions){reference-type="eqref" reference="eq:valid-works-positions"}. The cost of $\pi$ is at most $c_H2^R\overline{w}^{4\mu}\xi^\eta$; by [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"} combined with the fact that $\xi{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace, \delta$, it follows that $\mathcal{C}(\pi) \le KA$. The event $\mathcal{X}_{\mathrm{high\textnormal{-}path}}$ ensures that the deviation of $\pi$ from the section $S_{0,x}$ is at most $3c_H\xi^\gamma$ where $\xi{\,\gg_{\star}\,}\zeta,\textnormal{\texttt{par}}\xspace, \delta$ and $\gamma<1$, so [\[eq:valid-works-costs\]](#eq:valid-works-costs){reference-type="eqref" reference="eq:valid-works-costs"} follows for any $\zeta>0$ fixed. Thus $$\mathbb{P}\big(\mathcal{X}_{(K,A)} \mid V,w_V\big) \ge \mathbb{P}\big(\mathcal{X}_{\mathrm{high\textnormal{-}path}} \mid V,w_V\big).$$ By Proposition [Proposition 43](#prop:path-from-hierarchy){reference-type="ref" reference="prop:path-from-hierarchy"} and the fact that $\xi{\,\gg_{\star}\,}q$, it follows that $$\mathbb{P}\big(\mathcal{X}_{(K,A)} \mid V,w_V\big) \ge 1 - 2e^{-(\log\log\xi)^{13}} \ge 1 - q.\qedhere$$ ◻ We shall now apply Lemma [Lemma 52](#lem:valid-works){reference-type="ref" reference="lem:valid-works"} to prove Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} (which covers the polylogarithmic regime). Here, there are two possible choices of parameters $(\gamma, z, \eta, R)$ for Lemma [Lemma 52](#lem:valid-works){reference-type="ref" reference="lem:valid-works"}: if $\alpha < 2$, then we are able to build a polylogarithmic-cost path using long-range edges between low-weight vertices (Claim [Claim 54](#claim:polylog-low-alpha){reference-type="ref" reference="claim:polylog-low-alpha"} below); if $\mu< \mu_{\log}$ then we are able to build a polylogarithmic-cost path using edges between high-weight vertices (Claim [Claim 55](#claim:polylog-low-mu){reference-type="ref" reference="claim:polylog-low-mu"} below). We then prove Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"} by applying whichever parameter setting constructs a lower-cost path (in Corollary [Corollary 56](#cor:polylog-combine){reference-type="ref" reference="cor:polylog-combine"}). In both regimes, we need the following algebraic fact. **Claim 53**. *Let $$\label{eq:R-def-long} R=R(\xi):= \Big\lceil\frac{\log\log\xi-(\log^{*4}\xi)^2}{\log\gamma^{-1}}\Big\rceil.$$ Then for all $\gamma \in (1/2,1)$ and $\xi{\,\gg_{\star}\,}\gamma$, it holds that $$\label{eq:xi-iteration-bounds} \xi^{\gamma^{R-1}} \in [e^{(\log^{*3}\xi)^2}, e^{\sqrt{\log\log\xi}}].$$* *Proof.* The value of $\xi$ is large, so using [\[eq:R-def-long\]](#eq:R-def-long){reference-type="eqref" reference="eq:R-def-long"} and that $\lceil x\rceil \le x+1$, $$\gamma^{R-1} \ge e^{-\log\log\xi + (\log^{*4}\xi)^2} = (\log^{*3}\xi)^{\log^{*4}\xi}/\log\xi \ge (\log^{*3}\xi)^2/\log \xi.$$ It follows that $\xi^{\gamma^{R-1}} \ge e^{(\log^{*3}\xi)^2}$, as required in [\[eq:xi-iteration-bounds\]](#eq:xi-iteration-bounds){reference-type="eqref" reference="eq:xi-iteration-bounds"}. Moreover, since $\xi{\,\gg_{\star}\,}\gamma$, it holds that $$\gamma^{R-1} \le e^{(\log^{*4}\xi)^2}/(\gamma^2\log\xi) \le e^{(\log^{*3}\xi)/2}/\log\xi = \sqrt{\log\log\xi}/\log\xi.$$ It follows that $\xi^{\gamma^{R-1}} \le e^{\sqrt{\log\log\xi}}$, as required in [\[eq:xi-iteration-bounds\]](#eq:xi-iteration-bounds){reference-type="eqref" reference="eq:xi-iteration-bounds"}. ◻ The next claim finds a $(K,A)$-valid parameter setting when $\alpha<2$, for polylogarithmic cost-bound $KA$. **Claim 54**. *Consider the reduced Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and fix $\varepsilon> 0$. When $\alpha < 2$, then writing $\Delta_\alpha := 1/(1-\log_2\alpha)$, the following assignment is $((\log \xi)^{\Delta_\alpha}, (\log\xi)^\varepsilon)$-valid for $\xi {\,\gg_{\star}\,}\varepsilon,\textnormal{\texttt{par}}\xspace$ and $0<\varepsilon'{\,\ll_{\star}\,}\varepsilon,\textnormal{\texttt{par}}\xspace$: $$\label{eq:choices-for-alpha} \gamma := \frac{\alpha}{2} + \varepsilon'; \qquad z := 0; \qquad \eta := 0; \qquad R:= \Big\lceil\frac{\log\log\xi-(\log^{*4}\xi)^2}{\log\gamma^{-1}}\Big\rceil.$$* *Proof.* We check the requirements in Definition [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} one-by-one. All the requirements of [\[eq:valid-trivial\]](#eq:valid-trivial){reference-type="eqref" reference="eq:valid-trivial"} and [\[eq:valid-R\]](#eq:valid-R){reference-type="eqref" reference="eq:valid-R"} are immediately satisfied except for $R \le (\log\log\xi)^2/4$, which follows from the definition since $\xi{\,\gg_{\star}\,}\gamma$ and $\gamma > 1/2$. Also since $\xi{\,\gg_{\star}\,}\gamma, \textnormal{\texttt{par}}\xspace$, [\[eq:valid-rounds\]](#eq:valid-rounds){reference-type="eqref" reference="eq:valid-rounds"} follows from Claim [Claim 53](#claim:valid-log-gamma-algebra){reference-type="ref" reference="claim:valid-log-gamma-algebra"} since $\overline{w}=\xi^{\gamma^{R-1}d/2}$ and $A=(\log \xi)^\varepsilon$. We now prove [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"}. Since $\xi{\,\gg_{\star}\,}\gamma$, we estimate $2^R$ using $R$ and $\gamma$ in [\[eq:choices-for-alpha\]](#eq:choices-for-alpha){reference-type="eqref" reference="eq:choices-for-alpha"}: $$\label{eq:polylog-low-alpha-1} 2^R \le 2^{\log\log\xi/\log\gamma^{-1}} = (\log \xi)^{\log 2/\log\gamma^{-1}}\!= (\log \xi)^{-\log 2/\log(\alpha/2+\varepsilon')}.$$ Since $\varepsilon'{\,\ll_{\star}\,}\varepsilon$, the exponent of $\log \xi$ on the rhs is $$\label{eq:polylog-low-alpha-2} \frac{\log 2}{-\log (\alpha/2+\varepsilon')} \le \frac{\log 2}{\log(2/\alpha)} + \frac{\varepsilon}{2} = \frac{1}{1-\log_2\alpha} + \frac{\varepsilon}{2} = \Delta_\alpha + \frac{\varepsilon}{2}.$$ Moreover, since $\eta=0$ in [\[eq:choices-for-alpha\]](#eq:choices-for-alpha){reference-type="eqref" reference="eq:choices-for-alpha"}, the other factor $\overline{w}^{4\mu}\xi^\eta$ in [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"} is at most (using Claim [Claim 53](#claim:valid-log-gamma-algebra){reference-type="ref" reference="claim:valid-log-gamma-algebra"} and $\xi {\,\gg_{\star}\,}\varepsilon$), $$\label{eq:polylog-low-alpha-3} \overline{w}^{4\mu}\xi^\eta = \xi^{2\mu d \gamma^{R-1}} \le e^{2\mu d \sqrt{\log\log\xi}} \le (\log\xi)^{\varepsilon/2}/\log\log\xi.$$ Then [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"} with $KA=(\log \xi)^{\Delta_\alpha+\varepsilon}$ follows from [\[eq:polylog-low-alpha-1\]](#eq:polylog-low-alpha-1){reference-type="eqref" reference="eq:polylog-low-alpha-1"}--[\[eq:polylog-low-alpha-3\]](#eq:polylog-low-alpha-3){reference-type="eqref" reference="eq:polylog-low-alpha-3"}. We next prove [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"}. Using the formula in [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}, with $z=0$ and $\eta=0$, and $\gamma=\alpha/2+\varepsilon'$, $$\Lambda(\eta, z):= 2d\gamma-\alpha(d-z)-z(\tau-1)+\big(0\wedge\beta(\eta-\mu z)\big) = 2d(\alpha/2 + \varepsilon') - \alpha d = 2d\varepsilon',$$ so $\Lambda(\eta,z) > 0$ as required. Since $z=0$, [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"} follows, so all criteria in Def. [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} are satisfied. ◻ The next claim finds a $(K,A)$-valid parameter setting when $\mu<\mu_{\log}$, for polylogarithmic cost bound $KA$. **Claim 55**. *Consider the reduced Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and fix $\varepsilon> 0$. When $\mu < \mu_{\log}$, then writing $\Delta_{\beta} := 1/(1-\log_2(\tau-1+\mu\beta))$, the following assignment is $((\log \xi)^{\Delta_\beta}, (\log\xi)^\varepsilon)$-valid for $\xi {\,\gg_{\star}\,}\varepsilon,\textnormal{\texttt{par}}\xspace$ and $\varepsilon'{\,\ll_{\star}\,}\varepsilon,\textnormal{\texttt{par}}\xspace$: $$\label{eq:choices-for-mu} \gamma := \frac{\tau-1+\mu\beta}{2} + \varepsilon'; \qquad z := d; \qquad \eta := 0; \qquad R:= \left\lceil\frac{\log\log\xi-(\log^{*4}\xi)^2}{\log\gamma^{-1}}\right\rceil.$$* *Proof.* First note that $\beta=\infty$ is not possible here, since in that case $\mu_{\log}=0$, see [\[eq:beta-infty-definitions\]](#eq:beta-infty-definitions){reference-type="eqref" reference="eq:beta-infty-definitions"}. We check the requirements in Definition [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} one-by-one. Since $\tau > 2$ and $\mu\beta\ge0$, we obtain $\gamma > 1/2 > 0$ above, and since $\mu < \mu_{\mathrm{log}} = (3-\tau)/\beta$ and $\varepsilon'{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ it also holds that $\gamma < 1$; thus all the requirements of [\[eq:valid-trivial\]](#eq:valid-trivial){reference-type="eqref" reference="eq:valid-trivial"} are satisfied. It is also immediate that [\[eq:valid-R\]](#eq:valid-R){reference-type="eqref" reference="eq:valid-R"} is satisfied except for $R \le (\log\log\xi)^2/4$, which follows from the definition in [\[eq:choices-for-mu\]](#eq:choices-for-mu){reference-type="eqref" reference="eq:choices-for-mu"} since $\xi{\,\gg_{\star}\,}\gamma$ and $\gamma > 1/2$. Since $\xi{\,\gg_{\star}\,}\gamma, \textnormal{\texttt{par}}\xspace$, [\[eq:valid-rounds\]](#eq:valid-rounds){reference-type="eqref" reference="eq:valid-rounds"} follows from Claim [Claim 53](#claim:valid-log-gamma-algebra){reference-type="ref" reference="claim:valid-log-gamma-algebra"} since $\overline{w}=\xi^{\gamma^{R-1}d/2}$ and $A=(\log \xi)^\varepsilon$ as in the previous claim. We now prove [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"}. Analogously to [\[eq:polylog-low-alpha-1\]](#eq:polylog-low-alpha-1){reference-type="eqref" reference="eq:polylog-low-alpha-1"}: $$\label{eq:polylog-high-alpha-1} 2^R \le 2^{\log\log\xi/\log\gamma^{-1}} = (\log \xi)^{\log 2/\log(1/\gamma)}.$$ Since $\varepsilon'{\,\ll_{\star}\,}\varepsilon$, now $\gamma$ is given in [\[eq:choices-for-mu\]](#eq:choices-for-mu){reference-type="eqref" reference="eq:choices-for-mu"} and $$\label{eq:polylog-high-alpha-2} \frac{\log 2}{\log (1/\gamma)} \le \frac{\log 2}{\log(2/(\tau-1+\mu\beta))} + \frac{\varepsilon}{2} = \frac{1}{1-\log_2(\tau-1+\mu\beta)} + \frac{\varepsilon}{2} = \Delta_{\beta} + \frac{\varepsilon}{2}.$$ Moreover, by Claim [Claim 53](#claim:valid-log-gamma-algebra){reference-type="ref" reference="claim:valid-log-gamma-algebra"} and since $\xi{\,\gg_{\star}\,}\varepsilon$, it holds that $$\label{eq:polylog-high-alpha-3} \overline{w}^{4\mu}\xi^\eta = \xi^{2\mu d \gamma^{R-1}} \le e^{2\mu d \sqrt{\log\log\xi}} \le (\log\xi)^{\varepsilon/2}/\log\log\xi.$$ Now [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"} follows immediately from [\[eq:polylog-high-alpha-1\]](#eq:polylog-high-alpha-1){reference-type="eqref" reference="eq:polylog-high-alpha-1"}--[\[eq:polylog-high-alpha-3\]](#eq:polylog-high-alpha-3){reference-type="eqref" reference="eq:polylog-high-alpha-3"}. We next prove [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"}. Using the formula in [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"}, with $z=d$ and $\eta=0$, and $\gamma$ as in [\[eq:choices-for-mu\]](#eq:choices-for-mu){reference-type="eqref" reference="eq:choices-for-mu"}, $$\Lambda(\eta, z) = 2d\gamma-\alpha(d-z)-z(\tau-1)+\big(0\wedge\beta(\eta-\mu z)\big) = 2d\gamma - d(\tau-1) - d\mu\beta = 2d\varepsilon',$$ so $\Lambda(\eta,z) > 0$ as required. This also remains true for $\alpha=\infty$ both formally with $\alpha(d-z)=\infty\cdot 0=0$ as well as intuitively, since $z=d$ means we use edges which are present with constant probability. It remains to prove $\Phi(\eta,z) > 0$. Using the formula in [\[eq:Phi-def\]](#eq:Phi-def){reference-type="eqref" reference="eq:Phi-def"}, and that $\gamma\wedge1/2=1/2$, $$\Phi(\eta, z) = \Big[d\gamma \wedge \frac{z}{2}\Big] + \Big[0 \wedge \beta\Big(\eta - \frac{\mu z}{2}\Big)\Big] = d(\gamma \wedge 1/2) - \beta\mu d/2 = d(1-\mu\beta)/2.$$ Since $\mu \le \mu_{\mathrm{log}} = (3-\tau)/\beta$, it follows that $\Phi(\eta,z) \ge d(\tau-2)/2$; since $\tau > 2$, [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"} follows. ◻ Comparing the definition of $\Delta_0$ in [\[eq:Delta_0\]](#eq:Delta_0){reference-type="eqref" reference="eq:Delta_0"} to those in Claims [Claim 54](#claim:polylog-low-alpha){reference-type="ref" reference="claim:polylog-low-alpha"} and [Claim 55](#claim:polylog-low-mu){reference-type="ref" reference="claim:polylog-low-mu"}, we recover here that $$\label{eq:delta-again} \Delta_0 = \frac{1}{1 - \log_2 (\min\{\alpha, \tau-1+\mu\beta\})}=\min\{\Delta_\alpha, \Delta_\beta\},$$ which formally remains true also when $\alpha = \infty$ or $\beta = \infty$ by [\[eq:alpha-infty-Delta_0\]](#eq:alpha-infty-Delta_0){reference-type="eqref" reference="eq:alpha-infty-Delta_0"}, or [\[eq:beta-infty-Delta_0\]](#eq:beta-infty-Delta_0){reference-type="eqref" reference="eq:beta-infty-Delta_0"}. Combining the two claims we obtain the following corollary: **Corollary 56**. *Consider the reduced Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, fix $\varepsilon> 0$. When either $\alpha\in(1,2)$ or $\mu\in(\mu_{\mathrm{expl}},\mu_{\log})$ or both hold, then there exists a setting of $(\gamma,z, \eta,R)$ (depending on $\varepsilon$) which is $((\log\xi)^{\Delta_0}, (\log\xi)^\varepsilon)$-valid for $\xi{\,\gg_{\star}\,}\varepsilon,\textnormal{\texttt{par}}\xspace$.* *Proof.* Recall that $\mu_{\mathrm{log}} = (3-\tau)/\beta$, so if $\mu_{\mathrm{expl}} < \mu < \mu_{\mathrm{log}}$ then $\beta < \infty$; thus we cannot have $\alpha = \beta = \infty$, and the formula [\[eq:delta-again\]](#eq:delta-again){reference-type="eqref" reference="eq:delta-again"} is valid whenever at least one of $\alpha, \beta$ is finite. We show that when the minimum in the denominator is $\alpha \le \tau-1+\mu\beta$, (so that $\Delta_0 = \Delta_\alpha$), then also $\alpha<2$ holds. Then, Claim [Claim 54](#claim:polylog-low-alpha){reference-type="ref" reference="claim:polylog-low-alpha"} directly gives a $((\log\xi)^{\Delta_\alpha},(\log\xi)^\varepsilon)$-valid parameter setting. There are two cases: either $\mu>\mu_{\log}$, then $\alpha < 2$ must hold by the hypothesis of the lemma; or $\mu<\mu_{\log}=(3-\tau)/\beta$, so $\alpha$ being the minimum gives that $\alpha< \tau-1+\mu_{\log}\cdot\beta=2$. Similarly, we show that when the minimum in the denominator is $\tau-1+\mu\beta<\alpha$, (so that $\Delta_0 = \Delta_\beta$), then also $\mu<\mu_{\log }$ holds. Then, Claim [Claim 55](#claim:polylog-low-mu){reference-type="ref" reference="claim:polylog-low-mu"} directly gives a $((\log\xi)^{\Delta_\beta},(\log\xi)^\varepsilon)$-valid parameter setting. There are again two cases: either $\alpha \ge 2$, then $\mu < \mu_{\mathrm{log}}$ must hold by the hypothesis of the lemma; or $\alpha<2$, so $\tau-1+\mu\beta$ being the minimum gives that $\tau-1 + \mu\beta<2$ and hence $\mu < (3-\tau)/\beta = \mu_{\mathrm{log}}$. ◻ Finally, we are ready to prove the first main result of Section [5](#sec:hierarchy){reference-type="ref" reference="sec:hierarchy"}: *Proof.* Immediate from combining Lemma [Lemma 52](#lem:valid-works){reference-type="ref" reference="lem:valid-works"} with Corollary [Corollary 56](#cor:polylog-combine){reference-type="ref" reference="cor:polylog-combine"}, where the required bounds on $\overline{w}$ in [\[eq:weight-crit-cor1\]](#eq:weight-crit-cor1){reference-type="eqref" reference="eq:weight-crit-cor1"} follow from [\[eq:valid-rounds\]](#eq:valid-rounds){reference-type="eqref" reference="eq:valid-rounds"}. ◻ We next apply Lemma [Lemma 52](#lem:valid-works){reference-type="ref" reference="lem:valid-works"} to prove Corollary [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"} that covers the polynomial regime. As with the proof of Corollary [\[cor:computations-polylog\]](#cor:computations-polylog){reference-type="ref" reference="cor:computations-polylog"}, we show that multiple possible choices of parameters are valid and choose the one which yields the lowest-cost path. We start with the case where $\alpha = \beta = \infty$. Recall the definition of $\eta_0$ from [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}, [\[eq:alpha-infty-definitions\]](#eq:alpha-infty-definitions){reference-type="eqref" reference="eq:alpha-infty-definitions"}, [\[eq:beta-infty-definitions\]](#eq:beta-infty-definitions){reference-type="eqref" reference="eq:beta-infty-definitions"} and [\[eq:alpha-beta-infty-definitions\]](#eq:alpha-beta-infty-definitions){reference-type="eqref" reference="eq:alpha-beta-infty-definitions"}. **Claim 57**. *Consider the reduced Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and fix $\varepsilon> 0$. When $\alpha=\beta=\infty$ and $\mu \in(\mu_{\mathrm{log}} ,\mu_{\mathrm{pol}}]$, i.e., $\eta_0=1\wedge d\mu$ in [\[eq:alpha-beta-infty-definitions\]](#eq:alpha-beta-infty-definitions){reference-type="eqref" reference="eq:alpha-beta-infty-definitions"}, then the following setting is $(\xi^{\eta_0},\xi^\varepsilon)$-valid whenever $\varepsilon'{\,\ll_{\star}\,}\varepsilon,\textnormal{\texttt{par}}\xspace$ (with $1/(\varepsilon')^2$ an integer), and $\xi{\,\gg_{\star}\,}\varepsilon,\varepsilon',\textnormal{\texttt{par}}\xspace$: $$\label{eq:choices-for-eta-inf-inf} \gamma := 1 - \varepsilon';\qquad z := d;\qquad \eta := \eta_0 + \sqrt{\varepsilon'};\qquad R := 1/(\varepsilon')^2.$$* *Proof.* Recall from [\[eq:alpha-beta-infty-definitions\]](#eq:alpha-beta-infty-definitions){reference-type="eqref" reference="eq:alpha-beta-infty-definitions"} that when $\alpha=\beta=\infty$, the values $\mu_{\mathrm{log}} = 0$, $\mu_{\mathrm{pol}} = 1/d$. We check the requirements in Definition [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} one-by-one. Both [\[eq:valid-trivial\]](#eq:valid-trivial){reference-type="eqref" reference="eq:valid-trivial"} and [\[eq:valid-R\]](#eq:valid-R){reference-type="eqref" reference="eq:valid-R"} are immediate. Since $\varepsilon' < 1/2$, it holds that $\gamma \in [e^{-2\varepsilon'}, e^{-\varepsilon'}]$; thus $\gamma^{R-1} \in [e^{-2/\varepsilon'}, e^{-1/(2\varepsilon')}]$ by the choice of $R$ in [\[eq:choices-for-eta-inf-inf\]](#eq:choices-for-eta-inf-inf){reference-type="eqref" reference="eq:choices-for-eta-inf-inf"}. Since $\xi{\,\gg_{\star}\,}\varepsilon'$ and $\varepsilon'{\,\ll_{\star}\,}\varepsilon, \textnormal{\texttt{par}}\xspace$, it follows that $\xi^{\gamma^{R-1}d} \in [e^{(\log^{*3}\xi)^2}, \xi^\varepsilon/\log\log\xi]$ as required by [\[eq:valid-rounds\]](#eq:valid-rounds){reference-type="eqref" reference="eq:valid-rounds"}. Moreover, using that $\overline{w}=\xi^{\gamma^{R-1}2/d}$ we estimate $2^R\overline{w}^{4\mu}\xi^\eta=2^R\xi^{\eta+2\mu d\gamma^{R-1}} \le \xi^{\varepsilon/3} \cdot \xi^{\eta_0} \cdot \xi^{\varepsilon/3} \le \xi^{\eta_0+\varepsilon}/\log\log \xi$ and [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"} holds. It remains to prove [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"}. Using the formula in  [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"} with $\gamma, \eta,z$ as in [\[eq:choices-for-eta-inf-inf\]](#eq:choices-for-eta-inf-inf){reference-type="eqref" reference="eq:choices-for-eta-inf-inf"}, and that $\mu\le \mu_{\mathrm{pol}}$, $$\begin{aligned} \Lambda(\eta, z) &= 2d\gamma-\alpha(d-z)-z(\tau-1)+\big(0\wedge\beta(\eta-\mu z)\big)\\ &= 2d(1-\varepsilon') - \infty\cdot 0 - d(\tau-1) + (0 \wedge \infty) = d(3-\tau-2\varepsilon'); \end{aligned}$$ since $\tau < 3$ and $\varepsilon'{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, $\Lambda(\eta,z)>0$ as required. Finally, using the formula in [\[eq:Phi-def\]](#eq:Phi-def){reference-type="eqref" reference="eq:Phi-def"} and that $\gamma\wedge 1/2=1/2$, we analogously obtain that $$\Phi(\eta, z) = \Big[d\gamma \wedge \frac{z}{2}\Big] + \Big[0 \wedge \beta\Big(\eta - \frac{\mu z}{2}\Big)\Big] = d(\gamma \wedge 1/2) + (0 \wedge \infty(d\mu/2 + \sqrt{\varepsilon'})) = d/2 > 0,$$ so [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"} follows. Hence, all criteria in Def. [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} are satisfied. ◻ When at least one of $\alpha, \beta$ is non-infinite, we can find two possible optimisers: one when $\mu<\mu_{\mu_{\mathrm{pol,\alpha}}}$ and one when $\mu<\mu_{\mathrm{pol, \beta}}$ hold in [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"}. We treat the two cases separately. Recall $\mu_{\mathrm{pol},\beta}=1/d+(3- \tau)/\beta$ and let $\eta_\beta:=d(\mu-\mu_{\log})$, the first term in the second row of [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}. **Claim 58**. *Consider the reduced Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and fix $\varepsilon> 0$. When $\alpha > 2$, $\mu\in(\mu_{\mathrm{log}},\mu_{\mathrm{pol},\beta}]$, then the following setting is $(\xi^{\eta_\beta},\xi^\varepsilon)$-valid for $\varepsilon'{\,\ll_{\star}\,}\varepsilon,\textnormal{\texttt{par}}\xspace$ (with $1/(\varepsilon')^2$ an integer) and $\xi{\,\gg_{\star}\,}\varepsilon,\varepsilon',\textnormal{\texttt{par}}\xspace$: $$\label{eq:choices-for-eta-mu} \gamma := 1 - \varepsilon';\qquad z := d;\qquad \eta := \eta_\beta + \sqrt{\varepsilon'};\qquad R := 1/(\varepsilon')^2.$$* *Proof.* The $\alpha=\beta=\infty$ case was treated in Claim [Claim 57](#claim:polynomial-infinite){reference-type="ref" reference="claim:polynomial-infinite"} with [\[eq:choices-for-eta-inf-inf\]](#eq:choices-for-eta-inf-inf){reference-type="eqref" reference="eq:choices-for-eta-inf-inf"} coinciding with [\[eq:choices-for-eta-mu\]](#eq:choices-for-eta-mu){reference-type="eqref" reference="eq:choices-for-eta-mu"}. We treat the cases when at least one of $\alpha,\beta$ is finite. We check the requirements in Definition [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} one-by-one. Both [\[eq:valid-trivial\]](#eq:valid-trivial){reference-type="eqref" reference="eq:valid-trivial"} and [\[eq:valid-R\]](#eq:valid-R){reference-type="eqref" reference="eq:valid-R"} are immediate. Since $\varepsilon'$ is small we may choose it $\varepsilon' < 1/2$, implying that $\gamma \in [e^{-2\varepsilon'}, e^{-\varepsilon'}]$; thus $\gamma^{R-1} \in [e^{-2/\varepsilon'}, e^{-1/(2\varepsilon')}]$. Since $\xi{\,\gg_{\star}\,}\varepsilon'$ and $\varepsilon'{\,\ll_{\star}\,}\varepsilon,\textnormal{\texttt{par}}\xspace$, it follows that $\xi^{\gamma^{R-1}d} \in [e^{(\log^{*3}\xi)^2}, \xi^\varepsilon/\log\log\xi]$ as required by [\[eq:valid-rounds\]](#eq:valid-rounds){reference-type="eqref" reference="eq:valid-rounds"}. Moreover, for [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"} we use that $\overline{w}=\xi^{\gamma^{R-1}2/d}$ and estimate $2^R\overline{w}^{4\mu}\xi^\eta = 2^R\xi^{\eta+2\mu d\gamma^{R-1}} \le \xi^{\varepsilon/3} \cdot \xi^{\eta_{\beta}} \cdot \xi^{\varepsilon/3} \le \xi^{\eta_{\beta}+\varepsilon}/\log\log \xi$ and so [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"} holds. It remains to prove [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"}. By their definition in [\[eq:choices-for-eta-mu\]](#eq:choices-for-eta-mu){reference-type="eqref" reference="eq:choices-for-eta-mu"}, $z=d$ and $\eta=\eta_\beta+\sqrt{\varepsilon'}$ where $\eta_\beta=d(\mu-\mu_{\log})=d(\mu-(3-\tau)/\beta)$, we compute $\eta - \mu z = \sqrt{\varepsilon'} - (3-\tau)d/\beta < 0$, since $\varepsilon'{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$. So, using the formula in [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"} with $\gamma, \eta,z$ as in [\[eq:choices-for-eta-mu\]](#eq:choices-for-eta-mu){reference-type="eqref" reference="eq:choices-for-eta-mu"}, $$\begin{aligned} \Lambda(\eta, z) &= 2d\gamma-\alpha(d-z)-z(\tau-1)+\big(0\wedge\beta(\eta-\mu z)\big)\\ &= 2d(1-\varepsilon') - d(\tau-1) + \beta \sqrt{\varepsilon'} - (3-\tau)d = \beta\sqrt{\varepsilon'} - 2d\varepsilon'. \end{aligned}$$ Since $\varepsilon'{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, it follows that $\Lambda(\eta,z) > 0$ as required by [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"}. This computation also remains valid both formally and intuitively when $\alpha=\infty$ and $\beta<\infty$, since $z=d$ and $\alpha(d-d)=0$ reflects the fact that the edges we use appear with constant probability each. When $\alpha<\infty$ and $\beta=\infty$, $\mu_{\log}=0$ and $\mu_{\mathrm{pol},\beta}=1/d$, hence $\eta-\mu z=d \mu+\sqrt{\varepsilon'} -\mu d=\sqrt{\varepsilon'}$, so the minimum in $0\wedge \beta(\eta-\mu z)=0$. Hence when $\beta=\infty$, since $\gamma=1-\varepsilon'$ and $\tau-1<2$, $$\begin{aligned} \label{eq:beta-comp-again} \Lambda(\eta, z) &= 2d\gamma-\alpha(d-d)-d(\tau-1)= d(2\gamma-(\tau-1))>0. \end{aligned}$$ Finally we treat $\Phi(\eta,z)>0$. When $\beta<\infty$, using the formula in [\[eq:Phi-def\]](#eq:Phi-def){reference-type="eqref" reference="eq:Phi-def"} and that $\gamma\wedge 1/2=1/2$, with parameters in [\[eq:choices-for-eta-mu\]](#eq:choices-for-eta-mu){reference-type="eqref" reference="eq:choices-for-eta-mu"} and $\eta=\eta_\beta+\sqrt{\varepsilon'}=\mu d - \frac{(3-\tau)d}{\beta}+\sqrt{\varepsilon'}$, we analogously obtain that $$\begin{aligned} \Phi(\eta,z) &= \Big[d\gamma \wedge \frac{z}{2}\Big] + \Big[0 \wedge \beta\Big(\eta - \frac{\mu z}{2}\Big)\Big] = \frac{d}{2} + \Big[0 \wedge \beta\Big(\sqrt{\varepsilon'} + \frac{\mu d}{2} - \frac{(3-\tau)d}{\beta}\Big)\Big]. \end{aligned}$$ In case the minimum on the rhs is at $0$, $\Phi(\eta,z)>0$ and so [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"} is satisfied. In case the minimum is at the other term, we use that $\mu > \mu_{\mathrm{log}} = (3-\tau)/\beta$, so $\mu d/2 > (3-\tau)d/(2\beta)$, so $$\Phi(\eta,z) \ge \frac{d}{2} - \beta\cdot \frac{(3-\tau)d}{2\beta} = \frac{(\tau-2)d}{2} > 0.$$ and so $\tau\in(2,3)$ ensures that [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"} holds again. The computation remains valid when $\alpha=\infty$ since $\Phi$ does not depend on $\alpha$. When $\alpha<\infty$ and $\beta=\infty$, the computation simplifies, and $\eta-\mu z/2>0$ holds since already $\eta-\mu z>0$ see above [\[eq:beta-comp-again\]](#eq:beta-comp-again){reference-type="eqref" reference="eq:beta-comp-again"}. Hence in this case $\Phi(\eta,z)=d\gamma\wedge z/2= d/2>0$. Hence, all criteria in Definition [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} are satisfied with the choice in [\[eq:choices-for-eta-mu\]](#eq:choices-for-eta-mu){reference-type="eqref" reference="eq:choices-for-eta-mu"}. ◻ The next claim finds minimisers whenever $\mu<\mu_{\mathrm{pol,\alpha}}$. Recall that $\mu_{\mathrm{pol},\alpha}=\tfrac{\alpha-(\tau-1)}{d(\alpha-2)}$ from [\[eq:mu_pol_log\]](#eq:mu_pol_log){reference-type="eqref" reference="eq:mu_pol_log"} and let $\eta_\alpha:=\mu/\mu_{\mathrm{pol},\alpha}$, the second term in the second row of [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}. **Claim 59**. *Consider the reduced Setting [\[set:hierarchy-common\]](#set:hierarchy-common){reference-type="ref" reference="set:hierarchy-common"}, and fix $\varepsilon> 0$. When $\alpha > 2$, $\mu\in(\mu_{\mathrm{log}}, \mu_{\mathrm{pol},\alpha}]$, then the following setting is $(\xi^{\eta_\alpha},\xi^\varepsilon)$-valid for $\varepsilon'{\,\ll_{\star}\,}\varepsilon,\textnormal{\texttt{par}}\xspace$ (with $1/(\varepsilon')^2$ and integer), and $\xi{\,\gg_{\star}\,}\varepsilon,\varepsilon',\textnormal{\texttt{par}}\xspace$: $$\label{eq:choices-for-eta-alpha} \gamma := 1 - \varepsilon';\qquad z := (\eta_\alpha + \sqrt{\varepsilon'})/\mu;\qquad \eta := \eta_\alpha + \sqrt{\varepsilon'};\qquad R := 1/(\varepsilon')^2.$$* *Proof.* We first show that $\alpha=\infty$, $\beta<\infty$ is not possible here. From [\[eq:alpha-infty-definitions\]](#eq:alpha-infty-definitions){reference-type="eqref" reference="eq:alpha-infty-definitions"} it follows that $\mu_{\mathrm{pol,\alpha}}=1/d$, while $\mu_{\log}=1/d+(3-\tau)/\beta$, so for all $\beta>0$ the strict inequality $\mu_{\log}> \mu_{\mathrm{pol},\alpha}$ holds and hence the interval for $\mu$ is empty when $\alpha=\infty$. Hence $\alpha<\infty$ is necessary for the conditions to be satisfied. We check the requirements of Definition [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} one-by-one. Using the formula for $\mu_{\mathrm{pol},\alpha}$ and $\tau <3$, we compute that $\eta_\alpha = \mu d(\alpha-2)/(\alpha-(\tau-1)) < \mu d$. Hence, since $\varepsilon'{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ for all sufficiently small $\varepsilon'$ the inequality $z\le d$ holds as required by [\[eq:valid-trivial\]](#eq:valid-trivial){reference-type="eqref" reference="eq:valid-trivial"}. The other conditions of [\[eq:valid-trivial\]](#eq:valid-trivial){reference-type="eqref" reference="eq:valid-trivial"} and [\[eq:valid-R\]](#eq:valid-R){reference-type="eqref" reference="eq:valid-R"} are immediate. Since $\gamma$ and $\eta$ is the same here and in Claim [Claim 58](#claim:polynomial-small-mu){reference-type="ref" reference="claim:polynomial-small-mu"}, [\[eq:valid-rounds\]](#eq:valid-rounds){reference-type="eqref" reference="eq:valid-rounds"} and [\[eq:valid-low-cost\]](#eq:valid-low-cost){reference-type="eqref" reference="eq:valid-low-cost"} hold by the same argument as in Claim [Claim 58](#claim:polynomial-small-mu){reference-type="ref" reference="claim:polynomial-small-mu"}. It remains to prove [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"}. Using the formula in  [\[eq:Lambda-def\]](#eq:Lambda-def){reference-type="eqref" reference="eq:Lambda-def"} with $\gamma, \eta,z$ as in [\[eq:choices-for-eta-alpha\]](#eq:choices-for-eta-alpha){reference-type="eqref" reference="eq:choices-for-eta-alpha"}, which implies that $\eta-\mu z=0$, $$\begin{aligned} \Lambda(\eta, z) &= 2d\gamma-\alpha(d-z)-z(\tau-1)+\big(0\wedge\beta(\eta-\mu z)\big)\\ &= d(2-\alpha) - 2\varepsilon'd + z(\alpha-(\tau-1)) + 0. \end{aligned}$$ This also remains valid both formally and intuitively when $\beta=\infty$ (with the convention that $\infty\cdot 0=0)$, since $\eta-\mu z=0$ reflects the fact that the random variable $L$ on the edge we use is constant order. We substitute $z=(\eta_\alpha+\sqrt{\varepsilon'})/\mu$ from [\[eq:choices-for-eta-alpha\]](#eq:choices-for-eta-alpha){reference-type="eqref" reference="eq:choices-for-eta-alpha"} and $\eta_\alpha=\mu d(\alpha-2)/(\alpha-(\tau-1))$: $$\begin{aligned} \Lambda(\eta, z) &= d(2-\alpha) + \eta_\alpha(\alpha-(\tau-1))/\mu + \sqrt{\varepsilon'}(\alpha-(\tau-1))/\mu - 2\varepsilon' d\\ &= \sqrt{\varepsilon'}(\alpha-(\tau-1))/\mu - 2\varepsilon'd, \end{aligned}$$ since the first two terms in the first row cancelled each other. Since $\varepsilon'{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, $\alpha>2$ and $\tau\in(2,3)$, $\alpha-(\tau-1)$ is positive, and so is $\mu>0$, so $\Lambda(\eta,z)>0$ as required by [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"}. Finally, by [\[eq:Phi-def\]](#eq:Phi-def){reference-type="eqref" reference="eq:Phi-def"} and since $z\le d$, $$\begin{aligned} \Phi(\eta,z) &= \Big[d\gamma \wedge \frac{z}{2}\Big] + \Big[0 \wedge \beta\Big(\eta - \frac{\mu z}{2}\Big)\Big] = \frac{z}{2} + 0 > 0, \end{aligned}$$ and so [\[eq:valid-lambda-phi\]](#eq:valid-lambda-phi){reference-type="eqref" reference="eq:valid-lambda-phi"} holds. This also remains true for $\beta=\infty$ since the minimum is at $0$, meaning we use edges with constant value $L$. Hence, all criteria in Definition [Definition 51](#def:valid){reference-type="ref" reference="def:valid"} are satisfied with the choice in [\[eq:choices-for-eta-alpha\]](#eq:choices-for-eta-alpha){reference-type="eqref" reference="eq:choices-for-eta-alpha"}. ◻ We are ready to prove Corollary [\[cor:computations-polynomial\]](#cor:computations-polynomial){reference-type="ref" reference="cor:computations-polynomial"}: *Proof.* Claim [Claim 57](#claim:polynomial-infinite){reference-type="ref" reference="claim:polynomial-infinite"} finds a setting of parameters that is $(|x|^{\eta_0}, |x|^{\varepsilon})$-valid whenever $\alpha=\beta=\infty$ and $\mu\le \mu_{\mathrm{pol}}=1/d$. When at least one of $\alpha,\beta$ is non-infinite, Claims [Claim 58](#claim:polynomial-small-mu){reference-type="ref" reference="claim:polynomial-small-mu"} and [Claim 59](#claim:polynomial-large-mu){reference-type="ref" reference="claim:polynomial-large-mu"} respectively find a setting of parameters that is $(|x|^{\eta_\beta}, |x|^{\varepsilon})$-valid whenever $\mu\le \mu_{\mathrm{pol,\beta}}$ and one that is $(|x|^{\eta_\alpha}, |x|^{\varepsilon})$-valid whenever $\mu\le \mu_{\mathrm{pol,\alpha}}$. By noting that $\eta_{\beta}\le 1$ exactly when $\mu<\mu_{\mathrm{pol}, \beta}$ and $\eta_{\alpha}\le 1$ exactly when $\mu\le \mu_{\mathrm{pol}, \alpha}$, we obtain that whenever $\mu\le\max\{\mu_{\mathrm{pol}, \alpha},\mu_{\mathrm{pol}, \beta}\}$, the two claims together find a parameter setting that is $(|x|^{\min\{\eta_\alpha, \eta_\beta\}}, |x|^{\varepsilon})$ valid. Since $\eta_0=\min\{\eta_\alpha, \eta_\beta\}$ in [\[eq:eta_0\]](#eq:eta_0){reference-type="eqref" reference="eq:eta_0"}, the proof from here is immediate by applying Lemma [Lemma 52](#lem:valid-works){reference-type="ref" reference="lem:valid-works"}, where the required bounds on $\overline{w}$ in [\[eq:weight-crit-cor2\]](#eq:weight-crit-cor2){reference-type="eqref" reference="eq:weight-crit-cor2"} follow from [\[eq:valid-rounds\]](#eq:valid-rounds){reference-type="eqref" reference="eq:valid-rounds"}. ◻ # Connecting the endpoints when $d=1$ {#app:1d-endpoints} In this section we provide the missing proofs from Section [6](#sec:endpoints){reference-type="ref" reference="sec:endpoints"} for $d=1$. We first show a simple variant of [@komjathy2020stopping Lemma 4.3]; this lemma says that any suitably high-weight vertex is very likely to lie at the start of an infinite weight-increasing path. **Lemma 60**. *Consider Setting [Setting 45](#set:joining-common){reference-type="ref" reference="set:joining-common"} with $d=1$. Let $\theta>1$ and $\delta\in(0,1)$ with $\delta,\theta-1{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, and let $M_0 {\,\gg_{\star}\,}\theta,\delta,\textnormal{\texttt{par}}\xspace$. Let $z \in \mathbb{R}$ (or $\mathbb{Z}$ for SFP), and for all $i\ge 0$ define, $M_i := M_0^{\theta^i}$, $R_i := M_i^{(1+\delta)(\tau-1)}$, and $I_i := [z, z+R_i]$. Let $\mathcal{A}_{\mathrm{inc}}(M_0,\theta,z)$ be the event that there is an infinite path $\pi_z = z_0z_1\dots$ in $G$ starting at $z=:z_0$ such that for all $i \ge 1$ we have $z_i \in (I_i\setminus I_{i-1}) \cap \mathcal{V}_{M_i}$. Then $$\label{eq:weight-increasing-11} \mathbb{P}(\neg\mathcal{A}_{\mathrm{inc}}(M_0,\theta,z) \mid z \in \mathcal{V}_{M_0}) \le \exp(-M_0^{\delta(\tau-1)/4}).$$ The bound remains true if we additionally condition on $y\in \mathcal{V}$ for any $y\in \mathbb{R}\setminus\{z\}$ for GIRG.* *Proof.* The proof is very similar to [@komjathy2020stopping Lemma 4.3], which uses a similar construction but in more than one dimension and with less control over the weights. For all $j \ge 1$, let $\mathcal{A}_{\mathrm{inc}}^j$ be the event that there is a path $\pi_z = z_0z_1\dots z_j$ in $G$ with $z_0 := z$ such that for all $i \in [j]$ we have $z_i \in (I_i\setminus I_{i-1})\cap \mathcal{V}_{M_i}$. Let $\mathcal{A}_{\mathrm{inc}}^0$ be the empty event. Then $$\label{eq:1d-infinite-path-Ainci} \mathbb{P}(\neg\mathcal{A}_{\mathrm{inc}}(M_0,\theta,z) \mid z \in \mathcal{V}_{M_0}) = \sum_{i=1}^\infty \mathbb{P}(\neg\mathcal{A}_{\mathrm{inc}}^i \mid \mathcal{A}_{\mathrm{inc}}^{i-1} \mbox{ and } z \in \mathcal{V}_{M_0}).$$ We now bound each term in the sum of [\[eq:1d-infinite-path-Ainci\]](#eq:1d-infinite-path-Ainci){reference-type="eqref" reference="eq:1d-infinite-path-Ainci"} above. Fix $i \ge 1$. Observe that $\mathcal{A}_{\mathrm{inc}}^{i-1}$ only depends on $G[I_{i-1}]$. Let $G'$ be a possible value (realisation) of $G[I_{i-1}]$ which implies $\mathcal{A}_{\mathrm{inc}}^{i-1}$. Then we can decompose the conditioning in [\[eq:1d-infinite-path-Ainci\]](#eq:1d-infinite-path-Ainci){reference-type="eqref" reference="eq:1d-infinite-path-Ainci"} by conditioning on events of the type $\mathcal{F}_i := \{G[I_{i-1}] = G'\} \cap \{z\in \mathcal{V}_{M_0}\}$ and later integrating over the possible realisations $G'$. Given $G'$ satisfying $\mathcal{A}_{\mathrm{inc}}^{i-1}$, fix the vertices $z_0,\dots,z_{i-1}$ ensuring $\mathcal{A}_{\mathrm{inc}}^{i-1}$. Let $\mathcal{A}_{\mathrm{vert}}^i$ be the event that $|(I_i\setminus I_{i-1}) \cap \mathcal{V}_{M_i}| \ge M_i^{\delta(\tau-1)/2}$; then $$\label{eq:1d-infinite-path-union-1} \mathbb{P}(\neg\mathcal{A}_{\mathrm{inc}}^i \mid \mathcal{F}_i) \le \mathbb{P}(\neg\mathcal{A}_{\mathrm{vert}}^i \mid \mathcal{F}_i) + \mathbb{P}(\neg\mathcal{A}_{\mathrm{inc}}^i \mid \mathcal{A}_{\mathrm{vert}}^i \cap \mathcal{F}_i).$$ By [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}, the number of vertices in $(I_i\setminus I_{i-1}) \cap \mathcal{V}_{M_i}$ is either a Poisson variable (for IGIRG) or a binomial variable (for SFP) with mean at least $$(R_i-R_{i-1}-1)\Big(\frac{\ell(M_i)}{M_i^{\tau-1}} - \frac{\ell(2M_i)}{(2M_i)^{\tau-1}} \Big).$$ Since $\ell$ is slowly-varying, $\tau > 2$, and $M_i > M_0 {\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$, it follows that $$\mathbb{E}\big[|(I_i\setminus I_{i-1}) \cap \mathcal{V}_{M_i}| \mid \mathcal{F}_i\big] \ge \frac{R_i}{2}\cdot \frac{\ell(M_i)}{4M_i^{\tau-1}} = \frac{\ell(M_i)M_i^{\delta(\tau-1)}}{8} \ge 2 M_i^{\delta(\tau-1)/2}.$$ In both IGIRG and SFP, it follows that $$\label{eq:1d-infinite-path-vert-prob} \mathbb{P}(\neg\mathcal{A}_{\mathrm{vert}}^i \mid \mathcal{F}_i) \le \exp(-M_i^{\delta(\tau-1)/2}).$$ We next lower-bound the probability that $z_{i-1}$ is connected to any given $z' \in (I_i\setminus I_{i-1}) \cap \mathcal{V}_{M_i}$. Let $(V,w_V)$ be a possible value of $\widetilde{\mathcal{V}}$ which implies $\mathcal{A}_{\mathrm{vert}}^i$, and suppose that $z' \in (I_i\setminus I_{i-1}) \cap \mathcal{V}_{M_i}$ for $\widetilde{\mathcal{V}} = (V,w_V)$. The distance between $z_{i-1}$ and $z'$ is at most $R_i$, and vertices have weight in $[M, 2M]$ in $\mathcal{V}_M$, so by [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"} (remembering that $d=1$), $$\begin{aligned} \begin{split}\label{eq:1d-infinite-path-conn-prob} \mathbb{P}(z_{i-1}z'\in\mathcal{E}\mid (V,w_V),\mathcal{F}_i) &\ge \underline{c} \cdot \min\Big\{1,\tfrac{M_{i-1}M_i}{R_i}\Big\}^\alpha\\ &= \underline{c}\cdot \min\Big\{1,M_0^{\theta^{i-1}[1+\theta-\theta (1+\delta)(\tau-1)]}\Big\}^\alpha. \end{split}\end{aligned}$$ Observe that $1+\theta > 2$, and that since $\theta-1,\delta{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ and $\tau < 3$ we have $\theta (1+\delta)(\tau-1) < 2$; thus the exponent on the rhs of [\[eq:1d-infinite-path-conn-prob\]](#eq:1d-infinite-path-conn-prob){reference-type="eqref" reference="eq:1d-infinite-path-conn-prob"} is positive and we obtain $$\mathbb{P}(z_{i-1}z'\in\mathcal{E}\mid (V,w_V),\mathcal{F}_i) \ge \underline{c}.$$ By $\mathcal{A}_{\mathrm{vert}}^i$ (defined above [\[eq:1d-infinite-path-union-1\]](#eq:1d-infinite-path-union-1){reference-type="eqref" reference="eq:1d-infinite-path-union-1"}) there are at least $M_i^{\delta(\tau-1)/2}$ such vertices $z'$, each joined to $z_{i-1}$ independently. Thus, $$\label{eq:1d-infinit-path-inc-prob} \mathbb{P}(\neg\mathcal{A}_{\mathrm{inc}}^i \mid \mathcal{A}_{\mathrm{vert}}^i \cap \mathcal{F}_i) \le \exp(-\log(1/\underline{c})M_i^{\delta(\tau-1)/2}).$$ Combining [\[eq:1d-infinite-path-union-1\]](#eq:1d-infinite-path-union-1){reference-type="eqref" reference="eq:1d-infinite-path-union-1"}, [\[eq:1d-infinite-path-vert-prob\]](#eq:1d-infinite-path-vert-prob){reference-type="eqref" reference="eq:1d-infinite-path-vert-prob"} and [\[eq:1d-infinit-path-inc-prob\]](#eq:1d-infinit-path-inc-prob){reference-type="eqref" reference="eq:1d-infinit-path-inc-prob"} and using $M_i \ge M_0 {\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace, \delta$ yields $$\mathbb{P}(\neg\mathcal{A}_{\mathrm{inc}}^i \mid \mathcal{F}_i) \le \exp(-M_i^{\delta(\tau-1)/2}) + \exp(-\log(1/\underline{c})M_i^{\delta(\tau-1)/2}) \le \exp(-M_i^{\delta(\tau-1)/3}).$$ Substituting this bound into [\[eq:1d-infinite-path-Ainci\]](#eq:1d-infinite-path-Ainci){reference-type="eqref" reference="eq:1d-infinite-path-Ainci"} and using $M_0 {\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace, \delta$ then yields the required bound of $$\mathbb{P}(\neg\mathcal{A}_{\mathrm{inc}}(M_0,\theta,z) \mid z \in \mathcal{V}_{M_0}) \le \sum_{i=1}^\infty \exp(-M_i^{\delta(\tau-1)/3}) \le \exp(-M_0^{\delta(\tau-1)/4}).\qedhere$$ The bound remains true if we additionally condition also on $y\in \mathcal{V}$: there is a unique index $i$ so that $y\in I_i\setminus I_{i-1}$. The number of points in this interval changes by one, but the concentration bound in [\[eq:1d-infinite-path-vert-prob\]](#eq:1d-infinite-path-vert-prob){reference-type="eqref" reference="eq:1d-infinite-path-vert-prob"} still remains valid under the conditioning. ◻ We now apply Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"} to prove the two remaining claims from Section [6](#sec:endpoints){reference-type="ref" reference="sec:endpoints"} when $d=1$. *Proof.* [\[proof:claim-rho\]]{#proof:claim-rho label="proof:claim-rho"} For $d \ge 2$ this result appears as [@komjathy2022one2 Lemma 3.10]. For $d=1$, we instead apply Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"}. Wlog, suppose $a<b$. Let $M_0,\theta>1$ and $0<\delta<1$ with $\delta,\theta-1{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ and $M_0{\,\gg_{\star}\,}\delta,\theta,\textnormal{\texttt{par}}\xspace$. Let $\mathcal{A}_{\mathrm{path}}(a)$ be the event that $a\in \mathcal{V}$ lies in an infinite component of $G[[a,\infty)]$, and let $\mathcal{A}_{\mathrm{path}}(b)$ be the event that $b$ lies in an infinite component of $G[(-\infty,b]]$. By Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"}, (and the last sentence there), we have $$\begin{aligned} \mathbb{P}(\neg\mathcal{A}_{\mathrm{path}}(a) \mid b \in \mathcal{V}, a \in \mathcal{V}_{M_0}) & \le \exp(-M_0^{\delta(\tau-1)/4}),\\ \mathbb{P}(\neg\mathcal{A}_{\mathrm{path}}(b) \mid a \in \mathcal{V}, b \in \mathcal{V}_{M_0}) & \le \exp(-M_0^{\delta(\tau-1)/4}). \end{aligned}$$ Thus by a union bound, $$\label{eq:two-in-Cinfty-0} \mathbb{P}(\mathcal{A}_{\mathrm{path}}(a) \cap \mathcal{A}_{\mathrm{path}}(b) \mid a,b\in\mathcal{V}) \ge \mathbb{P}(a,b \in \mathcal{V}_{M_0} \mid a,b\in\mathcal{V}) - 2\exp(-M_0^{\delta(\tau-1)/4}).$$ By [\[eq:power_law\]](#eq:power_law){reference-type="eqref" reference="eq:power_law"}, since $M_0{\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$, $\tau>2$, and $\ell$ is slowly-varying, we have $$\mathbb{P}(a,b\in\mathcal{V}_{M_0}\mid a,b\in\mathcal{V}) = \Big(\frac{\ell(M_0)}{M_0^{\tau-1}} - \frac{\ell(2M_0)}{(2M_0)^{\tau-1}}\Big)^2 \ge \Big(\frac{\ell(M_0)}{4M_0^{\tau-1}}\Big)^2 \ge \frac{1}{M_0^{3(\tau-1)}}.$$ Since $M_0 {\,\gg_{\star}\,}\delta,\textnormal{\texttt{par}}\xspace$, it follows from [\[eq:two-in-Cinfty-0\]](#eq:two-in-Cinfty-0){reference-type="eqref" reference="eq:two-in-Cinfty-0"} that $$\mathbb{P}(\mathcal{A}_{\mathrm{path}}(a) \cap \mathcal{A}_{\mathrm{path}}(b) \mid a,b\in\mathcal{V}) \ge M_0^{-3(\tau-1)} - 2\exp(-M_0^{\delta(\tau-1)/4}) > M_0^{-3(\tau-1)}/2;$$ since $\mathcal{C}_\infty$ is a.s. unique, the result therefore follows by taking $\rho := M_0^{-3(\tau-1)}/2$. ◻ *Proof.* [\[proof:lem:new_external\]]{#proof:lem:new_external label="proof:lem:new_external"} We write $r_M=:r$. Since $\mathcal{H}_M = B_{2r}(z) \cap \mathcal{V}_M$, i.e, all vertices in $\mathcal{V}_M$ in $B_{2r}(z)$ belong to $\mathcal{H}_M$, also all vertices in $B_{r^{1/3}}(y)\cap \mathcal{V}_M$ are in $\mathcal{H}_\infty$ for all $y\in B_r(z)$, so the event $\mathcal{A}_\mathrm{dense}(\mathcal{H},\mathcal{V}_M,r,z)$ always occurs by definition, as required by [\[eq:dense-near-1d\]](#eq:dense-near-1d){reference-type="eqref" reference="eq:dense-near-1d"}. We next prove [\[eq:linear-again-1d\]](#eq:linear-again-1d){reference-type="eqref" reference="eq:linear-again-1d"} by dividing $B_{2r}(z)$ into sub-interval "cells" and proving that each cell is whp both connected and joined to each of its adjacent cells in $G_M$. To this end, let $R := M^{2/d}/\sqrt{d}=M^2$, let $i_{\max} := \lceil 2r/R\rceil$ and $i_{\min} := -i_{\max}$. For all $i \in [i_{\min},i_{\max}]$, let $y_i := z+i\cdot R$ and $Q^{\scriptscriptstyle{(i)}} := [y_i,y_i+R)$; thus $Q^{\scriptscriptstyle{(i_{\min})}},\dots,Q^{\scriptscriptstyle{(i_{\max})}}$ partition $[z-R\lceil 2r/R \rceil,z+(R+1)\lceil 2r/R \rceil) \supset B_{2r}(z)$. Let $\mathcal{A}_{\mathrm{path}}$ be the event that $G_M[Q^{\scriptscriptstyle{(i_{\min})}}],\dots,G_M[Q^{\scriptscriptstyle{(i_{\max})}}]$ are connected graphs containing at most $2R$ vertices and that for all $i \in [i_{\min},i_{\max}-1]$ there is at least one edge in $G_M$ from $Q^{\scriptscriptstyle{(i)}}$ to $Q^{\scriptscriptstyle{(i+1)}}$. If $\mathcal{A}_{\mathrm{path}}$ occurs, then for all $a,b \in B_{2r}(z)\cap \mathcal{V}_M$ there is a path $\pi_{a,b}$ from $a$ to $b$ in $G_M$ intersecting at most $\lfloor |a-b|/R\rfloor+2\le |a-b|/R+2$ many cells; since each cell contains at most $2R$ vertices and each edge in $G_M$ has cost at most $M^{3\mu}$, and since $2RM^{3\mu} = \kappa$, it follows that $$\mathcal{C}(\pi_{a,b}) \le (|a-b|/R+2) \cdot 2R \cdot M^{3\mu} = 2(|a-b|+2R) \cdot M^{3\mu} \le \kappa|a-b| + 2\kappa.$$ Moreover, the deviation of $\pi_{a,b}$ is at most the size of the box-length, i.e., $R$, and $R< \kappa$, i.e., it does not depend on $|a-b|$. With $\zeta=0$ and $C=2\kappa$, we have just shown that $$\label{eq:new-external-1d-path} \mathcal{A}_{\mathrm{linear}}(\mathcal{H}_M,\mathcal{H}_M,r_M,\kappa,0,2\kappa,z) \subseteq \mathcal{A}_{\mathrm{path}}.$$ We now bound $\mathbb{P}(\mathcal{A}_{\mathrm{path}})$ below. The same approach of dividing $\mathbb{R}^d$ into cells is used in [@komjathy2022one2], and for both IGIRG and SFP [@komjathy2022one2 Corollary 3.9(i)] lower bounds the probability that the conditions of $\mathcal{A}_{\mathrm{path}}$ hold for a single cell $Q^{\scriptscriptstyle{(i)}}$ by $1-e^{-M^{3-\tau-\varepsilon}}$ for some $\varepsilon{\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ with $M {\,\gg_{\star}\,}\varepsilon$ (by coupling to an Erdős-Rényi graph). Combining [@komjathy2022one2 Corollary 3.9(i)] with a union bound over the at most $2\cdot\lceil 2r/R\rceil+1$ cells yields that $$\mathbb{P}(\mathcal{A}_{\textrm{path}}) \ge 1-(2\cdot\lceil 2r/R\rceil+1) \cdot e^{-M^{3-\tau-\varepsilon}} \ge 1 -5re^{-M^{3-\tau-\varepsilon}}.$$ Since $r = e^{(\log M)^2}$ and $M {\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$, the $e^{-M^{3-\tau-\varepsilon}}$ term dominates, and together with [\[eq:new-external-1d-path\]](#eq:new-external-1d-path){reference-type="eqref" reference="eq:new-external-1d-path"} and $M{\,\gg_{\star}\,}c,q$ we obtain $\mathbb{P}(\mathcal{A}_{\mathrm{linear}}(\mathcal{H}_M,\mathcal{H}_M,r,\kappa,0,2\kappa,z)) \ge 1-q/10,$ and we have proved [\[eq:linear-again-1d\]](#eq:linear-again-1d){reference-type="eqref" reference="eq:linear-again-1d"} as required. The argument conditioned on $\mathcal{F}_{0,x}$ is identical; note in particular that [@komjathy2022one2 Corollary 3.9(i)] explicitly allows for planted vertices. It remains to bound $\mathbb{P}(\mathcal{A}_{\mathrm{near}}(\mathcal{H}_M,C_M,C_M,z))$ conditioned on $z \in \mathcal{C}_\infty$, see [\[eq:a-near\]](#eq:a-near){reference-type="eqref" reference="eq:a-near"} for the definition of $\mathcal{A}_{\mathrm{near}}$. Here, we replaced the 'usual' radius $r_M=\exp((\log M)^2)$ by $C_M=M^{2(\tau-1)+3\mu}\ll r_M$, i.e., we can find a path from $z$ to a vertex with weight $M$ within a much smaller radius from $z$ that $r_M$ would give. We first dominate $\mathcal{A}_{\mathrm{near}}(\mathcal{H}_M,C_M,C_M,z)$ below by events $\mathcal{A}_1$ to $\mathcal{A}_4$ defined as follows. Let $\rho {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ be as in Claim [\[claim:two-in-Cinfty\]](#claim:two-in-Cinfty){reference-type="ref" reference="claim:two-in-Cinfty"}, and define $M_0 > 0$ satisfying $M {\,\gg_{\star}\,}M_0{\,\gg_{\star}\,}q,\rho,\textnormal{\texttt{par}}\xspace$, and let $r_0 := M_0^{2(\tau-1)}$ coming from Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"}. By Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"} we know that a.s. $\mathcal{C}_\infty$ contains a vertex in $\mathcal{V}_{M_0}$, and let $v_0$ be an (arbitrarily-chosen) closest such vertex to $z$ in Euclidean distance. We define the following events: 1. [\[item:a1\]]{#item:a1 label="item:a1"} $\mathcal{A}_1$: there is a path $\pi_{z,v_0}$ from $z$ to $v_0$ with $\mathcal{C}(\pi_{z,v_0}) \le C_M/2$ and $\mathcal{V}(\pi_{z,v_0}) \subseteq B_{C_M}(z)$; 2. [\[item:a2\]]{#item:a2 label="item:a2"} $\mathcal{A}_2$: $B_{r_0}(z)$ contains a vertex in $\mathcal{V}_{M_0}\cap \mathcal{C}_\infty$, i.e., $v_0 \in B_{r_0}(z)$; 3. [\[item:a3\]]{#item:a3 label="item:a3"} $\mathcal{A}_3$: every vertex $x \in B_{r_0}(z) \cap \mathcal{V}_{M_0}$ has an associated path $\pi_{x\to\mathcal{V}_M}$ from $x$ to some vertex in $\mathcal{V}_M$ with $\mathcal{V}(\pi_{x\to \mathcal{V}_M}) \subset B_{C_M}(z)$; and 4. [\[item:a4\]]{#item:a4 label="item:a4"} $\mathcal{A}_4$: $\mathcal{A}_2$ and $\mathcal{A}_3$ both occur and $\mathcal{C}(\pi_{v_0\to\mathcal{V}_M}) \le M^{3\mu} \le C_M/2$. Observe that if $\mathcal{A}_1$, $\mathcal{A}_2$, $\mathcal{A}_3$ and $\mathcal{A}_4$ all occur then concatenating $\pi_{z,v_0}$ and $\pi_{v_0\to\mathcal{V}_M}$ yields the path required by $\mathcal{A}_{\mathrm{near}}(\mathcal{H}_M,C_M,C_M,z)$; thus $$\begin{aligned} \mathbb{P}(\neg\mathcal{A}_{\mathrm{near}}(\mathcal{H}_M,C_M,C_M,z) \mid z \in \mathcal{C}_\infty) &\le \sum_{i=1}^3 \mathbb{P}(\neg\mathcal{A}_i\mid z \in\mathcal{C}_\infty) + \mathbb{P}(\neg\mathcal{A}_4\mid \mathcal{A}_2,\mathcal{A}_3, z \in \mathcal{C}_\infty). \end{aligned}$$ By Claim [\[claim:two-in-Cinfty\]](#claim:two-in-Cinfty){reference-type="ref" reference="claim:two-in-Cinfty"}, it follows that $$\begin{aligned} \begin{split}\label{eq:external-1d-union} &\mathbb{P}(\neg\mathcal{A}_{\mathrm{near}}(\mathcal{H}_M,C_M,C_M,z) \mid z \in \mathcal{C}_\infty) \le \mathbb{P}(\neg\mathcal{A}_1\mid z \in\mathcal{C}_\infty) + \\ &\qquad\qquad \mathbb{P}(\neg\mathcal{A}_2\mid z \in\mathcal{V})/\rho +\mathbb{P}(\neg\mathcal{A}_3\mid z \in \mathcal{V})/\rho + \mathbb{P}(\neg\mathcal{A}_4\mid \mathcal{A}_2,\mathcal{A}_3,z \in \mathcal{V})/\rho. \end{split} \end{aligned}$$ We first bound $\mathbb{P}(\neg\mathcal{A}_1\mid z\in\mathcal{C}_\infty)$ in (C[\[item:a1\]](#item:a1){reference-type="ref" reference="item:a1"}). Given that we fixed $v_0$, let $\pi_{z,v_0}$ be an (arbitrarily-chosen) cheapest path from $z$ to $v_0$; such a path must exist whenever $z \in \mathcal{C}_\infty$. Since $\mathcal{C}(\pi_{z,v_0})$ and $\inf\{R>0\colon \mathcal{V}(\pi_{z,v_0})\subseteq B_R(z)\}$ are a.s. finite random variables and since $C_M {\,\gg_{\star}\,}q,\textnormal{\texttt{par}}\xspace, M_0$, we can choose $C_M$ sufficiently large so that $$\label{eq:external-1d-A1} \mathbb{P}(\neg\mathcal{A}_1 \mid z \in \mathcal{C}_\infty) \le q/40.$$ We next bound $\mathbb{P}(\neg\mathcal{A}_2\mid z \in \mathcal{V})$ in (C[\[item:a2\]](#item:a2){reference-type="ref" reference="item:a2"}). The event $\mathcal{A}_2$ occurs if and only if $B_{r_0}(z) \cap \mathcal{V}_{M_0} \cap \mathcal{C}_\infty \ne \emptyset$. Similarly as before, $|B_{r_0}(z) \cap \mathcal{V}_{M_0}|$ is either a Poisson variable (in IGIRG) or a binomial variable (in SFP) with mean $$\label{eq:mean-vertices-in-r0} \mathbb{E}[|B_{r_0}(z) \cap \mathcal{V}_{M_0}| \mid z \in \mathcal{V}] \ge r_0\Big(\frac{\ell(M_0)}{M_0^{\tau-1}} - \frac{\ell(2M_0)}{(2M_0)^{\tau-1}}\Big) \ge 2M_0^{(\tau-1)/2},$$ where we used $M_0 {\,\gg_{\star}\,}\textnormal{\texttt{par}}\xspace$ and the value of $r_0=M_0^{2(\tau-1)}$ for the second inequality. In particular, by Chernoff's bound, $$\begin{aligned} \label{eq:external-1d-Markov1} \mathbb{P}(|B_{r_0}(z) \cap \mathcal{V}_{M_0}| < M_0^{(\tau-1)/2} \mid z \in \mathcal{V}) \le \exp(-M_0^{(\tau-1)/8}). \end{aligned}$$ Let $\theta > 1$ and $\delta \in (0,1)$ satisfy $\delta,\theta\! -\! 1 {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ and $M_0 {\,\gg_{\star}\,}\delta,\theta$. Recall the event $\mathcal{A}_{\mathrm{inc}}(M_0,\theta,x)$ about having an infinite weight-increasing path from Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"}; this event implies $\{x\in\mathcal{C}_\infty\}$. So by [\[eq:weight-increasing-11\]](#eq:weight-increasing-11){reference-type="eqref" reference="eq:weight-increasing-11"} in Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"}, $\mathbb{P}(x \notin \mathcal{C}_\infty \mid x\in\mathcal{V}_{M_0}) \le \exp(-M_0^{\delta(\tau-1)/4})$ for some $\delta {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$ with $M_0 {\,\gg_{\star}\,}\delta$. By translation invariance, this implies $$\begin{aligned} \mathbb{P}(x \notin \mathcal{C}_\infty \mid x\in B_{r_0}(z)\cap\mathcal{V}_{M_0}, z\in\mathcal{V}) \le \exp(-M_0^{\delta(\tau-1)/4}). \end{aligned}$$ Hence, the expected number of vertices in $\mathcal{V}_{M_0}$ outside the infinite component is at most $$\begin{aligned} &\mathbb{E}[|(B_{r_0}(z) \cap \mathcal{V}_{M_0})\setminus\mathcal{C}_\infty| \mid z \in \mathcal{V}] \le \mathbb{E}[|B_{r_0}(z) \cap \mathcal{V}_{M_0}| \mid z \in \mathcal{V}] \cdot \exp(-M_0^{\delta(\tau-1)/4}) \\ &\quad\quad \le \mathbb{E}[|B_{r_0}(z) \cap \mathcal{V}| \mid z \in \mathcal{V}] \cdot \exp(-M_0^{\delta(\tau-1)/4}) \le (2r_0+1)\exp(-M_0^{\delta(\tau-1)/4}), \end{aligned}$$ which is a crude upper bound. It follows by Markov's inequality that $$\begin{aligned} \begin{split}\label{eq:external-1d-Markov2} \mathbb{P}(|(B_{r_0}(z) \cap \mathcal{V}_{M_0}) \setminus \mathcal{C}_\infty| \ge M_0^{(\tau-1)/2}/2 \mid z\in \mathcal{V}) &\le \frac{(2r_0+1)\exp(-M_0^{\delta(\tau-1)/4})}{M_0^{(\tau-1)/2}/2} \\ & \le \exp(-M_0^{\delta(\tau-1)/8}), \end{split}\end{aligned}$$ where we used $r_0 = M_0^{2(\tau-1)}$ and $M_0 {\,\gg_{\star}\,}\delta,\textnormal{\texttt{par}}\xspace$. By a union bound over [\[eq:external-1d-Markov1\]](#eq:external-1d-Markov1){reference-type="eqref" reference="eq:external-1d-Markov1"} and [\[eq:external-1d-Markov2\]](#eq:external-1d-Markov2){reference-type="eqref" reference="eq:external-1d-Markov2"}, $$\begin{aligned} \begin{split}\label{eq:external-1d-A2} \mathbb{P}(\neg\mathcal{A}_2 \mid z \in \mathcal{V}) &\le \exp(-M_0^{(\tau-1)/8}) + \exp(-M_0^{\delta(\tau-1)/8})\le \rho q/40, \end{split} \end{aligned}$$ for all sufficiently large $M_0$. We next bound $\mathbb{P}(\neg\mathcal{A}_3\mid z \in \mathcal{V})$ in (C[\[item:a3\]](#item:a3){reference-type="ref" reference="item:a3"}). For all $x \in B_{r_0}(z) \cap \mathcal{V}_{M_0}$, let $\mathcal{A}_3(x)$ be the event that $x$ has an associated path $\pi_{x\to \mathcal{V}_M}$ as in $\mathcal{A}_\mathrm{3}$. We restrict this path to be a weight-increasing path as in Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"}. Let $\theta > 1$ and $\delta \in (0,1)$ satisfy $\delta,\theta - 1 {\,\ll_{\star}\,}\textnormal{\texttt{par}}\xspace$, and require that $M_0 {\,\gg_{\star}\,}\delta,\theta$. For sufficiently large $M$, we may choose $M_0$ such that $i := (\log \log M -\log \log M_0)/ \log \theta \le \log M$ is an integer, so $M_0^{\theta^i} = M$. Then by Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"}, for any given $x \in B_{r_0}(z)$, the weight increasing path reaches a vertex of weight $M$ at radius $R_i=M_0^{\theta^i(\tau-1)(1+\delta)}=M^{(\tau-1)(1+\delta)} < M^{2(\tau-1)+3\mu}=C_M$, and so the path is contained in $B_{C_M}(z)$, and we obtain $$\label{eq:weight-increasing-from-x} \begin{aligned} \mathbb{P}(\neg\mathcal{A}_3(x) \mid x \in \mathcal{V}_{M_0},z \in \mathcal{V}) &\le \mathbb{P}(\neg\mathcal{A}_\mathrm{inc}(M_0, \theta, x) \mid x \in \mathcal{V}_{M_0},z \in \mathcal{V})\\&\le \exp(-M_0^{\delta(\tau-1)/4}). \end{aligned}$$ It follows by a union bound that $$\begin{aligned} \mathbb{P}(\neg\mathcal{A}_3 \mid z \in \mathcal{V}) \le \mathbb{P}(|B_{r_0}(z) \cap \mathcal{V}_{M_0}| \ge r_0 \mid z \in \mathcal{V}) + r_0 \exp(-M_0^{\delta(\tau-1)/4}). \end{aligned}$$ As before $|B_{r_0}(z) \cap \mathcal{V}_{M_0}|$ is either a Poisson variable (in IGIRG) or a binomial variable (in SFP) with mean bounded from above $$\begin{aligned} \mathbb{E}(|B_{r_0}(z) \cap \mathcal{V}_{M_0}| \mid z \in \mathcal{V}) \le 2r_0\Big(\frac{\ell(M_0)}{M_0^{\tau-1}} - \frac{\ell(2M_0)}{(2M_0)^{\tau-1}}\Big) \le r_0 M_0^{-(\tau-1)/2}. \end{aligned}$$ Therefore $\mathbb{P}(|B_{r_0}(z) \cap \mathcal{V}_{M_0}| \ge r_0 \mid z \in \mathcal{V}) \le 2^{-r_0}$ and we get $$\begin{aligned} \begin{split}\label{eq:external-1d-A3} \mathbb{P}(\neg\mathcal{A}_3 \mid z \in \mathcal{V}) & \le 2^{-r_0}+r_0 \exp(-M_0^{\delta(\tau-1)/4}) \le \rho q/40, \end{split} \end{aligned}$$ where the second inequality holds because because $r_0 = M_0^{2(\tau-1)}$ and $M_0 {\,\gg_{\star}\,}\delta, \rho, q, \textnormal{\texttt{par}}\xspace$. Finally we bound the last term in [\[eq:external-1d-union\]](#eq:external-1d-union){reference-type="eqref" reference="eq:external-1d-union"}, (see $\mathcal{A}_4$ in (C[\[item:a4\]](#item:a4){reference-type="ref" reference="item:a4"})). Conditioned on the realisation of $G$, any path $\pi_{v_0\to \mathcal{V}_M}$ that satisfies the weight-increasing path property in [\[eq:weight-increasing-from-x\]](#eq:weight-increasing-from-x){reference-type="eqref" reference="eq:weight-increasing-from-x"} has at most $\log M$ edges and all vertex weights at most $M$. So, its expected cost is at most $$\mathbb{E}[\mathcal{C}(\pi_{v_0\to \mathcal{V}_M})]\le |\mathcal{E}(\pi_{v_0\to \mathcal{V}_M})| \cdot M^{2\mu}\mathbb{E}[L] \le M^{2\mu}\mathbb{E}[L]\log M \le \rho q M^{3\mu}/40$$ since $M {\,\gg_{\star}\,}\rho, q, \textnormal{\texttt{par}}\xspace$. Thus by Markov's inequality, the probability that the cost of this path is larger than $M^{3\mu}$ is at most $\rho q/40$. $$\label{eq:external-1d-A4} \mathbb{P}(\neg\mathcal{A}_4 \mid \mathcal{A}_2,\mathcal{A}_3, z \in \mathcal{V}) \le \rho q/40.$$ The result therefore follows on substituting the bounds [\[eq:external-1d-A1\]](#eq:external-1d-A1){reference-type="eqref" reference="eq:external-1d-A1"}, [\[eq:external-1d-A2\]](#eq:external-1d-A2){reference-type="eqref" reference="eq:external-1d-A2"}, [\[eq:external-1d-A3\]](#eq:external-1d-A3){reference-type="eqref" reference="eq:external-1d-A3"}, and [\[eq:external-1d-A4\]](#eq:external-1d-A4){reference-type="eqref" reference="eq:external-1d-A4"} into [\[eq:external-1d-union\]](#eq:external-1d-union){reference-type="eqref" reference="eq:external-1d-union"}. The argument conditioned on $\mathcal{F}_{y,z}$ is identical; note in particular that in applying Lemma [Lemma 60](#lem:1d-infinite-path){reference-type="ref" reference="lem:1d-infinite-path"}, we may assume wlog that $y < z$ by symmetry. ◻ [^1]: Delft University of Technology, j.komjathy\@tudelft.nl [^2]: University of Bristol, john.lapinskas\@bristol.ac.uk [^3]: ETH Zürich, johannes.lengler\@inf.ethz.ch [^4]: ETH Zürich, ulysse.schaller\@inf.ethz.ch U.S. was supported by the Swiss National Science Foundation \[grant number 200021_192079\]. [^5]: They have also been called EGIRG, where E stands for extended [@komjathy2020explosion]. [^6]: *If we take an IGIRG and rescale the underlying space $\mathbb R^d$ by a factor $\lambda$, then we obtain a random graph which satisfies all conditions of IGIRGs except that the density of the Poisson point process is $\lambda^{-d}$ instead of one. Thus it is no restriction to assume density one.* [^7]: For $\tau >3$, an infinite component only exists for high enough edge density, which is captured by $h$ in [\[eq:connection_prob\]](#eq:connection_prob){reference-type="eqref" reference="eq:connection_prob"}. [^8]: The phase is called *explosive* since the size of the the cost-ball of radius $r$ jumps from finite to infinite at some random finite threshold, called the *explosion time*. [^9]: For $\mu=0, L\equiv 1$, the cost-distance $d_{\mathcal{C}}(x,y)$ then equals the *graph-distance* between $x$ and $y$. [@komjathy2022one2] contains as special cases the linear lower bound on graph-distances by Berger [@berger2004lower] for long-range percolation (LRP) and by Deprez, Hazra, and Wüthrich [@deprez2015inhomogeneous] for SFP, see [@komjathy2022one2 Proposition 2.4].

arxiv_math

--- abstract: | We show that, for a class of locally solid topologies on vector lattices, a topologically convergent net has an embedded sequence that is unbounded order convergent to the same limit. Our result implies, and often improves, many of the known results in this vein in the literature. A study of metrisability and submetrisability of locally solid topologies on vector lattices is included. address: - Yang Deng; School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan, 611130 PR China - Marcel de Jeu; Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, the Netherlands; and Department of Mathematics and Applied Mathematics, University of Pretoria, Corner of Lynnwood Road and Roper Street, Hatfield 0083, Pretoria, South Africa author: - Yang Deng - Marcel de Jeu bibliography: - general_bibliography.bib title: Embedded unbounded order convergent sequences in topologically convergent nets --- # Introduction and overview {#sec:introduction_and_overview} Let $(X,\Sigma,\mu)$ be a measure space, and let $\mathrm{L}^0(X,\Sigma,\mu)$ denote the vector lattice of measurable functions on $X$, with identification of two functions that are $\mu$-almost everywhere equal. It is a classical result that a sequence in $\mathrm{L}^0(X,\Sigma,\mu)$ that is (globally) convergent in measure has a subsequence that is convergent almost everywhere to the same limit; see [@aliprantis_burkinshaw_PRINCIPLES_OF_REAL_ANALYSIS_THIRD_EDITION:1998 Theorem 19.4] or [@folland_REAL_ANALYSIS_SECOND_EDITION:1999 Theorem 2.30], for example. Another result in a similar vein, but then for nets, is the following. If $\mu$ is $\sigma$-finite, and if $(f_\alpha)_{\alpha\in A}$ is a net in $\mathrm{L}^0(X,\Sigma,\mu)$ that is (locally) convergent in measure, then there exist indices $\alpha_1\leq\alpha_2\leq\alpha_3\leq\dotsc$ such that the sequence $(f_{\alpha_n})_{n\geq 1}$ converges almost everywhere to the same limit; see [@deng_de_jeu:2022a Theorem 7.11]. In this case, there is the following way to rephrase the statement: if $(f_\alpha)_{\alpha\in A}$ is a net in $\mathrm{L}^0(X,\Sigma,\mu)$ that is convergent in the uo-Lebesgue topology of $\mathrm{L}^0(X,\Sigma,\mu)$, then there exists indices $\alpha_1\leq\alpha_2\leq\alpha_3\leq\dotsc$ such that the sequence $(f_{\alpha_n})_{n\geq 1}$ is unbounded order convergent to the same limit. There are quite a few other results in the literature, stating that a net (resp. a sequence), which is convergent in a particular locally solid topology on a vector lattice, has such an 'embedded sequence' (resp. a subsequence) that is unbounded order convergent to the same limit.[^1] In this paper, we establish a result[^2] which asserts precisely this, but then for a reasonably large class of locally solid topologies all at once. It has many of the known results in this vein as special cases, often in an improved form. This paper is organised as follows. [2](#sec:preliminaries){reference-type="ref" reference="sec:preliminaries"} contains the necessary definitions, notations, conventions, and preliminary results. We elaborate a little on the countable sup property, since this is often a sufficient condition for the applications in the final [5](#sec:applications){reference-type="ref" reference="sec:applications"}. [3](#sec:metrisable_and_submetrisable_topologies_on_vector_lattices){reference-type="ref" reference="sec:metrisable_and_submetrisable_topologies_on_vector_lattices"} contains a study of metrisability and submetrisability of locally solid topologies on vector lattices. The topologies to which our key result in [4](#sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets){reference-type="ref" reference="sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets"} applies are what we call the locally interval complete solidly submetrisable topologies,[^3] and we take some effort to show that topologies of practical interest fall in this category. This is the case, for example, for Fatou topologies on a Dedekind complete vector lattice that have full carriers (which is automatic in the presence of the countable sup property). Due to further properties of Fatou topologies and unbounded order convergence, our result then effectively applies to Fatou topologies with full carriers on arbitrary vector lattices, in particular to Fatou topologies on vector lattices with the countable sup property.[^4] [4](#sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets){reference-type="ref" reference="sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets"} contains our core result regarding the existence of embedded unbounded order convergent sequences.[^5] In the final [5](#sec:applications){reference-type="ref" reference="sec:applications"}, we combine the material on locally solid topologies from [3](#sec:metrisable_and_submetrisable_topologies_on_vector_lattices){reference-type="ref" reference="sec:metrisable_and_submetrisable_topologies_on_vector_lattices"} with the basic theorem from [4](#sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets){reference-type="ref" reference="sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets"}, yielding several results on the existence of embedded unbounded order convergent sequences in topologically convergent nets. For a uo-Lebesgue topology, such an embedded unbounded order convergent sequence is again convergent in this topology, resulting in stronger statements in this case. We also give applications to weak closures and to the equality of adherences and closures related to various convergence structures. During [5](#sec:applications){reference-type="ref" reference="sec:applications"}, we point out how some of our theorems specialise to (improved versions of) known results. # Preliminaries {#sec:preliminaries} In this section, we collect the necessary definitions, notations, conventions, and preliminary results. All vector spaces are over the real numbers. A linear topology on a vector space is supposed to be Hausdorff. Neighbourhoods of points need not be open. We recall that a topological vector space is metrisable if and only if zero has a countable neighbourhood basis; see [@schaefer_wolff_TOPOLOGICAL_VECTOR_SPACES_SECOND_EDITION:1999 Theorem I.6.1], for example. ## Vector lattices {#subsec:vector_lattices} All vector lattices are supposed to be Archimedean. For a vector lattice $E$, we let ${E}^{\thicksim}$ denote its order dual, and ${E}^\thicksim_{\mathrm{oc}}$ its order continuous dual. A net $(x_\alpha)_{\alpha\in A}$ in a vector lattice $E$ is said to be *order convergent in $E$ to $x\in E$* (denoted by $x_\alpha \xrightarrow{\mathrm{o}}x$) if there exists a net $(y_\beta)_{\beta\in B}$ in $E$ such that $y_\beta\downarrow 0$ and with the property that, for every $\beta_0\in B$, there exists an $\alpha_0\in A$ such that $\lvert x_\alpha - x\rvert\leq y_{\beta_0}$ for all $\alpha\in A$ with $\alpha\geq \alpha_0$. A net $(x_\alpha)_{\alpha\in A}$ in $E$ is said to be *unbounded order convergent* (or *uo-convergent* for short) *in $E$ to $x\in E$* (denoted by $x_\alpha \xrightarrow{\mathrm{uo}}x$) if $\lvert x_\alpha - x\rvert\wedge \lvert y\rvert\xrightarrow{\mathrm{o}}0$ in $E$ for all $y\in E$. Order convergence implies unbounded order convergence to the same limit. A subset $S$ of a vector lattice $E$ is *order closed* if $x\in S$ whenever a net $(x_\alpha)_{\alpha\in A}$ in $S$ and $x\in E$ are such that $x_\alpha\xrightarrow{\mathrm{o}}x$. Let $F$ be a vector sublattice of a vector lattice $E$. We recall that $F$ is said to be *order dense in $E$* if, for every $x>0$ in $E$, there exists a $y\in F$ with $0<y\leq x$; *majorising in $E$* if, for every $x\in E$, there exists a $y\in F$ with $x\leq y$; and *a regular vector sublattice of $E$* if $x_\alpha\xrightarrow{\mathrm{o}}x$ in $E$ whenever a net $(x_\alpha)_{\alpha\in A}$ in $F$ and $x\in F$ are such that $x_\alpha\xrightarrow{\mathrm{o}}x$ in $F$. Ideals and order dense vector sublattices are regular vector sublattices; see [@gao_troitsky_xanthos:2017 p.653], for example. Furthermore, if $F$ is a regular vector sublattice of $E$, $(x_\alpha)_{\alpha\in A}$ is a net in $F$, and $x\in F$, then $x_\alpha\xrightarrow{\mathrm{uo}}x$ in $F$ if and only if $x_\alpha\xrightarrow{\mathrm{uo}}x$ in $E$. This property even characterises the regular vector sublattices among the vector sublattices; see [@gao_troitsky_xanthos:2017 Theorem 3.2]. A non-empty subset $A$ of a vector lattice $E$ is said to be an *order basis* of $E$ when the band generated by $A$ equals $E$, i.e., when $A^\mathrm{d}=\left\{0\right\}$; cf. [@luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971 Definition 28.1]. If $F$ is an order dense vector sublattice of $E$, then it is easy to see that an order basis of $F$ is also an order basis of $E$. Let $E$ be a vector lattice. For $x\in E$, the ideal and the band in $E$ that are generated by $x$ are denoted by $I_x$ and $B_x$, respectively; the ideal and the band in the Dedekind completion $E^\delta$ of $E$ that are generated by $x$ are denoted by $I_x^\delta$ and $B_x^\delta$, respectively. Using the fact that the Dedekind completion of a vector lattice $E$ is the unique (up to isomorphism) Dedekind complete vector lattice that contains $E$ as an order dense and majorising vector sublattice, it is easy to see that $I_x^\delta$ is the Dedekind completion of $I_x$. ## The countable sup property {#subsec:countable_sup_property} A vector lattice $E$ has the *countable sup property* when every subset of $E$ that has a supremum in $E$ contains an at most countable subset with the same supremum. In some sources, such as [@luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971] and [@zaanen_INTRODUCTION_TO_OPERATOR_THEORY_IN_RIESZ_SPACES:1997], $E$ is then said to be *order separable*. Since it will become apparent in [5](#sec:applications){reference-type="ref" reference="sec:applications"} that having the countable sup property is a condition that is sufficient for several of the results in that section to hold, we use the opportunity to argue that, in practice, quite a few vector lattices have this property. If $E$ and $F$ are vector lattices such that there exists a strictly positive linear operator $T\colon E\to F$, then $E$ has the countable sup property when $F$ has; see [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 1.45]. In particular, if $E$ has a strictly positive linear functional, then it has the countable sup property. Hence every separable Banach lattice has the countable sup property, as a consequence of [@aliprantis_burkinshaw_POSITIVE_OPERATORS_SPRINGER_REPRINT:2006 Exercise 4.1.4]. It is easy to see from [@luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971 Theorem 23.2.(iii)] that a Banach lattice with an order continuous norm also has the countable sup property. For a measure space $(X,\Sigma,\mu)$ where $\mu$ is semi-finite, $\mathrm{L}^0(X,\Sigma,\mu)$ has the countable sup property if and only if $\mu$ is $\sigma$-finite; see [@kandic_taylor:2018 Proposition 6.5]. When $E$ has the countable sup property, then so does every vector sublattice of $E$; this follows from [@luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971 Theorem 23.5]. There is a conditional converse: suppose that $F$ is a vector sublattice of a vector lattice $E$, and that $F$ has the countable sup property. If $F$ has a countable order basis, and if $F$ is order dense in $E$, then $E$ has the countable sup property. Indeed, by [@kandic_taylor:2018 Theorem 6.2], the universal completion $F^{\mathrm u}$ of $F$ has the countable sub property. Since $F^{\mathrm u}$ has $E$ as a vector sublattice by [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 7.23], $E$ also has the countable sup property. ## Locally solid topologies on vector lattices {#subsec:locally_solid_topologies_on_vector_lattices} A *locally solid* topology on a vector lattice $E$ is a linear topology on $E$ such that zero has a neighbourhood basis consisting of solid subsets of $E$. In this case, $(E,\tau)$ is called a *locally solid vector lattice*. An *o-Lebesgue topology on $E$* is a locally solid topology $\tau$ on $E$ with the property that $x_\alpha\xrightarrow{\tau}x$ whenever a net $(x_\alpha)_{\alpha\in A}$ in $E$ and $x\in E$ are such that $x_\alpha\xrightarrow{\mathrm{o}}x$.[^6] A *uo-Lebesgue topology on $E$* is a locally solid topology $\tau$ on $E$ with the property that $x_\alpha\xrightarrow{\tau}x$ whenever a net $(x_\alpha)_{\alpha\in A}$ in $E$ and $x\in E$ are such that $x_\alpha\xrightarrow{\mathrm{uo}}x$. A *Fatou topology on $E$* is a locally solid topology on $E$ such that zero has a neighbourhood basis consisting of order closed solid subsets of $E$. A uo-Lebesgue topology is an o-Lebesgue topology, and an o-Lebesgue topology is a Fatou topology (see [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Lemma 4.2]). A vector lattice admits at most one (Hausdorff) uo-Lebesgue topology; see [@conradie:2005] or [@taylor:2019 Theorem 5.5]. Suppose that $E$ is a vector lattice, and that $\tau_F$ is a locally solid topology on an order dense ideal $F$ in $E$. Take a solid $\tau_F$-neighbourhood of zero in $F$ and $y$ in $F$, and define the solid subset $U_{V,y}$ of $E$ by setting $$U_{V,y}\coloneqq\{x\in E:\lvert x\rvert\wedge \lvert y\rvert\in V\}.$$ As $V$ runs through a $\tau_F$-neighbourhood basis at zero that consists of solid subsets of $F$, and as $y$ runs over $F$, the $U_{V,y}$ form a neighbourhood basis at zero for a locally solid topology on $E$. This topology is called the *unbounded topology on $E$ that is generated by $\tau_F$*, and it is denoted by $\mathrm{u}_F\tau_F$. We refer to [@deng_de_jeu:2022a Theorem 3.1] for details on this construction, which refines ideas in [@taylor:2019] that are also already implicit in [@conradie:2005]. When $F=E$, where $E$ has a locally solid topology $\tau$, we simply write $\mathrm{u}\tau$ for $\mathrm{u}_E\tau$. For a Banach lattice $E$ with its norm topology $\tau$, $\mathrm{u}\tau$ was introduced in [@deng_o_brien_troitsky:2017]. The unbounded norm topology $\mathrm{u}\tau$ on $E$ is then usually referred to as the un-topology. For a Banach lattice $F$ with its norm topology $\tau_F$ which is an order dense ideal in a vector lattice $E$, $\mathrm{u}_F\tau_F$ was introduced in [@kandic_li_troitsky:2018]. The unbounded topology $\mathrm{u}_F\tau_F$ on $E$ is then usually referred to as the un-$F$-topology. Let $(E,\tau)$ be a locally solid vector lattice, and let $(V_n)_{n\geq 1}$ be a sequence of $\tau$-neighbourhoods of zero. We say that the sequence is *normal* if $$V_{n+1}+V_{n+1}\subseteq V_n$$ for all $n$, and we let $\mathcal{N}$ denote the collection of all normal sequences consisting of solid $\tau$-neighbourhoods of zero. The *carrier of $\tau$*, denoted by $C_\tau$, is defined by setting $$C_\tau\coloneqq\bigcup \left\{N^\mathrm{d}: \text{there exists }(V_n)_{n\geq 1} \in \mathcal{N} \text{ such that }N=\bigcap_{n\geq 1} V_n\right\}.$$ Clearly, if $\tau$ is metrisable, then $C_\tau=E$. We shall need the following result in the proof of [Corollary 36](#res:subsequence_uo-lebesgue_induced){reference-type="ref" reference="res:subsequence_uo-lebesgue_induced"}. **Lemma 1**. *Let $E$ be a vector lattice, and let $F$ be an order dense ideal in $E$. Suppose that $\tau_F$ is a metrisable locally solid topology on $F$. Then $F\subseteq C_{u_F\tau_F}$.* *Proof.* Choose a normal sequence $(V_n)_{n\geq 1}$ of solid $\tau_F$-neighbourhoods of zero in $F$ such that $\bigcap_{n\geq 1}V_n=\{0\}$. For $y\in F$, the sequence $(U_{V_n,y})_{n\geq 1}$ is a sequence of solid $\mathrm{u}_F\tau_F$-neighbourhoods of zero. It is easily seen that this is a normal sequence, and also that $\bigcap_{n\geq 1} U_{V_n,y}=\{y\}^\mathrm{d}$, where the disjoint complement is taken in $E$. Hence $y\in C_{\mathrm{u}_F\tau_F}$. ◻ ## Embedded sequences in nets The main theme of this paper is the existence of unbounded order convergent sequences that are embedded in topologically convergent nets in the sense of the following definition. **Definition 2**. Let $S$ be a non-empty set, let $A$ be a directed set, and let $(x_\alpha)_{\alpha\in A}$ be a net in $S$. Suppose that $\alpha_1,\alpha_2,\alpha_3,\dotsc$ in $A$ are such that $\alpha_1\leq\alpha_2\leq\alpha_3,\dotsc$, and such that $\alpha_1<\alpha_2<\alpha_3,\dotsc$ when $A$ has no largest element. Then the sequence $(x_{\alpha_n})_{n\geq 1}$ is said to be *embedded in the net $(x_\alpha)_{\alpha\in A}$*. It is not required that the $\alpha_n$ form a cofinal subset of $A$, only that the map $n\mapsto\alpha_n$ be increasing, and strictly increasing when $A$ has no largest element. The embedded sequences in a given sequence are its subsequences. [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} asserts the existence of embedded unbounded order convergent sequences in topologically convergent nets, where the indices $\alpha_n$ for the sequences can even be found in a specific way. A particular case is the existence of unbounded order convergent subsequences of topologically convergent sequences. When the pertinent topology is metrisable, these two properties are equivalent. We now proceed to show this. **Lemma 3**. *Let $(E,\tau)$ be a topological vector lattice, where $\tau$ is metrisable. The following are equivalent:* 1. *[\[part:nets_and_sequences_no_largest_element_1\]]{#part:nets_and_sequences_no_largest_element_1 label="part:nets_and_sequences_no_largest_element_1"} for every directed set $A$ that has no largest element, for every net $(x_ \alpha)_{\alpha\in A}$ and $x$ in $E$ such that $x_\alpha\xrightarrow{\tau}x$, one can choose any $\widetilde\alpha_1\in A$; then find an $\alpha_1\in A$; then choose any $\widetilde\alpha_2\in A$; then find an $\alpha_2\in A$; etc., such that:* 1. *$\alpha_1<\alpha_2<\alpha_3<\dotsc$;* 2. *$\alpha_n>\widetilde\alpha_n$ for $n\geq 1$;* 3. *$x_{\alpha_n}\xrightarrow{\mathrm{uo}}x$ as $n\to\infty$.* 2. *[\[part:nets_and_sequences_no_largest_element_2\]]{#part:nets_and_sequences_no_largest_element_2 label="part:nets_and_sequences_no_largest_element_2"} every sequence in $E$ that is $\tau$-convergent to an element $x$ of $E$ has a subsequence that is uo-convergent to $x$.* *Proof.* It is clear that [\[part:nets_and_sequences_no_largest_element_1\]](#part:nets_and_sequences_no_largest_element_1){reference-type="ref" reference="part:nets_and_sequences_no_largest_element_1"} implies [\[part:nets_and_sequences_no_largest_element_2\]](#part:nets_and_sequences_no_largest_element_2){reference-type="ref" reference="part:nets_and_sequences_no_largest_element_2"}. We prove that [\[part:nets_and_sequences_no_largest_element_2\]](#part:nets_and_sequences_no_largest_element_2){reference-type="ref" reference="part:nets_and_sequences_no_largest_element_2"} implies [\[part:nets_and_sequences_no_largest_element_1\]](#part:nets_and_sequences_no_largest_element_1){reference-type="ref" reference="part:nets_and_sequences_no_largest_element_1"}. Suppose that an index set $A$ with no largest element, a net $(x_ \alpha)_{\alpha\in A}$ in $E$, and $x\in E$ are such that $x_\alpha\xrightarrow{\tau}x$. Take a countable $\tau$-neighbourhood basis $V_1,V_2,V_3,\dotsc$ of $x$. Choose any $\widetilde\alpha_1\in A$. Since $A$ has no largest element, there exists an index $\alpha_1^\prime>\widetilde\alpha_1$ such that $x_\alpha\in V_1$ whenever $\alpha\geq\alpha_1^\prime$. For $k\geq 2$, we inductively choose any $\widetilde\alpha_k$ in $A$, and then find an index $\alpha_k^\prime$ with $\alpha_k^\prime>\widetilde\alpha_{k}$ and $\alpha_k^\prime>\alpha_{k-1}^\prime$, such that $x_\alpha\in V_k$ whenever $\alpha\geq\alpha_k^\prime$. Then $x_{\alpha_k^\prime}\xrightarrow{\tau}x$ as $k\to\infty$. By assumption, there is a subsequence $(x_{\alpha_{k_n}^\prime})_{n\geq 1}$ such that $x_{\alpha_{k_n}^\prime}\xrightarrow{\mathrm{uo}}x$ as $n\to\infty$. For $n\geq 1$, set $\alpha_n\coloneqq \alpha_{k_n}^\prime$. Then $\alpha_1,\alpha_2,\alpha_3,\dotsc$ are as required. ◻ When the index set $A$ of a convergent net $(x_\alpha)_{\alpha\in A}$ has a largest element $\alpha_{\mathrm{largest}}$, its limit equals $x_{\alpha_{\mathrm{largest}}}$. Taking this into account, we have the following less precise consequence of [Lemma 3](#res:nets_and_sequences_no_largest_element){reference-type="ref" reference="res:nets_and_sequences_no_largest_element"}. **Corollary 4**. *Let $(E,\tau)$ be a topological vector lattice, where $\tau$ is metrisable. The following are equivalent:* 1. *[\[part:nets_and_sequences_general_1\]]{#part:nets_and_sequences_general_1 label="part:nets_and_sequences_general_1"} for every net $(x_\alpha)_{\alpha\in A}$ and $x$ in $E$ such that $x_\alpha\xrightarrow{\tau}x$, there exist indices such that $\alpha_1\leq\alpha_2\leq\alpha_3,\dotsc$, and such that $\alpha_1<\alpha_2<\alpha_3<\dotsc$ when $A$ has no largest element, with $x_{\alpha_n}\xrightarrow{\mathrm{uo}}x$ as $n\to\infty$* 2. *[\[part:nets_and_sequences_general_2\]]{#part:nets_and_sequences_general_2 label="part:nets_and_sequences_general_2"} every sequence in $E$ that is $\tau$-convergent to an element $x$ of $E$ has a subsequence that is uo-convergent to $x$.* Obviously, there is a more general principle that underlies [Corollary 4](#res:nets_and_sequences_general){reference-type="ref" reference="res:nets_and_sequences_general"}, as its proof merely exploits the countability of a neighbourhood basis. We refrain from attempting to phrase this, but we do note that its analogues for, e.g., order convergence or relative uniform convergence are clearly also true. ## Measure spaces When $(X,\Sigma, \mu)$ is a measure space, we let $\mathrm{L}^0(X,\Sigma,\mu)$ denote the vector lattice of measurable functions on $X$, with identification of two functions that are $\mu$-almost everywhere equal. Let $(f_\alpha)_{\alpha\in A}$ be a net in $\mathrm{L}_0(X,\Sigma,\mu)$, and let $f\in \mathrm{L}^0(X,\Sigma,\mu)$. Then $(f_\alpha)_{\alpha\in A}$ is said to *converge globally in measure to $f$* if $$\lim_\alpha \mu\big(\left\{t\in X: \lvert f_\alpha(t)-f(t)\rvert\geq \varepsilon\right\}\big)=0$$ for every $\varepsilon>0$, and to *converge locally in measure to $f$* (denoted by $f_\alpha\xrightarrow{\mu^\ast} f$) if $$\lim_\alpha \mu\big(\left\{t\in A: \lvert f_\alpha(t)-f(t)\rvert\geq \varepsilon\right\}\big)=0$$ for every $\varepsilon>0$ and every measurable subset $A$ of $X$ with finite measure. We recall that a measure space $(X,\Sigma,\mu)$ is called *semi-finite* when, for every $A\in \Sigma$ with $\mu(A)=\infty$, there exists a measurable subset $A_0$ of $A$ such that $0<\mu(A_0)<\infty$. Clearly, $\sigma$-finite measure spaces are semi-finite. # Metrisable and locally solidly submetrisable locally solid topologies {#sec:metrisable_and_submetrisable_topologies_on_vector_lattices} This section starts with a general metrisation theorem for locally solid vector lattices in [3.1](#subsec:metrisable_locally_solid_topologies){reference-type="ref" reference="subsec:metrisable_locally_solid_topologies"}. It is of some interest in its own right, and essential to a number of proofs in the remainder of [3](#sec:metrisable_and_submetrisable_topologies_on_vector_lattices){reference-type="ref" reference="sec:metrisable_and_submetrisable_topologies_on_vector_lattices"}. In [3.2](#subsec:locally_solidly_metrisable_locally_solid_topologies){reference-type="ref" reference="subsec:locally_solidly_metrisable_locally_solid_topologies"}, we consider locally solid submetrisability of general locally solid topologies. The final [3.3](#subsec:norm_unbounded_norm_lebesgue_and_fatou){reference-type="ref" reference="subsec:norm_unbounded_norm_lebesgue_and_fatou"} specialises to norm, unbounded norm, o-Lebesgue, and Fatou topologies. ## Metrisable locally solid topologies {#subsec:metrisable_locally_solid_topologies} Suppose that a locally solid topology on a vector lattice is metrisable. In this case, a compatible metric can be chosen with convenient properties, as is shown by the following result. We shall need only the case of locally solid topologies, but we also include the case of locally convex-solid topologies for the sake of completeness. **Theorem 5**. *Let $(E,\tau)$ be a locally solid vector lattice. If $\tau$ is metrisable, then there exists a metric $d\colon E\times E\to[0,1]$ on $E$ with the following properties:* 1. *[\[part:metrisation_theorem_1\]]{#part:metrisation_theorem_1 label="part:metrisation_theorem_1"} $d$ is compatible with $\tau$;* 2. *[\[part:metrisation_theorem_2\]]{#part:metrisation_theorem_2 label="part:metrisation_theorem_2"} $d$ is translation invariant;* 3. *[\[part:metrisation_theorem_3\]]{#part:metrisation_theorem_3 label="part:metrisation_theorem_3"} $d\left(0,x\right)\leq d\left(0,y\right)$ for $x,y\in E$ such that $\lvert x\rvert\leq\lvert y\rvert$;* 4. *[\[part:metrisation_theorem_4\]]{#part:metrisation_theorem_4 label="part:metrisation_theorem_4"} the metric balls $\left\{x\in E: d(0,x)<r\right\}$ and $\left\{x\in E: d(0,x)\leq r\right\}$ are solid for all $r\geq 0$.* *If, in addition, $\tau$ is locally convex, then there exists a metric $d\colon E\times E\to[0,1]$ on $E$ satisfying [\[part:metrisation_theorem_1\]](#part:metrisation_theorem_1){reference-type="ref" reference="part:metrisation_theorem_1"}--[\[part:metrisation_theorem_4\]](#part:metrisation_theorem_4){reference-type="ref" reference="part:metrisation_theorem_4"}, as well as:* 1. *[\[part:metrisation_theorem_5\]]{#part:metrisation_theorem_5 label="part:metrisation_theorem_5"} the metric balls $\left\{x\in E: d(c,x)<r\right\}$ and $\left\{x\in E: d(c,x)\leq r\right\}$ are convex for all $c\in E$ and $r\geq 0$.* *Proof.* We recall how a metric on a metrisable topological vector space $E$ can be found that satisfies [\[part:metrisation_theorem_1\]](#part:metrisation_theorem_1){reference-type="ref" reference="part:metrisation_theorem_1"} and [\[part:metrisation_theorem_2\]](#part:metrisation_theorem_2){reference-type="ref" reference="part:metrisation_theorem_2"}. We start by choosing a neighbourhood basis $(V_n)_{n\geq 1}$ of (not necessarily open) balanced neighbourhoods of zero such that $$\label{eq:local_base_inclusions} V_{n+1}+V_{n+1}\subseteq V_n$$ for $n\geq 1$. For each non-empty finite subset $H$ of $\left\{1,2,\dotsc\right\}$, we set $V_H\coloneqq \sum_{n\in H} V_n$ and $p_H\coloneqq\sum_{n\in H}2^{-n}$. For $x\in E$, set $$\lvert x\rvert\coloneqq \begin{cases} 1&\text{ if } x \text{ is not in any of the } V_H;\\ \inf_H \left\{p_H: x\in V_H\right\}&\text{ otherwise}. \end{cases}$$ As is shown in [@schaefer_wolff_TOPOLOGICAL_VECTOR_SPACES_SECOND_EDITION:1999 Proof of Theorem I.6.1], one then obtains a metric $d$ on $E$ satisfying [\[part:metrisation_theorem_1\]](#part:metrisation_theorem_1){reference-type="ref" reference="part:metrisation_theorem_1"} and [\[part:metrisation_theorem_2\]](#part:metrisation_theorem_2){reference-type="ref" reference="part:metrisation_theorem_2"} by defining $d\left(x,y\right)\coloneqq \lvert x-y\rvert$ for $x,y\in E$. In the case of a locally solid vector lattice, one can choose the $V_n$ such that they are also solid. Since a finite sum of solid subsets is solid as a consequence of (a precise form of) the Riesz decomposition (see [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 1.10]), it is then clear that [\[part:metrisation_theorem_3\]](#part:metrisation_theorem_3){reference-type="ref" reference="part:metrisation_theorem_3"} holds, which obviously implies [\[part:metrisation_theorem_4\]](#part:metrisation_theorem_4){reference-type="ref" reference="part:metrisation_theorem_4"}. Suppose that $\tau$ is locally solid as well as locally convex. In this case, the $V_n$ can be chosen to be solid as well as convex, as a consequence of the fact that the convex hull of a solid set is solid (see [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 1.11]). The resulting metric $d$ from the construction satisfies [\[part:metrisation_theorem_1\]](#part:metrisation_theorem_1){reference-type="ref" reference="part:metrisation_theorem_1"}--[\[part:metrisation_theorem_4\]](#part:metrisation_theorem_4){reference-type="ref" reference="part:metrisation_theorem_4"}; we now show that the convexity of the $V_n$ yields [\[part:metrisation_theorem_5\]](#part:metrisation_theorem_5){reference-type="ref" reference="part:metrisation_theorem_5"}. By the translation invariance of $d$, it suffices to take $c=0$ in [\[part:metrisation_theorem_5\]](#part:metrisation_theorem_5){reference-type="ref" reference="part:metrisation_theorem_5"}. Since the $V_n$ are convex, so are the $V_H$. Furthermore, if $H_1, H_2$ are nonempty subsets of $\left\{1,2,\dotsc\right\}$ such that $p_{H_1}\leq p_{H_2}$, then a moment's thought shows that $V_{H_1}\subseteq V_{H_2}$ as a consequence of [\[eq:local_base_inclusions\]](#eq:local_base_inclusions){reference-type="ref" reference="eq:local_base_inclusions"}. Together with the density of the dyadic rationals, this implies that $$\left\{x\in E: d\left(0,x\right)\leq r\right\}= \begin{cases} \bigcap_{H:p_H>r} V_H&\text{ if }0\leq r< 1;\\ E&\text{ if }r\geq 1;\\ \end{cases}$$ which is convex. Since $\left\{x\in E: d\left(0,x\right)<r\right\}=\bigcup_{0\leq s<r }\left\{x\in E: d\left(0,x\right)\leq s\right\}$ is a nested union of convex sets, all open balls are convex as well. ◻ ## Locally solidly submetrisable locally solid topologies {#subsec:locally_solidly_metrisable_locally_solid_topologies} We recall that a linear topology on a topological vector space is called *submetrisable* if it is finer than a metrisable linear topology on the space. In [@kandic_taylor:2018], metrisability and submetrisability of minimal and of unbounded locally solid topologies are studied. In this section, we are mainly concerned with a *local* version of submetrisability, and then for general locally solid topologies. **Definition 6**. Let $(E,\tau)$ be a locally solid vector lattice. We say that $\tau$ is: 1. [\[part:l.c.m.t.\_1\]]{#part:l.c.m.t._1 label="part:l.c.m.t._1"} *solidly submetrisable* if it is finer than some metrisable locally solid topology on $E$;[^7] 2. [\[part:l.c.m.t.\_2\]]{#part:l.c.m.t._2 label="part:l.c.m.t._2"} *locally solidly submetrisable* if, for every $x\in E$, the restricted topology $\tau\vert_{B_x}$ is solidly submetrisable; 3. [\[part:l.c.m.t.\_3\]]{#part:l.c.m.t._3 label="part:l.c.m.t._3"} *locally interval complete solidly submetrisable* if, for every $x\in E^+$, there exists a metrisable locally solid topology $\widetilde\tau_x$ on $B_x$ such that: 1. [\[part:l.c.m.t.\_3_a\]]{#part:l.c.m.t._3_a label="part:l.c.m.t._3_a"} the restricted topology $\tau\vert_{B_x}$ is finer than $\widetilde\tau_x$; 2. [\[part:l.c.m.t.\_3_b\]]{#part:l.c.m.t._3_b label="part:l.c.m.t._3_b"} every $\widetilde\tau_x$-Cauchy sequence in the order interval $[0,x]$ is $\widetilde\tau_x$-convergent to an element of $[0,x]$. Clearly, when a locally solid topology is solidly submetrisable or locally interval complete solidly submetrisable, it is locally solidly submetrisable. When the vector lattice has a weak order unit, being solidly submetrisable and being locally solidly submetrisable are equivalent. The locally solid topologies to which the basic [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} applies are the locally interval complete solidly submetrisable ones. **Remark 7**. 1. It follows from [Theorem 5](#res:metrisation_theorem){reference-type="ref" reference="res:metrisation_theorem"} that the metrics figuring in the three parts of [Definition 6](#def:l.c.m.t.){reference-type="ref" reference="def:l.c.m.t."} can be chosen to be translation invariant and such that they are 'lattice metrics' as in part [\[part:metrisation_theorem_3\]](#part:metrisation_theorem_3){reference-type="ref" reference="part:metrisation_theorem_3"} of that theorem. In the case of a locally interval complete solidly submetrisable topology $\tau$, they then make $[0,x]$ into a complete metric space as a consequence of their translation invariance; see [@rudin_FUNCTIONAL_ANALYSIS_SECOND_EDITION:1991 p.20-21]. These observations will frequently be used in the sequel. 2. The completeness in part [\[part:l.c.m.t.\_3\]](#part:l.c.m.t._3){reference-type="ref" reference="part:l.c.m.t._3"} is required only for the single interval $[0,x]$, but we have refrained from including this into the terminology. This minimal requirement is already sufficient for the proof of the key [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} to be valid. In fact, as the case of the un-topology will show, a more 'natural' requirement, such as requiring that all order intervals in $B_x$ be $\widetilde\tau_x$-complete, could well be asking too much; see [Remark 14](#rem:proof_for_un-topology){reference-type="ref" reference="rem:proof_for_un-topology"}. For a locally solid topology, being metrisable and being solidly submetrisable are actually equivalent, as is shown by our next result, [Proposition 8](#res:redundancy){reference-type="ref" reference="res:redundancy"}.[^8] Hence there is a certain redundancy in the parts [\[part:l.c.m.t.\_1\]](#part:l.c.m.t._1){reference-type="ref" reference="part:l.c.m.t._1"} and [\[part:l.c.m.t.\_2\]](#part:l.c.m.t._2){reference-type="ref" reference="part:l.c.m.t._2"} of [Definition 6](#def:l.c.m.t.){reference-type="ref" reference="def:l.c.m.t."}, where we could equally well have spoken of 'submetrisable' and 'locally submetrisable'. This redundancy is, however, not present in part [\[part:l.c.m.t.\_3\]](#part:l.c.m.t._3){reference-type="ref" reference="part:l.c.m.t._3"}, since completeness may be lost when passing to a finer topology, as is done in [Proposition 8](#res:redundancy){reference-type="ref" reference="res:redundancy"}. For reasons of uniformity, we have, therefore, still included the solidness in the terminology in all parts of [Definition 6](#def:l.c.m.t.){reference-type="ref" reference="def:l.c.m.t."}. **Proposition 8**. *Let $(E,\tau)$ be a locally solid vector lattice. Suppose that there exists a metrisable linear topology $\tau^\ast$ on $E$ that is coarser than $\tau$. Then there exists a metrisable locally solid topology $\widetilde\tau$ on $E$ that is coarser than $\tau$ and finer than $\tau^\ast$. If $\tau$ is an o-Lebesgue topology [(]{.upright}resp. a uo-Lebesgue topology[)]{.upright}, then any such $\widetilde\tau$ is an o-Lebesgue topology [(]{.upright}resp. a uo-Lebesgue topology[)]{.upright}. If $\tau$ is a Fatou topology, then $\widetilde\tau$ can be chosen to be a Fatou topology.* *Proof.* Take a metrisable linear topology $\tau^\ast$ on $E$ that is coarser than $\tau$, and let $V_1,V_2,\dotsc$ be a countable neighbourhood basis at zero for $\tau^\ast$. Since $\tau^\ast$ is coarser than $\tau$ and $\tau$ is locally solid, we can choose solid $\tau$-neighbourhoods $U_1,U_2,\dotsc$ of zero such that $U_n\subseteq V_n$ for all $n$. If $\tau$ is a Fatou topology, then we choose such $U_n$ that are also order closed. Set $\widetilde V_1\coloneqq U_1\subseteq V_1$. Then $\widetilde V_1$ is a solid $\tau$-neigbourhood of zero, which is order closed if $\tau$ is a Fatou topology. There exists a solid $\tau$-neighbourhood $A_2$ of zero such that $A_2 + A_2\subseteq \widetilde V_1$. If $\tau$ is a Fatou topology, then we choose such an $A_2$ that is order closed. Set $\widetilde V_2\coloneqq A_2\cap U_2\subseteq U_2\subseteq V_2$. Then $\widetilde V_2$ is a solid $\tau$-neighbourhood of zero, which is order closed if $\tau$ is a Fatou topology. Furthermore, $\widetilde V_2+\widetilde V_2\subseteq A_2+A_2\subseteq \widetilde V_1$. Continuing this way, we construct a normal sequence $\widetilde V_1,\widetilde V_2,\dotsc$ of solid $\tau$-neighbourhoods of zero such that $\widetilde V_n\subseteq V_n$, and which can be chosen to be order closed if $\tau$ is a Fatou topology. Since $\left\{0\right\}\subseteq\bigcap_{n\geq 1}\widetilde V_n\subseteq \bigcap_{n\geq 1} V_n=\left\{0\right\}$, it is now clear from [@kelley_namioka_LINEAR_TOPOLOGICAL_SPACES_SECOND_CORRECTED_PRINTING:1976 Theorem 5.1] that the $\widetilde V_n$ form a neighbourhood basis at zero for a linear topology on $E$ which is obviously coarser than $\tau$, finer than $\tau^\ast$, metrisable, locally solid, and a Fatou topology if $\tau$ is. The statement that $\widetilde\tau$ is an o-Lebesgue topology or a uo-Lebesgue topology when $\tau$ is, is obviously already true for any locally solid topology that is coarser than $\tau$. ◻ The following is an easy consequence of [Proposition 8](#res:redundancy){reference-type="ref" reference="res:redundancy"} and the fact that the inclusion map from a regular vector sublattice into the superlattice preserves order convergence and unbounded order convergence. It will be used in the proof of [Theorem 15](#res:when_fatou_is_submetrisable_dedekind){reference-type="ref" reference="res:when_fatou_is_submetrisable_dedekind"}. **Corollary 9**. *Let $(E,\tau)$ be a locally solid vector lattice, and let $F$ be a regular vector sublattice of $E$ such that there exists a metrisable linear topology $\tau^\ast$ on $F$ that is coarser than $\tau|_F$. Then there exists a metrisable locally solid topology $\widetilde\tau$ on $F$ that is coarser than $\tau|_F$ and finer than $\tau^\ast$. If $\tau$ is an o-Lebesgue topology [(]{.upright}resp. a uo-Lebesgue topology[)]{.upright}, then $\tau|_F$ and any such $\widetilde\tau$ are o-Lebesgue topologies [(]{.upright}resp. uo-Lebesgue topologies[)]{.upright}. If $\tau$ is a Fatou topology, then so is $\tau|_F$, and in this case $\widetilde\tau$ can be chosen to be a Fatou topology.* In the definition of local submetrisability, we could also have used a seemingly weaker condition, as is shown by the following result that will be used in the proof of [Theorem 15](#res:when_fatou_is_submetrisable_dedekind){reference-type="ref" reference="res:when_fatou_is_submetrisable_dedekind"}. **Proposition 10**. *Let $(E,\tau)$ be a locally solid vector lattice. Take $x\in E$. The following are equivalent:* 1. *[\[part:tau_I\_x_B\_x_1\]]{#part:tau_I_x_B_x_1 label="part:tau_I_x_B_x_1"} $\tau|_{B_x}$ is solidly submetrisable;* 2. *[\[part:tau_I\_x_B\_x_2\]]{#part:tau_I_x_B_x_2 label="part:tau_I_x_B_x_2"} $\tau|_{I_x}$ is solidly submetrisable.* *Proof.* It is clear that [\[part:tau_I\_x_B\_x_1\]](#part:tau_I_x_B_x_1){reference-type="ref" reference="part:tau_I_x_B_x_1"} implies [\[part:tau_I\_x_B\_x_2\]](#part:tau_I_x_B_x_2){reference-type="ref" reference="part:tau_I_x_B_x_2"}. For the converse, suppose that there exists a locally solid topology $\tau_x^\ast$ on $I_x$ that is coarser than $\tau\vert_{I_x}$, and which is metrisable by means of a metric $d_x^\ast$. By [Theorem 5](#res:metrisation_theorem){reference-type="ref" reference="res:metrisation_theorem"}, we may suppose that $d_x^\ast$ is translation invariant, and that $d_x^\ast\left(0,y_1\right)\leq d_x^\ast\left(0,y_2\right)$ for $y_1,y_2\in I_x$ with $\lvert y_1\rvert\leq\lvert y_2\rvert$. Define $\widetilde d_x\colon B_x\times B_x\to[0,\infty)$ by setting $\widetilde{d}_x\left(z_1, z_2\right)\coloneqq d_x^\ast\left(0, \lvert z_1-z_2\rvert\wedge \lvert x\rvert\right)$ for $z_1,z_2\in B_x$. Arguing as in the proof of [@kandic_marabeh_troitsky:2017 Theorem 3.2], it is easily seen that $\widetilde{d}_x$ is a translation invariant metric on $B_x$, and that the metric balls $\left\{z\in B_x:\widetilde d_x\left(0,z\right)<r\right\}$ are solid subsets of $B_x$ for $r>0$. Take $z\in B_x$. Then $\lvert\lambda z\rvert\xrightarrow{\tau} 0$ as $\lambda\to 0$. Since $\tau$ is locally solid, we also have that $\lvert\lambda z\rvert\wedge\lvert x\rvert\xrightarrow{\tau}0$. Hence $\lvert\lambda z\rvert\wedge\lvert x\rvert\xrightarrow{\tau|_{I_x}}0$, and then also $\lvert\lambda z\rvert\wedge\lvert x\rvert\xrightarrow{\tau_x^\ast}0$. This shows that $\widetilde d_x\left(0,\lambda z\right)\to 0$ as $\lambda\to 0$, so that the open balls $\left\{z\in B_x:\widetilde d_x\left(0,z\right)<r\right\}$ are absorbing subsets of $B_x$. It is now easy to see that these open balls satisfy the conditions in [@kelley_namioka_LINEAR_TOPOLOGICAL_SPACES_SECOND_CORRECTED_PRINTING:1976 Theorem 5.1] to be a neighbourhood basis at zero for a linear topology $\widetilde\tau_x$ on $B_x$, which is obviously locally solid. We are left to show that $\widetilde\tau_x$ is coarser than $\tau\vert_{B_x}$. Let $(x_\alpha)_{\alpha\in A}$ be a net in $B_x$ such that $x_\alpha\xrightarrow{\tau\vert_{B_x}} 0$ . Then $x_\alpha\xrightarrow{\tau} 0$ in $E$, so that $\lvert x_\alpha\rvert\wedge\lvert x\rvert\xrightarrow{\tau} 0$ in $E$ as $\tau$ is locally solid. Hence $\lvert x_\alpha\rvert\wedge\lvert x\rvert\xrightarrow{\tau|_{I_x}} 0$, and then also $\lvert x_\alpha\rvert\wedge\lvert x\rvert\xrightarrow{\tau_x^\ast} 0$ in $I_x$. The definition of $\widetilde{d}_x$ then shows that $x_\alpha\xrightarrow{\widetilde \tau_x} 0$ in $B_x$, as desired. ◻ For a locally solid topology, being locally solidly submetrisable is a slightly weaker property than having full carrier. This is shown by the following. **Proposition 11**. *Let $(E,\tau)$ be a locally solid vector lattice.* 1. *[\[part:when_is_submetrisable_1\]]{#part:when_is_submetrisable_1 label="part:when_is_submetrisable_1"} If $C_\tau=E$, then $\tau$ is locally solidly submetrisable.* 2. *[\[part:when_is_submetrisable_2\]]{#part:when_is_submetrisable_2 label="part:when_is_submetrisable_2"} Suppose that $E$ has the principal projection property. Then $C_\tau=E$ if and only if $\tau$ is locally solidly submetrisable.* *Proof.* We prove [\[part:when_is_submetrisable_1\]](#part:when_is_submetrisable_1){reference-type="ref" reference="part:when_is_submetrisable_1"}. Take $x\in{E}^+$. Since $C_\tau=E$, there exists a normal sequence $(V_n)_{n\geq 1}$ of solid neighbourhoods of 0 in $E$ such that $x\in\left(\bigcap_{n\geq 1}V_n\right)^\mathrm{d}$. Consider the normal sequence $(V_n\cap B_x)_{n\geq 1}$ of solid subsets of $B_x$. Suppose that $y\in\bigcap_{n\geq 1}\left(V_n\cap B_x\right)=\left(\bigcap_{n\geq 1} V_n\right)\cap B_x$. Then $y\perp x$, so that $y\perp B_x$. As also $y\in B_x$, we see that $y=0$. It now follows from [@kelley_namioka_LINEAR_TOPOLOGICAL_SPACES_SECOND_CORRECTED_PRINTING:1976 Theorem 5.1] that the $V_n\cap B_x$ are a neighbourhood basis at zero for a linear topology on $B_x$, which is clearly locally solid, metrisable, and coarser than $\tau|_{B_x}$. We prove [\[part:when_is_submetrisable_2\]](#part:when_is_submetrisable_2){reference-type="ref" reference="part:when_is_submetrisable_2"}. In view of [\[part:when_is_submetrisable_1\]](#part:when_is_submetrisable_1){reference-type="ref" reference="part:when_is_submetrisable_1"}, we need only show that $C_\tau=E$ when $\tau$ is locally solidly submetrisable and $E$ has the principal projection property. Take $x\in E$. Since $\tau$ is locally solidly submetrisable, there exists a locally solid topology $\widetilde\tau_x$ on $B_x$ which is coarser than $\tau|_{B_x}$ and which is metrisable by some metric $d_x$. From [Remark 7](#rem:invariant){reference-type="ref" reference="rem:invariant"} we see that we may suppose that $d_x$ is translation invariant and such that the metric balls $O_{n,x}\coloneqq\left\{y\in B_x: d_x\left(0,y\right)<2^{-n}\right\}$ in $B_x$ are solid in $B_x$ (and then also in $E$) for $n=1,2,\dotsc$. The translation invariance of $d_x$ implies that the $O_{n,x}$ form a normal sequence of solid $\widetilde\tau_x$-neighbourhoods of zero in $B_x$. Since $\widetilde\tau_x$ is coarser than $\tau|_{B_x}$, there exists, for each $n$, a solid $\tau$-neighbourhood $U_n$ of zero in $E$ such that $U_n\cap B_x\subseteq O_{n,x}$. Take $y\in U_n$. We can write $y=y_1+y_2$ with $y_1\in B_x$ and $y_2\in B_x^\mathrm{d}$ because $E$ has the principal projection property. From $\lvert y_1\rvert\leq \lvert y\rvert$ and the solidness of $U_n$ in $E$, it follows that $y_1\in U_n \cap B_x\subseteq O_{n,x}$. We thus see that $y=y_1+y_2\in O_{n,x}+B_x^\mathrm{d}$, and we conclude that $U_n\subseteq O_{n,x}+B_x^\mathrm{d}$. Hence the $O_{n,x}+B_x^\mathrm{d}$ are $\tau$-neighbourhoods of zero in $E$. They are solid in $E$ because $B_x^\mathrm{d}$ and the $O_{n,x}$ are, and they form a normal sequence of subsets of $E$ since the $O_{n,x}$ form a normal sequence in $B_x$. Set $N\coloneqq \bigcap_{n\geq 1} \big(O_{n,x}+ B_x^\mathrm{d}\big)$. We claim that $N=B_x^\mathrm{d}$, which will finish the proof since then clearly $x\in N^\mathrm{d}\subseteq C_\tau$. To establish the claim, take $y\in \bigcap_{n\geq 1} \big(O_{n,x}+ B_x^\mathrm{d}\big)$. For each $n$, there exist $u_n\in O_{n,x}$ and $v_n\in B_x^\mathrm{d}$ such that $y=u_n+v_n$. Then $u_n-u_m=v_m-v_n$ for all $m,n$. Since $u_n-u_m\in B_x$ and $v_m-v_n\in B_x^\mathrm{d}$, it follows that $u_n=u_m$ and $v_m=v_n$ for all $m,n$. We thus see that there exist $u\in B_x$ and $v\in B_x^\mathrm{d}$ such that $y=u+v$, and where $u\in\bigcap_{n\geq 1} O_{n,x}$. Since the latter intersection equals $\left\{0\right\}$, it follows that $y\in B_x^\mathrm{d}$. This shows that $\bigcap_{n\geq 1} \big(O_{n,x}+ B_x^\mathrm{d}\big)\subseteq B_x^\mathrm{d}$. As the reverse inclusion is clear, our claim has been established and the proof is complete. ◻ We conclude this section with a small result which is an immediate consequence of the fact that every order interval in a locally solid vector lattice is closed in the topology. **Proposition 12**. *Let $(E,\tau)$ be a locally solid vector lattice such that $\tau$ is metrisable and complete. Then $\tau$ is locally interval complete solidly submetrisable* ## Norm, unbounded norm, o-Lebesgue, and Fatou topologies {#subsec:norm_unbounded_norm_lebesgue_and_fatou} In this section, we consider a number of particular locally solid topologies and study when they are locally (interval complete) solidly submetrisable. We start with the norm and unbounded norm topology on a Banach lattice. **Proposition 13**. *Let $E$ be a Banach lattice. Then the norm topology and the unbounded norm topology on $E$ are locally interval complete solidly submetrisable.* *Proof.* We denote the norm topology on $E$ by $\tau$. It is clear from [Proposition 12](#res:metrisable_is_OK){reference-type="ref" reference="res:metrisable_is_OK"} that $\tau$ is locally interval complete solidly submetrisable. We turn to the locally solid unbounded norm topology $\mathrm{u}\tau$. Take $x\in{E}^+$, and define $d_x\colon B_x\times B_x\to[0,\infty)$ by setting $$d_x\left(x_1,x_2\right)\coloneqq\left\Vert\lvert x_1-x_2\rvert\wedge (2x)\right\Vert$$ for $x_1, x_2\in B_x$. As in the proof of [@kandic_marabeh_troitsky:2017 Theorem 3.2], it is easily seen that $d_x$ is a metric on $B_x$. The facts that $d_x$ is translation invariant and that the open balls $\left\{y\in B_x: d_x\left(0,y\right)<r\right\}$ for $r>0$ are solid make it simple to verify that these open balls verify the conditions in [@kelley_namioka_LINEAR_TOPOLOGICAL_SPACES_SECOND_CORRECTED_PRINTING:1976 Theorem 5.1] to be a neighbourhood basis at zero for a linear topology $\widetilde\tau_x$ on $B_x$, which is then obviously locally solid. It is immediate from the definition of the $\mathrm{u}\tau$-topology that $\mathrm{u}\tau|_{B_x}$ is finer than $\widetilde\tau_x$. Finally, the fact that $d_x\left(x_1,x_2\right)=\lVert x_1-x_2\rVert$ for $x_1,x_2\in[0,x]$ makes it clear that $([0,x],d_x)$ is a complete metric space. ◻ **Remark 14**. The proof for the unbounded norm topology can easily be modified to yield a more general result: for $x\in{E}^+$ and $y,z\in I_x$ with $y\leq z$, there exists a metric $d_{y,z}$ on $B_x$ that is coarser than $\tau|_{B_x}$ and such that $([y,z], d_{y,z})$ is a complete metric space. Indeed, after choosing $\lambda\geq 0$ such that $[y,z]\subseteq[-\lambda x,\lambda x]$, defining $d_{y,z}\left(x_1,x_2\right)\coloneqq\left\Vert\lvert x_1-x_2\rvert\wedge (2\lambda x)\right\Vert$ for $x_1,x_2\in B_x$ yields such a metric. There does not appear to be an obvious way (if it is at all possible) to produce a metric on $B_x$ that induces a locally solid topology on $B_x$ which is coarser than $\mathrm{u}\tau|_{B_x}$, and such that *all* order intervals in $I_x$ are complete metric spaces, let alone all order intervals in $B_x$. We now turn to o-Lebesgue and Fatou topologies. If $(E,\tau)$ is a locally solid vector lattice, where $\tau$ is a Fatou topology, then we know from [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 4.12] that there exists a unique Fatou topology $\tau^\delta$ on the Dedekind completion $E^\delta$ of $E$ extending $\tau$. When $\tau$ is an o-Lebesgue topology, so is $\tau^\delta$. An inspection of the proof shows that $\tau^\delta$ is metrisable when $\tau$ is. Furthermore, if $\tau_1$ and $\tau_2$ are Fatou topologies on $E$ such that $\tau_1\subseteq\tau_2$, then $\tau_1^\delta\subseteq\tau_2^\delta$. These two observations are used in the proof of the following. **Theorem 15**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is a Fatou topology. Let $\tau^\delta$ be the extension of $\tau$ to a Fatou topology on the Dedekind completion $E^\delta$ of $E$. The following are equivalent:* 1. *[\[dedekind1\]]{#dedekind1 label="dedekind1"} $\tau$ is locally solidly submetrisable;* 2. *[\[dedekind4\]]{#dedekind4 label="dedekind4"} $\tau^\delta$ is locally solidly submetrisable;* 3. *[\[dedekind5\]]{#dedekind5 label="dedekind5"} $\tau^\delta$ is locally interval complete solidly submetrisable;* 4. *[\[dedekind2\]]{#dedekind2 label="dedekind2"} $C_\tau=E$;* 5. *[\[dedekind3\]]{#dedekind3 label="dedekind3"} $C_{\tau^\delta}=E^\delta$.* *When $\tau$ is an o-Lebesgue topology, then [\[dedekind1\]](#dedekind1){reference-type="ref" reference="dedekind1"}--[\[dedekind3\]](#dedekind3){reference-type="ref" reference="dedekind3"} are also equivalent to:* 1. *[\[dedekind6\]]{#dedekind6 label="dedekind6"} $E$ has the countable sup property.* *Proof.* We see from [Proposition 11](#res:when_is_submetrisable){reference-type="ref" reference="res:when_is_submetrisable"} that [\[dedekind2\]](#dedekind2){reference-type="ref" reference="dedekind2"} $\Rightarrow$ [\[dedekind1\]](#dedekind1){reference-type="ref" reference="dedekind1"} and that [\[dedekind3\]](#dedekind3){reference-type="ref" reference="dedekind3"} $\Leftrightarrow$ [\[dedekind4\]](#dedekind4){reference-type="ref" reference="dedekind4"}. It is clear that [\[dedekind5\]](#dedekind5){reference-type="ref" reference="dedekind5"} $\Rightarrow$ [\[dedekind4\]](#dedekind4){reference-type="ref" reference="dedekind4"}. The combination of [Corollary 9](#res:can_be_lebesgue_fatou){reference-type="ref" reference="res:can_be_lebesgue_fatou"} and [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 4.28] shows that [\[dedekind4\]](#dedekind4){reference-type="ref" reference="dedekind4"} $\Rightarrow$ [\[dedekind5\]](#dedekind5){reference-type="ref" reference="dedekind5"}. The proof of the equivalence of the first five properties will be complete once we show that [\[dedekind1\]](#dedekind1){reference-type="ref" reference="dedekind1"} $\Rightarrow$ [\[dedekind4\]](#dedekind4){reference-type="ref" reference="dedekind4"} and that[\[dedekind3\]](#dedekind3){reference-type="ref" reference="dedekind3"} $\Rightarrow$ [\[dedekind2\]](#dedekind2){reference-type="ref" reference="dedekind2"}. We prove that [\[dedekind1\]](#dedekind1){reference-type="ref" reference="dedekind1"} $\Rightarrow$ [\[dedekind4\]](#dedekind4){reference-type="ref" reference="dedekind4"}. The first step is to show that $\tau^\delta|_{I_x^\delta}$ is solidly submetrisable for $x\in E$, as follows. We see from [Corollary 9](#res:can_be_lebesgue_fatou){reference-type="ref" reference="res:can_be_lebesgue_fatou"} that there exists a metrisable Fatou topology $\widetilde\tau_x$ on $I_x$ such that $\widetilde\tau_x\subseteq \tau|_{I_x}$. Since $I_x^\delta$ is the Dedekind completion of $I_x$, there exists a unique Fatou topology $\left(\widetilde\tau_x\right)^\delta$ (resp. $\left(\tau|_{I_x}\right)^\delta$) on $I_x^\delta$ that extends $\widetilde\tau_x$ (resp. $\tau|_{I_x}$). From the observations prior to the theorem, we see that $\left(\widetilde\tau_x\right)^\delta$ is metrisable and that $\left(\widetilde\tau_x\right)^\delta\subseteq \left(\tau|_{I_x}\right)^\delta$. Now we note that $\tau^\delta|_{I_x^\delta}$ is also a Fatou topology on $I_x^\delta$, and that $\left(\tau^\delta|_{I_x^\delta}\right)|_{I_x}=\tau^\delta|_{I_x}=\left(\tau^\delta|_E\right)|_{I_x}=\tau|_{I_x}$. Since there is only one Fatou topology on $I_x^\delta$ extending $\tau|_{I_x}$, we conclude that $\left(\tau|_{I_x}\right)^\delta=\tau^\delta|_{I_x^\delta}$. Hence $\left(\widetilde\tau_x\right)^\delta\subseteq \tau^\delta|_{I_x^\delta}$, which shows that $\tau^\delta|_{I_x^\delta}$ is solidly submetrisable, as desired. For the second step, take $x\in E^\delta$. Since $E$ is majorising in $E^\delta$, there exists an $y\in E$ such that $I_x^\delta\subseteq I_y^\delta$. As we know that $\tau^\delta|_{I_y^\delta}$ is solidly submetrisable, the same is then clearly true for $\tau^\delta_{I_x^\delta}$. [Proposition 10](#res:tau_I_x_B_x){reference-type="ref" reference="res:tau_I_x_B_x"} then yields that $\tau^\delta$ is solidly submetrisable. We prove that [\[dedekind3\]](#dedekind3){reference-type="ref" reference="dedekind3"} $\Rightarrow$ [\[dedekind2\]](#dedekind2){reference-type="ref" reference="dedekind2"}. Using [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Exercise 4.5] in the first step, we have $$C_\tau=C_{\tau^\delta}\cap E=E^\delta\cap E=E,$$ as desired. Hence the first five properties are equivalent. When $\tau$ is an o-Lebesgue topology, the equivalence of [\[dedekind2\]](#dedekind2){reference-type="ref" reference="dedekind2"} and [\[dedekind6\]](#dedekind6){reference-type="ref" reference="dedekind6"} is a part of [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 4.26]. ◻ **Remark 16**. 1. [\[rem:fatou_topology_major_theorem_1\]]{#rem:fatou_topology_major_theorem_1 label="rem:fatou_topology_major_theorem_1"} The equivalence of all parts [\[dedekind1\]](#dedekind1){reference-type="ref" reference="dedekind1"}--[\[dedekind6\]](#dedekind6){reference-type="ref" reference="dedekind6"} in [Theorem 15](#res:when_fatou_is_submetrisable_dedekind){reference-type="ref" reference="res:when_fatou_is_submetrisable_dedekind"} does not hold for arbitrary Fatou topologies, not even when the vector lattice is Dedekind complete. Indeed, if $A$ is an uncountable set, then, in view of [Proposition 12](#res:metrisable_is_OK){reference-type="ref" reference="res:metrisable_is_OK"}, the (Fatou) supremum norm topology on $\ell^\infty(A)$ is locally interval complete solidly submetrisable, but $\ell^\infty(A)$ does not have the countable sup property. 2. [\[rem:fatou_topology_major_theorem_2\]]{#rem:fatou_topology_major_theorem_2 label="rem:fatou_topology_major_theorem_2"} In view of [Proposition 8](#res:redundancy){reference-type="ref" reference="res:redundancy"}, it follows from [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 5.33] that a vector lattice has the countable sup property when it admits a submetrisable o-Lebesgue topology. This can now be improved: as a consequence of [Theorem 15](#res:when_fatou_is_submetrisable_dedekind){reference-type="ref" reference="res:when_fatou_is_submetrisable_dedekind"}, [Proposition 10](#res:tau_I_x_B_x){reference-type="ref" reference="res:tau_I_x_B_x"}, and [Proposition 8](#res:redundancy){reference-type="ref" reference="res:redundancy"}, a vector lattice has the countable sup property when it admits an o-Lebesgue topology such that its restrictions to all principal ideals are submetrisable. **Example 17**. Let $A$ be a non-empty set. We provide the Dedekind complete vector lattice $\ell^\infty(A)$ with the o-Lebesgue topology $\tau$ that is generated by the family of lattice seminorms $p_a$ for $a\in A$, defined by setting $p_a(x)\coloneqq\lvert x(a)\rvert$ for $x\in\ell^\infty(A)$. Then [Theorem 15](#res:when_fatou_is_submetrisable_dedekind){reference-type="ref" reference="res:when_fatou_is_submetrisable_dedekind"} shows that this topology of pointwise convergence is locally (interval complete) solidly submetrisable if and only if $\ell^\infty(A)$ has the countable sup property, i.e., if and only if $A$ is countable. In view of [Theorem 15](#res:when_fatou_is_submetrisable_dedekind){reference-type="ref" reference="res:when_fatou_is_submetrisable_dedekind"} and, in particular, of its role as a stepping stone for [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} in the current paper, it is desirable to have sufficient conditions for a Fatou topology $\tau$ to have full carrier $C_\tau$. This is evidently so when $\tau$ is metrisable, and also when the vector lattice has the countable sup property (see [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 4.17]). We shall now proceed to show that a Fatou topology $\tau$ has full carrier when $C_\tau$ has a countable order basis. We need the following preparatory result, for which we recall that a $\sigma$-Dedekind complete vector lattice has the principal projection property; see [@luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971 Theorem 25.1], for example. **Lemma 18**. *Let $E$ be a $\sigma$-Dedekind complete vector lattice with a countable order basis $\left\{e_n:n\geq 1\right\}$ such that $0\leq e_1\leq e_2\leq \dotsc$. Denote the corresponding order projections onto the principal bands generated by the $e_n$ by $P_{e_n}$. Then $x=\sup_{n\geq 1} P_{e_n}x$ for $x\in {E}^+$.* *Proof.* Take $x\in{E}^+$. Since the $e_n$ are increasing, so are the $P_{e_n}$. As $0\leq P_{e_n} x\uparrow\leq x$ and $E$ is $\sigma$-Dedekind complete, $\sup_{n\geq 1} P_{e_n} x$ exists. From the fact that $0\leq x-\sup_{n\geq 1} P_{e_n} x\leq x - P_{e_k}x$ for $k\geq 1$, we see that $(x-\sup_{n\geq 1} P_{e_n} x)\perp e_k$ for $k\geq 1$. Hence $x-\sup_{n\geq 1} P_{e_n} x=0$. ◻ **Proposition 19**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is a Fatou topology. Suppose that $C_\tau$ has a countable order basis. Then $C_\tau=E$.* *Proof.* We first suppose that $E$ is $\sigma$-Dedekind complete. According to [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 4.17], $C_\tau$ is an order dense $\sigma$-ideal in $E$. Let $\left\{e_n: n\geq 1\right\}$ be a countable order basis of $C_\tau$. By passing to $\left\{\sum_{i=1}^n \lvert e_i\rvert : n\geq 1\right\}$, we may suppose that $0\leq e_1\leq e_2\leq\dotsc$. Since $C_\tau$ is order dense in $E$, $\left\{e_n: n\geq 1\right\}$ is also an order basis of $E$. Let $P_{e_n}$ denote the order projections from $E$ onto the principal bands generated by the $e_n$. Take $x\in{E}^+$. Then [Lemma 18](#res:countable_order_basis_and_sigma-Dedekind_complete){reference-type="ref" reference="res:countable_order_basis_and_sigma-Dedekind_complete"} yields that $x=\sup_{n\geq 1} P_{e_n}x$ in $E$. The formula for principal band projections and the fact that $C_\tau$ is a $\sigma$-ideal in $E$ imply that $P_{e_n}x\in C_\tau$. Using once more the fact that $C_\tau$ is a $\sigma$-ideal in $E$, we see that $x\in C_\tau$. This concludes the proof when $E$ is $\sigma$-Dedekind complete. Now we consider arbitrary $E$, where we let $\tau^\delta$ denote the unique extension of $\tau$ to a Fatou topology on $E^\delta$. Let $\left\{e_n:n\geq 1\right\}$ be a countable order basis of $C_\tau$. Then $\left\{e_n:n\geq 1 \right\}\subseteq C_\tau\subseteq C_{\tau^\delta}$ because $C_\tau=E\cap C_{\tau^\delta}$ by [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Exercise 4.5]. Since $E$ is order dense in $E^\delta$, and since [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 4.17] still shows that $C_\tau$ is order dense in $E$, $C_\tau$ is order dense in $E^\delta$. Hence $\left\{e_n:n\geq 1 \right\}$ is also an order basis of $E^\delta$, and then in particular of $C_\tau^\delta$. The first part of the proof then shows that $C_{\tau^\delta}=E^\delta$, which implies that $C_\tau=E\cap C_{\tau^\delta}=E\cap E^\delta=E$, as required. ◻ **Remark 20**. For an o-Lebesgue topology $\tau$, one can do better than [Proposition 19](#res:C_tau=E_c.o.basis){reference-type="ref" reference="res:C_tau=E_c.o.basis"}. As seen from [@kandic_taylor:2018 Lemma 6.8], the fact that $C_\tau$ has a countable order basis then not only implies that $\tau$ has full carrier (which is equivalent to $E$ having the countable sup property), but also that $E$ itself has a countable order basis. We combine our sufficient conditions for a Fatou topology to have full carrier with a part of [Theorem 15](#res:when_fatou_is_submetrisable_dedekind){reference-type="ref" reference="res:when_fatou_is_submetrisable_dedekind"} in the following result, formulated in a way ready to be combined with [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"}. **Theorem 21**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is a Fatou topology. Suppose that $C_\tau=E$, which is certainly the case when $E$ has the countable sup property, when $C_\tau$ has a countable order basis, and when $\tau$ is metrisable. Let $\tau^\delta$ be the extension of $\tau$ to a Fatou topology on the Dedekind completion $E^\delta$ of $E$. Then $\tau^\delta$ is locally interval complete solidly submetrisable.* # Embedded unbounded order convergent sequences in topologically convergent nets {#sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets} The following theorem, which will be combined with the results from [3](#sec:metrisable_and_submetrisable_topologies_on_vector_lattices){reference-type="ref" reference="sec:metrisable_and_submetrisable_topologies_on_vector_lattices"} in [5](#sec:applications){reference-type="ref" reference="sec:applications"}, is the core result of this paper. Although the statements in its final paragraph will most likely be sufficient in the majority of applications, we still formulate it in its most precise form. [Corollary 23](#res:subsequences_several_convergences){reference-type="ref" reference="res:subsequences_several_convergences"} may give evidence that it is worth doing so; see also [Remark 24](#rem:flexibility){reference-type="ref" reference="rem:flexibility"}. **Theorem 22**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is locally interval complete solidly submetrisable, and let $F$ be a regular vector sublattice of $E$. Let $(x_\alpha)_{\alpha\in A}$ be a net in $F$ and let $x\in F$ be such that $x_\alpha\xrightarrow{\tau}x$ in $E$.* *Suppose that $A$ has a largest element $\alpha_{\mathrm{largest}}$. Then $x=x_{\alpha_{\mathrm{largest}}}$.* *Suppose that $A$ has no largest element. In this case, one can choose any $\widetilde\alpha_1\in A$; then find an $\alpha_1\in A$; then choose any $\widetilde\alpha_2\in A$; then find an $\alpha_2\in A$; etc., such that:* 1. *[\[indices_1\]]{#indices_1 label="indices_1"} $\alpha_1<\alpha_2<\alpha_3<\dotsc$;* 2. *[\[indices_2\]]{#indices_2 label="indices_2"} $\alpha_n>\widetilde\alpha_n$ for $n\geq 1$;* 3. *[\[indices_3\]]{#indices_3 label="indices_3"} $x_{\alpha_n}\xrightarrow{\mathrm{uo}}x$ as $n\to\infty$ in $F$.* *In particular, for arbitrary $A$, there exist indices $\alpha_1\leq\alpha_2\leq\alpha_3\leq\dotsc$ such that $x_{\alpha_n}\xrightarrow{\mathrm{uo}}x$ in $F$; if $A$ has no largest element, then there exist indices $\alpha_1<\alpha_2<\alpha_3<\dotsc$ such that $x_{\alpha_n}\xrightarrow{\mathrm{uo}}x$ in $F$.* *Proof.* It is clear that $x=x_{\alpha_{\mathrm{largest}}}$ when $A$ has a largest element $\alpha_{\mathrm{largest}}$. Suppose that $A$ does not have a largest element. Using the linearity of $\tau$-convergence and uo-convergence; that $\lvert x_\alpha\rvert\xrightarrow{\tau}0$ when $x_\alpha\xrightarrow{\tau}0$; that uo-convergence in $E$ of a sequence (or net) in the regular vector sublattice $F$ of $E$ to an element of $F$ implies uo-convergence in $F$ to the same limit; and that $x_\alpha\xrightarrow{\mathrm{uo}}0$ when $\lvert x_\alpha\rvert\xrightarrow{\mathrm{uo}}0$, it is easy to see that it is sufficient to consider the case of a net $(x_\alpha)_{\alpha\in A}$ in ${E}^+$ such that $x_\alpha\xrightarrow{\tau}0$ in $E$. After this reduction, we choose any index $\widetilde\alpha_1\in A$. Since $A$ has no maximal element, we can find an index $\alpha_1$ such that $\alpha_1>\widetilde\alpha_1$. By assumption (see also [Remark 7](#rem:invariant){reference-type="ref" reference="rem:invariant"}), we can find a translation invariant metric $d_1$ on $B_{x_{\alpha_1}}$ such that $d_1\left(0,y\right)\leq d_1\left(0,z\right)$ for all $y,z\in B_{x_{\alpha_1}}$ with $\lvert y\rvert\leq\lvert z\rvert$, and also such that: 1. the metric topology on $B_{x_{\alpha_1}}$ induced by $d_1$ is coarser than $\tau\vert_{B_{x_{\alpha_1}}}$; 2. $([0,x_{\alpha_1}],d_1)$ is a complete metric space. For $n=2,3,\dotsc$, we shall now inductively choose any index $\widetilde\alpha_n$, and then find an index $\alpha_n$, together with a translation invariant metric $d_n$ on $B_{x_{\alpha_n}}$ such that $d_n\left(0,y\right)\leq d_n\left(0,z\right)$ for all $y,z\in B_{x_{\alpha_n}}$ with $\lvert y\rvert\leq\lvert z\rvert$, and also such that: 1. [\[property_1\]]{#property_1 label="property_1"} the metric topology on $B_{x_{\alpha_n}}$ induced by $d_n$ is coarser than $\tau\vert_{B_{x_{\alpha_n}}}$; 2. [\[property_2\]]{#property_2 label="property_2"} $([0,x_{\alpha_n}],d_n)$ is a complete metric space; 3. [\[property_3\]]{#property_3 label="property_3"} $\alpha_n>\alpha_{n-1}$ and $\alpha_n>\widetilde\alpha_n$; 4. [\[property_4\]]{#property_4 label="property_4"} $d_k\left(0,x_{\alpha_n}\wedge x_{\alpha_k}\right)\leq \dfrac{1}{2^n}$ for $k=1,2,\dotsc,n-1$. We start with $n=2$. Choose any index $\widetilde\alpha_{2}$. From the fact that $x_\alpha\xrightarrow{\tau}0$ in $E$ we see that also $x_\alpha\wedge x_{\alpha_1}\xrightarrow{\tau}0$ because $\tau$ is locally solid. Since $x_{\alpha}\wedge x_{\alpha_1}\in B_{x_{\alpha_1}}$ for $\alpha\in A$, and since $\tau\vert_{B_{x_{\alpha_1}}}$ is finer than the metric topology on $B_{x_{\alpha_1}}$ induced by $d_1$, we conclude that $d_1\left(0,x_\alpha \wedge x_{\alpha_1}\right)\to 0$. Hence we can find an $\alpha_2\in A$ with $\alpha_2>\alpha_1$ and $\alpha_2>\widetilde\alpha_2$ such that $d_1\left(0,x_{\alpha_2}\wedge x_{\alpha_1}\right)\leq \dfrac{1}{2^2}$. This takes care of [\[property_3\]](#property_3){reference-type="ref" reference="property_3"} and [\[property_4\]](#property_4){reference-type="ref" reference="property_4"}. Since $\tau$ is locally interval complete solidly submetrisable, we can find a translation invariant metric $d_2$ on $B_{x_{\alpha_2}}$ such that $d_2\left(0,y\right)\leq d_2\left(0,z\right)$ for all $y,z\in B_{x_{\alpha_2}}$ with $\lvert y\rvert\leq\lvert z\rvert$, and also such that [\[property_1\]](#property_1){reference-type="ref" reference="property_1"} and [\[property_2\]](#property_2){reference-type="ref" reference="property_2"} are satisfied. This completes the choices for $n=2$. Suppose that $n\geq 2$ and that we have already found $\alpha_2,\dotsc,\alpha_n$ and $d_2,\dotsc,d_n$ satisfying [\[property_1\]](#property_1){reference-type="ref" reference="property_1"}--[\[property_4\]](#property_4){reference-type="ref" reference="property_4"}. Choose any index $\widetilde\alpha_{n+1}$. Then $\alpha_{n+1}$ and $d_{n+1}$ can be found by essentially the same argument as for $n=2$. Indeed, the fact that $x_\alpha\xrightarrow{\tau}0$ implies that $x_{\alpha}\wedge x_{\alpha_k}\xrightarrow{\tau}0$ for $k=1,\dotsc,n$. Since $x_{\alpha}\wedge x_{\alpha_k}\in B_{x_{\alpha_k}}$ for $\alpha\in A$ and $k=1,\dotsc,n$, and since $\tau\vert_{B_{x_{\alpha_k}}}$ is finer than the metric topology on $B_{x_{\alpha_k}}$ induced by $d_k$ for $k=1,\dotsc,n$, we conclude that $d_k\left(0,x_\alpha \wedge x_{\alpha_k}\right)\to 0$ for $k=1,\dotsc,n$. Hence we can find an $\alpha_{n+1}\in A$ with $\alpha_{n+1}>\alpha_n$ and $\alpha_{n+1}>\widetilde\alpha_{n+1}$ such that $d_k\left(0,x_{\alpha_{n+1}}\wedge x_{\alpha_k}\right)\leq \dfrac{1}{2^{n+1}}$ for $k=1,2,\dotsc,n$. This takes care of [\[property_3\]](#property_3){reference-type="ref" reference="property_3"} and [\[property_4\]](#property_4){reference-type="ref" reference="property_4"}. Since $\tau$ is locally interval complete solidly submetrisable, we can find a translation invariant metric $d_{n+1}$ on $B_{x_{\alpha_{n+1}}}$ such that $d_{n+1}\left(0,y\right)\leq d_{n+1}\left(0,z\right)$ for all $y,z\in B_{x_{\alpha_{n+1}}}$ with $\lvert y\rvert\leq\lvert z\rvert$, and also such that [\[property_1\]](#property_1){reference-type="ref" reference="property_1"} and [\[property_2\]](#property_2){reference-type="ref" reference="property_2"} are satisfied. This completes the induction step. Now that the $\alpha_n$ and the $d_n$ as required have been found, we fix a $k\geq 1$. From [\[property_4\]](#property_4){reference-type="ref" reference="property_4"} we know that $d_k\left(0,x_{\alpha_i}\wedge x_{\alpha_k}\right)\leq \dfrac{1}{2^i}$ for $i\geq k+1$. Noticing that $\bigvee_{i=r}^s x_{\alpha_i}\wedge x_{\alpha_k}\in B_{x_{\alpha_k}}$ for $1\leq r\leq s$, using the translation invariance of $d_k$ on $B_{x_{\alpha_k}}$, the fact that $y\vee z-y\leq z$ for positive elements $y,z$ of a vector lattice, as well as the monotonicity of $d_k\left(0,\,\cdot\,\right)$ on $B_{x_{\alpha_k}}^+$, we see from this that, for $p>q\geq n\geq k$, $$\begin{aligned} d_k\left(\bigvee_{i=n}^p x_{\alpha_i}\wedge x_{\alpha_k},\bigvee_{i=n}^q x_{\alpha_i} \wedge x_{\alpha_k}\right)&=d_k\left(0,\bigvee_{i=n}^p x_{\alpha_i}\wedge x_{\alpha_k}-\bigvee_{i=n}^q x_{\alpha_i}\wedge x_{\alpha_k}\right) \\&\leq d_k\left(0,\bigvee_{i=q+1}^p x_{\alpha_i}\wedge x_{\alpha_k}\right) \\&\leq d_k\left(0,\sum_{i=q+1}^p x_{\alpha_i}\wedge x_{\alpha_k}\right) \\&\leq \sum_{i=q+1}^p d_k\left(0,x_{\alpha_i}\wedge x_{\alpha_k}\right) \\&\leq\sum_{i=q+1}^\infty d_k\left(0,x_{\alpha_i}\wedge x_{\alpha_k}\right) \\&\leq \sum_{i=q+1}^\infty \dfrac{1}{2^i}=\dfrac{1}{2^q}. \end{aligned}$$ For a fixed $n\geq k$, this implies that the sequence $\left(\bigvee_{i=n}^p x_{\alpha_i}\wedge x_{\alpha_k}\right)_{p=n}^\infty$ is a Cauchy sequence in the complete metric space $([0,x_{\alpha_k}], d_k)$. Hence its limit $l_{n,k}$ as $p\to\infty$ exists in the topology that $d_k$ induces on $B_{x_{\alpha_k}}$. It follows from the local solidness of this topology that $l_{n,k}\downarrow_{n; n\geq k}$, and also that $0\leq x_{\alpha_n}\wedge x_{\alpha_k}\leq l_{n,k}$ for $n\geq k$ ; see [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 2.21]. Furthermore, for $p\geq n\geq k+1$, we have $$\begin{aligned} d_k\left(0,\bigvee_{i=n}^p x_{\alpha_i}\wedge x_{\alpha_k}\right)&\leq d_k\left(0,\sum_{i=n}^p x_{\alpha_i}\wedge x_{\alpha_k}\right) \\ &\leq \sum_{i=n}^p d_k\left(0, x_{\alpha_i}\wedge x_{\alpha_k}\right)\\ &\leq \sum_{i=n}^p \frac{1}{2^i}\\ &<\frac{1}{2^{n-1}}.\end{aligned}$$ We conclude that $d_k\left(0,l_{n,k}\right)\leq 1/2^{n-1}$ for $n\geq k+1$, so that the sequence $\left (l_{n,k}\right)_{n=k+1}^\infty$ converges to zero in the topology that $d_k$ induces on $B_{x_{\alpha_k}}$. As this topology is locally solid and the sequence is decreasing, [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 2.21 (c)] shows that $l_{n,k}\downarrow_{n;n\geq k+1} 0$ in $B_{x_{\alpha_k}}$. Since we had already observed that $0\leq x_{\alpha_n}\wedge x_{\alpha_k}\leq l_{n,k}$ for $n\geq k$, we can now conclude that $x_{\alpha_n}\wedge x_{\alpha_k}\xrightarrow{\mathrm{o}}0$ in $B_{x_{\alpha_k}}$ as $n\to\infty$. Because $B_{x_{\alpha_k}}$ is a regular vector sublattice of $E$, we see that also $x_{\alpha_n}\wedge x_{\alpha_k}\xrightarrow{\mathrm{o}}0$ in $E$ as $n\to\infty$. After having covered one fixed $k\geq 1$, we now let $B$ be the band in $E$ that is generated by $\left\{x_{\alpha_k}:k=1,2,\dotsc\right\}$. Clearly, $(x_{\alpha_n})_{n=1}^\infty\subseteq B$, and we have just shown that $x_{\alpha_n}\wedge x_{\alpha_k}\xrightarrow{\mathrm{o}}0$ in $E$ for $k=1,2,\dotsc$. By [@deng_de_jeu:2022a Proposition 7.4], this implies that $x_{\alpha_n}\xrightarrow{\mathrm{uo}}0$ in $E$, as desired. This completes the proof of the existence of indices $\alpha_1,\alpha_2,\alpha_3,\dotsc$ satisfying [\[indices_1\]](#indices_1){reference-type="ref" reference="indices_1"}--[\[indices_3\]](#indices_3){reference-type="ref" reference="indices_3"} in the case that $A$ has no largest element. The statements in the final paragraph follow by taking all $\alpha_n$ equal to $\alpha_{\mathrm{largest}}$ if $A$ has a largest element $\alpha_{\mathrm{largest}}$, and by taking all $\widetilde\alpha_{n}$ equal to a fixed element of $A$ if $A$ has no largest element. ◻ The precise formulation regarding the $\widetilde\alpha_n$ and the $\alpha_n$ in [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} allows one to combine several convergences, as follows. **Corollary 23**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is locally interval complete solidly submetrisable, and let $F$ be a regular vector sublattice of $E$. Let $(x_\alpha)_{\alpha\in A}$ be a net in $F$ and let $x\in F$ be such that $x_\alpha\xrightarrow{\tau}x$ in $E$. Suppose, furthermore, that $\tau_1,\dotsc,\tau_k$ are metrisable linear topologies on $E$, and that $x_1,\dotsc,x_k\in E$ are such that $x_\alpha\xrightarrow{\tau_i} x_i$ for $i=1,\dotsc,k$.* *Suppose that $A$ has a largest element $\alpha_{\mathrm{largest}}$. Then $x=x_1=\dotsb=x_k=x_{\alpha_{\mathrm{largest}}}$.* *Suppose that $A$ has no largest element. In this case, there exist indices $\alpha_1<\alpha_2<\alpha_3<\dotsc$ in $A$ such that:* 1. *$x_{\alpha_n}\xrightarrow{\mathrm{uo}}x$ as $n\to\infty$ in $F$.* 2. *$x_{\alpha_n}\xrightarrow{\tau_i} x_i$ as $n\to\infty$ for $i=1,\dotsc,k$.* *Consequently, for arbitrary $A$, there exist indices $\alpha_1\leq\alpha_2\leq\alpha_3\leq\dotsc$ such that $x_{\alpha_n}\xrightarrow{\mathrm{uo}}x$ in $F$ as well as $x_{\alpha_n}\xrightarrow{\tau_i} x_i$ for $i=1,\dotsc,k$; when $A$ has no largest element, one can choose the $\alpha_n$ to be strictly increasing.* *Proof.* When $A$ has a largest element, all is clear. Suppose that this not the case. For $i=1,\dotsc,k$, let $\{V_n^i:n\geq 1\}$ be a $\tau_i$-neighbourhood base at $x_i$. For each $n$, there exists an $\widetilde\alpha_n$ such that $x_{\alpha}\in V_n^i$ for all $\alpha\geq\widetilde\alpha_n$ and all $i=1,\dots,k$. We now choose these $\widetilde\alpha_n$ in [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"}. ◻ **Remark 24**. [Corollary 23](#res:subsequences_several_convergences){reference-type="ref" reference="res:subsequences_several_convergences"} does still not use the precise formulation of [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} to its full extent. The choice of the $\widetilde\alpha_n$ in [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} is flexible, in the sense that one can let the chosen value of $\widetilde\alpha_{n+1}$ depend on the then already known values of $\alpha_1,\dotsc,\alpha_n$ and $\widetilde\alpha_1,\dotsc,\widetilde\alpha_n$. In the proof of [Corollary 23](#res:subsequences_several_convergences){reference-type="ref" reference="res:subsequences_several_convergences"}, however, all $\widetilde\alpha_n$ that are used in the concluding application of [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} are already given in advance. This additional flexibility is presently still awaiting an application in presumably more delicate proofs. For the ease of reference in the remainder of the paper, we include the following rephrasing of the (less precise) final paragraph of [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} in the terminology of [Definition 2](#def:embedded_sequence){reference-type="ref" reference="def:embedded_sequence"}. **Theorem 25**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is locally interval complete solidly submetrisable, and let $F$ be a regular vector sublattice of $E$. Let $(x_\alpha)_{\alpha\in A}$ be a net in $F$ and let $x\in F$ be such that $x_\alpha\xrightarrow{\tau}x$ in $E$. Then $(x_\alpha)_{\alpha\in A}$ contains an embedded sequence that is uo-convergent to $x$ in $F$.* *In particular, every sequence in $F$ that is $\tau$-convergent to an element of $F$ has a subsequence that is uo-convergent to the same limit in $F$.* **Remark 26**. When $\tau$ in [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"} is, in addition, metrisable, [Corollary 4](#res:nets_and_sequences_general){reference-type="ref" reference="res:nets_and_sequences_general"} shows that the existence of an embedded sequence in an arbitrary $\tau$-convergent net that is uo-convergent to the same limit is equivalent to the existence of a subsequence of every $\tau$-convergent sequence that is uo-convergent to the same limit. It follows from [Lemma 3](#res:nets_and_sequences_no_largest_element){reference-type="ref" reference="res:nets_and_sequences_no_largest_element"} that they are both also equivalent to the existence of embedded sequences in nets in the more precise formulation as in [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"}. # Applications {#sec:applications} In this section, we combine [\[sec:metrisable_and_submetrisable_topologies_on_vector_lattices,sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets\]](#sec:metrisable_and_submetrisable_topologies_on_vector_lattices,sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets){reference-type="ref" reference="sec:metrisable_and_submetrisable_topologies_on_vector_lattices,sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets"}. While doing so, we take some care pointing out how our results relate to the existing literature. It will become apparent that the presence of the countable sup property is a sufficient condition for several results in this section to hold. We refer to [2.2](#subsec:countable_sup_property){reference-type="ref" reference="subsec:countable_sup_property"} for some additional material on this property. For the sake of simplicity, the results that are to follow are based on [Theorem 25](#res:subsequence_simplified){reference-type="ref" reference="res:subsequence_simplified"} where we take $F=E$. We remark, however, that the more precise formulation in [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"}, involving a regular vector sublattice and a more precise statement on the indices for the embedded sequence, is also always valid.[^9] We start with a case where an application of the results in [4](#sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets){reference-type="ref" reference="sec:embedded_unbounded_order_convergent_sequences_in_convergent_nets"} does, in fact, not yield an optimal statement. Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is metrisable and complete. The combination of [Proposition 12](#res:metrisable_is_OK){reference-type="ref" reference="res:metrisable_is_OK"} and [Theorem 25](#res:subsequence_simplified){reference-type="ref" reference="res:subsequence_simplified"} shows that every $\tau$-convergent net has an embedded sequence that is uo-convergent to the same limit. A much stronger statement is true, however. **Proposition 27**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is metrisable and complete. Then every $\tau$-convergent net in $E$ has an embedded sequence that is relatively uniformly convergent to the same limit.* *Proof.* According to [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Exercise 5.8], every $\tau$-convergent sequence has a subsequence that is relatively uniformly convergent to the same limit. Now apply the version of [Corollary 4](#res:nets_and_sequences_general){reference-type="ref" reference="res:nets_and_sequences_general"} for relative uniform convergence. ◻ For the un-topology on a Banach lattice, we have the following. The proof of part [\[part:subsequence_un_2\]](#part:subsequence_un_2){reference-type="ref" reference="part:subsequence_un_2"} is essentially that of [@deng_o_brien_troitsky:2017 Theorem 4.4], but formulated in such a way that it can be used in a wider context; see [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"}. **Theorem 28**. *Let $E$ be a Banach lattice. Then:* 1. *[\[part:subsequence_un_0\]]{#part:subsequence_un_0 label="part:subsequence_un_0"} every un-convergent net in $E$ has an embedded sequence that is uo-convergent to the same limit.* *If $E$ has an order continuous norm, then:* 1. *[\[part:subsequence_un_1\]]{#part:subsequence_un_1 label="part:subsequence_un_1"} every un-convergent net in $E$ has an embedded sequence that is uo-convergent as well as un-convergent to the same limit;* 2. *[\[part:subsequence_un_2\]]{#part:subsequence_un_2 label="part:subsequence_un_2"} a sequence in $E$ is un-convergent to $x\in E$ if and only if every subsequence has a further subsequence that is uo-convergent to $x$.* *Proof.* The statement in [\[part:subsequence_un_0\]](#part:subsequence_un_0){reference-type="ref" reference="part:subsequence_un_0"} is immediate from [Proposition 13](#res:un_is_l.c.sub){reference-type="ref" reference="res:un_is_l.c.sub"} and [Theorem 25](#res:subsequence_simplified){reference-type="ref" reference="res:subsequence_simplified"}. Suppose that $E$ has an order continuous norm. In this case, the un-topology is the uo-Lebesgue topology on $E$ (see [@deng_o_brien_troitsky:2017 Proposition 2.5], for example), which makes clear that [\[part:subsequence_un_1\]](#part:subsequence_un_1){reference-type="ref" reference="part:subsequence_un_1"} follows from [\[part:subsequence_un_0\]](#part:subsequence_un_0){reference-type="ref" reference="part:subsequence_un_0"}. The forward implication in [\[part:subsequence_un_2\]](#part:subsequence_un_2){reference-type="ref" reference="part:subsequence_un_2"} is immediate from [\[part:subsequence_un_0\]](#part:subsequence_un_0){reference-type="ref" reference="part:subsequence_un_0"}. For the converse implication, suppose that a sequence $(x_n)_{n\geq 1}$ has the property that every subsequence has a further subsequence that is uo-convergent to $x$, but that $(x_n)_{n\geq 1}$ is not un-convergent to $x$. In this case, there exists an un-neighbourhood $V$ of $x$ and a subsequence of $(x_n)_{n\geq 1}$ that stays outside $V$. By assumption, there is a further subsequence that is uo-convergent to $x$. Since the un-topology is the uo-Lebesgue topology on $E$, this further subsequence is eventually in $V$. This is a contradiction. ◻ **Remark 29**. Part [\[part:subsequence_un_0\]](#part:subsequence_un_0){reference-type="ref" reference="part:subsequence_un_0"} of [Theorem 28](#res:subsequence_un){reference-type="ref" reference="res:subsequence_un"} improves [@deng_o_brien_troitsky:2017 Proposition 4.1], where it is shown that every un-convergent *sequence* in a Banach lattice has a subsequence that is uo-convergent to the same limit. The statement in [\[part:subsequence_un_1\]](#part:subsequence_un_1){reference-type="ref" reference="part:subsequence_un_1"} is [@deng_o_brien_troitsky:2017 Corollary 3.5]. The statement in [\[part:subsequence_un_2\]](#part:subsequence_un_2){reference-type="ref" reference="part:subsequence_un_2"} is [@deng_o_brien_troitsky:2017 Theorem 4.4]. We now turn to Fatou topologies with full carrier, for which we have the following basic result. **Theorem 30**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is a Fatou topology. Suppose that $C_\tau=E$, which is certainly the case when $E$ has the countable sup property, when $C_\tau$ has a countable order basis, and when $\tau$ is metrisable. Then every $\tau$-convergent net in $E$ has an embedded sequence that is uo-convergent to the same limit.* *Proof.* Suppose that $(x_\alpha)_{\alpha\in A}$ is a net in $E$ and that $x_\alpha\xrightarrow{\tau}x$ for some $x\in E$. If we let $\tau^\delta$ denote the extension of $\tau$ to a Fatou topology on $E^\delta$, then also $x_\alpha\xrightarrow{\tau^\delta}x$ in $E^\delta$. The combination of [Theorem 21](#res:when_fatou_is_locally_interval_complete_submetrisable_countability){reference-type="ref" reference="res:when_fatou_is_locally_interval_complete_submetrisable_countability"} and [Theorem 25](#res:subsequence_simplified){reference-type="ref" reference="res:subsequence_simplified"} yields that $(x_\alpha)_{\alpha\in A}$ has an embedded sequence that is uo-convergent to $x$ in $E^\delta$. As $E$ is a regular vector sublattice of $E^\delta$, this embedded sequence is also uo-convergent to $x$ in $E$. ◻ **Remark 31**. 1. The special case of [Theorem 30](#res:subsequence_fatou){reference-type="ref" reference="res:subsequence_fatou"} where $C_\tau$ has a countable order basis is [@kandic_taylor:2018 Theorem 6.7]. 2. According to [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 4.19], an order bounded convergent net in a Fatou topology on a vector lattice with the countable sup property has an embedded sequence that is order convergent to the same limit. This is clear from [Theorem 30](#res:subsequence_fatou){reference-type="ref" reference="res:subsequence_fatou"}, since an order bounded uo-convergent sequence is order convergent. It is, in fact, more generally true when the topology has full carrier. Before continuing with our applications, we note that, for o-Lebesgue topologies, there is a converse to [Theorem 30](#res:subsequence_fatou){reference-type="ref" reference="res:subsequence_fatou"}. In fact, we have the following. **Theorem 32**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is an o-Lebesgue topology. The following are equivalent:* 1. *[\[part:equivalence_for_o-lebesgue_topologies_1\]]{#part:equivalence_for_o-lebesgue_topologies_1 label="part:equivalence_for_o-lebesgue_topologies_1"} $C_\tau=E$;* 2. *[\[part:equivalence_for_o-lebesgue_topologies_2\]]{#part:equivalence_for_o-lebesgue_topologies_2 label="part:equivalence_for_o-lebesgue_topologies_2"} $E$ has the countable sup property;* 3. *[\[part:equivalence_for_o-lebesgue_topologies_3\]]{#part:equivalence_for_o-lebesgue_topologies_3 label="part:equivalence_for_o-lebesgue_topologies_3"} every $\tau$-convergent net in $E$ has an embedded sequence that is uo-convergent to the same limit;* 4. *[\[part:equivalence_for_o-lebesgue_topologies_4\]]{#part:equivalence_for_o-lebesgue_topologies_4 label="part:equivalence_for_o-lebesgue_topologies_4"} every increasing $\tau$-convergent net in ${E}^+$ has an embedded sequence that is uo-convergent to the same limit.* *Proof.* The equivalence of [\[part:equivalence_for_o-lebesgue_topologies_1\]](#part:equivalence_for_o-lebesgue_topologies_1){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_1"} and  [\[part:equivalence_for_o-lebesgue_topologies_2\]](#part:equivalence_for_o-lebesgue_topologies_2){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_2"}, taken from the literature, was already in [Theorem 15](#res:when_fatou_is_submetrisable_dedekind){reference-type="ref" reference="res:when_fatou_is_submetrisable_dedekind"}. [Theorem 30](#res:subsequence_fatou){reference-type="ref" reference="res:subsequence_fatou"} shows that  [\[part:equivalence_for_o-lebesgue_topologies_1\]](#part:equivalence_for_o-lebesgue_topologies_1){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_1"} implies [\[part:equivalence_for_o-lebesgue_topologies_3\]](#part:equivalence_for_o-lebesgue_topologies_3){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_3"}, and it is clear that [\[part:equivalence_for_o-lebesgue_topologies_3\]](#part:equivalence_for_o-lebesgue_topologies_3){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_3"} implies [\[part:equivalence_for_o-lebesgue_topologies_4\]](#part:equivalence_for_o-lebesgue_topologies_4){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_4"}. We show that [\[part:equivalence_for_o-lebesgue_topologies_4\]](#part:equivalence_for_o-lebesgue_topologies_4){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_4"} implies [\[part:equivalence_for_o-lebesgue_topologies_2\]](#part:equivalence_for_o-lebesgue_topologies_2){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_2"}. Suppose that $(x_\alpha)_{\alpha\in A}$ is a net in ${E}^+$ and that $x_\alpha\uparrow x$ for some $x\in{E}^+$. Then $x_\alpha\xrightarrow{\tau}x$. By assumption, there exists an embedded sequence $(x_{\alpha_n})_{n\geq 1}$ such that $x_{\alpha_n}\xrightarrow{\mathrm{uo}}x$ as $n\to\infty$. Since the sequence is order bounded, we even have $x_{\alpha_n}\xrightarrow{\mathrm{o}}x$. As the $\alpha_n$ are increasing, so are the $x_{\alpha_n}$, and we conclude that $x_{\alpha_n}\uparrow x$. Now [@luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971 Theorem 23.2.(iii)] shows that $E$ has the countable sup property. ◻ **Remark 33**. 1. We see that, for o-Lebesgue topologies, the existence of embedded uo-convergent subsequences in the restricted class of topologically convergent nets in [\[part:equivalence_for_o-lebesgue_topologies_4\]](#part:equivalence_for_o-lebesgue_topologies_4){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_4"} already implies this existence in general topologically convergent nets in the more precise form in [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"}. 2. For arbitrary Fatou topologies, [Theorem 32](#res:equivalence_for_o-lebesgue_topologies){reference-type="ref" reference="res:equivalence_for_o-lebesgue_topologies"} does not hold. Indeed, the combination of part [\[rem:fatou_topology_major_theorem_1\]](#rem:fatou_topology_major_theorem_1){reference-type="ref" reference="rem:fatou_topology_major_theorem_1"} of [Remark 16](#rem:fatou_topology_major_theorem){reference-type="ref" reference="rem:fatou_topology_major_theorem"} and [Theorem 30](#res:subsequence_fatou){reference-type="ref" reference="res:subsequence_fatou"} shows that  [\[part:equivalence_for_o-lebesgue_topologies_3\]](#part:equivalence_for_o-lebesgue_topologies_3){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_3"} does not imply  [\[part:equivalence_for_o-lebesgue_topologies_2\]](#part:equivalence_for_o-lebesgue_topologies_2){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_2"} within this larger class of topologies. For Fatou topologies, it would be satisfactory if, for example, [\[part:equivalence_for_o-lebesgue_topologies_1\]](#part:equivalence_for_o-lebesgue_topologies_1){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_1"} could still be shown to be equivalent to [\[part:equivalence_for_o-lebesgue_topologies_3\]](#part:equivalence_for_o-lebesgue_topologies_3){reference-type="ref" reference="part:equivalence_for_o-lebesgue_topologies_3"}, or perhaps to the more precise version of the existence of embedded uo-convergent sequences as in [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"}. In support of this, we can, however, presently only mention that the (metrisable) Fatou topology in part [\[rem:fatou_topology_major_theorem_1\]](#rem:fatou_topology_major_theorem_1){reference-type="ref" reference="rem:fatou_topology_major_theorem_1"} of [Remark 16](#rem:fatou_topology_major_theorem){reference-type="ref" reference="rem:fatou_topology_major_theorem"} indeed has full carrier. Continuing with our applications, we now specialise within the Fatou topologies to the uo-Lebesgue topologies. Employing arguments as in the proof of the parts [\[part:subsequence_un_1\]](#part:subsequence_un_1){reference-type="ref" reference="part:subsequence_un_1"} and [\[part:subsequence_un_2\]](#part:subsequence_un_2){reference-type="ref" reference="part:subsequence_un_2"} of [Theorem 28](#res:subsequence_un){reference-type="ref" reference="res:subsequence_un"}, and recalling that, for o-Lebesgue topologies, having full carrier is equivalent to the vector lattice having the countable sup property, one obtains the following result. **Theorem 34**. *Let $(E,\tau)$ be a locally solid vector lattice, where $\tau$ is a uo-Lebesgue topology. Suppose that $C_\tau=E$ or, equivalently, that $E$ has the countable sup property; this is certainly the case when $C_\tau$ has a countable order basis, and when $\tau$ is metrisable. Then:* 1. *[\[part:subsequence_uo-lebesgue_1\]]{#part:subsequence_uo-lebesgue_1 label="part:subsequence_uo-lebesgue_1"} every $\tau$-convergent net in $E$ has an embedded sequence that is uo-convergent as well as $\tau$-convergent to the same limit;* 2. *[\[part:subsequence_uo-lebesgue_2\]]{#part:subsequence_uo-lebesgue_2 label="part:subsequence_uo-lebesgue_2"} a sequence in $E$ is $\tau$-convergent to $x\in E$ if and only if every subsequence has a further subsequence that is uo-convergent to $x$.* **Remark 35**. 1. Suppose that $E$ is a Banach lattice, and let $\tau$ denote its norm topology. Then $C_{\mathrm{u}\tau}=E$ by [Lemma 1](#res:carrier_of_induced_unbounded_topology){reference-type="ref" reference="res:carrier_of_induced_unbounded_topology"}. If $E$ has an order continuous norm (which implies that it has the countable sup property), then $\mathrm{u}\tau$ is the uo-Lebesgue topology on $E$. We thus see that [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"} yields the parts [\[part:subsequence_un_1\]](#part:subsequence_un_1){reference-type="ref" reference="part:subsequence_un_1"} and [\[part:subsequence_un_2\]](#part:subsequence_un_2){reference-type="ref" reference="part:subsequence_un_2"} of [Theorem 28](#res:subsequence_un){reference-type="ref" reference="res:subsequence_un"} again, albeit not via the shortest route. 2. It follows from [@kandic_taylor:2018 Lemma 6.8 and Corollary 6.9] that part [\[part:subsequence_uo-lebesgue_2\]](#part:subsequence_uo-lebesgue_2){reference-type="ref" reference="part:subsequence_uo-lebesgue_2"} of [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"} holds when $E$ has the countable sup property and a countable order basis. [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"}, however, shows that $E$ having the countable sup property is alone already sufficient for part [\[part:subsequence_uo-lebesgue_2\]](#part:subsequence_uo-lebesgue_2){reference-type="ref" reference="part:subsequence_uo-lebesgue_2"} to hold. 3. For the sake of completeness we recall from [@kandic_taylor:2018 Theorem 4.9] that a uo-Lebesgue topology on a vector lattice is metrisable if and only if the vector lattice has the countable sup property and a countable order basis. [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"} has the following consequence. **Corollary 36**. *Let $E$ be a vector lattice, and suppose that $F$ is an order dense ideal in $E$ admitting an o-Lebesgue topology $\tau_F$. Then $\mathrm{u}_F\tau_F$ is a uo-Lebesgue topology on $E$. Suppose that at least one of the following is satisfied:* 1. *[\[part:subsequence_uo-lebesgue_induced_hypothesis_1\]]{#part:subsequence_uo-lebesgue_induced_hypothesis_1 label="part:subsequence_uo-lebesgue_induced_hypothesis_1"} $E$ has the countable sup property;* 2. *[\[part:subsequence_uo-lebesgue_induced_hypothesis_2\]]{#part:subsequence_uo-lebesgue_induced_hypothesis_2 label="part:subsequence_uo-lebesgue_induced_hypothesis_2"} $\tau_F$ is metrisable and $F$ has a countable order basis [(]{.upright}which is always the case when $\tau_F$ is a metrisable uo-Lebesgue topology[)]{.upright};* *Then $C_{\mathrm{u}_F\tau_F}=E$. Consequently:* 1. *[\[part:subsequence_uo-lebesgue_induced_1\]]{#part:subsequence_uo-lebesgue_induced_1 label="part:subsequence_uo-lebesgue_induced_1"} every $\mathrm{u}_F\tau_F$-convergent net in $E$ has an embedded sequence that is uo-convergent as well as $\mathrm{u}_F\tau_F$-convergent to the same limit;* 2. *[\[part:subsequence_uo-lebesgue_induced_2\]]{#part:subsequence_uo-lebesgue_induced_2 label="part:subsequence_uo-lebesgue_induced_2"} a sequence in $E$ is $\mathrm{u}_F\tau_F$-convergent to $x\in E$ if and only if every subsequence has a further subsequence that is uo-convergent to $x$.* *Proof.* It follows from [@deng_de_jeu:2022a Proposition 4.1] that $\mathrm{u}_F\tau_F$ is a uo-Lebesgue topology on $E$. By [@kandic_taylor:2018 Theorem 4.9], $F$ has a countable order basis when $\tau_F$ is a metrisable uo-Lebesgue topology. Suppose that $\tau_F$ is metrisable and that $F$ has a countable order basis. According to [Lemma 1](#res:carrier_of_induced_unbounded_topology){reference-type="ref" reference="res:carrier_of_induced_unbounded_topology"}, we have $F\subseteq C_{\mathrm{u}_F\tau_F}$. Since $F$ is then an order dense vector sublattice of $C_{\mathrm{u}_F\tau_F}$, $C_{\mathrm{u}_F\tau_F}$ also has a countable order basis. Now we can apply [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"}. ◻ **Remark 37**. 1. When $E$ is a vector lattice containing a Banach lattice $F$ with an order continuous norm as an order dense ideal, [Corollary 36](#res:subsequence_uo-lebesgue_induced){reference-type="ref" reference="res:subsequence_uo-lebesgue_induced"} specifies to a considerable improvement of [@kandic_li_troitsky:2018 Theorem 9.5]. An inspection of [@kandic_li_troitsky:2018 Example 9.6] shows that the condition in [\[part:subsequence_uo-lebesgue_induced_hypothesis_2\]](#part:subsequence_uo-lebesgue_induced_hypothesis_2){reference-type="ref" reference="part:subsequence_uo-lebesgue_induced_hypothesis_2"} that $F$ have a countable order basis cannot be omitted. 2. A vector lattice $E$ with the property that ${E}^\thicksim_{\mathrm{oc}}$ separates its points admits an o-Lebesgue topology; see [@deng_de_jeu:2022a Lemma 5.1], for example. With this in mind, it is easy to see that [Corollary 36](#res:subsequence_uo-lebesgue_induced){reference-type="ref" reference="res:subsequence_uo-lebesgue_induced"} implies [@deng_de_jeu:2022a Theorems 7.8 and 7.13]. We continue with a consequence of [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"} in measure theory. **Corollary 38**. *Let $(X,\Sigma,\mu)$ be a semi-finite measure space, and let $E$ be the ideal in $\mathrm{L}^0(X,\Sigma,\mu)$ consisting of [(]{.upright}equivalence classes of[)]{.upright} measurable functions with $\sigma$-finite supports. Then $E$ admits a uo-Lebesgue topology, which is the topology of local convergence in measure, and $E$ has the countable sup property. Consequently:* 1. *[\[part:subsequence_measure_spaces_1\]]{#part:subsequence_measure_spaces_1 label="part:subsequence_measure_spaces_1"} every $\mu^\ast$-convergent net in $E$ has an embedded sequence that is almost everywhere convergent as well as $\mu^\ast$-convergent to the same limit;* 2. *[\[part:subsequence_measure_spaces_2\]]{#part:subsequence_measure_spaces_2 label="part:subsequence_measure_spaces_2"} a sequence in $E$ is $\mu^\ast$-convergent to $f\in E$ if and only if every subsequence has a further subsequence that is almost everywhere convergent to $f$.* *Proof.* It was already observed in [@conradie:2005 p.292] that $\mathrm{L}^0(X,\Sigma,\mu)$ has a uo-Lebesgue topology, and that it is the topology of local convergence in measure. A more detailed statement and its proof can be found as [@deng_de_jeu:2022a Theorem 6.1]. The restriction of this topology to the regular vector sublattice $E$ is a uo-Lebesgue topology on $E$. A moment's thought shows that the $\sigma$-finiteness of the supports of the elements of $E$ implies that every disjoint system of positive elements of $E$ which is bounded from above in $E$ is at most countable. Hence $E$ has the countable sup property by [@luxemburg_zaanen_RIESZ_SPACES_VOLUME_I:1971 Theorem 29.3(vi)]. We can now apply [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"}. The proof is then completed by using that, for arbitrary measure spaces, uo-convergence of sequences in regular vector sublattices of $\mathrm{L}^0(X,\Sigma,\mu)$ is equivalent to convergence almost everywhere; see [@gao_troitsky_xanthos:2017 Remark 3.4]. ◻ **Remark 39**. 1. When $\mu$ is $\sigma$-finite, one has $E=\mathrm{L}^0(X,\Sigma,\mu)$ in [Corollary 38](#res:subsequence_measure_spaces){reference-type="ref" reference="res:subsequence_measure_spaces"}. In this case, part [\[part:subsequence_measure_spaces_1\]](#part:subsequence_measure_spaces_1){reference-type="ref" reference="part:subsequence_measure_spaces_1"} yields [@deng_de_jeu:2022a Theorem 7.11], and part [\[part:subsequence_measure_spaces_2\]](#part:subsequence_measure_spaces_2){reference-type="ref" reference="part:subsequence_measure_spaces_2"} yields [@fremlin_MEASURE_THEORY_VOLUME_2:2003 245K]. 2. For the sake of completeness we recall (as was already mentioned in [1](#sec:introduction_and_overview){reference-type="ref" reference="sec:introduction_and_overview"}) that, for arbitrary measure spaces, a sequence in $\mathrm{L}^0(X,\Sigma,\mu)$ that is (globally) convergent in measure has a subsequence that converges almost everywhere to the same limit; see [@aliprantis_burkinshaw_PRINCIPLES_OF_REAL_ANALYSIS_THIRD_EDITION:1998 Theorem 19.3] or [@folland_REAL_ANALYSIS_SECOND_EDITION:1999 Theorem 2.30], for example. This goes back to F. Riesz in [@riesz:1909], correcting an earlier statement by Lebesgue in [@lebesgue_LECONS_SUR_LES_SERIES_TRIGONOMETRIQUES:1906]. We continue with an application of [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"} to weak closures. For this, we need some terminology. Suppose that $E$ is a vector lattice, and that $I$ is an ideal in ${E}^\thicksim_{\mathrm{oc}}$ that separates the points of $E$. In this case, the lattice semi-norms $\rho_\varphi$, defined for $\varphi\in I$ by setting $\rho_\varphi(x)\coloneqq\lvert\varphi\rvert(\lvert x\rvert)$ for $x\in E$, define a locally convex o-Lebesgue topology $\lvert\sigma\rvert(E,I)$ on $E$. The uo-Lebesgue topology $\mathrm{u}\lvert\sigma\rvert(E,I)$ on $E$, resulting from this so-called absolute weak topology $\lvert\sigma\rvert(E,I)$ on $E$ that is induced by $I$, plays a part in the following result. **Corollary 40**. *Let $E$ be a vector lattice, and suppose that ${E}^\thicksim_{\mathrm{oc}}$ has an ideal $I$ that separates the points of $E$. Suppose that $C_{\mathrm{u}\lvert\sigma\rvert(E,I)}=E$ or, equivalently, that $E$ has the countable sup property; this is certainly the case when $C_{\mathrm{u}\lvert\sigma\rvert(E,I)}$ has a countable order basis, and when $\mathrm{u}\lvert\sigma\rvert(E,I)$ is metrisable. Let $S$ be a convex subset of $E$, and take $x\in \overline{S}^{\sigma(E,I)}$. Then there exists a sequence in $S$ that is uo-convergent to $x$.* *Proof.* By Kaplan's theorem (see [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 2.33], for example), the topological duals of the locally convex topological vector spaces $(E, \lvert\sigma\rvert(E,I))$ and $(E, \sigma(E,I))$ are both equal to $I$. Hence $\overline{S}^{\sigma(E,I)}=\overline{S}^{\lvert\sigma\rvert(E,I)}$, so that there exists a net $(x_\alpha)_{\alpha\in A}$ in $S$ with $x_\alpha\xrightarrow{\lvert\sigma\rvert(E,I)} x$. Then certainly $x_\alpha\xrightarrow{\mathrm{u}\lvert\sigma\rvert(E,I)} x$. We can now apply [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"}. ◻ **Remark 41**. In [@gao_leung_xanthos:2018 Theorem 4.1], the existence of a uo-convergent sequence as in [Corollary 40](#res:convex){reference-type="ref" reference="res:convex"} was established under the single hypothesis that the ideal $I$ in ${E}^\thicksim_{\mathrm{oc}}$ contain a strictly positive linear functional. This is a special case of [Corollary 40](#res:convex){reference-type="ref" reference="res:convex"}. Indeed, if a vector lattice has a strictly positive linear functional $\varphi$, then it has the countable sup property by [@aliprantis_burkinshaw_LOCALLY_SOLID_RIESZ_SPACES_WITH_APPLICATIONS_TO_ECONOMICS_SECOND_EDITION:2003 Theorem 1.45]. Furthermore, the principal ideal in ${E}^{\thicksim}$ generated by $\varphi$ then already separates the points of $E$ by [@deng_de_jeu:2022a Proposition 2.1(2)], so this is then certainly true for $I$. We refer to [@gao_leung_xanthos:2018] for further elaborations on this special case. We continue with an application of [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"} to adherences and closures. As a preparation, we recall some notation from [@deng_de_jeu:2022b Section 8]. Let $A$ be a subset of a vector lattice $E$. We write $$a_{\sigma\text{-}\mathrm{uo}}(A)=\left\{x\in E: \text{ there exists a sequence }(x_n)_{n\geq 1}\text{ in } A \text{ with }x_n\xrightarrow{\mathrm{uo}}x \text{ in } E\right\}$$ and $$a_{\mathrm{uo}}(A)=\left\{x\in E: \text{ there exists a net }(x_\alpha)_\alpha\text{ in } A \text{ with }x_\alpha\xrightarrow{\mathrm{uo}}x \text{ in } E\right\}$$ for the *$\sigma$-uo-adherence* and *uo-adherence* of $A$ in $E$, respectively. When $\tau$ is a topology on $E$, one similarly defines the $\sigma$-$\tau$-adherence and the $\tau$-adherence of $A$, the latter simply being its $\tau$-closure $\overline{A}^\tau$. A subset $A$ of $E$ is said to be *$\sigma$-uo-closed* when $a_{\sigma\text{-}\mathrm{uo}}(A)=A$. The $\sigma$-uo-closed subsets of $E$ are the closed subsets of a topology on $E$, which is called the $\sigma$-uo-topology. The closure of a subset $A$ in this topology is denoted by $\overline{A}^{\sigma\textup{-}\mathrm{uo}}$. We have $\overline{a_{\sigma\text{-}\mathrm{uo}}(A)}^{\sigma\textup{-}\mathrm{uo}}=\overline{A}^{\sigma\textup{-}\mathrm{uo}}$, and $A$ is closed in the $\sigma$-uo-topology if and only if $a_{\sigma\text{-}\mathrm{uo}}(A)=\overline{A}^{\sigma\textup{-}\mathrm{uo}}$. We refer to [@deng_de_jeu:2022b Lemmas 2.3 and 2.4] for details. Similar statements hold for uo-adherences and $\sigma$-$\tau$-adherences and the associated topologies. The conclusion and the proof of the following result are completely analogous to those of [@deng_de_jeu:2022b Theorem 8.1]. Its hypotheses are considerably weaker, however, since we now have [Theorem 34](#res:subsequence_uo-lebesgue){reference-type="ref" reference="res:subsequence_uo-lebesgue"} available, rather than [@deng_de_jeu:2022b Theorem 7.8] with its much more restrictive hypotheses. **Theorem 42**. *Let $E$ be a vector lattice that admits a uo-Lebesgue topology $\tau$, and let $A$ be a subset of $E$. Suppose that $C_\tau=E$ or, equivalently, that $E$ has the countable sup property; this is certainly the case when $C_\tau$ has a countable order basis, and when $\tau$ is metrisable. Then the following seven subsets of $E$ are all equal:* 1. *$a_{\sigma\text{-}\tau}(A)$ and $\overline{A}^{\sigma\text{-}\tau}$;* 2. *$a_{\sigma\text{-}\mathrm{uo}}(A)$ and $\overline{A}^{\sigma\textup{-}\mathrm{uo}}$;* 3. *$a_{\mathrm{uo}}(A)$ and $\overline{A}^{\mathrm{uo}}$;* 4. *$\overline{A}^\tau$.* *In particular, the $\sigma$-$\tau$-topology, the $\sigma$-uo-topology, and the uo-topology on $E$ all coincide with $\tau$.* On combining [Corollary 38](#res:subsequence_measure_spaces){reference-type="ref" reference="res:subsequence_measure_spaces"} and [Theorem 42](#res:seven_sets_equal){reference-type="ref" reference="res:seven_sets_equal"}, and recalling again from [@gao_troitsky_xanthos:2017 Remark 3.4] that uo-convergence of sequences in regular vector sublattices of $\mathrm{L}^0(X,\Sigma,\mu)$ is the same as almost everywhere convergence, we obtain the following. **Corollary 43**. *Let $(X,\Sigma,\mu)$ be a semi-finite measure space, and let $E$ be the ideal in $\mathrm{L}^0(X,\Sigma,\mu)$ consisting of [(]{.upright}equivalence classes of[)]{.upright} measurable functions with $\sigma$-finite supports. Let $A$ be a subset of $E$. Then $A$ is a closed subset of $E$ in the topology of local convergence in measure if and only if it contains the almost everywhere limits of sequences in $A$.* **Remark 44**. For a $\sigma$-finite measure, [Corollary 43](#res:subset_fremlind_improved){reference-type="ref" reference="res:subset_fremlind_improved"} is [@fremlin_MEASURE_THEORY_VOLUME_2:2003 245L(b)]. [^1]: See [Definition 2](#def:embedded_sequence){reference-type="ref" reference="def:embedded_sequence"} for a precise definition of embedded sequences. [^2]: See [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"}. [^3]: See [Definition 6](#def:l.c.m.t.){reference-type="ref" reference="def:l.c.m.t."}. [^4]: See [Theorem 30](#res:subsequence_fatou){reference-type="ref" reference="res:subsequence_fatou"}. [^5]: See [Theorem 22](#res:subsequence){reference-type="ref" reference="res:subsequence"}. [^6]: In the literature, what we call an o-Lebesgue topology is simply called a Lebesgue topology. Now that uo-Lebesgue topologies, with a completely analogous definition, have become objects of a more extensive study, it seems consistent to also add a prefix to the original term. [^7]: In [@kandic_taylor:2018], $\tau$ is then called *Riesz submetrisable*. The present terminology may be a little more suggestive. [^8]: In the terminology of [@kandic_taylor:2018], it shows that, for locally solid topologies, submetrisability coincides with Riesz submetrisability. [^9]: With the exception of [Proposition 27](#res:subsequence_frechet){reference-type="ref" reference="res:subsequence_frechet"}, where there appears to be no obviously valid statement for regular vector sublattices.

arxiv_math

--- abstract: | For any prime $p>5$ we construct a Calabi-Yau threefold $X$ defined over finite extension $K$ of $\mathbb{Q}_p$ such that every model of $X$ over $\operatorname{Spec}\mathcal{O}_K$ has singular special fiber, yet the Galois action on the $\ell$-adic cohomology group $H^3_{\acute{e}t}(X,\mathbb{Q}_\ell)$ is unramified for $\ell\neq p$ and crystalline for $\ell=p$. This provides a counterexample to the analogue of the Néron-Ogg-Shafarevich criterion for three-dimensional Calabi-Yau manifolds. author: - Tymoteusz Chmiel, Marcin Oczko date: February 2020 title: | Counterexample\ to Néron--Ogg--Shafarevich criterion\ for Calabi--Yau threefolds --- # Introduction A *Calabi-Yau manifold* is a smooth projective variety $X$ with trivial canonical bundle and vanishing cohomology groups $H^i(X,\mathcal{O}_X)$ for $0<i<\text{dim}\,X$. Usually one considers Calabi-Yau manifolds defined over $\mathbb{C}$. On the other hand, in this paper we are concerned with arithmetic properties of Calabi-Yau manifolds and the field of definition is either $p$-adic or finite. One-dimensional Calabi-Yau manifolds are *elliptic curves* and two-dimensional ones are called *K3 surfaces*. This paper is concerned with *Calabi-Yau threefolds* and a phenomenon that does not appear in lower dimensions: the étale cohomology fails to detect bad reduction. Therefore, the natural generalization of the Néron-Ogg-Shafarevich criterion does not hold in dimension $3$. Néron-Ogg-Shafarevich criterion states that an elliptic curve $E$ over a $p$-adic field has good reduction if and only if the Galois representation $H^1_{\acute{e}t}(E,\mathbb{Q}_\ell)$ is unramified for $\ell\neq p$ and crystalline for $\ell=p$ (see [@Ogg]). A version of this criterion holds for $K3$ surfaces: $K3$ surface $S$ admitting a semi-stable model has potentially good reduction if and only if the Galois representation $H^2_{\acute{e}t}(S,\mathbb{Q}_\ell)$ is unramified for $\ell\neq p$ and crystalline for $\ell=p$ (see [@LM; @BLL]). It means that for Calabi-Yau varieties of dimension $\leq 2$ good reduction is detected by the Galois action on the middle cohomology. We show it is not the case in dimension three: **Main Theorem 1**. *Let $p>5$ be a prime, $K:=\mathbb{Q}_p[\sqrt{p}]$ and denote by $G_K$ the absolute Galois group of $K$. There exists smooth Calabi-Yau threefold $Y_p$ defined over $K$ such that:* - *the Galois representation $$G_K\rightarrow \operatorname{Aut}\left( H^3_{\acute{e}t}(Y_p,\mathbb{Q}_\ell )\right)$$ is unramified for $\ell \neq p$ and crystalline for $\ell =p$;* - *$Y_p$ does not have a smooth model over $\operatorname{Spec}\mathcal{O}_L$ for any finite field extension $L/K$.* The Galois action on the étale cohomology of a $p$-adic variety is an arithmetic analogue of the monodromy action on the rational cohomology of a complex manifold. In [@CvS] Cynk and van Straten constructed a family of complex Calabi-Yau threefolds over the punctured unit disc $\Delta^*$ such that the monodromy action of the fundamental group $\pi_1(\Delta^*,t)\simeq \mathbb{Z}$ on $H^3(X_{t},\mathbb{C})$ is trivial, yet the family cannot be completed to a smooth family over the entire unit disc $\Delta$. To prove our Main Theorem we repurpose their example for mixed characteristic. In fact, the variety $Y_p$ is defined over $\mathbb{Q}[\sqrt{p}]\hookrightarrow\mathbb{C}$ and is a smooth member of the family of Cynk-van Straten. Thus our result proves the conjecture stated at the end of [@CvS]. The structure of the paper is as follows. In section [2](#sec:construction){reference-type="ref" reference="sec:construction"} we present a detailed construction of the manifold $Y_p$. In section [3](#sec:crystalline){reference-type="ref" reference="sec:crystalline"} we prove that the Galois action on its cohomology is crystalline. In section [4](#sec:no-good){reference-type="ref" reference="sec:no-good"} we show that $Y_p$ does not admit a smooth integer model over any finite extension of the base field $K$. # Construction {#sec:construction} ## Double octics Our example is constructed as a desingularization of a double cover of $\mathbb P^3$ branched along a union of eight planes with no $k$-fold lines for $k \geq 4$ and no $l$-fold points for $l \geq 6$. Double covers of $\mathbb P^3$ branched along such plane arrangements were introduced in [@CYB] and are called *double octics*; their branching locus is called an *octic arrangement*. Double octics provide a rich source of examples of Calabi-Yau threefolds: **Theorem 1** (Theorem 2.1, [@CYB]). *Let $D \subset \mathbb P^3$ be an octic arrangement. There exists a sequence of blow-ups $\sigma = \sigma_1 \circ ... \circ \sigma_s \colon \widetilde{\mathbb P^3} \rightarrow\mathbb P^3$ and a smooth even divisor $D^* \subset \widetilde{\mathbb P^3}$ such that $\sigma_*(D^*) = D$ and the double covering of $\widetilde{\mathbb P^3}$ branched along $D^*$ is a smooth Calabi-Yau threefold.* We apply Theorem [Theorem 1](#resolution of octic){reference-type="ref" reference="resolution of octic"} to the following family of double octics: $$\label{double-octic} X_t := \Big\{u^2 = xy (x + y) z (x + 2 y + z + tv) v (y + z + v) (x + y + z + (t - 1) v)\Big\}$$ Here $t\in\mathbb P^1$ is a parameter of the family and $(u:x:y:z:v) \in \mathbb P(4,1,1,1,1)$ are coordinates of the weighted projective space containing double octics $X_t$. We label eight planes defining $X_t$ as $P_1,...,P_8$, in the same order as their equations appear in ([\[double-octic\]](#double-octic){reference-type="ref" reference="double-octic"}). Note that we suppress the dependence on $t$ from notation. Each $X_t$ is a double cover of $\mathbb{P}^3$ branched along the octic arrangement $P_1\cup\cdots\cup P_8$. For generic value of $t\ (\neq 0,1,2,\infty)$ double octic $X_t$ has the following singularities: - a single triple line $l_3:=\{x=y=0\}$; - 25 double lines; - six fourfold points, none of them lying on the triple line $l_3$; - five fourfold points on the triple line $l_3$. When $t=0$, two fourfold points $(0:0:0:1)$ and $(0:0:-t:1)$ coincide to create a fivefold point lying on the planes $P_1,P_2,P_3,P_4,P_5$. Other singularities remain the same. Let $P\subset X_0$ be the plane spanned by $P_4 \cap P_5$ and $l_3$. Four planes $P_1,P_2,P_3,P$, all passing through the line $l_3$, define four points in $\mathbb P^1$ identified with the space of planes through $l_3$. We define elliptic curve $E$ as the double cover of $\mathbb P^1$ branched at these four points; the $j$-invariant of $E$ is equal to 1728. This elliptic curve will appear later in the cohomology of the special fiber of the family (see Theorem [Theorem 3](#th:coh){reference-type="ref" reference="th:coh"}). Fix a prime $p > 5$. Equations of $X_p$ have coefficients in $\mathbb{Z}\subset \mathbb{Z}_p$, hence they define a scheme $\mathcal{X}\rightarrow\operatorname{Spec}\mathbb{Z}_p$ whose generic fiber is $X_p$. Note that the variety $X_p$ is a generic fiber of the family $X_t$ over $\operatorname{Spec}\mathbb{Z}$ and its reduction modulo $p$ is the same as reduction of $X_0$. We shall denote $X_\infty:=X_p$. Our goal is to construct a scheme $\mathcal{Y}$ whose generic fiber $Y_\infty$ is a resolution of $X_\infty$ as described in Theorem [Theorem 1](#resolution of octic){reference-type="ref" reference="resolution of octic"}, while the central fiber $Y_0$ is a simple normal crossing divisor with a rigid Calabi-Yau threefold as one of its irreducible components. We start resolving the generic fiber of $\mathcal{X}$ by the following sequence of blow-ups: 1. Let $\sigma_1 \colon \mathbb P^{(1)} \rightarrow\mathbb P^3$ be the blow-up in the triple line $l_3$ and in all fourfold points which do not lie on $l_3$. Let $\mathcal{D}^{(1)}$ be the strict transform of the branching locus $\mathcal{D}$ plus the exceptional divisor $H = \mathbb P^1 \times \mathbb P^1$ of the triple line. We define $\mathcal{X}^{(1)}$ as the double cover of $\mathbb P^{(1)}$ branched along $\mathcal{D}^{(1)}$. The branch divisor of $\mathcal{X}^{(1)}$ contains 8 new double lines coming from intersections of $H$ with the 8 planes in the branching divisor; denote these lines as $m_i := P_i \cap H,i=1, ... ,8$. The lines $m_1,m_2,m_3$ are in one ruling of $H$, while the other 5 lines are in the second ruling. In local coordinates near the line $m_1$, the scheme $\mathcal{X}^{(1)}$ is given by equation $$u^2 = xyz(x+2xy+z+p)F(u,x,y,z,w)$$ In this coordinates $x$ corresponds to the exceptional divisor, while $y,\ z$ and $(x+2xy+z+p)$ to strict transforms of $P_1,\ P_4$ and $P_5$ respectively. Factor $F$ with $F(\textbf{0})\neq 0$, corresponds to the other surfaces in the branching divisor. 2. Let $\sigma_2 \colon \mathbb P^{(2)} \rightarrow\mathbb P^{(1)}$ be the blow-up of $\mathbb P^{(1)}$ in all double lines other than $m_4,\ m_5$ and $P_4 \cap P_5$. Let $\mathcal{D}^{(2)}$ be the strict transform of $\mathcal{D}^{(1)}$. Define $\mathcal{X}^{(2)}$ as the double cover of $\mathbb P^{(2)}$ branched along $\mathcal{D}^{(2)}$. Over the line $m_1$, which in local coordinates is given by $x=y=0$, this blow-up can be described in two affine charts as $$u^2 = yz(x + 2x^2y+z+p) \quad u^2 = xz(xy+2xy^2 +z+p)$$ 3. Let $\sigma_3 \colon \mathbb P^{(3)} \rightarrow\mathbb P^{(2)}$ be the blow-up of $\mathbb P^{(3)}$ in double curves $m_4,\ m_5$ and $P_4 \cap P_5$, which are the only remaining singularities of the generic fiber. Let $\mathcal{D}^{(3)}$ be the strict transform of $\mathcal{D}^{(2)}$ and let $\mathcal{X}^{(3)}$ be a double cover of $\mathbb P^{(3)}$ branched along $\mathcal{D}^{(3)}$. Consider again the affine charts above $m_1$. In the first one, the branch divisor is a simple normal crossing with smooth components both in the generic, and in the degenerate fiber. Thus by blowing up intersections of the components of the branch divisor we will not introduce any singularities. This is not the case for the second chart, since the intersection $z = xy+2xy^2 + z +p =0$ is singular $\mod p$. The blow-up in this chart is a blow-up in the ideal $(xz,x(xy+2xy^2 +z+p),z(xy+2xy^2 +z+p),u)$. To compute this blow-up we take a closure of the graph of the following map. $$(x,y,z,t,u) \rightarrow(X,Y,Z,T) = (xz,x(xy+2xy^2 +z+p),z(xy+2xy^2 +z+p),u)$$ By explicit computations we verify that the generic fiber $X^{(3)}_\infty$ is smooth, and the central fiber $X^{(3)}_0$ is singular along the line $L = \{x=z=u=X=Y=Z=0\}$, with four pinch points lying on this line. 4. Let $\mathcal{X}^{(4)}$ be the blow-up of $\mathcal{X}^{(3)}$ in the ideal $(p,I(L))$. This blow-up transforms the degenerate fiber $X^{(3)}_0$ into a union of a *rigid* Calabi-Yau threefold $R_0$ and a conic bundle $V_0\rightarrow L$, intersecting transversally along a conic bundle. Note that the $\mathbb{P}^2$-bundle lies over the double line $L$, and thus appears with multiplicity two. In suitable coordinates, $\mathcal{X}^{(4)}$ can be written as $FG^2=p$, where $F$ stands for local equations of the rigid Calabi-Yau threefold $R_0$ and $G$ stands for local equations of the $\mathbb{P}^2$-bundle $V_0$. To obtain reduced components in the degenerate fiber, first we take a pullback of the family by $p \mapsto p ^2$, obtaining $\mathcal{X}^{(5)}:=\mathcal{X}^{(4)}\times_{\mathbb{Z}_p}\operatorname{Spec}\mathbb{Z}_p[\sqrt{p}]$. Let $\pi$ be a uniformizer of $\mathbb{Q}[\sqrt{p}]$. Using the previous notation we can write the equation of $\mathcal{X}^{(5)}$ in the form $FG^2 = p^2$. We take the double cover of $\mathcal{X}^{(5)}$ branched along $F=0$. It embeds into the zero locus of $FG^2 = \pi^2,\ u^2=F$. The latter is reducible, so consider one of its irreducible components; call it $\mathcal{Y}$. Its equations are $FG = \pi, u^2=X$, there is a morphism $\mathcal{X}^{(4)} \rightarrow\mathcal{Y}$ sending $(F,G,\pi,u) \mapsto (F^2,G,\pi^2)$. The scheme $\mathcal{Y}$ is a semi-stable degeneration of $\mathcal{X}$. Its generic fiber is isomorphic to that of $\mathcal{X}^{(4)}$ and the special fiber $Y_0$ has two components: as before there is a smooth Calabi-Yau threefold $R_0$, while the $\mathbb{P}^2$-bundle $V_0\rightarrow L$ was replaced by a quadric bundle $Q_0$ over $\mathbb P^1$ with four singular fibers over four pinch-points. The generic fiber $Y_\infty$ is isomorphic over $\mathbb{Q}[\sqrt{p}]$ to the generic fiber of $\mathcal{X}^{(4)}$. This construction yields the following result: **Theorem 2**. *There exists a scheme $\mathcal{Y}\rightarrow\operatorname{Spec}\mathbb{Z}_p[\sqrt{p}]$ sch that the generic fiber $Y_\infty\rightarrow\operatorname{Spec}\mathbb{Q}_p[\sqrt{p}]$ is isomorphic to $X_\infty$ and the special fiber $Y_0\rightarrow\operatorname{Spec}\mathbb{F}_p$ is a union of rigid Calabi-Yau threefold $R_0$ and a quadric bundle $Q_0$, intersecting transversally in a conic bundle.* The irreducible components $R_0$ and $Q_0$ have natural lifts to characteristic zero (see [@CYK; @CvS]); we denote them by $R_\infty$ and $Q_\infty$. A key observation in the up-coming proof that the Galois action on $H^3_{\acute{e}t}(Y_\infty,\mathbb{Q}_p)$ is crystalline is the following identification: **Theorem 3**. *There is an isomorphism $$H^3_{\acute{e}t}(Q_\infty,\mathbb{Q}_p) \simeq H^1_{\acute{e}t}(E,\mathbb{Q}_p)(-1),$$ where $E$ is an elliptic curve with $j$-invariant $1728$.* *Proof.* Let $Q:=Q_\infty$ and let $\mathbb{Q}_Q$ denote the locally constant sheaf on $Q$. The second page of the Leray spectral sequence for the morphism $\pi \colon Q \rightarrow L$, $L\simeq\mathbb{P}^1$, is: The sheaves $R^1\pi_*\mathbb{Q}_Q$ and $R^3\pi_*\mathbb{Q}_Q$ are zero sheaves and consequently the appropriate rows in the spectral sequence vanish. This sequence degenerates on the second page; hence $H^1(L, R^2\pi_*\mathbb{Q}_Q) \simeq H^3_{\acute{e}t}(Q,\mathbb{Q}_p)$. We need to show that $H^1(L, R^2\pi_*\mathbb{Q}_Q) \simeq H^1_{\acute{e}t}(E,\mathbb{Q}_p)(-1)$, where elliptic curve $E$ is the double cover of $L$ branched along four pinch points of $Q$. We denote the covering map by $\tau:E \rightarrow L$. Sections of the sheaf $R^2\pi_*\mathbb{Q}_Q$ are generated by rulings on the fibers of $Q$. Let $P \in Q$ be a point which does not lie on a singularity of a fiber, and let $l(P)$ denote the lines through $P$ contained in the fiber $\pi^{-1}(\pi(P))$. When $P$ lies on a smooth fiber, there are two such lines. When $P$ lies on a singular fiber, there is one such line with multiplicity two. As a bundle of quadric surfaces over $L$, $Q$ is given by a degree two polynomial over the field of rational functions on $L$. By Tsen's theorem (see [@Tsen]) there exists a section $s \colon L \rightarrow Q$. Let $V := \{(l(s(P)),P) \colon P \in L\}$. The variety $V$ is a bundle of singular conics over $L$. Its generic fiber consists of two intersecting lines, and its degenerate fibers are double lines. Since the degenerate fibers of $V$ lie over the same points of $L$ as the degenerate fibers of $Q$, the bundles have the same determinants. Moreover, $R^2\pi_*\mathbb{Q}_Q = R^2\pi_*\mathbb{Q}_V$. Consider the fiber product $V \times_{L} E$. The map $\tau:E\rightarrow L$ is a double cover branched over the vanishing locus of the determinant of $V$. Since the generic fiber of $V \times_{L} E\rightarrow L$ consists of two identical pairs of intersecting lines, the determinant of pullback $\tau^*V = V \times_{L} E$ is a square and $V \times_{L} E$ is reducible. Let $S$ be one of its irreducible components. Since $E$ is irreducible, the projection $\alpha \colon S \rightarrow E$ is surjective and the fiber of $S\rightarrow L$ over a generic point consists of two lines. The other projection $\beta \colon S \rightarrow V$ is also surjective, hence $R^2\pi_*\mathbb{Q}_V = R^2(\pi \beta)_*\mathbb{Q}_S = R^2(\tau \alpha)_*\mathbb{Q}_S = \tau_* \mathbb{Q}_E \otimes H^2(\mathbb P^1) = \tau_* \mathbb{Q}_E(-1)$. ◻ # Crystalline cohomology {#sec:crystalline} Let us fix some notation for the rest of the paper. $K$ is a $p$-adic field containing $\sqrt{p}$, $\mathcal{O}_K$ is its ring of integers, $\mathbb{F}_q$ is its residue field. For a scheme $\mathcal{A}\rightarrow\operatorname{Spec}\mathcal{O}_K$, we denote by $A_\infty\rightarrow\operatorname{Spec}K$ its generic fiber and by $A_0\rightarrow \operatorname{Spec}\mathbb{F}_q$ its special fiber. $\mathcal{X}$ is the family of double octics defined by ([\[double-octic\]](#double-octic){reference-type="ref" reference="double-octic"}); $\mathcal{Y}$ is as in Theorem [Theorem 2](#th:con){reference-type="ref" reference="th:con"}; $\mathcal{R}$ is the rigid Calabi-Yau threefold and $\mathcal{Q}$ is the quadric bundle, both described in Theorem [Theorem 2](#th:con){reference-type="ref" reference="th:con"} and paragraphs below it. In this section, we prove the first part of our Main Theorem: the Galois representation $$G_{\mathbb{Q}_{\sqrt{p}}}\longrightarrow \operatorname{Aut}\left( H^3_{\acute{e}t}(Y_\infty,\mathbb{Q}_p )\right)$$ is crystalline. In particular, the action on the $\ell$-adic cohomology for $\ell\neq p$ is unramified. For the sake of completeness let us state the definitions. **Definition 4**. *Let $K$ be a $p$-adic field with residue field $k$, $G_K$ and $G_k$ their absolute Galois groups and $I_K$ the inertia subgroup, i.e. the kernel of the natural homomorphism $G_K\rightarrow G_k$. Let $V$ be a $\mathbb{Q}_\ell$-adic vector space.* *A representation $\rho: G_K\rightarrow \mathrm{GL}(V)$ is called *unramified* if the inertia group acts trivially: $\rho(I_K)=\operatorname{Id}_V$.* *Assume $\ell=p$. A representation $\rho: G_K\rightarrow \mathrm{GL}(V)$ is called *crystalline* if the equality $\dim_{K_0}\left(B_{cris}\otimes_{\mathbb{Q}_p}V\right)^{G_K}=\dim_{\mathbb{Q}_p}V$ holds, where $B_{cris}$ is the Fontaine-Illusie-Kato crystalline module and $K_0$ is the field of fractions of Witt vectors.* In our proof we follow the strategy from [@CvS], adapting complex geometry arguments to the $p$-adic setting. For example, existence of the limiting mixed Hodge structure at a singular point of a Picard-Fuchs operator corresponds to the existence of *logarithmic structure* $H^j_{log-cryst}(Y_0/W)$ on $Y_0$. The exact definition of this structure can be found in [@Kato]. Let $S:=(\operatorname{Spec}(\mathbb{F}_q),\mathbb{N}\oplus\mathbb{F}_q^*)$, let $W=W(\mathbb{F}_q)$ be the Witt ring and $K_0$ its field of fractions. Let $Z_0/S$ be a proper SNCL variety of pure dimension, and finally let $Z_0^{(j)}$ denote the disjoint union of $j$-fold intersections of the irreducible components of $Z_0$. Proof of our Main Theorem relies on the *$p$-adic Clemens-Schmidt sequence*: **Lemma 5** (Theorem 3.6, [@Nakka]). *In the setup as above, the spectral sequence $$E^{-k,h+k}_1=\oplus_{j\geq \max\{-k,0\}}H^{h-2j-k}_{cryst}(Z_0^{(2j+k+1)}/W)(-j-k)$$ $$\Longrightarrow H^h_{log-cryst}(Z_0/W)$$ degenerates at $E_2$ modulo torsions.* Lemma [Lemma 5](#l:nakka){reference-type="ref" reference="l:nakka"} allows us to compute $H^h_{log-cryst}(Z_0/W)$ but we need to compute $H^3_{\acute{e}t}(Z_\infty,\mathbb{Q}_p)$. For this purpose we use a semi-stable comparison theorem between étale cohomology and logarithmic structure. It works with coefficients in the ring of semi-stable periods $B_{st}$. This ring contains $B_{cris}$ and $B_{cris}=B_{st}^{N=0}$, where $N$ is the monodromy operator. In general, we have inequalities $\dim_{K_0}\left(B_{cris}\otimes_{\mathbb{Q}_p}V\right)^{G_K}\leq \dim_{K_0}\left(B_{st}\otimes_{\mathbb{Q}_p}V\right)^{G_K}\leq\dim_{\mathbb{Q}_p}V$. **Lemma 6** (Theorem 0.2, [@Tsuji]). *In the setup as above, there is an isomorphism $$B_{st}\otimes_{\mathbb{Q}_p}H_{\acute{e} t}(Z_\infty,\mathbb{Q}_p)\simeq B_{st}\otimes_{W}H_{log-cryst}(Z_0/W)$$ compatible with all the natural structures.* Using lemmas [Lemma 5](#l:nakka){reference-type="ref" reference="l:nakka"} and [Lemma 6](#l:tsuji){reference-type="ref" reference="l:tsuji"}, as well as the geometric description from section [2](#sec:construction){reference-type="ref" reference="sec:construction"}, we can restate the calculations from section 3.2 of [@CvS] in mixed characteristic. Thus we obtain the following result: **Proposition 7**. *There is a direct sum decomposition $$B_{st}\otimes_{\mathbb{Q}_p}H_{\acute{e} t}^3(Y_\infty,\mathbb{Q}_p)\simeq B_{st}\otimes_{\mathbb{Q}_p}\left(H_{\acute{e} t}^3(R_\infty,\mathbb{Q}_p)\oplus H_{\acute{e} t}^1(E,\mathbb{Q}_p)(-1)\right),$$ where $R_\infty$ is a rigid Calabi-Yau threefold and $E$ is an elliptic curve.* *Proof.* Recall that $Y_\infty$ is the generic fiber of the projective, semi-stable scheme $\mathcal{Y}\rightarrow\operatorname{Spec}\mathcal{O}_K$ with central fiber $Y_0$. In particular, $Y_0$ is a proper SNCL variety. In the notation of lemma [Lemma 6](#l:tsuji){reference-type="ref" reference="l:tsuji"}, we have $Y^{(1)}_0=R_0\cup Q_0$, $C_0:=Y^{(2)}_0=R_0\cap Q_0$ is a conic bundle, $Y_0^{(j)}=\varnothing$ for $j\geq 3$. The natural lifts $\mathcal{R},\mathcal{Q}$ and $\mathcal{C}$ are all smooth. Three applications of lemma [Lemma 6](#l:tsuji){reference-type="ref" reference="l:tsuji"} reveal that the first page of the $p$-adic Clemens-Schmidt spectral sequence reads: By lemma [Lemma 5](#l:nakka){reference-type="ref" reference="l:nakka"} this spectral sequence degenerates at the second page and converges to $B_{st}\otimes H^h_{log-cryst}(Y_0/W)$. Thus the latter has $B_{st}\otimes\left(H_{\acute{e}t}^3(R_\infty,\mathbb{Q}_p)\oplus H^3_{\acute{e}t}(Q_\infty,\mathbb{Q}_p)\right)$ as a direct summand. By Theorem [Theorem 3](#th:coh){reference-type="ref" reference="th:coh"} there is an elliptic curve $E$ such that $H^3_{\acute{e}t}(Q_\infty,\mathbb{Q}_p)\simeq H^1_{\acute{e}t}(E,\mathbb{Q}_p)(-1)$. In particular, $b_3(Q_\infty)=b_1(E)=2$. We can check equality $b_3(Y_\infty)=4$ in characteristic zero, where it is part of the main Theorem of [@CvS]. Since $R_\infty$ is rigid, $b_3(R_\infty)=2$. Lemma [Lemma 6](#l:tsuji){reference-type="ref" reference="l:tsuji"} and dimension count imply that $B_{st}\otimes H^h_{log-cryst}(Y_0/W)\simeq B_{st}\otimes\left(H_{\acute{e}t}^3(R_\infty,\mathbb{Q}_p)\oplus H^3_{\acute{e}t}(Q_\infty,\mathbb{Q}_p)\right)$. Using lemma [Lemma 6](#l:tsuji){reference-type="ref" reference="l:tsuji"} again gives the claim. ◻ Now we prove the first part of the Main Theorem. **Theorem 8**. *The Galois action on the cohomology group $H^3_{\acute{e}t}(Y_\infty,\mathbb{Q}_p)$ is crystalline.* *Proof.* Put $H_Y:=H^3_{\acute{e}t}(Y_\infty,\mathbb{Q}_p)$, $H_R:=H^3_{\acute{e}t}(R_\infty,\mathbb{Q}_p)$ and $H_E:=H^1_{\acute{e}t}(E,\mathbb{Q}_p)(-1)$. By the previous proposition there is an isomorphism $B_{st}\otimes_{\mathbb{Q}_p} H_Y\simeq B_{st}\otimes_{\mathbb{Q}_p} (H_R\oplus H_E)$. Using the fact that this decomposition respects the monodromy action, we get $B_{cris}\otimes_{\mathbb{Q}_p} H_Y=(B_{st}\otimes_{\mathbb{Q}_p} H_Y)^{N=0}\simeq(B_{st}\otimes_{\mathbb{Q}_p} (H_R\oplus H_E))^{N=0}=(B_{st}\otimes_{\mathbb{Q}_p} H_R)^{N=0}\oplus (B_{st}\otimes_{\mathbb{Q}_p}H_E)^{N=0}=(B_{cris}\otimes_{\mathbb{Q}_p} H_R)\oplus (B_{cris}\otimes_{\mathbb{Q}_p}H_E)$. Galois representations $H_R$ and $H_E$ are crystalline because $R_\infty$ and $E$ have smooth models. Thus $\dim_{K_0}\left(B_{cris}\otimes_{\mathbb{Q}_p} H_R\right)^{G_K}=\dim_{K_0}\left(B_{cris}\otimes_{\mathbb{Q}_p} H_E\right)^{G_K}=2$ and $\dim_{K_0}\left(B_{cris}\otimes_{\mathbb{Q}_p} H_Y\right)^{G_K}=\dim_{K_0}\left(B_{cris}\otimes_{\mathbb{Q}_p} H_R\right)^{G_K}+\dim_{K_0}\left(B_{cris}\otimes_{\mathbb{Q}_p} H_E\right)^{G_K}=4=b_3(Y_\infty)=\dim_{\mathbb{Q}_p}(H_Y)$. Hence $H_Y$ is crystalline. ◻ # Lack of a smooth model {#sec:no-good} Given Theorem [Theorem 8](#th:cryst){reference-type="ref" reference="th:cryst"}, showing that $Y_\infty$ does not admit potentially good reduction will conclude the proof of our Main Theorem. In characteristic zero, bad reduction means that the family over the punctured unit disc cannot be completed to a smooth family. To prove it one can use general theory of degenerations of Calabi-Yau threefolds (see [@CvS]). In mixed characteristic we must take a different approach. Let us recall the definitions: **Definition 9** (Definition 2.1, [@BLL]). * * *A variety $V$ defined over a $p$-adic field $K$ has *potentially good reduction* if there exists a finite field extension $L/K$ with the ring of integers $\mathcal{O}_L$ and a smooth proper algebraic space $\mathcal{V}\rightarrow\operatorname{Spec}\mathcal{O}_L$ flat over $\mathcal{O}_L$ such that $V_\infty$ is isomorphic to $V$ over $L$ and the special fiber $V_0$ is a scheme.* *A variety has *bad reduction* if it does not have potentially good reduction.* Note that we allow smooth models over fields larger than the field of definition. Thus our notion of bad reduction means *bad reduction over* $\overline{K}$, not only over $K$. Furthermore, we work in the category of *algebraic spaces* rather than schemes, since in the latter the Néron-Ogg-Shafarevich criterion can fail even for $K3$ surfaces (see [@Matsumoto]). In the proof we use several general lemmas. Recall that a variety is *ruled* if it is birational to the product $\mathbb{P}^1\times V$ for some variety $V$. **Lemma 10**. *If $A$ is a Calabi-Yau variety, then it is not ruled.* *Proof.* The geometric genus $p_g(V):=h^{n,0}(V)$ of a variety is a birational invariant (Theorem 8.19, [@Hart]). Since we have $p_g(\mathbb{P}^1\times V)=0$ and $p_g(A)=1$, the claim follows. ◻ **Lemma 11** (Proposition 3, [@Ab]). *Let $f:X\rightarrow Y$ be a proper birational morphism with $Y$ regular. Then every irreducible component of the exceptional locus of $f$ is ruled.* **Lemma 12** (Corollary 5.7.14, [@RG]). *Let $Y$ be a coherent algebraic space, $X \rightarrow Y$ a separated $Y$-algebraic space of finite type and $U\subset X$ an open subspace quasi-projective over $Y$. Then there exists an $U$-admissible blow-up $\pi \colon \widetilde{X} \rightarrow X$ such that $\widetilde{X}$ is quasi-projective over $Y$.* **Lemma 13**. *Let $f:A\rightarrow B$ be a birational map of Calabi-Yau threefolds defined over a finite field. There exist open subsets $U\subset A$ and $V\subset B$ such that $\textnormal{codim}_A (A\setminus U),\ \textnormal{codim}_B (B\setminus V)\geq 2$ and $f\big| _U:U\rightarrow V$ is an isomorphism.* *Proof.* Proof is the same as that of Proposition 3.1 in [@batyrev_1999] for the complex case. ◻ **Lemma 14**. *Let $\mathcal{A}$ and $\mathcal{B}$ be projective, normal, irreducible schemes over $\textnormal{Spec}\ \mathcal{O}_K$ such that the generic fibers $A_\infty,B_\infty$ are isomorphic. Moreover, let us assume that $\mathcal{B}$ is regular. Then every non-ruled component of the special fiber $A_0$ is birational to a component of $B_0$.* *Proof.* Any isomorphism $f:A_\infty\rightarrow B_\infty$ induces a birational map $\overline{f}:\mathcal{A}\rightarrow\mathcal{B}$ of schemes over $\operatorname{Spec}\mathcal{O}_K$. Consider the normalization of the (closure of the) graph $\Gamma_f$ of $\overline{f}$: $$\mathcal{Z}:={\overline{\Gamma_f}}^\nu$$ for the normalization morphism $\nu:=\nu_V:V^\nu\rightarrow V$ as in Definition 29.54.1 of [@stacks]. Since $\overline{\Gamma_f}\subset \mathcal{A}\times \mathcal{B}$, we have natural projections $\pi_A:\overline{\Gamma_f}\rightarrow \mathcal{A}$, $\pi_B:\overline{\Gamma_f}\rightarrow \mathcal{B}$. Consider the exceptional locus $E_A\subset \mathcal{Z}$, resp. $E_B$, of the birational morphism $f_A:=\pi_A\circ\nu_{\overline{\Gamma_f}}:\mathcal{Z}\rightarrow \mathcal{A}$, resp. $f_B:=\pi_B\circ\nu_{\overline{\Gamma_f}}:\mathcal{Z}\rightarrow \mathcal{B}$. Let $A_0'$ be a non-ruled component of $A_0$. Since $E_A=\{z\in \mathcal{Z}: \dim \left(f^{-1}(f(z))>0\right)\}$ and $\operatorname{codim}_\mathcal{A} A_0'=1$, it follows that $A_0'\setminus f_A(E_A)\neq\varnothing$. As $f_A$ is an isomorphism away from $E_A$, we conclude that $A_0'$ and $Z_0':=f_A^{-1}\left(A'_0\setminus f_A(E_A)\right)$ are birational. In particular, $Z_0'$ is not ruled. However, by lemma [Lemma 11](#lem:ab){reference-type="ref" reference="lem:ab"} every irreducible component of the exceptional locus $E_B$ is ruled, hence $Z'_0\not\subset E_B$ by Zariski's Main Theorem [@Zariski]. It follows that $Z_0'\setminus E_B$ is birational to some irreducible component of $B_0$, which concludes the proof. ◻ Now we are ready to prove the remaining part of the Main Theorem: **Theorem 15**. *Calabi-Yau threefold $Y_\infty$ has bad reduction.* *Proof.* Assume that $Y_\infty$ has good reduction, w.l.o.g over $K$. Let $\mathcal{C}\rightarrow\operatorname{Spec}\mathcal{O}_K$ be a smooth algebraic space such that $Y_\infty\simeq C_\infty$. The special fiber $Y_0$ has two irreducible components $R_0$ and $Q_0$, as in Theorem [Theorem 2](#th:con){reference-type="ref" reference="th:con"}, while $C_0$ is a smooth Calabi-Yau threefold defined over a finite field such that $b_3(C_0)=4$. By lemma [Lemma 12](#lem:chow){reference-type="ref" reference="lem:chow"} there exists a blow-up $\pi:\mathcal{D}\rightarrow\mathcal{C}$ at a closed subscheme of $C_0$ such that $\mathcal{D}$ is projective over $\text{Spec} \, \mathcal{O}_L$ and the generic fiber $D_\infty$ is isomorphic to $C_\infty \simeq Y_\infty$. The special fiber $D_0$ contains a component birational to $C_0$. Let $\mathcal{E}$ be the normalization of $\mathcal{D}$. Schemes $\mathcal{E}$ and $\mathcal{Y}$ are projective, irreducible and normal. $R_0$ is a Calabi-Yau threefold, and $E_0$ contains a (smooth) Calabi-Yau component $E_0'$; both are not ruled by lemma [Lemma 10](#lem:ruled){reference-type="ref" reference="lem:ruled"}. It follows from lemma [Lemma 14](#lem:iso-bi){reference-type="ref" reference="lem:iso-bi"} that the special fibers' components $E_0'$ and $R_0$ are birational. By lemma [Lemma 13](#lem:batyrev){reference-type="ref" reference="lem:batyrev"} there exist closed subsets $F\subset E_0',G\subset R_0$ of dimension at most $1$ such that $E_0'\setminus F$ and $R_0\setminus G$ are isomorphic. In particular, we have the equality of zeta functions: $$\label{eq:Z} Z(E_0',s)\cdot Z(G,s)=Z(R_0,s)\cdot Z(F,s)$$ For any rational function $f\in\mathbb{Q}(T)$ we define the multi-sets (counted with multiplicities): $$\mathcal{Z}(f):=f^{-1}(\{0\}),\quad \mathcal{Z}_i(f):=\mathcal{Z}(f)\cap\{ x\in\overline{\mathbb{Q}}: \forall_{\iota:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C} }\ |\iota(x)|=q^{-\tfrac{i}{2}}\}$$ By the Weil conjectures, functions $Z(E_0',s),\ Z(R_0,s)$ are rational (see [@Dwork]). By the Weil conjectures for (possibly singular) curves, so are $Z(F,s)$ and $Z(G,s)$ (see [@AP]). Moreover, all roots of $Z(F,s)$ and $Z(G,s)$ have absolute value $1$ or $\sqrt{q}^{-1}$. Hence, $$\mathcal{Z}\left(Z(F,s)\right)=\mathcal{Z}_{0}\left(Z(F,s)\right)\cup \mathcal{Z}_{1}\left( Z(F,s)\right)\ \textnormal{and}\ \mathcal{Z}\left(Z(G,s)\right)=\mathcal{Z}_{0}\left(Z(G,s)\right)\cup \mathcal{Z}_{1}\left( Z(G,s)\right)$$ Combining this with ([\[eq:Z\]](#eq:Z){reference-type="ref" reference="eq:Z"}), we obtain $$\mathcal{Z}_3\Big(Z(E_0',s)\Big)=\mathcal{Z}_3\Big(Z(E_0',s)\cdot Z(F,s)\Big)=\mathcal{Z}_3\Big(Z(R_0,s)\cdot Z(G,s)\Big)=\mathcal{Z}_3\Big(Z(R_0,s)\Big)$$ However, by the Riemann hypothesis (see [@Deligne]) we have $|\mathcal{Z}_3(Z(E_0',s))|=b_3(E_\infty)=b_3(Y_\infty)=4$ and $|\mathcal{Z}_3(Z(R_0,s))|=b_3(R_\infty)=2$, a contradiction. ◻ 10 *Stacks project*. <https://stacks.math.columbia.edu/>. Accessed: 2023-06-21. S. Abhyankar, *On the valuations centered in a local domain*, American Journal of Mathematics, 78 (1956), pp. 321--348. Y. Aubry and M. Perret, *A Weil theorem for singular curves*, Contemporary mathematics, (1996), pp. 1--7. V. Batyrev, *Birational Calabi--Yau n-folds have equal betti numbers*, (1999), p. 1--12. B. Chiarellotto, C. Lazda, and C. Liedtke, *A Néron-Ogg-Shafarevich criterion for K3 surfaces*, Proceedings of the London Mathematical Society, 119 (2017). S. Cynk and B. Kocel-Cynk, *Classification of double octic Calabi-Yau threefolds with $h^{1,2} \leq 1$ defined by an arrangement of eight planes*, Communications in Contemporary Mathematics, 22 (2018). S. Cynk and T. Szemberg, *Double covers of $\mathbb{P}^3$ and Calabi-Yau varieties*, Banach Center Publications, 44 (1999), pp. 93--101. S. Cynk and D. van Straten, *A special Calabi--Yau degeneration with trivial monodromy*, Communications in Contemporary Mathematics, 24 (2022). P. Deligne, *La conjecture de Weil : I*, Publications Mathématiques de l'IHÉS, 43 (1974), pp. 273--307. B. Dwork, *On the Rationality of the Zeta Function of an Algebraic Variety*, American Journal of Mathematics, 82 (1960), pp. 631--648. R. Hartshorne, *Algebraic Geometry*, Graduate texts in mathematics, Springer-Verlag, 1977. K. Kato, *Logarithmic structures of Fontaine-Illusie*, Algebraic Analysis, Geometry and Number Theory, 154 (1989), p. 191--224. C. Liedtke and Y. Matsumoto, *Good reduction of K3 surfaces*, Compositio Mathematica, 154 (2018), p. 1--35. Y. Matsumoto, *Good reduction criterion for K3 surfaces*, Math. Z. 279, 241--266 (2015)., 13 (1971). Y. Nakkajima, *p-adic weight spectral sequences of log varieties*, Journal of Mathematical Sciences-the University of Tokyo, 12 (2005), pp. 513--661. A. Ogg, *Elliptic curves and wild ramification*, American Journal of Mathematics, 89 (1967), pp. 1--21. M. Raynaud and L. Gruson, *Critères de platitude et de projectivité. techniques de "platification" d'un module*, Inventiones Mathematicae, 13 (1971). C. Tsen, *Divisionsalgebren über funktionenkörpern*, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, (1933), pp. 335--339. T. Tsuji, *p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case*, Inventiones mathematicae, 137 (1999), pp. 233--411. O. Zariski, *p-adic foundations of a general theory of birational correspondences*, Trans. Amer. Math. Soc, 53 (3) (1943), pp. 490--542.

arxiv_math

--- abstract: | In this paper, we obtain the sharp bounds of the second Hankel determinant of logarithmic inverse coefficients for the strongly starlike and strongly convex functions of order alpha. address: - Vasudevarao Allu, School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar-752050, Odisha, India. - Amal Shaji, School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar-752050, Odisha, India. author: - Vasudevarao Allu - Amal Shaji title: Second Hankel Determinant for Logarithmic Inverse Coefficients of Strongly Convex and Strongly Starlike Functions --- # Introduction {#Introduction} Let $\mathcal{H}$ denote the class of analytic functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$. Here $\mathcal{H}$ is a locally convex topological vector space endowed with the topology of uniform convergence over compact subsets of $\mathbb{D}$. Let $\mathcal{A}$ denote the class of functions $f\in \mathcal{H}$ such that $f(0)=0$ and $f'(0)=1$. Let $\mathcal{S}$ denote the subclass of $\mathcal{A}$ consisting of functions which are univalent (*i.e., one-to-one*) in $\mathbb{D}$. If $f\in\mathcal{S}$ then it has the following series representation $$\label{f} f(z)= z+\sum_{n=2}^{\infty}a_n z^n, \quad z\in \mathbb{D}.$$ The *Logarithmic coefficients* $\gamma_{n}$ of $f\in \mathcal{S}$ are defined by, $$\label{amal-1} F_{f}(z):= \log\frac{f(z)}{z}=2\sum\limits_{n=1}^{\infty}\gamma_{n}z^{n}, \quad z \in \mathbb{D}.$$ The logarithmic coefficients $\gamma_{n}$ play a central role in the theory of univalent functions. A very few exact upper bounds for $\gamma_{n}$ seem to have been established. The significance of this problem in the context of Bieberbach conjecture was pointed by Milin[@milin] in his conjecture. Milin [@milin] has conjectured that for $f\in \mathcal{S}$ and $n\ge 2$, $$\sum\limits_{m=1}^{n}\sum\limits_{k=1}^{m}\left(k|\gamma_{k}|^{2}-\frac{1}{k}\right)\le 0,$$ which led De Branges, by proving this conjecture, to the proof of Bieberbach conjecture [@De; @Branges-1985]. For the Koebe function $k(z)=z/(1-z)^{2}$, the logarithmic coefficients are $\gamma_{n}=1/n$. Since the Koebe function $k$ plays the role of extremal function for most of the extremal problems in the class $\mathcal{S}$, it is expected that $|\gamma_{n}|\le1/n$ holds for functions in $\mathcal{S}$. But this is not true in general, even in order of magnitude. By differentiating [\[amal-1\]](#amal-1){reference-type="eqref" reference="amal-1"} and the equating coefficients we obtain $$\label{gamma} \begin{aligned} & \gamma_{1}=\frac{1}{2}a_{2}, \\[2mm] & \gamma_{2}=\frac{1}{2}(a_{3}-\frac{1}{2}a_{2}^{2}),\\[2mm] & \gamma_{3}=\frac{1}{2}(a_{4}-a_{2}a_{3}+\frac{1}{3}a_{2}^{3}). \end{aligned}$$ If $f\in \mathcal{S}$, it is easy to see that $|\gamma_{1}|\le 1$, because $|a_2| \leq 2$. Using the Fekete-Szeg$\ddot{o}$ inequality [@Duren-book-1983 Theorem 3.8] for functions in $\mathcal{S}$ in (1.4), we obtain the sharp estimate $$|\gamma_{2}|\le\frac{1}{2}\left(1+2e^{-2}\right)=0.635\ldots.$$ For $n\ge 3$, the problem seems much harder, and no significant bound for $|\gamma_{n}|$ when $f\in \mathcal{S}$ appear to be known. In 2017, Ali and Allu[@vasu2017] obtained the initial logarithmic coefficients bounds for close-to-convex functions. The problem of computing the bound of the logarithmic coefficients is also considered in [@cho; @PSW20; @vasu-2018; @Thomas-2016] for several subclasses of close-to-convex functions.\ For $q,n \in \mathbb{N}$, the Hankel determinant $H_{q,n}(f)$ of Taylor's coefficients of function $f \in \mathcal{A}$ of the form [\[f\]](#f){reference-type="eqref" reference="f"} is defined by $$H_{q,n}(f) = \begin{vmatrix} a_n & a_{n+1} & \cdots & a_{n+q-1} \\ a_{n+1} & a_{n+2} & \cdots & a_{n+q} \\ \vdots & \vdots & \ddots &\vdots \\ a_{n+q-1} & a_{n+q} & \cdots & a_{n+2(q-1)} \end{vmatrix}.$$ The Hankel determinant for various order is also studied recently by several authors in different contexts; for instance see [@Pom66; @Pom67; @ALT]. One can easily observe that the Fekete-Szegö functional is the second Hankel determinant $H_{2,1}(f)$. Fekete-Szeg$\ddot{o}$ then further generalized the estimate $|a_3 - \mu a_2 ^2|$ with $\mu$ real for $f$ given by [\[f\]](#f){reference-type="eqref" reference="f"} (see [@Duren-book-1983 Theorem 3.8]).\ Let $g$ be the inverse function of $f\in \mathcal{S}$ defined in a neighborhood of the origin with the Taylor series expansion $$\label{inverse} g(w)=f^{-1}(w)=w+\sum_{n=2}^{\infty}A_n w^n,$$ where we may choose $|w| < 1/4$, as we know from Koebe's $1/4$-theorem. Using variational method, Löwner [@Lowner] obtained the sharp estimate: $$|A_n| \leq K_n \quad \text{for each}\,\,\, n \in \mathbb{N}$$ where $K_n = (2n)!/(n!(n + 1)!)$ and $K(w) = w + K_2w_2 + K_3w_3 + \cdots$ is the inverse of the Koebe function. There has been a good deal of interest in determining the behaviour of the inverse coefficients of $f$ given in [\[f\]](#f){reference-type="eqref" reference="f"} when the corresponding function $f$ is restricted to some proper geometric subclasses of $\mathcal{S}$.\ Let $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ be a function in class $\mathcal{S}$. Since $f(f^{-1})(w)=w$ and using [\[inverse\]](#inverse){reference-type="eqref" reference="inverse"}, it follows that $$\label{inversetaylor} \begin{aligned} & A_{2}=-a_2,\\[2mm] & A_{3}=-a_3+2a_2^2,\\[2mm] & A_{4}=-a_4+5a_2a_3-5a_2^3. \end{aligned}$$\ The notion of logarithmic inverse coefficients, *i.e.*, logartithmic coefficients of inverse of $f$, was proposed by ponnusamy *et al.* [@samyinverselog]. The *logarithmic inverse coefficients* $\Gamma_n$, $n \in \mathbb{N}$, of $f$ are defined by the equation $$\label{Gamma} F_{f^{-1}}(w):= \log\frac{f^{-1}(w)}{w}=2\sum\limits_{n=1}^{\infty}\Gamma_{n}w^{n}, \quad |w|<1/4.$$ By differentiating [\[Gamma\]](#Gamma){reference-type="eqref" reference="Gamma"} together with [\[inversetaylor\]](#inversetaylor){reference-type="eqref" reference="inversetaylor"}, we get $$\label{Gammaaa} \begin{aligned} & \Gamma_{1}=-\cfrac{1}{2}a_2,\\[2mm] & \Gamma_2=-\cfrac{1}{2}a_3+\cfrac{3}{4}a_2^2,\\[2mm] & \Gamma_3=-\cfrac{1}{2}a_4+2a_2a_3-\cfrac{5}{3}a_2^3. \end{aligned}$$ In [@samyinverselog] Ponnusamy *et al.* found the sharp upper bound for the logarithmic inverse coefficients for the class $\mathcal{S}$. In fact ponnusamy *et al.* [@samyinverselog] proved that when $f\in \mathcal{S}$, $$|\Gamma_n| \leq \frac{1}{2n} \left( \begin{matrix} 2n \\ n \end{matrix} \right), \quad n \in \mathbb{N}$$ and equality holds only for the Koebe function or one of its rotations. Further, ponnusamy *et al.* [@samyinverselog] obtained sharp bound for the initial logarithmic inverse coefficients for some of the important geometric subclasses of $\mathcal{S}$.\ Recently, Kowalczyk and Lecko [@adam] together have proposed the study of the Hankel determinant whose entries are logarithmic coefficients of $f \in \mathcal{S}$, which is given by $$H_{q,n}(F_f/2) = \begin{vmatrix} \gamma_n & \gamma_{n+1} & \cdots & \gamma_{n+q-1} \\ \gamma_{n+1} & \gamma_{n+2} & \cdots & \gamma_{n+q} \\ \vdots & \vdots & \ddots &\vdots \\ \gamma_{n+q-1} & \gamma_{n+q} & \cdots & \gamma_{n+2(q-1)} \end{vmatrix}.$$ Kowalczyk and Lecko [@adam] have obtained the sharp bound of the second Hankel determinant of $F_f/2$, *i.e.,* $H_{2,1}(F_f/2)$ for starlike and convex functions. The problem of computing the sharp bounds of $H_{2,1}(F_f/2)$ has been considered by many authors for various subclasses of $\mathcal{S}$ (See [@vibhuthi; @vibhuthi2; @adam2; @Mundalia]).\ In this paper, we consider the notion of the second Hankel determinant for logarithmic inverse coefficients. Let $f \in \mathcal{S}$ given by [\[f\]](#f){reference-type="eqref" reference="f"}, then the second Hankel determinant of $F_{f^{-1}}/2$ by using [\[Gammaaa\]](#Gammaaa){reference-type="eqref" reference="Gammaaa"}, is given by $$\label{hankel} \begin{aligned} H_{2,1}(F_{f^{-1}}/2) &=\Gamma_1\Gamma_3-\Gamma_{2}^2 \\[2mm] &=\frac{1}{4}\left(A_2A_4-A_3^2+\frac{1}{4}A_2^4\right)\\[2mm] &=\frac{1}{48}\left(13a_2^4-12a_2^2a3-12a_3^2+12a_2a_4\right). \end{aligned}$$\ It is now appropriate to remark that $H_{2,1}(F_{f^{-1}}/2)$ is invariant under rotation, since for $f_{\theta}(z):=e^{-i \theta} f\left(e^{i \theta} z\right), \theta \in \mathbb{R}$ when $f \in \mathcal{S}$ we have $$\label{invariance} H_{2,1}(F_{f_{\theta}^{-1}}/2)=\frac{e^{4 i \theta}}{48}\left(13a_2^4-12a_2^2a3-12a_3^2+12a_2a_4\right)=e^{4 i \theta} H_{2,1}(F_{f^{-1}}/2) .$$ In this paper we find sharp upper bounds for $|H_{2,1}(F_{f^{-1}}/2)|$ when $f$ belongs to the class of strongly convex or strongly starlike function of order alpha. Given $\alpha \in (0,1]$, a function $f \in \mathcal{A}$ is called strongly convex of order $\alpha$ if $$\label{convexdef} \left|\text{arg} \left(1+\cfrac{zf''(z)}{f'(z)}\right)\right|< \cfrac{\pi \alpha}{2}$$ The set of all such functions denoted by $\mathcal{K}_{\alpha}$. Also, a function $f \in \mathcal{A}$ is called strongly starlike of order $\alpha$ if $$\label{stardef} \left|\text{arg} \left(\cfrac{zf'(z)}{f(z)}\right)\right|< \cfrac{\pi \alpha}{2}$$ The set of all such functions denoted by $\mathcal{S}^*_{\alpha}$. The notion of strongly starlike functions was introduced by Stankiewicz [@Stankiewicz1] and s independently by Brannan and Kirwan[@Brannan]. An external geometric characterisation of strongly starlike functions was proposed by Stankiewicz [@Stankiewicz2]. Brannan and Kirwan[@Brannan] found a geometrical condition called $\delta-$ visibility which is sufficient for functions to be starlike. # Preliminary Results In this section, we present the key lemmas which will be used to prove the main results of this paper. Let $\mathcal{P}$ denote the class of all analytic functions $p$ having positive real part in $\mathbb{D}$, with the form $$\label{p} p(z)=1+c_{1} z+c_{2} z^{2}+c_{3} z^{3}+ \cdots .$$ A member of $\mathcal{P}$ is called a Carathéodory function. It is known that $\left|c_{n}\right| \leq 2, n \geq 1$ for a function $p \in \mathcal{P}$. By using [\[convexdef\]](#convexdef){reference-type="eqref" reference="convexdef"} and [\[stardef\]](#stardef){reference-type="eqref" reference="stardef"}, functions in the classes $\mathcal{S}^*_{\alpha}$ and $\mathcal{K}_{\alpha}$ can be represented interms of functions in Carathéodory class $\mathcal{P}$. To prove our main result, we need some preliminary lemmas. The first one is known as Caratheodory's lemma and the second one is due to Libera and Zlotkiewicz. **Lemma 1**. *[@Duren-book-1983][\[pp\]]{#pp label="pp"} For a function $p \in \mathcal{P}$ of the form [\[p\]](#p){reference-type="eqref" reference="p"}, the sharp inequality holds for each $n\geq 1$. Equality holds for the function $p(z)=1+z/1-z$.* **Lemma 2**. *[@LZ1; @LZ2][\[caratheodary\]]{#caratheodary label="caratheodary"} If $p \in \mathcal{P}$ is of the form [\[p\]](#p){reference-type="eqref" reference="p"} with $c_1 \geq 0$. Then there exits $z,w \in \mathbb{D}$ such that $$2c_{2}=c_1^2+(4-c_1^2)z$$ and* *$$4c_3=c_1^3+2(4-c_1^2)c_1z-c_1(4-c_1^2)z+2(4-c_1^2)(1-|x|^2)w.$$* Next we recall the following well-known result due to Choi *et al.* [@choi]. Lemma [Lemma 3](#y(a,b,c)){reference-type="ref" reference="y(a,b,c)"} plays an important role in the proof of our main results. **Lemma 3**. *Let $A, B, C$ be real numbers and* *$$Y(A, B, C):=\max _{z \in \overline{\mathbb{D}}}\left(\left|A+B z+C z^{2}\right|+1-|z|^{2}\right) .$$* *(i) If $A C \geq 0$, then* *$$Y(A, B, C)= \begin{cases}|A|+|B|+|C|, & |B| \geq 2(1-|C|), \\[2mm] 1+|A|+\cfrac{B^{2}}{4(1-|C|)}, & |B|<2(1-|C|) .\end{cases}$$* *(ii) If $A C<0$, then* *$$Y(A, B, C)= \begin{cases}1-|A|+\cfrac{B^{2}}{4(1-|C|)}, & -4 A C\left(C^{-2}-1\right) \leq B^{2} \wedge|B|<2(1-|C|), \\[2mm] 1+|A|+\cfrac{B^{2}}{4(1+|C|)}, & B^{2}<\min \left\{4(1+|C|)^{2},-4 A C\left(C^{-2}-1\right)\right\}, \\[2mm] R(A, B, C), & \text { otherwise, }\end{cases}$$* *where* *$$R(A, B, C)= \begin{cases}|A|+|B|+|C|, & |C|(|B|+4|A|) \leq|A B|, \\[2mm] -|A|+|B|+|C|, & |A B| \leq|C|(|B|-4|A|), \\[2mm] (|A|+|C|) \sqrt{1-\cfrac{B^{2}}{4 A C}}, & \text { otherwise. }\end{cases}$$* # Main Results {#sec3} We prove the following sharp inequality for second hankel determinant of inverse logarithmic coefficient for $\mathcal{K}_\alpha$. **Theorem 4**. *Let $f\in\mathcal{K}_{\alpha}$ given by [\[f\]](#f){reference-type="eqref" reference="f"} then $$\label{thm1} |H_{2,1}(F_{f^{-1}}/2)|\leq \begin{cases} \cfrac{\alpha^2}{36}, & 0< \alpha \leq 1/3,\\[5mm] \cfrac{\alpha^2(17+18\alpha+13\alpha^2)}{144(4+6\alpha+\alpha^2)}, & 1/3 < \alpha \leq 1. \end{cases}$$ The inequality is sharp.* *Proof.* Fix $\alpha \in (0,1]$ and let $f\in \mathcal{K}_{\alpha}$ be of the form [\[f\]](#f){reference-type="eqref" reference="f"}. Then by [\[convexdef\]](#convexdef){reference-type="eqref" reference="convexdef"}, $$\label{3.1.1} 1+\cfrac{zf''(z)}{f'(z)}=(p(z))^\alpha$$ for some $p \in \mathcal{P}$ of the form [\[p\]](#p){reference-type="eqref" reference="p"}. By comparing the coefficients on both the sides of [\[3.1.1\]](#3.1.1){reference-type="eqref" reference="3.1.1"}, we obtain $$\label{3.1.2} \begin{aligned} & a_2=\cfrac{\alpha}{2}\,\,\alpha c_1, \\[2mm] & a_3=\frac{\alpha}{12}(2c_2+(3\alpha-1)c_1^2), \\[2mm] & a_4=\frac{\alpha}{144}\left(12c_3+6(5\alpha-2)c_1c_2+(17\alpha^2-15\alpha+4)c_1^3 \right). \end{aligned}$$ Hence by [\[hankel\]](#hankel){reference-type="eqref" reference="hankel"}, we have $$H_{2,1}(F_{f^{-1}}/2)=\cfrac{\alpha^2}{2304}\left[24c_1c_3-16c_2^2-4(2+3\alpha) c_1^2 c_2+(4+6\alpha+\alpha^2)c_1^4 \right].$$ Since the class $\mathcal{K}_\alpha$ is invariant under rotation\[add one sentensece about lemma\] and [\[invariance\]](#invariance){reference-type="eqref" reference="invariance"} holds, without loss of generality we can assume that $c_1 = c$, where $0 \leq c \leq 2$. Now using Lemma [\[caratheodary\]](#caratheodary){reference-type="ref" reference="caratheodary"} and straight forward computation $$\label{mainconvex} \begin{aligned} H_{2,1}(F_{f^{-1}}/2) =&\frac{\alpha^2}{2304}\left[(2+\alpha^2)c^4-6\alpha(4-c^2)c^2 z-2(4-c^2)(8+c^2)z^2\right.\\[2mm] &\left. +12c(4-c^2)(1-|z|^2)w\right]. \end{aligned}$$ Now we consider different cases on $c$.\ **Case 1.** Suppose that $c=0$. Then from [\[mainconvex\]](#mainconvex){reference-type="eqref" reference="mainconvex"}, for $\alpha \in (0,1]$, $$\label{convexc=0} |\Gamma_1\Gamma_3-\Gamma_2^2|=\cfrac{\alpha^2}{36}\,\, |z|^2 \leq \cfrac{\alpha^2}{36}$$ **Case 2.** Suppose that $c=2$. Then from [\[mainconvex\]](#mainconvex){reference-type="eqref" reference="mainconvex"}, for $\alpha \in (0,1]$, $$\label{convexc=2} |\Gamma_1\Gamma_3-\Gamma_2^2| \leq \cfrac{\alpha^2(2+\alpha^2)}{144}$$ **Case 3.** Suppose that $c\in (0,2)$. Since $|y|\leq 1$, from [\[mainconvex\]](#mainconvex){reference-type="eqref" reference="mainconvex"} we obtain $$\label{3cc} \begin{aligned} |\Gamma_1\Gamma_3-\Gamma_2^2|& \leq \frac{\alpha^2}{2304}\left[\left|(2+\alpha^2)c^4-6\alpha(4-c^2)c^2 z-2(4-c^2)(8+c^2)z^2\right|\right.\\[2mm] &\left.\,\,\,\, +12c(4-c^2)(1-|z|^2)w\right]\\[2mm] &=\cfrac{\alpha^2}{192}(c(4-c^2))\left[|A+Bz+Cz^2|+1-|z|^2\right] \end{aligned}$$ where $$A:=\cfrac{c^3(2+\alpha^2)}{12(4-c^2)},\,\,\,\,\,\,\, B:-\cfrac{c \alpha}{2},\,\,\,\,\,\,\, C:=-\cfrac{8+c^2}{6c}.$$ Since $A C < 0$, we apply case (ii) ofLemma [Lemma 3](#y(a,b,c)){reference-type="ref" reference="y(a,b,c)"}. **3(a).** Note that the inequality $$-4 A C\left(\frac{1}{C^{2}}-1\right) \leq B^{2}$$ is equivalent to $$\cfrac{c^4(7\alpha^2-4)+8c^2(8+13\alpha^2))}{8+c^2}\geq 0$$ which is evidently holds for $c \in (0,2).$ Moreover, the inequality $|B|<2(1-|C|)$ is equivalent to $c\,(16-2c+c^2(2+3\alpha))< 0$ which is false for $c \in(0,2)$.\ **3(b).** Since $$4(1+|C|)^2=\cfrac{c^4+52c^2+64}{9c^2} > 0,$$ and $$-4 A C\left(C^{-2}-1\right)=-\cfrac{(2+\alpha^2)c^2(16-c^2)}{18(8+c^2)}<0.$$ We see that $$\min \left\{4(1+|C|)^{2},-4 A C\left(C^{-2}-1\right)\right\}=-4 A C\left(C^{-2}-1\right),$$ and from case $3(a),$ we know that $$-4 A C\left(C^{-2}-1\right) \leq B^{2}.$$ Therefore, the inequality $B^{2} < \min \left\{4(1+|C|)^{2},-4 A C\left(C^{-2}-1\right)\right\}$ does not holds for $0<c<2$.\ **3(c).** Next observe that the inequality, $$|C|(|B|+4|A|) -|A B|=\cfrac{192\alpha^2+8(8-3\alpha+4\alpha^2)c^2+(8-12\alpha+4\alpha^2-3\alpha^3)c^4}{4-c^2} \leq 0$$ is equivalent to $$\label{3c} \varphi_1(c^2) \leq 0$$ where $$\varphi_1(t)=192\alpha^2+8(8-3\alpha+4\alpha^2)x+(8-12\alpha+4\alpha^2-3\alpha^3)x^2,\,\,\,\,\,\, t\in (0,4).$$ Note that $\varphi_1(0)>0$ and $\varphi_1(4)>0$. Also we can easily seen that $\varphi_1'(t)>0$ for $t \in (0,4)$. So $\varphi_1$ is increasing and hence $\varphi_1(t) >0$ in $t\in (0,4)$. Thus we deduce that the inequality [\[3c\]](#3c){reference-type="eqref" reference="3c"} is false.\ **3(d).** Next note that the inequality $$\label{3d1} |AB|-|C|(|B|-4|A|)=\cfrac{\alpha c^4(2+\alpha)^2}{24(4-c^2)}-\cfrac{8+c^2}{6c}\left(\cfrac{\alpha c}{2}-\cfrac{c^3(2+\alpha)^2}{3(4-c^2)}\right) \leq 0$$ is equivalent to $$\label{3d2} \varphi_2(u^2)\leq 0,$$ where $$\varphi_2(u)=(8+12\alpha+4\alpha^2+3\alpha^3)u^2+8(8+3\alpha+4\alpha^2)u-192\alpha, \,\,\,\,\,\,\, u \in (0,4).$$ We see that the discriminant $\Delta := 64(64+144\alpha+217\alpha^2+72\alpha^3+52\alpha^4)>0.$ Thus we consider, $$u_{1,2}=\cfrac{4(-(4\alpha^2+3\alpha+8)\mp\sqrt{52\alpha^4+72\alpha^3+217\alpha^2+144\alpha+64}}{3\alpha^3+4\alpha^2+12\alpha+8}.$$ It is clear that $u_1 <0$. Moreover $0<u_2<4$ holds. Indeed both inequalities $u_2>0$ and $u_2<4$ are equivalent to the evidently true inequalities $$8+12\alpha+4\alpha^2+3\alpha^3 >0$$ and $$192+33\alpha+264\alpha^2+264\alpha^3+102\alpha^4+48\alpha^5+9 \alpha^6 >0$$ Thus [\[3d2\]](#3d2){reference-type="eqref" reference="3d2"} and so [\[3d1\]](#3d1){reference-type="eqref" reference="3d1"} is valid only when $$0<c\leq \tilde{c}=\sqrt{u_2}.$$ Thus by [\[3cc\]](#3cc){reference-type="eqref" reference="3cc"} and Lemma [Lemma 3](#y(a,b,c)){reference-type="ref" reference="y(a,b,c)"}, $$\begin{aligned} |\Gamma_1\Gamma_3-\Gamma_2^2| &\leq \cfrac{\alpha^2}{192}c(4-c^2)(-|A|+|B|+|C|)\\[2mm] &=\cfrac{\alpha^2}{2304}(64-(8-24\alpha)c^2-(4+6\alpha+\alpha^2)c^4)\\[2mm] &=\Phi(c^2) \end{aligned}$$ where $$\Phi(t)=\cfrac{\alpha^2}{2304}(64-(8-24\alpha)t-(4+6\alpha+\alpha^2)t^2), \,\,\,\,\,\,\,\, t \in (0,u_2).$$ Since $$\Phi'(t)=\cfrac{\alpha^2}{2304}((8-24\alpha)-2(4+6\alpha+\alpha^2)c),\,\,\,\,\,\, t \in (0,u_2),$$ we see that $0<\alpha \leq 1/3$, the funtion $\Phi$ is decreasing and so $$\Phi(t)\leq\Phi(0)=\cfrac{\alpha^2}{36}, \,\,\,\,\, 0\leq t\leq u_2$$ In the case $1/3 < \alpha \leq 1$, $$t_0=\cfrac{4(3\alpha-1)}{4+6\alpha+\alpha^2}$$ is a unique critical point of $\Phi$. Clearly $t_0>0$. It remains to check whether $t_0<u_2$, which is equivalent to $$\begin{aligned} \delta(\alpha)=& 117\alpha^8+240\alpha^7-149\alpha^6 -1212\alpha^5-4344\alpha^4 \\[2mm] &-6288\alpha^3-4464\alpha^2 -1920\alpha-448 <0, \,\,\,\,\, \alpha \in (1/3,1], \end{aligned}$$ and since $$\delta(\alpha)\leq -149\alpha^6 -1212\alpha^5-4344\alpha^4 -6288\alpha^3-4464\alpha^2 -1920\alpha-91 <0$$ for $\alpha \in (1/3,1]$, we deduce that $t_0 <u_2$.\ Thus for $1/3 < \alpha \leq 1$, we have $$\Phi(t) \leq \Phi(t_0)=\cfrac{\alpha^2(17+18\alpha+13\alpha^2)}{144(4+6\alpha+\alpha^2)}, \,\,\,\,\,\, 0<t<u_2.$$ We can conclude that, for $0<c\leq\tilde{c}$ $$\label{3cconclusion} |\Gamma_1\Gamma_3-\Gamma_2^2| \leq \begin{cases} \cfrac{\alpha^2}{36}, & 0< \alpha \leq 1/3,\\[4mm] \cfrac{\alpha^2(17+18\alpha+13\alpha^2)}{144(4+6\alpha+\alpha^2)}, & 1/3 < \alpha \leq 1. \end{cases}$$ **3(d).** We now consider the last case in Lemma [Lemma 3](#y(a,b,c)){reference-type="ref" reference="y(a,b,c)"}, which in view of 3(d) holds for $\tilde{c}<c<2$. Then by [\[3cc\]](#3cc){reference-type="eqref" reference="3cc"}, we have $$\label{lastconvex} \begin{aligned} |\Gamma_1\Gamma_3-\Gamma_2^2| &\leq \cfrac{\alpha^2}{192}\,\,c\,\,(4-c^2)(|C|+|A|)\sqrt{1-\cfrac{B^2}{4AC}}\\[2mm] &=\cfrac{\alpha^2}{2304\sqrt{2(2+\alpha^2)}}\left(64-8c^2+\alpha^2c^4\right)\sqrt{\cfrac{32+52\alpha^2+c^2(4-7\alpha^2)}{8+c^2}}\\[2mm] &=\cfrac{\alpha^2}{2304\sqrt{2(2+\alpha^2)}}g_1(c^2)\sqrt{g_2(c^2)} \end{aligned}$$ where $$g_1(x)=64-8x+\alpha^2x^2$$ and $$g_2(x)=\cfrac{32+52\alpha^2+c^2(4-7\alpha^2)}{8+c^2}$$ It is easily seen that $g_1$ and $g_2$ are decreasing on $(u _2,4)$, and so from [\[lastconvex\]](#lastconvex){reference-type="eqref" reference="lastconvex"} we obtain $$\label{3dd} |\Gamma_2\Gamma_3-\Gamma_1^2| \leq\cfrac{\alpha^2}{2304\sqrt{2(2+\alpha^2)}}g_1(u_2)\sqrt{g_2(u_2)}=\Psi(u_2)$$ **Case 4.** It remains to compare the bounds in [\[convexc=0\]](#convexc=0){reference-type="eqref" reference="convexc=0"},[\[convexc=2\]](#convexc=2){reference-type="eqref" reference="convexc=2"},[\[3cconclusion\]](#3cconclusion){reference-type="eqref" reference="3cconclusion"} and [\[3dd\]](#3dd){reference-type="eqref" reference="3dd"}. The inequality $$\cfrac{\alpha^2(2+\alpha^2)}{144} \leq \cfrac{\alpha^2}{36}, \,\,\,\,\, \alpha \in(0,1]$$ is trivial. Since the function is $\Phi$ is decreasing in $(u_2,4)$ and $\Phi(u_2)=\Psi(u_2)$, the inequality $$\cfrac{\alpha^2(2+\alpha^2)}{144}\leq \cfrac{\alpha^2(17+18\alpha+13\alpha^2)}{144(4+6\alpha+\alpha^2)}$$ is true for $1/3 < \alpha \leq 1$. Finally the inequality $$\cfrac{\alpha^2}{36}\leq \cfrac{\alpha^2(17+18\alpha+13\alpha^2)}{144(4+6\alpha+\alpha^2)}, \,\,\,\,\, \alpha \in (1/3,1]$$ is equivalent to $$9\alpha^2-6\alpha+1\geq0$$ which is evidently true for $1/3 < \alpha \leq 1$.\ Thus summarizing the results in cases 1-4, we see that [\[thm1\]](#thm1){reference-type="eqref" reference="thm1"} holds.\ We now proceed to prove the equality part. When $\alpha \in (0,1/3]$, equality holds for the function $f \in \mathcal{A}$ given by [\[3.1.1\]](#3.1.1){reference-type="eqref" reference="3.1.1"} with $p$ given by $$p(z)=\cfrac{1+z}{1-z}, \,\,\,\,\,\,\,\, z \in \mathbb{D}.$$ When $\alpha \in (1/3,1]$, set $$\tau:=\sqrt{t_0}=\sqrt{\cfrac{4(3\alpha-1)}{(4+6\alpha+\alpha^2)}}.$$ Since $\tau \leq 2$, the function $$\tilde{p}(z)=\cfrac{1-\tau z+z^2}{1-z^2}, \,\,\,\,\, z \in \mathbb{D},$$ belongs to $\mathcal{P}$. Thus the function $f$ given by [\[3.1.1\]](#3.1.1){reference-type="eqref" reference="3.1.1"}, where $p$ replaced by $\tilde{p}$ belongs to $\mathcal{K}_\alpha$ and is extremal function for the second inequality in [\[thm2\]](#thm2){reference-type="eqref" reference="thm2"} where . This completes the proof. ◻ For $\alpha=1$, we get the following result[@amal]. **Corollary 5**. *If $f \in \mathcal{K}$, then $$|H_{2,1}(F_{f}/2)|\leq \cfrac{1}{33}.$$* We next find the sharp bound for the second Hankel determinant of inverse logarithmic coefficient of functions in $\mathcal{S}^*_\alpha$. **Theorem 6**. *Let $f\in\mathcal{S}^*_{\alpha}$ given by [\[f\]](#f){reference-type="eqref" reference="f"} then $$\label{thm2} |H_{2,1}(F_{f^{-1}}/2)|\leq \begin{cases} \cfrac{\alpha^2}{4}, & 0< \alpha < 1/5,\\[4mm] \cfrac{\alpha^2(2+5\alpha+15\alpha^2)}{7+30\alpha+35\alpha^2}, & 1/5 \leq \alpha \leq \alpha', \\[4mm] \cfrac{\alpha^2}{36}(4+35\alpha^2), & \alpha'<\alpha \leq 1. \end{cases}$$ where $\alpha'=0.390595...$ is the unique root in $(0,1)$ of the equation $44+60\alpha+155\alpha^2-1050\alpha^3-1225\alpha^4 =0$. The inequality [\[thm2\]](#thm2){reference-type="eqref" reference="thm2"} is sharp.* *Proof.* Fix $\alpha \in (0,1]$ and let $f\in \mathcal{S}^*_{\alpha}$ be of the form [\[f\]](#f){reference-type="eqref" reference="f"}. Then by [\[stardef\]](#stardef){reference-type="eqref" reference="stardef"}, $$\label{3.3.1} \cfrac{zf'(z)}{f(z)}=(p(z))^\alpha$$ for some $p \in \mathcal{P}$ of the form [\[p\]](#p){reference-type="eqref" reference="p"}. By comparing the coefficients on both sides of [\[3.3.1\]](#3.3.1){reference-type="eqref" reference="3.3.1"}, we obtain $$\label{3.3.2} \begin{aligned} & a_2=\alpha c_1, \\[2mm] & a_3=\frac{\alpha}{4}(2c_2+(3\alpha-1)c_1^2), \\[2mm] & a_4=\frac{\alpha}{36}\left(12c_3+6(5\alpha-2)c_1c_2+(17\alpha^2-15\alpha+4)c_1^3 \right). \end{aligned}$$ Hence by [\[hankel\]](#hankel){reference-type="eqref" reference="hankel"}, we have $$H_{2,1}(F_{f^{-1}}/2)=\cfrac{\alpha^2}{576}\left((7+30\alpha+35\alpha^2) c1^4-(1+5\alpha) c_1^2c_2-36c_2^2+48 c_1c_3\right).$$ Since the class $\mathcal{S}^*_{\alpha}$ is invariant under rotation, without loss of generality, we can assume that $c_1 = c,$ where $0 \leq c \leq 2$. Therefore, by Lemma [\[caratheodary\]](#caratheodary){reference-type="ref" reference="caratheodary"}, for some $c \in [0,2]$ and $z,w \in \mathbb{D}$ we have $$\label{main} \begin{aligned} H_{2,1}(F_{f^{-1}}/2) =&\frac{\alpha^2}{576}\left[(4+35\alpha^2)c^4-30\alpha(4-c^2)c^2 z-3(4-c^2)(12+c^2)z^2\right.\\[2mm] &\left. +24c(4-c^2)(1-|z|^2)w\right]. \end{aligned}$$ Now we have the following cases on $c$.\ **Case 1:** Suppose that $c=0$. Then by [\[main\]](#main){reference-type="eqref" reference="main"}, we obtain $$\label{starlikec=0} |H_{2,1}(F_{f^{-1}}/2)|=\frac{1}{4}|z^2|\alpha^2\leq\frac{\alpha^2}{4}.$$ **Case 2:** Suppose that $c=2$. Then by [\[main\]](#main){reference-type="eqref" reference="main"}, we obtain $$\label{convexc=2} |H_{2,1}(F_{f^{-1}}/2)|=\frac{\alpha^2}{36}(4+35\alpha^2).$$ **Case 3:** Suppose that $c \in (0,2)$. Applying the triangle inequality in [\[main\]](#main){reference-type="eqref" reference="main"} and by using the fact that $|w| \leq 1$, we obtain $$\label{case3main} \begin{aligned} H_{2,1}(F_{f^{-1}}/2) &=\frac{\alpha^2}{576}\left[\left|(4+35\alpha^2)c^4-30\alpha(4-c^2)c^2 z-3(4-c^2)(12+c^2)z^2\right|\right.\\[2mm] &\left. \,\,\,\, + 24c(4-c^2)(1-|z|^2)w\right] \\[2mm] &\leq \frac{\alpha^2}{24}c(4-c^2)\left[\left| A+B z+C z^2 \right|+1-|z^2|\right], \end{aligned}$$ where $$A:=\frac{c^3 (4 + 35 \alpha^2)}{24(4-c^2)}, \quad B:=-\frac{5}{4} \alpha c,\quad C:=-\frac{12+c^2}{8c}.$$ Since $AC < 0$, so we can apply case (ii) of Lemma [Lemma 3](#y(a,b,c)){reference-type="ref" reference="y(a,b,c)"}.\ **3(a).** Note that the inequality $$-4 A C\left(\frac{1}{C^{2}}-1\right) \leq B^{2}$$ is equivalent to $$\cfrac{c^2(36+540\alpha^2+c^2(10\alpha^2-1))}{12+c^2}\geq 0$$ which evidently holds for $c \in (0,2).$ Moreover, the inequality $|B|<2(1-|C|)$ is equivalent to $12 + c^2 (1 + 5 \alpha)-8c<0$ which is not true for $c \in(0,2)$.\ **3(b).** Since $$-4 A C\left(C^{-2}-1\right)=\cfrac{c^2(-36+c^2)(4+35\alpha^2)}{48(12+c^2)} < 0,$$ and $$4(1+|C|)^2=\frac{(12-8c+c^2)^2}{16c^2}>0.$$ So we get $$\min \left\{4(1+|C|)^{2},-4 A C\left(\frac{1}{C^{2}}-1\right)\right\}=-4 A C\left(\frac{1}{C^{2}}-1\right),$$ and from $3(a),$ we know that $$-4 A C\left(\frac{1}{C^{2}}-1\right) \leq B^{2}.$$ Therefore, the inequality $B^{2} < \min \left\{4(1+|C|)^{2},-4 A C\left(\frac{1}{C^{2}}-1\right)\right\}$ does not holds for $0<c<2$.\ **3(c).** Note that the inequality $$|C|(|B|+4|A|) \leq |A B|$$ is equivalent to $$720\alpha+24c^2(4-5\alpha+35\alpha^2)-c^4(-8+35\alpha-70\alpha^2+175\alpha^3) \leq 0.$$ Consider the function $\phi_1:(0,4)\rightarrow \mathbb{R}$ defined by $$\phi_1(x)=720\alpha+24x(4-5\alpha+35\alpha^2)-x^2(-8+35\alpha-70\alpha^2+175\alpha^3)$$ Clearly $\phi_1(0)=720 \alpha > 0$ and $\phi_1(4)=16(32-20\alpha+280\alpha^2-175\alpha^3>0$. It can be shown that $\phi_1'(x)>0$. Hence $\phi_1$ is increasing and consequently we concluded that $\phi_1(x)>0$. Therefore the inequality $|C|(|B|+4|A|) \leq |A B|$ is false. **3(d).** Note that the inequality $$\label{eq1} \begin{aligned} &|A B|-|C|(|B|-4|A|)\\[2mm] &= \cfrac{5c^4\alpha (4+35\alpha^2)}{96(4-c^2)}-\cfrac{12+c^2}{8c}\left(\cfrac{5 c \alpha}{4}-\cfrac{c^3(4+35\alpha^2}{6(4-c^2}\right) \end{aligned}$$ is equivalent to $$\label{eq2} \delta(c^2)\geq 0,$$ where $$\delta(t)=720 \alpha-24(4+5\alpha+35\alpha^2)t-(8+35\alpha+70\alpha^2+175\alpha^3)t^2, \,\,\, t \in (0,4)$$ We see that for $\alpha \in (0,1]$ $$4+5\alpha+35\alpha^2 >0, \,\, 8+35 \alpha+70\alpha^2+175\alpha^3 >0,$$ and the discriminant $\Delta:=2304(4+20 \alpha+120\alpha^2+175\alpha^3+525\alpha^4]>0$ for $\alpha \in (0,1].$\ Thus we consider $$t_{1,2}=-\cfrac{12\left(4+5\alpha+35\alpha^2\pm 2\sqrt{4+20\alpha+120\alpha^2+175\alpha^3+525\alpha^4}\right)}{8+35\alpha+70\alpha^2+175\alpha^3}$$ It is clear that $t_1 <0$ and so it remains to check if $0<t_2<4$. The inequality $t_2>0$ is equivalent to $$4+5\alpha+35\alpha^2-2\sqrt{4+20\alpha+120\alpha^2+175\alpha^3+525\alpha^4}<0$$ which is true for $\alpha \in (0,1]$. Further the inequality $t_2 <4$ can be written as $$6\sqrt{4+20\alpha+120\alpha^2+175\alpha^3+525\alpha^4}<5(4+10\alpha+35\alpha^2+35\alpha^3),$$ which is also true for $\alpha \in (0,1]$. There for [\[eq2\]](#eq2){reference-type="eqref" reference="eq2"} and so [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"} is valid for $$0<c\leq c^*:=\sqrt{t_2}.$$ Then by Lemma 2 and [\[eq1\]](#eq1){reference-type="eqref" reference="eq1"}, for $0<c\leq c^*$, we have $$\begin{aligned} |\Gamma_1\Gamma_3-\Gamma_2^2| & \leq \cfrac{\alpha^2}{24}c(4-c^2)\left(-|A|+|B|+|C|\right)\\[2mm] &=\cfrac{\alpha^2}{576}\left(144-24(1-5\alpha)c^2-(70+30\alpha+35\alpha^2)c^4\right)\\[2mm] &=\cfrac{\alpha^2}{576}\phi_2(c^2) \end{aligned}$$ where $$\phi_2(s)=144-24(1-5\alpha)s-(70+30\alpha+35\alpha^2)s^2, \,\,\,\, 0 < s \leq t_1.$$\ we note that $\phi'(s)=0$ only when $s=s_0:=12(5\alpha-1)/(7+30\alpha+35\alpha^2)$ and it is easy to see that $s_0>0$ only if $\alpha>1/5$. Further $s_0 \leq t_2$ is true if $\beta(\alpha)<0$,\ where $$\begin{aligned} \beta(x)=&-96-1060x-7040x^2-29975x^3-77525x^4\\ &-124250x^5-82250x^6+153125x^7+459375x^8, \,\,\,\, 0 < x \leq 1 \end{aligned}$$ So we get $0<s_0<t_2$ if and only if $1/5<\alpha\leq \alpha_0$, where $\alpha_0$ is the unique positive real root of the polynomial $\beta(x)$ lies between $0$ and $1$. Since the leading coefficient of $\phi(x)$ is negative, for $1/5 < \alpha < \alpha_0$, we get $$\label{alpha1} \begin{aligned} |\Gamma_1\Gamma_3-\Gamma_2^2| &\leq \cfrac{\alpha^2}{576}\phi_2(s_0)\\[2mm] &=\alpha^2\left(\cfrac{2+5\alpha+15\alpha^2}{7+30\alpha+35\alpha^2}\right). \end{aligned}$$ For $0<\alpha \leq 1/5$, it is clear that $\phi_2$ is decreasing, so we get $$\label{alpha2} \begin{aligned} |\Gamma_1\Gamma_3-\Gamma_2^2| &\leq \cfrac{\alpha^2}{576}\,\,\phi_2(0)\\[2mm] &=\cfrac{\alpha^2}{4}. \end{aligned}$$ For $\alpha_0<\alpha \leq 1$, it is clear that $\phi_2$ is increasing, so we get $$\label{alpha3} \begin{aligned} |\Gamma_1\Gamma_3-\Gamma_2^2| &\leq \cfrac{\alpha^2}{576} \,\, \phi_2(t_1) \end{aligned}$$ **3(e).** Next suppose that $c \in (c^*,2)$. Then by Lemma [Lemma 3](#y(a,b,c)){reference-type="ref" reference="y(a,b,c)"}, we have $$\begin{aligned} |\Gamma_1\Gamma_3-\Gamma_2^2| &\leq \frac{\alpha^2}{576}c(4-c^2)(|C|+|A|) \sqrt{1-\frac{B^{2}}{4 A C}}\\ &= \cfrac{\alpha^2}{288\sqrt{4+35\alpha^2}}\left(144-24c^2+(1+35\alpha^2)c^4\right)\sqrt{\cfrac{12+180\alpha^2+(1-10\alpha^2)c^2}{12+c^2}}\\ &=\cfrac{\alpha^2}{288\sqrt{4+35\alpha^2}} \,\,\,h_1(c^2)\sqrt{h_2(c^2)} \end{aligned}$$ where $h_1$ and $h_2$ are defined by $$h_1(x)=144-24x+(1+35\alpha^2)x^2$$ and $$h_2(x)=\cfrac{12+180\alpha^2+(1-10\alpha^2)x}{12+x}.$$ Now we consider the function $F(x)=h_1(x)\sqrt{h_2(x)}$ on $(t_1,4)$ and we show that $F$ is convex function. It is enough to show that $F''(x)\geq 0.$\ Since $$\begin{aligned} F''(x)(h_2(x))^{3/2}&=h_1''(x)(h_2(x))^2+h_1'(x)h_2(x)h_2'(x)-\frac{1}{4}h_1(x)(h_2'(x))^2+\frac{1}{2}h_1(x)h_2(x)h_2''(x)\\ &=\cfrac{1}{(12+x)^4}\left[2592(8+820\alpha^2+14075\alpha^4+63000\alpha^6) \right.\\ &\left.\,\,\,\,-432(-16-890\alpha^2-4525\alpha^4+31500\alpha^6)x \right.\\ &\left.\,\,\,\,- 54(-16-540\alpha^2+1475\alpha^4+31500\alpha^6)x^2\right.\\ &\left.-\,\,\,\, 6(8+195\alpha^2-2925\alpha^4+1750\alpha^6)x^3+ (1-10\alpha^2)^2(1+35\alpha^2)x^4\right]\\ &=\cfrac{1}{(12+x)^2}G(x) \end{aligned}$$ It is easy to see that $h_2$ is a positive decreasing function in $(t_2,4)$. We show that our assertion is true by proving that $G(x)\geq 0$ for $x \in (t_2, 4).$\ Let $x \in (t_2,4)$ is fixed and $$H(\alpha)=m_0+m_1\alpha^2+m_2\alpha^4+m_3\alpha^6$$ where $$\begin{aligned} &m_0:=20736+6912x+864x^2+48x^3+x^4, \\[2mm] &m_1:=2125440+384480x+29160x^2+1170x^3+15x^4, \\[2mm] &m_2:=36482400+1954800x-79650x^2-17550x^3-600x^4, \\[2mm] &m_3:=163296000-13608000x-146750x^2+10500x^3+3500x^4.\\ \end{aligned}$$ Clearly $m_0>0$ and $m_1>0$. For $x \in (x_1,4)$, we have $$\begin{aligned} m_2&=36482400+1954800x-79650x^2-17550x^3-600x^4, \\[2mm] &> 1200(28276+1629x)>0 \end{aligned}$$ Similarly, we have $$\begin{aligned} m_3&=163296000-13608000x-146750x^2+10500x^3+3500x^4.\\[2mm] &>3500 (24408+3x^3+x^4)>0 \end{aligned}$$ So we get $H(\alpha) >0$ for $\alpha \in (0,1]$ and $x \in (t_2,4)$, which therefore shows that $G(x) \geq 0.$ So we can conclude that $$\begin{aligned} F(x)&\leq \,\, \max \{F(t_2),F(4)\} \\[2mm] &\leq \begin{cases} F(t_2),& 0< \alpha \leq \alpha^*\\[2mm] F(4), & \alpha^* \leq \alpha \leq 1 \end{cases} \end{aligned}$$ So we get, for $c \in (c^*,2)$ $$\label{alpha4} |\Gamma_1\Gamma_3-\Gamma_2^2| \leq \begin{cases} \cfrac{\alpha^2}{288\sqrt{4+35\alpha^2}}\,\,F((c^*)^2), & 0< \alpha \leq \alpha^*\\[4mm] \cfrac{\alpha^2}{36}(4+35\alpha^2), & \alpha^* \leq \alpha \leq 1 \end{cases}$$ **Case 4.** Now we compare the bounds in [\[alpha1\]](#alpha1){reference-type="eqref" reference="alpha1"},[\[alpha2\]](#alpha2){reference-type="eqref" reference="alpha2"},[\[alpha3\]](#alpha3){reference-type="eqref" reference="alpha3"} and [\[alpha4\]](#alpha4){reference-type="eqref" reference="alpha4"}. The inequality $$\cfrac{\alpha^2}{36}(4+35\alpha^2) \leq \cfrac{\alpha^2}{4}$$ is true for $0<\alpha\leq 1/\sqrt{7}$. So we can conclude that for $0<\alpha<1/5$, $$|H_{2,1}(F_{f^{-1}}/2)|\leq \cfrac{\alpha^2}{4}.$$ The inequality $$\cfrac{\alpha^2}{4}\leq \cfrac{\alpha^2(2+5\alpha+15\alpha^2)}{7+30\alpha+35\alpha^2}$$ is equivalent to the evidently true inequality $(1-5\alpha)^2\geq 0.$ A tedious long calculation shows that the following inequality $$\cfrac{\alpha^2}{288\sqrt{4+35\alpha^2}}\,\,F((c^*)^2) \leq \cfrac{\alpha^2(2+5\alpha+15\alpha^2)}{7+30\alpha+35\alpha^2},$$ is true for $0<\alpha\leq 1$. The inequality $$\cfrac{\alpha^2}{36}(4+35\alpha^2) \leq \cfrac{\alpha^2(2+5\alpha+15\alpha^2)}{7+30\alpha+35\alpha^2}$$ is equivalent to $$44+60\alpha+155\alpha^2-1050\alpha^3-1225\alpha^4 \geq 0,$$ which is true for $0\leq \alpha \leq \alpha'$, where $\alpha'$ is the unique positive real root of the polynomial $44+60\alpha+155\alpha^2-1050\alpha^3-1225\alpha^4$ lies between $0$ and $1$. With further long computations, we can show that the inequality $$\cfrac{\alpha^2}{576} \,\, \phi_2(t_1) \leq \cfrac{\alpha^2}{36}(4+35\alpha^2)$$ is true for $2/3 \leq \alpha \leq 1$. By summarizing the above cases we get [\[thm2\]](#thm2){reference-type="eqref" reference="thm2"} holds.\ In order to show that the inequalities in [\[thm2\]](#thm2){reference-type="eqref" reference="thm2"} are sharp. For $0< \alpha <1/5$, consider the function $$p(z)=\cfrac{1+z^2}{1-z^2}, \,\,\,\,\,\, z \in \mathbb{D}.$$ It is obvious that $p$ is in $\mathcal{P}$ with $c_1=c_3=0$ and $c_2=2$. The corresponding function $f \in \mathcal{S}^*_\alpha$ is described by [\[3.3.1\]](#3.3.1){reference-type="eqref" reference="3.3.1"}. Hence by [\[3.3.2\]](#3.3.2){reference-type="eqref" reference="3.3.2"}, it follows that $a_2=a_4=0$ and $a_3=\alpha$. From [\[hankel\]](#hankel){reference-type="eqref" reference="hankel"}, we obtain $$|H_{2,1}(F_{f^{-1}}/2)|=\cfrac{\alpha^2}{4}.$$ For $1/5 \leq \alpha \leq \alpha'$, consider the function $f \in \mathcal{A}$ given by [\[3.3.1\]](#3.3.1){reference-type="eqref" reference="3.3.1"} with $p$ given by $$p(z)=\cfrac{1-\zeta z+z^2}{1-z^2}, \,\,\,\,\,\,\, z \in \mathbb{D},$$ where $\zeta:=\sqrt{s_0}=\sqrt{12(5\alpha-1)/(7+30\alpha+35\alpha^2)}.$ By simple computation we can show that $$|H_{2,1}(F_{f^{-1}}/2)|= \cfrac{\alpha^2(2+5\alpha+15\alpha^2)}{7+30\alpha+35\alpha^2}.$$ For the last case, $\alpha' <\alpha \leq 1$ equality holds for the function $f \in \mathcal{A}$ of the form [\[3.3.1\]](#3.3.1){reference-type="eqref" reference="3.3.1"} with $p(z)=1+z/1-z$. This completes the proof. ◻ For $\alpha=1$, we get the estimate for the class $\mathcal{S}^*$ of starlike functions [@vibhuthi2]. **Corollary 7**. *If $f \in \mathcal{S}^*$, then $$|H_{2,1}(F_{f^{-1}}/2)|\leq \cfrac{13}{12}.$$* **Acknowledgment:** The research of the first named author is supported by SERB-CRG, Govt. of India and the second named author's research work is supported by UGC-SRF. 99 M. F. Ali and V. Allu, On logarithmic coefficients of some close-to-convex functions, *Proc. Amer. Math. Soc.* **146** (2017), 1131--1142. M. F. Ali and V. Allu, Logarithmic coefficients of some close-to-convex functions, *Bull. Aust. Math. Soc.* **95** (2017), 228-- 237. V. Allu, A. Lecko, and D. K. Thomas, Hankel, Toeplitz and Hermitian-Toeplitz Determinants for Ozaki Close-to-convex Functions, *Mediterr. J. Math.* **19** (2022), DOI:10.1007/s00009-021-01934-y. V. Allu,V. Arora and A. 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arxiv_math

--- abstract: | Motivated by the problem of detecting a change in the evolution of a network, we consider the preferential attachment random graph model with a *time-dependent* attachment function. Our goal is to detect whether the attachment mechanism changed over time, based on a single snapshot of the network and without directly observable information about the dynamics. We cast this question as a hypothesis testing problem, where the null hypothesis is a preferential attachment model with a constant affine attachment parameter $\delta_0$, and the alternative hypothesis is a preferential attachment model where the affine attachment parameter changes from $\delta_0$ to $\delta_1$ at an unknown changepoint time $\tau_n$. For our analysis we focus on the regime where $\delta_0$ and $\delta_1$ are fixed, and the changepoint occurs close to the observation time of the network (i.e., $\tau_n = n - c n^\gamma$ with $c>0$ and $\gamma \in (0, 1)$). This corresponds to the relevant scenario where we aim to detect the changepoint shortly after it has happened. We present two tests based on the number of vertices with minimal degree, and show that these are asymptotically powerful when $\tfrac{1}{2} < \gamma < 1$. We conjecture that there is no powerful test based on the final network snapshot when $\gamma<\tfrac{1}{2}$. The first test we propose requires knowledge of $\delta_0$. The second test is significantly more involved, and does not require the knowledge of $\delta_0$ while still achieving the same performance guarantees. Furthermore, we prove that the test statistics for both tests are asymptotically normal, allowing for accurate calibration of the tests. This is demonstrated by numerical experiments, that also illustrate the finite sample test properties. address: - Department of Mathematics and Computer Science, University of Florence - Department of Mathematics and Computer Science, Eindhoven University of Technology author: -   -   -   -   bibliography: - library.bib title: | Detecting a late changepoint\ in the preferential attachment model --- , # Introduction {#sec:changepoint_detect_introduction} One of the most celebrated successes of complex network theory has been the recognition that simple *dynamical* random graph models with local connection rules are able to successfully explain important macroscopic features observed in real-world networks. The preferential attachment model and its generalizations are perhaps the most successful of such models. Barabási and Albert [@Barabasi1999] proposed this model to explain the occurrence of power-law degree sequences, which are often observed in real-world networks such as the world wide web [@Broder2000; @Adamic2000] or internet [@Faloutsos1999], biological networks [@Jeong2000; @Farkas2003; @Middendorf2005], collaboration networks of movie actors [@Albert2000; @Gao2011], and citation networks [@Redner1998; @Newman2001; @Barabasi2002; @Wang2008]. Furthermore, the typical distance between vertices in the preferential attachment model is small [@DomHofHoo10] (see also [@VanderHofstad2020 Chapter 8] and references therein). This is called the *small-world* phenomenon [@Watts1998; @Watts1999]. The preferential attachment model considers the entire evolution of a network by adding vertices one by one using a simple *preferential attachment* rule. Informally, as new vertices are added to the graph, they are more likely to attach to vertices that already have a large degree, hence further increasing the degree of these vertices. This formalism essentially creates a paradigm where "the rich get richer", which is often invoked to explain the wide-spread inequality in socio-economic contexts [@perk:2014]. Accordingly, the degree of the oldest vertices grows as new vertices attach to the graph. On the other hand, the degree of the last few vertices to join is typically quite small. Since its introduction in [@Barabasi1999], the preferential attachment model has received a tremendous amount of attention thanks to its early explanatory successes. The structural properties of the model are investigated formally in [@Bollobas2001a; @Bollobas2004], see also [@VanderHofstad2017; @VanderHofstad2020] for a detailed overview on this model and many of its properties. In our work we are interested in situations where the growth dynamics of the network do not remain constant over time, but have a change at some point. This captures a situation where a major event could cause a change in the subsequent evolution of the network. To model this, we consider a time-inhomogeneous affine preferential attachment model, where a new vertex $v_t$ that enters the graph at time $1 \leq t \leq n$ connects to a pre-existing vertex with degree $k$ with probability proportional to $f(k) = k + \delta(t)$. We consider the hypothesis testing problem where $\delta(t) = \delta_0$ remains constant under the null hypothesis, whereas under the alternative the affine attachment parameter $\delta(t)$ changes from $\delta_0$ to $\delta_1$ at an unknown moment $\tau_n$, called a *changepoint*. For our work we are particularly interested in scenarios where the change occurs very late, and affects only a very small part of the graph. Specifically, in the regime we are interested in, the changepoint has the form $\tau_n = n - c n^\gamma$ with $c > 0$ and $\gamma \in (0, 1)$, as explained in Section [2](#sec:model){reference-type="ref" reference="sec:model"}. From a practical standpoint this is relevant when one wants to detect the change as quickly as possible. ## Related work Our work nicely complements earlier results [@Bhamidi2018; @Banerjee2018] that focus on the detection of a changepoint in the setting of preferential attachment trees, where every vertex that enters the graph connects to $m = 1$ other vertices. There are also some differences. First, our results consider the more general case of preferential attachment graphs, where vertices may enter the graph with $m \geq 1$ edges. The other difference is that we focus on a *late* changepoint $\tau_n = n - c n^\gamma$, whereas [@Bhamidi2018; @Banerjee2018] focus on a changepoint that happens at a linear time $\ensuremath{\mathrm{O}}(n)$ or even $\ensuremath{\mathrm{o}}(n)$. Thus, in our setting a much smaller number of vertices enter the graph after the changepoint, making it harder to detect. The authors of [@Cirkovic2022] propose a likelihood-ratio testing procedure to detect a changepoint in a preferential attachment tree and the associated changepoint estimator. Crucially the methods in [@Cirkovic2022] rely on the knowledge of the entire network evolution. This is not the case for our test, which only requires a snapshot of the network at the final time. The authors of [@Cirkovic2022] extend their test to detect multiple changepoints by applying two general techniques (namely, Screening and Ranking, and Binary Segmentation) to decompose the multiple-changepoints problem into a sequence of single changepoint problems. This work, however, is still in the scenario where the changepoint occurs at a linear time, in stark contrast with the regime we investigate. Although different from this work, there has been much interest in understanding and detecting the effect of an initial seed graph on the evolution of the preferential attachment tree [@Bubeck2015; @Curien2015; @Bubeck2017; @Bubeck2017a; @Marchand2020]. Here one starts with a given initial graph at time $t = 1$ and then grows the remaining tree according to a preferential (or uniform) attachment. The goal is to estimate the initial seed graph based on an observation of the fully developed graph at a much later time. Finally, changepoint detection and related inference questions have also received much attention in the setting of dynamic stochastic block models [@Wang2014; @Wang2018; @Pensky2019; @Zhao2019; @Bhattacharjee2020]. In those works the aim is primarily to understand the evolution of the network's community structure. # Model {#sec:model} We formalize the problem of detecting a changepoint in a dynamical network as a hypothesis testing problem on random graphs. We first explain the model that we use in general, and then define concrete versions of this model for the null and alternative hypothesis. This model has parameters $m\in\ensuremath{\mathbb{N}}$ and $\delta:\ensuremath{\mathbb{N}}\to(-m,\infty)$ and produces a sequence of undirected graphs without loops. Let $G_n = (V_n, E_n)$ be an undirected graph, where $V_n = \{v_0, \ldots, v_n\}$ denotes the vertex set and $E_n \subseteq \{(i,j) : i, j \in V_n\}$ denotes a random set of edges. Note that $G_n$ has $n+1$ vertices. For $v\in V_n$ let $D_{v}(G_n)$ denote the degree of vertex $v$ in the graph $G_n$. There exist various versions of the preferential attachment model, each following slightly different conventions for adding new vertices. Here we consider the following model: the first graph $G_1$, also called the seed graph, consists of two vertices $v_0$ and $v_1$ connected by $m$ edges. For $t>1$, the graph $G_t$ is constructed by taking $G_{t-1}$ and adding one extra vertex $v_t$, that is connected to the vertices in $G_t$ by exactly $m$ edges. In the model we consider this process is better described by introducing a number of intermediate steps, described by graphs $G_{t,0}, G_{t,1}, \ldots G_{t,m}$, with vertex set $V_t=\{v_0,\ldots,v_t\}$. Begin by defining $G_{t,0}$ to be identical to $G_{t-1}$ together with an isolated vertex $v_t$. The graph $G_{t,1}$ is obtained by adding an edge between $v_t$ and one of the vertices in $V_{t-1}$ with probability proportional to $D_v(G_{t,0})+\delta(t)$. In general, for $i\in[m]$, we proceed by sampling vertex $v_{t,i}\in\{v_0,\ldots,v_{t-1}\}$ with conditional probability $$\label{eq:attachment_function_general} \ensuremath{\mathbb{P}}\left(v_{t,i} = v\mid G_{t,i-1}\right)=\frac{D_{v}(G_{t,i-1})+\delta(t)}{\sum_{j=0}^{t-1} D_{j}(G_{t,i-1})+\delta(t)}\ ,$$ and constructing $G_{t,i}$ by adding the edge $\{v_{t,i},v_t\}$ to $G_{t,i-1}$. Finally, define $G_t=G_{t,m}$. Note that the degree of $v_t$ in $G_t$ is exactly $m$. The above model is rather general, as it allows for quite a bit of flexibility in terms of the function $\delta(t)$, as the only requirement is that $\delta(t)>-m$ to ensure that [\[eq:attachment_function_general\]](#eq:attachment_function_general){reference-type="eqref" reference="eq:attachment_function_general"} is indeed a valid probability. A classical choice is to take $\delta(t)$ as a constant, and the properties of the ensuing graphs are well studied (e.g., see [@VanderHofstad2017; @VanderHofstad2020] and the references therein). However, we are interested in knowing when it is possible to distinguish graphs generated by a model where $\delta(t)$ is constant versus graphs generated by a model where $\delta(t)$ is a step function. The latter models a preferential attachment evolution, where at some point the characteristics of the attachment process change. Consider the above model, and let $G_n$ denote the "last" graph obtained. This is our only observation, i.e., we do not have access to the sequence $\{G_1,\ldots,G_{n-1}\}$. In particular, the order of the vertices is unknown to us. Clearly the distribution of this random graph is parameterized by $m$ and $\delta$. Since $G_n$ has exactly $n+1$ vertices and $nm$ edges, we have knowledge of $m$ (so this is not an unknown parameter). Therefore the only unknown parameter is the function $\delta$. Our goal is to determine when one can find evidence in the final graph that the growth dynamics of the network has changed at some point. This can be rather naturally formulated as an hypothesis testing problem. Namely, we would like to conduct the following binary hypothesis test: under the null hypothesis (denoted by $H_0$) we assume $\delta(t)=\delta_0>-m$ for all $t\in\ensuremath{\mathbb{N}}$. Under the alternative hypothesis (denoted by $H_1$) we assume $\delta$ is a step function, namely $$\delta(t) = \mathds 1\{t \leq \tau_n\}\delta_0 + \mathds 1\{t > \tau_n\}\delta_1\ ,$$ for some $\delta_0\neq\delta_1$ with $\delta_0,\delta_1>-m$, and $\tau_n\in\ensuremath{\mathbb{N}}$ with $\tau_n \leq n$. Our main research goal is to determine when it is possible to distinguish the two hypotheses, based solely on $G_n$ (where we do not know the order in which the various vertices have arrived). For our first result we consider $\delta_0$ to be known, but we then relax this assumption and devise a test that does not require this knowledge while retaining the same asymptotic power characterization. In both cases we use the parameterization $\tau_n=n-cn^\gamma$, where $c>0$ and $\gamma\in(0,1)$, and obviously $c$ and $\gamma$ are unknown. Note that the alternative model does coincide with the null model when either $\delta_1=\delta_0$ or $\tau_n=n$. Furthermore, since $\delta$ is a step-function, the attachment rule in [\[eq:attachment_function_general\]](#eq:attachment_function_general){reference-type="eqref" reference="eq:attachment_function_general"} can be further simplified to get, for $v\in\{v_0,\ldots,v_{t-1}\}$, $$\label{eq:attachment_function_alt} \ensuremath{\mathbb{P}}\left(v_{t,i} = v\mid G_{t,i-1}\right)=\left\{\begin{array}{ll} \frac{D_{v}(G_{t,i-1})+\delta_0}{2(t-1)m + t\delta_0 + (i-1)} & \quad\text{if } t\leq \tau_n,\\\\ \frac{D_{v}(G_{t,i-1})+\delta_1}{2(t-1)m + t\delta_1 + (i-1)} & \quad\text{if } t>\tau_n. \end{array}\right. \ $$ ## Assumptions and notation Throughout this paper, when limits are unspecified, they are taken as the graph size $n \to \infty$. We recall that we consider the parameterization $\tau_n=n-cn^\gamma$. All the parameters $m$, $\delta_0$, $\delta_1$, $c$, and $\gamma$ are assumed to remain constant as a function of $n$. We use the subscripts $0$ and $1$ in the expectation and probability operators to indicate whether we are considering the null or alternative hypothesis. Finally, we also use standard asymptotic notation: $a_n = \ensuremath{\mathrm{O}}(b_n)$ when $| a_n / b_n |$ is bounded, $a_n = \ensuremath{\mathrm{\Omega}}(b_n)$ when $b_n = \ensuremath{\mathrm{O}}(a_n)$, $a_n = \ensuremath{\mathrm{o}}(b_n)$ when $a_n / b_n \to 0$, $a_n=\omega(b_n)$ when $b_n=o(a_n)$, and $a_n \asymp b_n$ when $a_n = (1 + \ensuremath{\mathrm{o}}(1)) b_n$. Also, we use the probabilistic versions of these: $a_n = \ensuremath{\ensuremath{\mathrm{O}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(b_n)$ when $| a_n / b_n |$ is stochastically bounded, $a_n = \ensuremath{\ensuremath{\mathrm{\Omega}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(b_n)$ when $b_n = \ensuremath{\ensuremath{\mathrm{O}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(a_n)$, and $a_n = \ensuremath{\ensuremath{\mathrm{o}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(b_n)$ when $a_n / b_n$ converges to $0$ in probability. # Minimal degree tests {#sec:minimal_degree_test} All the tests we consider use the information contained in low-degree vertices, in particular the number of vertices of minimal degree. Although this is a rather simple idea, the number of minimal degree vertices is substantially affected by the presence of a changepoint, even at late stages in the growth of the graph. ## Powerful test for known $\delta_0$ We begin with the assumption that $\delta_0$ is known. Although this might seem unrealistic, it provides a wealth of information about the properties of the test statistic we consider, and paves the way for more general results when $\delta_0$ is unknown. To define our test we first introduce some notation. To reduce the notational burden we identify the set of vertices $\{v_0,\ldots,v_n\}$ with $[n]\coloneqq \{0,1,\ldots,n\}$. Furthermore, let $D_v(t)\coloneqq D_v(G_t)$ denote the degree of vertex $v$ in graph $G_t$, and let $N_k(t)$ be the number of vertices of degree $k$ in the graph $G_t$, that is, $$N_k(t) \coloneqq \sum_{v\in[t]} \mathds 1\{D_v(t) = k\}\ .$$ Since each vertex is attached to at least $m$ other vertices, in our model we naturally have that $N_k(n) = 0$ for $k < m$, and $N_m(n)$ denotes the number of vertices with minimal degree. The latter quantity plays a crucial role in our test. It is well-known that in the classical preferential attachment model, which corresponds to our null model, the number of vertices of degree $k\geq m$ is highly concentrated [@Deijfen2007; @VanderHofstad2017]. In particular, $N_k(n)$ is well concentrated around $n p_k(\delta_0)$ where $p_k=p_k(\delta_0)$ satisfies the recursion $$\label{eqn:recursion_p_k} p_k=\frac{k-1+\delta_0}{2+\delta_0/m}p_{k-1}-\frac{k+\delta_0}{2+\delta_0/m}p_k\ ,$$ for $k>m$ with $$\label{eq:limiting_degree_distribution} p_m(\delta_0) = \frac{2 + \delta_0 / m}{m + \delta_0 + 2 + \delta_0 / m}\ .$$ This recursion is easily solved, giving rise to the following expression for $p_k(\delta_0)$: $$\label{eqn:p_k} p_k(\delta_0)\coloneqq(2+\delta_0/m)\frac{\Gamma(k+\delta_0)\Gamma(m+2+\delta_0+\delta_0/m)}{\Gamma(m+\delta_0)\Gamma(k+3+\delta_0+\delta_0/m)}\ .$$ Thus, $p_k(\delta_0)$ is the limiting degree distribution of the random graph $G_n$ under the null model. We are now able to introduce our test statistic, that simply compares the number of minimal degree vertices to its asymptotic expected value under the null model, as $$T(G_n)\coloneqq N_m(n) - n p_m(\delta_0)\ .$$ If the observed value of $T(G_n)$ is significantly different from zero, then we have evidence to reject the null model. This brings us to the first result, characterizing when such a test is powerful. Specifically we introduce a test based on this statistic that is guaranteed to have type-I error that is at most $\alpha$ (asymptotically), and that is asymptotically powerful provided $\gamma>\tfrac{1}{2}$. In other words, under the alternative hypothesis the type-II error converges to zero provided $\gamma>\tfrac{1}{2}$. When $\gamma<\tfrac{1}{2}$, the test is powerless, and when $\gamma=\tfrac{1}{2}$, the type-II error is bounded by a constant that depends on the specific model parameters. Although this result indicates when the test is powerful or powerless, it provides only a conservative upper bound on the type I error. It follows from [@Baldassarri2021] that this test statistic is asymptotically normal, and therefore we can calibrate this test to guarantee that the type I error is $(1+\ensuremath{\mathrm{o}}(1))\alpha$ as $n\to\infty$. We do this in Section [3.3](#sec:asymptotically_calibrated_tests){reference-type="ref" reference="sec:asymptotically_calibrated_tests"}. The proof of Theorem [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"} is given in Section [7](#sec:proof_known){reference-type="ref" reference="sec:proof_known"}. **Theorem 1** (Asymptotically powerful test: known $\delta_0$). *Consider the test that rejects the null hypothesis for large values of $T(G_n)$. Namely, define the test $$\label{eq:minimal_degree_test_known_delta} \psi(G_n)\coloneqq \mathds 1{\left\{|T(G_n)|\geq m \sqrt{8n \log(2 / \alpha)}\right\}}\ ,$$ where $\alpha\in(0,1)$. The type-I error of this test is asymptotically bounded by $\alpha$, i.e., $$\ensuremath{\mathbb{P}}_0\left(\psi(G_n)\neq 0\right)\leq (1 + \ensuremath{\mathrm{o}}(1)) \alpha\ .$$ Furthermore, the type-II error of this test satisfies $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{P}}_1\left(\psi(G_n)= 0\right)}\\ &\leq \left\{\begin{array}{ll} \ensuremath{\mathrm{o}}(1) & \text{ when } \gamma > \tfrac{1}{2} ,\\ (2 + \ensuremath{\mathrm{o}}(1)) \exp\left(-\left(\left(\frac{c |1 - p_m(\delta_0) / p_m(\delta_1)|}{m \sqrt{8}} - \sqrt{\log(2 / \alpha)}\right) \vee 0\right)^2\right) & \text{ when } \gamma = \tfrac{1}{2}, \end{array}\right.\ \end{aligned}$$ and $\ensuremath{\mathbb{P}}_1\left(\psi(G_n)= 0\right)\geq(1+\ensuremath{\mathrm{o}}(1))(1-\alpha)$ when $\gamma<\tfrac{1}{2}$.* The proof of this result is based on the following observations. Under the null model it is known that $\ensuremath{\mathbb{E}}_0(N_m(n))-np_m(\delta_0)=\ensuremath{\mathrm{O}}(1)$. Under the alternative model we can show that $$\ensuremath{\mathbb{E}}_1\left[N_m(n)\right] - np_m(\delta_0)= (1+\ensuremath{\mathrm{o}}(1)) \eta(\delta_0,\delta_1) n^\gamma\ ,$$ where $$\eta(\delta_0,\delta_1)\coloneqq c (1 - p_m(\delta_0) / p_m(\delta_1))\ .\label{eqn:shift_T}$$ Therefore there is a substantial difference in the expected values of the test statistic under the null and alternative models. Note that both $\ensuremath{\mathbb{E}}_1\left[N_m(n)\right]$ and $np_m(\delta_0)$ have the same order of magnitude $\ensuremath{\mathrm{O}}(n)$, so the above result characterizes the second-order behavior of $\ensuremath{\mathbb{E}}_1\left[N_m(n)\right]$ and thus is somewhat delicate. Besides characterizing the mean of $N_m(n)$, we must also characterize the fluctuations of $N_m(n)$ around it. These are of small order, and controlled by a rather standard application of Azuma-Hoeffding's inequality. Specifically $N_m-\ensuremath{\mathbb{E}}[N_m(n)]=\ensuremath{\ensuremath{\mathrm{O}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(\sqrt{n})$. This result holds both under the null and alternative hypothesis, showing that it is possible to construct a powerful test when $\gamma>\tfrac{1}{2}$. As the proposed test is powerless when $\gamma<\tfrac{1}{2}$, one might wonder if there is any test that can have power in that situation. Although a formal answer to this question is still open, we conjecture that no test can be powerful in that scenario: *Conjecture 2* (Powerless tests when $\gamma<\tfrac{1}{2}$). Consider the case $\gamma<\tfrac{1}{2}$. We conjecture the following: 1. All tests based on the vertex degrees $\{D_v(n)\}_{v\in[n]}$ are powerless. 2. All tests based on $G_n$ are powerless. Obviously the statement (ii) implies (i). The main motivation for (i) is that when $\gamma<\tfrac{1}{2}$ the number of vertices with degree $k$ will deviate from $np_k(\delta_0)$ by at most $\ensuremath{\mathrm{O}}(n^\gamma)$. These deviations become smaller when $k$ gets larger. On the other hand, the fluctuations of $N_k(n)$ around its mean will also become smaller, but should always be of higher order than this mean-shift. Actually, the results in [@Baldassarri2021] characterize the joint distribution of the degree counts under the null model and show this is asymptotically a multi-variate normal distribution. As shown in Lemma [Lemma 14](#lem:joint_CLT){reference-type="ref" reference="lem:joint_CLT"} these degree counts are also asymptotically normal, with the same covariance matrix, but a rather small mean-shift. We conjecture this shift is small enough so to imply the total variation distance between the two distributions is close to zero, implying (i). The conjecture (ii) is significantly stronger, stating that higher-level information contained on the edge structure of $G_n$ will not be helpful for this testing problem. This is expected given that the attachment dynamics are only driven by the vertex degrees, but proving such a statement requires a careful formalization of this insight. ## Powerful test for unknown $\delta_0$ The knowledge of $\delta_0$ was crucial for the test above, as it gives a benchmark to compare $N_m(n)$ against, namely $np_m(\delta_0)$. Without this knowledge we must essentially estimate, from $G_n$, the value of $\delta_0$. For such an approach to be fruitful a candidate estimator must be "close enough" to $\delta_0$ both under the null and alternative models. Gao and van der Vaart in [@Gao2017] consider the problem of estimating $\delta_0$ when the preferential attachment function is *constant*, meaning that we are under the null model in our formulation. The authors proposed a maximum likelihood estimator based on $G_n$, and showed it consistently estimates $\delta_0$ and is asymptotically normal. A natural idea is to start by considering this estimator as well, and understand how its properties change when $G_n$ is generated under the alternative model. As done in [@Gao2017], to avoid the usual issues at the boundary of the parameter space we make an extra assumption that the range of possible values for $\delta_0$ and $\delta_1$ is known: *Assumption 3* (Containment of $\delta_0,\delta_1$). Let $-m<\delta_{\min}\leq \delta_{\max}$ be known, and assume that $\delta_0,\delta_1\in(\delta_{\min},\delta_{\max})$. As shown in [@Gao2017], under the null model the (normalized) log-likelihood function $\iota_n:[\delta_{\min},\delta_{\max}]\to\ensuremath{\mathbb{R}}$ is given by $$\begin{aligned} \iota_n(\delta) \coloneqq{}& \frac{1}{n+1}\left(\sum_{k=1}^\infty \log(k+\delta) \left(N_{>k}(n)-(n+1)\mathds 1\{k<m\}\right)\ -\ \sum_{t=2}^n\sum_{i=1}^m \log S_{t,i-1}(\delta)\right)\notag\\ ={}& \frac{1}{n+1}\sum_{k=m}^\infty \log(k+\delta) N_{>k}(n)\ -\ \frac{1}{n+1}\sum_{t=2}^n\sum_{i=1}^m \log S_{t,i-1}(\delta)\ ,\end{aligned}$$ where $S_{t,i-1}(\delta)\coloneqq t\delta+2m(t-1)+(i-1)$ and $N_{>k}(n)\coloneqq \sum_{j>k} N_j(n)$. The maximum-likelihood estimator is defined as $$\label{eqn:MLE} \hat\delta_n\coloneqq\mathop{\mathrm{arg\tinyspace{}max}}_{\delta\in[\delta_{\min},\delta_{\max}]} \iota_n(\delta)\ .$$ Equivalently (for large $n$) we can define $\hat\delta_n$ as the solution in $\delta\in[\delta_{\min},\delta_{\max}]$ of $\frac{\partial}{\partial \delta} \iota_n(\delta)\coloneqq \iota'_n(\delta)=0$. Although not obvious, this definition coincides with [\[eqn:MLE\]](#eqn:MLE){reference-type="eqref" reference="eqn:MLE"} for large $n$, since it is shown in [@Gao2017] that the solution of $\iota'_n(\delta)=0$ exists and is unique for large enough $n$ with high probability. Note that the score function is given by $$\label{eqn:iota_n_prime} \iota'_n(\delta)=\frac{1}{n+1}\sum_{k=m}^\infty \frac{1}{k+\delta}N_{>k}(n)\ -\ \frac{1}{n+1}\sum_{t=2}^n\sum_{i=1}^m \frac{t}{S_{t,i-1}(\delta)}\ .$$ Motivated by this estimator we consider the test statistic $$\label{eqn:test_statistic_unknown} Q(G_n)\coloneqq N_m(n)-np_m(\hat\delta_n)\ .$$ This is analogous to the previously considered statistic, with the exception that $\delta_0$ is replaced by the above estimator. A test based on the above statistic will only be sensible if $\hat\delta_n$ is a good surrogate for $\delta_0$, under both the null and alternative models. When $\tfrac{1}{2}<\gamma<1$ this is indeed the case, and we show that $\hat\delta_n$ is a consistent estimator of $\delta_0$ under both the null and alternative models. However, consistency is not enough, and it is necessary to carefully characterize the rate of convergence of this estimator under the alternative model. It turns out that the deviations of $N_m(n)$ and $np_m(\hat\delta_n)$ around their respective means have exactly the same order, but with *different* leading constants. A careful characterization of those constants is the crucial result leading to our main result: **Theorem 4** (Asymptotically powerful test, unknown $\delta_0$). *Consider Assumption [Assumption 3](#ass:parameter_space){reference-type="ref" reference="ass:parameter_space"} and the test statistic defined in [\[eqn:test_statistic_unknown\]](#eqn:test_statistic_unknown){reference-type="eqref" reference="eqn:test_statistic_unknown"}. Let $a_n$ be a positive diverging sequence such that $a_n=\omega(\sqrt{n}\log n)$ and $a_n=n^{\tfrac{1}{2}+o(1)}$ and define the test $$\phi(G_n)\coloneqq \mathds 1{\left\{|Q(G_n)|\geq a_n\right\}}\ .$$ The type-I error of this test converges to zero as $n\to\infty$, i.e., $$\ensuremath{\mathbb{P}}_0\left(\phi(G_n)\neq 0\right)=\ensuremath{\mathrm{o}}(1)\ .$$ Furthermore, when $\delta_0\neq\delta_1$ and $\tfrac{1}{2}<\gamma<1$ this test has vanishing type II error, i.e., as $n\to\infty$, $$\ensuremath{\mathbb{P}}_1\left(\phi(G_n)=0\right)=\ensuremath{\mathrm{o}}(1)\ .$$* The proof of the theorem is deferred to Section [8](#sec:proof_unknown){reference-type="ref" reference="sec:proof_unknown"} and it is rather involved. It builds upon some of the results used to prove Theorem [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"}, namely the characterization of $N_m(n) - np_m(\delta_0)$. However, it does require a very careful characterization of $np_m(\hat\delta_n)-np_m(\delta_0)$. It turns out that both quantities have essentially the same order of magnitude, namely $\ensuremath{\ensuremath{\mathrm{O}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(n^{\gamma})$. However, the leading constants are different, and this fact allows the test to be powerful in the regime $\tfrac{1}{2}<\gamma<1$. The cases $\gamma=\tfrac{1}{2}$ and $\gamma=1$ are special. For $\gamma=\tfrac{1}{2}$, one can expect Gaussian fluctuations of $N_m(n)$ under the null and alternative hypotheses to compete with the resulting change in expectations of $N_m(n)$, so the Type-II error can not be expected to vanish, as it does for $\gamma\in (\tfrac{1}{2},1)$. For $\gamma=1$, on the other hand, both the maximal and minimal degree tests seem to perform well, but it is unclear which performs best, or whether there even is a better test available. ## Asymptotically calibrated tests {#sec:asymptotically_calibrated_tests} The two theorems above characterize the regime when the proposed tests are powerful. However, they fall short of providing guidelines to properly calibrate the tests. Particularly, in a fixed significance testing framework, for any $\alpha\in(0,1)$ we would like to ensure that under the null model the type-I error is approximately $\alpha$. Theorem [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"} provides only an rather conservative asymptotic upper bound on the type-I error, due to the worst-case nature of the Azuma-Hoeffding inequality. To introduce our calibrated test, we define $$\begin{aligned} % \label{w-var-def} w(\delta_0,m)&\coloneqq \frac{m^2 (m+\delta_0) (1 + m + \delta_0) (2m+\delta_0)}{(\delta_0 + 2m(1+m+\delta_0)) (\delta_0 + m(2+m+\delta_0))^2}\ .\end{aligned}$$ Furthermore, let $z_\alpha$ denote the right-quantile function of the standard normal distribution[^1]. In particular, $z_{\alpha/2}>0$ when $\alpha\in(0,1)$. We are now ready to present the asymptotically calibrated test when $\delta_0$ is assumed known. **Theorem 5** (Asymptotically calibrated test for known $\delta_0$). *Let $\alpha\in(0,1)$ and define the test $$\label{eq:asymptotically_calibrated_minimal_degree_test_known_delta} \psi_{\rm cal}(G_n)\coloneqq \mathds 1{\left\{|T(G_n)|\geq \sqrt{n w(\delta_0,m)}z_{\alpha/2}\right\}}\ .$$ As $n\to\infty$, the type-I error of this test converges to $\alpha$: $$\ensuremath{\mathbb{P}}_0\left(\psi_{\rm cal}(G_n)\neq 0\right)\to \alpha\ .$$ Furthermore, when $\delta_0\neq\delta_1$ and $\tfrac{1}{2}<\gamma<1$ this test has vanishing type-II error as $n\rightarrow \infty$: $$\begin{aligned} \ensuremath{\mathbb{P}}_1\left(\psi_{\rm cal}(G_n)= 0\right)\to 0\ .\end{aligned}$$* A statement regarding partial power when $\gamma=\frac{1}{2}$ is also possible, but not particularly insightful. To define the next test we require some additional notation. Let $$\begin{aligned} % \label{v-var-def} v(\delta_0,m)&\coloneqq \sum_{k=m}^\infty \frac{m p_k(\delta_0)}{(k+\delta_0)(2m+\delta_0)} - \frac{m}{(2m+\delta_0)^2}\ ,%\\ %\end{aligned}$$ and $$\begin{aligned} \label{u-var-def} u(\delta_0,m)&\coloneqq - \frac{m^4}{v(\delta_0,m) (\delta_0 + m(2 + m + \delta_0))^4}\ .\end{aligned}$$ The next theorem presents the asymptotically calibrated test for the unknown $\delta_0$ case: **Theorem 6** (Asymptotically calibrated test, unknown $\delta_0$). *Consider Assumption [Assumption 3](#ass:parameter_space){reference-type="ref" reference="ass:parameter_space"}, let $\alpha\in(0,1)$, and define the test $$\phi_{\rm cal}(G_n)\coloneqq \mathds 1{\left\{|Q(G_n)|\geq \sqrt{n (w(\hat{\delta}_n,m)+u(\hat{\delta}_n,m)) } z_{\alpha/2} \right\}}\ .$$ As $n\to\infty$, the type-I error of this test converges to $\alpha$: $$\ensuremath{\mathbb{P}}_0\left(\phi_{\rm cal}(G_n)\neq 0\right)\to\alpha\ .$$ Furthermore, when $\delta_0\neq\delta_1$ and $\tfrac{1}{2}<\gamma<1$ this test has vanishing type-II error as $n\rightarrow \infty$: $$\ensuremath{\mathbb{P}}_1\left(\phi_{\rm cal}(G_n)=0\right)\to0\ .$$* The proofs of Theorems [Theorem 5](#thm:asymptotically_calibrated_minimal_degree_test_known_delta){reference-type="ref" reference="thm:asymptotically_calibrated_minimal_degree_test_known_delta"} and [Theorem 6](#thm:asymptotically_calibrated_minimal_degree_test_unknown_delta){reference-type="ref" reference="thm:asymptotically_calibrated_minimal_degree_test_unknown_delta"} are an immediate consequence of the asymptotic normality of the test statistics as discussed in the next section, together with the consistency of $\hat\delta_n$ as an estimator of $\delta_0$. # Asymptotic normality of test statistics In this section we characterize the asymptotic distribution of the proposed test statistics which allowed us to calibrate the tests in Section [3.3](#sec:asymptotically_calibrated_tests){reference-type="ref" reference="sec:asymptotically_calibrated_tests"}. When $\delta_0$ is known, the situation is relatively simple. Under the null model and when $m=1$, it is known that $N_m(n)$ with $k \geq m$ admits a central limit theorem [@Samorodnitsky2016]. In particular, this shows that $N_m(n)$ is asymptotically normally distributed. Furthermore, [@Baldassarri2021] extends these results to the general case $m\geq 1$, making them applicable to our setting. For the case of unknown $\delta_0$ the situation is a bit more complicated. Recall that our test statistic is $Q(G_n) = N_m(n)-n p_m(\hat\delta_n)$. It is known from [@Gao2017] that, under the null model, $\hat\delta_n$ is asymptotically normal. This does not, however, immediately imply that $Q(G_n)$ is also asymptotically normal under the null model. The results in [@Baldassarri2021] establish that $(N_m(n),N_{m+1}(n),\ldots)$ is asymptotically normal (under the null model), strongly hinting at asymptotic normality of $Q(G_n)$. Furthermore, even under the alternative model, one might expect asymptotic normality of the test statistics, with exactly the same asymptotic variance, as the number of vertices that enter after the (late) changepoint is too small to change the asymptotic variance. Let $\mathcal{N}(\mu,\sigma^2)$ denote the normal distribution with mean $\mu$ and variance $\sigma^2$, and let $\xrightarrow{\smash{\raisebox{-1.5pt}{$\scriptstyle{}D$}}}$ denote convergence in distribution. The following theorem, proved in Section [9](#sec:proof_asymptotic_normality_test_statistics){reference-type="ref" reference="sec:proof_asymptotic_normality_test_statistics"}, establishes the asymptotic properties of the test statistics. **Theorem 7** (Asymptotic normality of test statistics). *Recall the definitions of $w$ and $u$ in [\[w-var-def\]](#w-var-def){reference-type="eqref" reference="w-var-def"} and [\[u-var-def\]](#u-var-def){reference-type="eqref" reference="u-var-def"} respectively. As $n\to\infty$, $$\begin{aligned} \label{eq:asymptotic_normality_known_delta_test_statistic_null_hypothesis}% \frac{T(G_n)}{\sqrt{n}} \xrightarrow{D} \mathcal{N}\left(0,w(\delta_0,m)\right)\ .\end{aligned}$$ Moreover, under Assumption [Assumption 3](#ass:parameter_space){reference-type="ref" reference="ass:parameter_space"} and the null model, as $n\to\infty$, $$\begin{aligned} % \frac{Q(G_n)}{\sqrt{n}}=\frac{N_m(n)-np_m(\hat\delta_n)}{\sqrt{n}}\ \xrightarrow{D} \mathcal{N}\left(0,w(\delta_0,m)+u(\delta_0,m)\right)\ .\label{eq:asymptotic_normality_unknown_delta_test_statistic}\end{aligned}$$ Furthermore, under the alternative model with $\gamma\in(0,1)$, as $n\to\infty$, $$\begin{aligned} \label{eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis}% \frac{T(G_n)-\ensuremath{\mathbb{E}}_1[T(G_n)]}{\sqrt{n}}\xrightarrow{D} \mathcal{N}\left(0,w(\delta_0,m)\right)\ .\end{aligned}$$ Moreover, under the alternative model with $\gamma\in(\tfrac{1}{2},1)$ and Assumption [Assumption 3](#ass:parameter_space){reference-type="ref" reference="ass:parameter_space"}, as $n\to\infty$, $$\begin{aligned} \label{eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis}% \frac{Q(G_n)-\ensuremath{\mathbb{E}}_1[Q(G_n)]}{\sqrt{n}}\xrightarrow{D} \mathcal{N}\left(0,w(\delta_0,m)+u(\delta_0,m)\right)\ .\end{aligned}$$* Note that [\[eq:asymptotic_normality_known_delta_test_statistic_null_hypothesis\]](#eq:asymptotic_normality_known_delta_test_statistic_null_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_known_delta_test_statistic_null_hypothesis"} has been proved in [@Baldassarri2021]. In the second statement [\[eq:asymptotic_normality_unknown_delta_test_statistic\]](#eq:asymptotic_normality_unknown_delta_test_statistic){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic"}, $u(\delta_0,m)$ captures the adjustment in the variability in the test statistic by using an estimate of $\delta_0$, instead of the actual value. The convergence results [\[eq:asymptotic_normality_known_delta_test_statistic_null_hypothesis\]](#eq:asymptotic_normality_known_delta_test_statistic_null_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_known_delta_test_statistic_null_hypothesis"} and [\[eq:asymptotic_normality_unknown_delta_test_statistic\]](#eq:asymptotic_normality_unknown_delta_test_statistic){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic"} provide an avenue for asymptotically exact calibration of the proposed test as given in Theorems [Theorem 5](#thm:asymptotically_calibrated_minimal_degree_test_known_delta){reference-type="ref" reference="thm:asymptotically_calibrated_minimal_degree_test_known_delta"} and [Theorem 6](#thm:asymptotically_calibrated_minimal_degree_test_unknown_delta){reference-type="ref" reference="thm:asymptotically_calibrated_minimal_degree_test_unknown_delta"} (equivalently, for computation of asymptotically exact $p$-values). To establish asymptotic normality under the alternative model, we exploit the corresponding statements under the null model, together with a correction term quantifying the effects due to the presence of a changepoint (partially relying on the arguments in Theorems [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"} and [Theorem 4](#thm:main2){reference-type="ref" reference="thm:main2"}). It is important to remark that, for the convergence result [\[eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis\]](#eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis"}, the assumption $\gamma>\tfrac{1}{2}$ appears to be merely technical, and it should be possible to drop it. Finally, note that the asymptotic variances remain the same, whether one is considering the null or alternative model, as expected. The proof of Theorem [Theorem 7](#thm:asymptotic_normality_test_statistics){reference-type="ref" reference="thm:asymptotic_normality_test_statistics"} is carried out in Section [9](#sec:proof_asymptotic_normality_test_statistics){reference-type="ref" reference="sec:proof_asymptotic_normality_test_statistics"}. # Numerical experiments In this section we examine the properties of the proposed tests using simulation. This serves a two-fold purpose, namely to empirically assess the validity of the theoretical guarantees given, as well as to investigate the finite-sample properties of the tests. To assess the finite sample properties of the asymptotically calibrated tests we conduct several numerical experiments. For simplicity we fix $\delta_0=0$ for all the experiments (this is the so-called linear preferential attachment model in [@Barabasi1999]). We take $m=5$, $c=1$, $\gamma=\frac{3}{4}$, and $\delta_1\in\{-1,0,1\}$. Note that $\delta_1=\delta_0=0$ corresponds to the null model. We consider graphs of different sizes, namely $n\in\{1000,2000,5000,10000,20000,50000,100000,200000\}$, and for each value of $n$ we generate $B=2000$ independent graphs using the preferential attachment model in [\[eq:attachment_function_general\]](#eq:attachment_function_general){reference-type="eqref" reference="eq:attachment_function_general"}. Specifically, for the three cases $\delta_1=\{-1,0,1\}$ and each value $n$ we obtain $\{g_n^{(b)}\}_{b=1}^B$ graphs. Note that as $n$ increases the relative distance between the changepoint and $n$ decreases considerably, and only a minute part of the graph is affected after the changepoint. For instance, for $n=5000$ there are only 594 vertices that join the graph after the changepoint, so about 12% of the vertices. However, for $n=200000$ only 9457 vertices join after the changepoint, a mere 4.7% of the total vertices. In Figure [\[fig:power\]](#fig:power){reference-type="ref" reference="fig:power"} we depict the power of the two proposed tests. For concreteness we consider fixed significance testing at level $\alpha=0.05$ (qualitatively the results are similar for other significance levels). We clearly see that both tests are well calibrated, even for small values of $n$. Also, as expected, the power increases as a function of $n$. As intuitively expected, the test that does not assume knowledge of $\delta_0$ has slightly less power than the test making use of that knowledge. Finally, there is an asymmetry of the power depending on whether $\delta_1>\delta_0$ or $\delta_1<\delta_0$, the later scenario leading to higher power. This is expected, as for smaller $\delta_1$ values the empirical degree distribution has heavier tails, and detecting the presence of a changepoint becomes easier. To illustrate the results of Theorem [Theorem 7](#thm:asymptotic_normality_test_statistics){reference-type="ref" reference="thm:asymptotic_normality_test_statistics"}, in Figure [\[fig:asym_norm\]](#fig:asym_norm){reference-type="ref" reference="fig:asym_norm"} we display normal quantile-quantile plots of the test statistics for very small values of $n$. We see that, even for $n=100$ (so, a graph with $nm=500$ edges) both test statistics are approximately normally distributed. In order to have a better assessment of the mean and variance of the test statistics we compute (for $\delta_1\in\{-1,0,1\}$) the first two empirical moments, and compare them with their asymptotic counterparts, both under the null and alternative hypothesis. Define $$\hat\mu^{\scriptscriptstyle(T)}_n \coloneqq \frac{1}{B}\sum_{b=1}^B T(g_n^{\scriptscriptstyle(b)})$$ and $$\hat v^{\scriptscriptstyle(T)}_n \coloneqq \frac{1}{B}\sum_{b=1}^B (T(g_n^{\scriptscriptstyle(b)})-\hat\mu^{\scriptscriptstyle(T)}_n)^2\ ,$$ and the analogous definitions of $\hat\mu^{\scriptscriptstyle(Q)}_n$ and $\hat v^{\scriptscriptstyle(Q)}_n$. In Table [\[tbl:asymptotic_mean_var\]](#tbl:asymptotic_mean_var){reference-type="ref" reference="tbl:asymptotic_mean_var"} we compare these values (adequately rescaled) with the expected asymptotic values. As can be seen, the empirical variance closely matches the asymptotic variance, both under the null and alternative models, even for small values of $n$. This is in agreement with Theorem [Theorem 7](#thm:asymptotic_normality_test_statistics){reference-type="ref" reference="thm:asymptotic_normality_test_statistics"}. For the mean of the statistics, one sees that the scaling by $n^\gamma$ and the corresponding leading constant is also accurate, but finite sample effects are more evident under the alternative model when $n$ is small. Finally, making use of Theorem [Theorem 7](#thm:asymptotic_normality_test_statistics){reference-type="ref" reference="thm:asymptotic_normality_test_statistics"} we can compare the empirical power with an estimate based on the asymptotic normality of the statistic. Namely, we know the test statistics are asymptotically normal with exactly the same variance, and a small mean-shift proportional to $n^\gamma(1+\ensuremath{\mathrm{o}}(1))$, where the proportionality constant is given by $\eta(\delta_0,\delta_1)$ from [\[eqn:shift_T\]](#eqn:shift_T){reference-type="eqref" reference="eqn:shift_T"} and $\alpha(\delta_0,\delta_1)$ from [\[eqn:shift_Q\]](#eqn:shift_Q){reference-type="eqref" reference="eqn:shift_Q"} for $T(G_n)$ and $Q(G_n)$ respectively. Based on this, one can get an estimate for the power of the tests for different values of $\delta_1$. This is shown by the dashed lines in Figure [\[fig:power\]](#fig:power){reference-type="ref" reference="fig:power"}. As one can see, although not terribly accurate, the estimates capture the exact behavior of the empirically observed power. This lack of accuracy is not unexpected, as all we know is that the mean-shifts are of the form $\text{const}(\delta_0,\delta_1,m) n^\gamma+\ensuremath{\mathrm{o}}(n^\gamma)$. However, the remainder term might still have an order only slightly smaller than $n^\gamma$, which will lead to rather poor power estimates for small values of $n$. # Discussion and open problems {#sec:changepoint_detect_discussion} In this section, we compare our results to the literature and state some open problems. ## Early changepoint In previous work [@Bhamidi2018; @Banerjee2018; @Cirkovic2022], the case of an *early* changepoint was considered for preferential attachment trees, i.e., for $m=1$. Thus, our work extends this setting from trees to graphs, as well as from an early changepoint to a late one. Arguably, the latter case is more relevant in practice, since one would rather detect a changepoint quickly, meaning, close to the time after which it occurs. This setting corresponds to a changepoint close to the time of observation of the final network. ## Dynamical graph observations It would be of interest to extend our results to a *dynamic* setting, where we detect the changepoint as the graph changes dynamically. Bear in mind though that we currently only assume that we observe the graph at the final time, and observing the graph dynamically thus provides much more information. Thus, it is an interesting extension to devise an appropriate setting where we only observe *partial information* on the network, while still detecting the changepoint dynamically. There are several settings that could be of interest. In the first, one observes the network snapshots only at multiples of $n^\gamma$. In the second, we assume that we only dynamically observe information about the degree counts, and not the entire network. We defer such problems to future work. ## Lower bounds Conjecture [Conjecture 2](#conj:lower_bounds){reference-type="ref" reference="conj:lower_bounds"} states that no test will be powerful when $\gamma<\tfrac{1}{2}$. Proving such lower bounds in the context of preferential attachment models is challenging, due to the latent nature of these models. However, part (i) of the conjecture might be approached by relying on the asymptotic normality characterization in Lemma [Lemma 14](#lem:joint_CLT){reference-type="ref" reference="lem:joint_CLT"}, together with bounds on $\ensuremath{\mathbb{E}}_1[N_k(n)]-\ensuremath{\mathbb{E}}_0[N_k(n)]$ obtained using the methods developed in this paper. ## Boundary case $\gamma=1$ Note that in Theorems [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"} and [Theorem 4](#thm:main2){reference-type="ref" reference="thm:main2"} the case $\gamma=1$ is excluded. This is in contrast with the results in [@Bhamidi2018; @Banerjee2018; @Cirkovic2022]. The proof of the two theorems relies on Proposition [Proposition 8](#prp:A){reference-type="ref" reference="prp:A"}, that explicitly excludes the case $\gamma=1$. It should be possible to extend that result for $\gamma=1$. Specifically, the current argument quantifies the contribution to $\ensuremath{\mathbb{E}}_1[N_m(n)]-np_m(\delta_0)$ made by vertices that arrived after the changepoint. Due to the late changepoint, most of those vertices will have degree $m$ in $G_n$. However, for $\gamma=1$, a small, but non-vanishing fraction of those vertices will have higher degree. Therefore, to extend the result, this needs to be quantified, and a slightly more refined argument will be needed, where the role of the parameter $c$ will become much more prevalent. Extending the result of Theorem [Theorem 4](#thm:main2){reference-type="ref" reference="thm:main2"} to this setting will likely be significantly more challenging, as it requires extending Proposition [Proposition 11](#prp:ana-diff-means){reference-type="ref" reference="prp:ana-diff-means"}, where the assumption that $\gamma<1$ was crucially used to bound the terms of order $\ensuremath{\mathrm{o}}(n^\gamma)$. On the other hand, when $\gamma=1$, our test statistic will likely not be a good practical choice (particularly when $c$ is large), and the statistics used in [@Bhamidi2018; @Banerjee2018] will likely lead to more powerful tests. ## Other test statistics Note that information about the presence of a changepoint is present not only in $N_m(n)$, but also on other counts of low-degree vertices. With this in mind, a test based on $(N_m(n),N_{m+1}(n))$ can be potentially more powerful (in a finite sample sense) than the test we proposed. Although we expect such tests to have exactly the same asymptotic performance, they can perform much better for finite $n$. An interesting avenue of research is to identify, in a principled way, statistics that lead to tests that have higher power than the ones proposed. ## Boundary case $\gamma=\tfrac{1}{2}$ When $\gamma=\tfrac{1}{2}$, the fluctuations of $N_m(n)$ around its mean under $H_0$ are of the same order as $\ensuremath{\mathbb{E}}_1[N_m(n)]-p_m(\delta_0)$. Moreover, since a central limit theorem holds for $N_m(n)-\ensuremath{\mathbb{E}}_1[N_m(n)]$ under $H_1$, with the same limiting variance as under $H_0$ (cf. Theorem [Theorem 7](#thm:asymptotic_normality_test_statistics){reference-type="ref" reference="thm:asymptotic_normality_test_statistics"}), it follows that, when $\gamma=\tfrac{1}{2}$, the type-II error of our test is strictly bounded away from zero. In other words, when $\gamma=\tfrac{1}{2}$, a large value of $N_m(n)-p_m$ can be explained either by a large deviation away from $\ensuremath{\mathbb{E}}_0[N_m]$ under $H_0$, or by a deviation around $\ensuremath{\mathbb{E}}_1[N_m]$ under $H_1$. # Powerful test for known $\delta_0$: Proof of Theorem [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"} {#sec:proof_known} The main idea of the proof is to decompose the test statistic $T(G_n)$ in two terms: $$\label{eqn:decomposition_T} T(G_n)=N_m(n)-np_m(\delta_0)=\underbrace{\ensuremath{\mathbb{E}}_\ell[N_m(n)]-np_m(\delta_0)}_{\coloneqq A}+\underbrace{N_m(n)-\ensuremath{\mathbb{E}}_\ell[N_m(n)]}_{\coloneqq B}\ ,$$ where $\ell\in\{0,1\}$. The characterization of the stochastic term $B$ is the same under both the null and alternative models, and follows a somewhat standard argument. Let $\ell\in\{0,1\}$ be fixed. Define the stochastic process $\{M_t\}_{t=1}^n$ such that $$M_t\coloneqq \ensuremath{\mathbb{E}}_\ell[N_m(n)|G_t]\ .$$ This is a Doob martingale [@VanderHofstad2017 Lemma 8.5], and such that $\ensuremath{\mathbb{E}}_\ell[N_m(n)]=M_1$ and $N_m(n)=M_n$. Furthermore, by [@VanderHofstad2017 Lemma 8.6], we know that, for every $t\in\{2,\ldots,n\}$, $|M_n-M_{n-1}|\leq 2m$ almost surely. Although strictly speaking these two lemmas were not stated for our model, their arguments do not depend on the specific sequence $\delta(t)$ in [\[eq:attachment_function_general\]](#eq:attachment_function_general){reference-type="eqref" reference="eq:attachment_function_general"}. Therefore, those results still hold and follow simply from the fact that every step in time we add precisely $m$ edges to the graph. With this in hand, we can directly apply the Azuma-Hoeffding inequality to get that, for any $x > 0$, $$\label{eqn:AH_Nm} \forall \ell\in\{0,1\}\quad \ensuremath{\mathbb{P}}_\ell\left(\left|N_m(n) - \ensuremath{\mathbb{E}}_\ell[N_m(n)]\right| \geq x\right) \leq 2 \mspace{1mu} \ensuremath{\mathrm{e}}^{-\frac{x^2}{8 m^2 n}}\ .$$ This completes the characterization of the term $B$ in [\[eqn:decomposition_T\]](#eqn:decomposition_T){reference-type="eqref" reference="eqn:decomposition_T"}. We now proceed by considering the type-I and type-II errors separately. ## *Type-I error* To control the term $A$ in [\[eqn:decomposition_T\]](#eqn:decomposition_T){reference-type="eqref" reference="eqn:decomposition_T"} under the null model we use [@Deijfen2007 Proposition 2.2] (see also [@VanderHofstad2017 Proposition 8.7]), which states that there exists a constant $C_0 = C_0(\delta_0, m)$ such that, for all $n \geq 1$, $$\label{eqn:bounded_difference} |\ensuremath{\mathbb{E}}_0[N_m(n)] - n \mspace{1mu} p_m(\delta_0)| \leq C_0\ .$$ Combining this with [\[eqn:AH_Nm\]](#eqn:AH_Nm){reference-type="eqref" reference="eqn:AH_Nm"} we see that the type-I error of the minimal degree test is bounded by $$\begin{aligned} \ensuremath{\mathbb{P}}_0(\psi(T(G_n)) \neq 0) &= \ensuremath{\mathbb{P}}_0\left(|N_m(n) - n \mspace{1mu} p_m(\delta_0)| \geq m \sqrt{8 n \log(2 / \alpha)}\right)\\ &\leq \ensuremath{\mathbb{P}}_0\left(|N_m(n) - \ensuremath{\mathbb{E}}_0[N_m(n)]| \geq m \sqrt{8 n \log(2 / \alpha)} - C_0\right)\\ &\leq 2 \exp\left(- \frac{(m \sqrt{8 n \log(2 / \alpha)} - C_0)^2}{8 m^2 n}\right) = (1 + \ensuremath{\mathrm{o}}(1)) \alpha \ .\end{aligned}$$ This shows that the type-I error is essentially at most $\alpha$, completing the first part of the proof. ## *Type-II error* Again we must control the term $A$ above, for which the following proposition is instrumental: **Proposition 8**. *Let $0<\gamma<1$. Then $$\begin{aligned} \ensuremath{\mathbb{E}}_1[N_m(n)]-np_m(\delta_0)&= (1+\ensuremath{\mathrm{o}}(1))\ cn^\gamma \left(1 - \frac{p_m(\delta_0)}{p_m(\delta_1)}\right)\\ &= (1+\ensuremath{\mathrm{o}}(1)) cn^\gamma (\delta_0-\delta_1)\frac{1}{(2+\delta_1/m)(m+\delta_0+2+\delta_0/m)}\ .\end{aligned}$$* *Proof.* Note that $$\begin{aligned} \label{eqn:decomposition_term_A} \ensuremath{\mathbb{E}}_1[N_m(n)]-np_m(\delta_0) &= \ensuremath{\mathbb{E}}_0[N_m(n)]-np_m(\delta_0)\\ &\hspace{-40pt}+\ensuremath{\mathbb{E}}_1[N_m(n)]-\ensuremath{\mathbb{E}}_1[N_m(\tau_n)] - (\ensuremath{\mathbb{E}}_0[N_m(n)]-\ensuremath{\mathbb{E}}_0[N_m(\tau_n)])\ .\end{aligned}$$ The equality holds as the law of $G_{\tau_n}$ is the same under the null and alternative models, and therefore $\ensuremath{\mathbb{E}}_0[N_m(\tau_n)]=\ensuremath{\mathbb{E}}_1[N_m(\tau_n)]$. The last two terms are controlled in a similar way. Let $\ell\in\{0,1\}$ and note that $$\begin{aligned} \ensuremath{\mathbb{E}}_\ell[N_m(n)]-\ensuremath{\mathbb{E}}_\ell[N_m(\tau_n)] &=\sum_{v\in[\tau_n]} (\ensuremath{\mathbb{P}}_\ell(D_v(n)=m)-\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=m))\\ % &\qquad + \sum_{v\in[n]\setminus[\tau_n]} \ensuremath{\mathbb{P}}_\ell(D_v(n)=m)\label{eqn:new_vertices}\ .\end{aligned}$$ In the above there is a contribution from vertices that arrived before the change-point (so-called "old" vertices), and vertices that arrived afterwards (the "new" vertices). The contribution by the new vertices $v\in[n]\setminus[\tau_n]$ is essentially the same regardless of the value of $\ell$, as $\ensuremath{\mathbb{P}}_\ell(D_v(n)=m)\approx 1$ when $v\in[n]\setminus[\tau_n]$. To see this, note that $\ensuremath{\mathbb{P}}_\ell(D_v(n)=m)=1-\ensuremath{\mathbb{P}}_\ell(D_v(n)>m)$ and that the event $D_v(n)>m$ can only occur if there is at least one vertex $v'>v$ attaching to $v$. Referring to [\[eq:attachment_function_alt\]](#eq:attachment_function_alt){reference-type="eqref" reference="eq:attachment_function_alt"} the probability of this happening is at most $(m+\delta_\ell)/((2m+\delta_{\ell})\tau_n-2m)$, and there are at most $m(n-\tau_n)$ possible edges that can lead to that connection. Therefore $$\ensuremath{\mathbb{P}}_\ell(D_v(n)=m)\geq 1-m(n-\tau_n)\frac{m+\delta_\ell}{(2m+\delta_\ell)\tau_n-2m}=1-\ensuremath{\mathrm{O}}(n^{\gamma-1})\ .$$ In conclusion (since $\ensuremath{\mathbb{P}}_\ell(D_v(n)=m)$ is bounded above by 1) $$\sum_{v\in[n]\setminus[\tau_n]} \ensuremath{\mathbb{P}}_\ell(D_v(n)=m)=c(n-\tau_n)(1+\ensuremath{\mathrm{O}}(n^{\gamma-1}))\ ,\qquad \ell\in\{0,1\}\ .$$ For the term involving the "old" vertices we use the following lemma, which will be used to the full extent for the proof of Theorem [Theorem 4](#thm:main2){reference-type="ref" reference="thm:main2"}: **Lemma 9**. *Let $v\in[\tau_n]$, $\gamma<1$ and $m\leq k = \ensuremath{\mathrm{o}}(n^{1-\gamma})$ and $\ell\in\{0,1\}$. Then $$\ensuremath{\mathbb{P}}_\ell(D_v(n)-D_v(\tau_n)>0\mid D_v(\tau_n)=k)=(1+\ensuremath{\mathrm{o}}(1))cn^{\gamma-1} m \frac{k+\delta_\ell}{2m+\delta_\ell}\ .$$ as $n\to\infty$.* *Proof.* Note that $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{P}}_\ell\left(D_v(n)-D_v(\tau_n)>0\mid D_v(\tau_n)=k\right)}\\ &= 1 - \ensuremath{\mathbb{P}}_\ell\left(D_v(n)-D_v(\tau_n)=0\mid D_v(\tau_n)=k\right)\\ &=1-\prod_{j\in[n]\setminus[\tau_n]} \prod_{i=1}^m \left(1-\frac{k+\delta_\ell}{j(2m+\delta_\ell)-2m+i-1}\right)\\ &=1-\exp\left(\sum_{j\in[n]\setminus[\tau_n]} \sum_{i=1}^m \log\left(1-\frac{k+\delta_\ell}{j(2m+\delta_\ell)-2m+i-1}\right)\right)\\ &=1-\exp\left(\sum_{j\in[n]\setminus[\tau_n]} \sum_{i=1}^m -(1+\ensuremath{\mathrm{o}}(1))\frac{k+\delta_\ell}{j(2m+\delta_\ell)-2m+i-1}\right)\label{eqn:step1}\\ &=1-\exp\left(-(1+\ensuremath{\mathrm{o}}(1))cmn^\gamma \frac{k+\delta_\ell}{n(2m+\delta_\ell)}\right)\\ &=1-\left(1-(1+\ensuremath{\mathrm{o}}(1))cmn^{\gamma-1} \frac{k+\delta_\ell}{2m+\delta_\ell}\right)\label{eqn:step2}\\ &=(1+\ensuremath{\mathrm{o}}(1))cmn^{\gamma-1} \frac{k+\delta_\ell}{2m+\delta_\ell}\ ,\end{aligned}$$ where in [\[eqn:step1\]](#eqn:step1){reference-type="eqref" reference="eqn:step1"} we relied on the fact that $k=\ensuremath{\mathrm{o}}(n)$ and for [\[eqn:step2\]](#eqn:step2){reference-type="eqref" reference="eqn:step2"} it is crucial that $k=\ensuremath{\mathrm{o}}(n^{1-\gamma})$. 0◻ With this lemma in hand, we clearly see that $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{P}}_\ell(D_v(n)=m)-\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=m)}\\ &= \ensuremath{\mathbb{P}}_\ell(D_v(n)-D_v(\tau_n)=0|D_v(\tau_n)=m)\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=m)\\ &= \left(1-\ensuremath{\mathbb{P}}_\ell(D_v(n)-D_v(\tau_n)>0|D_v(\tau_n)=m)\right)\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=m)\\ &= \left(1-(1+\ensuremath{\mathrm{o}}(1))cn^{\gamma-1}m\frac{m+\delta_\ell}{2m+\delta_\ell}\right)\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=m)\ .\end{aligned}$$ Putting the two results together we get $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{E}}_\ell[N_m(n)]-\ensuremath{\mathbb{E}}_\ell[N_m(\tau_n)]}\\ &= \left(1-(1+\ensuremath{\mathrm{o}}(1))cn^{\gamma-1}m\frac{m+\delta_\ell}{2m+\delta_\ell}\right)\ensuremath{\mathbb{E}}_\ell[N_m(\tau_n)]+c(n-\tau_n)(1+\ensuremath{\mathrm{O}}(n^{\gamma-1}))\ .\end{aligned}$$ Note that this results characterizes what happens both under the null and alternative models. Hence this result, together with [\[eqn:decomposition_term_A\]](#eqn:decomposition_term_A){reference-type="eqref" reference="eqn:decomposition_term_A"} and the fact that $\ensuremath{\mathbb{E}}_0[N_m(\tau_n)]=\ensuremath{\mathbb{E}}_1[N_m(\tau_n)]$, yields $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{E}}_1[N_m(n)]-np_m(\delta_0)}\\ &= \ensuremath{\mathbb{E}}_0[N_m(n)]-np_m(\delta_0)\\ &\qquad +(1+\ensuremath{\mathrm{o}}(1))cn^{\gamma-1}m\left(\frac{m+\delta_1}{2m+\delta_1}-\frac{m+\delta_0}{2m+\delta_0}\right)\ensuremath{\mathbb{E}}_0[N_m(n)]+\ensuremath{\mathrm{O}}(n^{2\gamma-1})\\ &=\ensuremath{\mathrm{O}}(1)+(1+\ensuremath{\mathrm{o}}(1))cn^{\gamma-1}(\delta_1-\delta_0)\frac{1}{(2+\delta_1/m)(2+\delta_0/m)}n\left(p_m(\delta_0)+\ensuremath{\mathrm{O}}(1/n)\right)\\ &\qquad+\ensuremath{\mathrm{O}}(n^{2\gamma-1})\\ &=(1+\ensuremath{\mathrm{o}}(1))cn^{\gamma}\frac{\delta_1-\delta_0}{(2+\delta_1/m)(m+\delta_0+2+\delta_0/m)}\ ,\end{aligned}$$ where $0<\gamma<1$, and we have again used [\[eqn:bounded_difference\]](#eqn:bounded_difference){reference-type="eqref" reference="eqn:bounded_difference"} to relate $\ensuremath{\mathbb{E}}_0[N_m(n)]$ to $np_m(\delta_0)$. ◻ Similarly as for the type-I error, we can use Proposition [Proposition 8](#prp:A){reference-type="ref" reference="prp:A"} together with the Azuma-Hoeffding inequality [\[eqn:AH_Nm\]](#eqn:AH_Nm){reference-type="eqref" reference="eqn:AH_Nm"} to get $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{P}}_1(T(G_n) \neq 1)}\\ &= \ensuremath{\mathbb{P}}_1\left(|N_m(n) - n p_m(\delta_0)| < m \sqrt{8 n \log(2 / \alpha)}\right)\\ &\leq \ensuremath{\mathbb{P}}_1\left(|N_m(n) - \ensuremath{\mathbb{E}}_1[N_m(n)]| > \left(|\ensuremath{\mathbb{E}}_1[N_m(n)]-n p_m(\delta_0)| - m \sqrt{8 n \log(2 / \alpha)}\right)\vee 0\right)\\ &\leq 2\exp\left(-\frac{\left(\left(|\ensuremath{\mathbb{E}}_1[N_m(n)]-n p_m(\delta_0)| - m \sqrt{8 n \log(2 / \alpha)}\right)\vee 0\right)^2}{8m^2n}\right)\ .\end{aligned}$$ Considering the cases $\gamma > \tfrac{1}{2}$ and $\gamma = \tfrac{1}{2}$ separately, this gives $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{P}}_1(T(G_n) \neq 1)}\\ &\leq \begin{cases} \ensuremath{\mathrm{o}}(1) & \text{when } \gamma > \tfrac{1}{2},\\ (2 + \ensuremath{\mathrm{o}}(1)) \mspace{2mu} \exp\left(-\Bigl(\Bigl(\frac{c \mspace{2mu} |1 - p_m(\delta_0) / p_m(\delta_1)|}{m \sqrt{8}} - \smash{\sqrt{\log(2 / \alpha)}}\Bigr) \vee 0\Bigr)^{\!2}\right) & \text{when } \gamma = \tfrac{1}{2}. \end{cases}\notag\end{aligned}$$ The case $\gamma<\tfrac{1}{2}$ does not follow immediately from the analysis above, as the characterization obtained by the Azuma-Hoeffding only provides an upper bound on the variability of the test statistic. However, in Theorem [Theorem 7](#thm:asymptotic_normality_test_statistics){reference-type="ref" reference="thm:asymptotic_normality_test_statistics"} it is shown that $(T(G_n)-\ensuremath{\mathbb{E}}_1[T(G_n)])/\sqrt{n}$ is asymptotically normal with mean 0 and variance $w(\delta_0,m)>0$. As shown above $\ensuremath{\mathbb{E}}_1[T(G_n)]/\sqrt{n}=\ensuremath{\mathrm{O}}(n^{\gamma-\tfrac{1}{2}})$. Therefore $\ensuremath{\mathbb{E}}_1[T(G_n)]/\sqrt{n}=\ensuremath{\mathrm{o}}(1)$ when $\gamma<\tfrac{1}{2}$, and therefore $T(G_n)/\sqrt{n}$ converges to the same distribution under the null and alternative models, meaning the type II error is asymptotically just the complement of the type I error. ◻ # Powerful test for unknown $\delta_0$: Proof of Theorem [Theorem 4](#thm:main2){reference-type="ref" reference="thm:main2"} {#sec:proof_unknown} To prove Theorem [Theorem 4](#thm:main2){reference-type="ref" reference="thm:main2"} we partially leverage on the results and analysis in Theorem [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"}. However, we must take into account that $\delta_0$ is not known, and rather we have only its estimate $\hat\delta_n$. The main idea is to decompose the test statistic $Q(G_n)$ as $$\begin{aligned} \label{eqn:three_terms} Q(G_n) &= N_m(n) - np_m(\hat \delta_n)\\ &=\underbrace{\ensuremath{\mathbb{E}}_\ell[N_m(n)]-np_m(\delta_0)}_{:=A}+\underbrace{np_m(\delta_0)-np_m(\tilde \delta_n)}_{:=B}\\ &\qquad + \underbrace{N_m(n)-\ensuremath{\mathbb{E}}_\ell[N_m(n)]+np_m(\tilde \delta_n)-np_m(\hat \delta_n)}_{:=C}\ ,\end{aligned}$$ where $\ell\in\{0,1\}$. In the above $\tilde \delta_n$ is a deterministic quantity, and it is formally defined below. At this moment one might simply think of it as a "population" version of $\hat\delta_n$. Clearly $A$ is already characterized by Proposition [Proposition 8](#prp:A){reference-type="ref" reference="prp:A"}. The bulk of the argument needed to prove Theorem [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"} is in the characterization of $B$, the second term. For this we need to understand how fast $\tilde \delta_n$ converges to $\delta_0$ as $n\to\infty$. It turns out that under the alternative model both $A$ and $B$ have the same first-order asymptotic behavior, but with different leading constants. This fact is crucial to ensure that the test is powerful. Finally $C$, the last term, appears complicated but it is very well concentrated around zero. Define $\tilde\delta_n$ as a solution of $\ensuremath{\mathbb{E}}_\ell[\iota'_n(\delta)]=0$ in $\delta\in[\delta_{\min},\delta_{\max}]$ (if more than one solution exists, then we choose an arbitrary one). Most of the analysis focuses on the alternative model, but the stated results apply to the null model simply by taking $\delta_1=\delta_0$. The following proposition gives a characterization of $\tilde\delta_n$ and the term $B$ above: **Proposition 10**. *Let $\tfrac{1}{2}< \gamma<1$. Under the alternative model $\tilde \delta_n$ converges to $\delta_0$ as $n\to\infty$. Furthermore, $$\begin{aligned} \label{eqn:convergence_tilde_delta} \tilde\delta_n-\delta_0&=(1+\ensuremath{\mathrm{o}}(1))n^{\gamma-1} c(\delta_1-\delta_0)\frac{2m+\delta_0}{2m+\delta_1}\ ,\end{aligned}$$ and $$n(p_m(\delta_0)-p_m(\tilde\delta_n)) = (1+\ensuremath{\mathrm{o}}(1)) cn^{\gamma} \frac{1}{(m+\delta_0+2+\delta_0/m)^2}(\delta_1-\delta_0)\frac{2m+\delta_0}{2m+\delta_1}\ .$$* The proof of this result is rather involved, due to the implicit nature of the definition of $\tilde\delta_n$. Note, however, that $\iota'_n(\delta)$ is a score function, therefore we have immediately that $\ensuremath{\mathbb{E}}_0[\iota'_n(\delta_0)]=0$. Under the alternative model $\tilde\delta_n$ will not be equal to $\delta_0$ and quantifying this deviation is crucial to our analysis. The following technical result, examining the difference between $\ensuremath{\mathbb{E}}_1[\iota'_n(\delta)]$ and $\ensuremath{\mathbb{E}}_0[\iota'_n(\delta)]$, is instrumental: **Proposition 11** (Analysis of differences of means). *Let $\tfrac{1}{2}<\gamma<1$. As $n\rightarrow \infty$, and uniformly for every $\delta\in[\delta_{\min},\delta_{\max}]$, $$(n+1)\left(\ensuremath{\mathbb{E}}_1[\iota'_n(\delta)]-\ensuremath{\mathbb{E}}_0[\iota'_n(\delta)]\right) = \kappa(\delta_1,\delta_0,\delta) n^\gamma (1+\ensuremath{\mathrm{o}}(1))\ ,$$ where $\kappa(\delta_1,\delta_0,\delta)$ equals $$\label{kappa-delta0-delta1-def} \kappa \coloneqq \kappa(\delta_1,\delta_0,\delta)=(\delta_1-\delta_0)\frac{cm}{(2m + \delta_1)(2m + \delta_0)}\left(-1+\sum_{k\geq m}\frac{2m+\delta}{k+\delta}p_k(\delta_0)\right)\ .$$* The proof of this result, which is long and rather technical, is given in the appendix. With this result in hand we are ready to prove Proposition [Proposition 10](#prp:B){reference-type="ref" reference="prp:B"}: *Proof of Proposition [Proposition 10](#prp:B){reference-type="ref" reference="prp:B"}.* We first establish that $\tilde\delta_n$ is well defined and that $\tilde\delta_n\to\delta_0$. Define $$\label{eqn:limit_iota_prime} \iota'(\delta)\coloneqq \sum_{k\geq m} \frac{p_{>k}(\delta_0)}{k+\delta}-\frac{1}{2+\delta/m}\ ,$$ where $p_{>k}(\delta_0)=\sum_{j>k} p_j(\delta_0)$. Intuitively, this should be the limit of $\iota'_n(\delta)$ as $n\to\infty$, as we see next. Note also that we can rewrite this expression in terms of $p_k$, by noticing (see [@Gao2017 Lemma 2]) that, for $k\geq m$ $$p_{>k}(\delta_0)=\frac{k+\delta_0}{2+\delta_0/m}p_k(\delta_0)\ .$$ Therefore, $$\iota'(\delta)=\frac{1}{2+\delta_0/m}\sum_{k\geq m} \frac{k+\delta_0}{k+\delta}p_k(\delta_0) -\frac{1}{2+\delta/m} \ .$$ [@Gao2017 Lemma 6] shows essentially that, as $n\rightarrow \infty$, $$\sup_{\delta\in[\delta_{\min},\delta_{\max}]} |\iota'_n(\delta)-\iota'(\delta)| \xrightarrow{\ensuremath{\mathbb{P}}_0} 0\ .$$ [@Gao2017 Lemma 6] is actually stated for a more general setting, where the number of edges added at each step is *random*. However, if the support of the distribution of the number of edges is bounded below by $m$, then following the steps in the proof of the lemma leads to the above result. Actually, in our setting we can state a slightly stronger result, namely convergence in $L_1$. First note that both $|\iota'_n(\delta)|$ and $|\iota'(\delta)|$ are bounded, uniformly in $\delta$ and $n$. To see this note that, regardless of the value $k$, $$kN_{>k}(n)=k\sum_{\ell>k}N_\ell(n)\leq \sum_{\ell>k}\ell N_\ell(n)\leq \sum_{\ell\geq m}\ell N_\ell(n)=2nm\ .$$ Therefore, $$\begin{aligned} 0 &\leq \frac{1}{n+1}\sum_{k=m}^\infty \frac{1}{k+\delta}N_{>k}(n) \leq \frac{1}{n+1}\sum_{k=m}^\infty \frac{1}{k(k+\delta)}2nm\\ &\leq \frac{2nm}{n+1}\sum_{k=m}^\infty \frac{1}{k(k+\delta_{\min})} \leq m\sum_{k=m}^\infty \frac{1}{k(k+\delta_{\min})}\\ &\leq m\left(\frac{1}{m(m+\delta_{\min})}+\sum_{k=1}^\infty \frac{1}{k^2}\right) = m\left(\frac{1}{m(m+\delta_{\min})}+\pi^2/6\right)\ .\end{aligned}$$ On the other hand, $$\begin{aligned} 0&\leq \frac{1}{n+1}\sum_{t=2}^n \sum_{i=1}^m \frac{t}{t\delta+2m(t-1)+(i-1)} \leq \frac{m}{n+1}\sum_{t=2}^n \frac{t}{t\delta_{\min}+2m(t-1)}\\ &\leq \frac{m(n-1)}{n+1} \frac{2}{2(m+\delta_{\min})} \leq \frac{2m}{2(m+\delta_{\min})}\ .\end{aligned}$$ These two results together imply that $|\iota'_n(\delta)|$ is almost surely uniformly bounded for all $\delta\in[\delta_{\min},\delta_{\max}]$. For $\iota'$ we must simply note that this is a continuous function defined in the compact set $[\delta_{\min},\delta_{\max}]$, and is therefore bounded. In conclusion, as $n\to\infty$, $$\begin{aligned} \ensuremath{\mathbb{E}}_0\left[\sup_{\delta\in[\delta_{\min},\delta_{\max}]} |\iota'_n(\delta)-\iota'(\delta)|\right]\to 0\ .\end{aligned}$$ This result, together with Proposition [Proposition 11](#prp:ana-diff-means){reference-type="ref" reference="prp:ana-diff-means"}, shows that this is also true under the alternative model, and therefore, as $n\to\infty$, $$\label{eqn:L1_convergence} \forall \ell\in\{0,1\}\quad \ensuremath{\mathbb{E}}_\ell\left[\sup_{\delta\in[\delta_{\min},\delta_{\max}]} |\iota'_n(\delta)-\iota'(\delta)|\right]\to 0\ .$$ By definition $\ensuremath{\mathbb{E}}_1[\iota'_n(\tilde\delta_n)]=0$, therefore we conclude that $\iota'(\tilde\delta_n)\to 0$. [@Gao2017 Lemma 4] shows that $\iota'$ has a unique zero at $\delta_0$, and $\iota'(\delta)>0$ for $\delta<\delta_0$ and $\iota'(\delta)<0$ for $\delta>\delta_0$. This immediately implies that $\tilde\delta_n\to\delta_0$ as $n\to\infty$, proving the first assertion in the proposition. To quantify the speed of convergence note first that $\ensuremath{\mathbb{E}}_0[\iota'_n(\delta_0)]=0$, since $\iota'_n$ is a score function. Furthermore, by definition $\ensuremath{\mathbb{E}}_1[\iota'_n(\tilde\delta_n)]=0$, therefore Proposition [Proposition 11](#prp:ana-diff-means){reference-type="ref" reference="prp:ana-diff-means"} implies that $$\begin{aligned} 0&=\ensuremath{\mathbb{E}}_1[\iota'_n(\tilde\delta_n)]-\ensuremath{\mathbb{E}}_0[\iota'_n(\tilde\delta_n)]+\ensuremath{\mathbb{E}}_0[\iota'_n(\tilde\delta_n)]\\ &=\kappa(\delta_1,\delta_0,\tilde\delta_n) n^{\gamma-1} (1+\ensuremath{\mathrm{o}}(1))+\ensuremath{\mathbb{E}}_0[\iota'_n(\delta_0)+\iota''_n(\bar\delta_n)(\tilde\delta_n-\delta_0)]\\ &=\kappa(\delta_1,\delta_0,\tilde\delta_n) n^{\gamma-1} (1+\ensuremath{\mathrm{o}}(1))+\ensuremath{\mathbb{E}}_0[\iota''_n(\bar\delta_n)](\tilde\delta_n-\delta_0)\ ,\label{eqn:pop_est}\end{aligned}$$ where $|\bar\delta_n-\delta_0|\leq |\tilde\delta_n-\delta_0|$. Clearly $\bar\delta_n\to\delta_0$. As shown in [@Gao2017], $\iota''_n(\delta)$ also converges in probability to $\iota''(\delta)$ uniformly in $\delta\in[\delta_{\min},\delta_{\max}]$. In fact, convergence holds also in $L^1$, since $\iota''_n(\delta)$ is uniformly bounded (following the type of argument used before showing that $\iota'_n$ is uniformly bounded). In addition, it is also shown in [@Gao2017] that $\iota''(\delta_0)<0$. Therefore, for $n$ large enough, $$\ensuremath{\mathbb{E}}_0[\iota''_n(\bar\delta_n)]=(1+\ensuremath{\mathrm{o}}(1))\iota''(\delta_0)<0\ .$$ We are now ready to show the second assertion in Proposition [Proposition 10](#prp:B){reference-type="ref" reference="prp:B"}. Note that $\kappa(\delta_1,\delta_0,\tilde\delta_n)\to \kappa(\delta_1,\delta_0,\delta_0)$ since $\kappa$ is a continuous function. Putting all this together and re-writing the expression [\[eqn:pop_est\]](#eqn:pop_est){reference-type="eqref" reference="eqn:pop_est"} we conclude that $$\begin{aligned} \tilde\delta_n-\delta_0&=(1+\ensuremath{\mathrm{o}}(1))n^{\gamma-1} \frac{\kappa(\delta_1,\delta_0,\delta_0)}{|\iota''(\delta_0)|}\ ,\end{aligned}$$ where $$\begin{aligned} \iota''(\delta_0)&=\frac{m}{(2m+\delta_0)^2}-\sum_{k\geq m}\frac{p_{>k}(\delta_0)}{(k+\delta_0)^2}\\ &=\frac{m}{(2m+\delta_0)^2}-\frac{m}{2m+\delta_0}\sum_{k\geq m}\frac{p_k(\delta_0)}{k+\delta_0}\ .\end{aligned}$$ Therefore, after trivial algebraic manipulation, we conclude that $$\begin{aligned} \tilde\delta_n-\delta_0&=(1+\ensuremath{\mathrm{o}}(1))n^{\gamma-1} \frac{c(\delta_1-\delta_0)}{2m+\delta_1}\frac{-1+\sum_{k\geq m}\frac{2m+\delta_0}{k+\delta_0}p_k(\delta_0)}{{\sum_{k\geq m} \frac{1}{k+\delta_0}p_k(\delta_0)}-\frac{1}{2m+\delta_0}}\\ &=(1+\ensuremath{\mathrm{o}}(1))n^{\gamma-1} c(\delta_1-\delta_0)\frac{2m+\delta_0}{2m+\delta_1}\ ,\end{aligned}$$ proving the second assertion in the proposition. For the last assertion note that $\delta\mapsto p_m(\delta)$ is a continuously differentiable function, so that $$p_m(\tilde\delta_n)=p_m(\delta_0)+p'_m(\bar\delta_n)(\tilde\delta_n-\delta_0)\ ,$$ where $|\bar\delta_n-\delta_0|\leq |\tilde\delta_n-\delta_0|$ and $$\label{eqn:p_m_prime} p'_m(\delta)=-\frac{1}{(m+\delta+2+\delta/m)^2}\ .$$ Clearly $\bar\delta_n\to \delta_0$, and $p'_m(\delta_0)\neq 0$ and so $$\begin{aligned} \lefteqn{n(p_m(\delta_0)-p_m(\tilde\delta_n))}\\ &= -(1+\ensuremath{\mathrm{o}}(1)) np'_m(\delta_0)(\tilde\delta_n-\delta_0)\\ &= (1+\ensuremath{\mathrm{o}}(1)) n^{\gamma} \frac{1}{(m+\delta_0+2+\delta_0/m)^2}c(\delta_1-\delta_0)\frac{2m+\delta_0}{2m+\delta_1}\ .\end{aligned}$$ ◻ In conclusion, the sum of the terms $A$ and $B$ in [\[eqn:three_terms\]](#eqn:three_terms){reference-type="eqref" reference="eqn:three_terms"} equals $\alpha(\delta_0,\delta_1)(1+\ensuremath{\mathrm{o}}(1))n^{\gamma}$ with $$\begin{aligned} \alpha(\delta_0,\delta_1)&=c \left[1-\frac{p_m(\delta_0)}{p_m(\delta_1)}+\frac{1}{(m+\delta_0+2+\delta_0/m)^2}(\delta_1-\delta_0)\frac{2m+\delta_0}{2m+\delta_1}\right]\\ &=c(\delta_0-\delta_1)\frac{m+\delta_0}{(2+\delta_1/m)(m+\delta_0+2+\delta_0/m)^2}\ .\end{aligned}$$ Clearly, this takes the value $0$ when $\delta_0=\delta_1$, and it is non-zero otherwise. To complete the proof of Theorem [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"} we need to characterize the term $C$ in [\[eqn:three_terms\]](#eqn:three_terms){reference-type="eqref" reference="eqn:three_terms"}. This has two components, the first one already studied in the proof of Theorem [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"}. For the second component we must characterize the deviations of $\hat\delta_n$ around $\tilde\delta_n$. Again, due to the implicit definition of the estimator this requires some care. The results are summarized in the following proposition, proven in the appendix: **Proposition 12**. *The estimator $\hat\delta_n$ is consistent, both under the null and alternative models. Specifically, $$\ensuremath{\mathbb{E}}_\ell[|\hat\delta_n-\delta_0|]\to 0\ ,$$ as $n\to\infty$, where $\ell\in\{0,1\}$. In addition, $$|\hat\delta_n-\tilde\delta_n|=\ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_\ell}\left(a_n/n\right)$$ where $a_n=\omega(\sqrt{n}\log n)$. Finally, $$n(p_m(\hat\delta_n)-p_m(\tilde\delta_n))=\ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_\ell}(a_n)\ .$$* With these results in hand we are ready to complete the proof of Theorem [Theorem 4](#thm:main2){reference-type="ref" reference="thm:main2"}. We proceed separately for the type I and type II error: ## *Type-I error* Recall the definition of $a_n$ in Theorem [Theorem 4](#thm:main2){reference-type="ref" reference="thm:main2"}, which satisfies $a_n=\omega(\sqrt{n}\log n)$ and $a_n=n^{\tfrac{1}{2}+o(1)}$. Refer to the decomposition of the test statistic $Q(G_n)$ in [\[eqn:three_terms\]](#eqn:three_terms){reference-type="eqref" reference="eqn:three_terms"}. Under the null model the term $B$ is equal to zero, and for the term $A$ we know (from [\[eqn:bounded_difference\]](#eqn:bounded_difference){reference-type="eqref" reference="eqn:bounded_difference"}) that $$|\ensuremath{\mathbb{E}}_0[N_m(n)] - n \mspace{1mu} p_m(\delta_0)| \leq C_0(\delta_0,m)\coloneqq C_0\ .$$ Therefore $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{P}}_0\left(|Q(G_n)|\geq a_n\right)}\\ &= \ensuremath{\mathbb{P}}_0\left(\left|\ensuremath{\mathbb{E}}_0[N_m(n)]-np_m(\delta_0)+N_m(n)-\ensuremath{\mathbb{E}}_0[N_m(n)]+np_m(\tilde \delta_n)-np_m(\hat \delta_n)\right|\geq a_n\right)\\ &\leq \ensuremath{\mathbb{P}}_0\left(\left|N_m(n)-\ensuremath{\mathbb{E}}_0[N_m(n)]+np_m(\tilde \delta_n)-np_m(\hat \delta_n)\right|\geq a_n-C_0\right)\\ &\leq \ensuremath{\mathbb{P}}_0\left(\left|N_m(n)-\ensuremath{\mathbb{E}}_0[N_m(n)]\right|\geq \frac{a_n-C_0}{2}\right) + \ensuremath{\mathbb{P}}_0\left(\left|np_m(\tilde \delta_n)-np_m(\hat \delta_n)\right|\geq \frac{a_n-C_0}{2}\right)\\ &=\ensuremath{\mathrm{o}}(1)\ ,\end{aligned}$$ where the final result follows from [\[eqn:AH_Nm\]](#eqn:AH_Nm){reference-type="eqref" reference="eqn:AH_Nm"} and the last statement in Proposition [Proposition 12](#prp:C2){reference-type="ref" reference="prp:C2"}, since $a_n=\omega(\sqrt{n}\log n)$. ## *Type-II error* Referring again to the decomposition in [\[eqn:three_terms\]](#eqn:three_terms){reference-type="eqref" reference="eqn:three_terms"} we see that the terms $A$ and $B$ now play a crucial role. Namely, we have shown that the sum of the two terms equals $\alpha(\delta_0,\delta_1)n^{\gamma}+\ensuremath{\mathrm{o}}(n^{\gamma})$, where $$\begin{aligned} \alpha(\delta_0,\delta_1)&=c \left[1-\frac{p_m(\delta_0)}{p_m(\delta_1)}+\frac{1}{(m+\delta_0+2+\delta_0/m)^2}(\delta_1-\delta_0)\frac{2m+\delta_0}{2m+\delta_1}\right]\\ &=c(\delta_0-\delta_1)\frac{m+\delta_0}{(2+\delta_1/m)(m+\delta_0+2+\delta_0/m)^2}\label{eqn:shift_Q}\ .\end{aligned}$$ Clearly, this takes the value $0$ when $\delta_0=\delta_1$, and it is non-zero otherwise. The characterization of the term $C$ remains exactly the same as under the null model. Therefore, $$\ensuremath{\mathbb{E}}_\ell[N_m(n)]-np_m(\delta_0)+np_m(\delta_0)-np_m(\tilde \delta_n)=\alpha(\delta_0,\delta_1)(1+\ensuremath{\mathrm{o}}(1))n^{\gamma}=\omega(a_n)\ ,$$ since $a_n=n^{\tfrac{1}{2}+\ensuremath{\mathrm{o}}(1)}$ and $\gamma>\tfrac{1}{2}$, and therefore $$\ensuremath{\mathbb{P}}_1(|Q(G_n)|\geq \tau_n)=1$$ concluding the proof of Theorem [Theorem 4](#thm:main2){reference-type="ref" reference="thm:main2"}. 0◻ # Asymptotic normality proofs {#sec:proof_asymptotic_normality_test_statistics} This section is dedicated to the proof of Theorem [Theorem 7](#thm:asymptotic_normality_test_statistics){reference-type="ref" reference="thm:asymptotic_normality_test_statistics"}. The proof for the null model relies on a somewhat involved application of a martingale central limit theorem to an appropriately constructed martingale. The proof for the alternative model hinges on the null model result, as well as further estimates of the effect of the last $n-\tau_n$ vertices on the distribution of $N_k(n)$ for $k\geq m$. Since one proof hinges on the other, and to avoid any confusion between the null and alternative models, we separate these two cases in two sections. Specifically, in Section [9.1](#sec:proof_asymptotic_normality_null_model){reference-type="ref" reference="sec:proof_asymptotic_normality_null_model"} we prove the asymptotic normality of test statistics under the null model, i.e., [\[eq:asymptotic_normality_unknown_delta_test_statistic\]](#eq:asymptotic_normality_unknown_delta_test_statistic){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic"}. In Section [9.2](#sec:proof_asymptotic_normality_alternative_model){reference-type="ref" reference="sec:proof_asymptotic_normality_alternative_model"} we prove the asymptotic normality of test statistics under the alternative model, i.e., [\[eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis\]](#eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis"} and [\[eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis\]](#eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis"}. ## Asymptotic normality under the null hypothesis {#sec:proof_asymptotic_normality_null_model} Let $\mathcal{N}(\mu,\Sigma)$ denote the normal distribution with mean $\mu$ and covariance matrix $\Sigma$. The following result establishes the joint asymptotic normality of $N_m(n)$ and $\hat\delta_n$, which can be them used to deduce [\[eq:asymptotic_normality_unknown_delta_test_statistic\]](#eq:asymptotic_normality_unknown_delta_test_statistic){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic"} by an application of the delta method: **Proposition 13** (Joint normality of count of degree $m$ vertices and estimator for $\delta_0$). *Under the null model, as $n\to\infty$, $$\begin{aligned} \label{eq:asymptotic_normality_known_delta_test_statistic} \sqrt{n}\begin{pmatrix} \frac{N_m(n)-np_m(\delta_0)}{n} \\ \hat\delta_n-\delta_0 \end{pmatrix} \xrightarrow{D} \mathcal{N}\left(\mspace{-3mu}\begin{pmatrix} 0\\0 \end{pmatrix},\Sigma(\delta_0,m)\right)\ ,\end{aligned}$$ where the covariance matrix is $$\begin{aligned} \label{eq:asymptotic_normality_covariance_matrix} \Sigma(\delta_0,m) \coloneqq \begin{pmatrix} w(\delta_0,m) & -\frac{b(\delta_0,m)}{v(\delta_0,m)}\\ -\frac{b(\delta_0,m)}{v(\delta_0,m)} & \frac{1}{v(\delta_0,m)} \end{pmatrix}\ ,\end{aligned}$$ where $w$ and $v$ are defined in [\[w-var-def\]](#w-var-def){reference-type="eqref" reference="w-var-def"} and [\[v-var-def\]](#v-var-def){reference-type="eqref" reference="v-var-def"}, and $b(\delta_0,m)\coloneqq \frac{m^2}{(\delta_0 + m(2 + m + \delta_0))^2}$.* *Proof of Proposition [Proposition 13](#prop:joint_asymptotic_normality_degree_counts_estimator_delta_zero){reference-type="ref" reference="prop:joint_asymptotic_normality_degree_counts_estimator_delta_zero"}.* The convergence of the marginals in [\[eq:asymptotic_normality_known_delta_test_statistic\]](#eq:asymptotic_normality_known_delta_test_statistic){reference-type="eqref" reference="eq:asymptotic_normality_known_delta_test_statistic"} follows from results in the literature, which we briefly recall here. From the general result in [@Baldassarri2021] it follows that, under the null model, $$\begin{aligned} % \frac{N_m(n)-np_m(\delta_0)}{\sqrt{n}}\ \xrightarrow{D} \mathcal{N}\left(0,w(\delta_0,m)\right)\ ,\end{aligned}$$ as $n\to\infty$. Moreover, from [@Gao2017] follows that, under the null model, $$\begin{aligned} \sqrt{n}(\hat\delta_n-\delta_0)\ \xrightarrow{D} \mathcal{N}\left(0,1/v(\delta_0,m)\right)\ ,\end{aligned}$$ as $n\to\infty$. To show joint convergence, we apply the multivariate martingale central limit theorem (MMCLT) to (a linear transformation of) the vector on the left-hand side of [\[eq:asymptotic_normality_known_delta_test_statistic\]](#eq:asymptotic_normality_known_delta_test_statistic){reference-type="eqref" reference="eq:asymptotic_normality_known_delta_test_statistic"}. For more details on the MMCLT, see, e.g., [@crimaldi2005convergence] and references therein. In fact, the asymptotic normality of the marginals has been proven respectively in [@Baldassarri2021] and [@Gao2017] by applying a martingale central limit theorem to appropriately constructed martingale difference sequences $X_{t,i}$ and $Y_{t,i}$, which we define precisely later on. To obtain our results, we apply the MMCLT to the (square-integrable) martingale difference array $(X_{t,i}, Y_{t,i})$. The MMCLT requires two ingredients. First, the components of the array must satisfy the Lindeberg condition. Since this is also an ingredient for the univariate martingale central limit theorem, it has already been verified for $X_{t,i}$ in [@Baldassarri2021 Section 3.2] and for $Y_{t,i}$ in [@Gao2017 Section 4]. Second, one should compute the asymptotic expression for the covariance matrix. Note that the asymptotic variance of $X_{t,i}$ (resp. $Y_{t,i}$) has already been computed explicitly in [@Baldassarri2021] (resp. [@Gao2017]). We begin by introducing the notation required to define the martingale difference array $(X_{t,i}, Y_{t,i})$, which, roughly speaking, is a linear transformation of $(N_m(t), \hat\delta_t)$. We then compute the asymptotic covariance matrix of $(X_{t,i}, Y_{t,i})$. Finally, we perform a linear transformation to deduce a joint central limit theorem for $(N_m(n), \hat\delta_n)$. To construct the martingale difference sequences, it is important to consider the "intermediate" $m$ steps in the preferential attachment process, at each time $t$. For $i=1,\ldots,m-1$, we denote by $N_k(s,i)$ the number of vertices of degree $k$ after the $i$-th edge has been attached at time $s$, excluding the vertex $s$. We set $N_k(s+1,0)\coloneqq N_k(s,m) = N_k(s)$. We further define $D_{s,i}\coloneqq D_{v_{s,i}}(G_{s,i-1})$. In other words, $D_{s,i}$ is the degree of the vertex which will be attached to when constructing $G_{s,i}$ from $G_{s,i-1}$. For compactness, we define the constants $$\begin{aligned} % W_{t,i} \coloneqq \prod_{s=1}^{t-1} \prod_{j=1}^{m} a_{s,j} \prod_{j=1}^{i} a_{t,j}\qquad \text{where }\qquad a_{t,i} \coloneqq 1 - \frac{m+\delta_0}{S_{t,i-1}} \ ,\end{aligned}$$ and where $$\begin{aligned} % S_{t,i} &\coloneqq t \delta_0 + 2(t-1)m + i\ .\end{aligned}$$ Define also $$\begin{aligned} % % \xi &\coloneqq \frac{m+\delta_0}{2m + \delta_0}\ ,\\ % \Lambda &\coloneqq \prod_{j=0}^{m-1} \frac{\Gamma\bigl(1-\frac{2m-j}{2m+\delta_0}\bigr)}{\Gamma\bigl(1-\frac{3m-j+\delta_0}{2m+\delta_0}\bigr)}\ .\end{aligned}$$ In the constants defined above, we omitted the dependence on $\delta_0$ and $m$ to facilitate readability of the computations that follow. The martingale difference sequences (from [@Baldassarri2021; @Gao2017a]) are defined as $$\begin{aligned} X_{t,i} &\coloneqq \left(\frac{N_m(t,i) - a_{t,i-1} N_m(t,i-1) + \mathds 1{\{i = m\}}}{W_{t,i-1}}\right) \frac{1}{n^{\tfrac{1}{2} + m \xi}}\ ,\\ % Y_{t,i} &\coloneqq \left(\frac{1}{D_{t,i} + \delta_0} - \frac{t}{S_{t,i-1}}\right) \frac{1}{n^{\tfrac{1}{2}}}\ .\end{aligned}$$ Here $X_{t,i}$ is the first component of the martingale difference array in [@Baldassarri2021], and $Y_{t,i}$ is the martingale difference defined in [@Gao2017]. Next, we compute the asymptotic covariance matrix. By [@Baldassarri2021 Section 3.1] and [@Gao2017 Section 4], as $n\rightarrow \infty$, $$\begin{aligned} \sum_{t=2}^n \sum_{i=1}^m \ensuremath{\mathbb{E}}[X_{t,i}^2 | G_{t,i-1}] &\xrightarrow{\ensuremath{\mathbb{P}}_0} \frac{1}{\Lambda^2} \frac{m}{1 + 2 m \xi} p_m \xi \bigl(1 - p_m \xi\bigr)\\ &= \frac{1}{\Lambda^2} \frac{m^2 (m+\delta_0) (1 + m + \delta_0) (2m+\delta_0)}{(\delta_0 + 2m(1+m+\delta_0)) (\delta_0 + m(2+m+\delta_0))^2} \ , \label{eq:asymptotic_variance_X}\\ % \sum_{t=2}^n \sum_{i=1}^m \ensuremath{\mathbb{E}}[Y_{t,i}^2 | G_{t,i-1}] &\xrightarrow{\ensuremath{\mathbb{P}}_0} \sum_{k=1}^\infty \frac{m p_k}{(k+\delta_0)(2m+\delta_0)} - \frac{m}{(2m+\delta_0)^2}\label{eq:asymptotic_variance_Y}\ .\end{aligned}$$ We are left to explicitly compute $\ensuremath{\mathbb{E}}[X_{t,i} Y_{t,i} | G_{t,i-1}]$. First, since $\ensuremath{\mathbb{E}}[Y_{t,i} | G_{t,i-1}] = 0$, $$\begin{aligned} \ensuremath{\mathbb{E}}[X_{t,i} Y_{t,i} | G_{t,i-1}] &= \ensuremath{\mathbb{E}}\left[\frac{1}{n^{\tfrac{1}{2} + m \xi}} \frac{N_m(t,i)}{W_{t,i-1}} Y_{t,i} \middle| G_{t,i-1}\right]\\ % &= \frac{1}{n^{1 + m \xi}} \frac{1}{W_{t,i-1}} \ensuremath{\mathbb{E}}\left[\frac{N_m(t,i)}{D_{t,i} + \delta_0} \middle| G_{t,i-1}\right] \\ &\quad- \frac{1}{n^{1 + m \xi}} \frac{1}{W_{t,i-1}} \frac{t}{S_{t,i-1}} \ensuremath{\mathbb{E}}\left[N_m(t,i) \middle| G_{t,i-1}\right]\ .\end{aligned}$$ To continue, we determine the above two (conditional) expectations. We start by computing $$\begin{aligned} \ensuremath{\mathbb{E}}\left[\frac{N_m(t,i)}{D_{t,i} + \delta_0} \middle| G_{t,i-1}\right] &= \frac{N_m(t,i-1) - 1 + \mathds 1{\{i = m\}}}{m + \delta_0} \ensuremath{\mathbb{P}}(D_{t,i} = m | G_{t,i-1})\\ &\quad{} + \sum_{k=m+1}^\infty \frac{N_m(t,i-1) + \mathds 1{\{i = m\}}}{k + \delta_0} \ensuremath{\mathbb{P}}(D_{t,i} = k | G_{t,i-1}) \notag\\ % &= \sum_{k=m}^\infty \frac{N_m(t,i-1) + \mathds 1{\{i = m\}}}{k + \delta_0} \ensuremath{\mathbb{P}}(D_{t,i} = k | G_{t,i-1})\\ % &\quad- \frac{1}{m + \delta_0} \ensuremath{\mathbb{P}}(D_{t,i} = m | G_{t,i-1})\\ % &= \frac{t}{S_{t,i-1}} \bigl(N_m(t,i-1) + \mathds 1{\{i = m\}}\bigr) - \frac{N_m(t,i-1)}{S_{t,i-1}} \,.\end{aligned}$$ Also, from [@Baldassarri2021], $$\ensuremath{\mathbb{E}}\left[N_m(t,i) \middle| G_{t,i-1}\right] = \left(1 - \frac{m+\delta_0}{S_{t,i-1}}\right) N_m(t,i-1) + \mathds 1{\{i = m\}} \,.$$ Combining the above, we get $$\begin{aligned} \ensuremath{\mathbb{E}}[X_{t,i} Y_{t,i} | G_{t,i-1}] &= \frac{1}{n^{1 + m \xi}} \frac{1}{W_{t,i-1}} \ensuremath{\mathbb{E}}\left[\frac{N_m(t,i)}{D_{t,i} + \delta_0} \middle| G_{t,i-1}\right] \\ % &\quad- \frac{1}{n^{1 + m \xi}} \frac{1}{W_{t,i-1}} \frac{t}{S_{t,i-1}} \ensuremath{\mathbb{E}}\left[N_m(t,i) \middle| G_{t,i-1}\right]\\ % &= \frac{1}{n^{1 + m \xi}} \frac{1}{W_{t,i-1}} \frac{t}{S_{t,i-1}} \bigg[N_m(t,i-1) + \mathds 1{\{i = m\}} - \frac{N_m(t,i-1)}{t}\\ % &\hspace{90pt}{} - \left(1 - \frac{m+\delta_0}{S_{t,i-1}}\right) N_m(t,i-1) - \mathds 1{\{i = m\}}\bigg] \notag\\ % &= \frac{1}{n^{1 + m \xi}} \frac{N_m(t,i-1)}{W_{t,i-1} S_{t,i-1}} \left[\frac{t}{S_{t,i-1}}(m+\delta_0) - 1\right] \,.\end{aligned}$$ To compute the limiting correlation we use that, uniformly over $i = 1, \ldots, m$, as $t\to\infty$, $$\frac{N_m(t,i)}{t} \xrightarrow{\ensuremath{\mathbb{P}}_0} p_m\,, \qquad \frac{S_{t,i}}{t} \to 2 + \delta_0\ .$$ Furthermore, tedious but straightforward computations show that, uniformly over $i = 1, \ldots, m$, as $t\to\infty$, $$W_{t,i} \asymp \frac{\Lambda}{t^{m \xi}}\ ,$$ where, crucially, $\Lambda$ is not a function of $t$. Hence, as $t\to\infty$, $$\ensuremath{\mathbb{E}}[X_{t,i} Y_{t,i} | G_{t,i-1}] = \frac{1}{n^{1 + m \xi}} \frac{t^{m \xi}}{\Lambda} \frac{p_m}{(2m + \delta_0)} (\xi - 1)(1+\ensuremath{\ensuremath{\mathrm{o}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(1)) \,.$$ Summing these terms and using $\sum_{t=1}^n t^{x-1} \asymp n^{x} / x$ as $n \to \infty$, gives $$\begin{aligned} \label{eq:asymptotic_covariance_final_result} \sum_{t=2}^n \sum_{i=1}^m \ensuremath{\mathbb{E}}[X_{t,i} Y_{t,i} | G_{t,i-1}] &\xrightarrow{\ensuremath{\mathbb{P}}_0} \frac{1}{\Lambda} \frac{m p_m}{(2m + \delta_0)} \frac{\xi - 1}{1 + m\xi}\\ % &\quad= - \frac{m^2}{(\delta_0 + m(2 + m + \delta_0))^2} \frac{1}{\Lambda}\ .\end{aligned}$$ In conclusion, putting together [\[eq:asymptotic_variance_X\]](#eq:asymptotic_variance_X){reference-type="eqref" reference="eq:asymptotic_variance_X"}, [\[eq:asymptotic_variance_Y\]](#eq:asymptotic_variance_Y){reference-type="eqref" reference="eq:asymptotic_variance_Y"}, and [\[eq:asymptotic_covariance_final_result\]](#eq:asymptotic_covariance_final_result){reference-type="eqref" reference="eq:asymptotic_covariance_final_result"} and applying the MMCLT (see, e.g., [@crimaldi2005convergence]) gives, as $n\to\infty$, $$\sqrt{n} \left(\mspace{-3mu} \begin{pmatrix} \frac{N_m(n)/n}{\Lambda} \\ \iota_n'(\delta_0) \end{pmatrix} - \begin{pmatrix} \frac{p_m}{\Lambda} \\ \iota'(\delta_0) \end{pmatrix}\mspace{-3mu} \right) \xrightarrow{D}\mathcal{N}\left(\mspace{-3mu}\begin{pmatrix} 0\\0 \end{pmatrix},\tilde{\Sigma}\right)\ ,\label{eq:asymptotic_normality_proof_intermediate_result}$$ where $$\widetilde{\Sigma} \coloneqq \begin{pmatrix} \frac{1}{\Lambda^2} \frac{m^2 (m+\delta_0) (1 + m + \delta_0) (2m+\delta_0)}{(\delta_0 + 2m(1+m+\delta_0)) (\delta_0 + m(2+m+\delta_0))^2} & -\frac{1}{\Lambda} \frac{m^2}{(\delta_0 + m(2 + m + \delta_0))^2}\\ -\frac{1}{\Lambda} \frac{m^2}{(\delta_0 + m(2 + m + \delta_0))^2} & v(\delta_0,m) \end{pmatrix}\ .$$ Recall that $v(\delta_0,m)$ is given in [\[v-var-def\]](#v-var-def){reference-type="eqref" reference="v-var-def"}. Note that the above central limit theorem result is not for $(N_m(n) / n, \hat{\delta}_n)$. This is because the martingale difference sequences are defined as a linear transformation of $(N_m(n) / n, \hat{\delta}_n)$. Therefore, next we apply a linear transformation to [\[eq:asymptotic_normality_proof_intermediate_result\]](#eq:asymptotic_normality_proof_intermediate_result){reference-type="eqref" reference="eq:asymptotic_normality_proof_intermediate_result"} to obtain a central limit theorem for $(N_m(n) / n, \hat{\delta}_n)$. To this end, observe that (see [@Gao2017 proof of Theorem 2]), $$\sqrt{n}(\hat{\delta}_n - \delta_0) = -\sqrt{n}\frac{(\iota_n'(\delta_0) - \iota'(\delta_0))}{\iota_n''(\bar\delta_n)} \,,$$ where $\bar\delta_n$ lies between $\delta_0$ and $\hat{\delta}_n$, and so $\iota_n''(\bar\delta_n) \xrightarrow{\ensuremath{\mathbb{P}}_0} \iota''(\delta_0) = -v(\delta_0,m)$. Using the above we obtain $$\begin{aligned} % \sqrt{n}(N_m(n)/n - p_m) &= \sqrt{n}\Lambda\left(\frac{N_m(n)/n}{\Lambda} -\frac{p_m}{\Lambda}\right)\\ \sqrt{n}(\hat{\delta}_n-\delta_0) &= \frac{\sqrt{n}}{v(\delta_0,m)}(\iota_n'(\delta_0) - \iota'(\delta_0))\ .\end{aligned}$$ In conclusion, using Slutsky's lemma, as $n\to\infty$, $$\begin{aligned} % \sqrt{n} \left(\mspace{-3mu} \begin{pmatrix} N_m(n)/n \\ \hat{\delta}_n \end{pmatrix} - \begin{pmatrix} p_m \\ \delta_0 \end{pmatrix}\mspace{-3mu} \right) \xrightarrow{D} \mathcal{N}\left(\mspace{-3mu}\begin{pmatrix} 0\\0 \end{pmatrix},\Sigma(\delta_0,m)\right)\ ,\end{aligned}$$ where $$\Sigma(\delta_0,m) \coloneqq \begin{pmatrix} \frac{m^2 (m+\delta) (1 + m + \delta) (2m+\delta)}{(\delta + 2m(1+m+\delta)) (\delta + m(2+m+\delta))^2} & -\frac{m^2}{(\delta + m(2 + m + \delta))^2} \frac{1}{v(\delta_0,m)}\\ -\frac{m^2}{(\delta + m(2 + m + \delta))^2} \frac{1}{v(\delta_0,m)} & \frac{1}{v(\delta_0,m)} \end{pmatrix}\ .$$ ◻ The proof of [\[eq:asymptotic_normality_unknown_delta_test_statistic\]](#eq:asymptotic_normality_unknown_delta_test_statistic){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic"} is now a simple consequence of Proposition [Proposition 13](#prop:joint_asymptotic_normality_degree_counts_estimator_delta_zero){reference-type="ref" reference="prop:joint_asymptotic_normality_degree_counts_estimator_delta_zero"} together with the delta method. Define the function $h(x, y) \coloneqq x - p_m(y)$. The gradient of $h(\cdot, \cdot)$ is given by $$\nabla h(x, y) = (1, -p_m'(y)) = \Bigl( 1,\; \frac{m^2}{(y + m(2 + m + y))^2} \Bigr),$$ where $p_m'(y)$ is given in [\[eqn:p_m\_prime\]](#eqn:p_m_prime){reference-type="eqref" reference="eqn:p_m_prime"}. Using $h(\cdot,\cdot)$, we can rewrite $$\label{eq:unknown_delta_test_statistic2} \frac{Q(G_n)}{\sqrt n} = \frac{N_m(n) - n \mspace{1mu} p_m(\hat{\delta}_n)}{\sqrt{n}} = \sqrt{n} \mspace{2mu} h\bigl(N_m(G_n)/n, \hat{\delta}_n\bigr)\,.$$ Hence, by applying the delta method we get that [\[eq:unknown_delta_test_statistic2\]](#eq:unknown_delta_test_statistic2){reference-type="eqref" reference="eq:unknown_delta_test_statistic2"} converges in distribution as $n\to\infty$ to a normal distribution with variance $$\begin{aligned} \label{eq:asymptotic_variance} &\lim_{n \to \infty} \mathrm{Var}\Big(\frac{N_m(G_n) - n \mspace{1mu} p_m(\hat{\delta}_n)}{\sqrt{n}}\Big)= \bigl(\nabla h(p_m(\delta_0), \delta_0)\bigr)^{\! T} \: \Sigma(\delta_0,m)\; \bigl(\nabla h(p_m(\delta_0), \delta_0)\bigr)\\ % &\quad= \frac{m^2 (m+\delta_0) (1 + m + \delta_0) (2m + \delta_0)}{(\delta_0 + 2m(1 + m + \delta_0)) (\delta_0 + m(2 + m + \delta_0))^2} - \frac{m^4}{v(\delta_0,m) (\delta_0 + m(2 + m + \delta_0))^4} \,. \notag\end{aligned}$$ where $\Sigma(\delta_0,m)$ and $v(\delta_0,m)$ are given by [\[eq:asymptotic_normality_covariance_matrix\]](#eq:asymptotic_normality_covariance_matrix){reference-type="eqref" reference="eq:asymptotic_normality_covariance_matrix"} and [\[v-var-def\]](#v-var-def){reference-type="eqref" reference="v-var-def"}, respectively. 0◻ ## Asymptotic normality under the alternative hypothesis {#sec:proof_asymptotic_normality_alternative_model} The main insight is that most of the contribution for the asymptotic distribution of the degree counts is due to the attachment process up to the changepoint. In fact, the asymptotic distribution of $(N_m(n),N_{m+1}(n),\ldots)$ is normal, both under the null and alternative models, with exactly the same covariance structure but with different means. Specifically, in [@Baldassarri2021] it was shown that under the null model $$\left(\frac{N_k(n)-\ensuremath{\mathbb{E}}_0[N_k(n)]}{\sqrt{n}}\right)_{k\geq m} \xRightarrow{D} \left(Z_k\right)_{k\geq m}\ ,$$ as $n\to\infty$, where the right-hand side is a zero-mean Gaussian process with covariance given in [@Baldassarri2021 Theorem 2.5], and the notation $(X_k(n))_{k\geq m} \xRightarrow{\smash{\raisebox{-1.0pt}{$\scriptstyle{}D$}}} (Z_k)_{k\geq m}$ means that for any $k\geq m$ we have $(X_m(n),\ldots,X_k(n)) \xrightarrow{\smash{\raisebox{-1.5pt}{$\scriptstyle{}D$}}} (Z_m,\ldots,Z_k)$ as $n\to\infty$ (i.e., the infinite vector converges in the product topology). The following lemma generalizes this result to our alternative model, immediately implying [\[eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis\]](#eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis"}. Furthermore, it provides a stepping stone towards the proof of [\[eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis\]](#eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis"}: **Lemma 14**. *Under the alternative model with $\gamma\in(0,1)$ $$\left(\frac{N_k(n)-\ensuremath{\mathbb{E}}_1[N_k(n)]}{\sqrt{n}}\right)_{k\geq m} \xRightarrow{D} \left(Z_k\right)_{k\geq m}\ ,$$ as $n\to\infty$ where the right-hand side is a zero-mean Gaussian process with covariance given in [@Baldassarri2021 Theorem 2.5].* *Proof.* Let $k\geq m$ be arbitrary. We must show that the standardized version of $\left(N_m(n),\ldots,N_k(n)\right)$ is asymptotically normal with the correct covariance structure. Let us first inspect the asymptotic marginal distributions of $N_k(n)$, as this illustrates the main premise of the argument, that can be extended easily by using an application of the Cramér-Wold device. Proceed by using the following decomposition $$\begin{aligned} \frac{1}{\sqrt{n}}\left(N_k(n)-\ensuremath{\mathbb{E}}_1[N_k(n)]\right)&=\frac{1}{\sqrt{n}}\left(\ensuremath{\mathbb{E}}_1\left[\left.N_k(n)\right|G_{\tau_n}\right]-\ensuremath{\mathbb{E}}_1\left[N_k(n)\right]\right)\\ &\qquad +\frac{1}{\sqrt{n}}\left(N_k(n)-\ensuremath{\mathbb{E}}_1\left[\left.N_k(n)\right|G_{\tau_n}\right]\right)\ .\label{eqn:decomposition}\end{aligned}$$ The second term in [\[eqn:decomposition\]](#eqn:decomposition){reference-type="eqref" reference="eqn:decomposition"} can be dealt with conveniently using the Azuma-Hoeffding inequality. Define the Doob martingale, for $t\in [n]\setminus [\tau_n]$, $$M_k(t)=\ensuremath{\mathbb{E}}_1\left[N_k(n)\mid G_t\right]\ .$$ As used before we know that almost surely $|M_k(t)-M_k(t-1)|\leq 2m$. Therefore, by the Azuma-Hoeffding inequality, $$\ensuremath{\mathbb{P}}_1\left(|N_k(n)-\ensuremath{\mathbb{E}}_1\left[\left.N_k(n)\right|G_{\tau_n}\right]|\geq x \right)\leq 2\ensuremath{\mathrm{e}}^{-\frac{x^2}{8m^2 (n-\tau_n)}}\ .$$ Thus, the second term in [\[eqn:decomposition\]](#eqn:decomposition){reference-type="eqref" reference="eqn:decomposition"} is $\ensuremath{\mathrm{O}}_{\ensuremath{\mathbb{P}}_1}(\sqrt{n-\tau_n}/\sqrt{n})=\ensuremath{\mathrm{O}}_{\ensuremath{\mathbb{P}}_1}(n^{(\gamma-1)/2})=\ensuremath{\mathrm{o}}(1)$ when $\gamma<1$. We now shift the focus to the first term in [\[eqn:decomposition\]](#eqn:decomposition){reference-type="eqref" reference="eqn:decomposition"}. Define $$N_k(\tau_n,n)=\sum_{v\in[\tau_n]} \mathds 1\{D_v(n)=k\}\ .$$ In words, this is the number of vertices of degree $k$ that were added to the graph $G_n$ up to the changepoint. Note that $$N_k(n)=N_k(\tau_n,n)+\sum_{v\in[n]\setminus[\tau_n]} \mathds 1\{D_v(n)=k\}\ .$$ Given the affine nature of the preferential attachment function (after time $\tau_n$) we know that $\sum_{v\in[n]\setminus[\tau_n]} \mathds 1\{D_v(n)=k\}$ is independent of $G_{\tau_n}$. Therefore, we can simplify the first term in [\[eqn:decomposition\]](#eqn:decomposition){reference-type="eqref" reference="eqn:decomposition"} as $$\begin{aligned} \frac{1}{\sqrt{n}}\left(\ensuremath{\mathbb{E}}_1\left[\left.N_k(n)\right|G_{\tau_n}\right]-\ensuremath{\mathbb{E}}_1\left[N_k(n)\right]\right) &= \frac{1}{\sqrt{n}}\left(\ensuremath{\mathbb{E}}_1\left[\left.N_k(\tau_n,n)\right|G_{\tau_n}\right]-\ensuremath{\mathbb{E}}_1\left[N_k(\tau_n,n)\right]\right)\ .\end{aligned}$$ At this point, it is useful to write $\ensuremath{\mathbb{E}}_1\left[\left.N_k(\tau_n,n)\right|G_{\tau_n}\right]$ in a slightly more explicit way. Note that $$\begin{aligned} \ensuremath{\mathbb{E}}_1\left[\left.N_k(\tau_n,n)\right|G_{\tau_n}\right] &=\ensuremath{\mathbb{E}}_1\left[\left. \sum_{v\in[\tau_n]} \mathds 1\{D_v(n)=k\}\right|G_{\tau_n}\right]\\ &=\sum_{v\in[\tau_n]} \sum_{j=m}^k \mathds 1\{D_v(\tau_n)=j\} \underbrace{\ensuremath{\mathbb{P}}_1\left(\left. D_v(n)=k\right|D_v(\tau_n)=j\right)}_{\coloneqq p_{j,k}(\tau_n,n)}\\ &=\sum_{j=m}^k N_j(\tau_n) p_{j,k}(\tau_n,n)\ .\end{aligned}$$ Using this we immediately see that $$\ensuremath{\mathbb{E}}_1[N_k(\tau_n,n)]=\sum_{j=m}^k \ensuremath{\mathbb{E}}_1[N_j(\tau_n)] p_{j,k}(\tau_n,n)=\sum_{j=m}^k \ensuremath{\mathbb{E}}_0[N_j(\tau_n)] p_{j,k}(\tau_n,n)\ .$$ In conclusion $$\begin{aligned} \lefteqn{\frac{1}{\sqrt{n}}\left(\ensuremath{\mathbb{E}}_1\left[\left.N_k(n)\right|G_{\tau_n}\right]-\ensuremath{\mathbb{E}}_1\left[N_k(n)\right]\right)}\\ &= \frac{1}{\sqrt{n}} \sum_{j=m}^k \left(N_j(\tau_n) -\ensuremath{\mathbb{E}}_0[N_j(\tau_n)]\right) p_{j,k}(\tau_n,n)\\ &= \sum_{j=m}^k \frac{1}{\sqrt{\tau_n}}\left(N_j(\tau_n) -\ensuremath{\mathbb{E}}_0[N_j(\tau_n)]\right) \sqrt{\frac{\tau_n}{n}}p_{j,k}(\tau_n,n)\ .\label{eqn:almost_final}\end{aligned}$$ To proceed all that is needed is to characterize $p_{j,k}(\tau_n,n)$. Lemma [Lemma 9](#lem:crucial){reference-type="ref" reference="lem:crucial"} immediately provides the necessary result, specifically $$p_{k,k}(\tau_n,n)=1-(1+\ensuremath{\mathrm{o}}(1))cn^{\gamma-1}m\frac{k+\delta_1}{2m+\delta_1}=1+\ensuremath{\mathrm{o}}(1)$$ and for $j<k$ $$p_{j,k}(\tau_n,n)\leq \ensuremath{\mathbb{P}}_1\left(\left. D_v(n)>k\right|D_v(\tau_n)=j\right)=\ensuremath{\mathrm{o}}(1)\ .$$ In conclusion, since $\sqrt{\tau_n/n}\to 1$ and $p_{k,l}(\tau_n,n)\to \mathds 1\{j=k\}$ as $n\to\infty$ we conclude that [\[eqn:almost_final\]](#eqn:almost_final){reference-type="eqref" reference="eqn:almost_final"} converges to the same normal distribution as $\frac{1}{\sqrt{n}}\left(N_k(n) -\ensuremath{\mathbb{E}}_0[N_k(n)]\right)$ under the null model. Owing to the linearity of [\[eqn:almost_final\]](#eqn:almost_final){reference-type="eqref" reference="eqn:almost_final"}, the same argument also shows that any finite linear combination of the (centered and rescaled) elements of $\left(N_m(n),\ldots,N_k(n)\right)$ is asymptotically normal with the appropriate variance. An application of the Cramér-Wold device then shows that, for any $k\geq m$, this vector converges in distribution to the desired finite-dimensional multi-variate Gaussian distribution. This concludes the proof. ◻ With this lemma at hand [\[eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis\]](#eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_known_delta_test_statistic_alternative_hypothesis"} is immediate. The second result [\[eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis\]](#eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis"} is, however, not a trivial consequence of the lemma, as the convergence in the product topology in Lemma [Lemma 14](#lem:joint_CLT){reference-type="ref" reference="lem:joint_CLT"} is unfortunately not sufficient to obtain the final result. To formally show [\[eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis\]](#eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis){reference-type="eqref" reference="eq:asymptotic_normality_unknown_delta_test_statistic_alternative_hypothesis"}, begin by recalling that $\iota'_n(\hat\delta_n)=0$. Expanding $\delta\mapsto\iota'_n(\delta)$ around $\delta_0$ we get $$\begin{aligned} \hat \delta_n-\delta_0&=-\frac{\iota'_n(\delta_0)}{\iota''_n(\bar \delta_n)}\ ,\end{aligned}$$ where $\bar \delta_n=\delta_0+\bar\zeta_n (\hat\delta_n-\delta_0)$ for some $\bar\zeta_n\in[0,1]$. Therefore, and considering also a Taylor expansion of $p_m(\hat\delta_n)$, we conclude that $$\begin{aligned} % Q(g_n) &= N_m(n)-np_m(\hat\delta_n)\\ &=N_m(n)-np_m(\delta_0)+n\frac{p_m'(\breve \delta_n)}{\iota''_n(\bar \delta_n)}\iota'_n(\delta_0)\ ,\end{aligned}$$ where $\breve \delta_n=\delta_0+\breve\zeta_n (\hat\delta_n-\delta_0)$ for some $\breve\zeta_n\in[0,1]$. Since $\hat\delta_n \xrightarrow{\ensuremath{\mathbb{P}}_1} \delta_0$ as $n\to\infty$, and using the uniform convergence results of $\iota''$ from [@Gao2017] (as already used in the proof of Proposition [Proposition 10](#prp:B){reference-type="ref" reference="prp:B"}) we conclude that $$\begin{aligned} % \frac{p'_m(\check\delta_n)}{{\iota''_n(\bar \delta_n)}} \xrightarrow{\ensuremath{\mathbb{P}}_1} \frac{p'_m(\delta_0)}{\iota''(\delta_0)}\neq 0.\end{aligned}$$ To aid the presentation we rewrite $Q(G_n)$ as $$\begin{aligned} % Q(G_n) &= N_m(n)-np_m(\delta_0)+n\frac{p'_m(\delta_0)}{\iota''(\delta_0)}\iota'_n(\delta_0)\\ &\qquad + n\left(\frac{p'_m(\breve\delta_0)}{\iota''_n(\bar\delta_0)}-\frac{p'_m(\delta_0)}{\iota''(\delta_0)}\right)\iota'_n(\delta_0)\ .\label{eq:Q_asympt_normal_second_term_rewritten}\end{aligned}$$ We argue that the last term is negligible, and therefore it suffices to characterize the asymptotic normality of the first two terms. Note that $N_m(n)$ and $\iota'_n(\delta_0)$ are not independent. To avoid unnecessarily cluttering the presentation we focus first on the asymptotic normality of $\iota'_n(\delta_0)$ and then argue that extending the analysis to the joint normality is straightforward. The following result generalizes [@Gao2017 Lemma 7] to the alternative model: **Lemma 15**. *Recall the definition of $v$ in [\[v-var-def\]](#v-var-def){reference-type="eqref" reference="v-var-def"}. Under the alternative model, $$\begin{aligned} % \sqrt{n}(\iota_n'(\delta_0) - \mathbb E_1[\iota_n'(\delta_0)]) \xrightarrow{D} \mathcal N(0, v(\delta_0,m))\ .\end{aligned}$$* *Proof.* We proceed similarly as in the proof of Lemma [Lemma 14](#lem:joint_CLT){reference-type="ref" reference="lem:joint_CLT"} by isolating the contributions of the vertices that join after the changepoint as follows $$\begin{aligned} % \sqrt{n}(\iota_n'(\delta_0) - \ensuremath{\mathbb{E}}_1[\iota_n'(\delta_0)]) &= \frac{\sqrt{n}}{n+1}\sum_{k\geq m}\frac1{k+\delta_0}(N_{>k}(n) - \ensuremath{\mathbb{E}}_1[N_{>k}(n)\vert G_{\tau_n}]) \\ % &\quad+ \frac{\sqrt{n}}{n+1}\sum_{k\geq m}\frac1{k+\delta_0}(\ensuremath{\mathbb{E}}_1[ N_{>k}(n)\vert G_{\tau_n}] - \ensuremath{\mathbb{E}}_1[ N_{>k}(n) ])\ .\label{eq:iota_prime_asymptotically_normal_decomposition}\end{aligned}$$ Following the exact same argument as in the proof of Lemma [Lemma 17](#lem:AZ_delta){reference-type="ref" reference="lem:AZ_delta"} the first term on the right-hand side is sufficiently small since $$\begin{aligned} % \ensuremath{\mathbb{P}}_1(n\vert \iota'_n(\delta_0) - \ensuremath{\mathbb{E}}_1[\iota'_n(\delta_0)\vert G_{\tau_n}]\vert \geq x) \leq 2 \exp\left(\frac{x^2}{2(n-\tau_n)c^2_{n,m}}\right)\ ,\end{aligned}$$ where $c_{n,m} = 2m\log(n)(1+o(1))$, and thus $$\begin{aligned} % \sqrt{n}(\iota'_n(\delta_0) - \ensuremath{\mathbb{E}}_1[\iota'_n(\delta_0)\vert G_{\tau_n}]) = \ensuremath{\mathrm{O}}_{\ensuremath{\mathbb{P}}}(n^{\gamma-1}\log(n)) = \ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_1}(1)\ .\end{aligned}$$ For the following term, it is convenient to introduce a convenient notation, similarly as done in the proof of Lemma [Lemma 14](#lem:joint_CLT){reference-type="ref" reference="lem:joint_CLT"}: $$\begin{aligned} % N_{>k}(\tau_n,n) := \sum_{v\in[\tau_n]}\mathds 1\{D_v(n)>k\}\ .\end{aligned}$$ Note that $$\begin{aligned} % \ensuremath{\mathbb{E}}_1[N_{>k}(n)\vert G_{\tau_n}] - \ensuremath{\mathbb{E}}_1[N_{>k}(n)] &= \ensuremath{\mathbb{E}}_1[N_{>k}(\tau_n, n)\vert G_{\tau_n}] - \ensuremath{\mathbb{E}}_1[N_{>k}(\tau_n, n)]\ ,\end{aligned}$$ and also $$\begin{aligned} \ensuremath{\mathbb{E}}_1[N_{>k}(n)\vert G_{\tau_n}] &= N_{>k}(\tau_n)+\sum_{v\in[\tau_n]} \mathds 1\{D_v(\tau_n)\leq k\}\underbrace{\ensuremath{\mathbb{P}}_1(D_v(n)>k\vert D_v(\tau_n)\leq k)}_{\coloneqq q_{k}(\tau_n,n)}\ .\end{aligned}$$ With this in hand, the second term on the right-hand side of [\[eq:iota_prime_asymptotically_normal_decomposition\]](#eq:iota_prime_asymptotically_normal_decomposition){reference-type="eqref" reference="eq:iota_prime_asymptotically_normal_decomposition"} is $$\begin{aligned} % &\frac{\sqrt{n}}{n+1}\sum_{k\geq m}\frac1{k+\delta_0}\left(\ensuremath{\mathbb{E}}_1[ N_{>k}(\tau_n, n)\vert G_{\tau_n}] - \ensuremath{\mathbb{E}}_1[ N_{>k}(\tau_n, n) ]\right) \\ % &= \frac{\sqrt{n\tau_n}}{n+1}\frac1{\sqrt{\tau_n}}\sum_{k\geq m}\frac1{k+\delta_0}\left( N_{>k}(\tau_n) - \ensuremath{\mathbb{E}}_1[ N_{>k}(\tau_n) ]\right)\\ % &\qquad+\frac{\sqrt{n}}{n+1}\sum_{k\geq m}\frac{q_{k}(\tau_n,n)}{k+\delta_0}\sum_{v\in[\tau_n]} \left(\mathds 1\{D_v(\tau_n)\leq k\} - \ensuremath{\mathbb{P}}_1(D_v(\tau_n)\leq k)\right)\\ &= \frac{\sqrt{n\tau_n}}{n+1}\frac1{\sqrt{\tau_n}}\sum_{k\geq m}\frac1{k+\delta_0}\left( N_{>k}(\tau_n) - \ensuremath{\mathbb{E}}_0[ N_{>k}(\tau_n) ]\right) \label{eq:iota_prime_asymptotically_normal_decomposition_main}\\ % &\qquad-\frac{\sqrt{n}}{n+1}\sum_{k\geq m}\frac{q_{k}(\tau_n,n)}{k+\delta_0}\sum_{v\in[\tau_n]} \left(\mathds 1\{D_v(\tau_n)> k\} - \ensuremath{\mathbb{P}}_0(D_v(\tau_n)> k)\right)\ .\label{eq:iota_prime_asymptotically_normal_decomposition_extra}\end{aligned}$$ The term [\[eq:iota_prime_asymptotically_normal_decomposition_main\]](#eq:iota_prime_asymptotically_normal_decomposition_main){reference-type="eqref" reference="eq:iota_prime_asymptotically_normal_decomposition_main"} converges in law to $\mathcal N(0,v(\delta_0,m))$ by [@Gao2017 Lemma 7]. To finalize our argument, we are left to prove that the term [\[eq:iota_prime_asymptotically_normal_decomposition_extra\]](#eq:iota_prime_asymptotically_normal_decomposition_extra){reference-type="eqref" reference="eq:iota_prime_asymptotically_normal_decomposition_extra"} is negligible. We do this by rewriting it as a Doob martingale difference, similarly as was done earlier in the proof. Let $$\begin{aligned} % M_t \coloneqq \frac{\sqrt{n}}{n+1}\ensuremath{\mathbb{E}}_1\left[\left.\sum_{k\geq m}\frac{q_{k}(\tau_n,n)}{k+\delta_0} N_{>k}(\tau_n)\right| G_t\right]\ .\end{aligned}$$ With this notation [\[eq:iota_prime_asymptotically_normal_decomposition_extra\]](#eq:iota_prime_asymptotically_normal_decomposition_extra){reference-type="eqref" reference="eq:iota_prime_asymptotically_normal_decomposition_extra"} is precisely $M_{\tau_n}-M_1$. Furthermore, the martingale differences are bounded by $$\begin{aligned} % \left| M_t-M_{t-1}\right| &\leq \frac{\sqrt{n}}{n+1}\sum_{k\geq m}\frac{2m}{k+\delta_0}q_k(\tau_n,n)\\ &= \frac{2m\sqrt{n}}{(n+1)\tau_n}\sum_{k\geq m}\sum_{v\in[\tau_n]}\frac{1}{k+\delta_0}\frac{\ensuremath{\mathbb{P}}_1(D_v(n)>k,D_v(\tau_n)\leq k)}{\ensuremath{\mathbb{P}}_1(D_v(\tau_n)\leq k)}\\ &\leq \frac{2m\sqrt{n}}{(n+1)\tau_n}\sum_{k\geq m}\sum_{v\in[\tau_n]}\frac{\ensuremath{\mathbb{P}}_1(D_v(n)>k,D_v(\tau_n)\leq k)}{k+\delta_0}\frac{1}{\ensuremath{\mathbb{P}}_0(D_v(\tau_n)\leq m)}\\ &\leq \frac{2mn^{-3/2}}{p_m(\delta_0)}(1+\ensuremath{\mathrm{o}}(1))\sum_{k\geq m}\sum_{v\in[\tau_n]}\frac{\ensuremath{\mathbb{P}}_1(D_v(n)>k,D_v(\tau_n)\leq k)}{k+\delta_0}\ .\end{aligned}$$ The double-summation is controlled in the proof of Proposition [Proposition 11](#prp:ana-diff-means){reference-type="ref" reference="prp:ana-diff-means"}, and it is the sum of the terms in expressions [\[eqn:series1\]](#eqn:series1){reference-type="eqref" reference="eqn:series1"} and [\[eqn:series2\]](#eqn:series2){reference-type="eqref" reference="eqn:series2"}. In conclusion $$\begin{aligned} % \left| M_t-M_{t-1}\right| &\leq \underbrace{\frac{2m^2 c}{(2m+\delta_0)p_m(\delta_0)}}_{\coloneqq \text{const}(m,\delta_0,c)}n^{\gamma-3/2}(1+\ensuremath{\mathrm{o}}(1))\ .\end{aligned}$$ Based on this and using the Azuma-Hoeffding inequality we conclude that, for any $x\geq 0$ $$\begin{aligned} \ensuremath{\mathbb{P}}\left(|M_{\tau_n}-M_1|\geq x\right) &\leq 2\exp\left\{-\frac{x^2}{(1+\ensuremath{\mathrm{o}}(1))\tau_n \left(\text{const}(m,\delta_0,c)n^{\gamma-3/2}\right)^2}\right\}\\ &\leq 2\exp\left\{-\frac{x^2}{(1+\ensuremath{\mathrm{o}}(1))\text{const}^2(m,\delta_0,c)n^{2(\gamma-1)}}\right\}\ .\end{aligned}$$ This implies that $M_{\tau_n}-M_1=\ensuremath{\mathrm{O}}_{\ensuremath{\mathbb{P}}_1}(n^{\gamma-1})=\ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_1}(1)$, showing that [\[eq:iota_prime_asymptotically_normal_decomposition_extra\]](#eq:iota_prime_asymptotically_normal_decomposition_extra){reference-type="eqref" reference="eq:iota_prime_asymptotically_normal_decomposition_extra"} is negligible, and concluding the proof of the lemma. ◻ Lemma [Lemma 15](#lem:iota_prime_normality){reference-type="ref" reference="lem:iota_prime_normality"} suffices to show that the last term in [\[eq:Q_asympt_normal_second_term_rewritten\]](#eq:Q_asympt_normal_second_term_rewritten){reference-type="eqref" reference="eq:Q_asympt_normal_second_term_rewritten"} gives a negligible contribution to the limit distribution of $(Q(G_n)-\ensuremath{\mathbb{E}}_1[Q(G_n)])/\sqrt{n}$, since $$\begin{aligned} \sqrt{n}\left(\frac{p'_m(\breve\delta_0)}{\iota''_n(\bar\delta_0)}-\frac{p'_m(\delta_0)}{\iota''(\delta_0)}\right)\iota'_n(\delta_0)-\ensuremath{\mathbb{E}}_1\left[\sqrt{n}\left(\frac{p'_m(\breve\delta_0)}{\iota''_n(\bar\delta_0)}-\frac{p'_m(\delta_0)}{\iota''(\delta_0)}\right)\iota'_n(\delta_0)\right]\\ =\ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_1}(1)\ensuremath{\mathrm{O}}_{\ensuremath{\mathbb{P}}_1}(1)=\ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_1}(1)\ .\end{aligned}$$ In conclusion $$\frac{Q(G_n)-\ensuremath{\mathbb{E}}_1[G_n]}{\sqrt{n}}=\frac{N_m(n)-\ensuremath{\mathbb{E}}_1[N_m(n)]}{\sqrt{n}}+\sqrt{n}\frac{p'_m(\delta_0)}{\iota''(\delta_0)}\left(\iota'_n(\delta_0)-\ensuremath{\mathbb{E}}_1[\iota'_n(\delta_0)]\right)+\ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_1}(1)\ .$$ Since $\iota'_n(\delta_0)$ and $N_m(n)$ are not independent we cannot directly rely on Lemma [Lemma 15](#lem:iota_prime_normality){reference-type="ref" reference="lem:iota_prime_normality"} and [Lemma 14](#lem:joint_CLT){reference-type="ref" reference="lem:joint_CLT"} to obtain the final result. However, using exactly the same type of argument leads to the following sequence of statements: $$\begin{aligned} \lefteqn{\frac{Q(G_n)-\ensuremath{\mathbb{E}}_1[Q(G_n)]}{\sqrt{n}}}\\ &= \ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_1}(1)+\frac{\ensuremath{\mathbb{E}}_1\left[\left.N_m(n)\right|G_{\tau_n}\right]-\ensuremath{\mathbb{E}}_1\left[N_m(n)\right]}{\sqrt{n}}\\ &\qquad+\frac{\sqrt{n}}{n+1}\sum_{k\geq m} \frac{1}{k+\delta_0}\left(\ensuremath{\mathbb{E}}_1\left[\left.N_{>k}(n)\right|G_{\tau_n}\right]-\ensuremath{\mathbb{E}}_1\left[N_{>k}(n)\right]\right)\\ &= \ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_1}(1)+\frac{N_m(\tau_n)-\ensuremath{\mathbb{E}}_0\left[N_m(\tau_n)\right]}{\sqrt{n}}\\ &\qquad+\frac{\sqrt{n}}{n+1}\sum_{k\geq m} \frac{1}{k+\delta_0}\left(N_{>k}(\tau_n)-\ensuremath{\mathbb{E}}_0\left[N_{>k}(\tau_n)\right]\right)\\ &\xrightarrow{D} \mathcal N(0, w(\delta_0,m)+u(\delta_0,m)),\end{aligned}$$ where the last statement follows from the joint convergence, after appropriate rescaling, of $N_m(\tau_n)$ and $\iota'_{\tau_n}$, which is guaranteed by [\[eq:asymptotic_normality_proof_intermediate_result\]](#eq:asymptotic_normality_proof_intermediate_result){reference-type="eqref" reference="eq:asymptotic_normality_proof_intermediate_result"} (cf. the definition of $\iota'_{n}$ in [\[eqn:iota_n\_prime\]](#eqn:iota_n_prime){reference-type="eqref" reference="eqn:iota_n_prime"}). # Proof of auxiliary results ## Proof of Proposition [Proposition 11](#prp:ana-diff-means){reference-type="ref" reference="prp:ana-diff-means"} {#proof-of-proposition-prpana-diff-means} Recall that $\delta\in[\delta_{\text{min}},\delta_{\text{max}}]$. Begin by noting that $$(n+1)\left(\ensuremath{\mathbb{E}}_1[\iota'_n(\delta)]-\ensuremath{\mathbb{E}}_0[\iota'_n(\delta)]\right) = \sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_1[N_{>k}(n)]-\ensuremath{\mathbb{E}}_0[N_{>k}(n)]}{k+\delta}\ .$$ The numerator in the above summand can be decomposed in a similar manner as used for the proof of Proposition [Proposition 8](#prp:A){reference-type="ref" reference="prp:A"}: $$\begin{aligned} \label{eqn:differences} \lefteqn{\ensuremath{\mathbb{E}}_1[N_{>k}(n)]-\ensuremath{\mathbb{E}}_0[N_{>k}(n)]}\\ &= \ensuremath{\mathbb{E}}_1[N_{>k}(n)]-\ensuremath{\mathbb{E}}_1[N_{>k}(\tau_n)]+\ensuremath{\mathbb{E}}_0[N_{>k}(\tau_n)]-\ensuremath{\mathbb{E}}_0[N_{>k}(n)]\\ &= \left(\ensuremath{\mathbb{E}}_1[N_{>k}(n)]-\ensuremath{\mathbb{E}}_1[N_{>k}(\tau_n)]\right)-\left(\ensuremath{\mathbb{E}}_0[N_{>k}(n)]-\ensuremath{\mathbb{E}}_0[N_{>k}(\tau_n)]\right)\ .\end{aligned}$$ The first equality holds as the law of $G_{\tau_n}$ is the same under the null and alternative models. Therefore, $$\begin{aligned} \label{eqn:main_decomposition} (n+1)\left(\ensuremath{\mathbb{E}}_1[\iota'_n(\delta)]-\ensuremath{\mathbb{E}}_0[\iota'_n(\delta)]\right) &= \sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_1[N_{>k}(n)]-\ensuremath{\mathbb{E}}_1[N_{>k}(\tau_n)]}{k+\delta}\\ &\qquad - \sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_0[N_{>k}(n)]-\ensuremath{\mathbb{E}}_0[N_{>k}(\tau_n)]}{k+\delta}\ .\end{aligned}$$ The treatment of the two terms is entirely analogous and it is done simultaneously. For the rest of the proof, let $\ell\in\{0,1\}$. Clearly $$\begin{aligned} \ensuremath{\mathbb{E}}_\ell[N_{>k}(n)]-\ensuremath{\mathbb{E}}_\ell[N_{>k}(\tau_n)] &= \sum_{v\in[\tau_n]} \ensuremath{\mathbb{P}}_\ell(D_v(n)>k)-\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)>k)\\ &\qquad + \sum_{v\in[n]\setminus [\tau_n]} \ensuremath{\mathbb{P}}_\ell(D_v(n)>k)\ .\end{aligned}$$ Like in the proof of Proposition [Proposition 8](#prp:A){reference-type="ref" reference="prp:A"} we distinguish the behavior of "old" vertices (that arrived before the change-point) from the remaining vertices (the "new" vertices). The contribution of the latter plays an insignificant role, as we see next. Note that, since $k\geq m$ and for $v\in [n]\setminus [\tau_n]$, the event $D_v(n)>k$ is only possible when there is a vertex $v'>v$ that attached to $v$. Referring to [\[eq:attachment_function_alt\]](#eq:attachment_function_alt){reference-type="eqref" reference="eq:attachment_function_alt"} we see that the probability of this happening is at most $(m+\delta_\ell)/((2m+\delta_\ell)\tau_n-2m)$. Since there are at most $m(n-\tau_n)$ possible edges that could attach we get the simple bound $$\sum_{v\in[n]\setminus [\tau_n]} \ensuremath{\mathbb{P}}_\ell(D_v(n)>k) \leq m(n-\tau_n)^2 \frac{m+\delta_\ell}{(2m+\delta_\ell)\tau_n-2m}=\ensuremath{\mathrm{O}}(n^{2\gamma-1})\ .$$ Using that result and the fact that the largest degree in $G_n$ is at most $nm$, this implies that $$\sum_{k=m}^{nm} \sum_{v\in[n]\setminus [\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(n)>k)}{k+\delta}\leq \ensuremath{\mathrm{O}}(n^{2\gamma-1})\sum_{k=m}^{nm}\frac{1}{k+\delta} \leq \ \ensuremath{\mathrm{O}}(n^{2\gamma-1}\log n) \ .$$ Therefore, $$\begin{aligned} \lefteqn{\sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_\ell[N_{>k}(n)]-\ensuremath{\mathbb{E}}_\ell[N_{>k}(\tau_n)]}{k+\delta}}\\ &= \sum_{k\geq m} \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(n)>k)-\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)>k)}{k+\delta} \ + \ \ensuremath{\mathrm{O}}(n^{2\gamma-1}\log n)\\ &= \sum_{k\geq m} \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)\leq k,D_v(n)>k)}{k+\delta} \ + \ \ensuremath{\mathrm{o}}(n^\gamma)\\ &= \sum_{k\geq m} \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=k,D_v(n)>k)}{k+\delta}\label{eqn:main_term}\\ &\qquad + \sum_{k> m} \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)<k,D_v(n)>k)}{k+\delta} \ + \ \ensuremath{\mathrm{o}}(n^\gamma)\label{eqn:extras}\ .\end{aligned}$$ The bulk of the analysis is therefore the characterization of the double summations above. This is somewhat delicate, and requires a good understanding of the behavior of $\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)\leq k,D_v(n)>k)$ for $v\in[\tau_n]$ and $k\geq m$. For an arbitrary vertex $v\in[\tau_n]$ and "small" $k$ we know that, most likely, the degree of the vertex will not change in $G_n$. Most of the contribution in the above expression will therefore be due to the attachment of a single edge to $v$. This reasoning does not apply when $k$ is "large", but in that case the denominator $k+\delta$ is large enough to make the contribution to the above summations negligible. Note that up to time $\tau_n$ both null model and alternative models coincide. Therefore $\frac1{\tau_n}\sum_{v\in[\tau_n]}\ensuremath{\mathbb{P}}_\ell(D_v(\tau)=k)= p_k(\delta_0)(1+\ensuremath{\mathrm{o}}(1))$, regardless of the value of $\ell$. To characterize the term [\[eqn:main_term\]](#eqn:main_term){reference-type="eqref" reference="eqn:main_term"} we proceed by truncating that series and using Lemma [Lemma 9](#lem:crucial){reference-type="ref" reference="lem:crucial"}. Define an auxiliary series $b_n=\lceil n^{1-\gamma}/\log n\rceil$. This is a divergent sequence of integers such that $b_n=\ensuremath{\mathrm{o}}(n^{1-\gamma})$. Then $$\begin{aligned} \lefteqn{\sum_{k\geq m} \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=k,D_v(n)>k)}{k+\delta}}\\ &= \sum_{k\geq m} \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(n)-D_v(\tau_n)>0\mid D_n(\tau_n)=k)\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=k)}{k+\delta}\\ &= \sum_{k=m}^{b_n} \sum_{v\in[\tau_n]} (1+\ensuremath{\mathrm{o}}(1))cn^{\gamma-1} m \frac{k+\delta_\ell}{2m+\delta_\ell}\frac{\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=k)}{k+\delta}\\ &\qquad + \sum_{k=b_n+1}^\infty \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(n)-D_v(\tau_n)>0\mid D_n(\tau_n)=k)\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=k)}{k+\delta}\\ &= (1+\ensuremath{\mathrm{o}}(1))cn^{\gamma-1} \frac{m}{2m+\delta_\ell} \tau_n \sum_{k=m}^{b_n} \frac{k+\delta_\ell}{k+\delta}p_k(\delta_0)\label{eqn:series1}\\ &\qquad + \sum_{k=b_n+1}^\infty \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(n)-D_v(\tau_n)>0\mid D_n(\tau_n)=k)\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=k)}{k+\delta}\label{eqn:series2}\ .\end{aligned}$$ The series in [\[eqn:series1\]](#eqn:series1){reference-type="eqref" reference="eqn:series1"} is convergent. For the series in [\[eqn:series2\]](#eqn:series2){reference-type="eqref" reference="eqn:series2"}, note that $$\begin{aligned} \lefteqn{\sum_{k=b_n+1}^\infty \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(n)-D_v(\tau_n)>0 \mid D_n(\tau_n)=k)\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=k)}{k+\delta}}\\ &\leq \sum_{k=b_n+1}^\infty \sum_{v\in[\tau_n]} \frac{k^2}{b_n^2}\frac{\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=k)}{k+\delta}\\ &= \frac{1}{b_n^2}\sum_{k=b_n+1}^\infty \sum_{v\in[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left(\frac{D^2_v(\tau_n)}{D_v(\tau_n)+\delta}\mathds 1\{D_v(\tau_n)=k\}\right)\\ &\leq \frac{1}{b_n^2}\sum_{k=1}^\infty \sum_{v\in[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left(\underbrace{\frac{D_v(\tau_n)}{D_v(\tau_n)+\delta}}_{\leq C_m}D_v(\tau_n)\mathds 1\{D_v(\tau_n)=k\}\right)\\ &=\frac{C_m}{b_n^2} \ensuremath{\mathbb{E}}_\ell\left(\sum_{v\in[\tau_n]} D_v(\tau_n)\right)\\ &=\frac{C_m}{b_n^2} 2m\tau_n=\ensuremath{\mathrm{O}}(n^{2\gamma-1}\log n)=\ensuremath{\mathrm{o}}(n^{\gamma})\ ,\end{aligned}$$ where $C_m:=\frac{m}{m+\delta_{\min}}$. In conclusion, $$\begin{aligned} \lefteqn{\sum_{k\geq m} \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)=k,D_v(n)>k)}{k+\delta}}\\ &= (1+\ensuremath{\mathrm{o}}(1))cn^{\gamma-1} \frac{m}{2m+\delta_\ell} \tau_n \sum_{k\geq m} \frac{k+\delta_\ell}{k+\delta}p_k(\delta_0)\ .\label{eq:D_tau_n_equals_k_D_n_greater_k_asymptotic}\end{aligned}$$ The characterization of [\[eqn:extras\]](#eqn:extras){reference-type="eqref" reference="eqn:extras"} is significantly more delicate. Note that $$\begin{aligned} \lefteqn{\sum_{k>m} \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)<k,D_v(n)>k)}{k+\delta}}\\ &=\sum_{v\in[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left[\sum_{k>m} \frac{1}{k+\delta}\mathds 1\{D_v(\tau_n)<k,D_v(n)>k\}\right]\\ &\leq\sum_{v\in[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left[\sum_{k>m} \frac{1}{D_v(\tau_n)+1+\delta}\mathds 1\{D_v(\tau_n)<k,D_v(n)>k\}\right]\\ &=\sum_{v\in[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left[\frac{1}{D_v(\tau_n)+1+\delta}\mathds 1\{D_v(n)-D_v(\tau_n)\geq 2\}\sum_{k=D_v(\tau_n)+1}^{D_v(n)-1} 1\right]\\ &=\sum_{v\in[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left[\mathds 1\{D_v(n)-D_v(\tau_n)\geq 2\}\frac{D_v(n)-D_v(\tau_n)-1}{D_v(\tau_n)+1+\delta}\right]\\ &=\sum_{v\in[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left[\mathds 1\{D_v(n)-D_v(\tau_n)\geq 1\}\frac{D_v(n)-D_v(\tau_n)-1}{D_v(\tau_n)+1+\delta}\right]\ .\label{eqn:key_formula}\end{aligned}$$ The key quantity to control is the expectation in [\[eqn:key_formula\]](#eqn:key_formula){reference-type="eqref" reference="eqn:key_formula"}. Let $\sigma_v$ denote the first time after $\tau_n$ when an edge is attached to vertex $v$. With this in hand we can bound [\[eqn:key_formula\]](#eqn:key_formula){reference-type="eqref" reference="eqn:key_formula"} as $$\begin{aligned} \lefteqn{\sum_{k>m} \sum_{v\in[\tau_n]} \frac{\ensuremath{\mathbb{P}}_\ell(D_v(\tau_n)<k,D_v(n)>k)}{k+\delta}}\\ &\leq \sum_{v\in[\tau_n]} \sum_{s\in[n]\setminus[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left[\mathds 1\{\sigma_v=s\}\frac{D_v(n)-D_v(s)}{D_v(\tau_n)+1+\delta}\right]\label{eqn:A}\ .\end{aligned}$$ The following lemma allows us to bound [\[eqn:A\]](#eqn:A){reference-type="eqref" reference="eqn:A"}: **Lemma 16**. *For $\ell\in \{0,1\}$ and $t\geq \tau>\tau_n$ $$\begin{aligned} &\ensuremath{\mathbb{E}}_\ell\left[D_v(t)+\delta_\ell \mid D_v(\tau)\right]\\ &\qquad=(D_v(\tau)+\delta_\ell)\prod_{j=1}^{t-\tau}\prod_{i=1}^m \left(1+\frac{1}{2(\tau+j-1)) m + \delta_\ell (\tau+j)+(i-1)}\right)\ . \end{aligned}$$* We postpone the proof of Lemma [Lemma 16](#lem:recursion_expected_degrees){reference-type="ref" reference="lem:recursion_expected_degrees"} to later on. With this in hand, for any $s\in[n]\setminus[\tau_n]$, $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{E}}_\ell\left[D_v(n)-D_v(s) \mid D_v(s)\right]}\\ &= \ensuremath{\mathbb{E}}_\ell\left[D_v(n)+\delta_\ell\mid D_v(s)\right]-(D_v(s)+\delta_\ell)\\ &=(D_v(s)+\delta_\ell)\left(\prod_{j\in[n]\setminus[\tau_n]} \prod_{i=1}^m \left(1+\frac{1}{j(2m+\delta_\ell)-2m+i-1}\right)\ -1\right)\\ &=(1+\ensuremath{\mathrm{o}}(1))(D_v(s)+\delta_\ell)(n-\tau_n)\frac{m}{(2m+\delta_\ell)\tau_n}\\ &\leq \ensuremath{\mathrm{O}}(1)(D_v(s)+\delta_\ell) n^{\gamma-1}\ .\end{aligned}$$ When $s=\sigma_v$ we know that $D_v(s)\leq D_n(\tau_n)+m$ (at time $s$ the first edge was attached to $v$, and therefore at most $m$ edges were attached to $v$ after all the intermediate steps). That means that $$\begin{aligned} \frac{D_v(s)+\delta_\ell}{D_v(\tau_n)+1+\delta} &\leq \frac{D_v(\tau_n)+m+\delta_\ell}{D_v(\tau_n)+1+\delta}\\ &\leq \frac{D_v(\tau_n)+m+\delta_{\max}}{D_v(\tau_n)+1+\delta_{\min}}\\ &\leq \frac{2m+\delta_{\max}}{m+1+\delta_{\min}} :=\text{const}\ ,\end{aligned}$$ where $\text{const}>0$ is simply a constant. Therefore, $$\begin{aligned} \lefteqn{\sum_{v\in[\tau_n]} \sum_{s\in[n]\setminus[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left[\mathds 1\{\sigma_v=s\}\frac{D_v(n)-D_v(s)}{D_v(\tau_n)+1+\delta}\right]}\\ &= \sum_{v\in[\tau_n]} \sum_{s\in[n]\setminus[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left[ \frac{1}{D_v(\tau_n)+1+\delta} \ensuremath{\mathbb{E}}_\ell\left[\mathds 1\{\sigma_v=s\}(D_v(n)-D_v(s))\mid D_v(s)\right]\right]\\ &= \ensuremath{\mathrm{O}}(1)\sum_{v\in[\tau_n]} \sum_{s\in[n]\setminus[\tau_n]} \ensuremath{\mathbb{E}}_\ell\left[ \frac{1}{D_v(\tau_n)+1+\delta} \ensuremath{\mathbb{E}}_\ell\left.\left[\mathds 1\{\sigma_v=s\}(D_v(s)+\delta_\ell) n^{\gamma-1}\right|D_v(s)\right]\right]\\ &\leq \ensuremath{\mathrm{O}}(n^{\gamma-1}) \sum_{v\in[\tau_n]} \sum_{s\in[n]\setminus[\tau_n]} \ensuremath{\mathbb{P}}_\ell\left(\sigma_v=s\right)\\ &= \ensuremath{\mathrm{O}}(n^{\gamma-1}) \sum_{v\in[\tau_n]} \ensuremath{\mathbb{P}}_\ell\left(D_v(n)-D_v(\tau_n)\geq 1\right)\\ &\leq \ensuremath{\mathrm{O}}(n^{\gamma-1}) \sum_{v\in[\tau_n]} (n-\tau_n)m\ensuremath{\mathbb{E}}_\ell\left[\frac{D_v(\tau_n)+\delta_\ell}{(2m+\delta_\ell)\tau_n+\delta_\ell}\right]\\ &= \ensuremath{\mathrm{O}}(n^{2\gamma-2}) \ensuremath{\mathbb{E}}_\ell\left[\sum_{v\in[\tau_n]} D_v(\tau_n)+\delta_\ell\right]= \ensuremath{\mathrm{O}}(n^{2\gamma-1})=\ensuremath{\mathrm{o}}(n^{\gamma})\ ,\end{aligned}$$ where the last inequality follows from the same reasoning used to obtain [\[eq:D_tau_n\_equals_k\_D_n\_greater_k\_asymptotic\]](#eq:D_tau_n_equals_k_D_n_greater_k_asymptotic){reference-type="eqref" reference="eq:D_tau_n_equals_k_D_n_greater_k_asymptotic"}, and the last step follows since $\gamma>\tfrac{1}{2}$. This means that the term in [\[eqn:extras\]](#eqn:extras){reference-type="eqref" reference="eqn:extras"} is of smaller order than the term in [\[eqn:main_term\]](#eqn:main_term){reference-type="eqref" reference="eqn:main_term"}. In conclusion, $$\begin{aligned} \sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_\ell[N_{>k}(n)]-\ensuremath{\mathbb{E}}_\ell[N_{>k}(\tau_n)]}{k+\delta}=(1+\ensuremath{\mathrm{o}}(1))cn^{\gamma} \frac{m}{2m+\delta_\ell} \sum_{k=m}^{\infty} \frac{k+\delta_\ell}{k+\delta}p_k(\delta_0)\ .\end{aligned}$$ We are now ready to go back to [\[eqn:main_decomposition\]](#eqn:main_decomposition){reference-type="eqref" reference="eqn:main_decomposition"} to get $$\begin{aligned} \lefteqn{(n+1)\left(\ensuremath{\mathbb{E}}_1[\iota'_n(\delta)]-\ensuremath{\mathbb{E}}_0[\iota'_n(\delta)]\right)}\\ &=\sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_1[N_{>k}(n)]-\ensuremath{\mathbb{E}}_1[N_{>k}(\tau_n)]}{k+\delta} - \sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_0[N_{>k}(n)]-\ensuremath{\mathbb{E}}_0[N_{>k}(\tau_n)]}{k+\delta}\\ &=(1+\ensuremath{\mathrm{o}}(1))\frac{cmn^\gamma}{(2m+\delta_0)(2m+\delta_1)}(\delta_1-\delta_0)\sum_{k\geq m}\frac{2m-k}{k+\delta}p_k(\delta_0)\\ &=(1+\ensuremath{\mathrm{o}}(1))\frac{cmn^\gamma}{(2m+\delta_0)(2m+\delta_1)}(\delta_1-\delta_0)\left(-1+\sum_{k\geq m}\frac{2m+\delta}{k+\delta}p_k(\delta_0)\right)\ ,\end{aligned}$$ as required and where in the last step we used the fact that $\sum_{k\geq m} p_k(\delta_0)=1$. 0◻ ## Proof of Lemma [Lemma 16](#lem:recursion_expected_degrees){reference-type="ref" reference="lem:recursion_expected_degrees"} {#proof-of-lemma-lemrecursion_expected_degrees} Note that between $t$ and $\tau$ the attachment function is affine with parameter $\delta_\ell$. Note also that the graph $G_t$ has precisely $t+1$ vertices and $mt$ edges. Let us describe what happens at each one of the intermediate steps. Let $v$ be a vertex in $G_{\tau}$ and let $D_v(\tau+1,i)$ denote its degree in the graph $G_{\tau+1,i}$, where $i\in\{1,\ldots,m\}$. Then, $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{E}}_\ell[D_v(\tau+1,i)+\delta_0\mid D_v(\tau+1,i-1)]}\\ &= D_v(\tau+1,i-1)+\delta_0+\ensuremath{\mathbb{E}}_\ell[D_v(\tau+1,i)-D_v(\tau+1,i-1)\mid D_v(\tau+1,i-1)]\\ &= D_v(\tau+1,i-1)+\delta_0+\frac{D_v(\tau+1,i-1)+\delta_0}{2\tau m + \delta_0(\tau+1)+(i-1)}\\ &= (D_v(\tau+1,i-1)+\delta_0)\left(1+\frac{1}{2\tau m + \delta_0(\tau+1)+(i-1)}\right)\ .\end{aligned}$$ Therefore, $$\begin{aligned} \ensuremath{\mathbb{E}}_{\ell}[D_v(\tau+1)+\delta_0\mid D_v(\tau)] &= \ensuremath{\mathbb{E}}_\ell[D_v(\tau+1,m)+\delta_0\mid D_v(\tau+1,0)]\\ &=(D_v(\tau)+\delta_0) \prod_{i=1}^m \left(1+\frac{1}{2\tau m + \delta_0(\tau+1)+(i-1)}\right)\ .\end{aligned}$$ Thus, in general, for $t>\tau$, $$\begin{aligned} \lefteqn{\ensuremath{\mathbb{E}}_\ell[D_v(t)+\delta_0\mid D_v(\tau)]}\\ &= \ensuremath{\mathbb{E}}_\ell\left[\ensuremath{\mathbb{E}}_{\ell}[D_v(t)+\delta_0\mid D_v(t-1)] \mid D_v(\tau)\right]\\ &= \ensuremath{\mathbb{E}}_\ell\left[\ensuremath{\mathbb{E}}_{\ell}[D_v(t)+\delta_0\mid D_v(t-1)] \mid D_v(\tau)\right]\\ &=\prod_{i=1}^m \left(1+\frac{1}{2(t-1) m + \delta_0 t+(i-1)}\right)\ensuremath{\mathbb{E}}_\ell\left[D_v(t-1)+\delta_0 \mid D_v(\tau)\right]\\ &\vdots\\ &=(D_v(\tau)+\delta_0)\prod_{j=1}^{t-\tau}\prod_{i=1}^m \left(1+\frac{1}{2(\tau+j-1)) m + \delta_0 (\tau+j)+(i-1)}\right)\ .\end{aligned}$$ 0◻ ## Proof of Proposition [Proposition 12](#prp:C2){reference-type="ref" reference="prp:C2"} {#proof-of-proposition-prpc2} To show consistency of $\hat\delta_n$ note that $\iota'_n(\hat\delta_n)=0$ by definition. Recalling [\[eqn:L1_convergence\]](#eqn:L1_convergence){reference-type="eqref" reference="eqn:L1_convergence"} we conclude immediately that $$\ensuremath{\mathbb{E}}[\iota'(\hat\delta_n)]\to 0\ .$$ [@Gao2017 Lemma 4] shows that $\iota'$ has a unique zero at $\delta_0$, and $\iota'(\delta)>0$ for $\delta<\delta_0$ and $\iota'(\delta)<0$ for $\delta>\delta_0$. This immediately implies that $\ensuremath{\mathbb{E}}_1[|\hat\delta_n-\delta_0|]$ as $n\to\infty$, proving the first assertion in the proposition. Note also that in Proposition [Proposition 10](#prp:B){reference-type="ref" reference="prp:B"} we have shown that $\tilde\delta_n\to\delta_0$, therefore we also have $\ensuremath{\mathbb{E}}_\ell[|\hat\delta_n-\tilde\delta_n|]\to 0$. To characterize the rate of convergence of $\hat\delta_n$ to $\tilde\delta_n$ we need the following lemma: **Lemma 17**. *For $\ell\in\{0,1\}$ and $x>0$ $$\ensuremath{\mathbb{P}}_\ell\left(\sup_{\delta\in[\delta_{\min},\delta_{\max}]}(n+1)\left|\iota'_n(\delta)-\ensuremath{\mathbb{E}}_\ell[\iota'_n(\delta)]\right|\geq x\right)\leq 2\ensuremath{\mathrm{e}}^{-\frac{x^2}{2nc^2_{n,m}}}\ ,$$ where $c_{n,m}=\sum_{k=m}^{nm} \frac{2m}{k+\delta_{\min}}\ .$* *Proof.* To prove this lemma we use a similar argument used in the proof of Theorem [Theorem 1](#thm:minimal_degree_test_known_delta){reference-type="ref" reference="thm:minimal_degree_test_known_delta"}, resorting to the Azuma-Hoeffding's inequality. Note that in the expression of $\iota'_n(\delta)$ in [\[eqn:iota_n\_prime\]](#eqn:iota_n_prime){reference-type="eqref" reference="eqn:iota_n_prime"} only the first term in not deterministic. Begin by constructing the Doob martingale $$M_t(\delta)=\sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_{\ell}\left[N_{>k}(n)\mid G_t\right]}{k+\delta}\ ,$$ where $t\in[n]$. Clearly $M_1(\delta)=\sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_\ell\left[N_{>k}(n)\right]}{k+\delta}$ and $M_n(\delta)=\sum_{k\geq m} \frac{N_{>k}(n)}{k+\delta}$, therefore $$M_n(\delta)-M_1(\delta)=(n+1)\left(\iota'_n(\delta)-\ensuremath{\mathbb{E}}_\ell[\iota'_n(\delta)]\right)\ .$$ Furthermore, at each timestep in the construction of $G_n$ we add $2m$ edges. Therefore $$|\ensuremath{\mathbb{E}}_\ell\left[N_{>k}(n)\mid G_t\right]-\ensuremath{\mathbb{E}}_\ell\left[N_{>k}(n)\mid G_{t-1}\right]\leq 2m\ ,$$ where $t\in\{2,\ldots,n\}$ and $m\leq k\leq nm$. As a result, $$|M_t(\delta)-M_{t-1}(\delta)| \leq \sum_{k=m}^{nm} \frac{2m}{k+\delta}\leq \sum_{k=m}^{nm} \frac{2m}{k+\delta_{\min}}\ .$$ Note that the bound on the martingale differences holds uniformly in $\delta$. With this in hand we can simply apply the Azuma-Hoeffding's inequality to get the desired result. ◻ Note that $c_{n,m}=2m(1+\ensuremath{\mathrm{o}}(1)\log n$ as $n\to\infty$. Let $a_n$ be an arbitrary sequence satisfying $a_n=\omega\left(\sqrt{n}\log n\right)$. The above lemma tells us that $$\ensuremath{\mathbb{P}}_\ell\left(\sup_{\delta\in[\delta_{\min},\delta_{\max}]}\ \left|\iota'_n(\delta)-\ensuremath{\mathbb{E}}_\ell[\iota'_n(\delta)]\right|\geq a_n/n\right)=\ensuremath{\mathrm{o}}(1)\ .$$ Now define $h_\ell:\delta\mapsto \ensuremath{\mathbb{R}}$ as $h_\ell(\delta)\coloneqq \ensuremath{\mathbb{E}}_\ell[\iota'_n(\delta)]$. We have in particular that $$\iota'_n(\hat\delta_n)-h_\ell(\hat \delta_n)=\ensuremath{\ensuremath{\mathrm{o}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(a_n/n)\ .$$ Using a Taylor expansion of $h_\ell$ allows us to characterize the difference between $\hat\delta_n$ and $\tilde\delta_n$. Let $h'_\ell(\delta)=\frac{\partial}{\partial \delta} h_\ell(\delta)=\ensuremath{\mathbb{E}}_\ell[\iota''_n(\delta)]$ and recall that $\iota_n(\hat\delta_n)=h_\ell(\tilde\delta_n)=0$ by definition. Then $$\begin{aligned} 0&= \iota_n(\hat\delta_n)-h_\ell(\tilde\delta_n)\\ &= \iota_n(\hat\delta_n)-h_\ell(\hat\delta_n)-h_\ell'(\bar\delta_n)(\tilde\delta_n-\hat\delta_n)\\ &= \ensuremath{\ensuremath{\mathrm{o}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(a_n/n)-h_\ell'(\bar\delta_n)(\tilde\delta_n-\hat\delta_n)\label{eqn:rate_delta_n}\ ,\end{aligned}$$ where $|\bar\delta_n-\hat\delta_n|\leq|\tilde\delta_n-\hat\delta_n|$. To proceed we must understand the behavior of $h'_\ell(\delta)=\ensuremath{\mathbb{E}}_\ell[\iota''_n(\delta)]$. As argued in the proof of Proposition [Proposition 10](#prp:B){reference-type="ref" reference="prp:B"}, thanks to the results in [@Gao2017] and the fact that $\iota''_n(\delta)$ and $\iota''(\delta)$ are uniformly bounded, $$\ensuremath{\mathbb{E}}_0\left[\sup_{\delta\in[\delta_{\min},\delta_{\max}]}\ |\iota''_n(\delta)-\iota''(\delta)|\right]\to 0\ .$$ Actually, this result also holds under the alternative hypothesis by using the following fact: **Lemma 18**. *Let $\tfrac{1}{2}<\gamma<1$. Then $\ensuremath{\mathbb{E}}_1[\iota''_n(\delta)]-\ensuremath{\mathbb{E}}_0[\iota''_n(\delta)]\to 0$ as $n\to\infty$ uniformly in $\delta\in[\delta_{\min},\delta_{\max}]$.* The proof of this result follows almost immediately from the arguments used to prove Proposition [Proposition 11](#prp:ana-diff-means){reference-type="ref" reference="prp:ana-diff-means"}. In addition, it is also shown in [@Gao2017] that $\iota''(\delta_0)<0$. Since $\bar\delta_n$ converges in probability to $\delta_0$ we therefore conclude that $$\ensuremath{\mathbb{E}}_\ell[\iota''_n(\bar\delta_n)]=(1+\ensuremath{\mathrm{o}}(1))\iota''(\delta_0)<0\ .$$ This, together with [\[eqn:rate_delta_n\]](#eqn:rate_delta_n){reference-type="eqref" reference="eqn:rate_delta_n"} implies that $$\hat\delta_n-\tilde\delta_n=\ensuremath{\ensuremath{\mathrm{o}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(a_n/n)\ ,$$ proving the second statement in the proposition. The third statement is a rather trivial consequence of the second statement by using a Taylor expansion of $p_m(\delta)$ around $\delta_0$ (recalling Equation [\[eqn:p_m\_prime\]](#eqn:p_m_prime){reference-type="eqref" reference="eqn:p_m_prime"}) to obtain $$n(p_m(\hat\delta_n)-p_m(\tilde\delta_n))=n p'_m(\delta_0)(1+\ensuremath{\ensuremath{\mathrm{o}}_{\scriptscriptstyle{}\ensuremath{\mathbb{P}}}}(1))(\hat\delta_n-\tilde\delta_n)=\ensuremath{\mathrm{o}}_{\ensuremath{\mathbb{P}}_\ell}(a_n)\ .$$ 0◻ ## Sketch proof of Lemma [Lemma 18](#lem:second_derivative_difference){reference-type="ref" reference="lem:second_derivative_difference"} {#sketch-proof-of-lemma-lemsecond_derivative_difference} Similarly to the proof of Proposition [Proposition 11](#prp:ana-diff-means){reference-type="ref" reference="prp:ana-diff-means"} note that $$\begin{aligned} \ensuremath{\mathbb{E}}_1[\iota''_n(\delta)]-\ensuremath{\mathbb{E}}_0[\iota''_n(\delta)]&=-\frac{1}{n+1}\sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_1[N_{>k}(n)]-\ensuremath{\mathbb{E}}_0[N_{>k}(n)]}{(k+\delta)^2}\ .\end{aligned}$$ This is quite similar to $$\begin{aligned} \ensuremath{\mathbb{E}}_1[\iota'_n(\delta)]-\ensuremath{\mathbb{E}}_0[\iota'_n(\delta)]=\frac{1}{n+1}\sum_{k\geq m} \frac{\ensuremath{\mathbb{E}}_1[N_{>k}(n)]-\ensuremath{\mathbb{E}}_0[N_{>k}(n)]}{k+\delta}\ .\end{aligned}$$ Note that $\sum_{k\geq m} \frac{1}{(k+\delta)^2}$ is a convergent series, unlike $\sum_{k\geq m} \frac{1}{k+\delta}$. By the same (and in fact somewhat simpler) arguments as in the proof of Proposition [Proposition 11](#prp:ana-diff-means){reference-type="ref" reference="prp:ana-diff-means"} we conclude therefore that $\ensuremath{\mathbb{E}}_1[\iota''_n(\delta)]-\ensuremath{\mathbb{E}}_0[\iota''_n(\delta)]=\ensuremath{\mathrm{o}}(n^{\gamma-1})$, as we wanted to show. 0◻ The work of RvdH was supported in part by the Netherlands Organisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003. [^1]: For $\alpha\in(0,1)$, let $z_\alpha$ be the unique solution of $\alpha=\int_{z_\alpha}^\infty \frac{1}{\sqrt{2\pi}}{\rm e}^{-\frac{z^2}{2}}dz$.

arxiv_math

--- abstract: | For a polarized toric variety $(X,L)$ of dimension $n\le4$, we give a lower bound of the $\Delta$-genus by using the vanishing number of adjoint bundle of a multiple of $L$. We show that for a polarized toric variety of dimension $n$ with nonvanishing adjoint bundle, the $\Delta$-genus is more than or equal to $n-1$. author: - Shoetsu Ogata and Riki Tabei title: An inequality for the $\Delta$-genus of toric varieties --- # Introduction {#introduction .unnumbered} A pair $(X,L)$ of a projective variety $X$ and an ample line bundle $L$ on $X$ is called a *polarized variety*. For a polarize variety $(X,L)$ of dimension $n$, Fujita (see [@Fj]) defined the $\Delta$-genus as $$\Delta(X,L) :=n + L^n -\dim \Gamma(X, L).$$ When $X$ is a toric variety, we call $(X,L)$ a polarized toric variety. **Theorem 1**. *Let $(X,L)$ be a polarized toric variety of dimension $n\le4$. Then we have $$\label{e:a1} \Delta(X,L) \ge n-\min\{k\ge1:\ \Gamma(X,L^{\otimes k}\otimes \omega_X)\not=0\},$$ where $\omega_X$ is the dualizing sheaf of $X$.* The statement of Theorem [Theorem 1](#t:A){reference-type="ref" reference="t:A"} is from Theorem [Theorem 6](#th:BN1){reference-type="ref" reference="th:BN1"}, Lemma [Lemma 4](#lem:5.1){reference-type="ref" reference="lem:5.1"}, Lemma [Lemma 5](#lem:5.2){reference-type="ref" reference="lem:5.2"} and Proposition [Proposition 4](#p:5){reference-type="ref" reference="p:5"}. In order to unify inequalities into this form, we have to restrict dimension $n\le4$ because of Example [Example 2](#ex:3.2){reference-type="ref" reference="ex:3.2"}. **Theorem 2**. *Let $(X,L)$ be a polarized toric variety of dimension $n\ge3$. Assume that $\Gamma(X,L^{\otimes n-2}\otimes\omega_X)=0$ and $\Gamma(X,L^{\otimes n-1}\otimes\omega_X)\not=0$. Then $\Delta(X,L)\ge1$. Moreover the equality holds if and only if $(X,L)$ is a Gorenstein toric Del Pezzo variety, that is, $X$ is a Gorenstein toric variety with $L^{n-1}\cong \omega_X^\vee$.* This theorem is given as Lemma [Lemma 4](#lem:5.1){reference-type="ref" reference="lem:5.1"}. **Remark 1**. *The assumption in Theorem [Theorem 2](#t:C){reference-type="ref" reference="t:C"} is necessary. See Example [Example 2](#ex:3.2){reference-type="ref" reference="ex:3.2"}. The polytope defines a polarized Gorenstein toric Fano variety $(X,L)$ not Del Pezzo with $\Delta(X,L)=1$.* **Theorem 3**. *Let $(X,L)$ be a polarized toric variety of dimension $n$ with $\Gamma(X,L\otimes\omega_X)\not=0$. Then we have $$\Delta(X,L) \ge n-1.$$ Moreover, when $3\le n$, the equality holds if and only if $(X,L)$ is the Gorenstein toric Fano variety which is a mirror to $(\mathbb{P}^n, {\cal O}(n+1))$.* This theorem is given as Proposition [Proposition 4](#p:5){reference-type="ref" reference="p:5"}. This research begins from the work of Tabei[@Tb]. We give his main result in the last section. # An origin of the inequality Let $X \subset \mathbb{P}^N$ be a projective variety of dimension $n$ and ${\cal I}_X$ the ideal sheaf of $X$. The *Castelnuovo-Mumford regularity* $\mbox{\rm reg}(X)$ of $X$ is defined by $$\mbox{\rm reg}(X):= \min\{r\ge1:\ H^i(\mathbb{P}^N, {\cal I}_X(r-i))=0 \quad \mbox{for all $i>0$}\}.$$ When $X$ is nondegenerate and irreducible, Eisenbud and Goto conjectured $$\label{e:1.1} \mbox{\rm reg}(X) \le \mbox{\rm deg} X -\mbox{\rm codim} X +1.$$ The conjecture has been proved true in several cases. In 2018, Peeva and McCullough gave a counterexample to this inequality. Here we consider the situation that $X$ is a toric variety and an embedding given by global sections of a very ample line bundle $L$, that is, $\Phi _L: X \hookrightarrow \mathbb{P}(\Gamma(X,L)^*)$. In this case, $\mbox{\rm reg}(X)$ is determined by two values: Set the bound of $k$-normality as $$\label{e:1.2} \kappa(X):=\min\{k_0\ge1:\ \Gamma(X,L)^{\otimes k} \to \Gamma(X,L^{\otimes k})\quad \mbox{is surjective for all $k\ge k_0$}\}.$$ And set $$\label{e:1.3} \lambda(X,L):=\min\{k\ge1:\ \Gamma(X, L^{\otimes k}\otimes\omega_X)\not=0\}.$$ Then we have $${\rm reg}(X)=\max\{\kappa(X)+1, \dim X+2-\lambda(X,L)\}.$$ From $\mbox{\rm reg}(X)\ge \dim X+2-\lambda(X,L)$, we have a weaker inequality $$\label{e:1.4} \dim \Gamma(X,L) -\deg X \le \lambda(X,L).$$ If $\dim X=n$, then we write $\deg X=L^n$, and all three values in the inequality ([\[e:1.4\]](#e:1.4){reference-type="ref" reference="e:1.4"}) can be defined for any ample line bundle $L$. Thus for a polarized toric variety $(X,L)$ of dimension $n$, we may consider an inequality $$\label{e:d1} \Delta(X,L) \ge n-\lambda(X,L)$$ ## Polarized toric variety For a proof of theorems we use combinatrics of lattice polytopes. So we recall basic notions of toric varieties and lattice polytopes. See, for example, Oda's book[@Od] or Fulton's book[@Ft]. Let $M$ be a free abelian group of rank $n$. Denote $M_{\mathbb{R}}\cong \mathbb{R}^n$ the extension of coefficients of $M$ to real numbers. Let $\mathbb{C}[M]$ be the group algebra over complex numbers, and $T=\mbox{\rm Spec}\ \mathbb{C}[M]$ the algebraic torus of dimension $n$. The character group $\mbox{\rm Hom}(T,\mathbb{C}^\times)$ of $T$ is isomorphic to $M$. For a lattice point $m\in M$, the corresponding character is written by $\chi^m$. A toric variety $X$ of dimension $n$ is a normal algebraic variety with an algebraic action of the algebraic torus $T$ of dimension $n$ such that $X$ has an open orbit isomorphic to $T$ and if we identify the open orbit with $T$ the action of $T$ is compatible with the multiplication of $T$. A lattice polytope $P$ in $M_{\mathbb{R}}$ is a convex hull of finite elements of $M$ in $M_{\mathbb{R}}$. A polarized toric variety $(X,L)$ of dimension $n$ corresponds to a lattice polytope $P\subset M_{\mathbb{R}}$ of dimension $n$. The correspondence implies the equalities $$\begin{aligned} \Gamma(X,L) &=& \bigoplus_{m\in P\cap M} \mathbb{C} \chi^m,\\ \Gamma(X,L\otimes\omega_X) &=& \bigoplus_{m\in \mbox{\scriptsize int}P\cap M} \mathbb{C} \chi^m.\end{aligned}$$ A twist $L^{\otimes k}$ corresponds to the multiple $kP$. The condition that the multiplication map $\Gamma(X,L)^{\otimes k} \to \Gamma(X,L^{\otimes k})$ is surjective is equivalent to the equality $$\label{e:1.6} \overbrace{(P\cap M) + \dots +(P\cap M)}^k = (kP)\cap M.$$ A lattice polytope $P\subset M_{\mathbb{R}}$ is *normal* if the equation ([\[e:1.6\]](#e:1.6){reference-type="ref" reference="e:1.6"}) holds for all $k\ge1$. Here we define *empty depth* of a lattice polytope $P$ as $$e(P) : = \begin{cases} \max\{k\ge1:\ \mbox{int}(kP)\cap M=\emptyset\} & \mbox{if $\mbox{int}P\cap M =\emptyset$},\\ 0& \mbox{if $\mbox{int}P\cap M \not=\emptyset$}. \end{cases}$$ Then we see $\lambda(X,L)=e(P)-1$. A lattice polytope $P$ of dimension $n$ has the normalized volume $n! \mbox{vol}(P)$, which we write as $v(P)$. Then the degree $L^n$ coincides with $v(P)$. The $\Delta$-genus $\Delta(X,L)$ is written as $$\Delta(P) = \dim P +v(P) -\sharp(P\cap M).$$ The inequality ([\[e:d1\]](#e:d1){reference-type="ref" reference="e:d1"}) is equivalent to $$\label{e:p1} \sharp(P\cap M) -v(P) \le e(P) +1.$$ ## The case of dimension two Let $P$ be a lattice polytope of dimension two. We know Pick's formula concerning the area and the number of lattice points in $P$ (see, for example, [@Od p.101]). **Theorem 4** (Pick). *Let $P\subset M_{\mathbb{R}}$ be a lattice polygon, that is , a lattice polytope of dimension two. Then we have $$\label{e:2.1} \mbox{\rm vol}(P) =\sharp (P\cap M) -\frac12\sharp(\partial P\cap M)-1.$$* Since $v(P)=2\mbox{vol}(P)$, we write the equality ([\[e:2.1\]](#e:2.1){reference-type="ref" reference="e:2.1"}) as $$\sharp(P\cap M) -v(P) = 2-\sharp(\mbox{int}P\cap M).$$ **Theorem 5** (Tabei[@Tb]). *Let $P\subset M_{\mathbb{R}}$ be a lattice polygon. Then we have $$\sharp(P\cap M) -v(P) \le e(P) +1.$$ Moreover, the equality holds if and only if $P$ is not basic and $\sharp(\mbox{\rm int}P\cap M)\le1$.* **Remark 2**. *Since a lattice polygon is always normal, the inequality ([\[e:d1\]](#e:d1){reference-type="ref" reference="e:d1"}) is equivalent to ([\[e:1.1\]](#e:1.1){reference-type="ref" reference="e:1.1"}).* **Proposition 1** (Tabei[@Tb], Koelman[@Ko]). *A lattice polygon $P$ with $e(P)=1$ is isomorphic to $$P=\mbox{\rm conv}\{0,(1,0),(0,a),(1,b)\}, \quad\mbox{where $a\ge1, b\ge0$ and $a+b\ge2$}$$ or $$P=\mbox{\rm conv}\{0, (0,2),(2,0)\}.$$* **Remark 3**. *A lattice polygon $P$ with $\sharp(\mbox{int}P\cap M)=1$ is called \"Fano polygon\". Fano polygons are classified and there are 16 polygons up to isomorphism. See [@Ks] or [@KN].* # Empty lattice simplices A lattice polytope $P\subset M_{\mathbb{R}}$ of dimension $n$ is called *empty lattice simplex* if $\sharp(P\cap M)=n+1$. Here we consider $e(P)$ and $k$-normality for an empty lattice polytope $P$. **Proposition 2**. *Let $P\subset M_{\mathbb{R}}$ be a lattice polytope of dimension $n$. If $$\mbox{\rm int}(nP)\cap M = \emptyset,$$ then $P$ is isomorphic to a basic $n$-simplex.* This fact is well known. We see that $$\mbox{$P$ is a basic $n$-simplex} \Longleftrightarrow e(P)=n.$$ We state a weaker condition than $k$-normality. The following two lemmas are given in [@OZ]. **Lemma 1**. *Let $P\subset M_{\mathbb{R}}$ be a lattice polytope of dimension $n$. If there exists an integer $r$ with $1\le r\le n-1$ such that $\mbox{\rm int}(rP)\cap M=\emptyset$, then for all $k\ge n-r$ we have $$(kP)\cap M +(P\cap M) =(k+1)P\cap M.$$* **Lemma 2**. *Let $P\subset M_{\mathbb{R}}$ be an empty lattice $n$-simplex. If $$\mbox{\rm int}(n-1)P\cap M=\emptyset,$$ then $P$ is basic.* **Lemma 3**. *Let $P\subset M_{\mathbb{R}}$ be an empty lattice $n$-simplex. Assume that all facets of $P$ are basic. If $$\mbox{\rm int}(rP)\cap M=\emptyset \quad \mbox{and $r\ge \frac{n}2$},$$ then $P$ is basic.* *Proof*. Lemma [Lemma 1](#l:3.1){reference-type="ref" reference="l:3.1"} saiys that $$(kP)\cap M +(P\cap M)= (k+1)P\cap M \quad \mbox{for $k\ge n-r$}.$$ From $r\ge \frac{n}2$, we have $n-r\le \frac{n}2$, hence $(n-r)P\cap M = \partial((n-r)P)\cap M$ from the assumption $\mbox{int}(rP)\cap M=\emptyset$ and $r\ge \frac{n}2$. Thus for an $m\in (n-r)P\cap M$ there is a facet $F\subset P$ such that $m\in (n-r)F$. Since $F$ is basic from the assumption, $m$ is a sum of $n-r$ elements of $F\cap M$. This means that for $k\le n-r$ we have $$(kP)\cap M=\overbrace{(P\cap M)+ \dots +(P\cap M)}^k=(kP)\cap M.$$ Therefor, $P$ is normal. A normal empty lattice $n$-simplex is basic. $\Box$. We calculate empty depth of several empty lattice simplices. **Example 1**. *Let $e_1, \dots, e_n$ be a basis of free abelian group $M$ of rank $n\ge3$. Consider an empty lattice $n$-simplex $$\Delta_2^n:=\mbox{\rm conv}\{0, e_1,e_2, e_1+e_2+2e_3, e_4, \dots, e_n\}.$$ This polytope is not normal. Since $$e_1+\dots+e_n\in \mbox{\rm int}(n-1)\Delta_2^n\cap M \setminus (n-2)\Delta_2^n,$$ we have $e(\Delta_2^n)=n-2$. Thus $$\sharp(\Delta_2^n \cap M) -v(\Delta_2^n)=n-1 =e(\Delta_2^n)+1.$$* **Example 2**. *For $n\ge4$ we consider another empty lattice $n$-simplex $$\Delta_{2,n}^n:=\mbox{\rm conv}\{0, e_1, \dots, e_{n-1}, e_1+\dots +e_{n-1}+2e_n\}.$$ Since $$e_1+\dots +e_n\in \frac{n}2 \Delta_{2,n}^n,$$ we have $e(\Delta_{2,n}^n)=[\frac{n}2]$. If $n\ge5$, then $$\sharp(\Delta_{2,n}^n \cap M) -v(\Delta_{2,n}^n)=n-1 > [\frac{n}2]+1 =e(\Delta_{2,n}^n).$$* **Remark 4**. *Example [Example 2](#ex:3.2){reference-type="ref" reference="ex:3.2"} implies that the inequaliyu ([\[e:p1\]](#e:p1){reference-type="ref" reference="e:p1"}) does not hold in general for $n\ge5$.* # $h^*$ polynomials For a lattice polytope $P\subset M_{\mathbb{R}}$ of dimension $n$ we consider a power series $$\varphi(P,t):= \sum_{k\ge0}\sharp ((kP)\cap M) t^k.$$ It is known that it has a form $$\label{e:4.1} \varphi(P,t) =\frac{h_0^*+h_1^*t+ \dots +h_n^*t^n}{(1-t)^{n+1}}$$ as a rational function ([@St]). Here $h_i^*\ge0$ and $h_0^*=1$. Moreover, it satisfies $$\label{e:v1} h_0^*+h_1^*+\dots +h_n^*=v(P).$$ **Definition 1**. *The polynomial $\sum_i h_i^*t^i=(1-t)^{n+1}\varphi(P,t)$ is called *$h^*$ polynomial* of $P$, and written as $h_P^*$. The degree of the polynomial is called *degree* of $P$, written as $\deg P$.* **Remark 5**. *For the basic $n$-simplex $\Delta^n$, we have $$\varphi(\Delta^n,t)=\frac1{(1-t)^{n+1}}.$$ Hence, $\deg(\Delta^n)=0$.* **Remark 6**. *For a lattice polytope $P$, we set $$\varphi^*(P,t):= \sum_{k\ge0} \sharp(\mbox{\rm int}(kP)\cap M) t^k.$$ From Ehrhart's reciprocity Theorem, we have $$\label{e:4.2} \varphi^*(P,t)=\frac{h_n^*t+\dots +h_0^*t^{n+1}}{(1-t)^{n+1}}.$$ In particular, we have $$h_n^*=\sharp(\mbox{\rm int}P\cap M).$$ From this we see $$\label{e:4.3} \deg P=n-e(P).$$* From this terminology Batyrev and Nill give results [@BN]. **Theorem 6** (Batyrev-Nill). *For a lattice polytope $P\subset M_{\mathbb{R}}$ of dimension $n$, the following two are equivalent;* - *$\deg P\le1$,* - *The equality $$\sharp(P\cap M) =v(P) +n$$ holds.* **Theorem 7** (Batyrev-Nill). *Let $e_1, \dots, e_n$ be a basis of a free abelian group $M$ of rank $n$. A lattice polytope $P\subset M_{\mathbb{R}}$ of dimension $n$ satisfies $deg P=1$ if and only if $$\begin{aligned} P&=& \mbox{\rm conv}\{0, e_1, \dots, e_{n-1}, e_1+a_1e_n< \dots, e_{n-1}+a_{n-1}e_n\}, a_ne_n\}\quad(n\ge2)\\ & & \mbox{where $a_1\ge \dots \ge a_{n-1}\ge0$, $a_n\ge1$ and $a_1+\dots +a_n\ge2$}\end{aligned}$$ or $$P=\mbox{\rm conv}\{0, 2e_1,2e_2, e_3, \dots, e_n\} \quad (n\ge3).$$ The first polytope is called Laurence prism and the second exceptional.* # The inequality In order to prove the inequality ([\[e:p1\]](#e:p1){reference-type="ref" reference="e:p1"}) we need some estimates of terms of $h^*$ polynomial. For a lattice polytope $P\subset M_{\mathbb{R}}$ of dimension $n$, we have $$\label{e:5.1} \sharp(P\cap M) = h_1^* +n+1$$ by comparing the terms of degree one in the equality ([\[e:4.1\]](#e:4.1){reference-type="ref" reference="e:4.1"}). From the equality ([\[e:4.2\]](#e:4.2){reference-type="ref" reference="e:4.2"}) we have $$\label{e:5.2} \sharp(\mbox{int}P\cap M) =h_n^*, \quad \sharp(\mbox{int}(2P)\cap M) = h_{n-1}^* +(n+1)h_n^*.$$ Since $\sharp(P\cap M) =\sharp(\partial P)\cap M +\sharp(\mbox{int}\ P\cap M)$ and $\sharp(\partial P\cap M)\ge n+1$, from the equalities ([\[e:5.1\]](#e:5.1){reference-type="ref" reference="e:5.1"}) and ([\[e:5.2\]](#e:5.2){reference-type="ref" reference="e:5.2"}) we have $$\label{e:5.3} h_1^* \ge h_n^*.$$ ## The case of $e\ge1$ Let $P\subset M_{\mathbb{R}}$ be a lattice polytope of dimension $n$ satisfying $e(P)=n-1$. Then $\deg h_P^*=1$ and the equality ([\[e:v1\]](#e:v1){reference-type="ref" reference="e:v1"}) is $$1+h_1^*=v(P).$$ From the equality ([\[e:5.1\]](#e:5.1){reference-type="ref" reference="e:5.1"}) we have $$\sharp(P\cap M)-v(P)=n.$$ This equality is just theorem [Theorem 6](#th:BN1){reference-type="ref" reference="th:BN1"} (2). **Lemma 4** (The case of $e=n-2$ and $n\ge3$). *Set $n\ge3$. Let $P\subset M_{\mathbb{R}}$ be a lattice polytope of dimension $n$ with $e(P)=n-2$. Then we have $$\sharp(P\cap M) -v(P) \le e(P)+1.$$ The equality holds if and only if $\sharp((n-1)P\cap M)=1$.* *Proof*. $h^*$ polynomial of $P$ is $$h_P^*=1+h_1^*t +h_2^*t^2.$$ Since $e(P)=n-2$ we have $h_2^*=\sharp(\mbox{int}(n-1)P\cap M)\ge1$. From ([\[e:5.1\]](#e:5.1){reference-type="ref" reference="e:5.1"}) and $$1+h_1^*+h_2^*=v(P),$$ we have $$\sharp(P\cap M) -v(P) = n-h_2^* = (n-2)+1 -(h_2^*-1)\le e(P)+1.$$ The equality holds if and only if $h_2^*=1$. $\Box$ **Remark 7**. *When the equality holds, $P$ defines a Gorenstein toric Del Pezzo variety. Batyrev and Juny classified Gorenstein toric Del Pezzo varieties [@BJ].* We need a general theory from Bruns and Herzog [@BH]. **Proposition 3** (Bruns-Herzog). *Let $P\subset M_{\mathbb{R}}$ be a lattice polytope of dimension $n$. Set $e(P)=e$ and $h_P^*=h_0^*+ h_1^*t+\dots +h_{n-e}^*t^{n-e}$. Then for $0\le l\le \frac{n-e}2$ we have $$\sum_{i=0}^l h_{n-e-i}^*\ge \sum_{i=0}^l h_i^*.$$* **Lemma 5** (The case of $e=n-3$ and $n\ge4$). *Set $n\ge4$. Let $P\subset M_{\mathbb{R}}$ be a lattice polytope of dimension $n$ with $e(P)=n-3$. Then we have $$\sharp(P\cap M) -v(P) \le e(P)+1$$ unless $n\ge5$ and $P$ is an empty lattice $n$-simplex isomorphic to $$\Delta_{2,5}^n:=\mbox{\rm conv}\{0,e_1, \dots, e_4, e_1+\dots + e_4+2e_5, e_6, \dots, e_n\}.$$* *Proof*. $h^*$ polynomial of $P$ is $$h_P^*=1+h_1^*t +h_2^*t^2+h_3^*t^3.$$ Since $e(P)=n-3$ we have $h_3^*=\sharp(\mbox{int}(n-2)P\cap M)\ge1$. From ([\[e:5.1\]](#e:5.1){reference-type="ref" reference="e:5.1"}) and $$1+h_1^*+h_2^*+h_3^*=v(P),$$ we have $$\sharp(P\cap M) -v(P) = n-h_2^* -h_3^*= (n-3)+1 -(h_2^*+h_3^*-2).$$ From Proposition [Proposition 3](#p:BH){reference-type="ref" reference="p:BH"} we have $$h_2^*+h_3^*\ge 1+h_1^*.$$ If $h_1^*\ge1$, then the inequality ([\[e:p1\]](#e:p1){reference-type="ref" reference="e:p1"}) holds. $h_1^*=0$ implies that $P$ is an empty lattice simplex. Set $n=4$. Then $e(P)=1$ and $\sharp((2P)\cap M)=h_3^*\ge1$. By comparing therms of degree two of ([\[e:p1\]](#e:p1){reference-type="ref" reference="e:p1"}) we have $$\sharp(2P\cap M) =\binom{n+2}2 +\binom{n+1}1 h_1^* +h_2^*.$$ If $P$ is an empty lattice simplex, then $h_1^*=0$ and $$\sharp(\partial(2P)\cap M)\ge \binom{n+2}2.$$ Hence $h_2^*\ge1$ and the inequality ([\[e:p1\]](#e:p1){reference-type="ref" reference="e:p1"}) holds when $n=4$. When $n\ge5$ the exception to the inequality is the case that $h_1^*=h_2^*=0$ and $h_3^*=1$, that is, $P$ is an empty lattice $n$-simplex with $v(P)=2$. Since $e(P)=n-3$, $P$ is isomorphic to $\Delta_{2,5}^n$. $\Box$ ## The case of $e=0$ **Example 3**. *Let $M$ be a free abelian group of rank $n$ with a basis $e_1, \dots, e_n$. Set $$\Delta_*^n:= \mbox{\rm conv}\{e_1, \dots, e_n.-e_1-\dots -e_n\}.$$ The origin is only lattice point in the interior of $\Delta_*^n$ and $v(\Delta_*^n)=n+1$. Its $h^*$ polynomial is $$h_{\Delta_*^n}^* = 1+ t+ \dots +t^n.$$ In particular, $h_1^*=\dots =h_n^*=1$ and $\sharp(\Delta_*^n\cap M)-v(\Delta_*^n)=(n+2)-(n+1)=1$.* Let $P\subset M_{\mathbb{R}}$ be a lattice polytope of dimension $n$ with $\mbox{int}P\cap M\not=\emptyset$. Then $h_1^*\ge h_n^*\ge1$ from the inequality ([\[e:5.3\]](#e:5.3){reference-type="ref" reference="e:5.3"}). Moreover from Hibi's lower bound theorem [@Hb] we have $h_i^*\ge h_1^*$ for $2\le i\le n-1$. **Proposition 4**. *Let $P\subset M_{\mathbb{R}}$ be a lattice polytope of dimension $n$ with $\mbox{\rm int}P\cap M\not=\emptyset$. Then we have $$\Delta(P)\ge n-1.$$ Moreover,when $n\ge3$ the equality holds if and only if $P\cong \Delta_*^n$.* *Proof*. When $n=1$ $\Delta(P)=0$. Set $n\ge2$. Set $h_P^*=\sum_{i=0}^n h_i^*t^i$. We note $h_1^*, \dots , h_n^*\ge1$. From $v(P)=1+h_1^*+\dots +h_n^*$ and $\sharp(P\cap M) =h_1^*+n+1$, we have $$\label{e:5.10} v(P)-\sharp(P\cap M)=(h_2^*-1)+\dots +(h_n^*-1)-1\ge -1.$$ Hence we have $$\Delta(P) = n+v(P)-\sharp(P\cap M) \ge n-1.$$ The equality holds if and only if $h_2^*=\dots=h_n^*=1$. In particular, $h_n^*=\sharp(\mbox{int}P\cap M)=1$. If $n\ge3$, then we know $h_2^*\ge h_1^*$. Since $h_1^*\ge h_n^*=1$, we have $h_1^*=1$, hence, $v(P)=n+1$ and $\sharp(P\cap M)=n+2$. This means $P\cong \Delta_*^n$. $\Box$ # The case of dimension three In this section we give the main result of Tabei[@Tb]. **Theorem 8** (Tabei). *Let $P\subset M_{\mathbb{R}}$ be a lattice polytope of dimension three. Then we have $$\sharp(P\cap M) -v(P)\le e(P)+1.$$ The lattice polytope $P$ satisfying the equality is one of the following:* - *When $e(P)=2$, $P$ is a Laurence prism or exceptional.* - *When $e(P)=1$,* - *16 cones over lattice polygons with one lattice point in their interiors.* - *15 polytopes those facets have no lattice points in their interiors.* - *When $e(P)=0$, $$P\cong \mbox{\rm conv}\{(1,0,0),(0,1,0),(0,0,1),(-1,-1,-1)\}.$$* 99 [V. Batyrev and D. Juny]{.smallcaps}, Classification of Gorenstein Toric Del Pezzo Varieties in arbitrary dimension, Moscow Math. J. **10**, 285--316 (2010). [V. Batyrev and B. Nill]{.smallcaps}, Multiples of Lattice Polytopes without interior Lattice points, Moscow Math. J. **7**, 195--207 (2007). [W. Bruns and J. Herzog]{.smallcaps}, Cohen-Macaulay rings, Cambridge University Press, 1997. [T. Fujita]{.smallcaps}, Classification theories of polarized varieties, London Mathematics Society Lecture Note Series **155**, Cambridge University Press (1990). [W. Fulton]{.smallcaps}, Introduction to toric varieties, Ann. of Math. Studies No.131, Princeton Univ. Press, 1993. [T. Hibi]{.smallcaps}, A lower bound theorem for Ehrhart polynomials of convex polytopes, Adv. Math. **105** (1994), 162--165. [A. M. Kasprzyk]{.smallcaps}, Canonical toric Fano threefolds, Canad. J. Math. **62** (2010), no. 6, 1293--1309. [A. M. Kasprzyk, M. Kreuzer and B. Nill]{.smallcaps}, On the combinatorial classification of toric log del pezzo surfaces, LMS Journal of Computation and Mathematics **13** (2010), 33--46. [R.J. Koelman]{.smallcaps}, The number of moduli of families of curves on toric surfaces, Proefschrift, Katholieke Universitet te Nijmengen, 1991. [T. Oda]{.smallcaps}, Convex Bodies and Algebraic Geometry, Ergebnisse der Math. **15** Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988. [S. Ogata and H.-L. Zhao]{.smallcaps}, A Characterization of Gorenstein toric Fano $n$-folds with index $n$ and Fujita's conjecture. Far East J. Math. Sci. **94** (2014), 65--88. [R.P. Stanley]{.smallcaps}, Decompositions of rational convex polytopes, Ann. Discrete Math. **6** (1980), 333--342. [R. Tabei]{.smallcaps}, Toric varieties with maximal regularity, Tohoku University Master these, 2022.

arxiv_math

--- abstract: | In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;d\sigma(y_1,y_2)\right|,$$ in dimensions $d=1,2$ and as an application, we deduce sharp endpoint estimates for the multilinear spherical maximal function. We also prove sharp $L^p-$estimates for the local spherical maximal function in all dimensions $d\geq 2$, thus resolving a question left open in the work of Jeong and Lee (https://doi.org/10.1016/j.jfa.2020.108629). We further study necessary conditions for the bilinear maximal function, $$\mathcal M (f,g)(x)=\sup_{t>0}\left|\int_{\mathbb S^{1}}f(x-ty)g(x+ty)\;d\sigma(y)\right|$$ to be bounded from $L^{p_1}(\mathbb R^2)\times L^{p_2}(\mathbb R^2)$ to $L^p(\mathbb R^2)$ and prove sharp results for a linearized version of $\mathcal M$. address: - | Ankit Bhojak\ Department of Mathematics\ Indian Institute of Science Education and Research Bhopal\ Bhopal-462066, India. - | Surjeet Singh Choudhary\ Department of Mathematics\ Indian Institute of Science Education and Research Bhopal\ Bhopal-462066, India. - | Saurabh Shrivastava\ Department of Mathematics\ Indian Institute of Science Education and Research Bhopal\ Bhopal-462066, India. - | Kalachand Shuin\ Reseearch Institute of Mathematics,\ Seoul National University, 08826\ Gwanak-RO 1, Seoul, Republic of Korea. author: - Ankit Bhojak - Surjeet Singh Choudhary - Saurabh Shrivastava - Kalachand Shuin bibliography: - biblio.bib title: Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions --- [^1] # Introduction and main results Let $\sigma$ be the surface measure on the $(d-1)$-dimensional unit sphere $\mathbb S^{d-1}$. The study of $L^p-$improving properties of the spherical averages defined as $$A_tf(x)=\int_{\mathbb S^{d-1}}f(x-ty)\;d\sigma(y),$$ was initiated by Littman [@Littman] and Strichartz [@Strichartz]. Consider the linear spherical maximal function defined as $$A_*f(x)=\sup_{t>0}\left|A_tf(x)\right|.$$ For $p>\frac{d}{d-1}$, the $L^p-$boundedness of $A_*$ was established by Stein [@MaximalFunctionsISphericalMeans] for dimension $d\geq3$ and Bourgain [@BourgainCircular] in dimension two. At the endpoint $p=\frac{d}{d-1}$, the restricted weak type inequality was proved by Bourgain for dimensions $d\geq3$. To study the case of dimension two, Oberlin [@OberlinLinearization] considered the following linearized version of the spherical maximal operator $$\widetilde Af(x)=\int_{\mathbb S^{d-1}}f(x-|x|y)\;d\sigma(y),$$ and showed that $\widetilde A$ maps $L^{2,1}({\mathbb {R}}^2)$ to $L^{2,\infty}({\mathbb {R}}^2)$ boundedly. However, using a modification of the Kakeya construction from [@Keich], it was finally shown in [@EndpointMappingPropertiesOfSphericalMaximalOperators] that the spherical maximal operator $A_*$ does not map $L^{2,1}({\mathbb {R}}^2)$ to $L^{2,\infty}({\mathbb {R}}^2)$. The local spherical maximal operator $A_{loc}f(x)=\sup_{1\leq t\leq2}\left|A_tf(x)\right|$ has also been widely studied. Schlag and Sogge [@Schlag; @SchlagSogge] showed that $A_{loc}$ is bounded from $L^p({\mathbb {R}}^d)$ to $L^q({\mathbb {R}}^d)$ when $d\geq 2$ and $\left(\frac{1}{p},\frac{1}{q}\right)$ lies in the interior of the closed convex hull generated by the points $(0,0),\;\left(\frac{d-1}{d},\frac{d-1}{d}\right),\;\left(\frac{d-1}{d},\frac{1}{d}\right)$ and $\left(\frac{d^2-d}{d^2+1},\frac{d-1}{d^2+1}\right)$ and unbounded if $\left(\frac{1}{p},\frac{1}{q}\right)$ lies outside the closed convex hull. Moreover, boundedness on the boundary of the hull was resolved by Lee [@Lee1] for all dimensions $d\geq2$. The sparse bounds for the operator $A_{loc}$ was obtained by Lacey [@SparseBoundsForSphericalMaximalFunction]. We also refer to [@SeegerVariation] for sparse domination of the corresponding $r-$variation operators related to the family of spherical averages. ## Part I In this part we deal with the well-studied bilinear variant of the linear spherical maximal function defined as $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;d\sigma(y_1,y_2)\right|.$$ This operator first appeared in the work [@GGIPS]. The optimal strong type bounds were obtained in [@MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators] by a method of "slicing\". The authors showed that for $d\geq 2$, the operator $\mathfrak{M}$ maps $L^{p_1}({\mathbb {R}}^d)\times L^{p_2}({\mathbb {R}}^d)$ to $L^{p}({\mathbb {R}}^d)$ if and only if $p>\frac{d}{2d-1}$ and $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$ except the points $(1,\infty,1)$ and $(\infty,1,1)$. Moreover, the restricted weak type estimate $\mathfrak{M}:L^{p_1,1}({\mathbb {R}}^d)\times L^{p_2,1}({\mathbb {R}}^d)\to L^{\frac{d}{2d-1},\infty}({\mathbb {R}}^d)$ holds for dimensions $d\geq3$; however, the endpoint boundedness in dimension two remained open. Now, we discuss the case of dimension one. The boundedness of $\mathfrak{M}$, when $d=1$, was studied in [@ChristZhou; @DosidisRamos]. They proved that $\mathfrak{M}:L^{p_1}({\mathbb {R}})\times L^{p_2}({\mathbb {R}})\to L^p({\mathbb {R}})$ for $p_1,p_2>2$ and $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Moreover, the weak type estimates fails at the lines $p_1=2$ and $p_2=2$. The boundedness of the lacunary analogue of $\mathfrak{M}$ was studied in [@Borges; @CLS] in dimensions $d\geq2$ and [@ChristZhou] in dimension one. Also see [@LeeShuin] for boundedness of bilinear maximal functions defined with degenerate surfaces. Our first main result addresses the restricted weak type bounds for $\mathfrak{M}$ at the respective endpoints in dimensions one and two. We have the following, **Theorem 1**. *Let $1\leq p_1,p_2\leq\infty$ with $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. The following is true,* 1. *For $1\leq p\leq\infty$, and either $p_1=2$ or $p_2=2$, we have $$\mathfrak{M}:L^{p_1,1}({\mathbb {R}})\times L^{p_2,1}({\mathbb {R}})\to L^{p,\infty}({\mathbb {R}}).$$[\[bilinearone\]]{#bilinearone label="bilinearone"}* 2. *For $1<p_1,p_2<2$, we have $$\mathfrak{M}:L^{p_1,1}({\mathbb {R}}^2)\times L^{p_2,1}({\mathbb {R}}^2)\to L^{\frac{2}{3},\infty}({\mathbb {R}}^2).$$[\[bilineartwo\]]{#bilineartwo label="bilineartwo"}* We now discuss some applications to multilinear spherical averages. Let $m\geq2$ and $f_1,f_2,\dots,f_m\in\mathcal{S}(\mathbb{R})$. Consider the multilinear spherical maximal function defined by $$\mathcal{S}^{m}(f_1,f_2,\dots,f_m)(x):=\sup_{t>0}\Big|\int_{\mathbb{S}^{m-1}}\prod^{m}_{i=1}f_i(x-ty_i)~d\sigma_{m-1}(\vec{y})\Big|,$$ where $d\sigma_{m-1}(\vec{y})$ is the normalized surface measure on the sphere $\mathbb{S}^{m-1}$. The corresponding single scale averaging operator was studied by Oberlin [@Oberlin88] and Bak and Shim [@BakShim]. Later, Shrivastava and Shuin [@ShrivastavaShuin] proved a complete $L^{p_1}({\mathbb {R}})\times L^{p_2}({\mathbb {R}})\times \cdots L^{p_m}({\mathbb {R}})\rightarrow L^{p}({\mathbb {R}})$ boundedness for Banach indices satisfying $\frac{1}{p}\leq \sum^{m}_{i=1}\frac{1}{p_i}$, $1\leq p_i,p\leq\infty$. Very recently, Dosidis and Ramos investigated the boundedness of the maximal function $\mathcal{S}^{m}$. They proved the following, **Theorem 2** ([@DosidisRamos]). *Let $m\geq2$, $1\leq p_i\leq\infty$ for $i=1,2,\dots,m$ and $\frac{1}{p}=\sum^{m}_{i=1}\frac{1}{p_i}$. Then there exists $C>0$ such that the following boundedness holds $$\begin{aligned} \label{eq1} \Vert \mathcal{S}^{m}(f_1,f_2,\dots,f_m)\Vert_{L^{p}(\mathbb{R})}\leq C\prod^{m}_{i=1}\Vert f_i\Vert_{L^{p_i}(\mathbb{R})}, \end{aligned}$$ if and only if* 1. *$\frac{1}{p}=\sum^{m}_{i=1}\frac{1}{p_i}<m-1$,* 2. *for every $i=1,2,\dots,m$, $\sum^{m}_{j=1,j\neq i}\frac{1}{p_j}<m-\frac{3}{2}$,* 3. *$(\frac{1}{p_1},\frac{1}{p_2},\dots,\frac{1}{p_m})\notin \{0,1\}^{m}\setminus\{(0,0,\dots,0)\}$.* *If $(\frac{1}{p_1},\frac{1}{p_2},\dots,\frac{1}{p_m})\in \{0,1\}^{m}\setminus\{(0,0,\dots,0)\}$, then weak type estimate holds, i.e. $$\Vert \mathcal{S}^{m}(f_1,f_2,\dots,f_m)\Vert_{L^{p,\infty}(\mathbb{R})}\leq C\prod^{m}_{i=1}\Vert f_i\Vert_{L^{p_i}(\mathbb{R})}$$ if and only if $(1)$ and $(2)$ both hold. Moreover, if for some $i\in\{1,2,\dots,m\}$, $\sum^{m}_{j=1,j\neq i}\frac{1}{p_j}=m-\frac{3}{2}$, then the estimate $\mathcal S^m:L^{p_1}({\mathbb {R}})\times\cdots\times L^{p_m}({\mathbb {R}})\to L^{p,\infty}({\mathbb {R}})$ does not hold.* Our next result concerns the restricted weak-type estimates for $\mathcal S^m$ on the boundary points of the convex hull defined in Theorem [Theorem 2](#Dosidis){reference-type="ref" reference="Dosidis"}. Namely, we have **Corollary 3**. *Let $m\geq2$, $1\leq p_i\leq\infty$ for $i=1,2,\dots,m$ and $\frac{1}{p}=\sum^{m}_{i=1}\frac{1}{p_i}$. Then there exists $C>0$ such that $$\begin{aligned} \label{eq1} \Vert \mathcal{S}^{m}(f_1,f_2,\dots,f_m)\Vert_{L^{p,\infty}(\mathbb{R})}\leq C\prod^{m}_{i=1}\Vert f_i\Vert_{L^{p_i,1}(\mathbb{R})} \end{aligned}$$ holds true for $(\frac{1}{p_1},\frac{1}{p_2},\dots,\frac{1}{p_m})$ belongs to the following closed line segments $$L_{k,j}=\left\{\left(\frac{1}{p_1},\frac{1}{p_2},\dots,\frac{1}{p_m}\right):0\leq \frac{1}{p_k}\leq\frac{1}{2}, \frac{1}{p_j}=\frac{1}{2},\frac{1}{p_i}=1,~\forall~i\neq k,j\right\},~~\forall~j,k\in\{1,2,\dots,m\}.$$* ## Local spherical maximal function In [@MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators], Jeong and Lee also studied the improving estimates for the local bilinear maximal operator, $$\mathfrak{M}_{loc}(f,g)(x):=\sup_{1<t<2}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;d\sigma(y_1,y_2)\right|.$$ They proved the following, **Theorem 4** ([@MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators]). *Let $d\geq2$, $1\leq p_1,p_2\leq\infty$, and $\frac{1}{2}<p<\infty$. Then the estimate $$\label{localbounds} \|\mathfrak{M}_{loc}\|_{L^{p_1}({\mathbb {R}}^d)\times L^{p_2}({\mathbb {R}}^d)\to L^{p}({\mathbb {R}}^d)}\lesssim 1,$$ holds for $\frac{1}{p}\leq\frac{1}{p_1}+\frac{1}{p_2}<\min\left\{\frac{2d-1}{d},1+\frac{d}{p},\frac{1}{p}+\frac{2(d-1)}{d}\right\}$. Conversely [\[localbounds\]](#localbounds){reference-type="eqref" reference="localbounds"} holds only if $\frac{1}{p}\leq\frac{1}{p_1}+\frac{1}{p_2}\leq\min\left\{\frac{2d-1}{d},1+\frac{d}{p}\right\}$. Furthermore, for $p=\infty$, [\[localbounds\]](#localbounds){reference-type="eqref" reference="localbounds"} holds if and only if $0\leq\frac{1}{p_1}+\frac{1}{p_2}\leq1$.* The above range is sharp for the strong type boundedness for $0<p<d$ and $\frac{2(d-1)}{d-2}<p<\infty$. Using a Knapp type example, the authors in [@BFOPZ] showed that the condition $\frac{1}{p_1}+\frac{1}{p_2}\leq\frac{d-1}{(d+1)p}+\frac{2d}{d+1}$ is necessary for the strong $(p_1,p_2,p)-$boundedness of $\mathfrak{M}_{loc}$; however, the sufficiency of the condition was left open. Our next result provides sharp bounds for $\mathfrak{M}_{loc}$, thus resolving the sufficiency issue in all dimensions $d\geq2$. To state our results, we require a few notations. We define the line segments $\ell^d_1,\ell^d_2,$,$\ell^d_3$ and quadrilateral $\mathscr{Q}^d$ as follows: $$\begin{aligned} \ell^d_1&=\left\{\left(x,y,z\right)\in [0,1]^2\times[0,2):z=x+y=\frac{2d-1}{d}\right\},\\ \ell^d_2&=\left\{\left(x,y,z\right)\in [0,1]^2\times[0,2):x+y=\frac{2d-1}{d}=\frac{d-1}{(d+1)}z+\frac{2d}{d+1}\right\},\\ \ell^d_3&=\left\{\left(x,y,z\right)\in [0,1]^2\times[0,2):x+y=\frac{d-1}{(d+1)}z+\frac{2d}{d+1}=1+dz\right\},\\ \mathscr{Q}^d&=\left\{\left(x,y,z\right)\in [0,1]^2\times[0,2):z<x+y=\frac{2d-1}{d}<\frac{d-1}{(d+1)}z+\frac{2d}{d+1}\right\}.\end{aligned}$$ **Theorem 5**. *Let $d\geq2$, $1\leq p_1,p_2\leq\infty$, and $\frac{1}{2}<p<\infty$. Then the following are true,* 1. *The estimate [\[localbounds\]](#localbounds){reference-type="eqref" reference="localbounds"} holds for $\frac{1}{p}\leq\frac{1}{p_1}+\frac{1}{p_2}\leq\min\left\{\frac{2d-1}{d},1+\frac{d}{p},\frac{d-1}{(d+1)p}+\frac{2d}{d+1}\right\}$ and $\left(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p}\right)\notin\ell^d_1\cup\ell^d_2\cup\ell^d_3\cup\mathscr{Q}^d$.* 2. *The restricted weak type inequality, $$\label{localboundsres} \|\mathfrak{M}_{loc}\|_{L^{p_1,1}({\mathbb {R}}^d)\times L^{p_2,1}({\mathbb {R}}^d)\to L^{p,\infty}({\mathbb {R}}^d)}\lesssim 1,$$ holds when* - *$d\geq 3$ and $\left(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p}\right)\in\ell^d_1\cup\ell^d_2\cup\ell^d_3$,* - *$d=2$, and $\left(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p}\right)\in\ell^2_1\cup\ell^2_2\cup\ell^2_3\setminus \left\{(1,\frac{1}{2},\frac{3}{2}),(\frac{1}{2},1,\frac{3}{2})\right\}$.* 3. *The restricted strong type inequality, $$\|\mathfrak{M}_{loc}\|_{L^{p_1,1}({\mathbb {R}}^d)\times L^{p_2,1}({\mathbb {R}}^d)\to L^{p}({\mathbb {R}}^d)}\lesssim 1,$$ holds when,* - *$d\geq 3$ and $\left(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p}\right)\in\mathscr{Q}^d$,* - *$d=2$, and $\left(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p}\right)\in\mathscr{Q}^2$, $p_1\neq1$ and $p_2\neq 1$.* **Remark 6**. *In [@BFOPZ], the authors obtained sparse domination of bilinear spherical maximal function in the range covered in Theorem [Theorem 4](#JeongLee){reference-type="ref" reference="JeongLee"}. We do not pursue a sparse domination of the bilinear spherical maximal function for the improved range obtained in Theorem [Theorem 5](#slicedbilinearimproving){reference-type="ref" reference="slicedbilinearimproving"} in this paper but aim to obtain it elsewhere.* To prove Theorem [Theorem 1](#fullmaximal){reference-type="ref" reference="fullmaximal"} [\[bilinearone\]](#bilinearone){reference-type="eqref" reference="bilinearone"} and [Theorem 5](#slicedbilinearimproving){reference-type="ref" reference="slicedbilinearimproving"}, we will rely on a modification of slicing argument in [@MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators]. In contrast to domination of $\mathfrak{M}$ by the product of Hardy-Littlewood and linear spherical maximal function, we will dominate $\mathfrak{M}$ by intermediate averaging operators defined below, $$\mathfrak{A}^rf(x)=\|A_tf(x)\|_{L^r([1,2],t^{d-1}dt)},\;1\leq r\leq\infty.$$ Observe that $\mathfrak{A}^1$ and $\mathfrak{A}^\infty$ are the local Hardy-Littlewood and local spherical maximal functions respectively. We also define the maximal operator $\mathfrak{A}^r_*$ as follows, $$\mathfrak{A}^r_*f(x)=\sup_{k\in{\mathbb Z}}\|A_{2^kt}f(x)\|_{L^r([1,2],t^{d-1}dt)},\;1\leq r\leq\infty.$$ We set $A=\left(\frac{1}{r},0\right),\;P=\left(\frac{r(d^2+1)}{d+1+r(d^2-d)},\frac{r'(d^2+1)}{d-1}\right),\;Q=\left(\frac{rd-r+1}{rd},\frac{1}{r'd}\right),$ and $R=\left(\frac{rd-r+1}{rd},\frac{rd-r+1}{rd}\right)$. We denote $QR$ to be the open line segment joining the points $Q$ and $R$. We have the following boundedness for $\mathfrak{A}^r$ and $\mathfrak{A}^r_*$, **Theorem 7**. *Let $d\geq2$ and $1 \leq p,q\leq \infty$. The following holds true,* 1. *For $1\leq r\leq\infty$ and $\frac{1}{q}\leq\frac{1}{p}\leq\min\{\frac{d}{q}+\frac{1}{r},\frac{d-1}{d}+\frac{1}{dr},\frac{d-1}{(d+1)q}+\frac{2}{(d+1)r}+\frac{d-1}{d+1}\}$ and $\left(\frac{1}{p},\frac{1}{q}\right)\notin\{P,Q,R\}\cup QR$, we have $$\label{localA} \|\mathfrak{A}^r\|_{L^p({\mathbb {R}}^d)\to{L^q({\mathbb {R}}^d)}}\lesssim 1.$$[\[stronglinearA\]]{#stronglinearA label="stronglinearA"} Moreover for $\left(\frac{1}{p},\frac{1}{q}\right)\in\{P,Q,R\}$, we have the restricted weak type inequality, $$\|\mathfrak{A}^r\|_{L^{p,1}({\mathbb {R}}^d)\to{L^{q,\infty}({\mathbb {R}}^d)}}\lesssim 1,$$ and for $\left(\frac{1}{p},\frac{1}{q}\right)\in QR$, we have the restricted strong type inequality $$\|\mathfrak{A}^r\|_{L^{p,1}({\mathbb {R}}^d)\to{L^{q}({\mathbb {R}}^d)}}\lesssim 1.$$* 2. *Conversely, the estimate [\[localA\]](#localA){reference-type="eqref" reference="localA"} holds only if $\frac{1}{q}\leq\frac{1}{p}\leq\min\{\frac{d}{q}+\frac{1}{r},\frac{d-1}{d}+\frac{1}{dr},\frac{d-1}{(d+1)q}+\frac{2}{(d+1)r}+\frac{d-1}{d+1}\}$.[\[necessarylinearAr\]]{#necessarylinearAr label="necessarylinearAr"}* 3. *For $1\leq r\leq\infty$, the operator $\mathfrak{A}^r_*$ maps $L^p({\mathbb {R}}^d)$ to $L^p({\mathbb {R}}^d)$ for $p>\frac{dr}{dr-r+1}$. Moreover $\mathfrak{A}^r_*$ is of restricted weak type $\left(\frac{dr}{dr-r+1},\frac{dr}{dr-r+1}\right)$ for $1\leq r<\infty$. [\[restrictedlinearA\]]{#restrictedlinearA label="restrictedlinearA"}* **Remark 8**. *It was pointed out in [@SeegerVariation] that the local spherical maximal function does not map $L^p({\mathbb {R}}^d)$ to $L^q({\mathbb {R}}^d)$ for when $d\geq 3$ and $\left(\frac{1}{p},\frac{1}{q}\right)$ lies in the line segment joining the points $\left(\frac{d-1}{d},\frac{d-1}{d}\right)$ and $\left(\frac{d-1}{d},\frac{1}{d}\right)$. An analogous result also holds for the averaging operators $\mathfrak A^r$ when $d\geq 2$ i.e. a strong type boundedness does not hold for $\mathfrak A^r$ when $\left(\frac{1}{p},\frac{1}{q}\right)$ lies in the line segment $QR$, this is a corollary of Proposition [Proposition 22](#restrictedexample){reference-type="ref" reference="restrictedexample"}.* We prove Theorem [Theorem 1](#fullmaximal){reference-type="ref" reference="fullmaximal"}, Corollary [Corollary 3](#mlinearspherical){reference-type="ref" reference="mlinearspherical"} and Theorem [Theorem 5](#slicedbilinearimproving){reference-type="ref" reference="slicedbilinearimproving"} in Section [2](#Sec:fullmaximal){reference-type="ref" reference="Sec:fullmaximal"}. The proof of Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"} is contained in Section [3](#Sec:localmaximal){reference-type="ref" reference="Sec:localmaximal"}. ## Part II Recently, for $0<\theta<2\pi$, Greenleaf et al [@GIKL] studied the bilinear spherical average $$\mathcal A_t^\theta (f,g)(x)=\int_{\mathbb S^{1}}f(x-ty)g(x-t\Theta y)\;d\sigma(y),\;t>0,$$ where $\Theta$ denotes the counter-clockwise rotation by an angle $\theta$. They proved the following, **Theorem 9** ([@GIKL]). *Let $0<\theta<2\pi$ and $1\leq p_1,p_2,p\leq\infty$. The operator $\mathcal A^\theta_1$ is bounded from $L^{p_1}({\mathbb {R}}^2)\times L^{p_2}({\mathbb {R}}^2)$ to $L^p({\mathbb {R}}^2)$ if* 1. *$\theta\neq\pi$ and $\left(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p}\right)$ lies in the closed convex hull generated by the vertices $(0,0,0),\;\left(\frac{2}{3},\frac{2}{3},1\right),\;\left(0,\frac{2}{3},\frac{1}{3}\right),\;\left(\frac{2}{3},0,\frac{1}{3}\right),\;(1,0,1),\;(0,1,1),$ and $\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$, or* 2. *$\theta=\pi$ and $\left(\frac{1}{p_1},\frac{1}{p_2},\frac{1}{p}\right)$ lies in the closed convex hull generated by the vertices $(0,0,0),\;\left(\frac{2}{3},\frac{2}{3},1\right),\;\left(0,\frac{2}{3},\frac{1}{3}\right),\;\left(\frac{2}{3},0,\frac{1}{3}\right),\;(1,0,1),$ and $(0,1,1)$.* The boundedness of $\mathcal A_1^\theta$ is sharp in the Hölder range of indices mentioned in the above theorem, i.e. when $p\geq 1$. However, it is unknown if the boundedness holds outside the closed convex hull in the above theorem for $p<1$. **Remark 10**. *In [@GIKL], the authors showed that $$\label{necessarycondition} \frac{3}{p_1}+\frac{3}{p_2}\leq 1+\frac{3}{p},$$ is a necessary condition for the operator $\mathcal{A}^\pi_1$ to be bounded from $L^{p_1}({\mathbb {R}}^2)\times L^{p_2}({\mathbb {R}}^2)$ to $L^{p}({\mathbb {R}}^2)$. In Section [5](#Sec:Necessary){reference-type="ref" reference="Sec:Necessary"}, we will provide a different example based on functions of product type, that works for dimensions $d\geq2$. In particular, we show that the condition [\[necessarycondition\]](#necessarycondition){reference-type="ref" reference="necessarycondition"} is in fact necessary for boundedness of $\mathcal{A}^\pi_1$ even when the functions are restricted to the space of functions of product type. We refer to [@Tanaka] for analogous results for the Kakeya maximal function acting on functions of product type.* The lacunary maximal function $\mathcal M_{lac}^\theta(f,g)(x)=\sup_{k\in{\mathbb Z}}\mathcal |A_{2^k}^\theta(f,g)(x)|$ was studied in [@CLS], where they obtained the following boundedness for $\mathcal M_{lac}^\theta$, **Theorem 11** ([@CLS]). *Let $0<\theta<2\pi$ and $1<p_1,p_2\leq\infty$, with $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Then the following estimate holds $$\Vert\mathcal{M}_{lac}^\theta(f,g)\Vert_{L^{p}(\mathbb{R}^{2})}\lesssim \Vert f\Vert_{L^{p_1}(\mathbb{R}^{2})}\Vert g\Vert_{L^{p_2}(\mathbb{R}^{2})}$$ for $p>\frac{3}{4}$.* In this article, we are concerned with the study of the full maximal function given by, $$\mathcal M^\theta (f,g)(x)=\sup_{t>0}\mathcal |A_t^\theta(f,g)(x)|.$$ We note that $\mathcal M^\theta$ is bounded from $L^{p_1}({\mathbb {R}}^2)\times L^{p_2}({\mathbb {R}}^2)$ to $L^p({\mathbb {R}}^2)$ for $2<p\leq\infty$ with $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Indeed, the boundedness is a consequence of bilinear interpolation, the linear estimates $A_*:L^p({\mathbb {R}}^2)\to L^p({\mathbb {R}}^2),\;p>2$ and the pointwise inequality $$\mathcal M^\theta(f,g)(x)\leq\min\{\|f\|_\infty A_*g(x),\|g\|_\infty A_*f(x)\}.$$ Our first main result concerns the restricted weak type non-inequality of the maximal function $\mathcal M^\theta$ at the endpoint boundaries $p_1=2$ and $p_2=2$ in dimension two. We have, **Theorem 12**. *Let $1<p_1,p_2\leq\infty,\;\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$ and $0<\theta<2\pi$. The operator $\mathcal{M}^\theta$ does not map $L^{p_1,1}({\mathbb {R}}^2)\times L^{p_2,1}({\mathbb {R}}^2)$ to $L^{p,\infty}({\mathbb {R}}^2)$ if $p_1\leq2$ or $p_2\leq2$. In particular, $\mathcal{M}^\theta$ is not of restriced weak type $(2,2,1)$.* Currently, we do not have a positive result for the operator $\mathcal M^\theta$ in the local $L^2$ range: $p_1,p_2>2$ and $1<p\leq2$. However, we will prove sharp boundedness results for the linearized version of $\mathcal M^\theta$ defined as $$\widetilde{\mathcal A}^\theta (f,g)(x)=\int_{\mathbb S^{1}}f(x-|x|y)g(x-|x|\Theta y)\;d\sigma(y).$$ More precisely, we have **Theorem 13**. *Let $2<p_1,p_2\leq\infty$ with $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Then $$\Vert\widetilde{\mathcal A}^\theta(f,g)\Vert_{L^{p}(\mathbb{R}^{2})}\lesssim \Vert f\Vert_{L^{p_1}(\mathbb{R}^{2})}\Vert g\Vert_{L^{p_2}(\mathbb{R}^{2})}.$$ Moreover, for $p_1=2$ or $p_2=2$ we have the following restricted weak type estimates, $$\Vert\widetilde{\mathcal A}^\theta(f,g)\Vert_{L^{p,\infty}(\mathbb{R}^{2})}\lesssim \Vert f\Vert_{L^{p_1,1}(\mathbb{R}^{2})}\Vert g\Vert_{L^{p_2,1}(\mathbb{R}^{2})}.$$* The proof is based on an appropriate change of variable in the polar co-ordinates and a multilinear version of Bourgain's interpolation trick (Lemma [\[interpolation\]](#interpolation){reference-type="ref" reference="interpolation"}). **Remark 14**. *We would like to remark that using our method of proof, one can recover the result of Oberlin [@OberlinLinearization] that the corresponding linear spherical operator $\widetilde A$ is also of restricted weak type $(2,2)$. This simplifies the proof of Oberlin [@OberlinLinearization].* The proofs of Theorem [Theorem 12](#Mfull){reference-type="ref" reference="Mfull"} and Theorem [Theorem 13](#linearized){reference-type="ref" reference="linearized"} are contained in Section [4](#Mfullsec){reference-type="ref" reference="Mfullsec"}. In Section [5](#Sec:Necessary){reference-type="ref" reference="Sec:Necessary"}, we also discuss some necessary conditions for $L^p-$ boundedeness of higher dimensional version of $\mathcal{M}^\theta$. # {#Sec:fullmaximal} ## Proof of Theorem [Theorem 1](#fullmaximal){reference-type="ref" reference="fullmaximal"} [\[bilinearone\]](#bilinearone){reference-type="eqref" reference="bilinearone"}: {#proof-of-theorem-fullmaximal-bilinearone} It is enough to show $\mathfrak{M}:L^{2,1}({\mathbb {R}})\times L^{2,1}({\mathbb {R}})\to L^{1,\infty}({\mathbb {R}})$. Observe that due to symmetry, it is enough to deal with the integral over the arc from $\theta=0$ to $\theta=\frac{\pi}{4}$ instead of integral over $\mathbb{S}^{1}$. By a change of variable we have, $$\begin{aligned} \mathfrak{M}(f,g)(x)\lesssim\sup_{t>0}\int_{y=0}^{\frac{1}{\sqrt{2}}}f(x-ty)g(x-t\sqrt{1-y^2})\;\frac{dy}{\sqrt{1-y^2}}+\text{ similar terms}\end{aligned}$$ By decomposing the interval $[0,\frac{1}{\sqrt{2}}]$ into dyadic annuli, we have $$\mathfrak{M}(f,g)(x)\lesssim\sum_{k=1}^{\infty}T_k(f,g)(x),$$ where the operator $T_k$ is defined by $$\begin{aligned} T_k(f,g)(x):&=\sup_{t>0}\int_{2^{-k-1}}^{2^{-k}}|f(x-ty)||g(x-t\sqrt{1-y^2})|\;dy\end{aligned}$$ We have the following pointwise inequality, $$\label{T_k} T_k(f,g)(x)\lesssim\min\;\left\{2^{\frac{k}{3}}M_3f(x)M_{\frac{3}{2}}g(x),2^{-\frac{k}{3}}M_\frac{3}{2}f(x)M_3g(x)\right\}.$$ Indeed by Cauchy-Schwartz inequality, we get $$\begin{aligned} &T_k(f,g)(x)\\ \lesssim&\sup_{t>0}\left(\int_{2^{-k-1}}^{2^{-k}}|f(x-ty)|^3\;dy\right)^\frac{1}{3}\;\sup_{t>0}\left(\int_{2^{-k-1}}^{2^{-k}}|g(x-t\sqrt{1-y^2})|^\frac{3}{2}\;dy\right)^\frac{2}{3}\\ =&2^{-\frac{k}{3}}\sup_{t>0}\left(\frac{1}{2^{-k}}\int_{2^{-k-1}}^{2^{-k}}|f(x-ty)|^3\;dy\right)^\frac{1}{3}\;\sup_{t>0}\left(\int_{\sqrt{1-2^{-2k}}}^{\sqrt{1-2^{-2k-2}}}|g(x-tz)|^\frac{3}{2}\;\frac{zdz}{\sqrt{1-z^2}}\right)^\frac{2}{3}\\ \lesssim& 2^{\frac{k}{3}}M_3f(x)M_{\frac{3}{2}}g(x).\end{aligned}$$ The other inequality in [\[T_k\]](#T_k){reference-type="eqref" reference="T_k"} follows similarly. Therefore, for a fixed $N\in{\mathbb N}$, we have $$\begin{aligned} \mathfrak{M}(f,g)(x)&\lesssim\sum_{k=1}^N2^{\frac{k}{3}}M_3f(x)M_{\frac{3}{2}}g(x)+\sum_{k=N+1}2^{-\frac{k}{3}}M_\frac{3}{2}f(x)M_3g(x\\ &\lesssim 2^{\frac{N}{3}}M_3f(x)M_{\frac{3}{2}}g(x)+2^{-\frac{N}{3}}M_\frac{3}{2}f(x)M_3g(x).\end{aligned}$$ Hence using the weak type bounds $M_p:L^p\to L^{p,\infty},\;p\geq 1,$ we obtain $$\begin{aligned} |\{x\in{\mathbb {R}}:\;\mathfrak{M}(\chi_F,\chi_G)(x)>\lambda\}|&\lesssim \frac{1}{\lambda}\left(2^{\frac{N}{3}}|F|^{\frac{1}{3}}|G|^\frac{2}{3}+2^{-\frac{N}{3}}|F|^{\frac{2}{3}}|G|^\frac{1}{3}\right)\\ &=\frac{1}{\lambda}|F|^\frac{1}{2}|G|^\frac{1}{2},\end{aligned}$$ where we have chosen $N=3\log_2(|F|^{\frac{1}{6}}|G|^{-\frac{1}{6}})$. 0◻ ## Proof of Corollary [Corollary 3](#mlinearspherical){reference-type="ref" reference="mlinearspherical"}: {#proof-of-corollary-mlinearspherical} In order to prove this theorem we invoke the slicing argument from [@MaximalEstimatesForTheBilinearSphericalAveragesAndTheBilinearBochnerRieszOperators]. Applying slicing argument we get for $m\geq3$, $$\begin{aligned} \mathcal{S}^{m}(f_1,f_2,\dots,f_m)(x)&=&\sup_{t>0}\Big|\int_{B^{m-2}(0,1)}\prod^{m-2}_{i=1}f_i(x-ty_i)\\ &&\hspace{15mm}\int_{r_y\mathbb{S}^{1}}f_{m-1}(x-ty_{m-1})f_{m}(x-ty_{m})d\sigma_{r_y}d\vec{y}\Big|,\end{aligned}$$ where $r_y=\sqrt{1-|\tilde{y}|^{2}}$, $\tilde{y}=(y_1,y_2,\dots,y_{m-2})$ and $d\vec{y}=\prod^{m-2}_{i=1}dy_i$. Now, applying a change of variable we get $$\begin{aligned} &&\mathcal{S}^{m}(f_1,f_2,\dots,f_m)(x)\\ &&=\sup_{t>0}\Big|\int_{B^{m-2}(0,1)}\prod^{m-2}_{i=1}f_i(x-ty_i)\int_{\mathbb{S}^{1}}f_{m-1}(x-tr_yy_{m-1})f_{m}(x-tr_yy_{m})d\sigma d\vec{y}\Big|\\ &&\lesssim \prod^{m-2}_{i=1}Mf_i(x)\mathfrak{M}(f_{m-1},f_m)(x).\end{aligned}$$ Here, $M$ denote the Hardy--Littlewood maximal function. Note that due to symmetry of the sphere $\mathbb{S}^{m-1}$ we can interchange the role of the functions $f_i$ for $i=1,2,\dots,m$ and deduce the following inequality $$\begin{aligned} \mathcal{S}^{m}(f_1,f_2,\dots,f_m)(x)\lesssim \mathfrak{M}(f_{j},f_k)(x)\prod^{m}_{i=1, i\neq j,k}Mf_i(x),\end{aligned}$$ for any $j,k\in \{1,2,\dots,m\}$. Therefore, using the estimates of Theorem [Theorem 1](#fullmaximal){reference-type="ref" reference="fullmaximal"} [\[bilinearone\]](#bilinearone){reference-type="eqref" reference="bilinearone"} we get the desired restricted weak-type estimates. 0◻ ## Proof of Theorem [Theorem 1](#fullmaximal){reference-type="ref" reference="fullmaximal"} [\[bilineartwo\]](#bilineartwo){reference-type="eqref" reference="bilineartwo"}: {#proof-of-theorem-fullmaximal-bilineartwo} We claim that the following holds, $$\label{domination} \mathfrak{M}(f,g)(x)\lesssim \mathfrak {A}^{r}_*(f)(x)\mathfrak {A}^{r'}_*(g)(x).$$ The Theorem [Theorem 1](#fullmaximal){reference-type="ref" reference="fullmaximal"} [\[bilineartwo\]](#bilineartwo){reference-type="eqref" reference="bilineartwo"} follows at once by the above claim along with Hölder's inequality and Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"} [\[restrictedlinearA\]](#restrictedlinearA){reference-type="eqref" reference="restrictedlinearA"}. We prove the above claim. Indeed, an application of the slicing argument implies that $$\begin{aligned} &&\mathfrak{M}(f,g)(x)\\ &=&\sup_{t>0}\left|\int_{0}^{1}\int_{\mathbb S^{d-1}}f(x-tsy_1)\;d\sigma(y_1)\int_{\mathbb S^{d-1}}g(x-t\sqrt{1-s^2}y_2)\;d\sigma(y_2)s^{d-1}(1-s^2)^{\frac{d-2}{2}}\;ds\right|\\ &\leq& \sup_{t>0}\left(\int_{0}^{1}\left|A_{ts}f(x)\right|^rs^{d-1}\;ds\right)^{\frac{1}{r}}\left(\int_{0}^{1}\left|A_{t\sqrt{1-s^2}}f(x)\right|^{r'}s(1-s^2)^{\frac{d-2}{2}}\;ds\right)^{\frac{1}{r'}}.\end{aligned}$$ Hence by a change of variable, it is enough to show that $\sup_{t>0}\left(\int_0^1|A_{ts}|^rs^{d-1}ds\right)^{\frac{1}{r}}\lesssim \mathfrak {A}^{r}_*(f)(x)$. Now ,we have $$\begin{aligned} \sup_{t>0}\left(\int_0^1|A_{ts}|^rs^{d-1}ds\right)^{\frac{1}{r}}&\leq\sup_{t>0}\left(\int_0^{\frac{1}{2}}|A_{ts}|^rs^{d-1}ds\right)^{\frac{1}{r}}+\sup_{t>0}\left(\int_{\frac{1}{2}}^1|A_{ts}|^rs^{d-1}ds\right)^{\frac{1}{r}}\\ &=\frac{1}{2^{\frac{d}{r}}}\sup_{t>0}\left(\int_0^1|A_{ts}|^rs^{d-1}ds\right)^{\frac{1}{r}}+\sup_{k\in{\mathbb Z}}\sup_{2^k\leq t\leq2^{k+1}}\left(\int_{\frac{1}{2}}^1|A_{ts}|^rs^{d-1}ds\right)^{\frac{1}{r}}.\end{aligned}$$ and the claim follows.0◻ ## Proof of Theorem [Theorem 5](#slicedbilinearimproving){reference-type="ref" reference="slicedbilinearimproving"}: {#proof-of-theorem-slicedbilinearimproving} The proof follows directly by Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"}. Indeed, by arguing similar to the inequality [\[domination\]](#domination){reference-type="eqref" reference="domination"}, we have $$\label{localdomination} \mathfrak{M}_{loc}(f,g)(x)\lesssim \mathfrak {A}^{r}(f)(x)\mathfrak {A}^{r'}(g)(x),$$ and by using appropriate Hölder's inequality along with the bounds for the operator $\mathfrak{A}^r$ obtained in Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"}, we obtain Thereom [Theorem 5](#slicedbilinearimproving){reference-type="ref" reference="slicedbilinearimproving"}.0◻ # Linear intermediary spherical functions: Proof of Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"} {#Sec:localmaximal} ## Proof of Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"}: {#proof-of-theorem-linearar} We employ a multiscale decomposition of the operator $\mathfrak{A}^{r}_*$. Let $\phi\in \mathcal S({\mathbb {R}}^d)$ be a function such that $\widehat\phi$ is supported in $B(0,2)$ and $\widehat\phi(\xi)=1$ for $\xi\in B(0,1)$. We define $\widehat{\phi_t}(\xi)=\widehat\phi(t\xi)$ and $\widehat{\psi}_t(\xi)= \widehat{\phi}(t\xi)-\widehat{\phi}(2t\xi)$. Then, we have the identity $$\begin{aligned} \label{identity} \widehat \phi(\xi)+\sum_{j=1}^\infty\widehat \psi_{2^{-j}}(\xi)=1,\;\xi\neq 0. \end{aligned}$$ Using this identity, we have the following pointwise inequalities, $$\mathfrak{A}^{r}f(x)\leq A^{r,0}_{1}f(x)+\sum_{j\geq1}A^{r,j}_{1}f(x),\;\;\text{and}\;\;\mathfrak{A}^{r}_*f(x)\leq A^{r,0}_{*}f(x)+\sum_{j\geq1}A^{r,j}_{*}f(x),$$ where $$A^{r,j}_{*}f(x)=\sup_{k\in{\mathbb Z}}A_{2^k}^{r,j}f(x)=\sup_{k\in{\mathbb Z}}\|A_{2^kt}(f\ast\psi_{2^{k-j}})(x)\|_{L^r([1,2],t^{d-1}dt)},$$ $$A^{r,0}_{*}f(x)=\sup_{k\in{\mathbb Z}}A_{2^k}^{r,0}f(x)=\sup_{k\in{\mathbb Z}}\|A_{2^kt}(f\ast\phi_{2^{k}})(x)\|_{L^r([1,2],t^{d-1}dt)}.$$ The operator $A_1^{r,j}$ has been studied extensively in [@SeegerVariation] to obtain variation estimates for spherical averages. We will require some $L^p$ estimates that are obtained by interpolating the bounds for the endpoint $r=1,\infty$. Some of our bounds are already proved in [@SeegerVariation], however we provide a proof for completeness. **Lemma 15**. *Let $d\geq 2$ and $j\in{\mathbb N}$. For $1\leq r\leq\infty$, we have the following estimates, $$\begin{aligned} \|A^{r,j}_{1}\|_{L^1({\mathbb {R}}^d)\to L^\infty({\mathbb {R}}^d)}&\lesssim 2^\frac{j}{r'},\label{L1-infty-r}\\ \|A^{r,j}_{1}\|_{L^1({\mathbb {R}}^d)\to L^1({\mathbb {R}}^d)}&\lesssim 2^\frac{j}{r'},\label{L1-1-r}\\ \|A^{r,j}_{1}\|_{L^r({\mathbb {R}}^d)\to L^r({\mathbb {R}}^d)}&\lesssim 2^{-j\left(\frac{d-1}{r'}\right)},\;\;\;\;\;\;1\leq r\leq2\label{Lr-r-r},\\ \|A^{r,j}_{1}\|_{L^r({\mathbb {R}}^d)\to L^{r'}({\mathbb {R}}^d)}&\lesssim 2^{-j\left(\frac{d-1}{r'}\right)},\;\;\;\;\;\;1\leq r\leq2\label{Lr-r'-r},\\ \|A^{r,j}_{1}\|_{L^2({\mathbb {R}}^d)\to L^2({\mathbb {R}}^d)}&\lesssim 2^{-j\left(\frac{d-2}{2}+\frac{1}{r}\right)},\;\;2\leq r\leq\infty\label{L2-2-r}. \end{aligned}$$* *Proof.* It is easy to see that $$\begin{aligned} \|A^{1,j}_{1}\|_{L^1({\mathbb {R}}^d)\to L^1({\mathbb {R}}^d)}&\lesssim 1,\label{L1-1-1}\\ \|A^{1,j}_{1}\|_{L^1({\mathbb {R}}^d)\to L^\infty({\mathbb {R}}^d)}&\lesssim 1\label{L1-infty-1}. \end{aligned}$$ By the kernel estimate $|\psi_{2^{-j}}*d\sigma_t(x)|\lesssim\frac{2^j}{(1+2^j||x|-t|)^N},$ for large $N$, we get $$\begin{aligned} \|A^{\infty,j}_{1}\|_{L^1({\mathbb {R}}^d)\to L^1({\mathbb {R}}^d)}&\lesssim 2^j,\label{L1-1-infty}\\ \|A^{\infty,j}_{1}\|_{L^1({\mathbb {R}}^d)\to L^\infty({\mathbb {R}}^d)}&\lesssim 2^j\label{L1-infty-infty}. \end{aligned}$$ The estimates [\[L1-infty-r\]](#L1-infty-r){reference-type="eqref" reference="L1-infty-r"} and [\[L1-1-r\]](#L1-1-r){reference-type="eqref" reference="L1-1-r"} follows by interpolating the bounds [\[L1-infty-1\]](#L1-infty-1){reference-type="eqref" reference="L1-infty-1"} with [\[L1-infty-infty\]](#L1-infty-infty){reference-type="eqref" reference="L1-infty-infty"} and [\[L1-1-1\]](#L1-1-1){reference-type="eqref" reference="L1-1-1"} with [\[L1-1-infty\]](#L1-1-infty){reference-type="eqref" reference="L1-1-infty"} respectively. Using the Fourier decay of the spherical measure $|\widehat{d\sigma}(\xi)|\lesssim (1+|\xi|)^{-\frac{d-1}{2}}$, we obtain $$\|A^{2,j}_{1}\|_{L^2({\mathbb {R}}^d)\to L^2({\mathbb {R}}^d)}\lesssim 2^{-j\left(\frac{d-1}{2}\right)}.\label{L2-2-2}$$ For $1\leq r \leq 2$, the estimate [\[Lr-r-r\]](#Lr-r-r){reference-type="eqref" reference="Lr-r-r"} follows directly from estimates [\[L1-1-1\]](#L1-1-1){reference-type="eqref" reference="L1-1-1"} and [\[L2-2-2\]](#L2-2-2){reference-type="eqref" reference="L2-2-2"}. Similarly, the estimate [\[Lr-r\'-r\]](#Lr-r'-r){reference-type="eqref" reference="Lr-r'-r"} follows from the estimates [\[L1-infty-1\]](#L1-infty-1){reference-type="eqref" reference="L1-infty-1"} and [\[L2-2-2\]](#L2-2-2){reference-type="eqref" reference="L2-2-2"}. Moreover, by Stein's proof of spherical maximal function [@MaximalFunctionsISphericalMeans], we have $$\|A^{\infty,j}_{1}\|_{L^2({\mathbb {R}}^d)\to L^2({\mathbb {R}}^d)}\lesssim 2^{-j\left(\frac{d-2}{2}\right)}.\label{L2-2-infty}$$ For $2\leq r \leq \infty$, the estimate [\[L2-2-r\]](#L2-2-r){reference-type="eqref" reference="L2-2-r"} follows by interpolating [\[L2-2-2\]](#L2-2-2){reference-type="eqref" reference="L2-2-2"} and [\[L2-2-infty\]](#L2-2-infty){reference-type="eqref" reference="L2-2-infty"}. ◻ We now state certain $L^p-$improving estimates for the single scale versions of the local spherical maximal operator $A_i^{\infty,j}$. For $d=2$, the estimates were obtained by Lee [@Lee1] by relying on local smoothing estimates and the case $d\geq3$ follows from the well-known Strichartz estimates. We refer to [@Lee1] for details. **Lemma 16** ([@Lee1]). *Let $1\leq r\leq \infty$. We have the following,* 1. *For $d\geq 3$, the following is true, $$\begin{aligned} \|A^{r,j}_{1}\|_{L^\frac{2r}{r+1}({\mathbb {R}}^d)\to L^{\frac{2r'(d+1)}{d-1}}({\mathbb {R}}^d)}&\lesssim 2^{-j\left(\frac{d^2-2d-1}{2r'(d+1)}\right)}\label{Lp03-q03-r}. \end{aligned}$$* 2. *For $\frac{1}{p}+\frac{3}{q}=1$ and $q>\frac{14}{3}$, we have $$\begin{aligned} \|A^{r,j}_{1}\|_{L^\frac{pr}{p+r-1}({\mathbb {R}}^2)\to L^{r'q}({\mathbb {R}}^2)}&\lesssim 2^{\frac{j}{r'}\left(1-\frac{5}{q}\right)}\label{Lp02-q02-r}. \end{aligned}$$* *Proof.* The estimate [\[Lp03-q03-r\]](#Lp03-q03-r){reference-type="eqref" reference="Lp03-q03-r"} follows by interpolating [\[L1-infty-1\]](#L1-infty-1){reference-type="eqref" reference="L1-infty-1"} and the Strichartz estimate [@Lee1] below, $$\|A^{\infty,j}_{1}\|_{L^2({\mathbb {R}}^d)\to L^{\frac{2(d+1)}{d-1}}({\mathbb {R}}^d)}\lesssim 2^{-j\left(\frac{d^2-2d-1}{2(d+1)}\right)}\label{L2-p0-infty}.$$ The estimate [\[Lp02-q02-r\]](#Lp02-q02-r){reference-type="eqref" reference="Lp02-q02-r"} follows by using the bound [\[L1-infty-1\]](#L1-infty-1){reference-type="eqref" reference="L1-infty-1"} and the bound below, which was obtained in [@Lee1] by local smoothing estimates, $$\|A^{\infty,j}_{1}\|_{L^p({\mathbb {R}}^2)\to L^{q}({\mathbb {R}}^2)}\lesssim 2^{j\left(1-\frac{5}{q}\right)}\label{Lp-q-infty}.$$ ◻ We now provide $L^p-$ bounds for the maximal operators $A^{r,j}_{*}$. The first is a direct consequence of estimates for the case $r=1,\infty$. **Lemma 17**. *Let $d\geq2$ and $j\geq 0$. Then for $f\in L^1_{loc}({\mathbb {R}}^d)$, we have $$A^{r,j}_{*}f(x)\lesssim 2^{\frac{j}{r'}}M_{HL}f(x),\quad a.e.\;x\in{\mathbb {R}}^d.$$* Now, using the estimates for single scale operators $A_1^{r,j}$, we will prove some $L^p-$estimates for the maximal operators $A^{r,j}_{*}$ with norm depending on $j$. To do that, we rely on a interpolation scheme based on a vector-valued argument. We state Lemma [Lemma 18](#vector){reference-type="ref" reference="vector"} and the proof can be obtained by arguments similar to Lemma 5.4 of [@BCSSNikodym]. **Lemma 18**. *Let $1\leq p_1,p_2\leq2$ be such that $$\begin{aligned} \|A^{r,j}_{1}\|_{L^{p_1}({\mathbb {R}}^d)\to L^{p_1,\infty}({\mathbb {R}}^d)}&\leq C_1,\\ \|A^{r,j}_{*}\|_{L^{p_2}({\mathbb {R}}^d)\to L^{p_2,\infty}({\mathbb {R}}^d)}&\leq C_2. \end{aligned}$$ Then, we have $$\|A^{r,j}_{*}\|_{L^{p}({\mathbb {R}}^d)\to L^{p}({\mathbb {R}}^d)}\lesssim C_1^\frac{p_1}{2} C_2^{1-\frac{p_1}{2}},\;\;\text{for}\;p=\frac{2p_2}{2+p_2-p_1}.$$* **Lemma 19**. *Let $d\geq2$. The following holds true.* - *Let $1\leq r\leq2$. Then $$\label{LpMj} \|A^{r,j}_{*}f(x)\|_{L^\frac{2}{3-r}({\mathbb {R}}^d)}\lesssim 2^{-j\left(\frac{d(r-1)}{2}-\frac{1}{r'}\right)}\|f\|_{L^{\frac{2}{3-r}}({\mathbb {R}}^d)}.$$* - *Let $2\leq r\leq\infty$. Then $$\label{L2Mj} \|A^{r,j}_{*}f(x)\|_{L^2({\mathbb {R}}^d)}\lesssim 2^{-j\left(\frac{d-2}{2}+\frac{1}{r}\right)}\|f\|_{L^{2}({\mathbb {R}}^d)}.$$* *Proof.* When $r\geq2$, the bound [\[L2Mj\]](#L2Mj){reference-type="eqref" reference="L2Mj"} follows from Littlewood-Paley theory and the estimate [\[L2-2-r\]](#L2-2-r){reference-type="eqref" reference="L2-2-r"}. For $1\leq r\leq2$, an application of Lemma [Lemma 18](#vector){reference-type="ref" reference="vector"} along with the estimates $\|A^{r,j}_{*}\|_{L^{1}\to L^{1,\infty}}\lesssim 2^\frac{j}{r'}$ (Lemma [Lemma 17](#L1Mj){reference-type="ref" reference="L1Mj"}) and [\[Lr-r-r\]](#Lr-r-r){reference-type="eqref" reference="Lr-r-r"} implies the bound [\[LpMj\]](#LpMj){reference-type="eqref" reference="LpMj"}. ◻ We will require the interpolation trick of Bourgain that provides a restricted weak type estimate from two strong type bounds with appropriate growth and decay, see [@Lee1] for details. **Lemma 20**. *[@Lee1][\[interpolation\]]{#interpolation label="interpolation"} Let $\epsilon_1,\epsilon_2>0$. Suppose that $\{T_j\}$ is a sequence of $n-$linear (or sublinear) operators such that for some $1\leq p^i_1,p^i_2<\infty$, $i=1,2,\dots,n$ and $1\leq q_1,q_2<\infty$, $$\begin{aligned} \Vert T_{j}(f^1,f^2,\dots,f^n)\Vert_{L^{q_1}({\mathbb {R}}^d)}&\leq M_12^{\epsilon_1 j}\prod^{n}_{i=1}\Vert f^i\Vert_{L^{p^{i}_1}({\mathbb {R}}^d)},\label{LeeLemma1}\\ \Vert T_{j}(f^1,f^2,\dots,f^n)\Vert_{L^{q_2}({\mathbb {R}}^d)}&\leq M_22^{-\epsilon_2 j}\prod^{n}_{i=1}\Vert f^i\Vert_{L^{p^{i}_2}({\mathbb {R}}^d)}.\label{LeeLemma2} \end{aligned}$$ Then $T=\sum_jT_j$ is bounded from $L^{p^1,1}({\mathbb {R}}^d)\times L^{p^2,1}({\mathbb {R}}^d)\times\cdots\times L^{p^{n},1}({\mathbb {R}}^d)$ to $L^{q,\infty}({\mathbb {R}}^d)$, i.e. $$\Vert T(f^1,f^2,\cdots,f^n)\Vert_{L^{q,\infty}({\mathbb {R}}^d)}\lesssim M^{\theta}_{1}M^{1-\theta}_{2}\prod^{n}_{i=1}\Vert f^i\Vert_{L^{p^i,1}({\mathbb {R}}^d)},\label{LeeLemma3}$$ where $\theta=\epsilon_2/(\epsilon_1+\epsilon_2)$, $1/q=\theta/q_1+(1-\theta)/q_2$, $1/r=\theta/r_1+(1-\theta)/r_2$ and $1/p^i=\theta/p^{i}_1+(1-\theta)/p^{i}_{2}$.* **Remark 21**. *We note that in the proof of Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"}, we use the above lemma for the case when $q_1=\infty$. This can be justified as follows. We obtain an intermediate strong type estimate with growth in $j$ using real interpolation with the estimates [\[LeeLemma1\]](#LeeLemma1){reference-type="eqref" reference="LeeLemma1"} and [\[LeeLemma2\]](#LeeLemma2){reference-type="eqref" reference="LeeLemma2"}, and apply Lemma [\[interpolation\]](#interpolation){reference-type="ref" reference="interpolation"} for the intermediate estimate and the bound [\[LeeLemma2\]](#LeeLemma2){reference-type="eqref" reference="LeeLemma2"}. This process results in the same restricted weak type estimate as [\[LeeLemma3\]](#LeeLemma3){reference-type="eqref" reference="LeeLemma3"}.* We now complete the proof of Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"}. **Proof of Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"} [\[stronglinearA\]](#stronglinearA){reference-type="eqref" reference="stronglinearA"}:** It is clear that $\|\mathfrak{A}^r\|_{L^\infty({\mathbb {R}}^d)\to L^\infty({\mathbb {R}}^d)}\lesssim1$ and $\|\mathfrak{A}^r\|_{L^r({\mathbb {R}}^d)\to L^\infty({\mathbb {R}}^d)}\lesssim1$. Hence, by real interpolation, it is enough to prove the restricted weak type estimate for $\mathfrak{A}^r$ at the points $P=\left(\frac{r(d^2+1)}{d+1+r(d^2-d)},\frac{r'(d^2+1)}{d-1}\right),\;Q=\left(\frac{rd-r+1}{rd},\frac{1}{r'd}\right),$ and $R=\left(\frac{rd-r+1}{rd},\frac{rd-r+1}{rd}\right)$ (see Figure [\[Fig:linearAr\]](#Fig:linearAr){reference-type="ref" reference="Fig:linearAr"}). To achieve that, we will use the Lemma [\[interpolation\]](#interpolation){reference-type="ref" reference="interpolation"} along with the estimates, $$\|A_1^{r,j}\|_{L^{p_1}({\mathbb {R}}^d)\to L^{q_1}}({\mathbb {R}}^d)\lesssim 2^{\epsilon_1 j},\;\;\; \|A_1^{r,j}\|_{L^{p_2}({\mathbb {R}}^d)\to L^{q_2}({\mathbb {R}}^d)}\lesssim 2^{-\epsilon_2 j},$$ with the necessary variables prescribed as in Table [\[interpolationexponents\]](#interpolationexponents){reference-type="ref" reference="interpolationexponents"}. $$\begin{aligned} \renewcommand{\arraystretch}{2} \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline d & \text{Range of }r & \left(\frac{1}{p},\frac{1}{q}\right) &{p_1}& {q_1} & \epsilon_1 & {p_2}& {q_2} & \epsilon_2\\ \hline \hline d=2 & [1,\infty] & P & 1 & \infty & \frac{1}{r'} \;\eqref{L1-infty-r}& \frac{19r}{12+7r} & \frac{19r'}{4} & \frac{1}{19r'}\;\;\;\;\;\;\;\eqref{Lp02-q02-r} \\ \hline d\geq 3 & [1,\infty] & P & 1 & \infty & \frac{1}{r'}\;\eqref{L1-infty-r} & \frac{2r}{r+1} & \frac{2r'(d+1)}{d-1} & \frac{d^2-2d-1}{2r'(d+1)}\;\;\eqref{Lp03-q03-r} \\ \hline d\geq 2 & [1,2] & Q & 1 & \infty & \frac{1}{r'}\;\eqref{L1-infty-r} & r & r' & \frac{d-1}{r'}\;\;\;\;\;\;\;\eqref{Lr-r'-r}\\ \hline d\geq 2 & [2,\infty] & Q & 1 & \infty & \frac{1}{r'}\;\eqref{L1-infty-r} & 2 & 2 & \frac{d-2}{2}+\frac{1}{r}\;\eqref{L2-2-r}\\ \hline d\geq 2 & [1,2] & R & 1 & 1 & \frac{1}{r'}\;\eqref{L1-1-r} & r & r & \frac{d-1}{r'}\;\;\;\;\;\;\;\eqref{Lr-r-r}\\ \hline d\geq 2 & [2,\infty] & R & 1 & 1 & \frac{1}{r'}\;\eqref{L1-1-r} & 2 & 2 & \frac{d-2}{2}+\frac{1}{r}\;\eqref{L2-2-r} \\ \hline \end{array} \end{aligned}$$ **Proof of Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"} [\[restrictedlinearA\]](#restrictedlinearA){reference-type="eqref" reference="restrictedlinearA"}:** It is enough to prove the restricted weak type inequality for $\mathfrak{A}_*^r$ at the point $\left(\frac{rd}{rd-r+1},\frac{rd}{rd-r+1}\right)$; For $p>\frac{rd}{rd-r+1}$, the $L^p$ boundedness for $\mathfrak A_*^r$ follows by interpolation. For $r\geq 2$, we rely on estimates [\[L2Mj\]](#L2Mj){reference-type="eqref" reference="L2Mj"} and $\mathfrak A_*^r:L^1({\mathbb {R}}^d)\to L^{1,\infty}({\mathbb {R}}^d)$ (Lemma [Lemma 17](#L1Mj){reference-type="ref" reference="L1Mj"}). However, the interpolation Lemma [\[interpolation\]](#interpolation){reference-type="ref" reference="interpolation"} is not applicable in this case. We can get around this by interpolating the weak $(1,1)$ estimate with the strong $(2,2)$ estimate to control the quantity $\|\mathfrak A_*^r\|_{L^{p_0}({\mathbb {R}}^d)\to L^{p_0}({\mathbb {R}}^d)}$ for some $1<p_0<\frac{rd}{rd-r+1}$. Now we can obtain the desired restricted weak type by using Lemma [\[interpolation\]](#interpolation){reference-type="ref" reference="interpolation"} for the strong $(p_0,p_0)$ and $(2,2)$ bounds. The case $1\leq r\leq 2$ can be resolved similarly by using the estimate [\[LpMj\]](#LpMj){reference-type="eqref" reference="LpMj"} instead of [\[L2Mj\]](#L2Mj){reference-type="eqref" reference="L2Mj"}. We leave the details to the reader.\ **Proof of Theorem [Theorem 7](#linearAr){reference-type="ref" reference="linearAr"} [\[necessarylinearAr\]](#necessarylinearAr){reference-type="eqref" reference="necessarylinearAr"}:** Let $\delta>0$ be a small number and $c>1$ be a fixed constant. We define $B(0,\delta)$ to be ball with center $0\in{\mathbb {R}}^d$ and radius $\delta$, and $S^{\delta}(0)$ to be the $\delta-$neighborhood of $S^{d-1}$, i.e. $\{x\in{\mathbb {R}}^d:||x|-1|<\delta\}$. We also define $R_1$ to be the rectangle of dimension $[-c\sqrt{\delta},c\sqrt{\delta}]^{d-1}\times[-c\delta,c\delta]$, centered at origin with smaller side along the direction of $e_d=(0,0,\dots,0,1)$ and $R_2$ to be the rectangle of dimension $[-\sqrt{\delta},\sqrt{\delta}]^{d-1}\times[1,2]$ centered at $(0,0,\dots,0,\frac{3}{2})$, with longer side along the direction of $e_d$. We define function $f$ with $\|f\|_{L^p}\simeq\delta^\alpha$ and a test set $E$ with $|E|\simeq\delta^\beta$ satisfying $$\|A_tf\|_{L^r([1,2])}\gtrsim \delta^\gamma, \;\;\;\text{for all}\;x\in E.$$ Since $\delta>0$ is a small number, the required necessary condition holds if $$\alpha \leq \frac{\beta}{q}+\gamma.$$ The functions and test sets are given in the Figure [\[examples-A\]](#examples-A){reference-type="ref" reference="examples-A"}.0◻ $$\begin{aligned} \renewcommand{\arraystretch}{2} \begin{array}{|l|c|c|c|c|c|} \hline \text{f (function)} & E \text{(test set)} & \alpha & \beta & \gamma & \text{Necessary Condition}\\ \hline \hline \chi_{S^{c\delta}(0)} & B(0,\delta) & \delta^{\frac{1}{p}} & \delta^{d} & \delta^{\frac{1}{r}} & \frac{1}{p}\leq\frac{d}{q}+\frac{1}{r}\\ \hline \chi_{B(0,c\delta)}& B(0,2)\setminus B(0,1) & \delta^{\frac{d}{p}} & 1 & \delta^{d-1+\frac{1}{r}} & \frac{d}{p}\leq d-1+\frac{1}{r}\\ \hline \chi_{R_1} & R_2 & \delta^{\frac{d+1}{2p}} & \delta^{\frac{d-1}{2}} & \delta^{\frac{d-1}{2}+\frac{1}{r}} & \frac{d+1}{2p}\leq\frac{d-1}{2q}+\frac{1}{r}+\frac{d-1}{2}\\ \hline \chi_{B(0,\frac{1}{\delta)}}& B(0,\frac{1}{\delta}) & \delta^{\frac{-d}{p}} & \delta^{-d} & 1 & \frac{1}{q}\leq \frac{1}{p}\\ \hline \end{array} \end{aligned}$$ As mentioned in Remark [Remark 8](#Rem:strongtype){reference-type="ref" reference="Rem:strongtype"}, we provide an example showing that $\mathfrak{A}^r$ does not map $L^p({\mathbb {R}}^d)$ to $L^q({\mathbb {R}}^d)$ when $\left(\frac{1}{p},\frac{1}{q}\right)$ lies on the line segment $QR$. **Proposition 22**. *The operator $\mathfrak{A}^{r}$ does not map $L^{\frac{dr}{dr-r+1},s}({\mathbb {R}}^d)$ to $L^{q}({\mathbb {R}}^d)$, for any $s>1$ and $q\geq1$.* *Proof.* Let $p_0=\frac{dr}{dr-r+1}$. For a small number $a>0$, we define $$f(x)=\sum_{i=1}^{N}4^{\frac{d}{p_0}i}\chi_{B(0,a4^{-i})}(x).$$ For $s>1$, we have the following bound on Lorentz space norm of $f$, $$\|f\|_{L^{p_0,s}}\lesssim N^{\frac{1}{s}}.$$ To see this, consider the sets $F_j=\bigg(4^{\frac{d}{p_0}j}\sum\limits_{k=0}^{j-1}4^{-\frac{d}{p_0}k},4^{\frac{d}{p_0}(j+1)}\sum\limits_{k=0}^{j}4^{-\frac{d}{p_0}k}\bigg]$ for $1\leq j\leq N-1$. If $t\in F_j$ , we have $$\{x\in{\mathbb {R}}^d:\;|f(x)|>t\}=B(0,a4^{-(j+1)}).$$ Denote $d_f(t)=|\{x\in{\mathbb {R}}^d:\;|f(x)|>t\}|$ and observe that, $$td_f(t)^\frac{1}{p_0}=t(a4^{-(j+1)d})^\frac{1}{p_0}\lesssim1.$$ Therefore, we have $$\begin{aligned} \|f\|_{L^{p_0,s}}&= p_0^\frac{1}{s}\left(\int_0^{4^{\frac{d}{p_0}}}[d_f(t)t]^s\frac{dt}{t}+\sum_{j=0}^{N-1}\int_{F_j}[d_f(t)t]^s\frac{dt}{t}\right)^\frac{1}{s}\\ &\lesssim \left(\int_0^{4^{\frac{d}{p_0}}}4^{-\frac{d}{p_0}}t^{s-1}\;dt+\sum_{j=0}^{N-1}\int_{F_j}\frac{dt}{t}\right)^\frac{1}{s}\\ &\leq \left(4^{\frac{d}{p_0}(s-1)}+\sum_{j=0}^{N-1}1\right)^\frac{1}{s}\lesssim N^\frac{1}{s}. \end{aligned}$$ Now for $x\in \{y:\;1\leq|y|\leq2\}$, we have $$\left(\int_1^2|A_tf(x)|^r\right)^\frac{1}{r}\gtrsim\sum_{i=1}^{N}4^{\frac{d}{p_0}i}4^{-\left(d-1+\frac{1}{r}\right)}=N.$$ If $\mathfrak{A}^{r}$ maps $L^{\frac{dr}{dr-r+1},s}({\mathbb {R}}^d)$ to $L^{q}({\mathbb {R}}^d)$, then $$\begin{aligned} N\lesssim\|\mathfrak{A}^{r}f\|_{L^{q}}\lesssim\|\mathfrak{A}^{r}f\|_{L^{p_0,s}\to L^{q}}\|f\|_{p_0,s}\lesssim \|\mathfrak{A}^{r}f\|_{L^{p_0,s}\to L^{q}}N^{\frac{1}{s}}. \end{aligned}$$ which is a contradiction for $s>1$. ◻ # Proof of Theorems [Theorem 12](#Mfull){reference-type="ref" reference="Mfull"} and [Theorem 13](#linearized){reference-type="ref" reference="linearized"} {#Mfullsec} ## Proof of Theorem [Theorem 12](#Mfull){reference-type="ref" reference="Mfull"}: {#proof-of-theorem-mfull} We will in fact provide a counterexample for the local maximal operator defined by $${\mathcal M}_{loc}^\theta (f,g)(x)=\sup_{t\in[1,2]} |\mathcal A_t^\theta(f,g)(x)|,$$ The proof is based on the Kakeya construction used in [@EndpointMappingPropertiesOfSphericalMaximalOperators] to show that the linear spherical maximal function does not map $L^{2,1}({\mathbb {R}}^2)\to L^{2,\infty}({\mathbb {R}}^2)$. Let $R_l$ be the collection of $\delta^{-1}$ overlapping rectangles of dimension $\delta\times \delta^2$ lying inside the cube $[-\delta,\delta]^2$ such that the longer side of $R_l$ is parallel to the vector $e^{i\delta l}$. We have $|\cup_l R_l|\sim \frac{\delta^{2}}{\log\frac{1}{\delta}}$. Also let $[1,2]=\cup_\nu I_\nu$, where $I_\nu$'s are $\delta^{-2}$ disjoint intervals of equal length. We denote $R_{l,\nu}$ to be the rectangle obtained by translating the rectangle $R_l$ by length $I_\nu$ along its shorter side. Also let $\widetilde R_{l,\nu}$ be the rectangle obtained by translating the rectangle $R_l$ by length $2I_\nu$ along its shorter side, followed by a counterclockwise rotation by the angle $\theta$. We define $$f(x)=\chi_{\bigcup\limits_l R_l}(x),\;\;\;\;g(x)=\chi_{\bigcup\limits_{l,\nu}\widetilde R_{l,\nu}}(x).$$ We have $\|f\|_{p_1}\sim(\frac{\delta^{2}}{\log\frac{1}{\delta}})^{\frac{1}{p_1}}$ and $\|g\|\sim1$. For $x\in\bigcup\limits_{l,\nu}R_{l,\nu}$, it follows that $\mathcal{M}_{loc}^\theta(f,g)(x)> c\delta$ for some absolute constant $c>0$. Thus, $$|\{{\mathcal{M}}_{loc}^\theta(f,g)(x)>c\delta\}|\gtrsim 1\gtrsim \frac{\delta^{-p(\frac{2}{p_1}-1)}\log\frac{1}{\delta}^{\frac{p}{p_1}}}{\delta^{p}}\|f\|_{p_1}^p\|g\|_{p_2}^p.$$ Therefore, we get that $$\|{\mathcal{M}}_{loc}^\theta\|_{L^{p_1,1}\times L^{p_2,1}\to L^{p,\infty}}=\infty~\text{for}~ p_1\leq2.$$ By symmetry, the same holds for $p_2\leq 2$. 0◻ ## Proof of Theorem [Theorem 13](#linearized){reference-type="ref" reference="linearized"}: {#proof-of-theorem-linearized} We will prove the theorem for the case $\theta=\pi$, the proof of other cases is similar. $$\begin{aligned} &\langle \widetilde{\mathcal A}^\pi(f,g),h\rangle\\ =&\int\int f(x+|x|y)g(x-|x|y)h(x)\;d\sigma(y)dx\\ =&\int_{r=0}^\infty\int_{\theta=0}^{2\pi}\int_{t=0}^{2\pi} f(r(e^{i\theta}+e^{it}))g(r(e^{i\theta}-e^{it}))h(re^{i\theta})\;dtrdrd\theta\\ =&\int_{r=0}^\infty\int_{\theta=0}^{2\pi}\int_{t=\theta}^{2\pi+\theta} f\left(2r\cos\left(\frac{\theta-t}{2}\right)e^{i\frac{\theta+t}{2}}\right)g\left(2r\sin\left(\frac{\theta-t}{2}\right)e^{i(\frac{\theta+t}{2}+\frac{\pi}{2})}\right)h(re^{i\theta})\;dtrdrd\theta\\ \end{aligned}$$ By the change of variable $u=\cos\left(\frac{\theta-t}{2}\right)$, the above term is equal to $$\begin{aligned} =&\int_{r=0}^\infty\int_{\theta=0}^{2\pi}\int_{u=-1}^1f(2rue^{i(\theta-\cos^{-1} u)})g(2r\sqrt{1-u^2}e^{i(\theta-\cos^{-1} u+\frac{\pi}{2})})h(re^{i\theta})\;2\frac{du}{\sqrt{1-u^2}}rdrd\theta\\ =&\int_{u=-1}^1\int_{{\mathbb {R}}^2}f(2ux)g(2\sqrt{1-u^2}\mathfrak{R}_{-\frac{\pi}{2}}x)h(\mathfrak{R}_{\cos^{-1} u}x)\;dx\frac{2du}{\sqrt{1-u^2}}, \end{aligned}$$ where $\mathfrak{R}_\phi x$ is the point obtained by rotating $x$ by angle $\phi$ counterclockwise. Applying Hölder's inequality and scaling, we get the above quantity is dominated by $$\begin{aligned} \lesssim& \int_{u=-1}^1\|f(2u\cdot)\|_{p_1}\|g(2\sqrt{1-u^2}\cdot)\|_{p_2}\|h\|_{p'}\frac{du}{\sqrt{1-u^2}}\\ \lesssim&\|f\|_{p_1}\|g\|_{p_2}\|h\|_{p'}\int_{u=0}^1u^{-\frac{2}{p_1}}(1-u^2)^{-\frac{1}{p_2}-\frac{1}{2}}\;du\\ \lesssim&\|f\|_{p_1}\|g\|_{p_2}\|h\|_{p'}\int_{t=0}^1t^{-\frac{1}{p_1}-\frac{1}{2}}(1-t)^{-\frac{1}{p_2}-\frac{1}{2}}\;dt, \end{aligned}$$ where the beta integral in the above quantity is finite for $p_1,p_2>2$ and the proof concludes. We now prove the restricted weak type estimates for $\widetilde{\mathcal A}^\pi$. We observe that $\widetilde{\mathcal A}^\pi:L^\infty({\mathbb {R}}^2)\times L^{2,1}({\mathbb {R}}^2)\to L^{2,\infty}({\mathbb {R}}^2)$ and $\widetilde{\mathcal A}^\pi:L^{2,1}({\mathbb {R}}^2)\times L^\infty({\mathbb {R}}^2)\to L^{2,\infty}({\mathbb {R}}^2)$ follows from the inequality, $$\widetilde{\mathcal A}^\pi(f,g)(x)\leq\min\{\|f\|_\infty \widetilde A(|g|)(x),\|g\|_\infty \widetilde A(|f|)(x)\}.$$ We prove the restricted weak type inequality at the endpoint $(2,2,1)$, the proof of the remaining endpoints are similar. We decompose the operator $\widetilde{\mathcal A}^\pi$ as follows $$\begin{aligned} \widetilde{\mathcal A}^\pi(f,g)(x)&=&\int_{|u|\leq1/2}f(2uxe^{-\iota\cos^{-1}u})g(2x\sqrt{1-u^2}e^{-\iota\cos^{-1}u+\iota\pi/2})~\frac{2du}{\sqrt{1-u^2}}\\ &+&\int_{1/2<|u|\leq1}f(2uxe^{-\iota\cos^{-1}u})g(2x\sqrt{1-u^2}e^{-\iota\cos^{-1}u+\iota\pi/2})~\frac{2du}{\sqrt{1-u^2}}\\ &:=&I_0(f,g)(x)+I_{1}(f,g)(x). \end{aligned}$$ We further decompose each of the two operators into infinitely many pieces as follows $$\begin{aligned} I_0(f,g)(x)&=&\sum_{j\geq1}\int_{2^{-j-1}<|u|\leq2^{-j}}f(2uxe^{-\iota\cos^{-1}u})g(2x\sqrt{1-u^2}e^{-\iota\cos^{-1}u+\iota\pi/2})~\frac{2du}{\sqrt{1-u^2}}\\ &:=&\sum_{j\geq1}I_{0,j}(f,g)(x). \end{aligned}$$ Note that the denominator $\sqrt{1-u^2}$ in the expression above behaves like a constant as $|u|\leq1/2$. We have $$\begin{aligned} \Vert I_{0,j}(f,g)\Vert_{L^{1}}&\lesssim& \int_{2^{-j-1}<|u|\leq2^{-j}}\Vert f(2u\cdot)\Vert_{L^{4}}\Vert g(2\sqrt{1-u^2}\cdot)\Vert_{L^{4/3}} ~du \\ &\lesssim&2^{-j/2}\Vert f\Vert_{L^{4}}\Vert g\Vert_{L^{4/3}}. \end{aligned}$$ On the other hand, $$\begin{aligned} \Vert I_{0,j}(f,g)\Vert_{L^{1}}&\lesssim& \int_{2^{-j-1}<|u|\leq2^{-j}}\Vert f(2u\cdot)\Vert_{L^{4/3}}\Vert g(2\sqrt{1-u^2}\cdot)\Vert_{L^{4}} ~du \\ &\lesssim&2^{j/2}\Vert f\Vert_{L^{4/3}}\Vert g\Vert_{L^{4}}. \end{aligned}$$ Applying Bourgain's interpolation trick (Lemma [\[interpolation\]](#interpolation){reference-type="ref" reference="interpolation"}) we get that the operator $I_0$ maps $L^{2,1}\times L^{2,1}$ to $L^{1,\infty}$. Next, consider a similar decomposition of $I_1(f,g)$ as follows $$\begin{aligned} I_1(f,g)(x)&=&\sum_{j\geq1}\int_{1-2^{-j}<|u|\leq 1-2^{-j-1}}f(2uxe^{-\iota\cos^{-1}u})g(2x\sqrt{1-u^2}e^{-\iota\cos^{-1}u+\iota\pi/2})~\frac{2du}{\sqrt{1-u^2}}\\ &:=&\sum_{j\geq1}I_{1,j}(f,g)(x). \end{aligned}$$ Now computing the $L^{1}-$norm we get $$\begin{aligned} \Vert I_{1,j}(f,g)\Vert_{L^{1}}&\lesssim& \int_{1-2^{-j}<|u|\leq1-2^{-j-1}}\Vert f(2u\cdot)\Vert_{L^{4}}\Vert g(2\sqrt{1-u^2}\cdot)\Vert_{L^{4/3}} ~\frac{du}{\sqrt{1-u^2}} \\ &\lesssim&2^{j/4}\Vert f\Vert_{L^{4}}\Vert g\Vert_{L^{4/3}}. \end{aligned}$$ And $$\begin{aligned} \Vert I_{1,j}(f,g)\Vert_{L^{1}}&\lesssim& \int_{1-2^{-j}<|u|\leq1-2^{-j-1}}\Vert f(2u\cdot)\Vert_{L^{4/3}}\Vert g(2\sqrt{1-u^2}\cdot)\Vert_{L^{4}} ~\frac{du}{\sqrt{1-u^2}} \\ &\lesssim&2^{-j/4}\Vert f\Vert_{L^{4/3}}\Vert g\Vert_{L^{4}}. \end{aligned}$$ Applying the interpolation lemma we get that the operator $I_1$ maps $L^{2,1}\times L^{2,1}$ to $L^{1,\infty}$. Finally, combining the estimates of both $I_0$ and $I_1,$ we get the desired result. 0◻ *Proof of Proposition [\[A1estimate\]](#A1estimate){reference-type="ref" reference="A1estimate"}.* We first claim that $\mathcal{A}_{1}$ satisfies $L^{\frac{d+1}{d}}\times L^{\frac{d+1}{d}}\to L^{1}$ boundedness. $$\begin{aligned} \|\mathcal A (f,g)\|_{1}&=&\int_{{\mathbb {R}}^d}|\int_{\mathbb S^{d-1}}f(x-y)g(x+y)\;d\sigma(y)|\;dx\\ &\leq& \int_{\mathbb S^{d-1}}\int_{{\mathbb {R}}^d}|f(x-y)||g(x+y)|\;dx\;d\sigma(y)\\ &=& \int_{{\mathbb {R}}^d}|f(x)|\int_{\mathbb S^{d-1}}|g(x+2y)|\;d\sigma(y)\;dx\\ &\leq& \|f\|_{\frac{d+1}{d}}\|\int_{\mathbb S^{d-1}}|g(x+2y)|\;d\sigma(y)\|_{d+1}\leq \|f\|_\frac{d+1}{d}\|g\|_{\frac{d+1}{d}}. \end{aligned}$$ Secondly, we claim that $\mathcal{A}_{1}$ satisfies $L^{\frac{d+1}{d}}\times L^{\infty}\to L^{d+1}$ and $L^{\infty}\times L^{\frac{d+1}{d}}\to L^{d+1}$ estimates. $$\begin{aligned} \|\mathcal A (f,g)\|_{d+1}&=&(\int_{{\mathbb {R}}^d}|\int_{\mathbb S^{d-1}}f(x-y)g(x+y)\;d\sigma(y)|^{d+1}\;dx)^{\frac{1}{d+1}}\\ &\leq& \|g\|_{\infty}\|\int_{\mathbb S^{d-1}}|f(x-y)|\;d\sigma(y)\|_{d+1}\leq \|f\|_\frac{d+1}{d}\|g\|_{\infty}. \end{aligned}$$ Finally, we claim that $\mathcal{A}_{1}$ satisfies $L^{1}\times L^{\infty}\to L^{1}$ and $L^{\infty}\times L^{1}\to L^{1}$ estimates. $$\begin{aligned} \|\mathcal A (f,g)\|_{1}&=&\int_{{\mathbb {R}}^d}|\int_{\mathbb S^{d-1}}f(x-y)g(x+y)\;d\sigma(y)|\;dx)\\ &\leq& \|g\|_{\infty}\|\int_{\mathbb S^{d-1}}|f(x-y)|\;d\sigma(y)\|_{1}\leq \|f\|_{1}\|g\|_{\infty}. \end{aligned}$$ Now, applying real interpolation we get the desired boundedness. ◻ # Necessary conditions for $\mathcal{M}$ in dimensions $d\geq2$ {#Sec:Necessary} In this section we provide some necessary conditions for the higher dimensional analogue $\mathcal{T}$ of the operator $\mathcal{A}_1^\pi$ defined as, $$\mathcal{T}(f,g)(x)=\int_{\mathbb S^{d-1}}f(x-y)g(x+y)\;d\sigma(y).$$ The first result concerns an generalization of the necessary condition [\[necessarycondition\]](#necessarycondition){reference-type="ref" reference="necessarycondition"} for $\mathcal{T}$. We note that the condition is obtained by considering functions of product type instead of examples generated by C. Fefferman boxes [@Fefferman], as was the case in [@GIKL]. **Proposition 23**. *Let $d\geq 2$ and $1\leq p_1,p_2<\infty$. Suppose $\mathcal{T}$ satisfies the following inequality, $$\|\mathcal{T}(f,g)\|_{L^p({\mathbb {R}}^d)}\lesssim\|f\|_{L^{p_1}({\mathbb {R}}^d)}\|g\|_{L^{p_2}({\mathbb {R}}^d)},$$ for functions $f,g$ of the form $f(x)=f_1(x_1)f_2(x_2)$ and $g(x)=g_1(x_1)g_2(x_2)$ where we write $x=(x_1,x_2)$ with $x_1\in{\mathbb {R}}^{d_1}$, $x_2\in{\mathbb {R}}^{d_2}$ and $d=d_1+d_2$. Then we have, $$\frac{d+1}{p_1}+\frac{d+1}{p_2}\leq d-1+\frac{d+1}{p}.$$* *Proof.* Consider the functions $$f(x)={||x_1|-1|^{-\frac{d_1\alpha_1}{p_1}}}{|x_2|^{-\frac{d_2\alpha_2}{p_1}}}\chi_{[0,1]^d}(x)\;\text{ and }\; g(x)={||x_1|-1|^{-\frac{d_1\beta_1}{p_2}}}{|x_2|^{-\frac{d_2\beta_2}{p_2}}}\chi_{[0,1]^d}(x).$$ Then $f\in L^{p_1}({\mathbb {R}}^d)$ if $\alpha_1<\frac{1}{d_1}$ and $\alpha_2<1$. Similarly, $g\in L^{p_2}({\mathbb {R}}^d)$ if $\beta_1<\frac{1}{d_1}$ and $\beta_2<1$. By the slicing argument and decomposing the interval $[0,1]$ into dyadic annulli, we get that $$\begin{aligned} &&\mathcal T (f,g)(x)\\ &=&\int_{B^{d_1}(0,1)}f(x_1-y_1)g(x_1+y_1)(1-|y_1|^2)^{\frac{d_2-2}{2}}\\ &&\hspace{10mm}\int_{\mathbb S^{d_2-1}}f(x_2-\sqrt{1-|y_1|^2}y_2)g(x_2+\sqrt{1-|y_1|^2}y_2)~d\sigma(y_2)dy_1\\ &=&\int_0^1(1-r^2)^{\frac{d_2-2}{2}}r^{d_1-1}\left(\int_{\mathbb S^{d_1-1}}f(x_1-ry_1)g(x_1+ry_1)~d\sigma(y_1)\right)\\ &&\hspace{10mm}\left(\int_{\mathbb S^{d_2-1}}f(x_2-\sqrt{1-r^2}y_2)g(x_2+\sqrt{1-r^2}y_2)~d\sigma(y_2)\right)dr\\ &=&\sum_{j\geq1}\int_{1-2^{-j+1}}^{1-2^{-j}}(1-r^2)^{\frac{d_2-2}{2}}r^{d_1-1}\left(\int_{\mathbb S^{d_1-1}}{||x_1-ry_1|-1|^{-\frac{d_1\alpha_1}{p_1}}}{||x_1+ry_1|-1|^{-\frac{d_1\beta_1}{p_2}}}~d\sigma(y_1)\right)\\ &&\hspace{10mm}\left(\int_{\mathbb S^{d_2-1}}{|x_2-\sqrt{1-r^2}y_2|^{-\frac{d_2\alpha_2}{p_1}}}{|x_2+\sqrt{1-r^2}y_2|^{-\frac{d_2\beta_2}{p_2}}}~d\sigma(y_2)\right)dr. \end{aligned}$$ Let $B_k=\{x=(x_1,x_2):2^{-k}\leq|x_1|\leq 2^{-k+1},2^{-\frac{k}{2}}\leq|x_2|\leq 2^{-\frac{k-1}{2}}$, then for large $k$ and $j>k$, we get that $$||x_1\pm ry_1|-1|\sim 2^{-k} ~~\text{and}~~ |x_2\pm\sqrt{1-r^2}y_2|\sim 2^{-\frac{k}{2}}.$$ We can see that $$\begin{aligned} \mathcal T (f,g)(x)&\geq& \sum_{j\geq k}\int_{1-2^{-j+1}}^{1-2^{-j}}(1-r^2)^{\frac{d_2-2}{2}}r^{d_1-1}\left(\int_{\mathbb S^{d_1-1}}{2^{k\frac{d_1\alpha_1}{p_1}}}{2^{k\frac{d_1\beta_1}{p_2}}}~d\sigma(y_1)\right)\\ &&\hspace{10mm}\left(\int_{\mathbb S^{d_2-1}}{2^{k\frac{d_2\alpha_2}{2p_1}}}{2^{k\frac{d_2\beta_2}{2p_2}}}~d\sigma(y_2)\right)dr\\ &\gtrsim& 2^{k(\frac{d_1\alpha_1}{p_1}+\frac{d_1\beta_1}{p_2}+\frac{d_2\alpha_2}{2p_1}+\frac{d_2\beta_2}{2p_2})}\sum_{j\geq k}2^{-j\frac{d_2}{2}}=2^{k(\frac{d_1\alpha_1}{p_1}+\frac{d_1\beta_1}{p_2}+\frac{d_2\alpha_2}{2p_1}+\frac{d_2\beta_2}{2p_2}-\frac{d_2}{2})}. \end{aligned}$$ Hence, we get that $$\begin{aligned} \|\mathcal T (f,g)\|_p^p&\geq& \sum_k\int_{B_k}|\mathcal T (f,g)(x)|^p~dx\\ &\gtrsim& \sum_k\int_{B_k}2^{kp(\frac{d_1\alpha_1}{p_1}+\frac{d_1\beta_1}{p_2}+\frac{d_2\alpha_2}{2p_1}+\frac{d_2\beta_2}{2p_2}-\frac{d_2}{2})}~dx\\ &=&\sum_k 2^{kp(\frac{d_1\alpha_1}{p_1}+\frac{d_1\beta_1}{p_2}+\frac{d_2\alpha_2}{2p_1}+\frac{d_2\beta_2}{2p_2}-\frac{d_2}{2})} 2^{-k(d_1+\frac{d_2}{2})}. \end{aligned}$$ The above sum is finite if $\frac{d_2+2}{2p_1}+\frac{d_2+2}{2p_2}\leq \frac{d_2}{2}+\frac{2d_1+d_2}{2p}$. This condition implies the necessary condition if we choose $d_1=1$ and $d_2=d-1$. ◻ We also record some $L^p-$ improving conditions for $\mathcal{T}$ in the following proposition. These conditions are higher dimensional analogues of results obtained in [@GIKL] by considering indicator functions of appropriate balls and annulus. **Proposition 24**. *Let $d\geq 2$ and $1\leq p_1,p_2<\infty$. Suppose $\mathcal{T}:L^{p_1}({\mathbb {R}}^d)\times L^{p_2}({\mathbb {R}}^d)\to L^p({\mathbb {R}}^d)$ boundedly, Then we have the following,* 1. *$\frac{d}{p_1}+\frac{1}{p_2}\leq 1+\frac{1}{p}$.* 2. *$\frac{1}{p_1}+\frac{d}{p_2}\leq 1+\frac{1}{p}$.* 3. *$\frac{1}{p}\leq \frac{1}{p_1}+\frac{1}{p_2}\leq \frac{d}{p}$.* We refer to Section 3 of [@GIKL] for details. # Acknowledgement {#acknowledgement .unnumbered} The authors thank Michael Lacey and Ben Krause for introducing the bilinear operator $\mathcal{A}_t^\theta$ to them. Ankit Bhojak and Saurabh Shrivastava acknowledge the financial support from Science and Engineering Research Board, Department of Science and Technology, Govt. of India, under the scheme Core Research Grant, file no. CRG/2021/000230. Surjeet Singh Choudhary is supported by CSIR(NET), file no.09/1020(0182)/2019- EMR-I for his Ph.D. fellowship. Kalachand Shuin is supported by NRF grant no. 2022R1A4A1018904 and BK 21 Post doctoral fellowship. The authors acknowledge the support and hospitality provided by the International Centre for Theoretical Sciences, Bangalore (ICTS) for participating in the program - Modern trends in Harmonic Analysis (code: ICTS/Mtha2023/06). [^1]:

arxiv_math

--- abstract: | We prove some weighted refined decoupling estimates. As an application, we give an alternative proof of the following result on Falconer's distance set problem by the authors in a companion work: if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 4$, then there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive Lebesgue measure. Aside from this application, the weighted refined decoupling estimates may be of independent interest. author: - Xiumin Du, Yumeng Ou, Kevin Ren, and Ruixiang Zhang bibliography: - main.bib title: Weighted refined decoupling estimates and application to Falconer distance set problem --- # Introduction In this paper, we prove some weighted refined decoupling estimates (see Theorems [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"} and [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}) and discuss their application to Falconer's distance set problem. ## Weighted refined decoupling estimates Here is the setup for refined decoupling estimates. Suppose that $S \subset \mathbb{R}^d$ is a compact and strictly convex $C^2$ hypersurface with Gaussian curvature $\sim 1$. For any $\epsilon>0$, suppose there exists $0<\beta\ll \epsilon$ satisfying the following. Suppose that the $R^{-1}$-neighborhood of $S$ is partitioned into $R^{-1/2} \times ... \times R^{-1/2} \times R^{-1}$ blocks $\theta$. For each $\theta$, let $\mathbb{T}_\theta$ be a set of finitely overlapping tubes of dimensions $R^{1/2 + \beta} \times \cdots \times R^{1/2+\beta}\times R$ with long axis perpendicular to $\theta$, let $G(\theta)\in \mathbb{S}^{d-1}$ denote this direction, and let $\mathbb{T} = \cup_\theta \mathbb{T}_\theta$. Each $T\in \mathbb T$ belongs to $\mathbb{T}_{\theta}$ for a single $\theta$, and we let $\theta(T)$ denote this $\theta$. We say that $f$ is microlocalized to $(T,\theta(T))$ if $f$ is essentially supported in $2T$ and $\hat{f}$ is essentially supported in $2\theta(T)$. Here is our first main result on weighted refined decoupling estimates. **Theorem 1**. *Suppose that $f=\sum_{T\in \mathbb{W}}f_T$, where $\mathbb W\subset \mathbb T$ and each $f_T$ is microlocalized to $(T,\theta(T))$. Let $Y$ be a union of $R^{1/2}$-cubes in $B^d_R$ each of which intersects at most $M$ tubes $T\in \mathbb W$. Denote $p_d=\frac{2(d+1)}{d-1}$. Then the following refined decoupling inequalities hold.* *(a) Let $p\geq 2$. Then $$\label{eq-RD-lp} \|f\|_{L^p(Y)} \lesssim_\epsilon R^{\gamma_d(p)+\epsilon} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p},$$ where $\gamma_d(p)=0$ when $2\leq p\leq p_d$, and $\gamma_d(p)=\frac{d-1}{4}-\frac{d+1}{2p}$ when $p\geq p_d$.* *(b) Let $p\leq p_d$, $\alpha\leq d$. Let $H:Y\to [0,1]$ be a function satisfying that $\int_Q H(x)\,dx \lesssim R^{\alpha/2}$ for any $R^{1/2}$-cube $Q$ in $Y$. Then $$\label{eq-RD-lp-R-alpha-0} \|f\|_{L^p(Y;Hdx)} \lesssim_\epsilon R^{\frac 12 (\alpha -d)(\frac 1p -\frac{1}{p_d})+\epsilon} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}.$$* The study of inequalities of this type originated from the work [@DGL], where linear and bilinear refined Strichartz estimates are established and applied to resolve Carleson's pointwise convergence problem of Schrödinger solutions in dimension $2+1$. Later, multilinear refined Strichartz estimates are proved in [@DGLZ] and play an important role in the final resolution of the pointwise convergence problem in all dimensions [@du2019sharp]. Refined decoupling inequalities are stronger versions of linear refined Strichartz estimates; they first appeared in [@guth2020falconer] and played a key role in recent study of the Falconer distance set problem [@guth2020falconer; @du2021improved]. See [@guth2020falconer] for a comparison between Bourgain--Demeter's decoupling theorem [@BDdecoupling] and the refined decoupling theorem. In the case that $2\leq p\leq p_d$, Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a) and the refined decoupling theorem in [@guth2020falconer Theorem 4.2] are equivalent. The main novelty is Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(b), which says that if we take the weighted $L^p$-norm $\|f\|_{L^p(Y; Hdx)}$ on the left-hand side of the decoupling inequality, where $H$ is a weight that satisfies the ball condition with exponent $\alpha$ at scale $R^{1/2}$, then we have an extra gain $R^{\frac 12 (\alpha -d)(\frac 1p -\frac{1}{p_d})}$ when $\alpha<d$ and $p<p_d$. We can also extend Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"} to intermediate dimensions. Let $2\leq m\leq d$. Denote $p_m=\frac{2(m+1)}{m-1}$. Let $R^{-1/2}\leq r\leq 1$, and $f=\sum_{T\in \mathbb{W}} f_T$, where each $f_T$ is microlocalized to $(T, \theta(T))$. We say that $f$ has **$(r,m)$-concentrated frequencies** if there is some $m$-dimensional subspace $V$ such that $$\textrm{Angle} (G(\theta(T)), V) \leq r, \quad \forall T\in \mathbb{W}\,.$$ Obviously, from the definition, all $f$ trivially has $(1,m)$-concentrated frequencies and also $(r,d)$-concentrated frequencies for any $r>0$. **Theorem 2**. *Suppose that $f=\sum_{T\in \mathbb{W}}f_T$, where $\mathbb W\subset \mathbb T$ and each $f_T$ is microlocalized to $(T,\theta(T))$. Let $Y$ be a union of $R^{1/2}$-cubes in $B^d_R$ each of which intersects at most $M$ tubes $T\in \mathbb W$. Then the following refined decoupling inequalities hold.* *(a) Let $2\leq m\leq d$, $p\geq 2$. Suppose that $f$ has $(R^{-1/2},m)$-concentrated frequencies. Then $$\label{eq-RD-lp-R} \|f\|_{L^p(Y)} \lesssim_\epsilon R^{\gamma_m(p)+\epsilon} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p},$$ where $\gamma_m(p)=0$ when $2\leq p\leq p_m$, and $\gamma_m(p)=\frac{m-1}{4}-\frac{m+1}{2p}$ when $p\geq p_m$.* *(b) Let $2\leq m\leq d$, $p\leq p_m$, $\alpha\leq d$. Suppose that $f$ has $(R^{-1/2},m)$-concentrated frequencies. Let $H:Y\to [0,1]$ be a function satisfying that $\int_Q H(x)\,dx \lesssim R^{\alpha/2}$ for any $R^{1/2}$-cube $Q$ in $Y$. Then $$\label{eq-RD-lp-R-alpha} \|f\|_{L^p(Y;Hdx)} \lesssim_\epsilon R^{\frac 12 (\alpha -d)(\frac 1p -\frac{1}{p_m})+\epsilon} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}.$$* *(c) Let $2\leq m\leq d$, $p_d\leq p\leq p_m$, $\alpha\leq d$. Suppose that $f$ has $(r,m)$-concentrated frequencies, where $R^{-1/2}\leq r\leq 1$. Let $H:Y\to [0,1]$ be a function satisfying that $\int_{Q'} H(x)\,dx \lesssim (\frac 1r)^\alpha$ for any $\frac 1r$-cube $Q'$ in $Y$. Then $$\label{eq-RD-lp-r-alpha} \|f\|_{L^p(Y;Hdx)} \lesssim_\epsilon R^\epsilon r^{(d-\alpha)(\frac 1p -\frac{1}{p_m})} (r^2 R)^{\frac{d-1}{4}-\frac{d+1}{2p}} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}.$$* Note that Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a)(b) is a special case of Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(a)(b) with $m=d$. Morally speaking, Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(a)(b) says that if $G(\theta)$'s are concentrated around a subspace, then one can apply decoupling in a lower dimensional space. Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(c) is obtained by a two-step decoupling process, combining Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(b) to first decouple frequencies into $r$-caps and Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a) to further decouple $r$-caps into smaller $R^{-1/2}$-caps. **Remark 3**. *(i). In our application to the Falconer distance set problem, the weight function $H$ satisfies the ball condition at all scales $\geq 1$. But in our proof of Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}, we only need the ball condition at a single scale: scale $R^{1/2}$ for part (b) and scale $\frac 1r$ for part (c).* *(ii). Though Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"} is stated for all $\alpha\leq d$, better result exists when $\alpha<d-m$. For $d-m\leq \alpha< d-\frac{m+1}{2}$, there may also be room to further improve Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}. We don't explore these directions in the current paper as they will not help in our application to Falconer's distance problem.* For large $\alpha$, the gain $R^{\frac 12 (\alpha -d)(\frac 1p -\frac{1}{p_m})}$ when $\alpha<d$ and $p<p_m$ in Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(b) is sharp. **Theorem 4**. *Let $2\leq m\leq d$, $d-\frac{m+1}{2}\leq \alpha\leq d$. Then there are $f, Y$ and $H$ satisfying the conditions of Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(b) such that for any $p>0$ there holds $$\label{eq-RD-eg} \|f\|_{L^p(Y;Hdx)} \gtrsim R^{\frac 12 (\alpha -d)(\frac 1p -\frac{1}{p_m})} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}.$$ In fact, the weight function $H$ here satisfies the ball condition at all scales up to $R^{1/2}$: $$\int_{\tilde Q} H(x)dx\lesssim s^\alpha, \quad \forall s\text{-cube } \tilde Q, \,\forall 0<s\leq R^{1/2};$$ moreover, if in addition $\alpha\geq m$, then $H$ satisfies the ball condition at all scales up to $R$.* **Remark 5**. *In our example for Theorem [Theorem 4](#thm: eg){reference-type="ref" reference="thm: eg"}, $(M, |\mathbb W|)$ is a fixed special pair of values. We expect it to be a difficult and interesting question to further enquire how sharp the gain $R^{\frac 12 (\alpha -d)(\frac 1p -\frac{1}{p_m})}$ is for other given choices of $(M, |\mathbb W|)$.* ## Application to Falconer's distance set problem Now let us see a classical question in geometric measure theory introduced by Falconer [@falconer1985hausdorff] in the early 80s. Let $E\subset\mathbb{R}^d$ be a compact set, its *distance set* $\Delta(E)$ is defined by $$\Delta(E):=\{|x-y|:x,y\in E\}\,.$$ **Conjecture 1**. *[\[Falconer\]]{.upright} Let $d\geq 2$ and $E\subset\mathbb{R}^d$ be a compact set. Then $${\dim_H}(E)> \frac d 2 \Rightarrow |\Delta(E)|>0.$$ Here $|\cdot|$ denotes the Lebesgue measure and ${\dim_H}(\cdot)$ is the Hausdorff dimension.* Falconer's conjecture remains open in all dimensions as of today. It has attracted a great amount of attention in the past decades. To name a few landmarks: in 1985, Falconer [@falconer1985hausdorff] showed that $|\Delta(E)|>0$ if ${\dim_H}(E)>\frac{d}{2}+\frac{1}{2}$. Bourgain [@bourgain1994distance] was the first to lower the threshold $\frac{d}{2}+\frac{1}{2}$ in dimensions $d=2, d=3$ and to use the theory of Fourier restriction in the Falconer problem. The thresholds were further improved by Wolff [@wolff1999decay] to $\frac{4}{3}$ in the case $d=2$, and by Erdoğan [@erdogan2005] to $\frac{d}{2}+\frac{1}{3}$ when $d\geq 3$. These records were only very recently rewritten: $$\begin{cases} \frac{5}{4}, &d=2, \quad\qquad\text{(Guth--Iosevich--Ou--Wang \cite{guth2020falconer})}\\ \frac{9}{5}, &d=3, \quad \qquad \text{(Du--Guth--Ou--Wang--Wilson--Zhang \cite{DGOWWZ})}\\ \frac d2+\frac 14+\frac{1}{8d-4}, &d\geq 3, \quad \qquad \text{(Du--Zhang \cite{du2019sharp})}\\ \frac{d}{2}+\frac 14, &d\geq 4 \text{ even}, \quad \text{(Du--Iosevich--Ou--Wang--Zhang \cite{du2021improved})}. \end{cases}$$ In this paper, we prove the following result on Falconer's conjecture using weighted refined decoupling estimates. Similar to [@guth2020falconer; @du2021improved], we in fact prove a slightly stronger version regarding the pinned distance set. **Theorem 6**. *Let $d\geq 3$ and $E\subset \mathbb{R}^d$ be a compact set. Suppose that $${\rm dim}_H(E)> \begin{cases}\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4},& d\geq 4,\\ \frac{3}{2}+\frac{1}{4}+\frac{17-12\sqrt{2}}{4},& d=3.\end{cases}$$Then, there is a point $x\in E$ such that $|\Delta_x(E)|>0$, where $$\Delta_x(E):=\{|x-y|:\, y\in E\}.$$* Theorem [Theorem 6](#thm: Leb){reference-type="ref" reference="thm: Leb"} improves the thresholds in [@DGOWWZ; @du2019sharp; @du2021improved] in all dimensions $d\geq 3$. The work [@DGOWWZ] uses the polynomial partitioning method developed by Guth [@guthPolynomial; @guthPolynomialII] and refined Strichartz estimates from [@DGL; @DGLZ]. The main ingredients in [@du2019sharp] are broad-narrow analysis, multilinear refined Strichartz estimates, Bourgain--Demeter's $l^2$ decoupling theorem, and a delicate induction on scales argument. In the current paper, we adapt the good tube/bad tube and refined decoupling method pioneered by [@guth2020falconer] for dimension $d = 2$ and continued in [@du2021improved] for even dimensions $d$. In both papers, Orponen's radial projection theorem [@orponen2018radial] plays a key role. However, the argument does not perform well for odd dimensions $d$, and the result of [@du2019sharp] provides a better bound for distance sets. The reason is that Orponen's radial projection theorem only works for sets with dimension $> d-1$, where $d$ is the dimension of the ambient space. To overcome this issue, [@du2021improved] projected the set onto a generic $(\frac{d}{2}+1)$-dimensional subspace of $\mathbb{R}^d$ (assuming $d$ is even). While this orthogonal projection trick works well in even dimensions, for odd dimensions we are forced to project to a $(\frac{d+1}{2})$-dimensional subspace instead, which creates some loss. To avoid this loss, a natural approach is to avoid the initial orthogonal projection; but then, we need a radial projection theorem that works for sets of dimension $\le d-1$. One new ingredient in this paper is a radial projection result, Theorem [Theorem 9](#conj:threshold){reference-type="ref" reference="conj:threshold"}, by the third author [@KevinRadialProj]. For each $\delta$-tube $T$, let $r(T)\in [\delta, 1]$ be the thickness of the smallest heavy plate containing $T$ (see Section [4.1](#sec: outline){reference-type="ref" reference="sec: outline"} for the precise definition). We can remove more bad parts (see Section [\[sec:bad\]](#sec:bad){reference-type="ref" reference="sec:bad"}) using Theorem [Theorem 9](#conj:threshold){reference-type="ref" reference="conj:threshold"} and give a new threshold [\[badthreshold\]](#badthreshold){reference-type="eqref" reference="badthreshold"} for bad tubes depending on $r(T)$. To deal with the varying values of $r(T)$, we apply weighted refined decoupling estimates in Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(c). In the case that $r(T)=\delta$, the threshold [\[badthreshold\]](#badthreshold){reference-type="eqref" reference="badthreshold"} is the same as the one obtained from combining Orponen's radial projection theorem and orthogonal projections; however, Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"} gives an extra gain when $r(T)$ is small. As $r(T)$ increases, the threshold [\[badthreshold\]](#badthreshold){reference-type="eqref" reference="badthreshold"} gets much better. **Remark 7**. *In a companion work [@DOKZFalconer], we provide an alternative proof of Theorem [Theorem 6](#thm: Leb){reference-type="ref" reference="thm: Leb"} (in fact, in [@DOKZFalconer] we can establish the dimensional threshold $\frac d2 +\frac 14 -\frac{1}{8d+4}$ in all dimensions $d\geq 3$). Compared with [@DOKZFalconer], in the current paper the construction of the good part $\mu_{1,g}$ is simpler and more intuitive so that the control of the bad part is much easier than that in [@DOKZFalconer]. On the other hand, the $L^2$ estimate for the good part is slightly complex, for which we need the new weighted refined decoupling.* ## Outline {#outline .unnumbered} In Section [2](#sec: dec){reference-type="ref" reference="sec: dec"}, we prove refined decoupling estimates in Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}. In Section [3](#sec: eg){reference-type="ref" reference="sec: eg"}, we present sharp examples for Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"} in the case of large fractal dimensions to prove Theorem [Theorem 4](#thm: eg){reference-type="ref" reference="thm: eg"}. In Section [4](#sec: app){reference-type="ref" reference="sec: app"}, we discuss an application of weighted refined decoupling estimates to Falconer distance set problem - proof of Theorem [Theorem 6](#thm: Leb){reference-type="ref" reference="thm: Leb"}. ## Notations. {#notations. .unnumbered} Throughout the article, we write $A\lesssim B$ if $A\leq CB$ for some absolute constant $C$; $A\sim B$ if $A\lesssim B$ and $B\lesssim A$; $A\lesssim_\varepsilon B$ if $A\leq C_\varepsilon B$; $A\lessapprox B$ if $A\leq C_\varepsilon R^\varepsilon B$ for any $\varepsilon>0, R>1$. For a large parameter $R$, ${\rm RapDec}(R)$ denotes those quantities that are bounded by a huge (absolute) negative power of $R$, i.e. ${\rm RapDec}(R) \leq C_N R^{-N}$ for arbitrarily large $N>0$. Such quantities are negligible in our argument. For subsets $E_1, E_2 \subset \mathbb{R}^d$, $\mathop{\mathrm{dist}}(E_1,E_2)$ is their Euclidean distance. For $A \subset X \times Y$ and $x \in X$, define the slice $A|_x = \{ y \in Y : (x, y) \in A \}$. Similar definition for $A|_y$, when $y\in Y$. We say a measure $\mu$ in $\mathbb{R}^d$ is an $\alpha$-dimensional measure with constant $C_\mu$ if it is a probability measure satisfying that $$\mu(B(x,t)) \leq C_\mu t^\alpha,\qquad \forall x\in \mathbb{R}^d,\, \forall t>0.$$ An $(r,m)$-plate $H$ in $\mathbb{R}^d$ is the $r$-neighborhood of an $m$-dimensional affine plane in the cube $[-10,10]^d$. More precisely, $$H=\{z\in[-10,10]^d: \mathop{\mathrm{dist}}(z, P_H) < r\},$$ where $P_H$ is an $m$-dimensional affine plane, which is called the central plane of $H$. A $C$-scaling of $H$ is $$CH=\{z\in[-10,10]^d: \mathop{\mathrm{dist}}(z, P_H) < Cr\}.$$ We say that an $(r,m)$-plate $H$ is $\gamma$-concentrated on $\mu$ if $\mu(H) \ge \gamma$. Let $\mathcal{E}_{r,m}$ be a set of $(r, m)$-plates with the following properties: - Each $(\frac{r}{2}, m)$-plate intersecting $B(0, 1)$ lies in at least one plate of $\mathcal{E}_{r,m}$; - For $s \ge r$, every $(s, m)$-plate contains $\lesssim\left( \frac{s}{r} \right)^{(m+1)(d-m)}$ many $(r, m)$-plates of $\mathcal{E}_{r,m}$. For example, when $m = 1$ and $d = 2$, we can simply pick $\sim r^{-1}$ many $r$-tubes in each of an $r$-net of directions. This generalizes to higher $m$ and $d$ via a standard $r$-net argument, see [@KevinRadialProj Section 2.2] for the details of its construction. **Acknowledgements 1**. *XD is supported by NSF DMS-2107729 (transferred from DMS-1856475), NSF DMS-2237349 and Sloan Research Fellowship. YO is supported by NSF DMS-2142221 and NSF DMS-2055008. KR is supported by a NSF GRFP fellowship. RZ is supported by NSF DMS-2207281 (transferred from DMS-1856541), NSF DMS-2143989 and the Sloan Research Fellowship.* # Weighted refined decoupling estimates - Proof of Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"} {#sec: dec} 0 In this section, we prove Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"} for the truncated paraboloid. The proof can be generalized for any compact and strictly convex $C^2$ hypersurface with Gaussian curvature $\sim 1$ by standard arguments. In particular, in the proof of part (a), we use the fact that under the assumption of $(R^{-1/2}, m)$-concentration, one can apply Bourgain--Demeter's $l^2$-decoupling in dimension $m$. See [@guthPolynomialII Lemma 9.3] and [@DGL Lemma 7.4] for justification of this fact in the case of the truncated paraboloid, and one can follow [@BDdecoupling Section 7] to generalize this fact to hypersurfaces as in the above. First, we present a slightly simplified proof of Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a) based on that in [@guth2020falconer]. *Proof of Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a).* Without loss of generality, we can assume that $$\label{Qconst} \| f \|_{L^p(Q)} \sim \textrm{ constant for all $R^{1/2}$-cubes $Q \subset Y$}.$$ Now we decompose $f$ as follows. We cover $\mathcal{S}$ with larger blocks $\tau$ of dimensions $R^{-1/4} \times \cdots \times R^{-1/4} \times R^{-1/2}$. For each $\tau$ we cover $B^d_R$ with cylinders $\Box$ with radius $R^{3/4}$ and length $R$, with the long axis perpendicular to $\tau$. Each cylinder $\Box$ is associated to a unique $\tau$, which we denote by $\tau(\Box)$. Then we define $$\mathbb W_\Box := \{ T \in \mathbb W: \theta(T) \subset \tau(\Box) \textrm{ and } T \cap B_R \subset \Box \}$$ and define $f_\Box := \sum_{T\in \mathbb W_\Box} f_T.$ Note that $\widehat f_\Box$ is essentially supported in $\tau(\Box)$. Next, write each $\Box$ as a union of cylinders running parallel to the long axis of $\Box$, with radius $R^{1/2}$ and length $R^{3/4}$. Let $Y_{\Box, M'}$ be the union of those cylinders that each intersect $\sim M'$ of the tubes $T \in \mathbb W_\Box$. Now we dyadically pigeonhole $M'$ so that $$\label{decentcube} \| f \|_{L^p(Q)} \lessapprox \left\| \sum_{\Box:\, Q \subset Y_{\Box, M'}} f_\Box \right\|_{L^p(Q)}$$ for a fraction $\approx 1$ of $Q \subset Y$. We fix this value of $M'$, and from now on we abbreviate $Y_\Box = Y_{\Box, M'}$. Denote the collection of cylinders $\Box$ by $\mathbb B$. We dyadically pigeonhole the cubes $Q \subset Y$ according to the number of $\Box \in \mathbb{B}$ so that $Q \subset Y_\Box$. We get a subset $Y' \subset Y$ so that for each cube $Q \subset Y'$, $Q \subset Y_\Box$ for $\sim M''$ choices of $\Box \in \mathbb{B}$, and $Q$ obeys [\[decentcube\]](#decentcube){reference-type="eqref" reference="decentcube"}. Moreover, by dyadic pigeonholing, we have $|Y'| \approx |Y|$. Since each cube $Q \subset Y$ has approximately equal $L^p$ norm, we also get $\| f \|_{L^p(Y')} \approx \| f \|_{L^p(Y)}$. We also note that $$\label{eq-M} M' M'' \lesssim M,$$ because a cube $Q \subset Y'$ belongs to $Y_\Box$ for $\sim M''$ different $\Box$, and if $Q \subset Y_\Box$, then it belongs to $T$ for $\sim M'$ different $T \in \mathbb{W}_\Box$. Note that an $R^{1/2}$-cube $Q$ lies in one cylinder $\Box$ associated to each cap $\tau$. So by applying Bourgain--Demeter's $l^2$-decoupling [@BDdecoupling] at scale $R^{1/2}$ to the RHS of [\[decentcube\]](#decentcube){reference-type="eqref" reference="decentcube"}, for each $Q\subset Y'$ we get $$\label{decex} \| f \|_{L^p(Q)} \lessapprox R^{\frac 12 \gamma_d(p)}\left( \sum_{\Box: Q\subset Y_\Box} \| f_{\Box} \|_{L^p(Q)}^2 \right)^{1/2}.$$ The next ingredient is induction on scales. After parabolic rescaling, the function $f_\Box$ with the decomposition $f_\Box = \sum_{T \in \mathbb{W}_\Box} f_T$ on the subset $Y_\Box$ is equivalent to the setup of the theorem at scale $R^{1/2}$ instead of scale $R$. So by induction on the radius, we get a version of our main inequality for each function $f_\Box$: $$\label{indbox} \| f_\Box \|_{L^p(Y_\Box)} \lesssim R^{\frac 12 (\gamma_d(p)+\epsilon)} (M')^{\frac{1}{2} - \frac{1}{p}} \left(\sum_{T \in \mathbb{W}_\Box} \| f_T \|_{L^p}^p \right)^{1/p}.$$ Now, combining all these ingredients [\[eq-M\]](#eq-M){reference-type="eqref" reference="eq-M"}, [\[decex\]](#decex){reference-type="eqref" reference="decex"} and [\[indbox\]](#indbox){reference-type="eqref" reference="indbox"}, we are ready to estimate $\|f\|_{L^p(Y)}$: $$\begin{aligned} \|f\|^p_{L^p(Y)} & \lessapprox \sum_{Q\subset Y'}\|f\|^p_{L^p(Q)} \\ &\lessapprox R^{\frac p2 \gamma_d(p)} \sum_{Q\subset Y'} \left( \sum_{\Box: Q\subset Y_\Box} \| f_{\Box} \|_{L^p(Q)}^2 \right)^{p/2}\\ &\lesssim R^{\frac p2 \gamma_d(p)} (M'')^{\frac p2 -1} \sum_{Q\subset Y'} \sum_{\Box: Q\subset Y_\Box} \| f_{\Box} \|_{L^p(Q)}^p\\ &\lesssim R^{\frac p2 \gamma_d(p)} (M'')^{\frac p2 -1} \sum_\Box \| f_{\Box} \|_{L^p(Y_\Box)}^p \\ &\lesssim R^{p \gamma_d(p)+\frac{p\epsilon}{2}} (M'M'')^{\frac p2 -1} \sum_\Box \sum_{T \in \mathbb{W}_\Box} \| f_T \|_{L^p}^p \\ &\lesssim R^{p \gamma_d(p)+\frac{p\epsilon}{2}} M^{\frac p2 -1} \sum_{T \in \mathbb{W}} \| f_T \|_{L^p}^p.\end{aligned}$$ Taking account of $\lessapprox$ throughout, we get $$\|f\|_{L^p(Y)} \lesssim R^{\gamma_d(p)+\frac{3\epsilon}{4}} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}.$$ This closes the induction and finishes the proof of Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a). *Proof of Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(a)*. The proof is almost identical to that of Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a). The only difference is that, when applying Bourgain--Demeter's $l^2$-decoupling at scale $R^{1/2}$, one uses the decoupling in dimension $m$ (instead of $d$) because of the $(R^{-1/2},m)$-concentration assumption (see [@guthPolynomialII Lemma 9.3] and [@DGL Lemma 7.4] for justifications of similar statements). Also, note that after parabolic rescaling, the function $f_\Box$ with the decomposition $f_\Box = \sum_{T \in \mathbb{W}_\Box} f_T$ on the subset $Y_\Box$ is equivalent to the setup of the theorem at scale $R^{1/2}$: the $f_\Box$ after rescaling has $(R^{-1/4},m)$-concentrated frequencies. *Proof of Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(b)*. Now we prove part (b) using the case $p=p_m$ of part (a). Without loss of generality, we can assume that $$|f_T|\sim \textrm{ constant for all $T\in \mathbb{W}$},$$ and each $R^{1/2}$-cube $Q$ in $Y$ intersects $\sim M$ tubes $T\in\mathbb{W}$. Denote the number of $R^{1/2}$-cubes in $Y$ by $N$, and let $W=|\mathbb{W}|$. Considering the incidence between $R^{1/2}$-cubes in $Y$ and tubes $T\in\mathbb{W}$, we get $$NM\lessapprox WR^{1/2}.$$ By the assumption that $|f_T|\sim$ constant for all $T\in \mathbb{W}$, we also get $$\label{eqn: part c} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^{p_m}}^{p_m}\right)^{\frac {1}{p_m}}\lessapprox \left(\frac{1}{WR^{(d+1)/2}}\right)^{\frac 1p -\frac{1}{p_m}} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}.$$ Combining all these ingredients together with the assumption that $\int_Q H(x)\,dx \lesssim R^{\alpha/2}$ for any $R^{1/2}$-cube $Q$ in $Y$, and applying the case $p=p_m$ of part (a) we get the following for any $p\leq p_m$: $$\begin{aligned} \|f\|_{L^p(Y;Hdx)} &\leq \left(\int_Y H\,dx\right)^{\frac 1p -\frac{1}{p_m}} \|f\|_{L^{p_m}(Y)}\\ &\lessapprox \left(\int_Y H\,dx\right)^{\frac 1p -\frac{1}{p_m}} M^{\frac 12 -\frac{1}{p_m}} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^{p_m}}^{p_m}\right)^{\frac {1}{p_m}}\\ &\lessapprox M^{\frac 12 -\frac 1p} \left(\frac{NR^{\alpha/2}M}{WR^{(d+1)/2}}\right)^{\frac 1p -\frac{1}{p_m}} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}\\ &\lesssim R^{\frac 12 (\alpha -d)(\frac 1p -\frac{1}{p_m})} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p},\end{aligned}$$ as desired. *Proof of Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(c)*. We prove part (c) by combining two steps of refined decoupling inequalities from (b) and Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a). Let $R^{-1/2}\leq r\leq 1$. Let $\tilde Y$ be a union of $r^{-2}$-cubes in $B_R$ such that each $r^{-2}$-cube $Q_1$ in $\tilde Y$ intersects some $R^{1/2}$-cube $Q$ in $Y$. Without loss of generality, we can assume that $$\label{Q1const} \| f \|_{L^p(Q_1;Hdx)} \sim \textrm{ constant for all $r^{-2}$-cubes $Q_1 \subset \tilde Y$},$$ and $$\| f \|_{L^p(Y;Hdx)} \lessapprox \| f \|_{L^p(\tilde Y;Hdx)} .$$ Now we decompose $f$ as follows. We cover $\mathcal{S}$ with blocks $\tau$ of dimensions $r \times \cdots \times r \times r^{2}$. For each $\tau$ we cover $B^d_R$ with cylinders $\Box$ with radius $rR$ and length $R$, with the long axis perpendicular to $\tau$. Each cylinder $\Box$ is associated to a unique $\tau$, which we denote by $\tau(\Box)$. Then we define $\mathbb W_\Box := \{ T \in \mathbb W: \theta(T) \subset \tau(\Box) \textrm{ and } T \cap B_R \subset \Box \}$ and define $f_\Box := \sum_{T\in \mathbb W_\Box} f_T.$ Note that $\widehat f_\Box$ is essentially supported in $\tau(\Box)$. Denote the collection of boxes $\Box$ by $\mathbb B$. Next, write each $\Box$ as a union of cylinders $\Box_1$ running parallel to the long axis of $\Box$, with radius $R^{1/2}$ and length $r^{-1}R^{1/2}$. Let $Y_{\Box, M_2}$ be the union of those cylinders that each intersect $\sim M_2$ of the tubes $T \in \mathbb W_\Box$. For each $r^{-2}$-cube $Q_1$ in $\tilde Y$, let $$f|_{Q_1}=\sum_{T_1\in \mathbb{T}[Q_1]} f_{T_1}$$ be the wave packet decomposition of $f|_{Q_1}$ at scale $r^{-2}$. Each tube $T_1\in \mathbb{T}[Q_1]$ has radius roughly $r^{-1}$ and length $r^{-2}$. Note that each $T_1$ is contained in a unique cylinder $\Box_1$ with radius $R^{1/2}$ and length $r^{-1}R^{1/2}$, which runs in the same direction as $T_1$. We denote this $\Box_1$ by $\Box_1(T_1)$. And this $\Box_1(T_1)$ is contained in a unique $\Box \in \mathbb B$, which runs in the same direction as $T_1$. We denote this $\Box$ by $\Box(T_1)$. Now we dyadically pigeonhole $M_2$ so that $$\label{decentcube1} \| f \|_{L^p(Q_1;Hdx)} \lessapprox \left\| \sum_{T_1\in\mathbb{T}[Q_1]:\, \Box_1(T_1)\subset Y_{\Box(T_1),M_2}} f_{T_1} \right\|_{L^p(Q_1;Hdx)}$$ for a fraction $\approx 1$ of $Q_1 \subset \tilde Y$. We fix this value of $M_2$, and from now on we abbreviate $Y_\Box = Y_{\Box, M_2}$. Next, write each $Q_1 \cap Y$ as a union of $r^{-1}$-cubes $Q'$. Let $Y_{Q_1, M_1}$ be the union of those $r^{-1}$-cubes that each intersect $\sim M_1$ of the tubes $T_1 \in \mathbb{T}[Q_1]$ with $\Box_1(T_1) \subset Y_{\Box(T_1)}$. Now we dyadically pigeonhole $M_1$ so that $$\label{decentcube2} \| f \|_{L^p(Q_1;Hdx)} \lessapprox \left\| \sum_{T_1\in\mathbb{T}[Q_1]:\, \Box_1(T_1)\subset Y_{\Box(T_1)}} f_{T_1} \right\|_{L^p(Y_{Q_1,M_1};Hdx)}$$ for a fraction $\approx 1$ of $Q_1 \subset \tilde Y$. We fix this value of $M_1$, and from now on we abbreviate $Y_{Q_1} = Y_{Q_1, M_1}$. Let $\tilde Y'$ be the collections of $Q_1$ satisfying [\[decentcube2\]](#decentcube2){reference-type="eqref" reference="decentcube2"}. Since $\| f \|_{L^p(Q_1;Hdx)} \sim \textrm{ constant }$ for all $Q_1 \subset \tilde Y$, we get $\|f\|_{L^p(Y;Hdx)}\approx\| f \|_{L^p(\tilde Y;Hdx)} \approx \| f \|_{L^p(\tilde Y';Hdx)}$. We also note that $$\label{M} M_1 M_2 \lesssim M,$$ because an $r^{-1}$ cube $Q' \subset Y_{Q_1}$ intersects $\sim M_1$ of the tubes $T_1 \in \mathbb{T}[Q_1]$ with $\Box_1(T_1) \subset Y_{\Box(T_1)}$, and each $\Box_1(T_1) \subset Y_{\Box(T_1)}$ intersects $\sim M_2$ of the tubes $T \in \mathbb{W}_{\Box(T_1)}$. Let $p_d\leq p\leq p_m$. By applying part (b) to the RHS of [\[decentcube2\]](#decentcube2){reference-type="eqref" reference="decentcube2"}, for each $r^{-2}$-cube $Q_1$ in $\tilde Y'$, we have $$\label{dec1} \| f \|_{L^p(Q_1;Hdx)} \lessapprox r^{(d-\alpha)(\frac 1p -\frac{1}{p_m})} M_1^{\frac 12 -\frac 1p} \left(\sum_{T_1\in\mathbb{T}[Q_1]:\, \Box_1(T_1)\subset Y_{\Box(T_1)}}\|f_{T_1}\|_{L^p}^p\right)^{\frac 1p}.$$ Also, after parabolic rescaling, the function $f_\Box$ with the decomposition $f_\Box = \sum_{T \in \mathbb{W}_\Box} f_T$ on the subset $Y_\Box$ is equivalent to the setup of Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a) at scale $r^2R$ instead of scale $R$. So by applying Theorem [Theorem 1](#thm-RD-lp-0){reference-type="ref" reference="thm-RD-lp-0"}(a) in the case $p\geq p_d$, we get $$\label{dec2} \| f_\Box \|_{L^p(Y_\Box)} \lessapprox (r^2 R)^{\frac{d-1}{4}-\frac{d+1}{2p}} M_2^{\frac{1}{2} - \frac{1}{p}} \left(\sum_{T \in \mathbb{W}_\Box} \| f_T \|_{L^p}^p \right)^{1/p}.$$ Now we are ready to estimate $\|f\|_{L^p(Y;Hdx)}$ by combining [\[M\]](#M){reference-type="eqref" reference="M"}, [\[dec1\]](#dec1){reference-type="eqref" reference="dec1"} and [\[dec2\]](#dec2){reference-type="eqref" reference="dec2"}: $$\begin{aligned} \|f\|^p_{L^p(Y;Hdx)} &\approx \sum_{Q_1\subset \tilde Y'}\| f \|^p_{L^p(Q_1;Hdx)} \\ &\lessapprox \sum_{Q_1\subset \tilde Y'}r^{(d-\alpha)(1 -\frac{p}{p_m})} M_1^{\frac p2 -1} \left(\sum_{T_1\in\mathbb{T}[Q_1]:\, \Box_1(T_1)\subset Y_{\Box(T_1)}}\|f_{T_1}\|_{L^p}^p\right) \\ &\sim r^{(d-\alpha)(1 -\frac{p}{p_m})} M_1^{\frac p2 -1} \sum_{\Box} \sum_{Q_1\subset \tilde Y'} \sum_{\underset{\Box_1(T_1)\subset Y_{\Box(T_1)},\, \Box(T_1) =\Box}{T_1\in\mathbb{T}[Q_1]}}\|f_{T_1}\|_{L^p}^p \\ & \lesssim r^{(d-\alpha)(1 -\frac{p}{p_m})} M_1^{\frac p2 -1} \sum_{\Box} \|f_\Box\|^p_{L^p(Y_\Box)}\\ & \lessapprox r^{(d-\alpha)(1 -\frac{p}{p_m})} (r^2 R)^{\frac{(d-1)p}{4}-\frac{d+1}{2}} (M_1M_2)^{\frac p2 -1} \sum_\Box \left(\sum_{T \in \mathbb{W}_\Box} \| f_T \|_{L^p}^p \right) \\ & \lesssim r^{(d-\alpha)(1 -\frac{p}{p_m})} (r^2 R)^{\frac{(d-1)p}{4}-\frac{d+1}{2}} M^{\frac p2 -1} \left(\sum_{T \in \mathbb{W}} \| f_T \|_{L^p}^p \right)\,.\end{aligned}$$ This concludes the proof of (c). # A sharp example in the case of large fractal dimensions - Proof of Theorem [Theorem 4](#thm: eg){reference-type="ref" reference="thm: eg"} {#sec: eg} We consider the following example obtained by adapting the one in [@BBCRV] to intermediate dimensions. Similar adaptions can be found in [@DKWZ; @du2019upper]. Let $c=1/1000$ be a fixed small constant, $0<\kappa<1/2$, and $2\leq m\leq d$. Denote $$x=(x_1,\cdots,x_d)=(x',x'',x_d)\in B^d(0,R)\,,$$ $$\xi=(\xi_1,\cdots,\xi_{d-1})=(\xi',\xi'')\in B^{d-1}(0,1)\,,$$ where $$x'=(x_1,\cdots,x_{d-m}), \quad x''=(x_{d-m+1},\cdots,x_{d-1}),$$ $$\xi'=(\xi_1,\cdots,\xi_{d-m}), \quad \xi''=(\xi_{d-m+1},\cdots,\xi_{d-1}).$$ For simplicity, we denote $B^d(0,r)$ by $B^d_r$, and write the interval $(-r,r)$ as $I_r$. Let $g(\xi)=\raisebox{0.7ex}{\(\chi\)}_\Omega(\xi)$, where the set $\Omega$ is defined by $$\label{Om} \Omega:=\left[B^{d-m}_{cR^{-1/2}} \times \left(2\pi R^{-\kappa} \mathbb{Z}^{m-1}+B^{m-1}_{cR^{-1}}\right)\right]\cap B^{d-1}(0,1)\,.$$ Take $f(x)=Eg(x)$, the Fourier extension of $g$ over the truncated paraboloid: $$\label{Ef} f(x):=Eg(x)=\frac{1}{(2\pi)^{d/2}} \int_{B^{d-1}(0,1)} e^{i(x'\cdot\xi'+x''\cdot \xi'' +x_d|\xi'|^2+x_d|\xi''|^2)}g(\xi)\,d\xi.$$ Next, we define a set $\Lambda$ in $B^d(0,R)$ by $$\label{La} \Lambda:=\left[B^{d-m}_{cR^{1/2}}\times \left(R^{\kappa}\mathbb{Z}^{m-1}+B^{m-1}_{c}\right)\times\left(\frac{1}{2\pi}R^{2\kappa}\mathbb{Z}+I_{c}\right)\right] \cap B^d_R\,.$$ And define $Y$ and $H$ by $$\label{Y H} Y:= B^{d-m}_{cR^{1/2}}\times B^m_R \quad \text{ and } \quad H:=\raisebox{0.7ex}{\(\chi\)}_\Lambda.$$ From the definition, it follows that $$\label{OmSize} |\Omega|\sim R^{(\kappa-1)(m-1)-(d-m)/2}\,.$$ and $$\label{sizeLa} |\Lambda|\sim R^{(d-m)/2+(1-\kappa)(m-1)+1-2\kappa}=R^{(d+m)/2-\kappa(m+1)}\,.$$ And it is straightforward (for example, see [@du2019upper proof of Lemma 3.1]) to check $$\label{xix} x'\cdot\xi'+x''\cdot \xi'' +x_d|\xi'|^2+x_d|\xi''|^2 \in 2\pi \mathbb Z + (-\frac{1}{100},\frac{1}{100})\,,$$ provided that $\xi\in\Omega$ and $x\in\Lambda$. **Claim 8**. *For $d-\frac{m+1}{2}\leq \alpha\leq d$, we can take $$\label{ka} \kappa:=\frac{d-\alpha}{2(m+1)}$$ such that $\kappa\leq 1/4$, and $f, Y$ and $H$ defined as in the above satisfy the conditions of Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(b), and for any $p>0$ there holds $$\label{eq-RD-eg'} \|f\|_{L^p(Y;Hdx)} \gtrsim R^{\frac 12 (\alpha -d)(\frac 1p -\frac{1}{p_m})} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}.$$* Indeed, from the constructions, we have the following properties: - $f$ has $(R^{-1/2},m)$-concentrated frequencies; - By [\[xix\]](#xix){reference-type="eqref" reference="xix"}, $|f(x)|\sim |\Omega|, \forall x\in \Lambda$; - The Fourier support of $f$ are partitioned into $\sim R^{\kappa(m-1)}$ many parabolic caps $\theta$, each of radius $R^{-1/2}$; - For each fixed $\theta$, let $f_\theta:=(\hat f|_\theta)^\vee$. Then, by a reason similar to [\[xix\]](#xix){reference-type="eqref" reference="xix"}, we get $|f_\theta|\sim R^{-(d-m)/2-(m-1)}$ on $Y$; - Take $\mathbb W := \{T\in \mathbb T: T\subset Y\}$. Then for each $R^{1/2}$-cube in $Y$, there are $M\sim R^{\kappa(m-1)}$ many $T\in \mathbb W$ passing through it; - For each $R^{1/2}$-cube $Q$ in $Y$, $$\int_Q H(x)\,dx =|Q\cap \Lambda|\sim R^{d/2-(m+1)\kappa}=R^{\alpha/2}\,,$$ by the choice of $\kappa$ in [\[ka\]](#ka){reference-type="eqref" reference="ka"}. Therefore, $$\|f\|_{L^p(Y;Hdx)} \sim |\Omega|\cdot |\Lambda|^{1/p}\,,$$ and $$\begin{split} &M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p} \sim M^{\frac 12 -\frac 1p} \left(\sum_{\theta}\|f_\theta\|_{L^p(Y)}^p\right)^{\frac 1p} \\ &\sim R^{\frac{\kappa(m-1)}{2}}R^{-\frac{d-m}{2}-(m-1)}R^{\frac{d+m}{2p}}\,, \end{split}$$ and thus $$\frac{\|f\|_{L^p(Y;Hdx)}}{M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}} \sim R^{\kappa(\frac{m-1}{2}-\frac{m+1}{p})}=R^{\frac 12 (\alpha -d)(\frac 1p -\frac{1}{p_m})}\,,$$ as desired. Moreover, by direct computation, one can verify the ball condition at all scales up to $R^{1/2}$: $$\int_{\tilde Q} H(x)dx =|\tilde Q \cap \Lambda| \lesssim s^\alpha, \quad \forall s\text{-cube } \tilde Q \text{ in } Y, \forall 0<s\leq R^{1/2};$$ and also at all scales up to $R$ if in addition $\alpha\geq m$. This completes the proof of Theorem [Theorem 4](#thm: eg){reference-type="ref" reference="thm: eg"}. # Application to Falconer distance set problem {#sec: app} 0 As an application of Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}, in this section we give a proof of Theorem [Theorem 6](#thm: Leb){reference-type="ref" reference="thm: Leb"} which is different from that in [@DOKZFalconer]. We first recall the following new radial projection theorem, which follows from [@KevinRadialProj Theorem 1.13]. **Theorem 9**. *Let $m \in \{ 1, 2, \cdots, d-1 \}$, $m-1 < \alpha \le m$, and fix $\eta, \varepsilon> 0$, and two $\alpha$-dimensional measures $\mu_1, \mu_2$ with constants $C_{\mu_1}, C_{\mu_2}$ supported on $E_1, E_2 \subset B(0, 1)$ respectively. There exists $\gamma > 0$ depending on $\eta, \varepsilon, \alpha, m$ such that the following holds. Fix $\delta < r < 1$. Let $A$ be the set of pairs $(x, y) \in E_1 \times E_2$ satisfying that $x$ and $y$ lie in some $\delta^\eta$-concentrated $(r,m)$-plate on $\mu_1 + \mu_2$. Then there exists a set $B \subset E_1 \times E_2$ with $\mu_1 \times \mu_2 (B) \le \delta^{\gamma}$ such that for every $x \in E_1$ and $\delta$-tube $T$ through $x$, we have $$\mu_2 (T \setminus (A|_x \cup B|_x)) \lesssim\frac{\delta^\alpha}{r^{\alpha-(m-1)}} \delta^{-\varepsilon}.$$ The implicit constant may depend on $\eta, \varepsilon, \alpha, m, C_{\mu_1}, C_{\mu_2}$.* Essentially, this theorem says that, up to a small loss, one can assume that the wave packets associated with the fractal measure supported on the set of interest all have small mass. We explore this idea in detail in the next subsections. ## Outline of the proof of Theorem [Theorem 6](#thm: Leb){reference-type="ref" reference="thm: Leb"} {#sec: outline} The setup of the proof is in the same line as [@guth2020falconer; @du2021improved]. To begin with, let $E\subset \mathbb{R}^d$ be a compact set with positive $\alpha$-dimensional Hausdorff measure, with $\frac{d}{2}<\alpha<\frac{d+1}{2}$. Without loss of generality, assume that $E$ is contained in the unit ball, and there are subsets $E_1, E_2\subset E$, each with positive $\alpha$-dimensional Hausdorff measure, and ${\rm dist}(E_1, E_2)\gtrsim 1$. Then there exist $\alpha$-dimensional probability measures $\mu_1$ and $\mu_2$ supported on $E_1$ and $E_2$ respectively, according to the classical Frostman lemma. To relate the measures to the distance set, we consider their pushforward measures under the distance map. For a fixed point $x\in E_2$, let $d^x:E_1\to \mathbb{R}$ be the pinned distance map given by $d^x(y):=|x-y|$. Then, the pushforward measure $d^x_\ast(\mu_1)$, defined as $$\int_{\mathbb{R}} \psi(t)\,d^x_\ast(\mu_1)(t)=\int_{E_1}\psi(|x-y|)\,d\mu_1(y),$$ is a natural probability measure that is supported on $\Delta_x(E_1)$. The idea is that we will construct another complex-valued measure $\mu_{1,g}$ that is the *good* part of $\mu_1$ with respect to $\mu_2$, and study its pushforward under the map $d^x$. To set things up, we recall the following decomposition of a function into microlocalized pieces, which has been used in [@guth2020falconer; @du2021improved]. We will eventually be choosing the following small parameters with the dependence $$0<\beta\ll \gamma\ll\eta \ll \varepsilon \ll\epsilon.$$ Fix a large parameter $R_0$, to be determined later, and consider a sequence of scales $R_j=2^j R_0$, $\forall j\geq 1$. In $\mathbb{R}^d$, cover the annulus $R_{j-1} \le |\omega| \le R_j$ by rectangular blocks $\tau$ with dimensions approximately $R_j^{1/2} \times \cdots \times R_j^{1/2} \times R_j$, with the long direction of each block $\tau$ being the radial direction. Choose a smooth partition of unity subordinate to this cover such that $$1 = \psi_0 + \sum_{j \ge 1, \tau} \psi_{j, \tau},$$ where $\psi_0$ is supported in the ball $B(0,2R_0)$. Let $\beta > 0$ be a sufficiently small constant that we will choose later (depending on $\eta, \varepsilon$). For each $(j, \tau)$, cover the unit ball in $\mathbb{R}^d$ with tubes $T$ of dimensions approximately $R_j^{-1/2 + \beta} \times\cdots \times R_j^{-1/2+\beta} \times 2$ with the long axis parallel to the long axis of $\tau$. The covering has uniformly bounded overlap, each $T$ intersects at most $C(d)$ other tubes. We denote the collection of all these tubes as $\mathbb{T}_{j, \tau}$. Let $\eta_T$ be a smooth partition of unity subordinate to this covering, so that for each choice of $j$ and $\tau$, $\sum_{T \in \mathbb{T}_{j, \tau}} \eta_T$ is equal to $1$ on the ball of radius $2$ and each $\eta_T$ is smooth. For each $T \in \mathbb{T}_{j, \tau}$, define an operator $$M_T f := \eta_T (\psi_{j, \tau} \hat f)^{\vee},$$ which, morally speaking, maps $f$ to the part of it that has Fourier support in $\tau$ and physical support in $T$. Define also $M_0 f := (\psi_0 \hat f)^{\vee}$. We denote $\mathbb{T}_j = \cup_{\tau} \mathbb{T}_{j, \tau}$ and $\mathbb{T} = \cup_{j \ge 1} \mathbb{T}_j$. Hence, for any $L^1$ function $f$ supported on the unit ball, one has the decomposition $$f = M_0 f + \sum_{T \in \mathbb{T}} M_T f+\text{RapDec}(R_0)\|f\|_{L^1}.$$ Fix parameters $\eta, \varepsilon>0$ (these parameters will be chosen depending on $\epsilon$ in Proposition [Proposition 13](#prop: l2){reference-type="ref" reference="prop: l2"}). Let $2T$ denote the concentric dilation of $T$ of twice the radius. For each $j\geq 1$ fixed, let $\delta=2 R_j^{-\frac{1}{2}+\beta}$ and consider a dyadic sequence of scales $r\in [C_0\delta, 1]$. Here, $C_0$ is a large constant depending on the separation of the sets $E_1, E_2$. For each fixed $r$, recall that $\mathcal{E}_{r,m}$ denotes a collection of essentially distinct $(r,m)$-plates such that every $(r/2, m)$-plate is contained in some element of $\mathcal{E}_{r,m}$. We further let $\mathcal{H}_r$ denote the subcollection of all $\delta^\eta$-concentrated $(r,m)$-plates of $\mathcal{E}_{r,m}$ on $\mu_1+\mu_2$. Here, $m\in \mathbb{Z}$ is the unique integer satisfying $m-1<\alpha\leq m$. For each tube $T \in \mathbb{T}_{j}$, define $$r(T):=\min\{{\rm dyadic }\,\,r\in [C_0 \delta,1]:\, \exists H\in \mathcal{H}_r \,\,{\rm s.t. }\,2T\subset H\}.$$Here, $r(T)$ captures the most efficient choice of the scale of heavy plate that $T$ is contained in. We say a tube $T \in \mathbb{T}_{j}$ is *bad* if $$\label{badthreshold} \mu_2 (4T) \ge \frac{\delta^{\alpha-2\varepsilon}}{r(T)^{\alpha-(m-1)}}.$$ Here, $\varepsilon$ is a fixed parameter and will be chosen in Proposition [Proposition 13](#prop: l2){reference-type="ref" reference="prop: l2"} below. A tube $T$ is *good* if it is not bad, and we define $$\mu_{1,g}:=M_0 \mu_1 + \sum_{T \in \mathbb{T}, T \textrm{ good}} M_T \mu_1.$$ We point out that $\mu_{1,g}$ is only a complex valued measure, and is essentially supported in the $R_0^{-1/2+\beta}$-neighborhood of $E_1$ with a rapidly decaying tail away from it. Theorem [Theorem 6](#thm: Leb){reference-type="ref" reference="thm: Leb"} will follow from the following two main estimates in the exact same way as in [@du2021improved]. We omit the details. **Proposition 10**. *Let $d\geq 3$, and $0<\alpha \leq d-1$. There exists a choice of $\beta>0$ and sufficiently large $R_0$ in the construction of $\mu_{1,g}$ in the above, such that there is a subset $E_2' \subset E_2$ so that $\mu_2(E_2') \ge 1 - \frac{1}{1000}$ and for each $x \in E_2'$, $$\| d^x_*(\mu_1) - d^x_*(\mu_{1,g}) \|_{L^1} < \frac{1}{1000}.$$* **Proposition 11**. *Let $d\geq 3$ and $$\alpha > \begin{cases}\frac{d}{2}+\frac{1}{4}-\frac{1}{4(2d+1)},& d\geq 4,\\ \frac{3}{2}+\frac{1}{4}+\frac{17-12\sqrt{2}}{4},& d=3,\end{cases}$$then there exist choices of $\varepsilon$ and $R_0$ so that for sufficiently small $\beta$ in terms of $\alpha$ in the construction of $\mu_{1,g}$ in the above, $$\int_{E_2} \| d^x_*(\mu_{1,g}) \|_{L^2}^2 d \mu_2(x) < + \infty.$$* We will prove these two propositions in the next two subsections. Before that, let us briefly explain the key new ideas here. In contrast to [@guth2020falconer; @du2021improved], where a similar framework were used to study the Falconer distance problem, our definition of good tubes here involves a new parameter $r(T)$, which captures the size of the smallest heavy plate containing tube $T$. Thanks to the new radial projection result (Theorem [Theorem 9](#conj:threshold){reference-type="ref" reference="conj:threshold"}), we are able to show that the bad tubes can be safely removed (Proposition [Proposition 10](#mainest1){reference-type="ref" reference="mainest1"}). To prove Proposition [Proposition 11](#mainest2){reference-type="ref" reference="mainest2"}, we make use of the weighted refined decoupling estimates to deal with varying values of $r(T)$. In the extreme case that $r(T)\sim R_{j}^{-1/2}$, the threshold for the good/bad tubes is the same as the one in [@du2021improved], however, in this case, from the fact that $T$ is contained in a thin $(R_{j}^{-1/2},m)$-plate, we get extra gain by applying the weighted refined decoupling estimate in Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}. At the other end of the spectrum, where $r(T)=1$, even though the weighted decoupling estimate is not going to help, the threshold for the good/bad tubes becomes much better than the one in [@du2021improved], allowing us to obtain improvement for the Falconer problem in this case as well. ## Removal of the bad region via the radial projection theorem In this section, we apply the radial projection theorem (Theorem [Theorem 9](#conj:threshold){reference-type="ref" reference="conj:threshold"}) to prove Proposition [Proposition 10](#mainest1){reference-type="ref" reference="mainest1"} [\[sec:bad\]]{#sec:bad label="sec:bad"}. From the exact same deduction as in the proof of [@du2021improved Proposition 2.1] (more precisely, Lemma 3.1), one has the bound $$\| d^x_*(\mu_1) - d^x_*(\mu_{1,g}) \|_{L^1}\lesssim \sum_{j\geq 1}R_j^{\beta d}\mu_1({\rm Bad}_j(x))+{\rm RapDec}(R_0),$$where $${\rm Bad_j}(x):=\bigcup_{T \in \mathbb{T}_j:\, x \in 2T \textrm{ and $T$ is bad}} 2T,\quad \forall j\geq 1.$$We also denote $$\begin{split} \text{Bad}_j:=&\{(y,x)\in E_1\times E_2:\, y\in \text{Bad}_j(x)\}\\ =&\{(y,x)\in E_1\times E_2:\, \exists \text{ bad }T\in \mathbb{T}_j \text{ s.t. }x,y\in 2T\}. \end{split}$$ The goal is then to obtain decay for $\mu_1({\rm Bad}_j(x))$, $\forall j\geq 1$. We have the following estimate, from which Proposition [Proposition 10](#mainest1){reference-type="ref" reference="mainest1"} follows. **Lemma 12**. *There exist sufficiently large $R_0$ and sufficiently small $\beta>0$ such that there is a subset $E_2' \subset E_2$ so that $\mu_2(E_2') \ge 1 - \frac{1}{1000}$ and for each $x \in E_2'$, $$\mu_1({\rm Bad}_j(x))\lesssim R_j^{-100\beta d}, \quad \forall j\geq 1.$$* *Proof.* Fix a scale $j\geq 1$. Our goal is to show that $$\label{eqn: mu1mu2} \mu_1\times \mu_2(\text{Bad}_j)\lesssim R_j^{-200\beta d}.$$ To see how this would imply the desired estimate, one first rewrites $$\mu_1\times \mu_2(\text{Bad}_j)=\int \mu_1(\text{Bad}_j(x))\,d\mu_2(x).$$Then, one can find a set $F_j \subset E_2$ with $\mu_2 (F_j) \le R_j^{-50 \beta d}$ such that $\mu_1 (\text{Bad}_j (x)) \le R_j^{-150 \beta d}$ for $x\in E_2\setminus F_j$. We take $E_2' := E_2 \setminus \bigcup_{j \ge 1} F_j$. Observe that $\mu_2 (E_2') \ge \mu(E_2) - \sum_{j \ge 1} R_j^{-50\beta d} > 1-R_0^{-\beta}>1-\frac{1}{1000}$ if $R_0$ is sufficiently large, hence the desired result would follow. To prove ([\[eqn: mu1mu2\]](#eqn: mu1mu2){reference-type="ref" reference="eqn: mu1mu2"}), one recalls that $\delta=2 R_j^{-\frac{1}{2}+\beta}$. For each bad $T\in \mathbb{T}_j$, let $r(T)$ be the parameter as in the definition of bad tubes. Since there are only $\sim \log \frac 1 \delta \sim \log R_j$ many choices of $r(T)$, one can assume that $r(T)$ are the same for bad $T\in \mathbb{T}_j$. One first decomposes $$\text{Bad}_j=\text{Bad}^1_j\cup \text{Bad}^2_j,$$where for $k=1,2$, $$\text{Bad}^k_j:=\{(y,x)\in E_1\times E_2:\, \exists T\in \mathcal{T}_j^k \text{ s.t. }x,y\in 2T\},$$with $$\mathcal{T}_j^1:=\{T\in \mathbb{T}_j:\, T \text{ is bad},\, C_0\delta\leq r(T)\leq 10C_0\delta\},$$ $$\mathcal{T}_j^2:=\{T\in \mathbb{T}_j:\, T \text{ is bad},\, r(T)> 10C_0\delta\}.$$ The part $\text{Bad}_j^1$ is simpler and doesn't require Theorem [Theorem 9](#conj:threshold){reference-type="ref" reference="conj:threshold"}. Indeed, all the bad tubes in this case satisfies that $\mu_2(4T)\gtrsim \delta^{m-1-2\varepsilon}$. Since $m-1<\alpha\leq m$, one can follow the exact same argument in [@du2021improved justification of (3.1)] to project the sets $E_1, E_2$ onto an $m$-dimensional subspace and apply the radial projection theorem of Orponen there. We omit the details. Next, to estimate $\text{Bad}_j^2(y)$, define $r=\frac{r(T)}{8}$ and apply the new radial projection theorem (Theorem [Theorem 9](#conj:threshold){reference-type="ref" reference="conj:threshold"}) with the $\delta$ and $r$ specified in the above, and with the roles of $x,y$ interchanged. Note that the parameter $\eta$ is the same as the one in the definition of $r(T)$. Then, one has a set $B\in E_1\times E_2$ and $\gamma>0$ such that $\mu_1\times \mu_2(B)\leq \delta^\gamma$ and that for each $y\in E_1$ and $\delta$-tube $\tilde{T}$ through $y$, one has $$\mu_2(\tilde{T}\setminus (A|_y \cup B|_y))\lesssim \frac{\delta^\alpha}{r^{\alpha-(m-1)}}\delta^{-\varepsilon}.$$Recall that here $$\begin{split} A:=\{(y,x)\in &E_1\times E_2:\,\\ &x,y \in H,\, \text{ for some $\delta^\eta$-concentrated $(r,m)$-plate $H$}\}. \end{split}$$ For any fixed $y\in E_1$ and $T\in \mathcal{T}^2_j(y):=\{T\in \mathcal{T}_j^2:\, y\in 2T\}$, we claim that $2T \cap A|_y=\emptyset$. Indeed, suppose there is a point $x\in A|_y$ such that $x,y\in 2T$. Then, one has that $x,y \in H$ for some $\delta^\eta$-concentrated $(r,m)$-plate $H$. Since $y\in E_1$ and $x\in E_2$ are separated, for sufficiently large constant $C_0$ (depending on ${\rm dist}(E_1, E_2)$), one has that $2T\subset 2H$. By the construction of collections $\{\mathcal{E}_{r,m}\}_r$, $2H$ is contained in some $H'\in \mathcal{E}_{4r,m}$. Obviously, $H'$ is $\delta^\eta$-concentrated, hence one concludes that $2T$ is contained in some $H'\in \mathcal{H}_{\frac{r(T)}{2}}$, which contradicts the definition of $r(T)$. (Note that the assumption $r(T)>10C_0\delta$ in this part is to guarantee that $\frac{r(T)}{2}\geq C_0 \delta$.) By taking $\beta$ sufficiently small depending on $\gamma$, it suffices to estimate $\mu_1\times \mu_2(\text{Bad}^2_j\setminus B)$. Write $$\mu_1\times \mu_2(\text{Bad}^2_j\setminus B)=\int \mu_2(\text{Bad}^2_j(y)\setminus (B|_y))\,d\mu_1(y)$$where $$\text{Bad}^2_j(y):=\{x\in E_2:\, (y,x)\in \text{Bad}^2_j\},\quad\forall y\in E_1.$$Hence $$\text{Bad}^2_j(y)\setminus (B|_y)=\bigcup_{T\in \mathcal{T}_j^2(y)} 2T\setminus B|_y.$$ For any $y\in E_1$ fixed, since each bad tube $T$ satisfies $\mu_2(4T)\geq \frac{\delta^{\alpha-2\varepsilon}}{r(T)^{\alpha-(m-1)}}$ and each point in $E_2$ is contained in at most $\sim R_j^{d\beta}$ many $4T$ with $y\in 2T$, one has that $$|\mathcal{T}_j(y)|\frac{\delta^{\alpha-2\varepsilon}}{r(T)^{\alpha-(m-1)}}\leq \int \sum_{T\in \mathcal{T}_j(y)} \chi_{4T} \,d\mu_2\lesssim R_j^{d\beta}\int \chi_{\bigcup_{T\in \mathcal{T}_j(y)} 4T}\,d\mu_2\lesssim R_j^{\beta d},$$hence $$|\mathcal{T}_j(y)|\lesssim R_j^{\beta d}\frac{r(T)^{\alpha-(m-1)}}{\delta^{\alpha-2\varepsilon}}.$$We now have the estimate $$\mu_2({\rm Bad}_j^2(y)\setminus (B|_y)) \lesssim \frac{\delta^\alpha}{r^{\alpha-(m-1)}}\delta^{-\varepsilon}|\mathcal{T}_j(y)|\lesssim R_j^{\beta d}\delta^{\varepsilon}.$$By choosing $\beta$ sufficiently small depending on $\varepsilon$, one has that the above is $\lesssim \delta^{1000\beta d}\lesssim R_j^{-200 \beta d}$. The proof is complete by integrating this estimate in $y$. ◻ ## $L^2$ bound of the good part via weighted decoupling estimates In this section, we prove Proposition [Proposition 11](#mainest2){reference-type="ref" reference="mainest2"}, applying weighted refined decoupling estimates. More precisely, we will apply Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(c). The argument below proceeds very similarly as in [@guth2020falconer; @du2021improved], except that we need to reduce to the situation where the function has concentrated frequencies in order to apply Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}. For $R>0$, let $\sigma_R$ denote the normalized surface measure on the sphere of radius $R$ in $\mathbb{R}^d$. Our main estimate of the section is the following. **Proposition 13**. *Let $\alpha>0$ and $R>10 R_0$, and let $\mu_{1,g}$ be as defined in the above. Then, for all $\epsilon>0$, and $\beta, \varepsilon$ sufficiently small depending on $\alpha, \epsilon$, there holds $$\label{eqn: good L2} \int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x) \lesssim R^{-\Gamma(d,\alpha)+\epsilon} R^{-(d-1)}\int |\hat\mu_1|^2\psi_R\, d\xi+{\rm RapDec}(R),$$where $$\Gamma(d,\alpha)=\begin{cases} \frac{d\alpha}{d+1},& d\geq 4,\\ \alpha-\frac{\alpha^2}{6},& d=3,\end{cases}$$and $\psi_R$ is a weight function which is $\sim 1$ on the annulus $R-1 \le |\xi| \le R+1$ and decays off of it. To be precise, we could take $$\psi_R(\xi) = \left( 1 + | R- |\xi|| \right)^{-100}.$$* It is a routine argument to check that Proposition [Proposition 13](#prop: l2){reference-type="ref" reference="prop: l2"} implies Proposition [Proposition 11](#mainest2){reference-type="ref" reference="mainest2"}, for instance see [@du2021improved Proof of Proposition 2.2]. We include it here for the sake of completeness. Observe first that $$d^x_*(\mu_{1,g})(t)=t^{d-1}\mu_{1,g}*\sigma_t(x)\,.$$ Since $\mu_{1,g}$ is essentially supported in the $R_0^{-1/2+\beta}$-neighborhood of $E_1$, for $x\in E_2$, we only need to consider $t\sim 1$. Hence, up to a loss of $\text{RapDec}(R_0)$ which is negligible in our argument, we have $$\begin{split} \int_{E_2} \|d_*^x(\mu_{1,g})\|_{L^2}^2\,d\mu_2(x)\lesssim &\int_0^\infty \int_{E_2}|\mu_{1,g}*\sigma_t(x)|^2\,d\mu_2(x)t^{d-1}\,dt\\ \sim & \int_0^\infty \int_{E_2}|\mu_{1,g}*\hat{\sigma}_R(x)|^2\,d\mu_2(x)R^{d-1}\,dR, \end{split}$$ where in the second step, we have used a limiting process and an $L^2$-identity proved by Liu [@LiuL2 Theorem 1.9]. For $R\leq 10 R_0$, we use a simple estimate following from orthogonality (for example, see [@DOKZFalconer Lemma 5.2]): $$\int_{E_2} |\mu_{1,g} * \hat{\sigma}_R (x)|^2 \, d\mu_2 (x) \lesssim(R+1)^{d-1} R^{-(d-1)} \,.$$ So the small $R$ contribution to $\int \| d_*^x (\mu_{1,g}) \|^2_{L^2} d\mu_2 (x)$ is $$\begin{aligned} \int_0^{10R_0} (R+1)^{d-1} \, dR \lesssim R_0^d. \end{aligned}$$ Applying Proposition [Proposition 13](#prop: l2){reference-type="ref" reference="prop: l2"} for each $R>10 R_0$ and dropping the rapidly decaying tail as we may, one can bound the large $R$ contribution to $\int \| d_*^x (\mu_{1,g}) \|^2_{L^2} d\mu_2 (x)$ by $$\begin{split} &\lesssim \int_{10 R_0}^\infty \int_{\mathbb{R}^d} R^{-\Gamma(d,\alpha)+\epsilon} \psi_R(\xi) | \hat \mu_1(\xi)|^2 \,d \xi dR \\ &\lesssim \int_{\mathbb{R}^d} |\xi|^{-\Gamma(d,\alpha)+\epsilon} | \hat \mu_1 (\xi)|^2 \,d \xi \sim I_{\beta} (\mu_1), \end{split}$$where $\beta=d-\Gamma(d,\alpha)+\epsilon$, by a Fourier representation for $I_\beta(\mu_1)$, the $\beta$-dimensional energy of $\mu_1$. One thus has $I_\beta(\mu_1)<\infty$ if $\beta<\alpha$, which is equivalent to $$\alpha > \begin{cases}\frac{d}{2}+\frac{1}{4}-\frac{1}{4(2d+1)},& d\geq 4,\\ \frac{3}{2}+\frac{1}{4}+\frac{17-12\sqrt{2}}{4},& d=3.\end{cases}$$The proof of Proposition [Proposition 11](#mainest2){reference-type="ref" reference="mainest2"} is thus complete. We are now ready to prove Proposition [Proposition 13](#prop: l2){reference-type="ref" reference="prop: l2"}. First, let's recall the following [@KevinRadialProj Lemma 7.5], which controls the total number of essentially distinct concentrated plates from a fixed scale. **Lemma 14**. *Let $m-1 < \alpha \le m$. There is $N=N(\alpha, m)$ such that the following holds: let $\nu$ be an $\alpha$-dimensional measure with constant $C_\nu \ge 1$ and let $\mathcal{H}=\{H\in\mathcal{E}_{r,m}: \nu(H) \ge a\}$. Then $|\mathcal{H}| \lesssim(\frac{C_\nu}{a})^N$.* *Proof of Proposition [Proposition 13](#prop: l2){reference-type="ref" reference="prop: l2"}.* Fix $\epsilon>0$ and $R>10R_0$. By definition, $$\mu_{1,g}* \hat\sigma_R=\sum_{R_j \sim R} \sum_\tau \sum_{T\in \mathbb{T}_{j, \tau}:\, T \textrm{ good}} M_T \mu_1 * \hat \sigma_R+{\rm RapDec}(R).$$Since the contribution of ${\rm RapDec}(R)$ is already taken into account in the statement of Proposition [Proposition 13](#prop: l2){reference-type="ref" reference="prop: l2"}, we may ignore the tail ${\rm RapDec}(R)$ in the argument below. The following reduction is the same as in [@du2021improved]. Let $\eta_1$ be a bump function adapted to the unit ball and define $$f_T = \eta_1 \left( M_T \mu_1 * \hat \sigma_R \right).$$It is easy to see that $f_T$ is microlocalized to $(T,\theta(T))$ (similarly as defined in Section [2](#sec: dec){reference-type="ref" reference="sec: dec"}, with $f$ essentially supported in $2T$ and $\hat{f}$ essentially supported in $2\theta(T)$, the block of dimension $R^{1/2}\times \cdots \times R^{1/2}\times 1$ in the partition of the $1$-neighborhood of $RS^{d-1}$ corresponding to the long direction of $T$). We now apply a series of dyadic pigeonholing to the integral to be estimated. First, there exists $\lambda>0$ such that $$\int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x)\lesssim \log R \int | f_\lambda (x) |^2 d\mu_2(x),$$where $$f_\lambda=\sum_{T\in \mathbb{W}_\lambda}f_T,\quad \mathbb{W}_\lambda:= \bigcup_{R_j\sim R}\bigcup_{\tau}\Big\{ T\in \mathbb{T}_{j,\tau}: T \text{ good }, \| f_T \|_{L^p} \sim \lambda \Big\}.$$ For each $R$, since there are only finitely many $R_j\sim R$, for the sake of brevity we will drop the first union in $\mathbb{W}_\lambda$ and assume without loss of generality that $R_j=R$ for all $T\in \mathbb{W}_\lambda$ from this point on. In addition, considering a sequence of dyadic scales $r\in [2C_0R^{-\frac{1}{2}+\beta},1]$, by dyadic pigeonholing, one can reduce to a subcollection $\mathbb{W}_{\lambda, r}$ for a fixed $r$, where $$\mathbb{W}_{\lambda, r}:=\{T\in \mathbb{W}_\lambda:\, r(T)= r\}.$$Here, recall that $$r(T):=\min\{{\rm dyadic }\,\,r\in [2C_0 R^{-\frac{1}{2}+\beta},1]:\, \exists H\in \mathcal{H}_r \,\,{\rm s.t. }\,2T\subset H\},$$where $\mathcal{H}_r$ denotes a collection of $(2R^{-\frac{1}{2}+\beta})^\eta$-concentrated $(r,m)$-plates of $\mathcal{E}_{r,m}$. Fixing such an $r$ from now on, one can further conclude from Lemma [Lemma 14](#lem:few_large_plates){reference-type="ref" reference="lem:few_large_plates"} that there are at most $\sim R^{\frac{N\eta}{2}}$ different $H\in \mathcal{H}_r$. Therefore, choosing $\eta \ll \frac{1}{N}$ (depending on $\varepsilon$), one can assume that there is a fixed $(r,m)$-plate $H\in \mathcal{H}_r$ such that $$\int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x)\lessapprox \int | f_{\lambda,r,H} (x) |^2 d\mu_2(x),$$where $$f_{\lambda, r,H}=\sum_{T\in \mathbb{W}_{\lambda,r,H}}f_T,\quad \mathbb{W}_{\lambda,r,H}:= \{ T\in \mathbb{W}_{\lambda,r}: 2T \subset H \}.$$ By dyadic pigeonholing again, one is able to find $M>0$ such that the right hand side of the inequality above is further dominated by $$\lessapprox\int_{Y_M} | f_{\lambda,r,H} (x) |^2 d\mu_2(x),$$where the region $Y_M=\bigcup_{q\in \mathcal{Q}_M}q$ is contained in the unit ball and $$\mathcal{Q}_M:=\{ R^{-1/2}\textrm{-cube } q: q \textrm{ intersects } \sim M \textrm{ tubes } T \in \mathbb{W}_{\lambda, r, H} \}.$$ Fix $p\in [p_d, p_m]$ and write $f=f_{\lambda,r,H}$, $Y=Y_M$, and $\mathbb{W}=\mathbb{W}_{\lambda, r, H}$ for the sake of brevity. We first claim that there holds the following $L^p$ estimate: $$\label{eqn: L2mu} \begin{split}&\|f\|_{L^p(Y;\,\mu_2)}\\ \lessapprox &r^{\left(\frac{1}{p}-\frac{1}{p_m}\right)(d-\alpha)}(r^2R)^{\frac{d-1}{4}-\frac{d+1}{2p}}R^{\frac{d-\alpha}{p}}\left(\frac{M}{|\mathbb{W}|}\right)^{\frac{1}{2}-\frac{1}{p}}\left(\sum_{T\in \mathbb{W}}\|f_T\|^2_{L^p}\right)^{1/2}. \end{split}$$ This is essentially a rescaled version of our new decoupling estimate, more precisely, Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(c). We will leave the justification of it to the end of the proof and proceed now assuming it holds true. Since $f$ only involves good wave packets, by considering the quantity $$\sum_{q\in \mathcal{Q}_M} \sum_{T\in \mathbb{W}: T\cap q\neq \emptyset} \mu_2(q),$$ one obtains that $$\label{eqn: count} M \mu_2 (Y) \lesssim | \mathbb{W} | \frac{R^{(-\frac{1}{2}+\beta)(\alpha-2\varepsilon)}}{r^{\alpha-(m-1)}}.$$ The quantity on the right hand side above follows from the threshold defining good wave packets. Hence, combining ([\[eqn: L2mu\]](#eqn: L2mu){reference-type="ref" reference="eqn: L2mu"}) and Hölder's inequality, one has that $$\begin{split} &\int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x)\lessapprox \int_Y |f(x)|^2\,d\mu_2(x)\\ \leq & \left(\int_Y |f(x)|^p\,d\mu_2(x)\right)^{\frac{2}{p}}\mu_2(Y)^{1-\frac{2}{p}}\\ \lessapprox & r^{2\left(\frac{1}{p}-\frac{1}{p_m}\right)(d-\alpha)}(r^2R)^{\frac{d-1}{2}-\frac{d+1}{p}}R^{\frac{2(d-\alpha)}{p}}\left(\frac{R^{(-\frac{1}{2}+\beta)(\alpha-2\varepsilon)}}{r^{\alpha-(m-1)}} \right)^{1-\frac{2}{p}}\sum_{T\in \mathbb{W}}\|f_T\|^2_{L^p}\\ =& r^{2\left(\frac{1}{p}-\frac{1}{p_m}\right)(d-\alpha)+d-1-\frac{2(d+1)}{p}-\left(1-\frac{2}{p}\right)(\alpha-m+1)}R^{(d-1-\alpha) (\frac{1}{2}+\frac{1}{p})+O_{\alpha}(\beta)+\varepsilon}\sum_{T\in \mathbb{W}}\|f_T\|^2_{L^p}. \end{split}$$From standard computation (see for instance [@du2021improved Proof of Lemma 4.1]), one has the simple bound $$\begin{split} \|f_T\|_{L^p}\lesssim & \|f_T\|_{L^\infty}|T|^{1/p}\lesssim \sigma_R(\theta(T))^{1/2}|T|^{1/p} \|\widehat{M_T\mu_1}\|_{L^2(d\sigma_R)}\\ = & R^{-(\frac{1}{2p}+\frac{1}{4})(d-1)+O_\alpha(\beta)}\|\widehat{M_T\mu_1}\|_{L^2(d\sigma_R)}. \end{split}$$Plugging this back into the above and applying orthogonality, one has that $$\begin{split} &\int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x)\\ \lessapprox & r^{2\left(\frac{1}{p}-\frac{1}{p_m}\right)(d-\alpha)+d-1-\frac{2(d+1)}{p}-\left(1-\frac{2}{p}\right)(\alpha-m+1)}\cdot \\ &\qquad\qquad\qquad R^{-\alpha (\frac{1}{2}+\frac{1}{p})+O_{\alpha}(\beta)+\varepsilon} \sum_{T\in \mathbb{W}} \|\widehat{M_T\mu_1}\|^2_{L^2(d\sigma_R)}\\ \lesssim & r^{\frac{2(d-\alpha)}{m+1}+m-2-\frac{2m}{p}}R^{-\alpha (\frac{1}{2}+\frac{1}{p})+O_{\alpha}(\beta)+\varepsilon} R^{-(d-1)}\int |\hat\mu_1|^2\psi_R\, d\xi. \end{split}$$ Thus, for sufficiently small $\beta, \varepsilon$ depending on $\alpha$ and $\epsilon$, $$\begin{split} &\int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x) \\ \lesssim & r^{\frac{2(d-\alpha)}{m+1}+m-2-\frac{2m}{p}}R^{-\alpha (\frac{1}{2}+\frac{1}{p})+\epsilon} R^{-(d-1)}\int |\hat\mu_1|^2\psi_R\, d\xi. \end{split}$$ Since $r$ can be anything ranging between $R^{-\frac{1}{2}+\beta}$ and $1$, one needs to maximize the right hand side of the above estimate over all possible values of $r$. To do this, we first rewrite $r=R^{-\gamma}$, where $0\leq \gamma\leq \frac{1}{2}$. Then, the estimate above becomes $$\label{eqn: L2gamma} \begin{split} &\int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x)\\ \lessapprox & R^{-\gamma\left(\frac{2(d-\alpha)}{m+1}+m-2-\frac{2m}{p}\right)-\alpha (\frac{1}{2}+\frac{1}{p})+\epsilon} R^{-(d-1)}\int |\hat\mu_1|^2\psi_R\, d\xi\\ =&R^{-\gamma\left(\frac{2(d-\alpha)}{m+1}+m-2\right)-\frac{\alpha}{2}+\epsilon}R^{\frac{2m\gamma-\alpha}{p}}R^{-(d-1)}\int |\hat\mu_1|^2\psi_R\, d\xi. \end{split}$$ First, we look at the case $d\geq 4$. Note that one can assume without loss of generality that $\frac{d}{2}<\alpha<\frac{d}{2}+\frac{1}{3}$, hence $m=\frac{d+1}{2}$ if $d$ is odd, and $m=\frac{d}{2}+1$ if $d$ is even. A quick calculation then shows that $\frac{2(d-\alpha)}{m+1}+m-2-\frac{2m}{p}>0$ for all $p\in [p_d, p_m]$. Therefore, the estimate is maximized at $\gamma=0$, becoming $$\begin{split} &\int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x)\\ \lessapprox & R^{-\alpha(\frac{1}{2}+\frac{1}{p})+\epsilon} R^{-(d-1)}\int |\hat\mu_1|^2\psi_R\, d\xi. \end{split}$$This bound can then be minimized at $p=p_d$, hence one concludes that $$\label{eqn: high dim} \int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x) \lessapprox R^{-\frac{d\alpha}{d+1}+\epsilon} R^{-(d-1)}\int |\hat\mu_1|^2\psi_R\, d\xi.$$ The computation in the case $d=3$ ($m=2$) is a bit more complicated. Observing ([\[eqn: L2gamma\]](#eqn: L2gamma){reference-type="ref" reference="eqn: L2gamma"}), one sees that in the case $\frac{\alpha}{4}\leq \gamma\leq \frac{1}{2}$, the optimal choice of $p$ is $p=p_m=6$, which gives $$\int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x) \lessapprox R^{-\gamma(\frac{4}{3}-\frac{2\alpha}{3})-\frac{2\alpha}{3}+\epsilon} R^{-2}\int |\hat\mu_1|^2\psi_R\, d\xi.$$This bound is the worst when $\gamma=\frac{\alpha}{4}$, at which it becomes $$\label{eqn: 3 dim} \int |\mu_{1,g}*\hat\sigma_R(x)|^2\,d\mu_2(x) \lessapprox R^{\frac{\alpha^2}{6}-\alpha+\epsilon} R^{-2}\int |\hat\mu_1|^2\psi_R\, d\xi.$$In the case $0\leq \gamma\leq \frac{\alpha}{4}$, ([\[eqn: L2gamma\]](#eqn: L2gamma){reference-type="ref" reference="eqn: L2gamma"}) is optimized at $p=p_d=4$, which is then maximized at $\gamma=\frac{\alpha}{4}$, producing the same bound as ([\[eqn: 3 dim\]](#eqn: 3 dim){reference-type="ref" reference="eqn: 3 dim"}). In sum, one concludes that the desired estimate ([\[eqn: good L2\]](#eqn: good L2){reference-type="ref" reference="eqn: good L2"}) holds true in all dimensions $d\geq 3$. We are left with the justification of estimate ([\[eqn: L2mu\]](#eqn: L2mu){reference-type="ref" reference="eqn: L2mu"}). First, note that $f$ has Fourier support in the $1$-neighborhood of the sphere of radius $R$, one has that $$\|f\|_{L^p(Y;\, \mu_2)}^p\lesssim \|f\|_{L^p(Y;\, \mu_2\ast \eta_{\frac{1}{R}})}^p=R^{-d}\int_{RY}|f(R^{-1}y)|^p \mu_2\ast \eta_{\frac{1}{R}}(R^{-1}y)\,dy,$$where $\eta_{\frac{1}{R}}$ is a bump function of integral $1$ that is essentially supported on the ball of radius $\frac{1}{R}$. The above quantity can be rewritten as $$\sim R^{d-\alpha}R^{-d}\|F\|^p_{L^p(RY;\, Hdy)}\,,$$ where $F(y):=f(R^{-1}y)$ and weight function $H(y):=c_1 R^{\alpha-d}\mu_2\ast \eta_{\frac{1}{R}}(R^{-1}y)$. It is easy to see that Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(c) applies. Indeed, $F$ obviously satisfies all the required conditions. From the observation that $\|\mu_2\ast \eta_{\frac{1}{R}}\|_{L^\infty}\lesssim R^{d-\alpha}$, one can choose a constant $c_1$ such that $H\leq 1$. Moreover, for any ball $Q'$ of radius $\frac{1}{r}$, the $\alpha$-dimensional condition of $\mu_2$ and the fact that $\int \eta_{\frac{1}{R}}=1$ imply that $\int_{Q'} H(y)\,dy\lesssim r^{-\alpha}$, (in fact, we have this ball condition for any radius $t>0$, but we only need this for $t=1/r$ in order to apply Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(c)). Therefore, Theorem [Theorem 2](#thm-RD-lp){reference-type="ref" reference="thm-RD-lp"}(c) yields that $$\begin{split} &\|f\|_{L^p(Y;\mu_2)}\\ \lessapprox &R^{\frac{d-\alpha}{p}}R^{-\frac{d}{p}} r^{(d-\alpha)(\frac 1p -\frac{1}{p_m})} (r^2 R)^{\frac{d-1}{4}-\frac{d+1}{2p}} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|F_T\|_{L^p}^p\right)^{\frac 1p}\\ =&R^{\frac{d-\alpha}{p}} r^{(d-\alpha)(\frac 1p -\frac{1}{p_m})} (r^2 R)^{\frac{d-1}{4}-\frac{d+1}{2p}} M^{\frac 12 -\frac 1p} \left(\sum_{T\in \mathbb W}\|f_T\|_{L^p}^p\right)^{\frac 1p}, \end{split}$$where $F_T(y):=f_T(R^{-1}y)$. Then, recalling that we have pigeonholed at the beginning to reduce to the case that all $\|f_T\|_{L^p}$ are roughly constant, one thus concludes that the right hand side in the above, up to a constant, coincides with the right hand side of the desired inequality ([\[eqn: L2mu\]](#eqn: L2mu){reference-type="ref" reference="eqn: L2mu"}). The proof is complete. ◻ Xiumin Du, Northwestern University, *xdu\@northwestern.edu*\ Yumeng Ou, University of Pennsylvania, *yumengou\@sas.upenn.edu*\ Kevin Ren, Princeton University, *kevinren\@princeton.edu*\ Ruixiang Zhang, UC Berkeley, *ruixiang\@berkeley.edu*

arxiv_math

--- abstract: | We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$, where $G_1$ is a $K$-linear group and $G_2$ is a $\mathbf{k}$-linear group. The analysis is carried out by studying the interaction of $G$ with four distinguished sorts: the valued field $K$, the residue field $\mathbf{k}$, the value group $\Gamma$, and the closed $0$-balls $K/\mathcal{O}$. address: - | Department of Mathematics\ University of Haifa\ 199 Abba Khoushy Avenue\ Haifa\ Israel - Department of Mathematics, Ben Gurion University of the Negev, Be'er-Sheva 84105, Israel - Department of Mathematics, University of Haifa, Haifa, Israel author: - Yatir Halevi - Assaf Hasson - Ya'acov Peterzil bibliography: - harvard.bib date: - - June 2022 title: Semisimple groups interpretable in various valued fields --- [^1] # Introduction We continue the study of groups interpretable in three classes of tame valued fields: $p$-adically closed fields (and their analytic expansions), power bounded $T$-convex expansions of o-minimal real closed fields, and $V$-minimal expansions of algebraically closed valued fields of equi-characteristic $0$. The tameness conditions in each of these classes have significant geometric implications on definable sets. For example, they imply a well behaved notion of dimension, generic differentiability of definable functions $f: K^n\to K$ with corresponding versions of Taylor's approximation theorem, and more (see, e.g., [@hensel-min]). For definable groups, expanding on Pillay's work in the o-minimal context [@Pi5] (and see also [@PilQp]), this gives rise to a rudimentary Lie theory ([@AcHa]). A group $G$ is *interpretable* in a structure ${\mathcal K}$ if its universe is the quotient of a definable set by a definable equivalence relation and multiplication is part of the induced structure. The powerful geometric tools described above are not directly available for the study of interpretable groups. Our general program aims, therefore, to exploit those tools (as well as tameness of the value group $\Gamma$, and the residue field $\textbf{k}$) to give structure theorems for interpretable groups using groups that are better understood by virtue of being definable in a small collection of well studied sorts. In our previous works, [@HaHaPeGps] and [@HaHaPeVF], we showed that any group $G$ interpretbale in ${\mathcal K}$ has \"infinitesimal\" type-definable subgroups definably isomorphic to groups that are (type)-definable in one of the four *distinguished* sorts: the valued field sort $K$, the value group, the residue field (when infinite) and the sort of closed $0$-balls $K/{\mathcal O}$. Our strategy here is to understand interpretable groups using these type-definable groups and their construction. In [@HaHaPeVF] we used this analysis to describe all interpretable fields in these family of structures. Here we use it to study *definably semisimple groups*, namely groups which contain no infinite definable normal abelian subgroups. Our main theorem (Theorem [Theorem 79](#T: main){reference-type="ref" reference="T: main"} below) is: **Theorem 1**. *Let ${\mathcal K}$ be either a power bounded $T$-convex field, a $V$-minimal field or a $p$-adically closed field. Let $G$ be an interpretable definably semisimple group in ${\mathcal K}$. Then there exists a finite normal subgroup $N\trianglelefteq G$ and two normal subgroups $H_1,H_2\trianglelefteq G/N$, such that* 1. *$H_1\cap H_2=\{e\}$, $H_1$ and $H_2$ centralize each other and $H_1\cdot H_2$ has finite index in $G/N$.* 2. *$H_1$ is definably isomorphic to a subgroup of $\mathop{\mathrm{GL}}_n(\textbf{k})$.* 3. *$H_2$ is definably semisimple and definably isomorphic to a subgroup of $\mathop{\mathrm{GL}}_n(K)$.* It may be worth pointing out, with regard to the formulation of the above theorem, that in our setting, definable semisimplicity is preserved under finite quotients (Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"}). we make use of this several times in the proof of the theorem. We have been informed by J. Gismatullin, I. Halupczok and D. Macpherson that in a recent unpublished work [@GisHalMac] they characterise simple groups *definable* in certain Henselian valued fields of characteristic $0$ (covering the classes of fields discussed in the present paper). Their work seems to combine with the present one to characterise definably simple groups interpretable in our settings. Our proof goes through a case by case reduction to one of the four distinguished sorts. This is based on [@HaHaPeGps], where we showed that after modding out by a finite subgroup, $G$ is *locally strongly internal* to one of the distinguished sorts $D$, namely there exists an infinite definable set $X\subseteq G$ and a definable injection $f:X\to D^k$, for some $k$. The main obstacle is to eliminate the cases when $D=\Gamma,K/{\mathcal O}$. In Proposition [Proposition 55](#P: Gamma){reference-type="ref" reference="P: Gamma"} we show that if $G$ is locally strongly internal to $\Gamma$ then it contains a definable normal finite index subgroup whose center is infinite, which prohibits $G$ from being definably semisimple. A more intricate result, Proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"}, allows us to conclude that a definably semisimple group $G$ cannot be locally strongly internal to $K/{\mathcal O}$. When $G$ is locally strongly internal to $K$ we use local differentiability of definable functions with respect to $K$, and basic Lie theory over $K$, to associate to $G$ an adjoint representation over $K$. When $D=\textbf{k}$ we either use similar methods, in the $T$-convex case, or use the theory of groups of finite Morley rank, in the V-minimal case, to complete the proof. Though the statement of Theorem [Theorem 1](#T: intro){reference-type="ref" reference="T: intro"} and some of the auxiliary results often hold in all settings regardless of whether ${\mathcal K}$ is $p$-adically closed, power bounded $T$-convex or V-minimal, some of the proofs depend on the specific context. E.g., o-minimality of the value group plays a crucial in our analysis of $\Gamma$-groups in the V-minimal and power bounded $T$-convex setting, and a rather different analysis -- albeit with a similar conclusion -- is needed for the $p$-adic case. **Remark 1**. We note that a-priori the notion of definable semisimplicity (more precisely, the existence of an infinite definable normal abelian subgroup) need not be elementary. Indeed, while the valued field sort in our settings is a geometric structure, so in particular has uniform finiteness (sometimes called "elimination of $\exists^\infty$") for definable families of subsets of $K^n$, the same might not be true in ${\mathcal K}^{eq}$. Johnson, [@JohCminimalexist], shows, in the V-minimal case, that ${\mathcal K}^{eq}$ does eliminate $\exists^\infty$ and using his methods we show the same for power bounded $T$-convex structures (see Section [Proposition 87](#P:exists-infty V-min or T-conv){reference-type="ref" reference="P:exists-infty V-min or T-conv"}). However, in the $p$-adically closed case this fails in ${\mathcal K}^{eq}$, as neither $\Gamma$ nor $K/{\mathcal O}$ have uniform finiteness. Nevertheless, one of the consequences of the present work is that definable semisimplicity is indeed an elementary property in all cases. **Remark 2**. In the power bounded $T$-convex case, our work makes use of results from James Tyne's PhD thesis, [@tynephd], which as far as we know, have not been published elsewhere. These results, together with the work of van den Dries, [@vdDries-Tconvex], imply that every definable subset of $K$ is a boolean combination of balls and intervals (first proven by Holly, [@holly], for real closed valued fields). In order to make the results available in print, we include in the appendix direct proofs. We note recent work on interpretable groups in $p$-adically closed fields, by Johnson, [@JohnTopQp], also together with Yao, [@JohnYao], [@JohnYaoAbelian], and with Guerrero, [@JohnGue]. Further work is needed in order to understand the relation between our methods and the model theoretic tools studied there, such as definable compactness, finitely satisfiable generics (fsg), definable $f$-generics (dfg), etc. *Acknowledgement* We would like to thank J. Gismatullin, I. Halupczok and D. Macpherson for sharing with us their unpublished work on simple groups definable in certain henselian fields. We also thank D. Macpherson for several conversations and useful suggestions, and E. Sayag for directing us to some useful references. # Preliminaries and Notation We set up some notation and terminology, and review some of the basic facts concerning the main objects of interest in the present paper. Throughout, structures are denoted by calligraphic capital letters, ${\mathcal M}$, ${\mathcal N}$, ${\mathcal K}$ etc., and their respective universes by the corresponding Latin letters, $M$, $N$ and $K$. Tuples from a structure ${\mathcal M}$ are always assumed to be finite, and are denoted by small Roman characters $a,b,c,\dots$. We apply the standard model theoretic abuse of notation writing $a\in M$ for $a\in M^{|a|}$. Variables will be denoted $x,y,z,\dots$ with the same conventions as above. We do not distinguish notationally between tuples and variables belonging to different sort, unless some ambiguity can arise. Capital Roman letters $A,B,C,\ldots$ usually denote small subsets of parameters from ${\mathcal M}$. As is standard in model theory, we write $Ab$ as a shorthand for $A\cup \{b\}$. In the context of definable groups we will, whenever confusion can arise, distinguish between, e.g., $Agh:=A\cup\{g,h\}$ and $A\,g\!\cdot \! h:=A\cup \{g\!\cdot\! h\}$. By a partial type we mean a consistent collection of formulas. Two partial types $\rho_1, \rho_2$ are equal, denoted $\rho_1=\rho_2$, if they are logically equivalent, i.e., if they have the same realizations in some sufficiently saturated elementary extension. All the definable sets we shall consider here have finite dp-rank, whose properties (such as sub-additivity, invariance under finite-to-finite correspondences, invariance under automorphisms etc.) we use freely. See the preliminaries sections of [@HaHaPeVF],[@HaHaPeGps] for a more detailed discussion. ## Valued fields Throughout ${\mathcal K}$ denotes an expansion of a valued field of characteristic $0$ in a language ${\mathcal L}$ expanding the language of valued rings. We assume ${\mathcal K}$ to be $(|{\mathcal L}|+2^{\aleph_0})^+$-saturated. Unless specifically written otherwise, we will always work in ${\mathcal K}^{\mathop{\mathrm{eq}}}$. **Henceforth, by "definable" we mean "definable in ${\mathcal K}^{\mathop{\mathrm{eq}}}$ using parameters", unless specifically mentioned otherwise**. In particular, we shall not use "interpretable" anymore. A more detailed review of standard definitions and notation can be found in [@HaHaPeGps §2]. For any valued field $(K,v)$, we let ${\mathcal O}$ denote its valuation ring, $\textbf{m}$ its maximal ideal and $\textbf{k}:={\mathcal O}/\textbf{m}$ the residue field. The value group is denoted $\Gamma$. In case of possible ambiguity, we may, for the sake of clarity, add a subscript (e.g., ${\mathcal O}_K$) to the above notation. A closed ball in $K$ is a set of the form $B_{\geq \gamma}(a):=\{x\in K: v(x-a)\geq \gamma\}$ and similarly $B_{>\gamma}(a)$ denotes the open ball of (valuative) radius $\gamma$ around $a$. We will use the fact that $v$ descends naturally to $K/{\mathcal O}\setminus \{0\}$ (by $v(a+{\mathcal O}):=v(a)$ for any $a\notin {\mathcal O}$), and use the same notation $B_{>\gamma}(x)$ and $B_{\ge \gamma}(x)$ for $x\in K/{\mathcal O}$ in the obvious way. We will, however, reserve **the term "ball" in $K/{\mathcal O}$, when ${\mathcal K}$ is $p$-adically closed, only to such sets where $\gamma<{\mathbb {Z}}$**. For $a=(a_1,\dots,a_n)\in K$ (or in $(K/{\mathcal O})^n$) we set $v(a)=\min_i\{v(a_i)\}$. A ball in $K^n$ (or in $(K/{\mathcal O})^n$) is an $n$-fold product of $K$-balls (or $(K/{\mathcal O})$-balls) of **equal radii**. When ${\mathcal K}$ is $p$-adically closed, it is elementarily equivalent to some finite extension $\mathbb{F}$ of $\mathbb{Q}_p$. By saturation, we may assume that $(K,v)$ is an elementary extension of $(\mathbb{F},v)$. Since its value group $\Gamma_{\mathbb{F}}$ is isomorphic to $\mathbb{Z}$, as ordered abelian groups, we identify $\Gamma_{\mathbb{F}}$ with $\mathbb{Z}$ and view it as a prime (and minimal) model for $\Gamma$. We denote ${\mathbb {Z}}_{Pres}$ the structure $({\mathbb {Z}}, +, <)$. ## The setting Unless otherwise stated, ${\mathcal K}$ is a saturated expansion of a valued field of one of three types (see [@HaHaPeGps] for definitions and more details): - A $V$-minimal expansion of an algebraically closed valued field of residue characteristic $0$. - A $T$-convex expansion of a real closed valued field, for an o-minimal power bounded theory $T$. - A $p$-adically closed field. **Remark 3**. Our proof for the $p$-adically closed case works, as written, in the context of $P$-minimal 1-h-minimal fields with definable Skolem functions in the valued field sort. These include models of the theory of $\mathbb Q_p^{an}$, the expansion of $\mathbb Q_p$ (or a finite extension thereof) by all convergent power series $f: {\mathcal O}^n\to \mathbb Q_p$ (any $n$). For the sake of clarity of exposition, we stick to the $p$-adically closed case. There are important similarities between the three setting. E.g., in all cases the structure ${\mathcal K}$ is dp-minimal, namely $\mathrm{dp\text{-}rk}({\mathcal K})=1$, so definable sets in ${\mathcal K}^{eq}$ have finite dp-rank. Also, in all cases the valued field sort is a geometric structure, carrying, moreover, the structure of an SW-uniformity. The latter introduced (without the name) by Simon and Walsberg, [@SimWal]. For our purposes it will suffice to know that a dp-minimal expansion of a topological group $G$ is an SW-uniformity if it has a definable basis for the topology, has no isolated points and every infinite definable subset has non-empty interior. However, there are also obvious differences. For example, the residue field is stable in the $V$-minimal case, o-minimal in the $T$-convex case and finite in the $p$-adic case. Thus, while the main theorems can be stated uniformly in all settings, some of the proofs will require us to specialize to the particular cases. ## The distinguished sorts As in our previous work, the analysis of definable quotients is carried out via a reduction to four *distinguished sorts*, $K, \Gamma, \textbf{k}$ and $K/{\mathcal O}$. They are all dp-minimal, except the finite $\textbf{k}$ in the $p$-adic case. Note that in all cases the sorts $K$, $\Gamma$ and $K/{\mathcal O}$ are partially ordered and therefore unstable. However, the residue field sort is unstable only in the $T$-convex case (in the $V$-minimal case it is a pure algebraically closed field, and in the $p$-adic case it is finite). Thus, when proofs mention the "unstable sorts" they refer to the distinguished sorts in all three cases except for $\textbf{k}$ in the $V$-minimal and $p$-adically closed settings. As noted above, in all settings the sort $K$ is an SW-uniformity, as is $\Gamma$ in the $V$-minimal and $T$-convex cases (it is in fact an ordered vector spaces so o-minimal) and $K/{\mathcal O}$ in the $T$-convex setting (it is weakly o-minimal). However, in all cases $K/{\mathcal O}$ is neither a geometric structure ($\mathop{\mathrm{acl}}(\cdot)$ in $K/{\mathcal O}$ does not satisfy the Steinitz Exchange Principle) nor is it stably embedded, leading to certain complications in some proofs. **Remark 4**. In [@HaHaPeGps §3] we study the structure of $K/{\mathcal O}$ in $p$-adically closed fields. In this context, it was helpful to work in a saturated model, expanding the language by constants for all elements of (a copy of) $\mathbb{F}$. Although the saturation assumption on ${\mathcal K}$ plays an important role in many of our proofs here, the main theorems of the present paper do not assume saturation. Thus, a copy of $\mathbb F$ cannot be expected to exist in all our models (let alone be named). Whenever needed, as part of the proof, we bridge this gap in the assumptions. ## Some specialised terminology We remind some terminology from [@HaHaPeGps] that is used throughout the paper: Assume that $S$ is definable in ${\mathcal K}$ and $D$ is one of the distinguished sorts. We say that $S$ is *locally almost strongly internal to $D$* if **in a sufficiently saturated elementary extension** there is a definable infinite set $X\subseteq S$ and a definable $m$-to-one map $f: X\to D^n$, for some $m,n\in {\mathbb{N}}$. The set $X$ is then called *almost strongly internal to $D$*. If we can find a definable injection $f:X\to D^n$ then $S$ is *locally strongly internal to $D$* and $X$ is *strongly internal to $D$*. We add "over $A$" to all the notions above if $S,X$ and the map $f$ are defined over a parameter set $A$. The starting point of our analysis is the following ([@HaHaPeGps Lemma 7.3, Lemma 7.6, Lemma 7.10]): **Fact 5**. *Every definable infinite set $S$ in ${\mathcal K}$ is locally almost strongly internal to $K$, $\textbf{k}$, $\Gamma$ or $K/{\mathcal O}$.* A *$D$-critical subset of $S$* is a definable $X\subseteq S$ of maximal dp-rank that is strongly internal to $D$. The *$D$-rank[^2]* of $S$ is the dp-rank of any $D$-critical $X\subseteq S$. *Almost $D$-critical sets* (and the corresponding almost $D$-rank) are defined by replacing "strongly internal to $D$" with "almost strongly internal to $D$". Clearly, the $D$-rank of $S$ is smaller or equal to its almost $D$-rank. We say that $S$ is *$D$-pure* if it is locally almost strongly internal to $D$ but not to any other distinguished sort. **Definition 6**. Let $X$ be an $A$-definable set in ${\mathcal K}$, $a\in X$ and $B\supseteq A$ a set of parameters. 1. The point $a$ is $B$-*generic* in $X$ (or, *generic in $X$ over $B$*) if $\mathrm{dp\text{-}rk}(a/B)=\mathrm{dp\text{-}rk}(X)$. 2. For an $A$-generic $a\in X$, a set $U\subseteq X$ is *a $B$-generic vicinity of $a$ in $X$* if $a\in U$, $U$ is $B$-definable, and $\mathrm{dp\text{-}rk}(a/B)=\mathrm{dp\text{-}rk}(X)$ (in particular, $\mathrm{dp\text{-}rk}(U)=\mathrm{dp\text{-}rk}(X)$). In order to overcome the failure of additivity of dp-rank, we introduced in [@HaHaPeGps] the notion of a $D$-group. In the present paper this notion can be used as a black box allowing us to seamlessly refer to results from [@HaHaPeGps]. However, for the sake of completeness, we give the definition: For $D$ one of the unstable distinguished sorts, an $A$-definable group $G$ is a *a $D$-group* if it locally strongly internal to $D$ and for every $X_1, X_2\subseteq G$ strongly internal to $D$, with $X_2$ $D$-critical, both defined over some $B\supseteq A$, and for every $(g,h)$ $B$-generic in $X_1\times X_2$, we have $$\mathrm{dp\text{-}rk}(g/B,g\cdot h)=\mathrm{dp\text{-}rk}(g/B).$$ We stress that, by definition, the notion of a $D$-group refers only to unstable $D$, namely all infinite sorts in our setting except $\textbf{k}$ in the $V$-minimal case. The following fact shows that a group $G$ almost strongly internal to an unstable sort $D$ is close to being a $D$-group. **Fact 7**. *[@HaHaPeGps Fact 4.25, Proposition 4.35][\[F: existence of finite normla to get D-group\]]{#F: existence of finite normla to get D-group label="F: existence of finite normla to get D-group"} Let $G$ be an infinite $A$-definable group in ${\mathcal K}$ locally almost strongly internal to an unstable distinguished sort $D$. Then there is an $A$-definable finite normal abelian subgroup $H\trianglelefteq G$ such that $G/H$ is a $D$-group. Moreover,* 1. *The almost $D$-rank and the $D$-rank of $G/H$ are equal (and equal to the almost $D$-rank of $G$).* 2. *$H$ is invariant under any definable automorphism of $G$ and is contained in any definable finite index subgroup of $G$.* Recall that every definable group in ${\mathcal K}$ is almost locally strongly internal to one of the distinguished sorts, hence the above fact applies whenever that sort is unstable. ## Vicinities and infinitesimal subgroups {#ss:infint and vicin} For $D$ a distinguished sort (not necessarily satisfying Exchange) and $a\in D^n$, we let $\dim_{\mathop{\mathrm{acl}}}(a)$ be the maximal length of a sub-tuple $a'\subseteq a$ such that no $a_i\in a'$ is in the $\mathop{\mathrm{acl}}$-closure of the rest of $a'$. We recall the following from [@HaHaPeGps]: **Definition 8**. A dp-minimal set $D$ is *vicinic* if it satisfies the following axioms: 1. $\dim_{\mathop{\mathrm{acl}}}=\mathrm{dp\text{-}rk}$; i.e. for any tuple $a\in D^n$ and set $A$, $\dim_{\mathop{\mathrm{acl}}}(a/A)=\mathrm{dp\text{-}rk}(a/A)$. 2. For any sets of parameters $A$ and $B$, for every $A$-generic elements $b\in D^n$, $c\in D^m$ and any $B$-generic vicinity $X$ of $b$ in $D^n$, there exists $C\supseteq A$ and a $C$-generic vicinity of $b$ in $X$ such that $\mathrm{dp\text{-}rk}(b,c/A)=\mathrm{dp\text{-}rk}(b,c/C)$. By [@HaHaPeGps Fact 4.6], all the unstable distinguished sorts in our settings are vicinic. Throughout this subsection, unless specifically stated otherwise, we let $D$ be one of them. Given a definable $D$-group $G$ in ${\mathcal K}$ the main technical result of [@HaHaPeGps] is the construction of the infinitesimal type-definable subgroup $\nu_D$. To achieve this, we introduce the notion of $D$-sets (in $G$). For completeness, we remind the somewhat technical definition. The fine details of the definition are unimportant for us here: **Definition 9**. [@HaHaPeGps Definition 4.16 and Remark 4.12] A definable set $X\subseteq G$ is a *$D$-set over $A$* if it is $D$-critical, witnessed by some $A$-definable function $f:X\to D^m$ and there exists a coordinate projection $\pi:f(X)\to D^n$, with $n=\mathrm{dp\text{-}rk}(X)$, such that for every $B\supseteq A$ and $B$-generic $a\in f(X)$, all elements of $\pi^{-1}(\pi(f(a)))$ have the same type over $B\pi(f(a))$. **Remark 10**. 1. If $G$ is a definable group locally strongly internal to $D$ then it always contains a $D$-set. See [@HaHaPeGps Remark 4.18]. 2. Note the following special case: if $X$ is $D$-critical, $f: X\to D^n$ a definable injection witnessing it, and [$n=\mathrm{dp\text{-}rk}(X)$]{.ul} then $X$ is a $D$-set. As we shall see, such an $X$ can always be found when $G$ is locally strongly internal to $D$. If $D$ is an SW-uniformity this follows from [@SimWal Proposition 4.6] and in the $p$-adically closed this follows from Lemma [Lemma 30](#L:local homeo K/O){reference-type="ref" reference="L:local homeo K/O"} when $D=K/{\mathcal O}$ and cell decomposition when $D=\Gamma$. See Lemma [Lemma 33](#L: Dsets){reference-type="ref" reference="L: Dsets"} for more information. **Definition 11**. Let $G$ be a $D$-group, $Z\subseteq G$ a $D$-set over $A$ and $d\in Z$ an $A$-generic point. The *infinitesimal vicinity of $d$ in $Z$*, denoted $\nu_Z(d)$, is the partial type consisting of all $B$-generic vicinities of $d$ in $Z$, as $B$ varies over all small parameter subsets of ${\mathcal K}$. By [@HaHaPeGps Lemma 4.20], the type $\nu_Z(d)$ is a filter-base, namely the intersection of any two generic vicinities of $d$ contains another. It follows that $\mathrm{dp\text{-}rk}(\nu_Z(d))$ equals the $D$-rank of $G$. We think of $\nu_Z(d)$ (and the type definable group $\nu_D$ defined below) both as a collection of formulas over ${\mathcal K}$ and a set of realization of the partial type in some monster model extending ${\mathcal K}$. We say that two such types are equal if they are logically equivalent. For a definable set $X$ we denote $\nu_Z(d)\vdash X$ if there is $Y\in \nu_Z(d)$ such that $Y\subseteq X$. By writing $\nu_Z(d)\vdash \nu_W(d')$ we mean that for all $X\in \nu_W(d')$ we have $\nu_Z(d)\vdash X$. **Fact 12**. *[@HaHaPeGps Proposition 5.8][\[F: properties of nu\]]{#F: properties of nu label="F: properties of nu"} Let $D$ be an unstable distinguished sort and let $G$ be a $D$-group.* 1. *Assume that $X\subseteq G$ is a $D$-set over $A$, then for every $A$-generic $a,b\in X$ the set $\nu_X(a)a^{-1}$ is a (type-definable) subgroup of $G$ and $\nu_X(a)a^{-1}=\nu_X(b)b^{-1}=a^{-1}\nu_X(a)$. We denote this group $\nu_X$.* 2. *If $X,Y\subseteq G$ are $D$-sets over $A$ then $\nu_X=\nu_Y$, and we can call it $\nu_D(G)$, *the infinitesimal type-definable subgroup of $G$ with respect to $D$*.* 3. *For every $g\in G({\mathcal K})$, we have $g\nu_D(G) g^{-1}=\nu_D(G)$.* Whenever the group $G$ is understood from context and there is no ambiguity, we denote $\nu_D(G)$ by $\nu_D$. **Remark 13**. Note that if $X\subseteq G$ is a $D$-set which happens to be a subgroup, then $\nu_D\vdash X$. **Lemma 14**. *Let $H\leq G$ be two definable $D$-groups, locally strongly internal to an unstable distinguished sort $D$. Then* 1. *$\nu_D(H)\vdash \nu_D(G)$.* 2. *If $H$ and $G$ have the same $D$-rank then $\nu_D(H)=\nu_D(G)$. In particular, this holds if $H$ has finite index in $G$.* *Proof.* Let $H\leq G$ be any subgroup, as in the statement. \(1\) Let $X_G\subseteq G$ be a $D$-set in $G$ and $X_H\subseteq H$ a $D$-set in $H$, all definable over a parameter set $A$. Let $(g,h)\in X_G\times X_H$ be generic over $A$, so $\nu_D(G)=g^{-1}\nu_{X_G}(g)$ and $\nu_D(H)=h^{-1}\nu_{X_H}(h)$. Let $V$ be a generic vicinity of $g$ and $U$ a generic vicinity of $h$. By [@HaHaPeGps Lemma 4.26], $U\cap hg^{-1} V$ is a generic vicinity of $h$, hence $$\nu_D(H)\vdash h^{-1}(U\cap hg^{-1}V)=h^{-1}U\cap g^{-1}V\subseteq g^{-1}V.$$ \(2\) Assume that $H$ and $G$ have the same $D$-rank, hence any $D$-set in $H$ is automatically a $D$-set in $G$. It now follows by definition that $\nu_D(H)=\nu_D(G)$. If $H$ has finite index in $G$ then it is easy to see that they have the same $D$-rank. ◻ The next lemma supports the intuition that the type-definable coset $g\cdot \nu_D(G)$ is an infinitesimal neighbourhood of $g$, for $g$ generic in a set locally strongly internal to $D$: **Lemma 15**. *Let $G$ be a $D$-group, $X\subseteq G$ an $A$-definable set strongly internal to $D$ over $A$, and $g\in X$ generic over $A$. Then $\mathrm{dp\text{-}rk}(X\cap g\cdot \nu_D )=\mathrm{dp\text{-}rk}(X)$.* *Proof.* Let $Z'$ be any $D$-set, definable over some parameter set $B'$. Find an element $g'\equiv_A g$ such that $\mathrm{dp\text{-}rk}(g'/AB')=\mathrm{dp\text{-}rk}(g/A)$. Applying an automorphism over $A$ we can move $g'$ to $g$ and $B'$ to some $B$. The image, $Z$, of $Z'$ under this automorphism, is definable over $B$ and $\mathrm{dp\text{-}rk}(g/AB)=\mathrm{dp\text{-}rk}(g/A)$. Renaming, we assume from now on, that $A=AB$. Fix an $A$-generic $h\in Z$ with $\mathrm{dp\text{-}rk}(g,h/A)=\mathrm{dp\text{-}rk}(X)+\mathrm{dp\text{-}rk}(Z)$. Thus, as $\nu_D=h^{-1}\nu_Z(h)$, we have to show that $\mathrm{dp\text{-}rk}(X\cap gh^{-1}\nu_Z(h))=\mathrm{dp\text{-}rk}(X)$. Let $Y\subseteq Z$ be some $B$-generic vicinity of $h$ (i.e. $Y\in \nu_Z(h)$), for some $B$; so it will suffice to prove that $\mathrm{dp\text{-}rk}(X\cap gh^{-1}Y)=\mathrm{dp\text{-}rk}(X)$. By [@HaHaPeGps Lemma 4.13], there exists $C\supseteq A$ and a $C$-generic vicinity $Y'\subseteq Y$ of $h$ such that $\mathrm{dp\text{-}rk}(g,h/A)=\mathrm{dp\text{-}rk}(g,h/C)$. So $(g,h)$ is $C$-generic in $X\times Y'$. It will be sufficient to prove that $\mathrm{dp\text{-}rk}(X\cap gh^{-1}Y')=\mathrm{dp\text{-}rk}(X)$; this is exactly [@HaHaPeGps Lemma 4.26]. ◻ **Lemma 16**. *Let $G$ be a definable group in ${\mathcal K}$, $H$ a finite normal subgroup and $f:G\to G/H$ the quotient map. Let $D$ be any of the distinguished sorts except the sort $K/{\mathcal O}$ in the $p$-adically closed case.* 1. *The almost $D$-ranks of $G$ and $G/H$ are equal.* 2. *The $D$-ranks of $G$ and $G/H$ are equal.* 3. *If, furthermore, $G$ is $D$-group (so $D$ is unstable) then so is $G/H$, and then $f(\nu_{\textbf{k}}(G))=\nu_{\textbf{k}}(G/H)$.* 4. *If the $D$-critical rank and the almost $D$-critical ranks of $G$ coincide, then the same is true for $G/H$.* *Proof.* We use the fact that in all cases covered in the statement $D$ is either an SW-uniformity or, when $D=\textbf{k}$ in the $V$-minimal case, it is stably embedded and eliminates finite imaginaries. For (1) and (2) we first note that for any (almost) $D$-critical set $X\subseteq G$, there exists an (almost) $D$-critical $Y\subseteq f(X)$ (with respect to $G/H$), with $\mathrm{dp\text{-}rk}(Y)=\mathrm{dp\text{-}rk}(X)$. Indeed, if $D$ is an SW-uniformity then this is [@HaHaPeGps Lemma 2.6] and if $D=\textbf{k}$ in the $V$-minimal case then it is [@HaHaPeGps Lemma 4.3]. This implies (1) and (2). \(3\) If $G$ is a $D$-group then $G/H$ is also locally strongly internal to $D$ by (1). Combined with [@HaHaPeGps Fact 4.25] it follows that $G/H$ is also a $D$-group. To show that $f(\nu_{\textbf{k}}(G))=\nu_{\textbf{k}}(G/H)$, let $X_0\subseteq G$ be a $D$-set. By the above, we may find a $D$-critical subset $Y_0\subseteq f(X_0)$. By [@HaHaPeGps Remark 4.18] there exists a $D$-set $Y\subseteq Y_0\subseteq G/H$. Setting $X=f^{-1}(Y)\subseteq X_0$, and since $X_0$ is a $D$-set so is $X_0$. We are now in the situation where $X$ and $Y=f(X)$ are both $D$-sets, with respect to $G$ and $G/H$, respectively. Assume everything is defined over some parameters set $A$. Let $a\in X$ be an $A$-generic in $X$, so $f(a)$ is an $A$-generic in $Y$. It suffices to prove that $f(\nu_X(a))=\nu_X(f(a))$. For this first note that if $U\subseteq X$ is a $B$-generic vicinity of $a$, for some $B\supseteq A$, then $f(U)$ is a $B$-generic vicinity of $f(a)$ since $f(a)\in \mathop{\mathrm{dcl}}(Aa)$ and $\mathrm{dp\text{-}rk}(U)=\mathrm{dp\text{-}rk}(f(U))$ as $f$ is finite-to-one. To show the other direction, let $V$ be a $B$-generic vicinity of $f(a)$ for some $B\supseteq A$, then $f^{-1}(V)$ is a $B$-generic vicinity of $a$ since $a\in\mathop{\mathrm{acl}}(Af(a)$ and $f(f^{-1}(V))=V$ because $f$ is surjective. \(4\) Assume that the almost $D$-critical and the $D$-critical ranks of $G$ coincide. By applying (1) twice we get that the same is true for $G/H$. ◻ ## Basics on groups definable in NIP theories Before the next corollary, we note the following application of Baldwin-Saxl ([@PoiGroups Lemma 1.3]). **Fact 17**. *Let $G$ be a group definable in a sufficiently saturated NIP structure and $\{H_i:i\in T\}$ a definable family of finite index subgroups of $G$. Then $\bigcap_{i\in T} H_i$ is a definable subgroup of finite index.* *Proof.* By Baldwin-Saxl, there is a finite bound on the index of finite intersections of the $H_i$. ◻ **Corollary 18**. *Let $G$ be a definable group in a sufficiently saturated NIP structure, $\{\lambda_t:t\in T\}$ a family of group automorphisms of $G$, and $X\subseteq G$, all definable over a parameter set $A$. Assume that for every $a\in X$, $C_G(a)$ has finite index in $G$. Then there exists an $A$-definable subgroup $G_1\subseteq C_G(X)$ of finite index in $G$ that is invariant under $\lambda_t$, for all $t\in T$.* *Proof.* By Fact [Fact 17](#F: Baldwin saxl){reference-type="ref" reference="F: Baldwin saxl"}, $C_G(X)$ has finite index in $G$. Applying this fact again to the intersection of the family $\{\lambda_t(C_G(X)):t\in T\}$ gives the desired conclusion. ◻ We need a couple of group theoretic observations on definable groups in our setting. We note for future reference that Lemma [Lemma 19](#L:groups 1){reference-type="ref" reference="L:groups 1"} and Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"} below do not require saturation of ${\mathcal K}$. **Lemma 19**. *Let $N$ be a definable group in ${\mathcal K}$ and $H\trianglelefteq N$ a definable normal subgroup, such that $N/H$ is abelian. For $k\in \mathbb N$, let $N^k=\{g^k:g\in N\}$. Then:* 1. *For every $k\in \mathbb N$, $N^k H$ is a normal subgroup of $N$ and $N/N^kH$ is finite.* 2. *If $H$ is finite and central in $N$, and $k=|H|$ then the set $N^k$ is contained in $Z(N)$ and $Z(N)$ has finite index in $N$.* *Proof.* (1) Since $N/H$ is abelian, for every $a,b\in N$, $ab=bah$ for some $h\in H$. Because $H$ is normal, for all $g\in G$ and $h\in H$ there is $h'\in H$ such that $hg=gh'$. It follows that $a^2b^2=(ab)^2h_1$, for $h_1\in H$, and by induction, $a^kb^k=(ab)^k h_0$, for some $h_0\in H$. Thus $N^kH$ is a subgroup, clearly normal in $N$. The order of every $g\in N/N^kH$ is at most $k$, thus $N/N^kH$ has bounded exponent. The group $N/N^kH$ is clearly also definable in ${\mathcal K}$, and by [@HaHaPeGps Theorem 7.4, Theorem 7.7 and Theorem 7.11] a definable group of bounded exponent must be finite. Thus, $N/N^kH$ must be finite. \(2\) Assume now that $k=|H|$ and $H$ is central. Since $G/H$ is abelian, For every $g, x\in N$ we have $g^{-1}xg=x h$ for some $h\in H$, and hence, since $H$ is central, $g^{-1}x^k g=(xh)^k=x^kh^k=x^k$. Thus $N^k\subseteq Z(N)$. It follows that $N^kH\subseteq Z(N)$, so by (1), $Z(N)$ has finite index in $N$. ◻ The proof of the next corollary is simpler when $H$ is central, but we need the more general statement: **Corollary 20**. *Let $G$ be a definable group in ${\mathcal K}$ and $H$ a finite normal subgroup of $G$, both defined over a parameter set $A$. Let $\{\lambda_t: t\in T\}$ be a definable family of group automorphisms of $G$ fixing $H$ setwise.* *If for some $B\supseteq A$ the group $G/H$ contains a $B$-definable normal abelian subgroup of dp-rank $k$ invariant under all the $\lambda_t$ then so does $G$. In particular, if $G$ is definably semisimple, then so is $G/H$.* *Proof.* For simplicity, let us call a set invariant under all the $\lambda_t$ $\Lambda$-invariant. By Lemma [Corollary 18](#Baldwin Saxl){reference-type="ref" reference="Baldwin Saxl"}, there exists a definable $\Lambda$-invarainat $G_1\trianglelefteq G$ of finite index such that $G_1\subseteq C_G(H)$. In particular, $G_1\cap H$ is central in $G_1$. We fix such $G_1$. Assume that $G/H$ has an infinite $\Lambda$-invariant definable abelian normal subgroup of the form $N/H$ for $N\trianglelefteq G$. It follows that $N$ is $\Lambda$-invariant. Let $N_1:=N\cap G_1$, an infinite normal subgroup of $G$ of finite index in $N$ and $H_1:=H\cap N_1$, a central subgroup of $N_1$. The quotient $N_1/H_1$ is isomorphic to $N_1H/H\subseteq N/H$ so is abelian. Note that $N_1$ is also $\Lambda$-invariant. By Lemma [Lemma 19](#L:groups 1){reference-type="ref" reference="L:groups 1"} (2), $Z(N_1)$ has finite index in $N_1$ and therefore $\mathrm{dp\text{-}rk}(Z(N_1))=\mathrm{dp\text{-}rk}(N_1)=\mathrm{dp\text{-}rk}(N)=\mathrm{dp\text{-}rk}(N/H)$. Because $N_1$ is $\Lambda$-invariabnt and normal in $G$ so is $Z(N_1)$. Hence, $Z(N_1)$ is a $\Lambda$-invariant definable normal abelian subgroup of $G$ of the same rank as $N_1/H$. Clearly, if $N/H$ is $B$-definable for some $B\supseteq A$ then so are $N_1$ and $Z(N_1)$. ◻ # Definable subgroups of $((K/{\mathcal O})^n,+)$ Let ${\mathcal K}$ be one of our valued fields. The purpose of this section is to describe the definable subgroups of $(K/{\mathcal O})^n$. When ${\mathcal K}$ is either power bounded $T$-convex or $V$-minimal those turn out to be definably isomorphic to a product of balls in $K/{\mathcal O}$. In this case we can also describe all their definable endomorphisms. When ${\mathcal K}$ is $p$-adically closed, the existence of finite subgroups creates obstructions (see Example [Example 22](#E:counter to K/O p-adic){reference-type="ref" reference="E:counter to K/O p-adic"}), nonetheless we will show that definable subgroups projects injectively onto subgroups of full dp-rank. ## ${\mathcal K}$ power-bounded $T$-convex or $V$-minimal {#ss:groups in K/O} We assume that ${\mathcal K}$ is either power bounded $T$-convex or $V$-minimal. Recall that for $a\in K\setminus {\mathcal O}$, $v(a+{\mathcal O})$ is well-defined, allowing us to refer to definable balls in $K/{\mathcal O}$. Below, we use the term *trivial ball* to refer to either $K$ (or $K/{\mathcal O}$) or $\{0\}$. We start with the following basic observation. **Lemma 21**. *Every definable subgroup $G$ of $(K,+)$ is a ball, possibly trivial. As a result, every definable subgroup of $K/{\mathcal O}$ is a (possibly trivial) ball.* *Proof.* Since $\pi: K\to K/{\mathcal O}$ is a group homomorphism, and the image of a ball (centred at $0$) under $\pi$ is again a ball, it suffices to show that the claim is true for definable subgroups of $(K,+)$. So let $G$ be a subgroup of $(K,+)$. Since $(K,+)$ is torsion-free, if $G$ is finite it is trivial. So we assume $G$ is infinite. Let $B$ be the union of all sub-balls of $G$ containing $0$. If $B=K$ then $G=K$ and we are done, so assume $B\neq K$. Because $\Gamma$ is definably complete, $B$ is a ball itself, possibly $\{0\}$. Since every infinite definable subset of $K$ has an interior, and $G$ is a group $B\neq \{0\}$. We will show that $G=B$. Assume for contradiction that $G\neq B$. In our settings, $B$ is a divisible group (indeed, the maps $x\mapsto nx$ send $B$ onto itself for all non-zero $n\in {\mathbb{N}}$), and since $(K,+)$ is torsion-free, it must be that $[G:B]=\infty$. This means that $G$ contains infinitely many disjoint maximal balls, cosets of $B$. Assume that $B$ is a closed ball. By the so-called (Cballs) property introduced in [@HaHaPeVF], which holds in our settings [@HaHaPeVF Proposition 5.6, Lemma 5.10], only finitely many translates of $B$ intersect $G$, so $G$ contains only finitely many cosets of $B$, contradiction. Assume then that $B$ is open. After re-scaling $G$, we may assume that $B=\textbf{m}$. Again, by (Cballs), $G$ intersects only finitely many closed $0$-balls. Consequently, ${\mathcal O}\cap G$ is an additive subgroup of $K$ containing infinitely many cosets of $\textbf{m}$. The image of ${\mathcal O}\cap G$ is, therefore, an infinite definable subgroup of $(\textbf{k},+)$. However, under our assumptions $\textbf{k}$ has no infinite definable proper subgroups, thus $G\cap {\mathcal O}={\mathcal O}$ contradicting the maximality of the ball $B=\textbf{m}$. Thus, $G=B$, with the desired conclusion. ◻ **Example 22**. The lemma above does not hold in the $p$-adically closed case. For example, consider a finite residual extension $K$ of $\mathbb{Q}_p$. Let $H$ be a non-trivial finite proper subgroup of $(\textbf{k}_K,+)$, then $G=\{g\in K:\mathrm{res}(g)\in H\}$ is a subgroup of $K$ that is not a ball. The following computation should be well known. **Fact 23**. *Let $B_1,B_2\subseteq K/{\mathcal O}$ be balls (possibly the whole of $K/{\mathcal O}$).* 1. *The point-set product $B_1\cdot B_2$ is also a ball.* 2. *If $0\notin B_2$ then their point-set quotient $B_1\cdot (B_2)^{-1}$ is also a ball.* *Proof.* $(1)$ Let $a\in B_1$ and $b\in B_2$. Since $B_1\cdot B_2-ab=(B_1-a)(B_2-b)+a(B_2-b)+b(B_1-a)$ the latter is a sum of subgroups of $K/{\mathcal O}$ so by Lemma [Lemma 21](#L:definable subgroup of K or K/O is a ball){reference-type="ref" reference="L:definable subgroup of K or K/O is a ball"} is a ball around $0$; so $B_1\cdot B_2$ is a ball around $ab$. $(2)$ By (1) it is enough to show that if $B$ is a ball and $0\notin B$ then $B^{-1}$ is also a ball. By replacing $B$ with $a^{-1}\cdot B$ for some $a\in B$ we may assume that $1\in B$ (and still $0\notin B$). But then a direct computation shows that $B^{-1}=B$ and we are done. ◻ **Lemma 24**. *Let $I,J, H\subseteq K$ be definable subgroups, $I\subseteq H\cap J$, and let $T:H/I\to K/J$ be a definable homomorphism. Then there is $d\in K$ such that, $d\cdot I\subseteq J$ and for every $x\in H$, $T(x+I)=d\cdot (x+I)+J$.* *Proof.* Since $I,J, H$ are definable subgroups of $K$, they are balls and so are their cosets, and because $T$ is a group homomorphism, the image under $T$ of a coset of $I$ is also a coset of a subgroup, so viewed as a subset of $K$ it is a ball. Given $x\in H\setminus I$, let $$S_x=\{w/z\in K:z\in x+I \wedge w\in T(x+I)\}$$ As a quotient of two balls $S_x$ is a ball, too (note that $0\notin x+I$ so Fact [Fact 23](#F:prod-balls){reference-type="ref" reference="F:prod-balls"} applies). For $d\in K$, let $$H_d=I\cup \{x\in H\setminus I :d\in S_x\}.$$ We claim that each $H_d$ is a subgroup of $K$ (and when $I=0$, possibly a singleton). To see this, let $H_d'=\{x\in H\setminus I :d\in S_x\}$; by definition $H_d'\cap I=\emptyset$. It follows directly from the definition of $H_d'$ that if $x_1\in I$ and $x_2\in H_d'$, then $x_1\pm x_2\in H_d'$. So it remains to show that if $x_1,x_2\in H_d'$ then $x_1-x_2\in H_d$. By assumption, $d\in S_{x_1}\cap S_{x_2}$, so we can write, $d=w_1/z_1=w_2/z_2$ with $w_i\in T(x_i+I)$ and $z_i\in x_i+I$. So $d(z_1-z_2)=w_1-w_2$. If $z_1-z_2\in I$ then $x_1+I=x_2+I$ so obviously $x_1-x_2\in H_d$. Otherwise, $d=(w_1-w_2)/(z_1-z_2)$, $z_1-z_2\in x_1-x_2+I$ and $w_1-w_2\in T(x_1+I)-T(x_2+I)=T(x_1-x_2+I)$. Hence, by Lemma [Lemma 21](#L:definable subgroup of K or K/O is a ball){reference-type="ref" reference="L:definable subgroup of K or K/O is a ball"}, $H_d$ is a ball around $0$. We use this fact now to show that the family $\{S_x:x\in K\}$ forms a chain of balls with respect to inclusion. Namely, we show that for $x_1,x_2\in H\setminus I$, if $v(x_1)\leq v(x_2)$ then $S_{x_1}\subseteq S_{x_2}$. Let $d\in S_{x_1}$. Since $H_d$ is a ball and $v(x_1)\leq v(x_2)$ then $x_1\in H_d$ implies that $x_2\in H_d$, i.e., $d\in S_{x_2}$. Since $V$-minimal and power bounded $T$-convex valued fields are $1$-h-minimal (see [@hensel-min Section 6]) they are definably spherically complete ([@hensel-min Lemma 2.7.1], namely the intersection of a definable chain of non-empty balls is non-empty. Thus, $\bigcap\limits_{x\in H\setminus I} S_x\neq \emptyset$, and we let $d$ be an element in the intersection. Let $\hat H_d=\{ z\in H: d\cdot z\in T(z+I)\}$. Since $T: H/I \to K/J$ is a homomorphism, $\hat H_d$ is a subgroup of $(K,+)$. By definition $H_d'\subseteq \hat H_d$ and as both $\hat H_d$ and $I$ are balls, either $I\subseteq \hat H_d$ or $\hat H_d\subseteq I$. Since $H_d'\cap I=\emptyset$ necessarily, $I\subseteq \hat H_d$ and thus $H_d\subseteq \hat H_d$. On the other hand, by the choice of $d$, for all $x\in H\setminus I$, $d\in S_x$, so $H=H_d=\hat H_d$. Finally, as $I\subseteq \hat H_d$, $d\cdot I\subseteq T(I)=J$. Thus $T(x+I)=d\cdot (x+I)+J$ for any $x\in H$. ◻ We are now ready to describe all definable subgroups of $K^n$ and the associated homomoprhisms. **Lemma 25**. *The following holds for all $n$:* *$(1)_n$ If $H\subseteq K^n$ is a definable subgroup of $K^n$ then there is $g\in \mathop{\mathrm{GL}}_n({\mathcal O})$ such that $g(H)$ is a cartesian product of balls, possibly trivial.* *$(2)_n$ If $H\subseteq K^n$ and $J\subseteq K$ are definable subgroups and $T:H\to K/J$ is a definable homomorphism then there are elements $\alpha_1,\ldots, \alpha_n\in K$ such that for all $x=(x_1,\ldots, x_n)\in H$, $$T(x_1,\ldots,x_n)=\alpha_1x_1+\cdots+\alpha_nx_n+J.$$* *Proof.* $(1)_1$ By Lemma [Lemma 21](#L:definable subgroup of K or K/O is a ball){reference-type="ref" reference="L:definable subgroup of K or K/O is a ball"}, every definable subgroup of $K$ is a ball, possibly trivial. $(2)_1$ This is Lemma [Lemma 24](#a general lemma){reference-type="ref" reference="a general lemma"} for $I=\{0\}$. We now proceed with the induction step, assuming $(1)_{n-1}, (2)_{n-1}$ and prove $(1)_{n}$: Let $\pi:K^n\to K^{n-1}$ be the projection onto the first $n-1$ coordinates. By $(1)_{n-1}$, we may assume that $\pi(H)=H_1\times \cdots \times H_{n-1}$, for balls $H_i\subseteq K$. Also, write $\ker(\pi)=H\cap (\{0\}^{n-1}\times K)$ as $\{0\}^{n-1}\times J$, for a definable subgroup $J\subseteq K$. Notice that for every $(a,b), (a,c)\in H\subseteq K^{n-1}\times K$ we have $b-c\in J$ and hence $H$ can be viewed as the graph of a function $T:\pi(H) \to K/J$, mapping $a$ to $b+J$, i.e. $$H=\{(a,b)\in K^n:a\in \pi(H)\, \wedge b\in T(a)\}.$$ By $(2)_{n-1}$, there are $\alpha_1,\ldots, \alpha_{n-1}\in K$, such that $T(x)=\sum_{i=1}^{n-1}\alpha_ix_i+J$. Hence, $$H=\{(x_1,\ldots, x_n)\in K^n: (x_1,\ldots, x_{n-1})\in \pi(H) \wedge x_n-\sum_{i=1}^{n-1} \alpha_i x_i\in J\}.$$ The groups $J$ and $\alpha_iH_i$, for $i=1,\ldots, n-1$, are subgroups of $(K,+)$, hence they are balls. Thus, for every $i=1,\ldots, n-1$, either $J\subseteq\alpha_i H_i$ or $\alpha_i H_i\subseteq J$. Note that if $\alpha_{i_0} H_{i_0}\subseteq J$ for some $i_0$ and $(x_1,\dots, x_{n-1})\in \pi(H)$ then $x_n-\sum\limits_{i\neq i_0} \alpha_i x_i\in J$ iff $x_n-\sum\limits_i \alpha_ix_i\in J$. So there is no harm assuming that $\alpha_i=0$ whenever $J\supseteq \alpha_i H_i$ and that $J\subseteq\alpha_i H_i$ whenever $\alpha_i\neq 0$. Also, we may assume that for some $i$, $\alpha_i\neq 0$, for otherwise $H=\pi(H)\times J$, and we are done. Fix $\alpha_1, \dots, \alpha_{n-1}$ as above. Permuting the coordinates, if needed, we may assume that $v(\alpha_1)\leq v(\alpha_j)$, for all $j=2,\ldots, n-1$. Thus, we can write $$H=\{(x_1,\ldots, x_n): (x_1,\ldots, x_{n-1})\in \pi(H)\,\wedge \frac{1}{\alpha_1}x_n-(x_1+\sum_{i=2}^{n-1} \frac{\alpha_i}{\alpha_1} x_i)\in \frac{1}{\alpha_1}J\}.$$ Let $S(x_2,\ldots, ,x_n)=\frac{1}{\alpha_1}x_n-\sum_{i=2}^{n-1} \frac{\alpha_i}{\alpha_1}x_i.$ Then $S:K^{n-1}\to K$ is a linear map defined over ${\mathcal O}$ and we have, $$\label{eq.1}H=\{(x_1,\ldots, x_n): (x_1,\ldots, x_{n-1})\in \pi(H)\,\wedge \, x_1 -S(x_2,\ldots, x_n) \in \frac{1}{\alpha_1}J\}$$ Let $\hat \pi(x_1,x_2,\ldots, x_n)=(x_2,\ldots, x_n)$ be the projection onto the last $n-1$ coordinates. **Claim 1**. *For every $\hat x=(x_2,\ldots, x_n)\in \hat \pi (H)$, we have $(S(\hat x),\hat x)\in H$.* Let $\hat x=(x_2,\dots,x_n)\in \hat \pi(H)$ and let $x_1=S(\hat x)$, then clearly $x_1-S(\hat x)=0\in \frac{1}{\alpha}J$, so by ([\[eq.1\]](#eq.1){reference-type="ref" reference="eq.1"}), it is sufficient to see that $(x_1,x_2,\ldots, x_{n-1})\in \pi(H)$. Since $\hat x\in \hat \pi(H)$, there exists $x_1'$ such that $(x_1',x_2,\ldots, x_n)\in H$. In particular, $x_2\in H_2,\ldots, x_{n-1}\in H_{n-1}$, so for $(x_1,\ldots, x_{n-1})$ to be in $\pi(H)$, we only need to verify that $x_1=S(\hat x)\in H_1$. By assumption, $(x_1',x_2,\ldots, x_{n-1}, x_{n})\in H$, so by ([\[eq.1\]](#eq.1){reference-type="ref" reference="eq.1"}), $x_1'\in H_1$ and $x_1'-S(\hat x)\in \frac{1}{\alpha_1} J$, so $S(\hat x)\in \frac{1}{\alpha_1}J+x_1'$. However, we assumed that $J\subseteq\alpha_1 H_1$ so $\frac{1}{\alpha_1}J\subseteq H_1$, and therefore $S(\hat x)\in H_1$, hence $(S(\hat x),\hat x)\in H$. We get that $$H=\{(x_1,x_2,\ldots, x_n):(x_2,\ldots, x_n)\in \hat \pi(H) \,\wedge\, x_1-S(x_2,\ldots, x_n)\in \frac{1}{\alpha_1}J\}.$$ So $H\cap (K\times \{0\}^{n-1})=\frac{1}{\alpha_1}J\times \{0\}^{n-1}$ and, in particular, the map $(x_1,\dots, x_n)\mapsto x_1-S(x_2,\dots, x_n)$ from $H$ to $\frac{1}{\alpha_1}J$ is surjective We now define $F:K^n\to K^n$ by $$F(x_1,x_2,\ldots, x_n)=(x_1-S(x_2,\ldots, x_n),x_2, \ldots x_n).$$ Then $F$ is over ${\mathcal O}$, and by a direct computation one sees that it has determinant $1$, hence $F\in \mathop{\mathrm{GL}}_n({\mathcal O})$. It follows from the definition of $F$ and the observation above that the restriction $F\restriction H$ is definable, injective and onto $\frac{1}{\alpha_1}J\times \hat \pi(H)$. By induction, there is $h\in \mathop{\mathrm{GL}}_{n-1}({\mathcal O})$ such that $h(\hat \pi (H))$ is a product of balls. Hence, there is $g\in \mathop{\mathrm{GL}}_n({\mathcal O})$ sending $H$ to a product of balls. This ends the proof of $(1)_n$. For $(2)_n$, we start with $T:H\to K/J$. with $H\subseteq K^n$, By $(1)_n$, we may assume that $H_1=V_1\times \cdots \times V_n$, for definable subgroups $V_i\subseteq K$. Thus, $$T(x_1,\ldots, x_n)=T(x_1,0,\ldots, 0)+\cdots +T(0,\ldots, 0,x_n),$$ with all elements still in $H$. The result follows from the case $n=1$. ◻ **Remark 26**. Lemma [Lemma 25](#end-groups2){reference-type="ref" reference="end-groups2"}(1) is inspired by the work of Hrushovski-Haskell-Macphereson on definable ${\mathcal O}$-submodules of $K^n$ in algebraically closed valued fields, [@HaHrMac1 Lemma 2.2.4]. In that work the authors prove that up to an automorphism in $\mathop{\mathrm{GL}}_n(K)$ every definable ${\mathcal O}$-submodule is a finite cartesian product of $K$, ${\mathcal O}$, $\mathfrak m$ and $\{0\}$. In our setting, if $G\subseteq K^n$ is a definable subgroup then it is an ${\mathcal O}$-submodule (the converse is clearly true), since $\{d\in {\mathcal O}:dG\subseteq G\}$ is a definable subgroup of $(K,+)$ containing $1$, so by Lemma [Lemma 21](#L:definable subgroup of K or K/O is a ball){reference-type="ref" reference="L:definable subgroup of K or K/O is a ball"}, it must be the whole of ${\mathcal O}$. Thus Lemma [Lemma 25](#end-groups2){reference-type="ref" reference="end-groups2"} (1) can be seen as a strengthening of [@HaHrMac1 Lemma 2.2.4] even in the ACVF$_{0,0}$ setting. We may now conclude: **Lemma 27**. *Let $H\subseteq(K/{\mathcal O})^n$ be a definable subgroup.* *$(1)$ There is an automorphism $T$ of $(K/{\mathcal O})^n$ such that $T(H)=H_1\times \dots \times H_n$, where each $H_i$ is either a, possibly trivial, ball.* *$(2)$ If $T:H\to K/{\mathcal O}$ is a definable homomorphism then there are scalars $d_1,\ldots, d_n\in {\mathcal O}$ such that for all $x=(x_1+{\mathcal O},\ldots, x_n+{\mathcal O})\in H$, $$T(x_1+{\mathcal O},\ldots,x_n+{\mathcal O})=d_1x_1+\cdots+d_nx_n+{\mathcal O}.$$* *Proof.* (1) Consider $\hat H\subseteq K^n$ the preimage of $H$ in $K^n.$ By Lemma [Lemma 25](#end-groups2){reference-type="ref" reference="end-groups2"}, there is $g\in \mathop{\mathrm{GL}}_n({\mathcal O})$ such that $g(\hat H)$ is a product of(possibly trivial) balls in $K$. Since $g\in \mathop{\mathrm{GL}}_n({\mathcal O})$, it descends to an automorphism of $(K/{\mathcal O})^n$ sending $H$ to a product of balls in $(K/{\mathcal O})$ (possibly trivial ones). For (2), we may assume that $H=V_1\times \cdots \times V_n$ for $V_i\subseteq K/{\mathcal O}$ and then $$T(x_1+{\mathcal O},\ldots, x_n+{\mathcal O})=T(x_1+{\mathcal O},0,\ldots,0)+\cdots+T(0,\ldots,0,x_n+{\mathcal O}),$$ with each element on the right inside $H$. We apply Lemma [Lemma 24](#a general lemma){reference-type="ref" reference="a general lemma"} with $I=J={\mathcal O}$, so there are $d_1,\ldots, d_n\in {\mathcal O}$ (because $d_i{\mathcal O}\subseteq {\mathcal O}$), such that $T(x_1+{\mathcal O},\ldots, x_n+{\mathcal O})=d\cdot x_1+\cdots+d_n\cdot x_n+{\mathcal O}$. ◻ Finally, we want: **Lemma 28**. *Let $H\subseteq(K/{\mathcal O})^n$ be a definable group and $T:H\to (K/{\mathcal O})^n$ a definable homomorphism. Then $T$ can be extended definably to an endomorphism of $(K/{\mathcal O})^n$.* *In addition, if $T$ is injective, then we can choose the extension to be an automorphism of $(K/{\mathcal O})^n$.* *Proof.* For the first part, we may think of $T$ in coordinates and apply Lemma [Lemma 25](#end-groups2){reference-type="ref" reference="end-groups2"}$(2)_n$ to each coordinate map, obtaining $L\in \mathrm{End}((K/{\mathcal O})^n)$ extending $T$. Assume now that $T$ is injective, and we shall see that so is $L$. By Lemma [Lemma 27](#K/O end-groups2){reference-type="ref" reference="K/O end-groups2"}$(1)_n$, after composing with a definable automorphism of $(K/{\mathcal O})^n$ we may assume that $H=B_1\times\dots\times B_n$, where each $B_i\subseteq K/{\mathcal O}$ is a ball around $0$ (possibly trivial). Assume first that, for all $i$, $B_i$ is not the zero ball. If $L$, the extension of $T$ provided above, were not injective then, after permutation of the coordinates, we may assume the projection of $\ker(L)$ into $B_1$ is infinite. But then, $\ker(L)\cap B_1\times \{0_{n-1}\}$ is nontrivial, contradicting the injectivity of $T$. So without loss of generality, we assume that $H=B_1\times\dots\times B_m\times \{0\}^{n-m}$ and that $B_i$ is non-trivial for $i\leq m$. Since $T$ is injective, $\mathrm{dp\text{-}rk}(T(H))=m=\mathrm{dp\text{-}rk}(H)$ and hence, after a definable automorphism of $(K/{\mathcal O})^n$ (the range) we may assume that $T(H)=C_1\times\dots\times C_m\times \{0\}^{n-m}$, where the $C_i\subseteq K/{\mathcal O}$ are balls with $r(C_i)<0$ (possibly $C_i=K/{\mathcal O}$). Setting $H_1=B_1\times\dots \times B_m$ and $H_2=C_1\times\dots\times C_m$, the map $T$ thus induces an injective isomorphism of $H_1$ and $H_2$, that, by what we have already noted, can be extended to a definable automorphism $L_1$ of $(K/{\mathcal O})^m$. Now, for $(x,y)\in (K/{\mathcal O})^m\times (K/{\mathcal O})^{n-m}$, let $S(x,y)=(L_1(x),y)$. This is an extension of $T$ to an automorphism of $(K/{\mathcal O})^n$. ◻ As a corollary, we obtain: **Corollary 29**. *Assume that $f:(K/{\mathcal O})^n\to (K/{\mathcal O})^n$ is a definable group automorphism. Then there is $g\in \mathop{\mathrm{GL}}_n({\mathcal O})$ such that for all $x\in K^n$, $f(x+{\mathcal O}^n)=gx+{\mathcal O}^n$. In particular, $T$ preserves the valuation.* *Proof.* By Lemma [Lemma 27](#K/O end-groups2){reference-type="ref" reference="K/O end-groups2"}(2), there exist $L_1,L_2\in M_n({\mathcal O})$ such that for every $x\in K^n$, $$f(x+{\mathcal O}^n)=L_1(x)+{\mathcal O}^n,\,\,\,\, f^{-1}(x+{\mathcal O}^n)=L_2(x)+{\mathcal O}^n.$$ It follows that for all $x\in K^n$, we have $L_1\circ L_2(x)-x\in {\mathcal O}^n$. It is easy to see that this forces the $K$-linear map $L_1\circ L_2(x)-x$ to be $0$. Thus, $L_2=L_1^{-1}$ and both belong to $\mathop{\mathrm{GL}}_n({\mathcal O})$. ◻ ## ${\mathcal K}$ $p$-adically closed In the present subsection, we assume that ${\mathcal K}$ is $p$-adically closed. As we have already seen, definable subgroups of $K/O$ need not be balls, so the analysis of definable subgroups of $(K/{\mathcal O})^n$ is more subtle than the V-minimal and the $T$-convex settings. We prove a weak version of Lemma [Lemma 27](#K/O end-groups2){reference-type="ref" reference="K/O end-groups2"}(1) that will suffice for our needs: **Lemma 30**. *If $H\leq (K/{\mathcal O})^n$ is a definable infinite subgroup, then there is a subgroup $H_1\leq H$, with $\mathrm{dp\text{-}rk}( H_1)=\mathrm{dp\text{-}rk}(H)=d$ such that $H_1$ projects invectively into some $(K/{\mathcal O})^{d}$.* *Proof.* First, note that every finite non-trivial subgroup of $K/{\mathcal O}$ contains $G_{-1}:=\{g\in K/{\mathcal O}:v(g)\geq -1\}$. Indeed, all torsion elements of $K/{\mathcal O}$ are of order $p^m$, for some natural number $m$ (see [@HaHaPeGps Lemma 3.10(1)] and also Fact [Fact 62](#F: torsion){reference-type="ref" reference="F: torsion"}). Thus, every finite group must contain $C_p$, the cyclic group of order $p$. The group of elements of order $p$ is exactly $G_{-1}$. We use induction on $n$, where the case $n=1$ is trivially true. Assume that $H\subseteq (K/{\mathcal O})^n$. For $i=1,\ldots, n$, let $\pi^i:(K/{\mathcal O})^n\to (K/{\mathcal O})^{n-1}$ denote the projection onto the remaining $n-1$ coordinates. Let $H^i=\ker(\pi^i\restriction H)$. Notice that the group product of the $H^i$ in $(K/{\mathcal O})^n$ is a direct product, thus if all $H^i$ are infinite then $\mathrm{dp\text{-}rk}(H)=n$ and we are done. Hence, we assume that one of the $H^i$'s must be finite. We use additional induction on $\min_i |H^i|$, for those $i$ such that $H^i$ is finite. If this minimum is $1$, then one of these projections $\pi^i$ is injective on $H$ and hence it is sufficient to prove the statement for $\pi^i(H)$, and then we may finish by induction. So we may assume that $1<\min_i |H^i|<\infty$. Assume next that all $H_i$ are finite (and non-trivial). Then they all contain $G_{-1}$. Consider the (surjective) definable map $$\sigma:(K/{\mathcal O})^n\to (K/{\mathcal O})^n\,\, \mbox{ defined by } \sigma(x)=p\cdot x.$$ By our assumption, $(G_{-1})^n= \ker(\sigma)\subseteq H$. Let $N=\sigma(H)\subseteq(K/O)^n$ and $N^i:=\ker(\pi^i\upharpoonright N)$. **Claim 2**. *For every $i$, $|N^i|<|H^i|$.* In fact, we shall see that $\sigma(H^i)=N^i$, with non-trivial kernel $G_{-1}$, so the result follows Let us see that for $i=1$. Clearly, if $(x,0,\ldots,0)\in H^1$ then $(px,0,\ldots,0)\in N^i$. Conversely, if $y=(y_1,0,\ldots, 0)\in N^1$ then there is $x=(x_1,x_2,\ldots, x_n)\in H$, such that $\sigma(x)=y$. This implies that $px_1=y_1$ and $x_2,\ldots, x_n\in G_{-1}$. Because $G_{-1}^n\subseteq H$ it follows that $x'=(x_1,0,\ldots,0)$ is also in $H$, so in $H^1$, and we have $\sigma(x')=y$ as we wanted, thus proving our claim. We can now apply induction to the group $N$ and conclude that there is $N_1\subseteq N\subseteq(K/{\mathcal O})^n$ of the same dimension as $N$, and an injective projection of $N_1$ onto some $(K/{\mathcal O})^d$, for $d=\mathrm{dp\text{-}rk}N$. We claim that in fact $N_1\subseteq H$ as well: Indeed, by the definition of $\sigma$ we clearly have $\sigma(N_1)\subseteq N_1$, and hence $\sigma^{-1}(\sigma(N_1))\subseteq\sigma^{-1}(N_1)$. However, because $\ker \sigma\subseteq H$, we have $\sigma^{-1}(N_1)\subseteq H$, so $N_1\subseteq \sigma^{-1}\sigma(N_1)\subseteq H$. Finally, $\ker \sigma$ is finite so $\mathrm{dp\text{-}rk}(N_1)=\mathrm{dp\text{-}rk}(H)$. This ends the proof when all $H^i$ are finite. Assume that one of the $H^i$, say $H^1$, is infinite. Let $\pi_1:(K/{\mathcal O})^n\to (K/{\mathcal O})^{n-1}$ be the projection on the first coordinate. Let $H_1=\ker \pi_1\cap H$. **Claim 3**. *$\mathrm{dp\text{-}rk}(H_1)=\mathrm{dp\text{-}rk}(H)-1$.* Indeed, for every $y\in \pi_1(H)\subseteq K/{\mathcal O}$, $\mathrm{dp\text{-}rk}(\pi_1^{-1}(y)\cap H)= \mathrm{dp\text{-}rk}(H_1)$, so by sub-additivity, $\mathrm{dp\text{-}rk}(H_1)\geq \mathrm{dp\text{-}rk}(H)-1$. Also, we have $H\supseteq H_1\oplus H^1$ (since $H_1\subseteq \{0\}\times K^{n-1}$ and $H_1\subseteq K/{\mathcal O}\times \{\bar 0_{n-1}\}$), so since $H^1$ is infinite then $\mathrm{dp\text{-}rk}(H_1)=\mathrm{dp\text{-}rk}(H)-1$. By identifying $H_1$ with a subgroup of $(K/{\mathcal O})^{n-1}$ we may apply induction and find a definable $H_1'\subseteq H_1$, $\mathrm{dp\text{-}rk}(H_1')=\mathrm{dp\text{-}rk}(H_1)$, and a projection, call it $\tau^k:(K/{\mathcal O})^{n-1}\to (K/{\mathcal O})^k$, onto some $k$ coordinates among the last $n-1$ ones, where $k=\mathrm{dp\text{-}rk}H_1$, such that $\tau^k\restriction H_1'$ is injective. The group $H':=H_1'\oplus H^1\subseteq H$ has the same rank as $H$ and the projection $\pi_1\times \tau^k:(K/{\mathcal O})^n\to (K/{\mathcal O})^{k+1}$ is injective on $H'$. This ends the proof of Lemma [Lemma 30](#L:local homeo K/O){reference-type="ref" reference="L:local homeo K/O"}. ◻ # Topology and dimension If $D$ is a distinguished sort which is an SW-uniformity, it follows from [@HaHaPeGps] (see below for details) that definable $D$-groups inherit a group topology, $\tau_D$, from $\nu_D$. On the other hand, since ${\mathcal K}$ is geometric, ${\mathcal K}^{eq}$ inherits a notion of dimension (that turns out to be non-trivial for $K$-groups). In the present section, we first recall the basic properties of the dimension induced from $K$ to ${\mathcal K}^{eq}$, and then study its relation with the topology $\tau_G$ in $K$-groups. ## Geometric dimension and equivalence relations A sufficiently saturated (one sorted) structure is *geometric* if $\mathop{\mathrm{acl}}(\cdot)$ satisfies Steinitz Exchange and the quantifier $\exists^\infty$ can be eliminated. Elimination of $\exists^\infty$, sometimes referred to as *uniform finiteness*, means that in definable families there is a uniform bound on the size of finite sets. In [@Gagelman], Gagleman shows that for geometric structures, the dimension associated with the $\mathop{\mathrm{acl}}(\cdot)$-pre-geometry can be extended naturally to imaginary sorts. In the present section, we review this extension of dimension and exploit it to show that in ${\mathcal K}$ the $K$-rank and the almost $K$-rank of definable sets coincide (compare with [@JohnTopQp Corollary 4.37]). Given a geometric structure ${\mathcal M}$, we remind Gagelman's extension of $\dim_{\mathop{\mathrm{acl}}}$ to ${\mathcal M}^{eq}$: Given a definable equivalence relation $E$ on $M^n$ set, and $A\subseteq{\mathcal M}^{eq}$ $$\dim^{eq}(a_E/A)=\max\{\dim(b/A)-\dim[a]:b\in [a]\},$$ where $\dim:=\dim_{\mathop{\mathrm{acl}}}$, the $E$-equivalence class of $a$ is $[a]\subseteq K^n$, $a_E:=a/E\in M^n/E$. For $Y\subseteq X/E$ defined over $A$, we define $$\dim^{eq}(Y)=\max\{\dim^{eq}(a_E/A):a_E\in Y\}.$$ For a concise summary of the properties of $\dim^{eq}$ we refer to [@JohnTopQp §2]. In the present text we will mostly use additivity of $\dim^{eq}$: For $a,b\in {\mathcal M}^{eq}$, $$\dim^{eq}(a,b/A)=\dim^{eq}(a/Ab)+\dim^{eq} (b/A).$$ Note that $\dim^{eq}$ coincides with $\dim_{\mathop{\mathrm{acl}}}$ on definable subsets of $M^n$, and on tuples in $M$, over parameters from $M$. There is, therefore, no ambiguity in extending the notation $\dim$ (instead of $\dim^{eq}$) to imaginary elements and definable sets. Note, however, that in this notation for a definable set $Y$, $\dim(Y)=0$ does not imply that $Y$ is finite, unless $Y\subseteq M^n$. E.g., $\dim(K/{\mathcal O})=\dim(\Gamma)=0$. Whenever ${\mathcal M}$ is in addition dp-minimal, dp-rank coincides with dimension on definable subsets of $M^n$ ([@Simdp Theorem 0.3]), a fact that we use without further mention. In our setting, as ${\mathcal K}$ is a geometric structure, this implies directly from the definitions that $\dim(X)\leq \mathrm{dp\text{-}rk}(X)$ for any definable set $X$ in ${\mathcal K}^{eq}$. Since dimension is preserved under definable finite-to-one functions, and infinite definable subsets of $K^n$ have positive dimension, it follows that if $X$ is locally almost strongly internal to $K$ then $\dim(X)>0$.\ The above observation allows us to show that, in our setting, the $K$-critical and the almost $K$-critical ranks coincide. We start with the following result [@PePiSt Lemma 3.8]. **Fact 31**. *Let ${\mathcal M}$ be a geometric structure and let $E$ be a definable equivalence relation on $M^n$. Then there exists a definable $S\subseteq M^n$ such that for every $x\in S$, $[x]\cap S$ is finite and $\dim(S)=dim(S/E)=\dim(M^n/E)$.* In the setting where ${\mathcal M}={\mathcal K}$ we can conclude the following: **Corollary 32**. *Let $Y$ be a definable set in ${\mathcal K}$ (so possibly in ${\mathcal K}^{eq}$). If $Y_0\subseteq Y$ is almost strongly internal to $K$ then there exists a definable subset $Y'\subseteq Y_0$ with $\mathrm{dp\text{-}rk}(Y')=\mathrm{dp\text{-}rk}(Y_0)$ that is strongly internal to $K$. Moreover, the following are equal* 1. *$\dim(Y)$* 2. *The $K$-rank of $Y$* 3. *The almost $K$-rank of $Y$.* *Proof.* We use the fact that, in our setting, the sort $K$ is a geometric SW-uniformity. The proof relies on the following claim. **Claim 4**. *For any $Z\subseteq Y$, there exists $Z_0\subseteq Z$ strongly internal to $K$ with $\mathrm{dp\text{-}rk}(Z_0)=\dim(Z)$.* Assume that $Z=X'/E$ for some $X'$. Let $S\subseteq X'$ be a definable set, as provided by Fact [Fact 31](#lemma1.1){reference-type="ref" reference="lemma1.1"}. I.e. $\dim(S)=\dim(Z)$ and $S$ intersects every $E$-class in a finite (possibly empty) set. Let $\pi:S\to S/E$ be the finite-to-one projection map; note that $S/E\subseteq Z$ and by [@Simdp Theorem 0.3(1)], $\mathrm{dp\text{-}rk}(S/E)=\mathrm{dp\text{-}rk}(S)=\dim(S)=\dim(X'/E)$. By [@HaHaPeGps Lemma 2.6(1)], as $K$ is an SW-uniformity, there exists a definable subset $Z_0\subseteq S/E\subseteq Z$ strongly internal to $M$ and satisfying $\mathrm{dp\text{-}rk}(Z_0)=\mathrm{dp\text{-}rk}(S/E)=\dim(Z)$. We now apply this claim to prove the statements of the corollary. First, let $Y_0$ be as in the statement; applying the claim for $Z=Y_0$, we get $Y'\subseteq Y_0$ strongly internal to $K$ with $\mathrm{dp\text{-}rk}(Y')=\dim(Y_0)$. But since $Y_0$ is almost strongly internal to $K$, $\mathrm{dp\text{-}rk}$ and $\dim$ also coincide on $Y_0$ so $\mathrm{dp\text{-}rk}(Y')=\dim(Y_0)=\mathrm{dp\text{-}rk}(Y_0)$. This result, assures that the $K$-rank and the almost $K$-rank of $Y$ are equal. To conclude, note that, since $\dim(Y)$ is obviously bounded below by the $K$-rank of $Y$, we only need to verify the other inequality. This is immediate by applying the claim to $Z=Y$. ◻ ## Topology Let $G$ be a definable group in ${\mathcal K}$, locally strongly internal to a fixed definable SW-uniformity $D$ (for example $D=K$). In particular, $D$ admits a definable basis for a topology. In this section, we review results from [@HaHaPeGps] on how to topologize $G$ using the $D$-topology. For $p$-adically closed fields, this was done using different techniques in [@JohnTopQp] for the case $D=K$. The group $G$ is automatically a $D$-group by [@HaHaPeGps Fact 4.25(1)]. By [@HaHaPeGps Proposition 5.8], there is a type-definable subgroup $\nu_D:=\nu_D(G)$ of $G$ definably isomorphic to an infinitesimal type-definable group in $D$. Specifically, given any $D$-critical set $X\subseteq G$, any definable injection $f:X\to D^n$ (for $n=\mathrm{dp\text{-}rk}(X)$) and any $c\in X$ generic over all the data we have: $$\label{eq: nuK} \nu_D=\{f^{-1}(U)c^{-1}: U\subseteq D^n \text{ definable open containing $f(c)$}\}.$$ Before proceeding with the description of $\nu_D$ we give the proof of the statement in Remark [Remark 10](#R: D-sets){reference-type="ref" reference="R: D-sets"}(2), assuring that such an $X$ can always be found. **Lemma 33**. *Let $D$ be an unstable distinguished sort in ${\mathcal K}$ and $G$ a ${\mathcal K}$-definable $D$-group. Then there exists a $D$-critical subset $X\subseteq G$ and a definable injection $f: X\to D^m$ for $m=\mathrm{dp\text{-}rk}(X)$. In particular, $X$ is a $D$-set.* *Proof.* If $D$ is an SW-uniformity this follows from [@SimWal Proposition 4.6], so we may assume that ${\mathcal K}$ is $p$-adically closed and $D$ is either $\Gamma$ or $K/{\mathcal O}$. If $D=\Gamma$ this follows from cell-decomposition in Presburger arithmetic (as referred to in the proof of Fact [Fact 45](#F:minimal fibers in Gamma){reference-type="ref" reference="F:minimal fibers in Gamma"}). If $D=K/{\mathcal O}$ then by [@HaHaPeGps Theorem 7.11(3)], there exists a definable subgroup $H\subseteq G$ with $\mathrm{dp\text{-}rk}(H)=n$, the $K/{\mathcal O}$-rank of $G$, definably isomorphic to a subgroup of $((K/{\mathcal O})^r,+)$ for some $r>0$. By Lemma [Lemma 30](#L:local homeo K/O){reference-type="ref" reference="L:local homeo K/O"}, we may assume, replacing $H$ with a subgroup of the same dp-rank that $r=n$. ◻ we now return to the assumption that $D$ is an SW uniformity. Note that $\nu_D$ is given as a definable collection of sets $\{U_t:t\in T\}$ which forms a filter-base: for every $t_1,t_2\in T$ there is $t_3\in T$ such that $U_{t_3}\subseteq U_{t_1}\cap U_{t_2}$. By [@HaHaPeGps Corollary 5.14], $G$ has a definable basis for a topology $\tau_D=\tau_{D}(G)$, making $G$ a non-discrete Hausdorff topological group. **For the rest of this section, all topological notions in $G$ refer to $\tau_D$**. A definable subset $X\subseteq G$ is open in this topology if and only if for all $a\in X$ $a\cdot \nu_D\subseteq X$. In particular, $\mathrm{dp\text{-}rk}(X)\ge \mathrm{dp\text{-}rk}(\nu_D)$, i.e., the dp-rank of any open definable subset of $G$ is at least the $D$-rank of $G$. Of course, it could be, for example, that $\mathrm{dp\text{-}rk}(G)>\mathrm{dp\text{-}rk}(\nu_D)$, so that definable open subsets need not all have the same dp-rank (but they all have the same $D$-rank). The next lemma shows that the topology $G$ inherits from $D$ shares some of its good properties. Toward that end, recall that the $D$-rank of a set $Z$ is the maximal dp-rank of a definable subset strongly internal to $D$. We let $\mathop{\mathrm{Fr}}(X)$, the frontier of $X$, denote $\mathop{\mathrm{cl}}(X)\setminus X$. **Lemma 34**. *If $X\subseteq G$ is definable, then the $D$-rank of $\mathop{\mathrm{Fr}}(X)$ is strictly smaller than the $D$-rank of $X$.* *Proof.* Let $d$ denote the $D$-rank of $\mathop{\mathrm{Fr}}(X)$ and let $X_1\subseteq \mathop{\mathrm{Fr}}(X)$ be $D$-critical over $A$. Fix an $A$-generic $g\in X_1$ and $Y\ni g$ a definable basic open set. In particular, we can choose $Y$ to be strongly internal to $D$. By definition of $\mathop{\mathrm{Fr}}(X)$, it follows that $\mathop{\mathrm{Fr}}(X)\cap Y= \mathop{\mathrm{Fr}}(X\cap Y)$. By Lemma [Lemma 15](#L:intersects largely){reference-type="ref" reference="L:intersects largely"}, $\mathrm{dp\text{-}rk}(X_1\cap Y)=\mathrm{dp\text{-}rk}(X_1)$. By the properties of SW-uniformities, ([@SimWal Propositioin 4.3, Lemma 2.3]), and since $X\cap Y$ can be identified with a subset of some $D^n$, $\mathrm{dp\text{-}rk}(\mathop{\mathrm{Fr}}(X\cap Y))<\mathrm{dp\text{-}rk}(X\cap Y)$. Thus, as $X_1\cap Y\subseteq \mathop{\mathrm{Fr}}(X\cap Y)$, $$d=\mathrm{dp\text{-}rk}(X_1)\leq \mathrm{dp\text{-}rk}(\mathop{\mathrm{Fr}}(X\cap Y))<\mathrm{dp\text{-}rk}(X\cap Y).$$ Since $X\cap Y$ is strongly internal to $D$ (as $Y$ was), its dp-rank is at most the $D$-rank of $X$, as needed. ◻ **Lemma 35**. *If $H$ is a definable subgroup of $G$ then $H$ is closed in $G$ and the following are equivalent:* 1. *$H$ is open,* 2. *the $D$-ranks of $H$ and $G$ are equal,* 3. *$\nu_D\vdash H$.* *Proof.* Because $G$ is a topological group, and a basis for the topology is definable, the closure of $H$, call it $H_1$ is also a definable subgroup. Therefore, If $H_1\setminus H\neq \emptyset$ then $H_1$ must contain a coset of $H$ thus the $D$-rank of $H_1\setminus H$ is at least that of $H$ contradicting Lemma [Lemma 34](#frontier){reference-type="ref" reference="frontier"}. So $H$ is closed in $G$. Now, assume that the $D$-ranks of $H$ and $G$ are equal. This implies (by definition of $\nu_D$), that $\nu_D\vdash H$. Since $\nu_D$ is open, and $H$ is a group, this implies that $H$ is open. Finally, as we have seen, if $H$ is open, then it contains $\nu_D$ as a subgroup, and therefore they have the same $D$-rank (since the $D$-rank of $\nu_D$ is maximal in $G$). ◻ **Definition 36**. For $G$ locally strongly internal to $D$, we let *the centralizer of the type $\nu_D$*, denoted by $C_G(\nu_D)$, be the set of all $g\in G$ such that for some definable $Y$ with $\nu_D\vdash Y$, $g$ commutes with all elements of $Y$. Since, as we noted, $\nu_D$ is given as a definable collection of sets $\{U_t:t\in T\}$, it follows that $C_G(\nu_D)$ is definable: $g\in C(\nu_D)$ if there exists $t\in T$ such that $g\in C_G(U_t)$. Moreover, by the filer-base property of the family, it is a subgroup of $G$. **Remark 37**. Let us note that, despite its name, if ${\mathcal K}\prec \hat {\mathcal K}$, and $g\in C_G(\nu_D)(\hat {\mathcal K})$ then $g$ does not necessarily centralize the set $\nu_D(\hat {\mathcal K})$. What we know is that there exists $t\in T(\hat {\mathcal K})$ such that $g$ centralizes $U_t(\hat {\mathcal K})$. with possibly $U_t\vdash \nu_D({\mathcal K})$. Recall that definable sets in o-minimal structures can be decomposed into finitely many definably connected sets (i.e. sets containing no non-trivial definable clopen sets). Thus, the same is true if $X\subseteq G$ is strongly internal to an o-minimal sort $D$. The result below will be useful in the sequel. **Lemma 38**. *Assume that $D$ is one of the o-minimal distinguished sorts. Assume that $X\subseteq G$ is definable, strongly internal to $D$ and $e\in X$. If $X$ is definably connected, then every $g\in C_G(\nu_D)$ centralizes $X$.* *Proof.* Let $g\in C_G(\nu_D)$. By definition $\nu_D\vdash C_G(g)$, so by Lemma [Lemma 35](#L:groups are closed, open iff full D-rank){reference-type="ref" reference="L:groups are closed, open iff full D-rank"}, $C_G(g)$ is a clopen subgroup of $G$. Now, $C_G(g)\cap X\neq \emptyset$ (as $e$ is in the intersection), so definable connectedness of $X$ implies $X\subseteq C_G(g)$. ◻ For the rest of this section we focus our attention on the case $D={\mathcal K}$ (so, in particular, it is an SW-uniformity), and the topology we discuss below is the one comping from $K$.\ We start with an immediate corollary of Lemma [Lemma 35](#L:groups are closed, open iff full D-rank){reference-type="ref" reference="L:groups are closed, open iff full D-rank"} and Corollary [Corollary 32](#C:every thing is strongly internal to K){reference-type="ref" reference="C:every thing is strongly internal to K"}. **Corollary 39**. *Let $G$ be a definable group and $H$ a definable subgroup. Then $H$ is open in $G$ if and only if $\dim(G)=\dim(H)$.* As the distinguished sorts, $\Gamma$, $\textbf{k}$ and $K/{\mathcal O}$ are $0$-dimensional, we get: **Lemma 40**. *A definable set $S$ is $K$-pure if and only if every definable $0$-dimensional $X\subseteq S$ is finite.* *Proof.* Assume that $X\subseteq S$ is infinite and $0$-dimensional. By Fact [Fact 5](#F: Reduction to sorts){reference-type="ref" reference="F: Reduction to sorts"}, $X$ (and hence also $S$) is locally almost strongly internal to some distinguished sort $D$. Namely, there is a definable infinite $X_1\subseteq X$ and a definable finite to one function $f:X_1\to f(X_1)\subseteq D^n$. Since $\dim(X_1)\ge \dim (f(X_1))$, necessarily $\dim (f(X_1))=0$ with $f(X_1)$ infinite. Hence, $D\neq K$, so $S$ is not $K$-pure. For the converse, assume that $S$ is not $K$-pure, witnessed by a definable infinite $X\subseteq S$ and a definable finite to one function $f:X\to D^n$ for some $D\neq K$. Since $\dim(D)=0$ for $D\neq K$, it follows that $\dim(f(X))=0$ and hence $\dim(X)=0$. So, $X$ is infinite and $0$-dimensional. ◻ For the sake of completeness, we note that the $\tau_K$-topology on $G$ is *locally Euclidean*, in the following sense: for every $g\in G$ there exists a definable open $U\ni g$, which is definably homeomorphic to an open subset of $K^{\dim(G)}$. Moreover, it is the unique such group topology on $G$. We prove: **Lemma 41**. *The $\tau_K$-topology on $G$ (taken to be discrete if $\dim G=0$) is locally Euclidean and if $\tau$ is any other locally Euclidean group topology on $G$ then $\tau=\tau_K$.* *In particular, if $K$ is a $p$-adically closed field, $\tau_K$ equals Johnson's admissible topology from [@JohnTopQp].* *Proof.* A non-discrete locally Euclidean topological group is, by definition, a $K$-group, so (by Corollary [Corollary 32](#C:every thing is strongly internal to K){reference-type="ref" reference="C:every thing is strongly internal to K"} $\dim(G)>0$ and since discrete $G$ groups are trivially locally Euclidean, we assume $\dim(G)>0$. Since the topology is invariant under translations, it is sufficient to find a single $g\in G$ at which the topology is locally Euclidean. If $n=\dim(G)>0$ then, by Lemma [Corollary 32](#C:every thing is strongly internal to K){reference-type="ref" reference="C:every thing is strongly internal to K"}, there exists a definable $X\subseteq G$, $\dim(X)=\dim(G)$, such that $X$ is strongly internal to $K$, over some $A$, and $\dim(G)$ is the $K$-rank of $G$. Given $g_1$ generic in $X$ over $A$, it follows from Equation [\[eq: nuK\]](#eq: nuK){reference-type="ref" reference="eq: nuK"} that there exists a definable $\tau_K$-open set $U$, $g_1\in U\subseteq X$, which is definably homeomorphic to an open set in $K^n$. Now, assume that $\tau_1,\tau_2$ are two locally Euclidean group topologies on $G$. Then for $g\in G$, there are definable $U_1,U_2\ni g$, $U_i$ a $\tau_i$-open set, and definable $f_i:U_i\to V_i\subseteq K^n$, such that each $f_i$ is a homeomorphism between $U_i$ with the $\tau_i$-topology and open $V_i$ with the $K^n$-topology. The map $f_2f_1^{-1}:f_1(U_1\cap U_2)\to V_2$, is a definable injection. However, in SW-uniformities, definable bijections are homeomorphisms at generic points, [@SimWal Corollary 3.8]. Thus, there is some $g_1\in U_1\cap U_2$ such that on $\tau_1, \tau_2$ open neighborhood of $g_1$, the two topologies agree. Thus, $\tau_1=\tau_2$. Since Johnson's admissible topology is locally Euclidean, the two topologies are the same. ◻ Using the exact same proof as above, one can show that for any distinguished sort $D$ which is an SW-uniformity, if $G$ is locally strongly internal to $D$ then every $g\in G$ has a $\tau_D$-open neighborhood which is definably homeomorphic to an open set in $D^m$, where $m$ is the $D$-rank of $G$. # The infinitesimal group $\nu_D$ and local (differentiable) groups {#S:infnit and local} In Section [2.5](#ss:infint and vicin){reference-type="ref" reference="ss:infint and vicin"} we gave an abstract description of $\nu_D(G)$ for an infinite definable $D$-group $G$ and an unstable distinguished sort $D$. In the present section, we collect -- for later use -- more specific information on the construction of $\nu_D(G)$, as $D$ ranges over the various distinguished sorts in the different settings we are interested in. Throughout, we fix an infinite group $G$ definable in ${\mathcal K}$. ## The sort of closed $0$-balls $K/O$ {#ss:nu in K/O} Let $G$ be an infinite definable $K/{\mathcal O}$-group. In each of our three settings, there exists a definable subgroup $H\subseteq G$ definably isomorphic to a subgroup of $((K/{\mathcal O})^m,+)$ for some $m>0$, such that $\mathrm{dp\text{-}rk}(H)$ is the $K/{\mathcal O}$-rank of $G$ [@HaHaPeGps Theorems 7.4(4), 7.7(4), 7.11(3)]. Let us see that we can choose $m$ to be equal to $\mathrm{dp\text{-}rk}(H)$. We prove this first for the case where ${\mathcal K}$ is either power bounded $T$-convex or $V$-minimal (so that $K/{\mathcal O}$ is an SW-uniformity). Let $f:H\to (K/{\mathcal O})^m$ be an injective group homomorphism, and let $n=\mathrm{dp\text{-}rk}(H)$. By [@SimWal Proposition 4.6], there exists an open set of the form $B_0+b\subseteq f(H)$, where $B_0\subseteq f(H)$ is an open ball around $0$, such that the projection $\pi:(K/{\mathcal O})^m\to (K/{\mathcal O})^n$ restricts to a bijection from $B_0+b$ onto an open ball in $(K/{\mathcal O})^n$. Therefore, $\pi$ also restricts to a bijection on $B_0$. We can now replace $H$ by the preimage of $B_0$ under $\pi\circ f$ and let $\pi\circ f$ be the new injective homomorphism, this time into $(K/{\mathcal O})^n$. The proof for ${\mathcal K}$ $p$-adically closed is similar, but uses Lemma [Lemma 30](#L:local homeo K/O){reference-type="ref" reference="L:local homeo K/O"} instead of [@SimWal Proposition 4.6]. Recall that the valuation descends to $K/{\mathcal O}$ and $(K/{\mathcal O})^n$, hence, so does the notion of a ball. However, we reserve the term "ball" for an infinite set, thus in the $p$-adically closed case we require the valuative radius to be infinitely negative, i.e., smaller than $n$ for all $n\in \mathbb{Z}$. We may now further assume that $H$ is definably isomorphic to a definable ball (of the same rank) centred at $0$: **Fact 42**. *For any $A$-definable set $X\subseteq (K/{\mathcal O})^n$ with $\mathrm{dp\text{-}rk}(X)=n$ and any $A$-generic $a\in X$, there exists a ball $B\subseteq X$ with $a\in B$.* *Proof.* If ${\mathcal K}$ is power-bounded $T$-convex or $V$-minimal then this is [@SimWal Corollary 2.7], and if ${\mathcal K}$ is $p$-adically closed this is [@HaHaPeGps Lemma 3.6]. ◻ We can now give, keeping the above notation and assumptions, a more specific description of the construction of $\nu_{K/{\mathcal O}}$: **Lemma 43**. *Let $f:H\to (K/{\mathcal O})^n$ be an $A$-definable injective homomorphism, $\mathrm{dp\text{-}rk}(H)=n$ the $K/{\mathcal O}$-rank of $G$. Then* *$$\nu_{K/{\mathcal O}}=\{f^{-1}(U): U\subseteq (K/{\mathcal O})^n \text{ is an open ball in $(K/{\mathcal O})^n$ centred at $0$}\}.$$* *Proof.* Let $\nu_1:=\{f^{-1}(U): U\subseteq (K/{\mathcal O})^n \text{ is an open ball in $(K/{\mathcal O})^n$ centred at $0$}\}$. By definition, $\nu_{K/{\mathcal O}}=\nu_H(c)c^{-1}$ for any $A$-generic $c\in H$. Let $H_1 :=f(H)\le (K/{\mathcal O})^n$. Since $\mathrm{dp\text{-}rk}(H_1)=n$, by Fact [Fact 42](#F:interior in K/O){reference-type="ref" reference="F:interior in K/O"}, we may assume, shrinking $H$ (but not its rank) if needed, that $H_1$ is a ball in $(K/{\mathcal O})^n$. We first show that $\nu_{K/{\mathcal O}}\vdash \nu_1$. Let $U\subseteq H_1$ be an open ball, $0\in U$. By [@HaHaPeVF Proposition 3.6] (if ${\mathcal K}$ is power-bounded $T$-convex or $V$-minimal) or [@HaHaPeGps Proposition 3.8] (if ${\mathcal K}$ is $p$-adically closed), there exists a ball $Y\subseteq U+f(c)$, $f(c)\in Y$, definable over some $B\supseteq A$ such that $\mathrm{dp\text{-}rk}(f(c)/B)=n$. Since $H_1$ is a subgroup, we have $Y\subseteq H_1$. Now, as $f$ is group homomorphism, $f^{-1}(Y-f(c))=f^{-1}(Y)c^{-1}\subseteq U$, $c\in f^{-1}(Y)$, and $\mathrm{dp\text{-}rk}(c/B)=n$. Thus, by the definition of $\nu_{K/{\mathcal O}}$, we have $\nu_{K/{\mathcal O}}\vdash f^{-1}(U)$, so $\nu_{K/{\mathcal O}}\vdash \nu_1$. Similarly, $\nu_1c\vdash \nu_{H_1}(c)$, so we conclude that $\nu_1\vdash \nu_{K/{\mathcal O}}$. ◻ ## The valuation group $\Gamma$. {#ss:infnit of gamma} When ${\mathcal K}$ is either power bounded $T$-convex our $V$-minimal, the valuation group $\Gamma$ is o-minimal, when it is $p$-adically closed, it is a model of Presburger arithmetic. In order to get a uniform treatment (and formulation of results) we make the following definition: **Definition 44**. A subset $B\subseteq \Gamma^n$ is called a $\Gamma$-box (around $a=(a_1,\dots,a_n)$) if it is of the following form: 1. (In the non $p$-adic case) $\prod_{i=1}^n (b_i,c_i)$ for some $b_i<a_i<c_i$ in $\Gamma$. 2. (In the $p$-adic case) A cartesian product of $n$-many sets of the form $(b_i,c_i)\cap \{x_i: x_i-a_i\in P_{m_i}\}$ where both intervals $(b_i,a_i)$ and $(a_i,c_i)$ are infinite and $P_{m_i}$ is the predicate for $m_i$-divisibility. **Fact 45**. *Let $Y\subseteq \Gamma^m$ be a definable set with $\mathrm{dp\text{-}rk}(Y)=n\leq m$. Then there exists a definable $Z\subseteq Y$ with $\mathrm{dp\text{-}rk}(Z)=n$ projecting injectively onto a $\Gamma$-box in $\Gamma^n$.* *Proof.* If ${\mathcal K}$ is power-bounded $T$-convex or $V$-minimal, $\Gamma$ is o-minimal and the result follows by cell-decomposition. If ${\mathcal K}$ is $p$-adically closed, then $\Gamma$ is a model of Presburger arithmetic. It also admits a cell-decomposition [@ClucPresburger] (see also [@OnVi Fact 2.4] for a more explicit formulation), and thus the result follows from the fact that dimension coincides with dp-rank ([@Simdp Theorem 0.3]). ◻ Using Fact [Fact 45](#F:minimal fibers in Gamma){reference-type="ref" reference="F:minimal fibers in Gamma"} and [@HaHaPeGps Lemma 4.2] repeatedly (as in the proof of Lemma [Lemma 43](#F: nu in K/O){reference-type="ref" reference="F: nu in K/O"} above) we get the following. **Lemma 46**. *Let $G$ be a definable $\Gamma$-group and $g:Y\to \Gamma^n$ be a definable injection with $\mathrm{dp\text{-}rk}(Y)=n$. Assume everything is defined over some parameter set $A$, and $c\in Y$ is $A$-generic. Then* *$$\nu_Y(c)=\{g^{-1}(U): U\subseteq \Gamma^n \text{ a $\Gamma$-box around $g(c)$}\}.$$* We can now conclude: **Lemma 47**. *Let $G$ be a definable $\Gamma$-group. There exists $X\subseteq G$, a $\Gamma$-critical set with $\nu_{\Gamma}\vdash X$, and $f:X\to \Gamma^n$ a definable injection satisfying:* 1. *$f(X)$ is a $\Gamma$-box around $0$,* 2. *$f(xy^{\pm 1})=f(x)\pm f(y)$ for any $x,y\in X$ with $xy^{\pm 1}\in X$ and* 3. *$\nu_\Gamma=\{f^{-1}(U): U\subseteq \Gamma^n \text{ a $\Gamma$-box around $0$}\}.$* *Proof.* By [@HaHaPeGps Theorems 7.4(3), 7.7(3), 7.11(2)], $\nu_\Gamma$ is definably isomorphic (as groups) to a type-definable subgroup of $(\Gamma^r,+)$ for some $r>0$, and using Fact [Fact 45](#F:minimal fibers in Gamma){reference-type="ref" reference="F:minimal fibers in Gamma"}, we may further assume that $r=n$. As this isomorphism is witnessed by an isomorphism of groups, the result follows by compactness and Lemma [Lemma 46](#L: inf neigh in Gamma){reference-type="ref" reference="L: inf neigh in Gamma"}. ◻ ## The valued field and the residue field {#ss: valued field and residue field} For this section $D$ is either the valued field $K$ or the residue field $\textbf{k}$ when ${\mathcal K}$ is power bounded $T$-convex. We first describe the infinitesimal group $\nu_D$ and then show how in these situations the type-definable group $\nu_D$ gives rise to a definable, differentiable local group with respect to either $K$ or $\textbf{k}$. ### Local differential groups {#sss:local differ group} Let ${\mathscr F}$ be an expansion of either a real closed field or a valued field with valuation $v$. Let us recall some standard definitions. We later apply them for when ${\mathscr F}=D$. **Definition 48**. Given $U\subseteq {\mathscr F}^n$ open, a map $f:U\to {\mathscr F}^m$ is *differentiable* at $x_0\in U$ if there exists a linear map $D_{x_0}f:{\mathscr F}^n\to {\mathscr F}^m$ such that: In the ordered case: $$\lim_{x\to x_0}\frac{|f(x)-f(x_0)-(D_{x_0}f)\cdot (x-x_0)|}{|x-x_0|}=0,$$ and in the valued case: $$\lim_{x\to x_0}\left[ v(f(x)-f(x_0)-(D_{x_0}f)\cdot (x-x_0))-v(x-x_0)\right]=\infty.$$ Also, in the valued setting, $f$ is called *strictly differentiable at $x_0$* if there exists a linear map $D_{x_0}f$ which satisfies: for all $\epsilon\in \Gamma$ there exists $\delta\in \Gamma$, such that for all $x_1,x_2\in B_{>\delta}(x_0)$, $$v(f(x_1)-f(x_2)-(D_{x_0}f)\cdot(x_1-x_2))-v(x_1-x_2)>\epsilon.$$ We are going to work extensively with the notion of *a local group*, so we first recall some additional definitions: **Definition 49**. A *local group with respect to ${\mathscr F}$* is a tuple $\mathcal{G}=(X,m,\iota,e)$ such that - $X$ is a topological space and there exists a homeomorphism $\varphi: U\to V$ between an open neighbourhood of $e$ in $X$ and an open $V\subseteq{\mathscr F}^n$, for some $n$. - the maps $m:X\times X\dashrightarrow X$ and $\iota:X\dashrightarrow X$ are continuous *partial* functions, with open domains containing $(e,e)$ and $e$, respectively. such that the following equalities hold **whenever both sides of the equations are defined**: 1. For any $x\in X$, $x=m(x,e)=m(e,x)$ 2. For any $x\in X$, $e=m(x,\iota(x))=m(\iota(x),x)$. 3. For all $x,y,z\in X$, $m(x,m(y,z))=m(m(x,y),z)$. The local group $\mathcal{G}$ is *differentiable* if $\varphi( m(\varphi^{-1}(x), \varphi^{-1}(y))$ and $\varphi(\iota(\varphi^{-1}(x))$ are differentiable. *Strictly differentiable* local groups are defined analogously. The local group $\mathcal{G}$ is *definable* in ${\mathscr F}$, if $X,m,\iota$ and $\varphi$ are definable. For $G$ a definable group, a *local subgroup with respect to ${\mathscr F}$* is a local subgroup with respect to ${\mathscr F}$ whose universe is a subset of $G$ and whose multiplication agrees with $G$-multiplication. **Definition 50**. Let $\mathcal{G}=(X,m, e, \iota)$ and $\mathcal{G}'=(X',m', e', \iota')$ be local groups. A *homomorphism* of local groups $f:\mathcal{G}\to\mathcal{G}'$ is a continuous function $f: U\to X'$, where $U\subseteq X$ is an open neighbourhood of $e$, such that $f(e)=e'$ and $f(m(x,y))=m(f(x), f(y))$ in a neighbourhood of $e$. Such an $f$ is a *local isomorphism* if, in addition, it is a homeomorphism onto its image. If ${\mathcal G}, {\mathcal G}'$ are (strictly) differentiable local groups, then such an $f$ is *(strictly) differentiable* if $\varphi'\circ f\circ \varphi^{-1}$ is (strictly) differentiable. For $G$ a definable group, a local subgroup ${\mathcal G}$ is called *normal in $G$* if for every $g\in G$, the map $x\mapsto x^g$ restricts to an automorphism of ${\mathcal G}$. In particular -- in the notation of local subgroups -- there for any definable open neighbourhood $U\subseteq X$ of $e$ there exists an open neighbourhood $V\subseteq X$ of $e$ such that $x^g$ maps $V$ into $U$. Assume further that every definable function in ${\mathscr F}$ is (strictly) *generically differentiable*, i.e. for every definable open $U\subseteq {\mathscr F}^n$, and definable $f:U\to {\mathscr F}$, the set of points $x\in U$ such that $f$ is not (strictly) differentiable at $x$ has empty interior. See [@HaHaPeVF Section 4.3] for more information. Now, if ${\mathcal G}, {\mathcal G}'$ as above are (strictly) differentiable local groups and $f:{\mathcal G}\to {\mathcal G}'$ is a definable homomorphism of local groups then $f$ is also (strictly) differentiable. Indeed, since definable functions are generically (strictly) differentiable with respect to ${\mathscr F}$, the corresponding map $\varphi'\circ f\circ \varphi$ is ${\mathscr F}$-(strictly) differentiable at a generic point, and then, using the local group structure, it is (strictly) differentiable n an open neighbourhood of $e$. **Definition 51**. Let $G$ be a definable group in ${\mathcal M}$ and let $\mathcal{G}=(X,\cdot,^{-1})$ be a differentiable normal local subgroup of $G$ with respect to ${\mathscr F}$, witnessed by a map $\varphi:X\to {\mathscr F}^n$. The *Adjoint map with respect to ${\mathscr F}$* is the map $\mathrm{Ad}^{{\mathcal G}}_{\mathscr F}:G\to \mathop{\mathrm{GL}}_n({\mathscr F})$, which assigns to every $g\in G$ the Jacobian matrix of the map $D_e(\varphi\circ \tau_g\circ\varphi^{-1})$. By the chain rule in ${\mathscr F}$, $\mathrm{Ad}^{\mathcal G}_{\mathscr F}$ is a group homomorphism. Note that while the matrix $D_e(\tau_g)$ may depend on the choice of $\varphi$ (up to conjugation), the definable group $\ker(\mathrm{Ad}^{\mathcal G}_{\mathscr F})$ does not. ### The infinitesimal group Under the assumptions of this section, the sort $D$ is an SW-uniformity expanding a field. Therefore, if $X\subseteq D^k$ is definable, $f: X\to D^m$ is a definable injection, then by possibly shrinking $X$, but not its rank, we may compose $f$ with a projection $\pi: X\to D^{\mathrm{dp\text{-}rk}(X)}$ such that $\pi\circ f (X)$ is a basic open set. Furthermore, every definable function in $D$ is generically differentiable with respect to $D$ in the o-minimal case and generically strictly differentiable in the valued case. Indeed, if $D=\textbf{k}$ in the power bounded $T$-convex case, then $\textbf{k}$ is a o-minimal so every definable function is generically differentiable. In the other cases, it follows from 1-$h$-minimality ([@AcHa Proposition 3.12]). **Fact 52**. *Let $G$ be a definable $D$-group, locally strongly internal to $D$ over $A$, witnessed by the injection $f:X\to D^n$, with $\mathrm{dp\text{-}rk}(X)=n$, the $D$-rank of $G$. Given $c\in X$, generic over $A$, $$\nu_D(G)=\{f^{-1}(U)c^{-1}:U\subseteq D^n \text{ open containing } f(c)\}.$$* *Proof.* By [@HaHaPeGps Proposition 5.6], for $c\in X$ $A$-generic $\nu_X(c)=f^{-1}(\mu(f(c))$, where $\mu(f(c))$ is the infinitesimal neighbourhood of $f(c)$ in the topology on $D$. The result now follows. ◻ **Lemma 53**. *Let $G$ be a definable $D$-group locally strongly internal to $D$.* *Then there exists a definable differentiable local normal subgroup ${\mathcal G}=(X,\cdot, ^{-1},e)$ with respect to the field $D$, with $\nu_D(G)\vdash X$. When $D=K$ the local group is strictly differentiable.* *If $G$ is definable over some ${\mathcal K}_0\prec {\mathcal K}$ then the local group and the map $\varphi:X\to D^n$ witnessing it can be found definable over ${\mathcal K}_0$.* *Proof.* Let $\nu_D=\nu_D(G)$. By Fact [Fact 52](#F: nu in K){reference-type="ref" reference="F: nu in K"}, $\nu_D\vdash X$, for some definable $\nu_D$-open set $X\subseteq G$, and there exists a definable injection $\varphi: X\to D^n$, with $\varphi(X)$ a definable open subset of $D^n$ and $n$ the $D$-rank of $G$ (indeed, in the notation of the above Fact, replace $Xc^{-1}$ by $X$). Let $\widehat {\mathcal K}\succ {\mathcal K}$ be a $|{\mathcal K}|^+$ saturated elementary extension. By [@HaHaPeGps Theorem 7.4(1,2), Theorem 7.7(1), Theorem 7.11(1)], $\nu_D(\widehat {\mathcal K})$ is a (differentiable) Lie group with respect to the structure induced by $\varphi$. Furthermore, since every definable function in the valued field case is generically strictly differentiable, a similar proof shows in this case that $\nu_D(\widehat {\mathcal K})$ is a strictly differentiable Lie group. Furthermore, $g\nu_D g^{-1}=\nu_D$ for any $g\in G({\mathcal K})$ (Fact [\[F: properties of nu\]](#F: properties of nu){reference-type="ref" reference="F: properties of nu"}). Using compactness, we can endow $X$ with the structure of a (strictly) differentiable local normal subgroup of $G$ with respect to the field $D$. Lastly, if $G$ is definable over ${\mathcal K}_0$ then since the existence of $X$ and $\varphi$ with the desired properties is first order, such can be found over ${\mathcal K}_0$ as well. ◻ Combining the last lemma with Definition [Definition 51](#N:Ad){reference-type="ref" reference="N:Ad"} we can find a definable group representation $\mathrm{Ad}_D^{\mathcal G}: G\to \mathop{\mathrm{GL}}_n(D)$, for $n$ the $D$-rank of $G$. As noted after Definition [Definition 51](#N:Ad){reference-type="ref" reference="N:Ad"}, the map $\mathrm{Ad}_D^{\mathcal G}$ depends on ${\mathcal G}$ (i.e. on $X$ and $\varphi$), only up to a change of coordinates. In particular, the group $\ker(\mathrm{Ad}^{\mathcal G}_D)$ does not depend on the choice of ${\mathcal G}$ and the image $\mathrm{Ad}_D^{\mathcal G}(G)$ is independent of ${\mathcal G}$, up to conjugation. As for the latter, since we do not care about the particular embedding in $\mathop{\mathrm{GL}}_n(D)$, the choice of ${\mathcal G}$ is unimportant, and **we will write, from now on, $\mathrm{Ad}_D(G)$ without specifying any choice of local subgroup ${\mathcal G}$.** For future reference we single out the following corollary of Lemma [Lemma 53](#L:existence of local group){reference-type="ref" reference="L:existence of local group"} and the above discussion: **Remark 54**. Given a $D$-group $G$ defined over a model ${\mathcal K}_0$ the subgroup $\ker(\mathrm{Ad}_D(G))$ is definable over ${\mathcal K}_0$. # Groups locally strongly internal to $\Gamma$ As above, ${\mathcal K}$ denotes a saturated model of one of our valued fields, $\Gamma$ its valued group. Since $\Gamma^n$ and $(K/{\mathcal O})^n$ are commutative, so are their (local) subgroups. In the present and the next section, we show that this is reflected in a strong sense in definable $\Gamma$-groups or $K/{\mathcal O}$-groups. For $\Gamma$-groups, we get a clean result: definable $\Gamma$-groups contain infinite definable normal abelian subgroups. We prove (keeping the notation and conversions of the previous sections): **Proposition 55**. *Assume that $G$ is a definable group locally strongly internal to $\Gamma$. Then $G$ contains a definable normal subgroup $G_1$ of finite index, defined over the same parameters as $G$, such that $\nu_\Gamma\vdash Z(G_1)$. In particular, $G$ contains a definable (over the same parameter set) infinite normal abelian subgroup.* The proof splits between the $p$-adic case (where $\Gamma$ is discrete) and the remaining cases (where $\Gamma$ is dense and o-minimal). ## ${\mathcal K}$ $p$-adically closed We assume that ${\mathcal K}$ is $p$-adically closed and thus $\Gamma$ is a model of Presburger arithmetic. Let $\mathbb{Z}$ be a prime (and minimal) model for $\Gamma$. We denote by $\mathbb{Z}_{Pres}$ the structure $(\mathbb{Z},+,<)$.\ Before proceeding to the proof of Proposition [Proposition 55](#P: Gamma){reference-type="ref" reference="P: Gamma"} in this setting, we need some preparatory results: **Lemma 56**. *For any definable family, $\{X_t\}_{t\in T}$, of subsets of $\Gamma^n$ the family $\{X_t\cap {\mathbb {Z}}^n\}_{t\in T}$ is definable in ${\mathbb {Z}}_{Pres}$.* *Proof.* Because ${\mathcal K}$ is $p$-adically closed, $\Gamma$ is stably embedded, so we may assume that $T\subseteq\Gamma^k$ for some $k$. Since in Presburger arithmetic types over $\mathbb{Z}$ are (uniformly) definable, the family $\{X_t\cap {\mathbb {Z}}^n:t\in T\}$ is definable in ${\mathbb {Z}}_{Pres}$. See [@CoVo Theorem 0.7] (and also [@delon-def]). ◻ **Lemma 57**. *Let $\{X_t:t\in T\}$ be a definable family of subsets of $\Gamma^n$ and assume that for all $t\in T$, $X_t\cap \mathbb{Z}^n$ contains a subgroup of $\mathbb{Z}^n$ of finite index. Then there is a uniform upper bound on $l(t)$, the minimal $l\in {\mathbb{N}}$ such that $X_t\cap {\mathbb {Z}}^n$ contains a subgroup ${\mathbb {Z}}^n$ of index $l$.* *Proof.* Assume towards a contradiction that there is no bound on $l(t)$ for $t\in T$. So the following type is consistent: $$\rho(t):=\{D\not\subseteq X_t: D\subseteq \mathbb{Z}^n \text{ finite, generating a definable subgroup of finite index}\},$$ contradicting the assumption. ◻ **Lemma 58**. 1. *Let $Y\subseteq \Gamma^n$ be a definable set. If $Y\cap \mathbb{Z}^n$ contains a subgroup of $\mathbb{Z}^n$ of finite index, then $\mathrm{dp\text{-}rk}(Y)=n$.* 2. *Every finite index subgroup $H\leq \Gamma^n$ is definable.* *Proof.* By Fact [Lemma 56](#F: def over Z){reference-type="ref" reference="F: def over Z"}, $Y\cap \mathbb{Z}^n$ is definable in ${\mathbb {Z}}_{Pres}$, as a subset of $\mathbb{Z}^n$. Since it contains a finite index subgroup, it has dp-rank $n$. Thus, we have $\mathbb Z_{pres}\prec \Gamma$ and $\mathrm{dp\text{-}rk}(Y\cap \mathbb Z^n)=n$. It follows by [@HaHaPeVF Lemma 3.10] that $\mathrm{dp\text{-}rk}(Y)=n$. For Clause (2) let $H\le G$ be a definable subgroup of finite index, and note that since $H$ has finite index, there is $k\in \mathbb N$ such that the map $x\mapsto kx$ sends $\Gamma^n$ into $H$. Because $k\Gamma^n$ has finite index in $\Gamma^n$, it follows that $H$ is a union of finitely many cosets of $k\Gamma^n$, $H$ is definable. ◻ Recall Definition [Definition 44](#D:box){reference-type="ref" reference="D:box"} of a $\Gamma$-box. **Lemma 59**. *Let $Y\subseteq\Gamma^n$ be a definable set such that $Y\cap {\mathbb {Z}}^n$ contains a subgroup $H$ of ${\mathbb {Z}}^n$ of finite index. Assume that $\{f_t\}_{t\in T}$ is a definable family of definable functions $f_t:Y \to Y$ such that for all $a,b\in Y$ with $a+b\in Y$, we have $f_t(a+b)=f_t(a)+f_t(b)$. Then:* 1. *For every $t\in T$, $f_t(H)\subseteq \mathbb{Z}^n$.* 2. *The family $\{f_t\restriction H:t\in T\}$ is uniformly definable in ${\mathbb {Z}}_{Pres}$ and therefore finite.* *Proof.* Assume everything is definable over some parameter set $A$. By stable embeddedness of $\Gamma$, the family $\{f_t: t\in T\}$ is uniformly definable in $\Gamma$ so we may assume that $T\subseteq\Gamma^k$. Since $H$ is a subgroup of finite index of $\mathbb{Z}^n$ it is generated by some finite set $\{m_1,\dots,m_s\}\subseteq\mathbb Z^n$. \(1\) Fix some $t\in T$. It suffices to prove that each coordinate function of $f_t$ sends $H$ into $\mathbb{Z}$. So we may assume $f_t:Y\to \Gamma$. Let $c\in Y$ be $A$-generic in $Y$. Since $\mathrm{dp\text{-}rk}(Y)=n$, it follows from cell decomposition, [@ClucPresburger Theorem 1], and [@OnVi Lemma 3.4] that there is an $A$-definable $n$-dimensional $\Gamma$-box, $B=\prod_i J_i\subseteq Y$, centred at $c=(c_1,\ldots, c_n)\in B$, such that $$(f_t\restriction B)(x)=\sum_i s_i\left( \frac{x-t_i}{k_i}\right) +\gamma,$$ with $\gamma\in \Gamma^n$, $s_i,t_i,k_i\in \mathbb N$ and $J_i=I_i\cap \{x-t_i\in P_{k_i}\}$, for some infinite interval $I_i$. By shrinking $B$, if needed (over the same parameters), we may assume that $B$ is a product of boxes of the form $I_i\cap P_k(x_i-t_i)$ (i.e., that $k_i=k$ for all $i$). Note that for every $\bar r\in\mathbb Z^n$, we have by the above description of $f_t$, that $f_t(c+k\bar r)-f_t(c)\in \mathbb{Z}$. In particular, if $m_i$, $1\leq i\leq s$, is any of the generators of $H$ we fixed earlier then we have $c, c+km_i$ and $km_i$ all in $Y$, so by the addivity assumptions, $$f_t(km_i)=f_t(c+k\bar r)-f_t(c)\in \mathbb Z.$$ However, since $f_t(km_i)=kf_t(m_i)$ this implies that $f_t( m_i)\in \mathbb{Z}$ and, as this is true of a set of generators of $H$, we see that $f_t(H)\subseteq \mathbb{Z}$, as claimed. \(2\) The first part of the claim is a consequence of Fact [Lemma 56](#F: def over Z){reference-type="ref" reference="F: def over Z"} using Lemma [Lemma 58](#F: full dp-rank, presburger){reference-type="ref" reference="F: full dp-rank, presburger"}. The second part follows from quantifier elimination in Presburger arithmetic, by noting that any definable family of group homomorphisms is finite (see also [@OnVi Fact 2.10]). ◻ We can now give the proof of Proposition [Proposition 55](#P: Gamma){reference-type="ref" reference="P: Gamma"} in $p$-adic case. We assume that $G$ is locally strongly internal to $\Gamma$. By Lemma [Lemma 47](#L: nu in Gamma){reference-type="ref" reference="L: nu in Gamma"} there are a definable $X\subseteq G$, with $\nu_{\Gamma}\vdash X$, and a definable function, $f:X\to \Gamma^n$, with $\mathrm{dp\text{-}rk}(X)=n$ for $n$ the $\Gamma$-rank of $G$. For simplicity of notation, identify $X$ with its image in $\Gamma^n$ and $e_G$ with $0_{\Gamma^n}$. We may further assume that, restricted to $X$, $G$-multiplication coincides with addition and the same for the inverse. By Lemma [Lemma 47](#L: nu in Gamma){reference-type="ref" reference="L: nu in Gamma"}, we may further assume that $\nu_{\Gamma}$ is the intersection of $\Gamma$-boxes around $0$. We fix one such $\Gamma$-box $B\subseteq X\subseteq \Gamma^n$, $\nu_\Gamma\vdash B$. By [@HaHaPeGps Proposition 5.8], $g\nu_{\Gamma} g^{-1}=\nu_{\Gamma}$ for every $g\in G$ and thus $\nu_{\Gamma}\vdash B^g\cap B$. By compactness, for every $g\in G$, there exists a $\Gamma$-box $B_0\subseteq B\cap B^g$ around $0$. By Lemma [Lemma 59](#L: family of functions, presburger){reference-type="ref" reference="L: family of functions, presburger"}(1), $B\cap \mathbb{Z}^n$ is a subgroup of $\mathbb{Z}^n$ of finite index (though $B^g$ need not be contained in $\Gamma^n$). By Lemma [Lemma 57](#L:bound on index){reference-type="ref" reference="L:bound on index"} there is some natural number $k$ such that for any $g\in G$, $B^g\cap B$ contains a box $B_g$ with $B_g\cap \mathbb{Z}^n$ a subgroup of index at most $k$ in ${\mathbb {Z}}^n$. Consequently, there exists some subgroup $H\subseteq \mathbb{Z}^n$ of finite index such that $H\subseteq B\cap B^g\cap \mathbb{Z}^n$ for all $g$. Let $Y=\bigcap\limits_{g\in G}B^g$. It is a definable set, contained in $B\subseteq \Gamma^n$, invariant under conjugation by all elements of $G$ and containing $H$. Let $Y_0:=Y\cap \mathbb{Z}^n$ (note that $H\subseteq Y_0$) and for every $g\in G$ let $\tau_g:Y\to Y$ denote the restriction of conjugation by $g$ to $Y$. By Lemma [Lemma 59](#L: family of functions, presburger){reference-type="ref" reference="L: family of functions, presburger"}(1), $\tau_g(H)\subseteq \mathbb{Z}^n$. By Lemma [Lemma 59](#L: family of functions, presburger){reference-type="ref" reference="L: family of functions, presburger"}(2), $\{\tau_g\restriction H\}_{g\in G}$ is a family of group homomorphisms uniformly definable in $\mathbb{Z}$, so it is finite. We may now replace $H$ by the (finite) intersection of all $\tau_g(H)$, and obtain another subgroup of ${\mathbb {Z}}^n$ of finite index. Thus, we may assume that $H$ is invariant under all $\tau_g$. Let $R$ be a finite set of generators for $H$ and let $E(g,h)$ be the definable equivalence relation on $G$ given by $d^g=d^h$ for all $d\in R$. Since addition on $H$ coincides with $\Gamma$=multiplication, and for all $g,h\in G$ both $\tau^g\restriction H$ and $\tau^h\restriction H$ are $\Gamma$-homomorphism preserving $H$,it follows that $E(g,h)$ holds if and only if $\tau_g\restriction H=\tau_h\restriction H$. The definable quotient $G/E$ can be identified with a finite subgroup of $\mathop{\mathrm{Aut}}(H)$, and the map $\sigma:G \to G/E$ is a definable group homomorphism. Its kernel, call it $G_1$, is a definable normal subgroup of $G$ of finite index, that -- by definition -- centralizes $H$, hence $H\subseteq Z(G_1)$. We claim that $\nu_{\Gamma}\vdash Z(G_1)$. By Lemma [Lemma 58](#F: full dp-rank, presburger){reference-type="ref" reference="F: full dp-rank, presburger"}(2), $H$ is definable in ${\mathbb {Z}}_{Pres}$ and $Z(G_1)$ contains all finite boxes of the form $[-a,a]^n\cap H$, for $a\in \mathbb N$. Since $H$ is definable, $Z(G_1)$ must contain a set of the form $I^n\cap H({\mathcal K})$, for an infinite interval $I\subseteq\Gamma$, so in particular, it contains a $\Gamma$-box. It follows that $\nu_{\Gamma}\vdash Z(G_1)$ and therefore $Z(G_1)$ is a definable infinite normal subgroup of $G$. 0◻$_{(\Gamma\text{ Presburger})}$\ We postpone the proof that $G_1$ can be taken to be definable over the same parameters as $G$ to the next section (since the proof is similar). ## ${\mathcal K}$ is power bounded $T$-convex or $V$-minimal We now assume that ${\mathcal K}$ is either power bounded $T$-convex or $V$-minimal, so that $\Gamma$ is an (o-minimal) ordered vector space. Recall Definition [Definition 44](#D:box){reference-type="ref" reference="D:box"} of a $\Gamma$-box. By the description of $\nu_\Gamma$ (Lemma [Lemma 47](#L: nu in Gamma){reference-type="ref" reference="L: nu in Gamma"}), there exists a definable subset $X\subseteq G$, with $\nu\vdash X$, definably isomorphic to a $\Gamma$-box (around $0$) in $\Gamma^n$. Identifying $X$ with its image, we assume (by compactness) that for every $x,y\in X$ with $xy^{\pm 1} \in X$ we have $xy^{\pm 1}=x\pm y$. Because $\Gamma$ is o-minimal, and $X$ is identified with a $\Gamma$-box in $\Gamma^n$, there is a definable neighbourhood base, $\{W_t: t\in T\}$, of $0$ in $X$. For every $g\in G$, let $\tau_g$ denote the map $x\mapsto x^g$, and for $g,h\in G$ write $g\sim h$ if $\tau_g$ and $\tau_h$ have the same germ at $0$, namely there exists an open neighbourhood $U\subseteq\Gamma^n$ of $0$, such that $\tau_g|U=\tau_h|U$. By the above, this is a definable equivalence relation. Let $\sigma$ be the definable function mapping $g\in G$ to $[g]_\sim$. It is a homomorphism of groups, with the group operation on the set of equivalence classes given by composition of germs. We know that for every $g\in G$, $\nu^g=\nu$ (as types), thus there is some $W_t\subseteq X$ such that $W_t^g\subseteq X$ is also a neighbourhood of $0$. So $\sigma(G)$ can be viewed as a definable family of definable germs on $X$. Since $\Gamma$ is a pure ordered vector space over a field $F$ (the field of exponents in the o-minimal $T$), it follows that $\sigma(G)$ is finite. Indeed, by [@vdDries §1.7 Corollary 7.6], each germ is the restriction of some $T\in \mathop{\mathrm{GL}}_n(F)$ to an open neighbourhood of $0$. Since each such $T$ is $\emptyset$-definable, a definable family of such germs must be finite. Hence, the definable group $G_1:=\ker(\sigma)$ has finite index in $G$. By definition, for every $g\in G_1$ there exists a $\tau_\Gamma$-open neighbourhood of $0$, on which $x^g=x$. Thus, $G_1\subseteq C_G(\nu_\Gamma)$ (recall Definition [Definition 36](#D: centralizer){reference-type="ref" reference="D: centralizer"}). Since $X\subseteq\Gamma^n$ is a $\Gamma$-box, it is definably connected, so we may apply Lemma [Lemma 38](#L:def connected){reference-type="ref" reference="L:def connected"} and conclude that $X\subseteq C_G(\nu_\Gamma)$ By Lemma [Lemma 14](#L:nu of subgroup){reference-type="ref" reference="L:nu of subgroup"}, $\nu_\Gamma\vdash G_1$. Thus, $\nu_\Gamma\vdash X\cap G_1\subseteq Z(G_1)$, as claimed. Since $G_1$ is normal in $G$ it follows that $Z(G_1)$ is a definable infinite abelian normal subgroup of $G$.\ Finally, let us verify that in both the current case, and in the $p$-adically closed case, we can replace $G_1$ with a subgroup defined over the same parameters as $G$. Without loss of generality, assume that $G$ is $\emptyset$-definable and let $\{G_s:s\in S\}$ be a $\emptyset$-definable family of normal subgroups of $G$ whose index in $G$ is $[G:G_1]$, and such that $G_1=G_s$ for some $s\in S$. We may further assume that for each $s\in S$, $Z(G_s)$ has a definable subset which is in definable bijection with a $\Gamma$-box (in $\Gamma^n$) around $0$. By Lemma [Lemma 46](#L: inf neigh in Gamma){reference-type="ref" reference="L: inf neigh in Gamma"}, $\nu_\Gamma \vdash Z(G_s)$. By Fact [Fact 17](#F: Baldwin saxl){reference-type="ref" reference="F: Baldwin saxl"}, $\bigcap_s G_s$ has finite index in $G$. It is $\emptyset$-definable, and its centre contains $\nu_\Gamma$. We have thus finished the proof of Proposition [Proposition 55](#P: Gamma){reference-type="ref" reference="P: Gamma"} in all cases. [\[section: Gamma\]]{#section: Gamma label="section: Gamma"} # Groups locally strongly internal to $K/{\mathcal O}$. We still assume ${\mathcal K}$ is a saturated model in one of our cases. In the present section, we extend the results of the previous section from $\Gamma$-groups to $K/{\mathcal O}$ groups. The result we get is somewhat weaker. Explicitly, we prove: **Proposition 60**. *Let ${\mathcal K}_0\prec {\mathcal K}$ be an elementary substructure, $G$ a ${\mathcal K}_0$-definable $K/{\mathcal O}$-group not locally strongly internal to $\textbf{k}$. Let $\mathcal A=\{\lambda_s:s\in S\}$ be a ${\mathcal K}_0$-definable family of automorphisms of $G$, fixing the partial type $\nu_{K/{\mathcal O}}$. Then there is a $K_0$-definable normal abelian subgroup $N\leq G$ which is stabilized under all of the $\lambda_g$ such that $\nu_{K/{\mathcal O}}\vdash N$. In particular, $\mathrm{dp\text{-}rk}(N)$ is at least the $K/{\mathcal O}$-rank of $G$.* **Remark 61**. For convenience of presentation, we chose in Proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"} a uniform statement for all cases. However, in fact, the results are slightly stronger in each case. For $p$-adically closed fields, the assumption that $G$ is not locally strongly internal to $\textbf{k}$ is vacuous, while in the remaining cases we obtain a group invariant under all definable automorphisms of $G$ (without the need to fix a family in advance). We say that a subgroup $H\leq G$ is $\mathcal A$-invariant if for every $s\in S$, $\lambda_s(H)=H$. Since the proposition does not make any assumptions on $\mathcal A$ we may assume that $\mathcal A$ contains the family of all conjugations by elements of $G$, and this an $\mathcal A$-invariant subgroup will be in particular normal in $G$. As in Section [\[section: Gamma\]](#section: Gamma){reference-type="ref" reference="section: Gamma"} , the proof splits between the $p$-adically closed case and the remaining cases. ## ${\mathcal K}$ is $p$-adically closed Since ${\mathcal K}$ is $P$-minimal and saturated, there is a finite extension, $\mathbb F$ of $\mathbb Q_p$ embedding (as a valued field) into $K$. We identify the image of some fixed such embedding with a valued subfield of $K$. Since the value group $\Gamma_{\mathbb{F}}$ is isomorphic to $\mathbb{Z}$, as ordered abelian groups, we identify $\Gamma_{\mathbb{F}}$ with $\mathbb{Z}$ and view it as a prime (and minimal) model for $\Gamma$. We denote ${\mathbb {Z}}_{Pres}$ the structure $({\mathbb {Z}}, +, <)$. The following fact is an easy consequence of the results of [@HaHaPeGps]: **Fact 62**. *Let ${\mathcal K}_0\equiv {\mathcal K}$, ${\mathcal K}_0$ not necessarily saturated, with ${\mathcal O}_0$ its valuation ring. Let $\mathrm{Tor}(K_0/{\mathcal O}_0)$ denote the torsion subgroup. Then* 1. *$\mathrm{Tor}(K_0/{\mathcal O}_0)=\{a\in K_0/{\mathcal O}_0:v(a)\in \mathbb{Z}\}$.* 2. *$\mathrm{Tor}(K_0/{\mathcal O}_0)$ is a finite direct sum of Prüfer $p$-groups and is isomorphic to $\mathbb{F}/{\mathcal O}_{\mathbb{F}}$. In particular, $\mathrm{Tor}(K_0/{\mathcal O}_0)$ is a $p$-group.* 3. *Every ball in $(K_0/{\mathcal O}_0)^n$ centred at $0$ contains $\mathrm{Tor}(K_0/{\mathcal O}_0)^n$ and the $p^k$-torsion points are exactly the points $b\in (K/{\mathcal O})^n$ with $v(b)\geq -k$.* *Proof.* We first argue in ${\mathcal K}$: Clause (1): If $v(a)=n\in \mathbb Z_{<0}$ then $p^n a\in {\mathcal O}$, so $a+{\mathcal O}\in \mathrm{Tor}({\mathcal K}/{\mathcal O})$. The reverse inclusion follows from [@HaHaPeGps Lemma 3.1](3). Clause (2): By [@HaHaPeGps Lemma 3.1](3), every torsion element of $(K/{\mathcal O})^n$ is in $(\mathbb F/{\mathcal O}_{\mathbb F})^n$, and with the previous clause (2) follows for ${\mathcal K}$ since $\mathbb F/{\mathcal O}_\mathbb F$ is isomorphic to a of Prüfer $p$-groups. Clause (3) follows from the structure of the Prüfer group.\ We now argue in $K_0$: By the basic properties of the Prüfer group, every proper subgroup (and in particular, the subgroup of $p^{k}$-torsion points) is a finite subgroup. Thus, $\mathrm{Tor}(K/{\mathcal O})\subseteq\mathop{\mathrm{acl}}(\emptyset)$, and because ${\mathcal K}$ is saturated enough, the results remain true in ${\mathcal K}_0$. ◻ **Lemma 63**. *Let $G$ be a definable $K/{\mathcal O}$-group. Let $H_1, H_2\leq G$ be definable subgroups, and $f_i:H_i\to (K/{\mathcal O})^n$ ($i=1,2$) definable group embeddings whose respective images are open balls in $(K/{\mathcal O})^n$, where $n$ is the $K/{\mathcal O}$-rank of $G$. Then $\mathrm{dp\text{-}rk}(H_1\cap H_2)=n$ and $$\mathrm{Tor}(H_1)=f_1^{-1}(\mathbb{F}/\mathcal{O}_{\mathbb{F}})=\mathrm{Tor}(H_2)=f_2^{-1}(\mathbb{F}/\mathcal{O}_{\mathbb{F}}).$$* *In particular, all definable subgroups of $G$ of dp-rank $n$ that can be definably embedded into $(K/{\mathcal O})^n$ share the same torsion subgroup.* *Proof.* The assumptions and the conclusions are invariant under naming new constants, so we may assume that $\mathbb{F}$ is named in ${\mathcal K}$ and so we may apply the results from [@HaHaPeGps]. By the construction of $\nu_{K/{\mathcal O}}$ (see Lemma [Lemma 43](#F: nu in K/O){reference-type="ref" reference="F: nu in K/O"} and Remark [Remark 13](#R: nu lives on any definable subgroup witnessing){reference-type="ref" reference="R: nu lives on any definable subgroup witnessing"}) we have $\nu_{K/{\mathcal O}}\vdash H_i$, $i=1,2$, hence $\nu_{K/{\mathcal O}}\vdash H_1\cap H_2$. By Lemma [Lemma 43](#F: nu in K/O){reference-type="ref" reference="F: nu in K/O"}, this implies that $\mathrm{dp\text{-}rk}(H_1\cap H_2)=n$. Since $f_i(H_i)$ is an open ball, for $i=1,2$, it follows from Fact [Fact 62](#F: torsion){reference-type="ref" reference="F: torsion"} that $\mathrm{Tor}(H_i)=f_i^{-1}((\mathbb{F}/{\mathcal O}_{\mathbb{F}})^n)$. As $\mathrm{dp\text{-}rk}(H_1\cap H_2)=n$ also $\mathrm{dp\text{-}rk}(f_i(H_1\cap H_2))=n$ for $i=1,2$, so by [@HaHaPeGps Lemma 3.6] $f_i(H_1\cap H_2)$ has non-empty interior, thus contains a sub-ball of $(K/{\mathcal O})^n$. Therefore, (since it is a group) it also contains a ball centred at $0$. Thus, $(\mathbb F/{\mathcal O}_{\mathbb F})^n\subseteq f_i(H_1\cap H_2)$ and hence $f_i^{-1}((\mathbb F/{\mathcal O}_{\mathbb F})^n)\subseteq H_1\cap H_2$. We conclude $$\mathrm{Tor}(H_1)=f_1^{-1}((\mathbb F/{\mathcal O}_{\mathbb F})^n)=f_2^{-1}((\mathbb F/{\mathcal O}_{\mathbb F})^n)=\mathrm{Tor}(H_2),$$ as needed. ◻ We can now prove Proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"} in the $p$-adic case. *Proof of proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"} in the $p$-adic case..* Recall that $\mathcal A=\{\lambda_s:s\in S\}$ is a definable family of automorphisms of $G$. First, we show that an infinite $\mathcal A$-invariant abelian subgroup of $G$ is definable in ${\mathcal K}$ and then we construct one that is definable over $K_0$ as needed. By Section [5.1](#ss:nu in K/O){reference-type="ref" reference="ss:nu in K/O"} we can find a definable subgroup $H_0$, $\nu_{K/{\mathcal O}}\vdash H_0\leq G$, that is definably isomorphic to an open ball in $(K/{\mathcal O})^n$ centred at $0$, where $n$ is the $K/{\mathcal O}$-rank of $G$. Let $f:H_0\to (K/{\mathcal O})^n$ be a group embedding witnessing this (note that $H_0$ and $f$ are not claimed to be ${\mathcal K}_0$-definable). Let $H=\bigcap\limits_{s\in S} H_0^{\lambda_s}$, where $H_0^{\lambda_s}=\lambda_s(H_0)$. It is a definable $\mathcal A$-invariant abelian subgroup, and by the previous lemma it is infinite, as claimed. We shall now replace $H$ by a group defined over $K_0$. By Lemma [Lemma 63](#L:full subgroups have the same torsion){reference-type="ref" reference="L:full subgroups have the same torsion"}, $\mathrm{Tor}(H_0^{\lambda_s})=f^{-1}((\mathbb F/{\mathcal O}_{\mathbb F})^n)$, for every $s\in S$. It follows, using compactness and saturation, that there is $r<\mathbb{Z}$ such that $B_{>r}(0)\subseteq f(H)$. Let $r_0$ be the minimal such $r$. Assume that $H$ and $f$ are definable over some $t_0\in {\mathcal K}$ and let $\{(H_t,f_t):t\in T\}$ be the corresponding $K_0$-definable family of subgroups of $G$ and definable group embeddings $f_t:H_t\to (K/{\mathcal O})^n$, such that $(H,f)= (H_{t_0},f_{t_0})$. Note that the statement that $H_{t_0}$ is $\mathcal A$-invariant is a first order property of $t_0$, defined over $K_0$. Thus we may assume that each $H_t$ is $\mathcal A$-invariant. Define $\eta:T\to \Gamma$ by $$\eta(t)=\min\{r\in \Gamma:B_{>r}(0)\subseteq f_t(H_t)\}.$$ In particular, $\eta(t_0)\leq r_0$ and by Lemma [Lemma 63](#L:full subgroups have the same torsion){reference-type="ref" reference="L:full subgroups have the same torsion"}, if $\eta(t), <\mathbb{Z}$ then $\hat H:=f_{t_0}^{-1}((\mathbb F/{\mathcal O}_{\mathbb F})^n)\subseteq H_t.$ Given $r \in \Gamma_{<0}$, let $$G(r):=\bigcap \{H_t:\eta(t)\leq r\}.$$ Because each $H_t$ is $\mathcal A$-invariant so is $G(r)$, and as noted above, $\hat H\subseteq G(r)$ for every $r\in \Gamma_{<0}$. The map $f_{t_0}$ restricts to an injective homomorphism from $G(r_0)$ into $(K/{\mathcal O})^n$, and since $\hat H\subseteq G(r_0)$, the set $\{r\in \Gamma: f_{t_0}^{-1}(B_{>r}(0))\subseteq G(r_0)\}$ contains $\mathbb Z$. It follows that there exists $r<\mathbb Z$ such that $f_{t_0}^{-1}(B_{>r}(0))\subseteq G(r_0)$ and therefore $\nu_{K/{\mathcal O}}\vdash G(r_0)$ (by Lemma [Lemma 43](#F: nu in K/O){reference-type="ref" reference="F: nu in K/O"}). The family $\{G(r):r\in \Gamma\}$ is definable over $K_0$ and, by its definition, it is increasing as $r$ tends to $-\infty$. Hence, the directed union $$N:=\bigcup\limits_{r\in \Gamma_{<0}} G(r)$$ is an abelian subgroup defined over $K_0$, $\mathcal A$-invariant and $\nu_{K/{\mathcal O}}\vdash N$. It follows that $\mathrm{dp\text{-}rk}(N)$ is at least the $K/{\mathcal O}$-rank of $G$ (note however that we do not claim that $N$ is strongly internal to $K/{\mathcal O}$). This concludes the proof of Proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"} in the $p$-adic case. ◻ We now proceed to the remaining cases. ## ${\mathcal K}$ is power bounded $T$-convex or $V$-minimal We assume that ${\mathcal K}$ is either power bounded $T$-convex or $V$-minimal. In both cases $K/{\mathcal O}$ is an SW-uniformity and ${\mathcal K}$ has residue characteristic $0$. Since $(K/{\mathcal O})^n$ is torsion-free we cannot use torsion elements as in the $p$-adic case, so we adopt a different approach. The key to our argument is the characterisation of definable groups and endomorphisms of $(K/{\mathcal O})^n$ from Section [3.1](#ss:groups in K/O){reference-type="ref" reference="ss:groups in K/O"}. The conclusion of Proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"}, in our case, will follow from the next proposition (recall that a ball containing $0$ in $K/{\mathcal O})^n$ is of the form $B^n$ for $B$ a ball in $K/{\mathcal O}$): **Proposition 64**. *Let $G$ be a definable group in ${\mathcal K}$ and let $H\subseteq G$ be an infinite definable subgroup, definably isomorphic to a ball in $(K/{\mathcal O})^n$. Let $\sigma$ be a definable automorphism of $G$ and let $H^\sigma:=\sigma(H)$. Then $H^\sigma\cdot H\subseteq G$ is in definable bijection with a set of the form $$H\times \prod B_i\times \prod C_i,$$ where each $B_i$ is a ball in $K/{\mathcal O}$ and each $C_i$ is a ball in $K/\textbf{m}$.* *Furthermore,* 1. *If the $\textbf{k}$-rank of $G$ is $0$ then there are no $C_i$ in the above description, so $H^\sigma\cdot H$ is strongly internal to $K/{\mathcal O}$.* 2. *If $H^\sigma\neq H$ then $\mathrm{dp\text{-}rk}(H^\sigma\cdot H)>\mathrm{dp\text{-}rk}(H)$.* *Proof.* We identify $H$ with its image in $(K/{\mathcal O})^n$ (but still write the group operations multiplicatively) and let $H_3=\{(a,b)\in H\times H: a^\sigma b=e\}$. **Claim 5**. *$H_3$ is a subgroup of $H\times H$ and $(H\times H)/H_3$ is in definable bijection with $H^\sigma\cdot H$.* Note that if $a^\sigma b=e$ then $a^\sigma$ and $b$ are in $H_0:=H\cap H^\sigma$, so they commute. To see that $H_3$ is a subgroup, assume that $a_1^\sigma b_1=a_2^\sigma b_2=e$ then $(a_2^{-1})^\sigma a_1^\sigma b_1b_2^{-1}=(a_1a_2^{-1})^\sigma (b_1b_2^{-1})=e$, so $(a_1a_2^{-1}, b_1b_2^{-1})\in H_3$. We claim that for $a,b\in H$, $a_1^\sigma b_1=a_2^\sigma b_2$ if and only if $(a_1,b_1)H_3=(a_2,b_2)H_3$, and therefore the map $(a,b)\mapsto a^\sigma b$ induces a well-defined bijection between $(H\times H)/H_3$ and $H^\sigma \cdot H$. Indeed, using the commutativity of $H^\sigma$, $$a_1^\sigma b_1=a_2^\sigma b_2\Leftrightarrow (a_2^\sigma)^{-1} a_1^\sigma b_1b_2^{-1}=e\Leftrightarrow a_1^\sigma (a_2^\sigma)^{-1}b_1b_2^{-1}=e\Leftrightarrow (a_1,b_1)H_3=(a_2,b_2)H_3.$$ The claim implies, in particular, that in order to compute $\mathrm{dp\text{-}rk}( (H^\sigma\cdot H)$ it will suffice to compute $\mathrm{dp\text{-}rk}\left( (H\times H)/H_3\right)$, to which we now turn our attention. By definition, $H_3$ is the graph of a definable injective partial function $T:H^\sigma\cap H\dashrightarrow H^\sigma\cap H$, $x\mapsto (x^{\sigma})^{-1}$, in particular $\mathop{\mathrm{dom}}(T)$ is a definable group. We want to study the map $T$. To do that we may work solely inside $(K/{\mathcal O})^n\times (K/{\mathcal O})^n \supseteq H\times H$ so we switch to additive notation. By Lemma [Lemma 28](#automorphism){reference-type="ref" reference="automorphism"}, there is a definable automorphism $f:(K/{\mathcal O})^n\to (K/{\mathcal O})^n$ extending $T$. By Corollary [Corollary 29](#C: same valuation){reference-type="ref" reference="C: same valuation"}, $f$ preserves the valuation, and as $H$ is a ball, we get that $f(H)=H$. Let us replace $f$ by $f\restriction H$. As $H$ is abelian, $x\mapsto -f(x)$ is again an automorphism. Consider the definable map $F:H\times H\to H\times H$: $F(x,y)=(x,y-f(x))$. Because $f$ is an automorphism of $H$, $F$ is an automorphism of $H\times H$. It maps $H_3$ onto a group of the form $H_1\times \{e\}$, where $H_1=\mathop{\mathrm{dom}}(T)$. Hence $$(H\times H)/H_3 \cong (H\times H)/(H_1\times \{e_H\}) \cong (H/H_1)\times H.$$ By Lemma [Lemma 27](#K/O end-groups2){reference-type="ref" reference="K/O end-groups2"}, there is a definable automorphism of $(K/{\mathcal O})^n$ mapping $H_1$ to a direct product of closed and open balls in $K/{\mathcal O}$ (or $K/{\mathcal O}$ or $\{0\}$). Since $H$ of the form $B^n$, for $B\subseteq K/{\mathcal O}$, this automorphism preserves $H$ (Corollary [Corollary 29](#C: same valuation){reference-type="ref" reference="C: same valuation"}). Consequently, we may assume that $$H_1=\prod B_i \times \prod C_i \times \prod \{0\},$$ where $B_i$ are closed balls and $C_i$ are open balls. Therefore, $H/H_1$ is definably isomorphic to $$\prod B/B_i\times \prod B/C_i\times \prod B.$$ Each $B/B_i$ is definably isomorphic to a ball in $K/{\mathcal O}$ (so strongly internal in $K/{\mathcal O}$) and each $B/C_i$ is definably isomorphic to ball in $K/\mathfrak m$ (so strongly internal to $K/\mathfrak m$. This gives the desired form. For $(1)$, if The $\textbf{k}$-rank of $G$ is $0$ then there are no open $C_i$ in the above description; so $H^\sigma\cdot H$ is strongly internal to $K/{\mathcal O}$. For $(2)$, if $H^\sigma\neq H$ then $H^\sigma \cap H\subsetneq H$ and in particular $H_1\subsetneq H$. Since $\Gamma$ is dense, $[H:H_1]=\infty$ so $\mathrm{dp\text{-}rk}(H/H_1)>0$ and thus $\mathrm{dp\text{-}rk}(H^\sigma\cdot H)>\mathrm{dp\text{-}rk}(H)$. ◻ We can now complete the proof of Proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"} when ${\mathcal K}$ is either power bounded or V-minimal. Let $G$ be an infinite ${\mathcal K}_0$-definable group whose $\textbf{k}$-rank is $0$. By Section [5.1](#ss:nu in K/O){reference-type="ref" reference="ss:nu in K/O"} we can find a definable subgroup $H\subseteq G$ definably isomorphic to an open ball in $(K/{\mathcal O})^n$ centred at $0$, where $n$ is the $K/{\mathcal O}$-rank of $G$. It follows from Proposition [Proposition 64](#a K/O result){reference-type="ref" reference="a K/O result"} and the choice of $H$ that $H$ is invariant under every definable automorphism of $G$. Indeed, assume towards contradiction that $H^\sigma\neq H$. Then by (1) of the proposition, $H^\sigma\cdot H$ is strongly internal to $K/{\mathcal O}$ and by (2) $\mathrm{dp\text{-}rk}(H^\sigma\cdot H)>\mathrm{dp\text{-}rk}(H)$, contradicting the fact that $\mathrm{dp\text{-}rk}(H)$ is the $K/{\mathcal O}$-rank of $G$. Thus, $H$ is infinite, normal and abelian. Since any non-zero subgroup of $(K/{\mathcal O})^n$ is infinite, the existence of such a subgroup $H$ is an elementary property, implies that such a group exists already in ${\mathcal K}_0$, as claimed. 0◻\ We end this section with an example illustrating that in Proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"} the assumption that the $\textbf{k}$-rank of $G$ is $0$, is essential. **Example 65**. We produce an example of a group $G$ of dp-rank $2$ that is locally strongly internal to $K/{\mathcal O}$ and $\textbf{k}$ but does not contain any infinite definable normal subgroup in definable bijection with a subgroup of $K/{\mathcal O}$. Let ${\mathcal K}$ be either a $V$-minimal valued field or a power-bounded $T$-convex valued field, and let $\gamma>0$ be some element of $\Gamma$. Let $B_{\geq \gamma}$ and be $B_{\geq -\gamma}$ the closed balls of respective radii $\gamma$ and $-\gamma$ around $0$. Pick any $\delta\in \Gamma$ with $2\delta>\gamma>\delta>0$, then $H=(1+B_{>\delta})/(1+B_{\geq \gamma})$ is a definable multiplicative group definably isomorphic (because of our choice of $\delta$) to the additive group $B_{>\delta}/B_{\geq \gamma}$ (via the map $a+B_{\geq \gamma}\mapsto (1+a)(1+B_{\geq \gamma})$). This latter group is obviously definably isomorphic to a subgroup of $K/{\mathcal O}$. Let $N=B_{>-\gamma}/\textbf{m}$ (which is strongly internal to $\textbf{k}$). Set $G=N\rtimes H$, where $H$ acts on $N$ by multiplication (it is well-defined) and the latter is a normal subgroup of $G$. We identify both these groups with their obvious images in $G$, namely we identify $g=\bar g+\mathfrak m\in N$ with $(\bar g+\mathfrak m,1+B_{\geq \gamma})$, and $a=\bar a(1+B_{\geq \gamma})\in H$ with $(\mathfrak m,\bar a(1+B_{\geq \gamma}))$. Direct computation gives that if $a\in H$ and $g\in N$ as above, $a^g=g^{-1}ag=(-\bar g+\bar a\bar g+\textbf{m},\bar a(1+B_{\geq \gamma}))$. It is not hard to see that for every $a\in H$ there is an element $g\in N$ with $a^g\notin H$, thus $\bigcap_{g\in G}H^g=\{e\}$ so $H$ does not contain any normal subgroup. On the other hand, for any $b=(\textbf{m},\bar b(1+B_\geq \gamma))\in H$, $a^gb=(\bar a(1-\bar g)+\textbf{m},\bar a\bar b(1+B_{\geq \gamma}))$, thus $$H^g\cdot H=\{(\bar a(1-\bar g)+\textbf{m},\bar b(1+B_{\geq \gamma})):\bar a,\bar b\in 1+B_{>\delta}\}.$$ We claim that $H^g\cdot H=B_{>\delta+v(g)}/\textbf{m}\times H$, leaving the computations to the reader. # Groups locally strongly internal to the residue field The results of the previous sections imply, in particular, that there are no definably semisimple groups locally strongly internal to $\Gamma$ (and in the $p$-adic case, nor to $K/{\mathcal O}$). This is, clearly, not the situation for groups locally strongly internal to the valued field or to the residue field. So our aim in the present and in the next section is to study such groups. We begin with the study of groups locally strongly internal to $\textbf{k}$, where ${\mathcal K}$ is either power-bounded $T$-convex or $V$-minimal. For the statement of the main result of this section, we need a weakening of definable semisimplicity: **Definition 66**. Let $G$ be a definable group. A definable normal subgroup $H\trianglelefteq G$ is $G$-*semisimple* if $H$ has no infinite abelian definable subgroups normal in $G$. Note that, in the above notation, if either $G$ or $H$ are definably semisimple, then $H$ is $G$-semisimple. We prove: **Proposition 67**. *Let $G$ be a definably semisimple group locally almost strongly internal to $\textbf{k}$. Then there exists a finite normal subgroup $N\trianglelefteq G$ and two normal subgroups $G_1,G_2\trianglelefteq G/N$, all defined over any model over which $G$ is defined, such that* 1. *$G_1\cap G_2=\{e\}$, $G_1,G_2$ centralize each other and $G_1\cdot G_2$ has finite index in $G/N$.* 2. *The almost $\textbf{k}$-rank of $G_1$ is $0$ and it is $G/N$-semisimple,* 3. *$G_2$ is definably semisimple, and it is definably isomorphic to a subgroup of $\mathop{\mathrm{GL}}_n(\textbf{k})$.* Recall that a group $G$ is *a definably connected* if it has no definable subgroups of finite index. Note that for $G$ an arbitrary definable group, if there exists a definably connected subgroup of finite index, then it is necessarily unique and denoted by $G^0$. Clearly, if $G^0$ exists then it is *definably characteristic in $G$*, namely invariant under all definable automorphisms of $G$. **Fact 68**. *[@PePiSt Fact 2.11][\[F:finite center, centerless\]]{#F:finite center, centerless label="F:finite center, centerless"} Let $G$ be a definably connected group definable in some structure ${\mathcal M}$.* 1. *If $N$ is a finite normal subgroup, then $N\subseteq Z(G)$.* 2. *If $Z(G)$ is finite, then $G/Z(G)$ is centerless.* The proof of Proposition [Proposition 67](#P: k){reference-type="ref" reference="P: k"} splits into two cases. ## $\textbf{k}$ is o-minimal In this subsection, we assume that ${\mathcal K}$ is power bounded $T$-convex, thus $\textbf{k}$ is an o-minimal expansion of a real closed field [@vdDries-Tconvex Theorem A]. We first need a lemma allowing us, under suitable assumptions, to transfer definable semisimplicity under definable group homomorphisms: **Lemma 69**. *Assume that $G$ is a definable group in ${\mathcal K}$, $B$ a definable group in $\textbf{k}$, and $f:G\to B$ a definable surjective homomorphism. Let $H\trianglelefteq G$ be a normal definable subgroup with $\ker(f\restriction H)$ finite. Then:* 1. *$H^0$ exists.* 2. *If $H$ is $G$-definably semisimple, then $H^0$ and $f(H^0)$ are definably semisimple.* *Proof.* (1) $f(H)$ is a definable group in the o-minimal structure $\textbf{k}$, so $f(H)^0$ exists. Since $\ker(f\restriction H)$ is finite, $H^0$ exists as well. Indeed, if not then there exists an infinite descending chain of finite index subgroups in $H$, which would give rise to a proper finite index subgroup of $f(H)^0$, contradiction. \(2\) Assume that $H$ is $G$-definably semisimple. Let $N=f(H^0)$; it is a definably connected component. If $N$ is definably semisimple then so is $H^0$, so it suffices to show that $N$ is definably semisimple. Assume towards a contradiction that $N$ contains an infinite definable abelian normal subgroup $A$. Recall that the *definable solvable radical of $N$* is the subgroup of $N$ generated by all definably connected solvable normal subgroups of $G$. It is itself definable because of dimension considerations, and clearly definably characteristic in $N$. Let $R$ be the definable solvable radical of $N$. The group $A^0$ is contained in $R$ so $R$ is infinite. By [@baro-solvconn Corollary 5.6], $R$ contains an infinite abelian definable subgroup $R_0$ that is definably characteristic in $N$. Let $A_1$ be the connected component of $f^{-1}(R_0)\cap H^0$. Since $R_0$ is a definably connected group, $f(A_1)=R_0$. We claim that $Z(A_1)$ is infinite. Indeed, if it were finite then, by Fact [\[F:finite center, centerless\]](#F:finite center, centerless){reference-type="ref" reference="F:finite center, centerless"}, the group $A_1/Z(A_1)$ is centerless. However, because $\ker(f\restriction A_1)$ is finite, it follows from the same fact that $\ker(f\restriction A_1)\subseteq Z(A_1)$. Thus, $A_1/Z(A_1)$ can also be written as a quotient of $f(A_1)=R_0$, so must be abelian, a contradiction. Since $R_0$ is a characteristic subgroup of $N=f(H^0)$ and $H^0$ is normal in $G$, the group $f^{-1}(R_0)\cap H^0$ is invariant under conjugation by elements of $G$; thus so are $A_1$ and $Z(A_1)$. Thus, $Z(A_1)$ is an infinite abelian definable subgroup of $H$ and normal in $G$, contradicting the definable $G$-semisimplicity of $H$. ◻ Assume that $G$ is locally strongly internal to $\textbf{k}$. Let $\mathrm{Ad}_{\textbf{k}}:G\to \mathop{\mathrm{GL}}_n(\textbf{k})$ be the adjoint map, as discussed at the end of Section [5](#S:infnit and local){reference-type="ref" reference="S:infnit and local"}. **Lemma 70**. *Let $G$ be locally strongly internal to $\textbf{k}$. Then,* 1. *$\ker(\mathrm{Ad}_{\textbf{k}})=C_G(\nu_\textbf{k})$* 2. *$\nu_\textbf{k}\vdash C_G(\ker(\mathrm{Ad}_{\textbf{k}}))$* *Proof.* Let $\nu=\nu_\textbf{k}$. \(1\) Let $g\in \ker(\mathrm{Ad}_{\textbf{k}})$. By [@OtPePi Lemma 3.2(ii)], two definable automorphisms of a definable subgroup $H$ in $\textbf{k}$ with the same differential at $e_H$ coincide on an open neighbourhood of $e_H$ in $H$. While the proof is stated for groups, the analysis hold for local groups as well. Hence, if $g\in \ker(\mathrm{Ad}_\textbf{k})$ then $\tau_g(x)=x$ on some $\tau_{\textbf{k}}$-open neighbourhood of $e$, so by definition $g \in C_G(\nu)$. The reverse inclusion is immediate from the definitions. \(2\) By cell decomposition in o-minimal structures, there is some definably connected definable $X\subseteq G$ for which $\nu\vdash X$. By Lemma [Lemma 38](#L:def connected){reference-type="ref" reference="L:def connected"}, $X\subseteq C_G(C_G(\nu))=C_G(\ker(\mathrm{Ad}_\textbf{k}))$, thus $\nu\vdash C_G(\ker (\mathrm{Ad}_{\textbf{k}}))$. ◻ **Proposition 71**. *Let $G$ be a definably semisimple group in ${\mathcal K}$, locally strongly internal to $\textbf{k}$. Let $H_1=\ker(\mathrm{Ad}_\textbf{k})$ and $H_2=C_G(H_1)$. Then* 1. *$H_1$ and $H_2$ are normal subgroups, $H_2^0$ is definably semisimple, $H_1\cap H_2$ is finite and $H_1$ and $H_2$ centralize each other.* 2. *$H_1\cdot H_2$ has finite index in $G$.* 3. *If the $\textbf{k}$-rank of $G$ equals the almost $\textbf{k}$-rank then $\mathrm{dp\text{-}rk}(H_2)$ equals the $\textbf{k}$-rank of $G$.* *Proof.* Let $\nu=\nu_\textbf{k}$. By Lemma [Lemma 70](#L:kerad in o-minimal){reference-type="ref" reference="L:kerad in o-minimal"}, $H_1=C_G(\nu)$ and $\nu\vdash H_2$. By definition, $H_1$ is a definable normal subgroup, and thus so is $H_2$. By the semisimplicity of $G$. the intersection of any definable normal subgroup $H$ with its centralizer is finite (otherwise, $Z(H)$ is infinite and normal in $G$). Thus $H_1\cap H_2$ is finite, and by definition $H_1$ and $H_2$ centralize each other. By Lemma [Lemma 69](#L:connected subgroup of defsemisimple){reference-type="ref" reference="L:connected subgroup of defsemisimple"}, $H_2^0$ is definably semisimple, completing the proof of (1). \(2\) Note that $$G/(H_1\cdot H_2)\cong \frac{G/H_1}{(H_1\cdot H_2)/H_1}\cong \frac{G/H_1}{H_2/(H_1\cap H_2)}\cong \mathrm{Ad}_\textbf{k}(G)/\mathrm{Ad}_\textbf{k}(H_2),$$ where $\mathrm{Ad}_\textbf{k}(G)$ is the image of $\mathrm{Ad}_\textbf{k}$ and $\mathrm{Ad}_\textbf{k}(H_2)$ is the image of $\mathrm{Ad}_\textbf{k}\restriction H_2$. Thus, we need to see that $\mathrm{Ad}_\textbf{k}(G)/\mathrm{Ad}_\textbf{k}(H_2)$ is finite. Since both images are subgroups of $\mathop{\mathrm{GL}}_n(\textbf{k})$, we may freely use properties of groups definable in o-minimal expansions of fields. By o-minimality, showing that $\mathrm{Ad}_\textbf{k}(G)/\mathrm{Ad}_\textbf{k}(H_2)$ is finite amounts to showing that $\dim_\textbf{k}(\mathrm{Ad}_\textbf{k}(G))=\dim_\textbf{k}(\mathrm{Ad}_\textbf{k}(H_2))$ (we use $\dim_\textbf{k}$ for the o-minimal dimension in $\textbf{k}$). So, it is sufficient to show that $\dim_\textbf{k}(\mathrm{Ad}_\textbf{k}(G))\leq \dim_\textbf{k}(\mathrm{Ad}_\textbf{k}(H_2))$. As $G$ is definably semisimple, $H_2$ is $G$-definably semisimple. Since, by (1), $\ker(\mathrm{Ad}_\textbf{k}\restriction H_2)$ is finite, $H_2^0$ and $\mathrm{Ad}_\textbf{k}(H_2^0)$ are definably semisimple by Lemma [Lemma 69](#L:connected subgroup of defsemisimple){reference-type="ref" reference="L:connected subgroup of defsemisimple"}. Let $\mathfrak{h}$ be the Lie algebra (in the sense of [@PePiSt-defsimple]) of the definably connected group $\mathrm{Ad}_\textbf{k}(H_2^0)$ with its $\textbf{k}$-differential structure. By [@PePiSt-defsimple Theorem 2.34], $\mathfrak{h}$ is a semisimple Lie algebra. Thus, by [@PePiSt-defsimple Claim 2.8], $\dim(\mathfrak{h})=\dim_\textbf{k}(\mathop{\mathrm{Aut}}(\mathfrak{h}))$ (we use the $\textbf{k}$-vector space dimension on the left and the fact that $\mathop{\mathrm{Aut}}(\mathfrak h)$ is definable in $\textbf{k}$). The group $\mathrm{Ad}_\textbf{k}(G)$ acts on $\mathrm{Ad}_\textbf{k}(H_2^0)$ by conjugation and thus also on $\mathfrak{h}$. We claim that the kernel of this action is trivial. Indeed, assume that for some $g\in G$, the action of $\mathrm{Ad}_\textbf{k}(g)$ on $\mathfrak h$ is the identity. By [@OtPePi Lemma 3.2(ii)], it follows that for all $x\in \mathrm{Ad}_\textbf{k}(H_2^0)$, $\mathrm{Ad}_\textbf{k}(g^{-1}xg)=\mathrm{Ad}_\textbf{k}(x)$, and hence for all $x\in H_2^0$, $g^{-1}xgx^{-1}\in \ker(\mathrm{Ad}_\textbf{k}\restriction H_2^0)$. Since $\ker(\mathrm{Ad}_\textbf{k}\restriction H_2^0)$ is finite, and $H_2^0$ is definably connected, it follows that for all $x\in H_2^0$, $g^{-1}xg=x$ and hence $g\in C_G(H_2^0)$. Because $\nu \vdash H_2$, then $g\in C_G(\nu)$, so by Lemma [Lemma 70](#L:kerad in o-minimal){reference-type="ref" reference="L:kerad in o-minimal"}, $g\in \ker(\mathrm{Ad}_\textbf{k})$ and hence $\mathrm{Ad}_\textbf{k}(g)=e$. We can therefore conclude that $\mathrm{Ad}_\textbf{k}(G)$ can be definably embedded into $\mathop{\mathrm{Aut}}(\mathfrak{h})$ hence we get that $\dim(\mathrm{Ad}_\textbf{k}(G))\leq \dim(\mathop{\mathrm{Aut}}(\mathfrak{h}))=\dim(\mathfrak{h})=\dim(\mathrm{Ad}_\textbf{k}(H_2^0))$, so $\mathrm{dp\text{-}rk}(\mathrm{Ad}_\textbf{k}(G))=\mathrm{dp\text{-}rk}(\mathrm{Ad}_{\textbf{k}}(H_2^0))=\mathrm{dp\text{-}rk}(\mathrm{Ad}_\textbf{k}(H_2))$, as required. \(3\) Because $\ker(\mathrm{Ad}_k)\cap H_2$ is finite, $H_2$ is almost strongly internal to $\textbf{k}$. Thus, the almost $\textbf{k}$-rank of $G$ is at least that of $H_2$. However, $\nu\vdash H_2$ so the $\mathrm{dp\text{-}rk}(H_2)$ is at least the $\textbf{k}$-rank of $G$. Because of the rank assumptions, we must have that $\mathrm{dp\text{-}rk}(H_2)$ is the $\textbf{k}$-rank of $G$. ◻ **Remark 72**. As was noted in Remark [Remark 54](#R: H1H2 over the same parameters){reference-type="ref" reference="R: H1H2 over the same parameters"}, the groups $H_1$ and $H_2$ appearing in the statement of Proposition [Proposition 71](#P:general H1,H2 into o-minimal){reference-type="ref" reference="P:general H1,H2 into o-minimal"} are definable over the same parameters as $G$. We isolate the following direct consequences: **Corollary 73**. *Let $G$ be locally strongly internal to $\textbf{k}$.* 1. *If $\ker(\mathrm{Ad}_\textbf{k})=G$ then $\nu_\textbf{k}\vdash Z(G)$. In particular, if $Z(G)$ is finite, then $\ker(\mathrm{Ad}_\textbf{k})$ is a proper subgroup of $G$.* 2. *If $G$ is definably simple (namely non-abelian and has no non-trivial definable normal subgroup) then $G$ is definably isomorphic to a definable subgroup of $\mathop{\mathrm{GL}}_n(\textbf{k})$.* *Proof.* (1) If $G=\ker(\mathrm{Ad}_\textbf{k})$ then by Lemma [Lemma 70](#L:kerad in o-minimal){reference-type="ref" reference="L:kerad in o-minimal"}(2), $\nu_\textbf{k}\vdash C_G(G)=Z(G)$. (2) Since $G$ is simple, either $\ker(\mathrm{Ad}_\textbf{k})=G$ or $\ker(\mathrm{Ad}_\textbf{k})=\{e\}$ Since $G$ is non-abelian, it follows from (1) that $\ker(\mathrm{Ad}_{\textbf{k}})$ must be equal to $\{e\}$. ◻ The proof of Proposition [Proposition 67](#P: k){reference-type="ref" reference="P: k"} when $\textbf{k}$ is o-minimal reduces to collecting what we have done so far: *Proof of Proposition [Proposition 67](#P: k){reference-type="ref" reference="P: k"} for o-minimal $\textbf{k}$.* Fix $G$ a definably semisimple group locally almost strongly internal to $\textbf{k}$. To prove (1) we need to find a finite normal $N\trianglelefteq G$ and definable $G_1, G_2\trianglelefteq G/N$ centralising each other with $G_1\cap G_2=\{e\}$. By Fact [\[F: existence of finite normla to get D-group\]](#F: existence of finite normla to get D-group){reference-type="ref" reference="F: existence of finite normla to get D-group"}, there exists a finite normal subgroup $N_1\trianglelefteq G$ such that $G/N_1$ is a $\textbf{k}$-group and the almost $\textbf{k}$-rank and the $\textbf{k}$-rank agree in $G/N_1$. Furthermore, $N_1$ is definable over any model over which $G$ is defined. By Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"} $G/N_1$ is definably semisimple, so -- in order to keep notation simpler -- we denote $G/N_1$ by $G$. By Lemma [Lemma 53](#L:existence of local group){reference-type="ref" reference="L:existence of local group"}, $G$ contains a definable normal differentiable local subgroup ${\mathcal G}$ with respect to $\textbf{k}$, with $\nu_{\textbf{k}}\vdash {\mathcal G}$. Then Proposition [Proposition 71](#P:general H1,H2 into o-minimal){reference-type="ref" reference="P:general H1,H2 into o-minimal"} provides us with two definable normal subgroups $H_1, H_2$ satisfying (1) of the proposition. By Remark [Remark 54](#R: H1H2 over the same parameters){reference-type="ref" reference="R: H1H2 over the same parameters"}, $H_1$ and $H_2$ are both definable over any model over which $G$ is defined. The group $N=H_1\cap H_2$ is a finite normal subgroup of $G$. Replace $G$ by $G/N$ and set $G_i:=H_i/N$. Then $G_1$ and $G_2$ satisfy (1) of the proposition. For (3) we need to show that $G_2$ is definably semisimple, and definably isomorphic to a $\textbf{k}$-linear group. The latter is clear, since $\mathrm{Ad}_\textbf{k}(G)$ is $\textbf{k}$-linear. For the first part, note that since $H_2^0$ is definably semisimple (by Proposition [Proposition 71](#P:general H1,H2 into o-minimal){reference-type="ref" reference="P:general H1,H2 into o-minimal"}), so is $H_2$ and thus so is $G_2$ by Lemma [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"}.\ It remains to prove (2), i.e., that the almost $\textbf{k}$-rank of $G_1$ is $0$ and that $G_1$ is $G/N$-semisimple. The latter part follows from the fact that $G_1$ is normal in the definably semisimple group $G$. So we only need to compute its almost $\textbf{k}$-rank. Assume toward a contradiction that $G_1$ is locally almost strongly internal to $\textbf{k}$, whence, a $\textbf{k}$-group. By applying Fact [\[F: existence of finite normla to get D-group\]](#F: existence of finite normla to get D-group){reference-type="ref" reference="F: existence of finite normla to get D-group"} to $G_1$, we get a finite normal subgroup $H\trianglelefteq G_1$ such that $G_1/H$ is locally strongly internal to $\textbf{k}$. Note that $H$ is normal in $G_1\cdot G_2$ as well. Since $G_1\cdot G_2$ has finite index in $G$, by Lemma [Lemma 14](#L:nu of subgroup){reference-type="ref" reference="L:nu of subgroup"}(2) $\nu_\textbf{k}(G_1\cdot G_2)=\nu_\textbf{k}(G)$, so $\nu_\textbf{k}(G_1\cdot G_2)\vdash G_2$ and thus $\nu_\textbf{k}(G_1\cdot G_2)/H\vdash G_2/H$. By Lemma [Lemma 16](#L:passage of D-group under finite-to-one){reference-type="ref" reference="L:passage of D-group under finite-to-one"}(3), $\nu_\textbf{k}(G_1\cdot G_2/H)\vdash G_2/H$ and by Lemma[Lemma 14](#L:nu of subgroup){reference-type="ref" reference="L:nu of subgroup"}(1), $\nu_\textbf{k}(G_1/H)\vdash \nu_\textbf{k}(G_1\cdot G_2/H)\vdash G_2/H$. On the other hand, obviously $\nu_\textbf{k}(G_1/H)\vdash G_1/H$ thus $(G_1\cap G_2)/H$ must be infinite, contradiction. ◻ ## Proof of Proposition [Proposition 67](#P: k){reference-type="ref" reference="P: k"} for $\textbf{k}$ an algebraically closed field. {#proof-of-proposition-p-k-for-textbfk-an-algebraically-closed-field.} Throughout this subsection ${\mathcal K}$ is assumed $V$-minimal, hence $\textbf{k}$ is a stably embedded pure algebraically closed field. In particular, $\textbf{k}$ is strongly minimal. Fix a ${\mathcal K}$-definable, definably semisimple group $G$. By [@HaHaPeGps Proposition 6.2], there exist definable subgroups $H_0\trianglelefteq H\trianglelefteq G$, with $H$ definably connected and $H_0$ finite such that $H/H_0$ is strongly internal to $\textbf{k}$. **Claim 6**. *There exists a definable group $H_0$ **normal in $G$** such that $H_0\trianglelefteq H$ and $H/H_0$ is strongly internal to $\textbf{k}$.* *Proof.* Amongst all finite definable normal subgroups of $H$ with $H/H_0$ strongly internal to $\textbf{k}$, choose $H_0$ of minimal cardinality and let $f:H\to H/H_0$ be the projection map. Let $g\in G$ and assume that $H_0^g\neq H_0$, so for $x\in H$, $x\mapsto (xH_0,xH_0^g)\in H/H_0\times H/H_0^g$ is a group homomorphism with kernel $H_0\cap H_0^g$. Thus, $H/(H_0\cap H_0^g)$ is strongly internal to $\textbf{k}$ contradicting the minimality assumption. ◻ Fix $H_0\trianglelefteq G$ as above and consider $H_1=H/H_0$. By [@BousHrWeil Theorem 1] it is a $\textbf{k}$-connected algebraic group. By a classical theorem of Rosenlicht [@rosenlicht Theorem 13], as $H_1$ is a connected algebraic group, $H_1/Z(H_1)$ is a $\textbf{k}$-linear group. As $G/H_0$ is definably semisimple and $H_1$ is normal in $G/H_0$, $Z(H_1)$ is finite. Since $H_1$ is connected $H/Z(H_1)$ is centerless. We now fix a finite $N\trianglelefteq G$, $H_0\subseteq N$, such that $H/N$ is a connected centerless $\textbf{k}$-linear group. Note that $G/N$ is still definably semisimple by Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"}. Below we work in $G/N$, and to simplify notation we still use $H$ for $H/N$. Note that, since $\textbf{k}$ has definable Morley Rank, the statement \"$H$ is a normal subgroup of $G$ strongly internal to $\textbf{k}$ whose Morley Rank equals the $\textbf{k}$-rank of $G$\" is definable in families, and we can choose $H$ to be definable over any model in which $G$ is defined. **Claim 7**. *$H$ has no infinite normal abelian subgroups, hence it is a semisimple algebraic group.* *Proof.* Assume towards contradiction that such a normal subgroup existed. Then its Zariski closure is an infinite normal abelian alegrbaic subgroup. Its (algebraic) connected component is contained in the solvable radical $R$ of $H$ which is therefore infinite as well. This radical contains an infinite abelian algebraic subgroup that is definably characteristic in $H$, and therefore is normal in $G$, contradicting our assumption. ◻ **Claim 8**. *The group $C_G(H)\cdot H$ has finite index in $G$.* *Proof.* The group $G$ acts on $H$ by conjugation and because $\textbf{k}$ is stably embedded, each action is $\textbf{k}$-algerbraic, so the map $f:g\mapsto \tau_g\restriction H$ sends $G$ into $\mathop{\mathrm{Aut}}(H)$ the group of all algebraic automorphisms of $H$ (recall that $\tau_g: (x\mapsto x^g)$). The kernel of the map is $C_G(H)$. Applying [@Humph Theorem 27.4], using the fact that $\textbf{k}$ is algebraically closed, we see that $\mathop{\mathrm{Aut}}(H)$ is the semi-direct product of $\mathrm{Int}(H)$, the inner automorphisms of $H$, and a finite group (we use here the fact that $H$ is assumed centreless). Since $f(H)=\mathrm{Int}(H)$, it follows that $f(G)$ has finite index in $\mathop{\mathrm{Aut}}(H)$ so $C_G(H)\cdot H$ must have finite index in $G$. ◻ We now let $G_1=C_G(H)$ and $G_2=H$. Since $G_1$ and $G_2$ centralize each other and $H_2$ is centerless, $G_1\cap G_2=\{e\}$. This ends the proof of (1). By construction, $G_2$ is a linear $\textbf{k}$-group. Assume towards a contradiction that $G_1$ is locally almost strongly internal to $\textbf{k}$ as well. By [@HaHaPeGps Proposition 6.2], there exists a finite definable normal subgroup $N'\trianglelefteq G_1$ such that $G_1/N'$ has a definable normal subgroup $B_1\trianglelefteq G_1/N'$ strongly internal to $\textbf{k}$. Since $G_1$ and $G_2$ intersect trivially, we may identify $G_2$ with $G_2/N'$. Moreover, the $\textbf{k}$-rank of $G/N$ and $G/N'$ are the same by Lemma [Lemma 16](#L:passage of D-group under finite-to-one){reference-type="ref" reference="L:passage of D-group under finite-to-one"}; so $G_2=H$ is still $\textbf{k}$-critical in $G/N'$. But then $B_1\cdot G_2 \cong B_1\times G_2$ is strongly internal to $\textbf{k}$, with $\mathrm{dp\text{-}rk}(B_1\cdot G_2)>\mathrm{dp\text{-}rk}(G_2)$, contradicting the fact that $H=G_2$ was $\textbf{k}$-critical in $G/N'$. Finally, we already saw that $G_2$ is definably semisimple. The fact that $G_1$ is $G/N$-semisimple, is immediate since $G/N$ is definably semisimple. This finishes the proof of Proposition [Proposition 67](#P: k){reference-type="ref" reference="P: k"} in the V-minimal case, and thus the proof of the proposition is now complete. # Pure $K$-groups In the notation of Section [5.3](#ss: valued field and residue field){reference-type="ref" reference="ss: valued field and residue field"}, for a $K$-group $G$ there exists an infinitesimal type-definable subgroup $\nu_K(G)$ inducing a definable homomorphism $\mathrm{Ad}_K:G\to \mathop{\mathrm{GL}}_n(K)$, for $n$ the $K$-rank of $G$. Recall that a definable group $G$ is *$K$-pure* if $G$ is locally strongly internal to $K$ but not locally almost strongly internal to $\Gamma$, to $\textbf{k}$ or to $K/{\mathcal O}$. In the present section we collect some basic facts concerning pure $K$-groups, as those appear naturally in our later analysis.\ For the following result, we observe that all the valued fields we consider are $1$-h-minimal. The exact definition is immaterial here. See [@hensel-min] and [@HaHaPeVF Section 4.5]. **Fact 74**. *[@AcHa Theorem 2][\[F:AcHa\]]{#F:AcHa label="F:AcHa"} Let ${\mathcal K}$ be a $1$-h-minimal field, ${\mathcal G}=(X,\cdot, ^{-1})$ a definable strictly differentiable local group with respect to ${\mathcal K}$ and $f:{\mathcal G}\dashrightarrow {\mathcal G}$ a definable strictly differentiable homomorphism of local groups. If $D_e(f)=\mathop{\mathrm{Id}}$ then $\{y\in \mathop{\mathrm{dom}}(f):f(y)=y\}$ contains a definable open neighbourhood of $e$* *Proof.* This is a theorem of Acosta and the second author, [@AcHa Theorem 2], implying that $\mathrm{dp\text{-}rk}\{y\in \mathop{\mathrm{dom}}(f) :f(y)=y\}=\mathrm{dp\text{-}rk}\mathop{\mathrm{dom}}(f)$, so contains a definable open subset; the result follows. ◻ We still use $\dim$ to denote the $\mathop{\mathrm{acl}}$-dimension in $K$ and the induced dimension on $K^{eq}$ and $\tau_K$ for the topology on $G$. **Proposition 75**. *Let $G$ be a definable group in ${\mathcal K}$ of positive dimension. If $g\in \ker(\mathrm{Ad}_K)$ then $\dim C_G(g)=\dim G$.* *Proof.* Let $\mathcal{G}=(X,\cdot,^{-1})$ be the definable strictly differentiable local group as provided by Lemma [Lemma 53](#L:existence of local group){reference-type="ref" reference="L:existence of local group"}. If $g\in \ker(\mathrm{Ad}_K)$ then by Fact [\[F:AcHa\]](#F:AcHa){reference-type="ref" reference="F:AcHa"}, the set $W:=\{x\in X: x^g=x\}\subseteq C_G(g)$ is open in $X$. Since $\dim (X)$ is the $K$-rank of $G$ (Corollary [Corollary 32](#C:every thing is strongly internal to K){reference-type="ref" reference="C:every thing is strongly internal to K"}), we get that $$\dim (G)=\dim(X)=\dim (W)\leq \dim C_G(g)\leq \dim(G) .\qedhere$$ ◻ The following is based on an analogous result of [@GisHalMac]: **Corollary 76**. *Let $G$ be a definable group, locally strongly internal to $K$ and let $g\in G$. If $G$ is $K$-pure and $\dim(C_G(g))=\dim(G)$ then $[G:C_G(g)]<\infty$. In particular, $[G:C_G(g)]<\infty$ for every $g\in \ker(\mathrm{Ad}_K)$.* *Proof.* The conjugacy class $g^G$ is in definable bijection with the imaginary sort $G/C_G(g)$. By additivity of dimension we get that $\dim(g^G)=\dim(G)-\dim(C_G(g))$. If $\dim(C_G(g))=\dim(G)$ then $\dim(g^G)=0$. By Lemma [Lemma 40](#l: pure 0dim){reference-type="ref" reference="l: pure 0dim"}, $g^G$ is finite, hence $[G:C_G(g)]$ is finite. ◻ # Definably semisimple groups We can finally prove the main results of the paper. Recall, first, that a definable group is *definably simple* if it is non-abelian and has no definable normal subgroups, it is *definably semisimple* if it has no definable infinite normal abelian subgroups. We point out that definable semisimplicity is not, a priori, an elementary property of groups definable in ${\mathcal K}^{eq}$, as ${\mathcal K}^{eq}$ may not eliminate the quantifier $\exists^\infty$. As we will see below, one of the corollaries of the present work is that in our setting, definable semisimplicity, is, in fact, elementary. i.e., if ${\mathcal K}_0\prec {\mathcal K}$ and $G$ is a $K_0$-definable group, such that $G$ is definably semisimple in $K_0$ then it remains so in ${\mathcal K}$. As before, ${\mathcal K}={\mathcal K}^{eq}$ is a sufficiently saturated valued field, either power-bounded $T$-convex, $V$-minimal or $p$-adically closed. Throughout the previous sections, we were working under the assumption that our definable group $G$ is a $D$-group (for some distinguished sort $D$). As shown in [@HaHaPeGps], this need not be the case as $G$ might no be locally strongly internal to any distinguished sort. The best we can obtain, in general, that if $G$ is locally *almost* strongly internal to $D$ and then there is a finite normal subgroup $H$ such that $G/H$ is a $D$-group (so in particular, locally strongly internal to $D$), Fact [\[F: existence of finite normla to get D-group\]](#F: existence of finite normla to get D-group){reference-type="ref" reference="F: existence of finite normla to get D-group"}. Fortunately, in our setting, Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"} assures that definable semisimplicity is preserved under finite quotients and under finite extensions. Before stating the first of the results, recall from [@JohnTopQp §9.3] that a topological group $G$ is *locally abelian* if there exists $W\ni e$, an open neighbourhood of $e$ in $G$, such that $xy=yx$ for all $x,y\in W$. The next theorem gives conditions under which a definable, infinite, abelian normal subgroup must exist in $G$. Recall that if $\dim(G)>0$ then by Corollary [Corollary 32](#C:every thing is strongly internal to K){reference-type="ref" reference="C:every thing is strongly internal to K"} it is locally strongly internal to $K$. **Theorem 77**. *Let $G$ be an infinite group definable over some ${\mathcal K}_0\prec {\mathcal K}$.* 1. *If $G$ is $K$-pure (so locally strongly internal to $K$) and locally abelian with respect to $\tau_K$ then there exists a definable abelian subgroup $G_1\trianglelefteq G$ of finite index, defined over ${\mathcal K}_0$. In particular, $G_1$ is open.* 2. 1. *If $G$ is locally almost strongly internal to $\Gamma$ then there exists a ${\mathcal K}_0$-definable infinite normal abelian subgroup $N\trianglelefteq G$, whose dp-rank is at least the almost $\Gamma$-rank of $G$.* 2. *If $G$ is locally almost strongly internal to $K/{\mathcal O}$ but not to $\textbf{k}$ then there exists a ${\mathcal K}_0$-definable infinite normal abelian subgroup $N\trianglelefteq G$ whose dp-rank is at least the $K/{\mathcal O}$-rank of $G$.* *Proof.* (1) Since $G$ is locally strongly internal to $K$, it is a topological group with respect to the $\tau_K$-topology. All topological notions below refer to $\tau_K$. Assume that $G$ is locally abelian. By Lemma [Lemma 53](#L:existence of local group){reference-type="ref" reference="L:existence of local group"}, there exists a local differentiable abelian subgroup ${\mathcal G}=(U,\cdot, ^{-1})$ of $G$. Let $\tau_g$ denote conjugation by $g$. As $\tau_g\upharpoonright U=\mathop{\mathrm{Id}}$ for all $g\in U$, we get that $U\subseteq \ker(\mathrm{Ad}_K)$. This gives $\dim(\ker(\mathrm{Ad}_K))=\dim(G)$. The proof that $G$ is abelian-by-finite is an adaptation of [@PiYao Proposition 2.3]. By Corollary [Corollary 76](#C:dugaldetal){reference-type="ref" reference="C:dugaldetal"}, since $G$ is $K$-pure, $[G:C_G(a)]<\infty$ for all $a\in U$. By Fact [Corollary 18](#Baldwin Saxl){reference-type="ref" reference="Baldwin Saxl"}, there is a definable normal subgroup of finite index $H_0\trianglelefteq G$ such that $H_0\leq C_G(U)$. For every $h\in H_0$, $U\subseteq C_G(h)$ hence $\dim C_G(h)=\dim G$, by Corollary [Corollary 76](#C:dugaldetal){reference-type="ref" reference="C:dugaldetal"} and $K$-purity, we have $[G:C_G(h)]<\infty$ for every $h\in H_0$. Thus, applying Fact [Corollary 18](#Baldwin Saxl){reference-type="ref" reference="Baldwin Saxl"} again, we see that $C_G(H_0)$ has finite index in $G$, so in particular, $G_1=C_G(H_0)\cap H_0$ has finite index in $G$ and is commutative. It follows that $G_1$ is open by Corollary [Corollary 39](#C: open iff full dim){reference-type="ref" reference="C: open iff full dim"}. The fact that $G_1$ is a definable, open, normal abelian, subgroup of index $k$ (some $k\in {\mathbb{N}})$, is first order, so we can find such $G_1$ defined over ${\mathcal K}_0$. \(2\) Assume now that $G$ is locally almost strongly internal to $D$, where $D=\Gamma$ or $D=K/{\mathcal O}$. By Fact [\[F: existence of finite normla to get D-group\]](#F: existence of finite normla to get D-group){reference-type="ref" reference="F: existence of finite normla to get D-group"} there exists $H\trianglelefteq G$ a finite normal subgroup such that $G/H$ is locally strongly internal to $D$ and a $D$-group. Moreover, the $D$-rank of $G/H$ is the almost $D$-rank of $G$, and $H$ is ${\mathcal K}_0$-definable. Also, if $G$ was not almost strongly internal to $\textbf{k}$ then neither is $G/H$. Assume that $D=\Gamma$. By Proposition [Proposition 55](#P: Gamma){reference-type="ref" reference="P: Gamma"}, we have $\nu_{\Gamma}(G/H)\vdash Z(G/H)$. In particular, $G/H$ contains a normal abelian subgroup whose dp-rank is at least the $\Gamma$-rank of $G/H$ (equivalently, the almost $\Gamma$-rank of $G$). By Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"}, $G$ contains a definable normal abelian subgroup of the same dp-rank. Assume that $G$ is locally almost strongly internal to $K/{\mathcal O}$ but not to $\textbf{k}$, so $G/H$ is locally strongly internal to $K/{\mathcal O}$ (but not to $\textbf{k}$) and its $K/{\mathcal O}$-rank equals the almost $K/{\mathcal O}$-rank of $G$. By Proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"}, as $G$ and $H$ are both ${\mathcal K}_0$-definable, there exists a ${\mathcal K}_0$-definable infinite normal abelian subgroup of $G/H$ whose dp-rank is at least the almost $\Gamma$-rank of $G/H$. By Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"}, $G$ contains a definable normal abelian group of the same rank. ◻ The following example shows that the assumption of $K$-purity is needed in Theorem [Theorem 77](#T:johnson question){reference-type="ref" reference="T:johnson question"}(1), in order for local commutativity to imply the existence of a definable open normal abelian subgroup: **Example 78**. Let ${\mathcal K}$ be a $p$-adiaclly closed field. Let ${\mathcal O}^\times$ denote the multiplicative group of ${\mathcal O}$. Consider the semi-direct product $G={\mathcal O}^\times \ltimes K/{\mathcal O}$, where $(a,b+{\mathcal O})\cdot (c,d+{\mathcal O})=(ac, b+ad+{\mathcal O})$. Then $\dim(G)=1$ and $\mathrm{dp\text{-}rk}(G)=2$. It is locally abelian, as witnessed by ${\mathcal O}^\times\times \{0\}$. We claim that $G$ has no definable open normal abelian subgroup. Assume, towards a contradiction, that $H$ is such, in particular by [@JohnTopQp Theorem 1.4(1)] $\dim(H)=\dim(G)$ so $\pi_1(H)$, the projection on the first coordinate, must be infinite. Let $(t,0)\in H$ for $t\neq 1$. Since the conjugation of $(t,0)$ by $(a,b+{\mathcal O})$ is $(t,b-bt+{\mathcal O})$, by letting $b$ vary we conclude that $\pi_2(H)$, the projection on the second coordinate, is equal to $K/{\mathcal O}$. Thus, $H=U\ltimes K/{\mathcal O}$ for some infinite definable subgroup $U$ of ${\mathcal O}^\times$. Every element of ${\mathcal O}^{\times}$ acts non-trivially on $K/{\mathcal O}$, thus $U\ltimes K/{\mathcal O}$ is not abelian unless $U=\{1\}$, proving that $H$ as required does not exist. On the other hand, note that $\{1\}\times K/{\mathcal O}$ is an infinite definable normal abelian subgroup (that is not open). Theorem [Theorem 77](#T:johnson question){reference-type="ref" reference="T:johnson question"} together with the above example answers a question of Johnson's [@JohnTopQp §9.3] on locally abelian groups in $p$-adically closed fields. We can now prove the main result of this paper. Note that below ${\mathcal K}_0$ is not assumed to be saturated. **Theorem 79**. *Let ${\mathcal K}_0$ be either a power bounded $T$-convex field, a $V$-minimal field or a $p$-adically closed field. Let $G$ be an infinite definable, definably semisimple group in ${\mathcal K}_0$. Then there exists a finite normal subgroup $N\trianglelefteq G$ and two normal subgroups $H_1,H_2\trianglelefteq G/N$, such that* 1. *$H_1\cap H_2=\{e\}$, $H_1$ and $H_2$ centralize each other and $H_2$ is definably semisimple.* 2. *$H_1\cdot H_2$ has finite index in $G/N$.* 3. *$H_1$ is definably isomorphic to a subgroup of $\mathop{\mathrm{GL}}_n(K_0)$* 4. *$H_2$ is definably isomorphic to a subgroup of $\mathop{\mathrm{GL}}_n(\textbf{k}_0)$.* *In the almost $\textbf{k}$-rank of $G$ is $0$ (e.g., in the $p$-adically closed case) then $H_1=G/N$.* *Proof.* Let ${\mathcal K}\succ {\mathcal K}_0$ be a sufficiently saturated elementary extension. Throughout the proof below, we use $G$ to denote $G({\mathcal K})$. As a first approximation we prove the existence of $N, H_1,H_2\subseteq G$ as above, all defined over $K_0$, satisfying (1), (2) and (4), such that $H_1$ is $K$-pure. We shall later show that after modding out by another finite subgroup $H_1$ becomes $K$-linear. We divide the proof into two cases: [(a) ${\mathcal K}_0$ is $V$-minimal or power bounded $T$-convex.]{.ul} In this case, either by [@JohCminimalexist §3] in the V-minimal case, or by Proposition [Proposition 87](#P:exists-infty V-min or T-conv){reference-type="ref" reference="P:exists-infty V-min or T-conv"} in the $T$-convex power bounded case, ${\mathcal K}^{eq}$ eliminates $\exists^\infty$ and therefore $G$ is definably semisimple. By Fact [\[F: existence of finite normla to get D-group\]](#F: existence of finite normla to get D-group){reference-type="ref" reference="F: existence of finite normla to get D-group"}, there exists a $K_0$-definable finite normal subgroup $N'\trianglelefteq G$ such that in $G/N'$ the almost $K/{\mathcal O}$-rank and the $K/{\mathcal O}$-rank agree (they may be zero); by Lemma [Lemma 16](#L:passage of D-group under finite-to-one){reference-type="ref" reference="L:passage of D-group under finite-to-one"} this still holds if we further quotient by finite normal subgroups. Replace $G$ by $G/N'$ (using Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"} which says it is still definably semisimple). Assume first that $G$ is locally almost strongly internal to $\textbf{k}$. By Proposition [Proposition 67](#P: k){reference-type="ref" reference="P: k"}, there is a finite normal subgroup $N_0\trianglelefteq G$ definable over $K_0$, and $K_0$-definable normal subgroups $H_1,H_2\trianglelefteq G/N_0$ such that $H_1\cap H_2=\{e\}$ and $H_1\cdot H_2$ has finite index in $G/N_0$. Furthermore, $H_2$ is $K_0$-definably isomorphic to a $\textbf{k}$-linear definably semisimple group and the almost $\textbf{k}$-rank of $H_1$ is $0$. Since $G$ is definably semisimple, so is $G/N_0$ (Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"}). Replace $G$ by $G/N_0$. If the almost $\textbf{k}$-rank of $G$ is $0$ then we just take $H_1=G$ and $H_2=\{e\}$. **Claim 9**. *The almost $K/{\mathcal O}$-rank of $H_1$ is $0$.* Assume towards contradiction that $H_1$ is almost locally strongly internal to $K/{\mathcal O}$. By Fact [\[F: existence of finite normla to get D-group\]](#F: existence of finite normla to get D-group){reference-type="ref" reference="F: existence of finite normla to get D-group"}, there exists a finite $N_1\trianglelefteq H_1$, invariant under conjugation in $G$ (namely normal in $G$), such that $H_1/N_1$ is locally strongly internal to $K/{\mathcal O}$. Notice that $G$ acts on $H_1/N_1$ by $\sigma_g(hN_1):=h^gN_1$. Since the almost $\textbf{k}$-rank of $H_1$ is $0$, so is the almost $\textbf{k}$-rank of $H_1/N_1$. We now apply Proposition [Proposition 60](#P: K/O){reference-type="ref" reference="P: K/O"} to $H_1/N_1$ and the family of automorphisms $\mathcal A=\{\sigma_g:g\in G\}$, and obtain a definable infinite normal abelian subgroup of $H_1/N_1$ which is $\mathcal A$-invariant. By Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"}, $H_1$ contains a definable infinite normal abelian subgroup which is invariant under conjugation in $G$, namely normal in $G$. This contradicts the semisimplicity of $G$. By Theorem [Theorem 77](#T:johnson question){reference-type="ref" reference="T:johnson question"}(2a), the almost $\Gamma$-rank of $G$ is $0$ and therefore the same is true for $H_1$. So $H_1$ is $K$-pure, as claimed. This completes the proof of our approximation to the theorem, when ${\mathcal K}$ is either $V$-minimal or power bounded $T$-convex. [(b) Assume now that ${\mathcal K}$ is $p$-adically closed.]{.ul} In this case, we just need to show that $G$ is $K$-pure (and then we take $H_1=G$). However, since ${\mathcal K}$ does not eliminate $\exists^\infty$ we cannot assume a-priori that it is definably semisimple. Again, by Theorem [Theorem 77](#T:johnson question){reference-type="ref" reference="T:johnson question"}(2a), the almost $\Gamma$-rank of $G$ is $0$, for otherwise $G$ would have a $K_0$-definable infinite normal abelian subgroup, whose $K_0$-points would contradict the definable semisimplicity of $G({\mathcal K}_0)$. Since the almost $\textbf{k}$-rank of $G$ is obviously $0$, it follows from Theorem [Theorem 77](#T:johnson question){reference-type="ref" reference="T:johnson question"} 2(b), that the almost $K/{\mathcal O}$-rank of $G$ must be $0$. Indeed, if not, then once again $G$ would contain an infinite $K_0$-definable normal abelian subgroup whose $K_0$-points would contradict the semisimplicity of $G({\mathcal K}_0)$. We therefore showed, in the $p$-adically closed case, that $G$ is $K$-pure. This ends the proof of the approximated statement in all cases.\ We now proceed with the proof of Theorem [Theorem 79](#T: main){reference-type="ref" reference="T: main"}. As we showed above, we have a finite $N\trianglelefteq G$, and $H_1, H_2\trianglelefteq G/N$. all defined over $K_0$, satisfying (1), (2), (4), with $H_1$ being $K$-pure (in particular, $H_1$ is locally strongly internal to $K$). In the $p$-adically closed case, we take $H_1=G/N$ and $H_2=\{e\}$. By Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"}, $G/N$ is still definably semisimple. For clarity of notation, we replace $G$ by $G/N$. Note that $\dim G=\dim H_1+\dim H_2$, and since $\dim H_2=0$, we have $\dim G=\dim H_1$. By Lemma [Lemma 14](#L:nu of subgroup){reference-type="ref" reference="L:nu of subgroup"}, $\nu_K(G)=\nu_K(H_1)$. By Lemma [Lemma 53](#L:existence of local group){reference-type="ref" reference="L:existence of local group"}, $G$ contains a definable, differentiable normal local subgroup, with respect to $K$, which we may assume to be contained in $H_1$. Thus we have an associated $K_0$-definable map $\mathrm{Ad}_K: G\to \mathop{\mathrm{GL}}_n(K)$, with $n=\dim H_1$. Let $\mathrm{Ad}_K^{H_1}=\mathrm{Ad}_K\restriction H_1$. **Claim 10**. *$\ker(\mathrm{Ad}_K^{H_1})$ is a finite normal subgroup of $G$.* Since $H_1$ is $K$-pure, by Corollary [Corollary 76](#C:dugaldetal){reference-type="ref" reference="C:dugaldetal"}, for every $h\in \ker(\mathrm{Ad}_K^{H_1})$, $C_G(h)$ has finite index in $H_1$. By Corollary [Corollary 18](#Baldwin Saxl){reference-type="ref" reference="Baldwin Saxl"}, there exists a ${\mathcal K}_0$-definable subgroup $\widetilde H_1\trianglelefteq H_1$ of finite index, that is also normal in $G$, such that $\widetilde H_1\leq C_{H_1}(\ker \mathrm{Ad}_K^{H_1})$ and thus $\widetilde H_1 \cap \ker(\mathrm{Ad}_K^{H_1})\subseteq Z(\widetilde H_1)$. Since $\ker(\mathrm{Ad}_K^{H_1})=\ker(\mathrm{Ad}_K)\cap H_1$ it is obviously normal in $G$. Thus, $\widetilde H_1\cap \ker(\mathrm{Ad}_K^{H_1})$ is a ${\mathcal K}_0$-definable normal abelian subgroup of $G$, so it must be finite by semisimplicity of $G({\mathcal K}_0)$. Finally, since $\widetilde H_1$ has finite index in $H_1$ it follows that $\ker(\mathrm{Ad}_K^{H_1})$ is finite, as claimed.[^3] Clearly, $H_1/\ker(\mathrm{Ad}_K^{H_1})$ is definably isomorphic, over $K_0$, to a subgroup of $\mathop{\mathrm{GL}}_n(K)$, with $n=\dim H_1$. Since $\ker(\mathrm{Ad}_K^{H_1})\cap H_2=\{e\}$, we can replace $G$ by $G/\ker(\mathrm{Ad}_K^{H_1})$ and obtain $H_1,H_2$ as needed. Since all the groups and maps are defined over $K_0$ the theorem now descends to $G({\mathcal K}_0)$ as well. This ends the proof of Theorem [Theorem 79](#T: main){reference-type="ref" reference="T: main"}. ◻ **Remark 80**. In Theorem [Theorem 79](#T: main){reference-type="ref" reference="T: main"} it is not claimed that $H_1$ is definably semisimple, though we believe it is true. We expect a standard proof using the tools developed in the unpublished paper [@GisHalMac] (and [@AcHa §6]). Note, however, that if $G$ in the theorem is definably connected or has almost $\textbf{k}$-rank $0$ then it follows easily that $H_1$ is definably semisimple. As a special case, we get: **Corollary 81**. *Let ${\mathcal K}_0$ be as above. If a group $G$, definable in ${\mathcal K}_0$, is definably simple, then it is definably isomorphic to either a $K_0$-linear group or a $\textbf{k}_0$-linear $H$.* We also have the following. **Corollary 82**. *Let ${\mathcal K}_0\prec {\mathcal K}$ be as above. Let $G$ a $K_0$-definable group. Then $G(K_0)$ is definably semisimple if and only if $G(K)$ is.* *Proof.* By Proposition [Proposition 87](#P:exists-infty V-min or T-conv){reference-type="ref" reference="P:exists-infty V-min or T-conv"} and [@JohCminimalexist §3], we may assume that ${\mathcal K}_0$ is $p$-adically closed. If $G(K)$ is definably semisimple, then so is $G(K_0)$. So we assume that $G(K_0)$ is definably semisimple and show that so is $G(K)$. By Theorem [Theorem 77](#T:johnson question){reference-type="ref" reference="T:johnson question"}(2), $G$ is $K$-pure; so by Theorem [Theorem 79](#T: main){reference-type="ref" reference="T: main"}, there exists a finite normal subgroup $H_0\trianglelefteq G$ with $G/H_0 ({\mathcal K}_0)$ definably isomorphic to a $K_0$-linear group. Note that $(G/H_0)({\mathcal K}_0)$ is definably semisimple by Corollary [Corollary 20](#C: semisimple in quotients){reference-type="ref" reference="C: semisimple in quotients"}. As ${\mathcal K}_0$ eliminates $\exists^\infty$ it follows that $(G/H_0)({\mathcal K})$ is definably semisimple as well. However, since $H_0$ is finite, $G({\mathcal K})$ is definably semisimple. ◻ # Auxiliary results on power-bounded $T$-convex valued fields In this appendix, we prove two results on power bounded $T$-convex valued fields. The first, stating that definable subsets of $K$ are finite boolean combinations of ball cuts, is due to Holly [@holly Theorem 4.8] in the case of RCVF. In full generality it was proved by Tyne, [@tynephd Page 94], but never published. Tyne's proof builds on a deep result, dubbed the valuation property (also not published in the required generality). As a service to the community, we provide an alternative, more direct proof. The second result shows, using a theorem of Johnson's [@JohCminimalexist], uniform finiteness for all imaginary sorts. From now on, ${\mathcal K}$ denotes a power bounded $T$-convex valued field. We remind some standard notation. ## Definable subsets of $K$ If $C\subseteq K$ is any convex set, by $x<C$ we mean that $x<y$ for all $y\in C$ and $x\leq C$ is defined similarly. For convex sets $C_1, C_2$ we write $C_1<C_2$ if $x<y$ for any $x\in C_1$ and $y\in C_2$, similarly $C_1\leq C_2$. By a *definable cut* in $K$ we mean a pair of disjoint non-empty definable convex sets ${\mathcal C}=(C_1,C_2)$, such that $C_1<C_2$ and $C_1\cup C_2=K$. A cut ${\mathcal C}$ is *realized* if either $C_1$ has a maximum or $C_2$ has minimum. For a definable function $f$ from $C_1$ (or some open interval containing it) to either $K$ or $\Gamma$ we say that $\lim_{x\to {\mathcal C}^-}f(x)=t_0$, if for every $t_1<t_0<t_2$ there exists $x\in C_1$ such that for all $x'>x$ in $C_1$, $t_1<f(x')<t_2$ (and likewise $\lim_{x\to {\mathcal C}^+}$). Following [@holly], we define: **Definition 83**. A definable cut ${\mathcal C}=(C_1,C_2)$ in $K$ is *a ball cut* if there is a ball $B$ (possibly a point) such that either $C_1=\{x\in K: x<B\}$ (and then $C_2=\{x\in K: B\leq x\}$), or $C_2=\{x\in K: B<x\}$ (and then $C_1=\{x\in K: x\leq B\}$. By o-minimality of $\Gamma$, for every definable set $X$ and $x\in X$ there exists a maximal ball around $x$ which is contained in $X$. We leave the following easy observation to the reader. **Lemma 84**. *Let $C\subseteq K$ be a convex definable subset and let $b_1,b_2,b_3$ be maximal balls in $C$ with $b_1<b_2<b_3$. Then $b_2$ is necessarily an open ball.* **Proposition 85**. *Every definable cut is a ball cut. In particular, every definable subset of $K$ is a boolean combination of balls and intervals.* *Proof.* Since every definable subset of $K$ is a finite union of convex sets [@TconvexI Corollary 3.14], it will suffice to prove the first clause of the statement. Fix, once and for all, an arbitrary definable unrealised cut ${\mathcal C}=(C_1,C_2)$. Denote for every $x\in C_1$ by $B_x$ the maximal ball in $C_1$ containing $x$ and let $r(x)\in \Gamma$ be its radius. Note that $r(x)$ is (weakly) increasing with $x$. We start with the following. **Claim 11**. *Keeping the above notation, if $r(x)$ stabilizes as $x\to \mathcal C^-$ then ${\mathcal C}$ is a ball cut.* Notice that $r(x)$ is (possibly weakly) increasing. Assume that $r(x)=r_0$ for sufficiently large $x$ in $C_1$. After re-scaling, assume that $r_0=0$. If $B_x$ is the same ball for all sufficiently large $x\in C_1$ then $\mathcal C$ is a ball cut, so assume that for every $x\in C_1$ there is some $x'>x$ in $C_1$ such that $B_x\neq B_{x'}$. By Lemma [Lemma 84](#L:at most two closed max balls){reference-type="ref" reference="L:at most two closed max balls"}, for all sufficiently large $x$, all the $B_x$ are open. Thus, for any $x\in C_1$, the closed ball $B_{\geq 0}(x)$ intersects $C_2$. As every ball is convex, we have $B_{\geq 0}(x_1)=B_{\geq 0}(x_2)$ for all sufficiently large elements of $C_1$; let $B$ be this closed ball. After translating, we may assume that $B={\mathcal O}$. As a result, the map $x\mapsto x+\textbf{m}$ maps $(B\cap C_1, B\cap C_2)$ into a cut in $\textbf{k}$. By o-minimality of $\textbf{k}$, this cut is realized, namely either the left side has a maximum or the ride side has a minimum. In the first case, $C_1$ has a right side ball and in the second case $C_2$ has a left side ball. By the claim, we may assume that $r(x)$ does not stabilise, as $x$ increases in $C_1$. Using definable Skolem functions, [@vdDries-Tconvex Remark 2.7], we find a definable $h:C_1\to K$ such that for all $x\in C_1$, $r(x)=v(h(x))$. Let ${\mathcal L}_{omin}$ be the language of the underlying o-minimal reduct (i.e., ${\mathcal L}_{omin}={\mathcal L}(T)$). By [@vdDries-Tconvex Corollary 2.8], there exists an ${\mathcal L}_{omin}$-definable function $\widehat h:I\to K$ such that $h=\hat h$ on an end segment of $C_1^-$, which we may assume equals to $I\cap C_1$. Since $\mathcal C$ is an unrealised cut and $I$ is an ${\mathcal L}_{omin}$-definable interval containing an end segment of $C_1$ then necessarily $I\cap C_2\neq \emptyset$. Shrinking $I$ (without losing the property that $I\cap C_i\neq 0$ for $i=1,2$) we may assume that $h$ is strictly monotone and continuous. By replacing, if needed, $h$ by $-h$ (and $\widehat h$ by $-\widehat h$) we may assume that $\widehat h$ is strictly decreasing.\ **Case 1:** $\lim\limits_{x\to {\mathcal C}^-} r(x)=\infty$. In this case $\lim\limits_{x\to {\mathcal C}^-}\widehat h(x)=0$. Thus, the function $\widehat h$, which is strictly decreasing and continuous, takes a convex set of the form $\{x\in C_1:x>c\}$, for some $c\in C_1\cap I$, onto an open interval $(0,d)$, with $d=\widehat h(c)$. Since $\widehat h$ is ${\mathcal L}_{omin}$-definable, so is its inverse function $\widehat h^{-1}\restriction(0,d)$. By o-minimality, and since $\hat h^{-1}$ is strictly decreasing and bounded, it takes the interval $(0,d)$ to an interval of the form $(c,a)$, for some $a\in K$, and therefore $a$ realizes the cut $\mathcal C$, contradicting our assumption.\ **Case 2:** $\lim\limits_{x\to {\mathcal C}^-} r(x)=r_0\in \Gamma$. Since $r(x)$ does not stabilize, then $r(x)=v(h(x))<r_0$ for all $x\in C_1$. After re-scaling, we may assume that $r_0=0$, so $v(\widehat h(x))<0$ for all $x\in C_1\cap I$ and $\lim\limits_{x\in {\mathcal C}^-}v(\widehat h(x))=0$. Thus, for all $x\in C_2\cap I$, we have $v(h_1(x))\geq 0$, and by continuity there must be an element $x\in C_2\cap I$ with $v(\widehat h(x))=0$. Hence, there is some $x_2\in C_2\cap I$ such that for all $x\in C_2$, if $x<x_2$ then $v(\widehat h(x))=0$. Consequently, $x\in C_2\cap I\iff \widehat h(x)\in {\mathcal O}$. Let $(C_1',C_2')$ be the ball cut $C_1'=\{y\in K:y\leq {\mathcal O}\}$ and let $J=\widehat h(I)$. Then $J\cap C_i'\neq \emptyset$, for $i=1,2$, and $\widehat h^{-1}$ is strictly decreasing (from $J$ to $I$). For simplicity, let $g=\widehat h^{-1}$. For any $y\in {\mathcal O}\cap J$, let $B_y\subseteq C_2$ be the maximal ball containing $g(y)\in C_2$, and denote its radius by $r'(y)$. We may assume that $y\mapsto B_y$ does not stabilise as $y\to \sup J$ (otherwise ${\mathcal C}$ is a ball cut, and we are done) and thus, by Lemma [Lemma 84](#L:at most two closed max balls){reference-type="ref" reference="L:at most two closed max balls"} the $B_y\subseteq C_2$ are open. By [@vdDries-Tconvex Proposition 4.2], $r'(y)$ stabilizes for sufficiently large $y\in J$. Since $g$ sends ${\mathcal O}\cap J$ to $C_2\cap I$, it follows that for some $c\in C_2$, all maximal balls $B\subseteq C_2$, with $B<c$, have the same radius. We can now conclude that $\mathcal C$ is a ball cut, using Claim [Claim 11](#C: ball cut){reference-type="ref" reference="C: ball cut"} (with the roles of $C_1$ and $C_2$ interchanged), thus finishing the proof of Proposition [Proposition 85](#ball interval){reference-type="ref" reference="ball interval"}. ◻ The fact that ${\mathcal K}$ is definably spherically complete is a consequence of $0$-h-minimality of ${\mathcal K}$, [@hensel-min Lemma 2.7.1]. The proof there is not hard, though it implicitly uses Tyne's theorem. We give here a different proof using the previous proposition. **Corollary 86**. *${\mathcal K}$ is definably spherically complete.* *Proof.* Let $\{B_t:t\in T\}$ be a definable chain of balls in $K$. Assume towards contradiction that $\bigcap_{t\in T} B_t= \emptyset$. Let $r(B_t)\in \Gamma$ be the valuative radius of $B_t$. We define two definable convex sets $C_1,C_2$ by $$C_1=\{x\in K: \exists t \,\, x<B_t\}\,\,;\,\, C_2=\{x\in K: \exists t \,\, B_t<x\}.$$ Since balls are convex, our assumption implies that ${\mathcal C}=(C_1,C_2)$ is a definable, unrealized, cut. By Proposition [Proposition 85](#ball interval){reference-type="ref" reference="ball interval"}, this is a ball cut. For simplicity (the other cases are similar), we assume that $C_1=\{ x\in K: x\leq B\}$ for some ball $B$. Translating and re-scaling, we may assume that $B$ is either ${\mathcal O}$ or $\mathfrak m$. Let $B_0=\bigcup\limits_{t\in T} B_t$. We define a function $r:B_0\to \Gamma$ by $r(x)=\sup\{r(B_t):x\in B_t\}$. Using definable Skolem functions, we find a definable function $h:B_0\to K$, such that $v(h(x))=r(x)$. Assume that $B={\mathcal O}$. By [@vdDries-Tconvex Proposition 4.2], the function $v(h(x))$, restricted to ${\mathcal O}$, eventually stabilizes as $x\to {\mathcal C}^-$. This implies that the chain of balls $B_t$ has a minimal element (there is a bijection between the balls and their radii), contradicting our assumption that the intersection of the chain is empty. Assume that $B=\mathfrak m$ and consider $h \restriction C_2$. Let ${\mathcal C}'=(C_1',C_2')$, where $C_1'=\{x\in K : x\leq {\mathcal O}\}$. As $x\to {\mathcal C}^+$, we get that $x^{-1}\to {\mathcal C}^-$, so applying [@vdDries-Tconvex Propostiion 4.2] to $h(x^{-1})$, we conclude that $v(h(x))$ must stabilize as $x\to {\mathcal C}^+$, again reaching a contradiction. ◻ ## Elimination of $\exists^\infty$ in the $T$-convex power bounded case We now show that ${\mathcal K}^{eq}$ eliminates $\exists^\infty$; the proof utilizes a criterion used by Johnson to prove a parallel result for $C$-minimal valued fields, see [@JohCminimalexist]. **Proposition 87**. *${\mathcal K}^{\mathop{\mathrm{eq}}}$ eliminates $\exists^\infty$.* *Proof.* We shall apply Johnson's criterion for eliminating $\exists^\infty$, [@JohCminimalexist]. By [@JohCminimalexist Theorem 2.3], it suffices to prove that if $X$ is a definable set in ${\mathcal K}^{eq}$ such that there exist a definable set $S\subseteq X\times K$ with the function $a\mapsto S_a:=\{b\in K: (a,b)\in S\}$ injective on $X$, then $\exists^{\infty}$ is eliminated on $X$. Namely, if $\{Y_t:t\in T\}$ is a definable family of subsets of $X$ then there is a bound on the size of those $Y_t$ that are finite. Let $X$ be such a definable set (with $S\subseteq X\times K$ as in the assumption). As ${\mathcal K}$ is weakly o-minimal (and saturated), there exists $k\in {\mathbb{N}}$ such that each $S_a$ is a finite union of at most $k$ convex sets. By partitioning $X$, we may assume that each $S_a$ consists of exactly $k$ convex sets. Let $X'=X\times \{1,\dots,k\}$ and let $S'\subseteq X'\times K$ the set satisfying that $S'_{a,i}$ is the $i$-th convex component of $S_a$. It is sufficient to prove that $\exists^\infty$ is eliminated on $X'$: Indeed, if $\exists^\infty$ is not eliminated on $X$ then there exists a definable family of subsets $\{Y_t: t\in T\}$ of $X$ and a sequence $\{t_n\}$, such that $|Y_{t_n}|$ is finite and tends to $\infty$. We define a family of finite subsets of $X'$ as follows: For $i=1,\dots, k$, let $$Y'_{t,i}=\{\text{ the $i$-th convex component of $S_a: a\in Y_t$}\}.$$ Since $|Y_{t_n}|\to \infty$ one of the $|Y_{ t_n,i}|$ must tend to $\infty$, thus $X'$ does not eliminate $\exists^\infty$. We replace $X$ by $X'$ and $S$ by $S'$, so we may assume that each $S_a$ is a convex subset of $K$. By Proposition [Proposition 85](#ball interval){reference-type="ref" reference="ball interval"}, every $S_a$ is a boolean combination of intervals and balls; so by convexity it must be of the form $B_1\square_1 x\square_2 B_2$, where each $B_i$ is either a point or a ball and $\square_i\in \{<,=,\leq\}$. Thus, every $S_a$ is coded by a pair of balls, so it is sufficient to treat the case where each $S_a$ is a ball, namely we may assume that $X$ is a set of balls. Let $\{Y_t:t\in T\}$ be a definable family of subsets of $X$. We claim that there is a bound on the size of the finite $Y_t$ in the family. We conclude the proof as in [@JohCminimalexist §3]. If a ball $b$ belongs to a finite $Y_t$ then it contains a ball $b'\in Y_t$ which is minimal with respect to inclusion. Thus, we may assume that for every $t\in T$, every ball in $Y_t$ contains a minimal ball in $Y_t$ (the set of all such $t$ is definable). For every $t\in Y_t$ the set of minimal balls in $Y_t$ is pairwise disjoint, and its number equals the number of convex components of $\bigcup\{b\in Y_t: b \mbox{ minimal} \}$ (we use here the fact that every ball in $K$ is convex and the union of two disjoint balls is not convex). Thus, by weak o-minimality, there is a bound on the number of minimal balls in $Y_t$ as $t$ ranges over $T$. Assume towards contradiction that the number of balls in those finite $Y_t$ is not uniformly bounded. Then, by the bound on the number of minimal balls in $Y_t$, there are chains of balls in $Y_t$, as $t$ varies, of unbounded size. This is impossible, as this would imply that the sets $\{r(B): B\in Y_t\}$ (where $r(B)$ is the valuative radius of $B$) are finite of unbounded size (as $t$ ranges over $T$). Since $\Gamma$ is o-minimal and stably embedded, definable families of finite subsets of unbounded size do not exist. ◻ [^1]: The first author was partially supported by ISF grant No. 555/21 and 290/19. The second author was supported by ISF grant No. 555/21. The third author was supported by ISF grant No. 290/19. [^2]: In [@HaHaPeGps] this was called the $D$-critical rank of $S$. [^3]: The argument given in the claim shows that for $K$-pure groups, the kernel of $\mathrm{Ad}$ has a (relatively) open normal abelian subgroup of finite index. This is true in particular for $p$-adic Lie groups definable in the $p$-adic field. Recently, [@helge-example], Glöckner constructed an example of a $1$-dimensional $p$-adic Lie group $G$ for which this fails. In fact, in his example $\ker (\mathrm{Ad}_K)=G$, but $G$ contains no open normal abelian subgroup.

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